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+{"text":"\\section{Introduction}\n\nIn engineering and management applications, one often has to collect data from unknown systems, learn their transition functions, and learn to make predictions and decisions. A critical precursor of decision making is to model the system from data. We study how to learn an unknown Markov model of the system from its state-transition trajectories. When the system admits a large number of states, recovering the full model becomes sample expensive.\n\nIn this paper, we focus on Markov processes where the transition matrix has a small rank.\nThe small rank implies that the observed process is governed by a low-dimensional latent process which we cannot see in a straightforward manner. It is a property that is (approximately) satisfied in a wide range of practical systems. Despite the large state space, the low-rank property unlocks potential of accurate learning of a full set of transition density functions based on short empirical trajectories. \n\n\\subsection{Motivating Examples}\n\nPractical state-transition processes with a large number of states often exhibit low-rank structures. For example, the sequence of stops made by a taxi turns out to follow a Markov model with approximately low rank structure \\citep{liu2012understanding, benson2017spacey}. \nFor another example, random walk on a lumpable network has a low-rank transition matrix \\citep{buchholz1994exact,e2008optimal}. The transition kernel with fast decaying eigenvalues has been also observed in molecular dynamics \\citep{rohrdanz2011determination}, which can be used to find metastable states, coresets and manifold structures of complicated dynamics \\citep{chodera2007automatic, coifman2008diffusion}.\n\n\nLow-rank Markov models are also related to dimension reduction for control systems and reinforcement learning. For example, the state aggregation approach for modeling a high-dimensional system can be viewed as a low-rank approximation approach \\citep{bertsekas1995dynamic, bertsekas1995neuro,singh1995reinforcement}. In state aggregation, one assumes that there exists a latent stochastic process $\\{z_t\\} \\subset [r]$ such that\n$\\mathbb{P}( s_{t+1}  \\mid s_t ) =  \\sum_z \\mathbb{P}( z_t =z \\mid  s_t) \\mathbb{P}( s_{t+1} \\mid z_t=z),$\nwhich is equivalent to a factorization model of the transition kernel $\\P$.\nIn the context of reinforcement learning, the nonnegative factorization model was referred to as the generalization to the rich-observation model \\citep{azizzadenesheli2016reinforcement1}. The low-rank structure allows us to model and optimize the system using significantly fewer observations and less computation. Effective methods for estimating the low-rank Markov model would pave the way to better understanding of process data and more efficient decision making.\n\n\n\n\n\n\n\n\n\n\n\n\n\\subsection{Our approach}\nWe propose to estimate the low-rank Markov model based on an empirical trajectory of states, whose length is only proportional to the total number of states. \nWe propose two approaches based on the maximum likelihood principle and low-rank optimization. The first approach uses a convex nuclear-norm regularizer to enforce the low-rank structure and a polyhedral constraint to ensure that optimization is over all probabilistic matrices. The second approach is to solve a rank-constrained optimization problem using difference-of-convex (DC) programming. For both approaches, we provide statistical upper bounds for the Kullback-Leibler (KL) divergence between the estimator and the true transition matrix as well as the $\\ell_2$ risk. We also provide an information-theoretic lower bound to show that the proposed estimators are nearly rate-optimal. Note that the low-rank estimation of the Markov model was considered in \\cite{zhang2018optimal} where a spectral method with total variation bound is given. In comparison, the novelty of our methods lies in the use of maximum likelihood principle and low-rank optimization, which allows us to obtain the first KL divergence bound for learning low-rank Markov models.\n\n\nOur second approach involves solving a rank constraint optimization problem over probabilistic matrices, which is a refinement of the convex nuclear-norm approach. Due to the non-convex rank constraint, the optimization problem is difficult - \u00a0to the best of our knowledge, there is no efficient approach that directly solves the rank-constraint problem.\u00a0In this paper, we develop a penalty approach to relax the rank constraint and transform the original problem into a DC (difference of convex functions) programming one. Furthermore, we develop a particular DC algorithm to solve the problem by initiating at the solution to the convex problem and successively refining the solution through solving a sequence of inner subproblems. Each subroutine is based on the multi-block alternating direction method of multipliers (ADMM). Empirical experiments show that the successive refinements through DC programming do improve the learning quality. As a byproduct of this research, we develop a new class of DC algorithms and a unified convergence analysis for solving non-convex non-smooth problems, which were not available in the literature to our best knowledge. \n\n\\subsection{Contributions and paper outline}\n\nThe paper provides a full set of solutions for learning low-rank Markov models. The main contributions are: (1) We develop statistical methods for learning low-rank Markov model with rate-optimal Kullback-Leiber divergence guarantee for the first time; (2) We develop low-rank optimization methods that are tailored to the computation problems for nuclear-norm regularized and rank-constrained M-estimation; (3) A byproduct is a generalized DC algorithm that applies to nonsmooth nonconvex optimization with convergence guarantee. \n\nThe rest of the paper is organized as follows. Section 2 surveys related literature. Section 3 proposes two maximum likelihood estimators based on low-rank optimization and establishes their statistical properties. Section 4 develops computation methods and establishes convergence of the methods. Section 5 presents the results of our numerical experiments.\n\n\n\n\\section{Related literature}\n\n\nModel reduction for complicated systems has a long history. It traces back to variable-resolution dynamic programming \\citep{moore1991variable} and state aggregation for decision process \\citep{sutton1998reinforcement}. In the case of Markov process, \\citep{deng2011optimal, deng2012model} considered low-rank reduction of Markov models with explicitly known transition probability matrix, but not the estimation of the reduced models. Low-rank matrix approximation has been proved powerful in analysis of large-scale panel data, with numerous applications including network analysis \\citep{e2008optimal}, community detection \\citep{newman2013spectral}, ranking \\citep{negahban2016rank}, product recommendation \\citep{keshavan2010matrix} and many more. The main goal is to impute corrupted or missing entries of a large data matrix. Statistical theory and computation methods are well understood in the settings where a low-rank signal matrix is corrupted with independent Gaussian noise or its entries are missed independently. \n\n\nIn contrast, our problem is to estimate the transition density functions from dependent state trajectories, where statistical theory and efficient methods are underdeveloped. When the Markov model has rank $1$, it becomes an independent process. In this case, our problem reduces to estimation of a discrete distribution from independent samples \\citep{steinhaus1957problem,lehmann2006theory,han2015minimax}. For a rank-$2$ transition matrix, \\cite{huang2016recovering} proposed an estimation method using a small number of independent samples. Very recently there have been some works on minimax learning of Markov chains. \\citet{HOP18} derived the minimax rates of estimating a Markov model in terms of a smooth class of $f$-divergences. They considered the family of $\\alpha$-minorated Markov chains, i.e., all the transition probabilities are greater than $\\alpha$. \\citet{WKo19} computed the finite-sample PAC-type minimax sample complexity of recovering the transition matrix from a state trajectory of a Markov chain, up to a tolerance in a total-variation-based (TV-based) metric. This TV-based metric does not belong to the family of the smooth $f$-divergences in \\citet{HOP18}, and their class of Markov models strictly contains the class of the $\\alpha$-minorated ones. Neither of these works considered low-rank Markov models though. \n\nThe closest work to ours is \\cite{zhang2018optimal}, in which a spectral method via truncated singular value decomposition was introduced and the upper and lower error bounds in terms of total variation were established. \\cite{yang2017dynamic} developed an online stochastic gradient method for computing the leading singular space of a transition matrix from random walk data. To our best knowledge, none of the existing works has analyzed efficient recovery of a low-rank Markov model with Kullback-Leiber divergence guarantee.\n\nHidden Markov Models (HMMs) are closely related with our low-rank Markov models. Note that the observation trajectory of an HMM is not necessarily Markovian. Therefore, an HMM can be regarded as a relaxed variant of low-rank Markov models. There have been many works on estimating HMM, in particular through spectral approaches, e.g., \\citet{hsu2012spectral} and \\citet{anandkumar2014tensor}. A critical difference is: States are not fully observable in HMM, but are fully observable in low-rank Markov models. Although HMM is more general, the low-rank Markov model is more suitable for dynamical processes where the state space is large but fully observable, for which we will establish tighter error bounds. \n\n\nOn the optimization side, we adopt DC programming to handle the rank constraint and replace it with the difference of two convex functions. DC programming was first introduced by \\cite{tao1997convex} and has become a prominent tool for handling a class of nonconvex optimization problems (see also \\cite{tao2005dc,le2012exact,le2017stochastic,lethi2018}). \nIn particular, \\citet{van2015convergence} and \\cite{wen2017proximal} considered the {majorized DC algorithm}, which motivated the optimization method developed in this paper.\nHowever, both \\cite{van2015convergence} and \\citet{wen2017proximal} used the majorization technique with restricted choices of majorants, and neither considered the introduction of the indefinite proximal terms. In addition, \\citet{wen2017proximal} further assumes the smooth part in the objective to be convex. \nIn comparison with the existing methods, our DC programming method applies to nonsmooth problems and is compatible with a more flexible and possibly indefinite proximal term.\n\nFinally, we would like to mention the probabilistic tools we used to derive the statistical results. Recent years have witnessed many works on measure concentration of dependent random variables, e.g., \\citet{Mar96, Kon07, KRa08, Pau15, JFS18}, etc. Nevertheless, these results do not suffice to establish the desired statistical guarantee, because exploiting low-rank structure requires studying the concentration of a matrix martingale in terms of the spectral norm, as shown in Lemma \\ref{lem:gradient}. The matrix Freedman inequality \\citep[][Corollary~1.3]{tropp2011freedman} turns out to be the right tool for analyzing the concentration of the matrix martingale. We also used an variant of Bernstein's inequality for general Markov chains \\citep[][Theorem~1.2]{JFS18} to derive an exponential tail bound for the status counts of the Markov chain ${\\mathcal X}$. \n\n\n\n\n\n\n\n\\section{Minimax rate-optimal estimation of low-rank Markov chains}\n\\label{sec:StatProperty}\n\nConsider an ergodic Markov chain ${\\mathcal X} = \\{X_0, X_1, \\ldots, X_n\\}$ on $p$ states ${\\mathcal S}=\\{s_j\\}_{j=1}^p$ with the transition probability matrix $\\mathbf{P}\\in\\mathbb{R}^{p\\times p}$ and stationary distribution $\\pi$, where $P_{ij} = \\mathbb{P}(X_1 = s_j|X_0 = s_i)$ for any $i, j \\in [p]$. Let $\\pi_{\\min} := \\min_{j \\in [p]} \\pi_j$ and $\\pi_{\\max} := \\max_{j \\in [p]} \\pi_j$. We quantify the distance between two transition matrices $\\mathbf{P}$ and $\\widehat{\\mathbf{P}}$ in Frobenius norm $\\|\\widehat{\\mathbf{P}} - \\mathbf{P}\\|_{\\rm F} = \\bigl\\{\\sum_{i,j  = 1}^p (\\widehat P_{ij} - P_{ij})^2\\bigr\\}^{1\/2}$ and Kullback--Leibler divergence  $D_{\\mathrm{KL}}(\\mathbf{P}, \\widehat{\\mathbf{P}}) = \\sum_{i, j=1}^p \\pi_iP_{ij}\\log(P_{ij}\/ \\widehat P_{ij})1_{\\{P_{ij} \\neq 0\\}}$.\nSuppose that the unknown transition matrix $\\mathbf{P}$ has a small rank $r \\ll p$. Our goal is to estimate the transition matrix $\\P$ via a state trajectory of length $n$. \n\n\\subsection{Spectral gap of nonreversible Markov chains}\nWe first introduce the {\\it right ${\\mathcal L}_2$-spectral gap} of $\\mathbf{P}$ \\citep{Fil91, JFS18}, a quantity that determines the convergence speed of the Markov chain ${\\mathcal X}$ to its invariant distribution $\\pi$. \nLet ${\\mathcal L}_2(\\pi):= \\{h \\in \\Re^p: \\sum_{j \\in [p]} h_j^2 \\pi_j < \\infty\\}$ be a Hilbert space endowed with the following inner product: \n\\[\n\\inn{h_1, h_2}_{\\pi} := \\sum_{j \\in [p]} h_{1j} h_{2j}\\pi_j. \n\\]\nThe matrix $\\mathbf{P}$ induces a linear operator on ${\\mathcal L}_2(\\pi)$: $h \\mapsto \\mathbf{P} h$, which we abuse $\\mathbf{P}$ to denote. Let $\\mathbf{P}^*$ be the adjoint operator of $\\mathbf{P}$ with respect to ${\\mathcal L}_2(\\pi)$:\n$$\\P^* = \\diag(\\pi)^{-1} \\P^\\top \\diag(\\pi).$$ \nNote that the following four statements are equivalent: (a) $\\P$ is self-adjoint; (b) $\\P^\\ast = \\P$; (c) the detailed balance condition holds: $\\pi_i P_{ij} = \\pi_j P_{ji}$; (d) the Markov chain is reversible. In our analysis, we do {\\it not} require the Markov chain to be reversible. We therefore introduce the {\\it additive reversiblization of $\\mathbf{P}$}: $(\\mathbf{P} + \\mathbf{P}^*) \/ 2$, which is a self-adjoint operator on ${\\mathcal L}_2(\\pi)$ and has the largest eigenvalue as 1. The right spectral gap of $\\mathbf{P}$ is defined as follows: \n\\begin{definition}[Right ${\\mathcal L}_2$-spectral gap]\n\tWe say  the right ${\\mathcal L}_2$-spectral gap of $\\mathbf{P}$ is $1 - \\rho_+ $ if \n\t\\[\n\t\\rho_+:= \\sup_{\\inn{h, 1}_{\\pi} = 0, \\inn{h, h}_{\\pi} = 1} \\frac{1}{2}\\inn{(\\mathbf{P} + \\mathbf{P}^*)h, h}_{\\pi} < 1, \n\t\\]\n\twhere $1$ in $\\inn{h, 1}$ refers to the all-one $p$-dimensional vector.\n\\end{definition}\nDefine the $\\epsilon$-mixing time of the Markov chain ${\\mathcal X}$ as\n\\[\n\\tau(\\epsilon) := \\min\\{t: \\max_{j \\in [p]} \\|(\\mathbf{P}^t)_{j\\cdot} - \\pi\\|_{\\mathrm{TV}} \\le \\epsilon\\}, \n\\]\nwhere $\\|(\\mathbf{P}^t)_{j\\cdot} - \\pi\\|_{\\mathrm{TV}} := 2 ^ {-1} \\lonenorm{(\\mathbf{P}^t)_{j\\cdot} - \\pi}$ is the total variation distance between $\\mathbf{P} ^ t_{j \\cdot}$ and $\\pi$. For reversible and ergodic Markov chains, \\citet[][Theorem~12.3]{LPe17} show that \n\\be\n\t\\label{eq:mixing_gap}\n\t\\tau(\\epsilon) \\le \\frac{1}{1 - \\rho_+}\\log\\biggl(\\frac{1}{\\epsilon \\pi_{\\min}}\\biggr), \n\\ee\nwhich implies that the larger the spectral gap is, the faster the Markov chain converges to the stationary distribution.\n\n\\subsection{Estimation methods and statistical results}\n\\label{sec:main_results}\n\nNow we are in position to present our methods and statistical results. Given the trajectory $\\{X_1,\\ldots,X_n\\}$, we count the number of times that the state $s_i$ transitions to $s_j$:\n\\[ n_{ij}: = \\left|\\{1\\le k\\le n \\mid \\, X_{k-1} = s_i, X_k = s_j\\}\\right|.\\]\nLet $n_i: = \\sum_{j=1}^{p} n_{ij}$ for $i=1,\\ldots, p$ and $n := \\sum_{i=1}^p n_i$. The averaged negative log-likelihood function of $\\P$ based on the state-transition trajectory $\\{x_0, \\ldots, x_n\\}$ is\n\\begin{equation}\\label{eq:log-likelihood}\n{\\ell_n}(\\P):= - \\frac{1}{n} \\sum_{k = 1}^n \\log (\\inn{\\mathbf{P}, \\mathbf{X}_k}) = -\\frac{1}{n} \\sum_{i=1}^{p}\\sum_{j=1}^{p} n_{ij}\\log(P_{ij}), \n\\end{equation}\nwhere $\\mathbf{X}_k := e_ie_j^\\top \\in \\Re^{p \\times p}$ if $x_k = s_i$ and $x_{k + 1} = s_j$. We first impose the following assumptions on $\\mathbf{P}$ and $\\pi$. \n\\begin{assumption}\n\t\\label{asp:1}\n\t(i) ${\\rm rank}(\\mathbf{P}) = r$; (ii) There exist some positive constants $\\alpha, \\beta > 0$ such that for any $1 \\le j, k \\le p$, $P_{jk} \\in \\{0\\} \\cup [\\alpha \/ p, \\beta \/ p]$. \n\\end{assumption}\n\\begin{remark}\nThe entrywise constraints on $\\mathbf{P}$ are imposed by our theoretical analysis and may not be necessary in practice. \n\tSpecifically, the upper and lower bounds for the nonzero entries of $\\mathbf{P}$\n\tensure that (i) the gradient of the log-likelihood $\\nabla \\ell_n(\\mathbf{P})$ is well controlled and exhibits exponential concentration around its population mean (see \\eqref{eq:log_likelihood} for the reason we need $\\alpha$ there); \n\t(ii) the converter between the $\\ell_2$-risk $\\fnorm{\\widehat \\mathbf{P} - \\mathbf{P}}$ ($\\fnorm{\\widehat \\mathbf{P}^r - \\mathbf{P}}$ resp.) and the KL-divergence $D_{\\mathrm{KL}}(\\mathbf{P}, \\widehat \\mathbf{P})$ ($D_{\\mathrm{KL}}(\\mathbf{P}, \\widehat \\mathbf{P}^r)$ resp.) depends on $\\alpha$ and $\\beta$, as per Lemma \\ref{lem:kl_to_l2}. The entry-wise upper and lower bounds are common in statistical analysis of count data, e.g., Poisson matrix completion \\citep[Equation (10)]{cao2016poisson}, Poisson sparse regression \\citep[Assumption 2.1]{jiang2015minimax}, point autoregressive model \\citep[Definition of $\\mathcal{A}_s$]{hall2016inference}, etc.\n\\end{remark\n\n\\begin{remark}\n\n\tIf we remove $0$ in the feasible range of $P_{jk}$, we obtain the $(\\alpha\/p)$-minoration condition: $P_{jk} \\ge \\alpha \/ p$ for all $j, k \\in [p]$. The $(\\alpha\/p)$-minoration condition implies strong mixing since combining \\citet[][pp. 237-238]{Bre99} and \\citet[][Lemma~2.2.2]{Kon07} yields $1 - \\rho_+ \\ge \\alpha$ and we can deduce that $\\tau(\\epsilon) \\le \\alpha^{-1}\\log\\{(\\epsilon \\pi_{\\min})^{-1}\\}$ given \\eqref{eq:mixing_gap}. \n\\end{remark}\n\nNext we propose and analyze a nuclear-norm regularized maximum likelihood estimator (MLE) of $\\mathbf{P}$ defined as follows: \n\\begin{equation}\n\\label{prob:convex-nuclear}\n\\begin{array}{rllll}\n\\widehat{\\P} := & \\argmin ~\\ell_n({\\mathbf{Q}}) + \\lambda \\nnorm{{\\mathbf{Q}}}\\\\\n\\mbox{s.t.} & {\\mathbf{Q}} 1_p = 1_p,\\quad \\alpha \/ p \\le Q_{ij}\\le \\beta \/ p,\\quad \\forall\\, 1\\le i,j\\le p, \n\\end{array}\n\\end{equation}\nwhere $\\lambda>0$ is a tuning parameter. Note that we cannot allow $\\mathbf{Q}$ to have zero entries as in Assumption \\ref{asp:1}, because otherwise we may have that $\\widehat P_{ij} = 0$ and $P_{ij} > 0$ for some $(i, j)$, violating the requirement of the definition of $D_{\\mathrm{KL}}(\\mathbf{P}, \\widehat\\mathbf{P})$. Our first theorem shows that with an appropriate choice of $\\lambda$, $\\widehat \\mathbf{P}$ exhibits a sharp statistical rate. For simplicity, from now on we say $a \\gtrsim b$ ($a \\lesssim b$) if there exists a universal constant $c > 0$ ($C > 0$) such that $a \\ge cb$ ($a \\le Cb$). \n\n\\begin{theorem}[Statistical guarantee for the nuclear-norm regularized estimator]\n\t\\label{thm:nuclear}\n\tSuppose the initial state $X_0$ is drawn from the stationary distribution $\\pi$ and Assumption \\ref{asp:1} holds. There exists a universal constant $C_1 > 0$, such that for any $\\xi > 1$, if we choose \n\t\\[\n\t\t\\lambda =  C_1 \\biggl\\{\\biggl(\\frac{\\xi p ^ 2\\pi_{\\max} \\log p}{n\\alpha}\\biggr)^{\\! 1 \/ 2} + \\frac{\\xi p\\log p}{n \\alpha}\\biggr\\}, \n\t\\]\n\tthen whenever $n\\pi_{\\max}(1 - \\rho_+) \\ge \\max\\{\\max(20, \\xi ^ 2) \\log p, \\log n\\}$, we have that \n\t\\[\n\t\\begin{aligned}\n\t\t\\mathbb{P} \\biggl( D_{\\mathrm{KL}}(\\mathbf{P}, \\widehat\\mathbf{P}) \\gtrsim \\frac{\\xi r\\pi_{\\max}\\beta ^ 2 p \\log p}{\\pi_{\\min}\\alpha ^ 3n} + \\frac{\\xi\\pi_{\\min}}{rp\\pi_{\\max}\\log p}\\biggr) \\lesssim e^{- \\xi} + p^{-(\\xi - 1)} + p^{-10},\n\t\\end{aligned}\n\t\\]\n\tand that \n\t\\[\n\t\t\\begin{aligned}\n\t\t\\mathbb{P} \\biggl( \\fnorm{\\widehat \\mathbf{P} - \\mathbf{P}} ^ 2 \\gtrsim \\frac{\\xi r\\pi_{\\max}\\beta ^ 4\\log p}{\\pi_{\\min} ^ 2\\alpha ^ 4n} + \\frac{\\xi\\beta ^ 2}{\\alpha rp ^ 2 \\pi_{\\max} \\log p}\\biggr) \\lesssim e^{- \\xi} + p^{-(\\xi - 1)} + p^{-10}. \n\t\t\\end{aligned}\n\t\\]\t\n\n\\end{theorem}\n\n\n\\begin{remark}\n\t\tWhen $n \\lesssim \\{rp \\pi_{\\max}(\\log p) \\beta\/ (\\pi_{\\min} \\alpha ^ {\\!3 \/ 2})\\} ^ 2$, the second terms of both the KL-Divergence and Frobenius-norm error bounds are dominated by the first terms respectively, so that \n\t\t\\be\n\t\t\t\\label{eq:hd_result}\n\t\t\tD_{\\mathrm{KL}}(\\mathbf{P}, \\widehat\\mathbf{P}) = O_{\\mathbb{P}}\\biggl(\\frac{r\\pi_{\\max}\\beta ^ 2 p \\log p}{\\pi_{\\min}\\alpha ^ 3n}\\biggr)~\\text{and}~\\fnorm{\\widehat\\mathbf{P} - \\mathbf{P}} ^ 2 = O_{\\mathbb{P}}\\biggl(\\frac{r\\pi_{\\max}\\beta ^ 4\\log p}{\\pi_{\\min} ^ 2\\alpha ^ 4n}\\biggr).\n\t\t\\ee\n\t\tWhen $\\alpha \\asymp \\beta$ and $\\pi_{\\max} \\asymp \\pi_{\\min}$, we have that $\\alpha, \\beta \\asymp 1$ and that $\\pi_{\\max}, \\pi_{\\min} \\asymp 1 \/ p$. Therefore, $D_{\\mathrm{KL}}(\\mathbf{P}, \\widehat \\mathbf{P}) = O_{\\mathbb{P}}(rp\\log p \/ n)$ and $\\fnorm{\\widehat \\mathbf{P} - \\mathbf{P}} ^ 2 = O_{\\mathbb{P}}(rp\\log p \/ n)$. These rates are consistent with those derived in the literature of low-rank matrix estimation \\citep{NWa11, KLT11}. For a big $n$, the current error bounds are sub-optimal: the second terms of the bounds are independent of $n$ and thus do not converge to zero as $n$ goes to infinity. These terms are due to the requirement of the uniform concentration of $\\widetilde D_{\\mathrm{KL}}(\\mathbf{P}, \\widehat\\mathbf{P})$, the empirical counterpart of $D_{\\mathrm{KL}}(\\mathbf{P}, \\widehat \\mathbf{P})$, to $D_{\\mathrm{KL}}(\\mathbf{P}, \\widehat \\mathbf{P})$ (see Lemma \\ref{lem:uniform_law}). We eliminate these trailing terms through an alternative proof strategy in Section \\ref{sec:alt}, though the resulting statistical rates have worse dependence on $\\alpha$ and $\\beta$ and are thus relegated to the appendix.\n\\end{remark}\n\n\\begin{remark}\n\tWhen $r = 1$, $\\mathbf{P}$ can be written as $1v^\\top$ for some vector $v\\in \\Re^{p}$, and then estimating $\\mathbf{P}$ essentially reduces to estimating a discrete distribution from multinomial count data. The first term of the upper bounds in Theorem \\ref{thm:nuclear} nearly matches (up to a log factor) the classical results of discrete distribution estimation $\\ell_2$ risks (see, e.g., \\citet[Pg. 349]{lehmann2006theory}). \n\n\\end{remark}\n\n\nNext we move on to the second approach -- using rank-constrained MLE to estimate $\\mathbf{P}$: \n\\begin{equation}\\label{prob:nonconvex-lowrank}\n\\begin{array}{rllll}\n\\widehat{\\P}^r  :=  &  \\argmin {\\ell_n}({\\mathbf{Q}}) \\\\\n \\mbox{s.t.} & {\\mathbf{Q}} 1_p = 1_p,\\quad \\alpha \/ p \\le Q_{ij}\\le \\beta \/ p,\\quad \\forall\\, 1\\le i,j\\le p,\\quad \\text{rank}({\\mathbf{Q}}) \\le r. \n\\end{array}\n\\end{equation}\nSimilarly to \\eqref{prob:convex-nuclear}, we cannot allow $\\mathbf{Q}$ to have zero entries. In contrast to $\\widehat \\mathbf{P}$, the rank-constrained MLE $\\widehat \\mathbf{P}^r$ enforces the prior knowledge ``$\\mathbf{P}$ is low-rank\" exactly without inducing any additional bias. It requires solving a non-convex and non-smooth optimization problem, for which we will provide an algorithm based on DC programming in Section \\ref{sec:6}. Here we first present its statistical guarantee. \n\n\\begin{theorem}[Statistical guarantee for the rank-constrained estimator]\n\t\\label{thm:rank}\n\t\tSuppose that Assumption \\ref{asp:1} holds and that $n\\pi_{\\max} (1 - \\rho_+) > \\max(20 \\log p, \\log n)$. There exist universal constants $C_1, C_2 > 0$ such that for any $\\xi > 0$, \n\t\t\\[\n\t\t\t\\mathbb{P}\\biggl\\{D_{\\mathrm{KL}}(\\mathbf{P}, \\widehat \\mathbf{P}^r) \\ge \\max\\biggl(\\frac{C_1 r\\pi_{\\max} \\beta^2 p \\log p }{\\pi_{\\min}\\alpha ^ 3n}, \\frac{\\xi \\pi_{\\min}}{rp \\pi_{\\max} \\log p}\\biggr)\\biggr\\} \\le C_2e^{- \\xi}, \n\t\t\\]\n\t\tand \n\t\t\\[\n\t\t\t\\mathbb{P}\\biggl\\{\\fnorm{\\widehat \\mathbf{P}^r - \\mathbf{P}} ^ 2 \\ge \\max\\biggl(\\frac{C_1 r\\pi_{\\max} \\beta^4 \\log p }{\\pi ^ 2_{\\min}\\alpha ^ 4n}, \\frac{\\xi \\beta ^ 2}{\\alpha rp ^ 2 \\pi_{\\max}\\log p}\\biggr)\\biggr\\} \\le C_2 e^{-\\xi}. \n\t\t\\]\n\\end{theorem}\n\n\\begin{remark}\n\tThe proof of the rank constrained method requires fewer inequality steps and is more straightforward than the that of the nuclear method. Although our upper bounds of the nuclear norm regularized method and the rank constrained one have the same rate, the difference of their proofs may implicitly suggest the advantage of the rank constrained method in the constant, as futher illustrated by our numerical studies.\n\\end{remark\n\nTo assess the quality of the established statistical guarantee, we further provide a lower bound result below. It shows that when $\\alpha, \\beta$ are constants, both estimators $\\widehat\\mathbf{P}$ and $\\widehat \\mathbf{P}^r$ are rate-optimal up to a logarithmic factor. Informally speaking, they are not improvable for estimating the class of rank-$r$ Markov chains. \n\\begin{theorem}[Minimax error lower bound for estimating low-rank Markov models]\n\t\\label{thm:lower_bound}\n\tConsider the following set of low-rank transition matrices\n\t$$\\Theta := \\bigl\\{\\mathbf{P}: \\forall j, k \\in [p], P_{jk} \\in \\{0\\} \\cup [\\alpha \/ p, +\\infty),~\\bP1_p = 1_p,~{\\rm rank}(\\mathbf{P}) \\le r\\bigr\\}.$$\n\tThere exists a universal constant $c > 0$ such that when $p(r -1) \\ge 192\\log 2$, we have\n\t\\[\n\t\t\\inf_{\\widehat{\\mathbf{P}}}\\sup_{\\mathbf{P}\\in \\Theta}\\mathbb{E}\\|\\widehat{\\mathbf{P}} - \\mathbf{P}\\|_F^2 \\geq \\frac{cp(r - 1)}{n\\alpha}. \n\t\\]\n\\end{theorem}\n\\begin{remark}\t\n\tTheorem \\ref{thm:lower_bound} shows that a smaller $\\alpha$ makes the estimation problem harder. It still remains to be an open problem whether $\\beta$ in Assumption \\ref{asp:1} should be in this minimax risk or not. \n\\end{remark}\n\nBesides the full transition matrix $\\mathbf{P}$, the leading left and right singular vectors of $\\mathbf{P}$, denoted by $\\bU, \\bV \\in \\mathbb{O}^{p\\times r}$, also play important roles in Markov chain analysis. For example, performing $k$-means on reliable estimate of $\\bU$ or $\\bV$ can give rise to state aggregation of the Markov chain \\citep{zhang2018optimal}. In the following, we further establish the statistical rate of estimating the singular subspace of the Markov transition matrix, based on the previous results. \n\n\\begin{theorem}\\label{thm:uv}\n\tUnder the setting of Theorem \\ref{thm:nuclear}, let $\\widehat{\\bU}, \\widehat \\bV \\in \\mathbb{O}^{p\\times r}$ be the left and right singular vectors of $\\widehat{\\mathbf{P}}$  respectively. Then there exist universal constants $C_1, C_2$, such that for any $\\xi > 0$, we have that \n\t\\begin{equation*}\n\t\\max\\left\\{\\|\\sin\\Theta(\\widehat{\\bU}, \\bU)\\|_F^2, \\|\\sin\\Theta(\\widehat{\\bV}, \\bV)\\|_F^2\\right\\} \n\t\\le  \\min\\biggl\\{\\max\\biggl(\\frac{C r\\pi_{\\max} \\beta^4\\log p }{\\pi ^ 2_{\\min}\\alpha ^ 4n\\sigma_r^2(\\mathbf{P})}, \\frac{\\xi \\beta ^ 2}{\\alpha rp ^ 2 \\pi_{\\max}(\\log p)\\sigma_r^2(\\mathbf{P}) }\\biggr), r\\biggr\\}\n\t\\end{equation*}\n\twith probability at least $1 - C_2(e^{- \\xi} + p^{-(\\xi - 1)} + p^{-10})$. Here, $\\sigma_r(\\mathbf{P})$ is the $r$-th largest singular value of $\\mathbf{P}$ and $\\|\\sin\\Theta(\\widehat{\\bU}, \\bU)\\|_F := (r - \\|\\widehat{\\bU}^\\top\\bU\\|_F^2)^{1\/2}$ is the Frobenius norm $\\sin \\Theta$ distance between $\\widehat{\\bU}$ and $\\bU$.\n\\end{theorem}\n\n\n\n\\subsection{Proof outline of Theorems \\ref{thm:nuclear}, \\ref{thm:rank}}\n\\label{sec:statistical_analysis}\nIn this section, we elucidate the roadmap to proving Theorems \\ref{thm:nuclear} and \\ref{thm:rank}. Complete proofs are deferred to the supplementary materials. We mainly focus on Theorem \\ref{thm:nuclear} for the nuclear-norm penalized MLE $\\widehat \\mathbf{P}$, as we use similar strategies to prove Theorem \\ref{thm:rank}. \n\nWe first show in the forthcoming Lemma \\ref{lem:large_lambda} that when the regularization parameter $\\lambda$ is sufficiently large, the statistical error $\\widehat \\boldsymbol{\\Delta} := \\widehat \\mathbf{P} - \\mathbf{P}$ falls in a restricted nuclear-norm cone. This cone structure is crucial to establishing strong  statistical guarantee for estimation of low-rank matrices with high-dimensional scaling \\citep{NWa11}. Define a linear subspace ${\\mathcal N} := \\{ \\mathbf{Q}: \\mathbf{Q} 1_p = 1_p \\}$ \nand denote the corresponding projection operator by $\\Pi_{{\\mathcal N}}$. In other words, for any $\\mathbf{Q} \\in {\\mathcal N}$ and any $j = 1, \\ldots, p$, the summation of all the entries in the $j$th row of $\\mathbf{Q}$ equals one. One can verify that for any $\\mathbf{Q} \\in \\Re^{p \\times p}$, $\\Pi_{{\\mathcal N}}(\\mathbf{Q}) = \\mathbf{Q} - \\mathbf{Q} 11^\\top \/ p$. Let $\\mathbf{P}=\\bU\\mathbf{D}\\bV^\\top$ be an SVD of $\\mathbf{P}$, where $\\bU, \\bV\\in \\Re^{p \\times r}$ are orthonormal and the diagonals of $\\mathbf{D}$ are in the non-increasing order. Define\n\\[\n\\begin{aligned}\n\t& {\\mathcal M}:=\\{\\mathbf{Q} \\in \\Re^{p \\times p}\\ |\\ \\text{row}(\\mathbf{Q})\\subseteq \\text{col}(\\bV), \\text{col}(\\mathbf{Q})\\subseteq \\text{col}(\\bU)\\}, \\\\\n\t& \\overline{{\\mathcal M}}^{\\perp}:=\\{\\mathbf{Q} \\in \\Re^{p \\times p}\\ |\\ \\text{row}(\\mathbf{Q})\\perp \\text{col}(\\bV), \\text{col}(\\mathbf{Q})\\perp \\text{col}(\\bU)\\},\n\\end{aligned}\n\\]\nwhere col$(\\cdot)$ and row$(\\cdot)$ denote the column space and row space respectively. We can write any $\\boldsymbol{\\Delta}\\in \\Re^{p \\times p}$ as\n\\[\n\\boldsymbol{\\Delta}=[\\bU, \\bU^\\perp]\\left[\n\\begin{array}{cc}\n\\boldsymbol{\\Gamma}}       \\def \\bDelta   {\\boldsymbol{\\Delta}_{11} & \\boldsymbol{\\Gamma}}       \\def \\bDelta   {\\boldsymbol{\\Delta}_{12} \\\\\n\\boldsymbol{\\Gamma}}       \\def \\bDelta   {\\boldsymbol{\\Delta}_{21} & \\boldsymbol{\\Gamma}}       \\def \\bDelta   {\\boldsymbol{\\Delta}_{22}\n\\end{array} \\right] [\\bV, \\bV^\\perp]^\\top.\n\\]\nDefine $\\boldsymbol{\\Delta}_{{\\mathcal W}}$ as the projection of $\\boldsymbol{\\Delta}$ onto any Hilbert space ${\\mathcal W}\\subseteq\\Re^{p \\times p}$. Then,\n\\be\n\\begin{aligned}\n\t\\boldsymbol{\\Delta}_{{\\mathcal M}} =\\bU\\boldsymbol{\\Gamma}}       \\def \\bDelta   {\\boldsymbol{\\Delta}_{11}{\\bV}^\\top,\\quad\\boldsymbol{\\Delta}_{\\overline{\\mathcal M}^\\perp} = \\bU^{\\perp} \\boldsymbol{\\Gamma}}       \\def \\bDelta   {\\boldsymbol{\\Delta}_{22}(\\bV^{\\perp})^{\\top}, \\quad \\boldsymbol{\\Delta}_{\\overline{\\mathcal M}} = [\\bU, \\bU^{\\perp}]\\left[\n\t\\begin{array}{cc}\n\t\t\\boldsymbol{\\Gamma}}       \\def \\bDelta   {\\boldsymbol{\\Delta}_{11} & \\boldsymbol{\\Gamma}}       \\def \\bDelta   {\\boldsymbol{\\Delta}_{12} \\\\\n\t\t\\boldsymbol{\\Gamma}}       \\def \\bDelta   {\\boldsymbol{\\Delta}_{21} & \\bzero\n\t\\end{array} \\right] [\\bV, \\bV^{\\perp}]^{\\top}.\n\\end{aligned}\n\\ee\nThe lemma below shows that $\\widehat \\boldsymbol{\\Delta} := \\widehat\\mathbf{P} - \\mathbf{P}$ falls in a nuclear-norm cone if $\\lambda$ is sufficiently large. \n\\begin{lemma}\n\t\\label{lem:large_lambda}\n\tIf $\\lambda\\ge 2 \\opnorm{\\Pi_{{\\mathcal N}}(\\nabla \\ell_n(\\mathbf{P}))}$ in \\eqref{prob:convex-nuclear}, then we have that \n\t\\[\n\t\t\\nnorm{ \\widehat\\boldsymbol{\\Delta}_{\\overline{\\mathcal M}^\\perp}} \\le 3\\nnorm{\\widehat\\boldsymbol{\\Delta}_{\\overline{\\mathcal M}}} + 4\\nnorm{\\mathbf{P}_{{\\mathcal M}^\\perp}}. \n\t\\]\n\tIn particular, when $\\mathbf{P} \\in {\\mathcal M}$, we have that $\\nnorm{\\mathbf{P}_{{\\mathcal M}^{\\perp}}} = 0$ and that\n\t\\be\n\t\\label{eq:converter}\n\t\\nnorm{\\widehat \\boldsymbol{\\Delta}} \\le \\nnorm{\\boldsymbol{\\Delta}_{\\overline{\\mathcal M}^{\\perp}}} + \\nnorm{\\boldsymbol{\\Delta}_{\\overline {\\mathcal M}}} \\le 4 \\nnorm{\\widehat \\boldsymbol{\\Delta}_{\\overline {\\mathcal M}}} \\le 4(2r)^{1 \/ 2}\\fnorm{\\widehat \\boldsymbol{\\Delta}}.\n\t\\ee\n\\end{lemma}\n\nLemma \\ref{lem:large_lambda} implies that the converting factor between the nuclear and Frobenius norms of $\\widehat \\boldsymbol{\\Delta}$ is merely $4(2r)^{1 \/ 2}$ when $\\P\\in \\mathcal{M}$, which is much smaller than the worst-case factor $p^{1 \/ 2}$ between nuclear and Frobenius norms of general $p$-by-$p$ matrices. This property of $\\widehat \\boldsymbol{\\Delta}$ is one cornerstone for establishing Theorem \\ref{thm:nuclear}.\n\n\n\\begin{comment}\n\\begin{remark}\n\tWhen $\\mathbf{P} \\in {\\mathcal M}$, $\\nnorm{\\mathbf{P}_{{\\mathcal M}^{\\perp}}} = 0$ and Lemma \\ref{lem:large_lambda} implies that \n\\be\n\t\\label{eq:converter}\n\t\\nnorm{\\widehat \\boldsymbol{\\Delta}} \\le \\nnorm{\\boldsymbol{\\Delta}_{\\overline{\\mathcal M}^{\\perp}}} + \\nnorm{\\boldsymbol{\\Delta}_{\\overline {\\mathcal M}}} \\le 4 \\nnorm{\\widehat \\boldsymbol{\\Delta}_{\\overline {\\mathcal M}}} \\le 4\\sqrt{2r}\\fnorm{\\widehat \\boldsymbol{\\Delta}}.\n\\ee\nThus the converting factor between the nuclear and Frobenius norms of $\\widehat \\boldsymbol{\\Delta}$ is merely $4\\sqrt{2r}$, which is much smaller than the worst-case factor $\\sqrt{p}$ between nuclear and Frobenius norms of general $p$-by-$p$ matrices. This property of $\\widehat \\boldsymbol{\\Delta}$ is one cornerstone for establishing Theorem \\ref{thm:nuclear} (see \\eqref{eq:stat_error_fnorm} for details).\n\\end{remark}\n\\end{comment}\n\n\nNext, we derive the rate of $\\opnorm{\\Pi_{{\\mathcal N}}(\\nabla \\ell_n(\\mathbf{P}))}$ to determine the order of $\\lambda$ that ensures the condition of Lemma \\ref{lem:large_lambda} to hold. \n\n\\begin{lemma}\n\t\\label{lem:gradient} \n\tUnder Assumption \\ref{asp:1}, whenever $n \\pi_{\\max} (1 - \\rho_+) \\ge 2 \\log p$, for any $\\xi > 1$,  \n\t\\[\n\t\t\\mathbb{P}\\biggl\\{ \\opnorm{\\Pi_{{\\mathcal N}}(\\nabla \\ell_n(\\mathbf{P}))} \\gtrsim \\biggl(\\frac{\\xi p ^ 2\\pi_{\\max}\\log p}{n\\alpha}\\biggr)^{1 \/ 2} + \\frac{\\xi p\\log p}{n \\alpha}\\biggr\\} \\le 4p^{-(\\xi - 1)} + \\exp\\biggl(- \\frac{n\\pi_{\\max} (1 - \\rho_+)}{2}\\biggr). \n\t\\]\n\\end{lemma}\n\n\\begin{remark}\n\tLemma \\ref{lem:gradient} is essentially due to concentration of a matrix martingale. Many existing results on measure concentration of dependent random variables \\citep{Mar96, Kon07, KRa08, Pau15} are not directly applicable because of the matrix structure of $\\nabla \\ell_n(\\mathbf{P})$. The main probabilistic tool we use here is the matrix Freedman inequality \\citep[][Corollary~1.3]{tropp2011freedman} that characterizes concentration behavior of a matrix martingale (See \\eqref{eq:matrix_freedman} for details). We notice two recent works, \\citet{WKo19} and \\citet{WKon19}, that use the same matrix Freedman inequality. Specifically, \\citet{WKon19} applied the matrix Freedman inequality to derive a confidence interval for the mixing time of a Markov chain based on its single trajectory, and \\citet{WKo19} used the same inequality to establish an upper bound for the sample complexity of learning a Markov chain. Finally, we also use an variant of Bernstein's inequality for general Markov chains \\citep[][Theorem~1.2]{JFS18} to derive an exponential tail bound for the status counts of the Markov chain ${\\mathcal X}$ (See \\eqref{eq:mc_bernstein} for details).\n\\end{remark}\n\n\n\nLet ${\\mathcal C} := \\{\\mathbf{Q} \\in \\mathbb{R}^{p \\times p}: \\nnorm{\\mathbf{Q} - \\mathbf{P}} \\le 4\\times 2^{1 \/2} \\fnorm{\\mathbf{Q} - \\mathbf{P}}, \\mathbf{Q} 1_p = 1_p, \\alpha \/ p \\le Q_{jk} \\le \\beta \/ p, \\forall (j, k) \\in  [p] \\times [p]\\}$. For any $\\mathbf{Q} \\in {\\mathcal C}$, define ${\\mathcal L}(\\mathbf{Q}) := \\mathbb{E} \\{- \\log (\\inn{\\mathbf{Q}, \\mathbf{X}_i})\\}$ and $\\ell_n(\\mathbf{Q}) := n^{-1}\\sum_{i = 1}^n -\\log(\\inn{\\mathbf{Q}, \\mathbf{X}_i})$. Recall that $D_{\\mathrm{KL}}(\\mathbf{P}, \\mathbf{Q}) = {\\mathcal L}(\\mathbf{Q}) - {\\mathcal L}(\\mathbf{P}) = \\sum_{i=1}^p \\pi_i D_{\\mathrm{KL}}(P_{i\\cdot}, Q_{i\\cdot}) = \\sum_{i = 1}^p \\sum_{j=1}^p \\pi_iP_{ij} \\log(P_{ij}\/Q_{ij})$. Define the empirical KL divergence of $\\mathbf{Q}$ from $\\mathbf{P}$ as\n\\[\n\\widetilde{D}_{\\mathrm{KL}}(\\mathbf{P},\\mathbf{Q}) := \\frac{1}{n}\\sum_{i=1}^n \\langle \\log (\\mathbf{P}) -\\log(\\mathbf{Q}), \\mathbf{X}_i\\rangle = \\ell_n(\\mathbf{Q}) - \\ell_n(\\mathbf{P}). \n\\]\nThe final ingredient of the analysis is the uniform converegence of $\\widetilde D_{\\mathrm{KL}}(\\mathbf{P}, \\mathbf{Q})$ to $D_{\\mathrm{KL}}(\\mathbf{P}, \\mathbf{Q})$ when $D_{\\mathrm{KL}}(\\mathbf{P}, \\mathbf{Q})$ is large. \n\\begin{lemma}\n\t\\label{lem:uniform_law}\n\tSuppose that $n\\pi_{\\max}(1 - \\rho_+) \\ge \\max(20\\log p, \\log n)$. For any $\\eta > \\pi_{\\min}  \/ (2\\pi_{\\max} rp \\log p)$, define ${\\mathcal C}(\\eta) := \\{\\mathbf{Q} \\in {\\mathcal C}: D_{\\mathrm{KL}}(\\mathbf{P}, \\mathbf{Q}) \\ge \\eta\\}$. Then there exist universal constants $C_1, C_2 > 0$ such that\n\t\\begin{equation}\\label{ineq:to-show-1}\n\t\\begin{aligned}\n\t\\mathbb{P}\\biggl\\{\\forall \\mathbf{Q} \\in {\\mathcal C}(\\eta), ~|\\widetilde D_{\\mathrm{KL}}(\\mathbf{P}, \\mathbf{Q}) - D_{\\mathrm{KL}}(\\mathbf{P}, \\mathbf{Q})| \\leq \\frac{1}{2}D_{\\mathrm{KL}}(\\mathbf{P}, \\mathbf{Q}) + &\\frac{C_1\\pi_{\\max} \\beta ^ 2rp\\log p}{\\pi_{\\min}\\alpha ^ 3n}\\biggr\\} \\\\\n\t& \\geq 1- C_2 \\exp\\biggl(- \\frac{\\eta \\pi_{\\max} rp \\log p}{\\pi_{\\min}}\\biggr).\n\t\\end{aligned}\n\t\\end{equation}\n\\end{lemma}\n\nTheorem \\ref{thm:nuclear} follows immediatley after one combines Lemmas \\ref{lem:large_lambda}, \\ref{lem:gradient} and \\ref{lem:uniform_law}. As for the rank-constrained MLE $\\widehat\\mathbf{P}^r$, let $\\widehat {\\boldsymbol{\\Delta}}(r) := \\widehat \\mathbf{P}^r - \\mathbf{P}$. Note that the rank constraint in \\eqref{prob:nonconvex-lowrank} implies that ${\\rm rank}(\\widehat \\boldsymbol{\\Delta}(r)) \\le 2r$. Thus, $\\nnorm{\\widehat\\boldsymbol{\\Delta}(r)} \\le (2r)^{1 \/ 2} \\fnorm{\\widehat\\boldsymbol{\\Delta}(r)}$ and Lemma \\ref{lem:uniform_law} remains applicable in the statistical anlaysis of $\\widehat \\mathbf{P}^r$. \n\n\n\n\n\n\n\n\n\n\n\n\\section{Computing Markov models using low-rank optimization}\n\nIn this section we develop efficient optimization methods to compute the proposed estimators for the low-rank Markov model. From now on, we drop the constraint that $\\alpha \/ p \\le Q_{ij}\\le \\beta \/ p$, which is used only to derive the statistical guarantees. In other words, $\\alpha$ and $\\beta$ are motivated by statistical theory, and do not need to be taken into account in the optimization. \n\n\\subsection{Optimization methods for the nuclear-norm regularized likelihood problem}\n\\label{sec:optNuc}\n\nWe first consider the nuclear-norm regularized likelihood problem \\eqref{prob:convex-nuclear}. It is a special case of the following linearly constrained optimization problem:\n\\begin{equation}\\label{prob:gen-convex-nuc}\n\\min \\, \\left\\{ g({\\mathbf{X}}) + c \\norm{{\\mathbf{X}}}_{*}\\,\\mid\\, {\\mathcal A}(\\mathbf{X}) = b \\right\\}, \n\\end{equation}\nwhere $g:\\Re^{p \\times p} \\to (-\\infty,+\\infty]$ is a closed, convex, but possibly non-smooth function, ${\\mathcal A}:\\Re^{p\\times p} \\to \\Re^m$ is a linear map, $b\\in\\Re^m$ and $c > 0$ are given data. If we take $\\alpha=0, \\beta=p$ in  problem \\eqref{prob:convex-nuclear}, it becomes a special case of the general problem \\eqref{prob:gen-convex-nuc} with $g({\\mathbf{X}}) = -\\ell_n({\\mathbf{X}}) + \\delta({\\mathbf{X}}\\ge 0)$, ${\\mathcal A}({\\mathbf{X}}) = {\\mathbf{X}} {\\bf 1}_p$, $b = {\\bf 1}_p$, and $\\delta(\\cdot)$ being the indicator function.\n\nDespite of its convexity, problem \\eqref{prob:gen-convex-nuc} is highly nontrivial due to the nonsmoothness of $g$ and the presence of the nuclear norm regularizer. Here, we propose to solve it via the dual approach.  \nThe dual of problem \\eqref{prob:gen-convex-nuc} is\n\\begin{equation} \n\\label{prob:D}\n\\begin{array}{rll}\n\\min  & g^*(-{\\bf\\Xi}) - \\inprod{b}{y}  \\\\\n\\mbox{s.t.} & {\\bf\\Xi} + {\\mathcal A}^*(y) + \\mathbb{S} = 0,\\quad \\norm{\\mathbb{S}}_2 \\le c,\n\\end{array}\n\\end{equation}\nwhere  $\\norm{\\cdot}_2$ denotes the spectral norm, and $g^*$ is the conjugate function of $g$ given by\n\\begin{align*} \ng^*({\\bf\\Xi}) {}\n= \\sum_{(i,j)\\in\\Omega} \\frac{n_{ij}}{n}(\\log \\frac{n_{ij}}{n} - 1 - \\log(-\\Xi_{ij})) + \\delta( \\Xi \\le 0) \\quad \\forall\\,{\\bf \\Xi}\\,\\in\\Re^{p\\times p}\n\\end{align*}\nwith $\\Omega = \\{(i,j) \\mid n_{ij} \\neq 0\\}$ and $\\overline \\Omega = \\{(i,j) \\mid n_{ij} = 0\\}$. \nGiven $\\sigma >0$, the augmented Lagrangian function ${\\mathcal L}_{\\sigma}$ associated with \\eqref{prob:D} is\n\\begin{align*}\n{\\mathcal L}_{\\sigma}({\\bf\\Xi}, y, \\mathbb{S}; {\\mathbf{X}}) = g^*(-{\\bf\\Xi}) - \\inprod{b}{y} + \\frac{\\sigma}{2} \\norm{{\\bf\\Xi} + {\\mathcal A}^*(y) + \\mathbb{S} + {\\mathbf{X}}\/\\sigma}^2 - \\frac{1}{2\\sigma}\\norm{{\\mathbf{X}}}^2.\n\\end{align*}\nWe consider popular ADMM type methods for solving problem \\eqref{prob:D} (A comprehensive numerical study has been conducted in \\citep{li2016schur} and justifies our procedure).\nSince there are three separable blocks in \\eqref{prob:D} (namely ${\\bf\\Xi}$, $y$, and $\\mathbb{S}$), the direct extended ADMM is not applicable. Indeed, it has been shown in \\citep{chen2016direct} that the direct extended ADMM for multi-block convex minimization problem is not necessarily convergent. Fortunately, the functions corresponding to block $y$ in the objective of \\eqref{prob:D} is linear. Thus we can apply the multi-block symmetric Gauss-Sediel based ADMM (sGS-ADMM) \\citep{li2016schur}.\nIn literature \\citep{chen2017efficient,ferreira2017semidefinite,lam2017fast,li2016schur,wang2018another}, extensive numerical experiments demonstrate that sGS-ADMM is not only convergent but also faster than the directly extended multi-block ADMM and its many other variants. Specifically, the algorithmic framework of sGS-ADMM for solving \\eqref{prob:D} is presented in Algorithm \\ref{alg:sGS-ADMM}. \n\n\\begin{algorithm} \n\t\\caption{An sGS-ADMM for solving \\eqref{prob:D}}\n\t\\label{alg:sGS-ADMM}\n\t\\begin{algorithmic}\n\t\t\\STATE {\\bfseries Input:} initial point $({\\bf\\Xi}^0, y^0, \\mathbb{S}^0, {\\mathbf{X}}^0)$, penalty parameter $\\sigma > 0$, maximum iteration number $K$, and the step-length $\\gamma \\in (0,(1+ \\sqrt{5})\/2)$\n\t\t\\FOR{$k=0$ {\\bfseries to} $K$}\n\t\t\\STATE $y^{k+\\frac{1}{2}} = \\argmin_{y} {\\mathcal L}_{\\sigma}({\\bf\\Xi}^k, y, \\mathbb{S}^k; {\\mathbf{X}}^k)$\n\t\t\\STATE ${\\bf\\Xi}^{k+1} = \\argmin_{{\\bf\\Xi}} {\\mathcal L}_{\\sigma}({\\bf\\Xi} ,y^{k+\\frac{1}{2}}, \\mathbb{S}^k; {\\mathbf{X}}^k) $\n\t\t\\STATE $y^{k+1} = \\argmin_{y} {\\mathcal L}_{\\sigma}({\\bf\\Xi}^{k+1}, y, \\mathbb{S}^k; {\\mathbf{X}}^k)$\n\t\t\\STATE $\\mathbb{S}^{k+1} = \\argmin_{\\mathbb{S}} {\\mathcal L}_{\\sigma}({\\bf\\Xi}^{k+1}, y^{k+1}, \\mathbb{S}, ; {\\mathbf{X}}^k)$\n\t\t\\STATE\n\t\t${\\mathbf{X}}^{k+1} = {\\mathbf{X}}^k + \\gamma\\sigma({\\bf\\Xi}^{k+1} + {\\mathcal A}^*(y^{k+1}) + \\mathbb{S}^{k+1})$\n\t\t\\ENDFOR\n\t\n\t\\end{algorithmic}\n\\end{algorithm}\n\nNext, we discuss how the $k$-th iteration of Algorithm \\ref{alg:sGS-ADMM} is performed:\n\\begin{description}\n\t\\item[{\\bf Computation of $y^{k+\\frac{1}{2}}$ and $y^{k+1}$}.]\n\tSimple calculations show that $y^{k+\\frac{1}{2}}$ and $y^{k+1}$ can be obtained by solving the following linear systems:\n\t\\begin{equation*}\n\t\\left\\{ \n\t\\begin{aligned}\n\t&y^{k+\\frac{1}{2}} = \\frac{1}{\\sigma}({\\mathcal A} {\\mathcal A}^*)^{-1}\\big( b - X^k - \\sigma({\\bf\\Xi}^k + \\mathbb{S}^k)\\big ), \\\\[5pt]\n\t&y^{k+1} =  \\frac{1}{\\sigma}({\\mathcal A} {\\mathcal A}^*)^{-1}\\big( b - X^k - \\sigma({\\bf\\Xi}^{k+1} + \\mathbb{S}^k)\\big ).\n\t\\end{aligned}\n\t\\right. \n\t\\end{equation*}\n\tIn our estimation problem, it is not difficult to verify that \n\t$ {\\mathcal A} {\\mathcal A}^* y = p y$ for any $y\\in \\Re^p$.\n\tThanks to this special structure, the above formulas can be further reduced to \n\t\\begin{equation*}\n\ty^{k+\\frac{1}{2}} = \\frac{1}{\\sigma p}\\big( b - {\\mathbf{X}}^k - \\sigma({\\bf\\Xi}^k + \\mathbb{S}^k)\\big) \\mbox{ and }\n\ty^{k+1} =  \\frac{1}{\\sigma p}\\big( b - {\\mathbf{X}}^k - \\sigma({\\bf\\Xi}^{k+1} + \\mathbb{S}^k)\\big).\n\t\\end{equation*}\n\t\\item [{\\bf Computation of ${\\bf \\Xi}^{k+1}$}.] \n\tTo compute ${\\bf \\Xi}^{k+1}$, we need to solve the following optimization problem:\n\n\t\\[\n\t\\min_{\\bf\\Xi} \\left\\{ g^*(-{\\bf\\Xi}) +\\frac{\\sigma}{2}\\norm{{\\bf\\Xi} + {\\bf R}^k}^2 \\right\\},\n\t\\]\n\twhere ${\\bf R}^k \\in \\Re^{p\\times p}$ is given. Careful calculations, together with the Moreau identity \\citep[Theorem 31.5]{rockafellar2015convex}, show that\n\t\\[\n\t{\\bf \\Xi}^{k+1} = \\frac{1}{\\sigma} [{\\mathbf{Z}}^k - \\sigma {\\bf R}^k]\\, \\mbox{ and } \\,\n\t{\\mathbf{Z}}^k = \\argmin_{{\\mathbf{Z}}} \\left\\{\\sigma g({\\mathbf{Z}}) + \\frac{1}{2}\\norm{{\\mathbf{Z}} - \\sigma {\\bf R}^k}^2 \\right\\}.\n\t\\]\n\tFor our estimation problem, i.e., $g({\\mathbf{X}}) = \\ell_n({\\mathbf{X}}) + \\delta( {\\mathbf{X}}\\ge 0)$, it is easy to see that \n\t$Z^k$ admits the following form:\n\t\\[\n\tZ^k_{ij} = \\frac{\\sigma R_{ij}^k + \\sigma\\sqrt{(R^k_{ij})^2 + 4 n_{ij}\/(n\\sigma)}}{2} \\quad \\mbox{if } (i,j)\\in \\Omega \\, \\mbox{ and } \\,\n\tZ^k_{ij} = \\sigma \\max(R_{ij}^k,0) \\quad \\mbox{if }  (i,j)\\in \\overline{\\Omega}.\n\t\\]\n\t\\item [{\\bf Computation of $\\mathbb{S}^{k+1}$}.]\n\tThe computation of $\\mathbb{S}^{k+1}$ can be simplified as:\n\t\\[\n\t\\mathbb{S}^{k+1} = \\argmin_{\\mathbb{S}} \\left\\{\n\t\\frac{\\sigma}{2} \\norm{\\mathbb{S} + {\\bf\\Xi}^{k+1} + {\\mathcal A}^* y^{k+1} + {\\mathbf{X}}^k\/\\sigma}^2 \\mid \n\t\\norm{\\mathbb{S}}_2 \\le c  \n\t\\right\\}.\n\t\\]\n\tLet ${\\bf W}_{k} := -({\\bf\\Xi}^{k+1} + {\\mathcal A}^* y^{k+1} + X^k\/\\sigma)$ admit the following singular value decomposition (SVD)\n\t$\n\t{\\bf W}_{k} = {\\bf U}_{k} {\\bf\\Sigma}_k {\\bf V}_k^\\top,\n\t$\n\twhere ${\\bf U}_{k}$ and ${\\bf V}_k$ are orthogonal matrices,\n\t${\\bf \\Sigma}_k = {\\rm Diag}(\\alpha_1^k,\\ldots, \\alpha_p^k)$ is\n\tthe diagonal matrix of singular values of $W_k$, with $\\alpha_1^k\\ge\\ldots\\ge \\alpha_p^k\\ge 0.$ Then, by Lemma 2.1 in \\citep{jiang2014partial}, we know that\n\t\\[\n\t\\mathbb{S}^{k+1} = {\\bf U}_k\\min({\\bf \\Sigma}_k,c){\\bf V}_k^\\top,\n\t\\]\n\twhere  $\\min({\\bf \\Sigma}_k,c) = {\\rm Diag}\\big(\\min(\\alpha_1^k,c),\\ldots, \\min(\\alpha_p^k,c)\\big)$.\n\n\tWe also note that in the implementation, only partial SVD, which is much cheaper than full SVD, is needed as $r\\ll p$.\n\\end{description}\nThe nontrivial convergence results and the sublinear non-ergodic iteration\ncomplexity of Algorithm \\ref{alg:sGS-ADMM} can be  obtained from \\cite{li2016schur} and \\cite{chen2017efficient}.\n{We put the convergence theorem and a sketch of the proof in the supplementary material.}\n\n\\subsection{Optimization methods for the rank-constrained likelihood problem}\n\\label{sec:6}\n\n\n\nNext we develop the optimization method for computing the rank-constrained likelihood maximizer from \\eqref{prob:nonconvex-lowrank}. In Subsection \\ref{sec:pen}, a penalty approach is applied to transform the original intractable rank-constrained problem into a DC programming problem. Then we solve this problem by a proximal DC (PDC) algorithm  in Subsection \\ref{subsec:proxdca}. \nWe also discuss the solver for the subproblems involved in the proximal DC algorithm. Lastly, a unified convergence analysis of a class of majorized indefinite-proximal DC (Majorized iPDC) algorithms is provided in Subsection \\ref{sec:DCA}.\n\n\\subsubsection{A penalty approach for problem \\eqref{prob:nonconvex-lowrank}. } \n\\label{sec:pen}\nRecall \\eqref{prob:nonconvex-lowrank} is intractable due to the non-convex rank constraint, we introduce a penalty approach to relax. We particularly study the following optimization problem:\n\\begin{equation}\\label{prob:gen-nonconvex-lowrank}\n\\min \\, \\left\\{ f({\\mathbf{X}}) \\,\\mid\\, {\\mathcal A}({\\mathbf{X}}) = b,\\, {\\rm rank}({\\mathbf{X}})\\le r \\right\\}, \n\\end{equation}\nwhere $f:\\Re^{p \\times p} \\to (-\\infty,+\\infty]$ is a closed proper convex, but possibly non-smooth, function.\nThe original rank-constraint maximum likelihood problem \\eqref{prob:nonconvex-lowrank} can be viewed as a special case of the general model \\eqref{prob:gen-nonconvex-lowrank}.\n\n\n\nGiven ${\\mathbf{X}}\\in\\Re^{p\\times p}$, let $\\sigma_1({\\mathbf{X}}) \\geq \\cdots \\geq \\sigma_p({\\mathbf{X}})\\geq 0$ be the singular values of ${\\mathbf{X}}$. Since ${\\rm rank}({\\mathbf{X}})\\le r$ if and only $\\sigma_{r+1}({\\mathbf{X}}) + \\ldots + \\sigma_{p}({\\mathbf{X}}) = \\|{\\mathbf{X}}\\|_\\ast - \\|{\\mathbf{X}}\\|_{(r)} = 0$ ($\\|{\\mathbf{X}}\\|_{(r)} = \\sum_{i=1}^r \\sigma_i({\\mathbf{X}})$ is the Ky Fan $r$-norm of ${\\mathbf{X}}$), \\eqref{prob:gen-nonconvex-lowrank} can be equivalently formulated as\n\\begin{equation*}\n\\min \\left\\{ f({\\mathbf{X}})  \\mid \\|{\\mathbf{X}}\\|_\\ast-\\|{\\mathbf{X}}\\|_{(r)}=0, \\, {\\mathcal A}({\\mathbf{X}}) = b \\right\\}.\n\\end{equation*}\nSee also \\citep[Equation (29)]{sun2010majorized}.\nThe penalized formulation of problem \\eqref{prob:gen-nonconvex-lowrank} is\n\\begin{equation}\\label{prob:nonconvex-pen-DC}\n\\mi\n\\left\\{ f({\\mathbf{X}}) +  c (\\|{\\mathbf{X}}\\|_\\ast - \\|{\\mathbf{X}}\\|_{(r)}) \\mid  {\\mathcal A}({\\mathbf{X}}) = b \\right\\},\n\\end{equation}\nwhere  $c>0$ is a penalty parameter. \nSince $\\norm{\\cdot}_{(r)}$ is convex, the objective in problem \\eqref{prob:nonconvex-pen-DC} is a difference\nof two convex functions: $f({\\mathbf{X}}) + c\\norm{{\\mathbf{X}}}_{*}$ and $c\\norm{{\\mathbf{X}}}_{(r)}$, i.e., \\eqref{prob:nonconvex-pen-DC} is a DC program.\n\nLet ${\\mathbf{X}}_c^*$ be an optimal solution to the penalized problem \\eqref{prob:nonconvex-pen-DC}. The following proposition shows that ${\\mathbf{X}}_c^\\ast$ is also the optimizer to \\eqref{prob:gen-nonconvex-lowrank} when it is low-rank.\n\\begin{proposition}\\label{prop:penlowrank}\n\tIf ${\\rm rank}({\\mathbf{X}}_c^*)\\le r$, then ${\\mathbf{X}}_c^*$ is also an optimal {solution} to the original problem \\eqref{prob:gen-nonconvex-lowrank}.\n\\end{proposition}\n\n\nIn practice, one can gradually increase the penalty parameter $c$ to obtain a sufficient low rank solution ${\\mathbf{X}}_c^*$. In our numerical experiments, we can obtain solutions with the desired rank with a properly chosen parameter $c$. \n\n\n\n\\subsubsection{A PDC algorithm for the penalized problem \\eqref{prob:nonconvex-pen-DC}.}\n\\label{subsec:proxdca}\n\n\n\nThe central idea of the DC algorithm \\citep{tao1997convex} is as follows: at each iteration, one approximates the concave part of the objective function by its affine majorant, then solves the resulting convex optimization problem. In this subsection, we present a variant of the classic DC algorithm for solving \\eqref{prob:nonconvex-pen-DC}. For the execution of the algorithm, we recall that the sub-gradient of Ky Fan $r$-norm at a point ${\\mathbf{X}}\\in\\Re^{p\\times p}$ \\citep{watson1993matrix} is\n\\[\\partial \\norm{{\\mathbf{X}}}_{(r)}=\\left\\{ \n{\\mathbf{U}} \\, {\\rm Diag}(q^*){\\mathbf{V}}^\\top \\, \\mid q^*\\in \\Delta \\right\\},\n\\\nwhere {$\\bU$ and $\\bV$ are the singular vectors of ${\\mathbf{X}}$,} and $\\Delta$ is the optimal solution set of the following problem\n\\begin{equation*}\n\\max_{q\\in \\Re^{p}} \\left\\{ \n\\sum_{i=1}^p \\sigma_i({\\mathbf{X}})q_i \\mid \n\\inprod{{\\bf 1}_p}{q} \\le r, \\, 0\\le q \\le 1\n\\right\\}.\n\\end{equation*}\nNote that one can efficiently obtain a component of $\\partial \\norm{{\\mathbf{X}}}_{(r)}$ by computing the\nSVD of $X$ and picking up the SVD vectors corresponding to the $r$ largest singular values.\nAfter these preparations, we are ready to state the PDC algorithm for problem \\eqref{prob:nonconvex-pen-DC} in Algorithm \\ref{alg:dc}.\nDifferent from the classic DC algorithm, an additional proximal term is added to ensure that solutions of subproblems \\eqref{subprob:mmalg} exist and the difference of two consecutive iterations converges. {See Theorem \\ref{thm:convergence-alg-MM} and Remark \\ref{remk:pdc} for more details.} \n\n\n\\begin{algorithm}\n\t\\caption{A PDC algorithm for solving \\eqref{prob:nonconvex-pen-DC}}\n\t\\label{alg:dc}\n\tGiven $c>0$, $\\alpha \\ge 0$, and the stopping tolerance $\\eta$, choose initial point ${\\mathbf{X}}^0\\in \\Re^{p\\times p}$.\n\tIterate the following steps for $k=0,1,\\ldots:$\n\t\n\t{\\bf 1.}  Choose  ${\\mathbf{W}}_k\\in \\partial \\norm{{\\mathbf{X}}^k}_{(r)}$. Compute\n\t\\begin{equation}\\label{subprob:mmalg}\n\t\\begin{split}\n\t\\mathbf{X}^{k+1}\\quad = \\quad & \\argmin f({\\mathbf{X}}) + c\\left(\\norm{{\\mathbf{X}}}_* - \\inprod{{\\mathbf{W}}_k}{{\\mathbf{X}} - {\\mathbf{X}}^k} - \\norm{{\\mathbf{X}}^k}_{(r)}\\right) \n\t+ \\frac{\\alpha}{2}\\norm{{\\mathbf{X}} - {\\mathbf{X}}^k}_F^2  \\\\\n\t& \\text{subject to } {\\mathcal A}({\\mathbf{X}}) = b.\n\t\\end{split}\n\t\\end{equation}\n\t{\\bf 2.} If $\\norm{{\\mathbf{X}}^{k+1} - {\\mathbf{X}}^k}_F\\le \\eta$, stop.\n\\end{algorithm}\nWe say that ${\\mathbf{X}}$ is a critical point of problem \\eqref{prob:nonconvex-pen-DC} if \n\\[\\partial (f({\\mathbf{X}}) + c\\norm{{\\mathbf{X}}}_* + \\delta({\\mathcal A}(X) = b) ) \\cap (c\\partial \\norm{{\\mathbf{X}}}_{(r)}) \\ne \\emptyset.\\]\nWe have the following convergence results for Algorithm \\ref{alg:dc}. \n\\begin{theorem}[Convergence of Algorithm \\ref{alg:dc}]\n\t\\label{thm:convergence-alg-MM}\n\tLet $\\{{\\mathbf{X}}^k\\}$ be the sequence generated by Algorithm \\ref{alg:dc} and $\\alpha \\ge 0$. Then $\\{ f({\\mathbf{X}}^k) + c(\\norm{{\\mathbf{X}}^k}_* - \\norm{{\\mathbf{X}}^k}_{(r)})\\}$ is a non-increasing sequence. If ${\\mathbf{X}}^{k+1} = {\\mathbf{X}}^k$ for some integer $k\\ge 0$, then ${\\mathbf{X}}^k$ is a critical point of \\eqref{prob:nonconvex-pen-DC}. Otherwise, it holds that\n\t\\begin{align*} \n\t&\\big(f({\\mathbf{X}}^{k+1}) + c(\\norm{{\\mathbf{X}}^{k+1}}_* - \\norm{{\\mathbf{X}}^{k+1}}_{(r)})\\big) \n\t- \\big( f({\\mathbf{X}}^k) + c(\\norm{{\\mathbf{X}}^k}_* - \\norm{{\\mathbf{X}}^k}_{(r)} )\\big)\n\t\\le {} -\\frac{\\alpha}{2}\\norm{{\\mathbf{X}}^{k+1} - {\\mathbf{X}}^k}^2_F.\n\t\\end{align*}\n\tMoreover, \n\tany accumulation point of the bounded sequence $\\{{\\mathbf{X}}^k\\}$ is a critical point of problem \\eqref{prob:nonconvex-pen-DC}.\n\tIn addition, if $\\alpha >0$, it holds that $\\lim_{k\\to \\infty}\\norm{{\\mathbf{X}}^{k+1} - {\\mathbf{X}}^k}_F = 0$.\n\\end{theorem}\n\n\\begin{remark}[Adjusting Parameters]\n\t\\label{remk:pdc}\n\t{\\rm In practice, a small $\\alpha >0$ is suggested to ensure strict decrease of the objective value and convergence \n\t\tof $\\{ \\norm{{\\mathbf{X}}^{k+1} - {\\mathbf{X}}^k}_F \\}$;\n\t\tif $f$ is strongly convex, one achieves these nice properties even if $\\alpha =0$ based on the results of Theorem \\ref{thm: MMconvergence}. The penalty parameter $c$ can be adaptively adjusted according to the rank of the sequence generated by Algorithm \\ref{alg:dc}.}\n\\end{remark}\n\n\\begin{remark}[Number of iterations of Algorithm \\ref{alg:dc}]\n\t\\label{prop:convergence-alg-MM}\n\tLet $\\eta >0$ be the stopping tolerance and $F^*$ be the optimal value of problem \\eqref{prob:nonconvex-pen-DC}. By using the inequality in Theorem \\ref{thm:convergence-alg-MM}, it can be shown that if $\\alpha >0$, then  Algorithm \\ref{alg:dc} terminates in no more than $K$ iterations, where\n\t\\[\n\tK = \\left\\lceil \\frac{2\\big(f(X^0) + c(\\norm{X^0}_* - \\norm{X^0}_{(r)}) - F^* \\big)}{\\alpha \\eta^2}\\right\\rceil + 1.\n\t\\] \n\\end{remark}\n\n\\begin{remark}[Statistical properties]\n\t\tThe statistical rate we derived in Theorem 2 does not carry over to the iterates of the DC algorithm here. Though we show in Theorem 5 that the DC algorithm can converge to a critical point, it remains unclear whethere this point is close to the global optimum and provably enjoys the statistical guarantee. Recently there have been many works conveying positive messages on the statistical properties of the non-convex optimization algorithms. For example, \\citet{LWa15} showed that any stationary point of the composite objective function they considered lies within statistical precision of the true parameter. We hope to establish similar theory for the proposed DC approach in future research.\n\\end{remark}\n\nNext, we discuss how to solve subproblems \\eqref{subprob:mmalg}. \\eqref{subprob:mmalg} is still a nuclear norm penalized convex optimization problem and is a special case of model \\eqref{prob:gen-convex-nuc} with $g({\\mathbf{X}}) = f({\\mathbf{X}}) + \\inprod{{\\mathbf{W}}}{{\\mathbf{X}}} + \\frac{\\alpha }{2}\\norm{{\\mathbf{X}}}_F^2$. Hence, Algorithm \\ref{alg:sGS-ADMM} can directly solve these subproblems efficiently. When Algorithm \\ref{alg:sGS-ADMM} is executed on this new function $g$, all computations, except for the update of $\\bf\\Xi$, have already been discussed in Section \\ref{sec:optNuc}. To update $\\bf\\Xi$ in the process of executing Algorithm \\ref{alg:sGS-ADMM} for solving \\eqref{subprob:mmalg} with $g({\\mathbf{X}}) = \\ell_n({\\mathbf{X}}) + \\delta({\\mathbf{X}}\\ge0) + \\inprod{{\\mathbf{W}}}{{\\mathbf{X}}} + \\frac{\\alpha }{2}\\norm{{\\mathbf{X}}}_F^2$, we need to solve the following minimization problem for given ${\\bf R}\\in\\Re^{p\\times p}$ and $\\sigma > 0$,\n\\[\n{\\mathbf{Z}}^* = \\argmin_{{\\mathbf{Z}}} \\left\\{ \\sigma g({\\mathbf{Z}}) + \\frac{1}{2} \\norm{{\\mathbf{Z}} - \\sigma{\\bf R}}^2\\right\\}.\n\\]  \n${\\mathbf{Z}}^*$ here can be calculated by\n\\begin{equation*}\nZ_{ij}^* = \n\\left\\{\n\\begin{aligned}\n{}&\\frac{(\\sigma R_{ij} - W_{ij}) + \\sigma\\sqrt{(R_{ij} - W_{ij}\/\\sigma)^2 + 4 (\\alpha + 1)n_{ij}\/(n\\sigma)}}{2(\\alpha + 1)} \\quad \\mbox{if } (i,j)\\in \\Omega; \\\\[5pt]\n{}&\\sigma \\max(R_{ij} - W_{ij}\/\\sigma,0) \\quad  \\mbox{if } (i,j)\\in \\overline{\\Omega}.\n\\end{aligned}\n\\right.\n\\end{equation*}\n\n\n\n\\subsubsection {A unified analysis for the majorized iPDC algorithm.}\n\\label{sec:DCA}\nDue to the presence of the proximal term $\\frac{\\alpha}{2}\\norm{{\\mathbf{X}} - {\\mathbf{X}}^k}^2$ in Algorithm \\ref{alg:dc}, the classical DC analyses cannot be applied directly. In this subsection, we provide a unified convergence analysis for the majorized indefinite-proximal DC (majorized iPDC) algorithm which includes Algorithm \\ref{alg:dc} as a special instance. \nLet $\\mathbb{X}$ be a finite-dimensional real Euclidean space endowed with inner product $\\inprod{\\cdot}{\\cdot}$ and induced norm $\\norm{\\cdot}$.\nConsider the following optimization problem\n\\begin{equation}\\label{eq:dca model}\n\\min_{x\\in \\mathbb{X}}\\; \\theta(x)\\triangleq g(x) + p(x) - q(x),\n\\end{equation}\nwhere $g:\\mathbb{X}\\to \\Re$ is a continuously differentiable function (not necessarily convex) with a Lipschitz continuous gradient and Lipschitz modulus $L_g >0$, i.e.,\n\\[\\norm{\\nabla f(x) - \\nabla f(x')} \\le L_g \\norm{x - x'}\\quad \\forall \\, x, x'\\in \\mathbb{X}, \\]\n$p:\\mathbb{X}\\to (-\\infty, +\\infty]$ and $q:\\mathbb{X}\\to(-\\infty, +\\infty]$ are two proper closed convex functions.\nIt is not difficult to observe that penalized problem \\eqref{prob:nonconvex-pen-DC} is a special instance of problem \\eqref{eq:dca model}. \nFor general model \\eqref{eq:dca model}, one can only expect the DC algorithm converges to a critical point $\\bar{x}\\in \\mathbb{X}$ of  \\eqref{eq:dca model} satisfying \n\\[ \\left(\\nabla g(\\bar{x}) + \\partial p(\\bar{x}) \\right) \\cap \\partial q(\\bar{x}) \\neq \\emptyset.\\]\n\n\n\nSince $g$ is continuously differentiable with Lipschitz continuous gradient, there exists a self-adjoint positive semidefinite linear operator ${\\mathcal G}:\\mathbb{X} \\to \\mathbb{X}$ such that for any\n$x, x'\\in \\mathbb{X}$,\n\\begin{equation*}\\label{ineq:majorization}\ng(x)\\le \\widehat{g}(x;x')\\triangleq g(x') + \\langle \\nabla g(x'), x-x'\\rangle  + \\frac{1}{2}\\|x - x'\\|^2_{\\mathcal{G}}.\n\\end{equation*}\n\n\\begin{algorithm}\n\t\\caption{A majorized indefinite-proximal DC algorithm for solving problem \\eqref{eq:dca model}}\n\t\\label{alg:dca-general}\n\tGiven initial point $x^0\\in \\mathbb{X}$ and stopping tolerance $\\eta$, choose a self-adjoint, possibly\n\tindefinite, linear operator ${\\mathcal T}:\\mathbb{X} \\to \\mathbb{X}$.\n\tIterate the following steps for $k=0,1,\\ldots:$\n\t\n\t{\\bf 1.} Choose $\\xi^k \\in \\partial q({x}^k)$. Compute\n\t\\begin{equation}\\label{eq:subproblem}\n\tx^{k+1} \\in \\argmin_{x\\in \\mathbb{X}} \\; \\left\\{ \\widehat{\\theta}(x;{x}^k) + \\frac{1}{2}\\|x - {x}^k\\|^2_{\\mathcal{T}}\\right\\}, \n\t\\end{equation}\n\twhere $\\widehat{\\theta}(x;{x}^k) \\triangleq \\widehat{g}(x;{x}^k) + p(x) - \\big(q({x}^k) + \\langle x-{x}^k, \\xi^k\\rangle \\big).$\n\t\n\t{\\bf 2.}  If $\\|{x}^{k+1}-{x}^k\\|\\le \\eta$, stop.\n\\end{algorithm}\n\n\n\nWe present the majorized iPDC algorithm for solving \\eqref{eq:dca model} in Algorithm \\ref{alg:dca-general} and provide the following convergence results. \n\\begin{theorem}[Convergence of iPDC]\\label{thm: MMconvergence}\n\tAssume that {$\\inf_{x\\in \\mathbb{X}}\\theta(x)>-\\infty$}. Let $\\{x^k\\}$ be the sequence generated by Algorithm \\ref{alg:dca-general}. \n\tIf $x^{k+1} = x^k$ for some $k\\ge 0$, then $x^k$ is a critical point of \\eqref{eq:dca model}. If $\\mathcal{G} + 2\\mathcal{T}\\succeq 0$, then\n\tany accumulation point of $\\{x^{k}\\}$, if exists, is a critical point of \\eqref{eq:dca model}. In addition, if $ \\mathcal{G} + 2\\mathcal{T}\\succ 0$, it holds that $\\displaystyle\\lim_{k\\to\\infty}\\|{x}^{k+1} - {x}^k\\| = 0$.\n\\end{theorem}\nThe proof of Theorem \\ref{thm: MMconvergence} is provided in the supplementary material.\n\n\\begin{remark}\nHere, we discuss the roles of linear operators ${\\mathcal G}$ and ${\\mathcal T}$.\nFirst, ${\\mathcal G}$ makes the subproblems \\eqref{eq:subproblem} in Algorithm \\ref{alg:dca-general} more amenable to efficient computations. Theorem \\ref{thm: MMconvergence} shows the algorithm is convergent if ${\\mathcal G} + 2{\\mathcal T} \\succeq 0$. This indicates that instead of adding the commonly used positive semidefinte or positive definite proximal terms, we allow ${\\mathcal T}$ to be indefinite for better practical performance. The computational benefit of using indefinite proximal terms has also been observed in \\citep{sun2010majorized,li2016majorized}.\nAs far as we know, Theorem \\ref{thm: MMconvergence} provides the first rigorous convergence proof of the DC algorithms with indefinite proximal terms. Second, ${\\mathcal G}$ and ${\\mathcal T}$ also help to guarantee that the solutions of the subproblems \\eqref{eq:subproblem} exist.\nSince ${\\mathcal G} + 2{\\mathcal T} \\succeq 0$ and ${\\mathcal G} \\succeq 0$, we have that $2{\\mathcal G} + 2{\\mathcal T} \\succeq 0$, i.e., ${\\mathcal G} + {\\mathcal T} \\succeq 0$.\nHence, ${\\mathcal G}+2{\\mathcal T} \\succeq 0$ (${\\mathcal G}+2{\\mathcal T} \\succ 0$) implies that subproblems \\eqref{eq:subproblem} \nare (strongly) convex. Third, the choices of ${\\mathcal G}$ and ${\\mathcal T}$ are very much problem dependent. The general principle is that ${\\mathcal G} + {\\mathcal T}$ should be as small as possible while  ensuring {$x^{k+1}$} is relatively easy to compute.\n\\end{remark}\n\n\n\n\n\n\n\n\n\n\n\n\\section{Simulation results}\n\\label{sec:num}\n\nIn this section, we conduct numerical experiments to validate our theoretical results. We first compare the proposed nuclear-norm regularized estimator and the rank-constrained estimator with previous methods in literature using synthetic data. We then use the rank-constrained method to analyze a dataset of Manhattan taxi trips to reveal citywide traffic patterns. All of our computational results are obtained by running {\\sc Matlab} (version 9.5) on a windows workstation (8-core, Intel Xeon W-2145 at 3.70GHz, 64 G RAM).\n\n\\subsection{Experiments with simulated data}\nWe randomly draw the transition matrix $\\P$ as follows. Let ${\\mathbf{U}}_0, {\\mathbf{V}}_0 \\in \\Re^{p\\times r}$ be random matrices with i.i.d. standard normal entries and let\n\\[\n\\widetilde {\\mathbf{U}}_{[i,:]} = ({\\mathbf{U}}_0 \\circ {\\mathbf{U}}_0)_{[i,:]} \/ \\norm{({\\mathbf{U}}_0)_{[i,:]}}_2^2 \\mbox{ and } \\widetilde {\\mathbf{V}}_{[:,j]} = ({\\mathbf{V}}_0 \\circ \\bV_0)_{[:,j]} \/ \\norm{({\\mathbf{V}}_0)_{[:,j]}}_2^2, \\quad i=1,\\ldots,p, j=1,\\ldots, r, \n\\]\nwhere $\\circ$ is the Hadamard product and $\\widetilde {\\mathbf{U}}_{[i,:]}$ denotes the $i$-th row of $\\widetilde{\\mathbf{U}}$. The transition matrix $\\P$ is obtained via $\\P = \\widetilde{\\mathbf{U}} \\widetilde{\\mathbf{V}}^\\top$. Then we simulate a Markov chain trajectory of length $n = {\\rm round} (krp \\log(p))$ on $p$ states, $\\{X_0,\\ldots, X_n\\}$, with varying values of $k$.\n\nWe compare the performance of four procedures: the nuclear norm penalized MLE, rank-constrained MLE, empirical estimator and spectral estimator. Here, the empirical estimator is the empirical count distribution matrix defined as follows: \n$$\\tilde{\\P} = \\left(\\tilde{\\P}_{ij}\\right)_{1\\leq i, j\\leq p}, \\quad \\tilde{\\P}_{ij} = \\left\\{\\begin{array}{ll}\n\\frac{\\sum_{k =1}^n 1_{\\{X_{k-1} = i, X_k = j\\}}}{\\sum_{k =1}^n 1_{\\{X_{k-1} = i\\}}}, & \\quad  \\text{when }\\sum_{k =1}^n 1_{\\{X_{k-1} = i\\}} \\geq 1;\\\\\n\\frac{1}{p}, & \\quad \\text{when } \\sum_{k =1}^n 1_{\\{X_{k-1} = i\\}} = 0.\n\\end{array}\\right.$$\nThe empirical estimator is in fact the unconstrained maximum likelihood estimator without taking into account the low-rank structure. The spectral estimator \\citep[Algorithm 1]{zhang2018optimal} is based on a truncated SVD. In the implementation of the nuclear norm penalized estimator, the regularization parameter $\\lambda$ in  \\eqref{prob:convex-nuclear} is set to be $C\\sqrt{{p\\log p\/n}}$ with constant $C$ selected by cross-validation.\nFor each method, let $\\widehat{{\\mathbf{U}}}$ and $\\widehat{{\\mathbf{V}}}$ be the leading $r$ left and right singular vectors of the resulting estimator $\\widehat \\P$. We measure the statistical performance of $\\widehat{\\mathbf{P}}$ through three quantities: \n\\begin{align*}\n \\eta_F := \\norm{\\P - \\widehat \\P}_F^2, ~~ \\eta_{KL}:= D_{\\mathrm{KL}}(\\mathbf{P},\\widehat\\mathbf{P}), \\mbox{ and } \\eta_{UV} := \\max\\bigl\\{ \\norm{\\sin \\Theta(\\widehat{{\\mathbf{U}}},{\\mathbf{U}})}_F^2, \\norm{\\sin \\Theta(\\widehat{{\\mathbf{V}}},{\\mathbf{V}})}_F^2 \\bigr\\}.\n\\end{align*} \n\n\nWe consider the following setting with $p=1000$, $r = 10$, and $k\\in [10, 100]$.\nThe results are plotted in Figure \\ref{figure:mlevsrank}.\nOne can observe from these results that for rank-constrained, nuclear norm penalized and spectral methods, $\\eta_F, \\eta_{KL}$ and $\\eta_{UV}$ converge to zero quickly as the number of the state transitions $n$ increases, while the statistical error of the empirical estimator decreases in a much slower rate. Among the three estimators in the zoomed plots (second rows of Figure \\ref{figure:mlevsrank}), the rank constrained estimator slightly outperforms the nuclear norm penalized estimator and the spectral estimator.\nThis observation is consistent with our algorithmic design: the nuclear norm minimization procedure is actually the initial step of Algorithm \\ref{alg:dc}; thus the rank-constrained estimator can be seen as a refined version of the nuclear norm regularized estimator. \n\n\nWe also consider the case where the invariant distribution $\\pi$ is ``imbalanced'', i.e., we construct $\\P$ such that $\\min_{i=1,\\ldots,p} \\pi_i$ is quite small and the appearance of some states is significantly less than the others. Specifically, given $\\gamma_1,\\gamma_2 >0$, we generate a diagonal matrix ${\\mathbf{D}}$ with i.i.d. beta-distributed (${\\rm Beta}(\\gamma_1, \\gamma_2)$) diagonal elements.\nAfter obtaining $\\widetilde {\\mathbf{U}}$ and $\\widetilde {\\mathbf{V}}$ in the same way as in the beginning of this subsection, we compute $\\widetilde \\P = \\widetilde {\\mathbf{U}} \\widetilde {\\mathbf{V}}^\\top {\\mathbf{D}}$. The ground truth transition matrix  $\\P$ is obtained after a normalization of $\\widetilde \\P$. Then, we simulate a Markov chain trajectory of length $n = {\\rm round}(krp\\log(p))$ on $p$ states.\nIn our experiment, we set $p = 1000$, $r = 10$, $k\\in [10,100]$, and $\\gamma_1 = \\gamma_2 = 0.5$. The detailed results are plotted in Figure \\ref{figure:betamlevsrank}. As can be seen from the figure, under the imbalanced setting, the rank-constrained, nuclear norm penalized and spectral methods perform much better than the empirical approach in terms of all the three statistical performance measures ($\\eta_F$, $\\eta_{KL}$ and $\\eta_{UV}$). In addition, the rank-constrained estimator exhibits a clear advantage over two other approaches.\n\n\n\\begin{figure*}\n\n\t\\begin{subfigure}{.33\\textwidth}\n\t\t\\label{figure:mle-etaF}\n\t\t\\includegraphics[width=\\linewidth]{figs\/\/normal\/\/mle-rankF-plot-n1000-r10-rolls10}\n\t\n\t\\end{subfigure}\n\t\\begin{subfigure}{.33\\textwidth}\n\t\t\\label{figure:mle-etaKL}\n\t\t\\includegraphics[width=\\linewidth]{figs\/\/normal\/\/mle-rankKL-plot-n1000-r10-rolls10}\n\t\n\t\\end{subfigure}\n\t\\begin{subfigure}{.33\\textwidth}\n\t\t\\label{figure:mle-etaUV}\n\t\t\\includegraphics[width=\\linewidth]{figs\/\/normal\/\/mle-rankUV-plot-n1000-r10-rolls10}\n\t\n\t\\end{subfigure}\n\n\t\\\\\n\t\n\t\\begin{subfigure}{.33\\textwidth}\n\t\t\\label{figure:svd-etaF}\n\t\t\\includegraphics[width=\\linewidth]{figs\/\/normal\/\/svd-rankF-plot-n1000-r10-rolls10}\n\t\n\t\\end{subfigure}\n\t\\begin{subfigure}{.33\\textwidth}\n\t\t\\label{figure:svd-etaKL}\n\t\t\\includegraphics[width=\\linewidth]{figs\/\/normal\/\/svd-rankKL-plot-n1000-r10-rolls10}\n\t\\end{subfigure}\n\t\\begin{subfigure}{.33\\textwidth}\n\t\t\\label{figure:svd-etaUV}\n\t\t\\includegraphics[width=\\linewidth]{figs\/\/normal\/\/svd-rankUV-plot-n1000-r10-rolls10}\n\t\n\t\\end{subfigure}\n\t\n\t\\caption{The first row compares the rank-constrained estimator, nuclear norm penalized estimator, spectral method, and empirical estimator in terms of $\\eta_F = \\norm{\\P - \\widehat \\P}_F^2, \\eta_{KL}= D_{\\mathrm{KL}}(\\mathbf{P},\\widehat\\mathbf{P})$, and $\\eta_{UV} = \\max\\bigl\\{ \\norm{\\sin \\Theta(\\widehat{{\\mathbf{U}}},{\\mathbf{U}},)}_F^2, \\norm{\\sin \\Theta(\\widehat{{\\mathbf{V}}},{\\mathbf{V}})}_F^2 \\bigr\\}$. The second row provides the zoomed plots of the first row without the empirical estimator. Here, $n = {\\rm round}(k  rp\\log p)$ with $p = 1,000$, $r = 10$ and $k$ ranging from $10$ to $100$. \n\t\n\t}\n\t\\label{figure:mlevsrank}\n\t\n\t\t\\begin{subfigure}{.33\\textwidth}\n\t\t\\label{figure:betamle-etaF}\n\t\t\\includegraphics[width=\\linewidth]{figs\/\/beta\/\/mle-rankF-beta-plot-n1000-r10-rolls10}\n\t\n\t\\end{subfigure}\n\t\\begin{subfigure}{.33\\textwidth}\n\t\t\\label{figure:betamle-etaKL}\n\t\t\\includegraphics[width=\\linewidth]{figs\/\/beta\/\/mle-rankKL-beta-plot-n1000-r10-rolls10}\n\t\n\t\\end{subfigure}\n\t\\begin{subfigure}{.33\\textwidth}\n\t\t\\label{figure:betamle-etaUV}\n\t\t\\includegraphics[width=\\linewidth]{figs\/\/beta\/\/mle-rankUV-beta-plot-n1000-r10-rolls10}\n\t\n\t\\end{subfigure}\n\n\t\\\\\n\t\n\t\\begin{subfigure}{.33\\textwidth}\n\t\t\\label{figure:betasvd-etaF}\n\t\t\\includegraphics[width=\\linewidth]{figs\/\/beta\/\/svd-rankF-beta-plot-n1000-r10-rolls10}\n\t\n\t\\end{subfigure}\n\t\\begin{subfigure}{.33\\textwidth}\n\t\t\\label{figure:betasvd-etaKL}\n\t\t\\includegraphics[width=\\linewidth]{figs\/\/beta\/\/svd-rankKL-beta-plot-n1000-r10-rolls10}\n\t\\end{subfigure}\n\t\\begin{subfigure}{.33\\textwidth}\n\t\t\\label{figure:betasvd-etaUV}\n\t\t\\includegraphics[width=\\linewidth]{figs\/\/beta\/\/svd-rankUV-beta-plot-n1000-r10-rolls10}\n\t\n\t\\end{subfigure}\n\t\n\t\\caption{The first row compares the rank-constrained estimator, nuclear norm penalized estimator, spectral method, and empirical estimator in terms of $\\eta_F = \\norm{\\P - \\widehat \\P}_F^2, \\eta_{KL}= D_{\\mathrm{KL}}(\\mathbf{P},\\widehat\\mathbf{P})$, and $\\eta_{UV} = \\max\\bigl\\{ \\norm{\\sin \\Theta(\\widehat{{\\mathbf{U}}},{\\mathbf{U}},)}_F^2, \\norm{\\sin \\Theta(\\widehat{{\\mathbf{V}}},{\\mathbf{V}})}_F^2 \\bigr\\}$ with imbalanced\n\t\tinvariant distribution. The second row provides the zoomed plots of the first row without the empirical estimator. Here, $n = {\\rm round}(k  rp\\log p)$ with $p = 1,000$, $r = 10$ and $k$ ranging from $10$ to $100$. \n\t}\n\t\\label{figure:betamlevsrank}\n\n\\end{figure*}\n\n\n\n\n\n\n\n\n\n\n\\subsection{Experiments with Manhattan Taxi data}\n\\label{sec:taxi}\n\nIn this experiment, we analyze a real dataset of $1.1\\times 10^7$ trip records of NYC Yellow cabs (Link: \\url{https:\/\/s3.amazonaws.com\/nyc-tlc\/trip+data\/yellow_tripdata_2016-01.csv}) in January 2016. Our goal is to partition the Manhattan island into several areas, in each of which the taxi customers share similar destination preference. This can provide guidance for balancing the supply and demand of taxi service and optimizing the allocation of traffic resources.\n\nWe discretize the Manhattan island into a fine grid and model each cell of the grid as a state of the Markov chain; each taxi trip can thus be viewed as a state transition of this \nMarkov chain \\citep{yang2017dynamic, benson2017spacey, liu2012understanding}. For stability concerns, our model ignores the cells that have fewer than $1,000$ taxi visits. Given that the traffic dynamics typically vary over time, \nwe fit the MC under three periods of a day, i.e., $06:00\\sim 11:59$ (morning), $12:00 \\sim 17:59$ (afternoon) and $18:00 \\sim 23:59$ (evening), where the number of the active states $p = 803$, $999$ and $1,079$ respectively. \nWe apply the rank-constrained likelihood approach to obtain the estimator $\\widehat \\mathbf{P}^r$ of the transition matrix, and then apply $k$-means to the left singular subspaces of $\\widehat{\\mathbf{P}}^r$ to classify all the states into several clusters.\nFigure \\ref{figure:lr4} presents the clustering result with $r = 4$ \nand $k = 4$ \nfor the three periods of a day.\n\nFirst of all, we notice that the locations within the same cluster are close with each other in geographical distance. This is non-trivial: we do not have exposure to GPS location in the clustering analysis. This implies that taxi customers in neighboring locations have similar destination preference, which is consistent with common sense. Furthermore, to track the variation of the traffic dynamics over time, \nFigure \\ref{figure:pb4} visualizes the distribution of the destination choice that is correspondent to the center of the green cluster in the morning, afternoon and evening respectively. We identify the varying popular destinations in different periods of the day and provide corresponding explanations in the following table: \n\\begin{table}[H]\n\t\\centering \n\t\\def\\arraystretch{0.7}\t\n\t\\begin{tabular}{c@{\\hskip 1cm}p{7cm}@{\\hskip 1cm}p{4cm}}\n\t\t\\hline\\hline\n\t\tTime & \\hfil Popular Destinations & \\hfil Explanation \\\\ \\hline\\hline\n\t\tMorning & New York--Presbyterian Medical Center,\\vspace{-.3cm} \\newline \\hfil 42--59 St. Park Ave, Penn Station & hospitals, workplaces,\\vspace{-.3cm} \\newline \\quad the train station \\\\ \\hline\n\t\tAfternoon & \\centering 66 St. Broadway & \\hfil lunch, afternoon break,\\vspace{-.3cm} \\newline short trips \\\\ \\hline\n\t\tEvening & \\centering Penn Station & go home \\\\\\hline\\hline \n\t\\end{tabular}\n\\end{table}\n\nFinally, it might be tempting to model the taxi trips by an HMM, where regions of Manhattan correspond to hidden states. However, such a region is always part of the current observation (i.e., location of taxi):  It is observable and is not a hidden state that has to be inferred from all past observations. As a result, although both HMM and the low-rank Markov model could apply to taxi trips, the low-rank Markov model is simpler and more accurate. \n\n\n\n\n\\begin{figure*}[t!]\n\t\\centering\n\n\n\n\n\n\t\\begin{subfigure}{.32\\textwidth\n\t\t\\label{figure:lrh1}\n\t\t\\includegraphics[width=\\linewidth]{figs\/\/taxi\/\/test-P-rank-r4hh2}\n\t\\end{subfigure} \n\t\\begin{subfigure}{.32\\textwidth\n\t\t\\label{figure:lrh2}\n\t\t\\includegraphics[width=\\linewidth]{figs\/\/taxi\/\/test-P-rank-r4hh3}\n\t\\end{subfigure}\n\n\t\\begin{subfigure}{.32\\textwidth\n\t\t\\label{figure:lrh3}\n\t\t\\includegraphics[width=\\linewidth]{figs\/\/taxi\/\/test-P-rank-r4hh4}\n\t\\end{subfigure}\n\t\\caption{\n\t\tThe meta-states compression of Manhattan traffic network via rank-constrained approach with $ r = 4$: mornings (left), afternoons (middle) and evenings (right). Each color or symbol represents a meta-state. One\n\t\tcan see the day-time state aggregation\n\t\tresults differ significantly from that of the evening time.}\n\t\\label{figure:lr4}\n\\end{figure*}\n\n\n\n\\begin{figure*}[tb]\n\t\\centering\n\t\\begin{subfigure}{.32\\textwidth\n\t\t\\label{figure:lr6h1}\n\t\t\\includegraphics[width=\\linewidth]{figs\/\/taxi\/\/P-cluster-rank-r4hh2}\n\t\\end{subfigure} \n\t\\begin{subfigure}{.32\\textwidth\n\t\t\\label{figure:lr6h2}\n\t\t\\includegraphics[width=\\linewidth]{figs\/\/taxi\/\/P-cluster-rank-r4hh3}\n\t\\end{subfigure}\n\n\t\\begin{subfigure}{.32\\textwidth\n\t\t\\label{figure:lr6h3}\n\t\t\\includegraphics[width=\\linewidth]{figs\/\/taxi\/\/P-cluster-rank-r4hh4}\n\t\\end{subfigure}\n\t\\caption{Visualization of the destination distributions corresponding to the pick-up locations in the green clusters in Figure \\ref{figure:lr4}: mornings (left), afternoons (middle) and evenings (right).}\n\t\\label{figure:pb4}\n\\end{figure*}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Conclusion}\n\\label{sec:conclusion}\nThis paper studies the recovery and state compression of low-rank Markov chains from empirical trajectories via a rank-constrained likelihood approach. We provide statistical upper bounds for the $\\ell_2$ risk and Kullback-Leiber divergence between the estimator and the true probability transition matrix for the proposed estimator. \nThen, a novel DC programming algorithm is developed to solve the associated rank-constrained optimization problem. The proposed algorithm non-trivially combines several recent optimization techniques, such as the penalty approach, the proximal DC algorithm, and the multi-block sGS-ADMM. We further study a new class of majorized indefinite-proximal DC algorithms for solving general non-convex non-smooth DC programming problems and provide a unified convergence analysis.\nExperiments on simulated data illustrate the merits of our approach.\n\n\n\\section{Introduction.}\\label{intro}\n\n\n\\input{main.tex}\n\n\n\n\n\n\n\n\n\\bibliographystyle{informs2014}\n\n\n\\section{Proof of Lemma \\ref{lem:restricted_strong_convexity}}\n\n\\section{Technical lemmas}\n\n\\begin{lemma}\n\t\t\\label{lem:kl_to_l2}\n\t\tGiven two discrete distributions $u, v \\in \\mathbb{R}^p$, if there exist $\\alpha, \\beta > 0$ such that $u_j \\in \\{0\\}\\cup [\\alpha \/ p, \\beta \/ p]$ and $v_j \\in [\\alpha \/ p, \\beta \/ p]$ for any $j \\in [p]$, then we have $$D_{\\mathrm{KL}}(u, v) \\ge \\{p\\alpha \/ (2\\beta ^ 2)\\} \\ltwonorm{u - v} ^ 2.$$ This implies that under Assumption \\ref{asp:1}, for any $\\mathbf{Q} \\in {\\mathcal C}$, $$\\fnorm{\\mathbf{P} - \\mathbf{Q}} ^ 2 \\le \\frac{2\\beta^2}{\\alpha\\pi_{\\min}p}D_{\\mathrm{KL}}(\\mathbf{P}, \\mathbf{Q}).$$\n\\end{lemma}\n\\begin{proof}{Proof of Lemma \\ref{lem:kl_to_l2}}\n\t\t\\noindent By the mean value theorem, for any $j \\in [p]$ such that $u_j \\neq 0$, there exists $\\xi_j \\in [\\alpha \/ p, \\beta \/ p]$ such that \n\t\t\\[\n\t\t\t\\log(v_j) - \\log(u_j) = \\frac{v_j - u_j}{u_j} - \\frac{(v_j - u_j) ^ 2}{2\\xi_j ^2}. \n\t\t\\]\n\t\tTherefore, \n\t\t\\[\n\t\t\t\\begin{aligned}\n\t\t\t\tD_{\\mathrm{KL}}(u, v) & = \\sum_{j: u_j \\neq 0} u_j\\log(u_j \/ v_j) = \\sum_{j: u_j \\neq 0} (u_j - v_j) + \\sum_{j: u_j \\neq 0} \\frac{(u_j - v_j) ^ 2}{2 \\xi_j ^ 2} \\\\\n\t\t\t\t& \\ge  1 - \\sum_{j: u_j \\neq 0} v_j + \\sum_{j: u_j \\neq 0} \\frac{p \\alpha(u_j - v_j) ^ 2}{2\\beta ^ 2} = \\sum_{j: u_j = 0} v_j - u_j + \\sum_{j: u_j \\neq 0} \\frac{p \\alpha(u_j - v_j) ^ 2}{2\\beta ^ 2} \\\\\n\t\t\t\t& \\ge  \\sum_{j: u_j = 0} \\frac{p(v_j - u_j) ^ 2}{\\beta} + \\sum_{j: u_j \\neq 0} \\frac{p \\alpha(u_j - v_j) ^ 2}{2\\beta ^ 2}  \\ge \\frac{p\\alpha}{2\\beta ^ 2} \\ltwonorm{u - v} ^ 2. \n\t\t\t\\end{aligned} \n\t\t\\]\n\t\tThen we have\n\t\t\\[\n\t\t\t\\fnorm{\\mathbf{P} - \\mathbf{Q}} ^ 2 = \\sum_{i \\in [p]} \\ltwonorm{P_{i\\cdot} - Q_{i\\cdot}} ^ 2 \\le \\sum_{i \\in [p]} \\frac{2\\beta ^ 2\\pi_{i}}{p\\alpha \\pi_{\\min}}D_{\\mathrm{KL}}(P_{i\\cdot}, Q_{i\\cdot}) = \\frac{2\\beta ^ 2}{p\\alpha \\pi_{\\min}}D_{\\mathrm{KL}}(\\mathbf{P}, \\mathbf{Q}). \n\t\t\\]\n\\end{proof}\n\\section{Proof of Theorem \\ref{thm:nuclear}}\n\n\\begin{proof}\n\t\\noindent Given the definition of $\\widehat \\mathbf{P}$, \n\t\\begin{equation}\\label{ineq:tilde_D-P-hat-P}\n\t\t\\widetilde{D}_{\\mathrm{KL}}(\\mathbf{P},\\widehat{\\mathbf{P}}) = \\frac{1}{n}\\sum_{i=1}^n \\langle\\log(\\mathbf{P}) - \\log(\\widehat{\\mathbf{P}}), \\mathbf{X}_i\\rangle = \\ell_n(\\widehat{\\mathbf{P}}) - \\ell_n(\\mathbf{P}) \\leq \\lambda (\\nnorm{\\widehat \\mathbf{P}} - \\nnorm{\\mathbf{P}}) \\le \\lambda \\nnorm{\\mathbf{P} - \\widehat \\mathbf{P}}.\n\t\\end{equation}\n\tThen we have \n\t\\be\n\t\\label{ineq:basic}\n\t\\begin{aligned}\n\t\tD_{\\mathrm{KL}}(\\mathbf{P}, \\widehat\\mathbf{P}) & = {\\mathcal L}(\\widehat\\mathbf{P}) - {\\mathcal L}(\\mathbf{P}) = {\\mathcal L}(\\widehat\\mathbf{P}) - \\ell_n(\\widehat\\mathbf{P}) + \\ell_n(\\widehat\\mathbf{P}) - \\ell_n(\\mathbf{P}) + \\ell_n(\\mathbf{P}) - {\\mathcal L}(\\mathbf{P}) \\\\\n\t\t& \\le {\\mathcal L}(\\widehat \\mathbf{P}) - \\ell_n(\\widehat \\mathbf{P}) + \\ell_n(\\mathbf{P}) - {\\mathcal L}(\\mathbf{P}) + \\lambda \\nnorm{\\mathbf{P} - \\widehat \\mathbf{P}} \\\\\n\t\t& = D_{\\mathrm{KL}}(\\mathbf{P}, \\widehat\\mathbf{P}) - \\widetilde D_{\\mathrm{KL}}(\\mathbf{P}, \\widehat\\mathbf{P}) + \\lambda \\nnorm{\\mathbf{P} - \\widehat \\mathbf{P}}. \n\t\\end{aligned}\n\t\\ee\n\tDefine ${\\mathcal E} := \\{\\lambda \\ge 2 \\opnorm{\\Pi_{{\\mathcal N}}(\\nabla \\ell_n(\\mathbf{P}))}\\}$. If ${\\mathcal E}$ holds, then by Lemma \\ref{lem:large_lambda} and then Lemma \\ref{lem:kl_to_l2}, we obtain that \n\t\\[\n\t\t\\begin{aligned}\n\t\t\tD_{\\mathrm{KL}}(\\mathbf{P}, \\widehat\\mathbf{P}) & \\le D_{\\mathrm{KL}}(\\mathbf{P}, \\widehat\\mathbf{P}) - \\widetilde D_{\\mathrm{KL}}(\\mathbf{P}, \\widehat\\mathbf{P}) + 4(2r)^{1 \/ 2}\\lambda \\fnorm{\\mathbf{P} - \\widehat \\mathbf{P}} \\\\\n\t\t\t& \t\\le  D_{\\mathrm{KL}}(\\mathbf{P}, \\widehat\\mathbf{P}) - \\widetilde D_{\\mathrm{KL}}(\\mathbf{P}, \\widehat\\mathbf{P}) + 8\\lambda \\beta\\biggl(\\frac{r D_{\\mathrm{KL}}(\\mathbf{P}, \\widehat \\mathbf{P})}{p\\pi_{\\min} \\alpha}\\biggr)^{\\!1\/2}. \n\t\t\\end{aligned}\n\t\\]\n\tFor any $\\xi > 1$, an application of Lemma \\ref{lem:uniform_law} with $\\eta = \\xi \\pi_{\\min}\/ (rp\\pi_{\\max}\\log p)$ yields\n\t\\[\n\t\t\\begin{aligned}\n\t\t\t\\mathbb{P}\\biggl[ \\biggl\\{D_{\\mathrm{KL}}(\\mathbf{P}, \\widehat \\mathbf{P}) \\le 16\\lambda \\beta\\biggl(\\frac{r D_{\\mathrm{KL}}(\\mathbf{P}, \\widehat \\mathbf{P})}{p\\pi_{\\min} \\alpha}\\biggr)^{\\!1\/2} +  \\frac{2C_1 r\\pi_{\\max}\\beta ^ 2 p\\log p}{\\pi_{\\min}\\alpha ^ 3n} + \\eta \\biggr\\} \\cap {\\mathcal E}\\biggr] \\ge 1 - C_2 e^{- \\xi} - \\mathbb{P}({\\mathcal E}^c), \n\t\t\\end{aligned}\n\t\\]\t\n\twhere $C_1$ and  $C_2$ are exactly the same constants as in Lemma \\ref{lem:uniform_law}. Some algebra yields that \n\t\\[\n\t\t\\begin{aligned}\n\t\t\t\\mathbb{P}\\biggl[ \\biggl\\{D_{\\mathrm{KL}}(\\mathbf{P}, \\widehat \\mathbf{P}) \\le \\frac{256\\lambda ^ 2 \\beta ^ 2 r}{p\\pi_{\\min} \\alpha} +  \\frac{2C_1 r\\pi_{\\max}\\beta ^ 2 p\\log p}{\\pi_{\\min}\\alpha ^ 3n} + \\eta \\biggr\\} \\cap {\\mathcal E}\\biggr] \\ge 1 - C_2 e^{- \\xi} - \\mathbb{P}({\\mathcal E}^c). \n\t\t\\end{aligned}\n\t\\]\t\t\n\tBy Lemma \\ref{lem:gradient}, there exists a universal constant $C_3 > 0$ such that if we choose \n\t\\[\n\t\t\\lambda = C_3\\biggl\\{\\biggl(\\frac{\\xi p ^ 2\\pi_{\\max} \\log p}{n\\alpha}\\biggr)^{1 \/ 2} + \\frac{\\xi p\\log p}{n \\alpha}\\biggr\\}, \n\t\\]\n\tthen for any $\\xi > 1$, whenever $n\\pi_{\\max}(1 - \\rho_+) \\ge \\max(20, \\xi ^ 2)\\log p$, we have that  \n\t\\[\n\t\t\\begin{aligned}\n\t\t\t\\mathbb{P} \\biggl( D_{\\mathrm{KL}}(\\mathbf{P}, \\widehat\\mathbf{P}^r) \\gtrsim \\frac{\\xi r\\pi_{\\max}\\beta ^ 2 p \\log p}{\\pi_{\\min}\\alpha ^ 3n} + \\frac{\\xi\\pi_{\\min}}{rp \\pi_{\\max}\\log p}\\biggr) \\lesssim e^{- \\xi} + p^{-(\\xi - 1)} + p^{-10},\n\t\t\\end{aligned}\n\t\\]\n\tas desired. The Frobenius-norm error bound follows immediately by applying Lemma \\ref{lem:kl_to_l2}. \n\\end{proof}\n\n\n\\section{Proof of Theorem \\ref{thm:rank}}\n\n\\begin{proof}\t\n\t\\noindent Given the definition of $\\widehat \\mathbf{P}^r$, \n\t\\begin{equation}\\label{ineq:tilde_D-P-hat-P}\n\t\\widetilde{D}_{\\mathrm{KL}}(\\mathbf{P},\\widehat{\\mathbf{P}}^r) = \\frac{1}{n}\\sum_{i=1}^n \\langle\\log(\\mathbf{P}) - \\log(\\widehat{\\mathbf{P}}^r), \\mathbf{X}_i\\rangle = \\ell_n(\\widehat{\\mathbf{P}}^r) - \\ell_n(\\mathbf{P}) \\leq 0.\n\t\\end{equation}\n\tThen we have\n\t\\be\n\t\\label{ineq:basic}\n\t\\begin{aligned}\n\t\tD_{\\mathrm{KL}}(\\mathbf{P}, \\widehat\\mathbf{P}^r) & = {\\mathcal L}(\\widehat\\mathbf{P}^r) - {\\mathcal L}(\\mathbf{P}) = {\\mathcal L}(\\widehat\\mathbf{P}^r) - \\ell_n(\\widehat\\mathbf{P}^r) + \\ell_n(\\widehat\\mathbf{P}^r) - \\ell_n(\\mathbf{P}) + \\ell_n(\\mathbf{P}) - {\\mathcal L}(\\mathbf{P}) \\\\\n\t\t& \\le {\\mathcal L}(\\widehat \\mathbf{P}^r) - \\ell_n(\\widehat \\mathbf{P}^r) + \\ell_n(\\mathbf{P}) - {\\mathcal L}(\\mathbf{P}) = D_{\\mathrm{KL}}(\\mathbf{P}, \\widehat\\mathbf{P}^r) - \\widetilde D_{\\mathrm{KL}}(\\mathbf{P}, \\widehat\\mathbf{P}^r). \n\t\\end{aligned}\n\t\\ee\n\tFor any $\\xi > 1$, an application of Lemma \\ref{lem:uniform_law} with $\\eta = \\pi_{\\min}\\xi \/ (rp\\pi_{\\max}\\log p)$ yields\n\t\\[\n\t\t\\mathbb{P}\\biggl\\{D_{\\mathrm{KL}}(\\mathbf{P}, \\widehat \\mathbf{P}^r) \\ge \\max\\biggl(\\frac{2C_1 r\\pi_{\\max} \\beta^2 p \\log p }{\\pi_{\\min}\\alpha ^ 3n}, \\frac{\\xi\\alpha ^ 2}{rp^ 2 \\pi_{\\max} \\log p}\\biggr)\\biggr\\} \\le C_2e^{- \\xi}, \n\t\\]\n\tas desired. The Frobenius-norm error bound immediately follows by Lemma \\ref{lem:kl_to_l2}. \n\t\n\\end{proof}\n\n\n\\section{Proof of Theorem \\ref{thm:lower_bound}}\n\n\\begin{proof}\n\t~\\hspace{-.5cm} To simplify the notation, assume without loss of generality that $p$ is a multiple of $4(r-1)$. For any $1\\leq k \\leq m$, consider\n\t\\begin{equation}\\label{eq:lower-bound-P^(k)2}\n\t\\begin{split}\n\t\\mathbf{P}^{(k)} = & \\begin{bmatrix}\n\t\\frac{2-\\alpha}{p} \\bone_{p\\times (p\/2)} ~~~~  \\frac{\\alpha}{p} \\bone_{p\\times (p\/2)} \n\t\\end{bmatrix} \\\\\n\t& + \\frac{\\eta(2-\\alpha)}{2p} \\begin{bmatrix}\n\t\\bzero_{(p\/2)\\times (p\/4)} & \\bzero_{(p\/2)\\times (p\/4)} & \\bzero_{(p\/2)\\times (p\/2)}\\\\\n\t~~ \\bR^{(k)} ~~ \\cdots  ~~~~~ \\bR^{(k)} & -\\bR^{(k)} ~~ \\cdots ~~ -\\bR^{(k)} & \\bzero_{(p\/4)\\times (p\/2)} \\\\\n\t\\underbrace{-\\bR^{(k)} ~~ \\cdots  ~~ - \\bR^{(k)}}_{l_0} & \\underbrace{~~\\bR^{(k)} ~~  \\cdots ~~~~~~  \\bR^{(k)}}_{l_0} & \\bzero_{(p\/4)\\times (p\/2)} \\\\\n\t\\end{bmatrix},\n\t\\end{split}\n\t\\end{equation}\n\twhere $l_0 = \\frac{p}{4(r - 1)}$, $\\bR^{(k)} \\in \\{0, 1\\}^{(p\/4)\\times (r-1)}$, and $\\eta$ is some positive value to be determined later. Let \n\t\\begin{equation}\n\t\\mu := \\biggl(\\frac{2-\\alpha}{p} 1_{p\/2}^\\top ~~ \\frac{\\alpha}{p} 1_{p\/2}^\\top\\biggr)^\\top.\n\t\\end{equation}\n\tFirst of all, regardless of the value of $\\bR^{(k)}$, one can see that for any $k \\in [m]$, \n\t\\begin{enumerate}\n\t\t\\item ${\\rm rank}(\\mathbf{P}^{(k)})\\leq r$; \n\t\t\\item $\\mu^\\top \\mathbf{P}^{(k)} = \\mu^\\top$, and hence $\\mu$ is the invariant distribution of $\\mathbf{P}^{(k)}$; \n\t\t\\item $\\mathbf{P}^{(k)} \\in \\Theta$. \n\t\\end{enumerate} \n\tLet $\\{\\bR^{(k)}\\}_{k = 1}^m$ be i.i.d. matrices of independent Rademacher entries, i.e., for any $k \\in [m]$, $\\{R^{(k)}_{ij}\\}_{i \\in [n], j\\in [d]}$ are independent Rademacher variables, and $\\{\\bR^{(k)}\\}_{k \\in [m]}$ are independent. For any $k \\neq l$,  one can see that $\\bigl\\{\\bigl|\\bR^{(k)}_{ij} - \\bR^{(l)}_{ij}\\bigr|\\bigr\\}$ are i.i.d. uniformly distributed on $\\{0, 2\\}$, and that \n\t$$\\mathbb{E}\\bigl|\\bR^{(k)}_{ij} - \\bR^{(l)}_{ij}\\bigr| = 1, \\quad {\\rm Var}\\bigl(\\bigl|\\bR^{(k)}_{ij} - \\bR^{(l)}_{ij}\\bigr|\\bigr)= 1,\\quad \\bigl|\\bigl|\\bR^{(k)}_{ij} - \\bR^{(l)}_{ij}\\bigr| - 1\\bigr| = 1.$$\n\tBy Bernstein's inequality \\citep[][Theorem~2.10]{BLM13}, for any $t>0$, \n\t\\begin{equation*}\n\t{\\mathbb{P}}\\biggl\\{\\biggl|\\bigl\\|\\bR^{(k)} - \\bR^{(l)}\\bigr\\|_1 - \\frac{p(r-1)}{4}\\biggr| \\ge \\biggl( \\frac{p(r - 1)t}{2}\\biggr)^{\\!1\/ 2} + t \\biggr\\} \\leq 2e^{-t}. \n\t\\end{equation*}\n\tLet $t = p(r - 1)\/ 64$ and $m = \\lfloor \\exp\\{p(r-1)\/128\\} \/ 2^{1 \/ 2} \\rfloor$. Since $p(r - 1) \\ge 192\\log 2$, we have that $m \\ge 2$. Then a union bound yields that\n\t\\begin{equation*}\n\t\\begin{split}\n\t{\\mathbb{P}}\\biggl(\\forall 1\\leq k < l \\leq m, ~ \\frac{p(r - 1)}{8} \\le \\bigl\\|\\bR^{(k)} - \\bR^{(l)}\\bigr\\|_1 \\le \\frac{3p(r - 1)}{8}\\biggr) \\ge 1 - 2m^2\\exp\\biggl(\\frac{-p(r - 1)}{64}\\biggr) > 0. \n\t\\end{split}\n\t\\end{equation*}\n\tHence, there exist $\\bR^{(1)},\\ldots, \\bR^{(m)} \\subseteq \\{-1, 1\\}^{(p \/ 4)\\times (r-1)}$ such that\n\t\\begin{equation}\\label{ineq:P^k-P^l}\n\t\\forall 1\\leq k < l \\leq m,~\\frac{p(r - 1)}{8} \\le \\bigl\\|\\bR^{(k)} - \\bR^{(l)}\\bigr\\|_1 \\le \\frac{3p(r - 1)}{8}, \n\t\\end{equation}\n\twhich, given that $\\fnorm{\\bR^{(j)} - \\bR^{(k)}} ^ 2 = 2\\|\\bR^{(j)} - \\bR^{(k)}\\|_1$, further implies that \n\t\\begin{equation}\n\t\\forall 1\\leq k < l \\leq m,~\\frac{p(r - 1)}{4} \\le \\fnorm{\\bR^{(k)} - \\bR^{(l)}}^2 \\le \\frac{3p(r - 1)}{4}. \n\t\\end{equation}\n\n\n\n\n\tNow we have that \n\t\\begin{equation*}\n\t\\begin{split}\n\t\\left\\|\\mathbf{P}^{(k)} - \\mathbf{P}^{(l)}\\right\\|_1 = \\frac{2l_0\\eta(2-\\alpha)}{p}  \\|\\bR^{(k)} - \\bR^{(l)}\\|_1 \\geq \\frac{\\eta p (2-\\alpha)}{16}, \\left\\|\\mathbf{P}^{(k)} - \\mathbf{P}^{(l)}\\right\\|_1 \\le \\frac{3\\eta p (2 - \\alpha)}{16},  \\\\\n\t\\fnorm{\\mathbf{P}^{(k)} - \\mathbf{P}^{(l)}} ^ 2 = \\frac{l_0\\eta ^ 2(2-\\alpha) ^ 2}{p ^ 2}  \\fnorm{\\bR^{(k)} - \\bR^{(l)}} ^ 2 \\geq \\frac{\\eta ^ 2(2-\\alpha) ^ 2}{16}, \\left\\|\\mathbf{P}^{(k)} - \\mathbf{P}^{(l)}\\right\\|_1 \\le \\frac{3\\eta ^ 2(2-\\alpha) ^ 2}{16}. \n\t\\end{split}\n\t\\end{equation*}\n\tBesides, \n\t\\begin{equation*}\n\t\\begin{split}\n\tD_{\\mathrm{KL}} ({\\mathcal X}^{(k)} || {\\mathcal X}^{(l)}) & = n\\sum_{i\\in [p]} \\pi_i D_{\\mathrm{KL}}\\bigl(\\mathbf{P}_{[i,:]}^{(k)},  \\mathbf{P}^{(l)}_{[i,:]}\\bigr)\n\t=  n\\sum_{i=(p\/2)+1}^{p} \\sum_{j=1}^{p\/2} \\frac{\\alpha}{p} P_{ij}^{(k)}\\log \\bigl(P_{ij}^{(k)}\/P_{ij}^{(l)}\\bigr) \\\\\n\t& = \\frac{2n\\alpha}{p}\\sum_{i=1}^{p\/4} \\frac{2-\\alpha}{2} D_{\\mathrm{KL}}\\bigl(u^{(k)}_i, u^{(l)}_i\\bigr),\n\t\\end{split}\n\t\\end{equation*}\n\twhere $u^{(k)}_i = \\frac{2}{p}1_{p\/2} + \\frac{\\eta}{p}\\left[\\bR^{(k)}_{[i,:]} ~ \\cdots ~ \\bR^{(k)}_{[i,:]} ~ -\\bR^{(k)}_{[i,:]} ~ \\cdots ~ -\\bR^{(k)}_{[i,:]}\\right]$ corresponds to a $(p\/2)$-dimensional distribution. By \\citet[][Lemma~4]{ZWa19}, we have that \n\t\\begin{equation*}\n\tD_{\\mathrm{KL}}\\bigl(u^{(k)}_i, u^{(l)}_i\\bigr) \\leq \\frac{3l_0\\eta ^ 2}{p}\\bigl\\|\\bR_{[i,:]}^{(k)} - \\bR_{[i,:]}^{(l)}\\bigr\\|_2^2 = \\frac{6 l_0\\eta^2}{p}\\bigl\\|\\bR_{[i,:]}^{(k)} - \\bR_{[i,:]}^{(l)}\\bigr\\|_1.\n\t\\end{equation*}\n\tTherefore, \n\t\\begin{equation*}\n\t\\begin{split}\n\tD_{\\mathrm{KL}}({\\mathcal X}^{(k)}, {\\mathcal X}^{(l)}) & = \\frac{6n\\alpha(2-\\alpha)l_0 \\eta ^ 2}{p ^ 2}\\sum_{i=1}^{p\/4} \\bigl\\|\\bR_{[i,:]}^{(k)} - \\bR_{[i,:]}^{(l)}\\bigr\\|_1 \\leq \\frac{12n\\alpha l_0\\eta^2}{p^2}\\|\\bR^{(k)} - \\bR^{(l)}\\|_1 \\\\\n\t& \\leq \\frac{12n\\alpha l_0 \\eta^2}{p^2} \\frac{3p(r-1)}{8} = \\frac{9n\\eta^2\\alpha}{8}.\n\t\\end{split}\n\t\\end{equation*}\n\n\tBy Fano's inequality \\citep[][Lemma~3]{yu1997assouad}, we have that\n\t\\begin{equation*}\n\t\\begin{split}\n\n\t\\inf_{\\widehat{\\mathbf{P}}} \\sup_{\\mathbf{P}\\in \\Theta} \\fnorm{\\widehat{\\mathbf{P}} - \\mathbf{P}} ^ 2 \\geq & \\inf_{\\widehat{\\mathbf{P}}} \\sup_{\\mathbf{P}\\in \\{\\mathbf{P}^{(1)}, \\ldots, \\mathbf{P}^{(m)}\\}} \\fnorm{\\hat{\\mathbf{P}} - \\mathbf{P}} ^ 2 \\geq \\frac{\\eta ^ 2(2-\\alpha) ^ 2}{16} \\left(1 - \\frac{9n\\eta^2\\alpha - \\log 2}{\\log m}\\right). \n\n\n\t\\end{split}\n\t\\end{equation*}\n\tThere exist universal constants $c_1, c_2 > 0$ such that when $p(r - 1) \\ge 192 \\log 2$, choosing $\\eta = c_1\\{p(r - 1) \/ (n \\alpha)\\}^{\\!1 \/ 2}$ yields that \n\t\\[\n\t\\inf_{\\widehat{\\mathbf{P}}} \\sup_{\\mathbf{P}\\in \\Theta} \\fnorm{\\widehat{\\mathbf{P}} - \\mathbf{P}} ^ 2 \\ge c_2\\frac{p(r - 1)}{n\\alpha}. \n\t\\]\n\t\\quad $\\square$\n\t\n\\end{proof}\n\n\n\\section{Proof of Theorem \\ref{thm:uv}}\n\\begin{proof}\n\t~Let $\\widehat{\\bU}_{\\perp}, \\widehat{\\bV}_{\\perp} \\in \\Re^{p\\times (p-r)}$ be the orthogonal complement of $\\widehat{\\bU}$ and $\\widehat{\\bV}$. Since $\\bU, \\bV, \\widehat{\\bU}$, and $\\widehat{\\bV}$ are the leading left and right singular vectors of $\\mathbf{P}$ and $\\widehat{\\mathbf{P}}$, we have\n\t\\begin{equation*}\n\t\\begin{split}\n\t\t\\|\\widehat{\\mathbf{P}} - \\mathbf{P}\\|_F \\geq & \\|\\widehat{\\bU}_{\\perp}^\\top(\\widehat{\\mathbf{P}} - \\bU\\bU^\\top \\mathbf{P})\\|_F = \\|\\widehat{\\bU}_{\\perp}^\\top \\bU\\bU^\\top\\mathbf{P}\\|_F \\geq \\|\\widehat{\\bU}^\\top_{\\perp} \\bU\\|_F \\sigma_r(\\bU^\\top \\mathbf{P}) = \\|\\sin\\Theta(\\widehat{\\bU}, \\bU)\\|_F \\sigma_r(\\mathbf{P}).\n\t\\end{split}\n\t\\end{equation*}\n\tSimilar argument also applies to $\\|\\sin\\Theta(\\widehat{\\bV}, \\bV)\\|$. Thus,\n\t$$\\max\\bigl(\\|\\sin\\Theta(\\widehat{\\bU}, \\bU)\\|_F, \\|\\sin\\Theta(\\widehat{\\bV}, \\bV)\\|_F\\bigr)\\le \\min\\biggl(\\frac{\\|\\widehat{\\mathbf{P}}-\\mathbf{P}\\|_F}{\\sigma_r(\\mathbf{P})}, r^{1 \/ 2}\\biggr).$$\n\tThe rest of the proof immediately follows from Theorem \\ref{thm:nuclear}.\n\\end{proof}\n\n\n\n\\section{Proof of Lemma \\ref{lem:large_lambda}}\n\n\\begin{proof}\n\t~By the inequality (52) in Lemma 3 in the Appendix of \\citet{negahban2012restricted}, we have for any $\\boldsymbol{\\Delta} \\in \\Re^{p \\times p}$, \n\t\\[\n\t\t\\nnorm{\\mathbf{P} + \\boldsymbol{\\Delta}} - \\nnorm{\\mathbf{P}} \\ge \\nnorm{\\boldsymbol{\\Delta}_{\\overline{\\mathcal M}^\\perp}} - \\nnorm{\\boldsymbol{\\Delta}_{\\overline {\\mathcal M}}} - 2\\nnorm{\\mathbf{P}_{{\\mathcal M}^\\perp}}. \n\t\\]\n\tBesides, \n\t\\[\n\t\t\\begin{aligned}\n\t\t\\ell_n(\\mathbf{P} + \\boldsymbol{\\Delta}) - \\ell_n(\\mathbf{P}) & \\ge \\inn{\\nabla \\ell_n(\\mathbf{P}), \\boldsymbol{\\Delta}} = \\inn{\\Pi_{{\\mathcal N}}(\\nabla\\ell_n(\\mathbf{P})), \\boldsymbol{\\Delta}} \\ge - |\\inn{\\Pi_{{\\mathcal N}}(\\nabla \\ell_n (\\mathbf{P})), \\boldsymbol{\\Delta}}| \\\\\n\t\t& \\ge - \\opnorm{\\Pi_{{\\mathcal N}}(\\nabla \\ell_n(\\mathbf{P}))} \\nnorm{\\boldsymbol{\\Delta} } \\ge - \\frac{\\lambda}{2} \\bigl(\\nnorm{\\boldsymbol{\\Delta}_{\\overline {\\mathcal M}}} + \\nnorm{\\boldsymbol{\\Delta}_{\\overline {\\mathcal M}^\\perp}} \\bigr). \n\t\t\\end{aligned}\n\t\\]\n\tBy the optimality of $\\widehat\\mathbf{P}$, $\\ell_n(\\widehat\\mathbf{P}) + \\lambda \\nnorm{\\widehat\\mathbf{P}} \\le \\ell_n(\\mathbf{P}) + \\lambda \\nnorm{\\mathbf{P}}$. Therefore, \n\t\\[\n\t\t\\lambda \\bigl(\\nnorm{\\boldsymbol{\\Delta}_{\\overline {\\mathcal M}}} + 2\\nnorm{\\mathbf{P}_{{\\mathcal M}^\\perp}} -\\nnorm{\\boldsymbol{\\Delta}_{\\overline{\\mathcal M}^\\perp}} \\bigr) \\ge \\lambda (\\nnorm{\\mathbf{P}} - \\nnorm{\\widehat\\mathbf{P}}) \\ge - \\frac{\\lambda}{2}\\bigl( \\nnorm{\\widehat\\boldsymbol{\\Delta}_{\\overline{\\mathcal M}}} + \\nnorm{ \\widehat\\boldsymbol{\\Delta}_{\\overline{\\mathcal M}^\\perp}}\\bigr), \n\t\\]\n\tfrom which we deduce that  \n\t\\[\n\t\\nnorm{ \\widehat\\boldsymbol{\\Delta}_{\\overline{\\mathcal M}^\\perp}} \\le 3\\nnorm{\\widehat\\boldsymbol{\\Delta}_{\\overline{\\mathcal M}}} + 4\\nnorm{\\mathbf{P}_{{\\mathcal M}^\\perp}}. \n\t\\]\n\\end{proof}\n\n\\section{Proof of Lemma \\ref{lem:gradient}}\n\n\\begin{proof}\n\t\t~Some algebra yields that \n\t\\be\n\t\\label{eq:log_likelihood}\n\t\\nabla \\ell_n(\\mathbf{Q} ) = \\frac{1}{n} \\sum\\limits_{i=1}^n -\\frac{\\mathbf{X}_i}{\\inn{\\mathbf{Q}, \\mathbf{X}_i}}. \n\t\\ee\n\tFor ease of notation, write $\\bZ_i := - \\mathbf{X}_i \/ \\inn{\\mathbf{P}, \\mathbf{X}_i}$. Note that $\\bZ_i$ is well-defined almost surely. Besides, \n\t\\[\n\t\\mathbb{E}(\\bZ_i | \\bZ_{i - 1}) = \\mathbb{E}(\\bZ_i | \\mathbf{X}_{i-1}) = \\sum\\limits_{j=1}^p -\\frac{e_{X_{i - 1}}e_{j}^\\top}{P_{X_{i - 1}, j}} P_{X_{i - 1}, j} = -e_{X_{i - 1}} 1^\\top. \n\t\\]\n\tThus $\\opnorm{\\bZ_i - \\mathbb{E}(\\bZ_i | \\bZ_{i - 1})} \\le p \/ \\alpha + \\sqrt{p} =: R < \\infty$. Define $\\mathbf{S}_k := \\sum_{i=1}^k \\bZ_i - \\mathbb{E}(\\bZ_i| \\bZ_{i - 1})$, then $\\{\\mathbf{S}_k\\}_{k = 1}^n$ is a matrix martingale. In addition, \n\t\\[\n\t\\begin{aligned}\n\t\\mathbb{E} & \\bigl\\{(\\bZ_i -  \\mathbb{E}(\\bZ_i | \\bZ_{i - 1}))^\\top (\\bZ_i - \\mathbb{E} (\\bZ_i | \\bZ_{i - 1})) | \\{\\mathbf{S}_k\\}_{k = 1}^{i -1}\\bigr\\}  = \\mathbb{E} \\bigl\\{(\\bZ_i -  \\mathbb{E}(\\bZ_i | \\bZ_{i - 1}))^\\top (\\bZ_i - \\mathbb{E} (\\bZ_i | \\bZ_{i - 1})) | \\bZ_{i - 1}\\bigr\\}  \\\\\n\t& = \\mathbb{E} (\\bZ^\\top_i \\bZ_i | \\bZ_{i - 1}) - \\mathbb{E}(\\bZ_i | \\bZ_{i - 1})^\\top \\mathbb{E} (\\bZ_i | \\bZ_{i - 1}) = \\biggl(\\sum_{j = 1}^p \\frac{e_je_j^\\top}{P_{X_{i - 1}, j}}\\biggr) -  11^\\top =: \\bW^{(1)}_i, \n\t\\end{aligned}\n\t\\]\n\tand similarly, \n\t\\[\n\t\\begin{aligned}\n\t\\mathbb{E}  \\bigl\\{(\\bZ_i -  \\mathbb{E}(\\bZ_i | \\bZ_{i - 1}))(\\bZ_i - \\mathbb{E} (\\bZ_i | \\bZ_{i - 1}))^\\top | \\{\\mathbf{S}_k\\}_{k = 1}^{i -1} \\bigr\\} =  \\biggl(\\sum\\limits_{j = 1}^ p \\frac{ e_{X_{i - 1}}e_{X_{i - 1}}^\\top}{P_{X_{i - 1}, j}} \\biggr) -  p e_{X_{i - 1}}e_{X_{i - 1}}^\\top =: \\bW^{(2)}_i. \t\n\t\\end{aligned}\n\t\\]\n\tWrite $\\opnorm{\\sum_{i=1}^n \\bW^{(1)}_i}$ as $W_n^{(1)}$, $\\opnorm{\\sum_{i = 1}^n \\bW^{(2)}_i}$ as $W_n^{(2)}$ and $\\max(W_n^{(1)}, W_n^{(2)})$ as $W_n$. By the matrix Freedman inequality \\citep[Corollary~1.3]{tropp2011freedman}, for any $t \\ge 0$ and $\\sigma^2 > 0$, \n\t\\be\n\t\\label{eq:matrix_freedman}\n\t\\mathbb{P} ( \\opnorm{\\mathbf{S}_n} \\ge t, W_n\\le \\sigma^2 ) \\le 2p \\exp\\biggl(-\\frac{t^2 \/2 }{\\sigma^2 + Rt \/ 3}\\biggr). \n\t\\ee\n\tNow we need to choose an appropriate $\\sigma^2$ so that $W_n \\le \\sigma^2$ holds with high probability. Note that $W^{(1)}_n \\le np(\\alpha^{-1} + 1)$ and $W_n^{(2)} \\le (p^2 \\alpha^{-1} - p) \\sup_{j\\in [p]} \\sum_{i = 1}^n 1_{\\{X_i = s_j\\}}$. In the following we derive a bound for $\\sup_{j\\in [p]} \\sum_{i = 1}^n 1_{\\{X_i = s_j\\}}$. For any $j \\in [p]$, by \\citet[Theorem~1.2]{JFS18}, which is a variant of Bernstein's inequality for Markov chains, we have that \n\t\\be\n\t\\label{eq:mc_bernstein}\n\t\\mathbb{P}\\biggl\\{\\frac{1}{n}\\sum_{i=1}^n (1_{\\{X_i = s_j\\}} - \\pi_j) > \\epsilon \\biggr\\} \\le \\exp\\biggl( -\\frac{n\\epsilon ^ 2}{2(A_1\\beta \/ p + A_2\\epsilon)}\\biggr), \n\t\\ee\n\twhere \n\t\\[\n\tA_1 = \\frac{1 + \\max(\\rho_+, 0)}{1 - \\max(\\rho_+, 0)}~~~\\textnormal{and}~~~A_2 = \\frac{1}{3}1_{\\{\\rho_{+} \\le 0\\}} +  \\frac{5}{1 - \\rho_+} 1_{\\{\\rho_+ > 0\\}}. \n\t\\]\n\tSome algebra yields that for any $\\xi > 0$, \n\t\\[\n\t\\mathbb{P}\\biggl\\{\\frac{1}{n}\\sum_{i=1}^n 1_{\\{X_i = s_j\\}} - \\pi_j > \\biggl(\\frac{4A_1\\xi}{np}\\biggr)^{1 \/ 2} + \\frac{4A_2\\xi}{n} \\biggr\\} \\le \\exp( - \\xi). \n\t\\]\n\tA union bound over $j \\in [p]$ yields that \n\t\\[\n\t\\mathbb{P}\\biggl\\{ \\sup_{j \\in [p]} \\frac{1}{n}\\sum_{i=1}^n (1_{\\{X_i = s_j\\}}- \\pi_j) > \\biggl(\\frac{4A_1\\xi \\log p}{np}\\biggr)^{1 \/ 2} + \\frac{4A_2\\xi \\log p}{n}\\biggr\\} \\le p^{-(\\xi - 1)}, \n\t\\]\n\twhich implies that \n\t\\[\n\t\\mathbb{P}\\biggl\\{\\sup_{j \\in [p]} \\frac{1}{n}\\sum\\limits_{i=1}^n  1_{\\{X_i = s_j\\}} > \\pi_{\\max} + \\biggl(\\frac{4A_1\\xi \\log p}{np}\\biggr)^{1 \/ 2} + \\frac{4A_2\\xi \\log p}{n}\\biggr\\} \\le p^{-(\\xi - 1)}. \n\t\\]\n\tTherefore, whenever $n \\pi_{\\max} (1 - \\rho_+) \\ge 2 \\log p$, we have that \n\t\\[\n\t\t\\mathbb{P}\\biggl(\\sup_{j \\in [p]} \\frac{1}{n} \\sum_{i = 1}^n 1_{\\{X_i = s_j\\}} \\gtrsim \\pi_{\\max}\\biggr) \\le \\exp\\biggl(- \\frac{n\\pi_{\\max} (1 - \\rho_+)}{2}\\biggr). \n\t\\]\n\tCombining this with the bounds of $W_n^{(1)}$ and $W_n^{(2)}$, we have that \n\t\\[\n\t\t\\mathbb{P}\\biggl(W_n \\ge \\frac{C_1np ^ 2\\pi_{\\max}}{\\alpha}\\biggr) \\le \\exp\\biggl(- \\frac{n\\pi_{\\max} (1 - \\rho_+)}{2}\\biggr), \n\t\\]\n\twhere $C_1$ is a universal constant. Now choosing $\\sigma^2 = C_1 np^2 \\pi_{\\max} \/ \\alpha$, we deduce that for any $t \\ge 0$, \n\t\\[\n\t\\begin{aligned}\n\t\t\\mathbb{P}(\\opnorm{\\mathbf{S}_n} \\ge t) & = \\mathbb{P}(\\opnorm{\\mathbf{S}_n} \\ge t, W_n \\le \\sigma^2 ) + \\mathbb{P}(\\opnorm{\\mathbf{S}_n} \\ge t, W_n > \\sigma^2 ) \\\\\n\t\t& \\le \\mathbb{P}(\\opnorm{\\mathbf{S}_n} \\ge t, W_n \\le \\sigma^2 ) + \\mathbb{P}(W_n > \\sigma^2 ) \\\\\n\t\t& \\le 2p \\exp\\biggl(-\\frac{t^2 \/2 }{\\sigma^2 + Rt \/ 3}\\biggr) + \\exp\\biggl(- \\frac{n\\pi_{\\max} (1 - \\rho_+)}{2}\\biggr). \n\t\\end{aligned}\n\t\\]\n\tEquivalently, for any $\\xi > 1$, \n\t\\[\n\t\t\\mathbb{P}\\biggl\\{\\biggl\\|\\frac{1}{n}\\mathbf{S}_n\\biggr\\|_2 \\gtrsim \\biggl(\\frac{\\xi p ^ 2\\pi_{\\max} \\log p}{n\\alpha}\\biggr)^{1 \/ 2} + \\frac{\\xi p\\log p}{n \\alpha}\\biggr\\} \\le 4p^{-(\\xi - 1)} + \\exp\\biggl(- \\frac{n\\pi_{\\max} (1 - \\rho_+)}{2}\\biggr). \n\t\\]\t\n\tFinally, observe that for any $i \\in [n]$, $\\Pi_{{\\mathcal N}}(\\mathbb{E}(\\bZ_i | \\bZ_{i - 1})) = \\Pi_{{\\mathcal N}}( -e_{X_{i - 1}} 1^\\top) = \\bzero$. Therefore, $\\Pi_{{\\mathcal N}}(\\nabla\\ell_n(\\mathbf{P})) = n^{-1}\\mathbf{S}_n$ and the final conclusion then follows. \t\n\\end{proof}\n\n\n\\section{Proof of Lemma \\ref{lem:uniform_law}}\n\\begin{proof}\n\t\\noindent We first split $\\mathcal{C}(\\eta)$ as the union of the sets \n\t\\begin{equation*}\n\t\\mathcal{C}_l := \\left\\{\\mathbf{Q} \\in \\mathcal{C}(\\eta): 2^{l-1}\\eta \\leq D_{\\mathrm{KL}}(\\mathbf{P}, \\mathbf{Q}) \\leq 2^l\\eta, ~ {\\rm rank}(\\mathbf{Q})\\leq r\\right\\}, \\quad l=1,2,3,\\ldots. \n\t\\end{equation*}\n\tDefine\n\t\\begin{equation*}\n\t\\begin{split}\n\t\\gamma_l = & \\sup_{\\mathbf{Q} \\in \\mathcal{C}_l} \\bigl|D_{\\mathrm{KL}}(\\mathbf{P}, \\mathbf{Q}) - \\widetilde D_{\\mathrm{KL}}(\\mathbf{P}, \\mathbf{Q})\\bigr| \\\\\n\t= & \\sup_{\\mathbf{Q} \\in \\mathcal{C}_l} \\biggl|\\frac{1}{n}\\sum_{i=1}^n \\langle \\log(\\mathbf{P}) - \\log(\\mathbf{Q}), \\mathbf{X}_i \\rangle - \\mathbb{E}\\langle \\log(\\mathbf{P}) - \\log(\\mathbf{Q}), \\mathbf{X}_i \\rangle\\biggr|.\n\t\\end{split}\n\t\\end{equation*}\n\tFirst, we wish to apply \\citet[][Theorem~7]{Ada08} to bound $|\\gamma_l - \\mathbb{E} \\gamma_l|$. Adamczak's bound entails the following asymptotic weak variance\n\t\\[\n\t\\sigma^2 := \\sup_{\\mathbf{Q}\\in{\\mathcal C}_l} {\\rm Var} \\biggl\\{\\sum_{i = S_1 + 1}^{S_2} \\langle \\log(\\mathbf{P}) - \\log(\\mathbf{Q}), \\mathbf{X}_i \\rangle - \\mathbb{E}\\langle \\log(\\mathbf{P}) - \\log(\\mathbf{Q}), \\mathbf{X}_i \\rangle \\biggr\\} \/ \\mathbb{E} T_2. \n\t\\]\n\tWe have that \n\t\\[\n\t\\begin{aligned}\n\t\\sigma^2  & \\le \\sup_{\\mathbf{Q}\\in{\\mathcal C}_l}\\mathbb{E} \\biggl[\\biggl\\{ \\sum_{i = S_1 + 1}^{S_2} \\langle \\log(\\mathbf{P}) - \\log(\\mathbf{Q}), \\mathbf{X}_i \\rangle - \\mathbb{E}\\langle \\log(\\mathbf{P}) - \\log(\\mathbf{Q}), \\mathbf{X}_i \\rangle \\biggr\\}^{\\!2}\\biggr] \/ \\mathbb{E} T_2\\\\\n\t& = \\frac{1}{2} \\sup_{\\mathbf{Q}\\in{\\mathcal C}_l} \\sum\\limits_{j = 1}^{\\infty} \\mathbb{E} \\biggl[\\biggl\\{ \\sum_{i = S_1 + 1}^{S_2} \\langle \\log(\\mathbf{P}) - \\log(\\mathbf{Q}), \\mathbf{X}_i \\rangle - \\mathbb{E}\\langle \\log(\\mathbf{P}) - \\log(\\mathbf{Q}), \\mathbf{X}_i \\rangle \\biggr\\}^{\\!2} 1_{\\{T_2 = j\\}}\\biggr] \\\\\n\t& \\le \\frac{1}{2}  \\sum\\limits_{j = 1}^{\\infty}  {4j^2\\log^ 2(\\beta \/ \\alpha) }\\mathbb{P}(T_2 = j) = {2 \\log^2(\\beta \/ \\alpha) \\mathbb{E} (T_2^2)} = {8\\log ^ 2(\\beta \/ \\alpha)}. \n\t\\end{aligned}\n\t\\]\n\tBy \\citet[][Theorem~7]{Ada08}, there exists a universal constant $K > 1$ such that for any $\\xi > 0$, \n\t\\[\n\t\\mathbb{P}\\biggl\\{ |\\gamma_l - \\mathbb{E} \\gamma_l| \\ge K \\mathbb{E}\\gamma_l + {2\\log(\\beta \/ \\alpha)}\\biggl(\\frac{2K\\xi }{n}\\biggr)^{\\! 1 \/ 2} +  \\frac{16 K\\log(\\beta \/ \\alpha)\\xi \\log n }{n}\\biggr\\} \\le K e^{- \\xi}.  \n\t\\]\n\tSince $n^{- 1\/ 2} \\ge 2 n^{-1}\\log n$ for any positive integer $n$, we have that\n\t\\be\n\t\\label{eq:gamma_l_uniform_concentration}\n\t\\mathbb{P}\\biggl\\{ |\\gamma_l - \\mathbb{E} \\gamma_l| \\ge K \\mathbb{E}\\gamma_l + {11K\\log(\\beta \/ \\alpha)}\\biggl(\\frac{\\xi }{n}\\biggr)^{\\! 1 \/ 2}\\biggr\\} \\le K e^{- \\xi}.  \t\n\t\\ee\n\tNext, we bound $\\mathbb{E} \\gamma_l$. Let $\\{\\varepsilon_i\\}_{i=1}^n$ be $n$ independent Rademacher random variables. By a symmetrization argument, \n\t\\begin{equation*}\n\t\\begin{split}\n\t\\mathbb{E}\\gamma_l = & \\mathbb{E}\\biggl(\\sup_{\\mathbf{Q} \\in \\mathcal{C}_l} \\biggl|\\frac{1}{n}\\sum_{i=1}^n \\langle \\log (\\mathbf{P}) - \\log(\\mathbf{Q}), \\mathbf{X}_i\\rangle - \\mathbb{E}\\langle \\log(\\mathbf{P}) - \\log(\\mathbf{Q}), \\mathbf{X}_i \\rangle\\biggr|\\biggr)\\\\\n\t\\leq & 2\\mathbb{E}\\biggl(\\sup_{\\mathbf{Q} \\in \\mathcal{C}_l}\\biggl|\\frac{1}{n}\\sum_{i=1}^n \\varepsilon_k \\langle \\log(\\mathbf{P}) - \\log(\\mathbf{Q}), \\mathbf{X}_i\\rangle\\biggr|\\biggr).\n\t\\end{split}\n\t\\end{equation*}\n\tLet $\\phi_i(t) = (\\alpha \/ p)\\{\\log(\\langle\\mathbf{P}, \\mathbf{X}_i\\rangle + t) - \\log(\\langle \\mathbf{P}, \\mathbf{X}_i\\rangle)\\}$. Then $\\phi_{i}(0) = 0$ and $|\\phi_i'(t)| \\leq 1$ for all $t$ such that $t + \\langle\\mathbf{P}, \\mathbf{X}_i\\rangle \\geq \\alpha\/p$. In other words, $\\phi_i$ is a contraction map for $t \\geq \\min_{j, k \\in [p]}(P_{jk} - \\alpha\/p)$. By the contraction principle (Theorem 4.12 in \\cite{ledoux2013probability}),\n\n\n\n\n\n\n\n\t\\begin{equation}\\label{ineq:expected-gamma_l}\n\t\t\\begin{split}\n\t\t\t\\mathbb{E}\\gamma_l \\leq & \\frac{2p}{\\alpha}\\mathbb{E}\\biggl(\\sup_{\\mathbf{Q} \\in \\mathcal{C}_l}\\biggl|\\frac{1}{n}\\sum_{i=1}^n\\varepsilon_i \\phi_i\\left(\\langle \\mathbf{Q}-\\mathbf{P}, \\mathbf{X}_i\\rangle\\right)\\biggr|\\biggr) \\leq  \\frac{4p}{\\alpha}\\mathbb{E}\\biggl(\\sup_{\\mathbf{Q} \\in \\mathcal{C}_l}\\biggl|\\frac{1}{n}\\sum_{i = 1}^n \\varepsilon_i\\langle \\mathbf{Q}-\\mathbf{P}, \\mathbf{X}_i\\rangle\\biggr|\\biggr)\\\\\n\t\t\t\\leq & \\frac{4p}{\\alpha}\\mathbb{E}\\biggl(\\sup_{\\mathbf{Q} \\in \\mathcal{C}_l}\\biggl\\|\\frac{1}{n}\\sum_{i = 1}^n \\varepsilon_i\\mathbf{X}_i\\biggr\\|_2 \\|\\mathbf{Q}-\\mathbf{P}\\|_\\ast\\biggr) \\leq  \\frac{4p}{\\alpha} \\mathbb{E}\\biggl\\|\\frac{1}{n}\\sum_{i = 1}^n \\varepsilon_i\\mathbf{X}_i\\biggr\\|_2  \\sup_{\\mathbf{Q} \\in \\mathcal{C}_l} \\|\\mathbf{Q} - \\mathbf{P}\\|_\\ast. \n\t\t\\end{split}\n\t\\end{equation}\t\n\tBy Lemma \\ref{lem:kl_to_l2}, \n\n\n\n\n\n\n\t\\begin{equation}\n\t\\label{ineq:Q-P-nuclear}\n\t\\begin{split}\n\t\t\\sup_{\\mathbf{Q} \\in \\mathcal{C}_l}  \\|\\mathbf{Q} - \\mathbf{P}\\|_\\ast & \\leq \\sup_{\\mathbf{Q} \\in \\mathcal{C}_l} (2r)^{1 \/ 2} \\fnorm{\\mathbf{Q} - \\mathbf{P}}  \\le 2\\beta \\biggl(\\frac{2^l \\eta r}{p\\alpha \\pi_{\\min}}\\biggr)^{\\!1 \/ 2}. \n\t\\end{split}\n\t\\end{equation}\t\n\n\tHence, the remaining task is to bound $\\mathbb{E}\\|n^{-1}\\sum_{i = 1}^n \\varepsilon_i\\mathbf{X}_i\\|$. From now on, we denote $\\varepsilon_i \\mathbf{X}_i$ by $\\bZ_i$. One can see that $(\\bZ_i)_{i = 1}^n$ is a martingale difference sequence. We wish to apply the matrix Freedman inequality \\citep[Corollary~1.3]{tropp2011freedman} to bound the average of $(\\bZ_i)_{i = 1}^n$. We have that\n\t\\begin{equation*}\n\t\\begin{split}\n\t\\biggl\\|\\sum_{i = 1}^n \\mathbb{E} \\bigl(\\bZ_i^\\top \\bZ_i \\vert X_{i - 1}\\bigr) \\biggr\\|_2 = & \\biggl\\|\\sum_{i=1}^n \\sum_{j=1}^p  P_{X_{i - 1}, j} (e_{X_{i - 1}} e_j^\\top)^\\top (e_{X_{i - 1}}e_j^\\top)\\biggr\\|_2 = \\biggl\\|\\sum_{j=1}^p  \\sum_{i=1}^n P_{X_{i - 1}, j} e_je_j^\\top\\biggr\\|_2 \\\\\n\t= & \\max_{j \\in [p]} \\sum_{i = 1}^n P_{X_{i - 1}, j} =: W_n^{(1)}\n\t\\end{split}\n\t\\end{equation*} \n\tand that \n\t\\begin{equation*}\n\t\\begin{split}\n\t\\biggl\\|\\sum_{i = 1}^n \\mathbb{E} \\bigl(\\bZ_i\\bZ_i^\\top\\vert X_{i - 1}\\bigr) \\biggr\\|_2 = & \\biggl\\|\\sum_{i=1}^n \\sum_{j=1}^p P_{X_{i - 1}, j} e_{X_{i - 1}} e_{X_{i - 1}}^\\top \\biggr\\|_2 = \\biggl\\|\\sum_{i=1}^n e_{X_{i - 1}} e_{X_{i - 1}}^\\top\\biggr\\|_2 \\\\\n\t= & \\max_{j \\in [p]} \\sum_{i = 1}^n 1_{\\{X_{i - 1} = j\\}} =: W_n^{(2)}. \n\t\\end{split}\n\t\\end{equation*}\t\n\tWe first bound $W_n^{(1)}$. Note that for any $j \\in [p]$, $\\mathbb{E} (P_{X_{i - 1}, j}) = \\pi_j $, and that\n\t\\[\n\t{\\rm Var}_{\\pi}(P_{X_{i - 1}, j}) = \\sum_{k = 1}^p \\pi_k (P_{kj} - \\pi_j)^2 = \\sum_{k = 1}^p \\pi_k P^2_{kj} - \\pi_j ^ 2 \\le \\pi_j(1 - \\pi_j). \n\t\\]\n\tBy a variant of Bernstein's inequality for Markov chains \\citep[Theorem~1.2]{JFS18}, we have that for any $j \\in [p]$, \n\t\\[\n\t\\label{eq:mc_bernstein}\n\t\\mathbb{P}\\biggl(\\frac{1}{n} \\sum_{i = 1}^n P_{X_{i - 1}, j} - \\pi_j > \\epsilon \\biggr) \\le \\exp\\biggl\\{ -\\frac{n\\epsilon ^ 2}{2(A_1\\pi_j + A_2\\epsilon)}\\biggr\\}, \n\t\\]\n\twhere\n\t\\[\n\tA_1 := \\frac{1 + \\max(\\rho_+, 0)}{1 - \\max(\\rho_+, 0)}~~~\\textnormal{and}~~~A_2 := \\frac{1}{3}1_{\\{\\rho_{+} \\le 0\\}} +  \\frac{5}{1 - \\rho_+} 1_{\\{\\rho_+ > 0\\}}. \n\t\\]\n\tA union bound yields that \n\t\\be\n\t\\label{eq:w1}\n\t\\mathbb{P}\\bigl\\{W_n^{(1)} \\ge n\\pi_{\\max} + (4nA_1 \\pi_{\\max}\\xi \\log p)^{1 \/ 2} + {4A_2\\xi \\log p} \\bigr\\} \\le p^{-(\\xi - 1)}. \n\t\\ee\n\tNext we bound $W_n^{(2)}$. Note that $W_n^{(2)} \\le \\max_{j \\in [p]} \\sum_{i =1}^n 1_{\\{X_{i - 1} = s_j\\}}$. Similarly, by \\citet[Theorem~1.2]{JFS18}, for any $j \\in [p]$, \n\n\n\t\\be\n\t\\label{eq:mc_bernstein}\n\t\\mathbb{P}\\biggl\\{\\frac{1}{n}\\sum_{i=1}^n 1_{\\{X_{i - 1} = s_j\\}} - \\pi_j > \\epsilon \\biggr\\} \\le \\exp\\biggl\\{ -\\frac{n\\epsilon ^ 2}{2(A_1\\pi_j + A_2\\epsilon)}\\biggr\\}, \n\t\\ee\n\tSome algebra yields that for any $\\xi > 0$, \n\t\\[\n\t\\mathbb{P}\\biggl\\{\\frac{1}{n}\\sum_{i=1}^n 1_{\\{X_{i - 1} = s_j\\}} - \\pi_j > \\biggl(\\frac{4A_1\\pi_j\\xi }{n}\\biggr)^{1 \/ 2} + \\frac{4A_2\\xi}{n} \\biggr\\} \\le \\exp( - \\xi). \n\t\\]\n\tBy a union bound over $j \\in [p]$, \n\t\\[\n\t\\mathbb{P}\\biggl\\{ \\max_{j \\in [p]} \\frac{1}{n}\\sum_{i=1}^n 1_{\\{X_{i - 1} = s_j\\}} > \\pi_{\\max} + \\biggl(\\frac{4A_1\\pi_{\\max}\\xi \\log p}{n}\\biggr)^{1 \/ 2} + \\frac{4A_2\\xi \\log p}{n}\\biggr\\} \\le p^{-(\\xi - 1)}, \n\t\\]\n\twhich further implies that \n\t\\be\n\t\\label{eq:w2}\n\t\\mathbb{P}\\bigl\\{ W_n^{(2)} \\ge n\\pi_{\\max} + (4nA_1 \\pi_{\\max}\\xi \\log p)^{1 \/ 2} + {4A_2\\xi \\log p} \\bigr\\} \\le p^{-(\\xi - 1)}. \n\t\\ee\n\tDefine $W_n := \\max(W_n^{(1)}, W_n^{(2)})$. Let $\\mathbf{S}_n :=  \\sum_{i = 1}^n \\varepsilon_i\\mathbf{X}_i$. By matrix Freedman's inequality \\citep[][Corollary~1.3]{tropp2011freedman}, for any $t \\ge 0$ and $\\sigma^2 > 0$, \n\t\\be\n\t\\label{eq:matrix_freedman}\n\t\\mathbb{P} ( \\opnorm{\\mathbf{S}_n} \\ge t, W_n\\le \\sigma^2 ) \\le 2p \\exp\\biggl(-\\frac{t^2 \/2 }{\\sigma^2 + t \/ 3}\\biggr). \n\t\\ee\n\tNow we need to choose an appropriate $\\sigma^2$ so that $W_n \\le \\sigma^2$ holds with high probability.\n\tGiven that $\\rho_+ > 0$ and $n\\pi_{\\max} \\ge 10\\xi \\log p \/ (1 - \\rho_+)$, combining \\eqref{eq:w1} and \\eqref{eq:w2} yields that \n\n\n\n\n\t\\be\n\t\\label{eq:w}\t\n\t\\mathbb{P}\\bigl( W_n \\ge 4n\\pi_{\\max}\\bigr) \\le 2p^{-(\\xi - 1)}. \n\t\\ee\n\tNow choosing $\\sigma^2 = 4n\\pi_{\\max}$ in \\eqref{eq:matrix_freedman}, we deduce that \n\t\\[\n\t\\begin{aligned}\n\t\\mathbb{P}(\\opnorm{\\mathbf{S}_n} \\ge t) & = \\mathbb{P}(\\opnorm{\\mathbf{S}_n} \\ge t, W_n \\le \\sigma^2 ) + \\mathbb{P}(\\opnorm{\\mathbf{S}_n} \\ge t, W_n > \\sigma^2 ) \\\\\n\t& \\le \\mathbb{P}(\\opnorm{\\mathbf{S}_n} \\ge t, W_n \\le \\sigma^2 ) + \\mathbb{P}(W_n > \\sigma^2 ) \\\\\n\t& \\le 2p \\exp\\biggl(-\\frac{t^2 \/2 }{\\sigma^2 + t \/ 3}\\biggr) + 2p^{-(\\xi - 1)}. \n\t\\end{aligned}\n\t\\]\n\tChoose $\\xi = n \\pi_{\\max} (1 - \\rho_+) \/ (10 \\log p)$. As long as $n\\pi_{\\max}(1 - \\rho_+)\\ge \\max (20 \\log p, \\log n)$, we have that \n\t\\be\n\t\\label{eq:expectation_sn_op}\n\t\\mathbb{E} \\biggl\\|\\frac{1}{n}\\mathbf{S}_n\\biggr\\|_2 \\lesssim \\biggl(\\frac{\\pi_{\\max}\\log p}{n}\\biggr)^{\\!1 \/2}. \t\t\n\n\n\t\\ee\n\tCombining \\eqref{ineq:expected-gamma_l}, \\eqref{ineq:Q-P-nuclear} and \\eqref{eq:expectation_sn_op} yields that\n\t\\begin{equation*}\n\t\\mathbb{E} \\gamma_l \\lesssim \\frac{\\beta}{\\alpha ^ {3 \/ 2}}\\biggl(\\frac{2^{l}\\eta \\pi_{\\max}rp\\log p}{\\pi_{\\min}n}\\biggr)^{\\!1 \/2}. \n\t\\end{equation*}\n\tThen combining this with \\eqref{eq:gamma_l_uniform_concentration} yields that   \n\t\\begin{equation*}\n\t\\mathbb{P}\\biggl\\{\\gamma_l \\gtrsim \\frac{\\beta}{\\alpha ^ {3 \/ 2}}\\biggl(\\frac{2^{l}\\eta \\pi_{\\max}rp\\log p}{\\pi_{\\min}n}\\biggr)^{\\!1 \/2} + \\log(\\beta \/ \\alpha)\\biggl(\\frac{\\xi }{n}\\biggr)^{\\! 1 \/ 2} \\hspace{0cm} \\biggr\\} \\lesssim e^{-\\xi}. \n\t\\end{equation*}\n\tLet $\\xi = 2^l \\eta \\pi_{\\max} rp \\log p \/ \\pi_{\\min}$. Then there exist universal constants $C_1, C_2 > 0$ such that\n\t\\begin{equation*}\n\t\\mathbb{P}\\biggl\\{\\gamma_l \\ge \\frac{C_1\\beta}{\\alpha ^ {3 \/ 2}}\\biggl(\\frac{2^{l}\\eta \\pi_{\\max}rp\\log p}{\\pi_{\\min}n}\\biggr)^{\\!1 \/2} \\biggr\\} \\le C_2\\exp\\biggl\\{- \\frac{(2l + 1) \\eta \\pi_{\\max} rp \\log p}{\\pi_{\\min}}\\biggr\\}. \t\n\t\\end{equation*}\n\tWe can thus deduce that there exists a universal constant $C_3 > 0$ such that \n\t\\begin{equation*}\n\t\\begin{split}\n\t&\\mathbb{P}\\biggl( |\\widetilde{D}_{\\mathrm{KL}}(\\mathbf{P}, \\mathbf{Q}) - D_{\\mathrm{KL}}(\\mathbf{P}, \\mathbf{Q})|> \\frac{1}{2}D_{\\mathrm{KL}}(\\mathbf{P}, \\mathbf{Q}) + \\frac{C_3\\pi_{\\max} \\beta ^ 2rp\\log p}{\\pi_{\\min}\\alpha ^ 3n} \\biggr)\\\\\n\t\\leq & \\sum_{l=0}^\\infty P\\left(\\exists \\mathbf{Q} \\in \\mathcal{C}_l, ~\\left|\\widetilde{D}_{\\mathrm{KL}}(\\mathbf{P}, \\mathbf{Q}) - D_{\\mathrm{KL}}(\\mathbf{P}, \\mathbf{Q})\\right|> 2^{l - 2}\\eta + \\frac{C_3\\pi_{\\max} \\beta ^ 2rp\\log p}{\\pi_{\\min}\\alpha ^ 3n}\\right)\\\\\n\t\\leq & \\sum_{l=0}^\\infty P\\biggl\\{\\gamma_l \\ge \\frac{C_1\\beta}{\\alpha ^ {3 \/ 2}}\\biggl(\\frac{2^l\\eta\\pi_{\\max}rp\\log p}{\\pi_{\\min}n}\\biggr)^{\\!1 \/2}\\biggr\\}\\\\\n\t\\leq & C_2 \\sum_{l=0}^\\infty \\exp\\biggl\\{- \\frac{(2l + 1) \\eta \\pi_{\\max} rp \\log p}{\\pi_{\\min}}\\biggr\\} \\le 2C_2 \\exp\\biggl(- \\frac{\\eta \\pi_{\\max} rp \\log p}{\\pi_{\\min}}\\biggr). \n\t\\end{split}\n\t\\end{equation*}\n\twhere we use the Cauchy-Schwarz inequality in the second step. \n\\end{proof}\n\n\n\\section{Alternative statistical error analysis}\n\\label{sec:alt}\n\n\\subsection{Main results}\n\nIn this section, we provide an alternative proof strategy that follows \\citet{NRW12} to bound the statistical error of $\\widehat \\mathbf{P}$ and $\\widehat \\mathbf{P} ^ r$. This strategy resolves the inconsistency issue of Theorems \\ref{thm:nuclear} and \\ref{thm:rank} when $n \\gg \\{rp \\pi_{\\max}(\\log p) \\beta\/ (\\pi_{\\min} \\alpha ^ {\\!3 \/ 2})\\} ^ 2$. For any $R>0$, define a constraint set ${\\mathcal C} (\\beta, R, \\kappa):= \\{\\boldsymbol{\\Delta} \\in \\Re^{p \\times p}: \\supnorm{\\boldsymbol{\\Delta}} \\le \\beta \/ p, \\fnorm{\\boldsymbol{\\Delta}}\\le R, \\nnorm{\\boldsymbol{\\Delta}} \\le \\kappa r^{1 \/ 2}\\fnorm{\\boldsymbol{\\Delta}}  \\}$. An important ingredient of this statistical analysis is the localized restricted strong convexity \\citep{NWa11, FLS18} of the loss function $\\ell_n(\\mathbf{P})$ near $\\mathbf{P}$. This property allows us to bound the distance in the parameter space by the difference in the objective function value. Define the first-order Taylor remainder term of the negative log-likelihood function $\\ell_n(\\mathbf{Q})$ around $\\mathbf{P}$ as\n\\[\n\t\\delta\\ell_n(\\mathbf{Q}; \\mathbf{P})  := \\ell_n(\\mathbf{Q}) - \\ell_n(\\mathbf{P}) - \\nabla\\ell_n(\\mathbf{P})^\\top (\\mathbf{Q} - \\mathbf{P}). \n\\]\nThe following lemma establishes the desired local restricted strong convexity. \n\\begin{lemma}\n\t\\label{lem:restricted_strong_convexity}\n\tUnder Assumption \\ref{asp:1}, there exists a universal constant $K$ such that for any $\\xi > 1$, it holds with probability at least $1 - K\\exp(-\\xi)$ that for any $\\boldsymbol{\\Delta} \\in {\\mathcal C}(\\beta, R, \\kappa)$, \n\t\\be\n\t\\begin{aligned}\n\t\\delta\\ell_n(\\mathbf{P} + \\boldsymbol{\\Delta}; \\mathbf{P}) \\ge \\frac{\\alpha ^2}{8\\beta^2}\\fnorm{\\boldsymbol{\\Delta}}^2 & - 8R\\biggl({\\frac{3K\\xi }{n}}\\biggr)^{1 \/ 2} - \\frac{8K\\xi\\alpha^2 \\log n }{\\beta^2 n} - \\frac{Kp\\kappa R}{\\beta}\\biggl(\\frac{r\\pi_{\\max}\\log p}{n}\\biggr) ^ {\\!1 \/ 2}. \n\\end{aligned}\n\t\\ee\n\\end{lemma}\n\nNow we present the statistical rates of $\\widehat \\mathbf{P}$ and $\\widehat \\mathbf{P} ^ r$. \n\n\\begin{theorem}[Alternative statistical guarantee for $\\widehat \\mathbf{P}$]\n\t\\label{thm:nuclear_alt}\n\tUnder the same assumptions of Theorem \\ref{thm:nuclear}, there exists a universal constant $C_1 > 0$, such that for any $\\xi > 1$, if we choose \n\t\\[\n\t\\lambda =  C_1 \\biggl\\{\\biggl(\\frac{\\xi p ^ 2\\pi_{\\max} \\log p}{n\\alpha}\\biggr)^{\\! 1 \/ 2} + \\frac{\\xi p\\log p}{n \\alpha}\\biggr\\}, \n\t\\]\n\tthen whenever $n\\pi_{\\max}(1 - \\rho_+) \\ge \\max\\{\\max(20, \\xi ^ 2) \\log p, \\log n\\}$, we have with probability at least $1 - K\\exp(-\\xi) - 4p^{-(\\xi - 1)} - p ^{-1}$ that \n\t\\[\n\t\\begin{aligned}\n\t\\fnorm{\\widehat\\mathbf{P} - \\mathbf{P}} \\lesssim \\frac{\\beta ^ 2}{\\alpha ^ 2}\\biggl(\\frac{\\xi r p ^ 2 \\pi_{\\max} \\log p}{n \\alpha}\\biggr) ^ {\\!1 \/ 2}~~\\text{and}~~D_{\\mathrm{KL}}(\\mathbf{P}, \\widehat\\mathbf{P}) \\lesssim \\frac{\\xi \\beta ^ 6 \\pi_{\\max} r p ^ 2  \\log p}{n \\alpha ^ 7},\n\t\\end{aligned}\n\t\\]\n\twhere $K$ is the same constant as in Lemma \\ref{lem:restricted_strong_convexity}. \n\\end{theorem}\n\n\n\\begin{theorem}[Alternative statistical guarantee for $\\widehat \\mathbf{P} ^ r$]\n\t\\label{thm:rank_alt}\n\tUnder the same assumptions of Theorem \\ref{thm:nuclear}, there exists a universal constant $C_1 > 0$, for any $\\xi > 1$, we have with probability at least $1 - K\\exp(-\\xi) - 4p^{-(\\xi - 1)} - p ^{-1}$ that \n\t\\[\n\t\\fnorm{\\widehat\\mathbf{P} ^ r- \\mathbf{P}} \\lesssim \\frac{\\beta ^ 2}{\\alpha ^ 2}\\biggl(\\frac{\\xi r p ^ 2 \\pi_{\\max} \\log p}{n \\alpha}\\biggr) ^ {\\!1 \/ 2} ~~\\text{and}~~D_{\\mathrm{KL}}(\\mathbf{P}, \\widehat\\mathbf{P} ^ r) \\lesssim \\frac{\\xi \\beta ^ 6 \\pi_{\\max} r p ^ 2  \\log p}{n \\alpha ^ 7}, \n\t\\]\n\twhere $K$ is the same constant as in Lemma \\ref{lem:restricted_strong_convexity}. \n\\end{theorem}\n\nOne can see from the theorems above that the derived error bounds converge to zero as $n$ goes to infinity. Nevertheless, their dependence on $\\alpha$ and $\\beta$ is worse than those in Theorems \\ref{thm:nuclear} and \\ref{thm:rank} when $n \\lesssim \\{rp \\pi_{\\max}(\\log p) \\beta\/ (\\pi_{\\min} \\alpha ^ {\\!3 \/ 2})\\} ^ 2$. This is why we do not present this result in the main text. \n\n\\subsection{Proof of Lemma \\ref{lem:restricted_strong_convexity}}\n\n\\begin{proof}\n\t~Given any $\\boldsymbol{\\Delta} \\in {\\mathcal C}(\\beta, R, \\kappa)$, it holds that for some $0\\le v \\le 1$ that \n\t\\be\n\t\\label{eq:rsc_first_step}\n\t\\begin{aligned}\n\t\t\\delta \\ell_n(\\mathbf{P} + \\boldsymbol{\\Delta}; \\mathbf{P}) & = \\frac{1}{2} \\text{vec}(\\boldsymbol{\\Delta})^\\top \\mathbf{H}_n(\\mathbf{P} + v\\boldsymbol{\\Delta})\\text{vec}(\\boldsymbol{\\Delta}) = \\frac{1}{2n} \\sum\\limits_{i=1}^n \\frac{\\inn{\\mathbf{X}_i, \\boldsymbol{\\Delta}}^2}{\\inn{\\mathbf{P} + v\\boldsymbol{\\Delta}, \\mathbf{X}_i}^2} \\\\\n\t\t& \\ge \\frac{1}{2n}\\sum\\limits_{i=1}^n \\frac{p ^ 2}{4\\beta^2} \\inn{\\boldsymbol{\\Delta}, \\mathbf{X}_i}^2. \n\t\\end{aligned}\n\t\\ee\n\tDefine \n\t\\[\n\t\\Gamma_n := \\sup_{\\boldsymbol{\\Delta} \\in {\\mathcal C}(\\beta, R, \\kappa)} \\biggl |\\frac{1}{n} \\sum\\limits_{i=1}^n \\inn{\\boldsymbol{\\Delta}, \\mathbf{X}_i}^2 - \\mathbb{E}(\\inn{\\boldsymbol{\\Delta}, \\mathbf{X}_i}^2) \\biggr |. \n\t\\]\n\tWe first bound the deviation of $\\Gamma_r$ from its expectation $\\mathbb{E} \\Gamma_r$. Note that $\\{\\mathbf{X}_i\\}_{i = 1}^n$ is a Markov chain on ${\\mathcal M} := \\{e_je_k^\\top\\}_{j, k = 1}^p$. Here we apply a tail inequality for suprema of unbounded empirical processes due to \\citet[][Theorem~7]{Ada08}. To apply this result, we need to verify that $\\{\\mathbf{X}_i\\}_{i = 1}^n$ satisfies the ``minorization condition'' as stated in Section 3.1 of \\citet{Ada08}. Below we characterize a specialized version of this condition. \n\t\n\t\\begin{condition}[Condition 1 (minorized condition).]\n\t\t\\label{con:minorization}\n\t\tWe say that a Markov chain ${\\mathcal X}$ on ${\\mathcal S}$ satisfies the minorized condition if there exist $\\delta > 0$, a set ${\\mathcal C}\\subset {\\mathcal S}$ and a probability measure $\\nu$ on ${\\mathcal S}$ for which $\\forall_{x \\in {\\mathcal C}} \\forall_{{\\mathcal A} \\subset {\\mathcal S}} \\mathbb{P}(x, {\\mathcal A}) \\ge \\delta \\nu({\\mathcal A})$ and $\\forall_{x \\in {\\mathcal S}} \\exists_{n \\in \\mathbb{N}} \\mathbb{P}^n(x, {\\mathcal C}) > 0$. \n\t\\end{condition} \n\tOne can verify that the Markov chain $\\{\\mathbf{X}_i\\}_{i = 1}^n$ satisfies Condition \\ref{con:minorization} with $\\delta = 1\/ 2$, ${\\mathcal C} = \\{e_1e_2^\\top\\}$ and $\\nu(e_je_k^\\top) = P_{jk}1_{\\{j = 2\\}}$ for $j, k \\in [p]$.\n\t\n\tNow consider a new Markov chain $\\{(\\widetilde \\mathbf{X}_i, R_i)\\}_{i=1}^n$ constructed as follows. Let $\\{R_i\\}_{i=1}^n$ be i.i.d. Bernoulli random variables with $\\mathbb{E} R_1 = \\delta$. For any $i \\in \\{0, \\ldots, n - 1\\}$, at step $i$, if $\\mathbf{X}_i \\notin {\\mathcal C}$, we sample $\\widetilde \\mathbf{X}_{i + 1}$ according to $\\mathbb{P}(\\widetilde \\mathbf{X}_i, \\cdot)$; otherwise, the distribution of $\\widetilde \\mathbf{X}_i$ depends on $R_i$: if $R_i = 1$, the chain regenerates in the sense that we draw $\\widetilde \\mathbf{X}_i$ from $\\nu$, and if $R_i = 0$, we draw $\\widetilde \\mathbf{X}_i$ from $(\\mathbb{P}(\\mathbf{X}_i, \\cdot) - \\delta \\nu(\\cdot)) \/ (1 - \\delta)$. One can verify that the sequence $\\{\\widetilde\\mathbf{X}_i\\}_{i=1}^n$ has exactly the same distribution as the original Markov chain $\\{\\mathbf{X}_i\\}_{i=1}^n$. Define $T_1 := \\inf\\{n > 0: R_n = 1\\}$ and $T_{i + 1} := \\inf\\{n > 0: R_{T_1 + \\ldots + T_i + n} = 1\\}$ for $i \\ge 0$. Note that $\\{T_i\\}_{i \\ge 0}$ are i.i.d. Geometric random variables with $\\mathbb{E} T_1 = 2$ and $\\|T_1\\|_{\\psi_1} \\le 4$. Let $S_0 := -1$, $S_j := T_1 + \\ldots + T_j$ and ${\\mathcal Y}_j := \\{ \\widetilde \\mathbf{X}_i\\}_{i = S_{j - 1} + 1}^{S_j}$ for $j \\ge 1$. Based on our construction, we deduce that $\\{{\\mathcal Y}_j\\}_{j \\ge 1}$ are independent. Thus we chop the original Markov chain $\\{X_i\\}_{i \\in [n]}$ into independent sequences. Finally, Adamazak's bound entails the following asymptotic weak variance\n\t\\[\n\t\\sigma^2 := \\sup_{\\boldsymbol{\\Delta}\\in{\\mathcal C}(\\beta, R, \\kappa)} {\\rm Var} \\biggl\\{\\sum_{i = S_1 + 1}^{S_2} \\inn{\\boldsymbol{\\Delta}, \\mathbf{X}_i}^2 - \\mathbb{E}(\\inn{\\boldsymbol{\\Delta}, \\mathbf{X}_i}^2) \\biggr\\} \/ \\mathbb{E} T_2. \n\t\\]\n\tWe have\n\t\\[\n\t\\begin{aligned}\n\t\\sigma^2  & \\le \\sup_{\\boldsymbol{\\Delta}\\in{\\mathcal C}(\\beta, R, \\kappa)}\\mathbb{E} \\biggl[\\biggl\\{ \\sum_{i = S_1 + 1}^{S_2} \\inn{\\boldsymbol{\\Delta}, \\mathbf{X}_i}^2 - \\mathbb{E}(\\inn{\\boldsymbol{\\Delta}, \\mathbf{X}_i}^2) \\biggr\\}^2\\biggr] \/ \\mathbb{E} T_2\\\\\n\t& = \\frac{1}{2} \\sup_{\\boldsymbol{\\Delta}\\in{\\mathcal C}(\\beta, R, \\kappa)} \\sum\\limits_{j = 1}^{\\infty} \\mathbb{E} \\biggl[\\biggl\\{ \\sum_{i = S_1 + 1}^{S_2} \\inn{\\boldsymbol{\\Delta}, \\mathbf{X}_i}^2 - \\mathbb{E}(\\inn{\\boldsymbol{\\Delta}, \\mathbf{X}_i}^2) \\biggr\\}^2 1_{\\{T_2 = j\\}}\\biggr] \\\\\n\t& \\le \\frac{1}{2}  \\sum\\limits_{j = 1}^{\\infty}  \\frac{j^2R^2\\beta^4}{p^4}\\mathbb{P}(T_2 = j) = \\frac{R^2 \\beta^4\\mathbb{E} (T_2^2)}{2p^4} = \\frac{3\\beta^4R^2}{p^4}. \n\t\\end{aligned}\n\t\\]\n\tBy \\citet[][Theorem~7]{Ada08}, there exists a universal constant $K$ such that for any $\\xi > 0$, \n\t\\be\n\t\\label{eq:1.7}\n\t\\mathbb{P}\\biggl\\{ |\\Gamma_n - \\mathbb{E} \\Gamma_n| \\ge K \\mathbb{E}\\Gamma_n + \\frac{R\\beta^2}{ p^2}\\biggl(\\frac{3K\\xi }{n}\\biggr)^{1 \/ 2} +  \\frac{64K\\xi\\alpha^2 \\log n }{np^2}\\biggr\\} \\le K \\exp(-\\xi).  \n\t\\ee\n\t\t\n\tNext, by the symmetrization argument and Ledoux-Talagrand contraction inequality \\citep{ledoux2013probability}, for $n$ independent and identically distributed Rademacher variables $\\{\\gamma_i\\}_{i=1}^n$, when $n \\pi_{\\max}(1 - \\rho_+) \\ge \\max(20\\log p, \\log n)$, we have that\n\t\\be\n\t\\label{eq:16}\n\t\\begin{aligned}\n\t\t\\mathbb{E} \\Gamma_n & \\le 2\\mathbb{E} \\sup_{\\substack{\\fnorm{\\boldsymbol{\\Delta}} \\le R, \\\\ \\boldsymbol{\\Delta} \\in {\\mathcal C}(\\beta, R, \\kappa)}} \\biggl | \\frac{1}{n} \\sum\\limits_{i=1}^n \\gamma_i \\inn{\\boldsymbol{\\Delta}, \\mathbf{X}_i}^2 \\biggr | \\le \\frac{8\\beta}{p}~\\mathbb{E} \\sup_{\\substack{\\fnorm{\\boldsymbol{\\Delta}} \\le R, \\\\ \\boldsymbol{\\Delta} \\in {\\mathcal C}(\\beta, R, \\kappa)}} \\biggl | \\inn{\\frac{1}{n}\\sum\\limits_{i=1}^n \\gamma_i \\mathbf{X}_i, \\boldsymbol{\\Delta}}  \\biggr| \\\\\n\t\t& \\le \\frac{8\\beta\\nnorm{\\boldsymbol{\\Delta}}}{p} ~\\mathbb{E} \\biggl\\| \\frac{1}{n}\\sum\\limits_{i=1}^n \\gamma_i \\mathbf{X}_i \\biggr \\|_{2}  \\le \\frac{8\\kappa\\beta r^{1 \/ 2} R}{p}\\mathbb{E} \\biggl\\| \\frac{1}{n}\\sum\\limits_{i=1}^n \\gamma_i \\mathbf{X}_i \\biggr\\|_2 \\le \\frac{8\\kappa\\beta R}{p}\\biggl(\\frac{r\\pi_{\\max}\\log p}{n}\\biggr)^{\\!1 \/ 2}, \n\t\\end{aligned}\n\t\\ee\n\twhere the penultimate inequality is due to the fact that $\\boldsymbol{\\Delta} \\in {\\mathcal C}(\\beta, R, \\kappa)$, and where the last inequality is due to \\eqref{eq:expectation_sn_op}. \n\t\n\tFinally, \n\t\\be\n\t\\mathbb{E} \\inn{\\boldsymbol{\\Delta}, \\mathbf{X}_i}^2 = \\sum\\limits_{1 \\le j,k \\le d} \\pi_j P_{jk}\\Delta^2_{jk} \\ge \\frac{\\alpha^2}{p^2} \\fnorm{\\boldsymbol{\\Delta}}^2. \n\t\\ee\n\tCombining all the bounds above, we have for any $\\xi > 1$, with probability at least $1 - K \\exp(-\\xi)$, \n\t\\be\n\t\\begin{aligned}\n\t\t\\delta\\ell_n(\\mathbf{P} + \\boldsymbol{\\Delta}; \\mathbf{P}) \\ge \\frac{\\alpha ^2}{8\\beta^2}\\fnorm{\\boldsymbol{\\Delta}}^2 & - 8R\\biggl({\\frac{3K\\xi }{n}}\\biggr)^{1 \/ 2} - \\frac{8K\\xi\\alpha^2 \\log n }{\\beta^2 n} - \\frac{Kp\\kappa R}{\\beta}\\biggl(\\frac{r\\pi_{\\max}\\log p}{n}\\biggr) ^ {\\!1 \/ 2}. \n\t\\end{aligned}\n\t\\ee\n\\end{proof}\n\n\\subsection{Proof of Theorem \\ref{thm:nuclear_alt}}\n\n\\begin{proof}\n\t~For a specific $R$ whose value will be determined later, we construct an intermediate estimator $\\widehat\\mathbf{P}_{\\eta} $ between $\\widehat\\mathbf{P}$ and $\\mathbf{P}$:\n\t\\[\n\t\\widehat \\mathbf{P}_{\\eta} = \\mathbf{P} + \\eta (\\widehat \\mathbf{P} - \\mathbf{P}), \n\t\\]\n\twhere $\\eta = 1$ if $\\fnorm{\\widehat \\mathbf{P} - \\mathbf{P}} \\le R$ and $\\eta  = R\/\\fnorm{\\widehat\\mathbf{P} - \\mathbf{P}}$ if $\\fnorm{\\widehat\\mathbf{P} - \\mathbf{P}} > R$. For any $\\xi > 1$, there exists a universal constant $C > 0$ such that when \n\t\\[\n\t\\lambda = C\\biggl\\{\\biggl(\\frac{\\xi p ^ 2\\pi_{\\max}\\log p}{n\\alpha}\\biggr)^{1 \/ 2} + \\frac{\\xi p\\log p}{n \\alpha}\\biggr\\}, \n\t\\] \n\twe have by Lemmas \\ref{lem:restricted_strong_convexity} and \\ref{lem:gradient} that with probability at least $1 - K\\exp(-\\xi) - 4p^{-(\\xi - 1)} - p ^{-1}$, \n\t\\be\n\t\\label{eq:stat_error_fnorm}\n\t\\begin{aligned}\n\t\t\\frac{\\alpha ^2}{8\\beta^2} & \\fnorm{\\boldsymbol{\\Delta}}^2 - 8R\\biggl({\\frac{3K\\xi }{n}}\\biggr)^{1 \/ 2} - \\frac{8K\\xi\\alpha^2 \\log n }{\\beta^2 n} - \\frac{Kp\\kappa R}{\\beta}\\biggl(\\frac{r\\pi_{\\max}\\log p}{n}\\biggr) ^ {\\!1 \/ 2}\\\\\n\t\t& \\le \\delta\\ell_n(\\widehat\\mathbf{P}_{\\eta}; \\mathbf{P}) \\le - \\inn{\\Pi_{{\\mathcal N}}(\\nabla{\\mathcal L}_n(\\mathbf{P})), \\widehat \\boldsymbol{\\Delta}_{\\eta}}  + \\lambda ( \\nnorm{\\mathbf{P}} - \\nnorm{\\widehat\\mathbf{P}_{\\eta}} ) \\\\\n\t\t& \\le - \\inn{\\Pi_{{\\mathcal N}}(\\nabla{\\mathcal L}_n(\\mathbf{P})), \\widehat \\boldsymbol{\\Delta}_{\\eta}}  + \\lambda\\nnorm{\\widehat\\boldsymbol{\\Delta}_{\\eta}} \\le (\\opnorm{\\Pi_{{\\mathcal N}}(\\nabla {\\mathcal L}_n(\\mathbf{P}))} + \\lambda) \\nnorm{\\widehat \\boldsymbol{\\Delta}_{\\eta}}    \\\\\n\t\t& \\le 8\\lambda \\nnorm{[\\widehat \\boldsymbol{\\Delta}_{\\eta}]_{\\overline{\\mathcal M}}} \\le 8\\lambda\\sqrt{r}\\fnorm{\\widehat\\boldsymbol{\\Delta}_{\\eta}},\n\t\\end{aligned}\n\t\\ee\n\twhere $K$ is the same universal constant as in Theorem \\ref{lem:restricted_strong_convexity}. Some algebra yields that \n\t\\be\n\t\\label{eq:5.24}\n\t\\begin{aligned}\n\t\t\\fnorm{\\widehat \\boldsymbol{\\Delta}_{\\eta}}^2  \\lesssim \\frac{\\beta ^ 2}{\\alpha ^ 2} \\max \\biggl\\{ \\frac{\\lambda ^ 2r\\beta ^ 2}{\\alpha ^ 2}, R\\biggl({\\frac{\\xi}{n}}\\biggr)^{1 \/ 2}, \\frac{\\xi \\alpha ^ 2 \\log n}{\\beta  ^ 2n}, \\frac{pR}{\\beta}\\biggl(\\frac{r\\pi_{\\max}\\log p}{n}\\biggr) ^ {\\!1 \/ 2}\\biggr\\}. \n\t\\end{aligned} \n\t\\ee\n\tLetting $R ^ 2$ be greater than the RHS of the inequality above, we can find a universal constant $C_4 > 0$ such that\n\t\\[\n\t\\begin{aligned}\n\tR \\ge \\frac{C_4\\beta ^ 2}{\\alpha ^ 2}\\biggl(\\frac{\\xi r p ^ 2 \\pi_{\\max} \\log p}{n \\alpha}\\biggr) ^ {\\!1 \/ 2} =: R_0. \n\t\\end{aligned}\n\t\\]\n\tChoose $R = R_0$. Therefore, $\\fnorm{\\widehat \\boldsymbol{\\Delta}_{\\eta}} \\le R$ and $\\widehat \\boldsymbol{\\Delta}_{\\eta} = \\widehat \\boldsymbol{\\Delta}$. We can thus reach the conclusion. As to the KL-Divergence, by \\citet[][Lemma~4]{zhang2018optimal}, we deduce that \n\t\\be\n\t\\label{eq:l2_to_kl}\n\tD_{\\mathrm{KL}}(\\widehat\\mathbf{P}, \\mathbf{P}) = \\sum\\limits_{j = 1}^p \\pi_j D_{\\mathrm{KL}}(\\mathbf{P}_{j\\cdot}, \\widehat\\mathbf{P}_{j\\cdot}) \\le \\sum\\limits_{j = 1}^p \\frac{\\beta^2 }{2 \\alpha^2}\\ltwonorm{\\mathbf{P}_{j\\cdot} - \\widehat\\mathbf{P}_{j\\cdot}}^2 = \\frac{\\beta^2}{2\\alpha^2} \\fnorm{\\widehat\\mathbf{P} - \\mathbf{P}}^2, \n\t\\ee\n\tfrom which we attain the conclusion.  \n\\end{proof}\n\n\\subsection{Proof of Theorem \\ref{thm:rank_alt}}\n\n\\begin{proof}\t\n\t~Define ${\\widehat\\boldsymbol{\\Delta}}(r) := \\widehat \\mathbf{P}^r - \\mathbf{P}$. Since ${\\rm rank}(\\mathbf{P}) \\le r$ and ${\\rm rank}(\\widehat\\mathbf{P}^r) \\le r$, ${\\rm rank}(\\widehat\\boldsymbol{\\Delta}(r))\\le 2r$. Thus $\\fnorm{\\widehat \\boldsymbol{\\Delta}(r)} \\le (2r)^{1 \/ 2} \\nnorm{\\widehat \\boldsymbol{\\Delta}(r)}$. \n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\tNow we follow the proof strategy of Theorem \\ref{thm:nuclear} to establish the statistical error bound for $\\widehat \\mathbf{P}^r$. Similarly, for a   specific $R > 0$ whose value will be determined later, we can construct an intermediate estimator $\\widehat\\mathbf{P}^r_{\\eta} $ between $\\widehat\\mathbf{P}^r$ and $\\mathbf{P}$:\n\t\\[\n\t\\widehat \\mathbf{P}^r_{\\eta} = \\mathbf{P} + \\eta (\\widehat \\mathbf{P}^r - \\mathbf{P}), \n\t\\]\n\twhere $\\eta = 1$ if $\\fnorm{\\widehat \\mathbf{P}^r - \\mathbf{P}} \\le R$ and $\\eta  = R\/\\fnorm{\\widehat\\mathbf{P}^r - \\mathbf{P}}$ if $\\fnorm{\\widehat\\mathbf{P}^r - \\mathbf{P}} > R$. Let $\\widehat \\boldsymbol{\\Delta}_\\eta(r) := \\widehat \\mathbf{P}^r_\\eta - \\mathbf{P}$. Since $\\widehat \\boldsymbol{\\Delta}_\\eta(r) \\in {\\mathcal C}(\\beta, R, \\sqrt{2})$, applying Lemma \\ref{lem:restricted_strong_convexity} yields that \n\t\\be\n\t\\begin{aligned}\n\t\t\\frac{\\alpha ^2}{8\\beta^2} & \\fnorm{\\boldsymbol{\\Delta}}^2 - 8R\\biggl({\\frac{3K\\xi }{n}}\\biggr)^{1 \/ 2} - \\frac{8K\\xi\\alpha^2 \\log n }{\\beta^2 n} - \\frac{Kp\\kappa R}{\\beta}\\biggl(\\frac{r\\pi_{\\max}\\log p}{n}\\biggr) ^ {\\!1 \/ 2}\\\\\n\t\t& \\le \\delta\\ell_n(\\widehat\\mathbf{P}^r_{\\eta}; \\mathbf{P}) \\le - \\inn{\\Pi_{{\\mathcal N}}(\\nabla\\ell_n(\\mathbf{P})), \\widehat \\boldsymbol{\\Delta}_{\\eta}(r)} \\le \\opnorm{\\Pi_{{\\mathcal N}}(\\nabla {\\mathcal L}_n(\\mathbf{P}))} \\nnorm{\\widehat \\boldsymbol{\\Delta}_{\\eta}(r)}\\\\\n\t\t& \\le \\sqrt{2r}\\opnorm{\\Pi_{{\\mathcal N}}(\\nabla {\\mathcal L}_n(\\mathbf{P}))} \\fnorm{\\widehat \\boldsymbol{\\Delta}_{\\eta}(r)}, \n\t\\end{aligned}\n\t\\ee\n\twhich futher implies that there exists $C_1$ depending only on $\\alpha$ and $\\beta$ such that \n\t\\[\n\t\\fnorm{\\widehat \\boldsymbol{\\Delta}_{\\eta}(r)}^2  \\le C_1 \\max \\biggl\\{r\\opnorm{\\Pi_{{\\mathcal N}}(\\nabla {\\mathcal L}_n(\\mathbf{P}))} ^ 2, R\\biggl({\\frac{\\xi}{n}}\\biggr)^{1 \/ 2}, \\frac{\\xi \\alpha ^ 2 \\log n}{\\beta  ^ 2n}, \\frac{pR}{\\beta}\\biggl(\\frac{r\\pi_{\\max}\\log p}{n}\\biggr) ^ {\\!1 \/ 2}\\biggr\\}. \n\t\\]\n\tBy a contradiction argument as in the proof of Theorem \\ref{thm:nuclear}, we can choose an appropriate $R$ large enough such that $\\widehat \\mathbf{P}^r_\\eta = \\widehat \\mathbf{P}^r $ and attain the conclusion. \n\t\n\\end{proof}\n\n\n\\section{Proof of Proposition \\ref{prop:penlowrank}\n\\begin{proof}\n\n\n\n\n\t~Since ${\\rm rank}({\\mathbf{X}}_c^*)\\le r$, we know that ${\\mathbf{X}}_c^*$ is in fact a feasible solution to the original problem (5) and $\\norm{{\\mathbf{X}}_c^*}_{*} - \\norm{{\\mathbf{X}}_c^*}_{(r)} = 0$. Therefore, for any feasible solution $X$ to\n\t(5), it holds that \n\t\\begin{align*} \n\tf({\\mathbf{X}}_c^*) ={}& f({\\mathbf{X}}_c^*) + c(\\norm{{\\mathbf{X}}_c^*}_{*} - \\norm{{\\mathbf{X}}_c^*}_{(r)})\\\\[5pt]\n\t\\le{}& f({\\mathbf{X}}) + c(\\norm{{\\mathbf{X}}}_* - \\norm{{\\mathbf{X}}}_{(r)})\n\t= f({\\mathbf{X}}).\n\t\\end{align*}\n\tThis completes the proof of the proposition.\n\\end{proof}\n\n\\section{Convergence and $o(1\/k)$ non-ergodic iteration complexity of Algorithm 1 (sGS-ADMM)}\n\nBefore deriving the desired results of Algorithm \\ref{alg:sGS-ADMM} for solving problem \\eqref{prob:D}, we present some notation and definitions for the subsequent analysis. Assume that the solution sets of \\eqref{prob:gen-convex-nuc} and \\eqref{prob:D} are nonempty. Then, the primal-dual solution pairs associated with problems \\eqref{prob:gen-convex-nuc} and \\eqref{prob:D} satisfy the following Karush-Kuhn-Tucker (KKT) system: \n\\begin{equation}\n\\label{KKT}\n0 \\in R({\\bf X}, {\\bf \\Xi}, {\\bf S}), \\quad {\\mathcal A}({\\bf X}) = b, \\quad \n{\\bf \\Xi} + {\\mathcal A}^*(y) + {\\bf S} = 0,\n\\end{equation}\nwith \n\\begin{equation*}\nR({\\bf X}, {\\bf \\Xi}, {\\bf S}): = \\begin{pmatrix}\n{\\bf \\Xi} + \\partial g({\\bf X}) \\\\[5pt]\n{\\bf X} + \\partial \\delta(\\norm{{\\bf S}}_\n\n2 \\le c)\n\\end{pmatrix}, \\quad ({\\bf X}, {\\bf \\Xi}, {\\bf S})\\in {\\rm dom}\\,g \\times \\Re^{p\\times p}\\times \\left\\{{\\bf S}\\in \\Re^{p\\times p}\\mid \\norm{{\\bf S}}_2 \\le c \\right\\}.\n\\end{equation*}\nDefine the KKT residual function $D:{\\rm dom}\\,g \\times \\Re^{p\\times p}\\times \\Re^n \\times \\left\\{{\\bf S}\\in \\Re^{p\\times p}\\mid \\norm{{\\bf S}}_2 \\le c \\right\\} \\to [0, +\\infty)$ as\n\\[\nD({\\bf X}, {\\bf \\Xi}, y, {\\bf S}):= {\\rm dist}^2(0, R({\\bf X}, {\\bf \\Xi}, {\\bf S})) + \\norm{{\\mathcal A}({\\bf X}) - b}^2 + \n\\norm{{\\bf \\Xi} + {\\mathcal A}^*(y) + {\\bf S}}^2.\n\\]\nWe say $({\\bf X}, {\\bf \\Xi}, y, {\\bf S})\\in {\\rm dom}\\,g \\times \\Re^{p\\times p}\\times \\Re^n \\times \\left\\{{\\bf S}\\in \\Re^{p\\times p}\\mid \\norm{{\\bf S}}_2 \\le c \\right\\}$ be an $\\epsilon$-approximate primal-dual solution pair for problems \\eqref{prob:gen-convex-nuc} and \\eqref{prob:D} if\n$D({\\bf X}, {\\bf \\Xi}, y, {\\bf S}) \\le \\epsilon$. We show in the following theorem the global convergence and the $o(1\/k)$ iteration complexity results of Algorithm sGS-ADMM.\n\\begin{theorem}\n\t\\label{thm:sGS-ADMM}\n\tSuppose that the solution sets of \\eqref{prob:gen-convex-nuc} and \\eqref{prob:D} are nonempty. Let $\\{({\\bf\\Xi}^k,y^k,\\mathbb{S}^k,{\\mathbf{X}}^k)\\}$ be the sequence generated by Algorithm \\ref{alg:sGS-ADMM}. If $\\tau\\in(0,(1+\\sqrt{5}\\,)\/2)$, then the sequence $\\{({\\bf\\Xi}^k,y^k,\\mathbb{S}^k)\\}$ converges to an optimal solution of \\eqref{prob:D} and $\\{{\\mathbf{X}}^k\\}$ converges to an optimal solution of \\eqref{prob:gen-convex-nuc}.\n\tMoreover, there exist a constant $\\omega >0$ such that \n\t\\[\n\t\\min_{1\\le i \\le k} \\left\\{ D({\\bf X}^k, {\\bf \\Xi}^k, y^k, {\\bf S}^k) \\right\\} \\le \\frac{\\omega}{k}, \\ \\forall\\, k\\ge 1, \\quad{\\rm and}  \\quad \n\t\\lim_{k\\to \\infty} \\left\\{ k \\min_{1\\le i \\le k} \\left\\{ D({\\bf X}^k, {\\bf \\Xi}^k, y^k, {\\bf S}^k) \\right\\} \\right\\}= 0.\n\t\\]\n\\end{theorem}\n\n\\begin{proof}\n\t~In order to use \\citep[Theorem 3]{li2016schur}, we need to write problem \\eqref{prob:D} as following\n\t\\begin{equation*} \n\t\\begin{array}{rll}\n\t\\min  & g^*(-{\\bf\\Xi}) - \\inprod{b}{y} + {\\delta( \\norm{\\mathbb{S}}_2 \\le c)}  \\\\\n\t\\mbox{s.t.} & {\\mathcal F} ({\\bf\\Xi}) + {\\mathcal A}_1^*(y) + {\\mathcal G}(\\mathbb{S})  = 0,\n\t\\end{array}\n\t\\end{equation*}\n\twhere ${\\mathcal F}, {\\mathcal A}_1$ and ${\\mathcal G}$ are linear operators such that for all $({\\bf \\Xi}, y, \\mathbb{S}) \\in \\Re^{p\\times p} \\times \\Re^n \\times \\Re^{p\\times p}$, ${\\mathcal F}({\\bf\\Xi}) = {\\bf \\Xi}$, \n\t${\\mathcal A}_1^*(y) = {\\mathcal A}^*(y)$ and ${\\mathcal G}(\\mathbb{S}) = \\mathbb{S}$. \n\tClearly, ${\\mathcal F} = {\\mathcal G} = {\\mathcal I}$  where ${\\mathcal I}:\\Re^{p\\times p} \\to \\Re^{p\\times p}$ is the identity map.\n\tTherefore, we have ${\\mathcal A}_1{\\mathcal A}_1^* \\succ 0$ and ${\\mathcal F}\\cF^* = {\\mathcal G}\\cG^* = {\\mathcal I} \\succ 0$. Hence, the assumptions and conditions in  \\citep[Theorem 3]{li2016schur} are satisfied. The convergence results thus follow directly. Meanwhile, the non-ergodic iteration complexity results follows from \\citep[Theorem 6.1]{chen2017efficient}.\n\\end{proof}\n\n\n\\section{Proof of Theorems  \\ref{thm:convergence-alg-MM} and \\ref{thm: MMconvergence}}\nWe only need to prove Theorem \\ref{thm: MMconvergence} \nas Theorem \\ref{thm:convergence-alg-MM} \nis a special incidence. To prove Theorem  \\ref{thm: MMconvergence}, \nwe first introduce the following lemma.\t\n\\begin{lemma}\\label{lemma:decrease}\n\tSuppose that $\\{ {x}^k \\}$ is the sequence generated by Algorithm 3.\n\tThen  $\\theta({x}^{k+1})\\le \\theta({x}^k) - \\frac{1}{2}\\|{x}^{k+1} - {x}^k\\|^2_{\\mathcal{G} + 2\\mathcal{T}}$.\n\\end{lemma}\n\\begin{proof}\n\n\n\n\n\n~For any $k\\geq 0$, by the optimality condition of problem (10) at\n\t${x}^{k+1}$, we know that  \n\tthere exist $\\eta^{k+1}\\in \\partial p({x}^{k+1})$  such that\n\t\\begin{equation*}\\label{eq:major-k-optimality}\n\t0 =  \\nabla   g ({x}^k) +   (\\mathcal{G} + \\mathcal{T})({x}^{k+1} -\n\tx^{k})  + \\eta^{k+1}-\\xi^k  = 0.\n\t\\end{equation*}\n\tThen for any $k\\ge 0$, we deduce\n\t\\begin{equation*}\n\t\\begin{array}{rl}\n\t& \\theta({x}^{k+1}) - \\theta({x}^k)\n\t\\le \\widehat{\\theta}({x}^{k+1};{x}^k) - \\theta({x}^k)\\\\[0.1in]\n\t= & p(x^{k+1}) - p(x^k) + \\langle {x}^{k+1} - {x}^k , \\nabla g({x}^k)-\\xi^k \\rangle +\n\t\\frac{1}{2} \\|{x}^{k+1} - {x}^k\\|^2_{\\mathcal{G}} \\\\[0.1in]\n\t\\le &  \\langle  \\nabla g(x^k)+\\eta^{k+1} -\\xi^k, {x}^{k+1} - {x}^k\\rangle+\n\t\\frac{1}{2} \\|{x}^{k+1} - {x}^k\\|^2_{\\mathcal{G}}\n\t\\\\[0.1in]\n\t=  &  - \\frac{1}{2}\\|x^{k+1} - x^k\\|^2_{\\mathcal{G} + 2\\mathcal{T}}.\n\t\\end{array}\n\t\\end{equation*}\n\tThis completes the proof of this lemma.\n\\end{proof}\n\nNow we are ready to prove Theorem \\ref{thm: MMconvergence}.\n\\begin{proof}\n\t~From the optimality condition at $x^{k+1}$, we have that \n\t\\[ 0 \\in \\nabla   g ({x}^k) +   (\\mathcal{G} + \\mathcal{T})({x}^{k+1} -\n\tx^{k})  +  \\partial p(x^{k+1})-\\xi^k.\\]\n\tSince $x^{k+1} = x^k$, this implies that \n\t\\[ 0 \\in \\nabla   g ({x}^k) +  \\partial p(x^{k})- \\partial q(x^k), \\]\n\ti.e., $x^k$ is a critical point.\n\tObserve that the sequence $\\{\\theta (x^{k})\\}$ is non-increasing since\n\t$${\\theta}(x^{k+1})  \\le \\widehat{\\theta}(x^{k+1}; x^{k})  \\le \\widehat{\\theta}(x^{k}; x^{k}) =\\theta(x^{k}), \\quad k\\geq 0.$$\n\tSuppose that there exists a subsequence $\\{x^{k_j}\\}$ that converging to $\\bar{x}$, i.e.,  one of the accumulation points of $\\{x^k\\}$.\n\tBy Lemma \\ref{lemma:decrease} and the assumption that $\\mathcal{G} + 2\\mathcal{T}\\succeq 0$, we know that for all $x\\in \\mathbb{X}$\n\t\\begin{align*}\n\t&\\widehat{\\theta}(x^{k_{j+1}};x^{k_{j+1}}) = \\theta(x^{k_{j+1}}) \\\\\n\t\\le &\\theta(x^{k_j+1})\\le \\widehat{\\theta}(x^{k_j+1};x^{k_j})\\le \\widehat{\\theta}(x;x^{k_j}).\n\t\\end{align*}\n\tBy letting $j\\to\\infty$ in the above inequality, we obtain that\n\t$$\n\t\\widehat{\\theta}(\\bar{x};\\bar{x})\\le  \\widehat{\\theta}(x;\\bar{x}).\n\t$$\n\tBy the optimality condition of $\\widehat{\\theta}(x; \\bar{x})$, we have that\n\tthere exists $\\bar{u}\\in \\partial p(\\bar{x})$ and $\\bar{v}\\in \\partial q(\\bar{x})$ such that\n\t$$\n\t0 \\in \\nabla g(\\bar{x}) + \\bar{u} - \\bar{v}.  \n\t$$\n\tThis implies that $\\left(\\nabla g(\\bar{x}) + \\partial p(\\bar{x})  \\right)\\cap \\partial q(\\bar{x})\\neq \\emptyset$.\n\tTo establish the rest of this proposition, \n\twe obtain from Lemma 1 that\n\t\\begin{align*}\n\t&\\lim_{t \\to + \\infty}\\frac{1}{2} \\sum_{i=0}^t \\|{x}^{k+1}-{x}^k\\|^2_{\\mathcal{G}+2\\mathcal{T}} \\\\\n\t\\le {}&\\liminf_{t\\to +\\infty} \\big( \\theta(x^0)\n\t-\\theta({x}^{k+1})\\big) \\le \\theta(x^0) < +\\infty \\,,\n\t\\end{align*}\n\twhich implies $  \\lim_{i\\to +\\infty} \\|{x}^{k+1} -\n\tx^{i}\\|_{\\mathcal{G}+2\\mathcal{T}} = 0.$ The proof of this theorem is thus complete by the positive definiteness of the operator $\\mathcal{G} + 2\\mathcal{T}$.\n\\end{proof}\n\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\subsubsection*{#1}}\n\n\\newcommand{S}{S}\n\\newcommand{s}{s}\n\\newcommand{\\pv}[1]{\\mathbf{#1}}\n\\newcommand{\\mathrm{t}}{\\mathrm{t}}\n\\newcommand{z_\\mr{diff}}{z_\\mr{diff}}\n\\newcommand{z_\\mr{same}}{z_\\mr{same}}\n\n\\newcommand{\\medskip}{\\medskip}\n\n\\newcommand{\\propertygroup}[2]{\\paragraph*{#1.~#2}}\n\\newcommand{\\property}[1]{\\emph{#1:}\\ }\n\n\\newcommand{\\dtxt}[1]{\\medskip\\begin{quote}%\n\\setlength{\\fboxsep}{2mm}%\n\\fbox{\\parbox{0.82\\textwidth}{\\sl #1}}%\n\\end{quote}\\medskip}\n\n\n\\newcommand{\\magn}[1]{|#1|}\n\\newcommand{\\pd}[1]{\\widetilde{#1}}\n\n\\newcommand{\\smp}[1]{\\Delta_{#1}}\n\\newcommand{\\ismp}[1]{\\Delta_{#1}^\\circ}\n\\newcommand{\\rstr}[2]{#1|_{#2}}\n\\newcommand{\\mr{supp}}{\\mr{supp}}\n\n\\newcommand{\\mr{c}}{\\mr{c}}\n\\newcommand{\\mr{d}}{\\mr{d}}\n\n\\newcommand{\\Dmax}[1]{D_\\mr{max}(#1)}\n\\newcommand{\\ptl}[1]{\\frac{\\partial}{\\partial #1}}\n\\newcommand{\\indprop}[1]{\\textbf{\\textsf{Q}}${}_{#1}$}\n\n\\newcommand{\\passage}[1]{\\paragraph{\\textit{#1}}}\n\n\\newcommand{\\lbl}[1]{\\label{#1}}\n\\renewcommand{\\iff}{\\Leftrightarrow}\n\\newcommand{\\trianglelefteq}{\\trianglelefteq}\n\\newcommand{\\approx}{\\approx}\n\\newcommand{\\simeq}{\\simeq}\n\\newcommand{\\um}[2]{U_{#1}^{#2}}\n\\newcommand{\\ip}[2]{\\langle #1, #2 \\rangle}\n\n\n\\section{Statement of the problem}\n\\lbl{sec:statement}\n\n\\passage{Basic definitions} Fix an integer $n \\geq 1$ throughout.  A\n\\demph{similarity matrix} is an $n\\times n$ symmetric matrix $Z$ with entries\nin the interval $[0, 1]$, such that $Z_{ii} = 1$ for all $i$.  A\n\\demph{(probability) distribution} is an $n$-tuple $\\pv{p} = (p_1, \\ldots,\np_n)$ with $p_i \\geq 0$ for all $i$ and $\\sum_{i = 1}^n p_i = 1$.\n\nGiven a similarity matrix $Z$ and a distribution $\\pv{p}$, thought of as a\ncolumn vector, we may form the matrix product $Z\\pv{p}$ (also a column\nvector), and we denote by $(Z\\pv{p})_i$ its $i$th entry.\n\n\\begin{lemma}   \\lbl{lemma:positive}\nLet $Z$ be a similarity matrix and $\\pv{p}$ a distribution.  Then $p_i \\leq\n(Z\\pv{p})_i \\leq 1$ for all $i \\in \\{1, \\ldots, n\\}$.  In particular, if $p_i\n> 0$ then $(Z\\pv{p})_i > 0$.\n\\end{lemma}\n\n\\begin{proof}\nWe have\n\\[\n(Z\\pv{p})_i \n=\n\\sum_{j = 1}^n Z_{ij} p_j\n=\np_i + \\sum_{j \\neq i} Z_{ij} p_j\n\\geq\np_i\n\\]\nand $(Z \\pv{p})_i \\leq \\sum_{j = 1}^n 1 p_j = 1$.\n\\hfill\\ensuremath{\\Box}\n\\end{proof}\n\nLet $Z$ be a similarity matrix and let $q \\in [0, \\infty)$.  The function\n$H_q^Z$ is defined on distributions $\\pv{p}$ by\n\\[\nH^Z_q(\\pv{p})\n=\n\\left\\{\n\\begin{array}{ll}\n\\displaystyle\n\\frac{1}{q - 1} \n\\left( 1 - \\sum_{i:\\,p_i > 0} p_i (Z\\pv{p})_i^{q - 1} \\right) &\n\\textrm{if } q \\neq 1   \\\\\n\\displaystyle\n- \\sum_{i:\\,p_i > 0} p_i \\log (Z\\pv{p})_i   &\n\\textrm{if } q = 1.\n\\end{array}\n\\right.\n\\]\nLemma~\\ref{lemma:positive} guarantees that these definitions are valid.  The\ndefinition in the case $q = 1$ is explained by the fact that $H_1^Z(\\pv{p}) =\n\\lim_{q\\to 1} H_q^Z(\\pv{p})$ (easily shown using l'H\\^opital's rule).  We\ncall $H_q^Z$ the \\demph{entropy of order $q$}.\n\n\\passage{Notes on the literature} The entropies $H_q^Z$ were introduced in\nthis generality by Ricotta and Szeidl in 2006, as an index of the diversity of\nan ecological community~\\cite{RS}.  Think of $n$ as the number of species,\n$Z_{ij}$ as indicating the similarity of the $i$th and $j$th species, and\n$p_i$ as the relative abundance of the $i$th species.  Ricotta and Szeidl used\nnot similarities $Z_{ij}$ but dissimilarities or `distances' $d_{ij}$; the\nformulas above become equivalent to theirs on putting $Z_{ij} = 1 - d_{ij}$.\n\nThe case $Z = I$ goes back further.  Something very similar to $H_q^I$, using\nlogarithms to base $2$ rather than base $e$, appeared in information theory in\n1967, in a paper of Havrda and Charv\\'at~\\cite{HC}.  Later, the entropies\n$H_q^I$ were discovered in statistical ecology, in a 1982 paper of Patil and\nTaillie~\\cite{PT}.  Finally, they were rediscovered in physics, in a 1988\npaper of Tsallis~\\cite{Tsa}.\n\nStill in the case $Z = I$, certain values of $q$ give famous quantities.  The\nentropy $H_1^I$ is Shannon entropy (except that Shannon used logarithms to\nbase~$2$).  The entropy $H_2^I$ is known in ecology as the Simpson or\nGini--Simpson index; it is the probability that two individuals chosen at\nrandom are of different species.\n\nFor general $Z$, the entropy of order $2$ is known as Rao's quadratic\nentropy~\\cite{Rao}.  It is usually stated in terms of the matrix with $(i,\nj)$-entry $1 - Z_{ij}$, that is, the matrix of dissimilarities mentioned\nabove. \n\nOne way to obtain a similarity matrix is to start with a finite metric space\n$\\{a_1, \\ldots, a_n\\}$ and put $Z_{ij} = e^{-d(a_i, a_j)}$.  Matrices of this\nkind are investigated in depth in~\\cite{MMS} and other papers cited therein.\nHere, metric spaces will only appear in two examples~(\\ref{eg:ultra}\nand~\\ref{eg:metric}).\n\n\\passage{The maximum entropy problem}  Let $Z$ be a similarity matrix and\nlet $q \\in [0, \\infty)$.  The maximum entropy problem is this:\n\\dtxt{%\nFor which distribution(s) $\\pv{p}$ is\n$H_q^Z(\\pv{p})$ maximal, and what is the maximum value?}\n\nThe solution is given in Theorem~\\ref{thm:main-entropy}.  The terms used in\nthe statement of the theorem will be defined shortly.  However, the following\nstriking fact can be stated immediately:\n\\dtxt{%\nThere is a distribution maximizing $H_q^Z$ for all\n$q$ simultaneously.}\nSo even though the entropies of different orders rank distributions\ndifferently, there is a distribution that is maximal for all of them.\n\nFor example, this fully explains the numerical coincidence noted in\nthe Results section of~\\cite{AKB}.\n\n\\passage{Restatement in terms of diversity}\nLet $Z$ be a similarity matrix.  For each $q \\in [0, \\infty)$, define a\nfunction $D_q^Z$ on distributions $\\pv{p}$ by\n\\begin{eqnarray*}\nD_q^Z(\\pv{p})   &\n=       &\n\\left\\{\n\\begin{array}{ll}\n\\left( 1 - (q - 1)H_q^Z(\\pv{p}) \\right)^\\frac{1}{1 - q}   &\n\\textrm{if } q \\neq 1   \\\\\ne^{H_1^Z(\\pv{p})} &\n\\textrm{if }q = 1\n\\end{array}\n\\right. \\\\\n        &=      &\n\\left\\{\n\\begin{array}{ll}\n\\displaystyle\n\\left(\n\\sum_{i:\\,p_i > 0}\np_i (Z\\pv{p})_i^{q - 1}\n\\right)^\\frac{1}{1 - q}         &\n\\textrm{if } q \\neq 1        \\\\\n\\displaystyle\n\\prod_{i:\\,p_i > 0}\n(Z\\pv{p})_i^{-p_i}      &\n\\textrm{if } q = 1.\n\\end{array}\n\\right.\n\\end{eqnarray*}\nWe call $D_q^Z$ the \\demph{diversity of order $q$}.  As for entropy, it is\neasily shown that $D_1^Z(\\pv{p}) = \\lim_{q \\to 1} D_q^Z(\\pv{p})$.\n\nThese diversities were introduced informally in~\\cite{EDC2}, and are explained\nand developed in~\\cite{MD}.  The case $Z = I$ is well known in several fields:\nin information theory, $\\log D_q^I$ is called the R\\'enyi entropy of order\n$q$~\\cite{Ren}; in ecology, $D_q^I$ is called the Hill number of order\n$q$~\\cite{Hill}; and in economics, $1\/D_q^I$ is the Hannah--Kay measure of\nconcentration~\\cite{HK}.\n\nThe transformation between $H_q^Z$ and $D_q^Z$ is invertible and\norder-preserving (increasing).  Hence the maximum entropy problem is\nequivalent to the maximum diversity problem:\n\\dtxt{%\nFor which distribution(s) $\\pv{p}$ is\n$D_q^Z(\\pv{p})$ maximal, and what is the maximum value?}\nThe solution is given in Theorem~\\ref{thm:main}.  It will be more convenient\nmathematically to work with diversity rather than entropy.  Thus, we prove\nresults about diversity and deduce results about entropy.\n\nWhen stated in terms of diversity, a further striking aspect of the solution\nbecomes apparent:\n\\dtxt{%\nThere is a distribution maximizing $D_q^Z$ for all\n$q$ simultaneously.  The maximum value of $D_q^Z$ is the same for all $q$.}\nSo every similarity matrix has an unambiguous `maximum diversity', the\nmaximum value of $D_q^Z$ for any $q$.\n\nA similarity matrix may have more than one maximizing distribution---but the\ncollection of maximizing distributions is independent of $q > 0$.  In other\nwords, a distribution that maximizes $D_q^Z$ for some $q$ actually maximizes\n$D_q^Z$ for all $q$ (Corollary~\\ref{cor:some-all}).\n\nThe diversities $D_q^Z$ are closely related to generalized means~\\cite{HLP},\nalso called power means.  Given a finite set $I$, positive real numbers\n$(x_i)_{i \\in I}$, positive real numbers $(p_i)_{i \\in I}$ such that $\\sum_i\np_i = 1$, and $t \\in \\mathbb{R}$, the \\demph{generalized mean} of $(x_i)_{i \\in\nI}$, weighted by $(p_i)_{i \\in I}$, of order $t$, is\n\\[\n\\left\\{\n\\begin{array}{ll}\n\\displaystyle\n\\left( \\sum_{i \\in I} p_i x_i^t \\right)^{1\/t}   &\n\\textrm{if } t \\neq 0   \\\\\n\\displaystyle\n\\prod_{i \\in I} x_i^{p_i}       &\n\\textrm{if } t = 0.\n\\end{array}\n\\right.\n\\]\nFor example, if $p_i = p_j$ for all $i, j \\in I$ then the generalized means of\norders $1$, $0$ and $-1$ are the arithmetic, geometric and harmonic means,\nrespectively.  \n\nGiven a similarity matrix $Z$ and a distribution $\\pv{p}$, take $I = \\{ i\n\\in \\{1, \\ldots, n\\} \\:|\\: p_i > 0\\}$.  Then $1\/D_q^Z(\\pv{p})$ is the\ngeneralized mean of $((Z\\pv{p})_i)_{i \\in I}$, weighted by $(p_i)_{i \\in I}$,\nof order $q - 1$.  We deduce the following.\n\n\\begin{lemma}   \\label{lemma:means}\nLet $Z$ be a similarity matrix and $\\pv{p}$ a distribution.  Then:\n\\begin{enumerate}\n\\item   \\label{part:cts}\n$D_q^Z(\\pv{p})$ is continuous in $q \\in [0, \\infty)$\n\\item   \\label{part:dec}\nif $(Z\\pv{p})_i =\n(Z\\pv{p})_j$ for all $i, j$ such that $p_i, p_j > 0$ then $D_q^Z(\\pv{p})$ is\nconstant over $q \\in [0, \\infty)$; otherwise, $D_q^Z(\\pv{p})$ is strictly\ndecreasing in $q \\in [0, \\infty)$\n\\item   \\label{part:infty}  \n$\\lim_{q \\to \\infty} D_q^Z(\\pv{p}) = 1\/\\max_{i: p_i > 0} (Z\\pv{p})_i$.\n\\end{enumerate}\n\\end{lemma}\n\n\\begin{proof}\nAll of these assertions follow from standard results on generalized means.\nContinuity is clear except perhaps at $q = 1$, where it follows from Theorem~3\nof~\\cite{HLP}.  Part~(\\ref{part:dec}) follows from Theorem~16 of~\\cite{HLP},\nand part~(\\ref{part:infty}) from Theorem~4.  \n\\hfill\\ensuremath{\\Box}\n\\end{proof}\n\nIn the light of this, we define\n\\[\nD_\\infty^Z(\\pv{p}) \n= \n1 \/ \\max_{i:\\,p_i > 0} (Z\\pv{p})_i.\n\\]\nThere is no useful definition of $H_\\infty^Z$, since $\\lim_{q \\to \\infty}\nH_q^Z(\\pv{p}) = 0$ for all $Z$ and $\\pv{p}$.\n\n\n\n\\section{Preparatory results}\n\\lbl{sec:prep}\n\nHere we make some definitions and prove some lemmas in preparation for solving\nthe maximum diversity and entropy problems.  Some of these definitions and\nlemmas can also be found in~\\cite{MMS} and~\\cite{AMSES}.\n\n\\emph{Convention:} for the rest of this work, unlabelled summations $\\sum$ are\nunderstood to be over all $i \\in \\{1, \\ldots, n\\}$ such that $p_i > 0$.\n\n\n\\passage{Weightings and magnitude}\n\n\\begin{defn}\nLet $Z$ be a similarity matrix.  A \\demph{weighting} on $Z$ is a column vector\n$\\pv{w} \\in \\mathbb{R}^n$ such that\n\\[\nZ\\pv{w} \n= \n\\left(\\!\\!\n\\begin{array}{c}\n1       \\\\\n\\vdots  \\\\\n1\n\\end{array}\n\\!\\!\\right).\n\\]\nA weighting $\\pv{w}$ is \\demph{non-negative} if $w_i \\geq 0$ for all\n$i$, and \\demph{positive} if $w_i > 0$ for all $i$. \n\\end{defn}\n\n\\begin{lemma}\nLet $\\pv{w}$ and $\\pv{x}$ be weightings on $Z$.  Then $\\sum_{i = 1}^n w_i =\n\\sum_{i = 1}^n x_i$.\n\\end{lemma}\n\n\n\\begin{proof} Write $\\pv{u}$ for the column vector $(1\\ \\cdots\\  1)^\\mathrm{t}$,\nwhere $(\\emptybk)^\\mathrm{t}$ means transpose.  Then \n\\[\n\\sum_{i = 1}^n w_i\n=\n\\pv{u}^\\mathrm{t} \\pv{w}\n=\n(Z\\pv{x})^\\mathrm{t} \\pv{w}\n=\n\\pv{x}^\\mathrm{t} (Z \\pv{w})\n=\n\\pv{x}^\\mathrm{t} \\pv{u}\n=\n\\sum_{i = 1}^n x_i,\n\\]\nusing symmetry of $Z$.\n\\hfill\\ensuremath{\\Box} \n\\end{proof}\n\n\\begin{defn}\nLet $Z$ be a similarity matrix on which there exists at least one weighting.\nIts \\demph{magnitude} is $\\magn{Z} = \\sum_{i = 1}^n w_i$, for any weighting\n$\\pv{w}$ on $Z$. \n\\end{defn}\n\nFor example, if $Z$ is invertible then there is a unique weighting $\\pv{w}$ on\n$Z$, and $w_i$ is the sum of the $i$th row of $Z^{-1}$.  So then\n\\[\n\\magn{Z} = \\sum_{i, j = 1}^n (Z^{-1})_{ij},\n\\]\nthe sum of all $n^2$ entries of $Z^{-1}$.  This formula also appears\nin~\\cite{SP}, \\cite{Shi} and~\\cite{POP}, for closely related reasons to do\nwith diversity and its maximization.\n\n\n\\passage{Weight distributions}\n\n\\begin{defn}\nLet $Z$ be a similarity matrix.  A \\demph{weight distribution} for $Z$ is a\ndistribution $\\pv{p}$ such that $(Z\\pv{p})_1 = \\cdots = (Z\\pv{p})_n$.\n\\end{defn}\n\n\\begin{lemma}   \\lbl{lemma:wt-distribs}\nLet $Z$ be a similarity matrix.  \n\\begin{enumerate}\n\\item If $Z$ admits a non-negative weighting then $\\magn{Z} > 0$.\n\\item If $\\pv{w}$ is a non-negative weighting on $Z$ then $\\pv{w}\/\\magn{Z}$ is\na weight distribution for $Z$, and this defines a one-to-one correspondence\nbetween non-negative weightings and weight distributions.\n\\item  \nIf $Z$ admits a weight distribution then $Z$ admits a weighting and $\\magn{Z}\n> 0$.  \n\\item   \\lbl{part:wt-distribs-reciprocal}\nIf $\\pv{p}$ is a weight distribution for $Z$ then $(Z\\pv{p})_i =\n1\/\\magn{Z}$ for all $i$.\n\\end{enumerate}\n\\end{lemma}\n\n\n\\begin{proof}\n\\begin{enumerate}\n\\item Let $\\pv{w}$ be a non-negative weighting.  Certainly $\\magn{Z} =\n\\sum_{i = 1}^n w_i \\geq 0$.  Since we are assuming that $n \\geq 1$, the vector\n$\\pv{0}$ is not a weighting, so $w_i > 0$ for some $i$.  Hence $\\magn{Z} > 0$.\n\n\\item The first part is clear.  To see that this defines a one-to-one\ncorrespondence, take a weight distribution $\\pv{p}$, writing $(Z\\pv{p})_i = K$\nfor all $i$.  Since $\\sum p_i = 1$, we have $p_i > 0$ for some $i$, and then\n$K = (Z\\pv{p})_i > 0$ by Lemma~\\ref{lemma:positive}.  The vector $\\pv{w} =\n\\pv{p}\/K$ is then a non-negative weighting.\n\nThe two processes---passing from a non-negative weighting to a weight\ndistribution, and vice versa---are easily shown to be mutually inverse.\n\n\\item Follows from the previous parts.\n\n\\item Follows from the previous parts.\n\\hfill\\ensuremath{\\Box}\n\\end{enumerate}\n\\end{proof}\n\nThe first connection between magnitude and diversity is this:\n\\begin{lemma}   \\lbl{lemma:mag-div}\nLet $Z$ be a similarity matrix and $\\pv{p}$ a weight distribution for $Z$.\nThen $D_q^Z(\\pv{p}) = \\magn{Z}$ for all $q \\in [0, \\infty]$.\n\\end{lemma}\n\n\\begin{proof}\nBy continuity, it is enough to prove this for $q \\neq 1, \\infty$.  In that\ncase, using Lemma~\\ref{lemma:wt-distribs}(\\ref{part:wt-distribs-reciprocal}),\n\\[\nD_q^Z(\\pv{p})\n=\n\\left(\n\\sum p_i (Z\\pv{p})_i^{q - 1}\n\\right)^\\frac{1}{1 - q}\n=\n\\left(\n\\sum p_i \\magn{Z}^{1 - q}\n\\right)^\\frac{1}{1 - q}\n=\n\\magn{Z},\n\\]\nas required.\n\\hfill\\ensuremath{\\Box}\n\\end{proof}\n\n\n\\passage{Invariant distributions}\n\n\\begin{defn}\nLet $Z$ be a similarity matrix.  A distribution $\\pv{p}$ is \\demph{invariant}\nif $D_q^Z(\\pv{p}) = D_{q'}^Z(\\pv{p})$ for all $q, q' \\in [0, \\infty]$.\n\\end{defn}\n\n\nSoon we will classify the invariant distributions.  To do so, we need some\nmore notation and a lemma.\n\nGiven a similarity matrix $Z$ and a subset $B \\subseteq \\{1, \\ldots, n\\}$, let\n$Z_B$ be the matrix $Z$ restricted to $B$, so that $(Z_B)_{ij} = Z_{ij}$ ($i,\nj \\in B$).  If $B$ has $m$ elements then $Z_B$ is an $m \\times m$ matrix, but\nit will be more convenient to index the rows and columns of $Z_B$ by the\nelements of $B$ themselves than by $1, \\ldots, m$.\n\nWe will also need to consider distributions on subsets of $\\{1, \\ldots, n\\}$.\nA distribution on $B \\subseteq \\{1, \\ldots, n\\}$ is said to be invariant, a weight\ndistribution, etc., if it is invariant, a weight distribution, etc., with\nrespect to $Z_B$.  Similarly, we will sometimes speak of `weightings on $B$',\nmeaning weightings on $Z_B$.  Distributions are understood to be on $\\{1,\n\\ldots, n\\}$ unless specified otherwise.\n\n\\begin{lemma}   \\lbl{lemma:absent}\nLet $Z$ be a similarity matrix, let $B \\subseteq \\{1, \\ldots, n\\}$, and let\n$\\pv{r}$ be a distribution on $B$.  Write $\\pv{p}$ for the distribution\nobtained by extending $\\pv{r}$ by zero.  Then $D_q^{Z_B}(\\pv{r}) =\nD_q^Z(\\pv{p})$ for all $q \\in [0, \\infty]$.  In particular, $\\pv{r}$ is\ninvariant if and only if $\\pv{p}$ is.\n\\end{lemma}\n\n\\begin{proof}\nFor $i \\in B$ we have $r_i = p_i$ and $(Z_B \\pv{r})_i = (Z\\pv{p})_i$.  The\nresult follows immediately from the definition of diversity of order $q$.\n\\hfill\\ensuremath{\\Box} \n\\end{proof}\n\nBy Lemma~\\ref{lemma:mag-div}, any weight distribution is invariant, and by\nLemma~\\ref{lemma:absent}, any extension by zero of a weight distribution is\nalso invariant.  We will prove that these are all the invariant distributions\nthere are.\n\nFor a distribution $\\pv{p}$ we write $\\mr{supp}(\\pv{p}) = \\{ i \\in \\{1, \\ldots,\nn\\} \\:|\\: p_i > 0\\}$, the \\demph{support} of $\\pv{p}$.\n\nLet $Z$ be a similarity matrix.  Given $\\emptyset \\neq B \\subseteq \\{1, \\ldots,\nn\\}$ and a non-negative weighting $\\pv{w}$ on $Z_B$, let $\\pd{\\pv{w}}$ be the\ndistribution obtained by first taking the weight distribution\n$\\pv{w}\/\\magn{Z_B}$ on $B$, then extending by zero to $\\{1, \\ldots, n\\}$.\n\n\n\\begin{propn}   \\lbl{propn:invt}\nLet $Z$ be a similarity matrix and $\\pv{p}$ a distribution.  The following\nare equivalent:\n\\begin{enumerate}\n\\item   \\lbl{part:invt-invt}\n$\\pv{p}$ is invariant\n\\item   \\lbl{part:invt-support}\n$(Z\\pv{p})_i = (Z\\pv{p})_j$ for all $i, j \\in \\mr{supp}(\\pv{p})$\n\\item   \\lbl{part:invt-wt-dist}\n$\\pv{p}$ is the extension by zero of a weight distribution on a nonempty\nsubset of $\\{1, \\ldots, n\\}$  \n\\item   \\lbl{part:invt-weighting}\n$\\pv{p} = \\pd{\\pv{w}}$ for some non-negative weighting $\\pv{w}$ on some\nnonempty subset of $\\{1, \\ldots, n\\}$.\n\\end{enumerate}\n\\end{propn}\n\n\\begin{proof}\n(\\ref{part:invt-invt}$\\,\\Rightarrow\\,$\\ref{part:invt-support}): Follows from\nLemma~\\ref{lemma:means}. \n\n(\\ref{part:invt-support}$\\,\\Rightarrow\\,$\\ref{part:invt-wt-dist}): Suppose\nthat~(\\ref{part:invt-support}) holds, and write $B = \\mr{supp}(\\pv{p})$.  The\ndistribution $\\pv{p}$ on $\\{1, \\ldots, n\\}$ restricts to a distribution\n$\\pv{r}$ on $B$.  This $\\pv{r}$ is a weight distribution on\n$B$, since for all $i \\in B$ we have $(Z_B \\pv{r})_i = (Z\\pv{p})_i$,\nwhich by~(\\ref{part:invt-support}) is constant over $i \\in B$.  Clearly\n$\\pv{p}$ is the extension by zero of $\\pv{r}$.\n\n(\\ref{part:invt-wt-dist}$\\,\\Rightarrow\\,$\\ref{part:invt-invt}): Suppose that $\\pv{p}$\nis the extension by zero of a weight distribution $\\pv{r}$ on a\nnonempty subset $B \\subseteq \\{1, \\ldots, n\\}$.  Then for all $q \\in [0, \\infty]$,\n\\[\nD_q^Z(\\pv{p})\n=\nD_q^{Z_B}(\\pv{r})\n=\n\\magn{Z_B}\n\\]\nby Lemmas~\\ref{lemma:absent} and~\\ref{lemma:mag-div} respectively; hence\n$\\pv{p}$ is invariant.\n\n(\\ref{part:invt-wt-dist}$\\iff$\\ref{part:invt-weighting}): Follows from\nLemma~\\ref{lemma:wt-distribs}.\n\\hfill\\ensuremath{\\Box}\n\\end{proof}\n\nThere is at least one invariant distribution on any given similarity matrix.\nFor we may choose $B$ to be a one-element subset, which has a unique\nnon-negative weighting $\\pv{w} = (1)$, and this gives the invariant\ndistribution $\\pd{\\pv{w}} = (0, \\ldots, 0, 1, 0, \\ldots, 0)$.\n\n\n\\passage{Maximizing distributions}\n\n\n\\begin{defn}    \\lbl{defn:max}\nLet $Z$ be a similarity matrix.  Given $q \\in [0,\n\\infty]$, a distribution $\\pv{p}$ is \\demph{$q$-maximizing} if $D_q^Z(\\pv{p})\n\\geq D_q^Z(\\pv{p}')$ for all distributions $\\pv{p}'$.  A distribution is\n\\demph{maximizing} if it is $q$-maximizing for all $q \\in [0, \\infty]$.\n\\end{defn}\n\nIt makes no difference to the definition of `maximizing' if we omit $q =\n\\infty$; nor does it make a difference to either definition if we replace\ndiversity $D_q^Z$ by entropy $H_q^Z$.\n\nWe will eventually show that every similarity matrix has a\nmaximizing distribution.\n\n\\begin{lemma}   \\lbl{lemma:max}\nLet $Z$ be a similarity matrix and $\\pv{p}$ an invariant distribution.  Then\n$\\pv{p}$ is $0$-maximizing if and only if it is maximizing.\n\\end{lemma}\n\n\\begin{proof}\nSuppose that $\\pv{p}$ is $0$-maximizing.  Then for all $q \\in [0, \\infty]$ and\nall distributions $\\pv{p}'$, \n\\[\nD_q^Z(\\pv{p}) \n=\nD_0^Z(\\pv{p})\n\\geq\nD_0^Z(\\pv{p}')\n\\geq\nD_q^Z(\\pv{p}'),\n\\]\nusing invariance in the first step and Lemma~\\ref{lemma:means} in the\nlast. \n\\hfill\\ensuremath{\\Box}\n\\end{proof}\n\n\\begin{lemma}   \\lbl{lemma:smaller-max}\nLet $Z$ be a similarity matrix and $B \\subseteq \\{1, \\ldots, n\\}$.  Suppose that\n$\\sup_{\\pv{r}} D_0^{Z_B}(\\pv{r}) \\geq \\sup_{\\pv{p}} D_0^Z(\\pv{p})$, where the\nfirst supremum is over distributions $\\pv{r}$ on $B$ and the second is over\ndistributions $\\pv{p}$ on $\\{1, \\ldots, n\\}$.  Suppose also that $Z_B$ admits\nan invariant maximizing distribution.  Then so does $Z$.\n\\end{lemma}\n(In fact, $\\sup_{\\pv{r}} D_0^{Z_B}(\\pv{r}) \\leq \\sup_{\\pv{p}} D_0^Z(\\pv{p})$\nin any case, by Lemma~\\ref{lemma:absent}.  So the `$\\geq$' in the statement\nof the present lemma could equivalently be replaced by `$=$'.)\n\n\\begin{proof}\nLet $\\pv{r}$ be an invariant maximizing distribution on $Z_B$.  Define a\ndistribution $\\pv{p}$ on $\\{1, \\ldots, n\\}$ by extending $\\pv{r}$ by zero.  By\nLemma~\\ref{lemma:absent}, $\\pv{p}$ is invariant.  Using\nLemma~\\ref{lemma:absent} again,\n\\[\nD_0^Z(\\pv{p})\n=\nD_0^{Z_B}(\\pv{r})\n=\n\\sup_{\\pv{r}'} D_0^{Z_B}(\\pv{r}')\n\\geq\n\\sup_{\\pv{p}'} D_0^Z(\\pv{p}'),\n\\]\nso $\\pv{p}$ is $0$-maximizing.  Then by Lemma~\\ref{lemma:max}, $\\pv{p}$ is\nmaximizing. \n\\hfill\\ensuremath{\\Box}\n\\end{proof}\n\n\n\\passage{Decomposition}\n\nLet $Z$ be a similarity matrix.  Subsets $B$ and $B'$ of $\\{1, \\ldots, n\\}$\nare \\demph{complementary} (for $Z$) if $B \\cup B' = \\{1, \\ldots, n\\}$, $B \\cap\nB' = \\emptyset$, and $Z_{i i'} = 0$ for all $i \\in B$ and $i' \\in B'$.  For\nexample, there exist nonempty complementary subsets if $Z$ can be expressed as\na nontrivial block sum\n\\[\n\\left(\\!\\!\n\\begin{array}{cc}\nX       &0      \\\\\n0       &X'\n\\end{array}\n\\!\\!\\right).\n\\]\n\nGiven a distribution $\\pv{p}$ and a subset $B \\subseteq \\{1, \\ldots, n\\}$ such that\n$p_i > 0$ for some $i \\in B$, let\n$\\rstr{\\pv{p}}{B}$ be the distribution on $B$ defined by \n\\[\n(\\rstr{\\pv{p}}{B})_i \n=\n\\frac{p_i}{\\sum_{j \\in B} p_j}.\n\\] \n\n\\begin{lemma}   \\lbl{lemma:complementary-basic}\nLet $Z$ be a similarity matrix, and let $B$ and $B'$ be nonempty complementary\nsubsets of $\\{1, \\ldots, n\\}$.  Then:\n\\begin{enumerate}\n\\item   \\lbl{part:complementary-weightings}\nFor any weightings $\\pv{v}$ on $Z_B$ and $\\pv{v}'$ on $Z_{B'}$,\nthere is a weighting $\\pv{w}$ on $Z$ defined by\n\\[\nw_i\n=\n\\left\\{\n\\begin{array}{ll}\nv_i    &\\textrm{if } i \\in B  \\\\\nv'_i   &\\textrm{if } i \\in B'.\n\\end{array}\n\\right. \n\\]\n\\item   \\lbl{part:complementary-invt}\nFor any invariant distributions $\\pv{r}$ on $B$ and $\\pv{r}'$ on $B'$, there\nexists an invariant distribution $\\pv{p}$ on $\\{1, \\ldots, n\\}$ such that\n$\\rstr{\\pv{p}}{B} = \\pv{r}$ and $\\rstr{\\pv{p}}{B'} = \\pv{r}'$.\n\\end{enumerate}\n\\end{lemma}\n\n\\begin{proof}\n\\begin{enumerate}\n\\item For $i \\in B$, we have\n\\[\n(Z\\pv{w})_i\n=\n\\sum_{j \\in B} Z_{ij} v_j + \\sum_{j \\in B'} Z_{ij} v'_j\n=\n\\sum_{j \\in B} (Z_B)_{ij} v_j\n=\n(Z_B \\pv{v})_i \n=\n1.\n\\]\nSimilarly, $(Z\\pv{w})_i = 1$ for all $i \\in B'$.  So $\\pv{w}$ is a weighting.\n\n\\item By Proposition~\\ref{propn:invt}, $\\pv{r} = \\pd{\\pv{v}}$ for some\nnon-negative weighting $\\pv{v}$ on some nonempty subset $C \\subseteq B$.\nSimilarly, $\\pv{r}' = \\pd{\\pv{v}'}$ for some non-negative weighting $\\pv{v}'$\non some nonempty $C' \\subseteq B'$.  By~(\\ref{part:complementary-weightings}),\nthere is a non-negative weighting $\\pv{w}$ on the nonempty set $C \\cup C'$\ndefined by \n\\[\nw_i\n=\n\\left\\{\n\\begin{array}{ll}\nv_i     &\\textrm{if } i \\in C   \\\\\nv'_i    &\\textrm{if } i \\in C'.\n\\end{array}\n\\right.\n\\]\nLet $\\pv{p} = \\pd{\\pv{w}}$, a distribution on $\\{1, \\ldots, n\\}$, which is\ninvariant by Proposition~\\ref{propn:invt}.  For $i \\in C$ we have\n\\begin{eqnarray*}\n(\\rstr{\\pv{p}}{B})_i    &=      &\n\\frac{p_i}{\\sum_{j \\in B} p_j}\n=\n\\frac{w_i\/\\magn{Z_{C \\cup C'}}}%\n{\\sum_{j \\in C} w_j\/\\magn{Z_{C \\cup C'}}}       \\\\\n        &=      &\n\\frac{w_i}{\\sum_{j \\in C} w_j}\n=\n\\frac{v_i}{\\sum_{j\\in C} v_j}   \\\\\n        &=      &\nr_i.\n\\end{eqnarray*}\nFor $i \\in B\\setminus C$ we have\n\\[\n(\\rstr{\\pv{p}}{B})_i \n=\n\\frac{p_i}{\\sum_{j \\in B} p_j}\n=\n0\n=\nr_i.\n\\]\nHence $\\rstr{\\pv{p}}{B} = \\pv{r}$, and similarly $\\rstr{\\pv{p}}{B'} =\n\\pv{r}'$.  \n\\hfill\\ensuremath{\\Box}\n\\end{enumerate}\n\\end{proof}\n\n\\begin{lemma}   \\lbl{lemma:complementary-D0}\nLet $Z$ be a similarity matrix, let $B$\nand $B'$ be complementary subsets of $\\{1, \\ldots, n\\}$, and let $\\pv{p}$ be a\ndistribution on $\\{1, \\ldots, n\\}$ such that $p_i > 0$ for some $i \\in B$ and\n$p_i > 0$ for some $i \\in B'$.  Then \n\\[\nD_0^Z(\\pv{p})\n=\nD_0^{Z_B}(\\rstr{\\pv{p}}{B}) + \nD_0^{Z_{B'}}(\\rstr{\\pv{p}}{B'}).\n\\]\n\\end{lemma}\n\n\\begin{proof}\nBy definition,\n\\[\nD_0^Z(\\pv{p})\n=\n\\sum \\frac{p_i}{(Z\\pv{p})_i}\n=\n\\sum_{i \\in B:\\ p_i > 0} \\frac{p_i}{(Z\\pv{p})_i}\n+\n\\sum_{i \\in B':\\ p_i > 0} \\frac{p_i}{(Z\\pv{p})_i}.\n\\]\nNow for $i \\in B$, \n\\[\np_i \n=\n\\left( \\sum_{j \\in B} p_j \\right)\n\\left( \\rstr{\\pv{p}}{B} \\right)_i\n\\]\nby definition of $\\rstr{\\pv{p}}{B}$, and \n\\[\n(Z\\pv{p})_i \n=\n\\sum_{j \\in B} Z_{ij} p_j + \\sum_{j \\in B'} Z_{ij} p_j\n=\n\\sum_{j \\in B} (Z_B)_{ij} p_j\n=\n\\left( \\sum_{j \\in B} p_j \\right)\n(Z_B \\rstr{\\pv{p}}{B})_i.\n\\]\nSimilar equations hold for $B'$, so\n\\begin{eqnarray*}\nD_0^Z(\\pv{p})   &\n=       &\n\\sum_{i \\in B:\\ p_i > 0} \n\\frac{(\\rstr{\\pv{p}}{B})_i}{(Z_B \\rstr{\\pv{p}}{B})_i}\n+\n\\sum_{i \\in B':\\ p_i > 0} \n\\frac{(\\rstr{\\pv{p}}{B'})_i}{(Z_{B'} \\rstr{\\pv{p}}{B'})_i}    \\\\\n        &=      &\nD_0^{Z_B}(\\rstr{\\pv{p}}{B}) + \nD_0^{Z_{B'}}(\\rstr{\\pv{p}}{B'}),\n\\end{eqnarray*}\nas required.\n\\hfill\\ensuremath{\\Box}\n\\end{proof}\n\n\n\\begin{propn}   \\lbl{propn:comp-invt-max}\nLet $Z$ be a similarity matrix and let $B$ and $B'$ be nonempty complementary\nsubsets of $\\{1, \\ldots, n\\}$.  Suppose that $Z_B$ and $Z_{B'}$ each admit an\ninvariant maximizing distribution.  Then so does $Z$.\n\\end{propn}\n\n\\begin{proof}\nChoose invariant maximizing distributions $\\pv{r}$ on $B$ and $\\pv{r}'$ on\n$B'$.  By\nLemma~\\ref{lemma:complementary-basic}(\\ref{part:complementary-invt}), there\nexists an invariant distribution $\\pv{p}$ on $\\{1, \\ldots, n\\}$ such that\n$\\rstr{\\pv{p}}{B} = \\pv{r}$ and $\\rstr{\\pv{p}}{B'} = \\pv{r}'$.  I claim that\n$\\pv{p}$ is maximizing.  Indeed, let $\\pv{s}$ be a distribution on $\\{1,\n\\ldots, n\\}$.  If $s_i > 0$ for some $i \\in B$ and $s_i > 0$ for some $i \\in\nB'$ then\n\\[\nD_0^Z(\\pv{s})\n=\nD_0^{Z_B}(\\rstr{\\pv{s}}{B}) + D_0^{Z_{B'}}(\\rstr{\\pv{s}}{B'}) \n\\leq\nD_0^{Z_B}(\\pv{r}) + D_0^{Z_{B'}}(\\pv{r}')\n\\]\nby Lemma~\\ref{lemma:complementary-D0}.  If not then without loss of\ngenerality, $s_i = 0$ for all $i \\in B'$; then $s_i > 0$ for some $i \\in B$,\nand\n\\[\nD_0^Z(\\pv{s})\n=\nD_0^{Z_B}(\\rstr{\\pv{s}}{B})\n\\leq\nD_0^{Z_B}(\\pv{r})\n\\leq \nD_0^{Z_B}(\\pv{r}) + D_0^{Z_{B'}}(\\pv{r}')\n\\]\nby Lemma~\\ref{lemma:absent}.  So in any case we have\n\\begin{eqnarray*}\nD_0^Z(\\pv{s})   \n        &\\leq   &\nD_0^{Z_B}(\\pv{r}) + D_0^{Z_{B'}}(\\pv{r}')       \\\\\n        &=      &\nD_0^{Z_B}(\\rstr{\\pv{p}}{B}) + D_0^{Z_{B'}}(\\rstr{\\pv{p}}{B'})   \\\\\n        &=      &\nD_0^Z(\\pv{p}),\n\\end{eqnarray*}\nusing Lemma~\\ref{lemma:complementary-D0} in the last step.  Hence $\\pv{p}$ is\n$0$-maximizing, and by Lemma~\\ref{lemma:max}, $\\pv{p}$ is maximizing.\n\\hfill\\ensuremath{\\Box}\n\\end{proof}\n\n\n\\passage{Positive definite similarity matrices}\nThe solution to the maximum diversity problem turns out to be simpler when the\nsimilarity matrix is positive definite and satisfies certain further\nconditions.  Here are some preparatory results.  They are not\nneeded for the proof of the main theorem~(\\ref{thm:main}) itself, but \nwill be used for the corollaries in Section~\\ref{sec:cors}.\n\n\n\\begin{lemma}   \\lbl{lemma:pos-def-basic}\nLet $Z$ be a positive definite similarity matrix.  Then $Z$ has a unique\nweighting and $\\magn{Z} > 0$.\n\\end{lemma}\n\n\\begin{proof}\nA positive definite matrix is invertible, so $Z$ has a unique weighting\n$\\pv{w}$.  By the definitions of magnitude and weighting,\n\\[\n\\magn{Z}\n=\n\\sum_{i = 1}^n w_i\n=\n\\pv{w}^\\mathrm{t} Z \\pv{w}.\n\\]\nBut $n \\geq 1$, so $\\pv{0}$ is not a weighting, so $\\pv{w} \\neq \\pv{0}$; then\nsince $Z$ is positive definite, $\\pv{w}^\\mathrm{t} Z \\pv{w} > 0$.\n\\hfill\\ensuremath{\\Box}\n\\end{proof}\n\n\n\\begin{lemma}   \\lbl{lemma:pos-def-sup}\nLet $Z$ be a positive definite similarity matrix.  Then \n\\[\n\\magn{Z}\n=\n\\sup_{\\pv{x}}\n\\frac{(\\sum_{i = 1}^n x_i)^2}{\\pv{x}^\\mathrm{t} Z \\pv{x}}\n\\]\nwhere the supremum is over all column vectors $\\pv{x} \\neq \\pv{0}$.  The\npoints at which the supremum is attained are exactly the nonzero scalar\nmultiples of the unique weighting on $Z$.\n\\end{lemma}\n\n\\begin{proof}\nSince $Z$ is positive definite, there is an inner product\n$\\ip{-}{-}$ on $\\mathbb{R}^n$ defined by\n\\[\n\\ip{\\pv{x}}{\\pv{y}} = \\pv{x}^\\mathrm{t} Z \\pv{y}\n\\]\n($\\pv{x}, \\pv{y} \\in \\mathbb{R}^n$).  The Cauchy--Schwarz inequality states that\nfor all $\\pv{x}, \\pv{y} \\in \\mathbb{R}^n$,\n\\[\n\\ip{\\pv{x}}{\\pv{x}} \\cdot \\ip{\\pv{y}}{\\pv{y}}\n\\geq\n\\ip{\\pv{x}}{\\pv{y}}^2\n\\]\nwith equality if and only if one of $\\pv{x}$ and $\\pv{y}$ is a scalar multiple\nof the other.  Let $\\pv{y}$ be the unique weighting on $Z$.  Then the\ninequality states that for all $\\pv{x} \\in \\mathbb{R}^n$,\n\\[\n\\pv{x}^\\mathrm{t} Z \\pv{x} \\cdot \\magn{Z}\n\\geq\n\\left(\\sum_{i = 1}^n x_i \\right)^2.\n\\]\nSince $\\pv{y} \\neq \\pv{0}$, equality holds if and only if $\\pv{x}$ is a scalar\nmultiple of $\\pv{y}$.  The result follows.  \n\\hfill\\ensuremath{\\Box}\n\\end{proof}\n\n\nA vector $\\pv{x}$ is \\demph{nowhere zero} if $x_i \\neq 0$ for all $i$.\n\n\\begin{propn}     \\lbl{propn:pos-def-sub}\nLet $Z$ be a positive definite similarity matrix and $B \\subseteq \\{1, \\ldots,\nn\\}$.  Then $Z_B$ is positive definite and $\\magn{Z_B} \\leq \\magn{Z}$.\nThe inequality is strict if $B$ is a proper subset and the unique weighting on\n$Z$ is nowhere zero.\n\\end{propn}\n\n\\begin{proof}\nSuppose without loss of generality that $B = \\{1, \\ldots, m\\}$, where $0 \\leq m\n\\leq n$.  \nLet $\\pv{y}$ be an $m$-dimensional column vector and write \n\\[\n\\pv{x} \n=\n(y_1, \\ldots, y_m, 0, \\ldots, 0)^\\mathrm{t}.\n\\]\nThen\n\\begin{equation}        \\label{eq:quad-form}\n\\pv{y}^\\mathrm{t} Z_B \\pv{y}\n=\n\\sum_{i, j = 1}^m y_i (Z_B)_{ij} y_j\n=\n\\sum_{i, j = 1}^n x_i Z_{ij} x_j\n= \n\\pv{x}^\\mathrm{t} Z \\pv{x}\n\\end{equation}\nand\n\\begin{equation}        \\label{eq:square}\n\\left( \\sum_{i = 1}^m y_i \\right)^2\n=\n\\left( \\sum_{i = 1}^n x_i \\right)^2.\n\\end{equation}\nBy~(\\ref{eq:quad-form}) and positive definiteness of $Z$, we have\n$\\pv{y}^\\mathrm{t} Z_B \\pv{y} \\geq 0$, with equality if and only if $\\pv{x} = 0$,\nif and only if $\\pv{y} = 0$.  So $Z_B$ is positive definite.  Then\nby~(\\ref{eq:quad-form}), (\\ref{eq:square}) and Lemma~\\ref{lemma:pos-def-sup},\n$\\magn{Z_B} \\leq \\magn{Z}$. \n\nNow suppose that $m < n$ and the weighting $\\pv{w}$ on $Z$ is nowhere zero.\nThe supremum in Lemma~\\ref{lemma:pos-def-sup} is attained only at nonzero\nscalar multiples of $\\pv{w}$; in particular, any vector $\\pv{x}$ at which it\nis attained satisfies $x_n \\neq 0$.  Let $\\pv{y}$ be the unique weighting\non $Z_B$ and let $\\pv{x}$ be the corresponding $n$-dimensional column vector,\nas above.  Since $x_n = 0$, we have\n\\[\n\\magn{Z_B}\n=\n\\frac{(\\sum_{i = 1}^m y_i)^2}{\\pv{y}^\\mathrm{t} Z_B \\pv{y}}\n=\n\\frac{(\\sum_{i = 1}^n x_i)^2}{\\pv{x}^\\mathrm{t} Z \\pv{x}}\n<\n\\magn{Z},\n\\]\nas required.  \n\\hfill\\ensuremath{\\Box}\n\\end{proof}\n\n\\begin{lemma}   \\lbl{lemma:scattered}\nLet $Z$ be a similarity matrix with $Z_{ij} < 1\/(n - 1)$ for all $i \\neq j$.\nThen $Z$ is positive definite, and the unique weighting on $Z$ is positive.\n\\end{lemma}\n\n\\begin{proof}\nTheorem~2 of~\\cite{AMSES} shows that $Z$ is positive definite.  Now,\nfor each $i \\in \\{1, \\ldots, n\\}$ and $r \\geq 0$, put\n\\[\nc_{i, r}\n= \n\\sum_{i = i_0 \\neq \\cdots \\neq i_r} \nZ_{i_0 i_1} Z_{i_1 i_2} \\cdots Z_{i_{r - 1} i_r},\n\\]\nwhere the sum is over all $i_0, \\ldots, i_r \\in \\{1, \\ldots, n\\}$ such\nthat $i_0 = i$ and $i_{s - 1} \\neq i_s$ whenever $1 \\leq s \\leq r$.  In\nparticular, $c_{i, 0} = 1$.  Write $\\gamma = \\max_{j \\neq k} Z_{jk}$.  Then\nfor all $r \\geq 0$,\n\\begin{equation}        \\label{eq:wt-alt}\nc_{i, r + 1} \n\\leq \n\\sum_{i = i_0 \\neq \\cdots \\neq i_r \\neq i_{r + 1}}\nZ_{i_0 i_1} Z_{i_1 i_2} \\cdots Z_{i_{r - 1} i_r} \\gamma\n=\n(n - 1)\\gamma \\cdot c_{i, r}.\n\\end{equation}\nHence $c_{i, r} \\leq ((n - 1)\\gamma)^r$ for all $r \\geq 0$; and $(n - 1)\\gamma\n< 1$, so the sum $w_i := \\sum_{r = 0}^\\infty (-1)^r c_{i, r}$ converges.\nAgain using~(\\ref{eq:wt-alt}), we have $c_{i, r + 1} < c_{i, r}$ for all $r$,\nso $w_i > 0$.\n\nIt remains to show that $\\pv{w} = (w_1, \\ldots, w_n)^\\mathrm{t}$ is a weighting.\nLet $i \\in \\{1, \\ldots, n\\}$.  Then\n\\begin{eqnarray*}\n(Z\\pv{w})_i     &=      &\nw_i + \\sum_{j \\neq i} Z_{ij} w_j \\\\\n        &=      &\nw_i + \n\\sum_{j \\neq i}  Z_{ij} \n\\sum_{r = 0}^\\infty (-1)^r\n\\sum_{j = j_0 \\neq \\cdots \\neq j_r} Z_{j_0 j_1} \\cdots Z_{j_{r-1} j_r}  \\\\\n        &=      &\nw_i - \n\\sum_{r = 0}^\\infty (-1)^{r + 1} c_{i, r + 1}   \\\\\n        &=      &\nw_i - (w_i - c_{i, 0}) \n\\\\\n        &=      &\n1,\n\\end{eqnarray*}\nas required.\n\\hfill\\ensuremath{\\Box}\n\\end{proof}\n\n\n\\pagebreak\n\n\n\\section{The main theorem}\n\n\n\\passage{Solution to the maximum diversity problem}\n\n\\begin{thm}[Main Theorem]     \\lbl{thm:main}\nLet $Z$ be a similarity matrix.  Then:\n\\begin{enumerate}\n\\item   \\lbl{part:main-value}\nFor all $q \\in [0, \\infty]$,\n\\begin{equation}        \\label{eq:sup-max}\n\\sup_{\\pv{p}} D_q^Z(\\pv{p})\n=\n\\max_B \\magn{Z_B}\n\\end{equation}\nwhere the supremum is over all distributions $\\pv{p}$ and the\nmaximum is over all subsets $B \\subseteq \\{1, \\ldots, n\\}$ such that $Z_B$\nadmits a non-negative weighting.\n\\item   \\lbl{part:main-place}\nThe maximizing distributions are precisely those of the form\n$\\pd{\\pv{w}}$, where $\\pv{w}$ is a non-negative weighting on a subset\n$B \\subseteq \\{1, \\ldots, n\\}$ such that $\\magn{Z_B}$ attains the\nmaximum~(\\ref{eq:sup-max}).  \n\\end{enumerate}\nIn particular, there exists a maximizing distribution, and the maximum\ndiversity of order $q$ is the same for all $q \\in [0, \\infty]$.\n\\end{thm}\n\nFor the definitions, including that of `maximizing distribution', see\nSection~\\ref{sec:prep}.\nThe proof is given later in this section.  First we make some remarks on\ncomputation and on maximum entropy. \n\nThe \\demph{maximum diversity} of a similarity matrix $Z$ is $\\Dmax{Z} :=\n\\sup_\\pv{p} D_q^Z(\\pv{p})$, which by Theorem~\\ref{thm:main} is independent of\nthe value of $q \\in [0, \\infty]$.\n\n\n\\passage{Remarks on computation}\nSuppose that we are given a similarity matrix $Z$ and want to compute its\nmaximizing distribution(s) and maximum diversity.  The theorem gives the\nfollowing algorithm.  For each of the $2^n$ subsets $B$ of $\\{1, \\ldots, n\\}$:\n\\begin{itemize}\n\\item perform some simple linear algebra to decide whether $Z_B$ admits a\nnon-negative weighting\n\\item if it does, tag $B$ as `good' and record the magnitude $\\magn{Z_B}$\n(the sum of the entries of any weighting).\n\\end{itemize}\nThe maximum of all the recorded magnitudes is the maximum\ndiversity $\\Dmax{Z}$.  For each good $B$ such that $\\magn{Z_B} =\n\\Dmax{Z}$, find all non-negative weightings $\\pv{w}$ on $Z_B$; the\ncorresponding distributions $\\pd{\\pv{w}}$ are the maximizing distributions.\n\nThis algorithm takes exponentially many steps.  However, each step is fast, so\nit might be possible to handle reasonably large values of $n$ in a reasonable\nlength of time.  Moreover, the results of Section~\\ref{sec:cors} may allow\nthe speed of the algorithm to be improved.\n\n\\passage{Solution to the maximum entropy problem}\nWe can translate the solution to the maximum \\emph{diversity} problem into a\nsolution to the maximum \\emph{entropy} problem.  The first part, giving the\nvalue of the maximum, becomes more complicated.  The second part, giving the\nmaximizing distribution(s), is unchanged.\n\n\\begin{thm}     \\lbl{thm:main-entropy}\nLet $Z$ be a similarity matrix.  Then:\n\\begin{enumerate}\n\\item   \nFor all $q \\in [0, \\infty)$,\n\\begin{equation}        \\label{eq:sup-max-ent}\n\\sup_{\\pv{p}} H_q^Z(\\pv{p})\n=\n\\left\\{\n\\begin{array}{ll}\n\\max_B \\frac{1}{q - 1} \\left( 1 - \\magn{Z_B}^{1 - q} \\right)    &\n\\textrm{if } q \\neq 1   \\\\\n\\max_B \\log \\magn{Z_B}  &\n\\textrm{if } q = 1\n\\end{array}\n\\right.\n\\end{equation}\nwhere the supremum is over all distributions $\\pv{p}$ and the\nmaxima are over all subsets $B \\subseteq \\{1, \\ldots, n\\}$ such that $Z_B$\nadmits a non-negative weighting.\n\\item\nThe maximizing distributions are precisely those of the form\n$\\pd{\\pv{w}}$, where $\\pv{w}$ is a non-negative weighting on a subset\n$B \\subseteq \\{1, \\ldots, n\\}$ such that $\\magn{Z_B}$ is maximal among all subsets\nadmitting a non-negative weighting.\n\\end{enumerate}\nIn particular, there exists a maximizing distribution.\n\\end{thm}\n\n\\begin{proof}\nThis follows almost immediately from Theorem~\\ref{thm:main}, using the\ndefinition of $D_q^Z$ in terms of $H_q^Z$.  Note that on the right-hand side\nof~(\\ref{eq:sup-max-ent}), the expressions $\\frac{1}{q - 1}(1 - \\magn{Z_B}^{1\n- q})$ and $\\log \\magn{Z_B}$ are increasing, injective functions of\n$\\magn{Z_B}$, so a subset $B$ maximizes any one of them if and only if it\nmaximizes $\\magn{Z_B}$.  \\hfill\\ensuremath{\\Box}\n\\end{proof}\n\nThe part of Theorem~\\ref{thm:main} stating that the maximum diversity of order\n$q$ is the same for all values of $q$ has no clean statement in terms of\nentropy.\n\n\n\\passage{Diversity of order zero}\n\nOur proof of Theorem~\\ref{thm:main} will depend on an analysis of the function\n$D_0^Z$, diversity of order zero.  The first step is to find its critical\npoints, and for that we need a technical lemma.\n\n\\begin{lemma}   \\lbl{lemma:skew}\nLet $m \\geq 1$, let $Y$ be an $m \\times m$ real skew-symmetric matrix, and let\n$\\pv{x} \\in (0, \\infty)^m$.  Suppose that $Y_{ij} \\geq 0$ whenever $i \\geq j$\nand that $\\sum_{j=1}^m Y_{ij} x_j$ is independent of $i \\in \\{1, \\ldots, m\\}$.\nThen $Y = 0$.\n\\end{lemma}\n\n\\begin{proof}\nThis is true for $m = 1$; suppose inductively that $m \\geq 2$.  We\nhave \n\\[\n\\sum_{j = 1}^m Y_{1j} x_j \n= \n\\sum_{j = 1}^m Y_{mj} x_j\n\\]\nwith $Y_{1j} x_j = -Y_{j1} x_j \\leq 0$ and $Y_{mj} x_j \\geq 0$ for all $j$;\nhence both sides are $0$ and $Y_{mj} x_j = 0$ for all $j$.  So for all $j$ we\nhave $Y_{mj} = 0$ (since $x_j > 0$) and $Y_{jm} = 0$ (by skew-symmetry).  Let\n$Y'$ be the $(m - 1) \\times (m - 1)$ matrix defined by $Y'_{ij} = Y_{ij}$.\nThen $Y'$ satisfies the conditions of the inductive hypothesis, so $Y' = 0$;\nthat is, $Y_{ij} = 0$ whenever $i, j < m$.  But we already have $Y_{ij} = 0$\nwhenever $i = m$ or $j = m$, so $Y = 0$, completing the induction.  \\hfill\\ensuremath{\\Box}\n\\end{proof}\n\nWrite\n\\[\n\\smp{n}\n=\n\\{ \\pv{p} \\in \\mathbb{R}^n \n\\:|\\: \n\\sum p_i = 1,\n\\ p_i \\geq 0 \n\\}\n\\]\nfor the space of distributions, and\n\\[\n\\ismp{n}\n=\n\\{ \\pv{p} \\in \\mathbb{R}^n \n\\:|\\: \n\\sum p_i = 1,\n\\ p_i > 0 \n\\}\n\\]\nfor the space of nowhere-zero distributions.  The function $D_0^Z$ on\n$\\smp{n}$ is given by\n\\[\nD_0^Z(\\pv{p})\n=\n\\sum \\frac{p_i}{(Z\\pv{p})_i}.\n\\]\n(Recall the standing convention that unlabelled summations are over all $i \\in\n\\{1, \\ldots, n\\}$ such that $p_i > 0$.)  It can be defined, using the same\nformula, for all $\\pv{p} \\in [0, \\infty)^n$.  It is then differentiable\non $(0, \\infty)^n$, where the summation is over all $i \\in \\{1, \\ldots, n\\}$.\n\n\\begin{propn}\nLet $Z$ be a similarity matrix and $\\pv{p} \\in \\ismp{n}$.  Then $\\pv{p}$ is a\ncritical point of $D_0^Z$ on $\\ismp{n}$ if and only if for all $i, j \\in \\{1,\n\\ldots, n\\}$,\n\\[\nZ_{ij} > 0 \\,\\Rightarrow\\, (Z\\pv{p})_i = (Z\\pv{p})_j.\n\\]\n\\end{propn}\n\n\\begin{proof}\nWe find the critical points of $D_0^Z$ on $\\ismp{n}$ using Lagrange\nmultipliers and the fact that $\\ismp{n}$ is the intersection of $(0,\n\\infty)^n$ with the hyperplane $\\{ \\pv{p} \\in \\mathbb{R}^n \\:|\\: \\sum p_i = 1\n\\}$.  Write $h(\\pv{p}) = \\sum p_i - 1$.\n\nFor $k, i \\in \\{1, \\ldots, n\\}$ and $\\pv{p} \\in (0, \\infty)^n$ we have\n$\\ptl{p_k} (Z\\pv{p})_i = Z_{ik}$, giving\n\\[\n\\ptl{p_k} \n\\left(\n\\frac{p_i}{(Z\\pv{p})_i}\n\\right)\n=\n\\left\\{\n\\begin{array}{ll}\n\\displaystyle\n\\frac{1}{(Z\\pv{p})_k^2} \\sum_{j \\neq k} Z_{kj} p_j      &\n\\textrm{if } k = i      \\\\\n\\displaystyle\n-\\frac{1}{(Z \\pv{p})_i^2} Z_{ik} p_i    &\n\\textrm{otherwise.}\n\\end{array}\n\\right.\n\\]\nFrom this and symmetry of $Z$ we deduce that for $k \\in \\{1, \\ldots, n\\}$ and\n$\\pv{p} \\in (0, \\infty)^n$,\n\\[\n\\ptl{p_k} D_0^Z(\\pv{p})\n=\n\\sum_{i = 1}^n Y_{ki} p_i\n\\]\nwhere\n\\[\nY_{ki} \n=\n\\left(\n\\frac{1}{(Z\\pv{p})_k^2} - \\frac{1}{(Z\\pv{p})_i^2} \n\\right)\nZ_{ki}.\n\\]\nOn the other hand, \n\\[\n\\ptl{p_k} h(\\pv{p})\n=\n1\n\\]\nfor all $k$.  A point $\\pv{p} \\in \\ismp{n}$ is a critical point of $D_0^Z$ on\n$\\ismp{n}$ if and only if there exists a scalar $\\lambda$ such that $(\\nabla\nD_0^Z)(\\pv{p}) = \\lambda (\\nabla h)(\\pv{p})$.  Hence $\\pv{p}$ is critical if\nand only if $\\sum_i Y_{ki} p_i$ is independent of $k \\in \\{1, \\ldots, n\\}$.\nSo the proposition is equivalent to the statement that, for $\\pv{p} \\in\n\\ismp{n}$, the sum $\\sum Y_{ki} p_i$ is independent of $k$ if and only if the\nmatrix $Y$ is $0$.\n\nThe `if' direction is trivial.  Conversely, suppose that $\\sum_i Y_{ki} p_i$\nis independent of $k$.  Assume without loss of generality that $(Z \\pv{p})_1\n\\geq \\cdots \\geq (Z\\pv{p})_n$.  Then Lemma~\\ref{lemma:skew} applies (taking\n$\\pv{x} = \\pv{p}$), and $Y = 0$.  \\hfill\\ensuremath{\\Box}\n\\end{proof}\n\nLet $\\sim$ be the equivalence relation on\n$\\{1, \\ldots, n\\}$ generated by $i \\sim j$ whenever $Z_{ij} > 0$.  Thus, $i\n\\sim j$ if and only if there is a chain $i = i_1, i_2, \\ldots, i_{r-1}, i_r =\nj$ with $Z_{i_t, i_{t+1}} > 0$ for all $t$.  Call $Z$ \\demph{connected} if $i\n\\sim j$ for all $i, j$.\n\n\\begin{cor}     \\lbl{cor:conn-crit}\nLet $Z$ be a connected similarity matrix.  Then every critical point of $D_0^Z$\nin $\\ismp{n}$ is a weight distribution.  \n\\hfill\\ensuremath{\\Box}\n\\end{cor}\n\nWe are aiming to show, among other things, that there is a $0$-maximizing\ndistribution: the function $D_0^Z$ on $\\smp{n}$ attains its supremum.  Its\nsupremum is finite, since by Lemma~\\ref{lemma:positive}, $D_0^Z(\\pv{p}) \\leq\nn$ for all distributions $\\pv{p}$.  If $D_0^Z$ is continuous on $\\smp{n}$ then\nit certainly does attain its supremum.  But in general, it is not.  For\nexample, if $Z = I$ then $D_0^Z(\\pv{p})$ is the cardinality of the support of\n$\\pv{p}$, which is not continuous in $\\pv{p}$.  We must therefore use another\nargument to establish the existence of a $0$-maximizing distribution.  The\nfollowing lemma will help us.\n\n\\begin{lemma}   \\lbl{lemma:seqs}\nLet $Z$ be a connected similarity matrix and let $(\\pv{p}^{k})_{k \\in \\nat}$\nbe a sequence in $\\smp{n}$.  Then $(\\pv{p}^k)_{k \\in \\nat}$ has a subsequence\n$(\\pv{p}^k)_{k \\in S}$ satisfying at least one of the following conditions:\n\\begin{enumerate}\n\\item   \\lbl{part:seqs-boundary}\nthere is some $i \\in \\{1, \\ldots, n\\}$ such that $p^k_i = 0$ for all $k\n\\in S$\n\\item   \\lbl{part:seqs-interior}\nthe subsequence $(\\pv{p}^k)_{k \\in S}$ lies in $\\ismp{n}$ and converges\nto some point of $\\ismp{n}$\n\\item   \\lbl{part:seqs-quotient}\nthe subsequence $(\\pv{p}^k)_{k \\in S}$ lies in $\\ismp{n}$, and there is\nsome $i \\in \\{1, \\ldots, n\\}$ such that\n$\\lim_{k \\in S} \\left(p^k_i\/(Z\\pv{p}^k)_i\\right) = 0$.\n\\end{enumerate}\n\\end{lemma}\nHere and in what follows, we treat sequences as families $(x_k)_{k \\in T}$\nindexed over some infinite subset $T$ of $\\nat$.  A subsequence of such a\nsequence therefore amounts to an infinite subset of $T$.\n\n\\begin{proof}\nIf there exist infinitely many pairs $(k, i) \\in \\nat \\times \\{1, \\ldots, n\\}$\nsuch that $p^k_i = 0$ then there is some $i \\in \\{1, \\ldots, n\\}$ such that\n$\\{k \\in \\nat \\:|\\: p^k_i = 0\\}$ is infinite.  Taking $S = \\{k \\in \\nat\n\\:|\\: p^k_i = 0\\}$ then gives condition~(\\ref{part:seqs-boundary}).  \n\nSuppose, then, that there are only finitely many such pairs.  We may choose a\nsubsequence $(\\pv{p}^k)_{k \\in Q}$ of $(\\pv{p}^k)_{k \\in \\nat}$ lying in\n$\\ismp{n}$.  Further, since $\\smp{n}$ is compact, we may choose a subsequence\n$(\\pv{p}^k)_{k \\in R}$ of $(\\pv{p}^k)_{k \\in Q}$ converging to some point\n$\\pv{p} \\in \\smp{n}$.  If $\\pv{p} \\in \\ismp{n}$\nthen $(\\pv{p}^k)_{k \\in R}$ satisfies~(\\ref{part:seqs-interior}).\n\nSuppose, then, that $\\pv{p} \\not\\in \\ismp{n}$; say $p_\\ell = 0$ where $\\ell\n\\in \\{1, \\ldots, n\\}$.  \n\nDefine a binary relation $\\trianglelefteq$ on $\\{1, \\ldots, n\\}$ by $i \\trianglelefteq j$ if\nand only if $(p^k_i\/p^k_j)_{k \\in R}$ is bounded (that is, bounded above).\nThen $\\trianglelefteq$ is reflexive and transitive, and if $i \\trianglelefteq j$ and $p_j = 0$\nthen $p_i = 0$.  Write $i \\approx j$ for $i \\trianglelefteq j \\trianglelefteq i$; then\n$\\approx$ is an equivalence relation.\n\nI claim that there exist $i, j \\in \\{1, \\ldots, n\\}$ with $Z_{ij} > 0$ and $i\n\\not\\trianglelefteq j$.  For if not, the equivalence relation $\\approx$ satisfies\n$Z_{ij} > 0 \\,\\Rightarrow\\, i \\approx j$, and since $Z$ is connected, $i \\approx j$\nfor all $i, j$.  But then $i \\trianglelefteq \\ell$ for all $i$, and $p_\\ell = 0$, so\n$p_i = 0$ for all $i$.  This contradicts $\\pv{p}$ being a distribution,\nproving the claim.\n\nNow without loss of generality, $Z_{1n} > 0$ and $1 \\not\\trianglelefteq n$.  So\n$(p^k_1\/p^k_n)_{k \\in R}$ is unbounded.  We may choose an infinite subset $S\n\\subseteq R$ such that $\\lim_{k \\in S} (p^k_1\/p^k_n) = \\infty$.  For all $k\n\\in S$ we have\n\\[\n\\frac{(Z\\pv{p}^k)_n}{p^k_n}\n\\geq\n\\frac{Z_{n1} p^k_1}{p^k_n}\n\\]\nwith $Z_{n1} = Z_{1n} > 0$, so\n\\[\n\\lim_{k \\in S}\n\\left(\n\\frac{(Z\\pv{p}^k)_n}{p^k_n}\n\\right)\n=\n\\infty,\n\\]\nand condition~(\\ref{part:seqs-quotient}) follows.\n\\hfill\\ensuremath{\\Box}\n\\end{proof}\n\n\\passage{Existence of a maximizing distribution} At the heart of\nTheorem~\\ref{thm:main} is the following result, from which we will deduce the\ntheorem itself.\n\n\\begin{propn}   \\lbl{propn:heart}\nEvery similarity matrix has a maximizing distribution, and every maximizing\ndistribution is invariant.  \n\\end{propn}\n\n\\begin{proof}\nLet $Z$ be a similarity matrix.  It is enough to prove that $Z$ admits an\ninvariant maximizing distribution: for if $\\pv{p}$ and $\\pv{p}'$ are both\nmaximizing then $D_q^Z(\\pv{p}) = D_q^Z(\\pv{p}')$ for all $q$, so $\\pv{p}$ is\ninvariant if and only if $\\pv{p}'$ is.\n\nThe result holds for $n = 1$.  Suppose inductively that $n \\geq 2$.\n\n\\emph{Case 1: $Z$ is not connected.} We may partition $\\{1, \\ldots, n\\}$ into\ntwo nonempty subsets, $B$ and $B'$, each of which is a union of\n$\\sim$-equivalence classes (where $\\sim$ is as defined before\nCorollary~\\ref{cor:conn-crit}).  Then $B$ and $B'$ are complementary, and by\ninductive hypothesis, $Z_B$ and $Z_{B'}$ each admit an invariant maximizing\ndistribution.  So by Proposition~\\ref{propn:comp-invt-max}, $Z$ admits one\ntoo. \n\n\\emph{Case 2: $Z$ is connected.}  Write $\\sigma = \\sup_{\\pv{p}}\nD_0^Z(\\pv{p})$.  We may choose a sequence $(\\pv{p}^k)_{k \\in \\nat}$ in\n$\\smp{n}$ with $\\lim_{k \\to \\infty} D_0^Z(\\pv{p}^k) = \\sigma$.  By\nLemma~\\ref{lemma:seqs}, at least one of the following three conditions holds.\n\\begin{enumerate}\n\\item There is a subsequence $(\\pv{p}^k)_{k \\in S}$ such that (without loss of\ngenerality) $p^k_n = 0$ for all $k \\in S$.  Write $B = \\{1, \\ldots, n - 1\\}$.\nDefine a sequence $(\\pv{r}^k)_{k \\in S}$ in $\\smp{n - 1}$ by \n\\[\n\\pv{r}^k = (p^k_1, \\ldots, p^k_{n - 1}).\n\\]\nThen for all $k \\in S$ we have $D_0^{Z_B}(\\pv{r}^k) = D_0^Z(\\pv{p}^k)$ (by\nLemma~\\ref{lemma:absent}), so $\\sup_{k \\in S} D_0^{Z_B}(\\pv{r}^k) = \\sigma$.\nThen by Lemma~\\ref{lemma:smaller-max} and inductive hypothesis, $Z$ admits an\ninvariant maximizing distribution.\n\n\\item There is a subsequence $(\\pv{p^k})_{k \\in S}$ in $\\ismp{n}$ convergent\nto some point $\\pv{p} \\in \\ismp{n}$.  Since $D_0^Z$ is continuous on\n$\\ismp{n}$,\n\\[\nD_0^Z(\\pv{p})\n=\n\\lim_{k \\in S} D_0^Z(\\pv{p}^k)\n=\n\\sigma.\n\\]\nSo $\\pv{p}$ is $0$-maximizing.  Now $\\pv{p}$ is a critical point of $D_0^Z$ on\n$\\ismp{n}$, and $Z$ is connected, so by Corollary~\\ref{cor:conn-crit},\n$\\pv{p}$ is a weight distribution.  By Lemma~\\ref{lemma:mag-div},\n$\\pv{p}$ is invariant; then by Lemma~\\ref{lemma:max}, $\\pv{p}$ is maximizing.\n\n\\item There is a subsequence $(\\pv{p}^k)_{k \\in S}$ in $\\ismp{n}$ such that\n(without loss of generality) $\\lim_{k \\in S} \\left( p^k_n \/ (Z\\pv{p}^k)_n\n\\right) = 0$.  Write $B = \\{1, \\ldots, n - 1\\}$.  Define a sequence\n$(\\pv{r}^k)_{k \\in S}$ in $\\ismp{n - 1}$ by $\\pv{r}^k = \\rstr{\\pv{p}^k}{B}$\n(which is possible because $\\pv{p}^k \\in \\ismp{n}$ and $n \\geq 2$).  Then for\nall $k \\in S$ and $i \\in B$,\n\\[\n\\frac{r^k_i}{(Z_B \\pv{r}^k)_i}\n=\n\\frac{p^k_i}{\\sum_{j = 1}^{n - 1} Z_{ij} p^k_j}\n=\n\\frac{p^k_i}{(Z\\pv{p}^k)_i - Z_{in} p^k_n}\n\\geq\n\\frac{p^k_i}{(Z\\pv{p}^k)_i}.\n\\]\nHence for all $k \\in S$,\n\\[\nD_0^{Z_B}(\\pv{r}^k)\n=\n\\sum_{i = 1}^{n - 1}\n\\frac{r^k_i}{(Z_B \\pv{r}^k)_i}\n\\geq\n\\sum_{i = 1}^{n - 1}\n\\frac{p^k_i}{(Z\\pv{p}^k)_i}\n=\nD_0^Z(\\pv{p}^k) - \\frac{p^k_n}{(Z\\pv{p}^k)_n}.\n\\]\nBut \n\\[\n\\lim_{k \\in S} \n\\left(\nD_0^Z(\\pv{p}^k) - \\frac{p^k_n}{(Z\\pv{p}^k)_n}\n\\right)\n=\n\\sigma - 0\n=\n\\sigma,\n\\]\nso $\\sup_{k \\in S} D_0^{Z_B}(\\pv{r}^k) \\geq \\sigma$.   Then by\nLemma~\\ref{lemma:smaller-max} and inductive hypothesis, $Z$ admits an\ninvariant maximizing distribution.\n\\end{enumerate}\nSo in all cases there is an invariant maximizing distribution, completing\nthe induction.\n\\hfill\\ensuremath{\\Box}\n\\end{proof}\n\n\\passage{Proof of the Main Theorem,~\\ref{thm:main}}\n\\begin{enumerate}\n\\item Let $q \\in [0, \\infty]$.  By Proposition~\\ref{propn:heart}, the supremum\n$\\sup_\\pv{p} D_q^Z(\\pv{p})$ is unchanged if $\\pv{p}$ is taken to run over only\nthe invariant distributions.  By Proposition~\\ref{propn:invt}, any invariant\ndistribution is of the form $\\pd{\\pv{w}}$ for some non-negative weighting\n$\\pv{w}$ on some nonempty subset $B \\subseteq \\{1, \\ldots, n\\}$.  Hence \n\\[\n\\sup_{\\pv{p}} D_q^Z(\\pv{p})\n=\n\\max_{B, \\pv{w}} D_q^Z(\\pd{\\pv{w}})\n\\]\nwhere the maximum is over all nonempty $B$ and non-negative weightings\n$\\pv{w}$ on $B$.  But for any such $B$ and $\\pv{w}$ we have \n\\[\nD_q^Z(\\pd{\\pv{w}}) \n=  \nD_q^{Z_B}(\\pv{w}\/\\magn{Z_B})\n=\n\\magn{Z_B}\n\\]\nby Lemmas~\\ref{lemma:absent} and~\\ref{lemma:mag-div} respectively.  Hence \n\\[\n\\sup_{\\pv{p}} D_q^Z(\\pv{p})\n=\n\\max_B \\magn{Z_B}\n\\]\nwhere the maximum is now over all nonempty $B \\subseteq \\{1, \\ldots, n\\}$ such that\nthere exists a non-negative weighting on $Z_B$.  And since $\\magn{\\emptyset} =\n0$, it makes no difference if we allow $B$ to be empty.\n\n\\item Any maximizing distribution is invariant, by\nProposition~\\ref{propn:heart}.  The result now follows from\nProposition~\\ref{propn:invt}.  \n\\hfill\\ensuremath{\\Box}\n\\end{enumerate}\n\n\n\n\n\\section{Corollaries and examples}\n\\lbl{sec:cors}\n\n\nHere we state some corollaries to the results of the previous section.  The\nfirst is a companion to Lemma~\\ref{lemma:max}.\n\n\\begin{cor}     \\lbl{cor:some-all}\nLet $Z$ be a similarity matrix and $q \\in (0, \\infty]$.  Then a distribution\nis $q$-maximizing if and only if it is maximizing.\n\\end{cor}\n\nIn other words, if a distribution is $q$-maximizing for \\emph{some}\n$q > 0$ then it is $q$-maximizing for \\emph{all} $q \\geq 0$.  The proof is\nbelow. \n\nHowever, a $0$-maximizing distribution is not necessarily maximizing.  Take $Z\n= I$, for example.  Then $D_0^Z(\\pv{p}) = \\sum_{i:\\,p_i > 0} 1 = $ cardinality\nof $\\mr{supp}(\\pv{p})$, so any nowhere-zero distribution is $0$-maximizing.  On\nthe other hand, only the uniform distribution $(1\/n, \\ldots, 1\/n)$ is\nmaximizing.  So the restriction $q \\neq 0$ cannot be dropped from\nCorollary~\\ref{cor:some-all}, nor can the word `invariant' be dropped\nfrom Lemma~\\ref{lemma:max}.\n\n\\begin{proof}\nLet $\\pv{p}$ be a $q$-maximizing distribution.  Then\n\\[\nD_q^Z(\\pv{p})\n=\n\\Dmax{Z}\n\\geq\nD_0^Z(\\pv{p})\n\\geq\nD_q^Z(\\pv{p}),\n\\]\nwhere the second inequality is by Lemma~\\ref{lemma:means}.  So we have\nequality throughout, and in particular $D_0^Z(\\pv{p}) = D_q^Z(\\pv{p})$.  But\n$q > 0$, so by Lemma~\\ref{lemma:means}, $\\pv{p}$ is invariant.  Hence for all\n$q' \\in [0, \\infty]$,\n\\[\nD_{q'}^Z(\\pv{p}) = D_q^Z(\\pv{p}) = \\Dmax{Z},\n\\]\nand therefore $\\pv{p}$ is maximizing.\n\\hfill\\ensuremath{\\Box} \n\\end{proof}\n\nThe importance of Corollary~\\ref{cor:some-all} is that if one has solved the\nproblem of maximizing entropy or diversity of any particular order $q > 0$,\nthen one has solved the problem of maximizing entropy and diversity of all\norders.  In the following example, we observe that for a certain class of\nsimilarity matrices, the problem of maximizing the entropy of order $2$ has\nalready been solved in the literature; we can immediately deduce a\nmore general maximum entropy theorem.\n\n\\begin{example} \\lbl{eg:graphs}\nLet $G$ be a finite reflexive graph.  Thus, $G$ consists of a finite set $\\{1,\n\\ldots, n\\}$ of vertices equipped with a reflexive symmetric binary relation\n$E$.  Such graphs correspond to similarity matrices $Z$ whose entries are all\n$0$ or $1$, taking $Z_{ij} = 1$ if $(i, j) \\in E$ and $Z_{ij} = 0$\notherwise.\n\nFor each $q \\in [0, \\infty]$ and distribution $\\pv{p}$, define\n$D_q^G(\\pv{p}) = D_q^Z(\\pv{p})$, the \\demph{diversity of order $q$} of\n$\\pv{p}$ with respect to $G$.  Thus,\n\\[\nD_q^G(\\pv{p})\n=\n\\left\\{\n\\renewcommand{\\arraystretch}{3}\n\\begin{array}{ll}\n\\displaystyle\n\\left( \\sum_{i:\\,p_i > 0} p_i\n\\left( \\sum_{j:\\,(i, j) \\in E} p_j \\right)^{q - 1}\n\\right)^\\frac{1}{1 - q} &\n\\textrm{if } q \\neq 1, \\infty   \\\\\n\\displaystyle\n\\prod_{i:\\,p_i > 0} \n\\left( \\sum_{j:\\,(i, j) \\in E} p_j \\right)^{-p_i}        &\n\\textrm{if } q = 1      \\\\\n\\displaystyle\n1\/\\max_{i:\\,p_i > 0} \\sum_{j:\\,(i, j) \\in E} p_j          &\n\\textrm{if } q = \\infty.\n\\end{array}\n\\renewcommand{\\arraystretch}{1}\n\\right.\n\\]\n\nA set $K \\subseteq \\{1, \\ldots, n\\}$ of vertices of $G$ is \\demph{discrete} if $(i,\nj) \\not\\in E$ whenever $i, j \\in K$ with $i \\neq j$.  Write $\\mr{d}(G)$\nfor the largest integer $d$ such that there exists a discrete set in $G$\nof cardinality $d$.  Also, given any\nnonempty set $B \\subseteq \\{1, \\ldots, n\\}$, write $\\pv{p}^B$ for the distribution\n\\[\n\\pv{p}^B_i\n=\n\\left\\{\n\\begin{array}{ll}\n1\/|B|   &\\textrm{if } i \\in B   \\\\\n0       &\\textrm{otherwise}.\n\\end{array}\n\\right.\n\\]\n\n\\emph{Claim:} For all $q \\in [0, \\infty]$,\n\\[\n\\sup_{\\pv{p}} D_q^G(\\pv{p}) = \\mr{d}(G),\n\\]\nand the supremum is attained at $\\pv{p}^K$ for any discrete set $K$ of\ncardinality $\\mr{d}(G)$.\n\n\\emph{Proof:} We use the following result of Berarducci, Majer and\nNovaga~\\cite{BMN}.  Let $G'$ be a finite \\emph{ir}reflexive graph with $n$\nvertices, that is, an irreflexive symmetric binary relation $E'$ on $\\{1,\n\\ldots, n\\}$.  A set $K$ of vertices of $G'$ is a \\demph{clique} (or complete\nsubgraph) if $(i, j) \\in E'$ whenever $i, j \\in K$ with $i \\neq j$.  Write\n$\\mr{c}(G')$ for the largest integer $c$ such that there exists a clique in\n$G'$ of cardinality $c$.  Their Proposition~4.1 states that\n\\[\n\\sup_{\\pv{p}} \\sum_{(i, j) \\in E'} p_i p_j\n=\n1 - \\frac{1}{\\mr{c}(G')}\n\\]\n(which they call the `capacity' of $G'$).  Their proof shows that the supremum\nis attained at $\\pv{p}^K$ for any clique $K$ of cardinality\n$\\mr{c}(G')$. \n\nWe are given a graph $G$.  Let $G'$ be its dual graph, with the same vertex-set\nand with edge-relation $E'$ defined by $(i, j) \\in E'$ if and only if $(i, j)\n\\not\\in E$.  Then $G'$ is irreflexive, a clique in $G'$ is the same as a\ndiscrete set in $G$, and $\\mr{c}(G') = \\mr{d}(G)$.  For any distribution\n$\\pv{p}$,\n\\[\n\\sum_{(i, j) \\in E'} p_i p_j \n=\n1 - \\sum_{(i, j) \\in E} p_i p_j\n=\nH_2^Z(\\pv{p}).\n\\]\nLet $K$ be a clique in $G'$ of maximal cardinality, that is, a discrete set in\n$G$ of maximal cardinality.  Then by~\\cite{BMN}, $\\pv{p}^K$ is 2-maximizing and\n\\[\nH_2^Z(\\pv{p}^K) \n= \n1 - \\frac{1}{\\mr{c}(G')} \n=\n1 - \\frac{1}{\\mr{d}(G)}.\n\\]\nBut it is a completely general fact that \n\\[\nH_2^Z(\\pv{p}) \n= \n1 - \\frac{1}{D_2^Z(\\pv{p})}\n\\]\nfor all $\\pv{p}$, directly from the definitions in\nSection~\\ref{sec:statement}.  Hence $\\mr{d}(G) = D_2^Z(\\pv{p}^K) =\nD_2^G(\\pv{p}^K)$, and the claim holds for $q = 2$.\nCorollary~\\ref{cor:some-all} now tells us that the claim holds for all $q \\in\n[0, \\infty]$.\n\nThis class of examples tells us that a similarity matrix may have several\ndifferent maximizing distributions, and that a maximizing distribution\n$\\pv{p}$ may have $p_i = 0$ for some values of $i$.  These phenomena have\nbeen observed in the ecological literature in the case $q = 2$ (Rao's\nquadratic entropy): see Pavoine and Bonsall~\\cite{PB} and references therein. \n\\end{example}\n\nComputing the maximum diversity is potentially slow, because in principle\none has to go through all $2^n$ subsets of $\\{1, \\ldots, n\\}$.  But if the\nsimilarity matrix satisfies some further conditions, a maximizing distribution\ncan be found very quickly:\n\n\\begin{cor}     \\lbl{cor:pos-def-div}\nLet $Z$ be a positive definite similarity matrix whose unique weighting\n$\\pv{w}$ is non-negative.  Then $\\Dmax{Z} = \\magn{Z}$.  Moreover,\n$\\pv{w}\/\\magn{Z}$ is a maximizing distribution, and if $\\pv{w}$ is\npositive then it is the unique such.\n\\end{cor}\n\n\\begin{proof}\nFollows immediately from Proposition~\\ref{propn:pos-def-sub} and\nTheorem~\\ref{thm:main}. \n\\hfill\\ensuremath{\\Box}\n\\end{proof}\n\n\\begin{example} \\lbl{eg:ultra}\nA similarity matrix $Z$ is \\demph{ultrametric} if $\\min\\{Z_{ij}, Z_{jk}\\} \\leq\nZ_{ik}$ for all $i, j, k$ and $Z_{ij} < 1$ for all $i \\neq j$.  As shown\nbelow, every ultrametric matrix is positive definite and its weighting\nis positive.  Hence its maximum diversity is its magnitude, it has a unique\nmaximizing distribution, and that distribution is nowhere zero.\n\nUltrametric matrices are closely related to \\demph{ultrametric spaces}, that\nis, metric spaces satisfying a stronger version of the triangle inequality:\n\\[\n\\max\\{ d(a, b), d(b, c) \\} \\geq d(a, c)\n\\]\nfor all points $a, b, c$.  Any finite metric space $A = \\{a_1, \\ldots, a_n\\}$\ngives rise to a similarity matrix $Z$ by putting $Z_{ij} = e^{-d(a_i, a_j)}$,\nand if the space $A$ is ultrametric then so is the matrix $Z$.\n\nUltrametric matrices also arise in the quantification of biodiversity.  Take a\ncollection of $n$ species, and suppose, for example, that we choose a\ntaxonomic measure of species similarity:\n\\[\nZ_{ij} =\n\\left\\{\n\\begin{array}{ll}\n1       &\\textrm{if } i = j     \\\\\n0.8     &\\textrm{if $i \\neq j$ but the $i$th and $j$th species are of the same\ngenus} \\\\\n0.6     &\\textrm{if the $i$th and $j$th species are of different genera but\nthe same family}        \\\\\n0       &\\textrm{otherwise.}\n\\end{array}\n\\right.\n\\]\nThis is an ultrametric matrix, so is guaranteed to have a unique maximizing\ndistribution.  That distribution is nowhere zero: maximizing diversity\ndoes not eradicate any species.  The same conclusion for general ultrametric\nmatrices was reached, in the case $q = 2$, by Pavoine, Ollier and\nPontier~\\cite{POP}. \n\nWe now prove that any ultrametric matrix $Z$ is positive definite with\npositive weighting.  That $Z$ is positive definite was also proved by Varga\nand Nabben~\\cite{VN}, and that the weighting is positive was also proved\nin~\\cite{POP}.  The following proof, which is probably not new either, seems\nmore direct.\n\nIf $n = 1$ then certainly $Z$ is positive definite and its weighting is\npositive.  \n\nSuppose inductively that $n \\geq 2$.  Write $z = \\min_{i, j} Z_{ij} < 1$.  By\nthe ultrametric property, there is an equivalence relation $\\simeq$ on\n$\\{1, \\ldots, n\\}$ defined by $i \\simeq j$ if and only if $Z_{ij} > z$.  We\nmay partition $\\{1, \\ldots, n\\}$ into two nonempty subsets, $B$ and $B'$, each\nof which is a union of $\\simeq$-equivalence classes; and without loss of\ngenerality, $B = \\{1, \\ldots, m\\}$ and $B' = \\{m + 1, \\ldots, n\\}$, where $1\n\\leq m < n$.  For all $i \\leq m$ and $j \\geq m + 1$ we have $Z_{ij} \\leq z$,\nthat is, $Z_{ij} = z$.  Hence \n\\[\nZ \n=\n\\left(\\!\\!\n\\begin{array}{cc}\nY               &z\\um{m}{n - m} \\\\\nz\\um{n - m}{m}  &Y'\n\\end{array}\n\\!\\!\\right)\n\\]\nwhere $Y$ is some $m \\times m$ matrix, $Y'$ is some $(n - m) \\times (n - m)$\nmatrix, and $\\um{k}{\\ell}$ denotes the $k \\times \\ell$ matrix all of whose\nentries are $1$.  Now $Y$ and $Y'$ are ultrametric with entries in $[z, 1]$,\nso the matrices\n\\[\nX = \\frac{1}{1 - z} (Y - z\\um{m}{m}),\n\\qquad\nX' = \\frac{1}{1 - z}(Y' - z\\um{n - m}{n - m})\n\\]\nare also ultrametric.  By inductive hypothesis, $X$ and $X'$ are\npositive definite and their respective weightings are positive.\n\nWe have\n\\begin{equation}        \\lbl{eq:ultra-inductive}\nZ  \n= \nz \\um{n}{n} + \n(1 - z)\n\\left(\\!\\!\n\\begin{array}{cc}\nX       &0      \\\\\n0       &X'\n\\end{array}\n\\!\\!\\right).\n\\end{equation}\nThe matrix $\\um{n}{n}$ is positive-semidefinite, since $\\pv{x}^\\mathrm{t}\n\\um{n}{n} \\pv{x} = (x_1 + \\cdots + x_n)^2$ for all $\\pv{x} \\in \\mathbb{R}^n$.\nAlso $\\left(\\!\\!\n\\begin{array}{cc}\nX       &0      \\\\\n0       &X'\n\\end{array}\n\\!\\!\\right)$ is positive definite, since $X$ and $X'$ are.  Finally, $z \\geq 0$\nand $1 - z > 0$.  It follows that $Z$ is positive definite.\n\nWrite $\\pv{v}$ and $\\pv{v}'$ for the weightings on $X$ and $X'$.  Put\n\\[\n\\pv{w}\n=\n\\frac{1}{z\\left(\\sum_{i = 1}^m v_i + \\sum_{j = 1}^{n - m} v'_j\\right) \n+ (1 - z)}\n\\left(\\!\\!\n\\begin{array}{c}\nv_1     \\\\ \n\\vdots  \\\\\nv_m     \\\\\nv'_1    \\\\\n\\vdots  \\\\\nv'_{n - m}\n\\end{array}\n\\!\\!\\right).\n\\]\nThe weightings $\\pv{v}$ and $\\pv{v'}$ are positive and $0 \\leq z \\leq 1$, so\n$\\pv{w}$ is positive.  And it is routine to verify,\nusing~(\\ref{eq:ultra-inductive}), that $\\pv{w}$ is the weighting on $Z$.\n\\end{example}\n\n\\begin{example} \\lbl{eg:metric}\nTake a metric space with three points, $a_1, a_2, a_3$, and put $Z_{ij}\n= e^{-d(a_i, a_j)}$.  This defines a $3 \\times 3$ similarity matrix $Z$ with\n$Z_{ij} < 1$ for all $i \\neq j$ and $Z_{ij} Z_{jk} \\leq Z_{ik}$ for all $i, j,\nk$.  We will show that $Z$ is positive definite and that its unique weighting\nis positive.  It follows that there is a unique maximizing distribution and\nthat the maximum diversity is $\\magn{Z}$.  We give explicit expressions for\nboth.\n\nFirst, Sylvester's Criterion states that a symmetric real $n\\times n$ matrix\nis positive definite if and only if for all $m \\in \\{1, \\ldots, n\\}$, the\nupper-left $m \\times m$ submatrix has positive determinant.  In this case:\n\\begin{itemize}\n\\item the upper-left $1 \\times 1$ matrix is $(1)$, which has determinant $1$\n\\item the upper-left $2 \\times 2$ matrix is $\\left(\\!\\! \\begin{array}{cc} 1\n&Z_{12}\\\\ Z_{12} &1 \\end{array} \\!\\!\\right)$, which has determinant $1 -\nZ_{12}^2 > 0$\n\\item the upper-left $3 \\times 3$ matrix is $Z$ itself, and\n\\begin{eqnarray*}\n\\det Z  &=      &\n1 - (Z_{12}^2 + Z_{23}^2 + Z_{31}^2) + 2 Z_{12} Z_{23} Z_{31}   \\\\\n        &=      &\n(1 - Z_{12})(1 - Z_{23})(1 - Z_{31}) \n+\n(1 - Z_{12})(Z_{12} - Z_{13} Z_{32})    \\\\\n        &       &\n{}+\n(1 - Z_{23})(Z_{23} - Z_{21} Z_{13})\n+\n(1 - Z_{31})(Z_{31} - Z_{32} Z_{21})    \\\\\n        &>      &\n0.\n\\end{eqnarray*}\n\\end{itemize}\nHence $Z$ is positive definite.  Next, it is easily checked that the\nunique weighting $\\pv{w}$ is given by $\\pv{w} = \\pv{v}\/\\det Z$, where, for\ninstance, \n\\begin{eqnarray*}\nv_1     &=      &\n1 - (Z_{12} + Z_{13}) + (Z_{13}Z_{32} + Z_{12}Z_{23}) - Z_{23}^2    \\\\\n        &=      &\n(1 - Z_{12})(1 - Z_{23})(1 - Z_{31})\n+\n(1 - Z_{23})(Z_{23} - Z_{21}Z_{13})     \\\\\n        &>      &\n0.\n\\end{eqnarray*}\nSince $\\det Z > 0$, the weighting $\\pv{w}$ is positive.  \n\nThe maximum diversity is $\\magn{Z} = w_1 + w_2 + w_3$, which is\n\\[\n1 +\n\\frac{2(1 - Z_{12})(1 - Z_{23})(1 - Z_{31})}%\n{1 - (Z_{12}^2 + Z_{23}^2 + Z_{31}^2) + 2 Z_{12}Z_{23}Z_{31}}.\n\\]\n(This expression was pointed out to me by Simon Willerton.)  The unique\nmaximizing distribution $\\pv{p}$ is given by $\\pv{p} = \\pd{\\pv{w}} =\n\\pv{w}\/\\magn{Z} = \\pv{v}\/(\\magn{Z}\\det Z)$, so\n\\[\np_1\n=\n\\frac{1 - (Z_{12} + Z_{13}) + (Z_{13}Z_{32} + Z_{12}Z_{23}) - Z_{23}^2}%\n{1 - (Z_{12}^2 + Z_{23}^2 + Z_{31}^2) + 2 Z_{12}Z_{23}Z_{31}\n+ 2(1 - Z_{12})(1 - Z_{23})(1 - Z_{31})}\n\\]\nand similarly for $p_2$ and $p_3$.\n\\end{example}\n\nExample~\\ref{eg:graphs} (graphs) shows that maximizing distributions\nsometimes contain some zero entries.  In ecological terms this means that\ndiversity is sometimes maximized by completely eradicating certain species,\nwhich may be contrary to acceptable practice.  For this and other reasons, we\nmight seek conditions under which some or all of the maximizing distributions\n$\\pv{p}$ satisfy $p_i > 0$ for all $i$.\n\n\\begin{cor}     \\lbl{cor:close-to-id}\nLet $Z$ be a similarity matrix such that $Z_{ij} < 1\/(n - 1)$ for all $i \\neq\nj$.  Then $\\Dmax{Z} = \\magn{Z}$.  Moreover, $Z$ has a unique weighting\n$\\pv{w}$, the unique maximizing distribution is $\\pv{p} = \\pv{w}\/\\magn{Z}$,\nand $p_i > 0$ for all $i$.\n\\end{cor}\n\n\\begin{proof}\nBy Lemma~\\ref{lemma:scattered}, $Z$ is positive definite and its unique\nweighting is positive.  Then apply Corollary~\\ref{cor:pos-def-div}.\n\\hfill\\ensuremath{\\Box}\n\\end{proof}\n\nThe extra hypothesis on $Z$ is strong, possibly too strong for the corollary\nto be of any use in ecology: when $n$ is large, it forces $Z$ to be very close\nto the identity matrix.  On the other hand, the ecological interpretation of\nCorollary~\\ref{cor:close-to-id} is clear: if we treat every species as highly\ndissimilar to every other, the distribution that\nmaximizes diversity conserves all of them.\n\n\n\n\\passage{Acknowledgements} Parts of this work were done during visits to the\nCentre de Recerca Matem\\`atica, Barcelona, and the School of Mathematics and\nStatistics at the University of Sheffield.  I am grateful to both institutions\nfor their hospitality, and to Eugenia Cheng, David Jordan and Joachim Kock for\nmaking my visits both possible and pleasant.  I thank Christina Cobbold,\nAndr\\'e Joyal and Simon Willerton for useful conversations, and Ji\\v{r}\\'\\i\\\nVelebil for tracking down a reference.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Non-ideal fabrication in fixed frequency\nqubits}\n\nLattices of coupled qubits are proposed to enable error-correction algorithms such as the `surface code' \\cite{Gambetta2017Build_s,fowler2012surface_s}. Qubits are arranged into a square grid with alternate qubits serving either data or error-checking functions. Bus-couplers provide interaction among adjacent qubits, with up to four qubits attached to each bus. A seven qubit-lattice thereby comprises 12 qubit pairs and a seventeen-qubit lattice comprises 34 pairs. However, single junction transmon qubits are challenging to fabricate at precisely set frequencies. Among dozens of identically-fabricated qubits, the frequencies typically have a spread of $\\sigma_f \\sim 200$ MHz \\cite{privcommsrosenblatt_s}. Such imprecision will inhibit functioning of qubit lattices. Considering a lattice of tansmon qubits of frequency $\\sim 5$ GHz and anharmonicity $\\delta\/2\\pi = -340$ MHz, and considering cross-resonance gate operations, we can estimate the number of undesired interactions among these pairs. Studies of the cross-resonance gate \\citep{divincenzo2013quantum_s} indicate that these gates will be dominated by undesirable interactions if the frequency separation $|\\Delta|$ between adjacent qubits is equal to zero, a degeneracy between $f_{01}$ of the qubits; equal to $-\\delta\/2\\pi$, a degeneracy between $f_{01}$ of one qubit and $f_{12}$ of the next; or if $|\\Delta| > -\\delta\/2\\pi$ (weak interaction leading to very slow gate operation). In a simple Monte Carlo model, we assign to all points in the lattice a random qubit frequency from a gaussian distribution around 5 GHz, and count the number of degenerate or weak-interaction pairs, taking a range of $\\pm (\\delta\/2\\pi)\/20$, or $\\pm 17$ MHz around each degeneracy. The results appearing in Table \\ref{table:MCModelCollisions} make it evident that the likelihood of frequency collisions increases as the lattice grows.\n\n\\begin{table}[h]\n\t\\centering\n\t\\begin{tabular}{c|c|c}\n\t\tNumber &  & Mean Number \\\\\n\t\tof QBs & $\\sigma_f$ & of Collisions \\\\ \\hline\n\t\t7 & $\\frac{1}{2}|\\delta\/2\\pi|$ & 2.3  \\\\ \n\t\t7 & $\\frac{3}{4}|\\delta\/2\\pi|$ & 3.6  \\\\ \n\t\t17 & $\\frac{1}{2}|\\delta\/2\\pi|$ & 6.6  \\\\ \n\t\t17 & $\\frac{3}{4}|\\delta\/2\\pi|$ & 10.6  \n\t\\end{tabular} \n\\caption{\\label{table:MCModelCollisions} Frequency-collision modeling in lattices of transmon qubits employing cross-resonance gates. Predicted number of bad gate pairs (`frequency collisions') in two different lattice sizes. 7-qubit lattice has 12 pairs and 17-qubit lattice has 34 pairs. Mean of distribution is 5 GHz and two different distribution widths $\\sigma_f$ are considered.}\n\\end{table}\n\n\n\\section{Device design and fabrication}\n\nThe device for sample A, shown in Fig. \\ref{fig:1}, has all\neight qubit\/cavities capacitively coupled to a common feedline through\nwhich individual qubit readout was achieved via a single microwave\ndrive and output line. Sample B, shown in Fig. \\ref{fig:1}, employs a design where all qubits have separate drive and readout microwave lines. As in Ref. \\cite{Takita2016Dem_s} and \\cite{ibmquantumexp_s}, this sample is designed as a lattice of coupled qubits for use in multi-qubit gate operations, although no such operations are presented in this paper. Coplanar-waveguide buses, half-wave resonant at $\\sim$6 GHz, span the space between the qubits. Each bus resonator couples together three adjacent qubits. As compared to Ref. \\cite{Takita2016Dem_s}, here the lattice comprises eight qubits and four buses instead of the seven qubits and two buses found in Ref. \\cite{Takita2016Dem_s}.\n\nBoth samples were fabricated using standard lithographic processing\nto pattern the coplanar waveguides, ground plane, and qubit capacitors\nfrom a sputtered Nb film on a Si substrate. In sample A the Nb films are 100 nm thick. In sample B they are 200 nm. The qubits were similar\nin design to \\citep{Sheldon_Procedure_2016_s,chow2014implementing_s,corcoles2015demonstration_s,Takita2016Dem_s} \nwith large transmon capacitor pads bridged by electron-beam patterned Al traces used to\ncreate Josephson junctions. Conventional shadow-evaporated double-angle Al-AlOx-Al was used to fabricate the junctions. Transmon capacitor pads in samples A and B have different size and separation, necessitating different SQUID loop geometries, as shown in Fig. \\ref{fig:1}. The SQUID loops for qubits on sample A were created by bridging the transmon capacitor pads with\ntwo separate $0.6-\\mu{\\rm m}$ wide Al traces and Josephson junctions, with the\nasymmetry in the junctions fabricated by increasing the width of one\njunction with respect to the other, while keeping the overlap fixed at $0.2\\, \\mu{\\rm m}$. The sum of the large and small junction areas was designed to be constant, independent of $\\alpha$.\nQubits on sample A had capacitor pads separated by $20\\, \\mu{\\rm m}$ and\nthe Al electrodes separated such that the SQUID loop area was roughly\n$400\\, \\mu{\\rm m^{2}}$. In sample B, the Nb capacitor pads were separated by $70\\, \\mu{\\rm m}$. The SQUID comprises a $\\sim 20 \\times 20\\, \\mu{\\rm m}^2$ Al loop of 2 $\\mu$m trace width, placed midway between the capacitor pads and joined to Nb leads extending from the pads. In sample B, the large and small junction differ in both width and overlap. In this sample, all SQUIDs of a given $\\alpha$ were fabricated identically but SQUIDs of different $\\alpha$ had different total junction area. \n\n\\begin{figure}[!b]\n\\includegraphics[width=1.0\\columnwidth]{Device_Image_Supp2}\n\n\\caption{(color online) Optical micrographs of samples including higher magnification images of qubits and SQUID loops. Sample B image is a chip of identical design to the ones used for measurements. In sample B image, labels indicate each qubit and its individual readout resonators, while unlabeled resonators are bus resonators. \n\\label{fig:1}}\n\\end{figure}\n\n\\section{Measurement setup}\n\nMeasurements of sample A were completed in a dilution\nrefrigerator (DR) at Syracuse University (SU), while sample B was measured in a DR at the IBM TJ Watson Research Center. Both samples were wire-bonded into holders designed to suppress microwave\nchip modes. Each sample was mounted to the mixing chamber of its respective DR and placed inside a cryoperm magnetic shield, thermally anchored at the mixing chamber. Both SU and IBM DRs had room-temperature $\\mu$-metal shields. Measurements for both samples were performed using standard cQED readout techniques \\citep{Reed_High_2010_s}.\n\nFor sample A, room-temperature microwave signals were supplied through attenuated coaxial\nlines, thermalized at each stage of the DR and filtered using 10\nGHz low pass filters (K\\&L) thermalized at the mixing chamber. We used\na total of 70 dB of attenuation on the drive-lines: 20 dB at $4\\, {\\rm K}$, 20\ndB at $0.7\\, {\\rm K}$ and 30 dB at the mixing chamber, with a base temperature of $30\\,{\\rm mK}$. Output measurement signals\nfrom the sample pass through another 10 GHz low-pass filter, a microwave\nswitch, and two magnetically shielded cryogenic isolators, all thermally anchored\nto the mixing chamber. In the case of sample A, the signal was amplified\nby a low-noise HEMT at $4\\, {\\rm K}$, passing through a Nb\/Nb superconducting\ncoaxial cable between the mixing chamber and $4\\, {\\rm K}$ stage. The signal was amplified further\nat room temperature before being mixed down to 10 MHz and digitized. The eight resonators, coupled to each qubit on sample A, had measured frequencies that ranged from $6.975 - 7.136\\, {\\rm GHz}$, separated by $20 - 25\\, {\\rm MHz}$. $\\kappa\/{2\\pi}$ linewidths for these resonators were on the order of a few hundreds of kHz.  \n\nFigure \\ref{fig:1} shows the layout of the sample B chip. The $\\alpha = 15$ asymmetric-SQUID transmon reported in the paper was located at position $Q_7$. It was read out through a coplanar waveguide resonator of frequency 6.559 GHz and linewidth $\\sim$ 300 kHz, and was found to have $f_{01}^{max} = 5.387$ GHz. The fixed-frequency transmon (5.346 GHz) at position $Q_2$ was read out through a 6.418 GHz resonator having linewidth $\\sim$ 300 kHz. Sample B qubits were measured via signal wiring similar to that presented in Refs. \\cite{Takita2016Dem_s,chow2014implementing_s,corcoles2015demonstration_s,sheldon2016characterizing_s}. Drive wiring included 10 dB of attenuation at 50 K, 10 dB at 4K, 6 dB at 0.7 K, 10 dB at 100 mK, and at the mixing-chamber plate 30 dB of attenuation plus a homemade `Eccosorb' low-pass filter. Drive signals entered a microwave circulator at the mixing plate. On one set of signal wiring, the 2nd port of the circulator passed directly to qubit $Q_7$. In another set of signal wiring, the second port of the circulator passed to several different qubits via a microwave switch. Signals reflected from the device passed back through the circulator to output and amplifier circuitry. Output circuitry comprised a low-pass Cu powder filter, followed by two cryogenic isolators in series, followed by an additional low-pass filter, followed by superconducting NbTi coaxial cable, followed by a low-noise HEMT amplifier at 4K and an additional low-noise amplifier at room temperature. Low-pass filters were intended to block signals above $\\sim$ 10 GHz. In the case of $Q_7$, additional amplification was afforded by a SLUG amplifier \\cite{HoverAPL2014_104_152601_s} mounted at the mixing stage, biased via two bias-tee networks and isolated from the sample by an additional cryogenic isolator. Output signals were mixed down to 5 MHz before being digitized and averaged. Mixing-plate thermometer indicated a temperature of $\\sim$ 15 to 20 mK during measurements. \n\nMagnetic flux was supplied to sample A via a 6-mm inner diameter superconducting\nwire coil placed $2\\,{\\rm mm}$ above the sample. A Stanford SRS SIM928 dc voltage source with a room-temperature $2\\,{\\rm k}\\Omega$ resistor in series supplied the bias current to the coil. The flux bias current passed through brass coaxial lines that were thermally anchored\nat each stage of the DR, with a $80~{\\rm MHz}$ $\\pi$-filter at 4K and a copper powder filter on the mixing chamber. In sample B, a similar wire-wound superconducting coil was mounted about 3 mm above the qubit chip and likewise driven from a SIM928 voltage source through a room-temperature $5\\,{\\rm k}\\Omega$ bias resistor. DC pair wiring (Cu above 4K within the fridge, NbTi below) was used to drive the coil. The coil had a self-inductance of 3.9 mH and mutual inductance to the SQUID loop of $\\sim$ 1 pH. The flux coil applied a dc flux through\nall qubits with the flux level being set just prior to qubit measurement\nand maintained at a constant level throughout the measurement. For each qubit, we measured $f_{01}$ as a function of coil current and fit this against Eq. (1) of our paper to enable scaling of $\\Phi_0$ and subtract any offset flux, as well as to determine $f_{01}^{max}$ and asymmetry $d$. We treat the sign of flux as arbitrary. \n\n\\section{Qubit Coherence}\n\nCoherence data for both samples was collected using an automated measurement algorithm. After applying a prescribed fixed flux, the system determined the qubit frequency from Ramsey fringe fitting, optimized $\\pi$ and $\\pi\/2$ pulses at this frequency, and measured coherence. $T_{2}^{*}$ measurements were completed at a frequency detuned from the qubit frequency, with the level of detuning optimized to provide a reasonable number of fringes for fitting. All raw coherence data was visually checked to confirm that a good quality measurement was achieved. If the automated tuning routine failed to find the frequency or properly scale the $\\pi$ and $\\pi\/2$ pulses, this point was omitted from the dataset.\nFor sample A, three $T_1$ measurements were made at each flux point followed by three $T_2^*$ measurements. At each flux point, the reported $T_1$ and $T_2^*$ values and error bars comprise the mean and standard deviation of the three measurements. The corresponding $\\Gamma_{\\phi}$ value is found from these mean values and its error bar is found by propagating the errors in $T_1$ and $T_2^*$ through via partial derivative and combining these in a quadrature sum. For sample B, at each flux point first $T_1$ was measured, then $T_2^*$, three times in succession. For this device the reported $T_1$ and $T_2^*$ values comprise the mean of the three measurements and the error bars are their standard deviation. Here the reported dephasing rate $\\Gamma_{\\phi}$ comprises the mean of the three values of $\\Gamma_{\\phi}=1\/T_{2}^{*}-1\/2T_{1}$ found from the three $T_1$, $T_2^*$ pairs, and the error bar is the standard deviation. \n\n\\begin{figure}\n\\includegraphics[width=0.75\\columnwidth]{T1_vs_Freq_SYR_IBM2}\n\n\\caption{$T_{1}$ vs. frequency measured for all qubits discussed in the main paper. Single points included for $T_{1}$ values measured for the fixed-frequency qubits. \n\\label{fig:2}}\n\n\\end{figure}\n\nFigure \\ref{fig:2} shows $T_{1}$ plotted versus qubit frequency, measured for the qubits discussed in our paper. We observe a trend\nof increasing  $T_{1}$ with decreasing qubit frequency. In sample A, each qubit's quality factor $\\omega T_{1}$ is roughly constant, consistent with dielectric loss and a frequency-independent loss tangent, as observed in other tunable superconducting qubits \\citep{barends2013coherent_s}. On sample B, $T_{1}$ decreases by about 10 $\\mu$s from the low to high end of the frequency range, consistent with Purcell loss to the readout resonator. In addition, fine structure is occasionally observed in Fig.\n\\ref{fig:2} where $T_{1}$ drops sharply at specific frequencies. These localized features in the $T_{1}$ frequency dependence are observed\nfor all tunable qubits that we have measured. These features, similar to those observed by \\citep{barends2013coherent_s}, are attributed to frequencies where a qubit transition is resonant with a two-level system defect on or near the qubit. Additionally, on sample B, at a few frequency points inter-qubit coupling affects relaxation. Where the $Q_7$ qubit is nearly degenerate to $Q_6$ (at $\\sim$5.33 GHz) and to $Q_8$ (at $\\sim$5.22 GHz), coupling via the adjacent buses produces an avoided crossing in the energy spectrum. This effect is barely noticeable in both the frequency curve of Fig. 2 of our paper as well as the relaxation data in Fig. \\ref{eq:2} here.\n\n\\begin{figure}\n\\includegraphics[width=0.75\\columnwidth]{Ramsey_vs_Phi_All2}\n\n\\caption{$T_{2}^{*}$ vs. flux measured for the qubits discussed in the main paper. $T_{2}^{*}$ measured for the fixed-frequency qubits on both samples is included with dashed lines to help guide the eye.\n\\label{fig:3}}\n\n\\end{figure}\n\nFigure \\ref{fig:3} shows $T_{2}^{*}$ plotted versus flux, measured\nfor the qubits discussed in our paper.\nFor the tunable qubits on\nsample A, $T_{2}^{*}$ is greatest at the qubit sweet-spots and decreases\naway from these sweet spots as $D_{\\Phi}$ increases.\nIn the $\\alpha = 15$ tunable qubit on sample B, $T_{2}^{*}$ is nearly constant over the measured half flux quantum range. The small frequency dependence observed in $T_{2}^{*}$ in sample B is consistent with the observed variation of $T_{1}$ with frequency, leading to the frequency-independent dephasing rate observed for this qubit\nin Fig. 3 of our paper.\n\n\\section{Relaxation Due to coupling to Flux Bias Line}\n\nWhile using two Josephson junctions to form a dc SQUID for the inductive element of a transmon allows its frequency to be tuned via magnetic flux, this opens up an additional channel for\nenergy relaxation via emission into the dissipative environment across the bias coil that is coupled to the qubit through a mutual inductance. This was first discussed by Koch et al \\citep{koch2007charge_s}. regarding\na near symmetrical split-junction transmon. We apply the same analysis\nhere to study the effect of increasing junction asymmetry on the qubit $T_{1}$ through this loss mechanism. For an asymmetric transmon, Koch et al. show in Eq. {(}2.17{)} of Ref. \\citep{koch2007charge_s} that the Josephson portion of the qubit Hamiltonian can be written in terms of a single phase variable with a shifted minimum that depends\nupon the qubit's asymmetry and the applied flux bias. \nBy linearizing this Hamiltonian about the static flux bias point for small noise amplitudes, Koch et al. compute the relaxation rate for a particular current noise power from the bias impedance coupled to the SQUID loop through a mutual inductance $M$.\nWe followed this same analysis for our qubit parameters, assuming harmonic oscillator wavefunctions for the qubit ground and excited state, and obtained the dependence of $T_{1}$ due to this mechanism as a function of bias flux.\nUsing our typical device\nparameters ($E_{J} = 20\\,{\\rm GHz}$, $E_{c} = 350\\,{\\rm MHz}$, $M = 2\\,{\\rm pH}$, $R =\n50~\\Omega$) we obtain the intrinsic loss for the asymmetries\ndiscussed in our paper, shown in Fig. \\ref{fig:4}. This analysis\nagrees with the results described in Ref. {[}4{]}. For a 10\\% junction asymmetry, this contribution results in a $T_{1}$ that varies between $25\\,{\\rm ms}$ and a few seconds. As the junction asymmetry is increased,\nthe minimum $T_{1}$ value, obtained at odd half-integer multiples of $\\Phi_{0}$, decreases slightly. However, even for our $\\alpha = 15$ qubit, the calculated value of $T_{1}$ due to this mechanism never falls below $10\\,{\\rm ms}$. Therefore, although increasing\njunction asymmetry does place an upper bound on $T_{1}$ of an asymmetric transmon, this level is two orders of magnitude larger than the measured $T_{1}$ in current state-of-the-art superconducting qubits due to other mechanisms.\n\n\\begin{figure}\n\n\\includegraphics[width=0.75\\columnwidth]{T1max_vs_phi}\n\n\\caption{Dependence of $T_{1}$  with flux for asymmetric transmons, calculated for the asymmetries discussed in the main paper, due to coupling to an external flux bias following the analysis of Koch et al \\citep{koch2007charge_s}. Though in the main paper our symmetric qubit was an $\\alpha = 1$, in this calculation we used $\\alpha = 1.1$ so that $T_{1}$ did not diverge at $\\Phi = 0$. \n\\label{fig:4}}\n\n\\end{figure}\n\nAlso in Ref. [4], Koch et al. described a second loss channel for a transmon related to coupling to the flux-bias line. In this case, the relaxation occurs due to the oscillatory current through the inductive element of the qubit -- independent of the presence of a SQUID loop -- coupling to the flux-bias line, described by an effective mutual inductance $M'$. This mutual vanishes when the Josephson element of the qubit and the bias line are arranged symmetrically. With a moderate coupling asymmetry for an on-chip bias line, Koch et al. estimate that the $T_{1}$ corresponding to this loss mechanism would be of the order of 70 ms. Because this mechanism does not directly involve the presence or absence of a SQUID loop for the inductive element, the asymmetry between junctions that we employ in our asymmetric transmons will not play any role here and this particular limit on $T_{1}$ should be no different from that for a conventional transmon. An additional potential relaxation channel may arise due to capacitive coupling to the flux-bias line, as discussed in Ref. \\cite{JohnsonDisserationYale2011_s}. However, this is expected to be negligible where a bobbin coil is used as in our experiments.\n\n\\section{Ramsey Decay Fitting}\n\nAs described in the main paper, our analysis of qubit dephasing rates used a purely exponential fit\nto all of the measured Ramsey decays. Here we discuss why this fitting approach is appropriate for all asymmetric qubits and a large portion of\nthe coherence data measured for the symmetric qubit.\n\nOf all the qubits measured in this study, the symmetric $\\alpha = 1$ qubit\nwas most impacted by flux noise away from the qubit sweet spot because of its large energy-band gradient. Therefore,\nto illustrate the impact that flux noise has upon the Ramsey decay\nenvelope we will consider the Ramsey measurements for this qubit on\nand off the sweet spot. Example measurements are shown at flux values of 0 and 0.3 ~$\\Phi_0$ in Fig. \\ref{fig:5}a and b, respectively. At each flux point,\nwe fit the Ramsey decay with both a purely exponential (Fig. \\ref{fig:5}a\nI) and purely Gaussian form (Fig \\ref{fig:5}a II), the residuals\nof each fit are included to compare the quality of fit in each case.\nAs has been discussed in the main paper, at the upper sweet-spot, where\n$D_{\\Phi} = 0$, non-flux dependent background-dephasing\nshould dominate and the Ramsey decay should be more readily fit using\nan exponential. Figure \\ref{fig:5}a shows that this is indeed the\ncase: the purely exponential fit provides a more precise fit to the\nRamsey decay, with the residuals to this fit being smaller over the entire range compared to those corresponding to the Gaussian fit. The Ramsey decay\nshown in Fig \\ref{fig:5}b was measured at a point where $D_{\\Phi}$\nwas the maximum measured for the $\\alpha = 1$ qubit. Here, it is clear that\na purely Gaussian form results in a better fit with smaller residuals than an exponential envelope. This indicates that, at this flux point, the $\\alpha = 1$ qubit is heavily impacted by low-frequency flux noise, as a purely 1\/f dephasing source would result in a Gaussian envolope for the decay \\citep{ithier2005decoherence_s}. Although a purely Gaussian fit form is useful\nfor illustrating the impact that flux noise has upon the Ramsey decay\nform, it is not an optimal quantitative approach for investigating dephasing in these qubits. This is because tunable transmons dephase not only due to flux noise with a roughly $1\/f$ power spectrum, but also due to other noise sources with different non-$1\/f$ power spectra \\citep{sears2012photon_s,schuster2005ac_s,gambetta2006qubit_s}. These other noise sources generally result in an exponential dephasing envelope. Also, dephasing has an intrinsic loss component that is always exponential in nature. Therefore, to accurately fit decay due to dephasing in these qubits, we must account for these exponential decay envelopes in any fitting approach that is not purely exponential.\n\n\\begin{figure}\n\n\\includegraphics[width=1\\columnwidth]{Ramsey_Decay_Fit}\n\\caption{Ramsey decay envolopes measured for the $\\alpha = 1$ qubit at a) the sweet-spot $\\Phi=0$ and b) $\\Phi=0.3 \\Phi_{0}$ where $D_{\\Phi}$ was the largest value measured for this qubit. At each flux point, the Ramsey decay envelopes are fit with both a purely exponential {(}I{)} and Gaussian {(}II{)} fit form. Functions fitted to the measured data {(}blue open circles{)} plotted as solid red lines.\n\\label{fig:5}}\n\n\\end{figure}\n\nTo account for the $T_{1}$ contribution to the Ramsey decay envelope in our non-exponential fitting, we take the average $T_{1}$ measured at each flux point\nand separate this from $T_{2}^{*}$ in the Ramsey fit function using\n$1\/T_{2}^{*} = 1\/T_{\\phi} + 1\/2T_{1}$. Therefore, instead\nof fitting a $T_{2}^{*}$ time, we fit $T_{\\phi} $\ndirectly. To fit the Ramsey using a Gaussian fit form, we square the\ndephasing exponent within the fitting function {[}Eq. {(}\\ref{eq:1}{)}{]}. We can go one step\nfurther by not forcing an explicit fit form to the dephasing exponent, but instead adding another fit parameter $\\gamma$ {[}Eq. {(}\\ref{eq:2}{)}{]}, which would be 1 for a pure exponential and 2 for a pure Gaussian. Although a fit that is not explicitly exponential or Gaussian is not motivated directly by a particular theoretical model, by fitting Ramsey decays with this free exponent\n$\\gamma$, we gain insight into the transition from flux-noise dominated dephasing at large $D_{\\Phi}$ to background dephasing near the sweet-spots. The two separate fit forms described above are given by the following decay functions:\n\n\\begin{equation}\nf_{Ramsey}(t)=A+B\\{\\cos{(\\omega t+\\delta)}\\exp{(-\\Gamma_{1}t\/2)}\\exp{[-(\\Gamma_{\\phi}t)^2]}\\}\\label{eq:1},\n\\end{equation}\n\\begin{equation}\nf_{Ramsey}(t)=A+B\\{\\cos{(\\omega t+\\delta)}\\exp{(-\\Gamma_{1}t\/2)}\\exp{[-(\\Gamma_{\\phi}t)^\\gamma}]\\}\\label{eq:2},\n\\end{equation}\nwhere A and B are magnitude and offset constants to adjust the arbitrary measured signal, $\\omega$ is the detuning from the qubit frequency with a phase offset $\\delta$, $\\Gamma_{1}$ is the intrinsic loss rate {(}$1\/T_{1}${)} and $\\Gamma_{\\phi}$ is the dephasing rate. Here, A, B, $\\omega$, $\\delta$, $\\Gamma_{\\phi}$, and $\\gamma$ are fit parameters. All other components are fixed with values determined using the methods discussed above.\n\n\\begin{figure}\n\n\\includegraphics[width=0.75\\columnwidth]{1to1_gamma_vs_Phi}\n\\caption{$\\gamma$ vs flux extracted from fits to the Ramsey measurements on the $\\alpha = 1$ qubit using Eq. \\ref{eq:2}.\n\\label{fig:6}}\n\n\\end{figure}\n\nThis behavior is illustrated in Fig. \\ref{fig:6}, where we plot $\\gamma$\nvs. flux extracted from fits to the Ramsey measurements on the $\\alpha = 1$\nqubit using Eq. {(}\\ref{eq:1}{)}. In the flux region between +\/- 0.1 $\\Phi_{0}$, $\\gamma \\approx 1$,\nindicating that the dephasing envelope is primarily exponential, and thus the dominant dephasing noise affecting the qubits here does not have a $1\/f$ spectrum. At flux bias points further away from the sweet-spot, $\\gamma$ shifts towards 2 as $D_{\\Phi}$ increases and appears\nto level off close to this value at flux biases above $\\sim 0.2\\,\\Phi_{0}$. Thus, in this bias regime, the dephasing envelope is primarily Gaussian and the dephasing noise influencing the qubits is predominantly low-frequency in nature with a $1\/f$-like spectrum \\citep{ithier2005decoherence_s,yoshihara2006decoherence_s}. \n\nWe can also vizualize this variable-exponent fit by plotting $\\gamma$ vs. $D_{\\Phi}$ rather than $\\Phi$, again, for the $\\alpha = 1$ qubit {(}Fig. \\ref{fig:7}{)}.\nIn this plot, $\\gamma$ approaches 2 for $D_{\\Phi}$ values around $6~\\rm GHz\/\\Phi_{0}$. We have also included vertical dashed lines on Fig. \\ref{fig:7} indicating the maximum $D_{\\Phi}$ values reached by the less tunable $\\alpha = 4$ and 7 qubits on sample A. Below these $D_{\\Phi}$ levels, $\\gamma$ is close to 1 implying that the decay envelope is nearly exponential, and thus justifying our use of an exponential decay for fitting the asymmetrical qubits in the main paper.\n\n\\begin{figure}\n\\includegraphics[width=0.75\\columnwidth]{1to1_gamma_vs_Grad}\n\n\\caption{$\\gamma$ vs $D_{\\Phi}$ extracted from fits to the Ramsey measurements on the $\\alpha = 1$ qubit using Eq. \\ref{eq:2}. Dashed lines included to indicate the maximum $D_{\\Phi}$ reached by the $\\alpha = 7$ {(}black dashed line{)} and $\\alpha = 4$ {(}blue dot-dashed line{)} qubits measured on sample A.\n\\label{fig:7}}\n\n\\end{figure}\n\nAs yet another approach to fitting the Ramsey decay envelopes, we can employ a function that separates the exponential from background-dephasing from the Gaussian form due to dephasing from noise with a low-frequency tail.\nFor this fit, along with separating\nout the $T_{1}$ contribution to the Ramsey decay envelope, we also determine the\nnon-flux dependent background-dephasing rate at the sweet-spot,\nthen use this rate as a fixed parameter in the fitting of our Ramsey measurements\nat any given flux point. We now have a composite Ramsey fit form that\nhas three components: a $T_{1}$ contribution and background dephasing component that are purely exponential and fixed by the fitting of separate measurements, plus a Gaussian component to capture the dephasing due to noise with a $1\/f$ spectrum. This leads to a composite fitting function of the form:\n\n\\begin{equation}\nf_{Ramsey}(t)=A+B\\{\\cos{(\\omega t+\\delta)}\\exp{(-\\Gamma_{1}t\/2)}\\exp{(-\\Gamma_{\\phi ,bkg}t)}\\exp{[-(\\Gamma_{\\phi}t)^2}]\\}\\label{eq:3},\n\\end{equation}\nwhere A and B are magnitude and offset constants to adjust the arbitrary measured signal, $\\omega$ is the detuning from the qubit frequency with a phase offset $\\delta$, $\\Gamma_{1}$ is the intrinsic loss rate {(}$1\/T_{1}${)}, $\\Gamma_{\\phi ,bkg}$ is the background dephasing rate measured at $D_{\\Phi}=0$ and $\\Gamma_{\\phi}$ is the fitted dephasing rate. Here, A, B, $\\omega$, $\\delta$, and $\\Gamma_{\\phi}$ are fit parameters. All other components are fixed with values determined using methods discussed above. Though this fit form well separates the different components to dephasing decay, it has one key deficiency: it assumes that the background dephasing rate is frequency independent, which is not necessarily justified, as the background dephasing mechanism may also vary with frequency. To calculate the total dephasing rate using this fit form, we add the constant background dephasing to the fitted $\\Gamma_{\\phi}$. \n\n\\begin{figure}\n\n\\includegraphics[width=0.75\\columnwidth]{1to1_RatevsGrad_4type_MAIN}\n\\caption{$\\Gamma_{\\phi}$ vs. $D_{\\Phi}$ calculated for the $\\alpha = 1$ qubit using the exponential,\nGaussian {[}Eq. {(}\\ref{eq:1}{)}{]}, $\\gamma$-exponent {[}Eq. {(}\\ref{eq:2}{)}{]}, and composite {[}Eq. {(}\\ref{eq:3}{)}{]}fitting forms.\n\\label{fig:8}}\n\\end{figure}\n\nTo understand how the explicit fitting form impacts the dephasing rate, in Fig. \\ref{fig:8} we plot $\\Gamma_{\\phi}$ vs. $D_{\\Phi}$ calculated for\nthe $\\alpha = 1$ qubit using the four different fitting forms: exponential, Gaussian {[}Eq. {(}\\ref{eq:1}{)}{]}, $\\gamma$-exponent {[}Eq. {(}\\ref{eq:2}{)}{]}, and composite {[}Eq. {(}\\ref{eq:3}{)}{]}. We first note that any differences in the rate of dephasing calculated at each point using the various fit methods\nare subtle and the fits are reasonably consistent with one another within the fit error bars and scatter. We do observe, though, that a purely exponential fit results in\na dephasing rate that is slightly higher than the values from the Guassian fits for all flux points, resulting in the largest slope and thus the highest effective flux-noise level. Therefore, we conclude that forcing a purely exponential fit to the Ramsey decay envelopes measured for qubits that are strongly influenced\nby $1\/f$ flux noise simply puts an upper bound on the absolute flux\nnoise strength. The $\\gamma$-exponent fitting approach provides a dephasing\nrate that agrees well with that extracted from the exponential fit\nform at low $D_{\\Phi}$ values where background-dephasing\nprocesses dominate. However, at higher $D_{\\Phi}$ values\nwhere the qubit is heavily impacted by $1\/f$ flux noise, the $\\gamma$-exponent\nfit provides better agreement with the Gaussian-fitted dephasing rate.\n\nThe composite fit is rigidly fixed in the $\\Gamma_{\\phi}$ axis by the value chosen\nto match the background dephasing rate, in this case chosen to match\nthe rate observed at the lowest $D_{\\Phi}$ for the pure\nexponential fit. For this reason, direct comparisons between this\nfit and the others at individual flux points is more difficult. Despite all of these potential issues, the slope of $\\Gamma_{\\phi}$ vs. $D_{\\Phi}$ is independent of the chosen background-dephasing\nrate. Therefore, this composite fit can be used to calculate a flux-noise level for this $\\alpha = 1$ qubit that takes into account both the exponential\nnature of non-flux dependent dephasing and the Gaussian nature of\n$1\/f$ flux-noise decay. Using the same methods outlined in our paper, where we \nspecified $\\Gamma_{\\phi}=2\\pi\\sqrt{A_{\\Phi}|\\ln{(2\\pi f_{IR}t)}|}D_{\\Phi}$, following the approach described in Ref. {[}\\citep{ithier2005decoherence_s}{]}, we use the slope of this composite fit\nto extract a $1\/f$ flux noise level of $A_{\\Phi}^{1\/2}=1.3~\\pm~0.2~\\mu \\Phi_0$. \nThis $\\sim10\\%$ reduction in the extracted flux-noise level for the $\\alpha = 1$ qubit compared to the purely exponential fit {(}$A_{\\Phi}^{1\/2}~=~1.4~\\pm~0.2~\\mu \\Phi_0${)} brings it closer to the flux-noise level extracted from the fits to the measurements on the $\\alpha = 7$ and 4 qubits: $1.3~\\pm~0.2~\\mu \\Phi_0$ and $1.2~\\pm~0.2~\\mu \\Phi_0$, respectively. The Ramsey measurements for these qubits were fit using a purely exponential fit form. It is important to note though, that the $\\sim10\\%$ reduction in the composite fit extracted flux-noise level for the $\\alpha = 1$ qubit is within the errors associated with our flux-noise calculations.\n\nTo conclude this fitting study, we have shown that:\n\\begin{enumerate}\n\\item The $\\alpha = 1$ qubit in this study has a Ramsey decay envelope that is more Gaussian\nin nature at high $D_{\\Phi}$ values where the dephasing of this qubit is strongly influenced by low-frequency flux noise.\n\\item Though we have discussed different fitting approaches that better\nmodel the Ramsey decay envelope of qubits influenced by $1\/f$ flux-noise,\nusing a purely exponential decay form for the Ramsey decay simply\nputs an upper bound on the extracted flux noise strength. Also, the value of the flux-noise level and the dephasing rates are comparable to those we obtained with the various other fitting approaches.\n\\item Using a Ramsey fit function that takes into account both the exponential\nnature of the $T_{1}$ contribution to the decay envelope and non-flux dependent dephasing, as well as the\nGaussian nature of dephasing due to $1\/f$ flux noise, allows us to calculate\na flux noise level for the $\\alpha = 1$ qubit that agrees well with the other,\nasymmetric qubits on the same sample. This is expected, as qubits of the same geometry on the same chip should experience similar flux noise \\citep{sendelbach2008magnetism_s}.\n\\end{enumerate}\n\n\\section{Dephasing Rate Discussion}\n\nIn Fig. \\ref{fig:9} we present dephasing rates for several additional qubits, plotted against $D_{\\Phi}$. These qubits were similar to those in our paper, but were prepared on additional chips and measured during additional cools of our cryostats. These data are not included in our paper for reasons of clarity and consistency. However, they are presented here to support the observations found in this study across all qubits measured in both of our labs. \n\n\\begin{figure}[!b]\n\n\\includegraphics[width=1.0\\columnwidth]{RatevsGrad_SUP2}\n\\caption{$\\Gamma_{\\phi}$ vs $D_{\\Phi}$ for qubits measured during this study that were not included in the main paper. $\\Gamma_{\\phi}$ for fixed-frequency qubits included as dashed lines. Type A\/B qubits were similar in design to those on sample A\/B, measured using similar methods and device designs as those described for the corresponding sample type.\n\\label{fig:9}}\n\\end{figure}\n\nThe first observation we make from Fig. \\ref{fig:9} is that a spread in background dephasing rates is measured between both fixed-frequency and tunable qubits. As discussed in our paper, these subtle variations in qubit dephasing rate are not unexpected and are commonly observed in multi-qubit devices \\citep{corcoles2015demonstration_s,chow2014implementing_s,Takita2016Dem_s}. While these variations in dephasing rate make the figure somewhat challenging to interpret, we can still draw the same conclusions for this data as those from our main paper. We still observe that the dephasing rate due to flux-noise increases linearly with $D_{\\Phi}$ for the lower asymmetry qubits. Again at lower $D_{\\Phi}$ values, below $\\sim 1\\,{\\rm GHz}\/\\Phi_0$, the rate of dephasing is constant within the experimental spread for all qubits. Here, it is important to note that, for several of the qubits shown here and those discussed in our paper, there are specific flux bias points for each qubit where the dephasing rate is anomalously high. These points almost always coincide with places where $T_{1}$ drops sharply at specific frequencies, presumably due to localized coupling to defects in these qubits. Again, this sharp frequency dependence in $T_{1}$ is not unusual for tunable superconducting qubits and is consistent with what others have observed \\citep{barends2013coherent_s}.  \n\nThe relatively flux-independent dephasing rate at low $D_{\\Phi}$ is particularly apparent in the 9:1 qubits we measured. Several of these qubits exhibited the lowest background depahsing rates we observed in our study, between 20 and 40~kHz. These dephasing rates are comparable to current state-of-the-art superconducting qubits \\cite{Takita2016Dem_s}. No fixed-frequency qubits were included on the same chips as these 9:1 asymmetric transmons, which prevents us from making a direct comparison with non-flux-noise-driven background dephasing rates as is done in the main paper. Nonetheless, for these 9:1 qubits, we can clearly see that the dephasing rate is essentially flux independent below $\\sim 1\\,{\\rm GHz}\/\\Phi_0$ even at these low background dephasing levels. This reinforces our statement that asymmetric qubits with a useful level of tunability can be incorporated into future fault-tolerant superconducting qubit devices, significantly aiding scalability in these systems. \n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nQuantum spin systems with energy gaps have attracted considerable attention\nbecause of their rich variety of interesting phenomena.\nIn these systems, the ground state is a nonmagnetic singlet state,\nand there exists a finite energy-gap between the ground and the excited magnetic states.\nUnder certain magnetic fields $H_{\\rm c}$, the energy of one of the magnetic excited states\nbecomes lower than that of the singlet state,\nwhere nonmagnetic-magnetic crossover occurs.\nRecently, special attention is paid to the magnetic transition\nthat develops just above $H_{\\rm c}$ in these spin-gap systems.~\\cite{Rice}\nA number of novel phenomena is reported in these fields;\nfor example, magnetic ordering of TlCuCl$_3$ in fields is interpreted as\nthe Bose-Einstein condensation (BEC) of magnons.~\\cite{Nikuni,Oosawa}\nFor the two dimensional dimer system SrCu$_2$(BO$_3$)$_2$,\nsuperlattice formation of localized triplet is observed.~\\cite{Kodama}\n\nThe Haldane systems, i.e., quasi-one-dimensional Heisenberg antiferromagnets with integer spins,\nalso are ones of the most  extensively studied spin gap systems.\nFor this class, unfortunately, field-induced magnetic ordering \nhas hardly been observed experimentally.\nThe archetypal Haldane-system, Ni(C$_2$H$_8$N$_2$)$_2$NO$_2$(ClO$_4$) (NENP),~\\cite{Renard}\nshows no evidence of field-induced ordering down to 0.2 K in fields up to 13 T.~\\cite{Kobayashi92}\nInstead, the existence of an energy gap was revealed even at $H_{\\rm c}$.~\\cite{Kobayashi92}\nThis is explained by the existence of a staggered field on Ni sites,\nwhich arises because the principal axis of the $g$ tensor in NENP tilts\nalternately.~\\cite{Chiba,Fujiwara}\nThis fact results in the slow crossover from nonmagnetic to magnetically polarized state in NENP,\npreventing from phase transition induced by fields.\n\nSo far, field-induced ordering in Haldane-systems was\nreported only for two cases: in Ni(C$_5$H$_{14}$N$_2$)$_2$N$_3$(PF$_6$)\nand Ni(C$_5$H$_{14}$N$_2$)$_2$N$_3$(ClO$_4$), abbreviated NDMAP and NDMAZ, respectively.\nField-induced transitions in these systems were demonstrated \nby specific heat~\\cite{Honda97,Honda98,Honda01,Kobayashi01}\nand neutron diffraction experiments.~\\cite{Chen,Zheludev01E}\nIn the ordered state, interestingly, unusual spin excitations are observed.\nESR~\\cite{Hagiwara03} and inelastic neutron-scattering experiments~\\cite{Zheludev03} on NDMAP\nhave revealed the existence of three distinct excitations in the ordered phase.\nThis feature is quite different from those in a conventional Neel state,\nwhere dominant excitations are the spin-wave modes.~\\cite{Zheludev02A}\nField-induced ordered phase in Haldane-systems is thus expected to illustrate\nmuch kind of novel physics, if much more examples are available.\n\nHere the compound PbNi$_2$V$_2$O$_8$ would be another candidate for the Haldane\nsystem where field-induced ordering can be observed experimentally.\nPbNi$_2$V$_2$O$_8$ has a tetragonal crystal structure with Ni$^{2+}$ ($S=1$) ions \nforming a chain along the $c$-axis.\nMagnetic susceptibility, high-field magnetization, and inelastic neutron scattering\nexperiments were performed and their results consistently suggest that this system\nis a Haldane-gap system.~\\cite{Uchiyama}\nThe spin-gap closes at $H^{\\parallel}_{\\rm c} = 14$ T and $H^{\\perp}_{\\rm c} = 19$ T,~\\cite{Uchiyama}\nwhere $H^{\\parallel}_{\\rm c}$ and $H^{\\perp}_{\\rm c}$ are the critical fields\napplied parallel and perpendicular to the chain ($c$-axis), respectively.\nThese values of $H_{\\rm c}$ are within experimentally accessible range.\nMoreover, PbNi$_2$V$_2$O$_8$ is reported to exhibit\nimpurity-induced magnetic transition around 3 K.~\\cite{Uchiyama}\nThis transition was found to be a long-range magnetic ordering \nby the neutron diffraction~\\cite{Lappas}\nas well as the specific heat measurements~\\cite{Masuda}.\nThese facts suggest the relatively large interchain coupling, $J_1$.\nIn fact, the $D-J_1$ plot~\\cite{Sakai90} (Sakai-Takahashi diagram) for this compound,\nwhere $D$ is the single-ion anisotropy, \nsuggests that PbNi$_2$V$_2$O$_8$ is in the spin-liquid (disordered) regime\nbut very close to the long-range ordered regime.~\\cite{Zheludev00}  \nHence, one can expect that applying fields beyond $H_{\\rm c}$\nwill result in the magnetic ordering.\nIn the present paper, we have investigated magnetic properties of PbNi$_2$V$_2$O$_8$\nat $H > H_{\\rm c}$ using temperature dependent measurements of magnetization in static fields\nup to 30 T,\nand have observed indication of magnetic ordering above $H_{\\rm c}$.\n\n\\section{Experimental}\nField-oriented powder sample of PbNi$_2$V$_2$O$_8$ was prepared \nas in the first report,~\\cite{Uchiyama}\nsince single crystalline samples are not yet available.\nPowder sample of PbNi$_2$V$_2$O$_8$ was synthesized by a solid state reaction\nfrom PbO (99.999\\% pure), NiO (99.99\\%) and V$_2$O$_5$ (99.99\\%).\nThey were mixed and heated in air, \nfirstly at 600$^\\circ$C and subsequently at 750$^\\circ$C for several days \nwith intermittent grindings. \n\nPowder X-ray diffraction (XRD) pattern agrees well with the calculated pattern\nbased on the structure refined by the neutron diffraction experiments,~\\cite{Lappas,Comment}\nand no second phase was detected.\nThe powder was aligned by a magnetic fields (6 T) in stycast.\nThe orientation was checked by the (004) XRD peak.\nThe result confirmed that the $c$-axis aligns parallel to the magnetic fields,\nas is reported previously.~\\cite{Uchiyama}\nIn the following, we refer to magnetization measured under fields\nparallel to the $c$-axis, as $M^{\\parallel}$, and\nto that under fields perpendicular to the $c$-axis, as $M^{\\perp}$.\n\nMagnetization was measured by an extraction method.\nMagnetic fields up to 15 T were generated by a superconducting magnet.\nFields higher than 15 T were generated by a hybrid magnet at the Tsukuba\nMagnet Laboratory.\nFor the measurements of magnetization as a function of magnetic fields,\nthe fields were swept at the rate about 0.3 Tesla per minute at the temperature of 1.5 K.\nFor the temperature-dependent measurements, magnetization was measured under\nconstant magnetic fields.\n\n\\section{Results and discussion}\n\\begin{figure}[t]\n\\begin{center}\n\\includegraphics[width=8cm]{Fig1.eps}\n\\caption{\\label{fig:epsart} Field dependence of the magnetization of the\nPbNi$_2$V$_2$O$_8$ powder samples.}\n\\end{center}\n\\end{figure}\n\\begin{figure*}[t]\n\\begin{center}\n\\includegraphics[width=14cm]{Fig2.eps}\n\\caption{\\label{fig:epsart} Temperature dependence of the magnetization of \nPbNi$_2$V$_2$O$_8$ for (a) $H \\parallel c$ and (b) $H \\perp c$.\nArrows indicate $T_{\\rm min}$ shown in the text.\n}\n\\end{center}\n\\end{figure*}\nFig.1 shows field dependence of the magnetization $M^{\\parallel}(H)$\nand $M^{\\perp}(H)$ measured at $T$ = 1.5 K.\nBoth the $M^{\\parallel}(H)$ and $M^{\\perp}(H)$ curves steeply increase above \nthe critical fields, $H_{\\rm c}^{\\parallel}$ = 19 T and $H_{\\rm c}^{\\perp}$ = 13.5 T, respectively. \nThese values correspond to the critical fields at which the Haldane gap closes,\nand are in good agreement with the previous report obtained by\nthe pulsed-field experiments.~\\cite{Uchiyama}\n\nHere the $M(H)$ just above $H_c$ increases almost linearly with $H$.\nThis behavior differs from the theoretical predictions,\nwhere $M(H)$ varies as proportional to $\\sqrt{H-H_{\\rm c}}$ for axially symmetric fields\n($H \\parallel c$).~\\cite{Affleck,Takahashi}\nOne of the reason of this discrepancy can be the finite temperature effect.\nThe $\\sqrt{H-H_{\\rm c}}$ dependence easily becomes invisible by a slight thermal excitation,\nas is observed in NDMAZ.~\\cite{Kobayashi01}\nOther origins can be imperfect powder orientation,\nthat can lead axially-asymmetric fields even for $H \\parallel c$.\nHowever, it is also possible that the linear $M(H)$ is intrinsic behavior of an\nantiferromagnetically ordered system.\nWhen the measured temperature is sufficiently low compared to the energy gap,\nthe system promptly enters the antiferromagnetic ordered regime above $H_c$,\nas is shown below.\n\n\nFig.2(a) shows temperature dependence of the magnetization measured under\nfields parallel to the $c$-axis, $M^{\\parallel}(T)$.\nFor $H < H^{\\parallel}_{\\rm c} = 19$ T, the $M^{\\parallel}(T)$ curves show no anomalies.\nBelow 5 K, $M^{\\parallel}(T)$ have small but finite values (0.02-0.03$\\mu_{\\rm B}$),\nand these are attributed to the saturation magnetization of impurity and\/or defects.\nFor $H$ = 22 T, there exists a cusp-like minimum at\naround $T_{\\rm min}$ = 6.4 K. With increasing fields, $T_{\\rm min}$ \nshifts to higher temperatures systematically.\n$T_{\\rm min}$ reaches to 11.5 K at $H$ = 30 T, as is seen in the figure.\n\nFig.2(b) shows temperature dependence of the magnetization \nmeasured under fields perpendicular to the $c$-axis, $M^\\perp(T)$.\nSimilar to the above results, $M^{\\perp}(T)$ also exhibits a cusp-like minimum\nfor $H > H^{\\perp}_{\\rm c}$ = 13.5T.\n\nIn both Fig.2 (a) and (b), $M(T)$ shows a sharp change in its slope at $T_{\\rm min}$.\nMoreover, below $T_{\\rm min}$, $M(T)$ increases with decreasing $T$ showing a convex curve,\nas is most clearly seen in $M^{\\perp}(T)$ at $H$ = 30 T (Fig.2 (b)).\nSuch $M(T)$ curves quite resemble those field-induced magnetic ordering \nof the coupled dimers TlCuCl$_3$.~\\cite{Oosawa,Nikuni}\nIn this compound, it is demonstrated that $T_{\\rm min}$ at which $M(T)$ has a minimum\nis the N\\'{e}el temperature,\nfrom the neutron diffraction~\\cite{Tanaka} and the specific heat measurements.~\\cite{Oosawa01}\nSimilarly, such cusp-like anomalies in $M(T)$ curves are shown to be \nthe ordering temperature \nin the $S=1\/2$ alternating chain Pb$_2$V$_3$O$_9$ (ref.~\\cite{Waki})\nand the quasi-2-dimensional BaCuSi$_2$O$_6$ (ref.~\\cite{Jaime}) \nfrom specific heat measurements.\nWe hence conclude that the data shown in Fig.2 also demonstrate the occurrence of field-induced\nmagnetic ordering with N\\'{e}el temperatures around $T_{\\rm min}$.\n\nIt is notable that the Haldane system NDMAP exhibits a minimum in $M(T)$ for $H > H_{\\rm c}$\nat temperatures much higher than $T_{\\rm N}$.~\\cite{Honda01,HondaJAP}\nThe origin of the minimum is not yet clear, and possibly related to a\ncrossover into the low-temperature Tomonaga-Luttinger(TL) liquid regime,\nas is predicted for non-interacting one-dimensional ladders.~\\cite{Wang,Wessel}\nThis is purely a one-dimensional phenomenon,\nand the three-dimensional ordering occurs at much lower temperatures.~\\cite{Wessel}\nFor those cases, $M(T)$ curves around $T_{\\rm min}$ are characterized by the relatively\nbroad minimum and the concave curve.~\\cite{Honda01,HondaJAP,Wessel}\nThis is in clear contrast with the cusp-like anomalies and the convex curve below $T_{\\rm min}$\nin the present study as well as in those reported for TlCuCl$_3$ etc.,\nwhich signals the 3-dimensional magnetic ordering.\nIt is of course important to perform other experiments in order \nto verify the magnetic ordering at $T_{\\rm min}$.\nThe lack of single crystalline samples makes it difficult\nto measure the specific heat of this anisotropic compound.\nWe   are then planning to measure the NMR spectra at high fields.\n\nIt may be interesting to compare the ordered-state induced by fields in PbNi$_2$V$_2$O$_8$\nwith that induced by impurity-doping in PbNi$_{2-x}$Mg$_x$V$_2$O$_8$.~\\cite{Uchiyama}\nIt is shown that the ordered-state of the latter has inhomogeneous distribution of magnetic moment,\nby ESR~\\cite{Smirnov} and ${\\rm \\mu}$SR experiments.~\\cite{Lappas}\nIn addition, the impurity-induced ordered state vanishes at fields higher than $H$ = 4 T,\nwhere the Haldane state with an energy gap recovers.~\\cite{Masuda}\nIn contrast, ordered state observed in the present experiments shows up only\nabove $H_{\\rm c}$.\nThe largest value of $T_{\\rm min}$ in the present study is $\\sim$10 K for $H$ = 30 T,\nwhich is much larger than the maximum value of $T_{\\rm N}$ induced by Mg-doping,\n3.3 K,~\\cite{Smirnov} or the value of $zJ_1 \\sim 0.03J \\simeq 3.1 $K,\nwith $z$ the number of nearest chains, and $J$ the intrachain coupling.~\\cite{Zheludev00}\nThis fact implies that the field-induced ordering occurs via the\ndeveloped antiferromagnetic-correlation along the chain,\nand the ordered moment induced by fields is distributed\nuniformly on the chain.\n\nIn Fig.3, the values of $T_{\\rm min}$ are plotted against the applied fields.\nThis corresponds to the magnetic phase diagram for PbNi$_2$V$_2$O$_8$.\nFor both of $H \\parallel c$ and $H \\perp c$, $T_{\\rm min}$ increases with fields.\nIt is notable that the phase boundaries for $H \\parallel c$ and $H \\perp c$ \ndo not cross each other at least within the field range measured.\nThis is in qualitative agreement with the theoretical calculation\nby Sakai,~\\cite{Sakai01} the $HT$ phase diagram calculated for a Haldane chain with\nnegative $D$.\nIndeed, $D\/J = -0.05$ is estimated for PbNi$_2$V$_2$O$_8$ from inelastic\nneutron scattering experiments.~\\cite{Zheludev00}\nIn contrast, it is reported that crossing of the phase boundaries occurs in the\nphase diagram of NDMAP and NDMAZ,~\\cite{Honda98,Kobayashi01}\nand is well explained by the theoretical calculation for positive $D$.~\\cite{Sakai00}\nThus, PbNi$_2$V$_2$O$_8$ is the first example of field-induced order in \nthe Haldane system with negative $D$.\n\n\\begin{figure}[tp]\n\\begin{center}\n\\includegraphics[width=8cm]{Fig3.eps}\n\\caption{\\label{fig:epsart} Magnetic phase diagram of PbNi$_2$V$_2$O$_8$\nsuggested from the present experiments (symbols).\n`AFO' and `Haldane-state' represent the antiferromagnetically ordered state\nand the nonmagnetic spin-singlet state with a Haldane-gap, respectively.\nOpen symbols represent the data for $H \\parallel c$, and filled ones are \nfor $H \\perp c$. Circles indicate $T_{\\rm min}$ determined from the $M(T)$ curves,\nwhile diamonds indicate $H_{\\rm c}$ estimated from the $M(H)$ curves.\n }\n\\end{center}\n\\end{figure}\nSince the early stage of the research of the Haldane systems,\nits magnetic state at $H > H_{\\rm c}$ has been discussed theoretically\nin terms of the BEC picture.~\\cite{Affleck,Sorensen}\nIn the following, we discuss on the possible condensed state in the ordered phase.\nIn Fig.2 (a), one can see that $M^{\\parallel}(T)$ increases \nbelow $T_{\\rm min}$ with decreasing $T$.\nSuch increase cannot be explained by a conventional mean-field theory,~\\cite{Tachiki}\nwhich predicts almost flat $M(T)$ below the ordered temperature.\nInstead, this increase is successfully explained by the magnon BEC theory,\nas to be due to the increase of magnon number as the condensation sets in.~\\cite{Nikuni}\nTo apply the magnon BEC theory to our case, it is essential that the rotational\nsymmetry around the magnetic field be conserved.~\\cite{Nikuni}\nIn the present case, $M^{\\parallel}(T)$ holds this restriction.\n\nIt is rather surprising that the $M^{\\perp}(T)$ also increases below $T_{\\rm min}$\nas is seen in Fig.2(b).\nHere $H$ is applied perpendicularly to $D$,\nthereby the rotational symmetry of the Hamiltonian around $H$ is broken.\nIn such cases, magnon BEC picture is not assured because the number of bosons \nis not conserved.~\\cite{Nikuni}\nIn fact, the $M^{\\perp}(T)$ of NDMAP does not increase below $T_{\\rm N}$\nbut becomes flat against $T$.~\\cite{Honda01,HondaJAP}\nSuch behavior is consistent with the Ising-like antiferromagnet that\nis predicted to develop for $H \\perp D$.~\\cite{Sakai00}\nFor the present system, the similarity of $M^{\\perp}(T)$ and $M^{\\parallel}(T)$\nmay be due to the relatively small $D$ ($D\/J = -0.05$).\nThis point should be studied more carefully.\n\nIt should be remarked, however, that the BEC picture requires some rigorous conditions.\nFirst, the concentration of magnons must be enough dilute.~\\cite{Nikuni}\nIn fact, experiments on KCuCl$_3$, isostructural of TlCuCl$_3$,\nshowed that the $M(T)$ curve becomes flat below $T_{\\rm N}$ for \nfields well above $H_{\\rm c}$.~\\cite{Oosawa02}\nThis behavior implies that for this dense magnon condition, \nmean field approximation is a better description.\nThe BEC picture should hence be applied only for the region $H - H_{\\rm C} \\sim 0$.\nMoreover, it is recently argued that some anisotropic interactions\narising from spin-orbit coupling like the Dzyaloshinsky-Moriya(DM) interaction \nand\/or the staggered $g$ effect can be qualitatively modify the BEC description,\neven if the interactions are very weak.~\\cite{Sirker04,Sirker05}\nRecent ESR measurements on TlCuCl$_3$ have indeed suggested the existence of \nsuch interactions.~\\cite{Glazkov}\nFor the present case, the screw-like crystal structure of PbNi$_2$V$_2$O$_8$\nmay cause the DM interaction, as is suggested to explain the weak-ferromagnetism \nin the isostructural SrNi$_2$V$_2$O$_8$.~\\cite{Zheludev00}\n\n\n\n\\section{Conclusions}\nWe have observed cusp-like anomaly at $T_{\\rm min}$ in the $M(T)$ curves for $H > H_{\\rm c}$.\nThe value of $T_{\\rm min}$ increases with applied fields.\nThese observations suggest the evolution of field-induced magnetic ordering\nin the Haldane chain system PbNi$_2$V$_2$O$_8$.\nMagnetic phase diagram of this system up to 30 T is presented.\nThe phase boundaries for $H \\parallel c$ and $H \\perp c$ do not\ncross each other, in qualitative agreement with the $HT$ phase diagram calculated theoretically for a Haldane \nsystem with $D<0$.~\\cite{Sakai01}\n\nIn the ordered phase, it is revealed that the magnetizations increase with decreasing $T$\nand the $M(T)$ have a convex curve for both the directions $H \\parallel c$ and $H \\perp c$.\nThese features may support that the magnon Bose-Einstein condensation picture\ncan be applicable as an approximation for Haldane gap systems at least for $H \\parallel c$.\nHowever, possible anisotropic effects including the Dzyaloshinsky-Moriya interaction\ncan modify the description of the ordered state significantly.\n\n\\begin{acknowledgments}\nN.T. gratefully acknowledges M. Hagiwara, A. Oosawa, M. Hase, and H. Kageyama for fruitful discussions.\nHe also thanks T. Waki and K. Yoshimura for informing their results, and \nK. Hashi and H. Shinagawa for the help of field-oriented sample preparation.\n\n\n\\end{acknowledgments}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nIdentifying and analyzing Cyber Threat Information (CTI) is an important part of validating security alerts and incidents~\\cite{islam2019multi, koyama2015security, menges2019unifying, mittal2019cyber}. Any piece of information that helps organizations identify, assess, and monitor cyber threats is known as CTI~\\cite{johnson2016guide}. To help a Security Operation Centre (SOC) in using CTI, existing approaches, such as a unifying threat intelligence platform~\\cite{islam2019multi, koyama2015security, menges2019unifying, MISPJournal}, aim to automatically gather and unify CTI relevant to security alerts and incidents. However, gathering CTI is not enough to perform validation tasks, as security teams need to analyze and understand CTI for defining response actions. Security teams write scripts and define rules to extract necessary information from CTI, and map alerts and incidents to CTI~\\cite{anstee2017great, elmellas2016knowledge, RFID2021, tounsi2018survey}. Whilst techniques such as defining rules and scripts can be automated, they do not help in identifying evolving threats and alerts ~\\cite{serketzis2019actionable, ward2017building, zhou2019ensemble}, because rules can only be defined for behavior of known threats. Thus, human understanding and resolution are required to identify, define and update CTI, rules and scripts for emerging threats to adapt changing contexts.\n\nThe vast volume of CTI makes it time-consuming for a human to analyze. Thus, to address the shortcoming of defining rules and scripts to use CTI, we present a novel framework, SmartValidator. SmartValidator identifies CTI and validates security alerts and incidents by leveraging Artificial Intelligence (AI) based automation techniques~\\cite{faiella2019enriching,  qamar2017data,serketzis2019actionable}. SmartValidator follows a systematic and structured approach for reducing the human cognitive burden to continuously monitor for changes (e.g., change in attack patterns and CTI) and define the automation strategies whenever changes occur. We focus on two aspects of automation: (i)~\\textit{automatic identification} of CTI for different alerts and (ii)~\\textit{automatic validation} of alerts using identified CTI. \n\nBy \\textit{automatic identification} of CTI, we mean identifying CTI from a wide variety of sources. The increasing presence and amount of CTI over the internet demands effective techniques to automate the identification of the required CTI for validation tasks~\\cite{faiella2019enriching,menges2019unifying,noor2019machine,qamar2017data}. The sources of CTI vary with differences in alerts and incidents~\\cite{EY2017,MISP2021,RFID2021}. Examples of CTI include Indicators of Compromise (IoC) (system artifacts or observables associated with an attack), Tactics Techniques Procedures (TTP) and threat intelligence reports~\\cite{johnson2016guide}. \n\nBy \\textit{automatic validation} of alerts and incidents, we refer to validating (i.e., prioritizing, and assessing the relevance or impact of) different types of alerts and incidents generated and identified by different detectors. In this context, by detector we mean any tools or systems used for detection of malicious activities. An organization deploys or develops different types of detectors that generate alerts upon detection of malicious activities. Examples of such detectors include Intrusion Detection Systems (IDS), vulnerability scanners and spam detectors. Validation of different types of security alerts and incidents requires extracting information from relevant CTI~\\cite{anstee2017great, elmellas2016knowledge, RFID2021, tounsi2018survey, WINKLER2017143}. For example, a network administrator or threat hunter writes scripts to search for CTI (e.g., information about the suspicious incident) and defines rules to validate an alert. \nThere are always cases for which automatic validation would not be suitable. For example, in our scenario, automated validation is not applicable for alerts and incidents that do not have associated CTIs; hence, such scenario would require a security team to perform manual analysis.\n\nThe massive volume and variations of CTI opens the door for automatic identification of patterns and gathering insights about CTI using Natural Language Processing (NLP) and Machine Learning (ML) techniques. For instance, Sonicwall has reported 9.9 billion malware attacks in its 2020 cyber threat report~\\cite{sonicwall2020}. The threat research team of Sonicwall has come across more than~1,200 new malware variants each day. Existing studies~\\cite{ibrahim2020challenges, le2019automated, noor2019machine,RFteam2018,  serketzis2019actionable, Struve2017, zahedi2018empirical, zhou2019ensemble} have highlighted  the power of AI to monitor, gather and analyze security intelligence. Recent advances have also been noticed in the use of NLP and ML techniques to extract patterns from threat data and gain insight about attacks and threats. The focus of these studies are application-specific, for example, detecting anomalies~\\cite{zhou2019ensemble} or automating vulnerability assessment~\\cite{le2019automated}, which need to be updated with changing CTI and organizational needs. These studies required knowledge of NLP and ML to build a model for performing the assessment or detection task. Most existing SOCs are managed SOCs (SOC as a service), which are subscription based~\\cite{Nick2020, ibrahim2020challenges}. They do not have dedicated data science or ML experts to design and update the AI based system based on their need. Considering this scenario \\textit{\"Can we design an efficient system to automate and assist the validation of security incidents and alerts with changing threat data and user needs\"?}.\n\nEvolving threat landscapes and changing needs of security teams demand a dynamic AI\/ML-based validation system which can be adapted at runtime.\nFor instance, if a security expert expresses interest to validate the maliciousness of a domain \"URL\", a prediction model is built by a data scientist team that classifies a URL as malicious or non-malicious. In an ML context, this task is known as a prediction or classification task. We propose three different layers to differentiate the tasks of threat data collection, validation and prediction model building. The purpose is to hide implementation complexity of data processing, prediction models and validators from security teams. Each layer is controlled and developed by experts with dedicated capabilities. Changing threat landscapes require SOCs to request new CTI and prediction models. One possible solution to this is to build and save prediction models for all possible attributes sets. However, building all possible models whenever changes occur will incur significant resource consumption (e.g., computation time). Most SOCs have limited resources; hence, instead of pre-building all possible combinations of prediction models, SmartValidator is designed to build the model based on a SOC's demands.\n\nWe have implemented a Proof of Concept (PoC) system to evaluate the effectiveness and efficiency of our proposed framework with two feature engineering approaches and eight ML algorithms. We have used an IoC form of CTI~\\cite{johnson2016guide} and collected open source threat intelligence (OSINT) from public websites and a CTI platform, MISP~\\cite{MISP2021, RFID2021}. The input of the developed system is a set of attributes from a security team. For example, a security team may want to investigate the \"\\textit{domain name}\" and \"\\textit{URL}\" to identify the \"\\textit{maliciousness}\" of an incident. The developed system takes these three attributes as input where \"\\textit{domain name}\" and \"\\textit{URL}\" are the observed attribute~($ob_{attrib}$) and \"\\textit{maliciousness}\" is the unknown attribute~($un_{attrib}$). The prediction models are built for classifying\/ predicting~$un_{attrib}$ based on~$ob_{attrib}$. \n\nTo capture changing contexts (i.e., security team requirements), we have considered five $un_{attrib}$: (i)~attack, (ii)~threat type, (iii)~name, (iv)~threat level and (v)~event. Eighteen different sets of~$ob_{attrib}$ (shown in Table~\\ref{tab:tableAttribute}) are provided to validate these five attributes to demonstrate the performance of the PoC with changing requirements. We have designed the PoC to select the suitable feature engineering approaches and ML algorithms at run time. \nSeven $ob_{attrib}$ are selected by the PoC to predict \\textit{attack} and 11 $ob_{attrib}$ sets are used to predict the remaining four attributes. Hence, the PoC provided a total 51 optimal prediction models for predicting five $un_{attrib}$ based on the preferred~18~$ob_{attrib}$. The results show that approximately~84\\% of the models have F1-scores above~0.72 and~75\\% of the models have F1-scores above~0.8. These results imply that SmartValidator is able to assist the automatic validation of threat alerts with a high level of confidence. Most of the models that were built with data gathered from the MISP platform can effectively predict~$un_{attrib}$ based on~$ob_{attrib}$ with a higher F1-score than the models that were built with CTI gathered from public websites. This demonstrates that trusted threat intelligence is more effective in validating alerts.\n\nThe results also demonstrate the efficiency of SmartValidator with dynamic changes in the preferred set of attributes. We pre-built all possible models, which required us to run~814 experiments. Given a maximum time limit of 48 hours and a memory limit of~100GB to build each prediction model, 20\\%~of the models failed to complete within the time limit and given memory. Hence, it shows the difficulties a security team would encounter in manually constructing each model. Results further reveal that building prediction models is a time-consuming process and requires expertise that can be automated through orchestrating different tasks. Saving the feature engineering approaches and ML algorithms helps SmartValidator to use them for predicting new attributes based on changing CTI and SOC requirements. Thus, constructing prediction models at run time based on a security team's preferred attributes sets reduces the overhead and resource consumption. The key contributions of this work are:\n\n\\vspace{-5pt}\n\n\\begin{itemize}\n\\item A novel AI-based framework, SmartValidator, that consists of three layers to effectively and efficiently identify and classify CTI for validating security alerts with changing CTI and security team requirements.\n\\vspace{-5pt}\n\\item A PoC system that automatically built 51 models to predict five different unknown attributes with~18~ observed attributes sets using two sources of OSINT.\n\\vspace{-5pt}\n\\item We demonstrated that SmartValidator can effectively select optimal prediction models to classify CTI where approximately 75\\% of optimal models have an F1-score of above~0.8.\n\\vspace{-5pt}\n\\item We showed the efficiency of SmartValidator by building prediction models based on security team demands which requires approximately 99\\%~less models to build, thus less resources and time consumption. \n\\vspace{-5pt}\n\\end{itemize}\n\n\n\nPaper organization: Section~\\ref{Sec:Motiv} presents a motivation scenario that highlights the need for SmartValidator. Section~\\ref{Sec:Preli} discusses the background knowledge about CTI. Section~\\ref{Sec:Framework} introduces the proposed framework, SmartValidator. Section~\\ref{section:POC} describes the large-scale experiment that is carried out for the evaluation of SmartValidator. Section~\\ref{Sec:Eval} demonstrates the effectiveness and efficiency of the proposed approach. Section~\\ref{Sec:RelatedWork}  discusses related work. Finally, section~\\ref{Sec:Conclu} concludes the paper with future works.\n\\vspace{-10pt}\n\n\\section{Motivation Scenario} \\label{Sec:Motiv}\n\n\\begin{figure*}\n    \\centering\n        \\subfloat[]{\\includegraphics[width=0.7\\textwidth]{figs\/Figure1a.pdf}\\label{fig:motiva}}\n        \\hfil\n        \\subfloat[]{\\includegraphics[width=0.8\\textwidth]{figs\/motivB.pdf}\\label{fig:motivb}}\n    \\caption{Motivation scenario illustrated (a) detection of malicious activities and validation of alerts with multiple detectors and sources of CTI respectively, (b) validation of same alerts for different set of preferences required two different validators and different CTI}\n    \\label{fig:Motiv}\n    \\vspace{-15pt}\n\\end{figure*}\n\nIn this section, we motivate an AI-based solution for alert validation through an example scenario. Figure~\\ref{fig:Motiv} shows a scenario where a SOC of an organization has deployed different types of detectors, validators and CTI to monitor and validate malicious behaviour in its network and business data. Figure~\\ref{fig:motiva} shows that three detectors (intrusion, phishing email, and vulnerability detectors) are deployed to detect suspicious and malicious activities of an organization. The information used by detectors varies with attack types. For example, the information that detectors use to identify an intrusion is different from identifying a phishing email\\footnote{https:\/\/github.com\/counteractive\/incident-response-plan-template\/blob\/master\/playbooks\/playbook-phishing.md} (Figure~\\ref{fig:motiva}). These detectors continuously monitor an organization's network and business data\\footnote{https:\/\/github.com\/rosenbet\/demisto\/tree\/master\/Playbooks} (e.g., emails, network traffic and business reports). \n\nMost detectors produce alerts upon detecting malicious activity that require a security team to act on it. These alerts require validation before analysing them for decision making\\footnote{https:\/\/www.incidentresponse.com\/playbooks\/}. In this paper, we consider a validator performs a task related to prioritizing and identifying the relevance or impact of alerts. Let us assume that an intrusion detector has detected a list of malicious IP addresses. A SOC has an alert validator to validate the maliciousness of the IPs~\\cite{Siemplify2019}. Figure~\\ref{fig:motiva} shows that validation of different types of alerts requires different forms of CTI. To validate IP maliciousness, blacklist and whitelist IP addresses are used. CTI further varies from organization to organization. Each security team has its own set of requirements (different attributes) to validate an alert. In this scenario, the security team rely on three types of CTI for alert validation. Considering different types of alerts have different attributes and require different sources of CTI, a SOC needs three validators to validate the alerts produced by three detectors. Figure \\ref{fig:motivb} illustrates the use of CTI to validate alerts.\n\nWe assume, an alert of type $A_i \\in A$ ($A$ is the set of alerts) is produced by detector $D_1$ (e.g., IDS). Each alert is represented with different attributes (or features). The function $F_{attrib}$($A_i$) provides attributes list of $A_i$. \n\n$F_{attrib}(A_i) \\rightarrow <f_1, \\ f_2,\\ f_3,\\ f_4,\\ f_5>$\n\\\\ where  $f_1 = \\text{IP}$, $f_2 = domain$, $f_3= \\text{URL}$, $f_4 = attack\\ type$, and $f_5  = threat\\ level$. Considering two different security roles of a SOC have different preferences and used different attributes to validate $A_i$, two validators $V_1$ and $V_2$  are built (Figure \\ref{fig:motivb}). For validator $V_1$, a security team prefers to validate \\textit{attack type} based on \\textit{\\text{IP}}, \\textit{domain} and \\textit{\\text{URL}}, thus\n\n$ob^1_{attrib} = <\\text{IP},\\ domain,\\ \\text{URL}>$ and \n\n$un^1_{attrib} = <\\text{IP}\\ maliciousness>$. \\\\\n For validator $V_2$, a security team's preference is to validate \\textit{threat level} using attributes \\textit{domain}, \\textit{URL} and \\textit{attack type}, thus,\n\n$ob^2_{attrib}= <\\text{IP},\\ domain,\\ \\text{URL},\\ attack\\ type>$ and \n\n$un^2_{attrib}= <\\text{URL}\\ threat\\ level>$.\n\nFor both cases, to perform validation, validators first extract $ob_{attrib}$  and if available $un_{attrib}$ from $A_i$ and then identify CTI with these attributes. In most cases, a security team provides CTI sources to a validator. In the next step, CTI that have $ob_{attrib}$ and $un_{attrib}$ are identified. As shown in Figure \\ref{fig:motivb}, validator $V_1$ extracts three attributes to validate \\textit{IP maliciousness} and validator $V_2$ extracts four attributes to validate \\textit{URL threat level}. $\\text{CTI}_1$ has the attributes that are required by validator ${V_1}$. On the other hand, $\\text{CTI}_2$ has the attributes required to investigate URL threat level by $V_2$. Therefore, threat data is extracted from $\\text{CTI}_1$ and $\\text{CTI}_2$ respectively for further investigation.\n\nThough $V_1$ and $V_2$ have investigated two different sets of attributes, the key steps (step~3.1 to step~3.4), as shown in Figure~\\ref{fig:motivb}, are the same. The tasks $V_1$ and $V_2$ can be formulated as an ML classification problem, where two different prediction models are required to be built. Building a prediction model involves pre-processing of data (e.g., $ob^1_{attrib}$), feature engineering, training and selecting a model, and then predicting a output ($un^1_{attrib}$). Many of the possible $ob_{attrib}$ of Cyber Threat Information (CTI) (e.g., domain, filename, description and comment) are textual features. Traditional categorical feature engineering or transformation approaches are not suitable to encode the textural features, hence require the application of NLP technique. To perform validation of $un^1_{attrib}$ and $un^2_{attrib}$ using $ob^1_{attrib}$ and $ob^2_{attrib}$, prediction models are required to be built where input of prediction models are security team preferences. Here, we consider the observed attributes and unknown attributes as SOCs' preferences\/ requirements. We assume, $AS$ as a set of a SOC's requirements, where \n\n $AS =  <ob_{attrib},\\ un_{attrib}> $\n\\\\ This mean for validator $V_1,\\ AS_1$ =  <$ob^1_{attrib}$ , $un^1_{attrib}$> \\& $AS_1 \\in AS$. For $ V_2, AS_2$ =  <$ob^2_{attrib}$, $un^2_{attrib}$> \\& $AS_2 \\in AS$\n\nConsidering emerging threat patterns, a SOC may deploy new detectors and update the existing rules of intrusion detectors to detect evolving anomalies. To validate the alerts of a new detector, new validators may be required. Thus, several changes can arise in the scenario of Figure~\\ref{fig:Motiv}. Following, we present the three scenarios that we consider in this work.\n\n\\vspace{-5pt}\n\n\\begin{itemize}\n    \\item Change in Alert: With changes in alerts types, variation can be seen in the attributes of alerts. For example, an alert of type $A_2$ may have a different set of attributes from $A_1$ such as timestamps, date, IP, organization, tools and comments. Depending on types of attack and detector, an alert attribute changes. Building prediction models with changing alerts might require incorporation of various types of pre-processing and features engineering approaches. \n    \\vspace{-5pt}\n    \\item Change in CTI: CTI are continuously changing with changing requirements from SOCs. In most cases, SOCs buy CTI from third parties where attributes provided by different vendors vary, or they build their own CTI platform. The validation of alerts relies on the attributes available in CTI.\n    \\vspace{-5pt}\n    \\item Changes in Preferred Attributes Set: Change in preferred attributes ($AS$) requires re-designing and building of prediction models. A SOC does not always have dedicated data scientists or experts to design and build prediction models. Even though the steps of model building are repetitive (e.g., pre-processing, feature engineering, model building and selection), few changes may be required for adaptation of the variations as existing solutions are not designed to automatically work with changing attributes.\n\\end{itemize}\n\\vspace{-5pt}\nTo address these changes, we propose SmartValidator to support the flexible design of a validator following a systematic and structured approach. The proposed framework can automatically construct prediction models to validate alerts\nwith changing requirements. \n\n\\vspace{-10pt}\n\n\\section{Preliminaries} \\label{Sec:Preli}\nThis section provides background information about CTI and MISP (an Open Source Threat Intelligence Platform).\n\n\\textbf{Indicator of Compromise: } IOCs provide characteristics of cyberattacks and threats. Based on IOCs, a security team decides whether a system is affected by a particular malware or not~\\cite{ anstee2017great, elmellas2016knowledge, tounsi2018survey}. Examples of IOCs include domain name, IP, file names and md5 file hashes. Three common categories of IOCs are network indicators, host-based indicators and email indicators. IP addresses, URLs and domain names are the most popular types of network indicators. The malicious file hash or signature, registry keys, malware name and dynamic link libraries are widely used host-based indicators. An email indicator may consist of a source email address, message objects, attachments, links and source IP addresses. The source of IOCs ranges from crowd-sourcing to government-sponsored sources. Just having threat data is not enough to fully understand the context or patterns of a cyberattack. For example, threat data may contain an IP address that is used only once to attack a network. Conversely, an associated URL in threat data might have been used many times. Therefore, threat intelligence must be extracted from the threat data with possible IOCs and their contextual information~\\cite{ anstee2017great, elmellas2016knowledge, ward2017building, tounsi2018survey}. Table~\\ref{tab:CTIsource} shows examples of some of these websites that provide threat feeds and are utilized for gathering OSINT. Table~\\ref{tab:CTIexample} shows examples of CTI publicly available in the malware domain website~\\cite{malwareDomain2021}. \n\n\\begin{table*}[h]\n \\caption{Description of each CTI sources with the Indicator of Compromise (IOCs) they contain}\n    \\begin{tabular}{ll}\n  \\toprule\n    \\textbf{Source} & \\textbf{Description}\\\\\n   \\midrule\nC\\&C Tracker \\cite{Ctracker2019} & Contains a list of C\\&C IPs (command  and control botnets), date, and a link to a manual which contains \\\\&  text description and false positive risk value. \\\\\nFeodo Tracker \\cite{Ftracker2019} & Tracks the Feodo Trojan. Contains IP, port, and date. \\\\\nMalware Domain List \\cite{malwareDomain2021} &  Contains a searchable list of malicious domains, IP, date, domain, reverse lookups \\\\ & and lists registrants. Mostly focused on phishing, Trojans, and exploit kits.\\\\\nRansomware Tracker \\cite{Ransomwware2019} & Provides overview of infrastructures used by Ransomware, status of URLs, IP address and \\\\ & domain names associated with Ransomware and various block list of malicious traffic. \\\\\nWHOIS data \\cite{WhoIS2019} & Provides a database of registered users and assignees of internet resources, which is widely used\\\\ &  for lookup of domain names.\\\\\nZeus Tracker \\cite{Ztracker2019} & Tracks domains of Zeus Command \\& Control servers.\\\\\nOpenPhish \\cite{OpenPhish2019} & A list of phishing URLs and their targeted brand. \\\\\n   \\bottomrule\n    \\end{tabular}\n    \\label{tab:CTIsource}\n\\vspace{-10pt}\n\\end{table*}\n\n\\begin{table*}[!hb]\n\\caption{An example of a list of observed malware domains with corresponding Indicator of Compromise (IOCs) from the website malwaredomainlist.com \\cite{malwareDomain2021}}\n    \\centering\n    \\begin{tabular}{llllll}\n    \\toprule\n       \\textbf{Date} & \\textbf{Domain} & \\textbf{IP} & \\textbf{Reserve Lookup} & \\textbf{Description} & \\textbf{ASN}\\\\\n\\midrule      \n       \n2017\/12\/04 18:50 & textspeier.de    & 104.27.163.228 & - & phishing\/ fraud & 13335 \\\\\n2017\/10\/26 13:48 & photoscape.ch\/ & 31.148.219.11 & knigazdorovya.com  & trojan & 14576 \\\\\n       &Setup.exe \\\\\n2017\/06\/02 08:38 & sarahdaniella.com\/swift & 63.247.140.224 & coriandertest. & trojan & 19271\\\\\n  & \/SWIFT\\%20\\$.pdf.ace & & hmdnsgroup.com & \\\\\n  \n2017\/05\/01 16:22 & amazon-sicherheit.kunden & 63.247.140.224 & hosted-by.blazingfast.io & phishing  & 49349\\\\\n& -ueberpruefung.xyz \\\\\n2017\/03\/20 10:13 & alegroup.info\/ntnrrhst & 185.61.138.74 &  mccfortwayne.org & Ransom, Fake & 197695\\\\\n&&&&.PCN, Malspam \\\\\n   \\bottomrule\n   \\end{tabular}\n    \\label{tab:CTIexample}\n    \\vspace{-10pt}\n\\end{table*}\n\n\\textbf{Threat Intelligence}: Threat Intelligence, also known as security threat intelligence, is an essential piece of CTI for a cybersecurity team. According to \\textit{Recorded Future}, \"\\textit{threat intelligence is the output of the analysis based on detection, identification, classification, collection and enrichment of relevant data and information}\"~\\cite{RFID2021}.Threat intelligence helps a security team understand what causes an attack and what needs to be done to defend against it by gathering contextual information about an attack. For example, security teams use threat intelligence to validate security incidents or alerts and enrich threat data to get more insights about a particular security incident~\\cite{anstee2017great, elmellas2016knowledge, tounsi2018survey, RFID2021, WINKLER2017143}. The gathered data is organized in a human-readable and structured form for further analysis~\\cite{anstee2017great, elmellas2016knowledge, faiella2019enriching, RF2019,  WINKLER2017143}. Open Source Intelligence (OSINT) is gathered from various websites (e.g., Zeus Tracker and Ransomware Tracker) that provide information about malware or blacklist domain\/IPs. \n\n\n\n\n\n\n\\textbf{Cyber Threat Intelligence Platform:} Threat intelligence platforms allow security communities to share and collaborate to learn more about the existing malware or threats. Using threat intelligence platforms, companies can improve their countermeasures against cyber-attacks and prepare detection and prevention mechanisms. In recent years, the cybersecurity communities have emphasized building common threat intelligence platforms to share threat information in a unified and structured way, and make CTI actionable~\\cite{faiella2019enriching, gao2018graph, menges2019unifying,mittal2019cyber, tounsi2018survey, ward2017building}. Various specifications and protocols such as STIX, TAXII, Cybox, CWE, CAPEC and CVE are widely used to describe and share threat information  through common platforms~\\cite{ Stixbarnum2012standardizing, barnum2012cybox, taxiiconnolly2014trusted, RamsdaleCTISurvey2019}. Trusted Automated Exchange of Indicator Information (TAXII)~\\cite{taxiiconnolly2014trusted} is developed as a protocol for exchanging threat intelligence represented in STIX format. Both STIX and TAXII are open source and have collaborative forums~\\cite{Stixbarnum2012standardizing,taxiiconnolly2014trusted}.  \n\n\\begin{figure}\n     \\includegraphics[scale = 0.35]{figs\/MISP.jpg}\n    \\caption{An example of MISP showing evoluation of a multi task Botnet}\n    \\label{fig:MISP}\n    \\vspace{-15pt}\n\\end{figure}\n\n\\textbf{MISP:} Malware Information Sharing Platform (MISP) is one of the most popular trusted threat intelligence platforms used by different industries to store and share CTI~\\cite{MISP2021, MISPJournal}. MISP is a framework for sharing and storing threat data in a structured way~\\cite{MISPJournal, azevedo2019pure}. MISP enables an organization to store both technical and non-technical information about attacks, threats and malware in a structured way. The relationship between malware and their IOCs are also available in MISP. Rules for network Intrusion Detection Systems (IDS) can also be generated from MISP, which can be imported into an IDS system, and hence improve the detection of anomalies and malicious behavior. A security team queries MISP for relevant data, and it shows the details of the attack. For example, Figure~\\ref{fig:MISP} shows the details of the evolution of a multitask Botnet. Table~\\ref{tab:MISPExample} and Table~\\ref{tab:MISPvaluePercentage} of Appendix~\\ref{app:A} show the key attributes gathered from MISP for this study and and the percentage of each attribute.\n\n\\vspace{-10pt}\n\n\\section{Proposed Framework} \\label{Sec:Framework}\nFigure~\\ref{fig:framework} provides an overview of our proposed framework, SmartValidator, that automates the identification and classification of CTI for validation of alerts. It comprises of three layers: (i)~threat data collection layer, (ii)~threat data prediction model building layer and (iii)~threat data validation layer. We consider each layer to have a separation of concerns so that while updating components of one layer, a security team does not need to worry about other layers.\n\n\\begin{figure*}\n    \\centering\n     \\includegraphics[width=0.85\\textwidth]{figs\/framework.pdf}\n    \\caption{An overview of SmartValidator for automated identification and validation (i.e., classification) of cyber threat data}\n    \\label{fig:framework}\n    \\vspace{-15pt}\n\\end{figure*}\n\n\\textit{Threat Data Collection layer} (section~\\ref{sub:4.1}): The threat data collection layer consists of a collector that automates the identification of CTI information from various sources (Figure~\\ref{fig:datalayer}). It further transforms the gathered CTI into a unified form (if required) and passes it to the threat data prediction model building layer.\n\n\\textit{Threat Data Prediction Model Building layer}(section~\\ref{sub:4.2}): This layer has a model builder that builds models for validation of alerts based on a SOC's requirements ($ob_{attrib}$, $un_{attrib}$) using the gathered CTI (Figure~\\ref{fig:predictionlayer}). Pre-processing of data, feature engineering, training and evaluation of prediction models are performed in this layer to generate candidate validation models. The candidate models and corresponding feature sets are saved with a SOC's preferences for use by the threat data validation layer at runtime to validate alert. \n\n\\textit{Threat Data Validation layer} (section~\\ref{sub:4.3}): The threat data validation layer takes alerts and a SOC's requirements as input to choose suitable prediction models for alert validation. SOC requirements are interpreted by an interpreter. Based on a SOC's requirements, attributes are extracted from alerts. The extracted attributes are used to choose suitable prediction model from the candidate list of saved models for predicting the unknown attributes ($un_{attrib}$) and perform validation of alerts based on observed attributes ($ob_{attrib}$).\n\nThe following sections elaborate the core components and functionalities of each layer of SmartValidator. \n\\vspace{-5pt}\n\n\\subsection{Threat Data Collection Layer}\\label{sub:4.1}\nValidation of a security alert requires the identification of relevant CTI for building a threat data prediction model. The purpose of the prediction model is to learn the pattern of CTI for automatic validation of alerts. Here, we have formulated the validation tasks as a classification task. For instance, to validate an IP maliciousness, a system needs to classify IPs as malicious or non malicious, which can be achieved through a prediction model. Similar to existing studies~\\cite{faiella2019enriching, tounsi2018survey, MISPJournal}, the data collection layer gathers CTI from multiple sources and combines them into a unified format. The data collection layer employs a collector to gather CTI data from an organization's preferred sources. Considering several types of CTI sources are used by an organization, deployment of various plugins, APIs and crawlers are required to collect CTI from these sources. Figure~\\ref{fig:datalayer} has shown the processing of three types of CTI data -  (i)~internet data, (ii)~business data and (iii)~external data. These are most commonly used CTI. Other forms of CTI can also be integrated by following standard data processing strategies. \n\n\\begin{figure}[htb]\n     \\includegraphics[width=0.5\\textwidth]{figs\/dataCollectionlayer.pdf}\n    \\caption{Threat data collection layer - IOCs are extracted and processed from three types of data and then combined into a unified form}\n    \\label{fig:datalayer}\n    \\vspace{-15pt}\n\\end{figure}\n\nFor processing internet data, the collector has web crawlers, scrapers and parsers to gather and process CTI data from web pages (Figure~\\ref{fig:datalayer}). A web crawler searches and identifies reliable sites that contain threat information and IOCs of various malware. Considering threat intelligence team  provides the relevant list of websites or keywords of their interest to gather CTI~\\cite{listThreatIntel2021}, a crawler crawls through the internet to search the relevant information. We propose a scraper as part of the collector for filtering out unnecessary information from crawled data. Crawling and scraping can be done on a variety of sources, such as RSS feeds, blogs and social media, but require different types of processing. A parser utilizes various information processing techniques to extract information from the output of a scraper and organise the data into a structured and language-agnostic format (e.g., markup language such as ontology). For forums or blog posts written in \\textbf{natural language}, a parser is required to extract threat information from sentences. NLP tools and techniques (e.g., Spacy and NLTK) are used to build a parser based on the structure of a document and information required by security team. \n\nTo get threat feeds from databases and CTI platforms, we design API calls and queries that are parts of the collector. Threat feeds can be gathered from both an organization's internal business data and external data (Figure~\\ref{fig:datalayer}). In this paper, we only gather external threat feeds. The collector can also query external data sources to find out missing information about available threat data. For example, after receiving an IP address, a query can be made to WHOIS query website to search for domain name. In this way, a collector gathers different sets of data, for example, blacklist and white list IP addresses, list of phishing websites and so on, from different types sources. The collected data is further combined into a unified form (e.g., dataset~P, Figure~\\ref{fig:datalayer}). To combine the data into a unified form, we first normalised the data, removed redundant information and then combined them. Examples of normalisation techniques include 1NF, 2NF and so forth. Depending on validation tasks (e.g., validation of IP maliciousness or validation of domain threat level), CTI is extracted and sent to the threat data prediction model building layer to build a validation model.\n\nWe consider organizations (e.g., government or financial) may have a dedicated threat intelligence teams or may use third party services to gather CTI. Any update related to the collection of CTI, such as adding or modifying CTI sources, deploying APIs or parsers to gather and extract information from these CTI, and inclusion or deletion of new data  collection and normalisation techniques, are performed in the threat data collection layer by dedicated threat intelligence team.\n\\vspace{-5pt}\n\n\\subsection{Threat Data Prediction Model Building Layer}\\label{sub:4.2}\nThe threat data prediction model building layer is designed to build ML-based classification models using CTI and SOC's preferences. For example, if a security team wants to validate the \\textit{maliciousness} of an IP considering the \\textit{IP}, \\textit{domain} and \\textit{URL}, an ML classification model is built that takes IP, domain and URL and predicts IP maliciousness. We consider these attributes (IP, domain, URL and IP maliciousness) as SOC's preference where $ob_{attrib} = \\{\\textit{IP}, domain, \\textit{URL}\\}$ and $un_{attrib} = \\{\\textit{IP}\\ maliciouness\\}$. SOC's preferences drive from organizational security requirements and alerts. \n\nFigure~\\ref{fig:predictionlayer} shows the core components and workflow of the threat data prediction model building layer. It comprises of a pre-processor that pre-processes CTI (step~1, Figure~\\ref{fig:predictionlayer}) for extracting features from it. The pre-processing techniques depend on the types of attributes (e.g., categorical or text-based). For example, IP maliciousness can be either malicious or non-malicious, which is categorical. On the other hand, domain name is text based, as shown in Table~\\ref{tab:CTIexample}.  Pre-processed data is passed to a feature engineering module where data is transformed into features (numeric form) (step 2, Figure~\\ref{fig:predictionlayer}), which are used as input for the ML algorithms.  The reason behind this is that ML algorithms can only work with numerical data~\\cite{Helge2020, RFteam2018, sabir2020machine, Struve2017}. Depending on the type, size and diversity of CTI, the data science team chooses a feature engineering approach. The first two steps of Figure~\\ref{fig:predictionlayer}~leverage simple NLP techniques for pre-processing and feature engineering. Categorical values can be directly transformed into features using label encoding or one-hot encoding and text values are transformed into features using count vectorizer and TFIDF (Term Frequency-Inverse Document Frequency) techniques. The associated text cleaning and pre-processing steps for each text attribute are discussed in Appendix B1. Common pre-processing techniques such as tokenization, stop words removal and lemmatization are performed before transforming text data into features. Hence, based on the alert attributes types, data science teams perform data pre-processing and select a feature engineering approach. \n\n\\begin{figure} [tbh]\n     \\includegraphics[scale = 0.5]{figs\/predictionlayer.pdf}\n    \\caption{Threat data prediction layer - prediction models are built to predict unknown attributes based on observed attributes}\n    \\label{fig:predictionlayer}\n    \\vspace{-15pt}\n\\end{figure}\n\nThe transformed data is split into training and testing datasets to build, select and evaluate an prediction model (step~3, Figure~\\ref{fig:predictionlayer}). ML algorithms are applied to train models based on CTI that connects and learns patterns in the data to derive a working model. Depending on the nature of the training data, different models are built by a data science team. Traditionally a set of ML algorithms are applied to find an algorithm suitable for a specific dataset and user requirements~\\cite{AHMED201619, Helge2020, sabir2020machine}. To investigate the effectiveness of prediction models for the validation task, we considered a set of ML algorithms (e.g., Decision Tree, Na\u00efve Bayes, K-Nearest Neighbours and Random Forest). The details of the PoC is discussed in section~\\ref{section:POC}. As most ML algorithms have a list of hyperparameters, validation techniques (e.g., k-fold cross-validation, random cross-validation and Bayesian optimisation) are incorporated to select  hyperparameters\\footnote{Hyperparameters are user-defined values that determine details about the ML classifier before training. For example, a decision tree requires tuning the value of variable depth, and k-nearest neighbours has a variable number of neighbours.} setting and feature engineering approach for a specific ML algorithm (step~4, Figure~\\ref{fig:predictionlayer}). \n\nThe built model's performance is evaluated using the testing dataset (step~5, Figure~\\ref{fig:predictionlayer}). Different types of performance metrics (e.g., precision, recall, accuracy and F1-score) are used to choose a model (also known as the optimal model) that provides the best performance (details in section~\\ref{sub:evaluationMetrics}). In this work, we mainly consider F1-score for evaluation, which is a score between~0 and~1. Higher F1-score indicates better performance of a model. Figure~\\ref{fig:predictionlayer} shows how the pre-processing techniques, feature engineering approaches and ML algorithms that are used to build threat data prediction models are stored for future model building process. Once an prediction model is found to have performed best, that model can be rebuilt using both training and testing datasets. The best models are saved with the evaluation score (step~6, Figure~\\ref{fig:predictionlayer}) for using it in the threat data validation layer.\n\n\\subsection{Threat Data Validation Layer}\\label{sub:4.3}\nWe design the threat data validation layer to (i)~collect a SOC's needs, (ii)~automatically orchestrate and request for CTI and prediction models and (iii)~validate alerts.\nFigure~\\ref{fig:framework} shows the validation layer comprises of an  \\textit{interpreter}, \\textit{orchestrator}, \\textit{data processor} and \\textit{predictor}. In this layer, security teams of SOCs provide their preferences, $AS$, as a set of requirements, where $AS$ = {<$ob_{attrib}, un_{attrib}>$}. Security teams may also provide a minimum threshold for F1-score, which we refer to as the confidence score of prediction models. The reason behind gathering confidence score is that the performance of prediction models will differ with variation in CTI, alerts and attributes sets. A security team might need a higher F1-score while dealing with safety critical data and sensitive information. For example, to identify IP maliciousness, a security team may request for a validation model with a minimum confidence score of 0.9. While categorizing comments or text messages as spam, model performance of 0.8 or above may be approved. Selection of F1-score values vary from application to application. Thus, instead of setting a fixed value, we consider providing security teams the flexibility to set the confidence score based on their application needs.  \n\nWe design Algorithm~\\ref{alg:Orchestrator} describing the key steps of the threat data validation layer. These steps are coordinated and orchestrated by the orchestrator. SOC preferences ($AS$ and confidence score) are the input of Algorithm~\\ref{alg:Orchestrator}. The \\textit{interpreter} receives the SOC requirements and extracts observed attributes $ob_{attrib}$, unknown attributes $un_{attrib}$ and confidence scores from that (line~3). The orchestrator checks model availability with F1-score above the confidence score for predicting $un_{attrib}$ based on $ob_{attrib}$ (line~4). If a model is available, the attributes are passed to the data processor, where it pre-processes and transforms the data based on the saved pre-processing and feature engineering approaches (lines~5-7). Finaly, the pre-processed and transformed data is sent to the predictor, which uses the available model to predict $un_{attrib}$ (line~8).\n\n\\begin{algorithm}[tbh]\n\\footnotesize\n    \\caption{Model building with orchestrator in threat data validation layer}\\label{alg:Orchestrator}\n    \\begin{algorithmic}[1]\n        \\State \\textbf{Input: } \\texttt{AS}  <$ob_{atrrib}, un_{attrib}$>, \\texttt{confidence score}\n        \\State \\textbf{Output: } \\texttt{predictedData}\n        \\State Interpret (\\texttt{AS, confidence score})\n        \\State  \\texttt{IsModels}= CheckModel(\\texttt{AS, confidence score}) \n        \\If{\\texttt{IsModels} true}\n            \\State \\texttt{model, featureEng} = getModel(\\texttt{AS,confidence score})\n            \\State \\texttt{processedData} = transformData(\\texttt{featureEng}, \\texttt{AS})\n            \\State \\texttt{predictedData} =  predictOutput(\\texttt{model,processedData})\n         \\Else\n            \\State IsData = CheckData(\\texttt{AS})\n            \\If{IsData true}\n                \\State \\texttt{CTIData} = RetrieveData(\\texttt{CTI, AS})\n                \\State  \\texttt{model} = buildModel(\\texttt{CTIData, AS, confidence score})\n                \\If{\\texttt{model} is built}\n                    \\State go to step 6\n                \\Else\n                    \\State go to step 19\n                \\EndIf\n            \\Else\n                \\State RequestData(\\texttt{AS})\n                \\State \\textbf{return NotApplicable} \n            \\EndIf\n        \\EndIf\n        \\State \\textbf{return} \\texttt{predictedData}\n     \\end{algorithmic}\n\\end{algorithm}\n\nIf a model is unavailable for the requested attributes set, e.g., for predicting~$un_{attrib}$ based on~$ob_{attrib}$, then the orchestrator requests the data collector module to gather the relevant CTI data for the preferred attribute sets (line~9). After identifying the relevant information, the data collector module sends the collected CTI data to the \\textit{model builder} to build an prediction model (lines~12-13). Using the CTI \\textit{data} from the data collection layer, the \\textit{model builder} follows the model building process as discussed in section~\\ref{sub:4.2} (lines~14-15). After building a model, it sends the model availability notification to the orchestrator. Then the same process of data processing and prediction is performed. If the requested data is unavailable, a notification is sent to a threat intelligence team and a SOC team to gather the required CTI and manually analyze the alerts, respectively (lines 18-19). To ensure that alerts are not ignored when models or CTIs are unavailable, SOC teams must keep informed that manual analysis is required.\n\nOur proposed framework, SmartValidator, streamlines gathering, identification and classification of CTI. SmartValidator allows a security team to swiftly make a response about incoming alerts. As most information is generated in a structured way, it can be easily pre-processed to share through a CTI platform such as MISP or Collective Intelligence Framework (CIF) to benefit diverse security teams. SmartValidator can be integrated with the existing security orchestration and automation process to validate alerts and thus work together with the existing security tools, such as Security Information and Event Management (SIEM) and Endpoint Detection and Response (EDR). Microsoft Azure Sentinel\\footnote{https:\/\/azure.microsoft.com\/en-au\/services\/azure-sentinel\/} and Splunk\\footnote{https:\/\/www.splunk.com\/} are examples of SIEM where Limacharlie\\footnote{https:\/\/www.limacharlie.io\/} and Google Chronicle\\footnote{https:\/\/cloud.google.com\/blog\/products\/identity-security\/introducing-chronicle-detect-from-google-cloud} are considered as EDR.\n\n\\vspace{-10pt}\n\\section{Experiment Design and Setup} \\label{section:POC}\n\nWe designed and implemented  a Proof of Concept (PoC) system to evaluate SmartValidator. We expected to demonstrate effectiveness of prediction models in validating security threat data and efficiency of building prediction models based on a SOC's requirements. The goal of the PoC is to identify the relevant CTI and build prediction models based on a list of SOC's requirements. Hence, we evaluated the PoC system based on the following two Research Questions (RQ).  \n\\vspace{-5pt}\n\\begin{itemize}\n    \\item RQ1. How effective is machine learning in classifying CTI for SmartValidator?\n     \n   \\vspace{-5pt}\n    \\item RQ2. How efficient is SmartValidator in selecting and building prediction models at runtime over pre-building all possible prediction models?\n\\end{itemize}\n\\vspace{-5pt}\nTo build the PoC system for SmartValidator, we implemented the core components of Figure \\ref{fig:framework}: the data collection layer presented in Figure \\ref{fig:datalayer}, the prediction layer presented in Figure \\ref{fig:predictionlayer} and an orchestrator for the validation layer described in section \\ref{sub:4.3}. \n\n\\subsection{SOC's Requirement}\n\nWe defined a set of attributes (validated attributes and observed attributes) as SOC's requirements to carry on the experiment. These attributes were mainly given by a team who were different than the one that implemented the prediction models. Thus, here we considered the team who provided the requirement as part of the SOC and the other team as part of the data science team. This setting gave us the option to evaluate a variety of different SOC's requirements, to appropriately assess the PoC system. In a practical scenario, these requirements would usually be defined by a security team. Among the various attributes, commonly validated attributes ($un_{attrib}$) are \\textbf{attack}, \\textbf{threat type} and \\textbf{threat level}. We considered them as the desired unknown attributes. Beside them, we also considered two others attributes \\textbf{name} and \\textbf{event} as the desired unknown attributes.\nExample of these attributes are shown in Table~\\ref{tab:MISPExample} in appendix~\\ref{app:A}. As shown in Table \\ref{tab:MISPExample}, an example of an event title (or event) is \"\\textit{OSINT Leviathan: Espionage actor spear phishes maritime and defence targets}\".\n\nAs observed attributes ($ob_{attrib}$) vary from security team to security team, we gathered~18 different sets of observed attributes from security team to validate the aforementioned $un_{attrib}$. As shown in Table~\\ref{tab:tableAttribute}, \\textit{IP information} (i.e., ASN, IP owner, country, and domain), \\textit{organization}, \\textit{comments about attributes}, \\textit{comments about attacks}, \\textit{event data}, \\textit{timestamp} and \\textit{category} are the attributes that we considered in the set of observed attributes. We selected these attributes from alert data that are also commonly used to validate alerts generated by different IDS. We also used the metadata of attributes such as URL, domain and filename. \n\\begin{table}[!hb]\n\\caption{List of the observed attribute sets}\n    \\centering\n    \\begin{tabular}{ll}\n    \\toprule\n       \\textbf{\\#} & \\textbf{List of attributes}\\\\\n\\midrule      \n   $ob^1_{attrib}$ & Date \\\\\n   $ob^2_{attrib}$ & Domain\\\\\n   $ob^3_{attrib}$ & IP, ASN, Owner, Country \\\\\n   $ob^4_{attrib}$ & Date, Domain \\\\\n   $ob^5_{attrib}$ & IP, ASN, Owner, Country, Domain \\\\\n   $ob^6_{attrib}$ & IP, ASN, Owner, Country, Date \\\\\n   $ob^7_{attrib}$ & IP, ASN, Owner, Country, Domain, Date \\\\\n   $ob^8_{attrib}$ & IP destination, Port, IP source, ASN, Owner, \\\\ & Country, Domain, File hash, Filename\\\\\n   $ob^9_{attrib}$ & IP destination, Port, IP source, ASN, Owner, \\\\ & Country, Domain, Description, Comment, File \\\\ & hash, Filename\\\\\n   $ob^{10}_{attrib}$ & IP destination, Port, IP source, ASN, Owner, \\\\ & Country, Domain, Description, Comment\\\\\n   $ob^{11}_{attrib}$ & IP destination, Port, IP source, ASN, Owner,\\\\ & Country, Domain, Date, Timestamp, File hash , \\\\ & Filename\\\\\n   $ob^{12}_{attrib}$ & IP destination, Port, IP source, ASN, Owner, \\\\ & Country, Domain, Date, Timestamp, Description,\\\\ &  Comment, File hash, Filename\\\\\n   $ob^{13}_{attrib}$ & IP destination, Port, IP source, ASN, Owner, \\\\ & Country, Domain, Date, Timestamp, Description,\\\\ &  Comment\\\\\n   $ob^{14}_{attrib}$ & IP destination, Port, IP source, ASN, Owner, \\\\ &Country, Domain, Date, Timestamp\\\\\n   $ob^{15}_{attrib}$ & Description, Comment, File hash, Filename\\\\\n   $ob^{16}_{attrib}$ & Date, Timestamp, File hash, Filename\\\\\n   $ob^{17}_{attrib}$ & Date, Timestamp, Description, Comment, \\\\ &File hash, filename\\\\\n   $ob^{18}_{attrib}$ & Date, Timestamp, Description, Comment\\\\\n   \\bottomrule\n   \\end{tabular}\n    \\label{tab:tableAttribute}\n    \\vspace{-10pt}\n\\end{table}\n\n\\subsection{Collecting CTI}\\label{subsec:5.2}\n\nWe gathered CTI from two types of sources \u2013 publicly available internet data and data from an OSINT platform, MISP. CTI gathered from these two sources is considered as dataset~1~($DS_1$) and dataset~2~($DS_2$), respectively. \n\n\\textbf{\\textit{Gathering CTI from websites}}: We obtained a list of publicly available websites from a GitHub CTI repository~\\cite{listThreatIntel2021} which are shown in Table~\\ref{tab:CTIsource}. We selected these websites because they provided malware RSS feeds and their access were not restricted (e.g., API limit). We built web crawlers and scrapers to gather and extract the key pieces of information from the selected websites. A parser was built to parse the information and stored it in a structured format (i.e., a CSV file). The gathered data had consistent tagging and was labelled with the malware used in the attack, for example, Zeus, Citadel or Ice~IX. $DS_1$~contained 4060 events and represented the data available through public CTI feeds from websites. \n\n\\textbf{\\textit{Gathering CTI from MISP}}: We selected the MISP platform as a threat intelligence platform due to its popularity amongst businesses and the abundance of labelled data. We first gathered the MISP default feeds that were written in JSON format and then built a parser to extract the key attributes from that. $DS_2$~contained~213,736 events and represented the data available to an organisation from a dedicated threat intelligence platform.\n\n\\textbf{\\textit{Gathering additional attributes:}} External information was gathered utilizing the parsed attributes of both~$DS_1$ and~$DS_2$. For example, the common features amongst each source were IP, domain and date. Additional attributes were gathered from WhoIS data (e.g., a database query of the RFC~3912 protocol) for each IP. We used the Python cymruwhois\\footnote{https:\/\/pypi.org\/project\/cymruwhois\/} module to search each IP in the WhoIS database, which returned the IPs ASN (i.e, a unique global identifier), owner, and country location. Besides, AlienVault forum was chosen as an external information source. We scraped the AlienVault forum updates using the Python module BeatifulSoup\\footnote{https:\/\/pypi.org\/project\/beautifulsoup4\/}. The AlienVault data, which were natural language text descriptions, was searched and extracted for the associated event and threat.\n\n\\subsection{Building Data Processor}\n\nWe built a data processor to clean, pre-process and transform the collected data for building ML-based validators. \n\n\\textbf{\\textit{Cleaning and pre-processing:}} We used the Python sci-kit learn libraries to pre-process the attribute values~\\cite{chen2020deep, scikitlearn2021}. We first cleaned the data by removing null values and removing events with missing information.  We observed missing information to be relatively infrequent, resulting in minimal information loss and a more robust model. For text values, we found two types of natural language features from the MISP data (i)~text attributes which are short paragraphs that describe an event in natural language and are often taken from blogs, and (ii)~comments. \n\nWe first analyzed the text and comments attribute to find a suitable processing and encoding technique. Thus, simple processing techniques were undertaken to decrease the dimensionality and remove any uninformative words (e.g., articles and prepositions). Each piece of text was stripped of all non-alphabetical characters, as numbers and special characters can rapidly increase dimensionality and rarely contain valuable information. The text was then stripped of any non-noun or non-proper noun tokens, as nouns are the most informative part of the text (e.g., attack names, attack types, and organisation). Finally, each word was lemmatized (i.e., changed to the base form of the word), so that similar words can be recognized. One of the key steps we followed was to tokenize the string values of attributes (i.e., domain, filename, hostname, URL) where patterns exit. We removed punctuation and special characters within a string to clean the data. We further split the text into small tokens based on a regular expression that tokenized a string using a given character. This separated each word within a value (e.g., value of domain or URL) and allowed a string to be tokenized. For example, we split the value of the URL in terms of \"\/\/\" and \".\". The tokenized data were then encoded as integers to create a numeric form of a feature vector.\n\n\\textbf{\\textit{Feature engineering:}} We encoded the categorical variables using one-hot encoding and label encoding. One-hot encoding considers each categorical value separately and represents each categorical variable as a column. Label encoding represents each categorical value as a unique integer. Table \\ref{tab:exampleEncoding} shows an example of one-hot encoding in the first table and label encoding in the second, for three types of attacks: phishing, DDoS and SQL injection. We used the labelEncoder() method of sci-kit learn to convert the string data into numerical values. An inbuilt function from the sci-kit-learn library, standardScaler(), was used to standardize the data. The function transformed data into a normalized distribution to remove outliers from the data, allowing for building more accurate prediction models. \nThe text variables (i.e., unstructured and structured natural language) did not conform to traditional one-hot or label encoding, as one-hot or label encoding interprets the text as a whole. Hence, we used two techniques: count vectorization and TFIDF as our feature engineering approach to encode text into numerical values. Count vectorization techniques stored each tokenized word as a column with its value being the number of times it appeared in each respective document.\nTable~\\ref{tab:exampleVectorizer} shows the examples of count vectorization of two sentences. The \\textbf{TFIDF} vectorizer\\footnote{https:\/\/scikit-learn.org\/stable\/modules\/generated\/sklearn.feature\\_\\\\extraction.text.TfidfVectorizer.html} worked similar to the count vectorization, except rather than storing counts it stored the TFIDF value of each word. TFIDF provided a metric for how 'important' a word is within a part of the text by comparing the term's frequency in a single document to the inverse of its frequency amongst all documents.\n\n\\begin{table}[h]\n\\caption{An example of one-hot encoding and label encoding for three types of attack}\n    \\centering\n   \n     \\begin{tabular}{|c c c|c |c p{.9cm} c|}\n    \\hline\n        Phishing & DDoS & SQL  & & & Attack & Encoded \\\\\n                 &      & injection & & & & attack\\\\\n        \\hline\n0&  1&  0&\t&   1&\tDDoS\t& 2\\\\\n1&\t0&\t0&\t&\t2&\tPhishing&\t1\\\\\n0&\t1&\t0&\t&\t3&\tDDoS&\t2\\\\\n1&\t0&\t0&\t&\t4&\tPhishing&\t1 \\\\\n1&\t0&\t0&\t&\t5&\tPhishing&\t1 \\\\\n0&\t0&\t1&\t&\t6&\tSQL&\t3\\\\\n& & & & & injection & \\\\\n\\hline\n        \n    \\end{tabular}\n    \\label{tab:exampleEncoding}\n    \\vspace{-10pt}\n\\end{table}\n\nFor the one-hot encoding setup, we used a simple count vectorizer, and for the label encoding setup we used a TFIDF vectorizer. It should be noted that whilst the dataset included text variables, the vast majority did not follow a natural language convention (e.g., domain or filename). Hence, more advanced NLP techniques, such as word embedding, cannot be accurately applied. The feature engineering schemes were saved for runtime use with the model building and prediction phase. Appendix~\\ref{app:B} summarizes the pre-processing techniques that we followed for different attributes.\n\n\\begin{table*}[!hb]\n\\caption{Count vectors for two sentences, S1: \"Fireball is malware\" and S2: \"Malware is any program that is harmful\"}\n    \\centering\n    \\begin{tabular}{cccccccc}\n    \\hline\n       &fireball &is &malware &any &program &that &harmful\\\\\n       \\hline\n       S1 & 1 & 1 & 1 & 0 & 0 & 0 & 0\\\\\n       \\hline\n       S2 & 0 & 2 & 1 & 1 & 1 & 1 & 1 \\\\\n       \\hline\n   \\end{tabular}\n    \\label{tab:exampleVectorizer}\n    \\vspace{-10pt}\n\\end{table*}\n\n\\subsection{Building the Validation Model} \\label{sub:buildingValidateModel}\nWe built prediction models following the traditional ML pipeline (i.e., selecting ML algorithms, building prediction models, performing hyperparameter tuning and evaluating the built model). We designed a model builder to build prediction models for various attribute sets $AS$ ($ob_{attrib}$ and $un_{attrib}$). We selected eight commonly used classification algorithms~\\cite{caruana2006}: Decision Tree (DT), Random Forest (RF), K-Nearest Neighbors (KNN), Support Vector Machine (SVM), Multi-Layer Perceptron (MLP), Ridge Classifier (RID), Na\u00efve Bayes (BAY) and eXtreme Gradient Boost (XGB)~\\cite{chen2016} to cover a wide range of classifier types. Appendix~\\ref{app:B} summarizes the ML algorithsm that we considered to build the PoC system. Bayesian Optimization was used to automatically tune each model~\\cite{snoek2012}. We used a straightforward train test split for evaluation, with~30\\% of the dataset hold out for testing. In a real world, setting the training data would be selected by the threat intelligence team, to ensure data  quality. The built model was optimised by performing hyperparameter tuning. The Python module sci-kit learn  was used to build the prediction models, as it is one of the popular and widely used libraries for building prediction models~\\cite{chen2020deep, scikitlearn2021, sabir2020machine}. \n\n\\subsection{ Developing the Orchestrator}\n\nWe designed and implemented a Python script to coordinate the data collector and model builder. The script worked as an orchestrator that automated the process from gathering SOC's requirements to predicting the outputs that is validating alerts. For example, we took the SOC's requirements as an attribute set, $<ob_{atrrib}, un_{attrib}>$ and the confidence score (a value between 0 and 1). The output of the script was the value of $un_{attrib}$ and F1-score. In this process, the script first checked whether a model was available to predict $un_{attrib}$ with $ob_{attrib}$. If a model was available, it then called the data processor to process $<ob_{atrrib},\\ un_{attrib}>$ and predict the value of $un_{attrib}$. \n\nIf a model was not found, it checked the availability of CTI with attributes $<ob_{atrrib},\\ un_{attrib}>$. If CTI was available, the model builder was invoked and models were built following the process of model building discussed in previous section. Here, the orchestrator used the saved feature engineering approaches and algorithm to train the model and then selected the model with the best F1-score as the optimal model. The scripts then checked the value of  the F1-score. If the F1-score was lower than the confidence score, it requested the data science team for building the model and returned that there is no model available to the security team. Otherwise, it notified the orchestrator about model availability and the next steps of data processing and predicting $un_{attrib}$ was followed and the value of $un_{attrib}$ was returned to the security team.\n\nIf required CTI data was not available, the orchestrator notified the security team about CTI unavailability. For example, the two CTI data we used did not have any vulnerability description and values. Now if we provided the vulnerability description as input and requested that we wanted to predict the severity, the orchestrator would return no data available. \n\n\\subsection{Evaluation Metrics} \\label{sub:evaluationMetrics}\n\nEvaluation metrics are needed to measure the success of a prediction model in validating security alerts and building prediction models on run time. This will determine the effectiveness and efficiency of SmartValidator. Accuracy, precision, recall and F1-score are the four commonly accepted evaluation metrics for evaluating a prediction models performance~\\cite{chen2020deep, sabir2020machine}. The correct and incorrect predictions are further calculated using number of (i)True Positive (TP) (refers to correct prediction of an attributes label), (ii) False Positive (FP) (indicates incorrect prediction of an attributes label), (iii) True Negative (TN) (refers to correct prediction that a threat does not have a particular label) and (iv) False Negative (FN) (indicates incorrect prediction that a threat does not have particular label). For example, if a model classifies a malicious IP address as non-malicious, it is calculated as a false positive. If it refers a malicious IP address as malicious it is calculated as a true positive. A true negative is when non-malicious IPs are not classified as malicious IPs. A false negative is if a malicious IP is not classified as such. Equation \\ref{equ:accuracy} to equation \\ref{equ:f1score} provides details of how accuracy, recall, precision and F1-score are calculated using TP, FP, TN and FN.\n\n\\vspace{-10pt}\n\n\\begin{equation}\\label{equ:accuracy}\n   \\text{Accuracy} =  \\frac{TP + TN}{TP+ TN+ FP+ FN }\n\\end{equation} \n\n\\begin{equation}\\label{equ:recall}\n   \\text{Recall}=  \\frac{TP}{TP + FN}      \n\\end{equation}\n\n \\begin{equation}\\label{equ:precision}\n     \\text{Precision}= \\frac{TP}{TP + FP}        \n \\end{equation}  \n\n\\begin{equation} \\label{equ:f1score}\n    \\text{F1-score}= \\frac{2 \\times(Recall \\times Precision )}{Recall + Precision}    \n\\end{equation}\n\nWe assessed the effectiveness of the prediction models in validating security alerts with F1-score because  accuracy (equation \\ref{equ:accuracy}) is not always a useful metric on its own. It does not capture the bias of the data. The recall (equation \\ref{equ:recall}) is a measure of robustness; it displays if a model is failing to predict the relevant samples, e.g., failing to classify IP maliciousness correctly. It is important for the PoC system to have high recall to ensure that no malicious events are misinterpreted or ignored. Precision is the model's ability to accurately predict the positive class (malicious events), shown in equation \\ref{equ:precision}. A low value for precision indicates a high amount of false positives. Thus, it is important to achieve high precision, as low precision would introduce the need for human validation of the output of SmartValidator. The F1-score can be considered the best metric for an overall evaluation, as it considers both precision and recall  (equation \\ref{equ:f1score}) together and evaluates each class separately. F1-score does not have any unit as this is the harmonic mean of precision and recall which do not have any unit as well.\n\nWe defined a confidence score between 0-1 to be used by the security team as a threshold value for prediction models. In our PoC, we compared the confidence score with the F1-score of the prediction model. If a model had a lower F1-score than the confidence score, the PoC discarded that model.\n\nWe defined computation time, as shown in equation \\ref{equ:comptime}, to evaluate the efficiency of building prediction models based on SOC's need. Computation time is the summation of the training time ($train_{time}$) and prediction time ({$predict_{time}$}). Training time is the time required to build a model and prediction time is the time required to predict unknown attributes using an optimal model.\n\n\\vspace{-15pt}\n\n\\begin{equation} \\label{equ:comptime}\n    computation_{time} = train_{time} + predict_{time}\n\\end{equation}\n\n\\vspace{-10pt}\n\n\\section {Evaluation and Results}\\label{Sec:Eval}\n\nIn this section, we present the results of the developed PoC of SmartValidator to show the  effectiveness and efficiency of a dynamic ML-based validator to automate and assist the validation of security alerts with changing threat data and SOC's needs.\n\n\\begin{table*}[thb]\n\\caption{Performance (F1-score) of different models for prediction of \\textit{attack} based on $ob^7_{attrib}$ using $DS_1$ and prediction of \\textit{threat type}, \\textit{threat level}, \\textit{name} and \\textit{event} based on $ob^{14}_{attrib}$ using $DS_2$}\n    \\centering\n  \n   \\begin{tabular}{|l|c|c|c|c|c|}\n\n     \\hline\n     \n     \\multirow{2}{*}{Model} & $DS_1\\ ob^7_{attrib}$ & \\multicolumn{4}{|c|}{$DS_2 ob^{14}_{attrib}$}  \\\\ [1ex]\n     \\cline{2-6}\n     \n     & attack & threat type & threat level & name & event  \\\\ \n     \n     \\hline\nDT+LE & 0.719\t& 0.941 & 0.998\t& 0.938\t& \\textbf{0.995 }\\\\\nRF+LE\t& 0.274\t& 0.727\t& 0.76\t& 0.689\t& 0.362\\\\\nKNN+LE\t& 0.594\t& 0.912\t&\t0.986\t&\t0.917\t&\t0.902\\\\\nGBAY+LE & 0.312\t& \t0.15\t&\t0.343\t&\t0.102\t&\t0.077\\\\\nRID+LE\t& 0.627\t& \t0.055\t&\t0.082\t&\t0.105\t&\t0.001\\\\\nSVM+LE\t& 0.401\t&\t0.135\t&\t0.654\t&\t0.295\t&\t0.283\\\\\nMLP+LE \t& \t0.546\t&\t0.762\t&\t-\t&\t0.878\t&\t-\\\\\nXGB+LE\t& \t\\textbf{0.787}\t&\\textbf{\t0.998}\t&\t\\textbf{0.999}\t&\t\\textbf{0.997}\t&\t-\\\\\n\\hline\nDT+OHE\t& \t0.587\t&\t0.917\t&\t0.988\t&\t0.905\t&\t0.926\\\\\nRF+OHE\t& \t0.501\t&\t0.916\t&\t0.948\t&\t0.912\t&\t0.87\\\\\nKNN+OHE\t& \t0.351\t&\t\\textbf{0.998}\t&\t\\textbf{0.998}\t&\t\\textbf{0.999}\t& \\textbf{0.996}\\\\\nGBAY+OHE\t& \t0.382\t&\t0.677\t&\t-\t&\t0.465\t&\t0.241\\\\\nRID+OHE\t& \t0.763\t&\t0.121\t&\t-\t&\t0.007 &\t- \\\\\nSVM+OHE\t& \t0.322\t&\t0.051\t&\t0.126\t&\t0.001\t&\t-\\\\\nMLP+OHE\t& \t0.762\t&\t0.061\t&\t-\t&\t0.079 &\t-\\\\\nXGB+OHE\t& \t\\textbf{0.784}\t&\t-\t&\t-\t&\t0.996\t&\t-\\\\\n     \\hline\n    \\end{tabular}\n    \\label{tab:performanceModel}\n    \\vspace{-10pt}\n\\end{table*}\n\n\\subsection{Evaluation of Effectiveness}\n\nWe evaluated the effectiveness of prediction models to answer RQ1 which is \\textit{\"How effective are prediction models in classifying CTI?\"}. \nSpecifically, we used two datasets $DS_1$ and $DS_2$, as described in Section \\ref{subsec:5.2}, for predicting five observed attributes ($ob_{attrib}$). We collected the datasets in a way so that each dataset was confirmed to have at least one observed value. Finally, all experiments were conducted on the collected datasets and different combinations of attributes sets. Based on self-defined SOC requirements, CTI datasets were selected and models were built. Optimal models were selected based on their effectiveness for a particular attribute set. Effectiveness is measured using the metrics described in Section \\ref{sub:evaluationMetrics}. We investigated the performance of the optimal models for classifying the threat data.\n\nWe found that 51 optimal models were returned by the PoC system based on the given requirements. Among them, seven of the models were built using $DS_1$ to predict \\textit{attack} that have used $ob^1_{attrib}$ to $ob^7_{attrib}$ and the other~44 models were build using $DS_2$ to predict the four other unknown attributes based on $ob^8_{attrib}$ to $ob^{18}_{attrib}$.\n\nThe performance of different ML algorithms and encoding methods are summarized in Table~\\ref{tab:performanceModel} for the two observed attributes sets \u2013 $ob^7_{attrib}$ and $ob^{14}_{attrib}$. The results shows that XGBoost (XGB) with Label Encoding (LE) achieved a near perfect F1 score while using $DS_2$. We further observed that while using One Hot Encoding (OHE), K-Nearest Neighbors (KNN) performed better than XGB. However, considering the time and memory constraints, XGB failed to train a model to predict \"\\textit{event}\" when LE was used as an encoding method. Some of the model building processes failed as they could not finish within the allocated memory and time limits that were 24 hours and 10GB for  $DS_1$ and 48 hours and 100GB for $DS_2$. These limits were set and tested to investigate and simulate computational resource limits.  \n\nFigure \\ref{fig:Effective} shows the evaluation score (F1-score) for different datasets, labels and encoding methods. Figure \\ref{fig:Effective(a)} and Figure \\ref{fig:Effective(b)} shows the comparison of the different classifiers when trained on $DS_1$ and $DS_2$, respectively. We first observe that ML algorithms generally performed better using $DS_2$ data, with the exception of the Ridge classifier. This finding demonstrates the importance of CTI data and information quality. prediction models require a large number of training examples to properly learn trends and patterns. We recommend utilizing data from CTI platforms such as MISP, as these platforms aggregate a large quantity of verified information from a variety of sources.\n\nFigure \\ref{fig:Effective(c)} shows the comparative classifier performance across both $DS_1$ and $DS_2$. We observe a large range in classifier performance. The variance in best classification algorithms further motivates the need for automated model building and selection. Some ML algorithms were not as effective for this classification task. Hence, if a human were to repetitively use one algorithm to build models for different set of attributes, the results would not be good (i.e., effective) for all models. It would also be time consuming to validate every model. The results demonstrate on average, the XGB classifier performed extremely well, but KNN, as well as other tree-based classifiers (DT and RF) also performed well. MLP classifiers also appear to perform well. As MLP utilized a simplistic artificial neural network, this potentially motivates the investigation of more sophisticated deep learning methods for future work \\cite{FERRAG2020102419}.\n\n\\begin{figure*}\n\\centering\n        \\subfloat[]{\\includegraphics[width=0.3\\textwidth]{figs\/ESDS1.pdf}\\label{fig:Effective(a)}}\n        \\subfloat[]{\\includegraphics[width=0.3\\textwidth]{figs\/ESDS2.pdf}\\label{fig:Effective(b)}}\n        \\subfloat[]{\\includegraphics[width=0.3\\textwidth]{figs\/EvaluationScore.pdf}\\label{fig:Effective(c)}}\n        \\hfil\n        \\subfloat[]{\\includegraphics[width=0.3\\textwidth]{figs\/ESlabel.pdf}\\label{fig:Effective(d)}}\n        \\subfloat[]{\\includegraphics[width=0.3\\textwidth]{figs\/ESEncoding.pdf}\\label{fig:Effective(e)}}\n        \\subfloat[]{\\includegraphics[width=0.3\\textwidth]{figs\/ESOptimalModel.pdf}\\label{fig:Effective(f)}}\n        \n    \\caption{Comparative analysis of different classifiers performance (F1-score) while (a) using dataset 1, $DS_1$, (b) using dataset 2, $DS_2$, and (c) using both datasets, (d) predicting different labels and (e) using different encoding methods. (f) number of optimal models for different evaluation score (F1-score)}\n    \\label{fig:Effective}\n    \\vspace{-15pt}\n\\end{figure*}\n\n\\begin{table}[]\n\\caption{Optimal classifier and encoding method with evaluation score (F1-score) for observed attributes $ob^1_{attrib}$ to $ob^7_{attrib}$ to predict \\textit{attack} using $DS_1$}\n\n    \\centering\n  \n   \\begin{tabular}{|c|c|c|}\n\n     \\hline\n     \n     \\multirow{2}{*}{Model} & \n     \\multicolumn{2}{|c|}{attack} \\\\\n     \\cline{2-3}\n     \n     & Optimal model & F1-score  \\\\ \n     \n     \\hline\n$ob^1_{attrib}$ &\tKNN + LE &\t0.572 \\\\[1ex]\n$ob^2_{attrib}$ &\tRID + OHE &\t0.537\\\\[1ex]\n$ob^3_{attrib}$ &\tSVM + OHE &\t0.623\\\\[1ex]\n$ob^4_{attrib}$ &\tXGB + OHE &\t0.728\\\\[1ex]\n$ob^5_{attrib}$ &\tRID + OHE &\t0.637\\\\[1ex]\n$ob^6_{attrib}$ &\tRID + OHE &\t0.772\\\\[1ex]\n$ob^7_{attrib}$ &\tXGB + LE &\t0.787\\\\[1ex]\n\n     \\hline\n    \\end{tabular}\n    \\label{tab:optimalModelDs1}\n    \\vspace{-10pt}\n\\end{table}\n\nWe further performed comparative analysis for predicting the five observed attributes in Figure \\ref{fig:Effective(d)}. Models generally performed well for all prediction tasks, but performed best when classifying \\textit{threat\\_level}. This is potentially because \\textit{threat\\_level} has the lowest dimensionality of the observed attributes, and hence ample training data for each class. Correspondingly, the \\textit{event} attribute had lower performance due to its high dimensionality. $DS_1$ data was used to predict \\textit{attack}. Performance is noticeably worse for this attribute,  with mean evaluation score of approximately 0.5. This further emphasizes the importance of CTI quality.\n\nThe relative effectiveness of the two encoding strategies is analyzed and shown in Figure \\ref{fig:Effective(e)}. These two encoding strategies also provide a comparison of the two NLP techniques that we considered, which are count vectorization and TFIDF vectorization. One-hot encoding appeared to typically outperform label encoding, but the results are relatively similar. This refers to the similar performance of count vectorization and TFIDF, where the count vectorizer performed slightly better, as seen under the one-hot encoding results. Similar to the previous observation, we found that depending on the type of attributes and algorithms, the performance of count vectorizer and TFIDF varies. However, the performance of classification with $DS_2$, irrespective of the encoding, built effective machine learning models \\cite{Helge2020, chen2020deep, islam2019multi, sabir2020machine}.  \n\nTable \\ref{tab:optimalModelDs1} shows the optimal model for predicting \\textit{attack} using $ob^1_{attrib}$ to $ob^7_{attrib}$ that were built using $DS_1$. Table \\ref{tab:optimalModelDs2} shows optimal model for predicting \\textit{threat type}, \\textit{threat level}, \\textit{name} and \\textit{event} based on $ob^8_{attrib}$ to $ob^{14}_{attrib}$ that were built using $DS_2$. Table \\ref{tab:optimalModelDs1} and Table \\ref{tab:optimalModelDs2} demonstrate that for different attribute sets different classifiers were seen to perform better. Thus, we cannot rely on a single algorithm. Table \\ref{tab:optimalModelDs1} shows that the IP extracted features that were in  $ob^3_{attrib}$ (i.e., IP, ASN, owner, country) performed the best singularly, out of the three available features. $ob^5_{attrib}$ which had date and IP features was also actually relatively similar in terms of predictive capability. However, by itself the IP features that is  $ob^2_{attrib}$ can only achieve a F1-Score of 0.623. Using more available features increased the effectiveness of the prediction model, with the largest feature set achieving the best F1-score of 0.787. The addition of the domain feature to the IP feature set which formed $ob^4_{attrib}$ did not significantly increase the effectiveness of the prediction model. This is likely due to the existing correlation between the IP and domain features. However, the date did appear to noticeably improve the F1-score. The large difference between the best and worst models F1-scores highlights the importance of proper feature engineering and model selection.\n\nAnalysing the results of Table \\ref{tab:optimalModelDs2}, we found that the optimal models generally performed very well, obtaining extremely good evaluation scores (F1-score). Several of the optimal models achieved a near perfect score on the testing dataset. The models also performed noticeably better than those trained on $DS_1$, further showcasing the need for good features and a large dataset. The models which used time-based features performed noticeably better than similar feature sets which did not. The time-based features likely had such strong predictive power as only a few MISP events were recorded close together. However, models without the temporal features were still able to achieve an F1-score of over 0.8. The file-based features also appeared to have weak predictive power as their inclusion or exclusion appeared to have very little impact on the evaluation score. The results of Table  \\ref{tab:optimalModelDs2} further reflects that the \\textit{event} label was harder to predict than other three labels. This was likely because \\textit{event} was the label with the highest number of classes. Interestingly, tree-based classifiers seemed to perform better for the \\textit{event} label. However, it should be noted that a lot of the more sophisticated models timed-out for the \\textit{event} label, so their results were not recorded. It can also be seen that the\\textit{ threat level} was the easiest to predict, likely because it had the lowest number of classes. Figure \\ref{fig:Effective(d)} displays the huge variance in F1-scores of trained models for different labels.\n\n\\begin{table*}[h]\n\\caption{Optimal models and encoding methods with evaluation score (f1-score) for attributes sets $ob^8_{attrib}$ to $ob^{18}_{attrib}$ to predict threat type, threat level, name and event}\n\n    \\centering\n  \n   \\begin{tabular}{|p{1.2cm}|l|p{.9cm}|l|p{.9cm}|l|p{.9cm}|l|p{.9cm}|}\n\n     \\hline\n     \n     \\multirow{2}{1.2cm}{Observed attributes} & \n     \\multicolumn{2}{c|}{threat type}  & \\multicolumn{2}{c|}{threat level} & \\multicolumn{2}{c|}{name} &\\multicolumn{2}{c|}{event}\\\\\n     \\cline{2-9}\n     \n     & Optimal model & F1-score  & Optimal model & F1-score  & Optimal model & F1-score  & Optimal model & F1-score  \\\\ \n     \\hline\n$ob^8_{attrib}$ &\tMLP + OHE&\t0.648&\tXGB + LE&\t0.68&\tRID + OHE&\t0.593&\tDT + LE&\t0.275\\\\[1ex]\n$ob^9_{attrib}$&\tSVM + OHE&\t0.854&\tXGB + LE&\t0.84&\tSVM + OHE&\t0.809&\tSVM + OHE&\t0.735\\\\[1ex]\n$ob^{10}_{attrib}$\t&SVM + OHE&\t0.859&\tSVM + OHE&\t0.915&\tSVM + OHE&\t0.864&\tSVM + OHE&\t0.763\\\\[1ex]\n$ob^{11}_{attrib}$\t&KNN + OHE&\t0.998&\tXGB + LE&\t0.999&\tKNN + OHE&\t0.999&\tDT + LE\t&0.898\\\\[1ex]\n$ob^{12}_{attrib}$&\tKNN + OHE&\t0.997&\tKNN + OHE&\t0.999&\tKNN + OHE&\t0.998&\tDT + LE\t&0.994\\\\[1ex]\n$ob^{13}_{attrib}$&\tKNN + OHE&\t0.998&\tXGB + LE&\t0.999&\tKNN + OHE&\t0.998&\tKNN + OHE&\t0.995\\\\[1ex]\n$ob^{14}_{attrib}$\t&KNN + OHE&\t0.998&\tXGB + LE&\t0.999\t&KNN + OHE&\t0.999&\tKNN + OHE&\t0.999\\\\[1ex]\n$ob^{15}_{attrib}$&\tSVM + OHE&\t0.873&\tSVM + OHE&\t0.862&\tSVM + OHE&\t0.813&\tSVM + OHE&\t0.725\\\\[1ex]\n$ob^{16}_{attrib}$&\tXGB + LE&\t0.998&\tXGB + LE&\t1&\tKNN + OHE&\t0.998&\tDT + LE\t&0.996\\\\[1ex]\n$ob^{17}_{attrib}$&\tXGB + LE&\t0.997&\tXGB + LE&\t0.999&\tKNN + OHE&\t0.999&\tRF + OHE&\t0.865\\\\[1ex]\n$ob^{18}_{attrib}$&\tXGB + LE&\t0.998&\tXGB + LE&\t1&\tKNN + OHE&\t0.998&\tDT + LE&\t0.996\\\\[1ex]\n\n     \n     \\hline\n\n     \\hline\n    \\end{tabular}\n    \\label{tab:optimalModelDs2}\n    \\vspace{-10pt}\n\\end{table*}\n\n\n\nWe further analyzed the classification confidence of the models of SmartValidator on the collected data. The evaluation results are visualized in Figure \\ref{fig:Effective(f)}. We consider different confidence scores (0.6 \u2013 0.9) as with the variation in attributes sets, preference of confidence score also varies. Figure \\ref{fig:Effective(f)} shows that at runtime, with confidence score of 0.8, 80\\% of the models that were built based on $DS_2$ fits the need of a SOC. These models were built based on the saved feature engineering and ML algorithms. Figure \\ref{fig:Effective(f)} provides an overview to the security team whether they can rely on a dataset where the performance is not up to their requirements. It shows with increase in the confidence score the number of models above the confidence score decreases. Obviously, it can be seen that the models on $DS_2$ classified the attributes with a much higher confidence. We found that one of the key reasons behind this is that $DS_1$ had comparatively less data elements than $DS_2$. Thus, with $DS_2$ capturing the variation in data and correlating them provided better results than $DS_1$. The results show that approximately 84\\% of the 51 optimal models had an F1-score (or confidence score) above 0.72 and 75\\% of the models had F1-score above 0.8. Most of the models that were built with data gathered from CTI platforms can effectively predict $un_{attrib}$  based on $ob_{attrib}$ with a higher F1-score than the models that were built with CTI gathered from public websites. \n\nIn summary, the MISP dataset (i.e., $DS_2$) was found to be a high-quality dataset that worked well with automated classification and thus validation of alerts. Hence, using attributes that are representative of possible threat data, prediction models can be built to effectively validate alerts with a substantial degree of accuracy, precision and recall. The results of $DS_2$ reflect that ML based validation models can be used to effectively validate the alerts with high quality CTI like $DS_2$. Model choice and alternatives are seen to be important steps to find the optimal models, as for different attributes sets the PoC has returned different models.\n\n\\vspace{-5pt}\n\\subsection{Evaluation of Efficiency} \\label{subsec:6.2}\n\nTo demonstrate the efficiency of SmartValidator, we answer RQ2 that is \"\\textit{How efficient is SmartValidator in selecting and building prediction models on runtime over pre-building prediction models?}\".\n\nIn SmartValidator, we propose to build the models at run time based on SOC's requirements instead of pre-building all possible models. We considered the time to build possible model combinations of the eight aforementioned classifier algorithms as a baseline to compare the efficiency of SmartValidator. We observed that it was infeasible to pre-build models for every possible combination of features. For $DS_1$, there were 62 possible feature combinations, and for $DS_2$ there were 8190. This would increase the number of experiments for $DS_2$ to 524,160. For predicting the five unknown attributes with 18 attribute sets we would only require to run 1440 experiments; only 0.26\\% of the total experiments. \nThus, for calculating the efficiency, that is the computation time, based on SOC's requirement the PoC ran a total of 816 experiments; 112 experiments for DS1 (8 ML algorithms $\\times$ 7 input attribute sets $\\times$ 1 output attribute $\\times$ 2 encoding methods) and 704 experiments for DS2 (8 ML algorithms $\\times$ 11 input attribute sets $\\times$ 4 output attributes $\\times$ 2 encoding methods) to test all combinations of the 18 observed attribute sets, prediction models, encoding methods and five unknown attributes (i.e., classification labels). \n\nWe considered the total time as a baseline to evaluate the efficiency of building the model at runtime. Here we attempted to simulate the resource limitations of model construction for a real-world environment. We considered $DS_1$ as a lightweight dataset and assigned the restrictions of 24 hours runtime and 10GB of memory, whereas $DS_2$ was a heavyweight dataset and assigned a 48 hours runtime limit and 100GB of memory. Any experiment that exceeded this run time or consumed too much memory would be aborted, as it was deemed impractical due to organisations' strict resources and fast response requirements \\cite{islam2019multi, sonicwall2020}. In this result section, we reported the efficiency in terms of time.\n\nFor $DS_1$, all 112 experiments completed successfully. However, for $DS_2$, 169 of the 704 experiments failed to finish whilst enforcing our experimental setup. The most common classifier model to time out were MLP and XGB, as these models had a significantly larger training time. For these failed jobs, 149 out of 169 jobs were either for the \\textit{event} or \\textit{threat level} label, as these datasets had many more valid entries, and thus also took more time and memory to train. Similarly, 116 of the failed jobs used one-hot encoding, as this encoding method was much less efficient than label encoding, due to every possible value adding a dimension to the encoded input. However, 53 of the label encoding experiments also failed due to the size of the text features. These features were encoded with very high dimensionality due to the lack of a natural language convention. To investigate this issue further in our future work, we plan to investigate efficient encoding methods through vocabulary size and dimensionality reduction.  \n\nTable \\ref{tab:timeDs1} and Table \\ref{tab:timeDs2} show the training time and prediction time of the optimal models that were built based on $DS_1$ and $DS_2$ respectively. The experimental results show that it is extremely inefficient to pre-build a large number of prediction models. For $DS_1$ the total training time was 61064 seconds (0.7 days), and for $DS_2$ the total training time was 7010279 seconds (81.1 days). The prediction time of a model was significantly faster than the training time, which further encouraged the use of validation models. On average, models made predictions in 0.6\\% of the training time for $DS_1$ (0.09 seconds), and 2.9\\% for $DS_2$ (17.29 seconds). \n\n\\begin{table}[h]\n\\caption{Training time and prediction time of optimal models in \\textbf{seconds} for attribute sets $ob^1_{attrib}$ to $ob^7_{attrib}$ to predict \\textit{attack} using $DS_1$}\n\n    \\centering\n  \n   \\begin{tabular}{|c|l|c|c|}\n\n     \\hline\n     \n     \\multirow{2}{*}{Model} & \n     \\multicolumn{3}{|c|}{attack} \\\\\n     \\cline{2-4}\n     \n     & Optimal model & Train time & Predict time \\\\ \n     \n     \\hline\n$ob^1_{attrib}$ &\tKNN + LE &\t758&\t0.676 \\\\[1ex]\n$ob^2_{attrib}$ &\tRID + OHE &\t0.90&\t0.002\\\\[1ex]\n$ob^3_{attrib}$ &\tSVM + OHE &\t482\t&0.001\\\\[1ex]\n$ob^4_{attrib}$ &\tXGB + OHE &\t2597&\t0.196\\\\[1ex]\n$ob^5_{attrib}$ &\tRID + OHE &\t5.6\t&0.001\\\\[1ex]\n$ob^6_{attrib}$ &\tRID + OHE &\t2.2\t&0.001\\\\[1ex]\n$ob^7_{attrib}$ &\tXGB + LE &\t1422.9&\t0.09\\\\[1ex]\n     \\hline\n    \\end{tabular}\n    \\label{tab:timeDs1}\n    \\vspace{-10pt}\n\\end{table}\n\n\\begin{table*}[h]\n\\caption{Training time and prediction time of optimal models in \\textbf{seconds} that were built to predict threat type, threat level, name and event based on attributes $ob^8_{attrib}$ to $ob^{18}_{attrib}$ using $DS_2$ }\n\n    \\centering\n  \n  \\resizebox{\\textwidth}{!}{%\n \n  \\begin{tabular}{|p{1.25cm}|p{1.45cm}|p{1cm}|p{.8cm}|p{1.45cm}|p{1cm}|p{.8cm}|p{1.45cm}|p{1cm}|p{.8cm}|p{1.45cm}|p{.8cm}|p{.8cm}|}  \n     \\hline\n     \n     \\multirow{2}{1.3cm}{Observed attributes} & \n     \\multicolumn{3}{c|}{threat type}  & \\multicolumn{3}{c|}{threat level} & \\multicolumn{3}{c|}{name} &\\multicolumn{3}{c|}{event}\\\\\n     \\cline{2-13}\n     \n     & Optimal model &\tTrain time  &\tPredict time\t& Optimal model &\ttrain Time&\tPredict time &\tOptimal model &\ttrain time &\tPredict time &\tOptimal model &\tTrain time&\tPredict time \\\\ \n     \\hline\n$ob^8_{attrib}$ & \tMLP+OHE& \t166623& \t0.18& \tXGB+LE& \t111788& \t56.86& \tRID+OHE\t& 10959\t& 0.036& \tDT+LE& \t289.87& \t0.199\\\\[1ex]\n$ob^9_{attrib}$& \tSVM+OHE& \t1141.1\t& 0.031& \tXGB+LE& \t144473& \t139.2& \tSVM+OHE& \t925.37& \t0.016& \tSVM+OHE& \t37846\t& 0.686\\\\[1ex]\n$ob^{10}_{attrib}$& \tSVM+OHE &\t1199.66 & \t0.024& \tSVM+OHE& \t804.92& \t0.006& \tSVM+OHE\t& 994.43& \t0.014& \tSVM+OHE& \t27004& \t0.657\\\\[1ex]\n$ob^{11}_{attrib}$\t& KNN+OHE\t& 2599.72\t& 23.1& \tXGB+LE& \t19190\t& 3.115& \tKNN+OHE& \t984.94& \t5.98& \tDT+LE\t& 371.38& \t0.204\\\\[1ex]\n$ob^{12}_{attrib}$& \tKNN+OHE& \t2592.79& \t22.11& \tKNN+OHE& \t16871& \t176.6& \tKNN+OHE& \t2795.24\t& 14.56\t& DT+LE& \t534.94& \t0.197\\\\[1ex]\n$ob^{13}_{attrib}$\t& KNN+OHE\t& 3177.3& \t32.04& \tXGB+LE\t& 27067& \t4.296& \tKNN+OHE& \t1178.39& \t9.311& \tKNN+OHE& \t18065& \t259.95\\\\[1ex]\n$ob^{14}_{attrib}$\t& KNN+OHE& \t2535.45& \t22.09& \tXGB+LE& \t16036& \t2.556& \tKNN+OHE& \t951.54& \t6.713& \tKNN+OHE& \t16731& \t224.98\\\\[1ex]\n$ob^{15}_{attrib}$\t& SVM+OHE\t& 1081.05\t& 0.015& \tSVM+OHE& \t739.58& \t0.004& \tSVM+OHE& \t1019.56& \t0.007& \tSVM+OHE& \t21275& \t0.369\\\\[1ex]\n$ob^{16}_{attrib}$\t& XGB+LE& \t10552.3& \t21.38\t& XGB+LE& \t4033.701& \t1.913& \tKNN+OHE& \t899.12&\t5.679&\tDT+LE&\t102.35& \t0.213\\\\[1ex]\n$ob^{17}_{attrib}$ & \tXGB+LE& \t19602.7& \t19.32& \tXGB+LE& \t13950.9& \t3.938& \tKNN+OHE& \t1005.59& \t6.64& \tRF+OHE& \t7036.7& \t15.78\\\\[1ex]\n$ob^{18}_{attrib}$\t& XGB+LE& \t14257.7& \t18.57& \tXGB+LE& \t8569.79\t& 3.175\t& KNN+OHE\t& 949.95& \t5.611& \tDT+LE\t& 142.76& \t0.22\\\\[1ex]\n     \n     \\hline\n\n     \\hline\n    \\end{tabular}\n    \\label{tab:timeDs2}\n    \\vspace{-10pt}\n    }\n\\end{table*}\n\n\\begin{figure*}\n\\centering\n        \\subfloat[]{\\includegraphics[width=0.32\\textwidth]{figs\/timeDS1.pdf}\\label{fig:efficiency(a)}}\n        \\subfloat[]{\\includegraphics[width=0.32\\textwidth]{figs\/timeDS2.pdf}\\label{fig:efficiency(b)}}\n        \\subfloat[]{\\includegraphics[width=0.32\\textwidth]{figs\/time.pdf}\\label{fig:efficiency(c)}}\n        \\hfil\n        \\subfloat[]{\\includegraphics[width=0.32\\textwidth]{figs\/TimeLabel.pdf}\\label{fig:efficiency(d)}}\n        \\subfloat[]{\\includegraphics[width=0.32\\textwidth]{figs\/timeEncoding.pdf}\\label{fig:efficiency(e)}}\n       \n    \\caption{Comparative analysis of time in \\textbf{seconds} required to train different ML based validation models with (a) dataset 1, (b) dataset 2, (c) both datasets, (d) different labels and (e) encoding methods}\n    \\label{fig:Efficiency}\n    \\vspace{-15pt}\n\\end{figure*}\n\nFigure \\ref{fig:Efficiency} shows the logarithmic distribution of the training time for different datasets, labels and encoding methods. The logarithmic distribution of training times for $DS_1$, $DS_2$ and both $DS_1$ and $DS_2$ is shown in Figure \\ref{fig:efficiency(a)}, Figure \\ref{fig:efficiency(b)} and Figure \\ref{fig:efficiency(c)}, respectively. It should be noted that Figure \\ref{fig:Efficiency} did not consider the run time of experiments which were timed out. As shown in Figure \\ref{fig:Efficiency}, the run times for more intensive models are skewed to the left. Noticeably, the Na\u00efve Bayes (GBAY) classifiers were trained near instantaneously, as these models did not require heavy fitting to the data. DT similarly was trained faster for both datasets, due to the simplicity of this model. Figure \\ref{fig:efficiency(a)} shows for $DS_1$, SVM, KNN and MLP required an average of 10-15 minutes to train. However, the XGB classifier took significantly the longest time to train with a median value of 36 minutes. For $DS_2$, the average overall training time was 217 minutes (shown in Figure \\ref{fig:efficiency(b)}) which was significantly larger than the average overall training time of 9 minutes for $DS_1$, due to the substantial dataset size increase. However, XGB had a significantly larger training time of over 8 hours. We observed that the runtime between RID and SVM was quite different even though both are linear classifiers (Figure \\ref{fig:efficiency(c)}). The SVM classifier took an average of 30 minutes to train on $DS_2$ due to hyperparameter optimization, whereas the RID classifier took an average of 10 seconds as they did not require any significant hyperparameters. These observations highlight the importance of SOC requirements, as we can see a trade-off between model performance and training time. XGB is the best performing model on average, but also exhibits the largest training time. Hence, SOC analysts would need to weigh model effectiveness against efficiency.\n\nFigure \\ref{fig:efficiency(d)} displays the training time for completed experiments for each predicted attribute. Models targeted towards predicting the \\textit{attack} attribute only took an average training time of 9 minutes, as they were trained using the much smaller $DS_1$ dataset in comparison to the other attributes that were trained using $DS_2$. Models took an average of 2 hours to train for the name attribute, in comparison to \\textit{threat\\_level}, \\textit{threat\\_type} and \\textit{event}, which took a mean time of around 4.5 hours. This could be because the training set was much smaller for the name attribute, as not as many CTI entries were assigned such information. Similarly, it should be noted that a large portion of the \\textit{event} and \\textit{threat\\_level} experiments timed out, as the training set was larger for these attributes, due to more valid entries.\n\nFigure \\ref{fig:efficiency(e)} reflects that the training time did not significantly differ between encoding methods. This is because the major encoding dimensionality came from the domain and filename features, which were treated as text attributes and were thus only one-hot encoded in our experiments. However, one-hot encoding usually exhibited larger training times, as it has much higher dimensionality and is thus less efficient. For $DS_1$, one-hot experiments took an extra 4.6 minutes on average (11.3 minutes vs 6.8 minutes). For $DS_2$, one-hot experiments took an extra 18.9 minutes on average (227.7 minutes vs 208.8 minutes).\n\nprediction models were built by experts (i.e., data scientists) who have the knowledge of ML technologies and pipeline to automatically validate the alerts. We observed that for changing SOC requirements, interaction or collaboration were required among the security team and the data science team where a security team specified the requirements and requested for the models they need. If the data required by the data science team were not available, they needed to request the threat intelligence team who gathered the requested information and updated the relevant list of information.\nHence, the model required redesigning and further actions needed to be performed to achieve the best prediction models.\nWhilst using the PoC based on SmartVadilator, these interaction could be minimized, by managing the interaction through the orchestrator. In this way, the orchestrator requested the model builder to build the required validation model. Constructing the models automatically based on a SOC's needs required less time and was more feasible than constructing possible model for all combination of attribute sets.\n\nEvaluating the efficiency of SmartValidator, we found that SmartValidator successfully identified and classified threat data required for alert validation. The same framework can be used to automate the validation of newly listed alerts, with new data sources. The data science team requires to map the suitable algorithm with suitable attributes sets and define the required data sources. \n\n\\vspace{-5pt}\n\\subsection{Discussion}\nWe consider the same attributes sets and CTI sources that are used for security incident and alert validation for three validation approaches. The validation approaches are (i) manual validation, (ii) pre-building prediction models and (iii) automatic construction of prediction models based on SOC's requirements \n\nIn manual validation, for each attribute or set of attributes, a security team first searched for attribute types and then looked for the relevant CTI availability. A security team used their previous experience to select the CTI to perform the validation. For example, to validate a malicious IP, a security team collected the blacklisted IP addresses and then looked for the IPs on the list. They further used WhoIS database to identify the relevant information about suspected IPs. The security team needed to manually write queries or call APIs to find and extract information for CTI and required knowledge about the underlying CTI sources. For similar types of alerts or changing context (i.e., change in CTI, alerts or  SOC's requirement), the same sequence of actions were repeated, that cost significant man-hours and required knowledge about the underlying plugin, API, CTI sources and so on. \n\nFor changing context while following  approach 2 (pre-building prediction models), the security team needed to request the data science team to train and build the possible prediction models for the new context. Section \\ref{subsec:6.2} reveals that to build prediction models each time a change occurs is not feasible. With automatic construction of prediction models, each time a SOC requested for a validation task, the models were built automatically. With changing context, the orchestrator coordinated the data collection and model building process that also freed the security team from coordinating and communicating with the data science team and also reduced the delay incurred due to communication. In this work, we have experimentally evaluated the performance of SmartValidator. We did not discuss the amount of time required by the security team to perform the validation activities or the time required for communication between the security team and data science team. This will also include the time gap between a request being made and the time for the security team to get the model. In future work, we plan to evaluate SmartValidator in a real SOC environment to\nfurther demonstrate how SmartValidator can be beneficial in a SOC environment. \nFor example, the effort required to manage the PoC system versus the cost saving from automation. Further we want to evaluate the maximum upfront cost to incorporate the PoC system in a real SOC environment.  \n\nFor automating construction of ML-based validation models, the PoC followed three major steps - i) collecting and processing the data, ii) training the classifier and iii) running the prediction model. Step 1 required a large amount of time  (in hours) as there were hundreds of thousands of data points to download and process, step 2 took a reasonable amount of time (in minutes) as the data were needed to be encoded and the choice of classifier were needed to train and optimise its hyperparameters. Step 3 was reasonably fast as it only needed time (in seconds) to apply a pre-trained model. These steps were bundled into an installable python package which could be made publicly available. We designed the PoC in a modular fashion so that it can be integrated into other network-enabled services to gain more information about network security. The system could easily be built for the future improvements.\n\nTo validate alerts coming from an IDS, the developed PoC system can be extended to first receive the IDS alerts over a network. After parsing the alert attributes (which would be similar to the attributes sets used in our PoC), the next steps are to transform the alert attributes into features, and then look for potential CTI to correlate the alerts information into patterns by building prediction models to predict suspicious behavior. Furthermore, the alerts can be validated using the models and the validated output can display security context of the network in a graphical user interface that is easy to understand. The PoC system can be enhanced to provide API that can be integrated as a part of a SOC's existing security system, such as middleware for EDR or SIEM. \n\nIn our experiment, we have selected two types of CTI \u2013 one is gathered from the websites, and another is from an OSINT platform MISP, which is widely used by industry as it contains high-quality data with enriched IOCs. We consider the confidence score of users to ensure that the models with low evaluation scores are not selected. We also merged multiple data sources to enrich CTI. The experimental results show that while using MISP, SmartValidator performs better than when using web data. We assert that this is due to the high quality of MISP.  The lower level of CTIs used in our experiment can be replaced with higher quality CTIs such as TTP, where SmartValidator will perform the same steps from identifying CTIs to building models and performing the validation. The prediction models can be enriched with advanced IOC and TTP with more details about the threats.\n\nThe key steps identified to improve the models automatically built through SmartValidator are effective feature encoding, hyperparameter optimization, data distribution, feature extraction, dimensionality reduction and classifier selection. Increasing the size of the dataset and number of features increases the F1-score. The dimensionality of the categorical variables needs to be decreased. It is worth noting that our investigation is by no means exhaustive; we adopt basic ML principles and NLP techniques to develop a simplistic PoC. We plan to investigate more ML techniques such as data balancing, normalization and feature selection with diverse types of CTI. Similarly, we intend to investigate more sophisticated NLP techniques, such as word embeddings that can capture semantic information of the natural language attributes.\n\nWe found that CTI datasets contain highly multi-variate categorical variables. Highly dimensional problems like this are likely to be linearly separable, as we can separate any d+1 points in a d-dimensional space with a linear classifier, regardless of how the points are labelled. We further found that overall ensemble classifiers such as XGB performed better than the other selected seven algorithms.\n\n\nAs shown in Figure \\ref{fig:framework}, we consider that dedicated expertise is required (e.g., a data scientist) to build prediction models, which in most cases is different from the SOC team using the models for validation task. The prediction models are built by experts (e.g., data scientists, ML experts, or developers) who are knowledgeable about ML libraries, feature engineering and algorithms. Considering a SOC's security team capability, the model building process can also be replaced with Automated Machine Learning (AutoML)\\footnote{https:\/\/www.automl.org\/automl\/} framework such as Google Cloud AutoML\\footnote{https:\/\/cloud.google.com\/automl}. AutoML frameworks are designed to provide ML as a service, where a security team is required to provide the pre-processed data and for some cases the transformed data. It considers multiple ML algorithms in a pipeline to evaluate the performance, perform hyperparameter tuning and validation in an attempt to improve the performance. An AutoML framework provides a list of optimal models. For example, TPOTClassifier\\footnote{http:\/\/epistasislab.github.io\/tpot\/api\/} is an automated ML classifier that is developed in an attempt to automate the ML pipeline in python. It explores prediction models configurations that a data analyst or security analyst may not consider and attempts to fine-tune the model to achieve the most optimized model. Hence, the model builder of our proposed SmartValidator can be developed following the process discussed in Figure~\\ref{fig:predictionlayer} or using an AutoML framework. Thus, depending on an organizations SOC capabilities they may use an AutoML framework instead of building the prediction models with assistance of data scientist.\n\nSOC teams overwhelmed with massive volume of alerts failed to respond to a security incident even they had the alerts and corresponding information in their CTIs\\footnote{\\url{https:\/\/www.trendmicro.com\/explore\/en_gb_soc-research}}. Hence, we assert that the manual and repetitive validation task can be automated through SmartValidator, whereas more critical or unknown alerts and incidents would still require human involvement. In future work, we plan to extend the PoC to provide more explainable output so that SOC can make decisions based on the validated output, where prediction model choice and alternatives would be captured with explanation.\n\n\\subsection{Limitation of SmartValidator}\n\n\nThe experimental results show the effectiveness of SmartValidator. However, we observed several cases in which SmartValidator is unable to perform the validation.\nFurthermore using CTI to validate alerts for unknown known, unknown or zero-day attacks might not always be practical. To empirically investigate this, we ran an experiment separately to explore the ability of SmartValidator to detect unknown alerts. For our experiment, we used the alert data of Snort and selected IP information that is $ob_{attrib}^3$ to predict \\textit{threat level}. To validate such information, we used SmartValidator to build a model trained exclusively with the MISP data ($DS_2$), so that the Snort alert data is almost entirely unseen. The Snort alert data contained 16317 distant IPs, for which only 78 of the IPs were seen in the our MISP training dataset. The prediction model achieved an F1-score of 0.307 which implies SmartValidator has some capability for prediction of unknown alerts, albeit limited. Even though the alerts and IPs are unseen, the prediction model was still able to detect some patterns inferred from the ASN, IP owner and country attributes. We assert that due to the intelligent nature and the ability to learn the underlying semantic patterns, the models built with SmartValidator have the potential to validate some unseen values of unknown attributes (i.e., unknown attributes for which the model is built depending on the observed attributes). If the alert data is entirely unseen, i.e., the IP, ASN, owner and country are all uncontained in the training data, then SmartValidator will predict an output based on the most common value in the training data.\n\nTo validate any unknown attributes (i.e., $un_{attrib}$) with specific values, SmartValidator always needs the observed attributes as input to build a model. Therefore, even if the given value of unknown attributes is not seen in the training data, it is still possible to make a correct prediction by learning the patterns from the model building phase. For example, malicious IPs can have the same domain and threat actors. Here, an IP that has not been seen before can be identified as malicious, observing the threat actors and the domain. However, it is always possible that a trained model can fail to correctly predict IPs, which is a limitation of our proposed approach. We observered there are the cases when the observed attributes are not representative for capturing and learning patterns about the unknown attributes. SmartValidator will not be applicable to automate the validation tasks in these scenarios. Quantifiable investigation of this limitation is out of the scope of this work; but it is an exciting area of exploration for future research.\n\nCTI is time-sensitive. Hence, SOC teams will need to update the model whenever a new CTI is available. The framework can further be extended to capture the timeliness of the CTI used for building the models and keep track of the models with up-to-date CTI. However, all the CTI will not be updated simultaneously; thus, changing all the models whenever there is an update in the CTI will not be feasible.  SmartValidator can be extended to handle this situation by retraining the model when new requests come and retraining and evaluating the available model built based on old CTI.\n\nThere can be bad quality of CTI sources which may affect the performance of SmartValidator. For example, it is possible that the models infer wrong patterns with bad quality data and consider legitimate or benign IPs as malicious. There is a need of empirical studies that focus on ensuring the quality of CTIs. However, this is not within the scope of this study. \n\n\n\n\\subsection{Threats to validity}\n\n\\textit{Construct Validity:} Our choice of data for our evaluation setup may not be suitable. We have considered CTI data to also be representative of alert attributes, as CTI is often generated from existing security alerts from external organizations. However, the classifiers have not yet been tested thoroughly with real-world internal business data. This evaluation will be attempted in future work.\n\n\\textit{Internal Validity:} A potential concern is that our models are not properly optimised. The hyperparameter tuning was performed for a specific set of configurations, as testing hyperparameters with all possible combination would take a large amount of time that may not justify here. Moreover, knowing all the combination of hyperparameters is quite im-possible. Similarly, the features that we have selected to train our models are non-exhaustive. The attribute sets selected were chosen based on the attributes used for validating alerts through security orchestration. The purpose was to show that SmartValidator can automate the construction of prediction models by identifying the CTI and the constructed prediction models can effectively validate alerts based on SOC's requirement. The attributes list might not reflect a complete list of attributes for validating certain alerts, but our system can be easily extended to several other attack scenarios.\n\n\\textit{External Validity:} Our experiments may not generalize to other datasets. The built classifiers and prediction models were evaluated based on simulated alerts attributes sets and publicly available CTIs such as MISP. \n\n\\vspace{-10pt}\n\n\\section{Related Work} \\label{Sec:RelatedWork}\n\nResearch trends are seen in the use of Machine Learning (ML) and Deep Learning (DL) in cybersecurity domain for detection and classification of cyberattacks. Most of the existing literature focused on using AI techniques such as NLP, ML and DL tools and techniques to identify and detect cyber attacks such as malware, network intrusion, data exploitation and vulnerabilities \\cite{AHMED201619, Helge2020,FERRAG2020102419, GAMAGE2020102767, GIBERT2020102526,  sabir2020machine, softVul9108283}. ML algorithms are used to extract knowledge from open source public repositories which are later used to analyze attacks, or validate alerts. Although automation has been achieved in detecting and analysis of attacks, validation of alerts and incidents still requires SOC's involvement \\cite{islam2019multi}.\n\nCTI is used by security experts of SOCs to analyze and validate alerts. To ease the use of CTI, researchers have been trying to come up with a unified structured for sharing CTI \\cite{menges2019unifying, tounsi2018survey}. STIX \\cite{Stixbarnum2012standardizing}, TAXII \\cite{taxiiconnolly2014trusted}, CyBox \\cite{barnum2012cybox} and UCO \\cite{menges2019unifying} are popular among them. Use of Artificial Intelligence (AI) is encouraged for identifying, gathering and extracting CTI objects \\cite{RF2019, qamar2017data, Struve2017}. Various AI tools and techniques are used for knowledge extraction, representation and analytics of CTI \\cite{brazhuk2019semantic, tounsi2018survey}.  For example, Zahedi et al. \\cite{zahedi2018empirical} has applied topic modelling techniques such as LDA to find the security relevant topics from open source repositories such as GitHub. Another example is a system used by EY \\cite{EY2017} who mined previous threat data and then analyze it to give information on threats. Using this information, they can respond to attacks and continuously monitor a system. They place data collectors at points of high movement in a network, like a server, where a system can continuously analyze data and keep the system safe. Attacks detected can be used to harvest IOCs and analyzed to discover security issues within the network. Recorded Future (widely known CTI service providers \\cite{RFID2021}) also elaborated on the fact that threat data can be found in a large variety of places such as Tweets, Facebook posts and emails. They also use AI to recognize patterns in email so that phishing emails can be found based on the information of the sender or file attached.\n\nRecent advances in CTI  domain have drawn attention to the use of the existing knowledge to automate the manual analysis of human experts and enrichment of quality CTIs \\cite{azevedo2019pure,edwards2017panning, noor2019machine, zhou2019ensemble}. For example,  vulnerability description of NVD like databases are being used to predict the severity, confidentiality and availability of threats.  Le et al. \\cite{le2019automated} have used NLP and traditional ML algorithms to perform automated vulnerability assessment using vulnerability description of open source public repositories. Noor et al. \\cite{noor2019machine} have used data provided by STIX and Mitre corporation to identify the documents related to attacks. Azevedo et al. have proposed a platform Pure to improve quality of CTI in the form of enriched IoCs by automatically correlating and aggregating the IoCs \\cite{azevedo2019pure}. They have evaluated the performance of the proposed platform with 34 OSINT feeds gathered from MISP.\n \nOne recent study by Recorded Future has laid out four ways of using AI techniques to extract CTI from a detected attack \\cite{Struve2017}. They have defined risk score metrics to identify malicious network activity. This extends the classification from being just about whether an attack has occurred and provides more in-depth information on the threat \\cite{RFteam2018, Struve2017}. Recorded Future has used NLP to increase the range of possible data sources by removing the limit on just structured information \\cite{Struve2017}. They utilize extracted text and perform classification for the language, topic and company. They have applied ML and NLP techniques to rank documents to identify malware data attacks. Their model also considers different classifications needed like scoring a risk value. They do not always use ML for scoring a risk value as they often have a rule-based system for the classifier to follow. \n\n\nUnlike the above-mentioned work, we propose SmartValidator to utilize NLP and ML techniques to assist in automating validation of security alerts and incident. To the best of our knowledge, this is the first attempt to use CTI, such as MISP data, to automate the classification and validation of security threat data based on a SOC's preferences. Here, we have investigated how effective ML algorithms are while classifying CTI to assist in alert validation.  Unlike the existing works, where possible prediction models are pre-built, here we propose to build the models on demand. We have demonstrated the efficiency of constructing prediction models dynamically. \n\\vspace{-10pt}\n\n\\section{Conclusion} \\label{Sec:Conclu}\n\nMany organizations are facing difficulty to keep pace with the changing threat landscape as security experts need to identify and analyze threat data in most circumstances. Without the indulgence of automation techniques, it is impossible to reduce the burden of analyzing the CTI to make a timely decision. \nIn this work, we propose a novel framework SmartValidator, to build an effective and efficient validation tool using CTI that automates the validation of the security alerts and incidents based on SOC's preferences. Different from the manual approaches, SmartValidator is designed in a way so that SOCs can add their requirements without worrying about collecting CTI and using CTI to build a validation model. SmartValidator consists of three layers: threat data collection, threat data prediction model building and threat data validation. Different teams are responsible for updating the components of different layers, thus freeing security teams from learning data processing and model building techniques. The validation task is designed as a classification problem that leverages existing NLP and ML techniques to extract features from CTI and learn patterns for alert validation. We developed a Proof of Concept (PoC) system to automatically extract features from CTI and build prediction models based on the preferences of SOCs. A SOC's preferences are collected as a set of attributes sets:~observed and unknown attributes, where the task of the PoC is to predict unknown attributes based on observed attributes.\n\nWe have demonstrated the effectiveness of SmartValidator by predicting \\textit{attack}, \\textit{events}, \\textit{threat type}, \\textit{threat level} and \\textit{name}. It collected and processed data from public websites and MISP. Next, CTI with preferred attributes sets were selected to build prediction models. Eight ML algorithms were ran to build and select the models with the highest F1-score. The best model was used to predict the unknown attributes and thus validate alerts. The developed PoC constructed validation models, and can be used to validate alerts generated by the threat detection tools and find the missing information to store the data in a structured format. The results show prediction models are effective in validating security alerts. Building prediction models at run time are more efficient then building prediction models for all possible attributes sets and~CTI. \n\nIn future work, we plan to extend the PoC system to reduce the amount of data that is sent to the SIEM tool, thus reducing the cost of data analysis. The system can also be extended to reduce the organizational dependence on human expertise to take actions against security threats such as blocking ports or identifying maliciousness of an incident. The proposed framework can assist an organization's security team to focus on decision making, rather than manually extracting and validating security alerts and incidents. The framework can also be leveraged to provide the benefit to choose CTI suitable for an organization's application rather than using generalized CTI.\n\\vspace{5pt}\n\n\\textbf{Acknowledgements}\n\nThis work is supported by Cyber Security Cooperative Research Centre (CSCRC), Australia.\n\n\\vspace{-10pt}\n\n\\bibliographystyle{cas-model2-names}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\n\nContinuous first-order logic is a generalization of first-order logic suitable for studying metric structures, which are mathematical structures with an underlying complete metric and with uniformly continuous functions and $\\mathbb{R}$-valued predicates. A rich class of metric structures arise from Banach spaces and expansions thereof. An active area of research in continuous logic is the characterization of inseparably categorical continuous theories.\nFor a general introduction to continuous logic, see \\cite{MTFMS}.\n\n\nIn the present work we will consider expansions of Banach spaces.  We introduce the notion of an indiscernible subspace. An indiscernible subspace is a subspace in which types of tuples of elements only depend on their quantifier-free type in the reduct consisting of only the metric and the constant $\\mathbf{0}$. Similarly to indiscernible sequences, indiscernible subspaces are always consistent with a Banach theory (with no stability assumption, see Theorem \\ref{thm:exist}), but are not always present in every model. We will show that an indiscernible subspace always takes the form of an isometrically embedded real Hilbert space wherein the type of any tuple only depends on its quantifier-free type in the Hilbert space. The notion of an indiscernible subspace is of independent interest in the model theory of Banach and Hilbert structures, and in particular here we use it to improve the results of Shelah and Usvyatsov in the context of types in the full language (as opposed to $\\Delta$-types). Specifically, in this context we give a shorter proof of Shelah and Usvyatsov's main result \\cite[Prop.\\ 4.13]{SHELAH2019106738}, we improve their result on the strong uniqueness of Morley sequences in minimal wide types \\cite[Prop.\\ 4.12]{SHELAH2019106738}, and we expand on their commentary on the ``induced structure'' of the span of a Morley sequence in a minimal wide type \\cite[Rem.\\ 5.6]{SHELAH2019106738}. This more restricted case is what is relevant to inseparably categorical Banach theories, so our work is applicable to the problem of their characterization.\n\nFinally, we present some relevant counterexamples and in particular we resolve (in the negative) the question of Shelah and Usvyatsov presented at the end of Section 5 of \\cite{SHELAH2019106738}, in which they ask whether or not the span of a Morley sequence in a minimal wide type is always a type-definable set.\n\n\\subsection{Background}\n\nFor $K \\in \\{\\mathbb{R}, \\mathbb{C}\\}$, we think of a $K$-Banach space $X$ as being a metric structure $\\mathfrak{X}$ whose underlying set is the closed unit ball $B(X)$ of $X$ with metric $d(x,y) = \\left\\lVert x - y \\right\\rVert$.\\footnote{For another equivalent approach, see \\cite{MTFMS}, which encodes Banach structures as many-sorted metric structures with balls of various radii as different sorts.}  This structure is taken to have for each tuple $\\bar{a} \\in K$  an $|\\bar{a}|$-ary predicate $s_{\\bar{a}}(\\bar{x}) = \\left\\lVert \\sum_{i<|\\bar{a}|} a_i x_i \\right\\rVert$, although we will always write this in the more standard form.\n Note that we evaluate this in $X$ even if $\\sum_{i<|\\bar{a}|} a_i x_i$ is not actually an element of the structure $\\mathfrak{X}$.  For convenience, we will also have a constant for the zero vector, $\\mathbf{0}$, and an $n$-ary function $\\sigma_{\\bar{a}}(\\bar{x})$ such that $\\sigma_{\\bar{a}}(\\bar{x}) = \\sum_{i<|\\bar{a}|} a_i x_i$ if it is in $B(X)$ and  $\\sigma_{\\bar{a}}(\\bar{x}) = \\frac{\\sum_{i<|\\bar{a}|} a_i x_i}{\\left\\lVert \\sum_{i<|\\bar{a}|} a_i x_i \\right\\rVert}$ otherwise. If $|a|\\leq 1$, we will write $ax$ for $\\sigma_{a}(x)$. Note that while this is an uncountable language, it is interdefinable with a countable reduct of it (restricting attention to rational elements of $K$). These structures capture the typical meaning of the ultraproduct of Banach spaces.  As is common, we will conflate $X$ and the metric structure $\\mathfrak{X}$ in which we have encoded $X$.\n\n\n\\begin{defn}\nA \\emph{Banach (or Hilbert) structure} is a metric structure which is the expansion of a Banach (or Hilbert) space. A \\emph{Banach (or Hilbert) theory} is the theory of such a structure. The adjectives \\emph{real} and \\emph{complex} refer to the scalar field $K$.\n\\end{defn}\n\n$C^{\\ast}$- and other Banach algebras are commonly studied examples of Banach structures that are not just Banach spaces.\n\n\n\n\n\n\n\nA central problem in continuous logic is the characterization of inseparably categorical countable theories, that is to say countable theories with a unique model in each uncountable density character. The analog of Morley's theorem was shown in continuous logic via related formalisms \\cite{ben-yaacov_2005, Shelah2011}, but no satisfactory analog of the Baldwin-Lachlan theorem or its precise structural characterization of uncountably categorical discrete theories in terms of strongly minimal sets is known. Some progress in the specific case of Banach theories has been made in \\cite{SHELAH2019106738}, in which Shelah and Usvyatsov introduce the notion of a wide type and the notion of a minimal wide type, which they argue is the correct analog of strongly minimal types in the context of inseparably categorical Banach theories.\n\n\\begin{defn}\nA type $p$ in a Banach theory is \\emph{wide} if its set of realizations consistently contain the unit sphere of an infinite dimensional real subspace.\n\nA type is \\emph{minimal wide} if it is wide and has a unique wide extension to every set of parameters.\n\\end{defn}\nIn \\cite{SHELAH2019106738}, Shelah and Usvyatsov were able to show that every Banach theory has wide complete types using the following classical concentration of measure results of Dvoretzky and Milman, which Shelah and Usvyatsov refer to as the Dvoretzky-Milman theorem.\n\n\\begin{fact}[Dvoretzky-Milman theorem] \\label{fact:DM-thm}\nLet $(X,\\left\\lVert \\cdot \\right\\rVert)$ be an infinite dimensional real Banach space with unit sphere $S$ and let $f:S \\rightarrow \\mathbb{R}$ be a uniformly continuous function. For any $k<\\omega$ and $\\varepsilon > 0$, there exists a $k$-dimensional subspace $Y \\subset X$ and a Euclidean norm\\footnote{A norm $\\vertiii{\\cdot}$  is \\emph{Euclidean} if it satisfies the parallelogram law, $2\\vertiii{a}^2 + 2 \\vertiii{b}^2 = \\vertiii{a+b}^2 + \\vertiii{a-b}^2,$ or equivalently if it is induced by an inner product.} $\\vertiii{\\cdot}$ on $Y$ such that for any $a,b \\in S\\cap Y$, we have $\\vertiii{a} \\leq \\left\\lVert a\\right\\rVert \\leq (1 + \\varepsilon)\\vertiii{a}$ and $|f(a) - f(b)| < \\varepsilon$.\\footnote{Fact \\ref{fact:DM-thm} without $f$ is (a form of) Dvoretsky's theorem.}\n\\end{fact}\n\n\nShelah and Usvyatsov showed that in a stable Banach theory every wide type has a minimal wide extension (possibly over a larger set of parameters) and that every Morley sequence in a minimal wide type is an orthonormal basis of a subspace isometric to a real Hilbert space. Furthermore, they showed that in an inseparably categorical Banach theory, every inseparable model is prime over a countable set of parameters and a Morley sequence in some minimal wide type, analogously to how a model of a discrete uncountably categorical theory is always prime over some finite set of parameters and a Morley sequence in some strongly minimal type.\n\nThe key ingredient to our present work is the following result, due to Milman.\n It extends the Dvoretzky-Milman theorem in a manner analogous to the extension of the pigeonhole principle by Ramsey's theorem.\\footnote{The original Dvoretzky-Milman result is often compared to Ramsey's theorem, such as when Gromov coined the term \\emph{the Ramsey-Dvoretzky-Milman phenomenon} \\cite{gromov1983}, but in the context of Fact \\ref{fact:main} it is hard not to think of the $n=1$ case as being analogous to the pigeonhole principle and the $n>1$ cases as being analogous to Ramsey's theorem.}\n\n\\begin{defn}\\label{defn:main-defn}\nLet $(X,\\left\\lVert \\cdot \\right\\rVert)$ be a Banach space. If $a_0,a_1,\\dots,a_{n-1}$ and $b_0,b_1,\\dots,\\allowbreak b_{n-1}$ are ordered $n$-tuples of elements of $X$, we say that $\\bar{a}$ and $\\bar{b}$ are \\emph{congruent} if $\\left\\lVert a_i - a_j\\right\\rVert=\\left\\lVert b_i - b_j \\right\\rVert$ for all $ i,j \\leq n$, where we take $a_{n}=b_{n}=\\mathbf{0}$. We will write this as $\\bar{a} \\cong \\bar{b}$.\n\\end{defn}\n\n\\begin{fact}[\\cite{zbMATH03376472}, Thm.\\ 3] \\label{fact:main}\nLet $S^\\infty$ be the unit sphere of a separable infinite dimensional real Hilbert space ${H}$ and let $f:(S^\\infty)^n \\rightarrow \\mathbb{R}$ be a uniformly continuous function. For any $\\varepsilon>0$ and any $k<\\omega$ there exists a $k$-dimensional subspace $V$ of $H$ such that for any $a_0,a_1,\\dots,a_{n-1},b_0,b_1,\\dots,b_{n-1}\\in S^\\infty$ with $\\bar{a} \\cong \\bar{b}$, $|f(\\bar{a})-f(\\bar{b})| < \\varepsilon$.\n\\end{fact}\n\nNote that the analogous result for inseparable Hilbert spaces follows immediately, by restricting attention to a separable infinite dimensional subspace. Also note that by using Dvoretsky's theorem and an easy compactness argument, Fact \\ref{fact:main} can be generalized to arbitrary infinite dimensional Banach spaces.\n\n\\subsection{Connection to Extreme Amenability}\n\n A modern proof of Fact \\ref{fact:main} would go through the extreme amenability of the unitary group of an infinite dimensional Hilbert space endowed with the strong operator topology, or in other words the fact that any continuous action of this group on a compact Hausdorff space has a fixed point, which was originally shown in \\cite{10.2307\/2374298}.  This connection is unsurprising. It is well known that the extreme amenability of $\\mathrm{Aut}(\\mathbb{Q})$ (endowed with the topology of pointwise convergence) can be understood as a restatement of Ramsey's theorem. It is possible to use this to give a high brow proof of the existence of indiscernible sequences in any first-order theory $T$:\n \n \\begin{proof}\nFix a first-order theory $T$. Let $Q$ be a family of variables indexed by the rational numbers. The natural action of $\\mathrm{Aut}(\\mathbb{Q})$ on $S_Q(T)$, the Stone space of types over $T$ in the variables $Q$, is continuous and so by extreme amenability has a fixed point. A fixed point of this action is precisely the same thing as the type of a $\\mathbb{Q}$-indexed indiscernible sequence over $T$, and so we get that there are models of $T$ with indiscernible sequences.\n \\end{proof}\n \n \n A similar proof of the existence of indiscernible subspaces in Banach theories (Theorem \\ref{thm:exist}) is possible, but requires an argument that the analog of $S_Q(T)$ is non-empty (which follows from Dvoretzky's theorem) and also requires more delicate bookkeeping to define the analog of $S_Q(T)$ and to show that the action of the unitary group of a separable Hilbert space is continuous. In the end this is more technical than a proof using Fact \\ref{fact:main} directly.\n\n\n\n\\section{Indiscernible Subspaces} \\label{sec:ind-subsp}\n\n\n\n\\begin{defn}\n Let $T$ be a Banach {theory}. Let $\\mathfrak{M}\\models T$ and let $A\\subseteq \\mathfrak{M}$ be some set of parameters.\n An \\emph{indiscernible subspace over $A$} is a real subspace $V$ of $\\mathfrak{M}$ such that for any $n<\\omega$ and any $n$-tuples $\\bar{b},\\bar{c} \\in V$, $\\bar{b} \\equiv_A \\bar{c}$ if and only if $\\bar{b} \\cong \\bar{c}$.\n\n{\\sloppy If $p$ is a type over $A$, then $V$ is an \\emph{indiscernible subspace in $p$ (over $A$)} if it is an indiscernible subspace over $A$ and $b\\models p$ for all $b\\in V$ with $\\left\\lVert b \\right\\rVert = 1$.}\n\n\n\\end{defn}\n\nNote that an indiscernible subspace is a real subspace even if $T$ is a complex Banach theory. Also note that an indiscernible subspace in $p$ is not literally contained in the realizations of $p$, but rather has its unit sphere contained in the realizations of $p$. It might be more accurate to talk about ``indiscernible spheres,'' but we find the subspace terminology more familiar.\n\nIndiscernible subspaces are very metrically regular.\n\n\\begin{prop}\nSuppose $V$ is an indiscernible subspace in some Banach structure. Then $V$ is isometric to a real Hilbert space.\n\nIn particular, a real subspace $V$ of a Banach structure is indiscernible over $A$ if and only if it is isometric to a real Hilbert space and for every $n<\\omega$ and every pair of $n$-tuples $\\bar{b},\\bar{c}\\in V$, $\\bar{b}\\equiv_A\\bar{c}$ if and only if for all $i,j<n$ $\\left<b_i,b_j\\right> = \\left<c_i,c_j\\right>$.\n\\end{prop}\n\\begin{proof}\nFor any real Banach space $W$, if $\\dim W \\leq 1$, then $W$ is necessarily isometric to a real Hilbert space. If $\\dim V \\geq 2$, let $V_0$ be a $2$-dimensional subspace of $V$. A subspace of an indiscernible subspace is automatically an indiscernible subspace, so $V_0$ is indiscernible. For any two distinct unit vectors $a$ and $b$, indiscernibility implies that for any $r,s\\in \\mathbb{R}$, $\\left\\lVert r a + s b\\right\\rVert = \\left\\lVert s a + r b\\right\\rVert$, hence the unique linear map that switches $a$ and $b$ fixes $\\left\\lVert \\cdot \\right \\rVert$. This implies that the automorphism group of $(V_0, \\left\\lVert \\cdot \\right \\rVert)$ is transitive on the $\\left\\lVert \\cdot \\right\\rVert$-unit circle. By John's theorem on maximal ellipsoids \\cite{MR0030135}, the unit ball of $\\left\\lVert \\cdot \\right \\rVert$ must be an ellipse, so $\\left\\lVert \\cdot \\right \\rVert$ is a Euclidean norm.\n\nThus every $2$-dimensional real subspace of $V$ is Euclidean and so $(V,\\left\\lVert \\cdot \\right \\rVert)$ satisfies the parallelogram law and is therefore a real Hilbert space.\n\nThe `in particular' statement follows from the fact that in a real Hilbert subspace of a Banach space, the polarization identity \\cite[Prop.\\ 14.1.2]{Blanchard2002} defines the inner product  in terms of a particular quantifier-free formula:\n\\begin{equation*}\n\\left<x, y\\right> = \\frac{1}{4}\\left( \\left\\lVert x + y \\right\\rVert ^2 - \\left\\lVert x - y \\right\\rVert^2 \\right).\\footnotemark \\qedhere\n\\end{equation*}\n\\end{proof}\n\\footnotetext{There is also a polarization identity for the complex inner product: $${\\left<x, y\\right>_{\\mathbb{C}} = \\frac{1}{4}\\left( \\left\\lVert x + y \\right\\rVert ^2 - \\left\\lVert x - y \\right\\rVert^2  + i\\left\\lVert x - iy \\right\\rVert^2 - i \\left\\lVert x + iy \\right\\rVert^2  \\right).}$$}\n\n \n\n\n\n\n\n\n\n\n\\subsection{Existence of Indiscernible Subspaces}\n\n\n\n\nAs mentioned in \\cite[Cor.\\ 3.9]{SHELAH2019106738}, it follows from Dvoretzky's theorem that if $p$ is a wide type and $\\mathfrak{M}$ is a sufficiently saturated model, then $p(\\mathfrak{M})$ contains the unit sphere of an infinite dimensional subspace isometric to a Hilbert space. We refine this by showing that, in fact, an indiscernible subspace can be found. \n\n\\begin{thm} \\label{thm:exist}\nLet $A$ be a set of parameters in a Banach {theory} $T$ and let $p$ be a wide type over $A$. For any $\\kappa$, there is $\\mathfrak{M} \\models T$ and a subspace $V\\subseteq \\mathfrak{M}$ of dimension $\\kappa$ such that $V$ is an indiscernible subspace in $p$ over $A$. In particular, any $\\aleph_0 + \\kappa+|A|$-saturated $\\mathfrak{M}$ will have such a subspace.\n\\end{thm}\n\\begin{proof}\nFor any set $\\Delta$ of $A$-formulas, call a subspace $V$ of a model $\\mathfrak{N}$ of $T_A$ \\emph{$\\Delta$-indiscernible in $p$} if every unit vector in $V$ models $p$ and for any $n<\\omega$ and any formula $\\varphi \\in \\Delta$ of arity $n$ and any $n$-tuples $\\bar{b},\\bar{c} \\in V$ with $\\bar{b} \\cong \\bar{c}$, we have $\\mathfrak{N}\\models \\varphi(\\bar{b}) = \\varphi(\\bar{c})$.\n\nSince $p$ is wide, there is a model $\\mathfrak{N}\\models T$ containing an infinite dimensional subspace $W$ isometric to a real Hilbert space such that for all $b\\in W$ with $\\left\\lVert b \\right\\rVert = 1$, $b\\models p$. This is an infinite dimensional $\\varnothing$-indiscernible subspace in $p$.\n\nNow for any finite set of $A$-formulas $\\Delta$ and formula $\\varphi$, assume that we've shown that there is a model $\\mathfrak{N}\\models T$ containing an infinite dimensional $\\Delta$-indiscernible subspace $V$ in $p$ over $A$. We want to show that there is a $\\Delta \\cup \\{\\phi\\}$-indiscernible subspace in $V$. By Fact \\ref{fact:main}, for every $k<\\omega$ there is a $k$-dimensional subspace $W_{k}\\subseteq V$ such that for any unit vectors $b_0,\\dots,b_{\\ell -1},c_0,\\dots,c_{\\ell-1}$ in $W_{k}$ with $\\bar{b}\\cong\\bar{c}$, we have that $|\\varphi^{\\mathfrak{N}}(\\bar{b})-\\varphi^{\\mathfrak{N}}(\\bar{c})| < 2^{-k}$. If we let $\\mathfrak{N}_k = (\\mathfrak{N}_k,W_k)$ where we've expanded the language by a fresh predicate symbol $D$ such that $D^{\\mathfrak{N}_k}(x)=d(x,W_k)$, then an ultraproduct of the sequence $\\mathfrak{N}_k$ will be a structure $(\\mathfrak{N}_\\omega,W_\\omega)$ in which $W_\\omega$ is an infinite dimensional Hilbert space.\n\n\\emph{Claim:} $W_\\omega$ is $\\Delta\\cup\\{\\varphi\\}$-indiscernible in $p$.\n\n\\emph{Proof of claim.} Fix an $m$-ary formula $\\psi \\in \\Delta \\cup \\{\\varphi\\}$ and let $f(k)=0$ if $\\psi \\in \\Delta$ and $f(k)=2^{-k}$ if $\\psi = \\varphi$. For any $k \\geq 2m$, fix $b_0,\\dots,b_{m-1},c_0,\\dots,c_{m-1}$ in the unit ball of $W_k$, there is a $2m$ dimensional subspace $W^\\prime \\subseteq W_k$ containing $\\bar{b},\\bar{c}$. By compactness of $B(W^\\prime)^m$ (where $B(X)$ is the unit ball of $X$), we have that for any $\\varepsilon > 0$ there is a $\\delta(\\varepsilon) > 0$ such that if $|\\left<b_i,b_j \\right> - \\left<c_i,c_j \\right>| < \\delta(\\varepsilon)$ for all $i,j < m$ then $|\\psi^{\\mathfrak{N}}(\\bar{b})-\\psi^{\\mathfrak{N}}(\\bar{c})| \\leq f(k) + \\varepsilon$. Note that we can take the function $\\delta$ to only depend on $\\psi$, specifically its arity and modulus of continuity, and not on $k$, since $B(W^\\prime)^m$ is always isometric to $B(\\mathbb{R}^{2m})^m$. Therefore, in the ultraproduct we will have $(\\forall i,j<m)|\\left<b_i,b_j \\right> - \\left<c_i,c_j \\right>| < \\delta(\\varepsilon) \\Rightarrow |\\psi^{\\mathfrak{N}}(\\bar{b})-\\psi^{\\mathfrak{N}}(\\bar{c})| \\leq \\varepsilon$ and thus  $\\bar{b}\\cong \\bar{c} \\Rightarrow \\psi^{\\mathfrak{N}_\\omega}(\\bar{b}) = \\psi^{\\mathfrak{N}_\\omega}(\\bar{c})$, as required. \\hfill $\\qed_{\\textit{Claim}}$\n\nNow for each finite set of $A$-formulas we've shown that there's a structure $(\\mathfrak{M}_\\Delta,V_\\Delta)$ (where, again, $V_\\Delta$ is the set defined by the new predicate symbol $D$) such that $\\mathfrak{M}_\\Delta \\models T_A$ and $V_\\Delta$ is an infinite dimensional $\\Delta$-indiscernible subspace in $p$. By taking an ultraproduct with an appropriate ultrafilter we get a structure $(\\mathfrak{M},V)$ where $\\mathfrak{M}\\models T_A$ and $V$ is an infinite dimensional subspace. $V$ is an indiscernible subspace in $p$ over $A$ by the same argument as in the claim.\n\nFinally note that by compactness we can take $V$ to have arbitrarily large dimension and that any subspace of an indiscernible subspace in $p$ over $A$ is an indiscernible subspace in $p$ over $A$, so we get the required result.\n\\end{proof}\n\n\n\n\n\n\nTogether with the fact that wide types always exist in Banach theories  with infinite dimensional models \\cite[Thm.\\ 3.7]{SHELAH2019106738}, we get a corollary.\n\n\\begin{cor} \\label{cor:ind-subsp}\nEvery Banach {theory} with infinite dimensional models has an infinite dimensional indiscernible subspace in some model. In particular, every such theory has an infinite indiscernible set, namely any orthonormal basis of an infinite dimensional indiscernible subspace.\n\\end{cor}\n\n\n\\section{Minimal{ }Wide Types}\n\n\n\nCompare the following Theorem \\ref{thm:main} with this fact in discrete logic: If $p$ is a minimal type (i.e.\\ $p$ has a unique global non-algebraic extension), then an infinite sequence of realizations of $p$ is a Morley sequence in $p$ if and only if it is an indiscernible sequence.\n\nHere we are using the definition of Morley sequence for (possibly unstable) $A$-invariant types: Let $p$ be a global $A$-invariant type, and let $B\\supseteq A$ be some set of parameters. A sequence $\\{c_i\\}_{i< \\kappa}$ is a \\emph{Morley sequence in $p$ over $B$} if for all $i< \\kappa$, $\\mathrm{tp}(c_i\/Bc_{<i}) = p \\upharpoonright Bc_{<i}$. Note that this definition of Morley sequence agrees with the standard definition for types that are stable in the sense of Lascar and Poizat (as described in \\cite[Def.\\ 4.1]{SHELAH2019106738}).\n\n\\begin{thm} \\label{thm:main}\nLet $p$ be a minimal{ }wide type over the set $A$. For $\\kappa\\geq \\aleph_0$, a set of realizations $\\{b_i\\}_{i<\\kappa}$ of $p$ is a Morley sequence in (the unique global minimal wide extension of) $p$ if and only if it is an orthonormal basis of an indiscernible subspace in $p$ over $A$. \n\\end{thm}\n\\begin{proof}\nAll we need to show is that an orthonormal basis of an indiscernible subspace in $p$ over $A$ is a Morley sequence in $p$. The converse will follow from the fact that all Morley sequences in a fixed invariant type of the same length have the same type along with the fact that minimal wide types have a unique global wide extension, which is therefore invariant.\n\nLet $V$ be an indiscernible subspace in $p$ over $A$. Let $\\{e_i\\}_{i<\\kappa}$ be an orthonormal basis of $V$. By construction, $\\mathrm{tp}(e_0\/A) = p$. Let $q$ be the global minimal wide extension of $p$. Assume that for some $j<\\kappa$ we've shown for all $i<j$ that $\\mathrm{tp}(e_i\/Ae_{<i}) = q \\upharpoonright Ae_{<i}$. Let $W = \\overline{\\mathrm{span}}(e_{\\geq j})$. Since $V$ is an indiscernible subspace over $A$, for all unit norm $b,c\\in W$, $b\\equiv_{Ae_{<j}} c$, so in particular $\\mathrm{tp}(b\/Ae_{<j})$ is wide. Since $p$ is minimal{ }wide we must have $\\mathrm{tp}(b\/Ae_{<j}) = q\\upharpoonright Ae_{<j}$. Therefore $\\{e_i\\}_{i<\\kappa}$ is a Morley sequence.\n\\end{proof}\n\nWhat is unclear at the moment is the answer to this question: \n\n\\begin{quest}\nIf $p$ is a minimal wide type over the set $A$, is it stable in the sense of \\cite[Def.\\ 4.1]{SHELAH2019106738}? In other words, is every type $q$ extending $p$ over a model $\\mathfrak{M}\\supseteq A$  a definable type?\n\\end{quest}\n\n\\section{Counterexamples} \\label{sec:count}\nHere we collect some counterexamples that may be relevant to any model theoretic development of the ideas presented in this paper.\n\n\\subsection{No Infinitary Ramsey-Dvoretzky-Milman Phenomena in General}\n\nUnfortunately some elements of the analogy between the  Ramsey-Dvoretzky-Milman Phenomenon and discrete Ramsey theory do not work. In particular, there is no extension of Dvoretzky's theorem, and therefore Fact \\ref{fact:DM-thm}, to $k \\geq \\omega$, even for a fixed $\\varepsilon>0$. \nRecall that a linear map $T:X\\rightarrow Y$ between Banach spaces is an \\emph{isomorphism} if it is a continuous bijection. This is enough to imply that $T$ is invertible and that both $T$ and $T^{-1}$ are Lipschitz. An analog of Dvoretzky's theorem for $k \\geq \\omega$ would imply that every sufficiently large Banach space has an infinite dimensional subspace isomorphic to Hilbert space, which is known to be false.  Here we will see as specific example of this.\n\n The following is a well known result in Banach space {theory} (for a proof see the comment after Proposition 2.a.2 in \\cite{Lindenstrauss1996}).\n\n\\begin{fact} \\label{fact:no-no}\nFor any distinct $X,Y \\in \\{\\ell_p: 1\\leq p < \\infty\\} \\cup \\{c_0\\}$, no subspace of $X$ is isomorphic to $Y$.\n\\end{fact}\n\nNote that, whereas Corollary \\ref{cor:ind-subsp} says that every Banach theory is consistent with the partial type of an indiscernible subspace, the following corollary says that this type can sometimes be omitted in arbitrarily large models (contrast this with the fact that the existence of an Erd\\\"os cardinal implies that you can find indiscernible sequences in any sufficiently large structure in a countable language \\cite[Thm.\\ 9.3]{Kanamori2003}).\n\n\\begin{cor} \\label{cor:no-no-cor}\nFor $p \\in [1,\\infty) \\setminus \\{2\\}$, there are arbitrarily large models of $\\mathrm{Th}(\\ell_p)$ that do not contain any infinite dimensional subspaces isomorphic to a Hilbert space. \n\\end{cor}\n\\begin{proof}\nFix $p \\in [1,\\infty) \\setminus \\{2\\}$ and $\\kappa \\geq \\aleph_0$.\n  Let $\\ell_p(\\kappa)$ be the Banach space of functions $f:\\kappa \\rightarrow \\mathbb{R}$ such that $\\sum_{i<\\kappa} |f(i)|^p < \\infty$. Note that $\\ell_p(\\kappa) \\equiv \\ell_p$.\\footnote{To see this, we can find an elementary sub-structure of $\\ell_p(\\kappa)$ that is isomorphic to $\\ell_p$: Let $\\mathfrak{L}_0$ be a separable elementary sub-structure of $\\ell_p(\\kappa)$. For each $i<\\omega$, given $\\mathfrak{L}_i$, let $B_i$ be the set of all $f \\in \\ell_p(\\kappa)$ that are the indicator function of a singleton $\\{i\\}$ for some $i$ in the support of some element of $\\mathfrak{L}_i$. $B_i$ is countable. Let $\\mathfrak{L}_{i+1}$ be a separable elementary sub-structure of $\\ell_p(\\kappa)$ containing $\\mathfrak{L}_i\\cup B_i$. $\\overline{\\bigcup_{i<\\omega}\\mathfrak{L}_{i+1}}$ is equal to the span of $\\bigcup_{i<\\omega} B_i$ and so is a separable elementary sub-structure of $\\ell_p(\\kappa)$ isomorphic to $\\ell_p$.}\n Pick a subspace $V \\subseteq \\ell_p(\\kappa)$. If $V$ is isomorphic to a Hilbert space, then any separable $V_0 \\subseteq V$ will also be isomorphic to a Hilbert space. There exists a countable set $A \\subseteq \\kappa$ such that $V_0 \\subseteq \\ell_p(A) \\subseteq \\ell_p(\\kappa)$. By Fact \\ref{fact:no-no}, $V_0$ is not isomorphic to a Hilbert space, which is a contradiction. Thus no such $V$ can exist.\n\\end{proof}\n\n\n\nEven assuming we start with a Hilbert space we do not get an analog of the infinitary pigeonhole principle (i.e.\\ a generalization of Fact \\ref{fact:DM-thm}). The discussion by H\\'ajeck and Mat\\v ej in \\cite[after Thm.\\ 1]{Hajek2018} of a result of Maurey \\cite{Maurey1995} implies that there is a Hilbert theory $T$ with a unary predicate $P$ such that for some $\\varepsilon>0$ there are arbitrarily large models $\\mathfrak{M}$ of $T$ such that for any infinite dimensional subspace $V \\subseteq \\mathfrak{M}$ there are unit vectors $a,b\\in V$ with $|P^{\\mathfrak{M}}(a)-P^{\\mathfrak{M}}(b)| \\geq \\varepsilon$.\n\nStability of a theory often has the effect of making Ramsey phenomena more prevalent in its models, so there is a natural question as to whether anything similar will happen here. Recall that a function $f:S(X)\\rightarrow \\mathbb{R}$ on the unit sphere $S(X)$ of a Banach space $X$ is \\emph{oscillation stable} if for every infinite dimensional subspace $Y \\subseteq X$ and every $\\varepsilon>0$ there is an infinite dimensional subspace $Z \\subseteq Y$ such that for any $a,b\\in S(Z)$, $|f(a)-f(b)|\\leq \\varepsilon$.\n\n\n\\begin{quest}\nDoes (model theoretic) stability imply oscillation stability? That is to say, if $T$ is a stable Banach theory, is every unary formula oscillation stable on models of $T$?\n\\end{quest}\n\n\\subsection{The (Type-)Definability of Indiscernible Subspaces and Complex Banach Structures} \\label{subsec:comp}\n\nA central question in the study of inseparably categorical Banach space theories is the degree of definability of the `minimal Hilbert space' that controls a given inseparable model of the theory. Results of Henson and Raynaud in \\cite{HensonRaynaud} imply that in general the Hilbert space may not be definable.  In \\cite{SHELAH2019106738}, Shelah and Usvyatsov ask whether or not the Hilbert space can be taken to be type-definable or a zeroset. In Example \\ref{ex:no-def} we present a simple, but hopefully clarifying, example showing that this is slightly too much to ask.\n\nIt is somewhat uncomfortable that even in complex Hilbert structures we are only thinking about \\emph{real} indiscernible subspaces rather than \\emph{complex} indiscernible subspaces. One problem is that Ramsey-Dvoretzky-Milman phenomena only deal with real subspaces in general. The other problem is that Definition \\ref{defn:main-defn} is incompatible with complex structure:\n\n\\begin{prop} \\label{prop:no-comp}\nLet $T$ be a complex Banach theory. Let $V$ be an indiscernible subspace in some model of $T$. For any non-zero $a\\in V$ and $\\lambda \\in \\mathbb{C} \\setminus \\{0\\}$, if $\\lambda a \\in V$, then $\\lambda \\in \\mathbb{R}$.\n\\end{prop}\n\\begin{proof}\nAssume that for some non-zero vector $a$, both $a$ and $ia$ are in $V$. We have that $(a,ia)\\equiv(ia,a)$, but $(a,ia)\\models d(ix,y)=0$ and $(ia,a)\\not\\models d(ix,y)=0$, which contradicts indiscernibility. Therefore we cannot have that both $a$ and $ia$ are in $V$. The same statement for $a$ and $\\lambda a$ with $\\lambda \\in \\mathbb{C}\\setminus \\mathbb{R}$ follows immediately, since $a,\\lambda a \\in V \\Rightarrow ia \\in V$. \n\\end{proof}\n\nIn the case of complex Hilbert space and other Hilbert spaces with a unitary Lie group action, this is the reason that indiscernible subspaces can fail to be type-definable. We will explicitly give the simplest example of this\n\n\n\n\n\\begin{ex} \\label{ex:no-def}\nLet $T$ be the theory of an infinite dimensional complex Hilbert space and let $\\mathfrak{C}$ be the monster model of $T$. $T$ is inseparably categorical, but for any partial type $\\Sigma$ over any small set of parameters $A$, $\\Sigma(\\mathfrak{C})$ is not an infinite dimensional indiscernible subspace (over $\\varnothing$).\n\n\\end{ex}\n\\begin{proof}\n$T$ is clearly inseparably categorical by the same reasoning that the theory of real infinite dimensional Hilbert spaces is inseparably categorical (being an infinite dimensional complex Hilbert space is first-order and there is a unique infinite dimensional complex Hilbert space of each infinite density character). \n\nIf $\\Sigma(\\mathfrak{C})$ is not an infinite dimensional subspace of $\\mathfrak{C}$, then we are done, so assume that $\\Sigma(\\mathfrak{C})$ is an infinite dimensional subspace of $\\mathfrak{C}$. Let $\\mathfrak{N}$ be a small model containing $A$. Since $\\mathfrak{N}$ is a subspace of $\\mathfrak{C}$, $\\Sigma(\\mathfrak{N}) = \\Sigma(\\mathfrak{C})\\cap \\mathfrak{N}$ is a subspace of $\\mathfrak{N}$. Let $v \\in \\Sigma(\\mathfrak{C})\\setminus \\Sigma(\\mathfrak{N})$. This implies that $v\\in \\mathfrak{C} \\setminus \\mathfrak{N}$, so we can write $v$ as $v_\\parallel+ v_\\perp$, where $v_\\parallel$ is the orthogonal projection of $v$ onto $\\mathfrak{N}$ and $v_\\perp$ is complex orthogonal to $\\mathfrak{N}$. Necessarily we have that $v_\\perp \\neq 0$. Let $\\mathfrak{N}^\\perp$ be the orthocomplement of $\\mathfrak{N}$ in $\\mathfrak{C}$. If we write elements of $\\mathfrak{C}$ as $(x,y)$ with $x\\in \\mathfrak{N}$ and $y\\in \\mathfrak{N}^\\perp$, then the maps  $(x,y)\\mapsto (x,-y)$, $(x,y)\\mapsto (x,iy)$, and $(x,y)\\mapsto(x,-iy)$ are automorphisms of  $\\mathfrak{C}$ fixing $\\mathfrak{N}$. Therefore $(v_\\parallel + v_\\perp) \\equiv_{\\mathfrak{N}} (v_\\parallel - v_\\perp) \\equiv_{\\mathfrak{N}} (v_\\parallel + iv_\\perp) \\equiv_{\\mathfrak{N}} (v_\\parallel -iv_\\perp)$, so we must have that $(v_\\parallel - v_\\perp),(v_\\parallel + iv_\\perp),( v_\\parallel - iv_\\perp) \\in \\Sigma(\\mathfrak{C})$ as well. Since $\\Sigma(\\mathfrak{C})$ is a subspace, we have that $b_\\perp \\in \\Sigma(\\mathfrak{C})$ and $ib_\\perp \\in \\Sigma(\\mathfrak{C})$. Thus by Proposition \\ref{prop:no-comp} $\\Sigma(\\mathfrak{C})$ is not an indiscernible subspace over $\\varnothing$.\n\\end{proof}\n\nThis example is a special case of this more general construction: If $G$ is a compact Lie group with an irreducible  unitary representation on $\\mathbb{R}^n$ for some $n$ (i.e.\\ the group action is transitive on the unit sphere), then we can extend this action to $\\ell_2$ by taking the Hilbert space direct sum of countably many copies of the irreducible unitary representation of $G$, and we can think of this as a structure by adding function symbols for the elements of $G$.\nThe theory of this structure will be totally categorical and satisfy the conclusion of Example \\ref{ex:no-def}.\n\nExample \\ref{ex:no-def} is analogous to the fact that in many strongly minimal theories the set of generic elements in a model is not itself a basis\/Morley sequence. The immediate response would be to ask the question of whether or not the unit sphere of the complex linear span (or more generally the `$G$-linear span,' i.e.\\ the linear span of $G\\cdot V$) of the indiscernible subspace in a minimal{ }wide type agrees with the set of realizations of that minimal{ }wide type, but this can overshoot:\n\n\\begin{ex} \\label{ex:bad-comp}\nConsider the structure whose universe is (the unit ball of) $\\ell_2 \\oplus \\ell_2$ (where we are taking $\\ell_2$ as a real Hilbert space), with a complex action $(x,y)\\mapsto (-y,x)$ and orthogonal projections $P_0$ and $P_1$ for the sets $\\ell_2 \\oplus \\{\\mathbf{0}\\}$ and $\\{\\mathbf{0}\\} \\oplus \\ell_2$, respectively. Let $T$ be the theory of this structure. This is a totally categorical complex Hilbert structure, but for any complete type $p$ and $\\mathfrak{M}\\models T $, $p(\\mathfrak{M})$ does not contain the unit sphere of a non-trivial complex subspace.\n\\end{ex}\n\\begin{proof}\n$T$ is bi-interpretable with a real Hilbert space, so it is totally categorical. For any complete type $p$, there are unique values of $\\left\\lVert P_0(x) \\right\\rVert$ and $\\left\\lVert P_1(x) \\right\\rVert$ that are consistent with $p$, so the set of realizations of $p$ in any model cannot contain $\\{\\lambda a\\}_{\\lambda \\in \\mathrm{U}(1)}$ for $a$, a unit vector, and $\\mathrm{U}(1) \\subset \\mathbb{C}$, the set of unit complex numbers. \n\\end{proof}\n\nThe issue, of course, being that, while we declared by fiat that this is a complex Hilbert structure, the expanded structure does not respect the complex structure.\n\nSo, on the one hand, Example \\ref{ex:bad-comp} shows that in general the unit sphere of the complex span won't be contained in the minimal{ }wide type. On the other hand, a priori the set of realizations of the minimal{ }wide type could contain more than just the unit sphere of the complex span, such as if we have an $\\mathrm{SU}(n)$ action. The complex (or $G$-linear) span of a set is of course part of the algebraic closure of the set in question, so this suggests a small refinement of the original question of Shelah and Usvyatsov:\n\n\\begin{quest}\nIf $T$ is an inseparably categorical Banach {theory}, $p$ is a  minimal{ }wide type, and $\\mathfrak{M}$ is a model of $T$ which is prime over an indiscernible subspace $V$ in $p$, does it follow that $p(\\mathfrak{M})$ is the unit sphere of a subspace contained in the algebraic closure of $V$?\n\\end{quest}\n\nThis would be analogous to the statement that if $p$ is a strongly minimal type in an uncountably categorical discrete theory and $\\mathfrak{M}$ is a model prime over a Morley sequence $I$ in $p$, then $p(\\mathfrak{M})\\subseteq \\mathrm{acl}(I)$.\n\n\n\n\n\n\n\n\n\n\\subsection{Non-Minimal Wide Types}\n\nThe following example shows, unsurprisingly, that Theorem \\ref{thm:main} does not hold for non-minimal{ }wide types.\n\n\\begin{ex}\nLet $T$ be the theory of (the unit ball of) the infinite Hilbert space sum $\\ell_2 \\oplus \\ell_2 \\oplus \\dots$, where we add a predicate $D$ that is the distance to $S^\\infty \\sqcup S^\\infty \\sqcup \\dots$, where $S^\\infty$ is the unit sphere of the corresponding copy of $\\ell_2$. This theory is $\\omega$-stable. The partial type $\\{D = 0\\}$ has a unique global non-forking extension $p$ that is wide, but the unit sphere of the linear span of any Morley sequence in $p$ is not contained in $p(\\mathfrak{C})$.\n\\end{ex}\n\\begin{proof}\nThis follows from the fact that on $D$ the equivalence relation `$x$ and $y$ are contained in a common unit sphere' is definable by a formula, namely \\[E(x,y) = \\inf_{z,w \\in D}(d(x,z)\\dotdiv 1) + (d(z,w)\\dotdiv 1) + (d(w,y)\\dotdiv 1),\\]\nwhere $a \\dotdiv b = \\max\\{a-b,0\\}$. If $x,y$ are in the same sphere, then let $S$ be a great circle passing through $x$ and $y$ and choose $z$ and $w$ evenly spaced along the shorter path of $S$. It will always hold that $d(x,z),d(z,w),d(w,y) \\leq 1$, so we will have $E(x,y)=0$. On the other hand, if $x$ and $y$ are in different spheres, then $E(x,y)= \\sqrt{2} -1$.\n\nTherefore a Morley sequence in $p$ is just any sequence of elements of $D$ which are pairwise non-$E$-equivalent and the unit sphere of the span of any such set is clearly not contained in $D$.\n\\end{proof}\n\n\n\n\\bibliographystyle{abbrv}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nRecurrent neural networks (RNNs) and their variants like long short-term memory (LSTM) have been empirically proven to be quite successful in structured prediction applications such as machine translation \\cite{nips-sutskever:14}, Chatbot \\cite{arxiv-vinyals:15}, parsing \\cite{emnlp-ballesteros:16},  summarization \\cite{emnlp-rash:15} and image caption generation \\cite{cvpr-vinyals:15} due to their capability to bridge arbitrary time lags. \nBy far the most popular strategy to train RNNs is via the maximum likelihood principle, which consists of maximizing the probability of the next word in the sequence given the current (recurrent) state and the previous ground truth word (also known as \\emph{teacher forcing}). \nAt inference time, truth previous words are unknown, and then are replaced by words predicted by the model itself.\n\n\\begin{figure}[t]\n \\small\n \\centering\n \\includegraphics[width=7.5cm]{sequence.pdf}\n \\caption{An example training sentence. For each time step, there are a single ground truth word (highlighted by grey color) and multiple semantically and syntactically similar words (highlighted by orange color). In our approach, the weighted embedding of those words at the time step $t$ is fed into the network at the next time step. Such design makes it possible for the model to explore more feasible combinations (e.g. ``I get few red graphs'' or ``We have some yellow peaches''), and can be considered as an approximation to the beam search in a less computationally demanding way.}\n \\label{fig:sequence}\n\\end{figure}\n\nThe models trained by such teacher forcing strategy suffer from at least three drawbacks. \nFirst, the discrepancy between training and inference, called \\emph{exposure bias} \\cite{iclr-ranzato:16}, can yield errors because the model is only exposed to the distribution of training data, instead of its own predictions at inference. \nAs a result the errors can accumulate quickly along the sequence to be generated. \nSecond, the training loss for RNNs is usually defined at the word-level using maximum likelihood estimation (MLE), while their performances are typically evaluated using discrete and non-differentiable sequence-level metrics, such BLUE \\cite{acl-papineni:02} or ROUGE \\cite{acl-lin:04}. \nWe call it \\emph{evaluation bias}. \nThird, the whole training process is not \\emph{fully differentiable} because the chosen input at each time step hinders the gradients back-propagating to all previous states in an end-to-end manner. \nAlthough Goodfellow et al \\shortcite{book-goodfellow:16} pointed out that the underlying graphical model of these RNNs is a complete graph, and no conditional independence assumption is made to simplify the sequence prediction, the individual output probabilities might be still conditioned on the last few predictions if we use the teacher forcing regimen \\cite{iclr-leblond:18}. \nNote that the chain rule can not be applied any longer where an incorrect prediction is replaced with its ground truth, which prevents the model from capturing long-term dependencies and recovering the probability of the whole sequence.\n\nMany approaches have been proposed to deal with the first two drawbacks of training RNNs by scheduled sampling \\cite{bengio2015scheduled}, adversarial domain adaptation \\cite{nips-goyal:16}, reinforcement learning \\cite{iclr-ranzato:16,iclr-bahdanau:17}, or learning to search (L2S) \\cite{emnlp-wiseman:16,iclr-leblond:18} while little attention has been given to tackling the last one. \nA fully differentiable solution is capable of back-propagating the gradients through the entire sequence, which alleviates the discrepancy between training and inference by feeding RNNs with the same form of inputs, and gently bridges the gap between the training loss defined at each word and the evaluation metrics derived from whole sequence. \nThanks to the chain rule for differentiation, the training loss of differentiable solution is indeed a sequence level loss that involves the joint probability of words in a sequence. \n\nWe propose a fully differentiable training algorithm for RNNs to addresses the issues discussed above. \nThe key idea is that at the time step $t+1$, the network takes as input a ``bundle'' of predicted words at the previous time step $t$ instead of a single ground truth. \nIntuitively, the input words at each time step need to be similar in terms of semantics and syntax, and they can be replaced each other without harming the structure of sentences (see Figure \\ref{fig:sequence}).\nThe mixture of the similar words can be represented by a convex hull formed by their representations, which could be viewed as regularization to the input of recurrent neural networks.\nThis design makes it possible for the model to explore more feasible combinations (possibly unseen sequences), and can be interpreted as a computationally efficient approximation to the beam search.\n\nWe also want the number of input words can vary for different time steps, unlike the end-to-end algorithm proposed by Ranzato et al \\shortcite{iclr-ranzato:16}, in which they propagated as input the predicted top-$k$ words, and the value of $k$ is fixed in advance.\nAlthough different numbers of $k$ can be tested to determine the optimal value, the number of proper words that can be used heavily depends on contexts in which they occur. \nFor a given $k$, we could introduce unnecessary noise if more words are taken as input, whereas we may prevent the model from exploring possible sequences if less words are involved. \nIn our architecture, an attention mechanism \\cite{nips-Vaswani:17} is applied to select candidate words, and their weighted embedding are fed into the network according to their attention scores. \nSmoothing the input by this way makes the whole process trainable and differentiable by using standard back-propagation. \n\n\n\\section{Related Work}\n\nUnlike Conditional Random Fields \\cite{icml-Lafferty:01} and other models that assume independence between outputs at different time steps, RNNs and their vriants are capable of representing the probability distribution of sequences in the relatively general form. However, the most popular strategy for training RNNs, known as ``teacher forcing'' takes the ground truth as input at each time step, and makes the later predictions partly conditioned on those inputs being fed back into the model. Such training strategy impairs their ability to learn rich distributions over entire sequences. Although some training methods, such as scheduled sampling \\cite{bengio2015scheduled} and ``professor forcing'' \\cite{nips-goyal:16} are proposed to encourage the states of recurrent networks to be the same as possible when the training and inference over multiple time steps, the neural networks using greedy predictions (top-$1$) by the arguments of the maxima (abbreviated argmax) is not fully differentiable. Discrete categorical variables are involved in those greedy prediction, and become the obstacle to permit efficient computation of parameter gradients.\n\nPerhaps the most natural and na\\\"{i}ve approach towards the differentiable training is that at time step $t + 1$ we take as input all the words whose contributions are weighted by their scores instead of the ground truth word, and the scores are the predicted distribution over words from the previous time step $t$. However, the output distribution at each time step is normally not sparse enough, and thus the input may blur by a large number of words semantically far from the ground truth. A simple remedy would be the end-to-end algorithm proposed by Ranzato et al \\shortcite{iclr-ranzato:16}, in which they propagated as input the top $k$ words predicted at the previous time step. The $k$ largest scoring words are weighted by their re-normalized scores (summing to one). Although different numbers of $k$ can be tested to choose the optimal value, the value of $k$ is fixed at each time step. It would better to learn to determine a particular $k$ for each time step, depending on a specific context. For a fixed $k$, we would introduce unnecessary noise if too many words are involved, whereas we may prevent the model from exploring possible sequences if too few words are chosen.\n\nJang et al \\shortcite{jang2017categorical} proposed Gumbel-Softmax that can make the output distributions over words more sparse by approximating discrete one-hot categorical distributions with their continuous analogues. Such approximate sampling mechanism for categorical variables can be integrated into RNNs, and trained using standard back-propagation by replacing categorical samples with Gumbel-Softmax ones. Theoretically, as the temperature $\\tau$ of Gumbel-Softmax distribution approaches $0$, samples from this distribution will become one-hot. In practice, they begin with a high temperature and anneal to a small but non-zero one. The temperature, however, can not be too close to $0$ (usually $\\tau \\ge 0.5$) because the variance of the gradients will be so large, which may result in the gradient explosion.\n\nIdeas coming from learning to search \\cite{emnlp-wiseman:16,iclr-leblond:18} or reinforcement learning, such as the REINFORCE \\cite{iclr-ranzato:16} and ACTOR-CRITIC \\cite{iclr-bahdanau:17}, have been used to derive training losses that are more closely related to the sequence-level evaluation metrics that we actually want to optimize. Those approaches side step the issues associated with discrete nature of the optimization by not requiring losses to be differentiable. While those approaches appear to be well suited to tackle the training problem of RNNs, they suffer from a very large action space which makes them difficult to learn or search efficiently, and thus slow to train the models, especially for natural language processing tasks normally with a large vocabulary size. We reduce the effect of search error by pursuing few next word candidates at each time step in a pseudo-parallel way. We note that those ideas and ours are complementary to each other and can be combined to improve performance further, although the extent of the improvements will be task dependent.\n\n\\section{Methodology}\n\nWe propose a fully differentiable method to train a sequence model, which allows the gradients to back-propagate through the entire sequence. \nFollowing \\citep{zhang2016generating,jang2017categorical,goyal2017differentiable}, the network takes as input the weighted sum of word embeddings at each time step, where the weights reflect how likely they are chosen in the previous step. \nThe following discussion is based on a general architecture that can be viewed as a variational encoder-decoder with an attention mechanism \\citep{bahdanau2014neural, kingma2014auto-encoding, bowman2016generating}. \nThe teacher forcing method, denoted as VAE-RNNSearch, is implemented with the above architecture, while we extend this architecture with a lexicon-based memory (Figure \\ref{fig:reg-model} shows the details, where the margin relaxed component should be ignored at present), called \\textbf{W}ord \\textbf{E}mbedding \\textbf{A}s \\textbf{M}emory (WEAM).\n\nGiven a source sentence $s_{1:M}$, $s_i \\in \\mathcal{V}^s$ where $\\mathcal{V}^s$ is the vocabulary of source language, a multi-layer BiLSTM encoder $f^s(\\cdot)$ is used to compute a high-level contextual word representation $e_i \\in \\mathbb{R}^d$ for each word $s_i$, where $d$ is the dimensionality of the embedded vector space,\n\\begin{equation} \\small\n    e_{1:M} = f^s(s_{1:M})\n\\end{equation}\nwhich is also the entries to be attended by the decoder. \nThe final state of BiLSTM is used to compute the mean and the log variance of the latent Gaussian distribution $\\mathcal{N}(\\mu, \\sigma^2)$ in VAE with a linear layer. \n\nIn the decoding process, a seed $c$ sampled for the latent Gaussian distribution is fed into a multi-layer LSTM decoder $f^t(\\cdot)$ with multi-hop attention. \nThe decoder aims at predicting the distributions $p_{1:N}$ of target words $t_{1:N}$ conditioned on the seed $c$ and the encoder output $e_{1:M}$,\n\\begin{equation} \\small\n\\label{eq:decoder}\n    p_{1:N}=f^t(c, e_{1:M})\n\\end{equation}\nwhere each $p_j\\in [0, 1]^{|\\mathcal{V}^t|}$ is the probability distribution of the $j$-th target word, and $\\mathcal{V}^t$ is the vocabulary of target language.\nThe words with the maximum probability are the predictions $\\tilde{t}_{1:N}$.\n\nSpecifically, we explain the decoding process in a recurrent manner. \nAbove all, the word embedding matrix $\\mathcal{M} \\in \\mathbb{R}^{|\\mathcal{V}^t| \\times d}$ is considered as a memory.\nThe recurrent function $g(\\cdot)$ is a multi-layer LSTM cell incorporated with multi-hop attention over encoder output $e_{1:M}$. \nAt timestamp $j$, the recurrent function $g(\\cdot)$ output a hidden representation $h_{j+1}$ by giving the previous states $h_{j}$ and the current word embedding $v_j$, and the hidden state is further used to computed the likelihood of the predictions\n\\begin{equation} \\small\n    h_{j+1} = g(h_j, v_j)\n\\end{equation}\n\\begin{equation} \\small\n    p_{j+1} = \\text{softmax}(\\mathcal{M}h_{j+1})    \n\\end{equation}\nFor the starting timestamp, $h_0$ is defined as the seed $c$, and $v_0$ is the word embedding of the end-of-sentence token. \nDuring the training process, $v_j$ used in VAE-RNNSearch is the ground truth word embedding $\\mathcal{M}(t_j)$ and that in WEAM is approximated by attention over word embeddings. \nThe attention score here could be obtained by reusing the predicted distribution of the next word, namely $p_j(\\cdot)$. \nThus the input word embedding $v_j$ is approximated by\n\\begin{equation} \\small\n\\label{eq:embed_atten}\n    v_j = p_j\\mathcal{M} = \\sum_{w\\in\\mathcal{V}^t}{p_j(w)\\mathcal{M}(w)}\n\\end{equation}\nFrom now on, we use ground truth words to denote words inputted into the network and target words to denote objective ones for the purpose of distinguishing their functions, although they refer to the same sequence.\n\nFinally, after the entire sequence is generated, the objective function of WEAM is computed from the predicted probability $p_j(\\cdot)$ and the target sequence $t_{1:N}$ as well as the latent Gaussian distribution, \n\\begin{equation} \\small\n\\begin{split}\n\\label{eq:loss}\n    L(\\theta) = & -\\sum_{j=1..N}{\\log p_j(t_j)} \\\\\n                & + D_{KL}(\\mathcal{N}(\\mu, \\sigma^2) || \\mathcal{N}(0,1))\n\\end{split}\n\\end{equation}\nIn Equation \\ref{eq:loss}, the first term is the cross-entropy between the predicted probability with the target one, and the second term is the KL divergence between the approximate latent Gaussian distribution and a prior normal distribution. \nEquipped with WEAM, the training process of text generation is fully differentiable.\n\nAt the inference, instead of using the attention-weighted embedding in Equation \\ref{eq:embed_atten}, we found it better to feed an exact word embedding into the neural network at each timestamp. \nAlthough the word embedding memory regularizes $v_{j+1}$ in a convex hull of word embeddings comparing with $h_j$ (Figure \\ref{fig:reg-weam}), there is still difference with the representations of genuine words in vocabulary, which is a finite set.\nThe bias could lead to minor mistakes, which accumulate as the prediction process goes on and is potentially harmful to future predictions.\nThus feeding an exact word embedding in inference would be helpful in generating a sequence by regularizing the input embedding and erasing the potential mistakes.\n\n\\begin{figure}[t]\n    \\centering\n    \\includegraphics[width=7.5cm]{model.pdf}\n    \\caption{\\small{M-WEAM architecture. It is based on a variational encoder-decoder with attention mechanism. (a) performs an attention operation over all the word embeddings to produce their probabilities (reflecting how likely they are chosen); (b) retrieves the words according to their probabilities (masked to make it sparse). By applying a margin relaxed method, we rule out the words having the lower probabilities than a threshold, and the remaining values are normalized to estimate the distribution of input words at the next time step.}}\n    \\label{fig:reg-model}\n\\end{figure}\n\n\\subsection{Sparse Word Distributions}\n\nAs discussed above, feeding an exact word embedding is a helpful property in inference, which is equivalent to setting a sparse distribution $p_j(\\cdot)$.\nHowever, WEAM model suffers from the long tail effect cause by the non-sparse $p_j(\\cdot)$.\nMost words in vocabulary are predicted with a very small but non-negligible probability, leading to a noisy $v_j$.\n\nA widely-used method to obtain a sparse distribution is rescaling the logits before presenting it to the final-layer classifier, equivalently\n\\begin{equation} \\small\n\\label{eq:rescaled-logit}\n    \\tilde{p}_j(k) = p_j(k)^{1\/\\tau}\n\\end{equation}\nwhere $\\tau$ is a rescaling factor decaying from one to zero \\citep{zhang2016generating, jang2017categorical}. \nIdeally, when $\\tau$ converges to zeros, the distribution $\\tilde{p}_j(\\cdot)$ converges to a categorical one.\nHowever, in practice, $\\tau$ cannot be set to a small value, which could lead to gradient explosion. \nThus an alternative strategy is setting a lowerbound for $\\tau$, which is empirically $0.5$ in \\citep{jang2017categorical}. \nThis value of $\\tau$ is far from the optimal number (i.e. $0$) in theory, and the sparse categorical distribution can be achieved in practice. \n\n\\begin{figure}[t]\n    \\centering\n    \\begin{subfigure}{0.23\\textwidth}\n        \\includegraphics[width=\\textwidth]{fullspace.png}\n        \\subcaption{WEAM}\n        \\label{fig:reg-weam}\n    \\end{subfigure}\n    \\begin{subfigure}{0.23\\textwidth}\n        \\includegraphics[width=\\textwidth]{subspace.png}\n        \\subcaption{M-WEAM}\n        \\label{fig:reg-rweam}\n    \\end{subfigure}\n    \\caption{The difference between (a) WEAM and (b) M-WEAM in the way they form the subspace in a convex hull defined by all the word embeddings, which can be viewed as a kind of regularizations. Blue points represent all the words in a vocabulary, whereas yellow ones are the unmasked words in M-WEAM. The simplex is the convex hull defined by the unmasked words chosen by the results of  the lexicon-based attention.}\n\\label{fig:reg}\n\\end{figure}\n\nWe propose a margin relaxed method to make a sparse distribution through restricting $v_j$ in a relatively smaller subspace (see Figure \\ref{fig:reg-rweam}). \nThe architecture of M-WEAM model is shown in Figure \\ref{fig:reg-model}. \nGiven a distribution $p_j(\\cdot)$, its sparse version $\\tilde{p}_j(\\cdot)$ is computed by ruling out the words with low probability,\n\\begin{equation} \\small\n    \\tilde{p}_j(w) = \\frac{p_j(w)mask_j(w)}{\\sum_{l}{p_j(w')mask_j(w')}}\n\\end{equation}\n\\begin{equation} \\small\n    mask_j(w) = \\begin{cases} 1 & p_j(w) > \\eta_j \\\\\n    0 & otherwise\n    \\end{cases}\n\\end{equation}\nwhere $\\eta_j$ is a threshold and could be determined by various ways.\nIts motivation is to alleviate the long tail effect by masking the ``long tail'' entries with close-to-zero probability.\nA possible choice is defined the threshold $\\eta_j$ as \n\\begin{equation} \\small\n\\label{eq:margin}\n    \\eta_j=e^{-\\epsilon}\\max_{k}p_j(k)\n\\end{equation}\nwhere $\\epsilon$ is a margin in log space.\nAs $\\epsilon$ converges to $0$, the $\\tilde{p}_j(k)$ converges to a categorical distribution as well.\nHowever, we do not intend to anneal $\\epsilon$ to zero, aiming at keep a mixture meaning of multiple semantically and syntactically similar words in the training process.\nIn vector space, the mixture meaning is represented as the convex hull shown in Figure \\ref{fig:reg-rweam}.\nAll the words forming this convex hull are reasonable predictions, and their representations are optimized in the training.\nIf $\\epsilon$ become very small, there is only one word, namely one point, to be optimized, leading to slower convergence.\nBesides, an input representation $v_j$ generated from a subspace would be more robust than that assigned to only one point, preventing the model from overfitting to a specific point.\nThe optimization over a subspace could be considered as a balance between that over a point and over the whole space in Figure \\ref{fig:reg-weam}.\n\nThe re-estimated probability $\\tilde{p}(\\cdot)$ can also be used in objective function Equation \\ref{eq:loss} following ``enough is enough'' principal.\nNote that the training objective is discriminating a target word with the others in vocabulary by increasing the margin between them.\nAs training progresses, the margin becomes larger implying that the target word can be easily pointed out from the vocabulary. \nKeeping optimizing the margin between the target word and all the others would be useless, and easily leads to overfitting.\nThe ``enough is enough'' principal means that a word requires no further optimization when its likelihood is much lower than the target one. \nThus the re-estimated probability $\\tilde{p}(\\cdot)$ is used in the loss function to prevent the words with low probability from over-optimization.\n\nWe denote the vanilla non-sparse version as WEAM, the one rescaling logits in Equation \\ref{eq:rescaled-logit} as R-WEAM, and the proposed margin relaxed one as M-WEAM. Note that R-WEAM is slightly different from that in \\citep{zhang2016generating, jang2017categorical} since word classifier and word embeddings are tied \\cite{inan2017tying}. \n\n\\subsection{Warm-up Strategy}\n\nAt the beginning of the training, the words generated by WEAM models are almost random. \nConditioned on such randomly predicted words, the model fails to benefit from the future training.\nIn order to alleviate this problem, the models are warmed up using the teacher forcing strategy. We present two warming up techniques to train WEAM-family models.\n\n\\subsubsection{Scheduled Sampling}\n\nScheduled sampling is a widely-used way to warm up WEAM-family models in training \\citep{bengio2015scheduled}.\nWhether to use the embedding of a ground truth word or its approximation in Equation \\ref{eq:embed_atten} at the next timestamp is randomly selected as:\n\\begin{equation} \\small \\label{eq:sample-i}\n\\begin{split}\n    v_{j+1} & =\\begin{cases}\n        \\mathcal{M}(w_{j+1}), & t=0 \\\\\n        \\sum_{w\\in\\mathcal{V}^t}{p_j(w)M(w)}, & t=1\n    \\end{cases} \\\\\n    t & \\sim \\mathcal{B}(\\text{steps}\/\\text{max\\_steps})\n\\end{split}\n\\end{equation}\nwhere $t$ is sampled from a Bernoulli distribution $\\mathcal{B}(\\cdot)$ whose expectation has a positive correlation with training progress.\nFrom a training perspective, the probability of choosing the approximated word embedding increases linearly from zero to one.\n\n\\subsubsection{Threshold}\n\nMasking words with low probability may cause a new problem that a target word may be masked and lose its opportunity for optimization.\nOnce a target word is ruled out of training, it may be ruled out forever since its low probability, leading to bad performances and slow convergence.\nAn alternative way to solve this problem is computing the threshold with the probability of the target word, and we determines the threshold in a probable way as:\n\\begin{equation} \\small \\label{eq:gold-margin}\n\\begin{split}\n    \\eta_j & =\\begin{cases}\n    e^{-\\epsilon} p_j(w_{j+1}), & r=0 \\\\\n    e^{-\\epsilon} \\max_{k}e^{-\\epsilon}p_j(k), & r=1\n    \\end{cases} \\\\\n    r & \\sim \\mathcal{B}(\\max(\\text{steps}\/\\text{max\\_steps}, \\xi))\n\\end{split}\n\\end{equation}\nwhere $r$ is similar to $t$ with lowerbound $\\xi$.\nEquation \\ref{eq:gold-margin} shows that the threshold is determined by the target word at the beginning of training whereas it is determined by the predictions when the algorithm almost converges.\nAdopting a lowerbound $\\xi$ also leads to an attractive property that a prediction and its re sptteiaevcrget word are replaceable to each other, and thus the model is expected to capturing the phenomenon of synonyms.\n\n\\section{Experiments}\n\nWe conducted three sets of experiments to demonstrate the effectiveness of M-WEAM on various tasks including machine translation, text summarization and dialogue generation. \n\nWe employed almost the same setting to all the tasks. \nA wrapped recurrent block was used for both the encoder and decoder, defined as a sequence of LSTM, dropout, residual connection and layer normalization. \nUni-directional LSTM was used in encoder while the bi-directional LSTM was used in decoder.\nThe multi-hop attention component was also wrapped in a similar way.\nWe set the dimensionality of vector space to $256$, layers of recurrent block to $3$, dropout rate to $0.3$, lowerbound of margin $\\epsilon$ to $1$, and $\\xi$ to $0.5$.\nThe models were optimized by Adam \\cite{kingma2014adam} with $\\beta_1=0.9, \\beta_2=0.999 $ and $\\text{eps}=10^{-8}$. \nLearning rate was scheduled following \\cite{feng2018neural}, and its maximum was set to $0.001$.\nLabel smoothing was also used with $0.1$ smoothing rate. \nSeparated vocabularies were constructed for source and target domains on translation and dialogue tasks whereas a shared one was used for summarization task.\nAll the models were implemented with PyTorch and trained on $1$ NVIDIA Titan Xp GPU.\n\n\n\\subsection{Machine Translation}\n\n\\begin{table}[b]\n    \\centering\n    \\begin{tabular}{l|c}\n        \\toprule\n        Model & BLEU \\\\\n        \\midrule\n        MIXER & $20.73$ \\\\\n        BSO & $23.83$ \\\\\n        $\\alpha$-soft annealing & $20.60$ \\\\\n        Actor-Critic & $27.49$ \\\\\n        SEARNN & $28.20$ \\\\\n        NPMT & $28.57$ \\\\\n        \\midrule\n        VAE-RNNSearch & $28.97$ \\\\\n        WEAM & $27.84$ \\\\\n        R-WEAM & $28.13$\\\\\n        \\midrule\n        M-WEAM & \\textbf{29.17} \\\\\n        \\bottomrule\n    \\end{tabular}\n    \\caption{Results on IWSLT14 German-English.}\n    \\label{tab:de-en}\n\\end{table}\n\nMachine translation task was experimented on two datasets, IWSLT14 German to English (De-En) \\citep{cettolo2014report} and IWSLT15 English to Vietnamese (En-Vi) \\citep{cettolo2015iwslt}. \nThe dataset was preprocessed basically following \\cite{huang2017towards, feng2018neural}. \nFor IWSLT14 De-En, the dataset roughly contains 153K training sentences, 7K development sentences and 7K testing sentences. \nWords whose frequency of occurrence is lower than 3 were replaced by an ``UNK'' token.\nFor IWSLT15 En-Vi, the dataset contains 133K translation pairs in training set, 1,553 in validation set (TED tst2012) and 1,268 in test set (TED tst2013).\nSimilar to De-En, words with frequency of occurrence lower than 5 were replaced by the ``UNK'' token.\n\n\\begin{table}[t]\n    \\centering\n    \\begin{tabular}{l|c}\n         \\toprule\n         Model & BLEU \\\\\n         \\midrule\n         Hard Monotonic & $23.00$ \\\\\n         NPMT & $26.91$ \\\\\n         \\midrule\n         VAE-RNNSearch & $28.66$ \\\\\n         WEAM & $27.14$ \\\\\n         R-WEAM & $26.79$ \\\\\n         \\midrule\n         M-WEAM & \\textbf{28.70} \\\\\n         \\bottomrule\n    \\end{tabular}\n    \\caption{Results on IWSLT15 English-Vietnamese.}\n    \\label{tab:en-vi}\n\\end{table}\n\nOn IWSLT14 De-En, we compared M-WEAM against VAE-RNNSearch \\citep{bowman2016generating}, WEAM, R-WEAM \\citep{zhang2016generating}, MIXER \\citep{iclr-ranzato:16}, BSO \\citep{emnlp-wiseman:16}, Actor-Critic \\citep{iclr-bahdanau:17}, SEARNN \\citep{iclr-leblond:18}, $\\alpha$-soft annealing \\cite{goyal2017differentiable}, and NPMT \\citep{huang2017towards}.\nOn IWSLT15 En-Vi, we compared M-WEAM against VAE-RNNSearch \\citep{bowman2016generating}, WEAM, R-WEAM \\citep{zhang2016generating}, Hard Monotonic \\citep{raffel2017online} and NPMT \\citep{huang2017towards}  .\n\n\nTable \\ref{tab:de-en} and Table \\ref{tab:en-vi} display the results on the German-English and English-Vietnamese translation task respectively. \nIt is can be seen that M-WEAM achieves decent results on the both tasks.\nComparing with differentiable models, M-WEAM seizes a notable gain of over $1$ BLEU point, outperforming WEAM and R-WEAM on both tasks. \nM-WEAM also achieves comparable results with its teacher forcing version.\nBesides, M-WEAM beats previous SOTA RNN-based models with a significant margin.\n\nInspired by \\cite{keskar2016large}, where they explored the neighborhood of the minima to measure the model's sensitivity, we conducted a similar experiment on IWSLT14 De-En to show the robustness of VAE-RNNSearch and WEAM family models. \nWe tested on the models by injecting noise into input embedding. \nThe noise was sampled from an unbiased Gaussian distribution with the standard deviation increasing from 0 to 1. \nAs the noise increases, models with sharper local minima should suffer more from performance degradation.  \n\nFigure \\ref{fig:sensitivity} shows how the performance of the compared models changes against noise. \nM-WEAM model surpasses the teacher forcing model by nearly 3 BLEU points when the standard deviation reaches 1. \nIt thus can be inferred that M-WEAM converges to a flatter minima than VAE-RNNSearch does.\nBesides, among all models, WEAM shows strongest anti-noise ability because it has been trained with massive noise due to the long tail effect. \nIn general, M-WEAM achieves a balance between performance and robustness.\n\n\\begin{figure}[t]\n    \\centering\n    \\includegraphics[width=7.5cm]{noise.pdf}\n    \\caption{Sensitivity test with different Gaussian noise levels. The $x$-axis is the standard deviation of the Gaussian noise. \n   \n    }\n    \\label{fig:sensitivity}\n\\end{figure}\n\n\n\n\\subsection{Abstractive Summarization}\n\nWe evaluated the models on Gigaword benchmark, and applied the same data preprocessing as \\citep{rush2015neural, chopra2016abstractive}.\nThe results are reported with F1 ROUGE scores, including ROUGE-1, ROUGE-2 and ROUGE-L.\n\nThe results of ABS \\citep{rush2015neural}, ABS+ \\citep{rush2015neural}, Feats \\citep{nallapati2016abstractive}, RAS-LSTM \\citep{chopra2016abstractive} and RAS-Elman \\citep{chopra2016abstractive} are extracted from the numbers reported by their authors. We implemented the VAE-RNNSearch \\citep{bowman2016generating} as well as WEAM, and report their results using our implementations. The results of ABS and ABS+ with the greedy search are unavailable, and thus we report those with the beam search instead. We are aware that SEASS \\citep{zhou2017selective} and DRGD \\citep{li2017deep} are two recent studies in which the higher ROUGE metrics were reported on Gigaword dataset. \nTheir results are not listed because those models are specifically tailored for the abstractive summarization task. SEASS is equipped with a selective network for salient language units, and DRGD uses a recurrent latent variable to capture the summary structure, while we focus on the general framework and training algorithm for the sequence-to-sequence tasks. \n\n\n\\begin{table}[t]\n    \\centering\n    \\begin{tabular}{l|c|c|c}\n        \\toprule\n        Model & R-1 & R-2 & R-L \\\\\n        \\midrule\n        ABS$^*$ & $29.55$ & $11.32$ & $26.42$ \\\\\n        ABS+$^*$ & $29.78$ & $11.89$ & $26.97$ \\\\\n        Feats & $32.67$ & $15.59$ & $30.64$ \\\\\n        RAS-LSTM & $31.71$ & $13.63$ & $29.31$ \\\\\n        RAS-Elman & $33.10$ & $14.45$ & $30.25$ \\\\\n       \n       \n        \\midrule\n        VAE-RNNSearch & $\\textbf{33.99}$ & $15.72$ & $\\textbf{31.67}$ \\\\\n        WEAM & $33.09$ & $15.05$ & $30.79$ \\\\\n        R-WEAM & $32.35$ & $14.52$ & $30.10$ \\\\\n        \\midrule\n       \n        M-WEAM & $33.87$ & $\\textbf{15.78}$ & $31.52$ \\\\\n        \\bottomrule\n    \\end{tabular}\n    \\caption{Results on Gigaword dataset. The models indicated with $^*$ use beam search to generate summaries whereas the others use the greedy search instead.}\n    \\label{tab:gigaword}\n\\end{table}\n\nTable \\ref{tab:gigaword} shows the results on Gigaword Test Set. \nAmong the fully differentiable WEAM family models, M-WEAM achieves the best performance, and outperform the WEAM and R-WEAM with at least $0.7$ improvement on the all ROUGE metrics.\nBy comparing with VAE-RNNSearch, M-WEAM achieves comparable results on all the ROUGE metrics. \nIt is worth noting that although M-WEAM performs slightly worse than VAE-RNNSearch on ROUGE-1, but it does better on ROUGE-2, which shows that M-WEAM is able to recover more longer patterns, thanks to its fully differentiable property.\n\n\\subsection{Dialogue}\n\nFinally, we evaluated the models on the task of single-turn dialogue, and used a subset of STC-weibo corpus \\cite{shang2015neural} as a dataset, where weibo is a popular Twitter-like microblogging service in China. \nThe complete corpus consists of roughly $4.4$M conversational post-response pairs.\nWe removed the sentences containing more than $10$ words and created a dataset with $907,809$ post-response pairs for fast training and testing.\nThe dataset was randomly split into training, validation, and testing sets, which contains $899$K, $10$K and $10$K pairs respectively. \nThe vocabulary was formed by the most frequent $40$K words in training data.\n\nBLEU score is a widely used metric in dialogue evaluation to measure the response relevance, whereas distinct-1\/2 is used to measure the response diversity. \nAs is shown in Table \\ref{tab:stc}, M-WEAM model achieves the highest performance in terms of BLEU, but performs worse than other models with distinct metrics. \nUnlike machine translation, a response can be taken as an acceptable answer to many posts (or questions) and the dataset contains a number of one-to-many cases. In order to recover the probability of entire sequences, the fully differentiable model with improved training efficiency tends to generate frequently-used sequences and $n$-grams, which hurts the diversity of the generated responses. We leave this issue for future research. \nThe high relevance but low diversity of responses generated by M-WEAM show that the proposed M-WEAM model is more desirable for the tasks requiring to generate exact sequences.\n\n\\begin{table}[t]  \n    \\small\n    \\centering\n    \\begin{tabular}{l|c|c|c}\n        \\toprule\n        Model & BLEU & distinct-1 & distinct-2 \\\\\n        \\midrule\n        VAE-RNNSearch & $5.95$ & $2.39$ & $19.01$ \\\\\n        WEAM & $5.67$ & \\textbf{2.50} & $19.52$ \\\\\n        R-WEAM & $5.73$ & $2.49$ & \\textbf{19.65} \\\\\n        \\midrule\n        M-WEAM & \\textbf{6.11} & $2.39$ & $16.31$ \\\\\n        \\bottomrule\n    \\end{tabular}\n    \\normalsize\n    \\caption{Automatic evaluation results on STC-weibo dataset. The best results are highlighted in bold font.}\n    \\label{tab:stc}\n\\end{table}\n\n\n\\section{Conclusion}\n\nWe proposed a fully differentiable training algorithm for RNNs to alleviate the discrepancy between training and inference (exposure bias), and to bridge the gap between the training loss defined at each word and the evaluation metrics derived from whole sequence (evaluation bias). \nThe fully differentiable property is achieved by feeding the network at each step with a ``bundle'' of the words carrying similar meaning instead of a single ground truth. \nIn our solution, the network is allowed to take different number of words as input at each time step, depending on the context. \nWe may have more candidate words at some positions, while less at others when trying to generate a sentence. \nExperiments on machine translation, abstractive summarization, and open-domain dialogue response generation tasks showed that the proposed architecture and training algorithm achieved the best or comparable performance, especially in BLUE or ROUGE-2 metrics defined on sequence level, reflecting the enhanced ability in capturing long-term dependencies and recovering the probability of the whole sequence. \nThe trained networks is also empirically proven to be more robust to noises, and perform constantly better other competitors with different noise levels.\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\\label{intro}\\setcounter{equation}{0}\n\t\n\tMany physical processes such as harvesting, natural disaster, shocks etc, cause abrupt changes in their states at certain time instant. These sudden changes occur for negligible time period and they are estimated in the form of instantaneous impulses. The theory of instantaneous impulsive systems has remarkable applications in several areas of science and engineering, for example, population dynamics, ecology, network control system with scheduling protocol etc., (cf. \\cite{NE,AM,TY}, etc). However, certain dynamics of evolution processes in pharmacotherapy cannot be modeled by instantaneous impulsive dynamical systems, for example, in hemodynamical equilibrium of a person, introduction of insulin into the bloodstream and the consequent absorption of  the body are gradual processes and  stay active for a finite time interval. Thus, we cannot describe this situation via instantaneous impulsive systems. Therefore, Hern\\'andez et. al. \\cite{EHD} introduced a new class of impulses termed as non-instantaneous impulses, which starts at an arbitrary fixed point and stays active on a finite time interval and they established the existence of solutions for such a class of impulsive differential equations. Later, Wang et. al. \\cite{JRW,JRY}, extended this model to two general classes of impulsive differential equations, which are very important in the study of dynamics of evolutionary processes in pharmacotherapy. For more details on the theory of non-instantaneous impulsive systems, we refer the interested readers to \\cite{YTJ,JMF}, etc, and the references therein. In addition to this, there are various real world phenomena, for example, neural network, inferred grinding models, ecological models, heat conduction in materials with fading memory etc. In these phenomena, the current state of a system is influenced by the past states. The dynamics of such processes are characterized by delay differential equations (finite or infinite), see for example, \\cite{XFL,LA,JW}, etc. In the application point of view, the functional evolution systems with state-dependent delays are more prevalent and adequate, see for instance, \\cite{AO,CNF}, etc, and the references therein. \n\t\n\tThe concept of controllability plays a vital role in the study of control systems. Controllability (exact or approximate) refers that the solution of a control system can steer from an arbitrary initial state to a desired final state by using some control function. In the infinite dimensional setting, in comparison with exactly controllable systems, approximate controllable systems are more extensive and have wide range of applications, cf. \\cite{M,TRR,TR,EZ}, etc. In the past two decades, the problem of approximate controllability of various kinds of systems (in Hilbert and Banach spaces) such as impulsive differential equations, functional differential equations, stochastic systems, Sobolev type evolution systems, etc,  is extensively studied  with the help of fixed point approach and produced excellent results, see for instance, \\cite{SM,SAM,FUX,AGK,M,ER}, etc. \n\t\n\tOn the other hand, fractional differential equations (FDEs), which involve fractional derivatives of the form $\\frac{d^{\\alpha}}{dt^{\\alpha}}$, where $\\alpha>0$ is not necessarily an integer, attained great importance due to their ability to model complex phenomena. They naturally appear  in diffusion processes, electrical engineering, viscoelasticity, control theory of dynamical systems, quantum mechanics, biological sciences, electromagnetic theory, signal processing, finance, economics, and many other fields (cf. \\cite{HlR,AAK,VM,IPF,TPJ,YAR,MST}, etc).  A comprehensive study on fractional calculus and FDEs are available in \\cite{AAK,IPF}, etc.  For the past few decades, FDEs in infinite dimensions seek incredible attention of many researchers and eminent contributions have been made both in theory as well as in applications. Several authors studied the existence and approximate controllability results for fractional order systems in Hilbert spaces, see for instance, \\cite{SNS,NIZ,SRY,JWY}, etc, and the references therein. \n\t\n\tThe study of approximate controllability of the fractional order control systems in Banach spaces has not got much attention in the literature. In \\cite{NMI}, Mahmudov developed sufficient conditions for the approximate controllability of the Sobolev type fractional evolution equations with Caputo derivative in separable reflexive Banach spaces using the Schauder fixed point theorem. Later in \\cite{MIN}, he studied the approximate controllability of the  fractional neutral evolution systems by taking infinite delay using Krasnoselkii's fixed-point theorem. After that Chalishazar et. al. \\cite{DNC} extended his work by considering instantaneous impulses and examined the approximate controllability in Banach spaces. \n\t\n\tThe articles  \\cite{PCY,FZM,RS,RGR}, etc claimed the approximate controllability of the factional order systems in general Banach spaces using resolvent operator condition. But, the resolvent operator defined in these works is valid only if the state space is a Hilbert space, whose dual is identified by the space itself (see, the resolvent operator definition in the expression \\eqref{2.1} and Remark \\ref{rem2.8}). Moreover, many papers deal with the fractional order impulsive systems with delays, see for instance, \\cite{ZT,ZH,YZO}, etc. In these works, the characterization of norm or seminorm defined in the phase space involves uniform norm, but the choice of such a norm or seminorm is not suitable in the impulsive case, for counter examples and more details, we refer the interested readers to \\cite{GU} (a detailed discussion on this problem is also available in \\cite{SAMT}). Many articles considered the fractional order impulsive systems with non-instantaneous impulses (cf. \\cite{RMS,Ad,Zyf,Zy} etc.). In theses works, the concept of the mild solution defined for the considered system is not realistic, a counter example and appropriate definition of the mild solution discussed in \\cite{Fe,JRY} (see Definition \\ref{def2.6} for the mild solution definition for system under our consideration). One of the main aims of this work is to resolve these issues.\n\t\n\tRecently, a few papers have been reported on the approximate controllability of the non-instantaneous impulsive systems with and without delays in Hilbert spaces, (cf. \\cite{RMS,SKS,SAJ}, etc). Dhayal et. al. \\cite{RMS} formulated the approximate controllability results for a class of fractional order non-instantaneous impulsive stochastic differential equations driven by fractional Brownian motion. In \\cite{SKS}, Kumar and Abdal derived sufficient conditions for the approximate controllability of non-instantaneous impulsive fractional semilinear measure driven control systems with infinite delay. Liu and his co-authors, in \\cite{SAJ},  investigated the approximate controllability of fractional differential equation with non-instantaneous impulses via iterative learning control scheme. To best of our knowledge, there is no work has been reported on the approximate controllability of the fractional order non-instantaneous impulsive systems with state-dependent delay in Banach spaces.\n\t\n\tMotivated from the above facts, in this work, we derive sufficient conditions for the approximate controllability of  fractional order non-instantaneous impulsive functional evolution equations with state-dependent delay in separable reflexive Banach spaces. Moreover, we properly define the resolvent operator in Banach spaces, which plays a crucial role in obtaining the aforementioned results (see the expression \\eqref{2.1} below). Proper motivation for the construction of different forms of feedback controls  for the fractional order semilinear systems available in the literature has been justified in this work (see Remarks \\ref{rem3.6} and \\ref{rem4.4} below). Furthermore, our paper modifies the phase space characterization to incorporate Guedda's observations in \\cite{GU}, by replacing the uniform norm on the phase space by integral norm for the impulsive differential equations (see Example \\ref{exm2.8}). \n\t\n\tWe consider the following fractional order non-instantaneous impulsive functional evolution equation with state-dependent delay: \n\t\\begin{equation}\\label{1.1}\n\t\\left\\{\n\t\\begin{aligned}\n\t^C\\mathrm{D}_{0,t}^{\\alpha}x(t)&=\\mathrm{A}x(t)+\\mathrm{B}u(t)+f(t,x_{\\rho(t, x_t)}), \\ t\\in\\bigcup_{k=0}^{m} (\\tau_k, t_{k+1}]\\subset J=[0,T], \\\\\n\tx(t)&=h_k(t, x(t_k^{-})),\\ t\\in(t_k, \\tau_k], \\ k=1,\\dots,m, \\\\\n\tx_{0}&=\\psi\\in \\mathfrak{B},\n\t\\end{aligned}\n\t\\right.\n\t\\end{equation}\n\twhere \\begin{itemize} \\item $^C\\mathrm{D}_{0,t}^{\\alpha}$ denotes a derivative in Caputo sense of order $\\alpha$ with $\\frac{1}{2}<\\alpha<1$, \\item  the operator $\\mathrm{A}:\\mathrm{D(A)}\\subset \\mathbb{X}\\to\\mathbb{X}$ is an infinitesimal generator of a $\\mathrm{C}_0$-semigroup $ \\mathcal{T}(t) $ on a separable reflexive Banach space $ \\mathbb{X}$ (having a strictly convex dual $\\mathbb{X}^*$), \\item  the linear operator $\\mathrm{B}:\\mathbb{U}\\to\\mathbb{X}$ is bounded with $\\left\\|\\mathrm{B}\\right\\|_{\\mathcal{L}(\\mathbb{U},\\mathbb{X})}= \\tilde{M}$ and the control function $u\\in \\mathrm{L}^{2}(J;\\mathbb{U})$, where $\\mathbb{U}$ is a separable Hilbert space, \\item  the function $ f:J\\times \\mathfrak{B}\\rightarrow \\mathbb{X} $, where $\\mathfrak{B}$ is a phase space, which will be specified in the subsequent sections, \\item for $k=1,\\ldots,m$, the functions $h_k:[t_k, \\tau_k]\\times\\mathbb{X}\\to\\mathbb{X}$ represent the non-instantaneous impulses and the fixed points $\\tau_k$ and $t_k$ satisfy $0=t_0=\\tau_0<t_1\\le \\tau_1\\le t_2<\\ldots<t_m\\le \\tau_m\\le t_{m+1}=T$,   the values  $x(t_k^+)$ and $x(t_k^-)$ stand for the right and left limit of $x(t)$ at the point $t=t_k,$ respectively and satisfy $x(t_k^-)=x(t_k),$ for $k=1,\\ldots,m$, \\item the function $x_{t}:(-\\infty, 0]\\rightarrow\\mathbb{X}$ with $x_{t}(\\theta)=x(t+\\theta)$ belongs to some abstract space $\\mathfrak{B}$, and the function $\\rho: J \\times \\mathfrak{B} \\rightarrow (-\\infty, T]$ is continuous.\n\t\\end{itemize}\n\t\n\tThe rest of the article is organized as follows: Section \\ref{pre} starts with some basic definitions of fractional calculus and then presents the phase space axioms. Moreover, we provide assumptions and important results to establish the existence and approximate controllability results of the system \\eqref{1.1}. Section \\ref{Linear} deals with the  approximate controllability  of  the fractional order linear control system corresponding to \\eqref{1.1}. For the approximate controllability results of the linear system, we first formulate the linear regulator problem to obtain the existence of an optimal control (Theorem \\ref{optimal}), and then derive the explicit expression of optimal control in Lemma \\ref{lem3.1}. With the help of this optimal control, we investigate the approximate controllability of the linear system \\eqref{3.2} in Theorem \\ref{lem4.2}. In section \\ref{semilinear}, we first show the existence of a mild solution of the system \\eqref{1.1}, by invoking Schauder's fixed point theorem. Then, we demonstrate the approximate controllability results of the system \\eqref{1.1} in Theorem \\ref{thm4.4}. A concrete example is presented in the final section to illustrate the developed results in previous sections.\n\n\t\\section{Preliminaries}\\label{pre}\\setcounter{equation}{0}\n\tLet us first recall some basic definitions and properties from fractional calculus. For a detailed study, the interested readers are referred to see \\cite{AAK,IPF}, etc. \tLet $\\mathrm{AC}([a,b];\\mathbb{R})$ denote the space of all absolutely continuous functions from $[a,b]$ to $\\mathbb{R}$ and $\\mathrm{AC}^n([a,b];\\mathbb{R}),$ for $n\\in\\mathbb{N},$ represent the space of all  functions $f:[a,b]\\to\\mathbb{R}$, which have continuous derivatives up to order $n-1$  with $f^{(n-1)}\\in\\mathrm{AC}([a,b];\\mathbb{R})$. \n\t\\begin{Def}\n\t\tThe \\emph{fractional integral} of a function $f:[a,b]\\to\\mathbb{R}$, $a,b\\in\\mathbb{R}$ with $a<b$, of order $q>0$ is defined as\n\t\t\\begin{align*}\n\t\tI_{a}^{q}f(t):=\\frac{1}{\\Gamma(q)}\\int_{a}^{t}\\frac{f(s)}{(t-s)^{1-q}}\\mathrm{d}s,\\ \\mbox{ for a.e. } \\  t\\in[a,b],\n\t\t\\end{align*}\n\t\twhere $f\\in\\mathrm{L}^1([a,b];\\mathbb{R})$ and $\\Gamma(\\alpha)=\\int_{0}^{\\infty}t^{\\alpha-1}e^{-t}\\mathrm{d}t$ is the Euler gamma function.\n\t\\end{Def}\n\t\\begin{Def\n\t\tThe \\emph{Riemann-Liouville fractional derivative} of a function $f:[a,b]\\to\\mathbb{R}$  of order $q>0$ is given as \n\t\t\\begin{align*}\n\t\t^L\\mathrm{D}_{a,t}^{q}f(t):=\\frac{1}{\\Gamma(n-q)}\\frac{d^n}{dt^n}\\int_{a}^{t}(t-s)^{n-q-1}f(s)\\mathrm{d}s,\\ \\mbox{ for a.e. }\\ t\\in[a,b],\n\t\t\\end{align*}\n\t\twith $n-1< q<n,$ and the function $f\\in\\mathrm{AC}^n([a,b];\\mathbb{R})$.\n\t\\end{Def}\n\t\\begin{Def\n\t\tThe \\emph{Caputo fractional derivative} of a function $f\\in\\mathrm{AC}^n([a,b];\\mathbb{R})$ of order $q>0$   is defined as \n\t\t\\begin{align*}\n\t\t^C\\mathrm{D}_{a,t}^{q}f(t):=\\ ^L\\mathrm{D}_{a,t}^{q}\\left[f(t)-\\sum_{p=1}^{n-1}\\frac{f^{(p)}(a)}{p!}(x-a)^p\\right],\\ \\mbox{ for a.e. } \\ t\\in[a,b].\n\t\t\\end{align*}\n\t\\end{Def}\n\n\n\n\t\\begin{rem}[\\cite{IPF}]\n\t\tIf a function $f\\in\\mathrm{AC}^n([a,b];\\mathbb{R})$, then the Caputo fractional derivative of $f$ can be written as\n\t\t\\begin{align*}\n\t\t^C\\mathrm{D}_{a,t}^{q}f(t)=\\frac{1}{\\Gamma(n-q)}\\int_{a}^{t}(t-s)^{n-q-1}f^{(n)}(s)\\mathrm{d}s,\\ \\mbox{ for a.e. }\\ \\ t \\in[a,b], \\ n-1< q<n.\n\t\t\\end{align*}\n\t\\end{rem}\n\t\n\t\n\tIn order to define the mild solution of the system \\eqref{1.1}, first we consider the one-dimensional stable probability density function (cf. \\cite{Ua} \\cite{MPG})\n\t\\begin{align*}\n\t\\phi_{q}(\\xi)=\\frac{1}{\\pi}\\sum_{n=1}^{\\infty}(-1)^{n-1}\\xi^{-nq -1}\\frac{\\Gamma(nq+1)}{n!}\\sin(n\\pi q),\\ \\xi\\in(0,\\infty), 0<q<1,\n\t\\end{align*}\n\tand for any $x\\in\\mathbb{X}$, we define\n\t\\begin{align*}\n\t\\mathcal{T}_{q}(t)x=\\int_{0}^{\\infty}\\!\\!\\varphi_{q}(\\xi)\\mathcal{T}(t^{q}\\xi)x\\mathrm{d}\\xi \\ \\mbox{ and }\\\n\t\\widehat{\\mathcal{T}}_{q}(t)x=q\\int_{0}^{\\infty}\\!\\!\\xi\\varphi_{q}(\\xi)\\mathcal{T}(t^{q}\\xi)x\\mathrm{d}\\xi,\n\t\\end{align*}\n\twhere $\\varphi_{q}(\\xi)=\\frac{1}{q}\\xi^{-1-\\frac{1}{q}}\\phi_{q}(\\xi^{-\\frac{1}{q}})$ and  $ \\mathcal{T}(t) $ is the $\\mathrm{C}_0$-semigroup. Note that $\\varphi_{q}$ is also a probability density function on $(0, \\infty)$, so that\n\t\\begin{align*}\n\t\\int_{0}^{\\infty}\\varphi_{q}(\\xi)\\mathrm{d}\\xi=1\\ \\mbox{ and }\\ \\int_{0}^{\\infty}\\xi\\varphi_q(\\xi)\\mathrm{d}\\xi=\\frac{1}{\\Gamma(1+q)},\n\t\\end{align*}\n\twhere the final expression is the first moment of $\\varphi_q$.  \n\t\n\tLet us provide some important properties of the operators $\\mathcal{T}_{q}(t)$ and $\\widehat{\\mathcal{T}}_{q}(t),$ for $t\\geq 0$.\n\t\\begin{lem}[\\cite{YZ}] \\label{lem2.5}The operators $\\mathcal{T}_{q}(t)$ and $\\widehat{\\mathcal{T}}_{q}(t)$ have the following properties:\n\t\t\\begin{enumerate}\n\t\t\t\\item [(i)] For any fixed $t\\ge0,$ the operators $\\mathcal{T}_{q}(t)$ and  $\\widehat{\\mathcal{T}}_{q}(t)$ are linear and bounded. Moreover\n\t\t\t\\begin{align*}\n\t\t\t\\left\\|\\mathcal{T}_{q}(t)\\right\\|_{\\mathcal{L}(\\mathbb{X})}\\le M \\ \\mbox{ and }\\  \\|\\widehat{\\mathcal{T}}_{q}(t)\\|_{\\mathcal{L}(\\mathbb{X})}\\le\\frac{M q}{\\Gamma(1+q)},\n\t\t\t\\end{align*} \n\t\t\twhere $M$ is a constant such that $\\left\\|\\mathcal{T}(t)\\right\\|_{\\mathcal{L(\\mathbb{X})}}\\leq M$.\n\t\t\t\\item [(ii)] The operators $\\mathcal{T}_{q}(t)$ and  $\\widehat{\\mathcal{T}}_{q}(t)$ are strongly continuous for $t\\ge0$.\n\t\t\t\\item [(iii)] If $\\mathcal{T}(t)$ is compact for $t>0$, then the operators $\\mathcal{T}_{q}(t)$ and  $\\widehat{\\mathcal{T}}_{q}(t)$ are also compact for $t>0$. \n\t\t\\end{enumerate}\n\t\\end{lem}\n\tLet us define the set \n\t\\begin{align*} \n\t\\mathrm{PC}(J;\\mathbb{X})&:=\\big\\{x:J \\rightarrow \\mathbb{X} : x\\vert_{t\\in I_k}\\in\\mathrm{C}(I_k;\\mathbb{X}),\\ I_k:=(t_k, t_{k+1}],\\ k=0,1,\\ldots,m \\ \\mbox{ and }\\ x(t_k^+)\\\\&\\qquad \\qquad \\mbox{ and }\\ x(t_k^-)\\ \\mbox{ exist for each }\\ k=1,\\ldots,m, \\ \\mbox{ and satisfy }\\ x(t_k)=x(t_k^-)\\big\\}, \n\t\\end{align*}\n\tendowed with the norm $\\left\\|x\\right\\|_{\\mathrm{PC}(J;\\mathbb{X})}:=\\sup\\limits_{t\\in J}\\left\\|x\\right\\|_{\\mathbb{X}}$.\n\t\n\tWe now introduce the concept of mild solution for the system \\eqref{1.1} (cf. \\cite{JRY}). \n\t\\begin{Def}[Mild solution]\\label{def2.6}\n\t\tA function $x(\\cdot;\\psi,u):(-\\infty, T]\\to\\mathbb{X}$   is said to be a  \\emph{mild solution} of \\eqref{1.1}, if  $x_0=\\psi\\in\\mathfrak{B}$ and $x\\vert_{J}\\in\\mathrm{PC}(J;\\mathbb{X})$ and satisfies the following:\n\t\t\\begin{equation}\\label{2.2}\n\t\tx(t)=\n\t\t\\begin{dcases}\n\t\t\\mathcal{T}_{\\alpha}(t)\\psi(0)+\\int_{0}^{t}(t-s)^{\\alpha-1}\\widehat{\\mathcal{T}}_{\\alpha}(t-s)\\left[\\mathrm{B}u(s)+f(s,x_{\\rho(s, x_s)})\\right]\\mathrm{d}s,\\ t\\in[0, t_1],\\\\\n\t\th_k(t, x(t_k^-)),\\ t\\in(t_k, \\tau_k],\\ k=1,\\ldots,m,\\\\\n\t\t\\mathcal{T}_{\\alpha}(t-\\tau_k)h_k(\\tau_k, x(t_k^-))-\\int_{0}^{\\tau_k}(\\tau_k-s)^{\\alpha-1}\\widehat{\\mathcal{T}}_{\\alpha}(\\tau_k-s)\\left[\\mathrm{B}u(s)+f(s,x_{\\rho(s, x_s)})\\right]\\mathrm{d}s\\\\\\quad+\\int_{0}^{t}(t-s)^{\\alpha-1}\\widehat{\\mathcal{T}}_{\\alpha}(t-s)\\left[\\mathrm{B}u(s)+f(s,x_{\\rho(s, x_s)})\\right]\\mathrm{d}s,\\ t\\in(\\tau_k,t_{k+1}],\\ k=1,\\ldots,m.\n\t\t\\end{dcases}\n\t\t\\end{equation}\n\t\\end{Def}\n\n\t\\subsection{Phase space}\n\tWe now provide the definition of phase space $\\mathfrak{B}$ introduced in \\cite{HY}, and suitably modify to incorporate the impulsive systems (cf. \\cite{VOj}). The linear space $\\mathfrak{B}$ equipped with the seminorm $\\left\\|\\cdot\\right\\|_{\\mathfrak{B}}$, consisting of all functions from $(-\\infty, 0]$ into $\\mathbb{X}$  and satisfying the following axioms:\n\t\\begin{enumerate}\n\t\t\\item [(A1)] If $x: (-\\infty, T]\\rightarrow \\mathbb{X}$ such that $x_{0}\\in \\mathfrak{B}$ and $x|_{J}\\in \\mathrm{PC}(J;\\mathbb{X})$. Then the following conditions hold:\n\t\t\\begin{itemize}\n\t\t\t\\item [(i)] $x_{t}\\in\\mathfrak{B}$ for $t\\in J$.\n\t\t\t\\item [(ii)] $\\left\\|x_{t}\\right\\|_{\\mathfrak{B}}\\leq \\Lambda(t)\\sup\\{\\left\\|x(s)\\right\\|_{\\mathbb{X}}: 0\n\t\t\t\\leq s\\leq t\\}+\\Upsilon(t)\\left\\|x_{0}\\right\\|_{\\mathfrak{B}},$  for $t\\in J$, where $\\Lambda, \\Upsilon:[0, \\infty)\\rightarrow [0, \\infty)$ are independent of $x$, the function $\\Lambda(\\cdot)$ is strictly positive and continuous, $\\Upsilon(\\cdot)$ is locally bounded.\n\t\t\n\t\t\\end{itemize}\n\t\t\\item [(A2)] The space $\\mathfrak{B}$ is complete. \n\t\\end{enumerate}  \n\tFor any $\\psi\\in \\mathfrak{B}$, the function $\\psi_{t}, \\ t\\leq 0,$ defined as $\\psi_{t}(\\theta)=\\psi(t+\\theta),\\  \\theta \\in (-\\infty, 0].$ Then for any function $x(\\cdot)$ satisfying the axiom (A1) with $x_{0}=\\psi$, we can extend the mapping $t\\mapsto x_{t}$ by setting $x_{t}=\\psi_{t}, \\  t\\leq0$, to the whole interval $(-\\infty, T]$. Moreover, let us  introduce a set \n\t\\begin{align*}\n\t\\mathcal{Q}(\\rho^-)&=\\{\\rho(s, \\varphi):\\  \\rho(s, \\varphi)\\leq 0, \\mbox{ for } (s, \\varphi)\\in J \\times \\mathfrak{B}\\}.\n\t\\end{align*} \n\tAssume that the function $t\\mapsto \\psi_{t}$,  defined from $\\mathcal{Q}(\\rho^-)$ into $\\mathfrak{B}$ is continuous, and there exists a continuous and bounded function $\\varTheta^{\\psi}: \\mathcal{Q}(\\rho^-)\\rightarrow (0, \\infty)$ such that  \n\t\\begin{align*}\n\t\\left\\|\\psi_{t}\\right\\|_{\\mathfrak{B}}&\\leq \\varTheta^{\\psi}(t)\\left\\|\\psi\\right\\|_{\\mathfrak{B}}.\n\t\\end{align*}\n\t\\begin{lem}[\\cite{HY}]{\\label{lem2.7}}\n\t\tLet $x:(-\\infty, T]\\rightarrow \\mathbb{X}$ be a function such that $x_0=\\psi$ and $x\\vert_J\\in\\mathrm{PC}(J;\\mathbb{X})$. Then \n\t\t\\begin{align*}\n\t\t\\left\\|x_s\\right\\|_{\\mathfrak{B}}&\\leq H_{1}\\left\\|\\psi\\right\\|_{\\mathfrak{B}}+H_{2}\\sup \\big\\{ \\left\\|x(\\theta)\\right \\|_{\\mathbb{X}}:\\theta \\in [0, \\max \\{0, s\\}] \\big\\}, \\ s\\in \\mathcal{Q}(\\rho^-) \\cup J,\n\t\t\\end{align*}\n\t\twhere $$H_{1}= \\sup\\limits_{t\\in \\mathcal{Q}(\\rho^-)} \\Theta^{\\psi}(t)  + \\sup\\limits_{t\\in J}\\Upsilon(t), \\;\\; H_{2}= \\sup\\limits_{t\\in J}\\Lambda(t).$$ \n\t\\end{lem}\n\t\\begin{Ex}\\label{exm2.8}\n\t\tLet us take $\\mathfrak{B}=\\mathrm{PC}_{r}\\times\\mathrm{L}^p_h(\\mathbb{X}), r\\ge0, 1\\le p<\\infty$, which consists of all functions $\\psi:(-\\infty,0]\\to\\mathbb{X}$ such that $\\psi\\vert_{[-r,0]}\\in \\mathrm{PC}([-r,0];\\mathbb{X}),$ Lebesgue measurable on $(-\\infty,-r)$ and $h\\|\\psi(\\cdot)\\|^p_{\\mathbb{X}}$ is Lebesgue integrable on $(-\\infty,-r]$. The seminorm in $\\mathfrak{B}$ is defined as \n\t\t\\begin{align}\n\t\t\\label{Bnorm}\\left\\|\\psi\\right\\|_{\\mathcal{P}}:=\\int_{-r}^{0}\\left\\|\\psi(\\theta)\\right\\|_\\mathbb{X}\\mathrm{d}\\theta+\\left(\\int_{-\\infty}^{-r}h(\\theta)\\left\\|\\psi(\\theta)\\right\\|^p_{\\mathbb{X}}\\mathrm{d}\\theta\\right)^{\\frac{1}{p}},\n\t\t\\end{align}\n\t\twhere the function $h:(-\\infty, 0]\\to\\mathbb{R}^+$ is locally bounded and Lebesgue integrable. Moreover, there exists a locally bounded function $H:(-\\infty,0]\\to\\mathbb{R}^+$ such that $h(t+\\theta)\\le H(t)h(\\theta),$ for all $t\\le0$ and $\\theta\\in(-\\infty,0)\\backslash \\mathcal{O}_t$ where $\\mathcal{O}_t\\subseteq(-\\infty,0)$ is a set with Lebesgue measure zero. \n\t\\end{Ex}\n\n\t\\subsection{Resolvent operator and assumptions}\n\tTo discuss the approximate controllability of the system \\eqref{1.1}, we first define the following operators:\n\t\\begin{equation}\\label{2.1}\n\t\\left\\{\n\t\\begin{aligned}\n\tL_0^Tu&:=\\int^{T}_{0}(T-t)^{\\alpha-1}\\widehat{\\mathcal{T}}_{\\alpha}(T-t)\\mathrm{B}u(t)\\mathrm{d}t,\\\\\n\t\\Phi_{0}^{T}&:=\\int^{T}_{0}(T-t)^{2(\\alpha-1)}\\widehat{\\mathcal{T}}_{\\alpha}(T-t)\\mathrm{B}\\mathrm{B}^*\\widehat{\\mathcal{T}}_{\\alpha}(T-t)^*\\mathrm{d}t,\\\\\n\t\\mathcal{R}(\\lambda,\\Phi_{0}^{T})&:=(\\lambda \\mathrm{I}+\\Phi_{0}^{T}\\mathcal{J})^{-1},\\ \\lambda > 0,\n\t\\end{aligned}\n\t\\right.\n\t\\end{equation}\n\twhere $\\mathrm{B}^{*}$ and $\\widehat{\\mathcal{T}}_{\\alpha}(t)^*$ denote the adjoint operators of $\\mathrm{B}$ and $\\widehat{\\mathcal{T}}_{\\alpha}(t),$ respectively. It is immediate that the operator $L_0^T$ is linear and bounded for $\\frac{1}{2}<\\alpha<1$. Moreover, the map $\\mathcal{J} : \\mathbb{X} \\rightarrow 2^{\\mathbb{X}^*}$ stands for the duality mapping, which is defined as \n\t\\begin{align*}\n\t\\mathcal{J}[x]&=\\{x^* \\in \\mathbb{X}^* : \\langle x, x^* \\rangle=\\left\\|x\\right\\|_{\\mathbb{X}}^2= \\left\\|x^*\\right\\|_{\\mathbb{X}^*}^2 \\}, \\mbox{ for all } x\\in \\mathbb{X},\n\t\\end{align*}\n\twhere  $\\langle \\cdot, \\cdot  \\rangle $  represents  a duality pairing between $\\mathbb{X}$ and $\\mathbb{X}^*$. Since the space $\\mathbb{X}$ is a reflexive Banach space, then  $\\mathbb{X}^*$ becomes strictly convex (see, \\cite{AA}), which implies that the mapping $\\mathcal{J}$ is bijective, strictly monotonic and demicontinuous, that is, $$x_k\\to x\\ \\mbox{ in }\\ \\mathbb{X}\\ \\mbox{ implies }\\ \\mathcal{J}[x_k] \\xrightharpoonup{w} \\mathcal{J}[x]\\ \\mbox{ in } \\ \\mathbb{X}^*\\ \\mbox{ as }\\ k\\to\\infty.$$ Moreover, the inverse mapping $\\mathcal{J}^{-1}:\\mathbb{X}^*\\to\\mathbb{X}$ is also duality mapping.\n\t\\begin{rem}\\label{rem2.8}\n\t\tIf $\\mathbb{X}$ is a separable Hilbert space (identified with its own dual), then the resolvent operator is defined as $\\mathcal{R}(\\lambda,\\Phi_{0}^{T}):=(\\lambda \\mathrm{I}+\\Phi_{0}^{T})^{-1},\\ \\lambda > 0$.\n\t\\end{rem}\n\t\\begin{lem}[Lemma 2.2 \\cite{M}]\\label{lem2.9}\n\t\tFor every $h\\in\\mathbb{X}$ and $\\lambda>0$, the equation\n\t\t\\begin{align}\\label{2.4}\\lambda z_{\\lambda}+\\Phi_{0}^{T}\\mathcal{J}[z_{\\lambda}]=\\lambda h,\\end{align}\n\t\thas a unique solution  $z_{\\lambda}(h)=\\lambda(\\lambda \\mathrm{I}+\\Phi_{0}^{T}\\mathcal{J})^{-1}(h)=\\lambda\\mathcal{R}(\\lambda,\\Phi_{0}^{T})(h)$ and \\begin{align}\\label{2.5}\n\t\t\\left\\|z_{\\lambda}(h)\\right\\|_{\\mathbb{X}}=\\left\\|\\mathcal{J}[z_{\\lambda}(h)]\\right\\|_{\\mathbb{X}^*}\\leq\\left\\|h\\right\\|_{\\mathbb{X}}.\n\t\t\\end{align}\n\t\t\\begin{proof}\n\t\t\tSince the non-negative operator $\\Phi_0^T$ is linear and bounded for $\\frac{1}{2}<\\alpha<1$, then proceeding similar way as in the proof of Lemma 2.2 \\cite{M}, one can obtain the results.\n\t\t\\end{proof}\n\t\\end{lem}\n\t\n\t\n\t\n\t\\begin{Def}[\\cite{FUX}]\n\t\tThe system \\eqref{1.1} is said to be \\emph{approximately controllable} on $ J $, for any initial function $\\psi\\in\\mathfrak{B}$, if the closure of reachable set is whole space $\\mathbb{X}$, that is, $\\overline{\\mathfrak{R}(T,\\psi)}=\\mathbb{X},$ where the reachable set is defined as \\begin{align*}\\mathfrak{R}(T,\\psi) = \\{x(T;\\psi,u): u(\\cdot) \\in \\mathrm{L}^{2}(J;\\mathbb{U})\\}.\\end{align*}\n\t\\end{Def}\n\tWe impose the following assumptions to investigate the approximate controllability of the system \\eqref{1.1}:\n\t\\begin{Ass}\\label{as2.1} \n\t\t\\begin{enumerate}\n\t\t\t\\item [\\textit{($H0$)}] For every $h\\in\\mathbb{X}$, $z_\\lambda=z_{\\lambda}(h)=\\lambda\\mathcal{R}(\\lambda,\\Phi_{0}^{T})(h) \\rightarrow 0$ as $\\lambda\\downarrow 0$ in strong topology, where $z_{\\lambda}(h)$ is a solution of the equation \\eqref{2.4}.\n\t\t\t\\item[\\textit{(H1)}] The $\\mathrm{C}_0$-semigroup of bounded linear operator $\\mathcal{T}(t)$ is compact for $t>0$ with bound $M\\geq 1,$ such that $\\|\\mathcal{T}(t)\\|_{\\mathcal{L(\\mathbb{X})}}\\leq M$.\n\t\t\t\\item [\\textit{(H2)}] \n\t\t\t\\begin{enumerate} \n\t\t\t\t\\item [(i)] Let $x:(-\\infty, T]\\rightarrow \\mathbb{X}$ be such that $x_0=\\psi$ and $x|_{J}\\in \\mathrm{PC}(J;\\mathbb{X}).$ The function $t\\mapsto f(t, x_{\\rho(t,x_{t})}) $ is strongly measurable on $J$  and the function $f(t,\\cdot): \\mathfrak{B}\\rightarrow \\mathbb{X}$ is continuous for a.e. $t\\in J$. Also, the map $t\\mapsto f(s,x_{t})$ is continuous on $\\mathcal{Q}(\\rho^-) \\cup J,$ for every $s\\in J$. \n\t\t\t\t\\item [(ii)] For each positive integer $r$, there exists a constant $\\alpha_1\\in[0,\\alpha]$ and a function $\\gamma_{r}\\in \\mathrm{L}^{\\frac{1}{\\alpha_1}}(J;\\mathbb{R^{+}})$, such that $$  \\sup_{\\left\\| \\psi\\right\\|_{\\mathfrak{B}}\\leq r} \\left\\|f(t, \\psi)\\right\\|_{\\mathbb{X}}\\leq\\gamma_{r}(t), \\mbox{ for a.e.} \\ t \\in J \\mbox{ and } \\psi\\in \\mathfrak{B},$$ with\n\t\t\t\t$$ \\liminf_{r \\rightarrow \\infty } \\frac {\\left\\|\\gamma_r\\right\\|_{\\mathrm{L}^{\\frac{1}{\\alpha_1}}(J;\\mathbb{R^+})}}{r} = \\beta< \\infty. $$ \n\t\t\t\\end{enumerate}\n\t\t\t\\item [\\textit{(H3)}] The impulses $ h_k:[t_k,\\tau_k]\\times\\mathbb{X}\\to\\mathbb{X}$, for  $k=1,\\dots,m$, are such that \n\t\t\t\\begin{itemize}\n\t\t\t\t\\item [(i)]The impulses $h_k(\\cdot,x):[t_k,\\tau_k]\\to\\mathbb{X}$ are continuous for each $x\\in\\mathbb{X}$. \n\t\t\t\t\\item [(ii)] Each $h_k(t,\\cdot):\\mathbb{X}\\to\\mathbb{X}$ is completely continuous, for all  $t\\in[t_k,\\tau_k]$.\n\t\t\t\t\\item[(iii)] $\\left\\|h_k(t,x) \\right\\|_{\\mathbb{X}} \\leq l_k, \\mbox{ for each } t\\in [t_k, \\tau_k] \\mbox{ and } x \\in \\mathbb{X},$ where $l_k$'s are positive constants.\n\t\t\t\\end{itemize} \n\t\t\\end{enumerate}\n\t\\end{Ass}\n\tThe following version of the discrete Gronwall-Bellman lemma (cf. \\cite{Ch}) is used in the sequel.  \n\t\\begin{lem}\\label{lem2.13}\n\tIf $\\{f_n\\}_{n=0}^{\\infty}, \\{g_n\\}_{n=0}^{\\infty}$ and $\\{w_n\\}_{n=0}^{\\infty}$ are non-negative sequences and $$f_n\\le g_n+\\sum_{k=0}^{n-1}w_kf_k,\\ \\text{ for }\\ n\\geq 0,$$  then $$f_n\\le g_n+\\sum_{k=0}^{n-1}g_kw_k\\exp\\left(\\sum_{j=k+1}^{n-1}g_j\\right),\\text{  for }\\ n\\geq 0.$$\n\t\\end{lem}\n\t\n\n\t\\section{Linear Control Problem} \\label{Linear}\\setcounter{equation}{0}\n\tThe present section is devoted for discussing the approximate controllability of the fractional order linear control problem corresponding to \\eqref{1.1}. To establish this result, we first obtain the existence of an optimal control by minimizing the cost functional given by\n\t\\begin{equation}\\label{3.1}\n\t\\mathcal{G}(x,u)=\\left\\|x(T)-x_{T}\\right\\|^{2}_{\\mathbb{X}}+\\lambda\\int^{T}_{0}\\left\\|u(t)\\right\\|^{2}_{\\mathbb{U}}\\mathrm{d}t,\n\t\\end{equation}\n\twhere $x(\\cdot)$ is the solution of the linear control system:\n\t\\begin{equation}\\label{3.2}\n\t\\left\\{\n\t\\begin{aligned}\n\t^CD_{0,t}^{\\alpha}x(t)&= \\mathrm{A}x(t)+\\mathrm{B}u(t),\\ t\\in J,\\\\\n\tx(0)&=\\zeta,\n\t\\end{aligned}\n\t\\right.\n\t\\end{equation}\n\twith the control $u\\in \\mathbb{U}$, $x_{T}\\in \\mathbb{X}$ and $\\lambda >0$. Since $\\mathrm{B}u\\in\\mathrm{L}^1(J;\\mathbb{X})$, the system \\eqref{3.2} has a unique mild solution $x\\in \\mathrm{C}(J;\\mathbb{X}) $ given by (see Corollary 2.2, Chapter 4, \\cite{P} and Lemma 4.68, Chapter 4, \\cite{Y})\n\t\\begin{align*}\n\tx(t)= \\mathcal{T}_{\\alpha}(t)\\zeta+\\int_{0}^{t}(t-s)^{\\alpha-1}\\widehat{\\mathcal{T}}_{\\alpha}(t-s)\\mathrm{B}u(s)\\mathrm{d}s,\n\t\\end{align*}\n\tfor any $u\\in\\mathscr{U}_{\\mathrm{ad}}=\\mathrm{L}^2(J;\\mathbb{U})$ (admissible control class).  Next, we define the \\emph{admissible class} $\\mathscr{A}_{\\mathrm{ad}}$  for the system \\eqref{3.2} as\n\t\\begin{align*}\n\t\\mathscr{A}_{\\mathrm{ad}}:=\\big\\{(x,u) :x\\mbox{ is \\mbox{the unique mild solution} of }\\eqref{3.2}  \\mbox{ with the control }u\\in\\mathscr{U}_{\\mathrm{ad}}\\big\\}.\n\t\\end{align*}\n\tFor any given control $u\\in\\mathscr{U}_{\\mathrm{ad}}$, the system \\eqref{3.2} has a unique mild solution, which ensures that the set $\\mathscr{A}_{\\mathrm{ad}}$ is nonempty. By using the definition of the cost functional, we can formulate the optimal control problem as:\n\t\\begin{align}\\label{3.3}\n\t\\min_{ (x,u) \\in \\mathscr{A}_{\\mathrm{ad}}}  \\mathcal{G}(x,u).\n\t\\end{align}\n\tIn the next theorem, we show the existence of an optimal pair for the problem \\eqref{3.3}.\n\t\\begin{theorem}[Existence of an optimal pair]\\label{optimal}\n\t\tFor a given $\\zeta\\in\\mathbb{X}$ and fixed $\\frac{1}{2}<\\alpha<1$, there exists a unique optimal pair  $(x^0,u^0)\\in\\mathscr{A}_{\\mathrm{ad}}$ for the problem \\eqref{3.3}.\n\t\\end{theorem}\n\t\\begin{proof}\n\t\tLet us assume $$\\mathcal{G} := \\inf \\limits _{u \\in \\mathscr{U}_{\\mathrm{ad}}}\\mathcal{G}(x,u).$$ Since $0\\leq \\mathcal{G} < +\\infty$, there exists a minimizing sequence $\\{u^n\\}_{n=1}^{\\infty} \\in \\mathscr{U}_{\\mathrm{ad}}$ such that $$\\lim_{n\\to\\infty}\\mathcal{G}(x^n,u^n) = \\mathcal{G},$$ where $x^n(\\cdot)$ is the unique mild solution of the system \\eqref{3.2}, corresponding to the control $u^n(\\cdot),$  for each $n\\in\\mathbb{N}$ with $x^n(0)=\\zeta$.\tNote that $x^n(\\cdot)$ satisfies\n\t\t\\begin{align}\\label{3.4}\n\t\tx^n(t)&=\\mathcal{T}_{\\alpha}(t)\\zeta+\\int_{0}^{t}(t-s)^{\\alpha-1}\\widehat{\\mathcal{T}}_{\\alpha}(t-s)\\mathrm{B}u^n(s)\\mathrm{d}s,\n\t\t\\end{align} \n\t\tfor $t\\in J$. Since  $0\\in\\mathscr{U}_{\\mathrm{ad}}$, without loss of generality, we may assume that $\\mathcal{G}(x^n,u^n) \\leq \\mathcal{G}(x,0)$, where $(x,0)\\in\\mathscr{A}_{\\mathrm{ad}}$. Using the definition of $\\mathcal{G}(\\cdot,\\cdot)$, we easily get\n\t\t\\begin{align}\\label{35}\n\t\t\\left\\|x^n(T)-x_{T}\\right\\|^{2}_{\\mathbb{X}}+\\lambda\\int^{T}_{0}\\left\\|u^n(t)\\right\\|^{2}_{\\mathbb{U}}\\mathrm{d}t\\leq \\left\\|x(T)-x_{T}\\right\\|^{2}_{\\mathbb{X}}\\leq 2\\left(\\|x(T)\\|_{\\mathbb{X}}^2+\\|x_T\\|_{\\mathbb{X}}^2\\right)<+\\infty.\n\t\t\\end{align}\n\t\tFrom the above estimate, it is clear that, there exists a large $L>0$ (independent of $n$), such that \n\t\t\\begin{align}\\label{3.5}\\int_0^T \\|u^n(t)\\|^2_{\\mathbb{U}} \\mathrm{d} t \\leq L < +\\infty .\\end{align}\n\t\tUsing the expression \\eqref{3.4}, we compute\n\t\t\\begin{align*}\n\t\t\\left\\|x^n(t)\\right\\|_{\\mathbb{X}}&\\le\\left\\|\\mathcal{T}_{\\alpha}(t)\\zeta\\right\\|_{\\mathbb{X}}+\\left\\|\\int_{0}^{t}(t-s)^{\\alpha-1}\\widehat{\\mathcal{T}}_{\\alpha}(t-s)\\mathrm{B}u^n(s)\\mathrm{d}s\\right\\|_{\\mathbb{X}}\\nonumber\\\\ &\\le\\left\\|\\mathcal{T}_{\\alpha}(t)\\right\\|_{\\mathcal{L}(\\mathbb{X})}\\left\\|\\zeta\\right\\|_{\\mathbb{X}}+\\int_{0}^{t}(t-s)^{\\alpha-1}\\|\\widehat{\\mathcal{T}}_{\\alpha}(t-s)\\|_{\\mathcal{L}(\\mathbb{X})}\\left\\|\\mathrm{B}\\right\\|_{\\mathcal{L}(\\mathbb{U},\\mathbb{X})}\\left\\|u^n(s)\\right\\|_{\\mathbb{U}}\\mathrm{d}s\\nonumber\\\\\n\t\t&\\le M\\left\\|\\zeta\\right\\|_{\\mathbb{X}}+\\frac{M\\tilde{M}\\alpha}{\\Gamma(1+\\alpha)}\\int_{0}^{t}(t-s)^{\\alpha-1}\\left\\|u^n(s)\\right\\|_{\\mathbb{U}}\\mathrm{d}s\n\t\t\\nonumber\\\\&\\le M\\left\\|\\zeta\\right\\|_{\\mathbb{X}}+\\frac{M\\tilde{M}\\alpha}{\\Gamma(1+\\alpha)}\\frac{T^{2\\alpha-1}}{2\\alpha-1}\\left(\\int_{0}^{t}\\left\\|u^n(s)\\right\\|_{\\mathbb{U}}^2\\mathrm{d}s\\right)^{\\frac{1}{2}}\\nonumber\\\\&\\le M\\left\\|\\zeta\\right\\|_{\\mathbb{X}}+\\frac{M\\tilde{M}\\alpha}{\\Gamma(1+\\alpha)}\\frac{T^{2\\alpha-1}L^{\\frac{1}{2}}}{2\\alpha-1}<+\\infty,\n\t\t\\end{align*}\n\t\tfor all  $t\\in J$ and $\\frac{1}{2}<\\alpha<1$. Since we know that $\\mathrm{L}^{2}(J;\\mathbb{X})$ is reflexive, then by applying the Banach-Alaoglu theorem, we can find a subsequence $\\{x^{n_k}\\}_{k=1}^{\\infty}$ of $\\{x^n\\}_{n=1}^{\\infty}$ such that \n\t\t\\begin{align}\\label{3.6}\n\t\tx^{n_k}\\xrightharpoonup{w}x^0\\ \\mbox{ in }\\ \\mathrm{L}^{2}(J;\\mathbb{X}), \\ \\mbox{ as }\\ k\\to\\infty. \n\t\t\\end{align}\n\t\tFrom the estimate \\eqref{3.5}, we also infer that the sequence $\\{u^n\\}_{n=1}^{\\infty}$ is uniformly bounded in the space $\\mathrm{L}^2(J;\\mathbb{U})$. Further, by using the Banach-Alaoglu theorem, there exists a subsequence, say,  $\\{u^{n_k}\\}_{k=1}^{\\infty}$ of $\\{u^n\\}_{n=1}^{\\infty}$ such that \n\t\t\\begin{align*}\n\t\tu^{n_k}\\xrightharpoonup{w}u^0\\ \\mbox{ in }\\ \\mathrm{L}^2(J;\\mathbb{U})=\\mathscr{U}_{\\mathrm{ad}}, \\ \\mbox{ as }\\ k\\to\\infty. \n\t\t\\end{align*}\n\t\tSince $\\mathrm{B}$ is a bounded linear operator from $\\mathbb{U}$ to $\\mathbb{X}$, then we have \n\t\t\\begin{align}\\label{3.7}\n\t\t\\mathrm{B}\tu^{n_k}\\xrightharpoonup{w}\\mathrm{B}u^0\\ \\mbox{ in }\\ \\mathrm{L}^2(J;\\mathbb{X}),\\ \\mbox{ as }\\ k\\to\\infty.\n\t\t\\end{align}\n\t\tMoreover, by using the above  convergences together with the compactness of the operator $(\\mathrm{Q}f)(\\cdot) =\\int_{0}^{\\cdot}(\\cdot-s)^{\\alpha-1}\\widehat{\\mathcal{T}}_{\\alpha}(\\cdot-s) f(s)\\mathrm{d}s:\\mathrm{L}^2(J;\\mathbb{X})\\rightarrow \\mathrm{C}(J;\\mathbb{X}) $ (see Lemma \\ref{lem2.12} below), we obtain\n\t\t\\begin{align*}\n\t\t\\left\\|\\int_{0}^{t}(t-s)^{\\alpha-1}\\widehat{\\mathcal{T}}_{\\alpha}(t-s)\\mathrm{B}u^{n_k}(s)\\mathrm{d}s-\\int_{0}^{t}(t-s)^{\\alpha-1}\\widehat{\\mathcal{T}}_{\\alpha}(t-s)\\mathrm{B}u(s)\\mathrm{d}s\\right\\|_{\\mathbb{X}}\\to0,\\ \\mbox{ as }\\  k\\to\\infty,\n\t\t\\end{align*}\n\t\tfor all $t\\in J$. We now estimate \n\t\t\\begin{align}\\label{3.8}\n\t\t\\left\\|x^{n_k}(t)-x^*(t)\\right\\|_{\\mathbb{X}}&=\\left\\|\\int_{0}^{t}(t-s)^{\\alpha-1}\\widehat{\\mathcal{T}}_{\\alpha}(t-s)\\mathrm{B}u^{n_k}(s)\\mathrm{d}s-\\int_{0}^{t}(t-s)^{\\alpha-1}\\widehat{\\mathcal{T}}_{\\alpha}(t-s)\\mathrm{B}u^{0}(s)\\mathrm{d}s\\right\\|_{\\mathbb{X}}\\nonumber\\\\&\\to 0,\\ \\mbox{ as } \\ k\\to\\infty, \\ \\mbox{for all }\\ t\\in J,\n\t\t\\end{align}\n\t\twhere \n\t\t\\begin{align*}\n\t\tx^*(t)=\\mathcal{T}_{\\alpha}(t)\\zeta+\\int_{0}^{t}(t-s)^{\\alpha-1}\\widehat{\\mathcal{T}}_{\\alpha}(t-s)\\mathrm{B}u^{0}(s)\\mathrm{d}s,\\ t\\in J.\n\t\t\\end{align*}\n\t\tIt is clear by the above expression, the function  $x^*\\in \\mathrm{C}(J;\\mathbb{X})$ is the unique mild solution of the equation \\eqref{3.2} with the control $u^{0}\\in\\mathscr{U}_{\\mathrm{ad}}$. Since the weak limit is unique, then by combining the convergences \\eqref{3.6} and \\eqref{3.8}, we obtain $x^*(t)=x^0(t),$ for all $t\\in J$. Hence, the function $x^0$ is the unique mild solution of the system \\eqref{3.2} with the control $u^{0}\\in\\mathscr{U}_{\\mathrm{ad}}$ and also the whole sequence  $x^n\\to x^0\\in\\mathrm{C}(J;\\mathbb{X})$. Consequently, we have  $(x^0,u^0)\\in\\mathscr{A}_{\\mathrm{ad}}$.\n\t\t\n\t\tIt remains to show that the functional $\\mathcal{G}(\\cdot,\\cdot)$ attains its minimum at  $(x^0,u^0)$, that is, \\emph{$\\mathcal{G}=\\mathcal{G}(x^0,u^0)$}. Since the cost functional $\\mathcal{G}(\\cdot,\\cdot)$ given in \\eqref{3.1} is continuous and convex (see Proposition III.1.6 and III.1.10,  \\cite{EI}) on $\\mathrm{L}^2(J;\\mathbb{X}) \\times \\mathrm{L}^2(J;\\mathbb{U})$, it follows that $\\mathcal{G}(\\cdot,\\cdot)$ is weakly lower semi-continuous (Proposition II.4.5, \\cite{EI}). That is, for a sequence \n\t\t$$(x^n,u^n)\\xrightharpoonup{w}(x^0,u^0)\\ \\mbox{ in }\\ \\mathrm{L}^2(J;\\mathbb{X}) \\times  \\mathrm{L}^2(J;\\mathbb{U}),\\ \\mbox{ as }\\ n\\to\\infty,$$\n\t\twe have \n\t\t\\begin{align*}\n\t\t\\mathcal{G}(x^0,u^0) \\leq  \\liminf \\limits _{n\\rightarrow \\infty} \\mathcal{G}(x^n,u^n).\n\t\t\\end{align*}\n\t\tHence, we obtain \n\t\t\\begin{align*}\\mathcal{G} \\leq \\mathcal{G}(x^0,u^0) \\leq  \\liminf \\limits _{n\\rightarrow \\infty} \\mathcal{G}(x^n,u^n)= \\lim \\limits _{n\\rightarrow \\infty} \\mathcal{G}(x^n,u^n) = \\mathcal{G},\\end{align*}\n\t\tand thus $(x^0,u^0)$ is a minimizer of the problem \\eqref{3.3}. Note that the cost functional given in \\eqref{3.1} is convex, the constraint \\eqref{3.2} is linear and the class $\\mathscr{U}_{\\mathrm{ad}}=\\mathrm{L}^2(J;\\mathbb{U})$ is convex, then the  optimal control obtained above is unique.\n\t\\end{proof}\n\t\n\t\n\tIn the following lemma, we prove the compactness of the operator $(\\mathrm{Q}f)(\\cdot) =\\int_{0}^{\\cdot}(\\cdot-s)^{\\alpha-1}\\widehat{\\mathcal{T}}_{\\alpha}(\\cdot-s) f(s)\\mathrm{d}s:\\mathrm{L}^2(J;\\mathbb{X})\\rightarrow \\mathrm{C}(J;\\mathbb{X}) ,\\ \\mbox{ for }\\ \\frac{1}{2}<\\alpha<1,$ where we assume that $\\mathbb{X}$ is a general Banach space. The case of $\\alpha=1$ is available in Lemma 3.2, \\cite{JYONG}. \n\t\\begin{lem}\\label{lem2.12}\n\t\tSuppose that Assumptions (H1) holds. Let the operator $\\mathrm{Q}:\\mathrm{L}^{2}(J;\\mathbb{X})\\rightarrow \\mathrm{C}(J;\\mathbb{X})$ be defined as\n\t\t\\begin{align}\n\t\t(\\mathrm{Q}\\psi)(t)= \\int^{t}_{0}(t-s)^{\\alpha-1}\\widehat{\\mathcal{T}}_{\\alpha}(t-s)\\psi(s)\\mathrm{d}s, \\ t\\in J,\\ \\frac{1}{2}<\\alpha<1.\n\t\t\\end{align}\n\t\tThen the operator $\\mathrm{Q}$ is compact.\n\t\\end{lem}\n\t\\begin{proof}\n\t\tWe prove that $\\mathrm{Q}$ is a compact operator by using  the infinite-dimensional version of Arzel\\'a-Ascoli theorem (see, Theorem 3.7, Chapter 2, \\cite{JYONG}). Let a closed and bounded ball $\\mathcal{B}_R$ in $\\mathrm{L}^{2}(J;\\mathbb{X})$ be defined as \n\t\t\\begin{align*}\n\t\t\\mathcal{B}_R=\\left\\{\\psi\\in \\mathrm{L}^{2}(J;\\mathbb{X}):\\left\\|\\psi\\right\\|_{\\mathrm{L}^{2}(J;\\mathbb{X})}\\leq R\\right\\}.\n\t\t\\end{align*}\n\t\tFor $s_1,s_2\\in J$  ($s_1<s_2$) and $\\psi\\in\\mathcal{B}_{R}$, we compute the following:\n\t\t\\begin{align}\n\t\t&\\left\\|(\\mathrm{Q}\\psi)(s_2)-(\\mathrm{Q}\\psi)(s_1)\\right\\|_{\\mathbb{X}}\\nonumber\\\\&\\le\\left\\|\\int_{0}^{s_1}\\left[(s_2-s)^{\\alpha-1}\\widehat{\\mathcal{T}}_{\\alpha}(s_2-s)-(s_1-s)^{\\alpha-1}\\widehat{\\mathcal{T}}_{\\alpha}(s_1-s)\\right]\\psi(s)\\mathrm{d}s\\right\\|_{\\mathbb{X}}\\nonumber\\\\&\\quad+\\left\\|\\int_{s_1}^{s_2}(s_2-s)^{\\alpha-1}\\widehat{\\mathcal{T}}_{\\alpha}(s_2-s)\\psi(s)\\mathrm{d}s\\right\\|_{\\mathbb{X}}\\nonumber\\\\&\\le\\left\\|\\int_{0}^{s_1}\\left[(s_2-s)^{\\alpha-1}-(s_1-s)^{\\alpha-1}\\right]\\widehat{\\mathcal{T}}_{\\alpha}(s_2-s)\\psi(s)\\mathrm{d}s\\right\\|_{\\mathbb{X}}\\nonumber\\\\&\\quad+\\left\\|\\int_{0}^{s_1}\\!\\!\\!(s_1-s)^{\\alpha-1}\\left[\\widehat{\\mathcal{T}}_{\\alpha}(s_2-s)-\\widehat{\\mathcal{T}}_{\\alpha}(s_1-s)\\right]\\psi(s)\\mathrm{d}s\\right\\|_{\\mathbb{X}}+\\left\\|\\int_{s_1}^{s_2}\\!\\!\\!(s_2-s)^{\\alpha-1}\\widehat{\\mathcal{T}}_{\\alpha}(s_2-s)\\psi(s)\\mathrm{d}s\\right\\|_{\\mathbb{X}}\\nonumber\\\\&\\le\\frac{M\\alpha}{\\Gamma(1+\\alpha)}\\int_{0}^{s_1}\\left|(s_2-s)^{\\alpha-1}-(s_1-s)^{\\alpha-1}\\right|\\left\\|\\psi(s)\\right\\|_{\\mathbb{X}}\\mathrm{d}s\\nonumber\\\\&\\quad+\\int_{0}^{s_1}(s_1-s)^{\\alpha-1}\\left\\|\\widehat{\\mathcal{T}}_{\\alpha}(s_2-s)-\\widehat{\\mathcal{T}}_{\\alpha}(s_1-s)\\right\\|_{\\mathcal{L}(\\mathbb{X})}\\left\\|\\psi(s)\\right\\|_{\\mathbb{X}}\\mathrm{d}s\\nonumber\\\\&\\quad+\\frac{M\\alpha}{\\Gamma(1+\\alpha)}\\int_{s_1}^{s_2}(s_2-s)^{\\alpha-1}\\left\\|\\psi(s)\\right\\|_{\\mathbb{X}}\\mathrm{d}s\\nonumber\\\\&\\le\\frac{MR\\alpha}{\\Gamma(1+\\alpha)}\\left(\\int_{0}^{s_1}\\left|(s_2-s)^{\\alpha-1}-(s_1-s)^{\\alpha-1}\\right|^2\\mathrm{d}s\\right)^{\\frac{1}{2}}\\nonumber\\\\&\\quad+\\int_{0}^{s_1}\\!\\!\\!(s_1-s)^{\\alpha-1}\\left\\|\\widehat{\\mathcal{T}}_{\\alpha}(s_2-s)-\\widehat{\\mathcal{T}}_{\\alpha}(s_1-s)\\right\\|_{\\mathcal{L}(\\mathbb{X})}\\left\\|\\psi(s)\\right\\|_{\\mathbb{X}}\\mathrm{d}s+\\frac{MR\\alpha}{\\Gamma(1+\\alpha)}\\left(\\frac{(s_2-s_1)^{2\\alpha-1}}{2\\alpha-1}\\right)^{\\frac{1}{2}}\\nonumber.\n\t\t\\end{align}\n\t\tIf $s_1=0,$ then by the above estimate, we deduce that \n\t\t\\begin{align*}\n\t\t\\lim_{s_2\\to 0^+}\\left\\|(\\mathrm{Q}\\psi)(s_2)-(\\mathrm{Q}\\psi)(s_1)\\right\\|_{\\mathbb{X}}=0,\\; \\mbox{ unifromly for } \\ \\psi\\in\\mathrm{L}^2(J;\\mathbb{X}).\n\t\t\\end{align*} \n\t\tFor $0<\\epsilon<s_1<T$, we have\n\t\t\\begin{align}\\label{2.8}\n\t\t&\\left\\|(\\mathrm{Q}\\psi)(s_2)-(\\mathrm{Q}\\psi)(s_1)\\right\\|_{\\mathbb{X}}\\nonumber\\\\&\\le\\frac{MR\\alpha}{\\Gamma(1+\\alpha)}\\left(\\int_{0}^{s_1}\\left|(s_2-s)^{\\alpha-1}-(s_1-s)^{\\alpha-1}\\right|^2\\mathrm{d}s\\right)^{\\frac{1}{2}}\\nonumber\\\\&\\quad+\\int_{0}^{s_1-\\epsilon}(s_1-s)^{\\alpha-1}\\left\\|\\widehat{\\mathcal{T}}_{\\alpha}(s_2-s)-\\widehat{\\mathcal{T}}_{\\alpha}(s_1-s)\\right\\|_{\\mathcal{L}(\\mathbb{X})}\\left\\|\\psi(s)\\right\\|_{\\mathbb{X}}\\mathrm{d}s\\nonumber\\\\&\\quad+\\int_{s_1-\\epsilon}^{s_1}(s_1-s)^{\\alpha-1}\\left\\|\\widehat{\\mathcal{T}}_{\\alpha}(s_2-s)-\\widehat{\\mathcal{T}}_{\\alpha}(s_1-s)\\right\\|_{\\mathcal{L}(\\mathbb{X})}\\left\\|\\psi(s)\\right\\|_{\\mathbb{X}}\\mathrm{d}s+\\frac{MR\\alpha}{\\Gamma(1+\\alpha)}\\left(\\frac{(s_2-s_1)^{2\\alpha-1}}{2\\alpha-1}\\right)^{\\frac{1}{2}}\\nonumber\\\\&\\le\\frac{MR\\alpha}{\\Gamma(1+\\alpha)}\\left(\\int_{0}^{s_1}\\left|(s_2-s)^{\\alpha-1}-(s_1-s)^{\\alpha-1}\\right|^2\\mathrm{d}s\\right)^{\\frac{1}{2}}\\nonumber\\\\&\\quad+\\sup_{s\\in[0,s_1-\\epsilon]}\\left\\|\\widehat{\\mathcal{T}}_{\\alpha}(s_2-s)-\\widehat{\\mathcal{T}}_{\\alpha}(s_1-s)\\right\\|_{\\mathcal{L}(\\mathbb{X})}\\int_{0}^{s_1-\\epsilon}(s_1-s)^{\\alpha-1}\\left\\|\\psi(s)\\right\\|_{\\mathbb{X}}\\mathrm{d}s\\nonumber\\\\&\\quad+\\frac{2M\\alpha}{\\Gamma(1+\\alpha)}\\int_{s_1-\\epsilon}^{s_1}(s_1-s)^{\\alpha-1}\\left\\|\\psi(s)\\right\\|_{\\mathbb{X}}\\mathrm{d}s+\\frac{MR\\alpha}{\\Gamma(1+\\alpha)}\\left(\\frac{(s_2-s_1)^{2\\alpha-1}}{2\\alpha-1}\\right)^{\\frac{1}{2}}\\nonumber\\\\&\\le\\frac{MR\\alpha}{\\Gamma(1+\\alpha)}\\left(\\int_{0}^{s_1}\\left|(s_2-s)^{\\alpha-1}-(s_1-s)^{\\alpha-1}\\right|^2\\mathrm{d}s\\right)^{\\frac{1}{2}}\\nonumber\\\\&\\quad+R\\sup_{s\\in[0,s_1-\\epsilon]}\\left\\|\\widehat{\\mathcal{T}}_{\\alpha}(s_2-s)-\\widehat{\\mathcal{T}}_{\\alpha}(s_1-s)\\right\\|_{\\mathcal{L}(\\mathbb{X})}\\left(\\frac{s_1^{2\\alpha-1}-\\epsilon^{2\\alpha-1}}{2\\alpha-1}\\right)^{\\frac{1}{2}}\\nonumber\\\\&\\quad+\\frac{2MR\\alpha}{\\Gamma(1+\\alpha)}\\left(\\frac{\\epsilon^{2\\alpha-1}}{2\\alpha-1}\\right)^{\\frac{1}{2}}+\\frac{MR\\alpha}{\\Gamma(1+\\alpha)}\\left(\\frac{(s_2-s_1)^{2\\alpha-1}}{2\\alpha-1}\\right)^{\\frac{1}{2}}.\n\t\t\\end{align}\n\t\tFrom Lemma \\ref{lem2.5}, we know that the operator $\\widehat{\\mathcal{T}}_{\\alpha}(t)$ is compact for $t>0$, which implies that the operator $\\widehat{\\mathcal{T}}_{\\alpha}(t)$ is continuous under the uniform operator topology (see Theorem 3.2, Chapter 2, \\cite{P}). Hence, using the arbitrariness of $\\epsilon$ and continuity of  $\\widehat{\\mathcal{T}}_{\\alpha}(t)$ in the uniform operator topology, the right hand side of the expression \\eqref{2.8} converges to zero as $|s_2-s_1| \\rightarrow 0$. Thus, $\\mathrm{Q}\\mathcal{B}_R$ is equicontinuous on $\\mathrm{L}^{2}(J;\\mathbb{X})$. \n\t\t\n\t\tNext, we show that $\\mathrm{V}(t):= \\left\\{(\\mathrm{Q}\\psi)(t):\\psi\\in \\mathcal{B}_R\\right\\},$  for all $t\\in J$ is relatively compact. For $t=0$, it is easy to check that the set $\\mathrm{V}(t)$ is relatively compact in $\\mathbb{X}$.  Let us take $ 0<t\\leq T$ be fixed and for given $\\eta$ with $ 0<\\eta<t$ and any $\\delta>0$, we define\n\t\t\\begin{align*}\n\t\t(\\mathrm{Q}^{\\eta,\\delta}\\psi)(t)&=\\alpha\\int_{0}^{t-\\eta}\\int_{\\delta}^{\\infty}\\xi(t-s)^{\\alpha-1}\\varphi_{\\alpha}(\\xi)\\mathcal{T}(t^{\\alpha}\\xi)\\psi(s)\\mathrm{d}\\xi\\mathrm{d}s\\nonumber\\\\&=\\mathcal{T}(\\eta^{\\alpha}\\delta)\\alpha\\int_{0}^{t-\\eta}\\int_{\\delta}^{\\infty}\\xi(t-s)^{\\alpha-1}\\varphi_{\\alpha}(\\xi)\\mathcal{T}(t^{\\alpha}\\xi-\\eta^{\\alpha}\\delta)\\psi(s)\\mathrm{d}\\xi\\mathrm{d}s.\n\t\t\\end{align*}\n\t\tSince the operator $\\mathcal{T}(\\cdot)$ is compact,  the set  $\\mathrm{V}_{\\eta,\\delta}(t)=\\{(\\mathrm{Q}^{\\eta,\\delta}_\\lambda \\psi)(t):\\psi\\in \\mathcal{B}_R\\}$ is relatively compact in $\\mathbb{X}$. Hence, there exist a finite $ x_{i}$'s, for $i=1,\\dots, n $ in $ \\mathbb{X} $ such that \n\t\t\\begin{align*}\n\t\t\\mathrm{V}_{\\eta,\\delta}(t) \\subset \\bigcup_{i=1}^{n}\\mathcal{S}(x_i, \\varepsilon\/2),\n\t\t\\end{align*}\n\t\tfor some $\\varepsilon>0$, where $\\mathcal{S}(x_i, \\varepsilon\/2)$ is an open ball centered at $x_i$ and of radius $\\varepsilon\/2$. Let us choose $\\delta>0$ and $\\eta>0$ such that \n\t\t\\begin{align*}\n\t\t\\left\\|(\\mathrm{Q}\\psi)(t)-(\\mathrm{Q}^{\\eta,\\delta}\\psi)(t)\\right\\|_{\\mathbb{X}}&\\le\\alpha\\left\\|\\int_{0}^{t}\\int_{0}^{\\delta}\\xi(t-s)^{\\alpha-1}\\varphi_{\\alpha}(\\xi)\\mathcal{T}(t^{\\alpha}\\xi-\\eta^{\\alpha}\\delta)\\psi(s)\\mathrm{d}\\xi\\mathrm{d}s\\right\\|_{\\mathbb{X}}\\nonumber\\\\&\\quad+\\alpha\\left\\|\\int_{t-\\eta}^{t}\\int_{\\delta}^{\\infty}\\xi(t-s)^{\\alpha-1}\\varphi_{\\alpha}(\\xi)\\mathcal{T}(t^{\\alpha}\\xi-\\eta^{\\alpha}\\delta)\\psi(s)\\mathrm{d}\\xi\\mathrm{d}s\\right\\|_{\\mathbb{X}}\\nonumber\\\\&\\le\\frac{MR\\alpha t^{2\\alpha-1}}{2\\alpha-1}\\int_{0}^{\\delta}\\xi\\varphi(\\xi)\\mathrm{d}\\xi+\\frac{MR\\alpha}{\\Gamma(1+\\alpha)}\\frac{\\eta^{2\\alpha-1}}{2\\alpha-1}\\le\\frac{\\varepsilon}{2}.\n\t\t\\end{align*}\n\t\tConsequently $$ \\mathrm{V}(t)\\subset \\bigcup_{i=1}^{n}\\mathcal{S}(x_i, \\varepsilon ). $$ Thus, for each $t\\in J$, the set $\\mathrm{V}(t)$ is relatively compact in $ \\mathbb{X}$. Then by invoking the  Arzela-Ascoli theorem, we conclude that the operator $\\mathrm{Q}$ is compact.\n\t\\end{proof}\n\tSince $\\mathbb{X}^*$ is strictly convex, the norm $\\|\\cdot\\|_{\\mathbb{X}}$ is Gateaux differentiable (cf. Fact 8.12, \\cite{MFb}). Furthermore, every separable Banach space admits an equivalent Gateaux differentiable norm (cf. Theorem 8.13, \\cite{MFb}). Since $\\mathcal{J}$ is single-valued,  the Gateaux derivative of $\\phi(x)=\\frac{1}{2}\\|x\\|_{\\mathbb{X}}^2$ is the duality map, that is, $$\\langle\\partial_x\\phi(x),y\\rangle=\\lim_{\\varepsilon \\to 0}\\frac{\\phi(x+\\varepsilon y)-\\phi(x)}{\\varepsilon}=\\frac{1}{2}\\frac{\\mathrm{d}}{\\mathrm{d}\\varepsilon}\\|x+\\varepsilon y\\|_{\\mathbb{X}}^2\\Big|_{\\varepsilon=0}=\\langle\\mathcal{J}[x],y\\rangle,$$ for $y\\in\\mathbb{X}$, where $\\partial_x\\phi(x)$ denotes the Gateaux derivative of $\\phi$ at $x\\in\\mathbb{X}$. In fact, since $\\mathbb{U}$ is a separable Hilbert space (identified with its own dual), by Theorem 8.24, \\cite{MFb}, we infer that $\\mathbb{U}$ admits a Fr\\'echet differentiable norm.  The explicit expression of the optimal control ${u}$ in the feedback form is obtained in the following lemma: \n\t\n\t\\begin{lem}\\label{lem3.1}\n\t\tLet $u$ be the optimal control satisfying \\eqref{3.3} and minimizing the cost functional \\eqref{3.1}. Then $u$ is given by \n\t\t\\begin{align*}\n\t\tu(t)=(T-t)^{\\alpha-1}\\mathrm{B}^*\\widehat{\\mathcal{T}}_{\\alpha}(T-t)^*\\mathcal{J}\\left[\\mathcal{R}(\\lambda,\\Phi_{0}^T)p(x(\\cdot))\\right],\\  t\\in [0, T),\\ \\lambda>0,\\ \\frac{1}{2}<\\alpha<1, \n\t\t\\end{align*}\n\t\twith\n\t\t\\begin{align*}\n\t\tp(x(\\cdot))=x_{T}-\\mathcal{T}_{\\alpha}(T)\\zeta.\n\t\t\\end{align*}\n\t\\end{lem}\n\t\\begin{proof}\n\t\tLet us first consider the functional \n\t\t\\begin{align*}\n\t\t\\mathcal{I}(\\varepsilon)=\\mathcal{G}(x_{u+\\varepsilon w},u+\\varepsilon w),\n\t\t\\end{align*}\n\t\twhere $(x,u)$ is the optimal solution of \\eqref{3.3} and $w\\in \\mathrm{L}^{2}(J;\\mathbb{U})$. Also the function  $x_{u+\\varepsilon w}$ is the unique mild solution of \\eqref{3.2} corresponding to the control $u+\\varepsilon w$. Then, it is immediate that \n\t\t\\begin{align*}\n\t\tx_{u+\\varepsilon w}(t)= \\mathcal{T}_{\\alpha}(t)\\zeta+\\int_{0}^{t}(t-s)^{\\alpha-1}\\widehat{\\mathcal{T}}_{\\alpha}(t-s)\\mathrm{B}(u+\\varepsilon w)(s)\\mathrm{d}s.\n\t\t\\end{align*}\n\t\tIt is clear $\\varepsilon=0$ is the critical point of $\\mathcal{I}(\\varepsilon)$. We now evaluate the first variation of the cost functional $\\mathcal{G}$ (defined in \\eqref{3.1}) as\n\t\t\\begin{align*}\n\t\t\\frac{\\mathrm{d}}{\\mathrm{d}\\varepsilon}\\mathcal{I}(\\varepsilon)\\Big|_{\\varepsilon=0}&=\\frac{\\mathrm{d}}{\\mathrm{d}\\varepsilon}\\bigg[\\left\\|x_{u+\\varepsilon w}(T)-x_{T}\\right\\|^{2}_{\\mathbb{X}}+\\lambda\\int^{T}_{0}\\left\\|u(t)+\\varepsilon w(t)\\right\\|^{2}_{\\mathbb{U}}\\mathrm{d}t\\bigg]_{\\varepsilon=0}\\nonumber\\\\\n\t\t&=2\\bigg[\\langle \\mathcal{J}(x_{u+\\varepsilon w}(T)-x_{T}), \\frac{\\mathrm{d}}{\\mathrm{d}\\varepsilon}(x_{u+\\varepsilon w}(T)-x_{T})\\rangle\\nonumber\\\\&\\qquad +2\\lambda\\int^{T}_{0}(u(t)+\\varepsilon w(t),\\frac{\\mathrm{d}}{\\mathrm{d}\\varepsilon}(u(t)+\\varepsilon w(t)))\\mathrm{d}t\\bigg]_{\\varepsilon=0}\\nonumber\\\\\n\t\t&=2\\left\\langle\\mathcal{J}(x(T)-x_T),\\int_0^T(T-t)^{\\alpha-1}\\widehat{\\mathcal{T}}_{\\alpha}(T-t)\\mathrm{B}w(t)\\mathrm{d}t \\right\\rangle+2\\lambda\\int_0^T(u(t),w(t))\\mathrm{d} t. \n\t\t\\end{align*}\n\t\tBy taking the first variation of the cost functional is zero, we deduce that\n\t\t\\begin{align}\\label{3.9}\n\t\t0&=\\left\\langle\\mathcal{J}(x(T)-x_T),\\int_0^T(T-t)^{\\alpha-1}\\widehat{\\mathcal{T}}_{\\alpha}(T-t)\\mathrm{B}w(t)\\mathrm{d}t\\right\\rangle+\\lambda\\int_0^T(u(t),w(t))\\mathrm{d} t\\nonumber\\\\&=\\int_0^T(T-t)^{\\alpha-1}\\left\\langle\\mathcal{J}(x(T)-x_T),\\widehat{\\mathcal{T}}_{\\alpha}(T-t)\\mathrm{B}w(t) \\right\\rangle\\mathrm{d}t+\\lambda\\int_0^T(u(t),w(t))\\mathrm{d} t\\nonumber\\\\&= \\int_0^T\\left((T-t)^{\\alpha-1}\\mathrm{B}^*\\widehat{\\mathcal{T}}_{\\alpha}(T-t)^*\\mathcal{J}(x(T)-x_T)+\\lambda u(t),w(t) \\right)\\mathrm{d}t,\n\t\t\\end{align}\n\t\twhere $(\\cdot,\\cdot)$ is the inner product in the Hilbert space $\\mathbb{U}$. Since $w\\in \\mathrm{L}^{2}(J;\\mathbb{U})$ is an arbitrary element (one can choose $w$ to be $(T-t)^{\\alpha-1}\\mathrm{B}^*\\widehat{\\mathcal{T}}_{\\alpha}(T-t)^*\\mathcal{J}(x(T)-x_T)+\\lambda u(t)$), it follows that the optimal control is given by\n\t\t\\begin{align}\\label{3.10}\n\t\tu(t)&= -\\lambda^{-1}(T-t)^{\\alpha-1}\\mathrm{B}^*\\widehat{\\mathcal{T}}_{\\alpha}(T-t)^*\\mathcal{J}(x(T)-x_T),\n\t\t\\end{align}\n\t\tfor a.e. $t\\in [0,T]$. Since by the relations \\eqref{3.9} and \\eqref{3.10}, it is clear that $u\\in\\mathrm{C}([0,T);\\mathbb{U})$. Using the above expression of the control, we find\n\t\n\t\t\\begin{align}\\label{3.11}\n\t\tx(T)&=\\mathcal{T}_{\\alpha}(T)\\zeta-\\int^{T}_{0}\\lambda^{-1}(T-s)^{2(\\alpha-1)}\\widehat{\\mathcal{T}}_{\\alpha}(T-s)\\mathrm{B}\\mathrm{B}^*\\widehat{\\mathcal{T}}_{\\alpha}(T-s)^*\\mathcal{J}(x(T)-x_T)\\mathrm{d}s\\nonumber\\\\\n\t\t&= \\mathcal{T}_{\\alpha}(T)\\zeta-\\lambda^{-1}\\Phi_{0}^T\\mathcal{J}\\left[x(T)-x_{T}\\right].\n\t\t\\end{align}\n\t\tLet us assume\n\t\t\\begin{align}\\label{3.12}\n\t\tp(x(\\cdot)):=x_{T}-\\mathcal{T}_{\\alpha}(T)\\zeta.\n\t\t\\end{align}\n\t\tCombining \\eqref{3.11} and \\eqref{3.12}, we have\n\t\t\\begin{align}\\label{3.13}\n\t\tx(T)-x_{T}&=-p(x(\\cdot))-\\lambda^{-1}\\Phi_{0}^T\\mathcal{J}\\left[x(T)-x_{T}\\right].\n\t\t\\end{align}\n\t\tFrom \\eqref{3.13}, one can easily deduce that \n\t\t\\begin{align}\\label{3.15}\n\t\tx(T)-x_T=-\\lambda\\mathrm{I}(\\lambda\\mathrm{I}+\\Phi_0^T\\mathcal{J})^{-1}p(x(\\cdot))=-\\lambda\\mathcal{R}(\\lambda,\\Phi_0^T)p(x(\\cdot)).\n\t\t\\end{align}\n\t\tFinally, from \\eqref{3.10}, we get  the expression for optimal control  as\n\t\t\\begin{align*}\n\t\tu(t)=(T-t)^{\\alpha-1}\\mathrm{B}^*\\widehat{\\mathcal{T}}_{\\alpha}(T-t)^*\\mathcal{J}\\left[\\mathcal{R}(\\lambda,\\Phi_{0}^T)p(x(\\cdot))\\right],\\ \\mbox{ for } \\ t\\in [0,T),\n\t\t\\end{align*}\n\t\twhich completes the proof. \n\t\\end{proof}\n\tNext, we examine the approximate controllability of the linear control system \\eqref{3.2} through the following lemma.  \n\t\\begin{lem}\\label{lem4.2}\n\t\tThe linear control system \\eqref{3.2} is approximately controllable on $J$ if and only if Assumption (H0) holds. \n\t\\end{lem}\n\tA proof of the above lemma can be obtained by proceeding similarly as in the proof of  Theorem 3.2, \\cite{SM}.\n\t\\begin{rem}\\label{rem3.4}\n\t\tIf Assumption (\\textit{H0}) holds, then by Theorem 2.3, \\cite{M}, we know that the operator $\\Phi_{0}^{T}$ is positive and vice versa. The positivity of $\\Phi_{0}^{T}$ is equivalent to $$ \\langle x^*, \\Phi_{0}^{T}x^*\\rangle=0\\Rightarrow x^*=0.$$ We know that \n\t\t\\begin{align}\n\t\t\\langle x^*, \\Phi_{0}^{T}x^*\\rangle =\\int_0^T\\left\\|(T-t)^{\\alpha-1}\\mathrm{B}^*\\widehat{\\mathcal{T}}_\\alpha(T-t)^*x^*\\right\\|_{\\mathbb{X}^*}^2\\mathrm{d}t.\n\t\t\\end{align}\n\t\tBy the above fact and  Lemma \\ref{lem4.2}, we infer that the approximate controllability of the linear system \\eqref{3.2} is equivalent to the condition $$\\mathrm{B}^*\\widehat{\\mathcal{T}}_\\alpha(T-t)^*x^*=0,\\ 0\\le t<T \\Rightarrow x^*=0.$$\n\t\\end{rem}\n\t\n\t\n\t\\begin{rem}\\label{rem3.6}\n\t\tInstead  \\eqref{3.1}, one can consider the following cost functional also: \n\t\t\\begin{equation}\\label{3.19}\n\t\t\\mathcal{G}(x,u)=\\left\\|x(T)-x_{T}\\right\\|^{2}_{\\mathbb{X}}+\\lambda\\int^{T}_{0}(T-t)^{\\alpha-1}\\left\\|u(t)\\right\\|^{2}_{\\mathbb{U}}\\mathrm{d}t. \n\t\t\\end{equation}\n\t\tFor $\\frac{1}{2}<\\alpha<1$,\tthe existence of optimal solution for the problem \\eqref{3.3} follows similarly as in the proof of Theorem \\ref{optimal}. For this, we have to replace \\eqref{35} by \n\t\t\\begin{align*}\n\t\tT^{\\alpha-1}\\lambda\\int^{T}_{0}\\left\\|u^n(t)\\right\\|^{2}_{\\mathbb{U}}\\mathrm{d}t\\leq  &\\left\\|x^n(T)-x_{T}\\right\\|^{2}_{\\mathbb{X}}+\\lambda\\int^{T}_{0}(T-t)^{\\alpha-1}\\left\\|u^n(t)\\right\\|^{2}_{\\mathbb{U}}\\mathrm{d}t\\nonumber\\\\&\\leq \\left\\|x(T)-x_{T}\\right\\|^{2}_{\\mathbb{X}}\\leq 2\\left(\\|x(T)\\|_{\\mathbb{X}}^2+\\|x_T\\|_{\\mathbb{X}}^2\\right)<+\\infty,\n\t\t\\end{align*}\n\t\tso that \\eqref{3.5} follows easily. Using the cost functional given in \\eqref{3.19}, calculations similar to Lemma \\ref{lem3.1} yields  the optimal control $u(\\cdot)$ as \t\\begin{align*}\n\t\tu(t)=\\mathrm{B}^*\\widehat{\\mathcal{T}}_{\\alpha}(T-t)^*\\mathcal{J}\\left[\\mathcal{R}(\\lambda,\\Phi_{0}^T)p(x(\\cdot))\\right],\\  t\\in [0, T),\\ \\lambda>0,\\ \\frac{1}{2}<\\alpha<1, \n\t\t\\end{align*}\n\t\twith\n\t\t$\tp(x(\\cdot))=x_{T}-\\mathcal{T}_{\\alpha}(T)\\zeta,$ and \n\t\t\\begin{equation}\\label{3.20}\n\t\t\\Phi_{0}^{T}=\\int^{T}_{0}(T-t)^{\\alpha-1}\\widehat{\\mathcal{T}}_{\\alpha}(T-t)\\mathrm{B}\\mathrm{B}^*\\widehat{\\mathcal{T}}_{\\alpha}(T-t)^*\\mathrm{d}t. \n\t\t\\end{equation}\n\t\\end{rem}\n\t\n\t\\section{Approximate Controllability of the Semilinear Impulsive System} \\label{semilinear}\\setcounter{equation}{0}\n\tThe purpose of this section is to investigate the approximate controllability of the fractional order  semilinear impulsive system \\eqref{1.1}. In order to acquire sufficient conditions on approximate controllability, we first show that for $\\lambda>0$ and $x_T\\in\\mathbb{X}$, there exists a mild solution of the system \\eqref{1.1} with the control function defined as \n\t\\begin{align}\\label{C}\n\tu^\\alpha_{\\lambda}(t)&=\\sum_{k=0}^{m}u^\\alpha_{k,\\lambda}(t)\\chi_{[\\tau_k, t_{k+1})}(t), \\ t\\in J,\\  \\frac{1}{2}<\\alpha<1,\n\t\\end{align}\n\twhere \n\t\\begin{align*}\n\tu^\\alpha_{k,\\lambda}(t)&=(t_{k+1}-t)^{\\alpha-1}\\mathrm{B}^*\\widehat{\\mathcal{T}}_{\\alpha}(t_{k+1}-t)^*\\mathcal{J}\\left[\\mathcal{R}(\\lambda,\\Phi_{\\tau_k}^{t_{k+1}})p_k(x(\\cdot))\\right],\n\t\\end{align*}\n\tfor $t\\in [\\tau_k, t_{k+1}),k=0,1,\\ldots,m$, with\n\t\\begin{align*}\n\tp_0(x(\\cdot))&=\\zeta_{0}-\\mathcal{T}_{\\alpha}(t_1)\\psi(0)-\\int^{t_1}_{0}(t_1-s)^{\\alpha-1}\\widehat{\\mathcal{T}}_{\\alpha}(t_1-s)f(s,\\tilde{x}_{\\rho(s,\\tilde{x}_s)})\\mathrm{d}s,\\nonumber\\\\\n\tp_k(x(\\cdot))&=\\zeta_{k}-\\mathcal{T}_{\\alpha}(t_{k+1}-\\tau_k)h_k(\\tau_k,\\tilde{x}(t_k^-))\\\\&\\quad+\\int_{0}^{\\tau_k}(\\tau_k-s)^{\\alpha-1}\\widehat{\\mathcal{T}}_{\\alpha}(\\tau_k-s)\\left[f(s,\\tilde{x}_{\\rho(s,\\tilde{x}_s)})+\\mathrm{B}\\sum_{j=0}^{k-1}u^\\alpha_{j,\\lambda}(s)\\chi_{[\\tau_j, t_{j+1})}(s)\\right]\\mathrm{d}s \\\\&\\quad-\\int_{0}^{t_{k+1}}(t_{k+1}-s)^{\\alpha-1}\\widehat{\\mathcal{T}}_{\\alpha}(t_{k+1}-s)f(s,\\tilde{x}_{\\rho(s,\\tilde{x}_s)})\\mathrm{d}s\t\\\\&\\quad-\\int_{0}^{\\tau_k}(t_{k+1}-s)^{\\alpha-1}\\widehat{\\mathcal{T}}_{\\alpha}(t_{k+1}-s)\\mathrm{B}\\sum_{j=0}^{k-1}u^\\alpha_{j,\\lambda}(s)\\chi_{[\\tau_j, t_{j+1})}(s)\\mathrm{d}s, \\ k=1,\\ldots,m,\n\n\t\\end{align*} \n\tand  $\\tilde{x}:(-\\infty,T]\\rightarrow\\mathbb{X}$  such that $\\tilde{x}(t)=\\psi(t), \\ t\\in(-\\infty,0]\\ \\tilde{x}(t)=x(t),\\ t\\in J=[0,T],$ and $\\zeta_{k}\\in \\mathbb{X}$ for $k=0,1,\\ldots,m$.\n\t\\begin{rem}\n\t\tSince the operator $\\Phi_{\\tau_k}^{t_{k+1}},$  for each $k=0,\\ldots,m,$ is non-negative, linear and bounded for $\\frac{1}{2}<\\alpha<1$, Lemma \\ref{lem2.9} is also valid for each $\\Phi_{\\tau_k}^{t_{k+1}},$  for $k=0,\\ldots,m$.\n\t\\end{rem}\n\tThe following theorem provides the existence of mild solution of the system \\eqref{1.1} with the control \\eqref{C}. \n\t\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\t\\begin{theorem}\\label{thm4.3}\n\t\tLet Assumptions (H1)-(H3) hold true. Then for every $ \\lambda>0 $ and  fixed $\\zeta_{k}\\in\\mathbb{X},$ for $k=0,1,\\ldots,m$, the system  \\eqref{1.1} with the control \\eqref{C} has at least one mild solution on $J$,  provided \n\t\t\\begin{align}\\label{cnd}\n\t\\frac{MH_{2}\\alpha\\beta}{\\Gamma(1+\\alpha)}\\frac{2T^{\\alpha-\\alpha_1}}{\\mu^{1-\\alpha_1}}\\left\\{1+\\frac{(m+1)(m+2)\\tilde{R}}{2}+\\frac{m(m+1)\\tilde{R}^2}{2}\\sum_{j=0}^{m-1}e^{\\frac{(m+j)(m-j-1)\\tilde{R}}{2}}\\right\\}<1,\n\t\t\\end{align}\n\t\twhere $\\tilde{R}=\\left(\\frac{M\\tilde{M}\\alpha}{\\Gamma(1+\\alpha)}\\right)^{2}\\frac{2T^{2\\alpha-1}}{\\lambda(2\\alpha-1)}$ and $\\mu=\\frac{\\alpha-\\alpha_1}{1-\\alpha_1}$.\n\t\\end{theorem}\n\t\\begin{proof}\n\t\tLet us take a set $\\mathrm{E}:=\\{x\\in\\mathrm{PC}(J;\\mathbb{X}) : x(0)=\\psi(0)\\}$ with the norm $\\left\\|\\cdot\\right\\|_{\\mathrm{PC}(J;\\mathbb{X})}$. For each $r>0$, we consider a set $\\mathrm{E}_{r}=\\{x\\in\\mathrm{E} : \\left\\|x\\right\\|_{\\mathrm{PC}(J;\\mathbb{X})}\\le r\\}$.\n\t\t\n\t\tFor $\\lambda>0$, let us define an operator $F_{\\lambda}:\\mathrm{E}\\to\\mathrm{E}$ such that\n\t\t\\begin{eqnarray}\\label{2}\n\t\t(F_{\\lambda}x)(t)=\\left\\{\n\t\t\\begin{aligned}\n\t\t&\\mathcal{T}_{\\alpha}(t)\\psi(0)+\\int_{0}^{t}(t-s)^{\\alpha-1}\\widehat{\\mathcal{T}}_{\\alpha}(t-s)\\left[\\mathrm{B}u^{\\alpha}_{\\lambda}(s)+f(s,\\tilde{x}_{\\rho(s, \\tilde{x}_s)})\\right]\\mathrm{d}s,\\ t\\in[0, t_1],\\\\\n\t\t&\th_k(t, \\tilde{x}(t_k^-)),\\ t\\in(t_k, \\tau_k],\\ k=1,\\ldots,m,\\\\\n\t\t&\\mathcal{T}_{\\alpha}(t-\\tau_k)h_k(\\tau_k,\\tilde{x}(t_k^-))-\\int_{0}^{\\tau_k}(\\tau_k-s)^{\\alpha-1}\\widehat{\\mathcal{T}}_{\\alpha}(\\tau_k-s)\\left[\\mathrm{B}u^{\\alpha}_{\\lambda}(s)+f(s,\\tilde{x}_{\\rho(s, \\tilde{x}_s)})\\right]\\mathrm{d}s\\\\&\\quad+\\int_{0}^{t}(t-s)^{\\alpha-1}\\widehat{\\mathcal{T}}_{\\alpha}(t-s) \\left[\\mathrm{B}u^{\\alpha}_{\\lambda}(s)+f(s,\\tilde{x}_{\n\t\t\t\\rho(s, \\tilde{x}_s)})\\right]\\mathrm{d}s,\\ t\\in(\\tau_k,t_{k+1}],\\ k=1,\\ldots,m,\n\t\t\\end{aligned}\n\t\t\\right.\n\t\t\\end{eqnarray}\n\t\twhere $u^{\\alpha}_{\\lambda}$ is given in \\eqref{C}. From the definition of $ F_{\\lambda}$, we infer that the system $\\eqref{1.1}$ has a mild solution, if the operator $ F_{\\lambda}$ has a fixed point. We divide the proof of the fact that the operator $F_{\\lambda}$ has a fixed point in the following steps. \n\t\t\\vskip 0.1in \n\t\t\\noindent\\textbf{Step (1): } \\emph{$ F_{\\lambda}(\\mathrm{E}_r)\\subset \\mathrm{E}_r,$ for some $ r $}. On the contrary, let us suppose that our claim is not true. Then for any $\\lambda>0$ and for all $r>0$, there exists $x^r\\in \\mathrm{E}_r,$ such that $\\left\\|(F_{\\lambda}x^r)(t)\\right\\|_\\mathbb{X}>r,$ for some $t\\in J$, where $t$ may depend upon $r$. \n\t\tFirst, by using Assumption \\ref{as2.1}, we estimate \n\t\t\\begin{align}\\label{4.4}\n\t\t\\left\\|p_0(x(\\cdot))\\right\\|_{\\mathbb{X}}&\\le\\left\\|\\zeta_0\\right\\|_{\\mathbb{X}}+\\left\\|\\mathcal{T}_{\\alpha}(t_1)\\psi(0)\\right\\|_{\\mathbb{X}}+\\int_{0}^{t_1}(t_1-s)^{\\alpha-1}\\left\\|\\widehat{\\mathcal{T}}_{\\alpha}(t_1-s)f(s,\\tilde{x}_{\\rho(s,\\tilde{x}_s)})\\right\\|_{\\mathbb{X}}\\mathrm{d}s\\nonumber\\\\&\\le\\left\\|\\zeta_0\\right\\|_{\\mathbb{X}}+M\\left\\|\\psi(0)\\right\\|_{\\mathbb{X}}+\\frac{M\\alpha}{\\Gamma(1+\\alpha)}\\int_{0}^{t_1}(t_1-s)^{\\alpha-1}\\gamma_{r'}(s)\\mathrm{d}s\\nonumber\\\\&\\le\\left\\|\\zeta_0\\right\\|_{\\mathbb{X}}+M\\left\\|\\psi(0)\\right\\|_{\\mathbb{X}}+\\frac{M\\alpha}{\\Gamma(1+\\alpha)}\\left(\\int_{0}^{t_1}(t_1-s)^{\\frac{\\alpha-1}{1-\\alpha_1}}\\mathrm{d}s\\right)^{1-\\alpha_1}\\left(\\int_{0}^{t_1}(\\gamma_{r'}(s))^{\\frac{1}{\\alpha_1}}\\mathrm{d}s\\right)^{\\alpha_1}\\nonumber\\\\&\\le\\left\\|\\zeta_0\\right\\|_{\\mathbb{X}}+M\\left\\|\\psi(0)\\right\\|_{\\mathbb{X}}+\\frac{M\\alpha}{\\Gamma(1+\\alpha)}\\frac{t_{1}^{\\alpha-\\alpha_1}}{\\mu^{1-\\alpha_1}}\\left\\|\\gamma_{r'}\\right\\|_{\\mathcal{L}^{\\frac{1}{\\alpha_1}}([0,t_1];\\mathbb{R^{+}})}\\nonumber\\\\&\\le\\left\\|\\zeta_0\\right\\|_{\\mathbb{X}}+M\\left\\|\\psi(0)\\right\\|_{\\mathbb{X}}+\\frac{M\\alpha}{\\Gamma(1+\\alpha)}\\frac{T^{\\alpha-\\alpha_1}}{\\mu^{1-\\alpha_1}}\\left\\|\\gamma_{r'}\\right\\|_{\\mathcal{L}^{\\frac{1}{\\alpha_1}}(J;\\mathbb{R^{+}})}\\nonumber\\\\&\\le\\left\\|\\zeta_0\\right\\|_{\\mathbb{X}}+M\\left\\|\\psi(0)\\right\\|_{\\mathbb{X}}+\\frac{M\\alpha}{\\Gamma(1+\\alpha)}\\frac{2T^{\\alpha-\\alpha_1}}{\\mu^{1-\\alpha_1}}\\left\\|\\gamma_{r'}\\right\\|_{\\mathcal{L}^{\\frac{1}{\\alpha_1}}(J;\\mathbb{R^{+}})}=N_0,\n\t\t\\end{align} \n\t\twhere $\\mu=\\frac{\\alpha-\\alpha_1}{1-\\alpha_1}$ and $r'=H_{1}\\left\\|\\psi\\right\\|_{\\mathfrak{B}}+H_{2}r$.\tFurther, using Assumption \\ref{as2.1}, we estimate \n\t\t\\begin{align*}\n\t\t&\\left\\|p_k(x(\\cdot))\\right\\|_{\\mathbb{X}}\\nonumber\\\\&\\le\\left\\|\\zeta_k\\right\\|_{\\mathbb{X}}+\\left\\|\\mathcal{T}_{\\alpha}(t-\\tau_k)h_k(\\tau_k,\\tilde{x}(t_k^-))\\right\\|_{\\mathbb{X}}+\\int_{0}^{\\tau_k}(\\tau_k-s)^{\\alpha-1}\\left\\|\\widehat{\\mathcal{T}}_{\\alpha}(\\tau_k-s)f(s,\\tilde{x}_{\\rho(s,\\tilde{x}_s)})\\right\\|_{\\mathbb{X}}\\mathrm{d}s\\nonumber\\\\&\\quad+\\int_{0}^{t_{k+1}}(t_{k+1}-s)^{\\alpha-1}\\left\\|\\widehat{\\mathcal{T}}_{\\alpha}(t_{k+1}-s)f(s,\\tilde{x}_{\\rho(s,\\tilde{x}_s)})\\right\\|_{\\mathbb{X}}\\mathrm{d}s\\nonumber\\\\&\\quad+\\int_{0}^{\\tau_k}(\\tau_k-s)^{\\alpha-1}\\left\\|\\widehat{\\mathcal{T}}_{\\alpha}(\\tau_k-s)\\mathrm{B}\\sum_{j=0}^{k-1}u^\\alpha_{j,\\lambda}(s)\\chi_{[\\tau_j,t_{j+1})}(s)\\right\\|_{\\mathbb{X}}\\mathrm{d}s\\nonumber\\\\&\\quad+\\int_{0}^{\\tau_k}(t_{k+1}-s)^{\\alpha-1}\\left\\|\\widehat{\\mathcal{T}}_{\\alpha}(t_{k+1}-s)\\mathrm{B}\\sum_{j=0}^{k-1}u^\\alpha_{j,\\lambda}(s)\\chi_{[\\tau_j, t_{j+1})}(s)\\right\\|_{\\mathbb{X}}\\mathrm{d}s\\nonumber\\\\&\\le\\left\\|\\zeta_k\\right\\|_{\\mathbb{X}}+Ml_k+\\frac{M\\alpha}{\\Gamma(1+\\alpha)}\\frac{\\tau_k^{\\alpha-\\alpha_1}}{\\mu^{1-\\alpha_1}}\\left\\|\\gamma_{r'}\\right\\|_{\\mathcal{L}^{\\frac{1}{\\alpha_1}}([0,\\tau_k];\\mathbb{R^{+}})}+\\frac{M\\alpha}{\\Gamma(1+\\alpha)}\\frac{t_{k+1}^{\\alpha-\\alpha_1}}{\\mu^{1-\\alpha_1}}\\left\\|\\gamma_{r'}\\right\\|_{\\mathcal{L}^{\\frac{1}{\\alpha_1}}([0,t_{k+1}];\\mathbb{R^{+}})}\\nonumber\\\\&\\quad+\\frac{1}{\\lambda}\\left(\\frac{M\\tilde{M}\\alpha}{\\Gamma(1+\\alpha)}\\right)^{2}\\sum_{j=0}^{k-1}\\left\\|p_j(x(\\cdot))\\right\\|_{\\mathbb{X}}\\int_{\\tau_j}^{t_{j+1}}(\\tau_k-s)^{\\alpha-1}(t_{j+1}-s)^{\\alpha-1}\\mathrm{d}s\\nonumber\\\\&\\quad+\\frac{1}{\\lambda}\\left(\\frac{M\\tilde{M}\\alpha}{\\Gamma(1+\\alpha)}\\right)^{2}\\sum_{j=0}^{k-1}\\left\\|p_j(x(\\cdot))\\right\\|_{\\mathbb{X}}\\int_{\\tau_j}^{t_{j+1}}(t_{k+1}-s)^{\\alpha-1}(t_{j+1}-s)^{\\alpha-1}\\mathrm{d}s\\nonumber\\\\&\\le\\left\\|\\zeta_k\\right\\|_{\\mathbb{X}}+Ml_k+\\frac{2M\\alpha}{\\Gamma(1+\\alpha)}\\frac{T^{\\alpha-\\alpha_1}}{\\mu^{1-\\alpha_1}}\\left\\|\\gamma_{r'}\\right\\|_{\\mathcal{L}^{\\frac{1}{\\alpha_1}}(J;\\mathbb{R^{+}})}\\nonumber\\\\&\\quad+\\frac{2}{\\lambda}\\left(\\frac{M\\tilde{M}\\alpha}{\\Gamma(1+\\alpha)}\\right)^{2}\\sum_{j=0}^{k-1}\\left\\|p_j(x(\\cdot))\\right\\|_{\\mathbb{X}}\\int_{\\tau_j}^{t_{j+1}}(t_{j+1}-s)^{2(\\alpha-1)}\\mathrm{d}s\\nonumber\\\\&\\le N_k+\\frac{2}{\\lambda}\\left(\\frac{M\\tilde{M}\\alpha}{\\Gamma(1+\\alpha)}\\right)^{2}\\sum_{j=0}^{k-1}\\left\\|p_j(x(\\cdot))\\right\\|_{\\mathbb{X}}\\frac{(t_{j+1}-\\tau_j)^{2\\alpha-1}}{2\\alpha-1}\\nonumber\\\\&\\le N_k+\\frac{2T^{2\\alpha-1}}{\\lambda(2\\alpha-1)}\\left(\\frac{M\\tilde{M}\\alpha}{\\Gamma(1+\\alpha)}\\right)^{2}\\sum_{j=0}^{k-1}\\left\\|p_j(x(\\cdot))\\right\\|_{\\mathbb{X}}\n\t\t\\nonumber\\\\&= N_k+\\tilde{R}\\sum_{j=0}^{k-1}\\left\\|p_j(x(\\cdot))\\right\\|_{\\mathbb{X}},\n\t\t\\end{align*}\n\t\twhere  $\\tilde{R}=\\left(\\frac{M\\tilde{M}\\alpha}{\\Gamma(1+\\alpha)}\\right)^{2}\\frac{2T^{2\\alpha-1}}{\\lambda(2\\alpha-1)}$ and $N_k=\\left\\|\\zeta_k\\right\\|_{\\mathbb{X}}+Ml_k+\\frac{2T^{\\alpha-\\alpha_1}}{\\mu^{1-\\alpha_1}}\\frac{M\\alpha}{\\Gamma(1+\\alpha)}\\left\\|\\gamma_{r'}\\right\\|_{\\mathrm{L}^{\\frac{1}{\\alpha_1}}(J;\\mathbb{R^+})}$ for $k=1,\\ldots,m.$ Applying the discrete Gronwall-Bellman lemma (Lemma \\ref{lem2.13}), we obtain \n\t\t\\begin{align}\\label{4.5}\n\t\t\\left\\|p_k(x(\\cdot))\\right\\|_{\\mathbb{X}}\\le&N_k+\\tilde{R}\\sum_{j=0}^{k-1}N_je^{\\frac{(k+j)(k-j-1)\\tilde{R}}{2}}=C_k, \\ \\mbox{for}\\ k=1,\\ldots,m.\n\t\t\\end{align}\n\t\tTaking $t\\in[0,t_1]$ and using the relation \\eqref{2.5}, Lemma \\ref{lem2.5} and Assumption \\ref{as2.1} \\textit{(H1)}-\\textit{(H3)}, we compute\n\t\t\\begin{align}\\label{4.21}\n\t\tr&<\\left\\|(F_{\\lambda}x^r)(t)\\right\\|_\\mathbb{X}\\nonumber\\\\&=\\left\\|\\mathcal{T}_{\\alpha}(t)\\psi(0)+\\int_{0}^{t}(t-s)^{\\alpha-1}\\widehat{\\mathcal{T}}_{\\alpha}(t-s)\\left[\\mathrm{B}u^{\\alpha}_{\\lambda}(s)+f(s,\\tilde{x}_{\\rho(s, \\tilde{x}_s)})\\right]\\mathrm{d}s\\right\\|_{\\mathbb{X}} \\nonumber\\\\&\\le\\left\\|\\mathcal{T}_{\\alpha}(t)\\psi(0)\\right\\|_{\\mathbb{X}}+\\int_0^t(t-s)^{\\alpha-1}\\left\\|\\widehat{\\mathcal{T}}_{\\alpha}(t-s)\\mathrm{B}u^{\\alpha}_{\\lambda}(s)\\right\\|_{\\mathbb{X}}\\mathrm{d}s\\nonumber\\\\&\\quad+\\int_0^t(t-s)^{\\alpha-1}\\left\\|\\widehat{\\mathcal{T}}_{\\alpha}(t-s)f(s,\\tilde{x}_{\\rho(s,\\tilde{x}_s)})\\right\\|_{\\mathbb{X}}\\mathrm{d}s\\nonumber\\\\&\\le  M\\left\\|\\psi(0)\\right\\|_{\\mathbb{X}}+\\frac{M\\tilde{M}\\alpha}{\\Gamma(1+\\alpha)}\\!\\!\\int_0^t\\!\\!(t-s)^{\\alpha-1}\\left\\|u^{\\alpha}_{\\lambda}(s)\\right\\|_{\\mathbb{U}}\\mathrm{d}s+\\frac{M\\alpha}{\\Gamma(1+\\alpha)}\\!\\!\\int_0^t\\!\\!(t-s)^{\\alpha-1}\\left\\|f(s,\\tilde{x}_{\\rho(s,\\tilde{x}_s)})\\right\\|_{\\mathbb{X}}\\mathrm{d}s\\nonumber\\\\&\\le M\\left\\|\\psi(0)\\right\\|_{\\mathbb{X}}+\\frac{1}{\\lambda}\\left(\\frac{M\\tilde{M}\\alpha}{\\Gamma(1+\\alpha)}\\right)^{2}\\left\\|p_0(x(\\cdot))\\right\\|_{\\mathbb{X}}\\int_{0}^{t}(t-s)^{\\alpha-1}(t_1-s)^{\\alpha-1}\\mathrm{d}s\\nonumber\\\\&\\quad+\\frac{M\\alpha}{\\Gamma(1+\\alpha)}\\int_0^t(t-s)^{\\alpha-1}\\gamma_{r'}(s)\\mathrm{d}s\\nonumber\\\\&\\le M\\left\\|\\psi(0)\\right\\|_{\\mathbb{X}}+\\left(\\frac{M\\tilde{M}\\alpha}{\\Gamma(1+\\alpha)}\\right)^{2}\\frac{t^{2\\alpha-1}}{\\lambda(2\\alpha-1)}\\left\\|p_0(x(\\cdot))\\right\\|_{\\mathbb{X}}\\nonumber\\\\&\\quad+\\frac{M\\alpha}{\\Gamma(1+\\alpha)}\\left(\\int_0^t(t-s)^{\\frac{\\alpha-1}{1-\\alpha_1}}\\mathrm{d}s\\right)^{1-\\alpha_1}\\left\\|\\gamma_{r'}\\right\\|_{\\mathrm{L}^{\\frac{1}{\\alpha_1}}([0,t];\\mathbb{R^{+}})}\\nonumber\\\\&\\le M\\left\\|\\psi(0)\\right\\|_{\\mathbb{X}}+\\left(\\frac{M\\tilde{M}\\alpha}{\\Gamma(1+\\alpha)}\\right)^{2}\\frac{T^{2\\alpha-1}}{\\lambda(2\\alpha-1)}\\left\\|p_0(x(\\cdot))\\right\\|_{\\mathbb{X}}+\\frac{M\\alpha}{\\Gamma(1+\\alpha)}\\frac{T^{\\alpha-\\alpha_1}}{\\mu^{1-\\alpha_1}}\\left\\|\\gamma_{r'}\\right\\|_{\\mathrm{L}^{\\frac{1}{\\alpha_1}}(J;\\mathbb{R^+})}\\nonumber\\\\&\\le M\\left\\|\\psi(0)\\right\\|_{\\mathbb{X}}+\\left(\\frac{M\\tilde{M}\\alpha}{\\Gamma(1+\\alpha)}\\right)^{2}\\frac{2T^{2\\alpha-1}}{\\lambda(2\\alpha-1)}\\sum_{j=0}^{m}\\left\\|p_j(x(\\cdot))\\right\\|_{\\mathbb{X}}+\\frac{M\\alpha}{\\Gamma(1+\\alpha)}\\frac{2T^{\\alpha-\\alpha_1}}{\\mu^{1-\\alpha_1}}\\left\\|\\gamma_{r'}\\right\\|_{\\mathrm{L}^{\\frac{1}{\\alpha_1}}(J;\\mathbb{R^+})}\\nonumber\\\\&\\le M\\left\\|\\psi(0)\\right\\|_{\\mathbb{X}}+\\tilde{R}\\sum_{j=0}^{m}C_j+\\frac{M\\alpha}{\\Gamma(1+\\alpha)}\\frac{2T^{\\alpha-\\alpha_1}}{\\mu^{1-\\alpha_1}}\\left\\|\\gamma_{r'}\\right\\|_{\\mathrm{L}^{\\frac{1}{\\alpha_1}}(J;\\mathbb{R^+})},\n\t\t\\end{align}\n\t\twhere $C_0=N_0$. For $t\\in(t_k,\\tau_k],\\ k=1,\\ldots,m$, we obtain  \n\t\t\\begin{align}\\label{4.22}\n\t\tr<\\left\\|(F_{\\lambda}x^r)(t)\\right\\|_\\mathbb{X}&\\le\\left\\|h_k(t, \\tilde{x}(t_k^-))\\right\\|_{\\mathbb{X}}\\nonumber\\\\&\\le l_k\\le l_k+\\tilde{R}\\sum_{j=0}^{m}C_j+\\frac{2T^{\\alpha-\\alpha_1}}{\\mu^{1-\\alpha_1}}\\frac{M\\alpha}{\\Gamma(1+\\alpha)}\\left\\|\\gamma_{r'}\\right\\|_{\\mathrm{L}^{\\frac{1}{\\alpha_1}}(J;\\mathbb{R^+})}.\n\t\t\\end{align}\n\t\tTaking $t\\in(\\tau_k,t_{k+1}], \\ k=1,\\dots,m$, we evaluate\n\t\t\\begin{align}\\label{4.23}\n\t\tr&<\\left\\|(F_{\\lambda}x^r)(t)\\right\\|_\\mathbb{X}\\nonumber\\\\&=\\bigg\\|\\mathcal{T}_{\\alpha}(t-\\tau_k)h_k(\\tau_k,\\tilde{x}(t_k^-))-\\int_{0}^{\\tau_k}(\\tau_k-s)^{\\alpha-1}\\widehat{\\mathcal{T}}_{\\alpha}(\\tau_k-s)\\left[\\mathrm{B}u^{\\alpha}_{\\lambda}(s)+f(s,\\tilde{x}_{\\rho(s,\\tilde{x}_s)})\\right]\\mathrm{d}s\\nonumber\\\\&\\quad+\\int_{0}^{t}(t-s)^{\\alpha-1}\\widehat {\\mathcal{T}}_{\\alpha}(t-s)\\left[\\mathrm{B}u^{\\alpha}_{\\lambda}(s)+f(s,\\tilde{x}_{\\rho(s,\\tilde{x}_s)})\\right]\\mathrm{d}s\\bigg\\|_{\\mathbb{X}}\\nonumber\\\\&\\le\\left\\|\\mathcal{T}_{\\alpha} (t)h_k(\\tau_k, \\tilde{x}(t_k^-))\\right\\|_{\\mathbb{X}}+\\int_{0}^{\\tau_k}(\\tau_k-s)^{\\alpha-1}\\left\\|\\widehat{\\mathcal{T}}_{\\alpha}(\\tau_k-s)\\mathrm{B}u^{\\alpha}_{\\lambda}(s)\n\t\t\\right\\|_{\\mathbb{X}}\\mathrm{d}s\\nonumber\\\\&\\quad+\\int_{0}^{\\tau_k}(\\tau_k-s)^{\\alpha-1}\\left\\|\\widehat{\\mathcal{T}}_{\\alpha}(\\tau_k-s)f(s,\\tilde{x}_{\\rho(s,\\tilde{x}_s)})\\right\\|_{\\mathbb{X}}\\mathrm{d}s+\\int_{0}^t(t-s)^{\\alpha-1}\\left\\|\\widehat{\\mathcal{T}}_{\\alpha}(t-s)\\mathrm{B}u^{\\alpha}_{\\lambda}(s)\\right\\|_{\\mathbb{X}}\\mathrm{d}s\\nonumber\\\\&\\quad +\\int_{0}^t(t-s)^{\\alpha-1}\\left\\|\\widehat{\\mathcal{T}}_{\\alpha}(t-s)f(s,\\tilde{x}_{\\rho(s, \\tilde{x}_s)})\\right\\|_{\\mathbb{X}}\\mathrm{d}s\n\t\t\\nonumber\\\\&\\le Ml_k+\\frac{M\\tilde{M}\\alpha}{\\Gamma(1+\\alpha)}\\int_{0}^{\\tau_k}(\\tau_k-s)^{\\alpha-1}\\left\\|u^{\\alpha}_{\\lambda}(s)\\right\\|_{\\mathbb{U}}\\mathrm{d}s+\\frac{M\\alpha}\n\t\t{\\Gamma(1+\\alpha)}\\int_{0}^{\\tau_k}(\\tau_k-s)^{\\alpha-1}\\left\\|f(s,\\tilde{x}_{\\rho(s, \\tilde{x}_s)})\\right\\|_{\\mathbb{X}}\\mathrm{d}s\\nonumber\\\\&\\quad+\\frac{M\\tilde{M}\\alpha}{\\Gamma(1+\\alpha)}\\int_{0}^{t}(t-s)^{\\alpha-1}\\left\\|u^{\\alpha}_{\\lambda}(s)\\right\\|_{\\mathbb{U}}\\mathrm{d}s+\\frac{M\\alpha}\n\t\t{\\Gamma(1+\\alpha)}\\int_{0}^{t}(t-s)^{\\alpha-1}\\left\\|f(s,\\tilde{x}_{\\rho(s, \\tilde{x}_s)})\\right\\|_{\\mathbb{X}}\\mathrm{d}s\\nonumber\\\\&\\le Ml_k+\\frac{1}{\\lambda}\\left(\\frac{M\\tilde{M}\\alpha}{\\Gamma(1+\\alpha)}\\right)^{2}\\sum_{j=0}^{k-1}\\left\\|p_j(x(\\cdot))\\right\\|_{\\mathbb{X}}\\int_{\\tau_j}^{t_{j+1}}(\\tau_k-s)^{\\alpha-1}(t_{j+1}-s)^{\\alpha-1}\\mathrm{d}s\\nonumber\\\\&\\quad+\\frac{M\\alpha}{\\Gamma(1+\\alpha)}\\left(\\int_{0}^{\\tau_k}(\\tau_k-s)^{\\frac{\\alpha-1}{1-\\alpha_1}}\\mathrm{d}s\\right)^{1-\\alpha_1}\\left\\|\\gamma_{r'}\\right\\|_{\\mathcal{L}^{\\frac{1}{\\alpha_1}}([0,\\tau_k];\\mathbb{R^{+}})}\\nonumber\\\\&\\quad+\\frac{1}{\\lambda}\\left(\\frac{M\\tilde{M}\\alpha}{\\Gamma(1+\\alpha)}\\right)^{2}\\Bigg[\\sum_{j=0}^{k-1}\\left\\|p_j(x(\\cdot))\\right\\|_{\\mathbb{X}}\\int_{\\tau_j}^{t_{j+1}}(t-s)^{\\alpha-1}(t_{j+1}-s)^{\\alpha-1}\\mathrm{d}s\\nonumber\\\\&\\qquad+\\left\\|p_k(x(\\cdot))\\right\\|_{\\mathbb{X}}\\int_{\\tau_k}^{t}(t-s)^{\\alpha-1}(t_{k+1}-s)^{\\alpha-1}\\mathrm{d}s\\Bigg]+\\frac{M\\alpha}{\\Gamma(1+\\alpha)}\\left(\\int_{0}^{t}(t-s)^{\\frac{\\alpha-1}{1-\\alpha_1}}\\mathrm{d}s\\right)^{1-\\alpha_1}\\left\\|\\gamma_{r'}\\right\\|_{\\mathcal{L}^{\\frac{1}{\\alpha_1}}([0,t];\\mathbb{R^{+}})}\\nonumber\\\\&\\le Ml_k+\\frac{2}{\\lambda}\\left(\\frac{M\\tilde{M}\\alpha}{\\Gamma(1+\\alpha)}\\right)^{2}\\sum_{j=0}^{k-1}\\left\\|p_j(x(\\cdot))\\right\\|_{\\mathbb{X}}\\int_{\\tau_j}^{t_{j+1}}(\\tau_k-s)^{\\alpha-1}(t_{j+1}-s)^{\\alpha-1}\\mathrm{d}s\\nonumber\\\\&\\quad+\\frac{M\\alpha}{\\Gamma(1+\\alpha)}\\left(\\int_{0}^{\\tau_k}(\\tau_k-s)^{\\frac{\\alpha-1}{1-\\alpha_1}}\\mathrm{d}s\\right)^{1-\\alpha_1}\\left\\|\\gamma_{r'}\\right\\|_{\\mathcal{L}^{\\frac{1}{\\alpha_1}}([0,\\tau_k];\\mathbb{R^{+}})}\\nonumber\\\\&\\quad+\\frac{1}{\\lambda}\\left(\\frac{M\\tilde{M}\\alpha}{\\Gamma(1+\\alpha)}\\right)^{2}\\left\\|p_k(x(\\cdot))\\right\\|_{\\mathbb{X}}\\int_{\\tau_k}^{t}(t-s)^{\\alpha-1}(t_{k+1}-s)^{\\alpha-1}\\mathrm{d}s\\nonumber\\\\&\\quad+\\frac{M\\alpha}{\\Gamma(1+\\alpha)}\\left(\\int_{0}^{t}(t-s)^{\\frac{\\alpha-1}{1-\\alpha_1}}\\mathrm{d}s\\right)^{1-\\alpha_1}\\left\\|\\gamma_{r'}\\right\\|_{\\mathcal{L}^{\\frac{1}{\\alpha_1}}([0,t];\\mathbb{R^{+}})}\\nonumber\\\\&\\le Ml_k+\\frac{2}{\\lambda}\\left(\\frac{M\\tilde{M}\\alpha}{\\Gamma(1+\\alpha)}\\right)^{2}\\sum_{j=0}^{k-1}\\left\\|p_j(x(\\cdot))\\right\\|_{\\mathbb{X}}\\int_{\\tau_j}^{t_{j+1}}(t_{j+1}-s)^{2(\\alpha-1)}\\mathrm{d}s\\nonumber\\\\&\\quad+\\frac{M\\alpha}{\\Gamma(1+\\alpha)}\\left(\\int_{0}^{\\tau_k}(\\tau_k-s)^{\\frac{\\alpha-1}{1-\\alpha_1}}\\mathrm{d}s\\right)^{1-\\alpha_1}\\left\\|\\gamma_{r'}\\right\\|_{\\mathcal{L}^{\\frac{1}{\\alpha_1}}([0,\\tau_k];\\mathbb{R^{+}})}\\nonumber\\\\&\\quad+\\frac{1}{\\lambda}\\left(\\frac{M\\tilde{M}\\alpha}{\\Gamma(1+\\alpha)}\\right)^{2}\\left\\|p_k(x(\\cdot))\\right\\|_{\\mathbb{X}}\\int_{\\tau_k}^{t}(t-s)^{2(\\alpha-1)}\\mathrm{d}s+\\frac{M\\alpha}{\\Gamma(1+\\alpha)}\\left(\\int_{0}^{t}(t-s)^{\\frac{\\alpha-1}{1-\\alpha_1}}\\mathrm{d}s\\right)^{1-\\alpha_1}\\left\\|\\gamma_{r'}\\right\\|_{\\mathcal{L}^{\\frac{1}{\\alpha_1}}([0,t];\\mathbb{R^{+}})}\\nonumber\\\\&\\le Ml_k+\\frac{2}{\\lambda}\\left(\\frac{M\\tilde{M}\\alpha}{\\Gamma(1+\\alpha)}\\right)^{2}\\sum_{j=0}^{k-1}\\left\\|p_j(x(\\cdot))\\right\\|_{\\mathbb{X}}\\frac{(t_{j+1}-\\tau_j)^{2\\alpha-1}}{2\\alpha-1}\\nonumber\\\\&\\quad+\\frac{M\\alpha}{\\Gamma(1+\\alpha)}\\left(\\int_{0}^{\\tau_k}(\\tau_k-s)^{\\frac{\\alpha-1}{1-\\alpha_1}}\\mathrm{d}s\\right)^{1-\\alpha_1}\\left\\|\\gamma_{r'}\\right\\|_{\\mathcal{L}^{\\frac{1}{\\alpha_1}}([0,\\tau_k];\\mathbb{R^{+}})}\\nonumber\\\\&\\quad+\\frac{1}{\\lambda}\\left(\\frac{M\\tilde{M}\\alpha}{\\Gamma(1+\\alpha)}\\right)^{2}\\left\\|p_k(x(\\cdot))\\right\\|_{\\mathbb{X}}\\frac{(t-\\tau_k)^{2\\alpha-1}}{2\\alpha-1}+\\frac{M\\alpha}{\\Gamma(1+\\alpha)}\\left(\\int_{0}^{t}(t-s)^{\\frac{\\alpha-1}{1-\\alpha_1}}\\mathrm{d}s\\right)^{1-\\alpha_1}\\left\\|\\gamma_{r'}\\right\\|_{\\mathcal{L}^{\\frac{1}{\\alpha_1}}([0,t];\\mathbb{R^{+}})}\\nonumber\\\\&\\le Ml_k+\\frac{2T^{2\\alpha-1}}{\\lambda(2\\alpha-1)}\\left(\\frac{M\\tilde{M}\\alpha}{\\Gamma(1+\\alpha)}\\right)^{2}\\sum_{j=0}^{k-1}\\left\\|p_j(x(\\cdot))\\right\\|_{\\mathbb{X}}\\nonumber\\\\&\\quad+\\frac{M\\alpha}{\\Gamma(1+\\alpha)}\\left(\\int_{0}^{\\tau_k}(\\tau_k-s)^{\\frac{\\alpha-1}{1-\\alpha_1}}\\mathrm{d}s\\right)^{1-\\alpha_1}\\left\\|\\gamma_{r'}\\right\\|_{\\mathcal{L}^{\\frac{1}{\\alpha_1}}([0,\\tau_k];\\mathbb{R^{+}})}+\\frac{2T^{2\\alpha-1}}{\\lambda(2\\alpha-1)}\\left(\\frac{M\\tilde{M}\\alpha}{\\Gamma(1+\\alpha)}\\right)^{2}\\left\\|p_k(x(\\cdot))\\right\\|_{\\mathbb{X}}\\nonumber\\\\&\\quad+\\frac{M\\alpha}{\\Gamma(1+\\alpha)}\\left(\\int_{0}^{t}(t-s)^{\\frac{\\alpha-1}{1-\\alpha_1}}\\mathrm{d}s\\right)^{1-\\alpha_1}\\left\\|\\gamma_{r'}\\right\\|_{\\mathcal{L}^{\\frac{1}{\\alpha_1}}([0,t];\\mathbb{R^{+}})}\\nonumber\\\\&\\le Ml_k+\\tilde{R}\\sum_{j=0}^{k}C_j+\\frac{2T^{\\alpha-\\alpha_1}}{\\mu^{1-\\alpha_1}}\\frac{M\\alpha}{\\Gamma(1+\\alpha)}\\left\\|\\gamma_{r'}\\right\\|_{\\mathrm{L}^{\\frac{1}{\\alpha_1}}(J;\\mathbb{R^+})}\\nonumber\\\\&\\le Ml_k+\\tilde{R}\\sum_{j=0}^{m}C_j+\\frac{2T^{\\alpha-\\alpha_1}}{\\mu^{1-\\alpha_1}}\\frac{M\\alpha}{\\Gamma(1+\\alpha)}\\left\\|\\gamma_{r'}\\right\\|_{\\mathrm{L}^{\\frac{1}{\\alpha_1}}(J;\\mathbb{R^+})}.\n\t\t\\end{align}\n\t\tUsing Assumption \\ref{as2.1} (\\textit{H2})(ii), we easily obtain\n\t\t\\begin{align*}\n\t\t\\liminf_{r \\rightarrow \\infty }\\frac {\\left\\|\\gamma_{r'}\\right\\|_{\\mathrm{L}^{\\frac{1}{\\alpha_1}}(J;\\mathbb{R^+})}}{r}&=\\liminf_{r \\rightarrow \\infty } \\left (\\frac {\\left\\|\\gamma_{r'}\\right\\|_{\\mathrm{L}^{\\frac{1}{\\alpha_1}}(J;\\mathbb{R^+})}}{r'}\\times\\frac{r'}{r}\\right)=H_2\\beta.\n\t\t\\end{align*}\n\t\tThus, dividing by $r$ in expressions \\eqref{4.21}, \\eqref{4.22}, \\eqref{4.23} and then passing $r\\to\\infty$, we obtain\n\t\t\\begin{align*}\n\t\t\\frac{MH_{2}\\alpha\\beta}{\\Gamma(1+\\alpha)}\\frac{2T^{\\alpha-\\alpha_1}}{\\mu^{1-\\alpha_1}}\\left\\{1+\\frac{(m+1)(m+2)\\tilde{R}}{2}+\\frac{m(m+1)\\tilde{R}^2}{2}\\sum_{j=0}^{m-1}e^{\\frac{(m+j)(m-j-1)\\tilde{R}}{2}}\\right\\}>1,\n\t\t\\end{align*}\n\t\twhich is  a contradiction to \\eqref{cnd}. Hence, for some  $ r>0$, $F_{\\lambda}(\\mathrm{E}_{r})\\subset \\mathrm{E}_{r}.$\n\t\t\\vskip 0.1in \n\t\t\\noindent\\textbf{Step (2): } \\emph{The operator $ F_{\\lambda}$ is continuous}. To achieve this goal, we consider a sequence $\\{{x}^n\\}^\\infty_{n=1}\\subseteq \\mathrm{E}_r$ such that ${x}^n\\rightarrow {x}\\mbox{ in }{\\mathrm{E}_r},$ that is,\n\t\t$$\\lim\\limits_{n\\rightarrow \\infty}\\left\\|x^n-x\\right\\|_{\\mathrm{PC}(J;\\mathbb{X})}=0.$$\n\t\tFrom Lemma \\ref{lem2.7}, we infer that\n\t\t\\begin{align*}\n\t\t\\left\\|\\tilde{x_{s}^n}- \\tilde{x_{s}}\\right\\|_{\\mathfrak{B}}&\\leq H_{2}\\sup\\limits_{\\theta\\in J}\\left\\|\\tilde{x^{n}}(\\theta)-\\tilde{x}(\\theta)\\right\\|_{\\mathbb{X}}=H_{2}\\left\\|x^{n}-x\\right\\|_{\\mathrm{PC}(J;\\mathbb{X})}\\rightarrow 0 \\ \\mbox{ as } \\ n\\rightarrow\\infty,\n\t\t\\end{align*}\n\t\tfor all $s\\in\\mathcal{Q}(\\rho^-)\\cup J$. Since $\\rho(s, \\tilde{x_s^k})\\in \\mathcal{Q}(\\rho^-)\\cup J,$ for all $k\\in\\mathbb{N}$, then we conclude that\n\t\t\\begin{align*}\n\t\t\\left\\|\\tilde{x^n}_{\\rho(s,\\tilde{x_s^k})}-\\tilde{x}_{\\rho(s,\\tilde{x_s^k})}\\right\\|_{\\mathfrak{B}}\\rightarrow 0 \\ \\mbox{ as } \\ n\\rightarrow\\infty, \\ \\mbox{ for all }\\ s\\in J\\  \\mbox{ and }\\ k\\in\\mathbb{N}.\n\t\t\\end{align*}\n\t\tIn particular, we choose $k=n$ and use the above convergence together with Assumption \\ref{as2.1} \\textit{(H1)} to  obtain\n\t\t\\begin{align}\\label{4.25}\n\t\t\\left\\|f(s, \\tilde{x^n}_{\\rho(s,\\tilde{x_s^n})})-f(s,\\tilde{x}_{\\rho(s,\\tilde{x_s})})\\right\\|_{\\mathbb{X}}&\\leq\\left\\|f(s, \\tilde{x^n}_{\\rho(s,\\tilde{x_s^n})})-f(s,\\tilde{x}_{\\rho(s,\\tilde{x_s^n})})\\right\\|_{\\mathbb{X}}\\nonumber\\\\&\\quad+\\left\\|f(s, \\tilde{x}_{\\rho(s,\\tilde{x_s^n})})-f(s,\\tilde{x}_{\\rho(s,\\tilde{x_s})})\\right\\|_{\\mathbb{X}}\\nonumber\\\\&\\to 0\\ \\mbox{ as }\\ n\\to\\infty, \\mbox{ uniformly for } \\ s\\in J. \n\t\t\\end{align}\n\t\tFrom the above convergence and the dominated convergence theorem, we evaluate\n\t\t\\begin{align}\\label{4.26}\n\t\t\\left\\|p_0(x^n(\\cdot))-p_0(x(\\cdot))\\right\\|_{\\mathbb{X}}&\\le\\left\\|\\int^{t_1}_{0}(t_1-s)^{\\alpha-1}\\widehat{\\mathcal{T}}_{\\alpha}(t_1-s)\\left[f(s,\\tilde{x^n}_{\\rho(s,\\tilde{x^n_s})})-f(s,\\tilde{x}_{\\rho(s,\\tilde{x_s})})\\right]\\mathrm{d}s\\right\\|_{\\mathbb{X}}\\nonumber\\\\&\\le\\frac{M\\alpha}{\\Gamma(1+\\alpha)}\\int^{t_1}_{0}(t_1-s)^{\\alpha-1}\\left\\|f(s,\\tilde{x^n}_{\\rho(s,\\tilde{x^n_s})})-f(s,\\tilde{x}_{\\rho(s,\\tilde{x_s})})\\right\\|_{\\mathbb{X}}\\mathrm{d}s\\nonumber\\\\&\\to 0\\ \\mbox{ as }\\ n\\to\\infty.\n\t\t\\end{align}\n\t\tUsing the convergences \\eqref{4.26} and the relation \\eqref{2.5}, we calculate\n\t\t\\begin{align*}\n\t\t\\left\\|\\mathcal{R}(\\lambda,\\Phi_{0}^{t_{1}})p_0(x^{n}(\\cdot))-\\mathcal{R}(\\lambda,\\Phi_{0}^{t_{1}})p_0(x(\\cdot))\\right\\|_{\\mathbb{X}}&=\\frac{1}{\\lambda}\\left\\|\\lambda\\mathcal{R}(\\lambda,\\Phi_{\\tau_k}^{t_{k+1}})\\left(p_0(x^{n}(\\cdot))-p_0(x(\\cdot))\\right)\\right\\|_{\\mathbb{X}}\\nonumber\\\\&\\leq\\frac{1}{\\lambda}\\left\\|p_0(x^{n}(\\cdot))-p_0(x(\\cdot))\\right\\|_{\\mathbb{X}}\\nonumber\\\\&\\to 0 \\ \\mbox{ as } \\ n \\to \\infty.\n\t\t\\end{align*}\n\t\tSince the mapping $\\mathcal{J}:\\mathbb{X}\\to\\mathbb{X}^{*}$  is  demicontinuous, it is easy to obtain\n\t\t\\begin{align}\\label{4.2}\n\t\t\\mathcal{J}\\left[\\mathcal{R}(\\lambda,\\Phi_{\\tau_k}^{t_{k+1}})p_0(x^{n}(\\cdot))\\right]\\xrightharpoonup{w}\\mathcal{J}\\left[\\mathcal{R}(\\lambda,\\Phi_{\\tau_k}^{t_{k+1}})p_0(x(\\cdot))\\right] \\ \\mbox{ as } \\ n\\to\\infty  \\ \\mbox{ in }\\ \\mathbb{X}^{*}.\n\t\t\\end{align}\n\t\tFrom Lemma \\ref{lem2.5}, we infer that the operator $\\widehat{\\mathcal{T}}_{\\alpha}(t)$ is compact for $t>0$. Therefore, the operator $\\widehat{\\mathcal{T}}_{\\alpha}(t)^*$ is also compact for $t>0$.   Hence, by using the compactness of this operator together with the weak convergence \\eqref{4.2}, one can obtain\n\t\t\\begin{align}\\label{4.1}\n\t\t&\\left\\|u^{n,\\alpha}_{0,\\lambda}(t)-u_{0,\\lambda}^{\\alpha}(t)\\right||_{\\mathbb{U}}\\nonumber\\\\&\\le\\left\\|(t_{1}-t)^{\\alpha-1}\\mathrm{B}^*\\widehat{\\mathcal{T}}_{\\alpha}(t_{1}-t)^*\\left[\\mathcal{J}\\left[\\mathcal{R}(\\lambda,\\Phi_{0}^{t_{1}})p_0(x^{n}(\\cdot))\\right]-\\mathcal{J}\\left[\\mathcal{R}(\\lambda,\\Phi_{0}^{t_{1}})p_0k(x(\\cdot))\\right]\\right]\\right\\|_{\\mathbb{U}}\\nonumber\\\\&\\le(t_{1}-t)^{\\alpha-1}\\tilde{M}\\left\\|\\widehat{\\mathcal{T}}_{\\alpha}(t_{1}-t)^*\\left[\\mathcal{J}\\left[\\mathcal{R}(\\lambda,\\Phi_{0}^{t_{1}})p_0(x^{n}(\\cdot))\\right]-\\mathcal{J}\\left[\\mathcal{R}(\\lambda,\\Phi_{0}^{t_{1}})p_0(x(\\cdot))\\right]\\right]\\right\\|_{\\mathbb{X}}\\nonumber\\\\&\\to 0\\ \\mbox{ as }\\ n\\to\\infty, \\ \\mbox{ for  each }\\ t\\in[0,t_{1}).\n\t\t\\end{align}\n\t\tSimilarly, for $k=1$, we compute\n\t\t\\begin{align}\\label{4.27}\n\t\t\\left\\|p_1(x^n(\\cdot))-p_1(x(\\cdot))\\right\\|_{\\mathbb{X}}&\\le\\left\\|\\mathcal{T}_{\\alpha}(t_{2}-\\tau_1)\\left[h_1(\\tau_1,\\tilde{x^n}(t_1^-))-h_1(\\tau_1,\\tilde{x}(t_1^-))\\right]\\right\\|_{\\mathbb{X}}\\nonumber\\\\&\\quad+\\left\\|\\int_{0}^{\\tau_{1}}(\\tau_{1}-s)^{\\alpha-1}\\widehat{\\mathcal{T}}_{\\alpha}(\\tau_{1}-s)\\left[f(s,\\tilde{x^n}_{\\rho(s,\\tilde{x^n_s})})-f(s,\\tilde{x}_{\\rho(s,\\tilde{x}_s)})\\right]\\mathrm{d}s\\right\\|_{\\mathbb{X}}\\nonumber\\\\&\\quad+\\left\\|\\int_{0}^{t_{2}}(t_{2}-s)^{\\alpha-1}\\widehat{\\mathcal{T}}_{\\alpha}(t_{2}-s)\\left[f(s,\\tilde{x^n}_{\\rho(s,\\tilde{x^n_s})})-f(s,\\tilde{x}_{\\rho(s,\\tilde{x}_s)})\\right]\\mathrm{d}s\\right\\|_{\\mathbb{X}}\\nonumber\\\\&\\quad+\\left\\|\\int_{0}^{\\tau_1}(\\tau_1-s)^{\\alpha-1}\\widehat{\\mathcal{T}}_{\\alpha}(\\tau_1-s)\\mathrm{B}\\left[u^{n,\\alpha}_{0,\\lambda}(s)-u^{\\alpha}_{0,\\lambda}(s)\\right]\\mathrm{d}s\\right\\|_{\\mathbb{X}} \\nonumber\\\\&\\quad+\\left\\|\\int_{0}^{\\tau_1}(t_2-s)^{\\alpha-1}\\widehat{\\mathcal{T}}_{\\alpha}(t_2-s)\\mathrm{B}\\left[u^{n,\\alpha}_{0,\\lambda}(s)-u^{\\alpha}_{0,\\lambda}(s)\\right]\\mathrm{d}s\\right\\|_{\\mathbb{X}} \\nonumber\\\\&\\le M\\left\\|h_1(\\tau_1,\\tilde{x^n}(t_1^-))-h_1(\\tau_1,\\tilde{x}(t_1^-))\\right\\|_{\\mathbb{X}}\\nonumber\\\\&\\quad+\\frac{M\\alpha}{\\Gamma(1+\\alpha)}\\int_{0}^{\\tau_{1}}(\\tau_{1}-s)^{\\alpha-1}\\left\\|f(s,\\tilde{x^n}_{\\rho(s,\\tilde{x^n_s})})-f(s,\\tilde{x}_{\\rho(s,\\tilde{x}_s)})\\right\\|_{\\mathbb{X}}\\mathrm{d}s\\nonumber\\\\&\\quad+\\frac{M\\alpha}{\\Gamma(1+\\alpha)}\\int_{0}^{t_{2}}(t_{2}-s)^{\\alpha-1}\\left\\|f(s,\\tilde{x^n}_{\\rho(s,\\tilde{x^n_s})})-f(s,\\tilde{x}_{\\rho(s,\\tilde{x}_s)})\\right\\|_{\\mathbb{X}}\\mathrm{d}s\\nonumber\\\\&\\quad+\\frac{M\\tilde{M}\\alpha}{\\Gamma(1+\\alpha)}\\int_{0}^{t_1}(\\tau_1-s)^{\\alpha-1}\\left\\|u^{n,\\alpha}_{0,\\lambda}(s)-u^{\\alpha}_{0,\\lambda}(s)\\right\\|_{\\mathbb{U}}\\mathrm{d}s\\nonumber\\\\&\\quad+\\frac{M\\tilde{M}\\alpha}{\\Gamma(1+\\alpha)}\\int_{0}^{t_1}(t_2-s)^{\\alpha-1}\\left\\|u^{n,\\alpha}_{0,\\lambda}(s)-u^{\\alpha}_{0,\\lambda}(s)\\right\\|_{\\mathbb{U}}\\mathrm{d}s\\nonumber\\\\&\\to0\\ \\mbox{ as }\\ n\\to\\infty, \n\t\t\\end{align}\n\t\twhere we used Assumption \\ref{as2.1} (\\textit{H3}),  convergences \\eqref{4.25}, \\eqref{4.1} and the dominated convergence theorem. Moreover, similar to the convergence \\eqref{4.1}, one can obtain\n\t\t\\begin{align*}\n\t\t&\\left\\|u^{n,\\alpha}_{1,\\lambda}(t)-u_{1,\\lambda}^{\\alpha}(t)\\right||_{\\mathbb{U}}\\to 0\\ \\mbox{ as }\\ n\\to\\infty, \\ \\mbox{ for  each }\\ t\\in[\\tau_1,t_{2}).\n\t\t\\end{align*}\n\t\tFurther, applying a similar analogy as above  for $k=2,\\ldots,m,$ one can compute \n\t\t\\begin{align*}\n\t\t&\\left\\|u^{n,\\alpha}_{k,\\lambda}(t)-u_{k,\\lambda}^{\\alpha}(t)\\right||_{\\mathbb{U}}\\to 0\\ \\mbox{ as }\\ n\\to\\infty, \\mbox{ for  each }\\ t\\in[\\tau_k,t_{k+1}), k=2,\\ldots,m.\n\t\t\\end{align*} \n\t\tTherefore, we have\n\t\t\\begin{align}\\label{4.29}\n\t\t\\left\\|u^{n,\\alpha}_{\\lambda}(t)-u_{\\lambda}^{\\alpha}(t)\\right||_{\\mathbb{U}}\\to 0\\ \\mbox{ as }\\ n\\to\\infty,\\ \\mbox{ for  each }\\ t\\in[\\tau_k,t_{k+1}), k=0,1,\\ldots,m.\n\t\t\\end{align}\n\t\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\tUsing the convergences \\eqref{4.25}, \\eqref{4.29} and the dominated convergence theorem, we arrive at \n\t\t\\begin{align*}\n\t\t\\left\\|(F_{\\lambda}x^n)(t)-(F_{\\lambda}x)(t)\\right\\|_{\\mathbb{X}}&\\le\\int_{0}^{t}(t-s)^{\\alpha-1}\\left\\|\\widehat{\\mathcal{T}}_{\\alpha}(t-s)\\mathrm{B}\\left[u^{n,\\alpha}_{\\lambda}(s)-u_{\\lambda}^{\\alpha}(s)\\right]\\right\\|_{\\mathbb{X}}\\mathrm{d}s\\nonumber\\\\&\\quad+\\int_{0}^{t}(t-s)^{\\alpha-1}\\left\\|\\widehat{\\mathcal{T}}_{\\alpha}(t-s)\\left[f(s,\\tilde{x^n}_{\\rho(s,\\tilde{x^n_s})})-f(s,\\tilde{x}_{\\rho(s,\\tilde{x}_s)})\\right]\\right\\|_{\\mathbb{X}}\\mathrm{d}s\\nonumber\\\\&\\le\\frac{M\\tilde{M}\\alpha}{\\Gamma(1+\\alpha)}\\int_{0}^{t}(t-s)^{\\alpha-1}\\left\\|u^{n,\\alpha}_{\\lambda}(s)-u_{\\lambda}^{\\alpha}(s)\\right\\|_{\\mathbb{U}}\\mathrm{d}s\\nonumber\\\\&\\quad+\\frac{M\\alpha}{\\Gamma(1+\\alpha)}\\int_{0}^{t}(t-s)^{\\alpha-1}\\left\\|f(s,\\tilde{x^n}_{\\rho(s, \\tilde{x^n_s})})-f(s,\\tilde{x}_{\\rho(s, \\tilde{x}_s)})\\right\\|_{\\mathbb{X}}\\mathrm{d}s\\nonumber\\\\&\\to 0 \\ \\mbox{ as }\\ n\\to\\infty, \\ \\mbox{ for }\\ t\\in[0,t_1].\n\t\t\\end{align*}\n\t\tSimilarly, for  $t\\in(\\tau_k,t_{k+1}],\\ k=1,\\ldots,m$, we deduce that \n\t\t\\begin{align*}\n\t\t\\left\\|(F_{\\lambda}x^n)(t)-(F_{\\lambda}x)(t)\\right\\|_{\\mathbb{X}}&\\le\\left\\|\\mathcal{T}_{\\alpha}(t-\\tau_k)\\left[h_k(\\tau_k,\\tilde{x^n}(t_k^-))-h_k(\\tau_k,\\tilde{x}(t_k^-))\\right]\\right\\|_{\\mathbb{X}}\\nonumber\\\\&\\quad+\\int_{0}^{\\tau_k}(\\tau_k-s)^{\\alpha-1}\\left\\|\\widehat{\\mathcal{T}}_{\\alpha}(\\tau_k-s)\\left[f(s,\\tilde{x^n}_{\\rho(s, \\tilde{x^n_s})})-f(s,\\tilde{x}_{\\rho(s, \\tilde{x}_s)})\\right]\\right\\|_{\\mathbb{X}}\\mathrm{d}s\\nonumber\\\\&\\quad+\\int_{0}^{\\tau_k}(\\tau_1-s)^{\\alpha-1}\\left\\|\\widehat{\\mathcal{T}}_{\\alpha}(\\tau_1-s)\\mathrm{B}\\left[u^{n,\\alpha}_{\\lambda}(s)-u_{\\lambda}^{\\alpha}(s)\\right]\\right\\|_{\\mathbb{X}}\\mathrm{d}s\\nonumber\\\\&\\quad+\\int_{0}^{t}(t-s)^{\\alpha-1}\\left\\|\\widehat{\\mathcal{T}}_{\\alpha}(t-s)\\mathrm{B}\\left[u^{n,\\alpha}_{\\lambda}(s)-u_{\\lambda}^{\\alpha}(s)\\right]\\right\\|_{\\mathbb{X}}\\mathrm{d}s\\nonumber\\\\&\\quad+\\int_{0}^{t}(t-s)^{\\alpha-1}\\left\\|\\widehat{\\mathcal{T}}_{\\alpha}(t-s)\\left[f(s,\\tilde{x^n}_{\\rho(s, \\tilde{x^n_s})})-f(s,\\tilde{x}_{\\rho(s, \\tilde{x}_s)})\\right]\\right\\|_{\\mathbb{X}}\\mathrm{d}s\\nonumber\\\\&\\le M\\left\\|h_k(\\tau_k,\\tilde{x^n}(t_k^-))-h_k(\\tau_k,\\tilde{x}(t_k^-))\\right\\|_{\\mathbb{X}}\\nonumber\\\\&\\quad+\\frac{M\\alpha}{\\Gamma(1+\\alpha)}\\int_{0}^{\\tau_k}(\\tau_k-s)^{\\alpha-1}\\left\\|f(s,\\tilde{x^n}_{\\rho(s, \\tilde{x^n_s})})-f(s,\\tilde{x}_{\\rho(s, \\tilde{x}_s)})\\right\\|_{\\mathbb{X}}\\mathrm{d}s\\nonumber\\\\&\\quad+ \\frac{M\\tilde{M}\\alpha}{\\Gamma(1+\\alpha)}\\int_{0}^{\\tau_k}(\\tau_k-s)^{\\alpha-1}\\left\\|u^{n,\\alpha}_{\\lambda}(s)-u_{\\lambda}^{\\alpha}(s)\\right\\|_{\\mathbb{U}}\\mathrm{d}s\\nonumber\\\\&\\quad+\\frac{M\\tilde{M}\\alpha}{\\Gamma(1+\\alpha)}\\int_{0}^{t}(t-s)^{\\alpha-1}\\left\\|u^{n,\\alpha}_{\\lambda}(s)-u_{\\lambda}^{\\alpha}(s)\\right\\|_{\\mathbb{U}}\\mathrm{d}s\\nonumber\\\\&\\quad+\\frac{M\\alpha}{\\Gamma(1+\\alpha)}\\int_{0}^{t}(t-s)^{\\alpha-1}\\left\\|f(s,\\tilde{x^n}_{\\rho(s, \\tilde{x^n_s})})-f(s,\\tilde{x}_{\\rho(s, \\tilde{x}_s)})\\right\\|_{\\mathbb{X}}\\mathrm{d}s\\nonumber\\\\&\\to 0 \\ \\mbox{ as }\\ n\\to\\infty.\n\t\t\\end{align*}\n\t\tMoreover, for $t\\in(t_k, \\tau_k],\\ k=1,\\ldots,m$, applying Assumption \\ref{as2.1} (\\textit{H3}), we obtain\n\t\t\\begin{align*}\n\t\t\\left\\|(F_{\\lambda}x^n)(t)-(F_{\\lambda}x)(t)\\right\\|_{\\mathbb{X}}&\\le\\left\\|h_k(t,\\tilde{x^n}(t_k^-))-h_k(t,\\tilde{x}(t_k^-))\\right\\|_{\\mathbb{X}}\\to 0 \\ \\mbox{ as }\\ n\\to\\infty.\n\t\t\\end{align*}\n\t\tHence, it follows that $F_{\\lambda}$ is continuous.\n\t\t\\vskip 0.1in \n\t\t\\noindent\\textbf{Step (3): } \\emph{$ F_{\\lambda}$ is a compact operator.} In order to prove this claim, we use the  well-known Ascoli-Arzela theorem. According to the infinite-dimensional version of the Ascoli-Arzela theorem (see, Theorem 3.7, Chapter 2, \\cite{JYONG}), it is enough to show that \n\t\t\\begin{itemize}\n\t\t\t\\item [(i)] the image of $\\mathrm{E}_r$ under $F_{\\lambda}$ is uniformly bounded (which is proved in Step I),\n\t\t\t\\item [(ii)] the image of $\\mathrm{E}_r$ under $F_{\\lambda}$ is equicontinuous,\n\t\t\t\\item [(iii)] for an arbitrary $t\\in J$, the set $\\mathrm{V}(t)=\\{(F_\\lambda x)(t):x\\in \\mathrm{E}_r\\}$ is relatively compact.\n\t\t\\end{itemize}\n\t\t\n\t\t\n\t\tFirst, we claim that the image of $\\mathrm{E}_r$ under $F_{\\lambda}$ is equicontinuous. For $s_1,s_2\\in[0,t_1]$ such that $s_1<s_2$ and $x\\in\\mathrm{E}_r$, we compute\n\t\t\\begin{align}\n\t\t&\\left\\|(F_{\\lambda}x)(s_2)-(F_{\\lambda}x)(s_1)\\right\\|_{\\mathbb{X}}\\nonumber\\\\&\\le\\left\\|\\left[\\mathcal{T}_{\\alpha}(s_2)-\\mathcal{T}_{\\alpha}(s_1)\\right]\\psi(0)\\right\\|_{\\mathbb{X}}+\\left\\|\\int_{s_1}^{s_2}(s_2-s)^{\\alpha-1}\\widehat{\\mathcal{T}}_{\\alpha}(s_2-s)f(s,\\tilde{x}_{\\rho(s,\\tilde{x}_s)})\\mathrm{d}s\\right\\|_{\\mathbb{X}}\\nonumber\\\\&\\quad+\\left\\|\\int_{s_1}^{s_2}(s_2-s)^{\\alpha-1}\\widehat{\\mathcal{T}}_{\\alpha}(s_2-s)\\mathrm{B}u^{\\alpha}_{\\lambda}(s)\\mathrm{d}s\\right\\|_{\\mathbb{X}}\\nonumber\\\\&\\quad+\\left\\|\\int_{0}^{s_1}\\left[(s_2-s)^{\\alpha-1}-(s_1-s)^{\\alpha-1}\\right]\\widehat{\\mathcal{T}}_{\\alpha}(s_2-s)\\mathrm{B}u^{\\alpha}_{\\lambda}(s)\\mathrm{d}s\\right\\|_{\\mathbb{X}}\\nonumber\\\\&\\quad+\\left\\|\\int_{0}^{s_1}(s_1-s)^{\\alpha-1}\\left[\\widehat{\\mathcal{T}}_{\\alpha}(s_2-s)-\\widehat{\\mathcal{T}}_{\\alpha}(s_1-s)\\right]\\mathrm{B}u^{\\alpha}_{\\lambda}(s)\\mathrm{d}s\\right\\|_{\\mathbb{X}}\\nonumber\\\\&\\quad+\\left\\|\\int_{0}^{s_1}\\left[(s_2-s)^{\\alpha-1}-(s_1-s)^{\\alpha-1}\\right]\\widehat{\\mathcal{T}}_{\\alpha}(s_2-s)f(s,\\tilde{x}_{\\rho(s,\\tilde{x}_s)})\\mathrm{d}s\\right\\|_{\\mathbb{X}}\\nonumber\\\\&\\quad+\\left\\|\\int_{0}^{s_1}(s_1-s)^{\\alpha-1}\\left[\\widehat{\\mathcal{T}}_{\\alpha}(s_2-s)-\\widehat{\\mathcal{T}}_{\\alpha}(s_1-s)\\right]f(s,\\tilde{x}_{\\rho(s,\\tilde{x}_s)})\\mathrm{d}s\\right\\|_{\\mathbb{X}}\\nonumber\\\\&\\le\\left\\|\\mathcal{T}_{\\alpha}(s_2)-\\mathcal{T}_{\\alpha}(s_1)\\right\\|_{\\mathcal{L}(\\mathbb{X})}\\left\\|\\psi(0)\\right\\|_{\\mathbb{X}}+\\frac{M\\alpha}{\\Gamma(1+\\alpha)}\\int_{s_1}^{s_2}(s_2-s)^{\\alpha-1}\\gamma_{r'}(s)\\mathrm{d}s\\nonumber\\\\&\\quad+\\frac{N_0}{\\lambda}\\left(\\frac{M\\tilde{M}\\alpha}{\\Gamma(1+\\alpha)}\\right)^{2}\\int_{s_1}^{s_2}(s_2-s)^{\\alpha-1}(t_1-s)^{\\alpha-1}\\mathrm{d}s\\nonumber\\\\&\\quad+\\frac{N_0}{\\lambda}\\left(\\frac{M\\tilde{M}\\alpha}{\\Gamma(1+\\alpha)}\\right)^{2}\\int_{0}^{s_1}\\left|(s_2-s)^{\\alpha-1}-(s_1-s)^{\\alpha-1}\\right|(t_1-s)^{\\alpha-1}\\mathrm{d}s\\nonumber\\\\&\\quad+\\frac{M\\tilde{M}^{2}N_{0}\\alpha}{\\lambda\\Gamma(1+\\alpha)}\\int_{0}^{s_1}(s_1-s)^{\\alpha-1}\\left\\|\\widehat{\\mathcal{T}}_{\\alpha}(s_2-s)-\\widehat{\\mathcal{T}}_{\\alpha}(s_1-s)\\right\\|_{\\mathcal{L}(\\mathbb{X})}(t_1-s)^{\\alpha-1}\\mathrm{d}s\\nonumber\\\\&\\quad+\\frac{M\\alpha}{\\Gamma(1+\\alpha)}\\int_{0}^{s_1}\\left|(s_2-s)^{\\alpha-1}-(s_1-s)^{\\alpha-1}\\right|\\gamma_{r'}(s)\\mathrm{d}s\\nonumber\\\\&\\quad+\\int_{0}^{s_1}(s_1-s)^{\\alpha-1}\\left\\|\\widehat{\\mathcal{T}}_{\\alpha}(s_2-s)-\\widehat{\\mathcal{T}}_{\\alpha}(s_1-s)\\right\\|_{\\mathcal{L}(\\mathbb{X})}\\gamma_{r'}(s)\\mathrm{d}s\\nonumber\\\\&\\le\\left\\|\\mathcal{T}_{\\alpha}(s_2)-\\mathcal{T}_{\\alpha}(s_1)\\right\\|_{\\mathcal{L}(\\mathbb{X})}\\left\\|\\psi(0)\\right\\|_{\\mathbb{X}}+\\frac{M\\alpha}{\\Gamma(1+\\alpha)}\\frac{(s_2-s_1)^{\\alpha-\\alpha_1}}{\\mu^{1-\\alpha_1}}\\left(\\int_{s_1}^{s_2}(\\gamma_{r'}(s))^{\\frac{1}{\\alpha_1}}\\mathrm{d}s\\right)^{\\alpha_1}\\nonumber\\\\&\\quad+\\frac{N_{0}}{\\lambda}\\left(\\frac{M\\tilde{M}\\alpha}{\\Gamma(1+\\alpha)}\\right)^{2}\\frac{(s_2-s_1)^{2\\alpha-1}}{2\\alpha-1}\\nonumber\\\\&\\quad+\\frac{N_{0}}{\\lambda}\\left(\\frac{M\\tilde{M}\\alpha}{\\Gamma(1+\\alpha)}\\right)^{2}\\int_{0}^{s_1}\\left|(s_2-s)^{\\alpha-1}-(s_1-s)^{\\alpha-1}\\right|(t_1-s)^{\\alpha-1}\\mathrm{d}s\\nonumber\\\\&\\quad+\\frac{M\\tilde{M}^{2}N_{0}\\alpha}{\\lambda\\Gamma(1+\\alpha)}\\int_{0}^{s_1}(s_1-s)^{\\alpha-1}\\left\\|\\widehat{\\mathcal{T}}_{\\alpha}(s_2-s)-\\widehat{\\mathcal{T}}_{\\alpha}(s_1-s)\\right\\|_{\\mathcal{L}(\\mathbb{X})}(t_1-s)^{\\alpha-1}\\mathrm{d}s\\nonumber\\\\&\\quad+\\frac{M\\alpha}{\\Gamma(1+\\alpha)}\\int_{0}^{s_1}\\left|(s_2-s)^{\\alpha-1}-(s_1-s)^{\\alpha-1}\\right|\\gamma_{r'}(s)\\mathrm{d}s\\nonumber\\\\&\\quad+\\int_{0}^{s_1}(s_1-s)^{\\alpha-1}\\left\\|\\widehat{\\mathcal{T}}_{\\alpha}(s_2-s)-\\widehat{\\mathcal{T}}_{\\alpha}(s_1-s)\\right\\|_{\\mathcal{L}(\\mathbb{X})}\\gamma_{r'}(s)\\mathrm{d}s.\n\t\t\\end{align}\n\t\tIf $s_1=0,$ then from the above estimate, we deduce that \n\t\t\\begin{align*}\n\t\t\\lim_{s_2\\to0^+}\\left\\|(F_{\\lambda}x)(s_2)-(F_{\\lambda}x)(s_1)\\right\\|_{\\mathbb{X}}=0\\ \\mbox{ unifromly for } \\ x\\in\\mathrm{E}_r.\n\t\t\\end{align*} \n\t\tFor $0<\\nu<s_1<t_1$, we have\n\t\t\\begin{align}\\label{4.31}\n\t\t&\\left\\|(F_{\\lambda}x)(s_2)-(F_{\\lambda}x)(s_1)\\right\\|_{\\mathbb{X}}\\nonumber\\\\&\\le\\left\\|\\mathcal{T}_{\\alpha}(s_2)-\\mathcal{T}_{\\alpha}(s_1)\\right\\|_{\\mathcal{L}(\\mathbb{X})}\\left\\|\\psi(0)\\right\\|_{\\mathbb{X}}+\\frac{M\\alpha}{\\Gamma(1+\\alpha)}\\frac{(s_2-s_1)^{\\alpha-\\alpha_1}}{\\mu^{1-\\alpha_1}}\\left(\\int_{s_1}^{s_2}(\\gamma_{r'}(s))^{\\frac{1}{\\alpha_1}}\\mathrm{d}s\\right)^{\\alpha_1}\\nonumber\\\\&\\quad+\\frac{N_{0}}{\\lambda}\\left(\\frac{M\\tilde{M}\\alpha}{\\Gamma(1+\\alpha)}\\right)^{2}\\frac{(s_2-s_1)^{2\\alpha-1}}{2\\alpha-1}\\nonumber\\\\&\\quad+\\frac{N_{0}}{\\lambda}\\left(\\frac{M\\tilde{M}\\alpha}{\\Gamma(1+\\alpha)}\\right)^{2}\\int_{0}^{s_1}\\left|(s_2-s)^{\\alpha-1}-(s_1-s)^{\\alpha-1}\\right|(t_1-s)^{\\alpha-1}\\mathrm{d}s\\nonumber\\\\&\\quad+\\frac{M\\tilde{M}^{2}N_{0}\\alpha}{\\lambda\\Gamma(1+\\alpha)}\\int_{0}^{s_1-\\nu}(s_1-s)^{\\alpha-1}\\left\\|\\widehat{\\mathcal{T}}_{\\alpha}(s_2-s)-\\widehat{\\mathcal{T}}_{\\alpha}(s_1-s)\\right\\|_{\\mathcal{L}(\\mathbb{X})}(t_1-s)^{\\alpha-1}\\mathrm{d}s\\nonumber\\\\&\\quad+\\frac{M\\tilde{M}^{2}N_{0}\\alpha}{\\lambda\\Gamma(1+\\alpha)}\\int_{s_1-\\nu}^{s_1}(s_1-s)^{\\alpha-1}\\left\\|\\widehat{\\mathcal{T}}_{\\alpha}(s_2-s)-\\widehat{\\mathcal{T}}_{\\alpha}(s_1-s)\\right\\|_{\\mathcal{L}(\\mathbb{X})}(t_1-s)^{\\alpha-1}\\mathrm{d}s\\nonumber\\\\&\\quad+\\frac{M\\alpha}{\\Gamma(1+\\alpha)}\\int_{0}^{s_1}\\left|(s_2-s)^{\\alpha-1}-(s_1-s)^{\\alpha-1}\\right|\\gamma_{r'}(s)\\mathrm{d}s\\nonumber\\\\&\\quad+\\int_{0}^{s_1-\\nu}(s_1-s)^{\\alpha-1}\\left\\|\\widehat{\\mathcal{T}}_{\\alpha}(s_2-s)-\\widehat{\\mathcal{T}}_{\\alpha}(s_1-s)\\right\\|_{\\mathcal{L}(\\mathbb{X})}\\gamma_{r'}(s)\\mathrm{d}s\\nonumber\\\\&\\quad+\\int_{s_1-\\nu}^{s_1}(s_1-s)^{\\alpha-1}\\left\\|\\widehat{\\mathcal{T}}_{\\alpha}(s_2-s)-\\widehat{\\mathcal{T}}_{\\alpha}(s_1-s)\\right\\|_{\\mathcal{L}(\\mathbb{X})}\\gamma_{r'}(s)\\mathrm{d}s\\nonumber\\\\&\\le\\left\\|\\mathcal{T}_{\\alpha}(s_2)-\\mathcal{T}_{\\alpha}(s_1)\\right\\|_{\\mathcal{L}(\\mathbb{X})}\\left\\|\\psi(0)\\right\\|_{\\mathbb{X}}+\\frac{M\\alpha}{\\Gamma(1+\\alpha)}\\frac{(s_2-s_1)^{\\alpha-\\alpha_1}}{\\mu^{1-\\alpha_1}}\\left(\\int_{s_1}^{s_2}(\\gamma_{r'}(s))^{\\frac{1}{\\alpha_1}}\\mathrm{d}s\\right)^{\\alpha_1}\\nonumber\\\\&\\quad+\\frac{N_{0}}{\\lambda}\\left(\\frac{M\\tilde{M}\\alpha}{\\Gamma(1+\\alpha)}\\right)^{2}\\frac{(s_2-s_1)^{2\\alpha-1}}{2\\alpha-1}\\nonumber\\\\&\\quad+\\frac{N_{0}}{\\lambda}\\left(\\frac{M\\tilde{M}\\alpha}{\\Gamma(1+\\alpha)}\\right)^{2}\\int_{0}^{s_1}\\left|(s_2-s)^{\\alpha-1}-(s_1-s)^{\\alpha-1}\\right|(t_1-s)^{\\alpha-1}\\mathrm{d}s\\nonumber\\\\&\\quad+\\frac{M\\tilde{M}^{2}N_{0}\\alpha}{\\lambda\\Gamma(1+\\alpha)}\\sup_{t\\in[0,s_1-\\nu]}\\left\\|\\widehat{\\mathcal{T}}_{\\alpha}(s_2-t)-\\widehat{\\mathcal{T}}_{\\alpha}(s_1-t)\\right\\|_{\\mathcal{L}(\\mathbb{X})}\\int_{0}^{s_1-\\nu}(s_1-s)^{\\alpha-1}(t_1-s)^{\\alpha-1}\\mathrm{d}s\\nonumber\\\\&\\quad+\\frac{2}{\\lambda}\\left(\\frac{M\\tilde{M}^{2}\\alpha}{\\Gamma(1+\\alpha)}\\right)^{2}\\frac{\\nu^{2\\alpha-1}}{2\\alpha-1}+\\frac{M\\alpha}{\\Gamma(1+\\alpha)}\\int_{0}^{s_1}\\left|(s_2-s)^{\\alpha-1}-(s_1-s)^{\\alpha-1}\\right|\\gamma_{r'}(s)\\mathrm{d}s\\nonumber\\\\&\\quad+\\sup_{t\\in[0,s_1-\\nu]}\\left\\|\\widehat{\\mathcal{T}}_{\\alpha}(s_2-t)-\\widehat{\\mathcal{T}}_{\\alpha}(s_1-t)\\right\\|_{\\mathcal{L}(\\mathbb{X})}\\int_{0}^{s_1-\\nu}(s_1-s)^{\\alpha-1}\\gamma_{r'}(s)\\mathrm{d}s\\nonumber\\\\&\\quad+\\frac{2M\\alpha}{\\Gamma(1+\\alpha)}\\frac{\\nu^{\\alpha-\\alpha_1}}{\\mu^{1-\\alpha_1}}\\left(\\int_{s_1-\\nu}^{s_1}(\\gamma_{r'}(s))^{\\frac{1}{\\alpha_1}}\\mathrm{d}s\\right)^{\\alpha_1}.\n\t\t\\end{align}\n\t\tSimilarly for $s_1,s_2\\in(\\tau_k,t_{k+1}], \\ k=1,\\ldots,m$ with $s_1<s_2$ and $x\\in\\mathrm{E}_r$, one can estimate \n\t\t\\begin{align}\\label{4.32}\n\t\t&\\left\\|(F_{\\lambda}x)(s_2)-(F_{\\lambda}x)(s_1)\\right\\|_{\\mathbb{X}}\\nonumber\\\\&\\le\\left\\|\\mathcal{T}_{\\alpha}(s_2-\\tau_k)-\\mathcal{T}_{\\alpha}(s_1-\\tau_k)\\right\\|_{\\mathcal{L}(\\mathbb{X})}l_k+\\frac{M\\alpha}{\\Gamma(1+\\alpha)}\\frac{(s_2-s_1)^{\\alpha-\\alpha_1}}{\\mu^{1-\\alpha_1}}\\left(\\int_{s_1}^{s_2}(\\gamma_{r'}(s))^{\\frac{1}{\\alpha_1}}\\mathrm{d}s\\right)^{\\alpha_1}\\nonumber\\\\&\\quad+\\frac{C_k}{\\lambda}\\left(\\frac{M\\tilde{M}\\alpha}{\\Gamma(1+\\alpha)}\\right)^{2}\\frac{(s_2-s_1)^{2\\alpha-1}}{2\\alpha-1}+\\frac{M\\tilde{M}\\alpha}{\\Gamma(1+\\alpha)}\\int_{0}^{s_1}\\left|(s_2-s)^{\\alpha-1}-(s_1-s)^{\\alpha-1}\\right|\\left\\|u_{\\lambda}^{\\alpha}(s)\\right\\|_{\\mathbb{U}}\\mathrm{d}s\\nonumber\\\\&\\quad+\\tilde{M}\\sup_{t\\in[0,s_1-\\nu]}\\left\\|\\widehat{\\mathcal{T}}_{\\alpha}(s_2-t)-\\widehat{\\mathcal{T}}_{\\alpha}(s_1-t)\\right\\|_{\\mathcal{L}(\\mathbb{X})}\\int_{0}^{s_1-\\nu}(s_1-s)^{\\alpha-1}\\left\\|u_{\\lambda}^{\\alpha}(s)\\right\\|_{\\mathbb{U}}\\mathrm{d}s\\nonumber\\\\&\\quad+\\frac{2C_k}{\\lambda}\\left(\\frac{M\\tilde{M}^{2}\\alpha}{\\Gamma(1+\\alpha)}\\right)^{2}\\frac{\\nu^{2\\alpha-1}}{2\\alpha-1}+\\frac{M\\alpha}{\\Gamma(1+\\alpha)}\\int_{0}^{s_1}\\left|(s_2-s)^{\\alpha-1}-(s_1-s)^{\\alpha-1}\\right|\\gamma_{r'}(s)\\mathrm{d}s\\nonumber\\\\&\\quad+\\sup_{t\\in[0,s_1-\\nu]}\\left\\|\\widehat{\\mathcal{T}}_{\\alpha}(s_2-t)-\\widehat{\\mathcal{T}}_{\\alpha}(s_1-t)\\right\\|_{\\mathcal{L}(\\mathbb{X})}\\int_{0}^{s_1-\\nu}(s_1-s)^{\\alpha-1}\\gamma_{r'}(s)\\mathrm{d}s\\nonumber\\\\&\\quad+\\frac{2M\\alpha}{\\Gamma(1+\\alpha)}\\frac{\\nu^{\\alpha-\\alpha_1}}{\\mu^{1-\\alpha_1}}\\left(\\int_{s_1-\\nu}^{s_1}(\\gamma_{r'}(s))^{\\frac{1}{\\alpha_1}}\\mathrm{d}s\\right)^{\\alpha_1}\\nonumber\\\\&\\le\\left\\|\\mathcal{T}_{\\alpha}(s_2-\\tau_k)-\\mathcal{T}_{\\alpha}(s_1-\\tau_k)\\right\\|_{\\mathcal{L}(\\mathbb{X})}l_k+\\frac{M\\alpha}{\\Gamma(1+\\alpha)}\\frac{(s_2-s_1)^{\\alpha-\\alpha_1}}{\\mu^{1-\\alpha_1}}\\left(\\int_{s_1}^{s_2}(\\gamma_{r'}(s))^{\\frac{1}{\\alpha_1}}\\mathrm{d}s\\right)^{\\alpha_1}\\nonumber\\\\&\\quad+\\frac{C_k}{\\lambda}\\left(\\frac{M\\tilde{M}\\alpha}{\\Gamma(1+\\alpha)}\\right)^{2}\\frac{(s_2-s_1)^{2\\alpha-1}}{2\\alpha-1}\\nonumber\\\\&\\quad+\\sum_{j=0}^{k-1}\\frac{C_j}{\\lambda}\\left(\\frac{M\\tilde{M}\\alpha}{\\Gamma(1+\\alpha)}\\right)^{2}\\int_{\\tau_j}^{t_{j+1}}\\left|(s_2-s)^{\\alpha-1}-(s_1-s)^{\\alpha-1}\\right|(t_{j+1}-s)^{\\alpha-1}\\mathrm{d}s\\nonumber\\\\&\\quad+\\frac{C_k}{\\lambda}\\left(\\frac{M\\tilde{M}\\alpha}{\\Gamma(1+\\alpha)}\\right)^{2}\\int_{\\tau_k}^{s_1}\\left|(s_2-s)^{\\alpha-1}-(s_1-s)^{\\alpha-1}\\right|(t_{k+1}-s)^{\\alpha-1}\\mathrm{d}s\\nonumber\\\\&\\quad+\\sup_{t\\in[0,s_1-\\nu]}\\left\\|\\widehat{\\mathcal{T}}_{\\alpha}(s_2-t)-\\widehat{\\mathcal{T}}_{\\alpha}(s_1-t)\\right\\|_{\\mathcal{L}(\\mathbb{X})}\\sum_{j=0}^{k-1}\\frac{C_j}{\\lambda}\\frac{M\\tilde{M}^{2}\\alpha}{\\Gamma(1+\\alpha)}\\int_{\\tau_j}^{t_{j+1}}(s_1-s)^{\\alpha-1}(t_{j+1}-s)^{\\alpha-1}\\mathrm{d}s\\nonumber\\\\&\\quad+\\sup_{t\\in[0,s_1-\\nu]}\\left\\|\\widehat{\\mathcal{T}}_{\\alpha}(s_2-t)-\\widehat{\\mathcal{T}}_{\\alpha}(s_1-t)\\right\\|_{\\mathcal{L}(\\mathbb{X})}\\frac{C_k}{\\lambda}\\frac{M\\tilde{M}^{2}\\alpha}{\\Gamma(1+\\alpha)}\\int_{\\tau_k}^{s_1-\\nu}(s_1-s)^{\\alpha-1}(t_{k+1}-s)^{\\alpha-1}\\mathrm{d}s\\nonumber\\\\&\\quad+\\frac{2C_k}{\\lambda}\\left(\\frac{M\\tilde{M}^{2}\\alpha}{\\Gamma(1+\\alpha)}\\right)^{2}\\frac{\\nu^{2\\alpha-1}}{2\\alpha-1}+\\frac{M\\alpha}{\\Gamma(1+\\alpha)}\\int_{0}^{s_1}\\left|(s_2-s)^{\\alpha-1}-(s_1-s)^{\\alpha-1}\\right|\\gamma_{r'}(s)\\mathrm{d}s\\nonumber\\\\&\\quad+\\sup_{t\\in[0,s_1-\\nu]}\\left\\|\\widehat{\\mathcal{T}}_{\\alpha}(s_2-t)-\\widehat{\\mathcal{T}}_{\\alpha}(s_1-t)\\right\\|_{\\mathcal{L}(\\mathbb{X})}\\int_{0}^{s_1-\\nu}(s_1-s)^{\\alpha-1}\\gamma_{r'}(s)\\mathrm{d}s\\nonumber\\\\&\\quad+\\frac{2M\\alpha}{\\Gamma(1+\\alpha)}\\frac{\\nu^{\\alpha-\\alpha_1}}{\\mu^{1-\\alpha_1}}\\left(\\int_{s_1-\\nu}^{s_1}(\\gamma_{r'}(s))^{\\frac{1}{\\alpha_1}}\\mathrm{d}s\\right)^{\\alpha_1}.\n\t\t\\end{align}\n\t\tMoreover, for $s_1,s_2\\in(t_k, \\tau_k], \\ k=1,\\ldots,m$ with $s_1<s_2$ and $x\\in\\mathrm{E}_r$, we have \n\t\t\\begin{align}\\label{4.33}\n\t\t\\left\\|(F_{\\lambda}x)(s_2)-(F_{\\lambda}x)(s_1)\\right\\|_{\\mathbb{X}}&\\le\\left\\|h_k(s_2,\\tilde{x}(t_k^-))-h_k(s_1,\\tilde{x}(t_k^-))\\right\\|_{\\mathbb{X}}.\n\t\t\\end{align}\n\t\tSimilar to the estimate \\eqref{2.8}, one can easily get that the right hand side of the expressions \\eqref{4.31} \\eqref{4.32}  and \\eqref{4.33} converge to zero as for the arbitrariness of $\\nu$ and $|s_2-s_1| \\rightarrow 0$.  Thus, the image of $\\mathrm{E}_r$ under $F_{\\lambda}$ is equicontinuous.\n\t\t\n\t\tNext,  we show that the set $\\mathrm{V}(t)=\\{(F_\\lambda x)(t):x\\in \\mathrm{E}_r\\}$ is relatively compact for each $t\\in J$. For $t=0$, it is easy to check the set  $\\mathrm{V}(t)$  is relatively compact in $\\mathrm{E}_r$.  Let us take $ 0<t\\leq t_1$ be fixed and for given $\\eta$ with $ 0<\\eta<t$ and any $\\delta>0$, we define\n\t\t\\begin{align*}\n\t&\t(F_{\\lambda}^{\\eta,\\delta}x)(t)\\nonumber\\\\&=\\int_{\\delta}^{\\infty}\\varphi_{\\alpha}(\\xi)\\mathcal{T}(t^{\\alpha}\\xi)\\psi(0)\\mathrm{d}\\xi +\\alpha\\int_{0}^{t-\\eta}\\int_{\\delta}^{\\infty}\\xi(t-s)^{\\alpha-1}\\varphi_{\\alpha}(\\xi)\\mathcal{T}((t-s)^{\\alpha}\\xi)\\mathrm{B}u^{\\alpha}_{\\lambda}(s)\\mathrm{d}\\xi\\mathrm{d}s\\nonumber\\\\&\\quad+\\alpha\\int_{0}^{t-\\eta}\\int_{\\delta}^{\\infty}\\xi(t-s)^{\\alpha-1}\\varphi_{\\alpha}(\\xi)\\mathcal{T}((t-s)^{\\alpha}\\xi)f(s,\\tilde{x}_{\\rho(s,\\tilde{x}_s)})\\mathrm{d}\\xi\\mathrm{d}s\\nonumber\\\\&=\\mathcal{T}(\\eta^{\\alpha}\\delta)\\bigg[\\int_{\\delta}^{\\infty}\\varphi_{\\alpha}(\\xi)\\mathcal{T}(t^{\\alpha}\\xi-\\eta^{\\alpha}\\delta)\\psi(0)\\mathrm{d}\\xi\\nonumber\\\\&\\qquad\\qquad+\\alpha\\int_{0}^{t-\\eta}\\int_{\\delta}^{\\infty}\\xi(t-s)^{\\alpha-1}\\varphi_{\\alpha}(\\xi)\\mathcal{T}((t-s)^{\\alpha}\\xi-\\eta^{\\alpha}\\delta)\\mathrm{B}u^{\\alpha}_{\\lambda}(s)\\mathrm{d}\\xi\\mathrm{d}s\\nonumber\\\\&\\qquad\\qquad+\\alpha\\int_{0}^{t-\\eta}\\int_{\\delta}^{\\infty}\\xi(t-s)^{\\alpha-1}\\varphi_{\\alpha}(\\xi)\\mathcal{T}((t-s)^{\\alpha}\\xi-\\eta^{\\alpha}\\delta)f(s,\\tilde{x}_{\\rho(s,\\tilde{x}_s)})\\mathrm{d}\\xi\\mathrm{d}s\\bigg]\\nonumber\\\\&=\\mathcal{T}(\\eta^{\\alpha}\\delta)y(t,\\eta,\\delta),\n\t\t\\end{align*}\n\t\twhere $y(\\cdot,\\cdot,\\cdot)$ is the term appearing inside the parenthesis. Using Assumption \\ref{as2.1}, one can calculate\n\t\t\\begin{align*}\n\t\t\\left\\|y(t,\\eta,\\delta)\\right\\|_{\\mathbb{X}}&\\le\\int_{\\delta}^{\\infty}\\varphi_{\\alpha}(\\xi)\\left\\|\\mathcal{T}(t^{\\alpha}\\xi-\\eta^{\\alpha}\\delta)\\psi(0)\\right\\|_{\\mathbb{X}}\\mathrm{d}\\xi\\nonumber\\\\&\\quad+\\alpha\\int_{0}^{t-\\eta}\\int_{\\delta}^{\\infty}\\xi(t-s)^{\\alpha-1}\\varphi_{\\alpha}(\\xi)\\left\\|\\mathcal{T}((t-s)^{\\alpha}\\xi-\\eta^{\\alpha}\\delta)\\mathrm{B}u^{\\alpha}_{\\lambda}(s)\\right\\|_{\\mathbb{X}}\\mathrm{d}\\xi\\mathrm{d}s\\nonumber\\\\&\\quad+\\alpha\\int_{0}^{t-\\eta}\\int_{\\delta}^{\\infty}\\xi(t-s)^{\\alpha-1}\\varphi_{\\alpha}(\\xi)\\left\\|\\mathcal{T}((t-s)^{\\alpha}\\xi-\\eta^{\\alpha}\\delta)f(s,\\tilde{x}_{\\rho(s,\\tilde{x}_s)})\\right\\|_{\\mathbb{X}}\\mathrm{d}\\xi\\mathrm{d}s\\nonumber\\\\&\\le\n\t\tM\\left\\|\\psi(0)\\right\\|_{\\mathbb{X}}+\\frac{N_0}{\\lambda}\\left(\\frac{M\\tilde{M}\\alpha}{\\Gamma(1+\\alpha)}\\right)^{2}\\frac{t^{2\\alpha-1}-\\eta^{2\\alpha-1}}{\\lambda(2\\alpha-1)}\\nonumber\\\\&\\quad\\times\\frac{M\\alpha}{\\Gamma(1+\\alpha)}\\left(\\frac{t^\\mu-\\eta^\\mu}{\\mu}\\right)^{1-\\alpha_1}\\left(\\int_{0}^{t-\\eta}(\\gamma_{r'}(s))^{\\frac{1}{\\alpha_1}}\\right)^{\\alpha_1}<+\\infty, \\ \\mbox{ for }\\ t\\in[0,t_1].\n\t\t\\end{align*}\n\t\tThe compactness of the operator $\\mathcal{T}(\\cdot)$  implies that the set  $\\mathrm{V}_{\\eta,\\delta}(t)=\\{(F^{\\eta,\\delta}_\\lambda x)(t):x\\in \\mathrm{E}_r\\}$ is relatively compact in $\\mathbb{X}$. Hence, there exist a finite $ x_{i}$'s, for $i=1,\\dots, n $ in $ \\mathbb{X} $ such that \n\t\t\\begin{align*}\n\t\t\\mathrm{V}_{\\eta,\\delta}(t) \\subset \\bigcup_{i=1}^{n}\\mathcal{S}(x_i, \\varepsilon\/2),\n\t\t\\end{align*}\n\t\tfor some $\\varepsilon>0$. Let us choose $\\delta>0$ and $\\eta>0$ such that \n\t\t\\begin{align*}\n\t\t&\\left\\|(F_{\\lambda}x)(t)-(F_{\\lambda}^{\\eta,\\delta}x)(t)\\right\\|_{\\mathbb{X}}\\nonumber\\\\&\\le\\left\\|\\int_{0}^{\\delta}\\varphi_{\\alpha}(\\xi)\\mathcal{T}(t^{\\alpha}\\xi)\\psi(0)\\mathrm{d}\\xi\\right\\|_{\\mathbb{X}}+\\alpha\\left\\|\\int_{0}^{t}\\int_{0}^{\\delta}\\xi(t-s)^{\\alpha-1}\\varphi_{\\alpha}(\\xi)\\mathcal{T}((t-s)^{\\alpha}\\xi)\\mathrm{B}u^{\\alpha}_{\\lambda}(s)\\mathrm{d}\\xi\\mathrm{d}s\\right\\|_{\\mathbb{X}}\\nonumber\\\\&\\quad+\\alpha\\left\\|\\int_{t-\\eta}^{t}\\int_{\\delta}^{\\infty}\\xi(t-s)^{\\alpha-1}\\varphi_{\\alpha}(\\xi)\\mathcal{T}((t-s)^{\\alpha}\\xi)\\mathrm{B}u^{\\alpha}_{\\lambda}(s)\\mathrm{d}\\xi\\mathrm{d}s\\right\\|_{\\mathbb{X}}\\nonumber\\\\&\\quad+\\alpha\\left\\|\\int_{0}^{t}\\int_{0}^{\\delta}\\xi(t-s)^{\\alpha-1}\\varphi_{\\alpha}(\\xi)\\mathcal{T}((t-s)^{\\alpha}\\xi)f(s,\\tilde{x}_{\\rho(s,\\tilde{x}_s)})\\mathrm{d}\\xi\\mathrm{d}s\\right\\|_{\\mathbb{X}}\\nonumber\\\\&\\quad+\\alpha\\left\\|\\int_{t-\\eta}^{t}\\int_{\\delta}^{\\infty}\\xi(t-s)^{\\alpha-1}\\varphi_{\\alpha}(\\xi)\\mathcal{T}((t-s)^{\\alpha}\\xi)f(s,\\tilde{x}_{\\rho(s,\\tilde{x}_s)})\\mathrm{d}\\xi\\mathrm{d}s\\right\\|_{\\mathbb{X}}\\nonumber\\\\&\\le M\\left\\|\\psi(0)\\right\\|_{\\mathbb{X}}\\int_{0}^{\\delta}\\varphi_{\\alpha}(\\xi)\\mathrm{d}\\xi+\\frac{M^2\\tilde{M}^2N_{0}\\alpha t^{2\\alpha-1}}{\\lambda(2\\alpha-1)(\\Gamma(\\alpha+1))}\\int_{0}^{\\delta}\\xi\\varphi_{\\alpha}(\\xi)\\mathrm{d}\\xi\n\t\t\\nonumber\\\\&\\quad+\\frac{N_0}{\\lambda}\\left(\\frac{M\\tilde{M}\\alpha}{\\Gamma(1+\\alpha)}\\right)^{2}\\int_{t-\\eta}^{t}(t-s)^{\\alpha-1}(t_1-s)^{\\alpha-1}\\mathrm{d}s\\nonumber\\\\&\\quad+\\frac{M\\alpha t^{\\alpha-\\alpha_1}}{\\mu^{1-\\alpha_1}}\\left\\|\\gamma_{r'}\\right\\|_{\\mathrm{L}^{\\frac{1}{\\alpha_1}}(J;\\mathbb{R^+})}\\int_{0}^{\\delta}\\xi\\varphi_{\\alpha}(\\xi)\\mathrm{d}\\xi+\\frac{M\\alpha}{(\\Gamma(1+\\alpha))^2}\\int_{t-\\eta}^{t}(t-s)^{\\alpha-1}\\gamma_{r'}(s)\\mathrm{d}s\\nonumber\\\\&\\le\\frac{\\varepsilon}{2}.\n\t\t\\end{align*}\n\t\tConsequently $$\\mathrm{V}(t)\\subset \\bigcup_{i=1}^{n}\\mathcal{S}(x_i, \\varepsilon ).$$\n\t\tThus, for each $t\\in [0,t_1]$, the set $\\mathrm{V}(t)$ is relatively compact in $ \\mathbb{X}$. Next, we take $ t\\in(\\tau_k,t_{k+1}],$ for $k=1,\\ldots,m$ and for given $\\eta$ with $ 0<\\eta<\\min\\{t-\\tau_k, \\tau_k\\}$ and any $\\delta>0$, we define\n\t\t\\begin{align*}\n\t\t&(F_{\\lambda}^{\\eta,\\delta}x)(t)\\nonumber\\\\&=\\int_{\\delta}^{\\infty}\\varphi_{\\alpha}(\\xi)\\mathcal{T}((t-\\tau_k)^{\\alpha}\\xi)h_k(\\tau_k,\\tilde{x}(t_k^-))\\mathrm{d}\\xi \\nonumber\\\\&\\quad-\\alpha\\int_{0}^{\\tau_k-\\eta}\\int_{\\delta}^{\\infty}\\xi(\\tau_k-s)^{\\alpha-1}\\varphi_{\\alpha}(\\xi)\\mathcal{T}((\\tau_k-s)^{\\alpha}\\xi)\\left[\\mathrm{B}u^{\\alpha}_{\\lambda}(s)+(s,\\tilde{x}_{\\rho(s,\\tilde{x}_s)})\\right]\\mathrm{d}\\xi\\mathrm{d}s\\nonumber\\\\&\\quad+\\alpha\\int_{0}^{t-\\eta}\\int_{\\delta}^{\\infty}\\xi(t-s)^{\\alpha-1}\\varphi_{\\alpha}(\\xi)\\mathcal{T}((t-s)^{\\alpha}\\xi)\\left[\\mathrm{B}u^{\\alpha}_{\\lambda}(s)+(s,\\tilde{x}_{\\rho(s,\\tilde{x}_s)})\\right]\\mathrm{d}\\xi\\mathrm{d}s\\nonumber\\\\&=\\mathcal{T}(\\eta^{\\alpha}\\delta)\\bigg[\\int_{\\delta}^{\\infty}\\varphi_{\\alpha}(\\xi)\\mathcal{T}((t-\\tau_k)^{\\alpha}\\xi-\\eta^{\\alpha}\\delta)h_k(\\tau_k,\\tilde{x}(t_k^-))\\mathrm{d}\\xi\\nonumber\\\\&\\quad+\\alpha\\int_{0}^{\\tau_k-\\eta}\\int_{\\delta}^{\\infty}\\xi(\\tau_k-s)^{\\alpha-1}\\varphi_{\\alpha}(\\xi)\\mathcal{T}((\\tau_k-s)^{\\alpha}\\xi-\\eta^{\\alpha}\\delta)\\left[\\mathrm{B}u^{\\alpha}_{\\lambda}(s)+(s,\\tilde{x}_{\\rho(s,\\tilde{x}_s)})\\right]\\mathrm{d}\\xi\\mathrm{d}s\\nonumber\\\\&\\quad+\\alpha\\int_{0}^{t-\\eta}\\int_{\\delta}^{\\infty}\\xi(t-s)^{\\alpha-1}\\varphi_{\\alpha}(\\xi)\\mathcal{T}((t-s)^{\\alpha}\\xi-\\eta^{\\alpha}\\delta)\\left[\\mathrm{B}u^{\\alpha}_{\\lambda}(s)+(s,\\tilde{x}_{\\rho(s,\\tilde{x}_s)})\\right]\\mathrm{d}\\xi\\mathrm{d}s\\bigg].\n\t\t\\end{align*}\n\t\tProceeding similarly for the case $t\\in[0,t_1]$, one can prove that the set $\\mathrm{V}(t),$ for $t\\in(\\tau_k,t_{k+1}],\\ k=1,\\ldots,m$ is relatively compact in $ \\mathbb{X}$. Moreover for $t\\in(t_k,\\tau_k], k=1\\ldots,m$, the fact that the set $\\mathrm{V}(t)$ is relative compact follows by the compactness of the impulses $h_k,$ for $k=1,\\ldots,m$. Therefore, the set $\\mathrm{V}(t)=\\{(F_\\lambda x)(t):x\\in \\mathrm{E}_r\\},$ for each $t\\in J$ is relatively compact in $\\mathbb{X}$.\n\t\t\n\t\tHence, by invoking the Arzela-Ascoli theorem, we conclude that the operator $ F_{\\lambda}$  is compact. Then \\emph{Schauder's fixed point theorem} yields that the operator $F_{\\lambda}$ has a fixed point in $\\mathrm{E}_{r}$, which is a mild solution of the system \\eqref{1.1}.\n\t\\end{proof}\n\t\n\tIn order to prove the approximate controllability of the system \\eqref{1.1}, we replace the assumption (\\textit{H2}) by the following stronger assumption: \n\t\\begin{enumerate}\\label{as}\n\t\t\\item [\\textit{(H4)}] The function $ f: J \\times \\mathfrak{B}_{\\alpha} \\rightarrow \\mathbb{X} $ satisfies the assumption (\\textit{$H2$})(i) and there exists a function $ \\gamma\\in \\mathrm{L}^{\\frac{1}{\\alpha_1}}(J;\\mathbb{R}^+)$ with $\\alpha_1\\in[0,\\frac{1}{2})$ such that $$ \\|f(t,\\psi)\\|_{\\mathbb{X}}\\leq \\gamma(t),\\ \\text{ for all }\\  (t,\\psi) \\in J \\times  \\mathfrak{B}_{\\alpha}. $$ \n\t\\end{enumerate}\n\t\\begin{theorem}\\label{thm4.4}\n\t\tSuppose that Assumptions (H0)-(H1), (H3)-(H4) and the condition \\eqref{cnd} of Theorem \\ref{thm4.3} are satisfied. Then the system \\eqref{1.1} is approximately controllable.\n\t\\end{theorem}\n\t\\begin{proof}\n\t\tBy using Theorem \\ref{thm4.3}, we infer that for every $\\lambda>0$ and $\\zeta_k\\in \\mathbb{X}$ for $k=0,1,\\ldots m$, there exists a mild solution $x^{\\lambda}\\in\\mathrm{E}_r$ such that\n\t\t\\begin{equation}\\label{M}\n\t\tx^{\\lambda}(t)=\\begin{dcases}\n\t\t\\mathcal{T}_{\\alpha}(t)\\psi(0)+\\int_{0}^{t}(t-s)^{\\alpha-1}\\widehat{\\mathcal{T}}_{\\alpha}(t-s)\\left[\\mathrm{B}u^{\\alpha}_{\\lambda}(s)+f(s,\\tilde{x^\\lambda}_{\\rho(s, \\tilde{x^\\lambda}_s)})\\right]\\mathrm{d}s,t\\in[0, t_1],\\\\\n\t\th_k(t, \\tilde{x^\\lambda}(t_k^-)),t\\in(t_k, \\tau_k],\\ k=1,\\ldots,m,\\\\\n\t\t\\mathcal{T}_{\\alpha}(t-\\tau_k)h_k(\\tau_k,\\tilde{x^\\lambda}(t_k^-))-\\int_{0}^{\\tau_k}(\\tau_k-s)^{\\alpha-1}\\widehat{\\mathcal{T}}_{\\alpha}(\\tau_k-s)\\left[\\mathrm{B}u^{\\alpha}_{\\lambda}(s)+f(s,\\tilde{x^\\lambda}_{\\rho(s,\\tilde{x^\\lambda}_s)})\\right]\\mathrm{d}s\\\\\\quad+\\int_{0}^{t}(t-s)^{\\alpha-1}\\widehat{\\mathcal{T}}_{\\alpha}(t-s)\\left[\\mathrm{B}u^{\\alpha}_{\\lambda}(s)+f(s,\\tilde{x^\\lambda}_{\\rho(s, \\tilde{x^\\lambda}_s)})\\right]\\mathrm{d}s,\\\\ \\qquad\\qquad\\qquad  t\\in(\\tau_k,t_{k+1}],\\ k=1,\\ldots,m,\n\t\t\\end{dcases}\n\t\t\\end{equation}\n\t\twith the control defined in \\eqref{C}. Next, we estimate\n\t\t\\begin{align}\\label{4.35}\n\t\tx^{\\lambda}(T)&=\\mathcal{T}_{\\alpha}(T-\\tau_m)h_{m}(\\tau_m,\\tilde{x^\\lambda}(t_m^-))\\nonumber\\\\&\\quad-\\int_{0}^{\\tau_m}(\\tau_m-s)^{\\alpha-1}\\widehat{\\mathcal{T}}_{\\alpha}(\\tau_m-s)\\left[\\mathrm{B}u^{\\alpha}_{\\lambda}(s)+f(s,\\tilde{x^\\lambda}_{\\rho(s,\\tilde{x^\\lambda}_s)})\\right]\\mathrm{d}s\\nonumber\\\\&\\quad+\\int_{0}^{T}(T-s)^{\\alpha-1}\\widehat{\\mathcal{T}}_{\\alpha}(T-s)\\left[\\mathrm{B}u^{\\alpha}_{\\lambda}(s)+f(s,\\tilde{x^\\lambda}_{\\rho(s,\\tilde{x^\\lambda}_s)})\\right]\\mathrm{d}s\\nonumber\\\\&=\\mathcal{T}_{\\alpha}(T-\\tau_m)h_{m}(\\tau_m,\\tilde{x^\\lambda}(t_m^-))\\nonumber\\\\&\\quad-\\int_{0}^{\\tau_m}(\\tau_m-s)^{\\alpha-1}\\widehat{\\mathcal{T}}_{\\alpha}(\\tau_m-s)\\left[\\mathrm{B}u^{\\alpha}_{\\lambda}(s)+f(s,\\tilde{x^\\lambda}_{\\rho(s,\\tilde{x^\\lambda}_s)})\\right]\\mathrm{d}s\\nonumber\\\\&\\quad+\\int_{0}^{T}(T-s)^{\\alpha-1}\\widehat{\\mathcal{T}}_{\\alpha}(T-s)f(s,\\tilde{x^\\lambda}_{\\rho(s,\\tilde{x^\\lambda}_s)})+\\int_{0}^{\\tau_m}(T-s)^{\\alpha-1}\\widehat{\\mathcal{T}}_{\\alpha}(T-s)\\mathrm{B}u^{\\alpha}_{\\lambda}(s)\\mathrm{d}s\\nonumber\\\\&\\quad+\\int_{\\tau_m}^{T}(T-s)^{2(\\alpha-1)}\\widehat{\\mathcal{T}}_{\\alpha}(T-s)\\mathrm{B}\\mathrm{B}^*\\widehat{\\mathcal{T}}_{\\alpha}(T-s)^*\\mathcal{J}\\left[\\mathcal{R}(\\lambda,\\Phi_{\\tau_m}^{T})p_m(x^\\lambda(\\cdot))\\right]\\mathrm{d}s\\nonumber\\\\&=\\mathcal{T}_{\\alpha}(T-\\tau_m)h_{m}(\\tau_m,\\tilde{x^\\lambda}(t_m^-))\\nonumber\\\\&\\quad-\\int_{0}^{\\tau_m}(\\tau_m-s)^{\\alpha-1}\\widehat{\\mathcal{T}}_{\\alpha}(\\tau_m-s)\\left[\\mathrm{B}u^{\\alpha}_{\\lambda}(s)+f(s,\\tilde{x^\\lambda}_{\\rho(s,\\tilde{x^\\lambda}_s)})\\right]\\mathrm{d}s\\nonumber\\\\&\\quad+\\int_{0}^{T}(T-s)^{\\alpha-1}\\widehat{\\mathcal{T}}_{\\alpha}(T-s)f(s,\\tilde{x^\\lambda}_{\\rho(s,\\tilde{x^\\lambda}_s)})+\\int_{0}^{\\tau_m}(T-s)^{\\alpha-1}\\widehat{\\mathcal{T}}_{\\alpha}(T-s)\\mathrm{B}u^{\\alpha}_{\\lambda}(s)\\mathrm{d}s\\nonumber\\\\&\\quad+\\Phi_{\\tau_m}^{T}\\mathcal{J}\\left[\\mathcal{R}(\\lambda,\\Phi_{\\tau_m}^{T})p_m(x^\\lambda(\\cdot))\\right]\\nonumber\\\\&=\\zeta_m-\\lambda\\mathcal{R}(\\lambda,\\Phi_{\\tau_m}^{T})p_m(x^\\lambda(\\cdot)).\n\t\t\\end{align}\n\t\tMoreover,  that fact that $ x^{\\lambda}\\in\\mathrm{E}_r $ is bounded implies that the  sequence $ x^{\\lambda}(t),$  for each $t\\in J$ is bounded in $\\mathbb{X}$. Then by the Banach-Alaoglu theorem, we can find a subsequence, still denoted as $ x^{\\lambda},$ such that \n\t\t\\begin{align*}\n\t\tx^\\lambda(t)\\xrightharpoonup{w}z(t) \\ \\mbox{ in }\\ \\mathbb{X} \\ \\ \\mbox{as}\\  \\ \\lambda\\to0^+,\\ t\\in J.\n\t\t\\end{align*}\n\t\tUsing the condition (\\textit{$H3$}) of Assumption \\ref{as2.1}, we obtain\n\t\t\\begin{align}\\label{4.19}\n\t\th_m(t,x^\\lambda(t_m^-))\\to h_m(t,z(t_m^-)) \\ \\mbox{ in }\\ \\mathbb{X} \\ \\ \\mbox{as}\\  \\ \\lambda\\to0^+.\n\t\t\\end{align}\n\t\tFurthermore, by using Assumption  \\textit{(H4)}, we get\n\t\t\\begin{align}\n\t\t\\int_{s_1}^{s_2}\\left\\|f(s,\\tilde{x^{\\lambda}}_{\\rho(s,\\tilde{x^{\\lambda}}_s)})\\right\\|_{\\mathbb{X}}^{2}\\mathrm{d}s&\\le \\int_{s_1}^{s_2}\\gamma^2(s)\\mathrm{d} s\\leq \\left(\\int_{s_1}^{s_2}\\gamma^{\\frac{1}{\\alpha_1}}(s)\\mathrm{d}s\\right)^{2\\alpha_1}(s_1-s_2)^{1-2\\alpha_1}<+\\infty, \\nonumber\n\t\t\\end{align}\n\t\tfor any $ s_1,s_2\\in[0,T]$ with $s_1<s_2$. Therefore, the sequence $ \\{f(\\cdot, \\tilde{x^{\\lambda}}_{\\rho(s,\\tilde{x^{\\lambda}}_s)}): \\lambda >0\\} $  in $ \\mathrm{L}^2([s_1,s_2]; \\mathbb{X})$ is bounded. By an application of the Banach-Alaoglu theorem, we can find a subsequence still denoted as $ \\{f(\\cdot, \\tilde{x^{\\lambda}}_{\\rho(s,\\tilde{x^{\\lambda}}_s)}): \\lambda > 0 \\}$ such that \n\t\t\\begin{align}\\label{4.36}\n\t\tf(\\cdot, \\tilde{x^{\\lambda}}_{\\rho(s,\\tilde{x^{\\lambda}}_s)})\\xrightharpoonup{w}f(\\cdot) \\ \\mbox{ in }\\ \\mathrm{L}^2([s_1,s_2];\\mathbb{X}).\n\t\t\\end{align}\n\t\tWe now calculate \n\t\t\\begin{align}\n\t\t&\\int_{0}^{\\tau_m}\\left\\|u^{\\alpha}_{\\lambda}(s)\\right\\|^2_{\\mathbb{U}}\\mathrm{d}s\\nonumber\\\\&=\\int_{0}^{t_1}\\left\\|u^{\\alpha}_{\\lambda}(s)\\right\\|^2_{\\mathbb{U}}\\mathrm{d}s+\\int_{t_1}^{\\tau_1}\\left\\|u^{\\alpha}_{\\lambda}(s)\\right\\|^2_{\\mathbb{U}}\\mathrm{d}s+\\int_{\\tau_1}^{t_2}\\left\\|u^{\\alpha}_{\\lambda}(s)\\right\\|^2_{\\mathbb{U}}\\mathrm{d}s+\\cdots+\\int_{t_m}^{\\tau_m}\\left\\|u^{\\alpha}_{\\lambda}(s)\\right\\|^2_{\\mathbb{U}}\\mathrm{d}s\\nonumber\\\\&=\\int_{0}^{t_1}\\left\\|u^{\\alpha}_{0,\\lambda}(s)\\right\\|^2_{\\mathbb{U}}\\mathrm{d}s+\\int_{\\tau_1}^{t_2}\\left\\|u^{\\alpha}_{1,\\lambda}(s)\\right\\|^2_{\\mathbb{U}}\\mathrm{d}s+\\cdots+\\int_{\\tau_{m-1}}^{t_m}\\left\\|u^{\\alpha}_{m-1,\\lambda}(s)\\right\\|^2_{\\mathbb{U}}\\mathrm{d}s\\nonumber\\\\&\\le\\left(\\frac{M\\tilde{M}\\alpha}{\\lambda\\Gamma(1+\\alpha)}\\right)^{2}\\frac{T^{2\\alpha-1}}{2\\alpha-1}\\sum_{j=0}^{m-1}C_j^2=C,\n\t\t\\end{align}\n\t\twhere $C_k,$ for $k=1,\\ldots,m-1$ are the same as given in \\eqref{4.5} and $C_0=N_0$ given in \\eqref{4.4}. Moreover, the above estimate ensures that the sequence $\\{u^\\alpha_{\\lambda}(\\cdot): \\lambda >0\\}$  in $ \\mathrm{L}^2([0,\\tau_m]; \\mathbb{U})$ is bounded. Further, by the Banach-Alaoglu theorem, we can find a subsequence, still denoted as $\\{u^\\alpha_{\\lambda}(\\cdot): \\lambda >0\\}$ such that \n\t\t\\begin{align}\\label{4}\n\t\tu^\\alpha_{\\lambda}(\\cdot)\\xrightharpoonup{w}u^\\alpha(\\cdot) \\ \\mbox{ in }\\ \\mathrm{L}^2([0,\\tau_m];\\mathbb{U}).\n\t\t\\end{align}\n\t\tNext, we compute\n\t\t\\begin{align}\\label{4.37}\n\t\t\\left\\|p_{m}(x^{\\lambda}(\\cdot))-\\omega\\right\\|_{\\mathbb{X}}&\\le\\left\\|\\mathcal{T}_{\\alpha}(T-\\tau_m)(h_k(\\tau_m,\\tilde{x^\\lambda}(t_m^-))-h_k(\\tau_m,z(t_m^-)))\\right\\|_{\\mathbb{X}}\\nonumber\\\\&\\quad+\\left\\|\\int_{0}^{\\tau_m}(\\tau_m-s)^{\\alpha-1}\\widehat{\\mathcal{T}}_{\\alpha}(\\tau_m-s)\\mathrm{B}\\left[u^\\alpha_{\\lambda}(s)-u^{\\alpha}(s)\\right]\\mathrm{d}s\\right\\|_{\\mathbb{X}}\\nonumber\\\\&\\quad+\\left\\|\\int_{0}^{\\tau_m}(\\tau_m-s)^{\\alpha-1}\\widehat{\\mathcal{T}}_{\\alpha}(\\tau_m-s)\\left[f(s,\\tilde{x^\\lambda}_{\\rho(s,\\tilde{x^\\lambda}_s)})-f(s)\\right]\\mathrm{d}s\\right\\|_{\\mathbb{X}}\\nonumber\\\\&\\quad+\\left\\|\\int_{0}^{\\tau_m}(T-s)^{\\alpha-1}\\widehat{\\mathcal{T}}_{\\alpha}(T-s)\\mathrm{B}\\left[u^\\alpha_{\\lambda}(s)-u^{\\alpha}(s)\\right]\\mathrm{d}s\\right\\|_{\\mathbb{X}}\\nonumber\\\\&\\quad+\\left\\|\\int_{0}^{T}(T-s)^{\\alpha-1}\\widehat{\\mathcal{T}}_{\\alpha}(T-s)\\left[f(s,\\tilde{x^\\lambda}_{\\rho(s,\\tilde{x^\\lambda}_s)})-f(s)\\right]\\mathrm{d}s\\right\\|_{\\mathbb{X}}\\nonumber\\\\&\\le\\left\\|\\mathcal{T}_{\\alpha}(T-\\tau_m)(h_k(\\tau_m,x^\\lambda(t_m^-))-h_k(\\tau_m,z(t_m^-)))\\right\\|_{\\mathbb{X}}\\nonumber\\\\&\\quad+\\left\\|\\int_{0}^{\\tau_m}(\\tau_m-s)^{\\alpha-1}\\widehat{\\mathcal{T}}_{\\alpha}(\\tau_m-s)\\mathrm{B}\\left[u^\\alpha_{\\lambda}(s)-u^{\\alpha}(s)\\right]\\mathrm{d}s\\right\\|_{\\mathbb{X}}\\nonumber\\\\&\\quad+\\left\\|\\int_{0}^{\\tau_m}(\\tau_m-s)^{\\alpha-1}\\widehat{\\mathcal{T}}_{\\alpha}(\\tau_m-s)\\left[f(s,\\tilde{x^\\lambda}_{\\rho(s,\\tilde{x^\\lambda}_s)})-f(s)\\right]\\mathrm{d}s\\right\\|_{\\mathbb{X}}\\nonumber\\\\&\\quad+\\frac{T^{2\\alpha-1}-(T-\\tau_m)^{2\\alpha-1}}{2\\alpha-1}\\left(\\int_{0}^{\\tau_m}\\left\\|\\widehat{\\mathcal{T}}_{\\alpha}(T-s)\\mathrm{B}\\left[u^\\alpha_{\\lambda}(s)-u^{\\alpha}(s)\\right]\\right\\|^2_{\\mathbb{X}}\\mathrm{d}s\\right)^{\\frac{1}{2}}\\nonumber\\\\&\\quad+\\left\\|\\int_{0}^{T}(T-s)^{\\alpha-1}\\widehat{\\mathcal{T}}_{\\alpha}(T-s)\\left[f(s,\\tilde{x^\\lambda}_{\\rho(s,\\tilde{x^\\lambda}_s)})-f(s)\\right]\\mathrm{d}s\\right\\|_{\\mathbb{X}}\\nonumber\\\\&\\quad\\to 0\\ \\mbox{as}\\ \\lambda\\to0^+, \n\t\t\\end{align}\n\t\twhere \n\t\t\\begin{align*}\n\t\t\\omega &=\\zeta_m-\\mathcal{T}_{\\alpha}(T-\\tau_m)h_m(\\tau_m,z(t_m^-))+\\int_{0}^{\\tau_m}(\\tau_m-s)^{\\alpha-1}\\widehat{\\mathcal{T}}_{\\alpha}(\\tau_m-s)\\left[\\mathrm{B}u^{\\alpha}(s)+f(s)\\right]\\mathrm{d}s\\nonumber\\\\&\\quad-\\int_{0}^{\\tau_m}(T-s)^{\\alpha-1}\\widehat{\\mathcal{T}}_{\\alpha}(T-s)\\mathrm{B}u^{\\alpha}(s)\\mathrm{d}s-\\int_{0}^{T}(T-s)^{\\alpha-1}\\widehat{\\mathcal{T}}_{\\alpha}(T-s)f(s)\\mathrm{d}s.\n\t\t\\end{align*}\n\t\tHere, we used the convergences \\eqref{4.19},\\eqref{4.36},\\eqref{4}, the dominated convergence theorem and the compactness of the operator $f(\\cdot)\\to\\int_{0}^{\\cdot}(\\cdot-s)^{\\alpha-1}\\widehat{\\mathcal{T}}_{\\alpha}(\\cdot-s) f(s)\\mathrm{d}s:\\mathrm{L}^2(J;\\mathbb{X})\\rightarrow \\mathrm{C}(J;\\mathbb{X})$, (see Lemma \\ref{lem2.12}).\n\t\t\tFinally, by using the equality \\eqref{4.35}, we evaluate \n\t\t\\begin{align}\n\t\t\\left\\|x^{\\lambda}(T)-\\zeta_m\\right\\|_{\\mathbb{X}}&\\le\\left\\|\\lambda\\mathcal{R}(\\lambda,\\Phi_{\\tau_m}^{T})p_m(x(\\cdot))\\right\\|_{\\mathbb{X}}\\nonumber\\\\&\\le\\left\\|\\lambda\\mathcal{R}(\\lambda,\\Phi_{\\tau_m}^{T})(p_m(x(\\cdot))-\\omega)\\right\\|_{\\mathbb{X}}+\\left\\|\\lambda\\mathcal{R}(\\lambda,\\Phi_{\\tau_m}^{T})\\omega\\right\\|_{\\mathbb{X}}\\nonumber\\\\&\\le\\left\\|\\lambda\\mathcal{R}(\\lambda,\\Phi_{\\tau_m}^{T})\\right\\|_{\\mathcal{L}(\\mathbb{X})}\\left\\|p_m(x(\\cdot))-\\omega\\right\\|_{\\mathbb{X}}+\\left\\|\\lambda\\mathcal{R}(\\lambda,\\Phi_{\\tau_m}^{T})\\omega\\right\\|_{\\mathbb{X}}.\n\t\t\\end{align}\n\t\tUsing the above inequality, \\eqref{4.37} and Assumption \\ref{as2.1} (\\textit{H0}), we obtain\n\t\t\\begin{align*}\n\t\t\\left\\|x^{\\lambda}(T)-\\zeta_m\\right\\|_{\\mathbb{X}}\\to0,\\ \\mbox{ as }\\ \\lambda\\to0^+,\n\t\t\\end{align*}\n\t\twhich ensures that the system \\eqref{1.1} is approximately controllable on $J$.\n\t\\end{proof}\n\t\\begin{rem}\\label{rem4.4}\n\t\tThe works \\cite{PCY,NIZ,SRY}, etc considered a different kind of control for the fractional order semilinear problems. If one follows Remark \\ref{rem3.6}, the controllability operator defined in \\eqref{2.1} changes to \\eqref{3.20}\n\t\tand  $u^\\alpha_{k,\\lambda}(\\cdot)$ appearing in the control defined in \\eqref{C} takes the form \n\t\t\\begin{align*}\n\t\tu^\\alpha_{k,\\lambda}(t)&=\\mathrm{B}^*\\widehat{\\mathcal{T}}_{\\alpha}(t_{k+1}-t)^*\\mathcal{J}\\left[\\mathcal{R}(\\lambda,\\Phi_{\\tau_k}^{t_{k+1}})p_k(x(\\cdot))\\right],\n\t\t\\end{align*}\n\t\tfor $t\\in [\\tau_k, t_{k+1}),k=0,1,\\ldots,m$. The control provided in \\eqref{C} is motivated from the linear regulator problem with the cost functional defined in \\eqref{3.1}.  The proof of Theorems \\ref{thm4.3} and \\ref{thm4.4} follows in a similar way with some obvious modifications in the calculations. \n\t\\end{rem}\n\t\\section{Application}\\label{application}\\setcounter{equation}{0}\n\tIn this section, we discuss a concrete example to verify the results developed in previous sections. \n\t\n\t\\begin{Ex}\\label{ex1} Let us take the following fractional heat equation with non-instantaneous impulses and delay:\n\t\t\\begin{equation}\\label{ex}\n\t\t\\left\\{\n\t\t\\begin{aligned}\n\t\t\\frac{\\partial^\\alpha z(t,\\xi)}{\\partial t^\\alpha}&=\\frac{\\partial^2z(t,\\xi)}{\\partial \\xi^2}+\\eta(t,\\xi)+\\int_{-\\infty}^{t}b(s-t)z(s-\\sigma(\\|z(t)\\|),\\xi)\\mathrm{d}s, \\\\&\\qquad \\qquad \\ t\\in\\bigcup_{k=0}^{m} (\\tau_k, t_{k+1}]\\subset J=[0,T], \\ \\xi\\in[0,\\pi], \\\\\n\t\tz(t,\\xi)&=h_k(t,z(t_k^-,\\xi)),\\ t\\in(t_k,\\tau_k],\\ k=1,\\ldots, m,\\ \\xi\\in[0,\\pi],\\\\\n\t\tz(t,0)&=0=z(t,\\pi), \\  t\\in [0, 1], \\\\\n\t\tz(\\theta,\\xi)&=\\psi(\\theta,\\xi), \\ \\xi\\in[0,\\pi], \\ \\theta\\leq0.\n\t\t\\end{aligned}\n\t\t\\right.\n\t\t\\end{equation}\n\t\twhere the function $\\eta:[0,1]\\times[0,\\pi]\\to[0,\\pi]$  is continuous in $t$ and the functions $\\sigma:[0,\\infty)\\to[0,\\infty)$ are also continuous.\n\t\\end{Ex}\n\t\\vskip 0.1 cm\n\t\\noindent\\textbf{Step 1:} \\emph{$\\mathrm{C}_0$-semigroup and phase space:} \n\tLet $\\mathbb{X}_p= \\mathrm{L}^{p}([0,\\pi];\\mathbb{R})$ with $p\\in[2,\\infty)$, and $\\mathbb{U}=\\mathrm{L}^{2}([0,\\pi];\\mathbb{R})$. Note that  $\\mathbb{X}_p$ is separable and reflexive with strictly convex dual $\\mathbb{X}_p^*=\\mathrm{L}^{\\frac{p}{p-1}}([0,\\pi];\\mathbb{R})$ and $\\mathbb{U}$ is separable. We define the linear operator  $\\mathrm{A}_p:\\mathrm{D}(\\mathrm{A}_p)\\subset\\mathbb{X}_p\\to\\mathbb{X}_p$  as\n\t\\begin{align*}\n\t\\mathrm{A}_pg(\\xi)= g''(\\xi),\n\t\\end{align*}\n\twhere $\\mathrm{D}(\\mathrm{A}_p)= \\mathrm{W}^{2,p}([0,\\pi];\\mathbb{R})\\cap\\mathrm{W}_0^{1,p}([0,\\pi];\\mathbb{R})$. Since we know that  $\\mathrm{C}_0^{\\infty}([0,\\pi];\\mathbb{R})\\subset\\mathrm{D}(\\mathrm{A}_p)$ and hence $\\mathrm{D}(\\mathrm{A}_p)$ is dense in $\\mathbb{X}_p$ and one can easily verify that the operator $\\mathrm{A}_p$ is closed. Next, we consider the following Sturm-Liouville system:\n\t\\begin{equation}\\label{59}\n\t\\left\\{\n\t\\begin{aligned}\n\t\\left(\\lambda\\mathrm{I}-\\mathrm{A}_p\\right)g(\\xi)&=l(\\xi), \\ 0<\\xi<\\pi,\\\\\n\tg(0)=g(\\pi)&=0.\n\t\\end{aligned}\n\t\\right.\n\t\\end{equation}\n\tOne can easily  rewrite the above system as \n\t\\begin{align}\\label{511}\n\t\\left(\\lambda\\mathrm{I}-\\Delta\\right)g(\\xi)&=l(\\xi),\n\t\\end{align}\n\twhere $\\Delta g(\\xi)=g''(\\xi)$. Multiplying both sides of \\eqref{511} by $g|g|^{p-2}$ and then integrating over $[0,\\pi]$, we obtain\n\t\\begin{align}\\label{536}\n\t\\lambda\\int_0^{\\pi}|g(\\xi)|^p\\mathrm{d}\\xi&+(p-1)\\int_0^{\\pi}|g(\\xi)|^{p-2}|f'(\\xi)|^2\\mathrm{d}\\xi=\\int_0^{\\pi}l(\\xi)g(\\xi)|g(\\xi)|^{p-2}\\mathrm{d}\\xi.\n\t\\end{align}\n\n\n\n\n\n\n\n\n\n\n\n\n\tApplying H\\\"older's inequality, we get\n\t\\begin{align*}\n\t\\lambda\\int_0^{\\pi}|g(\\xi)|^p\\mathrm{d}\\xi&\\leq \\left(\\int_0^{\\pi}|g(\\xi)|^p\\mathrm{d}\\xi\\right)^{\\frac{p-1}{p}}\\left(\\int_0^{\\pi}|l(\\xi)|^p\\mathrm{d}\\xi\\right)^{\\frac{1}{p}}.\n\t\\end{align*}\n\tThus, we have \n\t\\begin{align*}\n\t\\|\\mathcal{R}(\\lambda,\\mathrm{A}_p)l\\|_{\\mathrm{L}^p}=\\|g\\|_{\\mathrm{L}^p}\\leq\\frac{1}{\\lambda}\\|l\\|_{\\mathrm{L}^p},\n\t\\end{align*}\n\tso that we obtain  \n\t\\begin{align}\n\t\\|\\mathcal{R}(\\lambda,\\mathrm{A}_p\\|_{\\mathcal{L}(\\mathrm{L}^p)}\\leq\\frac{1}{\\lambda}.\n\t\\end{align}\n\tHence, by applying the Hille-Yosida theorem, we obtain that the operator $\\mathrm{A}_p$ generate a strongly continuous semigroup $\\{\\mathcal{T}_p(t):t\\ge0\\}$ of bounded linear operators. \n\t\n\tMoreover, the infinitesimal generator $\\mathrm{A}_p$ and the semigroup $\\mathcal{T}_p(t)$ can be written as\n\t\\begin{align}\n\t\\mathrm{A}_pg&= \\sum_{n=1}^{\\infty}-n^{2}\\langle g, w_{n} \\rangle  w_{n},\\ g\\in \\mathrm{D}(\\mathrm{A}_p),\\nonumber\\\\\n\t\\mathcal{T}_p(t)g&= \\sum_{n=1}^{\\infty}\\exp\\left(-n^2t\\right)\\langle g, w_{n} \\rangle  w_{n},\\ g\\in\\mathbb{X}_p,\n\t\\end{align}\n\twhere, $w_n(\\xi)=\\sqrt{\\frac{2}{\\pi}}\\sin(n\\xi)$ are the normalized eigenfunctions corresponding to the eigenvalues $\\lambda_n=-n^2\\ (n\\in\\mathbb{N})$ of the operator $\\mathrm{A}_p$ and $\\langle g,w_n\\rangle :=\\int_0^{\\pi}g(\\xi)w_n(\\xi)\\mathrm{d}\\xi$. Further, the resolvent operator $\\mathcal{R}(\\lambda,\\mathrm{A}_p)$ is compact (see \\cite{MTM} for more details). Therefore, the generated semigroup $\\mathcal{T}_p(t)$ is compact for $t>0$. Thus, the condition (\\textit{H1}) of Assumption \\ref{as2.1} holds.\n\t\n\tWe now define the following operators\n\t\\begin{align}\n\t\\mathcal{T}_{\\alpha,p}(t)g&=\\int_{0}^{\\infty}\\varphi_{\\alpha}(\\xi)\\mathcal{T}_p(t^{\\alpha}\\xi)g(\\xi)\\mathrm{d}\\xi=\\int_{0}^{\\infty}\\varphi_{\\alpha}(\\xi)\\sum_{n=1}^{\\infty}\\exp\\left(-n^2t^\\alpha\\xi\\right)\\langle g, w_{n} \\rangle  w_{n}(\\xi)\\mathrm{d}\\xi.\\\\\n\t\\widehat{\\mathcal{T}}_{\\alpha,p}(t)g&=\\alpha\\int_{0}^{\\infty}\\xi\\varphi_{\\alpha}(\\xi)\\mathcal{T}_p(t^{\\alpha}\\xi)g(\\xi)\\mathrm{d}\\xi=\\alpha\\int_{0}^{\\infty}\\xi\\varphi_{\\alpha}(\\xi)\\sum_{n=1}^{\\infty}\\exp\\left(-n^2t^\\alpha\\xi\\right)\\langle g, w_{n} \\rangle  w_{n}(\\xi)\\mathrm{d}\\xi, \\label{5.7}\n\t\\end{align}\n\tfor all $g\\in\\mathbb{X}_p$.  \n\t\n\tLet us take $\\mathfrak{B}=\\mathrm{PC}_{0}\\times\\mathrm{L}^1_h(\\mathbb{X})$ with  $h(\\theta)=e^{\\nu\\theta}$, for some $\\nu>0$ (see Example \\ref{exm2.8}). Proceeding similar arguments as in section 5,  \\cite{SSM}, one can verify that the space $\\mathfrak{B}=\\mathrm{PC}_{0}\\times\\mathrm{L}^1_h(\\mathbb{X})$  is a phase space, which satisfies the axioms (A1) and (A2) with $\\Lambda(t) =\\int_{-t}^{-0}h(\\theta)\\mathrm{d}\\theta$ and $\\Upsilon(t)= H(-t)$.\n\tWe define  $K:=\\sup\\limits_{\\theta\\in(-\\infty,0]}\\frac{|b(-\\theta)|}{h(\\theta)}$.\n\t\\vskip 0.1 cm \n\t\\noindent\\textbf{Step 2:} \\emph{Abstract formulation and approximate controllability.}\n\tLet us define $$x(t)(\\xi):=z(t,\\xi),\\ \\mbox{ for }\\ t\\in J\\ \\mbox{ and }\\ \\xi\\in[0,\\pi],$$ and the bounded linear operator $\\mathrm{B}:\\mathbb{U}\\to\\mathbb{X}_p$ as  $$\\mathrm{B}u(t)(\\xi):=\\eta(t,\\xi)=\\int_{0}^{\\pi}K(\\zeta,\\xi)u(t)(\\zeta)\\mathrm{d}\\zeta, \\ t\\in J,\\ \\xi\\in [0,\\pi],$$ where $K\\in\\mathrm{C}([0,\\pi]\\times[0,\\pi];\\mathbb{R})$ with $K(\\zeta,\\xi)=K(\\xi,\\zeta),$ for all $\\zeta,\\xi\\in [0,\\pi]$. We assume that the operator $\\mathrm{B}$ is one-one.\n\tLet us estimate \n\t\\begin{align*}\n\t\\left\\|\\mathrm{B}u(t)\\right\\|_{\\mathbb{X}_p}^p=\\int_{0}^{\\pi}\\left|\\int_{0}^{\\pi}K(\\zeta,\\xi)u(t)(\\zeta)\\mathrm{d}\\zeta\\right|^p\\mathrm{d}\\xi.\n\t\\end{align*}\n\tApplying the Cauchy-Schwarz inequality, we have\n\t\\begin{align*}\n\t\\left\\|\\mathrm{B}u(t)\\right\\|_{\\mathbb{X}_p}^p&\\le\\int_{0}^{\\pi}\\left[\\left(\\int_{0}^{\\pi}|K(\\zeta,\\xi)|^2\\mathrm{d}\\zeta\\right)^{\\frac{1}{2}}\\left(\\int_{0}^{\\pi}|u(t)(\\zeta)|^2\\mathrm{d}\\zeta\\right)^{\\frac{1}{2}}\\right]^{p}\\mathrm{d}\\xi\\\\&=\\left(\\int_{0}^{\\pi}|u(t)(\\zeta)|^2\\mathrm{d}\\zeta\\right)^{\\frac{p}{2}}\\int_{0}^{\\pi}\\left(\\int_{0}^{\\pi}|K(\\zeta,\\xi)|^2\\mathrm{d}\\zeta\\right)^{\\frac{p}{2}}\\mathrm{d}\\xi.\n\t\\end{align*}\n\tSince the kernel $K(\\cdot,\\cdot)$ is continuous, we arrive at\n\t\\begin{align*}\n\t\\left\\|\\mathrm{B}u(t)\\right\\|_{\\mathbb{X}_p}\\le C\\left\\|u(t)\\right\\|_{\\mathbb{U}},\n\t\\end{align*}\n\tso that we get \n\t$\t\\left\\|\\mathrm{B}\\right\\|_{\\mathcal{L}(\\mathbb{U};\\mathbb{X}_p)}\\le C.$\n\tHence, the operator $\\mathrm{B}$ is bounded. Moreover, the symmetry of the kernel  implies that the operator $\\mathrm{B}=\\mathrm{B}^*$ (self adjoint). For example, one can take $K(\\xi,\\zeta)=1+\\xi^2+\\zeta^2,\\ \\mbox{for all}\\ \\xi, \\zeta\\in [0,\\pi]$.\n\tThe function $\\psi:(-\\infty,0]\\rightarrow\\mathbb{X}$ is given as\n\t\\begin{align}\n\t\\nonumber \\psi(t)(\\xi)=\\psi(t,\\xi),\\ \\xi\\in[0,\\pi].\n\t\\end{align}\t \n\tNext, the functions $f, \\rho:J\\times \\mathfrak{B}\\to\\mathbb{X}$ are defined as\n\t\\begin{align}\n\t\\nonumber f(t,\\psi)\\xi&:=\\int_{-\\infty}^{0}b(-\\theta)\\psi(\\theta,\\xi)\\mathrm{d}\\theta,\\\\\n\t\\nonumber\\rho(t,\\psi):&=t-\\sigma(\\|\\psi(0)\\|_{\\mathbb{X}}),\n\t\\end{align}\t\n\tfor $\\xi\\in[0,\\pi]$. Clearly, $f$ is continuous and uniformly bounded by $K$. These facts guarantee that the function $f$ satisfied the condition \\textit{$(H2)$} of Assumption \\ref{as2.1} and the condition \\textit{$(H4)$}.\n\t\n\tMoreover, the impulse functions $h_k:[t_k,\\tau_k]\\times\\mathbb{X}\\to\\mathbb{X},$ for $k=1,\\ldots,m,$ are defined as \n\t\\begin{align*}\n\th_k(t,x)\\xi:=\\int_{0}^{\\pi}\\rho_k(t,\\xi,z)\\cos^2(x(t_k^-)z)\\mathrm{d}z, \\ \\mbox{ for }\\ t\\in(t_k,\\tau_k],\n\t\\end{align*}\n\twhere, $\\rho_k\\in\\mathrm{C}([0,1]\\times[0,\\pi]^2;\\mathbb{R})$. It is easy to verify that the impulses $h_k,$ for $k=1,\\ldots,m,$ satisfy the condition \\textit{$(H3)$} of Assumption \\ref{as2.1}.\n\t\n\t\n\tThe system \\eqref{ex} can be transformed into the abstract form \\eqref{1.1} by using the above substitutions and it satisfies  Assumption \\ref{as2.1} \\textit{$(H1)$-$(H3)$} and Assumption (\\textit{$H4$}). Moreover, it remains to verify that the associated linear system of the equation \\eqref{1.1} is approximately controllable. In order to prove this, we consider\n\t$$(T-t)^{\\alpha-1}\\mathrm{B}^*\\widehat{\\mathcal{T}}_{\\alpha,p}(T-t)^*x^*=0,\\ \\mbox{ for any}\\ x^*\\in\\mathbb{X}^*,\\ 0\\le t<T.$$ Therefore, we have $$\\mathrm{B}^*\\widehat{\\mathcal{T}}_{\\alpha,p}(T-t)^*x^*=0,\\ \\mbox{ for any}\\ x^*\\in\\mathbb{X}^*,\\ 0\\le t<T,$$ and since the operator $\\mathrm{B}^*=\\mathrm{B}$ is one-one, then we obtain\n\t$$\\widehat{\\mathcal{T}}_{\\alpha,p}(T-t)^*x^*=0,\\ \\mbox{ for all } \\ t\\in [0,T).$$\n\tFurther, we have\n\t\\begin{align*}\n\t\\widehat{\\mathcal{T}}_{\\alpha,p}(T-t)^*x^*&=\\int_{0}^{\\infty}\\xi\\varphi_{\\alpha}(\\xi)\\sum_{n=1}^{\\infty}\\exp\\left(-n^2(T-t)^\\alpha\\xi\\right)\\langle x^*, w_{n} \\rangle  w_{n}\\mathrm{d}\\xi=0\\nonumber\\\\&\\implies\\langle x^*, w_{n} \\rangle =0, \\ \\mbox{ for all }\\  n\\in\\mathbb{N},\n\t\\end{align*}\nwhich implies $x^*=0$.\tHence, the approximately controllability of the linear system follows by using Lemma \\ref{lem4.2} and Remark \\ref{rem3.4}.  Finally, by invoking Theorem \\ref{thm4.4}, we deduce that the semilinear system \\eqref{1.1} (equivalent to the system \\eqref{ex}) is approximately controllable.\n\t\n\n\n\n\n\n\t\n\t\n\t\\medskip\\noindent\n\t{\\bf Acknowledgments:} The first author would like to thank Council of Scientific and Industrial Research, New Delhi, Government of India (File No. 09\/143(0931)\/2013 EMR-I), for financial support to carry out his research work and Department of Mathematics, Indian Institute of Technology Roorkee (IIT Roorkee), for providing stimulating scientific environment and resources. M. T. Mohan would  like to thank the Department of Science and Technology (DST), Govt of India for Innovation in Science Pursuit for Inspired Research (INSPIRE) Faculty Award (IFA17-MA110). \n\t","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\n\\section{Introduction}\n\\label{sec:int}\nTo date it is possible to perform most collider-physics studies with  fully-differential NLO+PS matching thanks to two general, well-developed, and widely applied procedures: M\\protect\\scalebox{0.8}{C}@N\\protect\\scalebox{0.8}{LO}\\xspace \\cite{Frixione:2002ik} and P\\protect\\scalebox{0.8}{OWHEG}\\xspace \\cite{Nason:2004rx}. \nBy fully differential matching, we understand that the matching is done point by point in both the Born- and real-emission phase spaces, with a parton shower that reflects the correct singular structure of the fixed-order calculation. In this sense, fully-differential matching requires the fixed-order expansion of the shower to develop the same singularities as the fixed-order calculation up to the matched order. At NLO, this is achieved by parton showers that exponentiate terms reducing to the universal DGLAP kernels in any collinear limit and the eikonal factor in soft limits, with the colour dependence in the soft limit requiring special attention~\\cite{Hoeche:2011fd}. As of today, no fully-differential matching method obeying these criteria is available at NNLO, although significant progress on including higher-order corrections to parton showers has been made \\cite{Li:2016yez,Hoche:2017iem,Hoche:2017hno,Dulat:2018vuy}.\n\nExisting NNLO+PS matching methods either extend existing merging schemes or utilise analytical resummation for the transition between the fixed-order and parton-shower realms.\nExamples of the first kind are UN$^2$LOPS\\xspace \\cite{Hoche:2014uhw}, which extends the UNLOPS\\xspace \\cite{Lonnblad:2012ix} scheme to the second order, and MiNNLO$_{\\mathrm{PS}}$\\xspace \\cite{Monni:2019whf,Re:2021vcb} as well as other extensions~\\cite{Hamilton:2013fea,Karlberg:2014qua,Astill:2016hpa,Astill:2018ivh,Hamilton:2012rf,Frederix:2015fyz,Hu:2021rkt} of the MiNLO\\xspace technique \\cite{Hamilton:2012np}.\nThe UN$^2$LOPS\\xspace scheme has recently been generalised to processes with an additional jet in the context of an UNLOPS\\xspace-based N$^3$LO+PS matching strategy \\cite{Prestel:2021vww}. The MiNLO\\xspace-based schemes may be seen as a hybrid approach, since they use a combination of analytical and numerical resummation.\nA noteworthy example of a scheme employing the latter approach is implemented in the G\\protect\\scalebox{0.8}{ENEVA}\\xspace framework \\cite{Alioli:2013hqa,Alioli:2021qbf}.\nWhile all of these have enabled impressive phenomenological studies \\cite{Hoche:2014uhw,Hoche:2014dla,Hoche:2018gti,Alioli:2015toa,Alioli:2016wqt,Cridge:2021hfr,Monni:2020nks,Lombardi:2020wju,Lombardi:2021rvg,Mazzitelli:2020jio,Buonocore:2021fnj} and provide pathways to matching precision calculations to event generators, they do not provide the same level of theoretical control as the fully-differential matching methods that are available at NLO.\n\nIn this letter, we present for the first time a fully-differential NNLO+PS matching scheme for final-state parton showers, restricting ourselves to the case of two coloured final-state particles. The new method combines NNLO antenna subtraction with the sector-antenna shower in V\\protect\\scalebox{0.8}{INCIA}\\xspace~\\cite{Brooks:2020upa}, suitably extended to include real-virtual and double-real corrections.\nA key aspect of the new technique is that the parton shower is employed only as an efficient Sudakov-weighted phase-space generator. It does not define the infrared subtraction terms that are key to M\\protect\\scalebox{0.8}{C}@N\\protect\\scalebox{0.8}{LO}\\xspace type matching strategies.\n\nThe letter is structured as follows. We review the matching method at NLO in \\cref{sec:strategyNLO} before extending it to the NNLO in \\cref{sec:strategyNNLO}, retaining a rather general notation. A numerical implementation in the V\\protect\\scalebox{0.8}{INCIA}\\xspace sector-antenna shower in P\\protect\\scalebox{0.8}{YTHIA}\\xspace 8.3~\\cite{Sjostrand:2014zea} is described in \\cref{sec:implementation}, featuring a more detailed description of the matching scheme. We conclude in \\cref{sec:conclusions} and provide an outlook on applications beyond the simple cases considered here.\n\n\\section{NLO Matching Strategy}\\label{sec:strategyNLO}\nOur matching strategy generalises the technique first developed in \\cite{Norrbin:2000uu}, which nowadays is referred to as the P\\protect\\scalebox{0.8}{OWHEG}\\xspace scheme \\cite{Nason:2004rx,Frixione:2007vw,Alioli:2010xd}.\nTo start with, it is thus useful to recap the NLO matching strategy, before moving on to the new NNLO technique.\n\nAt NLO, the expected value of an infrared-safe observable $O$ defined on a two-particle final state process with a colourless initial state is given by\n\\begin{align}\n   &\\avg{O}_\\mathrm{NLO} = \\int \\, \\ensuremath{\\,\\text{d}} \\Phi_2 \\left[ {\\mathrm{B}}(\\Phi_2) + {\\mathrm{V}}(\\Phi_2) + {\\mathrm{I}}^\\mathrm{NLO}_{{\\mathrm{S}}}(\\Phi_2) \\right] O(\\Phi_2) \\nonumber \\\\\n   &+\\int \\, \\ensuremath{\\,\\text{d}} \\Phi_3 \\left[ {\\mathrm{R}}(\\Phi_3) O(\\Phi_3) - {\\mathrm{S}}^\\mathrm{NLO}(\\Phi_3) O(\\Phi_2(\\Phi_3)) \\right] \\, ,\n\\label{eq:expvalNLO}\n\\end{align}\nwhere ${\\mathrm{B}}$ and ${\\mathrm{V}}$ denote the Born cross section and virtual correction, differential in the two-particle phase space $\\Phi_2$. Similarly, ${\\mathrm{R}}$ denotes the real-radiation cross section differential in the three-particle phase space $\\Phi_3$, and ${\\mathrm{S}}^\\mathrm{NLO}$ denotes the differential NLO subtraction term in the antenna subtraction method, with its integral over the antenna phase space given by ${\\mathrm{I}}^\\mathrm{NLO}_{\\mathrm{S}}$.\\footnote{Other well-established NLO subtraction schemes such as FKS~\\cite{Frixione:1995ms} or dipole subtraction~\\cite{Catani:1996vz} may equally well be employed here.}\nIn order to achieve a Born-local cancellation of the subtraction term upon integration over the real-emission phase space,\nthe observable acting on ${\\mathrm{S}}^\\mathrm{NLO}$ must be evaluated at the reduced phase-space point $\\Phi_2(\\Phi_3)$, where the precise mapping from the\nthree-parton to the two-parton state depends on the subtraction scheme.\nWe can invert this mapping and factorise the phase space into the 2-particle (Born) phase space $\\Phi_2$, and the one-particle radiation phase space $\\Phi_{+1}$,\n\\begin{equation}\n    \\ensuremath{\\,\\text{d}}\\Phi_{3} = \\ensuremath{\\,\\text{d}}\\Phi_{2}\\times\\ensuremath{\\,\\text{d}}\\Phi_{+1} \\, .\n\\end{equation}\nBy defining a Born-local NLO weight,\n\\begin{multline}\\label{eq:local_kfac_nlo}\n    k_\\mathrm{NLO}(\\Phi_2) :=  1 + \\frac{{\\mathrm{V}}(\\Phi_2)}{{\\mathrm{B}}(\\Phi_2)} + \\frac{{\\mathrm{I}}^\\mathrm{NLO}_{{\\mathrm{S}}}(\\Phi_2)}{{\\mathrm{B}}(\\Phi_2)} \\\\\n    + \\int \\ensuremath{\\,\\text{d}} \\Phi_{+1}\\, \\left[\\frac{{\\mathrm{R}}(\\Phi_2,\\Phi_{+1})}{{\\mathrm{B}}(\\Phi_2)} - \\frac{{\\mathrm{S}}^\\mathrm{NLO}(\\Phi_2,\\Phi_{+1})}{{\\mathrm{B}}(\\Phi_2)}\\right]\n\\end{multline}\n\\cref{eq:expvalNLO} can be rewritten as\n\\begin{multline}\\label{eq:born_local_nlo}\n    \\avg{O}_\\mathrm{NLO} = \\int \\ensuremath{\\,\\text{d}} \\Phi_2 \\, {\\mathrm{B}}(\\Phi_2)\\, \\Bigg[ k_\\mathrm{NLO}(\\Phi_2) O(\\Phi_2)\\\\\n    + \\int \\ensuremath{\\,\\text{d}} \\Phi_{+1}\\,\\frac{{\\mathrm{R}}(\\Phi_2,\\Phi_{+1})}{{\\mathrm{B}}(\\Phi_2)} \\left( O(\\Phi_2,\\Phi_{+1}) - O(\\Phi_2) \\right) \\Bigg] \\, .\n\\end{multline}\n\nThe parton-shower evolution, on the other hand, is described by a generating functional, the shower operator, recursively defined for an infrared-safe observable $O$ by\n\\begin{align}\n    {\\mathcal{S}}_n(t,O) &= \\Delta_n(t,\\ensuremath{t_\\mathrm{c}}) O(\\Phi_n) \\label{eq:showerOperator}\\\\\n    & + \\int^t_{\\ensuremath{t_\\mathrm{c}}}\\ensuremath{\\,\\text{d}}\\Phi_{+1}\\, {\\mathrm{A}}^{(0)}_n(\\Phi_{+1}) \\Delta_n(t,t') {\\mathcal{S}}_{n+1}(t',O) \\, , \\nonumber\n\\end{align}\nwhere ${\\mathrm{A}}^{(0)}_n(\\Phi_{+1})$ is the sum of all leading-order antenna functions\\footnote{We refer to NLO antenna subtraction terms as LO antenna functions.} competing for the next branching $IK \\mapsto ijk$ off the $n$-parton configuration,\n\\begin{align}\\label{eq:ps_partitioning}\n    &\\int^t_{\\ensuremath{t_\\mathrm{c}}}\\ensuremath{\\,\\text{d}}\\Phi_{+1}\\, {\\mathrm{A}}^{(0)}_{n\\mapsto n+1}(\\Phi_{+1}) \\\\\n    &\\qquad := \\int^t_{\\ensuremath{t_\\mathrm{c}}} \\sum\\limits_{j\\in \\{n\\mapsto n+1\\}}A_{j\/IK}^{(0)}\\left(\\Phi_{+1}^j\\right) \\, \\ensuremath{\\,\\text{d}}\\Phi_{+1}^j \\, , \\nonumber \\\\\n    &\\qquad = \\sum\\limits_{j\\in \\{n\\mapsto n+1\\}} \\int^t_{\\ensuremath{t_\\mathrm{c}}}\\, \\frac{\\ensuremath{\\alpha_{\\mathrm{s}}}(t)}{4\\uppi} {\\mathcal{C}}_{j\/IK} \\bar{A}^{(0)}_{j\/IK}(t,\\zeta,\\phi) \\ensuremath{\\,\\text{d}} t \\ensuremath{\\,\\text{d}} \\zeta \\frac{\\ensuremath{\\,\\text{d}} \\phi}{2\\uppi}\\nonumber\n\\end{align}\nwith the sum and shower variables left implicit in our notation. \nNote that when working in the sector antenna framework~\\cite{Kosower:1997zr,Kosower:2003bh,\n Larkoski:2009ah,Lopez-Villarejo:2011pwr,Brooks:2020upa}, \nEq.~\\eqref{eq:ps_partitioning} implicitly defines a partitioning\nof the real-emission term along with the associated subtractions in Eq.~\\eqref{eq:local_kfac_nlo}.\nThis is crucial to avoid double-counting of radiative corrections generated by the parton shower.\nThe associated Sudakov factor is given by\n\\begin{equation}\n    \\Delta_n^\\mathrm{LO}(\\ensuremath{t_0},t) = \\exp{-\\int^{\\ensuremath{t_0}}_{t}\\ensuremath{\\,\\text{d}}\\Phi_{+1}\\, {\\mathrm{A}}_{n\\mapsto n+1}^{(0)}(\\Phi_{+1})} \\, .\n\\end{equation}\n\nTaking only the first shower emission into account, the expected value of the observable $O$ at LO is given by\n\\begin{multline}\n    \\avg{O}_{\\mathrm{LO}+\\mathrm{PS}} = \\int \\ensuremath{\\,\\text{d}} \\Phi_2 \\, {\\mathrm{B}}(\\Phi_2)\\, \\Big[\\Delta(\\ensuremath{t_0},\\ensuremath{t_\\mathrm{c}}) O(\\Phi_2) \\nonumber \\\\\n    + \\int \\ensuremath{\\,\\text{d}} \\Phi_{+1}\\, {\\mathrm{A}}^{(0)}_{2\\mapsto3}(\\Phi_{+1}) \\Delta(\\ensuremath{t_0},t) O(\\Phi_2,\\Phi_{+1}) \\Big] \\, .\n\\end{multline}\nThis implies that upon the replacement\n\\begin{align}\n    {\\mathrm{B}}(\\Phi_2) &\\to k_\\mathrm{NLO}(\\Phi_2) {\\mathrm{B}}(\\Phi_2) \\nonumber\\\\\n    {\\mathrm{A}}^{(0)}_{2\\mapsto 3} &\\to w^\\mathrm{LO}_{2\\mapsto3}(\\Phi_2,\\Phi_{+1}) {\\mathrm{A}}^{(0)}_{2\\mapsto 3}\n\\end{align}\nwhere we have defined the $2\\mapsto 3$ LO matrix-element correction factor,\n\\begin{equation}\n    w^\\mathrm{LO}_{2\\mapsto3}(\\Phi_2,\\Phi_{+1}) = \\frac{{\\mathrm{R}}(\\Phi_2,\\Phi_{+1})}{{\\mathrm{A}}^{(0)}_{2\\mapsto3}(\\Phi_{+1}){\\mathrm{B}}(\\Phi_2)} \\, ,\n\\label{eq:LOMEC2to3}\n\\end{equation}\nthe following matching formula is NLO accurate up to terms appearing at order $\\ensuremath{\\alpha_{\\mathrm{s}}}^2$\n\\begin{align}\n    \\avg{O}_{\\mathrm{NLO}+\\mathrm{PS}} = \\int \\ensuremath{\\,\\text{d}} \\Phi_2 \\, {\\mathrm{B}}(\\Phi_2)\\, k_\\mathrm{NLO}(\\Phi_2)\\Big[\\Delta(\\ensuremath{t_0},\\ensuremath{t_\\mathrm{c}}) O(\\Phi_2) \\nonumber \\\\\n    + \\int^{\\ensuremath{t_0}}_{\\ensuremath{t_\\mathrm{c}}} \\ensuremath{\\,\\text{d}} \\Phi_{+1}\\,  w^\\mathrm{LO}_{2\\mapsto3}{\\mathrm{A}}_{2\\mapsto3}(\\Phi_{+1}) \\Delta(\\ensuremath{t_0},t) O(\\Phi_2,\\Phi_{+1}) \\Big] \\, .\n\\end{align}\nThis can be seen by expanding the result to order $\\ensuremath{\\alpha_{\\mathrm{s}}}$.\n\n\\section{NNLO Matching Strategy} \\label{sec:strategyNNLO}\nWe now turn to the main result of this work, the definition of a fully-differential NNLO matching strategy for processes with two coloured final-state particles. It is applicable to both decays of colour singlets as well as scattering processes as long as all initial-state particles are colourless, for instance as in $e^+e^- \\to jj$.\n\nIn the antenna formalism, the expected value for an infrared-safe observable of a process with two coloured final-state particles is given at NNLO by\n\\begin{align}\n    \\avg{O}_\\mathrm{\\small{NNLO}}\n    &= \\int \\ensuremath{\\,\\text{d}}\\Phi_2 \\, \\Big[ {\\mathrm{B}}(\\Phi_2) + {\\mathrm{V}}(\\Phi_2) + {\\mathrm{I}}^\\mathrm{NLO}_{{\\mathrm{S}}}(\\Phi_2) \\nonumber \\\\\n    &\\qquad + {\\mathrm{V\\kern-0.15em V}}(\\Phi_2) + {\\mathrm{I}}_{{\\mathrm{T}}}(\\Phi_2) + {\\mathrm{I}}_{{\\mathrm{S}}}(\\Phi_2)\\Big]\\, O(\\Phi_2) \\nonumber \\\\\n    &+ \\int \\ensuremath{\\,\\text{d}}\\Phi_3 \\,  \\Big[ {\\mathrm{R}}(\\Phi_2,\\Phi_{+1})O(\\Phi_2,\\Phi_{+1}) \\nonumber \\\\\n    &\\qquad - {\\mathrm{S}}^\\mathrm{NLO}(\\Phi_2,\\Phi_{+1})O(\\Phi_2) \\nonumber \\\\\n    &\\qquad + {\\mathrm{R\\kern-0.15emV}}(\\Phi_2,\\Phi_{+1})O(\\Phi_2,\\Phi_{+1}) \\nonumber \\\\\n    & \\qquad + {\\mathrm{T}}(\\Phi_2,\\Phi_{+1},O) \\Big] \\nonumber \\\\\n    &+ \\int \\ensuremath{\\,\\text{d}}\\Phi_4 \\,  \\Big[{\\mathrm{R\\kern-0.15em R}}(\\Phi_2,\\Phi_{+2})O(\\Phi_2,\\Phi_{+2}) \\nonumber \\\\\n    &\\qquad - {\\mathrm{S}}(\\Phi_2,\\Phi_{+2},O) \\Big] \\, ,\n\\label{eq:expvalNNLO}\n\\end{align}\nwhere ${\\mathrm{R\\kern-0.15em R}}$ is the differential double-real radiation cross section and ${\\mathrm{R\\kern-0.15emV}}$ and ${\\mathrm{V\\kern-0.15em V}}$ denote the differential virtual (one-loop) and double-virtual (two-loop) corrections to the real-radiation cross section ${\\mathrm{R}}$ and the Born cross section ${\\mathrm{B}}$, respectively.\nIn this context, the term ${\\mathrm{S}}^\\mathrm{NLO}$ denotes the differential NLO real antenna subtraction term, ${\\mathrm{S}}$ denotes the differential NNLO double-real antenna subtraction term \\cite{GehrmannDeRidder:2004tv,GehrmannDeRidder:2005aw,GehrmannDeRidder:2005hi,GehrmannDeRidder:2005cm},\n\\begin{multline}\n    {\\mathrm{S}}(\\Phi_2,\\Phi_{+2},O) = {\\mathrm{S}}^{a}(\\Phi_2,\\Phi_{+2})O(\\Phi_2) + {\\mathrm{S}}^{b}(\\Phi_3,\\Phi_{+1})O(\\Phi_3) \\\\\n    - {\\mathrm{S}}^c(\\Phi_2,\\Phi_{+1},\\Phi_{+1}')O(\\Phi_2)\n\\label{eq:SNNLORR}\n\\end{multline}\nand ${\\mathrm{T}}$ denotes the differential NNLO real-virtual antenna subtraction term \\cite{GehrmannDeRidder:2004tv,GehrmannDeRidder:2005aw,GehrmannDeRidder:2005hi,GehrmannDeRidder:2005cm},\n\\begin{multline}\n    {\\mathrm{T}}(\\Phi_2,\\Phi_{+1},O) = {\\mathrm{T}}^{a}(\\Phi_2,\\Phi_{+1})O(\\Phi_2) + {\\mathrm{T}}^{b}(\\Phi_3)O(\\Phi_3) \\\\\n    - {\\mathrm{T}}^{c}(\\Phi_2,\\Phi_{+1})O(\\Phi_2)\n\\label{eq:TNNLORV}\n\\end{multline}\nTheir integrated counterparts are given by ${\\mathrm{I}}^\\mathrm{NLO}_{{\\mathrm{S}}}$, ${\\mathrm{I}}_{{\\mathrm{T}}}$, and ${\\mathrm{I}}_{{\\mathrm{S}}}$.\nIn this context, terms labelled with superscript $a$ constitute the double-real\/real-virtual subtraction terms \nwith compensating terms labelled with a superscript $c$ that remove spurious single-unresolved singularities.\nThe single-unresolved singularities are captured by the NLO subtraction terms of the $+1$-jet calculation, labelled with superscript $b$,\n\\begin{equation}\n    {\\mathrm{S}}^b(\\Phi_3,\\Phi_{+1}') \\equiv {{\\mathrm{S}}}^\\mathrm{NLO}(\\Phi_3,\\Phi_{+1}')\\, , \n    \\quad {\\mathrm{T}}^b(\\Phi_3) \\equiv {\\mathrm{I}}_{{\\mathrm{S}}}^\\mathrm{NLO}(\\Phi_3) \\, .\n\\label{eq:subtTermsNLO1jet}\n\\end{equation}\nThe terms labeled with superscript $b$ and $c$ cancel independently in eq.~\\eqref{eq:expvalNNLO}. \nThey are constructed such as to make the integrals individually infrared finite and thus amenable\nto evaluation with Monte-Carlo methods.\n\nAs for the NLO case, we define a Born-local weight,\n\\begin{align}\\label{eq:born_local_nnlo_kfactor}\n    k_\\mathrm{NNLO}(\\Phi_2) &:= 1 + \\frac{{\\mathrm{V}}(\\Phi_2)}{{\\mathrm{B}}(\\Phi_2)} + \\frac{{\\mathrm{I}}^\\mathrm{NLO}_{{\\mathrm{S}}}(\\Phi_2)}{{\\mathrm{B}}(\\Phi_2)} \\\\\n    &+ \\frac{{\\mathrm{V\\kern-0.15em V}}(\\Phi_2)}{{\\mathrm{B}}(\\Phi_2)} + \\frac{{\\mathrm{I}}_{{\\mathrm{T}}}(\\Phi_2)}{{\\mathrm{B}}(\\Phi_2)} + \\frac{{\\mathrm{I}}_{{\\mathrm{S}}}(\\Phi_2)}{{\\mathrm{B}}(\\Phi_2)} \\nonumber \\\\\n    &+ \\int \\ensuremath{\\,\\text{d}}\\Phi_{+1}\\, \\Big[\\frac{{\\mathrm{R}}(\\Phi_2,\\Phi_{+1})}{{\\mathrm{B}}(\\Phi_2)} - \\frac{{\\mathrm{S}}^\\mathrm{NLO}(\\Phi_2,\\Phi_{+1})}{{\\mathrm{B}}(\\Phi_2)} \\nonumber \\\\\n    &\\qquad+\\frac{{\\mathrm{R\\kern-0.15emV}}(\\Phi_2,\\Phi_{+1})}{{\\mathrm{B}}(\\Phi_2)} + \\frac{{\\mathrm{T}}(\\Phi_2,\\Phi_{+1})}{{\\mathrm{B}}(\\Phi_2)} \\Big] \\nonumber \\\\\n    &+ \\int \\ensuremath{\\,\\text{d}}\\Phi_{+2}\\, \\Big[\\frac{{\\mathrm{R\\kern-0.15em R}}(\\Phi_2,\\Phi_{+2})}{{\\mathrm{B}}(\\Phi_2)} - \\frac{{\\mathrm{S}}(\\Phi_2,\\Phi_{+2})}{{\\mathrm{B}}(\\Phi_2)} \\Big]\\nonumber \\, ,\n\\end{align}\nwhich will be used to construct the NNLO matching formula, and which can be used to perform the fixed-order computation\nin complete analogy to eq.~\\eqref{eq:born_local_nlo}.\nHere, $\\ensuremath{\\,\\text{d}}\\Phi_{+2}$ is the two-particle radiation phase space that enters the factorised $n+2$-particle phase space\n\\begin{equation}\n    \\ensuremath{\\,\\text{d}}\\Phi_{n+2} = \\ensuremath{\\,\\text{d}}\\Phi_{n} \\times \\ensuremath{\\,\\text{d}}\\Phi_{+2}\\,.\n\\end{equation}\nWe shall further need to distinguish between an ordered and unordered component of the two-particle radiation phase space, according to the following partition of unity:\n\\begin{align}\n    \\ensuremath{\\,\\text{d}}\\Phi_{+2} &= \\theta(t'-t)\\ensuremath{\\,\\text{d}}\\Phi_{+2} + \\theta(t-t')\\ensuremath{\\,\\text{d}}\\Phi_{+2} \\, ,\\nonumber \\\\\n    &= \\ensuremath{\\,\\text{d}}\\Phi_{+2}^> + \\ensuremath{\\,\\text{d}}\\Phi_{+2}^< \\, .\n\\end{align}\nThe ordered part $\\ensuremath{\\,\\text{d}}\\Phi_{+2}^<$ corresponds to the region accessible to strongly-ordered shower paths $\\ensuremath{t_0} > t > t'$, whereas the unordered part $\\ensuremath{\\,\\text{d}}\\Phi_{+2}^>$ is inaccessible to strongly-ordered showers because of the larger intermediate scale $\\ensuremath{t_0} > t' > t$. We will use V\\protect\\scalebox{0.8}{INCIA}\\xspace's sector criterion, cf.\\ sec.~3.3 in \\cite{Brooks:2020upa}, to distinguish between the two, cf.~\\cref{subsec:2to4}.\n\nIn order to be able to match the NNLO calculation with the shower, the shower needs to incorporate virtual corrections to ordinary $2\\to 3$ branchings as well as new $2\\to 4$ branchings, accounting for the simultaneous emission of two particles. These new shower terms correspond to the real-virtual and double-real corrections in the NNLO calculation. In addition, we need to incorporate the corresponding parton-shower counterterms.\nWe start by defining the two-particle NLO Sudakov as \\cite{Li:2016yez}\n\\begin{multline}\n    \\Delta_2^\\mathrm{NLO}(\\ensuremath{t_0},t)  \\\\\n    = \\exp\\Bigg\\{-\\int^{\\ensuremath{t_0}}_{t}\\ensuremath{\\,\\text{d}}\\Phi_{+1}\\, {\\mathrm{A}}_{2\\mapsto3}^{(0)}(\\Phi_{+1}) w^\\mathrm{NLO}_{2\\mapsto3}(\\Phi_2,\\Phi_{+1}) \\Bigg\\}  \\\\\n    \\times \\exp\\Bigg\\{-\\int^{\\ensuremath{t_0}}_{t}\\ensuremath{\\,\\text{d}}\\Phi_{+2}^>\\, {\\mathrm{A}}_{2\\mapsto 4}^{(0)}(\\Phi_{+2})w^\\mathrm{LO}_{2\\mapsto4}(\\Phi_2,\\Phi_{+2}) \\Bigg\\} \\, ,\n\\label{eq:nloSudakov}\n\\end{multline}\nwhere we have introduced the $2\\mapsto4$ LO matrix-element correction factor,\n\\begin{equation}\n    w^\\mathrm{LO}_{2\\mapsto4}(\\Phi_2,\\Phi_{+2}) = \\frac{{\\mathrm{R\\kern-0.15em R}}(\\Phi_2,\\Phi_{+2})}{{\\mathrm{A}}^{(0)}_{2\\mapsto4}(\\Phi_{+2}){\\mathrm{B}}(\\Phi_2)}\n\\label{eq:LOMEC2to4}\n\\end{equation}\nand the $2\\mapsto3$ NLO matrix-element correction factor $w^\\mathrm{NLO}_{2\\mapsto3}(\\Phi_{+1})$, which we write in terms of a second order correction to the LO $2\\mapsto 3$ MEC in \\cref{eq:LOMEC2to3},\n\\begin{multline}\n    w^\\mathrm{NLO}_{2\\mapsto3}(\\Phi_2,\\Phi_{+1}) = w^\\mathrm{LO}_{2\\mapsto3}(\\Phi_2,\\Phi_{+1})\\\\\n    \\times\\big(1+\\tilde{w}^\\mathrm{FO}_{2\\mapsto3}(\\Phi_2,\\Phi_{+1})\n    +\\tilde{w}^\\mathrm{PS}_{2\\mapsto3}(\\Phi_2)\\big)\\,.\n\\label{eq:NLOMEC2to3}\n\\end{multline}\nThe coefficients $\\tilde{w}$ are given by matching the $\\Order{\\ensuremath{\\alpha_{\\mathrm{s}}}^2}$ terms\nin the expansion of the truncated shower approximation to the fixed-order result \nin \\cref{eq:expvalNNLO} \\cite{Li:2016yez,Hartgring:2013jma}.\nWe find the fixed-order contribution\n\\begin{multline}\n    \\tilde{w}^\\mathrm{FO}_{2\\mapsto3}(\\Phi_2,\\Phi_{+1})=\\\\\n    \\frac{{\\mathrm{R\\kern-0.15emV}}(\\Phi_2,\\Phi_{+1})}{{\\mathrm{R}}(\\Phi_2,\\Phi_{+1})} + \\int^{t}_{0}\\ensuremath{\\,\\text{d}}\\Phi_{+1}^\\prime\\, \\frac{{\\mathrm{R\\kern-0.15em R}}(\\Phi_2,\\Phi_{+1},\\Phi_{+1}^\\prime)}{{\\mathrm{R}}(\\Phi_2,\\Phi_{+1})} \\\\\n    -\\left(\\frac{{\\mathrm{V}}(\\Phi_2)}{{\\mathrm{B}}(\\Phi_2)}+\\int^{\\ensuremath{t_0}}_{0}\\ensuremath{\\,\\text{d}}\\Phi_{+1}'\\,\\frac{{\\mathrm{R}}(\\Phi_2,\\Phi_{+1}')}{{\\mathrm{B}}(\\Phi_2)}\\right)\\;,\n\\label{eq:VMEC2to3}\n\\end{multline}\nand the second-order parton-shower matching term\n\\begin{multline}\n    \\tilde{w}^\\mathrm{PS}_{2\\mapsto3}(\\Phi_2)\n    = \\frac{\\ensuremath{\\alpha_{\\mathrm{s}}}}{2\\pi}\\ln\\frac{\\kappa^2\\mu_{\\rm S}^2}{\\ensuremath{\\mu}_\\mathrm{R}^2}\\\\\n    + \\int^{\\ensuremath{t_0}}_{t}\\ensuremath{\\,\\text{d}}\\Phi_{+1}'\\, {\\mathrm{A}}_{2\\mapsto3}^{(0)}(\\Phi_{+1}')w^\\mathrm{LO}_{2\\mapsto3}(\\Phi_2,\\Phi_{+1}')\\;.\n\\end{multline}\nThe factor $\\kappa$ is a constant and $\\mu_\\mathrm{S}^2$ is the parton-shower renormalisation scale.\nThe two are conventionally chosen such that the logarithmic structure of eq.~\\eqref{eq:VMEC2to3}\nis reproduced, which leads to $\\mu_\\mathrm{S}=p_\\perp$ and $\\kappa^2=\\exp\\{K\/\\beta_0\\}$, with $K$ \nthe two-loop cusp anomalous dimension~\\cite{Kodaira:1981nh,Davies:1984hs,Davies:1984sp,Catani:1988vd}. \nThis is known as the CMW scheme~\\cite{Catani:1990rr}.\n\nNote that in \\cref{eq:nloSudakov}, the integral over ${\\mathrm{A}}_{2\\mapsto 4}^{(0)}$ is defined over \nthe range $[t, \\ensuremath{t_0}]$, since the ``ordered'' contribution $t'<t$ has been reabsorbed \ninto $\\tilde w^\\mathrm{FO}_{2\\mapsto3}(\\Phi_{+1})$.\n\nIt should be emphasised that we do not require the NLO three-jet calculation to be provided externally \nbut include the correction directly in the shower evolution. This means that, different to \nthe situation in traditional merging approaches, this correction is exponentiated into a Sudakov factor.\nUp to the first emission, this agrees with the treatment in~\\cite{Hoche:2017iem,Dulat:2018vuy}\nand implicitly includes the contribution from higher-order matching terms and collinear mass \nfactorization counterterms that are needed to recover the NLO DGLAP splitting functions.\n\nIn addition, we need the $3$-particle Sudakov, which we describe at LO,\n\\begin{multline}\n    \\Delta_3^\\mathrm{LO}(t,t') \\\\\n    = \\exp\\Bigg\\{-\\int^{t}_{t'}\\ensuremath{\\,\\text{d}}\\Phi^\\prime_{+1}\\, {\\mathrm{A}}_{3\\mapsto4}^{(0)}(\\Phi^\\prime_{+1}) w^\\mathrm{LO}_{3\\mapsto4}(\\Phi_3,\\Phi_{+1}^\\prime) \\Bigg\\} \\, .\n\\end{multline}\nwith the $3\\mapsto 4$ LO matrix-element correction factor,\n\\begin{equation}\n    w^\\mathrm{LO}_{3\\mapsto4}(\\Phi_3,\\Phi_{+1}^\\prime) \n    = \\frac{{\\mathrm{R\\kern-0.15em R}}(\\Phi_2,\\Phi_{+2})}{{\\mathrm{A}}^{(0)}_{3\\mapsto4}(\\Phi_{+1}^\\prime){\\mathrm{R}}(\\Phi_2,\\Phi_{+1})} \\, .\n\\label{eq:LOMEC3to4}\n\\end{equation}\n\nUp to the second emission, the shower operator is thus given by\n\\begin{align}\n    &{\\mathcal{S}}_2(\\ensuremath{t_0},O) = \\Delta_2^\\mathrm{NLO}(\\ensuremath{t_0},\\ensuremath{t_\\mathrm{c}})O(\\Phi_2) \\label{eq:showerOpNLO} \\\\\n    &+ \\int^{\\ensuremath{t_0}}_{\\ensuremath{t_\\mathrm{c}}}\\ensuremath{\\,\\text{d}}\\Phi_{+1}\\, {\\mathrm{A}}_{2\\mapsto3}^{(0)}(\\Phi_{+1})w^\\mathrm{NLO}_{2\\mapsto3}\\Delta^\\mathrm{NLO}_2(\\ensuremath{t_0},t) \\nonumber \\\\\n    &\\quad \\times \\Big(\\Delta_3^\\mathrm{LO}(t,\\ensuremath{t_\\mathrm{c}})O(\\Phi_2,\\Phi_{+1})\\nonumber \\\\\n    &\\qquad + \\int^t_{\\ensuremath{t_\\mathrm{c}}}\\ensuremath{\\,\\text{d}}\\Phi_{+1}^\\prime\\, {\\mathrm{A}}^{(0)}_{3\\mapsto4}(\\Phi^\\prime_{+1})  w^\\mathrm{LO}_{3\\mapsto4}(\\Phi_3,\\Phi_{+1}^\\prime)O(\\Phi_3,\\Phi_{+1}^\\prime) \\Big) \\nonumber \\\\\n    &+ \\int^{\\ensuremath{t_0}}_{\\ensuremath{t_\\mathrm{c}}}\\ensuremath{\\,\\text{d}}\\Phi_{+2}^>\\, {\\mathrm{A}}_{2\\mapsto4}^{(0)}(\\Phi_{+2})w^\\mathrm{LO}_{2\\mapsto4} (\\Phi_2,\\Phi_{+2})O(\\Phi_2,\\Phi_{+2}) \\nonumber\n\\end{align}\nand our final NNLO+PS matching formula takes the simple form:\n\\begin{equation}\n    \\avg{O}_{\\mathrm{NNLO}+\\mathrm{PS}} = \\int \\ensuremath{\\,\\text{d}} \\Phi_2\\, {\\mathrm{B}}(\\Phi_2) k_\\mathrm{NNLO}(\\Phi_2) {\\mathcal{S}}_2(\\ensuremath{t_0},O) \\, .\n\\label{eq:nnlops}\n\\end{equation}\n\nWhen expanding the truncated shower operator ${\\mathcal{S}}_2$ in \\cref{eq:nnlops} up to order $\\ensuremath{\\alpha_{\\mathrm{s}}}^2$, NNLO accuracy is recovered for the observable $O(\\Phi_2)$, while $O(\\Phi_3)$ and $O(\\Phi_4)$ achieve NLO and LO accuracy, respectively.\nThis is true, because the combination of the iterated $2\\mapsto 3\\mapsto 4$ and the direct $2\\mapsto 4$ contributions to \\cref{eq:showerOpNLO} yields the correct double-real correction ${\\mathrm{R\\kern-0.15em R}}$ in \\cref{eq:expvalNNLO} by means of the LO MEC factors in \\cref{eq:LOMEC2to3,eq:LOMEC3to4,eq:LOMEC2to4}.  Moreover the NLO correction \\cref{eq:NLOMEC2to3} recovers the correct real and real-virtual corrections ${\\mathrm{R}}$ and ${\\mathrm{R\\kern-0.15emV}}$ in \\cref{eq:expvalNNLO} by means of \\cref{eq:LOMEC2to3} and \\cref{eq:VMEC2to3}.\n\n\\begin{comment}\n\\todo{SH: I think the following paragraph is not needed anymore. We could comment earlier on when we describe the counterterms. One really does not have a choice if one wants to avoid a large logarithm in the NLO weight.}\nWe want to close this section by elaborating upon the renormalisation scales in our calculation. While the fixed-order calculation is renormalised at the scale of the hard process, $\\ensuremath{\\mu}_\\mathrm{R}^2$, the scales in the real-radiation contributions are dictated by the parton-shower resummation, meaning that the strong coupling is evaluated at the emission scales $t \\equiv p_\\perp^2$.\nThis is reflected in our calculation above by evaluating all antenna functions at the shower scales,\n\\begin{equation}\n    {\\mathrm{A}}^{(\\ell)}_{n\\mapsto n+m}(\\Phi_{+m}) \\equiv {\\mathrm{A}}^{(\\ell)}_{n\\mapsto n+m}(\\Phi_{+m}; p_{\\perp,n+m}^2) \\, ,\n\\end{equation}\nwhereas all matrix-element correction factors and the Born-local (N)NLO weights are evaluated at the renormalisation scale of the hard process,\n\\begin{align}\n    w^\\mathrm{MEC}_{n\\mapsto n+m}(\\Phi_n,\\Phi_{+m}) &\\equiv w^\\mathrm{MEC}_{n\\mapsto n+m}(\\Phi_n,\\Phi_{+m}; \\ensuremath{\\mu}_\\mathrm{R}^2) \\, , \\\\\n    k_\\mathrm{(N)NLO}(\\Phi_2) &\\equiv k_\\mathrm{(N)NLO}(\\Phi_2; \\ensuremath{\\mu}_\\mathrm{R}^2) \\, .\n\\end{align}\nThis allows the calculation to be reorganised in a way that all logarithms containing scale hierarchies can be reabsorbed in a multiplicative factor.\n\\end{comment}\n\n\\section{Numerical Implementation} \\label{sec:implementation}\nIn this section, we want to present all necessary components of an implementation of our NNLO matching strategy. These are:\n\\begin{itemize}\n    \\item a framework to calculate the Born-local NNLO $K$-factors in Eq.~\\eqref{eq:born_local_nnlo_kfactor}\n    \\item a shower filling the strongly-ordered \\cite{Brooks:2020bhi} and unordered \\cite{Li:2016yez} regions of the single- and double-emission phase space\n    \\item tree-level MECs in strongly-ordered \\cite{Fischer:2017yja} and unordered \\cite{Giele:2011cb} shower paths\n    \\item NLO MECs in the first emission \\cite{Hartgring:2013jma}\n\\end{itemize}\nWith the exception of the first point, (process-dependent) implementations of these components existed in previous V\\protect\\scalebox{0.8}{INCIA}\\xspace versions (not necessarily simultaneously), and have been described in detail in the various references.\nWe have (re-)implemented all components in a semi-automated~\\footnote{Semi-automated here refers to the fact that antenna subtraction terms are explicitly implemented for each class of processes.} fashion in the V\\protect\\scalebox{0.8}{INCIA}\\xspace antenna shower in P\\protect\\scalebox{0.8}{YTHIA}\\xspace 8.3. We access loop matrix elements via a novel M\\protect\\scalebox{0.8}{CFM}\\xspace \\cite{Campbell:1999ah,Campbell:2011bn,Campbell:2015qma,Campbell:2019dru} interface presented in \\cite{Campbell:2021vlt} and tree-level matrix elements via a new run-time interface \\cite{ComixInterface} to the C\\protect\\scalebox{0.8}{OMIX}\\xspace matrix element generator \\cite{Gleisberg:2008fv} in S\\protect\\scalebox{0.8}{HERPA}\\xspace \\cite{Gleisberg:2008ta,Sherpa:2019gpd}.\n\nOur NNLO matching algorithm can be summarised in the following steps:\n\\begin{enumerate}\n    \\item[1.] Generate a phase space point according to the Born cross section ${\\mathrm{B}}(\\Phi_2)$.\n    \\item[2.] Calculate the Born-local NNLO factor $k_\\mathrm{NNLO}(\\Phi_2)$ and reweight the phase space point by the result. \n    \\item[3.] Let the phase-space maximum given by the invariant mass of the two Born partons define the starting scale for the shower, $t_\\mathrm{now} = t_0(\\Phi_2)$.\n    \\item[4.] Starting from the current shower scale, $t_\\mathrm{now}$, let the $2\\mapsto 3$ and $2\\mapsto 4$ showers compete for the highest branching scale. \n    \\item[5.] Update the current shower scale to be that of the winning branching, $t_\\mathrm{now} = \\mathrm{max}(t_{2\\mapsto 3},t_{2\\mapsto 4})$.\n    \\item[6a.] If the winning branching is a $2\\mapsto 3$ branching, calculate the accept probability including the NLO MEC $w^\\mathrm{NLO}_{2\\mapsto3}$.  \n    \\begin{itemize}\n        \\item  If rejected, continue from step 4.\n        \\item  If accepted, continue with a LO shower from the resulting three-particle configuration, starting from $t_\\mathrm{now}$ and including the LO MEC $w^\\mathrm{LO}_{3\\mapsto 4}$ when calculating accept probabilities for the $3\\mapsto4$ step. \n    \\end{itemize}\n    When a $3\\mapsto 4$ branching is accepted (or the shower cutoff scale is reached), continue with step 7.\n    \\item[6b.] If the winning branching is a $2\\mapsto 4$ branching, calculate the accept probability including the LO MEC $w^\\mathrm{LO}_{2\\mapsto4}$. \n    \\begin{itemize}\n        \\item If rejected, continue from step 4. \n        \\item If accepted, continue with step 7.\n    \\end{itemize}\n    \\item[7.] Continue with a standard (possibly uncorrected) shower from the resulting four-particle configuration, starting from $t_\\mathrm{now}$. \n\\end{enumerate}\nIt should be emphasised that the matrix-element correction factors make this algorithm independent of the splitting kernels (i.e.\\ antenna functions in our case) up to the matched order and the shower merely acts as an efficient Sudakov-weighted phase-space generator. Hence, if the algorithm is stopped after step 6, an NNLO-matched result is obtained, which can be showered by any other parton shower, just as is the case for P\\protect\\scalebox{0.8}{OWHEG}\\xspace NLO matching. Note, that there remains a dependence on the ordering variable, which has to be properly accounted for.\n\n\\subsection{NNLO Kinematics}\\label{subsec:kinematics}\nFor both, the unordered shower contributions and the Born-local NNLO weight, new kinematic maps are needed to reflect their direct $2\\mapsto 4$, i.e.\\ unordered or double-unresolved, nature. We utilise that the $n$-particle phase space measure\nmay be factorised into the product of a $2\\mapsto 3$ antenna phase space and the $n-1$-particle phase space measure, as well as into the product of a $2\\mapsto 4$ antenna phase space and the $n-2$-particle phase space.\nThis allows us to write the $2\\mapsto 4$ antenna phase space as the product of two $2\\mapsto 3$ antenna phase spaces,\n\\begin{multline}\n    \\ensuremath{\\,\\text{d}} \\Phi_{+2} (p_I+p_K; p_i, p_{j_1}, p_{j_2}, p_{k}) \\\\\n    = \\ensuremath{\\,\\text{d}} \\Phi_{+1} (p_I+p_K; \\hat{p}_i, \\hat{p}_{j}, p_{k}) \\\\\n    \\times \\ensuremath{\\,\\text{d}} \\Phi_{+1} (\\hat{p}_i+\\hat{p}_j; p_i, p_{j_1}, p_{j_2}) \\, ,\n\\label{eq:PS2to4}\n\\end{multline}\ncorresponding to the kinematic mapping\n\\begin{equation}\n    p_I + p_K = \\hat{p}_i + \\hat{p}_j + p_k = p_i + p_{j_1} + p_{j_2} + p_k \\, ,\n\\end{equation}\neffectively representing a tripole map \\cite{Gehrmann-DeRidder:2003pne}. In line with the phase space factorisation, the kinematic mapping is then constructed as an iteration of two on-shell $2\\mapsto 3$ antenna maps given in sec.~2.3 in \\cite{Brooks:2020upa}. \n\nWe have tested the validity of our kinematic maps by comparing V\\protect\\scalebox{0.8}{INCIA}\\xspace's phase-space mappings (double-gluon emission and gluon-emission-plus-splitting) to a flat sampling via R\\protect\\scalebox{0.8}{AMBO}\\xspace.\n\n\\subsection{Unordered Shower Contributions}\\label{subsec:2to4}\n\\begin{figure*}[t]\n    \\centering\n    \\includegraphics[width=0.4\\textwidth]{main205-1.pdf}\n    \\includegraphics[width=0.4\\textwidth]{main205-2.pdf}\n    \\includegraphics[width=0.4\\textwidth]{main205-3.pdf}\n    \\includegraphics[width=0.4\\textwidth]{main205-4.pdf}\n    \\caption{Ratio of the evolution variable of the four-parton and three-parton configuration $\\log(p_{\\perp,4}^2\/p_{\\perp,3}^2)$ in $e^+e^-\\to 4j$. The region $> 0$ corresponds to unordered contributions not reached by strongly-ordered showers.}\n    \\label{fig:orderingZDecays}\n\\end{figure*}\n\nAn important part of our proposal is the inclusion of double-unresolved radiation in the shower evolution.\nTo this end, we employ the sector-antenna framework \\cite{Brooks:2020upa} and amend it by direct $2\\mapsto 4$ branchings as described in \\cite{Li:2016yez}. \nIn the sector-shower approach, each branching is restricted to the region in phase space where it minimises the resolution variable, defined for final-state clusterings by\n\\begin{equation}\n    Q^2_{\\mathrm{res},j} = \\begin{cases} \\frac{s_{ij}s_{jk}}{s_{IK}} & \\text{if } j \\text{ is a gluon} \\\\ s_{ij} \\sqrt{\\frac{s_{jk}}{s_{IK}}} & \\text{if } (i,j) \\text{ is a quark-antiquark pair} \\end{cases}\n\\label{eq:resVar}\n\\end{equation}\nThis is achieved by a ``sectorisation'' of phase space according the partition of unity,\n\\begin{equation}\n    1 = \\sum\\limits_j \\Theta^\\mathrm{sct}_{j\/IK} = \\sum\\limits_j \\theta\\left(\\min\\limits_{i}\\left\\{Q^2_{\\mathrm{res},i}\\right\\} - Q^2_{\\mathrm{res},j}\\right) \\, ,\n\\label{eq:sectorVetoLO}\n\\end{equation}\nwhich is implemented in the shower evolution as an explicit veto for each trial branching.\nSince only a single branching kernel contributes per colour-ordered phase space point, sector antenna functions have to incorporate the full singularity structure associated with the respective sector. At LO, this amounts to including both the full single-collinear and single-soft limits in the antenna function. The full set of V\\protect\\scalebox{0.8}{INCIA}\\xspace's LO sector antenna functions is collected in \\cite{Brooks:2020upa}.\n\nBy construction, the default sector shower generates only strongly-ordered sequences\\footnote{This is different to virtually any other strongly-ordered shower, where recoil effects introduce unordered sequences. Such phase space points are vetoed in a sector shower.}, as the sector veto ensures that each emission is the softest (or most-collinear) in the post-branching configuration. The inclusion of direct $2\\mapsto 4$ branchings (which look unordered from an iterated $2\\mapsto 3$ point of view) in the sector shower is facilitated by extending the sector decomposition in \\cref{eq:sectorVetoLO} by an ordering criterion,\n\\begin{align}\n    1 &= \\sum\\limits_j \\left[\\Theta^<_{j\/IK}\\Theta^\\mathrm{sct}_{j\/IK} + \\Theta^>_{j\/IK}\\Theta^\\mathrm{sct}_{j\/IK}\\right]\\\\\n    &= \\underbrace{\\sum\\limits_j \\theta\\left(\\hat{p}_{\\perp,\\hat{j}}^2 - p_{\\perp,j}^2\\right)\\Theta^\\mathrm{sct}_{j\/IK}}_{2\\mapsto 3 \\text{ (strongly ordered)}} + \\underbrace{\\sum\\limits_j \\theta\\left(p_{\\perp,j}^2 - \\hat{p}_{\\perp,\\hat{j}}^2\\right)\\Theta^\\mathrm{sct}_{j\/IK}}_{2\\mapsto4 \\text{ (unordered)}} \\nonumber \n\\end{align}\nwhere $p_\\perp^2$ denotes V\\protect\\scalebox{0.8}{INCIA}\\xspace's transverse-momentum ordering variable and hatted variables denote the intermediate node in a sequence $IL \\mapsto \\hat{i} \\hat{j} \\hat{\\ell} \\mapsto i j k \\ell$. Here, the scales $p_\\perp^2$ and $\\hat p_\\perp^2$ are uniquely defined by the ordering variable of the sector-shower emission, i.e., that emission which minimises \\cref{eq:resVar}. Direct $2\\mapsto 4$ emissions are thus restricted to the unordered region of the double-emission phase space, denoted as $\\ensuremath{\\,\\text{d}}\\Phi_{+2}^>$ in \\cref{eq:nloSudakov} and defined as\n\\begin{equation}\n    \\ensuremath{\\,\\text{d}}\\Phi_{+2}^> = \\sum\\limits_{j} \\Theta^>_{j\/IK} \\Theta^\\mathrm{sct}_{j\/IK} \\ensuremath{\\,\\text{d}} \\Phi_{+2}^j \\, .\n\\end{equation}\n\nFor $2\\to 4$ emissions off quark-antiquark and gluon-gluon antennae, we use the double-real antenna functions in \\cite{GehrmannDeRidder:2004tv,GehrmannDeRidder:2005aw,GehrmannDeRidder:2005cm}. We note that NLO quark-gluon antenna functions appear in the Standard Model at lowest order for three final-state particles and are hence not of interest for our test case of $e^+e^-\\to jj$. We wish to point out, however, that the NLO quark-gluon antenna functions in \\cite{GehrmannDeRidder:2005hi,GehrmannDeRidder:2005cm} contain spurious singularities which have to be removed before a shower implementation is possible.\n\nAs a validation, we show in \\cref{fig:orderingZDecays} the ratio of the four-jet to three-jet evolution variable for $e^+e^- \\to 4j$ at $\\sqrt{s} = 240~\\giga e\\volt$. To focus on the perturbative realm, the shower evolution is constrained to the region between $\\ensuremath{t_0} = s$ and $\\ensuremath{t_\\mathrm{c}} = (5~\\giga e\\volt)^2$. The region $>0$ corresponds to the unordered part of phase space to which strongly-ordered showers cannot contribute. Due to the use of sector showers, there is a sharp cut-off at the boundary between the ordered and unordered region, as the sector criterion ensures that the last emission is always the softest and therefore, no recoil effects can spoil the strong ordering of the shower. As expected, the inclusion of direct $2\\to 4$ branchings gives access to the unordered parts of phase space, a crucial element of our matching method.\n\n\\subsection{LO Matrix-Element Corrections}\nIn order for the shower expansion to match the fixed-order calculation, we need (iterated) $2\\mapsto 3$ tree-level MECs and (direct) $2\\mapsto4$ tree-level MECs. Both take a particularly simple form in the sector-antenna framework, as will be shown below.\n\nAt leading-colour, tree-level MECs to the ordered sector shower can be constructed as \\cite{LopezVillarejo:2011ap,Fischer:2017yja}\n\\begin{align*}\n    w_{2\\mapsto 3,i}^{\\mathrm{LO},\\mathrm{LC}}(\\Phi_2,\\Phi_{+1})\n    &= \\frac{{\\mathrm{R}}^\\mathrm{LC}_{i}(\\Phi_2,\\Phi_{+1})}{\\sum_j \\Theta^\\mathrm{sct}_{j\/IK} A^\\mathrm{sct}_{j\/IK}(p_i,p_j,p_k){\\mathrm{B}}(\\Phi_2)} \\, ,\\\\\n    w_{3\\mapsto 4,i}^{\\mathrm{LO},\\mathrm{LC}}(\\Phi_3,\\Phi_{+1})\n    &= \\frac{{\\mathrm{R\\kern-0.15em R}}^\\mathrm{LC}_{i}(\\Phi_3,\\Phi_{+1})}{\\sum_j \\Theta^\\mathrm{sct}_{j\/IK} A^\\mathrm{sct}_{j\/IK}(p_i,p_j,p_k){\\mathrm{R}}^\\mathrm{LC}_i(\\Phi_3)} \\, ,\n\\end{align*}\nwhere\n\\begin{align*}\n    {\\mathrm{B}}(\\Phi_2) &= \\abs{{\\mathcal{M}}_2^{(0)}(p_1,p_2)}^2 \\, , \\\\\n    {\\mathrm{R}}^\\mathrm{LC}_{i}(\\Phi_3) &= \\abs{{\\mathcal{M}}_3^{(0)}(\\sigma_i\\{p_1,p_2,p_3\\})}^2 \\, , \\\\\n    {\\mathrm{R\\kern-0.15em R}}^\\mathrm{LC}_{i}(\\Phi_4) &= \\abs{{\\mathcal{M}}_4^{(0)}(\\sigma_i\\{p_1,p_2,p_3,p_4\\})}^2 \\, ,\n\\end{align*}\ndenote squared leading-colour colour-ordered amplitudes with the index $i$ denoting the respective permutation $\\sigma_i$ (the number of permutations depends on the process). The sector veto $\\Theta^\\mathrm{sct}_{j\/IK}$ ensures that only the most singular term contributes in the denominators, rendering the fraction exceptionally simple.\n\nDirect $2\\mapsto 4$ branchings can be corrected in an analogous way, replacing the sum over $2\\mapsto3$ antenna functions with a sum of $2\\mapsto4$ ones,\n\\begin{multline*}\n    w_{2\\mapsto 4,i}^{\\mathrm{LO},\\mathrm{LC}}(\\Phi_2,\\Phi_{+2})\n    \\\\\n    = \\frac{{\\mathrm{R\\kern-0.15em R}}^\\mathrm{LC}_{i}(\\Phi_2,\\Phi_{+2})}{\\sum_{\\{j,k\\}} \\Theta^\\mathrm{sct}_{jk\/IL} A^\\mathrm{sct}_{jk\/IL}(p_i,p_j,p_k,p_\\ell){\\mathrm{B}}(\\Phi_2)} \\, ,\n\\end{multline*}\n\nThe full-colour matrix element can be recovered on average by multiplication with a full-colour to leading-colour-summed matrix-element weight,\n\\begin{align}\n    w_{2\\mapsto 3,i}^{\\mathrm{LO}} &= w_{2\\mapsto 3,i}^{\\mathrm{LO},\\mathrm{LC}}\n    \\times \\frac{{\\mathrm{R}}(\\Phi_2,\\Phi_{+1})}{\\sum_j {\\mathrm{R}}^\\mathrm{LC}_{j}(\\Phi_2,\\Phi_{+1})} \\, , \\\\\n    w_{3\\mapsto 4,i}^{\\mathrm{LO}} &= w_{3\\mapsto 4,i}^{\\mathrm{LO},\\mathrm{LC}}\n    \\times \\frac{{\\mathrm{R\\kern-0.15em R}}(\\Phi_3,\\Phi_{+1})}{\\sum_j {\\mathrm{R\\kern-0.15em R}}^\\mathrm{LC}_{j}(\\Phi_3,\\Phi_{+1})} \\, , \\\\\n    w_{2\\mapsto 4,i}^{\\mathrm{LO}} &= w_{2\\mapsto 4,i}^{\\mathrm{LO},\\mathrm{LC}}\n    \\times \\frac{{\\mathrm{R\\kern-0.15em R}}(\\Phi_2,\\Phi_{+2})}{\\sum_j {\\mathrm{R\\kern-0.15em R}}^\\mathrm{LC}_{j}(\\Phi_2,\\Phi_{+2})} \\, .\n\\end{align}\n\nFor gluon splittings, multiple histories contribute even in the sector shower, because all permutations of quark lines have to be taken into account. To ensure that the MEC factors remain finite for final states with multiple quark pairs, an additional quark-projection factor has to be included. Since we only deal with a maximum of two quark pairs, it is given by\n\\begin{equation}\n    \\rho_j = \\frac{A^\\mathrm{sct}_{j_q\/g_IX_K}(\\bar q_i, q_j, X_k)}{\\sum_{j}A^\\mathrm{sct}_{j_q\/g_IX_K}(\\bar q_i, q_j, X_k)}\n\\end{equation}\nfor $2\\to 3$ branchings and\n\\begin{equation}\n    \\rho_j = \\frac{A^\\mathrm{sct}_{j_qk_{\\bar q}\/X_IY_L}(X_i, q_j,\\bar q_k, Y_\\ell)}{\\sum_{j}A^\\mathrm{sct}_{j_qk_{\\bar q}\/X_IY_L}(X_i, q_j, \\bar q_k, Y_\\ell)}\n\\end{equation}\nfor $2\\mapsto 4$ branchings.\n\n\\subsection{NLO Matrix-Element Corrections}\nMaking the antenna subtraction terms explicit, the fixed-order correction to the NLO matrix-element correction \\cref{eq:NLOMEC2to3} reads\n\\begin{align}\n    &\\tilde{w}^\\mathrm{FO}_{2\\mapsto3}(\\Phi_2,\\Phi_{+1}) = \\frac{{\\mathrm{R\\kern-0.15emV}}(\\Phi_2,\\Phi_{+1})}{{\\mathrm{R}}(\\Phi_2,\\Phi_{+1})} + \\frac{{\\mathrm{I}}^\\mathrm{NLO}(\\Phi_2,\\Phi_{+1})}{{\\mathrm{R}}(\\Phi_2,\\Phi_{+1})}\n    \\label{eq:VMEC2to3Numerical} \\\\\n    &\\, + \\int^{t}_{0}\\ensuremath{\\,\\text{d}}\\Phi_{+1}^\\prime\\, \\left[ \\frac{{\\mathrm{R\\kern-0.15em R}}(\\Phi_2,\\Phi_{+1},\\Phi_{+1}^\\prime)}{{\\mathrm{R}}(\\Phi_2,\\Phi_{+1})} - \\frac{{\\mathrm{S}}^\\mathrm{NLO}(\\Phi_2,\\Phi_{+1},\\Phi_{+1}^\\prime)}{{\\mathrm{R}}(\\Phi_2,\\Phi_{+1})} \\right] \\nonumber\\\\\n    &\\, - \\Bigg(\\frac{{\\mathrm{V}}(\\Phi_2)}{{\\mathrm{B}}(\\Phi_2)} + \\frac{{\\mathrm{I}}^\\mathrm{NLO}(\\Phi_2)}{{\\mathrm{B}}(\\Phi_2)} \\nonumber \\\\\n    &\\qquad + \\int^{\\ensuremath{t_0}}_{0}\\ensuremath{\\,\\text{d}}\\Phi_{+1}'\\, \\left[\\frac{{\\mathrm{R}}(\\Phi_2,\\Phi_{+1}^\\prime)}{{\\mathrm{B}}(\\Phi_2)} - \\frac{{\\mathrm{S}}^\\mathrm{NLO}(\\Phi_2,\\Phi_{+1}^\\prime)}{{\\mathrm{B}}(\\Phi_2)} \\right]\\Bigg)\\, , \\nonumber\n\\end{align}\nwith the differential NLO antenna subtraction terms ${\\mathrm{S}}^\\mathrm{NLO}(\\Phi_2,\\Phi_{+1}^\\prime)$, ${\\mathrm{S}}^\\mathrm{NLO}(\\Phi_2,\\Phi_{+1},\\Phi_{+1}^\\prime)$ and their integrated counterparts ${\\mathrm{I}}^\\mathrm{NLO}_{{\\mathrm{S}}}(\\Phi_2)$, ${\\mathrm{I}}^\\mathrm{NLO}_{{\\mathrm{S}}}(\\Phi_2,\\Phi_{+1})$ cf.~\\cref{eq:expvalNNLO,eq:subtTermsNLO1jet}.\nBased on the argument of the last subsection, we construct the full-colour NLO matrix-element correction as\n\\begin{align}\n   w^\\mathrm{NLO}_{2\\mapsto 3,i}(\\Phi_2,\\Phi_{+1}) &= w^{\\mathrm{LO},\\mathrm{LC}}_{2\\mapsto3,i}(\\Phi_2,\\Phi_{+1}) \\frac{{\\mathrm{R}}(\\Phi_2,\\Phi_{+1})}{\\sum_j {\\mathrm{R}}^\\mathrm{LC}_{j}(\\Phi_2,\\Phi_{+1})} \\nonumber \\\\ \n   &\\quad \\times (1 + \\tilde{w}^\\text{FO}_{2\\mapsto 3}(\\Phi_2,\\Phi_{+1}) + \\tilde{w}^\\text{PS}_{2\\mapsto 3}(\\Phi_2)) \\, .\n\\end{align}\n\nThe integration over the radiation phase spaces denoted $\\Phi_{+1}^\\prime$ in \\cref{eq:VMEC2to3Numerical} is done numerically, utilising antenna kinematics to map $3$-parton configurations to $4$-parton configurations (similarly for $2$-parton configurations). This phase-space generation approach will be described in detail in the next subsection in the context of the NNLO Born weight.\nNote that the radiation phase space $\\Phi_{+1}$ in \\cref{eq:VMEC2to3Numerical} is generated by the shower.\n\n\\subsection{NNLO Born Weight}\nThe Born-local NNLO weight can be calculated numerically using a ``forward-branching'' phase-space generation approach \\cite{Frixione:2007vw,Hoche:2010pf,Alioli:2010xd,Giele:2011tm,Figy:2018imt}, which has previously been applied to unweighted NLO event generation, using Catani-Seymour dipole subtraction \\cite{Campbell:2012cz}. The application to NNLO corrections to $e^+e^- \\to 2j$ using antenna subtraction has been outlined in \\cite{Weinzierl:2006ij}.\n\nGiven a Born phase space point, the real-radiation phase space is generated by uniformly sampling the shower variables $(t,\\zeta,\\phi)$ for each antenna, which represent integration channels in this context. As for the shower evolution, every phase space point is restricted to the sector in which the emission(s) correspond to the most-singular clusterings. \nThe momenta of the Born$+1j$ point are constructed according to the same kinematic map as the shower uses, summarised in sec.~2.3 in \\cite{Brooks:2020bhi}. Since antenna functions are azimuthally averaged, they do not cancel spin-correlations in collinear gluon branchings locally. To obtain a point-wise pole cancellation, the subtracted real correction ${\\mathrm{R}}-{\\mathrm{S}}$ can be evaluated on two correlated phase space points,\n\\begin{equation*}\n    \\left\\{ \\left(t,\\zeta,\\phi\\right), \\left(t,\\zeta,\\phi+\\uppi\/2\\right)\\right\\} \\,\n\\end{equation*}\nwhich cancels the collinear spin correlation exactly, as it is proportional to $\\cos(2\\phi)$.\nTo obtain double-real radiation phase space points for the subtracted double-real correction ${\\mathrm{R\\kern-0.15em R}}-{\\mathrm{S}}$, this procedure can be iterated, yielding four angular-correlated phase space points which cancel spin correlations in double single-collinear and triple-collinear limits.\nDue to the bijective nature of the sector-antenna framework, each $3$- or $4$-particle phase-space point obtained in this way can be mapped back uniquely to its $2$-particle origin, making the NNLO weight exactly Born-local. For $e^+e^- \\to 2j$ this procedure is identical to the one in \\cite{Weinzierl:2006ij}.\n\n\\begin{figure*}[t]\n    \\centering\n    \\includegraphics[width=0.4\\textwidth]{spiketest-nnlo-tc.pdf}\n    \\includegraphics[width=0.4\\textwidth]{spiketest-nnlo-ds.pdf}\\\\\n    \\includegraphics[width=0.4\\textwidth]{nnlotest-134.pdf}\n    \\includegraphics[width=0.4\\textwidth]{nnlotest-243.pdf}\n    \\caption{Test of the convergence of the double-real subtraction term ${\\mathrm{S}}(\\Phi_{2},\\Phi_{+2},O)$ in \\cref{eq:SNNLORR} in $e^+e^-\\to q g g \\bar q$. \\textsl{Top row}: progression of weight distributions from $x=10^{-2}$ to $x=10^{-4}$ in the triple-collinear limit ($s_{134}\/s_{1234} < x$) and double-soft limit ($s_{134}s_{234}\/s_{1234}^2 < x$). \\textsl{Bottom row}: trajectories $x\\cdot s_{134}$, $x\\cdot s_{234}$, $x\\to 0$ approaching the two triple collinear limits. Phase space points are not azimuthally averaged.}\n    \\label{fig:subtractionNNLO}\n\\end{figure*}\n\nWe have implemented the NNLO antenna subtraction terms for processes with two massless final-state jets, cf.~e.g.~\\cite{GehrmannDeRidder:2004tv}, in V\\protect\\scalebox{0.8}{INCIA}\\xspace in a semi-automated fashion. \nAs a validation, we illustrate the convergence of the double-real radiation subtraction term \\cref{eq:SNNLORR} in the triple-collinear and double-soft limits for the process $e^+e^-\\to q g g \\bar q$ in \\cref{fig:subtractionNNLO}. Phase space points are sampled according to the kinematic map in \\cref{subsec:kinematics} and we do not make use of the azimuthal averaging alluded to above.\n\nIt should be noted that a numerical calculation of the Born-local NNLO weight is not necessary for colour-singlet decays, as the inclusive $K$-factors are well known from analytical calculations, cf.~e.g.~\\cite{Chetyrkin:1996ela,GehrmannDeRidder:2004tv} for $Z\\to q\\bar q$ (with massless quarks), \\cite{Gorishnii:1990zu,Chetyrkin:1996sr,Baikov:2005rw,DelDuca:2015zqa} for $H \\to b\\bar b$ (with massless $b$s), and \\cite{Chetyrkin:1997iv,GehrmannDeRidder:2005aw} for $H\\to gg$ (in the Higgs effective theory).\n\n\\section{Conclusions and Outlook} \\label{sec:conclusions}\nWe have presented a technique to match final-state parton showers fully-differentially to next-to-next-to-leading order calculations in processes with two final-state jets. To our knowledge, this is the first method of its kind.\n\nWe have outlined a full-fledged numerical implementation in the V\\protect\\scalebox{0.8}{INCIA}\\xspace antenna shower in the P\\protect\\scalebox{0.8}{YTHIA}\\xspace 8.3 event generator. Phenomenological studies employing our strategy will be presented in separate works.\n\nWe want to close by noting that, while we here focused on the simplest case of two massless final-state jets, the use of the NNLO antenna subtraction formalism facilitates its adaption to more complicated processes such as $e^+ e^- \\to t\\bar t$ or $e^+e^-\\to 3j$. Considering the latter, spurious singularities in the quark-gluon NNLO antenna subtraction terms need to be removed before exponentiation in the shower.\nFor future work, an extension of our method to processes with coloured initial states can be envisioned, given the applicability of the NNLO antenna subtraction to hadronic collisions.\n\n\\section*{Acknowledgements} \nWe thank Aude Gehrmann-de Ridder and Thomas Gehrmann for providing us with FORM files of their antenna functions.\nWe thank Philip Ilten for the development of a general matrix-element generator interface for P\\protect\\scalebox{0.8}{YTHIA}\\xspace 8.3, which allowed us to interface C\\protect\\scalebox{0.8}{OMIX}\\xspace in this work.\nCTP is supported by the Monash Graduate Scholarship, the Monash International Postgraduate Research Scholarship, and the J.L.~William~Scholarship.\nHTL is supported by the U.S. Department of Energy under Contract No. DE-AC02-06CH11357 and the National Science Foundation under Grant No. NSF-1740142.\nThis research was supported by Fermi National Accelerator Laboratory (Fermilab), a U.S. Department of Energy, Office of Science, HEP User Facility. Fermilab is managed by Fermi Research Alliance, LLC (FRA), acting under Contract No. DE-AC02-07CH11359.  \nThis work was further partly funded by the Australian Research Council via Discovery Project DP170100708 \u2014 \"Emergent Phenomena in Quantum Chromodynamics\".\nThis work was also supported in part by the European Union's Horizon 2020 research and innovation programme under the Marie Sk\\l{}odowska-Curie grant agreement No 722104 \u2013 MCnetITN3.\n\n\\bibliographystyle{elsarticle-num}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nNext generation 5G cellular systems will encompass frequencies from around 500 MHz all the way to around 100 GHz. For the development of new 5G systems to operate in bands above 6 GHz, there is a need for accurate radio propagation models for these bands which are not fully modeled by existing channel models below 6 GHz, as previous generations were designed and evaluated for operation at frequencies only as high as 6 GHz. One important example is the recently developed 3D-Indoor Hotspot (InH) channel model~\\cite{3GPP36873}. This paper is a summary of key results provided in a much more detailed white paper by the authors found at the link in~\\cite{5GSIG}, in addition to a 3GPP-style outdoor contribution in~\\cite{5GSIG_VTC16}. The 3GPP 3D channel model provides additional flexibility for the elevation dimension, thereby allowing modeling two dimensional antenna systems, such as those that are expected in next generation system deployments. It is important for future system design to develop a new channel model that will be validated for operation at higher frequencies (e.g., up to 100 GHz) and that will allow accurate performance evaluation of possible future technical specifications in indoor environments. Furthermore, the new models should be consistent with the models below 6 GHz. In some cases, the requirements may call for deviations from the modeling parameters or methodology of the existing models, but these deviations should be kept to a bare minimum and only introduced when necessary for supporting the 5G simulation use cases.\n\nThere are many existing and ongoing campaign efforts worldwide targeting 5G channel measurements and modeling. They include METIS2020~\\cite{METIS2015}, COST2100\/COST~\\cite{COST2100}, IC1004~\\cite{COSTic1004}, ETSI mmWave~\\cite{ETSI2015}, NIST 5G mmWave Channel Model Alliance~\\cite{NIST}, MiWEBA~\\cite{MiWEBA2014}, mmMagic~\\cite{mmMagic}, and NYU WIRELESS~\\cite{Rap13a,Rap15a,Rap15b,Mac15a}. METIS2020, for instance, has focused on 5G technologies and has contributed extensive studies in terms of channel modelling. Their target requirements include a wide range of frequency bands (up to 86 GHz), very large bandwidths (hundreds of MHz), fully three dimensional and accurate polarization modelling, spherical wave modelling, and high spatial resolution. The METIS channel models consist of a map-based model, stochastic model, and a hybrid model which can meet requirement of flexibility and scalability.\n\nThe COST2100 channel model is a geometry-based stochastic channel model (GSCM) that can reproduce the stochastic properties of multiple-input\/multiple output (MIMO) channels over time, frequency, and space. On the other hand, the 5G mmWave Channel Model Alliance is newly established and will establish guidelines for measurement calibration and methodology, modeling methodology, as well as parameterization in various environments and a database for channel measurement campaigns. NYU WIRELESS has conducted and published extensive urban propagation measurements at 28, 38, 60, and 73 GHz for both outdoor and indoor channels, and has created large-scale and small-scale channel models and concepts of \\emph{time cluster spatial lobes} (TCSL) to model multiple multipath time clusters that are seen to arrive in particular directions~\\cite{Rap15a,Rap13a,Rap15b,Samimi15a,Samimi15b,Samimi15c,Nie13a}.\n\nThis paper presents a brief overview of the indoor channel properties for bands up to 100 GHz based on extensive measurements and results across a multitude of bands. In addition we present a preliminary set of channel parameters suitable for indoor 5G simulations that are capable of capturing the main properties and trends.\n\n\\section{Requirements For New Channel Model}\n\nThe requirements of the new channel model that will support 5G operation across frequency bands up to 100 GHz are outlined below:\n\\begin{enumerate}\n\\item The new channel model should preferably be based on the existing 3GPP 3D channel model~\\cite{3GPP36873} but with extensions to cater for additional 5G modeling requirements and scenarios, for example:\n\t\\begin{enumerate}\n\t\t\\item Antenna arrays, especially at higher-frequency millimeter-wave bands, will very likely be 2D and dual-polarized both at the access point (AP) and at the user equipment (UE) and will hence need properly-modeled azimuth and elevation angles of departure and arrival of multipath components.\n\t\t\\item Individual antenna elements will have antenna radiation patterns in azimuth and elevation and may require separate modeling for directional performance gains.  Furthermore, polarization properties of the multipath components need to be accurately accounted for in the model.  \n\t\\end{enumerate}\n\\item The new channel model must accommodate a wide frequency range up to 100 GHz.  The joint propagation characteristics over different frequency bands will need to be evaluated for multi-band operation, e.g., low-band and high-band carrier aggregation configurations. \n\\item The new channel model must support large channel bandwidths (up to 2 GHz), where:\n\t\\begin{enumerate}\n\t\t\\item The individual channel bandwidths may be in the range of 100 MHz to 2 GHz and may support carrier aggregation.\n\t\t\\item The operating channels may be spread across an assigned range of several GHz.\n\t\\end{enumerate}\n\\item The new channel model must support a range of large antenna arrays, in particular:\n\t\\begin{enumerate}\n\t\t\\item Some large antenna arrays will have very high directivity with angular resolution of the channel down to around 1.0 degree.\n\t\t\\item 5G will consist of different array types, e.g., linear, planar, cylindrical and spherical arrays, with arbitrary polarization.\n\t\t\\item The array manifold vector can change significantly when the bandwidth is large relative to the carrier frequency. As such, the wideband array manifold assumption is not valid and new modeling techniques may be required. It may be preferable, for example, to model departure\/arrival angles with delays across the array and follow a spherical wave assumption instead of the usual plane wave assumption.\n\t\\end{enumerate}\n\\item The new channel model must accommodate mobility, in particular (for outdoor models, although mentioned here for consistency):\n\t\\begin{enumerate}\n\t\t\\item The channel model structure should be suitable for mobility up to 350 km\/hr.\n\t\t\\item The channel model structure should be suitable for small-scale mobility and rotation of both ends of the link in order to support scenarios such as device to device (D2D) or vehicle to vehicle (V2V).\n\t\\end{enumerate}\n\\item The new channel model must ensure spatial\/temporal\/frequency consistency, in particular:\n\t\\begin{enumerate}\n\t\t\\item The model should provide spatial\/temporal\/frequency consistencies which may be characterized, for example, via spatial consistence, inter-site correlation, and correlation among frequency bands. \n\t\t\\item The model should also ensure that the channel states, such as line-of-sight (LOS)\/non-LOS (NLOS) for outdoor\/indoor locations, the second order statistics of the channel, and the channel realizations change smoothly as a function of time, antenna position, and\/or frequency in all propagation scenarios. \n\t\t\\item The spatial\/temporal\/frequency consistencies should be supported for simulations where the channel consistency impacts the results (e.g. massive MIMO, mobility and beam tracking, etc.).  Such support could possibly be optional for simpler studies.\n\t\\end{enumerate}\n\\item The new channel model must be of practical computational complexity, in particular:\n\t\\begin{enumerate}\n\t\t\\item The model should be suitable for implementation in single-link simulation tools and in multi-cell, multi-link radio network simulation tools. Computational complexity and memory requirements should not be excessive. The 3GPP 3D channel model~\\cite{3GPP36873} is seen, for instance, as a sufficiently accurate model for its purposes, with an acceptable level of complexity. Accuracy may be provided by including additional modeling details with reasonable complexity to support the greater channel bandwidths, and spatial and temporal resolutions and spatial\/temporal\/frequency consistency, required for millimeter-wave modeling.\n\t\t\\item The introduction of a new modeling methodology (e.g. Map based model) may significantly complicate the channel generation mechanism and thus substantially increase the implementation complexity of the system-level simulator. Furthermore, if one applies a completely different modeling methodology for frequencies above 6 GHz, it would be difficult to have meaningful comparative system evaluations for bands up to 100 GHz.\n\t\\end{enumerate}\n\\end{enumerate}\n\n\\section{Indoor Deployment Scenarios - Indoor (InH): Open and Closed Office, Shopping Mall}\nThe indoor scenario includes open and closed offices, corridors within offices and shopping malls as examples. The typical office environment has open cubicle areas, walled offices, open areas, corridors, etc., where the partition walls are composed of a variety of materials like sheetrock, poured concrete, glass, cinder block, etc. For the office environment, APs are generally mounted at a height of 2-3 m either on the ceilings or walls, with UEs at heights between 1.2 and 1.5 m. Shopping malls are generally 2-5 stories high and often include an open area (``atrium\"). In the shopping-mall environment, APs are generally mounted at a height of approximately 3 m on the walls or ceilings of the corridors and shops, with UEs at heights between 1.2 and 1.5 m. The density of APs may range from one per floor to one per room, depending on the frequency band and output power. A typical indoor office and shopping mall scenario are shown in Figures~\\ref{fig:InHsc} and~\\ref{fig:SMsc}, respectively. \n\\begin{figure}[b!]\n\t\\centering\n\t\\includegraphics[width = 3.7in]{InHsc.eps}\n\t\\caption{Typical Indoor Office.}\n\t\\label{fig:InHsc}\n\\end{figure}\n\\begin{figure}[b!]\n\t\\centering\n\t\\includegraphics[width = 3.7in]{SMsc.eps}\n\t\\caption{Indoor Shopping Malls.}\n\t\\label{fig:SMsc}\n\\end{figure}\n\n\\section{Characteristics of the InH Channel from 6 GHz to 100 GHz}\nMeasurements over a wide range of frequencies have been performed by the co-authors of this paper. In the following sections we outline the main observations per scenario with some comparisons to the existing 3GPP models for below 6 GHz (e.g.~\\cite{3GPP36873}).  \n\nIn LOS conditions, multiple reflections from walls, floor, and ceiling give rise to waveguiding. Measurements in both office and shopping mall scenarios show that path loss exponents, based on a 1 m free space reference distance, are typically below 2 in LOS conditions, leading to more favorable path loss than predicted by Friis' free space path loss formula. The strength of the waveguiding effect is variable and the path loss exponent appears to increase very slightly with increasing frequency, possibly due to the relation between the wavelength and surface roughness. \n\nMeasurements of the small scale channel properties such as angular spread and delay spread have shown remarkable similarities between channels over a very wide frequency range. It appears as if the main multipath components are present at all frequencies though with some smaller variations in amplitudes.\n\nRecent work shows that polarization discrimination ranges between 15 and 25 dB for indoor millimeter wave channels~\\cite{Karttunen15a}, with greater polarization discrimination at 73 GHz than at 28 GHz~\\cite{Mac15a}.\n\n\\section{Penetration Inside Buildings}\nMeasurements have been reported for penetration loss for various materials at 2.5, 28, and 60 GHz for indoor scenarios~\\cite{Rap15a,Rap13a,Anderson02a,Zhao13a}, although all materials were not measured for the same frequencies. For easy comparisons, walls and drywalls were lumped together into a common dataset and different types of clear class were lumped together into a common dataset with normalized penetration loss shown in Figure~\\ref{fig:NYUpenetration}. It was observed that clear glass has widely varying attenuation (20 dB\/cm at 2.5 GHz, 3.5 dB\/cm at 28 GHz, and 11.3 dB\/cm at 60 GHz). For mesh glass, penetration was observed to increase as a function of frequency (24.1 dB\/cm at 2.5 GHz and 31.9 dB\/cm at 60 GHz), and a similar trend was observed with whiteboard penetration increasing as frequency increased. At 28 GHz, indoor tinted glass resulted in a penetration loss of 24.5 dB\/cm. Walls showed very little attenuation per cm of distance at 28 GHz (less than 1 dB\/cm)~\\cite{5GSIG}. Furthermore, a simple parabolic model as a function of frequency for low-loss and high-loss building penetration is given in~\\cite{5GSIG_VTC16}.\n\\begin{figure}\n\t\\centering\n\t\\includegraphics[width = 3.7in]{NYUpenetration.eps}\n\t\\caption{2.5 GHz, 28 GHz, and 60 GHz normalized material penetration losses from indoor measurements with common types of glass and walls were lumped together into common datasets~\\cite{Rap15b,Anderson02a,Zhao13a}.}\n\t\\label{fig:NYUpenetration}\n\\end{figure}\n\n\\section{Path loss, Shadow Fading, LOS, and Blockage Modeling}\n\\subsection{LOS Probability}\nThe definition of LOS used in this paper is discussed in this sub-section together with other LOS models.  The LOS state is determined by a map-based approach, i.e., by considering the transmitter (AP) and receiver (UE) positions and whether any buildings or walls block the direct path between the AP and the UE. The impact of objects not represented in the map such as chairs, desks, office furniture, etc. is modelled separately using shadowing\/blocking terms. An attractive feature of this LOS definition is that it is frequency independent, as only walls are considered in the definition.  \n\nSince the 3GPP 3D model~\\cite{3GPP36873} does not include an indoor scenario for LOS-probability, and the indoor hotspot scenario in e.g. the IMT advanced model~\\cite{ITU-M.2135-1} differs from the office scenario considered in this paper, an investigation on the LOS probability for indoor office has been conducted based on ray-tracing simulations. Different styles of indoor office environments were investigated, including open-plan office with cubical area, closed-plan office with corridor and meeting room, and also a hybrid-plan office with both open and closed areas. It has been verified that the following model fits the propagation in indoor office environment the best, of the three models evaluated:\n\\begin{equation}\\label{eq1}\nP_{LOS} = \\begin{cases}\n1, & d\\leq1.2\\text{ m}\\\\\n\\exp(-(d-1.2)\/4.7), & 1.2<d<6.5\\text{ m}\\\\\n\\exp(-(d-6.5)\/32.6)\\cdot 0.32, & d\\geq6.5\\text{ m}\n\\end{cases}\n\\end{equation}\nThe verification results are shown in Table~\\ref{tbl:InHpLOS} and Figure~\\ref{fig:InHpLOS}. The LOS probability model used in ITU IMT-Advanced evaluation~\\cite{ITU-M.2135-1} and WINNER II~\\cite{WINNERII} are also presented here for comparison. For the ITU and WINNER II model, parameterization results based on new data vary a lot from the original model. The results show that the new model has a good fit to the data in an average sense and can be used for 5G InH scenario evaluations. However, note the high variability between different deployments and degrees of openness in the office area.\n\\begin{table*}\n\\caption{Comparison of the LOS probability models for the InH environment}\\label{tbl:InHpLOS}\n\\centering\n\\renewcommand{\\arraystretch}{1.6}\n\\begin{center}\n\t\\scalebox{1}{\n\t\t\\fontsize{8}{8}\\selectfont\n\t\t\\begin{tabular}{|c||c|c|c|}\n\t\t\t\\hline\n\t\t\tModels\t& Original\t& Updated\/New\t& MSE \\\\ \\hline \\hline\n\t\t\tITU Model\t& \\parbox{6.5cm}{\\begin{equation*}\\label{eq2} P_{LOS} = \\begin{cases} 1, & d\\leq18\\text{ m}\\\\ \\exp(-(d-18)\/27), & 18\\text{ m}<d<37\\text{ m}\\\\ 0.5, & d\\geq37\\text{ m} \\end{cases} \\end{equation*}} & \\parbox{6.5cm}{\\begin{equation*}\\label{eq3} P_{LOS} = \\begin{cases} 1, & d\\leq1.1\\text{ m}\\\\ \\exp(-(d-1)\/4.9), & 1.1\\text{ m}<d<9.8\\text{ m}\\\\ 0.17, & d\\geq9.8\\text{ m} \\end{cases} \\end{equation*}} & 0.0499 \\\\ \\hline\n\t\t\t\\scriptsize WINNER II model (B3)\t& \\parbox{6.5cm}{\\begin{equation*}\\label{eq4} P_{LOS} = \\begin{cases} 1, & d\\leq10\\text{ m}\\\\ \\exp(-(d-10)\/45), & d>10\\text{ m}\\end{cases}\\end{equation*}} & \\parbox{6.5cm}{\\begin{equation*}\\label{eq5} P_{LOS} = \\begin{cases} 1, & d\\leq1\\text{ m}\\\\ \\exp(-(d-1)\/9.4), & d>1\\text{ m}\\end{cases}\\end{equation*}} & 0.0572  \\\\ \\hline\n\t\t\t\\scriptsize WINNER II model (A1)\t& \\parbox{6.7cm}{\\begin{equation*}\\tiny\\label{eq6} P_{LOS} = \\begin{cases} 1, & d\\leq2.5\\text{ m}\\\\ 1-0.9(1-(1.24-0.61\\log_{10}(d))^3)^{1\/3}, & d>2.5\\text{ m}\\end{cases}\\end{equation*}} & \\parbox{6.7cm}{\\begin{equation*}\\tiny\\label{eq7} P_{LOS} = \\begin{cases} 1, & d\\leq2.6\\text{ m}\\\\ 1-0.9(1-(1.16-0.4\\log_{10}(d))^3)^{1\/3}, & d>2.6\\text{ m}\\end{cases}\\end{equation*}} & 0.0473  \\\\ \\hline\n\t\t\t\\scriptsize New Model\t& N\/A & \\parbox{7.5cm}{\\begin{equation*}\\label{eq8}\n\t\t\tP_{LOS} = \\begin{cases} 1, & d\\leq1.2\\text{ m}\\\\ \\exp(-(d-1.2)\/4.7), & 1.2\\text{ m}<d<6.5\\text{ m}\\\\ \\exp(-(d-6.5)\/32.6)\\cdot 0.32, & d\\geq6.5\\text{ m} \\end{cases} \\end{equation*}} & 0.0449 \\\\ \\hline\n\t\t\\end{tabular}}\n\t\\end{center}\n\\end{table*}\n\\begin{figure}\n\t\\centering\n\t\\includegraphics[width = 3.7in]{InHpLOS.eps}\n\t\\caption{Indoor office LOS probability for the three models considered.}\n\t\\label{fig:InHpLOS}\n\\end{figure}\n\n\\subsection{Path Loss Models}\nTo adequately assess the performance of 5G systems, multi-frequency path loss (PL) models, LOS probability, and blockage models will need to be developed across the wide range of frequency bands and for operating scenarios. Three PL models are considered in this paper; namely the close-in (CI) free space reference distance PL model~\\cite{Rap15b,Anderson02a,Sun15a} the close-in free space reference distance model with frequency-dependent path loss exponent (CIF)~\\cite{Mac15a}, and the Alpha-Beta-Gamma (ABG) PL model~\\cite{Mac15a,Piersanti12a,Hata80a,Mac13a}. These models are described in the following text and are then applied to various scenarios.\n\nTable~\\ref{tbl:InHPL} shows the parameters of the CI, CIF, and ABG path loss models for different environments for omnidirectional antennas. It may be noted that the models presented here are multi-frequency models, and the parameters are invariant to carrier frequency and can be applied across the 0.5-100 GHz band.\n\\begin{equation}\\label{eq9}\n\\mathrm{PL}^{CI}(f,d)[\\mathrm{dB}]=\\textrm{FSPL}(f,1\\;\\text{m})+10n\\log_{10}\\left(\\frac{d}{1\\;\\text{m}}\\right)+X_{\\sigma}^{CI}\n\\end{equation}\nwhere $f$ is the frequency in Hz, $n$ is the PLE, $d$ is the distance in meters, $X_{\\sigma}^{CI}$ is the shadow fading (SF) with $\\sigma$ in dB, and the free space path loss (FSPL) at 1 m, with frequency $f$ is given as:\n\\begin{equation}\\label{eq10}\n\\textrm{FSPL}(f,1\\;\\text{m})=20\\log_{10}\\left(\\frac{4\\pi f}{c}\\right),\n\\end{equation}\nwhere $c$ is the speed of light.\n\nThe ABG PL model~\\cite{Sun16a,Thomas16a,Piersanti12a,Mac15a} is given as:\n\\begin{equation}\\label{eq11}\n\\begin{split}\n\\mathrm{PL}^{\\textrm{ABG}}(f,d)[\\mathrm{dB}]=10\\alpha\\log_{10}(d)+\\beta\\\\\n+10\\gamma\\log_{10}(f)+X^{\\textrm{ABG}}_{\\sigma}\n\\end{split}\n\\end{equation}\nwhere $\\alpha$ captures how the PL increase as the transmit-receive distance (in meters) increases, $\\beta$ is a floating offset value in dB, $\\gamma$ attempts to capture the PL variation over the frequency $f$ in GHz, and $X^{\\textrm{ABG}}_{\\sigma}$  is the SF term with standard deviation in dB.\n\nThe CIF PL model is an extension of the CI model~\\cite{Mac15a}, and uses a frequency-dependent path loss exponent given by:\\begin{equation}\\label{eq9}\n\\begin{split}\n\\mathrm{PL}^{\\textrm{CIF}}(f,d)[\\mathrm{dB}]=\\textrm{FSPL}(f,\\text{1 m})+\\\\10n\\Bigg(1+b\\left(\\frac{f-f_0}{f_0}\\right)\\Bigg)\\log_{10}\\left(\\frac{d}{\\text{1 m}}\\right)+X^{\\textrm{CIF}}_{\\sigma}\n\\end{split}\n\\end{equation}\nwhere $n$ denotes the path loss exponent (PLE), and $b$ is an optimization parameter that captures the slope, or linear frequency dependency of the path loss exponent that balances at the centroid of the frequencies being modeled (e.g., path loss increases as $f$ increases when $b$ is positive). The term $f_0$ is a fixed reference frequency, the centroid of all frequencies represented by the path loss model~\\cite{Mac15a}, found as the weighed sum of measurements from different frequencies, using the following equation:\n\\begin{equation}\\label{eq10}\nf_0 = \\frac{\\sum_{k=1}^Kf_k N_K}{\\sum_{k=1}^K N_K}\n\\end{equation}\nwhere $K$ is the number of unique frequencies, and $N_k$ is the  number of path loss data points corresponding to the $k^{th}$  frequency $f_k$. The input parameter $f_0$ represents the weighted frequencies of all measurement (or Ray-tracing) data applied to the model. The CIF model reverts to the CI model when $b$ = 0 for multiple frequencies, or when a single frequency $f$ = $f_0$ is modelled. For InH, a dual-slope path loss model might provide a good fit for different distance zones of the propagation environment. Frequency dependency is also observed in some of the indoor measurement campaigns conducted by co-authors. For NLOS, both a dual-slope ABG and dual-slope CIF model can be considered for 5G performance evaluation (they each require 5 modeling parameters to be optimized), and a single-slope CIF model (that uses only 2 optimization parameters) may be considered as a special case for InH-Office~\\cite{Mac15a}. The dual-slope may be best suited for InH-shopping mall or large indoor distances (greater than 50 m). The dual slope InH large scale path loss models are given in~\\eqref{eq11} (ABG) and~\\eqref{eq12} (CIF).\n\\begin{figure*}[t!]\n\t\\begin{equation}\\label{eq11}\n\t\\resizebox{0.9\\hsize}{!}{$%\n\t\t\\mathrm{PL}_{Dual}^{ABG}(f,d)[\\mathrm{dB}] = \\begin{cases} \\alpha_1\\cdot10\\log_{10}(d)+\\beta_1+\\gamma\\cdot10\\log_{10}(f), & 1\\text{ m}<d\\leq d_{BP}\\\\ \\alpha_1\\cdot10\\log_{10}(d_{BP})+\\beta_1+\\gamma\\cdot10\\log_{10}(f)+\\alpha_2\\cdot 10\\log_{10}\\left(\\frac{d}{d_{BP}}\\right), & d>d_{BP}\n\t\\end{cases}$}\n\t\\end{equation}\n\\end{figure*}\n\\begin{figure*}\n\t\\begin{equation}\\label{eq12}\n\t\\resizebox{0.94\\hsize}{!}{$%\n\t\t\\mathrm{PL}_{Dual}^{CIF}(f,d)[\\mathrm{dB}] = \\begin{cases} \\textrm{FSPL}(f,\\text{1 m})+10n_1\\left(1+b_1\\left(\\frac{f-f_0}{f_0}\\right)\\right)\\log_{10}\\left(\\frac{d}{\\text{1 m}}\\right), & 1\\text{ m}<d\\leq d_{BP}\\\\\n\t\t\\textrm{FSPL}(f,\\text{1 m})+10n_1\\left(1+b_1\\left(\\frac{f-f_0}{f_0}\\right)\\right)\\log_{10}\\left(\\frac{d_{BP}}{\\text{1 m}}\\right)+10n_2\\left(1+b_2\\left(\\frac{f-f_0}{f_0}\\right)\\right)\\log_{10}\\left(\\frac{d}{d_{BP}}\\right), & d>d_{BP}\n\t\t\\end{cases}$}\n\t\\end{equation}\n\t\\hrulefill\n\n\t\\vspace*{4pt}\n\\end{figure*}\n\nIn the CI PL model, only a single parameter, the path loss exponent (PLE), needs to be determined through optimization to minimize the SF standard deviation over the measured PL data set~\\cite{Rap15b,Sun15a,Sun16a}. In the CI PL model there is an anchor point that ties path loss to the FSPL at 1 m, which captures frequency-dependency of the path loss, and establishes a uniform standard to which all measurements and model parameters may be referred. In the CIF model there are 2 optimization parameters ($n$ and $b$), and since it is an extension of the CI model, it also uses a 1 m free-space close-in reference distance path loss anchor. In the ABG PL model there are three parameters which need to be optimized to minimize the standard deviation (SF) over the data set~\\cite{Mac15a,Sun16a}. Closed form expressions for optimization of the model parameters for the CI, CIF, and ABG path loss models are given in~\\cite{Mac15a}, where it was shown that indoor channels experience an increase in the PLE value as the frequency increases, whereas the PLE is not very frequency dependent in outdoor UMa or UMi scenarios~\\cite{Mac15a,Rap15b,Sun15a,Sun16a,Thomas16a}. The CI, CIF, and ABG models, as well as cross-polarization forms and closed-form expressions for optimization are given for indoor channels in~\\cite{Mac15a}.\n\nAnother important issue related to path loss is shadow fading. For InH, the distance dependency and frequency dependency were investigated for both indoor office and shopping mall. For the LOS propagation condition, the frequency and distance dependency is weak. But for the NLOS propagation condition, frequency and distance dependency is more apparent as indicated in Table 7 of~\\cite{5GSIG}.\n\n\\begin{table}\n\t\\caption{InH and Shopping Mall Path Loss Models for LOS and NLOS.}\\label{tbl:InHPL}\n\t\\centering\n\t\\renewcommand{\\arraystretch}{1.6}\n\t\\begin{center}\n\t\\scalebox{0.82}{\n\t\t\\fontsize{8}{8}\\selectfont\n\t\t\\begin{tabular}{|p{3cm}|p{3cm}|p{3cm}|}\n\t\t\t\\hline\n\t\t\tScenario & CI\/CIF Model Parameters & ABG Model Parameters \\\\ \\hline \\hline\n\t\t\tInH-Indoor-Office-LOS & $n$=1.73, $\\sigma_{\\mathrm{SF}}$=3.02 dB & N\/A \\\\ \\hline\n\t\t\tInH-Indoor-Office-NLOS single slope (FFS)  & $n$=3.19, $b$=0.06, $f_0$=24.2 GHz, $\\sigma_{\\mathrm{SF}}$=8.29 dB & $\\alpha$=3.83, $\\beta$=17.30, $\\gamma$=2.49, $\\sigma_{\\mathrm{SF}}$=8.03 dB \\\\ \\hline\n\t\t\tInH-Indoor-Office-NLOS dual slope & $n_1$=2.51, $b_1$=0.12, $f_0$=24.1 GHz, $n_2$=4.25, $b_2$=0.04, $d_{BP}$=7.8 m, $\\sigma_{\\mathrm{SF}}$=7.65 dB & $\\alpha_1$=1.7, $\\beta_1$=33.0, $\\gamma$=2.49, $d_{BP}$=6.90 m, $\\alpha_2$=4.17, $\\sigma_{\\mathrm{SF}}$=7.78 dB \\\\ \\hline\n\t\t\tInH-Shopping Malls-LOS & $n$=1.73, $\\sigma_{\\mathrm{SF}}$=2.01 dB & N\/A \\\\ \\hline\n\t\t\tInH-Shopping Malls-NLOS single slope (FFS)  & $n$=2.59, $b$=0.01, $f_0$=39.5 GHz, $\\sigma_{\\mathrm{SF}}$=7.40 dB & $\\alpha$=3.21, $\\beta$=18.09, $\\gamma$=2.24, $\\sigma_{\\mathrm{SF}}$6.97 dB \\\\ \\hline\n\t\t\tInH-Shopping Malls-NLOS dual slope & $n_1$=2.43, $b_1$=0.01, $f_0$=39.5 GHz, $n_2$=8.36, $b_2$=0.39, $d_{BP}$= 110 m, $\\sigma_{\\mathrm{SF}}$=6.26 dB & $\\alpha_1$=2.9, $\\beta_1$=22.17, $\\gamma$=2.24, $d_{BP}$=147.0 m, $\\alpha_2$=11.47, $\\sigma_{\\mathrm{SF}}$=6.36 dB \\\\ \\hline\n\t\t\\end{tabular}}\n\t\\end{center}\n\\end{table}\n\n\\section{Fast Fading Modeling}\nFor InH scenarios, an investigation of fast fading modelling has been conducted based on both measurement and ray-tracing. Both indoor office and shopping mall environments have been investigated at frequencies including 2.9 GHz, 3.5 GHz, 6 GHz, 14 GHz, 15 GHz, 20 GHz, 28 GHz, 29 GHz, 60 GHz, and 73 GHz. Some preliminary analysis on large-scale channel characteristics have been summarized in~\\cite{5GSIG}. Although it is still too early to apply these results to the full frequency range up to 100 GHz, these preliminary investigations have provided insight into the difference induced by the largely extended frequency range. The preliminary analysis in~\\cite{5GSIG} illustrates the frequency dependency of large-scale channel characteristics across the measured frequency range.\n\\section{Conclusion}\nThe basis for this paper is the open literature in combination with recent and ongoing propagation channel measurements performed by a majority of the co-authors of this paper, some of which are as of yet unpublished. The InH propagation models are somewhat different from the outdoor UMi and UMa models in that the indoor channels are more frequency-dependent than outdoor channels, leading to the ABG and CIF frequency-dependent NLOS path loss models. In LOS conditions, waveguiding effects were observed in all frequencies measured, leading to path loss exponents less than the theoretical value of $n$ = 2 in LOS. The preceding tables give an overview of these recent measurement activities in different frequency bands and scenarios, in addition to further information provided in~\\cite{5GSIG}.\n\\section*{Acknowledgment}\nThe authors would like to thank Jianhua Zhang\\textsuperscript{b} and Yi Zheng\\textsuperscript{c} who are also contributing authors of this manuscript and 5G white paper~\\cite{5GSIG}.\n\n\\bibliographystyle{IEEEtran}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction: Motivation \\& Outline}\n\\label{s:intro}\n\nAll natural optical media are to some extent variable in space, often in such a complex way that they are best represented with statistics.  In nuclear engineering, there is increasing interest in pebble-bed reactors where the core is made of many small spheres that contain both fuel and moderator material.  In contrast with classic reactor designs, their detailed 3D geometry (i.e., how the spheres stack) is quite random.  Earth's cloudy atmosphere is another instance of a very clumpy 3D optical medium.  These are just two examples from vastly different disciplines where a good theory for stochastic transport would be a valuable asset.\n\nBroadly speaking, three kinds of model have been proposed to account for unresolved spatial variability in a transport medium.\n\\begin{itemize}\n\\item\nThe most natural approach is ``homogenization'' where one seeks \\emph{effective} material properties that can be used in the solution of a transport problem for a uniform medium, but would make an accurate prediction of the behavior of the heterogeneous stochastic medium.  The homogenized material properties will depend on statistical quantities (means, variances, correlations, etc.) that characterize the stochastic medium of interest.  Examples for the cloudy atmosphere are in \\cite{DavisEtAl1990,Cahalan1994,Cairns2000}.\n\\item\nAn alternative is to develop new transport equations to solve either analytically or numerically.  Examples for the cloudy atmosphere are in \\cite{AvasteVainikko1974,Stephens1988b,Davis2006}.  Interestingly, the early paper by Avaste and Vyanikko \\cite{AvasteVainikko1974} proposes a binary mixture model that has a long and ongoing history of application to nuclear engineering, going at least back to the seminal papers by Levermore, Pomraning et al. \\cite{LevermorePomraning1986,LevermoreEtAl1988}.  This approach is at least conceptually more difficult than the previous one since new methods must be found to solve the new transport equations.\n\\item\nA third approach, of intermediate complexity, is to linearly combine the answers of a number of computations for uniform media in order to approximate the answer for the spatially heterogeneous stochastic medium.  Examples of application to the cloudy atmosphere are in \\cite{Mullamaa1972,RonnholmEtAl1980,StephensEtAl_TTSP91,CahalanEtAl1994a,Barker1996a,Barker_etal08}.\n\\end{itemize}\nIn our experience, homogenization will work well for weaker kinds of variability and\/or higher tolerance for error.  A model derived using the second approach, such as the one proposed in the following pages, should be more broadly applicable.  Models of the 3rd kind can be competitive, largely due to their straightforward implementation.\n\nIn the following, we will primarily keep clouds and atmospheric optics in mind, but the generalized transport model we propose may prove to be more broadly applicable.  Accordingly, we will talk about radiative transfer (RT) and RT equations (RTEs), but the entirety of this work can be thought of as transport theory as defined by the linear Boltzmann equation.\n\n\\paragraph{Outline:}\nThe remainder of this article is organized as follows.\nSection~\\ref{s:dDim_sRTE} introduces our notations and states the standard RTE and boundary conditions for homogeneous---or random but ``homogenized''---plane-parallel media in $d$ spatial dimensions ($d=1,2,3$).\nSection~\\ref{s:dDim_gRTE} introduces our ansatz leading to a new class of generalized RTEs in integral form with power-law transmission laws.  Therein, we first see how non-exponential transmission laws arise from the statistics of stochastic media, with an emphasis on the role of spatial correlations, as exemplified by the Earth's turbulent and cloudy atmosphere.\nIn Section~\\ref{s:2stream_MarCh}, the $d=1$ case gets special attention.  In the framework of standard RT, it is formally identical to the well-known two-stream model.  Turning to generalized RT, we derive ab initio a deterministic numerical solution in $d=1$, and use it to investigate internal radiation fields.  The new generalized RT solver is based on Markov chain formalism, traditionally a tool for random walk theory (including its application to Monte Carlo methods in transport).  A technical Appendix details the computational methodology used in the Markov chain code.\nSection~\\ref{s:DiffusionLimits} revisits the behavior of diffuse transmission in the absence of absorption for standard and generalized RT in the diffusion limit (i.e., asymptotically large transport optical depth).  New numerical experiments in $d=2$ validate the theoretical prediction based on self-similar L\\'evy flights.  This reduced dimensionality is easier to comprehend graphically, and also may have applications in transport phenomena on random surfaces.\nIn Section~\\ref{s:ReciprocityViolation}, we use the single scattering limit in $d=3$ to show that generalized RT is not reciprocal under a switch of sources and detectors.  This violation of angular reciprocity is in fact observed in the Earth's cloudy atmosphere---the original motivation and application of the generalized RT model.\nIn the final Section~\\ref{s:concl}, we present our conclusions and an outlook on practical applications of our theoretical and computational advances, including a connection with recent work atmospheric spectroscopy \\cite{ConleyCollins2011}.\n\n\n\\section{Standard Radiative Transport in $d$ Spatial Dimensions}\n\\label{s:dDim_sRTE}\n\n\\subsection{RTE for Homogeneous---or Homogenized---Media in Integro-Differential Form}\n\nLet $I(z,\\mathbf{\\Omega})$ denote the steady-state radiance field at level $z$ in a \\emph{uniform} $d$-dimensional plane-parallel optical medium of thickness $H$,\n\\begin{equation}\n\\text{M}_d(H) = \\{ \\vect{x}\\in\\mathbb{R}^d, 0<z<H \\},\n\\end{equation}\npropagating in direction $\\mathbf{\\Omega}$ on the $d$-dimensional sphere,\n\\begin{equation}\n\\Xi_d = \\{ \\mathbf{\\Omega}\\in\\mathbb{R}^d, \\|\\mathbf{\\Omega}\\|=1 \\}.\n\\end{equation}\n$I(z,\\mathbf{\\Omega})$ has physical units of radiant power per unit of $d$-dimensional ``area'' per $d$-dimensional ``solid angle.''  Table~\\ref{t:Definitions} gives explicit definitions of $\\vect{x}$, $\\mathbf{\\Omega}$, and other properties introduced further on for $d = 1,2,3$.\n\nDenoting the extinction coefficient (expressed in m$^{-1}$) by $\\sigma$, $I(z,\\mathbf{\\Omega})$ is a function of exactly $d$ variables that verifies the linear transport equation\n\\begin{equation}\n\\left[ \\Omega_z\\frac{\\mathrm{d}\\;}{\\mathrm{d} z} + \\sigma \\right]\nI(z,\\mathbf{\\Omega}) = S(z,\\mathbf{\\Omega}) + q(z,\\mathbf{\\Omega}),\n\\label{e:nD_RTE_uniform}\n\\end{equation}\nwhere $S(z,\\mathbf{\\Omega})$ is the (unknown) source \\emph{function} for multiple scattering and $q(\\vect{x},\\mathbf{\\Omega})$ is the (specified) source \\emph{term}.  These quantities have the physical units of [$I$] further divided by a unit of length, hence radiant power per unit of $d$-dimensional ``volume,'' instead of ``area.''  Specifically, we have\n\\begin{equation}\nS(z,\\mathbf{\\Omega}) = \\sigma_\\text{s}\\int\\limits_{\\Xi_d}\np(\\mathbf{\\Omega}^\\prime\\cdot\\mathbf{\\Omega})I(z,\\mathbf{\\Omega}^\\prime)\\mathrm{d}\\mathbf{\\Omega}^\\prime,\n\\label{e:nD_RTE_SourceFunction}\n\\end{equation}\nwhere $p(\\mathbf{\\Omega}^\\prime\\cdot\\mathbf{\\Omega})$ is the phase function (PF) in units of inverse $d$-dimensional solid angle, which we assume is only a function of the scattering angle $\\theta_\\text{s} = \\cos^{-1}\\mathbf{\\Omega}^\\prime\\cdot\\mathbf{\\Omega}$.  As an important example, we have listed in Table~\\ref{t:Definitions} values for the PF when scattering is isotropic.  The quantity $\\sigma_\\text{s}$, appearing in (\\ref{e:nD_RTE_SourceFunction}), is the scattering coefficient in m$^{-1}$.  Combining (\\ref{e:nD_RTE_uniform}) and (\\ref{e:nD_RTE_SourceFunction}) leads to the RTE in $d$ dimensions in standard \\emph{integro-differential} form.\n\nA popular approach for modeling RT in stochastic media is to use ``homogenized'' optical properties $\\sigma$, $\\sigma_\\text{s}$ and $p(\\mathbf{\\Omega}^\\prime\\cdot\\mathbf{\\Omega})$.  This means that, rather than simple averages over the $d$-dimensional spatial variability of actual optical properties, an \\emph{effective} value is taken that somehow captures the average impact of the spatial fluctuations on $I(z,\\mathbf{\\Omega})$, itself a spatial average radiance field.  The effective optical properties will depend on a subset of their respective means, variances, possibly higher-order moments, auto-correlations, cross-correlations, and so on.\n\nApart from previously mentioned physics-based homogenization techniques in RT for the cloudy atmosphere \\cite{DavisEtAl1990,Cahalan1994,Cairns2000}, rigorous mathematical methods have been brought to bear on this still challenging problem; see, e.g., \\cite{Allaire1992,DumasGolse2000,BalJing2010}.  However, these studies focus on highly oscillatory optical media.  Such high-frequency (``noisy'') stochastic media were independently investigated by Davis and Mineev-Weinstein \\cite{DavisMineev11} that are predicated on power-law (scaling) statistics.  They used averaging methods akin to those described further on (in \\S\\ref{s:nonExpTran}), but with the necessary modifications to account for noise-like spatial variability.  Specifically, these authors assumed media where the extinction coefficient fluctuations have a wavenumber spectrum\n\\begin{equation}\nE_\\sigma(k) \\sim k^{-\\beta},\n\\label{e:PowerLaw_EnergySpectum}\n\\end{equation}\nover a broad range of scales (i.e., $1\/k$) that overlaps with radiatively relevant ones, including $H$ and the mean free path (defined rigorously further on for variable media).  They found that in cases of white- or blue-noise media ($\\beta \\le 0$), homogenization will likely work, being enabled by approximately exponential mean transmission laws.  Otherwise, that is, in cases of pink- or red-noise media ($0 < \\beta \\le 1$), it will not work since exponential decay is a poor approximation to the mean transmission law.  Media with $\\beta > 1$ are not noise-like---they have a stochastic continuity property---and are discussed in \\S\\ref{s:nonExpTran}.\n\nIn the present study, we are exclusively interested in the response of uniform or stochastic media to irradiation from an external source.  If this source is collimated (highly concentrated into a single direction $\\mathbf{\\Omega}_0$, with $\\Omega_{0z} > 0$), then we can take\n\\begin{equation}\nq(z,\\mathbf{\\Omega}) = F_0 \\exp(-\\sigma z\/\\mu_0) \\sigma_\\text{s} p(\\mathbf{\\Omega}_0\\cdot\\mathbf{\\Omega})\n\\label{e:nD_RTE_SourceTerm}\n\\end{equation}\nin the uniform case, where $F_0$ (in W\/m$^{d-1}$) is its uniform areal density.  We also introduce here\n\\begin{equation*}\n\\mu_0 = \\cos\\theta_0 = \\Omega_{z0}.\n\\end{equation*}\nNote that we have oriented the $z$-axis positively in the direction of the incoming flow of solar radiation, as is customary in atmospheric optics.  The meaning of each factor in (\\ref{e:nD_RTE_SourceTerm}) is clear: the incoming flux $F_0$ at $z = 0$ is attenuated exponentially (Beer's law) along the oblique path to level $z$ where it is scattered with probability $\\sigma_\\text{s}$ per unit of path length and, more specifically, into direction $\\mathbf{\\Omega}$ according to the PF value for $\\theta_\\text{s} = \\cos^{-1}\\mathbf{\\Omega}_0\\cdot\\mathbf{\\Omega}$.  In this case, $I(z,\\mathbf{\\Omega})$ is the diffuse radiation (i.e., scattered once or more).\n\nThe appropriate boundary conditions (BCs) for the diffuse radiance that obeys (\\ref{e:nD_RTE_uniform})--(\\ref{e:nD_RTE_SourceTerm}) will express that none is coming in from the top of the medium, $I(0,\\mathbf{\\Omega}) = 0$ for $\\Omega_z > 0$.  At the lower ($z = H$) boundary, we will take\n\\begin{equation}\nI(H,\\mathbf{\\Omega}) = F_-(H) \/ c_d,\n\\label{e:nD_RTE_lowerBC_Lambertian}\n\\end{equation}\nwhere\n\\begin{equation}\nF_-(H) = \\rho \\, F_+(H),\n\\label{e:rho_Lambertian}\n\\end{equation}\nfor all $\\Omega_z < 0$, where $\\rho$ is the albedo of the partially ($0 < \\rho < 1$) or totally ($\\rho = 1$) reflective surface; we have also introduced the downwelling (subscript ``$+$'') and upwelling (subscript ``$-$'') hemispherical fluxes\n\\begin{equation}\n\\begin{array}{l}\nF_+(z) = \\int\\limits_{\\Omega_z>0} \\Omega_z I(z,\\mathbf{\\Omega})\\mathrm{d}\\mathbf{\\Omega}\n       + \\mu_0 F_0 \\mathrm{e}^{-\\sigma z\/\\mu_0}, \\\\\nF_-(z) = \\int\\limits_{\\Omega_z<0}|\\Omega_z|I(z,\\mathbf{\\Omega})\\mathrm{d}\\mathbf{\\Omega}.\n\\end{array}\n\\label{e:hemispherical_fluxes}\n\\end{equation}\nThis surface reflectivity model is, for simplicity, Lambertian (isotropically reflective), and the numerical constant $c_d = \\int_{\\Omega_z>0}\\Omega_z\\mathrm{d}\\mathbf{\\Omega}$ is given in Table~\\ref{t:Definitions} for $d = 1,2,3$.  Naturally, we will also consider a black (purely absorbing) surface in (\\ref{e:nD_RTE_lowerBC_Lambertian}) by setting $\\rho = 0$.\n\nAlternatively, we can view $I(z,\\mathbf{\\Omega})$ as total (uncollided \\emph{and} once or more scattered) radiance, and assume $q(z,\\mathbf{\\Omega}) \\equiv 0$ inside M$_d(H)$.  Radiation sources will then be represented in the expression of boundary conditions (BCs).  The upper ($z = 0$) BC expresses either diffuse or collimated incoming radiation.  In the former case, we have\n\\begin{equation}\nI(0,\\mathbf{\\Omega}) = F_0 \/ c_d,\n\\label{e:nD_RTE_upperBC_diffuse}\n\\end{equation}\nfor any $\\mathbf{\\Omega}$ with $\\Omega_z > 0$.  In the latter case, we have\n\\begin{equation}\nI(0,\\mathbf{\\Omega}) = F_0 \\delta(\\mathbf{\\Omega}-\\mathbf{\\Omega}_0),\n\\label{e:nD_RTE_upperBC_collimated}\n\\end{equation}\nfor $\\Omega_z > 0$.  To reconcile (\\ref{e:nD_RTE_SourceTerm}) with the above BC, we notice that\n\\begin{equation}\nI_0(z,\\mathbf{\\Omega}) = F_0 \\exp(-\\sigma z\/\\mu_0) \\delta(\\mathbf{\\Omega}-\\mathbf{\\Omega}_0)\n\\label{e:uncollided}\n\\end{equation}\nis the solution of the ODE in (\\ref{e:nD_RTE_uniform}) when the r.-h. side vanishes identically (no internal sources, nor scattering), and we use (\\ref{e:nD_RTE_upperBC_collimated}) as the initial condition.  This uncollided radiance becomes the source of diffuse radiation immediately after scattering, hence its role in (\\ref{e:nD_RTE_SourceTerm}).\n\nIn (\\ref{e:uncollided}), $s = z\/\\mu_0$ is simply the oblique path covered by the radiation in the medium from its source at $z = s = 0$ to the location where it is detected, or scattered, or absorbed, or even escapes the medium ($s \\ge H\/\\mu_0$).  From the well-known properties of the exponential probability distribution, this makes the mean free path (MFP) $\\ell$ between emission, scattering or absorption events equal to the the e-folding distance $1\/\\sigma$.\n\nQuantities of particular interest in many applications, including atmospheric remote sensing, are radiances at the boundaries that describe out-going radiation: $I(0,\\mathbf{\\Omega})$ with $\\Omega_z \\le 0$; $I(H,\\mathbf{\\Omega})$ with $\\Omega_z \\ge 0$.  Normalized (outgoing, hemispherical) boundary fluxes,\n\\begin{eqnarray}\n\\label{e:reflectivity}\nR &=& \\frac{F_-(0)}{\\mu_0 F_0}, \\\\\n\\label{e:transmittivity}\nT &=& \\frac{F_+(H)}{\\mu_0 F_0},\n\\end{eqnarray}\nare also of interest, particularly, in radiation energy budget computations.  In (\\ref{e:reflectivity})--(\\ref{e:transmittivity}), the denominator is in fact $F_+(0)$ from (\\ref{e:hemispherical_fluxes}).  Therefore, for the diffuse illumination pattern in (\\ref{e:nD_RTE_upperBC_diffuse}), we only need to divide by $F_0$.\n\nFinally, a convenient non-dimensional representation of out-going radiances, at least at the upper boundary, uses the ``bidirectional reflection factor'' (BRF) form:\n\\begin{equation}\nI_\\text{BRF}(\\mathbf{\\Omega}) = \\frac{c_d I(0,\\mathbf{\\Omega})}{\\mu_0 F_0},\n\\label{e:BRF_form}\n\\end{equation}\nfor $\\mu < 0$.  This is the ``effective'' albedo $\\rho$ of the medium, i.e., as defined in (\\ref{e:nD_RTE_lowerBC_Lambertian})--(\\ref{e:rho_Lambertian}), but with $z=0$ rather than $z=H$, knowing $I(0,\\mathbf{\\Omega})$ and hence $F_+(0) = \\mu_0 F_0$.\nUnlike the optical property $\\rho$ in (\\ref{e:rho_Lambertian}) and the radiative response $R$ in (\\ref{e:reflectivity}), $I_\\text{BRF}(\\mathbf{\\Omega})$ is not restricted by energy conservation to the interval $[0,1]$.\n\n\\begin{table}[ht]\n\\caption{{\\bf Definitions for }$d = 1,2,3$}\n\\label{t:Definitions}\n\\begin{center}\n\\begin{tabular}{|l||c|c|c|}\n\\hline\n$d$ & 1 & 2 & 3 \\\\\n\\hline\\hline\n$\\vect{x}$ & $z$ & $(x,z)^\\text{T}$ & $(x,y,z)^\\text{T}$ \\\\\n\\hline\n$\\mathrm{d}\\vect{x}$ & $\\mathrm{d} z$ & $\\mathrm{d} x\\mathrm{d} z$ & $\\mathrm{d} x\\mathrm{d} y\\mathrm{d} z$ \\\\\n\\hline\n$\\mathbf{\\Omega}$   & $\\pm 1$ & $(\\sin\\theta,\\cos\\theta)^\\text{T}$ & $(\\sin\\theta\\cos\\phi,\\sin\\theta\\sin\\phi,\\cos\\theta)^\\text{T}$ \\\\\n\\hline\n$\\mathrm{d}\\mathbf{\\Omega}$ & n\/a$^\\dag$ & $\\mathrm{d}\\theta$ & $\\mathrm{d}\\cos\\theta\\mathrm{d}\\phi$ \\\\\n\\hline\n$c_d$ in (\\ref{e:nD_RTE_lowerBC_Lambertian}), (\\ref{e:nD_RTE_upperBC_diffuse}) & 1 & 2 & $\\pi$ \\\\\n\\hline\n[$F_0$] in (\\ref{e:nD_RTE_SourceTerm}), (\\ref{e:nD_RTE_upperBC_diffuse})--(\\ref{e:BRF_form}) & W & W\/m & W\/m$^2$ \\\\\n\\hline\n[$I$] & W & W\/m\/rad & W\/m$^2$\/sr \\\\\n\\hline\n[$S$] = [$q$] & W\/m & W\/m$^2$\/rad & W\/m$^3$\/sr \\\\\n\\hline\n$p_{d,\\text{iso}} = p_0(\\mu_\\text{s})$ & 1\/2 [-] & 1\/2$\\pi$ [rad$^{-1}$] & 1\/4$\\pi$ [sr$^{-1}$] \\\\\n\\hline\n$p_g(\\mu_\\text{s})$ & $\\frac{1+g\\mu_\\text{s}}{2}$ & $\\left(\\frac{1}{2\\pi}\\right)\\frac{1-g^2}{1+g^2-2g\\mu_\\text{s}}$ & $\\left(\\frac{1}{4\\pi}\\right)\\frac{1-g^2}{(1+g^2-2g\\mu_\\text{s})^{3\/2}}$ \\cite{HG_41}\\\\\n\\hline\n$\\chi_d$ in (\\ref{e:dD_diffusion_BCs}) & 1 & $\\pi$\/4 & 2\/3 \\\\\n\\hline\n\\end{tabular} \\\\\n\\end{center}\n{\\footnotesize $\\null^\\dag$In $d=1$, angular integrals become sums over the up ($\\mu = -1$) and down ($\\mu = +1$) directions, or only downward in (\\ref{e:nD_RTE_lowerBC_Lambertian}).} \\\\\n{\\footnotesize N.B. In all cases, we use $\\mu_\\text{s} = \\cos\\theta_\\text{s}$ to denote $\\mathbf{\\Omega}\\cdot\\mathbf{\\Omega}^\\prime$, the scalar product of the ``before'' and ``after'' scattering direction vectors.}\n\\end{table}\n\nActually, in the familiar $d = 3$ dimensions, all of the above is known as ``1D'' RT theory since only the spatial dimensions with any form of variability count.  If $\\sigma$, $\\sigma_\\text{s}$ and $p(\\cdot)$ depend on $z$, it is still 1D RT.  One can even remove from further consideration the former quantity by adopting the standard change of variables, $z \\mapsto \\tau = \\int_0^z \\sigma(z^\\prime) \\mathrm{d} z^\\prime$.  In this case, $z \\mapsto \\tau = \\sigma z$ (depth in units of MFP $\\ell = 1\/\\sigma$), then (\\ref{e:nD_RTE_uniform})--(\\ref{e:nD_RTE_SourceFunction}) become\n\\begin{equation}\n\\left[ \\mu\\frac{\\mathrm{d}\\;}{\\mathrm{d}\\tau} + 1 \\right]\nI(\\tau,\\mathbf{\\Omega}) = \\omega \\int\\limits_{\\Xi_d}\np(\\mathbf{\\Omega}^\\prime\\cdot\\mathbf{\\Omega})I(\\tau,\\mathbf{\\Omega}^\\prime)\\mathrm{d}\\mathbf{\\Omega}^\\prime + q(\\tau,\\mathbf{\\Omega}),\n\\label{e:1D_RTE}\n\\end{equation}\nwhere $\\mu$ denotes $\\Omega_z$ ($=\\cos\\theta$ if $d>1$) and $\\omega = \\sigma_\\text{s}\/\\sigma$ is the single scattering albedo (SSA).  We have assumed that $\\omega$ and $p(\\cdot)$ are independent of $z$, hence of $\\tau$, for simplicity as well as consistency with the notion of a homogenized optical medium.\n\nAnother important non-dimensional property is the total optical thickness of the medium M$_3(H)$, namely, $\\tau^\\star = \\sigma H = H\/\\ell$.  BCs for (\\ref{e:1D_RTE}) are expressed as in (\\ref{e:nD_RTE_lowerBC_Lambertian})--(\\ref{e:nD_RTE_upperBC_collimated}) but at $\\tau = 0,\\tau^\\star$.\n\nFinally, we adopt the Henyey--Greenstein (H--G) PF model $p_g(\\mu_\\text{s})$ expressed in the penultimate row of Table~\\ref{t:Definitions}.  Its sole parameter is the asymmetry factor $g = \\int_{\\Xi_d}\\mathbf{\\Omega}^\\prime\\cdot\\mathbf{\\Omega} p(\\mathbf{\\Omega}^\\prime\\cdot\\mathbf{\\Omega})\\mathrm{d}\\mathbf{\\Omega}$.  The whole 1D RT problem is then determined entirely by the choice of four quantities, $\\{\\omega,g;\\tau^\\star;\\rho\\}$, plus $\\mu_0$ if $d > 1$.\n\n\\subsection{Integral Forms of the $d$-Dimensional RTE}\n\nHenceforth, we take $q(\\tau,\\mathbf{\\Omega}) \\equiv 0$ in (\\ref{e:nD_RTE_uniform}) and, consequently, $I(\\tau,\\mathbf{\\Omega})$ is total (uncollided and scattered) radiation and the upper BC is (\\ref{e:nD_RTE_upperBC_collimated}).  We will also assume in the remainder that $\\rho = 0$ in the lower BC, cf. (\\ref{e:nD_RTE_lowerBC_Lambertian})--(\\ref{e:rho_Lambertian}), which then becomes simply $I(\\tau^\\star,\\mathbf{\\Omega}) = 0$ for $\\mu < 0$.  These assumptions are not essential to our goal of generalizing RT theory to account for spatial heterogeneity with long-range correlations, but they do simplify many of the following expressions that are key to the discussion.\n\nNow suppose that we somehow know $S(\\tau,\\mathbf{\\Omega})$ in (\\ref{e:nD_RTE_uniform}), with $q(\\tau,\\mathbf{\\Omega}) \\equiv 0$.  It is then straightforward to compute $I(\\tau,\\mathbf{\\Omega})$ everywhere.  We simply use upwind integration or ``sweep:''\n\\begin{equation}\nI(\\tau,\\mathbf{\\Omega}) =\n\\left\\{ \\begin{array}{ll}\n\\int\\limits_0^\\tau S(\\tau^\\prime,\\mathbf{\\Omega})\\mathrm{e}^{-(\\tau-\\tau^\\prime)\/\\mu}\\frac{\\mathrm{d}\\tau^\\prime}{\\mu}\n+ I(0,\\mathbf{\\Omega})\\mathrm{e}^{-\\tau\/\\mu},\n& \\text{if }\\mu > 0, \\\\\n\\int\\limits_\\tau^{\\tau^\\star}\nS(\\tau^\\prime,\\mathbf{\\Omega})\\mathrm{e}^{-(\\tau^\\prime-\\tau)\/|\\mu|}\\frac{\\mathrm{d}\\tau^\\prime}{|\\mu|}\n+ I(\\tau^\\star,\\mathbf{\\Omega})\\mathrm{e}^{-(\\tau^\\star-\\tau)\/|\\mu|},\n& \\text{otherwise,}\n\\end{array} \\right.\n\\label{e:S_sweep}\n\\end{equation}\nwhere the boundary contributions are specified by the BCs.  When these BCs express an incoming collimated beam at $\\tau = 0$, cf. (\\ref{e:nD_RTE_upperBC_collimated}), and an absorbing surface at $\\tau = \\tau^\\star$, cf. (\\ref{e:nD_RTE_lowerBC_Lambertian})--(\\ref{e:rho_Lambertian}) with $\\rho = 0$, this simplifies to\n\\begin{equation}\nI(\\tau,\\mathbf{\\Omega}) =\n\\left\\{ \\begin{array}{ll}\n\\int\\limits_0^\\tau S(\\tau^\\prime,\\mathbf{\\Omega})\\mathrm{e}^{-(\\tau-\\tau^\\prime)\/\\mu}\\frac{\\mathrm{d}\\tau^\\prime}{\\mu}\n+ I_0(\\tau,\\mathbf{\\Omega}),\n& \\text{if }\\mu > 0, \\\\\n\\int\\limits_\\tau^{\\tau^\\star}\nS(\\tau^\\prime,\\mathbf{\\Omega})\\mathrm{e}^{-(\\tau^\\prime-\\tau)\/|\\mu|}\\frac{\\mathrm{d}\\tau^\\prime}{|\\mu|},\n& \\text{otherwise,}\n\\end{array} \\right.\n\\label{e:simpleS_sweep}\n\\end{equation}\nwhere $I_0(\\tau,\\mathbf{\\Omega})$ is uncollided radiance from (\\ref{e:uncollided}) with $z = \\tau\/\\sigma$.\n\nWith this formal solution of the integro-differential RTE in hand, we can substitute the definition of $S(\\tau,\\mathbf{\\Omega})$ in terms of $I(\\tau,\\mathbf{\\Omega})$ expressed in (\\ref{e:nD_RTE_SourceFunction}), and obtain an integral form of the RTE:\n\\begin{equation}\nI(\\tau,\\mathbf{\\Omega}) = \\int\\limits_{\\Xi_d}\\int\\limits_0^{\\tau^\\star} \\mathcal{K}(\\tau,\\mathbf{\\Omega};\\tau^\\prime,\\mathbf{\\Omega}^\\prime) I(\\tau^\\prime,\\mathbf{\\Omega}^\\prime) \\mathrm{d}\\tau^\\prime\\mathrm{d}\\mathbf{\\Omega}^\\prime\n+ Q_I(\\tau,\\mathbf{\\Omega}),\n\\label{e:nD_integral_RTE}\n\\end{equation}\nwhere\n\\begin{equation}\nQ_I(\\tau,\\mathbf{\\Omega}) = \\exp(-\\tau\/\\mu_0) \\delta(\\mathbf{\\Omega}-\\mathbf{\\Omega}_0).\n\\label{e:Q_integral_RTE}\n\\end{equation}\nThis is simply the uncollided radiance field $I_0(\\tau,\\mathbf{\\Omega})$ from (\\ref{e:simpleS_sweep}) and (\\ref{e:uncollided}) where, without loss of generality, we henceforth take $F_0 = 1$.  The kernel of the integral RTE is given by\n\\begin{equation}\n\\mathcal{K}(\\tau,\\mathbf{\\Omega};\\tau^\\prime,\\mathbf{\\Omega}^\\prime) =\n\\omega p_g(\\mathbf{\\Omega}\\cdot\\mathbf{\\Omega}^\\prime)\n\\Theta\\left(\\frac{\\tau-\\tau^\\prime}{\\mu}\\right)\n\\frac{\\exp(-|\\tau-\\tau^\\prime|\/|\\mu|)}{|\\mu|},\n\\label{e:K_integral_RTE}\n\\end{equation}\nwhere $\\Theta(x)$ is the Heaviside step function ($=1$ if $x \\ge 0$, $=0$ otherwise).  It enforces the causal requirement of doing upwind sweeps.\n\nConversely, one can substitute (\\ref{e:simpleS_sweep}) into (\\ref{e:nD_RTE_SourceFunction}), with the adopted change of spatial coordinate ($z\\mapsto\\tau$) leading to $\\sigma_\\text{s}\\mapsto\\omega$.  That yields the so-called ``ancillary'' integral RTE:\n\\begin{equation}\nS(\\tau,\\mathbf{\\Omega}) = \\int\\limits_{\\Xi_d}\\int\\limits_0^{\\tau^\\star} \\mathcal{K}(\\tau,\\mathbf{\\Omega};\\tau^\\prime,\\mathbf{\\Omega}^\\prime) S(\\tau^\\prime,\\mathbf{\\Omega}^\\prime) \\mathrm{d}\\tau^\\prime\\mathrm{d}\\mathbf{\\Omega}^\\prime\n+ Q_S(\\tau,\\mathbf{\\Omega}),\n\\label{e:nD_ancillary_IRTE}\n\\end{equation}\nwhere\n\\begin{equation}\nQ_S(\\tau,\\mathbf{\\Omega}) = \\omega p_g(\\mathbf{\\Omega}\\cdot\\mathbf{\\Omega}_0) \\exp(-\\tau\/\\mu_0).\n\\label{e:Q_ancillary_IRTE}\n\\end{equation}\nThe kernel is the same as given in (\\ref{e:K_integral_RTE}).  However, if there were spatial variations in the optical properties, SSA $\\omega$ and\/or PF $p(\\cdot)$, then the kernels would differ in that (\\ref{e:Q_integral_RTE}) would use the starting point and (\\ref{e:nD_ancillary_IRTE}) the end point of the transition; see, e.g., \\cite{DavisKnyazikhin2005}.\n\nIf (\\ref{e:nD_integral_RTE}) is written in operator language as $I = K I + Q_I$, then it is easy to verify that the Neumann series is a constructive approach for the solution: $I = \\sum_{n=0}^\\infty I_n$, where $I_{n+1} = K I_n$, hence\n\\begin{equation}\nI = \\sum_{n=0}^\\infty K^n Q_I = (E-K)^{-1} Q_I,\n\\label{e:IRTE_soln}\n\\end{equation}\nwhere $E$ is the identity operator.  This applies equally to the estimation of $S$ as a solution of (\\ref{e:nD_ancillary_IRTE}).  Once $S(\\tau,\\mathbf{\\Omega})$ is a known quantity, one can obtain the readily observable quantity $I(\\tau,\\mathbf{\\Omega})$ using (\\ref{e:simpleS_sweep}).\n\n\\noindent\n{\\it Comment on Angular Reciprocity:}\n\n\\noindent\nNote that $\\mathcal{K}(\\tau,\\mathbf{\\Omega};\\tau^\\prime,\\mathbf{\\Omega}^\\prime)$ in (\\ref{e:K_integral_RTE}) is invariant when we replace $(\\tau,\\mathbf{\\Omega};\\tau^\\prime,\\mathbf{\\Omega}^\\prime)$ with $(\\tau^\\prime,-\\mathbf{\\Omega}^\\prime;\\tau,-\\mathbf{\\Omega})$, i.e., swap positions in the medium and switch the direction of propagation.  This leads to reciprocity of the radiance fields for plane-parallel slab media under the exchange of sources and detectors \\cite{Chandra1950}.  In our case, we consider radiance escaping the medium in reflection ($\\tau = 0$) or transmission ($\\tau = \\tau^\\star$) since the source is external.  Focusing on reflected radiance in BRF form (\\ref{e:BRF_form}), reciprocity reads as\n\\begin{equation}\nI_\\text{BRF}(0,\\mathbf{\\Omega};\\mathbf{\\Omega}_0) = I_\\text{BRF}(0,-\\mathbf{\\Omega}_0;-\\mathbf{\\Omega}),\n\\label{e:R_reciprocity}\n\\end{equation}\nwhere the second angular argument reads as a parameter (from upper BC) rather than an independent variable.\nSimilarly, we have $I_\\text{BRF}(\\tau^\\star,\\mathbf{\\Omega};\\mathbf{\\Omega}_0) = I_\\text{BRF}(\\tau^\\star,-\\mathbf{\\Omega}_0;-\\mathbf{\\Omega})$ in transmittance, using the same BRF-type normalization.\n\nWe can verify transmissive reciprocity explicitly on $I_0 = Q_I$ in (\\ref{e:Q_integral_RTE}) for uncollided radiance. Reflective reciprocity can be verified less trivially using singly-scattered radiance $I_1 = K I_0 = K Q_I$.  Based on (\\ref{e:Q_integral_RTE})--(\\ref{e:K_integral_RTE}), this leads to\n\\begin{equation}\nI_1(0,\\mathbf{\\Omega};\\mathbf{\\Omega}_0) =\n\\omega p_g(\\mathbf{\\Omega}\\cdot\\mathbf{\\Omega}_0)\n\\int\\limits_0^{\\tau^\\star} \\exp(-\\tau^\\prime\/\\mu_0)\n\\exp(-\\tau^\\prime\/|\\mu|) \\mathrm{d}\\tau^\\prime\/|\\mu|.\n\\label{e:1_scatter_sweep}\n\\end{equation}\nFrom there, (\\ref{e:BRF_form}) yields\n\\begin{equation}\n\\frac{c_d}{\\mu_0} I_1(0,\\mathbf{\\Omega};\\mathbf{\\Omega}_0) = c_d\n\\frac{\\omega p_g(\\mathbf{\\Omega}\\cdot\\mathbf{\\Omega}_0)}{\\mu_0+|\\mu|}\n\\left( 1-\\exp\\left[-\\tau^\\star\\left(\\frac{1}{\\mu_0}+\\frac{1}{|\\mu|}\\right)\\right] \\right),\n\\label{e:1_scatter_R}\n\\end{equation}\nwith $\\mu_0 > 0$ and $\\mu < 0$.  Noting that $-\\mu > 0$ and $-\\mu_0 < 0$, (\\ref{e:1_scatter_R}) verifies (\\ref{e:R_reciprocity}).  The same can be shown for transmitted radiance.\n\n\n\\section{Generalized Radiative Transport in $d$ Spatial Dimensions}\n\\label{s:dDim_gRTE}\n\n\\subsection{Emergence of Non-Exponential Transmission Laws in the Cloudy Atmosphere}\n\\label{s:nonExpTran}\n\n\\subsubsection{Two-Point Correlations in Clouds According to In-Situ Probes}\n\nWe refer to Davis and Marshak \\cite{DavisMarshak04} and Davis \\cite{Davis2006} for a detailed accounts of the optical variability we expect---and indeed observe \\cite[and references therein]{Davis_etal99}---in the Earth's turbulent cloudy atmosphere.  See also Kostinski \\cite{Kostinski2001} for an interestingly different approach.\n\nThe important---almost defining---characteristic of this variability is that it prevails over a broad range of scales, which translates statistically into auto-correlation properties with long ``memories.''  The traditional metric for 2-point correlations in turbulent media is the $q^\\text{th}$-order structure function \\cite{MoninYaglom1975}\n\\begin{equation}\n\\text{SF}_q(r) = \\overline{|f(\\vect{x}+\\vect{r})-f(\\vect{x})|^q},\n\\label{e:qSF_def}\n\\end{equation}\nwhere $f(\\vect{x})$ is a spatial variable of interest, $\\vect{r}$ is a spatial increment of magnitude $r$, and the overscore denotes spatial or ensemble averaging.  Structure functions are the appropriate quantities to use for fields that are non-stationary but have stationary increments.\\footnote{\nFollowing many others, we borrow here the terminology of time-series analysis since the proper language of statistical ``homogeneity'' might be confused with structural homogeneity, a usage we've already introduced in the above.} \nStationarity of the increments in $f(\\vect{x})$ means that the ensemble average on the right-hand side of (\\ref{e:qSF_def}) depends only on $\\vect{r}$.  Further assuming statistical isotropy, for simplicity, it will depend only on $r$.  The norm of wavelet coefficients have become popular alternatives to the absolute increment in $f(\\vect{x})$ used in (\\ref{e:qSF_def}) \\cite{Farge92,MuzyEtAl1994}.\n\nAs expected for all turbulent phenomena, in-situ observations in clouds invariably show that \\cite{Davis_etal94,Davis_etal96,Marshak_etal97,Davis_etal99}\n\\begin{equation}\n\\overline{|f(\\vect{x}+\\vect{r})-f(\\vect{x})|^q} \\sim r^{\\zeta_q},\n\\label{e:qSF_scaling}\n\\end{equation}\nfor $r$ ranging from meters to kilometers, where $\\zeta_q$ is generally a ``multiscaling'' or ``multifractal'' property, meaning that $\\zeta_q\/q$ is not a constant.  Physically, this means that knowledge of one statistical moment, such as variance SF$_2(r)$, of the absolute increments cannot be used to predict all others based on dimensional analysis.  Otherwise, it is deemed ``monoscaling'' or ``monofractal.''\n\nIt has long been known theoretically---and well-verified empirically---that $\\zeta_2 = 2\/3$ when $f$ is a component of the wind \\cite{Kolmogorov41}, temperature or a passive scalar density \\cite{Obukhov1949,Corrsin1951}, when the turbulence is statistically homogeneous and isotropic.  This is equivalent \\cite{MoninYaglom1975} to stating that energy spectra of these various quantities in turbulence are power-law with an exponent $\\beta = -5\/3$ in (\\ref{e:PowerLaw_EnergySpectum}).  It can also be shown theoretically that $\\zeta_q$ is necessarily a convex function, a prediction that has also been amply verified empirically, although in practice the convexity is relatively weak.\n\nAt scales smaller than meters, cloud liquid water content (LWC) undergoes, according to reliable in situ measurements in marine stratocumulus, an interesting transition toward higher levels of variability than expected from the scaling in (\\ref{e:qSF_scaling}) \\cite{Davis_etal99}.  Specifically, sharp quasi-discontinuities associated with positively-skewed deviations occur at random points\/times in the transect through the LWC field sampled by airborne instruments.  These jumps are believed to be a manifestation of the random entrainment of non-cloudy air into the cloud \\cite{GerberEtAl01}.\n\nAt sufficiently large scales, $\\overline{|f(\\vect{x}+\\vect{r})-f(\\vect{x})|^q}$ ceases to increase with $r$ as $f(\\vect{x})$ become independent (decorrelates) from itself at very large distances.  For non-negative properties, such as the extinction coefficient or particle density, this decoupling has to happen at least at the scale $r$ where the absolute increments (fluctuations) become commensurate in magnitude with the positive mean of the property.  This rationalizes the upper limit of the scaling range for cloud LWC or droplet density at the scale of several kilometers.  In the cloudy atmosphere, decorrelation can happen sooner in the vertical than in the horizontal in cases of strong stratification, i.e., stratus and strato-cumulus scenarios versus broken cumulus generated by vigorous convection.\n\nIn atmospheric RT applications to be discussed next, the outer limit of $r$ only needs to be on the order of whatever scale it takes to reach significant optical distances.  That can be less than cloud thickness in stratus\/strato-cumulus cases, or can stretch to the whole troposphere (cloudy part of the atmosphere) when convection makes the dynamics more 3D than 2D.\n\n\\subsubsection{Statistical Ramifications for Cloud Optical Properties}\n\nOur present goal is to quantify the impact of unresolved random spatial fluctuations of $\\sigma(\\vect{x})$ on macroscopic transport properties such as large-scale boundary fluxes or remotely observable radiances, spatially averaged in the instrument's field-of-view.  In view of the importance of sweep operations in $d$-dimensional RT, we also need to understand the statistics of integrals of $\\sigma(\\vect{x})$ over a range of distances $s$ in an arbitrary direction $\\mathbf{\\Omega}$.  This is the optical path along a straight line between points $\\vect{x}$ and $\\vect{x}+s\\mathbf{\\Omega}$:\n\\begin{equation}\n\\tau(\\vect{x},\\vect{x}+s\\mathbf{\\Omega}) = \\int\\limits_0^s \\sigma(\\vect{x}+s\\mathbf{\\Omega})\\mathrm{d} s.\n\\label{e:optical_distance}\n\\end{equation}\nBetter still, we need to characterize statistically the direct transmission factor $\\exp[-\\tau(\\vect{x},\\vect{x}+s\\mathbf{\\Omega})]$ that is used systematically in the upwind sweep operation.  Assuming stationarity and isotropy, we define\n\\begin{equation}\n\\overline{T}(s) = \\overline{\\exp[-\\tau(\\vect{x},\\vect{x}+s\\mathbf{\\Omega})]} = \\overline{\\exp[-\\sigma_\\text{avr}(\\vect{x},\\mathbf{\\Omega};s)s]},\n\\label{e:mean_Tdir}\n\\end{equation}\nwhere $\\sigma_\\text{avr}(\\vect{x},\\mathbf{\\Omega};s)$ is the average extinction encountered by radiation propagating uncollided between $\\vect{x}$ and $\\vect{x}+s\\mathbf{\\Omega}$:\n\\begin{equation}\n\\sigma_\\text{avr}(\\vect{x},\\mathbf{\\Omega};s) = \\frac{1}{s}\\int\\limits_0^s\\sigma(\\vect{x}+s^\\prime\\mathbf{\\Omega})\\mathrm{d} s^\\prime\n\\label{e:1ptScaleInv}\n\\end{equation}\nThis is essentially a coarse version of the random field $\\sigma(\\vect{x})$, smoothed over a given scale $s$.  What behavior do we expect it to have?\n\nBeing at the core a material density (times a collision cross-section), increments of $\\sigma(\\vect{x})$ will follow (\\ref{e:qSF_scaling}) when $s$ is varied, but the segment-mean $\\sigma_\\text{avr}(\\vect{x},\\mathbf{\\Omega};s)$ will not depend much on the scale $s$.  Indeed, comparing values of $\\sigma_\\text{avr}(\\vect{x},\\mathbf{\\Omega};s)$ for different values of $s$ is really just saying, with linear transects, that the notion of a material density can be defined.  That simple proposition is usually stated as a volumetric statement: the amount of material in a volume $s^d$ is proportional to that volume, and the proportionality factor is called ``density.''\n\nEven more fundamentally, $s$-independence of (\\ref{e:1ptScaleInv}), at least in the $s \\to 0$ limit,\\footnote{\nThis limit is to be understood physically as to the scale where noise-like fluctuations occur, which is at least the inter-particle distance in a cloud but could be larger \\cite{Davis_etal99,GerberEtAl01}.}\nis tantamount to saying that $\\sigma(\\vect{x}) = \\sigma_\\text{avr}(\\vect{x},\\mathbf{\\Omega};0)$ is indeed a ``function,'' i.e., the symbol ``$\\sigma(\\vect{x})$'' represents a number.  This is a natural consequence of increments that vanish at least on average with $s$, a property known as ``stochastic continuity,'' which incidentally does not exclude a countable number of discontinuities (e.g., sharp cloud edges in the atmosphere).\nAll of these ramifications come with the above-mentioned long-range correlations described by (\\ref{e:qSF_scaling}) for finite positive values of $\\zeta_q$.\n\nA counter-example of such intuitive $s$-independent behavior for averages is the spatial equivalent of white noise.  Indeed, if $\\sigma(\\vect{x})$ on the right-hand side of (\\ref{e:1ptScaleInv}) could somehow represent white noise (in the discrete world, just a sequence of uncorrelated random numbers), then the left-hand side is just an estimate of its mean value based on as many samples as there are between 0 and $s$.  As $s\\to\\infty$, this estimate is known to converge to the mean (law of large numbers) in $1\/\\sqrt{s}$ and, moreover, the PDF of $\\sigma_\\text{avr}(\\vect{x},\\mathbf{\\Omega};s)$ is a Gaussian with variance $\\propto s$ (central limit theorem).  This is a reminder that one should not denote white (or other) noises as a function $f(\\vect{x})$ but rather as a distribution that exists only under integrals: $f(\\vect{x})\\mathrm{d}\\vect{x}$ is better, $f(\\vect{x},\\mathrm{d}\\vect{x})$ is the best.  That remark is key to Davis and Mineev-Weinstein's \\cite{DavisMineev11} generalization of RT to rapidly fluctuating extinction fields, possibly with anti-correlations across scales.\n\n\\subsubsection{Representation of Spatial Fluctuations with Gamma PDFs}\n\nTurbulent density fields are often found to have log-normal distributions and cloud LWC is no exception.  However, Gamma distributions can be used to approximate log-normals in terms of skewness, and are much easier to manipulate.  Barker et al. \\cite{Barker1996b} showed that Gamma distributions with a broad range of parameters can fit histograms of cloud optical depth reasonably well.  Now, letting $\\vect{x} = (\\vec{x},z)^\\text{T}$, cloud optical depth is simply $\\sigma(\\vec{x},z)$ averaged spatially over $z$ from 0 to the fixed cloud (layer) thickness $H$, then re-multiplied by $H$, and histograms were cumulated over $\\vec{x}$; in this case, many large cloudy images where $\\vec{x}$ designates a 30-m LandSat pixel.  Barker \\cite{Barker1996a} proceeded to apply this empirical finding to evaluate unresolved variability effects using the standard two-stream approximation for scattering media, as defined further on, in (\\ref{e:2st_model}), for a representative case.  We have used it previously for uncollided radiation \\cite{Davis2006}, and do the same here.\n\nIn summary, assume that the random number $\\sigma_\\text{avr}(\\vect{x},\\mathbf{\\Omega};s)$ in (\\ref{e:mean_Tdir})--(\\ref{e:1ptScaleInv}) is indeed statistically independent of $s$, at least over a range of values that matter for RT.  This range of $s$ should encompass small, medium and large propagation distances between emission, scattering, absorption or escape events.  These events can unfold in the whole medium, or else in portions of it we might wish to think of in isolation, e.g., clouds in the atmosphere.  If $\\overline{\\sigma}$ is the spatially-averaged extinction coefficient, over some or all of the transport space $(\\vect{x},\\mathbf{\\Omega})$, then we require the range of statistical independence on $s$ to go from vanishingly small to several times $1\/\\overline{\\sigma}$.   Were the medium homogeneous, this last quantity would be the particle's MFP, but is in general an underestimate of the true MFP \\cite{DavisMarshak04}, as illustrated further on in the specific case of interest here.\n\nFollowing Barker et al. \\cite{Barker1996b}, we now assume that the variability of $\\sigma_\\text{avr}(\\vect{x},\\mathbf{\\Omega};s)$, for fixed $s$, can be approximated with an ensemble of Gamma-distributed values:\n\\begin{equation}\n\\Pr\\{\\sigma,\\mathrm{d}\\sigma\\} = \\frac{(a\/\\overline{\\sigma})^a}{\\Gamma(a)} \\, \\sigma^{a-1}\\exp(-a\\sigma\/\\overline{\\sigma}) \\, \\mathrm{d}\\sigma,\n\\label{e:Gamma_pdf}\n\\end{equation}\nwhere $\\overline{\\sigma}$ is the mean and $a$ is the variability parameter\n\\begin{equation*}\na = \\frac{1}{\\overline{\\sigma^2}\/\\overline{\\sigma}^2-1},\n\\end{equation*}\nan important quantity that varies from 0$^+$ to $\\infty$ since $\\overline{\\sigma^2} \\ge \\overline{\\sigma}^2$ (Schwartz's inequality).  If $\\sigma_\\text{avr}(\\vect{x},\\mathbf{\\Omega};s)$ is $\\Gamma$-distributed for fixed $s$, then so is their product $\\tau(\\vect{x},\\vect{x}+s\\mathbf{\\Omega})$ in (\\ref{e:optical_distance}).\n\nEquation (\\ref{e:mean_Tdir}) then reads as the Laplace characteristic function of this Gamma PDF supported by the positive real axis:\n\\begin{equation}\nT_a(s) = \\int\\limits_0^\\infty\\exp(-\\sigma s)\\Pr\\{\\sigma,\\mathrm{d}\\sigma\\} = \\frac{1}{(1+\\overline{\\sigma}s\/a)^a}.\n\\label{e:Gamma_Tdir}\n\\end{equation}\nIn the limit $a\\to\\infty$, variance $\\overline{\\sigma^2}-\\overline{\\sigma}^2$ vanishes as the PDF in (\\ref{e:Gamma_pdf}) becomes degenerate, i.e., $\\delta(\\sigma-\\overline{\\sigma})$.  We then retrieve Beer's law: $T_\\infty(s) = \\exp(-\\overline{\\sigma}s)$.\nFor an explicit model-based derivation of (\\ref{e:Gamma_Tdir}) where the exponent $a$ is expressed in terms of the statistical parameters, see Davis and Mineev-Weinstein's \\cite{DavisMineev11} study of scale-invariant media in the ``red-noise'' limit ($\\beta \\to 1$) where the correlations are at their longest.\n\nThe direct transmission law in (\\ref{e:Gamma_Tdir}) with a power-law tail thus generalizes the standard law of exponential decay for the cumulative probability of radiation to reach a distance $s$ (or \\emph{mean} optical distance $\\tau(s)$) from a source without suffering a collision in the material.  Figure~\\ref{f:new_Tdir} illustrates both the positively skewed PDFs for $\\sigma$, at fixed $s$, in (\\ref{e:Gamma_pdf}) and the generalized transmission laws in (\\ref{e:Gamma_Tdir}) for selected values of $a$ that we will use further on in numerical experiments.  In the middle panel, we can see that direct transmission probability at $\\tau = 1$ increases from $1\/\\mathrm{e} = 0.368\\cdots$ to almost 1\/2 going from $a = \\infty$ (Beer's exponential law) to a power-law with $a = 1.2$.  The rightmost panel shows that there is still appreciable probability of direct transmission when $a < \\infty$ at large optical distances where radiation is all but extinguished in the standard $a = \\infty$ case.\n\n\\begin{figure}[ht]\n\\begin{center}\n\\includegraphics[width=5.5in]{new_Tdir.pdf}\n\\end{center}\n\\caption[new_Tdir]\n{\\label{f:new_Tdir}\nLeft to right: Gamma PDFs in (\\ref{e:Gamma_pdf}) for the spatial variability of $\\sigma$ at a fixed scale $s$, normalized to its ensemble mean $\\overline{\\sigma}$, for indicated values of $a$; generalized transmission laws $T_a(\\tau)$ in (\\ref{e:Gamma_Tdir}) associated with PDFs with $\\tau = \\overline{\\sigma}s$; same as middle panel but in semi-log axes and a broader range of $\\tau$.}\n\\end{figure}\n\nNow, viewing $s$ as a random variable that is crucial to transport theory, we have\n\\begin{equation}\nT_a(s) = \\Pr\\{\\text{step}>s\\} = \\int\\limits_s^\\infty p_a(s)\\mathrm{d} s.\n\\label{e:Tdir_asPDF}\n\\end{equation}\nThe PDF for a random step of length $s$ is therefore\n\\begin{equation}\np_a(s) = \\left|\\frac{\\mathrm{d} T_a}{\\mathrm{d} s}\\right|(s) = -\\,\\frac{\\mathrm{d} T_a}{\\mathrm{d} s}(s) = \\frac{\\overline{\\sigma}}{(1+\\overline{\\sigma}s\/a)^{a+1}}.\n\\label{e:Gamma_stepPDF}\n\\end{equation}\n\nIn the case of particle transport, we know that the MFP for the $a = \\infty$ case (uniform optical media) is $\\ell_\\infty = 1\/\\overline{\\sigma}$.  What is it for finite $a$ (variable optical media)?  One finds\n\\begin{equation*}\n\\ell_a = \\langle s \\rangle_a = \\int\\limits_0^\\infty s \\, \\mathrm{d} T_a(s)\n        = \\int\\limits_0^\\infty s \\, p_a(s)\\mathrm{d} s\n        = \\frac{a}{a-1} \\, \\ell_\\infty,\n\\end{equation*}\nwhich is larger than $\\ell_\\infty$ and indeed diverges as $a \\to 1^+$.  Generally speaking, the step moment $\\langle s^q \\rangle_a$ is convergent only as long as $-1< q < a$.  This immediately opens interesting questions (addressed in depth elsewhere \\cite{DavisMarshak1997} and briefly discussed further on) about the diffusion limit of this variability model when $a \\le 2$, i.e., when the 2$^\\text{nd}$-order moment of the step distribution is divergent.\n\nFigure~\\ref{f:RWs_g0_inset} demonstrates how RT unfolds in $d=2$ inside boundless conservatively scattering media where $\\overline{\\sigma}$ is unitary.  The media are either uniform or stochastic but spatially-correlated in such a way the ensemble average transmission law is of the power-law form in (\\ref{e:Gamma_Tdir}).  We consider media with exponential transmission ($a = \\infty$, uniform case) and power-law transmission laws with $a$ = 10, 5, 2, 1.5, and 1.2.  Scattering is assumed to isotropic and we follow the random walk of a transported particle for 100 scatterings.  For illustration, the same scattering angles are used in each of the 6 instances.  For the exponential case, the random free paths are generated using the standard rule: $s = -\\log\\xi\/\\overline{\\sigma}$, where $\\xi$ ia a uniform random variable on the interval (0,1).  For the power-law cases, we use\n\\begin{equation}\ns = a \\times (\\xi^{-1\/a}-1)\/\\overline{\\sigma}\n\\label{e:Gamma_step}\n\\end{equation}\nAs for the random scattering angles, we use the same sequence of 100 values of $2\\pi\\times\\xi$.\n\nIn the inset fi Fig.~\\ref{f:RWs_g0_inset}, we see that all the traces start at the same point in the same direction.  Physically, we can imagine an electron bound to a crystal surface hoping between holes associated with random defects \\cite{LuedtkeLandman1999}.  Certain heterogeneous predator-prey and scavenging problems can also lead to 2D transport processes with a mix of small and large jumps \\cite{BuldyrevEtAl2001}.  We can immediately appreciate how the MFP increases as $a \\to 1$.  At the same time, the increasing frequency of large jumps enables the cumulative traces to end further and further from their common origin as $a$ decreases from $\\infty$ to nearly unity.\n\n\\begin{figure}[ht]\n\\begin{center}\n\\includegraphics[width=5.5in]{RWs_g0_inset.pdf}\n\\end{center}\n\\caption[RWs_g0_inset]\n{\\label{f:RWs_g0_inset}\nSix traces of random walks in $d=2$ dimensions with 100 isotropic scatterings and step sequences that follow power-law cumulative probabilities (\\ref{e:Gamma_Tdir}) and PDFs (\\ref{e:Gamma_stepPDF}).  Both scatterings and steps use the same sequences of uniform random variables.  Values of $a$ are $\\infty$ (exponential law), 10, 5, 2, 1.5 and 1.2.  The two last ones are asymptotically self-similar L\\'evy-stable flights (steps with divergent variance), the three former are asymptotically self-similar Gaussian walks (steps with finite variance), and for $a=2$ it is a transition case (steps with log-divergent variance).  The inset is a $\\times 3$ zoom into the commun origin of the 6 traces.  More discussion in main text.}\n\\end{figure}\n\nOne can read Fig.~\\ref{f:RWs_g0_inset} with atmospheric optics in mind, albeit with each isotropic scattering representing $\\approx$$(1-g)^{-1}$ forward scatterings \\cite{Davis2006}.  Recalling that $g$ is in the range 0.75 to 0.85 in various types of clouds and aerosols, this translates to 4 to 7 scatterings before directional memory is lost.  For all values of $a$ there is a wide distribution of lengths of jumps between scatterings.  However, for large values of $a$, the distance covered by a cluster of small steps can equally well be covered with one larger jump.  This behavior is characteristic of solar radiation trapped in a single opaque cloud.  In contrast, for the smallest values of $a$, it is increasingly unlikely that a cluster of smaller steps can rival in scale a single large jump.  This behavior is typical of solar radiation that is alternatively trapped in clouds and bouncing between them.  In other words, we are looking at a 2D version of a typical trace of a multiply scattered beam of sunlight in a 3D field of broken clouds.\n\n\\subsection{$d$-Dimensional Generalized RTE in Integral Form}\n\nOur goal is now to formulate the transport equations that describe RT in media when, as in Fig.~\\ref{f:RWs_g0_inset}, we transition from an exponential direct transmission law to a power-law counterpart.\n\nOur starting point is the integral form of the $d$-dimensional plane-parallel RTE in (\\ref{e:nD_integral_RTE}); alternatively, (\\ref{e:nD_ancillary_IRTE}) paired with (\\ref{e:simpleS_sweep}).  These formulations are sufficiently general to describe RT and other linear transport processes.  It gets specific to the standard form of RT theory only when we look at the make up of the kernel $\\mathcal{K}$ in (\\ref{e:K_integral_RTE}), the source terms $Q_I$ in (\\ref{e:Q_integral_RTE}) and $Q_S$ in (\\ref{e:Q_ancillary_IRTE}).\n\nTherein, we find exponential functions that describe the propagation part of the transport.  Specifically, we identify\n\\begin{equation}\nT_\\infty(\\tau) = \\exp(-\\tau)\n\\label{e:T_Beer_cum}\n\\end{equation}\nin $Q_I$, assuming $\\mu_0 = 1$ for the present discussion.  The subscript $\\infty$ notation is consistent with our usage in (\\ref{e:Gamma_Tdir}) with $\\overline{\\sigma} = 1$, which is implicit in a non-dimensionalized 1D RT.  This is Beer's classic law of direct transmission, the hallmark of homogeneous optical media where $\\tau(s) = \\sigma s$ with, in the present setting, $\\sigma \\equiv \\overline{\\sigma}$ (i.e., a degenerate probability distribution for $\\sigma$).\n\nIn $Q_I$, $T_\\infty(\\tau)$ is therefore at work as the cumulative probability in (\\ref{e:Tdir_asPDF}).  In the kernel $\\mathcal{K}$, as well as in $Q_S$, we again find exponential functions, but here we interpret them as a PDF:\n\\begin{equation}\n\\left|\\frac{\\mathrm{d} T_\\infty}{\\mathrm{d} \\tau}\\right| = -T_\\infty(|\\tau-\\tau^\\prime|) = \\exp(-|\\tau-\\tau^\\prime|),\n\\label{e:T_Beer_pdf}\n\\end{equation}\nagain assuming $\\mu = 1$ for the present discussion.  The fact that the (\\ref{e:T_Beer_cum}) and (\\ref{e:T_Beer_pdf}) are identical functions is of course a defining property of the exponential.\n\nWhat makes us assign the ``cumulative probability'' interpretation of $\\exp(-\\tau)$ to its use in $Q_I$, and the ``PDF'' interpretation of $\\exp(-|\\tau-\\tau^\\prime|)$ to its use in $Q_S$ and $\\mathcal{K}$?  The clue is the foundational transport physics.  In $Q_I = I_0$, the uncollided radiation is simply detected at optical distance $\\tau$.  It could have gone deeper into the medium before suffering a scattering, an absorption, a reflection, or escaping through the lower boundary.  In $\\mathcal{K}$ however, it is used to obtain $I_n$ from $I_{n-1}$, as previously demonstrated for $n = 1$, cf. (\\ref{e:1_scatter_sweep}).  In this case, the radiation must be stopped between $\\tau$ and $\\tau+\\delta\\tau$.  It is a probability density that is invariably associated with the differential $\\mathrm{d}\\tau$.  Similarly in $Q_S$, the transport process is to stop the the propagation in a given layer and, moreover, it is specifically by a scattering event.\n\nIn order to account for unresolved random-but-correlated spatial variability of extinction $\\sigma(\\vect{x})$, we propose for the integral forms of the $d$-dimensional plane-parallel RTE the following \\emph{generalization}: use\n\\begin{equation}\n\\mathcal{K}(\\tau,\\mathbf{\\Omega};\\tau^\\prime,\\mathbf{\\Omega}^\\prime) =\n\\omega p_g(\\mathbf{\\Omega}\\cdot\\mathbf{\\Omega}^\\prime)\n\\Theta\\left(\\frac{\\tau-\\tau^\\prime}{\\mu}\\right)\n\\frac{|\\Dot{T}_a(|\\tau-\\tau^\\prime|\/|\\mu|)|}{|\\mu|},\n\\label{e:K_integral_gRTE}\n\\end{equation}\nrather than (\\ref{e:K_integral_RTE}), with\n\\begin{equation}\nQ_S(\\tau,\\mathbf{\\Omega}) = \\omega p_g(\\mathbf{\\Omega}\\cdot\\mathbf{\\Omega}_0) |\\Dot{T}_a(\\tau\/\\mu_0)|\n\\label{e:Q_ancillary_gIRTE}\n\\end{equation}\nfor (\\ref{e:nD_ancillary_IRTE}), and\n\\begin{equation}\nQ_I(\\tau,\\mathbf{\\Omega}) = T_a(\\tau\/\\mu_0) \\delta(\\mathbf{\\Omega}-\\mathbf{\\Omega}_0)\n\\label{e:Q_integral_gRTE}\n\\end{equation}\nfor (\\ref{e:nD_integral_RTE}), where $a$ can have any strictly positive value, including $\\infty$.  We use the overdot notation in (\\ref{e:K_integral_gRTE})--(\\ref{e:Q_ancillary_gIRTE}) to denote the derivative of a function of a single variable, which is the case here when $\\overline{\\sigma}$ is combined with $s$ to form $\\tau$ in (\\ref{e:Gamma_Tdir}) and (\\ref{e:Gamma_stepPDF}), and $a$ is viewed as a fixed parameter.\n\n\\subsection{Are There Integro-Differential Counterparts of Generalized Integral RTEs?}\n\\label{s:id_gRTEs}\n\nIn short, the $d$-dimensional stochastic transport model we propose is simply to replace $T_\\infty(\\tau)$ in (\\ref{e:T_Beer_cum}) with $T_a(\\tau)$ for finite $a$, which we equate with $T_a(s)$ in (\\ref{e:Gamma_Tdir}) when $\\overline{\\sigma}=1$ (thus $s=\\tau$).  This logically requires the use of $\\dot{T}_a(\\tau)$ obtained similarly from $\\mathrm{d} T_a(s)\/\\mathrm{d} s$ in (\\ref{e:Gamma_stepPDF}).  We thus have a well-defined transport problem using an integral formulation, to be solved analytically or numerically.  Now, is there an integro-differential counterpart?\n\nWe do not yet have an answer to this question.  One path forward to address it is to follow the steps of Larsen and Vasques \\cite{LarsenVasques11} who start with the classic RT\/linear Boltzmann equation in integro-differential form and transform it into a ``non-classical'' one by introducing a special kind of time-dependence that is essentially reset to epoch 0 at every scattering.  Non-exponential free path distributions are thus accommodated, and a modified diffusion limit is derived in cases where $\\langle s^2 \\rangle$ is greater than $2\\langle s \\rangle^2$, its value for the exponential distribution, but not too much larger.  Traine and co-authors \\cite{TaineEtAl2010} have also proposed a ``generalized'' RTE for large-scale transport through random (porous) media; this model uses an empirical counterpart of our parametric non-exponential transmission law in some parts of the computation, but retains the standard integro-differential form for the final estimation of radiance using the upwind sweep operator in (\\ref{e:S_sweep}).\n\nAnother path forward is to essentially define new differential (or more likely pseudo-differential) operators as those from which the new integral operator in (\\ref{e:K_integral_gRTE}) follows. This amounts to broadening the definition of the Green function,\n$G(\\tau,\\mathbf{\\Omega};\\tau_\\star,\\mathbf{\\Omega}_\\star) =\nT_a(|\\tau-\\tau_\\star|\/|\\mu|)\\delta(\\mathbf{\\Omega}-\\mathbf{\\Omega}_\\star)$,\nfor 1D RT in the absence of scattering, previously with $a = \\infty$, now with arbitrary values, and assigning a role to $\\partial G\/\\partial\\tau$.  This more formal approach seems to us less promising in terms of physical insights---a judgment that may be altered if a rigorous connection to the concept of fractional derivatives \\cite{MillerRoss1993} can be established.  These pseudo-differential operators have indeed found many fruitful applications in statistical physics \\cite{MetzlerKlafter2000,West2003}.\n\nAlthough out of scope for the present study, there is an implicit time-dependence aspect to generalized (as well as standard) RT even if the radiance fields are steady in time.  The best way to see this is to return to the inset in Fig.~\\ref{f:RWs_g0_inset}.  The highlighted region (between gray brackets) shows in essence how standard and generalized 2D RT unfolds for solar illumination of a medium of optical thickness $\\approx$11 at an angle of $\\approx$30$^\\circ$ from zenith.  The smaller the value of $a$, the shorter the path of the light inside the medium.  The number of scatterings decreases from 25 ($a=\\infty$) to 10 ($a=1.2$).  The flight time for sunlight to cross the cloudy portion of the atmosphere---at most from near-ground level to the troposphere (10 to 15~km altitude, depending on latitude)---cannot be measured directly.  However, it can be estimated statistically via oxygen spectroscopy \\cite{Pfeil_etal98}.  Pfeilsticker, Scholl et al. \\cite{Pfeil99,Scholl_etal06}, as well as Min, Harrison et al. \\cite{MinHarrison99,Min_etal01,Min_etal04}, have found that the more variable the atmosphere at a given mean optical thickness, the shorter the top-to-ground paths on average.  This finding offers a degree of validation of generalized RT for applications to the Earth's cloudy atmosphere.\n\nIn the remainder of this study, we derive analytical and numerical solutions of the generalized RTE in (\\ref{e:nD_integral_RTE}) with (\\ref{e:Q_integral_gRTE})--(\\ref{e:K_integral_gRTE}), and then apply them to specific topics where standard and generalized RT differ significantly.\n\n\n\\section{Deterministic Numerical Solution in $d = 1$: The Markov Chain Approach}\n\\label{s:2stream_MarCh}\n\nIn \\S\\ref{s:dDim_sRTE}, we stated that once we adopted the H--G PF in Table~\\ref{t:Definitions} the whole 1D RT problem is determined entirely (in the absence of surface reflection) by three numbers, $\\{\\omega,g;\\tau^\\star\\}$ for a given $d$ = 1, 2, or 3, with the possible addition of $\\mu_0$ when $d > 1$.  To this small parameter set, we now add the exponent $a$ of the power-law direct transmission function that distinguishes standard RT (exponential limit, $a\\to\\infty$) from its generalized counterpart ($0 < a < \\infty$).  The complete parameter set is therefore $\\{\\omega,g,a;\\tau^\\star(;\\mu_0)\\}$.\n\n\\subsection{Exact Solution of the Standard RTE in $d = 1$}\n\nThe ``$d=1$'' (\\emph{literal} 1D) version of 1D RT has in fact a vast literature of its own since as it is formally identical to the two-stream RT model \\cite{Schuster1905,KubelkaMunk1931}, a classic approximation for (standard) RT in $d=3$ space.  This simplified RT model is still by far the most popular way to compute radiation budgets in climate and atmospheric dynamics models \\cite{MeadorWeaver80}.  We note that there is no longer an angular integral to compute in the $d$-dimensional RTE in (\\ref{e:1D_RTE}).  It is understood to be replaced everywhere by a sum over two directions: ``up'' and ``down.'' Correspondingly, scattering can only be through an angle of $0$ or $\\pi$ rad: $\\mu_\\text{s} = \\pm 1$, respectively.  The $d=1$ RT problem at hand thus takes the form of a pair of coupled ODEs:\n\\begin{equation}\n\\left( \\pm\\frac{\\mathrm{d}\\;}{\\mathrm{d}\\tau} + 1 \\right) I_{\\pm}(\\tau) =\n\\omega \\left[p_{+}I_{\\pm}(\\tau)+p_{-}I_{\\mp}(\\tau)\\right] + q_{\\pm}(\\tau)\n\\label{e:2st_model}\n\\end{equation}\nwith $p_{\\pm} = (1 \\pm g)\/2$ (cf. Table~\\ref{t:Definitions}) and $q_{\\pm}(\\tau) = \\omega p_{\\pm} \\exp(-\\tau)$.  This system of coupled ODEs is subject to BCs $I_{+}(0) = I_{-}(\\tau^\\star) = 0$ when $\\rho = 0$ (otherwise $I_{-}(\\tau^\\star) = \\rho I_{+}(\\tau^\\star)$).\n\nLet us use\n\\begin{equation}\nI_{\\pm}(\\tau) = \\frac{J(\\tau) \\pm F(\\tau)}{2}\n\\label{e:1D_diffusion}\n\\end{equation}\nto recast the diffuse radiance field in the above 2-stream model, where\n\\begin{eqnarray}\n\\label{e:J_2st_def}\nJ(\\tau) &=& I_+(\\tau) + I_-(\\tau), \\\\\n\\label{e:F_2st_def}\nF(\\tau) &=& I_+(\\tau) - I_-(\\tau),\n\\end{eqnarray}\nare respectively the \\emph{scalar} and \\emph{vector} fluxes.\n\nBy summing the two ODEs in (\\ref{e:2st_model}), we find an expression of radiant energy conservation:\n\\begin{equation}\n\\mathrm{d} F\/\\mathrm{d}\\tau = -(1-\\omega)J + \\omega\\exp(-\\tau).\n\\label{e:2st_conservation}\n\\end{equation}\nDifferencing (\\ref{e:2st_model}) yields\n\\begin{equation}\nF(\\tau) = (-\\mathrm{d} J\/\\mathrm{d}\\tau + \\omega g \\mathrm{e}^{-\\tau})\/(1-\\omega g).\n\\label{e:2st_model_mFick}\n\\end{equation}\nThe 1st term on the right-hand side (and the only one that survives after the 2nd one has decayed at large $\\tau$) is a non-dimensional version of Fick's law, a reminder that diffusion theory is exact in $d=1$.  Using (\\ref{e:2st_model_mFick}) in (\\ref{e:2st_conservation})\nleads to a 1D screened Poisson equation for $J(\\tau)$:\n\\begin{equation*}\n\\left[ -\\frac{\\mathrm{d}^2\\;}{\\mathrm{d}\\tau^2} + (1-\\omega)(1-\\omega g) \\right] J(\\tau) =\n\\omega \\left[ 1+(1-\\omega)g \\right] \\exp(-\\tau),\n\\end{equation*}\nsubject to BCs, $J(0)+F(0) = J(\\tau^\\star)-F(\\tau^\\star) = 0$ when $\\rho = 0$ (black surface).  Factoring in (\\ref{e:2st_model_mFick}), these are always of the 3rd (Robin) type.\n\nWhen $\\omega = 1$ (no absorption), the solution of the above pair of ODEs and BCs is\n\\begin{eqnarray}\n\\label{e:J_2st_NoAbs}\nJ(\\tau) &=& 1+R(\\tau^\\star)\\times\\left(1-\\frac{\\tau}{\\tau^\\star\/2}\\right)-\\exp(-\\tau), \\\\\n\\label{e:F_2st_NoAbs}\nF(\\tau) &=& T(\\tau^\\star)-\\exp(-\\tau).\n\\end{eqnarray}\nWe have used here boundary-escaping radiances\n\\begin{eqnarray}\n\\label{e:R_2st_NoAbs}\nR(\\tau^\\star) &=& I_{-}(0) = 1-T(\\tau^\\star), \\\\\n\\label{e:T_2st_NoAbs}\nT(\\tau^\\star) &=& I_{+}(\\tau^\\star)+\\exp(-\\tau^\\star) = \\frac{1}{1+(1-g)\\tau^\\star\/2},\n\\end{eqnarray}\nin the above representation of the solution.  When $\\omega < 1$, somewhat more complex expressions result in the form of 2nd-order rational functions of $\\exp(-k\\tau)$, where $k = 1\/\\sqrt{(1-\\omega)(1-\\omega g)}$, with polynomial coefficients dependent on $\\omega$, $g$ and $\\exp(-\\tau)$.  All these classic results will be used momentarily to verify the new Markov chain numerical scheme.\n\n\\subsection{Markov Chain (MarCh) Scheme}\n\nWe now adapt our ``Markov Chain'' (MarCh) formulation of standard RT in $d=3$ dimensions \\cite{Xu_EtAl_MarCh11,Xu_EtAl_MarChSurf11,Xu_EtAl_MarChLin12} to the present $d=1$ setting for generalized RT.  MarCh is an under-exploited alternative to the usual methods of solving the plane-parallel RT problem first proposed by Esposito and House \\cite{Esposito_House78,Esposito79}.  It differs strongly from many of the usual approaches: discrete ordinates, spherical harmonics, adding\/doubling, matrix-operator and kindred techniques.  It has more in common with source iteration (successive orders of scattering, or Gauss-Seidel iteration), and even with Monte Carlo (MC).  In short, we can say that MarCh is an efficient deterministic solution of a discretized version of the integral RTE solved by MC.  We illustrate in $d=1$ for simplicity, but also for previously articulated reasons, that there may be an acute need for generalized RT in the 2-stream approximation in climate and, generally speaking, atmospheric dynamical modeling.\n\nThe generalized ancillary integral RTE is expressed in generic form in (\\ref{e:nD_ancillary_IRTE}) with the kernel in (\\ref{e:K_integral_gRTE}) and the source term in (\\ref{e:Q_ancillary_gIRTE}).  In $d = 1$, it yields a system of two coupled integral equations for the two possible directions in $S_\\pm(\\tau)$:\n\\begin{eqnarray}\nS_\\pm(\\tau) &=& \\omega \\left[\np_\\pm \\int\\limits_0^\\tau\nS_{+}(\\tau^\\prime) \\, \\left|\\dot{T}_a(\\tau-\\tau^\\prime)\\right| \\, \\mathrm{d}\\tau^\\prime +\np_\\mp \\int\\limits_\\tau^{\\tau^\\star}\nS_{-}(\\tau^\\prime) \\, \\left|\\dot{T}_a(\\tau^\\prime-\\tau)\\right| \\, \\mathrm{d}\\tau^\\prime\n\\right] \\nonumber \\\\\n&+& Q_{S\\pm}(\\tau),\n\\label{e:AncIRTE_1D}\n\\end{eqnarray}\nwhere\n\\begin{equation}\nQ_{S\\pm}(\\tau) = \\omega p_\\pm \\left|\\dot{T}_a(\\tau)\\right|.\n\\label{e:AncIRTE_1D_Q_Order1}\n\\end{equation}\nWe recognize here the operator form of the integral equation, $S = K S + Q_S$, which can be solved by Neumann series expansion, similarly to (\\ref{e:IRTE_soln}):\n\\begin{equation}\nS = Q_S + K Q_S + K^2 Q_S + \\cdots = (E-K)^{-1}Q_S.\n\\label{e:Neumann_S}\n\\end{equation}\n\nAs detailed in the Appendix, the pair of simultaneous integral RTEs in (\\ref{e:AncIRTE_1D}), given (\\ref{e:AncIRTE_1D_Q_Order1}),\nare finely discretized in $\\tau$ (200 layers with $\\Delta\\tau = 0.05$, hence $\\tau^\\star = 10$), with careful attention to accuracy in the evaluation of the integrals using finite summations.  The resulting  matrix problem is large but tractable.  It can be solved using either a truncated series expansion of matrix multiplies or the full matrix inversion depending on problem parameters (primarily, $\\tau^\\star$) and the desired accuracy.\n\nAs usual when working with the ancillary integral RTE, we finish computing radiance \\emph{detected} inside the medium using the formal solution, as in (\\ref{e:simpleS_sweep}) but in $d = 1$ format, and with the appropriate generalized transmission law:\n\\begin{equation}\n\\left\\{ \\begin{array}{l}\nI_{+}(\\tau) = \\int\\limits_0^\\tau\nS_{+}(\\tau^\\prime) \\, T_a(\\tau-\\tau^\\prime) \\, \\mathrm{d}\\tau^\\prime\n+ T_a(\\tau),\n\\\\\nI_{-}(\\tau) = \\int\\limits_\\tau^{\\tau^\\star}\nS_{-}(\\tau^\\prime) \\, T_a(\\tau^\\prime-\\tau) \\, \\mathrm{d}\\tau^\\prime,\n\\end{array} \\right.\n\\label{e:FormalSoln_1D_escape}\n\\end{equation}\nwith $q_{-}$.  Indeed, ``detection'' implies that the radiation reaches a level, but could have gone further.  A special case of detection is radiation \\emph{escaping} the medium at a boundary: $I_{+}(\\tau^\\star)$ or $I_{-}(0)$, which can also be obtained from known values of $S_\\pm$ using one or another of the expressions in (\\ref{e:FormalSoln_1D_escape}).  At any rate, it is the ``cumulative probability'' version of the transmission law that is needed here.\nIn short, after implementing (\\ref{e:Neumann_S}), the final step of the numerical computation is to derive radiances $I_\\pm(\\tau)$ everywhere (it is required) from the known source function $S_\\pm(\\tau)$ using a discretized version of (\\ref{e:FormalSoln_1D_escape}).\n\nIn the Appendix, the discrete-space version of the above problem is derived directly from an analogy with random walk theory using Markov chain formalism: present state, state transition probabilities, probability of stagnation, of absorption (including escape), starting position\/direction of walkers, and so on.  Although intimately related to all these concepts, which are used extensively in MC modeling, the new model is deterministic since it uses normal rather than random quadrature rules.  We naturally call it the Markov Chain (MarCh) approach to RT.  In a recent series of papers \\cite{Xu_EtAl_MarCh11,Xu_EtAl_MarChSurf11,Xu_EtAl_MarChLin12}, we have brought it to bear on aerosol remote sensing on Earth (in $d=3$), so far only with $a=\\infty$, but including polarization.\n\n\\subsection{Illustration with Internal Fields}\n\nTo demonstrate our MarCh code for generalized transport in $d=1$, we focus on uniform or stochastic media with $\\tau^\\star = 10$ irradiated by a unitary source at its upper ($\\tau = 0$) boundary, here, to the left of each panel in Fig.~\\ref{f:MarCh}.  We first assume conservative ($\\omega = 1$) and isotropic ($g = 0$) scattering.  The outcome is plotted in the top two panels in the $d=1$ equivalent of a decomposition in Fourier modes (in $d=2$) or spherical harmonic modes (in $d=3$).  Specifically, we have scalar flux $J = I_{+}+I_{-}$ in the left column and (negative) vector flux $-F = I_{-}-I_{+}$ in the right column.  In the middle row, $g$ is raised from 0 to 0.8.  In the bottom row, $\\omega$ is then lowered from unity to 0.98.  In all of these scenarios, $a$ was varied, the selected values being 1\/2, 1, 3\/2, 2, 10, and $\\infty$; the latter choice is designated as ``Beer's law'' and the others as ``power laws'' in the Figure.\n\nWhen $\\omega = 1$, radiant energy conservation requires that total net flux $F(\\tau)+T_a(\\tau)$ be constant across the medium, and equal to $T(1,g,a;\\tau^\\star)$.  This was verified numerically for all values of $a$ and both values of $g$.  In the right hand panels, we see that indeed $-F(\\tau) = -T(1,g,a;\\tau^\\star)+T_a(\\tau)$; see (\\ref{e:F_2st_NoAbs}) and (\\ref{e:T_2st_NoAbs}) for the case of $a = \\infty$ and $\\omega = 1$.\n\nFor reference, Table~\\ref{t:MarCh_results} gives $R$, $T_\\text{dif}$, $T_a(\\tau^\\star)$, and absorbtance $A = 1-(R+T) = 1-R-T_\\text{dif}-T_a(\\tau^\\star)$ for our three choices of $\\{\\omega,g\\}$ and all values of $a$.  All the entries in Table~\\ref{t:MarCh_results} were verified to all the expressed digits using a custom MC code for generalized transport in $d = 1$.  Apart from the fact that there is no oblique illumination, nor is there a distinction between collimated and diffuse illumination, the key difference between a MC for $d=1$ and $d>1$ is how to select a scattering angle.  In $d>1$, it is a continuous random variable but in $d=1$ the forward versus backward scattering decision is made based on a Bernoulli trial.\n\nAs another element of verification for the MarCh code, we recognize in the upper and middle left-hand panels of Fig.~\\ref{f:MarCh} the characteristic result for $J(\\tau)$ in the case of standard transport theory ($a\\to\\infty$) in the absence of absorption ($\\omega = 1$), namely, the linear decrease modulated by an exponential expressed in (\\ref{e:J_2st_NoAbs}).\n\nIn standard transport theory in $d = 1$, or using the 2-stream\/diffusion approximation for higher dimensions, the linear decrease of $J(\\tau)$ when $\\omega = 1$ follows directly from the constancy of $F(\\tau)$, assuming they include both diffuse and uncollided radiation; see (\\ref{e:2st_model_mFick}) and (\\ref{e:F_2st_NoAbs}), but without the exponential terms.  An interesting finding here is that, although $F(\\tau)+T_a(\\tau)$ is constant for all values of $a$, the linear decrease of $J(\\tau)+T_a(\\tau)$ does not generalize from $a = \\infty$ to $a < \\infty$.  We conclude that generalized RT conserves energy, as it should, both globally ($A+R+T = 1$) and locally, as expressed in\n\\begin{equation}\n\\mathrm{d} F\/\\mathrm{d}\\tau = -(1-\\omega)J(\\tau) + \\omega \\dot{T}_a(\\tau),\n\\label{e:Gen_2st_model_EnCons}\n\\end{equation}\nwhich is follows directly from (\\ref{e:2st_model})--(\\ref{e:F_2st_def}) when $a = \\infty$.  It is not obvious how to derive (\\ref{e:Gen_2st_model_EnCons}) for the generalized transport model described by one or another of its integral RTEs, even in $d=1$.  In contrast, Fick's law in (\\ref{e:2st_model_mFick}), which relates $F$ to $\\mathrm{d} J\/\\mathrm{d}\\tau$, can only be exact when $a = \\infty$ and, moreover, when $d = 1$.\n\nAnother interesting numerical finding is that when $\\omega = 1$ and $a = \\infty$, $T$ depends only on the scaled optical thickness $(1-g)\\tau^\\star$, as is readily seen in (\\ref{e:T_2st_NoAbs}).  This means that, by similarity, an isotropically scattering medium ($g = 0$) with $\\tau^\\star = 10$ has the same total transmittance $T$ as a forward scattering medium with (say) $g = 0.8$ and $\\tau^\\star = 50$.  Formally, $T(1,g,\\infty;\\tau^\\star)$ is only a function of $(1-g)\\tau^\\star$.  More generally, allowing $\\omega \\le 1$, we have\n\\begin{equation}\nF(\\omega,g,\\infty;\\tau^\\star) \\equiv\nf_F\\left(\\frac{1-\\omega}{1-\\omega g},(1-\\omega g)\\tau^\\star\\right)\n\\end{equation}\nfor $F = A,R,T$, where the first argument on the r.-h. side is known as the similarity parameter \\cite{King1987}.  This is not the case when $a < \\infty$.\n\n\\begin{figure}[ht]\n\\begin{center}\n\\includegraphics[width=4in]{MarCh.pdf}\n\\end{center}\n\\caption[MarCh]\n{\\label{f:MarCh}\nInternal radiance fields $J = I_{+}+I_{-}$ (right) and $-F = -I_{+}+I_{-}$ (left) computed using the new MarCh scheme for $d = 1$ described in the Appendix.  $J(\\tau)$ in (\\ref{e:J_2st_def}) and $-F(\\tau)$ in (\\ref{e:F_2st_def}) are plotted as a function of optical depth $\\tau$ into a medium with $\\tau^\\star = 10$, the unitary source being at $\\tau = 0$, for selected values of $a$.  The standard exponential law obtained when $a\\to\\infty$ is designated as ``Beer's law.''  In the top two rows, no absorption is included but the phase function is varied: $p_{+} = 1\/2$ ($g = 0$) on top; $p_{+} = 0.9$ ($g = 0.8$) in the middle.  In the bottom row, again $g = 0.8$ but $\\omega$ is reduced from unity to 0.98.}\n\\end{figure}\n\n\\begin{table}[ht]\n\\caption{Boundary fluxes $R$, $T_\\text{dif}$, $T_\\text{dir} = T_a(\\tau^\\star)$, and absorbtance $A$ for the $d = 1$ stochastic medium with $\\tau^\\star = 10$ used in Fig.~\\ref{f:MarCh}.}\n\\label{t:MarCh_results}\n\\begin{center}\n\\begin{tabular}{|c|c|c||c|c|c|c|c|}\n\\hline\n$\\omega$ & $g$ & $a$ & $R$ & $T_\\text{dif}$ & $T_\\text{dir}$ & $T$ & $A$ \\\\\n\\hline\\hline\n1.00 & 0.0 & $\\infty$ & 0.833 & 0.167 & 0.000 & 0.167 & 0.000 \\\\\n\\hline\n1.00 & 0.0 & 10.      & 0.814 & 0.185 & 0.001 & 0.186 & 0.000 \\\\\n\\hline\n1.00 & 0.0 & 2.0      & 0.727 & 0.245 & 0.028 & 0.273 & 0.000 \\\\\n\\hline\n1.00 & 0.0 & 1.5      & 0.693 & 0.260 & 0.047 & 0.307 & 0.000 \\\\\n\\hline\n1.00 & 0.0 & 1.0      & 0.632 & 0.277 & 0.091 & 0.368 & 0.000 \\\\\n\\hline\n1.00 & 0.0 & 0.5      & 0.500 & 0.282 & 0.218 & 0.500 & 0.000 \\\\\n\\hline\\hline\n1.00 & 0.8 & $\\infty$ & 0.500 & 0.500 & 0.000 & 0.500 & 0.000 \\\\\n\\hline\n1.00 & 0.8 & 10.      & 0.475 & 0.524 & 0.001 & 0.525 & 0.000 \\\\\n\\hline\n1.00 & 0.8 & 2.0      & 0.378 & 0.594 & 0.028 & 0.622 & 0.000 \\\\\n\\hline\n1.00 & 0.8 & 1.5      & 0.345 & 0.608 & 0.047 & 0.655 & 0.000 \\\\\n\\hline\n1.00 & 0.8 & 1.0      & 0.290 & 0.619 & 0.091 & 0.710 & 0.000 \\\\\n\\hline\n1.00 & 0.8 & 0.5      & 0.192 & 0.590 & 0.218 & 0.808 & 0.000 \\\\\n\\hline\\hline\n0.98 & 0.8 & $\\infty$ & 0.422 & 0.401 & 0.000 & 0.401 & 0.176 \\\\\n\\hline\n0.98 & 0.8 & 10.      & 0.406 & 0.432 & 0.001 & 0.433 & 0.161 \\\\\n\\hline\n0.98 & 0.8 & 2.0      & 0.335 & 0.524 & 0.028 & 0.552 & 0.112 \\\\\n\\hline\n0.98 & 0.8 & 1.5      & 0.309 & 0.546 & 0.047 & 0.593 & 0.098 \\\\\n\\hline\n0.98 & 0.8 & 1.0      & 0.264 & 0.568 & 0.091 & 0.659 & 0.077 \\\\\n\\hline\n0.98 & 0.8 & 0.5      & 0.179 & 0.557 & 0.218 & 0.775 & 0.046 \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\end{table}\n\n\n\\section{Diffusion Study in $d = 2$: Theory and Monte Carlo Simulation}\n\\label{s:DiffusionLimits}\n\n\\subsection{Theoretical Predictions}\n\nIn this section, we focus on $d$ = 2 spatial dimensions, partly for simplicity (fidelity with Fig.~\\ref{f:RWs_g0_inset}, where nothing is happening outside of the depicted $(x,z)^\\text{T}$-plane), partly because there are previously-mentioned two-dimensional transport processes on real substrates (including random ones where a stochastic model is in order).  We focus specifically on non-absorbing media ($\\omega$ = 1) over an absorbing lower boundary ($\\rho$ = 0).  Moreover, we will assume an isotropic source at the upper boundary, that is, BC in (\\ref{e:nD_RTE_upperBC_diffuse}) with $F_0$ = 1.\n\nWe will investigate transmitted fluxes, both direct and diffuse, their total $T(g,a;\\tau^\\star)$ being defined in (\\ref{e:nD_RTE_upperBC_diffuse}), but ignoring $\\mu_0$.  We start with a review of the standard $a=\\infty$ case.\n\nIn \\S\\ref{s:dDim_sRTE}, the exact expression for $T(g,\\infty;\\tau^\\star)$ is given in (\\ref{e:T_2st_NoAbs}) for $d=1$ where the diffusion ODE model is mathematically exact.  In $d>1$, diffusion is only a physically reasonable approximation to plane-parallel RT for very opaque highly scattering media.  In lieu of (\\ref{e:1D_diffusion}), it is based on the 1st-order truncation \\begin{equation}\nI(\\tau,\\mathbf{\\Omega}) \\approx \\frac{J(\\tau) + d \\times F_z(\\tau) \\mu}{\\Xi_d},\n\\end{equation}\nand, in lieu of the first entry in the next-to-last row of Table~\\ref{t:Definitions}, we take\n\\begin{equation}\np_g(\\mu_\\text{s}) \\approx \\frac{1+d \\times g \\mu_\\text{s}}{\\Xi_d}.\n\\end{equation}\nThis leads to Laplace\/Helmholtz or Poisson ODEs for $J(\\tau)$, respectively for boundary and volume expressions for the sources.  In plane-parallel slab geometry, the BCs are again Robin-type.  When homogeneous (hence sources in the volume), they are conventionally expressed as\n\\begin{equation*}\n\\left[ J - \\frac{\\chi_d}{(1-\\omega g)} \\frac{\\mathrm{d} J}{\\mathrm{d}\\tau} \\right]_{\\tau = 0} = 0, \\;\n\\left[ J + \\frac{\\chi_d}{(1-\\omega g)} \\frac{\\mathrm{d} J}{\\mathrm{d}\\tau} \\right]_{\\tau = \\tau^\\star} = 0,\n\\label{e:dD_diffusion_BCs}\n\\end{equation*}\nwhere $\\chi_d$ is the extrapolation length, i.e., boundary values of $J\/|\\mathrm{d} J\/\\mathrm{d} z|$, expressed in transport MFPs, that is,\n\\begin{equation}\n\\ell_\\text{t} = 1\/(1-\\omega g)\\sigma.\n\\end{equation}\nClassic values for $\\chi_d$ are listed in Table~\\ref{t:Definitions} (last row).  In the absence of absorption and using boundary sources, total transmission is\n\\begin{equation}\nT(g,\\infty;\\tau^\\star) \\approx \\frac{1}{1+\\tau_\\text{t}^\\star\/2\\chi_d},\n\\label{e:dD_diffusion_Ttot}\n\\end{equation}\nwhere $\\tau_\\text{t}^\\star = (1-g)\\tau^\\star = H\/\\ell_\\text{t}$ is the scaled optical thickness.  This expression is identical to (\\ref{e:T_2st_NoAbs}) for $d=1$ ($\\chi_1 = 1$), but here we use $\\chi_2 = \\pi\/4$.\n\nDiffusion theory for $a < \\infty$ cases is in a far worse state since we do not know yet how to formulate generalized RT in integro-differential form.  What is known is the asymptotic scaling of $T(g,a;\\tau^\\star)$ with respect to $\\tau_\\text{t}^\\star$.  Based on the appropriate truncation of the Sparre-Anderson law of first returns \\cite{Sparre-Anderson1953}, Davis and Marshak \\cite{DavisMarshak1997} showed that\n\\begin{equation}\nT(g,a;\\tau^\\star) \\propto {\\tau_\\text{t}^\\star}^{-\\alpha\/2},\n\\label{e:scaling_diffusion_Ttot}\n\\end{equation}\nwhere $\\alpha = \\min\\{2,a\\}$ is the L\\'evy index.  Recall that $a$ is the generally non-integer value of the lowest order moment of $\\langle s^q \\rangle$ that is divergent for the power-law step distribution in (\\ref{e:Gamma_Tdir}).  Then one of two outcomes occurs:\n\\begin{itemize}\n\\item\nIf $a \\ge 2$, hence $\\alpha = 2$, then the position of the random walk in Fig.~\\ref{f:RWs_g0_inset} is Gaussian (central limit theorem), and standard diffusion theory applies.  As can be seen from (\\ref{e:dD_diffusion_Ttot}), the scaling exponent in (\\ref{e:scaling_diffusion_Ttot}) is indeed (negative) $\\alpha\/2 = 1$.\n\\item\nIf $a < 2$, hence $\\alpha = a$, then the position of the random walk in Fig.~\\ref{f:RWs_g0_inset} is L\\'evy-stable (generalized central limit theorems), and the diffusion process is ``anomalous.''\n\\end{itemize}\nThe predicted scaling in (\\ref{e:scaling_diffusion_Ttot}) will occur for any spatial dimensionality.\n\n\\subsection{Numerical Results}\n\nExtensive numerical computations were performed in $d=2$ spatial dimensions using a straightforward Monte Carlo scheme.  The goal was to estimate $T(g,a;\\tau^\\star)$ for a wide range of $\\tau$ (0.125 to 4096), two choices for $g$ (0 and 0.85), and a representative selection of values for $a$: 1.2, 1.5, 2, 5, 10, and $\\infty$.  We used (\\ref{e:Gamma_step}) to sample the distance to the next collision.\n\nThe key idiosyncrasies of Monte Carlo simulation of RT in $d=2$ are for the two procedures for generating random angles:\n\\begin{itemize}\n\\item\nAt the departure point of the trajectory, an isotropic source in the angular half-space ($|\\theta| < \\pi\/2$) uses $\\sin\\theta_0 = 1-2\\xi$ (where $\\xi$ is a uniform random variable on [0,1]) and $\\cos\\theta_0 = \\sqrt{1-\\sin^2\\theta_0}$.\n\\item\nIf $g \\ne 0$, directional correlation is implemented by computing $\\theta_{n+1} = \\theta_n + \\theta_\\text{s}$ where $\\theta_\\text{s} = 2\\tan^{-1}\\left[ \\tan[(\\xi-1\/2)\\pi]\\times(1-g)\/(1+g) \\right]$\nbased on the corresponding H--G PF from Table~\\ref{t:Definitions} for $d = 2$.\n\\end{itemize}\nThe remaining operations (boundary-crossing detection and tallies) are similar in $d$ = 1,2,3.\n\nFigure~\\ref{f:2D_T_vs_tau_t} shows our results for $T(g,a;\\tau^\\star)$ as a function of scaled optical thickness $\\tau_\\text{t}^\\star = (1-g)\\tau^\\star$ in a log-log plot.  We notice the similarity of $T(0,\\infty;\\tau^\\star)$ and $T(0.85,\\infty;\\tau^\\star)$ using the scaled optical thickness, as predicted in (\\ref{e:dD_diffusion_Ttot}): $T(g,\\infty;\\tau^\\star) \\sim T((1-g)\\tau^\\star)$ when $(1-g)\\tau^\\star \\gg 1$.  Specifically, the two transmission curves overlap when plotted against $(1-g)\\tau^\\star$, at least for large values.  In contrast, we see clear numerical evidence that generalized RT does not have such asymptotic similarity in $T(g,a;\\tau^\\star)$, as was previously anticipated when examining internal radiation fields in $d=1$.  More precisely, the scaling exponent in (\\ref{e:scaling_diffusion_Ttot}) is, as indicated, independent of $g$ but the prefactor (and approach to the asymptote) is.  In Fig.~\\ref{f:2D_T_vs_tau_t}, we have estimated the exponents numerically, and they are close to the predicted value, $\\min\\{1,a\/2\\}$.\n\nIn summary, our modest diffusion theoretical result in (\\ref{e:scaling_diffusion_Ttot}) for generalized RT is well verified numerically, and we have gained some guidance about what to expect for a more comprehensive theory.\n\n\\begin{figure}[ht]\n\\begin{center}\n\\includegraphics[width=5.33in]{2D_T_vs_tau_t.pdf}\n\\end{center}\n\\caption[2D_T_vs_tau_t]\n{\\label{f:2D_T_vs_tau_t}\n2D Monte Carlo evaluations of $T(g,a;\\tau^\\star)$ versus transport (or ``scaled'') optical thickness $\\tau^\\star_\\text{t}$ in log-log axes for $g$ = 0 (solid black) 0.85 (dotted gray) and for $a$ = 1.2, 1.5, 2, 5, 10, and $\\infty$ (from top down).  Asymptotic scaling exponents are estimated numerically using the last two values of $\\tau^\\star$ and compared with theoretical predictions in the main text.}\n\\end{figure}\n\n\n\\section{Single Scattering in $d = 3$: Violation of Angular Reciprocity}\n\\label{s:ReciprocityViolation}\n\nWe first revisit the closed-form expression we derived in standard RT for the single scattering approximation in (\\ref{e:1_scatter_R})\nfor radiances escaping the upper boundary.  We remarked that they have the reciprocity property that reversing $\\mathbf{\\Omega}_0$ (source) and $\\mathbf{\\Omega}$ (detector) in both sign and in order gives the same answer.\n\nHere, we need to evaluate\n\\begin{equation}\nI_1(0,\\mathbf{\\Omega};\\mathbf{\\Omega}_0) =\n\\omega p_g(\\mathbf{\\Omega}\\cdot\\mathbf{\\Omega}_0)\n\\int\\limits_0^{\\tau^\\star} T_a(\\tau^\\prime\/\\mu_0) \\,\n|\\Dot{T}_a(\\tau^\\prime\/|\\mu|)| \\, \\mathrm{d}\\tau^\\prime\/|\\mu|.\n\\label{e:1_scatter_gRT_reflection}\n\\end{equation}\nFrom there, (\\ref{e:BRF_form}) yields for the BRF form\n\\begin{equation}\n\\begin{array}{l}\n\\frac{\\pi}{\\mu_0} I_1(0,\\mathbf{\\Omega};\\mathbf{\\Omega}_0) = \\pi \\omega p_g(\\mathbf{\\Omega}\\cdot\\mathbf{\\Omega}_0)\n\\times \\frac{a}{\\mu_0}\n\\left( \\frac{\\mu_0}{|\\mu|} \\right)^a\n\\left( 1-\\frac{\\mu_0}{|\\mu|} \\right)^{-2a} \\\\\n\\quad\\quad\\quad\\quad\\quad\n\\times \\left[\n \\mathrm{B}\\left(2a,1-a;1-\\frac{\\mu_0}{|\\mu|}\\right)\n-\\mathrm{B}\\left(2a,1-a;a\\,\\frac{1-\\mu_0\/|\\mu|}{a+\\tau^\\star\/|\\mu|}\\right)\n\\right],\n\\end{array}\n\\label{e:1_scatter_gR}\n\\end{equation}\nwith $-1 \\le \\mu < 0$ and $0 < \\mu_0 \\le 0$), and where we use the incomplete Euler Beta function: $\\mathrm{B}(x,y;z) = \\int_0^z t^{x-1} (1-t)^{y-1} \\mathrm{d} t$.\n\nTo demonstrate that this complex expression violates the reciprocity relation in (\\ref{e:R_reciprocity}) and by how much, we have plotted in Fig.~\\ref{f:NonRecip_gRT} the ratio of $I_1(0,-\\mathbf{\\Omega}_0;-\\mathbf{\\Omega})\/|\\mu|$ to $I_1(0,\\mathbf{\\Omega};\\mathbf{\\Omega}_0)\/\\mu_0$ for a small value of $\\tau^\\star$ compatible with the single scattering approximation used in (\\ref{e:1_scatter_gRT_reflection}).  This ratio is independent of the SSA, $\\omega$, and of azimuthal angle, $\\phi$ (assuming $\\phi_0 = 0$), in $d = 3$ since it appears only in the evaluation of the PF via $\\mathbf{\\Omega}_0\\cdot\\mathbf{\\Omega} = \\mu_0\\mu+\\sqrt{1-\\mu_0^2}\\sqrt{1-\\mu^2}\\cos\\phi$.  As expected, the violation is stronger for smaller values of $a$ ($a$ = 1.2 and $a$ = 10 are displayed).\n\nThis violation of reciprocity is a desirable attribute of stochastic RT modeling at least in atmospheric applications.  It is indeed consistent with real-world satellite observations of reciprocity violation uncovered by DiGirolamo et al. \\cite{DiGirolamo_EtAl_1998} in spatially variable cloud scenes inside a relatively broad field of view, and readily replicated with numerical Monte Carlo simulations.  These findings were soon explained theoretically by Leroy \\cite{Leroy2001}.  This provides an element of validation of the new model and, by the same token, invalidates for atmospheric applications all models for RT in stochastic media based on either homogenization or linear mixing.\n\nIt is important to realize that this reciprocity violation is related ({\\it i}) to the uniform illumination of the scene and ({\\it ii}) to the spatial averaging that is inherent in the observations that the new model is designed to predict.  Indeed, at the scale of a collimated source at a single point in space and a collimated receiver aimed at another direction at another point in any medium, spatially variable or not, there is a fundamental principle of reciprocity as long as the PF has it, $p(\\mathbf{\\Omega}^\\prime\\to\\mathbf{\\Omega}) = p(-\\mathbf{\\Omega}\\to-\\mathbf{\\Omega}^\\prime)$, and Helmholtz's reciprocity principles will guaranty that property under most circumstances.  Starting from there, Case \\cite{Case1957} showed that invariance under arbitrary horizontal translation  is also required to extend this (internal) ``Green's function'' reciprocity to Chandrasekhar's \\cite{Chandra1950} (external) reciprocity relations for plane-parallel slabs.\n\n\\begin{figure}[ht]\n\\begin{center}\n\\includegraphics[width=2.5in]{NonRecip_weak.pdf}\n\\includegraphics[width=2.5in]{NonRecip_strong.pdf}\n\\end{center}\n\\caption[NonRecip_gRT]\n{\\label{f:NonRecip_gRT}\nContour plots of $I_1(0,-\\mathbf{\\Omega}_0;-\\mathbf{\\Omega})\/|\\mu| \\div I_1(0,\\mathbf{\\Omega};\\mathbf{\\Omega}_0)\/\\mu_0$ as functions of $\\mu_0$ and $\\mu$ for $\\tau^\\star = 0.1$ (a small value consistent with the adopted single scattering approximation) and $a=10$ (left), $a=1.2$ (right).}\n\\end{figure}\n\n\n\\section{Conclusions \\& Outlook}\n\\label{s:concl}\n\nWe have surveyed a still small but growing literature on radiative transfer (equivalently, mono-group linear transport) theory where there is no requirement for the direct transmission law---hence the propagation kernel---to be exponential in optical distance.  In particular, we gather the evidence from the atmospheric radiation and turbulence\/cloud literatures that a better choice of transmission law \\emph{on average} would have a power-law tail, at least for solar radiative transfer in large domains with a strong but unresolved variability of clouds and aerosols that is shaped by turbulent dynamics.  Long-range spatial correlations in the fluctuations of the extinction coefficient in the stochastic medium are essential to the emergence of power-law transmission laws, and such correlations are indeed omnipresent in turbulent media such as cloudy airmasses as well as entire cloud fields.\n\nFrom there, we modified the integral form of the radiative transfer equation to accomodate such power-law kernels.  This leads to a \\emph{generalized} linear transport theory parameterized by the power-law exponent.  This new model reverts to the standard one where exponential transmission prevails in the limit where the characteristic power-law exponent increases without bound.  In the new theory however, the physics dictate that there are two specific roles for the transmission function, which lead to different but related expressions.  There is no such formal distinction in standard transport theory.  However, when the origins of the exponentials are carefully scrutinized from a transport physics perspective, their different functionalities become apparent.\n\nThe new transport theory, possibly with some restrictions, is likely to be one instance of the new ``non-classical'' class of transport models investigated recently by Larsen and Vasques \\cite{LarsenVasques11}.  These authors were primarily motivated by fundamental questions about neutron multiplication processes in pebble-bed nuclear reactors.  We do not anticipate long-range spatial correlations in these reactors so the relevant transmission laws are more likely to be modified exponentials such as found by Davis and Mineev-Weinstein \\cite{DavisMineev11} in media with very high frequency fluctuations.\n\nWe presented a unified formulation for standard and generalized transport theory in $d = 1,2,3$ spatial dimensions and their associated direction spaces.  The present study first adds to previous ones the capability of a new deterministic computational scheme for solving the generalized linear transport equation, which does not have at present an integro-differential form, only an integral one.  We thus address the stochastic transport problem at hand, so far only in $d=1$, using a Markov chain formalism.  It is used to explore internal intensity and flux fields where numerical results shed light on questions of similarity and diffusion.  Diffusion theory and the space-angle similarity captured in the scaled or ``transport'' mean-free-path are exact in $d=1$ for standard transport---not so in generalized transport.  In $d>1$, diffusion is only an approximation applicable to opaque scattering media, and the associated similarity is only asymptotic (large optical thickness regimes).\n\nNew numerical simulations presented here in $d=2$ confirm and qualify the violation of similarly.  They also confirm previous predictions about the asymptotic scaling of diffuse transmission, which is anomalous or ``L\\'evy-like'' if the characteristic exponent is less than 2.  L\\'evy flights are now attracting considerable interest in laboratory as well as atmospheric optics \\cite{BarthelemyEtAl2008}.  Finally, the generalized transport problem is solved in $d=3$ in the single scattering approximation.  This solution is used to highlight the violation of angular reciprocity in generalized radiative transfer.  This is yet another distinction between standard transport theory, including homogenization-based models for transport in stochastic media, and the new class of generalized transport models.  This non-reciprocity is in fact observed in the Earth's cloudy atmosphere using reflected sunlight, and is therefore a desirable attribute for stochastic transport of solar radiation.\n\nA logical next step is to implement the Markov chain solution in $d = 3$.  Monte Carlo-based predictions of the angular patterns for radiance escaping the medium on the upper (obliquely illuminated) boundary are already available for the verification process.  In $d = 3$, there may be an interest in adding light polarization capability and linearizing the model with respect to the new spatial variability parameter $a$, the exponent that controls the power law tail of the direct transmission law.\n\nFinally, we draw attention to a serendipitous development in the atmospheric radiation literature.  While our ongoing theoretical and computational work on unresolved\/stochastic \\emph{spatial} variability of the extinction coefficient in turbulent scattering media has lead to the parameterized class of power-law propagation kernels described herein, Conley and Collins \\cite{ConleyCollins2011} have independently arrived at the very same power-law parameterization for the problem of unresolved \\emph{spectral} variability of the absorption coefficient due to all manner of molecules in the Earth's atmosphere that might contribute to the thermal and solar radiative processes that build up the greenhouse effect.\n\nThis opens tantalizing questions about novel \\emph{unified} formulations of the challenging problem of radiation transport in clumpy 3D scattering media that are permeated with spatially uniform but spectrally variable absorbing gases.  A first step in that direction is to assume that the scattering elements of the optical medium are in fact spatially uniform.  However, our generalized radiation transport model for multiple scattering based on power-law transmission between scatterings\/absorptions still applies, and it can be invoked to capture the impact of purely spectral variability on the overall radiation transport from sources to sinks\/detectors.\n\n\n\n\\newpage\n\\section{Appendix: Markov Chain Formalism for Generalized Radiative Transfer}\n\\label{s:append}\nThis page (35) is intentionally left blank, to be replaced by corresponding Appendix page.  N.B. arXiv users will find a PDF version of the Appendix as an ancillary file accessible from the abstract page.\n\n\\newpage\nThis page (36) is intentionally left blank, to be replaced by corresponding Appendix page.  N.B. arXiv users will find a PDF version of the Appendix as an ancillary file accessible from the abstract page.\n\n\\newpage\nThis page (37) is intentionally left blank, to be replaced by corresponding Appendix page.  N.B. arXiv users will find a PDF version of the Appendix as an ancillary file accessible from the abstract page.\n\n\\newpage\nThis page (38) is intentionally left blank, to be replaced by corresponding Appendix page.  N.B. arXiv users will find a PDF version of the Appendix as an ancillary file accessible from the abstract page.\n\n\\newpage\nThis page (39) is intentionally left blank, to be replaced by corresponding Appendix page.  N.B. arXiv users will find a PDF version of the Appendix as an ancillary file accessible from the abstract page.\n\n\\newpage\n\n\n\n\\section*{Acknowledgments}\n\nThe authors are thankful for sustained support from NASA's Radiation Sciences Programs managed by Hal Maring and Lucia Tsaoussi.  We also acknowledge many fruitful discussions with Howard Barker, Bob Cahalan, Bill Collins, Martin Frank, Lee Harrison, Alex Kostinski, Ed Larsen, Shaun Lovejoy, Alexander Marshak, Qilong Min, Klaus Pfeilsticker, Anil Prinja, Ray Shaw, and Bob West.  We also thank the Editors and two anonymous reviewers for many insightful comments about the submitted manuscript.  This research was carried out at the Jet Propulsion Laboratory, California Institute of Technology, under a contract with the National Aeronautics and Space Administration.\n\n\n\\textcolor{white}{\\cite{Marchuk_etal80}}\n\n\n\\newpage\n\n\\bibliographystyle{unsrt}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nA tremendous volume of credit card transactions is conducted daily, especially with the COVID-19 pandemic. Nevertheless, this financial activity necessities many robust resources, in terms of CPU, RAM, storage and human expertise, to detect fraudulent payments.  For the year 2019, the Nelson Report estimated the worldwide financial loss from credit card fraud to \\$28.65 billion ~\\cite{CCFstat2019}. Over the last ten years, users that revealed at least one credit card fraud soared by 71\\% in Canada ~\\cite{CCFstat2021}.  Credit-card fraud classification is complex to tackle due to the following challenges:  {\\bf (1)} transactions are generated continuously and speedily. In this data stream context, fraud must be detected in real-time to avoid losses on the customers' side, {\\bf (2)} Machine Learning Algorithms (MLAs) require storing a very large number of transactions to conduct batch learning. However, financial transactions must not be accumulated in an enormous quantity by the fraud detection models because of data sensitivity and confidentiality, {\\bf (3)} MLAs cannot adapt previous knowledge to newly available transactions to improve their accuracy, making the detection models obsolete and unreliable in the long run, and {\\bf (4)} the architectures of all past incremental learning approaches are pre-determined \\cite{DBLP:journals\/ci\/AnowarS21}, \\cite{DBLP:conf\/flairs\/AnowarS20}, thus, the accuracy may not improve with new data. \n\nMost of the credit-card fraud detection studies employed conventional MLAs that are however inadequate for this specific environment. In the industry, a new fraud prediction system is created from scratch for every number of days to learn the new behavior ~\\cite{DBLP:conf\/dsaa\/LebichotPBSHO20}. However, re-training is very time-consuming, and the learned knowledge is completely lost. To overcome the three challenges above, we develop a new adaptive learning algorithm that learns frequently and efficiently from transaction chunks or mini-batches.  According to the incremental training frequency, we decide how much data to collect in each chunk. More the frequency is shorter,  less sensitive data is accumulated. In our study, a chunk contains the payment transactions of one day, which is still large. We only process one chunk at a time in the short-term memory and discard it after each model adaptation, without storing it.\n\n\\medskip\n\nThe proposed adaptive approach combines transfer learning and incremental feature learning (a new paradigm). Thanks to transfer learning, we extract valuable features from the original ones and reuse the new features for the subsequent transaction chunks. For instance, in the image processing area, the first layers of the neural network extract fundamental features that can be reused in another image processing task. Following the same reasoning, we use the first layers to collect more beneficial features and then add a new network to utilize those features. By doing so, we take advantage of the previous chunks' knowledge. \n\nOur new Incremental Feature Learning (IFL) algorithm adapts gradually to the new transaction chunks by  {\\bf (1)} preserving the previously learned knowledge and {\\bf (2)} dynamically adjusting the network architecture for each new chunk to achieve the highest performance during training.  IFL expands the network topology by adding new hidden layers and units during each adaptation phase.  Determining the most suitable model's architecture leading to the best performance is a complex problem. Looking for the optimal number of units in a hidden layer is already tricky for static MLAs.  Our IFL approach adds hidden units one by one until the model does not converge anymore. Nevertheless, as we are changing the network architecture to increase the performance, we may over-fit the resulting model.  Hence, we utilize a validation chunk during training to avoid over-fitting after each extension.  More precisely, only the weights of the new hidden units are updated each time as the previous units are frozen to store the previous knowledge. Thus, the less computational time is required to conduct learning for each new chunk. In this way, our IFL approach will always outperform other incremental learning approaches as the former continuously adapts its architecture to reach optimal accuracy. The architecture is permanently fixed for past incremental approaches, and the accuracy may not improve when new training chunks are provided.  Developing such an algorithm is very challenging, requiring a deep investigation of the building blocks and libraries of MLA toolkits, such as creating a new hidden unit, adding a new connection, and freezing the weights of a old connection (not to be re-optimized).\n\n\\medskip\n\nThere is only one study that developed an IFL approach, but very elementary ~\\cite{DBLP:journals\/npl\/GuanL01}. Still, this paper does not provide details on the algorithm design and its implementation. Instead, our paper presents the steps of our new IFL approach that learns progressively from newly available chunks. Through a concrete example, we illustrate step by step the sophisticated behavior of our approach. Moreover, using a real credit-card fraud dataset, we create training, testing, and validation chunks and handle the highly imbalanced learning problem. We build four fraud classification models for the experiments: (1) the initial model trained on the first-day transaction chunk, (2) the initial model trained on the second-day transaction chunk, (3) the re-fitted model trained on the second-day chunk, and (4) the final optimal model trained on the second day and produced with our new approach. We thoroughly evaluate and compare all these learned models on unseen data.  \n\\section{Related Work}\nWe review recent research on detecting credit card fraud and highlight its weaknesses. The majority of studies conducted batch learning, such as ~\\cite{Najadat2020} that explored deep learning, like BiLSTM and BiGRU, and classical learning, such as Decision Tree, Ada Boosting, Logistic Regression, Random Forest, Voting and Naive Base. Since the fraud dataset is highly imbalanced, the authors adopted random under-sampling, over-sampling and SMOTE. The hybrid of over-sampling, BiLSTM and BiGRU lead to the highest accuracy.  Another work  ~\\cite{nguyen2020deep} also assessed several MLAs, including LSTM, 2-D CNN, 1-D CNN, Random Forest, ANN and SVM, using different data sampling methods on three credit-card datasets. LSTM and 1-D CNN combined with SMOTE returned the best results.  We believe LSTM can be a good option for incremental learning since this algorithm can remember past data and therefore creates predictions using the current inputs and past data, leading to a better response to the environmental changes.  In both papers, LSTMs and the other models were trained on very large datasets, requiring storing sensitive information forever.  Nevertheless, since user transactions are available incrementally, conventional MLAs are inappropriate for streaming data. Our proposed method aims to address the real credit-card fraud classification context. \n\n\\smallskip \n\nIn \\cite{DBLP:conf\/smc\/AnowarS20}, the authors first utilized SMOTE-ENN to handle a highly imbalanced credit-card fraud dataset and then divided the dataset into multiple training chunks to simulate incoming data. They proposed an ANN-based incremental learning approach that learns gradually from new chunks using an incremental memory model. For adjusting the model each time, the memory consists of one past chunk (so that data are not forgotten immediately) and one recent chunk (to conduct the model adaptation). The authors demonstrated that incremental learning is superior to static learning. However, using two chunks every time can be expensive computationally. Also, since the ANN topology is fixed, the model cannot adapt to significant changes in the chunk patterns. In our study, the ANN architecture is dynamic to build an optimal fraud detection model. Instead of using two chunks simultaneously, leading to storing more data, we use only one chunk. With transfer learning, we take advantage of the previous chunk without storing it.\n\n\\smallskip \n\nIn ~\\cite{Barris2020}, the authors introduced an incremental Gradient Boosting Trees (GBT) approach, which is just an ensemble of decision trees, to minimize the loss function gradually. The ensemble is updated for each new group of transactions by appending a new Decision Tree to create a more robust GBT model.  The authors developed three models:  static GBT (batch learning), re-trained GBT (re-training all the transactions by including the\ninvestigated ones), and incremental GBT (ensemble).  All these methods necessitate storing a tremendous quantity of user transactions. The authors divided the credit-card dataset into several sub-datasets w.r.t time (month) and evaluated the three models for each month (four months in total). Although the re-trained GBT (performed with 1.6 million transactions) achieved the best performance, it is 3000 times slower than the incremental approach.  Since re-training is too time-consuming, we aim to improve the accuracy and training time for learning incrementally. \n\\section{Credit-Card Fraud Dataset Preparation}\nWe select a public, anonymized credit-card fraud dataset consisting of 284807 users' purchasing transactions that occurred during two days in September 2013 ~\\cite{CCFD2016}. We may mention that credit-card fraud datasets are lacking due to data privacy. The 2-class dataset possesses 30 predictive features obtained with the feature extraction method PCA; the two features Time and Amount were not transformed as their actual values are essential. Time represents the seconds passed between the current and the first transactions in the dataset. After examining the dataset, only $0.172\\%$ (492) of the transactions is fraudulent. We found duplicated fraud data (19) that we eliminate. We are in the presence of a highly imbalanced dataset where the count of legitimate data is much higher than the count of fraudulent data. In this situation, classifiers will be biased towards the normal class and will misclassify the fraud class. \n\n\\smallskip\n\nWe divide the dataset into two chunks according to the transaction time: the first chunk is related to the first day and the second chunk to the second day. We use the first chunk to perform transfer learning, and the second chunk IFL. Next, using the stratified method, we split first chunk into 70\\% training and 30\\% testing and the second chunk into 70\\% training, 15\\% validation and 15\\% testing. Table \\ref{ccfd} (a) presents the original dataset's statistics. As given in Table \\ref{ccfd} (a), the training chunks are highly imbalanced, therefore we adopt the famous over-samping method SMOTE with the new class distribution ratio of 1.3. We employ over-sampling to keep all the fraud data as they are rare events. For example, the training chunk2 has an imbalance ratio of 1:689 that we re-balance with a new ratio 1.3.  Table \\ref{ccfd} (b) exposes the re-sampled train, validation and test chunks. Since Time is irrelevant for the classification task, we remove it.\n\n\\begin{figure}[H]\n\\centering\n\\includegraphics[width=1.00\\linewidth]{figures\/DataTable.png}\n\\caption{Original and Re-Sampled Training and Testing Chunks}\n\\end{figure}  \n\n\n\n\n\\section{Transfer Learning and Incremental Feature Learning}\nTransfer learning is usually adopted in image processing, where an existing classifier is used to detect new objects, as the classifier can efficiently extract new features from images.  In our fraud classification application, based on the initial model trained on the first-day transaction chunk, our algorithm utilizes the transformed features to train on the second-day chunk with the optimal number of hidden units.  Instead of re-training the entire model for each new chunk, we only train the newly added units and freeze all the previous layers. This strategy will significantly decrease the training time so that the classifier identifies fraud activities much faster, especially that our application requires real-time responses. The most apparent difference of incremental learning from conventional learning is that the former does not assume the availability of sufficient training data, as data are available over time ~\\cite{DBLP:journals\/npl\/GuanL01}. So, based on these facts, improving learning for each newly supplied chunk is an excellent approach.\\\\\n\nAlgorithm 1 first conducts transfer learning and then IFL. We first build the initial network with four fully connected layers, where the first layer has the input features, the first hidden layer 500 units, the second hidden layer ten units, and the output layer one unit for the binary classification. The study ~\\cite{DBLP:conf\/smc\/AnowarS20}, which validated an example-based incremental learning approach with the same credit-card dataset, determined that 500 units as the best number for the first hidden layer. So, we train this first network with the mentioned topology on the first chunk. Then, after removing the output unit, we refit the initial model using previous knowledge and second-day chunk.  Now,  the second hidden layer represents the transformed features (called tSubset), i.e., the new dimensional space more relevant to the target class because it is obtained with previous knowledge.  Here, tSubset contains the values corresponding to the transformed features. For example, assume we have 100,000 transactions in the second chunk. Using the refitted model, we convert each transaction of 29 values to ten new values. So, our transformed tSubset will contain 100,000 new data with 10 new features.  \n\n\\medskip\nWe actually extend the initial network with two sub-networks; the first NN considers the high-level features, and the second NN the low-level features. We leverage both of them to predict the output more efficiently.  Also, the study ~\\cite{Wang2014EEGES} showed that creating sub-networks according to the features' relevancy can lead to a better outcome. We add the second sub-NN to improve more the fraud detection model performance. A similar idea is used in the incremental approach defined in ~\\cite{DBLP:journals\/npl\/GuanL01} by connecting the original inputs directly to the output unit with the same motivation. Another paper ~\\cite{DBLP:journals\/isci\/WangWYDC21} explained and compared the performance results of connecting the inputs to the output unit directly.  These are the reasons we use high-level features for creating the first sub-NN and then low-level features for creating the second sub-NN. Thus, we can gradually improve the accuracy using newly available data without forgetting the previous knowledge (stored in the previous weights). Moreover, the training time will be much lower as we only train the new chunk and one unit only per epoch. Our approach can be used for any number of chunks by appending a new hidden layer to the first sub-NN.\n\n\\medskip\n\nAlgorithm 2 shows how to extend a sub-NN with new hidden units and connections. The main challenge is determining the number of units that should be added to attain the best performance. In transfer learning, we add a predetermined number of layers and units to the previously trained model.  It can lead to fewer layers\/units; therefore, the resulting model will return poor predictions. It can also lead to unnecessary layers\/units, which only increases the learning time. Since the numbers of layers and units are predetermined, all the units are trained together and not one at a time like our proposed method. So, the training time will be very time-consuming. We determine the needed number of units in our work by adding and training them one by one. This approach is highly efficient computationally. Moreover, instead of computing the gradient descent for all the new units at once for each epoch, which is time-consuming, we utilize the ``patience\" parameter as the early stopping callbacks. Based on the convergence threshold, this parameter checks whether the error is reducing or not in a certain number of epochs. In the experiments, if the accuracy does not increase in ten epochs, then the learning will stop. \n\n\\newpage\n\n\\begin{footnotesize}\n\\begin{algorithm}[H]\n{\\bf Inputs:} trainChunk1, trainChunk2, validChunk2, threshold, nEpoch \\\\\n\\KwResult{optimal predictive model} \n\\medskip\n{\\color{blue} (*Initial Model with chunk1*)}\\\\\niniNN $\\gets$ build network with four fully connected layers;\\\\\niniModel $\\gets$ train initNN on trainChunk1; \\\\\n\n{\\color{blue} (*Refitted Model with chunk2*)}\\\\\nrefModel $\\gets$ delete output unit of initModel;\\\\\nrefModel $\\gets$ feedforward trainChunk2 to refModel using past weights; \\\\\n\n{\\color{blue} (*Transformed Feature Sub-dataset*)}\\\\\ntGroup $\\gets$  extract set of transformed features;\\\\\ntSubset $\\gets$  create dataset w.r.t tGroup and trainChunk2;\\\\\n\n{\\color{blue} (*Incremental Feature Learning*)} \\\\\n{\\color{red} (*Sub-NN Training and Extension for Transformed Features*)} \\\\\nsubNN1 $\\gets$ create network (2 hidden units and 1 output unit);\\\\\nsubNN1 $\\gets$ connect subNN1 to output layer of refModel ;\\\\\nmodel1  $\\gets$ train subNN1 with tSubset by freezing past weights;\\\\\nextModel1= {\\bf Algorithm2}(model1, tSubset, validChunk2, threshold, nEpoch);\\\\\n\n{\\color{red} (*Sub-NN Training and Extension for Input Features*)} \\\\\nsubNN2 $\\gets$ create network (2 hidden units);\\\\\nsubNN2 $\\gets$ connect hidden units to output unit of extModel1 ;\\\\\nmodel2 $\\gets$ freeze weights of previous units except hidden and output units;\\\\\nmodel2 $\\gets$ train subNN2 with trainChunk2;\\\\\nextModel2 $\\gets$ {\\bf Algorithm2}(model2, trainChunk2, validChunk2, threshold, nEpoch);\\\\\nreturn extModel2;\\\\\n\\caption{Transfer Learning and Incremental Feature Learning}\n\\end{algorithm}\n\n\\begin{algorithm}[H]\n{\\bf Inputs:} {model; trainData, validData, threshold, nEpoch}\\\\\n\\KwResult{extended model} \n\\medskip\npreValAcc $\\gets$ $0$;\n\\While{True}{\ncurValAcc $\\gets$ compute accuracy of model using validData;\\\\\n\\While{(curValAcc is converging with patience of nEpoch)}{\n     model $\\gets$ train non-frozen weights with trainData;\\\\\n     curValAcc $\\gets$ compute accuracy of model using validData;\\\\\n}\n\n\\If{(curValAcc - preValAcc $<$ threshold) and (preValAcc $\\neq$ 0)}{\n Break;\n}\npreValAcc $\\gets$ curValAcc;\nmodel $\\gets$ add new hidden unit to hidden layer;\\\\\nmodel $\\gets$ freeze weights of previous units except new hidden and output units;}\nreturn model;\n\\caption{Model Extension with New Hidden Units until No Convergence}\n\\end{algorithm}\n\\end{footnotesize}\n\n\\subsection{A Concrete Example}\nBased on the credit-card fraud dataset, we illustrate the behavior of our new algorithm in Figures 2, 3, and 4, where the green color denotes new connections and units, and red the frozen part. Figure 2 creates the initial network, trains it on chunk1, and after deleting the output unit, utilizes previously learned weights, and creates the refitted model by training with chunk2. Figure 3 adds the first sub-NN and keeps training on data of the ten features by including new hidden units and connections and freezing past parameters until the performance is not converging anymore.  Figure 4 creates the second sub-NN using the original features, connects it to the past output unit, freezes previous weights, and trains it on chunk2 until the model does not convergence anymore. \n\n\n\n\n\n\n\n\\begin{figure}[h]\n\n\\centering\n\\subfloat[create initial model on chunk1]{\n\\includegraphics[width=0.40\\linewidth]{figures\/baseModel.png}\n}\n\\subfloat[transform inputs on chunk2 using past weights]{\n\\includegraphics[width=0.36\\linewidth]{figures\/baseModelNoOutput.png}\n} \n\\caption{Transfer Learning}\n\n\\end{figure}\n\n\\begin{figure*}[h]\n\\centering\n\\subfloat[train subNN1 \\& freeze past NN]{\n\\includegraphics[width=0.45\\linewidth]{figures\/baseModelOneLayerAdded1.png}\n}\n\\subfloat[add hidden unit(s) until no convergence]{\n\\includegraphics[width=0.50\\linewidth]{figures\/baseModelOneLayerAdded2.png}\n} \n\\caption{Network training and expansion using transformed features}\n\n\\end{figure*}\n\n\\begin{figure*}[h]\n\\centering\n\\subfloat[connect subNN2 \\& freeze past NN]{\n\\includegraphics[width=0.45\\linewidth]{figures\/subNetAdded1.png}} \n\\subfloat[add hidden unit(s) until no convergence]{\n\\includegraphics[width=0.45\\linewidth]{figures\/subNetAdded2.png}}\n\\caption{Network training and expansion using original inputs}\n\n\\end{figure*}\n\n\\section{Validation}\n\\subsection{Experiment Setup}\nSince the 29 input features of our fraud dataset have different ranges of values and with significant discrepancies between the features, we first normalize them into the same range. Then, we train the initial network (four fully connected layers) on chunk1 data using different ranges to determine the most appropriate feature scale. The range of [-5, +5] returns the best performance. Furthermore, we tune the network hyperparameters, such as the learning rate to 0.001, batch size to 1024, epoch to 100, convergence threshold to 0.01, and patience to 10. We utilize five quality metrics for evaluating the predictive performance: Precision, Recall, F1-score, FNR, and Time (in seconds). Due to the randomness of NNs, we run ten training sessions and ten corresponding testing sessions. We consider the average of test sessions for comparing different fraud detection models.  \n\n\\subsection{Feature Transformation}\nUsing the trained model's knowledge obtained with the first-day transaction chunk, we transform the input features by feed-forwarding the second-day chunk to the first model. The new features are now more relevant to the target class, and after checking their correlation values, we find them much higher. Figure \\ref{disNew} exposes the distribution of the ten transformed features.\n\n\\begin{figure}[h]\n\\centering\n\\includegraphics[width=0.9\\linewidth]{figures\/D20.png}\n\\caption{Distribution of Transformed Features}\n\n\\end{figure}\n\n\\subsection{Performance Evaluation}\nWe first build two initial models, the first one is trained on train chunk1 and assessed on test chunk1 (presented in Table 1), and the second one using train chunk2 and test chunk 2 (presented in Table 2). According to the 10-round testing results of the two models, F1-score decreased by 4.6\\% for day2 transactions. This expected decrease is due to the much higher number of instances in chunk1 (60\\% of data) compared to chunk2 (40\\% of data). Moreover, by preserving past knowledge, we re-fit the initial model (trained on chunk1) using train chunk2 to conduct transfer learning and evaluate its accuracy on test chunk2.  In this case, F-score decreased by 5.1\\%. Changes in data patterns may have caused this decrease \\cite{DBLP:conf\/dsaa\/LebichotPBSHO20}. For instance, if a concept drift occurred in the credit card data, the model performance decreases ~\\cite{DBLP:journals\/jnca\/AbdallahMZ16}.  Lastly, we develop the final optimal model using our incremental feature learning approach. F1-score improved by 9\\% using only one chunk.  Also, Recall, a significant metric in fraud detection, is augmented with 9.8\\%, which means we have fewer false negatives in the second chunk. Also, as we are training one hidden unit at a time, the training time is less than the re-fitting time. Therefore, we can conclude that the final model outperforms the initial and re-fitted models using only one chunk. \n\nAnother essential metric for fraud detection is the False Negative Rate (FNR), which refers to the rate of fraudulent transactions detected as normal ones. According to the average of the ten experiments, as exposed in Figure 6, the initial-model FNR on chunk1 is $0.149$, the initial-model FNR on chunk2 is $0.219$, the re-fitting-model FNR is $0.228$, and the final-model FNR is $0.13$. These values show that our model caches more fraudulent cases when fed with new chunks, reducing FNR over time. \n\nFinancial transactions cannot be stored for a long time by the fraud classifiers because of privacy issues.  In this case, we train the model incrementally chunk by chunk and discard each chunk right away. As we observe, the proposed approach can enhance the performance using each chunk at a time and with less computational cost. \n\n\n                                                                                                                                      \n\n\n\\begin{figure}[H]\n\\centering\n\\includegraphics[width=1.00\\linewidth]{figures\/t1.png}\n\\caption{Training and Testing on Day1 Transactions}\n\n\\end{figure}  \n\n\\begin{figure}[H]\n\\centering\n\\includegraphics[width=1.00\\linewidth]{figures\/t2.png}\n\\caption{Training and Testing of Different Models on Day2 Transactions}\n\n\\end{figure}  \n\n\n\n\\begin{figure}[H]\n\\centering\n\\includegraphics[width=0.65\\linewidth]{figures\/FNR_CHART_EDITED.png}\n\\caption{FNR Comparison on the Four Incremental Models }\n\n\\end{figure}  \n\n\\section{Conclusion}\nFor tackling the actual behavior of the credit-card fraud detection environment, we introduced a new classification algorithm that adjusts gradually and efficiently to incoming chunks based on transfer learning and IFL. Transfer learning preserves past knowledge and utilizes it to adapt to subsequent chunks.  IFL extends the network topology incrementally by finding the optimal number of hidden units for detecting fraud in each chunk. Our hybrid approach improves the predictive accuracy without the necessity of accumulating a substantial volume of data and spending too much time on training. Our new approach can be employed on everyday credit card transactions to prevent the performance from decreasing using current and past knowledge.\n\n\n\n\\section{Introduction}\nNowadays, a large number of real-world applications, such as fraud detection and e-commerce, are subject to changes in their data distribution over time. In the context of online learning,  data space may change over time, with new classes or new features. Such non-stationary datasets are split into two groups: concept drift where new classes appear in the data \\cite{}, and virtual concept drift where new features are included in the data \\cite{}. The latter received very little attention in the literature because it is a complicated problem to solve.  For instance, in fraud detection applications, the feature space may change as a response to the new fraud strategies devised by fraudsters. If the fraud model does not self-adjust to the new fraud features, it will become unreliable in the long run. In an evolving application where the feature space changes over time, it is vital to develop an adaptive machine learning algorithm that can adjust to the new features incrementally, instead to re-training the model from scratch. Machine learning algorithms are suitable only for static application where the feature space is fixed and does not change. To fill the huge gap in the literature,  we are interested in exploring Incremental Feature Learning (IFL) that consists of adding one feature or a group of features at a time into the predictive algorithm. \\\\ \n \nNeural Networks are the only learning method that can support IFL by modifying its architecture as features are introduced gradually.  \\\\ \n\n{\\bf PLAN- IDEA}\n\\begin{itemize}\n\\item Cross validation for training\n\\item split to training and testing \n\\item entropy instead of correlation or not\n\\item compare between feature orderings (descending, ascending and no grouping)\n\\item the whole new set of features is consistent; when we add, we check consistency? \n\\item may remove existing features when we add new ones \n\\item Xavier initialize rs compare with zero one\n\\item can we take the average of weights?????\n\\item KPCA for every new feature group\n\\end{itemize}\n\n\n\\section{Related Work}\n\n\\section{IFL Approach}\nAlgorithm 1 exposes the steps of our IFL process to create dynamically the final neural network (fully connected) by incorporating the feature groups incrementally. After determining the feature groups and ordering them, we split the original dataset into the appropriate training sub-datasets. At the beginning, we first create the first NN (with 3 layers) and train it on the corresponding sub-dataset. A hidden layer always begins with two units.  As long as the model's training loss is not converging, we keep adding new hidden units and their required connections (dense) to increase accuracy and decrease the loss. A model converges when the difference between the current and previous models' errors is greater than a certain threshold. We re-train the current network for each new hidden unit by optimizing only the new connections' weights and the output unit's weights and freezing all the other weights. \\\\ \n\nSubsequently, we create a new sub-network (with two fully connected layers) for each remaining feature group and keep adding hidden units and dense connections to the sub-network until the loss function does not converge anymore.  For each newly added sub-network, we consider all the previous sub-networks features and the current feature group. As presented in Algorithm 2, we develop a new idea on adding a new hidden unit to the current sub-network using the ``concatenate layer\", a building block defined in the KERAS toolkit. We first transform each hidden unit into a layer to be able to disable the optimization part of that unit.  Next, we add the new unit as a layer. Then, we merge all the hidden layers using the concatenate layer. The resulting final model will possess all the dataset's features in the input layer,  one hidden layer with many units, and one output unit since we are dealing with binary classification and one-value regression problems. \\\\\n\nWhen creating a new hidden unit, we adopt the random function with the standard deviation of 0.01 for the weight initialization, and we set the weights of biases to zero. Also, since want the new sub-networks and the new units to affect the output smoothly and without any sudden gap, we therefore set the weights of the new connections to the output unit to zero when creating them for the first time.\n\n\\begin{algorithm}[h]\n \\KwData{Training Dataset, Threshold}\n \\KwResult{Optimal Predictive Model} \n \\medskip\n {\\color{red} (*Dataset preparation*)}\\\\\n Create groups of features w.r.t. relevancy to target class; \\\\\n Sort groups in descending order and save into list; \\\\\n Divide dataset into sub-datasets w.r.t. created groups  \\\\\n \\smallskip\n \n {\\color{red} (*Build and optimize first sub-network*)} \\\\\n Create first sub-network for first group (1 input layer, 2 hidden units and 1 output unit);\\\\\n\\While{(\\textbf{error is converging})}{\n    Train non-frozen weights of sub-network on first sub-dataset;\\\\\n      \\If{\\textbf{(error is not converging)}}{\n         \\If{(\\textbf{(PreviousError - currentError) $<$ threshold})}{\n             Break;\n         }\n         Add new hidden unit to sub-network (*call Algorithm 2*);\\\\\n         Freeze weights of previous units except new hidden and output units;\\\\\n      }\n      Store current error\n}\n\\smallskip\nDelete first group from list; \\\\ \n{\\color{red} (*Build next sub-networks, merge and optimize them*)} \\\\ \n \\While{(list of groups is not empty)}{\n Create sub-network for current group (1 input layer and 2 hidden units);\\\\\n Connect hidden units to output unit;\\\\\n Freeze weights of previous units except new and output units\\\\\n \\While{(\\textbf{error is converging})}{\n     Train non-frozen weights on proper sub-dataset;\\\\\n     \\If{(\\textbf{error is not converging})}{\n         \\If{((\\textbf{PreviousError - currentError) $<$ threshold})}{\n             Break;\n         }\n         Add new hidden unit to sub-network (*call Algorithm 2*);\\\\\n         Freeze weights of previous units except new hidden and output units;\\\\\n      }\n      Store current error\n }\nDelete current group from list;}\\\\  \\\\ \n\\caption{Incremental Feature Learning with Constructive NNs}\n\\end{algorithm}\n\n\\begin{algorithm}[]\n{\\bf Input:} {Sub-network}\\\\\n \\KwResult{Extended sub-network with one hidden unit and dense connections} \n\\medskip\nConvert each hidden unit to a layer\\\\\nCreate one hidden unit as a layer with desired activation function\\\\\nConnect fully new hidden unit to input layer\\\\\nMerge all the hidden layers using concatenate layer\\\\\n\\caption{Adding New Hidden Unit to Current Sub-network}\n\\end{algorithm}\n\n\\bigskip\n\nImplementing such an IFL algorithm is a challenging task that required a deep investigation of the finely granular building blocks of the KERAS toolkit and its libraries, as shown below: \n\\begin{itemize}\n\\item There is a Boolean variable assigned to each unit named \\textit{trainable} that can be set to FALSE so that the weights of its connections are not updated during training. When it is set to TRUE, the weights are then re-optimized.\n\\item The API called ``Functional\" allows creating custom connections between units. We utilize this mode to be able to merge two sub-networks by creating new connections. \n\\item In order to fuse several layers in Algorithm 2, we use the ``concatenation layer\" concept. This layer does not contain any trainable parameters, so it will not impact the feed-forward or back-propagation processes because it is only used to merge layers. \n\\item To determine when to stop training the current model, we also utilize the ``patience\" parameter as the early stopping callbacks. This parameter checks whether the error is reducing or not in a certain number of epochs. As an example, let us set the patience parameter to $100$; in this case, if the error does not decrease in 100 epochs, then the training would be stopped.\n\\end{itemize}\n\n\\newpage\n\\newpage\n\n\\section{Case Study 1: Credit-Card Fraud}\nWe select a robust credit-card fraud dataset consisting of 284807 user transactions that occurred during two days in September 2013 (https:\/\/www.kaggle.com\/mlg-ulb\/creditcardfraud). The 2-class dataset has 30 numerical robust features as they were obtained with the feature extraction method PCA. However, two features Time and Amount were not transformed. Time is irrelevent for the classification task, as it represents the number of seconds elapsed between the current transaction and the first transaction in the dataset. Nevertheless, the features come with different scales as shown in Figure Y that depicts three examples of feature distributions. {\\bf How come after applying PCA, we have these distributions- let me ask my other PhD student??? perhaps not normalized before PCA???}. \n\n\\medskip\n\nAfter examining the dataset, only $0.172\\%$ (492) of the data is fraudulent. We found duplicated fraud data (19) that we eliminate. We are in presence of a highly imbalanced dataset with a class distribution ratio of normal to fraud samples equal to {\\bf 1:?????.} \n \n \n \n\\begin{table}[]\n\\begin{tabular}{|c|c|c|}\n\\hline\n\\multicolumn{3}{|c|}{ Imbalanced  Dataset}                     \\\\ \\hline\nNormal                          & Fraud                        & Total                       \\\\ \\hline\n\\multicolumn{1}{|c|}{283253} & \\multicolumn{1}{c|}{473} & \\multicolumn{1}{c|}{283726} \\\\ \\hline\n\\multicolumn{3}{|c|}{Re-Balanced Dataset(1:3)}                                                      \\\\ \\hline\nNormal                            & Fraud                        & Total                       \\\\ \\hline\n283253                       & 93473                   & 376726                      \\\\ \\hline\n\\end{tabular}\n\\end{table}\n\n\\begin{table}[]\n\\begin{tabular}{|c|c|c|}\n\\hline\n\\multicolumn{3}{|c|}{Training (60\\%)}                                                       \\\\ \\hline\nNormal                       & Fraud                          & Total                       \\\\ \\hline\n\\multicolumn{1}{|l|}{170064} & \\multicolumn{1}{l|}{55971} & \\multicolumn{1}{l|}{226035} \\\\ \\hline\n\\multicolumn{3}{|c|}{Validation (10\\%)}                                                  \\\\ \\hline\nNormal                          & Fraud                          & Total                       \\\\ \\hline\n28345                        & 9328                      & 37673                       \\\\ \\hline\n\\multicolumn{3}{|c|}{Testing (30\\%)}                                                        \\\\ \\hline\nNormal                        & Fraud                          & Total                       \\\\ \\hline\n\\multicolumn{1}{|l|}{84844} & \\multicolumn{1}{l|}{28174} & \\multicolumn{1}{l|}{113018} \\\\ \\hline\n\\end{tabular}\n\\end{table}\n\\end{center}\n\n\n\\subsection{Feature Grouping}\n\n\n\n\\section{Case Study 2: Multi-class dataset}\n\n\\section{Conclusion and Future Work}\n\\end{document}\n\n\\href{https:\/\/www.kaggle.com\/gauravduttakiit\/creditcard-fraud-detection-by-logistic-regression}{https:\/\/www.kaggle.com\/gauravduttakiit\/creditcard-fraud-detection-by-logistic-regression}\n\n   \\item Correlation heat-map:\n    \\begin{figure}[]\n    \\centering\n    \\caption{Correlation Heatmap}\n    \\includegraphics[scale=.90]{Correlation heatmap.png}\n    \\label{Correlation Heatmap}\n    \\end{figure}\n\\end{itemize}\n\\begin{center}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nIn the classical approach in the $n$-dimensional Euclidean space $\\mathbb{R}^n$ as the iteration of the operator nabla, $\\Delta= div\\,grad\\,=\\nabla^2$, the Laplacian, can be seen as the composition of two differential operators each of first order of differentiation. The gradient acting on scalars and the divergence on vectors. The intrinsic definitions of these operators are both based in the geometry of the Euclidean space. In fact, the gradient of a scalar function $u$ at a point $P$ in $\\mathbb{R}^n$, is the vector with the direction of the maximal growth of $u$ from $P$ and length equal to the rate of growth of $u$ in that maximal direction. The divergence of a vector field $F$ at a point $P$ of the space is given by the outer flow of $F$, from $P$, per unit volume. Sometimes some problems of potential analysis are posed on sets without any a priori geometric or algebraic structure and there is no other way to measure a gradient of a potential $u$, than the mere difference $u(y)-u(x)$ for any two different points $x$ and $y$ of the domain of $u$. A classical instance of this situation is provided by the Kirchhoff laws in the theory of electric circuits. A resistive circuit of $n$ nodes, $\\set{1,2,\\ldots, n}$ can be schematically seen as a weighted graph. Assume that $R_{ij}$ is the electrical resistance of the connection between nodes $i$ and $j$ of the circuit. If we admit for each $R_{ij}$ any nonnegative value  including $+\\infty$, and $\\Phi_{ij}$ is the potential difference between nodes $i$ and $j$, from Ohm's law, the sum of all current intensities is given at each node $i=1,2,\\ldots,n$, by $(K\\Phi)_i=\\sum_{j=1}^n \\frac{\\Phi_{ij}}{R_{ij}}=\\sum_{j=1}^n w_{ij}\\Phi_{ij}$, where $w_{ij}=\\frac{1}{R_{ij}}$. A function $\\Phi$ defined on the edges of the graph will satisfy the first Kirchhoff law if $K\\Phi=0$. The first Kirchhoff law can be seen as a conservation law in the sense that at each node of the electrical circuit all incoming electrical currents have to compensate with all out-coming electrical currents. In this sense the operator $K$ acting on functions $\\Phi$ defined on the edges of the graph, can be seen as a divergence operator. When $\\Phi$ is $\\nabla u$, the ``naive'' gradient of $u$, $(\\nabla u)_{ij}=u_j-u_i$ and $u$ is any function defined on the nodes of the graph, we have the Laplacian operator $\\Delta = K\\nabla$. In particular, $u$ is harmonic ($\\Delta u=0$) if and only if $u$ satisfies the mean value identity at each node, i.e. $u_i =\\sum_{j=1}^n w_{ij}u_j$, whose global or local characters depend on the concentration of the nonvanishing terms of the matrix $w_{ij}$. For a complete analysis see \\cite{DoyleSnellBook84}.\n\nLet us start by some basic abstract definitions. In the following sections we shall profusely exemplify and illustrate the general framework.\n\nLet $X$ be a locally compact Hausdorff space. Set $\\mathscr{C}_c(X)$ and $\\mathscr{C}_c(X\\times X)$ to denote the spaces of real  valued continuous functions with compact support, defined on $X$ and $X\\times X$ respectively. We shall use capital Greek letters $\\Phi=\\Phi(x,y)$ to denote real functions defined on $X\\times X$ and small Greek letters $\\varphi=\\varphi(x)$ to denote real functions defined on $X$. Let $\\mu$ be a Borel probability measure on $X$ and $\\pi$ be a Borel probability measure on $X\\times X$. For a given $\\Phi\\in\\mathscr{C}_c(X\\times X)$ we say that a function $\\psi$ defined on $X$ is a \\textbf{Kirchhoff divergence of $\\Phi$ with respect to $\\mu$ and $\\pi$} if the equation\n\\begin{equation}\\label{eq:KichDivOperator}\n\\int_X \\varphi(x)\\psi(x) d\\mu(x) = \\iint_{X\\times X} \\varphi(x)\\Phi(x,y) d\\pi(x,y)\n\\end{equation}\nholds for every $\\varphi$ in $\\mathscr{C}_c(X)$. It is not difficult to provide examples showing that solutions of \\eqref{eq:KichDivOperator} may not exist and may not be unique. For nonexistence take $X=[0,1]$, $\\mu=\\delta_0$ the Dirac delta at the origin, $d\\pi= dx dy$ the area in the unit square and $\\Phi\\equiv 1$. To show a case of non-uniqueness take again $X=[0,1]$, $\\mu=\\delta_0$, $\\Phi\\equiv 1$ and $\\pi=\\delta_0\\times \\delta_0$.\n\nGiven a finite measure $\\pi$ on $X\\times X$, as usual, we denote with $\\pi^i$, $i=1,2$; the two marginal distributions of $\\pi$, for $B$ a Borel set in $X$, $\\pi^1(B)=\\pi(B\\times X)$ and $\\pi^2(B)=\\pi(X\\times B)$.\n\nRadon-Nikodym Theorem provides a simple criteria for existence and uniqueness, up to null sets, of a solution $\\psi$ of \\eqref{eq:KichDivOperator} which shall be enough for our further work. Given $\\Phi\\in\\mathscr{C}_c(X\\times X)$ and $\\pi$ a probability on $X\\times X$, we write $\\pi_\\Phi$ to denote the measure $d\\pi_\\Phi(x,y)=\\Phi(x,y)d\\pi(x,y)$ and $\\pi^1_\\Phi$ and $\\pi^2_\\Phi$ to denote the first and second marginals of $\\pi_\\Phi$.\n\\begin{proposition}\n\tLet $X$ be a locally compact Hausdorff space. Let $\\mu$ be a Borel probability on $X$, $\\pi$ a Borel probability on $X\\times X$ and $\\Phi\\in\\mathscr{C}_c(X\\times X)$. If the first marginal $\\pi^1_\\Phi$ of $\\pi_\\Phi$ is absolutely continuous with respect to $\\mu$, then, the Radon-Nikodym derivative of $\\pi^1_\\Phi$ with respect to $\\mu$, $\\frac{d\\pi^1_\\Phi}{d\\mu}$, solves \\eqref{eq:KichDivOperator}.\n\\end{proposition}\n\\begin{proof}\nNotice that since $\\pi_{\\Phi}^1$ is the first marginal of $\\pi_{\\Phi}$, we have that $\\int_X\\varphi(x) d\\pi_{\\Phi}^1(x)= \\iint_{X\\times X}\\varphi(x) d\\pi_{\\Phi}(x,y)=\\iint_{X\\times X}\\varphi(x)\\Phi(x,y) d\\pi(x,y)$ for every $\\varphi\\in\\mathscr{C}_c(X)$. On the other hand, since we are assuming $\\pi_{\\Phi}^1<<\\mu$, we have\n\\begin{equation*}\n\\iint_{X\\times X}\\varphi(x)\\Phi(x,y) d\\pi(x,y) = \\int_X\\varphi(x)d\\pi^1_\\Phi(x) = \\int_X\\varphi(x)\\frac{d\\pi_{\\Phi}^1(x)}{d\\mu} d\\mu(x)\n\\end{equation*}\nfor every $\\varphi\\in\\mathscr{C}_c(X)$. Which gives \\eqref{eq:KichDivOperator} with $\\psi=\\frac{d\\pi_{\\Phi}^1}{d\\mu}$, as desired.\n\\end{proof}\n\nA probability measure $\\pi$ on $X\\times X$ is said to be a coupling (see \\cite{Villanibook}) between the probability measures $\\mu$ and $\\nu$ on $X$ if $\\pi^1=\\mu$ and $\\pi^2=\\nu$. If $\\pi$ is a given probability on $X\\times X$, then $\\pi^1_\\Phi$ is absolutely continuous with respect to $\\pi^1=\\mu$ and also $\\pi^2_\\Phi$ is absolutely continuous with respect to $\\pi^2=\\nu$, no matter what is the particular $\\Phi\\in\\mathscr{C}_c(X\\times X)$.\n\n\\begin{proposition}\n\tLet $\\mu$ and $\\nu$ be two probability measures on the Borel sets of $X$ and let $\\pi$ be a coupling of $\\mu$ and $\\nu$. Then for every $\\Phi\\in\\mathscr{C}_c(X\\times X)$, the measures $\\pi^1_\\Phi$ and $\\pi^2_\\Phi$ are absolutely continuous with respect to $\\mu$ and $\\nu$ respectively.\n\\end{proposition}\n\\begin{proof}\n\tSince $\\Phi$ is bounded, we have that for every Borel set $B$ in $X$,\n\\begin{align*}\n\t\\abs{\\pi^1_\\Phi (B)} & = \\abs{\\pi_\\Phi (B\\times X)}\\\\\n\t& = \\abs{\\iint_{B\\times X}\\Phi(x,y) d\\pi(x,y)}\\\\\n\t&\\leq \\norm{\\Phi}_\\infty \\pi^1_\\Phi(B)\\\\\n\t&= \\norm{\\Phi}_\\infty \\mu(B).\n\t\\end{align*}\nAlso, $\\abs{\\pi^2_\\Phi (B)}\\leq \\norm{\\Phi}_\\infty \\nu(B).$\n\\end{proof}\nThe above propositions prove the following statement.\n\\begin{theorem}\\label{thm:RadonNikodymFirstMarginal}\n\tLet $X$ be a locally compact Hausdorff space. Let $\\pi$ be a Borel probability measure on $X\\times X$. Let $\\Phi\\in\\mathscr{C}_c(X\\times X)$. Then the Radon-Nikodym derivative $\\frac{d\\pi^1_\\Phi}{d\\mu}$ solves equation \\eqref{eq:KichDivOperator} with $\\mu=\\pi^1$, the first marginal of $\\pi$, and $\\pi^1_\\Phi$ is the first marginal of $\\pi_{\\Phi}(A)=\\iint_A \\Phi d\\pi$, $A$ Borel set in $X\\times X$.\n\\end{theorem}\nWe shall use the notation $Kir_\\pi \\Phi$ for the solution $\\frac{d\\pi^1_\\Phi}{d\\mu}$\n of \\eqref{eq:KichDivOperator} in this case. Notice that $\\Phi$ continuous and bounded suffice for the above results. In particular, if $f:X\\to \\mathbb{R}$ is continuous and bounded, so is $F(x,y)=f(y)-f(x)$, and we can define a Laplacian type operator based on Kirchhoff divergence by\n \\begin{equation*\n \\Delta_\\pi f = Kir_\\pi F.\n \\end{equation*}\n Hence, at least formally, the solution for the heat conduction problem\n \\begin{equation*}\n (P)\\,\\left\\{\n \\begin{array}{ll}\n \\frac{\\partial u}{\\partial t} = \\Delta_\\pi u,\\, & t>0, x\\in X\\\\\n u(x,0)=g(x) \\, & x\\in X\n \\end{array}\n \\right.\n \\end{equation*}\n is given by\n \\begin{equation*}\n u(x,t) = (e^{t \\Delta_\\pi}g)(x).\n \\end{equation*}\n Or, when a spectral resolution of the operator $\\Delta_\\pi$, in terms of a sequence of eigenvalues $\\lambda_i$ and eigenfunctions $\\psi_i$, is available\n \\begin{equation*}\n u(x,t) = \\sum_i e^{t \\lambda_i} \\proin{g}{\\psi_i} \\psi_i (x)\n \\end{equation*}\n with $\\proin{g}{\\psi_i} = \\int_X g(x)\\psi_i (x) d\\mu(x)$.\n\n Along the next sections we aim to explore the above abstract setting for some particular couplings.\n\n\\section{The finite case. Weighted graphs}\\label{sec:FiniteCase}\n\nLet $X=\\mathcal{V}=\\set{1,2,\\ldots,n}$ be the set of vertices of a weighted graph $\\mathcal{G}=(\\mathcal{V},E,w)$, where $E=\\set{(i,j): i\\in\\mathcal{V}, j\\in\\mathcal{V}}=X\\times X$ is the set of all edges of $\\mathcal{G}$, and $w:E\\to\\mathbb{R}^+\\cup\\{0\\}$ is the nonnegative weight of each edge, with $w_{ij}=w_{ji}$, $w_{ii}=0$ for every $i$, $w_{ij}>0$, $i\\neq j$ and $\\sum_{i=1}^n\\sum_{j=1}^n w_{ij}=1$. The weight $w$ determines the probability measure $\\pi$ by $\\pi(A)=\\sum_{(i,j)\\in A}w_{ij}$ for a subset $A$ of $E$. Hence the first marginal $\\mu$ of $\\pi$ is given by the weights $\\mu_i=\\sum_{j=1}^n w_{ij}$. In other words, $\\mu(B)=\\sum_{i\\in B}\\mu_i=\\sum_{i\\in B}\\sum_{j=1}^n w_{ij}=\\pi(B\\times X)$. So that\n\\begin{equation*}\n\\int_X \\varphi d\\mu = \\sum_{i=1}^{n} \\mu_i \\varphi_i, \\textrm{ and }\n\\end{equation*}\n\\begin{equation*}\n\\iint_{X\\times X} \\Phi d\\pi =\\sum_{j=1}^n\\sum_{i=1}^n w_{ij}\\Phi_{ij}.\n\\end{equation*}\nEquation \\eqref{eq:KichDivOperator} takes the form\n\\begin{equation*}\n\\sum_{i=1}^n \\mu_i\\varphi_i \\psi_i =\\int_X \\varphi \\psi d\\mu=\n\\iint_{X\\times X} \\varphi(x)\\Phi(x,y) d\\pi =\\sum_{j=1}^n\\sum_{i=1}^n w_{ij}\\varphi_i\\Phi_{ij} = \\sum_{i=1}^n\\varphi_i \\left(\\sum_{j=1}^n w_{ij}\\Phi_{ij}\\right),\n\\end{equation*}\nwhich should hold for every $\\varphi$. Hence\n$\\mu_i\\psi_i = \\sum_{j=1}^n w_{ij}\\Phi_{ij}$,\nor, since each $\\mu_i$ is positive\n\\begin{equation*\n\\psi_i =(Kir\\, \\Phi)_i =\\frac{1}{\\mu_i}\\sum_{j=1}^n w_{ij}\\Phi_{ij}=\\frac{1}{\\sum_{i=1}^n w_{ij}}\\sum_{j=1}^n w_{ij}\\Phi_{ij}.\n\\end{equation*}\nAnd the corresponding Laplace's operator of a function $f$ defined on the vertices, is given by\n\\begin{equation*}\n(\\Delta_\\pi f)_i =\\frac{1}{\\mu_i}\\sum_{j=1}^n w_{ij}(f_j - f_i).\n\\end{equation*}\nThe harmonic functions in this setting are those that satisfy the mean value identity\n\\begin{equation*}\nf_i = \\frac{1}{\\sum_{j=1}^n w_{ij}}\\sum_{j=1}^n w_{ij} f_j.\n\\end{equation*}\nIn matrix notation, $\\Delta_\\pi=D^{-1}W-I$, where $D = \\textrm{diagonal}(\\mu_1,\\ldots,\\mu_n)$, and $W=(w_{ij})$. The diffusion problem\n\\begin{equation*}\n\\left\\{\n\\begin{array}{ll}\n\\frac{\\partial u}{\\partial t} = \\Delta_\\pi u,\\, & t>0\\\\\nu(i,0)=f_i \\, &i=1,\\ldots,n\n\\end{array}\n\\right.\n\\end{equation*}\nhas the solution $\\overline{u}(t)=e^{t\\Delta_\\pi}\\overline{f}$, $\\overline{f}=(f_1,\\ldots,f_n)$.\n\nWe have $e^{t\\Delta_\\pi}=e^{-t}e^{tD^{-1}W}$.\nThe general theory of Markov chains can be applied to the analysis of the steady state for the solution $\\overline{u}(t)$ of $(P)$. See \\cite{Roberts1976discrete}. A $n\\times n$ transition matrix $B$ is said to correspond to a regular Markov chain, if some power of $B$ has only positive elements. The Fundamental Limit Theorem for regular Markov chains proves that there exist a Markov matrix $M$ with all rows equal to $m=(m_1,\\ldots,m_n)$, with $m_i>0$ for every $i=1, \\ldots, n$ and $\\sum_{i=1}^n m_i=1$, such that\n\\begin{equation*\n\\lim_{k\\to\\infty} B^k = M.\n\\end{equation*}\nThe next result is a particular instance of convergence to equilibrium (see \\cite{NorrisBook97}).\n\\begin{proposition}\\label{propo:limitMarkovmatrix}\nFor $n\\geq 3$, let $W=(w_{ij})$ be a nonnegative $n\\times n$ matrix such that $w_{ii}=0$ for every $i=1,\\ldots,n$, $w_{ij}=w_{ji}>0$ for each $i\\neq j$ and $\\sum_{i=1}^n\\sum_{j=1}^n w_{ij}=1$. Let $\\pi$ be the probability measure defined on $\\{1,\\ldots,n\\}^2$ by $\\pi(A)=\\sum_{(i,j)\\in A} w_{ij}$. Given a function $\\overline{f}=(f_1,\\ldots,f_n)$ defined on the vertices $\\mathcal{V}$ of the weighted graph $\\mathcal{G}$, set $\\overline{u}(t)= e^{t\\Delta_\\pi}\\overline{f}$ to denote the solution of $(P)$ with $t>0$. Then\n\\begin{equation*}\n\\lim_{t\\to\\infty} \\overline{u}(t) = \\left(\\sum_{j=1}^n m_j f_j\\right)\\overline{1},\n\\end{equation*}\nwhere $\\overline{1}=(1,\\ldots,1)$ and $\\overline{m}=(m_1,\\ldots,m_n)$ is the constant row of the Markov limit matrix\n\\begin{equation*}\n(D^{-1}W)^{\\infty} = \\lim_{k\\to\\infty} (D^{-1}W)^k = \\begin{pmatrix}\n\\overline{m}\\\\\n\\vdots\\\\\n\\overline{m}\n\\end{pmatrix}\n\\end{equation*}\n\\end{proposition}\n\n\\begin{proof}\nNotice first that $\\widetilde{W}=D^{-1}W$ is the matrix of a Markov chain positive entries except for its diagonal terms. Hence $\\widetilde{W}$ is a regular Markov chain. From the Fundamental Limit Theorem for regular Markov chains, we have that\n\\begin{equation*}\n\\lim_{k\\to\\infty} \\widetilde{W}^k = M = \\begin{pmatrix}\n\\overline{m}\\\\\n\\vdots\\\\\n\\overline{m}\n\\end{pmatrix}\n\\end{equation*}\nwith $\\overline{m}=(m_1,\\ldots,m_n)$, $m_i>0$ for every $i=1,\\ldots,n$ and $\\sum_{i=1}^n m_i =1$. On the other hand,\n\\begin{equation*}\n\\overline{u}(t) = e^{-t}\\sum_{l\\geq 0}\\frac{t^l}{l!}\\widetilde{W}^l\\overline{f},\n\\end{equation*}\nfor $t>0$. With the standard notation for norms in $\\mathbb{R}^n$, we have that for every $\\varepsilon>0$ there exists an integer $L$ such that for $l>L$ we have $\\norm{\\widetilde{W}^l\\overline{f}-M\\overline{f}}<\\frac{\\varepsilon}{2}$. Hence\n\\begin{align*}\n\\norm{\\overline{u}(t)-\\left(\\sum_{j=1}^n m_j f_j\\right)\\overline{1}}\n& = \\norm{\\overline{u}(t)-M\\overline{f}}\\\\\n&= \\norm{e^{-t}\\sum_{l\\geq 0}\\frac{t^l}{l!}(\\widetilde{W}^l\\overline{f}-M\\overline{f})}\\\\\n& \\leq e^{-t}\\sum_{l=0}^L\\frac{t^l}{l!}\\norm{\\widetilde{W}^l\\overline{f}-M\\overline{f}} +\n\\frac{\\varepsilon}{2}e^{-t}\\sum_{l\\geq L+1}\\frac{t^l}{l!}\\\\\n&<\\varepsilon\n\\end{align*}\nfor $t$ large enough.\n\\end{proof}\n\n\n\n\\section{Markov coupling}\nLet $X$ be a locally compact Hausdorff space. Let $\\mu$ and $\\nu$ be two probabilities on the Borel subsets of $X$. Let $\\pi$ be a probability on the Borel sets of $X\\times X$ that is absolutely continuous with respect to $\\mu\\times \\nu$.\n\\begin{lemma}\\label{lem:couplingMarkov}\nLet $X$, $\\mu$, $\\nu$ and $\\pi$ be as described. Then $\\pi$ is a coupling for $\\mu$ and $\\nu$ if and only if $K=\\frac{d\\pi}{d(\\mu\\times\\nu)}$ is a Markov kernel in the sense that\n\\begin{enumerate}[(i)]\n\\item $\\int_X K(x,y) d\\mu(x)=1$ for $\\nu$ almost every $y\\in X$, and\n\\item $\\int_X K(x,y) d\\nu(y)=1$ for $\\mu$ almost every $x\\in X$.\n\\end{enumerate}\n\\end{lemma}\n\\begin{proof}\nAssume first that $\\pi$ is a coupling for $\\mu$ and $\\nu$. Then for every $B$, Borel set in $X$, we have\n\\begin{equation*}\n\\mu(B)=\\pi(B\\times X)=\\iint_{B\\times X} \\frac{d\\pi}{d(\\mu\\times \\nu)} d\\mu d\\nu\n=\\int_B\\left(\\int_X K(x,y) d\\nu(y)\\right) d\\mu(x).\n\\end{equation*}\nHence $\\int_X K(x,y) d\\nu(y) =1$ for $\\mu$ almost every $x\\in X$. Identity $(i)$ follows the same argument. Assume now that $(i)$ and $(ii)$ hold, then by Fubini's theorem\n\\begin{equation*}\n\\pi(B\\times X)=\\iint_{B\\times X} d\\pi=\\iint_{B\\times X} K(x,y) d\\mu(x) d\\nu(y)=\\int_B\\left(K(x,y) d\\nu(y)\\right) d\\mu(x)=\\mu(B).\n\\end{equation*}\n\\end{proof}\n\nThe next statement provides the Kirchhoff divergence operator associated to these type of Markov couplings.\n\\begin{theorem}\n\tLet $X$ be a locally compact Hausdorff topological space. Let $\\mu$ and $\\nu$ be two given Borel probability measures on $X$. Let $\\pi$ be the coupling for $\\mu$ and $\\nu$ given by\n\t\\begin{equation*}\n\t\\pi(A) = \\iint_A K(x,y) d\\mu(x) d\\nu(y)\n\t\\end{equation*}\n\twith $K$ satisfying $(i)$ and $(ii)$ in Lemma~\\ref{lem:couplingMarkov} and $A$ any Borel set in $X\\times X$. Then, for $\\Phi\\in\\mathscr{C}_c(X\\times X)$ we have\n\t\\begin{equation*}\n\tKir_\\pi \\Phi(x) = \\int_{y\\in X} K(x,y) \\Phi(x,y) d\\nu(y).\n\t\\end{equation*}\n\\end{theorem}\n\n\\begin{proof}\n\tThe measure $\\pi_{\\Phi}$ induced by $\\Phi\\in\\mathscr{C}_c(X\\times X)$ is now given by $\\pi_{\\Phi}(A)=\\iint_A \\Phi K d\\mu d\\nu$. Its first marginal is\\begin{equation*}\n\t\\pi_{\\Phi}^1(B) = \\pi_{\\Phi}(B\\times X)=\\int_B\\left(\\int_{y\\in X}\\Phi(x,y) K(x,y) d\\nu(y)\\right) d\\mu(x).\n\t\\end{equation*}\n\tHence\n\t\\begin{equation*}\n\tKir_\\pi \\Phi(x)=\\frac{d\\pi_{\\Phi}^1}{d\\mu} = \\int_{y\\in X}\\Phi(x,y)K(x,y) d\\nu(y),\n\t\\end{equation*}\n\tas desired.\n\\end{proof}\n\nFor a function $f\\in\\mathscr{C}_c(X)$, the function $\\Phi(x,y)=f(y)-f(x)$ is continuous and bounded and the operator $Kir_\\pi$ is well defined on $\\Phi$. The Laplacian of $f$ in this setting is, then\n\\begin{equation*}\n\\Delta_\\pi f(x) = \\int_{y\\in X} (f(y)-f(x)) K(x,y) d\\nu(y) = \\int_{y\\in X} K(x,y) f(y) d\\nu(y) - f(x).\n\\end{equation*}\nOr, in terms of operators, $\\Delta_\\pi=\\mathcal{K}-I$, with $\\mathcal{K}f(x)=\\int_{y\\in X} K(x,y)f(y) d\\nu(y)$.\nIn this setting, the corresponding diffusion (P), is given by\n$u(x,t) =e^{t\\Delta_\\pi} f(x)$,\nwith $e^{t\\Delta_\\pi}=\\sum_{k\\geq 0}\\frac{t^k}{k!}(\\mathcal{K}-I)^k= e^{-t}\\sum_{m\\geq 0}\\frac{t^m}{m!}\\mathcal{K}^m$.\n\nAs before, the existence and structure of steady states for $t\\to\\infty$ depends on the particular dynamics of the sequence $\\{\\mathcal{K}^j: j\\geq 0\\}$ of iterations of the operator $\\mathcal{K}$ induced by the kernel $K$ and the measure $\\nu$ on $X$. Next we explore the case of dyadic Markov kernels, where the spectral analysis can be explicitly carried through Haar type systems built on dyadic type families on abstract settings.\n\nLet $X=[0,1)$ with the usual distance. Let $\\mathcal{D}=\\cup_{j\\geq 0}\\mathcal{D}^j$, $\\mathcal{D}^j=\\{I^j_k=[k2^{-j},(k+1)2^{-j}): k=0,1,\\ldots,2^j-1\\}$ be the family of standard dyadic intervals in $[0,1)$. For $j\\geq 1$, $I=I^j_k\\in\\mathcal{D}^j$ the Haar function $h_I$ is given by $h_I(x)=2^{j\/2}h^0_0(2^j x-k)$, with $h^0_0(x)=\\mathcal{X}_{[0,\\tfrac{1}{2})}(x)-\\mathcal{X}_{[\\tfrac{1}{2},1)}(x)$. The system $\\mathscr{H}\\cup \\{\\mathcal{X}_{[0,1)}\\}$, with $\\mathscr{H}=\\{h_I: I\\in\\mathcal{D}\\}$, provides an orthonormal basis for $L^2([0,1), dx)$. The dyadic family $\\mathcal{D}$ provides also a natural metric structure on $[0,1)$. For $x$ and $y$ in $[0,1)$, set $\\delta(x,y)=\\inf\\{\\abs{I}: I\\in\\mathcal{D} \\textrm{ with } x,y\\in I\\}$. Then $\\delta$ is an ultrametric on $[0,1)$ whose balls are the dyadic intervals. In fact, $B_{\\delta}(x,r)=\\{y\\in [0,1): \\delta(x,y)<r\\}=[0,1)$ for $r\\geq 1$ and $B_{\\delta}(x,r)=I$, where $x\\in I\\in\\mathcal{D}^j$ and $2^{-j}<r\\leq 2^{-j+1}$.\n\nLet $d\\mu=d\\nu=dx$ on the Borel subsets of $[0,1)$ and let us consider a special type of non trivial couplings. We say that an absolutely continuous coupling $\\pi$ of $dx$ with itself is a dyadic coupling if $K(x,y)=\\tfrac{d\\pi}{dx dy}=\\varphi(\\delta(x,y))$ for some nonnegative real function defined on $[0,1]$. Notice that $K(x,y)=\\varphi(\\delta(x,y))$ is a symmetric kernel since $\\delta$ is symmetric. On the other hand since $\\pi$ is a coupling probability measure, Lemma~\\ref{lem:couplingMarkov} implies that $\\int_{[0,1)}\\varphi(\\delta(x,y)) dy=1$ for every $x\\in [0,1)$. Let us mention at this point that this type of kernels have been considered in \\cite{AiGoMo18CLT} regarding some extension of the Central Limit Theorem. Set $\\mathcal{M}_\\delta (dx)$ to denote the class of the all kernels $K$ of the form $\\varphi\\circ \\delta$ with $\\int_{[0,1)}K(x,y)dy=1$.\n\nThe next statement contains some basic properties of these Markov kernel. Its proof can be found in \\cite{AiGoMo18CLT}.\n\\begin{proposition}\n\t\\quad\n\\begin{enumerate}[(1)]\n\t\\item $K\\in\\mathcal{M}_{\\delta}(dx)$ if and only if $K(x,y)=\\sum_{j\\geq 0}\\alpha_j 2^j \\mathcal{X}_{(0,2^{-j}]}(\\delta(x,y))$ with\n\t\\begin{enumerate}[(a)]\n\t\t\\item $\\sum_{j\\geq 0} \\abs{\\alpha_j}<\\infty$,\n\t\t\\item $\\sum_{l\\leq j}\\alpha_l 2^l\\geq 0$, for every $j\\geq 0$,\n\t\t\\item $\\sum_{j\\geq 0} \\alpha_j = 1$.\n\t\\end{enumerate}\n\\item For $K\\in\\mathcal{M}_\\delta (dx)$ the operator $\\mathcal{K}f(x)=\\int_{[0,1)}K(x,y)f(y)dy=\\int_{[0,1)}\\varphi(\\delta(x,y)) dy$ has the special resolution $\\mathcal{K}(\\mathcal{X}_{[0,1)})=\\mathcal{X}_{[0,1)]}$ and $\\mathcal{K}h=\\lambda_h h$, for $h\\in\\mathscr{H}$, with $\\lambda_h=\\sum_{j\\geq j(h)}\\alpha_j$, where $j(h)$ is the scale parameter of the support of $h$.\n\\item Since each $\\lambda_h$ depends only on the scale of $j$ of $h$ we denote the sequence by $\\{\\lambda_j: j\\geq 0\\}$. With this notation $\\lambda_0=1$ and $\\lambda_j\\to 0$ as $j\\to\\infty$.\n\\end{enumerate}\n\\end{proposition}\n\n\nFrom the above proposition we readily obtain the spectral analysis for the Laplacian $\\Delta_\\pi$. As a consequence we obtain the Haar-Fourier approach to the solution of diffusion in this setting.\n\n\\begin{proposition}\\label{thm:heatsolutioncoupling}\n\tLet $K\\in\\mathcal{M}_\\delta(dx)$ as before. Then\n\t\\begin{enumerate}\n\t\t\\item $\\Delta_\\pi\\mathcal{X}_{[0,1)}=0$ and $\\Delta_\\pi h= (\\lambda_h -1)h$, $h\\in\\mathscr{H}$;\n\t\t\\item for $f\\in L^2([0,1), dx)$, the function $u(x,t)=\\int_{[0,1)} f + \\sum_{h\\in\\mathscr{H}}e^{t(\\lambda_h -1)}\\proin{f}{h} h(x)$ solves the problem\n\\begin{equation*}\n(P)\\,\\left\\{\n\\begin{array}{ll}\n\\frac{\\partial u}{\\partial t} = \\Delta_\\pi u,\\, & t>0, x\\in [0,1)\\\\\nu(x,0)=f(x) \\, & x\\in [0,1).\n\\end{array}\n\\right.\n\\end{equation*}\n\\end{enumerate}\t\n\\end{proposition}\n\n\nThe formula provided by $(2)$ in Proposition~\\ref{thm:heatsolutioncoupling}  shows that the steady state of this diffusion is the mean value of the initial condition, in agreement with the discrete case in Proposition~\\ref{propo:limitMarkovmatrix}, when $m_j=\\tfrac{1}{n}$ for every $j$.\n\n\\section{Deterministic coupling}\nLet $X$ be a locally compact Hausdorff topological space. Let $T$ be a Borel measurable mapping on $X$. Let $\\mu$ be a Borel probability on $X$. Set $G:X\\to X\\times X$ be given by $G(x)=(x,T(x))$. For $A$ a Borel set in $X\\times X$, define $\\pi(A)=\\mu(G^{-1}(A))$. Hence, the first marginal $\\pi^1$ of $\\pi$ is $\\mu$ and the second is $\\nu(B)=\\mu(T^{-1}(B))$. See \\cite{Villanibook} for details.\n\\begin{theorem}\nLet $X$, $T$ and $\\pi$ as above. Then for $\\Phi\\in\\mathscr{C}_c(X\\times X)$ we have\n\\begin{equation*}\nKir_\\pi \\Phi(x) = \\Phi(x,T(x)).\n\\end{equation*}\n\\end{theorem}\n\\begin{proof}\nIn order to apply Theorem~\\ref{thm:RadonNikodymFirstMarginal}, we have to compute the first marginal $\\pi^1_\\Phi$ of $\\pi_\\Phi$. Notice first that since for every Borel set $A$ in $X\\times X$,\n\\begin{equation*}\n\\iint_{X\\times X}\\mathcal{X}_A(x,y) d\\pi(x,y) = \\pi(A) = \\mu(G^{-1}(A))=\\int_X \\mathcal{X}_A(x,T(x)) d\\mu(x),\n\\end{equation*}\nwe also have the formula\n$\\iint_{X\\times X}\\sigma(x,y) d\\pi(x,y) = \\int_X \\sigma(x,T(x)) d\\mu(x)$\nfor simple functions $\\sigma$ and also for bounded measurable functions. Hence\n\\begin{equation*}\n\\pi_\\Phi(A)=\\iint_A \\Phi d\\pi=\\iint_{X\\times X}\\mathcal{X}_A \\Phi d\\pi = \\int_X \\mathcal{X}_A(x,T(x)) \\Phi(x,T(x)) d\\mu(x).\n\\end{equation*}\nSo that\n\\begin{equation*}\n\\pi^1_\\Phi(B)=\\pi_\\Phi(B\\times X)=\\int_B \\Phi(x,T(x)) d\\mu(x).\n\\end{equation*}\nHence\n\\begin{equation*}\nKir_\\pi \\Phi(x)=\\frac{d\\pi^1_\\Phi(x)}{d\\mu}=\\Phi(x,T(x)).\n\\end{equation*}\n\\end{proof}\n\nFor a function $f\\in\\mathscr{C}_c(X)$ the corresponding Laplacian operator is then given by\n\\begin{equation*}\n\\Delta_\\pi f(x) = f(T(x))-f(x).\n\\end{equation*}\nOr, in operational form\n\\begin{equation*}\n\\Delta_\\pi = \\tau - I,\n\\end{equation*}\nwhere $\\tau f = f\\circ T$ and $I$ is the identity.\n\nLet us now consider the diffusion problem (P) in our current situation of the determi\\-nis\\-tic transport of $\\mu$ through $T$.\n A way to get the solution of (P) in this setting is provided by the explicit computation of $e^{t\\Delta_\\pi}=\\sum_{k=0}^{\\infty}\\frac{t^k}{k!}\\Delta^k_\\pi$.\n\n\\begin{lemma}\\label{lemma:formulaLaplacianSemigroupIterated}\nLet $X$, $\\mu$, $T$ and $\\pi$ be as before, then\n\\begin{equation*}\ne^{t\\Delta_\\pi}f = e^{-t}\\sum_{l\\geq 0}\\frac{t^l}{l!} (f\\circ T^l)\n\\end{equation*}\nfor every $f\\in \\mathscr{C}_c(X)$.\n\\end{lemma}\n\\begin{proof}\nSince $f$ is bounded, we see that the series above is absolutely convergent and that the $L^\\infty$-norm of the right hand side is bounded by the $L^\\infty$-norm of $f$. Since\n\\begin{equation*}\\label{eq:laplacianIterated}\n\\Delta^k_\\pi f = \\sum_{l=0}^k \\binom{k}{l}(-1)^{k-l} f\\circ T^l,\n\\end{equation*}\nwe have\n\\begin{align*}\ne^{t\\Delta_\\pi} f &= \\sum_{k\\geq 0}\\frac{t^k}{k!}\\Delta_\\pi^k f\\\\\n&=  \\sum_{k\\geq 0}\\frac{t^k}{k!}\\sum_{l=0}^k \\binom{k}{l}(-1)^{k-l} f\\circ T^l\\\\\n&= \\sum_{l\\geq 0}f\\circ T^l\\sum_{k\\geq l}\\frac{(-1)^{k-l}}{k!}\\binom{k}{l} t^k\\\\\n&= \\sum_{l\\geq 0}\\frac{t^l}{l!}f\\circ T^l\\sum_{k\\geq l}\\frac{(-1)^{k-l}}{(k-l)!} t^{k-l}\\\\\n&= e^{-t}\\sum_{l\\geq 0}\\frac{t^l}{l!}f\\circ T^l.\n\\end{align*}\n\\end{proof}\nHence in the case of the Laplacian provided by a deterministic coupling of measures, the solution of the initial problem for the equation with initial data $f$, has a really wide diversity of steady states depending on the dynamics induced by the iterated system $\\{T^l: l\\geq 0\\}$. In the next examples we only aim to illustrate this fact. For the case of ergodic mappings $T$, where $\\lim_{t\\to\\infty}u(x,t)$ is a mean value of the initial condition, we mention a preliminary result due to F.~J.~ Mart\\'in-Reyes \\cite{MartinReyespreprint}. Nevertheless, perhaps more interesting from the point of view of the steady state as a classifier of coupling and transports are some particular non-ergodic cases as those considered in the next particular cases.\n\\begin{proposition}\nLet $X=[-\\tfrac{1}{2},\\tfrac{1}{2}]$, $d\\mu_1=dx$, $T(x)= -x$. Hence for every continuous function $f$ defined on $[-\\tfrac{1}{2},\\tfrac{1}{2}]$, its even part $f_e$ is the steady state of $e^{t\\Delta}f$. Precisely\n\\begin{equation*}\ne^{t\\Delta}f\\to f_e = \\frac{f\\circ T + f}{2}, \\quad t\\to \\infty,\n\\end{equation*}\nuniformly on $[-\\tfrac{1}{2},\\tfrac{1}{2}]$.\n\\end{proposition}\n\\begin{proof}\nLet us apply Lemma~\\ref{lemma:formulaLaplacianSemigroupIterated} in the current setting. Note that $f\\circ T^l(x)=f((-1)^lx)$. Hence, with $f=f_e+f_o$ and $f_o=\\frac{f(x)-f(-x)}{2}$ the odd part of $f$,\n\\begin{align*}\ne^{t\\Delta}f(x) &= e^{-t}\\sum_{l\\geq 0}\\frac{t^l}{l!} f((-1)^l x)\\\\\n&= e^{-t}\\sum_{l\\geq 0}\\frac{t^l}{l!}\\left(f_e(x) + f_o((-1)^l x)\\right)\\\\\n&= e^{-t}\\left(\\sum_{l\\geq 0}\\frac{t^l}{l!}\\right) f_e(x) + e^{-t}\\left(\\sum_{l\\geq 0}\\frac{(-t)^l}{l!}\\right)f_o(x)\\\\\n&= f_e(x) + e^{-2t} f_o(x).\n\\end{align*}\n\\end{proof}\n\nThe next result deals with the Cantor function in $[0,1]$.\n\\begin{proposition}\n\tLet $X=[0,1]$, $T$ the Cantor function and $\\mu$ the Hausdorff probability measure supported in the Cantor set contained in $[0,1]$. Then $\\pi=\\mu\\circ G^{-1}$, with $G(x)=(x,Tx)$ is a coupling between $\\mu$ and $dx$ in $[0,1]$. Then, the steady state of the solution of (P) is given by the function $g=f\\circ T$.\n\\end{proposition}\n\\begin{proof}\n\tSince $T$ is continuous, the uniform probability $\\mu$ on the Cantor set is given on intervals by  $\\mu([a,b))=T(b)-T(a)$. Let $\\nu=\\mu\\circ T^{-1}$. For $[c,d]$ in $[0,1]$ with $c$ and $d$ that do not belong to the set of dyadic numbers $\\{k2^{-j}: j\\geq 0; k=0,\\ldots,2^j-1\\}$ we have that $T^{-1}(\\{c\\})$ and $T^{-1}(\\{d\\})$ are singletons, or $\\alpha=T^{-1}(c)$, $\\beta=T^{-1}(d)$. Then $\\nu([c,d])=\\mu(T^{-1}([c,d]))=\\mu([T^{-1}(c),T^{-1}(d)])=T(T^{-1}(d))-T(T^{-1}(c))=d-c$. Since the complement of the dyadic numbers of $[0,1]$ is dense in $[0,1]$, we get that $d\\nu=dx$. On the other hand, the first and second marginals of $\\pi$ are $\\mu$ and $\\nu$ respectively. The solution of\n\t\\begin{equation*}\n\t\\left\\{\n\t\\begin{array}{ll}\n\t\\frac{\\partial u}{\\partial t} = \\Delta_\\pi u,\\, & x\\in [0,1), t>0, \\\\\n\tu(x,0)=f(x) \\, & x\\in [0,1).\n\t\\end{array}\n\t\\right.\n\t\\end{equation*}\nwith $f$ continuous on $[0,1]$ is given by $u(x,t)=e^{-t}\\sum_{l\\geq 0} \\frac{t^l}{l!} f\\circ T^l(x)$. Let us compute $T^l(x)$ for $x\\in [0,1]$. Assume that $x\\in L^j_k$ the $k$-th middle third deleted in the $j$-th approximation of the Cantor set. The central point of $L^j_k$ is $k2^{-j}$ and $T(k2^{-j})=k2^{-j}$. Hence $T(x)=k2^{-j}$, $T^2(x)=T(k2^{-j})=k2^{-j}$. So that $T^l(x)=x$ for $l=0$ and $T^l(x)=k2^{-j}$ for every $l\\geq 1$. Thus\n\\begin{align*}\nu(x,t) &= e^{-t}\\left[f(x)+ \\left(\\sum_{l\\geq 1}\\frac{t^l}{l!}\\right)f(k2^{-j})\\right]\\\\\n&= e^{-t} f(x) + e^{-t}(e^t-1) f(k2^{-j})\\\\\n&= e^{-t} [f(x)-f(k2^{-j})] + f(k2^{-j}).\n\\end{align*}\t\nWhich tends to $f(k2^{-j})=f(T(x))$ for $t\\to\\infty$.\n\\end{proof}\n\n\n\\providecommand{\\bysame}{\\leavevmode\\hbox to3em{\\hrulefill}\\thinspace}\n\\providecommand{\\MR}{\\relax\\ifhmode\\unskip\\space\\fi MR }\n\\providecommand{\\MRhref}[2]{%\n\t\\href{http:\/\/www.ams.org\/mathscinet-getitem?mr=#1}{#2}\n}\n\\providecommand{\\href}[2]{#2}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{To do's}\n\t\n\n\n\t\n\t\n\t\n\t\n\n\n\n\n\n\n\n\n\n    \\maketitle\n\n\n\\section{Introduction}\nIn~\\cite{BBM_01_Another_look}, Bourgain, Brezis and Mironescu established the following  (BBM) difference quotient representation of Sobolev spaces. Let $\\Omega \\subset \\mathbf{R}^n$ be an open set with Lipschitz boundary and let $u \\in W^{1,p}(\\Omega)$, then \n\\[\n\t\\lim_{\\varepsilon \\to 0^+} \\int_\\Omega\\int_\\Omega \\frac{|u(y) - u(x)|^p}{|y - x|^p} \\, \\rho_\\varepsilon(y-x) \\;\\mathrm{d} y \\;\\mathrm{d} x= K_{p,n} \\int_\\Omega |\\nabla u|^p,\n\\]\nwhere $K_{p,n}$ is a constant depending solely on $p \\in [1,\\infty)$ and the spatial dimension $n$. Here,\n$\\{\\rho_\\varepsilon\\}_{\\varepsilon > 0}\\subset L^1(\\mathbf{R}^n)$ is a family of probability non-negative radial functions, that is,   \n\\begin{equation}\\label{eq:eps1}\n\\|\\rho_\\varepsilon\\|_{L^p(\\mathbf{R}^n)}  = 1,\n\\end{equation}\nwhich approximates the Dirac mass at zero in the sense that\n\\begin{equation}\\label{eq:eps}\n\\lim_{\\varepsilon \\to 0^+} \\|\\rho_\\varepsilon\\|_{L^p(\\mathbf{R}^n \\setminus B_\\delta)} \n= 0 \\qquad \\text{for all} \\; \\delta > 0.\n\\end{equation}\nTheir result states that the converse also holds in the range $1 < p < \\infty$. In this regard, D\\'avila~\\cite{Davila_02} showed that a related representation holds for $BV(\\Omega)$ in the limiting case $p = 1$. \nIt is well-known that a map $u \\in L^1_\\mathrm{loc}(\\mathbf{R}^n)$ belongs to the homogeneous Sobolev space $u \\in \\dot W^{1,p}(\\mathbf{R}^n)$ if and only if \n\\begin{equation}\\label{eq:pointwiseD}\n\t\\liminf_{h \\to 0^+} \\|\\delta_{h\\xi} u \\|_{L^p} < \\infty \\quad \\text{for all directions $\\xi \\in \\mathbb{S}^{n-1}$},\n\\end{equation}\nwhere \n\\[\n\t\\delta_{\\xi} u(x)\\coloneqq \\frac{u(x + \\eta) - u(x)}{|\\eta|}, \\qquad x \\in \\mathbf{R}^n,\n\\] \nis the $|\\eta|$-scale point-wise difference quotient of $u$ on the direction $\\frac \\eta{|\\eta|}$. When~\\eqref{eq:pointwiseD} holds, then the limit in~\\eqref{eq:pointwiseD} exists and \n\\[\n\tD_h u \\longrightarrow Du \\; \\text{strongly in $L^p(\\mathbf{R}^n)^n$, \\, as $h \\to^+ 0$.}\n\\]\n\n\nMotivated by the aforementioned BBM criterion and by the results contained in~\\cite{Du_13} criterion, where the first rigorous non-local vector calculus was developed for the gradient, curl, and divergence operators in Hilbert spaces, Mengesha and Spector~\\cite{MS} proved that the non-local gradient \\begin{align*}\n\t\\mathfrak{D}_{\\varepsilon} u & \\coloneqq n \\left(\\pv \\int  \\delta_{\\xi} u \\, \\frac{\\xi}{|\\xi|} \\, \\rho_\\varepsilon(\\xi)  \\;\\mathrm{d} \\xi\\right) \n\\end{align*}\n``localizes'' towards the classical gradient $D$ in the spirit of the Bourgain--Brezis--Mironescu characterization of Sobolev space above. More precisely,  they showed that if $u$ belongs to a topological function space $(X,\\tau)$, then \n\\[\n\t\\mathfrak D_\\varepsilon u \\stackrel{\\tau}\\longrightarrow Du\n\\] \nfor a wide selection of relevant topological pairs $(X,\\tau)$; including the Sobolev space $\\dot W^{1,p}(\\mathbf{R}^n)$, and the space of functions with bounded variation $\\dot {BV}(\\mathbf{R}^n)$ endowed with area convergence topology (which is finer than the strict topology of measures). In particular, they obtained new derivative-free characterizations of Sobolev and BV spaces in the spirit of (BBM).  This work is motivated by several the introduction of the peridynamics model of continuum mechanics, which is a model defined over nonlocal integral operators describing physically relevant quantities and laws. For a more detailed bibliographical account of this model, we refer the reader to~\\cite{DM_16,MS} and references therein. \n\n\n\nThe objective of this paper is twofold: Firstly,  we provide a simple formula indenting the natural distributional radial nonlocal operators $\\mathfrak{A}_\\rho$ associated with any general constant-coefficient first-order differential operator $\\mathcal{A}$, in terms of their associated principal symbol. Secondly, \nwe introduce the nonlocal spherical operators $\\mathscr A_s$, for which we establish smooth, distributional, $L^p$ and area-convergence localizations. With the aid of the spherical kernels, we are able to give a straightforward proof of the localization of the nonlocal radial operators \\emph{on spaces of distributions}; for $\\mathscr D, \\mathscr D'$, and $L^p$ topologies as well as the \\emph{area-convergence of measures}. In particular, we obtain a derivative-free characterization of $L^p$ and measure-valued anisotropic gradients. \n\nOur findings contain certain generalizations of the Du and Mengesha~\\cite{DM_16}, where localization results were already introduced and established \\emph{on function spaces}, for nonlocal operators arising from general cubic third-order tensors; with respect to the strong $L^p$ topology and the \\emph{coarser strict topology of measures} (amongst other interesting topologies). In this regard, our symbolic calculus formulas appear to substantially improve upon the transparency of the framework developed in~\\cite{DM_16}.  Lastly, it is worth mentioning that our methods \\emph{differ significantly} from the ones developed in previous works, as they are based on the study of localization concerning spherical surface kernels (cf. {\\scshape Section}~\\ref{sec:spherical}). An essential advantage of this is that our proofs can be easily adapted to find localizations for kernels associated with iso-symmetric shapes (such as cubes, regular polygons, or rings) upon a simple modification of the kernels. This even applies to non-symmetric profiles (in this case, the localization would yield approximations for certain non-isotropic norms).\n\n\n\\section{Set up and main results}\n\nWe consider a general constant coefficient first-order homogeneous distributional differential operator $\\mathcal{A} : \\mathscr D'(\\mathbf{R}^n;V) \\to \\mathscr D'(\\mathbf{R}^n;W)$, where $V,W$ are finite-dimensional inner product spaces (identified with their own duals).  Such operators can always be written in the form. {\n\\begin{equation}\\label{eq:A}\n\t\\mathcal{A} = \\sum_{i = 1}^n A_i \\partial_i\\;, \\qquad A_i \\in \\mathrm{Hom}(V,W).\n\\end{equation}\nThe gradient, divergence, curl, symmetric gradient, exterior differential, del-var are examples of such general operators.  We recall that, in this form and up to a complex constant, the principal symbol associated with $\\mathcal{A}$ is given by the $\\mathrm{Hom}(V,W)$-valued linear polynomial \n\\[\n\t\\mathbb{A}(\\xi) \\coloneqq \\sum_{i = 1}^n \\xi_iA_i\\,,\\qquad \\xi \\in \\mathbf{R}^n.\n\\] \nIn light of our previous discussion, we formally define the nonlocal $\\mathcal{A}$-gradient is defined as\n\\begin{align*}\n\t\\mathfrak{A}_\\varepsilon u(x)  =   \\, \\pv  \\, n \\int \\frac{\\mathbb{A}(h)[u(x) - u(x)]}{|y-x|^2}\\, \\rho_\\varepsilon(y-x) \\, dy\n\\end{align*}\nwhere $u : \\mathbf{R}^n \\to V$ is a sufficiently well-behaved vector field.\nBy looking at each coordinate separately and invoking the results discussed in the introduction, we deduce that if  $u \\in \\dot W^{1,p}(\\mathbf{R}^n; V)$, then\n\\begin{equation}\\label{eq:first}\n\t\\mathfrak{A}_\\varepsilon u \\longrightarrow \\mathcal{A} u \\quad \\text{strongly in $L^p$}.\n\\end{equation}\nIn general, the requirement $Du \\in L^p$ is \\emph{too restrictive} to deduce this $L^p$ localization, as one would expect the $p$-integrability of $\\mathcal{A} u$ to be necessary and sufficient for its validity (cf. {\\scshape Theorem}~\\ref{thm:Lp}). Indeed, the Calderon--Zygmund estimate $\\|D u\\|_{L^p} \\cong \\|\\mathcal{A} u\\|_{L^p}$ only holds for elliptic operators in the range $1 < p < \\infty$.  The situation worsens for $p=1$, where the requirement $Du \\in L^1$ is  restrictive, even for elliptic operators (cf.~\\cite{KK_16,Ornstein_62}). \n\n\\subsection{Nonlocal calculus} Motivated by this discussion, our main object of study will be the following nonlocal spherical and radial (difference quotient) operators. \n\n\\begin{definition}[Nonlocal spherical operator]For a given test function map $u\\in \\mathscr D(\\mathbf{R}^n;V)$ and $s >0$, we define the \\emph{$s$-spherical operator} \n\\begin{align*}\n\t\\mathscr A_s u(x) & \\coloneqq \\frac{n}{s}  \\fint_{\\partial B_s} \\Omega_\\mathbb{A}(\\omega) [\\delta_\\omega u(x)] \\, dS(\\omega) \\\\[1em]\n\t& \\phantom{:}= \\frac{n}{s}  \\fint_{\\partial B_s} \\mathbb{A}\\left(\\frac{\\omega}{s}\\right) \\left[\\frac{u(x+ \\omega) - u(x)}{s}\\right] \\, dS(\\omega), \\qquad x \\in \\mathbf{R}.\n\\end{align*}\nHere, $S$ denotes the uniform $(n-1)$-dimensional Hausdorff measure on $\\partial B_s$ and where $\\Omega_\\mathbb{A}$ is the zero-homogeneous profile of $\\mathbb{A}$.\n\\end{definition}\n\nHereinafter $\\rho : \\mathbf{R}^n \\to [0,\\infty)$ will denote an integrable radial Borel function with profile $\\hat \\rho : \\mathbf{R} \\to [0,\\infty)$. With a slight abuse of notation (between $\\rho$ and $\\varepsilon$), we define a nonlocal operator associated with $\\mathcal{A}$ through the radial map $\\rho$.\n\n\n\\begin{definition}[Nonlocal radial operator]\n For a given test vector field $u \\in\\mathscr D(\\mathbf{R}^n;V)$, we define the \\emph{nonlocal operator}\n\t\\begin{align*}\\label{eq:rho}\n\t\t\\mathfrak{A}_\\rho u(x) & \\coloneqq \\pv \\, n  \\int  \\Omega_\\mathbb{A}(h)[\\delta_{h} u(x)] \\, \\rho(h)\\, dh \\\\[1em]\n\t& \\phantom{:}= \\pv \\, n  \\int  \\frac{\\mathbb{A}(y-x)[u(y) - u(x)]}{|y-x|^2} \\, \\, \\rho(y -x)dy, \\quad x \\in \\mathbf{R}^n.\n\t\\end{align*}\n\\end{definition}\nThese nonlocal operators are linked through the Bochner integral identity\n\\[\n\t\\mathfrak{A}_\\rho u = \\int_0^\\infty n \\omega_n \\hat \\rho(s) s^{n-1} \\mathscr A_s u \\, ds.\n\\]\nIt is straightforward to verify that both $\\mathscr A_s$ and $\\mathfrak{A}_\\rho$ are bounded integral  operators from $\\mathscr D(\\mathbf{R}^n;V)$ into $\\mathscr D(\\mathbf{R}^n;W)$ ---this will be formally discussed in {\\scshape Sections}~\\ref{sec:spherical} and~\\ref{sec:radial}--- and  are also stable under integration by parts:\n\\begin{proposition}[Integration by parts]\nLet $u \\in \\mathscr D(\\mathbf{R}^n;V)$ and $\\varphi \\in \\mathscr D(\\mathbf{R}^n,W)$ be test functions. Then\n\\[\n\t\\int \\mathscr A_s u \\cdot \\varphi = \\int u \\cdot \\mathscr A_s^* \\varphi\n\\]\nand\n \\[\n \t\\int \\mathfrak{A}_\\rho u \\cdot \\varphi = \\int u \\cdot \\mathfrak{A}^*_\\rho \\varphi\n \\] \n Here $\\mathscr A_s^*$ and\n  $\\mathfrak{A}^*_\\rho$ are the respective spherical and nonlocal operators associated with $\\mathcal{A}^* = - \\sum_{i = 1}^n A^t_i \\partial_i$, the formal adjoint of $\\mathcal{A}$. \n\\end{proposition}\nUsing duality, it is, therefore, possible to \\emph{uniquely and continuously} extend  both $\\mathscr A_s$ and $\\mathfrak{A}_\\rho$  to all distributions taking values on the vector-space $V$:\n\n\\begin{definition}[Distributional operators] Let $T \\in \\mathscr D'(\\mathbf{R}^n;V)$. The distributional spherical and nonlocal $\\mathcal{A}$-gradient are defined respectively as the distributions acting on test functions $\\varphi \\in \\mathscr D(\\mathbf{R}^n;W)$ as\n\\[\n\t\\mathscr A_s T(\\varphi) = T(\\mathscr A_s^* \\varphi) \\qquad \\text{and} \\qquad\n\t\\mathfrak{A}_\\rho T(\\varphi) = T(\\mathfrak{A}_\\rho^* \\varphi).\n\\]\n\\end{definition}\nBy construction, it stems that $\\mathscr A_s,\\mathfrak{A}_\\rho :  \\mathscr D'(\\mathbf{R}^n;V) \\to  \\mathscr D'(\\mathbf{R}^n;W)$ are bounded linear operators. Given that we shall often work directly with distributions, it will be useful to work with extended $L^p$-norms:\n\n\\begin{definition}[Extended $L^p$ norms] Let $1 \\le p \\le \\infty$ and let $q$ be the the H\\\"older conjugate of $p$. For a distribution $T\\in \\mathscr D'(\\mathbf{R}^n;V)$, we define the extended $L^p$-norm of $T$ as\n\t\\begin{align*}\n\t\t\\|T\\|_{p} & = \\left(\\int |T|^p\\right)^\\frac{1}{p}  \\coloneqq \\sup \\set{ T[\\varphi]\\,, }{\\varphi \\in \\mathscr D(\\mathbf{R}^n;V), \\|\\varphi\\|_{L^q} \\le 1}\n\t\\end{align*}\n\twhere $q$ is the H\\\"older conjugate of $p$ satisfying $p^{-1} + q^{-1} = 1$.\n\t\\end{definition}\n\t\n\\begin{remark}If $p \\in (1,\\infty]$, then $\\|T\\|_p$ is finite if and only if $T$ can be respresented by a $p$-integrable map (bounded map when $p = \\infty)$. Similarly, $\\|T\\|_1$ is finite if and only if $T$ can be represented by a measure with bounded total variation. \n\\end{remark\n\t\nOur first result establishes extended $L^p$-norm bounds and localizations of the spherical operators in terms of the $\\mathcal{A}$-gradient (its proof is contained in {\\scshape Section}~\\ref{sec:spherical}):\n\n\\begin{theorem}[Spherical localization]\\label{thm:spherical_localization}\\ \\begin{enumerate}\n\\setlength{\\itemsep}{10pt}\\setlength{\\parskip}{10pt}\n\\item If $u \\in \\mathscr D(\\mathbf{R}^n;V)$, then \n\\[\n\t\\mathscr A_s u \\stackrel{\\mathscr D}\\longrightarrow \\mathcal{A} u \\qquad \\text{as $s \\to 0^+$}.\n\\]\nand\n\\[\n\\int f(\\mathscr A_s u) \\le \\int f(\\mathcal{A} u)\n\\]\nfor all convex functions $f : W \\to \\mathbf{R}$. \n\n\\item If $T \\in \\mathscr D'(\\mathbf{R}^n;V)$, then\n\\[\n\t\\mathscr A_s T \\stackrel{\\mathscr D'}\\longrightarrow \\mathcal{A} T \\qquad \\text{as $s \\to 0^+$}.\n\\]\nand\n\\[\n\t\\|\\mathscr A_s T\\|_p \\le \\|\\mathcal{A} T\\|_p \\quad \\text{for all $p \\in [1,\\infty]$,}\n\\]\n\n\n\\item If $\\mathcal{A} T \\in L^p(\\mathbf{R}^n;W)$ for some $p \\in [1,\\infty]$, then, in fact,\n\\[\n\\mathscr A_s T \\in (C \\cap L^p)(\\mathbf{R}^n;W).\n\\]\nIn this case, and for $p \\in [1,\\infty)$, the distributional convergence improves to the strong convergence\n\\[\n\t\\mathscr A_s T \\stackrel{L^p}\\longrightarrow \\mathcal{A} T \\qquad \\text{as $s \\to 0^+$}.\n\\]\n\\item If $\\mathcal{A} T\\in \\mathcal{M}(\\mathbf{R}^n;W)$, then \n\\[\n\\mathscr A_s T \\in (L^\\infty \\cap L^1)(\\mathbf{R}^n;W) \n\\]\nand the absolutely continuous measures $\\mathscr A_s T \\, \\mathscr L^n$ converge to $\\mathcal{A} u$ in the area sense of measures as $s \\to 0$, that is,\n\\[\n\t\\lim_{s \\to 0^+} \\int f(\\mathscr A_s T) =  \\int f(\\mathcal{A}^a T) + \\int f^\\infty(\\mathcal{A}^s T) \n\\]\nfor all convex maps $f: W \\to \\mathbf{R}$ with linear growth at infinity. (\\,\\footnote{Here, \n\\[\n\tf^\\infty(w) \\coloneqq \\lim_{t \\to \\infty} \\frac {f(tw)}{t}, \\qquad w \\in W,\n\\]\nis the recession function of $F$ and\n\\[\n\t\\mathcal{A} u = \\mathcal{A}^a u\\, \\mathscr L^n +  \\mathcal{A}^s  u, \\qquad |\\mathcal{A}^s u| \\perp \\mathscr L^n,\n\\]\nis the Lebesgue--Radon--Nykod\\'ym decomposition of $\\mathcal{A} u$.})\n\\end{enumerate}\n\\end{theorem}\n\nThanks to a simple measure-theoretic lemma discussed in the Appendix, we obtain the following representation for the spherical operators when acting on locally summable maps (cf. Eqn.~\\ref{eq:rep}).\n\\begin{lemma}[Convolution represenation]\\label{lem:GG}\n\tLet $u \\in L^1_\\mathrm{loc}(\\mathbf{R}^n;V)$ and further assume that $\\mathcal{A} u \\in \\mathcal{M}(\\mathbf{R}^n;W)$. Then,  \n\t\\[\n\t\t\\mathscr A_s u(x) =  \\frac{\\mathcal Au(B_s(x))}{\\omega_n s^n}.\n\t\\]\nat almost every $x \\in \\mathbf{R}^n$.\n\\end{lemma}\n\nThe localization result for the spherical operators builds directly into a swift proof of the estimates and localization results for nonlocal operators, which will be described next. (The proof of this result is a direct consequence of Eqn.~\\eqref{eq:p_radial} and {\\scshape Proposition}~\\ref{prop:12}.) \n\n\\begin{theorem}\\label{thm:Lp_bounds} Let $1 \\le p < \\infty$ and let $u \\in \\mathscr D'(\\mathbf{R}^n;V)$. Then\n\\[\n\t\\|\\mathfrak{A}_\\rho u \\|_p  \\le  \\|\\rho\\|_{L^1}^\\frac{1}{p} \\|\\mathcal{A} u \\|_p.\n\\]\nIf moreover $\\|\\mathcal{A} u\\|_p < \\infty$, then\n\\[\n\\mathfrak{A}_\\rho u \\in L^p(\\mathbf{R}^n;W)\n\\]\nand the extended $L^1$-norm estimate\n\\[\n\t\\int |f(\\mathfrak{A}_\\rho u)| \\,d x \\le \\int |f(\\mathcal{A}^a u)| \\,d x +  \\int |f^\\infty(d\\mathcal{A}^s u)|\\,\n\\]\nholds for all convex maps $F : W \\to \\mathbf{R}$. \n\\end{theorem}\n\n\n\n\nFor locally summable maps with locally bounded  measure $\\mathcal{A}$-gradients, the operator $\\mathfrak{A}_\\rho$ is expressed, as one would expect, by the principal value integral (this follows from {\\scshape Lemma}~\\ref{lem:11}):\n\\begin{lemma}\nLet $u \\in  L^1_\\mathrm{loc}(\\mathbf{R}^n;V)$ with\n$\\|\\mathcal{A} u\\|_p < \\infty$.  Then, the limit\n\\[\n\t\\pv \\, n \\int  \\frac{\\mathbb{A}(y-x)[u(y) - u(x)]}{|y-x|^2} \\, dy \n\\]\nexists and coincides with $\\mathfrak{A}_\\rho u(x)$ for almost every $x \\in \\mathbf{R}^n$. Here, $u(x)$ is understood as the Lebesgue representative of $u$ at $x$. \n\\end{lemma}\n\nOur first main localization result concerns both an \\emph{energy criterion}  and an approximation result for the extended $L^p$-norms of the distribution $\\mathcal{A} u$, in terms of $\\{\\mathfrak{A}_{\\varepsilon} u\\}_\\varepsilon$, where $\\{\\rho_\\varepsilon\\}_{\\varepsilon > 0}$ is a family of radial probability functions satisfying~\\eqref{eq:eps1}-\\eqref{eq:eps}. The precise statement, which follows by gathering the results contained in Eqn.~\\eqref{eq:distributional_convergence}, {\\scshape Lemma}~\\ref{lem:11}\n{\\scshape} Proposition~\\ref{prop:12}, is the following:\n\\begin{lemma}\\label{lem:Lp_extended}\nLet $\\varphi \\in \\mathscr D(\\mathbf{R}^n;V)$ and let $u \\in \\mathscr D'(\\mathbf{R}^n;V)$ a. Then\n\\[\n\t\\mathfrak{A}_\\varepsilon \\varphi \\stackrel{\\mathscr D}\\longrightarrow \\mathfrak{A} \\varphi \\qquad \\text{and}\\qquad \\mathfrak{A}_\\varepsilon u \\stackrel{\\mathscr D'}\\longrightarrow \\mathfrak{A} u.\n\\]\nThe extended $L^p$-norm limit \n\\[\n\t\\lim_{\\varepsilon \\to 0^+} \\int |\\mathfrak{A}_{\\varepsilon} u|^{p}  \\in [0,\\infty]\n\\]\t\n always exists. \n \n If moreover $u \\in L^1_\\mathrm{loc}(\\mathbf{R}^n;V)$, then\n\t\\[\n\t\t  \\lim_{\\varepsilon \\to 0^+} \\, \\int \\bigg| \\pv n  \\int\\frac{\\mathbb{A}(y-x)[u(y) - u(x)]}{|y-x|^2} \\, \\rho_\\varepsilon(y-x) \\;\\mathrm{d} y\\,\\bigg|^p = \\int |\\mathcal{A} u|^p .\n\t\\]\n\\end{lemma} \nThis result and {\\scshape Corollary}~\\ref{cor:2} directly imply the following localization (and characterization) criterion: \n\n\\begin{theorem}[$L^p$-localization]\\label{thm:Lp}\nLet $1 < p < \\infty$ and let $u \\in \\mathscr D'(\\mathbf{R}^n;V)$. Then\n$\\mathcal{A} u \\in L^p(\\mathbf{R}^n;W)$ if and only if \n\\[\n\t\\liminf_{\\varepsilon \\to 0^+}   \\|\\mathfrak{A}_{\\varepsilon} u\\|_{p}  < \\infty.\n\\]\nMoreover, in this case\n\\[\n\t\\mathfrak A_{\\varepsilon} u \\longrightarrow \\mathcal{A} u \\quad \\text{strongly in $L^p(\\mathbf{R}^n;W)$.}\n\\]\n\\end{theorem}\n\n\\begin{remark}\nFor $p = 1$, it can be shown that the assertion still holds in the following sense: if  $\\mathcal{A} u \\in L^1(\\mathbf{R}^n;W)$, then the  \nrepresentation of {\\scshape Theorem}~\\ref{thm:Lp} exists and agrees with the  $L^1$-norm of $\\mathcal{A} u$ (cf. Corollary~\\ref{cor:L1_sufficiency} below). In general, the converse fails due to concentration effects appearing on the difference quotient integral.\n\\end{remark}\n\t Our second main result covers this case precisely when $\\mathcal{A} u$ is a vector-valued Radon measure.\n\tNotice that, with our definition of extended norm, it holds that a $W$-valued distribution $T$ belongs to $\\mathcal{M}_b(\\mathbf{R}^n;W)$, the space of $W$-valued Radon measures with bounded total variation, if and only if $\\|T\\|_1$, in which case it also holds $\\|T\\|_1 = |T|(\\mathbf{R}^n)$. In particular, a direct consequence of {\\scshape Theorem}~\\ref{thm:Lp_bounds} and {\\scshape Lemma}~\\ref{lem:Lp_extended} is the following criterion and approximation result:\n\n\\begin{theorem}[Area convergence localization]\\label{thm:M}\nLet $u \\in L^1_\\mathrm{loc}(\\mathbf{R}^n;V)$. Then $\\mathcal{A} u\\in \\mathcal{M}_b(\\mathbf{R}^n;W)$ if and only if \n\\[\n\t\\liminf_{\\varepsilon \\to 0^+}   \\|\\mathfrak{A}_{\\varepsilon} u\\|_1  < \\infty.\n\\] \nMoreover, in this case\n\\[\n\t(\\mathfrak A_{\\varepsilon} u) \\mathscr L^n  \\longrightarrow \\mathcal{A} u \\quad \\text{in the area sense of measures.}\n\\]\nIn particular, \n\\[\n\t\\lim_{\\varepsilon \\to 0} \\int f(\\mathfrak{A}_\\varepsilon u) \\, dx  = \\int f(\\mathcal{A}^a u) \\, dx + f^\\infty(\\frac{\\mathcal{A} u}{|\\mathcal{A} u|}) \\, d|\\mathcal{A}^s u|\n\\]\nfor all continuous integrands $f: W \\to \\mathbf{R}$ with linear growth such that the limit \n\\[\n\tf^\\infty(A) \\coloneqq \\lim_{t \\to \\infty} \\frac{f(tA)}{t} \n\\]\nexists for all $A \\in W$.\n\\end{theorem}\n\nThe particular advantage of the area convergence of measures (over the usual strict convergence) is that it discriminates the appearance of $L^1$ concentrations. In particular, we obtain the following refinement of {\\scshape Theorem}~\\ref{thm:M}:\n\n\\begin{corollary}\\label{cor:L1_sufficiency} \nIf $\\mathcal{A} u \\in L^1(\\mathbf{R}^n;W)$, then\n\\[\n\\mathfrak A_{\\varepsilon} u \\longrightarrow \\mathcal{A} u \\quad \\text{strongly in $L^1$.}\n\\]\n\\end{corollary}\n\n\n\\begin{remark}[More general shapes]\nThe results presented here can be deduced for more general shapes other than the sphere and radial symmetry. It is straightforward from our methods that the same results can be obtained by working with any boundary $\\partial U$, where $U$ is a simply connected Lipschitz set containing $0 \\in \\mathbf{R}^n$. In this particular case, the modified surface kernel takes the form \n\\[\n\t\\mathscr A_{U,s} u(x) =  \\frac{1}{s^n\\mathscr L^n(U)} \\int_{\\partial (sU)} \\mathbb{A}(\\nu(\\omega)) \\left[\\frac{u(x+ \\omega) - u(x)}{s}\\right] \\, dS(\\omega).\n\\]\nwhere $\\nu$ is the exterior normal to $sU$ at $\\partial (sU)$.\n\\end{remark}\n\n\\section{The spherical surface kernels}\\label{sec:spherical} \nIn this section, we discuss localizations concerning spherical kernels. As we shall see below, working with these lower-dimensional measure kernels arises naturally from the Gauss-Green theorem. The analysis of these kernels will be fundamental in establishing similar estimates and localizations for the radial kernels discussed in the introductions.   We shall from now on consider  the uniform spherical surface measure \n\\[\nm_s = \\frac{ \\mathcal{H}^{n-1} \\mres \\partial B_s}{|B_s|}, \\qquad s > 0.\n\\]\nTo it, we associate a \\emph{surface} kernel $K_s :\\mathbf{R}^n \\setminus \\{0\\} \\to \\mathrm{Hom}(V,W)$ by setting\n\\[\n\tK_s(d\\omega) \\coloneqq \\Omega_\\mathbb{A}(\\omega) \\, m_s(d\\omega),\n\\]\nwhere $\\Omega_\\mathbb{A}$ is the zero-homogeneous angular part of the symbol $\\mathbb{A}$ map, i.e.,  \n\\[\n\t\\Omega_\\mathbb{A}(x) \\coloneqq \\mathbb{A}\\left(\\frac{x}{|x|}\\right), \\qquad x \\in \\mathbf{R}^n - \\{0\\}.\n\\]\nFor each $s>0$, $K_s$ defines a finite $\\mathrm{Hom}(V,W)$-valued measure on $\\mathbf{R}^n$ and its therefore a tensor-valued compactly supported distribution. When applied on a test map $u \\in \\mathscr D(\\mathbf{R}^n;V)$, the Gauss--Green theorem yields the identity\n\\begin{align*}\n\tK_s \\star u(x)  & =  \\frac 1{\\omega_n s^n} \\int_{\\partial B_s(x)}  \\Omega_\\mathbb{A}(y-x)u(y) \\, dS(y) = \\frac{\\mathcal{A} u(B_s(x))}{\\omega_n s^n}.\n\\end{align*}  \nThe cancelation property \n\\begin{equation}\\label{eq:cancel}\n\t\\int_{\\mathbf{S}^{n-1}} \\mathbb{A}(\\omega) \\,dS(\\omega) = 0 \n\\end{equation}\nconveys the identity \n\\[\n\t\\mathscr A_s u = K_s \\star u \\qquad \\text{for all $\\varphi \\in \\mathscr D(\\mathbf{R}^n;V)$}\n\\]\nso that, in particular, we can reduce the study of $\\mathscr A_s$ to the study of the $K_s$-convolution operator. Notice that a direct consequence of this identity is the integration by parts formula\n\\[\n\t\\int \\mathscr A_s \\varphi \\cdot \\psi = \\int (K_s \\star \\varphi) \\cdot \\psi = \\int \\varphi \\cdot (\\tilde K_s^t \\star \\psi) = \\int \\mathscr \\varphi \\cdot \\mathscr A^*_s \\psi\n\\]\nfor all test functions $\\varphi \\in \\mathscr D(\\mathbf{R}^n;V)$ and $\\psi \\in \\mathscr D(\\mathbf{R}^n;W)$. Here, the operator $(\\tilde\\,\\sbullet\\,)$ denotes the reflection operation $x \\mapsto -x$ so that indeed $\\tilde K_s^t(h) = \\Omega_{\\mathbb{A}^*} \\, dm_s$. \n\tBy standard functional analysis theory, the convolution $K_s \\star T$ defines a $W$-valued distribution on $\\mathbf{R}^n$ for all $T \\in \\mathscr D'(\\mathbf{R}^n;V)$ by setting\n\\[\n\tK_s \\star T (\\varphi)\\coloneqq T(\\tilde K_s^t \\star \\varphi) \\quad \\forall \\varphi \\in \\mathscr D(\\mathbf{R}^n;W).\n\\]\nHere, the operator $(\\tilde\\,\\sbullet\\,)$ denotes the operation $x \\mapsto -x$. The previous convolution representation holds for all distributions $T$ such that $\\mathcal{A} T$ is a Radon measure. \nIndeed, a standard approximation argument implies that, in fact,\n\\[\nK_s \\star T \\in L^\\infty \\quad \\text{with} \\quad \\|K_s \\star T\\|_\\infty \\le \\frac{ |\\mathcal{A} u|(\\mathbf{R}^n)}{\\omega_n s^n}.\n\\]\nTherefore, by standard mollifier theory, we get that\n\\[\n\tK_s \\star T(x) = \\frac 1{|B_s|} \\lim_{\\varepsilon \\to 0^+} (\\mathcal{A}  T)_\\varepsilon(B_s(x)) \\quad \\text{for a.e. $x \\in \\mathbf{R}^n$,}\n\\]\nwhere $(\\,\\sbullet\\,)_\\varepsilon$ denotes mollification with a standard mollifier at scale $\\varepsilon$.\nLemma~\\ref{lem:Leb} implies that $|\\mathcal{A} T|(\\partial B_s(x)) = 0$ for almost every $x \\in \\mathbf{R}^n$ and therefore also the right-hand side above converges to $\\mathcal{A} T(B_s(x))\/|B_s|$ almost everywhere on $\\mathbf{R}^n$, or equivalently,\n\\begin{equation}\\label{eq:rep}\n\tK_s \\star T = \\frac{\\mathcal{A} T(B_s(\\,\\sbullet\\,))}{|B_s|} \\quad \\text{as maps in $L^\\infty$}.\n\\end{equation}\n\n\\begin{remark}In general, $K_s \\star u$ needs not to be continuous for $u \\in L^1_\\mathrm{loc}(\\mathbf{R}^n;V)$ with $\\mathcal{A} u \\in \\mathcal{M}(\\mathbf{R}^n;W)$. Indeed,  the vector field on the plane given by\n\\[\n\tu(x_1,x_2) = \\frac1{2\\pi} \\left(\\frac{(x_1,x_2 + 1)}{|(x_1,x_2+1)|^2} - \\frac{(x_1+1,x_2)}{|(x_1+1,x_2)|^2}\\right),\n\\]\nsatisfies $\\diverg u = \\delta_{(0,1)} - \\delta_{(1,0)}$. In particular, the map $x \\mapsto \\diverg u(B_1(x))$ is not continuous at $0 \\in \\mathbf{R}^2$. \\end{remark}\n\nWe devise the following direct properties from this discussion, which validate the assertions contained in {\\scshape Theorem}~\\ref{thm:spherical_localization}.\n\\begin{enumerate}[(i)]\n\\item If $u \\in \\mathscr D(\\mathbf{R}^n;V)$, then \n\\[\n\tK_s \\star u \\stackrel{\\mathscr D}\\longrightarrow \\mathcal{A} u \\qquad \\text{as $s \\to 0^+$}.\n\\]\nIndeed, by the properties of differentiation for convolutions, it suffices to observe the validity of the supremum norm estimate\n\\[\n\t|\\partial^\\alpha (K_s \\star u - \\mathcal{A} u)(x)| \\le \\fint_{B_s(x)} |\\mathcal{A} \\partial^\\alpha u(y) - \\mathcal{A} \\partial^\\alpha u(x)| \\, dy \\le s\\|D^k(\\mathcal{A} u)\\|_\\infty\n\\]\nwhere $k = |\\alpha| + 1$.\n\\item If $T \\in \\mathscr D'(\\mathbf{R}^n;V)$, then a byproduct of the previous convergence is the distributional convergence\n\\[\n\tK_s \\star T \\stackrel{\\mathscr D'}\\longrightarrow \\mathcal{A} T \\qquad \\text{as $s \\to 0^+$}.\n\\]\nIndeed, observing that $-\\mathbb{A}^t$ is the principal symbol associated with the formal adjoint $\\mathcal{A}^*$ of $\\mathcal{A}$, we deduce that\n\\[\n\tK_s \\star T (u) = T (\\tilde K_s^t \\star  u) \\longrightarrow T(\\mathcal{A}^* u) = \\mathcal{A} T(u).\n\\]\n\\item If $u \\in \\mathscr D(\\mathbf{R}^n;V)$, then Jensen's inequality give\n\\begin{equation}\\label{eq:jensen1}\n\tF(K_s \\star u) \\le   F(\\mathcal{A} u) \\star \\frac{\\chi_{B_s}}{|B_s|} \n\\end{equation}\nfor all convex maps $F: W \\to \\mathbf{R}$. Young's convolution inequality conveys the estimate \n\\[\n\\|F(K_s \\star u)\\|_{L^1} \\le \\|F(\\mathcal{A} u)\\|_{L^1} \\quad \\text{for all $p \\in [1,\\infty]$.}\n\\]\nIn particular, \n\\[\n\t\\|K_s \\star u\\|_{L^p} \\le \\|\\mathcal{A} u\\|_{L^p}  \\quad \\text{for all $1 \\le p < \\infty$.}\n\\]\n\\item The lower semicontinuity of the extended $L^p$-norms under distributional convergence and  the previous estimate gives rise to the  extended estimate\n\\[\n\t\\forall p \\in [1,\\infty], \\quad \\|K_s \\star T\\|_p \\le \\|\\mathcal{A} T\\|_p \\quad \\text{for all $T \\in \\mathscr D'(\\mathbf{R}^n;V)$}.\n\\]\nIn particular, again from lower semicontinuity and distributional convergence, we obtain\n\\[\n\t\\limsup_{s \\to 0^+} \\|K_s \\star T\\|_p \\le \\|\\mathcal{A} T\\|_p \\le \\liminf_{s \\to 0^+} \\|K_s \\star T\\|_p\\,,\n\\]\nwhich means that the limit exists and equals $\\|\\mathcal{A} T\\|_p$. \n\\item If $\\mathcal{A} T \\in L^p$ for some $p \\in [1,\\infty]$, then the fact that $|\\mathcal{A} u|(\\partial B_s(x)) = 0$ for all $x \\in \\mathbf{R}^n$ implies that\n\\[\nK_s \\star T \\in C(\\mathbf{R}^n;W) \\quad \\forall s > 0.\n\\]\nMoreover, for $1 < p < \\infty$, the convergence of the $L^p$ norms implies that the distributional converge improves to the strong convergence\n\\[\n\tK_s \\star T  \\stackrel{L^p}\\longrightarrow \\mathcal{A} T \\quad \\text{as $s \\to 0^+$.}\n\\]\n\n\\item Lastly, if $\\mathcal{A} T \\in \\mathcal{M}$, then~\\eqref{eq:rep} and the fact that the approximation by convolution for measures implies convergence with respect to the area functional gives\n\\[\n\t(K_s \\star T) \\, \\mathscr L^n  \\stackrel{\\mathrm{area}}\\longrightarrow \\mathcal{A} T \\quad \\text{as $s \\to 0^+$.}\n\\]\n\\end{enumerate}\n\n\n\n\\section{The radial kernels}\\label{sec:radial}\nLet $\\rho(x) = \\hat \\rho(|x|)$ be a non-negative radial summable function and consider the kernel $\\mathcal{K}_\\rho \\in C^\\infty(\\mathbf{R}^n - \\{0\\};\\mathrm{Hom}(V,W))$ defined by\n\\[\n\t\\mathcal{K}_\\rho(z) \\coloneqq n \\, \\Omega_\\mathbb{A}\\left(\\frac{z}{|z|}\\right)\\rho(z), \\qquad z \\in \\mathbf{R}^n \\setminus \\{0\\}.\n\\]\nThe mean-value zero property \n\\[\n\t\\int_{\\partial B_s(0)} \\mathcal{K}_\\rho(\\omega) \\, dS(\\omega) = 0 \\qquad \\text{for all $s >0$} \n\\]\nensures that the \\emph{principal value} convolution integral \n\\begin{align*}\n\t\\mathfrak{A}_\\rho u(x) & \\phantom{:}= \\pv \\int  \\mathcal{K}_\\rho(y-x) u(y) \\, dy \\\\\n\t& \\coloneqq \\lim_{\\delta \\to 0} \\int_{\\mathbf{R}^n \\setminus B_\\delta(x)} \\mathcal{K}_\\rho(y-x) u(y) \\, dy\n\\end{align*}\nis well-defined for all test maps $u \\in \\mathscr D(\\mathbf{R}^n;V)$ and all $x \\in \\mathbf{R}^n$, independently of the infinitesimal sequence $\\delta \\to 0^+$. Indeed, once more, the cancelation property~\\eqref{eq:cancel} gives\n\\[\n\t\t\\mathfrak{A}_\\rho u(x)  =  \\pv \\int \\mathcal{K}_\\rho(y-x)[u(y) - u(x)] \\, dy,\n\\]\nSince the integrand on the right-hand is dominated by a multiple of $\\|Du\\|_\\infty \\tau^x \\rho$, it follows that the limit in the principal value integral exists and $\\mathfrak{A}_\\rho u \\in L^\\infty(\\mathbf{R}^n;W)$,  with\n\\[\n\\|\\mathfrak{A}_\\rho u\\|_{L^\\infty} \\le n \\|\\Omega_\\mathbb{A}\\|_{L^\\infty} \\|Du\\|_{L^\\infty} \\|\\rho\\|_{L^1}.\n\\]\nMoreover, the absolute integrability of the integrand above and Fubini's theorem give\n\\begin{align*}\n\t\\dpr{\\mathfrak{A}_\\rho u, \\varphi} & = \\int\\int \\dpr{u(x + h) - u(x),\\mathcal{K}_\\rho(h)^t \\varphi(x)} \\, dx \\, dh \\\\\n\t& = \\int \\int \\dpr{u(z),  \\mathcal{K}_\\rho(h)^t [\\varphi(z - h) - \\varphi(z)]} \\, dz \\, dh \\\\\n\t& = -\\int \\int \\dpr{u(z),  \\mathcal{K}_\\rho(y - z)^t [\\varphi(y) - \\varphi(z)]} \\, dy \\, dz = \\dpr{u,\\mathfrak{A}^*_\\rho \\varphi},\n\\end{align*}\nfor all $\\varphi \\in \\mathscr D(\\mathbf{R}^n;W)$. This shows that the formal adjoint $(\\mathfrak{A}_\\rho)^*$ of $\\mathfrak{A}_\\rho$ is precisely $\\mathfrak{A}^*_\\rho$. By a standard duality argument, we may extend $\\mathfrak{A}_\\rho$ uniquely and continuously to all distributions $T \\in \\mathscr D'(\\mathbf{R}^n;V)$ by setting\n\\[\n\t\\mathfrak{A}_\\rho T(\\varphi) \\coloneqq T(\\mathfrak{A}_\\rho^* \\varphi), \\qquad \\varphi \\in \\mathscr D(\\mathbf{R}^n;W).\n\\]\nIn particular, this shows that, for a family $\\{\\rho_\\varepsilon\\}_{\\varepsilon > 0}$ of probability non-negative radial functions satisfying~\\eqref{eq:eps1}-\\eqref{eq:eps}, it holds\n\\begin{equation}\\label{eq:distributional_convergence}\n\t\\mathfrak A{\\rho_\\varepsilon} T \\stackrel{\\mathscr D'}\\longrightarrow \\mathcal{A} T \\qquad \\text{as $\\varepsilon \\to 0^+$}\n\\end{equation}\nfor all $T \\in \\mathscr D'(\\mathbf{R}^n;V)$.\n\nLet $p \\in [1,\\infty]$ and let $q$ be its associated H\\\"older conjugate exponent. If $T \\in \\mathscr D'(\\mathbf{R}^n;W)$, then, using that the radial integral defining $\\mathfrak{A}^* \\varphi$ is absolutely convergent as a Bochner integral of smooth maps, we deduce from the spherical kernel estimates the following one:\n\\begin{align*}\n\t\\mathfrak{A}_\\rho T(\\varphi) & = T(\\mathfrak{A}_\\rho^* \\varphi) \\\\\n\t& = T\\left(-\\int_0^\\infty n\\omega_n\\hat\\rho(r)r^{n-1}  \\tilde K_r^t  \\star \\varphi \\, dr \\right) \\\\\n\t& =  \\int_0^\\infty n\\omega_n\\hat\\rho(r)r^{n-1} T( K_r^* \\star \\varphi) \\, dr \\\\\n\t& =  \\int_0^\\infty n\\omega_n\\hat\\rho(r)r^{n-1} (K_r  \\star T)(\\varphi)  \\, dr \\\\\n\t\t& \\le  \\int_0^\\infty n\\omega_n\\hat\\rho(r)r^{n-1} \\|K_r\\star T\\|_p\\|\\varphi\\|_{L^q} \\, dr \\le \\|\\rho\\|_{L^1} \\|\\mathcal{A} T\\|_p\\|\\varphi\\|_{L^q}.\n\\end{align*} \nTaking the supremum over all $\\varphi$ with $\\|\\varphi\\|_{L^q} = 1$ gives\n\\begin{equation}\\label{eq:p_radial}\n\\|\\mathfrak{A}_\\rho T\\|_p \\le \\|\\rho\\|_{L^1} \\|\\mathcal{A} T\\|_p \\quad \\text{for all $T \\in \\mathscr D'(\\mathbf{R}^n;V)$}.\n\\end{equation} \n\n\\begin{lemma}\\label{lem:11}\nLet $u \\in  L^1_\\mathrm{loc}(\\mathbf{R}^n;V)$. Then\n\\[\n\\|\\mathfrak{A}_\\rho u\\|_p^p \\le \\liminf_{\\delta \\to 0} \\int \\left| \\int_{\\mathbf{R}^n \\setminus B_\\delta(x)} K_\\rho(y-x)u(y)\\, dy \\right|^p. \n\\] \nIf moreover $\\|\\mathcal{A} u\\|_p < \\infty$, then $\\mathfrak{A}_\\rho u \\in L^p(\\mathbf{R}^n;W)$, the limit above exists, commutes with the integral and\n\\[\n\t\\mathfrak{A}_\\rho u(x) = \\pv \\int K_\\rho(y-x)u(y) \\, dy \\qquad \\text{for almost every $x \\in \\mathbf{R}^n$.}\n\\]\n\\end{lemma}\n\\begin{remark}\nNotice that $\\mathfrak{A}_\\rho u$ is absolutely continuous even if $\\mathcal{A} u$ is only a finite measure.\n\\end{remark}\n\\begin{proof}[Proof of Lemma~\\ref{lem:11}]\nFirst, we show that $\\mathfrak{A}_\\rho u$ is given by the principal value convolution $\\pv \\mathcal{K}_\\rho \\star u$. By construction, we have that $\\mathfrak{A}_\\rho u \\in \\mathcal{M}(\\mathbf{R}^n;W)$. Since the integrand \n\\[\n(x,y) \\mapsto \\dpr{K_\\rho(y-x)u(y),\\varphi(x) - \\varphi(y)}\n\\]\nis absolutely integrable on $\\mathbf{R}^n \\times \\mathbf{R}^n$, we deduce from Fubini's theorem and a few changes of variables that, for any infinitesimal sequence $\\delta \\to 0$, it holds\n\\begin{align*}\n\t\\dpr{u , \\mathfrak{A}^*_\\rho \\varphi} & =   \\int \\pv \\int \\dpr{ K_\\rho(y-x) u(y),\\varphi(x)} \\, dx \\, dy \\\\\n\t& =  \\int\\int \\dpr{ K_\\rho(y-x) u(y),\\varphi(x) - \\varphi(y)} \\, dx \\, dy \\\\\n\t& =  \\int\\int\\dpr{K_\\rho(h)u(y),\\varphi(y - h) - \\varphi(y)} \\, dh \\, dy.\n\\end{align*}\nNow, the right-hand side can be expressed as the limit \n\\begin{align*} \n\t& = \\lim_{\\delta \\to 0} \\int_{B_\\delta(0)^c}\\int\\dpr{K_\\rho(h)[u(y) - u(y-h)],\\varphi(y - h) } \\, dy \\, dh \\\\\n\t& =  \\lim_{\\delta \\to 0}\\int \\int_{B_\\delta(0)^c} \\dpr{K_\\rho(h)[u(y) - u(y-h)],\\varphi(y-h) } \\, dh \\, dy. \\\\\n\t\t& =\\lim_{\\delta \\to 0}  \\int \\int_{B_\\delta(y)^c} \\dpr{K_\\rho(y-x)[u(y) - u(x)],\\varphi(x) } \\, dx \\, dy \\\\\n\t\t\t& =\\lim_{\\delta \\to 0}  \\int\\dpr{ \\int_{B_\\delta(x)^c} K_\\rho(y-x)u(y)\\, dy ,\\varphi(x) }  \\, dx \\\\\n\t& \\le \\|\\varphi\\|_{L^q} \\lim_{\\delta \\to 0}   \\left(\\int \\left| \\int_{\\mathbf{R}^n \\setminus B_\\delta(x)} K_\\rho(y-x)u(y)\\, dy \\right|^p \\right)^\\frac{1}{p}.\n\\end{align*}\nTaking the supremum over all $\\varphi \\in \\mathscr D(\\mathbf{R}^n;W)$ with $\\|\\varphi\\|_{L^q} = 1$, where $q$ is the H\\\"older conjugate exponent of $p$, we conclude that \n\\[\n\t\\|\\mathfrak{A}_\\rho u\\|_p^p \\le \\liminf_{\\delta \\to 0}  \\int \\left| \\int_{\\mathbf{R}^n \\setminus B_\\delta(x)} K_\\rho(y-x)u(y) \\, dy \\right|^p.\n\\]\nLet us now assume that $|\\mathcal{A} u|$ is a finite measure. Since for every $r > 0$ the function $x \\mapsto |\\mathcal{A} u|(B_r(x))$ is measurable, it follows that the non-negative function defined by\n\\[\n\t  f(x) \\coloneqq \\int_0^\\infty \\hat \\rho(r) r^{-1}|\\mathcal{A} u|(B_{r}(x))\\varphi(x) \\, dr \\in [0,\\infty], \\qquad x \\in \\mathbf{R}^n,\n\\]\nis also Borel measurable on $\\mathbf{R}^n$. Moreover,  \n Fubini's theorem and Young's convolution inequality convey that $f$ is $p$-integrable  with $\\|f\\|_{L^p} \\le \\|\\rho\\|_{L^1} \\|\\mathcal{A} u\\|_{p}$. \nLet us now define the measurable maps \n\\[\n\tg_\\delta(x) \\coloneqq \\int_{\\mathbf{R}^n \\setminus B_\\delta(x)} \\mathcal{K}_\\rho(y-x)u(y)\\, dy, \\quad x \\in \\mathbf{R}^n.\n\\]\nBy Lemma~XX and by the generalized Gauss--Green theorem it follows that $f$ dominates the family $\\{|g_\\delta|^p\\}_{\\delta > 0}$. Since also \n\\[\ng_\\delta(x) \\stackrel{\\delta \\to 0}\\longrightarrow \\pv \\int K_\\rho(y-x)u(y) \\, dy \\quad \\text{for almost every $x \\in \\mathbf{R}^n$},\n\\] \nthe Dominated convergence theorem implies that $\\mathfrak{A}_\\rho u = \\pv (\\mathcal{K}_\\rho \\star u)$.\nThis finishes the proof.\n\\end{proof}\n\n\n\\begin{proposition}\\label{prop:12}\nLet $u \\in \\mathscr D'(\\mathbf{R}^n;V)$ and further suppose that $\\mathcal{A} u \\in \\mathcal{M}_b(\\mathbf{R}^n;W)$. If $\\rho$ is a probability density, then\n\\[\n\t\\|F(\\mathfrak{A}_\\rho u)\\|_{1} \\le \\|F(\\mathcal{A}^a u)\\|_1 +\\|F^\\infty(\\mathcal{A}^s u)\\|_1\n\\]\nfor all convex functions $F : W \\to \\mathbf{R}$. In particular, \n\\[\n\t\\lim_{\\varepsilon \\to 0} \\int |F(\\mathfrak{A}_{\\rho_\\varepsilon} u)| \\, dx = \\int F(\\mathcal{A}^a u)\\, dx + \\int F^\\infty\\left(\\frac{\\mathcal{A} u}{|\\mathcal{A} u|}\\right) \\, |\\mathcal{A} u|(dx)\n\\]\nwhenever $F$ has linear growth at infinity. \n\\end{proposition}\n\n\n\\begin{proof}\nBy the discussion above, we infer that  $F(\\mathfrak{A}_\\rho u) \\in L^1_\\mathrm{loc}$. Let $K$ be a compact set of $\\mathbf{R}^n$. Appealing to Jensen's inequality and Young's convolution inequality, we deduce that\n\\begin{align*}\n\t\\int_K F(\\mathfrak{A}_\\rho u(x)) \\, dx & = \\int_K F\\left(\\int_0^\\infty  K_s \\star u(x) \\, n\\omega_n \\hat \\rho(s) s^{n-1} \\, ds \\right) \\, dx \\\\\n\t& \\le \\int_K \\int_0^\\infty F(K_s \\star u) \\, n\\omega_n \\hat \\rho(s) s^{n-1} \\, ds \\, dx\\\\\n\t& = \\int_0^\\infty n\\omega_n \\hat \\rho(s) s^{n-1} \\,ds \\int_K F(\\mathcal{A} u \\star \\frac{\\chi_{B_s}}{|B_s|}) \\, dx.\n\\end{align*}\nThe inequality $F(a + b) \\le F(a) + F^\\infty(b)$ and subsequent applications of Jensen's and Young's inequality permit us to estimate the right-hand side above by\n\\begin{align*}\n\\int_K & F(\\mathcal{A}^a u \\star \\frac{\\chi_{B_s}}{|B_s|}) + F^\\infty(\\mathcal{A}^s u \\star \\frac{\\chi_{B_s}}{|B_s|}) \\, dx \\\\ \n& \\le \\int_K F(\\mathcal{A}^a u) \\star \\frac{\\chi_{B_s}}{|B_s|} \\, dx + \\int_K F^\\infty(\\mathcal{A}^s u) \\star \\frac{\\chi_{B_s}}{|B_s|} \\, dx \\\\\n& \\quad \\le \\|F(\\mathcal{A}^a u)\\|_1 + \\|F^\\infty(\\mathcal{A}^s u)\\|_1.\n\\end{align*}\nThe first assertion follows this estimate by taking a sequence of compact sets $K_j \\nearrow \\mathbf{R}^n$. When $F$ has linear growth at infinity, the recession function $F^\\infty$ is a positively homogeneous convex function taking values on $\\mathbf{R}$. Notice that \n\\[\n\tF^\\infty (\\mu) \\coloneqq F^\\infty(g_\\mu) |\\mu| \\in \\mathcal{M}(\\mathbf{R}^n), \\qquad \\forall \\mu \\in  \\mathcal{M}(\\mathbf{R}^n;W).\n\\]\nBy~\\eqref{eq:distributional_convergence}, Reschetnyak's theorem (which holds under distributional convergence) and the previous proposition, we conclude that\n\\begin{align*}\n\t\\limsup_{\\varepsilon \\to 0} \\|F(\\mathfrak{A}_{\\rho_\\varepsilon} u)\\|_1 & \\le \\|F(\\mathcal{A}^a u)\\|_{1} + |F^\\infty(\\mathcal{A}^s u)|(\\mathbf{R}^n) \\\\\n\t& \\le \\liminf_{\\varepsilon \\to 0} \\|F(\\mathfrak{A}_{\\rho_\\varepsilon} u)\\|_1.\n\\end{align*}\nThis proves the second assertion.\n\\end{proof}\n\n\n\n\n\n\n\\begin{corollary}\\label{cor:2}\nLet $u \\in  L^1_\\mathrm{loc}(\\mathbf{R}^n;V)$ and let $1 \\le p < \\infty$.\n\\begin{enumerate}\n\\item If $\\mathcal{A} u \\in L^p$, then\n\\[\n\t\\mathfrak{A}_{\\rho_\\varepsilon}u \\stackrel{L^p}\\longrightarrow \\mathcal{A} u \\quad \\text{as $\\varepsilon \\to 0$}.\n\\]\n\\item If $\\mathcal{A} u \\in \\mathcal{M}(\\mathbf{R}^n;W)$, then\n\\[\n\t\\mathfrak{A}_{\\rho_\\varepsilon} u \\, \\mathscr L^d \\stackrel{area}\\longrightarrow \\mathcal{A} u \\quad \\text{as $\\varepsilon \\to 0$}.\n\\]\n\\end{enumerate}\n\\end{corollary}\n\\begin{proof} By letting $F(t) = t^p$ in the previous proposition, we obtain that\n\\begin{equation}\\label{eq:extendedlimit} \n\t\\lim_{\\varepsilon \\to 0} \\|\\mathfrak{A}_{\\rho_\\varepsilon} u\\|_p =\\|\\mathcal{A} u\\|_p \\in [0,\\infty].\n\\end{equation}\nFor $1 < p \\le \\infty$, it follows that $\\mathcal{A} u \\in L^p(\\mathbf{R}^n;V)$ if and only if $\\liminf_{\\varepsilon \\to 0} \\|\\mathfrak{A}_{\\rho_\\varepsilon} u\\|_p < \\infty$. \nFor $1 < p < \\infty$, the Brezis--Lieb lemma implies that the distributional convergence in~\\eqref{eq:distributional_convergence} improves to the convergence  \n\\[\n\t\\mathfrak{A}_{\\rho_\\varepsilon} u \\longrightarrow \\mathcal{A} u \\quad \\text{strongly in $L^p$}.\n\\] \nThis proves (1) for $p > 1$. \nFor $p = 1$, the previous proposition implies that $\\mathfrak{A}_{\\rho_\\varepsilon} u$ indeed converges in measure to $\\mathcal{A} u$. This proves (2). We are left to show (1) for $p =1$. Notice that when $\\mathcal{A} u \\in L^1$, then the singular part $\\mathcal{A}^s u$ vanishes, and hence from the area convergence, we get\n\\[\n\t\\int \\sqrt{1 + |\\mathfrak{A}_{\\rho_\\varepsilon} u|^2} \\, dx \\longrightarrow \\int \\sqrt{1 + |\\mathcal{A} u|^2} \\, dx.\n\\]\nSince $\\sqrt{1 + |\\,\\sbullet\\,|^2}$ is uniformly strictly convex and larger or equal than $|\\,\\sbullet\\,|$, it follows that $\\{\\mathfrak{A}_{\\rho_\\varepsilon} u\\}_\\varepsilon$ is an equi-bounded family of $L^1$. The Dunford--Pettis criterion implies that the distributional convergence~\\eqref{eq:distributional_convergence} lifts to $L^1$ strong convergence. \n\\end{proof}\n\n\n\\subsection*{Acknowledgements}This project has received funding from the Fonds National de la Recherche Scientifique - FNRS under fellowship No. 40005112 (Compensated compactness and fine properties of PDE-constrained measures).\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction} \nIn propositional or i.i.d. data, marginal probabilities are independent of the number of entities for which data are available. For example, if we start with a test set of 100 data points with fully observed attributes, and we are told that another 1,000 test cases are available, this information does not change a model's predictions on the original 100, because all data points are independent. Several authors have observed that for relational data, this is not true \\cite{lauritzen2017random,shalizi2013consistency,poole2014population,jain2010adaptive}: additional entities, or nodes in a network, are potentially related, and therefore their mere presence can be inferentially relevant.\n\nThe dependence on domain size raises several difficulties. (1) Counter-intuitiveness: if we are interested in making predictions about the 100 members of a Facebook group, learning that Facebook has gained another 1,000 users in the last hour should not change our predictions about the group of interest, unless we have specific information about the new users. (2) Complexity of Inference: We typically learn from fairly large networks and apply the results of learning to draw inferences about relatively small sub-networks. If the presence of other nodes is relevant to the marginal probabilities associated with the sub-network of interest, inference becomes computationally challenging, as it involves a large summation over the possible states of many nodes outside the sub-network. \n(3) Complexity of Learning: The observed network for learning is typically embedded in a much larger network. If marginal predictions of the observed sub-network depend on the size of the embedding network, learning should not treat the sub-network as a closed world to be analyzed in isolation. \n\nOur aim in this paper is to connect dependence on domain size with the classic concept of {\\em projectivity} from statistical theory \\cite{shalizi2013consistency}. An SRL model typically defines a family of probability distributions over relational structures, one for each domain size $n$. Projectivity requires that  the distribution for size $m$ agrees with the marginal distribution obtained from any large domain $n > m$. This ensures that the distributions for different domain sizes are consistent with each other, and that predictions about a sub-network do not depend on the presence of outside nodes.\n\nMost (if not all) SRL modeling frameworks support the construction of non-projective models, and in many cases, this is an important source of model expressivity. However, taking into consideration the significant benefits in terms of computational complexity and robustness of inference and learning results, it is worthwhile to explore projective fragments of SRL languages. After establishing the formal framework for defining and analyzing projectivity in an SRL context (Sections~\\ref{sec:background},\\ref{sec:projectivity}), we identify syntactic constraints for three prototypical SRL frameworks that ensure projectivity (Section~\\ref{sec:fragments}). We finally take a closer look at how projectivity can justify maximum likelihood learning from observed sub-networks (Section~\\ref{sec:learning}).  \n\n\n\\paragraph{Related Work.} Several SRL works have addressed changing domain sizes. \\citeauthor{poole2014population} discuss the impact of domain size and inference~\\citeyear{poole2014population}.  \nConvergence of inference~\\cite{jaeger1998convergence} and learning~\\cite{Xiang2011} in the domain size limit has been investigated.~\\citeauthor{kuzelka2018relational} discuss the validity of learning from a sub-network embedded in a larger network. They focus on models that satisfy a given set of marginal probabilities, not projectivity.  Statisticians have examined the projectivity of network models from the exponential family \\cite{shalizi2013consistency,lauritzen2017random}, but have not considered other SRL models.\n\n\\section{Background}\n\\label{sec:background}\n\n\nThe following definitions provide a framework for talking about SRL systems in general.\n\nA relational \\emph{signature} $S$\\ contains relation symbols of varying arities. We write \n$r\/k$ for a relation $r$ of arity $k\\geq 0$.  \nA \\emph{possible world} $\\omega$\\ (for $S$)  is given by a finite domain $D=\\{d_1,\\ldots,d_n\\}$\\ on \nwhich extensions of the relations in $S$\\ are defined. We then refer to $n$ as the \\emph{size} of \n$\\omega$. Moreover, we generally assume that $D=\\{0,\\ldots,n-1\\}=:[n]$. Then $\\Omega^{(S,n)}$ denotes the set of \nall possible worlds for the signature $S$ over the domain $[n]$. \nThis can be abbreviated as  $\\Omega^{(n)}$ when the signature is understood from the context. \n\n$\\omega\\in \\Omega^{(n)}$ can also be identified with a truth value assignment \n\\begin{displaymath}\n  \\omega: r(\\boldi) \\mapsto \\{\\emph{true},\\emph{false}\\}\n\\end{displaymath}\nfor all ground atoms $r(\\boldi)$, where $r\/k\\in S$ and $\\boldi=(i_1,\\ldots,i_k)\\in [n]^k$.\n\nWe denote with $\\Delta\\Omega^{(n)}$ the set of all probability distributions on $\\Omega^{(n)}$. \nA \\emph{random relational structure model (rrsm)} is a family of distributions \n$\\{Q^{(n)} \\in\\Delta\\Omega^{(n)} \\mid n\\in\\Nset\\}$. The term RRSM, which was first\nintroduced in~\\cite{JaegerMI02}, is intended to emphasize that this is a multi-relational generalization of\nthe classical concept of random graph models (or \\emph{random networks}~\\cite{lauritzen2017random}).  \n\nFor this paper we limit ourselves to the consideration of \\emph{exchangeable} RRSMs: a single distribution\n$Q^{(n)}$ is exchangeable, if $Q^{(n)}(\\omega)=Q^{(n)}(\\omega')$ whenever $\\omega$ and $\\omega'$ are isomorphic. An \nRRSM is exchangeable, when all $Q^{(n)}$ ($n\\in\\Nset$) are exchangeable. \nExchangeability of $Q^{(n)}$ implies that \n$Q^{(n)}(r(\\boldi)=\\emph{true})=Q^{(n)}(r(\\boldj)=\\emph{true})$ when $\\boldj = \\pi(\\boldi)$ for some permutation $\\pi$ of $[n]$.\n\nRRSMs defined by typical SRL frameworks are necessarily exchangeable when the model specification does not\nmake use of any constants referring to specific domain elements, and is not conditioned on a pre-defined\nstructure on the domain. Examples of RRSMs that do not satisfy this condition are temporal models such as\nhidden Markov models, where the model is conditioned on a linear ordering $0<1<\\cdots< n-1$ of the domain, \nand the marginal distributions at different timepoints $t\\neq t'$ are ususally different. \n\n\n\\subsection{SRL Systems}\n\nIn this section we briefly review three different SRL frameworks  that can be used to define\nRRSMs. We do not give complete definitions of syntax and semantics for these frameworks here, but\nonly describe their main characteristics by way of examples. The selected systems are representatives\nfor three distinct modeling paradigms.\n\n\\subsubsection{Relational Bayesian Networks}\n\nRelational Bayesian networks (RBNs)~\\cite{Jaeger97UAI} are a representative of frameworks that\nare closely linked to directed graphical models. Other approaches in this group include \nprobabilistic relational models~\\cite{friedman1999learning}, Bayesian logic programs~\\cite{kersting2000interpreting}\nand relational logistic regression~\\cite{kazemi2014relational}. \n\nAn RBN associates with each relation $r\\in S$ a functional expression that defines the probabilities\nof ground atoms $r(\\boldi)$ conditional on other ground atoms. Functional expressions can be formed \nby combining and nesting basic constructs, which support Boolean conditions, mixtures of distributions, \nand aggregation of dependencies using the central tool of \\emph{combination functions}. \n\nFor example, the two formulas \n\n\\begin{displaymath}\n  \\begin{array}{lll}\n    \\emph{red}(X) & \\leftarrow  & 0.3 \\\\\n    \\emph{black}(X) & \\leftarrow  & {\\tt if}\\ \\emph{red}(X):0\\ {\\tt else}:\\ 0.5\n  \\end{array}\n\\end{displaymath}\ndefine Boolean attributes \\emph{red} and \\emph{black}, such that no element can be both \\emph{red}\nand \\emph{black}. One can then condition a binary \\emph{edge} relation on the colors of the nodes:\n\\begin{displaymath}\n  \\begin{array}{lll}\n    \\emph{edge}(X,Y) & \\leftarrow &  {\\tt if}\\  \\emph{red}(X)\\, \\&\\, \\emph{red}(Y): 0.7 \\\\\n    & & {\\tt else\\ if}\\  \\emph{black}(X)\\, \\&\\, \\emph{black}(Y): 0.4 \\\\\n    & & {\\tt else}:\\ 0.05 \\\\\n  \\end{array}\n\\end{displaymath}\n\nTogether, the three formulas for \\emph{red}, \\emph{black} and \\emph{edge} represent a simple \nstochastic block model~\\cite{snijders1997estimation}. \nThe main expressive power of RBNs derives from \\emph{combination functions} that \ncan condition a ground atom on properties of other domain entities:\n\\begin{displaymath}\n  \\begin{array}{lll}\n   t(X)  & \\leftarrow & \\emph{noisy-or}\\{ {\\tt if}\\ \\emph{edge}(X,Y)\\, \\&\\, \\emph{red}(Y):0.2  \\mid Y  \\}\n  \\end{array}\n\\end{displaymath}\nThis formula makes the probability of the attribute $t$ for $X$ dependent on the number of \\emph{red}\nentities $Y$ that $X$ is connected to via an \\emph{edge}. Each such entity causes $t(X)$ to be\ntrue with probability 0.2, and the different causes $Y$ are modeled as independent via \nthe \\emph{noisy-or} combination function.\n\nAn RBN specification defines an RRSM, provided that for each $n$ the formulas induce an \nacyclic dependency structure on the atoms of $\\Omega^{(S,n)}$. The distribution $Q^{(n)}$ can then be \nrepresented by a Bayesian network whose nodes are the ground atoms $r(\\boldi)$, and where \n$r'(\\boldi')$ is a parent of $r(\\boldi)$, if the truth value of $r'(\\boldi')$ is needed to \nevaluate the probability formula for $r(\\boldi)$. \n\n\\subsubsection{Markov Logic Networks}\n\nMarkov Logic networks (MLNs)~\\cite{RicDom06} are the multi-relational generalization of exponential \nrandom graph models~\\cite{frank1986markov,wasserman1996logit}. An MLN consists of a set\nof weighted formulas. As an example, let $S=\\{\\emph{red}\/1,\\emph{edge}\/2\\}$, and define\nthe following two weighted formulas:\n\\begin{displaymath}\n  \\begin{array}{lr}\n    \\phi_1(X,Y) :\\equiv \\emph{edge}(X,Y)\\wedge \\emph{red}(X) \\wedge \\emph{red}(Y) & 1.2 \\\\\n    \\phi_2(X,Y) :\\equiv \\emph{edge}(X,Y)\\wedge \\emph{red}(X) \\wedge \\neg \\emph{red}(Y) & -0.2 \\\\\n  \\end{array}\n\\end{displaymath}\nThese two weighted formulas represent a homophily model for the \\emph{red} attribute \nin a graph. Each pair $(i,j)\\in [n]^2$ contributes a weight of 1.2 (-0.2) to a possible \nworld $\\omega\\in\\Omega^{(S,n)}$ if $\\phi_1(i,j)$ ($\\phi_2(i,j)$) is true in $\\omega$. The probability\n$Q^{(n)}(\\omega)$ then is the normalized sum of all weights contributed by all the weighted formulas, \nand all possible substitutions of domain elements for the variables in the formulas.\n\n$Q^{(n)}$ can be represented by a Markov network whose nodes are the ground atoms $r(\\boldi)$, and\nwhere two atoms $r(\\boldi),r'(\\boldi')$ are connected by an edge if these two atoms \nappear jointly in a grounding of one of the weighted formulas.\n\n\n\\subsubsection{ProbLog}\n\nProbLog~\\cite{DeRKimToi2007,kimmig2011implementation} is a representative of RRSMs that \nare closely linked to logic programming. Other frameworks in this class are Prism~\\cite{Sato95}, and\nindependent choice logic~\\cite{Poole08}. A ProbLog model consists of \na set of  \\emph{labeled facts}, which are atoms with a probability label attached:\n\\begin{equation}\n\\label{eq:problog1}\n    0.8::\\ \\emph{red}(X)\n\\end{equation}\ntogether with \\emph{background knowledge} consisting of (non-probabilistic) definite clauses:\n\\begin{equation}\n\\label{eq:problog2}\n    \\emph{edge}(X,Y) :-\\ \\emph{red}(X),\\emph{red}(Y).\n\\end{equation}\nWe note that~\\cite{kimmig2011implementation} emphasize the use of ground atoms in the labeled \nfacts. Since our interest is with  generic, domain-independent RRSMs, we restrict attention\nto ProbLog models that do not contain domain constants.\n\nThe ProbLog model consisting of (\\ref{eq:problog1}) and (\\ref{eq:problog2}) defines the \nprobability of $\\omega\\in\\Omega^{(S,n)}$ as 0 if $\\omega$ does not satisfy the property that\ntwo elements are connected by an edge if and only if they both are red (i.e., \n$\\omega$ must be a minimal model of  (\\ref{eq:problog2})). If $\\omega$ satisfies \n (\\ref{eq:problog2}), then its probability is $0.8^{|\\emph{red}(\\omega)|}0.2^{n-|\\emph{red}(\\omega)|} $, \nwhere $\\emph{red}(\\omega)$ denotes the set of elements  $i\\in [n]$ for which \n$\\emph{red}(i)$\\ is true in $\\omega$.   \n\nIn order to also make rules such as (\\ref{eq:problog2}) probabilistic, one can introduce\nlatent relations that represent whether rules in the background knowledge become applicable.\nIn our example, we can add a new relation \\emph{rule}\/2,  an additional labeled fact\n\\begin{equation}\n\\label{eq:problog3}\n    0.5::\\ \\emph{rule}(X,Y),\n\\end{equation} \nand modify the background knowledge to \n\\begin{equation}\n\\label{eq:problog4}\n    \\emph{edge}(X,Y) :-\\ \\emph{red}(X),\\emph{red}(Y),\\emph{rule}(X,Y).\n\\end{equation}\nNow the resulting ProbLog model is a (partial) stochastic block model, where with probability\n0.5 two red nodes are connected by an edge (and still no other than pairs of red nodes could \nbe connected; further labeled facts and rules can be added to also model connections between\nnon-red nodes). \n\n\n\n\n\\section{Projectivity}\n\\label{sec:projectivity}\n\nFor\n$Q^{(n)}\\in\\Delta\\Omega^{(n)}$  and a subset $\\{i_0,\\ldots,i_{m-1}\\}\\subseteq [n]$ we \ndenote with  $Q^{(n)}\\downarrow  \\{i_0,\\ldots,i_{m-1}\\}$ the marginal distribution on the sub-structures\ninduced by $\\{i_0,\\ldots,i_{m-1}\\}$, or, equivalently, the marginal distribution on the ground atoms\n$r(\\boldi)$ with $\\boldi\\in  \\{i_0,\\ldots,i_{m-1}\\}^k$. When $Q^{(n)}$ is exchangeable, then the induced\ndistribution is independent of the actual choice of the elements $i_h$, and we may assume that\n$\\{i_0,\\ldots,i_{m-1}\\}=\\{0,\\ldots,m-1\\}$. We therefore can limit attention to the \nmarginalizations\n\\begin{equation}\n  \\label{eq:marginalizeQn}\n  Q^{(n)}\\downarrow [m] \\in \\Delta \\Omega^{(m)}.\n\\end{equation}\n\n\n\nBased on \\cite{shalizi2013consistency} we define several different versions of ``projectivity'' for \nprobabilistic relational models. Our definitions are more restricted than the one given by Shalizi \nand Rinaldo~(\\citeyear{shalizi2013consistency}) in that \nwe specifically tailor the definitions to  random relational structure models.\nFor convenience, we also include the condition of exchangeability in the following \ndefinitions, even though exchangeability and projectivity are not necessarily linked.\n\n\n\\begin{definition}\n\\label{def:proj1}\n  A RRSM is \\emph{projective}, if \n  \\begin{itemize}\n  \\item every $Q^{(n)}$ is exchangeable\n  \\item for every $n$, every $m<n$:\n    $Q^{(n)}\\downarrow [m]=Q^{(m)}$.\n  \\end{itemize}\n\\end{definition}\n\n\\begin{example}\n  A simple classical example for a projective RRSM is the Erd\\\"os-R\\'{e}nyi random graph model with\nedge probability $p$, where\n$S=\\{e\/2\\}$, and for $\\omega\\in\\Omega^{(n)}$: $Q^{(n)}(\\omega)=p^{|e(\\omega)|}(1-p)^{n^2-|e(\\omega)|}$, \nwhere $ |e(\\omega)| $ denotes the number of edges in $\\omega$.\n\nAn example of a very different nature is the RRSM where for every $n$: \n\\begin{equation}\n  \\label{eq:clique_empty}\n  Q^{(n)}= \\frac{1}{2}\\mathbbm 1_{K_n} +  \\frac{1}{2}\\mathbbm 1_{E_n}\n\\end{equation}\n where $K_n$\\ is the complete and $E_n$\\ the empty graph over $n$, and \n$\\mathbbm 1_{\\omega}$ denotes the distribution that assigns probability one to $\\omega$. \n\nNot projective are sparse \nrandom graph models, where the edge probability is a decreasing function of the domain size $n$, e.g. \n$Q^{(n)}(e(i,j))=\\theta\/n$\\ for some parameter $\\theta$. \n\\end{example}\n\n\nDefinition~\\ref{def:proj1} relates to a single RRSM. In the context of learning, we are initially only \ngiven a space of candidate models. The models in this space are typically characterized by a structural \ncomponent, and a set of numerical parameters. In the following, we assume that the learning problem only\nconsists of optimizing over a fixed set of numerical parameters, i.e., structure learning is outside \nthe scope of our considerations. \n\nConcretely, let $\\Theta\\subseteq \\Rset^k$ be a parameter space, so that each $\\boldtheta\\in\\Theta$ defines\na distribution $Q^{(n)}_{\\boldtheta}\\in\\Delta\\Omega^{(n)}$. Then  $\\{Q^{(n)}_{\\boldtheta}\\mid \\boldtheta\\in\\Theta, n\\in\\Nset\\}$\nis a  \\emph{parametric family of RRSMs}.\n\n\nThis leads to the following  definition of projectivity, which is essentially the one\nof~\\cite{shalizi2013consistency}.\n\\begin{definition}\n\\label{def:projective}\n  A parametric family of RRSMs is \\emph{projective}, if  $\\{Q^{(n)}_{\\boldtheta}\\mid n\\in\\Nset\\}$ is\nprojective for every $\\boldtheta\\in\\Theta$.\n\\end{definition}\n\nWe can weaken this as follows:\n\\begin{definition}\n   A parametric family of rrms is \\emph{structurally projective} if \n   \\begin{itemize}\n   \\item  every $Q^{(n)}_{\\boldtheta}$ is exchangeable\n   \\item for every $n,\\boldtheta$, and $m<n$:  there \nexists $\\boldtheta'\\in \\Theta$ such that  $Q^{(n)}_{\\boldtheta}\\downarrow [m] =Q^{(m)}_{\\boldtheta'}$.\n   \\end{itemize}\n\\end{definition}\n\nThis concept of structural projectivity corresponds to  \\emph{weak consistency} in the sense of \n\\cite{lauritzen2017random} restricted to exchangeable distributions.\n\nAn example for structurally projective models are sparse random graph models with edge probabilities \n$\\theta\/n$: here we then have $Q^{(n)}_{\\theta}\\downarrow I=Q^{(m)}_{\\theta'}$ with $\\theta'=\\theta m\/n$\n\n\\begin{example}\n\\label{ex:mlnnotsp}\n  MLNs are not structurally projective: consider the MLN\n  \\begin{displaymath}\n    \\begin{array}{ll}\n      a(X),e(X,Y) & w\n    \\end{array}\n  \\end{displaymath}\ndefining probability distributions $Q^{(n)}_w$ depending on the weight parameter $w$.\nConsider the  conditional probability \n\\begin{equation}\n\\label{eq:qna0}\n  q(n,w):=Q^{(n)}_w(a(0)| \\neg e(0,0),\\neg e(0,1),\\neg e(1,0),\\neg e(1,1)).\n\\end{equation}\nIf $w>0$, then $q(n,w)$ is increasing in $n$ (and in $w$). However, for \n$n=2$, we have $q(n,w)=1\/2$, regardless of the value of $w$. Thus, for a suitable combination\nof $n,w$ where $q(n,w)>1\/2$, we cannot find a value $w'$ such that\n$Q^{(n)}_q\\downarrow [2] = Q^{(2)}_{w'}$.\n\\end{example}\n\nThe lack of structural projectivity sets some limits to the approach of defining the\nweights in an MLN as functions of the domain size $n$~\\cite{jain2010adaptive} in \norder to compensate for the domain dependence of the model. \n\n\\begin{example}\n  RBNs are not structurally projective: similarly to Example~\\ref{ex:mlnnotsp}, we can construct\na counterexample as follows. Consider the RBN\n\\begin{displaymath}\n  \\begin{array}{lll}\n   \\emph{edge}(X,Y) & \\leftarrow & 0.5 \\\\\n   a(X)  & \\leftarrow & \\emph{noisy-or}\\{ {\\tt if}\\ \\emph{edge}(X,Y):\\theta  \\mid Y  \\}\n  \\end{array}\n\\end{displaymath}\ndefining probability distributions $Q^{(n)}_{\\theta}$ depending on the probability parameter $\\theta$.\nDefining $q(n,\\theta)$ as in (\\ref{eq:qna0}), we obtain that $q(2,\\theta)=0$, regardless of $\\theta$,\nbut $q(n,\\theta)>0$ for $n\\geq 3$ and $\\theta>0$. \n\\end{example}\n\n\n\\section{Projective Fragments of SRL Systems}\n\\label{sec:fragments}\n\nIn this section we identify for the three representative SRL frameworks introduced in \nSection~\\ref{sec:background}, restrictions on model structure that give rise to projective models. \n\nFor the propositions stated in this section we give short proof sketches. Full proofs would\nneed to refer to complete formal specifications of syntax and semantics of the various \nframeworks, which we have omitted in Section~\\ref{sec:background}. Even the proof sketches we\nprovide may appeal to some facts about the individual frameworks that were not explicitly\nspelled out in Section~\\ref{sec:background}.\n\n\n\n\\begin{proposition}\n  An RBN defines a projective RRSM if it does not contain any combination functions.\n\\end{proposition}\n\n\\begin{proof}[Sketch]\nConsider the Bayesian network representation $B^{(n)}$ \nof $Q^{(n)}$, and a parent child pair $ r'(\\boldi') \\rightarrow r(\\boldi)$ in $B^{(n)}$.\nIf the probability for the relation $r$ is defined without the use of combination functions, \nthen all constants appearing in $\\boldi'$ must also be contained in $\\boldi$. This implies \nthat  the sub-network for the ground atoms over $[m]$ ($m\\leq n$) is \nan upward-closed sub-graph of $B^{(n)}$ whose structure and probability parameters do not\ndepend on $n$. This means that the marginal distributions on the ground atoms over  $[m]$\ndoes not depend on $n$.\n\\end{proof}\n\nEven though combination functions are the main source for the expressive power of RBNs, one\ncan still encode some relevant models with the combination function free fragment. \nOne example are the stochastic\nblock models as shown in Section~\\ref{sec:background}. Another example are temporal models \nsuch as dynamic Bayesian networks or hidden Markov models. However, as mentioned in Section~\\ref{sec:background}, these models are not exchangeable, and therefore outside the scope of this paper.\n\n\n\n\\begin{proposition}\n\\label{prop:mlnfrag}\nAn MLN  defines a projective RRSM  if its formulas $\\phi_i$ satisfy the property that\nany two atoms appearing in $\\phi_i$ contain exactly the same variables. \n\\end{proposition}\n\n\\begin{proof}[Sketch]\nIf the condition of the proposition holds, then \nthe Markov network representation $M^{(n)}$ of $Q^{(n)}$  decomposes  \ninto a system of disconnected sub-networks, where each sub-network contains ground\natoms over a fixed set of domain elements, with a cardinality at most equal to the maximal \narity of any $r\\in S$.  The structure and parameterization of sub-networks containing\nonly the atoms for domain elements from $[m]$  ($m\\leq n$)  are the same for all $n$, and they define a \nmarginal distribution that is independent of the nodes contained in the other sub-networks, i.e., \nindependent of $n$. \n\\end{proof}\n\nAlternatively, Proposition~\\ref{prop:mlnfrag} can also be proven by an application of \nTheorem 1 of~\\cite{shalizi2013consistency}.\nOur MLN example from Section~\\ref{sec:background} does not satisfy the restriction of \nProposition~\\ref{prop:mlnfrag}. A somewhat synthetic example that satisfies the condition is\n\n\\begin{displaymath}\n  \\begin{array}{lr}\n   \\emph{red}(X)\\wedge \\emph{edge}(X,X)  & -1.5 \\\\\n   \\emph{edge}(X,Y)\\wedge\\emph{edge}(Y,X) & 0.8 \\\\ \n  \\end{array}\n\\end{displaymath}\n\nThis MLN represents a model according to which red nodes are unlikely to have self-loops, and\nthe edge relation tends to be symmetric. It is an open question whether our projective MLN fragment\ncontains more natural and practically relevant classes of models. \n\n\n\n\\begin{proposition}\n\\label{prop:problogproj}\n  A ProbLog model defines a projective RRSM, if all the background knowledge clauses satisfy \nthe property that the body of the clause does not contain any variables that are not contained in the\nhead of the clause.\n\\end{proposition}\n\n\\begin{proof}[Sketch]\nIf the stated property holds, then a proof for a ground atom $r(\\boldi)$ can only contain\nground atoms $r'(\\boldi')$ with $\\boldi'\\subseteq \\boldi$. The probability that $r(\\boldi)$ is\nprovable when $\\boldi\\subset [m]$ then only depends on the probabilities of ground labeled facts\nwith arguments from $[m]$. The joint distribution of these ground facts is independent of $n$. \n\\end{proof}\n\nThe ProbLog example of Section~\\ref{sec:background} satisfies the condition of \nProposition~\\ref{prop:problogproj}. \n\n\\subsection{Discussion}\nOur conditions for the projective RBN and ProbLog structures are very\nsimilar, and it seems that both\nfragments support more or less the same types of models. \nThe projectivity conditions are essentially limitations on probabilistic dependencies. For\nthe frameworks that (implicitly) encode a directed sampling process for possible worlds (RBN, ProbLog), \nthe limit on the dependency is that when a ground atom $r(\\boldi)$ is sampled, its distribution \ncannot depend on ground atoms containing elements other than included in $\\boldi$. However, \n$r(\\boldi)$, in turn, may influence the sampling probabilities of other atoms containing elements \nnot in $\\boldi$. As a consequence, both in the RBN and the ProbLog model of Section~\\ref{sec:background},\nthe value of $\\emph{red}(i)$ influences the probability for $\\emph{edge}(i,j)$, and as a result,\nthe two variables $\\emph{red}(i), \\emph{edge}(i,j)$ are not independent. \n\nDue to the undirected nature of MLNs, one there cannot impose an ``upstream only'' limitation on\nprobabilistic dependencies. As a result, the restriction imposed in Proposition~\\ref{prop:mlnfrag}\nimplies that a when a model defines two random variables  $\\emph{red}(i), \\emph{edge}(i,j)$ ($i\\neq j$),\nthese  must be independent.\n\n\\section{Projectivity and Inference}\n\\label{sec:inference}\n\nIn an inference scenario, we are given an RRSM $\\{Q^{(n)}\\mid n\\in \\Nset\\}$, a domain of size $n$, and\na query, which for the sake of concreteness we assume to be of the form\n\\begin{displaymath}\n  P(r(\\boldi)\\mid (\\neg)r_1(\\boldi_1),\\ldots,(\\neg)r_h(\\boldi_h)) = ?,\n\\end{displaymath}\nwhere the $(\\neg)r_j(\\boldi_j)$ are observed evidence literals. Let $I:=\\boldi\\cup\\cup_{j=1}^h \\boldi_j$, \nand $m:=|I|$. \nTo answer the query, all we need is the marginal distribution \n$Q^{(n)}\\downarrow I$. If the model is projective, then this marginal is equivalent to $Q^{(m)}$, and \nindependent of $n$. In practice this means, that when inference is performed by grounding the\nrelational model over a concrete domain, that we here only need to ground the model over the\ndomain that contains exactly the entities mentioned in the query. \n\nApart from the clear computational advantages that projectivity affords, it also leads to \nrobustness with regard to domains that are only  incompletely known or observed: in many cases\nit can be difficult to specify exactly the size of the domain in which our observed entities $I$ ``live''.\n(what is the domain of a person's social network?). Here projectivity means that our inferences\nare not impacted by missing or imprecise information about the domain.\n\n\n\\section{Projectivity and Learning}\n\\label{sec:learning}\n\nIn a learning setting, we are interested in estimating the\nparameter $\\boldtheta$ for a given parametric family of RRSMs. We assume that\nthe training data consists of one or several observed possible worlds, in the latter \ncase possibly worlds of varying sizes. Like~\\cite{Xiang2011} and\n\\cite{kuzelka2018relational} we mostly focus on the scenario where the training data \nconsists of a single possible world $\\omega\\in\\Omega^{(m)}$, and we estimate $\\boldtheta$ by maximizing\nthe likelihood\n\n\\begin{equation}\n  \\label{eq:thetalik}\n  L(\\boldtheta|\\omega) = Q^{(m)}_{\\boldtheta}(\\omega).\n\\end{equation}\nFor simplicity we here restrict attention to pure maximum likelihood inference, but our considerations\nare also pertinent for penalized likelihood or Bayesian inference approaches.\n\nFor this learning problems, too, the dependence on the domain size $n$ is an important concern.\nConsider, for example, the problem of learning a model for an evolving (social) network. \nIs the model we learn today, while the network is of size $m$, still a good model when the\nnetwork has evolved to size $n>m$? Moreover, the data from which we learn may not be  complete,\nand not contain the data about all the entities that are actually present in the relational \ndomain. Then we try to learn a model for a domain of size $n$, from data corresponding to \na domain of smaller size $m<n$~\\cite{kuzelka2018relational}.\n\nThe general question we want to address, therefore,  is the following: \nlet $\\boldtheta^*\\in\\Theta$ be the parameter estimate we obtain by maximizing\n(\\ref{eq:thetalik}) for some $\\omega\\in\\Omega^{(m)}$. Let \n$\\omega'\\in\\Omega^{(n)}$ such that $\\omega'\\downarrow I = \\omega$ for \nsome $I\\subset [n]$ of size $m$, and $\\boldtheta^{**}$ the estimate obtained \nby maximizing $Q^{(n)}_{\\boldtheta}(\\omega')$. What can we say about the relationship \nbetween $\\boldtheta^*$ and $\\boldtheta^{**}$?\n\n\nTo obtain more precise statements of this overall question, we first consider the scenario where we do\nnot know $\\omega'$, but we do know $\\omega'$'s size $n$. Then, given the partial observation\n$\\omega$, we can maximize likelihood  in the appropriate space \n$\\{Q^{(n)}_{\\boldtheta}| \\boldtheta\\in\\Theta\\}$ via the marginal likelihood function \n\\begin{equation}\n  \\label{eq:thetafvlik}\n  ML^{(n)}(\\boldtheta|\\omega):=Q^{(n)}_{\\boldtheta}( \\{ \\omega'\\in\\Omega^{(n)}: \\omega' \\downarrow [m]=  \\omega \\}) \n\\end{equation}\n\nThe following then is immediate:\n\\begin{proposition}\nIf the parametric family $\\{Q^{(n)}_{\\boldtheta}\\mid \\boldtheta\\in\\Theta, n\\in \\Nset\\}$ is projective, then\nthe likelihood functions (\\ref{eq:thetalik}) and (\\ref{eq:thetafvlik}) are identical.\n\\end{proposition}\n\n\nEven in the context of a projective family, however, learning from $\\omega$ that is a sub-sample \nof the ``true'' world $\\omega'$ usually is\nproblematic: the use of the marginal likelihood function\n(\\ref{eq:thetafvlik}) only is justified, when the data in $\\omega$ is \n\\emph{missing at random}~\\cite{Rubin76}. This will be \nthe case when $\\omega$ is the induced sub-structure of  $m$  uniformly, randomly selected elements \nfrom the domain $[n]$ \nof $\\omega'$ (\\emph{induced subgraph sampling}~\\cite[Chapter 5]{kolaczyk2009statistical}). However, a more\nrealistic scenario for observing a substructure of a possible world $\\omega'$ is by mechanisms such as\n\\emph{traceroute} or \\emph{snowball} sampling~\\cite[Chapter 5]{kolaczyk2009statistical}). \nWhen $\\omega$ is observed through such a mechanism, then\n(\\ref{eq:thetafvlik}) is not an appropriate likelihood function.\n\n\\subsection{Projectivity and Maximum Likelihood Learning}\nIn the following, we make the\nidealized assumption\nthat $\\omega\\in\\Omega^{(m)}$ was observed through induced subgraph sampling\nfrom a larger world $\\omega'\\in\\Omega^{(n)}$.\nIn conjunction with projectivity,  the use of \n(\\ref{eq:thetalik}) then is justified, giving us the parameter estimate $\\theta^*$. \nCan we derive some guarantees for the quality of $\\theta^*$ as an approximation\nof the estimate $\\theta^{**}$ one would obtain by maximizing \n$Q^{(n)}_{\\boldtheta}(\\omega')$?\n\nClearly, for any specific  pair $\\omega,\\omega'$,  no\nguarantees can be given, because $\\omega$ could be a very un-representative \nsub-structure of $\\omega'$. We therefore can only ask whether such guarantees can\nbe given in expectation, where  expectation is\nwith respect to induced subgraph sampling of $\\omega$ from $\\omega'$. \n\nThis question still has two \ndistinct precise formalizations, expressed by the following two equalities:\n\n\n\\begin{eqnarray}\n  E_{\\omega}  [ \\argmax_{\\boldtheta} \\emph{log}L^{(m)}(\\boldtheta|\\omega)  ] \n =  \\argmax_{\\boldtheta} \\emph{log}L^{(n)}(\\boldtheta|\\omega') & & \n\\label{eq:unbiased1}\\\\\n  \\argmax_{\\boldtheta} E_{\\omega} [ \\emph{log}L^{(m)}(\\boldtheta|\\omega)]  \n =  \\argmax_{\\boldtheta} \\emph{log}L^{(n)}(\\boldtheta|\\omega') & &\n\\label{eq:unbiased2}\n\\end{eqnarray}\n\nWe here turn to the log-likelihood, because  the expectation in the left-hand side\nexpression of  (\\ref{eq:unbiased2}) is more meaningful when applied to the log-likelihood.\nFor all other expressions in  (\\ref{eq:unbiased1}) and (\\ref{eq:unbiased2}) the application\nof the \\emph{log} makes no difference. In cases where the log-likelihood functions need \nnot have a unique maximum, the $\\argmax_{\\boldtheta}$ should be read as returning\nthe set of all maxima.\n\nCondition (\\ref{eq:unbiased1}) basically says that estimating $\\boldtheta$ from an $m$-element induced \nsubstructure of $\\omega'$ is an \\emph{unbiased} estimator for the estimate one would have obtained from \n$\\omega'$. Condition (\\ref{eq:unbiased2}) essentially expresses a statistical \\emph{consistency} property: if one\ntakes repeated size $m$ samples $\\omega_1,\\ldots,\\omega_N$, and maximizes the sample log-likelihood\n$\\frac{1}{N}\\sum_{i=1}^N   \\emph{log}L^{(m)}(\\boldtheta|\\omega_i)$ then, for large $N$, this will become equivalent\nto maximizing $L^{(n)}(\\boldtheta|\\omega')$. We note though, that (\\ref{eq:unbiased2}) alone does not\ndirectly guarantee this consistency. Additional regularity conditions on the likelihood function will be \nneeded. \n \nThe following is a cautionary example that shows that even for projective models, neither \n(\\ref{eq:unbiased1}) nor (\\ref{eq:unbiased2}) need hold.\n\n\n\\begin{example}\nConsider the following RBN:\n  \\begin{eqnarray}\n    \\emph{red}(X) & \\leftarrow  & \\theta \\label{eq:rbncounterex1} \\\\\n    \\emph{edge}(X,Y) & \\leftarrow  & \\theta \\label{eq:rbncounterex2}\n  \\end{eqnarray}\nIn this RBN there is a common parameter $\\theta$ that denotes both the probability for the\nattribute $a\/1$, and the edge relation $\\emph{edge}\/2$.   This form of parameter sharing by different\nprobability formulas is supported by RBNs and their learning tools, even though it seems to be\nof limited use in practice. This RBN clearly is projective. \n\nNow consider the possible world $\\omega'\\in\\Omega^{(n)}$ where $a(i)$ is true for $i=0,\\ldots,n\/2-1$, and\nfalse for $i=n\/2,\\ldots,n-1$, and where $\\emph{edge}(i,j)$ is false for all $i,j$. Let $m=2$. \nA random $\\omega\\in\\Omega^{(2)}$ drawn from $\\omega'$ then is with equal probability one of the 4 \nworlds with $(r(0),r(1))$, $(r(0),\\neg r(1))$, $(\\neg r(0),r(1))$, or $(\\neg r(0),\\neg r(1))$ (and \nno edges). Writing $l\\theta$ for $\\emph{log}\\theta$, and $l\\bar{\\theta}$ for \n$\\emph{log}(1-\\theta)$, we obtain as the expected log-likelihood:\n\\begin{multline}\n  E_{\\omega} [ \\emph{log}L^{(2)}(\\boldtheta|\\omega)]= \\\\\n \\frac{1}{4}\n((4l\\bar{\\theta}+2l\\theta)+ 2(4l\\bar{\\theta}+l\\theta +l\\bar{\\theta})+(4l\\bar{\\theta}+2l\\bar{\\theta}))\n\\label{eq:lik1}\n\\end{multline}\nwhere the  recurring first term $4l\\bar{\\theta}$ accounts for the absence of the four possible edges (including\npossible self-loops) in all possible sampled worlds. Maximizing this gives\n\\begin{displaymath}\n  \\theta^*=\\argmax_{\\boldtheta} E_{\\omega} [ \\emph{log}L^{(2)}(\\boldtheta|\\omega)] = 1\/6.\n\\end{displaymath}\nThe log-likelihood function induced by $\\omega'$ is\n\\begin{equation}\n\\label{eq:lik2}\n  \\emph{log}L^{(n)}(\\boldtheta|\\omega') = n^2 l\\bar{\\theta} + (n\/2)l\\theta +  (n\/2)\\bar{\\theta}  \n\\end{equation}\nhich is maximized by\n\\begin{displaymath}\n  \\theta^{**}=\\argmax_{\\boldtheta} \\emph{log} L^{(n)}(\\boldtheta|\\omega') = \\frac{n\/2}{n^2+n}=\\frac{1}{2(n+1)}\n\\end{displaymath}\nThus, (\\ref{eq:unbiased2}) does not hold. We also compute\n\\begin{displaymath}\n   E_{\\omega}  [ \\argmax_{\\boldtheta} L^{(2)}(\\boldtheta|\\omega)  ] =\n   \\frac{1}{4}(\\frac{1}{3}+ \\frac{1}{6}+ \\frac{1}{6}+ 0  ) =\\frac{1}{6},\n\\end{displaymath}\nand see that (\\ref{eq:unbiased1}) also fails (in general, other than in this example,  \nthe left-hand sides of  (\\ref{eq:unbiased1})  and (\\ref{eq:unbiased2}) need not be identical).  \n\nFinally, now suppose that the two formulas (\\ref{eq:rbncounterex1}) and  (\\ref{eq:rbncounterex2}) each\nare defined by distinct parameters $\\theta_r$ and $\\theta_e$, respectively. Then the likelihood function\n(\\ref{eq:lik1}) becomes\n\\begin{displaymath}\n   \\frac{1}{4}\n((4l\\bar{\\theta}_e+2l\\theta_r)+ 2(4l\\bar{\\theta}_e+l\\theta_r +l\\bar{\\theta}_r)+(4l\\bar{\\theta}_e+2l\\bar{\\theta}_r)),\n\\end{displaymath}\nand (\\ref{eq:lik2}) turns into\n\\begin{displaymath}\n   n^2 l\\bar{\\theta}_e + (n\/2)l\\theta_r +  (n\/2)\\bar{\\theta}_r,  \n\\end{displaymath}\nboth of which are maximized by $\\theta_e=0$ and $\\theta_r=1\/2$.\n\\end{example}\n\nWe now proceed to establish conditions under which (\\ref{eq:unbiased2}) is guaranteed to hold.\nWe first\nrestrict attention to models where the sufficient statistics for \n$Q^{(n)}_{\\boldtheta}(\\omega)$ are given by induced substructure counts, defined next.\n\n\\begin{definition}\n  Let $\\omega\\in\\Omega^{(n)}$ and $k\\leq n$. Let $\\tilde{\\omega}\\in\\Omega^{(k)}$. The \n\\emph{ordered substructure count} \nof $\\tilde{\\omega}$ in $\\omega$ is defined as \n\\begin{displaymath}\n  |\\{\\boldi: \\omega\\downarrow \\boldi \\equiv \\tilde{\\omega}   \\}| =: C_{\\tilde{\\omega}}(\\omega)\n\\end{displaymath}\nwhere \n\\begin{itemize}\n\\item $\\boldi=(i_0,\\ldots,i_{k-1})$ ranges over the $\\frac{n!}{(n-k)!}$ \ntuples of $k$ distinct elements from $[n]$\n\\item $\\omega\\downarrow\\boldi$ is the  sub-structure induced by $\\boldi$ in $\\omega$\n\\item $ \\equiv $ stands for isomorphism under the mapping $i_j \\mapsto j$\n\\end{itemize}\n\\end{definition}\n\nWe refer to this as ordered substructure count, because it corresponds to taking an ordered sample\nof $k$ elements from $\\omega$, and matching the induced substructure against $\\tilde{\\omega}$ under\nthe unique mapping $i_j \\mapsto j$ defined by the order of $\\boldi$. \nThe ordered substructure counts are closely related to the ``Model B'' of~\\cite{kuzelka2018relational} for\ndefining the probability of a formula, and \\emph{homomorphism densities} \nfor the \nconvergence analysis of graph sequences~\\cite{borgsa2008convergent}.  \n\nCollecting all ordered substructure counts for worlds $\\tilde{\\omega}$ up to size $k$, we obtain:\n\n\n\\begin{definition}\n  The \\emph{complete $k$-count statistics} of $\\omega$ is the set\n\\begin{displaymath}\n  C_k(\\omega):=\\cup_{1\\leq l\\leq k}\\{ C_{\\tilde{\\omega}}(\\omega)\\mid \\tilde{\\omega} \\in\\Omega^{(l)} \\}\n\\end{displaymath}\n\\end{definition}\n\nClearly, there is redundancy in the complete $k$-count statistics, since the substructure counts \nfor worlds of size $l<k$ can be derived from the substructure counts for the worlds of \nsize $k$. However, it is convenient to also directly include the counts for smaller substructures\nexplicitly in our statistics. The following definition describes a special case of statistical\nsufficiency, which is suitable in our context.\n\n\\begin{definition}\n  Let $k\\geq 1$. A parametric family of RRSMs is \\emph{determined by $k$-count statistics},\nif for all $n$, $Q^{(n)}_{\\boldtheta}(\\omega)$ depends on $\\omega$ only as a function of\n$C_k(\\omega)$. \n\\end{definition}\n\n\nIt is\nstraightforward that $k$-count statistics are sufficient for MLNs that contain\nat most $k$ variables in each formula. For RBNs,  $k$-count statistics are not sufficient \nin general, but they are sufficient for the restricted class of projective RBNs identified in \nSection~\\ref{sec:fragments}. For ProbLog the situation is a bit more involved, as discussed \nin Example~\\ref{ex:consistency-problog} below.\n\nAs a first step to exploit determination by $k$-count statistics to ensure (\\ref{eq:unbiased2}), we \nnote that induced substructure sampling provides an unbiased estimator for the\nordered substructure counts of $\\omega'$ (normalized to substructure \\emph{frequencies}):\n\n\\begin{lemma}\n\\label{lem:suffstats}\n  Let $\\omega'\\in\\Omega^{(n)}$, $k\\leq m\\leq n$, and $\\tilde{\\omega}\\in\\Omega^{(k)}$. Then\n  \\begin{displaymath}\n    E_{\\omega}[\\frac{(m-k)!}{m!}  C_{\\tilde{\\omega}}(\\omega)]=\\frac{(n-k)!}{n!}C_{\\tilde{\\omega}}(\\omega'),\n  \\end{displaymath}\nwhere the expectation is with respect to induced substructure sampling from $\\omega'$ of \nworlds $\\omega\\in\\Omega^{(m)}$.\n\\end{lemma}\n\nNext, we consider a particular form of the likelihood function determined\nby $k$-count statistics:\n\n\\begin{definition}\n\\label{def:linsep}\n  We say that the log-likelihood function is \\emph{linear and separable} in the $k$-count statistics, if\nfor $\\omega\\in\\Omega^{(m)}$:\n  \\begin{equation}\n  \\label{eq:linsep}\n    \\emph{log}L^{(m)}(\\boldtheta\\mid \\omega) =\n    \\sum_{l=1}^k c(m,l) \\sum_{\\tilde{\\omega}\\in\\Omega^{(l)}}  \n    C_{\\tilde{\\omega}}(\\omega)\\cdot f_{\\tilde{\\omega}}(\\boldtheta[\\tilde{\\omega}])\n  \\end{equation}\nwhere\n\\begin{itemize}\n\\item $ c(m,l)$ is a constant depending only on $m$ and $l$\n\\item $f_{\\tilde{\\omega}}(\\boldtheta[\\tilde{\\omega}])$ is a function of a subset of \nparameters $\\boldtheta[\\tilde{\\omega}]\\subseteq \\boldtheta$, such that the following \\emph{separability}\nproperty holds: if $\\tilde{\\omega}\\in\\Omega^{(l)}$, $\\tilde{\\omega}'\\in\\Omega^{(l')}$ with\n$l\\neq l'$, then $\\boldtheta[\\tilde{\\omega}] \\cap  \\boldtheta[\\tilde{\\omega}']=\\emptyset$.\n\\end{itemize}\n\\end{definition}\n\n\n\\begin{proposition}\n\\label{prop:linseplik}\n  If the likelihood function is linear and separable in the $k$-count\nstatistics, then (\\ref{eq:unbiased2}) holds.\n\\end{proposition}\n\n\\begin{proof}\n Due to the separability condition, the maximization of $\\emph{log}L^{(m)}$ can be divided into $k$ \nindependent maximization problems of the functions\n\\begin{multline*}\n\\emph{log}L^{(m)}_l(\\boldtheta\\mid \\omega) :=\n  c(n,l) \\sum_{\\tilde{\\omega}\\in\\Omega^{(l)}}  \n    C_{\\tilde{\\omega}}(\\omega)\\cdot f_{\\tilde{\\omega}}(\\boldtheta[\\tilde{\\omega}])\\hspace{5mm}\\\\\n    (l=1,\\ldots,k),\n\\end{multline*}\nfor disjoint sets of parameters $\\boldtheta[l]:=\\cup_{\\tilde{\\omega}\\in\\Omega^{(l)}}\\boldtheta[\\tilde{\\omega}]$. \n\nThen with Lemma~\\ref{lem:suffstats}:\n\\begin{multline}\n  E_{\\omega}[\\emph{log}L^{(m)}_l(\\boldtheta\\mid \\omega)]= \\\\\n   c(m,l)\\frac{m!}{(m-l)!}  \\sum_{\\tilde{\\omega}\\in\\Omega^{(l)}}  E_{\\omega}[\\frac{(m-l)!}{m!} C_{\\tilde{\\omega}}(\\omega)]\\cdot f_{\\tilde{\\omega}}(\\boldtheta[\\tilde{\\omega}])=\\\\\n c(m,l)\\frac{m!}{(m-l)!}  \\sum_{\\tilde{\\omega}\\in\\Omega^{(l)}}  \\frac{(n-l)!}{n!} C_{\\tilde{\\omega}}(\\omega)\\cdot f_{\\tilde{\\omega}}(\\boldtheta[\\tilde{\\omega}])=\\\\\n \\frac{c(m,l)}{c(n,l)}\\frac{m! (n-l)!}{(m-l)! n!} \\emph{log}L^{(n)}_l(\\boldtheta\\mid \\omega')],\n\\end{multline}\nfrom which  (\\ref{eq:unbiased2}) follows.\n\\end{proof}\n\n\\subsection{Directed Model Examples}\nFor ProbLog and RBNs we have to consider two different scenarios: the first scenario is that all \nrelations that appear in the model are also observed in the possible worlds $\\omega$ that constitute our training data. Under complete observability, (\\ref{eq:unbiased2})  holds in both models as follows. \n\n\\begin{example}\n\\label{ex:consistency-rbn}\n  Using Proposition~\\ref{prop:linseplik} we can show that (\\ref{eq:unbiased2}) holds for \nRBNs without combining functions, under the following two additional conditions: \n\\begin{description}\n\\item[(i)] $m \\geq k$, where $k$ is  the largest arity\nof relations in the underlying signature $S$;\n\\item[(ii)] the RBN does not contain \nmultiple occurrences of the same parameter.\n\\end{description}\nCondition (i) in conjunction with the restriction to the projective fragment \nensures that the model is determined by $k$-count statistics. Condition (ii) ensures the \nseparability property. \n\\end{example}\n\nRelated to the previous example is Schulte's \\citeyear{Schulte2011} pseudo-likelihood function for \nfirst-order Bayesian networks, which also satisfies the conditions of Definition~\\ref{def:linsep}.\n  \n\n\\begin{example}\n\\label{ex:consistency-problog}\nUnder complete observability,\nparameter learning for ProbLog becomes trivial, since\nthe maximum likelihood parameters for the labeled facts are just obtained as the empirical \nfrequencies with which groundings of the relations in the labeled facts are true. In particular, then \nthe likelihood function is determined by $k$-count statistics, with $k$ the maximal arity of any\nrelation in labeled facts, and (\\ref{eq:unbiased2})\ntrivially holds, even without the restriction to projective ProbLog.\n\n\nHowever, more realistically, the ProbLog model will contain also some synthetic, unobserved \nrelations, as shown in (\\ref{eq:problog3}) and (\\ref{eq:problog4}).  \nIn that case, the likelihood function induced by a possible world \n$\\omega$ for the observable relations becomes a sum of products of parameters, which does not \ndecompose as (\\ref{eq:linsep}).\n\\end{example}\n\n\n\\section{Conclusion} The domain-size dependence of marginal probabilities is a  property that causes difficulties for both inference and learning. This dependence has been examined in statistical theory using the concept of a projective family of distributions, whose inferences do not depend on domain size. This paper considered whether common SRL models define projective families. \nThe paper gives sufficient conditions on the model structures for three different types of SRL frameworks that ensure projectivity. The conditions are quite restrictive, which is evidence that projectivity is difficult to achieve in SRL models. For learning, we examined conditions under which maximum likelihood parameter estimation is valid when it is applied to an observed sub-network drawn from a larger network of known size. \n\nWe believe that the projectivity of SRL models is an important and fruitful topic for future research. Open questions include the following. (1) Are there examples of domains for which our MLN projectivity condition is natural (all atoms in a formula share the same first-order variables)? (2) For RBNs, are there conditions on combining rules (rather than model structure) that ensure projectivity? For example, under which conditions does the combining rule \\emph{average} define projective models? (3) Do our learning results for maximum likelihood learning carry over to learning with pseudo-likelihood functions? (4) Are suitable fragments of relational dependency networks projective \\cite{Neville2007}?\n\n\\bibliographystyle{aaai}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\n\nLet $f : X \\to S$ be a smooth projective family defined over $\\mathcal{O}_{K, N} = \\mathcal{O}_{K}[N^{-1}]$, with $K \\subset \\mathbb{C}$ a number field, $\\mathcal{O}_{K}$ its ring of integers, and $N$ any non-zero element of $\\mathbb{Z}$. In a recent paper \\cite{LV}, Lawrence and Venkatesh proposed a strategy for bounding the dimension of the Zariski closure $\\overline{S(\\mathcal{O}_{K,N})}^{\\textrm{Zar}}$ of $S(\\mathcal{O}_{K,N})$. Since for any point $s \\in S(\\mathcal{O}_{K,N})$ the fibre $X_{s}$ has good reduction away from $N$, the problem of bounding the dimension of $\\overline{S(\\mathcal{O}_{K,N})}^{\\textrm{Zar}}$ can be interpreted as Shafarevich-type problem for the family $f$. In particular, the Shaferevich conjectures for curves, abelian varieties, and K3 surfaces are equivalent to showing that $\\dim \\overline{S(\\mathcal{O}_{K,N})}^{\\textrm{Zar}} = 0$ for particular choices of $f$.\n\nThe Lawrence-Venkatesh strategy has been implemented to reprove various classical diophantine finiteness results, including the Mordell conjecture \\cite{LV}, the finiteness of $S$-units \\cite{LV}, and Seigel's theorem \\cite{noordman2021siegels}. However it is of particular interest to apply the strategy in situations where $\\dim S > 1$. In this situation the problem becomes much harder, because one needs some understanding of the monodromy of the family $f$ over essentially arbitrary geometrically irreducible subvarieties $Z \\subset S_{\\mathbb{C}}$. To explain what we mean, let $\\mathbb{V} = R^{i} f_{*} \\mathbb{Z}$ and define for each such $Z \\subset S_{\\mathbb{C}}$:\n\\begin{defn}\nThe algebraic monodromy group $\\mathbf{H}_{Z}$ of $Z$ is the identity component of Zariski closure of the monodromy representation associated to $\\restr{\\mathbb{V}}{Z^{\\textrm{nor}}}$, where $Z^{\\textrm{nor}} \\to Z$ is the normalization.\n\\end{defn}\n\\noindent The variation $\\mathbb{V}$ determines a flag variety $\\ch{L}$, on which the group $\\mathbf{H}_{Z}$ can be said to act after identifying $\\ch{L}$ with the variety of Hodge flags on some fibre $\\mathbb{V}_{s}$ for $s \\in Z(\\mathbb{C})$. Let $\\varphi : S \\to \\Gamma \\backslash D$ be a period map determined by $\\mathbb{V}$ with $D \\subset \\ch{L}$ the complex submanifold of polarized Hodge flags. Then the key quantity relevant for the Lawrence-Venkatesh method is\n\\begin{equation}\n\\Delta = \\min_{Z, s} \\left[ \\dim (\\mathbf{H}_{Z} \\cdot F^{\\bullet}_{s}) - \\dim \\varphi(Z) \\right] , \n\\end{equation}\nwhere the minimum is taken over all positive-dimensional geometrically irreducible subvarieties $Z \\subset S_{\\mathbb{C}}$, points $s \\in Z(\\mathbb{C})$, and where $F^{\\bullet}_{s}$ is the Hodge flag on $\\mathbb{V}_{s}$.\\footnote{We not here that $\\varphi(Z)$ is analytically constructible, for instance by applying the main result of \\cite{OMINGAGA}, so its dimension makes sense.} In particular, the Lawrence-Venkatesh method produces an integer $k$, and shows that if $\\Delta \\geq k$, then the Shafarevich conjecture holds for $f$.\n\nFor successful applications of the Lawrence-Venkatesh strategy for the Shafarevich problem in situations when $\\dim S > 1$ we know of only the paper \\cite{lawrence2020shafarevich} of Lawrence and Sawin, who are able to apply this strategy beyond the first induction step to prove a Shafarevich conjecture for hypersurfaces lying inside a fixed abelian variety $A$. Their methods require the auxilliary use of a Tannakian category associated to $A$, and it seems unclear what to do without this abelian variety structure present.\n\nOur main result is as follows:\n\n\\begin{thm}\n\\label{mainthm}\nConsider a smooth projective family $f : X \\to S$ defined over $\\mathcal{O}_{K,N}$ and with $S$ smooth and quasi-projective, and given an integer $d$, define\n\\begin{equation}\n\\Delta_{d} = \\min_{Z, s} \\left[ \\dim (\\mathbf{H}_{Z} \\cdot F^{\\bullet}_{s}) - \\dim \\varphi(Z) \\right] , \n\\end{equation}\nwhere the minimum ranges over all geometrically-irreducible subvarieties $Z \\subset S_{\\mathbb{C}}$ with $\\dim Z > d$ and points $s \\in Z(\\mathbb{C})$. Then there exists an effective procedure which outputs an infinite sequence of integers\n\\[ \\kappa(1) \\leq \\kappa(2) \\leq \\cdots \\leq \\kappa(r) \\leq \\cdots < \\Delta_{d} \\]\nsuch that for some $r = r_{0}$ we have $\\kappa(r_{0}) = \\Delta_{d} - 1$.\n\nIf moreover the period map $\\varphi$ is quasi-finite, one can determine $r_{0}$.\n\\end{thm}\n\n\\begin{rem}\nLet us make absolutely clear what is meant by the term ``effective procedure'' in \\autoref{mainthm}. We mean that there exists an infinite-time algorithm (for instance, a non-halting Turing machine), which outputs a sequence of integers $\\{ \\kappa(r) \\}_{r = 1}^{\\infty}$, with the integer $\\kappa(r)$ being outputted at some finite time $t_{r}$ depending on $r$. Moreover, one also has at time $t_{r}$ a proof that $\\kappa(r) < \\Delta_{d}$. Therefore, after time $t_{r}$, one can stop the algorithm and use the bound $\\kappa(r) < \\Delta_{d}$ as input to the Lawrence-Venkatesh method. One is also guaranteed that there is some $r_{0}$ so that at time $t_{r_{0}}$ the bound $\\kappa(r_{0}) < \\Delta_{d}$ is best possible, however one does not necessarily have a method to determine $r_{0}$ unless $\\varphi$ is quasi-finite. Finally, the algorithm can be described entirely in terms of algebro-geometric computations involving algebraically constructible sets, and implicit in the proof is a description of how to implement it.\n\\end{rem}\n\nThere is no fundamental obstruction which requires us to restrict to quasi-finite $\\varphi$ for the second part of \\autoref{mainthm}. Rather, the second part of \\autoref{mainthm} references some delicate arguments in \\cite{urbanik2021sets} which are only enough to handle the quasi-finite case directly, and recalling enough of the machinery of \\cite{urbanik2021sets} to carry out the argument for the general case would lead us too far astray from the main ideas. We note that one does not actually need to determine the integer $r_{0}$ referenced in \\autoref{mainthm} to apply the machinery of Lawrence and Venkatesh: one wants to be able to compute the best possible lower bound for $\\Delta_{d}$, but one is not required to actually prove that the bound one has is optimal in order to deduce diophantine finiteness results.\n\n\\subsection{The Approach of Lawrence and Venkatesh}\n\nWe begin with a preliminary observation. To show that $\\dim \\overline{S(\\mathcal{O}_{K,N})}^{\\textrm{Zar}} \\leq d$, it suffices to show, for any irreducible subscheme $T \\subset S$ of dimension $> d$ and defined over $\\mathcal{O}_{K,N}$, that the Zariski closure $\\overline{T(\\mathcal{O}_{K,N})}^{\\textrm{Zar}}$ is a proper algebraic subscheme of $T$. We therefore fix such a subscheme $T \\subset S$ with $\\dim T > d$, and seek to show that $\\dim \\overline{T(\\mathcal{O}_{K,N})}^{\\textrm{Zar}} < \\dim T$.\n\nFix a prime $p \\in \\mathbb{Z}$ not dividing $N$, and let $t \\in T(\\mathcal{O}_{K,N})$ be a point. It is conjectured that for any $i$ the representation of $\\textrm{Gal}(\\overline{K}\/K)$ on $H^{i}_{\\textrm{\\'et}}(X_{t,\\overline{K}}, \\mathbb{Q}_{p})$ is semisimple. If this result were to be known for all such $t$, an argument of Faltings shows that for $t \\in T(\\mathcal{O}_{K,N})$ there are at most finitely many possibilities for the isomorphism class of the representation of $\\textrm{Gal}(\\overline{K}\/K)$ on $H^{i}_{\\textrm{\\'et}}(X_{t,\\overline{K}}, \\mathbb{Q}_{p})$. To establish the non Zariski-density of $T(\\mathcal{O}_{K,N})$ it would then suffice to show that the fibres of the map\n\\[ t \\in T(\\mathcal{O}_{K,N})  \\hspace{1em} \\xrightarrow{\\tau} \\hspace{1em} \\big\\{ \\textrm{Gal}(\\overline{K}\/K)\\textrm{-rep. on }H^{i}_{\\textrm{\\'et}}(X_{t,\\overline{K}}, \\mathbb{Q}_{p}) \\big\\}\\hspace{0.5em} \\big\/ \\hspace{0.5em} \\textrm{iso}.  \\]\nare not Zariski dense. As explained by Lawrence and Venkatesh in \\cite{LV}, this is essentially the original argument for the Mordell conjecture due to Faltings.\n\nThe problem with applying this strategy in general is twofold. First, the semisimplicity of $H^{i}_{\\textrm{\\'et}}(X_{t,\\overline{K}}, \\mathbb{Q}_{p})$ is not known, and for most choices of $f$ appears out of reach. Secondly, without a good geometric interpretation of the \\'etale cohomology $H^{i}_{\\textrm{\\'et}}(X_{t,\\overline{K}}, \\mathbb{Q}_{p})$ it is difficult to understand $\\tau$. The key insight in the paper of Lawrence and Venkatesh is that one may potentially overcome both problems by passing to a $p$-adic setting where they are more managable.\n\nInstead of considering the global Galois representation $\\rho_{t} : \\textrm{Gal}(\\overline{K}\/K) \\to H^{i}_{\\textrm{\\'et}}(X_{t,\\overline{K}}, \\mathbb{Q}_{p})$, we consider its semisimplification $\\rho^{\\textrm{ss}}_{t}$, and restrict $\\rho_{t}$ along the map $\\textrm{Gal}(\\overline{K_{v}}\/K_{v}) \\to \\textrm{Gal}(\\overline{K}\/K)$ induced by a fixed embedding $\\overline{K} \\hookrightarrow \\overline{K_{v}}$ to obtain $\\rho_{t,v}$, where $v$ is a fixed place above $p$. The functors of $p$-adic Hodge theory tell us that the representation $\\rho_{t,v}$ determines a triple $(H^{i}_{\\textrm{dR}}(X_{t,K_{v}}), \\phi_{t}, F^{\\bullet}_{t})$, where $F^{\\bullet}_{t}$ is the Hodge filtration on de Rham cohomology and $\\phi_{t}$ is the crystalline Frobenius at $t$. If we somehow manage to consider the data $(H^{i}_{\\textrm{dR}}(X_{t,K_{v}}), \\phi_{t}, F^{\\bullet}_{t})$ up to ``semisimplification'' (in the sense that we identify such triples when the associated \\emph{global} Galois representations have isomorphic semisimplifications), our problem is then to study the fibres of the map\n\\[ t \\in T(\\mathcal{O}_{K,N}) \\hspace{0.5em} \\xrightarrow{\\tau_{p}} \\hspace{0.5em} \\big\\{\\textrm{``semisimplifications'' of } (H^{i}_{\\textrm{dR}}(X_{t,K_{v}}), \\phi_{t}, F^{\\bullet}_{t}) \\big\\} \\hspace{0.5em} \\big\/ \\hspace{0.5em} \\textrm{iso}. \\]\nand show they lie in a Zariski closed subscheme of smaller dimension.\n\nNext, we make the elementary observation that to bound the dimension of the Zariski closure of $T(\\mathcal{O}_{K,N})$, it suffices to cover $T(\\mathcal{O}_{K,v})$ by finitely many $v$-adic disks $D_{1}, \\hdots, D_{k}$ and bound the dimension of the Zariski closure of $D_{i} \\cap T(\\mathcal{O}_{K,N})$ for each $i$; here $\\mathcal{O}_{K,v}$ is the ring of $v$-adic integers. It can then be shown that there exists such a cover for which the Hodge bundle $\\mathcal{H} = R^{i} f_{*} \\Omega^{\\bullet}_{X\/S}$ can be trivialized rigid-analytically over each $D_{i}$, moreover with respect to each such trivialilzation the Frobenius operator $\\phi_{t}$ is independent of $t \\in D_{i}$. The problem then reduces to studying a varying filtration $F^{\\bullet}_{t}$ on a fixed vector space $V_{p} = H^{i}_{\\textrm{dR}}(X_{t_{0}})$ for some $t_{0} \\in D_{i}$. In particular, we obtain a rigid-analytic map\n\\[ D_{i} \\hspace{0.5em} \\xrightarrow{\\psi_{p}} \\hspace{0.5em} \\underbrace{\\{ \\hspace{0.3em} \\textrm{Hodge filtrations on }V_{p} \\hspace{0.3em} \\}}_{\\ch{L}_{p}} , \\]\nand those points of $\\ch{L}_{p}$ arising from points $t \\in T(\\mathcal{O}_{K,N}) \\cap D_{i}$ lie inside finitely many subvarieties $O_{i1}, \\hdots, O_{i\\ell}$ of $\\ch{L}_{p}$ corresponding to the finitely many possible isomorphism classes of $\\rho^{\\textrm{ss}}_{t}$.\n\nWe are now faced with the problem of understanding the intersections $\\psi_{p}(D_{i}) \\cap O_{ij}$, and showing that their inverse images under $\\psi_{p}^{-1}$ lie in an algebraic subscheme of smaller dimension. One part of this problem is to understand the dimensions of the varieties $O_{ij}$, and to show that they are sufficiently small: this step is carried out successfully for the families of hypersurfaces studied both by Lawrence-Venkatesh and Lawrence-Sawin, and appears to be managable in general. The more difficult object to control is $\\psi_{p}(D_{i})$, for which we need to understand the variation of the filtration $F^{\\bullet}$ over $D_{i}$. This, in turn, is governed by the Gauss-Manin connection $\\nabla : \\mathcal{H} \\otimes \\Omega^{1}_{T} \\to \\mathcal{H}$, which exists universally over $S$ after possibly increasing $N$; we may adjust $p$ so that it does not divide $N$ if necessary. The fact that $\\nabla$ exists universally over $\\mathcal{O}_{K,N}$ means that the same system of differential equations satisfied by $\\psi_{p}$ at $t \\in T(\\mathcal{O}_{K,N}) \\cap D_{i}$ is also satisfied by a Hodge-theoretic period map $\\psi : B \\to \\ch{L}$ on a sufficiently small analytic neighbourhood $B \\subset T(\\mathbb{C})$ containing $t$, where $\\ch{L}$ is a variety of Hodge flags. In particular, one can prove that the Zariski closures of $\\psi_{p}(D_{i})$ and $\\psi(B)$ have the same dimension.\n\nThe final step, which is to show that the Zariski closure of $T(\\mathcal{O}_{K,N}) \\cap D_{i}$ in $T$ has smaller dimension, is completed as follows. The Ax-Schanuel Theorem \\cite{AXSCHAN} for variations of Hodge structures due to Bakker-Tsimerman shows that if $V$ is an algebraic subvariety of $\\ch{L}$ of dimension at most $\\dim \\overline{\\psi(B)}^{\\textrm{Zar}} - \\dim \\psi(B)$,\\footnote{The dimension $\\dim \\psi(B)$ can once again, at least for open neighbourhoods $B$ with a sufficiently mild geometry, be made sense of the dimension of a locally constructible analytic set. Alternatively one can replace $\\dim \\psi(B)$ with $\\dim \\varphi(Z)$, where $\\varphi$ is as before and $Z \\subset T_{\\mathbb{C}}$ is a component containing $B$.} then the inverse image under $\\psi$ of the intersection $\\psi(B) \\cap V$ lies in an algebraic subvariety of $T_{\\mathbb{C}}$ of smaller dimension. Choosing an isomorphism $\\mathbb{C} \\cong \\overline{K_{v}}$ one can transfer this fact to the same statement for the map $\\psi_{p}$ and in particular for $V = O_{ij}$. Our problem is finally reduced to giving a lower bound for the difference $\\dim \\overline{\\psi(B)}^{\\textrm{Zar}} - \\dim \\psi(B)$. Our main result now reads:\n\n\\begin{thm}\n\\label{mainthm2}\nDefine the quantity\n\\[ \\Delta_{d} := \\min_{Z, \\psi} \\left[ \\dim \\overline{\\psi(B)}^{\\textrm{Zar}} - \\dim \\psi(B) \\right] , \\]\nwhere $Z$ ranges over all irreducible complex algebraic subvarieties $Z \\subset S_{\\mathbb{C}}$ of dimension greater than $d$, and where $\\psi$ is any complex analytic period map determined by the variation of Hodge structures $\\mathbb{V} = R^{i} f_{*} \\mathbb{Z}$ and defined on a neighbourhood $B \\subset Z(\\mathbb{C})$. Then there exists an effective procedure which outputs an infinite sequence of lower bounds\n\\[ \\kappa(1) \\leq \\kappa(2) \\leq \\cdots \\leq \\kappa(r) \\leq \\cdots < \\Delta_{d} \\]\nsuch that for some $r = r_{0}$ we have $\\kappa(r_{0}) = \\Delta_{d} - 1$.\n\nIf moreover the period map $\\varphi$ is quasi-finite, one can determine $r_{0}$.\n\\end{thm}\n\n We note that by \\cite[Lem. 4.10(ii)]{urbanik2021sets} it is also a consequence of the Bakker-Tsimerman Theorem that when $Z$ is geometrically irreducible, we have $\\overline{\\psi(B)}^{\\textrm{Zar}} = \\mathbf{H}_{Z} \\cdot \\psi(t)$ for any point $t \\in Z(\\mathbb{C})$, which recovers the statement of \\autoref{mainthm}. \n\n\\subsection{Basic Idea of the Method}\n\\label{methodsketch}\n\nWe may observe that the computation of the bound described in \\autoref{mainthm2} is a purely Hodge-theoretic problem, i.e., it concerns only properties of the integral variation $\\mathbb{V} = R^{i} f_{*} \\mathbb{Z}$ of Hodge structures on $S_{\\mathbb{C}}$. Let $\\mathcal{Q} : \\mathbb{V} \\otimes \\mathbb{V} \\to \\mathbb{Z}$ be a polarization of $\\mathbb{V}$, and let $(V, Q)$ be a fixed polarized lattice isomorphic to one (hence any) fibre of $(\\mathbb{V}, \\mathcal{Q})$; as it causes no harm, we will assume that $V = \\mathbb{Z}^{m}$ for some $m$, and therefore sometimes write $\\textrm{GL}_{m}$ for $\\textrm{GL}(V)$. Let $D$ be the complex manifold parametrizing polarized Hodge structures on $(V, Q)$ with the same Hodge numbers as $(\\mathbb{V}, \\mathcal{Q})$. A point $h \\in D$ we may view as a morphism $h : \\mathbb{S} \\to \\textrm{GL}(V)_{\\mathbb{R}}$, where $\\mathbb{S}$ is the Deligne torus, and the Mumford-Tate group $\\textrm{MT}(h)$ is the $\\mathbb{Q}$-Zariski closure of $h(\\mathbb{S})$. \n\nTo present our method, we introduce some terminology:\n\n\\begin{notn}\nWe denote by $\\ch{L}$ the $\\mathbb{Q}$-algebraic variety of all Hodge flags on the lattice $V$, not necessarily polarized. We note that $D$ is an open submanifold of a closed $\\mathbb{Q}$-algebraic subvariety $\\ch{D} \\subset \\ch{L}$.\n\\end{notn}\n\n\\begin{defn}\nGiven two subvarieties $W_{1}, W_{2} \\subset \\ch{L}$, we say that $W_{1} \\sim_{\\textrm{GL}} W_{2}$ if there exists $g \\in \\textrm{GL}_{m}(\\mathbb{C})$ such that $g \\cdot W_{1} = W_{2}$. Given a variety $W \\subset \\ch{L}$, we call the equivalence class $\\mathcal{C}(W)$ under $\\sim_{\\textrm{GL}}$ a \\emph{type}. The dimension of a type $\\mathcal{C}(W)$ is the dimension of $W$.\n\\end{defn}\n\n\\begin{defn}\nWe say that a type $\\mathcal{C}$ is \\emph{Hodge-theoretic} if $\\mathcal{C} = \\mathcal{C}(W)$, where $W = N(\\mathbb{C}) \\cdot h$ for $h \\in D$ and $N$ a $\\mathbb{Q}$-algebraic normal subgroup of $\\textrm{MT}(h)$.\n\\end{defn}\n\n\\vspace{0.5em}\n\nThe first step in our algorithm is:\n\n\\begin{quote}\n\\textbf{Step One:} Compute a finite list of types $\\mathcal{C}_{1}, \\hdots, \\mathcal{C}_{\\ell}$ such that every Hodge-theoretic type appears somewhere in the list.\n\\end{quote}\n\n\\noindent When we say to compute a type $\\mathcal{C}$, we mean to compute a representative $W \\subset \\ch{L}$ such that $\\mathcal{C} = \\mathcal{C}(W)$. That there are only finitely many Hodge-theoretic types is shown in \\autoref{finmantypes} below. \n\n\nThe problem given in Step One is solved in \\cite[Prop. 5.4]{urbanik2021sets}; we will say little about it here. It is related to the problem of classifying Mumford-Tate groups up to conjugacy by $\\textrm{GL}_{m}(\\mathbb{C})$, for which one can use a constructive version of the proof in \\cite[Thm. 4.14]{hodgelocivoisin}. It is also similar to the problem of classifying Mumford-Tate domains as studied in \\cite[Chap. VII]{GGK}. We note that the methods of \\cite[Chap. VII]{GGK}, when they can be carried out effectively, result in an approach for which $\\mathcal{C}_{1}, \\hdots, \\mathcal{C}_{\\ell}$ will be exactly the set of Hodge-theoretic types.\n\n\\vspace{0.5em}\n\nThe second step is more involved, and is the crux of our method. To describe it we need to introduce some terminology. \n\n\\begin{defn}\n\\label{locperdef}\nA \\emph{local period map} is a map $\\psi : B \\to \\ch{L}$ obtained as a composition $\\psi = q \\circ A$, where:\n\\begin{itemize}\n\\item[(i)] The set $B \\subset S(\\mathbb{C})$ is a connected analytic neighbourhood on which $\\mathbb{V}$ is constant and $F^{k} \\mathcal{H}$ is trivial for each $k$, where $\\mathcal{H} = \\mathbb{V} \\otimes \\mathcal{O}_{\\an{S}}$.\n\\item[(ii)] The map $A : B \\to \\textrm{GL}_{m}(\\mathbb{C})$ is a varying filtration-compatible period matrix over $B$. More precisely, there exists a basis $v^{1}, \\hdots, v^{m}$ for $\\mathcal{H}(B)$, compatible with the filtration in the sense that $F^{k} \\mathcal{H}(B)$ is spanned by $v^{1}, \\hdots, v^{i_{k}}$ for some $i_{k}$, and a flat frame $b^{1}, \\hdots, b^{m}$ for $\\mathbb{V}_{\\mathbb{C}}(B)$, such that $A(s)$ is the change-of-basis matrix from $v^{1}_{s}, \\hdots, v^{m}_{s}$ to $b^{1}_{s}, \\hdots, b^{m}_{s}$.\n\\item[(iii)] The map $q : \\textrm{GL}_{m} \\to \\ch{L}$ sends a matrix $M$ to the Hodge flag $F_{M}^{\\bullet}$ defined by the property that $F^{k}_{M}$ is spanned by the first $i_{k}$ columns.\n\\end{itemize}\n\\end{defn}\n\n\\noindent To summarize the preceding definition: a local period map is exactly a period map on $B$ except one does not necessarily compute periods with respect to the integral lattice $\\mathbb{V}(B) \\subset \\mathbb{V}_{\\mathbb{C}}(B)$ but is instead allowed to consider periods with respect to a more general complex flat frame. There is a natural $\\textrm{GL}_{m}(\\mathbb{C})$-action on the set of germs of local period maps at a point $s \\in S(\\mathbb{C})$, where $M \\in \\textrm{GL}_{m}(\\mathbb{C})$ acts on the map $\\psi = q \\circ A$ to give $M \\cdot \\psi = q \\circ (M \\cdot A)$. This action corresponds exactly to a change of the flat frame $b^{1}, \\hdots, b^{m}$, and all germs of local period maps at $s$ lie in a single $\\textrm{GL}_{m}(\\mathbb{C})$-orbit.\n\n\nThe construction of a local period map $\\psi : B \\to \\ch{L}$ involves picking a basis $b^{1}, \\hdots, b^{m}$ of $\\mathbb{V}_{\\mathbb{C}}(B)$, and hence choosing an isomorphism $\\mathbb{V}_{\\mathbb{C}}(B) \\simeq \\mathbb{C}^{m}$. When working with a local period map, we will always assume that such a basis has been choosen, and hence identify subgroups of $\\textrm{GL}(\\mathbb{V}_{\\mathbb{C}}(B))$ with subgroups of $\\textrm{GL}_{m}(\\mathbb{C})$. In particular, if $Z \\subset S_{\\mathbb{C}}$ is a geometrically irreducible subvariety which intersects $B$, we have an induced action of $\\mathbf{H}_{Z}$ on $\\ch{L}$.\n\nLastly, we need:\n\n\\begin{defn}\nGiven two types $\\mathcal{C}_{1}$ and $\\mathcal{C}_{2}$, we say that $\\mathcal{C}_{1} \\leq \\mathcal{C}_{2}$ if there exists $W_{i} \\subset \\ch{L}$ for $1 = 1, 2$ such that $\\mathcal{C}_{i} = \\mathcal{C}(W_{i})$ and $W_{1} \\subset W_{2}$.\n\\end{defn}\n\n\\begin{defn}\n\\label{vartypedef}\nGiven a local period map $\\psi : B \\to \\ch{L}$ and a geometrically irreducible subvariety $Z \\subset S_{\\mathbb{C}}$ intersecting $B$ at $s$, we call $\\mathcal{C}(\\overline{\\psi(B \\cap Z)}^{\\textrm{Zar}}) = \\mathcal{C}(\\mathbf{H}_{Z} \\cdot \\psi(s))$ the \\emph{type} of $Z$, and denote it by $\\mathcal{C}(Z)$.\n\\end{defn}\n\n\\noindent For well-definedness, see \\autoref{welldeflem} below. From Step One, we have computed a finite list $\\mathcal{C}_{1}, \\hdots, \\mathcal{C}_{\\ell}$ of types containing all types that can arise from the variation $\\mathbb{V}$. Our next task is then:\n\n\\begin{quote}\n\\textbf{Step Two:} For each type $\\mathcal{C}_{i}$ appearing in the list, compute a differential system $\\mathcal{T}(\\mathcal{C}_{i})$ on $S$ characterized by the property that an algebraic subvariety $Z \\subset S_{\\mathbb{C}}$ is an integral subvariety for $\\mathcal{T}(\\mathcal{C}_{i})$ if and only if $\\mathcal{C}(Z) \\leq \\mathcal{C}_{i}$, and determine the dimension of a maximal integral subvariety for this system.\n\\end{quote}\n\n\\noindent We explain precisely what we mean by ``differential system'' in \\autoref{secthree}; actually our method does something more subtle than Step Two due to the fact that we can only approximate $\\mathcal{T}(\\mathcal{C}_{i})$ up to some finite order, but for expository purposes this is the essential point. After this, we will see the problem is reduced to analyzing which of the differential systems $\\mathcal{T}(\\mathcal{C}_{i})$ admit algebraic solutions of ``exceptional'' dimension, which can be carried out using tools from functional transcendence.\n\n\\subsection{Acknowledgements}\n\nThe author thanks Brian Lawrence, Akshay Venkatesh, and Will Sawin for comments on a draft of this manuscript.\n\n\n\\section{Algebraic Monodromy Orbits up to Conjugacy}\n\nIn this section we describe an effective method for solving ``Step One'' as posed in \\autoref{methodsketch}. We will also prove some preliminary facts about types used in the introduction, and we continue with the notation established there. We will work in the context of a general polarizable integral variation of Hodge structure $\\mathbb{V}$ on the complex algebraic variety $S$, not necessarily coming from a projective family as in the introduction.\n\n\\subsection{Basic Properties of Types}\n\n\\begin{lem}\n\\label{finmantypes}\nFor any geometrically irreducible subvariety $Z \\subset S$ and any local period map $\\psi : B \\to \\ch{L}$ with $Z \\cap B$ non-empty, we have\n\\[ \\overline{\\psi(Z \\cap B)}^{\\textrm{Zar}} = \\mathbf{H}_{Z} \\cdot \\psi(s) , \\]\nfor any point $s \\in Z(\\mathbb{C})$.\n\\end{lem}\n\n\\begin{proof}\nIt suffices to show that \n\\[ \\overline{\\psi(C)}^{\\textrm{Zar}} = \\mathbf{H}_{Z} \\cdot \\psi(s) , \\]\nfor each analytic component $C \\subset Z \\cap B$ separately, with $s$ a point of $C$. By acting on $\\psi$ by an element of $\\textrm{GL}_{m}(\\mathbb{C})$, the claim can be reduced to the situation where the periods which determine $\\psi$ are computed with respect to a basis for the integral lattice $\\mathbb{V}(B)$, and then the claim follows from \\cite[Lem. 4.10(ii)]{urbanik2021sets}.\n\\end{proof}\n\n\\begin{lem}\n\\label{welldeflem}\nThe equivalence class under $\\sim_{\\textrm{GL}}$ of $\\overline{\\psi(B \\cap Z)}^{\\textrm{Zar}}$ is independent of $\\psi$; i.e., the type of $Z \\subset S$ is well-defined.\n\\end{lem}\n\n\\begin{proof}\nLet $p : Z^{\\textrm{sm}} \\to Z$ be a smooth resolution, and consider the variation $p^{*} \\mathbb{V}$. From the fact that germs of local period maps on $Z^{\\textrm{sm}}$ with respect to the variation $p^{*} \\mathbb{V}$ factor through germs of local period maps on $S$, we may reduce to the same problem for $Z^{\\textrm{sm}}$ and the variation $p^{*} \\mathbb{V}$, i.e., we may assume $Z = S$. By analytically continuing a fixed local period map $\\psi$ to the universal covering $\\widetilde{S} \\to S$, we learn from the irreducibility of $\\widetilde{S}$ that at each point $s \\in S$, there exists a local period map $\\psi_{s} : B_{s} \\to \\ch{L}$ such that $\\overline{\\psi_{s}(B_{s})}^{\\textrm{Zar}} = \\overline{\\psi(B)}^{\\textrm{Zar}}$. Since the Zariski closure of $\\psi_{s}(B_{s})$ is determined by the germ of $\\psi_{s}$ at $s$, and because all germs of local period maps at $s$ lie in a single $\\textrm{GL}_{m}(\\mathbb{C})$-orbit, the result follows.\n\\end{proof}\n\n\\begin{lem}\n\\label{fintypesarise}\nThere are only finitely many Hodge-theoretic types.\n\\end{lem}\n\n\\begin{proof}\nWe observe that the problem reduces to the following: show they are finitely many $\\textrm{GL}_{m}(\\mathbb{C})$-equivalence classes of pairs $(h, N)$, where\n\\begin{itemize}\n\\item[(i)] $h \\in D$ is a polarized Hodge structure; and\n\\item[(ii)] $N$ is a $\\mathbb{Q}$-algebraic connected normal subgroup of $\\textrm{MT}(h)$;\n\\end{itemize}\nwhere we regard $\\textrm{GL}_{m}(\\mathbb{C})$ as acting on $h$ through its action on $\\ch{L}$, and on $N$ by conjugation. Note that two such equivalent pairs will generate orbits in $\\ch{L}$ equivalent under $\\sim_{\\textrm{GL}}$. Since the groups $\\textrm{MT}(h)$ are reductive and have finitely many connected normal algebraic factors, this reduces to the same problem for pairs of the form $(h, \\textrm{MT}(h))$. We recall that $D$ is an open submanifold of $\\ch{D}$, the flag variety of flags satisfying the first Hodge-Riemann bilinear relation (the isotropy condition), and that $\\ch{D}$ is an algebraic subvariety of $\\ch{L}$. We then use the fact that there are finitely many Mumford-Tate groups up to $\\textrm{GL}_{m}(\\mathbb{C})$-conjugacy (see \\cite[Thm. 4.14]{hodgelocivoisin}), and that for a fixed Mumford-Tate group $M$ the Hodge structures in $D$ with Mumford-Tate contained in $M$ lie inside finitely many $M(\\mathbb{C})$-orbits in $\\ch{D}$, see \\cite[VI.B.9]{GGK}.\n\\end{proof}\n\n\\subsection{Computing Types up to Conjugacy}\n\nIn this section we give some references for carrying out Step One as described in the introduction.\n\n\\begin{prop}\n\\label{MTgroupequivalgo}\nThere exists an algorithm to compute subvarieties $W_{1}, \\hdots, W_{\\ell} \\subset \\ch{L}$ such that the set of Hodge-theoretic types is a (possibly proper) subset of $\\{ \\mathcal{C}(W_{1}), \\hdots, \\mathcal{C}(W_{\\ell}) \\}$.\n\\end{prop}\n\n\\begin{proof}\nThis is solved in \\cite[Prop. 5.4]{urbanik2021sets}.\n\\end{proof}\n\nLet us comment briefly on a different approach to Step One given in \\cite[Chap. VII]{GGK}. In \\cite[Chap. VII]{GGK}, the authors describe a method for classifying both Mumford-Tate groups and Mumford-Tate domains (orbits of points $h \\in D$ under $\\textrm{MT}(h)(\\mathbb{R})$ and $\\textrm{MT}(h)(\\mathbb{C})$). Given an appropriate such classification, one can easily solve Step One by computing the decompositions of the groups $\\textrm{MT}(h)$ that arise into $\\mathbb{Q}$-simple factors. The method of \\cite[Chap. VII]{GGK} is to first classify CM Hodge structures $h_{\\textrm{CM}} \\in D$, and then give a criterion for deciding when a Lie subalgebra of $\\mathfrak{gl}(V)$ corresponds to a Mumford-Tate group generating a Mumford-Tate domain containing $h_{\\textrm{CM}}$. They carry out this classification procedure successfully when $\\dim V = 4$, and so for variations with Hodge numbers $(2, 2)$ and $(1, 1, 1, 1)$.\n\nThe method given for classifying CM Hodge structures given in \\cite[Chap. VII]{GGK} is to observe that CM Hodge structures up to isogeny are determined by certain data associated to embeddings of CM fields, and hence the first step of the procedure in \\cite[Chap. VII]{GGK} is to ``classify all CM fields of rank up to [$\\dim V$] by [their] Galois group''. We are not aware of an effective method for carrying out this step.\\footnote{The paper \\cite{dodson1984structure} gives a potential approach by giving a method to classify certain abstract structures associated with Galois groups of CM fields. However one still needs to determine which such structures are actually associated to a concrete CM field.} It is also not clear to us precisely the sense in which the term ``classify'' is being used; i.e., we do not know what form the data of a ``classification of CM Hodge structures'' takes, and consequently what form the resulting classification of Mumford-Tate domains will have. For this reason, we were unable to apply the methods of \\cite[Chap. VII]{GGK} to prove \\autoref{MTgroupequivalgo}. \n\n\n\n\n\\section{Differential Tools and a Jet Criterion}\n\\label{secthree}\n\nIn this section we introduce a collection of effectively computable algebro-geometric correspondences which can be used for studying systems of differential equations on $S$ induced by the variation $\\mathbb{V}$, and then use it to solve the main problem. We have already carried out most of the work in two preceding papers \\cite{periodimages} and \\cite{urbanik2021sets}, so we will first need to collect some results. In this section we assume that $S$ is a $K$-variety for $K \\subset \\mathbb{C}$ a number field, and that $\\mathbb{V}$ is a polarizable integral variation of Hodge structure on $S_{\\mathbb{C}}$ such that the vector bundle $\\mathcal{H} = \\mathbb{V} \\otimes_{\\mathbb{Z}} \\mathcal{O}_{\\an{S_{\\mathbb{C}}}}$, the filtration $F^{\\bullet}$, and the connection $\\nabla : \\mathcal{H} \\to \\mathcal{H} \\otimes \\Omega^{1}_{S}$ all admit $K$-algebraic models. Moreover, we assume that we may effectively compute a description of these objects in terms of finitely-presented $K$-modules over an affine cover of $S$; for a justification of this assumption in the situation where $\\mathbb{V}$ comes from a smooth projective $K$-algebraic family $f : X \\to S$ see \\cite[\\S2]{urbanik2021sets}.\n\n\\subsection{The Constructive Period-Jet Correspondence}\n\nOur algebro-geometric correspondences will be formulated using the language of \\emph{jets}. Let $A^{d}_{r} = K[t_{1}, \\hdots, t_{d}]\/\\langle t_{1}, \\hdots, t_{d} \\rangle^{r+1}$, and define $\\mathbb{D}^{d}_{r} = \\textrm{Spec} \\hspace{0.15em}  A^{d}_{r}$ to be the $d$-dimensional disk of order $r$; we suppress the field $K$ in the notation. A \\emph{jet space} associated to a space $X$ is a space which parametrizes maps $\\mathbb{D}^{d}_{r} \\to X$. More formally, for $X$ a finite-type $K$-scheme, we have:\n\n\n\n\\begin{defn}\n\\label{jetspacedef}\nWe define $J^{d}_{r} X$ to be the scheme representing the contravariant functor $\\textrm{Sch}_{K} \\to \\textrm{Set}$ given by \\vspace{-0.2em}\n\\[ T \\mapsto \\textrm{Hom}_{K}(T \\times_{K} \\mathbb{D}^{d}_{r}, X), \\hspace{1.5em} [T \\to T'] \\mapsto [\\textrm{Hom}_{K}(T' \\times_{K} \\mathbb{D}^{d}_{r}, X) \\to \\textrm{Hom}_{K}(T \\times_{K} \\mathbb{D}^{d}_{r}, X)] , \\]\nwhere the natural map $\\textrm{Hom}_{K}(T' \\times_{K} \\mathbb{D}^{d}_{r}, X) \\to \\textrm{Hom}_{K}(T \\times_{K} \\mathbb{D}^{d}_{r}, X)$ obtained by pulling back along $T \\times_{K} \\mathbb{D}^{d}_{r} \\to T' \\times_{K} \\mathbb{D}^{d}_{r}$. \n\\end{defn}\n\n\\vspace{0.5em}\n\n\\noindent That the functor defining $J^{d}_{r} X$ in \\autoref{jetspacedef} is representable is handled by \\cite[\\S2]{periodimages}. Moreover, $J^{d}_{r}$ is itself a functor, sending a map $g : X \\to Y$ to the map $J^{d}_{r} g : J^{d}_{r} X \\to J^{d}_{r} Y$ that acts on points by post-composition. For $X$ an analytic space, there is an analogous construction that appears in \\cite[\\S2.3]{periodimages}. If $K \\subset \\mathbb{C}$ is a subfield, this construction is compatible with analytification.\n\nThe purpose of introducing jets is the following result, proven in \\cite{urbanik2021sets}, building on \\cite{periodimages}.\n\n\\begin{thm}\n\\label{jetcorresp}\nFor each $d, r \\geq 0$, a variation of Hodge structure $\\mathbb{V}$ on $S$ gives rise to a canonical map \n\\[ \\eta^{d}_{r} : J^{d}_{r} S \\to \\textrm{GL}_{m} \\backslash J^{d}_{r} \\ch{L} , \\] \nof algebraic stacks characterized by the property that for any local period map $\\psi : B \\to \\ch{L}$ and any jet $j \\in J^{d}_{r} B$ we have $\\psi \\circ j = \\eta^{d}_{r}(j)$ modulo $\\textrm{GL}_{m}(\\mathbb{C})$.\n\nMoreover, if the data $(\\mathcal{H}, F^{\\bullet}, \\nabla)$ associated to the variation $\\mathbb{V}$ admits a $K$-algebraic model, the map $\\eta^{d}_{r}$ is defined over $K$, and there exists an algorithm to compute the $\\textrm{GL}_{m}$-torsor $p^{d}_{r} : \\mathcal{P}^{d}_{r} \\to J^{d}_{r} S$ and the $\\textrm{GL}_{m}$-invariant map $\\alpha^{d}_{r} : \\mathcal{P}^{d}_{r} \\to J^{d}_{r} \\ch{L}$ which defines $\\eta^{d}_{r}$ from a presentation of the data $(\\mathcal{H}, F^{\\bullet}, \\nabla)$ in terms of finitely-presented $K$-modules.\n\\end{thm}\n\nWe note that the computability of the torsor $\\mathcal{P}^{d}_{r}$ in \\autoref{jetcorresp} has in particular the following consequence: if $\\mathcal{S} \\subset (\\textrm{GL}_{m} \\backslash J^{d}_{r} \\ch{L})(\\mathbb{C})$ is a subset which is the image under the quotient of a constructible $L$-algebraic set $\\mathcal{F} \\subset J^{d}_{r} \\ch{L}$, where $K \\subset L$ is a computable extension, then we can compute $(\\eta^{d}_{r})^{-1}(\\mathcal{S})$ by computing $p^{d}_{r}((\\alpha^{d}_{r})^{-1}(\\mathcal{F}))$. Thus if we define \\vspace{0.5em}\n\\begin{defn}\n\\label{Tconstdef}\nFor a constructible $L$-algebraic set $\\mathcal{F} \\subset J^{d}_{r} \\ch{L}$, with $K \\subset L$ an extension, we write\n\\[ \\mathcal{T}^{d}_{r}(\\mathcal{F}) := (\\eta^{d}_{r})^{-1}(\\textrm{GL}_{m} \\cdot \\mathcal{F}) . \\]\nMoreover, for a type $\\mathcal{C} = \\mathcal{C}(W)$, we will write either $\\mathcal{T}^{d}_{r}(\\mathcal{C})$ or $\\mathcal{T}^{d}_{r}(W)$ for the set $\\mathcal{T}^{d}_{r}(J^{d}_{r} W)$.\n\\end{defn} \\vspace{0.3em}\n\\noindent then the main consequence of the preceding discussion for our situation is the following, which is immediate from what we have said:\n\\begin{prop}\n\\label{diffconstprop}\nFor each $d, r \\geq 0$ there exists an algorithm which, given a constructible $L$-algebraic set $\\mathcal{F} \\subset J^{d}_{r} \\ch{L}$ with $K \\subset L$ a computable extension, computes $\\mathcal{T}^{d}_{r}(\\mathcal{F}) \\subset J^{d}_{r} S$. \\qed\n\\end{prop}\n\n\\subsection{Jet Conditions and Types}\n\nLet us now try to understand how computing the ``differential constraints'' induced by types $\\mathcal{C}(W)$ as in \\autoref{diffconstprop} can help us carry out Step Two of \\autoref{methodsketch}. Let $\\Gamma = \\textrm{Aut}(V, Q)(\\mathbb{Z})$, and let $\\varphi : S_{\\mathbb{C}} \\to \\Gamma \\backslash D$ be the canonical period map which sends a point $s \\in S(\\mathbb{C})$ to the isomorphism class of the polarized Hodge structure on $\\mathbb{V}_{s}$. By \\cite{OMINGAGA}, the map $\\varphi$ factors as $\\iota \\circ p$, where $p : S_{\\mathbb{C}} \\to T$ is a dominant map of algebraic varieties and $\\iota$ is a closed embedding of analytic spaces; it follows that for each subvariety $Z \\subset S_{\\mathbb{C}}$ the dimension of the image $\\varphi(Z)$ makes sense as the dimension of a constructible algebraic set.\n\nFix a sequence of compatible embeddings \n\\[ \\textrm{Spec} \\hspace{0.15em}  K = \\mathbb{D}^{0}_{r} \\xhookrightarrow{\\iota_{0}} \\mathbb{D}^{1}_{r} \\xhookrightarrow{\\iota_{1}} \\mathbb{D}^{2}_{r} \\xhookrightarrow{\\iota_{2}} \\mathbb{D}^{3}_{r} \\xhookrightarrow{\\iota_{3}} \\mathbb{D}^{4}_{r} \\xhookrightarrow{\\iota_{4}} \\cdots \\]\nof formal disks. By acting on points via pullback, we obtain natural transformations of functors $\\textrm{res}^{d}_{e} : J^{d}_{r} \\to J^{e}_{r}$ which produce maps $J^{d}_{r} X \\to J^{e}_{r} X$ that take jets $j : \\mathbb{D}^{d}_{r} \\to X$ to their restrictions $j \\circ \\iota_{d-1} \\circ \\cdots \\circ \\iota_{e}$. We are now ready to present the key proposition for our method:\n\n\\begin{defn}\nFor a scheme $X$ (resp. analytic space $X$) denote by $J^{d}_{r,nd} X \\subset J^{d}_{r} X$ the subscheme (resp. the analytic subspace) parametrizing those maps $j : \\mathbb{D}^{d}_{r} \\to X$ which are injective on the level of tangent spaces. We call such $j$ \\emph{non-degenerate} jets.\n\\end{defn}\n\n\\begin{prop}\n\\label{mainjetprop}\nLet $\\mathcal{S}$ be a set of types containing all the Hodge-theoretic types, and let $e$ and $k$ be non-negative integers. Then the following are equivalent:\n\\begin{itemize}\n\\item[(i)] there exists a geometrically irreducible subvariety $Z \\subset S_{\\mathbb{C}}$ with $\\dim Z > d$, $\\dim \\varphi(Z) \\geq e$, and such that $\\dim \\mathcal{C}(Z) - \\dim \\varphi(Z) \\leq k$;\n\\item[(ii)] there exists $\\mathcal{C} \\in \\mathcal{S}$ with $\\dim \\mathcal{C} - e \\leq k$, and such that the intersection \n\\[ \\mathcal{K}^{d}_{r}(\\mathcal{C}, e, k) := \\mathcal{T}^{d+1}_{r}(\\mathcal{C}) \\cap \\mathcal{T}^{d+1}_{r}((\\textrm{res}^{d+1}_{e})^{-1}(J^{e}_{r,nd} \\ch{L})) \\cap J^{d+1}_{r,nd} S \\] \nis non-empty for each $r \\geq 0$.\n\\end{itemize}\n\\end{prop}\n\n\\begin{rem}\nIn the situation that the variation $\\mathbb{V}$ admits a local Torelli theorem, one can drop the distinction between $\\dim Z$ and $\\dim \\varphi(Z)$ and consider instead the intersections $\\mathcal{T}^{d+1}_{r}(\\mathcal{C}) \\cap J^{d+1}_{r,nd} S$ in part (ii), ignoring the middle term.\n\\end{rem} \\vspace{0.5em}\n\nThe rest of this section we devote to proving \\autoref{mainjetprop}, identifying $S$ with $S_{\\mathbb{C}}$ for ease of notation. To begin with, let us check that (i) implies (ii) by applying the definitions. If $g : S' \\to S$ is an \\'etale cover and we consider the variation $\\mathbb{V}' = g^{*} \\mathbb{V}$, then the maps $\\eta^{d}_{r}$ and $\\eta'^{d}_{r}$ obtained from \\autoref{jetcorresp} are related by $\\eta'^{d}_{r} = \\eta^{d}_{r} \\circ (J^{d}_{r} g)$. Choosing a finite index subgroup $\\Gamma' \\subset \\Gamma$ and passing to such a cover, we can reduce to the case where we have a period map $\\varphi : S \\to \\Gamma \\backslash D$ with $D \\to \\Gamma \\backslash D$ a local isomorphism. Applying \\cite{OMINGAGA} the map $\\varphi : S \\to \\Gamma \\backslash D$ factors as $\\varphi = \\iota \\circ p$, where $p : S \\to T$ is a dominant map of algebraic varieties and $\\iota$ is an analytic closed embedding. Then via $p$, the variety $Z$ is dominant over a closed subvariety $Y \\subset T$ of dimension $\\dim \\varphi(Z) \\geq e$. Shrinking $S$ (and hence $Z$) we may assume that $Z$ is smooth, and that $Z$ is surjective onto a dense open subset $Y^{\\circ} \\subset Y$. Shrinking $S$ even further we may assume that $Z \\to Y^{\\circ}$ is smooth. The smoothness of $Z \\to Y^{\\circ}$ implies in particular that the induced jet space maps $J^{d}_{r} Z \\to J^{d}_{r} Y^{\\circ}$ for all choices of $d$ and $r$ are surjective.\n\nWe may choose neighbourhoods $B \\subset S(\\mathbb{C})$ and $U \\subset D$ such that $\\restr{\\pi}{U} : U \\to \\pi(U)$ is an isomorphism, both $B \\cap Z$ and $\\pi(U) \\cap Y^{\\circ}$ are non-empty, and we have a local lift $\\psi : B \\to U$ of $\\varphi$. Choose a jet $\\sigma \\in J^{e}_{r,nd} (Y^{\\circ} \\cap \\pi(U))$ and lift it along $p$ to a jet $\\widetilde{\\sigma} \\in J^{e}_{r,nd} (Z \\cap B)$ landing at the point $s \\in S(\\mathbb{C})$. Using the fact that the germ $(Z, s)$ is smooth of dimension $\\dim Z > d$ the jet $\\widetilde{\\sigma}$ can be extended to a jet $j \\in J^{d+1}_{r, nd} (Z \\cap B)$ such that $\\textrm{res}^{d+1}_{e} (j) = \\widetilde{\\sigma}$, and hence $\\textrm{res}^{d+1}_{e} (\\varphi \\circ j) = \\sigma$. From the fact that $\\restr{\\varphi}{B} = \\pi \\circ \\psi$ and the defining property of the map $\\eta^{d}_{r}$ it follows that $j$ lies inside $\\mathcal{T}^{d+1}_{r}((\\textrm{res}^{d+1}_{e})^{-1}(J^{e}_{r,nd} \\ch{L})) \\cap J^{d+1}_{r,nd} S$. We can then take $\\mathcal{C} = \\mathcal{C}(Z)$, and the fact that $j$ factors through $Z$ implies that $j \\in \\mathcal{T}^{d+1}_{r}(\\mathcal{C})$ as well.\n\nTo prove the reverse implication, we review some preliminary facts relating to jets.\n\n\\begin{defn}\nWe say a sequence $\\{ j_{r} \\}_{r \\geq 0}$ with $j_{r} \\in J^{d}_{r} X$ is \\emph{compatible} if the projections $J^{d}_{r} X \\to J^{d}_{r-1} X$ map $j_{r}$ to $j_{r-1}$.\n\\end{defn}\n\n\\begin{lem}\n\\label{compseqlem}\nSuppose that $\\mathcal{T}_{r} \\subset J^{d}_{r} X$ is a collection of non-empty constructible algebraic sets such that the projections $J^{d}_{r} X \\to J^{d}_{r-1} X$ map $\\mathcal{T}_{r}$ into $\\mathcal{T}_{r-1}$. Then there exists a compatible sequence $\\{ j_{r} \\}_{r \\geq 0}$ with $j_{r} \\in \\mathcal{T}_{r}$ for all $r \\geq 0$.\n\\end{lem}\n\n\\begin{proof}\nSee \\cite[Lem. 5.3]{periodimages}.\n\\end{proof}\n\n\\begin{defn}\nGiven a variety $Z$ (algebraic or analytic) and $z \\in Z$ a point, we denote by $(J^{d}_{r} Z)_{z}$ the fibre above $z$ of the natural projection map $J^{d}_{r} Z \\to Z$.\n\\end{defn}\n\n\\begin{lem}\n\\label{factorthrough}\nIf $g : (Z, z) \\to (Y, y)$ is a map of analytic germs with $\\dim (Z, z) = d$ and $(Z, z)$ smooth, we have an infinite compatible family $j_{r} \\in (J^{d}_{r,nd} Z)_{z}$, and $g \\circ j_{r} \\in (J^{d}_{r} X)_{y}$ for some germ $(X, y) \\subset (Y, y)$ and all $r \\geq 0$, then $g$ factors through the inclusion $(X, x) \\subset (Y, y)$. \n\\end{lem}\n\n\\begin{proof}\nSee \\cite[Lem. 4.5]{urbanik2021sets}.\n\\end{proof}\n\n\\begin{lem}\n\\label{jetdimlem}\nSuppose that $X$ is an algebraic variety (resp. analytic space) and $x \\in X$ is a point for which the fibre $(J^{d}_{r,nd} X)_{x}$ above $x$ is non-empty for all $r \\geq 0$. Then the germ $(X, x)$ has dimension at least $d$. \n\\end{lem}\n\n\\begin{proof}\nSee \\cite[Prop. 2.7]{periodimages}.\n\\end{proof}\n\n\\begin{proof}[Proof of \\ref{mainjetprop}:]\nBy what we have said, we are reduced to showing that (ii) implies (i). The statement is unchanged by replacing $S$ with a finite \\'etale covering $g : S' \\to S$ and the variation $\\mathbb{V}$ with $g^{*} \\mathbb{V}$; as before this does not affect the hypothesis (ii) since the maps $\\eta^{d}_{r}$ and $\\eta'^{d}_{r}$ associated to $S$ and $S'$ are related by $\\eta'^{d}_{r} = \\eta^{d}_{r} \\circ (J^{d}_{r} g)$. Choosing a finite index subgroup $\\Gamma' \\subset \\Gamma$ and choosing $g$ so the monodromy of $g^{*} \\mathbb{V}$ lies in $\\Gamma'$ we may reduce to the case where $D \\to \\Gamma \\backslash D$ is a local isomorphism. Moreover, taking a futher finite \\'etale cover we may apply \\cite[Cor. 13.7.6]{CMS} to reduce to the case where $\\varphi$ is proper; this requires possibly extending $S'$ to a variety $S''$ by adding a closed subvariety at infinity, but as long as we are careful to work only with jets that factor through $S'$ our proof will produce a variety $Z$ intersecting $S'$; in particular, we now assume that $\\varphi : S \\to \\Gamma \\backslash D$ is proper but redefine the sets $\\mathcal{K}^{d}_{r}$ to equal\n\\[ \\mathcal{T}^{d+1}_{r}(\\mathcal{C}) \\cap \\mathcal{T}^{d+1}_{r}((\\textrm{res})^{d+1}_{e})^{-1}(J^{d}_{r,nd} \\ch{L})) \\cap J^{d+1}_{r,nd} S^{\\circ} , \\]\nfor some open subvariety $S^{\\circ} \\subset S$.\n\nApplying the main result of \\cite{OMINGAGA}, the map $\\varphi$ once again factors as $\\varphi = \\iota \\circ p$ with $p : S \\to T$ a dominant (now proper) map of algebraic varieties. We can then consider the Stein factorization $S \\xrightarrow{q} U \\xrightarrow{r} T$ of $p$; note that $q$ is proper with connected fibres, $U$ is normal, and $r$ is finite. One can define the type of a subvariety $Y \\subset U$ exactly as in \\autoref{vartypedef} with respect to the period map $U \\to \\Gamma \\backslash D$. From \\autoref{compseqlem} above, the assumption (ii) entitles us to a compatible sequence $\\{ j_{r} \\}_{r \\geq 0}$ of jets such that $j_{r} \\in \\mathcal{K}^{d}_{r}(\\mathcal{C}, e, k)$ for all $r \\geq 0$. Let us write $\\mathcal{C} = \\mathcal{C}(W)$ for some subvariety $W \\subset \\ch{L}$.\n\nBy construction, the jets $\\sigma_{r} = \\textrm{res}^{d+1}_{e} j_{r}$ are non-degenerate, and remain so after composing with any local period map $\\psi : B \\to D$ for which $\\sigma_{r}$ factors through $B$. This in particular implies (since $D \\to \\Gamma \\backslash D$ is a local isomorphism) that the jets $\\varphi \\circ \\sigma_{r}$ are non-degenerate, and hence so are the jets $q \\circ \\sigma_{r}$. Let $Y \\subset U$ be the smallest algebraic subvariety such that $q \\circ j_{r} \\in J^{d+1}_{r} Y$ for all $r$. We observe that there exists a component $Z$ of $q^{-1}(Y)$ of dimension at least $d+1$ that contains the image of the jets $\\{ j_{r} \\}_{r \\geq 0}$: one can see this by picking a neighbourhood of $j_{0}$ of the form $\\mathbb{C}^{\\ell} \\times \\mathbb{C}^{d+1}$ such that $j_{r}$ is constant on the first factor, and applying \\autoref{factorthrough} above to see that the restriction of $q$ to $\\{ 0 \\} \\times \\mathbb{C}^{d+1}$ factors through $Y$. Moreover, we must have $q(Z) = Y$ by minimality, and by applying \\autoref{jetdimlem} to the non-degenerate sequence $\\{ q \\circ \\sigma_{r} \\}_{r \\geq 0}$ that $\\dim Y \\geq e$. Since $r$ is finite, this means $\\dim \\varphi(Z) \\geq e$. From the fact that local period maps on $S$ factor through local period maps on $U$ we learn that $\\mathcal{C}(Z) = \\mathcal{C}(Y)$, so the result will follow if we can show that $\\dim \\mathcal{C}(Y) - \\dim Y \\leq k$. For ease of notation let us now write $\\tau_{r} = q \\circ j_{r}$.\n\nFix a local lift $\\psi : B \\to D$ of the period map $U \\to \\Gamma \\backslash D$ with $B \\subset U(\\mathbb{C})$ an analytic ball such that the jets $\\tau_{r}$ factor through $B$. Consider the set $\\mathcal{G}_{r} \\subset \\textrm{GL}_{m}(\\mathbb{C})$ consisting of those $g \\in \\textrm{GL}_{m}(\\mathbb{C})$ for which $\\psi \\circ \\tau_{r} \\in g \\cdot (J^{d+1}_{r} W)$. Then for each $r$ the set $\\mathcal{G}_{r}$ is algebraically-constructible, and using the fact that $j_{r} \\in \\mathcal{T}^{d+1}_{r}(W)$ the set $\\mathcal{G}_{r}$ is necessarily non-empty. Let $g_{\\infty}$ be an element of this intersection. Extend $\\psi$ to a lift $\\widetilde{\\varphi}_{Y} : \\widetilde{Y} \\to D$ of $Y \\to \\Gamma \\backslash D$ to the universal covering. Then $\\widetilde{\\varphi}_{Y}(\\widetilde{Y}) \\subset D$ is a closed analytic set containing the jets $\\psi \\circ \\tau_{r}$, and hence the non-degenerate jets $\\textrm{res}^{d+1}_{e}(\\psi \\circ \\tau_{r})$. Letting $A \\subset \\widetilde{\\varphi}_{Y}(\\widetilde{Y}) \\cap (g_{\\infty} \\cdot W)$ be the minimal analytic germ through which $\\psi \\circ \\tau_{r}$ (and hence $\\textrm{res}^{d+1}_{e}(\\psi \\circ \\tau_{r})$) factors, it follows from \\autoref{jetdimlem} that $A$ has dimension at least $e$.\n\nConsider the Zariski closure $V \\subset Y$ of $\\psi^{-1}(A)$. We claim that $V = Y$. Because $Y$ was chosen minimal containing the compatible family of jets $\\{ \\tau_{r} \\}_{r \\geq 0}$, it suffices to show that each $\\tau_{r}$ factors through $V$. Consider the component of $q^{-1}(B)$ containing $j_{0}$; by choosing coordinates we may assume $q^{-1}(B) \\subset \\mathbb{C}^{\\ell} \\times \\mathbb{C}^{d+1}$ is an open neighbourhood and identify $j_{0}$ with the origin. After a further change of coordinates we may assume $j_{r}$ is constant on the first factor, and let $F = q^{-1}(B) \\cap (\\{ 0 \\} \\times \\mathbb{C}^{d+1})$. Applying \\autoref{factorthrough} we find that $\\psi(q(F)) \\subset A$, and hence $q(F) \\subset V$. Using proper base change the map $q^{-1}(B) \\to B$ is proper, so $q(F)$ is an analytic subvariety of $B$, and by construction the jets $\\{ \\tau_{r} \\}_{r \\geq 0}$ factor through it, hence through $V$.\n\nWe are now ready to apply the Bakker-Tsimerman transcendence theorem; the jets are no longer needed. It follows from the structure theorem for period mappings \\cite[III.A]{GGK}, the closed analytic set $\\widetilde{\\varphi}_{Y}(\\widetilde{Y})$ lies inside an orbit $\\ch{D}' = \\mathbf{H}_{Y} \\cdot \\psi(\\tau_{0})$ of the algebraic monodromy of $Y$. Consider the graph $E$ of $\\widetilde{\\varphi}_{Y}$ in $Y \\times \\ch{D}'$. Then as $A$ has dimension $e$ and $\\psi^{-1}(A)$ is Zariski dense, there exists a component $C$ of $E \\cap (Y \\times (\\ch{D}' \\cap g_{\\infty} \\cdot W))$ of dimension at least $e$ and projecting to a Zariski dense subset of $Y$. Applying the main theorem of \\cite{AXSCHAN} we learn that\n\\begin{align*}\n\\textrm{codim}_{Y \\times \\ch{D}'} (Y \\times (\\ch{D}' \\cap g_{\\infty} \\cdot W)) + \\textrm{codim}_{Y \\times \\ch{D}'} E &\\leq \\textrm{codim}_{Y\n \\times \\ch{D}'} C \\\\\n(\\dim \\ch{D}' - \\dim W) + \\dim \\ch{D}' &\\leq \\dim Y + \\dim \\ch{D}' - \\dim C \\\\\n\\dim \\ch{D}' - \\dim Y &\\leq \\dim W - \\dim C \\\\\n\\dim \\mathcal{C}(Y) - \\dim Y &\\leq \\dim \\mathcal{C} - e \\\\\n&\\leq k\n\\end{align*}\n\\end{proof}\n\n\\section{Main Results}\n\n\\subsection{Computing Bounds on $\\Delta_{d}$}\n\n\\subsubsection{Computing Lower Bounds}\n\nLet us explain the significance of \\autoref{mainjetprop} in proving \\autoref{mainthm}, i.e., giving an effective method to compute bounds for \\[ \\Delta_{d} = \\min_{\\dim Z > d} [ \\dim \\mathcal{C}(Z) - \\dim \\varphi(Z) ] , \\]\nwhere we have used \\autoref{mainthm2} and \\autoref{vartypedef} to give this equivalent expression for $\\Delta_{d}$. Since $\\Delta_{d}$ is a integer bounded by $\\dim D$, giving an effective method to compute it amounts to developing a procedure to decide, for any integer $k$, whether we have $\\Delta_{d} \\leq k$. This in turn amounts to deciding, for any integer $0 \\leq e \\leq \\dim \\varphi(S)$, whether (ii) holds in \\autoref{mainjetprop}.\n\nLet us take $\\mathcal{S}$ to be the set up types computed by Step One, and let us suppose that in fact $\\Delta_{d} > k$. Then by the equivalence in \\autoref{mainjetprop}, we should find that for any $\\mathcal{C} \\in \\mathcal{S}$ with $\\dim \\mathcal{C} - e \\leq k$, there must be some $r = r(\\mathcal{C}, e, k)$ such that $\\mathcal{K}^{d}_{r}(\\mathcal{C}, e, k)$ is empty. Moreover, verifying that such an $r$ exists for each such $\\mathcal{C}$ and $e$ proves, again by the same equivalence, that $\\Delta_{d} > k$. Consequently, we obtain the following result, which is the first half of \\autoref{mainthm}:\n\n\\begin{prop}\n\\label{lowerboundcomp}\nBy computing the sets $\\mathcal{K}^{d}_{r}(\\mathcal{C}, e, k)$ described in \\autoref{mainjetprop} in parallel, we may compute a non-decreasing sequence of lower bounds\n\\[ \\kappa(1) \\leq \\kappa(2) \\leq \\cdots \\leq \\kappa(r) \\leq \\cdots < \\Delta_{d} \\]\nsuch that for some $r = r_{0}$ we have $\\kappa(r_{0}) = \\Delta_{d} - 1$.\n\\end{prop}\n\n\\begin{proof}\nAt the $r$'th stage we compute all the sets $\\mathcal{K}^{d}_{r}(\\mathcal{C}, e, k)$ for all applicable choices of $\\mathcal{C}$, $e$ and $k$, and set $\\kappa(r)$ to be the smallest $k$ for which all the sets $\\mathcal{K}^{d}_{r}(\\mathcal{C}, e, k)$ are empty. From the discussion preceding the Proposition, the result follows.\n\\end{proof}\n\n\\subsubsection{Computing Upper Bounds}\n\n\\autoref{lowerboundcomp} does not actually give an algorithm for computing $\\Delta_{d}$, since no way is given to decide when $r = r_{0}$. For applications to the Lawrence-Venktesh method this doesn't matter: one wants to be able to compute an optimal lower bound for $\\Delta_{d}$, but one does not actually have to prove that this lower bound actually equals $\\Delta_{d}$ in order to apply the diophantine finiteness machinery. Nevertheless, let us explain how one can do this in the case where $\\varphi$ is quasi-finite; under this assumption, we may drop the distinction between $\\dim Z$ and $\\dim \\varphi(Z)$, and we are instead interested in computing\n\\[ \\min_{\\dim Z > d} \\, \\left[ \\dim \\mathcal{C}(Z) - \\dim Z \\right] . \\]\n\n What is needed is the following:\n\n\\begin{prop}\n\\label{upperboundcomp}\nSuppose that $S$ is quasi-projective and $\\varphi$ is quasi-finite. Then there exists a procedure that outputs an infinite sequence of upper bounds\n\\[ \\tau(1) \\geq \\tau(2) \\geq \\cdots \\geq \\tau(i) \\geq \\cdots \\geq \\Delta_{d} \\] \nsuch that for some $i = i_{0}$ we have $\\tau(i) = \\Delta_{d}$.\n\\end{prop}\n\n\\noindent Given both \\autoref{lowerboundcomp} and \\autoref{upperboundcomp} we obtain an algorithm for computing $\\Delta_{d}$ by running both procedures in parallel and terminating when $\\kappa(r) + 1 = \\tau(i)$. \n\n\\subsection{Finding Varieties that Exhibit $\\Delta_{d}$}\n\nIn this section we prove \\autoref{upperboundcomp}, assuming throughout that $S$ is quasi-projective and $\\varphi$ is quasi-finite. Let us fix a projective compactification $S \\subset \\overline{S}$ of $S$ and consider the Hilbert scheme $\\textrm{Hilb}(\\overline{S})$. There exist algorithms, for instance by working with the Pl\\\"uker coordinates of the appropriate Grassmannian, for computing any finite subset of components of $\\textrm{Hilb}(\\overline{S})$. By \\cite[Lem. 5.10]{urbanik2021sets} we obtain the same fact for the open locus $\\textrm{Var}(S) \\subset \\textrm{Hilb}(\\overline{S})$ consisting of just those points $[\\overline{Z}]$ for which $Z = S \\cap \\overline{Z}$ is a non-empty geometrically irreducible algebraic subvariety of $S$. What we will show is that there exists a procedure which outputs an infinite sequence $\\{ \\mathcal{W}_{i} \\}_{i = 1}^{\\infty}$ of constructible algebraic loci $\\mathcal{W}_{i} \\subset \\textrm{Var}(S)$, with the following two properties:\n\\begin{itemize}\n\\item[(i)] for each $i$, the type $\\mathcal{C}(Z)$ and dimension $\\dim Z$ are constant over all $[Z] \\in \\mathcal{W}_{i}$;\n\\item[(ii)] there exists some $i = i_{0}$ such that \n\\[ \\Delta_{d} = \\dim \\mathcal{C}(Z) - Z , \\]\nfor some (hence any) point $[Z] \\in \\mathcal{W}_{i}$.\n\\end{itemize}\nGiven such an algorithm the problem of computing the bound $\\tau(i)$ that appears in \\autoref{upperboundcomp} reduces to choosing a point $[Z] \\in \\mathcal{W}_{i}$, computing $\\dim \\mathcal{C}(Z) - \\dim Z$, and setting \n\\[ \\tau(i) := \\textrm{min} \\{ \\tau(i-1), \\dim \\mathcal{C}(Z) - \\dim Z \\} . \\] \n(We note that the problem of computing $\\dim \\mathcal{C}(Z)$ from $Z$ and the restriction $\\restr{(\\mathcal{H}, F^{\\bullet}, \\nabla)}{Z}$ of the algebraic data on $S$ is solved for us by \\cite[Lem. 5.8]{urbanik2021sets} by taking the family $g$ in the statement of \\cite[Lem. 5.8]{urbanik2021sets} to be a trivial family; we will say little about this problem here.)\n\nIn fact, an algorithm for computing the sets $\\mathcal{W}_{i}$ has already been given in a previous paper by the author. We begin by recalling the necessary background. We regard $S$ as a complex algebraic variety in what follows. Given two (geometrically) irreducible subvarieties $Z_{1}, Z_{2} \\subset S$ with $Z_{1} \\subset Z_{2}$, the algebraic monodromy group $\\mathbf{H}_{Z_{1}}$ may be naturally regarded as a subgroup of $\\mathbf{H}_{Z_{2}}$ (after choosing a base point $s \\in Z_{1}(\\mathbb{C})$). Using this we define:\n\\begin{defn}\nAn irreducible complex subvariety $Z \\subset S$ is said to be \\emph{weakly special} if it is maximal among such subvarieties for its algebraic monodromy group.\n\\end{defn}\nThe key fact is then the following:\n\\begin{lem}\nFor each integer $d > 0$, there exists a weakly special subvariety $Z \\subset S$ such that \n\\[ \\Delta_{d} = \\dim \\mathcal{C}(Z) - \\dim Z . \\]\n\\end{lem}\n\n\\begin{proof}\nBy \\cite[Prop. 4.18]{urbanik2021sets}, the condition that $Z$ be weakly special is equivalent to $Z$ being a maximal irreducible complex subvariety of $S$ of type $\\mathcal{C}(Z)$. Thus if we have any $Y$ which is not weakly special, there exists a weakly special $Z$ properly containing $Y$ with $\\mathcal{C}(Y) = \\mathcal{C}(Z)$, hence\n\\[ \\dim \\mathcal{C}(Z) - \\dim Z < \\dim \\mathcal{C}(Y) - \\dim Y . \\]\nIt follows that the value of $\\Delta_{d}$ can only be achieved by a weakly special variety.\n\\end{proof}\n\n\\begin{proof}[Proof of \\ref{upperboundcomp}]\nBy the \\emph{degree} of a subvariety $Z \\subset S$ we will mean the degree of its closure $\\overline{Z}$ inside $\\overline{S}$. For any integer $b$, denote by $\\textrm{Var}(S)_{b} \\subset \\textrm{Var}(S)$ the finite-type subscheme parametrizing varieties of degree at most $b$. Denote by $\\mathcal{W} \\subset \\textrm{Var}(S)$ the locus of weakly special subvarieties. Then given an integer $b$, the algorithm that appears in \\cite[Thm. 5.15]{urbanik2021sets} computes the intersection $\\mathcal{W} \\cap \\textrm{Var}(S)_{b}$ as a constructible algebraic locus.\n\nLet us describe the algorithm appearing in \\cite[Thm. 5.15]{urbanik2021sets} more precisely. Consider the types $\\mathcal{C}_{1}, \\hdots, \\mathcal{C}_{\\ell}$ computed by Step One, and define for each such type $\\mathcal{C}_{j}$ the locus\n\\[ \\mathcal{W}(\\mathcal{C}_{j}) := \\{ [Z] \\in \\textrm{Var}(S) : \\mathcal{C}(Z) \\leq \\mathcal{C}_{j} \\} . \\] \nIt is shown in \\cite[Prop. 4.31]{urbanik2021sets} that for each $j$ the locus $\\mathcal{W}(\\mathcal{C}_{j})$ is closed algebraic. We can then consider the sublocus $\\mathcal{W}(\\mathcal{C}_{j})_{\\textrm{opt}} \\subset \\mathcal{W}(\\mathcal{C}_{j})$ consisting of just those components $C \\subset \\mathcal{W}(\\mathcal{C}_{j})$ for which a generic point $[Z] \\in C$ satisfies $\\mathcal{C}(Z) = \\mathcal{C}_{j}$.\n\nIn \\cite[Prop. 5.14]{urbanik2021sets}, an algorithm is given for computing $\\mathcal{W}(\\mathcal{C}_{j})_{\\textrm{opt}} \\cap \\textrm{Var}(S)_{b}$ for each $j$. Using this, one can compute all the finitely many closed algebraic loci $C_{1}, \\hdots, C_{i_{b}}$ which arise as a component of $\\mathcal{W}(\\mathcal{C}_{j})_{\\textrm{opt}} \\cap \\textrm{Var}(S)_{b}$ for some $j$. The problem of computing $\\mathcal{W} \\cap \\textrm{Var}(S)_{b}$ is reduced to computing constructible algebraic conditions on each component $C_{i} \\subset \\mathcal{W}(\\mathcal{C}_{j})_{\\textrm{opt}}$ which define the locus $\\mathcal{W}_{i} \\subset C_{i}$ of points $[Z] \\in C_{i}$ that are weakly special of type $\\mathcal{C}_{j}$. This is taken care of by the proof of \\cite[Thm. 5.15]{urbanik2021sets}. By construction, the points in $\\mathcal{W}_{i}$ all have the same type and the same dimension, so we complete the proof by computing these loci for increasing values of $b$.\n\\end{proof}\n\n\n\\section{Application to Lawrence-Venkatesh}\n\nWe now show how the bound of \\autoref{mainthm} can be used to establish diophantine finiteness results. Similar arguments appear in \\cite{LV} and \\cite{lawrence2020shafarevich}, but as they are not precisely adapted to our setup, we give our own version. We recall the situation: we have a smooth projective family $f : X \\to S$ over the smooth base $S$, with everything defined over $\\mathcal{O}_{K,N}$.\\footnote{Note in particular we are assuming now that $S$ is smooth over $\\mathcal{O}_{K,N}$, which we can achieve by increasing $N$ if necessary.} The relative algebraic de Rham cohomology $\\mathcal{H} = R^{i} f_{*} \\Omega^{\\bullet}_{X\/S}$ gives a model for the Hodge bundle $\\mathbb{V} \\otimes \\mathcal{O}_{\\an{S}}$, where $\\mathbb{V} = R^{i} f_{*} \\mathbb{Z}$. By a result of Katz and Oda \\cite{katz1968}, the flat connection associated to the local system $\\mathbb{V}_{\\mathbb{C}}$ by the Riemann-Hilbert correspondence admits a model $\\nabla : \\mathcal{H} \\to \\Omega^{1}_{S} \\otimes \\mathcal{H}$ after possibly increasing $N$. Likewise, we may also assume the Hodge filtration $F^{\\bullet}$ gives a filtration of $\\mathcal{H}$ by vector subbundles. \n\nFix a prime $p$ not dividing $N$, and a place $v$ of $K$ above $p$. Then for each integral point $s \\in S(\\mathcal{O}_{K,N})$, we have a Galois representation $\\rho_{s} : \\textrm{Gal}(\\overline{K}\/K) \\to \\textrm{Aut}(H^{i}_{\\textrm{\\'et}}(X_{\\overline{K}, s}, \\mathbb{Q}_{p}))$, and an argument of Faltings \\cite[Lem 2.3]{LV} shows that the semisimplifications of the representations $\\rho_{s}$ belong to a finite set of isomorphism classes. From crystalline cohomology, each $s \\in S(\\mathcal{O}_{K,N})$, viewed as a point of $S(\\mathcal{O}_{K,v})$ where $\\mathcal{O}_{K,v}$ is the $v$-adic ring of integers, gives rise to a triple $(H^{i}_{\\textrm{dR}}(X_{s}), \\phi_{s}, F^{\\bullet}_{s})$ where $\\phi_{s}$ is the crystalline Frobenius. Moreover, using the functor $D_{\\textrm{cris}}$ of $p$-adic Hodge theory \\cite[Expos\\'e III]{fontaine1994corps}, the triple $(H^{i}_{\\textrm{dR}}(X_{s}), \\phi_{s}, F^{\\bullet}_{s})$ is determined up to isomorphism by the restriction $\\rho_{s,v}$ along the map $\\textrm{Gal}(\\overline{K_{v}}\/K_{v}) \\to \\textrm{Gal}(\\overline{K}\/K)$ determined by a fixed embedding $\\overline{K} \\hookrightarrow \\overline{K_{v}}$. We denote by $\\mathcal{I}(s)$ all those triples $(V, \\phi, F^{\\bullet})$ which are of the form $D_{\\textrm{cris}}(\\restr{\\rho}{\\mathbb{Q}_{p}})$, where $\\rho$ is a global Galois representation whose semisimplification is isomorphic to the semisimplification of $\\rho_{s}$. \n\n\nRecall that we have fixed the integral lattice $V = \\mathbb{Z}^{m}$, where $m$ is the dimension of the cohomology of the fibres of $f$, and a $\\mathbb{Q}$-algebraic flag variety $\\ch{L}$ of Hodge flags on $V$. In what follows we write $V_{p}$ for $V \\otimes \\mathbb{Q}_{p}$, and $\\ch{L}_{p}$ for $\\ch{L}_{\\mathbb{Q}_{p}}$. Then the key idea of the Lawrence-Venkatesh method is the following:\n\n\\begin{prop}\n\\label{LVprop}\nSuppose that for each $s \\in S(\\mathcal{O}_{K,N})$, whenever we have an endomorphism $\\phi_{s} : V_{p} \\to V_{p}$ and a flag $F^{\\bullet}_{s}$ on $V_{p}$ such that $(V_{p}, \\phi_{s}, F^{\\bullet}_{s})$ represents $\\mathcal{I}(s)$, the Hodge flags $F^{\\bullet}$ on $V_{p}$ for which $(V_{p}, \\phi_{s}, F^{\\bullet}) \\in \\mathcal{I}(s)$ lie in an algebraic subvariety $O_{s} \\subset \\ch{L}_{p}$ satisfying $\\Delta_{d} \\geq \\dim O_{s}$. Then $\\dim \\overline{S(\\mathcal{O}_{K,N})}^{\\textrm{Zar}} \\leq d$.  \n\\end{prop}\n\nTo prove \\autoref{LVprop} we will need a rigid-analytic version of the Bakker-Tsimerman transcendence theorem, which we will see can be deduced formally from the complex analytic one. To set things up, let us revisit the term \\emph{local period map}, this time in the rigid analytic setting (c.f. \\autoref{locperdef}). We will denote by $\\mathbb{C}_{p}$ the completion of the algebraic closure $\\overline{K_{v}}$. In what follows we sometimes identify algebraic varieties with their rigid-analytifications when the context is clear.\n\n\\begin{defn}\n\\label{padiclocperdef}\nLet $K_{p}$ be a local field containing $K_{v}$, let $\\an{S_{K_{p}}}$ be the rigid-analytification of the base-change $S_{K_{p}}$ of $S$, and suppose that $B_{p} \\subset \\an{S_{K_{p}}}$ is an affinoid subdomain. Then a (rigid-analytic) local period map $\\psi : B_{p} \\to \\an{\\ch{L}_{K_{p}}}$ is a rigid-analytic map obtained as a composition $\\psi = \\an{q_{K_{p}}} \\circ A_{p}$, where:\n\\begin{itemize}\n\\item[(i)] The rigid analytifications $F^{k} \\an{\\mathcal{H}_{K_{p}}}$ are all trivial on $B_{p}$.\n\\item[(ii)] The map $A_{p} : B_{p} \\to \\an{\\textrm{GL}_{m, K_{p}}}$ is a varying filtration-compatible $p$-adic period matrix over $B_{p}$. More precisely, there exists a basis $v^{1}, \\hdots, v^{m}$ for $\\an{\\mathcal{H}_{K_{p}}}(B_{p})$, compatible with the filtration in the sense that $F^{k} \\an{\\mathcal{H}_{K_{p}}}(B_{p})$ is spanned by $v^{1}, \\hdots, v^{i_{k}}$ for some $i_{k}$, and a flat (for $\\an{\\nabla_{K_{p}}}$) frame $b^{1}, \\hdots, b^{m}$ such that $A_{p}$ gives a varying change-of-basis matrix from $v^{1}, \\hdots, v^{m}$ to $b^{1}, \\hdots, b^{m}$.\n\\item[(iii)] The map $q : \\textrm{GL}_{m} \\to \\ch{L}$ is the map that sends a matrix $M$ to the Hodge flag $F^{\\bullet}_{M}$ defined by the property that $F^{k}_{M}$ is spanned by the first $i_{k}$ columns.\n\\end{itemize}\n\\end{defn}\n\nTo prove \\autoref{LVprop} we will need a version of the Bakker-Tsimerman transcendence result for rigid-analytic local period maps, which we prove by formally transferring the same result for complex analytic local period maps. To avoid certain minor pathologies that can occur in the complex analytic case we will restrict to local period maps $\\psi : B \\to \\ch{L}$ which are definable in the structure $\\mathbb{R}_{\\textrm{an}, \\textrm{exp}}$; for background on definability and definable analytic spaces we refer to \\cite{van1996geometric} and \\cite{OMINGAGA}. We note that this is not a serious restriction: given any local period map $\\psi$ and any point $s \\in B$ there exists a definable restriction of $\\psi$ to a neighbourhood of $s$, a fact which is for instance easily deduced from \\cite[Prop. 4.27]{urbanik2021sets}.\n\n\\begin{lem}\n\\label{padicaxschanlem}\n~\\begin{itemize}\n\\item[(i)] Suppose that $\\psi : B \\to \\an{\\ch{L}_{\\mathbb{C}}}$ is a definable analytic local period map on $\\an{S_{\\mathbb{C}}}$. Let $V \\subset \\ch{L}_{\\mathbb{C}}$ be an algebraic subvariety satisfying $\\Delta_{d} \\geq \\dim V$. Then $\\psi^{-1}(V)$ lies in an algebraic subvariety of $S_{\\mathbb{C}}$ of dimension at most $d$.\n\\item[(ii)] Suppose that $\\psi_{p} : B_{p} \\to \\an{\\ch{L}_{\\mathbb{C}_{p}}}$ is a rigid-analytic local period map on $\\an{S_{\\mathbb{C}_{p}}}$. Let $V_{p} \\subset \\ch{L}_{p, \\mathbb{C}_{p}}$ be an algebraic subvariety satisfying $\\Delta_{d} \\geq \\dim V_{p}$. Then $\\psi_{p}^{-1}(V_{p})$ lies in an algebraic subvariety of $S_{\\mathbb{C}_{p}}$ of dimension at most $d$.\n\\end{itemize}\n\\end{lem}\n\n\\paragraph{Proof of \\autoref{padicaxschanlem}(i):} ~ \\\\\n\n\\vspace{-0.5em}\n\nThis is an application of the Bakker-Tsimerman transcendence theorem. Let $Z \\subset S_{\\mathbb{C}}$ be the Zariski closure of $\\psi^{-1}(V)$. We assume for contradiction that $\\dim Z > d$, and let $Z_{0} \\subset Z$ be a component of maximal dimension. Let $\\varphi : \\an{S_{\\mathbb{C}}} \\to \\Gamma \\backslash D$ be the canonical period map with $\\Gamma = \\textrm{Aut}(V,Q)(\\mathbb{Z})$. The statement is invariant under replacing $\\psi$ with a $\\textrm{GL}_{m}(\\mathbb{C})$-translate $g \\cdot \\psi$ and $V$ with $g \\cdot V$, so we may assume that $\\psi$ is a local lift of $\\varphi$. Arguing as in \\cite[Cor. 13.7.6]{CMS} we may assume that $\\varphi$ is proper, hence the image $T = \\varphi(S)$ is algebraic by \\cite{OMINGAGA}, and we may consider the Stein factorization $S \\xrightarrow{q} U \\xrightarrow{r} T$ of the map $S \\to T$. \n\nLet $Y = q(Z_{0})$, and note that $\\dim Y = \\dim \\varphi(Z_{0})$. By assumption we have $\\Delta_{d} \\leq \\dim \\mathcal{C}(Z_{0}) - \\dim \\varphi(Z_{0}) = \\dim \\mathcal{C}(Y) - \\dim Y$, where the type of $Y$ is taken with respect to the period map $U \\to \\Gamma \\backslash D$. Moreover, this continues to hold if we replace $Y$ with a smooth resolution $Y'$. The variation of Hodge structure on $S$ descends to $U$, and hence shrinking $B$ if necessary we may factor $\\psi$ through a definable local lift on $U$. By pulling back along the resolution we obtain a definable local lift $\\psi' : B' \\to D$ of the period map $\\varphi' : Y' \\to \\Gamma \\backslash D$ such that $\\psi'^{-1}(V)$ is Zariski dense in $Y'$. We are reduced to the following situation: we have smooth variety $Y'$ with a period map $\\varphi' : Y' \\to \\Gamma \\backslash D$, a local lift $\\psi' : B' \\to D$ such that $\\psi'^{-1}(V)$ is Zariski dense, and such that $\\dim \\mathcal{C}(Y') - \\dim Y' \\geq \\dim V$. \n\nWe now contradict the Bakker-Tsimerman theorem. In particular, we may extend the local lift $\\psi'$ to a lift $\\widetilde{\\varphi'} : \\widetilde{Y'} \\to D$ of $\\varphi'$ to the universal cover, and consider the graph $W \\subset Y' \\times \\ch{D}'$ of the map $\\widetilde{\\varphi'}$, where $\\ch{D}'$ is the orbit $\\mathbf{H}_{Y'} \\cdot \\psi'(y)$ for some $y \\in Y'(\\mathbb{C})$. We then have that\n\\begin{align*}\n\\textrm{codim}_{Y' \\times \\ch{D}'} (Y' \\times (V \\cap \\ch{D}')) + \\textrm{codim}_{Y' \\times \\ch{D}'} W &\\geq \\dim \\ch{D}' - \\dim V + \\dim \\ch{D}' \\\\\n&= \\dim \\mathcal{C}(Y') - \\dim V + \\dim \\mathcal{C}(Y') \\\\\n&\\geq \\dim Y' + \\dim \\mathcal{C}(Y') .\n\\end{align*}\nSince $\\psi'$ is definable, $\\psi'^{-1}(V)$ has finitely many components, and hence there exists an analytic component $C \\subset B'$ of $\\psi'^{-1}(V)$ such that $C$ is Zariski dense in $Y'$. Let $\\widetilde{C} \\subset Y' \\times \\ch{D}'$ be its graph under $\\psi'$. If $\\dim Y' = 0$ there is nothing to show, so we may assume that $\\dim \\mathcal{C}(Y') > 0$. Hence we find that $\\dim Y' + \\dim \\mathcal{C}(Y') > \\textrm{codim}_{Y' \\times \\ch{D}'} \\widetilde{C}$, and by the Bakker-Tsimerman theorem \\cite{AXSCHAN} the component $C$ lies in a proper subvariety of $Y'$, giving a contradiction. \\qed\n\n\\vspace{1em}\n\nTo prove \\autoref{padicaxschanlem}(ii) we first translate \\autoref{padicaxschanlem}(i) into a claim about rings of formal power series. In particular let $\\psi : B \\to \\ch{L}$ be a local period map with $V \\subset \\ch{L}$ an algebraic subvariety, and choose a point $s \\in B$ such that $\\psi(s) = t \\in V$. Then $\\psi$ induces a map on formal power series rings $\\widehat{\\psi}^{\\sharp} : \\widehat{\\mathcal{O}}_{\\ch{L}_{\\mathbb{C}}, t} \\to \\widehat{\\mathcal{O}}_{S_{\\mathbb{C}}, s}$. The claim of \\autoref{padicaxschanlem}(i) then says that if $I_{V} \\subset \\mathcal{O}_{\\ch{L}_{\\mathbb{C}}, t}$ is the ideal defining $V$ with extension $\\widehat{I}_{V}$ inside $\\widehat{\\mathcal{O}}_{\\ch{L}_{\\mathbb{C}}, t}$, then the ideal generated by $\\widehat{\\psi}^{\\sharp}(\\widehat{I}_{V})$ contains an ideal $\\widehat{I}_{Z}$ which is the extension of an ideal $I_{Z} \\subset \\mathcal{O}_{S_{\\mathbb{C}}, s}$ defining the germ of a subvariety of dimension at most $d$.\n\n\n\\paragraph{Proof of \\autoref{padicaxschanlem}(ii):} ~ \\\\\n\n\\vspace{-0.5em}\n\nThe claim is Zariski-local on $S$, so we can in particular assume that the bundles $F^{k} \\mathcal{H}$ for varying $k$ are algebraically trivial over $S$, that $S$ is affine, and by smoothness that $\\Omega^{1}_{S}$ is free. By definition, the map $\\psi : B_{p} \\to \\an{\\ch{L}_{\\mathbb{C}_{p}}}$ is associated to the following data: a filtration-compatible frame $v^{1}, \\hdots, v^{m}$, where $v^{1}, \\hdots, v^{i_{k}}$ spans $F^{k} \\an{\\mathcal{H}_{\\mathbb{C}_{p}}} (B_{p})$, and a flat frame $b^{1}, \\hdots, b^{m}$ spanning $\\an{\\mathcal{H}_{\\mathbb{C}_{p}}}(B_{p})$, where flatness means $\\an{\\nabla_{\\mathbb{C}_{p}}} b_{i} = 0$ for all $1 \\leq i \\leq m$. This data satisfies the property that $\\psi = \\an{q_{\\mathbb{C}_{p}}} \\circ A_{p}$, where $A_{p}$ is the change-of-basis matrix from the frame  $v^{1}, \\hdots, v^{m}$ to the frame $b^{1}, \\hdots, b^{m}$, and $q$ is the map $\\an{\\textrm{GL}_{m,\\mathbb{C}_{p}}} \\to \\an{\\ch{L}_{\\mathbb{C}_{p}}}$ sending a matrix to the Hodge flag it represents. We note that changing the frame $v^{1}, \\hdots, v^{m}$ to another filtration-compatible frame $v'^{1}, \\hdots, v'^{m}$ does not change the local period map: such a change has the effect of replacing the map $A_{p}$ with $A_{p} \\cdot C$, where $C$ is a varying matrix over $B_{p}$ whose right-action on $A_{p}$ preserves the span of the first $i_{k}$ columns for each $k$, and hence $q \\circ (A_{p} \\cdot C) = q \\circ A_{p}$. We threfore lose no generality by assuming the filtration-compatible frame is the restriction to $B_{p}$ of an algebraic filtration-compatible frame over $S$. \n\nThe affinoid neighbourhood $B_{p}$ is of the form $\\textrm{Sp} \\, T$, where $T$ is an affinoid $\\mathbb{C}_{p}$-algebra. The inverse image $\\psi^{-1}(V)$ is then a closed affinoid subdomain of $B_{p}$, i.e., it corresponds to an ideal $I \\subset T$ such that $\\psi^{-1}(V)$ may be identified with $\\textrm{Sp} \\, T\/I$. If $R$ is the coordinate ring of $S_{\\mathbb{C}_{p}}$, then the map $B_{p} \\hookrightarrow \\an{S_{\\mathbb{C}_{p}}} \\to S_{\\mathbb{C}_{p}}$ induces a map $\\iota : R \\to T$, and the claim to be shown is that there exists an ideal $J \\subset R$ defining a subvariety of dimension at most $d$ such that $\\iota(J) \\subset I$. The ring $T$ is Noetherian, so the ideal $I$ admits a primary decomposition. Taking radicals, we obtain finitely many prime ideals $I_{1}, \\hdots, I_{\\ell}$ containing $I$ such that the problem reduces, for each $1 \\leq j \\leq \\ell$, to finding $J_{j} \\subset R$ defining subvarieties of dimension at most $d$ such that $\\iota(J_{j}) \\subset I_{j}$ for each $j$. The analytification map $\\an{S_{\\mathbb{C}_{p}}} \\to S_{\\mathbb{C}_{p}}$ is bijective onto the set of closed points of $S_{\\mathbb{C}_{p}}$ and induces isomorphisms on completed local rings [see whatever]. It follows that if we choose a maximal ideal $\\mathfrak{m} \\subset T$ containing $I_{j}$ we obtain a commuting diagram\n\\begin{center}\n\\begin{tikzcd}\nR \\arrow[r,\"\\iota\"] \\arrow[d, hook] & \\arrow[d, hook] T \\\\\n\\widehat{R}_{\\iota^{-1}(\\mathfrak{m})} \\arrow[r,\"\\sim\"] & \\widehat{\\mathcal{O}}_{B_{p}, \\mathfrak{m}} ,\n\\end{tikzcd}\n\\end{center}\nwhere the bottom arrow is an isomorphism of completed local rings, and the vertical arrows are injections. In particular, if we denote by $\\widehat{I}_{j}$ the extension of $I_{j}$ in $\\widehat{\\mathcal{O}}_{B_{p}, \\mathfrak{m}}$, it suffices to show that $\\widehat{\\iota}(J_{j}) \\subset \\widehat{I}_{j}$, where $\\widehat{\\iota}$ is the composition of the left and bottom arrow; here we have used the fact that $I_{j} = \\widehat{I}_{j} \\cap T$. \n\nFix an isomorphism $\\tau : \\mathbb{C}_{p} \\xrightarrow{\\sim} \\mathbb{C}$, which we choose to preserve the embeddings $K \\subset \\mathbb{C}$ and $K \\subset \\mathbb{C}_{p}$. Using the model for $S$ over $K$, the isomorphism $\\tau$ allows us to identify $R$ with the coordinate ring of $S_{\\mathbb{C}}$, the ideal $\\iota^{-1}(\\mathfrak{m})$ with a complex point $s \\in S(\\mathbb{C})$, the ring $\\widehat{R}_{\\iota^{-1}(\\mathfrak{m})}$ with the completed local ring $\\widehat{\\mathcal{O}}_{S_{\\mathbb{C}}, s}$. Let $t_{p}$ be the image of the point corresponding to $\\mathfrak{m}$ under $\\psi$, and let $t$ be the composition $t_{p} \\circ \\tau^{-1}$. Applying the isomorphism $\\tau$ at the level of formal power series, the rigid-analytic local period map $\\psi$ induces a map \n\\[ \\widehat{\\mathcal{O}}_{\\ch{L}_{\\mathbb{C}}, t} \\xrightarrow{\\tau} \\widehat{\\mathcal{O}}_{\\ch{L}_{\\mathbb{C}_{p}}, t_{p}} \\xrightarrow{\\widehat{\\psi}^{\\sharp}} \\widehat{\\mathcal{O}}_{B_{p}, \\mathfrak{m}} \\xrightarrow{\\sim} \\widehat{R}_{\\iota^{-1}(\\mathfrak{m})} \\xrightarrow{\\tau} \\widehat{\\mathcal{O}}_{S_{\\mathbb{C}}, s} , \\]\nwhose composition we denote by $\\widehat{\\eta}$. In what follows we identify the ideals $\\widehat{I}_{j}$ with their images in $\\widehat{\\mathcal{O}}_{S_{\\mathbb{C}}, s}$; by construction they are the extensions along $\\widehat{\\eta}$ of an ideal in $\\widehat{\\mathcal{O}}_{\\ch{L}_{\\mathbb{C}}, t}$ associated to the base-change of $V$ using $\\tau$. By part (i) of this theorem and our reformulation of it in terms of completed local rings, it suffices to show that $\\widehat{\\eta}$ is induced by a complex analytic local period map defined on a neighbourhood of $s$.\n\nRecall that we have a decomposition $\\psi = \\an{q_{\\mathbb{C}_{p}}} \\circ A_{p}$, where $q$ is the rigid-analytification of a $\\mathbb{Q}$-algebraic map, and $A_{p}$ gives a varying change-of-basis matrix between a filtration-compatible frame $v^{1}, \\hdots, v^{m}$ and a rigid-analytic flat frame. Recall also that we have chosen $v^{1}, \\hdots, v^{m}$ so that it is the rigid-analytification of a $K$-algebraic filtration-compatible frame $w^{1}, \\hdots, w^{m}$ over $S$. Using the decomposition $\\psi = \\an{q_{\\mathbb{C}_{p}}} \\circ A_{p}$ and the isomorphism $\\tau$ we may factor $\\widehat{\\eta}$ as $\\widehat{q} \\circ \\widehat{\\kappa}$, where $\\widehat{\\kappa} : \\widehat{\\mathcal{O}}_{S_{\\mathbb{C}}, s} \\to \\widehat{\\mathcal{O}}_{\\textrm{GL}_{m, \\mathbb{C}}, P}$ is the base-change under $\\tau$ of the map induced by $A_{p}$. From our definition of local period map in \\autoref{locperdef}, it suffices to show that $\\widehat{\\kappa}$ is induced by a varying change-of-basis matrix $A : B \\to \\an{\\textrm{GL}_{m,\\mathbb{C}}}$ from $w^{1}, \\hdots, w^{m}$ to a complex-analytic flat frame.\n\nThe result will follow from the fact that $A$ and $A_{p}$ satisfy a common set of $K$-algebraic differential equations whose solutions are uniquely determined by the period matrix they assign to a point in $B$. To see this, let us write $\\nabla w^{i} = \\sum_{j = 1}^{m} c_{ij} \\otimes w^{j}$ for $K$-algebraic sections $c_{ij} \\in \\Omega^{1}_{S_{K}}$. Suppose then that $b^{k} = \\sum_{i = 1}^{m} f_{ik} w^{i}$ is a flat frame on some complex analytic or rigid-analytic neighbourhood. We then have that\n\\begin{align*}\n\\nabla b^{k} &= \\nabla \\left( \\sum_{i = 1}^{m} f_{ik} w^{i} \\right) \\\\\n&= \\sum_{j = 1}^{m} df_{jk} \\otimes w^{j} + \\sum_{i = 1}^{m} f_{ik} \\left( \\sum_{j = 1}^{m} c_{ij} \\otimes v^{j} \\right) \\\\\n&= \\sum_{j = 1}^{m} \\left( df_{jk} + \\sum_{i = 1}^{m} f_{ik} c_{ij} \\right) \\otimes v^{j} ,\n\\end{align*}\nfrom which we see that $b^{k}$ giving a flat frame is equivalent to $f_{jk}$ satisfying the system of differential equations $df_{jk} = -\\sum_{i = 1}^{m} f_{ik} c_{ij}$ for all $1 \\leq j, k \\leq m$. If we choose a trivialization $dz_{1}, \\hdots, dz_{n}$ of $\\Omega^{1}_{S_{K}}$, we may write the $c_{ij}$ in terms of their coefficients $c_{ij,\\ell}$ with respect to this trivialization, and the same system of differential equations becomes \n\\begin{equation}\n\\label{diffeq}\n\\partial_{\\ell} f_{jk} = -\\sum_{i} f_{ik} c_{ij,\\ell} ;\n\\end{equation} \nhere the operator $\\partial_{\\ell}$ is defined using the dual basis to $dz_{1}, \\hdots, dz_{n}$. By differentiating \\autoref{diffeq} and substituting the lower-order differential equations into the higher-order ones, we obtain, for each sequence $\\{ \\ell_{i} \\}_{i = 1}^{e}$ with $1 \\leq \\ell_{i} \\leq n$ and $e \\geq 1$, a set of $K$-algebraic polynomials $\\xi_{\\ell_{1}, \\hdots, \\ell_{e}; jk}$ in the functions $f_{uv}$ for $1 \\leq u, v \\leq m$ with coefficients in the coordinate ring of $S_{K}$ such that \n\\begin{equation}\n\\label{diffeqpolys}\n\\partial_{\\ell_{1}} \\cdots \\partial_{\\ell_{e}} f_{jk} = \\xi_{\\ell_{1}, \\hdots, \\ell_{e}; jk}([f_{uv}]) .\n\\end{equation}\n\nBecause $S_{K}$ is smooth, given a point $s$ of $S_{K}$ the functions $z_{1} - s_{1}, \\hdots, z_{n} - s_{n}$, where $s_{i}$ is the value of $z_{i}$ on $s$, induce a coordinate system in the local and formal power series rings associated to $S_{K}$ at $s$. In these coordinates, the map $A_{p}$ is given by $f^{-1}_{p}$, where $f_{p} = [f_{uv}]$ is a rigid-analytic matrix-valued solution to the differential equations \\autoref{diffeq}. The formal map $\\widehat{\\kappa}$ obtained using the isomorphism $\\tau$ then satisfies the same set of differential equations, and in particular its derivatives of all orders at $s$ are determined using \\autoref{diffeqpolys} by the initial condition $f^{-1}(s) = P$. If we then construct an analytic solution to the differential  system in \\autoref{diffeq} in a neighbourhood of $s$ satisfying $f^{-1}(s) = P$, the resulting analytic map induces the map $\\widehat{\\kappa}$ on formal power series rings. It follows that $\\widehat{\\eta}$ is induced by a local period map, which completes the proof. \\qed\n\n\\paragraph{Proof of \\autoref{LVprop}:} ~ \\\\\n\n\\vspace{-0.5em}\n\nWe denote by $\\mathcal{O}_{K,(v)}$ the ring of integers localized at the prime ideal $\\mathfrak{p}$ of $\\mathcal{O}_{K}$ corresponding to $v$. We begin by showing that (base changes to $K_{v}$ of) the points of $S(\\mathcal{O}_{K,(v)})$ lie inside finitely many distinguished open affinoids $B_{p} \\subset \\an{S_{K_{v}}}$ admitting local period maps $\\psi_{p} : B_{p} \\to \\an{\\ch{L}_{K_{v}}}$. This reduces to showing that there are finitely many distinguished open affinoids $B_{p} \\subset \\an{S_{K_{v}}}$ containing the points in $S(\\mathcal{O}_{K,(v)})$ over which $\\an{\\mathcal{H}_{K_{v}}}$ admits a rigid-analytic flat frame. We may cover $S$ by finitely many open subschemes $U \\subset S$ such that $\\Omega^{1}_{U}$ and the bundles $F^{k} \\mathcal{H}$ for varying $k$ are all trivial. Then any point $s \\in S(\\mathcal{O}_{K,(v)})$ factors through some element of this cover, so we may reduce to the case where $\\Omega^{1}_{S}$ and the bundles $F^{k} \\mathcal{H}$ are all trivial.\n\nProceeding as in the proof of \\autoref{padicaxschanlem}, we can choose algebraic functions $z_{1}, \\hdots, z_{n}$ on $S$ such that $d z_{1}, \\hdots, d z_{n}$ trivializes $\\Omega^{1}_{S}$. We obtain differential equations as in (\\ref{diffeqpolys}), where the polynomials $\\xi_{\\ell_{1}, \\hdots, \\ell_{e}; jk}$ are functions in the coordinate ring $R$ of $S \\times \\textrm{GL}_{m}$, and so in particular we may view them after base-changing as elements of $R_{\\mathcal{O}_{K,v}}$, where $\\mathcal{O}_{K,v}$ is the ring of $v$-adic integers. Choose a point $\\overline{s_{0}} \\in S(\\mathcal{O}_{K,N}\/\\mathfrak{p} \\mathcal{O}_{K,N})$. Then as $S$ has (by assumption) good reduction modulo $\\mathfrak{p}$, we obtain by \\cite[IV. 18.5.17]{EGA} a lift $s_{0} \\in S(\\mathcal{O}_{K,v})$. Choosing an initial condition $P \\in \\textrm{GL}_{m}(\\mathcal{O}_{K,v})$, we may use (\\ref{diffeqpolys}) to construct a map $\\psi_{s_{0}} = \\an{q_{K_{v}}} \\circ f^{-1}$, where the partial derivatives of $f$ at $s_{0}$ are given by evaluating the polynomials $\\xi_{\\ell_{1}, \\hdots, \\ell_{e}; jk}$ at $(s_{0}, P)$. As the coefficients of the power series defining $\\psi_{s_{0}}$ lie in $\\mathcal{O}_{K,v}$, the map $\\psi_{s_{0}}$ is defined on a residue disk $B_{p, \\overline{s_{0}}}$ of radius $|p|^{1\/[K_{v} : \\mathbb{Q}_{p}]}$, where $|\\cdot|$ is the absolute value on $\\mathbb{Q}_{p}$. Varying $s_{0}$ over the finitely many elements of $S(\\mathcal{O}_{K,N}\/\\mathfrak{p} \\mathcal{O}_{K,N})$, we obtain the desired cover.\n\nNow we wish to show that $\\dim S(\\mathcal{O}_{K,N})^{\\textrm{Zar}} \\leq d$. Recall that for each $s \\in S(\\mathcal{O}_{K,N})$, we have a set $\\mathcal{I}(s) \\subset S(\\mathcal{O}_{K,N})$ of points whose associated Galois representations have isomorphic semisimplifications. As there are finitely many possibilities for the semisimplification, it suffices to consider the sets $\\mathcal{S}(s) \\subset S(\\mathcal{O}_{K,N})$ defined by\n\\[ \\mathcal{S}(s) = \\{ s' \\in S(\\mathcal{O}_{K,N}) : \\mathcal{I}(s) = \\mathcal{I}(s') \\textrm{ and } s \\equiv s' \\textrm{ mod } \\mathfrak{p} \\} , \\]\nand show that $\\dim \\overline{\\mathcal{S}(s)}^{\\textrm{Zar}} \\leq d$. In particular, we can consider the Zariski closure of just those elements of $\\mathcal{S}(s)$ whose associated points in $S(\\mathcal{O}_{K,v})$ lie inside one of the neighbourhoods $B_{p,\\overline{s_{0}}}$ constructed above on which we have a local period map $\\psi_{s_{0}} : B_{p, \\overline{s_{0}}} \\to \\an{\\ch{L}_{K_{v}}}$. The hypothesis of the proposition tells us that the image under $\\psi_{s_{0}}$ of the points in $\\mathcal{S}(s)$ lie in a subvariety $O_{s} \\subset \\ch{L}_{\\mathbb{C}_{p}}$ satisfying $\\dim \\Delta_{d} \\geq \\dim O_{s}$; here we use the fact that the flat frame on $B_{p, \\overline{s_{0}}}$ is compatible with the Frobenius endomorphism (c.f. the discussion in \\cite[\\S3]{LV}). Base-changing to $\\mathbb{C}_{p}$ and applying \\autoref{padicaxschanlem} above, we find that $\\dim \\overline{\\psi_{s_{0}}^{-1}(O_{s})}^{\\textrm{Zar}} \\leq d$, hence the result.\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction and Background}\\label{sec:introduction}\n\nAs a fundamental tool in discrete mathematics,  Lov\\'asz extension  has been deeply connected to  submodular analysis \\cite{Choquet54,Lovasz}, and has been applied in many  areas like  combinatorial  optimization, game  theory, matroid theory, stochastic processes, electrical networks, computer vision and machine learning \\cite{F05-book}. There are many generalizations, such as the disjoint-pair Lov\\'asz extension and the Lov\\'asz extension on distributive lattices \\cite{F05-book,Murota03book}.  Recent developments include quasi-Lov\\'asz extension on some algebraic structures and fuzzy mathematics, applications of Lov\\'asz extensions to  graph cut problems and computer science, as well as Lov\\'asz-softmax loss in deep learning.\n\n\n\n\n\\vspace{0.16cm}\n\nWe shall start by looking  at the original Lov\\'asz extension. For simplicity, we shall work throughout this paper with a finite and nonempty set $V=\\{1,\\cdots,n\\}$ and its power set $\\mathcal{P}(V)$. Also, we shall sometimes work on $\\mathcal{P}(V)^k:=\\{(A_1,\\cdots,A_k):A_i\\subset V,\\,i=1,\\cdots,k\\}$ and $\\mathcal{P}_k(V):=\\{(A_1,\\cdots,A_k)\\in\\ensuremath{\\mathcal{P}}(V)^k:A_i\\cap A_j=\\varnothing,\\,\\forall i\\ne j\\}$, as well as some restricted family $\\ensuremath{\\mathcal{A}}\\subset \\mathcal{P}(V)^k$. We denote the cardinality of a set $A$ by $\\#A$.\nGiven a function $f:\\mathcal{P}(V)\\to \\ensuremath{\\mathbb{R}}$, one identifies every $A\\in \\mathcal{P}(V){\\setminus\\{\\varnothing\\}}$ with its indicator vector $\\vec1_A\\in \\ensuremath{\\mathbb{R}}^V=\\ensuremath{\\mathbb{R}}^n$. The Lov\\'asz extension  extends the domain of $f$ to the whole Euclidean space\\footnote{Some other versions in the literature only extend the domain to the cube $[0,1]^V$ or the nonnegative orthant $\\ensuremath{\\mathbb{R}}_{\\ge0}^V$. In fact, many works on Boolean lattices  identify  $\\ensuremath{\\mathcal{P}}(V)$ with the discrete cube $\\{0,1\\}^n$.} $\\ensuremath{\\mathbb{R}}^V$. There are several equivalent\nexpressions:\n\n\\begin{itemize}\n\\item For $\\vec x =(x_1,\\dots ,x_n)\\in \\mathbb{R}^n$, let $\\sigma:V\\cup\\{0\\}\\to V\\cup\\{0\\}$ be a bijection such that $ x_{\\sigma(1)}\\le x_{\\sigma(2)} \\le \\cdots\\le x_{\\sigma(n)}$ and $\\sigma(0)=0$, where $x_0:=0$. The Lov\\'asz extension of $f$ is defined by\n\\begin{equation}\\label{eq:Lovasum}\nf^{L}(\\vec x)=\\sum_{i=0}^{n-1}(x_{\\sigma(i+1)}-x_{\\sigma(i)})f(V^{\\sigma(i)}(\\vec x)),\n\\end{equation}\nwhere   $V^0(\\vec x)=V$ and $V^{\\sigma(i)}(\\vec x):=\\{j\\in V: x_{j}> x_{\\sigma(i)}\\},\\;\\;\\;\\; i=1,\\cdots,n-1$. We can  write \\eqref{eq:Lovasum} in an integral form as\n\\begin{align}\\label{eq:Lovaintegral}\nf^{L}(\\vec x)&=\\int_{\\min\\limits_{1\\le i\\le n}x_i}^{\\max\\limits_{1\\le i\\le n}x_i} f(V^t(\\vec x))d t+f(V)\\min_{1\\le i\\le n}x_i\n\\end{align}\n\nwhere $V^t(\\vec x)=\\{i\\in V:  x_i>t\\}$. If we apply the   M\\\"obius transformation, this becomes\n \\begin{equation}\\label{eq:LovaMobuis} f^{L}(\\vec x)=\\sum\\limits_{A\\subset V}\\sum\\limits_{B\\subset A}(-1)^{\\#A-\\#B}f(B)\\bigwedge\\limits_{i\\in A} x_i,\\end{equation}\nwhere $\\bigwedge\\limits_{i\\in A} x_i$ is the minimum over $\\{x_i:i\\in A\\}$.\n\\end{itemize}\nIt is easy to see  that $f^L$ is positively one-homogeneous, PL (piecewise linear) and Lipschitz continuous \\cite{Lovasz,Bach13}. Also, $f^L(\\vec x+t\\vec 1_V)=f^L(\\vec x)+tf(V)$, $\\forall t\\in\\ensuremath{\\mathbb{R}}$, $\\forall \\vec x\\in\\ensuremath{\\mathbb{R}}^V$, {and $f^L(\\vec1_A)=f(A)$ for any $A\\in\\ensuremath{\\mathcal{P}}(V)\\setminus\\{\\varnothing\\}$. The definition of $f^L$ does not involve the datum $f(\\emptyset)$, and thus by convention, it is natural to reset $f(\\emptyset)=0$ to match the equality  $f^L(\\vec0)=0$, unless stated otherwise. For convenience, we say that $f:\\ensuremath{\\mathcal{P}}(V)\\to\\ensuremath{\\mathbb{R}}$ is a constant (resp., positive) function if $f$ is constant (resp., positive) on $\\ensuremath{\\mathcal{P}}(V)\\setminus\\{\\varnothing\\}$}. \nMoreover, a continuous function $F:\\ensuremath{\\mathbb{R}}^V\\to \\ensuremath{\\mathbb{R}}$ is the Lov\\'asz extension of some $f:\\ensuremath{\\mathcal{P}}(V)\\to\\ensuremath{\\mathbb{R}}$ if and only if $F(\\vec x+\\vec y)=F(\\vec x)+F(\\vec y)$ whenever $(x_i-x_j)(y_i-y_j)\\ge0$, $\\forall i,j\\in V$.\n\n\n\\vspace{0.13cm}\n\nIn this paper, we shall use the Lov\\'asz extension and its variants to study\nthe interplay between discrete and continuous aspects in  topics such as  convexity, optimization and spectral  theory.\n\n\n\n\n\\vspace{0.13cm}\n\n\\textbf{Submodular and convex functions}\n\n\nSubmodular function have emerged as a powerful concept in discrete optimization, see  Fujishige's monograph \\cite{F05-book} and {Bach's works \\cite{Bach13,Bach19}}.   We also refer the readers to some recent related works regarding \nsubmodular functions on  hypergraphs  \\cite{LM18,LM18-,LHM20}.  We recall\n  that a discrete function $f:\\ensuremath{\\mathcal{A}}\\to \\ensuremath{\\mathbb{R}}$  defined on  an algebra $\\ensuremath{\\mathcal{A}}\\subset\\mathcal{P}(V)$ (i.e., $\\ensuremath{\\mathcal{A}}$ is  closed under  union and  intersection)  is submodular if $f(A)+f(B)\\ge f(A\\cup B)+f(A\\cap B)$, $\\forall A,B\\in\\ensuremath{\\mathcal{A}}$. The Lov\\'asz extension turns a  submodular into a convex function, and we can hence minimize the former by minimizing the latter:\n\\begin{theorem}[Lov\\'asz \\cite{Lovasz}]\n\t$f:\\mathcal{P}(V)\\to\\mathbb{R}$ is submodular $\\Leftrightarrow$ $f^L$ is convex.\n\\end{theorem}\n\n\\begin{center}\n\t\\begin{tikzpicture}[node distance=6cm]\n\t\n\t\\node (convex) [startstop] {  Submodularity };\n\t\n\t\\node (submodular) [startstop, right of=convex, xshift=1.6cm]  { Convexity  };\n\t\n\t\\draw [arrow](convex) --node[anchor=south] { \\small Lov\\'asz extension } (submodular);\n\t\\draw [arrow](submodular) --node[anchor=north] {  } (convex);\n\t\\end{tikzpicture}\n\\end{center}\n\n\\begin{theorem}[Lov\\'asz \\cite{Lovasz}]If $f:\\mathcal{P}(V)\\to\\mathbb{R}$ is submodular with $f(\\varnothing)=0$, then\n\t$$\\min\\limits_{A\\subset V}f(A)=\\min\\limits_{\\vec x\\in [0,1]^V}f^L(\\vec x).$$\n\\end{theorem}\n\n\\begin{center}\n\t\\begin{tikzpicture}[node distance=6cm]\n\t\n\t\\node (convex) [process] {\n\t\tSubmodular optimization  };\n\t\n\t\\node (submodular) [process, right of=convex, xshift=1.6cm]  {\n\t\tConvex programming   };\n\t\n\t\\draw [arrow](convex) --node[anchor=south] { \\small Lov\\'asz extension } (submodular);\n\t\\draw [arrow](submodular) --node[anchor=north] { \\small} (convex);\n\t\\end{tikzpicture}\n\\end{center}\n\n\nThus, submodularity can be seen  as some kind of\n `discrete convexity', and that naturally lead to  many generalizations, such as bisubmodular, $k$-submodular, L-convex and M-convex, see \\cite{F05-book,Murota03book}.\nMoreover, the following classical result characterizes the class of all functions which can be expressed as Lov\\'asz extensions of  submodular functions. \n \\begin{theorem}\n  Theorem 7.40 in \\cite{Murota03book}]\n \tA one-homogeneous function $F:\\ensuremath{\\mathbb{R}}^V\\to \\ensuremath{\\mathbb{R}}$ is a Lov\\'asz extension of some submodular function if and only if $F(\\vec x+t\\vec 1_V)=F(\\vec x)+tF(\\vec 1_V)$, $\\forall t\\in\\ensuremath{\\mathbb{R}}$, $\\forall \\vec x\\in\\ensuremath{\\mathbb{R}}^V$, and $F(\\vec x)+F(\\vec y)\\ge F(\\vec x\\vee \\vec y)+F(\\vec x\\wedge \\vec y)$, where the $i$-th components of $\\vec x\\vee \\vec y$ and $\\vec x\\wedge \\vec y$ are\n \t$(\\vec x\\vee \\vec y)_i=\\max\\{x_i,y_i\\}$ and $(\\vec x\\wedge \\vec y)_i=\\min\\{x_i,y_i\\}$.\n \\end{theorem}\n One may want to extend such a result to the  bisubmodular  or more general cases.\n In that direction, we shall obtain some   results such as Proposition \\ref{pro:bisubmodular-continuous} and Theorem \\ref{thm:submodular-L-equivalent} in Section \\ref{sec:SubmodularityConvexity}. It is also worth noting that Bach investigated an interesting generalization of submodular functions by a generalized Lov\\'asz extension \\cite{Bach19}. \n\n\nSo far,  research has mainly  focused  on `discrete convex' functions, leading to\n`Discrete Convex Analysis' \\cite{Murota98,Murota03book}, whereas the  discrete non-convex setting which is quite  popular in modern sciences has not yet received that much attention.\n\n\\vspace{0.16cm}\n\n\\textbf{Non-submodular cases}\n\nObviously, the non-convex case is so diverse and general that it cannot be directly studied by standard submodular tools.  Although some publications show several results on non-submodular (i.e., non-convex) minimization based on Lov\\'asz extension \\cite{HS11}, so far, these only work for special minimizations over the whole power set.  Here, we shall find applications for discrete optimization and nonlinear spectral graph theory by employing the multi-way Lov\\'asz extension on enlarged and restricted  domains. \n\n\n\\vspace{0.16cm}\n\nIn summary, we are going to initiate the study of diverse continuous extensions in non-submodular settings. This paper develops a systematic framework for many aspects around the topic. We  establish a universal discrete-to-continuous framework via multi-way extensions, by systematically utilizing integral representations. \nIn \\cite{JZ-prepare21}, we establish the links between discrete Morse theory and continuous Morse theory via the original Lov\\'asz extension. \nWe shall now discuss some connections with other various fields.\n\n\\vspace{0.19cm}\n\n\\textbf{Connections with  combinatorial optimization}\n\nBecause of the wide range of applications of discrete mathematics in computer science,  combinatorial optimization has been much studied from the mathematical perspective.\nIt is known that any combinatorial optimization can be equivalently expressed as a continuous optimization via convex (or concave) extension, but often,\nthere is the difficulty that one cannot  write down an equivalent continuous\nobject function in closed form.\nFor practical purposes, it would be very helpful if one could  transfer a\ncombinatorial optimization problem to an explicit and simple equivalent\ncontinuous optimization problem in closed form. Formally, in many concrete situations, it would be useful if one could get an identity of the  form\n\\begin{equation}\\label{eq:D-to-C-formal}\\min\\limits_{(A_1,\\cdots,A_k)\\in \\ensuremath{\\mathcal{A}}\\cap \\ensuremath{\\mathrm{supp}}(g)}\\frac{f(A_1,\\cdots,A_k)}{g(A_1,\\cdots,A_k)}=\\inf\\limits_{\\psi\\in {\\mathcal D}(\\ensuremath{\\mathcal{A}})}\\frac{\\widetilde{f}(\\psi)}{\\widetilde{g}(\\psi)}.\\end{equation}\nwhere $f,g:\\ensuremath{\\mathcal{A}}\\to [0,\\infty)$,  ${\\mathcal D}(\\ensuremath{\\mathcal{A}})$ is a feasible domain determined by $\\ensuremath{\\mathcal{A}}$ only, $\\mathrm{supp}(g)$ is the support of $g$, and $\\widetilde{f}$ and $\\widetilde{g}$ are suitable continuous extensions of $f$ and $g$.\n\nSo far, only situations where  $f:\\ensuremath{\\mathcal{P}}(V)\\to \\ensuremath{\\mathbb{R}}$ or $f:\\ensuremath{\\mathcal{P}}_2(V)\\to\\ensuremath{\\mathbb{R}}$ have been investigated systematically \\cite{HS11,CSZ18}, and what is lacking are situations with restrictions, that is,  incomplete data.\n\nAlso, to the best of our knowledge, the known results in the literature do not work for  combinatorial optimization directly on set-tuples.  But most of combinatorial optimization problems should be formalized in the form of set-tuples, and only a few can be represented in set form or disjoint-pair form. Whenever one can find an equivalent Lipschitz function for a combinatorial problem in the field of discrete optimization, this makes  useful tools available and leads to new connections.\nThat is, one wishes to establish   a {\\sl discrete-to-continuous transformation} like the operator $\\sim$ in \\eqref{eq:D-to-C-formal}.\nWe will show in Section \\ref{sec:CC-transfer} that the Lov\\'asz extension and its variants \nare suitable choices for such a transformation\n(see Theorems \\ref{thm:tilde-fg-equal}, \\ref{thm:tilde-H-f} and Proposition \\ref{pro:fraction-f\/g} for details).\n\n\\vspace{0.15cm}\n\n\nTo reach these goals, we need to systematically study various generalizations  of the  Lov\\'asz extension. More precisely, we shall work with the following  two different multi-way forms:\n\n\\begin{enumerate}[(1)]\n\n\\item Disjoint-pair version:\n    for a function $f:\\ensuremath{\\mathcal{P}}_2(V)\\to\\ensuremath{\\mathbb{R}}$, its disjoint-pair\n  Lov\\'asz extension is defined as\n\\begin{equation}\\label{eq:disjoint-pair-Lovasz-def-integral}\nf^{L}(\\vec x)=\\int_0^{\\|\\vec x\\|_\\infty} f(V_+^t(\\vec x),V_-^t(\\vec x))dt,\n\\end{equation}\nwhere $V_\\pm^t(\\vec x)=\\{i\\in V:\\pm x_i>t\\}$, $\\forall t\\ge0$. For $\\ensuremath{\\mathcal{A}}\\subset\\ensuremath{\\mathcal{P}}_2(V)$ and $f:\\ensuremath{\\mathcal{A}}\\to\\ensuremath{\\mathbb{R}}$, the feasible domain ${\\mathcal D}_\\ensuremath{\\mathcal{A}}$ of the disjoint-pair Lov\\'asz extension is $\\{\\vec x\\in\\ensuremath{\\mathbb{R}}^V:(V_+^t(\\vec x),V_-^t(\\vec x))\\in\\ensuremath{\\mathcal{A}},\\forall t\\ge0\\}$. \n\nIt should be noted that the disjoint-pair Lov\\'asz extension introduced by\nQi \\cite{Qi88} has been systematically  investigated by Fujishige \\cite{Fujishige14,F05-book} and Murota \\cite{Murota03book} in the context of  discrete convex analysis (or the theory of submodular functions). The \nintegral formulation \\eqref{eq:disjoint-pair-Lovasz-def-integral}, however,   is \nmore convenient to obtain a closed formula of the equivalent continuous\noptimization problem for a combinatorial optimization problem. Moreover, the references and the present paper \nfocus on different aspects, with the exception of the submodularity theorem (i.e., $f$ is bisubmodular iff $f^L$ is convex).\n\n\\item $k$-way version:\nfor a function $f:\\mathcal{P}(V)^k\\to \\ensuremath{\\mathbb{R}}$, the {\\sl simple $k$-way Lov\\'asz extension} $f^L:\\ensuremath{\\mathbb{R}}^{kn}\\to \\ensuremath{\\mathbb{R}}$ is defined as\n\\begin{equation}\\label{eq:Lovasz-Form-1}\nf^L(\\vec x^1,\\cdots,\\vec x^k)=\\int_{\\min \\vec x}^{\\max \\vec x}f(V^t(\\vec x^1),\\cdots,V^t(\\vec x^k))dt+ f(V,\\cdots,V)\\min\\vec x,\n\\end{equation}\nwhere $V^t(\\vec x^i)=\\{j\\in V:x^i_j>t\\}$, $\\min\\vec x=\\min\\limits_{i,j} x^i_j$ and $\\max\\vec x=\\max\\limits_{i,j} x^i_j$.  For $\\ensuremath{\\mathcal{A}}\\subset\\ensuremath{\\mathcal{P}}^k(V)$ and $f:\\ensuremath{\\mathcal{A}}\\to\\ensuremath{\\mathbb{R}}$, we take  ${\\mathcal D}_\\ensuremath{\\mathcal{A}}=\\{\\vec x\\in\\ensuremath{\\mathbb{R}}^{kn}_{\\ge0}:(V^t(\\vec x^1),\\cdots,V^t(\\vec x^k))\\in\\ensuremath{\\mathcal{A}},\\forall t\\in\\ensuremath{\\mathbb{R}}\\}$ as a feasible domain of the $k$-way Lov\\'asz extension $f^L$. \n\nBy the Lov\\'asz extension of  submodular functions on distributive\n  lattices \\cite{F05-book,Murota03book},  our $k$-way version\n  \\eqref{eq:Lovasz-Form-1} can be reduced to the classical version on\n  distributive lattices. Our main purposes and key results, however,  are\n  different from that approach. In fact, we mainly aim to deal with discrete\n  fractional programming by the $k$-way  Lov\\'asz extension, while those references concentrate on  submodularity and  convex optimization. \n\\end{enumerate}\n\n All these multi-way Lov\\'asz extensions satisfy the optimal identity Eq.~\\eqref{eq:D-to-C-formal}\n\n\\begin{introthm}[Theorem \\ref{thm:tilde-H-f} and Proposition \\ref{pro:fraction-f\/g}]\\label{thm:tilde-fg-equal}\nGiven two  functions $f,g:\\ensuremath{\\mathcal{A}}\\to [0,+\\infty)$, let $\\tilde{f}$ and $\\tilde{g}$ be two real\nfunctions on ${\\mathcal D}_\\ensuremath{\\mathcal{A}}$ satisfying $\\tilde{f}(\\vec1_{A_1,\\cdots,A_k})=f(A_1,\\cdots,A_k)$ and $\\tilde{g}(\\vec1_{A_1,\\cdots,A_k})=g(A_1,\\cdots,A_k)$. Then Eq.~\\eqref{eq:D-to-C-formal} holds if $\\tilde{f}$ and $\\tilde{g}$ further possess the properties (P1) or (P2) below. Correspondingly, if $\\tilde{f}$ and $\\tilde{g}$ fulfil (P1') or (P2),\nthere similarly holds\n$$\\max\\limits_{(A_1,\\cdots,A_k)\\in \\ensuremath{\\mathcal{A}}\\cap\\ensuremath{\\mathrm{supp}}(g)}\\frac{f(A_1,\\cdots,A_k)}{g(A_1,\\cdots,A_k)}=\\sup\\limits_{\\psi\\in {\\mathcal D}_\\ensuremath{\\mathcal{A}}\\cap\\ensuremath{\\mathrm{supp}}(\\widetilde{g})}\\frac{\\widetilde{f}(\\psi)}{\\widetilde{g}(\\psi)}.$$\nHere the optional additional conditions of $\\tilde{f}$ and $\\tilde{g}$ are:\n\n(P1) $\\tilde{f}\\ge f^L$ and $\\tilde{g}\\le g^L$.\\;\\;\\;  (P1') $\\tilde{f}\\le f^L$ and $\\tilde{g}\\ge g^L$.\n\n(P2)\n$\\tilde{f}=((f^\\alpha)^L)^{\\frac1\\alpha}$ and $\\tilde{g}=((g^\\alpha)^L)^{\\frac1\\alpha}$, where $\\alpha>0$.\n\nHere $f^L$ is either the original or the disjoint-pair or the $k$-way Lov\\'asz extension.\n\\end{introthm}\n\nTheorem \\ref{thm:tilde-fg-equal}\nshows that by the multi-way Lov\\'asz extension, the  combinatorial\noptimization in quotient form can be transformed to  fractional\nprogramming. And based on this fractional optimization, we propose an\neffective local convergence scheme, which relaxes the Dinkelbach-type\niterative scheme and mixes the inverse power method and the steepest descent\nmethod. Furthermore, many other  continuous iterations, such as\nKrasnoselski-Mann iteration, and the stochastic subgradient method, could be directly applied here. We refer the readers to \\cite{JostZhang} for another development on  equalities between  discrete and continuous optimization problems via various generalizations of Lov\\'asz extension. \n\n\nThe power of Theorem \\ref{thm:tilde-fg-equal} is embodied in many new examples and applications including  Cheeger-type problems, various\nisoperimetric constants and max $k$-cut problems (see Subsections \\ref{sec:max-k-cut}, \\ref{sec:boundary-graph-1-lap} and \\ref{sec:variantCheeger}).\nAnd moreover, we find that not only combinatorial optimization,\nbut also some combinatorial invariants like the independence number and the chromatic number, can\nbe transformed into a continuous representation by this scheme.\n\n\\begin{introthm}[Sections \\ref{sec:independent-number} and \\ref{sec:chromatic-number}]\n\\label{thm:graph-numbers}\nFor an unweighted and undirected simple graph $G=(V,E)$ with $\\#V=n$, its independence number can be represented as\n$$\\alpha(G)=\\max\\limits_{\\vec x\\in \\ensuremath{\\mathbb{R}}^n\\setminus\\{\\vec 0\\}}\\frac{\\sum\\limits_{\\{i,j\\}\\in E}(|x_i-x_j|+|x_i+x_j|)- 2\\sum\\limits_{i\\in V}(\\deg_i-1)|x_i|}{2\\|\\vec x\\|_\\infty},$$\nwhere $\\deg_i=\\#\\{j\\in V:\\{j,i\\}\\in E\\}$, $i\\in V$, and its chromatic number is\n\\begin{equation*}\n\\gamma(G)= n^2-\\max\\limits_{\\vec x\\in\\ensuremath{\\mathbb{R}}^{n^2}\\setminus\\{\\vec 0\\}}\\sum\\limits_{k\\in V}\\frac{n\\sum\\limits_{\\{i,j\\}\\in E}(|x_{ik}-x_{jk}|+|x_{ik}+x_{jk}|)+2n\\|\\vec x^{k}\\|_{\\infty}-2n\\deg_k\\|\\vec x^{k}\\|_1- 2\\|\\vec x^{,k}\\|_{\\infty}}{2\\|\\vec x\\|_\\infty},\n\\end{equation*}\nwhere $\\vec x=(x_{ki})_{k,i\\in V}$, $\\vec x^{k}=(x_{k1},\\cdots,x_{kn})$ and $\\vec x^{,k}=(x_{1k},\\cdots,x_{nk})^T$.\nThe maximum matching number of $G$ can be expressed as\n$$\\max\\limits_{\\vec y\\in\\ensuremath{\\mathbb{R}}^E\\setminus\\{\\vec 0\\}}\\frac{\\|\\vec y\\|_1^2}{\\|\\vec y\\|_1^2-2\\sum_{e\\cap e'=\\varnothing}y_ey_{e'}}.$$\n\\end{introthm}\n\n \\vspace{0.15cm}\n\n\n\n\nThere are some equivalent continuous  reformulations of the maxcut problem and\nthe  independence number of a graph in the literature. However, a continuous  reformulation of the coloring number has not yet been proposed. The main reason seems to be the complexity of coloring a graph. Hence, it is very difficult to discover a continuous form of the coloring number by direct  observation. \n\n\\begin{introthm}[Theorem \\ref{thm:tilde-H-f}]\\label{thm:tilde-fg-equal-PQ}\nGiven   functions $f_1,\\cdots,f_n:\\ensuremath{\\mathcal{A}}\\to[0,+\\infty)$,and  $p$-homogeneous functions $P,Q:[0,+\\infty)^n\\to[0,+\\infty)$, we have\n$$\\max\\limits_{A\\in\\ensuremath{\\mathcal{A}}}\\frac{P(f_1(A),\\cdots,f_n(A))}{Q(f_1(A),\\cdots,f_n(A))}=\\sup\\limits_{x\\in{\\mathcal D}_\\ensuremath{\\mathcal{A}}}\\frac{P(f_1^L(\\vec x),\\cdots,f_n^L(\\vec x))}{Q(f_1^L(\\vec x),\\cdots,f_n^L(\\vec x))}$$\nif $P^{\\frac1p}$ is \nsubadditive  and  $Q^{\\frac1p}$ is superadditive. One can replace  `max' by `min' if $P^{\\frac1p}$ is \nsuperadditive  and  $Q^{\\frac1p}$ is subadditive.\n\\end{introthm}\n\nTheorems \\ref{thm:tilde-fg-equal},  \\ref{thm:tilde-fg-equal-PQ} and \\ref{thm:tilde-H-f}  can be seen as natural and nontrivial generalizations of the related original works by   Hein's group \\cite{HS11}.\n\n\\vspace{0.1cm}\n\n\\textbf{Connections with  spectral graph theory}\n\nSpectral graph theory aims to derive properties of a (hyper-)graph from its eigenvalues and eigenvectors. Going beyond the linear case, nonlinear spectral graph theory is developed in terms of discrete geometric analysis and difference equations on (hyper-)graphs.\nEvery discrete eigenvalue problem can be formulated as a variational problem  for an\nobjective functional, a Rayleigh-type quotient.\nIn some cases, this\nfunctional is natural and easy to obtain,\nsince one may compare the discrete version with its original continuous analog in geometric analysis. However, in other\nsituations, there is no such analog. Fortunately, we find a unified framework based on multi-way Lov\\'asz extension to produce appropriate objective functions from a combinatorial problem (see Sections\n\\ref{sec:CC-transfer} and \\ref{sec:eigenvalue}).\n\nMore precisely, for a combinatorial problem with a  discrete objective function of the form $\\frac{f(A)}{g(A)}$, we might obtain some correspondences by studying the  \nset-valued eigenvalue problem\n$$\n \\nabla f^L (\\vec x)\\bigcap \\lambda\\nabla  g^L (\\vec x) \\ne\\varnothing\n$$\nwhich is simply called the eigenvalue problem of the function pair $(f^L,g^L)$.  \nHereafter we use $\\nabla$ to denote the (Clarke) sub-gradient operator acting on Lipschitz functions.\n\\begin{center}\n\\begin{tikzpicture}[node distance=6cm]\n\n\\node (graph) [startstop] {  combinatorial  quantities };\n\n\\node (spectrum) [startstop, right of=convex, xshift=2.6cm]  { eigenvalues and eigenvectors };\n\n\\draw [arrow](graph) --node[anchor=south] { \\small Spectral graph theory} (spectrum);\n\\draw [arrow](spectrum) --node[anchor=north] { } (graph);\n\\end{tikzpicture}\n\\end{center}\n\n\n\n\n\\vspace{0.16cm}\nWe shall consider the following three concepts:\n\\begin{itemize}\n\\item {\\sl Eigenvectors and eigenvalues}:\\;\\;\nThe set-valued eigenvalue problems above are usually written as $\\vec0\\in \\nabla f^L (\\vec x)-\\lambda\\nabla  g^L (\\vec x)$ by using the Minkowski summation of convex sets. We call $\\lambda$ an eigenvalue and $\\vec x$ an eigenvector associated to $\\lambda$.\n\n\\item {\\sl Critical points and critical values}:\\;\\;\nThe set of critical points\n$\\left\\{\\vec x\\left|0\\in\\nabla \\frac{f^L(\\vec x)}{g^L(\\vec x)}\\right.\\right\\}$\n and the corresponding critical values.\n\n\n\\item {\\sl Minimax critical values} (i.e., {\\sl variational eigenvalues in Rayleigh quotient form}):\\;\\;\nThe Lusternik-Schnirelman theory tells us that the min-max\nvalues\n\\begin{equation}\\label{eq:def-c_km}\n\\lambda_{m}=\\inf_{\\Psi\\in \\Gamma_m}\\sup\\limits_{\\vec x \\in \\Psi}\\frac{f^L(\\vec x)}{g^L(\\vec x)}\n\\end{equation}\nare critical values of $f^L(\\cdot)\/g^L(\\cdot)$. Here $\\Gamma_m$ is a class of certain topological objects at level $m$, e.g., the family of subsets with L-S category (or Krasnoselskii's $\\mathbb{Z}_2$-genus) not smaller than $m$.\n\\end{itemize}\n\n  There are the following relations between these three classes:\n$$\\{\\text{Eigenvalues in Rayleigh quotient}\\}\\subset\\{\\text{Critical values}\\}\\subset\\{\\text{Eigenvalues}\\}.$$\nFor linear spectral theory, the above three classes  coincide. However, for the non-smooth spectral theory derived by Lov\\'asz extension, we only have the inclusion relations.\n\n\nWe have the following result on the eigenvalue problem for the  disjoint-pair Lov\\'asz extension, while  for the results on the original Lov\\'asz extension, we refer to  Section \\ref{sec:eigenvalue} for details. \n\\begin{introthm}\n\\label{introthm:eigenvalue}\nGiven $f,g:\\ensuremath{\\mathcal{P}}_2(V)\\to\\ensuremath{\\mathbb{R}}$, then every eigenvalue of $(f^L,g^L)$ has an eigenvector of the form $\\vec 1_A-\\vec1_B$. Moreover, we have the following claims:\n\\begin{itemize}\n\\item If $2f(A,B)=f(A,V\\setminus A)+f(V\\setminus B,B)$ and $2g(A,B)=g(A,V\\setminus A)+g(V\\setminus B,B)$  for any $(A,B)\\in \\ensuremath{\\mathcal{P}}_2(V)\\setminus\\{(\\varnothing,\\varnothing)\\}$, then every eigenvalue of $(f^L,g^L)$ has an eigenvector of the form  $\\vec 1_A-\\vec1_{V\\setminus A}$.\n\\item If $g=\\mathrm{Const}$,  then for any $A\\subset V$, $\\vec 1_A-\\vec1_{V\\setminus A}$ is an eigenvector.\n\\item If $f(A,B)=\\hat{f}(A)+\\hat{f}(B)$ and $g(A,B)=\\hat{g}(A)+\\hat{g}(B)$  for some  symmetric function  $\\hat{f}:\\ensuremath{\\mathcal{P}}(V)\\to\\ensuremath{\\mathbb{R}}$ (i.e., $\\hat{f}(A)=\\hat{f}(V\\setminus A)$, $\\forall A$) and non-decreasing submodular function  $\\hat{g}:\\ensuremath{\\mathcal{P}}(V)\\to\\ensuremath{\\mathbb{R}}_+$,  then the second \neigenvalue $\\lambda_2$ of $(f^L,g^L)$ equals \n$$\\min\\limits_{\\vec x\\bot\\vec 1}\\frac{f^L(\\vec x)}{\\min\\limits_{t\\in\\ensuremath{\\mathbb{R}}}g^L(\\vec x-t\\vec 1)}=\\min\\limits_{A\\in\\mathcal{P}(V)\\setminus\\{\\varnothing,V\\}}\\frac{\\hat{f}(A)}{\\min\\{\\hat{g}(A),\\hat{g}(V\\setminus A)\\}}=\\min\\limits_{(A,B)\\in\\mathcal{P}_2(V)\\setminus\\{(\\varnothing,\\varnothing )\\}}\\max\\{\\frac{\\hat{f}(A)}{\\hat{g}(A)},\\frac{\\hat{f}(B)}{\\hat{g}(B)}\\}.$$\n\\end{itemize}\n\n\n\\end{introthm}\n\nThis generalizes  recent results on the\ngraph 1-Laplacian and Cheeger's constant \\cite{HeinBuhler2010,TVhyper-13,Chang16,CSZ15,CSZ17}. { And as a new application, we show  that the   min-cut problem and the   max-cut problem are equivalent to solving  the \nsmallest  nontrivial (i.e., the second) eigenvalue and the largest eigenvalue of a certain nonlinear eigenvalue problem (see Theorem \\ref{thm:mincut-maxcut-eigen}).\n}\n\n\\vspace{0.16cm}\n\n\\textbf{Applications to   frustration in signed network }\n\nAs a key measure for analysing signed networks, the frustration index on a signed graph  quantifies  how far a signature  is\nfrom being balanced (see Section \\ref{sec:frustration}). Computing the frustration index  is NP-hard, and few algorithms have been proposed  \\cite{ArefWilson19,ArefMasonWilson20}. \n\n\n\nConsidering a signed graph $(V,E_+\\cup E_-)$ with  $E_+$ (resp. $E_-$)\nthe set of positive (resp. negative)  edges,   based on the  disjoint-pair Lov\\'asz  extension, we obtain an equivalent continuous optimization of the frustration index (or the line index of balance \\cite{Harary59}):\n$$\\#E_-+\\min\\limits_{ x\\ne0}\\frac{\\sum_{\\{i,j\\}\\in E_+}|x_i-x_j|-\\sum_{\\{i,j\\}\\in E_-}|x_i-x_j|}{2\\|\\vec x\\|_\\infty}.$$\nThis new reformulation can be computed via typical algorithms in continuous optimization. \n\nAlso, we propose the  eigenvalue problem\n\\begin{equation}\\label{eq:frustration-eigen}\n\\nabla\\left(\\sum_{\\{i,j\\}\\in E_+}|x_i-x_j|+\\sum_{\\{i,j\\}\\in E_-}|x_i+x_j|\\right)\\bigcap\\lambda \\nabla\\|\\vec x\\|_\\infty \\ne\\varnothing    \n\\end{equation}\nand we show an iterative scheme for searching the frustration index based on the smallest eigenvalue of the nonlinear eigenvalue problem \\eqref{eq:frustration-eigen}. See Section \\ref{sec:frustration} for details and  more results.  \n\n\\vspace{0.16cm}\n\n Since the  transformation of a combinatorial optimization to a\ncontinuous optimization or a nonsmooth  eigenvalue problem usually leads\n  to a quotient,\nthe task for  fractional programming then becomes to compute  an optimal value or an eigenvector. In Section \\ref{sec:algo}, we present a general  algorithm which is available to compute the resulting continuous reformulations arising in Theorems \\ref{thm:tilde-fg-equal}, \\ref{thm:graph-numbers}, \\ref{thm:tilde-fg-equal-PQ} and \\ref{introthm:eigenvalue}. \n\n In summary, we present a systematic study for  constructing  nonlinear\neigenvalue problems and equivalent continuous  reformulations for\ncombinatorial quantities, which capture the key properties of the original   combinatorial problems. This is helpful to increase understanding of certain combinatorial problems by the corresponding  eigenvalue problems and the  equivalent continuous  reformulations. The following picture summarizes the relations between the various concepts developed and studied in this paper.\n\n\n\\begin{figure}[H]\n\\centering\n\\begin{tikzpicture}[node distance=4.5cm]\n\n\\node (CQ) [startstop] {\\begin{tabular}{l}\nCombinatorial\\\\ Quantities\n\\end{tabular}};\n\n\\node (DO) [process, right of=CQ, xshift=-1cm]  { \\begin{tabular}{l}\n Discrete\\\\\n Optimization\n\\end{tabular}};\n\n\\node (CO) [startstop, right of=DO, yshift=0cm, xshift=2.6cm] {\\begin{tabular}{l}\n Continuous\\\\\n Optimization\n\\end{tabular} };\n\\node (DM) [io1, below of=CQ, xshift=0.1cm,yshift=2cm] {\\begin{tabular}{l}\n Discrete Morse theory\n\\end{tabular}\n };\n\\node (F\/G) [process, right of=DM, xshift=2.2cm] {\\begin{tabular}{l}\n(topological) Morse theory    \\\\\n  (metric) critical point theory\n\\end{tabular}\n};\n\\node (FG) [startstop, below of=F\/G, yshift=2.6cm,xshift=3.3cm] {\n\\begin{tabular}{r}\n(Nonlinear) Spectral theory\\\\\n\\end{tabular}};\n\n\\node (algorithm) [io1, right of=F\/G, xshift=2cm] {\\begin{tabular}{l}\n Continuous\\\\\n Programming\\\\\n\\& Algorithm\n\\end{tabular}};\n\\node (submodular) [io2, above of=DO, yshift=-2.6cm] {Submodularity};\n\\node (convexity) [io2, right of=submodular, xshift=5.2cm] {Convexity};\n\\draw [arrow](CQ) --node[anchor=south] { } (DO);\n\\draw [arrow](DO) --node[anchor=south] { } (CQ);\n\\draw [arrow](DO) --node[anchor=south]{Discrete-to-Continuous }node[anchor=north] {extension} (CO);\n\\draw [arrow](DM) -- node[anchor=south] {\\cite{JZ-prepare21}}node[anchor=north] { Part I } (F\/G);\n\\draw [arrow](FG) -- node[anchor=south] {} (algorithm);\n\\draw [arrow](F\/G) -- node[anchor=south] {  } (FG);\n\\draw [arrow](CO) -- node[anchor=south] {  } (algorithm);\n\\draw [arrow](CO) -- node[anchor=south] {  } (F\/G);\n\\draw [arrow](CO) -- node[anchor=south] {  } (FG);\n\\draw [arrow](submodular) -- node[anchor=south] {  Lov\\'asz extension } (convexity);\n\\draw [arrow](submodular) -- node[anchor=south] { } (DO);\n\\draw [arrow](convexity) -- node[anchor=south] { } (submodular);\n\\draw [arrow](convexity) -- node[anchor=south] { } (algorithm);\n\\end{tikzpicture}\n\\caption{\\label{fig:flowchart} The relationship between the aspects\nstudied in our work.}\n\\end{figure}\n\n\n\n\\begin{notification}\nSince this paper contains  many interacting parts and relevant results, some notions and concepts may have slightly distinct\nmeanings in different sections, but this will be stated at the beginning of each section.\n\\end{notification}\n\n\n\n\n\\section{A preliminary:  Lov\\'asz extension and submodular functions}\n\\label{sec:Lovasz}\n\n\n\nWhile most of the results on submodularity are known in the field of discrete\nconvex analysis, we present some details in a simple manner, which should be helpful to understand our main results in Section \\ref{sec:main}.\n\n\nWe first formalize some important results about the original Lov\\'asz extension.\n\n\n\\begin{defn}\nTwo vectors $\\vec x$ and $\\vec y$ are {\\sl comonotonic} if $(x_i-x_j)(y_i-y_j)\\ge0$, $\\forall i,j\\in \\{1,2,\\cdots,n\\}$.\n\nA function $F:\\ensuremath{\\mathbb{R}}^n\\to\\ensuremath{\\mathbb{R}}$ is  {\\sl comonotonic additive} if $F(\\vec x+\\vec y)=F(\\vec x)+F(\\vec y)$ for any comonotonic pair $\\vec x$ and $\\vec y$.\n\\end{defn}\nThe following proposition shows that a function is comonotonic additive if and only if it can be expressed as the Lov\\'asz extension of some function.\n\\begin{pro}\n\\label{pro:comonotonic-additivity}\n$F:\\ensuremath{\\mathbb{R}}^n\\to\\ensuremath{\\mathbb{R}}$ is the Lov\\'asz extension $F=f^L$ of some function $f:\\ensuremath{\\mathcal{P}}(V)\\to\\ensuremath{\\mathbb{R}}$ if and only if $F$ is comonotonic additive.\n\\end{pro}\n\nRecall the following known results:\n\n\\begin{theorem}[Lov\\'asz]\\label{thm:Lovasz}\nThe following conditions are equivalent: (1) $f$ is submodular; (2) $f^L$ is convex; (3) $f^L$ is submodular.\n\\end{theorem}\n\n\\begin{remark}\nThe fact that $f$ is submodular if and only if  $f^L$ is submodular is provided by Propositions 7.38 and 7.39 in \\cite{Murota03book}. We shall give a detailed proof for a generalized version of Theorem \\ref{thm:Lovasz} (see Theorem \\ref{thm:submodular-L-equivalent}).   \n\\end{remark}\n\n\n\\begin{theorem}\nMurota]\\label{thm:Chateauneuf-Cornet}\n$F:\\ensuremath{\\mathbb{R}}^n\\to\\ensuremath{\\mathbb{R}}$ is the Lov\\'asz extension $F=f^L$ of some submodular $f:\\ensuremath{\\mathcal{P}}(V)\\to\\ensuremath{\\mathbb{R}}$ if and only if $F$ is positively one-homogeneous, submodular and $F(\\vec x+t\\vec 1)=F(\\vec x)+tF(\\vec 1)$.\n\\end{theorem}\n\n\\begin{remark}\nTheorem \\ref{thm:Chateauneuf-Cornet} was   originally  proved by establishing a  one-to-one correspondence between positively  \nhomogeneous L-convex functions and submodular  functions  (see \n  Theorem 7.40 in Murota's book  \\cite{Murota03book}). An alternative proof is\n  given in   \\cite{CC18}.\n\\end{remark}\n\n\nWe shall  establish these results for  the disjoint-pair version and the $k$-way version of the Lov\\'asz extension.\n\n\\subsection{Disjoint-pair and $k$-way Lov\\'asz extensions}\n\nSince it is natural to set $f(\\varnothing,\\varnothing)=0$, one may write \\eqref{eq:disjoint-pair-Lovasz-def-integral} as\n\\begin{equation}\\label{eq:disjoint-pair-Lovasz-def-integral2}\nf^{L}(\\vec x)=\\int_0^{\\infty} f(V_+^t(\\vec x),V_-^t(\\vec x))dt,\n\\end{equation}\n\\begin{equation}\\label{eq:disjoint-pair-Lovasz-def}\nf^{L}(\\vec x)=\\sum_{i=0}^{n-1}(|x_{\\sigma(i+1)}|-|x_{\\sigma(i)}|)f(V_{\\sigma(i)}^+(\\vec x),V_{\\sigma(i)}^-(\\vec x)),\n\\end{equation}\nwhere $\\sigma:V\\cup\\{0\\}\\to V\\cup\\{0\\}$ is a bijection such that $|x_{\\sigma(1)}|\\le |x_{\\sigma(2)}| \\le \\cdots\\le |x_{\\sigma(n)}|$ and $\\sigma(0)=0$, where $x_0:=0$, and\n$$V_{\\sigma(i)}^\\pm(\\vec x):=\\{j\\in V:\\pm x_j> |x_{\\sigma(i)}|\\},\\;\\;\\;\\; i=0,1,\\cdots,n-1.$$\nWe regard $\\ensuremath{\\mathcal{P}}_2(V)=3^V$ as $\\{-1,0,1\\}^n$ by identifying the disjoint pair $(A,B)$ with the ternary (indicator) vector $\\vec 1_A-\\vec1_B$.\n\nOne may compare the original and the disjoint-pair Lov\\'asz extensions by writing \\eqref{eq:disjoint-pair-Lovasz-def-integral} as\n\\begin{equation}\\label{eq:disjoint-pair-form}\n\\int_{\\min_i |x_i|}^{\\max_i |x_i|} f(V_+^t(\\vec x),V_-^t(\\vec x))dt+\\min_i |x_i| f(V_+,V_-),\n\\end{equation}\nwhere $V_\\pm=\\{i\\in V:\\pm x_i>0\\}$. Note that \\eqref{eq:disjoint-pair-form} is very similar to \\eqref{eq:Lovaintegral}.\n\n\\begin{defn}\nGiven $V_i=\\{1,\\cdots,n_i\\}$, $i=1,\\cdots,k$, and a function $f:\\mathcal{P}(V_1)\\times \\cdots\\times \\mathcal{P}(V_k)\\to \\ensuremath{\\mathbb{R}}$, the  $k$-way Lov\\'asz extension $f^L: \\ensuremath{\\mathbb{R}}^{V_1}\\times\\cdots\\times \\ensuremath{\\mathbb{R}}^{V_k}\\to \\ensuremath{\\mathbb{R}}$ can be written as\n\\begin{align*}\nf^L(\\vec x^1,\\cdots,\\vec x^k)&=\\int_{\\min \\vec x}^{\\max \\vec x}f(V^t_1(\\vec x^1),\\cdots,V^t_k(\\vec x^k))dt+ f(V_1,\\cdots,V_k)\\min\\vec x\\\\&\n=\\int_{-\\infty}^0(f(V^t_1(\\vec x^1),\\cdots,V^t_k(\\vec x^k))-f(V_1,\\cdots,V_k))dt + \\int_0^{+\\infty}f(V^t_1(\\vec x^1),\\cdots,V^t_k(\\vec x^k)) d t\n\\end{align*}\nwhere $V^t_i(\\vec x^i)=\\{j\\in V_i:x^i_j>t\\}$, $\\min\\vec x=\\min\\limits_{i,j} x^i_j$ and $\\max\\vec x=\\max\\limits_{i,j} x^i_j$.\n\\end{defn}\n\n\\begin{defn}[$k$-way analog for disjoint-pair Lov\\'asz extension]\n\n   Given  $V_i=\\{1,\\cdots,n_i\\}$, $i=1,\\cdots,k$, and a function $f:\\mathcal{P}_2(V_1)\\times \\cdots\\times \\mathcal{P}_2(V_k)\\to \\ensuremath{\\mathbb{R}}$, define $f^L: \\ensuremath{\\mathbb{R}}^{V_1}\\times\\cdots\\times \\ensuremath{\\mathbb{R}}^{V_k}\\to \\ensuremath{\\mathbb{R}}$ by\n   $$f^L(\\vec x^1,\\cdots,\\vec x^k)=\n   \\int_0^{\\|\\vec x\\|_\\infty} f(V_{1,t}^+(\\vec x^1),V_{1,t}^-(\\vec x^1),\\cdots,V_{k,t}^+(\\vec x^k),V_{k,t}^-(\\vec x^k))dt\n   $$\n   where $V_{i,t}^\\pm(\\vec x^i)=\\{j\\in V_i:\\pm x^i_j>t\\}$, $\\|\\vec x\\|_\\infty=\\max\\limits_{i=1,\\cdots,k} \\|\\vec x^i\\|_\\infty$. We can replace $\\|\\vec x\\|_\\infty$ by $+\\infty$ if we set $f(\\varnothing,\\cdots,\\varnothing)=0$.\n\\end{defn}\n\\vspace{0.19cm}\n\nSome basic properties of the multi-way Lov\\'asz extension are shown below.\n\\begin{pro}\\label{pro:multi-way-property}\nFor the multi-way Lov\\'asz extension $f^L(\\vec x)$, we have\n\\begin{enumerate}[(a)]\n\\item  $f^L(\\cdot)$ is positively one-homogeneous, piecewise linear, and Lipschitz continuous.\n\\item $(\\lambda f)^L=\\lambda f^L$, $\\forall\\lambda\\in\\ensuremath{\\mathbb{R}}$.\n\n\\end{enumerate}\n\\end{pro}\n\n\\begin{pro}\\label{pro:setpair-property}\nFor the disjoint-pair Lov\\'asz extension $f^L(\\vec x)$, we have\n\\begin{enumerate}[(a)]\n\\item $f^L$ is Lipschitz continuous, and $|f^L(x)-f^L(y)|\\le 2\\max\\limits_{(A,B)\\in \\ensuremath{\\mathcal{P}}_2(V)}f(A,B) \\|x-y\\|_1$, $\\forall x,y\\in \\mathbb{R}^n$. Also,  $|f^L(x)-f^L(y)|\\le 2\\sum\\limits_{(A,B)\\in \\ensuremath{\\mathcal{P}}_2(V)}f(A,B)  \\|x-y\\|_\\infty$, $\\forall x,y\\in \\mathbb{R}^n$.\n\\item $f^L(-\\vec x)=\\pm f^L(\\vec x)$, $\\forall \\vec x\\in\\ensuremath{\\mathbb{R}}^V$ if and only if $f(A,B)=\\pm f(B,A)$, $\\forall (A,B)\\in \\ensuremath{\\mathcal{P}}_2(V)$.\n\\item\\label{pro:pro-c} $f^L(\\vec x+\\vec y)=f^L(\\vec x)+f^L(\\vec y)$ whenever $V_\\pm(\\vec y)\\subset V_\\pm(\\widetilde{\\vec x})$, where $\\widetilde{\\vec x}$ has components $\\widetilde{ x}_i=\\begin{cases}x_i,&\\text{ if }|x_i|=\\|\\vec x\\|_\\infty,\\\\ 0,&\\text{ otherwise}.\\end{cases}$\n\\end{enumerate}\n\\end{pro}\n\\begin{proof} (a) and (b) are actually known results and their proofs are elementary.\n(c) can be derived from the definition \\eqref{eq:disjoint-pair-Lovasz-def}.\n\\end{proof}\n\n\\begin{defn\n\\label{def:associate-piece}\nTwo vectors $\\vec x$ and $\\vec y$ are said to be absolutely comonotonic \nif $x_iy_i\\ge0$, $\\forall i$, and $(|x_i|-|x_j|)(|y_i|-|y_j|)\\ge0$, $\\forall i,j$\n\\end{defn}\n\n\\begin{pro}\\label{pro:setpair-character}\nA continuous function $F$ is a disjoint-pair Lov\\'asz extension of some function $f:\\ensuremath{\\mathcal{P}}_2(V)\\to\\ensuremath{\\mathbb{R}}$, if and only if\n$F(\\vec x)+F(\\vec y)=F(\\vec x+\\vec  y)$ whenever $\\vec x$ and $\\vec y$ are absolutely comonotonic.\n\\end{pro}\n\n\\begin{proof} By the definition of the disjoint-pair Lov\\'asz extension (see \\eqref{eq:disjoint-pair-Lovasz-def}),\nwe know that $F$ is a disjoint-pair Lov\\'asz extension of some function $f:\\ensuremath{\\mathcal{P}}_2(V)\\to\\ensuremath{\\mathbb{R}}$ if and only if\n$\\lambda F(\\vec x)+(1-\\lambda)F(\\vec y)=F(\\lambda\\vec x+(1-\\lambda)\\vec  y)$ for all absolutely comonotonic\nvectors $\\vec x$ and $\\vec y$, $\\forall \\lambda\\in[0,1]$. Therefore, we only need to prove the sufficiency part.\n\nFor $\\vec x\\in\\ensuremath{\\mathbb{R}}^V$, since $s\\vec x$ and $t\\vec x$ with $s,t\\ge 0$ are absolutely comonotonic,\n $F(s\\vec x)+F(t\\vec x)=F((s+t)\\vec x)$, which yields a Cauchy equation on the half-line. Thus the continuity assumption implies the linearity of $F$ on the ray $\\ensuremath{\\mathbb{R}}^+\\vec x$, which implies the property $F(t\\vec x)=tF(\\vec x)$, $\\forall t\\ge 0$, and hence $\\lambda F(\\vec x)+(1-\\lambda)F(\\vec y)=F(\\lambda\\vec x+(1-\\lambda)\\vec  y)$ for any absolutely comonotonic\n vectors $\\vec x$ and $\\vec y$, $\\forall \\lambda\\in[0,1]$. This completes the proof.\n\\end{proof}\n\nFor relations between the original and the disjoint-pair Lov\\'asz extensions, we further have\n\\begin{pro}\\label{pro:setpair-original}\n For $h:\\ensuremath{\\mathcal{P}}(V)\\to [0,+\\infty)$ with $h(\\varnothing)=0$, and $f:\\ensuremath{\\mathcal{P}}_2(V)\\to [0,+\\infty)$ with $f(\\varnothing,\\varnothing)=0$ \\footnote{In fact, if $h(\\varnothing)\\ne 0$ or $f(\\varnothing,\\varnothing)\\ne0$, one may change the value and it does not affect the related  Lov\\'asz extension.}, we have:\n\\begin{enumerate}[(a)]\n\\item If $f(A,B)=h(A)+h(V\\setminus B)-h(V)$, $\\forall (A,B)\\in \\ensuremath{\\mathcal{P}}_2(V)$, then   $f^L=h^L$.\n\\item  If $f(A,B)=h(A)+h(B)$ and $h(A)=h(V\\setminus A)$, $\\forall (A,B)\\in \\ensuremath{\\mathcal{P}}_2(V)$, then $f^L=h^L$.\n\\item  If $f(A,B)=h(A)$, $\\forall (A,B)\\in \\ensuremath{\\mathcal{P}}_2(V)$, then $f^L(\\vec x)=h^L(\\vec x)$, $\\forall \\vec x\\in[0,\\infty)^V$.\n\\item If $f(A,B)=h(A\\cup B)$, $\\forall (A,B)\\in \\ensuremath{\\mathcal{P}}_2(V)$, then  $f^L(\\vec x)=h^L(\\vec x^++\\vec x^-)$.\n\\item  If $f(A,B)=h(A)\\pm h(B)$, $\\forall (A,B)\\in \\ensuremath{\\mathcal{P}}_2(V)$, then $f^L(\\vec x)=h^L(\\vec x^+)\\pm h^L(\\vec x^-)$.\n\\end{enumerate}\nHere $\\vec x^\\pm:=(\\pm \\vec x)\\vee \\vec0$.\n\\end{pro}\n\n\\begin{proof}\n\\begin{enumerate}[(a)]\n\\item Note that\n\\begin{align*}\nf^L(\\vec x)&=\\int_0^{\\|\\vec x\\|_\\infty}f(V^+_t(\\vec x),V^-_t(\\vec x))dt\n=\\int_0^{\\|\\vec x\\|_\\infty}(h(V_t(\\vec x))+h(V_{-t}(\\vec x))-h(V))dt\\\\\n&=\\int_{-\\|\\vec x\\|_\\infty}^{\\|\\vec x\\|_\\infty}h(V_t(\\vec x))dt- \\|\\vec x\\|_\\infty h(V)\n=\\int_{x_{\\sigma(1)}}^{x_{\\sigma(n)}}h(V_t(\\vec x))dt +x_{\\sigma(1)} h(V) = h^L(\\vec x),\n\\end{align*}\nwhere we use $\\|x\\|_\\infty=\\max\\{-x_{\\sigma(1)},x_{\\sigma(n)}\\}$ and $h(\\varnothing)=0$.\n\\item This is a direct consequence of (a) since  $h(V)=h(\\varnothing)=0$ and $h(B)=h(V\\setminus B)$.\n\\item For any $\\vec x\\in \\ensuremath{\\mathbb{R}}^V$ with $x_i\\ge0$, we note that $f^L(\\vec x)=\\int_0^{\\|\\vec x\\|_\\infty}h(V^+_t(\\vec x))dt=\\int_0^{\\max x_i}h(V_t(\\vec x))dt=\\int_{\\min x_i}^{\\max x_i}h(V_t(\\vec x))dt+\\min x_i h(V)=h^L(\\vec x)$.\n\\item\nSimilar to (c), one can check that $f^L(\\vec x)=h^L(\\vec x^++\\vec x^-)$.\n\\item It is straightforward.\n\\end{enumerate}\n\\end{proof}\n\nIn the sequel, we will not distinguish the original and the disjoint-pair Lov\\'asz extensions, since the reader can infer it\nfrom the domains ($\\ensuremath{\\mathcal{P}}(V)$ or $\\ensuremath{\\mathcal{P}}_2(V)$). Sometime we work on $\\ensuremath{\\mathcal{P}}(V)$ only, and in this situation, the disjoint-pair Lov\\'asz extension acts on the redefined  $f(A,B)=h(A\\cup B)$ as Proposition \\ref{pro:setpair-original} states.\n\nThe next result is useful for the application on graph coloring.\n\n\\begin{pro}\\label{pro:separable-summation}\nFor the  simple $k$-way Lov\\'asz extension of $f:\\ensuremath{\\mathcal{P}}(V_1)\\times\\cdots\\times\\ensuremath{\\mathcal{P}}(V_k)\\to \\ensuremath{\\mathbb{R}}$ with the separable summation form $f(A_1,\\cdots,A_k):=\\sum_{i=1}^kf_i(A_i)$, $\\forall (A_1,\\cdots,A_k)\\in \\ensuremath{\\mathcal{P}}(V)^k$, we have $f^L(\\vec x^1,\\cdots,\\vec x^k)=\\sum_{i=1}^kf_i^L(\\vec x^i)$, $\\forall (\\vec x^1,\\cdots,\\vec x^k)$.\n\nFor $f:\\ensuremath{\\mathcal{P}}_2(V_1)\\times\\cdots \\times\\ensuremath{\\mathcal{P}}_2(V_k)\\to \\ensuremath{\\mathbb{R}}$ with the form $f(A_1,B_1\\cdots,A_k,B_k):=\\sum_{i=1}^kf_i(A_i,B_i)$, $\\forall (A_1,B_1,\\cdots,A_k,B_k)\\in \\ensuremath{\\mathcal{P}}_2(V_1)\\times\\cdots \\times \\ensuremath{\\mathcal{P}}_2(V_k)$, there similarly holds $f^L(\\vec x^1,\\cdots,\\vec x^k)=\\sum_{i=1}^kf_i^L(\\vec x^i)$.\n\\end{pro}\n\n\\subsection{Submodularity and Convexity}\\label{sec:SubmodularityConvexity}\nIn this subsection, we give new analogs of Theorems \\ref{thm:Lovasz} and \\ref{thm:Chateauneuf-Cornet} for the disjoint-pair Lov\\'asz extension and the $k$-way Lov\\'asz extension. The major difference to existing results in the literature is that we work with the restricted or the enlarged domain of a function.\n\nLet's first recall the standard concepts of submodularity:\n\\begin{enumerate}[({S}1)]\n\\item A discrete function $f:\\ensuremath{\\mathcal{A}}\\to \\ensuremath{\\mathbb{R}}$   is submodular if $f(A)+f(B)\\ge f(A\\cup B)+f(A\\cap B)$, $\\forall A,B\\in\\ensuremath{\\mathcal{A}}$, where $\\ensuremath{\\mathcal{A}}\\subset\\mathcal{P}(V)$ is an algebra  (i.e., $\\ensuremath{\\mathcal{A}}$ is  closed under  union and intersection).\n\\item A continuous function $F:\\ensuremath{\\mathbb{R}}^n\\to \\ensuremath{\\mathbb{R}}$ is submodular if\n $F(\\vec x)+F(\\vec y)\\ge F(\\vec x\\vee \\vec y)+F(\\vec x\\wedge \\vec y)$, where $(\\vec x\\vee \\vec y)_i=\\max\\{x_i,y_i\\}$ and $(\\vec x\\wedge \\vec y)_i=\\min\\{x_i,y_i\\}$, $i=1,\\cdots,n$. For a sublattice ${\\mathcal D}\\subset\\ensuremath{\\mathbb{R}}^n$ that  is closed under $\\vee$ and $\\wedge$, one can define  submodularity in the same way.\n\\end{enumerate}\n\n\\begin{notification} The discussion about algebras of sets can be reduced to lattices.\nClassical submodular functions on a sublattice of the Boolean lattice\n$\\{0,1\\}^n$ and their continuous versions on $\\ensuremath{\\mathbb{R}}^n$ are presented in (S1) and\n(S2), respectively. Bisubmodular functions on a graded sub-poset (partially\nordered set)  of   $\\{-1,0,1\\}^n$ are defined in \\eqref{eq:2-submodular} below.\n\\end{notification}\n\n    Now, we\n    recall the concept of bisubmodularity and introduce its continuous version.\n    \\begin{enumerate}\n\\item[(BS1)] A discrete function $f:\\mathcal{P}_2(V)\\to \\ensuremath{\\mathbb{R}}$   is bisubmodular if $\\forall\\,(A,B),(C,D)\\in \\ensuremath{\\mathcal{P}}_2(V)$\n\\begin{equation}\\label{eq:2-submodular}\nf(A,B)+f(C,D)\\ge f((A\\cup C)\\setminus (B\\cup D),(B\\cup D)\\setminus(A\\cup C))+f(A\\cap C,B\\cap D).\n\\end{equation}\nOne can denote $A\\vee B=((A_1\\cup B_1)\\setminus (A_2\\cup B_2),(A_2\\cup B_2)\\setminus (A_1\\cup B_1))$ and $A\\wedge B=(A_1\\cap B_1,A_2\\cap B_2)$, where $A=(A_1,A_2)$, $B=(B_1,B_2)$. For a subset  $\\ensuremath{\\mathcal{A}}\\subset \\mathcal{P}_2(V)$ that is closed under $\\vee$ and $\\wedge$, the bisubmodularity of $f:\\ensuremath{\\mathcal{A}}\\to\\ensuremath{\\mathbb{R}}$ can be expressed as $f(A)+f(B)\\ge f(A\\vee B)+f(A\\wedge B)$, $\\forall A,B\\in\\ensuremath{\\mathcal{A}}$.\n\\end{enumerate}\n\nIf we were to continue the definition of submodularity stated in (S2), we would obtain nothing new.  Hence, the proof of Theorem \\ref{thm:Chateauneuf-Cornet} cannot directly apply to our situation. To overcome this issue, we need to provide a matched definition of bisubmodularity for functions on $\\ensuremath{\\mathbb{R}}^n$, and an appropriate and careful modification of the translation linearity\ncondition.\n\n\\begin{enumerate}\n \\item[(BS2)] A continuous function $F:\\ensuremath{\\mathbb{R}}^n\\to \\ensuremath{\\mathbb{R}}$ is bisubmodular if\n $F(x)+F(y)\\ge F(x\\vee y)+F(x\\wedge y)$, where\n $$(x \\vee y)_i=\\begin{cases}\\max\\{x_i,y_i\\},&\\text{ if } x_i,y_i\\ge0,\\\\\n  \\min\\{x_i,y_i\\},&\\text{ if } x_i,y_i\\le0, \\\\\n  0,&\\text{ if } x_iy_i<0,\\end{cases}\\;\\;\\;\\;\\;\\;\\;(x \\wedge y)_i=\\begin{cases}\\min\\{x_i,y_i\\},&\\text{ if } x_i,y_i\\ge0,\\\\\n  \\max\\{x_i,y_i\\},&\\text{ if } x_i,y_i\\le0, \\\\\n  0,&\\text{ if } x_iy_i<0.\\end{cases}$$\n\\end{enumerate}\n\n\\begin{pro}\\label{pro:bisubmodular-continuous}\nA function $F:\\ensuremath{\\mathbb{R}}^V\\to \\ensuremath{\\mathbb{R}}$ is a disjoint-pair Lov\\'asz extension of a bisubmodular function if and only if $F$ is (continuously) bisubmodular (in the sense of (BS2)) and for any $\\vec x\\in\\ensuremath{\\mathbb{R}}^V ,\\,t\\ge0$,\n{ \\linespread{0.95} \\begin{enumerate}\n\\item[] $F(t\\vec x)=tF(\\vec x)$ (positive homogeneity);\n\\item[] $F(\\vec x+t\\vec 1_{V^+,V^-})\\ge F(\\vec x)+F(t\\vec 1_{V^+,V^-})$ for some\\footnote{This is some kind of `translation linearity' if we adopt the assumption $F(\\vec x+t\\vec 1_{V^+,V^-})= F(\\vec x)+F(t\\vec 1_{V^+,V^-})$. } $V^\\pm\\supset V^\\pm(\\vec x)$ with $V^+\\cup V^-=V$.\n\\end{enumerate} }\n\nHenceforth, $\\vec 1_{A,B}$ is defined as $\\vec 1_A-\\vec 1_B$ for simplicity.\n\\end{pro}\n\nThe proof is a modification of the previous  version on the original Lov\\'asz extension for  submodular functions. \n\n\\begin{proof}\nTake the discrete function $f$ defined as $f(A_1,A_2)=F(\\vec 1_{A_1,A_2})$. One can check the bisubmodularity of $f$ directly. Fix an $\\vec x\\in\\ensuremath{\\mathbb{R}}^n$ and let $\\sigma:V\\cup\\{0\\}\\to V\\cup\\{0\\}$ be a bijection such that $|x_{\\sigma(1)}|\\le |x_{\\sigma(2)}| \\le \\cdots\\le |x_{\\sigma(n)}|$ and $\\sigma(0)=0$, where $x_0:=0$, and\n$$V_{\\sigma(i)}^\\pm=V_{\\sigma(i)}^\\pm(\\vec x):=\\{j\\in V:\\pm x_j> |x_{\\sigma(i)}|\\},\\;\\;\\;\\; i=0,1,\\cdots,n-1.$$\nAlso, we denote $\\vec x_{V_{\\sigma(i)}^+,V_{\\sigma(i)}^-}=\\vec x * \\vec 1_{V_{\\sigma(i)}^+\\cup V_{\\sigma(i)}^-}$ (i.e., the restriction of $\\vec x$ onto $V_{\\sigma(i)}^+\\cup V_{\\sigma(i)}^-$,  with other components $0$), where $\\vec x*\\vec y:=(x_1y_1,\\cdots,x_ny_n)$.\n\nFor simplicity, in the following formulas, we identify $\\sigma(i)$ with $i$ for all $i=0,\\cdots,n$.\n\nIt follows from $|x_{i+1}|\\vec 1_{V_{i}^+,V_{i}^-}\\bigvee \\vec x_{V_{i+1}^+,V_{i+1}^-}=\\vec x_{V_{i}^+,V_{i}^-}$ and\n$$|x_{i+1}|\\vec 1_{V_{i}^+,V_{i}^-}\\bigwedge \\vec x_{V_{i+1}^+,V_{i+1}^-}= |x_{i+1}|\\vec 1_{V_{i+1}^+,V_{i+1}^-}$$ that\n\\begin{align*}\nf^{L}(\\vec x)&=\\sum_{i=0}^{n-1}(|x_{i+1}|-|x_{i}|)f(V_{i}^+,V_{i}^-)\n\\\\&=\\sum_{i=0}^{n-1}|x_{i+1}|\\left(f(V_{i}^+,V_{i}^-)-f(V_{i+1}^+,V_{i+1}^-)\\right)\n\\\\&=\\sum_{i=0}^{n-1}\\left\\{F\\left(|x_{i+1}|\\vec 1_{V_{i}^+,V_{i}^-}\\right)-F\\left(|x_{i+1}|\\vec 1_{V_{i+1}^+,V_{i+1}^-}\\right)\\right\\}\n\\\\&\\ge\\sum_{i=0}^{n-1}\\left\\{F\\left(\\vec x_{V_{i}^+,V_{i}^-}\\right)-F\\left(\\vec x_{V_{i+1}^+,V_{i+1}^-}\\right)\\right\\}=F(\\vec x).\n\\end{align*}\nOn the other hand,\n\\begin{align*}\nf^{L}(\\vec x)&=\\sum_{i=0}^{n-1}(|x_{i+1}|-|x_{i}|)f(V_{i}^+,V_{i}^-)\n=\\sum_{i=0}^{n-1}F\\left((|x_{i+1}|-|x_{i}|)\\vec 1_{V_{i}^+,V_{i}^-}\\right)\n\\\\&=\\sum_{i=0}^{n-2}\\left\\{F((|x_{i+1}|-|x_{i}|)\\vec 1_{V_{i}^+,V_{i}^-})-F((|x_{i+1}|-|x_{i}|)\\vec 1_{V^+,V^-})\\right\\}\n\\\\&\\;\\;\\;\\;\\;+\\left\\{\\sum_{i=0}^{n-2}F((|x_{i+1}|-|x_{i}|)\\vec 1_{V^+,V^-})\\right\\}+F\\left((|x_n|-|x_{n-1}|)\\vec 1_{V^+_{n-1},V^-_{n-1}}\\right)\n\\\\&\\le \\sum_{i=0}^{n-2}\\left\\{F\\left(\\vec x_{V_{i}^+,V_{i}^-}-|x_{i}|\\vec 1_{V^+,V^-}\\right)-F\\left(\\vec x_{V_{i+1}^+,V_{i+1}^-}-|x_{i+1}|\\vec 1_{V^+,V^-}+(|x_{i+1}|-|x_{i}|)\\vec 1_{V^+,V^-}\\right)\\right\\}\n\\\\&\\;\\;\\;\\;\\;+\\left\\{\\sum_{i=0}^{n-2}(|x_{i+1}|-|x_{i}|)F(\\vec 1_{V^+,V^-})\\right\\}+F\\left((|x_n|-|x_{n-1}|)\\vec 1_{V^+_{n-1},V^-_{n-1}}\\right)\n\\\\&\\le \\sum_{i=0}^{n-2}\\left(F(\\vec x_{V_{i}^+,V_{i}^-}-|x_{i}|\\vec 1_{V^+,V^-})-F(\\vec x_{V_{i+1}^+,V_{i+1}^-}-|x_{i+1}|\\vec 1_{V^+,V^-})\\right)+F\\left((|x_n|-|x_{n-1}|)\\vec 1_{V^+_{n-1},V^-_{n-1}}\\right)\n\\\\&=F(\\vec x)\n\\end{align*}\naccording to $(|x_{i+1}|-|x_{i}|)\\vec 1_{V^+,V^-}\\bigwedge (\\vec x_{V_{i}^+,V_{i}^-}-|x_{i}|\\vec 1_{V^+,V^-})= (|x_{i+1}|-|x_{i}|)\\vec 1_{V_{i}^+,V_{i}^-}$ and $$(|x_{i+1}|-|x_{i}|)\\vec 1_{V^+,V^-}\\bigvee (\\vec x_{V_{i}^+,V_{i}^-}-|x_{i}|\\vec 1_{V^+,V^-})=\\vec x_{V_{i+1}^+,V_{i+1}^-}-|x_{i+1}|\\vec 1_{V^+,V^-}+(|x_{i+1}|-|x_{i}|)\\vec 1_{V^+,V^-}$$\nfor $i=0,\\cdots,n-2$, as well as $\\vec x_{V_{n-1}^+,V_{n-1}^-}-|x_{n-1}|\\vec 1_{V^+,V^-}= (|x_n|-|x_{n-1}|)\\vec 1_{V^+_{n-1},V^-_{n-1}}$.  Therefore, we have  $F(\\vec x)=f^L(\\vec x)$. The proof is completed.\n\\end{proof}\n\n\n\\begin{pro}\\label{pro:setpair-character2}\nA continuous function $F$ is a disjoint-pair Lov\\'asz extension of some function $f:\\ensuremath{\\mathcal{P}}_2(V)\\to\\ensuremath{\\mathbb{R}}$ if and only if $F(\\vec x\\vee c\\vec 1_{V^+,V^-})+F(\\vec x-\\vec x\\vee c\\vec 1_{V^+,V^-})=F(\\vec x)$ (or $F(\\vec x\\wedge c\\vec 1_{V^+,V^-})+F(\\vec x-\\vec x\\wedge c\\vec 1_{V^+,V^-})=F(\\vec x)$), for some  $V^\\pm\\supset V^\\pm(\\vec x)$  with  $V^+\\cup V^-=V$, $\\forall c\\ge0$ and $\\vec x\\in\\ensuremath{\\mathbb{R}}^n$.\n\\end{pro}\n\n\\begin{proof}\nWe only need to prove that the condition implies the absolutely comonotonic additivity of $F$,\nand then apply Proposition \\ref{pro:setpair-character}. Note that the property $F(\\vec x\\vee c\\vec 1)+F(\\vec x-\\vec x\\vee c\\vec 1)=F(\\vec x)$ implies a  summation form of $F$ which agrees with the form of the disjoint-pair Lov\\'asz extension. Then using the absolutely comonotonic additivity, we get the desired result.\n\\end{proof}\n\nThe $k$-way submodularity can be naturally defined as (S1) and (S2):\n\\begin{enumerate}\n\\item[({KS})] Given a tuple $V=(V_1,\\cdots,V_k)$ of finite sets and $\\ensuremath{\\mathcal{A}}\\subset\\{(A_1,\\cdots,A_k):A_i\\subset V_i,\\,i=1,\\cdots,k\\}$,\na discrete function $f:\\ensuremath{\\mathcal{A}}\\to \\ensuremath{\\mathbb{R}}$   is $k$-way submodular if $f(A)+f(B)\\ge f(A\\vee B)+f(A\\wedge B)$, $\\forall A,B\\in\\ensuremath{\\mathcal{A}}$, where $\\ensuremath{\\mathcal{A}}$ is a lattice under the corresponding  lattice operations join $\\vee$ and meet $\\wedge$ defined by $A\\vee B=(A_1\\cup B_1,\\cdots, A_k\\cup B_k)$ and $A\\wedge B=(A_1\\cap B_1,\\cdots, A_k\\cap B_k)$.\n\\end{enumerate}\n\n\n\\begin{theorem}\\label{thm:submodular-L-equivalent}\nUnder the assumptions and notations in (KS) above, ${\\mathcal D}_\\ensuremath{\\mathcal{A}}$ is also closed under $\\wedge$ and $\\vee$, with $\\wedge$ and $\\vee$ as in (S2).\n Moreover, the following statements are equivalent:\n \\begin{enumerate}[a)]\n \\item $f$ is $k$-way submodular on $\\ensuremath{\\mathcal{A}}$;\n\\item the $k$-way Lov\\'asz extension $f^L$  is convex on each convex subset of ${\\mathcal D}_\\ensuremath{\\mathcal{A}}$;\n\\item the $k$-way Lov\\'asz extension $f^L$  is submodular on ${\\mathcal D}_\\ensuremath{\\mathcal{A}}$.\n\\end{enumerate}\n\nIf one replaces (KS) and (S2) by (BS1) and (BS2) respectively for the bisubmodular setting, then all the above results hold analogously.\n\\end{theorem}\n\nThe proof is a slight  variation of the  original version by  Lov\\'asz,  and\nis provided for convenience. \n\n\\begin{proof}\nNote that $V^t(\\vec x)\\vee V^t(\\vec y)=V^t(\\vec x\\vee \\vec y)$ and $V^t(\\vec x)\\wedge V^t(\\vec y)=V^t(\\vec x\\wedge \\vec y)$, where $V^t(\\vec x):=(V^t(\\vec x^1),\\cdots,V^t(\\vec x^k))$, $\\forall t\\in\\ensuremath{\\mathbb{R}}$.\nSince $\\vec x\\in{\\mathcal D}_\\ensuremath{\\mathcal{A}}$ if and only if $V^t(\\vec x)\\in \\ensuremath{\\mathcal{A}}$, $\\forall t\\in\\ensuremath{\\mathbb{R}}$, and $\\ensuremath{\\mathcal{A}}$ is a lattice, ${\\mathcal D}_\\ensuremath{\\mathcal{A}}$ must be\na lattice that is closed under the operations $\\wedge$ and $\\vee$.\nAccording to the $k$-way Lov\\'asz extension \\eqref{eq:Lovasz-Form-1},\nwe may write\n$$f^L(\\vec x)=\\int_{-N}^Nf(V^t(\\vec x))dt-Nf(V)\n$$\nwhere $N>\\|\\vec x\\|_\\infty$ is a sufficiently large number\\footnote{Here we set $f(\\varnothing,\\cdots,\\varnothing)=0$}.\nNote that $\\vec 1_A\\vee \\vec 1_B=\\vec 1_{A\\vee B}$ and $\\vec 1_A\\wedge \\vec 1_B=\\vec 1_{A\\wedge B}$.\nCombining the above results, we immediately get\n$$f(A)+f(B)\\ge f(A\\vee B)+ f(A\\wedge B) \\; \\Leftrightarrow \\; f^L(\\vec x)+f^L(\\vec y)\\ge f^L(\\vec x\\vee \\vec y)+f^L(\\vec x\\wedge \\vec y),$$\nwhich proves (a) $\\Leftrightarrow$ (c). Note that for  $\\vec x\\in{\\mathcal D}_\\ensuremath{\\mathcal{A}}$, $f^L(\\vec x)=\\sum\\limits_{A\\in \\ensuremath{\\mathcal{C}}(\\vec x)}\\lambda_Af(A)$ for a unique chain $\\ensuremath{\\mathcal{C}}(\\vec x)\\subset\\ensuremath{\\mathcal{A}}$ that is determined by $\\vec x$ only, and\nthe extension $f^{\\mathrm{convex}}(\\vec x):=\\inf\\limits_{\\{\\lambda_A\\}_{A\\in\\ensuremath{\\mathcal{A}}}\\in\\Lambda(\\vec x)}\\sum\\limits_{A\\in \\ensuremath{\\mathcal{A}}}\\lambda_Af(A)$ is convex on each convex subset of ${\\mathcal D}_\\ensuremath{\\mathcal{A}}$, where $\\Lambda(\\vec x):=\\{\\{\\lambda_A\\}_{A\\in\\ensuremath{\\mathcal{A}}}\\in\\ensuremath{\\mathbb{R}}^\\ensuremath{\\mathcal{A}}:\\sum\\limits_{A\\in\\ensuremath{\\mathcal{A}}}\\lambda_A\\vec 1_A=\\vec x,\\,\\lambda_A\\ge 0\\text{ whenever }A\\ne V\\}$. We only need to prove $f^L(\\vec x)=f^{\\mathrm{convex}}(\\vec x)$ if and only if $f$ is submodular. In fact, along a standard idea proposed in Lov\\'asz's original paper \\cite{Lovasz}, one could prove that for a (strictly) submodular function, the set $\\{A:\\lambda_A^*\\ne0\\}$ must be a chain, where $\\sum\\limits_{A\\in \\ensuremath{\\mathcal{A}}}\\lambda_A^*f(A)=f^{\\mathrm{convex}}(\\vec x)$ achieves the minimum over $\\Lambda(\\vec x)$, and  one can then easily check that it agrees with $f^L$. The converse can be proved\nin a standard way: $f(A)+f(B)=f^L(\\vec 1_A)+f^L(\\vec 1_B)\\ge 2f^L(\\frac{1}{2}(\\vec 1_A+\\vec 1_B))=f(\\vec 1_A+\\vec 1_B)=f(\\vec 1_{A\\vee B}+\\vec 1_{A\\wedge B})=f(\\vec 1_{A\\vee B})+f(\\vec 1_{A\\wedge B})=f(A\\vee B)+f(A\\wedge B)$. Now, the proof is completed.\n\nFor the  bisubmodular case, the above reasoning can be repeated  with minor differences.\n\\end{proof}\n\\label{pro:how-to-be-k-way-Lovasz-submodular}\n\n\n\n\\begin{remark}\nWe show some examples about how both convexity and continuous\nsubmodularity can be satisfied. In fact, it is easy to see that the $l^p$-norm $\\|\\vec x\\|_p$ is both convex and continuously\nsubmodular on $\\ensuremath{\\mathbb{R}}_+^n$, while the $l^1$-norm $\\|\\vec x\\|_1$ is convex and continuously\nsubmodular on the whole  $\\ensuremath{\\mathbb{R}}^n$. Besides, an elementary proof shows that a one-homogeneous continuously\nsubmodular function on $\\ensuremath{\\mathbb{R}}_+^2$ must be convex.\n\\end{remark}\n\n\n\n\n\n\n\\section{Main  results on  optimization and eigenvalue problems}\n\\label{sec:main}\n\nWe uncover the links between  combinatorial optimizations and continuous  programmings as well as eigenvalue\nproblems  in a general setting.\n\\subsection{Combinatorial and continuous optimization}\n\\label{sec:CC-transfer}\n\nAs we have told in the introduction, the application of the Lov\\'asz extension to non-submodular optimization meets with several difficulties, and in this section, we start attacking those. First, we set up some useful results.\n\n\\begin{notification}\\label{notification:fL}\nIn this section, $\\ensuremath{\\mathbb{R}}_{\\ge0}:=[0,\\infty)$ is the set of all non-negative numbers. We use $f^L$ to denote the multi-way Lov\\'asz extension which can be either the original or the disjoint-pair or the $k$-way Lov\\'asz extension.\n\\end{notification}\n\n\\begin{theorem}\\label{thm:tilde-H-f}\n Given set functions $f_1,\\cdots,f_n:\\ensuremath{\\mathcal{A}}\\to \\ensuremath{\\mathbb{R}}_{\\ge0}$, and a zero-homogeneous function $H:\\ensuremath{\\mathbb{R}}^n_{\\ge0}\\setminus\\{\\vec 0\\}\\to\\ensuremath{\\mathbb{R}}\\cup\\{+\\infty\\}$\n with $H(\\vec a+\\vec b)\\ge\\min\\{H(\\vec a),H(\\vec b)\\}$, $\n \\forall \\vec a,\\vec b\\in\\ensuremath{\\mathbb{R}}^n_{\\ge0}\\setminus\\{\\vec 0\\}$, we have\n \\begin{equation}\\label{eq:H-minimum}\n \\min\\limits_{A\\in \\ensuremath{\\mathcal{A}}'}H(f_1(A),\\cdots,f_n(A))=\\inf\\limits_{\\vec x\\in {\\mathcal D}'} H(f^L_1(\\vec x),\\cdots,f^L_n(\\vec x)),\\end{equation}\nwhere $\\ensuremath{\\mathcal{A}}'=\\{A\\in\\ensuremath{\\mathcal{A}}: (f_1(A),\\cdots,f_n(A))\\in\\mathrm{Dom}(H)\\}$, ${\\mathcal D}'=\\{\\vec x\\in{\\mathcal D}_\\ensuremath{\\mathcal{A}}\\cap\\ensuremath{\\mathbb{R}}_{\\ge0}^V:\\,(f^L_1(\\vec x),\\cdots,f^L_n(\\vec x))\\in\\mathrm{Dom}(H)\\}$ and $\\mathrm{Dom}(H)=\\{\\vec a\\in \\ensuremath{\\mathbb{R}}^n _{\\ge0}\\setminus\\{\\vec 0\\}: H(\\vec a)\\in\\ensuremath{\\mathbb{R}}\\}$\n\\end{theorem}\n\n\\begin{proof}\nBy the property of $H$, $\\;\\forall t_i\\ge0\\,,n\\in\\mathbb{N}^+,\\, a_{i,j}\\ge0,i=1,\\cdots,m,\\,j=1,\\cdots,n$,\n\\begin{align*}H\\left(\\sum_{i=1}^m t_i a_{i,1},\\cdots,\\sum_{i=1}^m t_i a_{i,n}\\right)&=H\\left(\\sum_{i=1}^m t_i \\vec a^i\\right)\\ge \\min_{i=1,\\cdots,m} H(t_i\\vec a^i) \\\\ &= \\min_{i=1,\\cdots,m} H(\\vec a^i)\n= \\min_{i=1,\\cdots,m} H(a_{i,1},\\cdots,a_{i,n}).\n\\end{align*}\nTherefore, in  the case of the original Lov\\'asz extension, for any $\\vec x\\in{\\mathcal D}'$,\n\\begin{align}\n&H\\left(f^L_1(\\vec x),\\cdots,f^L_n(\\vec x)\\right)  \\label{eq:psi-H}\n\\\\ =\\, & H\\left(\\int_{\\min \\vec x}^{\\max  \\vec x}f_1(V^t(\\vec x))dt+ f_1(V(\\vec x))\\min \\vec x,\\cdots,\\int_{\\min \\vec x}^{\\max \\vec x}f_n(V^t(\\vec x))dt+ f_n(V(\\vec x))\\min \\vec x\\right)\\notag\n\\\\ =\\, & H\\left(\\sum_{i=1}^m (t_i-t_{i-1}) f_1(V^{t_{i-1}}(\\vec x)),\\cdots,\\sum_{i=1}^m (t_i-t_{i-1}) f_n(V^{t_{i-1}}(\\vec x)) \\right)\\notag\n\\\\ \\ge\\, & \\min_{i=1,\\cdots,m} H\\left(f_1(V^{t_{i-1}}(\\vec x)),\\cdots,f_n(V^{t_{i-1}}(\\vec x)) \\right)\\notag\n\\\\ \\ge\\, &  \\min\\limits_{A\\in \\ensuremath{\\mathcal{A}}'}H(f_1(A),\\cdots,f_n(A))\\label{eq:min-H}\n\\\\ =\\, &  \\min\\limits_{A\\in \\ensuremath{\\mathcal{A}}'}H(f_1^L(\\vec1_A),\\cdots,f_n^L(\\vec1_A))\\notag\n\\\\ \\ge\\, &  \\inf\\limits_{\\vec x\\in {\\mathcal D}'} H(f^L_1(\\vec x),\\cdots,f^L_n(\\vec x)).\\label{eq:inf-H}\n\\end{align}\nCombining \\eqref{eq:psi-H} with \\eqref{eq:min-H}, we have $\\inf\\limits_{\\vec x\\in {\\mathcal D}'} H(f^L_1(\\vec x),\\cdots,f^L_n(\\vec x))\\ge \\min\\limits_{A\\in \\ensuremath{\\mathcal{A}}'}H(f_1(A),\\cdots,f_n(A))$, and then together with \\eqref{eq:min-H} and \\eqref{eq:inf-H}, we get the reverse inequality. Hence, \\eqref{eq:H-minimum} is proved for the original Lov\\'asz extension $f^L$. For multi-way settings, the proof is similar.\n\\end{proof}\n\n\\begin{remark}\\label{remark:thm-H} Duality: If one replaces $H(\\vec a+\\vec b)\\ge\\min\\{H(\\vec a),H(\\vec b)\\}$ by $H(\\vec a+\\vec b)\\le\\max\\{H(\\vec a),H(\\vec b)\\}$, then\n   \\begin{equation}\\label{eq:H-maximum}\n   \\max\\limits_{A\\in \\ensuremath{\\mathcal{A}}'}H(f_1(A),\\cdots,f_n(A))=\\sup\\limits_{\\vec x\\in {\\mathcal D}'} H(f^L_1(\\vec x),\\cdots,f^L_n(\\vec x)).\\end{equation}\n   The proof of the  identity \\eqref{eq:H-maximum} is similar to that of \\eqref{eq:H-minimum}, and thus we omit it.\n   \\end{remark}\n   \\begin{remark}\nA function $H:[0,+\\infty)^n\\to \\overline{\\ensuremath{\\mathbb{R}}}$ has the (MIN) property\nif\n$$H\\left(\\sum_{i=1}^m t_i \\vec a^i\\right)\\ge \\min_{i=1,\\cdots,m} H(\\vec a^i),\\;\\forall t_i>0\\,,m\\in\\mathbb{N}^+,\\,\\vec a^i\\in [0,+\\infty)^n.$$\nThe (MAX) property is formulated analogously.\n\nWe can verify that the (MIN) property is equivalent to the\nzero-homogeneity and $H(\\vec x+\\vec y)\\ge\\min\\{H(\\vec x),H(\\vec y)\\}$. A similar correspondence holds for the (MAX) property.\n\\end{remark}\n\n\\begin{remark}Theorem \\ref{thm:tilde-H-f} shows that if $H$ has the (MIN)  or (MAX) property, then the corresponding combinatorial optimization is equivalent to a continuous optimization by means of  \n the multi-way Lov\\'asz extension.\n Here are some examples:\n \n Given $c,c_i\\ge 0$ with $\\sum_i c_i>0$, let $H(f_1,\\cdots,f_n)=\\frac{c_1f_1+\\cdots+c_nf_n-c\\sqrt{f_1^2+\\cdots+f_n^2}}{f_1+\\cdots+f_n}$. Then $H$ satisfies the (MIN)  property, and by Theorem \\ref{thm:tilde-H-f}, we have\n $$\\min\\limits_{A\\in \\ensuremath{\\mathcal{A}}'}\\frac{\\sum_i c_if_i(A)-c\\sqrt{\\sum_if_i^2(A)}}{\\sum_i f_i(A)}=\\inf\\limits_{\\psi\\in {\\mathcal D}'}\\frac{\\sum_i c_if_i^L(\\psi)-c\\sqrt{\\sum_i (f_i^L(\\psi))^2}}{\\sum_i f_i^L(\\psi)}.$$\n\nTaking $H(f_1,\\cdots,f_n)=\\frac{(c_1f_1^p+\\cdots+c_nf_n^p)^{\\frac1p}}{f_1+\\cdots+f_n}$ for some $p> 1$, then $H$ satisfies the (MAX)  property, and by Theorem \\ref{thm:tilde-H-f}, there holds $$\\max\\limits_{A\\in \\ensuremath{\\mathcal{A}}'}\\frac{(\\sum_i c_if_i(A)^p)^{\\frac1p}}{\\sum_i f_i(A)}=\\sup\\limits_{\\psi\\in {\\mathcal D}'}\\frac{(\\sum_i c_if_i^L(\\psi)^p)^{\\frac1p}}{\\sum_i f_i^L(\\psi)}.$$\n\n\n\\begin{proof}[Proof of Theorem \\ref{thm:tilde-fg-equal-PQ}]\nWithout loss of generality, we may assume that $P(f_1,\\cdots,f_n)$ is one-homogeneous and subaddtive, while $Q(f_1,\\cdots,f_n)$ is one-homogeneous and superadditive on $(f_1,\\cdots,f_n)\\in\\ensuremath{\\mathbb{R}}_{\\ge0}^n$. \n\nThen $H(f_1,\\cdots,f_n)=\\frac{P(f_1,\\cdots,f_n)}{Q(f_1,\\cdots,f_n)}$ is zero-homogeneous on $[0,+\\infty)^n$,   and \n$$H(\\vec f+\\vec g)=\\frac{P(\\vec f+\\vec g)}{Q(\\vec f+\\vec g)}\\le \\frac{P(\\vec f)+P(\\vec g)}{Q(\\vec f)+Q(\\vec g)}\\le \\max\\{\\frac{P(\\vec f)}{Q(\\vec f)},\\frac{P(\\vec g)}{Q(\\vec g)}\\}=\\max\\{H(\\vec f),H(\\vec g)\\}$$\nwhere $\\vec f=(f_1,\\cdots,f_n)$ and $\\vec g=(g_1,\\cdots,g_n)$.\n\n\n\nThen the proof is completed by Theorem \\ref{thm:tilde-H-f} \n (and Remark \\ref{remark:thm-H}).\n\\end{proof}\n\n\n\\begin{example}\\label{exam:maxcut1}\nGiven a finite graph $(V,E)$, for $\\{i,j\\}\\in E$, let $f_{\\{i,j\\}}(A)=1$ if $A\\cap\\{i,j\\}=\\{i\\}$ or $\\{j\\}$, and $f_{\\{i,j\\}}(A)=0$ otherwise. Let $g(A)=|A|$ for $A\\subset V$. It is clear that $\\frac{\\left(\\sum_{\\{i,j\\}\\in E}f_{\\{i,j\\}}^p\\right)^{\\frac1p}}{g}$ satisfies the condition of  Theorem \\ref{thm:tilde-fg-equal-PQ}. Thus, we  derive that\n$$\\max\\limits_{A\\ne\\varnothing}\\frac{|\\partial A|^{\\frac1p}}{|A|}=\\max\\limits_{A\\ne\\varnothing}\\frac{(\\sum_{\\{i,j\\}\\in E}f_{\\{i,j\\}}^p(A))^{\\frac1p}}{g(A)} =\\max\\limits_{x\\in\\ensuremath{\\mathbb{R}}_+^V}\\frac{(\\sum_{\\{i,j\\}\\in E}|x_i-x_j|^p)^{\\frac1p}}{\\sum_{i\\in V}x_i}=\\max\\limits_{x\\ne 0}\\frac{(\\sum_{\\{i,j\\}\\in E}|x_i-x_j|^p)^{\\frac1p}}{\\sum_{i\\in V}|x_i|}.$$\nSimilarly, we have\n$$\\max\\limits_{A\\ne\\varnothing}|\\partial\nA|^{\\frac1p}=\\max\\limits_{x\\in\\ensuremath{\\mathbb{R}}_+^V}\\frac{(\\sum_{\\{i,j\\}\\in\n    E}|x_i-x_j|^p)^{\\frac1p}}{\\max\\limits_{i\\in V}x_i}=\\max\\limits_{x\\ne\n  0}\\frac{(\\sum_{\\{i,j\\}\\in E}|x_i-x_j|^p)^{\\frac1p}}{2\\|\\vec\n  x\\|_\\infty},$$\nwhich gives a  continuous representation of the Max-Cut problem. \nThe last equality holds due to the following reason: letting $F(\\vec x)=(\\sum_{\\{i,j\\}\\in\n    E}|x_i-x_j|^p)^{\\frac1p}$, we can check that  \n$\\max\\limits_{x\\in\\ensuremath{\\mathbb{R}}_+^V}\\frac{F(\\vec x)}{\\max_{i\\in V}x_i}$ achieves its maximum at some characteristic vector $\\vec 1_A$, and then $\\vec 1_A-\\vec 1_{V\\setminus A}$ is a maximizer of $\\frac{F(\\vec x)}{2\\|\\vec\n  x\\|_\\infty}$ on $\\ensuremath{\\mathbb{R}}^V\\setminus\\vec0$. \n  \n \n  Similarly,    $\\max\\limits_{x\\ne\n  0}\\frac{F(\\vec x)}{2\\|\\vec\n  x\\|_\\infty}$ achieves its maximum at $\\vec 1_A-\\vec 1_{V\\setminus A}$ for some $A$, and then $\\vec1_A$ indicates a maximizer of $\\frac{F(\\vec x)}{\\max_{i\\in V}x_i}$ on the first orthant $\\ensuremath{\\mathbb{R}}^V_+$. We need the factor 2  because $F(\\vec 1_A-\\vec 1_{V\\setminus A})=2F(\\vec 1_A)$. \n  \nIt should be noted that the two equivalent continuous reformulations are derived by the original and disjoint-pair Lov\\'asz extensions in the following two ways: $$\\max\\limits_{A\\ne\\varnothing}|\\partial\nA|^{\\frac1p}=\\max\\limits_{A\\in\\ensuremath{\\mathcal{P}}(V)\\setminus\\{\\varnothing\\}}\\frac{(\\sum_{\\{i,j\\}\\in E}f_{\\{i,j\\}}^p(A))^{\\frac1p}}{1} = \\max\\limits_{x\\in\\ensuremath{\\mathbb{R}}_+^V}\\frac{(\\sum_{\\{i,j\\}\\in\n    E}|x_i-x_j|^p)^{\\frac1p}}{\\max_{i\\in V}x_i}$$\n    where we use  $f^L_{\\{i,j\\}}(\\vec x)=|x_i-x_j|$ and $1^L=\\max_{i\\in V}x_i$; \n      $$\\max\\limits_{A\\ne\\varnothing}|\\partial\nA|^{\\frac1p}=\\max\\limits_{(A,B)\\in\\ensuremath{\\mathcal{P}}_2(V)\\setminus\\{ (\\varnothing,\\varnothing)\\}}\\frac{\\left(\\sum_{\\{i,j\\}\\in E}(f_{\\{i,j\\}}(A)+f_{\\{i,j\\}}(B))^p\\right)^{\\frac1p}}{2} =\\max\\limits_{x\\ne\n  0}\\frac{(\\sum_{\\{i,j\\}\\in E}|x_i-x_j|^p)^{\\frac1p}}{2\\|\\vec\n  x\\|_\\infty}$$\n  where we use the fact that the disjoint-pair Lov\\'asz extension of $(A,B)\\mapsto f_{\\{i,j\\}}(A)+f_{\\{i,j\\}}(B)$ is $|x_i-x_j|$ and the disjoint-pair Lov\\'asz extension of $(A,B)\\mapsto 1$ is $\\|\\vec x\\|_\\infty$.\n\\end{example}\n\n\n\\begin{example}\\label{exam:maxcut2}\nThere are many other equalities that can be obtained by  Theorem \\ref{thm:tilde-fg-equal-PQ}, such as:\n$$\\min\\limits_{A\\ne\\varnothing,V}\\frac{|\\partial A|}{|A|^{\\frac1p}}=\\min\\limits_{x\\in\\ensuremath{\\mathbb{R}}^V:\\,\\min x=0}\\frac{\\sum_{\\{i,j\\}\\in E}|x_i-x_j|}{(\\sum_{i\\in V}x_i^p)^{\\frac1p}}$$\nand \n$$\\max\\limits_{(A,B)\\in\\ensuremath{\\mathcal{P}}_2(V)\\setminus\\{(\\varnothing,\\varnothing)\\}}\\frac{2|E(A,B)|^{\\frac1p}}{\\vol(A\\cup B)}= \\max\\limits_{x\\ne 0}\\frac{(\\sum_{\\{i,j\\}\\in E}(|x_i|+|x_j|-|x_i+x_j|)^p)^{\\frac1p}}{\\sum_{i\\in V}\\deg_i|x_i|}$$\nwhenever $p\\ge 1$. Here, $\\vol A =\\sum_{i\\in A} \\deg_i$.\\\\ \nThe last equality shows a variant of the dual Cheeger constant. A slight modification gives \n$$\\max\\limits_{A\\in\\ensuremath{\\mathcal{P}}(V)}2|\\partial A|^{\\frac1p}=\\max\\limits_{(A,B)\\in\\ensuremath{\\mathcal{P}}_2(V)}2|E(A,B)|^{\\frac1p}=\\max\\limits_{x\\ne 0}\\frac{\\left(\\sum_{\\{i,j\\}\\in E}(|x_i|+|x_j|-|x_i+x_j|)^p\\right)^{\\frac1p}}{\\|\\vec x\\|_\\infty}$$\nshowing a new continuous formulation of the Maxcut problem.\n\\end{example}\n\nTaking $n=2$ and $H(f_1,f_2)=\\frac{f_1}{f_2}$ in Theorem \\ref{thm:tilde-H-f}, then such an $H$ satisfies both (MIN) and (MAX) properties. So, we get\n$$\\min\\limits_{A\\in \\ensuremath{\\mathcal{A}}'}\\frac{f_1(A)}{f_2(A)}=\\inf\\limits_{\\psi\\in {\\mathcal D}'}\\frac{f_1^L(\\psi)}{f_2^L(\\psi)},\\;\\;\\;\\text{ and } \\;\\; \\max\\limits_{A\\in \\ensuremath{\\mathcal{A}}'}\\frac{f_1(A)}{f_2(A)}=\\sup\\limits_{\\psi\\in {\\mathcal D}'}\\frac{f_1^L(\\psi)}{f_2^L(\\psi)}.$$\nIn fact, we can get more:\n\\end{remark}\n\n\n\\begin{pro}\\label{pro:fraction-f\/g}\n Given two  functions $f,g:\\ensuremath{\\mathcal{A}}\\to [0,+\\infty)$, let $\\tilde{f},\\tilde{g}:{\\mathcal D}_\\ensuremath{\\mathcal{A}}\\to \\ensuremath{\\mathbb{R}}$ satisfy $\\tilde{f}\\ge f^L$, $\\tilde{g}\\le g^L$, $\\tilde{f}(\\vec1_{A})=f(A)$ and $\\tilde{g}(\\vec1_{A})=g(A)$. Then\n$$\\min\\limits_{A\\in \\ensuremath{\\mathcal{A}}\\cap\\ensuremath{\\mathrm{supp}}(g)}\\frac{f(A)}{g(A)}=\\inf\\limits_{\\psi\\in {\\mathcal D}_\\ensuremath{\\mathcal{A}}\\cap\\ensuremath{\\mathrm{supp}}(\\tilde{g})}\\frac{\\widetilde{f}(\\psi)}{\\widetilde{g}(\\psi)}.$$\n\nIf we replace the condition $\\tilde{f}\\ge f^L$ and $\\tilde{g}\\le g^L$  by $\\tilde{f}\\le f^L$ and $\\tilde{g}\\ge g^L$, then $$\\max\\limits_{A\\in \\ensuremath{\\mathcal{A}}\\cap\\ensuremath{\\mathrm{supp}}(g)}\\frac{f(A)}{g(A)}=\\sup\\limits_{\\psi\\in {\\mathcal D}_\\ensuremath{\\mathcal{A}}\\cap\\ensuremath{\\mathrm{supp}}(\\tilde{g})}\\frac{\\widetilde{f}(\\psi)}{\\widetilde{g}(\\psi)}.$$\n\nFor any $\\alpha\\ne0$,  then $\\tilde{f}=((f^\\alpha)^L)^{\\frac1\\alpha}$ and\n$\\tilde{g}=((g^\\alpha)^L)^{\\frac1\\alpha}$ satisfy the above two\nidentities. \n\\end{pro}\n\n\\begin{proof}\nIt is obvious that\n $$\\inf\\limits_{\\psi\\in {\\mathcal D}_\\ensuremath{\\mathcal{A}}\\cap\\ensuremath{\\mathrm{supp}}(\\tilde{g})}\\frac{\\widetilde{f}(\\psi)}{\\widetilde{g}(\\psi)}\\le\\min\\limits_{A\\in \\ensuremath{\\mathcal{A}}\\cap\\ensuremath{\\mathrm{supp}}(g)} \\frac{\\widetilde{f}(\\vec1_{A})}{\\widetilde{g}(\\vec1_{A})} = \\min\\limits_{A\\in \\ensuremath{\\mathcal{A}}\\cap\\ensuremath{\\mathrm{supp}}(g)}\\frac{f(A)}{g(A)}.$$\n On the other hand, for any $\\psi\\in{\\mathcal D}_\\ensuremath{\\mathcal{A}}\\cap\\ensuremath{\\mathrm{supp}}(\\tilde{g})$, $g^L(\\psi)\\ge \\tilde{g}(\\psi)>0$. Hence, there exists $t\\in (\\min \\widetilde{\\beta}\\psi-1,\\max \\widetilde{\\beta}\\psi+1)$ satisfying $g(V^t(\\psi))>0$. Here $\\widetilde{\\beta}\\psi=\\psi$ (resp., $|\\psi|$), if $f^L$ represents either the original or the $k$-way Lovasz extension of $f$ (resp., either the disjoint-pair or the $k$-way disjoint-pair Lovasz extension). So, the set $W(\\psi):=\\{t\\in\\ensuremath{\\mathbb{R}}: g(V^t(\\psi))>0\\}$ is nonempty. Since $\\{V^t(\\psi):t\\in W(\\psi)\\}$ is finite, there exists $t_0\\in W(\\psi)$ such that $\\frac{f(V^{t_0} (\\psi))}{g(V^{t_0} (\\psi))}=\\min\\limits_{t\\in W(\\psi)}\\frac{f(V^{t} (\\psi))}{g(V^{t} (\\psi))}$. Accordingly, $f(V^{t} (\\psi))\\ge \\frac{f(V^{t_0} (\\psi))}{g(V^{t_0} (\\psi))}g(V^{t} (\\psi))$ for any $t\\in W(\\psi)$, and thus\n  $$f(V^{t} (\\psi))\\ge Cg(V^{t} (\\psi)),\\;\\;\\;\\text{ with }\\;\\;C=\\min\\limits_{t\\in W(\\psi)}\\frac{f(V^{t} (\\psi))}{g(V^{t} (\\psi))}\\ge0,$$\n  holds for any $t\\in\\ensuremath{\\mathbb{R}}$ (because $g(V^{t} (\\psi))=0$ for $t\\in\\ensuremath{\\mathbb{R}}\\setminus W(\\psi)$ which means that the above inequality automatically holds).\n  Consequently,\n \\begin{align*}&\\tilde{f}(\\psi)\\ge f^L(\\psi)\n\\\\ =\\,& \\int_{\\min \\widetilde{\\beta}\\psi}^{\\max  \\widetilde{\\beta}\\psi}f(V^t(\\psi))dt+ f(V(\\psi))\\min \\widetilde{\\beta}\\psi\n\\\\ \\ge\\,& C \\int_{\\min \\widetilde{\\beta}\\psi}^{\\max  \\widetilde{\\beta}\\psi}g(V^t(\\psi))dt+ g(V(\\psi))\\min \\widetilde{\\beta}\\psi.\n\\\\ =\\,&C g^L(\\psi)\\ge C\\tilde{g}(\\psi).\n \\end{align*}\n It follows that\n $$\\frac{\\widetilde{f}(\\psi)}{\\widetilde{g}(\\psi)}\\ge C\\ge \\min\\limits_{A\\in \\ensuremath{\\mathcal{A}}\\cap\\ensuremath{\\mathrm{supp}}(g)}\\frac{f(A)}{g(A)}$$\n and thus the proof is completed. The dual case is similar.\n\nFor $\\alpha>0$, we can simply suppose $\\ensuremath{\\mathrm{supp}}(g)=\\ensuremath{\\mathcal{A}}$. Then \n \\begin{align*}\n \\min\\limits_{A\\in \\ensuremath{\\mathcal{A}}}\\frac{f(A)}{g(A)}=\\min\\limits_{A\\in \\ensuremath{\\mathcal{A}}}\\frac{ (f^\\alpha)^{\\frac1\\alpha}(A)}{(g^\\alpha)^{\\frac1\\alpha}(A)}=\\left(\\min\\limits_{A\\in \\ensuremath{\\mathcal{A}}}\\frac{f^\\alpha(A)}{ g^\\alpha(A)}\\right)^{\\frac1\\alpha}\n = \\left(\\inf\\limits_{\\psi\\in {\\mathcal D}_\\ensuremath{\\mathcal{A}}}\\frac{( f^\\alpha)^L(\\psi)}{(g^\\alpha)^L(\\psi)}\\right)^{\\frac1\\alpha}=\\inf\\limits_{\\psi\\in {\\mathcal D}_\\ensuremath{\\mathcal{A}}}\\frac{(( f^\\alpha)^L)^{\\frac1\\alpha}(\\psi)}{(( g^\\alpha)^L)^{\\frac1\\alpha}(\\psi)}.\n \\end{align*}\n \nFor $\\alpha<0$, we may suppose without loss of generality that $g(A)>0$ and $f(A)>0$ for any $A\\in\\ensuremath{\\mathcal{A}}$. Then, in this case,  \n \\begin{align*}\n \\min\\limits_{A\\in \\ensuremath{\\mathcal{A}}}\\frac{f(A)}{g(A)}=\\min\\limits_{A\\in \\ensuremath{\\mathcal{A}}}\\frac{ (f^\\alpha)^{\\frac1\\alpha}(A)}{(g^\\alpha)^{\\frac1\\alpha}(A)}=\\left(\\max\\limits_{A\\in \\ensuremath{\\mathcal{A}}}\\frac{f^\\alpha(A)}{ g^\\alpha(A)}\\right)^{\\frac1\\alpha}\n= \\left(\\sup\\limits_{\\psi\\in {\\mathcal D}_\\ensuremath{\\mathcal{A}}}\\frac{( f^\\alpha)^L(\\psi)}{(g^\\alpha)^L(\\psi)}\\right)^{\\frac1\\alpha}=\\inf\\limits_{\\psi\\in {\\mathcal D}_\\ensuremath{\\mathcal{A}}}\\frac{(( f^\\alpha)^L)^{\\frac1\\alpha}(\\psi)}{(( g^\\alpha)^L)^{\\frac1\\alpha}(\\psi)}.\n \\end{align*}\n \n This completes the proof.\n\\end{proof}\n\nIt is worth noting that in Proposition  \\ref{pro:fraction-f\/g}, $\\ensuremath{\\mathcal{A}}$ can be a family of some set-tuples, and $f^L$ is the  multi-way Lov\\'asz extension of the corresponding  $f$. We point out that we can replace the Lov\\'asz extension $f^L$ by any other extension $f^E$ with the property that $f^E\/g^E$ achieves its minimum and maximum at some $0$-$1$ vector $\\vec 1_A$ for some $A\\in\\ensuremath{\\mathcal{A}}$. \n Similarly, we have:\n\n\\begin{pro}\\label{pro:maxconvex}\nLet $f,g:\\ensuremath{\\mathcal{A}}\\to [0,+\\infty)$ be two set functions and $f:=f_1-f_2$ and $g:=g_1-g_2$ be decompositions of differences of submodular functions.\n\nLet $\\widetilde{f}_2,\\widetilde{g}_1$ be the restriction of  positively one-homogeneous convex functions onto ${\\mathcal D}_\\ensuremath{\\mathcal{A}}$, with $f_1(A)= \\widetilde{f}_1(\\vec1_{A})$ and $g_2(A)= \\widetilde{g}_2(\\vec1_{A})$.\nDefine $\\widetilde{f}=f_1^L-\\widetilde{f}_2$ and $\\widetilde{g}=\\widetilde{g}_1-g_2^L$. Then,\n$$\\min\\limits_{A\\in \\ensuremath{\\mathcal{A}}\\cap\\ensuremath{\\mathrm{supp}}(g)}\\frac{f(A)}{g(A)}=\\min\\limits_{\\vec x\\in {\\mathcal D}_\\ensuremath{\\mathcal{A}}\\cap\\ensuremath{\\mathrm{supp}}(\\widetilde{g})}\\frac{\\widetilde{f}(\\vec x)}{\\widetilde{g}(\\vec x)}.$$\n\\end{pro}\n\n\\begin{remark}\nHirai et al introduce the generalized Lov\\'asz extension of $f:\\mathcal{L}\\to\\overline{\\ensuremath{\\mathbb{R}}}$ on a graded poset $\\mathcal{L}$ (see \\cite{Hirai18,HH17-arxiv}). Since $f^L(\\vec x)=\\sum_i\\lambda_i f(\\vec p_i)$ for $\\vec x=\\sum_i\\lambda_i \\vec p_i$ lying in the orthoscheme complex $K(\\mathcal{L})$, the same results as stated in Theorem \\ref{thm:tilde-H-f} and Proposition \\ref{pro:fraction-f\/g} hold for such a generalized Lov\\'asz extension $f^L$.   Propositions \\ref{pro:fraction-f\/g} and \\ref{pro:maxconvex} are also generalizations of Theorem 3.1 in \\cite{HS11} and Theorem 1 (b) in \\cite{BRSH13}.\n\\end{remark}\n\nAlthough the continuous representations translate the original problems into  equivalently difficult\noptimization problems,\nwe should point out that the continuous reformulations ensure that many fast algorithms in continuous programming  can be applied directly to certain  combinatorial optimization problems.  For example, the fractional form of the equivalent continuous optimizations shown in  Theorem \\ref{thm:tilde-fg-equal} as well as Propositions \\ref{pro:fraction-f\/g} and \\ref{pro:maxconvex}  implies that we can directly adopt the Dinkelbach iteration in Fractional Programming \\cite{SI83} to solve them. In addition, since the equivalent continuous formulation is Lipschitz, we can also adopt the \nstochastic subgradient method  \\cite{Davis19-FoCM} to solve certain discrete optimization problems directly.\n\nTables~\\ref{tab:L-one} and \\ref{tab:L-two} and Propositions \\ref{pro:discrete:one-to-two}, \\ref{pro:discrete:one-to-k} and \\ref{pro:discrete:two-to-k} present a general correspondence between set or set-pair functions and their Lov\\'asz extensions. We shall make use of several of those in Section \\ref{sec:examples-Applications}. Note that the first four lines in Table \\ref{tab:L-one} for the original Lov\\'asz extension, and the first five lines in Table \\ref{tab:L-two} for the disjoint-pair Lov\\'asz extension are known (see \\cite{HS11,CSZ18}).\n\n\\begin{table}\n\\centering\n\\caption{\\small Original Lov\\'asz extension of some objective functions.}\n\\begin{tabular}{|l|l|}\n               \\hline\n               Set function $f(A)=$ & Lov\\'asz extension $f^L(\\vec x)=$  \\\\\n               \\hline\n                 $\\#E(A,V\\setminus A)$ & $\\sum\\limits_{\\{i,j\\}\\in E}|x_i-x_j|$\\\\\n                 \\hline\n               $C$&$C\\max_i x_i$\\\\\n               \\hline\n               $\\vol(A)$ & $\\sum_i \\deg_i x_i$\\\\\n               \\hline\n               $\\min\\{\\vol(A),\\vol(V\\setminus A)\\}$& $\\min\\limits_{t\\in \\mathbb{R}}\\|\\vec x-t \\vec 1\\|_1$\\\\\n               \\hline\n               $\\#A\\cdot\\#(V\\setminus A)$ & $\\sum\\limits_{i,j\\in V}|x_i-x_j|$ \\\\\n               \\hline\n               $\\#V(E(A,V\\setminus A))$ & $\\sum\\limits_{i=1}^n(\\max\\limits_{j\\in N(i)}x_j-\\min\\limits_{j\\in N(i)}x_j)$\n               \\\\\n               \\hline\n            \\end{tabular}\n             \\label{tab:L-one}\n\\end{table}\n\n\\begin{table}\n\\centering\n\\caption{\\small Disjoint-pair Lov\\'asz extension of several objective functions.}\n\\begin{tabular}{|l|l|}\n               \\hline\n               Objective function $f(A,B)=$ & Disjoint-pair Lov\\'asz extension $f^L(\\vec x)=$  \\\\\n               \\hline\n                 $\\#E(A,V\\setminus A)+\\#E(B,V\\setminus B)$& $\\sum\\limits_{\\{i,j\\}\\in E}|x_i-x_j|$\\\\\n                 \\hline\n               $\\#E(A,B)$&$\\frac12\\left(\\sum\\limits_{i\\in V}\\deg_i|x_i|- \\sum\\limits_{\\{i,j\\}\\in E}|x_i+x_j|\\right)$\\\\\n               \\hline\n               $C$&$C\\|\\vec x\\|_\\infty$\\\\\n               \\hline\n               $\\vol(A)+\\vol(B)$&$\\sum\\limits_{i\\in V}\\deg_i|x_i|$\\\\\n               \\hline\n               $\\min\\{\\vol(A),\\vol(V\\setminus A)\\}+\\min\\{\\vol(B),\\vol(V\\setminus B)\\}$&$\\min\\limits_{\\alpha\\in \\mathbb{R}}\\|(x_1,\\cdots,x_n)-\\alpha \\vec 1\\|$\\\\\n               \\hline\n              $ \\# E(A\\cup B,A\\cup B)$ & $\\sum_{i\\sim j}\\min\\{|x_i|,|x_j|\\}$ \\\\\n              \\hline\n               $\\# (A\\cup B)\\cdot\\# E(A\\cup B,A\\cup B)$ & $\\sum_{k\\in V,i\\sim j}\\min\\{|x_k|,|x_i|,|x_j|\\}$\\\\ \\hline\n$\\#(A\\cup B)\\cdot\\#(V\\setminus (A\\cup B))$ & $\\sum_{i>j}||x_i|-|x_j||$ \\\\ \\hline\n            \\end{tabular}\n             \\label{tab:L-two}\n\\end{table}\n\n\n\\begin{pro}\\label{pro:discrete:one-to-two}\nSuppose $f,g:\\ensuremath{\\mathcal{P}}(V)\\to [0,+\\infty)$ are two set functions with $g(A)>0$ for any $A\\in \\ensuremath{\\mathcal{P}}(V)\\setminus\\{\\varnothing\\}$.\nThen $$\\min\\limits_{A\\in \\ensuremath{\\mathcal{P}}(V)\\setminus\\{\\varnothing\\}}\\frac{f(A)}{g(A)}=\\min\\limits_{(A,B)\\in \\ensuremath{\\mathcal{P}}(V)^2\\setminus\\{(\\varnothing,\\varnothing)\\}}\\frac{f(A)+f(B)}{g(A)+g(B)}=\\min\\limits_{(A,B)\\in \\ensuremath{\\mathcal{P}}_2(V)\\setminus\\{(\\varnothing,\\varnothing)\\}}\\frac{f(A)+f(B)}{g(A)+g(B)},$$\nwhere the right identity needs additional assumptions like\n$f(\\varnothing)=g(\\varnothing)=0$\\footnote{This setting is natural, as the\n  Lov\\'asz extension doesn't use the datum on $\\varnothing$.} or  that $f$\n  and $g$ are symmetric.\\footnote{A function $f:\\ensuremath{\\mathcal{P}}(V)\\to\\ensuremath{\\mathbb{R}}$ is {\\sl symmetric} if $f(A)=f(V\\setminus A)$, $\\forall A\\subset V$.}\nReplacing $f(B)$ and $g(B)$ by $f(V\\setminus B)$ and $g(V\\setminus B)$, all the above identities hold without any additional assumption. Clearly, replacing `min' by `max', all statements still hold.\n\\end{pro}\n\n\\begin{pro}\\label{pro:discrete:one-to-k}\nSuppose $f,g:\\ensuremath{\\mathcal{P}}(V)\\to [0,+\\infty)$ are two set functions with $g(A)>0$ for any $A\\in \\ensuremath{\\mathcal{P}}(V)\\setminus\\{\\varnothing\\}$.\nThen $$\\min\\limits_{A\\in \\ensuremath{\\mathcal{P}}(V)}\\frac{f(A)}{g(A)}=\\min\\limits_{(A_1,\\cdots,A_k)\\in \\ensuremath{\\mathcal{P}}(V)^k}\\frac{\\sum_{i=1}^kf(A_i)}{\\sum_{i=1}^kg(A_i)}=\\min\\limits_{(A_1,\\cdots,A_k)\\in \\ensuremath{\\mathcal{P}}(V)^k}\\sqrt[k]{\\frac{\\prod_{i=1}^kf(A_i)}{\\prod_{i=1}^kg(A_i)}}=\\min\\limits_{(A_1,\\cdots,A_k)\\in \\ensuremath{\\mathcal{P}}_k(V)}\\frac{\\sum_{i=1}^kf(A_i)}{\\sum_{i=1}^kg(A_i)},$$\nwhere the last identity needs additional assumptions like $f(\\varnothing)=g(\\varnothing)=0$.\n\\end{pro}\n\n\\begin{pro}\\label{pro:discrete:two-to-k}\nSuppose $f,g:\\ensuremath{\\mathcal{P}}_2(V)\\to [0,+\\infty)$ are two set functions with $g(A,B)>0$ for any $(A,B)\\in \\ensuremath{\\mathcal{P}}_2(V)\\setminus\\{(\\varnothing,\\varnothing)\\}$.\nThen $$\\min\\limits_{A\\in \\ensuremath{\\mathcal{P}}_2(V)}\\frac{f(A,B)}{g(A,B)}=\\min\\limits_{(A_1,B_1,\\cdots,A_k,B_k)\\in \\ensuremath{\\mathcal{P}}_2(V)^k}\\frac{\\sum_{i=1}^kf(A_i,B_i)}{\\sum_{i=1}^kg(A_i,B_i)}=\\min\\limits_{(A_1,B_1,\\cdots,A_k,B_k)\\in \\ensuremath{\\mathcal{P}}_{2k}(V)}\\frac{\\sum_{i=1}^kf(A_i,B_i)}{\\sum_{i=1}^kg(A_i,B_i)},$$\nwhere the last identity needs additional assumptions like $f(\\varnothing,\\varnothing)=g(\\varnothing,\\varnothing)=0$\\footnote{This setting is natural, as the  disjoint-pair Lov\\'asz extension doesn't use the information on $(\\varnothing,\\varnothing)$.}.\n\\end{pro}\n\nTogether with Propositions \\ref{pro:setpair-original} and \\ref{pro:discrete:one-to-two}, one may directly transfer the data from  Table \\ref{tab:L-one} to Table \\ref{tab:L-two}. Similarly, by employing Propositions \\ref{pro:separable-summation}, \\ref{pro:discrete:one-to-k} and \\ref{pro:discrete:two-to-k}, the $k$-way Lov\\'asz extension of some special functions can be transformed to the  original and the disjoint-pair versions.\n\n\\begin{pro}\\label{pro:Lovasz-f-pre}\nFor any $a<b$, and for any $f$, \n$$ \\min\\limits_{x\\in [a,b]^n}f^L(\\vec x)= \\min\\limits_{x\\in \\{a,b\\}^n}f^L(\\vec x)=af(V)+(b-a)\\min\\limits_{A\\subset V}f(A).$$\nClearly, we can replace all `min' by `max'.\n\\end{pro}\n\n\\begin{proof}\nSince $f^L$ is linear on each piece $\\triangle_\\sigma\\cap [a,b]^n:=\\{\\vec x\\in\n\\ensuremath{\\mathbb{R}}^n:a\\le x_i\\le b,x_{\\sigma(1)}\\le \\cdots\\le x_{\\sigma(n)}\\}$, where\n$\\sigma:\\{1,\\cdots,n\\}\\to \\{1,\\cdots,n\\}$ is a permutation, the maximum and\nminimum of $f^L$ on $\\triangle_\\sigma\\cap [a,b]^n$ can be reached at some\nvertices of the simplex  $\\triangle_\\sigma\\cap [a,b]^n$. Note that the\nvertices of $\\triangle_\\sigma\\cap [a,b]^n$ are included in $\\{a,b\\}^n$ (i.e.,\nthe vertices of the hypercube $ [a,b]^n$). Thus, the maximum and minimum of\n$f^L$ on $ [a,b]^n$ can be attained at some points in  $\\{a,b\\}^n$. By the definition of Lov\\'asz extension, it is easy to check that $f^L(b\\vec1_B+a\\vec1_{V\\setminus B})=af(V)+(b-a)f(B)$.  The proof is completed.\n\\end{proof}\n\n\n\n\\subsection{Eigenvalue problems for Lov\\'asz extension}\n\\label{sec:eigenvalue}\n\nIn this section, we give the proof of Theorem \\ref{introthm:eigenvalue} by\nestablishing some  properties on the nonlinear eigenvalue problem of the\nfunction pair $(f^L,g^L)$.\n\\begin{defn}\n $\\lambda \\in \\ensuremath{\\mathbb{R}}$ is called an eigenvalue, and $\\vec x \\in\n\\ensuremath{\\mathbb{R}}^n\\setminus\\{\\vec0\\}$ a corresponding eigenvector of the\nfunction pair $(f^L,g^L)$ if\n$$\\vec0\\in \\nabla f^L (\\vec x)- \\lambda\\nabla  g^L (\\vec x).$$\nWe then also call $(\\lambda, \\vec x)$ an eigenpair. \n    \\end{defn}\n\n\n\n\\begin{pro}\\label{pro:Lovasz-eigen-pre}\nIn the setting of the original Lov\\'asz extension, for any eigenvalue $\\lambda$ of $(f^L,g^L)$, there exists $A\\in\\ensuremath{\\mathcal{P}}(V)\\setminus\\{\\varnothing\\}$  such that $\\lambda=f(A)\/g(A)$ and $\\vec 1_A$ is a corresponding eigenvector.  Indeed, every eigenvalue has an  eigenvector in   $\\{a,b\\}^n$, for given distinct real numbers $a$ and $b$. \nMoreover, we have:\n\\begin{itemize}\n    \\item if $g(V)\\ne0$, then $\\lambda=f(V)\/g(V)$ is the only possible eigenvalue of $(f^L,g^L)$; \n    \\item if $g(V)=0$ and $(f^L,g^L)$ has at least one eigenvalue, then $f(V)=0$ and in this case, $(f^L,g^L)$ may have many distinct eigenvalues.\n\\end{itemize}\n\\end{pro}\n\\begin{proof}\n We need the following  basic statement.\n\n\\textbf{Argument}. For a piecewise linear function $F:\\ensuremath{\\mathbb{R}}^n\\to\\ensuremath{\\mathbb{R}}$ with finite pieces, if $F$ is linear on a convex subset  $\\Omega$, then $\\nabla F(\\vec x)\\subset \\nabla F(\\vec y)$ for any relative interior point $\\vec x$ of $\\Omega$ and any relative boundary point $\\vec y$ in $ \\overline{\\Omega}$. \n\n\\vspace{0.16cm}\n\nSuppose that $(\\lambda,\\vec x)$ is an eigenpair of $(f^L,g^L)$. Since $f^L$ is one-homogeneous and  linear along the direction $\\vec 1$, we can assume without loss of generality that  $\\vec x$ lies in the interior of the simplex $\\triangle$ with vertices $\\vec 1_{A_1},\\cdots,\\vec1_{A_k}$, where $A_1,\\cdots,A_k$ are upper level sets of $\\vec x$.  Applying  the above argument to the piecewise linear functions $f^L$ and $g^L$, we immediately get $\\vec0\\in\\nabla f^L(\\vec x)-\\lambda \\nabla g^L(\\vec x)\\subset \\nabla f^L(\\vec 1_{A_i})-\\lambda \\nabla g^L(\\vec 1_{A_i})$ for any $i=1,\\cdots,k$, meaning that each $\\vec 1_{A_i}$ is an eigenvector of  $(f^L,g^L)$. \n\nMoreover, by the definition of Lov\\'asz extension, we can check that $\\forall \\vec v\\in\\nabla f^L(\\vec x)$, $\\sum_{i\\in V} v_i=f(V)$. Therefore, $\\nabla f^L(\\vec x)\\cdot\\vec1=f(V)$, $\\forall \\vec x$. Hence, for any eigenpair $(\\lambda,\\vec x)$, $f(V)-\\lambda g(V)=(\\nabla f^L(\\vec x)-\\lambda\\nabla g^L(\\vec x) )\\cdot\\vec1=0$, implying that $\\lambda=f(V)\/g(V)$ if $g(V)\\ne 0$; and $f(V)=0$ if $g(V)= 0$.\n\nFor the case of $f(V)=g(V)= 0$, we may take a look at the example\n$f(A)=|\\partial A|$ and $g(A)=\\min\\{\\vol(A),\\vol(V\\setminus A)\\}$ defined on a\nsimple graph $G=(V,E)$. Then the eigenvalue problem of $(f^L,g^L)$ reduces to\nthe problem for the  graph 1-Laplacian. And it is known that the 1-Laplacian may have many different  eigenvalues \\cite{CSZ17}.\n\\end{proof}\n\n\\begin{pro}\\label{pro:Lovasz-eigen}\nIn the setting of the  disjoint-pair  Lov\\'asz extension, for any eigenvalue $\\lambda$ of $(f^L,g^L)$, there exists $(A,B)\\in\\ensuremath{\\mathcal{P}}_2(V)\\setminus\\{(\\varnothing,\\varnothing)\\}$  such that $\\lambda=f(A,B)\/g(A,B)$ and $\\vec 1_A-\\vec1_B$ is a corresponding eigenvector.  If we further assume that $2f(A,B)=f(A,V\\setminus A)+f(V\\setminus B,B)$ and $2g(A,B)=g(A,V\\setminus A)+g(V\\setminus B,B)$ for any $(A,B)\\in \\ensuremath{\\mathcal{P}}_2(V)\\setminus\\{(\\varnothing,\\varnothing)\\}$, then for any eigenvalue $\\lambda$ of $(f^L,g^L)$, there exists $A\\subset V$  such that $\\lambda=f(A,V\\setminus A)\/g(A,V\\setminus A)$ and $\\vec 1_A-\\vec1_{V\\setminus A}$ is a corresponding eigenvector.\n\\end{pro}\n\n\\begin{proof}\n The general result on  the disjoint-pair  Lov\\'asz extension is similar to \n Proposition \\ref{pro:Lovasz-eigen-pre}, and thus we omit the proof. \n \nLet's focus on the special case that $2f(A,B)=f(A,V\\setminus A)+f(V\\setminus B,B)$ and $2g(A,B)=g(A,V\\setminus A)+g(V\\setminus B,B)$  for any $(A,B)\\in \\ensuremath{\\mathcal{P}}_2(V)\\setminus\\{(\\varnothing,\\varnothing)\\}$. We shall prove that in this case, \ntypical eigenvectors can be taken from  $\\{-1,1\\}^n$. \n\n\\vspace{0.16cm}\n\n\\textbf{Claim}. For any $A_1\\subset \\cdots\\subset A_k$ with $(A_1,A_k)\\ne (\\varnothing,V)$, $f^L$ is linear on  $\\mathrm{conv}\\{\\vec1_{A_i}-\\vec1_{V\\setminus A_i}:i=1,\\cdots,k\\}$. \n\n\\textbf{Proof}. The absolute  comonotonicity of the disjoint-pair Lov\\'asz extension implies that $f^L$ is linear on $\\mathrm{conv}\\{\\vec1_{A_i}-\\vec1_{B_i}:i=1,\\cdots,k\\}$ whenever $A_1\\subset \\cdots\\subset A_k$ and $B_1\\subset \\cdots\\subset B_k$ and $A_k\\cap B_k=\\varnothing$. Thus, for any $\\sigma,\\tau:\\{1,\\cdots,k\\}\\to \\{1,\\cdots,k\\}$  with $\\sigma(1)\\le \\cdots\\le \\sigma(k)\\le\\tau(k)\\le\\cdots\\le\\tau(1)$, $f^L$ is linear on $\\mathrm{conv}\\{\\vec1_{A_{\\sigma(i)}}-\\vec1_{V\\setminus A_{\\tau(i)}}:i=1,\\cdots,k\\}$. Note that\n\\begin{align*}\n\\mathrm{conv}\\{\\vec1_{A_i}-\\vec1_{V\\setminus A_i}:i=1,\\cdots,k\\}&=\\bigcup_{\\sigma,\\tau} \\mathrm{conv}\\{\\vec1_{A_{\\sigma(i)}}-\\vec1_{V\\setminus A_{\\tau(i)}}:i=1,\\cdots,k\\} \n\\\\&=\\bigcup_{l=1}^k\\bigcup_{\\sigma(1)=1,\\sigma(k)=\\tau(k)=l,\\tau(1)=k}\\mathrm{conv}\\{\\vec1_{A_{\\sigma(i)}}-\\vec1_{V\\setminus A_{\\tau(i)}}:i=1,\\cdots,k\\}\n\\end{align*}\nwhere in the second line we can  further assume that each $\\mathrm{conv}\\{\\vec1_{A_{\\sigma(i)}}-\\vec1_{V\\setminus A_{\\tau(i)}}:i=1,\\cdots,k\\}$ is a $(k-1)$-dim simplex, and there are exactly $\\sum_{l=1}^k{k-1\\choose l-1}=2^{k-1}$ simplexes of dimension $(k-1)$.  \nSince\n\\begin{align*}\n2f^L(\\vec1_{A_{\\sigma(i)}}-\\vec1_{V\\setminus A_{\\tau(i)}})\n&=2f(A_{\\sigma(i)},V\\setminus A_{\\tau(i)})\n\\\\&=f(A_{\\sigma(i)},V\\setminus A_{\\sigma(i)})+f(A_{\\tau(i)},V\\setminus A_{\\tau(i)})\n\\\\&=\nf^L(\\vec1_{A_{\\sigma(i)}}-\\vec1_{V\\setminus A_{\\sigma(i)}})+f^L(\\vec1_{A_{\\tau(i)}}-\\vec1_{V\\setminus A_{\\tau(i)}})\n\\end{align*}\nand $\\vec1_{A_{\\sigma(i)}}-\\vec1_{V\\setminus A_{\\sigma(i)}}+\\vec1_{A_{\\tau(i)}}-\\vec1_{V\\setminus A_{\\tau(i)}}=2(\\vec1_{A_{\\sigma(i)}}-\\vec1_{V\\setminus A_{\\tau(i)}})$, $f^L$ must be linear on the segment $ \\mathrm{conv}\\{\\vec1_{A_{\\sigma(i)}}-\\vec1_{V\\setminus A_{\\sigma(i)}},\\vec1_{A_{\\tau(i)}}-\\vec1_{V\\setminus A_{\\tau(i)}}\\}$. Therefore, one can check that $f^L$ is linear on the simplex  $\\mathrm{conv}\\{\\vec1_{A_i}-\\vec1_{V\\setminus A_i}:i=1,\\cdots,k\\}$. \n\n\\vspace{0.16cm}\n\nGiven an eigenvalue $\\lambda$, let $\\vec 1_A-\\vec 1_B$ be a corresponding eigenvector for some $(A,B)\\in\\ensuremath{\\mathcal{P}}_2(V)\\setminus\\{(\\varnothing,\\varnothing)\\}$. By the above claim and argument, it can be verified that  both $\\vec1_A-\\vec 1_{V\\setminus A}$ and  $\\vec 1_{V\\setminus B}-\\vec1_B$ are eigenvectors w.r.t. $\\lambda$.  \n\\end{proof}\n\n\\begin{remark}\nIn Proposition \\ref{pro:Lovasz-eigen}, the  condition $2f(A,B)=f(A,V\\setminus A)+f(V\\setminus B,B)$ for any $(A,B)\\in \\ensuremath{\\mathcal{P}}_2(V)$ is natural and easy to satisfy. Below, we provide two examples satisfying the condition. \n\nExample 1. $f(A,B)=\\hat{f}(A)+\\hat{f}(B)$ for some $\\hat{f}:\\ensuremath{\\mathcal{P}}(V)\\to\\ensuremath{\\mathbb{R}}$ with $\\hat{f}(A)=\\hat{f}(V\\setminus A)$ for any $A\\subset V$. \n In this case,  $f^L(\\vec x)=\\hat{f}^L(\\vec x)$. \n\nExample 2. $f(A,B)=1$ whenever $A\\cup B\\ne\\varnothing$. In this case,  $f^L(\\vec x)=\\|\\vec x\\|_\\infty$.  \n\nOne may observe  that in the above examples, $f$ is symmetric, i.e.,\n$f(A,B)=f(B,A)$, but it is not a sufficient condition for Proposition\n\\ref{pro:Lovasz-eigen}. In fact, for symmetric functions  $f$ and $g$ on\n$\\ensuremath{\\mathcal{P}}_2(V)$,  not every  eigenvalue has an eigenvector possessing the form\nof $\\vec 1_A-\\vec1_{V\\setminus A}$. In fact, taking $f(A,B)=\\#E(A,V\\setminus\nA)+\\#E(B,V\\setminus B)$ and $g(A,B)=\\vol(A\\cup B)$, we have $f^L(\\vec\nx)=\\sum_{\\{i,j\\}\\in E}|x_i-x_j|$  and $g^L(\\vec x)=\\sum_{i\\in\n  V}\\deg(i)|x_i|$. Letting $V=\\{1,2,3\\}$ and $E=\\{\\{1,2\\},\\{2,3\\},\\{1,3\\}\\}$,\nit is known in \\cite{Chang16} that $\\vec 1_A-\\vec1_{V\\setminus A}$ cannot be\nan eigenvector w.r.t. the largest eigenvalue of the  1-Laplacian, $\\forall A\\subset V$.\n\\end{remark}\n\n We conclude the following  result, which asserts that every\nvector\nin $\\{-1,1\\}^n$ is an eigenvector of $(f^L,\\|\\cdot\\|_\\infty)$ if $f^L$ is nonnegative,  where $f^L$ indicates the disjoint-pair Lov\\'asz extension of $f$. \n\n\\begin{pro}\\label{pro:set-pair-infty-norm}\nLet $f:\\ensuremath{\\mathcal{P}}_2(V)\\to[0,+\\infty)$ be nonnegative. Then, for any $A\\subset V$,  $\\vec 1_A-\\vec 1_{V\\setminus A}$ is an eigenvector of  $(f^L,\\|\\cdot\\|_\\infty)$.\n\\end{pro}\n\n\\begin{proof}\nBy the definition of the eigenvalue problem of $(f^L,\\|\\cdot\\|_\\infty)$, we only need to prove that $\\nabla f^L(\\vec x)\\cap ((-\\ensuremath{\\mathbb{R}}^n_{\\mathrm{sign}(x)})\\cup \\ensuremath{\\mathbb{R}}^n_{\\mathrm{sign}(x)}) \\ne \\varnothing$ for any $\\vec x\\in\\{-1,1\\}^n$, where $\\ensuremath{\\mathbb{R}}^n_{\\mathrm{sign}(x)}:=\\{\\vec y\\in\\ensuremath{\\mathbb{R}}^n:y_ix_i\\ge 0,\\forall i\\}=\\mathrm{cone}(\\nabla \\|\\vec x\\|_\\infty)$. We first assume that $f$ is positive-definite, i.e., $f(A,B)>0$ whenever $(A,B)\\ne(\\varnothing,\\varnothing)$, and we shall apply the following argument about polar cones to this case. \n\n\\vspace{0.16cm}\n\n\\textbf{Argument}. Let $C$ and $\\Omega$ be two convex cones in $\\ensuremath{\\mathbb{R}}^n$ such that $\\Omega\\cap ((-C)\\cup C)=\\{\\vec0\\}$. Then $\\Omega^*\\cap C^*\\ne\\{\\vec 0\\}$ and $\\Omega^*\\cap (-C^*)\\ne\\{\\vec 0\\}$, where $C^*$ indicates the polar cone of $C$. \n\nProof: Indeed, $\\Omega^*\\cap C^*=(\\Omega\\cup C)^*\\supset (\\Omega+ C)^*$, where $\\Omega+ C$ is the Minkowski summation of $C$ and $\\Omega$. If $\\Omega+ C=\\ensuremath{\\mathbb{R}}^n$, then for any $-\\vec c\\in (-C)\\setminus\\{\\vec0\\}$, there exist $\\vec a\\in\\Omega\\setminus\\{\\vec0\\}$ and $\\vec c'\\in C\\setminus\\{\\vec0\\}$ such that $\\vec a+\\vec c'=-\\vec c$. This implies $\\vec a=-\\vec c'-\\vec c\\in -C\\setminus\\{\\vec0\\}$, which contradicts  the condition that $\\Omega\\cap(-C)=\\{\\vec0\\}$. Therefore, the convex cone $\\Omega+ C$ is not the whole space $\\ensuremath{\\mathbb{R}}^n$, which implies that $(\\Omega+ C)^*\\ne\\{\\vec0\\}$. Consequently, $\\Omega^*\\cap C^*\\ne\\{\\vec 0\\}$ and similarly, $\\Omega^*\\cap (-C^*)\\ne\\{\\vec 0\\}$. The proof is completed.\n\n\\vspace{0.16cm}\n\nSuppose on the contrary, that $\\nabla f^L(\\vec x)\\cap ((-\\ensuremath{\\mathbb{R}}^n_{\\mathrm{sign}(x)})\\cup \\ensuremath{\\mathbb{R}}^n_{\\mathrm{sign}(x)}) =\\varnothing$ for some $\\vec x\\in\\{-1,1\\}^n$. \nFixing such an $\\vec x$, then $\\mathrm{cone}(\\nabla f^L(\\vec x))\\cap ((-\\ensuremath{\\mathbb{R}}^n_{\\mathrm{sign}(x)})\\cup \\ensuremath{\\mathbb{R}}^n_{\\mathrm{sign}(x)}) =\\{\\vec0\\}$, and by the above  argument, we have  $\\mathrm{cone}^*(\\nabla f^L(\\vec x))\\cap\\ensuremath{\\mathbb{R}}^n_{\\mathrm{sign}(x)}=\\mathrm{cone}^*(\\nabla f^L(\\vec x))\\cap(-\\ensuremath{\\mathbb{R}}^n_{\\mathrm{sign}(x)})^* \\ne\\{\\vec0\\}$. \n\nHowever, since $f$ is positive-definite, it is known \nthat  $\\mathrm{cone}^*(\\nabla f^L(\\vec x))\\subset T_x(\\{\\vec y:f^L(\\vec y)\\le f^L(\\vec x)\\})$, meaning that $ T_x(\\{\\vec y:f^L(\\vec y)\\le f^L(\\vec x)\\})\\cap \\ensuremath{\\mathbb{R}}^n_{\\mathrm{sign}(x)}\\ne\\{\\vec0\\}$, where $T_x$ represents the tangent cone at $\\vec x$. Now, suppose $\\vec x=\\vec 1_{A_n}-\\vec 1_{B_n}$ with $A_n\\sqcup B_n=V$. Every  permutation  $\\sigma:\\{1,\\cdots,n\\}\\to \\{1,\\cdots,n\\}$ determines a  sequence $\\{(A_i,B_i):i=1,\\cdots,n\\}\\subset \\ensuremath{\\mathcal{P}}_2(V)\\setminus\\{(\\varnothing,\\varnothing)\\}$ by the iterative construction:   $A_1\\cup B_1=\\{\\sigma(1)\\}$ and  $A_{i+1}\\cup B_{i+1}=A_i\\cup B_i\\cup \\{\\sigma(i+1)\\}$, $i=1,\\cdots,n-1$. \n\nSince $f(A_i,B_i)>0$, $f^L(\\vec x^i)=1$ where $\\vec x^i:=(\\vec1_{A_i}-\\vec1_{B_i})\/f(A_i,B_i)$, $i=1,\\cdots,n$. Also, $T_{x^n}\\{\\vec y:f^L(\\vec y)\\le 1\\}=T_x(\\{\\vec y:f^L(\\vec y)\\le f^L(\\vec x)\\})$.\nWithout loss of generality, we may assume that $\\vec x=\\vec x^n$.\n\nThe definition of $f^L$ yields that $\\mathrm{conv}(\\vec0,\\vec x^1,\\cdots,\\vec x^n)\\subset \\{\\vec y:f^L(\\vec y)\\le 1\\}$.  We denote by $\\triangle_\\sigma=\\mathrm{conv}(\\vec0,\\vec x^1,\\cdots,\\vec x^n)$ since the  construction of $\\vec x^1,\\cdots,\\vec x^n$ depends on the permutation  $\\sigma$. \nFor any $\\vec y=\\sum_{i=1}^nt_i\\vec x^i\\in\\mathrm{conv}(\\vec0,\\vec x^1,\\cdots,\\vec x^n)\\setminus \\{\\vec x\\}$, \n$(\\vec y-\\vec x)_{\\sigma(n)}x_{\\sigma(n)}=-(1-t_n)x_{\\sigma(n)}^2<0$, and thus  $\\vec y-\\vec x\\not\\in \\ensuremath{\\mathbb{R}}^n_{\\mathrm{sign}(x)}$.\nHence, $T_x(\\triangle_\\sigma)\\cap \\ensuremath{\\mathbb{R}}^n_{\\mathrm{sign}(x)}=\\{\\vec0\\}$. It follows from the fact  $T_x(\\{\\vec y:f^L(\\vec y)\\le 1\\})=\\bigcup_{\\sigma}T_x(\\triangle_\\sigma)$ that $T_x(\\{\\vec y:f^L(\\vec y)\\le 1\\})\\cap \\ensuremath{\\mathbb{R}}^n_{\\mathrm{sign}(x)}=\\{\\vec0\\}$. This is a contradiction. \n\n\\vspace{0.16cm}\n\nNow we turn to the general case that $f\\ge 0$. Take a sequence $\\{f_n\\}_{n\\ge 1}$ of  positive-definite functions on $\\ensuremath{\\mathcal{P}}_2(V)$ such that $f_n\\to f$ as $n$ tends to $+\\infty$. Then it can be verified that for any $\\vec v_n\\in \\nabla f_n^L(\\vec x)$, all limit  points of $\\{\\vec v_n\\}_{n\\ge 1}$ belong to $\\nabla f^L(\\vec x)$. Now, there exist $\\vec u_n\\in \\nabla \\|\\vec x\\|_{\\infty}$ and $\\lambda_n=f^L_n(\\vec x)\/\\|\\vec x\\|_{\\infty}>0$ such that $\\lambda_n\\vec u_n\\in \\nabla f^L_n(\\vec x)$. Then for any limit point $\\vec u$ of $\\{\\vec u_n\\}_{n\\ge 1}$, $\\vec u\\in  \\nabla \\|\\vec x\\|_{\\infty}$ and $\\lambda\\vec u\\in \\nabla f^L(\\vec x)$ where $\\lambda=\\lim\\limits_{n\\to+\\infty}\\lambda_n$. Therefore, $(\\lambda,\\vec x)$ is an eigenpair of $(f^L,\\|\\cdot\\|_\\infty)$.\n\nThe proof is completed. \n\\end{proof}\n\nBy Propositions \\ref{pro:Lovasz-eigen} and  \\ref{pro:set-pair-infty-norm}, we have \n\\begin{cor}\nIf $2f(A,B)=f(A,V\\setminus A)+f(V\\setminus B,B)$ for any $(A,B)\\in \\ensuremath{\\mathcal{P}}_2(V)$, then the set of  eigenvalues of $(f^L,\\|\\cdot\\|_\\infty)$ coincides with $\\{f(A,V\\setminus A):A\\subset V\\}$, and every vector in $\\{-1,1\\}^n$ is an eigenvector.\n\\end{cor}\n\n\\begin{remark}\nThe proof of  Proposition  \\ref{pro:set-pair-infty-norm} heavily depends on\nthe property  that $N_v(X)=-T_v(X)$ for any vertex $v$ of the  hypercube  $X:=\\{\\vec x:\\|\\vec x\\|_\\infty\\le1\\}$.  Characterizing  the class of  polytopes satisfying   $N_v=-T_v$ for any vertex $v$  remains an open problem, where $N_v$ is the normal cone at $v$ and $T_v$ is the tangent  cone at $v$. \n\\end{remark}\n\nMotivated by Propositions \\ref{pro:Lovasz-eigen} and  \\ref{pro:set-pair-infty-norm}, we suggest a combinatorial eigenvalue problem for $(f,g)$ as follows:\n\nGiven $A\\subset V$ and a permutation  $\\sigma:\\{1,\\cdots,n\\}\\to \\{1,\\cdots,n\\}$, there exists a unique sequence $\\{(A_i^\\sigma,B_i^\\sigma):i=1,\\cdots,n\\}\\subset \\ensuremath{\\mathcal{P}}_2(V)\\setminus\\{(\\varnothing,\\varnothing)\\}$   satisfying $A_1^\\sigma\\subset \\cdots \\subset A_n^\\sigma=A$, $B_1^\\sigma\\subset \\cdots \\subset B_n^\\sigma=V\\setminus A$, $A_1^\\sigma\\cup B_1^\\sigma=\\{\\sigma(1)\\}$ and  $A_{i+1}^\\sigma\\cup B_{i+1}^\\sigma=A_i^\\sigma\\cup B_i^\\sigma\\cup \\{\\sigma(i+1)\\}$, $i=1,\\cdots,n-1$.\nLet $\\vec u^{A,\\sigma}\\in\\ensuremath{\\mathbb{R}}^n$ be defined by\n$$u^{A,\\sigma}_i=\\begin{cases} f(A_{\\sigma^{-1}(i)}^\\sigma,B_{\\sigma^{-1}(i)}^\\sigma)-f(A_{\\sigma^{-1}(i)-1}^\\sigma,B_{\\sigma^{-1}(i)-1}^\\sigma)\n,&\\text{ if }i\\in A,\\\\\nf(A_{\\sigma^{-1}(i)-1}^\\sigma,B_{\\sigma^{-1}(i)-1}^\\sigma)-f(A_{\\sigma^{-1}(i)}^\\sigma,B_{\\sigma^{-1}(i)}^\\sigma)\n,&\\text{ if }i\\not\\in A.\\end{cases}\n$$\nDenote by $S(f,A)=\\{\\vec u^{A,\\sigma}:\\sigma\\in S_n\\}$ and\n$$\\nabla f(A,B):=\\mathrm{conv}\\left(\\bigcup\\limits_{\\tilde{A}:\\,A\\subset \\tilde{A}\\subset V\\setminus B}S(f,\\tilde{A})\\right),\\;\\;\\forall (A,B)\\in\\ensuremath{\\mathcal{P}}_2(V)$$\nwhere $S_n$ is the permutation group over $\\{1,\\cdots,n\\}$.\n\n\\begin{defn}[Combinatorial eigenvalue problem]\nGiven $f,g:\\ensuremath{\\mathcal{P}}_2(V)\\to \\ensuremath{\\mathbb{R}}$, the combinatorial eigenvalue problem of $(f,g)$ is to find $\\lambda\\in\\ensuremath{\\mathbb{R}}$ and $(A,B)\\in \\ensuremath{\\mathcal{P}}_2(V)\\setminus\\{(\\varnothing,\\varnothing)\\}$ such that $\\nabla f(A,B)\\cap \\lambda \\nabla g(A,B)\\ne\\varnothing$.\n\\end{defn}\n\nSince it can be verified that $\\nabla f^L(\\vec1_A-\\vec1_B)=\\nabla f(A,B)$, Proposition \\ref{pro:Lovasz-eigen} (or Theorem \\ref{introthm:eigenvalue}) implies that the combinatorial eigenvalue problem for $(f,g)$ is equivalent to the nonlinear eigenvalue problem of $(f^L,g^L)$.\n\n{ By Propositions \\ref{pro:Lovasz-eigen-pre} and \\ref{pro:Lovasz-eigen}, for a pair of functions $f$ and $g$ on $\\ensuremath{\\mathcal{P}}(V)$ (resp.,  $\\ensuremath{\\mathcal{P}}_2(V)$), every eigenvalue of the function pair   $(f^L,g^L)$ generated by Lov\\'asz extension has an eigenvector of the form $\\vec1_A$ (resp., $\\vec1_A-\\vec1_B$) for some $A\\in\\ensuremath{\\mathcal{P}}(V)\\setminus\\{\\varnothing\\}$ (resp., $(A,B)\\in\\ensuremath{\\mathcal{P}}_2(V)\\setminus\\{(\\varnothing,\\varnothing)\\}$). We may call such a set $A$ (resp., $(A,B)$) an  eigen-set of $(f,g)$. And, we are interested in the eigen-sets and the corresponding eigenvalues, which  encode the \nkey information about the data structure generated by the function pair $(f,g)$.     }  \n\nNext, we study the second eigenvalue of the function pair $(f^L,g^L)$, which\nis closely related to a combinatorial Cheeger-type constant  of the form \n$$\\ensuremath{\\mathrm{Ch}}(f,g):=\\min\\limits_{A\\in\\mathcal{P}(V)\\setminus\\{\\varnothing,V\\}}\\frac{f(A)}{\\min\\{g(A),g(V\\setminus A)\\}}$$\nwhere $f:\\ensuremath{\\mathcal{P}}(V)\\to\\ensuremath{\\mathbb{R}}$ is symmetric, i.e., $f(A)=f(V\\setminus A)$, $\\forall A$, and $g:\\ensuremath{\\mathcal{P}}(V)\\to\\ensuremath{\\mathbb{R}}_+$ is  submodular and non-decreasing. \n\n\\begin{pro}\\label{eq:Cheeger-identity}\nLet $f_s,g_s:\\ensuremath{\\mathcal{P}}_2(V)\\to\\ensuremath{\\mathbb{R}}$ be defined by $f_s(A,B)=f(A)+f(B)$ and $g_s(A,B)=g(A)+g(B)$. Then \n$$\\ensuremath{\\mathrm{Ch}}(f,g)=\\text{the second eigenvalue of the function pair }(f_s^L,g_s^L)\\;(\\text{or equivalently }(f_s,g_s)).$$\n\\end{pro}\n\nWe need the following  auxiliary proposition.\n\\begin{pro}\\label{pro:original-vs-disjoint-pair}Suppose that $g:\\ensuremath{\\mathcal{P}}(V)\\to\\ensuremath{\\mathbb{R}}_+$ is non-decreasing, i.e., $g(A)\\le g(B)$ whenever $A\\subset B$. \nLet $G:\\ensuremath{\\mathbb{R}}^n\\to\\ensuremath{\\mathbb{R}}$ be the disjoint-pair Lov\\'asz extension of the function $(A,B)\\mapsto g(A)+g(B)$.   Then the  Lov\\'asz extension of the function $A\\mapsto\\min\\{g(A),g(V\\setminus A)\\}$ is $\\min\\limits_{t\\in\\ensuremath{\\mathbb{R}}}G(\\vec x-t\\vec1)$.\n\\end{pro}\n\\begin{proof}\nDenote by $g_{m}(A)=\\min\\{g(A),g(V\\setminus A)\\}$ and $g_s(A,B)=g(A)+g(B)$, where $g_{m}^L$ is the original Lov\\'asz extension of  $g_{m}$, and $g_s^{L}$ is the disjoint-pair Lov\\'asz extension of $g_s$. \nSince $g$ is non-decreasing, $g(V^t(\\vec x))$ must be non-increasing on $t\\in \\ensuremath{\\mathbb{R}}$, i.e., $g(V^{t_1}(\\vec x))\\ge g(V^{t_2}(\\vec x))$ whenever $t_1\\le t_2$. Hence, there exists $t_0\\in\\ensuremath{\\mathbb{R}}$ such that $g(V^t(\\vec x))\\ge g(V\\setminus V^t(\\vec x))$, $\\forall t\\le t_0$; and   $g(V^t(\\vec x))\\le g(V\\setminus V^t(\\vec x))$, $\\forall t\\ge t_0$.\nThen \\begin{align*}\ng_{m}^L(\\vec x)&= \\int_{\\min \\vec x}^{\\max \\vec x}g_{m}(V^t(\\vec x))dt+\\min\\vec x g_{m}(V)\n\\\\&=\\int_{\\min \\vec x}^{t_0}g(V\\setminus V^t(\\vec x))dt+\\int_{t_0}^{\\max\\vec x}g(V^t(\\vec x))dt\n\\\\&=\\int_{\\min (\\vec x-t_0\\vec1)}^0g(V\\setminus V^t(\\vec x-t_0\\vec1))dt+\\int_{0}^{\\max(\\vec x-t_0\\vec1)}g(V^t(\\vec x-t_0\\vec1))dt\n\\\\&=\\int_{-\\|\\vec x-t_0\\vec1\\|_\\infty}^0g(V\\setminus V^t(\\vec x-t_0\\vec1))dt+\\int_{0}^{\\|\\vec x-t_0\\vec1\\|_\\infty}g(V^t(\\vec x-t_0\\vec1))dt\n\\\\&=\\int_{0}^{\\|\\vec x-t_0\\vec1\\|_\\infty}g(V^t(\\vec x-t_0\\vec1))+g(V\\setminus V^{-t}(\\vec x-t_0\\vec1))dt\n\\\\&=g_{s}^{L}(\\vec x-t_0\\vec1)=\\min\\limits_{t\\in\\ensuremath{\\mathbb{R}}}g_s^{L}(\\vec x-t\\vec1).\n\\end{align*}\nThe proof is completed. \n\\end{proof}\n\n\\begin{proof}[Proof of Proposition \\ref{eq:Cheeger-identity}]\nSince $f$ is symmetric, by Proposition \\ref{pro:setpair-original}, $f_s^L(\\vec x)=f^L(\\vec x)=f_m^L(\\vec x)$. \n\nSince $g$ is positive,  submodular and non-decreasing, it is not difficult to check that\n$g_s$ is bisubmodular.  Thus, by the equivalence of submodularity and convexity, $g_s^L$ is a convex function. \nTherefore, we have\n$$\\min\\limits_{\\vec x\\bot\\vec 1}\\frac{f_s^L(\\vec x)}{\\min\\limits_{t\\in\\ensuremath{\\mathbb{R}}}g_s^L(\\vec x-t\\vec 1)}=\\min\\limits_{\\text{nonconstant }\\vec x\\in\\ensuremath{\\mathbb{R}}_+^n}\\frac{f_s^L(\\vec x)}{\\min\\limits_{t\\in\\ensuremath{\\mathbb{R}}}g_s^L(\\vec x-t\\vec 1)}=\\min\\limits_{\\vec x\\in\\ensuremath{\\mathbb{R}}_+^n:\\min\\vec x=0}\\frac{f_m^L(\\vec x)}{g_m^L(\\vec x)}=\\min\\limits_{A\\ne\\varnothing,V}\\frac{f_m(A)}{g_m(A)}=\\ensuremath{\\mathrm{Ch}}(f,g),$$\nwhere the first equality is based on the fact that $\\vec x\\mapsto f_s^L(\\vec x)=f^L(\\vec x)$ and $\\vec x\\mapsto\\min\\limits_{t\\in\\ensuremath{\\mathbb{R}}}g_s^L(\\vec x-t\\vec 1)$ are translation invariant along $\\vec1$, the second equality is derived by Proposition \\ref{pro:original-vs-disjoint-pair}, and the third one follows from Theorem \\ref{thm:tilde-fg-equal}.\n\\end{proof}\nFinally, we prove that for any $A,B\\ne \\varnothing$ with $A\\cap B=\\varnothing$,  $$\\max\\{\\frac{f(A)}{g(A)},\\frac{f(B)}{g(B)}\\}\\ge\\min\\{\\frac{f(A)}{\\min\\{g(A),g(V\\setminus A)\\}},\\frac{f(B)}{\\min\\{g(B),g(V\\setminus B)\\}}\\}.$$\nSuppose the contrary, and keep $f(A)=f(V\\setminus A)$ in mind. Then, we have $g(A)>g(V\\setminus A)$ and $g(B)>g(V\\setminus B)$, implying $g(A)+g(B)>g(V\\setminus A)+g(V\\setminus B)$.\nSince $A\\subset V\\setminus B$ and $g$ is non-decreasing, one has $g(A)\\le g(V\\setminus B)$. Similarly, $g(B)\\le g(V\\setminus A)$, which leads to a contradiction. \n\nCombining all the results and discussions in this section, we complete the proof of Theorem \\ref{introthm:eigenvalue}. \n\n\n\\subsection{A relaxation of a Dinkelbach-type scheme}\n \\label{sec:algo}\n\nWe would like to establish an iteration framework for finding minimum and maximum eigenvalues.  These extremal eigenvalues play significant roles in optimization theory. They can be found via the  so-called Dinkelbach iterative scheme \\cite{D67}. This will provide  a good starting point for an appropriate\niterative algorithm for the resulting fractional programming. Actually, the equivalent continuous optimization has a fractional form, but such kind of fractions have been hardly touched in the field of fractional programming \\cite{SI83}, where optimizing the ratio of a concave function to a convex one is usually considered. For convenience, we shall work in  a normed space $X$ in this subsection.\n\nFor a convex function $F:X\\to \\mathbb{R}$,  its sub-gradient (or sub-derivative) $\\nabla F(\\vec x)$ is defined as the collection of $\\vec u\\in X^*$ satisfying  $F(\\vec y)-F(\\vec x)\\ge \\langle \\vec u,\\vec y-\\vec x\\rangle,\\;\\forall \\vec y\\in X$, where $X^*$ is the dual of $X$ and $\\langle \\vec u,\\vec y-\\vec x\\rangle$ is the action of $\\vec u$ on $\\vec y-\\vec x$.\nThe concept of a sub-gradient has been extended to Lipschitz functions. This is called the  Clarke derivative \\cite{Clarke}:\n $$\\nabla F(\\vec x)=\\left\\{\\vec u\\in X^*\\left|\\limsup_{\\vec y\\to \\vec x, t\\to 0^+}\\frac{F(\\vec y+t\\vec h)-F(\\vec y)}{t}\\ge \\langle \\vec u,\\vec h\\rangle,\\forall \\vec h\\in X\\right.\\right\\}.$$\n And it can even be  generalized to the class of lower semi-continuous functions \\cite{DM94,D10}.\n\n\\begin{theorem}[Global convergence of a Dinkelbach-type  scheme \\cite{D67}] \\label{thm:global convergence}\nLet $S$ be a compact set and let $F,G:S\\to\\mathbb{R}$ be two continuous functions with $G(\\vec x)>0$, $\\forall \\vec x\\in S$. Then the sequence $\\{r^k\\}$ generated by the two-step iterative scheme\n\\begin{numcases}{}\n\\vec x^{k+1}=\\argopti\\limits_{\\vec  x\\in S} \\{F(\\vec x)-r^k G(\\vec x)\\}, \\label{iter0-1}\n\\\\\nr^{k+1}=\\frac{F( \\vec x^{k+1})}{G(\\vec x^{k+1})},\n\\label{iter0}\n\\end{numcases}\n from any initial point $\\vec  x^0\\in S$, converges monotonically to a global optimum of $F(\\cdot)\/G(\\cdot)$,\nwhere `opti' is `min' or `max'.\n\\end{theorem}\n\n\\begin{cor}\nIf $F\/G$ is a zero-homogeneous continuous function, then the iterative scheme \\eqref{iter0-1}\\eqref{iter0} from any initial point $\\vec  x^0$ converges monotonically to a global optimum on the cone spanned by $S$ (i.e., $\\{t\\vec x: t>0, \\vec x\\in S\\}$).\n\\end{cor}\n\nWe note that Theorem \\ref{thm:global convergence} generalizes Theorem 3.1 in \\cite{CSZ15} and Theorem 2 in \\cite{CSZ18}.  Since it is a Dinkelbach-type iterative algorithm in the field of fractional programming, we omit the proof\n\nMany minimization problems in the field of fractional programming possess the form\n$$\n\\min\\,\\frac{\\text{convex }F}{\\text{concave }G},\n$$\nwhich is not necessary for a  convex programming problem. The original Dinkelbach iterative scheme turns the ratio form to the inner problem \\eqref{iter0-1} with the form lik\n$$\n\\min \\; (\\text{convex }F-\\text{concave }G),\n$$\nwhich is indeed a convex programming problem. However, most of our examples are in the form\n$$\n\\min\\frac{\\text{convex }F}{\\text{convex }G},\n$$\ni.e., both the numerator and the denominator of the fractional object function are convex.\nSince the difference of two convex functions may not be convex, the inner\nproblem \\eqref{iter0-1} is no longer a convex optimization problem and hence might be very difficult to solve.\n\n\nIn other practical applications, we may encounter optimization problems of the form\n$$\n\\min\\frac{\\text{convex }F_1 - \\text{convex }F_2}{\\text{convex }G_1- \\text{convex }G_2}.\n$$\nThis is NP-hard in general. Fortunately, we can construct an effective relaxation of \\eqref{iter0-1}.\n\n\nThe starting point of the relaxation step is the following  classical fact:\n\\begin{pro}\\label{pro:difference-two-submodular}\nFor any function $f:\\ensuremath{\\mathcal{A}}\\to \\ensuremath{\\mathbb{R}}$, there are two submodular functions $f_1$ and $f_2$ on $\\ensuremath{\\mathcal{A}}$ such that $f=f_1-f_2$.\n\\end{pro}\n\nAlthough this is  an old result,  for readers' convenience, we present a short\nproof below. \n\n\\begin{proof}\nTaking $g$ to be a strict submodular function and letting $$\\delta=\\min\\limits_{A,A'\\in \\ensuremath{\\mathcal{A}}}\\left(g(A)+g(A')-g(A\\vee A')-g(A\\wedge A')\\right)>0.$$ Set $f_2=Cg$ and $f_1=f+f_2$ for a sufficiently large $C>0$. It is clear that $f_2$ is strict submodular and $f_1$ is submodular. So, $f=f_1-f_2$, which completes the proof.\n\\end{proof}\n\nThanks to Proposition \\ref{pro:difference-two-submodular}, any discrete function can be expressed as the difference of two submodular functions. Since the Lov\\'asz extension of a submodular function is convex, every Lov\\'asz extension function is the difference of two convex functions. \n\nThen, for the fractional programming derived by Theorem \\ref{thm:tilde-fg-equal} (or Propositions  \\ref{pro:fraction-f\/g} and \\ref{pro:maxconvex}),  both the numerator\nand denominator can be rewritten as the differences of two convex functions. This  implies  that a simple and efficient\nalgorithm can be obtained via further relaxing the Dinkelbach iteration by\ntechniques in DC Programming \\cite{HT99}. It should be noted that the following recent works (especially the papers by Hein's group  \\cite{HeinBuhler2010,HS11,TVhyper-13,TMH18}) motivated us to investigate more on this direction:\n\\begin{enumerate}\n\\item  The efficient generalization of the inverse power method proposed by Hein et al  \\cite{HeinBuhler2010} and the extended steepest descent \nmethod by Bresson et al  \\cite{BLUB12}  deal with fractional programming  in the same spirit. For more relevant papers by Hein's group, we refer to \\cite{HS11} for the RatioDCA method, and  \\cite{TMH18} for the generalized RatioDCA technique.\n\\item In \\cite{MaeharaMurota15,MMM18}, the authors address difference convex programming (DC programming)\nfor discrete convex functions, in which  an algorithm and a convergence result similar to Theorem \\ref{th:gsd} are presented. \n\n\\item A simple iterative algorithm based on the continuous reformulation by the disjoint-pair Lov\\'asz extension provides the best cut values for maxcut on a G-set among all existing continuous algorithms \\cite{SZZmaxcut}. \n\\end{enumerate}\n In view of these recent  developments, and in order to enlarge the scope of fractional programming and RatioDCA method, it is helpful to study this\n  aspect by general formulations (see also Remark \\ref{remark:very-general-RatioDCA} for the most general form).   \n  Thus,  \nwe begin to establish a method based on convex programming for solving $\\mathtt{i}\n\\frac{F(\\vec x)}{G(\\vec x)}$ with $F=F_1-F_2$ and $G=G_1-G_2$ being two nonnegative functions, where $F_1,F_2,G_1,G_2$ are four  nonnegative convex functions on $X$. Let $\\{H_{\\vec y}(\\vec x):\\vec y\\in X\\}$ be a family of convex differentiable functions on $X$ with $H_{\\vec y}(\\vec x)\\ge H_{\\vec y}(\\vec y)$, $\\forall \\vec x\\in X$. Consider the following three-step iterative scheme\n\\begin{subequations}\n\\label{iter1}\n\\begin{numcases}{}\n\\vec  x^{k+1}\\in \\argmin\\limits_{\\vec x\\in \\mathbb{B}} \\{F_1(\\vec x)+r^k G_2(\\vec x) -(\\langle \\vec u^k,\\vec x\\rangle+r^k \\langle \\vec v^k,\\vec x\\rangle) + H_{\\vec x^k}(\\vec x)\\}, \\label{eq:twostep_x2}\n\\\\\nr^{k+1}=F( \\vec x^{k+1})\/G( \\vec x^{k+1}),\n\\label{eq:twostep_r2}\n\\\\\n \\vec u^{k+1}\\in\\nabla F_2( \\vec x^{k+1}),\\;\n \\vec v^{k+1}\\in\\nabla G_1( \\vec x^{k+1}),\n\\label{eq:twostep_s2}\n\\end{numcases}\n\\end{subequations}\nwhere $\\mathbb{B}$ is a convex body containing $\\vec 0$ as its inner point.  The following slight modification  \n\\begin{subequations}\n\\label{iter2}\n\\begin{numcases}{}\n\\vec  y^{k+1}\\in \\argmin\\limits_{\\vec x\\in X} \\{F_1(\\vec x)+r^k G_2(\\vec x) -(\\langle \\vec u^k,\\vec x\\rangle+r^k \\langle \\vec v^k,\\vec x\\rangle) + H_{\\vec x^k}(\\vec x)\\}, \\label{eq:2twostep_x2}\n\\\\\nr^{k+1}=F( \\vec y^{k+1})\/G( \\vec y^{k+1}),~~ \\vec x^{k+1}=\\partial \\mathbb{B}\\cap\\{t\\vec y^{k+1}:t\\ge 0\\} \n\\label{eq:2twostep_r2}\n\\\\\n \\vec u^{k+1}\\in\\nabla F_2( \\vec x^{k+1}),\\;\n \\vec v^{k+1}\\in\\nabla G_1( \\vec x^{k+1}),\n\\label{eq:2twostep_s2}\n\\end{numcases}\n\\end{subequations}\n is available when $F\/G$ is zero-homogeneous and \\eqref{eq:2twostep_x2} has a\n solution. In  \\eqref{eq:2twostep_r2}, $\\vec x^{k+1}$ indicates  the\n normalization of $\\vec y^{k+1}$ w.r.t. the convex body $\\mathbb{B}$; in  particular, $\\vec x^{k+1}:=\\vec y^{k+1}\/\\|\\vec y^{k+1}\\|_2$ if we let  $\\mathbb{B}$ be the unit ball.     \n These schemes  mixing the inverse power (IP) method and steepest descent (SD) method can be well used in computing special eigenpairs of $(F,G)$. Note that the inner problem \\eqref{eq:twostep_x2} (resp.  \\eqref{eq:2twostep_x2}) is a convex optimization and thus many algorithms in convex programming are applicable. We should note that the above schemes provide a generalization of the RatioDCA  technique in \\cite{HS11}.\n\n\\begin{theorem}[Local convergence for the  mixed IP-SD scheme]\\label{th:gsd}\nThe sequence $\\{r^k\\}$ generated by the iterative scheme \\eqref{iter1} (resp. \\eqref{iter2}) from any initial point $\\vec x^0\\in \\mathrm{supp}(G)\\cap \\mathbb{B}$ (resp. $\\vec x^0\\in \\mathrm{supp}(G)$) converges monotonically, where $\\mathrm{supp}(G)$ is the support of $G$.\n\nNext we further assume that $X$ is of finite dimension. If one of the following  additional  conditions  holds, then  $\\lim_{k\\to+\\infty} r^k=r^*$ is an eigenvalue of the function pair $(F,G)$ \nin the sense that it fulfills $\\vec0\\in \\nabla F_1(\\vec x^*)-\\nabla F_2(\\vec x^*)-r^*\\left(\\nabla G_1(\\vec x^*)-\\nabla G_2(\\vec x^*)\\right)$,  where $\\vec x^*$ is a  cluster point of $\\{\\vec x^k\\}$.\n\n\\begin{itemize}\n\\item[Case 1.] For the  scheme \\eqref{iter1}, $F_2$ and $G_1$ are one-homogeneous, and  $F_1$ and $G_2$ are $p$-homogeneous with $p\\ge 1$, and $H_{\\vec x}=\\text{const}$, $\\forall \\vec x\\in\\mathbb{B}$. \n\\item[Case 2.1.]  For the  scheme \\eqref{iter2}, $F_1$, $F_2$, $G_1$ and $G_2$ are $p$-homogeneous with  $p>1$.\n\\item[Case 2.2.]  For the  scheme \\eqref{iter2}, $F_1$, $F_2$, $G_1$ and $G_2$\n  are one-homogeneous, and $H_{\\vec x}(\\vec x)$ is a continuous\n  function of $\\vec x\\in \\mathbb{B}$ and  $\\forall M>0$,  $\\exists C>0$ such that   $H_{\\vec x}(\\vec y)>M \\|\\vec y\\|_2$  whenever $\\vec x\\in\\mathbb{B}$ and  $\\|\\vec y\\|_2\\ge C$. \n\\end{itemize}\n\n\\end{theorem}\n\n\nTheorem \\ref{th:gsd} partially generalizes  Theorem 3.4 in  \\cite{CSZ15},\nand it is indeed an extension of both the IP and the SD method \\cite{BLUB12,CP11,M16,HeinBuhler2010}. \n\n\n\\begin{proof}[Proof of Theorem \\ref{th:gsd}]\n\nIt will be helpful to divide this proof into several parts and steps:\n\n\\begin{enumerate}\n\\item[Step 1.] We may assume $G(\\vec x^k)>0$ for any $k$.\nIn fact, the initial point $\\vec x^0$ satisfies $G(\\vec x^0)> 0$. We will show $F(\\vec x^1)=0$ if $G(\\vec x^1)=0$ and thus the iteration should be terminated at $\\vec x^1$. This tells us that we may assume $G(\\vec x^k)>0$ for all $k$ before the termination of the iteration.\n\nNote that\n\\begin{align*}&F_1(\\vec x^1)+r^0 G_2(\\vec x^1) -(\\langle \\vec u^0,\\vec x^1\\rangle+r^0 \\langle \\vec v^0,\\vec x^1\\rangle) + H_{\\vec x^0}(\\vec x^1)\\\\ \\le~& F_1(\\vec x^0)+r^0 G_2(\\vec x^0) -(\\langle \\vec u^0,\\vec x^0\\rangle+r^0 \\langle \\vec v^0,\\vec x^0\\rangle) + H_{\\vec x^0}(\\vec x^0),\n\\end{align*}\nwhich implies\n\\begin{align*}&F_1(\\vec x^1)-F_1(\\vec x^0)+r^0 (G_2(\\vec x^1)-G_2(\\vec x^0)) + H_{\\vec x^0}(\\vec x^1)-H_{\\vec x^0}(\\vec x^0)\\\\ \\le~& \\langle \\vec u^0,\\vec x^1-\\vec x^0\\rangle+r^0 \\langle \\vec v^0,\\vec x^1-\\vec x^0\\rangle\\le F_2(\\vec x^1)-F_2(\\vec x^0) +r^0 (G_1(\\vec x^1)-G_1(\\vec x^0)),\n\\end{align*}\ni.e.,\n\\begin{align}F(\\vec x^1)-F(\\vec x^0)+  H_{\\vec x^0}(\\vec x^1)-H_{\\vec x^0}(\\vec x^0)&\\le r^0 (G(\\vec x^1)-G(\\vec x^0))\\label{eq:important-inequality}\\\\&=-r^0G(\\vec x^0)=-F(\\vec x^0).\\notag\n\\end{align}\nSince the equality holds, we have $F(\\vec x^1)=0$, $H_{\\vec x^0}(\\vec x^1)=H_{\\vec x^0}(\\vec x^0)$, $\\langle \\vec u^0,\\vec x^1-\\vec x^0\\rangle=F_2(\\vec x^1)-F_2(\\vec x^0)$ and $\\langle \\vec v^0,\\vec x^1-\\vec x^0\\rangle=G_1(\\vec x^1)-G_1(\\vec x^0)$. So this step is finished.\n\n\n\n\\item[Step 2.] $\\{r^k\\}_{k=1}^\\infty$ is monotonically decreasing and hence convergent.\n\nSimilar to \\eqref{eq:important-inequality} in Step 1, we can arrive at\n$$F(\\vec x^{k+1})-F(\\vec x^k)+  H_{\\vec x^k}(\\vec x^{k+1})-H_{\\vec x^k}(\\vec x^k) \\le r^k (G(\\vec x^{k+1})-G(\\vec x^k)),$$\nwhich leads to\n$$F(\\vec x^{k+1})\\le r^k G(\\vec x^{k+1}).$$\nSince $G(\\vec x^{k+1})$ is assumed to be positive,\n$r^{k+1}=F(\\vec x^{k+1})\/G(\\vec x^{k+1})\\le r^k$.\n Thus, there exists $r^*\\in [r_{\\min},r^0]$ such that $\\lim\\limits_{k\\to+\\infty}r^k=r^*$.\n\\end{enumerate}\n\nIn the sequel, we assume that the  dimension of $X$ is finite.\n\n\\begin{enumerate}\n\\item[Step 3.]   $\\{\\vec x^k\\}$, $\\{\\vec u^k\\}$ and $\\{\\vec v^k\\}$ are sequentially compact.\n\nIn this setting, $\\mathbb{B}$ must be compact. In consequence, there exist $k_i$, $r^*$, $\\vec x^*$,  $\\vec x^{**}$, $\\vec u^*$ and $\\vec v^*$ such that $\\vec x^{k_i}\\to \\vec x^*$, $\\vec x^{k_i+1}\\to \\vec x^{**}$, $\\vec u^{k_i}\\to \\vec u^*$ and $\\vec v^{k_i}\\to \\vec v^*$, as $i\\to +\\infty$.\n\n Clearly, the statements in Steps 1, 2 and 3 are also available for the  scheme \\eqref{iter2}.\n\n\\item[Step 4.] For the  scheme \\eqref{iter1},   $\\vec x^*$ is a minimum of $F_1(\\vec x)+r^* G_2(\\vec x) -(\\langle \\vec u^*,\\vec x\\rangle+r^* \\langle \\vec v^*,\\vec x\\rangle) + H_{\\vec x^*}(\\vec x)$ on $\\mathbb{B}$.  For the  scheme  \\eqref{iter2}, under the additional assumptions introduced in  Case 2.1 or Case  2.2, $\\vec x^*$ is a minimum of $F_1(\\vec x)+r^* G_2(\\vec x) -(\\langle \\vec u^*,\\vec x\\rangle+r^* \\langle \\vec v^*,\\vec x\\rangle) + H_{\\vec x^*}(\\vec x)$ on $X$. \n\nLet $g(r,\\vec y,\\vec u,\\vec v)=\\min\\limits_{  \\vec x\\in \\mathbb{B}} \\{F_1(\\vec x)+r G_2(\\vec x) -(\\langle \\vec u,\\vec x\\rangle+r \\langle \\vec v,\\vec x\\rangle) + H_{\\vec y}(\\vec x)\\}$. It is standard to verify that $g(r,\\vec y,\\vec u,\\vec v)$ is continuous on $\\mathbb{R}^{1}\\times X\\times X^*\\times X^*$ according to the compactness of $\\mathbb{B}$.\n\nSince $g(r^{k_i},\\vec x^{k_i},\\vec u^{k_i},\\vec v^{k_i})=r^{k_i+1}$, taking $i\\to+\\infty$, one obtains $g(r^*,\\vec x^*,\\vec u^*,\\vec v^*)=r^*$.\n\nBy Step 3, $\\vec x^{**}$ attains the minimum of $F_1(\\vec x)+r^* G_2(\\vec x) -(\\langle \\vec u^*,\\vec x\\rangle+r^* \\langle \\vec v^*,\\vec x\\rangle) + H_{\\vec x^*}(\\vec x)$ on $\\mathbb{B}$. Suppose the contrary, that $\\vec x^*$ is not a minimum of $F_1(\\vec x)+r^* G_2(\\vec x) -(\\langle \\vec u^*,\\vec x\\rangle+r^* \\langle \\vec v^*,\\vec x\\rangle) + H_{\\vec x^*}(\\vec x)$ on $\\mathbb{B}$. Then\n\\begin{align*}&F_1(\\vec x^{**})+r^* G_2(\\vec x^{**}) -(\\langle \\vec u^*,\\vec x^{**}\\rangle+r^* \\langle \\vec v^*,\\vec x^{**}\\rangle) + H_{\\vec x^*}(\\vec x^{**})\\\\ <~& F_1(\\vec x^*)+r^* G_2(\\vec x^*) -(\\langle \\vec u^*,\\vec x^*\\rangle+r^* \\langle \\vec v^*,\\vec x^*\\rangle) + H_{\\vec x^*}(\\vec x^*),\n\\end{align*}\nand thus $F(\\vec x^{**})<r^* G(\\vec x^{**})$ (similar to Step 1), which implies $G(\\vec x^{**})>0$ and $F(\\vec x^{**})\/ G(\\vec x^{**})<r^*$. This is a contradiction. Consequently, $  \\vec x^*$ is a minimizer of $F_1(\\vec x)+r^* G_2(\\vec x) -(\\langle \\vec u^*,\\vec x\\rangle+r^* \\langle \\vec v^*,\\vec x\\rangle) + H_{\\vec x^*}(\\vec x)$ on $\\mathbb{B}$.\n\n On the scheme  \\eqref{iter2}, we refer to   Cases 2.1 and 2.2 below for details.  \n\n\nNext, we will\nverify that $(r^*,\\vec x^*)$ is an eigenpair  under certain  additional conditions. \n\n\\item[Case 1.]  On  the scheme \\eqref{iter1}, $F_2$ and $G_1$ are one-homogeneous, and  $F_1$ and $G_2$ are $p$-homogeneous with $p\\ge 1$, and $H_{\\vec x}=Const$, $\\forall \\vec x\\in\\mathbb{B}$. \n\nSince $H_{\\vec x^k}=Const$,  the above claim shows that $\\vec x^*$ is a minimizer of $F_1(\\vec x)+r^* G_2(\\vec x) -(\\langle \\vec u^*,\\vec x\\rangle +r^* \\langle \\vec v^*,\\vec x\\rangle)$ on $\\mathbb{B}$. Also, since $F_2$ and $G_1$ are one-homogeneous, the Euler identity on homogeneous functions gives $F_1(\\vec x^*)+r^* G_2(\\vec x^*) -(\\langle \\vec u^*,\\vec x^*\\rangle +r^* \\langle \\vec v^*,\\vec x^*\\rangle)=F_1(\\vec x^*)+r^* G_2(\\vec x^*) -(F_2(\\vec x^*)+r^* G_1(\\vec x^*))=F(\\vec x^*)-r^* G(\\vec x^*)=0$. Thus, $F_1(\\vec x)+r^* G_2(\\vec x) -(\\langle \\vec u^*,\\vec x\\rangle+r^* \\langle \\vec v^*,\\vec x\\rangle)\\ge 0$, $\\forall \\vec x\\in \\mathbb{B}$, and the equality holds when $\\vec x=\\vec x^*$.  \n\n\nSince $\\mathbb{B}$ contains $0$ as its inner point, we have $\\{\\alpha \\vec x: \\vec x\\in \\mathbb{B},\\alpha\\ge 1\\}=X$.\n Keeping  $\\alpha\\ge 1$ and $p\\ge 1$ in mind, for any $\\alpha\\ge 1$ and $\\vec x\\in \\mathbb{B}$,\n \\begin{align*}\n &F_1(\\alpha \\vec x)+r^* G_2(\\alpha \\vec x) -(\\langle \\vec u^*,\\alpha \\vec x\\rangle+r^* \\langle \\vec v^*,\\alpha \\vec x\\rangle)\n  \\\\ =~& \\alpha\\left(F_1( \\vec x)+r^* G_2( \\vec x) -(\\langle \\vec u^*, \\vec x\\rangle+r^* \\langle \\vec v^*, \\vec x\\rangle)\\right)+(\\alpha^p-\\alpha)(F_1( \\vec x)+r^* G_2( \\vec x))\n  \\\\ (\\text{by Step 4})~\\ge~& (\\alpha^p-\\alpha)(F_1( \\vec x)+r^* G_2( \\vec x)) \\ge 0.\n\n\n \\end{align*}\n\nConsequently, $\\vec x^*$ is a minimizer of $F_1(\\vec x)+r^* G_2(\\vec x) -(\\langle \\vec u^*,\\vec x\\rangle+r^* \\langle \\vec v^*,\\vec x\\rangle)$ on $X$, and thus\n\\begin{align*}\n\\vec 0 &\\in \\nabla|_{  \\vec x=  \\vec x^*} \\left(F_1(\\vec x)+r^* G_2(\\vec x) -(\\langle \\vec u^*,\\vec x\\rangle+r^* \\langle \\vec v^*,\\vec x\\rangle) \\right)\n\\\\&=\\nabla F_1(\\vec x^*)+r^* \\nabla G_2(\\vec x^*) -\\vec u^*-r^*\\vec v^*\n\\\\& \\subset \\nabla F_1(\\vec x^*)-\\nabla F_2(\\vec x^*)+r^*\\nabla G_2(\\vec x^*)-r^*\\nabla G_1(\\vec x^*).\n\\end{align*}\n\n\\item[Case 2.1.] On the  scheme \\eqref{iter2}, $F_1$, $F_2$, $G_1$ and $G_2$ are $p$-homogeneous with  $p>1$. \n\nDenote by $B:X\\to[0,+\\infty)$ the unique convex and one-homogeneous function satisfying $B(\\partial\\mathbb{B})=1$. Then the normalization of $\\vec x$ in  \\eqref{eq:2twostep_r2} can be expressed as $\\vec x\/B(\\vec x)$.\n\nThe compactness of $\\{\\vec x:B(\\vec x)\\le 1\\}$ and the upper semi-continuity and compactness of subderivatives imply that $\\bigcup_{\\vec x:B(\\vec x)\\le 1}\\nabla F_2(\\vec x)$ and $\\bigcup_{\\vec x:B(\\vec x)\\le 1}\\nabla G_1(\\vec x)$ are bounded sets. So, we have a uniform constant $C_1>0$ such that $\\|\\vec u\\|_2+r^*\\|\\vec v\\|_2\\le C_1$, $\\forall \\vec u\\in \\nabla F_2(\\vec x)$, $ \\vec v\\in \\nabla G_1(\\vec x)$, $\\forall \\vec x\\in\\mathbb{B}$. Let $C_2>0$ be such that  $\\|\\vec x\\|_2\\le C_2B(\\vec x)$, and  $C_3=\\min\\limits_{B(\\vec x)=1} F_1(\\vec x)>0$ (here we assume without loss of generality that $F_1(\\vec x)>0$ whenever $\\vec x\\ne \\vec 0$). For any $\\vec x$ with $B(\\vec x)\\ge \\max\\{2,(2C_1C_2\/C_3)^{\\frac{1}{p-1}}\\}$, and for any $\\vec x^*\\in \\mathbb{B}$, $ \\vec u^*\\in \\nabla F_2(\\vec x^*)$, $ \\vec v^*\\in \\nabla G_1(\\vec x^*)$, \n\\begin{align*}\n&F_1(\\vec x)+r^* G_2(\\vec x) -(\\langle \\vec u^*,\\vec x\\rangle+r^*\\langle \\vec v^*,\\vec x\\rangle) + H_{\\vec x^*}(\\vec x)\n\\\\ =~&   B(\\vec x)^p F_1(\\frac{\\vec x}{B(\\vec x)})+r^*B(\\vec x)^p G_2(\\frac{\\vec x}{B(\\vec x)}) -(\\|\\vec x\\|_2\\langle \\vec u^*,\\frac{\\vec x}{\\|\\vec x\\|_2}\\rangle+r^*\\|\\vec x\\|_2\\langle \\vec v^*,\\frac{\\vec x}{\\|\\vec x\\|_2}\\rangle) + H_{\\vec x^*}(\\vec x)\n\\\\ \\ge~& B(\\vec x)^pF_1(\\frac{\\vec x}{B(\\vec x)})-\\|\\vec x\\|_2(\\|\\vec u^*\\|_2+r^*\\|\\vec v^*\\|_2 ) + H_{\\vec x^*}(\\vec x^*)\n\\\\ \\ge~& B(\\vec x)^pC_3-C_2C_1B(\\vec x) + H_{\\vec x^*}(\\vec x^*)=B(\\vec x)(B(\\vec x)^{p-1}C_3-C_2C_1) + H_{\\vec x^*}(\\vec x^*)\n> H_{\\vec x^*}(\\vec x^*)\\\\ >~& -(p-1)(F_2(\\vec x^*)+r^* G_1(\\vec x^*))+ H_{\\vec x^*}(\\vec x^*)\n\\\\ =~& F_1(\\vec x^*)+r^* G_2(\\vec x^*) -(\\langle \\vec u^*,\\vec x^*\\rangle+r^*\\langle \\vec v^*,\\vec x^*\\rangle) + H_{\\vec x^*}(\\vec x^*)\n\\end{align*}\nwhich means that the  minimizers of $F_1(\\vec x)+r^* G_2(\\vec x) -(\\langle \\vec u^*,\\vec x\\rangle+r^*\\langle \\vec v^*,\\vec x\\rangle) + H_{\\vec x^*}(\\vec x)$ exist and they always lie in the bounded set $\\{\\vec x:B(\\vec x)< \\max\\{2,(2C_1C_2\/C_3)^{\\frac{1}{p-1}}\\}\\}$. Since $B(\\vec x^k)=1$, $\\{\\vec y^k\\}$ must be a bounded sequence.  There exists  $\\{k_i\\}\\subset \\{k\\}$ such that $\\vec x^{k_i} \\to \\vec x^*$, $\\vec y^{k_i+1}\\to \\vec y^{**}$, $\\vec x^{k_i+1} \\to \\vec x^{**}$ for some $\\vec x^*$, $\\vec y^{**}$ and $\\vec x^{**}=\\vec y^{**}\/B(\\vec y^{**})$. Similar to Step 4 and  Case 1,  $\\vec x^*$ is a minimizer of $F_1(\\vec x)+r^* G_2(\\vec x) -(\\langle \\vec u^*,\\vec x\\rangle+r^* \\langle \\vec v^*,\\vec x\\rangle)+H_{\\vec x^*}(\\vec x)$ on $X$, and thus\n\\begin{align*}\n\\vec 0 &\\in \\nabla|_{  \\vec x=  \\vec x^*} \\left(F_1(\\vec x)+r^* G_2(\\vec x) -(\\langle \\vec u^*,\\vec x\\rangle+r^* \\langle \\vec v^*,\\vec x\\rangle)+H_{\\vec x^*}(\\vec x) \\right)\n\\\\&=\\nabla F_1(\\vec x^*)+r^* \\nabla G_2(\\vec x^*) -\\vec u^*-r^*\\vec v^*\n\\subset \\nabla F_1(\\vec x^*)-\\nabla F_2(\\vec x^*)+r^*\\nabla G_2(\\vec x^*)-r^*\\nabla G_1(\\vec x^*)\n\\end{align*}\n\n\\item[Case 2.2.] On the scheme \\eqref{iter2},  $F_1$, $F_2$, $G_1$ and $G_2$ are one-homogeneous; $H_x(\\vec x)$ is continuous of $\\vec x\\in \\mathbb{B}$ and for any $M>0$, there exists $C>0$ such that $H_{\\vec x}(\\vec y)>M\\cdot B(\\vec y)$ whenever $\\vec x\\in\\mathbb{B}$ and  $B(\\vec y)\\ge C$. \n\nTaking $M=C_1C_2+2$ in which the constants  $C_1$ and $C_2$ are introduced in Case 2.1, there exists $C>\\max\\{\\max\\limits_{x\\in \\mathbb{B}}H_x(\\vec x),1\\}$ such that $H_{\\vec x^*}(\\vec x)\\ge M\\cdot B(\\vec x)$ whenever $\\vec x^*\\in \\mathbb{B}$ and $B(\\vec x)\\ge C$.  \n\nSimilar to Case 2.1, for any $\\vec x^*\\in \\mathbb{B}$,  $\\vec x\\in X$ with $B(\\vec x)\\ge C$, and $\\forall \\vec u^*\\in \\nabla F_2(\\vec x^*)$, $ \\vec v^*\\in \\nabla G_1(\\vec x^*)$, \n\\begin{align*}\n&F_1(\\vec x)+r^* G_2(\\vec x) -(\\langle \\vec u^*,\\vec x\\rangle+r^*\\langle \\vec v^*,\\vec x\\rangle) + H_{\\vec x^*}(\\vec x)\n\\\\>~& B(\\vec x)(C_3-C_2C_1) + (C_1C_2+2)\\cdot B(\\vec x)\n\\ge 2 B(\\vec x) >H_{\\vec x^*}(\\vec x^*)\n\\\\ =~& F_1(\\vec x^*)+r^* G_2(\\vec x^*) -(\\langle \\vec u^*,\\vec x^*\\rangle+r^*\\langle \\vec v^*,\\vec x^*\\rangle) + H_{\\vec x^*}(\\vec x^*).\n\\end{align*}\nThe remaining part can refer to  Case 2.1.\n   \\end{enumerate}\n  \n\\end{proof}\n\n\\begin{remark}\\label{remark:very-general-RatioDCA}\nAs some direct extensions of the so-called {\\sl generalized RatioDCA} in \\cite{TMH18}, we have the following modified schemes:\n\\begin{subequations}\n\\label{iter1-}\n\\begin{numcases}{}\n\\vec  x^{k+1}\\in \\argmin\\limits_{\\vec x\\in \\mathbb{B}} F_1(\\vec x)+r^k G_2(\\vec x) -(\\langle \\vec u^k,\\vec x\\rangle+r^k \\langle \\vec v^k,\\vec x\\rangle) + H_{\\vec x^k}(\\vec x)\\text{ if }r^k\\ge0, \\label{eq:twostep_x2-}\n\\\\\n\\vec  x^{k+1}\\in \\argmin\\limits_{\\vec x\\in \\mathbb{B}} G_1(\\vec x)-\\langle \\vec w^k,\\vec x\\rangle-\\frac{1}{r^k}( F_1(\\vec x) - \\langle \\vec u^k,\\vec x\\rangle) + H_{\\vec x^k}(\\vec x)\\text{ if }r^k<0, \\label{eq:twostep_x22-}\n\\\\\nr^{k+1}=F( \\vec x^{k+1})\/G( \\vec x^{k+1}),\n\\label{eq:twostep_r2-}\n\\\\\n \\vec u^{k+1}\\in\\nabla F_2( \\vec x^{k+1}),\\;\n \\vec v^{k+1}\\in\\nabla G_1( \\vec x^{k+1}),\\;\\vec w^{k+1}\\in\\nabla G_2( \\vec x^{k+1})\n\\label{eq:twostep_s2-}\n\\end{numcases}\n\\end{subequations}\nand\n\\begin{subequations}\n\\label{iter2-}\n\\begin{numcases}{}\n\\vec  y^{k+1}\\in \\argmin\\limits_{\\vec x\\in X} F_1(\\vec x)+r^k G_2(\\vec x) -(\\langle \\vec u^k,\\vec x\\rangle+r^k \\langle \\vec v^k,\\vec x\\rangle) + H_{\\vec x^k}(\\vec x)\\text{ if }r^k\\ge0, \\label{eq:2twostep_x2-}\n\\\\\n\\vec  y^{k+1}\\in \\argmin\\limits_{\\vec x\\in X} G_1(\\vec x)-\\langle \\vec w^k,\\vec x\\rangle-\\frac{1}{r^k}( F_1(\\vec x) - \\langle \\vec u^k,\\vec x\\rangle) + H_{\\vec x^k}(\\vec x)\\text{ if }r^k<0, \\label{eq:2twostep_x22-}\n\\\\\nr^{k+1}=F( \\vec y^{k+1})\/G( \\vec y^{k+1}),~~ \\vec x^{k+1}=\\partial \\mathbb{B}\\cap\\{t\\vec y^{k+1}:t\\ge 0\\} \n\\label{eq:2twostep_r2-}\n\\\\\n \\vec u^{k+1}\\in\\nabla F_2( \\vec x^{k+1}),\\;\n \\vec v^{k+1}\\in\\nabla G_1( \\vec x^{k+1}),\n\\label{eq:2twostep_s2-}\n\\end{numcases}\n\\end{subequations}\nin which the previous assumption $F_1-F_2\\ge 0$ in \\eqref{iter1} and \\eqref{iter2}  has been removed. For these modifications, a convergence property like Theorem \\ref{th:gsd} still holds.\n\n\\end{remark}\n\n\n\n\\begin{remark}\nTheorem \\ref{th:gsd} shows the local convergence of a general relaxation of  Dinkelbach's algorithm  in the spirit of DC programming.\n The DC programming  consists in  minimizing $F-G$ where $F$ and $G$ are convex functions. \nAs described in \\cite{MaeharaMurota15,MMM18}, both the original DC algorithm and its discrete version can be written as the simple iteration: $\\vec u^k\\in \\nabla G(\\vec x^k)$, $\\vec x^{k+1}\\in \\nabla F^\\star(\\vec u^k)$, where $F^\\star$ is the Fenchel conjugate of $F$. It is known that such an iteration is equivalent to the following scheme  \\begin{subequations}\n\\label{F-Giter1}\n\\begin{numcases}{}\n\\vec  x^{k+1}\\in \\argmin\\limits_{\\vec x} F(\\vec x) -\\langle \\vec u^k,\\vec x\\rangle, \\label{eq:F-Gtwostep_x2}\n\\\\ \\vec u^{k+1}\\in\\nabla G( \\vec x^{k+1}).\n\\label{eq:F-Gtwostep_s2}\n\\end{numcases}\n\\end{subequations}\nMoreover,\na slight variation of the above scheme by adding a normalization step\n\\begin{subequations}\n\\label{FGiter1}\n\\begin{numcases}{}\n  \\hat{\\vec x}^{k+1}\\in \\argmin\\limits_{\\vec x\\in\\ensuremath{\\mathbb{R}}^n} F(\\vec x) -\\langle \\vec u^k,\\vec x\\rangle, \\label{eq:FGtwostep_x2}\n\\\\ \\vec x^{k+1}=  \\hat{\\vec x}^{k+1}\/G(\\hat{\\vec x}^{k+1})^{\\frac1p}\n\\\\ \\vec u^{k+1}\\in\\nabla G( \\vec x^{k+1}).\n\\label{eq:FGtwostep_s2}\n\\end{numcases}\n\\end{subequations}\ncan be used to solve the fractional programming $\\min F\/G$, where $F$ and $G$ are convex and $p$-homogeneous with $p>1$. This scheme is nothing but\nAlgorithm 2 in \\cite{HeinBuhler2010}.\nIn fact, we can say more about it. \n\\end{remark}\n\\begin{pro}\nLet $F$ and $G$ be convex,   $p$-homogeneous and positive-definite functions on $\\ensuremath{\\mathbb{R}}^n$, where $p>1$.  Then, for any  initial point $\\vec x^0$, the sequence of the pairs $\\{(r^k,\\vec x^k)\\}_{k\\ge1}$ produced by the following scheme\n\\begin{subequations}\n\\label{iter1FG}\n\\begin{numcases}{}\n\\hat{\\vec x}^{k+1}\\in \\argmin\\limits_{\\vec x\\in \\ensuremath{\\mathbb{R}}^n} F(\\vec x) -a_k\\langle \\vec u^k,\\vec x\\rangle, \\label{eq:twostep_x2FG}\n\\\\\n\\vec x^{k+1}= b_{k+1}  \n\\hat{\\vec x}^{k+1}\\; (\\mathrm{scaling}),\\;\\; r^{k+1}=F( \\vec x^{k+1})\/G( \\vec x^{k+1}),\n\\label{eq:twostep_r2FG}\n\\\\\n \\vec u^{k+1}\\in\\nabla G( \\vec x^{k+1}),\n\\label{eq:twostep_s2FG}\n\\end{numcases}\n\\end{subequations}\nconverges to an eigenpair $(r^*,\\vec x^*)$ of $(F,G)$ in the sense that $\\lim\\limits_{k\\to+\\infty}r^k= r^*$ and $\\vec x^*$ is a limit point of $\\{\\vec x^k\\}_{k\\ge1}$,  whenever $a_k,b_k>0$ as well as  both $\\{a_k\\}_{k\\ge 1}$ and  $\\{b_k\\hat{\\vec x}^k\\}_{k\\ge 1}$\n are  bounded away from $0$ and $\\infty$.\n\\end{pro}\n\nThe proof is very similar to the original proof of Theorem 3.1 in  \\cite{HeinBuhler2010},  with an additional trick  like the proof of Case 2.1 in Theorem \\ref{th:gsd}. It can be regarded as a supplement of both Theorem 3.1 in  \\cite{HeinBuhler2010} and Theorem \\ref{th:gsd}. It is also interesting that the scheme is stable under   perturbations of $a_k$ and $b_k$. Besides, it can be seen that the resulting eigenvalue $r^*$ should be  independent of the choice of $a_k$ and $b_k$. In fact, $r^*$ only depends on the initial data and the choice of subgradient $\\vec u^k$. \nThe assumption that  $F$ is positive-definite can be removed in some sense. Indeed, if $r^k\\le0$ for some $k$, we can modify \\eqref{eq:twostep_x2FG} as $\\hat{\\vec x}^{k+1}\\in \\argmin\\limits_{\\vec x\\in \\ensuremath{\\mathbb{R}}^n} F(\\vec x) -r^kG(\\vec x)$ or $\\hat{\\vec x}^{k+1}\\in \\argmin\\limits_{\\vec x\\in \\ensuremath{\\mathbb{R}}^n} G(\\vec x) -\\frac{1}{r^k}F(\\vec x)$ when $r^k<0$. Then $\\{r^k\\}$  converges to the global minimum of $F\/G$.\n\n\nAnother solver for the continuous optimization $\\min\\frac{F(\\vec x)}{G(\\vec x)}$ is\n the stochastic subgradient method:\n\\begin{equation*}\n\\vec x^{k+1}=\\vec x^k-\\alpha_k(\\vec y^k+\\vec\\xi^k),\\;\\;\\;\\vec y^k\\in\\nabla\\frac{F(\\vec x^k)}{G(\\vec x^k)},\n\\end{equation*}\nwhere $\\{\\alpha_k\\}_{k\\ge1}$ is a step-size sequence and $\\{\\vec\\xi^k\\}_{k\\ge1}$ is now a sequence of random variables (the ``noise'') on some probability space. Theorem 4.2 in \\cite{Davis19-FoCM} shows that under some natural assumptions, almost surely, every limit point of the stochastic subgradient iterates $\\{\\vec x^k\\}_{k\\ge1}$ is critical for $F\/G$, and the function values $\\{\\frac{F}{G}(\\vec x^k)\\}_{k\\ge1}$ converge.\n\n\\section{Examples and Applications}\n\\label{sec:examples-Applications}\n\n\n\\subsection{Submodular vertex cover and multiway partition problems}\nAs a first immediate application of Theorem \\ref{thm:tilde-fg-equal}, we obtain an easy way to rediscover the famous identity by Lov\\'asz, and the two typical submodular optimizations -- submodular vertex cover and multiway partition problems.\n\\begin{example}\nThe identity $\\min\\limits_{A\\in\\ensuremath{\\mathcal{P}}(V)}f(A)=\\min\\limits_{\\vec x\\in[0,1]^V}f^L(\\vec x)$ discovered by Lov\\'asz in his original paper \\cite{Lovasz} can be obtained by our result. In fact, $$\\min\\limits_{A\\in\\ensuremath{\\mathcal{P}}(V)}f(A)=\\min\\limits_{A\\in\\ensuremath{\\mathcal{P}}(V)}\\frac{f(A)}{1}=\\min\\limits_{\\vec x\\in [0,\\infty)^V}\\frac{f^L(\\vec x)}{\\max\\limits_{i\\in V}x_i}=\\min\\limits_{\\vec x\\in [0,1]^V}\\frac{f^L(\\vec x)}{\\max\\limits_{i\\in V}x_i}=\\min\\limits_{\\vec x\\in [0,1]^V,\\max\\limits_i x_i=1}f^L(\\vec x).$$\nChecking this is easy:\n if $f\\ge 0$, then $\\min\\limits_{\\vec x\\in [0,1]^V,\\max\\limits_i x_i=1}f^L(\\vec x)=0$; if $f(A)<0$ for some $A\\subset V$, then $\\min\\limits_{\\vec x\\in [0,1]^V,\\max\\limits_i x_i=1}f^L(\\vec x)=\\min\\limits_{\\vec x\\in [0,1]^V}f^L(\\vec x)$.\n\\end{example}\n\n\\paragraph{Vertex cover number }\nA vertex cover (or node cover) of a graph is a set of vertices such that each edge of the graph is incident to at least one vertex of the set. The vertex cover number is the minimal cardinality of a  vertex cover. Similarly, the independence number of a graph is the maximal number of vertices not connected by edges. The sum of the vertex cover number and the independence number is  the  cardinality of the vertex set.\n\nBy a variation of the Motzkin-Straus theorem and Theorem \\ref{thm:graph-numbers}, the vertex cover number thus has at least two equivalent continuous representations similar to the independence number.\n\n\\paragraph{Submodular vertex cover problem}\nGiven a graph $G=(V,E)$, and a submodular function $f:\\ensuremath{\\mathcal{P}}(V)\\to[0,\\infty)$,  find a vertex cover $S\\subset V$ minimizing $f(S)$.\n\nBy Theorem \\ref{thm:tilde-fg-equal},\n$$\\min\\{f(S):S\\subset V,\\,S\\text{ is a vertex cover}\\}=\\min\\limits_{\\vec x\\in{\\mathcal D}}\\frac{f^L(\\vec x)}{\\|\\vec x\\|_\\infty}=\\min\\limits_{\\vec x\\in \\widetilde{{\\mathcal D}}}f^L(\\vec x)$$\nwhere ${\\mathcal D}=\\{\\vec x\\in[0,\\infty)^V:V^t(\\vec x)\\text{ vertex cover},\\,\\forall t\\ge0\\}=\\{\\vec x\\in[0,\\infty)^V:x_i+x_j>0,\\forall\\{i,j\\}\\in E,\\,\\{i:x_i=\\max_j x_j\\}\\text{ vertex cover}\\}$,  and $\\widetilde{{\\mathcal D}}=\\{\\vec x\\in{\\mathcal D}: \\|\\vec x\\|_\\infty=1\\}=\\{\\vec x\\ge\\vec 0:x_i+x_j\\ge 1,\\forall\\{i,j\\}\\in E,\\,\\{i:x_i=\\max_j x_j\\}\\text{ vertex cover}\\}$. Note that\n$$\\mathrm{conv}(\\widetilde{{\\mathcal D}})=\\{\\vec x:x_i+x_j\\ge 1,\\forall\\{i,j\\}\\in E,\\,x_i\\ge 0,\\forall i\\in V\\}.$$\n\nTherefore, $\\min\\limits_{\\vec x\\in \\mathrm{conv}(\\widetilde{{\\mathcal D}})}f^L(\\vec x)\\le \\min\\{f(S):\\text{ vertex cover }S\\subset V\\}$, which rediscovers the convex programming relaxation.\n\n\\paragraph{Submodular multiway partition problem}\nThis problem  is about to minimize $\\sum_{i=1}^k f(V_i)$ subject to $V=V_1\\cup\\cdots\\cup V_k$, $V_i\\cap V_j=\\varnothing$, $i\\ne j$, $v_i\\in V_i$, $i=1,\\cdots,k$, where $f:\\ensuremath{\\mathcal{P}}(V)\\to\\ensuremath{\\mathbb{R}}$ is a submodular function.\n\n Letting $\\ensuremath{\\mathcal{A}}=\\{\\text{ partition }(A_1,\\cdots,A_k)\\text{ of }V:A_i\\ni a_i,\\,i=1,\\cdots,k\\}$, by Theorem \\ref{thm:tilde-fg-equal},\n $$\\min\\limits_{(A_1,\\cdots,A_k)\\in\\ensuremath{\\mathcal{A}}}\\sum_{i=1}^k f(A_i)=\\inf\\limits_{\\vec x\\in {\\mathcal D}_\\ensuremath{\\mathcal{A}}}\\frac{\\sum_{i=1}^k f^L(\\vec x^i)}{\\|\\vec x\\|_\\infty}=\\inf\\limits_{\\vec x\\in {\\mathcal D}'}\\sum_{i=1}^k f^L(\\vec x^i),$$\n  where ${\\mathcal D}_\\ensuremath{\\mathcal{A}}=\\{\\vec x\\in [0,\\infty)^{kn}: (V^t(\\vec x^1),\\cdots,V^t(\\vec x^k))\\text{ is a partition}, V^t(\\vec x^i)\\ni a_i,\\forall t\\ge 0\\}=\\{\\vec x\\in [0,\\infty)^{kn}: \\vec x^i=t1_{A_i},A_i\\ni a_i,\\forall t\\ge 0\\}$, and ${\\mathcal D}'=\\{(\\vec x^1,\\cdots,\\vec x^k):\\vec x^i\\in [0,\\infty)^V,\\,\\vec x^i=\\vec 1_{A_i},A_i\\ni a_i\\}$. Note that $$\\mathrm{conv}({\\mathcal D}')=\\{(\\vec x^1,\\cdots,\\vec x^k):\\sum_{v\\in V} x^i_v=1,x^i_{a_i}=1,x^i_v\\ge0\\}.$$ So one rediscovers the corresponding convex programming relaxation $\\min\\limits_{\\vec x\\in \\mathrm{conv}({\\mathcal D}')}\\sum_{i=1}^k f^L(\\vec x^i)$.\n  \n\\subsection{Min-cut and Max-cut}\n\nGiven an undirected weighted   graph $(V,E,w)$, the  min-cut problem  \n$$\\min\\limits_{S\\ne\\varnothing,V}|\\partial S|:=\\min\\limits_{S\\ne\\varnothing,V}|E(S,V\\setminus S)|=\\min\\limits_{S\\ne\\varnothing,V}\\sum\\limits_{i\\in S,j\\in V\\setminus S}w_{ij}$$\n and the \nmax-cut problem  $$\\max\\limits_{S\\ne\\varnothing,V}|\\partial S|:=\\max\\limits_{S\\ne\\varnothing,V}|E(S,V\\setminus S)|=\\max\\limits_{S\\ne\\varnothing,V}\\sum\\limits_{i\\in S,j\\in V\\setminus S}w_{ij}$$\nhave been  investigated systematically.\n\n\n\n\\begin{theorem}\\label{thm:mincut-maxcut-eigen}Let $(V,E,w)$ be a weighted  undirected graph. \nThen, we have the equivalent continuous optimization formulations for the min-cut and max-cut problems:\n$$\\min\\limits_{S\\ne\\varnothing,V}|\\partial S|=\\min\\limits_{\\min _ix_i+\\max_i x_i=0}\\frac{\\sum_{ij\\in E}w_{ij}|x_i-x_j|}{2\\|\\vec x\\|_\\infty}=\\tilde{\\lambda}_2,$$\n$$\\max\\limits_{S\\ne\\varnothing,V}|\\partial S|= \\max\\limits_{\\vec x\\ne \\vec0}\\frac{\\sum_{ij\\in E}w_{ij}|x_i-x_j|}{2\\|\\vec x\\|_\\infty}=\\tilde{\\lambda}_{\\max},$$\nwhere $\\tilde{\\lambda}_2$ and $\\tilde{\\lambda}_{\\max}$ are the second \n(i.e., the smallest  nontrivial)  eigenvalue and the largest eigenvalue of the nonlinear eigenvalue problem:\n\\begin{equation}\\label{eq:mincut-maxcut-eigen}\n\\vec0\\in \\nabla\\sum_{ij\\in E}w_{ij}|x_i-x_j|-\\lambda\\nabla 2\\|\\vec x\\|_\\infty.    \n\\end{equation}\\end{theorem}\n\n\\begin{proof}\nWe only prove the min-cut case. It is clear that\n$$ \\min\\limits_{S\\ne\\varnothing,V}|\\partial S|= \\min\\limits_{A,B\\ne\\varnothing,A\\cap B=\\varnothing}\\frac{ |\\partial A|+|\\partial B|}{2}$$\nLet $\\ensuremath{\\mathcal{A}}=\\{(A,B)\\in \\ensuremath{\\mathcal{P}}_2(V):A,B\\ne\\varnothing\\}$. Then ${\\mathcal D}_\\ensuremath{\\mathcal{A}}=\\{\\vec x\\in\\ensuremath{\\mathbb{R}}^n: \\max_i x_i=-\\min_i x_i>0\\}$, and by Theorem  \\ref{thm:tilde-fg-equal}, $$\\min\\limits_{S\\ne\\varnothing,V}|\\partial S|=\\min\\limits_{(A,B)\\in\\ensuremath{\\mathcal{A}}}\\frac{ |\\partial A|+|\\partial B|}{2}=\\min\\limits_{\\vec x\\in{\\mathcal D}_\\ensuremath{\\mathcal{A}}}\\frac{\\sum_{ij\\in E}w_{ij}|x_i-x_j|}{2\\|\\vec x\\|_\\infty}. $$ \n In addition,  according to Theorem \\ref{introthm:eigenvalue}, the set of the eigenvalues of $(f^L,g^L)$ coincides with \n $$\\left\\{\\frac{f^L(\\vec 1_A-\\vec 1_{V\\setminus A})}{g^L(\\vec 1_A-\\vec 1_{V\\setminus A})}:A\\subset V\\right\\}=\\left\\{\\frac{f(A,V\\setminus A)}{g(A,V\\setminus A)}:A\\subset V\\right\\}=\\left\\{\\frac{|\\partial A|+|\\partial (V\\setminus A)|}{2}:A\\subset V\\right\\}=\\{|\\partial A|:A\\subset V\\},$$\n where $f(A,B)=|\\partial A|+|\\partial B|$ and $g(A,B)=2$. In consequence, $\\min\\limits_{S\\ne\\varnothing,V}|\\partial S|$ is the second \neigenvalue  of $(f^L,g^L)$. The proof is completed.\n\\end{proof}\n\nEq.~\\eqref{eq:mincut-maxcut-eigen} shows the first nonlinear eigenvalue problem which possesses two nontrivial eigenvalues that are equivalent to two important graph optimization problems, respectively. \n\nIn addition, by our results, we  present a lot of equivalent continuous optimizations for the maxcut problem (see Examples \\ref{exam:maxcut1} and \\ref{exam:maxcut2}):\n\\begin{align*}\n\\max\\limits_{S\\subset V}|\\partial S|&=\\max\\limits_{x\\ne\n  0}\\frac{ \\sum_{\\{i,j\\}\\in E}w_{ij}(|x_i|+|x_j|-|x_i+x_j|)^p }{(2\\|\\vec\n  x\\|_\\infty)^p}\n\\\\&=\\max\\limits_{x\\ne\n  0}\\frac{ \\sum_{\\{i,j\\}\\in E}w_{ij}|x_i-x_j|^p }{(2\\|\\vec\n  x\\|_\\infty)^p}=\\max\\limits_{\\|\\vec\n  x\\|_\\infty\\le\\frac12 }  \\sum_{\\{i,j\\}\\in E}w_{ij}|x_i-x_j|^p  \n\\end{align*}\nfor any $p\\ge1$.  \n\n\\subsection{Max $k$-cut problem}\n\\label{sec:max-k-cut}\nThe max $k$-cut problem is to determine a graph $k$-cut by solving\n\\begin{equation}\\label{eq:maxk}\n\\mathrm{MaxC}_k(G)=\\max_{\\text{partition }(A_1,A_2,\\ldots,A_k)\\text{ of }V}\\sum_{i\\ne j}|E(A_i,A_j)|=\\max_{(A_1,A_2,\\ldots,A_k)\\in \\mathcal{C}_{k}(V)}\\sum_{i=1}^k|\\partial A_i|,\n\\end{equation}\nwhere $\\mathcal{C}_{k}(V)=\\{(A_1,\\ldots,A_k)\\big|A_i\\cap A_j = \\varnothing, \\bigcup_{i=1}^{k} A_i= V \\}$,  and $\\partial A_i:=E(A_i,V\\setminus A_i)$. We may write \\eqref{eq:maxk} as\n$$ \\mathrm{MaxC}_k(G)=\\max_{(A_1,A_2,\\ldots,A_{k-1})\\in \\mathcal{P}_{k-1}(V)}\\sum_{i=1}^{k-1}|\\partial A_i|+|\\partial (A_1\\cup\\cdots\\cup A_{k-1})|.$$\nTaking $f_k(A_1,\\cdots,A_k)=\\sum_{i=1}^{k}|\\partial A_i|+|\\partial (A_1\\cup\\cdots\\cup A_{k})|$, the $k$-way Lov\\'asz extension is $$f^L_k(\\vec x^1,\\cdots,\\vec x^k)=\\sum_{i=1}^k\\sum_{i\\sim j}|x^k_i-x^k_j|+\\sum_{j\\sim j'}\\left|\\max\\limits_{i=1,\\cdots,k} x^i_j-\\max\\limits_{i=1,\\cdots,k} x^i_{j'}\\right|.$$\nApplying  Theorem \\ref{thm:tilde-fg-equal}, we have\n$$ \\mathrm{MaxC}_{k+1}(G)=\\max\\limits_{\\vec x^i\\in\\ensuremath{\\mathbb{R}}^n_{\\ge0}\\setminus\\{\\vec0\\},\\,\\ensuremath{\\mathrm{supp}}(\\vec x^i)\\cap \\ensuremath{\\mathrm{supp}}(\\vec x^j)=\\varnothing}\\frac{\\sum_{i=1}^k\\sum_{i\\sim j}|x^k_i-x^k_j|+\\sum_{j\\sim j'}\\left|\\max\\limits_{i=1,\\cdots,k} x^i_j-\\max\\limits_{i=1,\\cdots,k} x^i_{j'}\\right|}{\\max\\limits_{i,j}x^i_j}$$\n\\subsection{Relative isoperimetric constants on a subgraph with boundary}\n\\label{sec:boundary-graph-1-lap}\nGiven a finite graph $G=(V,E)$ and a subgraph, we\nconsider the Dirichlet and Neumann eigenvalue problems for the corresponding 1-Laplacian. For $A\\subset V$, put $\\overline{A}=A\\cup \\delta A$, where $\\delta A$ is the set of points in  $A^c$ that are adjacent to some points in $A$ (see Fig.~\\ref{fig:AdeltaA}).\n\n\\begin{figure}[!h]\\centering\n\\begin{tikzpicture}[scale=0.69\n\\draw (0,0) to (1,0);\n\\draw (0,0) to (0,1);\n\\draw (1,0) to (2,0);\n\\draw (1,0) to (1,1);\n\\draw (1,1) to (0,1);\n\\draw (0,2) to (0,1);\n\\draw (0,2) to (1,2);\n\\draw (2,0) to (2,1);\n\\draw (2,2) to (1,2);\n\\draw (2,2) to (2,1);\n\\draw (1,1) to (1,2);\n\\draw (1,1) to (2,1);\n\\draw (0,0) to (-0.9,0);\n\\draw (0,0) to (0,-0.9);\n\\draw (1,0) to (1,-0.9);\n\\draw (2,0) to (2,-0.9);\n\\draw (0,2) to (-0.9,2);\n\\draw (0,1) to (-0.9,1);\n\\draw[densely dotted] (-2,2) to (-1.1,2);\n\\draw[densely dotted] (-2,1) to (-1.1,1);\n\\draw[densely dotted] (-2,0) to (-1.1,0);\n\\draw[densely dotted] (4,2) to (3.1,2);\n\\draw[densely dotted] (4,1) to (3.1,1);\n\\draw[densely dotted] (4,0) to (3.1,0);\n\\draw[densely dotted] (2,-2) to (2,-1.1);\n\\draw[densely dotted] (1,-2) to (1,-1.1);\n\\draw[densely dotted] (0,-2) to (0,-1.1);\n\\draw[densely dotted] (2,4) to (2,3.1);\n\\draw[densely dotted] (1,4) to (1,3.1);\n\\draw[densely dotted] (0,4) to (0,3.1);\n\\draw[densely dotted] (-0.1,3) to (-1,3);\n\\draw[densely dotted] (-0.1,-1) to (-1,-1);\n\\draw[densely dotted] (2.1,3) to (3,3);\n\\draw[densely dotted] (2.1,-1) to (3,-1);\n\\draw[densely dotted] (3,-0.1) to (3,-1);\n\\draw[densely dotted] (-1,-0.1) to (-1,-1);\n\\draw[densely dotted] (3,2.1) to (3,3);\n\\draw[densely dotted] (-1,2.1) to (-1,3);\n\\draw[dashed] (-1,1.9) to (-1,1.1);\n\\draw[dashed] (-1,0.9) to (-1,0.1);\n\\draw[dashed] (3,1.9) to (3,1.1);\n\\draw[dashed] (3,0.9) to (3,0.1);\n\\draw[dashed] (1.9,-1) to (1.1,-1);\n\\draw[dashed] (0.9,-1) to (0.1,-1);\n\\draw[dashed] (1.9,3) to (1.1,3);\n\\draw[dashed] (0.9,3) to (0.1,3);\n\\node (00) at (0,0) {$\\bullet$};\n\\node (10) at (1,0) {$\\bullet$};\n\\node (11) at (1,1) {$\\bullet$};\n\\node (01) at (0,1) {$\\bullet$};\n\\node (02) at (0,2) {$\\bullet$};\n\\node (20) at (2,0) {$\\bullet$};\n\\node (12) at (1,2) {$\\bullet$};\n\\node (21) at (2,1) {$\\bullet$};\n\\node (22) at (2,2) {$\\bullet$};\n\\node (03) at (0,3) {$\\circ$};\n\\node (30) at (3,0) {$\\circ$};\n\\node (13) at (1,3) {$\\circ$};\n\\node (31) at (3,1) {$\\circ$};\n\\node (23) at (2,3) {$\\circ$};\n\\node (32) at (3,2) {$\\circ$};\n\\node (01') at (0,-1) {$\\circ$};\n\\node (1'0) at (-1,0) {$\\circ$};\n\\node (11') at (1,-1) {$\\circ$};\n\\node (1'1) at (-1,1) {$\\circ$};\n\\node (21') at (2,-1) {$\\circ$};\n\\node (1'2) at (-1,2) {$\\circ$};\n\\draw (2.9,0) to (2,0);\n\\draw (0,2.9) to (0,2);\n\\draw (2.9,1) to (2,1);\n\\draw (1,2.9) to (1,2);\n\\draw (2.9,2) to (2,2);\n\\draw (2,2.9) to (2,2);\n\\end{tikzpicture}\n\\caption{\\label{fig:AdeltaA} In this graph, let  $A$ be the set of solid\n  points, $\\delta A$ the set of hollow points. We only consider the edges\n   for which one vertex is in $A$ and the other in $\\overline{A}$ (solid lines). We will ignore the dashed lines in $\\delta A$, and the dotted lines outside $\\overline{A}$.}\n\\end{figure}\n\nGiven $S\\subset \\overline{A}$, denote the boundary of $S$ relative to $A$ by\n$$\\partial_A S=\\{(u,v)\\in E:u\\in S\\cap A,v\\in \\delta A\\setminus S\\text{ or }u\\in S, v\\in A\\setminus S\\}.$$\nIf $S\\subset A$, then $\\partial_A S=\\{(u,v)\\in E:u\\in S, v\\in \\overline{A}\\setminus S\\}$.~\n\nThe Cheeger (cut) constant of the subgraph $A$ of $G$ is defined as\n$$h(A)=\\min_{S\\subset \\overline{A}}\\frac{|\\partial_A S|}{\\min\\{\\vol(A\\cap S),\\vol(A\\setminus S)\\}}.$$\nA set pair $(S,\\overline{A}\\setminus S)$ that achieves the Cheeger constant is called a Cheeger cut.\n\nThe Cheeger isoperimetric constant\\footnote{Some authors call it the Dirichlet isoperimetric constant.} of $A$ is defined as\n$$h_1(A)=\\min_{S\\subset  A}\\frac{|\\partial_A S|}{\\vol(S)},$$\nwhere a set $S$ achieving the Cheeger isoperimetric constant is called a Cheeger set. In the sequel, we fix  $A\\subset V$,  and we write $h(G)$ and $h_1(G)$ instead of $h(A)$ and $h_1(A)$, respectively.\n\nAccording to our generalized Lov\\'asz extension, we have\n\\begin{equation}\\label{eq:Dirichlet-Cheeger-h_1}\nh_1(G\n=\\inf_{\\vec x\\in \\mathbb{R}^n\\setminus\\{0\\}}\\frac{\\sum_{i\\sim j} |x_i-x_j|+\\sum_{i\\in A} p_i|x_i|}{\\sum_{i\\in A}d_i|x_i|}\n\\end{equation}\nand\n$$h(G\n=\\inf_{\\vec x\\in \\mathbb{R}^n\\setminus\\{0\\}}\\frac{\\sum_{i\\sim j,i,j\\in A}|x_i-x_j|+\\sum_{i\\sim j,i\\in A,j\\in\\delta A}|x_i-x_j|}{\\inf_{c\\in \\mathbb{R}}\\sum_{i\\in A}d_i|x_i-c|}.$$\n\nNote that the term on the right hand side of \\eqref{eq:Dirichlet-Cheeger-h_1} can be written as\n$$\n\\inf\\limits_{ \\vec x|_{V\\setminus S}=0,\\, \\vec x\\ne 0}\\mathcal{R}_1(x)\n$$\nwhich is called the {\\sl Dirichlet $1$-Poincare constant} (see \\cite{OSY19}) over $S$,\nwhere $$\\mathcal{R}_1(\\vec x):=\\frac{\\sum\\limits_{\\{i,j\\}\\in E} |x_i-x_j|}{\\sum_i d_i|x_i|}$$\nis called the $1$-Rayleigh quotient of $\\vec x$.\n\nWe can consider the corresponding spectral problems.\n\\begin{itemize}\n\\item Dirichlet eigenvalue problem:\n$$\\begin{cases}\n\\Delta_1 \\vec x\\cap \\mu D \\Sgn \\vec x\\ne\\varnothing,& \\text{ in } A\\\\\n\\vec x = 0,&\\text{ on }  \\delta A\n\\end{cases}$$  where $D$ is the diagonal matrix of the vertex degrees,  \nthat is,\n$$\\begin{cases}\n(\\Delta_1 \\vec x)_i-\\mu d_i\\Sgn x_i\\ni 0,&i\\in A\\\\\nx_i=0,&i\\in\\delta A\n\\end{cases}$$~\nwhose component form is: $\\exists$~$c_i\\in \\Sgn(x_i)$,~$z_{ij}\\in \\Sgn(x_i-x_j)$ satisfying $z_{ji}=-z_{ij}$ and\n$$\\sum_{j\\sim i} z_{ij}+ p_ic_i\\in \\mu d_i\\Sgn(x_i),~i\\in A,$$\nin which $p_i$ is the number of neighbors of $i$ in $\\delta A$.\n\\item Neumann eigenvalue problem: There exists $c_i\\in \\Sgn(x_i)$,~$z_{ij}\\in \\Sgn(x_i-x_j)$ with $z_{ji}=-z_{ij}$ such that\n$$\n\\begin{cases}\n\\sum_{j\\sim i,j\\in \\overline{A}}z_{ij}-\\mu d_i c_i=0,&i\\in A\\\\\n\\sum_{j\\sim i,j\\in A}z_{ij}=0,&i\\in\\delta A.\n\\end{cases}\n$$\n\\end{itemize}\nFor a graph $G$ with boundary, we use $\\Delta_1^D(G)$ and $\\Delta_1^N(G)$ to denote the Dirichlet 1-Laplacian and the Neumann 1-Laplacian, respectively. Then\n\\begin{pro}\n$$h_1(G)=\\lambda_1(\\Delta_1^D(G))\\;\\text{ and }\\;h(G)=\\lambda_2(\\Delta_1^N(G)).$$\n\\end{pro}\n\\begin{figure}\\centering\n\\begin{tikzpicture}[scale=0.69\n\\draw (0,0) to (2,0);\\draw (0,0) to (-0.9,0);\n\\draw (0,0) to (1,1);\n\\draw (0,0) to (1,-1);\n\\draw (1,1) to (1,-1);\\draw (1,1) to (1,1.9);\n\\draw (1,1) to (2,0);\n\\draw (1,-1) to (2,0);\\draw (1,-1) to (1,-1.9);\n\\draw (3,0) to (2,0);\\draw (3,0) to (3,0.9);\n\\draw (3,0) to (4,0);\\draw (3,0) to (3,-0.9);\n\\draw (6,0) to (4,0);\n\\draw (5,1) to (4,0);\n\\draw (5,-1) to (4,0);\n\\draw (5,1) to (6,0);\n\\draw (5,-1) to (6,0);\n\\draw (5,-1) to (5,1);\n\\draw (6,0) to (7,0);\n\\draw (8,0) to (7,0);\n\\draw (8,0) to (10,0);\n\\draw (8,0) to (9,1);\n\\draw (8,0) to (9,-1);\n\\draw (10,0) to (9,1);\n\\draw (10,0) to (9,-1);\n\\draw (9,1) to (9,-1);\n\\draw (10,0) to (10.9,0);\n\\draw (5,1) to (5,1.9);\n\\draw (5,1) to (5,-1.9);\n\\draw (9,1) to (9,1.9);\n\\draw (9,1) to (9,-1.9);\n\\draw (7,0) to (7,0.9);\n\\draw (7,0) to (7,-0.9);\n\\node (00) at (0,0) {$\\bullet$};\n\\node (11) at (1,1) {$\\bullet$};\n\\node (1-1) at (1,-1) {$\\bullet$};\n\\node (20) at (2,0) {$\\bullet$};\n\\node (30) at (3,0) {$\\bullet$};\n\\node (40) at (4,0) {$\\bullet$};\n\\node (60) at (6,0) {$\\bullet$};\n\\node (51) at (5,1) {$\\bullet$};\n\\node (5-1) at (5,-1) {$\\bullet$};\n\\node (70) at (7,0) {$\\bullet$};\n\\node (80) at (8,0) {$\\bullet$};\n\\node (100) at (10,0) {$\\bullet$};\n\\node (91) at (9,1) {$\\bullet$};\n\\node (9-1) at (9,-1) {$\\bullet$};\n\\node (-10) at (-1,0) {$\\circ$};\n\\node (110) at (11,0) {$\\circ$};\n\\node (12) at (1,2) {$\\circ$};\n\\node (1-2) at (1,-2) {$\\circ$};\n\\node (31) at (3,1) {$\\circ$};\n\\node (31) at (3,-1) {$\\circ$};\n\\node (71) at (7,1) {$\\circ$};\n\\node (7-1) at (7,-1) {$\\circ$};\n\\node (52) at (5,2) {$\\circ$};\n\\node (5-2) at (5,-2) {$\\circ$};\n\\node (92) at (9,2) {$\\circ$};\n\\node (9-2) at (9,-2) {$\\circ$};\n\\end{tikzpicture}\n\\caption{\\label{fig:k-nodal-domain} In this example, there are $3$ nodal domains of an eigenvector corresponding to the first Dirichlet eigenvalue of the graph 1-Laplacian. Each nodal domain is the vertex set of the $4$-order complete subgraph shown in the figure. }\n\\end{figure}\n\nFor a connected graph, the first eigenvector of $\\Delta_1^N(G)$ is constant\nand it has only one nodal domain while the first eigenvector of $\\Delta_1^D(G)$\n may have any number of nodal domains.\n \\begin{pro}\n For any $k\\in \\mathbb{N}^+$, there exists a connected graph $G$ with boundary such that its Dirichlet 1-Laplacian $\\Delta_1^D(G)$ has a first eigenvector (corresponding to $\\lambda_1(\\Delta_1^D(G))$) with exactly $k$ nodal domains; and its Neumann 1-Laplacian $\\Delta_1^N(G)$ possesses a second eigenvector (corresponding to $\\lambda_2(\\Delta_1^N(G))$) with exactly $k$ nodal domains.\n \\end{pro}\n\n\n \\begin{figure}\\centering\n \\begin{tikzpicture}[scale=0.8]\n\\node (1) at (0,1) {$\\bullet$};\n\\node (2) at (1,0) {$\\bullet$};\n\\node (3) at (1,1) {$\\bullet$};\n\\node (4) at (2,2) {$\\bullet$};\n\\node (5) at (3,1) {$\\bullet$};\n\\node (6) at (4,1) {$\\bullet$};\n\\node (7) at (3,0) {$\\bullet$};\n\\node (8) at (1,3) {$\\bullet$};\n\\node (9) at (1,4) {$\\bullet$};\n\\node (10) at (0,3) {$\\bullet$};\n\\node (11) at (3,3) {$\\bullet$};\n\\node (12) at (3,4) {$\\bullet$};\n\\node (13) at (4,3) {$\\bullet$};\n\\draw (0,1) to (1,1);\n\\draw (1,0) to (1,1);\n\\draw (1,1) to (2,2);\n\\draw (2,2) to (3,1);\n\\draw (3,1) to (4,1);\n\\draw (3,1) to (3,0);\n\\draw (2,2) to (1,3);\n\\draw (1,3) to (1,4);\n\\draw (1,3) to (0,3);\n\\draw (2,2) to (3,3);\n\\draw (3,3) to (3,4);\n\\draw (3,3) to (4,3);\n\\end{tikzpicture}\n\\caption{\\label{fig:second-eigen-k-nodal-domain} In this example, there are $4$ nodal domains of an eigenvector corresponding to the second Neumann eigenvalue of the graph 1-Laplacian. Each nodal domain is the vertex set of the $3$-order subgraph after removing the center vertex and its edges.}\n\\end{figure}\n\n\n\\subsection{Independence number}\\label{sec:independent-number}\nThe independence number $\\alpha(G)$ of an unweighted and undirected simple graph $G$ is  the largest cardinality of a subset of vertices in $G$, no two of which are adjacent. It can be seen as an optimization problem $\\max\\limits_{S\\subset V \\text{ s.t. }E(S)=\\varnothing}\\#S$. However, such a graph optimization is not global, and the feasible domain seems to be very complicated.\nBut we may simply  multiply by a truncated term $(1-\\#E(S))$.\nThe independence number can then be expressed as a global optimization on the power set of vertices:\n\\begin{equation}\\label{eq:independent-multiple}\n\\alpha(G)=\\max\\limits_{S\\subset V}\\#S(1-\\#E(S)),\n\\end{equation}\nand thus the Lov\\'asz extension can be applied.\n\\begin{proof}[Proof of Eq.~\\eqref{eq:independent-multiple}] Since $G$ is simple, $\\#S$ and $\\#E(S)$ take values in the natural numbers. Therefore,\n$$\n\\#S(1-\\#E(S))\\;\\; \\begin{cases}\\le 0,&\\text{ if } E(S)\\ne\\varnothing\\text{ or }S=\\varnothing,\\\\\n\\ge 1,&\\text{ if } E(S)=\\varnothing\\text{ and }S\\ne\\varnothing.\\end{cases}\n$\nThus, $\\max\\limits_{S\\subset V}\\#S(1-\\#E(S))=\\max\\limits_{S\\subset V \\text{ s.t. }E(S)=\\varnothing}\\#S=\\alpha(G)$.\n\\end{proof}\nHowever, Eq.~\\eqref{eq:independent-multiple} is  difficult to calculate. By the disjoint-pair Lov\\'asz extension, it equals \n$$\n\\alpha(G)=\\max\\limits_{\\vec x\\ne \\vec 0}\\frac{\\|\\vec x\\|_1-\\sum\\limits_{k\\in V,i\\sim j}\\min\\{|x_k|,|x_i|,|x_j|\\}}{\\|\\vec x\\|_\\infty},\n$$\nbut we don't know how to further simplify it.\n\nFortunately, there is a known  representation of the independence number as follows, and  we present a proof for convenience. \n\\begin{pro}The independence number $\\alpha(G)$ of a finite simple graph $G=(V,E)$ satisfies\n\\begin{equation}\\label{eq:independent-difference}\n\\alpha(G)=\\max\\limits_{S\\subset V}\\left(\\#S-\\#E(S)\\right).\n\\end{equation}\n\\end{pro}\n\n\\begin{proof\nLet $A$ be an independent set of $G$, then $\\alpha(G)=\\#A=\\#A-\\#E(A)\\le \\max\\limits_{S\\subset V}\\left(\\#S-\\#E(S)\\right)$ because there is no edge connecting points in $A$.\n\nLet $B\\subset V$ satisfy $ \\#B-\\#E(B) =\\max\\limits_{S\\subset V}\\left(\\#S-\\#E(S)\\right)$. Assume the induced subgraph $(B,E(B))$ has $k$ connected components, $(B_i,E(B_i))$, $i=1,\\cdots,k$. Then $B=\\sqcup_{i=1}^k B_i$ and $E(B)=\\sqcup_{i=1}^k E(B_i)$. Since $(B_i,E(B_i))$ is connected, $\\# B_i\\le \\# E(B_i)+1$ and  equality holds if and only if $(B_i,E(B_i))$ is a tree.\nNow taking $B'\\subset B$ such that $\\#(B'\\cap B_i)=1$, $i=1,\\cdots,k$, then $B'$ is an independent set and thus  \\begin{align*}\n\\alpha(G)&\\ge \\#B'=k=\\sum_{i=1}^k 1\\ge \\sum_{i=1}^k (\\# B_i- \\# E(B_i))\n= \\sum_{i=1}^k\\# B_i- \\sum_{i=1}^k\\# E(B_i)\\\\&=\\#(\\cup_{i=1}^kB_i)-\\#(\\cup_{i=1}^kE(B_i)) = \\#B-\\#E(B)=\\max\\limits_{S\\subset V}\\left(\\#S-\\#E(S)\\right).\n\\end{align*}\nAs a result, Eq.~\\eqref{eq:independent-difference} is proved.\n\\end{proof}\n\nAccording to Lov\\'asz extension, we get\n\\begin{equation}\\label{eq:independent-continuous}\n\\alpha(G)=\\max\\limits_{\\vec x\\ne \\vec 0}\\frac{\\|\\vec x\\|_1-\\sum\\limits_{i\\sim j}\\min\\{|x_i|,|x_j|\\}}{\\|\\vec x\\|_\\infty}.\n\\end{equation}\nBy the elementary identities: $\\sum_{i\\sim j}|x_i+x_j|+\\sum_{i\\sim j}|x_i-x_j|=2\\sum_{i\\sim j}\\max\\{|x_i|,|x_j|\\}=\\sum_{i\\sim j}\\left||x_i|-|x_j|\\right|+\\sum_i\\mathrm{deg}_i|x_i|$ and $\\sum_i\\mathrm{deg}_i|x_i|=\\sum_{i\\sim j}\\max\\{|x_i|,|x_j|\\}+\\sum_{i\\sim j}\\min\\{|x_i|,|x_j|\\}$,  Eq.~\\eqref{eq:independent-continuous} can be reduced to\n\\begin{equation}\\label{eq:independent-continuous-1}\n\\alpha(G)=\\max\\limits_{\\vec x\\ne \\vec 0}\\frac{2\\|\\vec x\\|_1+I^-(\\vec x)+I^+(\\vec x)- 2\\|\\vec x\\|_{1,\\mathrm{deg}}}{2\\|\\vec x\\|_\\infty},\n\\end{equation}\nwhere $I^\\pm(\\vec x)=\\sum_{i\\sim j}|x_i\\pm x_j|$ and $\\|\\vec x\\|_{1,\\mathrm{deg}}=\\sum_i\\mathrm{deg}_i|x_i|$. One would like to write Eq.~\\eqref{eq:independent-continuous-1} as\n\\begin{equation}\\label{eq:independent-continuous-2}\n\\alpha(G)=\\max\\limits_{\\vec x\\ne \\vec 0}\\frac{I^-(\\vec x)+I^+(\\vec x)- 2\\|\\vec x\\|_{1,\\mathrm{deg}'}}{2\\|\\vec x\\|_\\infty},\n\\end{equation}\nwhere $\\|\\vec x\\|_{1,\\mathrm{deg}'}=\\sum\\limits_{i\\in V}(\\mathrm{deg}_i-1)|x_i|$.\n\n\n\\begin{remark}\nThe maximum clique number can be reformulated in a similar way. \nIn addition, we refer to \\cite{BBcontinuous05,BBcontinuous06,SBBB20} for some other continuous formulations of the independence number. \n\\end{remark}\n\n\n\n\\paragraph{Chromatic number of a  perfect graph}\nBerge's strong perfect graph conjecture has been proved in \\cite{Annals06}.  A graph $G$ is perfect if for every induced subgraph $H$ of $G$, the chromatic number of $H$ equals the size of the largest clique of $H$. The complement of every perfect graph is perfect.\n\n\nSo for a  perfect graph, we have an easy way to calculate the  chromatic number. In a general simple graph, we refer to Section \\ref{sec:chromatic-number} for transforming the chromatic number.\n\n\\paragraph{Maximum matching }\nA matching $M$ in $G$ is a set of pairwise non-adjacent edges, none of which are loops; that is, no two edges share a common vertex.\nA maximal matching  is one with the largest possible number of edges.\n\nConsider the line graph $(E,R)$ whose vertex set $E$ is the edge set of $G$, and whose edge set is $R=\\{\\{e,e'\\}:e\\cap e'\\not=\\varnothing,\\,e,e'\\in E\\}$. Then the maximum matching number of $(V,E)$ coincides with the independence number of $(E,R)$. So, we have an equivalent continuous optimization for a maximum matching problem.\n\nHall's Marriage Theorem  provides a characterization of bipartite graphs which have a perfect matching and the Tutte theorem provides a characterization for arbitrary graphs.\n\nThe Tutte-Berge formula says that the size of a maximum matching of a graph is\n$$\\frac12\\min\\limits_{U\\subset V}\\left(\\#V+\\#U-\\#\\text{ odd connected components of }G|_{V\\setminus U}\\right).$$\nCan one  transform the above discrete optimization problem into an explicit continuous optimization via some extension?\n\n\\paragraph{$k$-independence number }\nThe independence number admits several generalizations:\n the maximum size of a set of vertices in a graph whose induced subgraph has maximum degree $(k-1)$ \\cite{CaroH13}; the size of the largest $k$-colourable subgraph \\cite{Spacapan11}; the\nsize of the largest set of vertices such that any two vertices in the set are at short-path distance larger than $k$ (see \\cite{Fiol97}). For the $k$-independence number involving short-path distance, one can easily transform it into the following two continuous representations:\n$$\\alpha_k= \\max\\limits_{\\vec x\\in\\ensuremath{\\mathbb{R}}^V\\setminus\\{\\vec 0\\}}\\frac{\\|\\vec x\\|_1^2}{\\|\\vec x\\|_1^2-2\\sum\\limits_{\\mathrm{dist}(i,j)\\ge k+1}x_ix_j} = \\max\\limits_{\\vec x\\in \\ensuremath{\\mathbb{R}}^n\\setminus\\{\\vec 0\\}}\\frac{\\sum\\limits_{\\mathrm{dist}(i,j)\\le k}(|x_i-x_j|+|x_i+x_j|)- 2\\sum\\limits_{i\\in V}(\\deg_{k}(i)-1)|x_i|}{2\\|\\vec x\\|_\\infty},$$\nwhere $\\deg_{k}(i)=\\#\\{j\\in V:\\mathrm{dist}(j,i)\\le k\\}$, $i=1,\\cdots,n$.\n\\subsection{Various and variant Cheeger problems}\n\\label{sec:variantCheeger}\n\nSeveral  Cheeger-type  constants on graphs have been proposed that are  different from  the classical one.\n\n\\paragraph{Multiplicative Cheeger constant\nFor instance\n$$h=\\min\\limits_{\\varnothing\\ne A\\subsetneqq V}\\frac{ \\#E(A,V\\setminus A) }{\\#A\\cdot\\#(V\\setminus A)}.$$\nIt is called the normalized cut problem which has many applications in image segmentation and spectral clustering \\cite{SM00,Luxburg07,HS11}.  By Proposition \\ref{pro:fraction-f\/g}, it is equal to\n$$\\min\\limits_{\\langle\\vec x,\\vec1\\rangle=0,\\vec x\\ne \\vec0}\\frac{\\sum_{i\\sim j}|x_i-x_j|}{\\sum_{i< j}|x_i-x_j|}.$$\n\n\\paragraph{Isoperimetric profile}\nThe isoperimetric profile $IP:\\mathbb{N}\\to [0,\\infty)$ is defined by\n$$IP(k)= \\inf\\limits_{A\\subset V,\\#A\\le k} \\frac{\\#E(A,V\\setminus A)}{\\#A}.$$\nThen by Lov\\'asz extension, it is equal to\n$$\\inf\\limits_{\\vec x\\in\\ensuremath{\\mathbb{R}}^V,\\,1\\le \\# \\ensuremath{\\mathrm{supp}}(\\vec x)\\le k}\\frac{\\sum_{\\{i,j\\}\\in E}|x_i-x_j|}{\\|\\vec x\\|_1}=\\min\\limits_{\\vec x\\in CH_k(\\ensuremath{\\mathbb{R}}^V)}\\frac{\\sum_{\\{i,j\\}\\in E}|x_i-x_j|}{\\|\\vec x\\|_1},$$\nwhere $CH_n:=\\{\\vec x\\in\\ensuremath{\\mathbb{R}}^V,\\,\\# \\ensuremath{\\mathrm{supp}}(\\vec x)\\le k\\}$ is the union of all $k$-dimensional coordinate hyperplanes in $\\ensuremath{\\mathbb{R}}^V$.\n\n\\paragraph{Modified Cheeger constant}\n\nOn a graph $G=(V,E)$, there are three definitions of the vertex-boundary of a subset $A\\subset V$:\n\\begin{align}\n& \\partial_{\\textrm{ext}} A:=\\{j\\in V\\setminus A\\,\\left|\\,\\{j,i\\}\\in E\\text{ for some }i\\in A\\right.\\} \\label{eq:ext-vertex-boundary}\\\\\n& \\partial_{\\textrm{int}} A:=\\{i\\in A\\,\\left|\\,\\{i,j\\}\\in E\\text{ for some }j\\in V\\setminus A\\right.\\}\\label{eq:int-vertex-boundary}\\\\\n& \\partial_{\\textrm{ver}} A:=\\partial_{\\textrm{out}} A\\cup \\partial_{\\textrm{int}} A=V(E(A,V\\setminus A))=V(\\partial_{\\textrm{edge}} A)\\label{eq:vertex-boundary}\n\\end{align}\nThe  {\\sl external vertex boundary} \\eqref{eq:ext-vertex-boundary} and the  {\\sl internal vertex boundary} \\eqref{eq:int-vertex-boundary} are introduced and studied recently in \\cite{Vigolo19tams,VigoloPHD}.  Research on metric measure space \\cite{HMT19} suggests to consider the {\\sl vertex boundary} \\eqref{eq:vertex-boundary}.\n\nDenote by $N(i)=\\{i\\}\\cup\\{j\\in V:\\{i,j\\}\\in E\\}$ the 1-neighborhood of $i$. Then the Lov\\'asz extensions of $\\#\\partial_{\\textrm{ext}} A$, $\\#\\partial_{\\textrm{int}} A$ and $\\#\\partial_{\\textrm{ver}} A$ are\n$$\\sum\\limits_{i=1}^n(\\max\\limits_{j\\in N(i)}x_j-x_i),\\;\\;\\;\\sum\\limits_{i=1}^n(x_i-\\min\\limits_{j\\in N(i)}x_j)\\;\\;\\text{ and }\\;\\;\\sum\\limits_{i=1}^n(\\max\\limits_{j\\in N(i)}x_j-\\min\\limits_{j\\in N(i)}x_j),$$ respectively.\nThey can be seen as the  `total variation' of $\\vec x$ with respect to $V$ in $G$, while the usual {\\sl edge boundary} leads to $\\sum\\limits_{\\{i,j\\}\\in E}|x_i-x_j|$ which is  regarded as the total variation of $\\vec x$ with respect to $E$ in $G$.  Their disjoint-pair Lov\\'asz extensions are $$\\sum_{i=1}^n \\max\\limits_{j\\in N(i)} |x_j|-\\|\\vec x\\|_1,\\;\\;\\;\\|\\vec x\\|_1-\\sum_{i=1}^n \\min\\limits_{j\\in N(i)} |x_j|,\\;\\;\\;\\sum_{i=1}^n \\left(\\max\\limits_{j\\in N(i)} |x_j|-\\min\\limits_{j\\in N(i)} |x_j|\\right).$$\nComparing with the graph $1$-Poincare profile (see \\cite{Hume17,HMT19,Hume19arxiv}) $$P^1(G):=\\inf\\limits_{\\langle\\vec x,\\vec 1\\rangle=0,\\vec x\\ne\\vec 0}\\frac{\\sum_{i\\in V} \\max\\limits_{j\\sim i}|x_i-x_j|}{\\|\\vec x\\|_1},$$\nwe easily get the following\n\\begin{pro}\n$$ \\frac12\\max\\{h_{\\mathrm{int}}(G),h_{\\mathrm{ext}}(G)\\}\\le P^1(G)\\le h_{\\mathrm{ver}}(G):=\\min\\limits_{A\\in\\ensuremath{\\mathcal{P}}(V)\\setminus\\{\\varnothing,V\\}}\n\\frac{\\#\\partial_{\\mathrm{ver}} A}{\\min\\{\\#(A),\\#(V\\setminus A)\\}}$$\nwhere $h_{\\mathrm{int}}(G)$,  $h_{\\mathrm{ext}}(G)$ and $h_{\\mathrm{ver}}(G)$ are modified Cheeger constants w.r.t. the type of vertex-boundary.\n\\end{pro}\n\\begin{proof}\nBy Theorem \\ref{introthm:eigenvalue}, $$h_{\\mathrm{ver}}(G)=\\min\\limits_{A\\in\\ensuremath{\\mathcal{P}}(V)\\setminus\\{\\varnothing,V\\}}\n\\frac{\\#\\partial_{\\mathrm{ver}} A}{\\min\\{\\#(A),\\#(V\\setminus A)\\}}=\\inf\\limits_{\\langle\\vec x,\\vec 1\\rangle=0,\\vec x\\ne\\vec 0}\\frac{\\sum_{i\\in V} \\max\\limits_{j\\sim i}|x_i-x_j|}{\\min\\limits_{t\\in\\ensuremath{\\mathbb{R}}}\\|\\vec x-t\\vec1\\|_1}\\ge P^1(G).$$\nOn the other hand, it is easy to check that $\\min\\limits_{t\\in\\ensuremath{\\mathbb{R}}}\\|\\vec x-t\\vec1\\|_1\\ge \\frac12 \\|\\vec x\\|_1$ whenever $\\langle\\vec x,\\vec 1\\rangle=0$. Thus, $h_{\\mathrm{ver}}(G)\\le 2P^1(G)$. The proof is then completed by noting that $\\max\\{h_{\\mathrm{int}}(G),h_{\\mathrm{ext}}(G)\\}\\le h_{\\mathrm{ver}}(G)$.\n\\end{proof}\n\n\\paragraph{Cheeger-like constant}\nSome further recent results  \\cite{JM19arxiv} can be also rediscovered via  Lov\\'asz extension.\n\nA main equality in \\cite{JM19arxiv} can be absorbed into the following identities:\n\\begin{align}\n\t      \\max_{\\text{edges }(v,w)}\\biggl(\\frac{1}{\\deg v}+\\frac{1}{\\deg w}\\biggr)\n&=\n\\max_{\\gamma:E\\rightarrow\\mathbb{R}}\\frac{\\sum_{v\\in V}\\frac{1}{\\deg v}\\cdot \\biggl|\\sum_{e_{\\text{in}}: v\\text{ input}}\\gamma(e_{\\text{in}})-\\sum_{e_{\\text{out}}: v\\text{ output}}\\gamma(e_{\\text{out}})\\biggr|}{\\sum_{e\\in E}|\\gamma(e)|} \\notag\n\\\\&=\\max_{\\hat{\\Gamma}\\subset\\Gamma \\text{ bipartite}} \\frac{\\sum_{v\\in V}\\frac{\\deg_{\\hat{\\Gamma}}(v)}{\\deg_\\Gamma (v)}}{|E(\\hat{\\Gamma})|}, \\label{eq:mainJM19}\n\\end{align}\nwhere the left quantity is called a Cheeger-like constant \\cite{JM19arxiv}.\n\nIn fact, given $c_i\\ge 0$, $i\\in V$,\n$$\\max\\limits_{\\{i,j\\}\\in E}(c_i+c_j)=\\max\\limits_{E'\\subset E}\\frac{\\sum_{\\{i,j\\}\\in E'}(c_i+c_j)}{\\# E'},$$\nand then via Lov\\'asz extension, one immediately gets that the above constant equals to\n$$\n\\max\\limits_{\\vec x\\in [0,\\infty)^{E}\\setminus\\{\\vec0\\}} \\frac{\\sum\\limits_{e=\\{i,j\\}\\in E}x_e(c_i+c_j)}{\\sum_{e\\in E}x_e}=\\max\\limits_{\\vec x\\in[0,\\infty)^{E}\\setminus\\{\\vec0\\}} \\frac{\\sum_{i\\in V}c_i\\sum_{e\\ni i}x_e}{\\sum_{e\\in E}x_e}=\\max\\limits_{\\vec x\\in \\ensuremath{\\mathbb{R}}^{E}\\setminus\\{\\vec0\\}} \\frac{\\sum_{i\\in V}c_i\\left|\\sum_{e\\ni i}x_e\\right|}{\\sum_{e\\in E}|x_e|}.\n$$\nThus, for any family $\\ensuremath{\\mathcal{E}}\\subset\\ensuremath{\\mathcal{P}}(E)$ such that $E'\\in \\ensuremath{\\mathcal{E}}$ $\\Rightarrow$ $E'\\supset \\{\\{e\\}:e\\in E\\}$, we have\n$$\\max\\limits_{\\{i,j\\}\\in E}(c_i+c_j)=\\max\\limits_{\\vec x\\in \\ensuremath{\\mathbb{R}}^{E}\\setminus\\{\\vec0\\}} \\frac{\\sum_{i\\in V}c_i\\left|\\sum_{e\\ni i}x_e\\right|}{\\sum_{e\\in E}|x_e|}=\\max\\limits_{E'\\in\\ensuremath{\\mathcal{E}}}\\frac{\\sum_{\\{i,j\\}\\in E'}(c_i+c_j)}{\\# E'},$$\nwhich recovers the interesting equality \\eqref{eq:mainJM19}  by taking $c_i=\\frac{1}{\\deg i}$ and $\\ensuremath{\\mathcal{E}}$ the collections of all edge sets of bipartite subgraphs.\n\nA similar simple trick gives\n\\begin{equation*}\n    \\min_{(v,w)}\\frac{\\bigl|\\mathcal{N}(v)\\cap \\mathcal{N}(w)\\bigr|}{\\max\\{\\deg v,\\deg w\\}}=\\min\\limits_{\\vec x\\in \\ensuremath{\\mathbb{R}}^{E}\\setminus\\{\\vec0\\}} \\frac{\\sum_{i\\in V}\\sum_{e\\ni i}\\left|x_e\\right|\\cdot\\#\\text{ triangles containing }e}{\\sum_{e=\\{i,j\\}\\in E}|x_e|\\max\\{\\deg i,\\deg j\\}}.\n\\end{equation*}\n\n\n\\subsection{Frustration in signed networks}\n\\label{sec:frustration}\nIn this section, we apply our theory to signed graphs, a concept first introduced by Harary \\cite{Harary55}.\n\\begin{defn}\n  A \\emph{signed graph} $\\Gamma$ consists of a vertex set $V$ and a set $E$ of undirected edges with a sign function\n\\begin{equation}\\label{sign1}\ns:E \\to \\{+1,-1\\}.\n\\end{equation}\nThe adjacency matrix of $(\\Gamma,s)$, is denoted by  $\\mathrm{A}^s:=(s_{ij})_{i,j\\in V}$, where $s_{ij}:=s(e)$ if $e=\\{i,j\\}\\in E$, and  $s_{ij}:=0$ otherwise.\n\\end{defn}\nWhen we replace the sign function $s$ by $-s$, we shall call the resulting graph \\emph{antisigned}.\n\\begin{defn}\n The signed cycle $C_m$ (consisting of $m$ vertices that are cyclically connected by $m$ edges) is \\emph{balanced} if \n\\begin{equation}\\label{sign3}\n\\prod_{i=1}^m s(e_i)=1.\n\\end{equation}\nA  signed graph $(\\Gamma,s)$ is \\emph{balanced}  if every cycle contained in it is balanced.\\\\\n$(\\Gamma,s)$ is \\emph{antibalanced}  if $(\\Gamma,-s)$ is balanced. \\\\\nThe frustration index of  a signed graph $\\Gamma=(V,E)$ is  \n\\begin{equation}\\label{eq:frustration}\n\\min_{x_i\\in \\{-1,1\\},\\forall i} \\sum_{\\{i,j\\}\\in E}|x_i-s_{ij}x_j|,\n\\end{equation}  where $s_{ij}\\in\\{-1,1\\}$  indicates the sign of the edge $(i,j)$.\n\\end{defn}\nThe frustration index then vanishes iff the graph is balanced.\\\\\n\\begin{defn}\n The  (normalized) Laplacian $\\Delta^s$ of a signed graph is defined by\n\\begin{equation}\n\\label{sign5}\n(\\Delta^s \\vec x)_i:= x_i-\\frac{1}{\\deg i}\\sum_{j \\sim i}s_{ij}x_j=\\frac{1}{\\deg i}\\sum_{j \\sim i}(x_i-s_{ij}x_j)\n\\end{equation}\nfor a vector $\\vec x\\in\\ensuremath{\\mathbb{R}}^V$.\n\\end{defn}\n\\begin{remark}\nThe Laplacian thus is of the form $\\Delta^s =\\mathrm{id}-\\mathrm{A}^s$, and \n when we change the signs of all the edges, that is, go from a signed graph to the corresponding antisigned graph, the operator becomes $\\Delta^{-s} =\\mathrm{id}+\\mathrm{A}^s$. Therefore, the eigenvalues simply  change from $\\lambda$ to $2-\\lambda$ (and therefore, also the ordering gets reversed). \n\\end{remark}\n\nBy Proposition \\ref{pro:Lovasz-eigen}, it is easy to verify that every eigenvalue of the function pair $(F,G)$ has an  eigenvector in  $\\{-1,0,1\\}^n$,  where $F(\\vec x)=\\sum_{\\{i,j\\}\\in E}|x_i-s_{ij}x_j|$ and $G(\\vec x)=\\|\\vec x\\|_\\infty$. One may  relax \\eqref{eq:frustration} as  \\begin{equation}\\label{eq:frustration-relax}\n\\min_{\\vec x\\in \\{-1,0,1\\}^n\\setminus\\{\\vec0\\}} \\sum_{(i,j)\\in E}|x_i-s_{ij}x_j|.    \n\\end{equation}\n\n\n\nThis suggests the  eigenvalue problem of $(F(\\vec x),\\|\\vec x\\|_\\infty)$ on a signed graph, \nwhere $F(\\vec x)=\\sum_{\\{i,j\\}\\in E}|x_i-s_{ij}x_j|$. Below,  we show some key properties.\n\n\n\n\n\n\n\n\\begin{itemize}\n\\item\nThe coordinate form of the eigenvalue problem $\\nabla \\sum_{\\{i,j\\}\\in E}|x_i-s_{ij}x_j| \\cap \\lambda\\nabla\\|\\vec x\\|_\\infty\\ne\\varnothing$ reads as\n\n $\\exists\\,z_{ij}\\in \\Sgn(x_i-s_{ij}x_j) \\mbox{ with }z_{ij}+s_{ij}z_{ji}=0$~such that\n\\begin{align}\n\\label{eq:1LinftyN1}&\\sum\\limits_{j\\sim i} z_{ij} = 0, &i\\in D_0(\\vec x),\\\\\n\\label{eq:1LinftyN2}&\\sum\\limits_{j\\sim i} z_{ij} \\in \\lambda\\ \\mathrm{sign}(x_i)\\cdot [0,1],& i\\in D_\\pm(\\vec x),\\\\\n\\label{eq:1LinftyN3}&\\sum\\limits_i^n \\big|\\sum\\limits_{j\\sim i} z_{ij} \\big|=\\lambda,&\n\\end{align}\nwhere $D_\\pm(\\vec x) =\\{i\\in V\\big| \\pm x_i = \\|\\vec x\\|\\}$,\nand $D_0(\\vec x)=\\{i\\in V\\big| |x_i|<\\|\\vec x\\|\\}$.\n\\item All eigenvalues are integers in  $\\{0,1,\\cdots,\\vol(V)\\}$. And each eigenvalue has an eigenvector in $\\{-1,0,1\\}^n$.\n\nProof: This is a direct consequence of Proposition \\ref{pro:Lovasz-eigen}. \n\n\\item The largest eigenvalue has an eigenvector in $\\{-1,1\\}^n$.\n\nProof: Let $\\vec 1_A-\\vec 1_B$ be an eigenvector  w.r.t. the largest eigenvalue. Note that $\\vec 1_A-\\vec 1_B=\\frac12(\\vec1_A-\\vec1_{V\\setminus A}+\\vec1_{V\\setminus B}-\\vec1_B)$. By the convexity of $F$, we have $F(\\vec 1_A-\\vec 1_B)\\le\\max\\{F(\\vec1_A-\\vec1_{V\\setminus A}),F(\\vec1_{V\\setminus B}-\\vec1_B)\\}$. Hence, either $\\vec1_A-\\vec1_{V\\setminus A}$ or $\\vec1_{V\\setminus B}-\\vec1_B$ is an eigenvector w.r.t. the largest eigenvalue. \n\n\\item \\textbf{The frustration index  is an eigenvalue}.\nHowever, in general, we don't know which eigenvalue the  frustration  index is.\n\n Proof: We shall check that for any $A\\subset V$, the binary vector $\\vec x:=\\vec1_A-\\vec1_{V\\setminus A}$ is an eigenvector w.r.t. the eigenvalue $\\lambda:=2(|E_+(A,V\\setminus A)|+|E_-(A)|+|E_-(V\\setminus A)|)$, where  $|E_+(A,V\\setminus A)|$ indicates the number of positive edges lying between  $A$ and $V\\setminus A$, while  $|E_-(A)|$ denotes the number of negative edges lying in $A$.  Indeed, $D_+(\\vec x)=A$ and $D_-(\\vec x)=V\\setminus A$.   For $i\\in A$, taking $z_{ij}=1$ if $s_{ij}x_j<0$; and $z_{ij}=0$ if $s_{ij}x_j>0$.  Similarly, for $i\\in V\\setminus A$, letting $z_{ij}=0$ if $s_{ij}x_j<0$; and $z_{ij}=-1$ if $s_{ij}x_j>0$. It is easy to see that $z_{ij}\\in \\mathrm{Sgn}(x_i-s_{ij}x_j)$ and $z_{ij}+s_{ij}z_{ji}=0$ for any edge $ij$. Next, we verify the conditions \\eqref{eq:1LinftyN2} and \\eqref{eq:1LinftyN3}.\n\nNote that  $\\sum_{j\\sim i}z_{ij}=\\#(\\{j\\in A:ij\\text{ is negative}\\}\\cup \\{j\\in V\\setminus A:ij\\text{ is positive}\\})\\in[0,\\lambda]$ for $i\\in A$, and  $\\sum_{j\\sim i}z_{ij}=-\\#(\\{j\\in A:ij\\text{ is positive}\\}\\cup \\{j\\in V\\setminus A:ij\\text{ is negative}\\})\\in[-\\lambda,0]$ for $i\\in V\\setminus A$. \nTherefore, $\\sum_{i\\in V}|\\sum_{j\\sim i}z_{ij}|=2(|E_+(A,V\\setminus A)|+|E_-(A)|+|E_-(V\\setminus A)|)=\\lambda$.\n\nIn particular, for $\\vec x\\in\\{-1,1\\}^n$ that realizes the frustration index, $\\vec x$ must be an eigenvector, and the frustration index is the corresponding eigenvalue. This fact can also be derived by  Proposition  \\ref{pro:set-pair-infty-norm}.  \n\n\\item We can use the the   Dinkelbach-type scheme in Section \\ref{sec:algo}  directly to  calculate the smallest eigenvalue. When we get an eigenvector $\\vec x$, we can take $\\vec 1_{D_+(\\vec x)}-\\vec 1_{D_-(\\vec x)}$ instead of $\\vec x$. \n\\item We construct a recursive method to approximate the frustration index:\n\\begin{itemize}\n\\item Input a signed graph $G$,  and use the   Dinkelbach-type algorithm to get a subpartition $(U_{+},U_{-})$ where $U_+=D_+(\\vec x)$ and $U_-=D_-(\\vec x)$ with $\\vec x$ being an eigenvector w.r.t. the smallest eigenvalue. \n\\item Let $G$ be the signed graph induced by $V\\setminus(U_+\\cup U_-)$, and let $(U_+',U_-')$ be the subpartition found by the  Dinkelbach-type  algorithmm; return $(U_+\\cup U_+',U_-\\cup U_-')$ or $(U_+\\cup U_-',U_-\\cup U_+')$, whichever is better.\n\\item Repeat the above process, until we get a partition $(V_+,V_-)$ of $V$, which derives an  approximate solution of the  frustration index. There are at most $n$ iterations.\n\\end{itemize}\nIn other words, the relaxation problem  \\eqref{eq:frustration-relax} can approximate the frustration index \\eqref{eq:frustration} in a recursive way.  This is inspired by the recursive spectral cut  algorithm for the   maxcut problem proposed by Trevisan  \\cite{Trevisan2012}.\n\\end{itemize}\n \n\n \nNext, we show some equivalent continuous representations of the  frustration index. Let $E_+$ (resp. $E_-$) collect all the positive (resp. negative) edges of $(V,E)$.\n  Note that up to a scale factor,  \\eqref{eq:frustration} is equivalent to solve $\\min\\limits_{A\\subset V}|E_+(A,V\\setminus A)|+|E_-(A)|+|E_-(V\\setminus A)|$, where $|E_+(A,V\\setminus A)|$ denotes the number of  positive edges between $A$ and $V\\setminus A$, while $|E_-(A)|$ indicates the number of negative edges in $A$. By Lov\\'asz extension,  the frustration index  is equivalent to\n$$|E_-|+\\min\\limits_{\\vec x\\ne0}\\frac{\\sum_{\\{i,j\\}\\in E_+}|x_i-x_j|+\\sum_{i\\in V}\\deg_i|x_i|-\\sum_{\\{i,j\\}\\in E_-}(|x_i-x_j|+|x_i+x_j|)}{\\|x\\|_\\infty}.$$ Also, \\eqref{eq:frustration} is equivalent to $|E_-|+\\min\\limits_{A\\subset V}(|E_+(A,V\\setminus A)|-|E_-(A,V\\setminus A)|)$, and by Lov\\'asz extension, the frustration index equals \n$$|E_-|+\\min\\limits_{\\vec x\\ne0}\\frac{\\sum_{\\{i,j\\}\\in E_+}|x_i-x_j|-\\sum_{\\{i,j\\}\\in E_-}|x_i-x_j|}{2\\|x\\|_\\infty}.$$\nOne can then apply  the   Dinkelbach-type scheme in Section \\ref{sec:algo}   straightforwardly  to  compute the  frustration index.\n\n \n\n\n\n\\begin{remark}\nWe should point out that the notion $|E_+(A)|$ (resp. $|E_-(A)|$) indicates the number of positive (resp. negative)  edges (unordered pairs) whose vertices are in $A$. Therefore, in   our paper, the values of  $|E_+(A)|$ and  $|E_-(A)|$ are \nhalf of those of \nAtay-Liu \\cite{AtayLiu}, in which they count the  ordered pairs. \n\\end{remark}\nBesides, by Theorem \\ref{thm:tilde-fg-equal-PQ} (or Theorem \\ref{thm:tilde-H-f}), we can derive another continuous formulation of the frustration index:\n$$\\min\\limits_{A\\subset V}|E_+(A,V\\setminus A)|+|E_-(A)|+|E_-(V\\setminus A)|=\\min\\limits_{x\\ne0}\\frac{\\sum\\limits_{\\{i,j\\}\\in E_+}|x_i-x_j|^\\alpha+\\sum\\limits_{\\{i,j\\}\\in E_-}(2\\|\\vec x\\|_\\infty-|x_i-x_j|)^\\alpha}{(2\\|x\\|_\\infty)^\\alpha}$$\nwhenever $0<\\alpha\\le 1$. It is interesting that by taking $\\alpha\\to 0^+$, we immediately get \n$$\\min\\limits_{A\\subset V}|E_+(A,V\\setminus A)|+|E_-(A)|+|E_-(V\\setminus A)|=\\min\\limits_{x\\ne0}\\sum\\limits_{\\{i,j\\}\\in E_+}\\mathrm{sign}(|x_i-x_j|)+\\sum\\limits_{\\{i,j\\}\\in E_-}\\mathrm{sign}(2\\|\\vec x\\|_\\infty-|x_i-x_j|).$$\n\n\\subsection{Modularity measure}\\label{sec:modularity-measure}\n\nFor a weighted graph $(V,(w_{ij})_{i,j\\in V})$, the   modularity measure \\cite{TMH18} is defined as\n$$Q(A)=\\sum_{i,j\\in A}w_{ij}-\\frac{\\vol(A)^2}{\\vol(V)},\\;\\;\\text{where }A\\subset V,$$\nand it satisfies the following equalities (see Theorem 3.7 and Theorem 3.9 in \\cite{TMH18}, respectively)\n\\begin{equation}\\label{eq:Q-modularity-measure}\n\\max\\limits_{A\\subset V}Q(A)=\\max\\limits_{x\\ne 0}\\frac{\\sum_{i,j\\in V}(\\frac{\\deg(i)\\deg(j)}{\\vol(V)}-w_{ij})|x_i-x_j|}{4\\|\\vec x\\|_\\infty}\n\\end{equation}\nand\n\\begin{equation}\\label{eq:Q\/mu-modularity-measure-mu}\n\\max\\limits_{A\\in\\ensuremath{\\mathcal{P}}(V)\\setminus\\{\\varnothing,V\\}}\\frac{Q(A)}{\\mu(A)\\mu(V\\setminus A)}=\\max\\limits_{\\sum_{i\\in V} \\mu_ix_i=0}\\frac{\\sum_{i,j\\in V}(\\frac{\\deg(i)\\deg(j)}{\\vol(V)}-w_{ij})|x_i-x_j|}{\\mu(V)\\sum_{i\\in V}\\mu_i|x_i|}.\n\\end{equation}\n\n\nIt is clear that  \\eqref{eq:Q-modularity-measure} can be obtained more directly by Theorem \\ref{thm:tilde-fg-equal}. We shall also state a new analog of \\eqref{eq:Q\/mu-modularity-measure-mu}: \n\\begin{equation}\\label{eq:Q\/mu-modularity-measure}\n\\max\\limits_{A\\in\\ensuremath{\\mathcal{P}}(V)\\setminus\\{\\varnothing,V\\}}\\frac{Q(A)}{\\mu(A)\\mu(V\\setminus A)}=\\max\\limits_{\\sum_{i\\in V} x_i=0}\\frac{\\sum_{i,j\\in V}(\\frac{\\deg(i)\\deg(j)}{\\vol(V)}-w_{ij})|x_i-x_j|}{\\sum_{i,j\\in V}\\mu_i\\mu_j|x_i-x_j|}\n\\end{equation}\nwhich can be derived straightforwardly by Theorem \\ref{thm:tilde-fg-equal}. \n\nBy Proposition \\ref{pro:Lovasz-f-pre}, we immediately obtain  Theorem 1 in \\cite{CRT20}, i.e., \nfor any $a,b>0$, \n$$ \\max\\limits_{-a\\le x_i\\le b,\\forall i}\\frac12\\sum_{i,j\\in V}(\\frac{\\deg(i)\\deg(j)}{\\vol(V)}-w_{ij})|x_i-x_j|=(a+b)\\max\\limits_{A\\subset V}Q(A).$$\n\n\n\\textbf{A relation with the frustration index}\n\n For a signed weighted graph with real weights $(w_{ij})_{i,j\\in V}$ and signs $s_{ij}=\\mathrm{sign}(w_{ij})$, we define the frustration index as \\begin{equation}\\label{eq:frustration-weight}\n\\min_{x_i\\in \\{-1,1\\},\\forall i} \\sum_{\\{i,j\\}}|w_{ij}|\\cdot|x_i-s_{ij}x_j|.\n\\end{equation} \n\nThe following result reveals an interesting relation between the modularity\nmeasure and the frustration index.\n\\begin{pro}For a weighted graph $(V,(w_{ij})_{i,j\\in V})$, let $\\tilde{w}_{ij}=w_{ij}-\\frac{\\deg(i)\\deg(j)}{\\vol(V)}$. \nIn the signed weighted graph $(V,(\\tilde{w}_{ij})_{i,j\\in V})$, \n $\\{i,j\\}$ is a positive (resp. negative) edge if $\\tilde{w}_{ij}>0$ (resp. $\\tilde{w}_{ij}<0$). Then, the frustration index of $(V,(\\tilde{w}_{ij})_{i,j\\in V})$ equals \n$2\\left(\\sum_{\\{i,j\\}:\\tilde{w}_{ij}<0}|\\tilde{w}_{ij}|-\\max\\limits_{A\\subset V}Q(A)\\right)$.\n\\end{pro}\n\n\\begin{proof}\nWe know from Section \\ref{sec:frustration} (or by Theorem \\ref{thm:tilde-fg-equal}) that the frustration index of $(V,(\\tilde{w}_{ij})_{i,j\\in V})$ equals \n$$2\\left(\\sum_{\\{i,j\\}:\\tilde{w}_{ij}<0}|\\tilde{w}_{ij}|+\\min\\limits_{\\vec x\\ne0}\\frac{\\sum_{i,j\\in V}\\tilde{w}_{ij}|x_i-x_j|}{4\\|x\\|_\\infty}\\right).$$\nThe proof is then completed by \\eqref{eq:Q-modularity-measure}.\n\\end{proof}\n\n\\subsection{Chromatic number}\\label{sec:chromatic-number}\nThe chromatic number (i.e., the smallest vertex coloring number) of a graph is the smallest number of colors needed to color the vertices\nso that no two adjacent vertices share the same color.\nGiven a simple connected graph $G=(V,E)$ with $\\#V=n$, its chromatic number $\\gamma(G)$ can be expressed as a global optimization on the $n$-power set of vertices:\n\\begin{equation}\\label{eq:coloring-number-sum}\n\\gamma(G)=\\min\\limits_{(A_1,\\cdots,A_n)\\in\\ensuremath{\\mathcal{P}}_n(V)}\\left\\{n\\sum_{i=1}^n\\#E(A_i)\n+\\sum_{i=1}^n\\sgn(\\#A_i)+n\\left(n-\\sum_{i=1}^n\\#A_i\\right)^2\\right\\}\n\\end{equation}\nand similarly, we get the following\n\\begin{pro}The chromatic number $\\gamma(G)$ of a finite simple graph $G=(V,E)$ satisfies\n\\begin{equation}\\label{eq:coloring-number}\n\\gamma(G)=\\min\\limits_{(A_1,\\cdots,A_n)\\in\\ensuremath{\\mathcal{P}}(V)^n}\\left\\{n\\sum_{i=1}^n\\#E(A_i)+\\sum_{i=1}^n\\sgn(\\#A_i)+n\\left(n-\\#\\bigcup_{i=1}^n A_i\\right)\\right\\}\n\\end{equation}\n\\end{pro}\n\n\\begin{proof\n Let $f:\\ensuremath{\\mathcal{P}}(V)^n\\to\\ensuremath{\\mathbb{R}}$ be defined by  $$f(A_1,\\cdots,A_n)=n\\sum_{i=1}^n\\#E(A_i)+\\sum_{i=1}^n\\sgn(\\#A_i)+n\\left(n-\\#\\bigcup_{i=1}^n A_i\\right).$$\n Let $\\{C_1,\\cdots,C_{\\gamma(G)}\\}$ be a proper coloring class of $G$, and set  $C_{\\gamma(G)+1}=\\cdots=C_n=\\varnothing$. Then we have $E(C_i)=\\varnothing$, $\\#\\cup_{i=1}^n C_i=n$,  $\\#C_i\\ge1$ for $1\\le i\\le \\gamma(G)$, and $\\#C_i=0$ for $i> \\gamma(G)$. In consequence, $f(C_1,\\cdots,C_n)=\\gamma(G)$. Thus, it suffices to prove $f(A_1,\\cdots,A_n)\\ge \\gamma(G)$ for any $(A_1,\\cdots,A_n)\\in\\ensuremath{\\mathcal{P}}(V)^n$.\n\nIf $\\bigcup_{i=1}^n A_i\\ne V$, then $f(A_1,\\cdots,A_n)\\ge n+1> \\gamma(G)$.\n\nIf there exist at least $\\gamma(G)+1$  nonempty sets $A_1,\\cdots,A_{\\gamma(G)+1}$, then $f(A_1,\\cdots,A_n)\\ge \\gamma(G)+1> \\gamma(G)$.\n\nSo we focus on the case that $\\bigcup_{i=1}^n A_i= V$ and $A_{\\gamma(G)+1}=\\cdots=A_n=\\varnothing$. If there further exists $i\\in\\{1,\\cdots,\\gamma(G)\\}$ such that $A_i=\\varnothing$, then by the definition of  the chromatic number, there is $j\\in \\{1,\\cdots,\\gamma(G)\\}\\setminus\\{i\\}$ with $E(A_j)\\ne \\varnothing$. So $f(A_1,\\cdots,A_n)\\ge n+1> \\gamma(G)$.\nAccordingly, each of $A_1,\\cdots,A_{\\gamma(G)}$ must be nonempty, and thus $f(A_1,\\cdots,A_n)\\ge\\gamma(G)$.\n\nAlso, when the equality $f(A_1,\\cdots,A_n)=\\gamma(G)$ holds, one may see from the above discussion that $A_1,\\cdots,A_{\\gamma(G)}$ are all independent sets of $G$ with $\\bigcup_{i=1}^n A_i\\ne V$.\n\n\\end{proof}\n\nNote that\n$$\\#\\bigcup_{i=1}^n V^t(\\vec x^i)=\\#\\{j\\in V:\\exists i \\text{ s.t. }x_{i,j}>t\\}=\\sum_{j=1}^n\\max\\limits_{i=1,\\cdots,n} 1_{x_{i,j}>t}=\\sum_{j=1}^n 1_{\\max\\limits_{i=1,\\cdots,n}x_{i,j}>t}$$\nSo the $n$-way Lov\\'asz extension of $\\#\\bigcup_{i=1}^n A_i$ is\n\\begin{align*}\n\\int_{\\min\\vec x}^{\\max\\vec x}\\#\\bigcup_{i=1}^n V^t(\\vec x^i)dt+\\min\\vec x\\#\\bigcup_{i=1}^n V(\\vec x^i)&=\\sum_{j=1}^n \\int_{\\min\\vec x}^{\\max\\vec x}1_{\\max\\limits_{i=1,\\cdots,n}x_{i,j}>t}dt+\\min\\vec x\\#V\n\\\\&=\\sum_{j=1}^n(\\max\\limits_{i=1,\\cdots,n}x_{i,j}-\\min\\vec x)+n\\min\\vec x\n\\\\&=\\sum_{j=1}^n\\max\\limits_{i=1,\\cdots,n}x_{i,j}\n\\end{align*}\nAnd the $n$-way disjoint-pair Lov\\'asz extension of $\\#\\bigcup_{i=1}^n A_i$ is $\\sum_{j=1}^n\\max\\limits_{i=1,\\cdots,n}|x_{i,j}|=\\sum_{j=1}^n\\|\\vec x^{,j}\\|_\\infty$.\n\nThe $n$-way Lov\\'asz extension of $\\sgn(\\#A_i)$ is\n\\begin{align*}\n\\int_{\\min\\vec x}^{\\max\\vec x}\\sgn(\\#V^t(\\vec x^i))dt+\\min\\vec x\\sgn(\\#V(\\vec x^i))&= \\int_{\\min\\vec x}^{\\max\\vec x^i}1dt+\\min\\vec x\\sgn(\\#V)\n\\\\&= \\max\\limits_{j=1,\\cdots,n}x_{i,j}-\\min\\vec x+\\min\\vec x= \\max\\limits_{j=1,\\cdots,n}x_{i,j}\n\\end{align*}\nand the $n$-way disjoint-pair  Lov\\'asz extension of $\\sgn(\\#A_i)$ is $\\|\\vec x^{i}\\|_\\infty$. Similarly, the $n$-way disjoint-pair Lov\\'asz extension of $\\#E(A_i)$ is $\\sum_{j\\sim j'}\\min\\{|x_{i,j}|,|x_{i,j'}|\\}$. Thus\n\\begin{align*}\nf^L(\\vec x)&=n\\sum_{i=1}^n\\sum_{j\\sim j'}\\min\\{|x_{i,j}|,|x_{i,j'}|\\} +\\sum_{i=1}^n\\|\\vec x^{i}\\|_\\infty+n\\left(n\\|\\vec x\\|_\\infty-\\sum_{j=1}^n\\|\\vec x^{,j}\\|_\\infty\\right)\n\\\\&\n=n\\sum_{i=1}^n\\left(\\|\\vec x^i\\|_{1,\\deg}-(I^+(\\vec x^i)+I^-(\\vec x^i))\/2\\right) +\\sum_{i=1}^n\\|\\vec x^{i}\\|_\\infty+n\\left(n\\|\\vec x\\|_\\infty-\\sum_{j=1}^n\\|\\vec x^{,j}\\|_\\infty\\right)\n\\\\&\n=n^2\\|\\vec x\\|_\\infty+n\\|\\vec x\\|_{1\\text{-}deg,1}+\\|\\vec x\\|_{\\infty,1}-nI_{\\pm,1}(\\vec x)-n\\|\\vec x\\|^{\\infty, 1}.\n\\end{align*}\nAccording to Proposition \\ref{pro:fraction-f\/g} on the multi-way Lov\\'asz extension, we get\n\\begin{equation}\\label{eq:chromatic-continuous}\n\\gamma(G)= n^2-\\sup\\limits_{\\vec x\\in\\ensuremath{\\mathbb{R}}^{n^2}\\setminus\\{\\vec 0\\}}\\frac{nI_{\\pm,1}(\\vec x)+n\\|\\vec x\\|^{\\infty, 1}-n\\|\\vec x\\|_{1\\text{-}deg,1}-\\|\\vec x\\|_{\\infty,1}}{\\|\\vec x\\|_\\infty}.\n\\end{equation}\n\n\\paragraph{Clique covering number }\n\nThe clique covering number of a graph $G$ is the minimal number of cliques in $G$ needed to cover the vertex set. It is equal to the chromatic number of the graph complement of $G$. Consequently, we can explicitly write down the continuous representation of a clique covering number by employing Theorem \\ref{thm:graph-numbers}.\n\n\n{ \\linespread{0.95} \\small ","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\\label{section 1}\n\nM. J. Golay introduced Golay complementary pairs (GCPs) in his work on multislit spectrometry \\cite{Golay51}. GCPs are sequence pairs having zero aperiodic autocorrelation sums (AACS) at all non-zero time shifts \\cite{Golay61}. Due to their ideal correlation sums, GCPs have found numerous applications in modern day communication systems \\cite{Davis1999,Paterson2000,Georghiades2001,farrel2003,abdi2007,lei2014}, Radar \\cite{Spano1996,Pezeshki2008}, etc. One of the main drawbacks of the GCPs are its availability for limited lengths \\cite{Borwein2000}. To overcome this drawback and to find the sequence pairs which depicts ``closest\" autocorrelation properties to that of the GCPs Fan \\textit{et al.} proposed Z-complementary pairs (ZCPs) in 2007 \\cite{Fan2007}. ZCPs are sequence pairs having zero AACS within a certain time-shift around the in-phase position \\cite{Fan2007}. In recent years lot of research has been done on the existence \\cite{Li2011}, systematic constructions \\cite{liu20141,liu2014,Adhikary2016,Adhikary2018,Adhikary2020,chen2017,Adhikary20201} and applications of ZCPs \\cite{Adhikary20191,chen20192}.\n\n\\subsection{Sequences pairs with zero periodic crosscorrelation zone}\nSince the autocorrelation of the sequence pair sum up to zero at all non-zero time-shifts (or time-shifts within a certain region in case of ZCPs) Golay sequences have been widely used to reduce peak-to-mean envelope power ratio in orthogonal frequency division multiplexing systems \\cite{Davis1999,Paterson2000}. However, the sequences own periodic autocorrelation plays an important role in some applications like synchronization and detection of the signal. Working in this direction, Gong \\textit{et al.} \\cite{Gong2013} investigated the periodic autocorrelation behaviour of a single Golay sequence in 2013. To be more specific, Gong \\textit{et al.} presented two constructions of Golay sequences of length $2^m$ each displaying a periodic zero autocorrelation zone (ZACZ) of $2^{m-2}$, and $2^{m-3}$, respectively, around the in-phase position \\cite{Gong2013}. In \\cite{Gong2013}, the authors also discussed the application of Golay sequences with large ZACZ for ISI channel estimation. Using Golay sequences with large ZACZ as channel estimation sequences (CES), the authors analysed the performance of\nGolay-sequence-aided channel estimation in terms of the error\nvariance and the classical Cramer-Rao lower bound (CRLB). The performance was also compared with the well known sequences (Frank-Zadoff-Chu sequences and $m$-sequences) which are generally used for ISI channel estimation. It was shown in \\cite{Gong2013} that when the channel impulse response (CIR) is within the ZACZ width then the variance of the Golay sequences attains the CRLB.\n\n\n\nInspired by the work of Gong \\textit{et al.} \\cite{Gong2013}, Chen \\textit{et al.} studied the zero cross-correlation zone among the Golay sequences in 2018 and proposed Golay-ZCZ sequence sets \\cite{Chen201811}. Golay-ZCZ sequence sets are sequence sets having periodic ZACZ for each sequences, periodic zero cross-correlation zone (ZCCZ) for any two sequences and also the aperiodic autocorrelation sum is zero for all non-zero time shifts. Specifically, Chen \\textit{et al.} gave a systematic construction of Golay-ZCZ sequence set consisting $2^k$ sequences, each of length $2^m$ and $\\min\\{ZACZ,ZCCZ\\}$ is $2^{m-k-1}$ \\cite{Chen201811}. However, the lengths of the GCPs with large ZCZs discussed in the works of Gong \\textit{et al.} and Chen \\textit{et al.} are all in the powers of two \\cite{Chen201811}. To the best of the authors knowledge, the problem of investigating the individual periodic autocorrelations of the GCPs and the periodic cross-correlations of the pairs when the length of the GCPs are non-power-of-two, remains largely open. An overview of of the previous works, which considers the periodic ZACZ of the individual sequences and ZCCZ of a GCP, is given in Table \\ref{Table duibi}.\n\n\\begin{table}\n  \\centering\n  \\caption{Golay sequences with periodic ZACZ and ZCCZ.}\n  \\label{Table duibi}\n\\begin{tabular}{|c|c|c|c|c|}\n  \\hline\n \n  Ref. & Length $N$ & Complementary sets size $M$ & ZACZ width & ZCCZ width  \\\\\\hline\n\n  \\hline\n\n  \\cite{Gong2013} & $2^m$ & $2$ & $2^{m-2}$ or $2^{m-3}$ & Not discussed  \\\\\n  \\hline\n\n  \\cite{Chen201811} & $2^m$ & $2^k$ & $2^{m-k-1}$ & $2^{m-k-1}$ \\\\\n  \\hline\n\n  Theorem 1 & $4N$ & $2$ & $N$ & $N$ \\\\\n  \\hline\n\\end{tabular}\n\\end{table}\n\nBased on the discussion on using Golay sequences as CES for ISI channel estimation in \\cite{Gong2013} it can be realised that our proposed constructions will add flexibility in choosing the Golay sequences of various lengths for using it as CES. Since,\nin practical scenarios, a longer training sequence will give\nrise to a higher training overhead, therefore, selection of the\ntraining length is a trade-off between channel estimation performance\nand training overhead. For example, let us consider\nthat in a practical scenario, the CIR is a length 10 vector. If only\nthe Golay sequences in \\cite{Gong2013} or \\cite{Chen201811} are considered, then one have to use a length 64 GCP which have a ZACZ width of 16. However, by our proposed constructions, a length 40 GCP which have a ZACZ width of 10 can be used as a CES as described in \\cite{Gong2013}. This will improve the system performance.\n\n\n\\subsection{Two-dimensional complementary array pairs}\nIn 1978, Ohyama \\textit{et al.} introduced two-dimensional (2D) sequence sets with zero side lobes in their work on image processing \\cite{Ohyama1978}. In 1985, H. D. Luke presented some iterative constructions of higher-dimensional complementary codes \\cite{Luke1985}. In 1990, Bomer \\textit{et al.} proposed perfect binary arrays \\cite{bomer1990}. In search of one-dimensional sequences with low autocorrelation magnitudes at non-zero time-shifts, in 2007, Jedwab and Parker made a remarkable progress by generalising the problem to multiple dimensions \\cite{Jedwab20071}. Instead of searching sequences with low correlation properties in one dimension, the authors analysed the possibility of existence of such sequence arrays in multiple dimensions in the hope of a larger existence patterns. In \\cite{Jedwab20071} the authors presented a systematic construction of $m$ dimensional Golay complementary array pairs (GCAPs) from an $m+1$ dimensional GCAPs. 2D- GCAPs are array pairs having the 2D autocorrelations of the constituent arrays sum up to zero for all non-zero time-shifts.\n\nGeneralizing the concept of ZCZ in 2D, Fan \\textit{et. al} introduced binary array set with ZCZ in 2001 \\cite{Tang2001}. In 2002 Hayashi proposed a class of 2D- binary sequences with ZCZ \\cite{Hayashi2004}. In 2010 Cheng \\textit{et al.} proposed another new class of class of 2D- binary sequences with ZCZ \\cite{Cheng2010}. Recently in 2019 Chen \\textit{et al.} proposed a systematic construction of 2D- Z-complementary array pairs (ZCAPs) \\cite{pai2019}. However, the behavior of autocorrelation\nof a single Golay array is still unknown. To the best of the authors knowledge, the problem of investigating the periodic 2D- autocorrelations of the constituent arrays of the 2D- GCAPs and the periodic 2D- cross-correlations of the 2D- GCAPs, remains largely open.\n\n\n\\subsection{Contributions}\n\nMotivated by the works of Gong \\textit{et al.} \\cite{Gong2013} and Chen \\textit{et al.} \\cite{Chen201811} we propose a systematic construction of GCPs of length non-power-of-two, where the individual sequences have a periodic ZACZ and the periodic cross-correlation of the sequence pairs also have a ZCCZ. We also extend the ideas to construct 2D-GCAPs.  To be more specific we make  the following contributions in this paper:\n\\begin{enumerate}\n\t\\item Assuming a GCP of length $N$ exists, we  systematically construct GCPs of length $4N$. The proposed GCPs have $Z_{\\min}=N+1$, where $Z_{\\min}=\\min\\{ZACZ,ZCCZ\\}$.\n\t\\item We  also systematically constructe a 2D GCAP of size $s_1\\times 4s_2$, assuming a 2D- GCAP of size $s_1 \\times s_2$ exists. The designed 2D- GCAPs have a $2D\\text{-}Z_{\\min}=s_1\\times (s_2+1)$, where $2D\\text{-}Z_{\\min}=\\min\\{2D\\text{-}ZACZ,2D\\text{-}ZCCZ\\}$.\n\t\\item We propose a systematic construction of 2D GCAP of size $4s_1\\times 4s_2$, assuming a 2D- GCAP of size $s_1 \\times s_2$ exists. The designed 2D- GCAPs have a $2D\\text{-}Z_{\\min}=(s_1+1)\\times (s_2+1)$, where $2D\\text{-}Z_{\\min}=\\min\\{2D\\text{-}ZACZ,2D\\text{-}ZCCZ\\}$.\n\\end{enumerate}\n\n\n\n\\subsection{Organization}\nThe rest of the paper is organized as follows. In Section \\ref{section 2}, some useful notations and preliminaries are recalled.\nIn Section \\ref{section 3}, a systematic construction for GCPs of lengths non-power-of-two with large periodic ZACZ and ZCCZ is proposed.\nIn Section \\ref{section 4}, we extended the construction to 2D- GCAPs.\nFinally, we conclude the paper in Section \\ref{section 5}.\n\n\n\n\n\n\n\n\n\\section{Preliminaries}\\label{section 2}\n\\begin{definition}\n\tLet $\\mathcal{A}=[A_{i,j}]$ and $\\mathcal{B}=[B_{i,j}]$, for $0\\leq i <L_1$ and $0\\leq j <L_2$, be two unimodular complex-valued 2D- arrays, each of size $L_1\\times L_2$. Then, for $-L_1<\\tau_1<L_1$ and $-L_2<\\tau_2<L_2$, the 2D- periodic crosscorrelation function is defined as follows:\n\t\\begin{equation}\n      R_{\\mathcal{A},\\mathcal{B}}(\\tau_1,\\tau_2)=\\sum\\limits_{i=0}^{L_1-1}\\sum\\limits_{j=0}^{L_2-1}A_{i,j}B^*_{(i+\\tau_1)\\pmod {L_1},(j+\\tau_2)\\pmod {L_2}},\n\t\\end{equation}\n\twhere $(^*)$ denotes the complex conjugate of $B_{(i+\\tau_1)\\pmod {L_1},(j+\\tau_2)\\pmod {L_2}}$. When $L_1=1$, then the above definition reduces to a 1D- periodic cross-correlation function. When $\\mathcal{A}=\\mathcal{B}$, it is called periodic autocorrelation function, and is denoted by $R_{\\mathcal{A}}$ for short.\n\\end{definition}\n\n\\begin{definition}\n\tFor $0\\leq i <L_1$ and $0\\leq j <L_2$, let $\\mathcal{A}=[A_{i,j}]$ and $\\mathcal{B}=[B_{i,j}]$ be two unimodular complex-valued 2D- arrays, each of size $L_1\\times L_2$. Then the 2D- aperiodic crosscorrelation function is defined as follows:\n\t\\begin{equation}\n\tC_{\\mathcal{A},\\mathcal{B}}(\\tau_1,\\tau_2)=\\begin{cases}\n\t\\sum\\limits_{i=0}^{L_1-1-\\tau_1}\\sum\\limits_{j=0}^{L_2-1-\\tau_2}A_{i,j}B^*_{(i+\\tau_1),(j+\\tau_2)}, & 0\\leq \\tau_1<L_1,0\\leq \\tau_2<L_2;\\\\\n\t\\sum\\limits_{i=0}^{L_1-1+\\tau_1}\\sum\\limits_{j=0}^{L_2-1-\\tau_2}A_{i-\\tau_1,j}B^*_{i,(j+\\tau_2)}, & -L_1< \\tau_1<0,0\\leq \\tau_2<L_2;\\\\\n\t\\sum\\limits_{i=0}^{L_1-1-\\tau_1}\\sum\\limits_{j=0}^{L_2-1+\\tau_2}A_{i,j-\\tau_2}B^*_{(i+\\tau_1),j}, & 0\\leq \\tau_1<L_1,-L_2< \\tau_2<0;\\\\\n\t\\sum\\limits_{i=0}^{L_1-1+\\tau_1}\\sum\\limits_{j=0}^{L_2-1+\\tau_2}A_{i-\\tau_1,j-\\tau_2}B^*_{i,j}, & -L_1< \\tau_1<0,-L_2< \\tau_2<0;\n\t\\end{cases}\n\t\\end{equation}\n\twhere $(^*)$ denotes the complex conjugate of the element. When $L_1=1$, then the above definition reduces to a 1D- aperiodic cross-correlation function. When $\\mathcal{A}=\\mathcal{B}$, it is called aperiodic autocorrelation function, and is denoted by $C_{\\mathcal{A}}$.\n\\end{definition}\n\n\n\n\n\n\n\n\nUsing the above definitions, we have the following property.\n\\begin{property}\nLet $\\mathcal{A}$ be a 2D- array with size $L_1\\times L_2$, then for $0\\leq \\tau_1<L_1$ and $0\\leq \\tau_2<L_2$, we have\n  \\begin{equation}\n  \\begin{split}\n  C_{\\mathcal{A}}(\\tau_{1},\\tau_2) &=C_{\\mathcal{A}}^*(-\\tau_{1},-\\tau_2);\\\\\n  R_{\\mathcal{A}}(\\tau_{1},\\tau_2) &=R_{\\mathcal{A}}^*(-\\tau_{1},-\\tau_2).\n  \\end{split}\n  \\end{equation}\n\\end{property}\n\n\n\\begin{definition}\\label{Definition-GCAP}\nA pair of 2D- arrays $\\mathcal{A}$ and $\\mathcal{B}$, each of size $L_1\\times L_2$ is called a Golay complementary array pair (GCAP) if\n\\begin{equation}\n  C_\\mathcal{A}(\\tau_1,\\tau_2)+C_\\mathcal{B}(\\tau_1,\\tau_2)=0, \\hbox{ for all } (\\tau_1,\\tau_2)\\neq(0,0).\n\\end{equation}\nIn particular, when $L_1=1$, GCAP is reduced to a Golay complementary pair (GCP), denoted by $(\\mathbf{a,b})$.\n\\end{definition}\n\n\\begin{definition}\\label{Definition-GAMate}\nLet $(\\mathcal{A,B})$ and $(\\mathcal{C,D})$ be two GCAPs, each of size $L_1\\times L_2$. Then\n$(\\mathcal{C,D})$ is called a Golay array mate of $(\\mathcal{A,B})$ if \\cite{Matsufuji2010}\n$$C_\\mathcal{A,C}(\\tau_1,\\tau_2)+C_\\mathcal{B,D}(\\tau_1,\\tau_2)=0,$$\nfor any $-L_1<\\tau_1<L_1$ and $-L_2<\\tau_2<L_2$.\nIn particular, when $L_1=1$, $(\\mathcal{A,B})$ reduces to a GCP and $(\\mathcal{C,D})$ reduces to mate of the GCP $(\\mathcal{A,B})$.\n\\end{definition}\n\n\\begin{lemma}\nLet $(\\mathcal{A,B})$ be GCAP of size $L_1\\times L_2$. Then one of the mate of $(\\mathcal{A,B})$ is as follows:\n\\begin{equation}\n\t(\\mathcal{C,D}) =(\\overleftarrow{\\mathcal{B^*}},-\\overleftarrow{\\mathcal{A^*}}),\n\\end{equation}\nwhere $\\overleftarrow{\\mathcal{A}}$ denote reverse of array $\\mathcal{A}$ at every dimension and $(^*)$ represents complex conjugation.\n\\end{lemma}\n\n\n\\begin{definition}\\label{Definition-GolayZCZ}\nLet ($\\mathcal{A}$,$\\mathcal{B}$) be a GCAP of size $L_1\\times L_2$. Let the 2D- periodic autocorrelation function of $\\mathcal{A}$ and $\\mathcal{B}$ have a 2D- ZACZ of $Z_{k_0}$ and $Z_{k_1}$, respectively. Also let the 2D- periodic cross-correlation function have a 2D- ZCCZ of $Z_{k_2}$ when the cross-correlation between $\\mathcal{A}$ and $\\mathcal{B}$ is considered. Then ($\\mathcal{A}$,$\\mathcal{B}$) is a GCAP having a 2D- periodic ZCZ of $Z_{\\min}=\\min\\{Z_{k_0},Z_{k_1},Z_{k_2}\\}$ and we denote it as\n$(L_1\\times L_2,Z_{\\min})$- GCAP. In other words, the 2D- array pair ($\\mathcal{A}$,$\\mathcal{B}$) is a $(L_1\\times L_2,Z_{\\min})$- GCAP, having 2D- periodic ZCZ $Z_{\\min}$ if\n\\begin{equation}\n\\begin{split}\n&\\text{C1: }C_\\mathcal{A}(\\tau_1,\\tau_2)+C_\\mathcal{B}(\\tau_1,\\tau_2)=0, \\hbox{ for all } (\\tau_1,\\tau_2)\\neq(0,0);\\\\\n&\\text{C2: }R_\\mathcal{A}(\\tau_1,\\tau_2)=R_\\mathcal{B}(\\tau_1,\\tau_2)=0, \\hbox{ for } |\\tau_1|<Z_{\\min},|\\tau_2|<Z_{\\min} \\text{ and }(\\tau_1,\\tau_2)\\neq(0,0);\\\\\n&\\text{C3: }R_{\\mathcal{A},\\mathcal{B}}(\\tau_1,\\tau_2)=0, \\hbox{ for } |\\tau_1|<Z_{\\min},|\\tau_2|<Z_{\\min},\n\\end{split}\n\\end{equation}\nwhere, $Z_{\\min}$ is defined as above. When $L_1=1$, we get the conditions for $(L_2,Z_{\\min})$- GCPs having 1D- periodic ZCZ width of $Z_{\\min}$.\n\\end{definition}\n\n\n\n\\section{New Construction of Golay Complementary Pair with ZCZ}\\label{section 3}\nIn this section we propose a new class of GCPs, where each of the sequences have a periodic ZACZ and also have a periodic ZCCZ when the cross-correlation between the pairs are considered. Please note that GCPs are 1D- GCAPs where $L_1=1$ and $L_2=N$ (in line with Section II). In this section, $\\mathbf{a}\\|\\mathbf{b}$ denotes the concatenation of sequences $\\mathbf{a}$ and $\\mathbf{b}$ and $Z_{\\min}$ is usually 1D- $Z_{\\min}$. Before introduce the construction, we need the following lemma.\n\n\\begin{lemma}\\label{lem2}\n\tLet $(\\mathcal{A,B})$ a be GCAP of size $s_1\\times s_2$ and $(\\mathcal{C,D})$ be one of the Golay array mate of $(\\mathcal{A,B})$. Let\n\t\\begin{equation}\n\t\\mathcal{P}=\n\t\\left[\n\t\\begin{array}{rrrr}\n\tx_1\\mathcal{A} & x_2\\mathcal{B} & x_3\\mathcal{A} & x_4\\mathcal{B} \\\\\n\t\\end{array}\n\t\\right],\n\t\\mathcal{Q}=\n\t\\left[\n\t\\begin{array}{rrrr}\n\tx_1\\mathcal{C} & x_2\\mathcal{D} & x_3\\mathcal{C} & x_4\\mathcal{D} \\\\\n\t\\end{array}\n\t\\right],\n\t\\end{equation}\n\twhere $x_1,x_2,x_3,x_4\\in\\{+1,-1\\}$. Then $(\\mathcal{P,Q})$ is a GCAP of size $s_1\\times 4s_2$, if the following condition holds:\n\t\\begin{equation}\n\t\tx_1x_2+x_3x_4=0.\n\t\\end{equation}\n\\end{lemma}\n\\begin{IEEEproof} Let us consider $0\\leq\\tau_1\\leq s_1-1$ and $0\\leq\\tau_2\\leq s_2-1$. Then we have\n\t\\begin{equation}\n\t\\begin{split}\n\tC_\\mathcal{P}(\\tau_1,\\tau_2)=&2\\big(C_\\mathcal{A}(\\tau_1,\\tau_2) +C_\\mathcal{B}(\\tau_1,\\tau_2)\\big)+(x_1x_2+x_3x_4)C_\\mathcal{B,A}^*(\\tau_1,s_2-\\tau_2) +x_2x_3\\rho_\\mathcal{A,B}^*(\\tau_1,s_2-\\tau_2), \\\\\n\tC_\\mathcal{Q}(\\tau_1,\\tau_2)=&2\\big(C_\\mathcal{C}(\\tau_1,\\tau_2) +C_\\mathcal{D}(\\tau_1,\\tau_2)\\big)+(x_1x_2+x_3x_4)C_\\mathcal{D,C}^*(\\tau_1,s_2-\\tau_2) +x_2x_3\\rho_\\mathcal{C,D}^*(\\tau_1,s_2-\\tau_2).\n\t\\end{split}\n\t\\end{equation}\n\tHence, for $0\\leq\\tau_1\\leq s_1-1$ and $0\\leq\\tau_2\\leq s_2-1$, we have\n\t\\begin{equation}\n\t\tC_\\mathcal{P}(\\tau_1,\\tau_2)+C_\\mathcal{Q}(\\tau_1,\\tau_2)=4\\big(C_\\mathcal{A}(\\tau_1,\\tau_2) +C_\\mathcal{B}(\\tau_1,\\tau_2)\\big).\n\t\\end{equation}\n\nConsider $0\\leq\\tau_1\\leq s_1-1$ and $s_2\\leq\\tau_2\\leq 2s_2-1$. Then one has\n\\begin{equation}\n\\begin{split}\nC_\\mathcal{P}(\\tau_1,\\tau_2)=& (x_1x_2+x_3x_4)C_\\mathcal{A,B}(\\tau_1,\\tau_2-s_2) +x_2x_3C_\\mathcal{B,A}(\\tau_1,\\tau_2-s_2) +x_1x_3C_\\mathcal{A}^*(\\tau_1,2s_2-\\tau_2)\\\\&\\hspace{1in} +x_2x_4C_\\mathcal{B}^*(\\tau_1,2s_2-\\tau_2), \\\\\nC_\\mathcal{Q}(\\tau_1,\\tau_2)=& (x_1x_2+x_3x_4)C_\\mathcal{C,D}(\\tau_1,\\tau_2-s_2) +x_2x_3C_\\mathcal{D,C}(\\tau_1,\\tau_2-s_2) +x_1x_3C_\\mathcal{C}^*(\\tau_1,2s_2-\\tau_2)\\\\&\\hspace{1in} +x_2x_4C_\\mathcal{D}^*(\\tau_1,2s_2-\\tau_2).\n\\end{split}\n\\end{equation}\n\nHence, for $0\\leq\\tau_1\\leq s_1-1$ and $s_2\\leq\\tau_2\\leq 2s_2-1$, we have\n\\begin{equation}\nC_\\mathcal{P}(\\tau_1,\\tau_2)+C_\\mathcal{Q}(\\tau_1,\\tau_2)=0.\n\\end{equation}\n\n\nConsider $0\\leq\\tau_1\\leq s_1-1$ and $2s_2\\leq\\tau_2\\leq 3s_2-1$. Then we have\n\\begin{equation}\n\\begin{split}\nC_\\mathcal{P}(\\tau_1,\\tau_2)=& x_1x_3C_\\mathcal{A}(\\tau_1,\\tau_2-2s_2) +x_2x_4C_\\mathcal{B}(\\tau_1,\\tau_2-2s_2) +x_1x_4C_\\mathcal{B,A}^*(\\tau_1,3s_2-\\tau_2), \\\\\nC_\\mathcal{Q}(\\tau_1,\\tau_2)=& x_1x_3C_\\mathcal{C}(\\tau_1,\\tau_2-2s_2) +x_2x_4C_\\mathcal{D}(\\tau_1,\\tau_2-2s_2) +x_1x_4C_\\mathcal{D,C}^*(\\tau_1,3s_2-\\tau_2).\n\\end{split}\n\\end{equation}\n\nHence, for $0\\leq\\tau_1\\leq s_1-1$ and $2s_2\\leq\\tau_2\\leq 3s_2-1$, we have\n\\begin{equation}\nC_\\mathcal{P}(\\tau_1,\\tau_2)+C_\\mathcal{Q}(\\tau_1,\\tau_2)=0.\n\\end{equation}\n\nConsider $0\\leq\\tau_1\\leq s_1-1$ and $3s_2\\leq\\tau_2\\leq 4s_2-1$. Then we have\n\\begin{equation}\n\\begin{split}\nC_\\mathcal{P}(\\tau_1,\\tau_2)=& x_1x_4C_\\mathcal{A,B}(\\tau_1,\\tau_2-3s_2), \\\\\nC_\\mathcal{Q}(\\tau_1,\\tau_2)=& x_1x_4C_\\mathcal{C,D}(\\tau_1,\\tau_2-3s_2),.\n\\end{split}\n\\end{equation}\n\nHence, for $0\\leq\\tau_1\\leq s_1-1$ and $3s_2\\leq\\tau_2\\leq 4s_2-1$, we have\n\\begin{equation}\nC_\\mathcal{P}(\\tau_1,\\tau_2)+C_\\mathcal{Q}(\\tau_1,\\tau_2)=0.\n\\end{equation}\n\nThe other cases can also be proved similarly.\n\\end{IEEEproof}\n\n\n\n\\begin{construction}\\label{Construction-seq-ZCZ}\nLet $(\\mathbf{a,b})$ be a GCP of length $N$ and $(\\mathbf{c,d})$ be the Golay mate of $(\\mathbf{a,b})$, then define\n\\begin{equation}\n  \\begin{split}\n    \\mathbf{p}=~&x_1\\cdot\\mathbf{a}~\\|~x_2\\cdot\\mathbf{b}~\\|~x_3\\cdot\\mathbf{a}~\\|x_4\\cdot\\mathbf{b}, \\\\\n    \\mathbf{q}=~&x_1\\cdot\\mathbf{c}~\\|~x_2\\cdot\\mathbf{d}~\\|~x_3\\cdot\\mathbf{c}~\\|x_4\\cdot\\mathbf{d},\n  \\end{split}\n\\end{equation}\nwhere $x_1,x_2,x_3,x_4\\in\\{+1,-1\\}$.\n\\end{construction}\n\n\\begin{theorem}\n  Let $(\\mathbf{p,q})$ be a sequence pair generated via Construction \\ref{Construction-seq-ZCZ}. Then $(\\mathbf{p,q})$ is a GCP of length $4N$ with $Z_{\\min}=N+1$, if $x_1,x_2,x_3,x_4$ if the following condition holds:\n  \\begin{equation}\n  \tx_1x_2+x_3x_4=0.\n  \\end{equation}\n\\end{theorem}\n\\begin{IEEEproof}\nUsing Lemma \\ref{lem2} for 1D cases, i.e., considering $s_1=1$ and $s_2=N$, we can prove that $(\\mathbf{p,q})$ is a GCP of length $4N$.\t\n\nNext, we have to prove the other two conditions of Definition \\ref{Definition-GolayZCZ}. Consider $1\\leq\\tau\\leq N$, then we have\n\\begin{equation}\n\\begin{split}\nR_\\mathbf{p}(\\tau)=& \\sum_{k=0}^{4N-1} p_kp_{k+\\tau}^*\\\\\n=& \\sum_{k=0}^{N-1-\\tau} a_ka_{k+\\tau}^* +\\sum_{k=N-\\tau}^{N-1} x_1a_kx_2^*b_{k-(N-\\tau)}^* +\\sum_{k=0}^{N-1-\\tau} b_kb_{k+\\tau}^* +\\sum_{k=N-\\tau}^{N-1} x_2b_kx_3^*a_{k-(N-\\tau)}^* +\\\\\n& \\sum_{k=0}^{N-1-\\tau} a_ka_{k+\\tau}^* +\\sum_{k=N-\\tau}^{N-1} x_3a_kx_4^*b_{k-(N-\\tau)}^* +\\sum_{k=0}^{N-1-\\tau} b_kb_{k+\\tau}^* +\\sum_{k=N-\\tau}^{N-1} x_4b_kx_1^*a_{k-(N-\\tau)}^* \\\\\n=&2\\big(\\rho_\\mathbf{a}(\\tau) +\\rho_\\mathbf{b}(\\tau)\\big). \\\\\n\\end{split}\n\\end{equation}\nSimilarly for $1\\leq\\tau\\leq N$, we have\n\\begin{equation}\n\\begin{split}\nR_\\mathbf{q}(\\tau)=&2\\big(C_\\mathbf{c}(\\tau) +C_\\mathbf{d}(\\tau)\\big), \\\\\nR_\\mathbf{p,q}(\\tau)=&2\\big(C_\\mathbf{a,c}(\\tau) +C_\\mathbf{b,d}(\\tau)\\big).\n\\end{split}\n\\end{equation}\nSince $(\\mathbf{a},\\mathbf{b})$ is a GCP and $(\\mathbf{c},\\mathbf{d})$ is one of the complementary mates of $(\\mathbf{a},\\mathbf{b})$, $R_\\mathbf{p}(\\tau)=R_\\mathbf{q}(\\tau)=0,\\hbox{ for all } 1\\leq\\tau\\leq N,$ and $R_\\mathbf{p,q}(\\tau)=0,\\hbox{ for all } 0\\leq\\tau\\leq N$.\n\nTherefore, $(\\mathbf{p,q})$ is a GCP of length $4N$ with $Z_{\\min}=N+1$.\n\\end{IEEEproof}\n\n\n\n\\begin{example}\\label{ex1 binary}\nLet $(\\mathbf{a,b})$ be a binary GCP of length 10, given by $\\mathbf{a}=(1,1,-1,1,1,1,1,1,-1,-1),\\mathbf{b}=(1,1,-1,1,-1,1,-1, -1,1,1)$.\nThen $(\\mathbf{c,d})=(\\overleftarrow{\\mathbf{b}^*},-\\overleftarrow{\\mathbf{a}^*})$ is a Golay mate of $(\\mathbf{a,b})$. Define\n\\begin{equation}\n  \\begin{split}\n    \\mathbf{p}=~&\\mathbf{a}~\\|~\\mathbf{b}~\\|~\\mathbf{a}~\\|-\\mathbf{b}, \\\\\n    \\mathbf{q}=~&\\mathbf{c}~\\|~\\mathbf{d}~\\|~\\mathbf{c}~\\|-\\mathbf{d}.\n  \\end{split}\n\\end{equation}\nThen $(\\mathbf{p,q})$ is a GCP of length $40$ with $Z_{\\min}=11$, because\n\\begin{equation}\n  \\begin{split}\n    \\big(R_\\mathbf{p}(\\tau)\\big)_{\\tau=0}^{39}=&(40,\\mathbf{0}_{10},-4,-8,4,8, -4,0,4,0,12,0,12,0,4,0,-4,8,4,-8,-4,\\mathbf{0}_{10}), \\\\\n    \\big(R_\\mathbf{q}(\\tau)\\big)_{\\tau=0}^{39}=&(40,\\mathbf{0}_{10},4,8,-4,-8, 4,0,-4,0,-12,0,-12,0,-4,0,4,-8,-4,8,4,\\mathbf{0}_{10}), \\\\\n    \\big(R_\\mathbf{p,q}(\\tau)\\big)_{\\tau=0}^{39}=&(\\mathbf{0}_{11},-4,-8,4, 16,4,0,4,-8,-4,0,4,-8,12,0,12,0,-4,8,4,\\mathbf{0}_{10}).\n  \\end{split}\n\\end{equation}\n\\end{example}\n\n\n\n\\begin{example}\\label{ex1}\nLet $\\mathbf{a}=(1,1,-1),\\mathbf{b}=(1,i,1)$, then $(\\mathbf{a,b})$ be a quadriphase GCP of length 3.\nThen $(\\mathbf{c,d})=(\\overleftarrow{\\mathbf{b}^*},-\\overleftarrow{\\mathbf{a}^*})$ is a Golay mate of $(\\mathbf{a,b})$. Define\n\\begin{equation}\n  \\begin{split}\n    \\mathbf{p}=~&\\mathbf{a}~\\|~\\mathbf{b}~\\|~\\mathbf{a}~\\|-\\mathbf{b} =(1,1,-1,1,i,1,1,1,-1,-1,-i,-1), \\\\\n    \\mathbf{q}=~&\\mathbf{c}~\\|~\\mathbf{d}~\\|~\\mathbf{c}~\\|-\\mathbf{d} =(1,-i,1,1,-1,-1,1,-i,1,-1,1,1).\n  \\end{split}\n\\end{equation}\nThen $(\\mathbf{p,q})$ is a GCP of length $12$ with $Z_{\\min}=4$, because\n\\begin{equation}\n  \\begin{split}\n    \\big(R_\\mathbf{p}(\\tau)\\big)_{\\tau=0}^{11}=&(12,0,0,0,-4,0,0,0,-4,0,0,0), \\\\\n    \\big(R_\\mathbf{q}(\\tau)\\big)_{\\tau=0}^{11}=&(12,0,0,0,4,0,0,0,4,0,0,0), \\\\\n    \\big(R_\\mathbf{p,q}(\\tau)\\big)_{\\tau=0}^{11}=&(0,0,0,0,-4,4-4i,4i,4+4i,4,0,0,0).\n  \\end{split}\n\\end{equation}\nThe periodic correlation magnitudes of $(12,4)$- GCP ($\\mathbf{p,q}$) are shown in Fig. \\ref{fig1}.\n\\begin{figure}[ht]\n  \\centering\n  \\includegraphics[width=.7\\textwidth]{Example1.pdf}\n  \\caption{A glimpse of the 1D- periodic correlations of the proposed GCP given in Example \\ref{ex1}.}\\label{fig1}\n\\end{figure}\n\\end{example}\n\n\\begin{remark}\n\tPlease note that binary GCP of length 40 and quadriphase GCP of length 12 with ZCZ has not been previously reported in the literature. By taking GCP $(\\mathbf{a,b})$ of length $2^m$, then length of $(\\mathbf{p,q})$ is $2^{m+2}$ and the generated GCP $(\\mathbf{p,q})$ is reduced to the GCP in \\cite{Gong2013}.\n\\end{remark}\n\n\n\n\\section{New Construction of Golay Complementary Array with ZCZ}\\label{section 4}\nIn this section we have proposed new sets of GCAPs, where each of the arrays have a 2D- periodic ZACZ and also have a 2D- periodic ZCCZ when the cross-correlation between the array pairs are considered. In this section $Z_{\\min}$ are usually 2D- $Z_{\\min}$.\n\\begin{construction}\\label{Construction-array-ZCZ1}\nLet $(\\mathcal{A,B})$ be a GCAP of size $s_1\\times s_2$ and $(\\mathcal{C,D})$ be the Golay array mate of $(\\mathcal{A,B})$, then define\n\\begin{equation}\n  \\mathcal{P}=\n  \\left[\n    \\begin{array}{rrrr}\n      x_1\\mathcal{A} & x_2\\mathcal{B} & x_3\\mathcal{A} & x_4\\mathcal{B} \\\\\n    \\end{array}\n  \\right],\n  \\mathcal{Q}=\n  \\left[\n    \\begin{array}{rrrr}\n      x_1\\mathcal{C} & x_2\\mathcal{D} & x_3\\mathcal{C} & x_4\\mathcal{D} \\\\\n    \\end{array}\n  \\right],\n\\end{equation}\nwhere $x_1,x_2,x_3,x_4\\in\\{+1,-1\\}$.\n\\end{construction}\n\n\\begin{theorem}\\label{Theorem-array-ZCZ1}\n  Let $(\\mathcal{P},\\mathcal{Q})$ be an array pair constructed via Construction \\ref{Construction-array-ZCZ1}. Then $(\\mathcal{P},\\mathcal{Q})$ is a GCAP of size $s_1\\times 4s_2$ with $Z_{\\min}=s_1\\times (s_2+1)$, if $x_1,x_2,x_3,x_4$ if the following condition holds:\n  \\begin{equation}\n  \tx_1x_2+x_3x_4=0.\n  \\end{equation}\n\\end{theorem}\n\\begin{IEEEproof}\n\tUsing Lemma \\ref{lem2}, we can show that $(\\mathcal{P,Q})$ is a GCAP of size $s_1\\times4s_2$.\n\t\n\t\nNext, we have to prove the other two conditions of Definition \\ref{Definition-GolayZCZ}. Consider\t$-(s_1-1)\\leq\\tau_1\\leq s_1-1,-(s_2-1)\\leq\\tau_2\\leq (s_2-1)$. Then we have\n\\begin{equation}\n\\begin{split}\nR_\\mathcal{P}(\\tau_1,\\tau_2)=&2\\big(C_\\mathcal{A}(\\tau_1,\\tau_2) +C_\\mathcal{B}(\\tau_1,\\tau_2) +C_\\mathcal{A}(\\tau_1-s_1,\\tau_2) +C_\\mathcal{B}(\\tau_1-s_1,\\tau_2)\\big), \\\\\nR_\\mathcal{Q}(\\tau_1,\\tau_2)=&2\\big(C_\\mathcal{C}(\\tau_1,\\tau_2) +C_\\mathcal{D}(\\tau_1,\\tau_2) +C_\\mathcal{C}(\\tau_1-s_1,\\tau_2) +C_\\mathcal{D}(\\tau_1-s_1,\\tau_2)\\big), \\\\\nR_\\mathcal{P,Q}(\\tau_1,\\tau_2)=&2\\big(C_\\mathcal{A,C}(\\tau_1,\\tau_2) +C_\\mathcal{B,D}(\\tau_1,\\tau_2) +C_\\mathcal{A,C}(\\tau_1-s_1,\\tau_2) +C_\\mathcal{B,D}(\\tau_1-s_1,\\tau_2)\\big),\n\\end{split}\n\\end{equation}\nSince $(\\mathcal{A},\\mathcal{B})$ is a GCAP and $(\\mathcal{C},\\mathcal{D})$ is one of the complementary mates of $(\\mathcal{A},\\mathcal{B})$, $R_\\mathcal{P}(\\tau_1,\\tau_2)=R_\\mathcal{Q}(\\tau_1,\\tau_2)=0,\\hbox{ for all } -(s_1-1)\\leq\\tau_1\\leq (s_1-1),-(s_2-1)\\leq\\tau_2\\leq (s_2-1) \\text{ except }(\\tau_1,\\tau_2)\\neq(0,0),$ and $R_\\mathcal{P,Q}(\\tau_1,\\tau_2)=0,\\hbox{ for all } -(s_1-1)\\leq\\tau_1\\leq s_1-1,-(s_2-1)\\leq\\tau_2\\leq (s_2-1)$.\n\nTherefore, $(\\mathcal{P,Q})$ is a GCAP with periodic $Z_{\\min}=s_1\\times (s_2+1)$.\n\\end{IEEEproof}\n\n\\begin{example}\\label{ex2}\n  Let $\\mathcal{A}=\\left[\\begin{array}{rrr} 1 & 1 & -1 \\\\ -1 & -i & -1 \\\\ \\end{array}\\right],\\mathcal{B}=\\left[\\begin{array}{rrr} 1 & 1 & -1 \\\\ 1 & i & 1 \\\\ \\end{array}\\right]$, then $(\\mathcal{A,B})$ is a quaternary GCAP of size $2\\times3$.\n  Then $(\\mathcal{C,D})= (\\overleftarrow{\\mathcal{B^*}},-\\overleftarrow{\\mathcal{A^*}})$ is a Golay array mate of $(\\mathcal{A,B})$. Define\n\\begin{equation}\n  \\mathcal{P}=\n  \\left[\n    \\begin{array}{rrrr}\n      \\mathcal{A} & \\mathcal{B} & \\mathcal{A} & \\mathcal{-B} \\\\\n    \\end{array}\n  \\right],\n  \\mathcal{Q}=\n  \\left[\n    \\begin{array}{rrrr}\n      \\mathcal{C} & \\mathcal{D} & \\mathcal{C} & \\mathcal{-D} \\\\\n    \\end{array}\n  \\right].\n\\end{equation}\nThen $(\\mathcal{P,Q})$ is a GCAP of size $2\\times12$ with $Z_{\\min}=2\\times4$.\nThe 2D- periodic correlation magnitudes of $(2\\times12,2\\times4)$- GCAP $(\\mathcal{P,Q})$ are shown in Fig. \\ref{fig2}.\n\\begin{figure}[ht]\n\t\\begin{subfigure}{.45\\textwidth}\n\t\t\\centering\n\t\n\t\t\\includegraphics[width=\\textwidth]{Ex2P.pdf}\n\t\t\\caption{2D- periodic autocorrelation of $\\mathcal{P}$}\n\t\t\\label{fig:sub-first}\n\t\\end{subfigure}\n\t\\begin{subfigure}{.45\\textwidth}\n\t\t\\centering\n\t\n\t\t\\includegraphics[width=\\textwidth]{Ex2Q.pdf}\n\t\t\\caption{2D- periodic autocorrelation of $\\mathcal{Q}$}\n\t\t\\label{fig:sub-second}\n\t\\end{subfigure}\n\\begin{subfigure}{.45\\textwidth}\n\t\\centering\n\n\t\\includegraphics[width=\\textwidth]{Ex2PQ.pdf}\n\t\\caption{2D- periodic cross-correlation of $\\mathcal{P}$ and $\\mathcal{Q}$}\n\t\\label{fig:sub-second}\n\\end{subfigure}\n\t\\caption{A glimpse of the 2D- periodic correlations of the GCAP given in Example \\ref{ex2}.}\n\t\\label{fig2}\n\\end{figure}\n\\end{example}\n\n\\begin{construction}\\label{Construction-array-ZCZ2}\nLet $(\\mathcal{A,B})$ be a GCAP of size $s_1\\times s_2$ and $(\\mathcal{C,D})$ be the Golay array mate of $(\\mathcal{A,B})$, then define\n\\begin{equation}\n  \\mathcal{P}=\n  \\left[\n    \\begin{array}{rrrr}\n      \\mathcal{A} & \\mathcal{B} & \\mathcal{A} & \\mathcal{-B} \\\\\n      \\mathcal{A} & \\mathcal{B} & \\mathcal{-A} & \\mathcal{B} \\\\\n      \\mathcal{A} & \\mathcal{B} & \\mathcal{A} & \\mathcal{-B} \\\\\n      \\mathcal{-A} & \\mathcal{-B} & \\mathcal{A} & \\mathcal{-B} \\\\\n    \\end{array}\n  \\right],\n  \\mathcal{Q}=\n  \\left[\n    \\begin{array}{rrrr}\n      \\mathcal{C} & \\mathcal{D} & \\mathcal{C} & \\mathcal{-D} \\\\\n      \\mathcal{C} & \\mathcal{D} & \\mathcal{-C} & \\mathcal{D} \\\\\n      \\mathcal{C} & \\mathcal{D} & \\mathcal{C} & \\mathcal{-D} \\\\\n      \\mathcal{-C} & \\mathcal{-D} & \\mathcal{C} & \\mathcal{-D} \\\\\n    \\end{array}\n  \\right].\n\\end{equation}\n\\end{construction}\n\n\\begin{theorem}\n  For the array pair $(\\mathcal{P},\\mathcal{Q})$ generated by Construction \\ref{Construction-array-ZCZ2}, it's a GCAP of size $4s_1\\times 4s_2$ with periodic zero correlation zone $(s_1+1)\\times (s_2+1)$.\n\\end{theorem}\n\\begin{IEEEproof}\nThe proof is similar to the one of Theorem \\ref{Theorem-array-ZCZ1}.\n\\end{IEEEproof}\n\n\\begin{example}\\label{ex3}\n  Let $\\mathcal{A}=\\left[\\begin{array}{rrr} 1 & 1 & -1 \\\\ -1 & -i & -1 \\\\ \\end{array}\\right],\\mathcal{B}=\\left[\\begin{array}{rrr} 1 & 1 & -1 \\\\ 1 & i & 1 \\\\ \\end{array}\\right]$, then $(\\mathcal{A,B})$ is a quaternary GCAP of size $2\\times3$.\n  Construct $(\\mathcal{C,D})= (\\overleftarrow{\\mathcal{B^*}},-\\overleftarrow{\\mathcal{A^*}})$ as a Golay array mate of $(\\mathcal{A,B})$. Define\n\\begin{equation}\n  \\mathcal{P}=\n  \\left[\n    \\begin{array}{rrrr}\n      \\mathcal{A} & \\mathcal{B} & \\mathcal{A} & \\mathcal{-B} \\\\\n      \\mathcal{A} & \\mathcal{B} & \\mathcal{-A} & \\mathcal{B} \\\\\n      \\mathcal{A} & \\mathcal{B} & \\mathcal{A} & \\mathcal{-B} \\\\\n      \\mathcal{-A} & \\mathcal{-B} & \\mathcal{A} & \\mathcal{-B} \\\\\n    \\end{array}\n  \\right],\n  \\mathcal{Q}=\n  \\left[\n    \\begin{array}{rrrr}\n      \\mathcal{C} & \\mathcal{D} & \\mathcal{C} & \\mathcal{-D} \\\\\n      \\mathcal{C} & \\mathcal{D} & \\mathcal{-C} & \\mathcal{D} \\\\\n      \\mathcal{C} & \\mathcal{D} & \\mathcal{C} & \\mathcal{-D} \\\\\n      \\mathcal{-C} & \\mathcal{-D} & \\mathcal{C} & \\mathcal{-D} \\\\\n    \\end{array}\n  \\right].\n\\end{equation}\nThen $(\\mathcal{P,Q})$ is a GCAP of size $8\\times12$ with $Z_{\\min}=3\\times4$.\nThe 2D- periodic correlation magnitudes of $(8\\times12,3\\times4)$- GCAP $(\\mathcal{P,Q})$ are shown in Fig. \\ref{fig3}.\n\\begin{figure}[ht]\n\t\\begin{subfigure}{.45\\textwidth}\n\t\t\\centering\n\t\n\t\t\\includegraphics[width=\\textwidth]{Ex3P1.pdf}\n\t\t\\caption{2D- periodic autocorrelation of $\\mathcal{P}$}\n\t\t\\label{fig:sub-first}\n\t\\end{subfigure}\n\t\\begin{subfigure}{.45\\textwidth}\n\t\t\\centering\n\t\n\t\t\\includegraphics[width=\\textwidth]{Ex3Q1.pdf}\n\t\t\\caption{2D- periodic autocorrelation of $\\mathcal{Q}$}\n\t\t\\label{fig:sub-second}\n\t\\end{subfigure}\n\t\\begin{subfigure}{.45\\textwidth}\n\t\t\\centering\n\t\n\t\t\\includegraphics[width=\\textwidth]{Ex3PQ.pdf}\n\t\t\\caption{2D- periodic cross-correlation of $\\mathcal{P}$ and $\\mathcal{Q}$}\n\t\t\\label{fig:sub-second}\n\t\\end{subfigure}\n\t\\caption{A glimpse of the 2D- periodic correlations of the GCAP given in Example \\ref{ex3}.}\n\t\\label{fig3}\n\\end{figure}\n\\end{example}\n\\section{Conclusion}\\label{section 5}\n\nIn this paper, we have made two  contributions. Firstly, we have systematically constructed GCPs of lengths non-power-of-two where each of the constituent sequences have a periodic ZACZ and the pair have a periodic ZCCZ. This partially fills the gap left by the previous remarkable works of Gong \\textit{et al.} and Chen \\textit{et. al}, in terms of the lengths of the GCP. Secondly, by extending our construction to higher dimensions, we construct 2D- GCAPs. For the first time, the periodic 2D-autocorrelations of the constituent arrays and the periodic 2D-cross-correlation of the 2D-GCAP has been analysed. All the proposed 2D- GCAPs have large periodic 2D autocorrelation and cross-correlation zones. An interesting future\nwork will be to design Golay complementary sequence sets, consisting sequences of length non-power of two, which will have periodic ZACZ and ZCCZ around the in-phase position.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n\\vskip 10 pt\n\nThis paper is based on three talks given at the PASI conference in Olinda, Brazil in the summer of\n2009. One point of the paper is  to introduce students to  one aspect of characteristic $p$\nmethods in commutative algebra. Such methods have been among the most powerful in the\nfield. The basic method of reduction to\ncharacteristic $p$ is used to prove results for arbitrary Noetherian rings containing a field; the field\ncan be characteristic $0$. Thus, even though one is often working in positive characteristic, ones main\ninterest might well be in rings containing the rationals. In the late 1980's, ``tight closure theory\" was\ndiscovered by M. Hochster and myself. This theory synthesized many existing reduction to characteristic $p$\nproofs into one theory, which has now grown to encompass a large number of different directions.  In this paper,\nwe'll concentrate on a result which Karen Smith has refered to as a ``crown jewel\" of\ntight closure theory, namely the fact that the absolute integral closure of a complete local domain\nof positive characteristic is Cohen-Macaulay. We first introduce basic concepts, and discuss some rather\namazing properties of absolute integral closures. Then we prove some classical theorems in dimension one.\nThe main part of the paper will give a recent proof of the Cohen-Macaulayness of the absolute integral closure\nin positive characteristic. Applications will be given in Section 6. Some further thoughts\nabout directions to go will be presented in the last section. \nFor unexplained results or definitions, we refer the reader to Eisenbud's book \\cite{Ei}.\n\\section{Basic Concepts}\n\n\nWe begin with a definition:\n\n\\begin{definition} {\\rm Let $R$ be a domain, and let $R\\subset S$, a ring. The \\it integral closure \\rm\nof $R$ in $S$ is the set of all elements $s\\in S$ which satisfy an equation of the form\n$$s^n + r_1s^{n-1} + ... + r_n = 0,$$ where $r_i\\in R$ for all $i$.}\n\\end{definition}\n\nThe set of elements in $S$ which are integral over $R$  form a ring, called the \\it integral closure of\n$R$ in $S$\\rm.  When $S$ is the fraction field of $R$, the resulting ring is called the\nintegral closure of $R$. The ring $R$ is said to be \\it integrally closed \\rm if \n$R$ is equal to its integral closure. For example, a well-known result in commutative algebra says that\nevery UFD is integrally closed. So, e.g., the integers are integrally closed. Another\nbasic result is states that if $R$ is integrally closed, so is\nthe ring of polynomials $R[X]$ as well as the ring of formal power series $R[[X]]$.  An obvious induction shows that also\n$R[X_1,...,X_n]$ and $R[[X_1,...,X_n]]$ are integrally closed for all $n\\geq 1$. \n\nThe main object we will study is given in the following definition.\n\n\\begin{definition} {\\rm Let $R$ be an integral domain with fraction field $K$. Let $\\overline{K}$ be a\nfixed algebraic closure of $K$. The integral closure of $R$ in $\\overline{K}$, denoted $R^+$, is\ncalled the \\it absolute integral closure \\rm of $R$.}\n\\end{definition}\n\nWe simply say ``R-plus\" to mean the absolute integral closure. This ring has been well-studied in\nseveral different contexts. \n\nFor other interesting results concerning $R^+$ which we will not be writing about, see\n\\cite{A},\\cite{AH}, and \\cite{Sm}.\n\n\\medskip\n\n\n\\medskip\n\n\\begin{disc} {\\rm Whenever we study $R^+$, we might as well assume that $R$ is integrally closed itself; if $S$ is\nthe integral closure of $R$, then $R\\subset S\\subset R^+$, and $S^+ = R^+$. More generally,\nif $S$ is \\it any \\rm finite integral extension of $R$ which is a domain, then the fraction field\nof $S$ is algebraic over the fraction field of $R$, and hence there is an isomorphic copy of $S$ which\ncontains $R$ and sits inside $R^+$. By an abuse of language, we'll just assume that $S\\subset R^+$. \nIn this case, $S^+ = R^+$.  Studying $R^+$ is basically studying all integral extensions of $R$\nat the same time.}\n\\end{disc}\n\n\\medskip\n\n\\begin{remark}  {\\rm Continuing the discussion from above, if $R$ is local and complete in the ${\\gothic{m}}$-adic topology and contains\na field, then $R$ is actually a finite integral extension of a formal power series ring over a\nfield $k$. Namely, the Cohen structure theorem gives that there exists a field $k$ inside\n$R$ such that the composition of maps $k\\rightarrow R\\rightarrow R\/{\\gothic{m}}$ is an isomorphism.\nMoreover, if $x_1,...,x_d$ is a system of parameters for $R$, then the complete local subring of $R$\ngenerated by these elements, $k[[x_1,...,x_d]] = A$ is isomorphic with a formal power series ring\nover $k$, and $R$ is module-finite over $A$.\nHence $A^+ = R^+$. \nThus, if we fix a copy of the residue field, say $k$,  sitting inside $R$,  there is really only one $R^+$ for each dimension; it is the absolute integral closure of\nthe formal power series ring over  the field $k$. In other words, for each dimension, we are really just studying\none ring, the absolute integral closure of the power series ring over $k$. There is a classical theorem\nwhich gives the structure of such a ring in dimension one over a field of characteristic $0$. See Theorem~\\ref{newton}.  } \n\\end{remark}\n\n\\medskip\n\n \\bigskip\n\n\\begin{disc} {\\rm It is not difficult to prove that if $R$ is an integral domain, then a domain $S$ integral\nover $R$ is isomorphic with $R^+$ if and only if monic polynomials over $S$ factor into monic linear polynomials over $S$.\nUsing this criterion, one can easily prove two very basic properites of $R^+$. \n\n\\medskip\n$\\bullet$ Firstly, if $q\\in \\text{Spec}(R)$, then $(R_q)^+\\cong (R^+)_q$, where the latter localization is at the multiplicatively closed set\n$R-q$.\n\n\\medskip\n\n$\\bullet$ Secondly, if $Q\\in \\text{Spec}(R^+)$, then $R^+\/Q\\cong (R\/(Q\\cap R))^+$.  Thus every quotient of $R^+$ by a prime ideal is itself\nthe absolute integral closure of an appropriate quotient of $R$.}\n\\end{disc}\n\nWe leave these statements as exercises below.\n\nThese properties not only allow us to mod out by primes and localize by changing the absolute integral closure\nin a similar way, they will give very strong properites. For example, the main result of Section 4 states\nthat the absolute integral closure of an excellent local domain in positive characteristic is Cohen-Macaulay\nin a suitable sense. The second property discussed above then says that the same is true modulo every prime\nideal of the absolute integral closure! This is remarkable.\n\nM. Artin \\cite{Ar} proved another amazing property of $R^+$:\n\n\\begin{thm} Let $R$ be a domain. If $P,Q$ are prime ideals in $R^+$, then either $P+Q = R^+$, or\n$P+Q$ is prime.\n\\end{thm}\n\nThis is very far from what happens in rings we typically study. For example in the polynomial ring $k[x,y]$,\nthe ideals $P = (x+y^2)$ and $Q = (x)$ are prime, but the sum is not. The proof we give is taken from\n\\cite{HH}.\n\n\\begin{proof}\nTo prove Artin's result, let $ab\\in P + Q$. Set $z= b-a$. Notice that $a^2 + za = ab = u+v$ for some $u\\in P, v\\in Q$.\nThe equation $X^2+zX= u$ has a solution $x\\in R$. Since $u\\in P$, it follows that either $x+z\\in P$ or $x\\in P$. But\n$(x^2+zx)-(a^2+za) = u-(u+v)\\in Q$, so either $x-a\\in Q$ or $x+a+z\\in Q$. In each of the four different cases, we\nobtain that either $a\\in P+Q$ or $b\\in P+Q$. For example, if $x+z\\in P$ and $x-a\\in Q$, then $b= (x+z)-(x-a)\\in Q$.\n\\end{proof}\n\\medskip\n\n\nNotice that if $R$ is complete and local, then every finite extension domain of $R$ is also local. In particular,\n$R^+$ is a local ring, i.e., has a unique maximal ideal. Therefore the possibility that $P+Q = R$ cannot happen\nin this case, and the sum of every set of prime ideals is prime.\n\n \\bigskip\n\n\n\\centerline{\\bf Exercises}\n\\bigskip\n\n1. Let $R$ be a domain with fraction field $K$, and let $L$ be an algebraic extension of $K$.\nProve that $L$ is algebraically closed if and only if every monic polynomial with coefficients in $R$\nhas a root in $L$.\n\n\\medskip\n\n2. Let $R$ be an integral domain. Prove that a domain $S$ integral over $R$ is isomorphic with $R^+$\nif and only if every monic polynomial over $S$ factors into monic linear polynomials over $S$.\n\n\\medskip\n\n3. Prove that Artin's theorem can be extended to primary ideals: the sum of any two primary ideals in $R^+$\nis again primary or the whole ring.\n\\medskip\n\n4.  Let $R$ be a domain of positive dimension. Prove that $R^+$ is not Noetherian.\n\n\\medskip\n\n5. Suppose that $R$ is an integral domain, and $R\\subseteq S\\subseteq R^+$. Prove that $R^+ = S^+$.\n\n\\medskip\n\n6. If $q\\in \\text{Spec}(R)$, then $(R_q)^+\\cong (R^+)_q$, where the latter localization is at the multiplicatively closed set\n$R-q$.\n\n\\medskip\n\n7. If $Q\\in \\text{Spec}(R^+)$, then $R^+\/Q\\cong (R\/(Q\\cap R))^+$.  Thus every quotient of $R^+$ by a prime ideal is itself\nthe absolute integral closure of an appropriate quotient of $R$.\n\n\\vskip.5truein\n\n\\section{Dimension One}\n\\bigskip\n\n Let $k$ be a field of characteristic $0$, and consider the formal power series ring $k[[t]]$, whose fraction field is\ndenoted $k((t))$. A famous theorem due to Puiseux, but apparently known to Newton, is that the algebraic closure of\nthe field $k((t))$ is the union of the fields $k((t^{1\/n}))$ for $n\\geq 1$.  A recent paper by Kedlaya (\\cite{K}) gives\na characterization of this algebraic closure in the case $k$ has positive characteristic; it is quite hard even to describe.\nChevalley  \\cite{C} pointed out that the Artin-Schreirer polynomial $x^p-x-t^{-1}$ has no root in the Newton-Puiseux field\n$\\bigcup_n k((t^{1\/n}))$. \nIn fact, as we shall see, there are remarkable differences in $R^+$ depending on the characteristic of the ground field.\n\nWe give a proof of the  Newton-Puiseux theorem, following a treatment in \\cite{No}.\n\n\\begin{thm}\\label{newton} Let $k$ be an algebraically closed field of characteristic $0$, and let $k((t))$ be the fraction field\nof the formal power series ring $k[[t]]$. Then an algebraic closure of $k((t))$ is the Newton-Puiseux field $\\cup_n k((t^{1\/n}))$.\n\\end{thm}\n\n\\begin{proof} We need to prove that every monic polynomial $P(z)\\in B[z]$ is reducible, where $B = \\cup_nk[[t^{1\/n}]]$.\nWrite $P = z^n+ a_1z^{n-1} + ... + a_n$, where $a_i\\in B$.  By using a transformation $z' = z+\\frac{1}{n}a_1$, we can\nassume without loss of generality that $a_1 = 0$. This is called the ``Tschirhausen transformation\". Furthermore, by replacing\n$k[[t]]$ by $k[[t^{1\/n}]]$ for some large $n$, we can change variables and assume that all $a_i\\in k[[t]]$.   Let $r_k = \\text{ord}(a_k)$,\nso that $a_k = t^{r_k}u_k(t)$, where $u_k(t)$ is a unit. Here ``ord\" denotes the $t$-adic order of an element. \n\nWe want to factor $P(z)$, and to do so we will use Hensel's lemma\\footnote{ The version of Hensel's lemma\nwe use is the following theorem:\n\\begin{thm} Let $(R,{\\gothic{m}})$ be a local ring which is complete in the ${\\gothic{m}}$-adic topology. If $f(T)$ is a monic polynomial\nwith coefficients in $R$ such that $f(T)$ factors modulo ${\\gothic{m}}$ into the product of two relatively prime monic\npolynomials of positive degree, then this factorization lifts to a factorization of $f(T)$ into two monic polynomials.\n\\end{thm}}; it suffices to factor $P(z)$ into two relatively prime polynomials\nafter going modulo $t$. This is always possible unless after reduction mod $t$, $P(z)$ becomes of the form $(t-\\alpha)^n$ for\nsome $\\alpha\\in k$. However, since $a_1 = 0$, the only possible such $\\alpha$ is $0$. Thus we are done unless for every $k$, $2\\leq k\\leq n$,\n$a_i(0) = 0$.  We want to make a change of variables where this does not occur. \n\nSet $z = t^ry$, for some rational $r$ to be chosen later. Substituting, we see that\n$P(z) = t^{rn}y^n + a_2t^{r(n-2)}y^{n-2} + ... + a_n$. We wish to factor out $t^{rn}$ from every term in such a way that\nat least one term becomes a unit. The power of $t$ dividing\nthe $i$th term is $t^{r_i}t^{(n-i)r}$, so what we need to do is to choose $r$ in such a way that \n$r_i + (n-i)r \\geq rn$, i.e., so that $r_i\\geq ri$. We set $r = \\text{min}\\{\\frac {r_i}{i}\\}$. Then we can rewrite\n$P(z) = t^{rn}Q(y,t)$, where $Q = y^n + b_2y^{n-1} + ... + b_n$. By again replacing $t$ by $t^{1\/m}$ for suitably large $m$,\nwe can assume that each $b_j\\in k[[t]]$. But now for at least one $b_i$, $b_i(0)\\ne 0$; this occurs for every term where\n$r_i\/i = r$. It follows that we can factor $Q$ over the residue field into relatively prime polynomials, and by Hensel's Lemma\nthis lifts to a factorization of $Q$ and hence also of $P$. \n\\end{proof} \n\n\nThis theorem has the following almost immediate corollary:\n\n\\begin{cor} Let $k$ be an algebraically closed field of characteristic $0$. Then $k[[t]]^+ = \\cup_nk[[t^{1\/n}]]$.\n\\end{cor}\n\n\\begin{proof} The ring on the right side of the equation is clearly integral over $k[[t]]$, and the fraction field\nof it is clearly the Newton-Puiseux field, which is an algebraic closure of $k((t))$.\nTo finish the proof of the corollary, we need to prove that $\\cup_nk[[t^{1\/n}]]$ is integrally closed. But since\nit is a union of discrete valuation rings, each integrally closed, it is also. \\end{proof}\n\n\\begin{example} {\\rm As an example, consider the\nequation $X^2 - X = t^{-1}$. Using the quadratic formula and\nTaylor series shows that the roots of this polynomial\nare a power series in $t^{-1\/2}$. One can solve recursively for the\ncoefficients. }\n\\end{example}\n\\medskip\n\nThe situation in positive characteristic is drastically different. We use the discussion in the  paper \\cite{K} in what follows.\n A \\it generalized power series \\rm is an\nexpression of the form, $\\sum_{i\\in \\mathbb Q} c_it^i$, with $c_i\\in k$, such that the set of $i$ with $x_i\\ne 0$\nis a well-ordered subset of $\\mathbb Q$, i.e., every non-empty subset has a least element. Such generalized power\nseries form a ring in a natural way; it makes sense to multiply and add them in the obvious way.\nAbyhankar pointed out that with this generalization of the idea of a power series,\nthe Chevalley polynomial has the root $t^{-1\/p} + t^{-1\/p^2}+...$.\n\nAre there enough generalized power series to obtain an algebraic closure of $k((t))$?  The following result was proved independently\nby Huang \\cite{Hua}, Rayner \\cite{Ray}, and \\c Stef\\u anescu \\cite{St}:\n\n\\begin{thm} Let $k$ be an algebraically closed field of positive characteristic, and let $K$ be the set of generalized power\n series of the form $f = \\sum_{i\\in S} c_it^i$,\nwith $c_i\\in k$, where the set $S$ has the following properties:\n\\begin{enumerate}[\\quad\\rm (1)]\n\\item  $S$ is a subset of $\\mathbb Q$ which depends on $f$.\n\\item  Every nonempty subset of $S$ has a least element.\n\\item  There exists a natural number $m$ such that every element of $mS$ has denominator a power of $p$.\n\\end{enumerate}\nThen $K$ is an algebraically closed field containing $k((t))$.\n\\end{thm}\n\n\nKedlaya \\cite{K} gives a construction  of the algebraic closure of $k((t))$ when $k$ is algebraically closed of positive characteristic.\nBy the above theorem, one needs to identify generalized power series which are algebraic over $k((t))$. \nHis result is quite complicated to even describe, but a glimpse of the issues which arise can be seen\nin the following result of Huang \\cite{Hua} and \\c Stef\\u anescu \\cite{St}:\n\n\\begin{thm} The series $\\sum_{i = 0}^{\\infty} c_it^{-1\/p^i} $ with $c_i\\in \\overline{F_p}$, the\nalgebraic closure of the field with $p$ elements, is algebraic over $\\overline{F_p}((t))$ if and only if \nthe sequence $\\{c_i\\}$ is eventually periodic.\n\\end{thm}\n\nA consequence of the main result of \\cite{K} is the following nice theorem, which perhaps can be\nproved directly.\n\n\\begin{thm} Let $k$ be an algebraically closed field of positive characteristic, and let $\\sum_i c_it^i$ be a generalized power series which is algebraic over $k((t))$. Then for\nevery real number $\\alpha$,  $\\sum_{i < \\alpha} c_it^i$ is also algebraic over $k((t))$.\n\\end{thm}\n\n\nWe now move away from the local case. \nAnother classical result in dimension one concerns the ring of all algebraic integers, $\\mathbb Z^+$.\nTo prove this theorem, we first recall some results about\nclass groups. \n\nLet $D$ be a ring of algebraic integers. This means that the fraction field $K$ of $D$ is a finite\nextension of $\\mathbb Q$, and $D$ is the integral closure of $\\mathbb Z$ in $K$. Necessarily\n$D$ is a one-dimensional integrally closed domain, i.e., a Dedekind domain. Moreover, a classical\nresult is that $D$ has a torsion class group.  In particular this means that every ideal $I$ in\n$D$ has a power which is a principal ideal. This fact leads to the following theorem\nabout $\\mathbb Z^+$.\n\n\n\\begin{thm}\\label{bezout} Every finitely generated ideal of\n$\\mathbb Z^+$ is principal (a domain with this property is said to be a B\\'ezout domain).\n\\end{thm}\n\n\\begin{proof} Let $I\\subset \\mathbb Z^+$ be generated by $t_1,...,t_m$. There is a finite field extension $L$\nof $\\mathbb Q$   containing all of these elements. The integral closure of $\\mathbb Z$\nin $L$, say $D$, is a Dedekind domain which contains $t_1,...,t_m$. Let $J$ be the ideal they\ngenerate in $D$. By the discussion above, some power of $J$ is principal, say $J^n = (d)$.\nWe claim then that $d^{\\frac{1}{n}}\\mathbb{Z}^+ = I$. Let $i\\in I$. We can write $i = \\sum_j e_jt_j$, where the\n$e_j\\in \\mathbb{Z}^+$. Again, there is a finite extension $T$ of $D$, with $T$ a Dedekind domain, such that \n$i\\in I\\cap T = (t_1,...,t_m)T = JT$. In a Dedekind domain there is unique factorization into prime ideals.\nWrite $JT = P_1^{a_1}\\cdots P_k^{a_k}$. Then $J^nT = dT = P_1^{na_1}\\cdots P_k^{na_k}$. By unique factorization,\nit follows that $d^{\\frac{1}{n}}T = P_1^{a_1}\\cdots P_k^{a_k} = JT$, so that $i\\in d^{\\frac{1}{n}}\\mathbb{Z}^+$.\n\\end{proof}\n\n\\bigskip\n\\bigskip\n\\section{Regular Sequences}\n\n\\medskip\n\nWe summarize some of the basic notions we will use to analyze $R^+$. \n\n\\begin{definition}{\\rm A sequence of elements $x_1,...,x_d$ in a ring $R$ is said to be\na \\it regular sequence \\rm if $rx_i\\in (x_0,...,x_{i-1})$ implies that\n$r\\in (x_0,...,x_{i-1})$ for all $1\\leq i\\leq d-1$ (here we set $x_0 = 0$), and\n$(x_1,...,x_d)R \\ne R$.}\n\\end{definition}\n\nAnother definition we will need is the following.\n\n\\begin{definition} {\\rm Let $(R,{\\gothic{m}})$ be a Noetherian local ring of dimension $d$. Elements\n$x_1,...,x_d$ are said to be a \\it system of parameters \\rm if the nilradical\nof the ideal they generate is ${\\gothic{m}}$.\nA Noetherian local ring is said to be \\it Cohen-Macaulay \\rm if some (equivalently) every\nsystem of parameters forms a regular sequence.}\n\\end{definition}\n\n\nWe are aiming at a theorem which shows that in characteristic $p$, $R^+$ has regular sequences having\nmaximal length. It turns out that this is equivalent to being flat in an appropriate sense. The next proposition\nproves this.\n\n\\medskip\n\n\\begin{prop} Let $A = k[[x_1,...,x_d]]$, where $k$ is a field of characteristic $p > 0$. Then the following two conditions\nare equivalent:\n\n(1) $x_1,...,x_d$ form a regular sequence on $A^+$ (by an abuse of language we say that $A^+$ is Cohen-Macaulay).\n\\newline\n(2) $A^+$ is flat over $A$.\n\\end{prop}\n\n\\begin{proof}\nWe prove the equivalence. First assume (1).\nIf $A^+$ is not flat over $A$, choose $i\\geq 1$ as large as possible so that\n$\\operatorname{Tor{}}_i^A(A\/P, A^+)\\ne 0$ for some prime $P$ in $A$. Such a choice is possible because $A$ is regular\nand large Tors vanish. If $y_1,...,y_s$ is a maximal regular sequence\nin $P$, then one can embed $A\/P$ in $A\/(y_1,...,y_s)$ with cokernel $C$. But since our assumption\nforces $\\operatorname{Tor{}}_{i+1}^R(C,A^+) =  0$ (as $C$ has a prime filtration), and $y_1,...,y_s$ form a\nregular sequence on $A^+$, we obtain that $\\operatorname{Tor{}}_i^A(A\/P, A^+) = 0$, a contradiction.\n\nTo see that $y_1,...,y_s$ form a regular sequence, extend them to a system of parameters, and let\n$B = k[[y_1,...,y_d]]$. Then $B^+ = A^+$, and our hypothesis says that the $y's$ form a regular\nsequence.\n\nAssume (2). Flat maps preserve regular sequence in general.\n\\end{proof}\n\n\nOur method of studying regular sequences relies on local cohomology. We only need the description below.\n\n For $x\\in R$, let $K^{\\bullet}(x;R)$ denote the\ncomplex $0\\rightarrow R\\rightarrow R_x\\rightarrow 0$, graded so that the degree $0$ piece of the complex is\n$R$, and the degree $1$ is $R_x$. If $x_1,...,x_n\\in R$, let $K^{\\bullet}(x_1,x_2,...,x_n;R)$ denote\nthe complex $K^{\\bullet}(x_1;R)\\otimes_R ...\\otimes_R K^{\\bullet}(x_n;R)$, where in general recall that\nif $(C^{\\bullet},d_C)$ and $(D^{\\bullet}, d_D)$ are complexes, then the tensor product\nof these complexes, $(C\\otimes_RD, \\Delta)$ is by definition the complex whose ith\ngraded piece is $\\sum_{j+k = i} C_j\\otimes D_k$ and whose differential is determined\nby the map from $ C_j\\otimes D_k\\rightarrow (C_{j+1}\\otimes D_k) \\oplus (C_j\\otimes D_{k+1})$\ngiven by $\\Delta(x\\otimes y) = d_C(x)\\otimes y + (-1)^k x\\otimes d_D(y)$.\n\nThe modules in this complex, called the Koszul cohomology complex, are\n$$0\\rightarrow R\\rightarrow \\oplus\\sum_i R_{x_i}\\rightarrow \\oplus\\sum_{i< j} R_{x_ix_j}\\rightarrow ...\\rightarrow R_{x_1x_2\\cdots x_n}\\rightarrow 0$$\nwhere the differentials are the natural maps induced from  localization, but with signs attached.\nIf $M$ is an $R$-module, we set $K^{\\bullet}(x_1,x_2,...,x_n; M) = K^{\\bullet}(x_1,x_2,...,x_n;R)\\otimes_RM$.\nWe denote the cohomology of $K^{\\bullet}(x_1,x_2,...,x_n; M)$ by $H^i_{I}(M)$, called\nthe $i$th local cohomology of $M$ with respect to $I = (x_1,...,x_d)$.\nIt is a fact that this module only depends on the ideal generated by the $x_i$ up to radical.\nWe summarize some useful information concerning these modules.\n\n\\begin{prop}\\label{propbasechange} Let $R$ be a Noetherian ring, $I$ and ideal and $M$ and $R$-module.\nLet $\\phi: R\\rightarrow S$ be a homomorphism and let $N$ be an $S$-module.\n\\begin{enumerate}[\\quad\\rm (1)]\n\\item If $\\phi$ is flat, then $H^j_I(M)\\otimes_RS\\cong H^j_{IS}(M\\otimes_RS)$. In particular,\nlocal cohomology commutes with localization and completion.\n\\item (Independence of Base) $H^j_I(N)\\cong H^j_{IS}(N)$, where the first local cohomology\nis computed over the base ring $R$.\n\\end{enumerate}\n\\end{prop}\n\n\\begin{proof} Choose generators $x_1,...,x_n$ of $I$. The first claim follows at once from the\nfact that $K^{\\bullet}(x_1,...,x_n;M)\\otimes_RS = K^{\\bullet}(\\phi(x_1),...,\\phi(x_n);M)\\otimes_RS)$, and that\n$S$ is flat over $R$, so that the cohomology of $K^{\\bullet}(x_1,...,x_n;M)\\otimes_RS$ is the cohomology\nof $K^{\\bullet}(x_1,...,x_n;M)$ tensored over $R$ with $S$.\n\nThe second claim follows from the fact that $$K^{\\bullet}(x_1,...,x_n;N) = K^{\\bullet}(x_1,...,x_n;R)\\otimes_RN =\n(K^{\\bullet}(x_1,...,x_n;R)\\otimes_RS)\\otimes_SN$$\n$$ = K^{\\bullet}(\\phi(x_1),...,\\phi(x_n);S)\\otimes_RN =\nK^{\\bullet}(\\phi(x_1),...,\\phi(x_n);N).$$\n\\end{proof}\n\\bigskip\n\n\\section{$R^+$ is Cohen-Macaulay in Positive Characteristic}\n\n\\medskip\n\nLet $R$ be a commutative ring containing a field of characteristic $p>0$, let $I\\subset R$\nbe an ideal, and let $R'$ be an $R$-algebra. The Frobenius ring homomorphism\n$f:R'\\stackrel{r\\mapsto r^p}{\\to}R'$ induces a map $f_*:H^i_I(R')\\to H^i_I(R')$ on all\nlocal cohomology modules of $R'$ called the action of the Frobenius on $H^i_I(R')$.\nFor an element $\\alpha\\in H^i_I(R')$ we denote $f_*(\\alpha)$ by $\\alpha^p$. This follows since\nthe Frobenius extends to localization of $R$ in the obvious way, and commutes with the\nmaps in the Koszul cohomology complex, which are simply signed natural maps.\n\nThe main result is that if $R$ is a local Noetherian domain which is a homomorphic image of a\nGorenstein local ring and has positive characteristic, then $R^+$ is Cohen-Macaulay in the\nsense that every system of parameters of $R$ form a regular sequence in $R^+$. To prove this\nresult we use the proof given in \\cite{HL}. The original proof, with slightly different assumptions, was given\nin 1992 in \\cite{HH}, as a result of developments from tight closure theory. Although tight closure has\nnow disappeared from the proof, it remains an integral part of the theory. A critical point is that we must find some\nway of annihilating nonzero local cohomology classes. \nThe next lemma is essentially the only way known to do this.\n\n\\begin{lemma} \\label{element} Let $R$ be a commutative Noetherian domain containing a field of\ncharacteristic $p>0$, let $K$ be the fraction field of $R$ and let $\\overline K$ be the algebraic closure\nof $K$. Let $I$ be an ideal of $R$ and let $\\alpha\\in H^i_I(R)$ be an element such that the elements\n$\\alpha, \\alpha^p,\\alpha^{p^2},\\dots,\\alpha^{p^t},\\dots$ belong to a finitely generated $R$-submodule of $H^i_I(R)$.\nThere exists an $R$-subalgebra $R'$ of $\\overline K$ (i.e. $R\\subset R'\\subset \\overline K$) that is finite\nas an $R$-module and such that the natural map $H^i_I(R)\\to H^i_I(R')$ induced by the natural\ninclusion $R\\to R'$ sends $\\alpha$ to 0.\n\\end{lemma}\n\\emph{Proof.} Let $A_t=\\sum_{i=1}^{i=t}R\\alpha^{p^i}$ be the $R$-submodule of $H^i_I(R)$ generated\nby $\\alpha,\\alpha^p,\\dots,\\alpha^{p^t}$. The ascending chain $A_1\\subset A_2\\subset A_3\\subset\\dots$\nstabilizes because $R$ is Noetherian and all $A_t$ sit inside a single finitely generated $R$-submodule\nof $H^i_I(R)$. Hence $A_s=A_{s-1}$ for some $s$, i.e. $\\alpha^{p^s}\\in A_{s-1}$. Thus there exists an\nequation $\\alpha^{p^s}=r_1\\alpha^{p^{s-1}}+r_2\\alpha^{p^{s-2}}+\\dots+r_{s-1}\\alpha$ with $r_i\\in R$\nfor all $i$. Let $T$ be a variable and let $g(T)=T^{p^s}-r_1T^{p^{s-1}}-r_2^{p^{s-2}}-\\dots-r_{s-1}T$.\nClearly, $g(T)$ is a monic polynomial in $T$ with coefficients in $R$ and $g(\\alpha)=0$.\n\nLet $x_1,\\dots, x_d\\in R$ generate the ideal $I$. Recall that we can calculate the local cohomology\nfrom the Koszul cohomology complex\n$C^{\\bullet}(R)$, \n$$0\\to C^0(R)\\to\\dots \\to C^{i-1}(R)\\stackrel{d_{i-1}}{\\to} C^i(R)\\stackrel{d_i}{\\to}\nC^{i+1}(R)\\to\\dots\\to C^d(R)\\to 0$$\nwhere $C^0(R)=R$ and $C^i(R)=\\oplus_{1\\leq j_1<\\dots<j_{i}\\leq d}R_{x_{j_1}\\cdots x_{j_{i}}}$,\nand $H^i_I(M)$ is the $i$th cohomology module of $C^{\\bullet}(R)$.\n\nLet $\\tilde \\alpha\\in C^i(R)$ be a cycle (i.e. $d_i(\\tilde \\alpha)=0$) that represents $\\alpha$.\nThe equality $g(\\alpha)=0$ means that $g(\\tilde \\alpha)=d_{i-1}(\\beta)$ for some $\\beta\\in C^{i-1}(R)$.\nSince $C^{i-1}(R)=\\oplus_{1\\leq j_1<\\dots<j_{i-1}\\leq d}R_{x_{j_1}\\cdots x_{j_{i-1}}}$, we may write\n$\\beta=(\\frac{r_{j_1,\\dots,j_{i-1}}}{x_{j_1}^{e_{1}}\\cdots x_{j_{i-1}}^{e_{i-1}}})$\nwhere $r_{j_1,\\dots,j_{i-1}}\\in R$, the integers $e_1,\\dots, e_{i-1}$ are non-negative, and\n$\\frac{r_{j_1,\\dots,j_{i-1}}}{x_{j_1}^{e_{1}}\\cdots x_{j_{i-1}}^{e_{i-1}}}\\in R_{x_{j_1}\\cdots x_{j_{i-1}}}$.\n\nConsider the equation $g(\\frac{Z_{j_1,\\dots,j_{i-1}}}{x_{j_1}^{e_{1}}\\cdots x_{j_{i-1}}^{e_{i-1}}})\n-\\frac{r_{j_1,\\dots,j_{i-1}}}{x_{j_1}^{e_{1}}\\cdots x_{j_{i-1}}^{e_{i-1}}}=0$ where\n$Z_{j_1,\\dots,j_{i-1}}$ is a variable. Multiplying this equation by\n$(x_{j_1}^{e_{1}}\\cdots x_{j_{i-1}}^{e_{i-1}})^{p^s}$ produces a monic polynomial equation in\n$Z_{j_1,\\dots,j_{i-1}}$ with coefficients in $R$. Let $z_{j_1,\\dots,j_{i-1}}\\in \\overline K$ be a root\nof this equation and let $R''$ be the $R$-subalgebra of $\\overline K$ generated by all the $z_{j_1,\\dots,j_{i-1}}$s,\ni.e. by the set $\\{z_{j_1,\\dots,j_{i-1}}|1\\leq j_1<\\dots<j_{i-1}\\leq d\\}$. Since each $z_{j_1,\\dots,j_{i-1}}$\nis integral over $R$ and there are finitely many $z_{j_1,\\dots,j_{i-1}}$s, the $R$-algebra $R''$ is finite\nas an $R$-module.\n\nLet $\\tilde{\\tilde\\alpha} =\n(\\frac{z_{j_1,\\dots,j_{i-1}}}{x_{j_1}^{e_{1}}\n\\cdots x_{j_{i-1}}^{e_{i-1}}})\\in C^{i-1}(R'')$. The natural inclusion $R\\to R''$ makes $C^{\\bullet}(R)$\ninto a subcomplex of $C^{\\bullet}(R'')$ in a natural way, and we identify $\\tilde \\alpha\\in C^i(R)$\nand $\\beta\\in C^{i-1}(R)$ with their natural images in $C^i(R'')$ and $C^{i-1}(R'')$ respectively.\nWith this identification, $\\tilde \\alpha\\in C^i(R'')$ is a cycle representing the image of $\\alpha$\nunder the natural map $H^i_I(R)\\to H^i_I(R'')$, and so is $\\overline \\alpha=\\tilde\\alpha-d_{i-1}(\\tilde{\\tilde\\alpha})\n\\in C^i(R'')$. Since $g(\\tilde{\\tilde\\alpha})=\\beta$ and $g(\\tilde \\alpha)=d_{i-1}(\\beta)$, we conclude that\n$g(\\overline \\alpha)=0$. Let $\\overline\\alpha=(\\rho_{j_1,\\dots,j_i})$ where $\\rho_{j_1,\\dots,j_i}\n\\in R''_{x_{j_1}\\cdots x_{j_i}}$. Each individual $\\rho_{j_1,\\dots,j_i}$ satisfies the equation\n$g(\\rho_{j_1,\\dots,j_i})=0$. Since $g(T)$ is a monic polynomial in $T$ with coefficients in $R$,\neach $\\rho_{j_1,\\dots,j_i}$ is an element of the fraction field of $R''$ that is integral over $R$.\nLet $R'$ be obtained from $R''$ by adjoining all the $\\rho_{j_1,\\dots,j_i}$.\ndirect sum of all such copies of $R'$. This subcomplex is exact because its cohomology groups are the\ncohomology groups of $R'$ with respect to the unit ideal. Since $\\overline \\alpha$ is a cycle and belongs to\nthis exact subcomplex, it is a boundary, hence it represents the zero element in $H^i_I(R')$. \\qed\n\n\n\\medskip\n\nWe recall that for a Gorenstein local ring $A$ of dimension $n$, local duality says that\nthere is an isomorphism of functors $D({\\rm Ext}^{n-i}_A(-, A))\\cong H^i_{\\mathfrak m}(-)$\non the category of finite $A$-modules, where $D={\\rm Hom}_A(-,E)$ is the Matlis duality\nfunctor (here $E$ is the injective hull of the residue field of $A$ in the category of\n$A$-modules). \n\n\\begin{thm}\\label{module}\\cite{HL}\nLet $R$ be a commutative Noetherian local domain containing a field of characteristic $p>0$, let\n$K$ be the fraction field of $R$ and let $\\overline K$ be the algebraic closure of $K$. Assume $R$ is a surjective\nimage of a Gorenstein local ring $A$. Let $\\mathfrak m$ be the maximal ideal of $R$. \nLet $i< \\dim R$\nbe a non-negative integer. There is an $R$-subalgebra $R'$ of $\\overline K$ (i.e. $R\\subset R'\\subset \\overline K$)\nthat is finite as an $R$-module and such that the natural map $H^i_{\\mathfrak m}(R)\\to H^i_{\\mathfrak m}(R')$\nis the zero map.\n\\end{thm}\n\n\\emph{Proof.} The proof comes from \\cite{HL}. Let $n=\\dim A$ and let $N={\\rm Ext}^{n-i}_A(R,A)$. \nClearly $N$ is a finite $A$-module.\n\nLet $d=\\dim R$. We use induction on $d$. For $d=0$ there is nothing to prove, so we assume that\n$d>0$ and the theorem proven for all smaller dimensions. Let $P\\subset R$ be a non-maximal prime ideal.\nWe claim there exists an $R$-subalgebra $R^P$ of $\\overline K$  such that\n$R^P$ is a finite $R$-module and for every $R^P$-subalgebra $R^*$ of $\\overline K$ (i.e. $R^P\\subset R^*\\subset \\overline K$)\nsuch that $R^*$ is a finite $R$-module, the image $\\mathcal I\\subset N$ of the natural map\n${\\rm Ext}^{n-i}_A(R^*,A)\\to N$ induced by the natural inclusion $R\\to R^*$ vanishes after localization at $P$,\ni.e. $\\mathcal I_P=0$. Indeed, let $d_P=\\dim R\/P$. Since $P$ is different from the maximal ideal,\n$d_P>0$. As $R$ is a surjective image of a Gorenstein local ring, it is catenary, hence the dimension\nof $R_{P}$ equals $d-d_P$, and $i<d$ implies $i-d_P<d-d_P=\\dim R_{P}$. By the induction hypothesis\napplied to the local ring $R_{P}$, \nthere is an $R_{P}$-subalgebra $\\tilde R$ of $\\overline K$, which is finite as an $R_{P}$-module, such that\nthe natural map $H^{i-d_P}_{P}(R_P)\\to H^{i-d_P}_{P}(\\tilde R)$ is the zero map.\nLet $\\tilde R=R_{P}[z_{1},z_{2},\\dots,z_{t}]$, where $z_{1},z_{2},\\dots,z_{t}\\in \\overline K$ are\nintegral over $R_{P}$. Multiplying, if necessary, each $z_{j}$ by some element of $R\\setminus P$,\nwe can assume that each $z_{j}$ is integral over $R$. We set $ R^P=R[z_{1},z_{2},\\dots,z_{t}]$.\nClearly, $R^P$ is an $R'$-subalgebra of $\\overline K$ that is finite as $R$-module.\n\nNow let $R^*$ be both\nan $R^P$-subalgebra of $\\overline K$ (i.e. $R^P\\subset R^*\\subset \\overline K$) and a finite $R$-module.\nThe natural inclusions $R\\to R^P\\to R^*$ induce natural maps\n$${\\rm Ext}^{n-i}_A(R^*,A)\\to {\\rm Ext}^{n-i}_A(R^P,A)\\to N.$$ This implies that\n$\\mathcal I\\subset \\mathcal J$, where $\\mathcal J$ is the image of the natural map\n$\\phi:{\\rm Ext}^{n-i}_A(R^P,A)\\to N$. Hence it is enough to prove that $\\mathcal J_{P}=0$.\nLocalizing this map at $P$ we conclude that $J_P$ is the image of the natural map\n$\\phi_P:{\\rm Ext}^{n-i}_{A_P}(\\tilde R,A_P)\\to {\\rm Ext}^{n-i}_{A_P}(R_P,A_P)$ induced by the\nnatural inclusion $R_P\\to \\tilde R$ (by a slight abuse of language we identify the prime ideal\n$P$ of $R$ with its full preimage in $A$). Let $D_P(-)={\\rm Hom}_{A_P}(-, E_P)$ be the Matlis duality\nfunctor in the category of $R_P$-modules, where $E_P$ is the injective hull of the residue field of\n$R_P$ in the category of $R_P$-modules. Local duality implies that $D_P(\\phi_P)$ is the natural map\n$H^{i-d_P}_{P}(R_P)\\to H^{i-d_P}_{P}(\\tilde R)$ which is the zero map by construction\n(note that $i-d_P=\\dim A_P-(n-i)$). Since $\\phi_P$ is a map between finite $R_P$-modules\nand $D_P(\\phi_P)=0$, it follows that $\\phi_P=0$. This proves the claim.\n\nSince $N$ is a finite $R$-module, the set of the associated primes of $N$ is finite.\nLet $P_1,\\dots,P_s$ be the associated primes of $N$ different from $\\mathfrak m$. For each $j$\nlet $R^{P_j}$ be an $R$-subalgebra of $\\overline K$ corresponding to $P_j$, whose existence is guaranteed\nby the above claim. Let $\\overline R=R[R^{P_1},\\dots, R^{P_s}]$ be the compositum of all the $R^{P_j}$, $1\\leq j\\leq s$.\nClearly, $\\overline R$ is an $R$-subalgebra of $\\overline K$. Since each $R^{P_j}$\nis a finite $R$-module, so is $\\overline R$. Clearly, $\\overline R$  contains every $R^{P_j}$. Hence the above\nclaim implies that $\\mathcal I_{P_j}=0$ for every $j$, where $\\mathcal I\\subset N$ is the image of the\nnatural map ${\\rm Ext}^{n-i}_A(\\overline R,A)\\to N$ induced by the natural inclusion $R\\to\\overline R$. It follows\nthat  not a single $P_j$ is an associated prime of $\\mathcal I$. But $\\mathcal I$ is a submodule of $N$,\nand therefore every associated prime of $\\mathcal I$ is an associated prime of $N$.\nSince $P_1,\\dots, P_s$ are all the associated primes of $N$ different from $\\mathfrak m$, we conclude that if\n$\\mathcal I\\ne 0$, then $\\mathfrak m$ is the only associated prime of $\\mathcal I$. Since $\\mathcal I$,\nbeing a submodule of a finite $R$-module $N$, is finite, and since $\\mathfrak m$ is the only associated\nprime of $\\mathcal I$, we conclude that $\\mathcal I$ is an $R$-module of finite length.\n\nWriting the natural map ${\\rm Ext}^{n-i}_A(\\overline R,A)\\to N$ as the composition of two maps\n$${\\rm Ext}^{n-i}_A(\\overline R,A)\\to \\mathcal I\\to N,$$ the first of which is surjective and the second injective,\nand applying the Matlis duality functor $D$, we get that the natural map\n$\\varphi:H^i_{\\mathfrak m}(R)\\to H^i_{\\mathfrak m}(\\overline R)$ induced by the inclusion $R\\to \\overline R$ is the\ncomposition of two maps $H^i_{\\mathfrak m}(R)\\to D(\\mathcal I)\\to H^i_{\\mathfrak m}(\\overline R)$, the first\nof which is surjective and the second injective. This shows that the image of $\\varphi$ is isomorphic to\n$D(\\mathcal I)$ which is an $R$-module of finite length since so is $\\mathcal I$. In particular, the image\nof $\\varphi$ is a finitely generated $R$-module. Let $\\alpha_1,\\dots, \\alpha_s\\in\nH^i_{\\mathfrak m}(\\overline R)$ generate Im$\\varphi$.\n\nThe natural inclusion $R\\to \\overline R$ is compatible with the Frobenius homomorphism, i.e. with\nthe raising to the $p$th power on $R$ and $\\overline R$. This implies that $\\varphi$ is compatible\nwith the action of the Frobenius $f_*$ on $H^i_{\\mathfrak m}(R)$ and $H^i_{\\mathfrak m}(\\overline R)$,\ni.e. $\\varphi(f_*(\\alpha))=f_*(\\varphi(\\alpha))$ for every $\\alpha\\in H^i_{\\mathfrak m}(R)$,\nwhich, in turn, implies that Im$\\varphi$ is an $f_*$-stable $R$-submodule of $H^i_{\\mathfrak m}(\\overline R)$,\ni.e. $f_*(\\alpha)\\in {\\rm Im}\\varphi$ for every $\\alpha\\in {\\rm Im}\\varphi$. We finish the proof\nby applying Lemma~\\ref{element} to each element of  a finite generating set $\\alpha_1,...,\n\\alpha_s$ of Im$\\varphi$.\nApplying Lemma~\\ref{element} we obtain a\n$\\overline R$-subalgebra $R_j$ of $\\overline K$ (i.e. $\\overline R\\subset R_j\\subset \\overline K$) such that\n$R'_j$ is a finite $R$-module and the natural map $H^i_{\\mathfrak m}(\\overline R)\\to H^i_{\\mathfrak m}(R_j)$\nsends $\\alpha_j$ to zero. Let $R'=R[R_1,\\dots, R_s]$ be the compositum of all the $R_j$.\nThen $R'$ is an $R$-subalgebra of $\\overline K$ and is a finite $R$-module since so is each $R_j$.\nThe natural map $H^i_{\\mathfrak m}(\\overline R)\\to H^i_{\\mathfrak m}(R')$ sends every $\\alpha_j$ to zero,\nhence it sends the entire Im$\\varphi$ to zero. Thus the natural map\n$H^i_{\\mathfrak m}(R)\\to H^i_{\\mathfrak m}(R')$ is zero. \\qed\n\n\\medskip\n\n\n\n\n\\begin{cor}\\label{corcm}\nLet $R$ be a commutative Noetherian local domain containing a field of characteristic $p>0$.\nAssume that $R$ is a surjective image of a Gorenstein local ring. Then the following hold:\n\n(a) $H^i_{\\mathfrak m}(R^+)=0$ for all $i<\\dim R$, where $\\mathfrak m$ is the maximal ideal of $R$.\n\n(b) Every system of parameters of $R$ is a regular sequence on $R^+$.\n\\end{cor}\n\n\\emph{Proof.} (a) $R^+$ is the direct limit of the finitely generated $R$-subalgebras $R'$, hence $H^i_{\\mathfrak m}(R^+)=\\varinjlim H^i_{\\mathfrak m}(R')$.\nBut Theorem~\\ref{module} implies that for each $R'$ there is $R''$ such that the map \n $H^i_{\\mathfrak m}(R')\\to H^i_{\\mathfrak m}(R'')$ in the inductive system is zero. Hence the limit is zero.\n\n(b) Let $x_1,..., x_d$ be a system of parameters of $R$. We prove that $x_1,...,x_j$ is a regular\nsequence on $R^+$ by induction on $j$. The case $j=1$ is clear, since $R^+$ is a domain.\nAssume that $j>1$ and $x_1,\\dots, x_{j-1}$ is a regular sequence on $R^+$. Set $I_t = (x_1,...,x_t)$.\nThe fact that $H^i_{\\mathfrak m}(R^+)=0$ for all $i<d$ and the short exact sequences\n$$0\\to R^+\/I_{t-1}R^+\\stackrel{\\rm x_t}{\\longrightarrow} R^+\/I_{t-1}R^+\\to R^+\/I_{t}R^+\\to 0$$ for\n$t\\leq j-1$ imply by induction on $t$ that $H^q_{\\mathfrak m}(R^+\/(x_1,...,x_{t})R^+)=0$ for $q<d-t$.\nIn particular, $H^0_{\\mathfrak m}(R^+\/(x_1,...,x_{j-1})R^+)=0$ since $0<d-(j-1)$.\nHence $\\mathfrak m$ is not an associated prime of $R^+\/(x_1,...,x_{j-1})R^+$. This implies that the\nonly associated primes of $R^+\/(x_1,...,x_{j-1})R^+$ are the minimal primes of $R\/(x_1,...,x_{j-1})R$.\nIndeed, if there is an embedded associated prime, say $P$, then $P$ is the maximal ideal of the ring\n$R_P$ whose dimension is bigger than $j-1$ and $P$ is an associated prime of\n$(R^+\/(x_1,...,x_{j-1})R^+)_P=(R_P)^+\/(x_1,...,x_{j-1})(R_P)^+$ which is impossible by the above.\nHence every element of $\\mathfrak m$ not in any minimal prime of $R\/(x_1,...,x_{j-1})R$, for example,\n$x_j$, is a regular element on $R^+\/(x_1,...,x_{j-1})R^+$.\\qed\n\n\\medskip\n\n\\begin{disc}\n{\\rm It is important to understand the huge differences between characteristic $p$, characteristic $0$, and\nmixed characteristic.\nSuppose that $k$ has characteristic $0$. Let $A$ be a complete Noetherian local integrally closed domain with residue field $k$,\nand fraction field $K$. \nIf $L$ is any finite field extension of $K$ and $B$ is the integral closure of $A$\nin $L$, then the reduced trace map\\footnote{The \\it reduced trace \\rm is defined as $\\frac{1}{n} Tr_{L\/K}$ where\n$L$ is a finite field extension of a field $K$, and $[L:K] = n$. This map fixes the ground field $K$. If\n$R$ is an integrally closed domain with fraction field $K$ and $S$ is the integral closure of $R$ in $L$,\nthen the reduced trace sends $S$ to $R$ and fixes $R$.} gives a splitting of $A$ from $B$, i.e., $B\\cong A\\oplus N$ as\nan $A$-module for some module $A$-module $N$. Then $H^i_{{\\gothic{m}}}(A)$ splits out of $H^i_{{\\gothic{m}}}(B)$, so\nthat the map $H^i_{{\\gothic{m}}}(A)\\to H^i_{{\\gothic{m}}}(B)$ is \\it never \\rm zero unless $H^i_{{\\gothic{m}}}(A) = 0$. Thus the\nexact opposite holds in characteristic $0$.}\n\\end{disc}\n\nWhat happens in mixed characteristic is a great mystery. One of the great results in recent years was that\nof Heitmann. He proved the following theorem:\n\n\\begin{thm} Let $(R,{\\gothic{m}})$ be a complete three-dimensional Noetherian local integrally closed domain of\nmixed characteristic $p\\in \\mathbb N$. (This means that $p\\in {\\gothic{m}}$.) Then for all $n\\geq 1$,\n$p^{\\frac {1}{n}}$ annihilates the local cohomology $H^2_{{\\gothic{m}}}(R^+)$.\n\\end{thm}\n\nHeitmann's theorem is slightly stronger than this, but this is the essential result of \\cite{He}. In characteristic\n$p$ we know this local cohomology module is zero, but this is not known in mixed characteristic.\n\nOne can hope that if $R$ has mixed characteristic $p$, then $R^+\/pR^+$ is Cohen-Macaulay. Of course, this\nring has positive characteristic. However, perhaps this is too much to hope for. The next best result would\nbe to conjecture that $R^+\/\\sqrt{pR^+}$ is Cohen-Macaulay, a question raised by Lyubeznik. He has a partial\nresult in this direction \\cite{Ly}:\n\n\\begin{thm} Let $(R,{\\gothic{m}})$ be a Noetherian local excellent domain of mixed characteristic $p$. Assume the\ndimension of $R$ is at least $3$.  Set $\\overline{R} = R\/\\sqrt{pR}$ and $\\overline{R^+} = R^+\/\\sqrt{pR^+}$.\nThen $H^1_{{\\gothic{m}}}(\\overline{R^+}) = 0$, and every part of a system of parameters $a,b$ of $\\overline{R}$\nform a regular sequence on $\\overline{R^+}$.\n\\end{thm}\n\n\n\n\\bigskip\n\n\nAnother interesting question is whether or not one must use inseparable extensions to trivialize local cohomology.\nIn fact, Anurag Singh \\cite{Si} found the following nice trick to change inseparable elements to separable.\n\n\\begin{prop} Let $R$ be an excellent domain of characteristic $p > 0$, and let $I$ be an ideal of $R$. Suppose that\n$z\\in R$ is such that $z^q\\in I^{[q]}$, where $ q = p^e$ is a power of $p$. Then there exists an integral domain\n$S$, which is a module-finite separable extension of $R$, such that $z\\in IS$.\n\\end{prop}\n\\bigskip\n\nThe point here is that there clearly a finite inseparable extension of $R$, say $T$, such that $z\\in IT$. Simply\ntake $qth$ roots of the elements $a_j$ such that $z = \\sum a_jx_j^q$ where $x_j\\in I$. \n\n\\begin{proof}  Write $z = \\sum_{1\\leq j\\leq n} a_jx_j^q$ where $x_j\\in I$ as above. Consider the equations for $2\\leq i\\leq n$,\n$$ U_i^q + U_ix_1^q-a_i = 0.$$\nThese are monic separable equations and therefore have roots $u_i$ in a separable field extension of the fraction field\nof $R$. Let $S$ be the integral closure of the ring $R[u_2,...,u_n]$. \nSince $R$ is excellent, $S$ is finite as an $R$-module. We claim that $z\\in IS$. Set\n$$u_1 = (z- \\sum_{2\\leq i\\leq n} x_iu_i)\/x_1.$$\nNote that $u_1$ is an element of the fraction field of $S$. Taking $qth$ powers we see that\n$$u_1^q = a_1 + \\sum_{2\\leq i\\leq n} u_ix_i^q.$$\nTherefore $u_1$ is integral over $S$. As $S$ is integrally closed, $u_1\\in S$. This implies that $$z = \\sum_{1\\leq i\\leq n} u_ix_i$$\nand so $z\\in IS$. \\end{proof}\n\n\\begin{disc}{\\rm  There is an interesting property pertaining to our main theorem. Suppose that\n$(R,{\\gothic{m}})$ is a complete local Noetherian domain of positive characteristic, and let $x_1,...,x_d$ form\na regular sequence. If $x_1,...,x_d$ is a system of parameters, and if $R$ is not Cohen-Macaulay, then\nthere is a non-trivial relation $r_1x_1+...+r_dx_d = 0$. Non-trivial means that it does not\ncome from the Koszul relations. Since $R^+$ is Cohen-Macaulay, we can trivialize this relation in $R^+$,\nand therefore in some finite extension ring $S$ of $R$, $R\\subseteq S\\subseteq R^+$. But Theorem~\\ref{module} does not say\nwhether or not there is a fixed finite extension ring $T$, $R\\subseteq T\\subseteq R^+$ in which all relations on all\nparameters of $R$ become simultaneously trivial. Even if such a ring $T$ exists, this does not mean $T$ is\nitself Cohen-Macaulay; new relations coming from elements of $T$ may be introduced. However, there is a finite\nextension which simultaneously trivializes all relations on systems of parameters. This fact has been proved by\nMelvin Hochster and Yongwei Yao \\cite{HY}.}\n\\end{disc}\n\n\\section{Applications}\n\\bigskip\n\nThe existence of a big Cohen-Macaulay algebra has a great many applications. In some sense\nit repairs the failure of a ring to be Cohen-Macaulay. Hochster proved and used the existence of big Cohen-Macaulay\nmodules (the word ``big\" refers to the fact the modules may not be finitely generated) to prove many of the homological\nconjectures. For a modern update, see \\cite{Ho1}. In general, if you can prove a theorem in the\nCohen-Macaulay case, you should immediately try to use $R^+$ to try to prove it in general.\nWe give several examples of this phenomena in this section.  As examples, we will prove some\nof the old homological conjectures using this approach; this is not new, but there are currently\na growing number of new homological conjectures, and it could be that characteristic $p$ methods\napply.\n\nOf course, some of the homological conjectures deal directly with systems of parameters. These are\neasy to prove once one has a Cohen-Macaulay module. For example, the next theorem gives the\nmonomial conjecture.\n\n\\begin{thm} Let $R$ be a local Noetherian ring of dimension $d$ and positive characteristic $p$.\nLet $x_1,...,x_d$ be a system of parameters. Then for all $t\\geq 1$, $(x_1\\cdots x_d)^t$ is not\nin the ideal generated by $x_1^{t+1},...,x_d^{t+1}$.\n\\end{thm}\n\n\\begin{proof} We use induction on the dimension $d$ of $R$. The case $d = 1$ is trivial.\nSuppose by way of contradiction that $d > 1$ and  $(x_1\\cdots x_d)^t\\in (x_1^{t+1},...,x_d^{t+1})$.\nThis is preserved after completion, and is further preserved after moding out a minimal prime $P$\nsuch that the dimension of the completion modulo $P$ is still $d$. After these operations,\nthe images of the elements $x_i$ still form a system of parameters as well.  Thus we may assume that\n$R$ is a complete local domain. We apply Theorem~\\ref{module} to conclude that $x_1,...,x_d$ is\na regular sequence in $R^+$. Write\n$$(x_1\\cdots x_d)^t = \\sum_i s_ix_i^{t+1},$$\nwhere $s_i\\in R$. Then $x_d^t((x_1\\cdots x_{d-1})^t - s_dx_d)\\in (x_1^{t+1},...,x_{d-1}^{t+1})$.\nSince the powers of the $x_i$ also form a regular sequence in $R^+$, we conclude that\n$(x_1\\cdots x_{d-1})^t - s_dx_d\\in (x_1^{t+1},...,x_{d-1}^{t+1})R^+$. It follows that\nthere is a Noetherian complete local domain $S$ containing $R$ and module-finite over $R$\nsuch that $(x_1\\cdots x_{d-1})^t\\in (x_1^{t+1},...,x_{d-1}^{t+1},x_d)S$. But now\n$(x_1\\cdots x_{d-1})^t$ is in the ideal $(x_1^{t+1},...,x_{d-1}^{t+1})$ in the ring $S\/x_dS$,\nwhich has dimension $d-1$. Our induction shows that this is impossible. \\end{proof}\n\nNext, we apply Theorem~\\ref{module} it to various intersection theorems. One of the first such intersection conjectures\nwas: \n\n\\begin{conj} Let $(R,{\\gothic{m}})$ be a local Noetherian ring, and let $M,N$ be two finitely\ngenerated nonzero $R$-modules such that $M\\otimes_RN$ has finite length. Then\n$$\\dim N \\leq \\text{pd}_R(M).$$\n\\end{conj}\n\nOf course there is nothing to prove if the projective dimension of $M$ is infinite.\nWe prove (see \\cite{Ho}):\n\n\\begin{thm}\\label{newintersection} Let $(R,{\\gothic{m}})$ be a local Noetherian ring of positive prime characteristic $p$, and let $M,N$ be two finitely\ngenerated nonzero $R$-modules such that $M\\otimes_RN$ has finite length. Then\n$$\\dim N \\leq \\text{pd}_R(M).$$\n\\end{thm}\n\n\\begin{proof}\nOne can begin by making some easy reductions. These types of reduction are very good practice\nin commutative algebra. First, note that the assumption that the tensor product has finite\nlength is equivalent to saying that $I+J$ is ${\\gothic{m}}$-primary, where $I = \\text{Ann}(N)$ and\n$J = \\text{Ann}(M)$. Then we can choose a prime $P$ containing $I$ such that $\\dim(R\/P) =\n\\dim (N)$, and observe that we can replace $N$ by $R\/P$ without loss of generality.\nIt is more difficult to change $M$, since the property of being finite projective dimension\ndoes not allow many changes.  \n\nLet's just suppose for a moment that $R\/P$ is Cohen-Macaulay. Since $P+J$ is ${\\gothic{m}}$-primary,\nwe can always choose $x_1,...,x_d\\in J$ whose images in $B = R\/P$ form a system of parameters\n(and thus are a regular sequence in $R\/P$). If the projective dimension of $M$\nis smaller than $d = \\dim(R\/P)$, then  $Tor_d^{R}(B\/(x_1,...,x_d)B, M) = 0$.  \nNotice that $Tor_0^{R}(B, M) \\ne 0$. We claim by induction that for $0\\leq i\\leq d$,\n$Tor_i^{R}(B\/(x_1,...,x_i)B, M) \\ne 0$. When $i=d$ we arrive at a contradiction.\nSuppose we know this for $i < d$.  Set $B _i = B\/(x_1,...,x_i)$. The short exact sequence\n$0\\to B_i\\to B_i\\to B_{i+1}\\to 0$ obtained by multiplication\nby $x_{i+1}$ on $B_i$ induces a map of Tors when tensored with $M$. Since all $x_i$ kill $M$,\nwe obtain a surjection of $Tor_{i+1}^{R}(B_{i+1}, M)$ onto $Tor_i^{R}(B_i, M)$.\nThis finishes the induction.\n\nOf course, we don't know that $R\/P$ is Cohen-Macaulay, and in general it won't be. But now\nsuppose that we are in positive characteristic. We can first complete $R$ before beginning\nthe proof. Now $R\/P$ is a complete local domain, and $S = (R\/P)^+$ is Cohen-Macaulay in\nthe sense that $x_1,..,x_d$ form a regular sequence in this ring. The same proof works\nverbatium, provided we know that $S\\otimes_RM\\ne 0$. But this is easy; it is even nonzero\nafter passing to the residue field of $S$.  \\end{proof}\n\n\nAs a corollary, we get a favorite of the old Chicago school of commutative algebra, \nthe zero-divisor conjecture (now a theorem):\n\n\n\\bigskip\n\n\\begin{thm} Let $(R,{\\gothic{m}})$ be a Noetherian local ring of characteristic $p$, and let\n$M$ be a nonzero finitely generated $R$-module having finite projective dimension. If\n$x$ is a non-zerodivisor on $M$, then $x$ is a non-zerodivisor on $R$.\n\\end{thm}\n\n\\begin{proof} This proof is taken from \\cite{PS}.  First observe that the statement of the theorem is equivalent to saying\nthat every associated prime of $R$ is contained in an associated prime of $M$. We induct on\nthe dimension of $M$ to prove this statement. If $\\dim M = 0$, then the only associated\nprime of $M$ is ${\\gothic{m}}$, which clearly contains every prime of $R$. Hence we may assume that\n$\\dim M > 0$. Let $P\\in \\text{Ass}(R)$. First suppose that there is a prime $Q\\in \\text{Supp}(M), Q\\ne {\\gothic{m}},$\nsuch that $P\\subseteq Q$. Then we can change the ring to $R_Q$ and the module to $M_Q$. By induction, $P_Q$ is\ncontained in an associated prime of $M_Q$, so lifting back gives us that $P$ is in an associated\nprime of $M$. We have reduced to the case in which $R\/P\\otimes_RM$ has finite length. By \nTheorem~\\ref{newintersection}, $\\dim (R\/P)\\leq \\text{pd}_R(M) =  \\text{depth}(R) - \\text{depth}(M)$. Since\n$P$ is associated to $R$,  $\\dim (R\/P)\\geq \\text{depth}(R)$ (exercise). It follows that\nthe depth of $M$ is $0$, and hence the maximal ideal is associated to $M$ (and contains $P$).\n\\end{proof} \n\n\\medskip\n\nFor a completely different type of application, we consider an old result of Grothendieck's concerning\nwhen the punctured spectrum of a local ring is connected.\nThere is a beautiful proof of Grothendieck's result in all characteristics due to Brodmann and Rung \\cite{BR}. The main\npoint here is that if $R$ is Cohen-Macaulay and the $x_i$ are parameters, then there is a very easy\nproof. It turns out that one can always assume that the $x_i$ are parameters, and then the proof\nof the Cohen-Macaulay case directly generalizes to one in characteristic $p$ using $R^+$. \nThe exact statement is:\n\n\n\\begin{thm} Let $(R,{\\gothic{m}})$ be a complete local Noetherian domain of dimension $d$, and let\n$x_1,...,x_k\\in {\\gothic{m}}$, where $k\\leq d-2$. Then the punctured spectrum of $R\/(x_1,...,x_k)$ is\nconnected.\n\\end{thm}\n\n\\begin{proof} We take the proof from \\cite{HH}.\nFirst assume that the $x_i$ are parameters.\nLet $I$ and $J$ give a disconnection of the punctured spectrum\nof $R\/(x_1,...,x_k)$. Choose elements $u+v$, $y+z$ which together with the $x_i$ form parameters\nsuch that $u,y\\in I$ and $v,z\\in J$. Modulo $(x_1,...,x_k)$ one has the relation\n$y(u+v) - u(y+z) = 0$. Since the parameters form a regular sequence in $R^+$, we obtain that\n$y\\in (y+z,x_1,...,x_k)R^+$. Similarly, $z\\in (y+z,x_1,...,x_k)R^+$. Write $y = c(y+z)$ modulo\n$(x_1,...,x_k)$ and $z = d(y+z)$ modulo $(x_1,...,x_k)$. Then $(1-c-d)(y+z)\\in (x_1,...,x_k)R^+$\nso that $1-c-d\\in (x_1,...,x_k)R^+$. At least one of $c$ or $d$ is a unit in $R^+$, say $c$. But then\n$y$ is not a zerodivisor modulo $(x_1,...,x_k)R^+$ and this implies that $J\\subseteq (x_1,...,x_k)R^+$.\nThen $I$ must be primary to $mR^+$ which is a contradiction since the height of $I$ is too small.\n\n\nIt remain to reduce to the case in which the $x_i$ are parameters.\n\nWe claim that any $k$-elements are up to radical in an ideal generated by $k$-elements which are\nparameters.\n The key point is to prove this for $k = 1$. Suppose that $x = x_1$ is given. If $x$\nalready has height one we are done. If $x$ is nilpotent, choose $y$ to be any parameter in $I$. So assume\nthat $x$ is not in every minimal prime. For $n\\gg 0$, $0:x^n = 0:x^{n+1}$, and changing $x$ to $x^n$,\nwe obtain that $x$ is not a zero divisor on $R\/(0:x)$. Then there is an element $s\\in 0:x$ such that\n$y = x +s$ has height one, and we may multiply $s$ by a general element of $I$ to obtain that\n$y\\in I$. But $xy = x^2$ so that $x$ is nilpotent on $(y)$. Inductively choose\n$y_1,...,y_{k-1}$ which are parameters such that the ideal they generate contains $x_1,...,x_{k-1}$ up\nto radical. Replace $R$ by $R\/(y_1,...,y_{k-1})$, and repeat the $k = 1$ step.\n\nNow if\n$I$ and $J$ disconnect the punctured spectrum of $R\/(x_1,...,x_k)$ choose any ideals $I'$ and $J'$\nof height at least $k$, not primary to the maximal ideal such that $I'$ contains $I$ up to radical\nand $J'$ contains $J$ up to radical. Choose parameters in $I'\\cap J'$ such that the $x_i$ are in the\nradical $K$ of these parameters. Then $K + I$ and $K + J$ disconnect the punctured spectrum of $R\/K$.\n\\end{proof}\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\subsection{General remarks}\nA large open problem of classical general relativity is the\ncharacterization of the structure of a spacetime by initial data. The\nflat case, Minkowski spacetime, is geodesically complete. To the\nother extreme the singularity theorems by R. Penrose and S. Hawking show\nthat the spacetime cannot be geodesically complete if the data are\nlarge \\cite{HaE73TL}.\n\\\\\nIn the last years there has been remarkable progress in describing\nwhat happens if one goes from data for flat space to large data: The\nfuture of small data evolving in accordance with\nthe Einstein equation with various matter models as sources, vacuum\nand Einstein-Maxwell-Yang-Mills, looks like the future of data for\nflat space \\cite{ChK93TG,Fr93as}. Nevertheless many problems are\nstill unsolved.\n\\\\\nThose results were significantly improved by D.~Christodoulou for\nspherically symmetric models with a massless Klein-Gordon scalar field as\nsource. He was able to relate properties of the initial data to\nproperties of singularities. But even in this case of high symmetry\nthe questions left are still numerous as numerical\nsimulations by M.~Choptuik show \\cite{Ch93ua}. He found very\ninteresting properties, the so called echoing effect, for models which\nare in the parameter space of initial data near to the boundary which\nseparates regular from singular spacetimes.\n\\\\\nIn this paper conformal techniques are used to analyze the\nhyperboloidal initial value problem with scalar fields as\nmatter models --- for data near Minkowskian data the future of the\ninitial value\nsurface possesses a smooth future null infinity and a regular timelike\ninfinity, for large data a smooth future null infinity exists for at\nleast some time. In the second part of the introduction more about\nconformal techniques and their application for a mathematical\ndescription of asymptotically flat spacetimes will be said.\n\\\\\nAlthough the primarily treated matter model is that of the conformally\ninvariant scalar field, whose equations can be written as\n\\begin{subequations}\n\\label{model}\n\\begin{eqnarray}\n\\label{Wllngl}\n  \\tilde{\\vphantom{\\phi}\\Box} \\mbox{$\\tilde{\\phi}$} - \\frac{\\mbox{$\\tilde{R}$}}{6} \\, \\mbox{$\\tilde{\\phi}$} & = & 0\n\\\\\n\\label{EinstPhys}\n  ( 1 - \\frac{1}{4} \\mbox{$\\tilde{\\phi}$}^2 ) \\, \\mbox{$\\tilde{R}$}_{ab} & = &\n       \\left(\n          (\\mbox{$\\tilde{\\nabla}$}_a \\mbox{$\\tilde{\\phi}$}) (\\mbox{$\\tilde{\\nabla}$}_b \\mbox{$\\tilde{\\phi}$}) - \\frac{1}{2} \\, \\mbox{$\\tilde{\\phi}$} \\, \\mbox{$\\tilde{\\nabla}$}_a \\mbox{$\\tilde{\\nabla}$}_b \\mbox{$\\tilde{\\phi}$}\n          - \\frac{1}{4} \\, \\mbox{$\\tilde{g}$}_{ab} (\\mbox{$\\tilde{\\nabla}$}^c \\mbox{$\\tilde{\\phi}$}) (\\mbox{$\\tilde{\\nabla}$}_c \\mbox{$\\tilde{\\phi}$})\n       \\right),\n\\end{eqnarray}\nthe results obtained apply to a larger class of scalar field models,\ngiven by the class of actions (\\ref{ScalarAction}), including the massless\nKlein-Gordon field, as shown in section \\ref{SkalarAequiv}. Note that\nan arbitrary factor can be absorbed into $\\mbox{$\\tilde{\\phi}$}{}$ which changes the\ncoefficients in (\\ref{EinstPhys}). My notational conventions are\ndescribed in the appendix, the $\\tilde{\\vphantom{H}}$ marks quantities\nin the physical spacetime (see definition \\ref{asymFlat}). The\nenergy-momentum tensor for the conformally invariant scalar field can\nbe written as\n\\begin{equation}\n\\label{TkonfS}\n  \\mbox{$\\tilde{T}$}_{ab} = (\\mbox{$\\tilde{\\nabla}$}_a \\mbox{$\\tilde{\\phi}$}) (\\mbox{$\\tilde{\\nabla}$}_b \\mbox{$\\tilde{\\phi}$}) - \\frac{1}{2} \\, \\mbox{$\\tilde{\\phi}$} \\, \\mbox{$\\tilde{\\nabla}$}_a \\mbox{$\\tilde{\\nabla}$}_b \\mbox{$\\tilde{\\phi}$}\n        + \\frac{1}{4} \\, \\mbox{$\\tilde{\\phi}$}^2 \\mbox{$\\tilde{R}$}_{ab} -\n        \\frac{1}{4} \\, \\mbox{$\\tilde{g}$}_{ab}\n          \\left( (\\mbox{$\\tilde{\\nabla}$}^c \\mbox{$\\tilde{\\phi}$}) (\\mbox{$\\tilde{\\nabla}$}_c \\mbox{$\\tilde{\\phi}$}) + \\frac{1}{6} \\, \\mbox{$\\tilde{\\phi}$}^2 \\mbox{$\\tilde{R}$}\n          \\right).\n\\end{equation}\n\\end{subequations}\nThe analytic investigation presented in this paper show the well\nposedness of the initial value problem in unphysical spacetime, which\nis a technical construct to ``compactified'' asymptotically flat\nspacetimes in analogy to the compactification of the plane of complex\nnumbers ($\\Bbb{R}^2$) into the Riemann sphere ($\\Bbb{S}^2$).\n\\\\\nOne goal\nof this work was making myself familiar with the system in unphysical\nspacetime as a preparation for numerical work showing\nthat the conformal techniques are well suited for a numerical\ninvestigation of global spacetime structure and gravitational\nradiation \\cite{HuXXIP,Hu93nu}. To lower the computational\nresources required these calculations have been done for spherical\nsymmetry. It is well known that spherically symmetric, uncharged vacuum\nmodels are Schwarzschild. The inclusion of matter removes\nthat obstacle, the spacetime may evolve dynamically. Furthermore there\nis no gravitational radiation in spherically symmetric models. Therefore\nthe matter model should also be a model for radiation. Scalar fields\nwith wave equations are choices for matter which model also radiation.\nThe conformally invariant scalar field has been\nchosen since the matter equations are form invariant under rescalings\nof the metric and an appropriate transformation of the scalar field\nas the name already suggests.\n\\\\\nThe scalar fields are interesting from the analytic viewpoint since\nfor the first time conformal techniques could be used for matter\nmodels whose energy-momentum tensor has non-vanishing trace.\n\\subsection{Asymptotically flat spacetimes}\nIn this paper a geometrical, coordinate independent definition of\nasymptotical flatness along the lines suggested by R.~Penrose\nwill be used. A more thorough discussion\nof the ideas and the interpretation can be found at various places in\nthe literature, e.g.\\ \\cite{Ge76as,Pe64ct}. The\ndefinitions of asymptotical flatness given in the literature differ\nslightly. The following will be used here:\n\\begin{Def}\n\\label{asymFlat}\n  A spacetime $(\\tilde M, \\mbox{$\\tilde{g}$}_{ab})$ is called {\\bf asymptotically\n    flat} if there is another ``unphysical'' spacetime $(M,g_{ab})$\n  with boundary \\mbox{$\\cal J$}{} and a smooth embedding by which $\\tilde M$\n  can be identified with $M-\\mbox{$\\cal J$}{}$ such that:\n  \\begin{enumerate}\n  \\item There is a smooth function $\\Omega$ on $M$ with\n    \\begin{equation*}\n      \\Omega \\mid_{\\tilde M} > 0 \\qquad \\mbox{and} \\qquad\n      g_{ab} \\mid_{\\tilde M} = \\Omega^2 \\mbox{$\\tilde{g}$}_{ab}.\n    \\end{equation*}\n  \\item On \\mbox{$\\cal J$}{}\n    \\begin{equation*}\n      \\Omega = 0 \\qquad \\mbox{and} \\qquad \\nabla_a \\Omega \\ne 0.\n    \\end{equation*}\n  \\item \\label{nullCompleteness} Each null geodesic in $(\\tilde\n    M,\\tilde g_{ab})$ acquires a past and a future endpoint on \\mbox{$\\cal J$}{}.\n  \\end{enumerate}\n\\end{Def}\nBecause of item~\\ref{nullCompleteness} null geodesically incomplete\nspacetimes like Schwarzschild are not asymptotically flat. The next\ndefinition includes those spacetimes which have only an\nasymptotically flat part:\n\\begin{Def}\n\\label{weakasymFlat}\n  A spacetime is called {\\bf weakly asymptotically flat} if\n  definition~\\ref{asymFlat} with the exception\n  of item~\\ref{nullCompleteness} is fulfilled.\n\\end{Def}\nDefinition~\\ref{asymFlat} and~\\ref{weakasymFlat} classify spacetimes,\nthey do not require that Einstein's equation is fulfilled. One would\nlike to know:\n\\\\\nAre they compatible with the Einstein equation with sources?\nNeither definition~\\ref{asymFlat} nor \\ref{weakasymFlat} is in an\ninitial value problem form: A given spacetime is or is not classified\nas asymptotically flat. But for a physical problem one would like to\ngive ``asymptotically flat data'' and have guaranteed that they evolve\ninto an at least weakly asymptotically flat spacetime.\n\\\\\nNevertheless the geometrically description was extremely helpful in\nanalyzing asymptotically flat spacetimes and it can be successfully\nused as guideline to construct a formalism which is better suited for\nanalyzing initial value problems. This method has been developed and\napplied to various matter sources by H.~Friedrich~\n\\cite{Fr81on,Fr83cp,Fr85ot,Fr86op,Fr86ot,Fr88os,Fr91ot}. In this paper\nit will be applied to general relativistic scalar field models.\n\\\\\nThe idea is to choose a spacelike initial value surface in the\nunphysical spacetime $(M,g_{ab})$ and to evolve it. The problems to\nbe faced are:\n\\\\\nFor Minkowski space the unphysical spacetime $(M,g_{ab})$ can be\nsmoothly extended with three points, future $(i^+)$ and past  $(i^-)$\ntimelike infinity, the end respectively the starting point of all\ntimelike geodesics of $(\\tilde M,\\mbox{$\\tilde{g}$}_{ab})$, and spacelike infinity\n$(i^0)$,\nthe\nend point of all spacelike geodesics of $(\\tilde M,\\mbox{$\\tilde{g}$}_{ab})$. The point $i^0$\ndivides \\mbox{$\\cal J$}{} into two disjunct parts, future $(\\mbox{$\\cal J$}^+)$ and past\n$(\\mbox{$\\cal J$}^-)$ null infinity. It is well known and has been discussed\nelsewhere that there are unsolved problems in smoothly\nextending a ``normal'' Cauchy hypersurface of $\\tilde M$ to $i^0$ if the\nspacetime has non-vanishing ADM mass. Certain curvature quantities\nblow up at $i^0$, reflecting the non-invariance of the mass under\nrescalings.\n\\\\\nBy choosing a spacelike (with respect to $g_{ab}$) hypersurface $S$\nnot intersecting $i^0$ but $\\mbox{$\\cal J$}^+$ ($\\mbox{$\\cal J$}^-$) we avoid the problems\nwith $i^0$. $S$ is called a hyperboloidal hypersurface --- the\ncorresponding initial value problem is called a hyperboloidal initial\nvalue (a detailed definition for the scalar field models is given in\nsection~\\ref{HypInitValProblSec}, definition~\\ref{HypInitValProbl}).\nThe domain of dependence $D(S)$ of\n$S$ will not contain the whole spacetime. The interior of $S$\ncorresponds to an everywhere spacelike hypersurface in the physical\nspacetime which approaches a null hypersurface $N$ asymptotically. If $N$\nis a light cone $L$ then the domain of dependence of $S$ is $L$. Therefore\nthe hyperboloidal initial value problem is well suited to describe the\nfuture (past) of data on the spacelike hypersurface $S$, e.~g.~a\nstellar object and the gravitational radiation caused by its\ntime evolution. It is not well suited to investigate the structure\nnear $i^0$.\n\\\\\nBut even for the hyperboloidal initial value problem there are\n``regularity'' problems at \\mbox{$\\cal J$}{}: Transforming the Einstein equation\nfrom physical to unphysical spacetime an equation ``singular'' for\n$\\Omega=0$ results. That problem is solved in this paper in analogy to\nH.~Friedrich's work. A new set of equations\nfor the unphysical spacetime will be derived, its equivalence to the\nEinstein equation on $\\tilde M$ proven. This new set of equations is\nused to prove the consistency of the hyperboloidal initial value\nproblem for scalar fields with (weakly) asymptotical flatness and the\nexistence of a regular future (past) timelike infinity\nfor data sufficiently close to data for Minkowski spacetime.\n\\section{Regularizing the unphysical field equations}\nA first attempt for equations determining $(M,g_{ab})$ is the rescaled\nform of the field equation in physical spacetime. A closer look at\nthe transformation of the Einstein tensor under rescalings\n$g_{ab}=\\Omega^2\\,\\mbox{$\\tilde{g}$}_{ab}$,\n\\begin{equation}\n\\label{GabTransf}\n  \\mbox{$\\tilde{G}$}_{ab} =\n  G_{ab}\n  + 2 \\, \\Omega^{-1} \\left(\n    \\nabla_a \\nabla_b \\Omega - (\\nabla^c \\nabla_c \\Omega ) \\, g_{ab}\n    \\right)\n  + 3 \\, \\Omega^{-2} ( \\nabla^c \\Omega ) ( \\nabla_c \\Omega ) \\, g_{ab},\n\\end{equation}\nshows that this first\nattempt fails. Either there are terms proportional to $\\Omega^{-2}$\nand $\\Omega^{-1}$, which need special care on the set ${\\cal I}$ of\npoints where $\\Omega=0$, including\n\\mbox{$\\cal J$}{}, which is part of $M$. Or alternatively, the highest (second\norder) derivatives of the metric, hidden in the Einstein tensor, are\nmultiplied by a factor of $\\Omega^2$ and then the principal part of\nthe second order equation for the metric components vanishes on\n${\\cal I}$. This behaviour of an equation will be called singular on\n${\\cal I}$.\n\\\\\nIn this section a system of equations without the singularity on\n${\\cal I}$ will be derived from the rescaled Einstein\nequation by introducing new variables and equations.\n\\\\\nThe set of equations together with the equations for the matter\nvariables may be a system with a very complicated principal part ---\nas it is the case for a conformally invariant scalar field as\nmatter model. A procedure is carried out to simplify the principal\npart to a form in which no equation contains both derivatives of\nmatter variables as well as derivatives of geometry variables and the\nprincipal part of the subsystem for the geometry variables is the same\nas for the vacuum case (``standard form''). All variables already\npresent in the vacuum case are called geometry variables.\n\\\\\nIt is shown that the procedure works for the conformally invariant\nscalar field. The procedure described does not use very restrictive\nassumptions --- it is very general --- and may work for most matter\nmodels, for which the unphysical matter equations can be regularized on\n${\\cal I}$.\n\\subsection{The geometry part of the system}\nAccording to the definition of asymptotical flatness\n(definition \\ref{asymFlat}) the unphysical spacetime is connected with the\nphysical spacetime through the rescaling\n\\begin{equation}\n  g_{ab} \\mid_{\\tilde M} = \\Omega^2 \\, \\mbox{$\\tilde{g}$}_{ab}.\n\\end{equation}\nThis rule also determines the transformation of the connection\nand the curvature.\n\\\\\nAdditionally the transformation of the matter variables $\\tilde\n\\Phi{}$ under rescaling must be specified,\n\\begin{equation}\n  \\Phi \\mid_{\\tilde M} = \\Phi[\\mbox{$\\tilde{g}$}_{ab},\\Omega,\\tilde \\Phi].\n\\end{equation}\nIt is assumed that $\\Phi{}$ has a smooth limit on \\mbox{$\\cal J$}{}, the\nrescaled equations for the matter variables are regular on ${\\cal\n  I} $\\footnote{In the general case it is not known how to achieve\n  that.},\nand there exists a tensor $T_{ab}$, which is independent of derivatives\nof $\\Omega{}$ and derivatives of curvature terms, fulfills\n\\begin{equation}\n  T_{ab} \\mid_{\\tilde M} = \\Omega^{-2}\\, \\mbox{$\\tilde{T}$}_{ab},\n\\end{equation}\nand has a limit on \\mbox{$\\cal J$}{} \\footnote{From the definition of\n  asymptotical flatness and the Einstein equation it follows that\n  $\\Omega^{-1}\\, \\mbox{$\\tilde{T}$}_{ab}$ has a limit on \\mbox{$\\cal J$}{}~\\cite{AsS80ni}. The\n  faster fall off and the requirements on the form of $T_{ab}$ have\n  technical reasons.}.\nThe conditions required may seem very restrictive but they\ncan be fulfilled for Yang-Mills fields~\\cite{Fr91ot} and for the\nconformally invariant scalar field.\n\\\\[0.1cm]\nThe Riemann tensor will be split into its irreducible parts, the\nconformal Weyl tensor\n\\begin{equation}\n\\label{ddef}\nC_{abc}{}^d=:\\Omega\\,d_{abc}{}^d,\n\\end{equation}\nthe trace free part $\\mbox{$\\hat{R}$}_{ab}$ of the Ricci tensor $R_{ab}$ and the\nRicci scalar $R$:\n\\begin{equation}\n\\label{Ralg}\n  R_{abcd} =\n  \\Omega \\, d_{abcd}\n  + g_{c[a} \\mbox{$\\hat{R}$}_{b]d} - g_{d[a} \\mbox{$\\hat{R}$}_{b]c}\n  + \\frac{1}{6} g_{c[a} g_{b]d} R,\n\\end{equation}\nA $\\hat{\\phantom{H}}$ is used as an indication for trace free parts of\n\ntensors.\n\\\\\nThe irreducible decomposition of the energy-momentum tensor is\n\\begin{equation}\n\\label{IrredT}\n  T_{ab} = \\mbox{$\\hat{T}$}_{ab} + \\frac{1}{4} \\, g_{ab} \\, T.\n\\end{equation}\nThe irreducible parts transform under rescalings according to\n$$\n  \\mbox{$\\hat{T}$}_{ab} = \\Omega^{-2} \\mbox{$\\tilde{\\hT}$}_{ab}\n$$\nand\n$$\n  T = \\Omega^{-4} \\mbox{$\\tilde{T}$}.\n$$\nThe vanishing of the divergence of $\\mbox{$\\tilde{T}$}_{ab}$ becomes\n\\begin{equation}\n\\label{transdivT}\n  0 = \\tilde\\nabla^a \\tilde T_{ab} =\n  \\Omega^4 \\, \\nabla^a \\mbox{$\\hat{T}$}_{ab} +\n  \\frac{1}{4} \\, \\Omega^4 \\, \\nabla_b T +\n  \\Omega^3 \\, T \\, \\nabla_b \\Omega.\n\\end{equation}\nFor energy-momentum tensors with non-vanishing trace equation\n(\\ref{transdivT}) as an equation for the components of the\nirreducible parts of the energy-momentum tensor $T_{ab}$\nis singular on ${\\cal I}$. Since (\\ref{transdivT}) should be in same way\npart of the matter equations problems in regularizing the matter\nequations are to be expected.\n\\subsubsection{A regular system}\nThe part of (\\ref{GabTransf}) proportional to $\\mbox{$\\Omega$}^{-2}$ is a pure\ntrace, thus the $\\mbox{$\\Omega$}^{-2}$ singularity is absent in the trace free equation.\nA decomposition into the trace and the trace free part moves the\nworst term into one equation.\n\\\\\n{}From the rescaling rule for the Ricci scalar and tensor,\n\\begin{equation}\n\\label{Rtrans}\n  \\mbox{$\\tilde{R}$} = \\Omega^2 R + 6 \\, \\Omega \\, \\nabla^a \\nabla_a \\Omega -\n        12 \\, (\\nabla^a \\Omega) \\, (\\nabla_a \\Omega),\n\\end{equation}\nand\n\\begin{eqnarray}\n  \\tilde{\\mbox{$\\hat{R}$}}_{ab} & := & \\mbox{$\\tilde{R}$}_{ab} - \\frac{1}{4} \\mbox{$\\tilde{g}$}_{ab} \\mbox{$\\tilde{R}$}\n\\nonumber \\\\\n   & = & \\mbox{$\\hat{R}$}_{ab} + 2 \\, \\Omega^{-1} \\nabla_a \\nabla_b \\Omega\n         - \\frac{1}{2} \\Omega^{-1} (\\nabla^c \\nabla_c \\Omega) \\,\n           g_{ab},\n\\end{eqnarray}\n$\\mbox{$\\tilde{G}$}_{ab} = \\mbox{$\\tilde{T}$}_{ab}$, ${\\mbox{$\\tilde{G}$}=-\\mbox{$\\tilde{R}$}}$, and ${\\mbox{$\\tilde{T}$} = \\mbox{$\\tilde{G}$}}$\nit follows\n\\begin{equation}\n\\label{SpGl}\n  \\Omega \\, R + 6 \\, \\nabla^a \\nabla_a \\Omega\n    - 12 \\, \\Omega^{-1} \\, (\\nabla^a \\Omega) \\, (\\nabla_a \\Omega)\n  = - \\, \\Omega^3 \\, T\n\\end{equation}\nand\n\\begin{equation}\n\\label{sfGl}\n  \\Omega \\, \\mbox{$\\hat{R}$}_{ab} + 2 \\, \\nabla_a \\nabla_b \\Omega -\n  \\frac{1}{2} (\\nabla^c \\nabla_c \\Omega) \\, g_{ab} =\n    \\Omega^3 \\, T_{ab}.\n\\end{equation}\nEquation (\\ref{SpGl}) can be dealt with by the following lemma:\n\\begin{Lemma}\n  From $\\mbox{$\\tilde{R}$}+\\mbox{$\\tilde{T}$}=0$ ($\\hat{=}$ (\\ref{SpGl})) at one point,\n  $\\hat{\\mbox{$\\tilde{G}$}}_{ab}=\\hat{\\mbox{$\\tilde{T}$}}_{ab}$ ($\\hat{=}$ (\\ref{sfGl})), and $\\mbox{$\\tilde{\\nabla}$}^b\n  {\\mbox{$\\tilde{T}$}_{ab}}=0$ $\\mbox{$\\tilde{R}$}+\\mbox{$\\tilde{T}$}=0$ follows everywhere.\n\\end{Lemma}\nProof:\n\\begin{equation*}\n  \\mbox{$\\tilde{\\nabla}$}^a \\mbox{$\\tilde{T}$}_{ab} = \\mbox{$\\tilde{\\nabla}$}^a \\mbox{$\\tilde{\\hT}$}_{ab} + \\frac{1}{4} \\mbox{$\\tilde{\\nabla}$}_b \\mbox{$\\tilde{T}$} = 0.\n\\end{equation*}\nCombined with\n\\begin{eqnarray*}\n  0 & = & \\mbox{$\\tilde{\\nabla}$}^a \\mbox{$\\tilde{G}$}_{ab} \\\\\n    & = & \\mbox{$\\tilde{\\nabla}$}^a \\tilde{\\mbox{$\\hat{G}$}}_{ab} + \\frac{1}{4} \\mbox{$\\tilde{\\nabla}$}_b \\mbox{$\\tilde{G}$}\n\\end{eqnarray*}\ngives\n\\begin{equation*}\n  \\mbox{$\\tilde{\\nabla}$}_b (\\mbox{$\\tilde{T}$} + \\mbox{$\\tilde{R}$}) = 0,\n\\end{equation*}\ni.e. $\\mbox{$\\tilde{T}$} + \\mbox{$\\tilde{R}$} $ is constant.\n\\\\\nEquation (\\ref{SpGl}) will not be used any longer since $\\mbox{$\\tilde{\\nabla}$}^b\n{\\mbox{$\\tilde{T}$}_{ab}}=0$ can be derived from the remaining equations, contract\n(\\ref{quaSysd}) or see the discussion following (\\ref{IntegrBed}).\n\\\\\nIn the following the Ricci scalar $R$ will be regarded as an\narbitrary, given function.\nIt fixes part of the gauge freedom on the transition from the\nphysical to the unphysical spacetime as follows:\nThe equations (\\ref{SpGl}) and (\\ref{sfGl}) are invariant under\nrescalings ${(g_{ab},\\Omega)}\\mapsto{(\\bar g_{ab},\\bar\n  \\Omega)}:={(\\Theta^2 g_{ab},\\Theta\\,\\Omega)}$ with $\\Theta>0$. All\nthe unphysical spacetimes $(M,\\Theta^2 g_{ab}, \\Theta\\,\\Omega)$ belong\nto the same physical spacetime $(\\tilde M,\\mbox{$\\tilde{g}$}_{ab})$.\n\\\\\nUnder the rescaling $\\bar g_{ab} = \\Theta^2\\,g_{ab}$, $R$ and $\\bar R$\nare connected by\n\\begin{equation}\n\\label{conformalGauge}\n  6\\, \\nabla^a \\nabla_a \\Theta = \\Theta R - \\Theta^3 \\bar R,\n\\end{equation}\nwhich is equation (\\ref{Rtrans}) where the covariant derivatives\n$\\nabla_a$ now corresponds to the unscaled metric. Solving\n(\\ref{conformalGauge}) for a spacetime ($M$, $g_{ab}$) and data\nfor $\\Theta{}$ and $\\dot \\Theta{}$ on\na spacelike surface $S$ we get at least locally a unphysical\nspace time with arbitrary Ricci scalar $\\bar{R}$.\n\\\\\nThere is still conformal gauge freedom left as every rescaling with\n$\\Theta>0$ and\n\\begin{equation}\n\\label{ConformalGaugeFix}\n  \\nabla^a \\nabla_a \\Theta = \\frac{1}{6} \\Theta\\,R\\,\\left(1-\\Theta^2\\right)\n\\end{equation}\nleaves the Ricci scalar unchanged.\n\\\\[2\\parskip]\nEquation (\\ref{sfGl}) serves as regular equation for\n$\\Omega{}$. Substituting ${\\omega = \\frac{1}{4} \\, \\nabla^c \\nabla_c\n  \\Omega}$ yields\n\\begin{equation}\n\\label{OmGl}\n  \\nabla_a \\nabla_b \\Omega = - \\, \\frac{1}{2} \\, \\Omega \\, \\mbox{$\\hat{R}$}_{ab}\n                      + \\omega \\, g_{ab}\n                      + \\frac{1}{2} \\, \\Omega^3 \\mbox{$\\hat{T}$}_{ab},\n\\end{equation}\nwhich is a second order equation for $\\Omega{}$.\n\\\\\nThe next step is to find equations for the metric and the quantities\nderived therefrom. Expressing the once contracted, second Bianchi\nidentity (${\\nabla_{[a} R_{bc]d}{}^a =0}$)  in terms of $\\mbox{$\\hat{R}$}_{ab}$ and\n$d_{abc}{}^d$\nresults in\n\\begin{equation}\n\\label{hRGl}\n  \\nabla_{[a} \\mbox{$\\hat{R}$}_{b]c} = - \\, \\frac{1}{12} \\, (\\nabla_{[a} R) g_{b]c}\n                          - (\\nabla_d \\Omega) \\, d_{abc}{}^d\n                          - \\Omega \\, \\nabla_d d_{abc}{}^d.\n\\end{equation}\nThe once contracted second Bianchi identity in the physical spacetime,\n$$\n  \\mbox{$\\tilde{\\nabla}$}_d \\mbox{$\\tilde{C}$}_{abc}{}^d =\n  - \\mbox{$\\tilde{\\nabla}$}_{[a} (\\mbox{$\\tilde{R}$}_{b]c} - \\frac{1}{6} \\mbox{$\\tilde{g}$}_{b]c} \\mbox{$\\tilde{R}$}),\n$$\ntogether with\n$$\n  \\Omega^{-1} \\mbox{$\\tilde{\\nabla}$}_d \\mbox{$\\tilde{C}$}_{abc}{}^d = \\nabla_d(\\Omega^{-1}\n                                    C_{abc}{}^d),\n$$\nand the Einstein equation in physical spacetime provide us with\nan equation for $d_{abc}{}^d$:\n\\begin{eqnarray}\n\\label{dGl}\n  \\nabla_d d_{abc}{}^d\n                       & = & - \\, \\Omega \\, \\nabla_{[a} \\mbox{$\\hat{T}$}_{b]c}\n                             - 3 \\, (\\nabla_{[a} \\Omega) \\, \\mbox{$\\hat{T}$}_{b]c}\n                             + g_{c[a} \\mbox{$\\hat{T}$}_{b]d} \\, (\\nabla^d \\Omega)\n                                                        \\nonumber  \\\\\n                       &   & + \\frac{1}{3} \\, (\\nabla_{[a} \\Omega) \\,\n                       T \\, g_{b]c}\n                             + \\frac{1}{12} \\Omega \\, (\\nabla_{[a}T)\n                               \\, g_{b]c}  \\; =: \\;  t_{abc}.\n\\end{eqnarray}\nWe can now derive the missing equation for $\\omega{}$ from the\nintegrability condition for (\\ref{OmGl}) and by substituting (\\ref{hRGl}):\n\\begin{eqnarray}\n\\label{omGl}\n  \\nabla_a \\omega & = & - \\frac{1}{2} \\mbox{$\\hat{R}$}_{ab} \\, \\nabla^b \\Omega\n                    - \\frac{1}{12} \\, R \\, \\nabla_a \\Omega\n                    - \\frac{1}{24} \\Omega \\, \\nabla_a R\n                    + \\frac{1}{2} \\Omega^2 \\, \\mbox{$\\hat{T}$}_{ab} \\, \\nabla^b \\Omega\n                                                       \\nonumber   \\\\\n                  &  &  - \\frac{1}{6} \\Omega^2 \\, (\\nabla_a \\Omega) \\, T\n                        - \\frac{1}{24} \\Omega^3 \\, \\nabla_a T.\n\\end{eqnarray}\nIn the following in addition to the abstract indices (small Latin\nletters) frame (underlined indices) and coordinate indices (Greek\nletters) are used. The used conventions are explained in the appendix\nin more detail.\n\\\\\nUsing $\\Omega_a := \\nabla_a \\Omega{}$, the frame  $e_{\\f{i}}{}^a$, and\nthe Ricci rotation coefficients $\\gamma^a{}_{\\f{i}\\f{j}}$ as further\nvariables, we get the following first order system of tensor equations\nfor  $\\Omega{}$, $\\Omega_a$, $\\omega$, $e_{\\f{i}}{}^a$,\n$\\gamma^a{}_{\\f{i}\\f{j}}$, $\\mbox{$\\hat{R}$}_{ab}$, and $d_{abc}{}^d$: \\footnote{\nThe symbol $E$ stands for equation, the first index reminds to the\nquantity for which a temporary equation will be formed by setting the\ntensor $E$ equal to $0$. The tensors providing the eventually used\nforms of\nthe\nequations are named with $\\N{}$ standing for null quantities.}\n\\begin{subequations}\n\\label{quaSys}\n\\begin{eqnarray}\n  \\label{quaSysOm}\n  \\label{NO}\n  \\N{\\Omega}_a = E\\I{\\Omega}_a & = & \\nabla_a \\Omega - \\Omega_a = 0   \\\\\n  \\label{quaSysDOm}\n  \\label{NDO}\n  \\N{D\\Omega}_{ab} = E\\I{D\\Omega}_{ab} & = &\n    \\nabla_a \\Omega_b + \\frac{1}{2} \\Omega \\mbox{$\\hat{R}$}_{ab}\n    - \\omega g_{ab} - \\frac{1}{2} \\Omega^3 \\mbox{$\\hat{T}$}_{ab} = 0           \\\\\n  \\label{No}\n  \\N{\\omega}_{ab} = E\\I{\\omega}_a & = &\n    \\nabla_a \\omega + \\frac{1}{2} \\mbox{$\\hat{R}$}_{ab} \\Omega^b\n    + \\frac{1}{12} R \\Omega_a + \\frac{1}{24} \\Omega \\nabla_a R\n    - \\frac{1}{2} \\Omega^2 \\mbox{$\\hat{T}$}_{ab} \\Omega^b           \\nonumber  \\\\\n    && \\qquad + \\frac{1}{6} \\Omega^2 \\Omega_a T\n    + \\frac{1}{24} \\Omega^3 \\nabla_a T     =  0                   \\\\\n  \\label{quaSysDe}\n  \\label{Ne}\n  \\N{e}^a{}_{bc} = E\\I{e}^a{}_{bc} & = & T^a{}_{bc}   = 0 \\\\\n  \\label{quaSysDgamma}\n  \\label{Ng}\n  \\N{\\gamma}_{abc}{}^d = E\\I{\\gamma}_{abc}{}^d & = &\n    R\\I{diff}_{abc}{}^d - R\\I{alg}_{abc}{}^d  = 0 \\\\\n  \\label{quaSysR}\n  E\\I{R}_{abc} & = & \\nabla_{[a} \\mbox{$\\hat{R}$}_{b]c}\n    + \\frac{1}{12} (\\nabla_{[a} R) g_{b]c} + \\Omega_d d_{abc}{}^d\n    + \\Omega \\, t_{abc} = 0  \\\\                         \n  \\label{quaSysd}\n  E\\I{d}_{abc} & = &  \\nabla_d d_{abc}{}^d  - t_{abc} = 0\n\\end{eqnarray}\n\\end{subequations}\nwhere (\\ref{quaSysDe}) means vanishing torsion $T^a{}_{bc}$, expressed\nin frame index form,\n\\begin{equation*}\n    T^{\\f{i}}{}_{\\f{j}\\f{k}} =\n    \\left( e_{\\f{j}}(e_{\\f{k}}{}^\\mu) -\n           e_{\\f{k}}(e_{\\f{j}}{}^\\mu) \\right)\n      e^{\\f{i}}{}_\\mu\n  + \\gamma^{\\f{i}}{}_{\\f{j}\\f{k}} - \\gamma^{\\f{i}}{}_{\\f{k}\\f{j}},\n\\end{equation*}\nand (\\ref{quaSysDgamma}) means that the curvature tensor in terms of\nthe Ricci rotation coefficients, in frame index form\n\\begin{eqnarray*}\n    R\\I{diff}_{\\f{i}\\f{j}\\f{k}}{}^{\\f{l}} & = &\n  e_{\\f{j}}(\\gamma^{\\f{l}}{}_{\\f{i}\\f{k}})\n    - e_{\\f{i}}(\\gamma^{\\f{l}}{}_{\\f{j}\\f{k}})\n  - \\gamma^{\\f{l}}{}_{\\f{i}\\f{m}} \\gamma^{\\f{m}}{}_{\\f{j}\\f{k}}\n  + \\gamma^{\\f{l}}{}_{\\f{j}\\f{m}} \\gamma^{\\f{m}}{}_{\\f{i}\\f{k}}\n  \\nonumber  \\\\\n  && \\qquad\n  + \\gamma^{\\f{m}}{}_{\\f{i}\\f{j}} \\gamma^{\\f{l}}{}_{\\f{m}\\f{k}}\n  + \\gamma^{\\f{m}}{}_{\\f{j}\\f{i}} \\gamma^{\\f{l}}{}_{\\f{m}\\f{k}}\n  - \\gamma^{\\f{l}}{}_{\\f{m}\\f{k}} T^{\\f{m}}{}_{\\f{j}\\f{i}},\n\\end{eqnarray*}\nshould equal the combination\n\\begin{equation*}\n  \\Omega \\, d_{abcd}\n  + g_{c[a} \\mbox{$\\hat{R}$}_{b]d} - g_{d[a} \\mbox{$\\hat{R}$}_{b]c}\n  + \\frac{1}{6} g_{c[a} g_{b]d} R =: R\\I{alg}_{abc}{}^d,\n\\end{equation*}\nwhich is the irreducible decomposition of a tensor with the symmetry\nof the Riemann tensor (\\ref{Ralg}). Hence (\\ref{quaSysDe}) and\n(\\ref{quaSysDgamma})\nensure that $R\\I{alg}_{abc}{}^d$ is the curvature tensor corresponding\nto the connection given by the Ricci rotation coefficients which again\nis the torsion free connection coming from the metric (frame).\n\\subsubsection{Complications by the matter terms}\nThe final goal is to use the terms $\\nabla_a \\Omega{}$,  $\\nabla_a\n\\Omega_b$, $\\nabla_a \\omega{}$, $\\left( e_{\\f{j}}(e_{\\f{k}}{}^\\mu) -\ne_{\\f{k}}(e_{\\f{j}}{}^\\mu) \\right)  e^{\\f{i}}{}_\\mu{}$,\\hfill\n${e_{\\f{j}}(\\gamma^{\\f{l}}{}_{\\f{i}\\f{k}})  -\ne_{\\f{i}}(\\gamma^{\\f{l}}{}_{\\f{j}\\f{k}})}$, $\\nabla_{[a} \\mbox{$\\hat{R}$}_{b]c}$, and\n$\\nabla_d d_{abc}{}^d$ in (\\ref{quaSys}) as principal\npart for the geometry variables of the system. I will call these terms\nleft side of the equations, the remaining terms right side. The left side\ndoes not contain the complete principle part of the system yet as the\nenergy momentum\ntensor $T_{ab}$ and its derivatives $\\nabla_{[a} T_{b]c}$ may contain\nderivatives of the matter and geometry variables.\n\\\\\nIn the case of the conformally invariant scalar field the field equation\n(\\ref{model}) remains invariant under the rescaling\n\\begin{equation*}\n  \\phi = \\Omega^{-1} \\, \\mbox{$\\tilde{\\phi}$},\n\\end{equation*}\ni.e.\n\\begin{equation*}\n  {\\vphantom{\\phi}\\Box} \\phi - \\frac{R}{6} \\, \\phi = 0.\n\\end{equation*}\nThe physical energy-momentum tensor $\\mbox{$\\tilde{T}$}_{ab}$ fulfills the assumed properties,\n\\begin{eqnarray*}\n  \\mbox{$\\tilde{T}$}_{ab} & = &  \\mbox{$\\Omega$}^2\n               \\left[\n                  (\\nabla_a \\phi) (\\nabla_b \\phi)\n                  - \\frac{1}{2} \\phi \\nabla_a \\nabla_b \\phi\n                  + \\frac{1}{4} \\phi^2 R_{ab}\n                  - \\frac{1}{4} g_{ab}\n                  \\left( (\\nabla^c \\phi) (\\nabla_c \\phi)\n                         + \\frac{1}{6} \\phi^2 R \\right)\n               \\right]  \\nonumber \\\\\n                   & =: &  \\mbox{$\\Omega$}^2 \\, T_{ab}.\n\\end{eqnarray*}\n\\\\\nThe mentioned complications in (\\ref{quaSys}) by the right sides are\nnow obvious. Firstly $\\nabla_{[a} T_{b]c}$ contains $\\nabla_{[a}\n\\nabla_{b]} \\nabla_c \\phi $ terms which are eliminated with the\nidentity $\\nabla_{[a} \\nabla_{b]} \\nabla_c \\phi{} = \\frac{1}{2}\nR_{abc}{}^d \\nabla_d \\phi $. To get rid of the second and first\norder derivatives of $\\phi $ we use the first order system\n\\begin{subequations}\n\\label{NM}\n\\begin{eqnarray}\n\\label{Np}\n  \\N{\\phi}_a & = & \\nabla_a \\phi - \\phi_a = 0\n\\\\\n\\label{NDp}\n  \\N{D\\phi}_{ab} & = &\n  \\nabla_a \\phi_b - \\hat{\\phi}_{ab} - \\frac{1}{4} \\phi_c{}^c \\, g_{ab} = 0\n\\\\\n\\label{NBp}\n  \\N{\\Box\\phi} & = & \\phi_a{}^a - \\frac{R}{6} \\, \\phi = 0\n\\\\\n\\label{NDDp}\n  \\N{DD\\phi}_{abc} & = &\n  \\nabla _{[a} \\hat{\\phi}_{b]c}\n  + \\frac{1}{6} \\, ( \\phi\\,\\nabla_{[a} R + R \\, \\phi_{[a} ) g_{b]c}\n  - \\frac{1}{2} \\, R\\I{alg}_{abc}{}^d \\phi_d = 0\n\\\\\n\\label{NDBp}\n  \\N{D\\Box\\phi}_a & = &\n  \\nabla _{a} \\phi_b{}^b\n  - \\frac{1}{6} \\, ( \\phi\\, \\nabla_a R + R \\, \\phi_{a} ) = 0.\n\\end{eqnarray}\n\\end{subequations}\nfor the variables $\\phi$, $\\phi_a$, the trace free symmetric tensor\n$\\hat{\\phi}_{ab}$ and the trace $\\phi_a{}^a$. The system is\nderived from $\\nabla_a\n\\left( {\\vphantom{\\phi}\\Box} \\phi - \\frac{R}{6} \\, \\phi \\right) =0$.\nSystem (\\ref{NM}) also serves as matter part of the system for the\nunphysical spacetime.\n$t_{abc}$ is now written in a form which does not contain any\nderivatives of matter variables explicitly.\n\\\\\n$\\nabla_{[a} T_{b]c}$ and thus $t_{abc}$ still contain\nderivatives $\\nabla_{[a} \\mbox{$\\hat{R}$}_{b]c}$ of the trace free Ricci tensor.\nBy combining (\\ref{quaSysR}) and (\\ref{quaSysd}) the\nderivatives of $\\mbox{$\\hat{R}$}_{ab}$ and $d_{abc}{}^d$ can be decoupled.\n(\\ref{quaSysR}) and (\\ref{quaSysd}) become\n\\begin{equation}\n\\label{quaSysRvar}\n  E'\\I{R}_{abc} = \\nabla_{[a} \\mbox{$\\hat{R}$}_{b]c}\n    + \\frac{1}{12} (\\nabla_{[a} R) g_{b]c} - \\Omega_d d_{abc}{}^d\n    + \\Omega m_{abc} = 0\n\\end{equation}\nand\n\\begin{equation}\n\\label{quaSysdvar}\n  E'\\I{d}_{abc} = \\nabla_d d_{abc}{}^d  - m_{abc} = 0,\n\\end{equation}\nwith\n\\begin{eqnarray*}\n  \\lefteqn{ m_{abc} = \\frac{1}{1-\\frac{1}{4} \\, \\Omega^2\\phi^2} \\qquad * }  \\\\\n  \\lefteqn{ \\Big( \\, \\Omega } && \\qquad\n  \\big[ \\frac{3}{2}  \\, \\phi_{[a} \\phi_{b]c}\n    - \\frac{1}{2}  \\, g_{c[a} \\phi_{b]d} \\phi^d\n    + \\frac{1}{4}  \\, \\phi  \\, \\Omega d_{abc}{}^d \\phi_d\n    + \\frac{1}{4}  \\, \\phi  \\, g_{c[a} \\mbox{$\\hat{R}$}_{b]}{}^d \\phi_d\n    - \\frac{3}{4}  \\, \\phi  \\, \\phi_{[a} \\mbox{$\\hat{R}$}_{b]c}        \\\\ && \\qquad \\quad\n    - \\frac{1}{12}  \\, \\phi  \\, \\phi_{[a} g_{b]c} R\n    + \\frac{1}{4}  \\, \\Omega  \\, \\phi^2 d_{abc}{}^d \\Omega_d  \\big]\n\\\\ && \\quad\n  - 3  \\, \\Omega_{[a} \\big[ \\phi_{b]} \\phi_c\n    - \\frac{1}{2}  \\, \\phi  \\, \\phi_{b]c}\n    + \\frac{1}{4}  \\, \\phi^2 \\mbox{$\\hat{R}$}_{b]c}\n    + \\frac{1}{36}  \\, \\phi^2 g_{b]c} R\n    - \\frac{1}{3}  \\, g_{b]c}  \\, \\phi^d \\phi_d \\big]\n\\\\ && \\quad\n  + \\Omega^d g_{c[a} \\big[ \\phi_{b]} \\phi_d\n    - \\frac{1}{2}  \\, \\phi  \\, \\phi_{bd}\n    + \\frac{1}{4}  \\, \\phi^2 \\mbox{$\\hat{R}$}_{b]d}    \\big]     \\quad     \\Big).\n\\end{eqnarray*}\nNote that $m_{abc}$ may become singular for\n$1-\\frac{1}{4}\\,\\Omega^2\\phi^2= 1-\\frac{1}{4}\\,\\mbox{$\\tilde{\\phi}$}^2=0$. In the\nEinstein equations for the physical spacetime (\\ref{EinstPhys})\n$\\mbox{$\\tilde{R}$}_{ab}$ carries a factor $1-\\frac{1}{4} \\mbox{$\\tilde{\\phi}$}^2$ too.\nWe will need later that\n\\begin{equation*}\n  \\N{m}_{abc} := t_{abc} - m_{abc} =\n  - \\frac{1}{4}  \\, \\Omega  \\, \\phi^2\n    \\left( \\N{R}_{abc} + \\frac{2}{3} \\, \\Omega \\, m_{[a|d|}{}^d \\,\n    g_{b]c}\n\\right),\n\\end{equation*}\nwhere $\\N{R}_{abc}$ is the null quantity representing the final form\nof the equation for $\\mbox{$\\hat{R}$}_{ab}$ (\\ref{NR}).\nThe final form of  the equation for $d_{abc}{}^d$ is\nobtained from (\\ref{quaSysdvar}) by replacing $E'\\I{d}_{abc}=0$ with\n\\begin{equation}\n\\label{Nd}\n  \\N{d}_{abc} := E'\\I{d}_{abc} + \\frac{2}{3} m_{[a|d|}{}^d g_{b]c} = 0.\n\\end{equation}\nThis gives $\\N{d}_{abc}$ the same index symmetry properties as the Weyl tensor.\nThat replacement does not change the equation\nsince $m_{ab}{}^b=0$ as will be seen later.\n\\\\\nAnalogously we replace (\\ref{quaSysRvar}) with\n\\begin{equation}\n\\label{NR}\n  \\N{R}_{abc} :=\n    E'\\I{R}_{abc} - \\frac{2}{3} \\, \\Omega \\, m_{[a|d|}{}^d g_{b]c} = 0,\n\\end{equation}\nthe contraction $\\N{R}_{ab}{}^b=0$ is then the contracted second\nBianchi identity.\n\\section{Evolution equations and constraints}\nIn the following I will assume a system $\\N{}=0$ of the form (\\ref{quaSysOm})\n-- (\\ref{quaSysDgamma}),  (\\ref{Nd}), (\\ref{NR}),  the geometry part,\nand a matter part, in the case of the scalar field model system\n(\\ref{NM}). $m_{abc}$ and $t_{abc}$ are assumed to differ only by terms\nexpressible as null quantities. The energy-momentum tensor\n$T_{ab}$, its derivatives $\\nabla_a T_{bc}$, $m_{abc}$, and $t_{abc}$\nare assumed to be expressed in variables and thus do not contain any\nexplicit derivative of variables. By these assumptions the\nprincipal part of the system has block form, the geometry block and\nthe matter block. The two blocks are coupled through the right sides.\n\\\\\nIn this chapter the system $\\N{}=0$ will be reduced to a system of\nsymmetric hyperbolic time evolution equations, the subsidiary system.\nSufficient conditions for the equivalence of the subsidiary system and\n$\\N{}=0$ are given as conditions on $m_{abc}$. If the system can be put\ninto the described block form there do not arise any more conditions\nfrom the geometry part of the system for any matter.\n\\\\\nThe explicit carry out is technical and lengthy, the idea can be\nsummarized as follows: All the equations of the system are\nregarded as null quantities. By requiring the vanishing of some of these null\nquantities and by choosing an appropriate gauge condition for the\ncoordinates and the frame we get a symmetric hyperbolic\nsubsidiary system of evolution equations.\nWhich null quantities to choose can best be seen by a decomposition into\nthe irreducible parts in the spinor calculus as performed\nin~\\cite{Fr91ot}.\n\\\\\nThe solution of this symmetric hyperbolic subsystem exists and is\nunique. To complete the proof we must show\nthat the solution obtained in this way is consistent with the rest of\nthe equations, i.e.\\ that all null quantities remain zero if they are\ninitially zero (``propagation of the constraints''). For that purpose a\nsymmetric hyperbolic system of time evolution equations for\nthe remaining null quantities is derived. Sufficient conditions for the\npropagation of the constraints are firstly the homogeneity of the evolution\nequations for the remaining null quantities in the null quantities\nsince then the unique solution of these evolution equations is the vanishing\nof all null quantities for all times if they vanish on the initial\nsurface and secondly that the domain of dependence of $S$ with respect\nto the equations for the propagation of the constraints is a\nsuperset of the domain of dependence of $S$ with respect to the\nsubsidiary system.\n\\subsection{A symmetric hyperbolic subsystem of evolution\n  equations}\n\\label{GaussGauge}\nIntroducing a timelike vector field $t^a$ not necessarily hypersurface\northogonal and its orthogonal projection tensor\n$h_{ab}:=g_{ab}-t_at_b\/(t_ct^c)$ allows to split the\nsystem of equations into two categories, the equations containing time\nderivatives and those containing no time derivatives (the\nconstraints). The equations with time derivatives provide a\nunder\/overdetermined system of evolution equations.\n\\\\\nThe system is\noverdetermined since for some quantities there are too many time\nevolution equations, e.g.\\ there are 12 time evolution equations from\n$\\N{R}_{abc}=0$\nfor 9 independent tensor components. 3 equations are a linear\ncombination of the other 9 equations and the constraints. An\nirreducible decomposition of the tensors $\\N{}$ is a systematic way to\nanalyze these dependencies. Since all the types of tensor index\nsymmetries appearing in the system have been thoroughly investigated\nin~\\cite{Fr91ot} I will only state which combinations are needed.\n\\\\\nThe system is underdetermined since there are 10 time evolution\nequations for the frame and the Ricci rotation coefficients missing.\nBy adding\n\\begin{equation}\n\\label{KoEichFrame}\n   e^{\\f{i}}{}_b \\, g^{\\f{j}\\f{k}} \\, e_{\\f{j}}{}^a \\left( \\nabla_a\n   e_{\\f{k}}{}^b \\right) = - F_{\\f{i}} =\n   \\gamma_{\\f{i}\\f{k}}{}^{\\f{k}}\n\\end{equation}\nand\n\\begin{equation}\n\\label{FrEichFrame}\n    \\partial_{\\f{k}} \\gamma^{\\f{i}\\f{k}\\f{j}}\n    + \\gamma^{\\f{i}\\f{k}\\f{j}} F_{\\f{k}}\n    + \\gamma^{\\f{l}\\f{k}\\f{i}} \\, \\gamma_{\\f{l}\\f{k}}{}^{\\f{j}}\n    - \\gamma^{\\f{l}\\f{k}\\f{j}} \\, \\gamma_{\\f{l}\\f{k}}{}^{\\f{i}}\n    = F^{\\f{i}\\f{j}},\n\\end{equation}\nthe system becomes complete. The freedom of giving ten functions\ncorresponds to the freedom of giving the lapse and the shift to\ndetermine the coordinates and the six parameters of the Lorentz group\nto determine the frame on every point. The gauge freedom is discussed\nin full detail in~\\cite{Fr85ot,Fr91ot}.\n\\\\\nA choice which makes the system especially simple for analytic\nconsiderations is a Gaussian coordinate and frame system defined as\nfollows. Give on the spacelike initial value surface $S$ coordinates\n$x^\\mu, \\mu = 1..3$, and 3 orthonormal vector fields $e_{\\f{i}}{}^a,\n\\f{i}=\\f{1}..\\f{3}$ in this hypersurface. The affine parameter of the\ngeodesics of the hypersurface orthonormal, timelike vector field\n$e_{\\f{0}}{}^a$ defines the\ntime coordinate $x^0=t$. The spacelike coordinates are transported by\nthese geodesics into a neighbourhood of the initial surface. By\ngeodesic transport of $e_{\\f{i}}{}^a, \\f{i}=\\f{1}\\ldots\\f{3}$, and\n$e_{\\f{0}}{}^a$  a frame is obtained in this neighbourhood. By\nconstruction we have\n$$\n  e_{\\f{0}}{}^0=1, \\quad e_{\\f{0}}{}^\\mu = 0 \\mbox{ for }\\mu = 1..3\n$$\nand\n$$\n  \\gamma^{\\f{i}}{}_{\\f{0}\\f{k}} = 0.\n$$\nIt is well known that Gaussian coordinates develop caustics if the\nenergy momentum tensor fulfills certain energy conditions, see\ne.g.~\\cite[lemma 9.2.1]{Wa84GR}. In the\nunphysical spacetime the $\\Omega{}$ terms provide a kind of unphysical\nenergy-momentum tensor.\nWhether this energy-momentum tensor fulfills the energy-momentum\nconditions is a difficult question and not known to the author.\nNevertheless the coordinates develop caustics as has been shown by\nnumerical calculations \\cite{Hu93nu}.\n\\\\\nThe following combinations give a symmetric hyperbolic\nsystem for the remaining variables as can be deduced from the\nconsiderations in~\\cite{Fr83cp}:\n\\begin{subequations}\n\\label{EvoSyst}\n\\begin{eqnarray}\n&& \\N{\\Omega}_{\\f{0}} = 0 , \\\\\n&& \\N{D\\Omega}_{\\f{0}b} = 0 , \\\\\n&& \\N{\\omega}_{\\f{0}} = 0 , \\\\\n&& \\N{e}^a{}_{b\\f{0}} = 0 , \\\\\n&& \\N{\\gamma}_{\\f{0}\\f{1}c}{}^d = 0 , \\\\\n&& g^{ab} \\N{R}_{\\f{i}ab} = 0 , \\qquad \\f{i} = \\f{1},\\f{2},\\f{3}, \\\\\n&& \\N{R}_{\\f{0}\\f{i}\\f{i}} = 0 , \\qquad \\f{i} = \\f{1},\\f{2},\\f{3}, \\\\\n&& \\N{R}_{\\f{0}\\f{i}\\f{j}} + \\N{R}_{\\f{0}\\f{j}\\f{i}} = 0 ,\n   \\qquad (\\f{i},\\f{j}) = (\\f{1},\\f{2}),(\\f{1},\\f{3}),(\\f{2},\\f{3}), \\\\\n&& \\N{d}_{\\f{2}\\f{1}\\f{2}} - \\N{d}_{\\f{3}\\f{1}\\f{3}} +\n   \\N{d}_{\\f{2}\\f{0}\\f{2}} - \\N{d}_{\\f{3}\\f{0}\\f{3}} = 0 , \\\\\n&& - \\N{d}_{\\f{1}\\f{0}\\f{2}} + \\N{d}_{\\f{1}\\f{2}\\f{1}} = 0 , \\\\\n&& \\N{d}_{\\f{1}\\f{0}\\f{1}} = 0 , \\\\\n&& \\N{d}_{\\f{1}\\f{0}\\f{2}} + \\N{d}_{\\f{1}\\f{2}\\f{1}} = 0 , \\\\\n&& - \\N{d}_{\\f{2}\\f{1}\\f{2}} +\n   \\N{d}_{\\f{3}\\f{1}\\f{3}} + \\N{d}_{\\f{2}\\f{0}\\f{2}} -\n   \\N{d}_{\\f{3}\\f{0}\\f{3}} = 0 , \\\\\n&& \\N{d}_{\\f{2}\\f{1}\\f{3}} + \\N{d}_{\\f{3}\\f{1}\\f{2}} + \\N{d}_{\\f{2}\\f{0}\\f{3}}+\n   \\N{d}_{\\f{3}\\f{0}\\f{2}} = 0 , \\\\\n&& - \\N{d}_{\\f{1}\\f{0}\\f{3}} + \\N{d}_{\\f{1}\\f{3}\\f{1}} = 0 , \\\\\n&& - \\N{d}_{\\f{1}\\f{2}\\f{3}} = 0 , \\\\\n&& - \\N{d}_{\\f{1}\\f{0}\\f{3}} - \\N{d}_{\\f{1}\\f{3}\\f{1}} = 0 , \\\\\n&& \\N{d}_{\\f{2}\\f{1}\\f{3}} + \\N{d}_{\\f{3}\\f{1}\\f{2}} -\n   \\N{d}_{\\f{2}\\f{0}\\f{3}} - \\N{d}_{\\f{3}\\f{0}\\f{2}} = 0 , \\\\\n&& \\N{\\phi}_{\\f{0}} = 0 , \\\\\n&& \\N{D\\phi}_{\\f{0}b} = 0 , \\\\\n&& g^{ab} \\N{DD\\phi}_{\\f{i}ab} = 0 , \\qquad \\f{i} = \\f{1},\\f{2},\\f{3},\n\\\\\n&& \\N{DD\\phi}_{\\f{0}\\f{i}\\f{i}} = 0 , \\qquad \\f{i} = \\f{1},\\f{2},\\f{3},\n\\\\\n&& \\N{DD\\phi}_{\\f{0}\\f{i}\\f{j}} + \\N{DD\\phi}_{\\f{0}\\f{j}\\f{i}} = 0\n, \\qquad (\\f{i},\\f{j}) = (\\f{1},\\f{2}),(\\f{1},\\f{3}),(\\f{2},\\f{3}), \\\\\n&& \\N{D\\Box\\phi}_{\\f{0}} = 0.  \\yesnumber\n\\end{eqnarray}\n\\end{subequations}\nTo see that the system is really symmetric hyperbolic one has to write\ndown the system explicitly. By an appropriate, in the explicit\nform of the system obvious definition of new variables, the system has\nthe structure\n\\begin{equation}\n\\label{symhypEvoSys}\n  \\underline{\\underline{A_t}}\\,\\partial_t \\underline{f} +\n  \\sum_{i=1}^3 \\underline{\\underline{A_{x^i}}}\\,\\partial_{x^i}\n  \\underline{f} +\n  \\underline{b}(\\underline{f},x^\\mu) = 0,\n\\end{equation}\nwith a diagonal matrix\n$\\underline{\\underline{A_t}}$, which is positive definite for\n$1-\\frac{1}{4}\\Omega^2\\phi^2>0$, and symmetric matrices\n$\\underline{\\underline{A_{x^i}}}$. $\\underline{f}$ is the vector build\nfrom the variables.\nAll the remaining equations are linear combinations of~(\\ref{EvoSyst})\nand constraints.  Since the explicit form of the constraints is not\nneeded I do not list them.\n\\\\\nAs the entries in $\\underline{\\underline{A_t}}$ coming from\n$\\N{R}=0$ and $\\N{d}=0$ vanish for $1-\\frac{1}{4}\\Omega^2\\phi^2=0$ the\nfollowing results apply only if $\\Omega^2\\phi^2<4$ everywhere on the\ninitial value surface $S$ and thus in a neighbourhood of $S$. The\nphysical Einstein equations have a corresponding singularity for\n$1-\\frac{1}{4}\\mbox{$\\tilde{\\phi}$}^2=0$ (see equation \\ref{EinstPhys}).\n\\subsection{A sufficient condition for the propagation of the\n  constraints}\nAccording to the analysis in~\\cite{Fr91ot}, involving the left hand\nside of the following identities, a symmetric\nhyperbolic system of evolution equations for the remaining null quantities\ncan be extracted from:\n\\begin{subequations}\n\\label{PropConstrSys}\n\\begin{equation}\n  \\nabla_{[a}\\N{\\Omega}_{b]} =\n  - \\frac{1}{2} T^c{}_{ab} \\nabla_c \\Omega - \\N{D\\Omega}_{[ab]}\n\\end{equation}\n\n\\begin{eqnarray}\n  \\lefteqn{ \\nabla_{[a}\\N{D\\Omega}_{b]c} = }   \\nonumber   \\\\ && \\qquad\n  \\frac{1}{2} \\N{\\gamma}_{abc}{}^d \\Omega_d\n  - \\frac{1}{2} T^d{}_{ab} \\nabla_d \\Omega_c\n  + \\frac{1}{2} \\Omega \\N{R}_{abc}\n  + \\frac{1}{2} \\mbox{$\\hat{R}$}_{c[b} \\N{\\Omega}_{a]}\n  - \\frac{3}{2} \\Omega^2 \\N{\\Omega}_{[a} \\mbox{$\\hat{T}$}_{b]c} \\nonumber \\\\ && \\qquad\n  - \\frac{1}{2} \\N{\\omega}_{[a} g_{b]c}\n  + \\frac{1}{3} \\Omega^2 m_{[a|d|}{}^d g_{b]c}\n  + \\frac{1}{3} \\Omega^2 \\N{m}_{[a|d|}{}^d g_{b]c}\n\\end{eqnarray}\n\n\\begin{eqnarray}\n  \\lefteqn{ \\nabla_{[a}\\N{\\omega}_{b]} = }  \\nonumber          \\\\ && \\qquad\n  - \\frac{1}{2} T^c{}_{ab} \\nabla_c \\omega\n  - \\frac{1}{24} \\Omega^3 T^c{}_{ab} \\nabla_c T\n  + \\frac{1}{24} \\N{\\Omega}_{[a} \\nabla_{b]} R\n  + \\frac{1}{8} \\Omega^2 \\N{\\Omega}_{[a} \\nabla_{b]} T  \\nonumber \\\\ && \\qquad\n  + \\frac{1}{2} ( \\mbox{$\\hat{R}$} _{c[b} - \\Omega^2 \\mbox{$\\hat{T}$}_{c[b} ) \\N{D\\Omega}_{a]}{}^c\n  + ( \\frac{1}{24} R + \\frac{1}{6} \\Omega^2 T ) \\N{D\\Omega}_{[ab]}\n  + \\frac{1}{2} \\Omega^c \\N{R}_{abc}             \\nonumber        \\\\ && \\qquad\n  + \\frac{1}{3} \\Omega \\, \\Omega_{[b} m_{a]c}{}^c\n  + \\frac{1}{2} \\Omega \\, \\Omega^c \\N{m}_{abc}\n\\end{eqnarray}\n\n\\begin{eqnarray}\n\\label{PropNR}\n  \\lefteqn{ \\nabla_{[a} \\N{R}_{bc]d} = }        \\nonumber  \\\\ && \\qquad\n  \\frac{1}{2} \\N{\\gamma}_{[abc]}{}^f \\mbox{$\\hat{R}$}_{fd}\n  + \\frac{1}{2} \\N{\\gamma}_{[ab|d]}{}^f \\mbox{$\\hat{R}$}_{c]f}\n  - \\frac{1}{2} T^f{}_{[ab} \\nabla_{|f|}\\mbox{$\\hat{R}$}_{c]d}\n  - \\frac{1}{24} T^f{}_{[ab} (\\nabla_{|f|} R ) g_{c]d} \\nonumber \\\\ && \\qquad\n  + \\N{D\\Omega}_{[a|f|} d_{bc]d}{}^f\n  + \\Omega^f ( \\N{d}_{[ca|d|} g_{b]f} - \\N{d}_{[ca|f|} g_{b]d} )\n  + \\N{\\Omega}_{[a} m_{bc]d}           \\nonumber                 \\\\ && \\qquad\n  - \\frac{2}{3} \\N{\\Omega}_{[a} m_{b|f|}{}^f g_{c]d}\n  - \\frac{1}{2} \\Omega^2 \\N{\\gamma}_{[abc]}{}^f \\mbox{$\\hat{T}$}_{fd}\n  - \\frac{1}{2} \\Omega^2 \\N{\\gamma}_{[ab|d|}{}^f \\mbox{$\\hat{T}$}_{c]f}\n  - \\frac{1}{2} \\Omega^2 T^f{}_{[ab} \\nabla_{|f|} \\mbox{$\\hat{T}$}_{c]d} \\nonumber\n  \\\\ &&\n\\qquad\n  - 3 \\Omega \\N{D\\Omega}_{[ab} \\mbox{$\\hat{T}$}_{c]d}\n  + \\Omega \\N{D\\Omega}_{[a}{}^f \\mbox{$\\hat{T}$}_{c|f|} g_{b]d}\n  + \\frac{1}{3} \\Omega \\N{D\\Omega}_{[ab} g_{c]d}\n  + \\frac{1}{12} \\Omega \\N{\\Omega}_{[a} ( \\nabla_b T ) g_{c]d}\n    \\nonumber  \\\\ && \\qquad\n  + \\frac{1}{12} \\Omega^2 T^f{}_{[ab} ( \\nabla_{|f|} T ) g_{c]d}\n  - 2 \\Omega_{[a} \\N{m}_{bc]d}\n  + \\Omega_f \\N{m}_{[ca}{}^f g_{b]d}      \\nonumber              \\\\ && \\qquad\n  - \\Omega \\nabla_{[a} \\N{m}_{bc]d}\n  + 2 \\Omega_{[a} m_{c|f|}{}^f g_{b]d}\n  - \\frac{2}{3} \\Omega ( \\nabla_{[a} m_{b|f|}{}^f ) g_{c]d}\n\\end{eqnarray}\n\n\\begin{eqnarray}\n  \\lefteqn{ \\nabla^{c} \\N{d}_{abc} = }  \\nonumber     \\\\ && \\qquad\n  \\frac{1}{2} \\N{\\gamma}^c{}_{da}{}^f d_{fbc}{}^d\n  + \\frac{1}{2} \\N{\\gamma}^c{}_{db}{}^f d_{afc}{}^d\n  + \\frac{1}{2} \\N{\\gamma}^c{}_{dc}{}^f d_{abf}{}^d\n  + \\frac{1}{2} \\N{\\gamma}^{cd}{}_{df} d_{abc}{}^f  \\nonumber   \\\\ && \\qquad\n  - \\frac{1}{2} T^{fcd} \\nabla_f d_{abcd}\n  - \\nabla^c m_{abc}\n  + \\frac{2}{3} \\nabla_{[b} m_{a]c}{}^c\n\\end{eqnarray}\n\n\\begin{equation}\n  \\nabla_{[a} \\N{e}^d{}_{bc]} =\n  \\N{\\gamma}_{[abc]}{}^d + \\N{e}^f{}_{[ab} \\N{e}^d{}_{c]f}\n\\end{equation}\n\n\\begin{eqnarray}\n\\label{dNgamma}\n  \\lefteqn{ \\nabla_{[f} \\N{\\gamma}_{ab]cd} = } \\nonumber \\\\ && \\qquad\n  - T^g{}_{[fa} R\\I{diff}_{b]gcd}\n  - \\N{\\Omega}_{[f} d_{ab]cd}\n  - \\Omega ( \\N{d}_{[bf|c|} g_{a]d} - \\N{d}_{[bf|d|} g_{a]c} )\n  \\nonumber \\\\ && \\qquad\n  - \\N{R}_{[fb|d|} g_{a]c}\n  + \\N{R}_{[fb|c|} g_{a]d}\n\\end{eqnarray}\n\n\\begin{equation}\n  \\nabla_{[a}\\N{\\phi}_{b]} =\n  - \\frac{1}{2} T^c{}_{ab} \\nabla_c \\phi - \\N{D\\phi}_{[ab]}\n\\end{equation}\n\n\\begin{equation}\n  \\nabla_{[a}\\N{D\\phi}_{b]c} =\n  \\frac{1}{2} \\N{\\gamma}_{abc}{}^d \\phi_d\n  - \\frac{1}{2} T^d{}_{ab} \\nabla_d \\phi_c\n  - \\N{DD\\phi}_{abc}\n  - \\frac{1}{4} \\N{D\\Box\\phi}_{[a} g_{b]c}\n\\end{equation}\n\n\\begin{equation}\n  \\nabla_a \\N{\\Box\\phi} =\n  \\N{D\\Box\\phi}_a\n  - \\frac{1}{6} R \\N{\\phi}_a\n\\end{equation}\n\n\\begin{eqnarray}\n\\label{PropNDDp}\n  \\lefteqn{ \\nabla_{[a} \\N{DD\\phi}_{bc]d} = }   \\nonumber  \\\\ && \\qquad\n  \\frac{1}{2} \\N{\\gamma}_{abc}{}^f \\hat{\\phi}_{fd}\n  + \\frac{1}{2} \\N{\\gamma}_{[ab|d|}{}^f \\hat{\\phi}_{c]f}\n  - \\frac{1}{2} T^f{}_{[ab} \\nabla_{|f|} \\hat{\\phi}_{c]d}\n  + \\frac{1}{6} \\N{\\phi}_{[a} ( \\nabla_b R ) g_{c]d}   \\nonumber \\\\ && \\qquad\n  - \\frac{1}{12} \\phi T^f{}_{[ab} g_{c]d} \\nabla_f R\n  + \\frac{1}{6} R \\N{D\\phi}_{[ab} g_{c]d}\n  + \\frac{1}{2} T^g{}_{[ab} R\\I{diff}_{c]gd}{}^f \\phi_f\n  - \\frac{1}{2} R_{[bc|d|}{}^f \\N{D\\phi}_{a]f}        \\nonumber \\\\ && \\qquad\n  + \\frac{1}{2} ( \\nabla_{[a} \\N{\\gamma}_{bc]d}{}^f ) \\phi_f,\n\\end{eqnarray}\nand\n\\begin{equation}\n  \\nabla_{[a} \\N{D\\Box\\phi}_{b]} =\n  - \\frac{1}{2} T^d{}_{ab} \\nabla_d \\phi_c{}^c\n  - \\frac{1}{6} \\N{\\phi}_{[a} \\nabla_{b]} R\n  - \\frac{1}{12} \\phi T^c{}_{ab} \\nabla_c R\n  - \\frac{1}{6} R \\N{D\\phi}_{[ab]}.\n\\end{equation}\n\\end{subequations}\nThe last term in (\\ref{PropNDDp}) is homogeneous in null quantities as\ncan be seen from (\\ref{dNgamma}),\nThe deviation of these equalities is even more lengthy than the\nequalities itself, but the essential ideas behind it can already be seen\nin the deviation of the first:\n\\begin{eqnarray*}\n  \\nabla_{[a}\\N{\\Omega}_{b]} & = &\n    \\nabla_{[a}  \\nabla_{b]}\\Omega - \\nabla_{[a}  \\Omega_{b]} \\\\\n  & = &\n    - \\frac{1}{2} T^c{}_{ab} \\nabla_c \\Omega - \\N{D\\Omega}_{[ab]}\n    + \\frac{1}{2} \\mbox{$\\hat{R}$}_{[ab]} \\Omega\n    - \\frac{1}{2} \\Omega^3 \\mbox{$\\hat{T}$}_{[ab]}\n    - \\omega g_{[ab]},\n\\end{eqnarray*}\nwith the last three terms vanishing since the tensors are symmetric. Note\nthat vanishing of the torsion, $T^c{}_{ab}=0$,  and\n$2\\,\\nabla_{[a}\\nabla_{b]}\\omega_c=R_{abc}{}^d \\omega_d$ cannot be\nused since they only\nhold if both the time evolution and the constraint equations for the frame\n$e_{\\f{i}}{}^\\mu $ and the Ricci rotation coefficients\n$\\gamma^a{}_{\\f{i}\\f{j}}$ hold everywhere.\n\\\\\nA sufficient set of conditions for homogeneity of the system derived\nin the null quantities is\n\\begin{subequations}\n\\label{IntegrBed}\n\\begin{equation}\n\\label{IntegrBed1}\n  m_{ab}{}^b = 0 \\bmod \\N{},\n\\end{equation}\n\\begin{equation}\n\\label{IntegrBed2}\n  \\nabla^c m_{abc} + \\frac{2}{3} \\nabla_{[a} m_{b]c}{}^c = 0 \\bmod \\N{},\n\\end{equation}\n\\begin{equation}\n\\label{IntegrBed3}\n  \\nabla_{[a} m_{b]c}{}^c = 0 \\bmod \\N{},\n\\end{equation}\nand\n\\begin{equation}\n\\label{IntegrBed4}\n  \\Omega \\nabla_{[a} \\N{m}_{bc]d} =\n     f \\, \\nabla_{[a} \\N{R}_{bc]d} \\bmod \\N{}, \\quad f \\ne -1.\n\\end{equation}\n\\end{subequations}\nA straightforward but long calculation shows that these conditions are\nfulfilled by the conformally invariant scalar field with\n$f=-\\frac{1}{4}\\Omega^2\\phi^2$. Equation  (\\ref{PropNR})\nbecomes singular for $1-\\frac{1}{4}\\Omega^2\\phi^2=0$.\n\\\\\nThe very technical integrability conditions (\\ref{IntegrBed}) have a\nvery simply interpretation. Replacing $m_{abc}$ with $t_{abc}$ --- they\nonly differ by null quantities --- the conditions\n(\\ref{IntegrBed1}--\\ref{IntegrBed3}) reduce to $\\mbox{$\\tilde{\\nabla}$}^b \\mbox{$\\tilde{T}$}_{ab} = 0$\nand ${\\mbox{$\\tilde{\\nabla}$}_b \\mbox{$\\tilde{\\nabla}$}^c \\mbox{$\\tilde{T}$}_{ac} = 0}$. Condition (\\ref{IntegrBed4}) is only\nof technical nature, it gives the principal part a\nsimple block form.\n\\\\\n{}From the considerations in \\cite{Fr91ot} also follows that the domain\nof dependence of $S$ with respect to the evolution equation of the\nconstraints includes the domain of dependence of $S$ with respect to\nthe subsidiary system.\n\\section{The hyperboloidal initial value problem}\n\\label{HypInitValProblSec}\nSo far a system of equations $(\\N{}=0)$ has been derived which contains\nfor at least one choice of gauge a symmetric hyperbolic subsystem of\nevolution equations. The remaining equations in the system --- either\nconstraints or a combination of constraints and time evolution\nequations --- will be satisfied for a solution of the evolution\nequations, if the constraints are satisfied by the initial data. If\nboth, the time evolution and the constraints, are fulfilled, $(\\tilde\nM,\\mbox{$\\tilde{g}$}_{ab},\\mbox{$\\tilde{\\phi}$})$ is a weakly asymptotically flat solution of the Einstein\nequation. This follows from the way the system $\\N{}=0$ for the\nunphysical spacetime has been derived.\n\\\\\nThe essential points in the proofs of the theorems in\n\\cite[chapter~10]{Fr91ot} are the symmetric hyperbolicity of the\nsubsidiary system and the form (\\ref{NDO}) of the equations for\n$\\Omega{}$. Therefore the same techniques can be used and the proofs\nwill not be repeated. The difference to\nthe model treated here lies in the derivation of the subsidiary system\nand the proof of the propagation of the constraints,\nwhich has been done in the previous chapters.\n\\subsection{The initial value problem}\nWe consider the following initial value problem:\n\\begin{Def}\n\\label{HypInitValProbl}\nA ``{\\bf hyperboloidal initial data set for the conformally invariant\nscalar field}'' consists of a pair $(\\bar S,f_0)$ such that:\n\\begin{enumerate}\n\\item $\\bar S = S \\cap \\partial S$ is a smooth manifold with boundary\n  $\\partial S$ diffeomorphic to the closed unit ball in $\\Bbb{R}^3$.\n  As coordinates on $S$ the pull backs of the natural coordinates\n  on $\\Bbb{R}^3$ are used.\n\\item $f_0$ is the vector $\\underline f$ of functions in system\n  (\\ref{EvoSyst}) written in the form (\\ref{symhypEvoSys}) at initial\n  time $t_0$.\n\\item The fields provided by $f_0$ have uniformly continuous derivatives\n  with respect to the coordinates of $S$ to all orders\\footnote{The\n    assumption about the smoothness of the data can certainly be\n    weakened from $C^\\infty{}$ to $C^n$ for sufficiently large $n$ but\n    then more technical effort would be needed in the proofs.}.\n\\item On $S$: $\\Omega>0$. On $\\partial S$: $\\Omega=0$ and $\\nabla_a\n  \\Omega{}$ is a future directed null vector.\n\\item The fields provided by $f_0$ satisfy the constraints following\n  from $\\N{}=0$ (\\ref{quaSys} and \\ref{NM}) and the gauge conditions.\n\\end{enumerate}\n\\end{Def}\nA point which deserves special notice is the existence of a\nhyperboloidal initial data set. The proof that\nthose data exist has to overcome two problems.\n\\\\\nFirstly,  the regularity of the solution on $\\partial S$ which is\nthe consistency of the data with asymptotical flatness.\nFor scalar field data with compact support regularity\nconditions are given in~\\cite{AnCXXxx,AnCA92ot}, which are sufficient\nfor the existence of a solution of the constraints near $\\partial S$.\n\\\\\nSecondly there is a problem with a possible singularity of\nequations in $\\N{}=0$ at {$1-\\frac{1}{4}\\,\\Omega^2\\phi^2=0$}.\n\\\\\nL.~Anderson and\nP.~Chrusc\\'{\\i}el are preparing a paper analyzing both\nproblems \\cite{AnCXXxx}.\n\\subsection{Theorems}\nThe ``theorems'' will be given in a form not containing every\ntechnical detail, since these technical details would make them\nlengthy and\ncan\nbe easily deduced from the theorems in~\\cite{Fr91ot} by replacing the\nYang-Mills matter with the (conformally invariant) scalar field.\n\\\\\nSince the constraints of $\\N{}=0$ propagate we have:\n\\begin{Theorem}\n\\label{physUnphysequi}\n  Any (sufficiently smooth) solution of the subsidiary system satisfying the\n  constraints on a spacelike hypersurface $\\bar S$ and\n  $1-\\frac{1}{4}\\,\\Omega^2\\phi^2>0$  defines in the\n  domain of dependence with respect to $g_{ab}$ of $\\bar S$ a\n  solution to the unphysical system. Thus $(\\tilde M,\\mbox{$\\tilde{g}$}_{ab},\\mbox{$\\tilde{\\phi}$})$ is\n  a weakly asymptotically flat solution of the Einstein equation.\n\\end{Theorem}\nSince the evolution equations are symmetric hyperbolic a unique\nsolution of the initial value problem exists for a finite time.\n{}From the combination with theorem (\\ref{physUnphysequi}) follows:\n\\begin{corollar}\n\\label{ExUni}\n  For every regular solution of the constraints\n  on $\\bar S$ with ${1-\\frac{1}{4}\\,\\Omega^2\\phi^2\\mid_{\\bar S} >0}$ exists\n  locally a unique, weakly asymptotically\n  flat solution of the Einstein equation.\n\\end{corollar}\nFor the Minkowski space we can extent $\\bar S$ and the solution of\nthe constraints beyond $\\partial S$ to $S'$ and\nget a solution in the unphysical spacetime which extents beyond\n$i^+$. The continuous dependence of the solution of symmetric\nhyperbolic systems on the data and the form of (\\ref{NO}),\n(\\ref{NDO}) and (\\ref{No}) (see the proof\nof theorem (10.2) in~\\cite{Fr91ot}) guarantees\nthat there is a solution covering the whole domain of dependence of\n$\\bar S$. Furthermore the proof there shows that\n$\\{p\\,|\\,\\Omega(p)=0\\}$ has an isolated critical point $i^+$, where\nall future directed timelike geodesics of $(\\tilde M,\\mbox{$\\tilde{g}$}_{ab})$ end, thus:\n\\begin{Theorem}\n  For a sufficient small deviation of the data from Minkowskian data the\n  solution of theorem~\\ref{ExUni} possesses a regular future null\n  infinity and a regular future timelike infinity.\n\\end{Theorem}\n\\section{The conformal equivalence of the scalar fields}\n\\label{SkalarAequiv}\nThis section shortly reviews the equivalence transformation between\nspacetime models with scalar matter under the viewpoint of solving\nhyperboloidal initial value problems. Other\naspects of this equivalence transformation, especially the generation\nof exact solutions, have been studied\nin~\\cite{AcWA93ce,Be74es,Be75bh,Kl93sf,KlK93ie,Pa91mw,XaD92eg}.\n\\subsection{Local equivalence of solutions}\n\\label{GenEquivalence}\nSpacetime models $(\\tilde M,\\tilde g_{ab},\\tilde\\phi)$\nwith scalar matter $\\tilde\\phi$ described by the action\n\\begin{equation}\n\\label{ScalarAction}\n  \\tilde S = \\int_{\\tilde M} \\left[ A(\\tilde\\phi) R - B(\\tilde\\phi)\n                    (\\tilde\\nabla_a\\tilde\\phi)\n                    (\\tilde\\nabla^a\\tilde\\phi)\n             \\right] \\> (-\\tilde g)^{\\frac{1}{2}} \\> d^4\\tilde x\n\\end{equation}\nwill be considered. Boundary terms in the action have been omitted,\n$\\mbox{$\\tilde{g}$}{}$ is the determinant of $\\mbox{$\\tilde{g}$}_{\\mu\\nu}$.\n\\\\\nBy varying the action $\\tilde S$ with respect to $\\mbox{$\\tilde{\\phi}$}{}$ and\n$\\mbox{$\\tilde{g}$}_{ab}$ the following field equations result:\n\\begin{subequations}\n\\label{allgSys}\n\\begin{eqnarray}\n\\label{allgWell}\n  B(\\mbox{$\\tilde{\\phi}$}) \\, \\tilde{\\vphantom{\\phi}\\Box} \\,\\tilde\\phi\n  + \\frac{1}{2} \\, \\frac{dB}{d\\tilde\\phi} \\,\n    \\left( \\mbox{$\\tilde{\\nabla}$}^a\\tilde\\phi \\right) \\, \\left( \\mbox{$\\tilde{\\nabla}$}_a\\tilde\\phi \\right)\n  + \\frac{1}{2} \\, \\frac{dA}{d\\tilde\\phi} \\, \\tilde R\n  \\quad & =  & \\quad 0 \\qquad \\qquad\n\\\\\n  A(\\tilde\\phi) \\, \\left( \\tilde R_{ab} - \\frac{1}{2} \\, \\tilde R\n  \\, \\tilde g_{ab} \\right)\n  + B(\\tilde \\phi) \\, \\left( \\frac{1}{2} \\,\n    \\left(\\mbox{$\\tilde{\\nabla}$}^c\\tilde \\phi\\right) \\,\n               \\left(\\mbox{$\\tilde{\\nabla}$}_c\\tilde \\phi\\right) \\mbox{$\\tilde{g}$}_{ab}\n    - \\left(\\mbox{$\\tilde{\\nabla}$}_a\\tilde \\phi\\right) \\,\n       \\left(\\mbox{$\\tilde{\\nabla}$}_b\\tilde \\phi\\right) \\right)\n\\nonumber\n\\\\\n\\label{allgGeo}\n  - \\left( \\mbox{$\\tilde{\\nabla}$}_a\\mbox{$\\tilde{\\nabla}$}_b A(\\tilde\\phi) \\right)\n   + \\left( \\mbox{$\\tilde{\\nabla}$}^c\\mbox{$\\tilde{\\nabla}$}_c A(\\tilde\\phi) \\right) \\tilde g_{ab}\n  \\quad & = & \\quad 0.\n\\end{eqnarray}\n\\end{subequations}\n$A$ and $B$ are assumed to be $C^\\infty$ functions.\nFor $ B \\neq 0$ the principal part of (\\ref{allgWell}) does not vanish\nand thus (\\ref{allgWell}) is a wave equation. For that reason I assume\n$B(\\tilde\\phi) > \\epsilon > 0$ for every $\\tilde\\phi$. (\\ref{allgGeo})\nis a second order equation for the metric if $A(\\mbox{$\\tilde{\\phi}$})\\ne{}0$.\n\\\\\nIn the spacetime region $\\tilde H := \\left\\{x \\in \\tilde{M} | \\, {\\rm\n    sign}(A(\\tilde\\Phi))>0 \\quad \\forall\n  \\tilde\\Phi \\in [\\mbox{$\\tilde{\\phi}$}_0,\\mbox{$\\tilde{\\phi}$}(x)]\\right\\}$ the trans\\-for\\-ma\\-tion\\footnote{The\n    choice of the parameter $\\mbox{$\\tilde{\\phi}$}_0$ reflects gauge freedom. Models\n    where there is no $\\mbox{$\\tilde{\\phi}$}_0$ with $A(\\mbox{$\\tilde{\\phi}$}_0) > 0$ will not be\n    considered.}\n\\begin{subequations}\n\\label{Transf}\n\\begin{eqnarray}\n\\label{allgphiTrans}\n  \\tilde{\\bar\\phi} & = & \\int_{\\tilde\\phi_0}^{\\tilde\\phi} \\frac{1}{A}\n    \\sqrt{ \\frac{3}{2} \\,\n    \\left(\\frac{dA}{d\\phi} \\right)^2 + A\\,B } \\quad\n  d\\phi\n\\\\\n\\label{allgTrans}\n  \\tilde {\\bar g}_{ab} & = & A \\, \\tilde g_{ab}\n\\end{eqnarray}\n\\end{subequations}\ngives a solution of the system (\\ref{allgSys}) with a massless\nKlein-Gordon field\n$\\widetilde{\\bar\\phi}$ as matter model corresponding to the choice\n$(A,B)=(1,1)$ and the equations\n\\begin{subequations}\n\\label{KGGl}\n\\begin{eqnarray}\n  \\widetilde{\\bar{\\vphantom{\\phi}\\Box}}\\widetilde{\\bar\\phi} & = & 0\n\\\\\n  \\widetilde{\\bar{R}}_{ab} - \\frac{1}{2} \\, \\widetilde{\\bar{R}} \\,\n  \\widetilde{\\bar{g}}_{ab}\n  & = & \\widetilde{\\bar{T}}_{ab}[{\\widetilde{\\bar{\\phi}}}]\n\\end{eqnarray}\nwith energy momentum tensor\n\\begin{equation}\n  \\widetilde{\\bar{T}}_{ab}[{\\widetilde{\\bar{\\phi}}}] =\n  (\\widetilde{\\bar\\nabla}_a\\widetilde{\\bar{\\phi}}) \\,\n  (\\widetilde{\\bar\\nabla}_b\\widetilde{\\bar{\\phi}})\n  - \\frac{1}{2} \\, (\\widetilde{\\bar\\nabla}_c\n                          \\widetilde{\\bar{\\phi}}) \\,\n                         (\\widetilde{\\bar\\nabla}^c\n                          \\widetilde{\\bar{\\phi}})\n                          \\, \\widetilde{\\bar g}_{ab}.\n\\end{equation}\n\\end{subequations}\n\\\\\n{}From the assumptions about $A$ and $B$ follows that the corresponding\nKlein-Gordon field will be unbounded approaching the\npart of the boundary of $\\tilde H$ where $A(\\mbox{$\\tilde{\\phi}$})\\rightarrow 0$.\nThe singularity in the Klein-Gordon field\nshows up at least in a singularity of the equations for $(\\tilde M,\\tilde\ng_{ab}, \\tilde \\phi)$.\n\\\\\nFor two of the scalar fields in the above class the field equations\nare very special, the already mention massless Klein-Gordon field\n$\\tilde{\\bar\\phi{}}$\n(\\ref{KGGl}) and the conformally invariant scalar field $\\mbox{$\\tilde{\\phi}$}{}$,\n$(A,B)=(1-\\frac{1}{4}\\mbox{$\\tilde{\\phi}$}^2,\\frac{3}{2})$ ($\\mbox{$\\tilde{\\phi}$}{}$ can be\nrescaled by an arbitrary factor).\n\\\\\nThe first, because the set of equations in the physical spacetime\nbecomes remarkable simple and has been analyzed intensely with\nanalytical (e.g.~\\cite{Ch91tf}) and numerical\n(e.g.~\\cite{Ch92CB}) methods for spacetimes with spherical symmetry.\n\\\\\nThe second, yielding the equations (\\ref{model}),\nbecause the matter equations are invariant under rescalings $g_{ab} =\n\\Omega^2 \\, \\mbox{$\\tilde{g}$}_{ab}$ and $\\phi = \\Omega^{-1} \\mbox{$\\tilde{\\phi}$}{}$.\n\\\\\nThe transformation between the two special cases is\n\\begin{subequations}\n\\begin{eqnarray}\n  \\tilde{\\B{\\phi}}    & = &\n    \\sqrt{6} \\, \\mbox{arctanh} \\frac{\\mbox{$\\tilde{\\phi}$}}{2}\n  \\\\\n  \\tilde{\\B{g}}_{ab}  & = & ( 1-\\frac{1}{4}\\mbox{$\\tilde{\\phi}$}^2 ) \\, \\mbox{$\\tilde{g}$}_{ab}\n\\end{eqnarray}\n\\end{subequations}\nwhich is a bijective mapping from\n$\\mbox{$\\tilde{\\phi}$} \\in ]-2,2[$ to $\\tilde{\\bar{\\phi}}\\in ]-\\infty,\\infty[$.\n\\\\\nDue to the following diagram, illustrating the above described\nrelations,  it is evident that there is a variable transformation\nregularizing\nthe\nunphysical equations for the Klein-Gordon field:\n\\begin{center}\n\\unitlength1cm\n\\begin{picture}(15,6.5)\n\\addcontentsline{lof}{figure}{{\\string\\numberline\\space{}Konforme\n    \\protect\"Aquivalenz von Skalarfeldern}}\n  %\n \n  \\put(0,0){\\makebox(2.5,2)[l]{\\shortstack[l]{unphysical\\\\spacetime}}}\n  \\put(3.5,0){\\framebox(4,2){%\n              \\shortstack[l]{conformal field $\\phi$,\\\\reg.\\ equations}}}\n  \\put(11.5,0){\\framebox(4,2){%\n              \\shortstack[l]{KG field $\\bar{\\phi}$,\\\\sing.\\ equations}}}\n  %\n \n  \\put(5.5,3.75){\\vector(0,-1){1.5}}\n    \\put(6,2.75){\\shortstack[l]{$\\phi=\\mbox{$\\tilde{\\phi}$}\/\\Omega$\\\\$g_{ab}=\\Omega^2\\mbox{$\\tilde{g}$}_{ab}$}}\n  \\put(13.5,3.75){\\vector(0,-1){1.5}}\n    \\put(11,2.75){\\shortstack[r]{$\\bar{\\phi}=\\tilde{\\bar{\\phi}}\/\\bar{\\Omega}$\\\\\n                        $\\bar{g}_{ab}=\\bar{\\Omega}^2\\tilde{\\bar{g}}_{ab}$}}\n  %\n \n  \\put(0,4){\\makebox(3,2)[l]{\\shortstack[l]{physical\\\\spacetime}}}\n  \\put(3.5,4){\\framebox(4,2){\\shortstack{conformal field $\\mbox{$\\tilde{\\phi}$}$\\\\\n    $\\mbox{$\\tilde{\\phi}$} \\in ]{-2},{2}[$}}}\n  \\put(11.5,4){\\framebox(4,2){\\shortstack{KG field\n        $\\tilde{\\bar{\\phi}}$\\\\ $\\tilde{\\bar{\\phi}}\\in\n        ]-\\infty,\\infty[$}}}\n  %\n \n  \\put(7.75,5){\\vector(1,0){3.5}}\n  \\put(8.5,5.2){\\mbox{$\\bar\\phi=f(\\mbox{$\\tilde{\\phi}$})$}}\n  \\put(8.5,4.6){\\mbox{$\\bar g_{ab}=\\omega^2\\mbox{$\\tilde{g}$}_{ab}$}}\n  %\n\\end{picture}\n\\end{center}\nBy mapping an arbitrary scalar field $\\tilde{\\bar{\\bar \\phi}}$ with\naction (\\ref{ScalarAction}) to the Klein-Gordon field $\\tilde{\\bar\n  \\phi}$ and then to the conformally invariant scalar field $\\tilde\n\\phi $ regular equations for $\\bar{\\bar \\phi}$ are obtained.\n\\subsection{The hyperboloidal initial value problem}\nSince $A(\\phi)>0$ on $\\bar S$ all scalar field models connected\nby transformation (\\ref{Transf}) to a\nhyperboloidal initial value problem with a conformally invariant\nscalar field as matter source are weakly asymptotically flat.\n\\\\\nFor a massless KG model there is a one parameter gauge freedom in the\nscalar field. If $\\tilde{\\bar{\\phi}}$ is a solution, then so is\n$\\tilde{\\bar{\\phi}}+\\tilde{\\bar{\\phi}}_0$ with\n$\\tilde{\\bar{\\phi}}_0=\\mbox{const} {}$, as the energy momentum tensor depends on\nderivatives of $\\tilde{\\bar{\\phi}}$ only. This can also be seen by\nmapping a Klein-Gordon model to a Klein-Gordon model with\n$\\tilde{\\bar{\\phi}}_0\\ne{}0$ and (\\ref{Transf}). The analogue holds for\nevery considered scalar field model.\nFor the hyperboloidal initial value problem $\\mbox{$\\tilde{\\phi}$} = \\Omega\n\\phi{}$, therefore $\\mbox{$\\tilde{\\phi}$}{}$ vanishes at \\mbox{$\\cal J$}{}, fixing the gauge in\n$\\mbox{$\\tilde{\\phi}$}{}$.\n\\\\\nIn definition (\\ref{HypInitValProbl}) $1-\\frac{1}{4} \\Omega^2\n\\phi^2 \\mid_{\\bar S} >0$ was assumed. But with the Bekenstein black\nhole~\\cite{Be75bh} a weakly asymptotically flat\nsolution is known where $A(\\mbox{$\\tilde{\\phi}$})$ vanishes on a regular\npart of the spacetime. In this case the transformation gives a\npossible extension of a massless Klein-Gordon scalar field solution beyond a\nsingularity -- the Klein-Gordon field $\\widetilde{\\bar\\phi}$ and the\nmetric $\\widetilde{\\bar g}_{ab}$ degenerate there.\n\\vspace{0.5cm}\nIt is a pleasure for me to thank Helmut Friedrich, Bernd Schmidt, and\nJ\\\"urgen Ehlers for the very helpful discussions during the grow of\nthis work which is part of my Ph.~D.\\ thesis.\n\n\\begin{appendix}\n\\section{Notation}\nThe signature of the Lorentzian metric $g_{ab}$ is $(-,+,+,+)$.\n\\\\\nWhenever possible I use abstract indices as described\nin~\\cite[chapter~2]{PeR84SA}. Small Latin letters denote abstract\nindices, underlined\nsmall Latin letters are frame indices. For the components of a tensor with\nrespect to coordinates small Greek letters are used. The frame\n$\\left(\\frac{\\partial}{\\partial x^\\mu}\\right)^a$ is constructed from\nthe coordinates $x^\\mu $, $e_{\\f{i}}{}^a$ denotes an arbitrary frame.\nIn this notation $v_a$ is a covector, $v_{\\f{i}}$ a scalar, namely\n$v_a\\, e_{\\f{i}}{}^a$.\n\\\\\n$v(f)$ is defined to be the action of the vector $v^a$ on the function\n$f$, i.e.\\ for every covariant derivative $\\nabla_a$: $t(f)=t^a\\,\\nabla_a f$.\n\\\\\nThe transformation between abstract, coordinate, and frame indices is\ndone by contracting with $e_{\\f{i}}{}^a$ and $e_{\\f{i}}{}^\\mu $. All\nindices may be raised and lowered with the metric $g_{AB}$ and the\ninverse $g^{AB}$. $g^{AC}\\,g_{CB} = \\delta^A{}_B$, $A$ and $B$ are\narbitrary indices, e.g.\\  $e_{\\f{i}a}=g_{ab}\\,e_{\\f{i}}{}^b$ and\n$e^{\\f{i}}{}_a=g^{\\f{i}\\f{j}}\\,e_{\\f{i}a}$.\n\\\\\nFor a frame $e_{\\f{i}}{}^a$ and a covariant derivative\n$\\nabla_a$ the Ricci rotation coefficients\nare defined as\n\\begin{equation*}\n  \\gamma^a{}_{\\f{i}\\f{j}} := e_{\\f{i}}{}^b \\nabla_b e_{\\f{j}}{}^a.\n\\end{equation*}\n{}From this definition follows\n\\begin{equation*}\n  e_{\\f{i}}{}^a \\, e^{\\f{j}}{}_b \\, (\\nabla_a t^b) =\n  e_{\\f{i}}(t^{\\f{j}}) + \\gamma^{\\f{j}}{}_{\\f{i}\\f{k}} \\, t^{\\f{k}}.\n\\end{equation*}\n\\\\\nWith respect to a coordinate frame $e_{\\mu}{}^a \\equiv\n\\left(\\frac{\\partial}{\\partial x^\\mu}\\right)^a$  the components\n$\\gamma^\\lambda{}_{\\mu\\nu}$ are the Christoffel symbols\n$\\Gamma^\\lambda{}_{\\mu\\nu}$.\n\\\\\nThe torsion $T^a{}_{bc}$ is defined by\n\\begin{equation*}\n  \\nabla_a\\nabla_b f - \\nabla_b\\nabla_a f = - T^c{}_{ab} \\, \\nabla_c f,\n\\end{equation*}\nthe Riemann tensor $R_{abc}{}^d$ by\n\\begin{equation*}\n  \\nabla_a\\nabla_b\\omega_c - \\nabla_b\\nabla_a\\omega_c =\n  R_{abc}{}^d \\, \\omega_d  - T^{d}{}_{ab} \\, \\nabla_d \\omega_c.\n\\end{equation*}\nContraction gives the Ricci tensor,\n\\begin{equation*}\n  R_{ab} = R_{acb}{}^c,\n\\end{equation*}\nand the Ricci scalar\n\\begin{equation*}\n  R = R_{ab}\\, g^{ab}.\n\\end{equation*}\nThe Einstein tensor is given by\n\\begin{equation*}\n  G_{ab} = R_{ab} - \\frac{1}{2} \\, R \\, g_{ab}.\n\\end{equation*}\n\\\\\nThe speed of light $c$ is set to $1$ as the gravitational constant\n$\\kappa $ in $G_{ab}=\\kappa\\, T_{ab}$.\n\\end{appendix}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\n\nPb and Sn $\\sqrt{3}\\times \\sqrt{3}$ adlayer structures on the (111)\nsurface of Ge have recently revealed a reversible charge density wave \n(CDW) transition to a low temperature reconstructed $3\\times 3$ phase.\n\\cite{nature,Modesti_1,Modesti_2,Avila,LeLay,Sn_carpinelli,asensio} \nA half-filled surface state band makes the high temperature phase metallic.\nThe low temperature phase is either metallic -- as seems to be the case\nfor Sn\/Ge(111) -- or weakly gapped, or pseudo-gapped, as suggested\nfor Pb\/Ge(111). \n\nRelated isoelectronic systems, like the $\\sqrt{3}$-adlayer of Si on\nthe (0001) surface of SiC \\cite{SiC} and on K\/Si(111):B,\\cite{KSi}\nshow a clear insulating behavior, with a large gap,\nno structural anomalies, no CDWs, and no transitions, at least to\nour present knowledge.\n\nThese adsorbate surfaces are altogether mysterious. \nThe very existence of a $\\sqrt{3}\\times \\sqrt{3}$ adsorbate phase, \nwith coverage 1\/3, is puzzling. \nFor isoelectronic Si on Si(111), or Ge on Ge(111), for instance, there exists \nno such phase. The stable low-coverage phase are $7\\times 7$ and $c(2\\times 8)$ \nrespectively, whose coverage is instead close to 1\/4. \nThey are made up of $2 \\times2$ basic building blocks, each with one adatom\nsaturating three out of four first-layer atoms, and one unsaturated\nfirst-layer atom, the ``restatom''. In this adatom-restatom block, \nthe nominally unpaired electron of the adatom and that of the restatom pair \noff together, giving rise to a stable, fully saturated, insulating surface. \nBy contrast, the $\\sqrt{3}\\times \\sqrt{3}$ phases\nwith 1\/3 coverage are very common for trivalent adsorbates, such as\nGa and In, and for pentavalent ones like As, on the same (111) surfaces.\nThese adatoms lack the unpaired electron, and can therefore lead to\na fully saturated insulating surface without the need for any restatoms. \n\nA $\\sqrt{3}\\times \\sqrt{3}$ adsorbate phase of {\\it tetravalent} adatoms  \nis bound by construction to possess one unpaired electron per adatom,\ngiving rise to a very destabilizing half-filled metallic surface state band.\nSeen in this crude light, it is a puzzle why this kind of coverage should  \nconstitute even only a locally stable state of the surface.\n\nLooking more closely, we may speculate that SiC(0001)\\cite{SiC} \nand K\/Si(111):B,\\cite{KSi} most likely Mott-Hubbard insulators,\n\\cite{santoro,Northrup,Anisimov,KSi} are perhaps ``stabilized'' by Coulomb \nrepulsions, so large to make it anyway difficult for electrons to move. \nFor the more innocent-looking, less correlated, Pb\/Ge(111) and Sn\/Ge(111), \nthis argument is less obvious, and the puzzle remains. The function of \nthe $3\\times 3$ CDW state -- whatever its real nature -- most likely \nserves the function of stabilizing these otherwise unstable surfaces at \nlow temperatures. Nonetheless, the CDW periodicity chosen by the surface CDW\n-- $3\\times 3$, meaning a $\\sqrt{3}\\times \\sqrt{3}$ super-cell of adatoms -- \nis not at all evident. In fact, it replaces a supposedly unstable state \ncharacterized by an odd number of electrons\/cell (three), with another \nwhere the electron number (nine) is, alas, odd again.\n\nBe all that as it may, there is little doubt that the main factor driving the\nphenomena on all these surfaces, appears to be precisely the half-filled--\nand extremely narrow -- surface state band.  We thus\nbegin with a discussion that in principle encompasses all the\n$\\sqrt{3}\\times \\sqrt{3}$ tetravalent adsorbed surfaces.\n\nWe believe the following points to be of general validity:\n\n{\\em i)\\\/} {\\bf Poor nesting}. \n\nTwo-dimensional Fermi surface (FS) nesting in the half-filled surface \nstates\\cite{tosatti} has been repeatedly invoked as the driving \nmechanism for the\nCDW instability in the case of Pb\/Ge,\\cite{nature,asensio} but excluded \nfor the case of Sn\/Ge.\\cite{Sn_carpinelli,Sn_scandolo} \nHowever, by inspecting either photoemission $k(E)$ data,\n\\cite{Modesti_2,Avila,LeLay,asensio} and existing \nfirst-principle (LDA) calculations\\cite{nature,Sandro,Sn_carpinelli}\nof the surface half-filled band (the``adatom dangling \nbond band''), we fail to detect a particularly good nesting of the \ntwo-dimensional FS at the surface Brillouin zone (BZ) corner \n${\\bf K}=(4\\pi\/3a,0)$. The wavevector-dependent susceptibility \ngenerated by the calculated band structure, in particular, has no \nespecially large value at this k-point, and rather peaks elsewhere \n(see inset in Fig.\\ \\ref{band_bz_chi0:fig}). \nTo be sure, there is nothing preventing in general a good nesting at \n${\\bf K}=(4\\pi\/3a,0)$, or any other k-point.\nHowever, insofar as the surface state band is really lying in a bulk gap at \neach single k-point, it should be with good accuracy \n-- by simple state counting and charge neutrality --\nprecisely half filled. \nThis implies that the filled and empty state areas should be equal. \nHypothetical Fermi surfaces with this kind of shape and good nesting at \n${\\bf K}=(4\\pi\/3a,0)$ do not appear to be compatible \nwith an integer electron number.\nWe thus believe lack of perfect nesting to be the case for both \nPb\/Ge as for Sn\/Ge.\n\nFig.\\ \\ref{band_bz_chi0:fig}, showing a tight binding fit to the LDA surface \nband dispersion for the test-case of Si(111)\/Si,\\cite{Sandro} as well as the\ncorresponding FS and Lindhard density response function $\\chi_o(\\bf q)$,\n\\[ \\chi_o(\\bf q) \\,=\\, \\int_{BZ} \\frac{d\\bf k}{(2\\pi)^2}\n\\; \\frac{ n_{\\bf k} - n_{{\\bf k}+{\\bf q}} }\n{\\epsilon_{{\\bf k}+{\\bf q}} - \\epsilon_{\\bf k} }  \\;, \\]\n$n_{\\bf k}$ and $\\epsilon_{\\bf k}$ being the occupation number and energy \nof an electron with Bloch momentum ${\\bf k}$, \nprovides a concrete illustration of these statements.\nWe note, in passing, that a strong nesting at ${\\bf K}$ is, on the contrary, \nautomatically guaranteed if the surface band acquires a uniform magnetization\nin such a way that the densities of up and down electrons become, \nrespectively, $2\/3$ and $1\/3$.\\cite{Sandro} \nThe majority spins would then fill the region external to the reduced BZ in \nFig.\\ \\ref{band_bz_chi0:fig}, and their FS would be strongly nested. \nThis suggestion, which turns out to be correct at the mean-field level,  \npoints into the direction of a possible role played by magnetism in these \nsystems.  \n\n{\\em ii)\\\/} {\\bf Importance of electron-electron interactions}. \n\nThe width $W$ of the surface band is relatively small:\n$W\\approx 0.5$ eV for Pb and Sn\/Ge(111), $W\\approx 0.3$ eV for SiC(0001). \nMoreover, this band is half-filled.\nThese facts call for a careful consideration of electron-electron\ninteractions, as well as of electron-phonon (e-ph), as possible sources of \ninstability. The importance of electron-electron interaction is \nunderlined by the different phenomenology\nof SiC(0001) and K\/Si(111):B with respect to Pb-Sn\/Ge(111). \nThe stronger insulating character of the former surfaces parallels\nclosely their stronger electron-electron \nrepulsions, connected both with more localized surface Wannier functions\n(see later on),\nand with reduced screening, due to larger bulk semiconducting gaps.\n\n{\\em iii)\\\/} {\\bf Weakness of LDA calculations for ground state prediction}. \n\nLDA electronic structure calculations -- an extremely well tested tool\nin many areas-- are certainly suitable for a weakly interacting system,\nsuch as the bulk semiconductor, or a passivated semiconductor surface. \nThey are less reliable, especially when they do not include spin, in \npredicting the stable state and the instabilities of a narrow band system. \nFor instance, the phenomenology of SiC(0001) -- suggesting a Mott-Hubbard \ninsulator -- is unreproducible by LDA. \nThe onset of a CDW on Sn\/Ge(111)is also not predicted by recent \nLDA calculations.\\cite{Sn_carpinelli,Sn_scandolo} While there is no\nreason to doubt the basic credibility of the one-electron band energies \nobtained from these Kohn-Sham equations, the mean-field treatment of \ninteractions, the screened local exchange, and especially \nthe neglect of magnetic\ncorrelations are the standard source of problems with LDA.  As a\nconsequence, it will be necessary to worry more substantially about\ninteractions, and to use methods which, even if mean-field, permit the\ninclusion of strong correlations, including magnetic effects.\n\n{\\em iv)\\\/} \n{\\bf Interaction-driven mechanisms for $3\\times 3$ CDW instabilities}.\n\nThere are several different couplings which the surface electrons, as they hop\nweakly between a surface adatom site and another, experience and can influence the \nformation of the CDW, or of an insulating ground state:\na) on-site, and nearest-neighbor (n.n.) inter-site electron-electron repulsion; \nb) on-site effective attraction (negative Hubbard-$U$ term) of electron-phonon\norigin.\n\nBecause of poor nesting, electron-phonon alone is unlikely to \ndrive the $3\\times 3$ CDW. \nAt weak coupling, the susceptibility peak in Fig.\\ \\ref{band_bz_chi0:fig} would\nrather drive an incommensurate periodicity. \nAt strong coupling, the frustration associated to the triangular lattice, \nwill favor, in general, a superconducting ground state over a \nCDW phase (see Appendix).\\cite{santos}\n\nOn the other hand, the electron-electron interaction, both on-site and, \nindependently, nearest neighbor, naturally suggests, as we shall see later, \nthe $3\\times 3$ surface periodicity, which is found experimentally. \n\nThe approach we will take is based on an extended Hubbard-Holstein model.\nIt is by necessity a ``non-first-principle'' approach and, as such,\nhas no strong predictive power. \nHowever, it is made more realistic by using parameters extracted \nfrom first-principle calculations, and we find it very helpful in clarifying \nthe possible scenarios as a function of the strength of electron-electron \ninteractions. \nBecause of this rather qualitative use, we will make no attempt to push the \naccuracy of treatment of this model to a very high level of sophistication. \nThe basic tool will be the unrestricted Hartree-Fock approximation. \nAlthough mean field, it allows magnetic solutions, favored by exchange which \nis unscreened. \n\n\\section{Model}\nEach tetravalent adatom on a (111) semiconductor surface carries a dangling\nbond -- an unpaired electron in an unsaturated orbital.\nIn the $\\sqrt{3}\\times\\sqrt{3}$ structure, the dangling bonds of the adatoms \ngive rise to a band of surface states which lies in the bulk semiconductor\ngap. \\cite{nature,Sandro} By electron counting, such a band is half-filled. \nOur basic starting point is the quantitatively accurate surface state band \ndispersion $\\epsilon_{\\bf k}$ which one calculates in gradient-corrected LDA,\n\\cite{nature,Sandro}. \nIt is shown in Fig.\\ \\ref{band_bz_chi0:fig} for the case of Si\/Si(111). \nThe solid and dashed lines in Fig.\\ \\ref{band_bz_chi0:fig} are tight-binding \nfits to the LDA results obtained by including, respectively, up to the $6^{th}$ \nand up to the $2^{nd}$ shell of neighbors.\nThe fit with hopping integrals $t_1,t_2,\\cdots,t_6$ is quite good.\nLess good, but qualitatively acceptable, is the fit obtained using only\nnearest neighbor (n.n.) and next-nearest neighbor (n.n.n.) hopping integrals\n$t_1$ and $t_2$.\nThe Fermi surface (FS) for the half-filled surface band is shown in the\nupper inset of Fig.\\ \\ref{band_bz_chi0:fig}.\nIt is important to stress that the FS does not show good nesting properties\nat the wavevector ${\\bf q}={\\bf K}$ (the BZ corner).\nThis feature is shared by all LDA calculations on similar \nsystems.\\cite{nature,Sandro,Sn_carpinelli}\nAlbeit small, the bandwidth $W$ of the surface band is much greater than one \nwould predict by a direct overlap of adatom dangling bonds, as the adatoms\nare very widely apart, for instance about $7\\AA$ on Ge(111). Hopping is\nindirect, and takes place from the adatom to the first-layer atoms\nunderneath, from that to a second-layer atom, then again to a first-layer\natom underneath the other adatom, and from there finally to other adatom\ndangling bond. \nThus, when expressed in terms of elementary hopping processes between \nhybrid orbitals, electron hopping between two neighboring adatom dangling \nbonds is fifth order.\nAs a result, the final dispersion of the surface state band strongly \nparallels that of the closest bulk band, the valence band. Correspondingly,\nhybridization effects of the dangling bond orbitals with first, second, and\neven third, bulk layer orbitals are strong, as shown by the extension \ninto the bulk of the Wannier orbital associated to the LDA surface band \n(Fig.\\ \\ref{wannier:fig}).\n\nIn spite of this, we can still associate to every adatom a Wannier orbital \nand write the effective Hamiltonian for the surface band as follows:\n\\begin{equation}\nH \\,=\\, \\sum_{{\\bf k}}^{BZ} \\sum_{\\sigma} \\epsilon_{\\bf k} \nc^{\\dagger}_{{\\bf k},\\sigma} c_{{\\bf k},\\sigma} \\,+\\, H_{\\rm ph} \\,+\\, \nH_{\\rm e-ph} \\,+\\, H_{\\rm int} \\;,\n\\end{equation}\nwhere $c^{\\dagger}_{{\\bf k},\\sigma}$ is the Fourier transform of \nthe Wannier orbital, namely the surface state in a Bloch picture.\nThe sum over the wavevectors runs over the surface BZ.\n$H_{\\rm int}$ includes correlation effects which are not correctly accounted \nfor within LDA, which we parametrize as follows:\n\\begin{equation}\nH_{\\rm int} = U \\sum_{{\\bf r}} n_{{\\bf r},\\uparrow} n_{{\\bf r},\\downarrow}\n+ \\frac{1}{2} \\sum_{{\\bf r}\\ne {\\bf r}'} V_{{\\bf r}-{\\bf r}'} \n(n_{{\\bf r}}-1) (n_{{\\bf r}'}-1) \\;.\n\\end{equation}\nHere $U$ is an effective repulsion (Hubbard-$U$) for two electrons on the same \nadatom Wannier orbital, and $V_{{\\bf r}-{\\bf r}'}$ is the \ndirect Coulomb interaction between different sites ${\\bf r}$ and \n${\\bf r}'$.\\cite{non-diag:nota}\nLet $V$ be the n.n.\\ value of $V_{{\\bf r}-{\\bf r}'}$, which is, clearly, the\nlargest term.\nWe have considered two models for $V_{{\\bf r}-{\\bf r}'}$: \na model (A) in which we truncate\n$V_{{\\bf r}-{\\bf r}'}$ to n.n., and a model (B) in which\n$V_{{\\bf r}-{\\bf r}'}$ has a long range Coulombic tail of the form\n\\[ V_{{\\bf r}-{\\bf r}'} \\,=\\, \\frac{a V}{|{\\bf r}-{\\bf r}'|} \\;, \\]\nwhere $a$ is the n.n.\\ distance.\nThe results for model B are qualitatively similar to those of A, \nand will be only briefly discussed later on. In other words, even if\nmost of the detailed results in this paper will be base on the n.n.\\ \n$V_{{\\bf r}-{\\bf r}'}$, their validity is more general. \n\nLDA estimates of the {\\it bare} coulomb repulsion $U_o$ and $V_o$ \nbetween two electrons respectively on the same and on neighboring Wannier \norbitals are -- for our test case of Si(111)\/Si -- of about $3.6$ eV and \n$1.8$ eV respectively.\\cite{Sandro} \nScreening effects by the the underlying bulk are expected to\nreduce very substantially these repulsive energies. An order of magnitude\nestimate for $U$ and $V$ is obtained by dividing their bare values\nby the image-charge screening factor, $(\\epsilon +1)\/2\\approx 6$, \nyielding, for Si, $U=0.6$ eV ($10 t_1$), and $V=0.3$ eV ($5 t_1$).\nCorresponding values would be somewhat smaller for Ge(111), in view\nof a very similar dispersion \\cite{Sn_scandolo} and of a ratio of about \n4\/3 between the dielectric constants of Ge and Si. SiC(0001), the opposite\nis true. The surface state band is extremely narrow, of order $0.3$ eV\n\\cite{pollmann}, while the bulk dielectric constant is only about $6.5$. \n\nAs for the e-ph interaction, in principle both the on-site\nWannier state energy and the hopping matrix elements between neighbors\ndepend on the positions of the adatoms. \nWithin the deformation potential approximation, we consider only a \nlinear dependence of the on-site energy from a single ionic coordinate \n(for instance, the height $z_{\\bf r}$ of the adatom measured \nfrom the equilibrium position), and take \n\\begin{equation} \\label{e-ph-ham:eqn}\nH_{\\rm e-ph} = -g \\sum_{{\\bf r}} z_{\\bf r} (n_{\\bf r}-1) \\;,\n\\end{equation}\nwith $g$ of the order of $\\approx 1$ eV\/$\\AA$. The free-phonon term will\nhave the usual form\n\\begin{equation}\nH_{\\rm ph} = \\sum_{\\bf k}^{BZ} \\hbar \\omega_{\\bf k}\n\\left( b^{\\dagger}_{\\bf k} b_{\\bf k} + \\frac{1}{2} \\right) \\;,\n\\end{equation}\nwhere $b_{\\bf k}$ is the phonon annihilation operator, and \n$\\hbar \\omega_{\\bf k}$ a typical phonon frequency of the system, which \nwe take to be about 30 meV, independent of {\\bf k}. \n\n\\section{Phase diagram: some limiting cases} \n\\label{pd_no_g:sec}\n\nPreliminary to the full treatment of Sect.\\ \\ref{hf:sec}, \nwe consider first the purely electronic problem in the absence of \ne-ph interaction. \nWe start the discussion from particular limiting cases for which \nwell-controlled statements, or at least intuitively clear ones, can be made,\nwithout the need of any new specific calculations.\nIn the Appendix we will also consider, because it is useful in connection \nwith the electron-phonon case, the unphysical limit of strong on-site \nattraction (large and negative $U$).\n\n\\subsection{Large positive $U$: the Mott insulator.} \\label{large_u:sec}\nFor $U\\gg V,W$, the system is deep inside the Mott insulating \nregime.\\cite{Anderson_SE} The charge degrees of freedom are frozen,\nwith a gap of order U. The only dynamics is in the spin degrees of freedom.\nWithin the large manifold of spin degenerate states with exactly one electron \nper site, the kinetic energy generates, in second order perturbation theory, \na Heisenberg spin-1\/2 antiferromagnetic effective Hamiltonian\ngoverning the {\\it spin\\\/} degrees of freedom,\n\\begin{equation}\nH_{\\rm eff} = \\sum_{(ij)} J_{ij} \\, {\\bf S}_{{\\bf r}_i} \\cdot\n{\\bf S}_{{\\bf r}_j} \\;,\n\\end{equation}\nwith $J_{ij}=4|t_{ij}|^2\/U$.\\cite{Anderson_SE}\n\nFor our test case of Si(111)\/Si, the values of the hoppings are such that\n$J_1 \\approx 20$ meV, $J_2\/J_1\\approx 0.12$ while the remaining couplings \n$J_3,\\cdots$ are very small.  \nAntiferromagnetism is frustrated on the triangular lattice. \nZero temperature long range order (LRO) -- if present -- should be of the \nthree-sublattice $120^o$-N\\'eel type, which can be also seen as a commensurate \nspiral spin density wave (s-SDW).\n\nBecause it does not imbalance charge, this state is not further affected by\nelectron-phonon coupling.\n\nIn summary, we expect for large values of $U$ a wide-gap Mott insulator with \na s-SDW (spins lying in a plane, forming $120^o$ angles), \na $3\\times 3$ {\\it magnetic} unit cell, but uniform charge (no CDW). \nThis is, most likely, the state to be found on the Si-terminated and\nC-terminated SiC(0001) surface at T=0 \\cite{Northrup,Anisimov}.\n\n\\subsection{Strong inter-site repulsion:\nan asymmetric CDW with three inequivalent sites.}\n\\label{large_v:sec}\nThe e-ph coupling can effectively reduce $U$, but not $V$. \nTherefore, it is of interest to consider the hypothetical regime\n$W<U\\ll V$.\nWhen the first-neighbor electron-electron repulsion $V$ is large the\nsystem, in order to minimize the interaction energy, will prefer a\n$3\\times 3$ CDW-like ground state, with two electrons on one sublattice (A), \na single electron on another sublattice (B), and\nzero electrons on the third sublattice (C) (see Fig.\\ \\ref{triang_cdw:fig}).\nThese states are still highly degenerate (in the absence of hopping) due to spin\ndegeneracy for the single unpaired electron on sublattice B.\nA gap $U$ separates these states from the lowest-energy excited\nconfigurations (see Fig.\\ \\ref{triang_cdw:fig}).\nThe spin degeneracy can be removed in second-order perturbation theory, \nowing to $t_2$, which leads to an effective spin-1\/2 Heisenberg Hamiltonian \nwithin sublattice B, \n\\begin{equation}\nH_{\\rm eff} = J \\sum_{(ij)}^{\\rm sublattice \\, B}\n{\\bf S}_{{\\bf r}_i} \\cdot {\\bf S}_{{\\bf r}_j} \\;,\n\\end{equation}\nwith a weak antiferromagnetic exchange constant $J=4t^2_2\/U$.\\cite{Anderson_SE}\nSummarizing, we expect in this regime a strong $3\\times 3$ asymmetric CDW \n(a-CDW) with three inequivalent sites ($\\phi_{\\rho}\\approx \\pi\/6$, see below), \nand a spiral $3\\sqrt{3}\\times 3\\sqrt{3}$ SDW, \ngoverning the unpaired electron spins, superimposed on it.  \nNotice that, while the charge periodicity is $3\\times 3$, the actual unit\ncell is larger, i.e., $3\\sqrt{3}\\times 3\\sqrt{3}$.\nDespite having the correct charge periodicity, namely  $3\\times 3$, this\na-CDW is not compatible with the experimental findings on Pb-Sn\/Ge,\nwhich is a symmetric CDW. \nWe conclude that the low-temperature CDW state of these systems is not \ncompletely dominated by $V$.\n\n\\section{Mean-field theory.} \\label{hf:sec}\nIn order to get a more complete picture of additional phases for smaller $U$, \nand of the possible phase diagram of the model we now turn to a quantitative \nmean field theory analysis.\nThe first issue is to include the possibility of magnetic correlations.\nFor small values of the interactions $U$ and $V$, the Stoner criterion can \nbe used to study the possible magnetic instabilities of the \nparamagnetic metal obtained from LDA calculations.\nThe charge and spin susceptibilities are given, within the random phase\napproximation,\\cite{Mahan} by \n\\begin{eqnarray}\n\\chi_C(\\bf q) &\\,=\\,&\n\\frac{2 \\chi_o({\\bf q}) }{1 + (U+2V_{\\bf q})\\chi_o({\\bf q})}\n\\nonumber \\\\\n\\chi_S(\\bf q) &\\,=\\,& \\frac{\\chi_o({\\bf q}) }{1 - U\\chi_o({\\bf q})}  \\;,\n\\end{eqnarray}\nwhere $\\chi_o$ is the non-interacting susceptibility per spin projection,\nand both factors of $2$ account for spin degeneracy.\nThe divergence of $\\chi_S$ is governed, in this approximation, by $U$ only.\nSince $\\chi_o({\\bf q})$ is finite everywhere, a finite $U$ is needed in\norder to destabilize the paramagnetic metal. \nThe wavevector ${\\bf q}^*$ at which $\\chi_S$ first diverges, by increasing $U$,\nis in general incommensurate with the underlying unit cell.\nThe instability is towards an incommensurate, metallic, \nspiral SDW.\\cite{KMurthy}\nFig.\\ \\ref{band_bz_chi0:fig} shows that, in our case, ${\\bf q}^*=(1.32 K,0)$ \n(with $K=4\\pi\/3a$, the BZ corner). We get $U_c^{HF}\/t_1\\approx 3.7$.\n(The other maximum of $\\chi_o$ at ${\\bf q}=(0.525 K,0)$\nis very close to the result obtained for the triangular lattice with n.n.\\\nhopping only.\\cite{KMurthy})\nAs for the charge susceptibility, a divergence can be caused only by\nan attractive Fourier component of the potential $V_{\\bf q}$. \n$V_{\\bf q}$ has a minimum at the BZ corners $\\pm {\\bf K}$, with\n$V_{\\pm {\\bf K}}=-3V$ for the n.n. model (A)\n($V_{\\pm {\\bf K}}\\approx -1.5422 V$ if a Coulomb tail is added, model B). \nThis minimum leads to an instability towards a $3\\times 3$ CDW as \n$(U+2V_{{\\bf K}})\\chi_o({\\bf K})=-1$, i.e., given our value of\n$\\chi_o({\\bf K})\\approx 0.2\/t_1$,\n$(U+2V_{{\\bf K}})\\approx -5t_1$.\nFor model A we get a transition, when $U=0$, at $V_c^{MF}\/t_1\\approx 0.83$.\n\nIn general, the small coupling paramagnetic metal is surrounded by an \nintermediate coupling region, where complicated incommensurate --\ngenerally metallic -- solutions occur. \nFor stronger $U$ and $V$, commensurate solutions are privileged.\\cite{KMurthy}\nIn view of the fact that a $3\\times 3$ CDW is experimentally relevant, we\nconcentrate our analysis on the simplest commensurate phases. \nThese are easy to study with a standard Hartree-Fock (HF) mean-field \ntheory.\nIn particular, we restrict ourselves to order parameters\nassociated with non-vanishing momentum space averages of the type\n$\\langle c^{\\dagger}_{{\\bf k},\\sigma} c_{{\\bf k},\\sigma '} \\rangle$ and \n$\\langle c^{\\dagger}_{{\\bf k},\\sigma} c_{{\\bf k}\\pm{\\bf K},\\sigma '}\\rangle$.  \nPossible non-vanishing order parameters are the uniform magnetization \ndensity ${\\bf m}$, \n\\begin{equation} \\label{m:eqn}\n{\\bf m} = \\frac{1}{N_s} \\sum_{\\bf k}^{BZ} \\sum_{\\alpha,\\beta}\n\\langle c^{\\dagger}_{{\\bf k},\\alpha} ({\\vec \\sigma})_{\\alpha\\beta}\nc_{{\\bf k},\\beta} \\rangle  = \\frac{2}{N_s} \\langle {\\bf S}_{\\rm tot} \\rangle \\;,\n\\end{equation}\nthe ${\\bf K}$-component of the charge density,\n\\begin{equation} \\label{rho:eqn}\n\\rho_{\\bf K} = \\frac{1}{N_s} \\sum_{\\bf k}^{BZ} \\sum_{\\sigma}\n\\langle c^{\\dagger}_{{\\bf k},\\sigma}\nc_{{\\bf k}-{\\bf K},\\sigma}  \\rangle \\;,\n\\end{equation}\nand the ${\\bf K}$-component of the spin density\n\\begin{equation}\n{\\bf S}_{\\bf K} = \\frac{1}{N_s} \\sum_{\\bf k}^{BZ} \\sum_{\\alpha,\\beta}\n\\langle c^{\\dagger}_{{\\bf k},\\alpha} \\frac{({\\vec \\sigma})_{\\alpha\\beta}}{2}\nc_{{\\bf k}-{\\bf K},\\beta} \\rangle \\;.\n\\end{equation}\nNote that only $\\rho_{\\bf K}$ and ${\\bf S}_{\\bf K}$ are $3\\times 3$ periodic.\nMoreover, $K$-components of bond order parameters of the type\n$\\langle c^{\\dagger}_{{\\bf r},\\sigma} c_{{\\bf r}',\\sigma '} \\rangle$ are\nautomatically included in the calculation.\n$\\rho_{\\bf K}$ and ${\\bf S}_{\\bf K}$ have phase freedom, and \nare generally complex: $\\rho_{\\bf K}=|\\rho_{\\bf K}| e^{i\\phi_{\\rho}}$, etc.\nThe role of the phase is clarified by looking at the real-space \ndistribution within the $3\\times 3$ unit cell. \nFor the charge, for instance, \n$\\langle n_{{\\bf r}_j}\\rangle=1+2|\\rho_{\\bf K}|\\cos{(2\\pi p_j\/3+\\phi_{\\rho})}$,\nwhere $p_j=0,1,2$, respectively, on sublattice A, B, and C. \nThe e-ph coupling is included but, after linearization, the\ndisplacement order parameter is not independent, and is given by\n$\\langle z_{\\bf K}\\rangle=(g\/M\\omega_{\\bf K}^2)\\rho_{\\bf K}$.\nOnly the phonon modes at $\\pm{\\bf K}$ couple directly to the CDW. \nThe phonon part of the Hamiltonian can be diagonalized by displacing\nthe oscillators at $\\pm{\\bf K}$. \nThis gives just an extra term in the electronic HF Hamiltonian of the form\n$\\Delta U (\\rho^*_{\\bf K} {\\hat \\rho}_{\\bf K} + {\\rm H.c.})$, \nwith an energy $\\Delta U=-g^2\/M\\omega^2_{\\bf K}$ which is the relevant \ncoupling parameter.  \nThis term acts, effectively, as a negative-$U$ contribution acting only\non the charge part of the electronic Hamiltonian.\n\nWith the previous choice of non-vanishing momentum space averages, the \nHartree-Fock Hamiltonian reads:\n\\begin{eqnarray} \\label{hf_ham:eqn}\nH_{\\rm H-F} &\\,=\\,&\n\\sum_{{\\bf k}}^{BZ} \\sum_{\\sigma} \\epsilon_{\\bf k} n_{{\\bf k},\\sigma}\n\\,-\\, U \\, {\\bf m} \\cdot {\\bf S}_{\\rm tot} \\nonumber \\\\\n&& + \\sum_{\\bf k}^{BZ} \\sum_{\\sigma}\n\\left\\{ \\left[ \\left( \\frac{U}{2} + V_{\\bf K} \n               -\\frac{g^2}{M\\omega^2_{\\bf K}} \\right) \\rho_{\\bf K}\n- \\sigma U S^z_{\\bf K} \\right]\nc^{\\dagger}_{{\\bf k},\\sigma} c_{{\\bf k}+{\\bf K},\\sigma} + {\\rm H.c.} \\right\\}\n\\nonumber \\\\\n&& - U \\sum_{\\bf k}^{BZ} \\left\\{\nS^+_{\\bf K} c^{\\dagger}_{{\\bf k},\\downarrow} c_{{\\bf k}+{\\bf K},\\uparrow} \\,+\\,\nS^-_{\\bf K} c^{\\dagger}_{{\\bf k},\\uparrow} c_{{\\bf k}+{\\bf K},\\downarrow}\n\\,+\\, {\\rm H.c.} \\right\\} \\nonumber \\\\\n&& + \\sum_{\\bf k}^{BZ} \\sum_{\\sigma}\n\\left\\{\nA^{(\\sigma\\sigma)}_{\\bf k}\nc^{\\dagger}_{{\\bf k},\\sigma} c_{{\\bf k},\\sigma}\n\\,+\\, \\left[ B^{(\\sigma\\sigma)}_{\\bf k}\nc^{\\dagger}_{{\\bf k},\\sigma} c_{{\\bf k}+{\\bf K},\\sigma} + {\\rm H.c.} \\right]\n\\right\\} \\nonumber \\\\\n&& + \\sum_{\\bf k}^{BZ} \\sum_{\\sigma}\n\\left\\{\nA^{({\\bar \\sigma}\\sigma)}_{\\bf k}\nc^{\\dagger}_{{\\bf k},{\\bar \\sigma}} c_{{\\bf k},\\sigma}\n\\,+\\, \\left[ B^{({\\bar \\sigma}\\sigma)}_{\\bf k}\nc^{\\dagger}_{{\\bf k},{\\bar \\sigma}} c_{{\\bf k}+{\\bf K},\\sigma} + {\\rm H.c.}\n\\right] \\right\\} \\;.\n\\end{eqnarray}\nThe last two terms originate exchange contributions due to the $V$-term;\n$A^{(\\sigma'\\sigma)}_{\\bf k}$ and $B^{(\\sigma'\\sigma)}_{\\bf k}$ are shorthands\nfor the following convolutions:\n\\begin{eqnarray} \\label{AB:eqn}\nA^{(\\sigma'\\sigma)}_{\\bf k} &\\,=\\,& - \\frac{1}{N_s} \\sum_{{\\bf k}'}^{BZ}\nV_{{\\bf k}-{\\bf k}'} \\,\n\\langle c^{\\dagger}_{{\\bf k}',\\sigma} c_{{\\bf k}',\\sigma'} \\rangle \\nonumber \\\\\nB^{(\\sigma'\\sigma)}_{\\bf k} &\\,=\\,& - \\frac{1}{N_s} \\sum_{{\\bf k}'}^{BZ}\nV_{{\\bf k}-{\\bf k}'} \\,\n\\langle c^{\\dagger}_{{\\bf k}'+{\\bf K},\\sigma} c_{{\\bf k}',\\sigma'} \\rangle \\;.\n\\end{eqnarray}\n\nThe BZ is divided into three regions: a reduced BZ (RBZ), and the two\nzones obtained by ${\\bf k}\\pm{\\bf K}$ with ${\\bf k}\\in$ RBZ.\nThe HF problem in Eq.\\ \\ref{hf_ham:eqn} reduces to the self-consistent \ndiagonalization of a $6\\times 6$ (including the spin) matrix for each \n${\\bf k}\\in$ RBZ.\n\n\\subsection{Landau theory} \\label{Landau:sec}\n\nThe mean-field solutions must be compatible with the symmetry of the\nproblem.\nBefore discussing the HF phase diagram we obtain, it is useful\nto present a few general phenomenological considerations based on a symmetry\nanalysis of the Landau theory built from the CDW order parameter\n$\\rho_{\\bf K}$ (a complex scalar), the SDW order parameter ${\\bf S}_{\\bf K}$\n(a complex vector), and the uniform magnetization ${\\bf m}$\n(a real vector).\\cite{Toledano}\nIn the absence of spin-orbit coupling, the possible contributions to the\nLaundau free energy $F$ allowed by symmetry, up to fourth order, have the form\n\\begin{eqnarray}\nF &\\,=\\,& \\frac{1}{2} a_{\\rho} |\\rho_{\\bf K}|^2\n    \\,+\\, \\frac{1}{2} a_{m}    |{\\bf m}|^2\n    \\,+\\, \\frac{1}{2} a_{s}    |{\\bf S}_{\\bf K}|^2\n    \\,+\\, F_{3} \\,+\\, F_{4} \\nonumber \\\\\nF_3 &\\,=\\,& ( B_{\\rho} \\rho_{\\bf K}^3 + {\\rm c.c.} )\n  \\,+\\, [ B_{\\rho s} \\rho_{\\bf K} ({\\bf S}_{\\bf K} \\cdot {\\bf S}_{\\bf K} )\n          \\,+\\, {\\rm c.c.} ] \\nonumber \\\\\nF_4 &\\,=\\,& b_{\\rho} |\\rho_{\\bf K}|^4 + b_{m} |{\\bf m}|^4\n   \\,+\\, b_s^{(1)} |{\\bf S}_{\\bf K}|^4\n   \\,+\\, b_s^{(2)} ({\\bf S}_{\\bf K} \\times {\\bf S}^*_{\\bf K})^2 \\nonumber \\\\\n&& +\\, b_{\\rho s} |\\rho_{\\bf K}|^2 |{\\bf S}_{\\bf K}|^2\n   +\\, b_{\\rho m} |\\rho_{\\bf K}|^2 |{\\bf m}|^2\n   +\\, b_{m s}^{(1)} |{\\bf m}|^2 |{\\bf S}_{\\bf K}|^2\n   +\\, b_{m s}^{(2)} ({\\bf m} \\cdot {\\bf S}_{\\bf K})\n                     ({\\bf m} \\cdot {\\bf S}^*_{\\bf K}) \\nonumber \\\\\n&& +\\, [b_{m s}^{(3)} ({\\bf m} \\cdot {\\bf S}_{\\bf K})\n                      ({\\bf S}_{\\bf K} \\cdot {\\bf S}_{\\bf K}) + {\\rm c.c.}]\n \\,+\\, [b_{\\rho m s} \\rho^2_{\\bf K}\n                     ({\\bf m} \\cdot {\\bf S}_{\\bf K}) + {\\rm c.c.}] \\;,\n\\end{eqnarray}\nwith $|{\\bf S}_{\\bf K}|^2=({\\bf S}_{\\bf K} \\cdot {\\bf S}^*_{\\bf K})$.\nNotice that third order invariants are present due to commensurability,\n$3{\\bf K}={\\bf G}$ (reciprocal lattice vector). Therefore, first order\ntransitions are generally possible.\\cite{Toledano}\n \nThis expansion suggests a number of additional comments:\n{\\it i\\\/}) A CDW can occur without accompanying magnetism, i.e.,  \n$\\rho_{\\bf K}\\ne 0$, while ${\\bf m}=0$ and ${\\bf S}_{\\bf K}=0$.\nThis is the case, as we shall see later, for the small $U$ region of the HF\nphase diagram.\n{\\it ii\\\/}) The possible SDW phases are either collinear (l-SDW)\n(for which $({\\bf S}_{\\bf K} \\times {\\bf S}^*_{\\bf K})=0$) or\ncoplanar.\\cite{Kivelson}\nThe latter have, with a suitable choice of the phases,\n${\\bf S}^x_{\\bf K}=|{\\bf S}_{\\bf K}|\\cos{\\alpha}$ and\n${\\bf S}^y_{\\bf K}=-i |{\\bf S}_{\\bf K}|\\sin{\\alpha}$,\nand can be generally described as a spiral SDW (s-SDW)\n\\begin{equation}\n\\langle {\\bf S}_{\\bf r} \\rangle \\,=\\, 2 |{\\bf S}_{\\bf K}| \\,\n[ {\\hat{\\bf x}} \\cos{\\alpha} \\, \\cos{({\\bf K}\\cdot {\\bf r})} -\n  {\\hat{\\bf y}} \\sin{\\alpha} \\, \\sin{({\\bf K}\\cdot {\\bf r})} ] \\;,\n\\end{equation}\nwith an eccentricity parameter $\\alpha\\ne 0,\\pi\/2$.\n($\\alpha=0$ or $\\pi\/2$ are actually l-SDW along the ${\\hat{\\bf x}}$\nor ${\\hat{\\bf y}}$ directions.)\n$\\alpha=\\pi\/4$ describes a circular spiral SDW.\nNow, the only possibility of having a SDW without CDW is via a circular spiral\nSDW ($\\alpha=\\pi\/4$). Indeed, the third order invariant\n$[B_{\\rho s} \\rho_{\\bf K} ({\\bf S}_{\\bf K} \\cdot {\\bf S}_{\\bf K})+{\\rm c.c.}]$\nvanishes by symmetry only for a circular spiral SDW, for which\n$({\\bf S}_{\\bf K} \\cdot {\\bf S}_{\\bf K})=0$; in all other cases, a SDW\nimplies -- if $B_{\\rho s}\\ne 0$ -- a CDW as well.\n{\\it iii\\\/}) The simultaneous presence of a SDW and a CDW implies, generally, \na finite magnetization $\\bf m$, via the fourth order invariant\n$[b_{\\rho m s} \\rho^2_{\\bf K} ({\\bf m} \\cdot {\\bf S}_{\\bf K})+{\\rm c.c.}]$,\nunless the phases of $\\rho$ and $S$ are such that \n$2\\phi_{\\rho}+\\phi_{\\sigma}=\\pi\/2+n\\pi$. \nThis happens in phase E of our phase diagram, which has therefore no \nuniform magnetization.\n{\\it iv\\\/}) The presence of a SDW leads, generally, to a finite uniform\nmagnetization as well, via the fourth order invariant\n$[b_{m s}^{(3)} ({\\bf m} \\cdot {\\bf S}_{\\bf K})\n({\\bf S}_{\\bf K} \\cdot {\\bf S}_{\\bf K}) + {\\rm c.c.}]$,\nunless the phase $\\phi_{\\sigma}$ is such that\n$3\\phi_{\\sigma}=\\pi\/2+m\\pi$.\n\n\\subsection{Phase diagram in the Hartree-Fock approximation} \\label{HF_pd:sec}\n\nWe present a brief summary of the mean-field HF calculations for \narbitrary $U$, $V$, and $g$, obtained by solving numerically\nthe self-consistent problem in Eqs.\\ \\ref{m:eqn}-\\ref{AB:eqn}.\nThe main phases present in the HF phase diagram are shown in \nFig.\\ \\ref{hf_pd:fig} for the case of $g=0$. \nThe effect of $g\\ne 0$ will be discussed further below.\n  \n{\\bf Phase A: Spiral SDW insulating phase.}\nThe circular spiral SDW (phase A) dominates the large $U$, small $V$ part of \nthe phase diagram, as expected from the Heisenberg model mapping at \n$U\\to\\infty$ (see sect.\\ \\ref{large_u:sec}). \nThis is the Mott insulator phase, probably relevant for SiC.\nIts HF bands are shown in Fig.\\ \\ref{hf_bands:fig}(a).\n\n{\\bf Phase A': Collinear SDW with $m^z=1\/3$ insulating phase.}\nThis is another solution of the HF equations in the large $U$, \nsmall $V$ region. \nIt is an insulating state characterized by a linear l-SDW plus a small \nCDW with $\\phi_{\\rho}=0$, accompanied by a magnetization $m^z=1\/3$ (phase A').\nThis collinear state lies above the s-SDW by only a small energy difference \n(of order $0.03 t_1$ per site), and could be stabilized by other factors\n(e.g., spin-orbit). \nA recent LSDA calculation for $\\sqrt{3}$-Si\/Si(111) has indicated this\nl-SDW as the ground state, at least if spins are forced to be \ncollinear.\\cite{Sandro}\nThe HF bands for this solution are shown in Fig.\\ \\ref{hf_bands:fig}(b), \nand are very similar to the LSDA surface band for Si\/Si(111). \nThe phase $\\phi_{\\rho}=0$ of the CDW order parameter corresponds \nto a real-space charge distribution in which one sublattice has a charge \n$1+2|\\rho_{\\bf K}|$, while the remaining two are equivalent and have \ncharges $1-|\\rho_{\\bf K}|$, compatible with the experimental findings on \nSn\/Ge(111) and Pb\/Ge(111). \nThe amplitude $|\\rho_{\\bf K}|$ of the CDW is in general quite small in this \nphase. \nIt should be noted, however, that a STM map is not simply a direct measure \nof the total charge density.\\cite{Selloni,Tosatti:high} \nThis will be discussed in sect.\\ \\ref{stm:sec}.\n\n{\\bf Phase B': Asymmetric CDW with $m^z=1\/3$ insulating phase.}\nBy increasing the n.n.\\ repulsion $V$, the energies of the s-SDW \nand of the l-SDW tend to approach, \nuntil they cross at a critical value $V_c$ of $V$. \nAt $U\/t_1=10$ we find $V_c\/t_1\\approx 3.3$ for model A, $V_c\/t_1\\approx 6.6$ \nfor model B.\nAs $V>V_c$, however, an insulating asymmetric CDW (a-CDW) prevails. \nThis is simply the spin collinear version of the non-collinear phase\ndescribed in Sect.\\ \\ref{large_v:sec}. \nFig.\\ \\ref{e_cdw:fig} shows the energy per site of the most relevant\nHF solutions at $U\/t_1=10$ as a function of $V$ for model B (Coulomb tail case).\nThe s-SDW and the l-SDW cross at $V_c\\approx 6.6 t_1$ where,\nhowever, the a-CDW insulating solution starts to be the favored one.\nThis large-$V$ solution has a large $CDW$ order parameter\nwith $\\phi_{\\rho}\\ne 0$ (mod. $2\\pi\/3$), a concomitant l-SDW, and $m^z=1\/3$.\nBy recalling the discussion in sect.\\ \\ref{large_v:sec}, we notice that\na state with a magnetization $m^z=1\/3$ and a l-SDW is the best HF solution once\na $3\\times 3$ restriction has been applied, since a spiral SDW on the\nsingly occupied sublattice would involve a larger periodicity (phase B).\n\n{\\bf Phase D: Symmetric non-magnetic CDW metallic phase.}\nFor small values of $U$ and $V$, or for large enough e-ph coupling\n$g$, a {\\em metallic\\\/} CDW with $\\phi_{\\rho}=0$ (m-CDW) is found. \n(See Fig.\\ \\ref{hf_bands:fig}(c) for the HF bands.)\nThis phase constitutes a candidate, alternative to the magnetic phase B', \nand compatible with the main experimental facts, which \nmight be relevant for the case of Pb\/Ge(111) and of Sn\/Ge(111). \nThe degree of metallicity of this phase is much reduced relative to the \nundistorted surface (pseudo-gap). \n\nWe stress that the e-ph interaction can stabilize the \n$\\phi_{\\rho}=0$ m-CDW also at relatively large $U$, by countering $U$ with a\nlarge negative $\\Delta U=-g^2\/M\\omega^2_{\\bf K}$.  \nWe demonstrate this in Fig.\\ \\ref{e_u8v2_ph:fig}, where we plot the energy\nper site as a function of $\\Delta U$ at $U\/t_1=8$ and $V\/t_1=2$, for the \nthree relevant HF solutions, i.e., the spiral SDW (phase A), the collinear\nSDW with $m^z=1\/3$ (phase A'), and the metallic non-magnetic CDW (phase D). \nThe spiral SDW is unaffected by the electron-phonon coupling. \nThe energy of the collinear SDW with $m^z=1\/3$ improves a little bit \nby increasing $g$, due to the small CDW amplitude of this phase. This effect\nis not large enough as to make this phase stable in any range of couplings.\nAt a critical value of $g$, the metallic non-magnetic CDW  \n(where the CDW order parameter is large, $|\\rho_{\\bf K}| \\sim 0.5$) wins over\nthe magnetic phases. \nThe Fourier transform of the lattice distortion at ${\\bf K}$ is given by \n$\\langle z_{\\bf K}\\rangle=(g\/M\\omega_{\\bf K}^2)\\rho_{\\bf K}\n=\\rho_{bf K} |\\Delta U|\/g$.\n\nA rough estimate shows that the order of magnitude\nof the electron-phonon coupling necessary to stabilize the CDW phase is not\nunreasonable.\nWith $g=1$ eV\/$\\AA$, $M_{Si}=28$, and $\\omega_{\\bf K}\\approx 30$ meV\nwe get $\\Delta U \\approx -3 t_1$, sufficient to switch from a \ns-SDW ground state to a m-CDW for $U\/t_1=8$ and $V\/t_1=2$. \nWith these values of the parameters we have $|\\rho_{\\bf K}| \\approx 0.43$, and\nwe estimate $|\\langle z_{\\bf K}\\rangle| \\approx 0.07 \\AA$. This corresponds,\nsince $\\langle z_{\\bf r} \\rangle \\sim 2\\cos({\\bf K}\\cdot {\\bf r}) \n|\\langle z_{\\bf K}\\rangle|$, to a total displacement between the adatom\ngoing up and the two going down of $\\Delta z = 3\n|\\langle z_{\\bf K}\\rangle| \\approx 0.2\\AA$. \n\nWe notice that values of $g$ much larger than those used in \nFig.\\ \\ref{e_u8v2_ph:fig} would eventually stabilize a superconducting \nground state (see Appendix).\n\n\\section{CDW order parameter and STM experiments} \\label{stm:sec}\n\nWe discuss, in the present section, the relationship between the\nCDW order parameter, as defined in Eq.\\ \\ref{rho:eqn}, and an STM map\nof the surface. \nAs the crudest approximation to the tunneling current for a given bias \n$V_{\\rm bias}$ we consider the integral of the charge density for \none-electron states within $V_{\\rm bias}$ from the Fermi level, \nweighted with barrier tunneling factor $T(V)$,\\cite{Selloni,Tosatti:high}\n\\begin{equation}\n\\label{stm:eqn}\nJ(V_{\\rm bias},{\\bf r}=x,y;z) \\approx \\int_0^{V_{\\rm bias}} dV  \\sum_{n{\\bf k}} \n|\\Psi_{n{\\bf k}}({\\bf r})|^2 \\delta (E_{n{\\bf k}}-E_F+V) T(V)  \\;.\n\\end{equation}\nThe tunneling factor leads to weighting prominently the states\nimmediately close to the Fermi level. \nIn view of the purely qualitative value of Eq.\\ (\\ref{stm:eqn}), \nwe have moreover decided to ignore $T(V)$ altogether and to account for \nits effect by reducing the bias voltage\n$V_{\\rm eff}$ in Eq.\\ (\\ref{stm:eqn}), to an effective value \n$V_{\\rm bias}^{\\rm eff}$. \nBy doing this, we have extracted an ``STM map'' for a point in phase A'\n($U\/t_1=9$ and $V=2$, model A)\n-- a spin-density waves where the amplitude of the CDW order parameter is \nrather small, $|\\rho_{\\bf K}|=0.039$ --\nand a point in phase D ($U\/t_1=4$ and $V=2$, model A) \n-- a pure CDW where the order parameter is quite large,\n$|\\rho_{\\bf K}|=0.4$. \nThe results for constant $z$, and $x,y$ moving from adatom A to B to C,\nare shown in Fig.\\ \\ref{stm:fig}(a) and (b), for the two cases. \nThe solid curves refer to positive bias (current flowing from the sample\nto the tip), probing occupied states close to the Fermi level. \nThe dashed curve refers to negative bias, probing unoccupied states.\nIn both cases a) and b), one of the three atoms yields a larger current at \npositive bias, while the other two atoms have larger currents at negative bias. \nThe insets show the predicted ``contrast'' between the two peak values,\n$(J_1-J_2)\/(J_1+J_2)$, $J_1$ and $J_2$ being in each case, respectively, \nthe largest and the smallest of the STM peak currents at the positions of \nthe adatoms.\nWe notice the following points: \ni) for the occupied states (positive bias) the pure CDW phase has, as expected,\na larger contrast than the magnetic phase. \nAs we neglect the tunneling factors $T(V)$, in the limit of large positive\neffective bias we recover the total asymmetry in the charge of the two \ninequivalent atoms, $(n_1-n_2)\/(n_1+n_2)$, indicated by a dashed horizontal \nline in the insets.  \nObserve that the way this large bias limit is reached is completely different \nfor the two cases a) and b): in the magnetic case a) the contrast overshoots \nat small biases attaining values substantially larger than the nominal CDW order\nparameter, and then goes to the limit $(n_1-n_2)\/(n_1+n_2)$ from above;\nin the pure CDW case b), on the contrary, the limit is reached monotonically\nfrom below. \nii) for empty states (negative bias) the contrast is even more surprising: at\nsmall bias it is very large in both cases a) and b). By increasing\nthe bias, the contrast for the pure CDW case tends monotonically to a large \nvalue, whereas the magnetic case shows a strong non monotonicity. \n\nThese results suggest that one should look more carefully, and quantitatively,\nat the behavior of the asymmetry between STM peak currents as a function\nof the bias, including the region of relatively small biases: the different\nbehavior of the asymmetry of the magnetic case versus the pure CDW case should \nbe marked enough -- and survive in a more refined analysis including $T(V)$ --\nas to make the STM map a good way of discriminating between the two scenarios. \n\n\\section{Discussion and conclusions} \\label{discussion:sec}\n\nWithin our model study we have learned that on the surfaces considered: \n\n(i) If $U$ and $V$ are ignored, there is no straight electron-phonon driven  \n$3\\times 3$ CDW surface instability. \nHowever, any phase involving a CDW, for example as\na secondary order parameter attached to a primary SDW, can take\nadvantage and gain some extra stabilization energy from a small surface\nlattice distortion, via electron-phonon coupling.\n\n(ii) Electron-electron repulsion and the two-dimensional Fermi Surface\nare capable of driving transitions of the undistorted metallic surface \nto a variety of states, that are either insulating or in any case less \nmetallic, some possessing the $3\\times 3$ periodicity. \n\n(iii) This can occur via two different mechanisms: a) the inter-site\nrepulsion $V$ can stabilize insulating or semi-metallic CDWs, without \na crucial involvement of spin degrees of freedom; b) the on-site \nrepulsion $U$ can produce essentially magnetic insulators with or without \na weak accompanying $3\\times 3$ CDW, as required by symmetry.\n\n(iv) For $U$ moderate of order $W$ and for smaller $V$, an interesting state \nis realized, with a large SDW and a small accompanying CDW. The state \nis either a small-gap insulator, or a semi-metal, and may or may not \nbe associated with a net overall magnetization, depending on the nature \n(linear or spiral, respectively) of the leading SDW. \n\n(v) For $U$ and $V$ both small but finite, a metallic CDW without any magnetism\nis obtained. The same phase can also be stabilized for larger values of $U$\nby the presence of a substantial electron-phonon coupling. \nWe stress that, in this case, $V$ is the coupling responsible for the\n$3\\times 3$ symmetry of the unit cell, whereas the role of the electron-phonon \ncoupling is that of destroying magnetism by effectively decreasing $U$.\nElectron-phonon coupling alone is not sufficient to justify a commensurate\n$3\\times 3$ CDW.\n\n(vi) Either of the phases in (iv) or (v) could be natural candidates for \nexplaining the weak $3\\times 3$ CDW seen experimentally on Sn-Pb\/Ge(111). \n\n(vii) Finally, for large $U$, small $V$ (in comparison with the bandwidth $W$)\nthe  Mott-Hubbard state prevails. It is a wide-gap insulator, with\na pure spiral SDW, with $3\\times 3$ overall periodicity, and coplanar \n$120^o$ long-range spin ordering at zero temperature. It possesses\nno net magnetization, and no accompanying CDW. \n\n(viii) The above is the kind of state which we expect to be realized on \nSiC(0001), and also possibly on K\/Si(111):B. \n\nAmong existing experiments, we have addressed particularly\nphotoemission and STM. Our calculated band structure for both the SDW\/CDW\nstate A' (iv) and the pure CDW state D (v) exhibit features which are similar \nto those found in photoemission from \nSn-Pb\/Ge(111).\\cite{Modesti_2,Avila,LeLay,asensio}  \nThe simulated STM images for the two kind of states are predicted to differ\nin their voltage dependence. \n\nFuture experiments are strongly called for, aimed at detecting whether\nmagnetic correlations are actually dominant, as we think is very likely, \non all these surfaces, or whether Sn-Pb\/Ge(111) are instead non-magnetic\nand electron-phonon driven. \nThe issue of whether magnetic long-range order -- which we definitely propose\nfor SiC(0001) and K\/Si(111):B at $T=0$, and also hypothesize for \nSn-Pb\/Ge(111) -- survives up to finite temperatures is one which we \ncannot settle at this moment. This due to the difficulty\nin estimating the surface magnetic anisotropy, without which order\nwould of course be washed out by temperature. In any case, it should be\npossible to pursue the possibility of either magnetism or incipient\nmagnetism using the appropriate spectroscopic tools.\n\nThis line of experimental research, although undoubtedly difficult, should \nbe very exciting since it might lead to the unprecedented discovery of \nmagnetic states at surfaces possessing no transition metal ions of any kind, \nsuch as these seemingly innocent semiconductor surfaces.  \n\nWe acknowledge financial support from INFM, through projects LOTUS and\nHTSC, and from EU, through ERBCHRXCT940438. \nWe thank S. Modesti, J. Lorenzana, M.C. Asensio, J. Avila, G. Le Lay, \nE.W. Plummer and his collaborators, for discussions.\n\n\\section{Appendix. Large negative $U$: a superconducting ground state.}\n\\label{neg_u:sec}\nThe limit of large negative $U$, $U\\to -\\infty$, is considered here\nto show that CDWs are not favored by on-site attraction alone. Instead,\na superconducting ground state is favored.\\cite{santos}\nTo see this, consider the real-space states which are the low energy\nconfigurations for $U\\to -\\infty$: they consist of $N_e\/2$ sites (if $N_e$\nis the number of electrons) each of which is occupied by a pair of electrons\nwith opposite spins.\nThe large degeneracy in this manifold of states is\n-- once again, like in the $U\\to \\infty$ case -- removed by kinetic energy\nin second order perturbation theory. By assigning a pseudo-spin-1\/2 state\nto each site (up, if occupied by a pair, down if empty) one can show that\nthe effective Hamiltonian is \\cite{santos}\n\\begin{equation}\nH_{\\rm eff} = -\\sum_{(ij)} \\frac{J^{\\perp}_{ij}}{2} \\,\n\\left( S^+_{{\\bf r}_i} S^-_{{\\bf r}_j} \\,+\\, {\\rm H.c.} \\right) \\,+\\,\n\\sum_{(ij)} J^{z}_{ij} \\, S^z_{{\\bf r}_i} S^z_{{\\bf r}_j} \\;,\n\\end{equation}\nwith $J^{\\perp}_{ij}=4|t_{ij}|^2\/|U|$ and $J^{z}_{ij}=J^{\\perp}_{ij}$.\nIf $V$-terms are added, $J^z$ is modified to\n$J^{z}_{ij}=J^{\\perp}_{ij}+4V_{ij}$.\nRestricting our consideration to the n.n. case, we are left with a\nn.n.\\ Heisenberg Hamiltonian with ferromagnetic xy-part and an \nantiferromagnetic z-part. The sign of the xy-part cannot be changed at will\nby a canonical transformation because the lattice is non-bipartite. \nThe result is that the order is in the plane (i.e., superconductivity wins)\nfor small $V$. Only if $V$ is large enough the CDW \n(i.e., order in the z-direction) will be favored. \n\nEntirely similar considerations apply to the case of strong electron-phonon\ncoupling, $g\\to \\infty$.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nIt has been difficult to read the recent financial news without finding mention of Collateralized Debt Obligations (CDO's).  These financial instruments\nprovide ways of aggregating risk from a large number of sources and reselling\nit in a number of parts, each part having different risk-reward characteristics.\nNotwithstanding the role of CDO's in the recent market meltdown, the near\nfuture will no doubt see the financial\nengineering community continuing to develop structured investment\nvehicles like CDO's.  Unfortunately, computational challenges in this area are formidable.\nThe main types of these assets have several common problematic features:\n\\begin{itemize}\n\\item they pool a large number of assets\n\\item they tranche the losses.\n\\end{itemize}\nThe ``problematic'' nature of this combination is that the trancheing procedure is nonlinear;\nand as is usual, the effect of a nonlinear transformation on a high-dimensional\nsystem is often difficult to understand.  Ideally, one would like a theory which gives,\nif not explicit answers, at least some guidance.  Lacking theory, one is\noften forced to search for models which are computationally\nfeasible, structurally robust, and which can be reasonably well-fitted to data.\n\nWe here consider a \\emph{large deviations} (cf. \\cite{MR1739680, MR1619036, MR758258}) analysis of certain aspects of synthetic CDO's.\nThe theory of large deviations is a collection of ideas which are\noften useful in studying rare events.  The rare events of interest here involve losses in\n(and hence pricing of)\ninvestment-grade (senior or super-senior) tranches of synthetic CDO's.\nWe would like to see how far we can take a rigorous analysis when we use\nmathematical tools, viz., large deviations, which are designed expressly to study rare events.\nThe theory of large deviations usually gives a very refined\nanalysis of rare events (more refined, for example, than one based on mean-variance calculations); what does this analysis look like for CDO's?\n\nIn the course of our analysis, we will see that\nlarge deviations theory provides a natural framework for studying large amounts\nof idiosyncratic randomness.  Moreover, the theory of large deviations\nprovides a way to compare rare events and see how they transform.\nWe believe this to be an important component of a larger\nanalysis of CDO's, particularly in cases where correlation comes from only a few sources (we will pursue a simple form of this idea in Subsection \\ref{S:Correlated}).  In a sequel to this paper we will consider the more challenging case of a heterogeneous pool of assets.\n\n\nThis is not the first attempt to apply large deviations to structured finance.\nLosses in pools of large assets like CDO's have been considered in \\cite{MR2022976}, \\cite{Glasserman}\\footnote{Glasserman in \\cite{Glasserman} makes important headway in understanding correlation.}, and \\cite{MR2384674} (see also \\cite{MR1644496} for another application of large deviations to finance).  Moreover, effects of tranching have been considered in\n\\cite{Vielex} and \\cite{YangHurdZhang}, both of which discuss saddlepoint\neffects of tranching once the distribution of the loss process is known.\nOur interest is to identify, as much as possible, exact asymptotic formulae\nfor the price of the CDO by focussing on the effects of large amounts\nof idiosyncratic randomness.  We find that if we interpret\nthe loss process as an occupation measure, \\emph{Sanov's theorem}\nsuggests how to proceed.\nFurthermore, it allows us to develop something of a bottom-up analysis\nwhich directly connects the CDO price to the default probababilities of the underlying bonds.  It also naturally leads to a number of calculations which reflect the\n\\emph{dynamics} of the default probabilities (as opposed\nto a snapshot of the default probabilities at expiry).\n\nFinally, the \\emph{ab initio} nature of our calculations bears note\\footnote{See in particular Remark \\ref{R:Comments} and the comments at the beginning of Section \\ref{S:HAS}.}.\nA number of models, such as the generalized Poission loss model \\cite{Brigo}, the Hawkes process \\cite{Giesecke} and others\n(cf. \\cite{Caflisch, FilOverSch}), which successfully capture some of the complexity of CDO's have been developed and implemented.\nOur approach is limited to investment-grade tranches, and hopefully will complement some of these models and contribute to their study.\n\n\\section{CDS to CDO---a Review}\nA standard review of credit default swaps and synthetic CDO's will help us fix notation, which comes from \\cite{Brigo}.\nLet's fix underlying probability triple $(\\Omega,\\mathscr{F},\\mathbb{P})$, where $\\mathbb{P}$ represents the risk-neutral probability measure and $\\mathbb{E}$ is the associated expectation operator..\n\n\\subsection{Credit Default Swaps}\\label{S:CDS}\nA Credit Default Swap (CDS) is a contract between a \\emph{protection seller}\nand a \\emph{protection buyer} based on the default of a reference bond\n(a \\emph{name}).  Under the contract, the protection seller pays the protection buyer $\\$1$\n(the \\emph{notional}) when the bond defaults\\footnote{We assume for simplicity no recovery.} (a nonnegative random time $\\tau$), as long as this\ndefault occurs before\\footnote{We require default to be \\emph{strictly} before expiry; that will save us some calculations resulting from potentially positive probability of default exactly at expiry.} the expiry of the contract (time $T$).  This is the\n\\emph{protection leg} of the contract.  In return, the protection buyer pays the\nprotection seller a premium $S$ at a finite collection $\\mathcal{T}$ of times (such that $t\\le T$ for all $t\\in \\mathcal{T}$ until\nthe default occurs.  This is the \\emph{premium} leg of the contract; see Figure \\ref{fig:CDS}.\n\\begin{figure}\n\\includegraphics[scale=0.8]{cds.pdf}\n\\caption{Credit Default Swap}\n\\label{fig:CDS}\n\\end{figure}\nTo write this mathematically, define the loss process\n\\begin{equation*} L^\\circ_t \\overset{\\text{def}}{=} \\chi_{\\{t\\ge \\tau\\}} = \\begin{cases} 1 &\\text{if $t\\ge \\tau$} \\\\ 0 &\\text{if $t<\\tau$}\\end{cases}\\end{equation*}\nfor all $t\\in \\mathbb{R}$ (of course then $L^\\circ_t=0$ for $t<0$).\nThe present value of the protection and premium legs are thus\n\\begin{gather*}e^{-\\textsf{R} \\tau}\\chi_{\\{\\tau< T\\}} = \\int_{s\\in [0,T)} e^{-\\textsf{R} s}dL^\\circ_s \\\\\nS\\sum_{t\\in \\mathcal{T}} e^{-\\textsf{R} t}\\chi_{\\{\\tau>t\\}}= S\\sum_{t\\in \\mathcal{T}}e^{-\\textsf{R} t}\\{1-\\chi_{\\{\\tau\\le t\\}}\\}= S\\sum_{t\\in \\mathcal{T}} e^{-\\textsf{R} t}\\left(1-L^\\circ_t\\right) \\end{gather*}\nwhere $\\textsf{R}$ is the riskless interest rate\\footnote{It is not difficult to see that\nthe maps $\\omega\\mapsto e^{-\\textsf{R} \\tau(\\omega)}\\chi_{\\{\\tau(\\omega)\\le T\\}}$ and $\\omega\\mapsto \\sum_{t\\in \\mathcal{T}} e^{-\\textsf{R} t}\\chi_{\\{\\tau(\\omega)>t\\}}$ are measurable maps from $\\Omega$ to $\\mathbb{R}$; thus the expectations make sense.}.  The value of $S$ is defined by requiring that the expectation of these two legs agree (under the risk-neutral measure).\n\n\\subsection{Synthetic CDO's}\n\nIt is an easy step to modify this notation to construct a synthetic CDO.  Consider $N$ credit default swaps (each one on a different name).  Each CDS has notional value $1\/N$, and the default of the $n$-th name occurs at a random nonnegative time $\\tau_n$.  The \\emph{notional} loss process is thus\n\\begin{equation*} L^{(N)}_t \\overset{\\text{def}}{=} \\frac{1}{N}\\sum_{n=1}^N \\chi_{\\{\\tau_n\\le t\\}} \\end{equation*}\nfor all $t\\in \\mathbb{R}$ (as in our above discussion of credit default swaps, $L^{(N)}_t=0$ for $t<0$).\nNote that $0\\le L^{(N)}_t\\le 1$ for all $t\\ge 0$.  Fix \\emph{attachment} and \\emph{detachment} points $\\alpha$ and $\\beta$ in $[0,1]$ such that $\\alpha<\\beta$.  We then define the \\emph{tranched} loss process $\\bar L^{(N)}$ as\n\\begin{equation*} \\bar L^{(N)}_t \\overset{\\text{def}}{=} \\frac{(L^{(N)}_t-\\alpha)^+-(L^{(N)}_t-\\beta)^+}{\\beta-\\alpha} = \\begin{cases} 0 &\\text{if $L^{(N)}_t<\\alpha$} \\\\\n\\frac{L^{(N)}_t-\\alpha}{\\beta-\\alpha} &\\text{if $\\alpha\\le L^{(N)}_t\\le \\beta$} \\\\\n1 &\\text{if $L^{(N)}_t\\ge \\beta$}\\end{cases} \\end{equation*}\nfor all $t\\in \\mathbb{R}$.  The protection and premium legs of a synthetic CDO are basically given by replacing the loss process $L^\\circ$ in a credit default swap with $\\bar L$.\nNamely, define\n\\begin{equation*}\\textbf{P}^{\\text{prot}}_N \\overset{\\text{def}}{=} \\int_{s\\in [0,T)} e^{-\\textsf{R} s}d\\bar L^{(N)}_s \\qquad\\text{and}\\qquad\n\\textbf{P}^{\\text{prem}}_N\\overset{\\text{def}}{=} \\sum_{t\\in \\mathcal{T}} e^{-\\textsf{R} t}\\left(1-\\bar L^{(N)}_t\\right); \\end{equation*}\n$S_N\\textbf{P}^{\\text{prem}}_N$ is the present value of the premium leg (where $S_N$ \nare the premiums) and $\\textbf{P}^{\\text{prot}}_N$ is the present value of the protection leg.\nThe protection leg thus makes payments when defaults occur, as long as at\nleast $\\alpha$ (in percent) of the names have already defaulted, and only\nas long as no more than $\\beta$ (in percent) of the names have defaulted.\nThese payments are proportioned so that they add up to at most $\\$1$.  The\npremium payments, on the other hand, are made only on the proportion\nof names which are still insured (i.e., which have not yet defaulted).\nThe premium $S_N$ should then be given by equating the risk-neutral expectation of two legs; i.e.,\n\\begin{equation}\\label{E:SN} S_N= \\frac{\\mathbb{E}[\\textbf{P}^{\\text{prot}}_N]}{\\mathbb{E}[\\textbf{P}^{\\text{prem}}_N]}. \\end{equation}\nNote that $L^{(N)}_t$ is measurable for each $t\\in \\mathbb{R}$.  Since $\\bar L^{(N)}_t$ is a continuous transformation of $L^{(N)}_t$, it is also measurable. Since $0\\le e^{-\\textsf{R} s}\\le 1$, $0\\le \\bar L\\le 1$, and $\\bar L$ is nondecreasing,  $\\textbf{P}^{\\text{prot}}_N$ and $\\textbf{P}^{\\text{prem}}_N$\nboth take values in $[0,1]$.  Moreover, the measurability of $\\bar L$ implies that $\\textbf{P}^{\\text{prot}}_N$ and $\\textbf{P}^{\\text{prem}}_N$ are measurable.  Thus both $\\mathbb{E}[\\textbf{P}^{\\text{prot}}_N]$ and $\\mathbb{E}[\\textbf{P}^{\\text{prem}}_N]$ are well-defined, finite, and nonnegative.\nOur goal is to evaluate $S_N$ when $N$ is large.  This will be accomplished in \\eqref{E:Sas}.\n\\begin{figure}\n\\includegraphics[scale=0.7]{lossprocesses.pdf}\n\\caption{Loss processes $L^{(N)}$ and $\\bar L^{(N)}$}\n\\label{fig:typ}\n\\end{figure}\n\n\\section{The Model}\\label{S:SM}\n\nLet's now think about the sources of randomness in the names.  Each name is affected by its own \\emph{idiosyncratic} randomness and by \\emph{systemic} randomness (which affects all of the names).  Assumedly, the systemic randomness,\nwhich corresponds to macroeconomic factors, is \\emph{low-dimensional}\ncompared to the number of names.  For example, there may be only a handful of macroeconomic factors which affect a pool of many thousands of names.  We can capture this functionality as\n\\begin{equation}\\label{E:structural} \\chi_{\\{\\tau_n<T\\}} = \\chi_A(\\xi^\\text{I}_n,\\xi^\\text{S}) \\end{equation}\nwhere the $\\{\\xi^\\text{I}_n\\}_{n\\in \\mathbb{N}}$ and $\\xi^\\text{S}$ are\nall independent random variables, and $A$ is some appropriate set in the product space of the sets where the $\\xi^\\text{I}_n$'s and $\\xi^\\text{S}$ take values.\nSince we want the defaults to be identically distributed, we may furthermore assume that the $\\xi^\\text{I}_n$'s are identically distributed.\n\nOur interest is to understand the implications of the structural model\n\\eqref{E:structural}.  We are not so much concerned with specific models for the $\\xi^\\text{I}_n$'s, the $\\xi^\\text{S}$, or the set $A$ but rather the structure of the rare losses in the investment-grade tranches.\nWe would also like to avoid, as much as possible, a detailed analysis\nof the parts of \\eqref{E:structural} since in practice what we have available\nto carry out pricing calculations is the price of credit default swaps for\nthe individual names; i.e. (after a transformation), $\\mathbb{P}\\{\\tau_N<T\\}$.\nThus we can't with certainty get our hands on the details of \\eqref{E:structural}.  There may in fact be several models of the type \\eqref{E:structural}\nwhich lead to the same ``price'' for the rare events involved in an\ninvestment-grade tranche.  If we can understand more about the structure\nof rare events in these tranches, we can understand which aspects of\n\\eqref{E:structural} are important (and then try to calibrate specific models\nusing that insight).\n\nRegardless of the details of \\eqref{E:structural}, we can make some headway.\nThe notional loss at time $T-$ will be given by\n\\begin{equation*} L^{(N)}_{T-} = \\frac{1}{N}\\sum_{n=1}^N \\chi_A(\\xi^\\text{I}_n,\\xi^\\text{S}). \\end{equation*}\nThe definition of an investment-grade tranche is that $\\mathbb{P}\\left\\{ L^{(N)}_{T-}>\\alpha\\right\\}$ is small.  Guided by Chebychev's inequality, lets' define\n\\begin{equation*} \\mu^{(N)} \\overset{\\text{def}}{=} \\frac{1}{N}\\sum_{n=1}^N \\mathbb{E}\\left[\\chi_A(\\xi^\\text{I}_n,\\xi^\\text{S})\\right] \\qquad \\text{and}\\qquad \\sigma^{(N)} \\overset{\\text{def}}{=} \\sqrt{\\mathbb{E}\\left[\\left(L^{(N)}_{T-}-\\mu^{(N)}\\right)^2\\right]}. \\end{equation*}\nIf $\\alpha>\\mu^{(N)}$, Chebychev's inequality gives us that\n\\begin{equation*} \\mathbb{P}\\left\\{ L^{(N)}_{T-}>\\alpha\\right\\} \\le \\frac{\\left(\\sigma^{(N)}\\right)^2}{\\left(\\alpha-\\mu^{(N)}\\right)^2}. \\end{equation*}\nIn order for this to be small, we would like that $\\sigma^{(N)}$ be small;\nthis is the point of pooling.  For any fixed value of $x$, the conditional\nlaw of $L^{(N)}_{T-}$ given that $\\xi^\\text{S}=x$ is the variance of \n$\\tfrac{1}{N}\\sum_{n=1}^N\\chi_A(\\xi^\\text{I}_n,x)$;\nthus the conditional variance of $L^{(N)}_{T-}$ given that $\\xi^\\text{S}=x$\nis at most of order $\\tfrac{1}{4N}$.  Hopefully, when we reinsert\nthe systemic randomness, the variance of $L^{(N)}$ will still be small,\nand we will indeed have an investment-grade tranche.\n\nIn fact, we can do better than Chebychev's inequality.\nBy again conditioning on $\\xi^\\text{S}$, we can write that\n\\begin{equation*} \\mathbb{P}\\left\\{ L^{(N)}_{T-}>\\alpha\\right\\} = \\mathbb{E}\\left[\\mathbb{P}\\left\\{ L^{(N) }_{T-}>\\alpha\\big|\\xi^\\text{S}\\right\\} \\right] \\end{equation*}\nThus the tranche will be investment-grade if $\\mathbb{P}\\left\\{ L^{(N) }_{T-}>\\alpha\\big|\\xi^\\text{S}=x\\right\\}$ is small for ``most'' values of $x$ (see Remark \\ref{R:systemic}).  As mentioned above,\nhowever, we know the law of $L^{(N)}_{T-}$ conditioned on $\\xi^\\text{S}$.\nNamely, \n\\begin{equation*} \\mathbb{P}\\left\\{ L^{(N) }_{T-}>\\alpha\\big|\\xi^\\text{S}=x\\right\\} = \n\\mathbb{P}\\left\\{ \\frac{1}{N}\\sum_{n=1}^N\\chi_A(\\xi^\\text{I}_n,x)>\\alpha\\right\\}. \\end{equation*}\nThis then clearly motivates a natural two-step approach.\nOur first step is to condition on the value of the systemic randomness \n(which we may think of as fixing a ``state of the world'' or a ``regime'') and concentrate\non how rare events occur due to idiosyncratic randomness (i.e., to effectively \\emph{suppress} the systemic randomness).  It will turn out that\nthis is in itself a fairly involved calculation.  Nevertheless, it is\nconnected with a classic problem in large deviations theory---\\emph{Sanov's theorem}.  With this in hand, we should\nthen be able to return to the original problem and average over the systemic randomness (in Subsection \\ref{S:Correlated}).  Some of the\nfiner details of these effects of correlation will appear in sequels\nto this paper.  Here we will restrict our interest in the effects of correlation\nto a very simple model (which is hopefully nevertheless illustrative).  \n\nDefine $I\\overset{\\text{def}}{=} [0,\\infty]$ and endow $I$ with its usual topology under which it is Polish and its usual ordering\n\\footnote{We endow $I$ with the usual topology and ordering.\n$I$ is the collection of nonnegative real numbers and a non-real ``point'',\nwhich we label as $\\infty$.  Define $\\wp:[0,\\pi\/2]\\to I$ as $\\wp(t) \\overset{\\text{def}}{=} \\tan(t)$ for $t\\in [0,\\pi\/2)$, and define $\\wp(\\pi\/2)\\overset{\\text{def}}{=} \\infty$.  Then $\\wp$\nis a bijection.  The topology\nand ordering of $I$ is that given by pushing the topology and ordering\nof $[0,\\pi\/2]$ forward through $\\wp$.  Thus $I$\nis Polish and in fact compact.}; each of the default times is an $I$-valued random variable.  Since we want to consider a countable\ncollection of default times, we will take\nour event space to be $\\Omega\\overset{\\text{def}}{=} I^\\mathbb{N}$ and\\footnote{As usual, for any topological space $\\mathsf{X}$, $\\mathscr{B}(\\mathsf{X})$ is the Borel sigma-algebra of subsets of $\\mathsf{X}$, and $\\mathscr{P}(\\mathsf{X})$ is the\ncollection of probability measures on $(\\mathsf{X},\\mathscr{B}(\\mathsf{X}))$.} we will take $\\mathscr{F}\\overset{\\text{def}}{=} \\mathscr{B}(I^\\mathbb{N})$.\nFix next $\\mu\\in \\mathscr{P}(I)$; we will want all of the names to be identically distributed with common law $\\mu$.\nTo reflect our initial working assumption\nthat the names are independent, we now let the risk neutral probability $\\mathbb{P}\\in \\mathscr{P}(I^\\mathbb{N})$ be defined\nby requiring that\n\\begin{equation*} \\mathbb{P}\\left(\\bigcap_{n=1}^N\\{\\tau_n\\in A_n\\}\\right) = \\prod_{n=1}^N \\mu(A_n). \\end{equation*}\nfor all $N\\in \\mathbb{N}$ and all $\\{A_n\\}_{n\\in \\mathbb{N}}\\subset \\mathscr{B}(I)$.\nWe also define, in the usual way,\n\\begin{equation*} F(t) \\overset{\\text{def}}{=} \\mu[0,t]. \\qquad t\\in I \\end{equation*}\nIn principle, one can recover $F$ from prices of credit default swaps.\n\n\\begin{example} Our setup includes both the Merton model and the reduced form model.  For the reduced form model, let $\\lambda:(0,\\infty)$ be the hazard rate and set\n\\begin{equation*} f(t) = \\lambda(t)\\exp\\left[-\\int_{s=0}^t \\lambda(s)ds\\right] \\qquad t\\in (0,\\infty)\\end{equation*}\nand let $F$ have density $f$.\nOn the other hand, for the Merton model with stock volatility $\\sigma$, risk-neutral drift $\\theta$, initial valuation $1$, and bankruptcy barrier $K\\in (0,1)$, we would have\n\\begin{equation*} f(t) = \\frac{\\ln (1\/K)}{\\sqrt{2\\pi \\sigma^2 t^3}}\\exp\\left[-\\frac{1}{2\\sigma^2 t}\\left(\\left(\\theta-\\frac{\\sigma^2}{2}\\right)t+\\ln \\frac{1}{K}\\right)^2\\right]. \\qquad t\\in (0,\\infty)\\end{equation*}\nAgain define $F$ by integrating $f$.\n\\end{example}\n\nWe can then rewrite the notional loss process as \n\\begin{equation*} L^{(N)}_t = \\frac{1}{N}\\sum_{n=1}^N\\chi_{[0,t]}(\\tau_n) = \\nu^{(N)}[0,t] \\end{equation*}\nwhere $\\nu^{(N)}$ is empirical distribution of the $\\tau_n$'s; i.e.,\n\\begin{equation}\\label{E:nudef} \\nu^{(N)} = \\frac{1}{N}\\sum_{n=1}^N \\delta_{\\tau_n}. \\end{equation}\nWe point out that $\\nu^{(N)}$ is a random element of $\\mathscr{P}(I)$ (i.e., a random measure\\footnote{Since the map $x\\mapsto \\delta_x$ is a measurable map from $I$ to $\\mathscr{P}(I)$, each map $\\omega\\mapsto \\delta_{\\tau_n(\\omega)}$ is a measurable map from $\\Omega$ to $\\mathscr{P}(I)$.  Thus for each $N$, the map $\\omega\\mapsto (\\delta_{\\tau_1(\\omega)},\\delta_{\\tau_2(\\omega)}\\dots \\delta_{\\tau_N(\\omega)})$ is a measurable map from $\\Omega$ to $(\\mathscr{P}(I))^N$.  Recalling the definition of the weak\ntopology as integration against continuous bounded functions, we then see that\nthe map $(\\mu_1,\\mu_2\\dots \\mu_N)\\mapsto \\frac{1}{N}\\sum_{n=1}^N \\mu_n$ is continuous and thus measurable as a map from $(\\mathscr{P}(I))^N$ to $\\mathscr{P}(I)$.  Hence $\\nu^{(N)}$ is indeed a $\\mathscr{P}(I)$-valued random variable.}).  This formulation\nis the starting point for our analysis and will lead to several insights.  In particular, the (weak) law of large numbers implies that for each $t>0$, \n\\begin{equation} \\label{E:convprob}\\lim_{N\\to \\infty}L^{(N)}_t=F(t). \\qquad \\text{(in probability)}\\end{equation}\nMore generally, $\\nu^{(N)}$ tends to $\\mu$ (in the Prohorov topology on $\\mathscr{P}(I)$); for every $\\varepsilon>0$,\n\\begin{equation*} \\lim_{N\\nearrow \\infty}\\mathbb{P}\\left\\{d_{\\mathscr{P}(I)}(\\nu^{(N)},\\mu)\\ge \\varepsilon\\right\\}=0. \\end{equation*}\nwhere $d_{\\mathscr{P}(\\mathbb{R}_+)}$ is the Prohorov metric \\cite{MR88a:60130}.\n\nConsider now an investment-grade tranche; i.e., a senior or super-senior tranche.  The attachment point for such a tranche should be set so that it is unlikely to suffer any defaults; i.e., it is unlikely that \n$\\textbf{P}^{\\text{prot}}_N$ is nonzero.   Clearly \n\\begin{equation}\\label{E:support} \\left\\{\\textbf{P}^{\\text{prot}}_N\\not = 0\\right\\} = \\{L^{(N)}_{T-}>\\alpha\\} = \\{\\nu^{(N)}[0,T)>\\alpha\\}, \\end{equation}\nand comparing this with \\eqref{E:convprob}, we see that a tranche will be investment-grade if and only an obvious requirement holds:\n\\begin{assumption}[Investment-grade]\\label{A:IG} We assume that\n\\begin{equation*} \\alpha>F(T-).\\end{equation*}\n\\end{assumption}\n\\noindent In this case, the valuation of such a tranche should depend in large part on how ``rare'' it is that \n$L^{(N)}_{T-}>\\alpha$.  As $N$ becomes large, \\eqref{E:convprob} means that in fact it becomes less and less likely that $L^{(N)}_{T-}>\\alpha$.  Note also that since $\\alpha<1$, this assumption implies that $F(T-)<1$.  This is natural; if $F(T-)=1$,\nthen all defaults must have occurred before $T$, essentially precluding the possibility of constructing an investment-grade tranche.\n\nCombining our comments after \\eqref{E:SN} about the structure of $\\textbf{P}^{\\text{prot}}_N$ and\n\\eqref{E:support}, we have that\n\\begin{equation}\\label{E:bounded} 0\\le \\textbf{P}^{\\text{prot}}_N\\le \\chi_{\\{\\nu^{(N)}[0,T)>\\alpha\\}}. \\end{equation}\nHence for an investment-grade tranche, $\\mathbb{E}[\\textbf{P}^{\\text{prot}}_N]$ is small if it is unlikely that $\\nu^{(N)}[0,T)>\\alpha$ (in other words, we don't have any competition between ``big'' values of $\\textbf{P}^{\\text{prot}}_N$ and ``small'' sets).\nNote also that \\eqref{E:convprob} implies that $\\lim_{N\\to \\infty}\\bar L^{(N)}_{T-}=0$ (in probability) so that in fact\n\\begin{equation}\\label{E:Premas}\\lim_{N\\to \\infty} \\mathbb{E}[\\textbf{P}^{\\text{prem}}_N] = \\sum_{t\\in \\mathcal{T}} e^{-\\textsf{R} t}. \\end{equation}\nIn other words, if losses are unlikely, all of the premiums will most likely be paid.\nThus the nontrivial part of $S_N$ comes from the protection leg, whose value is small.\n\nLet's now step into the world of large deviations, which tells us how to \nstudy rare events.  The asymptotics of $\\nu^{(N)}$ is exactly the subject of \n\\emph{Sanov's} theorem \\cite{MR1619036}, which states that $\\nu^{(N)}$ has\na large deviations principle with rate function given by relative entropy with respect to $\\mu$; i.e., with rate function\n\\begin{equation*} H(\\mu'|\\mu) = \\begin{cases} \\int_{t\\in I}\\ln \\frac{d\\mu'}{d\\mu}(t)\\mu'(dt) &\\text{if $\\mu'\\ll \\mu$} \\\\\n\\infty &\\text{else.}\\end{cases}\\end{equation*}\nInformally, for any $A\\in \\mathscr{B}(\\mathscr{P}(I))$,\n\\begin{equation}\\label{E:san} \\mathbb{P}\\left\\{\\nu^{(N)}\\in A\\right\\} \\overset{N\\nearrow \\infty}{\\asymp} \\exp\\left[-N\\inf_{\\mu'\\in A}H(\\mu'|\\mu)\\right]. \\end{equation}\nSince large deviations is not in the mainstream of financial mathematics (see, however, \\cite{MR1644496})\nwe have summarized some of its foundations in Subsection \\ref{S:LD}.  Combining \\eqref{E:bounded} with Sanov's theorem, we conjecture that for large $N$\n\\begin{equation*} \\mathbb{E}\\left[\\textbf{P}^{\\text{prot}}_N\\right] \\le \\mathbb{P}\\left\\{\\nu^{(N)}[0,T)>\\alpha\\right\\} \\overset{N\\nearrow \\infty}{\\asymp} \\exp\\left[-N \\mathfrak{I}(\\alpha)\\right] \\end{equation*}\nwhere\n\\begin{equation*} \\mathfrak{I}(\\alpha) \\overset{\\text{def}}{=} \\inf\\left\\{H(\\mu'|\\mu): \\mu'[0,T)\\ge \\alpha\\right\\}. \\end{equation*}\nAlthough this looks intimidating (it is an infinite-dimensional minimization problem), in fact it has an easy solution and an explicit minimizer.\nFor $\\alpha_1$ and $\\alpha_1$ in $[0,1]$, define\n\\begin{equation*} \\hbar(\\alpha_1,\\alpha_2) \\overset{\\text{def}}{=} \\begin{cases} \\alpha_1\\ln \\frac{\\alpha_1}{\\alpha_2} + (1-\\alpha_1)\\ln \\frac{1-\\alpha_1}{1-\\alpha_2} &\\text{for $\\alpha_1$ and $\\alpha_2$ in $(0,1)$} \\\\\n\\ln \\frac{1}{\\alpha_2} &\\text{for $\\alpha_1=1$, $\\alpha_2\\in (0,1)$} \\\\\n\\ln \\frac{1}{1-\\alpha_2} &\\text{for $\\alpha_1=0$, $\\alpha_2\\in [0,1)$} \\\\\n\\infty &\\text{else.}\\end{cases}\\end{equation*}\n\\begin{proposition}\\label{P:explicit} We have that\n\\begin{equation*} \\mathfrak{I}(\\alpha) = \\hbar(\\alpha,F(T-)) = H(\\tilde \\mu^*_\\alpha|\\mu), \\end{equation*}\nwhere\n\\begin{equation}\\label{E:dacc} \\tilde \\mu^*_\\alpha(A) = \\mu(A\\cap [0,T))\\frac{\\alpha}{F(T-)} + \\mu(A\\cap [T,\\infty])\\frac{1-\\alpha}{1-F(T-)} \\end{equation}\nfor all $A\\in \\mathscr{B}(I)$.\\end{proposition}\n\\noindent The proof of this is given Section \\ref{S:Proofs}.\nIn fact, the formula for $\\mathfrak{I}$ is what we would expect from considering only $L^{(N)}_{T-}$.  We can think of $L^{(N)}_{T-}$ as counting the normalized number of\nheads in a collection of i.i.d. coin flips, where the probability of heads\n(i.e., defaults before time $T$)\nfor each coin is $F(T-)$.  The likelihood that the normalized number of heads\nis approximately $\\alpha$ is given, via Sanov's theorem, by relative entropy\nof a coin flip with bias $\\alpha$ with respect to a coin with bias $F(T-)$\n(see the comments after Theorem \\ref{T:measurechange}).\n\nWe are almost ready to state our main theorem.\nWe need one last assumption.\n\\begin{assumption}\\label{A:density} We assume that $F(T')<F(T)$ for all $T'\\in [0,T)$.\n\\end{assumption}\n\\noindent In other words, $F$ cannot be flat to the left of $T$.  Thus $F(T)$ is positive (viz., for $T'\\in [0,T)$, $F(T)>F(T')\\ge 0$); this is natural, since if $F(T)=0$, then there is no possibility of any defaults by time $T$.\nSecondly, if $F$ is flat right before $T$, then any defaults by time $T$ must in fact have occurred earlier, so we can effectively reduce the time interval of interest to a smaller one.\nBy disallowing such a flat, we ensure that there is some likelihood of\ndefaults right before $T$, allowing us to carry out a quantitative analysis\nof $L^{(N)}$ right before time $T$ (see the proof of Lemma \\ref{L:TTimes}).\n\nThe goal of this paper is to formalize the asymptotics conjectured above.\nSet\n\\begin{equation}\\label{E:vkapdef} \\varkappa\\overset{\\text{def}}{=} \\ln \\left(\\frac{\\alpha}{1-\\alpha}\\frac{1-F(T-)}{F(T-)}\\right) = \\ln \\left(\\frac{\\frac{1}{F(T-)}-1}{\\frac{1}{\\alpha}-1}\\right). \\end{equation}\nIn light of Assumption \\ref{A:IG}, the second formula ensures that $\\varkappa>0$.\n\\begin{theorem}[Main]\\label{T:Main} We have that\n\\begin{multline*} \\mathbb{E}\\left[\\textbf{P}^{\\text{prot}}_N\\right] = \\frac{e^{-\\textsf{R} T}\\exp\\left[-\\varkappa\\left(\\lceil N\\alpha \\rceil -N\\alpha\\right)\\right]}{N^{3\/2}(\\beta-\\alpha)\\sqrt{2\\pi\\alpha(1-\\alpha)}}\\left\\{ \\frac{\\alpha(1-\\alpha)F(T-)(1-F(T-))}{(\\alpha-F(T-))^2} \\right.\\\\\n\\left. +\\left(\\lceil N\\alpha \\rceil-N\\alpha\\right)\\frac{\\alpha(1-F(T-))}{\\alpha-F(T-)} + \\mathcal{E}(N)\\right\\} \\exp\\left[-N \\mathfrak{I}(\\alpha)\\right] \\end{multline*}\nwhere $\\lim_{N\\to \\infty}\\mathcal{E}(N)=0$.\\end{theorem}\n\\noindent We can recognize a number of effects here.  Firstly, the $e^{-\\textsf{R} T}$\nterm reflects the fact that while by assumption losses in the CDO are unlikely, the least unlikely way for them to occur is right before expiry.  The term $\\beta-\\alpha$ in the denominator reflects the tranche width; note that we are looking at large $N$-approximations here; if we were to first take asymptotics as the tranche\nwidth tends to zero, we would probably capture some different effects\n(but we expect that the exponentially small entropy term would still appear).\nThe $\\sqrt{2\\pi\\alpha(1-\\alpha)}$ reflects something like a Gaussian correction\nterm (it directly comes from the calculations of Section \\ref{S:Proofs}).  The $N^{3\/2}$\nis a combination of two things.  Part of it ($N^{1\/2}$) also comes from the Gaussian correction.  The rest ($N$) comes from the actual size of the protection\nleg payments $\\textbf{P}^{\\text{prot}}_N$ once the attachment point has been reached.\nThe unsightly term $\\lceil N\\alpha \\rceil - N\\alpha$ comes from an unavoidable granularity in our problem; the loss process can only take on values in $\\mathbb{Z}\/N$.  We expect this granularity to disappear if the notional loss takes on a continuum of values.  This would be the case, for example, with random recoveries (cf. \\cite{AndersonSidenius}).  Of course, by taking $\\alpha$ to be a multiple of $1\/N$, we can make this granularity disappear---at the cost of making our calculations\nlook more restrictive than they actually are.\n\nFinally, we explicitly point out that our analysis is \\emph{asymptotic} as\nthe number $N$ of names becomes large.  We cannot say anything specific about\nany finite $N$.  This is analogous to the law of large numbers; the law of\nlarge numbers cannot, for example, give information about any finite number\nof coin flips, but rather is useful in framing one's thoughts when one\nhas ``many'' coin flips.\n\nCombining \\eqref{E:Premas} and Theorem \\ref{T:Main}, we see that the asymptotic behavior of the premium $S_N$ is given by\n\\begin{equation}\\label{E:Sas} \\begin{aligned} S_N &= \\frac{1}{N^{3\/2}}\\frac{e^{-\\textsf{R} T}\\exp\\left[-\\varkappa\\left(\\lceil N\\alpha \\rceil -N\\alpha\\right)\\right]}{\\left\\{ \\sum_{t\\in \\mathcal{T}} e^{-\\textsf{R} t}\\right\\} (\\beta-\\alpha)\\sqrt{2\\pi\\alpha(1-\\alpha)}}\\left\\{ \\frac{\\alpha(1-\\alpha)F(T-)(1-F(T-))}{(\\alpha-F(T-))^2}\\right.\\\\\n&\\qquad \\left. +\\left(\\lceil N\\alpha \\rceil-N\\alpha\\right)\\frac{\\alpha(1-F(T-))}{\\alpha-F(T-)}  + \\mathcal{E}'(N)\\right\\} \\exp\\left[-N \\mathfrak{I}(\\alpha)\\right] \\end{aligned}\\end{equation}\nwhere $\\lim_{N\\to \\infty}\\mathcal{E}'(N)=0$.  \n\nTo close this section, we plot some ``theoretical'' prices as a function of the number $N$. By ``theoretical'', we mean the\nquantity \n\\begin{multline*} S^*_N \\overset{\\text{def}}{=} \\frac{\\exp\\left[-\\varkappa\\left(\\lceil N\\alpha \\rceil -N\\alpha\\right)\\right]}{N^{3\/2}\\sqrt{\\alpha(1-\\alpha)}}\\left\\{ \\frac{\\alpha(1-\\alpha)F(T-)(1-F(T-))}{(\\alpha-F(T-))^2} \\right.\\\\\n\\left.+\\left(\\lceil N\\alpha \\rceil-N\\alpha\\right)\\frac{\\alpha(1-F(T-))}{\\alpha-F(T-)}\\right\\} \\exp\\left[-N \\mathfrak{I}(\\alpha)\\right] \\end{multline*}\nWe have here set $\\mathcal{E}'\\equiv 0$ in \\eqref{E:Sas} and have removed the prefactor\n\\begin{equation*} \\frac{e^{-\\textsf{R} T}}{\\left\\{ \\sum_{t\\in \\mathcal{T}} e^{-\\textsf{R} t}\\right\\} (\\beta-\\alpha)\\sqrt{2\\pi}}. \\end{equation*}\n\\begin{figure}[b]\n\\includegraphics[width=5in, height=3in]{prices.pdf}\n\\caption{$S^*_N$ for several values of $\\alpha$}\n\\label{fig:prices}\n\\end{figure}\n\n\\subsection{Correlation}\\label{S:Correlated}\nWe can now introduce a simple model of correlation without too much trouble.  Assume that $\\xi^\\text{S}$ takes values in a finite set $\\mathsf{X}$.  Fix $\\{p(x);\\, x\\in \\mathsf{X}\\}$ such that $\\sum_{x\\in \\mathsf{X}}p(x)=1$ and $p(x)>0$ for all $x\\in \\mathsf{X}$;\nwe will assume that $\\xi^\\text{S}$ takes on the value $x$ with probability $p(x)$.  We can think of the set $\\mathsf{X}$ as the collection of possible states of the world.  If we believe in \\eqref{E:structural}, we should then be in the previous\ncase if we condition on the various values of $\\xi^\\text{S}$.  To formalize this,\nfix a $\\{\\mu(\\cdot,x)\\}_{x\\in \\mathsf{X}}\\subset \\mathscr{P}(I)$.  Fix a probability measure $\\mathbb{P}$ such that\n\\begin{equation}\\label{E:corrprob} \\mathbb{P}\\left(\\bigcap_{n=1}^N\\{\\tau_n\\in A_n\\}\\right) = \\sum_{x\\in\\mathsf{X}}\\left\\{ \\prod_{n=1}^N \\mu(A_n,x)\\right\\} p(x). \\end{equation}\nfor all $\\{A_n\\}_{n\\in \\mathbb{N}}\\subset \\mathscr{B}(I)$.\n\nTo adapt the previous calculations to this case, we need the analogue of Assumptions \\ref{A:IG} and \\ref{A:density}.\nNamely, we need that $\\max_{x\\in \\mathsf{X}}\\mu([0,T),x)<\\alpha$ and also that $\\mu([0,T'],x)<\\mu([0,T],x)$ for all $T'\\in [0,T)$ and all $x\\in \\mathsf{X}$.\n\n\\begin{remark}\\label{R:systemic} The requirement that $\\max_{x\\in \\mathsf{X}}\\mu([0,T),x)<\\alpha$ is a particularly unrealistic one.  It means that the tranche losses will be rare\nfor \\emph{all} values of the systemic parameter.  In any truly applicable\nmodel, the losses will come from a combination of bad values of the\nsystemic parameter and from tail events in the pool of idiosyncratic randomness\n(i.e., we need to balance the size of $\\mathbb{P}\\left\\{ L^{(N) }_{T-}>\\alpha\\big|\\xi^\\text{S}=x\\right\\}$ against the distribution of $\\xi^\\text{S}$).\nOne can view our effort here as study which focusses primarily on tail events in the pool of idiosyncratic randomness.\nAny structural model which attempts to study losses due to both idiosyncratic and systemic randomness will most likely involve calculations which are similar\nin a number of ways to ours here.  We will explore this issue elsewhere.\n\\end{remark}\n\nFor each $x\\in \\mathsf{X}$, define\n\\begin{equation*} \\varkappa_x\\overset{\\text{def}}{=} \\ln \\left(\\frac{\\alpha}{1-\\alpha}\\frac{1-\\mu([0,T),x)}{\\mu([0,T),x)}\\right) = \\ln \\left(\\frac{\\frac{1}{\\mu([0,T),x)}-1}{\\frac{1}{\\alpha}-1}\\right). \\end{equation*}\nThen\n\\begin{multline*} \\mathbb{E}\\left[\\textbf{P}^{\\text{prot}}_N\\right] = \\frac{e^{-\\textsf{R} T}}{N^{3\/2}(\\beta-\\alpha)\\sqrt{2\\pi\\alpha(1-\\alpha)}}\\sum_{x\\in \\mathsf{X}}\\left(\\exp\\left[-\\varkappa_x\\left(\\lceil N\\alpha \\rceil -N\\alpha\\right)\\right]\\right.\\\\\n\\times \\left.\\left\\{ \\frac{\\alpha(1-\\alpha)\\mu([0,T),x)\\left(1-\\mu([0,T),x)\\right)}{\\left(\\alpha-\\mu([0,T),x)\\right)^2} +\\left(\\lceil N\\alpha \\rceil-N\\alpha\\right)\\frac{\\alpha\\left(1-\\mu([0,T),x)\\right)}{\\alpha-\\mu([0,T),x)} + \\mathcal{E}(N)\\right\\} \\right.\\\\\n\\left.\\times \\exp\\left[-N \\hbar(\\alpha,\\mu([0,T),x))\\right]p(x)\\right) \\end{multline*}\nwhere $\\lim_{N\\to \\infty}\\mathcal{E}_x(N)=0$ for all $x\\in \\mathsf{X}$.  Similarly we have that\n\\begin{multline*} S_N = \\frac{1}{N^{3\/2}}\\frac{e^{-\\textsf{R} T}}{\\left\\{ \\sum_{t\\in \\mathcal{T}} e^{-\\textsf{R} t}\\right\\} (\\beta-\\alpha)\\sqrt{2\\pi\\alpha(1-\\alpha)}}\\sum_{x\\in \\mathsf{X}}\\left(\\exp\\left[-\\varkappa_x\\left(\\lceil N\\alpha \\rceil -N\\alpha\\right)\\right]\\right.\\\\\n\\left. \\times \\left\\{ \\frac{\\alpha(1-\\alpha)\\mu([0,T),x)\\left(1-\\mu([0,T),x)\\right)}{\\left(\\alpha-\\mu([0,T),x)\\right)^2}+\\left(\\lceil N\\alpha \\rceil-N\\alpha\\right)\\frac{\\alpha\\left(1-\\mu([0,T),x)\\right)}{\\alpha-\\mu([0,T),x)}  + \\mathcal{E}'(N)\\right\\}\\right.\\\\\n\\left.\\times \\exp\\left[-N \\hbar(\\alpha,\\mu([0,T),x))\\right]p(x)\\right) \\end{multline*}\nwhere $\\lim_{N\\to \\infty}\\mathcal{E}'_x(N)=0$ for all $x\\in \\mathsf{X}$.\nIf we further assume that there is a unique $x^*\\in \\mathsf{X}$ such that $\\min_{x\\in \\mathsf{X}}  \\mathfrak{I}(\\alpha,\\mu([0,T),x))=\\mathfrak{I}(\\alpha,\\mu([0,T),x^*))$, we furthermore\nhave that\n\\begin{align*} \\mathbb{E}\\left[\\textbf{P}^{\\text{prot}}_N\\right] &= \\frac{e^{-\\textsf{R} T}}{N^{3\/2}(\\beta-\\alpha)\\sqrt{2\\pi\\alpha(1-\\alpha)}}\\exp\\left[-\\varkappa_{x^*}\\left(\\lceil N\\alpha \\rceil -N\\alpha\\right)\\right]\\\\\n&\\qquad \\times \\left\\{ \\frac{\\alpha(1-\\alpha)\\mu([0,T),x^*)\\left(1-\\mu([0,T),x^*)\\right)}{\\left(\\alpha-\\mu([0,T),x^*)\\right)^2} \n+\\left(\\lceil N\\alpha \\rceil-N\\alpha\\right)\\frac{\\alpha\\left(1-\\mu([0,T),x^*)\\right)}{\\alpha-\\mu([0,T),x^*)} + \\mathcal{E}(N)\\right\\} \\\\\n&\\qquad \\times \\exp\\left[-N \\hbar(\\alpha,\\mu([0,T),x^*))\\right]p(x^*)\\\\\nS_N &= \\frac{1}{N^{3\/2}}\\frac{e^{-\\textsf{R} T}}{\\left\\{ \\sum_{t\\in \\mathcal{T}} e^{-\\textsf{R} t}\\right\\} (\\beta-\\alpha)\\sqrt{2\\pi\\alpha(1-\\alpha)}}\\exp\\left[-\\varkappa_{x^*}\\left(\\lceil N\\alpha \\rceil -N\\alpha\\right)\\right]\\\\\n&\\qquad \\times \\left\\{ \\frac{\\alpha(1-\\alpha)\\mu([0,T),x^*)\\left(1-\\mu([0,T),x^*)\\right)}{\\left(\\alpha-\\mu([0,T),x^*)\\right)^2}+\\left(\\lceil N\\alpha \\rceil-N\\alpha\\right)\\frac{\\alpha\\left(1-\\mu([0,T),x^*)\\right)}{\\alpha-\\mu([0,T),x^*)}  + \\mathcal{E}'(N)\\right\\}\\\\\n&\\qquad \\times \\exp\\left[-N \\hbar(\\alpha,\\mu([0,T),x^*))\\right]p(x^*) \\end{align*}\nwhere $\\lim_{N\\to \\infty}\\mathcal{E}(N)=0$ and $\\lim_{N\\to \\infty}\\mathcal{E}'(N)=0$.\n\n\n\nNote that we can use this methodology to approximately study Gaussian\ncorrelations.  Fix a positive $M\\in \\mathbb{N}$ and define $x_i\\overset{\\text{def}}{=} \\tfrac{i}{M}$\nfor $i\\in \\{-M^2,-M^2+1\\dots M^2\\}$; set $\\mathsf{X} \\overset{\\text{def}}{=} \\{x_i\\}_{i=-M^2}^{M^2}$.\nDefine\n\\begin{equation*} \\Phi(x) \\overset{\\text{def}}{=} \\int_{t=-\\infty}^x\\frac{1}{\\sqrt{2\\pi}}\\exp\\left[-\\frac{t^2}{2}\\right]dt \\qquad x\\in \\mathbb{R} \\end{equation*}\nas the standard Gaussian cumulative distribution function.\nDefine\n\\begin{equation*} p(x_i) \\overset{\\text{def}}{=} \\begin{cases} \\Phi\\left(x_i+\\frac{1}{2M}\\right)-\\Phi\\left(x_i-\\frac{1}{2M}\\right) &\\qquad \\text{if $i\\in \\{-M^2+1,\\dots M^2-1\\}$} \\\\\n\\Phi\\left(x_{-M^2}+\\frac{1}{2M}\\right) &\\qquad \\text{if $i=-M^2$}\\\\\n1-\\Phi\\left(x_{M^2}-\\frac{1}{2M}\\right) &\\qquad \\text{if $i=M^2$}\\end{cases}\\end{equation*}\nIf we have a pool of $N$ names with common probability of default $p$ by time $T$ and we want to consider a Gaussian copula with\ncorrelation $\\rho>0$ (the case $\\rho<0$ can be dealt with similarly),\nwe would take the $\\mu(\\cdot,x_i)$'s such that\n\\begin{equation*} \\mu([0,T),x_i) \\overset{\\text{def}}{=} \\Phi\\left(\\frac{\\Phi^{-1}(p)-\\rho x_i}{\\sqrt{1-\\rho^2}}\\right). \\end{equation*}\nThis is related to the calculations of \\cite{Glasserman} and \\cite{MR2384674};\nthose calculations are asymptotically related to our calculations.  We shall explore the connection with these two papers elsewhere.  We note, by way of contrast with \\cite{Glasserman} and \\cite{MR2384674}, that our efforts give a good picture of the \\emph{dynamics} of the loss process prior to expiry.\nWe also note that our model of \\eqref{E:corrprob} is entirely comfortable with non-Gaussian correlation.  Note also that one could also (by discretization) allow the systemic parameter $\\xi^\\text{S}$ to be path-valued.\n\n\\subsection{Large Deviations}\\label{S:LD}\nWe shall here give a very short summary of the main ideas of large deviations; see \\cite{MR1619036} for a comprehensive treatment.  The basic observation behind\nthe theory is that a sum of exponentials behaves like largest-growing\nexponential.  For example,\n\\begin{equation*} e^{-3N} + e^{7N} + e^{4N} = e^{7N}\\left\\{ 1 + e^{-10N} + e^{-3N}\\right\\} \\asymp e^{-7N}. \\end{equation*}\nHere ``$\\asymp$'' means ``having the same exponential growth''; in other words,\n$A_N\\asymp B_N$ if $\\lim_{N\\to \\infty}\\tfrac1N\\ln A_N=\\lim_{N\\to \\infty}\\tfrac1N\\ln B_N$.\n\\emph{Laplace asymptotics} extends this to integrals.  This is a relevant place\nto start the study of rare events if we consider a collection $\\{X_n\\}_{n\\in \\mathbb{N}}$\nof random variables whose laws are of the form\n\\begin{equation}\\label{E:laplace} \\mathbb{P}\\{X_N\\in A\\} \\overset{\\text{def}}{=} \\int_{x\\in A}c_N\\exp\\left[-N \\phi(x)\\right]dx \\qquad A\\in \\mathscr{B}(\\mathbb{R}) \\end{equation}\nfor some $\\phi\\in C(\\mathbb{R})$ and some normalization constant $c_N$ (e.g., if we take $\\phi(x) = \\tfrac12(x-1)^2$ and $c_N = 1\/\\sqrt{2\\pi\/N}$, then $X_N$ will be a\nnormal random variable with mean $1$ and variance $\\tfrac{1}{N}$).  If we\nassume that $\\phi$ has nice enough growth properties (so that the integrals\nin \\eqref{E:laplace} are well-defined and $c_N$ has subexpontial growth) ,\nthen Laplace asymptotics states that\n\\begin{equation}\\label{E:expasymp} \\mathbb{P}\\{X_N\\in A\\} \\asymp \\exp\\left[-N \\inf_{x\\in A}\\phi(x)\\right] \\end{equation}\nfor ``nice'' enough sets $A$.  By taking $A=\\mathbb{R}$, we see that\nwe must have that $\\inf_{x\\in \\mathbb{R}}\\phi(x)=0$.  If this minimum is achieved at a single point $x^*$, then by taking $A$ as the complement of a neighborhood of $x^*$\nwe have that $X_N\\to x^*$ in probability, so $\\{X_N\\in A\\}$ is a rare\nevent for any nice enough set $A$ not containing $x^*$.\n\nOne of the main aspects of large deviations theory is something of an inverse problem.  Can we have \\eqref{E:expasymp} even without \\eqref{E:laplace}?\nIn some cases, yes.  Fix $\\theta\\in \\mathbb{R}$\nand consider the limiting rate of growth of the logarithmic moment\ngenerating function; we have that\n\\begin{equation*} \\lim_{N\\to \\infty}\\frac1N\\ln \\mathbb{E}\\left[\\exp\\left[\\theta N X_N\\right]\\right] = \\lim_{N\\to \\infty}\\frac1N\\ln \\int_{x\\in \\mathbb{R}}c_N\\exp\\left[N\\left\\{ \\theta x-\\phi(x)\\right\\}\\right]dx = \\sup_{x\\in \\mathbb{R}}\\left\\{ \\theta x-\\phi(x)\\right\\}. \\end{equation*}\nThe key realization is that the right-hand side is the \\emph{Legendre-Fenchel transform} of $\\phi$, and that if $\\phi$ has nice convexity properties,\nwe can recover $\\phi$ from $M$ by taking the Legendre-Fenchel transform\nagain; i.e., \n\\begin{equation*} \\phi(x) = \\sup_{\\theta\\in \\mathbb{R}}\\left\\{ \\theta x-M(\\theta)\\right\\}. \\end{equation*}\nThe strength of this chain of arguments is that the moment generating function is\nwell-defined (but of course possibly infinite) regardless of whether\n$X_N$ is discrete or continuous.  It even makes sense when $X_N$ \ntakes values in an infinite-dimensional topological linear space $\\mathsf{X}$\nif we replace multiplication by $\\theta$ with the action of a linear functional\non $\\mathsf{X}$.  The rigorous definition of a large deviations principle is as follows \\cite{MR758258}.  We say that $\\{X_n;\\, n\\in \\mathbb{N}\\}$ (which we now assume\nto take values in a topological space $\\mathsf{X}$) has a large deviations principle with rate function $\\mathcal{I}:\\mathsf{X}\\to [0,\\infty]$ if the following three requirements hold:\n\\begin{itemize}\n\\item For every $s\\ge 0$, $\\{x\\in \\mathsf{X}: \\mathcal{I}(x)\\le s\\}$ is a compact subset of $\\mathsf{X}$.\n\\item For every open subset $G$ of $\\mathsf{X}$,\n\\begin{equation*} \\varliminf_{N\\to \\infty}\\frac{1}{N}\\ln \\mathbb{P}\\{X_n\\in G\\} \\ge - \\inf_{x\\in G}\\mathcal{I}(x). \\end{equation*}\n\\item For every closed subset $F$ of $\\mathsf{X}$,\n\\begin{equation*} \\varlimsup_{N\\to \\infty}\\frac{1}{N}\\ln \\mathbb{P}\\{X_n\\in F\\} \\le - \\inf_{x\\in F}\\mathcal{I}(x). \\end{equation*}\n\\end{itemize}\n\nReturning to our focus, which is Sanov's theorem applied to \\eqref{E:nudef},\nwe have that for any $\\phi\\in C(I)$ (the dual of $\\mathscr{P}(I)$),\n\\begin{equation*} \\lim_{N\\to \\infty}\\frac1N\\ln \\mathbb{E}\\left[\\exp\\left[N\\int_{t\\in I}\\phi(t)\\nu^{(N)}(dt)\\right]\\right] = \\ln \\int_{t\\in I}e^{\\phi(t)}\\mu(dt) \\end{equation*}\nand we can then show that\n\\begin{equation}\\label{E:dual} H(\\mu'|\\mu) = \\sup_{\\phi\\in C(I)}\\left\\{ \\int_{t\\in I}\\phi(t)\\mu'(dt) - \\ln \\int_{t\\in I}e^{\\phi(t)}\\mu(dt)\\right\\}. \\qquad \\mu'\\in \\mathscr{P}(I)\\end{equation}\nThis suggests that indeed we should have \\eqref{E:san} as interpreted as a large\ndeviations principle (Sanov's theorem).\n\n\\section{A Measure Transformation}\\label{S:Tilt}\nOne of the things which naturally occurs in proofs of large deviations\nprinciples is a \\emph{measure change} under which the unlikely event becomes more likely---the cost\nof this change of measure is exactly the desired exponential rate of decay (see \\cite{MR1619036}).\nLet's see what this looks like in our situation (see \\cite{MR1619036} for a more\ncomplete motivation of measure changes in large deviations).  Define\n\\begin{equation*} \\phi^*_\\alpha(t) = \\ln \\frac{d\\tilde \\mu^*_\\alpha}{d\\mu}(t) = \\ln \\frac{\\alpha}{F(T-)}\\chi_{[0,T)}(t) + \\ln \\frac{1-\\alpha}{1-F(T-)}\\chi_{[T,\\infty]}(t)\\qquad t\\in I \\end{equation*}\n(note that since $\\mathfrak{I}(\\alpha)<\\infty$, $H(\\tilde \\mu^*_\\alpha|\\mu)<\\infty$, so $\\tilde \\mu^*_\\alpha \\ll \\mu$).  \nIt is easy to verify that\n\\begin{equation*} H(\\tilde \\mu^*_\\alpha|\\mu) = \\int_{t\\in I}\\phi^*_\\alpha(t)\\tilde \\mu^*_\\alpha(dt) - \\ln \\int_{t\\in I}e^{\\phi^*_\\alpha(t)}\\mu(dt); \\end{equation*}\nthus $\\phi^*_\\alpha$ is the extremal in the variational representation \\eqref{E:dual} for $H(\\tilde \\mu^*_\\alpha|\\mu)$ (if we allow ourselves\nto extend the supremum over $C(I)$ to the collection of bounded measurable functions; it turns out that this is allowable).\nIn our analysis of $\\nu^{(N)}$ of \\eqref{E:nudef}, $\\phi^*_\\alpha$\nwill naturally give us an optimal way to ``tilt'' our original\nprobability measure so that it becomes likely that $\\nu^{(N)}[0,T)\\approx \\alpha$.  The penalty for doing this is exactly $\\mathfrak{I}(\\alpha)$.\n\\begin{theorem}\\label{T:measurechange}  We have that\n\\begin{equation*} \\mathbb{E}[\\textbf{P}^{\\text{prot}}_N] = I_Ne^{-N\\mathfrak{I}(\\alpha)} \\end{equation*}\nfor all positive integers $N$, where\n\\begin{equation}\\label{E:IDef} I_N\\overset{\\text{def}}{=} \\tilde \\mathbb{E}_N\\left[\\textbf{P}^{\\text{prot}}_N\\exp\\left[-\\varkappa\\gamma_N\\right]\\chi_{\\{\\gamma_N>0\\}}\\right] \\end{equation}\nwhere in turn\n\\begin{equation}\\label{E:MIX}\\begin{aligned} \\tilde \\mathbb{P}_N(A) &\\overset{\\text{def}}{=} \\mathbb{E}\\left[\\chi_A\\prod_{n=1}^N \\frac{d\\tilde \\mu^*_\\alpha}{d\\mu}(\\tau_n)\\right] \\qquad A\\in \\mathscr{F} \\\\\n\\gamma_N&= \\sum_{n=1}^N \\left\\{ \\chi_{[0,T)}(\\tau_n)-\\alpha\\right\\} =N(L_{T-}^{(N)}-\\alpha)\\end{aligned}\\end{equation}\nUnder $\\tilde \\mathbb{P}_N$, $\\{\\tau_1,\\tau_2\\dots \\tau_N\\}$ are independent and identically distributed with common law $\\tilde \\mu^*_\\alpha$.\n\\end{theorem}\n\\begin{proof}  Set\n\\begin{equation*}\n\\Gamma_N =  N\\left\\{ \\int_{t\\in I}\\phi^*_\\alpha(t)\\nu^{(N)}(dt)-\\int_{t\\in I}\\phi^*_\\alpha(t)\\tilde \\mu^*_\\alpha(dt)\\right\\}\\end{equation*}\nThen\n\\begin{equation*} \\mathbb{E}[\\textbf{P}^{\\text{prot}}_N] = \\frac{\\mathbb{E}\\left[\\textbf{P}^{\\text{prot}}_N \\exp\\left[-\\Gamma_N\\right]\\exp\\left[\\Gamma_N\\right]\\right]}{\\mathbb{E}\\left[\\exp\\left[\\Gamma_N\\right]\\right]}\\mathbb{E}\\left[\\exp\\left[\\Gamma_N\\right]\\right]. \\end{equation*}\nNote that\n\\begin{align*} \\int_{t\\in I}\\phi^*_\\alpha(t)\\tilde \\mu^*_\\alpha(dt)&= \\mathfrak{I}(\\alpha)\\\\\n\\exp\\left[N \\int_{t\\in I}\\phi^*_\\alpha(t)\\nu^{(N)}(dt)\\right]&= \\exp\\left[\\sum_{n=1}^N\\ln \\frac{d\\tilde \\mu^*_\\alpha}{d\\mu}(\\tau_n)\\right] = \\prod_{n=1}^N \\frac{d\\tilde \\mu^*_\\alpha}{d\\mu}(\\tau_n) \\\\\n\\mathbb{E}\\left[\\exp\\left[\\Gamma_N\\right]\\right]&= e^{-N\\mathfrak{I}(\\alpha)}\\mathbb{E}\\left[\\prod_{n=1}^N \\frac{d\\tilde \\mu^*_\\alpha}{d\\mu}(\\tau_n)\\right] = e^{-N\\mathfrak{I}(\\alpha)} \\end{align*}\n(these equalities in fact reflect some of the basic properties of\nlarge deviations measure transformations and are intimately related with\nthe fact that $\\phi^*_\\alpha$ solves the variational problem \\eqref{E:dual} associated with $H(\\tilde \\mu^*_\\alpha|\\mu)$).\nWe also clearly have that\n\\begin{equation*}\\frac{\\mathbb{E}\\left[\\chi_A\\exp\\left[\\Gamma_N\\right]\\right]}{\\mathbb{E}\\left[\\exp\\left[\\Gamma_N\\right]\\right]}\\\\\n= \\frac{\\mathbb{E}\\left[\\chi_A\\exp\\left[N\\int_{t\\in I}\\phi^*_\\alpha(t)\\nu^{(N)}(dt)\\right]\\right]}{\\mathbb{E}\\left[\\exp\\left[N\\int_{t\\in I}\\phi^*_\\alpha(t)\\nu^{(N)}(dt)\\right]\\right]} = \\tilde \\mathbb{P}_N(A) \\end{equation*}\nfor all $A\\in \\mathscr{F}$.  The properties of $\\tilde \\mathbb{P}_N$ are clear from the explicit formula.\nWe next check that\n\\begin{multline*}\\Gamma_N=N\\left\\{ \\ln \\frac{\\alpha}{F(T-)} \\nu^{(N)}[0,T) + \\ln \\frac{1-\\alpha}{1-F(T-)}\\nu^{(N)}[T,\\infty] \n-\\frac{\\alpha}{F(T-)} \\alpha - \\ln \\frac{1-\\alpha}{1-F(T-)}(1-\\alpha)\\right\\}\\\\\n=N\\ln \\frac{\\alpha}{F(T-)} \\left\\{ \\nu^{(N)}[0,T)-\\alpha\\right\\} +N \\ln \\frac{1-\\alpha}{1-F(T-)}\\left\\{ \\nu^{(N)}[T,\\infty]-(1-\\alpha)\\right\\} = \\varkappa\\gamma_N. \\end{multline*}\nFinally, we see that $\\textbf{P}^{\\text{prot}}_N$ is nonzero only if $\\gamma_N>0$; we have explicitly included this\nin the expression for $I_N$.\n\\end{proof}\n\\noindent We note here that\n\\begin{equation*} \\tilde \\mathbb{E}_N\\left[L^{(N)}_{T-}\\right] = \\tilde \\mu^*_\\alpha[0,T)=\\alpha\\qquad \\text{and}\\qquad \\tilde \\mathbb{E}_N\\left[\\left(L^{(N)}_{T-}-\\alpha\\right)^2\\right] = \\frac{\\alpha(1-\\alpha)}{N^2}\\le \\frac{1}{4N^2}, \\end{equation*}\nso by Chebychev's inequality, we have that \n\\begin{equation*} \\lim_{N\\to \\infty}\\tilde \\mathbb{P}_N\\left\\{ \\left|L^{(N)}_{T-}-\\alpha\\right|\\ge \\varepsilon\\right\\} = 0 \\end{equation*}\nfor every $\\varepsilon>0$.  In other words, $L^{(N)}_{T-}$ tends to the attachment point $\\alpha$ under the sequence $(\\tilde \\mathbb{P}_N)_{N\\in \\mathbb{N}}$ of probability measures\nand thus loss is not a rare event under $\\tilde \\mathbb{P}_N$ as $N\\nearrow \\infty$.\n\nWe also note that we need to understand the appropriate\nchange of measure for the empirical measure $\\nu^{(N)}$ (as opposed to the\nchange of measure for the empirical sum $L^{(N)}_{T-}$) since $\\textbf{P}^{\\text{prot}}_N$ involves the dynamics of the loss process (and not just the probability of loss).\n\n\\section{Asymptotic Analysis}\nWhere do we now stand?  If we can show that $I_N$ has no exponential growth or decay (comparable to $e^{-N\\mathfrak{I}(\\alpha)}$) then we have successfully identified the asymptotic behavior of $\\mathbb{E}[\\textbf{P}^{\\text{prot}}_N]$; we will have decomposed it into\nan exponentially small part and a prefactor which is of order 1 as $N\\nearrow \\infty$.  Our goal now is to organize our thoughts about the prefactor,\nand in particular to actually extract the asymptotics of Theorem \\ref{T:Main};\ni.e., to ``do the math''.\n\nLooking at the expression \\eqref{E:IDef} for $I_N$, we see that the dominant part of $I_N$ will be where $\\gamma_N$ is order\\footnote{actually, it will be where $\\gamma_N\\ll \\sqrt{N}$} 1; if $\\gamma_N\\gg 1$, then $\\exp[-\\varkappa \\gamma_N]$ will be very small so the contribution to $I_N$ will be negligible (recall here that $\\textbf{P}^{\\text{prot}}_N$ is bounded).  This suggests\nwe organize the formula for $I_N$ based on the values of $\\gamma_N$.\nNote that the range of $\\gamma_N$ when it is positive is\n$\\mathcal{S}_N\\overset{\\text{def}}{=} \\{n-N\\alpha: \\text{$n\\in \\mathbb{Z}$ and $N\\alpha\\le n\\le N$}\\}$.\n\\begin{definition} For each $N$, let $H_N:\\mathcal{S}_N\\to [0,1]$ be such that\n\\begin{equation*} H_N(\\gamma_N)= \\tilde \\mathbb{E}_N\\left[\\textbf{P}^{\\text{prot}}_N\\big|\\gamma_N\\right] \\end{equation*}\non $\\{\\gamma_N>0\\}$.\n\\end{definition}\n\\noindent Then we have that\n\\begin{equation*} I_N = \\tilde \\mathbb{E}_N\\left[H_N(\\gamma_N)\\chi_{\\{\\gamma_N>0\\}}\\exp\\left[-\\varkappa \\gamma_N\\right]\\right]. \\end{equation*}\n\\noindent It turns out that $H_N$ has very nice asymptotics.\n\\begin{lemma}\\label{L:hasymp} For all $N$, we have that\n\\begin{equation*} H_N(s) = \\frac{e^{-\\textsf{R} T}s\\left\\{ 1 + \\mathcal{E}_1(s,N)\\right\\}}{(\\beta-\\alpha)N} \\end{equation*}\nwhere\n\\begin{equation*} \\varlimsup_{N\\nearrow \\infty}\\sup_{\\substack{s\\in \\mathcal{S}_N \\\\ s\\le N^{1\/4}}}|\\mathcal{E}_1(s,N)|=0. \\end{equation*}\n\\end{lemma}\n\\noindent We will prove this in Section \\ref{S:HAS}.\n\nThe next step is to understand the distribution of $\\gamma_N$.\n\\begin{lemma}\\label{L:probasymp} We have that\n\\begin{equation*} \\tilde \\mathbb{P}_N\\{\\gamma_N=s\\} = \\frac{1+\\mathcal{E}_2(s,N)}{\\sqrt{2\\pi N\\alpha(1-\\alpha)}} \\end{equation*}\nfor all $N$ and all $s\\in \\mathcal{S}_N$, where\n\\begin{equation*} \\varlimsup_{N\\nearrow \\infty}\\sup_{\\substack{s\\in \\mathcal{S}_N \\\\ s\\le N^{1\/4}}}|\\mathcal{E}_2(s,N)|=0. \\end{equation*}\n\\end{lemma}\n\\noindent We will prove this in Section \\ref{S:Proofs}.  Using this result, we can now start our proof of Theorem \\ref{T:Main}.  Set\n\\begin{align*} \\tilde I_{1,N} &\\overset{\\text{def}}{=} \\sum_{\\substack{s\\in \\mathcal{S}_N\\\\s\\le N^{1\/4}}}(s-\\alpha) e^{-\\varkappa s}\\\\\n\\tilde I_{2,N} &\\overset{\\text{def}}{=} \\exp\\left[-\\varkappa\\left(\\lceil N\\alpha \\rceil -N\\alpha\\right)\\right]\\left\\{ \\frac{e^{-\\varkappa}}{(1-e^{-\\varkappa})^2} +\\frac{\\lceil N\\alpha \\rceil-N\\alpha}{1-e^{-\\varkappa}}\\right\\} \\end{align*}\nWe thus expect that\n\\begin{equation*} I_N \\approx \\frac{e^{-\\textsf{R} T}\\tilde I_{1,N}}{N^{3\/2}(\\beta-\\alpha)\\sqrt{2\\pi \\alpha(1-\\alpha)}}. \\end{equation*}\nWe then claim that $\\tilde I_{1,N} \\approx \\tilde I_{2,N}$.  As a preliminary\nto showing this, let's recall some calculations about geometric series.\nFor $\\lambda>0$ and each positive integer $n$,\n\\begin{equation*} \\sum_{j=0}^n e^{-\\lambda j}= \\frac{1}{1-e^{-\\lambda}}- \\frac{e^{-\\lambda(n+1)}}{1-e^{-\\lambda}}. \\end{equation*}\nDifferentiating with respect to $\\lambda$, we get that\n\\begin{equation*} \\sum_{j=0}^n j e^{-\\lambda j}  = \\frac{e^{-\\lambda}}{(1-e^{-\\lambda})^2} -e^{-\\lambda(n+1)}\\frac{n(1-e^{-\\lambda}) + 1}{(1-e^{-\\lambda})^2}. \\end{equation*}\nLet's bound the error terms in these expressions.  Note that $\\sup_{x>0}x e^{-x} = e^{-1}$.  For $\\lambda>0$ we have that\n\\begin{align*} \\left|e^{-\\lambda(n+1)}\\frac{n(1-e^{-\\lambda}) + 1}{(1-e^{-\\lambda})^2}\\right|\n&\\le e^{-\\lambda(n+1)}\\frac{n+1}{(1-e^{-\\lambda})^2}\\\\\n&= 2\\left\\{ \\frac{\\lambda}{2} (n+1)\\exp\\left[-\\frac{\\lambda}{2}(n+1)\\right]\\right\\} \\frac{\\exp\\left[-\\frac{\\lambda}{2}(n+1)\\right]}{\\lambda\\left(1-e^{-\\lambda}\\right)^2} \\\\\n&\\le 2e^{-1} \\frac{\\exp\\left[-\\frac{\\lambda}{2}(n+1)\\right]}{\\lambda\\left(1-e^{-\\lambda}\\right)^2} \\end{align*}\nand similarly\n\\begin{equation*} \\left|\\frac{e^{-\\lambda(n+1)}}{1-e^{-\\lambda}}\\right|\n= 2 e^{-\\lambda n\/2}\\left(1-e^{-\\lambda}\\right)\\left\\{ \\frac{\\lambda}{2}e^{-\\lambda\/2}\\right\\} \\frac{\\exp\\left[-\\frac{\\lambda}{2}(n+1)\\right]}{\\lambda\\left(1-e^{-\\lambda}\\right)^2}\\\\\n\\le 2 e^{-1}\\frac{\\exp\\left[-\\frac{\\lambda}{2}(n+1)\\right]}{\\lambda\\left(1-e^{-\\lambda}\\right)^2}. \\end{equation*}\nObserve now that\n\\begin{equation*} \\lfloor N\\alpha + N^{1\/4}\\rfloor - \\lceil N\\alpha \\rceil + 1\n\\ge N\\alpha + N^{1\/4} -1-N\\alpha -1+1 =N^{1\/4}-1  \\end{equation*}\nfor all $N\\in \\mathbb{N}$.\nCombining things and recalling that $\\varkappa>0$, we see that for all $N\\in \\mathbb{N}$,\n\\begin{equation} \\label{E:QQ} \\begin{aligned} \\tilde I_{1,N} &= \\sum_{\\substack{j\\in \\mathbb{Z}\\\\0\\le j-N\\alpha \\le N^{1\/4}}}(j-N\\alpha)\\exp\\left[-\\varkappa (j-N\\alpha)\\right]\\\\\n&= \\sum_{j=\\lceil N\\alpha \\rceil}^{\\lfloor N\\alpha + N^{1\/4}\\rfloor}(j-N\\alpha)\\exp\\left[-\\varkappa (j-N\\alpha)\\right]\\\\\n&= \\sum_{j=0}^{\\lfloor N\\alpha +N^{1\/4}\\rfloor-\\lceil N\\alpha \\rceil}(j+\\lceil N\\alpha \\rceil-N\\alpha)\\exp\\left[-\\varkappa (j+\\lceil N\\alpha \\rceil-N\\alpha)\\right]\\\\\n&= \\exp\\left[-\\varkappa (\\lceil N\\alpha \\rceil-N\\alpha)\\right]\\left\\{ \\sum_{j=0}^{\\lfloor N\\alpha +N^{1\/4}\\rfloor-\\lceil N\\alpha \\rceil}j e^{-\\varkappa j} + \\left(\\lceil N\\alpha \\rceil-N\\alpha\\right)\\sum_{j=0}^{\\lfloor N\\alpha +N^{1\/4}\\rfloor-\\lceil N\\alpha \\rceil}e^{-\\varkappa j}\\right\\} \\\\\n&= \\exp\\left[-\\varkappa(\\lceil N\\alpha \\rceil-N\\alpha)\\right]\\left\\{ \\frac{e^{-\\varkappa}}{(1-e^{-\\varkappa})^2} + \\left(\\lceil N\\alpha \\rceil-N\\alpha\\right)\\frac{1}{1-e^{-\\varkappa}} + \\mathcal{E}_3(N)\\right\\}\n\\end{aligned}\\end{equation}\nwhere\n\\begin{equation}\\label{E:QQQ} |\\mathcal{E}_3(N)| \\le 4e^{-1}\\frac{\\exp\\left[-\\frac{\\varkappa}{2}(N^{1\/4}-1)\\right]}{\\varkappa \\left(1-e^{-\\varkappa}\\right)^2}. \\end{equation}\nAs a consequence, we furthermore have that\n\\begin{multline*}\\left|\\tilde I_{1,N}\\right|\n\\le \\frac{e^{-\\varkappa}}{\\left(1-e^{-\\varkappa}\\right)^2} + \\frac{1}{1-e^{-\\varkappa}} + 4e^{-1}\\frac{\\exp\\left[-\\frac{\\varkappa}{2}(N^{1\/4}-1)\\right]}{\\varkappa \\left(1-e^{-\\varkappa}\\right)^2}\\\\ \n\\le \\frac{\\varkappa\\left(e^{-\\varkappa}+1-e^{-\\varkappa}\\right) + 4e^{-1}}{\\varkappa\\left(1-e^{-\\varkappa}\\right)^2}\n\\le \\frac{4e^{-1}(1+\\varkappa)}{\\varkappa\\left(1-e^{-\\varkappa}\\right)^2}. \\end{multline*}\nFrom \\eqref{E:vkapdef}, we have that\n\\begin{equation*} e^{-\\varkappa} = \\frac{1-\\alpha}{\\alpha}\\frac{F(T-)}{1-F(T-)}\\qquad \\text{and}\\qquad 1-e^{-\\varkappa}= \\frac{\\alpha-F(T-)}{\\alpha(1-F(T-))}. \\end{equation*}\nso\n\\begin{equation*} \\frac{e^{-\\varkappa}}{(1-e^{-\\varkappa})^2} = \\frac{1-\\alpha}{\\alpha}\\frac{F(T-)}{1-F(T-)}\\frac{\\alpha^2(1-F(T-))^2}{(\\alpha-F(T-))^2} = \\frac{\\alpha(1-\\alpha)F(T-)(1-F(T-))}{(\\alpha-F(T-))^2}. \\end{equation*}\n\nWe can finally prove our desired result.\n\\begin{proof}[Proof of Theorem \\ref{T:Main}]\nWe have that\n\\begin{equation*} I_N = \\frac{\\tilde I_{2,N}}{N^{3\/2}(\\beta-\\alpha)\\sqrt{2\\pi \\alpha(1-\\alpha)}} + \\sum_{j=1}^5 \\tilde \\mathcal{E}_j(N)\\end{equation*}\nwhere\n\\begin{align*}\n\\tilde \\mathcal{E}_1(N)&\\overset{\\text{def}}{=} \\tilde \\mathbb{E}_N\\left[\\textbf{P}^{\\text{prot}}_N e^{-\\varkappa \\gamma_N}\\chi_{\\{\\gamma_N>N^{1\/4}\\}}\\right] \\\\\n\\tilde \\mathcal{E}_2(N)&\\overset{\\text{def}}{=} \\sum_{\\substack{s\\in \\mathcal{S}_N\\\\s\\le N^{1\/4}}}\\frac{H_N(s)e^{-\\varkappa s}\\mathcal{E}_2(s,N)}{\\sqrt{2\\pi N\\alpha(1-\\alpha)}} \\\\\n\\tilde \\mathcal{E}_3(N)&\\overset{\\text{def}}{=} \\frac{e^{-\\textsf{R} T}}{\\beta-\\alpha}\\sum_{\\substack{s\\in \\mathcal{S}_N\\\\s\\le N^{1\/4}}}\\frac{se^{-\\varkappa s}\\mathcal{E}_1(s,N)}{N^{3\/2}\\sqrt{2\\pi \\alpha(1-\\alpha)}}\\\\\n\\tilde \\mathcal{E}_4(N)&\\overset{\\text{def}}{=} \\frac{e^{-\\textsf{R} T}}{\\beta-\\alpha}\\frac{\\exp\\left[-\\varkappa\\left(\\lceil N\\alpha \\rceil-N\\alpha\\right)\\right]\\mathcal{E}_3(N)}{N^{3\/2}\\sqrt{2\\pi \\alpha(1-\\alpha)}}\\end{align*}\nThen there is a $K_1>0$ such that \n\\begin{equation*} |\\tilde \\mathcal{E}_1(N)|\\le \\frac{1}{K_1}e^{-K_1 N^{1\/4}}\\qquad \\text{and}\\qquad |\\tilde \\mathcal{E}_4(N)|\\le \\frac{1}{K_1}e^{-K_1 N^{1\/4}}\\end{equation*}\nfor all $N\\in \\mathbb{N}$.\nFurthermore, we can fairly easily see that there is a $K_2$ such that\n\\begin{equation*}\n|\\tilde \\mathcal{E}_2(N)|\\le \\frac{K_2\\tilde I_{1,N}}{N^{3\/2}} \\sup_{\\substack{s\\in \\mathcal{S}_N \\\\ s\\le N^{1\/4}}}|\\mathcal{E}_2(s,N)|\\qquad \\text{and}\\qquad |\\tilde \\mathcal{E}_3(N)|\\le \\frac{K_2 \\tilde I_{1,N}}{N^{3\/2}}\\sup_{\\substack{s\\in \\mathcal{S}_N \\\\ s\\le N^{1\/4}}}|\\mathcal{E}_1(s,N)| \\end{equation*}\nfor all $N\\in \\mathbb{N}$ (note from \\eqref{E:QQ} and \\eqref{E:QQQ} that $\\tilde I_{1,N}$ is uniformly bounded in $N$).  Combine things together to get the stated result.\\end{proof}\n\n\\begin{remark}\\label{R:Comments}  Several comments are in order about the analysis of this section.\n\nFirstly, we re-emphasize that we first identified the law of $L^{(N)}_{T-}$ and then studied the law of $L^{(N)}$ right before $T$.\nFor investment-grade tranches, only this last part of $L^{(N)}$ should be\nof interest. For an investment-grade tranche, losses in general should be\nrare events; losses significantly before expiry should be \\emph{very} rare\nevents.  This would follow from a detailed analysis of the measure transformation\nof Section \\ref{S:Tilt}.\n\nSecondly, our analysis here suggests that in more realistic models\n(i.e., not i.i.d. names), the first order of business should be a thorough\nstudy of the law of $L^{(N)}_{T-}$.  This is somewhat appealing; by time $T$,\nvarious transients will assumedly have died out, and some sort of macroscopic\nanalysis may be available.\n\nThe third point of interest is the asymptotics of Lemma \\ref{L:probasymp}.\nThis does \\emph{not} directly reflect a Poisson distribution for $L^{(N)}_{T-}$.\nA number of other studies of CDO's have modelled the loss process as a Poisson\nprocess; an interesting question would thus be to try to find a limiting regime of our calculations which leads to Poisson statistics.\n\\end{remark}\n\nFinally, it would not be hard to use the measure change of Section \\ref{S:Tilt}\nand calculations similar to those of this section\nto compute the expected loss given default.  We will leave that to the reader.\n\n\\section{Proof of Lemma \\ref{L:hasymp}}\\label{S:HAS}\nWe here prove Lemma \\ref{L:hasymp}.  To do so, we need to develop a clear picture of the dynamics of $L^{(N)}$.  We note that the calculations of this section, though technical, provide a \\emph{direct} link to the distribution of the default times.\n\nFirst of all, we recall that the definition of $\\bar L^{(N)}$ implies that $\\bar L^{(N)}$ is nonzero only where $L^{(N)}$ exceeds $\\alpha$; since $L^{(N)}$ is nondecreasing, this will in fact be an interval.  Set\n\\begin{align*} \\tau^\\alpha_N &\\overset{\\text{def}}{=} \\inf\\{r>0: \\bar L^{(N)}_r>0\\} = \\inf\\{r>0: L^{(N)}_r>\\alpha\\} \\\\\n\\tau^\\beta_N &\\overset{\\text{def}}{=} \\sup\\{r>0: \\bar L^{(N)}_r<\\beta-\\alpha\\} = \\sup\\{r>0: L^{(N)}_r<\\beta\\} \\end{align*}\nA typical graph of $\\bar L^{(N)}$ is given in Figure \\ref{fig:typ}.\nNext note that on $\\{\\gamma_N>0\\}$,\n\\begin{equation}\\label{E:AA} \\textbf{P}^{\\text{prot}}_N = \\int_{s\\in [\\tau^\\alpha_N,\\tau^\\beta_N]\\cap[0,T)}e^{-\\textsf{R} s}d\\bar L^{(N)}_s. \\end{equation}\nIf $\\gamma_N=s$ for some $s\\in \\mathcal{S}_N$, where $s\\le N^{1\/4}$, then (recall the second line of \\eqref{E:MIX}) $L^{(N)}_{T-}=\\alpha+\\frac{s}{N}$ and $\\frac{s}{N}\\ll 1$; thus $L^{(N)}_{T-}$ is close to $\\alpha$.  Hence $\\tau^\\beta_N>T$ (at least if $N>(\\beta-\\alpha)^{-4\/3}$)\nand $\\tau^\\alpha_N$ should be close to $T$; it should only take a short amount\nof time for $L^{(N)}$ to increase the extra distance (which is at most $s\/N$) past $\\alpha$.\n\\begin{lemma}\\label{L:TTimes} We have that\n\\begin{equation*} \\varlimsup_{N\\nearrow \\infty}\\sup_{\\substack{s\\in \\mathcal{S}_N \\\\ s\\le N^{1\/4}}}\\tilde \\mathbb{E}_N\\left[T-\\tau^\\alpha_N\\bigg|\\gamma_N\\right]\\chi_{\\{\\gamma_N=s\\}}=0. \\end{equation*}\n\\end{lemma}\n\\noindent Let's rigorously put all of these thoughts together.  Assume\nthat $N> (\\beta-\\alpha)^{-4\/3}$ and $0<\\gamma_N\\le N^{1\/4}$.  Then\n$0\\le \\tau^\\alpha_N\\le T\\le \\tau^\\beta_N$ and $0\\le L^{(N)}_{\\tau^\\alpha_N-}\\le L^{(N)}_{T-}= \\alpha+\\tfrac{\\gamma_N}{N}<\\beta$.  Hence\n\\begin{equation*} \\int_{s\\in [\\tau^\\alpha_N,\\tau^\\beta_N]\\cap [0,T)}e^{-\\textsf{R} s}d\\bar L^{(N)}_s\n= e^{-\\textsf{R} T}\\left(\\bar L^{(N)}_{T-}-\\bar L^{(N)}_{\\tau^\\alpha_N-}\\right) + \\int_{s\\in [\\tau^\\alpha_N,T)}\\left(e^{-\\textsf{R} s}-e^{-\\textsf{R} T}\\right) d\\bar L^{(N)}_s. \\end{equation*}\nNote that\n\\begin{equation*} \\bar L^{(N)}_{T-}=\\frac{L^{(N)}_{T-}-\\alpha}{\\beta-\\alpha} = \\frac{1}{\\beta-\\alpha}\\frac{\\gamma_N}{N} \\qquad \\text{and}\\qquad \\bar L^{(N)}_{\\tau^\\alpha_N-}=\\frac{\\left(L^{(N)}_{\\tau^\\alpha_N-}-\\alpha\\right)^+}{\\beta-\\alpha}. \\end{equation*}\nThus\n\\begin{equation*} \\int_{s\\in [\\tau^\\alpha_N,\\tau^\\beta_N]\\cap [0,T)}e^{-\\textsf{R} s}d\\bar L^{(N)}_s = \\frac{e^{-\\textsf{R} T}}{\\beta-\\alpha}\\frac{\\gamma_N}{N} + \\textsc{\\tiny E}_N \\end{equation*}\nwhere\n\\begin{equation*} \\textsc{\\tiny E}_N = -e^{\\textsf{R} T}\\frac{\\left(L^{(N)}_{\\tau^\\alpha_N-}-\\alpha\\right)^+}{\\beta-\\alpha} + \\int_{s\\in [\\tau^\\alpha_N,T)}e^{-\\textsf{R} s}\\left\\{ 1- e^{-\\textsf{R} (T-s)}\\right\\} d\\bar L^{(N)}_s. \\end{equation*}\nIf $\\tau^\\alpha_N>0$, then $L^{(N)}_{\\tau^\\alpha_N-}\\le \\alpha$.  Thus\n\\begin{equation*} \\left(L^{(N)}_{\\tau^\\alpha_N-}-\\alpha\\right)^+ \\le \\frac{\\gamma_N}{N}\\chi_{\\{\\tau^\\alpha_N=0\\}}\n=\\frac{\\gamma_N}{N}\\chi_{\\{T-\\tau^\\alpha_N=T\\}}\n\\le \\frac{1}{T}\\frac{\\gamma_N}{N}\\left(T-\\tau^\\alpha_N\\right). \\end{equation*}\nSimilarly,\n\\begin{multline*} 0\\le \\int_{s\\in [\\tau^\\alpha_N,T)}e^{-\\textsf{R} s}\\left\\{ 1- e^{-\\textsf{R} (T-s)}\\right\\} d\\bar L^{(N)}_s \\le \\textsf{R}(T-\\tau^\\alpha_N)\\left(\\bar L^{(N)}_{T-}-\\bar L^{(N)}_{\\tau^\\alpha_N-}\\right)\n\\le \\frac{\\textsf{R}}{\\beta-\\alpha}(T-\\tau^\\alpha_N)\\left(L^{(N)}_{T-}-\\alpha\\right)\\\\\n= \\frac{\\textsf{R}}{\\beta-\\alpha}(T-\\tau^\\alpha_N)\\frac{\\gamma_N}{N} \\end{multline*}\n(we use here the fact that $\\bar L_{\\tau^\\alpha_N-}\\ge 0$ and that $e^{-x}\\ge 1-x$ for all $x\\ge 0$).\nCombining things, we get that\n\\begin{equation*} |\\textsc{\\tiny E}_N|\\le \\frac{1}{\\beta-\\alpha}\\left\\{ \\frac{1}{T}+\\textsf{R}\\right\\}(T-\\tau^\\alpha_N)\\frac{\\gamma_N}{N} \\end{equation*}\non $\\left\\{ 0<\\gamma_N\\le N^{1\/4}\\right\\}$ if $N>(\\beta-\\alpha)^{-4\/3}$.\nWe then have\n\\begin{proof}[Proof of Lemma \\ref{L:hasymp}] For $s\\in \\mathcal{S}_N$ such that $s\\le N^{1\/4}$, we have that\n\\begin{equation*} \\mathcal{E}_1(s,N)= (\\beta-\\alpha)e^{\\textsf{R} T}\\frac{\\tilde \\mathbb{E}_N\\left[\\textsc{\\tiny E}_N\\big|\\gamma_N\\right]}{\\frac{\\gamma_N}{N}}\\chi_{\\{\\gamma_N=s\\}} \\le e^{\\textsf{R} T}\\left\\{ \\frac{1}{T} + \\textsf{R}\\right\\} \\mathbb{E}_N[T-\\tau^\\alpha_N|\\gamma_N]\\chi_{\\{\\gamma_N=s\\}} \\end{equation*}\nif $N>(\\beta-\\alpha)^{-4\/3}$.\nCombine \\eqref{E:AA}, the preceding calculations, and Lemma \\ref{L:TTimes}. \\end{proof}\n\nWe now need to prove Lemma \\ref{L:TTimes}.  This is a moderately complex step.\nThe first problem is that by conditioning on $\\gamma_N$, we are conditioning\non the value of $L^{(N)}$ near the \\emph{endpoint} of the interval $[0,T)$\nof interest.  The second problem is that we have a large amount\nof randomness; $L^{(N)}$ can be decomposed into $N$ (independent) processes,\none corresponding to each name.\n\nWe shall resolve these issues by using the martingale problem to decompose\n$L^{(N)}$ into a (reverse-time) zero-mean martingale and a term of bounded\nvariation\\footnote{Much of our notation will thus be in reverse time.}.  We will use a martingale inequality to show that the martingale\npart is small.  Thus the behavior of $L^{(N)}$ near $T$ will be given by\nthe bounded-variation part, which we can analyze via straightforward calculations.\n\nDefine now\n\\begin{equation*} Z^{(n)}_t \\overset{\\text{def}}{=} \\chi_{\\{\\tau_n< T-t\\}}=\\chi_{(t,\\infty]}(T-\\tau_n) \\qquad t\\in [0,T) \\end{equation*}\nfor each positive integer $n$ (note that the $Z^{(n)}$'s are right-continuous).\nAlso define $\\mathscr{G}_t \\overset{\\text{def}}{=} \\sigma\\{Z^{(n)}_s: 0\\le s\\le t,\\, n\\in \\{1,2\\dots\\}\\}$\nfor all $t\\in [0,T)$.  Observe that\n\\begin{equation*} L^{(N)}_{t-} = \\frac{1}{N}\\sum_{n=1}^N \\chi_{[0,t)}(\\tau_n)=\\frac{1}{N}\\sum_{n=1}^N Z^{(n)}_{T-t}. \\end{equation*}\nfor all $t\\in (0,T]$.\n\n\nLet's now localize in time.  Let $T^*\\in (0,T)$ be such that $F((T-T^*)-)>0$;\nAssumption \\ref{A:density} ensures that this is possible.\nFor all $t\\in [0,T^*]$, define\n\\begin{align*} \nA^{(n)}_t &= -\\int_{r\\in [T-t,T)} \\frac{1}{F(r)} Z^{(n)}_{(T-r)-}dF_r\\\\\nM^{(n)}_t&\\overset{\\text{def}}{=} Z^{(n)}_t-\\chi_{\\{\\tau_n<T\\}}-A^{(n)}_t \\end{align*}\n(essentially, $A^{(N)}$ is the integral of the hazard function).\nFor future reference, we calculate that for any $t\\in [0,T)$,\n\\begin{equation*} Z^{(n)}_{(T-t)-} = \\lim_{s\\searrow t}Z^{(n)}_{T-s} = \\lim_{s\\searrow t}\\chi_{\\{\\tau_n<s\\}} = \\lim_{s\\searrow t}\\chi_{(0,s)}(\\tau_n) = \\chi_{(0,t]}(\\tau_n). \\end{equation*}\nNote that by definition of $T^*$, \n\\begin{equation}\\label{E:Abound} \\left|\\frac{1}{F(r-)}Z^{(n)}_{(T-r)-}\\right|\\le \\frac{1}{F((T-T^*)-)}<\\infty \\end{equation}\nfor all $r\\in [T-T^*,T)$; thus $A^{(n)}$ is well-defined, finite, right-continuous, and it has left-hand limits.\n\\begin{lemma}\\label{L:wmx} For every $n\\in \\{1,2\\dots N\\}$, $M^{(n)}$ is a $\\tilde \\mathbb{P}_N$-zero-mean-martingale with\nrespect to $\\{\\mathscr{G}_t;\\, t\\in [0,T^*]\\}$; i.e., for $0\\le s\\le t\\le T^*$,\n$\\tilde \\mathbb{E}_N[M^{(n)}_t|\\mathscr{G}_s]=M^{(n)}_s$.\n\\end{lemma}\n\\begin{proof} Fix $n$ as specified.   Clearly $M^{(n)}$ is adapted to $\\{\\mathscr{G}_t;\\, t\\in [0,T^*]\\}$.\nBy \\eqref{E:Abound}, we have that $A^{(n)}$ is also bounded, so $M^{(n)}_t$ is $\\tilde \\mathbb{P}_N$-integrable for each $t\\in [0,T^*]$.\n\nWe next compute some transition probabilities.  Fix $s$ and $t$ in $[0,T^*]$ such that $s\\le t$.  Then $(t,\\infty]\\subset (s,\\infty]$, so $Z^{(n)}_t\\le Z^{(n)}_s$; hence $Z^{(n)}$ is nonincreasing.  This implies that\n\\begin{equation} \\label{E:Zmon} \\{Z^{(n)}_s=0\\}\\subset \\{Z^{(n)}_t=0\\} \\qquad \\text{and}\\qquad  \\{Z^{(n)}_t=1\\}\\subset \\{Z^{(n)}_s=1\\}. \\end{equation}\n\nFix $0\\le s_1<s_2\\dots s_n\\le s$ and $\\{z_n\\}_{n=1}^n\\subset \\{0,1\\}$.  From\n\\eqref{E:Zmon}, we immediately have that\n\\begin{align*} &\\tilde \\mathbb{P}_N\\left\\{ Z^{(n)}_t=0,\\, Z^{(n)}_s=0,\\, Z^{(n)}_{s_1}=z_1,\\, Z^{(n)}_{s_2}=z_2\\dots Z^{(n)}_{s_n}=z_n\\right\\} \\\\\n&\\qquad =\\tilde \\mathbb{P}_N\\left\\{ Z^{(n)}_s=0,\\, Z^{(n)}_{s_1}=z_1,\\, Z^{(n)}_{s_2}=z_2\\dots Z^{(n)}_{s_n}=z_n\\right\\} \\\\\n&\\tilde \\mathbb{P}_N\\left\\{ Z^{(n)}_t=1,\\, Z^{(n)}_s=0,\\, Z^{(n)}_{s_1}=z_1,\\, Z^{(n)}_{s_2}=z_2\\dots Z^{(n)}_{s_n}=z_n\\right\\} =0. \\end{align*}\nA similar computation which also uses the definition of $Z^{(n)}$ gives us that\n\\begin{multline*} \\tilde \\mathbb{P}_N\\left\\{ Z^{(n)}_t=1,\\, Z^{(n)}_s=1,\\, Z^{(n)}_{s_1}=z_1,\\, Z^{(n)}_{s_2}=z_2\\dots Z^{(n)}_{s_n}=z_n\\right\\} \\\\\n= \\tilde \\mathbb{P}_N\\left\\{ Z^{(n)}_t=1,\\, Z^{(n)}_{s_1}=z_1,\\, Z^{(n)}_{s_2}=z_2\\dots Z^{(n)}_{s_n}=z_n\\right\\}\n=\\tilde \\mathbb{P}_N\\{\\tau_n< T-t\\} \\prod_{k=1}^n \\delta_1(\\{z_k\\})\\end{multline*}\nA final computation (again using the definition of $Z^{(n)}$) gives us that\n\\begin{equation*} \\tilde \\mathbb{P}_N\\left\\{ Z^{(n)}_t=0,\\, Z^{(n)}_s=1,\\, Z^{(n)}_{s_1}=z_1,\\, Z^{(n)}_{s_2}=z_2\\dots Z^{(n)}_{s_n}=z_n\\right\\} =\\tilde \\mathbb{P}_N\\{T-t\\le \\tau_n< T-s\\} \\prod_{k=1}^n \\delta_1(\\{z_k\\})\\end{equation*}\n\nWith some manipulations, and using the fact that the $Z^{(n)}$'s are $\\tilde \\mathbb{P}_N$-independent, we get that\n\\begin{equation}\\label{E:condprob}\\begin{aligned} \\tilde \\mathbb{P}_N\\left\\{ Z^{(n)}_t=0\\big| \\mathscr{G}_s\\right\\} &= \\chi_{\\{0\\}}(Z^{(n)}_s)+\\frac{\\tilde \\mathbb{P}_N\\{T-t\\le \\tau_n< T-s\\}}{\\tilde \\mathbb{P}_N\\{\\tau_n<T-s\\}}\\chi_{\\{1\\}}(Z^{(n)}_s)\\\\\n\\tilde \\mathbb{P}_N\\left\\{ Z^{(n)}_t=1\\big| \\mathscr{G}_s\\right\\} &= \\frac{\\tilde \\mathbb{P}_N\\{\\tau_n< T-t\\}}{\\tilde \\mathbb{P}_N\\{\\tau_n< T-s\\}}\\chi_{\\{1\\}}(Z^{(n)}_s). \\end{aligned}\\end{equation}\nSince $s<T^*$, \n\\begin{equation*} \\tilde \\mathbb{P}_N\\{\\tau_n< T-s\\} = \\frac{\\alpha}{F(T)}\\mathbb{P}\\{ \\tau_n<T-s\\}  = \\frac{\\alpha}{F(T)}F((T-s)-)\\ge \\frac{\\alpha}{F(T)}F((T-T^*)-)>0; \\end{equation*}\nthus the expressions on the right of \\eqref{E:condprob} are well-defined.\nProceeding, we compute that\n\\begin{equation*} \\tilde \\mathbb{E}_N[Z^{(n)}_t|\\mathscr{G}_s] = \\frac{\\tilde \\mathbb{P}_N\\{\\tau_n< T-t\\}}{\\tilde \\mathbb{P}_N\\{\\tau_n< T-s\\}}Z^{(n)}_s \\end{equation*}\nand hence\\footnote{Under normalization, $\\mu$ and $\\tilde \\mu^*_\\alpha$ agree\non $\\mathscr{B}[0,T)$.}\n\\begin{equation}\\label{E:LZ} \\tilde \\mathbb{E}_N[Z^{(n)}_t|\\mathscr{G}_s]-Z^{(n)}_s = \\frac{\\tilde \\mathbb{P}_N\\{\\tau_n< T-t\\}-\\tilde \\mathbb{P}_N\\{\\tau_n< T-s\\}}{\\tilde \\mathbb{P}_N\\{\\tau_n< T-s\\}}Z^{(n)}_s =\\frac{F((T-t)-)-F((T-s)-)}{F((T-s)-)}Z^{(n)}_s.\\end{equation}\n\nAgain fix $s$ and $t$ in $[0,T^*]$ such that $s\\le t$. For each positive integer $m$, define $r^m_k \\overset{\\text{def}}{=} s+(k\/m)(t-s)$ for $k\\in \\{0,1\\dots m\\}$.  Using \\eqref{E:LZ}, we can write that $Z^{(n)}_t-Z^{(n)}_s = \\mathcal{A}_m+\\mathcal{M}_m$ where\n\\begin{equation*} \\mathcal{A}_m = \\sum_{k=0}^{m-1}\\left\\{ \\tilde \\mathbb{E}_n\\left[Z^{(n)}_{r^m_{k+1}}\\big|\\mathscr{G}_{r^m_k}\\right]-Z^{(n)}_{r^m_k}\\right\\} \\qquad \\text{and}\\qquad\n\\mathcal{M}_m = \\sum_{k=0}^{m-1}\\left\\{ Z^{(n)}_{r^m_{k+1}}-\\tilde \\mathbb{E}_n\\left[Z^{(n)}_{r^m_{k+1}}\\big|\\mathscr{G}_{r^m_k}\\right]\\right\\}. \\end{equation*}\nFor $C\\in \\mathscr{G}_s$,\n\\begin{equation}\\label{E:miss} \\tilde \\mathbb{E}_N\\left[\\left\\{ Z^{(n)}_t-Z^{(n)}_s-\\mathcal{A}_m\\right\\} \\chi_C\\right]\n=\\tilde \\mathbb{E}_N\\left[\\mathcal{M}_m\\chi_C\\right]=0. \\end{equation}\n\nWe now need to show that $\\tilde \\mathbb{P}_N$-a.s.,\n\\begin{equation}\\label{E:ggoal} \\lim_{m\\nearrow \\infty}\\mathcal{A}_m = -\\{A^{(n)}_t-A^{(n)}_s\\} \\end{equation}\nThis will require a bit of care.  We first rewrite $\\mathcal{A}_m$ as a integral;\n\\begin{equation*} \\mathcal{A}_m = -\\sum_{k=0}^{m-1}\\int_{r\\in [T-r^m_{k+1},T-r^m_k)}\\frac{1}{F((T-r^m_k)-)}Z^{(n)}_{r^m_k}dF_r = \\int_{r\\in [T-t,T-s)}\\phi^m(r,\\tau_n)dF_r \\end{equation*}\nwhere\n\\begin{equation*} \\phi^m(r,t') \\overset{\\text{def}}{=} \\sum_{k=0}^{m-1}\\chi_{[T-r^m_{k+1},T-r^m_k)}(r)\n\\frac{1}{F((T-r^m_k)-)}\\chi_{\\{t'< T-r^m_k\\}} \\end{equation*}\nfor all $r\\in [T-t,T-s)$ and $t'\\in I$.\nDefining\n\\begin{equation*} \\tilde \\phi(r,t') \\overset{\\text{def}}{=} \\frac{1}{F(r-)}\\chi_{(0,r)}(t') \\end{equation*}\nfor all $r\\in [T-t,T-s]$ and $t'\\in I$, we thus have that\n\\begin{equation*} \\phi^m(r,t') =\\sum_{k=0}^{m-1}\\chi_{[T-r^m_{k+1},T-r^m_k)}(r)\\tilde \\phi(T-r^m_k,t') \\end{equation*}\nfor all $r\\in [T-t,T-s)$ and $t'\\in I$.  For $r\\in [T-t,T-s)$, $F(r-)\\ge F((T-t)-)\\ge F((T-T^*)-)>0$; thus $\\tilde \\phi$ and the $\\phi^m$'s are all uniformly bounded.  It is fairly easy to see that \n\\begin{equation*} \\lim_{m\\to \\infty}\\phi^m(r,t') = \\frac{1}{F(r)}\\chi_{(0,r]}(t') \\end{equation*}\nfor all $r\\in [T-t,T-s)$ and all $t'\\in I$.  Thus by dominated convergence,\n\\begin{equation*} \\lim_{m\\to \\infty} \\mathcal{A}_m = -\\int_{r\\in [T-t,T-s)}\\frac{1}{F(r)}\\chi_{(0,r]}(\\tau_n)dF_r, \\end{equation*}\nand \\eqref{E:ggoal} follows.\n\nTaking the limit in \\eqref{E:miss}, we now have that\n\\begin{equation*} \\tilde \\mathbb{E}_N\\left[\\left\\{ (Z^{(n)}_t-A_t(\\tau_n))-(Z^{(n)}_s-A_s(\\tau_n))\\right\\} \\chi_C\\right]=0, \\end{equation*}\nwhich is the martingale property.\nFinally, since $M^{(n)}_0=0$, we have that $M^{(n)}$ is zero-mean.\n\\end{proof}\n\nLet's now recombine things.  Set\n\\begin{equation*} \\tilde M^{(N)}_t \\overset{\\text{def}}{=} \\frac{1}{N}\\sum_{n=1}^N M^{(n)}_t \\qquad \\text{and}\\qquad \\tilde A^{(N)}_t \\overset{\\text{def}}{=} \\frac{1}{N}\\sum_{n=1}^N A_t(\\tau_n) \\end{equation*}\nfor $t\\in [0,T)$.  Note that\n\\begin{equation}\\label{E:gmeas} L^{(N)}_{T-} = \\frac{1}{N}\\sum_{n=1}^N Z^{(n)}_0. \\end{equation}\nWe next rewrite $\\tau^\\alpha_N$ as a stopping time with respect to $\\{\\mathscr{G}_t;\\, t\\in [0,T)\\}$.  Set\n\\begin{equation*} \\varrho^\\alpha_N \\overset{\\text{def}}{=} \\inf\\left\\{ t\\in [0,T): L^{(N)}_{(T-t)-}\\le \\frac{\\lfloor N\\alpha \\rfloor}{N}\\right\\}\\wedge T = \\inf\\left\\{ t\\in [0,T): \\frac{1}{N}\\sum_{n=1}^N Z^{(n)}_t\\le \\frac{\\lfloor N\\alpha \\rfloor}{N}\\right\\}\\wedge T; \\end{equation*}\nthen\\footnote{If $L^{(N)}_0>\\alpha$, then $\\varrho^\\alpha_N=T$ and $\\tau^\\alpha_N=0$.\nAssume next that $L^{(N)}_0\\le \\alpha$.  Since $L^{(N)}$ is piecewise-constant and right-continuous,\nwe must have that $\\tau^\\alpha_N>0$.  At time $\\tau^\\alpha_N$, we have that\n$L^{(N)}_{\\tau^\\alpha_N}>\\alpha$ and $L^{(N)}_{\\tau^\\alpha_N-}\\le \\alpha$;\nsee Figure \\ref{fig:typ}.  Since $L^{(N)}$ takes values only in $\\mathbb{Z}\/N$, we\nhave that $L^{(N)}_{\\tau^\\alpha_N-}\\le \\tfrac{\\lfloor N\\alpha\\rfloor}{N}$\nand $L^{(N)}_{\\tau^\\alpha_N}\\ge \\tfrac{\\lfloor N\\alpha\\rfloor+1}{N}$.\nThus $\\rho^\\alpha_N=T-\\tau^\\alpha_N$, as claimed.} $\\varrho^\\alpha_N=T-\\tau^\\alpha_N$.  Furthermore, $\\varrho^\\alpha_N$ is a $\\{\\mathscr{G}_t;\\, t\\in [0,T)\\}$-stopping time.\n\nIt will help to truncate $\\varrho^\\alpha_N$ at $T^*$; set\n$\\tilde \\varrho^\\alpha_N \\overset{\\text{def}}{=} \\varrho^\\alpha_N\\wedge T^*$; this is also a $\\{\\mathscr{G}_t;\\, t\\in [0,T)\\}$-stopping time and $\\tilde \\varrho^\\alpha_N\\le T^*$.\nThus\n\\begin{equation*} L^{(N)}_{(T-\\tilde \\varrho^\\alpha_N)-} = L^{(N)}_{T-} + \\tilde A^{(N)}_{\\tilde \\varrho^\\alpha_N} + \\tilde M^{(N)}_{\\tilde \\varrho^\\alpha_N}. \\end{equation*}\nIf $\\gamma_N>0$, then $L^{(N)}_{T-}>\\alpha$, and since $\\tilde \\varrho^\\alpha_N\\le \\varrho^\\alpha_N$, we have that $L^{(N)}_{(T-\\tilde \\varrho^\\alpha_N)-}\\ge \\frac{\\lfloor N\\alpha \\rfloor}{N}$ and consequently\n\\begin{multline*} - \\tilde A^{(N)}_{\\tilde \\varrho^\\alpha_N} = L^{(N)}_{T-} - L^{(N)}_{(T-\\tilde \\varrho^\\alpha_N)-} + \\tilde M^{(N)}_{\\tilde \\varrho^\\alpha_N}\n\\le L^{(N)}_{T-} - \\frac{\\lfloor N\\alpha \\rfloor}{N} + \\tilde M^{(N)}_{\\tilde \\varrho^\\alpha_N}\n\\le L^{(N)}_{T-} - \\alpha + \\frac{1}{N}+ |\\tilde M^{(N)}_{\\tilde \\varrho^\\alpha_N}|\\\\\n\\le \\gamma_N + \\frac{1}{N}+ |\\tilde M^{(N)}_{\\tilde \\varrho^\\alpha_N}|. \\end{multline*}\n\nLet's now use the fact that $\\frac{1}{N}\\sum_{n=1}^N Z^{(n)}_{\\tilde \\varrho^\\alpha_N}\\ge \\frac{\\lfloor N\\alpha \\rfloor}{N}$ to bound $\\tilde A^{(N)}_{\\tilde \\varrho^\\alpha_N}$.  As we pointed out in the proof of Lemma \\ref{L:wmx}, the $Z^{(n)}$'s are nonincreasing.  Also, $F\\le 1$.  Thus for $N\\ge 2\/\\alpha$ (which implies that $\\lfloor N\\alpha \\rfloor\/N\\ge \\alpha\/2$),\nwe have the following string of inequalities.\n\\begin{multline*} -\\tilde A^{(N)}_{\\tilde \\varrho^\\alpha_N} \n=\\int_{r\\in [T-t,T)}\\frac{1}{F(r-)}\\left(\\frac{1}{N}\\sum_{n=1}^N Z^{(n)}_{(T-r)-}\\right)dF_r \n\\ge \\left(\\frac{1}{N}\\sum_{n=1}^N Z^{(n)}_{\\varrho^\\alpha_N-}\\right)\\int_{r\\in [T-\\tilde \\varrho^\\alpha_N,T)}dF_r\\\\\n\\ge \\left(\\frac{1}{N}\\sum_{n=1}^N Z^{(n)}_{\\varrho^\\alpha_N}\\right)\\left\\{ F(T-)-F((T-\\tilde \\varrho^\\alpha_N)-)\\right\\} \\ge \\frac{\\alpha}{2}\\mathfrak{f}(\\tilde \\rho^\\alpha_N) \\end{multline*}\nwhere we have defined\n\\begin{equation*} \\mathfrak{f}(\\varepsilon) \\overset{\\text{def}}{=} F(T-)-F((T-\\varepsilon)-) \\end{equation*}\nfor all $\\varepsilon\\in (0,T)$.  Note that $\\lim_{\\varepsilon \\searrow 0}\\mathfrak{f}(\\varepsilon)=0$, and, thanks to Assumption \\ref{A:density}, $\\mathfrak{f}(\\varepsilon)>0$ for $\\varepsilon\\in (0,T)$.\nThus\n\\begin{equation} \\label{E:CW} \\mathfrak{f}\\left(\\tilde \\varrho^\\alpha_N\\right)\\chi_{\\{\\gamma_N>0\\}}\\le \\frac{2}{\\alpha}\\left\\{ \\gamma_N + \\frac{1}{N} + \\left|\\tilde M^{(N)}_{\\tilde \\varrho^\\alpha_N}\\right|\\right\\}\\chi_{\\{\\gamma_N>0\\}}\n\\le \\frac{2}{\\alpha}\\left\\{ \\gamma_N^+ + \\frac{1}{N} + \\left|\\tilde M^{(N)}_{\\tilde \\varrho^\\alpha_N}\\right|\\right\\}\\end{equation}\nif $N>2\/\\alpha$.\n\\begin{proof}[Proof of Lemma \\ref{L:TTimes}]  We begin by taking conditional expectations of \\eqref{E:CW}.  Note that $\\gamma_N$ is $\\mathscr{G}_0$-measurable (see \\eqref{E:gmeas}).\nWe have\n\\begin{equation*} \\tilde \\mathbb{E}_N\\left[\\mathfrak{f}\\left(\\tilde \\varrho^\\alpha_N\\right)\\big|\\mathscr{G}_0\\right]\\chi_{\\{\\gamma_N>0\\}} \\le \\frac{2}{\\alpha}\\left\\{ \\gamma_N^+ + \\frac{1}{N} + \\tilde \\mathbb{E}_N\\left[\\left|\\tilde M^{(N)}_{\\tilde \\varrho^\\alpha_N}\\right|\\bigg| \\mathscr{G}_0\\right]\\right\\} . \\end{equation*}\nBy Jensen's inequality,\n\\begin{equation*} \\tilde \\mathbb{E}_N\\left[\\left|\\tilde M^{(N)}_{\\tilde \\varrho^\\alpha_N}\\right|\\bigg| \\mathscr{G}_0\\right]\\le \\tilde \\mathbb{E}_N\\left[\\left(\\tilde M^{(N)}_{\\tilde \\varrho^\\alpha_N}\\right)^2\\bigg| \\mathscr{G}_0\\right]^{1\/2} \\end{equation*}\n$\\tilde \\mathbb{P}_N$-a.s.\nWe can now use optional sampling;\n\\begin{multline*} \\tilde \\mathbb{E}_N\\left[\\left(\\tilde M^{(N)}_{\\tilde \\varrho^\\alpha_N}\\right)^2\\bigg| \\mathscr{G}_0\\right] \\le \\tilde \\mathbb{E}_N\\left[\\left(\\tilde M^{(N)}_{T_2^*}\\right)^2\\bigg| \\mathscr{G}_0\\right]\n= \\frac{1}{N^2}\\sum_{n=1}^N \\tilde \\mathbb{E}_N\\left[\\left(M^{(n)}_{T_2^*}\\right)^2\\bigg| \\mathscr{G}_0\\right]\\\\\n\\le \\frac{3}{N^2}\\sum_{n=1}^N \\left\\{ \\tilde \\mathbb{E}_N\\left[\\left(Z^{(n)}_{T_2^*}\\right)^2\\bigg| \\mathscr{G}_0\\right]+\\tilde \\mathbb{E}_N\\left[\\chi^2_{\\{\\tau_n<T\\}}\\bigg| \\mathscr{G}_0\\right]\n+\\tilde \\mathbb{E}_N\\left[\\left(A^{(n)}_{T^*}\\right)^2\\bigg| \\mathscr{G}_0\\right]\\right\\}\\\\\n\\le \\frac{3}{N}\\left\\{ 2 + \\frac{1}{F^2((T-T^*)-)}\\right\\}  \\end{multline*}\n$\\tilde \\mathbb{P}_N$-a.s.\nWe have used here the fact that the $M^{(n)}$'s are independent, the explicit\nformula for $M^{(n)}$, and \\eqref{E:Abound}.\nSummarizing thus far, we have that\n\\begin{equation*}  \\tilde \\mathbb{E}_N\\left[\\mathfrak{f}\\left(\\tilde \\varrho^\\alpha_N\\right)\\big|\\mathscr{G}_0\\right]\\chi_{\\{\\gamma_N>0\\}}\\le \\frac{2}{\\alpha}\\left\\{ \\left(L^{(N)}_{T-}-\\alpha\\right)^+ + \\frac{1}{N}  + \\sqrt{\\frac{3}{N}\\left\\{ 2 + \\frac{1}{F^2((T-T^*)-)}\\right\\} }\\right\\} \\end{equation*}\n$\\tilde \\mathbb{P}_N$-a.s.  As we pointed out earlier, $\\sigma\\{\\gamma_N\\} = \\sigma\\{L^{(N)}_{T-}\\} \\subset \\mathscr{G}_0$, so by iterated conditioning, we next have that\n\\begin{equation*}  \\tilde \\mathbb{E}_N\\left[\\mathfrak{f}\\left(\\tilde \\varrho^\\alpha_N\\right)\\big|\\gamma_N\\right]\\chi_{\\{\\gamma_N>0\\}}\\le \\frac{2}{\\alpha}\\left\\{ \\gamma_N^+ + \\frac{1}{N}  + \\sqrt{\\frac{3}{N}\\left\\{ 2 + \\frac{1}{F^2((T-T^*)-)}\\right\\}}\\right\\} \\end{equation*}\n\nFix now $\\varepsilon\\in (0,T^*)$.  If $\\tilde \\varrho^\\alpha_N<\\varepsilon$, then in fact $T-\\tau^\\alpha_N = \\tilde \\varrho^\\alpha_N<\\varepsilon$.  On the other hand, if $\\tilde \\varrho^\\alpha_N\\ge \\varepsilon$, then \n\\begin{equation*} \\gamma_N = L^{(N)}_{T-}-\\alpha \\ge \\frac{\\lfloor N\\alpha \\rfloor + 1}{N}-\\alpha >0; \\end{equation*}\nthus\n\\begin{equation*} \\chi_{\\{\\tilde \\varrho^\\alpha_N>\\varepsilon\\}} \\le \\frac{1}{\\mathfrak{f}(\\varepsilon)}\\mathfrak{f}(\\tilde \\varrho^\\alpha_N)\\chi_{\\{\\gamma_N>0\\}}. \\end{equation*}\nHence\n\\begin{multline*} \\tilde \\mathbb{E}_N\\left[T-\\tau^\\alpha_N\\bigg|\\gamma_N\\right]\n\\le \\varepsilon + T \\frac{\\tilde \\mathbb{E}_N\\left[\\mathfrak{f}\\left(\\tilde \\varrho^\\alpha_N\\right)\\chi_{\\{\\gamma_N>0\\}}\\bigg| L^{(N)}_{T-}\\right]}{\\mathfrak{f}(\\varepsilon)} \\\\\n\\le \\varepsilon + \\frac{2T}{\\alpha\\mathfrak{f}(\\varepsilon)}\\left\\{ \\gamma_N^+ + \\frac{1}{N} + \\sqrt{\\frac{3}{N}\\left\\{ 2 + \\frac{1}{F^2((T-T^*)-)}\\right\\}}\\right\\} \\end{multline*}\n$\\tilde \\mathbb{P}_N$-a.s. \nIn other words,\n\\begin{equation*} \\sup_{\\substack{s\\in \\mathcal{S}_N \\\\ s\\le N^{1\/4}}}\\tilde \\mathbb{E}_N\\left[T-\\tau^\\alpha_N\\big| \\gamma_N\\right]\\chi_{\\{\\gamma_N=s\\}}\\le \n\\varepsilon + \\frac{2T}{\\alpha\\mathfrak{f}(\\varepsilon)}\\left\\{ \\frac{1}{N^{3\/4}} + \\frac{1}{N} + \\sqrt{\\frac{3}{N}\\left\\{ 2 + \\frac{1}{F^2((T-T^*)-)}\\right\\} }\\right\\}. \\end{equation*}\nLet $N\\nearrow \\infty$ and then let $\\varepsilon\\searrow 0$.\n\\end{proof}\n\n\\section{Proofs}\\label{S:Proofs}\n\nWe here give the deferred proofs.\n\n\\begin{proof}[Proof of Lemma \\ref{L:probasymp}]  To begin, recall Stirling's formula.  Let $\\tilde \\mathcal{E}_1:(-1,\\infty)\\to \\mathbb{R}$ be defined by\n\\begin{equation*} \\Gamma(x+1) \\overset{\\text{def}}{=} \\int_{u=0}^\\infty u^x e^{-u}du = \\left(\\frac{x}{e}\\right)^x\\sqrt{2\\pi x}\\left\\{ 1+\\tilde \\mathcal{E}_1(x)\\right\\}; \\end{equation*}\nfor all $x>-1$; then $\\lim_{x\\to \\infty}|\\tilde \\mathcal{E}_1(x)|=0$.\nThen for any $s=n-N\\alpha\\in \\mathcal{S}_N$,\n\\begin{multline*} \\tilde \\mathbb{P}_N\\left\\{ \\gamma_N=s\\right\\}\n= \\tilde \\mathbb{P}_N\\left\\{ \\text{$n$ of the $\\tau$'s are in $[0,T)$ and $N-n$ are in $[T,\\infty)$}\\right\\} \\\\\n= \\binom{N}{n}\\alpha^n (1-\\alpha)^{N-n}\n= \\frac{\\Gamma(N+1)}{\\Gamma(N\\alpha+s+1)\\Gamma(N(1-\\alpha)-s+1)}\\alpha^{N\\alpha+s}(1-\\alpha)^{N(1-\\alpha)-s}\n= A(N)B(s,N) \\end{multline*}\nwhere\n\\begin{align*} A(N) &=  \\frac{\\Gamma(N+1)}{\\Gamma(N\\alpha+1)\\Gamma(N(1-\\alpha)+1)}\\alpha^{N\\alpha}(1-\\alpha)^{N(1-\\alpha)}\\\\\nB(s,N) &= \\frac{\\Gamma(N\\alpha+1)\\Gamma(N(1-\\alpha)+1)}{\\Gamma(N\\alpha+s+1)\\Gamma(N(1-\\alpha)-s+1)}\\alpha^s(1-\\alpha)^{-s}. \\end{align*}\nLet's now use Stirling's formula.\nWe have\n\\begin{align*} A(N)&= \\frac{\\left(\\frac{N}{e}\\right)^N}{\\left(\\frac{N\\alpha}{e}\\right)^{N\\alpha}\\left(\\frac{N(1-\\alpha)}{e}\\right)^{N(1-\\alpha)}}\\frac{\\sqrt{2\\pi N}}{\\sqrt{2\\pi N\\alpha}\\sqrt{2\\pi N(1-\\alpha)}}\\alpha^{N\\alpha}(1-\\alpha)^{N(1-\\alpha)} \\\\\n&\\qquad \\times \\frac{1+\\tilde \\mathcal{E}_1(N)}{\\{1+\\tilde \\mathcal{E}_1(N\\alpha)\\}\\{1+\\tilde \\mathcal{E}_1(N(1-\\alpha))\\}} \\\\\n&=\\frac{1}{\\sqrt{2\\pi N\\alpha(1-\\alpha)}}\\frac{1+\\tilde \\mathcal{E}_1(N)}{\\{1+\\tilde \\mathcal{E}_1(N\\alpha)\\}\\{1+\\tilde \\mathcal{E}_1(N(1-\\alpha))\\}} \\end{align*}\nThus\n\\begin{equation*} A(N)=\\frac{1+\\tilde \\mathcal{E}_2(N)}{\\sqrt{2\\pi N\\alpha(1-\\alpha)}}. \\end{equation*}\nwhere $\\lim_{N\\to \\infty}\\tilde \\mathcal{E}_2(N)=0$.\nTo find the asymptotics of $B$, we first let $\\tilde \\mathcal{E}_3:(-1,\\infty)$ be such that\n\\begin{equation*} \\ln (1+x) = x+\\tilde \\mathcal{E}_3(x). \\qquad x>-1 \\end{equation*}\nThen there is a $K>0$ such that $|\\tilde \\mathcal{E}_3(x)|\\le K x^2$ for all $x\\in (-1\/2,1\/2)$.\nAgain using Stirling's formula, we have that\n\\begin{align*} B(s,N) &= \\frac{\\left(\\frac{N\\alpha}{e}\\right)^{N\\alpha}}{\\left(\\frac{N\\alpha+s}{e}\\right)^{N\\alpha+s}}\\frac{\\left(\\frac{N(1-\\alpha)}{e}\\right)^{N(1-\\alpha)}}{\\left(\\frac{N(1-\\alpha)-s}{e}\\right)^{N(1-\\alpha)-s}}\\sqrt{\\left(\\frac{N\\alpha}{N\\alpha+s}\\right)\\left(\\frac{N(1-\\alpha)}{N(1-\\alpha)-s}\\right)}\\alpha^s (1-\\alpha)^{-s}\\\\\n&\\qquad \\times \\frac{1+\\tilde \\mathcal{E}_1(N\\alpha)}{\\left\\{ 1+\\tilde \\mathcal{E}_1(N\\alpha+s)\\right\\}\\left\\{ 1+\\tilde \\mathcal{E}_1(N(1-\\alpha)-s)\\right\\}}\\\\\n      &= \\left(\\frac{\\alpha}{\\alpha+\\frac{s}{N}}\\right)^{N\\alpha+s}\\left(\\frac{1-\\alpha}{1-\\alpha-\\frac{s}{N}}\\right)^{N\\alpha-s}\\frac{1}{\\sqrt{\\left(1+\\frac{s}{\\alpha N}\\right)\\left(1-\\frac{s}{N(1-\\alpha)}\\right)}}\\\\\n&\\qquad \\times \\frac{1+\\tilde \\mathcal{E}_1(N\\alpha)}{\\left\\{ 1+\\tilde \\mathcal{E}_1(N\\alpha+s)\\right\\}\\left\\{ 1+\\tilde \\mathcal{E}_1(N(1-\\alpha)-s)\\right\\}}\\\\\n& = \\frac{1}{\\left(1+\\frac{s}{N\\alpha}\\right)^{N\\alpha+s}\\left(1-\\frac{s}{N(1-\\alpha)}\\right)^{N(1-\\alpha)-s}} \\frac{1}{\\sqrt{\\left(1+\\frac{s}{\\alpha N}\\right)\\left(1-\\frac{s}{N(1-\\alpha)}\\right)}}\\\\\n&\\qquad \\times \\frac{1+\\tilde \\mathcal{E}_1(N\\alpha)}{\\left\\{ 1+\\tilde \\mathcal{E}_1(N\\alpha+s)\\right\\}\\left\\{ 1+\\tilde \\mathcal{E}_1(N(1-\\alpha)-s)\\right\\}}\\\\\n&= \\exp\\left[-(N\\alpha+s)\\left(\\frac{s}{N\\alpha} + \\tilde \\mathcal{E}_2\\left(\\frac{s}{N\\alpha}\\right)\\right)- (N(1-\\alpha)-s)\\left(-\\frac{s}{N(1-\\alpha)} + \\tilde \\mathcal{E}_2\\left(\\frac{-s}{N(1-\\alpha)}\\right)\\right)\\right]\\\\\n&\\qquad \\times \\frac{1}{\\sqrt{\\left(1+\\frac{s}{\\alpha N}\\right)\\left(1-\\frac{s}{N(1-\\alpha)}\\right)}}\\times \\frac{1+\\tilde \\mathcal{E}_1(N\\alpha)}{\\left\\{ 1+\\tilde \\mathcal{E}_1(N\\alpha+s)\\right\\}\\left\\{ 1+\\tilde \\mathcal{E}_1(N(1-\\alpha)-s)\\right\\}} \\\\\n&= \\exp\\left[-\\tilde \\mathcal{E}_4(s,N)\\right]\\frac{1}{\\sqrt{\\left(1+\\frac{s}{\\alpha N}\\right)\\left(1-\\frac{s}{N(1-\\alpha)}\\right)}}\\\\\n&\\qquad \\times \\frac{1+\\tilde \\mathcal{E}_1(N\\alpha)}{\\left\\{ 1+\\tilde \\mathcal{E}_1(N\\alpha+s)\\right\\}\\left\\{ 1+\\tilde \\mathcal{E}_1(N(1-\\alpha)-s)\\right\\}} \\end{align*}\nwhere\n\\begin{equation*} \\tilde \\mathcal{E}_4(s,N) \\overset{\\text{def}}{=} \\frac{s^2}{N}\\left(\\frac{1}{\\alpha}+\\frac{1}{1-\\alpha}\\right) + (N\\alpha+s)\\tilde \\mathcal{E}_3\\left(\\frac{s}{N\\alpha}\\right)+(N(1-\\alpha)-s)\\tilde \\mathcal{E}_3\\left(-\\frac{s}{N(1-\\alpha)}\\right). \\end{equation*}\n\nLet's now combine things together.  We have that\n\\begin{equation*} \\tilde \\mathbb{P}_N\\{\\gamma_N=s\\}-\\frac{1}{\\sqrt{2\\pi N\\alpha(1-\\alpha)}} = \\frac{\\tilde \\mathcal{E}_5(s,N)-1}{\\sqrt{2\\pi N\\alpha(1-\\alpha)}} \\end{equation*}\nwhere\n\\begin{align*} \\tilde \\mathcal{E}_5(s,N)&=\\frac{1+\\tilde \\mathcal{E}_1(N)}{\\{1+\\tilde \\mathcal{E}_1(N\\alpha)\\}\\{1+\\tilde \\mathcal{E}_1(N(1-\\alpha))\\}}\\exp\\left[-\\tilde \\mathcal{E}_4(s,N)\\right]\\frac{1}{\\sqrt{\\left(1+\\frac{s}{\\alpha N}\\right)\\left(1-\\frac{s}{N(1-\\alpha)}\\right)}}\\\\\n&\\qquad \\times\\frac{1+\\tilde \\mathcal{E}_1(N\\alpha)}{\\left\\{ 1+\\tilde \\mathcal{E}_1(N\\alpha+s)\\right\\}\\left\\{ 1+\\tilde \\mathcal{E}_1(N(1-\\alpha)-s)\\right\\}}. \\end{align*}\nNote that if $|s|\\le N^{1\/4}$, then\n\\begin{equation*} \\left|\\frac{s}{N\\alpha}\\right|\\le \\frac{1}{\\alpha N^{3\/4}} \\qquad \\text{and}\\qquad \\left|\\frac{s}{N(1-\\alpha)}\\right|\\le \\frac{1}{(1-\\alpha) N^{3\/4}}. \\end{equation*}\nThus if $|s|\\le N^{1\/4}$, then for $N$ large enough\n\\begin{equation*} |\\tilde \\mathcal{E}_4(s,N)| \\le \\frac{K}{\\sqrt{N}}. \\end{equation*}\nThe claimed statement follows.\n\\end{proof}\n\n\nLet's now start to prove Proposition \\ref{P:explicit}.\nFirst, define\n\\begin{equation*} \\mathfrak{I}_\\circ(\\alpha') \\overset{\\text{def}}{=} \\inf\\left\\{ H(\\tilde \\mu|\\mu): \\tilde \\mu\\in \\mathscr{P}(I), \\tilde \\mu[0,T)=\\alpha'\\right\\} \\end{equation*}\nfor all $\\alpha'\\in [0,1]$.\n\\begin{lemma}\\label{L:hear} We have that\n\\begin{equation*} \\mathfrak{I}_\\circ(\\alpha') = \\hbar(\\alpha',F(T-))=H(\\tilde \\mu^*_{\\alpha'}|\\mu), \\end{equation*}\nwhere $\\tilde \\mu_{\\alpha'}^*$ is given by \\eqref{E:dacc}.\n\\end{lemma}\n\\begin{proof}\nFix $\\mu'\\in \\mathscr{P}(I)$ such that $\\mu'[0,T)=\\alpha'$.\nIf $\\mu'$ is not absolutely continuous\nwith respect to $\\mu$, then $H(\\mu'|\\mu)=\\infty$; thus we assume\nthat $\\mu'$ is absolutely continuous with respect to $\\mu$.\nDefine\n\\begin{equation*} f(x) \\overset{\\text{def}}{=} \\begin{cases} x \\ln x &\\text{for $x>0$} \\\\\n0 &\\text{if $x=0$.} \\end{cases}\\end{equation*}\nThen $f$ is convex on $[0,\\infty)$.  Recall that Assumptions \\ref{A:density} and \\ref{A:IG} imply that $F(T)\\in (0,1)$.  Thus we can write (using Jensen's inequality) that\n\\begin{multline*} H(\\mu'|\\mu) = \\int_{t\\in I}f\\left(\\frac{d\\mu'}{d\\mu}(t)\\right)\\mu(dt)\\\\\n=\\mu[0,T)\\int_{t\\in [0,T)}f\\left(\\frac{d\\mu'}{d\\mu}(t)\\right)\\frac{\\mu(dt)}{\\mu[0,T)} + \\mu[T,\\infty]\\int_{t\\in [T,\\infty]}f\\left(\\frac{d\\mu'}{d\\mu}(t)\\right)\\frac{\\mu(dt)}{\\mu[T,\\infty]} \\\\\n\\ge \\mu[0,T)f\\left(\\int_{t\\in [0,T)}\\frac{d\\mu'}{d\\mu}(t)\\frac{\\mu(dt)}{\\mu[0,T)}\\right) + \\mu[T,\\infty]f\\left(\\int_{t\\in [T,\\infty]}\\frac{d\\mu'}{d\\mu}(t)\\frac{\\mu(dt)}{\\mu[T,\\infty]}\\right) \\\\\n= \\mu[0,T)f\\left(\\frac{\\mu'[0,T)}{\\mu[0,T)}\\right) + \\mu[T,\\infty]f\\left(\\frac{\\mu'[T,\\infty]}{\\mu[T,\\infty]}\\right) = \\hbar(\\mu'[0,T),\\mu[0,T)). \\end{multline*}\nWe have equality here if and only if $\\mu$-a.s.\n\\begin{equation*} \\frac{d\\mu'}{d\\mu}(t) = C_1\\chi_{[0,T)}(t) + C_2\\chi_{[T,\\infty]}(t), \\end{equation*}\nwhich holds if and only if $\\mu' = \\tilde \\mu^*_{\\alpha'}$.  Collecting things together,\nwe have the claimed result.\\end{proof}\n\n\\begin{proof}[Proof of Proposition \\ref{P:explicit}]\nIn light of Lemma \\ref{L:hear}, we need to show that\n\\begin{equation} \\label{E:minc} \\inf_{\\alpha'\\ge \\alpha}\\mathfrak{I}_\\circ(\\alpha')=\\mathfrak{I}(\\alpha). \\end{equation}\nTo do this, we observe from that $\\mathfrak{I}_\\circ$ is differentiable and that\n\\begin{equation*} \\mathfrak{I}_\\circ'(\\alpha') = \\ln \\left(\\frac{\\alpha'}{1-\\alpha'}\\frac{1-F(T-)}{F(T-)}\\right)=\\ln \\left(1+\\frac{\\alpha'-F(T-)}{F(T-)(1-\\alpha')}\\right) \\end{equation*}\nfor all $\\alpha'\\in (0,1)$.   Thus $\\mathfrak{I}'_\\circ(\\alpha')>0$ if $\\alpha'>F(T-)$.  Assumption \\ref{A:IG} then gives us \\eqref{E:minc}.  Combining our arguments, we get the claimed result.\n\\end{proof}\n\n\\section{Appendix: Measurability}\nWe here verify that $\\textbf{P}^{\\text{prot}}_N$ is measurable.\nLet $D_+$ be the collection of nondecreasing functions $\\phi:\\mathbb{R}\\to [0,1]$ which are right-continuous and have left-hand limits and for which $\\phi(t)=0$ for $t<0$.\nFor $\\mu'\\in \\mathscr{P}(I)$ and $t\\ge 0$, define $\\iota(\\mu')(t) \\overset{\\text{def}}{=} \\mu'[0,t]$ for $t\\ge 0$ and $\\iota(\\mu')(t)=0$ for $t<0$.  In particular, $L^{(N)}=\\iota(\\nu^{(N)})$.  It is clear that $\\iota:\\mathscr{P}(I)\\to D_+$ and is a bijection (note that $\\mu'\\{\\infty\\} = 1- \\lim_{t\\nearrow \\infty}\\iota(\\mu')(t)$; this\nallows us to recover $\\mu'\\{\\infty\\}$ when writing down the inverse of $\\iota$).  We can then topologize $D_+$ by pushing the topology of $\\mathscr{P}(I)$ forward through $\\iota$; thus $\\iota$ is continuous.   We also note that $\\{\\ell_n\\}_{n=1}^\\infty \\subset D_+$ converges to $\\ell\\in D_+$ if and only if $\\lim_{n\\to \\infty}\\ell_n(t)=\\ell(t)$\nfor all points $t\\in [0,\\infty)$ at which $\\ell$ is continuous.  Thus $L^{(N)}$ is a $D_+$-valued random variable.  We next define\n$\\Phi^\\circ_1:D_+\\to D_+$ as \n\\begin{equation*} \\Phi^\\circ_1(\\phi)(t) \\overset{\\text{def}}{=} \\frac{(\\phi(t)-\\alpha)^+-(\\phi(t)-\\beta)^+}{\\beta-\\alpha} \\qquad t\\in \\mathbb{R} \\end{equation*}\nfor all $\\phi\\in D_+$.  Thus $\\bar L^{(N)}=\\Phi^\\circ_1(L^{(N)})$.  By the above characterization of convergence in $D_+$,\nwe see that $\\Phi^\\circ_1$ is continuous; thus $\\bar L^{(N)}$ is also a $D_+$-valued\nrandom variable.  Finally, define $\\Phi^\\circ_2 \\overset{\\text{def}}{=} D_+\\to \\mathbb{R}$ as \n\\begin{equation*} \\Phi^\\circ_2(\\ell) \\overset{\\text{def}}{=} \\int_{s\\in [0,T)} e^{-\\textsf{R} s}d\\ell(s) = e^{-\\textsf{R} T}\\ell(T-)+ \\textsf{R} \\int_{s\\in (0,T)} e^{-\\textsf{R} s}\\ell(s)ds \\end{equation*}\nfor all $\\ell\\in D_+$ (we define the $d\\ell$ integral as a Lebesgue-Stieltjes\nintegral).  Then $\\textbf{P}^{\\text{prot}}_N = \\Phi^\\circ_2(\\bar L^{(N)})$.  We claim that\n$\\textbf{P}^{\\text{prot}}_N$ is measurable (from $D_+$ to $\\mathbb{R}$).\nLet $\\zeta\\in C^\\infty(\\mathbb{R};[0,1])$ be such that $\\zeta(t)=1$\nfor $t\\le -1$ and $\\zeta(t)=0$ for $t\\ge 0$.  For each positive integer $n$\nand each $\\ell\\in D_+$, set\n\\begin{equation*} \\tilde \\Phi^n_2(\\ell) \\overset{\\text{def}}{=} \\int_{s\\in \\mathbb{R}} e^{-\\textsf{R} s}\\zeta(n(s-T))d\\ell(s) = \\int_{s=0}^\\infty e^{-\\textsf{R} s}\\left\\{ \\textsf{R} \\zeta(n(s-T))- n\\dot \\zeta(n(s-T))\\right\\} \\ell(s)ds. \\end{equation*}\nClearly $\\tilde \\Phi^n_2:D_+\\to \\mathbb{R}$ is continuous.  Furthermore, by dominated convergence, $\\lim_{n\\to \\infty}\\tilde \\Phi^n_2(\\ell) = \\Phi^\\circ_2(\\ell)$ for each $\\ell$ (i.e., pointwise on $D_+$).  Being the pointwise limit of continuous functions, $\\Phi^\\circ_2$ is thus measurable.\n\nCombining all of these arguments, we conclude that $\\textbf{P}^{\\text{prot}}_N$ is indeed a $\\mathbb{R}$-valued random variable.  Clearly $\\bar L^{(N)}_T\\le 1$, so $0\\le\\textbf{P}^{\\text{prot}}_N\\le 1$.\n\n\\bibliographystyle{alpha}\n\\def$'$} \\def\\cprime{$'$} \\def\\cprime{$'$} \\def\\cprime{$'${$'$} \\def$'$} \\def\\cprime{$'$} \\def\\cprime{$'$} \\def\\cprime{$'${$'$} \\def$'$} \\def\\cprime{$'$} \\def\\cprime{$'$} \\def\\cprime{$'${$'$} \\def$'$} \\def\\cprime{$'$} \\def\\cprime{$'$} \\def\\cprime{$'${$'$}\n  \\def\\polhk#1{\\setbox0=\\hbox{#1}{\\ooalign{\\hidewidth\n  \\lower1.5ex\\hbox{`}\\hidewidth\\crcr\\unhbox0}}}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{sec.intro}\n\nThe chemostat is a biotechnological process of continuous culture developed in the 50s \\citep{monod1950a,novick1950a} and which is at the heart of several industrial applications as well as laboratory devices \\citep{smith1995a}. Bioreactors operating under this mode are maintained under perfect mixing conditions and usually at large bacterial population sizes.\n\nThese features allow such processes to be modeled by ordinary (deterministic) differential systems since, in large populations and under certain conditions, demographic randomness can be neglected. Moreover, perfect mixing conditions permit us to neglect the spatial distribution and express these models in terms of mean concentration in the chemostat. In its simplest version, the chemostat model is expressed as a system of two coupled ordinary differential equations respectively for biomass and substrate concentrations \\citep{smith1995a}. This approach extends to the case of several bacterial species and several substrates. The simplicity of such models makes possible the development of efficient tools for automatic control and the improvement of the associated biotechnological processes. \nHowever, it is increasingly necessary to develop models beyond the standard assumption of perfect mixing with a bacterial population possessing uniform characteristics. For this purpose, several paths are available which take into account the different sources of randomness or the structuring of the bacterial population and its discrete nature. All these aspects have been somewhat neglected in previous models.\n\nIn addition, the  recent development of so-called ``omics'' approaches such as genomics and large-scale DNA sequencing technology are the basis of a renewed interest in chemostat techniques \\citep{hoskisson2005a}. These bacterial cultures may also be considered as laboratory models to study selection phenomena and evolution in bacterial ecosystems.\n\nBeyond classical models based on systems of (deterministic) ordinary differential equations (ODE) which neglect any structuring of the bacterial populations, have also appeared in the 60's and 70's bacterial growth models structured  in size or mass based on integro-differential equations (IDE) \\citep{fredrickson1967a,ramkrishna1979a}, \u200b\u200bsee also the monograph \\cite{ramkrishna2000a} on these so-called population balance equations for  growth-frag\\-men\\-tation models.\n\n\nVarious research papers have been devoted to the stochastic modeling of the chemostat.\n\\cite{crump1979a} propose a model of pure jump for the biomass growth coupled with a differential equation for the substrate evolution. \\cite{stephanopoulos1979a} propose a model with randomly fluctuating dilution rate, hence the noise is rather of an environmental nature, whereas in the previous model it is rather demographic. \\cite{grasman2005a} propose a chemostat model at three trophic levels where randomness appears only in the upper trophic level. \\cite{imhof2005a} also propose a  stochastic chemostat model but, as in previous models, the noise is simply ``added'' to the classical deterministic model. In contrast, in \\cite{campillo2011chemostat} article the demographic noise emerges from a description of the dynamics at the microscopic level.\n\nIn recent years, many models for the evolution in chemostats have been proposed either using integro-differential equations \\citep{diekmann-odo2005a,mirrahimi2012a,mirrahimi2012b} or individual-based models (IBM) \\citep{champagnat2013a}.\n\nThere are also computer models, for example \\cite{lee-min-woo2009a} propose an IBM  structured in mass for a `` batch'' culture process. In this model, as in \\citep{champagnat2013a} the dynamics of the substrate is described by deterministic differential equations. \nIndeed, the difference in scale between a bacterial cell and a substrate molecule guaranties that, at the scale of the bacterial population, the dynamics of the substrate can be correctly represented by the fluid limit model, while the dynamics of the bacterial population is discrete and random.\n\n\nWe focus here on an individual-based model (IBM) of the chemostat. In contrast to deterministic models with continuous variables, in IBMs all variables, or at least some of them, are stochastic and discrete. These models are generally cumbersome in terms of simulation and difficult to analyze mathematically, but they can be useful in accounting for phenomena inaccessible in earlier models. The majority of IBMs are described initially in natural languages with simple rules. From there they are described as  computer models in terms of algorithms, it is this approach that is often termed IBM.\nNonetheless, they can also be described mathematically using a Markov process.\nThe advantage of this approach is to allow the mathematical analysis of the IBM.\nIn particular, as we will see here,  the convergence of the IBM to an integro-differential model can be demonstrated.\n The latter approach has been developed in a series of papers: for a simple model of position \\citep{fournier2004a}, for the evolution of trait structured population \\citep{champagnat2006b}, which is then extended to take into account the age of individuals \\citep{vietchitran2006a,vietchitran2008a}. More recently \\citet{champagnat2013a} proposed a chemostat model with multiple resources where the bacterial population has a genetic trait subject to evolution.\n\n \nIn the context of a growth-fragmentation model, \\cite{hatzis1995a}  proposed an IBM, without  substrate variables, and draw a parallel between this model and an integro-differential model.\n\n\n\n\n\\medskip\n\nIn Section \\ref{sec.model} we introduce the IBM where each individual in the bacterial population is explicitly represented by its mass. We describe the phenomenon which the model will take into account at a microscopic scale:   \nindividual cell growth,  cell division, up-take \n(substrate and bacteria are constantly withdrawn from the chemostat vessel), as well as the individual consumption described as a coupling with the ordinary differential equation which models the dynamics of the substrate. Then we describe the associated exact Monte Carlo algorithm, noting that this algorithm is asynchronous in time, i.e. different events occur at random instants which are not predetermined.\n\nIn Section \\ref{sec.notations} we  introduce some notation, then in Section \\ref{sec.processus.microscopique} we construct the stochastic process associated with the IBM as a Markov process with values in the space of finite measures over the state-space of masses.\n\nIn Section \\ref{sec.convergence} we prove the convergence, in large population limit, of the IBM towards an integro-differential equation of the population-balance equation type \\citep{fredrickson1967a,ramkrishna1979a,ramkrishna2000a}  coupled with an equation for the dynamics of the substrate. Finally in Section \\ref{sec.simulations}  we present several numerical simulations.\n\n\n\\section{The model}\n\\label{sec.model}\n\n\\subsection{Description of the dynamics}\n\nWe consider an \\emph{individual-based model (IBM) structured in mass} where the\nbacterial population is represented as individuals growing in a perfectly mixed vessel of volume $V$ (l). Each individual is solely characterized by its mass $x\\in\\X \\eqdef [0,{\\textrm{max}}]$, this model does not take into account spatialization. At time $t$ the system is characterized by the pair:\n\\begin{align}\n\\label{eq.xi}\n  (S_{t},\\nu_{t})\n\\end{align}\nwhere\n\\begin{enumerate}\n\n\\item $S_{t}$ is the \\emph{substrate concentration} (mg\/l) which is assumed to be uniform in the vessel;\n\n\\item $\\nu_{t}$ is the \\emph{bacterial population}, that is $N_{t}$ individuals\nand the mass of the individual number $i$ will be denoted  $x^i_{t}$ (mg)  \nfor $i=1,\\dots,N_{t}$. \nIt will be convenient to represent the population $\\{x^i_{t}\\}_{i=1,\\dots,N_{t}}$ at time $t$ as the following punctual measure:\n\\begin{align}\n\\label{eq.nu}\n  \\nu_t(\\rmd x)=\\sum_{i=1}^{N_t}\\delta_{x_t^i}(\\rmd x)\\,.\n\\end{align}\n\\end{enumerate}\n\n\nThe dynamics of the chemostat combines \\emph{discrete evolutions}, cell division and bacterial up-take, as well as \\emph{continuous evolutions}, the growth of each individual and the dynamics of the substrate. We now describe the four components of the dynamics, first the discrete ones and then the continuous ones which occur between the discrete ones.\n\n\n\n\\begin{enumerate}\n\\vskip0.8em\n\\item{\\textbf{Cell division}} --\n\\emph{Each individual of mass $x$ divides at rate $\\lambda(s,x)$ into two\nindividuals of respective masses  $\\alpha\\,x$ and $(1-\\alpha)\\,x$:\n\\begin{center}\n\\includegraphics[width=5cm]{fig_division1.pdf}\n\\end{center}\nwhere $\\alpha$ is distributed according to a given probability distribution  $Q(\\rmd\\alpha)$ on $[0,1]$,\nand  $s$ is the substrate concentration.} \n\n\\medskip\n\nFor instance, the function $\\lambda(s,x)$ does not depend on  the substrate concentration $s$ and could be of the following form which will be used in the simulation presented in Section \\ref{sec.simulations}:\n\\begin{center}\n\\includegraphics[width=4cm]{fig_function_lambda.pdf}\n\\end{center} \nThus, below a certain mass $m_{\\textrm{\\tiny\\rm div}}$ it is assumed that the cell cannot divide. There are models where the rate also depends on the concentration $s$, see for example  \\citep{daoutidis2002a,henson2003b}.\n\n\\medskip\n\nWe suppose that the distribution $Q(\\rmd\\alpha)$ is symmetric with respect to $\\frac{1}{2}$, i.e. $Q(\\rmd\\alpha)=Q(1-\\rmd \\alpha)$. It also may admit a density  $Q(\\rmd \\alpha)=q(\\alpha)\\,\\rmd \\alpha$ with the same symmetry:\n\\begin{center}\n\\includegraphics[width=4cm]{fig_division2.pdf}\n\\end{center}\n\nThus, the  division kernel of an individual of mass $x$ is $ K(x,\\rmd y) = Q(\\frac{1}{x}\\,\\rmd y)$ with support $[0,x]$. In the case of perfect mitosis, an individual of mass $x$ is divided into two individuals of masses $\\frac x 2$ and then $Q(\\rmd \\alpha)=\\delta_{1\/2}(\\rmd \\alpha)$. \n\n\\medskip\n\n\\emph{It is therefore assumed that, relative to their mass, the division kernel is the same for all individuals. This allows us to reduce the model to a single division kernel. More complex scenarios can also be investigated.} \n\n\n\\vskip0.8em\n\\item{\\textbf{Up-take}} --\n\\emph{Each individual is withdrawn from the chemostat at rate~$D$.}\nOne places oneself in the framework of a \\emph{perfect mixing} hypothesis, where individuals are uniformly distributed in the volume $V$ independently from their mass. During a time step $\\delta$, a total volume of $D\\,V\\,\\delta$ is withdrawn from the chemostat:\n\\begin{center}\n\\includegraphics[width=10cm]{fig_removal_rate.pdf}\n\\end{center}\nand therefore, if we assume that all individuals have the same volume considered as negligible, during this time interval $\\delta$, an individual has a probability $D\\,\\delta$ to be withdrawn from the chemostat, $D$ is the dilution rate. This rate could possibly depend on the mass of the individual.\n\n\n\\end{enumerate}\nWhen the division of an individual occurs, the size of the population instantaneously jumps   from $N_{t}$ to $N_{t}+1$; when an individual is withdrawn from the vessel, the size of the population jumps instantaneously from $N_{t}$ to $N_{t}-1$; between each discrete event the size $N_{t}$ remains constant and the chemostat evolves according to the following two continuous mechanisms:\n\\begin{enumerate}\n\\setcounter{enumi}{2}\n\\vskip0.8em\n\\item {\\textbf{Growth of each individual}} --\n\\emph{Each individual of mass $x$ growths at speed $\\rho_{\\textrm{\\tiny\\rm\\!\\! g}}(S_{t},x)$}:\n\\begin{align}\n\\label{eq.masses}\n   \\dot x^i_{t} \n   =\n   \\rho_{\\textrm{\\tiny\\rm\\!\\! g}}(S_{t},x^i_{t})\\,, \\quad i=1,\\dots,N_{t}\n\\end{align}\nwhere $\\rho_{\\textrm{\\tiny\\rm\\!\\! g}}:\\R^2_{+}\\mapsto\\R_{+}$ is given. For the simulation we will consider the following Gompertz model:\n\\begin{align*}\n   \\rho_{\\textrm{\\tiny\\rm\\!\\! g}}(s,x) \\eqdef r(s)\\,\\log\\Big(\\frac{{\\textrm{max}}}{x}\\Big)\\,x\n\\end{align*}\nwhere the growth rate $r(s)$ depends on the substrate concentration according to the Monod kinetics:\n\\begin{align*}\n    r(s)=r_{\\textrm{\\tiny\\rm max}}\\,\\frac{s}{k_r+s}\n\\end{align*}\nhere ${\\textrm{max}}$ is the maximum weight that an individual can reach. In Section\n\\ref{subsec.modeles.deterministes} we also present an example of a function $\\rho_{\\textrm{\\tiny\\rm\\!\\! g}}(s,x)$ linear in $x$ which will lead to the classical model of chemostat.\n\n\n\\vskip0.8em\n\\item{\\textbf{Dynamic of the substrate concentration}} -- \n\\emph{The substrate concentration evolves according to the ordinary differential equation:\n\\begin{align}\n\\label{eq.substrat}\n  \\dot S_t = \\rho_{\\textrm{\\tiny\\rm\\!\\! s}}(S_{t},\\nu_{t})\n\\end{align}}\nwhere\n\\begin{align*}\n  \\rho_{\\textrm{\\tiny\\rm\\!\\! s}} (s,\\nu)\n  &\\eqdef\n  D({\\mathbf s}_{\\textrm{\\tiny\\rm in}}-s)-k\\, \\mu(s,\\nu)\\,,\n\\\\\n  \\mu(s,\\nu)\n  &\\eqdef\n  \\frac{1}{V} \\int_{\\XX} \\rho_{\\textrm{\\tiny\\rm\\!\\! g}}(s,x)\\,\\nu(\\rmd x)\n  =    \n  \\frac{1}{V} \\sum_{i=1}^{N} \\rho_{\\textrm{\\tiny\\rm\\!\\! g}}(s,x^i)\n\\end{align*}\nwith $\\nu=\\sum_{i=1}^N \\delta_{x^i}$;\n$D$ is the dilution rate (1\/h), \n${\\mathbf s}_{\\textrm{\\tiny\\rm in}}$ is the input concentration (mg\/l),\n$k$ is the stoichiometric coefficient (inverse of the yield coefficient),\nand  $V$ is the representative volume (l).\nMass balance leads to Equation \\eqref{eq.substrat} and the initial condition $S_{0}$ may be random.\n\\end{enumerate}\n\nTo ensure the existence and uniqueness of solutions of the ordinary differential equations \\eqref{eq.masses} and \\eqref{eq.substrat}, we assume\nthat application $\\rho_{\\textrm{\\tiny\\rm\\!\\! g}}(s,x)$ is Lipschitz continuous w.r.t. $s$ uniformly in $x$:\n\\begin{align}\n\\label{hyp.rhog.lipschitz}\n  \\bigl|\\rho_{\\textrm{\\tiny\\rm\\!\\! g}}(s_1,x)-\\rho_{\\textrm{\\tiny\\rm\\!\\! g}}(s_2,x)\\bigr|\n  \\leq \n  k_g \\,|s_1-s_2|\n\\end{align}\nfor all $s_1,\\,s_2 \\geq 0$ and all $x\\in\\XX$. It is further assumed that:\n\\begin{align}\n\\label{hyp.rhog.borne}\n  0\n  \\leq\n  \\rho_{\\textrm{\\tiny\\rm\\!\\! g}}(s,x)\n  \\leq \n  \\bar g\n\\end{align}\nfor all $(s,x)\\in\\RR_+ \\times \\X$, and that \n in the absence of substrate the bacteria do not grow:\n\\begin{align}\n\\label{hyp.rhog.nulle.en.0}\n  \\rho_{\\textrm{\\tiny\\rm\\!\\! g}}(0,x)\n  =\n  0\n\\end{align}\nfor all $x\\in \\X$. To ensure that the mass of a bacterium  \nstays between $0$ and ${\\textrm{max}}$, it is finally assumed that:\n\\begin{align}\n\\label{hyp.rhog.nulle.en.x=mmax}\n  \\rho_{\\textrm{\\tiny\\rm\\!\\! g}}(s,{\\textrm{max}})\n  =\n  0\n\\end{align}\nfor any $s\\geq 0$.\n\n\n\\subsection{Algorithm}\n\n\\begin{algorithm}\n\\begin{center}\n\\begin{minipage}{14cm}\n\\begin{algorithmic}\n\\STATE $t\\ot 0$\n\\STATE sample $(S_0,\\nu_0=\\sum_{i=1}^{N_{0}}\\delta_{x^i_{t}})$\n\\WHILE {$t\\leq t_{\\textrm{\\tiny\\rm max}}$}\n  \\STATE $N \\ot \\crochet{\\nu_{t},1}$\n  \\STATE $\\tau \\ot (\\bar\\lambda+D)\\,N$\n  \\STATE $\\Delta t \\sim {\\textrm{\\rm Exp}}(\\tau)$\n  \\STATE integrate the equations for the mass \\eqref{eq.masses} \n    and the substrate \\eqref{eq.substrat} over $[t,t+\\Delta t]$\n  \\STATE $t \\ot t+\\Delta t$\n  \\STATE draw $x$ uniformly in $\\{x^i_{t}\\,;\\,i=1,\\dots,N_{t}\\}$  \n  \\STATE $u\\sim U[0,1]$ \n  \\IF {$u\\leq \\lambda(S_{t},x)\/(\\bar\\lambda+D)$}\n      \\STATE $\\alpha \\sim Q$\n      \\STATE $\\nu_{t} \\ot \\nu_{t} -\\delta_{x}+\\delta_{\\alpha\\,x}\n              +\\delta_{(1-\\alpha)\\,x}$\n      \\COMMENT{division}\n  \\ELSIF{$u\\leq (\\lambda(S_{t},x)+D)\/(\\bar\\lambda+D)$}\n      \\STATE $\\nu_{t} \\ot \\nu_{t} -\\delta_{x}$\n      \\COMMENT{up-take}\n  \\ENDIF\n\\ENDWHILE\n\\end{algorithmic}\n\\end{minipage}\n\\end{center}\n\\caption{\\itshape ``Exact'' Monte Carlo  simulation of the individual-based model:\napproximations only lie in the numerical integration of the ODEs and in \nthe pseudo-random numbers generators.}\n\\label{algo.ibm}\n\\end{algorithm}\n\nIn the model described above, the division rate $\\lambda(s,x)$ depends on the concentration of substrate $s$ and on the mass  $x$ of each individual which continuously evolves according to the system of coupled ordinary differential equations \\eqref{eq.masses} and \\eqref{eq.substrat}, so to simulate the division of the cell we make use of a rejection sampling technique. It is assumed that there exists $\\bar\\lambda<\\infty$ such that:\n\\[\n  \\lambda(s,x)\\leq \\bar\\lambda\n\\]\nhence an upper bound for the rate of event, division and up-take combined, at the population level is given by:\n\\[\n    \\tau \\eqdef (\\bar\\lambda+D)\\,N\\,.\n\\]\nAt time $t+\\Delta t$ with $\\Delta t\\sim {\\textrm{\\rm Exp}}(\\tau)$, we determine if an event has occurred and what is its type by acceptance\/rejection.\nTo this end, the masses of the $N$ individuals and the substrate concentration evolve according to the coupled ODEs \\eqref{eq.masses} and \\eqref{eq.substrat}.\nThen we chose uniformly at random an individual within the population $\\nu_{(t+\\Delta t)^-}$, that is the population at time  $t+\\Delta t$ before any possible event, let $x_{(t+\\Delta t)^-}$ denotes its mass, then:\n\\begin{enumerate}\n\n\\item With probability:\n\\[\n   \\frac{\\bar\\lambda}{(\\bar\\lambda+D)}\n\\]\nwe determine if there has been division by acceptance\/rejection:\n\\begin{itemize}\n\\item division occurs, that is:\n\\begin{align}\n\\label{eq.event.division}\n   \\nu_{t+\\Delta t}\n   =\n   \\nu_{(t+\\Delta t)^-}\n   -\\delta_{x_{(t+\\Delta t)^-}}  \n   +\\delta_{\\alpha\\,x_{(t+\\Delta t)^-}}  \n   +\\delta_{(1-\\alpha)\\,x_{(t+\\Delta t)^-}}  \n   \\qquad\n   \\textrm{with }\\alpha\\sim Q\n\\end{align}\nwith probability $\\lambda(S_{t},x_{(t+\\Delta t)^-})\/\\bar\\lambda$;\n\\item \nno event occurs with probability $1-\\lambda(S_{t},x_{(t+\\Delta t)^-})\/\\bar\\lambda$.\n\\end{itemize}\nIn conclusion, the event \\eqref{eq.event.division} occurs with probability:\n\\[\n  \\frac{\\lambda\\bigl(S_{t},x_{(t+\\Delta t)^-}\\bigr)}{\\bar\\lambda}\n  \\,\\frac{\\bar\\lambda}{(\\bar\\lambda+D)} \n  = \n  \\frac{\\lambda\\bigl(S_{t},x_{(t+\\Delta t)^-}\\bigr)}{(\\bar\\lambda+D)}\\,.\n\\]\n\n\\item With probability:\n\\[\n   \\frac{D}{(\\bar\\lambda+D)}\n   =\n   1-\\frac{\\bar\\lambda}{(\\bar\\lambda+D)}\n\\]\nthe individual is withdrawn, that is:\n\\begin{align}\n\\label{eq.event.soutirage}\n   \\nu_{t+\\Delta t}\n   =\n   \\nu_{(t+\\Delta t)^-}\n   -\\delta_{x_{(t+\\Delta t)^-}}  \n\\end{align}\n\\end{enumerate}\nFinally, the events and the associated probabilities are:\n\\begin{itemize}\n\\item \ndivision \\eqref{eq.event.division} with probability\n$\\lambda(S_{t},x_{(t+\\Delta t)^-})\/(\\bar\\lambda+D)$,\n\\item\nup-take \\eqref{eq.event.soutirage} with probability \n${D}\/{(\\bar\\lambda+D)}$\n\\end{itemize}\nand no event (rejection) with the remaining probability.\nThe details are given in Algorithm~\\ref{algo.ibm}.\n\nTechnically, the numbering of individuals is as follows: at the initial time individuals are numbered from $1$ to $N$, in case division the daughter cell $\\alpha \\,x$ keeps the index of the parent cell and the daughter cell $(1-\\alpha)\\,x$ takes the index $N+1$; in case of the up-take, the individual $N$ acquires  the index of the withdrawn cell.\n\n\n\\section{Notations}\n\\label{sec.notations}\n\nBefore proposing an explicit mathematical description of the process $(\\nu_{t})_{t\\geq 0}$ we introduce some notations.\n\n\n\\subsection{Punctual measures}\n\n\nNotation \\eqref{eq.nu} designating the bacterial population seems somewhat abstract but it will bridge the gap between the ``discrete'' -- counting punctual measures --\nand the ``continuous'' -- continuous measures of the population densities --  in the context of the asymptotic large population analysis. Indeed for any measure $\\nu(\\rmd x)$ defined on $\\R_{+}$ and any function $\\varphi:\\R_{+}\\mapsto\\R$, we define:\n\\[\n  \\crochet{\\nu,\\varphi}\n  \\eqdef\n  \\int_{\\R_{+}}\\varphi(x)\\,\\nu(\\rmd x)\\,.\n\\]\nThis notation is valid for continuous measures as well as for punctual measures $\\nu_{t}(\\rmd x)$ defined by \\eqref{eq.nu}, in the latter case \n$\\crochet{\\nu_{t},\\varphi} = \\sum_{i=1}^{N_{t}} \\varphi(x^i_{t})$.\n\nPractically, this notation allows us to link to macroscopic quantities, e.g. \nat time $t$ the \\emph{population size} is:\n\\[\n  N_{t} = \\crochet{\\nu_{t},1}\n\\]\nand the  \\emph{total biomass} is:\n\\[\n  X_{t} \\eqdef \\crochet{\\nu_{t},I}\n  = \\sum_{i=1}^{N_{t}} x^i_{t}\n\\]\nwhere $1(x)\\equiv 1$ and $I(x)\\equiv x$. Finally:\n\\[\n   x\\in\\nu_{t}=\\sum_{i=1}^{N_{t}}\\delta_{x^i_{t}}(\\rmd x)\n\\]\nwill denote any individual among $\\{x^1_{t},\\dots,x^{N_{t}}_{t}\\}$.\n\nThe set of finite and positive measures on $\\X$ is denoted  $\\M_{F}(\\XX)$,\nand $\\M(\\XX)$ is the subset of punctual finite measures on $\\X$:\n\\[\n  \\M(\\XX)\n  \\eqdef\n  \\left\\{\n    \t\\sum_{i=1}^N \\delta_{x^i} \\;; \\; N \\in \\NN, \\, x^i \\in \\X \n  \\right\\}\n\\]\nwhere by convention $\\sum_{i=1}^0\\delta_{x^i}$ is the null measure.\n\n\n\\subsection{Growth flow}\n\n\nLet:\n\\[\n\\begin{array}{rrcl}\n   A_{t} &: \n     \\R_{+}\\times\\MM(\\XX)  \n     &\\xrightarrow[]{\\mbox{}\\hskip1em} & \n     \\R_{+}\\times\\MM(\\XX)\n     \\\\\n   &(s,\\nu)&\\xrightarrow[]{\\mbox{}\\hskip1em} &\n   A_{t} (s,\\nu)\n\\end{array}\n\\]\nbe the differential flow associated with the couple system of ODEs \\eqref{eq.substrat}--\\eqref{eq.masses} apart from any event (division or up-take), i.e.:\n\\begin{align}\n\\label{eq.flot.A}\n   A_{t}(s,\\nu)\n   =\n   \\Biggl(\n   \t\tA^0_{t}(s,\\nu) \\;,\\; \\sum_{i=1}^N\\delta_{A^i_{t}(s,\\nu)}\n   \\Biggr)\n   \\quad\\textrm{with }\\nu=\\sum_{i=1}^N\\delta_{x^i}\n   \\,.\n\\end{align}\nwhere $A^0_{t}(s,\\nu)$ and $(A^i_{t}(s,\\nu)\\,;\\,i=1,\\dots,N)$ are the coupled solutions of \\eqref{eq.substrat}--\\eqref{eq.masses} taken at time $t$ from the initial condition $(s,\\nu)$, that is:\n\\begin{align*}\n  \\frac{\\rmd}{\\rmd t} \n   A_{t}^0(s,\\nu)\n   &= \\rho_{\\textrm{\\tiny\\rm\\!\\! s}}\\Bigl(A_{t}^0(s,\\nu),\\sum_{i=1}^N \\delta_{A_{t}^i(s,\\nu)}\\Bigr)\n\\\\\n   &= D\\,({\\mathbf s}_{\\textrm{\\tiny\\rm in}}-A_{t}^0(s,\\nu))\n      -\\frac{k}{V} \\sum_{i=1}^N \\rho_{\\textrm{\\tiny\\rm\\!\\! g}}(A_{t}^0(s,\\nu),A_{t}^i(s,\\nu))\\,,\n   &\n   A_{0}^0(s,\\nu)=s\\,,\n\\\\[0.5em]\n  \\frac{\\rmd}{\\rmd t} \n   A_{t}^i(s,\\nu)\n   &=\n   \\rho_{\\textrm{\\tiny\\rm\\!\\! g}}(A_{t}^0(s,\\nu),A_{t}^i(s,\\nu))\\,,\n   &\n   A_{0}^i(s,\\nu)=x^i\n\\end{align*}\nfor  $i=1,\\dots,N$. Hence the flow $A_{t}(s,\\nu)$ depends implicitly on the  size $N=\\crochet{\\nu,1}$ of the population $\\nu$. \n\n\\medskip\n\nThe stochastic process $(\\nu_{t})_{t\\geq 0}$ features a jump dynamics (division and up-take) and follows the dynamics of the flow $A_{t}$ between the jumps.\nWe can therefore generalize a  well-known formula for the pure jump process:\n\\begin{align}\n\\label{eq.dyn.saut.flot}\n  \\Phi(S_{t},\\nu_{t})\n  &=\n  \\Phi(A_{t}(S_{0},\\nu_{0}))\n  +\n  \\sum_{u\\leq t} \n    \\bigl[ \n       \\Phi(A_{t-u}(S_{u},\\nu_{u}))-\\Phi(A_{t-u}(S_{u},\\nu_{{{u^{\\raisebox{-1pt}{\\scriptsize\\scalebox{0.5}{$-$}}}}}}))\n    \\bigr]  \n  \\,,\\quad t\\geq 0\n\\end{align}\nfor any function  $\\Phi$ defined on $\\R\\times\\MM(\\XX)$. \n\nThe sum $ \\sum_{u\\leq t}$ contains only a finite number of terms as the process $(\\nu_{t})_{t\\geq 0}$ admits only a finite number of jumps over any finite time interval. Indeed, the number of jumps in the process $(\\nu_{t})_{t\\geq 0}$ is bounded by a linear birth and death process with \\textit{per capita} birth rate $\\bar\\lambda$\nand \\textit{per capita} death rate $D$ \\citep{allen2003a}.\n\n\n\n\n\n\n\n\\section{Microscopic process}\n\\label{sec.processus.microscopique}\n\nLet $(S_0,\\nu_0)$ denote the initial condition of the process, it is a random variable with values in $\\R_+ \\times \\M(\\X)$.\n\n\nThe equation \\eqref{eq.dyn.saut.flot} includes information on the flow, i.e. the dynamics between the jumps, but no information on the jumps themselves. To obtain an explicit equation for $(S_t,\\nu_t)_{t\\geq 0}$ we introduce Poisson random measures which manage the incoming of new individuals by cell division on the one hand, and the withdrawal of individuals by up-take on the other. To this end we consider two punctual Poisson random  measures  $N_1(\\mathrm{d} u, \\mathrm{d} j, \\mathrm{d} \\alpha, \\mathrm{d} \\theta)$ and $N_2(\\mathrm{d} u, \\mathrm{d} j)$\nrespectively defined on $\\RR_+ \\times  \\NN^* \\times \\X \\times [0,1]$ and $\\RR_+ \\times  \\NN^*$ with respective intensity measures:\n\\begin{align*}\n   n_1(\\mathrm{d} u, \\mathrm{d} j, \\mathrm{d} \\alpha, \\mathrm{d} \\theta)\n   &\\eqdef\n   \\bar\\lambda\\, \\mathrm{d} u \\,\\Big(\\sum_{k \\geq 0} \n       \\delta_k(\\mathrm{d} j)\\Big) \\, Q(\\mathrm{d} \\alpha) \\, \\mathrm{d} \\theta\\,,\n\\\\\n   n_2(\\mathrm{d} u, \\mathrm{d} j)\n   &\\eqdef\n   D\\,\\mathrm{d} u \\,\\Big(\\sum_{k \\geq 0} \\delta_k(\\mathrm{d} j)\\Big)\\,.\n\\end{align*}\nSuppose that $N_1$, $N_{2}$, $S_{0}$ and $\\nu_{0}$ are mutually independent.\nLet $(\\mathcal{F}_t)_{t \\geq 0}$ be the canonical filtration generated by  $(S_0,\\nu_0)$, $N_1$ and $N_2$.\nAccording to \\eqref{eq.dyn.saut.flot}, for any function  $\\Phi$ defined on  \n$\\R\\times\\MM(\\XX)$:\n\\begin{align}\n\\nonumber\n  &\\Phi(S_{t},\\nu_{t})\n  =\n  \\Phi(A_{t}(S_{0},\\nu_{0}))\n\\\\\n\\nonumber\n  &\n  \\quad+\n  \\iiiint\\limits_{[0,t]\\times\\N^*\\times[0,1]^2}\n        1_{\\{j\\leq N_{{{u^{\\raisebox{-1pt}{\\scriptsize\\scalebox{0.5}{$-$}}}}}}\\}} \\, \n        1_{\\{0\\leq \\theta \\leq \\lambda(S_{u},x_{{u^{\\raisebox{-1pt}{\\scriptsize\\scalebox{0.5}{$-$}}}}}^j)\/\\bar\\lambda\\}}\\,\n        \\bigl[\\Phi(A_{t-u}(S_{u},\\nu_{{{u^{\\raisebox{-1pt}{\\scriptsize\\scalebox{0.5}{$-$}}}}}}\n                -\\delta_{x^j_{{u^{\\raisebox{-1pt}{\\scriptsize\\scalebox{0.5}{$-$}}}}}}\n                +\\delta_{\\alpha\\,x^j_{{u^{\\raisebox{-1pt}{\\scriptsize\\scalebox{0.5}{$-$}}}}}}\n                +\\delta_{(1-\\alpha)\\,x^j_{{u^{\\raisebox{-1pt}{\\scriptsize\\scalebox{0.5}{$-$}}}}}}))\n  \\\\[-0.7em]&\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\n\\nonumber\n             -\\Phi(A_{t-u}(S_{u},\\nu_{{{u^{\\raisebox{-1pt}{\\scriptsize\\scalebox{0.5}{$-$}}}}}}))\\bigr]\\,\n\t\t \t\tN_1(\\mathrm{d} u, \\mathrm{d} j, \\mathrm{d} \\alpha, \\mathrm{d} \\theta) \n\\\\[0.2em]\n\\label{eq.Phi.S.nu}\n  &\\quad+\n\t\t\\iint\\limits_{[0,t]\\times\\N^*} \n\t\t\t1_{\\{j\\leq N_{{{u^{\\raisebox{-1pt}{\\scriptsize\\scalebox{0.5}{$-$}}}}}}\\}} \\, \n\t\t\t \\bigl[\\Phi(A_{t-u}(S_{u},\\nu_{{{u^{\\raisebox{-1pt}{\\scriptsize\\scalebox{0.5}{$-$}}}}}}-\\delta_{x^j_{{u^{\\raisebox{-1pt}{\\scriptsize\\scalebox{0.5}{$-$}}}}}}))\n\t\t\t   -\\Phi(A_{t-u}(S_{u},\\nu_{{{u^{\\raisebox{-1pt}{\\scriptsize\\scalebox{0.5}{$-$}}}}}}))\\bigr]\n\t\t\t \t\t\t \\,\tN_2(\\mathrm{d} u, \\mathrm{d} j)\\,.\n\\end{align}\nIn particular, we obtain the following equation for the couple  $(S_{t},\\nu_{t})$:\n\\begin{align}\n\\nonumber\n  &(S_{t},\\nu_{t})\n  =\n  A_{t}(S_{0},\\nu_{0})\n\\\\\n\\nonumber\n  &\n  \\qquad+\n  \\iiiint\\limits_{[0,t]\\times\\N^*\\times[0,1]^2}\n        1_{\\{j\\leq N_{{{u^{\\raisebox{-1pt}{\\scriptsize\\scalebox{0.5}{$-$}}}}}}\\}} \\, \n        1_{\\{0\\leq \\theta \\leq \\lambda(S_{u},x_{{u^{\\raisebox{-1pt}{\\scriptsize\\scalebox{0.5}{$-$}}}}}^j)\/\\bar\\lambda\\}}\\,\n        \\bigl[A_{t-u}(S_{u},\\nu_{{{u^{\\raisebox{-1pt}{\\scriptsize\\scalebox{0.5}{$-$}}}}}}\n                -\\delta_{x^j_{{u^{\\raisebox{-1pt}{\\scriptsize\\scalebox{0.5}{$-$}}}}}}\n                +\\delta_{\\alpha\\,x^j_{{u^{\\raisebox{-1pt}{\\scriptsize\\scalebox{0.5}{$-$}}}}}}\n                +\\delta_{(1-\\alpha)\\,x^j_{{u^{\\raisebox{-1pt}{\\scriptsize\\scalebox{0.5}{$-$}}}}}})\n\\\\[-0.7em]\n\\nonumber\n&\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\n             -A_{t-u}(S_{u},\\nu_{{{u^{\\raisebox{-1pt}{\\scriptsize\\scalebox{0.5}{$-$}}}}}})\\bigr]\\,\n\t\t \t\tN_1(\\mathrm{d} u, \\mathrm{d} j, \\mathrm{d} \\alpha, \\mathrm{d} \\theta) \n\\\\[0.2em]\n\\label{def.proc.S.nu}\n  &\\qquad+\n\t\t\\iint\\limits_{[0,t]\\times\\N^*} \n\t\t\t1_{\\{j\\leq N_{{{u^{\\raisebox{-1pt}{\\scriptsize\\scalebox{0.5}{$-$}}}}}}\\}} \\, \n\t\t\t \\big[A_{t-u}(S_{u},\\nu_{{{u^{\\raisebox{-1pt}{\\scriptsize\\scalebox{0.5}{$-$}}}}}}-\\delta_{x^j_{{u^{\\raisebox{-1pt}{\\scriptsize\\scalebox{0.5}{$-$}}}}}})\n\t\t\t   -A_{t-u}(S_{u},\\nu_{{{u^{\\raisebox{-1pt}{\\scriptsize\\scalebox{0.5}{$-$}}}}}})\\big]\n\t\t\t \t\t\t \\,\tN_2(\\mathrm{d} u, \\mathrm{d} j)\\,.\n\\end{align}\n\n\\medskip\n\nFrom now on, we consider test functions $\\Phi$ of the form:\n\\[\n   \\Phi(s,\\nu)\n   =\n   F(s,\\crochet{\\nu,f})\n\\]\nwith $F \\in C^{1,1}(\\RR^+ \\times \\RR)$ and $f \\in C^1(\\X)$.\n\n\n\\begin{lemma}\n\\label{lem.gen.inf}\nFor any $t>0$:\n\\begin{align}\n\\nonumber\n  &\n  F(S_t,\\crochet{\\nu_t, f})\n  = \n  F(S_0,\\crochet{\\nu_0, f})\n\\\\\n\\nonumber\n  &\\quad\n   + \\int_0^t  \\left[\n\t \t\t\\rho_{\\textrm{\\tiny\\rm\\!\\! s}} (S_u,\\nu_u) \\, \n\t \t\t\\partial_s F\\left(S_u, \\, \\left<\\nu_u, f\\right>\\right)\n\t \t\t+ \\left< \\nu_u , \\, \\rho_{\\textrm{\\tiny\\rm\\!\\! g}}(S_u,.)\\,f'\\right> \\,\n\t \t\t\\partial_x F\\left(S_u, \\, \\left<\\nu_u, f\\right>\\right)\n\t \t\\right] \\mathrm{d} u \n\\\\\n\\nonumber  \n  &\\quad\n  + \\iiiint\\limits_{[0,t]\\times\\N^*\\times[0,1]^2} \n\t\t1_{\\{j\\leq N_{{{u^{\\raisebox{-1pt}{\\scriptsize\\scalebox{0.5}{$-$}}}}}}\\}} \\,\n\t\t1_{\\{0\\leq \\theta \\leq \\lambda(S_{u},x_{{u^{\\raisebox{-1pt}{\\scriptsize\\scalebox{0.5}{$-$}}}}}^j)\/\n\t\t\t\t\\bar\\lambda\\}}\\, \n\t\t\\bigl[\n\t\t\tF(S_u,\\crochet{\\nu_{{{u^{\\raisebox{-1pt}{\\scriptsize\\scalebox{0.5}{$-$}}}}}}\n\t\t\t\t\t-\\delta_{x_{{u^{\\raisebox{-1pt}{\\scriptsize\\scalebox{0.5}{$-$}}}}}^j}\n\t\t\t\t\t+\\delta_{\\alpha \\, x_{{u^{\\raisebox{-1pt}{\\scriptsize\\scalebox{0.5}{$-$}}}}}^j}\n\t\t\t\t\t+\\delta_{(1-\\alpha)\\, x_{{u^{\\raisebox{-1pt}{\\scriptsize\\scalebox{0.5}{$-$}}}}}^j}, f} \n\t\t\t)\n        \\\\[-0.9em] \n        \\nonumber\n        & \\qquad\\qquad\\qquad\\qquad\\qquad\\quad\n                     \\qquad\\qquad\\qquad\\qquad\\qquad\n\t\t\t-\n\t\t\tF(S_u, \\crochet{\\nu_{{{u^{\\raisebox{-1pt}{\\scriptsize\\scalebox{0.5}{$-$}}}}}}, f}) \n\t    \\bigr] \n\t    \\;\n\t\t N_1(\\mathrm{d} u, \\mathrm{d} j, \\mathrm{d} \\alpha, \\mathrm{d} \\theta)  \n\\\\\n\\label{eq.F.f}\n  &\\quad  \n  + \\iint\\limits_{[0,t]\\times\\N^*} \n\t  1_{\\{j\\leq N_{{{u^{\\raisebox{-1pt}{\\scriptsize\\scalebox{0.5}{$-$}}}}}}\\}} \\, \n\t  \\bigl[\n\t\tF(S_u,  \\crochet{\\nu_{{{u^{\\raisebox{-1pt}{\\scriptsize\\scalebox{0.5}{$-$}}}}}}-\\delta_{x_{{u^{\\raisebox{-1pt}{\\scriptsize\\scalebox{0.5}{$-$}}}}}^j}, f})\n\t    -\n\t    F\\left(S_u, \\, \\left<\\nu_{{{u^{\\raisebox{-1pt}{\\scriptsize\\scalebox{0.5}{$-$}}}}}}, f\\right>\\right) \n\t  \\bigr] \\, N_2(\\mathrm{d} u, \\mathrm{d} j).\n\\end{align}\n\\end{lemma}\n\n\n\\begin{proof}\nFrom \\eqref{eq.Phi.S.nu}:\n\\begin{align*}\n  &\n  \\crochet{\\nu_t, f}\n  = \n  \\sum_{i=1}^{N_0} f(A_t^i(S_0,\\nu_0))\n\\\\\n  &\\quad\n  + \\iiiint\\limits_{[0,t]\\times\\N^*\\times[0,1]^2}\n     1_{\\{j\\leq N_{{{u^{\\raisebox{-1pt}{\\scriptsize\\scalebox{0.5}{$-$}}}}}}\\}} \\, \n     1_{\\{0\\leq \\theta \\leq \\lambda(S_{u},x_{{{u^{\\raisebox{-1pt}{\\scriptsize\\scalebox{0.5}{$-$}}}}}}^j)\/\\bar\\lambda\\}}\\,\n\\\\[-0.5em]\n\t& \\qquad\\qquad\\qquad\n     \\times\\Bigl[\n   \t    \\sum_{i=1}^{N_{{{u^{\\raisebox{-1pt}{\\scriptsize\\scalebox{0.5}{$-$}}}}}}+1}\n\t\t \tf(A_{t-u}^i(S_u,\n\t\t \t\t\\nu_{{{u^{\\raisebox{-1pt}{\\scriptsize\\scalebox{0.5}{$-$}}}}}}-\\delta_{x_{{{u^{\\raisebox{-1pt}{\\scriptsize\\scalebox{0.5}{$-$}}}}}}^j}\n\t\t\t\t+\\delta_{\\alpha\\,x_{{{u^{\\raisebox{-1pt}{\\scriptsize\\scalebox{0.5}{$-$}}}}}}^j}\n\t\t \t    +\\delta_{(1-\\alpha)\\,x_{{{u^{\\raisebox{-1pt}{\\scriptsize\\scalebox{0.5}{$-$}}}}}}^j}))\n\\\\[-1em]\n\t& \\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\n\t\t-\\sum_{i=1}^{N_{{{u^{\\raisebox{-1pt}{\\scriptsize\\scalebox{0.5}{$-$}}}}}}}\n\t\t \tf(A_{t-u}^i(S_u,\\nu_{{{u^{\\raisebox{-1pt}{\\scriptsize\\scalebox{0.5}{$-$}}}}}}))\n    \\Bigl] \\,\n    N_1(\\mathrm{d} u, \\mathrm{d} j, \\mathrm{d} \\alpha, \\mathrm{d} \\theta) \n\\\\\n  &\\quad\n  + \\iint\\limits_{[0,t]\\times\\N^*} \n\t\t1_{\\{j\\leq N_{{{u^{\\raisebox{-1pt}{\\scriptsize\\scalebox{0.5}{$-$}}}}}}\\}} \\, \n\t\t\\Bigl[\n\t\t  \\sum_{i=1}^{N_{{{u^{\\raisebox{-1pt}{\\scriptsize\\scalebox{0.5}{$-$}}}}}}-1}\n\t\t \t    f(A_{t-u}^i(S_u,\n\t\t \t\t\t\\nu_{{{u^{\\raisebox{-1pt}{\\scriptsize\\scalebox{0.5}{$-$}}}}}}-\\delta_{x_{{{u^{\\raisebox{-1pt}{\\scriptsize\\scalebox{0.5}{$-$}}}}}}^j}))\n\t\t  -\n\t\t  \\sum_{i=1}^{N_{{{u^{\\raisebox{-1pt}{\\scriptsize\\scalebox{0.5}{$-$}}}}}}} f(A_{t-u}^i(S_u,\\nu_{{{u^{\\raisebox{-1pt}{\\scriptsize\\scalebox{0.5}{$-$}}}}}})) \n\t\t\\Bigl] \\, N_2(\\mathrm{d} u, \\mathrm{d} j).\n\\end{align*}\nAccording to the chain rule formula, for any  $\\nu=\\sum_{i=1}^{N}\\delta_{x^i}$:\n\\begin{align*} \n  f(A_{t-u}^i(s,\\nu))\n  & = \n  f(x^i)\n  +\n  \\int_u^t \t\\rho_{\\textrm{\\tiny\\rm\\!\\! g}}(A_{\\tau-u}^0(s,\\nu),A_{\\tau-u}^i(s,\\nu)) \\, f'(A_{\\tau-u}^i(s,\\nu))\n\t\t \\,\\mathrm{d} \\tau\n\\\\\n  & = \n  f(x^i)\n  +\n  \\int_u^t \\varphi(A_{\\tau-u}^0(s,\\nu),A_{\\tau-u}^i(s,\\nu)) \\,\\mathrm{d} \\tau\n\\end{align*}\nfor $i \\leq N$, with:\n\\begin{align*}\n  \\varphi(s,x) \\eqdef \\rho_{\\textrm{\\tiny\\rm\\!\\! g}}(s,x) \\, f'(x)\\,.\n\\end{align*}\nHence:\n\\begin{align*}\n  &\\crochet{\\nu_t, f}\n  = \n  \\crochet{\\nu_0, f} \n\\\\\n  & \\qquad \n  + \\iiiint\\limits_{[0,t]\\times\\N^*\\times[0,1]^2}\n\t\t1_{\\{j\\leq N_{{u^{\\raisebox{-1pt}{\\scriptsize\\scalebox{0.5}{$-$}}}}}\\}} \\, \n\t\t1_{\\{0\\leq \\theta \\leq \\lambda(S_{u},x_{{u^{\\raisebox{-1pt}{\\scriptsize\\scalebox{0.5}{$-$}}}}}^j)\/\\bar\\lambda\\}}\n\\\\[-0.8em]\n  &\\qquad\\qquad\\qquad\\qquad\\qquad\n  \\times  \\,\n\t\t\t \\left[\n\t\t\t    f(\\alpha \\, x_{{u^{\\raisebox{-1pt}{\\scriptsize\\scalebox{0.5}{$-$}}}}}^j)\n\t\t\t    + f((1-\\alpha) \\, x_{{u^{\\raisebox{-1pt}{\\scriptsize\\scalebox{0.5}{$-$}}}}}^j)\n\t\t\t\t- f(x_{{u^{\\raisebox{-1pt}{\\scriptsize\\scalebox{0.5}{$-$}}}}}^j)\n\t\t     \\right] \\, \n  N_1(\\mathrm{d} u, \\mathrm{d} j, \\mathrm{d} \\alpha, \\mathrm{d} \\theta) \n\\\\\n  &\\qquad  \n  - \\iint\\limits_{[0,t]\\times\\N^*}  \n\t\t\t1_{\\{j\\leq N_{{u^{\\raisebox{-1pt}{\\scriptsize\\scalebox{0.5}{$-$}}}}}\\}} \\, f(x_{{u^{\\raisebox{-1pt}{\\scriptsize\\scalebox{0.5}{$-$}}}}}^j) \\,\n\t\t\t \\,\tN_2(\\mathrm{d} u, \\mathrm{d} j)\n  +T_0+T_1+T_2\n\\end{align*}\nwhere:\n\\begin{align*}\n  T_0 \n  &\\eqdef\n  \\sum_{i=1}^{N_0} \\int_0^t \\varphi(A_\\tau^0(S_0,\\nu_0), A_\\tau^i(S_0,\\nu_0)) \\, \\rmd\\tau\t\n\\\\[1em]\n  T_1\n  &\\eqdef \n  \\iiiint\\limits_{[0,t]\\times\\N^*\\times[0,1]^2}\n    1_{\\{j\\leq N_{{u^{\\raisebox{-1pt}{\\scriptsize\\scalebox{0.5}{$-$}}}}}\\}} \\,\n    1_{\\{0\\leq \\theta \\leq \\lambda(S_{u},x_{{u^{\\raisebox{-1pt}{\\scriptsize\\scalebox{0.5}{$-$}}}}}^j)\/\\bar\\lambda\\}} \\, \n\\\\[-0.7em]\n  &\\qquad\\qquad\\qquad\n  \\times\\int_u^t \\Bigl[\n\t \\sum_{i=1}^{N_{{u^{\\raisebox{-1pt}{\\scriptsize\\scalebox{0.5}{$-$}}}}}+1}\n\t\t \t\\varphi(A_{\\tau-u}^0(S_u,\n\t\t \t\t\t\\nu_{{u^{\\raisebox{-1pt}{\\scriptsize\\scalebox{0.5}{$-$}}}}}-\\delta_{x_{{u^{\\raisebox{-1pt}{\\scriptsize\\scalebox{0.5}{$-$}}}}}^j}\n\t\t\t\t\t+\\delta_{\\alpha \\, x_{{u^{\\raisebox{-1pt}{\\scriptsize\\scalebox{0.5}{$-$}}}}}^j}\n\t\t \t\t\t+\\delta_{(1-\\alpha) \\, x_{{u^{\\raisebox{-1pt}{\\scriptsize\\scalebox{0.5}{$-$}}}}}^j}),\n\\\\[-1em]\n  &\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad \n\t\t       A_{\\tau-u}^i(S_u,\\nu_{{u^{\\raisebox{-1pt}{\\scriptsize\\scalebox{0.5}{$-$}}}}}\n\t\t           -\\delta_{x_{{u^{\\raisebox{-1pt}{\\scriptsize\\scalebox{0.5}{$-$}}}}}^j}\n\t\t           +\\delta_{\\alpha \\, x_{{u^{\\raisebox{-1pt}{\\scriptsize\\scalebox{0.5}{$-$}}}}}^j}\n\t\t \t\t   +\\delta_{(1-\\alpha) \\, x_{{u^{\\raisebox{-1pt}{\\scriptsize\\scalebox{0.5}{$-$}}}}}^j}))\n\\\\[-0.6em]\n  &\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\n\t -\\sum_{i=1}^{N_{{u^{\\raisebox{-1pt}{\\scriptsize\\scalebox{0.5}{$-$}}}}}}\n\t\t \\varphi(A_{\\tau-u}^0(S_u,\\nu_{{u^{\\raisebox{-1pt}{\\scriptsize\\scalebox{0.5}{$-$}}}}}),A_{\\tau-u}^i(S_u,\\nu_{{u^{\\raisebox{-1pt}{\\scriptsize\\scalebox{0.5}{$-$}}}}}))\n\t\t\t \\Bigr]\\,\\mathrm{d} \\tau \t\n\\\\\n\t&\\qquad\\qquad\\qquad\n\t\\times N_1(\\mathrm{d} u, \\mathrm{d} j, \\mathrm{d} \\alpha, \\mathrm{d} \\theta)\n\\\\[1em]\n  T_2\n  &\\eqdef  \n  \\iint\\limits_{[0,t]\\times\\N^*} 1_{\\{j\\leq N_{{u^{\\raisebox{-1pt}{\\scriptsize\\scalebox{0.5}{$-$}}}}}\\}}\n  \\; \\int_u^t\n\t\\Bigl[ \\sum_{i=1}^{N_{{u^{\\raisebox{-1pt}{\\scriptsize\\scalebox{0.5}{$-$}}}}}-1}\n\t\t \t\\varphi(A_{\\tau-u}^0(S_u,\n\t\t \t\t\t\\nu_{{u^{\\raisebox{-1pt}{\\scriptsize\\scalebox{0.5}{$-$}}}}}-\\delta_{x_{{u^{\\raisebox{-1pt}{\\scriptsize\\scalebox{0.5}{$-$}}}}}^j}), A_{\\tau-u}^i(S_u,\n\t\t \t\t\t\\nu_{{u^{\\raisebox{-1pt}{\\scriptsize\\scalebox{0.5}{$-$}}}}}-\\delta_{x_{{u^{\\raisebox{-1pt}{\\scriptsize\\scalebox{0.5}{$-$}}}}}^j}))\n\\\\[-0.6em]\n  &\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\n\t\t\t-\\sum_{i=1}^{N_{{u^{\\raisebox{-1pt}{\\scriptsize\\scalebox{0.5}{$-$}}}}}}\n\t\t \t\\varphi(A_{\\tau-u}^0(S_u,\\nu_{{u^{\\raisebox{-1pt}{\\scriptsize\\scalebox{0.5}{$-$}}}}}),A_{\\tau-u}^i(S_u,\\nu_{{u^{\\raisebox{-1pt}{\\scriptsize\\scalebox{0.5}{$-$}}}}})) \n\t\t\t \\Bigr] \\, \\mathrm{d} \\tau\n  \\;  N_2(\\mathrm{d} u, \\mathrm{d} j).\n\\end{align*}\nFubini's theorem applied to $T_1$ and $T_2$ leads to:\n\\begin{align*}\n  T_1\n  &= \n  \\int_0^t \\iiiint\\limits_{[0,\\tau]\\times\\N^*\\times[0,1]^2}\n   1_{\\{j\\leq N_{{u^{\\raisebox{-1pt}{\\scriptsize\\scalebox{0.5}{$-$}}}}}\\}} \\, \n   \\times 1_{\\{0\\leq \\theta \\leq \\lambda(S_{u},x_{{u^{\\raisebox{-1pt}{\\scriptsize\\scalebox{0.5}{$-$}}}}}^j)\/\\bar\\lambda\\}} \\, \n\\\\[-0.7em]\n  &\\qquad\\qquad\\qquad\n  \\times \\Bigl[\n\t\t\\sum_{i=1}^{N_{{u^{\\raisebox{-1pt}{\\scriptsize\\scalebox{0.5}{$-$}}}}}+1}\n\t\t \t\\varphi(A_{\\tau-u}^0(S_u,\n\t\t \t\t\t\\nu_{{u^{\\raisebox{-1pt}{\\scriptsize\\scalebox{0.5}{$-$}}}}}\n\t\t\t\t\t-\\delta_{x_{{u^{\\raisebox{-1pt}{\\scriptsize\\scalebox{0.5}{$-$}}}}}^j}\n\t\t\t\t\t+\\delta_{\\alpha \\, x_{{u^{\\raisebox{-1pt}{\\scriptsize\\scalebox{0.5}{$-$}}}}}^j}\n\t\t \t\t\t+\\delta_{(1-\\alpha) \\, x_{{u^{\\raisebox{-1pt}{\\scriptsize\\scalebox{0.5}{$-$}}}}}^j}),\n        \\\\[-1em]\n        &\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\n        A_{\\tau-u}^i(S_u,\n             \\nu_{{u^{\\raisebox{-1pt}{\\scriptsize\\scalebox{0.5}{$-$}}}}}-\\delta_{x_{{u^{\\raisebox{-1pt}{\\scriptsize\\scalebox{0.5}{$-$}}}}}^j}\n                +\\delta_{\\alpha \\, x_{{u^{\\raisebox{-1pt}{\\scriptsize\\scalebox{0.5}{$-$}}}}}^j}\n\t\t \t\t+\\delta_{(1-\\alpha) \\, x_{{u^{\\raisebox{-1pt}{\\scriptsize\\scalebox{0.5}{$-$}}}}}^j}))\n\\\\[-0.7em]\n  &\\qquad\\qquad\\qquad\\qquad\\qquad\n\t\t-\\sum_{i=1}^{N_{{u^{\\raisebox{-1pt}{\\scriptsize\\scalebox{0.5}{$-$}}}}}}\n\t\t \t\\varphi(A_{\\tau-u}^0(S_u,\\nu_{{u^{\\raisebox{-1pt}{\\scriptsize\\scalebox{0.5}{$-$}}}}}),A_{\\tau-u}^i(S_u,\\nu_{{u^{\\raisebox{-1pt}{\\scriptsize\\scalebox{0.5}{$-$}}}}}))\n  \\Bigr] \n\\\\\n  &\\qquad\\qquad\\qquad\n  \\times N_1(\\mathrm{d} u, \\mathrm{d} j, \\mathrm{d} \\alpha, \\mathrm{d} \\theta) \\;\\mathrm{d} \\tau\n\\\\[1em]\n  T_2\n  &\\eqdef\n  \\int_0^t \\iint\\limits_{[0,\\tau]\\times\\N^*} \n  1_{\\{j\\leq N_{{u^{\\raisebox{-1pt}{\\scriptsize\\scalebox{0.5}{$-$}}}}}\\}} \\, \n  \\Bigl[\n\t\t\\sum_{i=1}^{N_{{u^{\\raisebox{-1pt}{\\scriptsize\\scalebox{0.5}{$-$}}}}}-1}\n\t\t \t\\varphi(A_{\\tau-u}^0(S_u,\n\t\t \t\t\t\\nu_{{u^{\\raisebox{-1pt}{\\scriptsize\\scalebox{0.5}{$-$}}}}}-\\delta_{x_{{u^{\\raisebox{-1pt}{\\scriptsize\\scalebox{0.5}{$-$}}}}}^j}), A_{\\tau-u}^i(S_u,\n\t\t \t\t\t\\nu_{{u^{\\raisebox{-1pt}{\\scriptsize\\scalebox{0.5}{$-$}}}}}-\\delta_{x_{{u^{\\raisebox{-1pt}{\\scriptsize\\scalebox{0.5}{$-$}}}}}^j}))\n    \\\\[-1em]\n\t& \\qquad\\qquad\\qquad\\qquad\\qquad\\qquad \n\t\t-\\sum_{i=1}^{N_{{u^{\\raisebox{-1pt}{\\scriptsize\\scalebox{0.5}{$-$}}}}}}\n\t\t \t\\varphi(A_{\\tau-u}^0(S_u,\\nu_{{u^{\\raisebox{-1pt}{\\scriptsize\\scalebox{0.5}{$-$}}}}}),A_{\\tau-u}^i(S_u,\\nu_{{u^{\\raisebox{-1pt}{\\scriptsize\\scalebox{0.5}{$-$}}}}}))\n   \\Bigr] \n   \\,\tN_2(\\mathrm{d} u, \\mathrm{d} j)\\; \\mathrm{d} \\tau\n\\end{align*}\nso, according to \\eqref{eq.Phi.S.nu}:\n\\begin{align*}\n  T_0+T_1+T_2\n  &= \n  \\int_0^t \\crochet{\\nu_\\tau,\\varphi(S_\\tau,.)} \\, \\mathrm{d}\\tau \\,.\t\t \n\\end{align*}\nFinally,\n\\begin{align*}\n  \\crochet{\\nu_t,f}\n  &=\n  \\crochet{\\nu_0, f} \n  + \n  \\int_0^t \\crochet{\\nu_u,\\rho_{\\textrm{\\tiny\\rm\\!\\! g}}(S_u,.)\\,f'} \\, \\mathrm{d} u \n\\\\\n  &\\quad  \n  + \\iiiint\\limits_{[0,t]\\times\\N^*\\times[0,1]^2} \n\t1_{\\{j\\leq N_{{u^{\\raisebox{-1pt}{\\scriptsize\\scalebox{0.5}{$-$}}}}}\\}} \\, \n\t1_{\\{0\\leq \\theta \\leq \\lambda(S_{u},x_{{u^{\\raisebox{-1pt}{\\scriptsize\\scalebox{0.5}{$-$}}}}}^j)\/\\bar\\lambda\\}}\\,\n\t\\bigl[f(\\alpha \\,x_{{u^{\\raisebox{-1pt}{\\scriptsize\\scalebox{0.5}{$-$}}}}}^j)+f((1-\\alpha)\\,x_{{u^{\\raisebox{-1pt}{\\scriptsize\\scalebox{0.5}{$-$}}}}}^j)\n\t\t\t\t-f(x_{{u^{\\raisebox{-1pt}{\\scriptsize\\scalebox{0.5}{$-$}}}}}^j)\\bigr] \n\\\\[-1em]\n  &\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\n  \\times N_1(\\mathrm{d} u, \\mathrm{d} j, \\mathrm{d} \\alpha, \\mathrm{d} \\theta) \n\\\\\n  &\\quad\n  - \\iint\\limits_{[0,t]\\times\\N^*}  \n\t\t\t1_{\\{j\\leq N_{{u^{\\raisebox{-1pt}{\\scriptsize\\scalebox{0.5}{$-$}}}}}\\}} \\, f(x_{{u^{\\raisebox{-1pt}{\\scriptsize\\scalebox{0.5}{$-$}}}}}^j) \\,\n\t\t\t \\,\tN_2(\\mathrm{d} u, \\mathrm{d} j).\n\\end{align*}\n\n\nSince $f$ and $f'$ are continuous and bounded (bounded as defined on a compact set), we can conclude this proof by using the It\u00f4 formula for stochastic integrals with respect to Poisson random measures \\citep{rudiger2006a} to develop the differential of $F(S_{t},\\crochet{\\nu_t,f})$ using Equation \\eqref{eq.substrat} and the previous equation.\n\\carre\n\\end{proof}\n\n\n\\bigskip\n\nConsider the compensated Poisson random  measures associated with $N_1$ and $N_2$:\n\\begin{align*}\n  \\tilde N_1(\\mathrm{d} u, \\mathrm{d} j, \\mathrm{d} y, \\mathrm{d} \\theta)\n  &\\eqdef\n  N_1(\\mathrm{d} u, \\mathrm{d} j, \\mathrm{d} y, \\mathrm{d} \\theta)\n  -\n  n_1(\\mathrm{d} u, \\mathrm{d} j, \\mathrm{d} y, \\mathrm{d} \\theta)\\,,\n\\\\\n \\tilde N_2(\\mathrm{d} u, \\mathrm{d} j)\n  &\\eqdef\n  N_2(\\mathrm{d} u, \\mathrm{d} j)-n_2(\\mathrm{d} u, \\mathrm{d} j)\\,.\n\\end{align*}\n\n\\medskip\n\nAs the integrands in the Poissonian integrals of \\eqref{eq.F.f} are predictable, one can make use of the result of \\citet[p. 62]{ikeda1981a}:\n\\begin{proposition}\n\\label{prop.martingale}\nLet:\n\\begin{align*}\n  M^1_t\n  &\\eqdef \n  \\iiiint\\limits_{[0,t]\\times\\N^*\\times[0,1]^2} \n     1_{\\{j\\leq N_{{u^{\\raisebox{-1pt}{\\scriptsize\\scalebox{0.5}{$-$}}}}}\\}} \\, \n     1_{\\{0\\leq \\theta \\leq \\lambda(S_{u},x_{{u^{\\raisebox{-1pt}{\\scriptsize\\scalebox{0.5}{$-$}}}}}^j)\/\\bar\\lambda\\}} \n\\\\[-1em]\n  &\\qquad\\qquad\\qquad\\qquad\n    \\times\n    \\left[\n\t F\\left(S_u,\\left<\\nu_{{u^{\\raisebox{-1pt}{\\scriptsize\\scalebox{0.5}{$-$}}}}}-\\delta_{x_{{u^{\\raisebox{-1pt}{\\scriptsize\\scalebox{0.5}{$-$}}}}}^j}\n\t\t\t +\\delta_{\\alpha \\, x_{{u^{\\raisebox{-1pt}{\\scriptsize\\scalebox{0.5}{$-$}}}}}^j}+\\delta_{(1-\\alpha)\\, x_{{u^{\\raisebox{-1pt}{\\scriptsize\\scalebox{0.5}{$-$}}}}}^j}, f\\right> \\right)\n\t\t\t\t-F\\left(S_u, \\, \\left<\\nu_{{u^{\\raisebox{-1pt}{\\scriptsize\\scalebox{0.5}{$-$}}}}}, f\\right>\\right) \n\t \\right]\n\\\\\n  &\\qquad\\qquad\\qquad\\qquad\n  \\times \\tilde N_1(\\mathrm{d} u, \\mathrm{d} j, \\mathrm{d} \\alpha, \\mathrm{d} \\theta)\\,,\n\\\\\n  M^2_t\n  &\\eqdef \n  \\iint\\limits_{[0,t]\\times\\N^*}\n\t 1_{\\{j\\leq N_{{u^{\\raisebox{-1pt}{\\scriptsize\\scalebox{0.5}{$-$}}}}}\\}} \\, \n\t \\left[\n\t  F\\left(S_u, \\left<\\nu_{{u^{\\raisebox{-1pt}{\\scriptsize\\scalebox{0.5}{$-$}}}}}-\\delta_{x_{{u^{\\raisebox{-1pt}{\\scriptsize\\scalebox{0.5}{$-$}}}}}^j}, f\\right>\\right)\n\t  -\n\t  F\\left(S_u, \\, \\left<\\nu_{{u^{\\raisebox{-1pt}{\\scriptsize\\scalebox{0.5}{$-$}}}}}, f\\right>\\right) \n\t \\right] \\,\t\\tilde N_2(\\mathrm{d} u, \\mathrm{d} j)\\,.\n\\end{align*}\nWe have the following properties of martingales:\n\\begin{enumerate}\n\n\\item if for any $t\\geq 0$:\n\\begin{align*}\n  &\\EE\\Bigl(\n    \\int_0^t \\int_{\\X} \\lambda(S_{u}, x) \\int_0^1\n\t \\bigl|\n         F(S_u,\\crochet{\\nu_u-\\delta_x+\\delta_{\\alpha \\, x}\n                +\\delta_{(1-\\alpha)\\,x},f})\n\\\\\n  &\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\n  -F(S_u,\\crochet{\\nu_u,f})\n\t\\bigr| \\, Q(\\mathrm{d} \\alpha)\\, \\nu_u(\\mathrm{d} x)\\,\\mathrm{d} u\n  \\Bigr) < + \\infty\n\\end{align*}\nthen  $(M^1_t)_{t\\geq 0}$ is a martingale;\n\\item if for any $t\\geq 0$\n\\begin{align*}\n  \\EE\\Bigl(\n    \\int_0^t \\int_{\\X} \n\t \\bigl|\n\t\tF(S_u,\\crochet{\\nu_u-\\delta_x,f})\n\t\t-\n\t\tF(S_u,\\crochet{\\nu_u,f})\n\t \\bigr|\n\t\t\t\\, \\nu_u(\\mathrm{d} x)\\,\\mathrm{d} u\n\t\\Bigr) < +\\infty\n\\end{align*}\nthen  $(M^2_{t})_{t\\geq 0}$ is a martingale;\n\n\\item if for any $t\\geq 0$\n\\begin{align*}\n  &\\EE\\Bigl(\n    \\int_0^t \\int_{\\X} \\lambda(S_{u}, x) \\int_0^1 \n\t  \\bigl|\n\t     F(S_u,\\crochet{\n\t        \\nu_u-\\delta_x+\\delta_{\\alpha \\, x}+\\delta_{(1-\\alpha)\\,x},f\n\t        })\n      \\\\ \n      &  \\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\n\t\t -\n\t\t F(S_u,\\crochet{\\nu_u,f})\n\t  \\bigr|^2 \\,\n\t  Q(\\mathrm{d} \\alpha)\\,\\nu_u(\\mathrm{d} x)\\,\\mathrm{d} u\n  \\Bigr)\n  < + \\infty\n\\end{align*}\nthen  $(M^1_{t})_{t\\geq 0}$ \nis a square integrable martingale and predictable quadratic variation:\n\\begin{align*}\n  &\\crochet{M^1}_t\t\n  \\eqdef\n  \\int_0^t \\int_{\\X} \\lambda(S_{u}, x) \\, \\int_0^1 \n    \\bigl[\n\t\tF(S_u,\n\t\t  \\crochet{\n\t\t    \\nu_u-\\delta_x+\\delta_{\\alpha \\, x}+\\delta_{(1-\\alpha)\\,x}\n\t\t    ,f})\n      \\\\ \n      &  \\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\n\t\t-\n\t\tF(S_u,\\crochet{\\nu_u,f})\n\t\\bigr]^2 \\,\n\tQ(\\mathrm{d} \\alpha)\\, \\nu_u(\\mathrm{d} x)\\,\\mathrm{d} u\\,;\n\\end{align*}\n\n\\item if for any $t\\geq 0$\n\\begin{align*}\n  \\EE\\Bigl(\\int_0^t \\int_{\\X} \n     \\left|\n\t    F(S_u,\\crochet{\\nu_u-\\delta_x,f})\n\t\t-\n\t\tF(S_u,\\crochet{\\nu_u,f})\n\t\\right|^2 \\, \\nu_u(\\mathrm{d} x)\\,\\mathrm{d} u\n  \\Bigr) < + \\infty\n\\end{align*}\nthen  $(M^2_{t})_{t\\geq 0}$ \nis a square integrable martingale and predictable quadratic variation:\n\\begin{align*}\n  \\left<M^2\\right>_t\t\n  \\eqdef\n  D \\, \\int_0^t \\int_{\\X} \n  \\left[\n    F(S_u,\\crochet{\\nu_u-\\delta_x,f})\n\t-\n\tF(S_u,\\crochet{\\nu_u,f})\n  \\right]^2 \\, \n  \\nu_u(\\mathrm{d} x)\\,\\mathrm{d} u\\,.\n\\end{align*}\n\\end{enumerate}\n\\end{proposition}\n\n\n\n\\bigskip\n\n\n\\begin{lemma}[Control of the population size]\n\\label{lem.taille.pop}\nLet $T>0$, if there exists $p\\geq 1$ such that $\\EE(\\crochet{\\nu_0,1}^p) <\\infty$, then:\n\\begin{align*}\n  \\EE\\left( \n    \\sup_{t \\in [0,T]} \\left< \\nu_t,1 \\right>^p \n  \\right) \n  \\leq C_{p,T}\n\\end{align*}\nwhere $C_{p,T}<\\infty $ depends only on $p$ and $T$.\n\\end{lemma}\n\n\\begin{proof}\nFor any $n \\in \\NN$, define the following stopping time:\n\\begin{align*}\n  \\tau_n \\eqdef \\inf \\{t \\geq 0, \\, N_t \\geq n \\}\\,.\n\\end{align*}\nLemma \\ref{lem.gen.inf} applied to $F(s,x)=x^p$ and $f(x)=1$ leads to:\n\\begin{multline*}\n  \\sup_{u\\in [0,t \\wedge \\tau_n]} \\crochet{\\nu_u,1}^p\n  \\leq \n  \\crochet{\\nu_0,1}^p\n\\\\\n  + \\iiiint\\limits_{[0,t\\wedge \\tau_n]\\times\\N^*\\times[0,1]^2} \n\t \t1_{\\{j\\leq N_{{u^{\\raisebox{-1pt}{\\scriptsize\\scalebox{0.5}{$-$}}}}}\\}}\\, \n\t\t1_{\\{0\\leq \\theta \\leq \\lambda(S_{u}, x_{{u^{\\raisebox{-1pt}{\\scriptsize\\scalebox{0.5}{$-$}}}}}^j)\/ \\bar\\lambda\\}}\\,\n\t \t\\bigl[ \n\t\t  (\\crochet{ \\nu_{{u^{\\raisebox{-1pt}{\\scriptsize\\scalebox{0.5}{$-$}}}}},1}+1)^p-\\crochet{\\nu_{{u^{\\raisebox{-1pt}{\\scriptsize\\scalebox{0.5}{$-$}}}}},1}^p \n\t\t\\bigr] \n\\\\[-1em]\n\t\\times N_1(\\mathrm{d} u, \\mathrm{d} j,\\mathrm{d} \\alpha,  \\mathrm{d} \\theta)\\,.\n\\end{multline*}\nFrom inequality  $(1+y)^p-y^p\\leq C_p\\,(1+y^{p-1})$ we get:\n\\begin{multline*}\n  \\sup_{u\\in [0,t \\wedge \\tau_n]} \\crochet{\\nu_u,1}^p\n  \\leq  \n  \\crochet{\\nu_0,1}^p \n\\\\\n  + C_p\\,\\iiiint\\limits_{[0,t\\wedge \\tau_n]\\times\\N^*\\times[0,1]^2} \n  1_{\\{j\\leq N_{{u^{\\raisebox{-1pt}{\\scriptsize\\scalebox{0.5}{$-$}}}}}\\}}\\,\n  1_{\\{0\\leq \\theta \\leq \\lambda(S_{u}, x_{{u^{\\raisebox{-1pt}{\\scriptsize\\scalebox{0.5}{$-$}}}}}^j)\/ \\bar\\lambda\\}}\\, \n  \\bigl[(1+\\crochet{\\nu_{{u^{\\raisebox{-1pt}{\\scriptsize\\scalebox{0.5}{$-$}}}}},1}^{p-1}\\bigr] \n\\\\[-1em]\n  \\times N_1(\\mathrm{d} u, \\mathrm{d} j,\\mathrm{d} \\alpha,  \\mathrm{d} \\theta)\\,.\n\\end{multline*}\nProposition \\ref{prop.martingale}, together with the inequality  $(1+y^{p-1})\\,y \\leq 2\\,(1+y^p)$ give:\n\\begin{align*}\n  \\EE \\Biggl(\n    \\sup_{u\\in [0,t \\wedge \\tau_n]} \\crochet{\\nu_u,1}^p \n  \\Biggr)\n  \\leq \n  \\EE(\\crochet{\\nu_0,1}^p)\n  + 2\\,\\bar\\lambda\\,C_p\\,\n    \\EE\\int_0^{t} \n\t\t\\bigl( 1+\\crochet{ \\nu_{u\\wedge \\tau_n},1}^p\\bigr)\\,\\mathrm{d} u\\,.\n\\end{align*}\nFubini's theorem and Gronwall's inequality allow us to conclude that for any $T< \\infty$:\n\\begin{align*}\n  \\EE\\Biggl(\n    \\sup_{t\\in [0,T \\wedge \\tau_n]} \\crochet{\\nu_t,1}^p \n  \\Biggr)\n  \\leq \n  \t\\Bigl(\\EE\\bigl(\\crochet{\\nu_0,1}^p\\bigr) + 2\\,\\bar\\lambda\\,C_p\\,T \\Bigr)  \\,\n  \t\\exp(2\\,\\bar\\lambda\\,C_p\\,T)\n  \\leq\n  C_{p,T}\n\\end{align*}\nwhere $C_{p,T}<\\infty$ as  $\\EE(\\crochet{\\nu_0,1}^p)<\\infty$.\n\n\nIn addition, the sequence of stopping times $\\tau_n$ tends to infinity,  otherwise there would exist $T_0<\\infty$ such that $\\PP(\\sup_n \\tau_n <T_0)=\\varepsilon_{T_0}>0$ hence $\\EE(\\sup_{t \\in [0,T_0 \\wedge \\tau_n]} \\crochet{\\nu_t,1}^p)\\geq \\varepsilon_{T_0}\\,n^p$  which contradicts the above inequality. Finally, Fatou's lemma gives:\n\\begin{align*} \n  \\EE\\Biggl( \\sup_{t \\in [0,T]} \\crochet{\\nu_t,1}^p \\Biggr)\n  =\n  \\EE \\Biggl( \n\t  \\liminf_{n \\to \\infty} \n\t    \\sup_{t\\in [0,T \\wedge \\tau_n]} \n\t    \\crochet{\\nu_t,1}^p\n  \\Biggr)\n  \\leq \n  \\liminf_{n \\to \\infty}\n   \\EE \\Biggl(\n     \\sup_{t\\in [0,T \\wedge \\tau_n]} \\crochet{\\nu_t,1}^p \n   \\Biggr)\n  \\leq \n  C_{p,T}\\,.\n\\end{align*}\n\\vskip-1em\\carre\n\\end{proof}\n\n\\bigskip\n\n\\begin{remark}\nIn particular, if $\\EE\\crochet{\\nu_0,1}<\\infty$ and if the function $ F $ is bounded, then by Lemma \\ref{lem.taille.pop} and Proposition \\ref{prop.martingale}, $(M^1_{t})_{t\\geq 0}$ and $(M^2_{t})_{t\\geq 0}$ are martingales.\n\\end{remark}\n\n\n\\begin{lemma}\n\\label{lem.integrabilite_Sigma}\nIf $\\EE\\crochet{\\nu_0,1}+\\EE(S_0)<\\infty$ then:\n\\begin{align*}\n  \\EE\\biggl(\\int_0^t |\\rho_{\\textrm{\\tiny\\rm\\!\\! s}} (S_u,\\nu_u) |\\,\\mathrm{d} u \\biggr)\n  \\leq\t\n  D\\, t\\,\\EE (S_0\\vee {\\mathbf s}_{\\textrm{\\tiny\\rm in}})\n  + \n  \\frac kV \\,\\bar{g}\\;\n  \\EE\\biggl( \\int_0^t\\crochet{\\nu_u,1} \\, \\mathrm{d} u\\biggr)\n  <\\infty\\,.\n\\end{align*}\n\\end{lemma}\n\n\\begin{proof}\nAs $S_u\\geq 0$ and $\\rho_{\\textrm{\\tiny\\rm\\!\\! g}}$ is a non negative function,\n\\begin{align*}\n  \\rho_{\\textrm{\\tiny\\rm\\!\\! s}} (S_u,\\nu_u)\n  \\leq \n  D \\, {\\mathbf s}_{\\textrm{\\tiny\\rm in}}\\,.\n\\end{align*}\nFurthermore, for any $(s,x)\\in\\RR_+ \\times \\X$,  $\\rho_{\\textrm{\\tiny\\rm\\!\\! g}}(s,x)\\leq \\bar g$, and \n$S_u\\leq S_0\\vee {\\mathbf s}_{\\textrm{\\tiny\\rm in}}$ so:\n\\begin{align*}\n  \\rho_{\\textrm{\\tiny\\rm\\!\\! s}} (S_u,\\nu_u)\n  \\geq \n  -D \\, (S_0\\vee {\\mathbf s}_{\\textrm{\\tiny\\rm in}})-\\frac kV \\, \\bar{g}\\,\\crochet{\\nu_u,1}\\,.\n\\end{align*}\nWe therefore deduce that:\n\\begin{align*}\n  \\int_0^t |\\rho_{\\textrm{\\tiny\\rm\\!\\! s}} (S_u,\\nu_u) |\\, \\mathrm{d} u\n  \\leq\t\n  D\\,t\\,(S_0\\vee {\\mathbf s}_{\\textrm{\\tiny\\rm in}})\n  + \n  \\frac kV \\, \\bar{g}\\, \\int_0^t \\crochet{\\nu_u,1}\\, \\mathrm{d} u\\,. \n\\end{align*}\nAccording to Lemma \\ref{lem.taille.pop}, the last term is integrable \nwhich concludes the proof.\n\\carre\n\\end{proof}\n\n\\bigskip\n\n\\begin{theorem}[Infinitesimal generator]\nThe process $(S_t,\\nu_t)_{t\\geq 0}$ is Markovian with values in  $\\R_{+}\\times\\MM(\\XX)$ and its infinitesimal generator is:\n\\begin{multline}\n  \\L \\Phi(s,\\nu)\n  \\eqdef \n  \\bigl(D({\\mathbf s}_{\\textrm{\\tiny\\rm in}}-s)-k\\,\\mu(s,\\nu) \\bigr)\\,\n  \\partial_s F(s, \\crochet{\\nu, f}) \t\t\n  +\n  \\crochet{\\nu ,\\rho_{\\textrm{\\tiny\\rm\\!\\! g}}(s,.) \\,  f'} \\,\n  \\partial_x F(s, \\crochet{\\nu, f}) \n\\\\\n  +  \n  \\int_{\\X} \\lambda(s,x) \\,\n     \\int_0^1 \t\t\t  \n\t   \\bigl[ \n\t      F(s,\n\t         \\crochet{\\nu-\\delta_x+\\delta_{\\alpha\\,x}\n\t\t    +\n\t\t    \\delta_{(1-\\alpha)\\,x},f}\n\t\t  ) \n\t\t  -\n\t\t  F(s, \\crochet{\\nu,f}) \n\t   \\bigr]\t\t\t\n\t\t\t  Q(\\mathrm{d} \\alpha) \\, \\nu(\\mathrm{d} x) \n\\\\\n\\label{eq.generateur.infinitesimal}\n\t\t+ D\\,  \\int_{\\X} \n\t\t\t \\bigl[\n\t\t\tF(s,\\crochet{\\nu-\\delta_x,f})\n\t\t\t-\n\t\t\tF(s,\\left<\\nu,f\\right>) \\bigr]\\,\n\t\t\t \\,\t\\nu(\\mathrm{d} x) \n\\end{multline}\nfor any $\\Phi(s,\\nu)=F(s,\\crochet{\\nu,f})$ with  $F \\in C_b^{1,1}(\\RR^+ \\times \\RR)$ and $f \\in C^{1}(\\X)$. Thereafter $\\L \\Phi(s,\\nu)$ is denoted \n$\\L F(s,\\crochet{\\nu,f})$.\n\\end{theorem}\n\n\\begin{proof}\nConsider deterministic initial conditions  $S_{0}=s\\in\\R_{+}$ and $\\nu_{0}=\\nu\\in\\MM(\\XX)$. \nAccording to Lemma \\ref{lem.gen.inf}:\n\\begin{align*}\n  &\n  \\EE\\bigl(F(S_t,\\crochet{\\nu_t, f})\\bigr)\n  =\n  F(s, \\, \\left<\\nu, f\\right>)\n  + \n  \\EE\\biggl(\\int_0^t \n\t   \\rho_{\\textrm{\\tiny\\rm\\!\\! s}} (S_u,\\nu_u) \\, \n\t   \\partial_s F(S_u, \\, \\crochet{\\nu_u, f})\\, \\mathrm{d} u \n    \\biggr)\n\\\\\n  &\\ \n  +\\EE \\biggl(\\int_0^t\n\t   \\crochet{ \\nu_u , \\rho_{\\textrm{\\tiny\\rm\\!\\! g}}(S_u,.)\\,f'} \\;\n\t   \\partial_x F\\left(S_u, \\, \\left<\\nu_u, f\\right>\\right)\\,\n\t   \\mathrm{d} u\n   \\biggr) \n\\\\\n  &\\ \n  + \\EE\\biggl(\\ \\ \n\t  \\iiiint\\limits_{[0,t]\\times\\N^*\\times[0,1]^2} \\!\\!\\!\\!\n\t\t 1_{\\{j\\leq N_{{u^{\\raisebox{-1pt}{\\scriptsize\\scalebox{0.5}{$-$}}}}}\\}} \\, \n\t\t 1_{\\{0\\leq \\theta \\leq \\lambda(S_{u}, x_{{u^{\\raisebox{-1pt}{\\scriptsize\\scalebox{0.5}{$-$}}}}}^j)\/\n\t\t\t\t\\bar\\lambda\\}}      \n\t\t\t\\Bigl[\n\t\t\t F\\bigl(\n\t\t\t   S_u\n\t\t\t   , \n\t\t\t   \\crochet{\\nu_{{u^{\\raisebox{-1pt}{\\scriptsize\\scalebox{0.5}{$-$}}}}}\n\t\t\t       -\\delta_{x_{{u^{\\raisebox{-1pt}{\\scriptsize\\scalebox{0.5}{$-$}}}}}^j}\n\t\t\t       + \\delta_{\\alpha \\, x_{{u^{\\raisebox{-1pt}{\\scriptsize\\scalebox{0.5}{$-$}}}}}^j}\n\t\t\t       + \\delta_{(1-\\alpha)\\, x_{{u^{\\raisebox{-1pt}{\\scriptsize\\scalebox{0.5}{$-$}}}}}^j}\n\t\t\t     , f}\n\t\t\t \\bigr)\n             \\\\[-1em]\n             &\\qquad\\qquad\\qquad\\qquad\\qquad\n              \\qquad\\qquad\\qquad\\qquad\\qquad \t      \n\t\t\t -\n\t\t\t F\\bigl(\n\t\t\t   S_u\n\t\t\t   , \n\t\t\t   \\crochet{\\nu_{{u^{\\raisebox{-1pt}{\\scriptsize\\scalebox{0.5}{$-$}}}}}, f}\\bigr) \\Bigr]\\,\n\t\t \t   N_1(\\mathrm{d} u, \\mathrm{d} j, \\mathrm{d} \\alpha, \\mathrm{d} \\theta)\n\t\t\t\\bigr)\n\\\\\n  &\\ \n  + \\EE\\biggl( \\ \\ \n     \\iint\\limits_{[0,t]\\times\\N^*} \n\t\t1_{\\{j\\leq N_{{u^{\\raisebox{-1pt}{\\scriptsize\\scalebox{0.5}{$-$}}}}}\\}} \\, \n\t\t\\Bigl[\n\t\t\t F(S_u, \\crochet{\\nu_{{u^{\\raisebox{-1pt}{\\scriptsize\\scalebox{0.5}{$-$}}}}}-\\delta_{x_{{u^{\\raisebox{-1pt}{\\scriptsize\\scalebox{0.5}{$-$}}}}}^j}, f})\n\t\t\t -F(S_u,\\crochet{\\nu_{{u^{\\raisebox{-1pt}{\\scriptsize\\scalebox{0.5}{$-$}}}}}, f}) \n\t    \\Bigr]\\,\n\t    N_2(\\mathrm{d} u,\\mathrm{d} j) \n\t \\biggr).\n\\end{align*}\nAs  functions $F$, $\\partial_sF$, $\\partial_x F$, $f'$ and $\\rho_{\\textrm{\\tiny\\rm\\!\\! g}}$  are bounded, from Lemmas \\ref{lem.taille.pop} and \\ref{lem.integrabilite_Sigma} and Proposition \\ref{prop.martingale}, the expectation the right side of the above equation are finite.\n\n\nFurthermore, from Proposition \\ref{prop.martingale}:\n\\begin{align*}\n  &\n  \\EE\\left(F\\left(S_t, \\, \\left<\\nu_t, f\\right>\\right)\\right)\n  =\n  F(s,\\crochet{\\nu, f})\n  + \n  \\EE (\\Psi(t))\n\\end{align*}\nwhere:\n\\begin{align*}\n & \\Psi(t)\n  \\eqdef\n   \t \\int_0^t \n\t \t\\rho_{\\textrm{\\tiny\\rm\\!\\! s}} (S_u,\\nu_u) \\, \n\t \t\\partial_s F(S_u, \\crochet{\\nu_u,f})\\, \\mathrm{d} u   \n\\\\\n  &\\quad \t\n  +\n  \\int_0^t\n\t \\crochet{\\nu_u,\\rho_{\\textrm{\\tiny\\rm\\!\\! g}}(S_u,.)\\,f'} \\;\n\t \\partial_x F(S_u, \\crochet{\\nu_u,f})\\,\n\t \\mathrm{d} u \n\\\\\n  &\\quad  \n  + \n     \\int_0^t \\int_{\\X} \\int_0^1 \n\t\t\\lambda(S_{u}, x)\\, \n\t\t\\left[\n\t\t\t F\\left(\n\t\t\t   S_u\n\t\t\t   ,\n\t\t\t   \\crochet{\\nu_u -\\delta_{x}\n\t\t\t   \t\t+\\delta_{\\alpha\\, x}\n\t\t\t   \t\t+\\delta_{(1-\\alpha)\\,x}\n\t\t\t   \\,,\\, \n\t\t\t   f}\n\t\t\t \\right)\n\t\t\t \\right.   \n\t\t\t\\\\[-0.7em]\n\t\t\t&\\qquad\\qquad\\qquad\\qquad\\qquad\n\t\t\t \\qquad\\qquad\\qquad\\qquad\n\t\t\t\\left. \n\t\t\t- F(S_u,\\crochet{\\nu_u,f}) \n\t\t\\right]\t\\,\n\t\tQ(\\mathrm{d} \\alpha)\\,\\nu_u(\\mathrm{d} x) \\, \\mathrm{d} u  \n\\\\\n  &\\quad  \n  + \n  D\\, \n    \\int_0^t \\int_{\\X} \n  \t\\bigl[\n\t\tF(S_u,\\crochet{\\nu_u-\\delta_{x}, f}) - F(S_u,\\crochet{\\nu_u, f}) \n\t\\bigr]\\,\n\t\\nu_u(\\mathrm{d} x) \\, \\mathrm{d} u\n\t\\,.\n\\end{align*}\nAlso:\n\\begin{align*}\n  &\n  \\frac{\\partial}{\\partial t} \n  \\Psi(t)\\Bigr|_{t=0}\n  =   \n  \\bigl(D({\\mathbf s}_{\\textrm{\\tiny\\rm in}}-s) - k\\,\\mu(s,\\nu)\\bigr) \\, \n  \\partial_s F(s, \\crochet{\\nu,f}) \n  + \\crochet{\\nu,\\rho_{\\textrm{\\tiny\\rm\\!\\! g}}(s,.)\\,f'} \\;\n\t\\partial_x F(s,\\crochet{\\nu,f}) \n\\\\\n  &\\qquad\n  + \\int_{\\X}\\int_0^1 \n\t  \\lambda(s, x) \\,  \n\t  \\Bigl[\n\t\tF\\left(\n\t\t  s\n\t\t  ,\n\t\t  \\crochet{\\nu-\\delta_x\n\t\t                +\\delta_{\\alpha \\, x}+\\delta_{(1-\\alpha)\\,x}, f\n\t\t  }\n\t\t\\right)\n\t\t-\n\t\tF(s,\\crochet{\\nu,f}) \n\t  \\Bigr]\\,\n\t  Q(\\mathrm{d}\\alpha) \\, \\nu(\\mathrm{d} x) \n\\\\\n  &\\qquad \n  + D\\,  \\int_{\\X} \n\t  \\Bigl[\n\t\t\tF(s,\\crochet{\\nu-\\delta_x,f})\n\t\t\t-\n\t\t\tF(s,\\crochet{\\nu,f}) \n\t  \\Bigr]\\,\n\t  \\nu(\\mathrm{d} x),\n\\end{align*}\nhence:\n\\begin{multline*}\n  \\biggl|\n    \\frac{\\partial}{\\partial t} \\Psi(t)\\Bigr|_{t=0}\n  \\biggr|\n  \\leq  \n  D\\,({\\mathbf s}_{\\textrm{\\tiny\\rm in}}+s)\n  +\n  \\left(\n\t \\frac kV\\,\\bar g  \\, \\norme{\\partial_s F}\n\t + \\bar g \\, \\norme{f'}\\,\\norme{\\partial_x F}\n\t  + 2\\,(\\bar\\lambda+D)\\, \\norme{F} \n  \\right) \\,\n  \\crochet{\\nu,1} \\, .\n\\end{multline*}\nThe right side of the last equation is finite.\nOne may apply the theorem of differentiation under the integral sign, hence the application\n$t\\mapsto \\EE(F(S_t, \\crochet{\\nu_t, f}))$ is differentiable at $t=0$ with derivative $\\L F(s,\\crochet{\\nu, f})$ defined by  \\eqref{eq.generateur.infinitesimal}.\n\\mbox{}\\hfil\\carre\n\\end{proof}\n\n\n\n\\begin{remark}\nWe define the washout time as the stopping time:\n\\begin{align*}\n  \\tau_{\\textrm{\\tiny\\rm w}}\n  \\eqdef\n  \\inf\\{t\\geq 0\\,;\\, N_{t}=\\crochet{\\nu_{t},1}=0\\}\n\\end{align*}\nwith the convention $\\inf\\emptyset=+\\infty$.\nBefore $\\tau_{\\textrm{\\tiny\\rm w}}$  the infinitesimal generator is given by \\eqref{eq.generateur.infinitesimal}, after this time $\\nu_{t}$ is the null measure, i.e. the chemostat does not contain any bacteria, and the infinitesimal generator is simply reduced to the generator associated with the ordinary  differential equation $\\dot S_{t}=D\\,({\\mathbf s}_{\\textrm{\\tiny\\rm in}}-S_{t})$ coupled with the null measure given by $\\crochet{\\nu_{t},f}=0$ for all $f$.\n\\end{remark}\n\n\\section{Convergence in distribution of the individual-based model}\n\\label{sec.convergence}\n\n\n\\subsection{Renormalization}\n\nIn this section we will prove that the coupled process of the  substrate concentration and  the bacterial population converges in distribution to a deterministic process in the space:\n\\[\n \\CC([0,T],\\R_{+}) \n \\times\n \\D([0,T],\\MM_{F}(\\X)) \n\\]\nequipped with the product metric: \n\\fenumi\\ the uniform norm on $\\CC([0,T],\\R_{+})$; \\fenumii\\ the Skorohod metric\non  $\\D([0,T],\\MM_{F}(\\X))$ where $\\MM_{F}(\\X)$ is equipped with the topology\nof the weak convergence of measures (see Appendix \\ref{appendix.skorohod}).\n\nRenormalization must have the effect that the density of the bacterial population must grow to infinity. To this end,  we first consider a growing volume, i.e. in the previous model the volume is replaced by:\n\\[\n  V_{n} = n\\,V\n\\]\nand $(S^n_{t},\\nu^n_{t})_{t\\geq 0}$ will denote the process \\eqref{def.proc.S.nu} where $V$ is replaced by $V_{n}$ and\n $x_t^{n,1},\\dots,x_t^{n,N_t^n}$ the $N_t^n$ individuals of $\\nu_t^n$; second\n we introduce the rescaled process:\n\\begin{align}\n\\label{def.renormalisation}\n  \\bar\\nu_t^n \\eqdef \\frac{1}{n}\\nu_t^n\\,,\\quad t\\geq 0\n\\end{align}\nand we suppose that:\n\\begin{align*}\n   \\bar\\nu_0^n \n   =\n   \\frac{1}{n}\\nu_0^n\n   \\xrightarrow[n\\to\\infty]{} \n   \\xi_0 \n   \\textrm{ in distribution in }\\MM_F(\\X) \\,.\n\\end{align*}\n$\\xi_0$ is the limit measure after renormalization of the population density at the initial time. It may be random, but we will assume without loss of generality that it is deterministic, moreover we suppose that $\\crochet{\\xi_{0},1}>0$.\n\n\\bigskip\n\nTherefore, this asymptotic consists in simultaneously letting the volume of chemostat  and the size of the initial population tend to infinity. \n\nAs the substrate concentration is maintained at the same value, it implies that the population tends to infinity. We will show that the rescaled process $(S^n_t,\\bar\\nu^n_t)_{t\\geq 0}$  defined by \\eqref{def.renormalisation}  converges in distribution to a process $(S_t,\\xi_t)_{t\\geq 0}$ introduced later.\n\n\n\n\n\\bigskip\n\nThe process  $(S^n_t,\\nu^n_t)_{t\\geq 0}$ is defined by:\n\\begin{align*}\n  \\dot S_t^n \n  &=\n  D\\,({\\mathbf s}_{\\textrm{\\tiny\\rm in}}-S_t^n)-\\frac k{V_{n}} \\, \n    \\int_{X} \\rho_{\\textrm{\\tiny\\rm\\!\\! g}}(S_t^n,x)\\,\\nu_t^n(\\mathrm{d} x) \n\\\\\n  &= \n  D({\\mathbf s}_{\\textrm{\\tiny\\rm in}}-S_t^n)-\\frac {k}{V} \\, \n    \\int_{X} \\rho_{\\textrm{\\tiny\\rm\\!\\! g}}(S_t^n,x)\\,\\bar\\nu_t^n(\\mathrm{d} x)\n  =\n  \\rho_{\\textrm{\\tiny\\rm\\!\\! s}}(S_{t}^n,\\bar\\nu_t^n)\n\\end{align*}\nand\n\\begin{align*}\n  \\bar\\nu_t^n\n  &= \n  \\frac 1n \\, \\sum_{j=1}^{n} \\delta_{A_t^j(S_0^n,\\nu_0^n)} \n\\\\\n  &\\qquad\n  + \\frac 1n \\, \n\t  \\iiiint\\limits_{[0,t]\\times\\N^*\\times[0,1]^2} \n\t\t 1_{\\{i\\leq N^n_{{u^{\\raisebox{-1pt}{\\scriptsize\\scalebox{0.5}{$-$}}}}}\\}} \\, \n\t\t 1_{\n\t\t     \\{0\\leq \\theta \\leq \n\t\t         \\lambda(S^n_u, x_{{u^{\\raisebox{-1pt}{\\scriptsize\\scalebox{0.5}{$-$}}}}}^{n,i})\/\\bar\\lambda\\}\n\t\t   }\\,\n  \\Bigl[\n\t-\\sum_{j=1}^{N^n_{{u^{\\raisebox{-1pt}{\\scriptsize\\scalebox{0.5}{$-$}}}}}} \n\t\t             \\delta_{A_{t-u}^j(S^n_u,\\nu^n_{{u^{\\raisebox{-1pt}{\\scriptsize\\scalebox{0.5}{$-$}}}}})}\n\\\\[-1em]\n  &\\qquad\\qquad\\qquad\\qquad\\qquad\n\t+\\sum_{j=1}^{N^n_{{u^{\\raisebox{-1pt}{\\scriptsize\\scalebox{0.5}{$-$}}}}}+1}\n\t\t           \t\\delta_{\n\t\t\t           A_{t-u}^j(S^n_u \\, ,\\,\n\t\t \t\t\t   \\nu^n_{{u^{\\raisebox{-1pt}{\\scriptsize\\scalebox{0.5}{$-$}}}}}-\\delta_{x_{{u^{\\raisebox{-1pt}{\\scriptsize\\scalebox{0.5}{$-$}}}}}^{n,i}}\n\t\t\t\t\t   +\\delta_{\\alpha x_{{u^{\\raisebox{-1pt}{\\scriptsize\\scalebox{0.5}{$-$}}}}}^{n,i}}\n\t\t \t\t\t   +\\delta_{(1-\\alpha) x_{{u^{\\raisebox{-1pt}{\\scriptsize\\scalebox{0.5}{$-$}}}}}^{n,i}})\n\t\t\t\t\t}\n  \\Bigr] \\;\n  N_1(\\mathrm{d} u, \\mathrm{d} i, \\mathrm{d} \\alpha, \\mathrm{d} \\theta)\n\\\\[0.5em]\n  &\\qquad\n  + \\frac 1n \\,  \n    \\iint\\limits_{[0,t]\\times\\N^*}\n    1_{\\{i\\leq N^n_{{u^{\\raisebox{-1pt}{\\scriptsize\\scalebox{0.5}{$-$}}}}}\\}} \\, \n    \\Bigl[\n\t\t-\\sum_{j=1}^{N^n_{{u^{\\raisebox{-1pt}{\\scriptsize\\scalebox{0.5}{$-$}}}}}}\n\t\t \t\\delta_{A_{t-u}^j(S^n_u,\\nu^n_{{u^{\\raisebox{-1pt}{\\scriptsize\\scalebox{0.5}{$-$}}}}})}\n\t\t+\\sum_{j=1}^{N^n_{{u^{\\raisebox{-1pt}{\\scriptsize\\scalebox{0.5}{$-$}}}}}-1}\n\t\t \t\\delta_{A_{t-u}^j(S^n_u,\n\t\t \t\t\t\\nu^n_{{u^{\\raisebox{-1pt}{\\scriptsize\\scalebox{0.5}{$-$}}}}}-\\delta_{x_{{u^{\\raisebox{-1pt}{\\scriptsize\\scalebox{0.5}{$-$}}}}}^{n,i}})}\n\t\\Bigr]\n\t\t\t \\,\tN_2(\\mathrm{d} u, \\mathrm{d} i)\n\\end{align*}\n\n\n\\begin{remark}\n\\label{remark.cv.nu.seule}\nDue to the structure of the previous system and specifically the above equation, \nit will be sufficient to prove the convergence in distribution of the component $\\bar\\nu_t^n$ to deduce also the convergence of the component  $S_{t}^n$.\n\\end{remark}\n\n\n\n\n\\subsection{Preliminary results}\n\n\n\n\\begin{lemma}\n\\label{lem.renorm}\nFor all $t>0$,\n\\begin{align*}\n  &\n  F(S_t^n,\\crochet{\\bar\\nu_t^n, f})\n  =\n  F(S_0^n,\\crochet{\\bar\\nu_0^n, f})\n\\\\\n  &\\qquad\n  + \n  \\int_0^t \\textstyle\\Bigl(\n    D\\,({\\mathbf s}_{\\textrm{\\tiny\\rm in}}-S_u^n)\n    - \\frac {k}{V}\\,\\int_X \\rho_{\\textrm{\\tiny\\rm\\!\\! g}}(S_u^n,x)\\,\\bar\\nu_u^n(\\mathrm{d} x) \n  \\Bigr)\n  \\;\n  \\partial_s F(S_u^n, \\, \\crochet{\\bar\\nu_u^n, f})\\,\\mathrm{d} u\n\\\\\n  &\\qquad \n  + \\int_0^t\n\t   \\crochet{\\bar\\nu_u^n , \\rho_{\\textrm{\\tiny\\rm\\!\\! g}}(S_u^n,.)\\,f'} \\;\n\t   \\partial_x F(S_u^n, \\, \\crochet{\\bar\\nu_u^n, f}) \\, \\mathrm{d} u \n\\\\\n  &\\qquad  \n  + n\\,\\int_0^t \\int_{\\X} \\lambda(S^n_u, x)\\, \n\t \\int_0^1\n\t   \\textstyle\n\t   \\Bigl[\n\t     F\\bigl(\n\t       S_u^n\n\t       \\,,\\,\n\t       \\crochet{\\bar\\nu_u^n,f} +\\frac 1n f(\\alpha \\, x )\n\t       +\\frac 1nf((1-\\alpha) \\, x)-\\frac 1nf(x)\n\t   \\bigr)  \n\\\\[-0.5em]\n  & \\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\n\t\t\t -F\\bigl(S_u^n, \\crochet{\\bar\\nu_u^n, f}\\bigr) \n\t   \\Bigr]\n\t   \\,Q(\\mathrm{d} \\alpha)\\, \\bar\\nu_u^n(\\mathrm{d} x)\\,\\mathrm{d} u  \n\\\\\t\t\t\n  &\\qquad  \n  + D\\,n\\, \n  \\int_0^t \\int_{\\X} \n  \\textstyle\\Bigl[\n\t F\\bigl(S_u^n, \\, \\crochet{\\bar\\nu_u^n, f}-\\frac 1nf(x) \\bigr)\n\t -\n\t F\\bigl(S_u^n, \\, \\crochet{\\bar\\nu_u^n, f} \\bigr) \n  \\Bigr]\\, \\bar\\nu_u^n(\\mathrm{d} x)\\,\\mathrm{d} u\n\t+ Z_t^{F,f,n}\n\\end{align*}\nwhere\n\\begin{align}\n\\nonumber\n  Z_t^{F,f,n}\n  &\\eqdef \n  \\iiiint\\limits_{[0,t]\\times\\N^*\\times[0,1]^2}\n\t1_{\\{i\\leq N_{{u^{\\raisebox{-1pt}{\\scriptsize\\scalebox{0.5}{$-$}}}}}^n\\}} \\, \n\t1_{\\{0\\leq \\theta \\leq \\lambda(S^n_u, x_{{u^{\\raisebox{-1pt}{\\scriptsize\\scalebox{0.5}{$-$}}}}}^{n,i})\/\\bar\\lambda\\}}\n\\\\[-1em]\n\\nonumber\n  &\\qquad\\qquad\\qquad\\qquad\n\t\\textstyle\n\t\\Bigl[\n\t\tF\\bigl(\n\t\t  S_u^n\n\t\t  \\,,\\, \n\t\t  \\crochet{\\bar\\nu_{{u^{\\raisebox{-1pt}{\\scriptsize\\scalebox{0.5}{$-$}}}}}^n, f}\n\t\t  + \\frac 1n f(\\alpha \\,x_{{u^{\\raisebox{-1pt}{\\scriptsize\\scalebox{0.5}{$-$}}}}}^{n,i})\n\t\t  +\\frac 1nf((1-\\alpha)\\,x_{{u^{\\raisebox{-1pt}{\\scriptsize\\scalebox{0.5}{$-$}}}}}^{n,i})\n\t\t  -\\frac 1nf(x_{{u^{\\raisebox{-1pt}{\\scriptsize\\scalebox{0.5}{$-$}}}}}^{n,i})\n\t\t\\bigr) \\, \n\\\\[-0.5em]\n\\nonumber\n  &\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\n      -F\\bigl(S_u^n, \\crochet{\\bar\\nu_{{u^{\\raisebox{-1pt}{\\scriptsize\\scalebox{0.5}{$-$}}}}}^n, f}\\bigr) \t\n\t\\Bigr]\t\t\t\n\t\\,\\tilde N_1(\\mathrm{d} u, \\mathrm{d} i, \\mathrm{d} \\alpha, \\mathrm{d} \\theta)  \n\\\\\n\\label{eq.Z,F,f}\n  & \\qquad \n  + \\iint\\limits_{[0,t]\\times\\N^*}\n  \t\\textstyle \n\t1_{\\{i\\leq N_{{u^{\\raisebox{-1pt}{\\scriptsize\\scalebox{0.5}{$-$}}}}}^n\\}} \\, \n\t\\Bigl[\n\t  F\\bigl(S_u^n, \\crochet{\\bar\\nu_{{u^{\\raisebox{-1pt}{\\scriptsize\\scalebox{0.5}{$-$}}}}}^n, f}\n\t         -\\frac 1nf(x_{{u^{\\raisebox{-1pt}{\\scriptsize\\scalebox{0.5}{$-$}}}}}^{n,i}) \\bigr)\n\t  -\n\t  F\\bigl(S_u^n, \\crochet{\\bar\\nu_{{u^{\\raisebox{-1pt}{\\scriptsize\\scalebox{0.5}{$-$}}}}}^n, f}\\bigr) \n\t\\Bigr] \n\t\\,\t\\tilde N_2(\\mathrm{d} u, \\mathrm{d} i)\n\\end{align}\n\\end{lemma}\n\n\\begin{proof}\nIt is sufficient to note that\n$\nF(S_t^n,\\crochet{ \\bar\\nu_t^n,f })\n\t = F(S_t^n,\\crochet{\\nu_t^n,\\frac{1}{n}f})\n$\nand to apply Lemma \\ref{lem.gen.inf}.\n\\carre\n\\end{proof}\n\n\\bigskip\n\n\\begin{lemma}\n\\label{lem.esp}\nIf $\\sup_{n\\in\\N} \\EE(\\crochet{\\bar\\nu_0^n,1}^p)<\\infty$ for some $p \\geq 1$, then:\n\\[\n  \\sup_{n\\in\\N} \\EE\\Biggl(\\sup_{u\\in [0,t]} \\crochet{\\bar\\nu_u^n,1}^p\\Biggr)\n  <\n  C_{t,p}\n\\]\nwhere $C_{t,p}$ depends only on $t$ and $p$.\n\\end{lemma}\n\n\\begin{proof}\nDefine the stopping time:\n\\[\t\n\t\\tau_N^n\n\t\\eqdef\n\t\\inf \\{t\\geq 0, \\crochet{\\bar\\nu_t^n,1} \\geq N \\}\\,.\n\\]\nAccording to Lemma \\ref{lem.renorm}:\n\\begin{align*}\n  &\n  \\sup_{u \\in [0, t \\wedge \\tau_N^n]} \\crochet{\\bar\\nu_u^n,1}^p\n  \\leq \n  \\crochet{\\bar\\nu_0^n,1}^p\n\\\\\n  &\\qquad\\qquad \n  + n\\,\\int_0^{t \\wedge \\tau_N^n} \\int_{\\X}\n\t\t\t\\lambda(S^n_u, x)\\,\n\t\t\t\\textstyle\n\t\t\t \\left[\\left(\\crochet{ \\bar\\nu_u^n,1} + \\frac{1}{n}\\right)^p\n\t\t\t -\\crochet{ \\bar\\nu_u^n,1}^p \\right]\\, \n\t\t\t\\bar\\nu_u(\\mathrm{d} x)\\,\\mathrm{d} u\n\\\\\n  & \\qquad\\qquad\n  + \\iiiint\\limits_{[0,t\\wedge \\tau_N^n]\\times\\N^*\\times[0,1]^2}\n\t\t1_{\\{i \\leq N_{{u^{\\raisebox{-1pt}{\\scriptsize\\scalebox{0.5}{$-$}}}}}^n\\}}\\,\n\t\t1_{\\{0\\leq \\theta \\leq \\lambda(S^n_u, x_{{u^{\\raisebox{-1pt}{\\scriptsize\\scalebox{0.5}{$-$}}}}}^{i,n})\/ \\bar\\lambda\\}}\n\\\\[-1em]\n  & \\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\n  \\times \\,\n\t\t\\textstyle\n\t\t\\left[\\left(\\crochet{\\bar\\nu_{{u^{\\raisebox{-1pt}{\\scriptsize\\scalebox{0.5}{$-$}}}}}^n,1} + \\frac{1}{n}\\right)^p\n\t\t\t -\\crochet{\\bar\\nu_{{u^{\\raisebox{-1pt}{\\scriptsize\\scalebox{0.5}{$-$}}}}}^n,1}^p \\right]\\, \n\t \\tilde N_1(\\mathrm{d} u, \\mathrm{d} i,\\mathrm{d} \\alpha,  \\mathrm{d} \\theta)\\,.\n\\end{align*}\nFrom the inequality \n$(1+y)^p-y^p \\leq C_p\\, (1+y^{p-1})$,\none can easily check that $(\\frac 1n+y)^p-y^p \\leq \n\\frac{C_p}{n}\\, (1+y^{p-1})$. Taking expectation in the previous inequality  and applying Proposition \\ref{prop.martingale} lead to:\n\\begin{align*}\n  \\EE \\Biggl( \n    \\sup_{u \\in [0, t \\wedge \\tau_N^n]} \\crochet{\\bar\\nu_u^n,1}^p \n  \\Biggr)\n  &\\leq \n  \\EE\\bigl(\\crochet{ \\bar\\nu_0^n,1}^p \\bigr)\n  + \n  \\EE \\int_0^{t \\wedge \\tau_N^n} \n    C_p\\,\n    \\bigl(1+\\crochet{\\bar\\nu_u^n,1}^{p-1}\\bigr)\n\t\\int_\\X \\lambda(S^n_u, x)\\,\\bar\\nu_u^n(\\mathrm{d} x)\\, \\mathrm{d} u\n\\\\\n\t&\\leq \n\t\\EE\\bigl(\\crochet{\\bar\\nu_0^n,1 }^p \\bigr)\n\t+ \n\t\\bar\\lambda \\, C_p\\, \n\t\\int_0^t \\EE \\left(\n\t  \\crochet{ \\bar\\nu_{u \\wedge \\tau_N^n}^n,1 }\n\t  +\n\t  \\crochet{\\bar\\nu_{u \\wedge \\tau_N^n}^n,1}^p \n\t\\right) \\, \\mathrm{d} u\\,.\n\\end{align*}\nAs:\n\\[\n  \\crochet{ \\bar\\nu_{u \\wedge \\tau_N^n}^n,1}  \n  +\n  \\crochet{ \\bar\\nu_{u \\wedge \\tau_N^n}^n,1}^p  \n  \\leq \n  2 \\, \\left(1+ \\crochet{\\bar\\nu_{u \\wedge \\tau_N^n}^n,1}^p \\right)\\,,\n\\]\nwe get:\n\\begin{align*}\n  \\EE \\Biggl( \n    \\sup_{u \\in [0, t \\wedge \\tau_N^n]} \\crochet{\\bar\\nu_u^n,1}^p \n  \\Biggr)\n  &\\leq \n  \\EE\\left(\\crochet{\\bar\\nu_0^n,1}^p \\right)\n\t\t\t+ 2\\, \\bar\\lambda \\, C_p\\, t\n\t\t\t+ 2\\, \\bar\\lambda \\, C_p\\, \\int_0^t \n\t\t\t\\EE \\left(\\sup_{u \\in [0, u \\wedge \\tau_N^n]} \n\t\t\t  \\crochet{\\bar\\nu_u^n,1}^p \\right) \\, \\mathrm{d} u\n\\end{align*}\nand from Gronwall's inequality we obtain:\n\\begin{align*}\n  \\EE \\Biggl( \n    \\sup_{u \\in [0, t \\wedge \\tau_N^n]} \\crochet{\\bar\\nu_u^n,1}^p \n  \\Biggr)\n  \\leq \n  \\Bigl(\n  \t\\EE(\\crochet{\\bar\\nu_0^n,1}^p ) + 2\\,\\bar\\lambda \\, C_p\\, t\n  \\Bigr)\n  \\,\n  \\exp(2\\,\\bar\\lambda \\, C_p\\, t)\\,.\n\\end{align*}\nThe sequence of stopping times $\\tau_N^n$ tends to infinity as $N$ tends to infinity \nfor the same reasons as those set  in the proof of Lemma \\ref{lem.taille.pop}.\nFrom Fatou's lemma we deduce:\n\\begin{align*}\n  \\EE \\Biggl(  \n       \\sup_{u \\in [0, t]} \\crochet{\\bar\\nu_u^n,1}^p \n  \\Biggr)\n  & =\n  \\EE \\Biggl( \n     \\liminf_{N \\to \\infty} \n       \\sup_{u \\in [0, t \\wedge \\tau_N^n]} \\crochet{\\bar\\nu_u^n,1}^p \n  \\Biggr)\n\\\\\n  &\\leq \n  \\liminf_{N \\to \\infty}\\EE \n     \\Biggl(\n        \\sup_{u \\in [0,t\\wedge\\tau_N^n]} \\crochet{\\bar\\nu_u^n,1}^p \n     \\Biggr)\n\\\\\n  &\\leq \n  \\Bigl(\n    \\EE\\left(\\crochet{\\bar\\nu_0^n,1}^p \\right) \n    + \n\t2\\, \\bar\\lambda \\, C_p\\, t\n  \\Bigr)\\, \n  \\exp(2\\,\\bar\\lambda\\,C_p\\, t)\n\\end{align*}\nand as $\\sup_n \\EE\\left(\\crochet{\\bar\\nu_0^n,1}^p \\right) <\\infty$, we deduce the proof of the lemma.\n\\carre\n\\end{proof}\n\n\\bigskip\n\n\\begin{corollary}\nLet $f \\in C^1(\\X)$, suppose that  $\\EE(\\crochet{\\bar\\nu_0^n, 1}^2)<\\infty$,\nthen for all  $t>0$:\n\\begin{align}\n\\nonumber\n  \\crochet{\\bar\\nu_t^n, f}\n  &=\n  \\crochet{\\bar\\nu_0^n, f}\n  + \\int_0^t \n\t \t\t\\left< \\bar\\nu_u^n , \\, \\rho_{\\textrm{\\tiny\\rm\\!\\! g}}(S_u^n,.)\\,f'\\right> \n\t \t\t\\, \\mathrm{d} u \n\\\\\n\\nonumber \n  &\\qquad\n  + \\int_0^t \\int_{\\X} \\lambda(S^n_u,x) \\, \n\t \\int_0^1  \n\t\t\t  \\bigl[ f(\\alpha \\, x)+f((1-\\alpha)\\,x)-f(x)\\bigr] \\, \n\t\t\t  Q(\\mathrm{d} \\alpha)\\,\\bar\\nu_u^n(\\mathrm{d} x) \\, \\mathrm{d} u\t\t\t  \n\\\\\t\t\n\\label{renormalisation}\n  &\\qquad  \n  - D\\, \\int_0^t \\int_{\\X} f(x) \\, \\bar\\nu_u^n(\\mathrm{d} x) \\, \\mathrm{d} u   \n  + Z_t^{f,n} \n\\end{align}\nwhere\n\\begin{align}\n\\nonumber\n  Z_t^{f,n}\n  &\\eqdef\n  \\frac 1n \\, \n  \\iiiint\\limits_{[0,t]\\times\\N^*\\times[0,1]^2} \n  \t1_{\\{i\\leq N_{{{u^{\\raisebox{-1pt}{\\scriptsize\\scalebox{0.5}{$-$}}}}}}^n\\}} \\, \n\t1_{\\{0\\leq \\theta \\leq \\lambda(S^n_u, x_{{{u^{\\raisebox{-1pt}{\\scriptsize\\scalebox{0.5}{$-$}}}}}}^{i,n})\/\\bar\\lambda\\}} \n\\\\[-1em]\n\\nonumber\n  &\\qquad\\qquad\\qquad\\qquad\t\t\n\t\\times \n\t\\left[\n\t   f(\\alpha\\,x_{{{u^{\\raisebox{-1pt}{\\scriptsize\\scalebox{0.5}{$-$}}}}}}^{i,n})\n\t   + f((1-\\alpha)\\,x_{{{u^{\\raisebox{-1pt}{\\scriptsize\\scalebox{0.5}{$-$}}}}}}^{i,n})\n\t   - f(x_{{{u^{\\raisebox{-1pt}{\\scriptsize\\scalebox{0.5}{$-$}}}}}}^{i,n})\n\t\\right] \\, \n\t\\tilde N_1(\\mathrm{d} u, \\mathrm{d} i, \\mathrm{d} \\alpha, \\mathrm{d} \\theta)  \n\\\\[0.5em]\n\\label{eq.Zfn}\n  &\\qquad\\qquad\\qquad\n  - \\frac 1n \\, \\iint\\limits_{[0,t]\\times\\N^*}\n\t\t\t1_{\\{i\\leq N_{{{u^{\\raisebox{-1pt}{\\scriptsize\\scalebox{0.5}{$-$}}}}}}^n\\}} \\, \n\t\t\tf(x_{{{u^{\\raisebox{-1pt}{\\scriptsize\\scalebox{0.5}{$-$}}}}}}^{i,n})\n\t\t\t \\,\t\\tilde N_2(\\mathrm{d} u, \\mathrm{d} i)\n\\end{align}\nis a martingale with the following predictable quadratic variation:\n\\begin{align} \n\\nonumber\n  \\crochet {Z^{f,n}}_t\n  &=\n  \\frac 1n  \n  \\int_0^t \\int_{\\X} \\lambda(S^n_u, x) \\, \n     \\int_0^1 \n        \\left[ f(\\alpha \\, x)+f((1-\\alpha) \\, x)-f(x) \\right]^2 \\, \n\t\t\t  Q(\\mathrm{d} \\alpha) \\,\\bar\\nu_u^n(\\mathrm{d} x) \\, \\mathrm{d} u  \n\\\\\n\\label{var_qua}\n\t&\\qquad\\qquad\\qquad\n\t + \\frac 1n \\, D\\, \\int_0^t \\int_{\\X} \n\t\t\tf(x)^2 \\, \\bar\\nu_u^n(\\mathrm{d} x) \\, \\mathrm{d} u\\,. \n\\end{align}\n\\end{corollary}\n\n\\begin{proof}\nEquation \\eqref{renormalisation} is obtained by applying Lemma \\ref{lem.renorm} \nwith  $F(s,x)=x$, then $Z^{f,n}$ is $Z^{F,f,n}$ defined by\n\\eqref{eq.Z,F,f}. Moreover as the random measures $\\tilde{N_1}$ and $\\tilde{N_2}$ are independent, we have:\n\\begin{align*}\n\\crochet {Z^{f,n}}_t\n\t = &\n\t \t\\frac 1{n^2} \\, \\crochet {M^1}_t\n\t \t+\\frac 1{n^2} \\, \\crochet {M^2}_t\n\\end{align*}\nwhere $M^1$ and $M^2$ are defined at Proposition \\ref{prop.martingale}. \nFrom this latter proposition and Lemma  \\ref{lem.esp} we deduce the proof of the corollary.\n\\carre\n\\end{proof}\n\n\\bigskip\n\n\\begin{remark}\nThe infinitesimal generator of the renormalized process $(S^n_t,\\bar\\nu^n_t)_{t\\geq 0}$ is:\n\\begin{multline*}\n  \\L^n \\Phi(s,\\nu)\n  \\eqdef \n  \\bigl(D({\\mathbf s}_{\\textrm{\\tiny\\rm in}}-s)-k\\,\\mu(s,\\nu) \\bigr)\\,\n  \\partial_s F(s, \\crochet{\\nu, f}) \t\t\n  +\n  \\crochet{\\nu ,\\rho_{\\textrm{\\tiny\\rm\\!\\! g}}(s,.) \\,  f'} \\,\n  \\partial_x F(s, \\crochet{\\nu, f}) \n\\\\\n  \\quad+  \n  n\\,\\int_{\\X} \\lambda(s,x) \\,\n     \\int_0^1 \t\t\t  \n\t   \\bigl[ \n\t      F\\bigl(s,\n\t         \\crochet{\\textstyle\\nu - \\frac 1n \\delta_x + \\frac 1n \\delta_{\\alpha\\,x} \n\t         + \\frac 1n \\delta_{(1-\\alpha)\\,x} \\, , \\, f}\n\t\t  \\bigr) \n\t\t  -\n\t\t  F(s, \\crochet{\\nu,f}) \n\t   \\bigr]\t\\;\t\t\n\t\t\t  Q(\\mathrm{d} \\alpha) \\, \\nu(\\mathrm{d} x) \n\\\\\n\t\\quad\n\t+ n\\, D\\, \\int_{\\X} \n\t \t\\bigl[\n\t\t\tF\\bigl(s,\\crochet{\\textstyle\\nu- \\frac 1n \\delta_x \\, , \\, f} \\bigr)\n\t\t\t-\n\t\t\tF(s,\\left<\\nu,f\\right>) \n\t\t\\bigr]\\,\n\t\t\t \\,\t\\nu(\\mathrm{d} x) \n\\end{multline*}\nfor any $\\Phi(s,\\nu)=F(s,\\crochet{\\nu,f})$ with $F \\in C_b^{1,1}(\\RR^+ \\times \\RR)$ and $f \\in C^{1}(\\X)$. \nNote that this generator has the same ``substrat'' part than that of the initial generator \n\\eqref{eq.generateur.infinitesimal} which again justifies the \nRemark \\ref{remark.cv.nu.seule}.\n\\end{remark}\n\n\n\\bigskip\n\n\nTo prove  the uniqueness of the solution of the limit IDE, we have\nto assume that the application $\\lambda(s,x)$ is Lipschitz continous w.r.t. $s$ uniformly in  $x$:\n\\begin{align}\n\\label{hyp.lambda.lipschitz}\n \\bigl|\\lambda(s_1,x)-\\lambda(s_2,x)\\bigr|\\leq k_\\lambda \\,|s_1-s_2|\n\\end{align}\nfor all $s_1,\\,s_2 \\geq 0$ and all $x\\in\\XX$.  \nThis hypothesis as well as Hypothesis \\ref{hyp.rhog.lipschitz} \nwill also be used to demonstrate the convergence of IBM, see Theorem \\ref{theo.cv.ibm}.\n\n\n\\subsection{Convergence result}\n\n\n\\begin{theorem}[Convergence of the IBM towards the IDE]\n\\label{theo.cv.ibm}\nUnder the assumptions described above, the process $(S_t^n,\\bar\\nu_t^n)_{t\\geq 0}$ converges in distribution in the product space \n$\\CC([0,T],\\R_{+}) \n \\times\n \\D([0,T],\\MM_{F}(\\X))$\ntowards the process $(S_t,\\xi_t)_{t\\geq 0}$ solution of:\n\\begin{align}\n\\label{eq.limite.substrat.faible}\n\tS_t  \n\t&= \n\tS_{0}\n\t+\n\t\\int_{0}^t\\biggl[\n\tD\\,({\\mathbf s}_{\\textrm{\\tiny\\rm in}}-S_u)-\\frac kV \\int_\\X \\rho_{\\textrm{\\tiny\\rm\\!\\! g}}(S_u,x)\\,\\xi_u(\\mathrm{d} x)\n\t\\biggr]\\,\\rmd u\\,,\n\\\\[1em]\n\\nonumber \n\t\\crochet{\\xi_t,f}\n\t&=\n\t\\crochet{\\xi_0,f}\n\t+\n\t\\int_{0}^t\\biggl[\n\t\\int_{\\X} \\rho_{\\textrm{\\tiny\\rm\\!\\! g}}(S_u,x)\\,f'(x) \\,\t\\xi_u(\\mathrm{d} x)\n\\\\\n\\nonumber \n\t&\\qquad\n\t+ \\int_{\\X} \\int_0^1\\lambda(S_u, x)\\,  \n\t\\Bigl[f(\\alpha\\, x)+ f((1-\\alpha)\\,x)-f(x)\\Bigr] \\, \t \t\t\t\n\t\t\tQ(\\mathrm{d} \\alpha) \\,\\xi_u(\\mathrm{d} x) \n\\\\\n\\label{eq.limite.eid.faible}\n\t&\\qquad\n\t- D\\, \\int_{\\X} f(x) \\,\t\\xi_u(\\mathrm{d} x) \n\t\\biggr]\\,\\rmd u\\,,\n\\end{align}\nfor any $f \\in C^{1}(\\X)$.\n\\end{theorem}\n\nThe proof is in three steps\\footnote{Note that our situation is simpler than that studied by \\cite{roelly1986a} and \\cite{meleard1993a} since in our case $\\XX$ is compact: in fact in our case the weak  topology -- the smallest topology which makes the applications $\\nu\\to\\crochet{\\nu,f}$ continuous for any $f$ continuous and bounded -- and the vague topology -- the smallest topology which makes the applications $\\nu\\to\\crochet{\\nu,f}$ continuous for all $f$ continuous with compact support -- are identical.}:\nfirst the uniqueness of the solution of the limit equation \\eqref{eq.limite.substrat.faible}-\\eqref{eq.limite.eid.faible}, \nsecond the tightness  (of the sequence of distribution) of  $\\bar\\nu^n$\nand lastly the convergence in distribution of the sequence.\n\n\n\n\\subsubsection*{Step 1: uniqueness of the solution of   \\eqref{eq.limite.substrat.faible}-\\eqref{eq.limite.eid.faible}}\n\nLet $(S_t,\\xi_t)_{t\\geq 0}$ be a solution of \\eqref{eq.limite.substrat.faible}-\\eqref{eq.limite.eid.faible}.\nWe first show that $(\\xi_t)_t$ is of finite mass for all $t\\geq 0$:\n\\begin{align*}\n\t\\crochet{\\xi_t, 1}\n\t& = \n\t\\crochet{\\xi_0, 1}\n\t+ \n\t\\int_0^t \\int_{\\X} \\int_0^1 \\lambda(S_u, x)\\, Q(\\mathrm{d} \\alpha) \\, \t \t\n\t\t\t\\xi_u(\\mathrm{d} x)\\, \\mathrm{d} u  \n\t\t- D\\, \\int_0^t \\int_{\\X} \\xi_u(\\mathrm{d} x) \\, \\mathrm{d} u\\\\ \n\t& \\leq  \\left<\\xi_0, 1\\right>+(\\bar \\lambda - D) \\int_0^t \\left<\\xi_u, 1\\right> \\, \\mathrm{d} u\n\\end{align*} \nand according to Gronwall's inequality: \n$\\crochet{\\xi_t, 1}  \\leq \\crochet{\\xi_0, 1}  \\, e^{(\\bar \\lambda - D)\\,t} \n\t< \\infty\n$.\n\nWe introduce the following norm on $\\MM_{F}(\\XX)$:\n\\[\n  \\normm{\\bar\\nu}\n  \\eqdef\n  \\sup \n\t\\Bigl\\{\n\t\t|\\crochet{\\bar\\nu,f}|\n\t\t\\,;\\,\n\t\tf \\in C^1(\\X),\\, \\norme{f}_\\infty \\leq 1,\\,\\norme{f'}_\\infty \\leq 1\n\t\\Bigr\\}\n\\]\nand consider two solutions  $(S^1_t,\\xi^1_t)_{t\\geq 0}$ and $(S^2_t,\\xi^2_t)_{t\\geq 0}$ \nof  \\eqref{eq.limite.substrat.faible}-\\eqref{eq.limite.eid.faible}. \n\n\nIt was previously shown that $\\xi^1_t$ and $\\xi^2_t$  are of finite mass on  $\\RR_+$, so we can define:\n\\[\n  C_t\n  \\eqdef \\sup_{0\\leq u \\leq t} \\crochet{\\xi^1_u+\\xi^2_u , 1}\n  \\,.\n\\]\nAccording to \\eqref{eq.limite.eid.faible}, for any  $f \\in C^1(\\X)$ such that \n$\\norme{f}_\\infty \\leq 1$ and $\\norme{f'}_\\infty \\leq 1$ we have:\n\\begin{align*}\n\t|\\crochet{\\xi^1_t-\\xi^2_t,f}|\n\t&\\leq \n\t\\int_0^t \\biggl|\\int_\\X\n\t\tf'(x) \\, \n\t\t\\Bigl[\t \t\t\t\n\t \t\\rho_{\\textrm{\\tiny\\rm\\!\\! g}}(S^1_u,x)\\, [\\xi^1_u(\\mathrm{d} x)-\\xi^2_u(\\mathrm{d} x)] \n\\\\\n\t\t&\\qquad\\qquad\\qquad\n\t\t- [\\rho_{\\textrm{\\tiny\\rm\\!\\! g}}(S^2_u,x)-\\rho_{\\textrm{\\tiny\\rm\\!\\! g}}(S^1_u,x)]\\, \\xi^2_u(\\mathrm{d} x)\n\t\t\\Bigr]\n\t \t\\biggr|\\,\\mathrm{d} u \n\\\\\n\t\t&\\qquad +\t\\int_0^t  \\biggl|\n\t\t\t\\int_\\X \\int_0^1 \n\t\t\t\t\\left[f(\\alpha\\,x)+f((1-\\alpha)\\,x)-f(x) \\right] \\,\n\t\t\t\tQ(\\mathrm{d} \\alpha)\n\\\\\n\t\t& \\qquad\\qquad\\qquad \n\t\t\t\\left[\n\t\t\t\t\\lambda(S^1_u, x) \\, [\\xi^1_u(\\mathrm{d} x)-\\xi^2_u(\\mathrm{d} x)]\n\t\t\t\t- [\\lambda(S^2_u, x)-\\lambda(S^1_u, x)]\\, \n\t\t\t\t\\xi^2_u(\\mathrm{d} x)\n\t\t\t\\right] \\biggr| \\, \\mathrm{d} u\n\\\\\n\t\t &\\qquad\n\t\t   + D \\, \\int_0^t \\biggl|\n\t\t\t\t\\int_\\X f(x) \\, (\\xi^1_u(\\mathrm{d} x)-\\xi^2_u(\\mathrm{d} x))\n\t\t\t\\biggr| \\, \\mathrm{d} u\n\\\\\n\t&\\leq\n\t(\\bar g +3\\, \\bar \\lambda +D)\\,\n\t\t\t\\int_0^t  \\normm{\\xi^1_u-\\xi^2_u} \\, \\mathrm{d} u \n\t\t\t+ C_t\\,( k_g + 3 \\, k_\\lambda) \\,\n\t\t\t \\int_0^t | S^1_u- S^2_u |\\, \\mathrm{d} u .\n\\end{align*}\nTaking the supremum over the functions $f$, we obtain:\n\\begin{align*}\n\t\\normm{\\xi^1_t-\\xi^2_t}\t\n\t&\\leq\n\t(\\bar g +3\\, \\bar \\lambda +D)\\,\n\t\\int_0^t  \\normm{\\xi^1_u-\\xi^2_u} \\, \\mathrm{d} u\n\t+ C_t\\,( k_g + 3 \\, k_\\lambda) \\,  \n\t\t\\int_0^t | S^1_u-S^2_u |\\, \\mathrm{d} u\n\t\\,.\n\\end{align*}\nMoreover, from \\eqref{eq.limite.substrat.faible} we get:\n\\begin{align*}\n\t&\n\t|S^1_t- S^2_t|\n\t\\leq  \n\tD \\, \\int_0^t |S^1_u- S^2_u|\\,\\mathrm{d} u  \n\\\\\n\t&\\quad\t\n\t+ \\frac kV \\int_0^t \\biggl|\n\t\t\\int_\\X \n\t\t\\Bigl(\t \t\t\t\n\t \t\t\t\\rho_{\\textrm{\\tiny\\rm\\!\\! g}}(S^1_u,x)\\, [\\xi^1_u(\\mathrm{d} x)-\\xi^2_u(\\mathrm{d} x)] \n\t \t\t\t- \n\t\t\t\t[\\rho_{\\textrm{\\tiny\\rm\\!\\! g}}(S^2_u,x)-\\rho_{\\textrm{\\tiny\\rm\\!\\! g}}(S^1_u,x)]\n\t\t\t\t\\, \\xi^2_u(\\mathrm{d} x)\n\t\t\\Bigr)\n\t\\biggr| \\, \\mathrm{d} u \n\\\\\n\t&\\quad\t\n\t\\leq \n\t\\Bigl(D+\\frac kV \\, C_t\\,k_g\\Bigr) \\, \n\t\\int_0^t |S^1_u- S^2_u| \\, \\mathrm{d} u\n\t+ \\frac kV \\,\\bar g \\int_0^t \\normm{\\xi^1_u-\\xi^2_u} \\, \\mathrm{d} u \\,.\n\\end{align*}\nWe define:\n\\begin{align*}\n\tM_t\n\t\\eqdef\n\t\\max \\left\\{\\bar g +3\\, \\bar \\lambda +D+ \\frac kV \\,\\bar g\n\t\t\\, , \\, C_t\\,( k_g + 3 \\, k_\\lambda)+D+\\frac kV \\, C_t\\,k_g\\right\\}\n\\end{align*}\nhence:\n\\begin{align*}\n\t\\normm{\\xi^1_t-\\xi^2_t}\n\t+|S^1_t-S^2_t|\n\t& \\leq\n\tM_t \\, \\int_0^t \\Bigl( \n\t\t\\normm{\\xi^1_u-\\xi^2_u}+|S^1_u-S^2_u|\t\n\t\\bigr) \\, \\mathrm{d} u \n\\end{align*}\nFinally from Gronwall's inequality we get \n$\\normm{\\xi^1_t-\\xi^2_t}+|S^1_t-S^2_t|=0$ for all $t\\geq 0$, hence  $\\xi^1_t = \\xi^2_t$ and $S^1_t=S^2_t$.\n\n\n\n\n\\subsubsection*{Step 2: tightness of $(\\bar\\nu^n)_{n\\geq 0}$}\n\nThe tightness of  \n$\\bar\\nu^n$ is equivalent to the fact that from any subsequence one can extract a subsequence that converges in distribution in the space $\\D([0,T],\\MM_{F}(\\X))$.\nAccording to \\citet[Th. 2.1]{roelly1986a} this amounts to proving the tightness \nof  $\\crochet{\\bar\\nu^n,f}$ in $\\D([0,T],\\R)$ for all $f$ in a set dense in \n$\\C(\\XX)$, here we will consider  $f\\in \\C^1(\\XX)$. To prove the latter result, it is sufficient to check the following Aldous-Rebolledo criteria \\citep[Cor. 2.3.3]{joffe1986a}:\n\\begin{enumerate}\n\\item \nThe sequence $(\\langle \\bar\\nu_t^n,f \\rangle)_{n\\geq 0}$ is tight for any $t\\geq 0$.\n\n\\item \nConsider the following semimartingale decomposition:\n\\[\n\t\\crochet{\\bar\\nu_t^n,f}\n\t=\n\t\\crochet{\\bar\\nu_0^n,f}\n\t+\n\tA_t^n+Z_t^n\\,.\n\\]\nwhere $A_t^n$ is of finite variation and $Z_t^n$ is a martingale.\nFor all $t>0$, $\\epsilon>0$, $\\eta>0$ there exists $n_{0}$ such that for any sequence \n$\\tau_n$ of stopping times with  $\\tau_{n}\\leq t$ we have:\n\\begin{align}\n\\label{eq.AR.2.1}\n  \\sup_{n\\geq n_0} \\sup_{\\theta\\in[0,\\delta]}\n  \\P\\Big(\n    \\big|\n       A^n_{\\tau_n+ \\theta} - A^n_{\\tau_n}\n    \\big|\n    \\geq \\eta\n  \\Big)\n  &\n  \\leq \\epsilon\\,,\n\\\\\n\\label{eq.AR.2.2}\n  \\sup_{n\\geq n_0} \\sup_{\\theta\\in[0,\\delta]}\n  \\P\\Big(\n    \\big|\n       \\crochet{Z^n}_{\\tau_n+ \\theta} - \\crochet{Z^n}_{\\tau_n}\n    \\big|\n    \\geq \\eta\n  \\Big)\n  &\n  \\leq \\epsilon\\,.\n\\end{align} \n\\end{enumerate}\n\n\n\\subsubsection*{\\it Proof of \\fenumi}\n\nFor any $K>0$,\n\\begin{align*}\n\t\\P\\bigl(|\\crochet{\\bar\\nu_t^n,f}| \\geq K\\bigr)\n\t&\\leq \n\t\\frac{1}{K}\\,\\norme{f}_\\infty \\,\n\t  \\sup_{n\\in\\N} \\E\\bigl(\\crochet{\\bar\\nu_t^n,1}\\bigr)\n\\end{align*}\nand using Lemma \\ref{lem.esp}, we deduce  \\fenumi.\n\n\n\n\\subsubsection*{\\it Proof of {\\rm(\\textit{ii})}}\n\n\\begin{align*}\n\tA_t^n\n\t&=  \n\t\\int_0^t  \\crochet{\\bar\\nu_u^n,\\rho_{\\textrm{\\tiny\\rm\\!\\! g}}(S_u^n,.)\\,f'} \\, \\mathrm{d} u \n\\\\\n\t&\\qquad\n\t+ \\int_0^t \\int_{\\X} \\int_0^1  \n\t\t\\lambda(S_u^n, x) \\,  \n\t\t\\bigl[ f(\\alpha\\,x)+f((1-\\alpha)\\,x)-f(x) \\bigr] \\, \n\t\tQ(\\mathrm{d} \\alpha) \\,\\bar\\nu_u^n(\\mathrm{d} x) \\, \\mathrm{d} u\n\\\\\n\t&\\qquad \n\t- D\\,\\int_0^t\\int_{\\X} f(x)\\,\\bar\\nu_u^n(\\mathrm{d} x) \\, \\mathrm{d} u\n\\end{align*}\nhence, according to Lemma \\ref{lem.esp}:\n\\begin{align*}\n\t\\EE | A_{\\tau_n+\\theta}^n-A_{\\tau_n}^n|\n\t&\\leq\n\t(\n\t\t\\norme{f'}_\\infty\\, \\bar{g}\n\t\t+ 3\\, \\norme{f}_\\infty\\,\\bar\\lambda\n\t\t+ D\\,\\norme{f}_\\infty\n\t)\n\t\\, C_{t,1}\\, \\theta\\,.\n\\end{align*}\nUsing  \\eqref{var_qua}, we also have:\n\\begin{align*}\n\t\\EE | \\crochet{Z^n}_{\\tau_n+\\theta}-\\crochet{Z^n}_{\\tau_n}|\n\t\\leq \n\t\\frac{1}{n}\\,\\left(9 \\, \\bar \\lambda+D \\right) \\, \n\t\t\\norme{f}_\\infty^2\\, C_{t,1} \\, \\theta\\,.\n\\end{align*}\nHence\n$\\EE| A_{\\tau_n+\\theta}^n-A_{\\tau_n}^n|+\\EE | \\crochet{Z^n}_{\\tau_n+\\theta}-\\crochet{Z^n}_{\\tau_n} | \n\t\\leq  C \\, \\theta$ \nand we obtain  \\fenumii\\ from the Markov inequality.\n\n\n\\bigskip\n\nIn conclusion, from the Aldous-Rebolledo criteria, the sequence $(\\bar\\nu^n)_{n\\geq 0}$ is tight.\n \n\n\n\n\\subsubsection*{Step 3: convergence of the sequence $(\\bar\\nu^n)_{n\\in\\N}$}\n\n\nTo conclude the proof of the theorem it is suffice to show that the sequence $(\\bar\\nu^n)_{n\\in\\N}$ has a unique accumulation point and that this point is equal to $\\xi$ described in Step 1. In order to characterize $\\xi$, the  solution of \\eqref{eq.limite.eid.faible}, we introduce, for any given $f \\in C^{1}(\\X)$, the following function defined for all\n$\\zeta\\in\\D([0,T],\\M_F(\\X))$:\n\\begin{align}\n\\nonumber\n  \\Psi_{t}(\\zeta)\n  &\\eqdef\n  \\crochet{\\zeta_{t},f}-\\crochet{\\zeta_{0},f}\n  -\\int_{0}^t \\biggl[\n\t\t\\int_{\\X} \\rho_{\\textrm{\\tiny\\rm\\!\\! g}}(S^\\zeta_u,x)\\,f'(x) \\,\t\\zeta_u(\\mathrm{d} x)\n\\\\\n\\nonumber\n\t&\\qquad\n\t+ \\int_{\\X} \\int_0^1\\lambda(S^\\zeta_u, x)\\,  \n\t\\bigl[f(\\alpha\\, x)+ f((1-\\alpha)\\,x)-f(x)\\bigr] \\, \t \t\t\t\n\t\t\tQ(\\mathrm{d} \\alpha) \\,\\zeta_u(\\mathrm{d} x) \n\\\\\n\\label{eq.Psi}\n\t&\\qquad\n\t- D\\, \\int_{\\X} f(x) \\,\t\\zeta_u(\\mathrm{d} x) \n\t\t\\biggr]\\,\\rmd u\n\\end{align}\nwhere $S_t^\\zeta$ is defined by:\n\\begin{align}\n\\label{eq.Psi.2}\n\tS_t^\\zeta\n\t&\\eqdef\n\tS_0\n\t+\n\t\\int_0^t \\Bigl(\n\t\tD\\,({\\mathbf s}_{\\textrm{\\tiny\\rm in}}-S_u^\\zeta)\n\t\t-\\frac kV \\int_\\X \\rho_{\\textrm{\\tiny\\rm\\!\\! g}}(S_u^\\zeta,x)\\,\\zeta_u(\\mathrm{d} x)\n\t\\Bigr) \\, \\mathrm{d} u\\,.\n\\end{align}\nHence, if $\\Psi_{t}(\\zeta)=0$ for all $t\\geq 0$ and all $f \\in C^{1}(\\X)$ then $(S^\\zeta,\\zeta)=(S,\\xi)$ where $(S,\\xi)$ is the unique solution of  \\eqref{eq.limite.substrat.faible}-\\eqref{eq.limite.eid.faible}.\n\n\\bigskip\n\nWe consider a subsequence $\\bar\\nu^{n'}$ of $\\bar\\nu^{n}$ which converges in distribution in the space  $\\D([0,T],\\M_F(\\X))$ and $\\tilde \\nu$ its limit.\n\n\\subsubsection*{\\it Sub-step 3.1: A.s. continuity of the limit~$\\tilde \\nu$.}\n\n\\begin{lemma}\n$\\tilde \\nu(\\omega)\\in \\CC([0,T],\\M_F(\\X))$ for  all $\\omega\\in\\Omega$ a.s.\n\\end{lemma}\n\n\\begin{proof}\nFor any  $f\\in\\C(\\X)$ such that $\\norme{f}_\\infty \\leq 1$:\n\\begin{align*}\n\t\\bigl|\n\t \t\\crochet{\\bar\\nu_t^{n'},f}-\\crochet{\\bar\\nu_{t^-}^{n'},f}\n\t\\bigr| \n\t\\leq \n\t\\frac{1}{n'}\\, \n\t\\bigl| \n\t\t\\crochet{\\nu_t^{n'},1}-\\crochet{\\nu_{t^-}^{n'},1}\n\t\\bigr|\\,.\n\\end{align*}\nBut $| \\crochet{\\nu_t^{n'},1}-\\crochet{\\nu_{t^-}^{n'},1}|$ represents \nthe difference between the number of individuals in $\\nu_t^{n'}$ and in $\\nu_{t^-}^{n'}$, which is at most 1. Hence:\n\\begin{align*}\n\t\\sup_{t \\in [0,T]} \\, \n\t\\normtv{\\bar\\nu_t^{n'}-\\bar\\nu_{t^-}^{n'}}\n\t\\leq \\frac{1}{n'}\n\\end{align*}\nwhich proves that the limit process $\\tilde\\nu$ is a.s. continuous\n\\cite[Th. 10.2 p. 148]{ethier1986a}  as the Prokhorov metric is dominated by the total variation metric.\n\\end{proof}\n\\carre\n\n\n\\subsubsection*{\\it Sub-step 3.2: Continuity of $\\zeta\\to\\Psi_{t}(\\zeta)$ in any $\\zeta$ continuous.}\n\n\n\\begin{lemma}\n\\label{lemma.Psi.continue}\nFor any given  $t\\in[0,T]$ and $f\\in C^{1}(\\X)$, the function  $\\Psi_{t}$ defined by\n\\eqref{eq.Psi} is continuous from  $\\DD([0,T],\\MM_{F}(\\X))$ with values in  $\\R$ in any point $\\zeta\\in\\CC([0,T],\\MM_{F}(\\X))$. \n\\end{lemma}\n\n\n\n\\proof\nConsider a sequence  $(\\zeta^{n})_{n\\in\\N}$ which converges towards  $\\zeta$ in\n$\\DD([0,T],\\MM_{F}(\\X))$ with respect to the Skorohod topology. As the limit $\\zeta$ is continuous we have that  $\\zeta^{n}$ converges to $\\zeta$ with the uniform topology:\n\\begin{align}\n\\label{eq.lemma.Psi.continue.a}\n   \\sup_{0\\leq t\\leq T} \n   d_{\\textrm{\\tiny\\rm PR}}({\\zeta_t^{n},\\zeta_{t}})\n   \\cv_{n\\to\\infty} 0\n\\end{align}\nwhere $d_{\\textrm{\\tiny\\rm PR}}$ is the Prokhorov metric (see Appendix \\ref{appendix.skorohod}). \n\n\nThe functions $\\lambda(s,x)$ and $\\rho_{\\textrm{\\tiny\\rm\\!\\! g}}(s,x)$ are Lipschitz continuous functions w.r.t. $s$ uniformly in $x$ and also bounded, see \\eqref{hyp.lambda.lipschitz} and \\eqref{hyp.rhog.lipschitz}, so from \\eqref{eq.Psi.2} we can easily check that:\n\\begin{align*}\n\t|S_t^{\\zeta^{n}}-S_t ^\\zeta|\n\t&\\leq\n\tC\\,\\int_0^t \\Bigl(\n\t\t|S_u^{\\zeta^{n}}-S_u ^\\zeta|\n\t\t+ \\Bigl|\n     \t\t\\int_\\X \\rho_{\\textrm{\\tiny\\rm\\!\\! g}}(S_u^{\\zeta^{n}},x)\\,\n\t\t\t\t[\\zeta^{n}_u(\\mathrm{d} x)-\\zeta_u(\\mathrm{d} x)]\n\\\\\n\t&\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\n     \t\t-\n\t\t\t\\int_\\X [\\rho_{\\textrm{\\tiny\\rm\\!\\! g}}(S_u^\\zeta,x)-\\rho_{\\textrm{\\tiny\\rm\\!\\! g}}(S_u^{\\zeta^{n}},x)]\n\t\t\t        \\,\\zeta_u(\\mathrm{d} x)\n\t\t\\Bigr|\n\t\\Bigr) \\, \\mathrm{d} u\n\\\\\n\t&\\leq\n\tC\\,\\int_0^t \\Bigl(\n\t\t|S_u^{\\zeta^{n}}-S_u ^\\zeta|\n\t\t+ |\\crochet{\\zeta^{n}_u-\\zeta_u,1}|\\Bigr) \\, \\mathrm{d} u\n\\end{align*}\nand the Gronwall's inequality leads to:\n\\begin{align}\n\\label {eq.lemma.Psi.continue.b} \n  |S_t^{\\zeta^{n}}-S_t ^\\zeta|\n  \\leq \n  C\\,\\int_{0}^t |\\crochet{\\zeta^n_{u}-\\zeta_{u},1}|\\,\\rmd u\\,.\n\\end{align}\nHere and in the rest of the proof the constant $ C $ will depend only on $ T $, $ f $ and on the parameters of the models.\nHence, from \\eqref{eq.Psi}:\n\\begin{align*}\n\t|\\Psi_t(\\zeta^{n})-\\Psi_t(\\zeta)|\n\t&\\leq \n\tC\\,\n\t\\Bigl[\n\t  \t|\\crochet{\\zeta_t^{n}- \\zeta_t,1}|\n\t\t+\n\t\t|\\crochet{\\zeta_0^{n}- \\zeta_0,1}|\n\\\\\n   &\\qquad\\qquad\\qquad\n\t\t+\n\t\t\\int_0^t |S_u^{\\zeta^{n}}-S_u ^\\zeta| \\, \\mathrm{d} u\n\t\t+\n\t\t\\int_0^t |\\crochet{\\zeta_u^{n}-\\zeta_u,1}| \\, \\mathrm{d} u\n\t\\Bigr]\n\\\\\n   &\\leq\n\t\tC\\,\\sup_{0\\leq t\\leq T}|\\crochet{\\zeta_t^{n}-\\zeta_t,1}|\\,.\n\\end{align*}\nLet $\\delta_{t}=d_{\\textrm{\\tiny\\rm PR}}(\\zeta_t^{n},\\zeta_t)$, by definition of the Prokhorov metric:\n\\begin{align*}\n  \\zeta_t^{n}(\\XX)-\\zeta_t(\\XX^{\\delta_{t}})&\\leq \\delta_{t}\\,,\n  &\n  \\zeta_t(\\XX)-\\zeta_t^{n}(\\XX^{\\delta_{t}})&\\leq \\delta_{t}\\,,\n\\end{align*}\nbut $\\XX^{\\delta_{t}}=\\XX$ hence $|\\zeta_t^{n}(\\XX)-\\zeta_t(\\XX)|\\leq \\delta_{t}$. Note finally that $|\\zeta_t^{n}(\\XX)-\\zeta_t(\\XX)| = |\\crochet{\\zeta_t^{n}-\\zeta_t,1}|$, so we get:\n\\[\n   |\\Psi_t(\\zeta^{n})-\\Psi_t(\\zeta)| \n   \\leq C\\,\n   \\sup_{0\\leq t\\leq T}d_{\\textrm{\\tiny\\rm PR}}({\\zeta_t^{n},\\zeta_t})\n\\]\nwhich tends to zero.\n\\carre\n\n\n\n\n\\subsubsection*{\\it Sub-step 3.3: Convergence in distribution  of $\\Psi_{t}(\\bar\\nu^{n'})$ to $\\Psi_{t}(\\tilde\\nu)$.}\n\nThe sequence  $\\bar\\nu^{n'}$ converges in distribution to  $\\tilde\\nu$ and $\\tilde\\nu(\\omega)\\in\\CC([0,T],\\MM_{F}(\\XX))$;\nmoreover the application  $\\Psi_{t}$ is continuous in any point of $\\CC([0,T],\\MM_{F}(\\XX))$, thus according to the continuous mapping theorem \\cite[Th. 2.7 p. 21]{billingsley1968a} we get:\n\\begin{align} \n\\label{eq.cv.loi.Psi}\n  \\Psi_{t}(\\bar\\nu^{n'})\n  \\xrightarrow[n\\to\\infty]{\\textrm{loi}}\n  \\Psi_{t}(\\tilde\\nu)\\,.\n\\end{align} \n\n\n\n\n\\subsubsection*{\\it Sub-step 3.4: $\\tilde\\nu=\\xi$ a.s.}\n\nFrom \\eqref{renormalisation}, for any $n\\geq 0$ we have:\n\\begin{align*}\n\t\\Psi_t(\\bar\\nu^n) =  Z_t^{f,n}\n\\end{align*}\nwhere $Z_t^{f,n}$ is defined by  \\eqref{eq.Zfn}. Also, \\eqref{var_qua} gives:\n\\begin{align*}\n\t\\EE(|Z_t^{f,n}|^2)\n\t&= \n\t\\EE \\crochet{Z^{f,n}}_t \n\t\\leq \\frac{1}{n}\\,(9\\,\\bar\\lambda+D) \\, \\norm{f}_{\\infty}^2\\,C_{t,1} \\, t\\,.\n\\end{align*}\nHence $\\Psi_{t}(\\bar\\nu^n)$ converges to $0$ in $L^2$ but also in $L^1$. \nFurthermore, we easily show that:\n\\begin{align*}\n|\\Psi_{t}(\\zeta)|\n\t\\leq & C_{f,t} \\sup_{0 \\leq u \\leq t} \\crochet{\\zeta_u,1}\n\\end{align*}\nmoreover, from Lemma \\ref{lem.esp}, $(\\Psi_{t}(\\bar\\nu^{n'}))_{n'}$ is uniformly\nintegrable. The dominated convergence theorem and \\eqref{eq.cv.loi.Psi} imply:\n\\[\n  0\n  = \\lim_{n' \\to \\infty} \\EE |\\Psi_t(\\bar\\nu^{n'})|\n  = \\EE |\\Psi_t(\\tilde \\nu)|\\,.\n\\]\nSo $\\Psi_t(\\tilde \\nu)=0$ a.s. and $\\tilde \\nu$ is a.s. equal to $\\xi$ where $(S,\\xi)$ \nis the unique solution of \\eqref{eq.limite.substrat.faible}-\\eqref{eq.limite.eid.faible}.\n\n\n\\bigskip\n\nThis last step concludes the proof of Theorem \\ref{theo.cv.ibm}.\n\n\n\n\\subsection{Links with deterministic models}\n\\label{subsec.modeles.deterministes}\n\n\nEquation \\eqref{eq.limite.eid.faible}  is actually a weak version of an \nintegro-differential equation that can be easily identified.\nIndeed suppose that the solution $\\xi_t$ of Equation \\eqref{eq.limite.eid.faible}\nadmits a density $p_t(x)\\,\\rmd x=\\xi_{t}(\\rmd x)$, and that $Q(\\rmd \\alpha)=q(\\alpha)\\,\\rmd \\alpha$, then the system of equations \\eqref{eq.limite.substrat.faible}-\\eqref{eq.limite.eid.faible} is a weak version of the following system:\n\\begin{align} \n\\label{eq.limite.substrat.fort}\n\t&\n\t\\frac{\\rmd}{\\rmd t} S_t  = \n\tD\\,({\\mathbf s}_{\\textrm{\\tiny\\rm in}}-S_t)-\\frac kV \\int_\\X \\rho_{\\textrm{\\tiny\\rm\\!\\! g}}(S_t,x)\\,p_t(x)\\,\\mathrm{d} x\\,,\n\\\\\n\\nonumber\n\t&\n\t\\frac{\\partial}{\\partial t} p_t(x)\n\t+\\frac{\\partial}{\\partial x} \\bigl( \\rho_{\\textrm{\\tiny\\rm\\!\\! g}}(S_t,x)\\,p_t(x)\\bigr)\n\t+ \\bigl(\\lambda(S_t, x)+D \\bigr)\\,p_t(x)\n\\\\\n\\label{eq.limite.eid.fort}\n  \t&\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\n\t=  \n\t2\\,\\int_\\X \n\t\\frac{\\lambda(S_t, z)}{z}\\,q\\left(\\frac xz \\right)\\,p_t(z)\\,\\mathrm{d} z\\,.\n\\end{align}\nIn fact, this is the population balance equation  introduced by \\cite{fredrickson1967a} and \\cite{ramkrishna1979a} for growth-fragmentation models.\n\n\\bigskip\n\nIt is easy to link  the model \\eqref{eq.limite.substrat.fort}-\\eqref{eq.limite.eid.fort}\nto the classic chemostat model. Indeed suppose that\nthe growth function $x \\mapsto \\rho_{\\textrm{\\tiny\\rm\\!\\! g}}(s,x)$ is proportional to $ x $, i.e.:\n\\[\n  \\rho_{\\textrm{\\tiny\\rm\\!\\! g}}(s,x)=\\tilde\\mu(s)\\,x\\,.\n\\]\nThe results presented now are formal insofar as a linear growth function \ndoes not verify the assumptions made in this article.\nWe introduce the bacterial concentration:\n\\[\n  Y_t \\eqdef \\frac1V \\,\\int_\\X x\\, p_t(x) \\, \\mathrm{d} x\\,.\n\\] \nAs $\\sup_{0\\leq t\\leq T}\\crochet{p_{t},1}<\\infty$, from \\eqref{eq.limite.eid.fort}:\n\\begin{multline*}\n\t\\frac{\\mathrm{d}}{\\mathrm{d} t} Y_t\n\t- \\frac1V \\int_\\X x \\, \n\t\t\t\\frac{\\partial}{\\partial x}\\bigl( \\rho_{\\textrm{\\tiny\\rm\\!\\! g}}(S_t,x)\\,p_t(x)\\bigr) \\, \\mathrm{d} x\n\t+ \\frac1V \\int_\\X x \\, \\lambda(S_t, x) \\, p_t(x)\\,\\mathrm{d} x\n\t+ D\\, Y_t\n\\\\\n\t= \\frac2V\\, \\int_\\X x \\, \n\t\t\t\t\\int_\\X \n\t\t\t\t\\frac{\\lambda(S_t, z)}{z}\\,q(x\/z)\\,p_t(z)\\,\n\t\t\t\t\\mathrm{d} z \\, \\mathrm{d} x\\,,\n\\end{multline*}\nbut\n\\begin{align*}\n&\\int_\\X x \\, \n\t\\int_\\X \n\t\t\\frac{\\lambda(S_t, z)}{z}\\,\n\t\tq(x\/z)\\,p_t(z)\\,\\mathrm{d} z \\, \\mathrm{d} x\n\t\t= \t\t\n\t\t\\int_\\X \\int_0^1\t\t\t \n\t\t\tz\\, \\lambda(S_t,z)\\, \\alpha \\,q\\left(\\alpha \\right)\\,p_t(z)\\,\\mathrm{d} \\alpha \\, \\mathrm{d} z\n\\\\\n\t\t&\\qquad\\qquad =  \\int_\\X \\int_0^1\t\t\t \n\t\t\tz\\, \\lambda(S_t, z)\\, \\alpha \\,q\\left(1-\\alpha \\right)\\,p_t(z)\\,\\mathrm{d} \\alpha \\, \\mathrm{d} z\n\t\t\t\\tag{by symmetry of $q$}\n\\\\\n\t\t&\\qquad\\qquad =\t\\int_\\X \\int_0^1\t\t\t \n\t\t\tz\\, \\lambda(S_t, z)\\, (1-\\alpha) \\,q\\left(\\alpha \\right)\\,p_t(z)\\,\\mathrm{d} \\alpha \\, \\mathrm{d} z\n\\\\\n\t\t&\\qquad\\qquad = - \\int_\\X \\int_0^1\t\t\t \n\t\t\tz\\, \\lambda(S_t, z)\\, \\alpha \\,q\\left(\\alpha \\right)\\,p_t(z)\\,\\mathrm{d} \\alpha \\, \\mathrm{d} z\n\t\t+ \\int_\\X \t\t\t \n\t\t\tz\\, \\lambda(S_t, z)\\,p_t(z)\\, \\mathrm{d} z\n\\end{align*}\nthus:\n\\begin{align*}\n 2\\, \\int_\\X x \\, \n\t\\int_\\X \n\t\\frac{\\lambda(S_t, z)}{z}\\,q(x\/z)\\,p_t(z)\\,\\mathrm{d} z \\, \\mathrm{d} x\n\t\t= \\int_\\X \t\t\t \n\t\t\tz\\, \\lambda(S_t, z)\\,p_t(z)\\, \\mathrm{d} z.\n\\end{align*}\nThe function $x\\mapsto p_t(x)$ is the population density at time $t$. On the one hand $p_0(x)$ has compact support. On the other hand the growth of each bacterium is defined by a differential equation whose right-hand side is bounded by a linear function in $x$, uniformly in $s$. Hence for all $t\\leq T$, we can uniformly bound the mass of all the bacteria and $p_{t}(x)$ has a compact support, i.e. there exists ${\\textrm{max}}$ such that the support of $p_{t}(x)$ is included in $[0,{\\textrm{max}}]$ with $p_{t}({\\textrm{max}})=0$, so we choose $\\X=[0,{\\textrm{max}}]$. Moreover $\\rho_{\\textrm{\\tiny\\rm\\!\\! g}}(S_t,0)=0$ hence:\n\\begin{align*}\n\t\\int_\\X \n\t\tx \\, \\frac{\\partial}{\\partial x}\\bigl( \\rho_{\\textrm{\\tiny\\rm\\!\\! g}}(S_t,x)\\,p_t(x)\\bigr) \n\t\\, \\mathrm{d} x\n\t= \n\t-\\int_\\X  \\rho_{\\textrm{\\tiny\\rm\\!\\! g}}(S_t,x)\\,p_t(x) \\, \\mathrm{d} x.\n\\end{align*}\nFinally:\n\\begin{align*}\n\t\\frac{\\mathrm{d}}{\\mathrm{d} t} Y_t \n\t& = \n\t\t\\frac1V \\int_\\X  \\rho_{\\textrm{\\tiny\\rm\\!\\! g}}(S_t,x)\\,p_t(x) \\, \\mathrm{d} x\n\t\t-D\\, Y_t\n\t= \\tilde\\mu(S_{t})\\,Y_{t} -D\\, Y_t\\,.\n\\end{align*}\nWe deduce that the concentrations $(Y_{t},S_{t})_{t\\geq 0}$  of biomass and substrate are  the solution of the following closed system of ordinary differential equations:\n\\begin{equation}\n\\label{eq.chemostat.edo}\n\t\\begin{split}\n\t\\dot Y_t \t& = \\bigl(\\tilde\\mu(S_t)-D \\bigr)\\, Y_t\\,,\n\t\\\\\n\t\\dot S_t\t& = D\\,({\\mathbf s}_{\\textrm{\\tiny\\rm in}}-S_t)-k \\,\\tilde\\mu(S_t)\\, Y_t\\,.\n\t\\end{split}\n\\end{equation}\nwhich is none other than the classic  chemostat equation \\citep{smith1995a}.\n\n\n\n\n\\section{Simulations}\n\\label{sec.simulations}\n\nIn this section we compare the behavior of the individual-based model (IBM) and two deterministic models: the integro-differential equation (IDE) \\eqref{eq.limite.substrat.fort}-\\eqref{eq.limite.eid.fort} and  classic chemostat model, represented by the ordinary differential equation \\eqref{eq.chemostat.edo} (ODE).\nSimulations of the IBM were performed following Algorithm \\ref{algo.ibm}.\nThe resolution of the integro-differential equation was made following the numerical scheme given in Appendix \\ref{appendix.schema.num}, with a discretization step in the mass space  of $\\Delta x = 2 \\times 10^{-7}$ and a discretization step in time of  $\\Delta t = 5 \\times 10^{-4}$.\n\n\n\\subsection{Simulation parameters}\n\nIn the simulations proposed in this section, the division rate of an individual is given by the following function:\n\\begin{align*}\n\t\\lambda(s,x)\n\t& =\n\t\\frac{\\bar\\lambda}{\\log \\bigl(({\\textrm{max}}-m_{\\textrm{\\tiny\\rm div}}) \\, p_\\lambda +1 \\bigr)} \n\t\\,\n\t\\log \\bigl((x-m_{\\textrm{\\tiny\\rm div}}) \\, p_\\lambda +1 \\bigr) \\, 1_{\\{x \\geq m_{\\textrm{\\tiny\\rm div}}\\}}\n\\end{align*}\nwhich does not depend on the substrate concentration.\n\n\nThe  division kernel $Q(\\mathrm{d} \\alpha)=q(\\alpha)\\,\\mathrm{d} \\alpha$ is given by a symmetric beta distribution:\n\\begin{align*}\n\tq(\\alpha)\n\t& = \n\t\\frac{1}{B(p_\\beta)}\\,\n\t\t\\bigl( \\alpha \\,(1-\\alpha) \\bigr)^{p_\\beta-1}\n\\end{align*}\nwhere $B(p_\\beta)\n\t= \\int_0^1 \\bigl( \\alpha \\,(1-\\alpha) \\bigr)^{p_\\beta-1} \\, \\mathrm{d} \\alpha$\nis a normalizing constant.\n\nIndividual growth follows a Gompertz model, with a growth rate depending on the substrate concentration:\n\\begin{align*}\n\tg(s,x)\n\t& = \n\tr_{\\textrm{\\tiny\\rm max}}\\,\\frac{s}{k_r+s} \\,\n\t\\log\\Big(\\frac{{\\textrm{max}}}{x}\\Big)\\,x\\,.\n\\end{align*}\nThe masses of individuals at the initial time are sampled according to the following probability density function:\n\\begin{align}\n\\label{eq.d}\nd(x)\n\t& =\t\n\t\t\\Biggl(\n\t\t\t\\frac{x-0.0005}{0.00025}\n\t\t\t\\,\\left(1-\\frac{x-0.0005}{0.00025}\\right)\n\t\t\\Biggr)^5 \\,\n\t\t1_{\\{0.0005 < x < 0.00075\\}}\\,.\n\\end{align}\nThis initial density will show a transient phenomenon that cannot be reproduced by the classical chemostat model described in terms of ordinary differential equations \\eqref{eq.chemostat.edo}, see Figure \\ref{fig.edo.ibm.eid}.\n\nThe simulations were performed using the parameters in Table  \\ref{table.parametres}. The parameters $V$, $N_0$ and $D$ will be specified for each simulation.\n\n\n\\begin{table}[h]\n\\begin{center}\n\\begin{tabular}{|c|c|}\n\t\\hline\n    Parameters & Values \\\\\n    \\hline\n    $S_0$\t\t\t\t&\t5 mg\/l \\\\\n    ${\\mathbf s}_{\\textrm{\\tiny\\rm in}}$\t\t\t\t&\t10 mg\/l \\\\\n    ${\\textrm{max}}$\t\t\t\t&\t0.001 mg \\\\\t\n    $m_{\\textrm{\\tiny\\rm div}}$\t\t\t\t&\t0.0004 mg \\\\\n\t$\\bar\\lambda$\t\t&\t1 h$^{-1}$\\\\\n    $p_\\lambda$\t\t\t&\t1000 \\\\\n    $p_\\beta$\t\t\t&\t7 \\\\\n    $r_{\\textrm{\\tiny\\rm max}}$\t\t\t\t&\t1 h$^{-1}$\\\\\n    $k_r$\t\t\t\t&\t10 mg\/l\\\\\n    $k$\t\t\t\t\t&\t1\\\\\n    \\hline\n\\end{tabular}\n\\end{center}\n\\caption{Simulation parameters.}\n\\label{table.parametres}\n\\end{table}\n\n\n\n\n\\subsection{Comparison of the IBM and the IDE}\n\nTo illustrate the convergence in large population asymptotic of the IBM to the IDE, we performed simulations at different levels of population size. To this end we vary the volume of the chemostat and the number of individuals at the initial time. We considered three cases:\n\\begin{enumerate}\n\\item small size: $V=0.05$ l and $N_0=100$,\n\\item medium size: $V=0.5$ l and $N_0=1000$,\n\\item large size: $V=5$ l and $N_0=10000$.\n\\end{enumerate} \nIn each of these three cases we simulate:\n\\begin{itemize}\n\\item 60 independent runs of the IBM;\n\\item the numerical approximation of \\eqref{eq.limite.substrat.fort}-\\eqref{eq.limite.eid.fort} using the finite difference schemes \ndetailed in Appendix \\ref{appendix.schema.num}\n\\end{itemize}\nwith the same initial biomass concentration distribution.\n\n \n\n\\bigskip\n\nThe convergence of IBM to EID is clearly illustrated\nin Figure \\ref{evol.taille.concentrations}  where the evolutions of the population size, of the biomass concentration, and of the substrate concentration are represented.\n\nIn Figure \\ref{fig.evol.eid}  the time evolution of the normalized mass distribution is depicted, i.e. the normalized solution of the IDE \\eqref{eq.limite.eid.fort}. We have represented the simulation until time $T=10$ (h) to illustrate the transient phenomenon due to the choice of the initial distribution \\eqref{eq.d}: after a few time iterations this  distribution is bimodal; the upper mode (large mass) grows in mass and disappears   before time $T=10$ (h). The lower mode (small mass) corresponds to the mass of the bacteria resulting from the division; the upper mode corresponds to the mass of the bacteria from the initial bacteria before their division.  Thus, the upper mode is set to disappear quickly by division or by up-take. The IBM realizes this phenomenon, see\nFigure \\ref{repartition.masse}. In contrast, the classical chemostat model presented below, see Equation \\eqref{eq.chemostat.edo}, cannot account for this phenomenon.\n\n\nFigure \\ref{repartition.masse} presents this  normalized mass distribution at three different instants, $t=1,\\,4,\\, 80$ (h),  and the simulation of the IDE is compared \nto 60 independent runs of the IBM, again for the three levels of population sizes described above. Depending on whether the population is large, medium or small, \nwe needed to adapt the number of bins of the histograms so that the resulting graphics are clear. The convergence of the IBM solution to the IDE in large population limit can be observed.\n\n\nIn conclusion, the IBM converges in large population limit to the IDE and variability ``around'' the asymptotic model is relatively large in small or medium population size; note that there is no reason why the IDE represents the mean value of the IBM.\n\n\n\\begin{figure}\n\\begin{center}\n\\begin{tabular}{ccc}\n\\includegraphics[width=4.5cm]{fig_simu130521160944_size.pdf}\n&\n\\includegraphics[width=4.5cm]{fig_simu130522173621_size.pdf}\n&\n\\includegraphics[width=4.5cm]{fig_simu130605175456_size.pdf}\n\\\\\n\\includegraphics[width=4.5cm]{fig_simu130521160944_biomass.pdf}\n&\n\\includegraphics[width=4.5cm]{fig_simu130522173621_biomass.pdf}\n&\n\\includegraphics[width=4.5cm]{fig_simu130605175456_biomass.pdf}\n\\\\\n\\includegraphics[width=4.5cm]{fig_simu130521160944_substrate.pdf}\n&\n\\includegraphics[width=4.5cm]{fig_simu130522173621_substrate.pdf}\n&\n\\includegraphics[width=4.5cm]{fig_simu130605175456_substrate.pdf}\n\\\\\n\\includegraphics[width=4.5cm]{fig_simu130521160944_phasePortrait.pdf}\n&\n\\includegraphics[width=4.5cm]{fig_simu130522173621_phasePortrait.pdf}\n&\n\\includegraphics[width=4.5cm]{fig_simu130605175456_phasePortrait.pdf}\n\\\\\n\\scriptsize small population size\n&\n\\scriptsize medium population size\n&\n\\scriptsize large population size\n\\\\\n\\scriptsize $V=0.05$ l, $N_0=100$\n&\n\\scriptsize $V=0.5$ l, $N_0=1000$\n&\n\\scriptsize $V=5$ l, $N_0=10000$\n\\end{tabular}\n\\end{center}\n\\vskip-1em\n\\caption{From top to bottom: time evolutions of the population size, the biomass concentration, the concentration substrate and  the concentrations phase portrait  for the three levels of population sizes (small, medium and large). The blue curves represent the trajectories of 60 independent runs of IBM. The green curve represents the mean value of these runs. The red curve represents the solution of the IDE. The rate $ D $ is 0.2 h$^{-1}$.}\n\\label{evol.taille.concentrations}\n\\end{figure}\n\n\n\n\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[trim=4.3cm 1.5cm 2.2cm 2.2cm,width=10cm]{fig_IDE_density_evolution.pdf}\n\\end{center}\n\\caption{Time evolution of the normalized mass distribution for the IDE \\eqref{eq.limite.eid.fort}: we represent the simulation until time $T=10$ (h) only to illustrate the transient phenomenon due to the choice of the initial distribution \\eqref{eq.d}. After a few iterations in time this distribution is bimodal, the upped mode growths in mass and disappears before $T=10$ (h).}\n\\label{fig.evol.eid}\n\\end{figure}\n\n\n\\begin{figure}\n\\begin{center}\n\\begin{tabular}{ccc}\n\\includegraphics[width=4.5cm]{fig_simu130521160944_density1.pdf}\n&\n\\includegraphics[width=4.5cm]{fig_simu130522173621_density1.pdf}\n&\n\\includegraphics[width=4.5cm]{fig_simu130605175456_density1.pdf}\n\\\\\n\\includegraphics[width=4.5cm]{fig_simu130521160944_density4.pdf}\n&\n\\includegraphics[width=4.5cm]{fig_simu130522173621_density4.pdf}\n&\n\\includegraphics[width=4.5cm]{fig_simu130605175456_density4.pdf}\n\\\\\n\\includegraphics[width=4.5cm]{fig_simu130521160944_density80.pdf}\n&\n\\includegraphics[width=4.5cm]{fig_simu130522173621_density80.pdf}\n&\n\\includegraphics[width=4.5cm]{fig_simu130605175456_density80.pdf}\n\\\\\n\\scriptsize small population size\n&\n\\scriptsize medium population size\n&\n\\scriptsize large population size\n\\\\\n\\scriptsize $V=0.05$ l, $N_0=100$\n&\n\\scriptsize $V=0.5$ l, $N_0=1000$\n&\n\\scriptsize $V=5$ l, $N_0=10000$\n\\end{tabular}\n\\end{center}\n\\caption{Mass distribution for the time $t=1$ (above), $t=4$ (middle) and $t=80$ (bottom) in small (left), medium (middle) and large (right) population size. For each graph, the blue histograms represent the empirical mass distributions of individuals for the 60 independent runs of IBM. In order to plot the histogram we have adapted the number of bins according to the population size. The red curve represents the mass distribution given by the IDE. The dilution rate $ D $ is 0.2 h$^{-1}$. Again the convergence of the IBM solution to the IDE in large population limit is observed.}\n\\label{repartition.masse}\n\\end{figure}\n\n\n\n\n\n\n\n\\subsection{Comparison of the IBM, the IDE and the ODE}\n\nWe now compare the IBM and the IDE to the classical chemostat model described by the system of ODE \\eqref{eq.chemostat.edo}. The function $\\tilde\\mu$ is the specific growth rate. The growth model in both the IBM and the IDE is of Monod type, so for the ODE model we also consider the classical Monod kinetics:\n\\begin{align}\n\\label{eq.edo.monod}\n\t\\tilde\\mu(S)\n\t& = \n\t\\mu_{\\max} \\, \\frac{S}{K_s+S}\\,.\n\\end{align}\nThe parameters of this Monod law are not given in the initial model and we use \na least squares method to determine the value of the parameters  $\\mu_{\\max}$ and $K_s$\nwhich minimize the quadratic distance between $(S_{t},X_{t})_{t\\leq T}$ given by \\eqref{eq.chemostat.edo} and  $(S_{t},X_{t}=\\int_{\\X} x\\,p_{t}(x)\\,\\rmd x)_{t\\leq T}$ given by \\eqref{eq.limite.substrat.fort}-\\eqref{eq.limite.eid.fort}.\n\nThe numerical integration of the ODE \\eqref{eq.chemostat.edo} presents no difficulties\nand is performed by the function \\texttt{odeint} of the module  \\texttt{scipy.integrate} of \\texttt{Python} with the default parameters.\n\n\\bigskip\n\nFirst we consider  a simulation based on the initial mass density $d(x)$  defined by \n\\eqref{eq.d}. With this initial density both the IDE and the IBM feature a transient phenomenon described in the previous section and illustrated in Figures  \\eqref{fig.evol.eid} et~\\eqref{repartition.masse}. \n\n\nFigure \\ref{fig.edo.ibm.eid} (left) shows a significant difference between the IBM and the IDE on the one hand and the ODE on the other hand, the latter model cannot account for the transient phenomenon. With the first two models, the individual bacteria are withdrawn uniformly and independently of their mass (large mass bacteria has the same probability of withdrawal as small mass bacteria) and \nat the beginning of the simulation there is  a decrease in biomass\nas at initial state  $d(x)$ has a substantial proportion of large bacteria mass.\nThe ODE is naturally not able to account for this phenomenon.\n\n\n\\begin{figure}\n\\begin{center}\n\\begin{tabular}{cc}\n\\includegraphics[width=5cm]{fig_simu130527183824_MC_biomasse.pdf}\n&\n\\includegraphics[width=5cm]{fig_simu130627163114_MC_biomasse.pdf}\n\\\\\n\\includegraphics[width=5cm]{fig_simu130527183824_MC_substrat.pdf}\n&\n\\includegraphics[width=5cm]{fig_simu130627163114_MC_substrat.pdf}\n\\\\\n\\includegraphics[width=6cm]{fig_simu130527183824_MC_phase.pdf}\n&\n\\includegraphics[width=6cm]{fig_simu130627163114_MC_phase.pdf}\n\\\\[0.7em]\n\\small initial density $d(x)$ \\eqref{eq.d}\n&\n\\small initial density $d'(x)$ \\eqref{eq.d'}\n\\end{tabular}\n\\end{center}\n\\caption{Time evolution of the biomass concentration (top), the substrate concentration (middle) and the concentration trajectories in the phase space (bottom) according to the initial mass distributions \\eqref{eq.d} (left) and \\eqref{eq.d'} (right). In blue, the trajectories of 60 independent runs of the IBM simulated with $V=3$ l and $N_0=20000$ (left), $N_0=25000$ (right); in green, the mean of the IBM runs; in red, the solution of IDE \\eqref{eq.limite.substrat.fort}-\\eqref{eq.limite.eid.fort}; in black, the solution of the ODE \\eqref{eq.chemostat.edo}. The latter is fitted by the least squares method on the IDE, the parameters of the Monod law \\eqref{eq.edo.monod} are $\\mu_{\\max}=0.341$ and $K_s = 2.862$ in the first case and $\\mu_{\\max}=0.397$ and $K_s = 3.996$ in the second.\nAs initial densities are different, $N_{0}$ is adapted so that the average initial biomass concentration is the same in both cases.\nThe dilution rate $D$ is 0.2 h$^{-1}$.  In the first case, the ODE gives no account for the transient phenomenon described in the previous section, see Figures \\eqref{fig.evol.eid} and \\eqref{repartition.masse}, while the IDE and the IBM give coherent account for it. In the second case the three models are consistent.}\n\\label{fig.edo.ibm.eid}\n\\end{figure}\n\nThis phenomenon no longer appears when uses the following density:\n\\begin{align}\n\\label{eq.d'}\nd'(x)\n\t& =\t\n\t\t\\Biggl(\n\t\t\t\\frac{x-0.00035}{0.0003}\n\t\t\t\\,\\left(1-\\frac{x-0.00035}{0.0003}\\right)\n\t\t\\Biggr)^5 \\,\n\t\t1_{\\{0.00035 < x < 0.00065\\}}\\,.\n\\end{align}\n\nIndeed, from Figure \\ref{fig.edo.ibm.eid} (right), there is no longer any biomass decay  at the beginning of the simulation and the different simulations are comparable,  \nthe ODE and the IDE match substantially.\n\n\n\n\\subsection{Study of the washout}\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=10cm]{fig_simu130704121134_MC_biomasse.pdf}\n\\end{center}\n\\caption{Time evolution of the biomass concentration. In blue, 1000 independent realizations of the IBM simulated with $V=0.5$ l and $N_0=30$;  in green, the mean of these runs; in red, the solution of the IDE; in black, the solution of the ODE with parameters values $\\mu_{\\max}=0.482$ and $K_s = 6.741$. The dilution rate $ D $ is   0.275 h$^{-1}$.  Among the 1000 independent runs of the IBM, 111 lead to washout while the deterministic models converge to an equilibrium with strictly positive biomass. The mean value of the  1000 runs of the IBM gives account for the washout probability while IDE and ODE models do not account for this question.}\n\\label{fig.lessivage}\n\\end{figure}\n\n\n\n\\begin{figure}[p]\n\\begin{center}\n\\includegraphics[width=9cm]{fig_simu130605093815_MC_biomasse.pdf}\\\\\n\\includegraphics[width=9cm]{fig_simu130605093815_loi_tps_extinction.pdf}\n\\caption{$\\blacktriangleright$ (Top) Evolution of biomass concentration between $t=20$ and $t=90$ h: blue, 1000 independent runs of the IBM; in green, the mean value of these runs; in red the solution of the IDE; in black, the solution to the ODE with parameters $\\mu_{\\max}=0.814$ and $K_s = 17.547$. The parameters are $V=10$ l and $N_0=10000$, the dilution rate $D$ is  0.5 h$^{-1}$. For both deterministic models, the size of the population decreases exponentially rapidly to 0 but remains strictly positive for any finite time. However, all the runs of the IBM reach washout in finite time $\\blacktriangleright$ (Bottom) empirical distribution of the washout time calculated from 7000 independent runs of the IBM and plotted using a time kernel regularization.}\n\\label{fig.lessivage2}\n\\end{center}\n\\end{figure}\n\n\nOne of the main differences between deterministic  and stochastic models lies in their way of accounting for the washout phenomenon  (or extinction phenomenon in the case of an ecosystem). With a sufficiently small dilution rate $D$, the solutions of the ODE \\eqref{eq.chemostat.edo} and the IDE  \\eqref{eq.limite.substrat.fort}-\\eqref{eq.limite.eid.fort}  converge to an equilibrium point with strictly positive biomass. In fact, the  washout is an unstable equilibrium point and apart from the line corresponding to the null biomass, the complete phase space corresponds to a basin of attraction leading to a solution with a strictly positive biomass asymptotic point. However, from Figure \\ref{fig.lessivage}, among the 1000 independent runs of the IBM, 111 of them converge to washout before time $t=1000$ h; so the probability of washout at this instant is approximately 11\\%. It may be noted that the IDE and the ODE do not correspond to the average value of the IBM since only the latter may reflect the washout in a finite time horizon.\n\nNow we consider a sufficiently large dilution rate, $D=0.5$ h$^{-1}$, corresponding to the washout conditions. Figure \\ref{fig.lessivage2} (left) presents the evolution of the biomass concentration in the different models. The runs of the IBM converge to the washout in finite time whereas both deterministic ODE and IDE models converge exponentially to washout without ever reaching it in finite time. Figure \\ref{fig.lessivage2} (right) shows the empirical distribution of the washout time calculated from 7000 independent runs of the IBM. This washout time features a relatively large variance.\n\n\n\n\n\n\\section*{Conclusion}\n\nWe proposed a hybrid chemostat model. On the one hand the bacterial population is modeled as an individual-based model: each bacterium is explicitly represented through its mass. On the other hand, the substrate concentration dynamics is represented as a conventional ordinary differential equation. Mechanisms acting on the bacteria are explicitly described: growth, division and up-take. Consumption of the substrate is also described and it is through this mechanism that bacteria interact.\n\n\nWe described an exact Monte Carlo simulation technique of the model. Our main result is   the convergence of the IBM to an integro-differential equation model when the population size tends to infinity. There is a convergence in distribution for coupled stochastic processes: the first one with c\u00e0dl\u00e0g trajectories takes values \u200b\u200bin the set of finite measures on the space of  masses, the second one with continuous trajectories takes values \u200b\u200bin the set of substrate concentration values. The integro-differential equation model limit has been known for many years as population-balance equation and is used in particular for growth-fragmentation models. The numerical tests have allowed us to illustrate this convergence: in large population, the integro-differential equation model accurately reflects the behavior of the IBM and thus, the randomness can be neglected. In small or medium population this randomness is not negligible. We also proposed a numerical test where the classical chemostat model, in terms of two coupled ordinary differential equations, cannot account for transient behavior observed with the integro-differential model as with the IBM. Finally, in the case of washout, thus in small population size, IBM gives account for the random washout time, whereas the conventional model as the integro-differential model merely offers a more limited vision of this phenomenon as an asymptotic convergence to washout,  never reached in finite time.\n\n\n\n\\bigskip\n\n\nIt would be interesting to propose extensive simulations of this model, we can consider several axes. First, our simulations are based on a division rate $\\lambda(s,x)$ which  does not depend on the substrate concentration~$s$. One can easily consider a rate depending on~$s$, there are indeed such examples \\citep[see e.g.][]{henson2003b}. Similarly, in our model the individual growth dynamics is deterministic, it would be appropriate to consider a stochastic individual growth dynamics. To progress in this direction it would  be necessary to consider specific cases studies where more explicit growth dynamics would be specified.\n\nIt would also be pertinent to develop  approximation methods based on moments,\nbut  these methods  are usually ad hoc and it would be worthwhile to rely on a rigorous and systematic approach.\n\n\nFinally, there are several relatively immediate extensions of the proposed model.\nAt first, one can imagine extending the model to the case of several species and several types of substrates.\nOne can also model the effects of aggregation of bacteria in the chemostat, for example in the form of flocculation with given rates for aggregation and fragmentation of flocs. We can also consider two classes of bacteria, attached bacteria and planktonic bacteria, to account for a biofilm dynamic.\n \n\n\n\n\n\n\\section*{Acknowledgements}\n\nThe authors are grateful to J\u00e9r\u00f4me Harmand and Claude Lobry for discussions on the model, to Pierre Pudlo and Pascal Neveu for their help concerning the programming of the IBM. This work is partially supported by the project ``Mod\u00e8les Num\u00e9riques pour les \u00e9cosyst\u00e8mes Microbiens'' of the French National Network of Complex Systems (RNSC call 2012). The work of Coralie Fritsch is partially supported by the Meta-omics of Microbial Ecosystems (MEM) metaprogram of INRA.\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction} \\label{sec:intro}\nThe data produced by functional Magnetic Resonance Imaging (fMRI) is high-dimensional and sometimes not suitable for analyzing the cognitive states \\cite{firat2015learning}. Learning efficient low-dimensional features from high-dimensional complex input spaces is crucial for the decoding of cognitive processes. In this paper, we explore deep learning algorithms in order to i) find a compact  representation of connectivity patterns embedded in fMRI signals, ii) detect natural groupings of these patterns and, iii) use these natural groups to extract brain networks to represent cognitive tasks.\n\nOur framework is built upon our previous work in the area \\cite{onal2016hierarchical}, where we decompose fMRI signals into various frequency sub-bands using their wavelet transforms. We further utilize the signals at different sub-bands to form multi-resolution brain networks. Recent studies have shown that brain networks formed by the correlation of voxel pairs' in fMRI signals provide more information for brain decoding compared to the temporal information of single voxels \\cite{lindquist2008statistical,richiardi2013machine}. Moreover, there has been a shift in the literature toward brain decoding algorithms that are based on the connectivity patterns in the brain motivated by the belief that these patterns provide more information about cognitive tasks than the isolated behavior of individual anatomic regions \\cite{shirer2012decoding,ekman2012predicting,onal2017new}. \n\nContrary to the methods suggested in \\cite{lindquist2008statistical,richiardi2013machine} where supervised learning algorithms are employed for brain decoding, in this paper, we investigate the common groupings in HCP task data set to find out if these natural groups correspond to the cognitive tasks. This approach enables us to find shared network representations of a cognitive task together with its variations across the subjects. Additionally, multi-resolution representation of the fMRI signals enables us to observe the variations of networks among different frequency sub-bands. \n\nAfter constructing the brain networks representing the connectivity patterns among the anatomic regions of the brain at each sub-level, a Stacked De-noising Auto-Encoder (SDAE) algorithm is employed to learn shared connectivity features associated with a task based on the estimated mesh networks at different sub-bands. We concatenate the learned connectivity patterns from several wavelet sub-bands and utilize them in a hierarchical clustering algorithm with a distance matrix based on their correlations. The main reason behind concatenation of the feature matrices is that the detected patterns in the brain at different frequencies provide complementary information in regard to the overall cognitive state of the brain. \n\nOur results show that the mesh network representation of cognitive tasks is superior compared to fMRI time-series representation. We observe that SDAE successfully learns a set of connectivity patterns which provide an increased clustering performance. The performances are further improved by fusing the learned representations from multiple time-resolutions. This shows that the modeling of the connectivity of brain in multiple sub-bands of the fMRI signal leads to diverse mesh networks carrying complementary information for representing the cognitive tasks. The high rand index $93\\%$ obtained at the output of the clustering algorithm proves the existence of natural groups with low within-cluster-variances and high between-class-variances among the tasks. \n\nIn order to analyze the similarities and distinctions among the network topologies of fMRI signal, we visualize the networks and their precisions at the cluster centers. The cluster precisions, indicate shared connectivities among the subjects, whereas the mesh networks at the cluster center show a representative network for each cognitive task. It is observed that there are high inter-subject variances in the mesh networks.\n\\section{Experimental Setup} \\label{sec:exp}\nWe use the fMRI data from HCP for $300$ subjects performing specific tasks. A subject performs seven distinct cognitive tasks during the experiment given in Table \\ref{tab:firsttable} \\cite{barch2013function}. Each task $t$ consists of $s_t$ scans of the brain volume representing the changes in the brain during the task (the underlying cognitive process). The duration and the number of scans are task-dependent but the same for all participants. The total number of scans is $S=1940$ and we use $R = 90$ anatomical regions of $116$ AAL after removing the anatomical regions in Cerebellum and Vermis. \n\\begin{table}\n\t\\centering\n\t\\resizebox{.80\\textwidth}{!}{\\begin{minipage}{\\textwidth}\n\t\t\t\\begin{tabular}{l*{7}{c}r}\n\t\t\t\t\\hline\n\t\t\t\t& Emotion & Gambling & Language & Motor & Relational & Social &  WM\\\\\n\t\t\t\t\\hline\n\t\t\t\tScans & 176 & 253 & 316 & 284  & 232 & 274 & 405\\\\\n\t\t\t\t\\hline\n\t\t\t\tDurations & 2:16 & 3:12 & 3:57 & 3:34& 2:56 & 3:27 & 5:01\\\\\n\t\t\t\\end{tabular}\n\t\\end{minipage}}\n\t\\caption{Scans per Task and the Duration for each Task (min:sec).}\n\t\\label{tab:firsttable}  \n\\end{table}\nRepresentative time-series data points are attained by spatially averaging the signals associated with voxels ($n$) residing in the same region ($r$) in the brain,\n\\begin{align*}\nX_r(t)=\\frac{1}{N} \\sum_{\\forall n \\in r}^{} X_n(t),\n\\end{align*}\nwhere $N$ represents the total number of voxels in region $r$.\n\\section{Hierarchical Multi-Resolution Mesh Networks (HMMNs)} \\label{sec:hmmns}\n\\begin{figure*}[!t]\n\t\\centering\n\t\\includegraphics[scale = 0.5]{Diagram}\n\t\\caption{An Overview of the Proposed Deep Learning Framework.}\n\t\\label{fig:Diagram}\n\\end{figure*}\nIn our work, we utilize the representative time series obtained for each anatomic region to build a set of local meshes. The local meshes estimated around each anatomic region are ensembled to form a mesh network. This is motivated by the fact that the structure of the brain is highly interconnected and that neurons influence each other based on the strengths of their synaptic connections \\cite{pantazatos2012decoding}. HMMNs model cognitive tasks by estimating a mesh network at each frequency sub-band. It is expected that the brain network at each sub-band provides supplementary information about the underlying brain activities. We will show that our modeling of the brain with HMMNs greatly enhances the brain decoding performance by allowing us to look at cognitive states of the brain regions in multiple time-resolutions.\n\nAs the first step, the representative time-series $X_r(t)$, for each anatomic region $r$ are decomposed into a set of signals in different time-resolutions. This allows us to estimate and analyze how the anatomical regions process information in different frequency resolutions \\cite{thompson2015frequency}. We adopt Discrete Wavelet Transform (DWT) as our main tool \\cite{bullmore2004wavelets}. We apply the DWT to $X_r(t)$ for all brain regions to decompose the signals into $l$ sub-bands where $l=1,2,...,11$ ($L=11$). At sub-band level $l$, we attain two sets of orthonormal components named as sets of approximation coefficients $\\mathcal{A}=\\{a_{r,l,k}\\}$ and detail coefficients $\\mathcal{D}=\\{d_{r,l,k}\\}$ where $k$ represents the location of the wavelet waveform in discrete-time \\cite{bullmore2004wavelets}. These coefficients then may be utilized to reconstruct the fMRI signals at each frequency level, yielding the total of $(2\\times L)+1$ fMRI time-series. Formally, the representative time-series at sub-band $j$ ($j\\in [0,1,...,2L]$) may be defined as,\n\\begin{align*}\nx_{j,r}(t)=\n\\begin{cases}\nX_r(t), & \\text{if}\\ j=0 \\\\\n\\sum_{k}^{}a_{r,l,k}\\Phi_{l,k}(t)~and~ l=j & \\text{if}\\ 1\\leq j\\leq L\\\\\n\\sum_{k}^{}d_{r,l,k}\\Psi_{l,k}(t)~and~ l=j-L+1 & \\text{if}\\ j > L\n\\end{cases}\n\\end{align*}\nwhere $\\Phi_{l,k}$ and $\\Psi_{l,k}$ are called the mother wavelet and the father wavelet. More details on our approach are given in \\cite{onal2016hierarchical}. \n\nNow, we can construct a mesh network at each sub-band to represent cognitive tasks in terms of the relationships among anatomic regions. The construction of these networks help us analyze the topological properties of the brain and extract connectivity patterns associated with a cognitive process at each sub-band. In order to demonstrate the benefits of our approach, we propose an unsupervised clustering framework which can successfully take advantage of the connectivity patterns and distinguish between different cognitive tasks at multiple sub-bands. For this purpose, we divide the entire experiment session ($S=1940$ number of scans) for a subject into unlabeled windows of length $w_i=30$ consisting of $30$ discrete scans, where $i=1,...,64$ for each subject for the entire experiment. The length of the window is determined empirically, as the shortest time-interval which provides the highest rand index, at the output of clustering. It is important to note that the windows are unlabeled and may consist of overlapping data points from different cognitive tasks. \n\nThe nodes of the mesh networks are connected to their $p$-nearest neighbors to form a star mesh around a region. The nearest $p$ neighbors for a certain node are the ones having the largest Pearson correlation coefficients with the node. For each mesh formed around an anatomic region $r$, the arc weights for the window $w_i$ are estimated at the sub-band $j$ using the following regularized linear model,\n\\begin{align} \\label{mesh}\nx_{j,w_i,r}=\\sum_{r^\\prime \\in N_p[r]}^{} a_{j,w_i,r,r^\\prime} x_{j,w_i,r^\\prime} + \\lambda |a_{j,w_i,r,r^\\prime}|^2 + \\epsilon_{j,w_i,r}\n\\end{align}\nwhere the regularization parameter is $\\lambda$. The mesh arc weights $a_{j,w_i,r,r^\\prime}$, defined in the $N_p[r]$ neighborhood of region $r$, are estimated by minimizing the error $\\epsilon_{j,w_i,r}$. $x_{j,w_i,r}$ is a vector representing the average voxel time-series in region $r$ at sub-band $j$ for the window $w_i$, such that,\n\\begin{align*}\nx_{j,w_i,r}=[x_{j,w_i,r}(1),x_{j,w_i,r}(2),...,x_{j,w_i,r}(30)].\n\\end{align*}\n\nThe relation defined in (\\ref{mesh}) is solved for each region $r$ with its neighbors separately. In other words, we obtain an independent local mesh around each region $r$. After estimating all the mesh arc weights, we put them under the vector $A_{j,w_i} = \\{a_{j,w_i,r,r^\\prime}\\}_{r,r^\\prime}^R$, called Mesh Arc Descriptors (MADs). We represent $G_{j,w_i}$ as an ensemble of all local meshes. Lastly, the mesh networks are estimated for the original fMRI signal, and its approximation and detail parts of different resolutions. Consequently, we form $2L + 1$ distinct mesh networks for the frequency sub-bands $\\{A_0,A_1,A_2,...,A_L,D_1,D_2,...,D_L\\}$.  \n\nThe multi-resolution mesh network for a subject is defined by a connectivity graph, $G_{j,w_i} = \\{ V, A_{j,w_i}: \\forall j \\}$, for each unlabeled window $w_i$ and for each sub-band $j$. The set of vertices $V$ corresponds to the ordered set of anatomic regions and is of size $R$. Vertex attributes are the time-series $x_{j,w_i,r}$ contained in the window $w_i$, at the sub-band $j$. The arc weights, $A_{j,w_i} = \\{a_{j,w_i,r,r^\\prime}\\}_{r,r^\\prime}^R$ between regions $r$ and $r^\\prime$, for each window $w_i$ are obtained from the local meshes of the representative time-series data points at sub-band $j$. This process results in $2L + 1$ distinct mesh networks represented by an adjacency matrix of size $R\\times R$ made up of ($\\forall_{r,r^\\prime} a_{j,w_i,r,r^\\prime}$) for each window $w_i$ ($i=1,...,64$). We concatenate the arc weights under a vector ($f_{j,w_i}$) of size $1 \\times R^2$ and embed the brain network for the window $w_i$ at sub-band $j$. This means that for each level $j$ and each subject, we represent the entire experiment by a large unlabeled matrix of size $64 \\times (R^2)$ i.e. $F_{j,sub_s}=[f_{j,w_1},...,f_{j,w_{64}}]^T$. Next, we will introduce a deep learning algorithm which learns a set of compact connectivity patters from the embedded brain networks and consequently, cluster the windows of similar connectivity patterns. Each cluster of similar connectivity patterns represent a specific cognitive task (see Table \\ref{tab:firsttable}).\n\\section{The Deep Learning Architecture} \\label{sec:dl}\nThe embedded mesh networks model the connectivity among the anatomic regions at different sub-bands of fMRI signal under each window $w_i$ for each subject. Next, we utilize a deep learning architecture to extract a set of compact connectivity patterns from the mesh networks. We will show that the learned connectivity patterns form natural clusters corresponding to cognitive states. To meet this goal, we design a multi-layer stacked de-noising sparse auto-encoder \\cite{poultney2007efficient}.\nFor each sub-band $j$, we train a SDAE that takes the windows in the embedded brain network associated with subject $sub_s$ i.e. $f_{j,w_i} \\in F_{j,sub_s}, i=1,...,64$ as its input, and produces a vector $y$ of size $1\\times 7$. Recall that there are a total of $7$ cognitive tasks. The learned features represent the connectivity patterns at sub-band $j$ for subject $sub_s$ as follows,\n\\begin{align*}\nY_\\theta(F_{j,sub_s})=S(W F_{j,sub_s}+B),\n\\end{align*} \n with the auto-encoder parameter set $\\theta=[W,B]$ where $W$ is the collection of weights $\\{W_i\\}_{1:4}$, $B$ is the collection of biases $\\{B_i\\}_{1:4}$ at each neuron and $S$ represents the activation function $\\arctan$. Our sparse auto-encoder design includes an input layer of size $f_{j,w_i}^T$ with three hidden layers $[500,64,21]$ and an output layer of size $7$ and the sparsity parameter $\\rho$. The output of each neuron $y_i$ may be represented as $y_i=\\sum_{j=1}^{n} w_jx_j + b_i$, where $n$ and $x_j$'s indicate the total number of neurons and the neurons' outputs from the previous layer. The objective function $J$ is to minimize the mean-squared loss function $L(W,B|F_{j,sub_s})$ in the presence of an $L_2$-Ridge regularization with parameter $\\lambda_2$ which adds stability and robustness to the learning process,\n\\begin{align*}\nJ=\\arg \\min_{w_i,b_i}\\{ L(W,B|F_{j,sub_s}) + \\lambda_2 ||W||_2^2 \\}.\n\\end{align*}\n\nIn order to deal with the possible noise in the input data points, we follow a dropout training procedure based on which at each learning epoch, $20\\%$ of the data points are removed. It has been shown that this de-noising procedure will control for over-fitting, as reported in \\cite{wager2013dropout}. After training the above auto-encoder, one can extract the feature matrices for subject $sub_s$ at sub-band $j$ to attain $({Y_{j,sub_s}^{(64 \\times 7)}})_f$. Our results will show that our proposed deep learning algorithm is capable of removing the large intra-variance amongst input data points and can give an effective representation of the brain networks in a low-dimensional space. This can be considered as a non-linear mapping model from a high-dimensional space to a low-dimensional space suited for clustering.\n\\section{Hierarchical Clustering}\nThe main objective behind our work is to design a data driven cluster analysis that is suitable for discriminating between distinct connectivity patterns associated with given cognitive states at different frequency levels. We perform a hierarchical clustering on a combination of features from different frequency levels attained from the deep learning algorithm to show that the framework is capable of detecting the cognitive tasks (given in Table \\ref{tab:firsttable}) based on their connectivity manifestation in the brain networks and their learned features after the deep learning architecture.\n\nThe clustering algorithm clusters a subject's brain features matrix $Y_f^{64 \\times (m \\times 7)}=[({Y_{1,sub_s}^{(64 \\times 7)}})_f,...,({Y_{m,sub_s}^{(64 \\times 7)}})_f]$ consisting of the concatenation of the feature matrices from $m$ different frequency levels selected from the the frequency sub-bands $\\{A_0,A_1,A_2,...,A_{11},D_1,D_2,...,D_{11}\\}$. This is to show that each frequency level carries complementary information in regard to cognitive tasks performed during the experiment. Given the $w=64$ discrete-time windows, the clustering algorithm attempts to divide the data points into $k=7$ clusters ($c_k$, $k=1,...,7$), by minimizing the following cost function,\n\\begin{align*}\nV=\\sum_{k=1}^{7} V_k = \\sum_{k=1}^{7}(\\sum_{y_j \\in c_k \\cap y_j\\in Y_f}^{}dis(y_j,c_k)),\n\\end{align*}\nwhere the distance matrix $dis(y_j,c_k)$ is based on the Pearson Correlation between data points which closely models the functional connectivity pattern in the brain from one task to another. The exact relation between the distance matrix and the correlation matrix is,\n\\begin{align*}\ndis(y_j,c_k)=1-Corr^2(y_j,y_k).\n\\end{align*}\n\nThe entries of the correlation matrix $Corr(y_j,y_k)$ indicate the degree to which window $y_j$ is correlated with window $y_k$. The above relation can capture the time-varying coupling between windows and consequently closely model the flow of change in brain features from one cognitive state to another \\cite{calhoun2014chronnectome}. Fig. \\ref{fig:Diagram} depicts the entire deep learning framework. After clustering the unlabeled windows into $7$ different clusters, in the next section, we will compare the resulting clusters with the labeled data points given in Table \\ref{tab:firsttable} in order to examine the performance of our proposed approach. We utilize Rand Index (RI) and Adjusted Rand Index (ARI) as performance measures for our algorithm \\cite{milligan1986study}.\n\\section{Experimental Results} \\label{sec:expres}\nIn this section, we test the validity of the suggested deep learning architecture in two groups of experiments. The first set of experiments measures the cluster validity by clustering the fMRI signal, mesh arc weights of single and multi-resolution signals and utilizing the measures of performance RI and ARI. The second group of experiments visualizes the mesh networks obtained across subjects and cognitive tasks to observe the inter-task and inter-subject variabilities. We perform within-subject clustering analyses based on the fMRI signals collected from $100$ subjects (described in Section \\ref{sec:exp}). The design parameters are selected empirically through a cross validation process based on performance. We search for the optimal design parameters based on the sets, $p \\in \\{10,20,30,40,50\\}$, $\\lambda \\in \\{16,32,128,256\\}$, $\\rho \\in \\{0.01,0.001,0.0001\\}$, and $\\lambda_2 \\in \\{0.00001,0.00055,0.0001\\}$. We select the design parameters, $p=40$ and $\\lambda=32$ for the mesh networks (Section \\ref{sec:hmmns}), $\\rho=0.001$ and $\\lambda_2=0.00055$ for the SDAE design (Section \\ref{sec:dl}) as optimal values. The RI and ARI values given in the tables for each experiment describe the average clustering performance for all $100$ subjects. \n\nTable \\ref{tab:table4} gives a performance comparison between the clustering of the raw fMRI data (i.e. representative time series of each anatomic region) and the clustering of the arc weights of mesh networks (MADs). Note that, clustering the MADs increases the rand index from $68\\%$ to $84\\%$. This substantial improvement shows that connectivity patterns are much more informative then the average voxel time-series.\n\nOur next analysis involves the representation power of individual frequency sub-bands, where we examine the performance of each sub-band in detecting MADs among anatomic regions for the given tasks. This may also be translated as the amount of complementary information each sub-band carries in regard to the functional connectivity of the given cognitive states. For further comparison, we cluster the data after attaining the MADs at each sub-band (Section \\ref{sec:hmmns}) and also after the deep learning architecture (Section \\ref{sec:dl}). The results are stated in Table \\ref{tab:table2}. The high rand indices for all individual sub-bands confirm the benefits of analyzing the fMRI signals in multiple time-resolutions as it shows that each sub-band carries important information in regard to the mesh network arc weights and the connectivity patterns learned at the output of stacked de-noising auto-encoders. This leads to a clustering performance between the range of $68-86\\%$ across all sub-bands. Note that, the sub-bands \\bf{A5} \\normalfont to \\bf{A11}\\normalfont, \\bf{D5} \\normalfont to \\bf{D7}\\normalfont~and \\bf{D9} \\normalfont to \\bf{D11}\\normalfont~show relatively higher performances indicating that these frequency bands are more informative then the rest. \n\nIn order to boost the RI and ARI values given in Table \\ref{tab:table2}, we fuse the learned connectivity patterns based on a combination of sub-bands to obtain a better representation. In our last set of clustering analyses, we examine the clustering performance by ensembling multiple sub-bands. RI and ARI values for the ensembled sub-bands, given in Table \\ref{tab:table3} point to a substantial increase compared to the best single sub-band clustering performance of $86\\%$ at sub-band A10 to a performance of $93\\%$ for the fusion of all sub-bands. This shows that not only the brain networks constructed at multiple time-resolutions provide complementary information for the clustering algorithm but that the proposed deep architecture is capable of detecting distinct connectivity patterns in the brain for a given cognitive task, independent of subjects.\n\nThe rather high ARI values in Table \\ref{tab:table3} confirm that utilizing the complementary information gained from different time-resolutions result in clusters with relatively low within-cluster variances and high between-cluster variances. This claim is backed by the high ARI values that result from combining the information from different sub-bands before clustering. Further, by increasing the number of subjects to $200$ in our data set, and by fusing the brain networks obtained from the entire $23$ sub-bands and clustering their connectivity pattern extracted by the SDAE platform, we are able to achieve the performance of $93\\%$ RI and $71\\%$ ARI. This experiment shows that increasing the number of subjects does not decrease the clustering performance.\n\\begin{figure*}[h]\n\t\\centering\n\t\\begin{subfigure}\n\t\t\\centering\n\t\t\\includegraphics[scale=0.2539]{Sub29_Task1_axial} \n\t\t\\includegraphics[scale=0.2539]{Sub29_Task2_axial} \n\t\t\\includegraphics[scale=0.2539]{Sub29_Task3_axial} \n\t\t\\includegraphics[scale=0.2539]{Sub29_Task4_axial} \n\t\t\\includegraphics[scale=0.2539]{Sub29_Task5_axial} \n\t\t\\includegraphics[scale=0.2539]{Sub29_Task6_axial} \n\t\t\\includegraphics[scale=0.2539]{Sub29_Task7_axial} \n\t\\end{subfigure}\n\t\\begin{subfigure}\n\t\t\\centering\n\t\t\\includegraphics[scale=0.2539]{Sub40_Task1_axial}\n\t\t\\includegraphics[scale=0.2539]{Sub40_Task2_axial}\n\t\t\\includegraphics[scale=0.2539]{Sub40_Task3_axial}\n\t\t\\includegraphics[scale=0.2539]{Sub40_Task4_axial}\n\t\t\\includegraphics[scale=0.2539]{Sub40_Task5_axial}\n\t\t\\includegraphics[scale=0.2539]{Sub40_Task6_axial}\n\t\t\\includegraphics[scale=0.2539]{Sub40_Task7_axial}\n\t\\end{subfigure}\n\t\\begin{subfigure}\n\t\t\\centering\n\t\t\\includegraphics[scale=0.2539]{Sub56_Task1_axial}\n\t\t\\includegraphics[scale=0.2539]{Sub56_Task2_axial}\n\t\t\\includegraphics[scale=0.2539]{Sub56_Task3_axial}\n\t\t\\includegraphics[scale=0.2539]{Sub56_Task4_axial}\n\t\t\\includegraphics[scale=0.2539]{Sub56_Task5_axial}\n\t\t\\includegraphics[scale=0.2539]{Sub56_Task6_axial}\n\t\t\\includegraphics[scale=0.2539]{Sub56_Task7_axial}\n\t\\end{subfigure}\n\t\\caption{Mesh Networks of 3 Subjects: the top row shows subject 29, the middle row shows subject  40, bottom row shows subject 56. The mesh networks at each row from left to right indicate tasks in the following order: emotion, gambling, language, motor, relational, social and working-memory.}\\label{fig:2}\t\t\n\\end{figure*}\n\\begin{figure*}[h]\n\t\\centering\n\t\\includegraphics[scale=0.252]{Pre_Task1_axial}\n\t\\includegraphics[scale=0.252]{Pre_Task2_axial}\n\t\\includegraphics[scale=0.252]{Pre_Task3_axial}\n\t\\includegraphics[scale=0.252]{Pre_Task4_axial}\n\t\\includegraphics[scale=0.252]{Pre_Task5_axial}\n\t\\includegraphics[scale=0.252]{Pre_Task6_axial}\n\t\\includegraphics[scale=0.252]{Pre_Task7_axial}\n\t\\includegraphics[scale=0.110]{Pre_Scale}\n\t\\caption{Precision of the Mesh Networks of a Subset of Subjects. The mesh networks from left to right indicate tasks in the following order: emotion, gambling, language, motor, relational, social and working-memory.}\n\t\\label{fig:Precision}\n\\end{figure*}\n\\begin{table}\n\t{\n\t\t\\centering\n\t\t\\resizebox{1.\\textwidth}{!}{\n\t\t\t\\begin{minipage}{\\textwidth}\n\t\t\t\t\\begin{tabular}{ | c | c | c | c | c | c | c|}\n\t\t \\hline\n\t\t& Rand & A. Rand & & Rand & A. Rand \\\\\n\t\t\\hline\n\t\tRaw fMRI Data &  0.68 & -0.07 &\n\t\tMAD   &  0.84  & 0.37  \\\\\n\t\t\\hline\t\n\t\t\t\t\\end{tabular}\n\t\t\\end{minipage}}\n\t}\n\t\\caption{Clustering Performance Comparison.}\n\t\\label{tab:table4}  \n\\end{table}\n\\begin{table}\n{\n\t\\centering\n\t\\resizebox{1.2\\textwidth}{!}{\n \\begin{minipage}{\\textwidth}\n\t\\begin{tabular}{ | c | c | c | c | c | c | }\n\t\\hline\n\tMAD & Rand & A. Rand & SDAE  & Rand & A. Rand\\\\\n\t\\hline\n\tA0 &  0.84 & 0.37 &  A0 &  0.78 & 0.11\\\\\n\t\\hline\n\tA1  &  0.83 & 0.34  &  A1  &  0.76  & 0.02\\\\\n\t\\hline\n\tD1 &  0.81 & 0.28 &  D1 &  0.75 & -0.04\\\\\n\t\\hline\n\tA2  &  0.77  & 0.15 &  A2  &  0.74  & -0.06\\\\\n\t\\hline\n\tD2 &  0.86 & 0.47 &  D2 &  0.76 & 0.11\\\\\n\t\\hline\n\tA3  &  0.75  & 0.12 &  A3  &  0.74  & 0.07\\\\\n\t\\hline\n\tD3 &  0.72  & 0.15 &  D3 &  0.74 & -0.34\\\\\n\t\\hline\n\tA4  &  0.68  & 0.06  &  A4  &  0.77  & 0.06\\\\\n\t\\hline\n\tD4 &  0.77 & 0.24  &  D4 &  0.78 & 0.15\\\\\n\t\\hline\n\tA5  &  0.68  & 0.08 &  A5  &  0.80  & 0.17\\\\\n\t\\hline\n\tD5 &  0.74 & 0.17 &  D5 &  0.80 & 0.16\\\\\n\t\\hline\n\tA6  &  0.75  & 0.18  &  A6  &  0.81  & 0.20\\\\\n\t\\hline\n\tD6  &  0.75  & 0.17 &  D6  &  0.80  & 0.20\\\\\n\t\\hline\n\tA7 &  0.87 & 0.50 &  A7 &  0.80 & 0.21\\\\\n\t\\hline\n\tD7  &  0.84  & 0.37  &  D7  &  0.82  & 0.26\\\\\n\t\\hline\n\tA8 &  0.85 & 0.37 &   A8 &  0.80 & 0.16\\\\\n\t\\hline\n\tD8  &  0.82  & 0.27  &  D8  &  0.79  & 0.14\\\\\n\t\\hline\n\tA9 & 0.85 & 0.39 &  A9 &  0.83 & 0.30\\\\\n\t\\hline\n\tD9  &  0.82 & 0.28 &  D9  &  0.80  & 0.12\\\\\n\t\\hline\n\tA10 &  0.82 & 0.29 &  A10 &  0.86 & 0.41\\\\\n\t\\hline\n\tD10  &  0.83  & 0.30  &  D10  & 0.84  & 0.20\\\\\n\t\\hline\n\tA11 &  0.79 & 0.20 &  A11 &  0.82 & 0.25\\\\\n\t\\hline\n\tD11  &  0.81  & 0.26  &  D11  &  0.83  & 0.29\\\\\n\t\\hline\n\t\\end{tabular}\n \\end{minipage}}\n\\caption{Clustering Performance for Individual Sub-bands.}\n\\label{tab:table2}  \t\n}\\end{table}\n\\begin{table}\n{\n\\centering\n\\resizebox{0.92\\textwidth}{!}{\n \\begin{minipage}{\\textwidth}\n\t\\begin{tabular}{ | l | c | c | l | c | c | }\n\t\\hline\n\tMAD & Rand & A. Rand & SDAE  & Rand & A. Rand \\\\\n\t\\hline\n\tAll Sub-bands & 0.91 & 0.64 &  All Sub-bands&  0.93 & 0.71\\\\\n\t\\hline\n\tSub-bands 7-9  &  0.92 & 0.66  &  Subbands 7-9&  0.90 & 0.59 \\\\\n\t\\hline\n\tSub-bands 7-11 &  0.92 & 0.66 & Subbands 7-11& 0.91 & 0.60 \\\\\n\t\\hline\n\tSub-bands 3-8  &  0.89  & 0.57 & Subbands 3-8&  0.91  & 0.64 \\\\\n\t\\hline\n\tSub-bands 3-11 &  0.90 & 0.59 & Subbands 3-11&  0.91 & 0.63 \\\\\n\t\\hline\n\t\\end{tabular}\n \\end{minipage}}\n}\n\\caption{Clustering Performance for Combinations of Sub-bands.}\n\\label{tab:table3}  \n\\end{table}\n\nFinally, we visualize the mesh networks obtained in the original fMRI signal to observe the inter-task and inter-subject variability of the brain networks. The motivation behind performing within-subject clustering rather than across-subject clustering in this study is the well-known inter-subject variability, which may prevent the clustering algorithm from finding natural groupings in the data. In order to illustrate the inter-subject variability, we plot the mesh networks of $3$ subjects in Fig. \\ref{fig:2} and Fig \\ref{fig:Precision} for each cognitive task. These subjects have the RI of $99\\%$, which indicates that the proposed model has successfully estimated the natural groupings  for each one of these $3$ subjects. The networks shown in the aforementioned figures represent the medoids of the clusters which correspond to each one of the $7$ different tasks. The mesh networks corresponding to each of the subjects are pruned by eliminating the mesh arc weights with values less then a threshold to reach $1\\%$ sparsity for simplification. A close analysis of the mesh networks corresponding to each task for the subjects shows that the mesh networks corresponding to the same task show small similarities across the $3$ subjects. This validates our prior claim on the existence of high inter-subject variabilities. To further investigate the inter-subject variability, we select a group of subjects with rand indices higher than $90\\%$ from the HCP task data set of $300$ individuals. We define the precision of the mesh network across the set of subjects as the inverse of variance and calculate this value for the selected subjects. Fig. \\ref{fig:Precision} shows the pruned precision of the mesh networks of the aforementioned set of subjects with $1\\%$ sparsity. The thickness and the colors of the edges are proportional to their corresponding precision values. One may observe from Fig. \\ref{fig:Precision} that the majority of the edges are thin-blue with only few of them thick-red. This indicates that the majority of the mesh network connections have high standard deviations across subjects.\n\\section{Conclusion} \\label{sec:con}\nIn this paper, we proposed a framework for constructing a set of brain networks in multiple time-resolutions in order to model the connectivity patterns amongst the anatomic regions for different cognitive states. We proposed an unsupervised deep learning architecture that utilized these brain networks in multiple frequency sub-bands to learn the natural groupings of connectivity patterns in the human brain for given cognitive tasks. We showed that our suggested deep learning algorithm is capable of clustering the representative groupings into their associated cognitive states. We examined our suggested architecture on a task data set from HCP and achieved the clustering performance of $93\\%$ Rand Index and $71\\%$ Adjusted Rand Index for $200$ subjects. Lastly, we visualized the mean values and the precisions of the mesh networks at each component of the cluster mixture. We showed that the mean mesh networks at cluster centers have high inter-subject variabilities.\n\\section*{Acknowledgment}\nWe would like to thank Dr. Itir Onal Erturul, Arman Afrasiabi and Omer Ekmekci of Middle East Technical University and Professor Mete Ozay of Tohoku University for supporting us throughout many fruitful discussions. The work is supported by TUBITAK (Scientific and Technological Research Council of Turkey) under the grant No: 116E091.\n\\bibliographystyle{IEEEtran}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\n\n\\section{Introduction}\n\\input{s_intro}\n\\section{Model problem and continuous setting} \\label{s_contiSetting}\n\\input{s_contiSetting}\n\\section{Numerical setting}\\label{sec::discretization}\n\\input{s_numSetting}\n\n\\newcommand{\\uu}{\\mathbf u}\n\\newcommand{\\pp}{\\mathbf p}\n\\newcommand{\\qq}{\\mathbf q}\n\\newcommand{\\yy}{\\mathbf y}\n\\newcommand{\\vv}{\\mathbf v}\n\\newcommand{\\ww}{\\mathbf w}\n\\newcommand{\\KK}{\\mathbf K}\n\\newcommand{\\MM}{\\mathbf M}\n\\newcommand{\\Amat}{\\mathbf A}\n\\newcommand{\\ff}{\\mathbf f}\n\\section{Numerical {topological-shape } derivative} \\label{sec::numTopShapeDer}\n\\input{s_numericalDerivatives}\n\n\\input{s_connection}\n\n\\section{Numerical Experiments}\\label{sec::numericalExamples}\n\\input{s_numericalExamples}\n\\section{Conclusions} \\label{sec::conclusion}\n\\input{s_conclusion}\n\n\n\n\n\\section*{Conflict of interest}\nThe authors declare that they have no conflict of interest.\n\n\n\\bibliographystyle{spmpsci}     \n\n\n\n\n\n\\section{Connection between continuous and discrete sensitivities} \\label{sec::connection}\n\nThe {topological-shape } derivative introduced in \\eqref{eq::topDerivative} and computed for model problem \\eqref{eq::ContinuousProblem} in Theorem \\ref{thm_numTopShapeDer_formula} represents a sensitivity of the discretized problem \\eqref{eq_discr_opti_problem}. In this section, we draw some comparisons with the classical topological and shape derivatives defined on the continuous level before discretization. While the purpose of this paper is to follow the idea \\textit{discretize-then-differentiate}, we consider the other way here for comparison reasons.\n\n\n\n\\subsection{Connections between continuous and discrete topological derivative} \\label{sec_connTD}\nFor comparison, we also illustrate the derivation of the continuous topological derivative according to \\eqref{eq_defTD} for problem \\eqref{eq::ContinuousProblem}. We use the same Lagrangian framework as introduced in Section \\ref{sec::Lagrangian}, see also \\cite{simplified}. Given a shape $\\Omega \\in \\mathcal A$, a point $z \\in D \\setminus \\partial \\Omega$, an inclusion shape $\\omega \\subset \\mathbb R^d$ with $0 \\in \\omega$ and $\\varepsilon \\geq 0$, we define the inclusion $\\omega_\\varepsilon = z + \\varepsilon \\omega$ and the perturbed Lagrangian\n\\begin{align*}\n    G(\\varepsilon, \\varphi, \\psi) := c_1 |\\Omega_\\varepsilon| + c_2 \\int_D \\tilde{\\alpha}_{\\Omega_\\varepsilon} | \\varphi - \\hat u|^2\\; \\mbox dx + \\int_D \\lambda_{\\Omega_\\varepsilon} \\nabla \\varphi \\cdot \\nabla \\psi + \\alpha_{\\Omega_\\varepsilon} \\varphi  \\psi \\;\\mbox dx - \\int_D f_{\\Omega_\\varepsilon} \\psi \\; \\mbox dx\n\\end{align*}\nwhere $\\Omega_\\varepsilon = \\Omega \\setminus \\omega_\\varepsilon$ for $z \\in \\Omega$ and $\\Omega_\\varepsilon = \\Omega \\cup \\omega_\\varepsilon$ for $z \\in D \\setminus \\overline \\Omega$. For simplicity, we only consider the latter case, \\textit{i.e.}\\, $z \\in D \\setminus \\overline \\Omega$ in the sequel.\n\nNoting that $u_\\varepsilon$, $\\varepsilon \\geq 0$, is the solution to the perturbed state equation with parameter $\\varepsilon$, the topological derivative can also be written as\n\\begin{align}\n   d \\mathcal J(\\Omega)(z) =  \\underset{\\varepsilon \\searrow 0}{\\mbox{lim }}\\frac{1}{|\\omega_\\varepsilon|} (G(\\varepsilon, u_\\varepsilon, p) - G(0, u_0, p))\n\\end{align}\nwith the solution to the unperturbed adjoint state equation $p$. As in Section \\ref{sec::Lagrangian}, this leads to the topological derivative consisting of the three terms\n\\begin{align*}\nd \\redPhiObjectiveFunction(\\phi)(z) = R_1(u, p) + R_2(u, p) + R_0(u, p)\n\\end{align*}\nwhere\n\\begin{align*}\nR_1(u, p) :=& \\underset{\\varepsilon \\searrow 0}{\\mbox{lim }} \\frac{1}{|\\Omega_\\varepsilon \\triangle \\Omega|}  \\int_0^1 [\\partial_u G(\\varepsilon, u_0 + s(u_\\varepsilon - u_0), p) - \\partial_u G(\\varepsilon, u, p)](u_\\varepsilon - u_0) \\mbox ds, \\\\\nR_2(u, p) :=& \\underset{\\varepsilon \\searrow 0}{\\mbox{lim }} \\frac{1}{|\\Omega_\\varepsilon \\triangle \\Omega|}[ \\partial_u G(\\varepsilon, u, p) - \\partial_u G(0, u, p)](u_\\varepsilon - u), \\\\\nR_0(u, p) := & \\underset{\\varepsilon \\searrow 0}{\\mbox{lim }} \\frac{1}{|\\Omega_\\varepsilon \\triangle \\Omega|} [ G(\\varepsilon, u, p) - G(0, u, p)],\n\\end{align*}\nprovided that these limits exist.\nIt is straightforwardly checked that for $z \\in D \\setminus \\overline \\Omega$\n\\begin{align*}\n    R_0(u,p) = c_1 + c_2 (\\tilde \\alpha_1 - \\tilde \\alpha_2) (u - \\hat u)^2(z) + (\\lambda_1 - \\lambda_2) \\nabla u(z) \\cdot \\nabla p(z) + (\\alpha_1 - \\alpha_2) u(z) p(z) - (f_1-f_2)(z) p(z).\n\\end{align*}\n\n\nFor the term $R_2(u,p)$, we obtain\n\\begin{align*}\n    R_2(u,p)\n    =& \\underset{\\varepsilon \\searrow 0}{\\mbox{lim }} \\frac{1}{|\\omega_\\varepsilon|} \\bigg[ 2 c_2 \\int_{\\omega_\\varepsilon} (\\tilde \\alpha_{1} - \\tilde \\alpha_2) (u_0 - \\hat u)(u_\\varepsilon - u_0) \\; dy+  \\int_{\\omega_\\varepsilon} (\\lambda_{1} - \\lambda_2) \\nabla (u_\\varepsilon - u_0) \\cdot \\nabla p \\; dy \\\\\n    & \\qquad + \\int_{\\omega_\\varepsilon} (\\alpha_{1} - \\alpha_2) (u_\\varepsilon - u_0) p \\; dy \n    \\bigg].\n\\end{align*}\nA change of variables $y = T_\\varepsilon(x) = z + \\varepsilon x$ yields\n\\begin{align} \\label{eq_R2omega}\n    \\begin{aligned}\n R_2(u,p) =&  \\underset{\\varepsilon \\searrow 0}{\\mbox{lim }} \\frac{1}{|\\omega|} \\bigg[  2 c_2 (\\tilde \\alpha_{1} - \\tilde \\alpha_2)\\int_{\\omega}  (u_0 - \\hat u)\\circ T_\\varepsilon (u_\\varepsilon - u_0) \\circ T_\\varepsilon  \\; dx \\\\ & +  (\\lambda_{1} - \\lambda_2) \\int_{\\omega}  (\\nabla (u_\\varepsilon - u_0) )\\circ T_\\varepsilon \\cdot (\\nabla p)\\circ T_\\varepsilon \\; dx  + (\\alpha_{1} - \\alpha_2) \\int_{\\omega}  (u_\\varepsilon - u_0)\\circ T_\\varepsilon \\, p\\circ T_\\varepsilon \\; dx \\bigg] \n    \\end{aligned}\n\\end{align}\n\nWe have a closer look at the diffusion term \n\\begin{align} \\label{eq_R2lambdaDef}\n    R_2^\\lambda(u,p):= \\underset{\\varepsilon \\searrow 0}{\\mbox{lim }} \\frac{1}{|\\omega|} (\\lambda_1 - \\lambda_2)\\int_\\omega (\\nabla(u_\\varepsilon - u_0))\\circ T_\\varepsilon \\cdot (\\nabla p)\\circ T_\\varepsilon \\; \\mbox dx.\n\\end{align}\nIn the continuous setting, we now define $K_\\varepsilon := \\frac{1}{\\varepsilon}(u_\\varepsilon - u_0)\\circ T_\\varepsilon$ and use the chain rule $(\\nabla \\varphi )\\circ T_\\varepsilon = \\frac1\\varepsilon \\nabla (\\varphi \\circ T_\\varepsilon)$ to obtain\n\\begin{align}\n    R_2^\\lambda (u,p)=& \\underset{\\varepsilon \\searrow 0}{\\mbox{lim }} \\frac{1}{|\\omega|} (\\lambda_1 - \\lambda_2) \\int_\\omega \\nabla K_\\varepsilon \\cdot (\\nabla p) \\circ T_\\varepsilon \\; \\mbox dx\n\\end{align}\nNext, one can show the weak convergence $\\nabla K_\\varepsilon \\rightharpoonup \\nabla K$ for $K \\in \\dot{BL}(\\mathbb R^2)$ being defined as the solution to the exterior problem\n\\begin{align*}\n    \\int_{\\mathbb R^d} \\lambda_\\omega \\nabla K \\cdot \\nabla \\psi \\; \\mbox dx = -(\\lambda_1 - \\lambda_2) \\int_\\omega \\nabla u(z) \\cdot \\nabla \\psi \\; \\mbox dx \\quad \\mbox{for all }\\psi \\in \\dot{BL}(\\mathbb R^2),\n\\end{align*}\nwhere $\\dot{BL}(\\mathbb R^2) := \\{v \\in H^1_{loc}(\\mathbb R^2): \\nabla v \\in L^2(\\mathbb R^2) \\}\/_{\\mathbb R}$ is a Beppo-Levi space.\nAssuming continuity of $\\nabla p$ around the point of perturbation $z$,\nit follows that \n\\begin{align} \\label{eq_R2lambda}\n    R_2^\\lambda(u,p) = \\frac{1}{|\\omega|} (\\lambda_1 - \\lambda_2) \\int_\\omega \\nabla K \\cdot \\nabla p(z)\\; \\mbox dx.\n\\end{align}\nIt can be shown that the other terms in \\eqref{eq_R2omega} vanish and thus $R_2(u,p) = R_2^\\lambda(u,p)$.\nFinally, it follows from the analysis in \\cite[Sec. 5]{simplified} that $R_1(u,p) + R_2(u,p) = \\frac{1}{|\\omega|}(\\lambda_1 - \\lambda_2) \\int_\\omega \\nabla K \\cdot \\nabla p(z) \\, \\mbox dx = R_2(u,p)$, thus $R_1(u,p) = 0$ and $d \\redPhiObjectiveFunction(\\Omega)(z)= R_0(u,p) + R_2^\\lambda(u,p)$.\n\n\\begin{remark}\nComparing the topological derivative formula obtained here with the sensitivity for interior nodes ${\\mathbf x}_k \\in \\mathfrak T^- \\cup \\mathfrak T^+$ obtained in Section \\ref{sec::numTopShapeDer}, we see that the term corresponding to $R_2^\\lambda(u,p)$, i.e., the term\n\\begin{align*}\n        \\underset{\\varepsilon \\searrow 0}{\\mbox{lim }} \\frac{1}{|\\Omega(\\phi_\\varepsilon)\\triangle \\Omega(\\phi)|}(\\KK_\\varepsilon - \\KK_0) (\\uu_\\varepsilon  - \\uu_0) \\cdot \\pp\n\\end{align*}\nin \\eqref{eq_R2discr}, vanishes in the discrete setting. This can be seen as follows:\nFor $u_\\varepsilon^h$, $\\varepsilon \\geq 0$, and $p^h \\in V_h$, we have the expansion in the finite element basis\n\\begin{align*}\n    u_\\varepsilon^h(x) = \\sum_{i=1}^M u_\\varepsilon^{(i)} \\varphi_i , \\qquad p^h(x) = \\sum_{i=1}^M p^{(i)} \\varphi_i .\n\\end{align*}\nIf we now plug in these discretized functions into \\eqref{eq_R2lambdaDef} and consider a fixed mesh size $h$, we get on the other hand\n\\begin{align*}\n    R_2^\\lambda(u^h, p^h) = &  \\underset{\\varepsilon \\searrow 0}{\\mbox{lim }} \\frac{1}{|\\omega|} (\\lambda_1 - \\lambda_2) \\int_\\omega (\\nabla(u_\\varepsilon^h - u_0^h))\\circ T_\\varepsilon \\cdot (\\nabla p^h)\\circ T_\\varepsilon \\; \\mbox dx \\\\\n    =&  \\underset{\\varepsilon \\searrow 0}{\\mbox{lim }} \\frac{1}{|\\omega|}(\\lambda_1 - \\lambda_2) \\sum_{i,j=1}^M (u_\\varepsilon^{(i)}-u_0^{(i)})p^{(j)}   \\int_\\omega (\\nabla \\varphi_i)(z+\\varepsilon x) \\cdot (\\nabla \\varphi_j)(z+\\varepsilon x) \\; \\mbox dx = \n\\end{align*}\nwhere we used the continuity of $\\varepsilon \\mapsto \\uu_\\varepsilon$ according to Lemma \\ref{lem_uepsu0}. Note that, since the mesh and the basis functions are assumed to be fixed and independent of $\\varepsilon$, unlike in the continuous setting, here the continuity of the normal flux of the discrete solution $u_\\varepsilon^h$ across the interface $\\partial \\omega_\\varepsilon$ is not preserved. We mention that, when using an extended discretization technique that accounts for an accurate resolution of the material interface (e.g. XFEM \\cite{MoesDolbowBelytschko1999} or CutFEM \\cite{BurmanClausHansboLarsonMassing2014}), the corresponding discrete sensitivities would include a term corresponding to $R_2^\\lambda(u,p)$.\n\\end{remark}\n\n\\subsection{Connection between continuous and discrete shape derivative}\nThe continuous shape derivative $dg(\\Omega)(V)$ for a PDE-constrained shape optimization problem given a shape $\\Omega \\in \\mathcal A$ and a smooth vector field $V$ can also be obtained via a Lagrangian approach. For our problem \\eqref{eq::ContinuousProblem}, it can be obtained as $dg(\\Omega)(V) = \\partial_t G(0, u, p)$ with\n\\begin{align*}\n    G(t, \\varphi, \\psi) :=& c_1 \\int_\\Omega \\xi(t) \\; \\mbox dx + c_2 \\int_D \\tilde{\\alpha}_{\\Omega}  | \\varphi - \\hat u\\circ T_t|^2 \\xi(t) \\; \\mbox dx \\\\\n    &+ \\int_D \\lambda_{\\Omega} A(t) \\nabla \\varphi \\cdot \\nabla \\psi + \\alpha_{\\Omega}  \\varphi  \\psi \\xi(t) \\;\\mbox dx- \\int_D f_{\\Omega} \\psi   \\xi(t)\\; \\mbox dx\n\\end{align*}\nwhere $T_t(x) = x + t V(x)$, $\\xi(t) = \\mbox{det}(\\partial T_t)$, $A(t) = \\xi(t) \\partial T_t^{-1} \\partial T_t^{-T}$,\nsee \\cite{Sturm2015} for a detailed description. In the continuous setting, the shape derivative reads in its volume form\n\\begin{align*}\n    d g(\\Omega)(V) = \\int_D \\mathcal S_1^{\\Omega} : \\partial V + \\mathcal S_0^\\Omega \\cdot V \\; \\mbox dx\n\\end{align*}\nwith $\\mathcal S_1^\\Omega$ and $\\mathcal S_0^\\Omega$ given in \\eqref{eq_S1} and \\eqref{eq_S0}, respectively. Under certain smoothness assumptions, it can be transformed into the Hadamard or boundary form\n\\begin{align} \\label{eq_dg_Hadamard}\n    d g(\\Omega)(V) = \\int_{\\partial \\Omega} L \\, (V\\cdot n) \\; \\mbox dS_x\n\\end{align}\nwith $L =( \\mathcal S_1^{\\Omega, \\text{in}} - \\mathcal S_1^{\\Omega, \\text{out}})n \\cdot n$ given by\n\\begin{align} \\label{eq_L}\n    L = c_1 + c_2(\\tilde \\alpha_1 - \\tilde \\alpha_2) (u - \\hat u)^2 + (\\alpha_1 - \\alpha_2) u p - (f_1 - f_2) p + L^\\lambda\n\\end{align}\nwhere $L^\\lambda$ is given by\n\\begin{align}\n    L^\\lambda :=& (\\lambda_1- \\lambda_2)  (\\nabla u \\cdot \\tau)(\\nabla p \\cdot\\tau) -  \\left(\\frac{1}{\\lambda_1} - \\frac{1}{\\lambda_2}\\right)( \\lambda_\\Omega \\nabla u \\cdot n)(  \\lambda_\\Omega \\nabla p \\cdot n) \\nonumber \\\\\n    =&\\lambda_1 \\nabla u_{\\text{in}} \\cdot \\nabla p_{\\text{in}} -  \\lambda_2 \\nabla u_{\\text{out}} \\cdot \\nabla p_{\\text{out}} - 2 \\lambda_1 (\\nabla u_{\\text{in}}\\cdot n) ( \\nabla p_{\\text{in}} \\cdot n) + 2 \\lambda_2 (\\nabla u_{\\text{out}}\\cdot n) ( \\nabla p_{\\text{out}} \\cdot n). \\label{eq_Llambda_inout}\n\\end{align}\nHere, $\\nabla u_{\\text{in}}, \\nabla p_{\\text{in}}$ and $\\nabla u_{\\text{out}}, \\nabla p_{\\text{out}}$ denote the restrictions of the gradients to $\\Omega$ and $D \\setminus \\overline \\Omega$, respectively, and $n$ denotes the unit normal vector pointing out of $\\Omega$.\nNote that, when using a finite element discretization which does not resolve the interface such that the gradients of the discretized state and adjoint variable are continuous, i.e. $\\nabla u_{h, \\text{in}} = \\nabla u_{h, \\text{out}}$ and $\\nabla p_{h, \\text{in}} = \\nabla p_{h, \\text{out}}$, \\eqref{eq_Llambda_inout} becomes\n\\begin{align} \\label{eq_Llambdah}\n    L^\\lambda_h = (\\lambda_1 - \\lambda_2) \\nabla u_h \\cdot \\nabla p_h - 2(\\lambda_1 - \\lambda_2) (\\nabla u_h \\cdot n)(\\nabla p_h \\cdot n)\n\\end{align}\n\n\n\nWe now discretize the continuous shape derivative formula \\eqref{eq_dg_Hadamard} for the vector field $V^{(k)}$ that is obtained from the perturbation of the level set function $\\phi$ in (only) node ${\\mathbf x}_k$, $k \\in \\mathfrak S$ fixed. For that purpose we fix $\\phi \\in S_h^1(D)$ and the corresponding domain $\\Omega(\\phi)$. Note that we consider $V^{(k)}$ to be supported only on the discretized material interface $\\partial \\Omega(\\phi) \\cap D$. We begin with the case of the pure volume cost function by setting $c_2=0$.\n\n\\begin{proposition} \\label{prop_SD_vol}\n    Let $c_2=0$ and ${\\mathbf x}_k \\in \\mathfrak S$ fixed. Let $V^{(k)}$ the vector field that corresponds to a perturbation of the value of $\\phi$ at position ${\\mathbf x}_k$. Then\n    \\begin{align*}\n        dg(\\Omega(\\phi))(V^{(k)}) = c_1 \\sum_{l \\in C_k} d_ka_l\n    \\end{align*}\n    where $d_k a_l$ is as in \\eqref{eq::derivativeArea}.\n\n\\end{proposition}\n\n\\begin{proof}\n    For $c_2=0$ we also have $p=0$ and thus $L = c_1$, i.e., we are in the case of pure volume minimization. From \\eqref{eq_dg_Hadamard} we know that $dg(\\Omega(\\phi))(V^{(k)}) = c_1 \\int_{\\partial \\Omega(\\phi)} V^{(k)} \\cdot n \\; \\mbox dS_x$.\n    First of all, we note that the vector field $V^{(k)}$ corresponding to a perturbation of $\\phi$ at node $x_k$ is only nonzero in elements $\\tau_l$ for $l \\in C_k$ with $C_k$ as defined in \\eqref{eq::areaChangeSet}. Thus, the shape derivative reduces to\n    \\begin{align*}\n        dg(\\Omega(\\phi))(V^{(k)}) = c_1 \\sum_{l \\in C_k} \\int_{\\tau_l \\cap \\partial \\Omega(\\phi)} V^{(k)}\\cdot n \\; \\mbox dS_x.\n    \\end{align*}\n    We compute the vector field $V^{(k)}$ and normal vector $n$ explicitly depending on the cut situation. Recall the sets $I_{\\pos_k}^{A_+}, I_{\\pos_k}^{A_-}, I_{\\pos_k}^{B_+}, I_{\\pos_k}^{B_-}, I_{\\pos_k}^{C_+}, I_{\\pos_k}^{C_-}$ introduced in \\eqref{eq_def_Ixk_ABCpm}, see also Figure \\ref{fig::sets}. \n    \n    Given two points $\\pp$ and $\\qq$ and their respective level set values $a$ and $b$ of different sign, $ab<0$, we denote the root of the linear interpolating function by\n    \\begin{align*}\n        \\yy(\\pp, \\qq, a, b) = \\pp -  \\frac{a}{b-a}(\\qq - \\pp)\n    \\end{align*}\n    and note that $\\frac{d}{da} \\yy(\\pp, \\qq, a, b) =  -  \\frac{b}{(b-a)^2}(\\qq - \\pp)$.\n\n    We begin with Configuration A. For an element index $l \\in I_{\\pos_k}^{A_+}\\cup I_{\\pos_k}^{A_-}$, we denote the vertices of element $\\tau_l$ by ${\\mathbf x}_{l_1}, {\\mathbf x}_{l_2}, {\\mathbf x}_{l_3}$ and assume their enumeration in counter-clockwise order with ${\\mathbf x}_k = {\\mathbf x}_{l_1}$. The corresponding values of the given level set function $\\phi$ are denoted by $\\phi_{l_1}, \\phi_{l_2}, \\phi_{l_3}$, respectively. We parametrize the line connecting the two roots of the perturbed level set function along the edges by\n    \\begin{align*}\n        p^A[\\varepsilon](s) = (1-s) \\yy({\\mathbf x}_{l_1}, {\\mathbf x}_{l_2}, \\phi_{l_1}+\\varepsilon, \\phi_{l_2}) +  s \\yy({\\mathbf x}_{l_1}, {\\mathbf x}_{l_3}, \\phi_{l_1}+\\varepsilon,\\phi_{l_3})\n    \\end{align*}\n    and obtain the vector field corresponding to the perturbation of $\\phi_{k} = \\phi_{l_1}$ along the line $\\tau_l \\cap \\partial \\Omega(\\phi)$ as\n    \\begin{align*}\n        \\hat V^A(s) = \\frac{d}{d\\varepsilon} p^A[\\varepsilon](s)|_{\\varepsilon = 0} = (1-s) \\frac{- \\phi_{l_2}}{(\\phi_{l_2}-\\phi_{l_1})^2} ({\\mathbf x}_{l_2} - {\\mathbf x}_{l_1}) + s \\frac{- \\phi_{l_3}}{ (\\phi_{l_3} - \\phi_{l_1})^2} ({\\mathbf x}_{l_3}-{\\mathbf x}_{l_1}).\n    \\end{align*}\n    Introducing the notation $d_{ki,kj} := |\\yy({\\mathbf x}_{l_k}, {\\mathbf x}_{l_i}, \\phi_{l_k}, \\phi_{l_i})-\\yy({\\mathbf x}_{l_k}, {\\mathbf x}_{l_j}, \\phi_{l_k}, \\phi_{l_j}) |$ for the length of the interface in element $\\tau_l$, the normed tangential vector along $\\tau_l \\cap \\partial \\Omega(\\phi)$ and the normed normal vector pointing out of $\\Omega(\\phi)$ are given by\n    \\begin{align*}\n        \\hat t^A =& \\frac{\\yy({\\mathbf x}_{l_1}, {\\mathbf x}_{l_3}, \\phi_{l_1}, \\phi_{l_3})  - \\yy({\\mathbf x}_{l_1}, {\\mathbf x}_{l_2}, \\phi_{l_1}, \\phi_{l_2}) }{d_{13,12}} \\\\% \\quad \\mbox{ with }  d_{1,3} := |\\yy({\\mathbf x}_{l_1}, {\\mathbf x}_{l_3}, \\phi_{l_1}, \\phi_{l_3})-\\yy({\\mathbf x}_{l_1}, {\\mathbf x}_{l_2}, \\phi_{l_1}, \\phi_{l_2}) | \\\\\n        \\hat n^{A_+} =& \\frac{\\phi_{l_1}}{d_{13,12}} \\left( \\frac{1}{\\phi_{l_2} - \\phi_{l_1}} R ({\\mathbf x}_{l_2} - {\\mathbf x}_{l_1})- \\frac{1}{\\phi_{l_3} - \\phi_{l_1}} R ({\\mathbf x}_{l_3} - {\\mathbf x}_{l_1}) \\right)\n        =-\\hat n^{A_-} \n    \\end{align*}\n    where $R$ denotes a 90 degree counter-clockwise rotation matrix, $R = \\begin{pmatrix} 0& -1\\\\ 1 & 0 \\end{pmatrix}$. Noting that $({\\mathbf x}_{l_3} - {\\mathbf x}_{l_1})^\\top R ({\\mathbf x}_{l_2} - {\\mathbf x}_{l_1}) = |\\mbox{det}J_l| = - ({\\mathbf x}_{l_2} - {\\mathbf x}_{l_1})^\\top R ({\\mathbf x}_{l_3} - {\\mathbf x}_{l_1})$ and $({\\mathbf x}_{l_j} - {\\mathbf x}_{l_1})^\\top R ({\\mathbf x}_{l_j} - {\\mathbf x}_{l_1}) = 0$, $j=2,3$, we get\n    \\begin{align} \\label{eq_VAnA}\n        \\hat V^A(s) \\cdot \\hat n^{A_+} = -\\frac{|\\mbox{det} J_l| \\phi_{l_1}}{d_{13,12}} \\frac{\\phi_{l_2}(\\phi_{l_3}-\\phi_{l_1}) + s \\phi_{l_1}(\\phi_{l_2} - \\phi_{l_3})}{(\\phi_{l_2}-\\phi_{l_1})^2(\\phi_{l_3}-\\phi_{l_1})^2} = - \\hat V^A(s) \\cdot \\hat n^{A_-} .\n    \\end{align}\n    Finally, by elementary computation we obtain for $l \\in I_{\\pos_k}^{A_+}$\n    \\begin{align*}\n        \\int_{\\tau_l \\cap \\partial \\Omega(\\phi)} V^{(k)} \\cdot n \\; \\mbox dS_x =& d_{13,12} \\int_0^1 \\hat V^A(t) \\cdot \\hat n^{A_+} \\; \\mbox dt \\\\\n        =&| \\mbox{det} J_l| \\phi_{l_1} \\frac{-\\phi_{l_2}\\phi_{l_3} + \\frac12 \\phi_{l_1}(\\phi_{l_2}+ \\phi_{l_3})}{(\\phi_{l_2}-\\phi_{l_1})^2(\\phi_{l_3}-\\phi_{l_1})^2} \n    \\end{align*}\n    and the same formula with a different sign for $l \\in I_{\\pos_k}^{A_-}$. Proceeding analogously, we obtain for $l \\in I_{\\pos_k}^{B_+}$ and $l \\in I_{\\pos_k}^{C_+}$\n    \\begin{align}\n        \\hat V^B(s) \\cdot \\hat n^{B_+} =& \\frac{|\\mbox{det} J_l|}{d_{23,21}} \\frac{(1-s) \\phi_{l_2}^2 }{(\\phi_{l_2} - \\phi_{l_1})^2 (\\phi_{l_3} - \\phi_{l_2})} = - \\hat V^B(s) \\cdot \\hat n^{B_-}, \\\\\n        \\hat V^C(s) \\cdot \\hat n^{C_+} = & \\frac{|\\mbox{det} J_l|}{d_{31,32}} \\frac{(1-s) \\phi_{l_3}^2}{(\\phi_{l_3}-\\phi_{l_1})^2 (\\phi_{l_2}-\\phi_{l_3})} = -\\hat V^C(s) \\cdot \\hat n^{C_-},    \\label{eq_VCnC}\n    \\end{align}\n    and further\n    \\begin{align*}\n         \\int_{\\tau_l \\cap \\partial \\Omega(\\phi)} V^{(k)} \\cdot n \\; \\mbox dS_x=& \\frac{|\\mbox{det}J_l|}{2} \\frac{\\phi_{l_2}^2}{(\\phi_{l_2}-\\phi_{l_1})^2(\\phi_{l_3}-\\phi_{l_2})}, \\quad l \\in I_{\\pos_k}^{B_+}, \\\\\n        \\int_{\\tau_l \\cap \\partial \\Omega(\\phi)} V^{(k)} \\cdot n \\; \\mbox dS_x =& \\frac{|\\mbox{det}J_l|}{2} \\frac{-\\phi_{l_3}^2}{(\\phi_{l_3}-\\phi_{l_1})^2(\\phi_{l_3}-\\phi_{l_2})}, \\quad l \\in I_{\\pos_k}^{C_+},\n    \\end{align*}\n    respectively. Again, the formulas for $l \\in I_{\\pos_k}^{B_-}$ and $l \\in I_{\\pos_k}^{C_-}$ just differ by a different sign.\n    \n    Finally, comparing the computed values with the formulas of $d_k a_l$ \\eqref{eq::derivativeArea} yields the claimed result.\n\\end{proof}\n\nIn view of Proposition \\ref{prop_SD_vol}, the definition in \\eqref{eq::shapeDerivativeAlternative} and Remark \\ref{remark::area}, we see that, in the case $c_2=0$, it holds\n\\begin{align}\n    \\hat d g(\\Omega(\\phi))(V^{(k)}) =  \\frac{c_1 \\sum_{l\\in C_k} d_ka_l}{\\sum_{l\\in C_k} d_k \\tilde a_l} = - c_1,\n\\end{align}\nwhich is in alignment with the first term of the formula in \\eqref{eq::formulaShapeDerivative}.\n\n\nNext, we consider the general PDE-constrained case where $c_2>0$.\n\\begin{proposition} \\label{prop_SD_pde}\n Let $c_1 = 0$ and ${\\mathbf x}_k \\in \\mathfrak S$ fixed. Let $V^{(k)}$ the vector field that corresponds to a perturbation of the value of $\\phi$ at position ${\\mathbf x}_k$. Then\n \\begin{align}  \\label{eq_formula_dg_pde}\n  \\begin{aligned}\n    dg(\\Omega(\\phi))(V^{(k)}) =& \\; \\; \\;\\; \\, \\Delta \\lambda\n\t\\sum_{l \\in C_k}  \\mathbf p_{l}^\\top \\mathbf k_{0,l} \\mathbf u_{l} \\,d_k a_{l} \n\t+ \\Delta \\alpha \\sum_{l \\in C_k}  \\mathbf p_{l}^\\top d_k\\mathbf m^I_{l} \\,\\mathbf u_{l}\n\t  \\\\\n\t &  - \\Delta f \\sum_{l \\in C_k} \\mathbf p_{l}^\\top d_k\\mathbf f_l^I \n\t+c_2 \\Delta \\tilde\\alpha \\sum_{l \\in C_k}  (\\mathbf u_{l}-\\hat{\\mathbf u}_{l})^\\top d_k\\mathbf m^I_{l} \\,(\\mathbf u_{l}-\\hat{\\mathbf u}_{l}) \\\\\n\t& -2 \\Delta \\lambda \\sum_{l \\in C_k}(\\nabla u_h \\cdot n^I)|_{\\tau_l \\cap \\partial \\Omega(\\phi)}(\\nabla p_h \\cdot n)|_{\\tau_l \\cap \\partial \\Omega(\\phi)} d_ka_l,\n    \\end{aligned}\n \\end{align}\nwhere we use the same notation as in Theorem \\ref{thm_numTopShapeDer_formula}. In particular, $d_k \\mathbf m^I_l$ and $d_k \\mathbf f^I_l$ depend on the cut situation, $I \\in \\{A^+, A^-, B^+, B^-, C^+, C^-\\}$ and are given explicitly in Appendix \\ref{app::shapeDerivative}.\n\\end{proposition}\n\\begin{proof}\nLet an element index $l \\in C_k$ fixed and $\\mathbf u_l = [u_{l_1}, u_{l_2}, u_{l_3}]^\\top$, $\\mathbf p_l = [p_{l_1}, p_{l_2}, p_{l_3}]^\\top$ contain the nodal values of the finite element functions $u_h$ and $p_h$ corresponding to the three vertices ${\\mathbf x}_{l_1}$, ${\\mathbf x}_{l_2}$, ${\\mathbf x}_{l_3}$ of element $l$, respectively. Also here, the ordering is in counter-clockwise direction starting with ${\\mathbf x}_{l_1} = {\\mathbf x}_k$.\nWe compute the shape derivative \\eqref{eq_dg_Hadamard} with $L$ given in \\eqref{eq_L} after discretization (i.e. after replacing the functions $u$, $p$, $\\hat u$ by finite element approximations $u_h$, $p_h$, $\\hat u_h$). In particular, the term $L^\\lambda$ is approximated by \\eqref{eq_Llambdah}.\nDepending on how the material interface $\\partial \\Omega(\\phi)$ cuts through element $\\tau_l$, the normal component of the vector field $V^{(k)}$ along the line $\\tau_l \\cap \\partial \\Omega(\\phi)$ is given in \\eqref{eq_VAnA}--\\eqref{eq_VCnC}.\nFor and $I \\in \\{A^+, A^-, B^+, B^-, C^+, C^-\\}$, it can be seen by elementary yet tedious calculations that\n\\begin{align*}\n    \\int_{\\tau_l \\cap \\partial \\Omega(\\phi)} p_h(x) V^{I}(x) \\cdot n^{I} \\, \\mbox dS_x =& \\; \\mathbf p_l^\\top d_k \\mathbf f_l^I,\\\\\n    \\int_{\\tau_l \\cap \\partial \\Omega(\\phi)} u_h(x) p_h(x) V^{I}(x) \\cdot n^{I} \\, \\mbox dS_x =& \\; \\mathbf p_l^\\top d_k \\mathbf m^I_l \\mathbf u_l,\\\\\n    \\int_{\\tau_l \\cap \\partial \\Omega(\\phi)} (u_h(x)-\\hat u_h(x))^2 V^{I}(x) \\cdot n^{I} \\, \\mbox dS_x =& \\; (\\mathbf u_l - \\hat{\\mathbf u}_l)^\\top d_k \\mathbf m^I_l (\\mathbf u_l - \\hat{\\mathbf u}_l),\n\\end{align*}\nwith $d_k \\mathbf m^I_l$ and $d_k \\mathbf f^I_l$ as given in Appendix \\ref{app::shapeDerivative}.\nExamplarily, we illustrate the calculation for the second of these terms for the cut situation $I = A^+$, see Figure \\ref{fig::sets}(a). Let $u_{l,12}$ and $u_{l,13}$ denote the values of the linear function $u_h|_{\\tau_l}$ at the intersection of the interface $\\partial \\Omega(\\phi)$ with the edges that connect the point ${\\mathbf x}_{l_1}$ with ${\\mathbf x}_{l_2}$ and ${\\mathbf x}_{l_1}$ with ${\\mathbf x}_{l_3}$, respectively. Note the relations $u_{l,12} = \\frac{u_{l_2} \\phi_{l_1} - u_{l_1} \\phi_{l_2}}{\\phi_{l_1}-\\phi_{l_2}}$ and $u_{l,13} = \\frac{u_{l_3} \\phi_{l_1} - u_{l_1} \\phi_{l_3}}{\\phi_{l_1}-\\phi_{l_3}}$. Analogously we define the values $p_{l,12}$ and $p_{l,13}$. The function $u_h$ along the line $\\tau_l \\cap \\partial \\Omega(\\phi)$ can now be written as $\\hat u_h(s) = u_{l,12} + s (u_{l,13} - u_{l,12})$, $s \\in [0,1]$ and we get\n\\begin{align*}\n    \\int_{\\tau_l \\cap \\partial \\Omega(\\phi)}& u_h(x) p_h(x) V^{I}(x) \\cdot n^{I} \\, \\mbox dS_x \\\\\n    &= d_{13,12} \\int_0^1 (u_{l,12} + s (u_{l,13} - u_{l,12})) (p_{l,12} + s (p_{l,13} - p_{l,12})) \\hat V^{A^+}(s) \\cdot \\hat n^{A^+} \\; \\mbox ds \\\\\n    &= \\mathbf p_l^\\top d_k \\mathbf m^{A^+}_l \\mathbf u_l\n\\end{align*}\nwhere $d_{13,12} = |\\tau_l \\cap \\partial \\Omega(\\phi)|$. The last equality is obtained by plugging in \\eqref{eq_VAnA} and straightforward (yet tedious) calculation.\nFinally, since $u_h$ and $p_h$ are linear and the normal vector is constant on $\\tau_l \\cap \\partial \\Omega(\\phi)$, we see that $L_h^\\lambda$ is constant and, using Proposition \\ref{prop_SD_vol}, we obtain\n\\begin{align*}\n    \\int_{\\tau_l \\cap \\partial \\Omega(\\phi)} L_h^\\lambda(x) V^I(x)\\cdot n^I \\; \\mbox dS_x =& L_h^\\lambda(x) \\int_{\\tau_l \\cap \\partial \\Omega(\\phi)}  V^I(x)\\cdot n^I \\; \\mbox dS_x \\\\\n    =&\\Delta \\lambda  \\left( \\mathbf p_l^\\top \\mathbf k_{0,l} \\mathbf u_l -2 (\\nabla u_h \\cdot n^I)|_{\\tau_l \\cap \\partial \\Omega(\\phi)}(\\nabla p_h \\cdot n)|_{\\tau_l \\cap \\partial \\Omega(\\phi)} \\right) d_ka_l.\n\\end{align*}\n\\end{proof}\n\nCombining the findings of Propositions \\ref{prop_SD_vol} and \\ref{prop_SD_pde} and dividing by $d_k \\tilde a$ (defined in \\ref{remark::area}), we obtain the resulting formula for the alternative definition of the shape derivative as defined in \\eqref{eq::shapeDerivativeAlternative}\n\\begin{align} \\label{eq_dhatg_discr}\n    \\begin{aligned}\n    \\hat d &g(\\Omega(\\phi))(V^{(k)}) = -c_1 + \\frac{1}{d_k \\tilde a} \\Bigg( \\Delta \\lambda\n\\sum_{l \\in C_k}  \\mathbf p_{l}^\\top \\mathbf k_{0,l} \\mathbf u_{l} \\,d_k a_{l} \n+ \\Delta \\alpha \\sum_{l \\in C_k}  \\mathbf p_{l}^\\top d_k\\mathbf m^I_{l} \\,\\mathbf u_{l}  - \\Delta f \\sum_{l \\in C_k} \\mathbf p_{l}^\\top d_k\\mathbf f_l^I \n    \\\\\n    & +c_2 \\Delta \\tilde\\alpha \\sum_{l \\in C_k}  (\\mathbf u_{l}-\\hat{\\mathbf u}_{l})^\\top d_k\\mathbf m^I_{l} \\,(\\mathbf u_{l}-\\hat{\\mathbf u}_{l})\n    -2 \\Delta \\lambda \\sum_{l \\in C_k}(\\nabla u_h \\cdot n^I)|_{\\tau_l \\cap \\partial \\Omega(\\phi)}(\\nabla p_h \\cdot n)|_{\\tau_l \\cap \\partial \\Omega(\\phi)} d_ka_l \\Bigg).\n    \\end{aligned}\n\\end{align}\n\\begin{remark}\n    Note that \\eqref{eq_dhatg_discr} is obtained by discretizing the continuous shape derivative \\eqref{eq_dg_Hadamard}--\\eqref{eq_Llambda_inout}. We see immediately that \\eqref{eq_dhatg_discr} resembles the formula for the discrete {topological-shape } derivative for nodes ${\\mathbf x}_k \\in \\mathfrak S$ \\eqref{eq::formulaShapeDerivative}. The only difference is the occurence of the last term in \\eqref{eq_dhatg_discr}, which is not accounted for when performing the sensitivity analysis in the discrete setting.\n\n\nNote that this term stems from the presence of $\\partial T_t^{-1} \\partial T_t^{-T}$ in the matrix $A(t)$, which, in turn, originates from two applications of the chain rule, $(\\nabla \\varphi) \\circ T_t = \\partial T_t^{-\\top} \\nabla (\\varphi \\circ T_t)$. \nSimilarly as in the case of the topological derivative in Section \\ref{sec_connTD}, the reason for this discrepancy is the fact that, for the given discretization scheme, the gradients of the finite element basis functions are constant on each element and thus $(\\nabla \\varphi) \\circ T_t = \\nabla \\varphi$ for small enough shape perturbation parameter $t$.\n\n\\end{remark}\n\n\\begin{remark}\n    As a second, conceptual difference between the classical continuous shape derivative defined by \\eqref{eq::shapeDerivative} and our discrete counterpart \\eqref{eq::formulaShapeDerivative}, we recall that in the definition \\eqref{eq::topDerivative}\n    we divided by the change of volume also in the case ${\\mathbf x}_k \\in \\mathfrak S$.\n\\end{remark}\n\n \n\n\\subsection{Classical topological derivative}\\label{sec::contiTopDerivative}\nLet $\\omega \\subset \\mathbb R^d$ with $0 \\in \\omega$.\nFor a point $z \\in \\Omega \\cup (D \\setminus \\overline \\Omega)$, let $\\omega_\\varepsilon := z + \\varepsilon \\omega$ denote a perturbation of the domain around $z$ of (small enough) size $\\varepsilon$ and of shape $\\omega$.\nThe continuous topological derivative of the shape function $\\redObjectiveFunction=\\redObjectiveFunction(\\Omega)$ is defined by \n\\begin{equation} \\label{eq_defTD}\nd\\redObjectiveFunction(\\Omega)(z) = \\begin{cases}\n\\lim_{\\varepsilon\\searrow 0}\\frac{\\redObjectiveFunction(\\Omega\\cup\\omega_\\varepsilon)-\\redObjectiveFunction(\\Omega)}{|\\omega_\\varepsilon|} \\quad \\text{for } z \\in D \\setminus \\overline \\Omega, \\\\ \n\\lim_{\\varepsilon\\searrow 0}\\frac{\\redObjectiveFunction(\\Omega\\setminus\\bar\\omega_\\varepsilon)-\\redObjectiveFunction(\\Omega)}{|\\omega_\\varepsilon|} \\quad \\text{for } z \\in \\Omega.\n\\end{cases}\n\\end{equation}\nNote that the topological derivative is not defined for points $z \\in \\partial \\Omega$ on the material interface.\nFor problem \\eqref{eq::ContinuousProblem} we obtain for $z \\in D \\setminus \\overline \\Omega$\n\\begin{align}\n    \\begin{aligned}\nd\\redObjectiveFunction_2(\\Omega)(z) =&c_1+c_2(\\tilde \\alpha_1 - \\tilde \\alpha_2)(u(z)-\\hat u(z))^2 \\\\\n&+  2\\lambda_2\\frac{\\lambda_1-\\lambda_2}{\\lambda_1+\\lambda_2}\\nabla u(z) \\cdot \\nabla p(z) +(\\alpha_1-\\alpha_2)u(z)p(z)- ( f_1 - f_2)p(z), \n    \\end{aligned}\n\\end{align} \nwhereas for  $z \\in \\Omega$\n\\begin{align}\n\\begin{aligned}\nd\\redObjectiveFunction_2(\\Omega)(z) =&-c_1+c_2 (\\tilde \\alpha_2 - \\tilde \\alpha_1)(u(z)-\\hat u(z))^2 \\\\\n&+ 2\\lambda_1\\frac{\\lambda_2-\\lambda_1}{\\lambda_1+\\lambda_2}\\nabla u(z) \\cdot \\nabla p(z) +(\\alpha_2-\\alpha_1)u(z)p(z)- ( f_2 - f_1)p(z),\n\\end{aligned}\n\\end{align}\nsee, e.g. \\cite{Amstutz_2006aa}.\n\n\\subsection{Classical shape derivative}\\label{sec::contiDerivative}\nWe recall the definition of the classical shape derivative as well as its formula for our model problem \\eqref{eq::ContinuousObjective}--\\eqref{eq::ContinuousProblem}. Given an admissible shape $\\Omega \\in \\mathcal A$ and a smooth vector field $V \\in C_c^\\infty(D)$ that is compactly supported in $D$, we define the perturbed domain\n\\begin{equation}\\label{eq::pertDomain}\n\\Omega_t = (\\text{id}+tV)(\\Omega),\n\\end{equation}\nfor a small perturbation parameter $t > 0$ where $\\text{id}:\\mathbb R^d \\rightarrow \\mathbb R^d$ denotes the identity operator. The classical shape derivative of $\\redObjectiveFunction$ at $\\Omega$ with respect to $V$ is then given by\n\\begin{equation}\\label{eq::shapeDerivative}\nd\\redObjectiveFunction(\\Omega)(V) = \\lim_{t\\searrow 0} \\frac{\\redObjectiveFunction(\\Omega_t)-\\redObjectiveFunction(\\Omega)}{t}\n\\end{equation}\nif this limit exists and the mapping $V \\mapsto d \\redObjectiveFunction(\\Omega)(V)$ is linear and continuous.\nUnder suitable assumptions it can be shown that this shape derivative admits the tensor representation\n\\begin{equation}\\label{eq::shapeDerivative_tensor}\nd\\redObjectiveFunction(\\Omega)(V) \n=\\int_D \\mathcal S_1^\\Omega : \\partial V + \\mathcal S_0^\\Omega \\cdot V dx,\n\\end{equation}\nfor some tensors $\\mathcal S_0^\\Omega \\in L^1(D, \\mathbb R^{d} )$, $\\mathcal S_1^\\Omega \\in L^1(D, \\mathbb R^{d\\times d} )$ \\cite{LaurainSturm2016}. Here, $\\partial V$ denotes the Jacobian of the vector field $V$. The structure theorem of Hadamard-Zol\\'esio \\cite[pp. 480-481]{DZ2} states that under certain smoothness assumptions the shape derivative of a shape function $\\redObjectiveFunction$ with respect to a vector field $V$ can always be written as an integral over the boundary of a scalar function $L$ times the normal component of $V$, i.e.,\n\\begin{align}\\label{eq::shapeDerivative_boundary}\nd\\redObjectiveFunction(\\Omega)(V) = \\int_{\\partial \\Omega} L \\,(V \\cdot n) \\; \\mbox ds\n\\end{align}\nwhere $n$ denotes the unit normal vector pointing out of $\\Omega$. For problem \\eqref{eq::ContinuousProblem} one obtains \\cite{LaurainSturm2016}\n\\begin{align} \\label{eq_S1}\n\\mathcal S_1^\\Omega =& (c_1 \\chi_\\Omega + c_2 \\tilde \\alpha_\\Omega (u-\\hat u)^2 + \\lambda_\\Omega \\nabla u \\cdot \\nabla p + \\alpha_\\Omega u p - f_\\Omega p) I - \\lambda_\\Omega \\nabla u \\otimes \\nabla p - \\lambda_\\Omega \\nabla p \\otimes \\nabla u, \\\\\n\\mathcal S_0^\\Omega =& -2 \\tilde \\alpha_\\Omega (u-\\hat u) \\nabla \\hat u \\label{eq_S0}\n\\end{align}\nwhere $I \\in \\mathbb R^{d,d}$ denotes the identity matrix, and\n\\begin{align*}\nL =& [(\\mathcal S_1^{\\Omega,\\text{in}} - \\mathcal S_1^{\\Omega,\\text{out}}) n ]\\cdot n \\\\\n=& c_1 + c_2 (\\tilde \\alpha_1- \\tilde \\alpha_2) (u-\\hat u)^2 + (\\alpha_1-\\alpha_2) u p - (f_1 - f_2) p  \\\\\n&+ (\\lambda_1- \\lambda_2)  (\\nabla u \\cdot \\tau)(\\nabla p \\cdot\\tau) -  \\left(\\frac{1}{\\lambda_1} - \\frac{1}{\\lambda_2}\\right)( \\lambda_\\Omega \\nabla u \\cdot n)(  \\lambda_\\Omega \\nabla p \\cdot n).\n\\end{align*}\nHere, $\\mathcal S_1^{\\Omega,\\text{in}}$ and $\\mathcal S_1^{\\Omega,\\text{out}}$ denote the restrictions of the tensor $\\mathcal S_1^{\\Omega}$ to $\\Omega$ and $D \\setminus \\Omega$, respectively. Furthermore, for two column vectors $a, b \\in \\mathbb R^d$, $a \\otimes b = a b^\\top \\in \\mathbb R^{d \\times d}$ denotes their outer product, $\\tau$ denotes the tangential vector and $p \\in H^1_0(D)$ is the solution to the adjoint equation\n\\begin{align*}\n    \\int_D \\lambda_\\Omega \\nabla v \\cdot \\nabla p + \\alpha_\\Omega v p \\; \\mbox dx = - 2c_2 \\int_D \\tilde \\alpha_\\Omega (u - \\hat u)v \\; \\mbox dx \\quad \\mbox{for all } v \\in H^1_0(D).\n\\end{align*}\n\n\n\n\nMoreover, motivated by the definition of the topological derivative \\eqref{eq_defTD} with the volume of the difference of the perturbed and unperturbed domains in the denominator, we introduce the alternative definition of a shape derivative\n\\begin{align} \\label{eq::shapeDerivativeAlternative}\n\\hat d \\redObjectiveFunction(\\Omega)(V) = \\underset{t \\searrow 0}{\\mbox{lim }} \\frac{\\redObjectiveFunction(\\Omega_t)-\\redObjectiveFunction(\\Omega)}{|\\Omega_t \\triangle \\Omega|},\n\\end{align}\nwith the symmetric difference of two sets $A \\triangle B := (A \\setminus \\overline B) \\cup (B \\setminus \\overline A)$. Note that the volume of the symmetric difference in \\eqref{eq::shapeDerivativeAlternative} can be written as\n\\begin{equation}\\label{eq::diffSets}\n|\\Omega_t \\triangle \\Omega| = \\int_D |\\chi_{\\Omega_t} - \\chi_{\\Omega}| \\;\\mathrm{d} {\\mathbf x}.\n\\end{equation}\n\n\n\\begin{lemma} \\label{LEM_SYMDIF}\n    Let $\\Omega$ and $V$ smooth. It holds\n    \\begin{align*}\n        \\underset{t \\searrow 0}{\\mbox{lim }}\\frac1t \\left \\lvert \\Omega_t \\triangle \\Omega \\right \\rvert = \\int_{\\partial \\Omega} |V \\cdot n| \\; \\mbox dS_x.\n    \\end{align*}\n\\end{lemma}\n\nThe proof is given in Appendix \\ref{app::proofSymDif}. \nFrom Lemma \\ref{LEM_SYMDIF}, we immediately obtain the following relation between $d \\redObjectiveFunction(\\Omega)(V)$ and $\\hat d \\redObjectiveFunction(\\Omega)(V)$.\n\n\\begin{corollary}\nSuppose that $\\redObjectiveFunction$ is shape differentiable at $\\Omega$ and that $\\Omega$ and $V$ are smooth and $\\int_{\\partial \\Omega} |V \\cdot n| \\; dS_x > 0$. Then it holds\n\\begin{align}\n\\hat d \\redObjectiveFunction(\\Omega)(V) = \\frac{ d \\redObjectiveFunction(\\Omega)(V)}{ \\int_{\\partial \\Omega} |V \\cdot n| \\; dS_x}.\n\\end{align}\n\\end{corollary}\n\\begin{proof}\n\tThis follows immediately from the definition of $\\hat d \\redObjectiveFunction(\\Omega)(V)$ by Lemma \\ref{LEM_SYMDIF} since\n\t\\begin{align}\n\t\\hat d \\redObjectiveFunction(\\Omega)(V) =  \\underset{t \\searrow 0}{\\mbox{lim }}\\frac{\\redObjectiveFunction(\\Omega_t)-\\redObjectiveFunction(\\Omega)}{|\\Omega_t \\triangle \\Omega|} =  \\frac{\\underset{t \\searrow 0}{\\mbox{lim }}\\frac{\\redObjectiveFunction(\\Omega_t)-\\redObjectiveFunction(\\Omega)}{t}}{\\underset{t \\searrow 0}{\\mbox{lim }}\\frac{|\\Omega_t \\triangle \\Omega|}{t}} =  \\frac{ d \\redObjectiveFunction(\\Omega)(V)}{\\int_{\\partial \\Omega} |V \\cdot n| \\; dS_x}. \n\t\\end{align}\t\n\t\n\\end{proof}\n\\begin{remark}The condition $\\int_{\\Omega} \\mbox{div}(V) \\; dx \\neq 0$ is a sufficient condition for $\\int_{\\partial \\Omega} |V \\cdot n| \\; dS_x > 0$, since\n\\begin{equation}\n0 < \\left|\\int_{\\Omega} \\mbox{div}(V) \\; dx \\right| = \\left|\\int_{\\partial \\Omega} V \\cdot n \\; dS_x \\right|< \\int_{\\partial \\Omega} |V \\cdot n| \\; dS_x.\n\\end{equation}\t\n\\end{remark}\n\n\n\n\\subsection{The continuous {topological-shape } derivative}\nHere and in the following, we assume that the domain $\\Omega$ is described by a level-set function $\\phi: D \\rightarrow \\mathbb R$ via\n\\begin{subequations}\n\\begin{align}\n\t\\phi({\\mathbf x}) < 0 &\\Longleftrightarrow {\\mathbf x} \\in \\Omega, \\label{eq::levelSet1} \\\\\n\t\\phi({\\mathbf x}) > 0 &\\Longleftrightarrow {\\mathbf x} \\in D \\setminus \\overline{\\Omega} \\label{eq::levelSet2}\\\\\n\t\\phi({\\mathbf x}) = 0 &\\Longleftrightarrow {\\mathbf x} \\in \\partial \\Omega \\cap D. \\label{eq::levelSet3}\n\\end{align}\n\\end{subequations}\nFor given $\\phi$, let $\\Omega(\\phi)$ denote the unique domain defined by \\eqref{eq::levelSet1}--\\eqref{eq::levelSet3}.\nIn this section, in contrast to the setting in Section \\ref{sec::contiDerivative}, we perturb $\\Omega$ indirectly by perturbing $\\phi$ such that\n$\\phi_\\varepsilon = \\operator_\\varepsilon \\phi$\nfor some operator $\\operator_\\varepsilon: C^0(D)\\rightarrow C^0(D)$ depending on $\\varepsilon \\geq 0$ with the property $\\Omega(\\operator_0 \\phi) = \\Omega(\\phi)$.\nLater on, in the discrete setting, we will distinguish between two different types of perturbation operators $\\operator_\\varepsilon$ corresponding to shape or topological perturbations of~$\\Omega$.\n\nLet, from now on, $\\redPhiObjectiveFunction(\\phi):= \\redObjectiveFunction(\\Omega(\\phi))$ denote the reduced cost function as a function of the level set function $\\phi$.\n This way, a continuous {topological-shape } derivative can be defined as \n\\begin{equation}\\label{eq::topologicalShapeDerivative}\n\td\\redPhiObjectiveFunction(\\phi) = \\lim_{\\varepsilon\\searrow 0}\\frac{\\redPhiObjectiveFunction(\\phi_\\varepsilon)-\\redPhiObjectiveFunction(\\phi)}{|\\Omega(\\phi_\\varepsilon) \\triangle \\Omega(\\phi)|}.\n\\end{equation}\nNote that this sensitivity depends on the choice of the perturbation operator $\\operator_\\varepsilon$, which can represent either a shape perturbation or a topological perturbation.\nWe will mostly be concerned with its discrete counterpart, which will be introduced in Section \\ref{sec::numTopShapeDer}.\nNote that, in the case of shape perturbations, due to the scaling $|\\Omega(\\phi_\\varepsilon)\\triangle \\Omega(\\phi)|$ instead of $\\varepsilon$ in the denominator the shape derivative is modified and does not correspond to \\eqref{eq::shapeDerivative} but rather to \\eqref{eq::shapeDerivativeAlternative}.\n\n\n\\paragraph{Relation to literature}\n\n\n\n\\newcommand{G\\^{a}teaux }{G\\^{a}teaux }\nThe sensitivity of shape functions with respect to perturbations of a level set function (representing a shape) was investigated in \\cite{laurain2018analyzing} for the case without PDE constraints. There, the author considers smooth level set functions and rigorously computes the G\\^{a}teaux  (semi-)derivative in the direction of a smooth perturbation of the level set function, both for the case of shape and topological perturbations. In the case of shape perturbations, it is shown that the G\\^{a}teaux  derivative coincides with the shape derivative \\eqref{eq::shapeDerivative} with respect to a suitably chosen vector field. On the other hand, a resemblance between the notions of G\\^{a}teaux  derivative and topological derivative is shown, yet the G\\^{a}teaux  derivative may vanish or not exist in cases where the topological derivative is finite.\nEvidently, this discrepancy results from the fact that the denominator in the definition of the G\\^{a}teaux  derivative is always of order one whereas it is of the order of the space dimension in the topological derivative.\n\nWhile the analysis for shape and topological perturbations is carried out separately in \\cite{laurain2018analyzing}, a more unified approach is followed in \\cite{delfour2018topological,Delfour_engcomp_2022}. In these publications, the idea is to consider sensitivities with respect to domain perturbations that are obtained by the dilation of lower-dimensional objects. Here, given a set $E \\subset \\mathbb R^d$ of dimension $k \\leq d$, the dilated set of radius $r$ is given by $E_r = \\{x \\in \\mathbb R^d: d_E(x) \\leq r\\}$ where $d_E(x)$ denotes a distance of a point $x$ to a set $E$. For instance, when $E$ is chosen as a single point, the dilated set is just a ball of radius $r$ around that point and performing a sensitivity analysis with respect to the volume of the dilated object leads to the topological derivative. On the other hand, when $E$ is chosen as the boundary of a domain, $E_r$ can be defined using a signed distance function and corresponds to a uniform expansion of the domain. Then, a similar procedure leads to the shape derivative with respect to a uniform expansion in normal direction  (i.e. $V = n$ in \\eqref{eq::pertDomain}--\\eqref{eq::shapeDerivative}).\nIn \\cite{delfour2018topological}, a sensitivity analysis for various choices of $E$ is carried out with respect to the volume of the perturbation. We note, however, that arbitrary shape perturbations are not covered and would require an extension of the theory. Comparing \\cite{laurain2018analyzing} and \\cite{delfour2018topological}, we observe that in the former paper only smooth perturbations of a level set function are admissible whereas, in the latter approach, domain perturbations by dilations can be interpreted as perturbations of level set functions by a (non-smooth) distance function.\n\nFinally, we mention \\cite{bernland2018acoustic} where a domain is represented by a discretized level set function and a shape sensitivity analysis is carried out with respect to a perturbation of the level set values close to the boundary. This procedure can be interpreted as an application of the idea of dilation to discretized shape optimization problems.\nAs the authors point out, this kind of shape sensitivity analysis is more natural compared to the standard approach based on domain transformations when employed in a level-set framework; an observation also made in \\cite[Sec. 3]{laurain2018analyzing}. The authors show numerical results for the shape optimization of an acoustic horn, but do not consider topological perturbations in this work.\n\n\\medskip\n\nAs it can be seen from \\eqref{eq::topologicalShapeDerivative}, our approach is related to the dilation concept since we also consider the sensitivity with respect to the volume of the domain perturbation $\\Omega_t \\triangle \\Omega$. In the following, we will investigate the {topological-shape } derivative in a discretized setting. Similarly to \\cite{bernland2018acoustic}, we will consider shape sensitivity analysis with respect to level set values on mesh nodes close to the boundary. Moreover, we will also be able to deal with topological perturbations and treat shape and topological updates in a unified way by a discretized version of \\eqref{eq::topologicalShapeDerivative}, called the numerical {topological-shape } derivative.\n\n\n\n\n\n\n\n\\subsection{Computation of the numerical {topological-shape } derivative for the area cost functional}\nBefore we compute the numerical {topological-shape } derivative \\eqref{eq::topDerivative} for the full model problem \\eqref{eq::ContinuousProblem}, we consider the case of the pure volume cost function and neglect the PDE constraint, i.e., we set $c_1 = 1, c_2 =0$ in \\eqref{eq::ContinuousObjective}.\n\nFor that purpose, we investigate\n\\begin{equation}\\label{eq::areaChange}\n\\delta_{k,\\eps} a  = |\\Omega(\\discreteOperator\\phi)|-|\\Omega(\\phi)|,\n\\end{equation}\nfor $\\discreteOperator \\in \\{ T^{-\\rightarrow+}_{k,\\varepsilon},\\,T^{+\\rightarrow-}_{k,\\varepsilon},\\,\\S \\}$. \nWe note that for the computation of \\eqref{eq::areaChange} only those \"cut elements\" are relevant which have a node ${\\mathbf x}_k$, \\textit{i.e.}\\,\n\\begin{equation}\\label{eq::areaChangeB}\n\\delta_{k,\\eps} a = {\\sum_{l\\in C_k} \n\\delta_{k,\\varepsilon}a_l} \t\\quad \\mbox{with }  \\delta_{k,\\varepsilon}a_l :=\n\\int_{\\tau_l} H(-\\discreteOperator\\phi) - H(-\\phi)  \\;\\mathrm{d} {\\mathbf x},\n\\end{equation}\nwhere $H(x)$ denotes the Heaviside step function and\n\\begin{equation}\\label{eq::areaChangeSet}\nC_k =  \\{l \\in I_{{\\mathbf x}_k} :  \\tau_l \\cap \\partial \\Omega(\\discreteOperator\\phi) \\neq \\emptyset \\}\n\\end{equation}\n is the set of all indices of elements adjacent to ${\\mathbf x}_k$ which are intersected by the perturbed interface. \\color{black}\n Note that, for $\\varepsilon>0$ small enough, $C_k$ does not depend on the concrete value of $\\varepsilon$. \\color{black}\nFor an element $\\tau_l$ with $l \\in I_{{\\mathbf x}_k}$ we denote the three vertices in counter-clockwise orientation by ${\\mathbf x}_{l_1}$, ${\\mathbf x}_{l_2}$, ${\\mathbf x}_{l_3}$ and assume that ${\\mathbf x}_k = {\\mathbf x}_{l_1}$. Moreover we denote $ \\phi_{l_j} := \\phi({\\mathbf x}_{l_j})$ and $\\tilde \\phi_{l_j} := \\discreteOperator\\phi({\\mathbf x}_{l_j})$ for $j=1,2,3$ and small enough $\\varepsilon$.\nIn the following, we will be interested in \n\\begin{align}\n    d_k a := \\sum_{l \\in C_k} d_k a_l \\quad \\mbox{ with } d_k a_l := \\underset{\\varepsilon \\searrow 0}{\\mbox{lim }} \\frac{\\delta_{k,\\varepsilon} a_l}{\\varepsilon^o}.\n\\end{align}\nwith $\\delta_{k,\\varepsilon} a_l$ defined in \\eqref{eq::areaChange}.\nWe consider six different sets (see Figure \\ref{fig::sets} for an illustration)\n\\begin{align}\\label{eq_def_Ixk_ABCpm}\n    \\begin{aligned}\nI_{{\\mathbf x}_k}^{A_+} = & \\{ l \\in I_{{\\mathbf x}_k}: \\tilde\\phi_{l_1} > 0, \\tilde\\phi_{l_2} < 0, \\tilde\\phi_{l_3} < 0 \\}, &&&\nI_{{\\mathbf x}_k}^{A_-} = & \\{ l \\in I_{{\\mathbf x}_k}: \\tilde\\phi_{l_1} < 0, \\tilde\\phi_{l_2} > 0, \\tilde\\phi_{l_3} > 0 \\}, \\\\\nI_{{\\mathbf x}_k}^{B_+} =& \\{ l \\in I_{{\\mathbf x}_k}: \\tilde\\phi_{l_1} < 0, \\tilde\\phi_{l_2} > 0, \\tilde\\phi_{l_3} < 0 \\}, &&&\nI_{{\\mathbf x}_k}^{B_-} =& \\{ l \\in I_{{\\mathbf x}_k}: \\tilde\\phi_{l_1} > 0, \\tilde\\phi_{l_2} < 0, \\tilde\\phi_{l_3} > 0 \\}, \\\\\nI_{{\\mathbf x}_k}^{C_+} =& \\{ l \\in I_{{\\mathbf x}_k}: \\tilde\\phi_{l_1} < 0, \\tilde\\phi_{l_2} < 0, \\tilde\\phi_{l_3} > 0 \\}, &&&\nI_{{\\mathbf x}_k}^{C_-} =& \\{ l \\in I_{{\\mathbf x}_k}: \\tilde\\phi_{l_1} > 0, \\tilde\\phi_{l_2} > 0, \\tilde\\phi_{l_3} < 0 \\},\n    \\end{aligned}\n\\end{align}\nsuch that \n\\begin{align*}\nC_k = I_{{\\mathbf x}_k}^{A_+} \\cup I_{{\\mathbf x}_k}^{A_-} \\cup I_{{\\mathbf x}_k}^{B_+} \\cup I_{{\\mathbf x}_k}^{B_-} \\cup I_{{\\mathbf x}_k}^{C_+} \\cup I_{{\\mathbf x}_k}^{C_-}\n\\end{align*}\nwith a direct sum on the right hand side. We can thus split the sum in \\eqref{eq::areaChangeB} into six parts,\n\\begin{align*}\n\\delta_{k,\\eps} a = \n\\mathcal{I}_{I_{{\\mathbf x}_k}^{A_+}} + \\mathcal{I}_{I_{{\\mathbf x}_k}^{A_-}}+\n\\mathcal{I}_{I_{{\\mathbf x}_k}^{B_+}} + \\mathcal{I}_{I_{{\\mathbf x}_k}^{B_-}}+\n\\mathcal{I}_{I_{{\\mathbf x}_k}^{C_+}} + \\mathcal{I}_{I_{{\\mathbf x}_k}^{C_-}}.\n\\end{align*}\n\\begin{figure}\n\t\\begin{subfigure}[t]{0.33\\textwidth}\n\t\\includegraphics{pic_setCk8.pdf}\n\t\\subcaption{Triangle is in $I_{{\\mathbf x}_k}^{A_+}$}\t\n\t\\end{subfigure}\\hfill\n\t\\begin{subfigure}[t]{0.33\\textwidth}\n\t\\includegraphics{pic_setCk4.pdf}\n\t\\subcaption{Triangle is in $I_{{\\mathbf x}_k}^{B_+}$}\n\t\\end{subfigure}\\hfill\n\t\\begin{subfigure}[t]{0.33\\textwidth}\n\t\\includegraphics{pic_setCk7.pdf}\n\t\\subcaption{Triangle is in $I_{{\\mathbf x}_k}^{C_+}$}\n\t\\end{subfigure}\n\n\t\\begin{subfigure}[t]{0.33\\textwidth}\n\t\\includegraphics{pic_setCk9.pdf}\n\t\\subcaption{Triangle is in $I_{{\\mathbf x}_k}^{A_-}$}\t\n\t\\end{subfigure}\\hfill\n\t\\begin{subfigure}[t]{0.33\\textwidth}\n\t\t\\includegraphics{pic_setCk5.pdf}\n\t\t\\subcaption{Triangle is in $I_{{\\mathbf x}_k}^{B_-}$}\t\t\n\t\\end{subfigure}\\hfill\n\t\\begin{subfigure}[t]{0.33\\textwidth}\n\t\\includegraphics{pic_setCk6.pdf}\n\t\\subcaption{Triangle is in $I_{{\\mathbf x}_k}^{C_-}$}\t\t\n\t\\end{subfigure}\n\t\\caption{Illustration of the sets $I_{{\\mathbf x}_k}^{A_+},~I_{{\\mathbf x}_k}^{A_-},~I_{{\\mathbf x}_k}^{B_+},~I_{{\\mathbf x}_k}^{B_-},~I_{{\\mathbf x}_k}^{C_+} ,~ I_{{\\mathbf x}_k}^{C_-}$. The nodal values of the level-set functions are indicated by $-$, $+$. The interface is drawn in red.}\n\t\\label{fig::sets}\n\\end{figure}\n\\begin{figure}\\centering\n\\begin{subfigure}[t]{0.49\\textwidth}\\centering\n\t\\includegraphics{pic_setCk1.pdf}\n\t\\subcaption{Triangles are in $I_{{\\mathbf x}_k}^{A_+}$}\n\t\\label{fig::IAtopA}\t\n\\end{subfigure}\n\\begin{subfigure}[t]{0.49\\textwidth}\\centering\n\t\\includegraphics{pic_setCk2.pdf}\n\t\\subcaption{Triangles are in $I_{{\\mathbf x}_k}^{A_-}$}\n\t\\label{fig::IAtopB}\t\t\n\\end{subfigure}\n\\caption{Illustration of $I_{{\\mathbf x}_k}^{A_+}$ and $I_{{\\mathbf x}_k}^{A_-}$ in the case of topological perturbations.}\n\\label{fig::topDir}\n\\end{figure}\n\\paragraph{Configuration A}\nFor $l \\in I_{{\\mathbf x}_k}^{A_+}$ we have\n\\begin{equation*}\n\\begin{aligned}\n\\int_{\\tau_{l}} H(-\\discreteOperator\\phi({\\mathbf x})) \\;\\mathrm{d} {\\mathbf x} &= \n \\frac{\\left|{\\textup{\\text{det}}}J_l\\right|}{2} \\left(1- \\int_{\\xi_1=0}^{\\ell_1} \\int_{\\xi_2=0}^{\\ell_2\\left(1-\\frac{\\xi_1}{\\ell_1}\\right)}  \\,d\\xi_2d\\xi_1 \\right) \\\\\n&=   \\frac{\\left|{\\textup{\\text{det}}}J_l\\right|}{2}(1-\\ell_1\\ell_2),\n\\end{aligned}\n\\end{equation*}\nwith\n\\begin{align*}\n\t\\ell_1 = \\frac{\\phi_{l_1}+\\varepsilon}{\\phi_{l_1}+\\varepsilon-\\phi_{l_2}}, \\qquad\n\t\\ell_2 = \\frac{\\phi_{l_1}+\\varepsilon}{\\phi_{l_1}+\\varepsilon-\\phi_{l_3}}.\n\\end{align*}\nTherefore,\n\\begin{equation}\n\\begin{aligned}\\label{eq::intAplus}\n\\mathcal{I}_{I_{{\\mathbf x}_k}^{A_+}}\n&= \\sum_{l \\in I_{{\\mathbf x}_k}^{A_+}} \\frac{\\left|{\\textup{\\text{det}}}J_l\\right|}2 \\left(-\\frac{(\\phi_{l_1}+\\varepsilon)^2}{(\\phi_{l_1}+\\varepsilon-\\phi_{l_2})(\\phi_{l_1}+\\varepsilon-\\phi_{l_3})}+\\frac{\\phi_{l_1}^2}{(\\phi_{l_1}-\\phi_{l_2})(\\phi_{l_1}-\\phi_{l_3})} \\right)\\\\\n&= \\sum_{l \\in I_{{\\mathbf x}_k}^{A_+}} \\frac{\\left|{\\textup{\\text{det}}}J_l\\right|}2 \\left(\\frac{\\varepsilon \\phi_{l_1} \\left[\\phi_{l_1}(\\phi_{l_2} + \\phi_{l_3})-2\\phi_{l_2}\\phi_{l_3}  \\right]+\\varepsilon^2\\left[\\phi_{l_1}(\\phi_{l_2} + \\phi_{l_3})-\\phi_{l_2}\\phi_{l_3} \\right]}{(\\phi_{l_1} - \\phi_{l_2})^2(\\phi_{l_1} - \\phi_{l_3})^2 + O(\\varepsilon)} \\right).\n\\end{aligned}\n\\end{equation}\nFor ${\\mathbf x}_k \\in \\mathfrak T^-$ and $\\discreteOperator = T^{-\\rightarrow+}_{k,\\varepsilon}$, it holds $I_{{\\mathbf x}_k}=I_{{\\mathbf x}_k}^{A_+}$ (see Figure \\ref{fig::IAtopA}) and we have to consider \\eqref{eq::intAplus} with $\\phi_{l_1} = 0$. Thus, we obtain\n\\begin{align\n\\mathcal{I}_{I_{{\\mathbf x}_k}^{A_+}}= -\n\\sum_{l \\in I_{{\\mathbf x}_k}^{A_+}}\\frac{\\varepsilon^2\\left|{\\textup{\\text{det}}}J_l\\right|}2 \\frac{\\phi_{l_2}\\phi_{l_3} }{ \\phi_{l_2}^2\\phi_{l_3}^2 + O(\\varepsilon)},\n\\end{align}\nand conclude for this case   \n\\begin{equation}\\label{eq::areaChangeTopAplus}\n d_{k}a = \\lim_{\\varepsilon\\searrow 0}\\frac{\\delta_{k,\\eps} a}{\\varepsilon^2} =- \\sum_{l \\in I_{{\\mathbf x}_k}^{A_+}}\\frac{\\left|{\\textup{\\text{det}}}J_l\\right|}{2\\phi_{l_2}\\phi_{l_3}}.\n\\end{equation}\nMoreover, for ${\\mathbf x}_k \\in \\mathfrak T^+$ and $\\discreteOperator = T^{+\\rightarrow-}_{k,\\varepsilon}$, it holds $I_{{\\mathbf x}_k} = I_{{\\mathbf x}_k}^{A_-}$ (see Figure \\ref{fig::IAtopB}) and we have for $l \\in I_{{\\mathbf x}_k}^{A_-}$,\n\\begin{align*}\n\\int_{\\tau_{l}} H(-\\discreteOperator\\phi({\\mathbf x})) \\;\\mathrm{d} {\\mathbf x} &= \n\\frac{\\left|{\\textup{\\text{det}}}J_l\\right|}{2} \\int_{\\xi_1=0}^{\\ell_1} \\int_{\\xi_2=0}^{\\ell_2\\left(1-\\frac{\\xi_1}{\\ell_1}\\right)}  \\,d\\xi_2d\\xi_1  =   \\frac{\\left|{\\textup{\\text{det}}}J_l\\right|}{2}\\ell_1\\ell_2,\n\\end{align*}\nwhich leads to \n\\begin{equation}\n\\begin{aligned}\\label{eq::intAminus}\n\\mathcal{I}_{I_{{\\mathbf x}_k}^{A_-}}\n&= -\\sum_{l \\in I_{{\\mathbf x}_k}^{A_-}} \\frac{\\left|{\\textup{\\text{det}}}J_l\\right|}2 \\left(\\frac{\\varepsilon \\phi_{l_1} \\left[\\phi_{l_1}(\\phi_{l_2} + \\phi_{l_3})-2\\phi_{l_2}\\phi_{l_3}  \\right]+\\varepsilon^2\\left[\\phi_{l_1}(\\phi_{l_2} + \\phi_{l_3})-\\phi_{l_2}\\phi_{l_3} \\right]}{(\\phi_{l_1} - \\phi_{l_2})^2(\\phi_{l_1} - \\phi_{l_3})^2 + O(\\varepsilon)} \\right).\n\\end{aligned}\n\\end{equation}\nTherefore, we obtain for this case\n\\begin{equation*\nd_{k}a = \\lim_{\\varepsilon\\searrow 0}\\frac{\\delta_{k,\\eps} a}{\\varepsilon^2} = \\sum_{l \\in I_{{\\mathbf x}_k}^{A_-}}\\frac{\\left|{\\textup{\\text{det}}}J_l\\right|}{2\\phi_{l_2}\\phi_{l_3}}.\n\\end{equation*}\nFinally, for ${\\mathbf x}_k \\in \\mathfrak S$ and $\\discreteOperator = \\S$ we deduce from \\eqref{eq::intAplus} \n\\begin{equation}\\label{eq::shapeA}\n\\lim_{\\varepsilon\\searrow 0}\\frac{\\mathcal{I}_{I_{{\\mathbf x}_k}^{A_+}}}{\\varepsilon} = \\sum_{l \\in I_{{\\mathbf x}_k}^{A_+}}\\frac{\\left|{\\textup{\\text{det}}}J_l\\right|\\;\\phi_{l_1} \\left[\\phi_{l_1}(\\phi_{l_2} + \\phi_{l_3}) - 2\\phi_{l_2}\\phi_{l_3} \\right]}{2(\\phi_{l_1} - \\phi_{l_2})^2(\\phi_{l_1} - \\phi_{l_3})^2},\n\\end{equation}\nand from \\eqref{eq::intAminus}\n\\begin{equation} \\label{eq_IAplus_o_eps}\n\\lim_{\\varepsilon\\searrow 0}\\frac{\\mathcal{I}_{I_{{\\mathbf x}_k}^{A_-}}}{\\varepsilon} = -\\sum_{l \\in I_{{\\mathbf x}_k}^{A_-}}\\frac{\\left|{\\textup{\\text{det}}}J_l\\right|\\;\\phi_{l_1} \\left[\\phi_{l_1}(\\phi_{l_2} + \\phi_{l_3})-2\\phi_{l_2}\\phi_{l_3}  \\right]}{2(\\phi_{l_1} - \\phi_{l_2})^2(\\phi_{l_1} - \\phi_{l_3})^2}.\n\\end{equation}\n\\paragraph{Configuration B}\nWe note that Configuration B can only occur for the case ${\\mathbf x}_k \\in \\mathfrak S$ and $\\discreteOperator = \\S$. For $l \\in I_{{\\mathbf x}_k}^{B_+}$ it holds\n\\begin{align}\\label{eq::configurationB}\n\\int_{\\tau_{l}} H(-\\S\\phi({\\mathbf x})) \\;\\mathrm{d} {\\mathbf x} = \\frac{\\left|{\\textup{\\text{det}}}J_l\\right|}2\\left(1- \\frac{\\phi_{l_2}^2}{(\\phi_{l_2}-\\phi_{l_3})(\\phi_{l_2}-\\phi_{l_1}-\\varepsilon)} \\right),\n\\end{align}\nand\n\\begin{equation*}\n\\begin{aligned}\n\\mathcal{I}_{I_{{\\mathbf x}_k}^{B_+}}\n&= \\sum_{l \\in I_{{\\mathbf x}_k}^{B_+}}\\frac{\\left|{\\textup{\\text{det}}}J_l\\right|\\phi_{l_2}^2}2 \\frac{(\\phi_{l_2}-\\phi_{l_3})(\\phi_{l_2}-\\phi_{l_1}-\\varepsilon)-(\\phi_{l_2}-\\phi_{l_3})(\\phi_{l_2}-\\phi_{l_1})}{(\\phi_{l_2}-\\phi_{l_3})(\\phi_{l_2}-\\phi_{l_1})(\\phi_{l_2}-\\phi_{l_3})(\\phi_{l_2}-\\phi_{l_1}-\\varepsilon)}\\\\\n&= \\sum_{l \\in I_{{\\mathbf x}_k}^{B_+}}\\frac{\\left|{\\textup{\\text{det}}}J_l\\right|\\phi_{l_2}^2}2 \\frac{-\\varepsilon(\\phi_{l_2}-\\phi_{l_3})}{(\\phi_{l_2}-\\phi_{l_3})^2(\\phi_{l_2}-\\phi_{l_1})(\\phi_{l_2}-\\phi_{l_1}-\\varepsilon)}.\n\\end{aligned}\n\\end{equation*}\nThus,\n\\begin{equation}\\label{eq::shapeB}\n\\lim_{\\varepsilon\\searrow 0}\\frac{\\mathcal{I}_{I_{{\\mathbf x}_k}^{B_+}}}{\\varepsilon} = -\\sum_{l \\in I_{{\\mathbf x}_k}^{B_+}} \\frac{\\left|{\\textup{\\text{det}}}J_l\\right|}2 \\frac{\\phi_{l_2}^2}{(\\phi_{l_2}-\\phi_{l_3})(\\phi_{l_2}-\\phi_{l_1})^2}.\n\\end{equation}\nFor the case $l \\in I_{{\\mathbf x}_k}^{B_-}$ we have\n\\begin{align}\\label{eq::configurationBminus}\n\t\\int_{\\tau_{l}} H(-\\S\\phi({\\mathbf x})) \\;\\mathrm{d} {\\mathbf x} = \\frac{\\left|{\\textup{\\text{det}}}J_l\\right|}2 \\frac{\\phi_{l_2}^2}{(\\phi_{l_2}-\\phi_{l_3})(\\phi_{l_2}-\\phi_{l_1}-\\varepsilon)},\n\\end{align}\nand obtain\n\\begin{equation*}\n\\lim_{\\varepsilon\\searrow 0}\\frac{\\mathcal{I}_{I_{{\\mathbf x}_k}^{B_-}}}{\\varepsilon} = \\sum_{l \\in I_{{\\mathbf x}_k}^{B_-}} \\frac{\\left|{\\textup{\\text{det}}}J_l\\right|}2 \\frac{\\phi_{l_2}^2}{(\\phi_{l_2}-\\phi_{l_3})(\\phi_{l_2}-\\phi_{l_1})^2}.\n\\end{equation*}\n\\paragraph{Configuration C}\nAnalogously as for Configuration B, we note that Configuration C can only occur for the case ${\\mathbf x}_k \\in \\mathfrak S$ and $\\discreteOperator = \\S$. We have  \n\\begin{equation}\\label{eq::shapeC}\n\\lim_{\\varepsilon\\searrow 0}\\frac{\\mathcal{I}_{I_{{\\mathbf x}_k}^{C_+}}}{\\varepsilon} = -\\sum_{l \\in I_{{\\mathbf x}_k}^{C_+}}\\frac{\\left|{\\textup{\\text{det}}}J_l\\right|}2 \\frac{\\phi_{l_3}^2}{(\\phi_{l_3}-\\phi_{l_2})(\\phi_{l_3}-\\phi_{l_1})^2},\n\\end{equation}\nand\n\\begin{equation}\n\\lim_{\\varepsilon\\searrow 0}\\frac{\\mathcal{I}_{I_{{\\mathbf x}_k}^{C_-}}}{\\varepsilon} = \\sum_{l \\in I_{{\\mathbf x}_k}^{C_-}}\\frac{\\left|{\\textup{\\text{det}}}J_l\\right|}2 \\frac{\\phi_{l_3}^2}{(\\phi_{l_3}-\\phi_{l_2})(\\phi_{l_3}-\\phi_{l_1})^2}.\n\\end{equation}\n\n\n\nSummarizing, we have shown the following result.\n\\begin{theorem}\\label{thm::areaDerivative}\nFor ${\\mathbf x}_k \\in \\mathfrak T^-$ we have\n\\begin{align}\\label{eq_derivAreaTminus}\n    d_k a = \\sum_{l \\in I_{{\\mathbf x}_k}^{A_+}} d_ka_l =  \\sum_{l \\in I_{{\\mathbf x}_k}^{A_+}}(-1)\\frac{\\left|{\\textup{\\text{det}}}J_l\\right|}{2\\phi_{l_2}\\phi_{l_3}}.\n\\end{align}\nFor ${\\mathbf x}_k \\in \\mathfrak T^+$ we have\n\\begin{align}\\label{eq_derivAreaTplus}\n    d_k a = \\sum_{l \\in I_{{\\mathbf x}_k}^{A_-}} d_ka_l =   \\sum_{l \\in I_{{\\mathbf x}_k}^{A_-}}\\frac{\\left|{\\textup{\\text{det}}}J_l\\right|}{2\\phi_{l_2}\\phi_{l_3}}.\n\\end{align}\nFor ${\\mathbf x}_k \\in \\mathfrak S$ we have\n\\begin{equation}\\label{eq::derivativeArea}\n\\begin{aligned}\n    d_k a =& \\sum_{l \\in C_{k}} d_k a_l\\\\=&\n    \\sum_{l \\in I_{{\\mathbf x}_k}^{A_+}}\\frac{\\left|{\\textup{\\text{det}}}J_l\\right|\\;\\phi_{l_1} \\left[\\phi_{l_1}(\\phi_{l_2} + \\phi_{l_3})-2\\phi_{l_2}\\phi_{l_3}  \\right]}{2(\\phi_{l_1} - \\phi_{l_2})^2(\\phi_{l_1} - \\phi_{l_3})^2} + \\sum_{l \\in I_{{\\mathbf x}_k}^{A_-}}  (-1)  \\frac{\\left|{\\textup{\\text{det}}}J_l\\right|\\;\\phi_{l_1} \\left[\\phi_{l_1}(\\phi_{l_2} + \\phi_{l_3}) - 2\\phi_{l_2}\\phi_{l_3} \\right]}{2(\\phi_{l_1} - \\phi_{l_2})^2(\\phi_{l_1} - \\phi_{l_3})^2} \\\\\n    &+\\sum_{l \\in I_{{\\mathbf x}_k}^{B_+}}  (-1)  \\frac{\\left|{\\textup{\\text{det}}}J_l\\right|}2 \\frac{\\phi_{l_2}^2}{(\\phi_{l_2}-\\phi_{l_3})(\\phi_{l_2}-\\phi_{l_1})^2}  +  \\sum_{l \\in I_{{\\mathbf x}_k}^{B_-}} \\frac{\\left|{\\textup{\\text{det}}}J_l\\right|}2 \\frac{\\phi_{l_2}^2}{(\\phi_{l_2}-\\phi_{l_3})(\\phi_{l_2}-\\phi_{l_1})^2} \\\\\n    &+\\sum_{l \\in I_{{\\mathbf x}_k}^{C_+}}  (-1)  \\frac{\\left|{\\textup{\\text{det}}}J_l\\right|}2 \\frac{\\phi_{l_3}^2}{(\\phi_{l_3}-\\phi_{l_2})(\\phi_{l_3}-\\phi_{l_1})^2}  +  \\sum_{l \\in I_{{\\mathbf x}_k}^{C_-}}\\frac{\\left|{\\textup{\\text{det}}}J_l\\right|}2 \\frac{\\phi_{l_3}^2}{(\\phi_{l_3}-\\phi_{l_2})(\\phi_{l_3}-\\phi_{l_1})^2}. \n\\end{aligned}\n\\end{equation}\n\\end{theorem}\n\n\n\n\n\n\\begin{remark}\\label{remark::area}\n\tThe corresponding computations for the denominators in \\eqref{eq::topDerivative}, \\textit{i.e.}\\, $|\\Omega(\\discreteOperator\\phi)\\triangle \\Omega(\\phi)|$, are closely related to the computations presented in this section for \\eqref{eq::areaChange}.\n\tDenoting\n\t\\begin{align*}\n        \\delta_{k,\\varepsilon} \\tilde a_l := \\int_{\\tau_l} | H(-\\discreteOperator\\phi) - H(-\\phi)|  \\;\\mathrm{d} {\\mathbf x}, \\quad d_k \\tilde a_l := \\underset{\\varepsilon \\searrow 0}{\\mbox{lim}} \\frac{\\delta_{k,\\varepsilon} \\tilde a_l}{\\varepsilon^o},\n\t\\end{align*}\n\twe get that \n\t\\begin{align} \\label{eq_dkatilde}\n        \\underset{\\varepsilon \\searrow 0}{\\mbox{lim }} \\frac{|\\Omega(\\discreteOperator\\phi) \\triangle  \\Omega(\\phi)|}{\\varepsilon^o} = \\sum_{l\\in C_k} d_k \\tilde a_l =: d_k \\tilde a\n\t\\end{align}\n\twhere $d_k \\tilde a_l = |d_k a_l|$ with the formulas for $d_k a_l$ given in \\eqref{eq_derivAreaTminus}--\\eqref{eq::derivativeArea}. It is obvious that, for ${\\mathbf x}_k \\in \\mathcal T^-$, $d_k a_l <0$ and for ${\\mathbf x}_k \\in \\mathcal T^+$, $d_k a_l >0$. Moreover, note that, for ${\\mathbf x}_k \\in \\mathcal S$, it holds $d_k a_l <0$ for all $l \\in C_k$. This yields that\n\t\\begin{align}\n        \\frac{d_k a_l}{d_k \\tilde a_l} = \\begin{cases} - 1 & \\mbox{if }k \\in \\mathcal S \\cup \\mathcal T^-, \\\\\n                          \\; \\;\\, 1 & \\mbox{if } k \\in \\mathcal T^+. \n                         \\end{cases}\n\t\\end{align}\n\\end{remark}\n\n\\begin{corollary} \\label{cor_dVol}\n    From Theorem \\ref{thm::areaDerivative} and Remark \\ref{remark::area}, it follows that the numerical {topological-shape } derivative of the volume cost function $\\mbox{Vol}(\\phi) := |\\Omega(\\phi)|$ is given by\n    \\begin{align*}\n        d \\mbox{Vol}(\\phi)({\\mathbf x}_k) = \\begin{cases} - 1 & \\mbox{if }k \\in \\mathcal S \\cup \\mathcal T^-, \\\\\n                          \\; \\;\\, 1 & \\mbox{if } k \\in \\mathcal T^+. \n                         \\end{cases}\n    \\end{align*}\n\n\\end{corollary}\n\\subsection{Computation of the numerical {topological-shape } derivative via Lagrangian framework}\\label{sec::Lagrangian}\n\\newcommand{{\\uu}}{{\\uu}}\nNext, we consider the computation of the numerical {topological-shape } derivative of an optimization problem that is constrained by a discretized PDE. For that purpose, we set $c_1 =0$ in \\eqref{eq::ContinuousObjective} and $J(\\phi,\\uu) := g(\\Omega(\\phi),\\uu)$. The discretized problem reads\n\\begin{align} \\label{eq_discOptiProb_J}\n\\underset{\\phi}{\\mbox{min }} J(\\phi, \\uu)&  \\\\\n\\mbox{s.t. } \\Amat_{\\phi}  \\uu &= \\ff_{\\phi} \\label{eq_discOptiProb_constr}\n\\end{align}\nand we are interested in the sensitivity of $J$ when the level set function $\\phi$ representing the geometry is replaced by a perturbed level set function $\\phi_{\\varepsilon}=\\discreteOperator\\phi$. \n The perturbed Lagrangian for \\eqref{eq_discOptiProb_J}--\\eqref{eq_discOptiProb_constr} with respect to a perturbation of $\\phi$ reads \n\\begin{align} \\label{eq_defLagDiscr}\nL(\\varepsilon, \\uu, \\vv) =& J(\\phi_\\varepsilon, \\uu) + \\Amat_\\varepsilon\\uu \\cdot \\vv - \\ff_\\varepsilon \\cdot \\vv\n\\end{align}\nwhere we use the abbreviated notation $\\Amat_\\varepsilon := \\Amat_{\\phi_\\varepsilon}$, and $\\ff_\\varepsilon := \\ff_{\\phi_\\varepsilon}$. Moreover, for $\\varepsilon \\geq 0$, we define the perturbed state $\\uu_\\varepsilon$ as the solution to\n\\begin{align*}\n0 = \\partial_\\vv L(\\varepsilon, \\uu_\\varepsilon, \\vv),\n\\end{align*}\ni.e. $\\uu_\\varepsilon$ is the solution to\n\\begin{align}  \\label{eq_defueps2}\n\\Amat_\\varepsilon \\uu_\\varepsilon = \\ff_\\varepsilon\n\\end{align}\nand the (unperturbed) adjoint state $\\pp$ as the solution to\n\\begin{align} \\label{eq_defAdjointDiscr}\n0 = \\partial_\\uu L(0, \\uu, \\pp),\n\\end{align}\nfor the state $\\uu$ given, i.e. $\\pp$ solves\n\\begin{align*}\n\\Amat^\\top \\pp = - \\partial_\\uu J(\\phi, \\uu).\n\\end{align*}\nNote that we use the notation $\\uu$ for $\\uu_{\\varepsilon = 0}$.\nThe numerical {topological-shape } derivative at the node ${\\mathbf x}_k$ can be computed as the limit\n\\begin{align*}\nd \\redPhiObjectiveFunction(\\phi)({\\mathbf x}_k) = \\underset{\\varepsilon \\searrow 0}{\\mbox{lim }} \\frac{1}{|\\Omega(\\phi_\\varepsilon) \\triangle \\Omega(\\phi)|} (J(\\phi_\\varepsilon, \\uu_\\varepsilon) - J(\\phi, \\uu) ).\n\\end{align*}\n\nWith the help of the Lagrangian \\eqref{eq_defLagDiscr}, we can rewrite the right hand side as\n\\begin{align*}\nJ(\\phi_\\varepsilon, \\uu_\\varepsilon) - J(\\phi,\\uu) = L(\\varepsilon, \\uu_\\varepsilon, \\pp) - L(0, {\\uu}, \\pp)\n\\end{align*}\nwhere we used that $\\uu_\\varepsilon$ solves \\eqref{eq_defueps2} for $\\varepsilon \\geq 0$. Following the approach used in \\cite{simplified}, we use the fundamental theorem of calculus as well as \\eqref{eq_defAdjointDiscr} to rewrite this as\n\\begin{align}\nJ(\\phi_\\varepsilon, \\uu_\\varepsilon) - J(\\phi,{\\uu}) =& \\int_0^1 [\\partial_\\uu L(\\varepsilon, {\\uu} + s(\\uu_\\varepsilon - {\\uu}), \\pp) - \\partial_\\uu L(\\varepsilon, \\uu, \\pp)](\\uu_\\varepsilon - {\\uu}) \\mbox ds \\\\\n&+[ \\partial_\\uu L(\\varepsilon, \\uu, \\pp) - \\partial_\\uu L(0, \\uu, \\pp)](\\uu_\\varepsilon - \\uu) \\\\\n&+ L(\\varepsilon, \\uu, \\pp) - L(0, \\uu, \\pp).\n\\end{align}\n\nThus the numerical {topological-shape } derivative can be obtained as the sum of three limits,\n\\begin{align*}\nd \\redPhiObjectiveFunction(\\phi)({\\mathbf x}_k) = R_1(\\uu, \\pp) + R_2(\\uu, \\pp) + R_0(\\uu, \\pp)\n\\end{align*}\nwhere\n\\begin{align*}\nR_1(\\uu, \\pp) :=& \\underset{\\varepsilon \\searrow 0}{\\mbox{lim }} \\frac{1}{|\\Omega(\\phi_\\varepsilon) \\triangle  \\Omega(\\phi)|}  \\int_0^1 [\\partial_\\uu L(\\varepsilon, {\\uu} + s(\\uu_\\varepsilon - {\\uu}), \\pp) - \\partial_\\uu L(\\varepsilon, \\uu, \\pp)](\\uu_\\varepsilon - {\\uu}) \\mbox ds, \\\\\nR_2(\\uu, \\pp) :=& \\underset{\\varepsilon \\searrow 0}{\\mbox{lim }} \\frac{1}{|\\Omega(\\phi_\\varepsilon) \\triangle  \\Omega(\\phi)|}[ \\partial_\\uu L(\\varepsilon, \\uu, \\pp) - \\partial_\\uu L(0, \\uu, \\pp)](\\uu_\\varepsilon - \\uu), \\\\\nR_0(\\uu, \\pp) := & \\underset{\\varepsilon \\searrow 0}{\\mbox{lim }} \\frac{1}{|\\Omega(\\phi_\\varepsilon) \\triangle  \\Omega(\\phi)|} [ L(\\varepsilon, \\uu, \\pp) - L(0, \\uu, \\pp)].\n\\end{align*}\n\nFor $J(\\phi_\\varepsilon, \\uu) = c_2 (\\uu - \\hat \\uu) \\tilde \\MM_\\varepsilon (\\uu - \\hat \\uu)$, where $\\tilde \\MM_\\varepsilon := \\tilde \\MM_{\\phi_\\varepsilon}$ represents the matrix $\\tilde \\MM$ defined in \\eqref{eq_defMtilde} with $\\Omega = \\Omega(\\phi_\\varepsilon)$, we get\n\\begin{align*}\nR_1(\\uu, \\pp) =& c_2 \\underset{\\varepsilon \\searrow 0}{\\mbox{lim }}  \\frac{1}{|\\Omega(\\phi_\\varepsilon) \\triangle  \\Omega(\\phi)|} (\\uu_\\varepsilon - {\\uu})^\\top \\tilde \\MM_\\varepsilon (\\uu_\\varepsilon - {\\uu}).\n\\end{align*}\nMoreover, \n\\begin{align} \\label{eq_R2discr}\nR_2(\\uu, \\pp) =& \\underset{\\varepsilon \\searrow 0}{\\mbox{lim }} \\frac{1}{|\\Omega(\\phi_\\varepsilon) \\triangle  \\Omega(\\phi)|} \\left[ 2 c_2 (\\tilde \\MM_\\varepsilon - \\tilde \\MM) (\\uu - \\hat \\uu)  \\cdot  (\\uu_\\varepsilon - {\\uu}) + (\\Amat_\\varepsilon - \\Amat) (\\uu_\\varepsilon  - {\\uu}) \\cdot \\pp \\right],\n\\end{align}\nand\n\\begin{align} \\label{eq_R0discr}\nR_0(\\uu, \\pp) =& \\underset{\\varepsilon \\searrow 0}{\\mbox{lim }} \\frac{1}{|\\Omega(\\phi_\\varepsilon) \\triangle \\Omega(\\phi)|} \\left[ c_2({\\uu} - \\hat \\uu)^\\top(\\tilde \\MM_\\varepsilon - \\tilde \\MM)  ({\\uu} - \\hat \\uu) + \\pp^\\top (\\Amat_\\varepsilon - \\Amat) \\uu - (\\ff_\\varepsilon - \\ff)\\cdot \\pp \\right].\n\\end{align}\n\n\\begin{lemma}\\label{lem_uepsu0} There exist constants $c>0, \\hat \\varepsilon >0$ such that for all $\\varepsilon \\in (0,\\hat\\varepsilon)$\n    \\begin{align*}\n        \\| \\uu_\\varepsilon - {\\uu} \\| \\leq c \\, \\varepsilon^o.\n    \\end{align*}\n    Here, $o=1$ in the case of a shape perturbation and $o=2$ in the case of a topological perturbation.\n\\end{lemma}\n\\begin{proof}    \n    Subtracting \\eqref{eq_defueps2} for $\\varepsilon = 0$ from the same equation with $\\varepsilon >0$, we get\n    \\begin{align*}\n         \\Amat_\\varepsilon (\\uu_\\varepsilon - \\uu) &= \\ff_\\varepsilon - \\ff - (\\Amat_\\varepsilon - \\Amat) \\uu \n    \\end{align*}\n    and thus, by the ellipticity of the bilinear form corresponding to $\\Amat_\\varepsilon$ and the triangle inequality, there is a constant $c>0$ such that for all $\\varepsilon>0$ small enough\n    \\begin{align}\n        \\| \\uu_\\varepsilon - \\uu \\| \\leq c ( \\| \\ff_\\varepsilon - \\ff \\| - \\| \\Amat_\\varepsilon - \\Amat \\|  \\| \\uu \\| ).\n    \\end{align}\n    For the difference between the perturbed and unperturbed right hand sides we have\n    \\begin{align*}\n        |(\\ff_\\varepsilon - \\ff)_i| \\leq& |(f_1 - f_2)| \\int_{\\Omega(\\phi_\\varepsilon) \\triangle \\Omega(\\phi)} |\\varphi_i(x)| \\; \\mbox dx \\\\\n        |(\\Amat_\\varepsilon - \\Amat)_{i,j}| \\leq & \\int_{\\Omega(\\phi_\\varepsilon) \\triangle \\Omega(\\phi)}  \\bigg( |(\\lambda_1 - \\lambda_2)| |\\nabla \\varphi_j(x)| |\\nabla \\varphi_i(x)|  +  |(\\alpha_1 - \\alpha_2)| |\\varphi_j(x)| |\\varphi_i(x)|  \\bigg)\\; \\mbox dx.\n        \\end{align*}\n    The result follows from the boundedness of $|\\varphi_i(x)|$ and $|\\nabla \\varphi_i(x)|$ together with \\eqref{eq_dkatilde} which implies the existence of $c>0$ such that $|\\Omega(\\phi_\\varepsilon) \\triangle \\Omega(\\phi)| \\leq c \\varepsilon^o$ (cf. Remark \\ref{remark::area}). \n\\end{proof}\nFrom Lemma \\ref{lem_uepsu0} it follows that the terms $R_1(\\uu, \\pp)$ and $R_2(\\uu, \\pp)$ vanish. We remark that this is in contrast to the continuous case, where asymptotic analysis shows that $R_2$ does not vanish. We will address this issue in more detail in Section \\ref{sec::connection}. \nThus, in the discrete setting we obtain $d \\redPhiObjectiveFunction(\\phi)({\\mathbf x}_k)=R_0(\\uu, \\pp)$, i.e.,\n\\begin{align}\\label{eq::derivativeTracking}\nd \\redPhiObjectiveFunction(\\phi)({\\mathbf x}_k) &= \n\\frac{\\mathbf p^\\top(d_k \\mathbf A \\,{\\uu} - d_k \\mathbf f)\n\t+c_2 ({\\uu} - \\hat \\uu)^\\top d_k\\tilde \\MM  ({\\uu} - \\hat \\uu)\n}{ d_k\\tilde a} \n\\end{align}\nwhere%\n\\begin{align} \\label{eq_derivativeLimits}\n\\begin{aligned}\nd_k \\Amat &= \\lim_{\\varepsilon\\searrow 0} \\frac{\\Amat_\\varepsilon- \\Amat}{\\varepsilon^o}, &&&\nd_k \\tilde \\MM& = \\lim_{\\varepsilon\\searrow 0} \\frac{\\tilde\\MM_\\varepsilon- \\tilde\\MM}{\\varepsilon^o}, \\\\\nd_k \\ff &= \\lim_{\\varepsilon\\searrow 0} \\frac{\\ff_\\varepsilon-\\ff}{\\varepsilon^o}, &&&\nd_k \\tilde a &= \\lim_{\\varepsilon\\searrow 0} \\frac{|\\Omega(\\phi_\\varepsilon)\\triangle \\Omega(\\phi)|}{\\varepsilon^o}, \n\\end{aligned}\n\\end{align}\nwith $o=1$ for ${\\mathbf x}_k\\in\\mathfrak S$ and $o =2$ for ${\\mathbf x}_k\\in\\mathfrak T^- \\cup \\mathfrak T^+$. To obtain \\eqref{eq::derivativeTracking}, we divided both numerator and denominator of \\eqref{eq_R0discr} by $\\varepsilon^o$ and used that the limit of the quotient coincides with the quotient of the limits provided both limits exist and the limit in the denominator does not vanish. Next we state the numerical {topological-shape } derivative of problem \\eqref{eq::ContinuousProblem} for arbitrary constant weights $c_1, c_2 \\geq 0$ in the cost function \\eqref{eq::ContinuousObjective}.\n\n\\begin{theorem}[Numerical {topological-shape } derivative] \\label{thm_numTopShapeDer_formula}\n\tFor $l=1,\\dots,N$, let $\\mathbf u_l = [u_{l_1},u_{l_2},u_{l_3}]^\\top$ and $\\mathbf p_l = [p_{l_1},p_{l_2},p_{l_3}]^\\top$ be the nodal values for element $\\tau_{l}$ of the solution and the adjoint, and \n\t\\begin{equation*}\n\t\\mathbf k_{0,l}[i,j] = \\left(J_l^{-1}\\nabla_\\xi \\psi_j\\right)^\\top \\left(J_l^{-1}\\nabla_\\xi \\psi_i\\right), \\quad i,j =1,\\dots,3.\n\t\\end{equation*}\n\tMoreover, $u_k = u({\\mathbf x}_k)$, $p_k = p({\\mathbf x}_k)$, and $\\hat u_k = \\hat u({\\mathbf x}_k)$.  \n\tFor ${\\mathbf x}_k \\in \\mathfrak T^-$ the numerical topological derivative reads\n\t\\begin{align}\\label{eq::formulaTopologicalDerivativePlus}\n\td\\redPhiObjectiveFunction(\\phi)({\\mathbf x}_k) = -c_1\n\t-\\Delta \\lambda\t\\frac{\\sum_{l \\in I_{\\pos_k}}  \\frac{\\mathbf p_{l}^\\top \\mathbf k_{0,l} \\mathbf u_{l}\\left|{\\textup{\\text{det}}}J_l\\right|}{\\phi_{l_2}\\phi_{l_3}}}{\\sum_{l \\in I_{\\pos_k}}\\frac{\\left|{\\textup{\\text{det}}}J_l\\right|}{\\phi_{l_2}\\phi_{l_3}} }\t-\\Delta \\alpha p_ku_k\t+\\Delta fp_k\t- c_2\\Delta \\tilde\\alpha(u_k-\\hat u_k)^2,\t\n\t\\end{align}\n\twhereas for ${\\mathbf x}_k \\in \\mathfrak T^+$ we have\n\t\\begin{align}\\label{eq::formulaTopologicalDerivativeMinus}\n\td\\redPhiObjectiveFunction(\\phi)({\\mathbf x}_k) = c_1 +\n\t\\Delta \\lambda\t\\frac{\\sum_{l \\in I_{\\pos_k}}  \\frac{\\mathbf p_{l}^\\top \\mathbf k_{0,l} \\mathbf u_{l}\\left|{\\textup{\\text{det}}}J_l\\right|}{\\phi_{l_2}\\phi_{l_3}}}{\\sum_{l \\in I_{\\pos_k}}\\frac{\\left|{\\textup{\\text{det}}}J_l\\right|}{\\phi_{l_2}\\phi_{l_3}} }\t+\\Delta \\alpha p_ku_k\t-\\Delta f p_k + c_2\\Delta \\tilde\\alpha(u_k-\\hat u_k)^2,\n\t\\end{align}\n\twith \n\t\\begin{align*}\n\t\\Delta \\alpha = \\alpha_1-\\alpha_2, \\quad\n\t\\Delta \\lambda = \\lambda_1-\\lambda_2, \\quad\n\t\\Delta f = f_1-f_2, \\quad\n\t\\Delta \\tilde\\alpha = \\tilde\\alpha_1-\\tilde\\alpha_2 .\n\t\\end{align*}\n\tFor ${\\mathbf x}_k \\in \\mathfrak S$ the numerical shape derivative reads\n\t\\begin{equation}\\label{eq::formulaShapeDerivative}\n\t\\begin{aligned}\n\td\\redPhiObjectiveFunction(\\phi)({\\mathbf x}_k) = -c_1 + & \n\t\\Delta \\lambda\n\t\\frac{\\sum_{l \\in C_k}  \\mathbf p_{l}^\\top \\mathbf k_{0,l} \\mathbf u_{l} \\,d_k a_{l}}{d_k\\tilde a}\n\t+\\Delta \\alpha\\frac{\\sum_{l \\in C_k}  \\mathbf p_{l}^\\top d_k\\mathbf m^I_{l} \\,\\mathbf u_{l}}{d_k\\tilde a} \n\t  \\\\\n\t &  -\\Delta f \\frac{\\sum_{l \\in C_k} \\mathbf p_{l}^\\top d_k\\mathbf f_l^I }{d_k\\tilde a}\n\t+c_2\\Delta \\tilde\\alpha \\frac{\\sum_{l \\in C_k}  (\\mathbf u_{l}-\\hat{\\mathbf u}_{l})^\\top d_k\\mathbf m^I_{l} \\,(\\mathbf u_{l}-\\hat{\\mathbf u}_{l})}{d_k\\tilde a},\n\t\\end{aligned}\n\t\\end{equation}\n\twhere the entries of the element matrix $d_k\\mathbf m_{l}^I$ and of the element vector $d_k\\mathbf f_{l}^I$ are dependent on the local cut situation (cases  $I=A^\\pm$, $B^\\pm$, $C^\\pm$) and are given in Appendix \\ref{app::shapeDerivative}. The values $d_k \\tilde a$ can be computed by \\eqref{eq::derivativeArea} considering Remark \\ref{remark::area}.\n\t\\color{black}\n\\end{theorem}\n\\begin{proof}\n\tWe evaluate \\eqref{eq::derivativeTracking} for ${\\mathbf x}_k \\in \\mathfrak T^-$ and $\\discreteOperator = T^{-\\rightarrow+}_{k,\\varepsilon}$. Thus, $o=2$ in \\eqref{eq_derivativeLimits}. We note that \n\t\\begin{equation}\n\t\\mathbf p^\\top d_k \\mathbf A \\,{\\uu} = \\sum_{l=1}^N  \\mathbf p_{l}^\\top (d_k\\mathbf m_{l} +d_k\\mathbf k_{l}) \\mathbf u_{l}.\n\t\\end{equation}\n\tWe have for $l\\in I_{\\pos_k}$\n\t\\begin{align*}\n\t\\mathbf k_{l}(\\phi_{\\varepsilon})-\\mathbf k_l(\\phi) &= \\mathbf k_{0,l} \\left|{\\textup{\\text{det}}}J_l\\right|\\bigg(  \\int_{\\xi_1=0}^1\\int_{\\xi_2=0}^{1-\\xi_1} \\lambda_1 H(-\\phi_{\\varepsilon}\\circ\\Phi_l) +\\lambda_2 H(\\phi_{\\varepsilon}\\circ\\Phi_l) \\;\\mathrm{d} \\xi_2 \\;\\mathrm{d} \\xi_1 \\\\ &\\quad-\\int_{\\xi_1=0}^1\\int_{\\xi_2=0}^{1-\\xi_1} \\lambda_1 H(-\\phi\\circ\\Phi_l) +\\lambda_2 H(\\phi\\circ\\Phi_l) \\;\\mathrm{d} \\xi_2 \\;\\mathrm{d} \\xi_1\\bigg) \\\\\n\t&=\\mathbf k_{0,l} \\left|{\\textup{\\text{det}}}J_l\\right|\\bigg(  \\lambda_1\\int_{\\xi_1=0}^1\\int_{\\xi_2=0}^{1-\\xi_1}  (H(-\\phi_{\\varepsilon}\\circ\\Phi_l) -H(-\\phi\\circ\\Phi_l))  \\;\\mathrm{d} \\xi_2 \\;\\mathrm{d} \\xi_1 \\\\ &\\quad+\\lambda_2\\int_{\\xi_1=0}^1\\int_{\\xi_2=0}^{1-\\xi_1}  (H(\\phi_\\varepsilon\\circ\\Phi_l) - H(\\phi\\circ\\Phi_l)) \\;\\mathrm{d} \\xi_2 \\;\\mathrm{d} \\xi_1\\bigg)\n\t\\\\\n\t&= \\mathbf k_{0,l}  (\\lambda_1 - \\lambda_2)\\delta_{k,\\eps} a_l\n\t\\end{align*}\n\tdue to \\eqref{eq::areaChangeB} because $|\\Omega_\\varepsilon|-|\\Omega| = -(|D \\setminus \\Omega_\\varepsilon|-|D \\setminus \\Omega|)$\n\tand with \\eqref{eq::areaChangeTopAplus} we obtain\n\t\\begin{align}\\label{eq::termA}\n\td_k\\mathbf k_l = \\underset{\\varepsilon \\searrow 0}{\\mbox{lim }} \\frac{\\mathbf k_l(\\phi_\\varepsilon) - \\mathbf k_l(\\phi)}{\\varepsilon^2}\n\t&= -\\Delta\\lambda\\frac{\\left|{\\textup{\\text{det}}}J_l\\right|}{2\\phi_{l_2}\\phi_{l_3}} \\mathbf k_{0,l}.\n\t\\end{align}\n\tDue to \n\t\\begin{equation*}\n\t\\int_{\\xi_1=0}^{l_1} \\int_{\\xi_2=0}^{l_2\\left(1-\\frac{\\xi_1}{l_1}\\right)} \n\t\\xi_1^a \\xi_2^b \\;\\mathrm{d} \\xi_2 \\;\\mathrm{d} \\xi_1 = \\frac{\\epsilon^{a+b+2}}{\\phi_{l_2}^{a+1}\\phi_{l_3}^{b+1} + O(\\epsilon^{a+b+2})}\n\t\\end{equation*}\n\tfor some $a,b \\in \\mathbb{N}$, we have\n\t\\begin{equation}\\label{eq::limitChangeMass}\n\t\\begin{aligned}\n\td_k\\mathbf m_l = \\lim_{\\varepsilon\\searrow 0} \\frac{\\mathbf m_{l}(\\phi_{\\varepsilon})-\\mathbf m_l(\\phi)}{\\varepsilon^2} &= - \\lim_{\\varepsilon\\searrow 0}\t \\frac{\\Delta\\alpha}{\\varepsilon^2}\n\t\\int_{\\xi_1=0}^{l_1} \\int_{\\xi_2=0}^{l_2\\left(1-\\frac{\\xi_1}{l_1}\\right)} \n\t\\psi_i(\\xi)\\psi_j(\\xi) \\left|{\\textup{\\text{det}}}J_l\\right|\\;\\mathrm{d} \\xi_2 \\;\\mathrm{d} \\xi_1 \\\\ &= \n\t\\begin{bmatrix}\n\t-\\frac{\\Delta\\alpha\\left|{\\textup{\\text{det}}}J_l\\right|}{2\\phi_{l_2}\\phi_{l_3}} & 0 & 0 \\\\\n\t0 & 0 & 0 \\\\\n\t0 & 0 & 0 \n\t\\end{bmatrix},\n\t\\end{aligned}\n\t\\end{equation}\n\tand conclude\n\t\\begin{align}\\label{eq::termB}\n\t\\sum_{l=1}^N \\mathbf p_{l}^\\top d_k\\mathbf m_{l}  \\mathbf u_{l} = -\\Delta\\alpha p_k u_k\\sum_{l \\in I_{\\pos_k}}\t\\frac{\\left|{\\textup{\\text{det}}}J_l\\right|}{2\\phi_{l_2}\\phi_{l_3}}.\n\t\\end{align}\n\tFurthermore, with \n\t\\begin{equation}\n\t\\begin{aligned}\\label{eq::limitChangeLoad}\n\td_k\\mathbf f_l =\\lim_{\\varepsilon\\searrow 0} \\frac{\\mathbf f_{l}(\\phi_{\\varepsilon})-\\mathbf f_l(\\phi)}{\\varepsilon^2} &= - \\lim_{\\varepsilon\\searrow 0}\t \\frac{\\Delta f}{\\varepsilon^2}\n\t\\int_{\\xi_1=0}^{l_1} \\int_{\\xi_2=0}^{l_2\\left(1-\\frac{\\xi_1}{l_1}\\right)} \n\t\\psi_i(\\xi) \\left|{\\textup{\\text{det}}}J_l\\right|\\;\\mathrm{d} \\xi_2 \\;\\mathrm{d} \\xi_1 \\\\ &= \n\t\\begin{bmatrix}\n\t-\\frac{\\Delta f\\left|{\\textup{\\text{det}}}J_l\\right|}{2\\phi_{l_2}\\phi_{l_3}}  \\\\\n\t0  \\\\\n\t0  \n\t\\end{bmatrix},\n\t\\end{aligned}\n\t\\end{equation}\n\tit follows that\n\t\\begin{equation}\\label{eq::termC}\n\t \\mathbf p^\\top d_k\\mathbf f = \\sum_{l=1}^N  \\mathbf p_{l}^\\top d_k\\mathbf f_{l} = -\\Delta f p_k \\sum_{l \\in I_{\\pos_k}}\t\\frac{\\left|{\\textup{\\text{det}}}J_l\\right|}{2\\phi_{l_2}\\phi_{l_3}}.\n\t\\end{equation}\n\tAnalogously to \\eqref{eq::limitChangeMass} we have\n\t\\begin{equation}\\label{eq::limitChangeTracking}\n\t\\begin{aligned}\n\td_k\\tilde{\\mathbf  m}_l  = \\lim_{\\varepsilon\\searrow 0} \\frac{\\tilde{\\mathbf m}_{l}(\\phi_{\\varepsilon})-\\tilde{\\mathbf  m}_l(\\phi)}{\\varepsilon^2} = \n\t\\begin{bmatrix}\n\t-\\frac{\\Delta\\tilde\\alpha\\left|{\\textup{\\text{det}}}J_l\\right|}{2\\phi_{l_2}\\phi_{l_3}} & 0 & 0 \\\\\n\t0 & 0 & 0 \\\\\n\t0 & 0 & 0 \n\t\\end{bmatrix},\n\t\\end{aligned}\n\t\\end{equation}\n\tand obtain\n\t\\begin{equation}\\label{eq::termD}\n\t\\begin{aligned}\n\t({\\uu} - \\hat \\uu)^\\top d_k\\tilde \\MM  ({\\uu} - \\hat \\uu) &= \\sum_{l \\in I_{\\pos_k}} (\\uu_{0,l} - \\hat \\uu_l)^\\top d_k\\tilde {\\mathbf m}_l  (\\uu_{0,l} - \\hat \\uu_l) \\\\ &= -\\Delta \\tilde\\alpha(u_k-\\hat u_k)^2 \\sum_{l \\in I_{\\pos_k}}\t\\frac{\\left|{\\textup{\\text{det}}}J_l\\right|}{2\\phi_{l_2}\\phi_{l_3}}.\t\n\t\\end{aligned}\n\t\\end{equation}\n\tIn the present situation, $d_k\\tilde a$ is given by the absolute value of \\eqref{eq::areaChangeTopAplus} (see also Remark \\ref{remark::area}),\n\t\\begin{equation}\\label{eq::changeAreaProof}\n\td_k\\tilde a = \\sum_{l \\in I_{{\\mathbf x}_k}^{A_+}}\\frac{\\left|{\\textup{\\text{det}}}J_l\\right|}{2\\phi_{l_2}\\phi_{l_3}}.\n\t\\end{equation}\n\tBy inserting \\eqref{eq::termA}, \\eqref{eq::termB}, \\eqref{eq::termC}, \\eqref{eq::termD}, and \\eqref{eq::changeAreaProof} in \\eqref{eq::derivativeTracking}, together with Corollary \\ref{cor_dVol}, we obtain the sought expression \\eqref{eq::formulaTopologicalDerivativePlus}. Formula \\eqref{eq::formulaTopologicalDerivativeMinus} can be obtained in an analogous way as \\eqref{eq::formulaTopologicalDerivativePlus}.\n\t\n\tThe formula in \\eqref{eq::formulaShapeDerivative} follows directly from \\eqref{eq::derivativeTracking} together with Corollary \\ref{cor_dVol}. The values of $d_k \\mathbf m^I_l$ and $d_k \\mathbf f^I_l$ for all possible cut situations $I \\in \\{A^+, A^-, B^+, B^-, C^+, C^-\\}$ are given in Appendix \\ref{app::shapeDerivative} and were computed using symbolic computer algebra tools. A mathematical derivation of these terms is omitted here for brevity.\n\\end{proof}\n\n\\subsection{Verification}\nThe implementation of the {topological-shape } derivative is verified by comparing the computed sensitivity values against numerical values obtained by three different approaches. These are (i) a finite difference test, (ii) an application of the complex step derivative \\cite{martins2003complex} and (iii) a test based on hyper-dual numbers developed in \\cite{fike2011development}. All tests are conducted for a fixed configuration.\n\nWe recall the definition of the {topological-shape } derivative \\eqref{eq::topDerivative} at a node ${\\mathbf x}_k$ of the mesh\n\\begin{align} \\label{eq_dJ_lim}\n    d\\redPhiObjectiveFunction(\\phi)({\\mathbf x}_k) = \\lim_{\\varepsilon\\searrow 0}\n        \\delta \\redPhiObjectiveFunction_{\\varepsilon}(\\phi)({\\mathbf x}_k) \\quad \\mbox{ with } \\quad\n    \\delta \\redPhiObjectiveFunction_{\\varepsilon}(\\phi)({\\mathbf x}_k) :=\n\\frac{\\redPhiObjectiveFunction({O_{k,\\varepsilon}} \\phi)-\\redPhiObjectiveFunction(\\phi)}\n{|\\Omega({O_{k,\\varepsilon}}\\phi) \\triangle \\Omega(\\phi)|}\n\\end{align}\nwhere $O_{k,\\varepsilon}$ represents the operator $T^{-\\rightarrow+}_{k,\\varepsilon}$, $T^{+\\rightarrow-}_{k,\\varepsilon}$, $S_{k,\\varepsilon}$ depending on whether the node ${\\mathbf x}_k$ is in $\\mathfrak T^-$, $\\mathfrak T^+$ or $\\mathfrak S$, respectively.\n\n\\subsubsection{Finite difference test}\nFor the finite difference (FD) test, we compute the errors\n\\begin{align} \\label{eq_def_errors_eSeT}\ne_S^{FD}(\\varepsilon) = \\sqrt{\\sum_{{\\mathbf x}_k \\in \\mathfrak S} (\\delta \\redPhiObjectiveFunction_{\\varepsilon}(\\phi)({\\mathbf x}_k) - d\\redPhiObjectiveFunction(\\phi)({\\mathbf x}_k))^2}, \\quad\ne_T^{FD}(\\varepsilon) = \\sqrt{\\sum_{{\\mathbf x}_k \\in \\mathfrak T^- \\cup \\mathfrak T^+} (\\delta \\redPhiObjectiveFunction_{\\varepsilon}(\\phi)({\\mathbf x}_k) - d\\redPhiObjectiveFunction(\\phi)({\\mathbf x}_k))^2}\n\\end{align}\nfor a decreasing sequence of values for $\\varepsilon$. \nThe results are shown in Figure \\ref{fig:verify}(a). We observe convergence of order $\\varepsilon$ up to a point where the cancellation error dominates. \n\\begin{figure}\n    \\begin{tabular}{ccc}\n        \\includegraphics[width=0.33\\linewidth]{verify-derivative-finite} & \\includegraphics[width=0.33\\linewidth]{verify-derivative-imag} &\n        \\includegraphics[width=0.33\\linewidth]{verify_derivative_hyper} \\\\\n        (a) & (b) & (c) \n    \\end{tabular}\n\\caption{(a) Results of the finite difference test. (b) Results obtained with the complex step derivative. (c) Results obtained with hyper-dual numbers.}\n    \\label{fig:verify}\n\\end{figure}\n\n\n\\subsubsection{Complex step derivative test}\nIn order to overcome this drawback of the finite difference test, we next consider a test based on the complex step (CS) derivative \\cite{martins2003complex}. For that purpose, using Remark \\ref{remark::area}, let us first rewrite \\eqref{eq_dJ_lim} as\n\\begin{align} \\label{eq_dJ_lim2}\n    d\\redPhiObjectiveFunction(\\phi)({\\mathbf x}_k) = \\frac{\\lim_{\\varepsilon\\searrow 0}\\frac{\\redPhiObjectiveFunction({O_{k,\\varepsilon}} \\phi)-\\redPhiObjectiveFunction(\\phi)}\n{\\varepsilon^o}}{\\lim_{\\varepsilon\\searrow 0}\\frac{|\\Omega({O_{k,\\varepsilon}}\\phi) \\triangle \\Omega(\\phi)|}{\\varepsilon^o}}\n =\\frac{1}{d_k \\tilde a} \\lim_{\\varepsilon\\searrow 0}\\frac{\\redPhiObjectiveFunction({O_{k,\\varepsilon}} \\phi)-\\redPhiObjectiveFunction(\\phi)}\n{\\varepsilon^o},\n\\end{align}\nwhere $o=1$ if ${\\mathbf x}_k \\in \\mathfrak S$ and $o=2$ if ${\\mathbf x}_k \\in \\mathfrak T^- \\cup \\mathfrak T^+$. Moreover,\nassuming a higher order expansion of the form\n\\begin{align} \\label{eq_expansion_op3}\n    \\redPhiObjectiveFunction(O_{k,\\varepsilon}\\phi) = \\redPhiObjectiveFunction(\\phi) \n    + \\varepsilon^o \\, d_k \\tilde a \\, d \\redPhiObjectiveFunction(\\phi)({\\mathbf x}_k)\n    + \\varepsilon^{o+1} \\, d_k \\tilde a \\, d^2 \\redPhiObjectiveFunction(\\phi)({\\mathbf x}_k)\n    + \\varepsilon^{o+2} \\, d_k \\tilde a \\, d^3 \\redPhiObjectiveFunction(\\phi)({\\mathbf x}_k) + \\mathfrak{o}(\\varepsilon^{o+2})\n\\end{align}\nwith some higher order sensitivities $\nd^2 \\redPhiObjectiveFunction(\\phi)({\\mathbf x}_k)$, $d^3 \\redPhiObjectiveFunction(\\phi)({\\mathbf x}_k)$ and assuming that \\eqref{eq_expansion_op3} also holds for complex-valued $\\varepsilon$, we can follow the idea of the complex step derivative \\cite{martins2003complex}: Setting $\\varepsilon = ih$ in \\eqref{eq_expansion_op3} with $h>0$ and $i$ the complex unit yields\n\\begin{align}\n    d \\redPhiObjectiveFunction(\\phi)({\\mathbf x}_k) = \\frac{\\text{Im}(\\redPhiObjectiveFunction(O_{k,ih}\\phi))}{h \\, d_k \\tilde a} + \\mathcal O(h^2)\n\\end{align}\nin the case $o=1$ where ${\\mathbf x}_k \\in \\mathfrak S$, and \n\\begin{align*}\n    d \\redPhiObjectiveFunction(\\phi)({\\mathbf x}_k) = \\frac{\\text{Re}(\\redPhiObjectiveFunction(O_{k,ih}\\phi)-\\redPhiObjectiveFunction(\\phi))}{-h^2 \\, d_k \\tilde a} + \\mathcal O(h^2)\n\\end{align*}\nin the case $o=2$ where ${\\mathbf x}_k \\in \\mathfrak T^- \\cup \\mathfrak T^+$. This means\n\\begin{align*}\n    d \\redPhiObjectiveFunction(\\phi)({\\mathbf x}_k) = \\delta \\redPhiObjectiveFunction_h^{CS}(\\phi)({\\mathbf x}_k) + \\mathcal O(h^2)\n\\end{align*}\nwith\n\\begin{align} \\label{eq_deltaJh_CS}\n    \\delta \\redPhiObjectiveFunction_h^{CS}(\\phi)({\\mathbf x}_k) := \\begin{cases}\n        \\frac{\\text{Re}(\\redPhiObjectiveFunction(T^{-\\rightarrow+}_{k, ih} \\phi)-\\redPhiObjectiveFunction(\\phi))}{-h^2 \\, d_k \\tilde a}, & {\\mathbf x}_k \\in \\mathfrak T^- ,\\\\\n        \\frac{\\text{Re}(\\redPhiObjectiveFunction(T^{+\\rightarrow-}_{k,ih} \\phi)-\\redPhiObjectiveFunction(\\phi))}{-h^2 \\, d_k \\tilde a}, & {\\mathbf x}_k \\in \\mathfrak T^+ ,\\\\\n        \\frac{\\text{Im}(\\redPhiObjectiveFunction( S_{k,ih} \\phi))}{h \\, d_k \\tilde a}, & {\\mathbf x}_k \\in \\mathfrak S.\n        \\end{cases}\n\\end{align}\nAnalogously to \\eqref{eq_def_errors_eSeT}, we define the summed errors $e_S^{CS}(h)$ and $e_T^{CS}(h)$ by just replacing $\\delta \\redPhiObjectiveFunction_\\varepsilon(\\phi)({\\mathbf x}_k)$ by $\\delta \\redPhiObjectiveFunction_h^{CS}(\\phi)({\\mathbf x}_k)$ defined above. Figure  \\ref{fig:verify}(b) shows the errors $e_S^{CS}$ and $e_T^{CS}$ for a positive, decreasing sequence of $h$ where we observe quadratic decay for both errors. While the error $e_S^{CS}$ corresponding to the shape nodes ${\\mathbf x}_k \\in \\mathfrak S$ decays to machine precision, the error $e_T^{CS}$ corresponding to the interior nodes ${\\mathbf x}_k \\in \\mathfrak T^- \\cup \\mathfrak T^+$ deteriorates at some point due to the cancellation error ocurring when subtracting $\\redPhiObjectiveFunction(\\phi)$ from $\\redPhiObjectiveFunction(O_{k,ih}\\phi)$ in \\eqref{eq_deltaJh_CS}.\n\n\\subsubsection{Test based on hyper-dual numbers}\nIn order to overcome this cancellation error also for the case of ${\\mathbf x}_k \\in \\mathfrak T^- \\cup \\mathfrak T^+$, we resort to hyper-dual (HD) numbers as introduced in \\cite{fike2011development}. Here, the idea is to consider numbers with three non-real components denoted by $E_1$, $E_2$ and $E_1 E_2$ with $E_1^2 = E_2^2 = (E_1E_2)^2=0$. Assuming that expansion \\eqref{eq_expansion_op3} holds up to order $o+1$ also for such hyper-dual values of $\\varepsilon$, we can set $\\varepsilon = h E_1 + h E_2 + 0 E_1E_2$ for some $h>0$. For $o=1$, considering only the first non-real part (i.e., the $E_1$-part) and exploiting that $E_1^2 = 0$, we obtain the equality\n\\begin{align} \\label{eq_dJHD_shape}\n    d \\redPhiObjectiveFunction(\\phi)({\\mathbf x}_k) = \\frac{E_1\\text{part}(\\redPhiObjectiveFunction(O_{k, h E_1 + h E_2}\\phi))}{h \\, d_k \\tilde a}\n\\end{align}\nfor ${\\mathbf x}_k \\in \\mathfrak S$. Similarly, with the same choice of $\\varepsilon$, by considering only the $E_1E_2$-part of the expansion and exploiting $E_1^2=E_2^2=E_1^2E_2^2=0$, we obtain for $o=2$\n\\begin{align} \\label{eq_dJHD_topo}\n    d \\redPhiObjectiveFunction(\\phi)({\\mathbf x}_k) = \\frac{E_1E_2\\text{part}(\\redPhiObjectiveFunction(O_{k, h E_1 + h E_2}\\phi))}{2 h^2 \\, d_k \\tilde a}\n\\end{align}\nfor ${\\mathbf x}_k \\in \\mathfrak T^- \\cup \\mathfrak T^+$. In this case, the corresponding summed errors $e_S^{HD}(h)$ and $e_T^{HD}(h)$ vanish for arbitrary $h \\in \\mathbb R$. This is also observed numerically since both \\eqref{eq_dJHD_shape} and \\eqref{eq_dJHD_topo} suffer neither from a truncation nor a cancellation error. Figure \\ref{fig:verify}(c) shows that the the obtained results agree up to machine precision with the derivatives obtained by \\eqref{eq::formulaTopologicalDerivativePlus}, \\eqref{eq::formulaTopologicalDerivativeMinus}, and the respective formula for the shape derivative \\eqref{eq::formulaShapeDerivative}.\n\\subsection{Application of optimization algorithm to model problem}\nFinally we show the use of the numerical {topological-shape } derivative computed in Section \\ref{sec::numTopShapeDer} within a level-set based topology optimization algorithm. We first state the precise model problem, before introducing the algorithm and showing numerical results.\n\\subsubsection{Problem setting}\\label{sec::numericalProblemSetting}\nWe consider the unit square $D=[0,1]^2$ and minimize the objective function \\eqref{eq::ContinuousObjective} with $c_1=0$ and $c_2=1$ subject to the PDE constraint in \\eqref{eq::ContinuousProblem}. The chosen problem parameters are shown in Table \\ref{tab::parameters}.\n\\begin{table}\n\t\\centering\n\t\\begin{tabular}{cccccccc}\n\t\t\\toprule\n\t\t$\\tilde\\alpha_1$ & $\\tilde\\alpha_2$ &\n\t\t$\\alpha_1$ &  $\\alpha_2$ &\n\t\t$\\lambda_1$ &  $\\lambda_2$ &\n\t\t$f_1$ &  $f_2$ \\\\\n\t\t1 & 0.9 & 1 & 0.2 & 1 & 0.6 & 1 & 0.5 \\\\\n\t\t\\bottomrule\n\t\\end{tabular}\n\t\\caption{Problem parameters for the numerical experiment}\n\t\\label{tab::parameters}\n\\end{table}\nWe consider a mixed Dirichlet-Neumann problem by choosing\n\\begin{equation*}\n\\Gamma_D = \\{(x,y)\\in\\partial D| y=0 \\mbox{ or } y=1\\} , \\qquad\n\\Gamma_N = \\partial D \\setminus \\Gamma_D,\n\\end{equation*}\nand $g_D(x,y) = y$, $g_N(x,y) = 0$.\nIn order to define a desired state $\\hat u$, we choose a level-set function $\\phi_d$ which implies a desired shape $\\Omega_d$, compute the corresponding solution $u^*$ to $\\phi_d$ and set $\\hat u := u^*$. Then, by construction, $(\\Omega_d, \\hat u)$ is also the solution of the design optimization problem.\nIn the numerical tests we used five different meshes with 145, 545, 2113, 8321, and 33025 nodes respectively.\nFor each mesh we obtain $\\phi_d$ by interpolation of\n\\begin{equation}\n\\bar\\phi_d(x,y) = \\left((x-0.3)^2+(y-0.4)^2-0.2^2\\right)\\left((x-0.7)^2+(y-0.7)^2-0.1^2\\right).\n\\end{equation}\nThis yields that $\\Omega$ are two (approximated) circles with radii $0.2$ and $0.1$ respectively, see Figure \\ref{fig:holes-meshes}.\n\\begin{figure}\n\t\\begin{subfigure}[t]{0.19\\textwidth}\n\t\t\\includegraphics[width=\\textwidth]{holes-meshes-3}\n\t\t\\subcaption{145 nodes}\n\t\\end{subfigure}\\hfill\n\t\\begin{subfigure}[t]{0.19\\textwidth}\n\t\t\\includegraphics[width=\\textwidth]{holes-meshes-4}\n\t\t\\subcaption{545 nodes}\n\t\\end{subfigure}\\hfill\n\t\\begin{subfigure}[t]{0.19\\textwidth}\n\t\t\\includegraphics[width=\\textwidth]{holes-meshes-5}\n\t\t\\subcaption{2113 nodes}\n\t\\end{subfigure}\\hfill\n\t\\begin{subfigure}[t]{0.19\\textwidth}\n\t\t\\includegraphics[width=\\textwidth]{holes-meshes-6}\n\t\t\\subcaption{8321 nodes}\n\t\\end{subfigure}\\hfill\n\t\\begin{subfigure}[t]{0.19\\textwidth}\n\t\t\\includegraphics[width=\\textwidth]{holes-meshes-7}\n\t\t\\subcaption{33025 nodes}\n\t\\end{subfigure}\\hfill\n\t\\caption{The different meshes and corresponding sought shapes used in the numerical experiments.}\n\t\\label{fig:holes-meshes}\n\\end{figure}\n\\subsubsection{Optimization algorithm}\\label{sec::algorithm}\n\\newcommand{i}{i}\nThe optimization algorithm we use to solve the problem introduced in Section \\ref{sec::numericalProblemSetting} is inspired by \\cite{amstutz2006new}. \n\n\\begin{definition} \\label{def_locOpti}\n    We say a level set function $\\phi \\in S_h^1(D)$ is locally optimal for the problem described by $\\mathcal J$ if\n    \\begin{align}\n        \\begin{cases}\n            d \\mathcal J(\\phi)({\\mathbf x}_k) \\geq 0 & \\text{for } {\\mathbf x}_k \\in \\mathfrak T^- \\cup \\mathfrak T^+,\\\\\n            d \\mathcal J(\\phi)({\\mathbf x}_k) = 0 & \\text{for } {\\mathbf x}_k \\in \\mathfrak S.\n        \\end{cases}\n    \\end{align}\n\\end{definition}\n\n\n\nWe introduce the generalized numerical {topological-shape } derivative $G_\\phi \\in S_h^1(D)$ with\n\\begin{equation} \\label{eq_defGphi}\n G_\\phi({\\mathbf x}_k) = \\begin{cases}\n -\\min(d\\redPhiObjectiveFunction(\\phi)({\\mathbf x}_k),0) \\quad &\\text{for } {\\mathbf x}_k \\in \\mathfrak T^-, \\\\[8pt]\n \\min(d\\redPhiObjectiveFunction(\\phi)({\\mathbf x}_k),0) \\quad &\\text{for } {\\mathbf x}_k \\in \\mathfrak T^+, \\\\[8pt]\n -d\\redPhiObjectiveFunction(\\phi)({\\mathbf x}_k) \\quad &\\text{for } {\\mathbf x}_k \\in \\mathfrak S.\n \\end{cases}\n\\end{equation}\nWith this definition, we immediately get the following optimality condition:\n\\begin{lemma}\n    Let $\\phi \\in S_h^1(D)$ and\n\t\\begin{equation} \\label{eq_optiCondition}\n\t\t G_\\phi({\\mathbf x}_k) = 0, \\quad \\text{for } k = 1,\\dots ,M.\n\t\\end{equation}\n\tThen $\\phi$ is locally optimal in the sense of Definition \\ref{def_locOpti}.\n\\end{lemma}\nThe update of the level-set function based on the information of the {topological-shape } derivative is done by spherical linear interpolation (see also \\cite{amstutz2006new})\n\\begin{equation} \\label{eq_update_slerp}\n\t\\phi_{i+1} = \\frac{1}{\\sin(\\theta_i)}\\left(\\sin((1-\\kappa_i)\\theta_i) \\phi_i + \\sin(\\kappa_i\\theta_i)\\frac{G_{\\phi_i}}{\\|G_{\\phi_i}\\|_{L_{2}(D)}} \\right),\n\\end{equation}\nwhere $\\theta_i = \\mbox{arc cos}(( \\phi_i, G_{\\phi_i})_{L^2(D)})$ is the angle between the given level set function $\\phi_i$ and the sensitivity $G_{\\phi_i}$ in an $L^2(D)$-sense. Here, $\\kappa_i \\in (0,1)$ is a line search parameter which is adapted such that a decrease in the objective function is achieved. Note that, by construction, the update \\eqref{eq_update_slerp} preserves the $L^2(D)$-norm, $\\|\\phi_{i+1}\\|_{L^2(D)} = \\|\\phi_{i}\\|_{L^2(D)}$. As in \\cite{amstutz2006new,gangl2020multi}, we can also show that $\\phi$ is evolving along a descent direction:\n\\begin{lemma}\n    Let $\\phi_i, \\phi_{i+1} \\in S_h^1(D)$ two subsequent iterates related by \\eqref{eq_update_slerp}. Then we have for ${\\mathbf x}_k \\in \\mathfrak T^-(\\phi_i) \\cup \\mathfrak T^+(\\phi_i)$\n    \\begin{align}\n        \\phi_{i}({\\mathbf x}_k) > 0 > \\phi_{i+1}({\\mathbf x}_k) \\Longrightarrow d \\mathcal J(\\phi_i)({\\mathbf x}_k) <0  \\label{eq_descentTplus}\\\\\n        \\phi_{i}({\\mathbf x}_k) < 0 < \\phi_{i+1}({\\mathbf x}_k) \\Longrightarrow d \\mathcal J(\\phi_i)({\\mathbf x}_k) <0 \\label{eq_descentTminus}\n    \\end{align}\n\\end{lemma}\n\\begin{proof}\n    Let ${\\mathbf x}_k \\in \\mathfrak T^+(\\phi_i)$, i.e. $\\phi_i({\\mathbf x}_k) > 0$ and assume that $\\phi_{i+1}({\\mathbf x}_k) < 0$. Since $\\mbox{sin}(\\theta)>0$ and $\\mbox{sin}(s\\theta)>0$ for all $\\theta \\in (0, \\pi)$ and $s\\in (0,1)$, it follows from \\eqref{eq_update_slerp} that $G_{\\phi_i}(x_k) < 0$ and thus, by \\eqref{eq_defGphi}, $d\\mathcal J(\\phi_i)({\\mathbf x}_k) < 0$ as claimed in \\eqref{eq_descentTplus}. An analogous argument yields \\eqref{eq_descentTminus}.\n\\end{proof}\n\nWe can also show that $G_\\phi$ constitutes a descent direction for ${\\mathbf x}_k \\in \\mathfrak S$.\n\\begin{lemma}\n    Let $\\phi \\in S^1_h(D)$ and suppose that\n    \\begin{align}   \\label{eq_limsubsuper}\n    \\underset{\\varepsilon \\searrow 0}{\\mbox{lim }} \\frac{\\mathcal J(S_{k,\\varepsilon} \\phi)-\\mathcal J(\\phi)}{|\\Omega(S_{k,\\varepsilon} \\phi) \\triangle \\Omega(\\phi)|} = - \\underset{\\varepsilon \\nearrow 0}{\\mbox{lim }} \\frac{\\mathcal J(S_{k,\\varepsilon} \\phi)-\\mathcal J(\\phi)}{|\\Omega(S_{k,\\varepsilon} \\phi) \\triangle \\Omega(\\phi)|}.\n    \\end{align}\n    Let ${\\mathbf x}_k \\in \\mathfrak S(\\phi)$ be fixed and let $\\phi^{\\kappa}$ be the level set function according to \\eqref{eq_update_slerp} with line search parameter $\\kappa \\in (0,1)$ that is updated only in ${\\mathbf x}_k$, i.e.,\n        $\\phi^{\\kappa} = a(\\kappa) \\phi + b(\\kappa) G_{\\phi} \\varphi_k$\n    with $a(\\kappa) = \\sin((1-\\kappa)\\theta) \/ \\sin(\\theta)$ and $b(\\kappa) = \\sin(\\kappa \\theta) \/ \\left( \\sin(\\theta) \\|G_{\\phi}\\|_{L_{2}(D)} \\right)$. Moreover assume that $|d \\mathcal J(\\phi)({\\mathbf x}_k)|>0$. Then there exists $\\overline \\kappa \\in (0,1)$ such that for all $\\kappa \\in (0,\\overline \\kappa)$\n    \\begin{align*}\n        \\mathcal J(\\phi^\\kappa) < \\mathcal J(\\phi).\n    \\end{align*}\n\\end{lemma}\n\n\\begin{proof}\n    Suppose that $0 > d \\mathcal J(\\phi)({\\mathbf x}_k)$. Then it follows from \\eqref{eq::topDerivative} that $\\mathcal J(\\phi + \\varepsilon \\varphi_k) < \\mathcal J(\\phi)$ for $\\varepsilon >0$ small enough.\n    Thus, since $a(\\kappa), b(\\kappa)>0$ for $\\theta \\in (0, \\pi)$ and $\\kappa \\in (0,1)$ and since $\\mathcal J(\\phi^\\kappa) = \\mathcal J(\\frac{1}{a(\\kappa)} \\phi^\\kappa)$, it follows\n    \\begin{align*}\n        \\mathcal J(\\phi^\\kappa) = \\mathcal J(\\phi + b(\\kappa)\/a(\\kappa) G_{\\phi} \\varphi_k) = \\mathcal J(\\phi_i - b(\\kappa)\/a(\\kappa) d \\mathcal J(\\phi_i)({\\mathbf x}_k) \\varphi_k) < \\mathcal J(\\phi_i)\n    \\end{align*}\n    for $\\kappa>0$ small enough since $-b(\\kappa)\/a(\\kappa) d\\mathcal J(\\phi_i)({\\mathbf x}_k)>0$ and $b(\\kappa)\/a(\\kappa) \\rightarrow 0$ as $\\kappa \\searrow 0$.\n    On the other hand, if $0 < d \\mathcal J(\\phi_i)({\\mathbf x}_k)$, it follows from \\eqref{eq::topDerivative} and \\eqref{eq_limsubsuper} that $\\mathcal J(\\phi + \\varepsilon \\varphi_k) < \\mathcal J(\\phi)$ for $\\varepsilon <0$ small enough and further for $\\kappa$ small enough\n    \\begin{align*}\n        \\mathcal J(\\phi^\\kappa)  = \\mathcal J(\\phi - b(\\kappa)\/a(\\kappa) d \\mathcal J(\\phi)({\\mathbf x}_k) \\varphi_k) < \\mathcal J(\\phi).\n    \\end{align*}\n\\end{proof}\n\n\\begin{remark}\n    In the continuous setting, the property corresponding to \\eqref{eq_limsubsuper} is fulfilled for smooth domains which can be seen as follows. Let $\\Omega_t^V = (\\textrm{id}+t V)(\\Omega)$ and note that $\\Omega_{-t}^V = \\Omega_t^{-V}$. Then, by Lemma \\ref{LEM_SYMDIF},\n    \\begin{align*}\n        \\underset{s \\nearrow 0}{\\mbox{lim }} \\frac{|\\Omega_s^V \\triangle \\Omega|}{s}\n     = -\\underset{t \\searrow 0}{\\mbox{lim }} \\frac{|\\Omega_{t}^{-V} \\triangle \\Omega|}{t}\n     = - \\int_{\\partial \\Omega} |V \\cdot n| \\; \\mbox d S_x = -\\underset{s \\searrow 0}{\\mbox{lim }} \\frac{|\\Omega_{s}^{V} \\triangle \\Omega|}{s}\n    \\end{align*}\n    and, assuming differentiability of $s \\mapsto \\mathfrak g(\\Omega_s^V)$,\n    \\begin{align*}\n        \\underset{s \\searrow 0}{\\mbox{lim }} \\frac{\\mathfrak g(\\Omega_s^V) - \\mathfrak g(\\Omega)}{|\\Omega_s^V \\triangle \\Omega|} =\n        \\frac{\\underset{s \\rightarrow 0}{\\mbox{lim }} \\left( \\mathfrak g(\\Omega_s^V) - \\mathfrak g(\\Omega)\\right) \/ s}{\\underset{s \\searrow 0}{\\mbox{lim }} |\\Omega_s^V \\triangle \\Omega| \/ s} = -\\underset{s \\nearrow 0}{\\mbox{lim }} \\frac{\\mathfrak g(\\Omega_s^V) - \\mathfrak g(\\Omega)}{|\\Omega_s^V \\triangle \\Omega|}.\n    \\end{align*}\n\n\n    In the discrete case, however, there may occur situations where the limits in \\eqref{eq_limsubsuper} do not coincide. This can be the case in particular in situations where $\\phi({\\mathbf x}_k) = 0$. We remark that this issue seemed not to cause problems in our numerical experiments.\n\n\\end{remark}\n\n\\begin{remark}\n    In practice it turned out to be advantageous to include a smoothing step of the level set function. Thus, we chose the following update strategy: We first set\n    \\begin{equation*}\n        \\psi = \\frac{1}{\\sin(\\theta_i)}\\left(\\sin((1-\\kappa_i)\\theta_i) \\phi_i + \\sin(\\kappa_i\\theta_i)\\frac{G_{\\phi_i}}{\\|G_{\\phi_i}\\|_{L_{2}(D)}} \\right),\n    \\end{equation*}\n    with the same notation as above before smoothing the level set function in $\\mathfrak T^-(\\psi) \\cup \\mathfrak T^+(\\psi)$ by\n    \\begin{equation}\n\t\\hat\\psi({\\mathbf x}_k) = \\begin{cases}\n\t\t\\frac{\\sum_{i\\in R_{{\\mathbf x}_k}} \\psi({\\mathbf x}_i) }{|R_{{\\mathbf x}_k}|} \\quad &\\text{for } {\\mathbf x}_k \\in \\mathfrak T^-(\\psi)\\cup\\mathfrak T^+(\\psi), \\\\[8pt]\n\t\t\\psi({\\mathbf x}_k) \\quad &\\text{for } {\\mathbf x}_k \\in \\mathfrak S.\n\t\\end{cases}\n    \\end{equation}\n    Finally, the level-set function is normalized and the next iterate is given by\n    \\begin{equation}\n        \\phi_{i+1} = \\frac{ \\hat\\psi}{\\| \\hat\\psi\\|_{L_{2}(D)}}.\n    \\end{equation}\n\\end{remark}\n\\color{black}\n\n\\subsubsection{Numerical results}\nAs an initial design for the optimisation, we take the empty set, $\\Omega = \\emptyset$. This is realized by choosing $\\phi_0 = 1\/\\|1\\|_{L_2(D)}$ as the initial level set function. We use the algorithm outlined in Section \\ref{sec::algorithm}\nto update this level set function. We terminated the algorithm after the fixed number of 800 iterations. The final as well as some intermediate configurations are illustrated in Figures  \\ref{fig:holes-pics2-3}-\\ref{fig:holes-pics2-7} for the five different levels of discretization.\n\\newcommand{\\myincludepic}[2]{\\begin{subfigure}{0.49\\linewidth}\\centering\\includegraphics[width=0.7\\linewidth,trim=0cm 0cm 0cm 0cm,clip]{holes-pics2-1e+20_1_1-#1}\\caption{#2 iteration}\n\\end{subfigure}}\n\\newcommand{100}{100}\n\\newcommand{\\picText}[1]{\\centering\n\t\\includegraphics[width=0.5\\linewidth]{colorbar}\n\t\\caption{Evolution of the level-set function for the #1 nodes mesh}}\n\\begin{figure\n\t\\myincludepic{3-1}{1}\n\t\\myincludepic{3-2}{2}\n\t\n\t\\myincludepic{3-10}{10}\n\t\\myincludepic{3-20}{20}\n\t\n\t\\myincludepic{3-100}{100}\n\t\\myincludepic{3-800}{800}\n\t\n\t\n\t\\picText{145}\n\t\\label{fig:holes-pics2-3}\n\\end{figure}\n\\begin{figure\n\t\\myincludepic{4-1}{1}\n\t\\myincludepic{4-2}{2}\n\t\n\t\\myincludepic{4-10}{10}\n\t\\myincludepic{4-20}{20}\n\t\n\t\\myincludepic{4-100}{100}\n\t\\myincludepic{4-800}{800}\n\t\\picText{545}\n\t\\label{fig:holes-pics2-4}\n\\end{figure}\n\\begin{figure\n\t\\myincludepic{5-1}{1}\n\t\\myincludepic{5-2}{2}\n\t\n\t\\myincludepic{5-10}{10}\n\t\\myincludepic{5-20}{20}\n\t\n\t\\myincludepic{5-100}{100}\n\t\\myincludepic{5-800}{800}\n\t\\picText{2113}\n\t\\label{fig:holes-pics2-5}\n\\end{figure}\n\\begin{figure\n\t\\myincludepic{6-1}{1}\n\t\\myincludepic{6-2}{2}\n\t\n\t\\myincludepic{6-10}{10}\n\t\\myincludepic{6-20}{20}\n\t\n\t\\myincludepic{6-100}{100}\n\t\\myincludepic{6-800}{800}\n\t\n\t\\picText{8321}\n\t\\label{fig:holes-pics2-6}\n\\end{figure}\n\\begin{figure\n\t\\myincludepic{7-1}{1}\n\t\\myincludepic{7-2}{2}\n\n\t\\myincludepic{7-10}{10}\n\t\\myincludepic{7-20}{20}\n\n\t\\myincludepic{7-100}{100}\n\t\\myincludepic{7-800}{800}\n\n\t\\picText{33025}\n\t\\label{fig:holes-pics2-7}\n\\end{figure}\n\\begin{figure\n\t\\centering\n\t\\begin{tabular}{cc}\n\t\t\\includegraphics[width=0.49\\linewidth]{holes_history_objective-1e+20_1_1}\n\t\t&\n\t\t\\includegraphics[width=0.49\\linewidth]{holes_history_derivative2-1e+20_1_1}\n\t\t\\\\\n\t\t(a) & (b) \n\t\\end{tabular}\n\t\\caption{Evolution of the objective function (a) and the norm of the {topological-shape } derivative (b) in course of the optimization.}\n\t\\label{fig:holeshistoryobjective-1}\n\\end{figure}\n\nWe observe that in all cases the two circles are recovered with high accuracy. In Figure \\ref{fig:holeshistoryobjective-1} the evolution of the objective function as well as of the norm of the generalized numerical {topological-shape } derivative is plotted. We observe that objective function decreases fast and after 800 iterations a reduction by a factor of approximately $10^{-5}-10^{-8}$ could be achieved. Moreover, we observe that the norm of the {topological-shape } derivative decreases continuously, more and more approaching the optimality condition \\eqref{eq_optiCondition}.\n\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\n\nRecently, a lot of interest has grown around the possibility of\napplying string inspired techniques to the non-perturbative\nregime of QCD. The starting point is the AdS\/CFT correspondence\n\\cite{Malda}, a conjectured duality between a maximally\nsupersymmetric strongly coupled conformal field theory and the\nsupergravity limit of type~IIB string theory, which involves\ntheories different from QCD. Further developments \\cite{adsqcd}\nhave tried to apply the correspondence to QCD, induced by the\nevidence of the existence of a window of energy in which QCD\nshows an approximate conformal behaviour \\cite{confqcd}. These\ndevelopments have taken different directions. The framework\nthrough which I move here is the so-called soft~wall model of\nAdS\/QCD \\cite{softwall}, a phenomenological model originally\nbuilt to holographically describe chiral symmetry breaking and\nthen adapted to several strong interaction processes. For a list\nof other approaches the reader can refer to \\cite{other}.\n\nIn the following, I discuss the scalar glueball sector and how\nthe spectrum and the two-point correlation function are\nrepresented in the soft wall model. Then, I comment on the\nresults, comparing them with current phenomenology and lattice\ndata.\n\n\\section{Framework}\n\nThe considered  model is defined in a $5d$ curved space (the\nbulk) with metric:\n\\begin{equation}\nds^2=g_{MN} dx^M\ndx^N=\\frac{R^2}{z^2}\\,\\big(\\eta_{\\mu\\nu}dx^{\\mu}dx^{\\nu}+dz^2\\big)\n\\label{metric}\n\\end{equation}\nwith $\\eta_{\\mu\\nu}=\\mbox{diag}(-1,+1,+1,+1)$; $R$ is the AdS\ncurvature radius, and the coordinate $z$  runs in the range\n$0\\leq z < +\\infty$. QCD is supposed to live on the boundary\n$z=0$, where the element $\\eta_{\\mu\\nu}dx^{\\mu}dx^{\\nu}$ describes a\nflat Minkowski space.\n\nIn addition to the AdS metric, the model is characterized by the\npresence of a background dilaton field:\n\\begin{equation}\n\\Phi(z)=(c z)^2 \\label{dilaton}\n\\end{equation}\nexponentially coupled to the fields, whose functional form is\nchosen in such a way to have linear Regge trajectories for light\nvector mesons \\cite{softwall}; $c$ is a dimensionful parameter\nsetting the scale of QCD quantities and it is of ${\\cal\nO}(\\Lambda_{QCD})$. It is the responsible of the breaking of\nconformal symmetry and it is fixed by the experimental slope of\nthe rho mesons trajectory.\n\n\n\n\\section{Scalar glueballs}\n\n$0^{++}$ glueballs can be described in QCD by the dimension four\noperator ${\\cal O}_S(x)=\\mbox{Tr}[\\beta(\\alpha_s)G^{a\\,\\mu\\nu}G_{\\mu\\nu}^a]$.\nIn the five dimensional theory its dual field is a massless\nscalar $Y(x,z)$ \\cite{glueballspectrum}, whose action is given\nby:\n\\begin{equation}\\label{action}\n  S=-\\displaystyle\\frac{1}{2k}\\int\n  d^5x\\sqrt{-g}\\,e^{-\\phi}g^{MN}(\\partial_MY)(\\partial_NY)\n\\end{equation}\nwhere $k$ is a parameter introduced to give the correct\ndimension to the action. The AdS\/CFT dictionary states that this\naction is equivalent to the QCD partition function, in which the\nsource of ${\\cal O}_S(x)$ is the boundary value $Y_0(x)$ of the\nfield $Y(x,z)$. The following relation can be written:\n\\begin{equation}\n  Y(x,z)=\\int d^4x^\\prime\n  K(x-x^\\prime,z)Y_0(x^\\prime)\\;\\;,\n\\end{equation}\nwhere the function $K(x-x^\\prime,z)$ is called bulk-to-boundary\npropagator, since it links the fields in the bulk with the\nsources on the boundary.\n\nIt is possible to obtain QCD correlation functions functionally\nderiving the action (\\ref{action}) with respect to $Y_0(x)$. The\ntwo-point function obtained in this way is, in the limit\n$q^2\\rightarrow+\\infty$ \\cite{glueballcorrel,forkel}:\n\\begin{eqnarray}\\label{piads}\n  \\Pi_{AdS}(q^2) & = & \\displaystyle\\frac{R^3}{k}\\biggl\\{q^4\\cdot\\displaystyle\\frac{1}{8}\\left[2-2\\gamma_E+\\ln4-\\ln(q^2\/\\nu^2)\\right]+\\nonumber\\\\\n  && +q^2\\left[-\\displaystyle\\frac{\\nu^2}{2}+\\displaystyle\\frac{c^2}{4}\\left(1-4\\gamma_E+2\\ln4-2\\ln(q^2\/\\nu^2)\\right)\\right]+\\\\\n  && -\\displaystyle\\frac{5c^4}{6}+\\displaystyle\\frac{2c^6}{3q^2}+{\\cal\n  O}\\left(\\displaystyle\\frac{1}{q^4}\\right)\\biggr\\}\\;\\;,\\nonumber\n\\end{eqnarray}\nto be compared with the QCD result \\cite{Paver}:\n\\begin{eqnarray}\\label{piqcd}\n  \\Pi_{QCD}(q^2) & = & 2\\biggl(\\displaystyle\\frac{\\beta_1}{\\pi}\\biggr)^2\\biggl(\\displaystyle\\frac{\\alpha_s}{\\pi}\\biggr)^2\\,q^4\\biggl(-\\ln{\\big(\\displaystyle\\frac{q^2}{\\nu^2}\\big)}+2\n  -\\displaystyle\\frac{1}{\\varepsilon^{\\prime}}\\biggr)+4\\beta_1^2\\,\\Big(\\frac{\\alpha_{s}}{\\pi}\\Big)\\langle{\\cal\n  C}_4\\rangle\\nonumber\\\\\n  &&+8\\beta_1^2\\Big(\\frac{\\alpha_s}{\\pi}\\Big)^2\\frac{\\langle{\\cal C}_6\\rangle}{q^2}+{\\cal\n  O}\\left(\\displaystyle\\frac{1}{q^4}\\right)\\; .\n\\end{eqnarray}\nMatching (\\ref{piads}) with (\\ref{piqcd})\n($\\frac{R^3}{8k}=2(\\beta_1\/\\pi)^2(\\alpha_s\/\\pi)^2$) one gets the fully\nanalytic form of the correlator. By casting it in the form:\n\\begin{equation}\n  \\Pi_{AdS}(q^2)=\\displaystyle\\sum_{n=0}^\\infty\\displaystyle\\frac{f_n^2}{q^2+m_n^2+i\\varepsilon}\n\\end{equation}\nit is possible to find the poles $m_n^2=4c^2(n+2)$ and the\nrelated residues $f_n^2=\\langle0|{\\cal\nO}_S(0)|n\\rangle^2=\\frac{8R^3c^6}{k}\\,(n+1)(n+2)$, corresponding\nto the mass spectrum and the decay constants of the scalar\nglueballs. The results for the lowest lying state are:\n\\begin{itemize}\n  \\item $M_{0^{++}}=1.089$~GeV\n  \\item $f_{0^{++}}=\\langle0|{\\cal\n  O}_S(0)|n=0\\rangle=0.763$~GeV$^3$~~~.\n\\end{itemize}\nThe $0^{++}$ glueball turns out to be heavier than the $\\rho$\nmeson, as expected by phenomenology, but slightly lighter with\nrespect to the results from lattice simulations\n\\cite{latticeglueball}.\n\nAnother point is that in the large $q^2$ expansion there is a\ndimension~two condensate, absent in QCD since there are no ways\nto construct scalar local gauge invariant quantities with that\ndimension \\cite{zakcond2}.\n\nFinally, the dimension four condensate $\\langle{\\cal\nC}_4\\rangle$ turns out to be negative in this picture, at odds\nwith the commonly used value\n$\\langle(\\alpha_s\/\\pi)G^2\\rangle\\simeq0.012$~GeV$^4$\n\\cite{khodjamirian}.\n\nA discussion on how to fix some of the problems exposed above\ncan be found in \\cite{glueballcorrel, scalarmesons}.\n\n\n\n\n\n\n\n\n\\acknowledgments I thank P.~Colangelo, F.~De~Fazio, F.~Giannuzzi\nand F.~Jugeau for collaboration.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nFor most of the history of exploring electromagnetic phenomena, it had been\nbelieved that knowledge of the electric and magnetic fields in a physical\nproblem is sufficient to define the problem. The scalar and vector potentials,\nwhose spatial and temporal derivatives yield the fields, had been regarded as\nauxiliary quantities that are useful but not essential. This conclusion was\napparently reinforced by the fact that the set of potentials to represent the\nfields is not unique. Subject to modest restrictions, there exist\ntransformations (called gauge transformations) to other sets of potentials\nthat produce the same fields.\n\nThis seemingly straightforward situation was upset by the Aharonov-Bohm effect\n\\cite{es,ab}. The simplest realization of this phenomenon is that an electron\nbeam passing outside a solenoid containing a magnetic field will be deflected,\neven though there is no field outside the solenoid. There is, however, a\npotential outside the solenoid that suffices to explain the deflection. The\neffect remained controversial until it was verified experimentally\n\\cite{tonomura}. This has the basic consequence that potentials are more\nfundamental than fields. The Aharonov-Bohm effect is founded on a single\nexplicitly quantum-mechanical phenomenon, and commentary about its\nsignificance has been in terms of quantum mechanics \\cite{furry,vaidman}.\n\nThe concept explored here is different from that of the Aharonov-Bohm effect,\nand much more consequential. It is shown that a change of gauge can introduce\na violation of basic symmetries, even when the usual constraints on allowable\ngauge transformations have been satisfied. Furthermore, these symmetry\nviolations can occur in classical physics as well as in quantum mechanics with\nexternal electromagnetic fields. The consequences of these results are\nprofound. There exist contrasting sets of potentials that yield exactly the\nsame fields, but where one set is consonant with physical requirements but\nanother is not. This proves directly that the selection of the proper set of\npotentials is the decisive matter, since the predicted fields are the same in\nboth cases. A corollary is that gauge transformations are not unitary\ntransformations. This contradicts the field-based assumption that gauge\ntransformations must preserve the values of measurable quantities. The\nassumption of unitarity (often implicit) underlies some of the influential\narticles that have been published on the subject of gauge choice.\n\nExamples employed here to demonstrate the primacy of potentials -- a charged\nparticle in interaction with a constant electric field, and a bound electron\nsubjected to a plane-wave field -- represent basic physics problems, unlike\nthe narrow specificity of the Aharonov-Bohm effect. An important feature\nrevealed by these examples is that, although the physical consequences of\nstatic or quasistatic-electric (QSE) fields are quite similar to plane-wave\neffects at the low field intensities that exist in the usual atomic, molecular\nand optical (AMO) physics processes, at the high field intensities now\nachievable with laser fields, they can be profoundly different. These\ndifferences have yet to be fully appreciated in the AMO literature, leading to\nmisconceptions that persist nearly forty years after the first laboratory\nobservation \\cite{ati} of explicit intense-field effects.\n\nA concept introduced here is that of a hierarchy of physical quantities. Since\npotentials are primary and fields are secondary, it follows that energies are\nprimary and forces are secondary. This ranking resolves the long-standing\nmystery about why the Schr\\\"{o}dinger equation cannot be written directly in\nterms of electric and magnetic fields, even though the fields were\nconventionally assumed to be basic physical quantities. All attempts to\nexpress the Schr\\\"{o}dinger equation directly in terms of fields have resulted\nin nonlocality \\cite{mandelstam,dewitt,belinfante,levy,rohrlich,priou}. This\napparent anomaly is one of the enduring puzzles of quantum mechanics. A\nhierarchy of physical quantities also serves to clarify the current confused\nsituation in the strong-laser community, where field-based intensity measures\nare employed that are inconsistent with energy-based criteria. One important\nexample of this misdirection is the introduction of the concept of the\n\\textquotedblleft critical electric field\\textquotedblrigh\n\\ \\cite{sauter,schwinger} into the discussion of strong laser effects, despite\nthe fact that lasers produce transverse fields and the critical field has\nwell-defined meaning only for longitudinal fields. The basic differences\nbetween transverse fields and longitudinal fields are also exhibited in the\nmacroscopic world in terms of the properties of extremely-low-frequency radio waves.\n\nThe range of applicability of the concepts examined here is very large, since\nit encompasses classical electromagnetism, and also relativistic and\nnonrelativistic quantum mechanics in which the electromagnetic field is\nregarded as an external classical field.\n\nThe limitation to external classical fields is significant, since it places\nthe present work outside the scope of a Yang-Mills theory \\cite{yangmills}.\nQuantum electrodynamics (QED) is a Yang-Mills theory, but standard QED does\nnot incorporate strong-field theory. Strong-field theories contain an apparent\nintensity-dependent \\textquotedblleft mass shift\\textquotedblright, discovered\nindependently by Sengupta \\cite{sengupta} and by the present author\n\\cite{hrdiss,hr62}. This mass shift can be explained in terms of a demand for\ncovariance in external fields \\cite{hrup}. The mass shift is a fundamental\nphenomenon in strong-field physics \\cite{hrje}, but it does not exist in the\ncontext of the quantized fields of QED \\cite{hrdiss,hr62,jehr}.\n\nSection II below discusses two basic examples that exhibit pairs of potential\nchoices that describe exactly the same fields, but where one set of potentials\nis physically acceptable and the other is not. One example is the simplest\npossible case: the classical interaction of a charged particle with a constant\nelectric field. Of the two possibilities for gauge choice, one contradicts\nNoether's Theorem \\cite{noether}. There is no such problem with the\nalternative gauge. The next example is the interaction of a charged particle\nwith a plane-wave field, such as the field of a laser. In this case, the key\nfactor is that the symmetry principle in question -- preservation of the\npropagation property of a plane-wave field -- is not often mentioned, even in\nthe context of very strong fields where this symmetry is crucial \\cite{hrup}.\nThe demand for the preservation of the propagation property imposes a strong\nlimitation on possible gauge transformations. In the presence of a\nsimultaneous scalar interaction, like a Coulomb binding potential, only the\nradiation gauge is possible \\cite{hrgge}. This limitation to a unique gauge\nexists in both classical and quantum domains. An important aspect of this\nproblem is that the widely-used dipole approximation in the description of\nlaser-caused effects suppresses this symmetry, thus masking the errors that\nfollow from ignoring this basic property. Both the constant-electric-field and\nthe propagating-field examples admit of only one possible gauge. This lack of\ngauge-equivalent alternatives is extremely important, since both situations\nrepresent commonplace physical environments. This is in contrast to the\nspecialized Aharonov-Bohm effect.\n\nAn immediate consequence of the demonstrated fact that some nominally valid\ngauge transformations can have unphysical consequences is that a gauge\ntransformation is not a unitary transformation. This is discussed in Section\nIII, where it is shown to be related to the construction of exact transition amplitudes.\n\nThe impossibility of writing the Schr\\\"{o}dinger equation directly in terms of\nelectric and magnetic fields is discussed in Section IV. This is further\nevidence of the basic nature of potentials, and it also supports the notion of\na hierarchy of physical phenomena. Quantum mechanics can be constructed from\nclassical mechanics when expressed in terms of system functions like the\nLagrangian or Hamiltonian, whereas a Newtonian form of classical mechanics has\nsuch an extension only by extrapolation to a desired result. System functions\nare related to energies, whereas Newtonian physics involves forces, and forces\nare directly connected with electric and magnetic fields, as shown by the\nLorentz force expression.\n\nThe ambiguities inherent in the view that the $\\mathbf{E}^{2}-\\mathbf{B}^{2}$\nand $\\mathbf{E}\\cdot\\mathbf{B}$ Lorentz invariants reliably characterize the\nelectrodynamic environment is another topic examined in Section IV. (The\nLorentz invariants, as are all electromagnetic quantities throughout this\npaper, are stated in Gaussian units.) This concept has an important failure\nwhen both invariants are zero, since it associates propagating plane-wave\nfields with the completely different constant crossed fields. The\ncommonly-held assumption that constant crossed fields are a zero-frequency\nlimit of plane-wave fields (see, for example, Refs. \\cite{nikrit66,ritus85}),\nis shown to be untenable.\n\nAnother topic in Section IV is the apparent dominance of the electric\ncomponent of the Lorentz force expression at low frequencies, a field-related\nconception that draws attention away from the rising importance of the\nmagnetic component of a propagating field as the frequency declines\n\\cite{hr75,hr82}. Inappropriate emphasis on the electric field has caused\nconceptual errors even in relativistic phenomena, as discussed in Section IV\nin the context of vacuum pair production. A potentials-related approach\nobviates this electric-field-dominance hazard. The concept of the\n\\textit{critical field} is often mentioned in connection with strong-laser\ninteractions \\cite{arb,ssb}. The critical field refers to that value of\nelectric field at which spontaneous pair production from the vacuum becomes\nsignificant. It has been applied to laser fields in terms of the electric\ncomponent of a plane-wave field. This is devoid of meaning for laser beams in\nvacuum because pure electric fields and plane-wave fields are disjoint\nconcepts, as is evident from Section II. The conservation conditions\napplicable to critical-field considerations cannot be satisfied by a laser.\nEven were the electric component of a laser field equal to the critical field,\npair production cannot occur because pair production from the vacuum by a\nlaser pulse cannot occur unless there is a counter-propagating field to\nprovide the necessary conservation of momentum \\cite{hrdiss,hr62,hr71,burke}.\nThe fact that photons convey momentum is incompatible with the concept of a\ncritical electric field for laser-induced processes.\n\nSection IV concludes with the practical problem of communicating with\nsubmerged submarines. This has been done under circumstances that emphasize\nhow different plane-wave fields are from QSE fields\n\nSection V explores the notion of a hierarchy of physical quantities.\nPotentials are directly related to energies, so they are identified as primary\nquantities. Fields are derived from potentials, so they are secondary. Forces\nare determined by fields and so forces are also secondary. The hierarchy\nconcept is related to classical mechanics in that Newtonian physics is couched\nin terms of forces, and so it is secondary to versions of classical mechanics\nbased on energy-based system functions like the Lagrangian and Hamiltonian.\nMechanics formulated with system functions infer Newtonian mechanics, but the\nconverse is not true.\n\n\\section{Symmetry violation}\n\nThe two examples presented have an important qualitative difference. The first\nexample -- a constant electric field -- is so elementary that the proper\nchoice of potentials is obvious, and there is no motivation to explore the\nproperties of the symmetry-violating alternative potentials. The next example\nis quite different in that the improper choice of potentials is very\nattractive to a laser-physics community that is accustomed to the dipole\napproximation. The requisite propagation property never appears within the\ndipole approximation, and its violation is thereby invisible.\n\nA preliminary step is to introduce the units and conventions employed in this\narticle, and to add some general remarks about terminology.\n\n\\subsection{Units and conventions}\n\nGaussian units are employed for all electromagnetic quantities. The\nexpressions for the electric field $\\mathbf{E}$ and magnetic field\n$\\mathbf{B}$ in terms of the scalar potential $\\phi$ and the 3-vector\npotential $\\mathbf{A}$ ar\n\\begin{equation}\n\\mathbf{E}=-\\mathbf{\\nabla}\\phi\\mathbf{-}\\frac{1}{c}\\partial_{t\n\\mathbf{A},\\qquad\\mathbf{B}=\\mathbf{\\nabla\\times A.} \\label{a\n\\end{equation}\nA gauge transformation generated by the scalar function $\\Lambda$ i\n\\begin{equation}\n\\widetilde{\\phi}=\\phi+\\frac{1}{c}\\partial_{t}\\Lambda,\\qquad\\widetilde\n{\\mathbf{A}}=\\mathbf{A-\\nabla}\\Lambda, \\label{b\n\\end{equation}\nwhere $\\Lambda$ must satisfy the homogeneous wave equatio\n\\begin{equation}\n\\left(  \\frac{1}{c^{2}}\\partial_{t}^{2}-\\mathbf{\\nabla}^{2}\\right)\n\\Lambda=\\partial^{\\mu}\\partial_{\\mu}\\Lambda=0. \\label{c\n\\end{equation}\nRelativistic quantities are expressed with the time-favoring Minkowski metric,\nwith the signature $\\left(  +---\\right)  $, where the scalar product of two\n4-vectors $a^{\\mu}$ and $b^{\\mu}$ i\n\\begin{equation}\na\\cdot b=a^{\\mu}b_{\\mu}=a^{0}b^{0}-\\mathbf{a\\cdot b}. \\label{e\n\\end{equation}\nThe 4-vector potential $A^{\\mu}$ incorporates the scalar and 3-vector\npotentials a\n\\begin{equation}\nA^{\\mu}:\\left(  \\phi,\\mathbf{A}\\right)  . \\label{g\n\\end{equation}\nIn 4-vector notation, the two gauge transformation expressions in Eq.\n(\\ref{b}) become the single expressio\n\\begin{equation}\n\\widetilde{A}^{\\mu}=A^{\\mu}+\\partial^{\\mu}\\Lambda. \\label{h\n\\end{equation}\nBoth the initial and gauge-transformed 4-vector potentials must satisfy the\nLorenz conditio\n\\begin{equation}\n\\partial^{\\mu}A_{\\mu}=0,\\qquad\\partial^{\\mu}\\widetilde{A}_{\\mu}=0. \\label{j\n\\end{equation}\nThe propagation 4-vector $k^{\\mu}$ consists of the propagation 3-vector\n$\\mathbf{k}$ as the space part, and the amplitude $\\left\\vert \\mathbf{k\n\\right\\vert =\\omega\/c$ as the time component\n\\begin{equation}\nk^{\\mu}:\\left(  \\omega\/c,\\mathbf{k}\\right)  . \\label{k\n\\end{equation}\nThe 4-vector $k^{\\mu}$ defines the light cone and, according to the rule\n(\\ref{e}), it is \\textquotedblleft self-orthogonal\\textquotedblright\n\\begin{equation}\nk^{\\mu}k_{\\mu}=\\left(  \\omega\/c\\right)  ^{2}-\\mathbf{k}^{2}=0, \\label{l\n\\end{equation}\nwhich is an important possibility in this non-Euclidean space.\n\nThe concept of transversality refers to the property of plane-wave fields\nexpressed in a relativistic context as \\textit{covariant transversality\n\\begin{equation}\nk^{\\mu}A_{\\mu}=0, \\label{m\n\\end{equation}\nin terms of the 4-potential $A^{\\mu}$. In many textbooks on classical\nelectromagnetic phenomena, transversality is defined as \\textit{geometrical\ntransversality\n\\begin{equation}\n\\mathbf{k\\cdot E}=0\\text{ and }\\mathbf{k\\cdot B}=0, \\label{n\n\\end{equation}\nin terms of the electric and magnetic fields. It can be shown that covariant\ntransversality infers geometrical transversality.\n\n\\subsection{Terminology}\n\nDespite the conclusion in this paper that potentials are more basic than\nfields, it is not possible to avoid the use of the term \\textquotedblleft\nfield\\textquotedblright\\ in a generic sense. For example, one important\nconclusion reached herein is that vector and scalar potentials provide more\ninformation than do electric and magnetic fields in the description of the\neffects of laser fields. In the preceding sentence, the term \\textquotedblleft\nlaser field\\textquotedblright\\ is used generically to identify the radiation\ncreated by a laser, despite the particular result that potentials are the\nbetter approach in the description of that radiation. A similar problem arises\nwhen it is concluded that the dipole approximation amounts to the replacement\nof the \\textquotedblleft transverse field\\textquotedblright\\ of a laser by the\nmore elementary \\textquotedblleft longitudinal field\\textquotedblright. In\neach of the phrases demarcated by quotation marks, the word \\textquotedblleft\nfield\\textquotedblright\\ is used in a generic sense to identify an\nelectromagnetic phenomenon.\n\n\\subsection{Constant electric field}\n\nThe problem of a particle of mass $m$ and charge $q$ immersed in a constant\nelectric field of magnitude $E_{0}$ is inherently one-dimensional. For present\npurposes, nothing is gained by going to three spatial dimensions. The problem\nis clearly one in which energy is conserved. By Noether's Theorem\n\\cite{noether}, the Lagrangian must be independent of time $t$, so that the\nconnection between the electric field and potentials given in Eq. (\\ref{a})\nmust depend only on the scalar potential $\\phi.$ Equation (\\ref{a}) can then\nbe integrated to give the potential\n\\begin{equation}\n\\phi=-xE_{0},\\qquad A=0, \\label{o\n\\end{equation}\nsince an additive constant of integration has no physical meaning. The\npotentials descriptive of this problem are unique, and given by Eq. (\\ref{o}).\n\nThe Lagrangian function is the difference of the kinetic energy $T$ and the\npotential energy $U$\n\\begin{align}\nL  &  =T-U\\label{p}\\\\\n&  =\\frac{1}{2}m\\overset{.}{x}^{2}+qxE_{0}. \\label{p1\n\\end{align}\nThe Lagrangian equation of motion i\n\\begin{equation}\n\\frac{d}{dt}\\frac{\\partial L}{\\partial\\overset{.}{x}}-\\frac{\\partial\nL}{\\partial x}=m\\overset{..}{x}-qE_{0}=0, \\label{q\n\\end{equation}\nwhich is just the elementary Newtonian equatio\n\\begin{equation}\nm\\overset{..}{x}=qE_{0}. \\label{r\n\\end{equation}\nThe simplest initial conditions for this problem -- initial position and\nvelocity set to zero -- lead to the solutio\n\\begin{equation}\nx=\\frac{qE_{0}}{2m}t^{2}. \\label{s\n\\end{equation}\nFrom Eqs. (\\ref{p}) and (\\ref{p1}), it follows tha\n\\begin{equation}\nT=\\frac{1}{2m}\\left(  qE_{0}t\\right)  ^{2},\\qquad U=-\\frac{1}{2m}\\left(\nqE_{0}t\\right)  ^{2},\\qquad T+U=0. \\label{t\n\\end{equation}\nThe anticipated conservation of energy holds true.\n\nDespite the uniqueness of the potentials of Eq. (\\ref{o}), there exists an\napparently proper gauge transformation generated by the functio\n\\begin{equation}\n\\Lambda=ctxE_{0}. \\label{u\n\\end{equation}\nThe gauge-transformed potentials ar\n\\begin{equation}\n\\widetilde{\\phi}=0,\\qquad\\widetilde{A}=-ctE_{0}, \\label{v\n\\end{equation}\nand the Lagrangian function is \\cite{hrjmo\n\\begin{equation}\n\\widetilde{L}=\\frac{1}{2}m\\overset{.}{x}^{2}-qtE_{0}\\overset{.}{x}. \\label{w\n\\end{equation}\nThe kinetic energy is unaltered ($\\widetilde{T}=T$), but the new potential\nenergy i\n\\begin{equation}\n\\widetilde{U}=qtE_{0}\\overset{.}{x}, \\label{x\n\\end{equation}\nwhich is explicitly time-dependent. The new equation of motion i\n\\begin{equation}\n\\frac{d}{dt}\\left(  \\frac{\\partial\\widetilde{L}}{\\partial\\overset{.}{x\n}\\right)  -\\frac{\\partial\\widetilde{L}}{\\partial x}=m\\overset{..}{x}-qE_{0}=0,\n\\label{y\n\\end{equation}\nwhich is identical to that found in the original gauge, so that the solution\nis the same as Eq. (\\ref{s}). However, the altered gauge has introduced a\nfundamental change. The gauge-transformed potential energy is evaluated a\n\\begin{equation}\n\\widetilde{U}=\\frac{1}{m}\\left(  qE_{0}t\\right)  ^{2}, \\label{z\n\\end{equation}\nso that\n\\begin{equation}\n\\widetilde{T}+\\widetilde{U}=\\frac{3}{2m}\\left(  qE_{0}t\\right)  ^{2}.\n\\label{aa\n\\end{equation}\nThe total energy is not conserved, as was presaged by the explicit time\ndependence of the gauge-transformed Lagrangian (\\ref{w}).\n\nHow did this happen? One constraint placed on gauge transformations (see, for\nexample, the classic text by Jackson \\cite{jackson}) is that the generating\nfunction must be a scalar function that satisfies the homogeneous wave\nequation, as in Eq. (\\ref{c}). This is satisfied by the function (\\ref{u}).\nThe only other condition is the Lorenz condition (\\ref{j}), which is satisfied\nby the potentials before and after transformation. However, there is no\ncondition that guarantees preservation of symmetries inherent in the physical\nproblem. It is not enough to employ appropriate fields; it is necessary to\nemploy the appropriate potentials to ensure that all aspects of the physical\nproblem are rendered properly.\n\nThis writer is unaware of any instance where inappropriate potentials have\nbeen accepted and employed in this exceedingly simple problem. The same cannot\nbe said for the next example.\n\n\\subsection{Plane-wave field}\n\nLaser fields are of central importance in contemporary physics, and laser\nfields are plane-wave fields. A plane-wave field is the only electromagnetic\nphenomenon that has the ability to propagate indefinitely in vacuum without\nthe continued presence of sources. In the typical laboratory experiments with\nlasers, the practical consequence of this ability to propagate without need\nfor sources means that all fields that arrive at a target can only be a\nsuperposition of plane-wave fields. Any contamination introduced by optical\nelements like mirrors or gratings can persist for only a few wavelengths away\nfrom such elements. On the scale of a typical laboratory optical table, this\nis negligible.\n\nPlane-wave fields propagate at the speed of light in vacuum; they are\nfundamentally relativistic. The 1905 principle of Einstein is basic: the speed\nof light is the same in all inertial frames of reference \\cite{einstein}. The\nmathematical statement of this principle is that any description of a\nplane-wave field can depend on the spacetime coordinate $x^{\\mu}$ only as a\nscalar product with the propagation 4-vector $k^{\\mu}$. The consequence of\nthis projection of the spacetime 4-vector onto the light cone is that any\nchange of gauge must be such as to be confined to the light cone. That is,\nwith the definitio\n\\begin{equation}\n\\varphi\\equiv k^{\\mu}x_{\\mu}, \\label{ab\n\\end{equation}\nthe field 4-vector must be such that\n\\begin{equation}\nA_{pw}^{\\mu}=A_{pw}^{\\mu}\\left(  \\varphi\\right)  , \\label{ac\n\\end{equation}\nwhere the subscript \\textit{pw} stand for \\textit{plane-wave}. When the gauge\ntransformation of Eq. (\\ref{h}) is applied, the gauge-altered 4-vector\npotential is confined by the condition (\\ref{ac}) to the form \\cite{hrgge\n\\begin{equation}\n\\widetilde{A}^{\\mu}=A^{\\mu}+k^{\\mu}\\Lambda^{\\prime}, \\label{ad\n\\end{equation}\nwhere the gauge-change generating function can itself depend on $x^{\\mu}$ only\nin the form of $\\varphi,$ and\n\\begin{equation}\n\\Lambda^{\\prime}=\\frac{d}{d\\varphi}\\Lambda\\left(  \\varphi\\right)  . \\label{ae\n\\end{equation}\nAs is evident from Eq. (\\ref{l}), transversality is maintained by the gauge\ntransformation (\\ref{ad}).\n\nA further limitation arises if an electron is subjected to a scalar binding\npotential in addition to the vector potential associated with the laser field.\nA relativistic Hamiltonian function for a charged particle in a plane-wave\nfield contains a term of the for\n\\begin{equation}\n\\left(  i\\hslash\\partial^{\\mu}-\\frac{q}{c}A^{\\mu}\\right)  \\left(\ni\\hslash\\partial_{\\mu}-\\frac{q}{c}A_{\\mu}\\right)  . \\label{ae1\n\\end{equation}\nThis occurs in the classical case, in the Klein-Gordon equation of quantum\nmechanics, and in the second-order Dirac equation of quantum mechanics\n\\cite{feynmangm,schweber}. The expansion of the expression in Eq. (\\ref{ae1})\ncontains the squared time par\n\\begin{equation}\n\\left(  i\\hslash\\partial_{t}-\\frac{q}{c}A^{0}\\right)  ^{2}. \\label{ae2\n\\end{equation}\nIf $A^{0}$ contains contributions from both a scalar potential and the time\npart of the plane-wave 4-vector potential, then executing the square in Eq.\n(\\ref{ae2}) would give a term containing the product of these two scalar\npotentials that is not physical; it does not occur in the reduction of\nrelativistic equations of motion to their nonrelativistic counterparts\n\\cite{hrgge}. This applies specifically to applications in AMO physics. That\nis, it must be true that \\cite{hr79,hrgge\n\\begin{equation}\nA_{pw}^{0}=\\phi_{pw}=0. \\label{af\n\\end{equation}\nThis means that gauge freedom vanishes. Only the \\textit{radiation gauge}\n(also known as \\textit{Coulomb gauge}) is possible. This is the gauge in which\nscalar binding influences are described by scalar potentials $\\phi$ and laser\nfields are described by 3-vector potentials $\\mathbf{A}$.\n\nConsider the gauge transformation generated by the function \\cite{hr79\n\\begin{equation}\n\\Lambda=-A^{\\mu}\\left(  \\varphi\\right)  x_{\\mu}. \\label{ag\n\\end{equation}\nThis leads to the transformed gaug\n\\begin{equation}\n\\widetilde{A}^{\\mu}=-k^{\\mu}x^{\\nu}\\left(  \\frac{d}{d\\varphi}A_{\\nu}\\right)  ,\n\\label{ah\n\\end{equation}\nwhich was introduced in Ref. \\cite{hr79} in an attempt to base the Keldysh\napproximation \\cite{keldysh} on plane-wave fields rather than on quasistatic\nelectric fields. The transformed 4-potential can also be written as\n\\cite{hr79\n\\begin{equation}\n\\widetilde{A}^{\\mu}=-\\frac{k^{\\mu}}{\\omega\/c}\\mathbf{r\\cdot E}\\left(\n\\varphi\\right)  , \\label{ai\n\\end{equation}\nthus suggesting a relativistic generalization of the nonrelativistic\n\\textit{length gauge} used by Keldysh and widely employed within the AMO\ncommunity. The problem with the $\\widetilde{A}^{\\mu}$ of Eq. (\\ref{ah}) or\n(\\ref{ai}) is that it violates the symmetry (\\ref{ac}) required of a\npropagating field. Nevertheless, this $\\widetilde{A}^{\\mu}$ satisfies the\nLorenz condition (\\ref{j}) and the transversality condition (\\ref{m}); and the\ngenerating function of Eq. (\\ref{ag}) satisfies the homogeneous wave equation\nof Eq. (\\ref{c}) \\cite{hr79}. That is, all the usual requirements for a gauge\ntransformation are met even though the transformed 4-vector potential\n$\\widetilde{A}^{\\mu}$ of Eq. (\\ref{ah}) or (\\ref{ai}) violates the symmetry\nrequired of a propagating field like a laser field.\n\nThis violation of a basic requirement for a laser field has unphysical and\nhence unacceptable consequences. The most obvious is that the covariant\nstatement of the all-important \\cite{hrup} ponderomotive energy $U_{p}$\nproduces a null result sinc\n\\begin{equation}\n\\widetilde{U}_{p}\\sim\\widetilde{A}^{\\mu}\\widetilde{A}_{\\mu}=0 \\label{aj\n\\end{equation}\nas a consequence of the self-orthogonality of the propagation 4-vector\n$k^{\\mu}$. The resemblance of Eq. (\\ref{ai}) to the length-gauge\nrepresentation of a quasistatic electric field suggests a tunneling model for\nthe relativistic case \\cite{vspopov,heidelberg}, which is inappropriate for\nstrong laser fields. Tunneling can occur only through interference between\nscalar potentials, and a strong laser field is inherently vector, not scalar.\n\nThe basic defect of the potentials (\\ref{ah}) or (\\ref{ai}) is violation of\nthe Einstein condition of the constancy of the speed of light in all Lorentz\nframes, despite the validity of the gauge transformation leading to those\npotentials. The importance of the physical situation in which this occurs is\nrobust evidence of the significance of the proper choice of potentials, since\nthe electric and magnetic fields attained from the unacceptable potentials\n(\\ref{ah}) or (\\ref{ai}) are exactly the same as those that follow from\npotentials that satisfy properly the condition (\\ref{ac}).\n\n\\section{Gauge transformations and unitarity}\n\nUnitary transformations in quantum physics preserve the values of physical\nobservables. It was shown above that not all gauge transformations produce\nphysically acceptable results. Therefore, gauge transformations are not\nunitary transformations. This conclusion is supported by the basic structure\nof transition amplitudes.\n\nTransition amplitudes without resort to perturbation theory are best expressed\nby S matrices. These are of two (equivalent) types. The \\textit{direct-time}\nor \\textit{post }amplitude i\n\\begin{equation}\n\\left(  S-1\\right)  _{fi}=-\\frac{i}{\\hslash}\\int_{-\\infty}^{\\infty}dt\\left(\n\\Phi_{f},H_{I}\\Psi_{i}\\right)  , \\label{aq\n\\end{equation}\nand the \\textit{time-reversed} or \\textit{prior} amplitude i\n\\begin{equation}\n\\left(  S-1\\right)  _{fi}=-\\frac{i}{\\hslash}\\int_{-\\infty}^{\\infty}dt\\left(\n\\Psi_{f},H_{I}\\Phi_{i}\\right)  . \\label{ar\n\\end{equation}\nThe indices $f$ and $i$ label the final and initial states. The $\\Phi$ states\nare non-interacting states and the $\\Psi$ states are fully interacting states\nsatisfying, respectively, the Schr\\\"{o}dinger equation\n\\begin{align}\ni\\hslash\\partial_{t}\\Phi &  =H_{0}\\Phi,\\label{as}\\\\\ni\\hslash\\partial_{t}\\Psi &  =\\left(  H_{0}+H_{I}\\right)  \\Psi, \\label{at\n\\end{align}\nwhere $H_{I}$ is the interaction Hamiltonian.\n\nIn a gauge transformation, the matrix elements within the time integrations in\nEqs. (\\ref{aq}) and (\\ref{ar}) transform a\n\\begin{align}\n\\left(  \\Phi_{f},H_{I}\\Psi_{i}\\right)   &  \\rightarrow\\left(  \\Phi\n_{f},\\widetilde{H}_{I}\\widetilde{\\Psi}_{i}\\right)  ,\\label{au}\\\\\n\\left(  \\Psi_{f},H_{I}\\Phi_{i}\\right)   &  \\rightarrow\\left(  \\widetilde{\\Psi\n}_{f},\\widetilde{H}_{I}\\Phi_{i}\\right)  . \\label{av\n\\end{align}\nBecause the noninteracting states are unaltered in a gauge transformation,\nthere is no necessary\\textit{ }equivalence between the two sides of the\nexpressions in Eqs. (\\ref{au}) and (\\ref{av}).\n\nThose authors that endorse the favored status of the length gauge\n\\cite{yang,kobesmirl,beckerss,lss,jbauer} \\textquotedblleft\nsolve\\textquotedblright\\ this problem by attaching a unitary operator to all\nstates, including non-interacting states: $\\widetilde{\\Phi}=U\\Phi$,\n$\\widetilde{\\Psi}=U\\Psi$. All $U$ and $U^{-1}$ operators exactly cancel in the\nmatrix element, and the transition amplitude is unchanged. This is what leads\nto the property \\textquotedblleft gauge-invariant formalism\\textquotedblrigh\n\\ sometimes ascribed to the length gauge. However, this procedure amounts to\nan identity or to a change of \\textit{quantum picture}, but not to a gauge transformation.\n\n\\section{Fundamental contrasts in the applicability of fields and potentials}\n\nThe first example to be presented is the very basic one of the impossibility\nof expressing the Schr\\\"{o}dinger equation directly in terms of electric and\nmagnetic fields, which should be possible if fields are truly more fundamental\nthan potentials. Other direct examples of difficulties posed by the assumption\nof the fundamental importance of fields are shown, many of them long employed\nunnoticed within the strong-field community.\n\n\\subsection{Schr\\\"{o}dinger equation}\n\nThe Schr\\\"{o}dinger equatio\n\\begin{equation}\ni\\hslash\\partial_{t}\\Psi\\left(  t\\right)  =H\\Psi\\left(  t\\right)  , \\label{ak\n\\end{equation}\nwhen viewed as a statement in a Hilbert space (that is, without selecting a\nrepresentation such as the configuration representation or the momentum\nrepresentation) states that the effect of rotating a state vector $\\Psi\\left(\nt\\right)  $ by the operator $H$ within the Hilbert space produces the same\neffect as differentiating the vector with respect to time (multiplied by\n$i\\hslash$). Time $t$ is an external parameter upon which the state vectors\ndepend, which accounts for why Eq. (\\ref{ak}) specifies $t$ as a label\nindependent of the Hilbert space. A unitary transformation preserves this\nequivalence. This can be stated a\n\\begin{equation}\ni\\hslash\\partial_{t}-\\widetilde{H}=U\\left(  i\\hslash\\partial_{t}-H\\right)\nU^{-1}. \\label{al\n\\end{equation}\nSince Eq. (\\ref{al}) can be written a\n\\begin{equation}\n\\widetilde{H}=UHU^{-1}+U\\left(  i\\hslash\\partial_{t}U^{-1}\\right)  ,\n\\label{am\n\\end{equation}\nthis shows explicitly that the Hamiltonian operator does not transform\nunitarily if there is any time dependence in $U$.\n\nAn important gauge transformation is that introduced by G\\\"{o}ppert-Mayer\n\\cite{gm}, widely employed in the AMO community. This transformation is given\nb\n\\begin{equation}\nU_{GM}=\\exp\\left(  \\frac{ie}{\\hslash c}\\mathbf{r\\cdot A}\\left(  t\\right)\n\\right)  , \\label{an\n\\end{equation}\nwhich depends explicitly on time when $\\mathbf{A}\\left(  t\\right)  $ describes\na laser field within the dipole approximation, meaning that Eq. (\\ref{am}) is consequential.\n\nThe fact that, in general\n\\begin{equation}\n\\widetilde{H}\\neq UHU^{-1}, \\label{ao\n\\end{equation}\nis the explanation for the curious result to be found in many papers (for\nexample, Refs. \\cite{yang,kobesmirl,beckerss,lss,jbauer}) that the\n$\\mathbf{r\\cdot E}$ potential is a preferred potential. If any other potential\nis employed in solving the Schr\\\"{o}dinger equation, then the claim is made\nthat a transformation factor must be employed even on a non-interacting state.\nThere is a logical contradiction inherent in the requirement that a\nnon-interacting state must incorporate a factor that depends on an\ninteraction, but the list of published papers that accept this premise is much\nlonger than the salient examples cited here. The underlying problem is the\nassumption that a gauge transformation transforms the Hamiltonian unitarily.\nThat problem exists in all of the references just cited, although it is\nusually submerged in complicated manipulations. It is especially clear in Ref.\n\\cite{jbauer}, where it is specified that all operators $O$ transform under a\ngauge transformation according to the unitary-transformation rul\n\\begin{equation}\n\\widetilde{O}=UOU^{-1},\\qquad U^{-1}=U^{\\dag}. \\label{ap\n\\end{equation}\nThat specification is applied to $H$, in violation of the condition\n(\\ref{am}), and to the interaction Hamiltonian $H_{I}$, with no explanation\nfor how it is possible to gauge-transform from the length-gauge interaction to\nany other gauge in view of the absence of operators in the scalar potential\n$\\mathbf{r\\cdot E}$. In the scheme proposed in Refs.\n\\cite{yang,kobesmirl,beckerss,lss,jbauer}, if the problem is initially\nformulated in the context of the $\\mathbf{r\\cdot E}$ potential, it is never\npossible to transform to any other gauge. This explains the use of the phrase\n\\textquotedblleft gauge-invariant formulation\\textquotedblright\\ with respect\nto $\\mathbf{r\\cdot E}$ to be found in some published works.\n\n\\subsection{Locality and nonlocality}\n\nFields are derived from potentials by the calculus process of differentiation,\nas exhibited in Eq. (\\ref{a}). Differentiation is carried out at a point in\nspacetime. It is \\textit{local}. If potentials are to be expressed from\nfields, that requires integration, consisting of information from a range of\nspacetime values; it is \\textit{nonlocal}. The fact that the Schr\\\"{o}dinger\nequation requires the local information from potentials, and cannot be\ndescribed by fields without inferring nonlocality in spacetime, is direct\nevidence that potentials are more fundamental than fields.\n\n\\subsection{Ambiguity in the electromagnetic field tensors}\n\nThe basic field tensor of electrodynamics is defined a\n\\begin{equation}\nF^{\\mu\\nu}=\\partial^{\\mu}A^{\\nu}-\\partial^{\\nu}A^{\\mu}. \\label{aj1\n\\end{equation}\nIt is important to note that this expression is in terms of the derivatives of\npotentials rather than the potentials themselves. Thus it is not surprising\nthat the Lorentz invariant found from the inner product of $F^{\\mu\\nu}$ with\nitself yields an expression in terms of fields\n\\begin{equation}\nF^{\\mu\\nu}F_{\\mu\\nu}=2\\left(  \\mathbf{B}^{2}-\\mathbf{E}^{2}\\right)  .\n\\label{aj2\n\\end{equation}\nA dual tensor can be defined a\n\\begin{equation}\nG^{\\mu\\nu}=\\frac{1}{2}\\epsilon^{\\mu\\nu\\rho\\lambda}F_{\\rho\\lambda}, \\label{aj3\n\\end{equation}\nwhere $\\epsilon^{\\mu\\nu\\rho\\lambda}$ is the completely asymmetric fourth-rank\ntensor. (The conventions of Jackson \\cite{jackson} are being employed.) The\ninner product of the basic and dual tensors gives a second Lorentz invariant\n\\begin{equation}\nG^{\\mu\\nu}F_{\\mu\\nu}=-4\\mathbf{B\\cdot E}, \\label{aj4\n\\end{equation}\nalso in terms of fields. The two Lorentz invariant\n\\begin{equation}\n\\mathbf{E}^{2}-\\mathbf{B}^{2},\\quad\\mathbf{E\\cdot B} \\label{aj5\n\\end{equation}\nare said to characterize the electrodynamic environment.\n\nAn important special case is that of transverse, propagating fields, wher\n\\begin{equation}\n\\mathbf{E}^{2}-\\mathbf{B}^{2}=0,\\quad\\mathbf{E\\cdot B}=0. \\label{aj6\n\\end{equation}\nThe properties (\\ref{aj6}) lead to radiation fields as sometimes being called\n\\textquotedblleft null fields\\textquotedblright. (The terms radiation field,\npropagating field, transverse field, plane-wave field, are here used\ninterchangeably.) Radiation fields propagate at the speed of light in vacuum,\nand they have the unique character that, after initial formation, they\npropagate indefinitely in vacuum without the presence of sources.\n\nHowever, the invariants (\\ref{aj6}) are not unique to radiation fields; they\napply also to \\textquotedblleft constant crossed fields\\textquotedblright.\nThat is, it is always possible to generate static electric and magnetic fields\nof equal magnitude that are perpendicular to each other, and will thus possess\nzero values for both of the Lorentz invariants of the electromagnetic field.\nConstant crossed fields do not propagate, and they cannot exist without the\npresence of sources. They are unrelated to radiation fields despite sharing\nthe same values of the Lorentz invariants. Most importantly, constant crossed\nfields cannot be considered as the zero-frequency limit of radiation fields,\nas they are sometimes described \\cite{nikrit66,ritus85}. All radiation fields\npropagate at the speed of light for all frequencies, no matter how low. There\nis no possible zero-frequency static limit \\cite{hrtun}.\n\nThere is no ambiguity when radiation fields and constant crossed fields are\nexpressed in terms of their potentials. Radiation-field potentials possess the\nperiodicity inherent in trigonometric dependence on the $\\varphi$ of Eq.\n(\\ref{ab}). This is unrelated to the $\\phi=-\\mathbf{r\\cdot E}_{0}$ potential\nof (\\ref{o}) for a constant electric field $\\mathbf{E}_{0}$, and\n$\\mathbf{A}=-\\left(  \\mathbf{r\\times B}_{0}\\right)  \/2$ for a constant\nmagnetic field $\\mathbf{B}_{0}$, both of which require source terms for their existence.\n\n\\subsection{Lorentz force}\n\nThe force exerted on a particle of charge $q$ moving with velocity\n$\\mathbf{v}$ in a field with electric and magnetic components $\\mathbf{E}$ and\n$\\mathbf{B}$ is given by the Lorentz force expressio\n\\begin{equation}\n\\mathbf{F}=q\\left(  \\mathbf{E}+\\frac{\\mathbf{v}}{c}\\times\\mathbf{B}\\right)  .\n\\label{aj7\n\\end{equation}\nIn a plane-wave field, the electric and magnetic fields are of equal\nmagnitude: $\\left\\vert \\mathbf{E}\\right\\vert =\\left\\vert \\mathbf{B}\\right\\vert\n$. Thus, under conditions where $\\left\\vert \\mathbf{v}\\right\\vert \/c\\ll1$, the\nmagnetic component of the force is minor as compared to the electric\ncomponent. The implication is that, as the field frequency declines, the\nmotion-related magnetic component reduces to an adiabatic limit. This concept\nof adiabaticity justifies the complete neglect of the magnetic field that is a\nkey element of the dipole approximation, applied within the AMO community in\nthe form\n\\begin{equation}\n\\mathbf{E}=\\mathbf{E}\\left(  t\\right)  ,\\quad\\mathbf{B}=0. \\label{aj8\n\\end{equation}\nThe adiabaticity line of reasoning suggests a so-called adiabatic limit where\nthe field frequency declines to zero, and the plane wave field behaves as a\nconstant crossed field that satisfies the plane-wave condition (\\ref{aj6}).\n\nThe entire line of reasoning that involves the concepts of adiabaticity,\nadiabatic limit, and a zero-frequency limit for plane-wave fields is\nfield-based and erroneous \\cite{hr101,hrtun}. When the problem is treated in\nterms of potentials, it becomes clear that $v\/c$ approaches unity for very\nstrong fields even when the frequency is very low, and the magnetic force\nbecomes equivalent to the electric force for very strong fields.\n\n\\subsection{Critical field}\n\nThe \\textquotedblleft critical field\\textquotedblright\\ in electrodynamics is\nrelated to the spontaneous breakdown of the vacuum into electron-positron\npairs. The critical field is defined as the electric field strength at which\nthe $\\pm mc^{2}$ limits for the rest energies of electron and positron in a\nparticle-hole picture are \\textquotedblleft tilted\\textquotedblright\\ by the\nelectric field to allow tunneling between positive and negative energy states\nwhen the spatial limits of the tunnel are separated by an electron Compton\nwavelength. This type of pair production is called Sauter-Schwinger pair\nproduction \\cite{sauter,schwinger}. It is fundamentally different from\nBreit-Wheeler pair production \\cite{bw,hrdiss,hr62} to be discussed below.\n\nThe reason for the fundamental difference is that Sauter-Schwinger pair\nproduction is a phenomenon due to electric fields and Breit-Wheeler is due to\nplane-wave fields. The distinction between pure electric fields and plane-wave\nfields could hardly be more clear, since both types of fields have unique\ngauge choices that are contrasting.\n\nThe critical field is often mentioned as a goal of strong-field laser\nfacilities, which is basically a \\textit{non-sequitur}. The critical field\napplies only to electric fields and lasers produce plane-wave fields. Even if\na laser field were sufficiently intense that its electric component had the\nmagnitude of the QSE critical field, pair production from the vacuum cannot\noccur because the photons of the laser field convey momentum. A\ncounter-propagating plane-wave field is necessary to satisfy momentum\nconservation as well as energy conservation in the production of pairs from\nthe vacuum. This is then the two-fields Breit-Wheeler process, which is\nunrelated to the single-field Sauter-Schwinger process.\n\n\\subsection{Pair production from the vacuum}\n\nFor many years the stated ultimate goal of large-laser programs was to achieve\na laser intensity such that the electric component of the laser is equal to\nthe critical field discussed in the preceding subsection This magnitude of\nelectric field corresponds to an intensity of about $4.6\\times10^{29}W\/cm^{2}$.\n\nThe problem is that the Schwinger limit applies \\textit{only} to electric\nfields. It does not apply to laser fields \\cite{hrspie}, as explained above.\nThe only way to produce momentum balance with a laser field while still\nproducing pairs from the vacuum is to have the laser beam collide with\noppositely-directed photons, as proposed in Ref. \\cite{hr71}. This was\npredicted to be done on a practical basis at a linear accelerator facility\nsuch as that at SLAC\\ (Stanford Linear Accelerator Center), with a laser\nintensity of only slightly greater than $10^{18}W\/cm^{2}$. An important note\nis that the prediction of Ref. \\cite{hr71} was for the use of the\nthen-important ruby laser at a different pulse length and a different energy\nof the energetic electron beam used to produce the countervailing photon\nfield. The predicted threshold of 25 photons from the laser field in Ref.\n\\cite{hr71} is altered to 5 photons for the parameters of the experiment that\nwas actually done at SLAC in 1997 \\cite{burke}. The theoretical prediction of\nan effective threshold intensity of about $10^{18}W\/cm^{2}$ is maintained\nbecause of the essential independence of the laser frequency that was remarked\nin Ref. \\cite{hr71}.\n\n(One \\textit{caveat} about the experiment is that it was reported as a\nhigh-order perturbative result, whereas it is readily shown to be at an\nintensity beyond the radius of convergence of perturbation theory \\cite{epjd}.\nA second problem is that it was described as \\textquotedblleft light-by-light\nscattering\\textquotedblright, which is a different process altogether. Feynman\ndiagrams of these processes have electron and positron as emergent particles\nin a pair production process, while light-by-light scattering has emergent photons.)\n\nThe striking difference between the $4.6\\times10^{29}W\/cm^{2}$ required to\nattain a laser electric component equal to the critical field and the actual\n$1.3\\times10^{18}W\/cm^{2}$ required for the SLAC experiment is evidence of a\nmisplaced focus on the electric field required for vacuum pair production. The\nrequired intensity of the laser field depends on the properties of the\ncounter-propagating field, but it is never as large as the Sauter-Schwinger\ncritical field.\n\n\\subsection{Low frequency limit of a plane-wave field}\n\nIt is conventional to view low-frequency laser-induced phenomena from the\nstandpoint of the Lorentz force expression of Eq. (\\ref{aj7}). In a plane-wave\nfield, the electric and magnetic fields are of equal magnitude: $\\left\\vert\n\\mathbf{E}\\right\\vert =\\left\\vert \\mathbf{B}\\right\\vert $. Thus, under\nconditions where $\\left\\vert \\mathbf{v}\\right\\vert \/c\\ll1$, the magnetic\ncomponent of the force is minor as compared to the electric component. The\nimplication is that, as the field frequency declines, the motion-related\nmagnetic component reduces to an adiabatic limit. This concept of adiabaticity\nappears to justify the complete neglect of the magnetic field that is a key\nelement of the dipole approximation, applied within the AMO community in the\nform given in Eq. (\\ref{aj8}). Adiabaticity suggests a so-called adiabatic\nlimit, where the field frequency declines to zero, and the plane wave field\nbehaves as a constant crossed field that satisfies the plane-wave condition\n(\\ref{aj6}).\n\nThe entire line of reasoning that involves the concepts of adiabaticity,\nadiabatic limit, and a zero-frequency limit for plane-wave fields is\nfield-based and erroneous \\cite{hr101,hrtun}. When the problem is treated in\nterms of potentials, it becomes clear that $v\/c$ approaches unity for very\nstrong fields. The magnetic force becomes equivalent to the electric force for\nvery strong fields.\n\nAnalysis from a potentials standpoint is in stark contrast to a fields-based approach.\n\nThe ponderomotive potential energy of a charged particle in a plane-wave field\nis a fundamental property of the particle \\cite{hrup}, and it becomes\ndivergent as the field frequency approaches zero. The immediate consequence is\nthat the limit $\\omega\\rightarrow0$ causes the dipole approximation to fail\n\\cite{hr101,hrtun}, and corresponds to an extremely relativistic environment\n\\cite{hrup}. This contradicts maximally the field-based conclusions.\n\nThe ponderomotive energy $U_{p}$ is given b\n\\begin{equation}\nU_{p}=\\frac{q^{2}}{2mc^{2}}\\left\\langle \\left\\vert A^{\\mu}A_{\\mu}\\right\\vert\n\\right\\rangle , \\label{aj9\n\\end{equation}\nwhere the angle brackets denote a cycle average, and the absolute value needs\nto be indicated because the 4-vector potential $A^{u}$ is a spacelike 4-vector\nand its square is thus negative with the metric being employed. The\nponderomotive energy is based on potentials. When expressed in terms of field\nintensity $I$, $U_{p}$ behaves a\n\\begin{equation}\nU_{p}\\sim I\/\\omega^{2}, \\label{aj10\n\\end{equation}\nwhich explains the relativistic property of the charged particle as the\nfrequency approaches zero.\n\n\\subsection{Extremely-low-frequency radio waves}\n\nA central issue in this article is that transverse fields and longitudinal\nfields are fundamentally different, even when they seem to have some\nproperties in common. An effective example is the matter of communicating with\nsubmerged submarines with extremely low frequency (ELF) radio waves. The U. S.\nNavy operated such a system \\cite{wikisanguine}, designed to communicate with\nsubmarines submerged at depths of the order of $100m$ over an operational\nrange of about half of the Earth's surface. The point of using extremely low\nfrequencies is the large skin depth in a conducting medium (seawater) that can\nbe achieved. What is most remarkable about the system is that the frequency of\n$76Hz$ that was used has a wavelength of about $4\\times10^{6}m$, which is\napproximately $0.62$ times the radius of the Earth. The system could convey an\nintelligible signal over about half of the Earth's surface. Considering the\nlength of the submarine (about $100m$) to be the size of the receiving\nantenna, this means that the wavelength to antenna-length ratio is about\n$4\\times10^{4}$. That is, the received radio wave is constant over the entire\nlength of the receiving antenna to a very high degree of accuracy. A\nnearly-constant electric field cannot be detected at a distance of half of our\nplanet away from its source; a nearly-constant electric field cannot penetrate\n$100m$ through a conducting medium; and so a nearly-constant electric field\ncannot convey an intelligible signal in the presence of all of these extremely\neffective barriers. A longitudinal field is fundamentally different from a\ntransverse field.\n\nThe proposed \\textquotedblleft local constant-field\napproximation\\textquotedblright\\ (LCFA) for relativistic laser effects\n\\cite{ritus85,heidrmp} is based on the presumed similarity of low-frequency\nlaser fields to constant crossed fields. This was shown earlier in this\nmanuscript to be a meaningless association. The application involved in the\ncommunication with submerged submarines is a practical demonstration that the\nLCFA is not a valid concept.\n\n\\section{Hierarchy of physical effects}\n\nIn basic calculus, if a function $f\\left(  x\\right)  $ is known, then so is\n$df\/dx$; differentiation is local. If the derivative is all that is known, the\nprocess of integration to find $f\\left(  x\\right)  $ requires the knowledge of\n$df\/dx$ over a range of values; integration is nonlocal. In electromagnetism,\npotentials are the analog of $f\\left(  x\\right)  $ and fields correspond to\n$df\/dx$. The example of the Schr\\\"{o}dinger equation shows that knowledge of\npotentials is primary and fields are secondary. This ordering can be extended\nto other physical quantities. For example, the Lorentz force expression in Eq.\n(\\ref{aj7}) relates forces directly to fields, meaning that forces are\nsecondary. A further implication is that Newtonian mechanics, expressed in\nterms of forces, is secondary to Lagrangian and Hamiltonian mechanics that are\nexpressed in terms of energies; that is, in terms of potentials. It is no\naccident that mechanics textbooks show that formalisms based on system\nfunctions like the Lagrangian or Hamiltonian infer the Newtonian formalism,\nbut to go in the reverse direction requires an extrapolation of concepts.\n\nThis ordering, or hierarchy, of physical quantities has consequences in\nproblems that go beyond simple electric-only or magnetic-only fields. In\nlaser-related problems where both electric and magnetic fields are involved,\nit is possible to arrive at invalid inferences if only secondary influences\nare regarded as controlling. An example is the common practice of viewing\nelectric fields as being the dominant quantities in long-wavelength\ncircumstances where the dipole approximation appears to be valid. This\ndisguises the fact that extremely low frequencies can lead to relativistic\nbehavior, where electric fields supply inadequate information\n\\cite{hr101,hrtun}, and inappropriate concepts such as adiabaticity exist.\n\nA cogent example is the critical field. A widely used \\textquotedblleft\nstrong-field QED parameter\\textquotedblright\\ is simply the ratio of the\nelectric field to the critical field. This has meaning only for\nstatic-electric or for QSE fields. For strong-field laser applications,\nrelevant intensity parameters are all ratios of energies. See, for example,\nthe section entitled \\textquotedblleft Measures of intensity\\textquotedblrigh\n\\ in Ref. \\cite{hrrev}. (The early, but still useful review article by Eberly\n\\cite{eberlyreview} is also instructive in this regard.) In particular, the\n\\textquotedblleft free-electron intensity parameter\\textquotedblrigh\n\\ $z_{f}=2U_{p}\/mc^{2}$ is known to measure the coupling between an electron\nand a nonperturbatively intense plane-wave field \\cite{hrdiss,hr62,hr62b,hrup\n, replacing the fine-structure constant of perturbative electrodynamics.\n\nA special insight arises when the $z_{f}$ parameter is related to the\nfine-structure constant $\\alpha$ \\cite{hrup}\n\\begin{equation}\nz_{f}=\\alpha\\rho\\left(  2\\lambda\\lambdabar_{C}^{2}\\right)  , \\label{aj11\n\\end{equation}\nwhere $\\rho$ is the number of photons per unit volume, and $2\\lambda\n\\lambdabar_{C}^{2}$ is approximately the volume of a cylinder of radius equal\nto the electron Compton wavelength $\\lambdabar_{C}$ and a length given by the\nwavelength $\\lambda$ of the laser field. That is, it appears that all of the\nphotons within the cylinder participate in the coupling between the electron\nand the laser field. The electron Compton wavelength is the usual interaction\ndistance for a free electron, but the wavelength can be a macroscopic\nquantity. The \\textquotedblleft strong-field physics\\textquotedblright\\ that\narises from the application of the Volkov solution to problems involving the\ninteraction of very intense radiation fields with matter\n\\cite{sengupta,hrdiss,hr62}, thus appears to be the bridge connecting quantum\nelectrodynamics with the classical electrodynamics of Maxwell.\n\nOther examples of the hazards of excessive dependence on secondary quantities\nhave been mentioned above in the context of the assumed dominance of the\nelectric component of the Lorentz force at low frequencies, and the inference\nfrom the Lorentz invariants that constant crossed fields are related to\npropagating fields.\n\n\\section{Failure of perturbation theory}\n\nPerturbation theory has been the cornerstone of QED since the Nobel-prize\nwinning work of Feynman, Schwinger, and Tomonaga. The radius of convergence of\nperturbation theory was examined in depth a long time ago \\cite{hrdiss}. The\nmotivation for the study was the demonstration by Dyson \\cite{dyson} that,\ndespite its remarkable quantitative success, standard covariant QED has a zero\nradius of convergence for a perturbation expansion. The question about whether\na strong-field theory based on the Volkov solution of the Dirac equation\n\\cite{volkov} could be convergent was the motivation for Ref. \\cite{hrdiss}.\nThe answer was affirmative, but with intensity-dependent singularities in the\ncomplex coupling-constant plane that limited the convergence. (This explains\nthe $z_{f}$ terminology, since $z_{f}$ was allowed to be complex, and the\nquantity $z$ is often used for complex numbers.) Physically, convergence fails\nwhenever the field intensity is high enough to cause a channel closing. That\nis, if a process requires a certain threshold energy to proceed in a\nfield-free process as measured by some photon number $n_{0}$, that threshold\nenergy will be increased due to the need for a free electron to possess the\nponderomotive energy $U_{p}$ in the field. When $U_{p}$ is large enough to\ncause $n_{0}$ to index up to $n_{0}+1$, that is called a channel-closing, and\nit marks a sufficient condition for the failure of perturbation theory. The\nupper limit on perturbation theory is therefore set b\n\\begin{equation}\nU_{p}<\\hslash\\omega,\\text{ \\ or \\ }I<4\\omega^{3}, \\label{aj12\n\\end{equation}\nwhere the last expression is in atomic units, using the equivalence\n$U_{p}=I\/4\\omega^{2}$ in atomic units, and $I$ is the laser intensity.\n\nAlthough the original convergence investigation was done for free electrons,\nthe same limit was found for atomic ionization \\cite{hr80}.\n\nThe relevance of using an index based on the ratio of the electric field to\nthe critical field as an \\textit{ad hoc }limit on perturbation theory for\nlaser effects is strongly questioned here. Such a basic matter as the\napplicability of perturbation methods to the treatment of the effects of\nradiation fields is governed by primary quantities like the energies $U_{p}$\nand $mc^{2}$, and not by secondary quantities like the magnitude of the\nelectric field.\n\n\\section{Overview}\n\nThe Aharonov-Bohm effect introduced a major change in electrodynamics because\nit showed that potentials are indeed more fundamental than fields. However,\nalthough the effect relates to a quantum phenomenon, it has had little effect\non the way in which quantum mechanics is employed. The ascendancy of\npotentials over fields as described above is much more consequential,\nespecially for strong-field phenomena. A simple summarizing statement is that\nelectromagnetic scalar and vector potentials convey more physical information\nthan the electric and magnetic fields derivable from them.\n\nOf special importance is the fact that two cases have been identified where\nthere can be only one acceptable set of potentials, and these two cases are of\nvery wide practical scope. One case is the constant electric field, which is\nexactly or approximately applicable to a wide variety of AMO and\ncondensed-matter applications. The unique acceptable potential for static\nelectric fields is the length-gauge, or $\\mathbf{r\\cdot E}$ potential, and\nthis is already widely employed for constant or slowly varying electric-field applications.\n\nThe other case with a unique allowable gauge is the propagating-field case,\nwhich is of profound importance in laser-based experiments. Such experiments\nconstitute an increasingly large proportion of AMO activities. An essential\nreminder is that only propagating fields can persist in the absence of\nsources, so that virtually all laser-based experiments occur in a\nsuperposition of propagating-field components. The radiation gauge (or Coulomb\ngauge) is the only gauge employable without risking the creation of hidden\nerrors due to improper gauge choice. It is unfortunate that, in strong-field\nlaser applications, the widespread use of the dipole approximation introduces\nprecisely that hazard of hidden errors that affects both practical\ncalculations and qualitative insights into the behavior of physical systems\nsubjected to laser fields.\n\nIt has been shown that gauge transformations are not, in general, unitary.\nThis has never previously been reported, and it can lead to further errors in\naddition to the important example explored in Section III above about the\nputative universality of the length gauge.\n\nThe concept of primary and secondary physical quantities has been introduced,\nwith over-dependence on a secondary quantity like the electric field having\nthe capability of leading to important misconceptions.\n\nThe criterion of a critical field for longitudinal fields has no relevance to\nthe transverse field of laser-induced processes.\n\nThe appearance of mixed quantum and classical quantities in ascertaining the\nlimits on perturbative methods in the application of strong-field theories\nidentifies strong-field physics as the bridge connecting quantum\nelectrodynamics and the classical electromagnetism of Maxwell.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n \nThe notion of minimal sets (in the sense of Almgren \\cite{Al76}, Reifenberg \\cite{Rei60}. See David \\cite{DJT}, Liang \\cite{topo}, etc..for other variants) is a way to try to solve Plateau's problem in the setting of sets. Plateau's problem, as one of the main interests in geometric measure theory, aims at understanding the existence, regularity and local structure of physical objects that minimize the area while spanning a given boundary, such as soap films. The result of this article is closely linked to two important aspects of this problem: the local behavior, and the local uniqueness of solutions.\n\nIt is known (cf. Almgren \\cite{Al76}, David \\& Semmes \\cite{DS00}) that a $d$-dimensional minimal set $E$ admits a unique tangent plane at almost every point $x$. In this case the local structure around such a point is very clear:  the set $E$ is locally a minimal surface (and hence a real analytic surface) around the point, due to the famous regularity result of Allard \\cite{All72}. \n\nSo we are mostly interested in what happens around points that admit no tangent plane, namely, the singular points. \n\nIn \\cite{DJT}, David proved that the blow-up limits (''tangent objects'') of $d$-dimensional minimal sets at a point are $d$-dimensional minimal cones (minimal sets that are cones in the means time). Blow-up limits of a set at a point reflect the asymptotic behavior of the set at infinitesimal scales around this point. As consequence, a first step to study local structures of minimal sets, is to classify all possible types of singularities--that is to say, minimal cones. \n\n\\subsection{Local behavior, and classification of singularities}\n\nThe plan for the list of $d$-dimensional minimal cones in $\\mathbb R^n$ is very far from clear. Even for $d=2$, we know very little, except for the case in $\\mathbb R^3$, where Jean Taylor \\cite{Ta} gave a complete classification in 1976, and the list is in fact already known a century ago in other circumstances (see \\cite{La} and \\cite{He}). They are, modulo isomorphism: a plane, a $\\mathbb Y$ set (the union of three half planes that meet along a straight line where they make angles of $120^\\circ$), and a $\\mathbb T$ set (the cone over the 1-skeleton of a regular tetrahedron centred at the origin). See the pictures below.\n\n\\centerline{\\includegraphics[width=0.16\\textwidth]{Y.pdf} \\hskip 2cm\\includegraphics[width=0.2\\textwidth]{T.pdf}}\n\\nopagebreak[4]\n\\hskip 4cm a $\\mathbb Y$ set\\hskip 3.9cm  a $\\mathbb T$ set\n\nBased on the above, a natural way to find new types of singularities, is by taking unions and products of known minimal cones.\n\nFor unions: The minimality of the union of two orthogonal minimal sets of dimension $d$ can be obtained easily from a well known geometric lemma (cf. for example Lemma 5.2 of \\cite{Mo84}). Thus one suspects that if the angle between two minimal sets is not far from orthogonal, the union of them might also be minimal.\n\nIn case of planes, the author proved in \\cite{2p} and \\cite{2ptopo}, that the almost orthogonal union of several $d$-dimensional planes is Almgren and topological minimal. When the number of planes is two, this is part of Morgan's conjecture in \\cite{Mo93} on the angle condition under which a union of two planes is minimal. \n\nAs for minimal cones other than unions of planes, since they are all with non isolated singularities (after the structure Theorem \\textcolor{blue}{2.22}), the situation is much more complicated, as briefly stated in the introduction of \\cite{2T}. Up to now we are able to treat a big part of 2 dimensional cases: in \\cite{2T} we prove that the almost orthogonal union of several 2-dimensional minimal cones in (in any ambient dimension) are minimal, provided that they are all measure and sliding stable, and satisfy some uniqueness condition. (The theorem is stated separately in the Almgren case and topological case in \\cite{2T}.) Moreover, this union stays measure and sliding stable, and satisfies the same uniqueness condition. This enables us to continue obtaining  infinitely many new families of minimal cones by taking a finite number of iterations of almost orthogonal unions.\n\nThe result makes good sense, because almost all known 2-dimensional minimal cones satisfy all the above conditions. The proof of this will be contained in the following papers : \n\nIn this article we prove that all 2-dimensional minimal cones in $\\mathbb R^3$ are topological and Almgren unique (Theorems \\textcolor{blue}{5.1, 5.2 and 5.6}). \n\nThen in \\cite{stablePYT} we prove that all 2-dimensional minimal cones in $\\mathbb R^n$ are measure stable, and all 2-dimensional minimal cones in $\\mathbb R^3$ satisfy the sliding stability. 2-dimensional minimal cones in $\\mathbb R^3$ are still sliding stable. \n\nBy Theorem \\textcolor{blue}{10.1} and Remark \\textcolor{blue}{10.5} of \\cite{2T}, the almost orthogonal unions of several planes in $\\mathbb R^n$ are also topological sliding and Almgren sliding stable.\n\nThe only known 2-dimensional minimal cone other than the aboves, is the set $Y\\times Y$, the product of two 1-dimensional $\\mathbb Y$ sets. The proof of its sliding stability and uniqueness are much more involved, so that we will treat it in a separate paper \\textcolor{blue}{\\cite{stableYXY}}.\n\nAs a small remark, compare to the unions, the case of product is much more mysterious. It is not known in general whether the product of two non trivial minimal cones stays minimal. We even do not know whether the product of a minimal cone with a line stays minimal. Moreover, if we consider the product of two concrete minimal cones (other than planes) one by one, up to now the only known result is the minimality of the product of two 1-dimensional $\\mathbb Y$ sets (cf. \\cite{YXY}). Among all singular minimal cones, 1-dimensional $\\mathbb Y$ sets are of simplest structure, but still, the proof of the minimality of their product is surprisingly hard.\n\n\\subsection{About uniqueness of solutions}\n\nAnother natural question about Plateau's problem is the uniqueness of solutions. \n\nIt is well known that solutions for Plateau's problem are in general not unique, even in codimension 1. A simplest example is the following: given two parallel circles in $\\mathbb R^3$, we can have three types of different solutions : the union of two disks, the part of catenoid, and the third type--a ''catenoid'' with a central disk. See the picture below. They admit different topologies, and they all exist in soap film experiments. \n\n\\centerline{\\includegraphics[width=0.2\\textwidth]{catenoid.jpg} \\hskip 2cm\\includegraphics[width=0.22\\textwidth]{catenodisk.jpg}}\n\\nopagebreak[4]\n\\hskip 3.7cm a catenoid\\hskip 2.7cm  a catenoid with a central disk\n\n\\bigskip\n\nOn the other hand, we know that around a regular point $x$ of a minimal set, the solution is locally unique, because the soap film is locally a minimal graph on a convex part of the tangent plane at $x$, and the uniqueness comes from calibrations for minimal graphs.\n\nThe advantage of considering local uniqueness is that we do not have to worry about topology. One can then ask whether this local uniqueness also holds for singular points. Since blow-up limits at singular points are minimal cones, a first step is to study whether each minimal cone is the unique solution, at least under a given topology.\n\nDue to the lack of information on the structure for minimal cones of dimension greater or equal to 3, we are kind of still far from a general conclusion. However, from the very little information we get, we can still give a positive answer for almost all known 2-dimensional minimal cones. See the account in the last subsection.\n\n\\subsection{Upper-semi-continuity, and the organization of the paper}\n\nBesides the main results about uniqueness, an indispensible intermediate step in the discuss for the uniqueness property is the upper-semi-continuity property for minimal sets with reasonable boundary regularity (Theorem \\textcolor{blue}{4.1}). It consists of saying that in many cases, when its boundary is not boo wild, a minimal set minimizes also the measure in the class of limits of deformations, which is much larger than the class of deformations. This property is helpful in various circumstances: when we have to carry on arguments using Hausdorff limits and some properties do not pass to the limit, the upper semi continuity can serve as a link. As an example, it plays a very important role throughout \\cite{2T}.\n\nThe organization of the rest of the article is the following: \n\nIn Section 2 we introduce basic definitions and preliminaries for minimal sets, and properties for 2-dimensional minimal cones. \n\nThe definitions of uniqueness and some related useful properties are proved in Seciton 3.\n\nIn section 4 we prove the upper-semi continuity property for minimal sets with relatively regular boundaries (Theorem \\textcolor{blue}{4.1}). This theorem guarantees in particular that the definition of uniqueness makes good sense for minimal cones, and many other minimal sets. It is also useful in many other circumstances, see \\cite{2T} for example.\n\nWe prove topological and Almgren uniqueness for each 2-dimensional minimal cones in $\\mathbb R^3$ in Section 5.\n\n\\textbf{Acknowledgement:} This work is supported by China's Recruitement Program of Global Experts, School of Mathematics and Systems Science, Beihang University. \n\n\n\\section{Definitions and preliminaries}\n\n\\subsection{Some useful notation}\n\n$[a,b]$ is the line segment with endpoints $a$ and $b$;\n\n$\\overrightarrow{ab}$ is the vector $b-a$;\n\n$R_{ab}$ denotes the half line issued from the point $a$ and passing through $b$;\n\n$B(x,r)$ is the open ball with radius $r$ and centered on $x$;\n\n$\\overline B(x,r)$ is the closed ball with radius $r$ and center $x$;\n\n${\\cal H} ^d$ is the Hausdorff measure of dimension $d$ ;\n\n$d_H(E,F)=\\max\\{\\sup\\{d(y,F):y\\in E\\},\\sup\\{d(y,E):y\\in F\\}\\}$ is the Hausdorff distance between two sets $E$ and $F$. \n\nFor any subset $K\\subset \\mathbb R^n$, the local Hausdorff distance in $K$ $d_K$ between two sets $E,F$ is defined as $d_K(E,F)=\\max\\{\\sup\\{d(y,F):y\\in E\\cap K\\},\\sup\\{d(y,E):y\\in F\\cap K\\}\\}$;\n\nFor any open subset $U\\subset \\mathbb R^n$, let $\\{E_n\\}_n$, $F$ be closed sets in $U$, we say that $F$ is the Hausdorff limit of $\\{E_n\\}_n$, if for any compact subset $K\\subset U$, $\\lim_n d_K(E_n,F)=0$;\n\n$d_{x,r}$ : the relative distance with respect to the ball $B(x,r)$, is defined by\n$$ d_{x,r}(E,F)=\\frac 1r\\max\\{\\sup\\{d(y,F):y\\in E\\cap B(x,r)\\},\\sup\\{d(y,E):y\\in F\\cap B(x,r)\\}\\}.$$\n\nFor any (affine) subspace $Q$ of $\\mathbb R^n$, and $x\\in Q$, $r>0$, $B_Q(x,r)$ stands for $B(x,r)\\cap Q$, the open ball in $Q$.\n\nFor any subset $E$ of $\\mathbb R^n$ and any $r>0$, we call $B(E,r):=\\{x\\in \\mathbb R^n: dist (x,E)<r\\}$ the $r$ neighborhood of $E$.\n\nIf $E$ is a $d$-rectifiable set, denote by $T_xE$ the tangent plane (if it exists and is unique) of $E$ at $x$.\n\n\\subsection{Basic definitions and notations about minimal sets}\n\nIn the next definitions, fix integers $0<d<n$. We first give a general definition for minimal sets. Briefly, a minimal set is a closed set which minimizes the Hausdorff measure among a certain class of competitors. Different choices of classes of competitors give different kinds of minimal sets.\n\n\\begin{defn}[Minimal sets]Let $0<d<n$ be integers. Let $U\\subset \\mathbb R^n$ be an open set. A relative closed set $E\\subset U$ is said to be minimal of dimension $d$ in $U$ with respect to the competitor class $\\mathscr F$ (which contains $E$) if \n\\begin{equation} {\\cal H} ^d(E\\cap B)<\\infty\\mbox{ for every compact ball }B\\subset U,\\end{equation}\nand\n\\begin{equation} {\\cal H} ^d(E\\backslash F)\\le {\\cal H} ^d(F\\backslash E)\\end{equation}\nfor any competitor $F\\in\\mathscr F$.\n\\end{defn}\n\n\\begin{defn}[Almgren competitor (Al competitor for short)] Let $E$ be relatively closed in an open subset $U$ of $\\mathbb R^n$. An Almgren competitor for $E$ is an relatively closed set $F\\subset U$ that can be written as $F=\\varphi_1(E)$, where $\\varphi_t:U\\to U,t\\in [0,1]$ is a family of continuous mappings such that \n\\begin{equation} \\varphi_0(x)=x\\mbox{ for }x\\in U;\\end{equation}\n\\begin{equation}\\mbox{ the mapping }(t,x)\\to\\varphi_t(x)\\mbox{ of }[0,1]\\times U\\mbox{ to }U\\mbox{ is continuous;}\\end{equation}\n\\begin{equation}\\varphi_1\\mbox{ is Lipschitz,}\\end{equation}\n  and if we set $W_t=\\{x\\in U\\ ;\\ \\varphi_t(x)\\ne x\\}$ and $\\widehat W=\\bigcup_{t\\in[0.1]}[W_t\\cup\\varphi_t(W_t)]$,\nthen\n\\begin{equation} \\widehat W\\mbox{ is relatively compact in }U.\\end{equation}\n \nSuch a $\\varphi_1$ is called a deformation in $U$, and $F$ is also called a deformation of $E$ in $U$.\n\\end{defn}\n\nFor future convenience, we also have the following more general definition:\n\n\\begin{defn}Let $U\\subset \\mathbb R^n$ be an open set, and let $E\\subset \\mathbb R^n$ be a closed set (not necessarily contained in $U$). We say that $E$ is minimal in $U$, if $E\\cap U$ is minimal in $U$. A closed set $F\\subset \\mathbb R^n$ is called a  deformation of $E$ in $U$, if $F=(E\\backslash U)\\cup \\varphi_1(E\\cap U)$, where $\\varphi_1$ is a deformation in $U$.\n\\end{defn}\n\nWe denote by ${\\cal F} (E,U)$ the class of all deformations of $E$ in $U$. In the article, we need to use Hausdorff limits in ${\\cal F} (E,U)$. However, if we regard elements of ${\\cal F} (E,U)$ as sets in $\\mathbb R^n$, and take the Hausdorff limit, the limit may have positive measure on $\\partial U\\backslash E$. In other words, sets in ${\\cal F} (E,U)$ may converge to the boundary. We do not like this. Hence we let $\\overline{\\cal F} (E,U)$ be the class of Hausdorff limits of sequnces in ${\\cal F} (E,U)$ that do not converge to the boundary. That is: we set\n\\begin{equation} \\begin{split}\\overline{\\cal F} (E,U)&=\\{F\\subset\\bar U: \\exists \\{E_n\\}_n\\subset{\\cal F} (E,U)\\mbox{ such that }d_K(E_n,F)\\to 0\\\\\n&\\mbox{ for all compact set }K\\subset \\mathbb R^n, \n\\textcolor{blue}{(2.1)}\\mbox{ holds for }F\\mbox{, and }{\\cal H} ^d(F\\cap\\partial U\\backslash E)=0\\}.\\end{split}\\end{equation}\n\nIt is easy to see that both classes are stable with respect to Lipschitz deformations in $U$. \n\n\\begin{defn}[Almgren minimal sets]\nLet $0<d<n$ be integers, $U$ be an open set of $\\mathbb R^n$. An Almgren-minimal set $E$ in $U$ is a minimal set defined in Definition \\textcolor{blue}{2.1} while taking the competitor class $\\mathscr F$ to be the class of all Almgren competitors for $E$.\\end{defn}\n\nFor the need of our future argument, we also have the following definition:\n\n\\begin{defn}Let $0<d<n$ be integers, $U$ be an open set of $\\mathbb R^n$. A closed set $E\\subset \\mathbb R^n$ is said to be Almgren minimal in $U$, if $E\\cap U$ is Almgren minimal in $U$.\n\\end{defn}\n\n\n\\begin{defn}[Topological competitors] Let $G$ be an abelian group. Let $E$ be a closed set in an open domain $U$ of $\\mathbb R^n$. We say that a closed set $F$ is a $G$-topological competitor of dimension $d$ ($d<n$) of $E$ in $U$, if there exists a convex set $B$ such that $\\bar B\\subset U$ such that\n\n1) $F\\backslash B=E\\backslash B$;\n\n2) For all Euclidean $n-d-1$-sphere $S\\subset U\\backslash(B\\cup E)$, if $S$ represents a non-zero element in the singular homology group $H_{n-d-1}(U\\backslash E; G)$, then it is also non-zero in $H_{n-d-1}(U\\backslash F;G)$.\nWe also say that $F$ is a $G$-topological competitor of $E$ in $B$.\n\nWhen $G=\\mathbb Z$, we usually omit $\\mathbb Z$, and say directly topological competitor.\n\\end{defn}\n\nAnd Definition \\textcolor{blue}{2.1} gives the definition of $G$-topological minimizers in a domain $U$ when we take the competitor class to be the class of $G$-topological competitors of $E$.\n\nThe simplest example of a $G$-topological minimal set is a $d-$dimensional plane in $\\mathbb R^n$.  \n\n\\begin{pro}[cf.\\cite{topo} Proposition 3.7 and Corollary 3.17]  \n\n$1^\\circ$ Let $E\\subset \\mathbb R^n$ be closed. Then for any $d<n$, and any convex set $B$, $B'$ such that $\\bar B'\\subset B^\\circ$, every Almgren competitor of $E$ in $B'$ is a $G$-topological competitor of $E$ in $B$ of dimension $d$.\n\n$2^\\circ$ All $G$-topological minimal sets are Almgren minimal in $\\mathbb R^n$.\n\\end{pro}\n\n\\begin{pro}[Topological competitors pass to the limit] Let $E$ be a closed set in an open domain $U$ of $\\mathbb R^n$, and let $B'$ be a convex set such that $\\bar B'\\subset U$. If $\\{F_n\\}$ is a sequence of $d$-$G$-topological competitors of $E$ in $B'$, and $F_n$ converge to $F$ in Hausdorff distance, then for any convex set $B$ such that $\\bar B'\\subset B\\subset\\bar B\\subset U$, $F$ is a $G$-topological competitor of $E$ in $B$. \n\\end{pro}\n\n\\noindent Proof.  Let us verify the two conditions in Definition \\textcolor{blue}{2.6}.\n\nSince $F_j$ converge to $F$, and $F_j\\backslash B'=E\\backslash B'$, hence $F\\backslash \\bar B'=E\\backslash \\bar B'$. Since $\\bar B'\\subset B$, we know that 1) holds;\n\nNow take any $n-d-1$-sphere $S\\subset U\\backslash (B\\cup E)$ that represents a non-zero element in $H_{n-d-1}(U\\backslash E;G)$. Since $B'\\subset B$, we know that $S\\subset U\\backslash (B'\\cup E)$. We know that each $F_j$ is a $d$-$G$-topological competitor for $E$ in $B$, hence $S$ also represents a non-zero element in $H_{n-d-1}(U\\backslash F_j;G)$.\n\nFor 2), suppose it does not holds. That is, $S$ represents a zero element in $H_{n-d-1}(U\\backslash F;G)$. As a result, there exists a singular $n-d$ chain $\\sigma$ in $U\\backslash F$, such that $\\partial \\sigma=S$. Then the support $|\\sigma|$ of $\\sigma$ is compact in $U\\backslash F$. Since $U\\backslash F$ is open, there exists $\\epsilon>0$ such that the $\\epsilon$-neighborhood $B(|\\sigma,\\epsilon)\\subset U\\backslash F$. As a result, since $F_j\\to F$, we know that for $j$ large enough, $F_j\\cap|\\sigma|=\\emptyset$. Hence $\\sigma$ is also a simplicial chain in $U\\backslash F_j$ for $j$ large. Then $\\partial\\sigma=S$ implies that $S$ represents a zero element in $H_{n-d-1}(U\\backslash F_j;G)$ for $j$ large. This contradicts the fact that $S$ represents a non-zero element in $H_{n-d-1}(U\\backslash F_j;G)$ for all $j$.\n\nHence 2) holds.\n\\hfill$\\Box$\\bigskip\n\n\\begin{defn}[reduced set] Let $U\\subset \\mathbb R^n$ be an open set. For every closed subset $E$ of $U$, denote by\n\\begin{equation} E^*=\\{x\\in E\\ ;\\ {\\cal H} ^d(E\\cap B(x,r))>0\\mbox{ for all }r>0\\}\\end{equation}\n the closed support (in $U$) of the restriction of ${\\cal H} ^d$ to $E$. We say that $E$ is reduced if $E=E^*$.\n\\end{defn}\n\nIt is easy to see that\n\\begin{equation} {\\cal H} ^d(E\\backslash E^*)=0.\\end{equation}\nIn fact we can cover $E\\backslash E^*$ by countably many balls $B_j$ such that ${\\cal H} ^d(E\\cap B_j)=0$.\n\n\\begin{rem}\n It is not hard to see that if $E$ is Almgren minimal (resp. $G$-topological minimal), then $E^*$ is also Almgren minimal (resp. $G$-topological minimal). As a result it is enough to study reduced minimal sets. An advantage of reduced minimal sets is, they are locally Ahlfors regular (cf. Proposition 4.1 in \\cite{DS00}). Hence any approximate tangent plane of them is a true tangent plane. Since minimal sets are rectifiable (cf. \\cite{DS00} Theorem 2.11 for example), reduced minimal sets admit true tangent $d$-planes almost everywhere.\n\\end{rem}\n\nIf we regard two sets to be equivalent if they are equal modulo ${\\cal H} ^d$-null sets, then a reduced set is always considered to be a good (in the sense of regularity) represent of its equivalent class. \n\n\\textbf{In the rest of the article, we only consider reduced sets.}\n\n\\begin{rem}The notion of (Almgren or $G$-topological) minimal sets does not depend much on the ambient dimension. One can easily check that $E\\subset U$ is $d-$dimensional Almgren minimal in $U\\subset \\mathbb R^n$ if and only if $E$ is Almgren minimal in $U\\times\\mathbb R^m\\subset\\mathbb R^{m+n}$, for any integer $m$. The case of $G$-topological minimality is proved in \\cite{topo} Proposition 3.18.\\end{rem}\n\n\\subsection{Regularity results for minimal sets}\n\nWe now begin to give regularity results for minimal sets. They are in fact regularity results for Almgren minimal sets, but they also hold for all $G$-topological minimizers, after Proposition \\textcolor{blue}{2.7}. By Remark \\textcolor{blue}{2.10}, from now on all minimal sets are supposed to be reduced. \n\n\\begin{defn}[blow-up limit] Let $U\\subset\\mathbb R^n$ be an open set, let $E$ be a relatively closed set in $U$, and let $x\\in E$. Denote by $E(r,x)=r^{-1}(E-x)$. A set $C$ is said to be a blow-up limit of $E$ at $x$ if there exists a sequence of numbers $r_n$, with $\\lim_{n\\to \\infty}r_n=0$, such that the sequence of sets $E(r_n,x)$ converges to $C$ for the local Hausdorff distance in any compact set of $\\mathbb R^n$.\n\\end{defn}\n\n\\begin{rem}\n $1^\\circ$ A set $E$ might have more than one blow-up limit at a point $x$. However it is not known yet whether this can happen to minimal sets. \n \n When a set $E$ admits a unique blow-up limit at a point $x\\in E$, denote this blow-up limit by $C_xE$.\n \n $2^\\circ$ Let $Q\\subset \\mathbb R^n$ be any subpace, denote by $\\pi_Q$ the orthogonal projection from $\\mathbb R^n$ to $Q$. Then it is easy to see that if $E\\subset \\mathbb R^n$, $x\\in E$, and $C$ is any blow-up limit of $E$ at $x$, then $\\pi_Q(C)$ is contained in a blow-up limit of $\\pi_Q(E)$ at $\\pi_Q(x)$. \n\\end{rem}\n\n\\begin{pro}[c.f. \\cite{DJT} Proposition 7.31]Let $E$ be a reduced Almgren minimal set in an open set $U$ of $\\mathbb R^n$, and let $x\\in E$. Then every blow-up limit of $E$ at $x$ is a reduced Almgren minimal cone $F$ centred at the origin, and ${\\cal H} ^d(F\\cap B(0,1))=\\theta(x):=\\lim_{r\\to 0} r^{-d}{\\cal H} ^d(E\\cap B(x,r)).$\\end{pro}\n\nAn Almgren minimal cone is just a cone which is also Almgren minimal. We will call them minimal cones throughout this paper, since we will not talk about any other type of minimal cones. \n\n\\begin{rem}$1^\\circ$ The existence of the density $\\theta(x)$ is due to the monotonicity of the density function $\\theta(x,r):=r^{-d}{\\cal H} ^d(E\\cap B(x,r))$ at any given point $x$ for minimal sets. See for example \\cite{DJT} Proposition 5.16.\n\n$2^\\circ$ After the above proposition, the set $\\Theta(n,d)$ of all possible densities for points in a $d$-dimension minimal set in $\\mathbb R^n$ coincides with the set of all possible densities for $d$-dimensional minimal cones in $\\mathbb R^n$. When $d=2$, this is a very small set. For example, we know that $\\pi$ is the density for a plane, $\\frac 32\\pi$ is the density for a $\\mathbb Y$ set, and for any $n$, and any other type of 2-dimensional minimal cone in $\\mathbb R^n$, its density should be no less than some $d_T=d_T(n)>\\frac 32\\pi$, by \\cite{DJT} Lemma 14.12.\n\n$3^\\circ$ Obviously, a cone in $\\mathbb R^n$ is minimal if and only if it is minimal in the unit ball, if and only if it is minimal in any open subset containing the origin.\n\n$4^\\circ$ For future convenience, we also give the following notation: let $U\\subset \\mathbb R^n$ be a convex domain containing the origin. A set $C\\subset U$ is called a cone in $U$, if it is the intersection of a cone with $U$.\n\\end{rem}\n\nWe now state some regularity results on 2-dimensional Almgren minimal sets. \n\n\\begin{defn}[bi-H\\\"older ball for closed sets] Let $E$ be a closed set of Hausdorff dimension 2 in $\\mathbb R^n$. We say that $B(0,1)$ is a bi-H\\\"older ball for $E$, with constant $\\tau\\in(0,1)$, if we can find a 2-dimensional minimal cone $Z$ in $\\mathbb R^n$ centered at 0, and $f:B(0,2)\\to\\mathbb R^n$ with the following properties:\n\n$1^\\circ$ $f(0)=0$ and $|f(x)-x|\\le\\tau$ for $x\\in B(0,2);$\n\n$2^\\circ$ $(1-\\tau)|x-y|^{1+\\tau}\\le|f(x)-f(y)|\\le(1+\\tau)|x-y|^{1-\\tau}$ for $x,y\\in B(0,2)$;\n\n$3^\\circ$ $B(0,2-\\tau)\\subset f(B(0,2))$;\n\n$4^\\circ$ $E\\cap B(0,2-\\tau)\\subset f(Z\\cap B(0,2))\\subset E.$  \n\nWe also say that B(0,1) is of type $Z$ for $E$.\n\nWe say that $B(x,r)$ is a bi-H\\\"older ball for $E$ of type $Z$ (with the same parameters) when $B(0,1)$ is a bi-H\\\"older ball of type $Z$ for $r^{-1}(E-x)$.\n\\end{defn}\n\n\\begin{thm}[Bi-H\\\"older regularity for 2-dimensional Almgren minimal sets, c.f.\\cite{DJT} Thm 16.1]\\label{holder} Let $U$ be an open set in $\\mathbb R^n$ and $E$ a reduced Almgren minimal set in $U$. Then for each $x_0\\in E$ and every choice of $\\tau\\in(0,1)$, there is an $r_0>0$ and a minimal cone $Z$ such that $B(x_0,r_0)$ is a bi-H\\\"older ball of type $Z$ for $E$, with constant $\\tau$. Moreover, $Z$ is a blow-up limit of $E$ at $x$.\n\\end{thm}\n\n\\begin{defn}[point of type $Z$] \n\n$1^\\circ$ In the above theorem, we say that $x_0$ is a point of type $Z$ (or $Z$ point for short) of the minimal set $E$. The set of all points of type $Z$ in $E$ is denoted by $E_Z$. \n\n$2^\\circ$ In particular, we denote by $E_P$ the set of regular points of $E$ and $E_Y$ the set of $\\mathbb Y$ points of $E$. Any 2-dimensional minimal cone other than planes and $\\mathbb Y$ sets are called $\\mathbb T$ type cone, and any point which admits a $\\mathbb T$ type cone as a blow-up is called a $\\mathbb T$ type point. Set $E_T=E\\backslash (E_Y\\cup E_P)$ the set of all $\\mathbb T$ type points of $E$.  Set $E_S:=E\\backslash E_P$ the set of all singular points in $E$.\n\\end{defn}\n\n\\begin{rem} Again, since we might have more than one blow-up limit for a minimal set $E$ at a point $x_0\\in E$, the point $x_0$ might be of more than one type (but all the blow-up limits at a point are bi-H\\\"older equivalent). However, if one of the blow-up limits of $E$ at $x_0$ admits the``full-length'' property (see Remark \\textcolor{blue}{\\ref{ful}}), then in fact $E$ admits a unique blow-up limit at the point $x_0$. Moreover, we have the following $C^{1,\\alpha}$ regularity around the point $x_0$. In particular, the blow-up limit of $E$ at $x_0$ is in fact a tangent cone of $E$ at $x_0$.\n\\end{rem}\n\n\\begin{thm}[$C^{1,\\alpha}-$regularity for 2-dimensional minimal sets, c.f. \\cite{DEpi} Thm 1.15]\\label{c1} Let $E$ be a 2-dimensional reduced minimal set in the open set $U\\subset\\mathbb R^n$. Let $x\\in E$ be given. Suppose in addition that some blow-up limit of $E$ at $x$ is a full length minimal cone (see Remark \\textcolor{blue}{\\ref{ful}}). Then there is a unique blow-up limit $X$ of $E$ at $x$, and $x+X$ is tangent to $E$ at $x$. In addition, there is a radius $r_0>0$ such that, for $0<r<r_0$, there is a $C^{1,\\alpha}$ diffeomorphism (for some $\\alpha>0$) $\\Phi:B(0,2r)\\to\\Phi(B(0,2r))$, such that $\\Phi(0)=x$ and $|\\Phi(y)-x-y|\\le 10^{-2}r$ for $y\\in B(0,2r)$, and $E\\cap B(x,r)=\\Phi(X)\\cap B(x,r).$ \n\nWe can also ask that $D\\Phi(0)=Id$. We call $B(x,r)$ a $C^1$ ball for $E$ of type $X$.\n\\end{thm}\n\n\\begin{rem}[full length, union of two full length cones $X_1\\cup X_2$]\\label{ful}We are not going to give the precise definition of the full length property. Instead, we just give some information here, which is enough for the proofs in this paper.\n\n$1^\\circ$ The three types of 2-dimensional minimal cones in $\\mathbb R^3$, i.e. the planes, the $\\mathbb Y$ sets, and the $\\mathbb T$ sets, all verify the full-length property (c.f., \\cite{DEpi} Lemmas 14.4, 14.6 and 14.27). Hence all 2-dimensional minimal sets $E$ in an open set $U\\subset\\mathbb R^3$ admits the local $C^{1,\\alpha}$ regularity at every point $x\\in E$. But this was known from \\cite{Ta}.\n\n$2^\\circ$ (c.f., \\cite{DEpi} Remark 14.40) Let $n>3$. Note that the planes, the $\\mathbb Y$ sets and the $\\mathbb T$ sets are also minimal cones in $\\mathbb R^n$. Denote by $\\mathfrak C$ the set of all planes, $\\mathbb Y$ sets and $\\mathbb T$ sets in $\\mathbb R^n$. Let $X=\\cup_{1\\le i\\le n}X_i\\in \\mathbb R^n$ be a minimal cone, where $X_i\\in \\mathfrak{C}, 1\\le i\\le n$, and for any $i\\ne j$, $X_i\\cap X_j=\\{0\\}$. Then $X$ also verifies the full-length property. \n\\end{rem}\n\n\\begin{thm}[Structure of 2-dimensional minimal cones in $\\mathbb R^n$, cf. \\cite{DJT} Proposition 14.1] Let $K$ be a reduced 2-dimensional minimal cone in $\\mathbb R^n$, and let $X=K\\cap \\partial B(0,1)$. Then $X$ is a finite union of great circles and arcs of great circles $C_j,j\\in J$. The $C_j$ can only meet at their endpoints, and each endpoint is a common endpoint of exactly three $C_j$, which meet with $120^\\circ$ angles. In addition, the length of each $C_j$ is at least $\\eta_0$, where $\\eta_0>0$ depends only on the ambient dimension $n$.\n\\end{thm}\n\nAn immediate corollary of the above theorem is the following:\n\n\\begin{cor}\n$1^\\circ$ If $C$ is a minimal cone of dimension 2, then for the set of regular points $C_P$ of $C$, each of its connected components is a sector. \n\n$2^\\circ$ Let $E$ be a 2-dimensional minimal set in $U\\subset \\mathbb R^n$. Then $\\bar E_Y=E_S$.\n\n$3^\\circ$ The set $E_S\\backslash E_Y$ is isolated. \\end{cor}\n\nAs a consequence of the $C^1$ regularity for regular points and $\\mathbb Y$ points, and Corollary \\textcolor{blue}{2.23}, we have\n\\begin{cor}Let $E$ be an 2-dimensional Almgren minimal set in a domain $U\\subset \\mathbb R^n$. Then \n\n$1^\\circ$ The set $E_P$ is open in $E$;\n\n$2^\\circ$ The set $E_Y$ is a countable union of $C^1$ curves. The endpoints of these curves are either in $E_T:=E_S\\backslash E_Y$, or lie in $\\partial U$.  \n\\end{cor}\n\n\nWe also have a similar quantified version of the $C^{1,\\alpha}$ regularity (cf. \\cite{DJT} Corollary 12.25). In particular, we can use the distance between a minimal set and a $\\mathbb P$ or a $\\mathbb Y$ cone to controle the constants of the $C^{1,\\alpha}$ parametrization. As a direct corollary, we have the following neighborhood deformation retract property for regular and $\\mathbb Y$ points:\n\n\\begin{cor}There exists $\\epsilon_2=\\epsilon_2(n)>0$ such that the following holds : let $E$ be an 2-dimensional Almgren minimal set in a domain $U\\subset \\mathbb R^n$. Then \n\n$1^\\circ$ For any $x\\in E_P$, and any co-dimension 1 submanifold $M\\subset U$ which contains $x$, such that $M$ is transversal to the tangent plane $T_xE+x$, if $r>0$ satisfies that $d_{x,r}(E, x+T_xE)<\\epsilon_2$, then ${\\cal H} ^1(B(x,r)\\cap M\\cap E)<\\infty$, and $B(x,r)\\cap M\\cap E$ is a Lipschitz deformation retract of $B(x,r)\\cap M$;\n\n$2^\\circ$ For any $x\\in E_Y$, and any co-dimension 1 submanifold $M\\subset U$ which contains $x$, such that $M$ is transversal to the tangent cone $C_xE+x$ and its spine, if $r>0$ satisfies that $d_{x,r}(E, x+C_xE)<\\epsilon_2$, then ${\\cal H} ^1(B(x,r)\\cap M\\cap E)<\\infty$, and $B(x,r)\\cap M\\cap E$ is a Lipschitz deformation retract of $B(x,r)\\cap M$.\n\\end{cor}\n\nAs for the regularity for minimal sets of higher dimensions, we know much less. But for points which admit a tangent plane (i.e. some blow up-limit on the point is a plane), we still have the $C^1$ regularity.\n\n\\begin{thm}[cf.\\cite{2p} Proposition 6.4]\\label{e1}For $2\\le d<n<\\infty$, there exists $\\epsilon_1=\\epsilon_1(n,d)>0$ such that if $E$ is a $d$-dimensional reduced minimal set in an open set $U\\subset\\mathbb R^n$, with $B(0,2)\\subset U$ and $0\\in E$. Then if $E$ is $\\epsilon_1$ near a $d-$plane $P$ in $B(0,1)$, then $E$ coincides with the graph of a $C^1$ map $f:P\\to P^\\perp$ in $B(0,\\frac 34)$. Moreover $||\\nabla f||_\\infty<1$.\n\\end{thm}\n\n\\begin{rem}\n$1^\\circ$ This proposition is a direct corollary of Allard's famous regularity theorem for stationary varifold. See \\cite{All72}.\n\n$2^\\circ$ After this proposition, a blow-up limit of a reduced minimal set $E$ at a point $x\\in E$ is a plane if and only if the plane is the unique approximate tangent plane of $E$ at $x$.\n\\end{rem}\n\nAfter Remark \\textcolor{blue}{2.27}, for any reduced minimal set $E$ of dimension $d$, and for any $x\\in E$ at which an approximate tangent $d$-plane exists (which is true for a.e. $x\\in E$), $T_xE$ also denotes the tangent plane of $E$ at $x$, and the blow-up limit of $E$ at $x$. \n\n\\section{Uniqueness: definitions and properties}\n\n\\begin{defn}\nLet $C$ be a $d$-dimensional reduced Almgren minimal set in a bounded domain $U$, we say that \n\n$1^\\circ$ $C$ is Almgren unique in $U$ if it is the only reduced set in $\\overline{\\cal F} (C,U)$ that attains the minimal measure. That is:\n\\begin{equation} \\forall \\mbox{ reduced set }E\\in \\overline{\\cal F} (C,U), {\\cal H} ^d(E)=\\inf_{F\\in \\overline {\\cal F} (C,U)}{\\cal H} ^d(F)\\Rightarrow E=C.\\end{equation}\n\n$2^\\circ$ $C$ is $G$-topological unique in $U$ if $C$ is $G$-topological minimal, and \n\\begin{equation} \\begin{split}\\mbox{For any reduced }d\\mbox{-dimensional }G-\\mbox{topological competitor }E\\mbox{ of }C\\mbox{ in }U,\\\\\n{\\cal H} ^d(E\\cap U)={\\cal H} ^d(C\\cap U), \\mbox{ implies }C=E;\\end{split}\\end{equation}\n\n$3^\\circ$ We say that a $d$-dimensional minimal set $C$ in $\\mathbb R^n$ is Almgren (resp. $G$-topological) unique, if it is Almgren (resp. $G$-topologial) unique in every bounded domain $U\\subset \\mathbb R^n$.\n\nWhen $G=\\mathbb Z$, we usually omit $\\mathbb Z$, and say directly topological unique.\n\\end{defn}\n\nFor minimal cones, we have immediately:\n\n\\begin{pro}[Unique minimal cones]Let $K$ be a $d$-dimensional Almgren minimal cone in $\\mathbb R^n$. Then it is Almgren (resp. $G$-topological) unique, if and only if it is Almgren (resp. $G$-topological) unique in some bounded convex domain $U$ that contains the origin. \n\\end{pro}\n\n\\noindent Proof.  By definition, the only if part is trivial. So let us prove the converse.\n\nSuppose that $K$ is a $d$-dimensional Almgren minimal cone in $\\mathbb R^n$, and is Almgren (resp. $G$-topological) unique in a bounded convex domain $U$ that contains the origin. Then since $K$ is a cone centered at the origin, $K$ is Almgren (resp. $G$-topological) unique in $rU$ for all $r>0$. Now for any other bounded domain $U'$, there exists $r$ such that $U'\\subset rU$, hence $K$ is Almgren (resp. $G$-topological) unique in $U'$.\\hfill$\\Box$\\bigskip\n\nLet us give some important remarks:\n\n\\begin{rem}\n\n$1^\\circ$ Note that for an arbitrary $d$-dimensional Almgren minimal set $C$ in $U$, by definition, $C$ only minimizes the measure in the class ${\\cal F} (C,U)$. Hence we do not necessarily have that \n\\begin{equation} {\\cal H} ^d(C)=\\inf_{F\\in \\overline {\\cal F} (C,U)}{\\cal H} ^d(F).\\end{equation}\nOn the other hand, this holds if $U$ is a uniformly convex domain. See Theorem \\textcolor{blue}{4.1} and Corollary \\textcolor{blue}{4.7}  in the next section. \n\n$2^\\circ$ As a corollary of the above term $1^\\circ$, and Proposition \\textcolor{blue}{3.2}, we know that if $K$ is a $d$-dimensinoal minimal cone in $\\mathbb R^n$, then \\textcolor{blue}{(3.3)} holds automatically. \n\n$3^\\circ$ The condition ${\\cal H} ^d(E)=\\inf_{F\\in \\overline{\\cal F} }{\\cal H} ^d(F)$ in \\textcolor{blue}{(3.1)} already implies that $E$ is itself a minimal set, since the class $\\overline{\\cal F} $ is stable under deformations. Also notice that ${\\cal H} ^d(E)=\\inf_{F\\in \\overline{\\cal F} }{\\cal H} ^d(F)$ is equivalent to the condition ${\\cal H} ^d(E)\\le\\inf_{F\\in \\overline{\\cal F} }{\\cal H} ^d(F)$ since $E\\in \\overline {\\cal F} $. Similarly, when $U$ is a convex domain, since the condition ${\\cal H} ^d(E\\cap U)={\\cal H} ^d(C\\cap U)$ in \\textcolor{blue}{(3.2)} implies that $E$ minimizes measure among all $G$-topological competitors for $C$, and all $G$-topological competitors for $E$ are $G$-topological competitors for $C$ for $U$ convex, hence $E$ is $G$-topological minimal in $U$.\n\n$4^\\circ$ If $C$ is an Almgren unique minimal set in $U$, $V\\subset U$ is a domain, then $C$ is also Almgren unique minimal in $V$.\n\\end{rem}\n\nThe next proposition shows that for minimal cones, $G$-topological uniqueness implies Almgren uniqueness: \n\n\\begin{pro}Let $K\\subset \\mathbb R^n$ be a $G$-topological unique minimal cone of dimension $d$. Then it is also Almgren unique.\n\\end{pro}\n\n\\noindent Proof.  Let $K$ be a $G$-topological unique minimal cone of dimension $d$ in $\\mathbb R^n$. By Proposition \\textcolor{blue}{3.2}, it is enough to prove that $K$ is Almgren unique in the unit ball $B$. \n\nLet $F\\in \\overline{\\cal F} (K,B)$, such that \n\\begin{equation} {\\cal H} ^d(F)=\\inf_{E\\in \\overline{\\cal F} (K,B)}{\\cal H} ^d(E)={\\cal H} ^d(K\\cap B),\\end{equation} \nthe last equality is by Remark \\textcolor{blue}{3.3} $1^\\circ$.\n\nBy definition of $\\overline{\\cal F} (K,B)$, there exists a sequence $F_j\\in {\\cal F} (K,B)$ that converge to $F$. By Proposition \\textcolor{blue}{2.7}, each set $F_j':=F_j\\cup (K\\backslash B)$ is a $G$-topological competitor for $K$ in $2B$.  Then by Proposition \\textcolor{blue}{2.8}, the limit $F'=F\\cup (K\\backslash B)$ is a $G$-topological competitor in $3B$. \n\nBy \\textcolor{blue}{(3.4)}, we know that \n\\begin{equation}\\begin{split}{\\cal H} ^d(F'\\cap 3B)&={\\cal H} ^d((F\\cup (K\\backslash B))\\cap 3B)={\\cal H} ^d(F)+{\\cal H} ^d(K\\cap 3B\\backslash B)\\\\\n&={\\cal H} ^d(K\\cap 3B)=\\inf_{E\\in \\overline{\\cal F} (K,3B)}{\\cal H} ^d(E),\\end{split}\\end{equation}\nwhere the last equality is again by Remark \\textcolor{blue}{3.3} $1^\\circ$.\n\nSince $K$ is $G$-topological unique, \\textcolor{blue}{(3.5)} implies that $F'=K$, which means that $F=K\\cap B$.\\hfill$\\Box$\\bigskip\n\n\\begin{pro}[Independent of ambient dimension]Let $K\\subset \\mathbb R^m$ be a $d$-dimensional Almgren minimal cone in $\\mathbb R^m$. If $K$ is Almgren (resp. $G$-topological) unique, then for all $n\\ge m$, $K$ is also Almgren (resp. $G$-topological) unique in $\\mathbb R^n$.\n\\end{pro}\n\n\\noindent Proof.  Fix any $n\\ge m$. Write $\\mathbb R^n=\\mathbb R^m\\times \\mathbb R^{n-m}$, and suppose, without loss of generality, that $K$ lives in $\\mathbb R^m\\times \\{0\\}$. Let $\\pi$ be the orthogonal projection from $\\mathbb R^n\\to \\mathbb R^m\\times \\{0\\}$. \n\nSuppose that $K$ is Almgren unique in $\\mathbb R^m$. We want to prove that $K$ is Almgren unique in $\\mathbb R^n$. Let $B_n$ denote the unit ball in $\\mathbb R^n$. Then by Proposition \\textcolor{blue}{3.2}, it is enough to prove that $K\\cap B_n$ is Almgren unique. So let $F\\in \\overline{\\cal F} (K,B_n)$, so that \n\\begin{equation}{\\cal H} ^d(F)=\\inf_{E\\in \\overline{\\cal F} (K,B_n)}{\\cal H} ^d(E)={\\cal H} ^d(K\\cap B_n)={\\cal H} ^d(K\\cap B_m).\\end{equation}  \n\nBy Remark \\textcolor{blue}{3.3, $3^\\circ$}, the condition \\textcolor{blue}{(3.6)} implies that $F$ is Almgren mininal in $B_n$. As a result, by the convex hull property of minimal sets, we know that $F$ must be included in the convex hull of $F\\cap \\partial B_n=K\\cap \\partial B_n=K\\cap \\partial B_m\\subset \\bar B_m.$\n\n As a result,  $F\\in \\overline{\\cal F} (K,B_m)$. By \\textcolor{blue}{(3.6)}, and the Almgren uniqueness of $K$, we know that $F$ must be $K\\cap B_m=K\\cap B_n$.\n \n The proof for the case of $G$-topological uniqueness is similar, and we leave it to the reader. \\hfill$\\Box$\\bigskip\n \n \\section{Upper-semi-continuity}\n \n In this section we prove the upper-simi-continuity property for minimal sets with reasonable boundary regularity. It consists of saying that in many cases, when its boundary is not boo wild, a minimal set minimizes also the measure in the class of limits of deformations. This serves as an indispensible part in the definition of uniqueness, as we have already seen in the last section (Remark \\textcolor{blue}{3.3}). This property also plays a very important role in \\textcolor{blue}{\\cite{2T}}. \n \n\\bigskip\n  \nFor each $k\\in \\mathbb N$, let $\\Delta_k$ denote the family of (closed) dyadic cubes of length $2^{-k}$. For $j\\le n$, let $\\Delta_{k,j}$ denote the set of all faces of elements in $\\Delta_k$. For each cube $Q$, denote by $\\Delta_j(Q)$ the set of all $j$-faces of $Q$. Set $|\\Delta_{k,j}|=\\cup_{\\sigma\\in \\Delta_{k,j}}\\sigma$ the $j$-skeleton of $\\Delta_k$.\n\n\\begin{thm}[upper semi continuity] Let $U\\subset \\mathbb R^n$ be a bounded convex domain, and $E$ be a closed set in $\\bar U$ with locally finite $d$-Hausdorff measure. Let $C$ denote the convex hull of $E$. Suppose that\n\n\\begin{equation} C\\cap\\partial U=E\\cap\\partial U\\end{equation}\nand\n\\begin{equation}\\mbox{There exists a bi Lipschitz map }\\psi: Q_0\\to C\\mbox{, such that }\\psi(E\\cap \\partial U)\\subset |\\Delta_{k,d-1}|.\\end{equation} Then\n\n$1^\\circ$ $\\inf_{F\\in \\overline{\\cal F} (E,U)}{\\cal H} ^d(F)=\\inf_{F\\in {\\cal F} (E,U)}{\\cal H} ^d(F)$;\n\n$2^\\circ$ If $E$ is a $d$-dimensional minimal set in $U$. Then \n\\begin{equation}{\\cal H} ^d(E)=\\inf_{F\\in \\overline{\\cal F} (E,U)}{\\cal H} ^d(F).\\end{equation}\n\\end{thm}\n\n\\begin{rem}\nThe conditions \\textcolor{blue}{(4.1) and (4.2)} can be relaxed, with essentially the same proof, but with more technical details. Here we only give proof under these two hypotheses, which is enough for purpose of use.\\end{rem}\n\n\\noindent Proof.  \n\n$1^\\circ$ Since ${\\cal F} (E,U)\\subset \\overline{\\cal F} (E,U)$, we have automatically $\\inf_{F\\in \\overline{\\cal F} (E,U)}{\\cal H} ^d(F)\\le\\inf_{F\\in {\\cal F} (E,U)}{\\cal H} ^d(F)$. So let us prove the converse. \n\nTo prove the converse, we first prove the following case: suppose that $\\partial E_0\\subset |\\Delta_{k_0,d-1}|$ for some $k_0\\in \\mathbb N$.\n\nLet $\\pi$ denote the shortest distance projection from $U$ to $C$. Then $\\pi$ is 1-Lipschitz, and hence for any set $F\\in U$, we have ${\\cal H} ^d(F)\\ge{\\cal H} ^d(\\pi(F))$. \n\nNow we need the following Theorem: \n \n  \\begin{thm}[Existence of minimal sets\\label{vincent}; c.f. \\cite{Fv}, Thm 6.1.7]\nLet $U\\subset\\mathbb R^n$ be an open domain, $0<d<n$, and let $\\mathfrak F$ be a class of non-empty sets relatively closed in $U$ and satisfying \\textcolor{blue}{(2.1)}, which is stable by deformations in $U$. Suppose that \n \\begin{equation} \\inf_{F\\in\\mathfrak F} H^d(F)<\\infty.\\end{equation}\n Then there exists $M>0$ (depends only on $d$ and $n$), a sequence $(F_k)$ of elements of $\\mathfrak F$, and a set $E$ of dimension $d$ relatively closed in $U$ that verifies \\textcolor{blue}{(2.1)}, such that:\n \n (1) There exists a sequence of compact sets $\\{K_m\\}_{m\\in\\mathbb N}$ in $U$ with $K_m\\subset K_{m+1}$ for all $m$ and $\\cup_{m\\in N}K_m=U$, such that \n \\begin{equation}\\lim_{k\\to\\infty}d_H(F_k\\cap K_m,E\\cap K_m)=0\\mbox{ for all }m\\in \\mathbb N;\\end{equation}\n \n (2) For all open sets $V$ such that $\\overline V$ is relatively compact in $U$, from a certain rank,\n \\begin{equation} F_k\\mbox{ is }(M,+\\infty)\\mbox{-quasiminimal in }V;\\end{equation}\n (See \\cite{DS00} for a precise definition.) \n \n (3) $H^d(E)\\le\\inf_{F\\in\\mathfrak F}H^d(F)$ ;\n \n (4) $E$ is minimal in $U$.\n\\end{thm}\n\nWe apply Theorem \\textcolor{blue}{4.3} to the class $\\mathfrak F$ of all Hausdorff limits of elements in ${\\cal F} (E, V)$, where $V=\\mathbb R^n\\backslash (E\\cap \\partial U)$. It is easy to see that $\\mathfrak F$ is stable by deformation in $V$, and by Hausdorff limit in $V$. As a result, there exists a set $F_0\\in \\mathfrak F$, such that $F_0$ is minimal in $V$, with\n\\begin{equation} {\\cal H} ^d(F_0)=\\inf_{F\\in \\mathfrak F}{\\cal H} ^d(F).\\end{equation}\n\nSet $E_0=\\pi(F_0)$. Then $E_0\\subset C$, and\n\\begin{equation} {\\cal H} ^d(E_0)={\\cal H} ^d(\\pi(F_0))\\le {\\cal H} ^d(F_0)=\\inf_{F\\in \\mathfrak F}{\\cal H} ^d(F).\\end{equation}\n It is easy to see that $E_0\\in \\mathfrak F$. \n \n So let $E_k$ be a sequence in ${\\cal F} (E,V)$ that converges to $E_0$ in $\\mathbb R^n$. Modulo projecting to $C$, we may also suppose that $E_k\\subset C$. Suppose $E_k=\\psi_k(E)$, where $\\psi_k$ is a deformation in $V$, for each $k$. Let $\\psi'_k(x)=\\pi\\circ\\psi_k(x)$ for $x\\in E$, $\\psi'_k=id$ on $\\mathbb R^n\\backslash C$ and $\\{x\\in \\mathbb R^n:\\psi(x)=x\\}$, and extend it to the whole $\\mathbb R^n$, such that $\\psi'_k(U)=U$. Then $\\psi'_k|_U$ is a map from $U$ to $U$, and it is homotopic to the identity through the line homotopy, since $U$ is convex. Thus $\\psi'_k|_U$ is a deformation in $U$, with $\\psi'_k|_U(E)=E_k$. Therefore $E_k\\in {\\cal F} (E,U)$, and hence $E_0\\in \\overline{\\cal F} (E,U)$. \n    \n    Note that \n$\\overline{\\cal F} (E,U)\\subset \\mathfrak F$, hence\n\\begin{equation} {\\cal H} ^d(E_0)\\le\\inf_{F\\in \\mathfrak F}{\\cal H} ^d(F)\\le \\inf_{F\\in \\overline{\\cal F} (E,U)}{\\cal H} ^d(F)\\le{\\cal H} ^d(E_0),\\end{equation}\ntherefore\n\\begin{equation} {\\cal H} ^d(E_0)=\\inf_{F\\in \\overline{\\cal F} (E,U)}{\\cal H} ^d(F).\\end{equation}\n\nOn the other hand, we know that ${\\cal F} (E_0,U)\\subset \\mathfrak F$ as well, hence \n\\begin{equation} {\\cal H} ^d(E_0)\\le \\inf_{F\\in \\mathfrak F}{\\cal H} ^d(F)\\le \\inf_{F\\in {\\cal F} (E_0,U)}{\\cal H} ^d(F),\\end{equation}\nwhich yields that $E_0$ is minimal in $U$.\n\nWe want to prove that when $E_k$ is sufficiently close to $E_0$, we can deform it into the union of $E_0$ and a set of very small measure, so that the measure after the deformation is less than $\\inf_{F\\in {\\cal F} (E,U)}{\\cal H} ^d(F)$, which cannot happen.\n\nThe construction of such a deformation is simlar to the construction in \\cite{GD03}: by minimality of $E_0$, around each regular point $x$ of $E_0$, there is a neighborhood retract to $E_0$ in some ball centered at $x$, with a uniform Lipschitz constant. We use a finite number of such balls to cover a big part of $E_0$, and the measure of $E_0$ which are not covered is very small.  When $E_k$ is close enough to $E_0$, a big part of $E_k$ is contained in the union of these balls, so we can deform $E_k$ onto $E_0$ in each of these balls, and then extend this deformation to the whole space, with the same Lipschitz constant. Outside these balls, since each $E_k$ is very close to $E_0$, we expect that measures of $E_k$ are comparable to the measure of $E_0$, and so the measures of the image of $E_k$ outside the above balls are still small.\n\nBut in our case, there is no reason why the measures of $E_k$ should be uniformly comparable to that of $E_0$ at small scales. This issue results in more works. In a word, we have to first deform $\\{E_k\\}$ into a new sequence $\\{E_k'\\}$, whose local measures can be controlled by that of $E_0$, and their limits are still $E_0$.\n\nNow let us give more details:\n\n Set \n\\begin{equation} Q'_k:=\\{Q\\in \\Delta_k: Q\\cap E_0\\ne\\emptyset\\},\\end{equation}\nand \n\\begin{equation} Q_k=\\{Q\\in \\Delta_k: \\exists Q'\\in Q'_k\\mbox{ such that }Q\\cap Q'\\ne\\emptyset\\},\\end{equation}\nthat is, $Q'_k$ is the family of elements in $\\Delta_k$ that are neighbors $E_0$, and we get $Q_k$ by adding another layer of cubes in $\\Delta_k$ to $Q_k$. Let $|Q_k|=\\cup_{Q\\in Q_k}Q$ be the union of elements in $Q_k$, and for each $j\\le n$, let $Q_{k,j}$ be the set of all $j$ faces of elements in $Q_k$, and let ${\\cal S} _{k,j}=\\cup_{\\sigma\\in Q_{k,j}}\\sigma$ denote the $j$-skeleton of $Q_k$. \n\nSet $\\partial E_0=E_0\\cap\\partial U$, and\n\\begin{equation} R_k:=\\{Q\\in \\Delta_k: \\exists Q'\\in \\Delta_k\\mbox{ such that } Q'\\cap \\partial E_0\\ne\\emptyset\\mbox{ and }Q\\cap Q'\\ne\\emptyset\\}.\\end{equation}\nLet $|R_k|=\\cup_{Q\\in R_k}Q$, and for each $j\\le n$, let $R_{k,j}$ be the set of all $j$ faces of elements in $R_k$, and let ${\\cal T}_{k,j}=\\cup_{\\sigma\\in R_{k,j}}\\sigma$ denote the $j$-skeleton of $R_k$. \n\nIt is easy to see that \n\\begin{equation} Q'_k\\subset Q_k,\\mbox{ and } R_k\\subset Q_k,\\end{equation}\nand hence\n\\begin{equation} |R_k|\\subset |Q_k|, R_{k,j}\\subset Q_{k,j}\\mbox{, and }{\\cal T}_{k,j}\\subset {\\cal S} _{k,j}\\mbox{ for all }j\\le n.\\end{equation}\n\nLet us first give some properties for the sets ${\\cal S} _{k,d}$ and ${\\cal T}_{k,d}$, where $d$ is the dimension of $E_0$.\n\n\\begin{pro}Suppose that $\\partial E_0\\subset {\\cal S} _{k_0,d-1}$ for some $k_0\\in \\mathbb N$, that is, $\\partial E_0$ is a union of dyadic $d-1$-faces. Then we have\n\n $1^\\circ$ $\\lim_{k\\to\\infty} {\\cal H} ^d({\\cal T}_{k,d})\\to 0$;\n \n $2^\\circ$ There exists $M>0$ which depends only on $n$ and $d$, such that for each $k>k_0$, and each $Q\\in Q_k$ and $Q^\\circ\\cap |R_{k-2}|=\\emptyset$, we have \n \\begin{equation} {\\cal H} ^d({\\cal S} _{k,d}\\cap Q)<M {\\cal H} ^d(E_0\\cap V(Q)),\\end{equation}\n where $V(Q)$ denotes the union of cubes that touch some cube that touches $Q$, that is:\n \\begin{equation} V(Q):=\\cup\\{Q'\\in \\Delta_k: \\mbox{ there exists }Q''\\in \\Delta_k\\mbox{ such that }Q''\\cap Q\\ne\\emptyset\\mbox{ and }Q''\\cap Q'\\ne\\emptyset\\}.\\end{equation}\n\\end{pro}\n\n\\noindent Proof.  $1^\\circ$ Since $\\partial E_0$ is $d-1$ rectifiable with finite $d-1$ Hausdorff measure, we apply \\cite{Fe} Theorem 3.2.29, and get\n\\begin{equation} {\\cal M}^{d-1}(\\partial E_0)={\\cal H} ^{d-1}(\\partial E_0)<\\infty,\\end{equation}\nwhere ${\\cal M}^{d-1}$ stands for the $d-1$-dimensional Minkowski content.\n\nBy definition of Minkowski content, we know that \n\\begin{equation} \\lim_{r\\to 0+} \\frac{{\\cal H} ^n(B(\\partial E_0, r))}{r^{n-d+1}}<\\infty,\\end{equation}\nand hence, when $k$ is large, we have\n\\begin{equation} {\\cal H} ^n(B(\\partial E_0, 2^{-k}))<C_02^{-k(n-d+1)}.\\end{equation}\n\nWe know that $|R_k|\\subset B(\\partial E_0, 2^{-k+3})$, hence for $k$ large, \n\\begin{equation} {\\cal H} ^n(|R_k|)\\le {\\cal H} ^n(B(\\partial E_0, 2^{-k+3}))<C_02^{(-k+3)(n-d+1)}=C_12^{-k(n-d+1)}.\\end{equation}\n \n On the other hand, \n \\begin{equation} {\\cal H} ^d({\\cal T}_{k,d})=\\sum_{\\sigma\\in R_{k,d}}{\\cal H} ^d(\\sigma)\\le\\sum_{Q\\in R_k}\\sum_{\\sigma\\in\\Delta_d(Q)}{\\cal H} ^d(\\sigma).\\end{equation}\n \n Now for each $Q\\in R_k$, we know that $\\sum_{\\sigma\\in\\Delta_d(Q)}{\\cal H} ^d(\\sigma)=\\alpha_{n,d}2^{-kd}$, where $\\alpha_{n,d}$ is the $d$-Hausdorff measure of the $d$-skeleton of a unit cube, which is a constant that depends only on $n$ and $d$. As a result, by \\textcolor{blue}{(4.23)}, \n \\begin{equation} {\\cal H} ^d({\\cal T}_{k,d})\\le \\sum_{Q\\in R_k}\\alpha_{n,d}2^{-kd}=\\alpha_{n,d}2^{-kd}\\sharp R_k,\\end{equation}\n where $\\sharp R_k$ is the number of cubes in $R_k$.\n \n Meanwhile, since the ${\\cal H} ^n$ measure of each cube in $R_k$ is $2^{-kn}$, we have, for $k$ large,\n \\begin{equation} \\sharp R_k=\\frac{{\\cal H} ^n(|R_k|}{2^{-kn}}\\le \\frac {C_12^{-k(n-d+1)}}{2^{-kn}}=C_12^{kd-k},\\end{equation}\n where the second inequality is by \\textcolor{blue}{(4.22)}. Combine with \\textcolor{blue}{(4.24)}, we get\n \\begin{equation}  {\\cal H} ^d({\\cal T}_{k,d})\\le\\alpha_{n,d}2^{-kd}\\times C_12^{kd-k}=C_1\\alpha_{n,d}2^{-k}\\to 0\\mbox{, as } k\\to \\infty,\\end{equation}\n which yields $1^\\circ$.\n \n $2^\\circ$ Fix any $Q\\in Q_k$, by definition, there exists $Q'\\in \\Delta_k$ such that $Q'\\cap E_0\\ne\\emptyset$ and $Q\\cap Q'\\ne\\emptyset$. Take $y\\in Q'\\cap E_0$, then by definition of $V(Q)$, $B(y,2^{-k})\\subset V(Q)$. On the other hand, since $Q^\\circ\\cap |R_{k-2}|=\\emptyset$, we know that $d(Q',\\partial E_0)>2^{-k+2}$. In particular, $d(y, \\partial E_0)>2\\times 2^{-k}$, which means $B(y,2\\times 2^{-k})\\subset \\mathbb R^n\\backslash \\partial E_0$. Since $E_0$ is minimal in $\\mathbb R^n\\backslash \\partial E_0$, by Ahlfors regularity for minimal sets (cf. \\cite{DS00} Proposition 4.1), \n \\begin{equation} C_2^{-1}2^{-kd}\\le {\\cal H} ^d(E_0\\cap B(y, 2^{-k}))\\le C_2 2^{-kd},\\end{equation}\n where $C_2$ is a constant that depends only on $n$ and $d$. As a result, we have\n \\begin{equation} {\\cal H} ^d({\\cal S} _{k,d}\\cap Q)=\\alpha(n,d)2^{-kd}\\le  C_2\\alpha(n,d){\\cal H} ^d(E_0\\cap B(y, 2^{-k}))\\le C_2\\alpha(n,d){\\cal H} ^d(E_0\\cap V(Q)).\\end{equation}\n \\hfill$\\Box$\\bigskip\n\nNext, let us construct the new sequence $E'_k$. Since $C$ is compact, we know that $d_C(E_k,E_0)\\to 0, k\\to \\infty$. As a result, modulo extracting a subsequence, we can suppose that $d_C(E_k, E_0)<2^{-k}$. Therefore, $E_k\\subset B(E_0,2^{-k})$. \n\n\\begin{pro}For each $\\epsilon>0$, there exists a sequence $\\{E'_k\\}_{k\\in \\mathbb N}\\subset {\\cal F} (E, \\mathbb R^n\\backslash \\partial E_0)$, such that \n\\begin{equation} E'_k\\subset B(E_0, 2^{-k+2}),\\end{equation} and for $k$ large, \n\\begin{equation}{\\cal H} ^d (E'_k\\backslash {\\cal S} _{k,d})<\\epsilon.\\end{equation}\nIn particular, $E'_k$ converge to $E_0$.\n\\end{pro}\n\n\\noindent Proof.  Fix any $k>k_0$. \n\nSince $E_k\\subset B(E_0,2^{-k})$, $E_k\\subset |Q_k|$. And we know that $E_k$ is a deformation of $E$, hence $E_k$ has locally finite d-Hausdorff measure. As a result, by a standard Federer-Fleming argument (cf. Section 4.2 of \\cite{Fe}, or Section 3 of \\cite{DS00}), there exists a Lipschitz map $\\varphi_k: |Q_k|\\to |Q_k|$ (the Lipschitz constant $L_k$ depends on $k$, and $L_k\\ge 1$), such that\n\n\\begin{equation} \\varphi_k(Q)\\subset Q,\\ \\forall Q\\in Q_k,\\end{equation}\n\\begin{equation} \\varphi_k(E_k)\\subset {\\cal S} _{k,d},\\end{equation}\nand\n\\begin{equation} \\varphi_k(x)=x, \\forall x\\in {\\cal S} _{k,d}.\\end{equation}\n\nIn particular, the sequence $\\varphi_k(E_k)\\subset {\\cal S} _{k,d}$. Note that $\\varphi_k(E_k)$ might not belong to ${\\cal F} (E,\\mathbb R^n\\backslash \\partial E_0)$, because the deformation $\\varphi_k$ may not satisfy the compactness condition \\textcolor{blue}{(2.6)}. So we need to do some slight modification.\n\nFix $\\epsilon>0$. Let $\\mu={\\cal H} ^d\\lfloor_{E_k}$, then $\\mu$ is a locally finite Hausdorff measure. In particular, we know that \n\\begin{equation} \\lim_{r\\to 0}\\mu(B(\\partial E_0, r))=\\mu(\\partial E_0)={\\cal H} ^d(E_k\\cap \\partial E_0)\\le {\\cal H} ^d(\\partial E_0)=0.\\end{equation}\n\nTake $r_k>0$ such that $\\mu(B(\\partial E_0, r_k))<(3L_k+2)^{-d}\\epsilon$, that is, ${\\cal H} ^d(E_k\\cap B(\\partial E_0, r_k))<(3L_k+2)^{-d}\\epsilon$.\n\nFor any $x\\in \\mathbb R^n$, set \n\\begin{equation} t_x=\\left\\{\\begin{array}{rcl}0&,\\  &x\\in B(\\partial E_0, \\frac 12 r_k)\\\\\n1 &,\\ &x\\in B(\\partial E_0, r_k)^C;\\\\\n\\frac{2 d(x,\\partial E_0)}{r_k}-1&,\\ &x\\in B(\\partial E_0, r_k)\\backslash B(\\partial E_0, \\frac 12r_k),\n\\end{array}\\right.\\end{equation}\nand set $f_k(x)=(1-t_x)x+t_x\\varphi_k(x)$.\n\n\nThen $f_k:\\mathbb R^n\\to [0,1]$ is $3L_k+2$-Lipschitz: in fact, for any $x,y$, suppose that $d(x, \\partial E_0)\\ge d(y,\\partial E_0)$, then we know that\n\n\\begin{equation}\\begin{split} ||f_k(x)-f_k(y)||&=||[(1-t_x)x+t_x\\varphi_k(x)]-[(1-t_y)y+t_y\\varphi_k(y)]||\\\\\n&=||(1-t_x)(x-y)+t_x(\\varphi_k(x)-\\varphi_k(y))+(t_x-t_y)(\\varphi_k(y)-y)||\\\\\n&\\le ||(1-t_x)(x-y)||+||t_x(\\varphi_k(x)-\\varphi_k(y))||+||(t_x-t_y)(\\varphi_k(y)-y)||\\\\\n&\\le (1-t_x)||x-y||+(t_x)L_k||x-y||+||(t_x-t_y)(\\varphi_k(y)-y)||\\\\\n&\\le L_k||x-y||+||(t_x-t_y)(\\varphi_k(y)-y)||.\n\\end{split}\\end{equation}\nTo estimate the second term, when $d(y,\\partial E_0)\\ge r_k$, we know that $t_x=t_y=1$, and this term vanishes. So suppose that $d(y,\\partial E_0)<r_k$.\nLet $z\\in \\partial E_0$ be such that $d(y,\\partial E_0)=d(z,y)$. Then we know that\n\\begin{equation} \\varphi_k(y)-y=\\varphi_k(y)-\\varphi_k(z)+\\varphi_k(z)-y.\\end{equation}\nSince $\\partial E_0\\subset {\\cal T}_{k,d}$, we know that $\\varphi_k$ is identity on $\\partial E_0$, and hence $\\varphi_k(z)=z$. Therefore\n\\begin{equation} \\begin{split}||\\varphi_k(y)-y||&=||\\varphi_k(y)-\\varphi_k(z)+(z-y)||\\le (1+L_k)||z-y||\\\\\n&=(1+L_k)d(y,\\partial E_0)\\le (1+L_k)r_k.\\end{split}\\end{equation}\nOn the other hand, since $d(x,\\partial E_0)\\ge d(y,\\partial E_0)$, we have $t_x\\ge t_y$, and hence\n\\begin{equation} 0\\le t_x-t_y\\le (\\frac{2 d(x,\\partial E_0)}{r_k}-1)-(\\frac{2 d(y,\\partial E_0)}{r_k}-1)=\\frac{2}{r_k}[d(x,\\partial E_0)-d(y,\\partial E_0)], \\end{equation}\nhence\n\\begin{equation} ||t_x-t_y||\\le \\frac{2}{r_k}||x-y||.\\end{equation}\nCombine \\textcolor{blue}{(4.38) and (4.40)}, we get\n\\begin{equation} ||(t_x-t_y)(\\varphi_k(y)-y)||\\le \\frac{2}{r_k}||x-y||\\times (1+L_k)r_k\\le 2(1+L_k)||x-y||.\\end{equation}\nTogether with \\textcolor{blue}{(4.36)}, we get\n\\begin{equation} ||f_k(x)-f_k(y)||\\le (3L_k+2)||x-y||.\\end{equation}\n\nLet $E'_k=f_k(E_k)$. Since $E_k\\in {\\cal F} (E,U)$, and $f_k$ is identity in a neighborhood of $\\partial E_0$, we know that $E'_k\\in {\\cal F} (E,\\mathbb R^n\\backslash \\partial E_0).$ \n\nBy definition, for $x\\in E_k$, we know that \n\\begin{equation} ||f(x)-x||=||(1-t_x)x+t_x\\varphi_k(x)-x||\\le ||\\varphi_k(x)-x||\\le 2^{-k+1},\\end{equation}\nthe last inequality is by \\textcolor{blue}{(4.32)}. Hence $E'_k=f_k(E_k)\\subset B(E_k, 2^{-k+1})\\subset B(E_0, 2^{-k+2})$, which yields \\textcolor{blue}{(4.29)}.\n\nMoreover, by definition of $f_k$, $f_k(E_k\\backslash B(\\partial E_0, r_k))\\subset {\\cal S} _{k,d}$, and hence\n\\begin{equation} {\\cal H} ^d(E'_k\\backslash {\\cal S} _{k,d})\\le {\\cal H} ^d(f_k(E_k\\cap B(\\partial E_0, r_k)))\\le (3L_k+2)^d{\\cal H} ^d(E_k\\cap B(\\partial E_0, r_k))<\\epsilon,\\end{equation}\nwhich gives \\textcolor{blue}{(4.30)}.\\hfill$\\Box$\\bigskip\n\nNow for $k$ large, we will deform a big part of it to $E_0$:  \n\n\\begin{pro}For $k$ large, for each $\\epsilon>0$, there exists $s_k>0$, and a deformation $h_k$ in $U$, such that $h_k=id$ in $B(\\partial E_0, s_k)$, and \n\\begin{equation} {\\cal H} ^d(h_k(E_k))<{\\cal H} ^d(E_0)+\\epsilon.\\end{equation}\n\\end{pro}\n\n\\noindent Proof.  Since $E_0$ is minimal in $\\mathbb R^n\\backslash \\partial E_0$, the set of regular points $E_{0P}$ of $E_0$ is of full measure: ${\\cal H} ^d(E\\backslash E_{0P})=0$.\nBy the $C^1$ regularity (Theorem \\textcolor{blue}{2.26}) for regular points, for each $x\\in E_{0P}$, there exists $r_x>0$, with $B(x,2r_x)\\subset U$, such that for all $r<r_x$, there is a Lipschitz deformation retraction $\\varphi_{x,r}$ from $B(x,r)\\to E\\cap B(x,r)$, with Lipschitz constant no more than 2. Note that ${\\cal H} ^d(E\\backslash E_P)=0$.\n\nThe family $\\{B(x,r):x\\in E_{0P}, r<r_x\\}$ forms a vitali cover for $E_{0P}$. \n\nTherefore, for any fixed $\\epsilon>0$, there exists a finite set of points $\\{x_j\\}_{1\\le j\\le m}\\subset E_{0P}$, and $r_j\\in (0,r_{x_j})$, such that the balls $B(x_j,r_j)$ are disjoint, $B(x_j, 2r_j)\\cap \\partial E_0=\\emptyset$, and ${\\cal H} ^d(E_{0P}\\backslash \\cup_{j=1}^n B(x_j, r_j))<\\frac{\\epsilon}{3M\\times 2^{d+1}\\times 5^n}$. Take $t_j<r_j$, so that ${\\cal H} ^d(E_{0P}\\backslash \\cup_{j=1}^n B(x_j, t_j))<\\frac{\\epsilon}{4M\\times 2^d\\times 5^n}$.\n\nLet $r=\\min_j r_j$, and $t=\\min_j (r_j-t_j)$. Define a Lipschitz map $g: (\\cup_{j=1}^n B(x_j, r_j))\\cup B(\\partial E_0, r)\\to E_0$, with $g(x)=\\varphi_{x_j,r_j}(x)$ when $x\\in B(x_j,r_j)$; $g(x)=x$ for $x\\in B(\\partial E_0,r)$ and for $x$ with $d(x,(\\cup_{j=1}^n B(x_j, r_j))\\cup E_0)>r$. Then $g$ is 2-Lipschitz, and we can extend it to a 2-Lipschitz map, still denoted by $g$, from $\\mathbb R^n$ to $\\mathbb R^n$. \n\n We would like to control the measure of ${\\cal H} ^d(E'_k\\backslash (\\cup_{j=1}^n B(x_j, r_j)))$. Since the major part of $E'_k$ is included in ${\\cal S} _{k,d}$, let us first estimate ${\\cal H} ^d({\\cal S} _{k,d}\\backslash (\\cup_{j=1}^n B(x_j, r_j)))$.\n \n Take any $Q\\in Q_k$ and $Q^\\circ\\cap |R_{k-2}|=\\emptyset$, then by Proposition \\textcolor{blue}{4.4} $2^\\circ$, we know that for $k>k_0$,\n \\begin{equation}  {\\cal H} ^d({\\cal S} _{k,d}\\cap Q)\\le M{\\cal H} ^d(E_0\\cap V(Q)).\\end{equation}\n \n Now if $k$ is such that $2^{-k}<\\frac 16 t$, for each $Q$ such that $Q\\backslash (\\cup_{j=1}^n B(x_j, r_j))\\ne\\emptyset$, we know that $d(Q, (\\cup_{j=1}^n B(x_j, t_j))>t-\\sqrt 2 2^{-k}$, and hence $d(V(Q), (\\cup_{j=1}^n B(x_j, t_j))>t-3\\times \\sqrt 2 2^{-k}>0$, that is $V(Q)\\cap (\\cup_{j=1}^n B(x_j, t_j)=\\emptyset$.  Hence we have\n \\begin{equation}\\begin{split}& {\\cal H} ^d({\\cal S} _{k,d}\\backslash (|R_{k-2}|\\cup (\\cup_{j=1}^n B(x_j, r_j))))\\\\\n &\\le\\sum\\{{\\cal H} ^d({\\cal S} _{k,d}\\cap Q):Q\\in Q_k, Q^\\circ\\cap |R_{k-2}|=\\emptyset\\mbox{, and }Q\\backslash (\\cup_{j=1}^n B(x_j, r_j))\\ne\\emptyset\\}\\\\\n &\\le \\sum\\{M{\\cal H} ^d(E_0\\cap V(Q)): Q\\in Q_k \\mbox{, and }V(Q)\\cap (\\cup_{j=1}^n B(x_j, t_j)=\\emptyset\\}\\\\\n &=M\\int_{E_0}\\sum\\{\\chi_{V(Q)}:Q\\in Q_k \\mbox{, and }V(Q)\\cap (\\cup_{j=1}^n B(x_j, t_j)=\\emptyset\\} d{\\cal H} ^d\\\\\n &\\le M\\int_{E_0\\backslash (\\cup_{j=1}^n B(x_j, t_j))}\\sum_{Q\\in Q_k}\\chi_{V(Q)}.\n \\end{split}\\end{equation}\n \n Note that $\\sum_{Q\\in Q_k}\\chi_{V(Q)}\\le\\sum_{Q\\in \\Delta_k}\\chi_{V(Q)}=5^n$, hence\n \\begin{equation} \\begin{split}{\\cal H} ^d({\\cal S} _{k,d}&\\backslash (|R_{k-2}|\\cup (\\cup_{j=1}^n B(x_j, r_j))))\\le M\\int_{E_0\\backslash (\\cup_{j=1}^n B(x_j, t_j))}\\sum_{Q\\in Q_k}\\chi_{V(Q)}\\\\\n &\\le 5^nM\\int_{E_0\\backslash (\\cup_{j=1}^n B(x_j, t_j))}d{\\cal H} ^d=5^nM{\\cal H} ^d(E_0\\backslash (\\cup_{j=1}^n B(x_j, t_j)))\\\\\n &<5^nM\\times \\frac{\\epsilon}{4M\\times 2^d\\times 5^n}=\\frac{\\epsilon}{4\\times 2^d}.\n \\end{split}\\end{equation}\n \nNext let us estimate ${\\cal H} ^d({\\cal S} _{k,d}\\cap |R_{k-2}|)$. For each $Q\\in \\Delta_{k-2}$, we know that\n\\begin{equation} {\\cal H} ^d({\\cal S} _{k,d}\\cap Q)=4^{n-d}{\\cal H} ^d(S_{k-2,d}\\cap Q),\\end{equation}\nhence\n \\begin{equation} {\\cal H} ^d({\\cal S} _{k,d}\\cap |R_{k-2}|)\\le\\sum_{Q\\in R_{k-2}}{\\cal H} ^d({\\cal S} _{k,d}\\cap Q)=4^{n-d}\\sum_{Q\\in R_{k-2}}{\\cal H} ^d({\\cal S} _{k-2,d}\\cap Q)\\le C_34^{n-d}{\\cal H} ^d({\\cal T}_{k-2,d}),\\end{equation}\n where $C_3=C_3(n,d)$ is the number of cubes $Q\\in \\Delta_k$ that share a same $d$-face. This is a constant that only depends on $n$ and $d$.\n \n By Proposition \\textcolor{blue}{4.4} $1^\\circ$, we know that for $k$ large, ${\\cal H} ^d({\\cal S} _{k,d}\\cap |R_{k-2}|)<\\frac{\\epsilon}{4\\times 2^d}.$\n \n \nNow by Proposition \\textcolor{blue}{4.5}, we take $E'_k=f_k(E_k)$ be such that\n\\begin{equation} {\\cal H} ^d(E'_k\\backslash {\\cal S} _{k,d})<\\frac{\\epsilon}{4\\times 2^d}.\\end{equation}\n\nThen for $k$ large, we have, by\n\\begin{equation} \\begin{split}{\\cal H} ^d(g(E'_k))\\le &{\\cal H} ^d(g({\\cal S} _{k,d}))+{\\cal H} ^d(g(E'_k\\backslash {\\cal S} _{k,d}))\\\\\n\\le &{\\cal H} ^d(g({\\cal S} _{k,d}\\cap (\\cup_{j=1}^n B(x_j, r_j))))+{\\cal H} ^d(g({\\cal S} _{k,d}\\cap |R_{k-2}|))\\\\\n&+{\\cal H} ^d(g({\\cal S} _{k,d}\\backslash (|R_{k-2}|\\cup (\\cup_{j=1}^n B(x_j, r_j)))))+{\\cal H} ^d(g(E'_k\\backslash {\\cal S} _{k,d}))\\\\\n\\le &{\\cal H} ^d(E_0)+2^d[{\\cal H} ^d({\\cal S} _{k,d}\\cap |R_{k-2}|)+{\\cal H} ^d({\\cal S} _{k,d}\\backslash (|R_{k-2}|\\cup (\\cup_{j=1}^n B(x_j, r_j))))\\\\\n&+{\\cal H} ^d(E'_k\\backslash {\\cal S} _{k,d})]\\\\\n\\le &{\\cal H} ^d(E_0)+2^d(\\frac{\\epsilon}{4\\times 2^d}+\\frac{\\epsilon}{4\\times 2^d}+\\frac{\\epsilon}{4\\times 2^d})={\\cal H} ^d(E_0)+\\frac 34\\epsilon.\n\\end{split}\\end{equation}\n\nNote that $g\\circ f_k$ is the identity map in a neighborhood $B(\\partial E_0,s_k)$ of $\\partial E_0$, with $s_k=\\min\\{r_k,r\\}$. But $g\\circ f_k$ might even not be a deformation in $\\mathbb R^n\\backslash \\partial E_0$.\n\nWe still have to modify this sequence $g\\circ f_k(E_k)$ to deformations of $E_k$ in $U$. For this purpose, let $C_k$ denote the convex hull of $C\\backslash B(\\partial E_0,s_k)$. Then $C_k$ is a compact subset of $U$: in fact, since $C\\cap \\partial U=\\partial E_0$, hence $d(C\\backslash B(\\partial E_0,s_k), \\partial U)>0$. But $C\\backslash B(\\partial E_0, s_k)\\subset U$, and $U$ is convex, hence $d(C_k, \\partial U)>0$.\n\nLet $\\pi_k$ be the shortest distance projection to $C_k$. We define $h_k: E_k\\to  (E_k\\cap B(\\partial E_0,s_k))\\cup C_k$: for $x\\in E_k\\cap B(\\partial E_0,s_k)$, $h_k(x)=x=g\\circ f_k(x)$, and for $x\\in E_k\\backslash B(\\partial E_0,s_k)$, let $h_k(x)=\\pi_k\\circ g\\circ f_k(x)$. It is easy to verify that $h_k$ is Lipschitz, $h_k=id$ outside $C_k$, and $h_k(C_k)\\subset C_k$. Moreover, we know that \n\\begin{equation} \\begin{split}{\\cal H} ^d(h_k(E_k))&\\le{\\cal H} ^d(h_k(E_k\\backslash B(\\partial E_0,s_k))+{\\cal H} ^d(h_k(E_k\\cap B(\\partial E_0,s_k)))\\\\\n&={\\cal H} ^d(\\pi_k\\circ g\\circ f_k(E_k\\backslash B(\\partial E_0,s_k))+{\\cal H} ^d(E_k\\cap B(\\partial E_0,s_k))\\\\\n&\\le {\\cal H} ^d(g\\circ f_k(E_k\\backslash B(\\partial E_0,s_k))+{\\cal H} ^d(E_k\\cap B(\\partial E_0,s_k))\\\\\n&\\le {\\cal H} ^d(g\\circ f_k(E_k)+{\\cal H} ^d(E_k\\cap B(\\partial E_0,s_k))\\\\\n&\\le {\\cal H} ^d(E_0)+\\frac 34\\epsilon+\\frac 14\\epsilon<{\\cal H} ^d(E_0)+\\epsilon.\n\\end{split}\\end{equation} \n \\hfill$\\Box$\\bigskip\n \n Note that after Proposition \\textcolor{blue}{4.6}, Theorem \\textcolor{blue}{4.1} $1^\\circ$ follows directly for the case when $\\partial E_0\\subset {\\cal S} _{k_0,d-1}$ for some $k_0\\in \\mathbb N$. Then $2^\\circ$ is a direct corollary of $1^\\circ$.\n \n For general case where \\textcolor{blue}{(4.2)} holds, we set \n\\begin{equation} Q'_k:=\\{Q\\in \\Delta_k: Q\\cap \\psi^{-1}(E_0)\\ne\\emptyset\\},\\end{equation}\nand \n\\begin{equation} Q_k=\\{Q\\in \\Delta_k: \\exists Q'\\in Q'_k\\mbox{ such that }Q\\cap Q'\\ne\\emptyset\\},\\end{equation}\n Let $|Q_k|=\\cup_{Q\\in Q_k}Q$, and for each $j\\le n$, let $Q_{k,j}$ be the set of all $j$ faces of elements in $Q_k$, and let ${\\cal S} _{k,j}=\\cup_{\\sigma\\in Q_{k,j}}\\sigma$ denote the $j$-skeleton of $Q_k$. \n\nSet \n\\begin{equation} R_k:=\\{Q\\in \\Delta_k: Q\\cap \\psi^{-1}(E_0\\cap U)\\ne\\emptyset\\}.\\end{equation}\nLet $|R_k|=\\cup_{Q\\in R_k}Q$, and for each $j\\le n$, let $R_{k,j}$ be the set of all $j$ faces of elements in $R_k$, and let ${\\cal T}_{k,j}=\\cup_{\\sigma\\in R_{k,j}}\\sigma$ denote the $j$-skeleton of $R_k$. \n \n Then we do all the constructions in $U$ with respect to $\\psi(Q_k)$, $\\psi(R_k)$. All the quantative properties of $Q_k$ and $R_k$ that are used in the proof above will hold also for $\\psi(Q_k)$ and $\\psi(R_k)$, since $\\psi$ is bi-Lipschitz. And the proof goes the same way.\n  \\hfill$\\Box$\\bigskip\n \n \\begin{cor}$1^\\circ$ Let $U\\subset \\mathbb R^n$ be a bounded convex domain, and $E$ is a closed set in $\\bar U$ with locally finite $d$-Hausdorff measure. Then the conclusion $1^\\circ$ and $2^\\circ$ of Theorem \\textcolor{blue}{4.1} hold in either of the following cases :\n \n $1^\\circ$ $U$ is uniformly convex, and \\textcolor{blue}{(4.2)} holds;\n \n $2^\\circ$ $d=2$, $E=K$ is a 2-dimensional minimal cone, and $U$ is a convex domain.\n\n \\end{cor} \n\n\\section{Uniqueness properties for 2-dimensional minimal cones in $\\mathbb R^3$}\n\nIn this section we prove the topological and Almgren uniqueness for all 2-dimensional minimal cones in $\\mathbb R^3$.\n\n\\subsection{Planes}\n\n\\begin{thm}A 2-dimensional linear plane $P$ is Almgren and $G$-topological unique in $\\mathbb R^n$ for all $n\\ge 3$, and all abelien group $G$.\n\\end{thm}\n\n\\noindent Proof.  Let $P\\subset \\mathbb R^n$ be a 2-dimensional plane containing the origin. By Proposition \\textcolor{blue}{3.2 and 3.4}, to prove that $P$ is Almgren and $G$-topological unique, it is enough to prove that $P$ is $G$-topological unique in the unit ball $B$.\n\nSuppose that $E$ is a reduced $G$-topological competitor for $P$ in $B$, so that\n\\begin{equation} {\\cal H} ^2(E\\cap B)={\\cal H} ^2(P\\cap B).\\end{equation}\nBy Remark \\textcolor{blue}{3.3 $3^\\circ$}, we know that $E$ is $G$-topological and hence Almgren minimal in $B$. By the convex hull property for Almgren minimal sets, $E\\cap B$ is contained in the convex hull of $E\\cap \\partial B=P\\cap \\partial B$, which is $P\\cap \\partial B$. Hence $E\\cap B\\subset P\\cap B$. Then since both $P$ and $E$ are reduced set, \\textcolor{blue}{(5.1)} gives that $E=P$. Hence $P$ is $G$-topological unique, and hence Almgren unique.\\hfill$\\Box$\\bigskip\n\n\\subsection{The $\\mathbb Y$ sets}\n\n\\begin{thm}Any 2-dimensional $\\mathbb Y$ set is Almgren and $G$-topological unique in $\\mathbb R^n$ for all $n\\ge 3$, and all abelien group $G$.\n\\end{thm}\n\n\\noindent Proof.  By Proposition \\textcolor{blue}{3.4 and 3.5}, it is enough to prove that $\\mathbb Y$ sets are $G$-topological unique in $\\mathbb R^3$.\n\nSo let $Y$ be a 2-dimensional $\\mathbb Y$ set in $\\mathbb R^3$. Modulo changing the coordinate system, we can suppose that the spine of $Y$ is the vertical line $Z=\\{(x,y,z)\\in \\mathbb R^3:x=y=0\\}$, and that the intersection of $Y$ with the horizontal plane $Q:=\\{z=0\\}$ is the union $Y_1$ of the three half lines $R_{oa_i},1\\le i\\le 3$, where $a_1=(1,0)$, $a_2=(-\\frac 12, \\frac{\\sqrt 3}{2})$, and $a_3=-\\frac 12, -\\frac{\\sqrt 3}{2})$. Then $Y=Y_1\\times Z$.\n\nBy Proposition \\textcolor{blue}{3.2}, it is enough to prove that $Y$ is $G$-topological unique in the cylinder $D:=B_Q(0,1)\\times (-1,1)$.\n\nFor $t\\in (-1,1])$, let $a_i^t=(a_i,t)\\in Q\\times (-1,1)$. \n\nLet $f:\\mathbb R^3\\to \\mathbb R:f(x,y,z)=z$. For any set $F\\subset \\mathbb R^3$, and each $t\\in \\mathbb R$, set $F_t=f^{-1}\\{t\\}\\cap F$ the slice of $F$ at level $t$. \n\nLet $\\wideparen{a_i^ta_j^t}$ denote the open minor arc of circle of $\\partial B_Q(0,1)\\times \\{t\\}=\\partial D_t$ between $a_i^t$ and $a_j^t$, $1\\le i\\ne j\\le 3$. Then they belong to $\\mathbb R^3\\backslash D$. Since $\\wideparen{a_i^ta_j^t}, 1\\le i<j\\le 3$ lie in 3 different connected components of $\\mathbb R^3\\backslash Y$, for any $G$-topological competitor $F$ of $Y$ in $D$, they also lie in 3 different connected components of $\\mathbb R^3\\backslash F$. In particular, they belong to 3 different connected components of $\\bar D_t\\backslash F_t$.\n\n\\begin{lem} If $F$ is a $G$-topological competitor for $Y$ of dimension 2 in $D$, then for each $t\\in (-1,1)$, $F_t\\cap D_t$ must connect the three points in $Y_t\\cap \\partial D_t=\\{a_i^t,1\\le i\\le 3\\}$, i.e.  the three points $a_i^t,1\\le i\\le 3$ lie in the same connected component of $(F_t\\cap B_t)\\cup \\{a_i^t,1\\le i\\le 3\\}$. \\end{lem}\n\n\\noindent Proof.  Take any $t\\in [-1,1]$. \n\nSuppose that the three points $a_i^t,1\\le i\\le 3$ do not belong to the same connected component of $(F_t\\cap D_t)\\cup \\{a_i^t,1\\le i\\le 3\\}$. Suppose for example the connected component $C_1$ of $(F_t\\cap D_t)\\cup \\{a_i^t,1\\le i\\le 3\\}$ contains $a_1^t$ but does not contain $a_2^t$ and $a^t_3$. Then there exists a curve $\\gamma:[0,1]\\to \\bar D_t$ with $\\gamma(0),\\gamma(1)\\in \\partial D_t$, which separates $C_1$ and $(F_t\\cap D_t)\\cup \\{a_i^t,1\\le i\\le 3\\}\\backslash C_1$. That is: \n$ \\gamma\\subset \\bar D_t\\backslash ((F_t\\cap D_t)\\cup \\{a_i^t,1\\le i\\le 3\\})$, and the sets $C_1$ and $\\{a_2^t, a_3^t\\}$ belong to different connected components of $\\bar D_t\\backslash \\gamma$.  \n\nAs  consequence, there exists $t_2, t_3\\in [0,1]$, such that $\\gamma(t_j)$ belong to the open minor arc of circle $\\wideparen{a_1^ta_j^t}$ of $\\partial D_t$ between $a_1^t, a_j^t$, $j=2,3$. As a result, $b_j:=\\gamma(t_j)$ belong to different connected components of $\\mathbb R^3_t\\backslash Y_t$, and hence they belong to different connected components of $\\mathbb R^3\\backslash Y$, since $Y=Y_t\\times\\mathbb R$. \n\nSince $Y$ is a cone, $b_j\\not\\in Y\\Rightarrow$ the segment $[b_j,2b_j]\\subset \\mathbb R^3\\backslash Y$. Note that $(b_j,2b_j]\\subset \\mathbb R^3\\backslash \\bar D$, and $Y\\backslash \\bar D=F\\backslash \\bar D$, hence $(b_j,2b_j]\\subset \\mathbb R^3\\backslash F$. Since $b_j\\in \\bar D_t\\backslash F_t$, we know that $b_j\\in \\mathbb R^3\\backslash F$ as well, hence $[b_j,2b_j]\\subset \\mathbb R^3\\backslash F$. \n\nLet $\\beta$ denote the curve $[2b_2,b_2]\\cup \\gamma([t_2,t_3])\\cup [b_3,2b_3]$. Then $\\beta\\subset \\mathbb R^3\\backslash F$, and it connects $2b_2$ and $2b_3$. Hence the two points $2b_2$ and $2b_3$ belong to the same connected components of $\\mathbb R^3\\backslash F$. \n\nOn the other hand, we know that $b_j,j=2,3$ belong to different connected components of $\\mathbb R^3\\backslash Y$. Since $[b_j,2b_j]\\subset \\mathbb R^3\\backslash Y$, $j=2,3$, we know that $2b_j,j=2,3$ belong to different connected components of $\\mathbb R^3\\backslash Y$. This contradicts the fact that $F$ is a $G$-topological competitor for $Y$ of codimension 1(which, by Remark 3.2 of \\cite{topo}, corresponds to Mumford-Shah competitors, as defined in \\cite{DJT} Section 19.)\\hfill$\\Box$\\bigskip\n\n\\begin{pro}Let $E\\subset \\bar B_Q(0,1)$ be a closed set with finite ${\\cal H} ^1$ measure, such that $E\\cap \\partial B_Q(0,1)=\\{a_1,a_2,a_3\\}$, and $a_i,1\\le i\\le 3$ belong to the same connected component of $E\\cup \\{a_1,a_2,a_3\\}$. Then \n\\begin{equation} {\\cal H} ^1(E)\\ge{\\cal H} ^1(Y_1\\cap \\bar B_Q(0,1)),\\end{equation}\nand equality holds if and only if $E=Y_1\\cap \\bar B_Q(0,1)$ modulo a ${\\cal H} ^1$-null set.\n\\end{pro}\n\n\\noindent Proof.  Let $B$ denote $\\bar B_Q(0,1)$ for short. Let $E$ be as in the statement. \n\nLet ${\\cal F} $ denote the class of all closed subsets $E_0$ (in fact, $E_0$ stands for the equivalent class of sets which are the same modulo ${\\cal H} ^1$-null sets) of $E$ such that $E_0$ is connected and $\\{a_1,a_2\\}\\subset E_0$. Define an order on ${\\cal F} $ as following: for sets $E_1,E_2\\in {\\cal F} $, $E_1\\le E_2\\Leftrightarrow E_1\\supset E_2$. Note that since we regards 2 sets $E_1$ and $E_2$ as the same if ${\\cal H} ^1(\\Delta(E_1,E_2))=0$, $E_1<E_2\\Leftrightarrow E_1\\supset E_2$ and ${\\cal H} ^1(E_1)>H^1(E_2)$.\n\nWe want to prove that ${\\cal F} $ admits a maximal element. So take a totally ordered subset ${\\cal F} _1$ of ${\\cal F} $. We will prove that ${\\cal F} _1$ admits a upper bound in ${\\cal F} $. \n\nLet $E_1$ be the intersection of all sets in ${\\cal F} _1: E_1=\\cap_{F\\in {\\cal F} _1}F$. Then $\\{a_1,a_2\\}\\subset E_1$, and for all $F\\in {\\cal F} _1$, $E_1\\subset F$. \n\nLet $H_1$ be a connected component of $F_0$ that contains $a_1$. As a connected component, it is closed in $F_0$. And since $F_0$ is closed, $H_1$ is closed. \n\nWe claim that $a_2\\in H_1$ as well. Otherwise, $a_2\\not\\in H_1$. Let $H_2=E_1\\backslash H_1$. Since both $H_i,i=1,2$ are compact, the distance $d$ between them is positive. Let $U:=B(H_1,\\frac d2)\\cap B$. Then $\\partial U$ is a compact Lipschitz curve, $a_1\\in U$, and $a_2\\in B\\backslash U$. Now for any $F\\in {\\cal F} _1$, it is connected, and contains $a_1$ and $a_2$. As a result, the set $I_F:=F\\cap \\partial U$ is non empty and closed. The family $I:=\\{I_F:F\\in {\\cal F} _1\\}$ is a class of closed set. Since ${\\cal F} _1$ is totally ordered, hence for any finite subsets $\\{F_1,\\cdots, F_k\\}\\subset {\\cal F} _1$, $\\cap_{i=1}^k F_k$ must be one of the $F_1,\\cdots, F_k$. Suppose, without loss of generality, that $\\cap_{i=1}^k F_i=F_1$. Then $\\cap_{i=1}^k I_{F_i}=I_{F_1}\\ne\\emptyset$. We have thus proved that the family $I$ has the finite intersection property. Since the elements in $I$ are subsets of the compact set $E$, we know that $\\cap_{F\\in {\\cal F} _1}I_F\\ne\\emptyset$. By definition, this means, that $E_1\\cap \\partial U\\ne\\emptyset$. But we have suppose that $E_1=H_1\\cup H_2$, and both $H_i,i=1,2$ do not meet $\\partial U$, contradiction.\n\nHence $a_2\\in H_1$, then $H_1\\in {\\cal F} $. Clearly $H_1\\ge F$ for all $F\\in {\\cal F} _0$, which yields that $H_1$ is an upper bound for ${\\cal F} _1$. \n\nWe have thus proved that ${\\cal F} _1$ admits un upper bound. This holds for all totally ordered subset ${\\cal F} _1$ of ${\\cal F} $. By Zorn's lemma, ${\\cal F} $ admits a maximal element $\\gamma$. \n\nWe claim that \n\\begin{equation}\\begin{split}\\forall p\\in \\gamma\\backslash\\{a_1,a_2\\}\\mbox{ there exists two connected sets }\\gamma_1\\mbox{ and }\\gamma_2,\\\\\n\\mbox{ such that }a_i\\in \\gamma_i\\subset \\gamma, i=1,2, \\mbox{, and }\\gamma_1\\cap\\gamma_2=\\{p\\}.\\end{split}\\end{equation}\n\nLet us prove the claim. Let $d=\\min\\{|p-a_1|,|p-a_2|\\}$. Then for all $0<r<d$, we know that $\\gamma\\cap B(p,r)$ connects $p$ to the boundary $\\partial B(p,r)$ (since $\\gamma$ connects $p$ and $a_i,i=1,2$). As a result, ${\\cal H} ^1(\\gamma\\cap B(p,r))\\ge r>0$. And hence the set $\\gamma\\backslash B(p,r)$ is a subset of $\\gamma$ with strictly smaller measure, and contains $a_i,i=1,2$. Since $\\gamma$ is a maximal element in ${\\cal F} $, we know that $ \\gamma\\backslash B(p,r)\\not\\in {\\cal F} $, hence is not connected, and do not contain any connected subset that contains both $a_1$ and $a_2$.\n\nAs a result, $a_1$ and $a_2$ lie in two different connected components $H_1^r$ and $H_2^r$ of $\\gamma\\backslash B(p,r)$ for each $r\\in (0,d)$. Note that for $0<s<r<d$ we have $H_i^r\\subset H_i^s, i=1,2$. So let $H_i=\\cup_{0<t<d}H_i^r$. Since the $H_i^r$ have a common point $a_i$, the set $H_i$ is connected and contains $a_i$. It is also easy to see that $H_1\\cap H_2=\\emptyset$.\n\nLet us prove that $p\\in \\bar H_i, i=1,2$. Take $H_1$ for example. Suppose $p\\not\\in H_1$. Then there exists $s>0$ such that $B(p,2s)\\cap H_1=\\emptyset$. Let $G=\\gamma\\backslash (H_1^s\\cup B(p,s))$. Then we have the disjoint union $\\gamma=H_1^s\\cup (\\gamma\\cap B(p,s))\\cup G$. Note that $H_1^s$ is a connected component of $\\gamma\\backslash B(p,s)$, hence the sets $G$ and $H_1^s$ are both relatively open in $H_1^s\\cup G$. As a result, there exists two disjoint open subsets $U_1$ and $U_2$, such that $H_1^s\\subset U_1$, and $G\\subset U_2$. Let $U_2'=U_2\\cup B(p,s)$, and $U_1'=U_1\\backslash \\bar B(p,s)$. Then $U_1'$ and $U_2'$ are disjoint open subsets, $H_1^s\\subset U_1'$, and $(\\gamma\\cap B(p,s))\\cup G\\subset U_2'$. This gives an open decomposition of $\\gamma$, which contradicts that fact that $\\gamma$ is connected.\n\nHence $p\\in \\bar H_i$, and thus $H_i\\cup\\{o\\}$ is connected, $i=1,2$. Let $\\gamma_i=H_i\\cup \\{p\\}$, and we get Claim \\textcolor{blue}{(5.3)}.\n\nNow since $a_1,a_2, a_3$ lie in the same connected component of $E$, we know that $\\gamma$ and $a_3$ lie in the same connected component $E_0$ of $E$. \n\nWe are going to define a connected set $\\gamma_3$, such that $a_3\\in \\gamma_3$, $\\gamma_3\\cup\\gamma$ is connected, and $\\gamma_3\\cap\\gamma$ is a single point.\n\nIf $a_3\\in \\gamma$, then we set $\\gamma_3=\\{a_3\\}$;\n\nOtherwise, we have $a_3\\not\\in\\gamma$. Let $\\gamma'=E_0\\backslash \\gamma$. Then $\\gamma'\\cup \\gamma$ is connected and $a_3\\in \\gamma'$. Let $\\gamma_4$ be the connected component of $\\gamma'$ that contains $a_3$. Then we claim that \n\\begin{equation}\\gamma_4\\cup \\gamma\\mbox{ is connected.}\\end{equation}\n\nIn fact, if $\\gamma_4=\\gamma'$ then it holds automatically; otherwise, if $\\gamma_4\\cup\\gamma$ is not connected, since both $\\gamma_4$ and $\\gamma$ are connected, they are the two connected components of $\\gamma_4\\cup \\gamma$, and hence there exists two disjoint open sets $U_1$ and $U_2$ of $\\mathbb R^3$ such that $\\gamma_4\\subset U_1$ and $\\gamma\\subset U_2$. Similarly since $\\gamma_4$ is a connected component of $\\gamma'$, there exists two disjoint open sets $U_3$ and $U_4$ of $\\mathbb R^3$ such that $\\gamma_4\\subset U_3$ and $\\gamma'\\backslash \\gamma_4\\subset U_4$. Then let $U=U_1\\cap U_2$, and $V=U_3\\cup U_4$. Then $U$ and $V$ are disjoint, and $\\gamma_4\\subset U$, $E_0\\backslash\\gamma_4=\\gamma\\cup \\gamma'\\backslash \\gamma_4\\subset V$. This contradicts that fact that $E_0$ is connected. Hence Claim \\textcolor{blue}{(5.4)} holds.\n\nAs a result, $\\bar\\gamma_4\\cap \\gamma\\ne\\emptyset$, because $\\gamma$ and $\\bar\\gamma_3$ are both closed, and their union is connected.\n\n\nTake $p\\in \\bar\\gamma_4\\cap\\gamma$, and set $\\gamma_3=\\gamma_4\\cup\\{p\\}$.  Then $\\gamma_3$ is connected, contains $a_3$, $\\gamma_3\\cup\\gamma$ is connected, and $\\gamma_3\\cap\\gamma=\\{p\\}$ is a single point.\n\nBy Claim \\textcolor{blue}{(5.3)}, there exists two connected sets $\\gamma_1$ and $\\gamma_2$, such that $\\gamma_1\\cap\\gamma_2=\\{p\\}$, and $a_i\\in \\gamma_i,i=1,2$.  \n\nTo summerize, we get 3 connected subsets $\\gamma_i,1\\le i\\le 3$ of $E$, such that $a_i\\in \\gamma_i$, ${\\cal H} ^1(\\gamma_i\\cap \\gamma_j)=0$ for $i\\ne j$, and $p\\in \\cap_{i=1}^3\\gamma_i$.\n\nSince each $\\gamma_i$ is connected and contains $a_i$ and $p$, we know that \n\\begin{equation} {\\cal H} ^1(\\gamma_i)\\ge {\\cal H} ^1([p,a_i]),1\\le i\\le 3,\\end{equation}\nand hence\n\\begin{equation} {\\cal H} ^1(E)\\ge {\\cal H} ^1(\\cup_{i=1}^3\\gamma_i)=\\sum_{i=1}^3{\\cal H} ^1(\\gamma_i)\\ge \\sum_{i=1}^3{\\cal H} ^1([p,a_i]).\\end{equation}\n\nObviously the point $p\\in \\bar B$. And it is well known that the quantity $\\sum_{i=1}^3{\\cal H} ^1([p,a_i])$ attains its minimum if and only if $p$ is the Fermat point of the triangle $\\Delta_{a_1a_2a_3}$, which is just the origin $o$. In this case, \n\\begin{equation} \\sum_{i=1}^3{\\cal H} ^1([0,a_i])={\\cal H} ^1(Y_1\\cap \\bar B).\\end{equation}\nThis leads to the conclusion of Proposition \\textcolor{blue}{5.4}.\\hfill$\\Box$\\bigskip\n\nNow let us return to the proof of Theorem \\textcolor{blue}{5.2}. Let $F$ be a reduced $G$-topological competitor of $Y$ in $D$, such that \n\\begin{equation} {\\cal H} ^2(F\\cap D)={\\cal H} ^2(Y\\cap D),\\end{equation}\nwe would like to show that $F=Y$.\n\nBy Lemma \\textcolor{blue}{5.3}, we know that $F_t$ connects the three points $a_i^t,1\\le i\\le 3$. Then Proposition \\textcolor{blue}{5.4} tells that \n\\begin{equation} {\\cal H} ^1(F_t\\cap D_t)\\ge {\\cal H} ^1(Y_t\\cap D_t).\\end{equation}\n\nWe apply the coarea formula (cf. \\cite{Fe} 3.2.22) to the Lipschitz function $f$, and the set $F\\cap D$, and get\n\\begin{equation} {\\cal H} ^2(F\\cap D)\\ge \\int_{-1}^1{\\cal H} ^1(F_t\\cap D_t)\\ge \\int_{-1}^1{\\cal H} ^1(Y_t\\cap D_t)={\\cal H} ^2(Y\\cap D).\\end{equation}\nThen \\textcolor{blue}{(5.8)} tells that\n\\begin{equation} {\\cal H} ^1(F_t\\cap D_t)= {\\cal H} ^1(Y_t\\cap D_t) \\mbox{ for a.e. }t\\in (0,1),\\end{equation}\nand hence \n\\begin{equation} F_t\\cap D_t=Y_t\\cap D_t\\mbox{ for a.e. }t\\in (0,1)\\end{equation}\nby Proposition \\textcolor{blue}{5.4}. Hence we know that $F\\cap D=Y\\cap D$ modulo ${\\cal H} ^2$-null sets. But $F$ is reduced, hence $F\\cap D=Y\\cap D$. Hence $Y$ is $G$-topological unique in $D$, and hence it is $G$-topological unique in $\\mathbb R^3$ (Proposition \\textcolor{blue}{3.2}), and hence in $\\mathbb R^n$ (Proposition \\textcolor{blue}{3.5}).\n\nBy Proposition \\textcolor{blue}{3.4}, $\\mathbb Y$ sets are also Almgren unique in $\\mathbb R^n$.\\hfill$\\Box$\\bigskip\n\n\\begin{rem}It is also possible to prove Theorem \\textcolor{blue}{5.2} by paired calibration (cf. \\cite{LM94} and \\cite{Br91}). In fact, we will use this method to prove the uniqueness for $\\mathbb T$ sets in $\\mathbb R^3$ in the next subsection, and interested readers can easily find a similar proof for $\\mathbb Y$ sets. The proof in this section is more elementary in some sense, mainly use elementary topology. \n\\end{rem}\n\n\\subsection{The $\\mathbb T$ sets}\n\n \\begin{thm}Any 2-dimensional $\\mathbb T$ set is Almgren and ($\\mathbb Z$-)topological unique in $\\mathbb R^n$ for all $n\\ge 3$.\n\\end{thm}\n\n\\noindent Proof.  By Proposition \\textcolor{blue}{3.4 and 3.5}, it is enough to prove that $\\mathbb T$ sets are topological unique in $\\mathbb R^3$.\n\nLet $T$ be a $\\mathbb T$ set centered at the origin in $\\mathbb R^3$. That is, $T$ is the cone over the 1-skeleton of a regular tetrahedron $C$ centered at the origin and inscribed in the closed unit ball $B$.\n\nBy Proposition \\textcolor{blue}{3.2}, to prove that $T$ is topological unique in $\\mathbb R^3$, it is enough to prove that $T$ is topological unique in $B$. So suppose that $E$ is a reduced topological competitor for $T$ in $B$, such that\n\\begin{equation} {\\cal H} ^2(E\\cap B)={\\cal H} ^2(T\\cap B).\\end{equation}\n\nBy Remark \\textcolor{blue}{3.3 $3^\\circ$}, we know that $E$ is minimal, and hence is rectifiable. Hence for almost all $x\\in E$, the tangent plane $T_xE$ exists.\n\nAs mentioned in the last subsection, our proof will profit from the paired calibration, so let use first give necessary details: \n\nDenote by $a_i,1\\le i\\le 4$ the four singular points of $T\\cap \\partial B$. Let $\\Omega_i,1\\le i\\le 4$ be the four equivalent connected spherical regions of $\\partial B\\backslash T$, $\\Omega_i$ being on the opposite of $a_i$. \n\nSince $E$ is a topological competitor for $T$ in $B$, we know that $\\partial B\\backslash E=\\partial B\\backslash T=\\cup_{i=1}^4\\Omega_i$, and the four $\\Omega_i$ live in different connected components of $B\\backslash E$.\n\nFor $1\\le i\\le 4$, let $C_i$ be the connected component of $B\\backslash E$ that contains $\\Omega_i$. Let $E_i=\\partial C_i\\backslash \\partial B=\\partial C_i\\backslash \\Omega_i$. Then we know that the four $C_i,1\\le i\\le 4$ are disjoint, and $E_i\\subset E$. Also note that $E_i\\cap \\Omega_i\\subset E\\cap\\partial B=T\\cap \\partial B$ is of ${\\cal H} ^2$ measure zero, hence we have the essentially disjoint unions\n\\begin{equation} \\partial C_i=E_i\\cup\\Omega_i,1\\le i\\le 4.\\end{equation}\n\nSince $C_i$ are disjoint regions in $\\mathbb R^3$, we know that for almost all $x\\in E$, they belong to at most two of the $E_i$'s. So for $i\\ne j$, let $E_{ij}=E_i\\cap E_j$. Let $E_{i0}$ denote $E_i\\backslash (\\cup_{j\\ne i}E_i)$, the set of points $x$ that belongs only to $E_i$. Let $F=\\cup_{1\\le i\\le 4}E_i\\subset E\\cap B$, then we have the disjoint union\n\\begin{equation} F=[\\cup_{1\\le i\\le 4}E_{i0}]\\cup[\\cup_{1\\le i<j\\le 4}E_{ij}].\\end{equation}\n\nFor points $x\\in \\partial C_i$, let $n_i(x)$ denote the normal vector pointing into the region $C_i$. Note that since $\\partial C_i\\subset E\\cup \\partial B$, it is rectifiable, and hence $n_i(x)$ is well defined for ${\\cal H} ^2$-a.e. $x\\in \\partial C_i$. Moreover, for $i\\ne j$, we have $n_i(x)=-n_j(x)$ for ${\\cal H} ^2$-a.e. $x\\in E_{ij}$. \n\nNow by Stoke's formula, we have, for $1\\le i\\le 4$,\n\\begin{equation} 0=\\int_{\\partial C_i}<a_i,n_i(x)>d{\\cal H} ^2(x)=\\int_{E_i}<a_i,n_i(x)>d{\\cal H} ^2(x)+\\int_{\\Omega_i}<a_i,n_i(x)>d{\\cal H} ^2(x),\\end{equation}\nand hence\n\\begin{equation} \\int_{E_i}<-a_i,n_i(x)>d{\\cal H} ^2(x)=\\int_{\\Omega_i}<a_i,n_i(x)>d{\\cal H} ^2(x)={\\cal H} ^2(\\pi_i(\\Omega_i)),\\end{equation}\nwhere $\\pi_i$ is the orthogonal projection from $\\mathbb R^3$ to the plane orthogonal to $a_i$, $1\\le i\\le 4$.\nWe sum over $i$, and get\n\\begin{equation} \\sum_{1\\le i\\le 4} \\int_{E_i}<-a_i,n_i(x)>d{\\cal H} ^2(x)=\\sum_{1\\le i\\le 4}{\\cal H} ^2(\\pi_i(\\Omega_i)).\\end{equation}\nFor the left-hand-side, by the disjoint union \\textcolor{blue}{(5.15)}, we have\n\\begin{equation} \\begin{split}&\\sum_{1\\le i\\le 4} \\int_{E_i}<-a_i,n_i(x)>d{\\cal H} ^2(x)\\\\\n=&\\sum_{1\\le i\\le 4}[ \\int_{E_{i0}}<-a_i,n_i(x)>d{\\cal H} ^2(x)+(\\sum_{i\\ne j}\\int_{E_{ij}}<-a_i,n_i(x)>d{\\cal H} ^2(x))\\\\\n=&\\sum_{1\\le i\\le 4}\\int_{E_{i0}}<-a_i,n_i(x)>d{\\cal H} ^2(x)+\\sum_{1\\le i<j\\le 4}\\int_{E_{ij}}(<-a_i,n_i(x)>+<-a_j,n_j(x)>)d{\\cal H} ^2(x)\\\\\n=&\\sum_{1\\le i\\le 4}\\int_{E_{i0}}<-a_i,n_i(x)>d{\\cal H} ^2(x)+\\sum_{1\\le i<j\\le 4}\\int_{E_{ij}}<n_j(x),a_i-a_j>d{\\cal H} ^2(x)\\\\\n\\le&\\sum_{1\\le i\\le 4}\\int_{E_{i0}}||a_i||d{\\cal H} ^2(x)+\\sum_{1\\le i<j\\le 4}\\int_{E_{ij}}||a_i-a_j||d{\\cal H} ^2(x)\\\\\n=&\\sum_{1\\le i\\le 4}|a_i|{\\cal H} ^2(E_{i0})+\\sum_{1\\le i<j\\le 4}||a_i-a_j||{\\cal H} ^2(E_{ij}).\\end{split}\\end{equation}\n\nNote that $||a_j||=1,1\\le j\\le 4$, and $||a_i-a_j||=\\frac{2\\sqrt 2}{\\sqrt 3}$, hence\n\\begin{equation}\\begin{split} \\sum_{j=1}^4\\int_{E_j}&<-a_i,n_i(x)>d{\\cal H} ^2(x)\\le \\sum_{1\\le i\\le 4}{\\cal H} ^2(E_{i0})+\\sum_{1\\le i<j\\le 4}\\frac{2\\sqrt 2}{\\sqrt 3}{\\cal H} ^2(E_{ij})\\\\\n&\\le \\frac{2\\sqrt 2}{\\sqrt 3}[\\sum_{1\\le i\\le 4}{\\cal H} ^2(E_{i0})+\\sum_{1\\le i<j\\le 4}{\\cal H} ^2(E_{ij})]\n=\\frac{2\\sqrt 2}{\\sqrt 3}{\\cal H} ^2(F)\\le \\frac{2\\sqrt 2}{\\sqrt 3}{\\cal H} ^2(E\\cap B),\\end{split}\\end{equation}\nwhere the second last equality is again because of the disjoint union \\textcolor{blue}{(5.15)}.\n\nAs a result, we have\n\\begin{equation} {\\cal H} ^2(E\\cap B)\\ge \\frac{\\sqrt 3}{2\\sqrt 2}\\sum_{1\\le i\\le 4}{\\cal H} ^2(\\pi_i(\\Omega_i)).\\end{equation}\n\nOn the other hand, either by chasing the condition of equality for the inequalities of \\textcolor{blue}{(5.19) and (5.20)} (since $T$ is a topological competitor for itself), or by a direct calculus, it is easy to see that\n\\begin{equation} {\\cal H} ^2(T\\cap B)= \\frac{\\sqrt 3}{2\\sqrt 2}\\sum_{1\\le i\\le 4}{\\cal H} ^2(\\pi_i(\\Omega_i)).\\end{equation}\n\nBy hypothesis \\textcolor{blue}{(5.13)}, we know that for the set $E$, equality in \\textcolor{blue}{(5.21)} holds, and hence all the inequalities in \\textcolor{blue}{(5.19) and (5.20)} are equalities, which implies, in particular, that\n\\begin{equation} \\begin{split}\\mbox{ For almost all }x\\in E_{ij}, &T_xE_{ij}\\perp v_i-v_j.\\mbox{ Denote by }P_{ij}\\mbox{ the plane}\\\\ \n\\mbox{perpendicular to }&v_i-v_j.\\mbox{ Then for almost all }x\\in E_{ij}, T_xE=T_xE_{ij}=P_{ij};\\end{split}\\end{equation} \n\\begin{equation} \\mbox{For all }j, {\\cal H} ^2(E_{j0})=0;\\end{equation}\n\\begin{equation} {\\cal H} ^2(E\\cap B\\backslash\\cup_{j=1}^4 E_j)=0.\\end{equation}\n\nNow since $E$ is minimal, if $x\\in E_P\\cap B^\\circ$ is a regular point of $E$, then by Theorem \\textcolor{blue}{2.20}, there exists $r=r(x)>0$ such that in $B(x,r)$, $E$ is the graph of a $C^1$ function from $T_xE$ to $T_xE^\\perp$, hence for all $y\\in E\\cap B(x,r)$, the tangent plane $T_yE$ exists, and the map $f:E\\cap B(x,r)\\to G(3,2): y\\mapsto T_yE$ is continuous. But by \\textcolor{blue}{(5.23)}, we have only six choices (which are isolated points in $G(3,2)$) for $T_yE$, hence $f$ is constant, and $T_yE=T_xE$ for all $y\\in E\\cap B(x,r)$. As a result, \n\\begin{equation} E\\cap B(x,r)=(T_xE+x)\\cap B(x,r)\\end{equation}\nis a disk parallel to one of the $P_{ij}$.\n\nStill by the $C^1$ regularity Theorem \\textcolor{blue}{2.20}, the set $E_P\\cap B^\\circ$ is a $C^1$ manifold, and is open in $E$. Thus we deduce that\n\\begin{equation} \\begin{split}\\mbox{Each connected component of }E_P\\cap B^\\circ\\mbox{ is part of a plane}\\\\\n\\mbox{ that is parallel to one of the }P_{ij}.\\end{split}\\end{equation}\n\nLet us look at $E_Y$. First, $E_Y\\ne\\emptyset$: otherwise, by \\textcolor{blue}{Corollary 2.23 $2^\\circ$}, $E\\cap B^\\circ=E_P\\cap B^\\circ$, and hence is a union of planes. But $E\\cap \\partial B$ does not coincide with any union of planes.\n\nTake any $x\\in E_Y$, then by the $C^1$ regularity around $\\mathbb Y$ points (Theorem \\textcolor{blue}{2.20} and Remark \\textcolor{blue}{2.21}), there exists $r=r(x)>0$ such that in $B(x,r)$, $E$ is image of a $C^1$ diffeomorphism $\\varphi$ of a $\\mathbb Y$-set Y, and $Y$ is tangent to $E$ at $x$. Denote by $L_Y$ the spine of $Y$, and by $R_i,1\\le i\\le 3$ the three open half planes of $Y$. Then $\\varphi(R_i),1\\le i\\le 3$ are connected subsets $E_P$, hence each of them is a part of a plane parallel to one of the $P_{ij},1\\le I<j\\le 4$. As consequence, $\\varphi(L_Y)\\cap B(x,r)$ is an open segment passing through $x$ and parallel to one of the spines $D_j,1\\le j\\le 4$ of $T$. Here $D_j$ is the intersection of the three $P_{ij}, i\\ne j$.\n\nAs a result, $E_Y\\cap B^\\circ$ is a union of open segments $I_1, I_2,\\cdots$, each of which is parallel to one of the $D_j, 1\\le j\\le 4$, and every endpoing is either a point on the boundary $\\partial B$, or a point of type $\\mathbb T$. Moreover, \n\\begin{equation} \\begin{split}\\mbox{For each }&x\\in E_Y\\mbox{ such that }T_xE_Y=D_j\\mbox{, there exists }r>0\\\\\n&\\mbox{such that, in }B(x,r), E\\mbox{ is a }\\mathbb Y-\\mbox{set whose spine is }x+D_j.\\end{split}\\end{equation} \n\nNext, since we are in dimension 3, the only other possible type of singular point is of type $\\mathbb T$. So we are going to discuss two cases: when there exists a $\\mathbb T$ points, or there is no $\\mathbb T$ points.\n\n\\textbf{Case 1:} There exists a point $x\\in E_T$. \n\n\\begin{lem} If there exists a point $x\\in E_T$, then $T\\cap B^\\circ =E$.\n\\end{lem}\n\n\\noindent Proof.  By the same argument as above, and by Theorem \\textcolor{blue}{2.20} and Remark \\textcolor{blue}{2.21}, the unique blow-up limit $C_xE$ of $E$ at $x$ must be the set $T$, and there exists $r>0$ such that in $B(x,r)$, $E$ coincides with $T+x$.\n As a result, for each segment $I_i$, at least one of its endpoints is in the unit sphere, because two parallel $\\mathbb T$-sets cannot be connected by a $\\mathbb Y$ segment.\n \n Hence all the segments $I_i$ touch the boundary $\\partial B$. That is, \n \\begin{equation} L_i\\backslash \\{\\{x\\}\\cup \\partial B)\\subset E_Y.\\end{equation}\n \n Denote by $L_i,1\\le i\\le 4$, the four spines of $T+x$. Then $L_i\\cap B^\\circ\\subset E_Y$, because $L_i\\cap B(x,r)$ is part of some $I_j\\subset E_Y$, which already has an endpoint $x$ that does not belong to $\\partial B$, hence the other endpoint must lie in $\\partial B$, which yields $I_j=L_i\\cap B^\\circ$.\n \n Now we take a one parameter family of open balls $B_s$ with radii $r\\le s\\le 1$, with $B_r=B(x,r)$, $B_1=B^\\circ$, such that\n \n $1^\\circ$ $B_s\\subsetneqq B_{s'}$ for all $s<s'$;\n \n $2^\\circ$ $\\cap_{1>t>s} B_t=\\bar B_s$ and $\\cup_{t<s}B_t=B_s$ for all $r\\le s\\le 1$.\n \n Set $R=\\inf\\{s>r, (T+x)\\cap B_s\\ne E\\}$. We claim that $R=1$.\n \n Suppose this is not true. By definition of $B_s$, we know that the four spines and the six faces of $T+x$ are never tangent to $\\partial B_s$ for any $r<s<1$. Then we know that $\\partial B_R\\cap (T+x)\\subset E_P\\cup E_Y$: in fact, if $y$ belong to one of the $L_i$, then by \\textcolor{blue}{(5.29)}, $y\\in L_i\\cap \\partial B_s\\subset E_Y$; otherwise, suppose $y$ does not lie in the four $L_i,1\\le i\\le 4$. Then $y$ belong to  $x+P_{ij}$ for some $i\\ne j$. As a result, for any $t>0$ small, we know that $E\\cap B(y,t)\\cap B_R=(x+P_{ij})\\cap B(y,t)\\cap B_R$. Note that the set $(x+P_{ij})\\cap B(y,t)\\cap B_R$ is almost a half disk when $t$ is sufficiently small, hence in particular, $E\\cap B(y,t)$ cannot coincide with a $\\mathbb Y$ set or a $\\mathbb T$ set$\\Rightarrow y\\in E_P$.\n \n If $y\\in E_P$, then $y\\in x+P_{ij}$ for some $i\\ne j$. Then $T_yE=P_{ij}$. By \\textcolor{blue}{(5.27)}, and the fact that $R<1$, there exists $r_y>0$ such that $B(y,r_y)\\subset B^\\circ$ and $E\\cap B(y,r_y)=(P_{ij}+y)\\cap B(y,r_y)$. In other words, \n \\begin{equation}\\mbox{ there exists }r_y>0\\mbox{ such that }E\\mbox{ coincides with }T+x\\mbox{ in }B(y,r_y).\\end{equation}\n \n If $y$ is a $\\mathbb Y$ point, then it lies in one of the $L_i$. By the same argument as above, using \\textcolor{blue}{(5.28)}, we also have \\textcolor{blue}{(5.30)}.\n \n Thus \\textcolor{blue}{(5.30)} holds for all $y\\in \\partial B_R\\cap (T+x)$. Since $\\partial B_R\\cap (T+x)$ is compact, we get an $r>0$, such that $E\\cap B(B_R,r)=(T+x)\\cap B(B_R,r)$. By the continuous condition $2^\\circ$ for the family $B_s$, there exists $R'\\in (R,1)$ such that $B_{R'}\\subset B(B_R,r)$. As consequence, $E\\cap B_{R'}=(T+x)\\cap B_{R'}$, this contradicts the definition of $R$.\n \n Hence $R=1$, and by definition of $R$, we have $(T+x)\\cap B^\\circ=E\\cap B^\\circ$. Since $E\\cap \\partial B=T\\cap \\partial B$, and $E$ is closed and reduced, $x$ must be the origin. Thus we get the conclusion of Lemma \\textcolor{blue}{5.7}.\\hfill$\\Box$\\bigskip\n \n \\textbf{Case 2:} $E_T=\\emptyset$. In this case, the same kind of argument as in Lemma \\textcolor{blue}{5.7} gives the following:\n \n \\begin{lem}Let $x$ be a $\\mathbb Y$ point in $E$ and $T_xE_Y=D_j$. Denote by $Y_j$ the $Y$ set whose spine is $D_j$ and whose three half planes lie in $P_{ij},i\\ne j$. Then  $(Y_j+x)\\cap B=E$.\n \\end{lem}\n \n But this is impossible, because $E\\cap \\partial B=T\\cap \\partial B$, which contains with no $(Y_j+x)\\cap \\partial B$ for any $x$ and $j$.\n \n Hence we have $E\\cap \\bar B=T\\cap \\bar B$, and thus $T$ is topological unique in $B$. We thus get Theorem \\textcolor{blue}{5.6}.\\hfill$\\Box$\\bigskip\n\n\n\n\\renewcommand\\refname{References}\n\\bibliographystyle{plain}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nSolvable lattice models related to Lie superalgebras \\cite{Ka} \nhave been received much attentions\n\\cite{KulSk,Kul,BS,KuR,DFI,Sa,ZBG,MR94}. \nTo construct eigenvalue formulae of transfer matrices \nfor such models is an important problem in mathematical \nphysics. To achieve this program,  \n Bethe ansatz has been often used. \n \nNowadays, there are many literatures \n(See for example \\cite{Kul,EK,EKS,FK,Ma,RM,PF,MR,ZB} \nand references therein.) \non Bethe ansatz analysis \nfor solvable lattice models related to Lie superalgebras. \nHowever most of them deal only with models related to \n simple representations like fundamental ones. \n Only a few people \\cite{Ma,PF,ZB} tried to deal with \n more complicated models such as fusion models \\cite{KRS} by \nBethe ansatz; while \nthere was no systematic study on this subject. \n\nTo break through such situations, we have \nrecently executed \\cite{T1,T2,T3,T4} an \nanalytic Bethe ansatz \\cite{R1,R2,BR,KS1,KOS,S2,KS2} systematically \nfor the type I Lie superalgebras $sl(r+1|s+1),C(s)$ \n cases. Namely, we have proposed a set of \n dressed vacuum forms (DVFs) and \n a class of functional relations for it. \n The purpose of this paper\n  is to develop our recent works\n to type II Lie superalgebras $B(r|s)=osp(2r+1|2s)$ \n ($r \\in {\\bf Z}_{\\ge 0}$, $s \\in {\\bf Z}_{\\ge 1}$) \n and $D(r|s)=osp(2r|2s)$ \n ($r \\in {\\bf Z}_{\\ge 2}$, $s \\in {\\bf Z}_{\\ge 1}$) cases. \n \n We can express \\cite{Kul,MR} the \n Bethe ansatz equation (BAE) (\\ref{BAE}) \n by using the representation theoretical data of \n  $B(r|s)$ \n ($r \\in {\\bf Z}_{\\ge 1}$, $s \\in {\\bf Z}_{\\ge 1}$) \n or $D(r|s)$ ($r \\in {\\bf Z}_{\\ge 2}$, $s \\in {\\bf Z}_{\\ge 1}$), \n as long as we adopt the distinguished simple root system \\cite{Ka}.  \nOn the other hand, $B(0|s)=osp(1|2s)$ is a peculiar one among \n the Lie superalgebras. \n In contrast to other Lie superalgebras, \n its simple root system is unique.  \n Corresponding to this fact, BAEs \n (\\ref{BAE1})-(\\ref{BAE4}) associated with the root system \n of $B(0|s)$ will be also unique.  \n Peculiarity of these BAEs is that so far \n a naive description in terms of \n the simple root system does not exist  \n for the $s$-th BAEs (\\ref{BAE3}) and (\\ref{BAE4}), \n which correspond to the odd root $\\alpha_{s}$\n  with  $(\\alpha_{s}|\\alpha_{s})\\ne 0$ (cf. \\cite{Kul,MR}).  \n \n We assume, as our starting point, above-mentioned \n BAEs (\\ref{BAE})-(\\ref{BAE4}) for \n $B(r|s)$ and $D(r|s)$, and \n  then carry out an analytic Bethe ansatz systematically \n to construct a class of DVFs . \nOn constructing DVFs, \n{\\em the pole-freeness under the BAE} \nand {\\em the top term hypothesis} \n \\cite{KS1,KOS} play important roles.\n\nWe introduce skew-Young (super) diagrams $\\lambda \\subset \\mu$, \nwhich are related to tensor-like representations\n\\footnote{In this paper, we do not deal with spinorial \nrepresentations.} of $B(r|s)$ or $D(r|s)$. \nOn these skew-Young (super) diagrams, we define \n a set of admissible tableaux  $B(\\lambda \\subset \\mu)$ \nwith some semi-standard like conditions. \nThere is a one-to-one correspondence between \nthese conditions for $B(0|s)$ case and  the \n  conditions for $A_{2s}^{(2)}$ case \\cite{KS2}.\n In addition, \n in contrast to $B(r|s)$ case, these conditions for $D(r|s)$ case \n have non-local nature. \n Next, we define a function ${\\cal T}_{\\lambda \\subset \\mu}(u)$  \n (\\ref{Tge1}) of a spectral parameter $u$ \n as summations over $B(\\lambda \\subset \\mu)$. \n It will provide us the spectra of a set of transfer \nmatrices for various fusion $B(r|s)$ \nor $D(r|s)$ \nvertex models.  It contains the top term \\cite{KS1,KOS},  \n which carries the highest weight of the irreducible \n representation of $B(r|s)$ or $D(r|s)$ \n labeled by a skew-Young (super) diagram \n $\\lambda \\subset \\mu$. \n  In particular, the simplest example of \n ${\\cal T}_{\\lambda \\subset \\mu}(u)$, that is, \n ${\\cal T}^{1}(u)={\\cal T}_{1}(u)={\\cal T}_{(1^{1})}(u)$\n  reduces to the eigenvalue formula of the transfer matrix \n  \\cite{MR} of \n  some vertex model related to the fundamental  \n representation of  $B(r|s)$ or $D(r|s)$ \n after some redefinitions.   \n  The BAEs (\\ref{BAE})-(\\ref{BAE4}) \n  are assumed common to all the DVFs \n for transfer matrices with various fusion types in the auxiliary space \n  as long as they act on a common quantum space. \n Therefore, we can prove the pole-freeness of \n ${\\cal T}^{a}(u)={\\cal T}_{(1^{a})}(u)$\n  for any $a \\in {\\bf Z}_{\\ge 0}$ under the common \n BAEs (\\ref{BAE})-(\\ref{BAE4}). \n  We further mention a determinant formula, \n  by which ${\\cal T}_{\\lambda \\subset \\mu}(u)$ can be expressed  \n  only by the fundamental functions $\\{ {\\cal T}^{a} \\} $ \n  and then pole freeness follows immediately. \n  A set of transfer matrix functional relations among DVFs \n   also follows \n  from this  formula.\n It will be a kind of the $T$-system \\cite{KNS1} \n(see also \\cite{BR,KP,KNS2,KS2,KOS,KNH,\nTK,KLWZ,Ma,PF,ZB,T1,T2,T3,T4,T5}). \n In particular for $B(0|s)$ case, \n  there is remarkable duality among DVFs  \n (see, Theorem \\ref{dual-th} and (\\ref{dual2})). \nOn constructing above-mentioned functional relations, \nthis duality among DVFs plays an important role.  \n\nThe outline of this paper is as follows.\nIn section 2, we briefly mention the Lie superalgebras \n$B(r|s)$ and $D(r|s)$. \nIn section 3, we execute an analytic Bethe ansatz \nbased on the BAEs (\\ref{BAE})-(\\ref{BAE4}) associated \nwith distinguished simple root systems. \n In Section 4, we discuss transfer matrix functional \n relations. \n Section 5 is devoted to summary and discussion. \n In appendix A.1-A.3, we prove the pole-freeness of DVFs. \nAppendix B provides generating series of \n${\\cal T}^{a}(u)$ and ${\\cal T}_{m}(u)={\\cal T}_{(m)}(u)$. \nIn this paper, we adopt similar notation in \\cite{KS1,KOS,T1,T2,T3,T4}. \n Finally we note that we can recover many formulae\n  in \\cite{KS1,KOS}  \n for $B_{r}$ or $D_{r}$, if we set $s=0$ and redefine the vacuum parts. \n\\setcounter{equation}{0}\n\\section{Lie superalgebras}\nIn this section, we briefly mention the \nLie superalgebras $B(r|s)$ and $D(r|s)$  \n(see for example \\cite{Ka,Ka2,BB,FJ,MSS,Je}). \n\nThere are several choices of simple root systems and \nthe simplest one is the distinguished simple root system.\n They read as follows: \\\\ \n\\begin{figure}\n    \\setlength{\\unitlength}{0.9pt}\n    \\begin{center}\n    \\begin{picture}(410,50) \n      \\put(10,20){\\circle{20}}\n      \\put(20,20){\\line(1,0){20}}\n      \\put(50,20){\\circle{20}}\n      \\put(60,20){\\line(1,0){20}}\n      \\put(90,20){\\line(1,0){10}}\n      \\put(110,20){\\line(1,0){10}}\n      \\put(130,20){\\line(1,0){20}}\n      \\put(160,20){\\circle{20}}\n      \\put(170,20){\\line(1,0){20}}\n      \\put(7,0){$\\alpha_{1}$}\n      \\put(46,0){$\\alpha_{2}$}\n      \\put(156,0){$\\alpha_{s-1}$}\n    %\n      \\put(192.929,12.9289){\\line(1,1){14.14214}}\n      \\put(192.929,27.07107){\\line(1,-1){14.14214}}\n      \\put(200,20){\\circle{20}}\n      \\put(210,20){\\line(1,0){20}}\n      \\put(240,20){\\circle{20}}\n      \\put(250,20){\\line(1,0){20}}\n      \\put(280,20){\\line(1,0){10}}\n      \\put(300,20){\\line(1,0){10}}\n      \\put(320,20){\\line(1,0){20}}\n      \\put(350,20){\\circle{20}}\n      \\put(358.8,25){\\line(1,0){32.4}}\n      \\put(358.8,15){\\line(1,0){32.4}}\n      \\put(400,20){\\circle{20}}\n      \\put(197,0){$\\alpha_{s}$}\n      \\put(236,0){$\\alpha_{s+1}$}\n      \\put(346,0){$\\alpha_{s+r-1}$}\n      \\put(397,0){$\\alpha_{s+r}$}\n      \\put(390,20){\\line(-1,1){14.14214}}\n      \\put(390,20){\\line(-1,-1){14.14214}}\n  \\end{picture}\n  \\end{center}\n  \\caption{Dynkin diagram for the Lie superalgebra \n  $B(r|s)=osp(2r+1|2s)$ ($r \\in {\\bf Z}_{\\ge 1}, \n  s \\in {\\bf Z}_{\\ge 1}$)\n   corresponding to the distinguished simple \n  root system: white circles denote even roots; \n  a gray (a cross) circle denotes an odd root $\\alpha$ with \n   $(\\alpha|\\alpha)=0$.}\n  \\label{dynkin-brs}\n\\end{figure}\n\\begin{figure}\n    \\setlength{\\unitlength}{0.8pt}\n    \\begin{center}\n    \\begin{picture}(250,50) \n      \\put(10,20){\\circle{20}}\n      \\put(20,20){\\line(1,0){20}}\n      \\put(50,20){\\circle{20}}\n      \\put(60,20){\\line(1,0){20}}\n      \\put(90,20){\\line(1,0){10}}\n      \\put(110,20){\\line(1,0){10}}\n      \\put(130,20){\\line(1,0){20}}\n      \\put(160,20){\\circle{20}}\n      \\put(170,20){\\line(1,0){20}}\n      \\put(7,0){$\\alpha_{1}$}\n      \\put(46,0){$\\alpha_{2}$}\n      \\put(156,0){$\\alpha_{s-2}$}\n    %\n      \\put(200,20){\\circle{20}}\n      \\put(240,20){\\circle*{20}}\n      \\put(208.8,25){\\line(1,0){32.4}}\n      \\put(208.8,15){\\line(1,0){32.4}}\n      \\put(197,0){$\\alpha_{s-1}$}\n      \\put(236,0){$\\alpha_{s}$}\n      \\put(230,20){\\line(-1,1){14.14214}}\n      \\put(230,20){\\line(-1,-1){14.14214}}\n  \\end{picture}\n  \\end{center}\n  \\caption{Dynkin diagram for the Lie superalgebra \n  $B(0|s)=osp(1|2s)$ ($s \\in {\\bf Z}_{\\ge 1}$):\n   a black circle denotes an odd root $\\alpha$ \n   with $(\\alpha|\\alpha)\\ne 0$.}\n  \\label{dynkin-b0s}\n\\end{figure}\n\\begin{figure}\n    \\setlength{\\unitlength}{0.9pt}\n    \\begin{center}\n    \\begin{picture}(420,50) \n      \\put(10,20){\\circle{20}}\n      \\put(20,20){\\line(1,0){20}}\n      \\put(50,20){\\circle{20}}\n      \\put(60,20){\\line(1,0){20}}\n      \\put(90,20){\\line(1,0){10}}\n      \\put(110,20){\\line(1,0){10}}\n      \\put(130,20){\\line(1,0){20}}\n      \\put(160,20){\\circle{20}}\n      \\put(170,20){\\line(1,0){20}}\n      \\put(7,0){$\\alpha_{1}$}\n      \\put(46,0){$\\alpha_{2}$}\n      \\put(156,0){$\\alpha_{s-1}$}\n    %\n      \\put(192.929,12.9289){\\line(1,1){14.14214}}\n      \\put(192.929,27.07107){\\line(1,-1){14.14214}}\n      \\put(200,20){\\circle{20}}\n      \\put(210,20){\\line(1,0){20}}\n      \\put(240,20){\\circle{20}}\n      \\put(250,20){\\line(1,0){20}}\n      \\put(280,20){\\line(1,0){10}}\n      \\put(300,20){\\line(1,0){10}}\n      \\put(320,20){\\line(1,0){20}}\n      \\put(350,20){\\circle{20}}\n      %\n      \\put(398,44){\\circle{20}}\n      \\put(359,25){\\line(2,1){30}}\n      \\put(398,-4){\\circle{20}}\n      \\put(359,16){\\line(2,-1){30}}\n      \\put(390,24){$\\alpha_{s+r-1}$}\n      \\put(393,-24){$\\alpha_{s+r}$}\n      %\n      \\put(197,0){$\\alpha_{s}$}\n      \\put(236,0){$\\alpha_{s+1}$}\n      \\put(343,0){$\\alpha_{s+r-2}$}\n      %\n  \\end{picture}\n  \\end{center}\n  \\caption{Dynkin diagram for the Lie superalgebra \n  $D(r|s)=osp(2r|2s)$ ($r \\in {\\bf Z}_{\\ge 2}, \n  s \\in {\\bf Z}_{\\ge 1}$)\n   corresponding to the distinguished simple \n  root system.}\n  \\label{dynkin-drs}\n\\end{figure}\n$B(r|s)$ ($r,s \\in {\\bf Z}_{\\ge 1}$) case \n(see Figure \\ref{dynkin-brs}):\n\\begin{eqnarray}\n   && \\alpha_{i} = \\delta_{i}-\\delta_{i+1} \n   \\quad i=1,2,\\dots,s-1, \\nonumber  \\\\ \n   && \\alpha_{s} =\\delta_{s}-\\epsilon_{1}, \\nonumber  \\\\  \n   && \\alpha_{s+j} = \\epsilon_{j}-\\epsilon_{j+1} ,\n    \\quad j=1,2,\\dots,r-1, \\\\ \n   && \\alpha_{s+r} =\\epsilon_{r}; \\nonumber\n \\end{eqnarray}\n$B(0|s)$ ($s \\in {\\bf Z}_{\\ge 1}$) case (see Figure \\ref{dynkin-b0s}):\n\\begin{eqnarray}\n   && \\alpha_{i} = \\delta_{i}-\\delta_{i+1} \n   \\quad {\\rm for} \\quad i=1,2,\\dots,s-1, \\nonumber  \\\\ \n   && \\alpha_{s} =\\delta_{s}; \\label{root-B0s}\n\\end{eqnarray}\n$D(r|s)$ ($r \\in  {\\bf Z}_{\\ge 2}$, $s \\in {\\bf Z}_{\\ge 1}$) case \n(see Figure \\ref{dynkin-drs}):\n\\begin{eqnarray}\n   && \\alpha_{i} = \\delta_{i}-\\delta_{i+1} \n   \\quad i=1,2,\\dots,s-1, \\nonumber  \\\\ \n   && \\alpha_{s} =\\delta_{s}-\\epsilon_{1}, \\nonumber  \\\\  \n   && \\alpha_{s+j} = \\epsilon_{j}-\\epsilon_{j+1} ,\n    \\quad j=1,2,\\dots,r-2, \\\\ \n   && \\alpha_{s+r-1} =\\epsilon_{r-1}-\\epsilon_{r}, \\nonumber \\\\ \n   && \\alpha_{s+r} =\\epsilon_{r-1}+\\epsilon_{r}, \\nonumber\n \\end{eqnarray}\nwhere  \n $\\epsilon_{1},\\dots, \\epsilon_{r};\\delta_{1},\\dots,\\delta_{s}$ \nare the bases of the dual space of the Cartan subalgebra \nwith the bilinear \nform $(\\ |\\ )$ such that \n\\footnote{We normalized the longest simple root as \n$|(\\alpha|\\alpha)|=2$.}\n\\begin{equation}\n (\\epsilon_{i}|\\epsilon_{j})=\\delta_{i\\, j}, \\quad \n (\\epsilon_{i}|\\delta_{j})=(\\delta_{i}|\\epsilon_{j})=0 , \\quad \n (\\delta_{i}|\\delta_{j})=-\\delta_{i\\, j} . \n\\end{equation}\n $\\{\\alpha_i \\}_{i \\ne s}$ are even roots and $\\alpha_{s}$ \nis an odd root. Note that $(\\alpha_{s} | \\alpha_{s})=0$ \nfor $B(r|s)$ ($r,s \\in {\\bf Z}_{\\ge 1}$) and $D(r|s)$ \n($r \\in  {\\bf Z}_{\\ge 2}$, $s \\in {\\bf Z}_{\\ge 1}$) cases, \nwhile $(\\alpha_{s} | \\alpha_{s}) \\ne 0$ for \n$B(0|s)$ ($s \\in {\\bf Z}_{\\ge 1}$) case.\n \nLet $\\lambda \\subset \\mu$ be a skew-Young (super) diagram labeled by \nthe sequences of non-negative integers \n$\\lambda =(\\lambda_{1},\\lambda_{2},\\dots)$ and \n$\\mu =(\\mu_{1},\\mu_{2},\\dots)$ such that\n$\\mu_{i} \\ge \\lambda_{i}: i=1,2,\\dots;$  \n$\\lambda_{1} \\ge \\lambda_{2} \\ge \\dots \\ge 0$;  \n$\\mu_{1} \\ge \\mu_{2} \\ge \\dots \\ge 0$ and \n$\\lambda^{\\prime}=(\\lambda_{1}^{\\prime},\\lambda_{2}^{\\prime},\\dots)$ \nbe the conjugate of $\\lambda $. \nIn particular, for $B(r|s)$ ($r,s \\in {\\bf Z}_{\\ge 1}$), \n$\\lambda=\\phi $, $\\mu_{r+1}\\le s$ case, \nthe Kac-Dynkin label \n$[b_{1},b_{2},\\dots , b_{s+r}]$ is related to \n the Young (super) diagram with shape \n $\\mu=(\\mu_{1},\\mu_{2},\\dots)$ as follows:  \\\\ \n\\begin{eqnarray}\n && b_{i} = \\mu_{i}^{\\prime}-\\mu_{i+1}^{\\prime} \n  \\qquad {\\rm for} \\qquad i\\in \\{1,2,\\dots,s-1\\}, \\nonumber \\\\ \n && b_{s} = \\mu_{s}^{\\prime}+\\eta_{1}, \\qquad  \\label{Kac-Dynkin} \\\\ \n && b_{s+j} = \\eta_{j}-\\eta_{j+1} \n  \\qquad {\\rm for} \\qquad j \\in \\{1,2,\\dots,r-1\\}, \\nonumber \\\\\n && b_{s+r} = 2\\eta_{r} \\nonumber,\n\\end{eqnarray} \n  where $\\eta_{i}=Max\\{\\mu_{i}-s,0 \\} $. \nFor $B(0|s)$ ($s \\in {\\bf Z}_{\\ge 1}$), \n$\\lambda=\\phi $ case, \nthe Kac-Dynkin label \n$[b_{1},b_{2},\\dots , b_{s}]$ is related to \n the Young (super) diagram with shape \n $\\mu=(\\mu_{1},\\mu_{2},\\dots)$ as follows:\n\\begin{eqnarray}\n && b_{i} = \\mu_{i}^{\\prime}-\\mu_{i+1}^{\\prime} \n  \\qquad {\\rm for} \\qquad i\\in \\{1,2,\\dots,s-1\\}, \\nonumber \\\\ \n && b_{s} = 2\\mu_{s}^{\\prime}.  \\label{Kac-Dynkin-b0s}\n\\end{eqnarray}  \nFor $D(r|s)$ case, we use only the Young (super) diagram with \nshape $\\mu=(1^{a})$ or $\\mu=(m^{1})$. \nThe Young (super) diagram with shape $\\mu=(1^{a})$ is \nrelated to the Kac-Dynkin label \n$[b_{1},b_{2},\\dots ,b_{s+r}]$ as follows:\n\\begin{eqnarray}\n && b_{j} = a\\delta_{j 1}. \\label{KD-tate}\n\\end{eqnarray} \nAnd the Young (super) diagram with shape \n $\\mu=(m^{1})$ is related to \n the Kac-Dynkin label \n$[b_{1},b_{2},\\dots ,  b_{s+r}]$ as follows:\n\\begin{eqnarray}\nb_{j}=\n \\left\\{\n  \\begin{array}{lll}\n   \\! \\delta_{j m} \n     & \\! {\\rm if} \\! & m \\in \\{1,2,\\dots, s \\},  \\\\ \n   \\! (m-s+1)\\delta_{j s}+(m-s)\\delta_{j s+1} \n     & \\! {\\rm if} \\! & r \\in {\\bf Z}_{\\ge 3},  \n         m \\in {\\bf Z}_{\\ge s+1}, \\\\ \n   \\! (m-s+1)\\delta_{j s}+(m-s)(\\delta_{j s+1}+\\delta_{j s+2}) \n     & \\! {\\rm if} \\! & r=2,  m \\in {\\bf Z}_{\\ge s+1}. \n  \\end{array}\n \\right. \n \\label{KD-yoko}\n\\end{eqnarray}\nAn irreducible representation of $B(0|s)$\nwith the Kac-Dynkin label $[b_{1},b_{2},\\dots ,  b_{s}]$\n is finite dimensional \\cite{Ka2} if and only if \n\\begin{eqnarray}\n&& b_{j} \\in {\\bf Z}_{\\ge 0} \\qquad {\\rm for} \\qquad \nj \\in \\{1,2,\\dots, s-1\\}, \\nonumber \\\\ \n&& b_{s} \\in  2{\\bf Z}_{\\ge 0}. \\label{finite}\n\\end{eqnarray}\nThe dimensionality of the irreducible representation \n $V[b_{1},b_{2},\\dots ,  b_{s}]$ of \n $B(0|s)$ with the highest weight \n labeled by the Kac-Dynkin label  $[b_{1},b_{2},\\dots ,  b_{s}]$ \n is given \\cite{Ka2,Je} as follows\n\\footnote{We assume that \n$b_{j}+b_{j+1}+\\cdots +b_{s-1}=0$ if $j=s$.} \n \\begin{eqnarray}\n\\hspace{-30pt} &&  {\\rm dim}V[b_{1},b_{2},\\dots,b_{s}]=\n   \\prod_{1\\le i < j \\le s}\n  \\frac{b_{i}+b_{i+1}+\\cdots +b_{j-1}+j-i}{j-i}\\nonumber \\\\\n\\hspace{-30pt}&& \\hspace{10pt} \\times  \n  \\frac{b_{i}+b_{i+1}+\\cdots +b_{j-1}+\n  2(b_{j}+b_{j+1}+\\cdots +b_{s-1})+b_{s}+2s-i-j+1}\n         {2s-i-j+1} \\nonumber \\\\\n\\hspace{-30pt}&& \\hspace{10pt}\\times  \\prod_{1 \\le k \\le s}\n  \\frac{2(b_{k}+b_{k+1}+\\cdots +b_{s-1})+b_{s}+2s-2k+1}\n         {2s-2k+1}.\n  \\label{dim}\n \\end{eqnarray}\n\\setcounter{equation}{0}\n\\section{Analytic Bethe ansatz} \nWe assume, as our starting point, the \nfollowing type of the Bethe ansatz equations\n\\footnote{In this paper, we deal with the case,\n as an example,  \n that the quantum spaces of the transfer matrices are fundamental \nrepresentations. } \n\\cite{RW,OW,Kul,RM,KOS}. \\\\ \n$B(r|s)$ ($r,s \\in {\\bf Z}_{\\ge 1}$) or $D(r|s)$ ($r \\in  {\\bf Z}_{\\ge 2}, s \\in {\\bf Z}_{\\ge 1}$) case:\n\\begin{eqnarray}\n- \\left\\{\n \\prod_{j=1}^{N}\n \\frac{\\Phi (u_k^{(a)}-w_{j}-1)}\n      {\\Phi(u_k^{(a)}-w_{j}+1)}\n   \\right\\}^{\\delta_{a 1}}\n   &=&(-1)^{{\\rm deg}(\\alpha_a)} \n    \\prod_{b=1}^{s+r}\\frac{Q_{b}(u_k^{(a)}+(\\alpha_a|\\alpha_b))}\n           {Q_{b}(u_k^{(a)}-(\\alpha_a|\\alpha_b))}. \\label{BAE} \n\\end{eqnarray}\n$B(0|s)$ ($s \\in {\\bf Z}_{\\ge 2}$) case: \n\\begin{eqnarray}\n\\hspace{-40pt}\n&& - \\prod_{j=1}^{N}\n\\frac{ \\Phi (u_k^{(1)}-w_{j}-1)}\n     { \\Phi (u_k^{(1)}-w_{j}+1)} \n = \n\\frac{Q_{1}(u_k^{(1)}-2)Q_{2}(u_k^{(1)}+1)}\n{Q_{1}(u_k^{(1)}+2)Q_{2}(u_k^{(1)}-1)}, \\label{BAE1}\n\\\\\n\\hspace{-40pt}\n&&-1= \\frac{Q_{a-1}(u_k^{(a)}+1)Q_{a}(u_k^{(a)}-2)Q_{a+1}(u_k^{(a)}+1)}\n      {Q_{a-1}(u_k^{(a)}-1)Q_{a}(u_k^{(a)}+2)Q_{a+1}(u_k^{(a)}-1)}\n      \\quad{\\rm for}\n       \\quad 2 \\le a \\le s-1, \\label{BAE2}\\\\ \n\\hspace{-40pt}\n&&1= \\frac{Q_{s-1}(u_k^{(s)}+1)Q_{s}(u_k^{(s)}+1)Q_{s}(u_k^{(s)}-2)}\n        {Q_{s-1}(u_k^{(s)}-1)Q_{s}(u_k^{(s)}-1)Q_{s}(u_k^{(s)}+2)}\n       \\label{BAE3} .\n\\end{eqnarray}\n$B(0|1)$ case: \n\\begin{eqnarray}\n\\hspace{-40pt}\n&& \\prod_{j=1}^{N}\n\\frac{ \\Phi (u_k^{(1)}-w_{j}-1)}\n     { \\Phi (u_k^{(1)}-w_{j}+1)} \n = \n\\frac{Q_{1}(u_k^{(1)}+1)Q_{1}(u_k^{(1)}-2)}\n     {Q_{1}(u_k^{(1)}-1)Q_{1}(u_k^{(1)}+2)}\n       \\label{BAE4}.\n\\end{eqnarray}\nHere $ Q_{a}(u)= \\prod_{j=1}^{N_{a}}\\Phi(u-u_j^{(a)})$; \n$ N \\in {\\bf Z }_{\\ge 0}$ is the number of \nthe lattice sites; $N_{a} \\in {\\bf Z }_{\\ge 0}$; \n$u_{j}^{(a)}, w_{j}\\in {\\bf C}$; $a,k \\in {\\bf Z}$ \n($a \\in \\{1,2,\\dots,s+r \\}$ ($r=0$ for $B(0|s)$ case); \n$\\ k \\in \\{1,2,\\dots, N_{a} \\}$); \n\\begin{eqnarray}\n    {\\rm deg}(\\alpha_a)&=&\\left\\{\n              \\begin{array}{@{\\,}ll}\n                0  & \\mbox{for even root} \\\\ \n                1 & \\mbox{for odd root} \n              \\end{array}\n              \\right. \\\\ \n             &=& \\delta_{a,s}. \\nonumber \n\\end{eqnarray}\n $\\Phi$ is a function, which has zero at $u=0$. \nFor example, $\\Phi(u)$ has the following form\n\\begin{equation}\n\\Phi(u)=u.\n\\end{equation}\nRemarkable enough, Bethe ansatz equations can be written \nin terms of root systems of  Lie algebras \\cite{RW,OW} or \nLie superalgebras \\cite{Kul,MR}. \nMartins and Ramos \\cite{MR} pointed out that \n$B(0|s)$ is an exception\n to this observation (see also \\cite{Kul}).  \nTo put it more precisely, an exception lies in \nthe right hand side of  (\\ref{BAE3}) and (\\ref{BAE4}), \nwhich correspond to \nthe odd root $\\alpha_{s}$ with $(\\alpha_{s}|\\alpha_{s}) \\ne 0$.  \nIn fact, one can derive (\\ref{BAE1}) and (\\ref{BAE2}) from \n(\\ref{BAE}) and (\\ref{root-B0s}); \nwhile cannot derive (\\ref{BAE3}) and (\\ref{BAE4}). \\\\ \n{\\em Remark:} \nThere are compact expressions of BAEs \nfor twisted quantum affine algebras \\cite{RW}. \n Moreover \nthe BAEs (\\ref{BAE1})-(\\ref{BAE4}) \nresemble to the BAEs for $A_{2s}^{(2)}$. \nThis resemblance will originate from resemblance between \n$B(0|s)^{(1)}$ and $A_{2s}^{(2)}$. \nThus there is a possibility that the BAEs \n(\\ref{BAE1})-(\\ref{BAE4}) are also \ncompactly written in terms of root system of the \nLie superalgebra $B(0|s)$ ($ s \\in {\\bf Z}_{\\ge 1} $). \nWe also point out that the expression (\\ref{BAE}) \nis not always valid for non-distinguished simple root systems.  \nIn fact we have confirmed for several cases that the Bethe \nansatz equations corresponding to the \n odd roots $\\alpha $ with $(\\alpha|\\alpha)\\ne 0$ \nhave similar structure to \n(\\ref{BAE3}) or (\\ref{BAE4}) \nby using the correspondence \\cite{T2} between \nthe particle-hole transformation \n and the (super) Weyl reflection. \n\nWe define the set \n\\begin{eqnarray}\n    &&  J=J_{+} \\cup J_{-},\n\\end{eqnarray}\nwhere \n\\begin{eqnarray}\nJ_{-}= \\{ 1,2,\\dots, s,\\overline{s},\\dots,\\overline{2},\n                  \\overline{1} \\} \\label{set} \n\\end{eqnarray}\nis common for $B(r|s)$ and $D(r|s)$; \nwhile $J_{+}$ is not:\n\\begin{eqnarray}\n    && J_{+}=\\{s+1,s+2,\\dots,s+r,\\overline{s+r},\\dots,\n       \\overline{s+2},\\overline{s+1} \\} \\cup \\{ 0 \\} \n       \\quad {\\rm for } \\quad B(r|s), \\nonumber \\\\ \n    && J_{+}=\\{s+1,s+2,\\dots,s+r,\\overline{s+r},\\dots,\n       \\overline{s+2},\\overline{s+1} \\} \n       \\quad {\\rm for } \\quad D(r|s). \\nonumber \n\\end{eqnarray}\nOn this set $J$, we define the total order \n\\begin{eqnarray} \n 1\\prec 2 \\prec \\cdots \\prec s+r \\prec 0 \n \\prec \\overline{s+r} \\prec \\cdots \\prec \\overline{2} \n \\prec \\overline{1} \\label{order}\n\\end{eqnarray}\n for $B(r|s)$ case, and the partial order \n\\begin{eqnarray} \n 1\\prec 2 \\prec \\cdots \\prec s+r-1 \\prec \n \\begin{array}{c} \n  s+r \\\\ \n      \\\\ \n  \\overline{s+r} \n \\end{array}\n \\prec \\overline{s+r-1} \\prec \\cdots \\prec \\overline{2} \n \\prec \\overline{1} \\label{order2}\n\\end{eqnarray}\nfor $D(r|s)$ case. In contrast to $B(r|s)$ case,\n there is no order between \n$s+r$ and $\\overline{s+r}$ for $D(r|s)$ case. \nWe also define the grading parameter as follows: \n\\begin{equation}\n      p(a)=\\left\\{\n              \\begin{array}{@{\\,}ll}\n                0  & \\mbox{for $a \\in J_{+}$},  \\\\ \n                1 & \\mbox{for $a \\in J_{-}$ }.\n              \\end{array}\n            \\right. \\label{grading}\n\\end{equation}\nFor $a \\in J $, we define \n\\footnote{\nIn this paper, we often abbreviate the spectral parameter $u$.}\n the following functions. \\\\ \n $B(r|s)$ ($r \\in {\\bf Z}_{\\ge 0},s \\in {\\bf Z}_{\\ge 1}$) case:\n\\begin{eqnarray} \n\\hspace{-20pt} && \\framebox{$a$}_{u}=\\psi_{a}(u)\n   \\frac{Q_{a-1}(u-a-1)Q_{a}(u-a+2)}\n        {Q_{a-1}(u-a+1)Q_{a}(u-a)}\n      \\quad  {\\rm for} \\quad 1 \\le a \\le s,\\nonumber \\\\\n\\hspace{-20pt} && \\framebox{$a$}_{u}=\\psi_{a}(u)\n   \\frac{Q_{a-1}(u-2s+a+1)Q_{a}(u-2s+a-2)}\n        {Q_{a-1}(u-2s+a-1)Q_{a}(u-2s+a)} \\nonumber \\\\ \n\\hspace{-20pt} && \\hspace{170pt}   \n{\\rm for} \\quad s+1 \\le a \\le s+r,\\nonumber \\\\\n\\hspace{-20pt} && \\framebox{$0$}_{u}=\\psi_{0}(u)\n   \\frac{Q_{s+r}(u-s+r+1)Q_{s+r}(u-s+r-2)}\n        {Q_{s+r}(u-s+r-1)Q_{s+r}(u-s+r)},\n      \\label{z+}   \\\\\n\\hspace{-20pt} && \\framebox{$\\overline{a}$}_{u}=\\psi_{\\overline{a}}(u)\n   \\frac{Q_{a-1}(u+2r-a-2)Q_{a}(u+2r-a+1)}\n        {Q_{a-1}(u+2r-a)Q_{a}(u+2r-a-1)}\n        \\nonumber \\\\ \n\\hspace{-20pt} && \\hspace{170pt} {\\rm for} \\quad \ns+1 \\le a \\le s+r, \\nonumber \\\\ \n\\hspace{-20pt} && \\framebox{$\\overline{a}$}_{u}=\\psi_{\\overline{a}}(u)\n   \\frac{Q_{a-1}(u-2s+2r+a)Q_{a}(u-2s+2r+a-3)}\n        {Q_{a-1}(u-2s+2r+a-2)Q_{a}(u-2s+2r+a-1)} \n  \\nonumber \\\\ \n\\hspace{-20pt} && \\hspace{170pt}  \n{\\rm for} \\quad 1 \\le a \\le s. \\nonumber \n\\end{eqnarray}\n$D(r|s)$ ($r \\in {\\bf Z}_{\\ge 2},s \\in {\\bf Z}_{\\ge 1}$) case: \n\\begin{eqnarray} \n\\hspace{-20pt} && \\framebox{$a$}_{u}=\\psi_{a}(u)\n   \\frac{Q_{a-1}(u-a-1)Q_{a}(u-a+2)}\n        {Q_{a-1}(u-a+1)Q_{a}(u-a)}\n      \\quad {\\rm for} \\quad 1 \\le a \\le s,\\nonumber \\\\\n\\hspace{-20pt} && \\framebox{$a$}_{u}=\\psi_{a}(u)\n   \\frac{Q_{a-1}(u-2s+a+1)Q_{a}(u-2s+a-2)}\n        {Q_{a-1}(u-2s+a-1)Q_{a}(u-2s+a)} \n        \\nonumber \\\\ \n\\hspace{-20pt} && \\hspace{160pt} \n {\\rm for} \\quad s+1 \\le a \\le s+r-2,\\nonumber \\\\\n\\hspace{-20pt} && \\framebox{$r+s-1$}_{u}=\\psi_{r+s-1}(u)\n   \\frac{Q_{s+r-2}(u-s+r)Q_{s+r-1}(u-s+r-3)}\n        {Q_{s+r-2}(u-s+r-2)Q_{s+r-1}(u-s+r-1)}\n        \\nonumber \\\\ \n\\hspace{-20pt} && \\hspace{160pt} \\times \n   \\frac{Q_{s+r}(u-s+r-3)}{Q_{s+r}(u-s+r-1)}\n ,\\nonumber \\\\\n\\hspace{-20pt} && \\framebox{$r+s$}_{u}=\\psi_{r+s}(u)\n   \\frac{Q_{s+r-1}(u-s+r+1)Q_{s+r}(u-s+r-3)}\n        {Q_{s+r-1}(u-s+r-1)Q_{s+r}(u-s+r-1)},\n      \\label{z++}   \\\\\n\\hspace{-20pt} && \n \\framebox{$\\overline{r+s}$}_{u}=\\psi_{\\overline{r+s}}(u)\n   \\frac{Q_{s+r-1}(u-s+r-3)Q_{s+r}(u-s+r+1)}\n        {Q_{s+r-1}(u-s+r-1)Q_{s+r}(u-s+r-1)},\n      \\nonumber   \\\\\n\\hspace{-20pt} && \n\\framebox{$\\overline{r+s-1}$}_{u}=\\psi_{\\overline{r+s-1}}(u)\n   \\frac{Q_{s+r-2}(u-s+r-2)Q_{s+r-1}(u-s+r+1)}\n        {Q_{s+r-2}(u-s+r)Q_{s+r-1}(u-s+r-1)}\n        \\nonumber \\\\ \n\\hspace{-20pt} && \\hspace{160pt} \n \\times \\frac{Q_{s+r}(u-s+r+1)}{Q_{s+r}(u-s+r-1)},\n        \\nonumber \\\\\n\\hspace{-20pt} && \\framebox{$\\overline{a}$}_{u}=\\psi_{\\overline{a}}(u)\n   \\frac{Q_{a-1}(u+2r-a-3)Q_{a}(u+2r-a)}\n        {Q_{a-1}(u+2r-a-1)Q_{a}(u+2r-a-2)}\n        \\nonumber \\\\ \n\\hspace{-20pt} && \\hspace{160pt}\n    {\\rm for} \\quad s+1 \\le a \\le s+r-2, \\nonumber \\\\ \n\\hspace{-20pt} && \\framebox{$\\overline{a}$}_{u}=\\psi_{\\overline{a}}(u)\n   \\frac{Q_{a-1}(u-2s+2r+a-1)Q_{a}(u-2s+2r+a-4)}\n        {Q_{a-1}(u-2s+2r+a-3)Q_{a}(u-2s+2r+a-2)} \n   \\nonumber \\\\ \n\\hspace{-20pt} && \\hspace{160pt}\n   {\\rm for} \\quad 1 \\le a \\le s.\\nonumber \n\\end{eqnarray}\nHere we assume $Q_{0}(u)=1$. \nThe vacuum parts of the functions \n$\\framebox{a}_{u}$ (\\ref{z+}) and (\\ref{z++})\n are given as follows. \\\\\nFor $B(r|s)$ ($r \\in {\\bf Z}_{\\ge 0},s \\in {\\bf Z}_{\\ge 1}$) case:\n\\begin{eqnarray}\n \\psi_{1}(u)&=&\\phi(u-2)\\phi(u-2s+2r-1), \\nonumber \\\\ \n \\psi_{a}(u)&=&\\phi(u)\\phi(u-2s+2r-1) \n  \\quad {\\rm for} \\quad \n   2 \\preceq a \\preceq \\overline{2}, \\nonumber \\\\\n \\psi_{\\overline{1}}(u)&=&\\phi(u)\\phi(u-2s+2r+1)\n   \\label{psi}.\n\\end{eqnarray}\n$D(r|s)$ ($r \\in {\\bf Z}_{\\ge 2},s \\in {\\bf Z}_{\\ge 1}$) case:\n\\begin{eqnarray}\n \\psi_{1}(u)&=&\\phi(u-2)\\phi(u-2s+2r-2), \\nonumber \\\\ \n \\psi_{a}(u)&=&\\phi(u)\\phi(u-2s+2r-2) \n \\quad {\\rm for} \\quad\n   2 \\preceq a \\preceq \\overline{2}, \\nonumber \\\\\n \\psi_{\\overline{1}}(u)&=&\\phi(u)\\phi(u-2s+2r) .\n   \\label{psi-d}\n\\end{eqnarray}\nHere \n\\begin{eqnarray}\n \\phi(u)=\\prod_{j=1}^{N}\\Phi(u-w_{j}).\n\\end{eqnarray}\nUnder the BAEs (\\ref{BAE})-(\\ref{BAE4}),\n we have: \n \\footnote{\n Here $Res_{u=a}f(u)$ denotes the residue of a function $f(u)$ at $u=a $.\n }\\\\  \n $B(r|s)$ ($r,s \\in {\\bf Z}_{\\ge 1}$) case:\n\\begin{eqnarray}\n\\hspace{-45pt}&& Res_{u=d+u_{k}^{(d)}}\n (\\framebox{$d$}_{u}+\\framebox{$d+1$}_{u})=0 \n  \\quad {\\rm for} \\quad 1\\le d \\le s-1 , \\label{res1} \\\\\n\\hspace{-45pt}&& Res_{u=s+u_{k}^{(s)}}\n(\\framebox{$s$}_{u}-\\framebox{$s+1$}_{u})=0  ,\n      \\\\\n\\hspace{-45pt}&& Res_{u=2s-d+u_{k}^{(d)}}\n (\\framebox{$d$}_{u}+\\framebox{$d+1$}_{u})=0 \n   \\quad {\\rm for} \\quad s+1\\le d \\le s+r-1 , \\label{res3} \\\\\n\\hspace{-45pt}&& Res_{u=s-r+u_{k}^{(s+r)}}\n(\\framebox{$s+r$}_{u}+\\framebox{$0$}_{u})=0  ,\n      \\\\\n\\hspace{-45pt}&& Res_{u=s-r+1+u_{k}^{(s+r)}}\n(\\framebox{$0$}_{u}+\\framebox{$\\overline{s+r}$}_{u})=0  ,\n      \\\\\n\\hspace{-45pt}&& Res_{u=d-2r+1+u_{k}^{(d)}}\n   (\\framebox{$\\overline{d+1}$}_{u}+\\framebox{$\\overline{d}$}_{u})=0 \n   \\quad {\\rm for} \\quad s+1 \\le d \\le s+r-1  ,\\label{res6} \\\\\n\\hspace{-45pt}&& Res_{u=s-2r+1+u_{k}^{(s)}}\n   (\\framebox{$\\overline{s+1}$}_{u}-\\framebox{$\\overline{s}$}_{u})=0,\n    \\\\\n\\hspace{-45pt}&& Res_{u=-d+2s-2r+1+u_{k}^{(d)}}\n   (\\framebox{$\\overline{d+1}$}_{u}+\\framebox{$\\overline{d}$}_{u})=0 \n \\quad {\\rm for} \\quad 1 \\le d \\le s-1  .  \\label{res8} \n\\end{eqnarray}\n$B(0|s)$ ($s \\in {\\bf Z}_{\\ge 1}$) case:\n\\begin{eqnarray}\n\\hspace{-45pt}&& Res_{u=d+u_{k}^{(d)}}\n (\\framebox{$d$}_{u}+\\framebox{$d+1$}_{u})=0 \n  \\quad {\\rm for} \\quad 1\\le d \\le s-1 , \\label{res1-b0s} \\\\\n\\hspace{-45pt}&& Res_{u=s+u_{k}^{(s)}}\n(\\framebox{$s$}_{u}-\\framebox{$0$}_{u})=0  ,\n \\\\\n \\hspace{-45pt}&& Res_{u=s+1+u_{k}^{(s)}}\n   (\\framebox{$0$}_{u}-\\framebox{$\\overline{s}$}_{u})=0,\n    \\\\\n\\hspace{-45pt}&& Res_{u=-d+2s+1+u_{k}^{(d)}}\n   (\\framebox{$\\overline{d+1}$}_{u}+\\framebox{$\\overline{d}$}_{u})=0 \n\\quad {\\rm for} \\quad 1 \\le d \\le s-1  .  \\label{res4-b0s} \n\\end{eqnarray}\n$D(r|s)$ ($r \\in {\\bf Z}_{\\ge 2},s \\in {\\bf Z}_{\\ge 1}$) case: \n\\begin{eqnarray}\n\\hspace{-45pt}&& Res_{u=d+u_{k}^{(d)}}\n (\\framebox{$d$}_{u}+\\framebox{$d+1$}_{u})=0 \n  \\quad {\\rm for} \\quad 1\\le d \\le s-1 , \\label{res1-d} \\\\\n\\hspace{-45pt}&& Res_{u=s+u_{k}^{(s)}}\n(\\framebox{$s$}_{u}-\\framebox{$s+1$}_{u})=0  ,\n      \\\\\n\\hspace{-45pt}&& Res_{u=2s-d+u_{k}^{(d)}}\n (\\framebox{$d$}_{u}+\\framebox{$d+1$}_{u})=0 \n  \\quad {\\rm for} \\quad s+1\\le d \\le s+r-1 , \\label{res3-d} \\\\\n\\hspace{-45pt}&& Res_{u=s-r+1+u_{k}^{(s+r)}}\n(\\framebox{$s+r-1$}_{u}+\\framebox{$\\overline{s+r}$}_{u})=0  ,\n      \\\\\n\\hspace{-45pt}&& Res_{u=s-r+1+u_{k}^{(s+r)}}\n(\\framebox{$s+r$}_{u}+\\framebox{$\\overline{s+r-1}$}_{u})=0  ,\n      \\\\\n\\hspace{-45pt}&& Res_{u=d-2r+2+u_{k}^{(d)}}\n   (\\framebox{$\\overline{d+1}$}_{u}+\\framebox{$\\overline{d}$}_{u})=0 \n \\quad {\\rm for} \\quad s+1 \\le d \\le s+r-1  ,\\label{res6-8} \\\\\n\\hspace{-45pt}&& Res_{u=s-2r+2+u_{k}^{(s)}}\n   (\\framebox{$\\overline{s+1}$}_{u}-\\framebox{$\\overline{s}$}_{u})=0,\n    \\\\\n\\hspace{-45pt}&& Res_{u=-d+2s-2r+2+u_{k}^{(d)}}\n   (\\framebox{$\\overline{d+1}$}_{u}+\\framebox{$\\overline{d}$}_{u})=0 \n    \\quad {\\rm for} \\quad 1 \\le d \\le s-1  .  \\label{res8-d} \n\\end{eqnarray}\nWe assign coordinates $(i,j)\\in {\\bf Z}^{2}$ \non the skew-Young  superdiagram $\\lambda \\subset \\mu$ \nsuch that the row index $i$ increases as we go downwards and the column \nindex $j$ increases as we go from the left to the right and that \n$(1,1)$ is on the top left corner of $\\mu$.\nWe define an admissible tableau $b$ \non the skew-Young superdiagram \n$\\lambda \\subset \\mu$ as a set of elements $b(i,j)\\in J$ \n labeled by the coordinates \n$(i,j)$ mentioned above, with the following rule. \\\\ \nThe admissible condition for $B(r|s)$ \n($r \\in {\\bf Z}_{\\ge 0}$, $s \\in {\\bf Z}_{\\ge 1}$): \n\\begin{enumerate}\n\\item\n\\begin{eqnarray*}\n  b(i,j) \\preceq b(i,j+1),   \n\\end{eqnarray*}\n\\item \n\\begin{eqnarray*}\n  b(i,j) \\preceq b(i+1,j), \n\\end{eqnarray*}\n\\item \n\\begin{eqnarray*} \n  b(i,j) \\prec b(i+1,j) \\quad {\\rm if}\\quad  b(i,j) \n \\in  J_{+} \\setminus \\{0 \\},  \n\\end{eqnarray*} \n\\item \n\\begin{eqnarray*}\n  b(i,j) \\prec b(i,j+1)  \\quad {\\rm if} \\quad  b(i,j) \n \\in  J_{-} \\cup \\{0 \\}.  \n\\end{eqnarray*}\n\\end{enumerate}\nThe admissible condition for $D(r|s)$ \n($r \\in {\\bf Z}_{\\ge 2}$, $s \\in {\\bf Z}_{\\ge 1}$); \n$\\lambda=\\phi$; $\\mu=(1^{a})$:\n\\begin{enumerate}\n\\item \n\\begin{eqnarray*}\n b(k,1) \\preceq b(k+1,1) \\quad \n {\\rm if} \\quad b(k+1,1) \\in J_{-} \n\\end{eqnarray*}\n\n\\item \n\\begin{eqnarray*}   \nb(k,1) \\prec b(k+1,1) \\quad \n {\\rm if} \\quad b(k+1,1) \\in J_{+} \n\\end{eqnarray*}\nunless \n\\begin{eqnarray*}\n (b(k,1),b(k+1,1))=(\\overline{s+r},s+r) \\quad {\\rm or} \n \\quad (s+r,\\overline{s+r}). \n\\end{eqnarray*}\n\\end{enumerate}\nThe admissible condition for $D(r|s)$ \n($r \\in {\\bf Z}_{\\ge 2}$, $s \\in {\\bf Z}_{\\ge 1}$); \n$\\lambda=\\phi$; $\\mu=(m^{1})$:\n\\begin{enumerate}\n\\item \n\\begin{eqnarray*}   \nb(1,k) \\preceq b(1,k+1) \\quad \n {\\rm if} \\quad b(1,k+1) \\in J_{+},  \n\\end{eqnarray*}\n\n\\item  \n\\begin{eqnarray*}\n b(1,k) \\prec b(1,k+1) \\quad \n {\\rm if} \\quad b(1,k+1) \\in J_{-},  \n\\end{eqnarray*}\n\n\\item \n$s+r$ and $\\overline{s+r}$ do not appear simultaneously.\n\\end{enumerate}\nLet $B(\\lambda \\subset \\mu)$ be \nthe set of admissible tableaux\n\\footnote{\nIn contrast to $B(r|s)$ case, the \nadmissible condition for $D(r|s)$ \n case has non-local nature. This property makes it \n difficult to extend the \nadmissible condition for $D(r|s)$ \n to more general skew-Young (super) diagrams.\n} \n on $\\lambda \\subset \\mu$. \nWe shall present a function ${\\cal T}_{\\lambda \\subset \\mu}(u)$ \nwith a spectral parameter $u\\in {\\rm C}$ and \n skew-Young superdiagrams $\\lambda \\subset \\mu$, \n which is a candidate of a set of DVFs \n for various fusion types in the \n auxiliary spaces\n \\footnote{We assume that they are finite dimensional modules of \n quantum affine superalgebras (or super Yangians) \\cite{Y1,Y2}. \n Thus ${\\cal T}_{\\lambda \\subset \\mu}(u)$ is expected to be a \n  kind of a (super) character of such algebras. \n  At present, we can not justify this speculations mathematically \n  in general, \n since we luck systematic representation theory of \n such algebras. \n We hope that mathematically satisfactory \n account on our formulae appear after the development of \n representation theory in the future. }\n  of transfer matrices of \n $B(r|s)$ or $D(r|s)$ vertex models. \nFor the skew-Young (super) diagrams $\\lambda \\subset \\mu$, \ndefine ${\\cal T}_{\\lambda \\subset \\mu}(u)$ as follows\n\\begin{equation}\n {\\cal T}_{\\lambda \\subset \\mu}(u)=\n\\sum_{b \\in B(\\lambda \\subset \\mu)}\n\\prod_{(i,j) \\in (\\lambda \\subset \\mu)}\n(-1)^{p(b(i,j))}\n\\framebox{$b(i,j)$}_{u-\\mu_{1}+\\mu_{1}^{\\prime}-2i+2j}\t\n\\label{Tge1},\n\\end{equation}\nwhere the product is taken over the coordinates $(i,j)$ on\n $\\lambda \\subset \\mu$.\n\nWe can express ${\\cal T}_{\\lambda \\subset \\mu}(u)$ \nas determinants over matrices, whose matrix elements are \n${\\cal T}^{a}$ or ${\\cal T}_{m}$ \n\\footnote{${\\cal T}_{m}^{a}(u):={\\cal T}_{(m^{a})}(u)$; \n${\\cal T}_{m}(u):={\\cal T}_{m}^{1}(u)$; \n${\\cal T}^{a}(u):={\\cal T}_{1}^{a}(u)$; \n${\\cal T}_{m}^{0}(u)={\\cal T}_{0}^{a}(u)=1$ for \n$m,a\\in {\\bf Z}_{\\ge 0}$; \n${\\cal T}_{m}^{a}(u)=0$ if \n$m\\in {\\bf Z}_{< 0}$ or $a\\in {\\bf Z}_{< 0}$. \nSee also Appendix B.}\n(cf. \\cite{KOS,BR}). \\\\ \nFor $B(r|s)$ ($r \\in {\\bf Z}_{\\ge 0}$, $s \\in {\\bf Z}_{\\ge 1}$) case, we have \n\\begin{eqnarray}\n\\hspace{-30pt}\n{\\cal T}_{\\lambda \\subset \\mu}(u)&=&{\\rm det}_{1 \\le i,j \\le \\mu_{1}}\n    ({\\cal T}^{\\mu_{i}^{\\prime}-\\lambda_{j}^{\\prime}-i+j}\n    (u-\\mu_{1}+\\mu_{1}^{\\prime}-\\mu_{i}^{\\prime}-\\lambda_{j}^{\\prime}+i+j-1))\n\t\\label{Jacobi-Trudi1} \\\\ \n\\hspace{-30pt}\t\n &=& {\\rm det}_{1 \\le i,j \\le \\mu_{1}^{\\prime}}\n    ({\\cal T}_{\\mu_{j}-\\lambda_{i}+i-j}\n    (u-\\mu_{1}+\\mu_{1}^{\\prime}+\\mu_{j}+\\lambda_{i}-i-j+1))\t.\n\t\\label{Jacobi-Trudi2} \n\\end{eqnarray}\nFor $D(r|s)$ ($r \\in  {\\bf Z}_{\\ge 2}$, $s \\in {\\bf Z}_{\\ge 1}$) case, we have \n\\begin{eqnarray}\n\\hspace{-60pt}&& \n{\\cal T}_{m}(u)={\\rm det}_{1 \\le i,j \\le m}\n    ({\\cal T}^{1-i+j}\n    (u-m+i+j-1))\n\t\\label{Jacobi-Trudi}. \n\\end{eqnarray}\nNote that the function ${\\cal T}^{1}(u)={\\cal T}_{1}(u)$ \n coincides with  \nthe eigenvalue formula of a $B(r|s)$ or $D(r|s)$ vertex model \nby the algebraic Bethe ansatz\n \\cite{MR} after some redefinitions. \n\nWe remark that if $\\Phi (-u) = \\pm \\Phi (u)$, \n$\\framebox{$a$}_{u}$ is transformed to \n$ \\framebox{$\\overline{a}$}_{u}$\n\\footnote{Here we interpret $\\overline{0}$ as $0$.}\nunder the following transformation. \\\\ \n$B(r|s)$ ($r \\in {\\bf Z}_{\\ge 0}, s \\in {\\bf Z}_{\\ge 1} $) case: \n\\begin{eqnarray}\n&& u \\to -(u+2r-2s-1), \\nonumber \\\\ \n&& u_{j}^{(a)} \\to -u_{j}^{(a)}, \\label{cross1} \\\\ \n&& w_{j} \\to -w_{j}. \\nonumber \n\\end{eqnarray}\n$D(r|s)$ ($r \\in  {\\bf Z}_{\\ge 2}, s \\in {\\bf Z}_{\\ge 1} $) case:\n\\begin{eqnarray}\n&& u \\to -(u+2r-2s-2), \\nonumber \\\\ \n&& u_{j}^{(a)} \\to -u_{j}^{(a)}, \\label{cross2} \\\\ \n&& w_{j} \\to -w_{j}. \\nonumber \n\\end{eqnarray}\n${\\cal T}_{m}(u)$ and ${\\cal T}^{a}(u)$ \nare  invariant under the transformations (\\ref{cross1}) \nor (\\ref{cross2}). This invariance may be\n viewed as a kind of crossing symmetry.\n\nNow we shall present examples of (\\ref{Tge1})  \nfor  $B(2|1), J_{-}=\\{1,\\overline{1} \\},\nJ_{+}=\\{2,3,0,\\overline{3},\\overline{2} \\}$ case: \n\\begin{eqnarray}\n{\\cal T}^{1}(u) &=&\n  -\\begin{array}{|c|}\\hline \n     \t1  \\\\ \\hline\n  \\end{array}\n +\\begin{array}{|c|}\\hline \n  \t2  \\\\ \\hline\n  \\end{array}\n +\\begin{array}{|c|}\\hline \n  \t3  \\\\ \\hline\n  \\end{array}\n +\\begin{array}{|c|}\\hline \n  \t0  \\\\ \\hline\n  \\end{array}\n +\\begin{array}{|c|}\\hline \n  \t\\stackrel{\\ }{\\overline{3}}  \\\\ \\hline\n  \\end{array}\n +\\begin{array}{|c|}\\hline \n  \t\\stackrel{\\ }{\\overline{2}}  \\\\ \\hline\n  \\end{array}\n -\\begin{array}{|c|}\\hline \n  \t\\stackrel{\\ }{\\overline{1}}  \\\\ \\hline\n  \\end{array}\n\\nonumber \\\\ \n&=&\n- \\phi(-2 + u)\\phi(1 + u)\\frac{Q_{1}(1 + u)}{Q_{1}(-1 + u)} \n \\nonumber \\\\ \n  &+& \n  \\phi(u)\\phi(1 + u)\\frac{Q_{1}(1 + u)Q_{2}(-2 + u)}{Q_{1}(-1 + u)Q_{2}(u)}\n   \\nonumber \\\\ \n   &+& \n  \\phi(u)\\phi(1 + u)\\frac{Q_{1}(u)Q_{2}(3 + u)}{Q_{1}(2 + u)Q_{2}(1 + u)} \n  \\nonumber \\\\\n  &+& \n  \\phi(u)\\phi(1 + u)\\frac{Q_{2}(2 + u)Q_{3}(-1 + u)}{Q_{2}(u)Q_{3}(1 + u)} \n  \\nonumber \\\\\n  &+&\n  \\phi(u)\\phi(1 + u)\\frac{Q_{2}(-1 + u)Q_{3}(2 + u)}{Q_{2}(1 + u)Q_{3}(u)} \n  \\nonumber \\\\\n  &+& \n  \\phi(u)\\phi(1 + u)\\frac{Q_{3}(-1 + u)Q_{3}(2 + u)}{Q_{3}(u)Q_{3}(1 + u)}\n  \\nonumber \\\\ \n  &-& \n  \\phi(u)\\phi(3 + u)\\frac{Q_{1}(u)}{Q_{1}(2 + u)} \n, \n\\label{t1-ex}\n\\end{eqnarray}\n\n\\begin{eqnarray}\n{\\cal T}^{2}(u) &=&\n   \\begin{array}{|c|}\\hline \n     \t1  \\\\ \\hline\n     \t1  \\\\ \\hline \n   \\end{array}\n  -\\begin{array}{|c|}\\hline \n     \t1  \\\\ \\hline\n     \t2  \\\\ \\hline \n   \\end{array}\n  -\\begin{array}{|c|}\\hline \n     \t1  \\\\ \\hline\n     \t3  \\\\ \\hline \n   \\end{array}\n  -\\begin{array}{|c|}\\hline \n     \t1  \\\\ \\hline\n     \t0  \\\\ \\hline \n   \\end{array}\n  -\\begin{array}{|c|}\\hline \n     \t1  \\\\ \\hline\n     \t\\stackrel{\\ }{\\overline{3}}  \\\\ \\hline \n   \\end{array}\n  -\\begin{array}{|c|}\\hline \n     \t1  \\\\ \\hline\n     \t\\stackrel{\\ }{\\overline{2}}  \\\\ \\hline \n   \\end{array}\n  +\\begin{array}{|c|}\\hline \n     \t1  \\\\ \\hline\n     \t\\stackrel{\\ }{\\overline{1}}  \\\\ \\hline \n   \\end{array}\n  +\\begin{array}{|c|}\\hline \n     \t2  \\\\ \\hline\n     \t3  \\\\ \\hline \n   \\end{array}\n  +\\begin{array}{|c|}\\hline \n     \t2  \\\\ \\hline\n     \t0  \\\\ \\hline \n   \\end{array}\n  +\\begin{array}{|c|}\\hline \n     \t2  \\\\ \\hline\n     \t\\stackrel{\\ }{\\overline{3}} \\\\ \\hline \n   \\end{array}\n   \\nonumber \\\\\n  &+& \\begin{array}{|c|}\\hline \n     \t2  \\\\ \\hline\n     \t\\stackrel{\\ }{\\overline{2}}  \\\\ \\hline \n   \\end{array}\n  -\\begin{array}{|c|}\\hline \n     \t2  \\\\ \\hline\n     \t\\stackrel{\\ }{\\overline{1}}  \\\\ \\hline \n   \\end{array}\n   +\\begin{array}{|c|}\\hline \n     \t3  \\\\ \\hline\n     \t0  \\\\ \\hline \n   \\end{array}\n  +\\begin{array}{|c|}\\hline \n     \t3  \\\\ \\hline\n     \t\\stackrel{\\ }{\\overline{3}}  \\\\ \\hline \n   \\end{array}\n  +\\begin{array}{|c|}\\hline \n     \t3  \\\\ \\hline\n     \t\\stackrel{\\ }{\\overline{2}}  \\\\ \\hline \n   \\end{array}\n  -\\begin{array}{|c|}\\hline \n     \t3  \\\\ \\hline\n     \t\\stackrel{\\ }{\\overline{1}}  \\\\ \\hline \n   \\end{array}\n  +\\begin{array}{|c|}\\hline \n     \t0  \\\\ \\hline\n     \t0  \\\\ \\hline \n   \\end{array}\n  +\\begin{array}{|c|}\\hline \n     \t0  \\\\ \\hline\n     \t\\stackrel{\\ }{\\overline{3}}  \\\\ \\hline \n   \\end{array}\n   +\\begin{array}{|c|}\\hline \n     \t0  \\\\ \\hline\n     \t\\stackrel{\\ }{\\overline{2}}  \\\\ \\hline \n   \\end{array}\n   -\\begin{array}{|c|}\\hline \n     \t0  \\\\ \\hline\n     \t\\stackrel{\\ }{\\overline{1}}  \\\\ \\hline \n   \\end{array}\n   \\nonumber \\\\\n   &+&\\begin{array}{|c|}\\hline \n     \t\\stackrel{\\ }{\\overline{3}}  \\\\ \\hline\n     \t\\stackrel{\\ }{\\overline{2}}  \\\\ \\hline \n   \\end{array}\n  -\\begin{array}{|c|}\\hline \n     \t\\stackrel{\\ }{\\overline{3}}  \\\\ \\hline\n     \t\\stackrel{\\ }{\\overline{1}}  \\\\ \\hline \n   \\end{array}\n  -\\begin{array}{|c|}\\hline \n     \t\\stackrel{\\ }{\\overline{2}}  \\\\ \\hline\n     \t\\stackrel{\\ }{\\overline{1}}  \\\\ \\hline \n   \\end{array}  \n  +\\begin{array}{|c|}\\hline \n     \t\\stackrel{\\ }{\\overline{1}}  \\\\ \\hline\n     \t\\stackrel{\\ }{\\overline{1}}  \\\\ \\hline \n   \\end{array}\n   \\nonumber \\\\\n&=& \\phi(-1+ u)\\phi(2+ u) \n\\Bigg(\n\\phi(-3 + u)\\phi(u)\\frac{Q_{1}(2 + u)}{Q_{1}(-2 + u)} \\nonumber \\\\ &+& \n \\phi(-1 + u)\\phi(2 + u)\\frac{Q_{1}(-1 + u)Q_{1}(2 + u)}\n    {Q_{1}(u)Q_{1}(1 + u)} \\nonumber \\\\ &+& \n \\phi(1 + u)\\phi(4 + u)\\frac{Q_{1}(-1 + u)}{Q_{1}(3 + u)} \\nonumber \\\\ &-& \n  \\phi(-1 + u)\\phi(u)\\frac{Q_{1}(2 + u)Q_{2}(-3 + u)}\n    {Q_{1}(-2 + u)Q_{2}(-1 + u)} \\nonumber \\\\ &-& \n  \\phi(1 + u)\\phi(2 + u)\\frac{Q_{1}(-1 + u)Q_{1}(2 + u)\n      Q_{2}(-1 + u)}{Q_{1}(u)Q_{1}(1 + u)Q_{2}(1 + u)} \\nonumber \\\\ &-& \n  \\phi(-1 + u)\\phi(u)\\frac{Q_{1}(-1 + u)Q_{1}(2 + u)Q_{2}(2 + u)}\n    {Q_{1}(u)Q_{1}(1 + u)Q_{2}(u)} \\nonumber \\\\ &+& \n  \\phi(u)\\phi(1 + u)\\frac{Q_{1}(-1 + u)Q_{1}(2 + u)Q_{2}(-1 + u)\n      Q_{2}(2 + u)}{Q_{1}(u)Q_{1}(1 + u)Q_{2}(u)Q_{2}(1 + u)} \\nonumber \\\\ &-& \n  \\phi(1 + u)\\phi(2 + u)\\frac{Q_{1}(-1 + u)Q_{2}(4 + u)}\n    {Q_{1}(3 + u)Q_{2}(2 + u)} \\nonumber \\\\ &+& \n  \\phi(u)\\phi(1 + u)\\frac{Q_{1}(2 + u)Q_{3}(-2 + u)}\n    {Q_{1}(u)Q_{3}(u)} \\nonumber \\\\ \n    &-& \\phi(-1 + u)\\phi(u)\\frac{Q_{1}(2 + u)\n      Q_{2}(1 + u)Q_{3}(-2 + u)}{Q_{1}(u)Q_{2}(-1 + u)Q_{3}(u)}\n       \\nonumber \\\\ &-& \n  \\phi(-1 + u)\\phi(u)\\frac{Q_{1}(2 + u)Q_{2}(-2 + u)Q_{3}(1 + u)}\n    {Q_{1}(u)Q_{2}(u)Q_{3}(-1 + u)} \\nonumber \\\\ &+& \n  \\phi(u)\\phi(1 + u)\\frac{Q_{1}(2 + u)Q_{2}(-2 + u)Q_{2}(-1 + u)\n      Q_{3}(1 + u)}{Q_{1}(u)Q_{2}(u)Q_{2}(1 + u)Q_{3}(-1 + u)} \\nonumber \\\\ \n      &-& \n  \\phi(-1 + u)\\phi(u)\\frac{Q_{1}(2 + u)Q_{3}(-2 + u)Q_{3}(1 + u)}\n    {Q_{1}(u)Q_{3}(-1 + u)Q_{3}(u)} \\nonumber \\\\ &+& \n  \\phi(u)\\phi(1 + u)\\frac{Q_{1}(2 + u)Q_{2}(-1 + u)Q_{3}(-2 + u)\n      Q_{3}(1 + u)}{Q_{1}(u)Q_{2}(1 + u)Q_{3}(-1 + u)Q_{3}(u)} \\nonumber \\\\ &-& \n  \\phi(1 + u)\\phi(2 + u)\\frac{Q_{1}(-1 + u)Q_{2}(3 + u)Q_{3}(u)}\n    {Q_{1}(1 + u)Q_{2}(1 + u)Q_{3}(2 + u)} \\nonumber \\\\ &+& \n  \\phi(u)\\phi(1 + u)\\frac{Q_{1}(-1 + u)Q_{2}(2 + u)Q_{2}(3 + u)\n      Q_{3}(u)}{Q_{1}(1 + u)Q_{2}(u)Q_{2}(1 + u)Q_{3}(2 + u)} \\nonumber \\\\ &+& \n  \\phi(u)\\phi(1 + u)\\frac{Q_{2}(3 + u)Q_{3}(-2 + u)Q_{3}(1 + u)}\n    {Q_{2}(1 + u)Q_{3}(-1 + u)Q_{3}(2 + u)} \\nonumber \\\\ &+& \n  \\phi(u)\\phi(1 + u)\\frac{Q_{2}(-2 + u)Q_{2}(3 + u)Q_{3}(u)\n      Q_{3}(1 + u)}{Q_{2}(u)Q_{2}(1 + u)Q_{3}(-1 + u)Q_{3}(2 + u)}\n  \\nonumber \\\\ \n   &+& \\phi(u)\\phi(1 + u)\\frac{Q_{1}(-1 + u)Q_{3}(3 + u)}\n    {Q_{1}(1 + u)Q_{3}(1 + u)} \\nonumber \\\\ &-& \n  \\phi(1 + u)\\phi(2 + u)\\frac{Q_{1}(-1 + u)Q_{2}(u)Q_{3}(3 + u)}\n    {Q_{1}(1 + u)Q_{2}(2 + u)Q_{3}(1 + u)} \\nonumber \\\\ &+& \n  \\phi(u)\\phi(1 + u)\\frac{Q_{3}(-2 + u)Q_{3}(3 + u)}\n    {Q_{3}(-1 + u)Q_{3}(2 + u)} \\nonumber \\\\ &+& \n  \\phi(u)\\phi(1 + u)\\frac{Q_{2}(-2 + u)Q_{3}(u)Q_{3}(3 + u)}\n    {Q_{2}(u)Q_{3}(-1 + u)Q_{3}(2 + u)} \\nonumber \\\\ &-& \n  \\phi(1 + u)\\phi(2 + u)\\frac{Q_{1}(-1 + u)Q_{3}(u)Q_{3}(3 + u)}\n    {Q_{1}(1 + u)Q_{3}(1 + u)Q_{3}(2 + u)} \\nonumber \\\\ &+& \n  \\phi(u)\\phi(1 + u)\\frac{Q_{1}(-1 + u)Q_{2}(2 + u)Q_{3}(u)\n      Q_{3}(3 + u)}{Q_{1}(1 + u)Q_{2}(u)Q_{3}(1 + u)Q_{3}(2 + u)}\n      \\Bigg),\n     \\label{t2-ex}\n\\end{eqnarray}\n\\begin{eqnarray}\n{\\cal T}_{2}(u) &=&\n-\\begin{array}{|c|c|} \\hline \n    1 & 2 \\\\ \\hline \n\\end{array}\n-\n\\begin{array}{|c|c|}\\hline\n    1 & 3  \\\\ \\hline\n\\end{array}\n-\n\\begin{array}{|c|c|}\\hline\n    1 & 0  \\\\ \\hline\n\\end{array}\n-\n\\begin{array}{|c|c|}\\hline\n    1 & \\stackrel{\\ }{\\overline{3}}  \\\\ \\hline\n\\end{array}\n-\n\\begin{array}{|c|c|}\\hline\n    1 & \\stackrel{\\ }{\\overline{2}}  \\\\ \\hline\n\\end{array}\n+\n\\begin{array}{|c|c|}\\hline\n    1 & \\stackrel{\\ }{\\overline{1}}  \\\\ \\hline\n\\end{array} \n\\nonumber \\\\ \n&+&\n\\begin{array}{|c|c|}\\hline\n    2 & 2  \\\\ \\hline\n\\end{array}\n+\n\\begin{array}{|c|c|}\\hline\n    2 & 3  \\\\ \\hline\n\\end{array}\n+\n\\begin{array}{|c|c|}\\hline\n    2 & 0  \\\\ \\hline\n\\end{array}\n+\n\\begin{array}{|c|c|}\\hline\n    2 & \\stackrel{\\ }{\\overline{3}}  \\\\ \\hline\n\\end{array} \n+\n\\begin{array}{|c|c|}\\hline\n    2 & \\stackrel{\\ }{\\overline{2}}  \\\\ \\hline\n\\end{array}\n-\n\\begin{array}{|c|c|}\\hline\n    2 & \\stackrel{\\ }{\\overline{1}}  \\\\ \\hline\n\\end{array}\n\\nonumber \\\\ \n&+&\n\\begin{array}{|c|c|}\\hline\n    3 & 3  \\\\ \\hline\n\\end{array}\n+\n\\begin{array}{|c|c|}\\hline\n    3 & 0  \\\\ \\hline\n\\end{array}\n+\n\\begin{array}{|c|c|}\\hline\n    3 & \\stackrel{\\ }{\\overline{3}}  \\\\ \\hline\n\\end{array}\n+\n\\begin{array}{|c|c|}\\hline\n    3 & \\stackrel{\\ }{\\overline{2}}  \\\\ \\hline\n\\end{array}\n-\n\\begin{array}{|c|c|}\\hline\n    3 & \\stackrel{\\ }{\\overline{1}}  \\\\ \\hline\n\\end{array}\n+\n\\begin{array}{|c|c|}\\hline\n    0 & \\stackrel{\\ }{\\overline{3}}  \\\\ \\hline\n\\end{array}\n\\nonumber \\\\ \n&+&\n\\begin{array}{|c|c|}\\hline\n    0 & \\stackrel{\\ }{\\overline{2}}  \\\\ \\hline\n\\end{array}\n-\n\\begin{array}{|c|c|}\\hline\n    0 & \\stackrel{\\ }{\\overline{1}}  \\\\ \\hline\n\\end{array}\n+\n\\begin{array}{|c|c|}\\hline\n    \\stackrel{\\ }{\\overline{3}} & \n    \\stackrel{\\ }{\\overline{3}}  \\\\ \\hline\n\\end{array}\n+\n\\begin{array}{|c|c|}\\hline\n    \\stackrel{\\ }{\\overline{3}} & \n    \\stackrel{\\ }{\\overline{2}}  \\\\ \\hline\n\\end{array}\n-\n\\begin{array}{|c|c|}\\hline\n    \\stackrel{\\ }{\\overline{3}} & \n    \\stackrel{\\ }{\\overline{1}}  \\\\ \\hline\n\\end{array}\n+\n\\begin{array}{|c|c|}\\hline\n    \\stackrel{\\ }{\\overline{2}} & \n    \\stackrel{\\ }{\\overline{2}}  \\\\ \\hline\n\\end{array}\n-\n\\begin{array}{|c|c|}\\hline\n    \\stackrel{\\ }{\\overline{2}} & \n    \\stackrel{\\ }{\\overline{1}}  \\\\ \\hline\n\\end{array}\n\\nonumber \\\\ \n&=& \\phi(u)\\phi(1+ u)\n\\Bigg(\n\\phi(-3 + u)\\phi(4 + u)\\frac{Q_{1}(u)Q_{1}(1 + u)}{Q_{1}(-2 + u)Q_{1}(3 + u)}\n \\nonumber \\\\ \n &-& \n  \\phi(-1 + u)\\phi(4 + u)\\frac{Q_{1}(u)Q_{1}(1 + u)Q_{2}(-3 + u)}\n    {Q_{1}(-2 + u)Q_{1}(3 + u)Q_{2}(-1 + u)} \\nonumber \\\\ &+& \n  \\phi(-1 + u)\\phi(2 + u)\\frac{Q_{1}(2 + u)Q_{2}(-3 + u)}\n    {Q_{1}(-2 + u)Q_{2}(1 + u)} \\nonumber \\\\ &-& \n  \\phi(-3 + u)\\phi(2 + u)\\frac{Q_{1}(2 + u)Q_{2}(-1 + u)}\n    {Q_{1}(-2 + u)Q_{2}(1 + u)} \\nonumber \\\\ &-& \n  \\phi(-1 + u)\\phi(4 + u)\\frac{Q_{1}(-1 + u)Q_{2}(2 + u)}\n    {Q_{1}(3 + u)Q_{2}(u)} \\nonumber \\\\ &+& \n    \\phi(-1 + u)\\phi(2 + u)\\frac{Q_{1}(-1 + u)\n      Q_{2}(4 + u)}{Q_{1}(3 + u)Q_{2}(u)} \\nonumber \\\\ &-& \n  \\phi(-3 + u)\\phi(2 + u)\\frac{Q_{1}(u)Q_{1}(1 + u)Q_{2}(4 + u)}\n    {Q_{1}(-2 + u)Q_{1}(3 + u)Q_{2}(2 + u)} \\nonumber \\\\ &+& \n  \\phi(-1 + u)\\phi(2 + u)\\frac{Q_{1}(u)Q_{1}(1 + u)Q_{2}(-3 + u)Q_{2}(4 + u)}\n    {Q_{1}(-2 + u)Q_{1}(3 + u)Q_{2}(-1 + u)Q_{2}(2 + u)} \\nonumber \\\\ &-& \n  \\phi(-1 + u)\\phi(4 + u)\\frac{Q_{1}(1 + u)Q_{2}(1 + u)Q_{3}(-2 + u)}\n    {Q_{1}(3 + u)Q_{2}(-1 + u)Q_{3}(u)} \\nonumber \\\\ &+& \n  \\phi(-1 + u)\\phi(2 + u)\\frac{Q_{1}(1 + u)Q_{2}(1 + u)Q_{2}(4 + u)\n      Q_{3}(-2 + u)}{Q_{1}(3 + u)Q_{2}(-1 + u)Q_{2}(2 + u)Q_{3}(u)} \\nonumber \\\\ &-& \n  \\phi(-1 + u)\\phi(4 + u)\\frac{Q_{1}(1 + u)Q_{2}(-2 + u)Q_{3}(1 + u)}\n    {Q_{1}(3 + u)Q_{2}(u)Q_{3}(-1 + u)} \\nonumber \\\\ &+& \n  \\phi(-1 + u)\\phi(2 + u)\\frac{Q_{1}(1 + u)Q_{2}(-2 + u)Q_{2}(4 + u)\n      Q_{3}(1 + u)}{Q_{1}(3 + u)Q_{2}(u)Q_{2}(2 + u)Q_{3}(-1 + u)} \\nonumber \\\\ &-& \n  \\phi(-1 + u)\\phi(4 + u)\\frac{Q_{1}(1 + u)Q_{3}(-2 + u)Q_{3}(1 + u)}\n    {Q_{1}(3 + u)Q_{3}(-1 + u)Q_{3}(u)} \\nonumber \\\\ &+& \n  \\phi(-1 + u)\\phi(2 + u)\\frac{Q_{1}(1 + u)Q_{2}(4 + u)Q_{3}(-2 + u)\n      Q_{3}(1 + u)}{Q_{1}(3 + u)Q_{2}(2 + u)Q_{3}(-1 + u)Q_{3}(u)} \\nonumber \\\\ &+& \n  \\phi(-1 + u)\\phi(2 + u)\\frac{Q_{2}(3 + u)Q_{3}(-2 + u)}\n    {Q_{2}(-1 + u)Q_{3}(2 + u)} \\nonumber \\\\ &-& \n  \\phi(-3 + u)\\phi(2 + u)\\frac{Q_{1}(u)Q_{2}(3 + u)Q_{3}(u)}\n    {Q_{1}(-2 + u)Q_{2}(1 + u)Q_{3}(2 + u)} \\nonumber \\\\ &+& \n  \\phi(-1 + u)\\phi(2 + u)\\frac{Q_{1}(u)Q_{2}(-3 + u)Q_{2}(3 + u)Q_{3}(u)}\n    {Q_{1}(-2 + u)Q_{2}(-1 + u)Q_{2}(1 + u)Q_{3}(2 + u)} \\nonumber \\\\ &+& \n  \\phi(-1 + u)\\phi(2 + u)\\frac{Q_{2}(-2 + u)Q_{3}(3 + u)}\n    {Q_{2}(2 + u)Q_{3}(-1 + u)} \\nonumber \\\\ &+& \n  \\phi(-1 + u)\\phi(2 + u)\\frac{Q_{2}(u)Q_{3}(-2 + u)Q_{3}(3 + u)}\n    {Q_{2}(2 + u)Q_{3}(-1 + u)Q_{3}(u)} \\nonumber \\\\ &-& \n  \\phi(-3 + u)\\phi(2 + u)\\frac{Q_{1}(u)Q_{2}(u)Q_{3}(3 + u)}\n    {Q_{1}(-2 + u)Q_{2}(2 + u)Q_{3}(1 + u)} \\nonumber \\\\ &+& \n  \\phi(-1 + u)\\phi(2 + u)\\frac{Q_{1}(u)Q_{2}(-3 + u)Q_{2}(u)Q_{3}(3 + u)}\n    {Q_{1}(-2 + u)Q_{2}(-1 + u)Q_{2}(2 + u)Q_{3}(1 + u)} \\nonumber \\\\ &+& \n  \\phi(-1 + u)\\phi(2 + u)\\frac{Q_{2}(u)Q_{2}(1 + u)Q_{3}(-2 + u)Q_{3}(3 + u)}\n    {Q_{2}(-1 + u)Q_{2}(2 + u)Q_{3}(u)Q_{3}(1 + u)} \\nonumber \\\\ &+& \n  \\phi(-1 + u)\\phi(2 + u)\\frac{Q_{2}(1 + u)Q_{3}(-2 + u)Q_{3}(3 + u)}\n    {Q_{2}(-1 + u)Q_{3}(1 + u)Q_{3}(2 + u)} \\nonumber \\\\ &-& \n  \\phi(-3 + u)\\phi(2 + u)\\frac{Q_{1}(u)Q_{3}(u)Q_{3}(3 + u)}\n    {Q_{1}(-2 + u)Q_{3}(1 + u)Q_{3}(2 + u)} \n    \\label{t21-ex}  \\\\ &+& \n  \\phi(-1 + u)\\phi(2 + u)\\frac{Q_{1}(u)Q_{2}(-3 + u)Q_{3}(u)Q_{3}(3 + u)}\n    {Q_{1}(-2 + u)Q_{2}(-1 + u)Q_{3}(1 + u)Q_{3}(2 + u)} \n\\Bigg)\n.  \\nonumber\n\\end{eqnarray}\nThanks to Theorem \\ref{th-tate} (see later) and \nthe relation (\\ref{Jacobi-Trudi1}),\n these DVFs are free of \npoles under the following BAE: \n\\begin{eqnarray}\n && \\frac{\\phi(u_{k}^{(1)}-1)}{\\phi(u_{k}^{(1)}+1)}\n =\\frac{Q_{2}(u_{k}^{(1)}-1)}\n       {Q_{2}(u_{k}^{(1)}+1)} \\quad {\\rm for } \\quad \n  1 \\le k \\le N_{1}, \\nonumber \\\\ \n && -1=\\frac{Q_{1}(u_{k}^{(2)}-1)\n       Q_{2}(u_{k}^{(2)}+2)Q_{3}(u_{k}^{(2)}-1)}\n         {Q_{1}(u_{k}^{(2)}+1)\n         Q_{2}(u_{k}^{(2)}-2)Q_{3}(u_{k}^{(2)}+1)} \n         \\quad {\\rm for } \\quad \n  1 \\le k \\le N_{2}, \\nonumber \\\\ \n && -1=\\frac{Q_{2}(u_{k}^{(3)}-1)Q_{3}(u_{k}^{(3)}+1)}\n         {Q_{2}(u_{k}^{(3)}+1)Q_{3}(u_{k}^{(3)}-1)},\n          \\quad {\\rm for } \\quad  \n    1 \\le k \\le N_{3}.\n\\label{BAE2-b21}\n\\end{eqnarray}\nNote that DVFs have so called \n{\\em Bethe-strap} structures \n\\cite{KS1,S2}, which bear resemblance to \nweight space diagrams.\n We have observed for many examples \n that ${\\cal T}_{\\lambda \\subset \\mu}(u)$ coincides with \n the Bethe-strap of the minimal connected component (cf \\cite{K2}) \n which\n  include the top term \\cite{KS1,KOS} \n as the examples in \n Figure \\ref{best1}, Figure \\ref{best2} and Figure \\ref{best3}\n\\footnote{Recently we have found curious terms \n (pseudo-top terms) in many Bethe-straps (cf. \\cite{T4}). \n However we have confirmed for several examples the fact that \n the pseudo-top terms do not influence on  \n connectivity of the Bethe straps (cf \\cite{KS1,KOS,K2}) \n in the whole.}.  \n The top term of ${\\cal T}_{\\lambda \\subset \\mu}(u)$ carries \n a $B(r|s)$ or $D(r|s)$ weight. \n For example, for $B(r|s)$, $\\lambda =\\phi$, $\\mu_{r+1}\\le s$ \n case, the term corresponding to the tableau \n\\begin{equation}\n b(i,j)=\n \\left\\{\n  \\begin{array}{llll}\n  j & {\\rm for } & 1 \\le i \\le \\mu_{j}^{\\prime} & 1 \\le j \\le s \\\\\n  i+s & {\\rm for } & 1 \\le i \\le \\mu_{j}^{\\prime} & s+1 \\le j \\le \\mu_{1}\n  \\end{array}\n \\right.\n\\end{equation}\ncarries the \n$B(r|s)$ weight with the Kac-Dynkin label (\\ref{Kac-Dynkin}) \nor (\\ref{Kac-Dynkin-b0s}). \n\nThe top term\n\\footnote{Here we omit the vacuum part.} \n\\cite{KS1} of the DVF (\\ref{Tge1}) for \n $D(r|s)$, $\\lambda =\\phi, \\mu=(1^{a})$ will be \n\\begin{eqnarray}\n \\left.\n  (-1)^{a}\n  \\begin{array}{|c|}\\hline \n     \t1  \\\\ \\hline\n        1  \\\\ \\hline\n     \t\\vdots \\\\ \\hline \n     \t1  \\\\ \\hline\n  \\end{array}\n  \\, \n \\right\\}\n \\! \n {\\tiny a} \\; \n  =(-1)^{a}\n\\frac{Q_{1}(u+a)}{Q_{1}(u-a)},\n\\label{top-tate}\n\\end{eqnarray}\nwhich carries the $D(r|s)$ weight with the Kac-Dynkin label \n in (\\ref{KD-tate}). \nThe top term \n\\footnote{Here we omit the vacuum part.} \n \\cite{KS1} of the DVF (\\ref{Tge1}) for \n $D(r|s)$, $\\lambda =\\phi, \\mu=(m^{1})$ will be \n\\begin{eqnarray}\n&&\n(-1)^{m}\n\\underbrace{\n \\begin{array}{|c|c|c|c|}\\hline \n     \t1 & 2 & \\cdots & m \\\\ \\hline\n  \\end{array}\n }_{m}\n=(-1)^{m} \\frac{Q_{m}(u+1)}{Q_{m}(u-1)} \n\\qquad {\\rm if} \\quad 1 \\le m \\le s,  \n\\nonumber \\\\ \n&& \n(-1)^{s}\n\\underbrace{\n \\begin{array}{|c|c|c|c|c|c|c|}\\hline \n     \t1 & 2 & \\cdots & s & s+1 & \\cdots & s+1 \\\\ \\hline\n  \\end{array}\n }_{m} \\label{top-yoko} \n  \\\\ \n&& \\quad =(-1)^{s} \\frac{Q_{s}(u+m-s+1)Q_{s+1}(u-m+s)}\n               {Q_{s}(u-m+s-1)Q_{s+1}(u+m-s)} \\nonumber \\\\ \n&& \\hspace{130pt} {\\rm if} \\quad r \\ge 3 \\quad \n{\\rm and} \\quad m \\ge s+1, \\nonumber \\\\ \n&& \\quad =(-1)^{s} \\frac{Q_{s}(u+m-s+1)Q_{s+1}(u-m+s)Q_{s+2}(u-m+s)}\n               {Q_{s}(u-m+s-1)Q_{s+1}(u+m-s)Q_{s+2}(u+m-s)} \n \\nonumber \\\\ \n&& \\hspace{130pt} {\\rm if} \\quad r =2 \\quad \n{\\rm and} \\quad m \\ge s+1, \\nonumber\n\\end{eqnarray}\nwhich carries the $D(r|s)$ weight    \nwith the Kac-Dynkin label in (\\ref{KD-yoko}). \n \n{\\em Remark}: \nThere is a supposition (cf \\cite{KOS,K2}) that the  \nauxiliary space of a transfer matrix  \nis a irreducible one as a representation space of the Yangian \n (or quantum affine algebra) \n if the Bethe strap of the DVF is connected\n\\footnote{Here the word \"connected\" means that any terms in DVF \n are connected directly (or indirectly) each other \nby the arrows like graphs in Figures \\ref{best1}-\\ref{best3}.}\n  in the whole. \n Then a natural question arise: \n \"Is the Bethe strap of \n  ${\\cal T}_{\\lambda \\subset \\mu}(u)$ (\\ref{Tge1}) always \n connected in the whole ?\"  The answer is no. \n In fact for $D(r|s)$ case, \n the Bethe strap of ${\\cal T}^{a}(u)$  \n is not connected \n if $0\\le r-s-1 \\le a \\le 2(r-s-1)$.\n  So it is desirable to \n extract the minimal connected component of the Bethe strap \n which contains the top term (\\ref{top-tate})  \n from ${\\cal T}^{a}(u)$.  \nA candidate is as follows: \n\\begin{eqnarray}\n {\\cal T}^{a}(u)-h^{a}(u){\\cal T}^{-a+2(r-s-1)}(u), \n\\end{eqnarray}\nwhere $h^{a}(u)=\\prod_{j=1}^{a+1-r+s}\n\\psi_{1}(u+a-2j+1)\\psi_{\\overline{1}}(u-a+2j-1)$. \n \nFor example, for $D(3|1)$ case, ${\\cal T}^{2}(u)$ consists \nof 31 terms and they divide into 30 terms whose Bethe strap \nis connected in the whole\n\\footnote{In this case, \n$\n-\\begin{array}{|c|}\\hline \n    2 \\\\ \\hline \n    \\overline{1} \\\\ \\hline \n\\end{array}\n$ \nis a pseudo-top term (cf \\cite{T4}).}\n and 1 isolated term \n$\nh^{2}(u)=\n\\begin{array}{|c|}\\hline \n    1 \\\\ \\hline \n    \\overline{1} \\\\ \\hline \n\\end{array}\n=\\{\\phi(u-1)\\phi(u+3)\\}^{2}.\n$ Thus  Bethe strap of ${\\cal T}^{2}(u)-h^{2}(u)$ \nis connected in the whole. \n On the other hand for $D(2|2)$ case, ${\\cal T}^{2}(u)$ \ndoes not have such an isolated term and that the Bethe strap \nis connected in the whole \n(in this case, ${\\cal T}^{2}(u)$ has 33 terms).  \nSo far this kind of an isolated term is peculiar to $D(r|s)$ case. \nIn fact, we have never yet observed such an isolated term \nin $B(r|s)$ case.\nSimilarly, \n the Bethe strap of ${\\cal T}_{m}(u)$  for $D(r|s)$ \n seems not to be connected \n if $0\\le s-r+1 \\le m \\le 2(s-r+1)$. \n A candidate for the minimal connected component of the Bethe strap \n which contains the top term (\\ref{top-yoko}) is \n\\begin{eqnarray}\n {\\cal T}_{m}(u)-h_{m}(u){\\cal T}_{-m+2(s-r+1)}(u), \n\\end{eqnarray}\nwhere $h_{m}(u)=\\prod_{j=1}^{m+r-s-1}\n\\psi_{j}(u-m+2j-1)\\psi_{\\overline{j}}(u+m-2j+1)$.\n \\rule{5pt}{10pt} \\\\ \n A remarkable resemblance between \n Bethe-straps for vector representations and crystal graphs \n \\cite{KN1,KN2} was pointed out in \\cite{KS1}. \n Whether such resemblance holds true for the Lie superalgebras \n in general or not will be an interesting question.   \nThere is a remarkable coincidence between \n  currents of deformed Virasoro algebra  and  DVFs \\cite{FR}. \n  Whether such coincidence holds true for the Lie superalgebras \n in general or not will be another interesting question. \n\n\\begin{figure}\n    \\setlength{\\unitlength}{1.2pt}\n    \\begin{center}\n    \\begin{picture}(250,40) \n      \\put(-8,3){$-$}\n      \\put(0,0){\\line(1,0){10}}\n      \\put(10,0){\\line(0,1){10}}\n      \\put(10,10){\\line(-1,0){10}}\n      \\put(0,10){\\line(0,-1){10}}\n      \\put(3,3){\\scriptsize{$1$}}\n    %\n      \\put(12,5){\\vector(1,0){20}}\n      \\put(15,9){\\scriptsize{$(1,1)$}}\n   \n      \\put(33,3){$$}\n      \\put(40,0){\\line(1,0){10}}\n      \\put(50,0){\\line(0,1){10}}\n      \\put(50,10){\\line(-1,0){10}}\n      \\put(40,10){\\line(0,-1){10}}\n      \\put(43,3){\\scriptsize{$2$}}\n    %\n      \\put(52,5){\\vector(1,0){20}}\n      \\put(55,9){\\scriptsize{$(2,0)$}}\n   \n      \\put(73,3){$$}\n      \\put(80,0){\\line(1,0){10}}\n      \\put(90,0){\\line(0,1){10}}\n      \\put(90,10){\\line(-1,0){10}}\n      \\put(80,10){\\line(0,-1){10}}\n      \\put(83,3){\\scriptsize{$3$}}\n    %\n      \\put(92,5){\\vector(1,0){20}}\n      \\put(95,9){\\scriptsize{$(3,-1)$}}\n   \n      \\put(113,3){$$}\n      \\put(120,0){\\line(1,0){10}}\n      \\put(130,0){\\line(0,1){10}}\n      \\put(130,10){\\line(-1,0){10}}\n      \\put(120,10){\\line(0,-1){10}}\n      \\put(123,3){\\scriptsize{$0$}}\n    %\n      \\put(132,5){\\vector(1,0){20}}\n      \\put(135,9){\\scriptsize{$(3,0)$}}\n   \n      \\put(153,3){$$}\n      \\put(160,0){\\line(1,0){10}}\n      \\put(170,0){\\line(0,1){10}}\n      \\put(170,10){\\line(-1,0){10}}\n      \\put(160,10){\\line(0,-1){10}}\n      \\put(163,3){\\scriptsize{$\\overline{3}$}}\n    %\n      \\put(172,5){\\vector(1,0){20}}\n      \\put(175,9){\\scriptsize{$(2,-1)$}}\n   \n      \\put(197,0){\\line(1,0){10}}\n      \\put(207,0){\\line(0,1){10}}\n      \\put(207,10){\\line(-1,0){10}}\n      \\put(197,10){\\line(0,-1){10}}\n      \\put(200,3){\\scriptsize{$\\overline{2}$}}\n    %\n      \\put(212,5){\\vector(1,0){20}}\n      \\put(215,9){\\scriptsize{$(1,-2)$}}\n   \n      \\put(232,3){$-$}\n      \\put(240,0){\\line(1,0){10}}\n      \\put(250,0){\\line(0,1){10}}\n      \\put(250,10){\\line(-1,0){10}}\n      \\put(240,10){\\line(0,-1){10}}\n      \\put(243,3){\\scriptsize{$\\overline{1}$}}\n  \\end{picture}\n  \\end{center}\n  \\caption{The  Bethe-strap structure of  \n   ${\\cal T}^{1}(u)$  (\\ref{t1-ex}) for \n   $B(2|1)=osp(5|2)$:  \n The pair $(a,b)$ denotes the common pole $u_{k}^{(a)}+b$ of the pair   \n of the tableaux connected by the arrow.   \n This common pole vanishes under the BAE (\\ref{BAE2-b21}).\n The leftmost tableau corresponds to the \n \\symbol{96}highest weight \\symbol{39}, \n which is called the {\\em top term}. \n Such a  correspondence between certain term in the DVF and a highest \n weight (to be more precise, a kind of Drinfel'd polynomial \n (cf \\cite{D,KOS}))\n  may be called {\\em top term hypothesis} \\cite{KS1,KOS}.}\n  \\label{best1}\n\\end{figure}\n\\begin{figure}\n    \\setlength{\\unitlength}{1pt}\n    \\begin{center}\n    \\begin{picture}(250,470) \n      \\put(120,0){\\line(1,0){10}}\n      \\put(130,0){\\line(0,1){20}}\n      \\put(130,20){\\line(-1,0){10}}\n      \\put(120,20){\\line(0,-1){20}}\n      \\put(120,10){\\line(1,0){10}}\n      \\put(123,12){\\scriptsize{$\\overline{1}$}}\n      \\put(123,2){\\scriptsize{$\\overline{1}$}}\n      \\put(114,9){$$}\n     \n      \\put(120,40){\\line(1,0){10}}\n      \\put(130,40){\\line(0,1){20}}\n      \\put(130,60){\\line(-1,0){10}}\n      \\put(120,60){\\line(0,-1){20}}\n      \\put(120,50){\\line(1,0){10}}\n      \\put(123,52){\\scriptsize{$\\overline{2}$}}\n      \\put(123,42){\\scriptsize{$\\overline{1}$}}\n      \\put(110,47){$-$}\n     \n      \\put(120,80){\\line(1,0){10}}\n      \\put(130,80){\\line(0,1){20}}\n      \\put(130,100){\\line(-1,0){10}}\n      \\put(120,100){\\line(0,-1){20}}\n      \\put(120,90){\\line(1,0){10}}\n      \\put(123,92){\\scriptsize{$\\overline{3}$}}\n      \\put(123,82){\\scriptsize{$\\overline{1}$}}\n      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\\put(167,149){\\tiny{$(3,-2)$}}\n     \n      \\put(85,357){\\vector(0,-1){14}}\n      \\put(87,349){\\tiny{$(3,0)$}}\n      %\n      \\put(165,357){\\vector(0,-1){14}}\n      \\put(167,349){\\tiny{$(3,1)$}}\n     \n      \\put(125,437){\\vector(0,-1){14}}\n      \\put(127,429){\\tiny{$(2,1)$}}\n      %\n      \\put(125,477){\\vector(0,-1){14}}\n      \\put(127,469){\\tiny{$(1,2)$}}\n     \n      \\put(94,160){\\vector(3,-1){62}}\n      \\put(124,152){\\tiny{$(1,-1)$}}\n      \\put(156,360){\\vector(-3,-1){62}}\n      \\put(117,357){\\tiny{$(1,0)$}}\n  \\end{picture}\n  \\end{center}\n  \\caption{The Bethe-strap structure of  \n  ${\\cal T}^{2}(u)$ (\\ref{t2-ex}) for \n   $B(2|1)$:  \n The topmost tableau corresponds to \n the {\\em top term}.}\n  \\label{best2}\n\\end{figure}\n\\begin{figure}\n    \\setlength{\\unitlength}{1.5pt}\n    \\begin{center}\n    \\begin{picture}(100,310) \n      \\put(40,0){\\line(1,0){20}}\n      \\put(60,0){\\line(0,1){10}}\n      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\\put(132,169){\\scriptsize{$(1,-3)$}}\n     \n      \\put(130,208){\\vector(0,-1){16}}\n      \\put(132,199){\\scriptsize{$(2,-2)$}}\n     \n      \\put(10,268){\\vector(0,-1){16}}\n      \\put(12,259){\\scriptsize{$(2,-1)$}}\n     \n      \\put(90,268){\\vector(0,-1){16}}\n      \\put(92,259){\\scriptsize{$(3,-2)$}}\n     \n      \\put(22,28){\\vector(1,-1){16}}\n      \\put(31,21){\\scriptsize{$(1,-3)$}}\n     \n      \\put(-18,88){\\vector(1,-1){16}}\n      \\put(-9,81){\\scriptsize{$(2,-2)$}}\n      %\n      \\put(62,88){\\vector(1,-1){16}}\n      \\put(71,81){\\scriptsize{$(1,-3)$}}\n     \n      \\put(22,238){\\vector(1,-1){16}}\n      \\put(31,231){\\scriptsize{$(3,-2)$}}\n      %\n      \\put(102,238){\\vector(1,-1){16}}\n      \\put(111,231){\\scriptsize{$(3,-1)$}}\n     \n      \\put(62,298){\\vector(1,-1){16}}\n      \\put(71,291){\\scriptsize{$(2,-1)$}}\n     \n      \\put(78,28){\\vector(-1,-1){16}}\n      \\put(56,21){\\scriptsize{$(2,0)$}}\n     \n      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\\put(6,176){\\scriptsize{$(2,1)$}}\n     \n      \\put(118,179){\\vector(-3,-1){55}}\n      \\put(86,176){\\scriptsize{$(1,2)$}}\n     \n      \\put(38,209){\\vector(-3,-1){55}}\n      \\put(6,206){\\scriptsize{$(2,1)$}}\n     \n      \\put(118,209){\\vector(-3,-1){55}}\n      \\put(86,206){\\scriptsize{$(1,2)$}}\n     \n      \\put(78,269){\\vector(-3,-1){55}}\n      \\put(46,266){\\scriptsize{$(1,2)$}}\n     \n  \\end{picture}\n  \\end{center}\n  \\caption{The Bethe-strap structure of \n  ${\\cal T}_{2}(u)$ (\\ref{t21-ex}) for $B(2|1)$:  \n The topmost  tableau corresponds to the {\\em top term}.}\n  \\label{best3}\n\\end{figure}\n \nWe can prove (see Appendix A.1-A.3) \nthe following Theorem, which\n is essential in the analytic Bethe ansatz.\n\\begin{theorem} \\label{th-tate}\nFor   \n$a\\in {\\bf Z}_{\\ge 0}$, ${\\cal T}^{a}(u)$  \n((\\ref{Tge1}) for $\\lambda =\\phi$, $\\mu =(1^{a})$) \nis free of poles under the condition that\nthe BAEs {\\rm (\\ref{BAE})-(\\ref{BAE4})} are valid\n\\footnote{\nWe consider the case that \nthe solutions $\\{ u_{j}^{(a)} \\}$ of the \nBAEs (\\ref{BAE})-(\\ref{BAE4}) \n have \\symbol{96}generic' distribution: \nWe assume that   \n$u_{i}^{(a)}-u_{j}^{(a)} \\ne (\\alpha_{a}|\\alpha_{a})$\n for any $i,j \\in \\{1,2,\\dots , N_{a}\\} $   \n and $a\\in \\{1,2,\\dots, s+r \\}$ ($i \\ne j$)\n in BAEs (\\ref{BAE})-(\\ref{BAE4}). \nMoreover we assume that the color $b$ pole \n(see Appendix A.1-A.3) of ${\\cal T}^{a}(u)$ \n and the color $c$ pole \n do not coincide each other if $b \\ne c$. \n  We will need separate \n consideration for the case where this assumption does not hold. \n We also note that similar assumption was assumed in \n \\cite{T1,T2,T3,T4}.\n}.\n\\end{theorem}\nIn proving Theorem \\ref{th-tate}, we use the following lemmas.\n\\begin{lemma} \\label{le-tate}\nFor  $r \\in {\\bf Z}_{\\ge 2}$ and $b\\in \\{s+1,s+2, \\dots,s+r-1\\}$,   \n\\begin{eqnarray}\n  \\begin{array}{|c|l}\\cline{1-1}\n    b & _{v}  \\\\ \\cline{1-1} \n    b+1 & _{v-2}  \\\\ \\cline{1-1}\n  \\end{array}\n  ,\\qquad \n  \\begin{array}{|c|l}\\cline{1-1}\n    \\stackrel{\\ }{\\overline{b+1}} & _{v}  \\\\ \\cline{1-1} \n    \\stackrel{\\ }{\\overline{b}} & _{v-2}  \\\\ \\cline{1-1}\n  \\end{array}\n\\end{eqnarray}\ndo not contain $Q_{b}$.\n\\end{lemma}\nFor $B(0|s)$ case, we use the following lemma: \n\\begin{lemma} \\label{le-yoko}\nFor  $b\\in \\{1,2, \\dots,s-1\\}$,   \n\\begin{eqnarray}\n  \\begin{array}{|c|c|}\n   \\multicolumn{2}{c}{\\quad} \\\\ \\hline \n     b & b+1 \\\\ \\hline \n    \\multicolumn{1}{c}{^{u}} & \n    \\multicolumn{1}{c}{^{u+2}}\n  \\end{array}\n  ,\\qquad \n  \\begin{array}{|c|c|}\n   \\multicolumn{2}{c}{\\quad} \\\\ \\hline \n     \\overline{b+1} & \\stackrel{\\ }{\\overline{b}} \\\\ \\hline \n    \\multicolumn{1}{c}{^{u}} & \n    \\multicolumn{1}{c}{^{u+2}}\n  \\end{array} \n\\end{eqnarray}\ndo not contain $Q_{b}$, and \n\\begin{eqnarray}\n  \\begin{array}{|c|c|c|}\n   \\multicolumn{3}{c}{\\quad} \\\\ \\hline \n     s & 0 & \\overline{s} \\\\ \\hline \n    \\multicolumn{1}{c}{^{u}} & \n    \\multicolumn{1}{c}{^{u+2}} &\n    \\multicolumn{1}{c}{^{u+4}}\n  \\end{array}\n\\end{eqnarray}\ndoes not contain $Q_{s}$.\n\\end{lemma}\nThen owing to the relation (\\ref{Jacobi-Trudi1}), \n$ {\\cal T}_{\\lambda \\subset \\mu}(u)$ for \n$B(r|s)$ is also free of poles under \nthe condition that the BAEs (\\ref{BAE})-(\\ref{BAE4})\n are valid.\t \n Similarly, \n owing to the relation (\\ref{Jacobi-Trudi}), \n$ {\\cal T}_{m}(u)$ for \n$D(r|s)$ is also free of poles under \nthe condition that the BAE (\\ref{BAE})\n is valid.\n\\setcounter{equation}{0}\n\\section{Functional relations among DVFs}\nNow we introduce the functional relations among DVFs.\n For $B(r|s)$ case, the following relation follows from \n the determinant formulae \n (\\ref{Jacobi-Trudi1}) or (\\ref{Jacobi-Trudi2}). \n\\begin{eqnarray}\n{\\cal T}_{m}^{a}(u-1){\\cal T}_{m}^{a}(u+1)\n={\\cal T}_{m-1}^{a}(u){\\cal T}_{m+1}^{a}(u)+\n{\\cal T}_{m}^{a-1}(u){\\cal T}_{m}^{a+1}(u) ,\n\\label{hirota}\n\\end{eqnarray}\nwhere $m,a \\in {\\bf Z}_{\\ge 1}$; \n${\\cal T}_{m}^{0}(u)={\\cal T}_{0}^{a}(u)=1$. \nThis functional relation (\\ref{hirota}) is a Hirota bilinear \ndifference equation \\cite{H} and \ncan be proved  by using the Jacobi identity. \n There is a constraint to (\\ref{hirota}) \nfollows from the relation \n(cf. \\cite{DM,MRi} for $sl(r|s)$ case): \\\\ \n {\\em ${\\cal T}_{\\lambda \\subset \\mu}(u) =0$ if \n $\\lambda \\subset \\mu$ contains $a \\times m$ rectangular subdiagram \n ($a$: the number of row, $m$: the number of column) with \n$m \\in {\\bf Z}_{\\ge 2s+2}$ and $a \\in {\\bf Z}_{\\ge 2r+1}$.} \\\\ \nIn particular, we have \n\\begin{eqnarray}\n{\\cal T}_{m}^{a}(u)=0 \\qquad {\\rm if } \\quad \n m \\in {\\bf Z}_{\\ge 2s+2} \\quad {\\rm and}\n  \\quad a \\in {\\bf Z}_{\\ge 2r+1}.\n\\label{cons}\n\\end{eqnarray}\nWe also note that the determinant formula (\\ref{Jacobi-Trudi}) \n for $D(r|s)$  reduces to the following functional relation:\n\\begin{equation}\n{\\cal T}^{1}(u-1){\\cal T}^{1}(u+1)\n={\\cal T}_{2}(u)+{\\cal T}^{2}(u) ,\n\\end{equation}\nif we set $m=2$.\n\nIn this section, we consider only $B(0|s)$ \n($s \\in {\\bf Z}_{\\ge 1}$) case from now on. \nNow we redefine the function ${\\cal T}_{\\lambda \\subset \\mu}(u)$\n as follows:\n\\begin{eqnarray}\n{\\cal T}_{\\lambda \\subset \\mu}(u):=\n{\\cal T}_{\\lambda \\subset \\mu}(u)\/\n\\{\n\\prod_{j=1}^{\\mu_{1}^{\\prime}}\n{\\cal F}_{\\mu_{j}-\\lambda_{j}}\n(u-\\mu_{1}+\\mu_{1}^{\\prime}+\\mu_{j}+\\lambda_{j}-2j+1)\n\\}, \n\\end{eqnarray}\nwhere \n\\begin{eqnarray}\n && {\\cal F}_{m}(u)=\n\\prod_{j=1}^{m-1}\\phi(u-m+2j+1)\\phi(u-2s-m+2j-2) \\nonumber \\\\ \n && \\hspace{160pt} {\\rm for} \\quad m \\in {\\bf Z}_{\\ge 2}, \n\\end{eqnarray}\nand \n\\begin{eqnarray} \n{\\cal F}_{1}(u)=1, \\qquad  \n{\\cal F}_{0}(u)=\\{\\phi(u+1)\\phi(u-2s-2)\\}^{-1}. \n\\end{eqnarray}\nIn particular, we have \n\\begin{eqnarray}\n{\\cal T}_{m}(u)&:=&{\\cal T}_{m}(u)\/{\\cal F}_{m}(u), \\\\ \n{\\cal T}_{0}^{a}(u)&=&\\prod_{j=1}^{a} {\\cal T}_{0}(u+a-2j+1)\n \\nonumber \\\\ \n &=& \\prod_{j=1}^{a} \\phi(u+a-2j+2)\\phi(u+a-2j-2s-1). \n\\end{eqnarray}\nThere is remarkable duality for ${\\cal T}_{m}(u)$. \n\\begin{theorem}\\label{dual-th}\nFor any $m \\in \\{0,1,\\dots,2s+1 \\}$, we have \n\\begin{eqnarray}\n {\\cal T}_{m}(u)={\\cal T}_{2s-m+1}(u).\n  \\label{dual1}\n\\end{eqnarray}\n\\end{theorem}\n{\\em Outline of the proof}: \nAt first, we consider the case\n that the vacuum parts are formally trivial. \nIn proving the relation (\\ref{dual1}), we use the following relations, \nwhich can be verified by direct computation. \n\\begin{eqnarray}\n  \\begin{array}{|c|}\n   \\multicolumn{1}{c}{\\quad} \\\\ \\hline \n     \\overline{a} \\\\ \\hline \n    \\multicolumn{1}{c}{^{u}}\n  \\end{array}\n  & \\times &\n  \\begin{array}{|c|c|c|c|}\n   \\multicolumn{4}{c}{\\quad} \\\\ \\hline \n     1 & 2 & \\cdots \\cdots & a \\\\ \\hline \n    \\multicolumn{1}{c}{^{u-2s-1}} & \n    \\multicolumn{1}{c}{^{u-2s+1}} &\n    \\multicolumn{1}{c}{} &\n    \\multicolumn{1}{c}{^{u+2a-2s-3}}\n  \\end{array}\n  \\nonumber \\\\ \n  &=&\n  \\begin{array}{|c|c|c|c|}\n   \\multicolumn{4}{c}{\\quad} \\\\ \\hline \n    1 & 2 & \\cdots \\cdots & a-1 \\\\ \\hline \n    \\multicolumn{1}{c}{^{u-2s+1}} & \n    \\multicolumn{1}{c}{^{u-2s+3}} &\n    \\multicolumn{1}{c}{} &\n    \\multicolumn{1}{c}{^{u+2a-2s-3}}\n  \\end{array}\n  \\label{modi1},\n\\end{eqnarray} \n\\begin{eqnarray}\n  \\begin{array}{|c|}\n   \\multicolumn{1}{c}{\\quad} \\\\ \\hline \n     a \\\\ \\hline \n    \\multicolumn{1}{c}{^{u}}\n  \\end{array}\n  & \\times &\n  \\begin{array}{|c|c|c|c|}\n   \\multicolumn{4}{c}{\\quad} \\\\ \\hline \n     \\stackrel{\\ }{\\overline{a}} & \n        \\cdots \\cdots & \\overline{2} & \n         \\stackrel{\\ }{\\overline{1}} \\\\ \\hline \n    \\multicolumn{1}{c}{^{u-2a+2s+3}} & \n    \\multicolumn{1}{c}{} &\n    \\multicolumn{1}{c}{^{u+2s-1}} &\n    \\multicolumn{1}{c}{^{u+2s+1}}\n  \\end{array}\n  \\nonumber \\\\ \n  &=&\n  \\begin{array}{|c|c|c|c|}\n   \\multicolumn{4}{c}{\\quad} \\\\ \\hline \n     \\stackrel{\\ }{\\overline{a-1}} & \n            \\cdots \\cdots & \\overline{2} & \n              \\stackrel{\\ }{\\overline{1}} \\\\ \\hline \n    \\multicolumn{1}{c}{^{u-2a+2s+3}} & \n    \\multicolumn{1}{c}{} &\n    \\multicolumn{1}{c}{^{u+2s-3}} &\n    \\multicolumn{1}{c}{^{u+2s-1}}\n  \\end{array}\n  \\label{modi}\n\\end{eqnarray} \nand  \n\\begin{eqnarray}\n\\begin{array}{|c|c|c|c|c|c|c|c|c|}\n   \\multicolumn{9}{c}{\\quad} \\\\ \\hline \n     1 & 2 & \\cdots  \\cdots & s & 0 & \n     \\stackrel{\\ }{\\overline{s}} & \\cdots \\cdots & \n     \\overline{2} & \\stackrel{\\ }{\\overline{1}} \\\\ \\hline \n    \\multicolumn{1}{c}{^{u-2s}} & \n    \\multicolumn{1}{c}{^{u-2s+2}} &\n    \\multicolumn{1}{c}{} &\n    \\multicolumn{1}{c}{^{u-2}} &\n    \\multicolumn{1}{c}{^{u}} &\n    \\multicolumn{1}{c}{^{u+2}} &\n    \\multicolumn{1}{c}{} & \n    \\multicolumn{1}{c}{^{u-2s-2}} &\n    \\multicolumn{1}{c}{^{u+2s}}\n\\end{array}\n  =1 \\label{const},\n\\end{eqnarray}\nwhere $a \\in \\{1,2,\\dots, s+1 \\}$ \n\\footnote{Here we assume \n$\\framebox{$s+1$}=\\framebox{$\\overline{s+1}$}=\\framebox{$0$}$.}; \n  the spectral parameter increases  (cf. (\\ref{Tge1})) \n  as we go from the left to the right on each tableau. \nWe will show that any term in ${\\cal T}_{m}(u)$ coincides with \na term in ${\\cal T}_{2s-m+1}(u)$. \nWe will consider the signs originated from \nthe grading parameter (\\ref{grading}) separately. \nAny term in ${\\cal T}_{m}(u)$ can be expressed by a tableau \n$b \\in B((m^{1}))$ such that \n$b(1,k)=i_{k}$ for $1\\le k \\le \\alpha $ \n($1 \\preceq i_{1} \\prec \\cdots \\prec i_{\\alpha} \\preceq s$); \n$b(1,k)=\\overline{j_{m-k+1}}$ for $\\alpha +1\\le k \\le m $\n ($0 \\preceq \\overline{j_{m-\\alpha}}\n  \\prec \\cdots \\prec \\overline{j_{1}} \\preceq \\overline{1}$); \n  $\\alpha \\in {\\bf Z}$. \nThe term corresponding to this tableau is given as follows\n\\footnote{(\\ref{mod0}) is $1$ if $m=0$.}: \n\\begin{eqnarray}\n\\hspace{-20pt}  && \\begin{array}{|c|c|c|c|c|c|}\n   \\multicolumn{6}{c}{\\quad} \\\\ \\hline \n     i_{1} & \\cdots \\cdots & i_{\\alpha} & \n     \\stackrel{\\ }{\\overline{j_{m-\\alpha}}} & \n       \\cdots \\cdots & \\overline{j_{1}} \n     \\\\ \\hline \n    \\multicolumn{1}{c}{^{u-m+1}} & \n    \\multicolumn{1}{c}{} &\n    \\multicolumn{1}{c}{^{u-m+2\\alpha -1}} &\n    \\multicolumn{1}{c}{^{u-m+2\\alpha +1}} & \n    \\multicolumn{1}{c}{} &\n    \\multicolumn{1}{c}{^{u+m-1}}\n  \\end{array}\n  \\label{mod0} \\\\ \n \\hspace{-20pt} &=&\n  \\begin{array}{|c|c|c|c|c|c|}\n   \\multicolumn{6}{c}{\\quad} \\\\ \\hline \n     i_{1} & \\cdots \\cdots & i_{\\alpha} & \n     \\stackrel{\\ }{\\overline{j_{m-\\alpha}}} & \n       \\cdots \\cdots & \\overline{j_{1}} \\\\ \\hline \n    \\multicolumn{1}{c}{^{u-m+1}} & \n    \\multicolumn{1}{c}{} &\n    \\multicolumn{1}{c}{^{u-m+2\\alpha -1}} &\n    \\multicolumn{1}{c}{^{u-m+2\\alpha +1}} & \n    \\multicolumn{1}{c}{} &\n    \\multicolumn{1}{c}{^{u+m-1}}\n  \\end{array}\n  \\nonumber \\\\ \n  \\hspace{-20pt} && \\times \n  \\begin{array}{|c|c|c|c|c|c|c|c|c|}\n   \\multicolumn{9}{c}{\\quad} \\\\ \\hline \n     1 & 2 & \\cdots \\cdots  & s & 0 & \n     \\overline{s} & \\cdots \\cdots & \n     \\stackrel{\\ }{\\overline{2}} & \\overline{1} \\\\ \\hline \n    \\multicolumn{1}{c}{^{u-m+2\\alpha -2s}} & \n    \\multicolumn{7}{c}{} &\n    \\multicolumn{1}{c}{^{u-m+2\\alpha +2s}}\n  \\end{array}\n  \\label{mod2} \\\\ \n  \\hspace{-20pt} &=& \n  \\begin{array}{|c|c|c|}\n   \\multicolumn{3}{c}{\\quad} \\\\ \\hline \n     i_{1} & \\cdots \\cdots & i_{\\alpha-1}  \\\\ \\hline \n    \\multicolumn{1}{c}{^{u-m+1}} & \n    \\multicolumn{1}{c}{} &\n    \\multicolumn{1}{c}{^{u-m+2\\alpha -3}} \n  \\end{array}\n  \\times \n  \\begin{array}{|c|c|c|}\n   \\multicolumn{3}{c}{\\quad} \\\\ \\hline  \n  \\stackrel{\\ }{\\overline{j_{m-\\alpha-1}}} & \n     \\cdots \\cdots & \\overline{j_{1}} \\\\ \\hline \n    \\multicolumn{1}{c}{^{u-m+2\\alpha +3}} & \n    \\multicolumn{1}{c}{} &\n    \\multicolumn{1}{c}{^{u+m-1}}\n  \\end{array}\n  \\nonumber \\\\ \n  \\hspace{-20pt} && \\times \n  \\begin{array}{|c|}\n   \\multicolumn{1}{c}{\\quad} \\\\ \\hline  \n  \\stackrel{\\ }{\\overline{j_{m-\\alpha}}} \\\\ \\hline \n    \\multicolumn{1}{c}{^{u-m+2\\alpha +1}}\n  \\end{array}\n  \\times \n  \\begin{array}{|c|c|c|c|}\n   \\multicolumn{4}{c}{\\quad} \\\\ \\hline \n     1 & 2 & \\cdots \\cdots & j_{m-\\alpha}  \\\\ \\hline \n    \\multicolumn{1}{c}{^{u-m+2\\alpha-2s}} & \n    \\multicolumn{2}{c}{} &\n    \\multicolumn{1}{c}{^{u-m+2\\alpha -2s+2j_{m-\\alpha}-2}} \n  \\end{array}\n  \\nonumber \\\\ \n  \\hspace{-20pt} && \\times \n  \\begin{array}{|c|c|c|c|c|c|c|}\n   \\multicolumn{7}{c}{\\quad} \\\\ \\hline \n     j_{m-\\alpha}+1 & \\cdots \\cdots & s & 0 & \\overline{s} &\n     \\cdots \\cdots & \n       \\stackrel{\\ }{\\overline{i_{\\alpha}+1}}  \\\\ \\hline \n    \\multicolumn{1}{c}{^{u-m+2\\alpha-2s+2j_{m-\\alpha}}} & \n    \\multicolumn{5}{c}{} &\n    \\multicolumn{1}{c}{^{u-m+2\\alpha +2s-2i_{\\alpha}}} \n  \\end{array}\n  \\nonumber \\\\ \n  \\hspace{-20pt} && \\times \n  \\begin{array}{|c|}\n   \\multicolumn{1}{c}{\\quad} \\\\ \\hline  \n  i_{\\alpha} \\\\ \\hline \n    \\multicolumn{1}{c}{^{u-m+2\\alpha -1}}\n  \\end{array}\n  \\times \n  \\begin{array}{|c|c|c|c|}\n   \\multicolumn{4}{c}{\\quad} \\\\ \\hline \n     \\stackrel{\\ }{\\overline{i_{\\alpha}}} & \n       \\cdots \\cdots & \\overline{2} & \\overline{1} \\\\ \\hline \n    \\multicolumn{1}{c}{^{u-m+2\\alpha-2i_{\\alpha}+2s+2}} & \n    \\multicolumn{2}{c}{} &\n    \\multicolumn{1}{c}{^{u-m+2\\alpha +2s}} \n  \\end{array}\n  \\label{mod3} \\\\ \n  \\hspace{-20pt} &=& \n  \\begin{array}{|c|c|c|}\n   \\multicolumn{3}{c}{\\quad} \\\\ \\hline \n     i_{1} & \\cdots \\cdots & i_{\\alpha-1}  \\\\ \\hline \n    \\multicolumn{1}{c}{^{u-m+1}} & \n    \\multicolumn{1}{c}{} &\n    \\multicolumn{1}{c}{^{u-m+2\\alpha -3}} \n  \\end{array}\n  \\times \n  \\begin{array}{|c|c|c|}\n   \\multicolumn{3}{c}{\\quad} \\\\ \\hline  \n  \\stackrel{\\ }{\\overline{j_{m-\\alpha-1}}} & \n    \\cdots \\cdots & \\overline{j_{1}} \\\\ \\hline \n    \\multicolumn{1}{c}{^{u-m+2\\alpha +3}} & \n    \\multicolumn{1}{c}{} &\n    \\multicolumn{1}{c}{^{u+m-1}}\n  \\end{array}\n  \\nonumber \\\\ \n  \\hspace{-20pt} && \n  \\times \n  \\begin{array}{|c|c|c|c|}\n   \\multicolumn{4}{c}{\\quad} \\\\ \\hline \n     1 & 2 & \\cdots \\cdots & j_{m-\\alpha}-1  \\\\ \\hline \n    \\multicolumn{1}{c}{^{u-m+2\\alpha-2s+2}} & \n    \\multicolumn{2}{c}{} &\n    \\multicolumn{1}{c}{^{u-m+2\\alpha -2s+2j_{m-\\alpha}-2}} \n  \\end{array}\n  \\nonumber \\\\ \n  \\hspace{-20pt} && \\times \n  \\begin{array}{|c|c|c|c|c|c|c|}\n   \\multicolumn{7}{c}{\\quad} \\\\ \\hline \n     j_{m-\\alpha}+1 & \\cdots \\cdots & s & 0 & \\overline{s} &\n     \\cdots \\cdots & \n       \\stackrel{\\ }{\\overline{i_{\\alpha}+1}}  \\\\ \\hline \n    \\multicolumn{1}{c}{^{u-m+2\\alpha-2s+2j_{m-\\alpha}}} & \n    \\multicolumn{5}{c}{} &\n    \\multicolumn{1}{c}{^{u-m+2\\alpha +2s-2i_{\\alpha}}} \n  \\end{array}\n  \\nonumber \\\\ \n  \\hspace{-20pt} && \\times  \n  \\begin{array}{|c|c|c|c|}\n   \\multicolumn{4}{c}{\\quad} \\\\ \\hline \n     \\stackrel{\\ }{\\overline{i_{\\alpha}-1}} & \n        \\cdots \\cdots & \\overline{2} &\n       \\overline{1} \\\\ \\hline \n    \\multicolumn{1}{c}{^{u-m+2\\alpha-2i_{\\alpha}+2s+2}} & \n    \\multicolumn{2}{c}{} &\n    \\multicolumn{1}{c}{^{u-m+2\\alpha +2s-2}} \n  \\end{array}\n   \\label{mod4} \\\\\n \\hspace{-20pt} &=& \n  \\cdots = \\begin{array}{|c|c|c|c|c|c|}\n   \\multicolumn{6}{c}{\\quad} \\\\ \\hline \n     J_{1} & \\cdots \\cdots & J_{s+1-m+\\alpha} & \n     \\stackrel{\\ }{\\overline{I_{s-\\alpha}}} & \n       \\cdots \\cdots & \\overline{I_{1}} \\\\ \\hline \n    \\multicolumn{1}{c}{^{u+m-2s}} & \n    \\multicolumn{1}{c}{} &\n    \\multicolumn{1}{c}{^{u-m+2\\alpha }} &\n    \\multicolumn{1}{c}{^{u-m+2\\alpha +2}} & \n    \\multicolumn{1}{c}{} &\n    \\multicolumn{1}{c}{^{u-m+2s}}\n  \\end{array},\n\\label{mod5}\n\\end{eqnarray}\nwhere $\\{J_{k}\\}= \\{1,2,\\dots, s,0 \\} \n\\setminus \\{j_{1},j_{2},\\dots,j_{m-\\alpha}\\}$\n ($1 \\preceq J_{1} \\prec \\cdots \\prec J_{s+1+\\alpha -m} \\preceq 0$); \n$\\{\\overline{I_{k}}\\}= \\{\\overline{s},\\overline{s-1},\\dots, \\overline{1} \\}\n \\setminus \\{\\overline{i_{\\alpha}},\\overline{i_{\\alpha-1}},\n \\dots,\\overline{i_{1}} \\}$\n ($\\overline{s} \\preceq \\overline{I_{s-\\alpha}}\n  \\prec \\cdots \\prec \\overline{I_{1}} \\preceq \\overline{1}$). \n (\\ref{mod2}) follows from (\\ref{const}); \n (\\ref{mod4}) follows from (\\ref{modi1}) and (\\ref{modi}). \n After repetition of procedures similar to \n (\\ref{mod3})-(\\ref{mod4}), we obtain (\\ref{mod5}).  \n Apparently, (\\ref{mod5}) is a term in ${\\cal T}_{2s-m+1}(u)$. \n Conversely, one can also show that any term in \n ${\\cal T}_{2s-m+1}(u)$ coincides \n with a term in ${\\cal T}_{m}(u)$. \n\nNoting the relation \n\\begin{eqnarray}\n\\hspace{-30pt}&&\\sum_{k=1}^{s+1-m+\\alpha}p(J_{k})+\n  \\sum_{k=1}^{s-\\alpha}p(\\overline{I_{k}}) \n  \\nonumber \\\\ \n\\hspace{-30pt}&& \\hspace{60pt}\n=\n\\left(\n \\sum_{k=1}^{s}p(k)+p(0)-\\sum_{k=1}^{m-\\alpha}p(j_{k})\n\\right)+\n\\left(\n \\sum_{k=1}^{s}p(\\overline{k})-\n \\sum_{k=1}^{\\alpha}p(\\overline{i_{k}})\n\\right) \\nonumber \\\\ \n\\hspace{-30pt}&& \\hspace{60pt}=\n2s-\\sum_{k=1}^{m-\\alpha}p(j_{k})\n-\\sum_{k=1}^{\\alpha}p(\\overline{i_{k}}) \n \\nonumber \\\\ \n\\hspace{-30pt}&& \\hspace{60pt} \n \\equiv \n\\sum_{k=1}^{m-\\alpha}p(j_{k})\n+\\sum_{k=1}^{\\alpha}p(\\overline{i_{k}}) \n\\quad {\\rm mod} \\quad 2, \n\\end{eqnarray}\nwe find that \n the overall sign for (\\ref{mod0}) coincides with that for (\\ref{mod5}). \n\nFinally, we comment on the vacuum parts. \nFrom now on, we assume that the vacuum parts are not trivial. \nEquivalence between the dress parts of ${\\cal T}_{m}(u)$ \nand ${\\cal T}_{2s-m+1}(u)$ has already been shown, \nso we have only to check that \nthe vacuum part of \n\\begin{eqnarray}\n  \\begin{array}{|c|c|c|c|c|c|}\n   \\multicolumn{6}{c}{\\quad} \\\\ \\hline \n     i_{1} & \\cdots \\cdots & i_{\\alpha} & \n     \\stackrel{\\ }{\\overline{j_{m-\\alpha}}} & \n       \\cdots \\cdots & \\overline{j_{1}} \n     \\\\ \\hline \n    \\multicolumn{1}{c}{^{u-m+1}} & \n    \\multicolumn{1}{c}{} &\n    \\multicolumn{1}{c}{^{u-m+2\\alpha -1}} &\n    \\multicolumn{1}{c}{^{u-m+2\\alpha +1}} & \n    \\multicolumn{1}{c}{} &\n    \\multicolumn{1}{c}{^{u+m-1}}\n  \\end{array}\n  \\ \/ {\\cal F}_{m}(u) \n\\end{eqnarray}\nis equivalent to that of \n\\begin{eqnarray}\n\\hspace{-10pt}\n\\begin{array}{|c|c|c|c|c|c|}\n   \\multicolumn{6}{c}{\\quad} \\\\ \\hline \n     J_{1} & \\cdots \\cdots & J_{s+1-m+\\alpha} & \n     \\stackrel{\\ }{\\overline{I_{s-\\alpha}}} & \n       \\cdots \\cdots & \\overline{I_{1}} \n     \\\\ \\hline \n    \\multicolumn{1}{c}{^{u+m-2s}} & \n    \\multicolumn{1}{c}{} &\n    \\multicolumn{1}{c}{^{u-m+2\\alpha }} &\n    \\multicolumn{1}{c}{^{u-m+2\\alpha +2}} & \n    \\multicolumn{1}{c}{} &\n    \\multicolumn{1}{c}{^{u-m+2s}}\n  \\end{array}\n \\ \/ {\\cal F}_{2s-m+1}(u). \n\\end{eqnarray}\nAll we have to do is to check this  \nby direct computation for the following \nfour cases: \n(i) $i_{1}=1$ and $\\overline{j_{1}}=\\overline{1}$ \n($1 \\prec J_{1}$ and $\\overline{I_{1}} \\prec \\overline{1}$); \n(ii) $i_{1}=1$ and $\\overline{j_{1}} \\prec \\overline{1}$ \n($J_{1}=1$ and $\\overline{I_{1}} \\prec \\overline{1}$);\n(iii) $1 \\prec i_{1}$ and $\\overline{j_{1}}=\\overline{1}$ \n($1 \\prec J_{1}$ and $\\overline{I_{1}}=\\overline{1}$);\n(iv) $1 \\prec i_{1}$ and $\\overline{j_{1}} \\prec \\overline{1}$ \n($J_{1}=1$ and $\\overline{I_{1}}=\\overline{1}$). \n\\rule{5pt}{10pt} \\\\ \n Owing to the relation (\\ref{Jacobi-Trudi2}), \nwe can generalize the relation (\\ref{dual1}) to  \n\\begin{eqnarray}\n {\\cal T}_{m}^{a}(u)={\\cal T}_{2s-m+1}^{a}(u) \n \\label{dual2} ,\n\\end{eqnarray}\nwhere $a \\in {\\bf Z}_{\\ge 1}$. \nTaking note on the relations (\\ref{dual2}) and (\\ref{cons}), \n we shall rewrite the functional relation (\\ref{hirota}) in a\n  \\symbol{96}canonical' form \nas the original $T$-system  for the simple Lie algebra \\cite{KNS1}. \nSet \n$T_{m}^{(a)}(u)={\\cal T}_{a}^{m}(u)$, \n$T_{2m}^{(s)}(u)={\\cal T}_{s}^{m}(u)$ and  \n$T_{0}^{(a)}(u)=T_{0}^{(s)}(u)=T_{m}^{(0)}(u)=1$\n for $a \\in \\{1,2,\\dots, s-1\\}$ and $m \\in {\\bf Z}_{\\ge 1}$, \nwhere the subscript $(n,a)$ of $T_{n}^{(a)}(u)$  \ncorresponds to the Kac-Dynkin label $[b_{1}, b_{2},\\dots ,b_{s}]$ \n for $b_{i}=n\\delta_{i a}$ (cf. (\\ref{Kac-Dynkin-b0s})).\n  Then we have \n\\begin{eqnarray}\n\\hspace{-40pt} \n&& T_{m}^{(a)}(u-1)T_{m}^{(a)}(u+1)\n=T_{m-1}^{(a)}(u)T_{m+1}^{(a)}(u)+\ng_{m}^{(a)}(u)T_{m}^{(a-1)}(u)T_{m}^{(a+1)}(u) \\nonumber \\\\ \n\\hspace{-40pt}\n&& \\hspace{190pt} {\\rm for } \\qquad a \\in \\{1,2,\\dots,s-2\\}, \n\\label{T-sys1}\\\\\n\\hspace{-40pt}\n && T_{m}^{(s-1)}(u-1)T_{m}^{(s-1)}(u+1)\n=T_{m-1}^{(s-1)}(u)T_{m+1}^{(s-1)}(u) \\nonumber \\\\ \n\\hspace{-40pt}\n&& \\hspace{160pt} \n+g_{m}^{(s-1)}(u)T_{m}^{(s-2)}(u)T_{2m}^{(s)}(u) ,\\\\\n\\hspace{-40pt}\n&& T_{2m}^{(s)}(u-1)T_{2m}^{(s)}(u+1)\n=T_{2m-2}^{(s)}(u)T_{2m+2}^{(s)}(u)+\ng_{2m}^{(s)}(u)T_{m}^{(s-1)}(u)T_{2m}^{(s)}(u), \\label{T-sys3}\n\\end{eqnarray}\nwhere $g_{m}^{(b)}(u)=\n\\{\\prod_{j=1}^{m}{\\cal T}_{0}(u+2j-m-1)\\}^{\\delta_{b,1}}$   \nif $s\\in {\\bf Z}_{\\ge 2}$; \n$g_{2m}^{(1)}(u)=\\prod_{j=1}^{m}{\\cal T}_{0}(u+2j-m-1)$ \nif $s=1$. \nNote that the function $g_{m}^{(b)}(u)$ obey the following relation \n\\begin{eqnarray}\n&& g_{m}^{(b)}(u+1)g_{m}^{(b)}(u-1)=\ng_{m+1}^{(b)}(u)g_{m-1}^{(b)}(u) \\quad {\\rm if } \\quad \n       s \\in {\\bf Z}_{\\ge 2}, \\nonumber \\\\\n&& g_{2m}^{(1)}(u+1)g_{2m}^{(1)}(u-1)=\ng_{2m+2}^{(1)}(u)g_{2m-2}^{(1)}(u) \\quad {\\rm if } \\quad s=1.\n\\end{eqnarray} \nThese functional relations (\\ref{T-sys1})-(\\ref{T-sys3}) \nwill be $B(0|s)$ version of \nthe $T$-system. Note that the subscript $n$ of \n$T_{n}^{(s)}(u)$ can take only even number (cf. (\\ref{finite})). \nBy construction, $T_{m}^{(a)}(u)$ can be expressed \nas a determinant over a \nmatrix whose matrix elements are only the fundamental functions \n$T_{1}^{(1)}$, \\dots, $T_{1}^{(s-1)}$, $T_{2}^{(s)}$\n  and $g_{1}^{(1)}$ for $s\\in {\\bf Z}_{\\ge 2}$; \n  $T_{2}^{(1)}$ and $g_{2}^{(1)}$ for $s=1$. \n  This can be summarized as follows:\n\\begin{theorem}\nFor $m\\in {\\bf Z}_{\\ge 1}$, \n\\begin{eqnarray}\n&& T_{m}^{(a)}(u)= \n   {\\rm det}_{1\\le i,j \\le m}({\\cal T}_{a+i-j}(u+m-i-j+1)) \n\\nonumber \\\\ && \\hspace{150pt}\n{\\rm for} {\\quad} a \\in \\{1,2, \\dots ,s-1\\}, \\\\\n&& T_{2m}^{(s)}(u)=\n   {\\rm det}_{1\\le i,j \\le m}({\\cal T}_{s+i-j}(u+m-i-j+1)) \n\\end{eqnarray}\nsolves (\\ref{T-sys1})-(\\ref{T-sys3}). \nHere ${\\cal T}_{a}(u)$ obeys the relation (\\ref{dual1}) and \nthe boundary condition \n\\begin{eqnarray}\n{\\cal T}_{a}(u)=\\left\\{\n\\begin{array}{lll}\n 0 &{\\rm if} & a <0, \\\\ \n g_{1}^{(1)}(u)& {\\rm if} &   \n   a=0 \\quad {\\rm and} \\quad s \\in {\\bf Z}_{\\ge 2}\\\\\n g_{2}^{(1)}(u)& {\\rm if} & \n   a=0 \\quad {\\rm and} \\quad s=1 \\\\ \n T_{1}^{(a)}(u) & {\\rm if} & a \\in \\{1,2, \\dots ,s-1\\} \\\\ \n T_{2}^{(s)}(u) & {\\rm if} & a=s ,\n\\end{array}\n\\right. \n\\end{eqnarray}\nwhere \n$g_{m}^{(a)}(u)=\\{ \\prod_{j=1}^{m}g_{1}^{(1)}\n                         (u+2j-m-1)\\}^{\\delta_{a,1}}$ \n                         if  $s \\in {\\bf Z}_{\\ge 2}$; \n$g_{2m}^{(1)}(u)=\\prod_{j=1}^{m}g_{2}^{(1)}(u+2j-m-1)$ \n                         if  $ s=1 $. \n\\end{theorem}\n{\\em Remark}: \nThese functional realtions (\\ref{T-sys1})-(\\ref{T-sys3}) \nresemble to the ones for $A_{2s}^{(2)}$ \\cite{KS2}. \nThis resemblance will originate from resemblance between \n$B(0|s)^{(1)}$ and $A_{2s}^{(2)}$. \n\nThere is a remarkable relation between \nthe number ${\\cal N}_{m}^{(a)}$ of the terms in \n$T_{m}^{(a)}(u)$ and the dimensionality (\\ref{dim}) \nof the Lie superalgebra $B(0|s)$. \n We conjecture that they are related each other as follows: \n\\begin{eqnarray}\n {\\cal N}_{m}^{(a)}&=& \\sum \n  {\\rm dim}V[k_{1},k_{2},\\dots, k_{a},0,\\dots,0]\n   \\quad {\\rm if } \\quad a \\in \\{1,2,\\dots, s-1 \\}, \\nonumber \\\\ \n{\\cal N}_{2m}^{(s)}&=& \\sum {\\rm dim}V[k_{1},k_{2},\\dots, k_{s-1},2k_{s}]\n ,\n\\end{eqnarray}\nwhere the summation is taken over non-negative \n integers $\\{ k_{j} \\}$ \nsuch that $k_{1}+k_{2}+\\cdots +k_{a} \\le m $ and \n$k_{j} \\equiv m \\delta_{j a}$ mod $2$. \n\\begin{table}\n\\begin{center}\n \\begin{tabular}{ccccc} \\hline \n $m$                 & 1 & 2 & 3 & 4 \\\\ \\hline \n ${\\cal N}_{m}^{(1)}$  & 5 & 15 & 35 & 70 \\\\  \n ${\\cal N}_{2m}^{(2)}$ & 10 & 50 & 175 & 490 \\\\ \\hline \n \\end{tabular} \n\\end{center}\n\\caption{The number ${\\cal N}_{m}^{(a)}$ of the terms \n in $T_{m}^{(a)}(u)$ for $B(0|2)$.}\n\\label{num-osp14} \n\\end{table}\n\\begin{table}\n\\begin{center}\n \\begin{tabular}{cccc} \\hline \n $[b_{1},b_{2}]$ & ${\\rm dim}V[b_{1},b_{2}]$ & \n $[b_{1},b_{2}]$ & ${\\rm dim}V[b_{1},b_{2}]$ \\\\ \\hline\n 0 0 & 1  & 0 2 & 10  \\\\ \n 1 0 & 5  & 0 4 & 35  \\\\ \n 2 0 & 14 & 0 6 & 84  \\\\ \n 3 0 & 30 & 2 2 & 81  \\\\ \\hline \n  \\end{tabular} \n\\end{center}\n\\caption{The dimensionality of the module  \n  $V[b_{1},b_{2}]$ for $B(0|2)$.}\n\\label{dim-osp14} \n\\end{table}\nFor example, for $B(0|2)$ case, we have \n(cf. Table \\ref{num-osp14} and Table \\ref{dim-osp14}): \n\\begin{eqnarray}\n {\\cal N}_{1}^{(1)}&=& {\\rm dim}V[1,0], \\nonumber \\\\ \n {\\cal N}_{2}^{(1)}&=& {\\rm dim}V[2,0]+{\\rm dim}V[0,0], \\nonumber \\\\ \n {\\cal N}_{3}^{(1)}&=& {\\rm dim}V[3,0]+{\\rm dim}V[1,0], \\nonumber \\\\ \n {\\cal N}_{2}^{(2)}&=& {\\rm dim}V[0,2], \\nonumber \\\\ \n {\\cal N}_{4}^{(2)}&=& {\\rm dim}V[0,4]+{\\rm dim}V[2,0]\n                              +{\\rm dim}V[0,0], \\nonumber \\\\ \n {\\cal N}_{6}^{(2)}&=& {\\rm dim}V[0,6]+{\\rm dim}V[2,2]\n                              +{\\rm dim}V[0,2].  \n\\end{eqnarray}\nThese relations seem to suggest a \nsuperization of the Kirillov-Reshetikhin formula \n\\cite{KR}, which gives the multiplicity of occurrence of \nthe irreducible representations of the Lie superalgebra \nin the Yangian module. \n\\setcounter{equation}{0}\n\\section{Summary and discussion}\nIn this paper, we have carried out an analytic Bethe ansatz \n based on the Bethe ansatz equations (\\ref{BAE})-(\\ref{BAE4})\n  with the distinguished\n simple root systems of the type II Lie superalgebras \n $B(r|s)$ and $D(r|s)$. We have proposed \n eigenvalue formulae of transfer matrices in \n DVFs related to a class of tensor-like representations, \n and shown their pole-freeness  under \n the BAEs (\\ref{BAE})-(\\ref{BAE4}). \n  The key is the top term hypothesis \n   and the pole-freeness under the BAE.  \n A class of functional relations has been proposed for the DVFs. \n In particular for $B(0|s)$ case, \n  remarkable duality among DVFs was found. \n By using this, a complete set of functional relations \n is written down for the DVFs labeled by rectangular Young \n (super) diagrams.   \nTo the author' knowledge, this paper is the first \n trial to construct {\\em systematic } theory \n of an analytic Bethe ansatz \n related to fusion $B(r|s)$ and $D(r|s)$ vertex models. \n \n In the present paper, we have executed an analytic Bethe ansatz \n only for tensor-like representations.  \n As for spnorial representations, details are under investigation. \n For example, in relation to the \n 64 dimensional typical representation of \n $B(2|1)$, \n we have confirmed the fact that the Bethe-strap generated by \n the following top term\n\\footnote{Here we omit the vacuum part.}\n\\begin{eqnarray}\n\\frac{Q_{1}(u+\\frac{5}{2})Q_{3}(u-\\frac{1}{2})}\n     {Q_{1}(u-\\frac{5}{2})Q_{3}(u+\\frac{1}{2})},\n\\end{eqnarray}\nwhich carries $B(2|1)$ weight with the Kac-Dynkin label \n$(\\frac{5}{2},0,1)$ consists of 64 terms. \n\nFor $D(r|s)$ case, we have proposed DVFs labeled by \n   Young (super) diagrams with one row or one column. \n   It is tempting to extend these DVFs to \n   general Young (super) diagrams. \n   However, this will be a difficult task \n   since we lack of tableaux sum expressions of \n   DVFs labeled by general\n    Young diagrams even for the non-superalgebra \n    $D_{r}$ case \\cite{KS1}.   \n    One way to bypass cumbersome tableaux sum expressions \n    is to construct a complete set of transfer matrix \n    functional relations (a hierarchy of $T$-system).  \n    By solving it, we will be able to calculate DVFs.\n\nIt is an interesting problem to  derive TBA equations \nfrom our $T$-system (\\ref{T-sys1})-(\\ref{T-sys3}). \nThis is accomplished by a similar procedure for \n $sl(r|s)$ case \\cite{JKS} (see also \\cite{FK99}). \n  We are going to report this in \n the near future.\n\\\\\n{\\bf Acknowledgment} \\\\  \nThe author would like to thank Prof. A. Kuniba for encouragement. \nThis work is supported by a Grant-in-Aid for \nJSPS Research   Fellows from the Ministry of Education, Science and \nCulture of Japan. \n\\setcounter{equation}{0}\n\\renewcommand{\\theequation}{A.1.\\arabic{equation}}\n\\section*{Appendix A.1  Outline of the proof of Theorem \n\\ref{th-tate}: $B(r|s)$ ($r,s \\in {\\bf Z}_{\\ge 1}$) case}\nFor simplicity, we assume that the vacuum parts \nare formally trivial from now on. \nWe prove that ${\\cal T}^{a}(u)$ \nis free of color $b$ poles, that is, \n$Res_{u=u_{k}^{(b)}+\\cdots}{\\cal T}^{a}(u)=0$ for any \n $ b \\in \\{1,2,\\dots, s+r \\} $\n under the condition that the BAE (\\ref{BAE}) is valid. \n The function $\\framebox{$c$}_{u}$  \n (\\ref{z+}) with $c \\in J $ has \n color $b$ poles only for $c=b$, $b+1$, \n$\\overline{b+1}$ or $\\overline{b}$ if \n$b\\in \\{1,2,\\dots,s+r-1 \\} $; \nfor $c=s+r$, $0$ or $\\overline{s+r}$ if $b=s+r$, \nso we shall trace only \n\\framebox{$b$}, \\framebox{$b+1$}, \n\\framebox{$\\overline{b+1}$} or \\framebox{$\\overline{b}$}\n for $b\\in\\{1,2,\\dots, s+r-1 \\}$; \n\\framebox{$s+r$}, \\framebox{$0$} \nor \\framebox{$\\overline{s+r}$} for $b=s+r$.\n Let $S_{k}$ be the partial sum of ${\\cal T}^{a}(u)$, \n which contains \n$k$ boxes among \\framebox{$b$}, \\framebox{$b+1$}, \n\\framebox{$\\overline{b+1}$} or \\framebox{$\\overline{b}$}\n for $b \\in \\{1,2,\\dots, s+r-1\\}$; \n\\framebox{$s+r$}, \\framebox{$0$} \nor \\framebox{$\\overline{s+r}$} for $b=s+r$.\n Evidently, $S_{0}$ does not have \ncolor $b$ poles. \n\n Now we examine $S_{1}$ which is a summation  \n of the tableaux (with sign) of the form \n\\begin{equation} \n\\begin{array}{|c|}\\hline\n    \\xi \\\\ \\hline \n    \\eta   \\\\ \\hline \n   \\zeta \\\\ \\hline\n\\end{array},\n\\end{equation}\nwhere \\framebox{$\\xi$} and \\framebox{$\\zeta$} are columns with \ntotal length $a-1$ and they do not involve $Q_{b}$. \n\\framebox{$\\eta$} is \\framebox{$b$}, \\framebox{$b+1$}, \n\\framebox{$\\overline{b+1}$} or \\framebox{$\\overline{b}$}\n for $b \\in \\{1,2,\\dots, s+r-1\\}$; \n\\framebox{$s+r$}, \\framebox{$0$} \nor \\framebox{$\\overline{s+r}$} for $b=s+r$. \n Thanks to the relations (\\ref{res1})-(\\ref{res8}), \n $S_{1}$ is free of color $b$ poles under the BAE (\\ref{BAE}). \n Hereafter we consider $S_{k}$  for $k\\ge 2$. \\\\ \n$\\bullet$ \nThe case\n\\footnote{This is void for $B(r|1)$ case.}\n $b \\in \\{1,2,\\dots,s-1 \\}$: \n$S_{k} (k\\ge 2)$ is a  \nsummation of the tableaux (with sign) of the form \n\\begin{eqnarray}\n&& \n\\sum_{n_{1}=0}^{k_{1}}\n\\sum_{n_{2}=0}^{k_{2}} \n\\begin{array}{|c|}\\hline \n    \\xi \\\\ \\hline\n    E_{1n_{1}}    \\\\ \\hline \n    \\eta \\\\ \\hline \n    E_{2n_{2}}  \\\\ \\hline \n    \\zeta \\\\ \\hline \n\\end{array}\n \\nonumber \\\\ \n&& =\n\\left(\n\\sum_{n_{1}=0}^{k_{1}} \n\\begin{array}{|c|}\\hline \n    E_{1n_{1}}    \\\\ \\hline \n\\end{array}\n\\right)\n\\left(\n\\sum_{n_{2}=0}^{k_{2}}\n\\begin{array}{|c|}\\hline \n    E_{2n_{2}}    \\\\ \\hline \n\\end{array}\n\\right)\n\\times \n\\begin{array}{|c|}\\hline \n    \\xi \\\\ \\hline \n\\end{array}\n\\times \n\\begin{array}{|c|}\\hline \n    \\eta \\\\ \\hline \n\\end{array}\n\\times \n\\begin{array}{|c|}\\hline \n    \\zeta \\\\ \\hline \n\\end{array}\n\\label{tableauxk2}, \n\\end{eqnarray} \nwhere \\framebox{$\\xi$}, \\framebox{$\\eta$} and \n\\framebox{$\\zeta$} are columns with total \nlength $a-k$, which do not contain \\framebox{$b$}, \n\\framebox{$b+1$}, \\framebox{$\\overline{b+1}$} and \n\\framebox{$\\overline{b}$}; \n\\framebox{$E_{1n_{1}}$} is a column \n\\footnote{We assume that \n$\\framebox{$E_{10}$}=\n\\begin{array}{|c|l}\\cline{1-1} \n    b+1 & _{v} \\\\ \\cline{1-1} \n   \\vdots & \\\\ \\cline{1-1} \n    b+1 & _{v-2k_{1}+2}\\\\ \\cline{1-1} \n \\end{array}\n$\n and  \n $\\framebox{$E_{1 k_{1}}$}=\n\\begin{array}{|c|l}\\cline{1-1} \n    b & _{v} \\\\ \\cline{1-1} \n   \\vdots & \\\\ \\cline{1-1} \n    b & _{v-2k_{1}+2}\\\\ \\cline{1-1} \n \\end{array}\n$. \\\\ \\ }\nof the form: \n\\begin{eqnarray}\n \\begin{array}{|c|l}\\cline{1-1} \n    b   & _v \\\\ \\cline{1-1} \n    \\vdots & \\\\ \\cline{1-1} \n    b & _{v-2n_{1}+2}\\\\ \\cline{1-1} \n    b+1 & _{v-2n_{1}} \\\\ \\cline{1-1} \n   \\vdots & \\\\ \\cline{1-1} \n    b+1 & _{v-2k_{1}+2}\\\\ \\cline{1-1} \n \\end{array} \n&=&\\frac{Q_{b-1}(v-b+1-2n_{1})Q_{b}(v-b+2)}\n      {Q_{b-1}(v-b+1)Q_{b}(v-b-2n_{1}+2)}  \n\\label{tableauxk3} \\\\ \n&& \\times  \n \\frac{Q_{b}(v-b-2k_{1})Q_{b+1}(v-b+1-2n_{1})}\n      {Q_{b}(v-b-2n_{1})Q_{b+1}(v-b+1-2k_{1})},\n\\nonumber\n\\end{eqnarray}\nwhere $v=u+h_{1}$: $h_{1}$ is some shift parameter \nand \\framebox{$E_{2 n_{2}}$} is a column\n\\footnote{We assume that \n$\\framebox{$E_{20}$}=\n\\begin{array}{|c|l}\\cline{1-1} \n    \\stackrel{\\ }{\\overline{b}} & _{w} \\\\ \\cline{1-1} \n   \\vdots & \\\\ \\cline{1-1} \n    \\stackrel{\\ }{\\overline{b}} & _{w-2k_{2}+2}\\\\ \\cline{1-1} \n \\end{array}\n$\n and  \n $\\framebox{$E_{2 k_{2}}$}=\n\\begin{array}{|c|l}\\cline{1-1} \n    \\stackrel{\\ }{\\overline{b+1}} & _{w} \\\\ \\cline{1-1} \n   \\vdots & \\\\ \\cline{1-1} \n    \\stackrel{\\ }{\\overline{b+1}} \n    & _{w-2k_{2}+2}\\\\ \\cline{1-1} \n \\end{array}\n$.}\n of the form:\n\\begin{eqnarray}\n \\begin{array}{|c|l}\\cline{1-1} \n    \\stackrel{\\ }{\\overline{b+1}} & _w \\\\ \\cline{1-1} \n    \\vdots & \\\\ \\cline{1-1} \n    \\stackrel{\\ }{\\overline{b+1}} & _{w-2n_{2}+2}\\\\ \\cline{1-1} \n    \\stackrel{\\ }{\\overline{b}} & _{w-2n_{2}} \\\\ \\cline{1-1} \n   \\vdots & \\\\ \\cline{1-1} \n    \\stackrel{\\ }{\\overline{b}} & _{w-2k_{2}+2}\\\\ \\cline{1-1} \n \\end{array} \n&=&\\frac{Q_{b-1}(w-2s+2r+b-2n_{2})}\n      {Q_{b-1}(w-2s+2r+b-2k_{2})}  \n\\label{tableauxk5} \\\\ \n&& \\times \n\\frac{Q_{b}(w-2s+2r+b-2k_{2}-1)}{Q_{b}(w-2s+2r+b-2n_{2}-1)}\n\\nonumber \\\\\n&& \\times  \n \\frac{Q_{b}(w-2s+2r+b+1)Q_{b+1}(w-2s+2r+b-2n_{2})}\n {Q_{b}(w-2s+2r+b-2n_{2}+1)Q_{b+1}(w-2s+2r+b)},\n\\nonumber\n\\end{eqnarray} \nwhere \n$w=u+h_{2}$: $h_{2}$ is some shift parameter; \n$k=k_{1}+k_{2}$\n\\footnote{We assume that $\\framebox{$E_{1 n_{1}}$}=1$\n (resp. $\\framebox{$E_{2 n_{2}}$}=1$) \nfor $k_{1}=0$ (resp. $k_{2}=0$). In this case, \n\\framebox{$E_{i n_{i}}$} does \nnot have poles.}. \n\nFor $b \\in \\{1,2,\\dots, s-1\\}$, \n\\framebox{$E_{1 n_{1}}$} has color $b$ poles at\n $u=-h_{1}+b+2n_{1}+u_{p}^{(b)}$ and \n $u=-h_{1}+b+2n_{1}-2+u_{p}^{(b)}$ \n  for $1 \\le n_{1} \\le k_{1}-1$; at  $u=-h_{1}+b+u_{p}^{(b)}$ \nfor $n_{1}=0$ ; at $u=-h_{1}+b+2k_{1}-2+u_{p}^{(b)}$\n for $n_{1}=k_{1}$\n \\footnote{\nWe  assume that these poles at \n$u=-h_{1}+b+2n_{1}+u_{i}^{(b)}$,\n  and \n $u=-h_{1}+b+2n_{1}-2+u_{q}^{(b)}$ \n do not coincide each other \n for any $i,q \\in \\{1,2,\\dots , N_{b}\\} $: \n namely $u_{i}^{(b)}-u_{q}^{(b)} \\ne 2$.}.  \nThe color $b$ residues at \n$u=-h_{1}+b+2n_{1}+u_{p}^{(b)}$  \n in \\framebox{$E_{1 n_{1}}$} and \\framebox{$E_{1 \\> n_{1}+1}$}\n cancel each other under the BAE (\\ref{BAE}). \n Thus, under the BAE\n  (\\ref{BAE}), $\\sum_{n_{1}=0}^{k_{1}}\\framebox{$E_{1 n_{1}}$}$ \n is free of color $b$ poles\n (see Figure \\ref{part-bs}).\n\n\\begin{figure}\n    \\setlength{\\unitlength}{1.5pt}\n    \\begin{center}\n    \\begin{picture}(270,40) \n     \\put(0,0){$\n       \\framebox{$E_{1 0}$} \n       \\stackrel{0}{\\longleftarrow }\n       \\framebox{$E_{1 1}$} \n       \\stackrel{2}{\\longleftarrow } \n       \\cdots \n       \\stackrel{2n-4}{\\longleftarrow }\n       \\framebox{$E_{1 n-1}$} \n       \\stackrel{2n-2}{\\longleftarrow }\n       \\framebox{$E_{1 n}$} \n       \\stackrel{2n}{\\longleftarrow } \n       \\framebox{$E_{1 n+1}$}\n       \\stackrel{2n+2}{\\longleftarrow } \n       \\cdots \n       \\stackrel{2k_{1}-2}{\\longleftarrow }\n       \\framebox{$E_{1 k_{1}}$}\n     $}  \n    \\end{picture}\n  \\end{center}\n  \\caption{Partial Bethe-strap structure of  \n $ E_{1 n}$ \n    for color $b$ poles ($1\\le b \\le s-1$):  \n The number $n$ on the arrow denotes the common color $b$ pole \n $-h_{1}+b+n+u_{k}^{(b)}$ \n of the pair of the tableaux connected by the arrow.   \n This common pole vanishes under the BAE (\\ref{BAE}).}\n  \\label{part-bs}\n\\end{figure}\n \n \\framebox{$E_{2 n_{2}}$} has color $b$ poles at\n $u=-h_{2}+2s-2r-b+2n_{2}-1+u_{p}^{(b)}$ and \n $u=-h_{2}+2s-2r-b+2n_{2}+1+u_{p}^{(b)}$ \n  for $1 \\le n_{2} \\le k_{2}-1$; at  \n  $u=-h_{2}+2s-2r-b+1+u_{p}^{(b)}$ \nfor $n_{2}=0$ ; at \n$u=-h_{2}+2s-2r-b+2k_{2}-1+u_{p}^{(b)}$\n for $n_{2}=k_{2}$. \n The color $b$ residues at \n$u=-h_{2}+2s-2r-b+2n_{2}+1+u_{p}^{(b)}$\n in \\framebox{$E_{2 n_{2}}$} and \\framebox{$E_{2, n_{2}+1}$}\n cancel each other under the BAE (\\ref{BAE}). \n Thus, under the BAE\n  (\\ref{BAE}), $\\sum_{n_{2}=0}^{k_{2}}\\framebox{$E_{2, n_{2}}$}$ \n is free of color $b$ poles. \n  So is $S_{k}$. \\\\ \n$\\bullet$ \nThe case $ b=s $ : $S_{k} (k\\ge 2)$ is a \nsummation of the tableaux (with sign) of the form \n\\begin{eqnarray}\n&& \\begin{array}{|c|}\\hline \n    D_{11}    \\\\ \\hline \n    \\eta \\\\ \\hline \n    D_{21}  \\\\ \\hline\n\\end{array}\n-\\begin{array}{|c|}\\hline \n    D_{11}    \\\\ \\hline \n    \\eta  \\\\ \\hline \n    D_{22}  \\\\ \\hline \n\\end{array}\n-\\begin{array}{|c|}\\hline\n    D_{12}   \\\\ \\hline \n    \\eta  \\\\ \\hline \n    D_{21}  \\\\ \\hline\n\\end{array}\n+\\begin{array}{|c|}\\hline \n    D_{12}  \\\\ \\hline \n    \\eta  \\\\ \\hline \n    D_{22}  \\\\ \\hline \n\\end{array} \\nonumber \\\\ \n&&=(\n\\begin{array}{|c|}\\hline \n    D_{11}  \\\\ \\hline \n\\end{array}\n-\\begin{array}{|c|}\\hline \n    D_{12}  \\\\ \\hline \n\\end{array}\n)(\n\\begin{array}{|c|}\\hline \n    D_{21}  \\\\ \\hline \n\\end{array}\n-\\begin{array}{|c|}\\hline \n    D_{22}  \\\\ \\hline \n\\end{array}\n)\n\\begin{array}{|c|}\\hline \n    \\eta \\\\ \\hline \n\\end{array}\n\\end{eqnarray} \nwhere \\framebox{$\\eta$} is a column with \nlength $a-k$, which does not contain \\framebox{$s$}, \n\\framebox{$s+1$}, \\framebox{$\\overline{s+1}$} and \n\\framebox{$\\overline{s}$}; \n\\framebox{$D_{11}$} is a column \n\\footnote{We assume that \n$\\framebox{$D_{11}$}=\\framebox{$s+1$}_{v}$ if $k_{1}=1$.}\nof the form: \n\\begin{equation}\n\\begin{array}{|c|l}\\cline{1-1}\n    s & _v  \\\\ \\cline{1-1} \n    \\vdots & \\\\  \\cline{1-1}\n    s & _{v-2k_{1}+4} \\\\ \\cline{1-1} \n    s+1 & _{v-2k_{1}+2} \\\\ \\cline{1-1} \n\\end{array}\n= \\frac{Q_{s-1}(v-s-2k_{1}+3)Q_{s}(v-s+2)Q_{s+1}(v-s-2k_{1}+1)}\n       {Q_{s-1}(v-s+1)Q_{s}(v-s-2k_{1}+2)Q_{s+1}(v-s-2k_{1}+3)}; \n\\label{tableauxk1-1}\n\\end{equation}\n\\framebox{$D_{12}$} is a column \nof the form:  \n\\begin{equation}\n\\begin{array}{|c|l}\\cline{1-1}\n    s & _v \\\\ \\cline{1-1}  \n   \\vdots & \\\\ \\cline{1-1} \n    s & _{v-2k_{1}+4}\\\\ \\cline{1-1}\n    s & _{v-2k_{1}+2}\\\\ \\cline{1-1} \n\\end{array}\n=\\frac{Q_{s-1}(v-s-2k_{1}+1)Q_{s}(v-s+2)}\n       {Q_{s-1}(v-s+1)Q_{s}(v-s-2k_{1}+2)}\n\\label{tableauxk1-2}, \n\\end{equation}\nwhere $v=u+h_{1}$: $h_{1}$ is some shift parameter; \n\\framebox{$D_{21}$} is a column\n\\footnote{We assume that \n$\\framebox{$D_{21}$}=\\framebox{$\\overline{s+1}$}_{w}$\n if $k_{2}=1$.}\n of the form: \n\\begin{eqnarray}\n\\begin{array}{|c|l}\\cline{1-1}\n    \\stackrel{\\ }{\\overline{s+1}} & _w  \\\\ \\cline{1-1} \n    \\stackrel{\\ }{\\overline{s}} & _{w-2} \\\\ \\cline{1-1}\n    \\vdots & \\\\  \\cline{1-1} \n    \\stackrel{\\ }{\\overline{s}} & _{w-2k_{2}+2} \\\\ \\cline{1-1} \n\\end{array}\n&=&\\frac{Q_{s-1}(w-s+2r-2)}{Q_{s-1}(w-s+2r-2k_{2})} \n\\label{tableauxk1-3} \\\\ \n&& \\times \n\\frac{Q_{s}(w-s+2r-2k_{2}-1)Q_{s+1}(w-s+2r)}\n     {Q_{s}(w-s+2r-1)Q_{s+1}(w-s+2r-2)}; \n\\nonumber \n\\end{eqnarray}\n\\framebox{$D_{22}$} is a column of the form:\n\\begin{equation}\n\\begin{array}{|c|l}\\cline{1-1}\n    \\stackrel{\\ }{\\overline{s}} & _w \\\\ \\cline{1-1} \n    \\stackrel{\\ }{\\overline{s}} & _{w-2} \\\\ \\cline{1-1}\n    \\vdots & \\\\ \\cline{1-1} \n    \\stackrel{\\ }{\\overline{s}} & _{w-2k_{2}+2}\\\\ \\cline{1-1} \n\\end{array}\n=\\frac{Q_{s-1}(w-s+2r)Q_{s}(w-s+2r-2k_{2}-1)}\n      {Q_{s-1}(w-s+2r-2k_{2})Q_{s}(w-s+2r-1)} \n       \\label{tableauxk1-4}, \n\\end{equation}\nwhere $w=u+h_{2}$: $h_{2}$ is some shift parameter; \n$k=k_{1}+k_{2}$\n \\footnote{Here we discussed the case for \n $k_{1}\\ge 1 $ and $k_{2}\\ge 1 $; \n the case for $k_{1}=0 $ or $k_{2}=0 $ can be treated similarly.}. \nObviously, the color $b=s$ residues at $v=s+2k_{1}-2+u_{j}^{(s)}$ \n in (\\ref{tableauxk1-1}) and \n (\\ref{tableauxk1-2}) cancel each other\n under the BAE (\\ref{BAE}). \nAnd the color $b=s$ residues\n at $w=s-2r+1+u_{j}^{(s)}$ \n in (\\ref{tableauxk1-3}) and\n (\\ref{tableauxk1-4}) cancel each other\n under the BAE (\\ref{BAE}).  \n Thus $S_{k}$ does not \n have color $s$ poles under the BAE (\\ref{BAE}). \\\\ \n$\\bullet$ \nThe case\n\\footnote{This is void for $B(1|s)$ case.}\n $b \\in \\{s+1,s+2,\\dots, s+r-1\\}$ : \nOwing to the admissibility conditions, \nwe have only to consider $S_{k}$ for \n$k=2,3,4$. \n\n$S_{2}$ is a \nsummation of the tableaux (with sign) of the form \n\\begin{eqnarray}\n&& \n\\begin{array}{|c|}\\hline \n    \\xi    \\\\ \\hline\n    b    \\\\ \\hline \n    b+1 \\\\ \\hline \n    \\zeta^{\\prime}  \\\\ \\hline\n\\end{array}\n+\n\\begin{array}{|c|}\\hline \n    \\xi^{\\prime}    \\\\ \\hline\n    \\stackrel{\\ }{\\overline{b+1}} \\\\ \\hline\n    \\stackrel{\\ }{\\overline{b}}   \\\\ \\hline\n    \\zeta  \\\\ \\hline\n\\end{array}\n\\end{eqnarray}\nand \n\\begin{eqnarray}\n&& \n\\begin{array}{|c|}\\hline \n    \\xi    \\\\ \\hline\n    b    \\\\ \\hline \n    \\eta \\\\ \\hline \n    \\stackrel{\\ }{\\overline{b}} \\\\ \\hline\n    \\zeta    \\\\ \\hline\n\\end{array}\n+\n\\begin{array}{|c|}\\hline \n    \\xi    \\\\ \\hline\n    b    \\\\ \\hline \n    \\eta \\\\ \\hline \n    \\stackrel{\\ }{\\overline{b+1}} \\\\ \\hline\n    \\zeta    \\\\ \\hline\n\\end{array}\n+\n\\begin{array}{|c|}\\hline \n    \\xi    \\\\ \\hline\n    b+1    \\\\ \\hline \n    \\eta \\\\ \\hline \n    \\stackrel{\\ }{\\overline{b}} \\\\ \\hline\n    \\zeta    \\\\ \\hline\n\\end{array}\n+\n\\begin{array}{|c|}\\hline \n    \\xi    \\\\ \\hline\n    b+1    \\\\ \\hline \n    \\eta \\\\ \\hline \n    \\stackrel{\\ }{\\overline{b+1}} \\\\ \\hline\n    \\zeta    \\\\ \\hline\n\\end{array}\n \\nonumber \\\\ \n&&=\n\\begin{array}{|c|}\\hline \n    \\xi \\\\ \\hline \n\\end{array}\n(\n\\begin{array}{|c|}\\hline \n    b  \\\\ \\hline \n\\end{array}\n+\n\\begin{array}{|c|}\\hline \n    b+1  \\\\ \\hline \n\\end{array}\n)\n\\begin{array}{|c|}\\hline \n    \\eta \\\\ \\hline \n\\end{array}\n(\n\\begin{array}{|c|}\\hline \n    \\stackrel{\\ }{\\overline{b}}  \\\\ \\hline \n\\end{array}\n+\n\\begin{array}{|c|}\\hline \n    \\stackrel{\\ }{\\overline{b+1}} \\\\ \\hline \n\\end{array}\n)\n\\begin{array}{|c|}\\hline \n    \\zeta \\\\ \\hline \n\\end{array},\n\\end{eqnarray} \nwhere \n\\{ \\framebox{$\\xi$}, \\framebox{$\\eta$}, \\framebox{$\\zeta$} \\}, \n\\{ \\framebox{$\\xi$}, \\framebox{$\\zeta^{\\prime}$} \\} and \n\\{ \\framebox{$\\xi^{\\prime}$}, \\framebox{$\\zeta$} \\}\n are  columns with total \nlength $a-2$, which do not contain \\framebox{$b$}, \n\\framebox{$b+1$}, \\framebox{$\\overline{b+1}$} and \n\\framebox{$\\overline{b}$}.  \nThus, owing to Lemma \\ref{le-tate},\nthe relations (\\ref{res3}) and (\\ref{res6}), $S_{2}$ does not \n have color $b$ poles under the BAE (\\ref{BAE}). \n \n$S_{3}$ is a \nsummation of the tableaux (with sign) of the form \n\\begin{eqnarray}\n&& \n\\begin{array}{|c|}\\hline \n    \\xi    \\\\ \\hline\n    b    \\\\ \\hline \n    \\eta \\\\ \\hline \n    \\stackrel{\\ }{\\overline{b+1}} \\\\ \\hline\n    \\stackrel{\\ }{\\overline{b}} \\\\ \\hline\n    \\zeta    \\\\ \\hline\n\\end{array}\n+\n\\begin{array}{|c|}\\hline \n    \\xi    \\\\ \\hline\n    b+1    \\\\ \\hline \n    \\eta \\\\ \\hline \n    \\stackrel{\\ }{\\overline{b+1}} \\\\ \\hline\n    \\stackrel{\\ }{\\overline{b}} \\\\ \\hline\n    \\zeta    \\\\ \\hline\n\\end{array}\n+\n\\begin{array}{|c|}\\hline \n    \\xi    \\\\ \\hline\n    b    \\\\ \\hline\n    b+1    \\\\ \\hline \n    \\eta^{\\prime} \\\\ \\hline \n    \\stackrel{\\ }{\\overline{b}} \\\\ \\hline\n    \\zeta    \\\\ \\hline\n\\end{array}\n+\n\\begin{array}{|c|}\\hline \n    \\xi    \\\\ \\hline\n    b    \\\\ \\hline\n    b+1    \\\\ \\hline \n    \\eta^{\\prime} \\\\ \\hline \n    \\stackrel{\\ }{\\overline{b+1}} \\\\ \\hline\n    \\zeta    \\\\ \\hline\n\\end{array}\n,\n\\end{eqnarray} \nwhere \\{ \\framebox{$\\xi$}, \\framebox{$\\eta$}, \\framebox{$\\zeta$} \\} \n and \\{ \\framebox{$\\xi$}, \\framebox{$\\eta^{\\prime }$},\n  \\framebox{$\\zeta$} \\} \nare  columns with \ntotal length $a-3$, which do not contain \\framebox{$b$}, \n\\framebox{$b+1$}, \\framebox{$\\overline{b+1}$} and \n\\framebox{$\\overline{b}$}.\nThus, owing to the relations (\\ref{res3}), (\\ref{res6}) and \nLemma \\ref{le-tate}, $S_{3}$ does not \n have color $b$ poles under the BAE (\\ref{BAE}). \n\n$S_{4}$ is a \nsummation of the tableaux (with sign) of the form \n\\begin{eqnarray}\n&& \n\\begin{array}{|c|}\\hline \n    \\xi    \\\\ \\hline\n    b    \\\\ \\hline \n    b+1    \\\\ \\hline\n    \\eta \\\\ \\hline \n    \\stackrel{\\ }{\\overline{b+1}} \\\\ \\hline\n    \\stackrel{\\ }{\\overline{b}} \\\\ \\hline\n    \\zeta    \\\\ \\hline\n\\end{array}\n,\n\\end{eqnarray} \nwhere \\framebox{$\\xi$}, \\framebox{$\\eta$} and \n\\framebox{$\\zeta$} are  columns with \ntotal length $a-4$, which do not contain \\framebox{$b$}, \n\\framebox{$b+1$}, \\framebox{$\\overline{b+1}$} and \n\\framebox{$\\overline{b}$}.\nThus, owing to the \nLemma \\ref{le-tate}, $S_{4}$ does not \n have color $b$ poles under the BAE (\\ref{BAE}).\n \\\\ \n$\\bullet$ \nThe case $b=s+r$ :  \n$S_{k}$ ($k\\ge 2$) is a \nsummation of the tableaux (with sign) of the form \n\\begin{eqnarray}\n\\begin{array}{|c|l}\\cline{1-1} \n    \\xi &   \\\\ \\cline{1-1}\n    0  & _{v}  \\\\ \\cline{1-1} \n    0  & _{v-2} \\\\ \\cline{1-1}\n    \\vdots &  \\\\ \\cline{1-1}\n    0  & _{v-2k+4} \\\\ \\cline{1-1}\n    0 & _{v-2k+2} \\\\ \\cline{1-1}\n    \\zeta   & \\\\ \\cline{1-1}\n\\end{array}\n+\n\\begin{array}{|c|l}\\cline{1-1} \n    \\xi &   \\\\ \\cline{1-1}\n    s+r  & _{v}  \\\\ \\cline{1-1} \n    0  & _{v-2} \\\\ \\cline{1-1}\n    \\vdots &  \\\\ \\cline{1-1}\n    0  & _{v-2k+4} \\\\ \\cline{1-1}\n    0 & _{v-2k+2} \\\\ \\cline{1-1}\n    \\zeta   & \\\\ \\cline{1-1}\n\\end{array}\n&+&\n\\begin{array}{|c|l}\\cline{1-1} \n    \\xi &   \\\\ \\cline{1-1}\n    0  & _{v}  \\\\ \\cline{1-1} \n    0  & _{v-2} \\\\ \\cline{1-1}\n    \\vdots &  \\\\ \\cline{1-1}\n    0  & _{v-2k+4} \\\\ \\cline{1-1}\n    \\stackrel{\\ }{\\overline{s+r}} & _{v-2k+2} \\\\ \\cline{1-1}\n    \\zeta   & \\\\ \\cline{1-1}\n\\end{array}\n+\n\\begin{array}{|c|l}\\cline{1-1} \n    \\xi &   \\\\ \\cline{1-1}\n    s+r  & _{v}  \\\\ \\cline{1-1} \n    0  & _{v-2} \\\\ \\cline{1-1}\n    \\vdots &  \\\\ \\cline{1-1}\n    0  & _{v-2k+4} \\\\ \\cline{1-1}\n    \\stackrel{\\ }{\\overline{s+r}} & _{v-2k+2} \\\\ \\cline{1-1}\n    \\zeta  &  \\\\ \\cline{1-1}\n\\end{array}\n \\nonumber \\\\ \n&& =\nA(v)B(v) \\times \\framebox{$\\xi$} \\times \\framebox{$\\eta$},\n\\end{eqnarray} \nwhere $v=u+h_{3}$: $h_{3}$ is some shift parameter; \n\\framebox{$\\xi$} and \n\\framebox{$\\zeta$} are columns with total \nlength $a-k$, which do not contain \\framebox{$s+r$}, \n\\framebox{$0$} and \n\\framebox{$\\overline{s+r}$};\n\\begin{eqnarray}\n A(v)&=&\\frac{Q_{s+r}(v-s+r+1)}{Q_{s+r}(v-s+r)} \n  \\nonumber \\\\ \n && +\\frac{Q_{s+r-1}(v-s+r+1)Q_{s+r}(v-s+r-1)}\n       {Q_{s+r-1}(v-s+r-1)Q_{s+r}(v-s+r)}, \\\\ \nB(v)&=& \\frac{Q_{s+r}(v-s+r-2k)}{Q_{s+r}(v-s+r-2k+1)} \n\\nonumber \\\\ \n && +\\frac{Q_{s+r-1}(v-s+r-2k)Q_{s+r}(v-s+r-2k+2)}\n       {Q_{s+r-1}(v-s+r-2k+2)Q_{s+r}(v-s+r-2k+1)}.\\nonumber \n\\end{eqnarray}\nOne can check $A(v)$ and $B(v)$ are free of color $s+r$\n poles under the BAE (\\ref{BAE}). \n Thus,  $S_{k}$ does not \n have color $s+r$ poles under the BAE (\\ref{BAE}). \n \\rule{5pt}{10pt} \\\\ \n {\\em Remark}: There is another proof for\n  Theorem \\ref{th-tate} by the determinant formula \n  (\\ref{Jacobi-Trudi2}): \n  for $b \\in \\{ 1,2,\\dots, s-1 \\}$, we prove \n  ${\\cal T}_{m}(u)$ is free of color $b$ poles, \n  and then the pole-freeness of ${\\cal T}^{a}(u)$ follows \n  from (\\ref{Jacobi-Trudi2}); while \n  for $b \\in \\{ s,s+1, \\dots, s+r \\}$, we prove \n  ${\\cal T}^{a}(u)$ is free of color $b$ poles in the \n  same way as the above-mentioned proof.  \n  An advantage of this another proof \n  is that we do not encounter an \n  awkward expression like (\\ref{tableauxk2}). \n  We note that similar idea is also applicable for \n  $sl(r|s)$ case \\cite{T1,T2,T3}. \n\\setcounter{equation}{0}\n\\renewcommand{\\theequation}{A.2.\\arabic{equation}}\n\\section*{Appendix A.2  Outline of the proof of Theorem \n\\ref{th-tate}: $B(0|s)$ ($s \\in {\\bf Z}_{\\ge 1}$) case}\n We will show that ${\\cal T}_{m}(u)$ \nis free of color $b$ poles, namely, \n$Res_{u=u_{k}^{(b)}+\\cdots}{\\cal T}_{m}(u)=0$ for any \n $ b \\in \\{1,2,\\dots, s \\} $\n under the condition that the BAE \n (\\ref{BAE1})-(\\ref{BAE4}) is valid. \n The function $\\framebox{$c$}_{u}$  \n (\\ref{z+}) with $c \\in J $ has \n color $b$ poles only for $c=b$, $b+1$, \n$\\overline{b+1}$ or $\\overline{b}$ if \n$b\\in \\{1,2,\\dots,s-1 \\} $; \nfor $c=s$, $0$ or $\\overline{s}$ if $b=s$, \nso we shall trace only \n\\framebox{$b$}, \\framebox{$b+1$}, \n\\framebox{$\\overline{b+1}$} or \\framebox{$\\overline{b}$}\n for $b\\in\\{1,2,\\dots, s-1 \\}$; \n\\framebox{$s$}, \\framebox{$0$} \nor \\framebox{$\\overline{s}$} for $b=s$.\n Let $S_{k}$ be the partial sum of ${\\cal T}_{m}(u)$, \n which contains \n$k$ boxes among \\framebox{$b$}, \\framebox{$b+1$}, \n\\framebox{$\\overline{b+1}$} or \\framebox{$\\overline{b}$}\n for $b\\in\\{1,2,\\dots, s-1 \\}$; \n\\framebox{$s$}, \\framebox{$0$} \nor \\framebox{$\\overline{s}$} for $b=s$. \n Apparently, $S_{0}$ does not have \ncolor $b$ poles. \n\n Now we examine $S_{1}$, which is a summation  \n of the tableaux (with sign) of the form \n\\begin{equation} \n\\begin{array}{|c|c|c|}\\hline\n    \\xi &  \\eta & \\zeta \n    \\\\ \\hline \n\\end{array}\n\\end{equation}\nwhere \\framebox{$\\xi$} and \\framebox{$\\zeta$} are rows with \ntotal length $m-1$ and they do not involve $Q_{b}$. \n\\framebox{$\\eta$} is \\framebox{$b$}, \\framebox{$b+1$}, \n\\framebox{$\\overline{b+1}$} or \\framebox{$\\overline{b}$}\n for $b\\in\\{1,2,\\dots, s-1 \\}$; \n\\framebox{$s$}, \\framebox{$0$} \nor \\framebox{$\\overline{s}$} for $b=s$. \n Owing to the relations (\\ref{res1-b0s})-(\\ref{res4-b0s}), \n $S_{1}$ is free of color $b$ poles under the BAEs \n (\\ref{BAE1})-(\\ref{BAE4}). \n From now on, we consider $S_{k}$  for $k\\ge 2$. \\\\ \n$\\bullet$ \nThe case\n\\footnote{This is void for $B(0|1)$ case.}\n $b\\in\\{1,2,\\dots, s-1 \\}$: Owing to the admissibility\n conditions, \nwe have only to consider $S_{2}$, $S_{3}$ and $S_{4}$. \n\n$S_{2}$ is a  \nsummation of the tableaux (with sign) of the form \n\\begin{eqnarray}\n \\begin{array}{|c|c|c|c|}\\hline \n   \\xi & b & b+1 & \\eta^{\\prime} \\\\\n    \\hline \n \\end{array}\n +\n \\begin{array}{|c|c|c|c|}\\hline \n   \\xi^{\\prime} & \\stackrel{\\ }{\\overline{b+1}} & \n               \\overline{b} & \\zeta\n     \\\\ \\hline \n \\end{array}\n\\end{eqnarray}\nand \n\\begin{eqnarray}\n && \n \\begin{array}{|c|c|c|c|c|}\\hline \n   \\xi & b & \\eta & \\stackrel{\\ }{\\overline{b}} & \\zeta\n    \\\\ \\hline \n \\end{array}\n+\n\\begin{array}{|c|c|c|c|c|}\\hline \n   \\xi & b & \\eta & \\stackrel{\\ }{\\overline{b+1}} & \\zeta\n    \\\\ \\hline \n \\end{array} \n \\nonumber \\\\ \n&& +\n \\begin{array}{|c|c|c|c|c|}\\hline \n   \\xi & b+1 & \\eta & \\stackrel{\\ }{\\overline{b}} & \\zeta\n    \\\\ \\hline \n \\end{array}\n+\n \\begin{array}{|c|c|c|c|c|}\\hline \n   \\xi & b+1 & \\eta & \\stackrel{\\ }{\\overline{b+1}} & \\zeta\n    \\\\ \\hline \n \\end{array} \n \\nonumber \\\\ \n&& =\n \\begin{array}{|c|}\\hline \n   \\xi \\\\ \\hline \n \\end{array} \n (\n \\begin{array}{|c|}\\hline \n   b \\\\ \\hline \n \\end{array}\n +\n \\begin{array}{|c|}\\hline \n   b+1 \\\\ \\hline \n \\end{array}\n )\n \\begin{array}{|c|}\\hline \n   \\eta \\\\ \\hline \n \\end{array}\n (\n \\begin{array}{|c|}\\hline \n   \\stackrel{\\ }{\\overline{b}} \\\\ \\hline \n \\end{array}\n +\n \\begin{array}{|c|}\\hline \n   \\stackrel{\\ }{\\overline{b+1}} \\\\ \\hline \n \\end{array}\n )\n \\begin{array}{|c|}\\hline \n   \\zeta \\\\ \\hline \n \\end{array}\n\\end{eqnarray}\nwhere (\\framebox{$\\xi$}, \\framebox{$\\eta^{\\prime}$}),\n(\\framebox{$\\xi^{\\prime}$}, \\framebox{$\\zeta$}) and \n(\\framebox{$\\xi$}, \\framebox{$\\eta$}, \\framebox{$\\zeta$}) \n are rows with total \nlength $m-2$, which do not contain \\framebox{$b$}, \n\\framebox{$b+1$}, \\framebox{$\\overline{b+1}$} and \n\\framebox{$\\overline{b}$}. \nThus, owing to Lemma \\ref{le-yoko},\nthe relations (\\ref{res1-b0s}) and (\\ref{res4-b0s}),\n $S_{2}$ does not \n have color $b$ poles under the BAE (\\ref{BAE1}) and (\\ref{BAE2}). \n\n $S_{3}$ is a  \nsummation of the tableaux (with sign) of the form \n\\begin{eqnarray}\n && \n \\begin{array}{|c|c|c|c|c|c|}\\hline \n   \\xi & b & b+1 & \\eta & \\stackrel{\\ }{\\overline{b}} & \\zeta\n    \\\\ \\hline \n \\end{array}\n+\n\\begin{array}{|c|c|c|c|c|c|}\\hline \n   \\xi & b & b+1 & \\eta & \\stackrel{\\ }{\\overline{b+1}} & \\zeta\n    \\\\ \\hline \n \\end{array} \n \\nonumber \\\\ \n&& =\n \\begin{array}{|c|c|c|c|}\\hline \n   \\xi & b & b+1 & \\eta \\\\ \\hline \n \\end{array} \n (\n \\begin{array}{|c|}\\hline \n   \\stackrel{\\ }{\\overline{b}} \\\\ \\hline \n \\end{array}\n +\n \\begin{array}{|c|}\\hline \n   \\stackrel{\\ }{\\overline{b+1}} \\\\ \\hline \n \\end{array}\n )\n \\begin{array}{|c|}\\hline \n   \\zeta \\\\ \\hline \n \\end{array}\n\\end{eqnarray}\nand \n\\begin{eqnarray}\n && \n \\begin{array}{|c|c|c|c|c|c|}\\hline \n   \\xi & b & \\eta^{\\prime} & \\stackrel{\\ }{\\overline{b+1}} \n     & \\overline{b} & \\zeta\n    \\\\ \\hline \n \\end{array}\n+\n\\begin{array}{|c|c|c|c|c|c|}\\hline \n   \\xi & b+1 & \\eta^{\\prime} & \\stackrel{\\ }{\\overline{b+1}}\n      & \\overline{b} & \\zeta\n    \\\\ \\hline \n \\end{array} \n \\nonumber \\\\ \n&& =\n \\begin{array}{|c|}\\hline \n   \\xi  \\\\ \\hline \n \\end{array} \n (\n \\begin{array}{|c|}\\hline \n   b \\\\ \\hline \n \\end{array}\n +\n \\begin{array}{|c|}\\hline \n   b+1 \\\\ \\hline \n \\end{array}\n )\n \\begin{array}{|c|c|c|c|}\\hline \n   \\eta^{\\prime} & \\stackrel{\\ }{\\overline{b+1}} \n     & \\overline{b} & \\zeta \\\\ \\hline \n \\end{array}\n\\end{eqnarray}\nwhere \n(\\framebox{$\\xi $}, \\framebox{$\\eta$}, \\framebox{$\\zeta$}) and \n(\\framebox{$\\xi$}, \\framebox{$\\eta^{\\prime}$}, \\framebox{$\\zeta$}) \n are rows with total \nlength $m-3$, which do not contain \\framebox{$b$}, \n\\framebox{$b+1$}, \\framebox{$\\overline{b+1}$} and \n\\framebox{$\\overline{b}$}. \nThus, owing to Lemma \\ref{le-yoko},\nthe relations (\\ref{res1-b0s}) and (\\ref{res4-b0s}), $S_{3}$ does not \n have color $b$ poles under the BAE (\\ref{BAE1}) and (\\ref{BAE2}).\n\n$S_{4}$ is a  \nsummation of the tableaux (with sign) of the form \n\\begin{eqnarray}\n && \n \\begin{array}{|c|c|c|c|c|c|c|}\\hline \n   \\xi & b & b+1 & \\eta & \\stackrel{\\ }{\\overline{b+1}} \n     & \\overline{b} & \\zeta\n    \\\\ \\hline \n \\end{array}\n\\end{eqnarray}\nwhere \n \\framebox{$\\xi$}, \\framebox{$\\eta$} and \\framebox{$\\zeta$} \n are rows with total \nlength $m-4$, which do not contain \\framebox{$b$}, \n\\framebox{$b+1$}, \\framebox{$\\overline{b+1}$} and \n\\framebox{$\\overline{b}$}. \nThus, owing to Lemma \\ref{le-yoko}, $S_{4}$ does not \n have color $b$ poles. \n \n$\\bullet$ The case $ b=s $ : \nOwing to the admissibility\n conditions, \nwe have only to consider $S_{2}$ and $S_{3}$. \n\n$S_{2}$ is a \nsummation of the tableaux (with sign) of the form \n\\begin{eqnarray}\n \\hspace{-20pt} && \n \\begin{array}{|c|c|c|c|}\\hline \n   \\xi & s & 0 & \\eta \n    \\\\ \\hline \n \\end{array}\n =\\frac{Q_{s-1}(v-s-1)Q_{s}(v-s+3)}{Q_{s-1}(v-s+1)Q_{s}(v-s+1)}\n \\times \n \\begin{array}{|c|}\\hline \n    \\xi \n    \\\\ \\hline \n \\end{array}\n \\times \n \\begin{array}{|c|}\\hline \n    \\eta \n    \\\\ \\hline \n \\end{array},\n \\\\ \n \\hspace{-20pt} && \n \\begin{array}{|c|c|c|c|}\\hline \n   \\xi & s & \\stackrel{\\ }{\\overline{s}} & \\eta \n    \\\\ \\hline \n \\end{array}\n =\\frac{Q_{s-1}(v-s-1)Q_{s}(v-s+2)}{Q_{s-1}(v-s+1)Q_{s}(v-s)}\n  \\nonumber \\\\ \n \\hspace{-20pt}&& \\hspace{70pt} \\times \n\\frac{Q_{s-1}(v-s+2)Q_{s}(v-s-1)}{Q_{s-1}(v-s)Q_{s}(v-s+1)}\n\\times \n \\begin{array}{|c|}\\hline \n    \\xi \n    \\\\ \\hline \n \\end{array}\n \\times \n \\begin{array}{|c|}\\hline \n    \\eta \n    \\\\ \\hline \n \\end{array}, \\\\\n \\hspace{-20pt}  && \n \\begin{array}{|c|c|c|c|}\\hline \n   \\xi & 0 & \\stackrel{\\ }{\\overline{s}} & \\eta \n    \\\\ \\hline \n \\end{array}\n =\\frac{Q_{s-1}(v-s+2)Q_{s}(v-s-2)}{Q_{s-1}(v-s)Q_{s}(v-s)}\n \\times \n \\begin{array}{|c|}\\hline \n    \\xi \n    \\\\ \\hline \n \\end{array}\n \\times \n \\begin{array}{|c|}\\hline \n    \\eta \n    \\\\ \\hline \n \\end{array}, \n\\end{eqnarray}\nwhere \n \\framebox{$\\xi$} and \\framebox{$\\eta$}  \n are rows with total length $m-2$, which do not contain  \n\\framebox{$s$}, \\framebox{$0$} and \n\\framebox{$\\overline{s}$}; $v=u+h$: $h$ is a shift parameter. \n The color $s$ residues at $u=u_{k}^{(s)}+s-1-h$ of \n the functions \n$\n\\begin{array}{|c|c|c|c|}\\hline \n   \\xi & s & 0 & \\eta \n    \\\\ \\hline \n \\end{array}\n$ and \n$\n\\begin{array}{|c|c|c|c|}\\hline \n   \\xi & s & \\stackrel{\\ }{\\overline{s}} & \\eta \n    \\\\ \\hline \n \\end{array}\n$ \ncancel each other under the BAE (\\ref{BAE3}) or (\\ref{BAE4}).\nThe color $s$ residues at $u=u_{k}^{(s)}+s-h$ of \n the functions \n$\n\\begin{array}{|c|c|c|c|}\\hline \n   \\xi & s & \\stackrel{\\ }{\\overline{s}} & \\eta \n    \\\\ \\hline \n \\end{array}\n$ \nand \n$\n \\begin{array}{|c|c|c|c|}\\hline \n   \\xi & 0 & \\stackrel{\\ }{\\overline{s}} & \\eta \n    \\\\ \\hline \n \\end{array}\n$\ncancel each other under the BAE (\\ref{BAE3}) or (\\ref{BAE4}). \nThus, $S_{2}$ does not \n have color $s$ poles under the BAE (\\ref{BAE3}) or (\\ref{BAE4}).\n \n$S_{3}$ is a  \nsummation of the tableaux (with sign) of the form \n\\begin{eqnarray}\n \\begin{array}{|c|c|c|c|c|}\\hline \n   \\xi & s & 0 & \\stackrel{\\ }{\\overline{s}} & \\eta \n    \\\\ \\hline \n \\end{array}\n\\end{eqnarray}\nwhere \n \\framebox{$\\xi$} and \\framebox{$\\eta$}  \n are rows with total \nlength $m-3$, which do not contain  \n\\framebox{$s$}, \\framebox{$0$} and \n\\framebox{$\\overline{s}$}. \nThus, owing to Lemma \\ref{le-yoko}, $S_{3}$ does not \n have color $s$ poles. \n \nThen $ {\\cal T}_{m}(u)$ is free of poles under \nthe condition that the BAEs \n(\\ref{BAE1})-(\\ref{BAE4}) are valid; \nowing to the relation (\\ref{Jacobi-Trudi2}), \nthis also hold true for \n$ {\\cal T}_{\\lambda \\subset \\mu}(u)$. In particular, \n the pole-freeness of ${\\cal T}^{a}(u)$ follows \n immediately.\n\\setcounter{equation}{0}\n\\renewcommand{\\theequation}{A.3.\\arabic{equation}}\n\\section*{Appendix A.3  Outline of the proof of Theorem \n\\ref{th-tate}: $D(r|s)$ \n($r \\in {\\bf Z}_{\\ge 2}$, $s \\in {\\bf Z}_{\\ge 1}$) case}\nWe prove that ${\\cal T}^{a}(u)$ \nis free of color $b$ poles, that is, \n$Res_{u=u_{k}^{(b)}+\\cdots}{\\cal T}^{a}(u)=0$ for any \n $ b \\in \\{1,2,\\dots, s+r \\} $\n under the condition that the BAE (\\ref{BAE}) is valid. \n The function $\\framebox{$c$}_{u}$  \n (\\ref{z++}) with $c \\in J $ has \n color $b$ poles only for $c=b$, $b+1$, \n$\\overline{b+1}$ or $\\overline{b}$ if \n$b\\in \\{1,2,\\dots,s+r-1 \\} $; \nfor $c=s+r-1$, $s+r$, $\\overline{s+r}$ or $\\overline{s+r-1}$ \nif $b=s+r$, \nso we shall trace only \n\\framebox{$b$}, \\framebox{$b+1$}, \n\\framebox{$\\overline{b+1}$} or \\framebox{$\\overline{b}$}\n for $b\\in\\{1,2,\\dots, s+r-1 \\}$; \n \\framebox{$s+r-1$}, \\framebox{$s+r$}, \\framebox{$\\overline{s+r}$} \n or \\framebox{$\\overline{s+r-1}$} \n for $b=s+r$.\n Let $S_{k}$ be the partial sum of ${\\cal T}^{a}(u)$, \n which contains \n$k$ boxes among \\framebox{$b$}, \\framebox{$b+1$}, \n\\framebox{$\\overline{b+1}$} or \\framebox{$\\overline{b}$}\n for $b\\in \\{1,2,\\dots,s+r-1 \\} $; \n\\framebox{$s+r-1$}, \\framebox{$s+r$}, \\framebox{$\\overline{s+r}$}\n or \\framebox{$\\overline{s+r-1}$} \nfor $b=s+r$.\n Apparently, $S_{0}$ does not have \ncolor $b$ poles. \n\n Next we consider $S_{1}$, which is a summation  \n of the tableaux (with sign) of the form \n\\begin{equation} \n\\begin{array}{|c|}\\hline\n    \\xi \\\\ \\hline \n    \\eta   \\\\ \\hline \n   \\zeta \\\\ \\hline\n\\end{array},\n\\end{equation}\nwhere \\framebox{$\\xi$} and \\framebox{$\\zeta$} are columns with \ntotal length $a-1$ and they do not contain $Q_{b}$. \n\\framebox{$\\eta$} is \\framebox{$b$}, \\framebox{$b+1$}, \n\\framebox{$\\overline{b+1}$} or \\framebox{$\\overline{b}$}\n for $b\\in \\{1,2,\\dots,s+r-1 \\} $; \n\\framebox{$s+r-1$}, \\framebox{$s+r$}, \\framebox{$\\overline{s+r}$} \n or \\framebox{$\\overline{s+r-1}$} for $b=s+r$. \n Owing to the relations (\\ref{res1-d})-(\\ref{res8-d}), \n $S_{1}$ is free of color $b$ poles under the BAE (\\ref{BAE}). \n From now on we consider $S_{k}$  for $k\\ge 2$. \\\\ \n$\\bullet$ \nThe case $b\\in \\{1,2,\\dots,s+r-2 \\} $: \nThe proof is similar to $B(r|s)$ ($r\\in {\\bf Z}_{\\ge 1}$) case, \nso we omit it. \\\\ \n$\\bullet$ \nThe case $b=s+r-1$ or $b=s+r$:  \n$S_{2n}$ ($k=2n$, $n \\in {\\bf Z}_{\\ge 2}$ \n\\footnote{The case $n=1$ can be treated similarly.}) is a \nsummation of the tableaux (with signs) of the form \n\\begin{eqnarray}\n&&\n\\begin{array}{|c|}\\hline \n    \\xi  \\\\ \\hline\n    s+r-1  \\\\ \\hline \n    \\stackrel{\\ }{\\overline{s+r}} \\\\ \\hline \n    s+r  \\\\ \\hline\n    \\vdots \\\\  \\hline\n    \\stackrel{\\ }{\\overline{s+r}} \\\\ \\hline\n    s+r  \\\\ \\hline\n    \\stackrel{\\ }{\\overline{s+r-1}}  \\\\ \\hline\n    \\zeta  \\\\ \\hline\n\\end{array}\n+\n\\begin{array}{|c|}\\hline \n    \\xi \\\\ \\hline\n    s+r-1  \\\\ \\hline \n    \\stackrel{\\ }{\\overline{s+r}} \\\\ \\hline \n    s+r  \\\\ \\hline\n    \\vdots \\\\ \\hline\n    \\stackrel{\\ }{\\overline{s+r}} \\\\ \\hline\n    s+r  \\\\ \\hline\n    \\stackrel{\\ }{\\overline{s+r}}  \\\\ \\hline\n    \\zeta  \\\\ \\hline\n\\end{array}\n+\n\\begin{array}{|c|}\\hline \n    \\xi    \\\\ \\hline\n    s+r   \\\\ \\hline\n    \\stackrel{\\ }{\\overline{s+r}} \\\\ \\hline \n    s+r   \\\\ \\hline\n    \\vdots \\\\ \\hline\n    \\stackrel{\\ }{\\overline{s+r}} \\\\ \\hline\n    s+r \\\\ \\hline\n    \\stackrel{\\ }{\\overline{s+r-1}} \\\\ \\hline\n    \\zeta  \\\\ \\hline\n\\end{array}\n+\n\\begin{array}{|c|l}\\cline{1-1} \n    \\xi & \\\\ \\cline{1-1}\n    s+r & _{v} \\\\ \\cline{1-1} \n    \\stackrel{\\ }{\\overline{s+r}}  & _{v-2}  \\\\ \\cline{1-1} \n    s+r  & _{v-4} \\\\ \\cline{1-1}\n    \\vdots &  \\\\ \\cline{1-1}\n    \\stackrel{\\ }{\\overline{s+r}}  & _{v-4n+6} \\\\ \\cline{1-1}\n    s+r & _{v-4n+4} \\\\ \\cline{1-1}\n    \\stackrel{\\ }{\\overline{s+r}} & _{v-4n+2} \\\\ \\cline{1-1}\n    \\zeta  &  \\\\ \\cline{1-1}\n\\end{array}\n\\nonumber \\\\ \n&&=\nA(v)B(v) \\times \\framebox{$\\xi$} \\times \\framebox{$\\zeta$}\n\\label{even-nul}\n\\end{eqnarray} \nand \n\\begin{eqnarray}\n&&\n\\begin{array}{|c|}\\hline \n    \\xi  \\\\ \\hline\n    \\stackrel{\\ }{\\overline{s+r}}  \\\\ \\hline \n    s+r  \\\\ \\hline\n    \\stackrel{\\ }{\\overline{s+r}} \\\\ \\hline \n    s+r  \\\\ \\hline\n    \\vdots \\\\  \\hline\n    \\stackrel{\\ }{\\overline{s+r}} \\\\ \\hline\n    s+r  \\\\ \\hline\n    \\stackrel{\\ }{\\overline{s+r}}  \\\\ \\hline\n    s+r  \\\\ \\hline\n    \\zeta  \\\\ \\hline\n\\end{array}\n+\n\\begin{array}{|c|}\\hline \n    \\xi \\\\ \\hline\n    s+r-1  \\\\ \\hline \n    s+r    \\\\ \\hline\n    \\stackrel{\\ }{\\overline{s+r}} \\\\ \\hline \n    s+r  \\\\ \\hline\n    \\vdots \\\\ \\hline\n    \\stackrel{\\ }{\\overline{s+r}} \\\\ \\hline\n    s+r  \\\\ \\hline\n    \\stackrel{\\ }{\\overline{s+r}}  \\\\ \\hline\n    s+r  \\\\ \\hline\n    \\zeta  \\\\ \\hline\n\\end{array}\n+\n\\begin{array}{|c|}\\hline \n    \\xi    \\\\ \\hline\n    \\stackrel{\\ }{\\overline{s+r}} \\\\ \\hline \n    s+r   \\\\ \\hline\n    \\stackrel{\\ }{\\overline{s+r}} \\\\ \\hline \n    s+r   \\\\ \\hline\n    \\vdots \\\\ \\hline\n    \\stackrel{\\ }{\\overline{s+r}} \\\\ \\hline\n    s+r \\\\ \\hline\n    \\stackrel{\\ }{\\overline{s+r}} \\\\ \\hline\n    \\stackrel{\\ }{\\overline{s+r-1}} \\\\ \\hline\n    \\zeta  \\\\ \\hline\n\\end{array}\n+\n\\begin{array}{|c|l}\\cline{1-1} \n    \\xi & \\\\ \\cline{1-1}\n    s+r-1 & _{v} \\\\ \\cline{1-1} \n    s+r  & _{v-2} \\\\ \\cline{1-1}\n    \\stackrel{\\ }{\\overline{s+r}} & _{v-4}  \\\\ \\cline{1-1} \n    s+r  & _{v-6} \\\\ \\cline{1-1}\n    \\vdots &  \\\\ \\cline{1-1}\n    \\stackrel{\\ }{\\overline{s+r}}  & _{v-4n+8} \\\\ \\cline{1-1}\n    s+r & _{v-4n+6} \\\\ \\cline{1-1}\n    \\stackrel{\\ }{\\overline{s+r}}  & _{v-4n+4} \\\\ \\cline{1-1}\n    \\stackrel{\\ }{\\overline{s+r-1}} & _{v-4n+2} \\\\ \\cline{1-1}\n    \\zeta  &  \\\\ \\cline{1-1}\n\\end{array}\n\\nonumber \\\\ \n&&=\nC(v)D(v) \\times \\framebox{$\\xi$} \\times \\framebox{$\\zeta$},\n\\end{eqnarray} \nwhere $v=u+h_{1}$: $h_{1}$ is some shift parameter; \n\\framebox{$\\xi$} and \n\\framebox{$\\zeta$} are columns with total \nlength $a-2n$, which do not contain \\framebox{$s+r-1$}, \n\\framebox{$s+r$}, \\framebox{$\\overline{s+r}$} and \n\\framebox{$\\overline{s+r-1}$};\n\\begin{eqnarray}\nA(v)&=&\\frac{Q_{s+r-1}(v-s+r+1)}{Q_{s+r-1}(v-s+r-1)} \n  \\nonumber \\\\ \n && +\\frac{Q_{s+r-2}(v-s+r)Q_{s+r-1}(v-s+r-3)}\n       {Q_{s+r-2}(v-s+r-2)Q_{s+r-1}(v-s+r-1)}, \\nonumber \\\\ \nB(v)&=& \\frac{Q_{s+r-1}(v-s+r-4n-1)}{Q_{s+r-1}(v-s+r-4n+1)} \n\\nonumber \\\\ \n && +\\frac{Q_{s+r-2}(v-s+r-4n)Q_{s+r-1}(v-s+r-4n+3)}\n          {Q_{s+r-2}(v-s+r-4n+2)Q_{s+r-1}(v-s+r-4n+1)},\n          \\nonumber \\\\ \n C(v)&=&\\frac{Q_{s+r}(v-s+r+1)}{Q_{s+r}(v-s+r-1)} \n  \\nonumber \\\\ \n && +\\frac{Q_{s+r-2}(v-s+r)Q_{s+r}(v-s+r-3)}\n       {Q_{s+r-2}(v-s+r-2)Q_{s+r}(v-s+r-1)}, \\\\ \nD(v)&=& \\frac{Q_{s+r}(v-s+r-4n-1)}{Q_{s+r}(v-s+r-4n+1)} \n\\nonumber \\\\ \n && +\\frac{Q_{s+r-2}(v-s+r-4n)Q_{s+r}(v-s+r-4n+3)}\n          {Q_{s+r-2}(v-s+r-4n+2)Q_{s+r}(v-s+r-4n+1)}.\n          \\nonumber \n\\end{eqnarray} \nApparently, $A(v)$ and $B(v)$ (resp. $C(v)$ and $D(v)$)  \ndo not contain $Q_{s+r}$ (resp. $Q_{s+r-1}$). \nOne can also check $A(v)$ and $B(v)$ \n(resp. $C(v)$ and $D(v)$) are free of color $s+r-1$ \n(resp. $s+r$) poles under the BAE (\\ref{BAE}).  \n\n$S_{2n+1}$ ($k=2n+1$, $n \\in {\\bf Z}_{\\ge 2}$ \n\\footnote{The case $n=1$ can be treated similarly.}) is a \nsummation of the tableaux (with signs) of the form \n\\begin{eqnarray}\n&&\n\\begin{array}{|c|}\\hline \n    \\xi  \\\\ \\hline\n    s+r \\\\ \\hline  \n    \\stackrel{\\ }{\\overline{s+r}} \\\\ \\hline \n    s+r  \\\\ \\hline\n    \\vdots \\\\  \\hline\n    \\stackrel{\\ }{\\overline{s+r}} \\\\ \\hline\n    s+r  \\\\ \\hline\n    \\stackrel{\\ }{\\overline{s+r}}  \\\\ \\hline\n    s+r  \\\\ \\hline\n    \\zeta  \\\\ \\hline\n\\end{array}\n+\n\\begin{array}{|c|}\\hline \n    \\xi \\\\ \\hline\n    s+r  \\\\ \\hline \n    \\stackrel{\\ }{\\overline{s+r}} \\\\ \\hline \n    s+r  \\\\ \\hline\n    \\vdots \\\\ \\hline\n    \\stackrel{\\ }{\\overline{s+r}} \\\\ \\hline\n    s+r  \\\\ \\hline\n    \\stackrel{\\ }{\\overline{s+r}}  \\\\ \\hline\n    \\stackrel{\\ }{\\overline{s+r-1}}  \\\\ \\hline\n    \\zeta  \\\\ \\hline\n\\end{array}\n+\n\\begin{array}{|c|}\\hline \n    \\xi    \\\\ \\hline\n    s+r-1   \\\\ \\hline\n    \\stackrel{\\ }{\\overline{s+r}} \\\\ \\hline \n    s+r   \\\\ \\hline\n    \\vdots \\\\ \\hline\n    \\stackrel{\\ }{\\overline{s+r}} \\\\ \\hline\n    s+r \\\\ \\hline\n    \\stackrel{\\ }{\\overline{s+r}} \\\\ \\hline\n    s+r \\\\ \\hline \n    \\zeta  \\\\ \\hline\n\\end{array}\n+\n\\begin{array}{|c|l}\\cline{1-1} \n    \\xi & \\\\ \\cline{1-1}\n    s+r-1 & _{v} \\\\ \\cline{1-1} \n    \\stackrel{\\ }{\\overline{s+r}} & _{v-2}  \\\\ \\cline{1-1} \n    s+r  & _{v-4} \\\\ \\cline{1-1}\n    \\vdots &  \\\\ \\cline{1-1}\n    \\stackrel{\\ }{\\overline{s+r}} & _{v-4n+6} \\\\ \\cline{1-1}\n    s+r & _{v-4n+4} \\\\ \\cline{1-1}\n    \\stackrel{\\ }{\\overline{s+r}} & _{v-4n+2} \\\\ \\cline{1-1}\n    \\stackrel{\\ }{\\overline{s+r-1}} & _{v-4n} \\\\ \\cline{1-1}\n    \\zeta  &  \\\\ \\cline{1-1}\n\\end{array}\n\\nonumber \\\\ \n&&=\nE(v)F(v) \\times \\framebox{$\\xi$} \\times \\framebox{$\\zeta$}\n\\end{eqnarray}\nand \n\\begin{eqnarray}\n&&\n\\begin{array}{|c|}\\hline \n    \\xi  \\\\ \\hline \n    \\stackrel{\\ }{\\overline{s+r}} \\\\ \\hline \n    s+r \\\\ \\hline  \n    \\stackrel{\\ }{\\overline{s+r}} \\\\ \\hline \n    s+r  \\\\ \\hline\n    \\vdots \\\\  \\hline\n    \\stackrel{\\ }{\\overline{s+r}} \\\\ \\hline\n    s+r  \\\\ \\hline\n    \\stackrel{\\ }{\\overline{s+r}} \\\\ \\hline\n    \\zeta  \\\\ \\hline\n\\end{array}\n+\n\\begin{array}{|c|}\\hline \n    \\xi \\\\ \\hline\n    s+r-1  \\\\ \\hline \n    s+r  \\\\ \\hline \n    \\stackrel{\\ }{\\overline{s+r}} \\\\ \\hline \n    s+r  \\\\ \\hline\n    \\vdots \\\\ \\hline\n    \\stackrel{\\ }{\\overline{s+r}} \\\\ \\hline\n    s+r  \\\\ \\hline\n    \\stackrel{\\ }{\\overline{s+r}}  \\\\ \\hline\n    \\zeta  \\\\ \\hline\n\\end{array}\n+\n\\begin{array}{|c|}\\hline \n    \\xi    \\\\ \\hline\n    \\stackrel{\\ }{\\overline{s+r}} \\\\ \\hline \n    s+r   \\\\ \\hline\n    \\stackrel{\\ }{\\overline{s+r}} \\\\ \\hline \n    s+r   \\\\ \\hline\n    \\vdots \\\\ \\hline\n    \\stackrel{\\ }{\\overline{s+r}} \\\\ \\hline\n    s+r \\\\ \\hline\n    \\stackrel{\\ }{\\overline{s+r-1}} \\\\ \\hline \n    \\zeta  \\\\ \\hline\n\\end{array}\n+\n\\begin{array}{|c|l}\\cline{1-1} \n    \\xi & \\\\ \\cline{1-1}\n    s+r-1 & _{v} \\\\ \\cline{1-1} \n    s+r & _{v-2} \\\\ \\cline{1-1} \n    \\stackrel{\\ }{\\overline{s+r}} & _{v-4}  \\\\ \\cline{1-1} \n    s+r  & _{v-6} \\\\ \\cline{1-1}\n    \\vdots &  \\\\ \\cline{1-1}\n    \\stackrel{\\ }{\\overline{s+r}} & _{v-4n+4} \\\\ \\cline{1-1}\n    s+r & _{v-4n+2} \\\\ \\cline{1-1}\n    \\stackrel{\\ }{\\overline{s+r-1}} & _{v-4n} \\\\ \\cline{1-1}\n    \\zeta  &  \\\\ \\cline{1-1}\n\\end{array}\n\\nonumber \\\\ \n&&=\nG(v)H(v) \\times \\framebox{$\\xi$} \\times \\framebox{$\\zeta$}\n\\end{eqnarray}\nwhere $v=u+h_{2}$: $h_{2}$ is some shift parameter; \n\\framebox{$\\xi$} and \n\\framebox{$\\zeta$} are columns with total \nlength $a-2n-1$, which do not contain \\framebox{$s+r-1$}, \n\\framebox{$s+r$}, \\framebox{$\\overline{s+r}$} and \n\\framebox{$\\overline{s+r-1}$};\n\\begin{eqnarray}\nE(v)&=&\\frac{Q_{s+r-1}(v-s+r+1)}{Q_{s+r-1}(v-s+r-1)} \n  \\nonumber \\\\ \n && +\\frac{Q_{s+r-2}(v-s+r)Q_{s+r-1}(v-s+r-3)}\n       {Q_{s+r-2}(v-s+r-2)Q_{s+r-1}(v-s+r-1)}, \\nonumber \\\\ \nF(v)&=& \\frac{Q_{s+r}(v-s+r-4n-3)}{Q_{s+r}(v-s+r-4n-1)} \n\\nonumber \\\\ \n && +\\frac{Q_{s+r-2}(v-s+r-4n-2)Q_{s+r}(v-s+r-4n+1)}\n          {Q_{s+r-2}(v-s+r-4n)Q_{s+r}(v-s+r-4n-1)},\n          \\nonumber \\\\ \n G(v)&=&\\frac{Q_{s+r}(v-s+r+1)}{Q_{s+r}(v-s+r-1)} \n  \\nonumber \\\\ \n && +\\frac{Q_{s+r-2}(v-s+r)Q_{s+r}(v-s+r-3)}\n       {Q_{s+r-2}(v-s+r-2)Q_{s+r}(v-s+r-1)}, \\\\ \nH(v)&=& \\frac{Q_{s+r-1}(v-s+r-4n-3)}{Q_{s+r-1}(v-s+r-4n-1)} \n\\nonumber \\\\ \n && +\\frac{Q_{s+r-2}(v-s+r-4n-2)Q_{s+r-1}(v-s+r-4n+1)}\n          {Q_{s+r-2}(v-s+r-4n)Q_{s+r-1}(v-s+r-4n-1)}.\n          \\nonumber \n\\end{eqnarray} \nApparently, $E(v)$ and $H(v)$ (resp. $F(v)$ and $G(v)$)  \ndo not contain $Q_{s+r}$ (resp. $Q_{s+r-1}$). \nOne can also check $E(v)$ and $H(v)$ \n(resp. $F(v)$ and $G(v)$) are free of color $s+r-1$ \n(resp. $s+r$) poles under the BAE (\\ref{BAE}). \n\n Thus,  $S_{k}$ have neither \n color $s+r-1$ poles nor \n color $s+r$ poles under the BAE (\\ref{BAE}). \n\\setcounter{equation}{0}\n\\renewcommand{\\theequation}{B.\\arabic{equation}}\n\\section*{Appendix B Generating series for \n${\\cal T}^{a}(u)$ and ${\\cal T}_{m}(u)$}\nThe functions ${\\cal T}^{a}(u)$ \nand ${\\cal T}_{m}(u)$ \n ($a,m \\in {\\bf Z }$; $u \\in {\\bf C }$) \n are determined by the following \n non-commutative generating series.  \\\\ \n$B(r|s)$ case: \n\\begin{eqnarray}\n\\hspace{-30pt} &&   (1+\\framebox{$\\overline{1}$}X)^{-1}\\cdots \n (1+\\framebox{$\\overline{s}$}X)^{-1} (1+\\framebox{$\\overline{s+1}$}X)\n\\cdots (1+\\framebox{$\\overline{s+r}$}X)(1-\\framebox{$0$}X)^{-1}\n  \\nonumber \\\\\n\\hspace{-30pt} && \\times  (1+\\framebox{$s+r$}X) \\cdots (1+\\framebox{$s+1$}X)\n (1+\\framebox{$s$}X)^{-1} \\cdots (1+\\framebox{$1$}X)^{-1}\n   \\nonumber \\\\        \n\\hspace{-30pt}  && \\hspace{40pt} =\\sum_{a=-\\infty}^{\\infty} \n         {\\cal T}^{a}(u+a-1)X^{a},\n   \\end{eqnarray}\n\\begin{eqnarray}\n\\hspace{-30pt} &&   (1-\\framebox{$1$}X)\\cdots \n (1-\\framebox{$s$}X) (1-\\framebox{$s+1$}X)^{-1}\n\\cdots (1-\\framebox{$s+r$}X)^{-1}(1+\\framebox{$0$}X)\n  \\nonumber \\\\\n\\hspace{-30pt} && \\times  (1-\\framebox{$\\overline{s+r}$}X)^{-1} \n\\cdots (1-\\framebox{$\\overline{s+1}$}X)^{-1}\n (1-\\framebox{$\\overline{s}$}X) \\cdots (1-\\framebox{$\\overline{1}$}X)\n   \\nonumber \\\\        \n\\hspace{-30pt}  && \\hspace{40pt} =\\sum_{m=-\\infty}^{\\infty} \n         {\\cal T}_{m}(u+m-1)X^{m}.\n\\end{eqnarray}   \n$D(r|s)$ case: \n\\begin{eqnarray}\n\\hspace{-30pt} &&   (1+\\framebox{$\\overline{1}$}X)^{-1}\\cdots \n (1+\\framebox{$\\overline{s}$}X)^{-1} (1+\\framebox{$\\overline{s+1}$}X)\n\\cdots (1+\\framebox{$\\overline{s+r}$}X)\n\\nonumber \\\\ \n\\hspace{-30pt} && \\times \n(1-\\framebox{$s+r$}X\\framebox{$\\overline{s+r}$}X)^{-1}\n  \\nonumber \\\\\n\\hspace{-30pt} && \\times  (1+\\framebox{$s+r$}X) \\cdots (1+\\framebox{$s+1$}X)\n (1+\\framebox{$s$}X)^{-1} \\cdots (1+\\framebox{$1$}X)^{-1}\n   \\nonumber \\\\        \n\\hspace{-30pt}  && \\hspace{40pt} =\\sum_{a=-\\infty}^{\\infty} \n         {\\cal T}^{a}(u+a-1)X^{a},\n\\end{eqnarray}\n\\begin{eqnarray}\n\\hspace{-30pt} &&   (1-\\framebox{$1$}X)\\cdots \n (1-\\framebox{$s$}X) (1-\\framebox{$s+1$}X)^{-1}\n\\cdots (1-\\framebox{$s+r-1$}X)^{-1} \\nonumber \\\\ \n\\hspace{-30pt} && \\times \n[(1-\\framebox{$s+r$}X)^{-1}+\n(1-\\framebox{$\\overline{s+r}$}X)^{-1}-1]\n  \\nonumber \\\\\n\\hspace{-30pt} && \\times  (1-\\framebox{$\\overline{s+r-1}$}X)^{-1} \n\\cdots (1-\\framebox{$\\overline{s+1}$}X)^{-1}\n (1-\\framebox{$\\overline{s}$}X) \\cdots (1-\\framebox{$\\overline{1}$}X)\n   \\nonumber \\\\        \n\\hspace{-30pt}  && \\hspace{40pt} =\\sum_{m=-\\infty}^{\\infty} \n         {\\cal T}_{m}(u+m-1)X^{m}.\n\\end{eqnarray}  \nHere $X$ is a shift operator $X=e^{2\\partial_{u}}$. \nIn particular,  we have ${\\cal T}^{0}(u)=1$; \n${\\cal T}_{0}(u)=1$; ${\\cal T}^{a}(u)=0$ for $a<0$; \n ${\\cal T}_{m}(u)=0$ for $m<0$. \n\\newpage\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nYang and Lee \\cite{yang} proposed a mechanism\nfor the occurrence of phase transitions in the thermodynamic limit\nand yielded an insight into the problem of the\nIsing ferromagnet at arbitrary temperature in an arbitrary nonzero external magnetic field\nby introducing the concept of the zeros of the grand partition function $Z(T,H)$\n(for fluid systems, $Z(T,\\mu)$)\nin the {\\it complex} magnetic-field (for fluid systems, fugacity) plane (Yang-Lee zeros).\nThey \\cite{lee} also formulated\nthe celebrated circle theorem which states that\nYang-Lee zeros of the Ising ferromagnet lie on the unit circle $x_0=e^{i\\theta}$\nin the complex $x=e^{-2H\/k_B T}$ plane.\nBelow and at the critical temperature $T_c$, Yang-Lee zeros cut the positive real axis\nat the ferromagnetic transition point $x_c=1$ ($H_c=0$) in the thermodynamic limit.\nThe spontaneous magnetization $m_0$ is determined by the density of Yang-Lee zeros\n$g(\\theta)$ on the positive real axis, i.e., $m_0=2\\pi g(\\theta=0)$.\nHowever, above the critical temperature, Yang-Lee zeros do not cut the positive real axis\nin the thermodynamic limit. There is a gap in the distribution of zeros\naround the positive real axis, that is, $g(\\theta)=0$ for $|\\theta|<\\theta_e$.\nWithin this gap, the free energy is analytic and there is no phase transition.\nThe Yang-Lee zeros at $\\theta=\\pm\\theta_e$ are called the Yang-Lee edge zeros whose\nlocations have never been known exactly for $T>T_c$ and $d\\ge2$.\nAt the critical temperature the gap disappears, i.e., $\\theta_e=0$.\nAs temperature changes from $T_c$ to $\\infty$, the Yang-Lee edge $\\theta_e$\nmoves from $\\theta=0$ to $\\theta=\\pi$ \\cite{kortman,creswick97,wang,kim98}.\n\nIn addition to Yang-Lee zeros,\nFisher showed that the thermal properties of the square-lattice Ising model for $H=0$\nare completely determined by the zeros of the canonical partition function $Z(T)$\nin the complex temperature plane (Fisher zeros) \\cite{fisher65,lu}.\nUntil now, Yang-Lee zeros and Fisher zeros\n(and other kinds of zeros for some problems)\nof numerous physical, chemical, biological, and mathematical systems\nhave been investigated extensively to understand their important properties \\cite{bena}.\n\nIf the density of zeros\n\\cite{lee,kortman,creswick97,fisher65,lu,bena,suzuki70,kenna94,kenna97,janke02,binek,binek2,kim04a,suzuki67}\nis found, the free energy, the equation of state,\nand all other thermodynamic functions can be obtained.\nHowever, very little is known about the actual form of the density of zeros.\nThe density of Yang-Lee zeros for the Ising ferromagnet has never been known exactly\nexcept for the one-dimensional case \\cite{lee}.\nSuzuki {\\it et al.} \\cite{suzuki70} obtained the exact grand\npartition functions of the Ising model on the $4\\times6$ square lattice and\nthe $3\\times3\\times3$ simple-cubic lattice, and calculated\nthe densities of Yang-Lee zeros for these small-size lattices.\nKortman and Griffiths \\cite{kortman} investigated\nthe densities of Yang-Lee zeros for the Ising ferromagnet\non the square lattice and the diamond lattice,\nbased on the high-field, high-temperature series expansion,\nand found that the density of zeros for the square-lattice Ising ferromagnet\ndiverges at Yang-Lee edge $\\theta_e$ in high temperatures.\nFor the $L\\times L$ square-lattice Ising model,\nCreswick and Kim \\cite{creswick97} obtained the exact grand partition functions up to $L=10$\nand the semi-exact grand partition functions up to $L=14$,\ncalculated the densities of Yang-Lee zeros for $T\\le T_c$,\nand evaluated the spontaneous magnetization $m_0(T)=2\\pi g(\\theta=0,T)$\nin the infinite-size limit.\nFrom Monte Carlo data, the density of Yang-Lee zeros\nnear the transition point has been studied\nfor the four-dimensional Ising ferromagnet \\cite{kenna94} and\nthe square-lattice XY ferromagnet \\cite{kenna97,janke02}.\nThe density of Yang-Lee zeros has also been investigated experimentally\nfor the two-dimensional Ising ferromagnet FeCl$_2$ in axial magnetic fields \\cite{binek,binek2}.\n\nUntil now, the divergence of the density of zeros at Yang-Lee edge\n(the so-called Yang-Lee edge singularity)\nfor the square-lattice Ising ferromagnet in high temperatures has been obtained only by\nthe series expansion \\cite{kortman,kurtze1,baker86}.\nFurthermore, for $T>T_c$, the finite-size effects of the density of Yang-Lee zeros\nfor the Ising ferromagnet have never been studied.\nIn this paper, we investigate the properties of the density of Yang-Lee zeros\nbased on the exact grand partition functions of the $L\\times L$ square-lattice Ising model\n($L=3\\sim16$).\nWe deal with the three different classes of phase transitions for the Ising ferromagnet,\n(1) first-order phase transition, (2) second-order phase transition,\nand (3) Yang-Lee edge singularity, in a unified framework using the densities of zeros\nfor finite-size systems.\nWe also study the density of Yang-Lee zeros\nand its divergence in high temperatures from the finite-size data.\n\n\n\\section{Density of zeros}\n\nFor a finite-size system of size $L$, the density of Yang-Lee zeros $g(\\theta,L)$\nat the critical temperature $T_c$ is given by \\cite{creswick97,bena}\n\\begin{equation}\ng(\\theta,L)=L^{-d+y_h}g(\\theta L^{y_h})\n\\end{equation}\nnear $\\theta=0$.\nEquation (1) implies\n\\begin{equation}\ng(\\theta)\\sim|\\theta|^{(d-y_h)\/y_h}=|\\theta|^{1\/\\delta}\n\\end{equation}\nfor the infinite system.\nAt the critical temperature the value of the magnetic scaling exponent $y_h$\nfor the Ising ferromagnet in two dimensions ($d=2$)\nis well known to be 15\/8, that is, $\\delta=15$.\nTherefore, the density of zeros vanishes at the critical point $H_c=0$ ($\\theta=0$).\nThis kind of behavior of the density of zeros at $T_c$ is the characteristic of\na second-order phase transition.\nOn the other hand, for $T<T_c$, we have the finite density of zeros at $\\theta=0$,\n$g(0)=m_0\/2\\pi$, which is the characteristic of a first-order phase transition.\n\nTo describe the density of zeros {\\it above} the critical temperature as well,\nwe propose that Eqs.~(1) and (2) can be generalized as\n\\begin{equation}\ng(\\Theta,L)=L^{-d+y_h}g(\\Theta L^{y_h})\n\\end{equation}\nand\n\\begin{equation}\ng(\\Theta)\\sim|\\Theta|^{(d-y_h)\/y_h},\n\\end{equation}\nwhere $\\Theta=\\theta-\\theta_e$.\nFor $T\\le T_c$, Eq.~(3) is the same as Eq.~(1) because of $\\theta_e=0$.\nThe form $g(\\Theta)\\sim|\\Theta|^\\sigma$ was suggested by Suzuki \\cite{suzuki67}\nwho assumed $\\sigma=1\/\\delta={1\\over15}$ for the two-dimensional Ising ferromagnet\neven above the critical temperature.\nAccording to Suzuki's assumption, the density of zeros above the critical temperature\nvanishes at Yang-Lee edge $\\theta_e$.\n\nUsing the high-temperature series expansion for the Ising ferromagnet on the square lattice,\nKortman and Griffiths \\cite{kortman} showed that the density of Yang-Lee zeros,\n$g(\\Theta)\\sim|\\Theta|^\\sigma$,\ndiverges with $\\sigma=-0.12(5)$ at $\\theta=\\theta_e$ for $T > 2T_c$,\nin disagreement with Suzuki's assumption.\nKurtze and Fisher \\cite{kurtze1} refined the estimation of $\\sigma$ by analyzing\nthe high-temperature series expansion for the classical $n$-vector model\nand the quantum Heisenberg model in the limit of infinite temperature,\nand reported $\\sigma=-0.163(3)$ in two dimensions (square and triangular lattices).\nBaker {\\it et al.} \\cite{baker86} analyzed the series expansion of much greater length for\nthe square-lattice Ising ferromagnet, and obtained $\\sigma=-0.1560(33)$\nat $y=e^{-2\\beta J}={2\\over3}$ ($T\\approx 5 J\/k_B$) and $\\sigma=-0.1576(34)$\nat $y={4\\over5}$ ($T\\approx 9 J\/k_B$) from integral approximants.\nRemarkably, the exponent $\\sigma$ has also been experimentally measured to be\n$-0.15(2)$ in the range $49K\\le T\\le53K$ and $-0.365$ at $T=34K$\nfor the triangular-lattice Ising ferromagnet FeCl$_2$ \\cite{binek2}.\n\nFisher \\cite{fisher78} renamed the edge zero as the Yang-Lee edge {\\it singularity}\nfor $T>T_c$, and proposed the idea that the Yang-Lee edge singularity\ncan be thought of as a new second-order phase transition with associated critical exponents.\nAbove the critical temperature the value of $\\sigma$ is known to be\n$\\sigma=-{1\\over6}$ in two dimensions \\cite{dhar},\nresulting in $y_h={12\\over5}$ (according to Eq.~(4)), clearly different from\nthe value of $y_h={15\\over8}$ for the Ising ferromagnet at the critical temperature.\nThe study of the Yang-Lee edge singularity has been extended to\nthe spherical model \\cite{kurtze2},\nthe quantum one-dimensional transverse Ising model \\cite{uzelac},\nthe hierarchical model \\cite{baker79},\nbranched polymers \\cite{parisi},\nthe Ising models on fractal lattices \\cite{southern},\nthe Ising systems with correlated disorder \\cite{tadic},\nfluid models with repulsive-core interactions \\cite{lai},\nand the antiferromagnetic Ising model \\cite{kim05}, etc.\n\n\n\\section{grand partition function and number of states}\n\nThe Ising model in an external magnetic field $H$ on a lattice\nwith $N_s$ sites and $N_b$ bonds is defined by the Hamiltonian\n\\begin{equation}\n{\\cal H}=J\\sum_{\\langle i,j\\rangle}(1-\\sigma_i\\sigma_j)+H\\sum_i(1-\\sigma_i),\n\\end{equation}\nwhere $J$ is the coupling constant,\n$\\langle i,j\\rangle$ indicates a sum over all nearest-neighbor pairs\nof lattice sites, and $\\sigma_i=\\pm1$.\nThe grand partition function of the Ising model is\n$$\nZ=\\sum_{\\{ \\sigma_n \\}} e^{-\\beta{\\cal H}},\n$$\nwhere $\\{ \\sigma_n \\}$ denotes a sum over $2^{N_s}$ possible spin configurations\nand $\\beta=(k_BT)^{-1}$.\nIf we define the number of states, $\\Omega(E,M)$,\nwith a given energy\n\\begin{equation}\nE={1\\over2}\\sum_{\\langle i,j\\rangle}(1-\\sigma_i\\sigma_j)\n\\end{equation}\nand a given magnetization\n\\begin{equation}\nM={1\\over2}\\sum_i(1-\\sigma_i),\n\\end{equation}\nwhere $E$ and $M$ are positive integers $0\\le E\\le N_b$ and $0\\le M\\le N_s$,\nthen the grand partition function can be written as\n\\begin{equation}\nZ(x,y)=\\sum_{E=0}^{N_b}\\sum_{M=0}^{N_s}\\Omega(E,M) y^E x^M,\n\\end{equation}\nwhere $y=e^{-2\\beta J}$ and $x=e^{-2\\beta H}$.\nFrom Eq.~(8) it is clear that $Z(x,y)$ is simply a polynomial in $x$ and $y$.\n\nThe {\\it exact} integer values for the number of states $\\Omega(E,M)$ of the Ising model\non the $L\\times L$ square lattices with cylindrical boundary conditions have been evaluated\nup to $L=10$ to study the density of Yang-Lee zeros for $T\\le T_c$\nusing the microcanonical transfer matrix ($\\mu$TM) \\cite{creswick97}.\nIn Ref.~[4], for $11\\le L\\le14$, we had to evaluate\nthe coefficients, for a fixed value of $y$,\n$$\n\\omega(M)=\\sum_E\\Omega(E,M)y^E\n$$\nas real numbers of finite precision due to memory limitations.\nThis semi-exact method worked well for $T\\le T_c$, and produced all zeros on the unit circle.\nHowever, for $T>T_c$ the number of zeros on the unit circle,\nobtained from $Z(x)=\\sum\\omega(M)x^M$, decreases as $T$ increases.\nTherefore, to study the density of Yang-Lee zeros for $T>T_c$ we need\nthe exact integer values for $\\Omega(E,M)$, not $\\omega(M)$.\nIn this work, the exact integer values for $\\Omega(E,M)$ are calculated up to $L=16$\n($2^{256}=1.158\\times10^{77}$ configurations).\nThe memory requirement during $\\mu$TM calculation\nis $2\\times4\\times(N_s\/2+1)\\times(N_b+1)\\times P_m\\times 2^L$ byte\nfor the two-dimensional Ising model,\nwhere $P_m$ is the maximum size for storing very long integer numbers.\nFor $L=16$, the memory requirement is $3.03\\times10^{11}$ byte.\n(For comparison, the memory requirement for $L=10$, the largest size calculated in Ref.~[4],\nis $3.19\\times10^8$ byte.\nThe largest size calculated before this work is $L=14$ whose memory requirement\nis $3.44\\times10^{10}$ byte \\cite{kim04b}.)\nThe largest number of states for $L=16$ is\n\\begin{eqnarray}\n\\Omega(249,128)=\n207964931539339552089271840638951892633 \\cr\n537987437849943362668628644700590272,\n\\end{eqnarray}\nwhich is approximately $2.08\\times10^{74}$.\n\n\n\\section{Numerical results}\n\nFor a finite-system of size $L$, the density of zeros (per site) may be defined as\n\\begin{equation}\ng(\\theta_k,L)={1\\over L^d}{1\\over{\\theta_{k+1}-\\theta_k}},\n\\end{equation}\nwhere $\\theta_k$ ($k=1,2,3,...$) is the argument of the $k$-th zero on the\nunit circle $x_0=e^{i\\theta}$ of $Z(x,y)$.\nThe index $k$ is counted from $x=1$ ($\\theta=0$) to $x=-1$ ($\\theta=\\pi$)\nfor the zeros with $\\theta\\ge0$. Hence, $\\theta_1(L)=\\theta_e(L)$.\nFigure 1 shows the density of Yang-Lee zeros for the $16\\times16$ Ising ferromagnet\nat various temperatures (in unit of $J\/k_B$).\nThe Yang-Lee zeros at $T=0$ are uniformly distributed on the unit circle \\cite{kim98},\nand the density of zeros is given by $g(\\theta)=1\/2\\pi$ ($=0.159...$).\nAs shown in Fig.~1, for a low temperature $T=1$, the density of zeros is\nnearly identical with $g(\\theta)=0.159$.\nThe spontaneous magnetization can be evaluated from the finite-size data for the density\nof zeros ($m_0(L)=2\\pi g(0,L)$, $L=3\\sim16$) using the BST extrapolation algorithm \\cite{bst}.\nFor $T=1$ the extrapolated value of the spontaneous magnetization is\n$m_0=0.9992757(3)$ in excellent agreement with the exact result $0.9992757519...$ \\cite{yang2}.\nAt the critical temperature $T_c$ ($=2.269...$), the maximum value of the density of zeros\nis 0.166096 at $\\theta=2.541$ where the density of zeros begins to decrease monotonically\nas $\\theta$ decreases.\nAt $T_c$ the density of zeros vanishes at $\\theta=0$ in the limit $L\\to\\infty$,\naccording to Eq.~(2).\nFor $T=3$ the density of zeros is similar to that at $T_c$, except for\na violation of the monotonic decrease of the density of zeros around $\\theta=0.2$.\nIt is interesting that the minimum value of the density of zeros at $T=3$\n($g(\\theta_e=0.126)=0.112$) is lower than that at $T_c$ ($g(0.023)=0.117$).\nTherefore, we may guess that the density of zeros for $T$ just above $T_c$, for example $T=3$,\nvanishes at Yang-Lee edge $\\theta_e$ in the infinite-size limit.\nAs shown in Fig.~1, at high temperatures such as $T=5$ and 10, the overall density of zeros\nbecomes higher because the Yang-Lee edge $\\theta_e$ tends to $\\pi$ as $T$ rises.\nFor $T=5$ the density of zeros has a local maximum $g=0.182$ near $\\theta_e=0.532$.\nFinally, for $T=10$, the density of zeros has the global maximum $g=0.279$\nnear $\\theta_e=1.161$.\nThis fact may be a precursor for the divergence of the density of zeros at $\\theta_e$\nin the infinite-size limit for high temperatures.\n\nFigure 2 shows the densities of Yang-Lee zeros for $L=12$ and 16 at different temperatures.\nWell below the critical temperature (for example, $T=1$), the densities of zeros\nfor $L=12$ and 16 are not distinguishable, as shown in Fig.~2(a).\nTherefore, the finite-size effect for the density of zeros is very small\nwell below $T_c$.\nAt and near the critical temperature (for example, $T=2.5$ in Fig.~2(a)),\nthe densities of zeros for $L=12$ and 16 are also indistinguishable,\nexcept for them near $\\theta=0$ where the density of zeros $g(L)$ decreases sharply\nas the system size $L$ increases.\nWell above the critical temperature (for example, $T=10$ in Fig.~2(b)),\nthe overall forms of the density of zeros for $L=12$ and 16 are almost identical.\nThe finite-size effect for the density of zeros at $T=10$ is relatively large,\ncompared to those at $T=1$ and 2.5.\nNear the Yang-Lee edge $\\theta_e=1.161$, the density of zeros is somewhat irregular,\nand it increases as $L$ increases.\n\nIf we set $\\Theta=0$ ($\\theta=\\theta_e(L)$) in Eq.~(3), we have\n\\begin{equation}\ng(0,L)=L^{-d+y_h}g(0),\n\\end{equation}\nfrom which we can define the magnetic scaling exponent\n\\begin{equation}\ny_h(L)=d+{{{\\rm ln}[g(0,L+1)\/g(0,L)]}\\over{\\rm ln}[(L+1)\/L]}\n\\end{equation}\nfor finite lattices.\nTable I shows the values of $y_h(L)$ at $T=1$, $T_c$, and 10, respectively.\nFor $T=1$, $y_h(L)$ changes very slightly around $d=2$ as $L$ increases.\nThis kind of behavior may result from the small finite-size effect\nfor the density of zeros well below the critical temperature.\nAt $T_c$ and $T=10$, $y_h(L)$ increases monotonically as $L$ increases.\nIn particular, at $T=10$, $y_h(L)$ increases quickly.\n\nTable II shows the extrapolated results for the magnetic scaling exponent $y_h$\nby the BST algorithm.\nFor $T=1$, the extrapolated result clearly indicates $y_h=d=2$ that is expected\nat a strong first-order phase transition \\cite{fisher82}.\nAt $T_c$, the extrapolated value of $y_h=1.8747(7)$ is in agreement with\nthe exact value of $y_h={15\\over8}=1.875$.\nFor $T\\ge3$, we obtained the extrapolated value of $y_h=2.4$\nwhich gives $\\sigma=-{1\\over6.0}$ (the characteristic of Yang-Lee edge singularity).\nAccording to Eq.~(4), the density of zeros diverges at Yang-Lee edge $\\theta_e$\nwhen $y_h>d$.\nFinally, we can easily distinguish the three different classes of phase transitions\n(first-order phase transition, second-order phase transition,\nand Yang-Lee edge singularity)\nby estimating $y_h$ from the densities of zeros on finite lattices.\n\nNear the critical temperature $T_c=2.269...$, we expect some crossovers\nbetween different classes.\nAt $T=2.2$ we obtained $y_h=1.98(15)$ that is close to $y_h=2$\nbut is not clearly distinguished from $y_h={15\\over8}$ due to the large error.\nAt $T=2.4$ and 2.5, the extrapolated values for $y_h$ are 2.33(23) and 2.89(72),\nrespectively, indicating the existence of Yang-Lee edge singularity.\nBut their errors are too large.\nIt is interesting that the densities of zeros for finite-size systems at $T=2.4$, 2.5, and 3\nare quite similar to those at $T_c$\nbut they give the values of $y_h$ ($>d=2$) completely different\nfrom $y_h={15\\over8}$ ($<d$).\nFor the first time, it is shown that, even in the range of $2.4\\le T \\le 4$,\nthe square-lattice Ising ferromagnet has the Yang-Lee edge singularity.\nAt $T=2.3$ we obtained $y_h=1.58(4)$ whose nearest value is $y_h={15\\over8}$ at $T_c$.\nThis value of $y_h=1.58(4)$ implies that the density of zeros for $T>T_c$\nvanishes at $\\theta_e$.\nHowever, if the data of much larger sizes can be used,\nthere will be a possibility of obtaining $y_h > 2$ even at $T=2.3$.\n\nThe finite-size data for the density of zeros $g(\\theta_e,L)$\n($L=3\\sim16$) in high temperatures have also been extrapolated to infinite size.\nIn high temperatures, it is convenient to deal with the inverse density of zeros\n$f(\\theta_e,L)\\equiv1\/g(\\theta_e,L)$ because\n$f(\\theta_e,L)$ vanishes with the scale factor $L^{-(y_h-d)}$, in the limit $L\\to\\infty$,\nbut $g(\\theta_e,L)$ diverges.\nTable III shows the extrapolated results for the inverse density of zeros $f(\\theta_e)$.\nFor $T\\ge5$ the extrapolated results imply the divergence of the density of zeros\nat $\\theta_e$.\nHowever, at $T=4$, we can not conclude anything only from the extrapolated result\nfor the inverse density of zeros because of the large error.\n\n\n\\section{phase transitions in small systems}\n\nIn the previous section, the density of zeros has been extracted\nfrom the locations of Yang-Lee zeros for finite sizes,\nusing Eq.~(10) \\cite{creswick97,suzuki70,kim04a}.\nRecently, two alternative approaches (JK approach \\cite{janke02,janke01,janke04,alves}\nand BMH approach \\cite{borrmann,alves}) have been proposed\nto extract the density of zeros from the locations of Fisher zeros\nin the complex temperature ($y=e^{-2\\beta J}$) plane for finite sizes\nand to study phase transitions in small systems.\nThese two approaches are based on the assumption that Fisher zeros\n$y_j=s_j+i t_j=y_c+i r_j\\exp(-i\\psi)$\nclose to the critical point $y_c$ lie on a single line for quite large $L$,\nwhere the index $j(=1,2,3,...)$ increases with distance $r_j$ from the critical point,\n$s_j={\\rm Re}(y_j)=y_c+r_j\\sin\\psi$, and $t_j={\\rm Im}(y_j)=r_j\\cos\\psi$.\n\nIn the JK approach \\cite{janke02,janke01,janke04,alves}, the density of zeros is defined as\n\\begin{equation}\n\\lambda_L(r)=L^{-d}\\sum_j\\delta(r-r_j(L)),\n\\end{equation}\nand the cumulative density of zeros $\\Lambda_L(r)\\equiv\\int_0^r \\lambda_L(r')dr'$\nbecomes a step function with\n\\begin{equation}\n\\Lambda_L(r)={j\\over L^d}\\ \\ \\ [{\\rm if}\\ r\\in(r_j,r_{j+1})]\n\\end{equation}\nfor a finite-size system of size $L$. Then the average cumulative density of zeros\nis given by\n\\begin{equation}\n\\Lambda_L(r_j)=\\frac{2j-1}{2L^d},\n\\end{equation}\nand it is expressed as\n\\begin{equation}\n\\Lambda_{\\infty}(r) = \\lambda_{\\infty}(0)r + a r^{w+1} + \\cdots\n\\end{equation}\nin the limit $L\\to\\infty$.\nHere, the density of zeros $\\lambda_{\\infty}(0)$ on the real axis is proportional\nto the latent heat $\\Delta e$.\nAt a first-order phase transition, the first term $\\lambda_{\\infty}(0)r$ becomes\ndominant for small $r$.\nAt a second-order phase transition, the second term $a r^{w+1}$\nis the leading one because of $\\lambda_{\\infty}(0)=0$.\nFinally, the cumulative density of zeros can be approximated as\n\\begin{equation}\n\\frac{2j-1}{2L^d}=\\Lambda_L(r_j)=a_1 [t_j(L)]^{a_2} + a_3\n\\end{equation}\nbecause the parameter $r_j$ for Fisher zeros close to the critical point $y_c$\nmay be taken to be the imaginary part $t_j(L)$ due to $y_c\\sim s_j(L)$.\nA value of $a_3$ inconsistent with zero indicates the absence of a phase transition.\nThe parameter $a_2$ determines the order of a phase transition.\nA first-order phase transition takes a value $a_2 \\sim 1$ for small $t$,\nwhereas a value of $a_2$ larger than $1$ indicates a second-order phase transition\nwhose specific-heat exponent is given by $\\alpha=2-a_2$.\nFor a first-order phase transition, the parameter $a_1$ corresponds to\nthe density of zeros $\\lambda_{\\infty}(0)$ on the real axis.\nThe parameter $a_2$ is also expressed as $a_2=d\/y_t$\nin terms of the thermal scaling exponent $y_t$.\nThe JK approach has also been applied to the Yang-Lee zeros of the square-lattice\nXY ferromganet \\cite{janke02} where the parameter $a_2$ is expressed as $a_2=d\/y_h$\nin terms of the magnetic scaling exponent $y_h$,\nthat is,\n\\begin{equation}\na_2={2d\\over d+2-\\eta}.\n\\end{equation}\nThe JK approach has also been generalized to study\nthe areal distributions of Fisher zeros \\cite{janke04}.\n\nOn the other hand, there are three important parameters\nin the BMH approach \\cite{borrmann,alves}.\nThe first parameter $t_1$ (the imaginary part of the first zero)\nbecomes 0 in the limit $L\\to\\infty$\nfor a true phase transition in the Ehrenfest sense \\cite{creswick97,kim98,kim04b,kims1,kims2}.\nIf there is a phase transition, we assume that the line density of zeros,\n\\begin{equation}\n\\phi(t_j)\\equiv{1\\over2}\\left({1\\over|y_j-y_{j-1}|}+{1\\over|y_{j+1}-y_j|}\\right),\n\\end{equation}\nfollows a simple power law $\\phi(t)\\sim t^{\\alpha_\\phi}$ for small $t$.\nThe second parameter $\\alpha_\\phi$ can be estimated as\n\\begin{equation}\n\\alpha_\\phi={\\ln\\phi(t_3)-\\ln\\phi(t_2)\\over\\ln t_3-\\ln t_2}\n\\end{equation}\nfrom the first four Fisher zeros ($y_1, y_2, y_3, y_4$).\nThe line of Fisher zeros also crosses the real axis\nat the critical point $y_c$ with the angle $\\psi$, yielding\nthe third parameter\n\\begin{equation}\n\\gamma_\\psi=\\tan\\psi={s_2-s_1\\over t_2-t_1}.\n\\end{equation}\nThe second and third parameters ($\\alpha_\\phi$ and $\\gamma_\\psi$) determine\nthe order of a phase transition.\nThe values of $\\alpha_\\phi=0$ ($\\alpha_\\phi$ can be less than zero for finite sizes)\nand $\\gamma_\\psi=0$ indicate a first-order phase transition.\nA second-order phase transition occurs for $0 < \\alpha_\\phi < 1$ and\narbitrary $\\gamma_\\psi$, and a higher-order phase transition for $\\alpha_\\phi > 1$\nand arbitrary $\\gamma_\\psi$.\nAlves {\\it et al.} \\cite{alves} have performed numerical comparisons\nof the JK and BMH approaches for the Fisher zeros\nof the two-dimensional four-state and five-state Potts models and of biological molecules,\nusing Monte Carlo data.\nTheir results implied that the JK approach is little better than the BMH approach.\n\nThe approach explained in the previous sections, which may be called as\nthe angular-density-of-zeros (ADOZ) approach, can also be used to study\nphase transitions in small systems. In Eq.~(4), the density of Yang-Lee zeros\nnear $\\Theta=0$ is given by $g(\\Theta)\\sim|\\Theta|^{(d-y_h)\/y_h}$.\nSimilarly, the density of Fisher zeros near $\\Theta=0$\nis also given by $g(\\Theta)\\sim|\\Theta|^{(d-y_t)\/y_t}$ \\cite{kim04a}.\nThe exponent $(d-y_t)\/y_t$ (or $(d-y_h)\/y_h$) corresponds to the parameter\n$a_2-1$ of the JK approach.\nAs in the BMH approach, the zero value of $t_1$\nin the limit $L\\to\\infty$ indicates the existence of a true phase transition.\nWhen there is no phase transition, Fisher (or Yang-Lee) edge singularity \\cite{kims2,kim02}\nexists if $y_t > d$ (or $y_h > d$).\nAs shown in Table I, the values of $y_h(L)$ at $T=10$ are clearly greater than $d=2$\nfor relatively large $L$, suggesting the existence of Yang-Lee edge singularity.\nWhen there is a phase transition, the value of $y_t$ (or $y_h$) determines\nthe order of the phase transition.\nTo calculate $y_t(L)$ (or $y_h(L)$), we need only the first two Fisher (or Yang-Lee)\nzeros according to Eqs.~(10) and (12).\nA value of $y_t\\approx d$ (or $y_h\\approx d$) means a first-order phase transition.\nThe values of $y_h(L)$ at $T=1$ in Table I clearly indicate a first-order phase transition.\nA second-order phase transition occurs for $d\/2\\le y_t<d$ (or $d\/2\\le y_h<d$),\nand a higher-order phase transition for $y_t<d\/2$ (or $y_h<d\/2$).\nThe values of $y_h(L)$ at $T_c$ in Table I indicate a second-order phase transition.\n\nNow the three (JK, BMH, and ADOZ) approaches are applied to Yang-Lee zeros\nof the square-lattice Ising ferromagnet.\nThe BMH approach is applied to Yang-Lee zeros for the first time.\nBecause the JK and BMH approaches assume a line distribution of zeros\nnear a transition point, we use Yang-Lee zeros from $L=8$.\nThe JK and BMH approaches have been compared for the approximate zeros of\nquite large (but still small) finite-size systems using Monte Carlo simulation \\cite{alves}.\nHere these approaches are compared for the exact zeros of very small finite-size systems\n($L=8$ to 16).\nBecause the BMH approach requires the first four zeros, we use only them\nin the applications of the JK and BMH approaches for a fair comparison.\nThe ADOZ approach requires the first two zeros of two different sizes,\nin total, four zeros (see Eq.~(12)).\n\nIn the identification of the orders of phase transitions in small systems,\nthe most important quantities are $a_2(L)$, $\\alpha_\\phi(L)$, and $y_h(L)$,\nrespectively, for the JK, BMH, and ADOZ approaches.\nFigure 3 shows the results of $a_2(L)$, $\\alpha_\\phi(L)$, and $y_h(L)$\nat $T=1$ (first-order phase transition), $T=2.2$ (weak first-order phase transition),\nand $T_c=2.269...$ (second-order phase transition) for the Yang-Lee zeros\nof the square-lattice Ising ferromagnet.\nHere, a weak first-order phase transition means a first-order phase transition\nwith very large (but finite) correlation length.\nAll the approaches clearly identify the first-order and second-order phase transitions.\nIn particular, the finite-size results at $T=1$ are perfect in that\nthey already reach the infinite-size values $a_2=1$, $\\alpha_\\phi=0$, and $y_h=2$.\nBut the three approaches fail to identify the weak first-order phase transition.\nAn advantage of the JK approach is that we can use the zeros from different sizes\ntogether in the estimation of $a_2$. The only first Yang-Lee zeros from $L=13$ to 16\n(in total, four zeros) are used to estimate $a_2$ at $T=2.2$ and $T_c$.\nAt $T_c$, we obtain $a_2=1.0874(4)$, close to the infinite-size value\n$a_2=d\/y_h=16\/15(=1.0666...)$.\nAt $T=2.2$, the result is $a_2=1.011(2)$.\nIt is remarkable because the JK approach almost identifies\nthe weak first-order phase transition only from very small systems up to $L=16$.\n\n\n\\section{Conclusion}\n\nWe have enumerated the exact integer values for the number of states $\\Omega(E,M)$ of the Ising model\non the $L\\times L$ square lattice up to $L=16$, from which the grand partition function\n$Z(T,H)=\\sum_{E,M}\\Omega(E,M) e^{-2\\beta(J E+H M)}$ is obtained and\nthe density of Yang-Lee zeros $g(\\theta,L)$ on the unit circle $x=e^{-2\\beta H}=e^{i\\theta}$\nis evaluated for the Ising ferromagnet.\nWe have compared the densities of Yang-Lee zeros at different temperatures for $L=16$.\nWe have also discussed the finite-size effects for the densities of zeros at different temperatures.\nThe three different classes of phase transitions for the Ising ferromagnet,\n(1) first-order phase transition, (2) second-order phase transition, and (3) Yang-Lee edge singularity\n(the divergence of the density of zeros at Yang-Lee edge in high temperatures)\nhave been clearly distinguished by estimating\nthe magnetic scaling exponent $y_h$ from the densities of zeros on finite lattices.\nThe Yang-Lee edge singularity that has been detected only by the series expansion until now\nfor the square-lattice Ising ferromagnet, has also been obtained\nusing the densities of zeros for finite-size systems.\nWe have also discussed the identification of the orders of phase transitions\nin small systems using the three (JK, BMH, and ADOZ) approaches to extract\nthe density of zeros from numerical data.\n\n\n\\begin{acknowledgments}\nThe author was supported by the Korea Research Foundation Grant\nfunded by the Korean Government (MOEHRD) (KRF-2005-005-J01103).\n\\end{acknowledgments}\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\n\nIt is well-known that the space of ordinary differential operators of\nthe form $\\pa^n + u_1 \\pa^{n-2} + \\ldots + u_{n-1}$ has a remarkable\nPoisson structure, often called the (second) Adler-Gelfand-Dickey\nbracket \\cite{A,GD}. Drinfeld-Sokolov reduction \\cite{DS} gives a\nnatural realization of this Poisson structure via the hamiltonian\nreduction of the dual space to the affine Kac-Moody algebra\n$\\sun$. Drinfeld and Sokolov \\cite{DS} have applied an analogous\nreduction procedure to the dual space of the affinization $\\G$ of an\narbitrary semisimple Lie algebra $\\g$. The Poisson algebra ${\\cal\nW}(\\g)$ of functionals on the corresponding reduced space is called\nthe classical ${\\cal W}$--algebra. Thus, one can associate a classical\n${\\cal W}$--algebra to an arbitrary semisimple Lie algebra $\\g$. In\nparticular, the classical ${\\cal W}$--algebra associated to $\\sw_2$ is\nnothing but the classical Virasoro algebra, i.e. the Poisson algebra\nof functionals on the dual space to the Virasoro algebra (see, e.g.,\n\\cite{FR}).\n\nIt is interesting that ${\\cal W}(\\g)$ admits another description as\nthe center of the universal enveloping algebra of an affine\nalgebra. More precisely, let $Z(\\widehat{{\\frak g}})_{-h^{\\vee }}$ be\nthe center of a completion of the universal enveloping algebra\n$U(\\G)_{-\\k}$ at the critical level $k=-h^{\\vee }$ (minus the dual\nCoxeter number). This center has a canonical Poisson structure. It was\nconjectured by V.~Drinfeld and proved by B.~Feigin and E.~Frenkel\n\\cite{FF,EF} that as Poisson algebra $Z(\\G)_{-\\k}$ is isomorphic to the\nclassical ${\\cal W}$--algebra ${\\cal W}(^L\\!\\g)$ associated with the\nLanglands dual Lie algebra $\\gL$ of $\\g$.\n\nIn \\cite{FR} two of the authors used this second realization of\n$\\W$--algebras to obtain their $q$--deformations. For instance the\n$q$--deformation $\\W_q(\\sw_n)$ of ${\\cal W}(\\sw_n)$ was defined as the\ncenter $Z_q(\\sun)$ of a completion of the quantized universal\nenveloping algebra $U_q(\\sun)_{-\\k}$. The Poisson structure on\n$Z_q(\\sun)$ was explicitly described in \\cite{FR} using results of\n\\cite{ext}. It was shown that the underlying Poisson manifold of\n$Z_q(\\sun)={\\cal W}_q(\\sw_n)$ is the space of $q$--difference\noperators of the form $D_q^n + t_1 D_q^{n-1} + \\ldots + t_{n-1} D_q +\n1$. Furthermore, in \\cite{FR} a $q$--deformation of the Miura\ntransformation, i.e. a homomorphism from $\\W_q(\\sw_n)$ to a\nHeisenberg-Poisson algebra was defined. The construction \\cite{FR} of\n${\\cal W}_q(\\sw_n)$ was followed by further developments: it was\nquantized \\cite{SKAO,ell,AKOS} and the quantum algebra was used in the\nstudy of lattice models \\cite{LP,Miwa}; Yangian analogue of ${\\cal\nW}_q(\\sw_2)$ was considered in \\cite{Hou}; $q$--deformations of the\ngeneralized KdV hierarchies were introduced \\cite{Fr}.\n\nIn this paper we first formulate the results of \\cite{FR} in terms of\nfirst order $q$--difference operators and $q$--gauge action. This\nnaturally leads us to a generalization of the Drinfeld-Sokolov scheme\nto the setting of $q$--difference operators. The initial Poisson\nmanifold is the loop group $LSL_n$ of $SL_n$, or more generally, the\nloop group of a simply-connected simple Lie group $G$. Much of the\nneeded Poisson formalism has already been developed by one of the\nauthors in \\cite{RIMS,dual}. Results of these works allow us to\ndefine a Poisson structure on the loop group, with respect to which the\n$q$--gauge action is Poisson. We then have to perform a reduction of\nthis Poisson manifold with respect to the $q$--gauge action of the\nloop group $LN$ of the unipotent subgroup $N$ of $G$.\n\nAt this point we encounter a new kind of anomaly in the Poisson\nbracket relations, unfamiliar from the linear, i.e. undeformed,\nsituation. To describe it in physical terms, recall that the reduction\nprocedure consists of two steps: (1) imposing the constraints and (2)\npassing to the quotient by the gauge group. An important point in the\nordinary Drinfeld-Sokolov reduction is that these constraints are of\nfirst class, according to Dirac, i.e., their Poisson bracket vanishes\non the constraint surface. In the $q$--difference case we have to\nchoose carefully the classical $r$--matrix defining the initial\nPoisson structure on the loop group so as to make all constraints\nfirst class. If we use the standard $r$--matrix, some of the\nconstraints are of second class, and so we have to modify it.\n\nIn this paper we do that in the case of $SL_2$. We show that there is\nessentially a unique classical $r$--matrix compatible with the\n$q$--difference Drinfeld-Sokolov scheme. To the best of our knowledge,\nthis classical $r$--matrix is new; it yields an elliptic deformation\nof the Lie bialgebra structure on the loop algebra of $\\sw_2$\nassociated with the Drinfeld ``new'' realization of quantized affine\nalgebras \\cite{nr}, \\cite{kh-t}. The result of the Drinfeld-Sokolov\nreduction is the $q$--deformation of the classical Virasoro algebra\ndefined in \\cite{FR}.\n\nWe also construct a finite difference version of the Drinfeld-Sokolov\nreduction in the case of $SL_2$. This construction gives us a discrete\nversion of the (classical) Virasoro algebra. We explain in detail the\nconnection between our discrete Virasoro algebra and the lattice\nVirasoro algebra of Faddeev--Takhtajan--Volkov \\cite{ft,v1,v2,fv}. We\nhope that our results will help to clarify further the meaning of the\ndiscrete Virasoro algebra and its relation to various integrable\nmodels.\n\nThe construction presented here can be generalized to the case of an\narbitrary simply-connected simple Lie group. This is done in the\nsecond part of the paper \\cite{SS} written by A.~Sevostyanov and one\nof us.\n\nThe paper is arranged as follows. In Sect.~2 we recall the relevant\nfacts of \\cite{DS} and \\cite{FR}. In Sect.~3 we interpret the results\nof \\cite{FR} from the point of view of $q$--gauge transformations.\nSect.~4 reviews some background material on Poisson structures on Lie\ngroups following \\cite{RIMS,dual}. In Sect.~5 we apply the results of\nSect.~4 to the $q$--deformation of the Drinfeld-Sokolov reduction in\nthe case of $SL_2$. In Sect.~6 we discuss the finite difference\nanalogue of this reduction and compare its results with the\nFaddeev-Takhtajan-Volkov algebra.\n\n\\subsubsection*{Acknowledgements.} E.~Frenkel thanks\nP.~Schapira for his hospitality at Universit\\'{e} Paris VI in June of\n1996, when this collaboration began. Some of the results of this paper\nhave been reported in E.~Frenkel's lecture course on Soliton Theory\ngiven at Harvard University in the Spring of 1996.\n\nThe research of E.~Frenkel was supported by grants from the Packard\nand Sloan Foundations, and by the NSF grants DMS 9501414 and DMS\n9304580. The research of N.~Reshetikhin was supported by the NSF grant\nDMS 9296120.\n\n\\section{Preliminaries}\n\n\\subsection{The differential Drinfeld-Sokolov reduction in the case of\n$\\sw_n$}    \\label{diff1}\n\nLet ${\\cal M}_n$ be the manifold of differential operators of the form\n\\beq\n\\label{L} L = \\pa^n + u_1(s) \\pa^{n-2} + \\ldots +\nu_{n-2}(s) \\pa + u_{n-1}(s),\n\\end{equation}\nwhere $u_i(s) \\in \\C((s))$.\n\nAdler \\cite{A} and Gelfand-Dickey \\cite{GD} have defined a remarkable\ntwo-parameter family of Poisson structures on ${\\cal M}_n$, with\nrespect to which the corresponding KdV hierarchy is hamiltonian. In\nthis paper we will only consider one of them, the so-called second\nbracket. There is a simple realization of this structure in terms of\nthe Drinfeld-Sokolov reduction \\cite{DS}, Sect.~6.5. Let us briefly\nrecall this realization.\n\nConsider the affine Kac-Moody algebra $\\sun$ associated to $\\sw_n$;\nthis is the central extension $$0 \\arr \\C K \\arr \\sun \\arr \\lsln \\arr\n0,$$ see \\cite{Kac}. Let $M_n$ be the hyperplane in the dual space to\n$\\sun$, which consists of linear functionals taking value $1$ on\n$K$. Using the differential $dt$ and the bilinear form $\\on{tr} AB$ on\n$\\sw_n$, we identify $M_n$ with the manifold of first order\ndifferential operators\n$$\\pa_s + A(s), \\quad \\quad A(s) \\in \\lsln.$$ The coadjoint action of\nthe Lie group $\\wh{SL}_n$ on $\\sun^*$ factors through the loop group\n$LSL_n$ and preserves the hyperplane $M_n$. The corresponding action\nof $g(s) \\in LSL_n$ on $M_n$ is given by \\beq \\label{coadjoint} g(s)\n\\cdot (\\pa_s + A(s)) = g(s)(\\pa_s + A(s))g(s)^{-1},\n\\end{equation} or $$A(s) \\mapsto g(s) A(s) g(s)^{-1} - \\pa_s g(s) \\cdot\ng(s)^{-1}.$$\n\nConsider now the submanifold $M^J_n$ of $M_n$ wich consists of\noperators $\\pa_s + A(s)$, where $A(s)$ is a traceless matrix of the form\n\\beq    \\label{upper}\n\\begin{pmatrix}\n* & * & * & \\hdots & * & * \\\\\n-1 & * & * & \\hdots & * & * \\\\\n0 & -1 & * & \\hdots & * & * \\\\\n\\hdotsfor{6} \\\\\n0 & 0 & 0 & \\hdots & * & * \\\\\n0 & 0 & 0 & \\hdots & -1 & *\n\\end{pmatrix}\n\\end{equation}\n\nTo each element ${\\cal L}$ of $M^J_n$ one can naturally attach an\n$n$th order scalar differential operator as follows. Consider the\nequation ${\\cal L} \\cdot \\Psi = 0$, where $$\\Psi = \\begin{pmatrix}\n\\Psi_n \\\\ \\Psi_{n-1} \\\\ \\hdots \\\\ \\Psi_1 \\end{pmatrix}.$$\nDue to the special form \\eqref{upper} of ${\\cal L}$, this equation is\nequivalent to an $n$th order differential equation $L \\cdot \\Psi_1 =\n0$, where $L$ is of the form \\eqref{L}. Thus, we obtain a map $\\pi:\nM^J_n \\arr \\M$ sending ${\\cal L}$ to $L$.\n\nLet $N$ be the subgroup of $SL_n$ consisting of the upper triangular\nmatrices, and $LN$ be its loop group. If $g \\in LN$ and $\\Psi$ is a\nsolution of ${\\cal L} \\cdot \\Psi = 0$, then $\\Psi'= g \\Psi$ is a\nsolution of ${\\cal L}' \\cdot \\Psi' = 0$, where ${\\cal L}' = g {\\cal L}\ng^{-1}$. But $\\Psi_1$ does not change under the action of\n$LN$. Therefore $\\pi({\\cal L}') = \\pi({\\cal L})$, and we see that\n$\\pi$ factors through the quotient of $M^J_n$ by the action of\n$LN$. The following proposition describes this quotient.\n\n\\begin{prop}[\\cite{DS}, Proposition 3.1]    \\label{free}\nThe action of $LN$ on $M^J_n$ is free, and each orbit contains a\nunique operator of the form\n\\beq    \\label{special}\n\\pa_s + \\begin{pmatrix}\n0 & u_1 & u_2 & \\hdots & u_{n-2} & u_{n-1} \\\\\n-1 & 0 & 0 & \\hdots & 0 & 0 \\\\\n0 & -1 & 0 & \\hdots & 0 & 0 \\\\\n\\hdotsfor{6} \\\\\n0 & 0 & 0 & \\hdots & 0 & 0 \\\\\n0 & 0 & 0 & \\hdots & -1 & 0\n\\end{pmatrix}.\n\\end{equation}\n\\end{prop}\n\nBut for ${\\cal L}$ of the form \\eqref{special}, $\\pi({\\cal L})$ is\nequal to the operator $L$ given by formula \\eqref{L}. Thus, we have\nidentified the map $\\pi$ with the quotient of $M^J_n$ by $LN$ and\nidentified ${\\cal M}_n$ with $M^J_n\/LN$.\n\nThe quotient $M^J_n\/LN$ can actually be interpreted as the result of\nhamiltonian reduction. The manifold $M_n$ has a canonical Poisson\nstructure, which is the restriction of the Lie-Poisson structure on\n$\\sun^*$ (such a structure exists on the dual space to any Lie\nalgebra). The coadjoint action of $LN$ on $M_n$ is hamiltonian with\nrespect to this structure. The corresponding moment map $\\mu: M_n \\arr\nL\\n_- \\simeq L\\n^*$ sends $\\pa_s + A(s)$ to the lower-triangular part\nof $A(s)$. Consider the one-point orbit of $LN$,\n$$J = \\begin{pmatrix} 0 & 0 & 0 & \\hdots & 0 & 0 \\\\ -1 & 0 & 0 &\n\\hdots & 0 & 0 \\\\ 0 & -1 & 0 & \\hdots & 0 & 0 \\\\ \\hdotsfor{6} \\\\ 0 & 0\n& 0 & \\hdots & 0 & 0 \\\\ 0 & 0 & 0 & \\hdots & -1 & 0\n\\end{pmatrix}.$$ Then $M^J_n = \\mu^{-1}(J)$. Hence $M_n$ is the\nresult of hamiltonian reduction of $M_n$ by $LN$ with respect to the\none-point orbit $J$.\n\nThe Lie-Poisson structure on $M_n$ gives rise to a canonical Poisson\nstructure on ${\\cal M}_n$, which coincides with the second\nAdler-Gelfand-Dickey bracket, see \\cite{DS}, Sect.~6.5. The Poisson\nalgebra of local functionals on ${\\cal M}_n$ is called the classical\n$\\W$--algebra associated to $\\sw_n$, and is denoted by $\\W(\\sw_n)$.\n\n\\begin{rem}    \\label{kostant}\nFor $\\al \\in \\C$, let $M_{\\al,n}$ be the hyperplane in the dual space\nto $\\sun$, which consists of linear functionals on $\\sun$ taking value\n$\\al$ on $K$. In the same way as above (for $\\al=1$) we identify\n$M_{\\al,n}$ with the space of first order differential operators\n$$\\al \\pa_s + A(s), \\quad \\quad A(s) \\in \\lsln.$$ The coadjoint action\nis given by the formula $$A(s) \\mapsto g(s) A(s) g(s)^{-1} - \\al \\pa_s\ng(s) \\cdot g(s)^{-1}.$$ The straightforward generalization of\n\\propref{free} is true for any $\\al \\in \\C$. In particular, for\n$\\al=0$ we obtain a description of the orbits in $M_n^J$ under the\nadjoint action of $LN$. This result is due to B.~Kostant \\cite{Ko}.\n\nDrinfeld and Sokolov \\cite{DS} gave a generalization of \\propref{free}\nwhen $\\sw_n$ is replaced by an arbitrary semisimple Lie algebra\n$\\g$. The special case of their result, corresponding to $\\al=0$,\nis also due to Kostant \\cite{Ko}.\\qed\n\\end{rem}\n\nThe Drinfeld-Sokolov reduction can be summarized by the following\ndiagram.\n\n\\setlength{\\unitlength}{1mm}\n\n\\begin{center}\n\\begin{picture}(125,40)(-35,-33)     \n\\put(0,-1){\\vector(0,-1){26}}\n\\put(62,-1){\\vector(0,-1){26}}\n\\put(9,3){\\vector(1,0){45}}\n\\put(9,-31){\\vector(1,0){45}}\n\\put(8.9,3.9){\\oval(1,1)[l]\n\\put(8.9,-30.1){\\oval(1,1)[l]\n\\put(-2,2){$M_n^J$}\n\\put(-23,-32){$\\M=M_n^J\/LN$}\n\\put(59,2){$M_n$}\n\\put(57,-32){$M_n\/LN$}\n\n\\end{picture}\n\\end{center}\n\n\nThere are three essential properties of the Lie-Poisson structure on\n$M_n$ that make the reduction work:\n\n\\begin{itemize}\n\n\\item[(i)] The coadjoint action of $LSL_n$ on $M_n$ is hamiltonian\nwith respect to this structure;\n\n\\item[(ii)] the subgroup $LN$ of $LSL_n$ is admissible in the sense that\nthe space $S$ of $LN$--invariant functionals on $M_n$ is a Poisson\nsubalgebra of the space of all functionals on $M_n$;\n\n\\item[(iii)] Denote by $\\mu_{ij}$ the function on $M_n$, whose\nvalue at $\\pa + A \\in M_n$ equals the $(i,j)$ entry of $A$. The ideal\nin $S$ generated by $\\mu_{ij} + \\delta_{i-1,j}, i>j$, is a Poisson\nideal.\n\n\\end{itemize}\n\nWe will generalize this picture to the $q$--difference case.\n\n\\subsection{The Miura transformation}\n\nLet $\\F$ be the manifold of differential operators of the\nform\n\\begin{equation}    \\label{cartan}\n\\pa_s +\n\\begin{pmatrix}\nv_1 & 0 & 0 & \\hdots & 0 & 0 \\\\\n-1 & v_2 & 0 & \\hdots & 0 & 0 \\\\\n0 & -1 & v_3 & \\hdots & 0 & 0 \\\\\n\\hdotsfor{6} \\\\\n0 & 0 & 0 & \\hdots & v_{n-1} & 0 \\\\\n0 & 0 & 0 & \\hdots & -1 & v_n\n\\end{pmatrix},\n\\end{equation}\nwhere $\\sum_{i=1}^n v_i = 0$.\n\nWe have a map $\\muu: \\F \\arr \\M$, which is the composition of the\nembedding $\\F \\arr M^J_n$ and the projection $\\pi: M^J_n \\arr \\M$.\n\nUsing the definition of $\\pi$ above, $\\muu$ can be described explicitly\nas follows: the image of the operator \\eqref{cartan} under $\\muu$ is\nthe $n$th order differential operator\n$$\\pa_s^n + u_1(s) \\pa_s^{n-2} + \\ldots + u_{n-1}(s) = (\\pa_s +\nv_1(s)) \\ldots (\\pa_s + v_n(s)).$$\n\nThe map $\\muu$ is called the Miura transformation.\n\nWe want to describe the Poisson structure on $\\F$ with respect to\nwhich the Miura transformation is Poisson. To this end, let us\nconsider the restriction of the gauge action (\\ref{coadjoint}) to the\nopposite triangular subgroup $LN_{-};$ let $\\ol{\\mu} :M_n\\rightarrow\nL {\\frak n}_+ \\simeq L{\\frak n}_{-}^{*}$ be the corresponding moment\nmap. The manifold $\\F$ is the intersection of two level surfaces,\n$\\F = \\mu^{-1}(J)\\cap \\ol{\\mu}^{-1}(0).$ It is easy to see that it\ngives a local cross-section for both actions (in other words, the\norbits of $LN$ and $LN_{-}$ are transversal to $F_n$). Hence $F_n$\nsimultaneously provides a local model for the reduced spaces\n$M_n=\\mu^{-1}(J)\/LN$ and $\\ol{\\mu}^{-1}(0)\/LN_{-}$. The Poisson\nbracket on $\\F$ that we need to define is the so-called Dirac\nbracket (see, e.g., \\cite{Flato}), where we may regard the matrix\ncoefficients of $\\ol{\\mu}$ as subsidiary conditions, which fix the\nlocal gauge. The computation of the Dirac bracket for the diagonal\nmatrix coefficients $v_i$ is very simple, since their Poisson brackets\nwith the matrix coefficients of $\\ol{\\mu}$ all vanish on $\\F$. The\nonly correction arises due to the constraint $\\sum_{i=1}^N v_i = 0$.\n\nDenote by $v_{i,n}$ the linear functional on $\\F$, whose value on the\noperator \\eqref{cartan} is the $n$th Fourier coefficient of\n$v_i(s)$. We obtain the following formula for the Dirac bracket on\n$\\F$:\n\\begin{align*}\n\\{ v_{i,n},v_{i,m} \\} &= \\frac{N-1}{N} n \\delta_{n,-m}, \\\\ \\{\nv_{i,n},v_{j,m} \\} &= - \\frac{1}{N} n \\delta_{n,-m}, \\quad i<j.\n\\end{align*}\n\nSince $\\F$ and $\\M$ both are models of the same reduced space, we\nimmediately obtain:\n\n\\begin{prop}[\\cite{DS}, Proposition 3.26]\nWith respect to this Poisson structure the map $\\muu: \\F \\arr \\M$ is\nPoisson.\n\\end{prop}\n\n\\subsection{The $q$--deformations of $\\W(\\sw_n)$ and Miura transformation}\n\\label{qdef}\n\nIn this section we summarize relevant results of \\cite{FR}.\n\nConsider the space ${\\cal M}_{n,q}$ of $q$--difference operators of\nthe form \\beq \\label{qL} L = D^n + t_1(s) D^{n-1} + \\ldots +\nt_{n-1}(s) D + 1,\n\\end{equation}\nwhere $t_i(s) \\in \\C((s))$ for each $i=1,\\ldots,n$, and $[D \\cdot\nf](s) = f(sq)$.\n\nDenote by $t_{i,m}$ the functional on $\\Mq$, whose value at $L$ is the\n$m$th Fourier coefficient of $t_i(s)$. Let $\\RN$ be the completion\nof the ring of polynomials in $t_{i,m}, i=1,\\ldots,N-1; m \\in \\Z$,\nwhich consists of finite linear combinations of expressions of the\nform \\beq\n\\label{qloc} \\sum_{m_1+\\ldots +m_k=M} c(m_1,\\ldots,m_k) \\cdot\nt_{i_1,m_1} \\ldots t_{i_k,m_k},\n\\end{equation}\nwhere $c(m_1,\\ldots,m_k) \\in \\C$. Given an operator of the form\n\\eqref{qL}, we can substitute the coefficients $t_{i,m}$ into an\nexpression like \\eqref{qloc} and get a number. Therefore elements of\n$\\RN$ define functionals on the space $\\Mq$.\n\nIn order to define the Poisson structure on $\\Mq$, it suffices to\nspecify the Poisson brackets between the generators $t_{i,m}$. Let\n$T_i(z)$ be the generating series of the functionals $t_{i,m}$:\n$$T_i(z) = \\sum_{m\\in\\Z} t_{i,m} z^{-m}.$$ We define the Poisson\nbrackets between $t_{i,m}$'s by the formulas \\cite{FR}\n\\begin{align} \\notag\n\\{ T_i(z),T_j(w) \\} = & \\sum_{m\\in\\Z} \\left( \\frac{w}{z}\n\\right)^m \\frac{(1-q^{im})(1-q^{m(N-j)})}{1-q^{mN}} T_i(z) T_j(w) \\\\\n\\label{p2} &+ \\sum_{r=1}^{\\on{min}(i,N-j)} \\delta \\left( \\frac{wq^r}{z}\n\\right) T_{i-r}(w) T_{j+r}(z) \\\\ \\notag &- \\sum_{r=1}^{\\on{min}(i,N-j)}\n\\delta\\left( \\frac{w}{zq^{j-i+r}} \\right) T_{i-r}(z) T_{j+r}(w), \\quad\ni\\leq j.\n\\end{align}\nIn these formulas $\\delta(x) = \\sum_{m\\in\\Z} x^m$, and we use the\nconvention that $t_0(z) \\equiv 1$.\n\n\\begin{rem}\nThe parameter $q$ in formula \\eqref{p2} corresponds to $q^{-2}$ in the\nnotation of \\cite{FR}.\\qed\n\\end{rem} \\medskip\n\nNow consider the space $\\Fq$ of $n$--tuples of $q$--difference operators\n\\beq    \\label{ntuple}\n(D+\\la_1(s),\\ldots,D+\\la_n(s)).\n\\end{equation}\nDenote by $\\la_{i,m}$ the functional on $\\Mq$, whose value is the\n$m$th Fourier coefficient of $\\la_i(s)$. We will denote by\n$\\La_i(z)$ the generating series of the functionals $\\la_{i,m}$:\n$$\\La_i(z) = \\sum_{m\\in\\Z} \\la_{i,m} z^{-m}.$$\n\nWe define a Poisson structure on $\\Fq$ by the formulas \\cite{FR}:\n\\begin{align}    \\label{pbg1}\n\\{ \\La_i(z),\\La_i(w) \\} & = \\sum_{m\\in\\Z} \\left( \\frac{w}{z} \\right)^m\n\\frac{(1-q^m)(1-q^{m(N-1)})}{1-q^{mN}} \\La_i(z) \\La_i(w), \\\\ \\label{pbg2}\n\\{ \\La_i(z),\\La_j(w) \\} & = - \\sum_{m\\in\\Z} \\left(\n\\frac{wq^{N-1}}{z} \\right)^m \\frac{(1-q^m)^2}{1-q^{mN}} \\La_i(z)\n\\La_j(w), \\quad i<j.\n\\end{align}\n\nNow we define the $q$--deformation of the Miura transformation as the\nmap $\\muu_q: \\Fq \\arr \\Mq$, which sends the $n$--tuple \\eqref{ntuple}\nto \\beq\n\\label{Li} L = (D+\\la_1(s))(D+\\la_2(sq^{-1})) \\ldots\n(D+\\la_N(sq^{-N+1})),\n\\end{equation}\ni.e.\n\\beq    \\label{formulai}\nt_i(s) = \\sum_{j_1 < \\ldots < j_i} \\la_{j_1}(s) \\la_{j_2}(sq^{-1})\n\\ldots \\la_{j_i}(sq^{-i+1}).\n\\end{equation}\n\n\\begin{prop}[\\cite{FR}]\nThe map $\\muu_q$ is Poisson.\n\\end{prop}\n\n\\subsection{$q$--deformation of the Virasoro algebra}\nHere we specialize the formulas of the previous subsection to the\ncase of $\\sw_2$ (we will omit the index $1$ in these formulas). We\nhave the following Poisson bracket on $T(z)$:\n\\beq    \\label{vira}\n\\{ T(z),T(w) \\} = \\sum_{m\\in\\Z} \\left( \\frac{w}{z}\n\\right)^m \\frac{1-q^m}{1+q^m} T(z) T(w) + \\delta \\left( \\frac{wq}{z}\n\\right) - \\delta \\left( \\frac{w}{zq} \\right).\n\\end{equation}\n\nThe $q$--deformed Miura transformation reads:\n\\beq    \\label{virmiura}\n\\La(z) \\mapsto T(z) = \\La(z) + \\La(zq)^{-1}.\n\\end{equation}\n\nThe Poisson bracket on $\\La(z)$:\n\\beq    \\label{virpois}\n\\{ \\La(z),\\La(w) \\} = \\sum_{m\\in\\Z} \\left( \\frac{w}{z}\n\\right)^m \\frac{1-q^m}{1+q^m} \\La(z) \\La(w).\n\\end{equation}\n\n\\section{Connection with $q$--gauge transformations}    \\label{qgauge}\n\nIn this section we present the results of \\cite{FR} in the form\nof $q$--difference Drinfeld-Sokolov reduction.\n\n\\subsection{Presentation via first order $q$--difference operators}\n\nBy analogy with the differential case, it is natural to consider the\nmanifold $M_{n,q}$ of first order difference operators $D + A(s)$,\nwhere $A(s)$ is an element of the loop group $LSL_n$ of $SL_n$. The\ngroup $LSL_n$ acts on this manifold by the $q$--gauge transformations\n\\beq \\label{qadjoint} g(s) \\cdot (D + A(s)) = g(sq)(D +\nA(s))g(s)^{-1},\n\\end{equation}\ni.e. $g(s) \\cdot A(s) = g(sq) A(s) g(s)^{-1}$.\n\nNow we consider the submanifold $M^J_{n,q} \\subset M_{n,q}$ which\nconsists of operators $D+A(s)$, where $A(s)$ is of the form\n\\eqref{upper}. It is preserved under the $q$--gauge action of the\ngroup $LN$.\n\nIn the same way as in the differential case, we define a map $\\pi_q:\nM^J_{n,q} \\arr \\Mq$, which sends each element of $M^J_{n,q}$ to an\n$n$th order $q$--difference operator $L$ of the form \\eqref{qL}. It is\nclear that the map $\\pi_q$ factors through the quotient of $M^J_{n,q}$\nby $LN$. Now we state the $q$--difference analogue of \\propref{free}.\n\n\\begin{lem}    \\label{qfree}\nThe action of $LN$ on $M^J_{n,q}$ is free and each orbit contains a\nunique operator of the form \\beq \\label{qcan} D +\n\\begin{pmatrix} t_1 & t_2 & t_3 & \\hdots & t_{n-1} & 1 \\\\ -1 & 0 & 0 &\n\\hdots & 0 & 0 \\\\ 0 & -1 & 0 & \\hdots & 0 & 0 \\\\ \\hdotsfor{6} \\\\ 0 & 0\n& 0 & \\hdots & 0 & 0 \\\\ 0 & 0 & 0 & \\hdots & -1 & 0\n\\end{pmatrix}.\n\\end{equation}\n\\end{lem}\n\n\\noindent {\\em Proof} is an exercise in elementary matrix algebra. For\n$\\al=1,\\ldots,n$, denote by $M^\\al_{n,q}$ the subset of matrices from\n$M^J_{n,q}$ satisfying the property that all entries in their rows $i=\n\\al+1,\\ldots,n$ are zero except for the $(i,i-1)$ entry equal to\n$-1$. We will prove that given $A(s) \\in M^\\al_{n,q}, \\al>1$, there\nexists $g(s) \\in LN$, such that $g(sq) A(s) g(s)^{-1} \\in\nM^{\\al-1}_{n,q}$. Since the condition is vacuous for $\\al=n$, i.e.\n$M^n_{n,q} = M^J_{n,q}$, this will imply that each $LN$--orbit in\n$M^J_{n,q}$ contains an element of the form \\eqref{qcan}.\n\nTo prove the statement for a given $\\al$, we will recursively\neliminate all entries of the $\\al$th row of $A(s)$ (except the\n$(\\al,\\al-1)$ entry), from right to left using elementary unipotent\nmatrices. Denote by $E_{i,j}(x)$ the upper unipotent matrix whose only\nnon-zero entry above the diagonal is the $(i,j)$ entry equal to\n$x$. At the first step, we eliminate the $(\\al,n)$ entry $A_{\\al,n}$\nof $A(s)$ by applying the $q$--gauge transformation \\eqref{qadjoint}\nwith $g(s) = E_{\\al-1,n}(-A_{\\al,n}(s))$. Then we obtain a new matrix\n$A'(s)$, which still belongs to $M^\\al_{n,q}$, but whose $(\\al,n)$\nentry is equal to $0$. Next, we apply the $q$--gauge transformation by\n$E_{\\al-1,n-1}(-A'_{\\al,n-1}(s))$ to eliminate the $(\\al,n-1)$ entry\nof $A'(s)$, etc. It is clear that at each step we do not spoil the\nentries that have already been set to $0$. The product of the\nelementary unipotent matrices constructed at each step gives us an\nelement $g(s) \\in LN$ with the desired property that $g(sq) A(s)\ng(s)^{-1} \\in M^{\\al-1}_{n,q}$.\n\nTo complete the proof, it suffices to remark that if $A(s)$ and\n$A'(s)$ are of the form \\eqref{qcan}, and $g(sq) A(s) g(s)^{-1} =\nA'(s)$ for some $g(s) \\in LN$, then $A(s)=A'(s)$ and $g(s)=1$.\\qed\n\nFor ${\\cal L}$ of the form \\eqref{special}, $p({\\cal L})$ equals the\noperator $L$ given by formula \\eqref{L}. Thus, we have identified the\nmap $\\pi_q$ with the quotient of $M^J_{n,q}$ by $LN$ and $\\Mq$ with\n$M^J_{n,q}\/LN$.\n\n\\begin{rem}\nIn the same way as above we can prove the following more general\nstatement. Let $R$ be a ring with an automorphism $\\tau$. It rives\nrise to an automorphism of $SL_n(R)$ denoted by the same\ncharacter. Define $M^J_{\\tau,n}$ as the set of elements of $SL_n(R)$\nof the form \\eqref{upper}. Let the group $N(R)$ act on\n$M^J_{\\tau,n}(R)$ by the formula $g \\cdot A = (\\tau \\cdot g) A\ng^{-1}$. Then this action of $N(R)$ is free, and the quotient is\nisomorphic to the set ${\\cal M}^J_{\\tau,n}(R)$ of elements of\n$SL_n(R)$ of the form \\eqref{qcan} (i.e. each orbit contains a unique\nelement of the form \\eqref{qcan}). Note that the proof is not sensible\nto whether $\\tau=\\on{Id}$ or not.\n\nWhen $\\tau=\\on{Id}$, this result is well-known. It gives the classical\nnormal form of a linear operator. Moreover, in that case R.~Steinberg\nhas proved that the subset ${\\cal M}^J_{\\on{Id},n}(K)$ of $SL_n(K)$,\nwhere $K$ is an algebraically closed field, is a cross-section of the\ncollection of regular conjugacy classes in $SL_n(K)$ \\cite{St},\nTheorem 1.4. Steinberg defined an analogous cross-section for any\nsimply-connected semisimple algebraic group \\cite{St}. His results can\nbe viewed as group analogues of Kostant's results on semisimple Lie\nalgebras \\cite{Ko} (cf. \\remref{kostant}). Steinberg's cross-section\nis used in the definition of the discrete Drinfeld-Sokolov reduction\nin the general semisimple case (see \\cite{SS}).\\footnote{We are\nindebted to B.~Kostant for drawing our attention to \\cite{St}}\\qed\n\\end{rem}\n\n\\subsection{Deformed Miura transformation via $q$--gauge action}\n\nLet us attach to each element of $\\Fq$ the $q$--difference operator\n\\beq    \\label{Lambda}\n\\La = D + \\begin{pmatrix}\n\\la_1(s) & 0 & \\hdots & 0 & 0 \\\\\n-1 & \\la_2(sq^{-1}) & \\hdots & 0 & 0 \\\\\n\\hdotsfor{5} \\\\\n0 & 0 & \\hdots & \\la_{n-1}(sq^{-n+2}) & 0 \\\\\n0 & 0 & \\hdots & -1 & \\la_n(sq^{-n+1})\n\\end{pmatrix},\n\\end{equation}\nwhere $\\prod_{i=1}^n \\la_i(sq^{-i+1}) = 1$.\n\nLet $\\wt{\\muu}_q: \\Fq \\arr \\Mq$ be the composition of the embedding\n$\\Fq \\arr M^J_{n,q}$ and $\\pi_q: M^J_{n,q} \\arr M^J_{n,q}\/LN \\simeq\n\\Mq$. Using the definition of $\\pi_q$ above, one easily finds that for\n$\\La$ given by \\eqref{Lambda}, $\\wt{\\muu}_q(\\La)$ is the operator\n\\eqref{qcan}, where $t_i(s)$ is given by formula \\eqref{formulai}.\n\nTherefore we obtain\n\n\\begin{lem}\nThe map $\\wt{\\muu}_q$ coincides with the $q$--deformed Miura\ntransformation $\\muu_q$.\n\\end{lem}\n\n\\begin{rem}    \\label{qchar}\nLet $G$ be a simply-connected semisimple algebraic group over $\\C$.\nLet $V_i$ be the $i$th fundamental representation of $G$ (in the case\n$G=SL_n$, $V_i = \\Lambda^i \\C^n$), and $\\chi_i: G \\arr \\C$ be the\ncorresponding character, $\\chi_i(g) = \\on{Tr}(g,V_i)$. Define a map\n$p: G \\arr \\C^N$ by the formula $p(g) =\n(\\chi_1(g),\\ldots,\\chi_n(g))$. By construction, $p$ is constant on\nconjugacy classes. In the case $G=SL_n$ the map $p$ has a\ncross-section $r: \\C^n \\arr SL_n(\\C)$:\n$$(a_1,\\ldots,a_n) \\mapsto \\begin{pmatrix} a_1 & a_2 & a_3 & \\hdots &\na_{n-1} & 1 \\\\ -1 & 0 & 0 & \\hdots & 0 & 0 \\\\ 0 & -1 & 0 & \\hdots & 0\n& 0 \\\\ \\hdotsfor{6} \\\\ 0 & 0 & 0 & \\hdots & 0 & 0 \\\\ 0 & 0 & 0 &\n\\hdots & -1 & 0\n\\end{pmatrix}.$$ The composition $r \\circ p$, restricted to\n$M^J_{n,1}$ coincides with the map $\\pi_1$. Moreover, $\\wt{\\muu}_1$\ncan be interpreted as the restriction of $p$ to the subset of $SL_n$\nconsisting of matrices of the form \\eqref{Lambda}. Hence $\\wt{\\muu}_1$\nsends $(\\la_1,\\ldots,\\la_n)$ to the elementary symmetric polynomials\n$$t_i = \\sum_{j_1 < \\ldots < j_i} \\la_{j_1} \\la_{j_2} \\ldots\n\\la_{j_i},$$ which are the characters of the fundamental\nrepresentations of $SL_n$. As we mentioned above, Steinberg has defined\nan analogue of the cross-section $r$ for an arbitrary simply-connected\nsemisimple algebraic group \\cite{St}.\n\nFormula \\eqref{formulai} means that in terms of $\\la_j(z)$ the\ngenerators $t_i(z)$ of $W_q(\\sw_n)$ can be thought of as\n$q$--deformations of the characters of fundamental representations of\n$SL_n$. It is interesting that the same interpretation is also\nsuggested by the definintion of $\\W_q(\\sw_n)$ as the center of a\ncompletion of the quantized universal enveloping algebra\n$U_q(\\sun)_{-\\k}$ \\cite{FR}. Namely, $t_i(z)$ is then defined as the\n($q$--deformed) trace of the so-called $L$--operator acting on\n$\\Lambda^i \\C^n$ considered as a representation of $U_q(\\sun)$, see\n\\cite{ext,FR} (note also that $t_i(z)$ is closely connected with a\ntransfer-matrix of the corresponding integrable spin model).\\qed\n\\end{rem}\n\nThus, we have now represented $\\Mq$ as the quotient of the submanifold\n$M^J_{n,q}$ of the manifold $M_{n,q}$ of first order $q$--difference\noperators by the action of the group $LN$ (acting by $q$--gauge\ntransformations). We have also interpreted the $q$--deformed Miura\ntransformation in these terms. In the next sections we discuss the\nPoisson structure on $M_{n,q}$, which gives rise to the Poisson\nstructure on $\\Mq$ given by explicit formula \\eqref{p2}.\n\n\\section{Poisson structures}    \\label{P}\n\n\\subsection{Overview}    \\label{over}\nIn view of the previous section, the following diagram is the\n$q$--difference analogue of the diagram presented at the end of\n\\secref{diff1}.\n\n\n\\begin{center}\n\\begin{picture}(125,40)(-30,-33)     \n\\put(0,-1){\\vector(0,-1){26}}\n\\put(62,-1){\\vector(0,-1){26}}\n\\put(9,3){\\vector(1,0){45}}\n\\put(9,-31){\\vector(1,0){45}}\n\\put(8.9,3.9){\\oval(1,1)[l]\n\\put(8.9,-30.1){\\oval(1,1)[l]\n\\put(-5,2){$M_{n,q}^J$}\n\\put(-26,-32){$\\Mq=M_{n,q}^J\/LN$}\n\\put(57,2){$M_{n,q}=(LG,\\,\\eta_*^q)$}\n\\put(57,-32){$M_{n,q}\/LN$}\n\n\\end{picture}\n\\end{center}\n\n\nAs in the differential case, in order to define a $q$--deformation of\nthe Drinfeld-Sokolov reduction we need to find a Poisson structure\n$\\eta_*^q$ on $M_{n,q}$ and a Poisson-Lie structure $\\eta$ on $LSL_n$\nsatisfying the following properties.\n\n\\begin{itemize}\n\n\\item[(i)] the action $LSL_n \\times M_{n,q} \\arr M_{n,q}$ by\n$q$--gauge transformations is Poisson;\n\n\\item[(ii)] the subgroup $LN$ of $LSL_n$ is admissible in the sense\nthat the algebra $S_q$ of $LN$--invariant functionals on $M_{n,q}$\nis a Poisson subalgebra of the algebra of all functionals on $M_{n,q}$;\n\n\\item[(iii)] Denote by $\\mu_{ij}$ the function on $M_{n,q}$, whose\nvalue at $D + A \\in M_{n,q}$ equals the $(i,j)$ entry of $A$. The\nideal in $S_q$ generated by $\\mu_{ij} + \\delta_{i-1,j}, i>j$, is a\nPoisson ideal.\n\n\\end{itemize}\n\nGeometrically, the last condition means that $\\Mq$ is a Poisson\nsubmanifold of the quotient $M_{n,q}\/LN$.\n\nFor the sake of completeness, we recall the notions mentioned\nabove. Let $M$ be a Poisson manifold, and $H$ be a Lie group, which is\nitself a Poisson manifold. An action of $H$ on $M$ is called Poisson\nif $H \\times M\\rightarrow M$ is a Poisson map (here we equip $H\\times\nM$ with the product Poisson structure). In particular, if the\nmultiplication map $H \\times H \\arr H$ is Poisson, then $H$ is called\na Poisson-Lie group.\n\nIn this section we describe the general formalism concerning problems\n(i)--(iii) above. Then in the next section we specialize to $M_{2,q}$\nand give an explicit solution of these problems.\n\n\\subsection{Lie bialgebras}    \\label{bialg}\nLet $\\g$ be a Lie algebra. Recall \\cite{Dr} that $\\g$ is called a Lie\nbialgebra, if $\\g^*$ also has a Lie algebra structure, such that the dual\nmap $\\delta: \\g \\arr \\Lambda^2 \\g$ is a one-cocycle. We will consider {\\em\nfactorizable} Lie bialgebras ($\\g,\\delta$) satisfying the following\nconditions:\n\n\\begin{itemize}\n\n\\item[(1)] There exists a linear map $r_+: \\g^* \\arr \\g$, such that both\n$r_+$ and $r_-= -r_+^*$ are Lie algebra homomorphisms.\n\n\\item[(2)] The endomorphism $t = r_+ -r_-$ is $\\g$-equivariant and\ninduces a linear isomorphism $\\g^*\\arr\\g$.\n\n\\end{itemize}\n\nInstead of the linear operator $r_+ \\in \\on{Hom}(\\g^*,\\g)$ one often\nconsiders the corresponding element $r$ of $\\g^{\\ot 2}$ (or a\ncompletion of $\\g^{\\ot 2}$ if $\\g$ is infinite-dimensional). The\nelement $r$ (or its image in the tensor square of a particular\nrepresentation of $\\g$) is called classical $r$--matrix. It satisfies\nthe classical Yang-Baxter equation:\n\\begin{equation}    \\label{yb}\n[r_{12},r_{13}] + [r_{12},r_{23}] + [r_{13},r_{23}] = 0.\n\\end{equation}\nIn terms or $r$, $\\delta(x)=[r,x], \\forall x \\in \\g$ (here $[a \\otimes\nb,x] = [a,x] \\otimes b + a \\otimes [b,x]$). The maps $r_\\pm: \\g^* \\arr\n\\g$ are given by the formulas: $r_+(y) = (y \\otimes \\on{id})(r),\nr_-(y) = - (\\on{id} \\otimes y)(r)$.\n\nProperty (2) above means that $r+\\sigma(r)$, where $\\sigma(a \\otimes\nb) = b \\otimes a$ is a non-degenerate $\\g$--invariant symmetric\nbilinear form on $\\g^*$.\n\nSet $\\g_\\pm = \\on{Im}(r_\\pm)$. Property (1) above implies that\n$\\g_\\pm \\subset \\g$ is a Lie subalgebra. The following statement is\nessentially contained in \\cite{BD} (cf. also \\cite{rmatr}).\n\n\\begin{lem}\nLet $(\\g, \\g^*)$ be a factorizable Lie bialgebra. Then\n\n(1) The subspace $\\n_\\pm = r_\\pm(\\on{Ker} r_\\mp)$ is a Lie ideal in\n$\\g_\\pm$.\n\n(2) The map $\\theta: \\g_+\/\\n_+\\arr\\g_-\/\\n_-$ which sends the\nresidue class of $r_+(X), X\\in\\g^*$, modulo $\\n_+$ to that of $r_-(X)$\nmodulo $\\n_-$ is a well-defined isomorphism of Lie algebras.\n\n(3) Let $\\D=\\g\\oplus\\g$ be the direct sum of two copies of\n$\\g$. The map $$i: \\g^* \\arr \\D, \\quad \\quad X \\mapsto\n(r_+(X),r_-(X))$$ is a Lie algebra embedding; its image $\\g^* \\subset\n\\D$ is $$\\g^* = \\{(X_+,X_-) \\in \\g_+ \\oplus \\g_- \\subset \\D | \\ol{X}_- =\n\\theta(\\ol{X}_+) \\},$$ where $\\ol{Y}_\\pm = Y \\on{mod} \\n_\\pm$.\n\\end{lem}\n\n\\begin{rem} The connection between our notation and that of\n\\cite{RIMS} is as follows: the operator $r \\in \\on{End}\\g$ of\n\\cite{RIMS} coincides with the composition of $r_+ + r_-$ up to the\nisomorphism $t = r_+ - r_-: \\g^* \\arr \\g$; the bilinear form used in\n\\cite{RIMS} is induced by $t$.\\qed\n\\end{rem}\n\n\\subsection{Poisson-Lie groups and gauge transformations}\n\\label{adj}\nLet ($G,\\eta$) (resp., \\newline ($G^*,\\eta^*$)) be a Poisson-Lie group\nwith factorizable tangent Lie bialgebra ($\\g,\\delta$) (resp.,\n($\\g^*,\\delta^*$)). Let $G_\\pm$ and $N_\\pm$ be the Lie subgroups of\n$G$ corresponding to the Lie subalgebras $\\g_\\pm$ and $\\n_\\pm$. We\ndenote by the same symbol $\\theta$ the isomorphism $G_+\/N_+ \\arr\nG_-\/N_-$ induced by $\\theta: \\g_+\/\\n_+ \\arr \\g_-\/\\n_-$. Then the group\n$G^*$ is isomorphic to $$\\{ (g_+,g_-) \\in G_+ \\times G_- |\n\\theta(\\ol{g}_+) = \\ol{g}_- \\},$$ and we have a map $i: G^* \\arr G$\ngiven by $i((g_+,g_-)) = g_+ (g_-)^{-1}$.\n\nExplicitly, Poisson bracket on ($G,\\eta$) can be written as follows:\n\\beq \\label{skl} \\{ \\vf,\\psi \\} = \\langle r,\\nb \\vf \\wedge \\nb \\psi -\n\\nb' \\vf \\wedge \\nb' \\psi \\rangle,\n\\end{equation}\nwhere for $x \\in G$, $\\nb \\vf(x), \\nb' \\vf(x) \\in \\g^*$ are defined by\nthe formulas:\n\\begin{align}    \\label{nabla}\n\\langle \\nb \\vf(x),\\xi \\rangle &= \\frac{d}{dt} \\vf\\left( e^{t\\xi} x\n\\right)|_{t=0},\\\\ \\langle \\nb' \\vf(x),\\xi \\rangle &= \\frac{d}{dt}\n\\vf\\left( x e^{t\\xi} \\right)|_{t=0},\n\\end{align}\nfor all $\\xi \\in \\g$. Analogous formula can be written for the\nPoisson bracket on ($G^*,\\eta^*$). In formula \\eqref{skl} we use the\nstandard notation $a \\wedge b = (a \\ot b - b \\ot a)\/2$.\n\nBy definition, the action of $G$ on itself by left translations is a\nPoisson group action. There is another Poisson structure $\\eta_*$ on\n$G$ which is covariant with respect to the adjoint action of $G$ on\nitself and such that the map $i: (G^*,\\eta^*) \\arr (G,\\eta_*)$ is\nPoisson. It is given by the formula \\beq \\label{another} \\{ \\vf,\\psi\n\\} = \\langle r,\\nb \\vf \\wedge \\nb \\psi + \\nb' \\vf \\wedge \\nb' \\psi\n\\rangle - \\langle r, \\nb' \\vf \\ot \\nb \\psi - \\nb' \\psi \\ot \\nb \\vf\n\\rangle.\n\\end{equation}\n\n\\begin{prop}    \\label{embed}\n(1) The map $i: G^* \\arr G$ is a Poisson map between the Poisson\nmanifolds ($G^*,\\eta^*$) and ($G,\\eta_*$);\n\n(2) The Poisson structure $\\eta_*$ on $G$ is covariant with respect to\nthe adjoint action, i.e. the map\n$$(G,\\eta) \\times (G,\\eta_*) \\arr (G,\\eta_*): (g,h) \\mapsto g h\ng^{-1}$$ is a Poisson map.\n\\end{prop}\n\nThese results are proved in \\cite{RIMS}, \\S~3 (see also \\cite{dual},\n\\S~2), using the notion of the Heisenberg double of $G$. Formula\n\\eqref{another} can also be obtained directly from the explicit\nformulas for the Poisson structure $\\eta^*$ and for the embedding $i$.\n\nMore generally, let $\\tau$ be an automorphism of $G$, such that the\ncorresponding automorphism of $\\g$ satisfies $(\\tau \\ot \\tau)(r) =\nr$. Define a twisted Poisson structure $\\eta_*^\\tau$ on $G$ by the\nformula\n\\begin{align}\n\\label{mainb1} \\{ \\vf,\\psi \\} &= \\langle r,\\nb \\vf \\wedge \\nb \\psi + \\nb'\n\\vf \\wedge \\nb' \\psi \\rangle \\\\ \\notag &- \\langle (\\tau \\otimes\n\\on{id})(r), \\nb' \\vf \\ot \\nb \\psi - \\nb' \\psi \\ot \\nb \\vf \\rangle,\n\\end{align}\nand the twisted adjoint action of $G$ on itself by the formula $g\n\\cdot h = \\tau(g) h g^{-1}$.\n\n\\begin{thm}    \\label{act1}\n{\\em The Poisson structure $\\eta_*^\\tau$ on $G$ is covariant with\nrespect to the twisted adjoint action, i.e. the map\n$$(G,\\eta) \\times (G,\\eta_*^\\tau) \\arr (G,\\eta_*^\\tau): (g,h) \\mapsto\n\\tau(g) h g^{-1}$$ is a Poisson map.}\n\\end{thm}\n\nThis result was proved in \\cite{RIMS}, \\S~3 (see also \\cite{dual},\n\\S~2), using the notion of the twisted Heisenberg double of $G$. We\nwill use \\thmref{act1} in two cases. In the first, $G$ is the loop\ngroup of a finite-dimensional simple Lie group $\\ol{G}$, and $\\tau$ is\nthe automorphism $g(s) \\arr g(sq), q \\in \\C^\\times$. In the second, $G\n= \\ol{G}^{\\Z\/N\\Z}$, and $\\tau$ is the automorphism $(\\tau(g))_i \\arr\ng_{i+1}$. In the first case twisted conjugations coincide with\n$q$--gauge transformations, and in the second case they coincide with\nlattice gauge transformations.\n\n\\subsection{Admissibility and constraints}    \\label{admcon}\n\nLet $M$ be a Poisson manifold, $G$ a Poisson Lie group and $G \\times M\n\\arr M$ be a Poisson action. A subgroup $H\\subset G$ is called\nadmissible if the space $C^\\infty(M)^H$ of $H$--invariant functions on\n$M$ is a Poisson subalgebra in the space $C^\\infty(M)$ of all\nfunctions on $M$.\n\n\\begin{prop}[\\cite{RIMS},Theorem 6]    \\label{admiss}\nLet $\\left( {\\frak g},{\\frak g}^{*}\\right) $ be the tangent\nLie bialgebra of $G.$ A connected Lie subgroup $H\\subset G$ with Lie algebra \n${\\frak h}\\subset {\\frak g}$ is admissible if ${\\frak h}^{\\perp }\\subset\n{\\frak g}^{*}$ is a Lie subalgebra.\n\\end{prop}\n\nIn particular, $G$ itself is admissible. Note that $H\\subset G$ is a\nPoisson subgroup if and only if ${\\frak h}^{\\perp }\\subset {\\frak\ng}^{*}$ is an ideal; in that case the tangent Lie bialgebra of $H$ is\n$\\left( {\\frak h},{\\frak g}^{*}\/{\\frak h}^{\\bot }\\right)$.\n\nLet $H\\subset G$ be an admissible subgroup, and $I$ be a Poisson ideal\nin $C^\\infty(M)^H$, i.e. $I$ is an ideal in the ring $C^\\infty(M)^H$,\nand $\\{ f,g \\} \\in C^\\infty(M)^H$ for all $f \\in I, g \\in\nC^\\infty(M)^H$. Then $C^\\infty(M)^H\/I$ is a Poisson algebra.\n\nMore geometrically, the Poisson structure on $C^\\infty(M)^H\/I$ can be\ndescribed as follows. Assume that the quotient $M\/H$ exists as a smooth\nmanifold. Then there exists a Poisson structure on $M\/H$ such that the\ncanonical projection $\\pi: M\\rightarrow M\/H$ is a Poisson map. Hamiltonian\nvector fields $\\xi_\\varphi ,\\varphi \\in \\pi ^{*}C^\\infty(M\/H),$ generate an\nintegrable distribution ${\\frak H}_\\pi$ in $TM$. The following result is\nstraightforward.\n\n\\begin{lem}\n\\label{poisson}\nLet $V\\subset M$ be a submanifold preserved by $H$. Then $V\/H$ is a\nPoisson submanifold of $M\/H$ if and only if $V$ is an integral\nmanifold of ${\\frak H}_\\pi.$\n\\end{lem}\n\nThe integrality condition means precisely that the ideal $I$ of all\n$H$--invariant functions on $M$ vanishing on $V$ is a Poisson ideal in\n$C^\\infty(M)^H$, and $C^\\infty(V\/H) = C^\\infty(V)^H$ $=\nC^\\infty(M)^H\/I$. If this property holds, we will say that the Poisson\nstructure on $M\/H$ can be restricted to $V\/H$. \n\n\\begin{center}\n\\begin{picture}(125,25)(-30,-18)     \n\\put(0,0){\\vector(0,-1){12}}\n\\put(50,0){\\vector(0,-1){12}}\n\\put(9,3){\\vector(1,0){35}}\n\\put(9,-16){\\vector(1,0){35}}\n\\put(8.9,3.9){\\oval(1,1)[l]\n\\put(8.9,-15.1){\\oval(1,1)[l]\n\\put(-1,2){$V$}\n\\put(-3,-17){$V\/H$}\n\\put(48,2){$M$}\n\\put(47,-17){$M\/H$}\n\n\\end{picture}\n\\end{center}\n\nThe Poisson structure on $V\/H$ can be described as follows. Let\n$N_V\\subset T^{*}M\\mid_V$ be the conormal bundle of $V$. Clearly,\n$T^{*}V\\simeq T^{*}M\\mid _V\/N_V.$ Let $\\varphi ,\\psi \\in C(V)^H$ and\n$\\overline{d\\varphi},\\overline{d\\psi}\\in T^{*}M\\mid _V$ be any\nrepresentatives of $d\\varphi,d\\psi \\in T^{*}V.$ Let $P_M\\in\n\\bigwedge^2T\\,M$ be the Poisson tensor on $M$.\n\n\\begin{lem}\n\\label{Reduce}\nWe have \n\\begin{equation}\n\\left\\{ \\varphi ,\\psi \\right\\} = \\left\\langle\nP_M,\\overline{d\\varphi }\\ot \\overline{d\\psi}\\right\\rangle ;\n\\label{Pbr}\n\\end{equation}\nin particular, the right hand side does not depend on the choice of\n$\\overline{d\\varphi},\\overline{d\\psi}.$\n\\end{lem}\n\n\\begin{rem}\nIn the case of Hamiltonian action (i.e. when the Poisson structure on\n$H$ is trivial), one can construct submanifolds $V$ satisfying the\ncondition of \\lemref{poisson} using the moment map. Although a similar\nnotion of the nonabelian moment map in the context of Poisson group\ntheory is also available \\cite{Lu}, it is less convenient. The reason\nis that the nonabelian moment map is ``less functorial'' than the\nordinary moment map. Namely, if $G\\times M\\rightarrow M$ is a\nHamiltonian action with moment map $\\mu_G: M\\rightarrow {\\frak\ng}^{*},$ its restriction to a subgroup $H\\subset G$ is also\nHamiltonian with moment $\\mu_H=p \\circ \\mu_G$ (here $p: {\\frak\ng}^{*}\\rightarrow {\\frak h}^{*}$ is the canonical projection). If $G$\nis a Poisson-Lie group, $G^{*}$ its dual, $G\\times M\\rightarrow M$ a\nPoisson group action with moment $\\mu_G: M\\rightarrow G^{*},$ and\n$H\\subset G$ a Poisson subgroup, the action of $H$ still admits a\nmoment map. But if $H\\subset G$ is only admissible, then the\nrestricted action does not usually have a moment map. This is\nprecisely the case which is encountered in the study of the\n$q$--deformed Drinfeld-Sokolov reduction.\\qed\n\\end{rem}\n\n\\section{The $q$--deformed Drinfeld-Sokolov reduction in the case of\n$SL_2$}\n\nIn this section we apply the general results of the previous section\nto formulate a $q$--analogue of the Drinfeld-Sokolov reduction when\n$G=SL_2$.\n\n\\subsection{Choice of $r$--matrix} Let $\\g = L\\sw_2$. We would like\nto define a factorizable Lie bialgebra structure on $\\g$ in such a way\nthat the resulting Poisson-Lie structure $\\eta$ on $LSL_2$ and the\nPoisson structure $\\eta_*^q$ on $M_{2,q}$ satisfy the conditions\n(ii)--(iii) of \\secref{over}.\n\nLet $\\{ E,H,F \\}$ be the standard basis in $\\sw_2$ and $\\{ E_n,H_n,F_n\n\\}$ be the corresponding (topological) basis of $L\\sw_2 = \\sw_2 \\ot\n\\C((s))$ (here for each $A \\in \\sw_2$ we set $A_n = A \\ot s^n \\in\nL\\sw_2$). Let $\\tau$ be the automorphism of $L\\sw_2$ defined by the\nformula $\\tau(A(s)) = A(sq)$ (we assume that $q$ is generic). We have:\n$\\tau \\cdot A_n = q^n A_n$. To be able to use \\thmref{act1}, the\n$r$--matrix $r \\in L\\sw_2^{\\ot 2}$ defining the Lie bialgebra\nstructure on $L\\sw_2$ has to satisfy the condition $(\\tau \\ot \\tau)(r)\n= r$. Hence the invariant bilinear form on $L\\sw_2$ defined by the\nsymmetric part of $r$ should also be $\\tau$--invariant.\n\nThe Lie algebra $L\\sw_2$ has a unique (up to a non-zero constant\nmultiple) invariant non-degenerate bilinear form, which is invariant\nunder $\\tau$. It is defined by the formulas\n$$(E_n,F_m) = \\delta_{n,-m}, \\quad \\quad (H_n,H_m) = 2\n\\delta_{n,-m},$$ with all other pairings between the basis elements\nare $0$. This fixes the symmetric part of the element $r$. Another\ncondition on $r$ is that the subgroup $LN$ is admissible. According to\n\\propref{admiss}, this means that $L\\n_+^\\perp$ should be a Lie\nsubalgebra of $L\\sw_2^*$.\n\nA natural example of $r$ satisfying these two conditions is given by\nthe formula: \\beq \\label{new} r_0 = \\sum_{n \\in \\Z} E_n \\ot F_{-n} +\n\\frac{1}{4} H_0 \\ot H_0 + \\pol \\sum_{n>0} H_n \\ot H_{-n}.\n\\end{equation}\nIt is easy to verify that this element defines a factorizable Lie\nbialgebra structure on $\\g$. We remark that this Lie bialgebra\nstructure gives rise to Drinfeld's ``new'' realization of the\nquantized enveloping algebra associated to $L\\sw_2$\n\\cite{nr,kh-t,ked}. As we will see in the next subsection, $r_0$ can\nnot be used for the $q$--deformed Drinfeld-Sokolov reduction. However,\nthe following crucial fact will enable us to perform the\nreduction. Let $L{\\frak h}$ be the loop algebra of the Cartan\nsubalgebra ${\\frak h}$ of $\\sw_2$.\n\n\\begin{lem}    \\label{defi}\nFor any $\\rho \\in \\wedge^2 L{\\frak h}$, $r_0 + \\rho$ defines a\nfactorizable Lie bialgebra structure on $L\\sw_2$, such that\n$L\\n_+^\\perp$ is a Lie subalgebra of $L\\sw_2^*$.\n\\end{lem}\n\nThe fact that $r_0 +\\rho$ still satisfies the classical Yang-Baxter\nequation is a general property of factorizable $r$--matrices\ndiscovered in \\cite{BD}. \\lemref{defi} allows us to consider the\nclass of elements $r$ given by the formula\n\\beq \\label{class1} r = \\sum_{n \\in\n\\Z} E_n \\ot F_{-n} + \\frac{1}{2} \\sum_{m,n \\in \\Z} \\phi_{n,m} \\cdot\nH_n \\ot H_m,\n\\end{equation}\nwhere $\\phi_{n,m} + \\phi_{m,n} = \\delta_{n,-m}$.  The condition $(\\tau\n\\ot \\tau)(r) = r$ imposes the restriction $\\phi_{n,m} = \\phi_n\n\\delta_{n,-m}$, so that \\eqref{class1} takes the form \\beq\n\\label{class} r = \\sum_{n \\in \\Z} E_n \\ot F_{-n} + \\frac{1}{2} \\sum_{n\n\\in \\Z} \\phi_n \\cdot H_n \\ot H_{-n},\n\\end{equation}\nwhere $\\phi_n + \\phi_{-n} = 1$.\n\n\\subsection{The reduction}\nRecall that $M_{2,q} = LSL_2 = SL_2((s))$ consists of $2 \\times 2$\nmatrices\n\\beq    \\label{two}\nM(s) = \\begin{pmatrix} a(s) & b(s) \\\\ c(s) & d(s) \\end{pmatrix}, \\quad\n\\quad ad - bc = 1.\n\\end{equation}\nWe want to impose the constraint $c(s) = -1$, i.e. consider the\nsubmanifold $M_{2,q}^J$ and take its quotient by the (free) action of\nthe group $$LN = \\left\\{ \\begin{pmatrix} 1 & x(s) \\\\ 0 & 1\n\\end{pmatrix} \\right\\}.$$ Let $\\eta$ be the Poisson-Lie structure on\n$LSL_2$ induced by $r$ given by formula \\eqref{class}.\n\nLet $\\eta_*^q$ be the Poisson structure on $M_{2,q}$ defined by\nformula \\eqref{mainb1}, corresponding to the automorphism $\\tau: g(s)\n\\arr g(sq)$. The following is an immediate corollary of \\thmref{act1},\n\\propref{admiss} and \\lemref{defi}.\n\n\\begin{prop}\n(1) The $q$--gauge action of ($LSL_2,\\eta$) on ($M_{2,q},\\eta^q_*$)\ngiven by formula $g(s) \\cdot M(s) = g(sq) M(s) g(s)^{-1}$ is Poisson;\n\n(2) The subgroup $LN \\subset LSL_2$ is admissible.\n\\end{prop}\n\nThus, we have satisfied properties (i) and (ii) of \\secref{over}. Now\nwe have to choose the remaining free parameters $\\phi_n$ so as to\nsatisfy property (iii).\n\nThe Fourier coefficients of the matrix elements of the matrix $M(s)$\ngiven by \\eqref{two} define functions on $M_{2,q}$. We will use the\nnotation $a_m$ for the $m$th Fourier coefficient of $a(s)$. Let\n$R_{2,q}$ be the completion of the ring of polynomials in $a_m, b_m,\nc_m, d_m, m \\in \\Z$, defined in the same way as the ring $\\RN$ of\nSect.~2.3. Let $S_{2,q} \\subset R_{2,q}$ be the subalgebra of\n$LN$--invariant functions. Denote by $I$ be the ideal of $S_{2,q}$\ngenerated by $\\{ c_n + \\delta_{n,0}, n\\in\\Z \\}$ (the defining ideal of\n$M_{2,q}^J$).\n\nProperty (iii) means that $I$ is a Poisson ideal of $s_{2,q}$, which\nis equivalent to the condition that $\\{ c_n,c_m \\} \\in I$, i.e. that\nif $\\{ c_n,c_m \\}$ vanishes on $M_{2,q}^J$. This condition means that\nthe Poisson bracket of the constraint functions vanishes on the\nconstraint surface, i.e. the constraints are of first class according\nto Dirac.\n\nLet us compute the Poisson bracket between $c_n$'s. First, we list the\nleft and right gradients for the functions $a_n,b_n,c_n,d_n$ (for this\ncomputation we only need the gradients of $c_n$'s, but we will soon\nneed other gradients as well). It will be convenient for us to\nidentify $L\\sw_2$ with its dual using the bilinear form introduced in\nthe previous section. Note that with respect to this bilinear form the\ndual basis elements to $E_n, H_n$, and $F_n$ are $F_{-n}, H_{-n}\/2$,\nand $E_{-n}$, respectively.\n\nExplicit computation gives (for shorthand, we write $a$ for $a(s)$,\netc.):\n$$\\nb a_m = s^{-m} \\begin{pmatrix} \\pol a & 0 \\\\ c & - \\pol a\n\\end{pmatrix}, \\quad \\quad \\nb b_m = s^{-m} \\begin{pmatrix} \\pol b & 0\n\\\\ d & - \\pol b \\end{pmatrix},$$\n$$\\nb c_m = s^{-m} \\begin{pmatrix} - \\pol c & a \\\\ 0 & \\pol c\n\\end{pmatrix}, \\quad \\quad \\nb d_m = s^{-m} \\begin{pmatrix} - \\pol d & b\n\\\\ 0 & \\pol b \\end{pmatrix},$$\n$$\\nb' a_m = s^{-m} \\begin{pmatrix} \\pol a & b \\\\ 0 & - \\pol a\n\\end{pmatrix}, \\quad \\quad \\nb' b_m = s^{-m} \\begin{pmatrix} - \\pol b\n& 0 \\\\ a & \\pol b \\end{pmatrix},$$\n$$\\nb' c_m = s^{-m} \\begin{pmatrix} \\pol c & d \\\\ 0 & - \\pol c\n\\end{pmatrix}, \\quad \\quad \\nb' d_m = s^{-m} \\begin{pmatrix} - \\pol d & 0\n\\\\ c & - \\pol d \\end{pmatrix}.$$\n\nNow we can compute the Poisson bracket between $c_n$'s using formula\n\\eqref{mainb1}: \\beq \\label{cbracket} \\{ c_m,c_k \\} = \\pol \\sum_{n \\in\n\\Z} \\left( \\phi_n - \\phi_{-n} + \\phi_n q^n - \\phi_{-n} q^{-n} \\right)\nc_{-n+m} c_{n+k}.\n\\end{equation}\n\nRestricting to $M_{2,q}^J$, i.e. setting $c_n = -\\delta_{n,0}$, we\nobtain:\n$$\\{ c_m,c_k \\}|_{M_{2,q}^J} = \\pol \\sum_{n \\in \\Z} \\left( \\phi_m -\n\\phi_{-m} + \\phi_m q^m - \\phi_{-m} q^{-m} \\right) \\delta_{m,-k}.$$ This\ngives us the following equation on $\\phi_m$'s: $$\\phi_m - \\phi_{-m} +\n\\phi_m q^m - \\phi_{-m} q^{-m} = 0.$$ Together with the previous condition\n$\\phi_m + \\phi_{-m} = 1$, this determines $\\phi_m$'s uniquely:\n\n\\begin{thm}    \\label{third}\n{\\em The Poisson structure $\\eta_*^q$ satisfies property (iii) of the\n$q$--deformed Drin\\-feld-Sokolov reduction if and only if} $$\\phi_n =\n\\frac{1}{1+q^n}.$$\n\\end{thm}\n\nConsider the $r$--matrix \\eqref{class1} with $\\phi_n = (1+q^n)^{-1}$.\nFor this $r$--matrix, the Lie algebras defined in section\n\\secref{bialg} are as follows: $\\g_\\pm = L{\\frak b}_\\mp, {\\frak n}_\\pm\n= L{\\frak n}_\\mp$, where ${\\frak n}_+ = \\C E, {\\frak n}_- = \\C F,\n{\\frak b}_\\pm = {\\frak h} \\oplus {\\frak n}_\\pm$. We have:\n$\\g_\\pm\/{\\frak n}_\\pm \\simeq L{\\frak h}$. The transformation $\\theta$\non $L{\\frak h}$ induced by this $r$--matrix is equal to $-\\tau$.\n\nExplicitly, on the tensor product of the two $2$--dimensional\nrepresentations of $\\sw_2((t))$, the $r$--matrix looks as follows:\n\\begin{equation}    \\label{explicit}\n\\begin{pmatrix} \\vff & 0 & 0 & 0 \\\\ 0 & - \\vff & \\delta \\left(\n\\frac{t}{s} \\right) & 0 \\\\ 0 & 0 & - \\vff & 0 \\\\ 0 & 0 & 0 & \\vff\n\\end{pmatrix},\n\\end{equation}\nwhere $$\\phi(x) = \\pol \\sum_{n \\in \\Z} \\frac{1}{1+q^n} x^n.$$ Note\nthat $2 \\pi \\phi(xq^{1\/2})$ coincides with the power series expansion\nof the Jacobi elliptic function $dn$ (delta of amplitude).\n\nNow we have satisfied all the necessary properties on the Poisson\nstructures and hence can perform the $q$--Drinfeld-Sokolov reduction\nof \\secref{qgauge} at the level of Poisson algebras. In the next\nsubsection we check that it indeed gives us the Poisson bracket\n\\eqref{vira} on the reduced space ${\\cal M}_{2,q} = M_{2,q}^J\/LN$.\n\n\\subsection{Explicit computation of the Poisson brackets}\n\nIntroduce the generating series $$A(z) = \\sum_{n \\in \\Z} a_n z^{-n},$$\nand the same for other matrix elements of $M(s)$ given by formula\n\\eqref{two}. We fix the element $r$ by setting $\\phi_n = (1+q^n)^{-1}$\nin formula \\eqref{class} in accordance with \\thmref{third}. Denote\n\\beq\n\\label{fi} \\vf(z) = \\sum_{n \\in \\Z} (\\phi_n - \\phi_{-n}) z^n = \\sum_{n \\in\n\\Z} \\frac{1-q^n}{1+q^n} z^n.\n\\end{equation}\nUsing the formulas for the gradients of the matrix elements given in\nthe previous section and formula \\eqref{mainb1} for the Poisson\nbracket, we obtain the following explicit formulas for the Poisson\nbrackets.\n\n\\begin{align*}\n\\{ A(z),A(w) \\} &= \\vf \\left( \\frac{w}{z} \\right) A(z) A(w), \\\\\n\\{ A(z),B(w) \\} &= - \\delta \\left( \\frac{w}{z} \\right) A(z) B(w), \\\\\n\\{ A(z),C(w) \\} &= \\delta \\left( \\frac{w}{z} \\right) A(z) C(w), \\\\\n\\{ A(z),D(w) \\} &= - \\vf \\left( \\frac{w}{z} \\right) A(z) D(w),\\\\\n\\{ B(z),B(w) \\} &= 0, \\\\\n\\{ B(z),C(w) \\} &= \\delta \\left( \\frac{w}{z} \\right) A(z) D(w) -\n\\delta \\left( \\frac{wq}{z} \\right) A(z) A(w), \\\\\n\\{ B(z),D(w) \\} &= - \\delta \\left( \\frac{wq}{z} \\right) A(z) B(w), \\\\\n\\{ C(z),C(w) \\} &= 0, \\\\\n\\{ C(z),D(w) \\} &= \\delta \\left( \\frac{w}{zq} \\right) A(z) C(w),\\\\\n\\{ D(z),D(w) \\} &= \\vf \\left( \\frac{w}{z} \\right) D(z) D(w) - \\delta\n\\left( \\frac{wq}{z} \\right) C(z) B(w) + \\delta \\left( \\frac{w}{zq}\n\\right) B(z) C(w).\n\\end{align*}\n\n\\begin{rem}\nThe relations above can be presented in matrix form as follows. Let\n$$L(z) = \\begin{pmatrix} A(z) & B(z) \\\\ C(z) & D(z) \\end{pmatrix},$$\nand consider the operators $L_1 = L \\ot \\on{id}, L_2 = \\on{id} \\ot L$\nacting on $\\C^2 \\ot \\C^2$. The $r$--matrix \\eqref{explicit} also acts\non $\\C^2 \\ot \\C^2$. Formula \\eqref{mainb1} can be written as follows:\n\\begin{align*}\n\\{ L_1(z),L_2(w) \\} &= \\pol r^- \\left( \\frac{w}{z} \\right) L_1(z)\nL_2(w) + \\pol L_1(z) L_2(w) r^- \\left( \\frac{w}{z} \\right) \\\\ &-\nL_1(z) r \\left( \\frac{wq}{z} \\right) L_2(w) + L_2(w) \\sigma(r) \\left(\n\\frac{zq}{w} \\right) L_1(z),\n\\end{align*}\nwhere $$r^- \\left( \\frac{w}{z} \\right) = r \\left( \\frac{w}{z} \\right)\n- \\sigma(r) \\left( \\frac{z}{w} \\right) = \\begin{pmatrix} \\pol \\vpp & 0\n& 0 & 0 \\\\ 0 & - \\pol \\vpp & \\delta \\left( \\frac{w}{z} \\right) & 0 \\\\\n0 & - \\delta \\left( \\frac{w}{z} \\right) & - \\pol \\vpp & 0 \\\\ 0 & 0 & 0\n& \\pol \\vpp\n\\end{pmatrix}.$$\\qed\n\\end{rem}\n\n\\subsection{Reduced Poisson structure}\nWe know that ${\\cal M}_{2,q} = M_{2,q}^J\/LN$ is isomorphic to\n$$\\left\\{\n\\begin{pmatrix} t(s) & 1 \\\\ -1 & 0 \\end{pmatrix} \\right\\}$$ (see\nSect.~3). The ring ${\\cal R}_{2,q}$ of functionals on ${\\cal M}_{2,q}$\nis generated by the Fourier coefficients of $t(s)$. In order to\ncompute the reduced Poisson bracket between them, we have to extend\nthem to $LN$--invariant functions on the whole $M_{2,q}$. Set \\beq\n\\wt{t}(s) = a(s) c(sq) + d(sq) c(s).\n\\end{equation}\nIt is easy to check that the Fourier coefficients $\\wt{t}_m$ of\n$\\wt{t}(s)$ are $LN$--invariant, and their restrictions to $M_{2,q}^J$\ncoincide with the corresponding Fourier coefficients of $t(s)$.\n\nLet us compute the Poisson bracket between $\\wt{t}_m$'s. Set\n$$\\wt{T}(z) = \\sum_{m \\in \\Z} \\wt{t}_m z^{-m}.$$ Using the explicit\nformulas above, we find\n\\begin{align} \\notag\n\\{ \\wt{T}(z),\\wt{T}(w) \\} &= \\vf \\left( \\frac{w}{z} \\right) \\wt{T}(z)\n\\wt{T}(w) \\\\ \\label{tildet} &+ \\delta \\left( \\frac{wq}{z} \\right)\n\\Delta(z) c(w) c(wq^2) - \\delta \\left( \\frac{w}{zq} \\right) \\Delta(w)\nc(z) c(zq^2),\n\\end{align}\nwhere $\\Delta(z) = A(z)D(z) - B(z)C(z) = 1$. Hence, restricting to\n$M_{2,q}^J$ (i.e. setting $c(z)=1$ in formula \\eqref{tildet}), we\nobtain:\n$$\\{ T(z),T(w) \\} = \\vf \\left( \\frac{w}{z} \\right) T(z) T(w) + \\delta\n\\left( \\frac{wq}{z} \\right) - \\delta \\left( \\frac{w}{zq} \\right).$$\nThis indeed coincides with formula \\eqref{vira}.\n\n\\begin{rem}\nConsider the subring $\\wt{S}_{2,q}$ of the ring $R_{2,q}$, generated by\n$c_m, \\wt{t}_m, m \\in \\Z$. The ring $\\wt{S}_{2,q}$ consists of\n$LN$--invariant functionals on $M_{2,q}$ and hence it can serve as a\nsubstitute for the ring of functions on $M_{2,q}\/LN$. Let us compute\nthe Poisson brackets in $\\wt{S}_{2,q}$. The Poisson brackets of\n$\\wt{t}_m$'s are given by formula \\eqref{tildet}, and by construction\n$\\{ c_m,c_k \\} = 0$. It is also easy to find that $\\{ c_m,\\wt{t}_k \\}\n= 0$. Hence $\\wt{S}_{2,q}$ is a Poisson subalgebra of $R_{2,q}$. Thus, the\n$q$--deformed Drinfeld-Sokolov reduction can be interpreted as\nfollows. The initial Poisson algebra is $R_{2,q}$. We consider its\nPoisson subalgebra $\\wt{S}_{2,q}$ generated by $c_m$'s and\n$\\wt{t}_m$'s. The ideal $I$ of $\\wt{S}_{2,q}$ generated by $\\{ c_m +\n\\delta_{m,0}, m \\in \\Z \\}$ is a Poisson ideal. The quotient\n$\\wt{S}_{2,q}\/I$ is isomorphic to the $q$--Virasoro algebra ${\\cal\nR}_{2,q}$ defined in Sect.~3.\\qed\n\\end{rem}\n\n\\subsection{$q$--deformation of Miura transformation}\n\nAs was explained in Sect.~3.2, the $q$--Miura transformation of\n\\cite{FR} is the map between two (local) cross-sections of the\nprojection $\\pi_q: M_{n,q}^J \\arr M_{n,q}^J\/LN$. In the case of\n$LSL_2$, the first cross-section $$\\left\\{ \\begin{pmatrix} \\la(s) & 0\n\\\\ -1 & \\la(s)^{-1} \\end{pmatrix} \\right \\}$$ is defined by the\nsubsidiary constraint $b(s)=0$, and the second $$\\left\\{\n\\begin{pmatrix} t(s) & 1 \\\\ -1 & 0 \\end{pmatrix} \\right\n\\}$$ is defined by the subsidiary constraint $d(s)=0$. The map between\nthem is given by the formula $$\\muu_q: \\la(s) \\mapsto t(s) = \\la(s) +\n\\la(sq)^{-1}.$$ Now we want to recover formula \\eqref{virpois} for the\nPoisson brackets between the Fourier coefficients $\\la_n$ of $\\la(s)$,\nwhich makes the map $\\muu_q$ Poisson.\n\nWe have already computed the Poisson bracket on the second (canonical)\ncross-section from the point of view of Poisson reduction. Now we need\nto compute the Poisson bracket between the functions $a_n$'s on the\nfirst cross-section, with respect to which the map $\\muu_q$ is\nPoisson. This computation is essentially similar to the one outlined\nin Sect.~3.2. The Poisson structure on the local cross-section is\ngiven by the Dirac bracket, which is determined by the choice of the\nsubsidiary conditions, which fix the cross-section.\n\nThe Dirac bracket has the following property (see\n\\cite{Flato}). Suppose we are given constraints $\\xi_n, n \\in I$, and\nsubsidiary conditions $\\eta_n, n \\in I$, on a Poisson manifold $M$,\nsuch that $\\{ \\xi_k,\\xi_l \\} = \\{ \\eta_k,\\eta_l \\} = 0, \\forall k,l\n\\in I$. Let $f, g$ be two functions on $M$, such that $\\{ f,\\xi_k \\}$\nand $\\{ g,\\xi_k \\}$ vanish on the common level surface of all $\\xi_k,\n\\eta_k$. Then the Dirac bracket of $f$ and $g$ coincides with their\nordinary Poisson bracket.\n\nIn our case, the constraint functions are $c_m+\\delta_{m,0}, m \\in\n\\Z$, and the subsidiary conditions $b_m, m \\in \\Z$, which fix the\nlocal model of the reduced space. We have: $\\{ b_m,b_k \\} = 0$, $\\{\nc_m,c_k \\} = 0$, and $\\{ a_m,b_k \\} = 0$, if we set $b_m=0, \\forall m\n\\in \\Z$. Therefore we are in the situation described above, and the\nDirac bracket between $a_m$ and $a_k$ coincides with their ordinary\nbracket. In terms of the generating function $A(z) = \\sum_{m \\in \\Z}\na_m z^{-m}$ it is given by the formula $$\\{ A(z),A(w) \\} = \\vf \\left(\n\\frac{w}{z} \\right) A(z) A(w),$$ which coincides with formula\n\\eqref{virpois}. Thus, we have proved the Poisson property of the\n$q$--deformation of the Miura transformation from the point of view of\nthe deformed Drinfeld-Sokolov reduction.\n\n\\section{Lattice Virasoro algebra}\n\nIn this section we consider the lattice counterpart of the\nDrinfeld-Sokolov reduction. Our group is thus $\\GG = (SL_2)^{\\Z\/N\\Z}$,\nwhere $N$ is an integer, and $\\tau$ is the automorphism of $G$, which\nmaps $(g_i)$ to $(g_{i+1})$. Poisson structures on $\\GG$ which are\ncovariant with respect to lattice gauge transformations $x_n \\mapsto\ng_{n+1} x_n g_n^{-1}$ have been studied already in \\cite{RIMS}\n(cf. also \\cite{AFS}). In order to make the reduction over the\nnilpotent subgroup $\\NN \\subset \\GG$ feasible, we have to be careful\nin our choice of the $r$--matrix.\n\n\\subsection{Discrete Drinfeld-Sokolov reduction}\nBy analogy with the continuous case, we choose the element $r$ defining the\nLie bialgebra structure on $\\g = \\sw_2^{\\oplus \\Z\/N\\Z}$ as follows: $$r =\n\\sum_{n \\in \\Z\/N\\Z} E_n \\ot F_n + \\frac{1}{4} \\sum_{m,n \\in \\Z\/N\\Z}\n\\phi_{n,m} H_n \\ot H_m,$$ where $\\phi_{n,m} + \\phi_{m,n} = 2\n\\delta_{m,n}$. It is easy to see that $r$ defines a factorizable Lie\nbialgebra structure on $\\g$. For \\thmref{act1} to be applicable, $r$ has to\nsatisfy the condition $(\\tau \\ot \\tau)(r) = r$, which implies that\n$\\phi_{n,m} = \\phi_{n-m}$.\n\nAn element of $\\GG$ is an $N$--tuple $(g_i)$ of elements of $SL_2$:\n$$g_k = \\begin{pmatrix} a_k & b_k \\\\ c_k & d_k \\end{pmatrix}.$$ We\nconsider $a_k,b_k,c_k,d_k, k \\in \\Z\/N\\Z$, as the generators of the\nring of functions on $\\GG$.\n\nThe discrete analogue of the Drinfeld-Sokolov reduction consists of\ntaking the quotient $\\MM=\\GG^J\/\\NN$, where $\\GG^J = (G^J)^{\\ZN}$,\n$$G^J = \\left\\{ \\begin{pmatrix} a & b \\\\ -1 & d \\end{pmatrix}\n\\right\\},$$ and $\\NN = N^{\\ZN}$, acting on $G^J$ by the formula \\beq\n\\label{dgauge} (h_i) \\cdot (g_i) = (h_{i+1} g_i h_i^{-1}).\n\\end{equation}\nIt is easy to see that $$\\MM \\simeq \\left \\{ \\begin{pmatrix} t_i &\n1 \\\\ -1 & 0 \\end{pmatrix}_{i \\in \\ZN} \\right\\}.$$\n\nThe element $r$ with $\\phi_{n,m} = \\phi_{n-m}, \\phi_k + \\phi_{-k} =\n2\\delta_{k,0}$, defines a Lie bialgebra structure on $\\g$ and Poisson\nstructures $\\eta, \\eta_*^\\tau$ on $\\GG$. According to \\thmref{act1},\nthe action of ($\\GG,\\eta$) on ($\\GG,\\eta_*^\\tau$) given by formula\n\\eqref{dgauge} is Poisson.\n\nAs in the continuous case, for the Poisson structure $\\eta_*^\\tau$ to\nbe compatible with the discrete Drinfeld-Sokolov reduction, we must\nhave:\n\\beq    \\label{vanish}\n\\{ c_n,c_m \\}|_{\\GG^J} = 0.\n\\end{equation}\nExplicit calculation analogous to the one made in the previous\nsubsection shows that \\eqref{vanish} holds if and only if $$\\phi_{n-1}\n+ 2\\phi_n + \\phi_{n+1} = 2 \\delta_{n,0} + 2 \\delta_{n+1,0}.$$ The\ninitial condition $\\phi_0 = 1$ and periodicity condition give us a\nunique solution: for odd $N$, $\\phi_k = (-1)^k$; for even $N$, $\\phi_k\n= (-1)^k \\left( 1 - \\dfrac{2k}{N} \\right)$. In what follows we\nrestrict ourselves to the case of odd $N$ (note that in this case the\nlinear operator $\\on{id}+\\tau$ is invertible).\n\nContinuing as in the previous subsection, we define $$\\wt{t}_n = a_n\nc_{n+1} + d_{n+1} c_n, \\quad \\quad n \\in \\ZN.$$ These are\n$\\NN$--invariant functions on $\\GG$. We find in the same way as in the\ncontinuous case:\n\\beq    \\label{wtt}\n\\{ \\wt{t}_n,\\wt{t}_m \\} = \\vf_{n-m} \\wt{t}_n \\wt{t}_m + \\delta_{n,m+1}\nc_m c_{m+2} - \\delta_{n+1,m} c_n c_{n+2},\n\\end{equation}\n$$\\{ \\wt{t}_n,c_m \\} = 0, \\quad \\quad \\{ c_n,c_m \\} = 0,$$ where\n$$\\vf_k = \\pol (\\phi_k - \\phi_{-k}) = \\left\\{ \\begin{array}{cc} 0, &\nk=0, \\\\ (-1)^k, & k\\neq 0.\n\\end{array} \\right.$$\nThe discrete Virasoro algebra $\\C[t_i]_{i \\in \\ZN}$ is the quotient of\nthe Poisson algebra $\\C[\\wt{t}_i,c_i]_{i \\in \\ZN}$ by its Poisson\nideal generated by $c_i+1, i \\in \\ZN$. From formula \\eqref{wtt} we\nobtain the following Poisson bracket between the generators $t_i$:\n\\beq\n\\label{dvir} \\{ t_n,t_m \\} = \\vf_{n-m} t_n t_m + \\delta_{n,m+1} -\n\\delta_{n+1,m}.\n\\end{equation}\n\nThe discrete Miura transformation is the map from the local\ncross-section $$\\left\\{ \\begin{pmatrix} \\la_n & 0 \\\\ -1 & \\la_n^{-1}\n\\end{pmatrix} \\right \\}$$ to $\\MM$,\n\\beq    \\label{dmiura}\n\\la_n \\mapsto t_n = \\la_n + \\la_{n+1}^{-1}.\n\\end{equation}\nIt defines a Poisson map $\\C[\\la_i^\\pm]_{i \\in \\ZN} \\arr \\C[t_i]_{i\n\\in \\ZN}$, where the Poisson structure on the latter is given by the\nformula \\beq \\label{dheis} \\{ \\la_n,\\la_m \\} = \\vf_{n-m} \\la_n \\la_m.\n\\end{equation}\n\n\\begin{rem}\nThe Poisson algebra $\\C[t_i]_{i \\in \\ZN}$ can be considered as a\nregularized version of the $q$--deformed Virasoro algebra when\n$q=\\ep$, where $\\ep$ is a primitive $N$th root of unity. Indeed, we\ncan then consider $t(\\ep^i), i \\in \\ZN$, as generators and truncate in\nall power series appearing in the relations, summations over $\\Z$ to\nsummations over $\\ZN$ divided by $N$. This means that we replace\n$\\phi(\\ep^n)$ given by formula \\eqref{vira} by\n$$\\wt{\\phi}(\\ep^n) = \\frac{1}{N} \\sum_{i \\in \\ZN}\n\\frac{1-\\ep^i}{1+\\ep^i} \\ep^{ni},$$ and $\\delta(\\ep^n)$ by\n$\\delta_{n,0}$. The formula for the Poisson bracket then becomes: $$\\{\nt(\\ep^n),t(\\ep^m) \\} = \\wt{\\phi}(\\ep^{m-n}) t(\\ep^n) t(\\ep^m) +\n\\delta_{n,m+1} - \\delta_{n+1,m}.$$ If we set $t(\\ep^i)=t_i$, we\nrecover the Poisson bracket \\eqref{dvir}, since it is easy to check\nthat $\\wt{\\phi}(\\ep^{m-n}) = \\vf_{n-m}$.\n\nOne can apply the same procedure to the $q$--deformed $\\W$--algebras\nassociated to $\\sw_N$ and obtain lattice Poisson algebras. It would be\ninteresting to see whether they are related to the lattice\n$\\W$--algebras studied in the literature, e.g., in \\cite{Be,Bo}. In\nthe case of $\\sw_2$, this connection is described in the next\nsubsection.\\qed\n\\end{rem}\n\n\\subsection{Connection with Faddeev-Takhtajan-Volkov algebra}\nThe Poisson structures \\eqref{dvir} and \\eqref{dheis} are nonlocal,\ni.e. the Poisson brackets between distant neighbors on the lattice\nare nonzero. However, one can define closely connected Poisson\nalgebras possessing local Poisson brackets; these Poisson algebras can\nactually be identified with those studied by L.~Faddeev,\nL.~Takhtajan, and A.~Volkov.\n\nLet us first recall some results of \\cite{FR} concerning the\ncontinuous case. As was explained in \\cite{FR}, one can associate a\ngenerating series of elements of the $q$--Virasoro algebra to an\narbitrary finite-dimensional representation of $\\sw_2$. The series\n$T(z)$ considered in this paper corresponds to the two-dimensional\nrepresentation. Let $T^{(2)}(z)$ be the series corresponding to the\nthree-dimensional irreducible representation of $sw_2$. We have the\nfollowing identity \\cite{FR}\n$$T(z) T(zq) = T^{(2)}(z) + 1,$$ which can be taken as the definition of\n$T^{(2)}(z)$. From formula \\eqref{virmiura} we obtain:\n\\begin{align*}\nT^{(2)}(z) &= \\La(z) \\La(zq) + \\La(z) \\La(zq^2)^{-1} + \\La(zq)^{-1}\n\\La(zq^2)^{-1} \\\\ &= A(z) + A(z) A(zq)^{-1} + A(zq)^{-1},\n\\end{align*}\nwhere\n\\beq    \\label{az}\nA(z) = \\La(z) \\La(zq)\n\\end{equation}\n(the series $A(z)$ was introduced in Sect.~7 of \\cite{FR}). From\nformula \\eqref{virpois} we find:\n$$\\{ A(z),A(w) \\} = \\left( \\delta \\left( \\frac{w}{zq} \\right) - \\delta\n\\left( \\frac{wq}{z} \\right) \\right) A(z) A(w).$$ It is also easy to find\n\\begin{align*}    \\label{three}\n\\{ T^{(2)}(z),T^{(2)}(w) \\} &= \\left( \\delta \\left( \\frac{w}{zq}\n\\right) - \\delta \\left( \\frac{wq}{z} \\right) \\right) \\left( T^{(2)}(z)\nT^{(2)}(w) - 1 \\right) \\\\ &+ \\delta \\left( \\frac{wq^2}{z} \\right) T(w)\nT(wq^3) - \\delta \\left( \\frac{w}{zq^2} \\right) T(z) T(zq^3).\n\\end{align*}\n\nWe can use the same idea in the lattice case. Let $\\nu_n = \\la_n\n\\la_{n+1}$; this is the analogue of $A(z)$. We have: \\beq \\label{nun}\n\\{ \\nu_n,\\nu_m \\} = (\\delta_{n+1,m} - \\delta_{n,m+1}) \\nu_n \\nu_m,\n\\end{equation}\nand hence $\\C[\\nu_i^\\pm]$ is a Poisson subalgebra of $\\C[\\la_n^\\pm]$\nwith local Poisson brackets. We can also define $t^{(2)}_n = t_n\nt_{n+1} - 1$. The Poisson bracket of $t^{(2)}_n$'s is local:\n\\begin{align}    \\label{t2}\n\\{ t^{(2)}_n,t^{(2)}_m \\} &= \\left( \\delta_{n+1,m} - \\delta_{n,m+1}\n\\right) \\left( t^{(2)}_n t^{(2)}_m - 1 \\right) \\\\ \\notag &+\n\\delta_{n,m+2} t_m t_{m+3} - \\delta_{n+2,m} t_n t_{n+3}.\n\\end{align}\nUnfortunately, it does not close on $t^{(2)}_n$'s, so that\n$\\C[t^{(2)}_i]$ is not a Poisson subalgebra of $\\C[t_i]$. But let us\ndefine formally\n\\beq    \\label{sn}\ns_n = \\frac{1}{1+t^{(2)}_n} = t_n^{-1} t_{n+1}^{-1} =\n\\frac{1}{(1+\\nu_n)(1+\\nu_{n+1}^{-1})}.\n\\end{equation}\nThen from formulas \\eqref{sn} and \\eqref{nun} we find:\n\\begin{align}    \\label{fad}\n\\{ s_n,s_m \\} = & s_n s_m \\big( (\\delta_{n+1,m} - \\delta_{n,m+1})(1 -\ns_n - s_m) - \\\\ \\notag & - s_{n+1} \\delta_{n+2,m} + s_{m+1}\n\\delta_{n,m+2} \\big).\n\\end{align}\nThus, the Poisson bracket closes among $s_n$'s and defines a Poisson\nstructure on $\\C[s_i]_{i\\in\\ZN}$.\n\nThe Poisson algebra $\\C[s_i]_{i\\in\\ZN}$ with Poisson bracket\n\\eqref{fad} was first introduced by Faddeev and Takhtajan in \\cite{ft}\n(see formula (54)). We see that it is connected with our version of\nthe discrete Virasoro algebra, $\\C[t_i]$, by a change of variables\n\\eqref{sn}. The Poisson algebra $\\C[\\nu_i^\\pm]$ and the Poisson map\n$\\C[\\nu_i^\\pm] \\arr \\C[s_n]$ given by formula \\eqref{sn} were\nintroduced by Volkov in \\cite{v1} (see formulas (2) and (23))\nfollowing \\cite{ft}; see also related papers \\cite{v2,fv}. This map is\nconnected with our version \\eqref{dmiura} of the discrete Miura\ntransformation by a change of variables.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{sec:intro}\nPre-trained Language Models (PLMs) have been successfully adapted to a wide range of Natural Language Processing (NLP) tasks using \\emph{prompt-based} learning~\\cite{Radford:2018,GPT-2,GPT3,petroni-etal-2019-language} such as sentiment classification~\\cite{gao-etal-2021-making}, natural language inference (NLI)~\\cite{schick-schutze-2021-exploiting,schick-schutze-2022-true}, relation extraction~\\cite{shin-etal-2020-autoprompt}, cross-lingual inference~\\cite{qi-etal-2022-enhancing}.\nHowever, manually writing prompts that generalize well is very challenging for several reasons such as (a) it might not always be possible to recruit domain-expert human annotators, \n(b) human annotators might not be able to cover all corner cases by writing prompts, and\n(c) there can be disagreements between human annotators regarding the coverage of a particular prompt.\nTo address these challenges, automatic learning of discrete prompts has been proposed such as AdvTrigger~\\cite{wallace-etal-2019-universal}, AutoPrompt~\\cite[\\textbf{AP};][]{shin-etal-2020-autoprompt}, WARP~\\cite{hambardzumyan-etal-2021-warp}, and RLPrompt~\\cite{DBLP:journals\/corr\/abs-2205-12548}.\n\n\n\\begin{table*}[t]\n    \\centering\n    \\small\n    \\begin{tabular}{lllc}\n        \\toprule\n         \\textbf{Relation} &  \\textbf{Method} & \\textbf{Prompt} & \\textbf{P@1} \\\\ \\midrule\n         \\textsf{native-language-of} (P103)   & Manual            & \\texttt{The native language of [X] is [Y]} & 74.54\\\\\n                                            & AP BERT   & \\texttt{[X]PA communerug speaks proper [Y]} & \\textbf{84.87}\\\\\n                                            & AP RoBERTa & \\texttt{[X]neau optionally fluent!?\\\" traditional [Y]} & 81.61\\\\ \\midrule\n         \\textsf{profession-of} (P106) & Manual & \\texttt{[X] is a [Y] by profession} & 0.73 \\\\\n                                            & AP BERT & \\texttt{[X] supporters studied politicians musician turned [Y]} & 15.83 \\\\\n                                            & AP RoBERTa & \\texttt{[X] (), astronomers businessman\u00b7former [Y]} & \\textbf{19.24} \\\\ \\midrule\n         \\textsf{music-played-by} (P136) & Manual & \\texttt{[X] plays [Y] music} & 0.7\\\\\n                                         & AP BERT & \\texttt{[X] freaking genre orchestra fiction acid [Y]} & \\textbf{59.95} \\\\\n                                          & AP RoBERTa & \\texttt{[X] blends postwar hostage drama sax [Y]} & 52.97 \\\\\n         \\bottomrule\n    \\end{tabular}\n    \\caption{Examples of prompts learnt by AP for the fact retrieval task for BERT and RoBERTa PLMs and the human-written manual prompts. T-REx relation ids are shown with brackets for each relation type. Precision@1 (P@1) scores are shown when each prompt is used in fact retrieval.}\n    \\label{tbl:autoprompt-examples}\n\\end{table*}\n\nAlthough discrete prompt learning methods have achieved good performance in numerous downstream tasks by automatically learnt prompts, such automatic prompts seem to be significantly different from the manually-written ones.\nFor example, \\autoref{tbl:autoprompt-examples} shows manually-written and AP-learnt prompts for fact retrieval~\\cite{petroni-etal-2019-language}.\nWe see that the AP-learnt prompts for BERT~\\cite{devlin-etal-2019-bert} and RoBERTa~\\cite{RoBERTa} outperform the manual prompts in precision\\@1 (P@1) scores.\nHowever, the AP-learnt prompts contain various counter-intuitive language constructs such as punctuation (e.g. `(', `?', `!', `)'), spelling errors (e.g. \\emph{commuenrug}) etc., which seem unrelated to the target relation.\nSimilar cases can be observed for AP-learnt prompts for other tasks as well (see Appendix in \\newcite{shin-etal-2020-autoprompt}).\nIt is unrealistic that a human annotator would be able to write such prompts even if they were able to see the same training instances as used by automatic methods.\n\nConsidering the fact that discrete prompt learning methods are trained in a few-shot setting where they use only a small number of training instances, the seemingly counter-intuitive nature of the discrete prompts learnt by automatic methods raises concerns about their robustness.\nFor example, \\emph{How will the performance of a target task change if we add small random perturbations to the prompts learnt by AP?} and \\emph{Whether the prompts learnt by AP generalize to out-of-domain data?}.\nTo study these issues, in this paper we evaluate the robustness of discrete prompts learnt by automatic prompt learning methods and compare that with manually-written prompts and direct fine-tuning of PLMs.\n\nAn evaluation of the robustness of discrete prompts is important for two main reasons.\nFirst, given that discrete prompt learning methods are learning those prompts from a small set of training instances, it is important that they cover the core patterns that generalize to the target task and not simply capture some random artefacts in the training samples.\nSecond, unlike embedding-based continuous prompts~\\cite{li-liang-2021-prefix,lester-etal-2021-power}, discrete prompts~\\cite{wallace-etal-2019-universal,shin-etal-2020-autoprompt,DBLP:journals\/corr\/abs-2205-12548} are represented in natural language and supposed to be interpretable.\nHowever, if a discrete prompt learning method is less robust, a seemingly harmless perturbation such as removing a punctuation character can significantly alter the performance of a downstream task.\n\nIn contrast to the numerous work that has used prompts for fine-tuning PLMs, to the best of our knowledge, the robustness of discrete prompts to random or adversarial perturbations has not been systematically studied.\nTo address this gap, we use AP as a concrete example of a widely-used method and evaluate its robustness under different types of carefully designed perturbations.\nHowever, we note that our perturbation techniques are not limited to AP and can be used for any discrete prompt learning method.\nWe compare the performance of AP-learnt prompts against fine-tuning using Manually-written Prompts (MP), and Head-based Fine-Tuning (HFT), where we fine-tune both the classifier head and the PLM parameters. \n\nFrom our evaluation, we find several interesting facts about the robustness of discrete prompts as summarized below.\n\\begin{itemize}\n    \\item Overall, when the number of training instances is increased, MP outperforms both AP and HFT on CB~\\cite{de23commitmentbank} and MNLI~\\cite{DBLP:conf\/naacl\/WilliamsNB18}, two independent benchmark datasets for NLI (\\autoref{sec:exp:datasize}).\n    In particular, the performance of AP on MNLI is much worse than that on CB. This is in contrast to the superior performance of AP on SICK-E~\\cite{DBLP:conf\/lrec\/MarelliMBBBZ14}, another NLI dataset, as reported by \\newcite{shin-etal-2020-autoprompt}.\n    \n    \\item Moreover, we see a performance drop when we use discrete prompts learnt from CB for MNLI and vice-versa (\\autoref{sec:exp-cross-data}).\n    These results indicate that the performance of discrete prompts learnt by AP is highly dataset-dependent and such discrete prompts do not generalize well across datasets.\n    \n    \\item Compared to MP, AP-learnt discrete prompts turn out to be highly sensitive to the ordering of prompt tokens (\\autoref{sec:exp:reorder}).\n    \n    \\item Random deletion of prompt tokens decreases performance in both AP and MP (\\autoref{sec:exp:deletion}).\n    \n    \\item  We create an adversarial NLI dataset from randomly-sampled test instances from MNLI and CB, and manually modify the hypothesis sentences with keeping the corresponding premise sentences unchanged, such that (a) the target label would not change, and (b) would reverse an entailment label to a contradiction (or vice-versa).\n    Both AP and MP remain relatively robust against the perturbations that do not change the target label, but the performance of MP drops significantly in the label-changing setting (\\autoref{sec:exp:input-perturbation}).\n    This shows that AP is relatively more robust against adversarial perturbations than MP, which explains AP's superior performance in various tasks. \n\\end{itemize}\n\n\n\\section{Related Work}\n\\paragraph{Prompting Methods:}\nPrompting or \\emph{in-context-learning} has received wide attention as an efficient method to extract knowledge from PLMs ~\\cite{GPT3,petroni-etal-2019-language,cui-etal-2021-template}.\nHowever, to manually write prompts one must possess task-specific domain knowledge.\nAs an alternative, methods that can automatically learn prompts from training data have been proposed.\nTwo distinct types of prompts have been learnt in prior work:\ndiscrete prompts (learns lexical sequences), and continuous prompts (learns embeddings).\nContinuous prompts~\\cite{li-liang-2021-prefix,lester-etal-2021-power} are parameter efficient because they learn generalizable task-specific embeddings, with performance comparable to PLM fine-tuning.\nHowever, continuous prompts cannot be learnt when a PLM is publicly unavailable and the only access to it is via an API~\\cite{GPT3}.\nMoreover, compared to discrete prompts, continuous prompts are difficult to interpret.\nLearning discrete prompts~\\cite{wallace-etal-2019-universal,shin-etal-2020-autoprompt,DBLP:journals\/corr\/abs-2205-12548} does not suffer from these limitations of continuous prompts and can be used with diverse NLP tasks.\nEspecially, fine-tuning massive PLMs has become computationally costly, which has made discrete prompt learning an attractive alternative.\n\n\n\\paragraph{Analysis of Prompting Methods:}\nPrior work has analyzed prompts from various viewpoints. \n\\citet{DBLP:conf\/naacl\/ScaoR21} studied the effect of training dataset size on fixed-prompt PLM fine-tuning and head-based fine-tuning and found that prompting is often worth 100s of instances on average across classification tasks.\n\\citet{kavumba-etal-2022-prompt} showed that the performance of prompt-based models varies significantly depending on the surface cues in the sentence.\n\\citet{lu-etal-2022-fantastically} found that ordering of task input significantly affects the performance.\n\\citet{utama-etal-2021-avoiding} focused on the reliance on lexical overlap in sentence pair classification and showed that prompt-based models fail to make predictions dependent on the lexical overlap.\nTo the best of our knowledge, the robustness of discrete prompts under different types of perturbations has not been studied in prior work, which is the main focus of this paper.\n\n\n\\section{Experiments}\n\\label{sec:experiments}\nLet us first describe experimental settings common to all experiments.\n\n\\paragraph{Prompting and Fine-Tuning Methods: }\nWe compared the following methods. \n\n\\begin{itemize}\n    \\item \\textbf{AutoPrompt}~\\cite[\\textbf{AP};][]{shin-etal-2020-autoprompt} is a representative method of discrete prompt learning.\n    The learning strategy is based on fill-in-the-blank task~\\cite{devlin-etal-2019-bert}.\n    First, a manually created prompt template (e.g., \\texttt{[X] <MASK> <T> ... <T> [Y]}) is given, and a prompt token (called a trigger token) is learnt by replacing \\texttt{<T>}, which is a special token representing a trigger token.\n    In the search for trigger tokens, the probability of \\texttt{<MASK>} is converted into class probability by using label tokens (e.g., \\{`\\emph{nobody}', `\\emph{nor}'\\} for contradiction~\\cite{shin-etal-2020-autoprompt}), and trigger tokens are searched by gradient-guided search~\\cite{wallace-etal-2019-universal} to find a candidate set consisting of trigger tokens from a vocabulary of the language model. \n    As a template for NLI, we used the one given by \\citet{shin-etal-2020-autoprompt}, and the prompt tokens were learnt from the training dataset. \n    In our experiments, we used the official implementation.\\footnote{\\url{https:\/\/github.com\/ucinlp\/autoprompt}}\n    \n    \\item \\textbf{Manually-written Prompts}~\\cite[\\textbf{MP};][]{schick-schutze-2021-exploiting} is a method for fine-tuning the entire masked language model with training data using manually-written prompts as the input and predicting the \\texttt{<MASK>} tokens for the labels (e.g., `\\emph{yes}' for entailment).\n    We used the template \\texttt{\\{hypothesis\\}? | <MASK>, \\{premise\\}} and  verbalizer (`\\emph{yes}' for entailment, `\\emph{no}' for contradiction, `\\emph{maybe}' for neutral) following prior work \\cite{schick-schutze-2021-exploiting,DBLP:conf\/naacl\/ScaoR21}.\n    \\citet{schick-schutze-2021-exploiting} proposed an ensemble-based method with multiple rounds of fine-tuning using different templates.\n    However, because a single template is used in AP, for a fair comparison in our experiments, we fine-tuned a PLM using one MP template.\n\n    \\item \\textbf{Head-based Fine-Tuning}~\\cite[\\textbf{HFT};][]{devlin-etal-2019-bert} fine-tunes the PLM with a classifier head.\n    We report the head-based results trained by \\newcite{DBLP:conf\/naacl\/ScaoR21}. \n    They trained HFT with a low learning rate ($10^{-5}$) and always with a large number of steps (at least 250), following the recommendations in prior work~\\cite{DBLP:conf\/iclr\/MosbachAK21,DBLP:conf\/iclr\/0007WKWA21}.\n    Note that HFT is not a prompt-based method, so it was excluded from some experiments on the robustness of discrete prompts.\n\\end{itemize}\n\n\\paragraph{Datasets:}\nWe used NLI as an evaluation task to compare the robustness of discrete prompting methods.\nThe NLI task has been used in multiple previous studies to evaluate and\/or propose novel prompt learning methods because it is a fundamental task related to many NLP applications~\\cite{shin-etal-2020-autoprompt,DBLP:conf\/naacl\/ScaoR21,DBLP:conf\/naacl\/WebsonP22}.\nIt is important to use the same NLI task and datasets in our experiments to facilitate fair comparisons and reach reproducible conclusions. \nWe used the two datasets: CommitmentBank \\cite[\\textbf{CB};][]{de23commitmentbank}\\footnote{\\url{https:\/\/super.gluebenchmark.com\/tasks}} (a corpus of short texts), and Multi-Genre Natural Language Inference Corpus \\cite[\\textbf{MNLI};][]{DBLP:conf\/naacl\/WilliamsNB18}\\footnote{\\url{https:\/\/cims.nyu.edu\/~sbowman\/multinli\/}} (a crowdsourced collection of sentence pairs for NLI).\nEach sentence pair is labelled with \\emph{entailment}, \\emph{neutral}, or \\emph{contradiction}.\n\n\\paragraph{PLM:}\nIn our experiments, we used the same pre-trained language model to evaluate AP, MP, and HFT equally.\nSpecifically, we used RoBERTa-large (355M parameters) \\footnote{\\url{https:\/\/huggingface.co\/roberta-large}}~\\cite{RoBERTa}, which has been used in much prior work in prompt learning~\\cite{shin-etal-2020-autoprompt,DBLP:conf\/naacl\/ScaoR21}.\nThe PLM was trained on five datasets, including BookCorpus\\footnote{\\url{https:\/\/yknzhu.wixsite.com\/mbweb}}, English Wikipedia\\footnote{\\url{https:\/\/en.wikipedia.org\/wiki\/English_Wikipedia}}, CC-News\\footnote{\\url{https:\/\/commoncrawl.org\/2016\/10\/news-dataset-available\/}}, OpenWebText\\footnote{\\url{https:\/\/github.com\/jcpeterson\/openwebtext}}, and Stories\\footnote{\\url{https:\/\/arxiv.org\/abs\/1806.02847}}.\nThe texts were tokenised using a byte-level Byte-Pair Encoding \\cite[BPE;][]{DBLP:conf\/acl\/SennrichHB16a} vocabulary of size 50,000.\n\n\\paragraph{Evaluating the Robustness of Prompts: }\nWe used \\emph{rate of degradation} (\\textbf{RoD})~\\cite{Meyers:2020} to evaluate robustness, which is defined as the decrease in accuracy of the target task due to the perturbations added to the prompt.\nIf the RoD of a model is small after the inclusion of a perturbation, the model is considered to be robust against that perturbation.\nSpecifically, we first calculate the respective accuracies $\\textrm{acc}_x$ and $\\textrm{acc}_{x^\\ast}$ on the same evaluation set for both prompt $x$ and its perturbated version $x^{\\ast}$.\nUsing the average accuracies $\\textrm{avg-acc}_x$ and $\\textrm{avg-acc}_{x^\\ast}$ over $M$ prompts ${x_1, ..., x_M}$, we calculate the RoD as $(\\textrm{avg-acc}_x - \\textrm{avg-acc}_{x^\\ast}) \/ \\textrm{avg-acc}_x = 1 - \\textrm{avg-acc}_{x^\\ast} \/ \\textrm{avg-acc}_x$.\n\n\n\\subsection{Effect of the Training Dataset Size}\n\\label{sec:exp:datasize} \nBefore moving on to robustness experiments, we first investigate the number of training instances on which AP and MP perform best, and used the best-performing AP and MP to evaluate their robustness in the subsequent experiments.\n\n\\paragraph{Experimental Settings:} \nWe gradually increased the size of the training dataset following the experimental setup of~\\citet{DBLP:conf\/naacl\/ScaoR21}. \nSpecifically, we experimented with randomly sampled subsets of the training dataset having varying numbers of instances in $\\{10, 15, 20, 30, 50, 70, 100, 150, 200\\}$.\nBecause the performance of few-shot learning methods often varies due to the high feature variance in the training data, we randomly sampled four subsets per each dataset size and used them independently for training the models\\footnote{NVIDIA RTX A5000 was mainly used.} (i.e. trigger tokens and label tokens for AP, or fine-tuned language model for MP and HFT) for each subset and report the average accuracy on the validation data for the four models ($M = 4$).\nWe used the matched (example from the same source as the training set) validation set for MNLI.\nFor CB, we held out 50 training instances for development as in \\citet{DBLP:conf\/naacl\/ScaoR21} and evaluated the original validation set as test data.\n\nWe searched for the optimal values for the following hyperparameters: the number of trigger tokens in \\{3, 5, 10\\}, the number of label tokens in \\{3, 5, 10\\}, and the number of tokens in a candidate set in \\{10, 50\\}. \nWe evaluated the test accuracy using the hyperparameters that had the highest accuracy on the validation data for each dataset size.\nIn the training of MP, we used AdamW optimizer~\\cite{DBLP:conf\/iclr\/LoshchilovH19} with an initial learning rate of $10^{-5}$ and a learning step of 1,000 following \\citet{DBLP:conf\/iclr\/MosbachAK21}.\n\n\n\\begin{table*}[t]\n    \\centering\n    \\small\n    \\begin{tabular}{ccccccc}\n        \\toprule\n        \\textbf{Method} & \\textbf{\\#Train} & \\textbf{Template} & \\textbf{\\#Prompt tokens} & \\textbf{\\#Label tokens per class} & \\multicolumn{2}{c}{\\textbf{Avg. accuracy}} \\\\\n        & & & & & \\textbf{CB} & \\textbf{MNLI} \\\\\n        \\midrule\n        AP & 200 & \\texttt{\\red{p} <MASK> \\blue{<T> ... <T>} \\red{h}} & 10 & 3 & 68.3 & 37.7 \\\\ \n        MP & 200 & \\texttt{\\red{h}\\blue{? |} <MASK>\\blue{,} \\red{p}} & 3 & 1 & \\uline{95.1} & \\uline{65.5} \\\\ \n        HFT & - & \\texttt{<CLS> \\red{p} <SEP> \\red{h}} & 0 & - & - & - \\\\ \n        \\bottomrule\n    \\end{tabular}\n    \\caption{\n    The average accuracy of the experiment with four training subsets of 200 instances.\n    \\red{Red} represents the task inputs, \\red{\\texttt{h}} represents the hypothesis, \\red{\\texttt{p}} represents the premise, \\blue{blue} represents the prompt tokens, and \\blue{\\texttt{<T>}} represents a trigger token.\n    Unreported values were marked with `-'. \n    }\n    \\label{tab:pre-train-best}\n\\end{table*}\n\n\n\\paragraph{Main Results:}\n\\autoref{fig:pre-train} shows the performance\\footnote{HFT results were obtained from \\citet{DBLP:conf\/naacl\/ScaoR21}, F1-macro for CB and accuracy for MNLI.} against the training dataset size.\nWe see that in both CB and MNLI \\textbf{MP is always superior to AP}.\nFor example, with a dataset of size 200, AP and MP achieved the best accuracy in CB, MP's accuracy was 92.7\\%, while that of AP was lower at 54.2\\%.\n\n\\begin{figure}[t]\n  \\centering\n    \\includegraphics[clip, width=8cm]{datapoints.pdf}\n    \\caption{\n    Performance of AutoPrompt (AP), Manually-written Prompt (MP), and Head-based Fine-Tuning (HFT) on the scale of dataset size for CB and MNLI.\n    Means and their 95\\% confidence intervals are plotted.\n    The accuracy of HFT for dataset size for CB was not plotted because the accuracy was not reported.\n    }\n    \\label{fig:pre-train}\n\\end{figure}\n\nOur results also suggest that \\textbf{the performance of discrete prompts learnt by AP is highly dataset dependent.}\n\\citet{shin-etal-2020-autoprompt} reported results for AP and HFT on SICK-E~\\cite{DBLP:conf\/lrec\/MarelliMBBBZ14}, which is another NLI dataset.\nThey concluded that AP was always superior to HFT up to training dataset sizes of 1,000 for the same RoBERTa-large PLM that we use.\nHowever, our experiments show the opposite trend (i.e. HFT is superior to AP).\nThis suggests that even if AP is superior to HFT on a given dataset, it is not guaranteed to be superior in a different dataset for the same task.\nThis may be due to the differences in the domain and annotation guidelines for each dataset.\nFor example, the accuracy of MNLI was quite low on AP, which contrasts with that of CB.\nThis result suggests that the discrepancies in domains and annotation guidelines make it difficult for AP to perform consistently.\n\n\\paragraph{Best Prompts:}\n\\autoref{tab:pre-train-best} shows the average accuracy of models trained on 200 instances that performed well in both CB and MNLI.\nNote that there are four training subsets for each dataset size, resulting in corresponding four trained AP prompts and four PLMs fine-tuned by MP. \\footnote{We show the four best prompts learnt by AP in \\autoref{sec:supplementary}.}\nIn the robustness evaluations in \\autoref{sec:exp:reorder} through \\autoref{sec:exp:input-perturbation}, we used these learnt APs and MPs.\nIn this paper, (a) trigger tokens learnt by AP, and (b) manually-written prompts excluding the task inputs and  mask tokens are collectively referred to as the \\emph{prompt tokens}.\n\n\n\\subsection{Token Reordering}\n\\label{sec:exp:reorder}\nAs seen from \\autoref{tbl:autoprompt-examples}, compared to MPs where the ordering of tokens in a prompt is manually determined, discrete prompts learnt by AP appear to have no obvious ordering among their tokens.\nTo empirically investigate the importance of the token order in a discrete prompt, we conduct an experiment where we randomly shuffle the prompt tokens and measure the effect on the downstream task performance.\n\n\\paragraph{Experimental Procedure:}\nGiven a discrete prompt, we first randomly reordered its prompt tokens (e.g. shaded in blue in \\autoref{tab:pre-train-best}).\nNext, we used the reordered prompt with the PLM to make entailment predictions for the test instances in the CB and MNLI datasets.\nFinally, the entailment prediction accuracy (Acc) obtained with the reordered prompts was computed.\nWe repeated this evaluation 10 times for each prompt and report the averaged values and the corresponding RoD values.\n\n\\paragraph{Main Results:}\nFrom \\autoref{tab:reorder_result} we see that the accuracy drops for both AP and MP when the prompt tokens are randomly reordered.\nIn particular, the accuracy of AP drops significantly compared to that of MP.\nFor example, the accuracy of AP on CB drops by ca. 14\\% due to token reordering, while that for MP drops only by ca. 2\\%.\nIntuitively, one would expect that changing the order of prompt tokens in MP would result in a significant drop in accuracy because the meaning of the prompts would change.\nHowever, we see that this is not the case. \nThis result shows that \\textbf{the discrete prompts learnt by AP strongly rely on the token order}.\n\n\\paragraph{Additional Analysis:}\nTo further analyze the relationship between the level of perturbation introduced by reordering prompt tokens in AP and its effect on the performance, we computed the token-level edit distance \\citep[Levenshtein distance;][]{levenshtein1966binary} between each prompt and its token-shuffled version as shown in \\autoref{fig:scatter_reodering}.\nFor all four AP prompts, we see that the accuracy drops when the perturbation noise (i.e. measured by edit distance) increases.\nThis reconfirms the lack of robustness in discrete prompts learnt by AP to the random shuffling of prompt tokens.\n\n\\begin{table}[t]\n    \\centering\n    \\small\n    \\begin{tabular}{clllccc}\n        \\toprule\n        \\textbf{Method} & \\textbf{Metrics} & \\textbf{CB} & \\textbf{MNLI} \\\\\n        \\midrule\n        \\multirow{2}{*}{AP} & Acc & \\textbf{54.2} & \\textbf{34.3} \\\\ \n                            & RoD & \\textbf{0.21} & \\textbf{0.10} \\\\ \n        \\midrule\n        \\multirow{2}{*}{MP} & Acc & \\uline{92.7} & \\uline{59.3} \\\\\n                            & RoD & \\uline{0.03} & \\uline{0.09} \\\\ \n        \\bottomrule\n    \\end{tabular}\n    \\caption{\n    Performance of reordered prompts.\n    Acc denotes accuracy; RoD denotes the RoD from before the reordering (\\autoref{tab:pre-train-best}).\n    The largest drops in accuracy are \\textbf{bolded} and the smallest drops are \\uline{underlined} for each method and dataset.\n    AP relies more strongly on word order than MP.\n    }\n    \\label{tab:reorder_result}\n\\end{table}\n\n\n\\subsection{Token Deletion} \n\\label{sec:exp:deletion}\nAs seen from \\autoref{tbl:autoprompt-examples}, the discrete prompts learnt by AP perform better than MP.\nHowever, it is often difficult to determine the importance of prompt tokens to the target task due to their lack of interpretability (e.g. prompt token `\\emph{neau}' in \\autoref{tbl:autoprompt-examples}).\nTo understand the significance of individual prompt tokens to the overall discrete prompt, we conducted an experiment where we systematically deleted one or more prompt tokens at various positions from a given discrete prompt and measure the drop (if any) in the performance of the NLI task.\n\n\\begin{figure}[t]\n  \\centering\n    \\includegraphics[clip, width=7.5cm]{reordering_edit_distance.png}\n    \\caption{Edit distance and accuracy of the reordered trigger tokens. We evaluated them on the validation data of CB. \n    The prompts numbered 0 through 3 each represent the four prompts learnt by AP (shown in \\autoref{tab:best-ap-prompts-cb}).\n    Note that a point with an edit distance of zero indicates accuracy with the original trigger token.\n    }\n    \\label{fig:scatter_reodering}\n    \\vspace{-3mm}\n\\end{figure}\n\n\\paragraph{Experimental Procedure:}\nWe evaluated two settings of prompt deletion: \\emph{single} and \\emph{multiple} token deletion.\nIn the single token deletion setting, we deleted one token at different positions in a given prompt.\nFor AP, we repeated this with each of the four discrete prompts (shown in \\autoref{tab:pre-train-best}) and report the average accuracy.\nIn the multiple token deletion setting, we delete $n \\in \\{1, 3, 5, 7\\}$ prompt tokens following three strategies:\n\\emph{Random-deletion} deletes $n$ prompt tokens randomly,\n\\emph{Front-deletion} deletes $n$ consecutive prompt tokens from the beginning of the prompt, and \\emph{Back-deletion} deletes $n$ tokens counted backwards from the end of the prompt.\nIn random-deletion, we ran 100 trials and report the average accuracy.\nAs in the previous experiments, we used four prompts for AP and report the averaged results.\n\n\\begin{table*}[t!]\n    \\setlength{\\tabcolsep}{1.9mm} \n    \\centering\n    \\small\n    \\begin{tabular}{cccccccccccccc}\n    \\toprule\n    \\multirow{2}{*}{\\textbf{Task}} & \\multirow{2}{*}{\\textbf{Method}} & \\multirow{2}{*}{\\textbf{Metrics}}  & \\multicolumn{10}{c}{\\textbf{Position of the deleted prompt token}} & \\multirow{2}{*}{\\textbf{Orig.}} \\\\\n    & & & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 \\\\\n    \\midrule\n    \\multirow{4}{*}{CB} & \n        \\multirow{2}{*}{AP} & Acc & 62.1 & \\textbf{61.6} & 63.4 & 59.4 & \\uline{65.6} & \\uline{65.6} & 62.1 & 63.8 & 62.1 & 62.9 & 68.3\\\\\n        & & RoD & 0.09 & \\textbf{0.10} & 0.07 & 0.13 & \\uline{0.04} & \\uline{0.04} & 0.09 & 0.07 & 0.09 & 0.08 & - \\\\\n        \\cmidrule(lr){2-14}\n        & \\multirow{2}{*}{MP} & Acc & 93.8 & \\textbf{93.3} & \\uline{96.0} & - & - & - & - & - & - & - & 95.1 \\\\\n        & & RoD & 0.01 & \\textbf{0.02} & \\uline{-0.01} & - & - & - & - & - & - & - & - \\\\\n        \\midrule\n    \\multirow{4}{*}{MNLI} & \n        \\multirow{2}{*}{AP} & Acc & \\uline{37.9} & 37.8 & \\textbf{36.6} & 37.5 & 37.5 & 37.2 & 37.5 & 37.4 & 37.5 & 37.1 & 37.7 \\\\\n        & & RoD & \\uline{-0.01} & 0.00 & \\textbf{0.03} & 0.01 & 0.01 & 0.01 & 0.01 & 0.01 & 0.01 & 0.02 & - \\\\\n        \\cmidrule(lr){2-14}\n        & \\multirow{2}{*}{MP} & Acc & 64.5 & \\uline{65.4} & \\textbf{55.4} & - & - & - & - & - & - & - & 65.5 \\\\\n        & & RoD & 0.02 & \\uline{0.00} & \\textbf{0.15} & - & - & - & - & - & - & - & - \\\\\n    \\bottomrule\n    \\end{tabular}\n    \\caption{\n    Average accuracy was obtained after deleting a single token at different positions of a given prompt.\n    The largest drops in accuracy over the deletion positions are \\textbf{bolded} and the smallest drops are \\uline{underlined} for each method and dataset.\n    Column `Orig.' shows the performance of the original prompt.\n    }\n    \\label{tab:deletion_one}\n\\end{table*}\n\n\\paragraph{Results:}\nFrom \\autoref{tab:deletion_one} we see that the \\textbf{accuracy of both AP and MP drops even when a single token is deleted} at specific positions.\nHowever, the observed trends differ in CB and MNLI.\nFor example, AP resulted in higher RoD values in CB compared to MNLI.\nThis shows that the robustness of AP under single token deletion heavily depends on the dataset.\n\\autoref{tab:deletion_multi} shows the results for the multiple token deletion setting.\nWe see that \\textbf{the performance of both AP and MP degrades when more tokens are deleted.}\nInterestingly, the accuracy drop in CB is very small for MP even when all prompt tokens are deleted (i.e., only the task inputs and \\texttt{<MASK>} were used as the input).\nThis suggests that the performance on CB is less reliant on the prompt tokens in MP.\n\n\n\\subsection{Cross-Dataset Evaluation}\n\\label{sec:exp-cross-data}\nGiven that discrete prompt learning methods such as AP learn prompts from a small set of training instances, it is important that the learnt prompts encode generalizable task-specific features and not random artefacts in the training sample used.\nTo study the transferability of the learnt discrete prompts from one dataset to another, we conduct a cross-dataset evaluation as described next.\n\n\\begin{table}[t!]\n    \\setlength{\\tabcolsep}{0.9mm} %\n    \\centering\n    \\small\n    \\begin{tabular}{ccccccccccccc}\n        \\toprule\n        \\multirow{2}{*}{\\textbf{Strategy}} & \\multirow{2}{*}{\\textbf{Method}} & \\multirow{2}{*}{\\textbf{Metrics}} & \\multicolumn{4}{c}{\\textbf{\\#Deleted Tokens}} & \\multirow{2}{*}{\\textbf{Orig.}} \\\\\n        & & & 1 & 3 & 5 & 7 \\\\\n        \\midrule\n        \\midrule\n        \\multicolumn{8}{c}{\\textbf{CB}} \\\\\n        \\midrule\n       \n        \\multirow{4}{*}{Random}\n            & \\multirow{2}{*}{AP}\n            & Acc & \\uline{56.7} & 56.0 & 55.4 & \\textbf{54.8} & 68.3 \\\\\n        & & RoD & \\uline{0.17} & 0.18 & 0.19 & \\textbf{0.20} & - \\\\\n            \\cmidrule(lr){2-8}\n            & \\multirow{2}{*}{MP}\n            & Acc & \\textbf{93.3} & \\uline{94.6} & - & - & 95.1\\\\\n        & & RoD & \\textbf{0.02} & \\uline{0.01} & - & - & - \\\\\n        \\midrule\n        \\multirow{4}{*}{Front}\n            & \\multirow{2}{*}{AP}\n            & Acc & \\uline{62.1} & \\textbf{49.1} & 57.6 & 57.6 & 68.3 \\\\\n        & & RoD & \\uline{0.09} & \\textbf{0.28} & 0.16 & 0.16 & - \\\\\n            \\cmidrule(lr){2-8}\n            & \\multirow{2}{*}{MP}\n            & Acc & \\textbf{93.8} & \\uline{94.6} & - & - & 95.1\\\\\n        & & RoD & \\textbf{0.01} & \\uline{0.01} & - & - & - \\\\\n        \\midrule\n        \\multirow{4}{*}{Back}\n            & \\multirow{2}{*}{AP}\n            & Acc & \\uline{62.9} & 57.6 & 55.8 & \\textbf{51.3} & 68.3 \\\\\n        & & RoD & \\uline{0.08} & 0.16 & 0.18 & \\textbf{0.25} & - \\\\\n            \\cmidrule(lr){2-8}\n            & \\multirow{2}{*}{MP}\n            & Acc & \\uline{96.0} & \\textbf{94.6} & - & - & 95.1 \\\\\n        & & RoD & \\uline{-0.01} & \\textbf{0.01} & - & - & - \\\\\n        \\midrule\n        \\midrule\n       \n        \\multicolumn{8}{c}{\\textbf{MNLI}} \\\\\n        \\midrule\n            \\multirow{4}{*}{Random}\n            & \\multirow{2}{*}{AP}\n            & Acc & \\textbf{35.8} & \\textbf{35.8} & 36.0 & \\uline{36.2} & 37.7 \\\\\n        & & RoD & \\textbf{0.05} & \\textbf{0.05} & 0.05 & \\uline{0.04} & - \\\\\n            \\cmidrule(lr){2-8}\n            & \\multirow{2}{*}{MP}\n            & Acc & \\uline{65.4} & \\textbf{52.6} & - & - & 65.5 \\\\\n        & & RoD & \\uline{0.0} & \\textbf{0.20} & - & - & - \\\\\n        \\midrule\n        \\multirow{4}{*}{Front}\n            & \\multirow{2}{*}{AP}\n            & Acc & \\uline{37.9} & 36.5 & 36.2 & \\textbf{36.0} & 37.7 \\\\\n        & & RoD & \\uline{-0.01} & 0.03 & 0.04 & \\textbf{0.05} & - \\\\\n            \\cmidrule(lr){2-8}\n            & \\multirow{2}{*}{MP}\n            & Acc & \\uline{64.5} & \\textbf{52.6} & - & - & 65.5 \\\\\n        & & RoD & \\uline{0.02} & \\textbf{0.20} & - & - & - \\\\\n        \\midrule\n        \\multirow{4}{*}{Back}\n            & \\multirow{2}{*}{AP}\n            & Acc & \\uline{37.1} & 36.7 & 35.7 & \\textbf{36.5} & 37.7 \\\\\n        & & RoD & \\uline{0.02} & 0.03 & 0.05 & \\textbf{0.03} & - \\\\\n            \\cmidrule(lr){2-8}\n            & \\multirow{2}{*}{MP}\n            & Acc & \\uline{55.4} & \\textbf{52.6} & - & - & 65.5 \\\\\n        & & RoD & \\uline{0.15} & \\textbf{0.20}  & - & - & - \\\\\n        \\bottomrule\n    \\end{tabular}\n    \\caption{\n    Average accuracy was obtained after deleting multiple tokens from a given prompt.\n    The largest drops in accuracy over the deleted tokens  are \\textbf{bolded} and the smallest drops are \\uline{underlined} for each strategy and method.\n    }\n    \\label{tab:deletion_multi}\n\\end{table}\n\n\\paragraph{Experimental Procedure:}\nWe used one NLI dataset (e.g. CB) to learn the prompts and then use them to make entailment predictions in another NLI dataset (e.g. MNLI).\nWe then measured the drop in accuracy using RoD for this cross-dataset transferability task with respect to the accuracy of test data from the same dataset.\n\n\\paragraph{Results:}\nAs seen from \\autoref{tab:cross-dataset-eval}, \\textbf{AP-based prompts do not generalize well across datasets.}\nFor both AP and MP, RoD is larger in the transfer from CB to MNLI than in the opposite direction.\nThis implies that MNLI is a better dataset for fine-tuning a PLM for NLI using discrete prompts.\n\n\\begin{table}[t]\n\\centering\n\\small\n\\begin{tabular}{lccccc}\n\\toprule\n\\multirow{2}{*}{\\textbf{Method}}& \\multicolumn{2}{c}{\\textbf{Test Dataset}} & \\multirow{2}{*}{\\textbf{RoD}}\\\\\n& \\textbf{CB} & \\textbf{MNLI} & \\\\\n\\midrule\nAP trained on CB   & 68.3 & 36.1  & \\uline{0.47} \\\\ \nAP trained on MNLI & 42.9 & 37.7  & \\uline{0.12} \\\\ \n\\midrule\nMP trained on CB   & 95.1 & 43.4 & \\textbf{0.54} \\\\ \nMP trained on MNLI & 43.8 & 65.5 & \\textbf{0.33} \\\\ \n\\bottomrule\n\\end{tabular}\n\\caption{\nAccuracy and RoD for the cross-dataset evaluation where a method (AP\/MP) is trained on one NLI dataset (CB\/MNLI) and the learnt prompts are used to make entailment predictions in a different NLI dataset.\n}\n\\label{tab:cross-dataset-eval}\n\\end{table}\n\n\n\\subsection{Adversarial Perturbations}\n\\label{sec:exp:input-perturbation}\nIntroducing carefully designed adversarial perturbations to the test instances such as modifications to sentences that might or might not alter the original target labels have been used as a technique for probing the robustness of models~\\cite{DBLP:journals\/corr\/GoodfellowSS14}. \nPrevious studies~\\cite{DBLP:journals\/corr\/SamantaM17,DBLP:conf\/aaai\/JinJZS20} have shown that pre-trained models can be easily fooled to make incorrect predictions with seemingly innocuous perturbations to the test instances. Therefore, we evaluate discrete prompt-based NLI models for their robustness against adversarially perturbated test instances.\n\n\\paragraph{Evaluation Dataset:}\nFor this purpose, we asked two annotators to manually edit hypothesis sentences in NLI test data considering two types of perturbations:\n(1) perturbations that do not change reference labels, and (2) perturbations that change reference labels.\nAn example is shown in  \\autoref{tab:perturbation-data}. \n\nFor the first type of perturbation, we edited a hypothesis sentence such that its relationship with the corresponding premise remains unchanged.\nFor the second type, we edited a hypothesis sentence such that its relationship (e.g., from \\emph{entailment} to \\emph{contradiction}) will be reversed.\nThe premise and hypothesis pairs were sampled from CB (validation set) and MNLI (test set).\nBecause there are ca. 10,000 test instances in MNLI and it is costly to manually edit sentences, we used 100 randomly-chosen sentence pairs covering MNLI and CB.\n\n\\begin{table}[t]\n    \\centering\n    \\small\n    \\begin{tabular}{lp{7em}p{5em}}\n        \\toprule\n        \\textbf{} & \\textbf{Hypothesis} & \\textbf{Label} \\\\ \n        \\midrule\n        Original & The Wither's only had daughters. & contradiction \\\\\n        \\midrule\n        Perturbation & \\\\\n        \\quad w\/o label changes & The Wither's did not have sons. & contradiction \\\\\n        \\quad w\/ label changes & The Wither's had a boy. & entailment \\\\\n        \\bottomrule\n    \\end{tabular}\n    \\caption{\n    Examples of our evaluation set consisting of task inputs with perturbations. \n    The premise sentence is `\\emph{The Wither's eldest boy, one of the four of the town militia, saluted in the old style with his stick sword.}'\n    } \n    \\label{tab:perturbation-data}\n\\end{table}\n\n\\iffalse\n\\begin{table}[t]\n    \\setlength{\\tabcolsep}{1mm} %\n    \\centering\n    \\begin{tabular}{lccccc}\n        \\toprule\n        & \\multicolumn{2}{c}{\\textbf{\\#Perturbation}} \\\\\n        & \\textbf{w\/o label changes} & \\textbf{w\/ label changes} \\\\ \n        \\midrule\n        CB & 50 & 49 \\\\ \n        MNLI & 48 & 47 \\\\ \n        \\bottomrule\n    \\end{tabular}\n    \\caption{\n    The dataset sizes (measured by the number of instances) for the adversarial dataset.\n    }\n    \\label{tab:perturbation-dataset-stat}\n\\end{table}\n\\fi\n\n\\paragraph{Experimental Procedure:}\nWe computed the RoD of average accuracies obtained with original and adversarial test instances.\nSpecifically, we used the AP prompts in \\autoref{tab:pre-train-best} under three settings:\n(a) original (without perturbations), \n(b) perturbations without label changes, \nand (c) perturbations with label changes.\nThen, we calculate RoD from (a) to (b) and (a) to (c) as shown in \\autoref{tab:task-input-perturbation}.\n\n\\begin{table}[t!]\n\\small\n\\setlength{\\tabcolsep}{0.5mm} %\n\\centering\n\\begin{tabular}{ccccc}\n\\toprule\n\\textbf{Perturbation} & \\textbf{Method} & \\textbf{Metrics} & \\textbf{CB} & \\textbf{MNLI} \\\\\n\\midrule\n\\multirow{4}{*}{Original}\n    & \\multirow{2}{*}{AP} \n      & Acc & 54.5 & 40.5 \\\\\n    & & RoD & - & - \\\\\n    \\cmidrule(l){2-5}\n    & \\multirow{2}{*}{MP} \n      & Acc & 95.5 & 71.0 \\\\\n    & & RoD & - & - \\\\\n\\midrule\n\\multirow{4}{*}{\\begin{tabular}{c}Perturbation \\\\w\/o label changes\\end{tabular}}\n    & \\multirow{2}{*}{AP} \n      & Acc & 55.5 & 43.2 \\\\\n    & & RoD & \\uline{-0.02} & \\uline{-0.07} \\\\\n    \\cmidrule(l){2-5}\n    & \\multirow{2}{*}{MP} \n      & Acc & 93.0 & 66.7 \\\\\n    & & RoD & \\textbf{0.03} & \\textbf{0.06} \\\\\n\\midrule\n\\multirow{4}{*}{\\begin{tabular}{c}Perturbation \\\\w\/ label changes\\end{tabular}}\n    & \\multirow{2}{*}{AP} \n      & Acc & 42.3 & 39.4 \\\\\n    & & RoD & \\uline{0.22} & \\uline{0.03} \\\\\n    \\cmidrule(l){2-5}\n    & \\multirow{2}{*}{MP} \n      & Acc & 41.8 & 61.2 \\\\\n    & & RoD & \\textbf{0.56} & \\textbf{0.14} \\\\\n\\bottomrule\n\\end{tabular}\n\\caption{\nAccuracy and RoD in prompts for task inputs that include perturbations. The RoD here is the rate of degradation in the average accuracy from the original without perturbations to perturbations without label changes or perturbations with label changes.\nThe largest drops in accuracy are \\textbf{bolded} and the smallest drops are \\uline{underlined} for each perturbation and method.\n}\n\\label{tab:task-input-perturbation}\n\\end{table}\n\n\n\\paragraph{Results:}\nOverall, we see that the RoD of AP is consistently smaller than that of MP in both CB and MNLI under both types of perturbations.\nHowever, it is also clear that the accuracy obtained with AP is much smaller than that with MP.\nFor the perturbations without label changes, both AP and MP show small RoD values, compared to those with label changes.\\footnote{w\/o label change modifications slightly increase the average length of a hypothesis and AP seems to better exploit this extra information for inference resulting in a slight improvement in accuracy (negative RoD).}\nThis shows that both AP and MP are relatively robust against modifications to the hypotheses that do not significantly alter the meaning.\nHowever, when stronger perturbations are introduced that would result in label changes, the accuracy of both AP and MP drops significantly. \\footnote{MP is less robust compared to AP, likely as a result of overfitting to strongly perturbed training data during fine-tuning the PLM.}\nThis is a concern because it shows that \\textbf{neither AP nor MP is sufficiently robust to correctly predict the target labels when the hypothesis sentences in test data are adversarially modified.}\n\n\n\\section{Conclusion}\nWe investigated the robustness of discrete prompts under different perturbations.\nWe found that although discrete prompts remain relatively robust against token deletion, it is highly sensitive to other types of perturbations such as token shuffling.\nFor adversarial perturbations to the input, discrete prompts were robust to weak perturbations without label changes, but AP was more robust than MP for perturbations with label changes.\nMoreover, they generalize poorly across different datasets annotated for NLI.\nWe hope our analysis will inspire future work to develop methods that learn both accurate as well as robust discrete prompts.\n\n\n\\section{Limitations}\nPossible limitations of this work are:\n\\begin{itemize}\n\\item We chose popular discrete prompt methods of AP and MP and did not investigate other methods in this work. Our analysis procedure can still be applied to other discrete prompts such as AvgTrigger~\\cite{wallace-etal-2019-universal}.\n\\item We chose RoBERTa-large following previous studies of HFT~\\cite{DBLP:conf\/naacl\/ScaoR21} and AP~\\cite{shin-etal-2020-autoprompt} for reproducible and identical comparisons with them. Other PLMs would lead to different results, but they can also be investigated in the same way as in this work.\n\\item This work focuses on NLI because it is a fundamental natural language understanding task and still difficult even with PLMs~\\cite{GPT3}. Other complex downstream tasks are worth investigating for a deeper understanding of prompt-based approaches in future work.\n\\item The results and conclusions are from the English datasets and would differ in other languages. However, our methodologies do not depend on English and can be applied to other languages as important future studies. \n\\item Since there was a performance gap between MP\/HFT and AP, the accuracies by the perturbations could be affected. However, this work does not aim to find the best prompt learning method but to analyze the robustness of discrete prompts for a deeper understanding of them.\n\\end{itemize}\n\n\n\\section{Ethical Considerations}\nOur adversarial dataset came from existing datasets of CB and MNLI.\nWe visually checked the instances in the data development and found no instances with ethical concerns.\n\nOne should also be aware of social biases (e.g. gender stereotypes) in PLM. \nRoBERTa, the PLM we used in our experiments, is known to have gender biases~\\cite{DBLP:journals\/corr\/abs-2105-05541}. \nSince we used it as-is in order to follow the experimental conditions of previous studies using RoBERTa, our current results are possibly influenced by such biases.\nHowever, the consideration of the prompt robustness of this work would not pose or magnify such ethical concerns.\n\n\n\\section{Acknowledgments}\nThis research was supported by the JSPS KAKENHI (Grants-in-Aid for Scientific Research) JP22H03654.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\n\n\\section*{Supplementary Material}\\label{supplementary_material}\n\n\n\\section{Introduction}\n\\label{introduction}\n\n\nSeveral studies about Hybrid Unmanned Aerial Underwater Vehicles (HUAUVs) have been published recently~\\cite{drews2014hybrid, neto2015attitude, da2018comparative, maia2017design, lu2019multimodal, mercado2019aerial, horn20, aoki2021}. These types of vehicles enable an interesting range of new applications due to their capability to operate both in the air and underwater. These include inspection and mapping of partly submerged areas in industrial facilities, search and rescue and others. Most of the literature in the field is still focused on vehicle design, with few published works on the theme of autonomous navigation~\\cite{bedin2021deep}. The ability to navigate in both environments and successfully transit from one to another imposes additional challenges that must be addressed.\n\nLately, approaches based on Deep-RL have been enhanced to address navigation-related tasks for a range of mobile vehicles, including ground mobile robots~\\cite{ota2020efficient}, aerial robots~\\cite{tong2021uav,grando2022double} and underwater robots~\\cite{carlucho2018}. Based on actor-critic methods and multi-layer network structures, these approaches have achieved interesting results in mapless navigation, obstacle avoidance, even including media transitioning for HUAUVs~\\cite{bedin2021deep, de2022depth}. However, the challenges faced by this kind of vehicle make these existing approaches still too limited, with poor generalization through different scenarios.\n\n\nIn this work, we present two new double-critic Deep-RL approaches in the context of HUAUVs to perform navigation-related tasks in a continuous state-space environment:  (1)~a deterministic approach based on Twin Delayed Deep Deterministic Policy Gradient (TD3)~\\cite{fujimoto2018addressing}; and\n (2)~a stochastic approach based on Soft Actor-Critic (SAC)~\\cite{haarnoja2018soft}. We show we are capable of training agents that are consistently better than state-of-the-art in generalizing through different simulated scenarios, with improved stability in mapless navigation, obstacle avoidance and medium transitions. Our evaluation tasks included both air-to-water and water-to-air transitions. We compared our methods with other single critic approaches and with an adapted version of a traditional Behavior-Based Algorithm (BBA)~\\cite{marino2016minimalistic} used in aerial vehicles. Fig.~\\ref{fig:simenv} shows a snapshot of our simulation environment.\n \n\\begin{figure}[tbp!]\n    \\vspace{-2mm}\n    \\centering\n    \\includegraphics[width=\\linewidth]{img\/sonar_v5.png}\n    \\caption{Our HUAUV underwater in the first scenario (left) and its respective sonar readings (right).}\n    \\label{fig:simenv}\n    \\vspace{-4mm}\n\\end{figure}\n\n\nThis work provides the following main contributions:\n\n\\begin{itemize}\n\n\\item We show that our agents present a robust capacity for generalization through different environments, achieving a good performance in a complex and completely unknown environment. The robot also performs the medium transition, being capable of arriving at the desired target and avoiding collisions.\n\n\\item We show that a Long Short Term Memory (LSTM) architecture can achieve better overall performance and capacity for generalization than the state-of-the-art Multi-Layer Perceptron (MLP) architectures\n\n\\end{itemize}\n\nThis work has the following structure: the related works are discussed in the following section (Sec. \\ref{related_works}). Following it, we present our methodology in Sec. \\ref{methodology}. The results are presented in Sec. \\ref{results} and discussed in Sec. \\ref{conclusion}.\n\n\\section{Related Work}\n\\label{related_works}\n\n\nFor more traditional types of vehicles, several works have been published demonstrating how efficiently Deep-RL can solve the mapless navigation problem~\\cite{tobin2017domain}. For a ground robot, Tai~\\emph{et al.}~\\cite{tai2017virtual} demonstrated a mapless motion planner based on the DDPG algorithm employing a 10-dimensional range finder combined with the relative distance to the target as inputs and continuous steering signals as outputs. Recently, Deep-RL methods have also been successfully used by Ota~\\emph{et al.}~\\cite{ota2020efficient}, de Jesus~\\emph{et al.}~\\cite{jesus2019deep,jesus2021soft} and others, to accomplish mapless navigation-related tasks for terrestrial mobile robots. Singh and Thongam~\\cite{singh2018mobile} demonstrated efficient near-optimal navigation for a ground robot in dynamic environments employing an MLP to perform speed control while choosing collision-free path segments.\n\nFor UAVs, Kelchtermans and Tuytelaars \\cite{kelchtermans2017hard} demonstrated how memory could help Deep Neural Networks (DNN) for navigation in a simulated room crossing task. Tong~\\emph{et al.}~\\cite{tong2021uav} showed better than state-of-the-art convergence and effectiveness in adopting a DRL-based method combined with a LSTM to navigate a UAV in highly dynamic environments, with numerous obstacles moving fast.\n\nWhen it comes to problems involving specifically mapless navigation for UAVs, few works examine the effectiveness of Deep-RL. Grando~\\emph{et al.}~\\cite{grando2020visual} explored a Deep-RL architecture, however, navigation was constrained to a 2D space. Rodriguez \\emph{et al.}~\\cite{rodriguez2018deep} employed a DDPG-based strategy to solve the problem of landing UAVs on a moving platform. Similar to our work, they employed RotorS framework~\\cite{furrer2016rotors} combined with the Gazebo simulator. Sampedro~\\emph{et al.}~\\cite{sampedro2019fully} proposed a DDPG-based strategy for search and rescue missions in indoor environments, utilizing real and simulated visual data. Kang~\\emph{et al.}~\\cite{kang2019generalization} also used visual information, although he focused on the subject of collision avoidance. In a go-to-target task, Barros~\\emph{et al.}~\\cite{2020arXiv201002293M} applied a SAC-based method for the low-level control of a UAV. Double critic-based Deep-RL approaches similar to the one proposed here have also been shown to yield good results\\cite{grando2022double}.\n\nThe HUAUV literature is still mostly concerned with vehicle design and modeling \\cite{drews2014hybrid, neto2015attitude, da2018comparative, maia2017design, lu2019multimodal, mercado2019aerial, horn20}. Two works have recently tackled the navigation problem with the medium transition of HUAUVs~\\cite{pinheiro2021trajectory}, \\cite{bedin2021deep}. Pinheiro~\\emph{et al.} \\cite{pinheiro2021trajectory} focused on smoothing the medium transition problem in a simulated model on MATLAB. Grando~\\emph{et al.}~\\cite{bedin2021deep} developed Deep-RL actor-critic approaches and a MLP architecture. These two works were developed using generic distance sensing information for aerial and underwater navigation. In contrast, our work relies on more realistic sensing data, with the simulated LIDAR and sonar being both based on real-world devices.\n\nThe HUAUV presented in this paper is based on Drews-Jr~\\emph{et al.} \\cite{drews2014hybrid} model, which Neto~\\emph{et al.}~\\cite{neto2015attitude} has largely expanded. Our work differs from the previously discussed works by only using the vehicle's relative localization data and not its explicit localization data. We also present Deep-RL approaches based on double critic techniques instead of single critic, with RNN structures instead of MLP, traditionally used for mapless navigation of mobile robots. We compare our approaches with state-of-the-art Deep-RL approaches and with a behavior-based algorithm \\cite{marino2016minimalistic} adapted for hybrid vehicles to show that our new methodology improves the overall capability to generalize through distinct environments.\n\n\\section{Methodology}\n\\label{methodology}\n\nIn this section, we describe our simulation environment, our hybrid vehicle, and the proposed Deep-RL, detailing the network structure for both deterministic and stochastic agents. We also introduce the task that the vehicle must accomplish autonomously and the respective reward function.\n            \n\\subsection{Deterministic Deep RL\n\nDeveloping on the DQN~\\cite{mnih2013playing}, Deep Deterministic Policy Gradient (DDPG)~\\cite{lillicrap2015continuous} employs an actor-network where the output is a vector of real values representing the chosen action, and a second neural network to learn the target function, providing stability and making it ideal for mobile robots~\\cite{jesus2019deep}. While it provides good results, DDPG still has its problems, like overestimating the Q-values, which leads to policy breaking. TD3~\\cite{fujimoto2018addressing} uses DDPG as its backbone, adding some improvements, such as clipped double-Q~learning with two neural networks as targets for the Bellman error loss functions, delayed policy updates, and Gaussian noise on the target action, raising its performance.    \n\nOur deterministic approach is based on the TD3 technique. The pseudocode can be seen in Algorithm~\\ref{alg:docrl_d}.\n\n\\begin{algorithm}[!htb]\n    \\algsetup{linenosize=\\tiny}\n    \\scriptsize\n    \\caption{Deep Reinforcement Learning Deterministic}\n    \\label{alg:docrl_d}\n    \\begin{algorithmic}[1]\n        \\STATE Initialize params of critic networks $\\theta_{1}$, $\\theta_{2}$ , and actor network $\\phi$\n        \\STATE Initialize params of target networks $\\phi^{\\prime}\\leftarrow\\phi$, $\\theta_{1}^{\\prime}\\leftarrow\\theta_{1}$, $\\theta_{2}^{\\prime}\\leftarrow\\theta_{2}$\n        \\STATE Initialize replay buffer $\\beta$\n        \\FOR{$ep = 1$ to $max\\_eps$}\n            \\STATE reset environment state\n            \\FOR{$t = 0$ to $max\\_steps$}\n                \\IF {$t < start\\_steps$}\n                    \\STATE $a_{t} \\leftarrow $ env.action\\_space.sample() \n                %\n                \\ELSE\n                    \\STATE $a_{t}\\leftarrow\\mu_{\\phi}(s_t)+\\epsilon,\\ \\epsilon\\sim \\mathcal{N}(0,OU)$\n                %\n                \\ENDIF\n                \n                \\STATE $s_{t+1}$, $r_{t}$, $d_{t}$, \\_ $\\leftarrow$ env.step($a_{t}$)\n                \n                \\STATE store the new transition $(s_{t}, a_{t}, r_{t}, s_{t+1}, d_{t})$ into $\\beta$\n                \n                \\IF{$t > start\\_steps$}\n                    \\STATE Sample mini-batch $B$ of $N$ transitions $(s_{t}, a_{t}, r_{t}, s_{t+1}, d_{t})$ from $\\beta$\n                    \n                    \\STATE $a'\\leftarrow\\mu_{\\phi^{\\prime}}(s^{\\prime})+\\epsilon,\\ \\epsilon\\sim clip(\\mathcal{N}(0,\\tilde{\\sigma}), -c,\\ c)$ \n                    \n                    \\STATE Computes target: \\\\ $Q_{t} \\leftarrow r+\\gamma*\\min_{i=1,2}Q_{\\theta_i}(s', a')$\n                    \n                   \n                    \n                    \\STATE Update double critics with one step gradient descent:\\\\\n                   $\\nabla_{\\theta_i} \\frac{1}{N} \\sum_{i \\in B}(Q_t - Q_{\\theta_{i}(s_{t},a_{t})})^2$  \\qquad for i=1,2\n                    \n                    \\IF {t \\% $policy\\_freq(t)$ == 0}\n                        \\STATE Update policy with one step gradient descent:\\\\             \n                        $\\nabla_{\\phi}\\frac{1}{N} \\sum_i[\\nabla_{a_{t}}Q_{\\theta_{1}}(s_{t},a_{t})\\vert _{a_{t}=\\mu(\\phi)}\\nabla_{\\phi}\\mu_{\\phi}(s_{t})]$\n                        \n                        Soft update for the target networks: \\\\\n                        \\STATE $\\phi^{\\prime}\\leftarrow\\tau\\phi+(1-\\tau)\\phi^{\\prime}$\n                        \\STATE $\\theta_{i}^{\\prime}\\leftarrow\\tau\\theta_{i}+(1-\\tau)\\theta_{i}^{\\prime}$ \\qquad for i=1,2\n                        \n                        \n                    \\ENDIF\n                \\ENDIF\n            \\ENDFOR\n        \\ENDFOR\n  \\end{algorithmic}\n\\end{algorithm}\n\nWe train for $max\\_steps$ steps in $max\\_eps$ episodes. Our approach starts by exploring random actions for the initial $start\\_steps$ steps. We use an LSTM as the actor-network $\\phi$ and $\\phi^{\\prime}$ as its target. The double critics are also LSTM networks, denoted by $\\theta_{1}$ and $\\theta_{2}$, with $\\theta_{1}^{\\prime}$ and $\\theta_{2}^{\\prime}$ as their targets. The learning of both networks happens simultaneously, addressing approximation error, reducing the bias, and finding the highest Q-values. The actor target chooses the action $a^{\\prime}$ based on the state $s^{\\prime}$, and we add Ornstein-Uhlenbeck noise to it. The double critic targets take the tuple ($s^{\\prime}$, $a^{\\prime}$) and return two Q-values as outputs, from which only the minimum of the two is considered. The loss is calculated with the Mean Squared Error of the approximate value from the target networks and the value from the critic networks. We use Adaptive Moment Estimation (Adam) to minimize the loss.\n\nWe update the policy network less frequently than the value network, taking into account a $policy\\_freq$ factor that increases over time by the following rule:\n\n\\begin{equation*} policy\\_freq(t)=\\left\\lfloor\\left(0.5- \\frac{t}{max\\_steps \\times 3}\\right)^{-1}\\right\\rfloor\\end{equation*}\n\n\\subsection{Stochastic Deep RL\n\nWe also introduce a bias-stochastic actor-critic algorithm based on SAC~\\cite{haarnoja2018soft}, that combines off-policy updates with a stochastic actor-critic method to learn continuous action space policies. It uses neural networks as approximation functions to learn a policy and two Q-values functions similarly to TD3. However, SAC utilizes the current stochastic policy to act without noise, providing better stability and performance, maximizing the reward and the policy's entropy, encouraging the agent to explore new states and improving training speed. We use the soft Bellman equation with neural networks as a function approximation to maximize entropy. The pseudocode can be seen in Algorithm \\ref{alg:docrl_s}.\n\n\\begin{algorithm}[!htb]\n    \\algsetup{linenosize=\\tiny}\n    \\scriptsize\n    \\caption{Deep Reinforcement Learning Stochastic}\n    \\label{alg:docrl_s}\n    \\begin{algorithmic}[1]\n        \\STATE Initialize params of critic networks $\\theta_{1}$, $\\theta_{2}$ , and actor network $\\phi$\n        \\STATE Initialize params of target networks $\\phi^{\\prime}\\leftarrow\\phi$, $\\theta_{1}^{\\prime}\\leftarrow\\theta_{1}$, $\\theta_{2}^{\\prime}\\leftarrow\\theta_{2}$\n        \\STATE Initialize replay buffer $\\beta$\n        \\FOR{$ep = 1$ to $max\\_eps$}\n            \\STATE reset environment state\n            \\FOR{$t = 0$ to $max\\_steps$}\n                \\IF {$t < start\\_steps$}\n                    \\STATE $a_{t} \\leftarrow $ env.action\\_space.sample() \n                %\n                \\ELSE\n                    \\STATE $a_t\\leftarrow \\text{sample from } \\pi_{\\phi}(\\cdot|s_t)$\n                %\n                \\ENDIF\n                \n                \\STATE $s_{t+1}$, $r_{t}$, $d_{t}$, \\_ $\\leftarrow$ env.step($a_{t}$)\n                \n                \\STATE store the new transition $(s_{t}, a_{t}, r_{t}, s_{t+1}, d_{t})$ into $\\beta$\n                \n                \\IF{$t > start\\_steps$}\n                    \\STATE Sample mini-batch $B$ of $N$ transitions $(s_{t}, a_{t}, r_{t}, s_{t+1}, d_{t})$ from $\\beta$\n\n                    \\STATE $\\tilde{a}_{t} \\leftarrow \\text{sample from } \\pi_{\\phi}(\\cdot|s_t)$\n                    \n                    \\STATE $double = ([min_{i=1,2}( Q_{\\theta'_{i}}({s_{t}},{\\tilde{a}_{t}}))-\\alpha \\log \\tilde{a}_{t})])$\n                    \n                    \\STATE $Q_t=r({s_{t}},{a_{t}})+\\gamma(1-d_{t})*double$ \n                    \n                    \\STATE Update double critics with one step gradient descent:\\\\\n                    $\\nabla_{\\theta_i} \\frac{1}{N} \\sum_{s_t \\in B}(Q_t - Q_{\\theta_{i}}(s_{t},a_{t}))^2 \\text{ for } i=1,2$\n                    \n                    \\IF {t \\% $policy\\_freq(t)$ == 0}\n                        \n                        \\STATE Update policy with one step gradient descent:\\\\\n                        $ \\nabla_{\\phi} \\frac{1}{N} \\sum_{s_t \\in B} ([min_{i=1,2}( Q_{\\theta_{i}}({s_{t}},{\\tilde{a}_{t}}))-\\alpha \\log \\tilde{a}_{t}])$\n                        \n                        \\STATE Soft update for the target networks: \\\\\n                        \\STATE $\\phi^{\\prime}\\leftarrow\\tau\\phi+(1-\\tau)\\phi^{\\prime}$\n                        \\STATE $\\theta_{i}^{\\prime}\\leftarrow\\tau\\theta_{i}+(1-\\tau)\\theta_{i}^{\\prime}$ \\qquad for i=1,2\n\n                    \\ENDIF\n                \\ENDIF\n            \\ENDFOR\n        \\ENDFOR\n  \\end{algorithmic}\n\\end{algorithm}\n\nLike before, here we train for ($max\\_steps$) steps in ($max\\_eps$) episodes as well, exploring random actions for the first ($start\\_steps$) steps. An LSTM structure was used for the policy network $\\phi$. After sampling a batch $B$ from the memory $\\beta$, we compute the targets for the Q-functions $Q_t({r_{t}},{s_{t+1}},{d_{t}})$, and update the Q-functions. Also, here we update the policy less frequently than the value network, using the same $policy\\_freq$ factor we used in our deterministic approach. \n\n\\subsection{Simulated Environments\n\nOur experiments were conducted on the Gazebo simulator together with ROS, using the RotorS framework \\cite{furrer2016rotors} to allow the simulation of aerial vehicles with different command levels, such as angular rates, attitude, location control and the simulation of wind with an Ornstein-Uhlenbeck noise. The underwater simulation is enabled by the UUV simulator \\cite{manhaes2016uuv}, which allows the simulation of hydrostatic and hydrodynamic effects, as well as thrusters, sensors, and external perturbations. With this framework, we define the vehicle's underwater model with parameters such as the volume,  additional mass, center of buoyancy, etc., as well as the characteristics of the underwater environment itself.\n\nWe developed two environments that simulate a walled water tank, with dimensions of 10$\\times$10$\\times$6 meters and a one-meter water column. The first environment has four cylindrical columns representing subsea drilling risers. The second environment simulates complex structures, like those found in sea platforms, and contains several elements, such as walls, half walls and pipes (Figure ~\\ref{fig:env2} ).\n\n\\begin{figure}[tbp!]\n    \\vspace{-2mm}\n    \\centering\n    \\includegraphics[width=\\linewidth]{img\/env_3_v2.png}\n    \\caption{Our HUAUV performing in the second scenario.}\n    \\label{fig:env2}\n    \\vspace{-4mm}\n\\end{figure}\n\n\\subsection{HUAUV Description}\n\n\nOur vehicle was based on the model presented by Drews-Jr \\emph{et al.} \\cite{drews2014hybrid}, Neto \\emph{et al.} \\cite{neto2015attitude} and \\emph{et al.} \\cite{horn2019study}. We described it using its actual mechanical settings, including inertia, motor coefficients, mass, rotor velocity, and others. A ROS package containing the vehicle's description plus the Deep-RL agents can be found in the \\nameref{supplementary_material}.\n\nThe vehicle sensing was optimized to mimic real-world LIDAR and Sonar. The described LIDAR is based on the UST 10LX model. It provides a 10 meters distance sensing with $270$\\degree~of range and $0.25$\\degree~of resolution, simulated using the plugin ray of Gazebo. Our simulated FLS sonar was based on the sonar simulation plugin developed by Cerqueira \\emph{et al.}~\\cite{cerqueira2017novel}. We described a FLS sonar with 20 meters of range, with a bin count of 1000 and a beam count of 256. The width and height angles of the beam were $90$\\degree~and $15$\\degree~, respectively. We obtained these values from the relative localization data using Rotors' geometric controller. In the real world, localization information can be obtained from a combination of standard localization sensing of hybrid vehicles like Global Positioning System (GPS) and Ultra Short Baseline (USBL).\n\n\\subsection{Network Structure and Rewarding System} \\label{secapproach}\n\n\n\n\nThe structure of both our approaches has a total of 26 dimensions for the state, 20 samples for the distance sensors, the three previous actions and three values related to the target goal, which are the vehicle's relative position to the target and relative angles to the target in the x-y plane and the z-distance plane. When in the air, 20 samples come from the LIDAR. We get these samples equally spaced by $13.5\\degree$ in the $270\\degree$ LIDAR. When underwater, the distance information comes from the Sonar. We also get 20 beams equally spaced among the total of 256, and we take the highest bin in each beam. This conversion based on the range gives us the distance towards the obstacle or the tank's wall \\cite{Santos18,Santos19}. The actions are scaled between $0$ and $0.25$ $m\/s$ for the linear velocity, from $-0.25$ $m\/s$ to $0.25$ $m\/s$ for the altitude velocity and from $-0.25$ to $0.25$ $rad$ for the $\\Delta$ yaw.\n\n\\subsubsection{Reward Function}\n\nWe proposed a binary rewarding function that yields a positive reward in case of success or a negative reward in case of failure or in case the episode ($ep$) ends at the 500 steps limit:\n\n\\vspace{-5mm}\n\\begin{equation}\nr(s_t, a_t)= \n\\begin{cases}\n    r_{arrive}           & \\text{if } d_t < c_d\\\\\n    r_{collide}          & \\text{if } min_x < c_o\\ ||\\ ep = 500\\\\\n   \n   \n   \n\\end{cases}\n\\end{equation}\n\nThe reward $r_{arrive}$ was set to 100, while the negative reward $r_{collide}$ was set to -10. Both $c_d$ and $c_o$ distances were set to $0.5$ meters.\n\\section{Experimental Results}\n\\label{results}\n\nIn this section, the results obtained during our evaluation are shown. During the training phase, we created a randomly generated goal towards which the agent should navigate. The agents trained for a maximum of 500 steps or until they collided with an obstacle or with the tank's border. In case of reaching the goal before the limit of episodes, a new random goal was generated, allowing the total amount of reward to eventually exceed 100. A learning rate of $10^{-3}$ was used, with a minibatch of 256 samples and the Adam optimizer for all approaches, including the compared methods. We limited the number of episodes trained to 1500 episodes. The limits for the episode number ($max\\_steps$) were used based on the stagnation of the maximum average reward received.\n\nFor each scenario and model, an extensive amount of statistics were collected. The task addressed is goal-oriented navigation considering medium transition, where the robot must navigate from a starting point to an endpoint. This task was addressed in two ways in our tests: (1) starting in the air, performing the medium transition and navigating to a target underwater; and the other way around, (2) starting underwater, performing the medium transition and navigating to a target in the air. We collected the statistics for each of our proposed models (Det. and Sto.) and compared them with the performance of the state-of-the-art deterministic (Det.) and stochastic (Sto.) Deep-RL methods for HUAUVs, \nas well as a behavior-based algorithm \\cite{marino2016minimalistic} .\nThese tasks were performed for 100 trials each and we recorded the total of successful trials, the average time for both underwater ($t\\_water$) and aerial ($t\\_air$) navigation and their standard deviations. \n\nThe models were all trained in the first environment and evaluated in both first (same as trained) and second (never seen) environments. We aim to outline one of the main contributions of this work, \\textit{i.e.} the robust capacity to generalize of our method across environments, in this case performing in a second, unknown and more complex environment. We set the initial position for the Air-Water (A-W) trials to (0.0, 0.0, 2.5) in the Gazebo Cartesian coordinates for the two scenarios. The target position used was (3.6, -2.4, -1.0). In both environments, the target was defined in a path with obstacles on the way. Table \\ref{table:mean_std} shows the results obtained for each environment for 100 navigation trials.\n\n\n\n\\begin{table}[bp!]\n\\vspace{-5mm}\n\\centering\n\\setlength{\\tabcolsep}{0.8pt}\n\\caption{Mean and standard deviation metrics over 100 navigation trials for all approaches in all scenarios.}\n\\label{table:mean_std}\n\\begin{tabular}{c c c c c} \n\\toprule\nEnv & Test & $t_{air}$ (s) & $t_{water}$ (s) & Success \\\\\n\\midrule\n1 & A-W Det. & $76.28$ $\\pm$ $63.20$ & $12.51$ $\\pm$ $20.71$ & 94 \\\\\n1 & A-W Sto. & $21.79$ $\\pm$ $4.57$ & $25.58$ $\\pm$ $5.70$ & $100$ \\\\\n1 & A-W Sto. Grando \\emph{et al.} \\cite{bedin2021deep} & $42.46$ $\\pm$ $62.94$ & $13.13$ $\\pm$ $15.15$ & $42$ \\\\\n1 &\\textbf{ A-W Det. Grando \\emph{et al.} \\cite{bedin2021deep}} & $\\textbf{13.84}$ $\\pm$ $\\textbf{2.11}$ & $\\textbf{5.44}$ $\\pm$ $\\textbf{1.73}$ & $\\textbf{100}$ \\\\\n1 & A-W BBA & $32.42$ $\\pm$ $1.79$ & $21.27$ $\\pm$ $0.18$ & $100$ \\\\\n1 & W-A Det. & $24.66$ $\\pm$ $10.06$ & $5.0$ $\\pm$ $0.71$ & $83$ \\\\\n1 & W-A Sto. & $79.73$ $\\pm$ $27.91$ & $5.41$ $\\pm$ $0.34$ & $100$ \\\\\n\n2 & \\textbf{A-W Det.} & $61.94$ $\\pm$ $45.29$ & $\\textbf{8.44}$ $\\pm$ $\\textbf{9.09}$ & $\\textbf{73}$ \\\\\n2 & \\textbf{A-W Sto.} & $\\textbf{14.89}$ $\\pm$ $\\textbf{1.120}$ & $18.48$ $\\pm$ $6.24$ & $\\textbf{94}$ \\\\\n2 & A-W Sto. Grando \\emph{et al.} \\cite{bedin2021deep} & - & - & $0$ \\\\\n2 & A-W Det. Grando \\emph{et al.} \\cite{bedin2021deep} & - & - & $0$ \\\\\n2 & A-W BBA & $39.69$ $\\pm$ $21.92$ & $11.32$ $\\pm$ $7.46$ & $28$ \\\\\n2 & \\textbf{W-A Det.} & $\\textbf{8.54}$ $\\pm$ $\\textbf{4.44}$ & $\\textbf{4.27}$ $\\pm$ $\\textbf{0.47}$ & $\\textbf{8}$ \\\\\n2 & \\textbf{W-A Sto.} & $\\textbf{15.43}$ $\\pm$ $\\textbf{13.39}$ & $\\textbf{6.60}$ $\\pm$ $\\textbf{1.75}$ & $\\textbf{10}$ \\\\\n2 & W-A Sto. Grando \\emph{et al.} \\cite{bedin2021deep} & - & - & $0$ \\\\\n2 & W-A Det. Grando \\emph{et al.} \\cite{bedin2021deep} & - & - & $0$ \\\\\n2 & W-A BBA & $34.3$ $\\pm$ $22.93$ & $6.13$ $\\pm$ $17.22$ & $8$ \\\\\n\\bottomrule\n\\end{tabular}\n\\end{table}\n\nWe also performed a complementary comparison in the second scenario. We used the models trained in the second environment to collect statistics. For a better analysis, we also performed a comparison between models in this second environment. First, we collected the data for Deterministic and Stochastic models trained only in the first environment for 1500 episodes (Env1), as shown before. Then, we trained these models for 500 more episodes in the second environment (Both). Lastly, we compared them with Deterministic and Stochastic trained only in the second environment for 1500 episodes. Table \\ref{table:Comparistion_lstm} shows the obtained results.\n\n\n\\begin{table}[tp!]\n\\centering\n\\setlength{\\tabcolsep}{2.5pt}\n\\caption{Mean and standard deviation metrics over 100 navigation trials tested in the second simulated environment, for both deterministic and stochastic models trained only in the first environment (Env1), in both first and second environments (Both), and only in the second environment (Env2).}\n\\label{table:Comparistion_lstm}\n\\begin{tabular}{c c c c c} \n\\toprule\nModel & $t_{air}$ (s) & $t_{water}$ (s) & Success \\\\\n\\midrule\n\\textbf{A-W Det. (Env1)} & $61.94$ $\\pm$ $45.29$ & $\\textbf{8.44}$ $\\pm$ $\\textbf{9.09}$ & $\\textbf{73}$ \\\\\n\\textbf{A-W Sto. (Env1)} & $\\textbf{14.89}$ $\\pm$ $\\textbf{1.120}$ & $18.48$ $\\pm$ $6.24$ & $\\textbf{94}$ \\\\\n\\textbf{A-W Det. (Both)} & $\\textbf{14.14}$ $\\pm$ $\\textbf{3.77}$ & $\\textbf{8.69}$ $\\pm$ $\\textbf{3.17}$ & $\\textbf{99}$ \\\\\n\\textbf{A-W Sto. (Both)} & $16.82$ $\\pm$ $2.12$ & $14.92$ $\\pm$ $3.60$ & $\\textbf{100}$ \\\\\nA-W Det. (Env2) & $23.17$ $\\pm$ $31.12$ & $32.53$ $\\pm$ $60.28$ & $21$ \\\\\nA-W Sto. (Env2) & $19.98$ $\\pm$ $15.99$ & $49.61$ $\\pm$ $27.86$ & $87$ \\\\\n\nW-A Det. (Env1) & $8.54$ $\\pm$ $4.44$ & $4.27$ $\\pm$ $0.47$ & $8$ \\\\\nW-A Sto. (Env1) & $15.43$ $\\pm$ $13.39$ & $6.60$ $\\pm$ $1.75$ & $10$ \\\\\n\\textbf{W-A Det. (Both)} & $\\textbf{25.09}$ $\\pm$ $\\textbf{38.86}$ & $\\textbf{4.62}$ $\\pm$ $\\textbf{0.51}$ & $\\textbf{34}$ \\\\\n\\textbf{W-A Sto. (Both)} & $33.41$ $\\pm$ $11.82$ & $11.40$ $\\pm$ $2.38$ & $\\textbf{83}$ \\\\\nW-A Det. (Env2) & - & - & $0$ \\\\\nW-A Sto. (Env2) & $3.73$ $\\pm$ $2.97$ & $30.47$ $\\pm$ $9.47$ & $1$ \\\\\n\\bottomrule\n\\end{tabular}\n\\vspace{-4mm}\n\\end{table}\n\n\n\\section{Discussion}\n\\label{discussion}\n\n\nThe evaluation shows an overall increase in performance in navigation through both environments. It is possible to see that our approaches achieve a consistent performance of 100 successful air-to-water navigation trials with also a consistent navigation time ($14.55 \\pm 0.87$ and $11.19 \\pm 2.86$). In this same scenario, the stochastic performed a little worse in air-to-water navigation but outperformed the deterministic approach in water-to-air navigation. \nIn the second scenario, we can see more clearly that a double-critic-based approach with an RNN structure also has a better ability to learn and generalize the environment, including the obstacles and the medium transition. While the state-of-the-art approaches with a MLP structure were not capable of performing the task, our approaches presented once again a consistent performance, especially in air-to-water navigation. Our approaches showed an excellent ability to learn the tasks and the environmental difficulties, not only the scenario itself. That was further addressed in our additional evaluation with agents trained in the first environment only, both first and second environments and the second environment only. Overall, we can conclude that double critic approaches with recurrent neural networks present a consistent ability to learn through scenarios and environments and to generalize between them. Also, our approaches outperformed the BBA algorithm \nin the rate of successful trials and average time in almost all situations.\n\nIt is important to mention that these approaches are extensively evaluated in a realistic simulation, including control issues and disturbances such as wind. Thus, the results indicate that our approach may achieve real-world application if the correct data from the sensing and the relative localization are correctly ensured. Finally, it is also possible to analyze that these new RNN-based approaches provided a more consistent average course of action throughout the environments.\n\n\n\n\n\\section{Conclusions}\n\\label{conclusion}\n\nThe evaluation shows an overall increase in performance in navigation through both environments. It is possible to see that our approaches achieve a consistent performance of 100 successful air-to-water navigation trials with also a consistent navigation time ($14.55 \\pm 0.87$ and $11.19 \\pm 2.86$). In this same scenario, the stochastic performed a little worse in air-to-water navigation but outperformed the deterministic approach in water-to-air navigation. \nIn the second scenario, we can see more clearly that a double-critic-based approach with an RNN structure also has a better ability to learn and generalize the environment, including the obstacles and the medium transition. While the state-of-the-art approaches with a MLP structure were not capable of performing the task, our approaches presented once again a consistent performance, especially in air-to-water navigation. Our approaches showed an excellent ability to learn the tasks and the environmental difficulties, not only the scenario itself. That was further addressed in our additional evaluation with agents trained in the first environment only, both first and second environments and the second environment only. Overall, we can conclude that double critic approaches with recurrent neural networks present a consistent ability to learn through scenarios and environments and to generalize between them. Also, our approaches outperformed the BBA algorithm \nin the rate of successful trials and average time in almost all situations.\n\nIt is important to mention that these approaches are extensively evaluated in a realistic simulation, including control issues and disturbances such as wind. Thus, the results indicate that our approach may achieve real-world application if the correct data from the sensing and the relative localization are correctly ensured. Finally, it is also possible to analyze that these new RNN-based approaches provided a more consistent average course of action throughout the environments.The evaluation shows an overall increase in performance in navigation through both environments. It is possible to see that our approaches achieve a consistent performance of 100 successful air-to-water navigation trials with also a consistent navigation time ($14.55 \\pm 0.87$ and $11.19 \\pm 2.86$). In this same scenario, the stochastic performed a little worse in air-to-water navigation but outperformed the deterministic approach in water-to-air navigation. \nIn the second scenario, we can see more clearly that a double-critic-based approach with an RNN structure also has a better ability to learn and generalize the environment, including the obstacles and the medium transition. While the state-of-the-art approaches with a MLP structure were not capable of performing the task, our approaches presented once again a consistent performance, especially in air-to-water navigation. Our approaches showed an excellent ability to learn the tasks and the environmental difficulties, not only the scenario itself. That was further addressed in our additional evaluation with agents trained in the first environment only, both first and second environments and the second environment only. Overall, we can conclude that double critic approaches with recurrent neural networks present a consistent ability to learn through scenarios and environments and to generalize between them. Also, our approaches outperformed the BBA algorithm \nin the rate of successful trials and average time in almost all situations.\n\nIt is important to mention that these approaches are extensively evaluated in a realistic simulation, including control issues and disturbances such as wind. Thus, the results indicate that our approach may achieve real-world application if the correct data from the sensing and the relative localization are correctly ensured. Finally, it is also possible to analyze that these new RNN-based approaches provided a more consistent average course of action throughout the environments.\n\n\n\nBy using physically realistic simulation in several water-tank-based scenarios, we showed that our approaches achieved an overall better capability to perform autonomous navigation, obstacle avoidance and medium transition than other approaches. Disturbances such as wind were successfully assimilated and good generalization through different scenarios was achieved. With our simple and realistic sensing approach that took into account only the range information, we presented overall better performance than the state-of-the-art and classical behavior-like algorithm. Future studies with our real HUAUV are on the way. \n\n\\section*{Acknowledgment}\n\n\nThe authors would like to thank the VersusAI team. This work was partly supported by the CAPES, CNPq and PRH-ANP.\n\n\\vspace{-2mm}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nAngular momentum (AM) is a fundamental parameter in the evolution of galaxies. A dark matter (DM) halo spinning up in the early universe is subject to the same tidal torques as the baryons at its centre, so the total AM of both components is linked, and the specific AM, $j = J\/M$ of the baryons is well matched to $j$ of the DM, (e.g \\cite[Catalan \\& Theuns 1996]{Catalan+1996}). \n$j$ is connected to photometric morphology via the stellar mass -- specific AM -- morphology plane, first shown by \\cite[Fall 1983]{Fall83}, such that galaxies with higher $M_*$ have higher $j$, modulo morphology, with the relation for earlier type galaxies offset to lower $j$. This has since also been shown by \\cite[Romanowsky \\& Fall (2012)]{RF12}, \\cite[Obreschkow \\& Glazebrook (2014)]{OG14}, \\cite[Cortese et al. (2016)]{Cortese+2016}, \\cite[Posti et al. (2018)]{Posti+2018}, and \\cite[Sweet et al. (2018)]{Sweet+2018}.\n\nAlthough the total $j$ for baryons and DM is linked, further physical processes affect the distribution of $j$ for baryons. \\cite[van den Bosch et al. (2001)]{vdb+01} studied the probability density function of $j$ normalised to the mean of the galaxy, \\pdf, and found that dwarf galaxies had a deficit of high-$j$ and of low-$j$ material with respect to the prediction for a DM halo. They attributed this to tidal stripping of the outer, rapidly-rotating material and feedback ejecting the inner, dispersion-dominated material respectively. \\cite[Sharma \\& Steinmetz (2005)]{SS05} then predicted the \\pdf\\\/ for baryonic components (see Fig.\\,\\ref{ss05}), where bulges, which are dominated by random motions, exhibit a peak at $j=0$, and disks have a \\pdf\\\/ of the form $x$exp$(-kx)$ due to their well-ordered rotation. Also see our updated predictions using the NIHAO simulations (Wang+ in prep)).\n\nThe \\pdf\\\/ encodes more physical information than photometry alone, so in this work I am investigating the utility of \\pdf\\\/ as a kinematic tracer of morphology and kinematic decomposition tool.\n\n\\begin{figure}\n\\begin{center}\n \\includegraphics[width=0.4\\linewidth]{SS05.png} \n \\caption{Predictions from \\cite[Sharma \\& Steinmetz 2005]{SS05} for baryonic galaxy components. The \\pdf\\\/ for bulge peaks at $j=0$, while disk components have an exponential profile.}\n   \\label{ss05}\n\\end{center}\n\\end{figure}\n\n\\section{\\pdf\\\/}\n\nI have constructed \\pdf\\\/ for a subset of the Calar Alto Legacy Integral Field Area survey (CALIFA, \\cite[Sanchez et al. 2012]{Sanchez+2012}), using observations of 25 galaxies where the stellar kinematics reach to three times the effective radius. I calculate $j_i = r_i \\times v_i$ in every spaxel $i$, where the velocity $v_i = sqrt( v_{i,circ}^2 + v_{i,disp}^2)$ includes the circular velocity $v_{i,circ}$ and dispersion $v_{i,disp}$ added in quadrature. The map of $j$ is then weighted by stellar surface density, as a proxy for mass, and the histogram plotted as the \\pdf\\\/. \n\nA spanning set of local examples is shown in Fig.\\,\\ref{examples}. The colour represents radial distance, with lighter colours indicating material nearer the centre. NGC 6063 is a late-type spiral galaxy with low bulge-to-total light ratio B\/T = 0.04. Its \\pdf\\\/ is broad and symmetric, and peaks near 1, reminiscent of the predictions by  \\cite[Sharma \\& Steinmetz (2005)]{SS05} for pure disks. NGC 2592 is an early type galaxy with large B\/T = 0.54; its \\pdf\\\/ is strongly-skewed and peaks near $j=0$ like the spheroidal components in \\cite[Sharma \\& Steinmetz (2005)]{SS05}. Intermediate between these two extremes, the \\pdf\\\/ for NGC 7653 (B\/T = 0.33) has characteristics of both disk and bulge. Unfortunately the spatial resolution, which dictates the number of bins, is not sufficient to resolve separate components in the \\pdf\\\/.\n\nI also show a clumpy disk galaxy at $z\\sim 1.5$, using a combination of OSIRIS adaptive optics integral field spectroscopy to mitigate the effects of beam smearing in the centre of the galaxy, with deeper KMOS seeing-limited data to trace the velocity field out to higher multiples of the effective radius, using a method described in Sweet et al. (in prep.). The shape is intermediate between the pure disk and bulge-dominated local examples, likely due to the typical high-$z$ morphology of dispersion-dominated clumps embedded in a strongly-rotating disk.\n\n\\begin{figure}\n\\begin{center}\n \\includegraphics[width=0.495\\linewidth]{6063.png} \n \\includegraphics[width=0.495\\linewidth]{2592.png} \n \\includegraphics[width=0.495\\linewidth]{7653.png} \n \\includegraphics[width=0.495\\linewidth]{COSMOS_127977.png} \n \\caption{Example \\pdf\\\/ for local galaxies in the CALIFA sample and one disk at $z\\sim 1.5$. Top left: late-type spiral NGC 6063 with low B\/T ratio has a broad, symmetric \\pdf\\\/ which peaks near 1. Top right: early type NGC 2592 with high B\/T ratio has a strongly-skewed \\pdf\\\/ which peaks nearer 0. Bottom left: NGC 7653 with moderate B\/T ratio has a \\pdf\\\/ which is intermediate between the two extremes. Bottom right: $z\\sim 1.5$ clumpy disk galaxy COSMOS 127977 also has an intermediate \\pdf\\\/.}\n   \\label{examples}\n\\end{center}\n\\end{figure}\n\nThere is an apparent trend whereby \\pdf that are more skewed and peak nearer $j=0$ correspond to galaxies that are earlier in type and have bigger bulges. This is quantified in the correlation between shape of \\pdf\\\/ (traced by skewness or kurtosis) and morphology (traced by Hubble type and B\/T ratio). For instance, Fig.\\,\\ref{t-skew} illustrates the relation between skewness and Hubble type. Much of the scatter in this correlation arises from the difficulties inherent in photometric classification of Hubble type and quantifying bulge-to-total ratio. Arguably, \\pdf\\\/ as a kinematic quantity encodes more physical information than photometry alone, so may be a more robust tracer of morphology.\n\n\\begin{figure}\n\\begin{center}\n \\includegraphics[width=0.45\\linewidth]{t-skew.png} \n \\caption{Correlation between morphology and shape of \\pdf\\\/. Galaxies with earlier Hubble type are more strongly negatively skewed.}\n   \\label{t-skew}\n\\end{center}\n\\end{figure}\n\nI also see that the bulge is linked with \\pdf\\\/.  Fig.\\,\\ref{bulgediskkurt} demonstrates that the shape of \\pdf\\\/ is moderately correlated with the surface brightness of the bulge, such that galaxies with bigger bulges have more strongly-tailed \\pdf. On the other hand, the \\pdf\\\/ shape is not at all correlated with the central surface brightness of the disk. This indicates that the size of the bulge is related to the distribution of $j$ within a galaxy, but disks of all sizes have similar \\pdf\\\/.\n\n\\begin{figure}\n\\begin{center}\n \\includegraphics[width=0.8\\linewidth]{bulgediskkurt.png} \n \\caption{Correlation between galaxy components and shape of \\pdf\\\/. Galaxies with bigger bulges have more strongly-tailed \\pdf\\\/, but the shape of \\pdf\\\/ is the same for disks of all sizes.}\n   \\label{bulgediskkurt}\n\\end{center}\n\\end{figure}\n\n\n\\section{Conclusion: Utility of PDF(j)}\nThe \\pdf\\\/ traces kinematic morphology of a galaxy. It encodes more physical information than photometry alone, and is a product of the evolutionary history of the galaxy. In future, as spatial resolution increases, I predict that \\pdf\\\/ will be useful to separate out kinematic components: thin disk from thick disk and bulge, clumps from bulges, and pseudobulges from classical bulges.\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nIn this paper, we study the quantitative long time dynamics for the spherically symmetric dispersive spacetimes satisfying the Einstein-scalar field equations. More precisely, these are spherically symmetric solutions $(\\calM,\\bfg,\\phi)$ to the Einstein-scalar field system, where $\\bfg$ is a Lorentzian metric and $\\phi$ is a real valued function on a $3+1$ dimensional manifold $\\calM$, such that $(\\calM,\\bfg)$ is future causally geodesically complete and $\\phi$ scatters locally in the scale-invariant bounded-variation (BV) norm. For these spacetimes, we establish a Price-law type decay for the scalar field $\\phi$, the Christoffel symbols associated to $\\bfg$ and all of their derivatives. To obtain the decay results, we do not need to assume any smallness of the initial data. Moreover, we show that the decay rates in this paper are sharp.\n\nThe spherically symmetric Einstein-scalar field system, being one of the simplest model of self-gravitating matter in this symmetry class, has been studied extensively both numerically and mathematically. In a seminal series of papers by Christodoulou \\cite{Christodoulou:1987ta}, \\cite{Christodoulou:1991}, \\cite{Christodoulou:1993bt}, \\cite{Christodoulou:1994}, \\cite{Christodoulou:1999}, he achieved a complete understanding of the singularity structure of spherically symmetric spacetime solutions to this system. The culmination of the results shows that generic\\footnote{in the BV class, i.e., the initial data for $\\partial_v(r\\phi)$ has bounded variation. More precisely, Christodoulou showed that the non-generic set of initial data has co-dimension at least two in the BV topology.} spherically symmetric initial data with one asymptotically flat end give rise to a spacetime whose global geometry is either dispersive (with a Penrose diagram represented by Figure 1) or contains a black hole region $\\mathcal B\\mathcal H$ which terminates in a spacelike curvature singularity $\\mathcal S$ (with a Penrose diagram represented by Figure 2). In particular, in either of these generic scenarios, the spacetime possesses a complete null infinity $\\mathcal I^+$ and thus obeys the weak cosmic censorship conjecture. Moreover, in either case the maximal Cauchy development of the data is inextendible with a $C^2$ Lorentzian metric and therefore also verifies the strong cosmic censorship conjecture. We refer the readers to \\cite{Kommemi} for a comprehensive discussion on general singularity structures for spherically symmetric spacetimes.\n\n\\begin{figure}[htbp] \\label{fig.disp}\n\\begin{center}\n \n\\includegraphics{fig_I.pdf}\n \n\\caption{The dispersive case}\n\\end{center}\n\\end{figure}\n\n\\begin{figure}[htbp]\n\\begin{center}\n \n\\includegraphics{fig_II.pdf}\n \n\\caption{The black hole case}\n\\end{center}\n\\end{figure}\n\nThe remarkable resolution of the cosmic censorship conjectures however gives very little information on the long time dynamics for these spacetimes except for the small data\\footnote{i.e., when the initial data is close to that of Minkowski space.} case  \\cite{Christodoulou:1993bt}. In particular, not much is known about the asymptotic decay of the scalar field as measured by a far-away observer at null infinity. In the dispersive case, Christodoulou showed that the Bondi mass decays to zero along null infinity without an explicit decay rate. In the black hole case, he showed that the Bondi mass approaches the mass of the black hole, from which one can infer the non-quantitative decay for the scalar field along null infinity \\cite{Christodoulou:1993bt}.\n \nThe long time dynamics in the case where the spacetime settles to a black hole was subsequently studied in the seminal work\\footnote{In fact, they considered the more general Einstein-Maxwell-scalar field equations.} of Dafermos-Rodnianski \\cite{DR} . They proved a quantitative decay rate for the scalar field (and its derivatives) in the spacetime including along null infinity $\\mathcal I^+$ and the event horizon $\\mathcal H^+$. The proof is based on the local conservation of energy, which is subcritical, together with techniques exploiting the conformal geometry of the spacetime and the celebrated red-shift effect along the event horizon. The result in particular justified, in a nonlinear setting, the heuristics of Price \\cite{Price}. It turns out that the quantitative decay rates, when combined with the results of \\cite{D}, also have interesting consequences for the strong cosmic censorship conjecture in the context of the spherically symmetric Einstein-Maxwell-scalar field system.\n\nIn this paper, we study the other generic scenerio, i.e., spherically symmetric dispersive spacetime solutions to the Einstein-scalar field system. Unlike in the black hole case, the monotonic Hawking mass is \\emph{supercritical} and provides no control over the dynamics of the solution. We thus do not expect to be able to obtain quantitative decay rates for large solutions without imposing extra assumptions. Instead, we assume \\emph{a priori} the non-quantitative decay of a \\emph{critical} quantity - the BV norm\\footnote{Solutions of bounded variation have been first studied by Christodoulou \\cite{Christodoulou:1993bt} and plays an important role in the proof of the cosmic censorship conjectures \\cite{Christodoulou:1999}.} - but only locally in a region where the area of the orbit of the symmetry group $SO(3)$ remains uniformly bounded. Under this assumption of local BV scattering, we show that the scalar field and all its derivatives decay with a quantitative rate, reminescent of the Price law decay rates in the black hole case. (We refer the readers to the statement of the main theorems in Section \\ref{sec.main.thm} for the precise rates that we obtain.) We prove, in particular, a quantitative decay rate for the scalar field along null infinity.\n\nOur results apply in particular to the class of solutions arising from initial data with small BV norm. Christodoulou \\cite{Christodoulou:1993bt} showed that these spacetimes are future causally geodesically complete. One can easily deduce from \\cite{Christodoulou:1993bt} that in fact these spacetimes satisfy the BV scattering assumption and therefore the solutions obey the quantitative decay estimates of our main theorem (see Theorem \\ref{thm:smallData}). On the other hand, our results do not require any smallness assumptions on the initial data. We conjecture that indeed our class of spacetimes contains those arising from large data:\n\\begin{conjecture}\\label{large.sol.conj}\nThere exists initial data of arbitrarily large BV norm whose maximal global development scatters locally in the BV norm.\n\\end{conjecture}\n\nIn addition to the upper bounds that we obtain in our main theorem, we also construct examples where we prove lower bounds for the solutions with the same rates as the upper bounds. In particular, there exists a class of initial data with compactly supported scalar field whose future development saturates the decay estimates in the main theorem. This shows that the decay rates are sharp. We note that the decay rate is also consistent with the numerical study of Biz\\'on-Chmaj-Rostworowski \\cite{BCR}.\n\nAs a corollary of the main result on decay, we show the following dichotomy: either the quantitative decay rates are satisfied or the solution blows up at infinity. The latter are solutions such that some scale-invariant spacetime norms become infinite (see precise definition in Definition \\ref{def.blow.up.infty}).\n\nThe decay result in this paper easily implies that the locally BV scattering solutions that we consider are stable against small, regular, \\emph{spherically symmetric} perturbations. More ambitiously, one may conjecture that\n\\begin{conjecture}\\label{stab.conj}\nSpherically symmetric locally BV scattering dispersive solutions to the Einstein-scalar field equations are stable against \\emph{non-spherically symmetric} perturbations.\n\\end{conjecture}\n\nConjecture \\ref{stab.conj}, if true, will generalize the monumental theorem on the nonlinear stability of Minkowski spacetime of Christodoulou-Klainerman \\cite{CK} (see also a simpler proof in \\cite{LR}). For nonlinear wave equations satisfying the null condition, it is known \\cite{Alinhac}, \\cite{Yang} that \\emph{large} solutions decaying sufficiently fast are globally stable against small perturbations. On the other hand, our main theorem shows a quantitative decay rate for spherically symmetric locally BV scattering dispersive spacetimes. Conjecture \\ref{stab.conj} can therefore be viewed as an attempt to generalize the results in \\cite{Alinhac}, \\cite{Yang} to the Einstein-scalar field system. We will address both Conjectures \\ref{large.sol.conj} and \\ref{stab.conj} in future works.\n\\\\\n\\\\\n\\noindent{\\bf Acknowledgements.} The authors would like to thank Mihalis Dafermos and Igor Rodnianski for valuable discussions. We also thank Jonathan Kommemi for providing the Penrose diagrams. \n\nJ. Luk is supported by the NSF Postdoctoral Fellowship DMS-1204493. S.-J. Oh is a Miller Research Fellow, and thanks the Miller Institute at UC Berkeley for the support.\n\n\\subsection{Outline of the paper}\nWe outline the remainder of the paper. In Section \\ref{sec.setup}, we discuss the set-up of the problem and in particular define the class of solutions considered in the main theorem, i.e., the locally BV scattering solutions. In Section \\ref{sec.main.thm}, we state the main theorems in the paper (Theorems \\ref{main.thm.1} and \\ref{main.thm.2}), their consequences and additional theorems on the optimality of the decay rates. In the same section, we outline the main ideas of the proof. In Sections \\ref{sec.anal.prop} and \\ref{sec.geom}, we explain the main analytic features of the equations and the geometry of the class of spacetimes that we consider.\n \nSections \\ref{sec.decay1} and \\ref{sec.decay2} consist of the main content of this paper. In Section \\ref{sec.decay1}, we prove the decay estimates for $\\phi$, $\\partial_v(r\\phi)$ and $\\partial_u(r\\phi)$, i.e., the first main theorem (Theorem \\ref{main.thm.1}). In Section \\ref{sec.decay2}, using in particular the results in Section \\ref{sec.decay1}, we derive the decay bounds for the second derivatives for $r\\phi$ and the metric components, i.e., the second main theorem (Theorem \\ref{main.thm.2}). \n\nIn the remaining sections of the paper, we turn to other theorems stated in Section \\ref{sec.main.thm}. In Section \\ref{sec.dichotomy}, we give a proof of the dichotomy alluded to above, i.e., either the quantitative decay holds or the spacetime blows up at infinity. In Section \\ref{sec:smallData}, we sketch a proof of a refinement of the conclusions of the main theorems in the small data case. Finally, in Section \\ref{sec.opt}, we prove optimality of the decay rates asserted by the main theorems.\n\n\\section{Set-up}\\label{sec.setup}\nIn this section, we define the set-up, formulate the equations in a double null coordinate system and explain the characteristic initial value problem. This will allow us to state the main theorem in the next section.\n\n\\subsection{Spherically Symmetric Einstein-Scalar-Field System (SSESF)} \\label{subsec:derivation}\nWe begin with a brief discussion on the derivation of the Spherically Symmetric Einstein-Scalar-Field System \\eqref{eq:SSESF} from the $(3+1)$-dimensional Einstein-scalar-field system. \n\nSolutions to the Einstein-scalar field equations can be represented by a triplet $(\\calM, \\bfg_{\\mu \\nu},\\phi)$, where $(\\calM, \\bfg_{\\mu \\nu})$ is a $(3+1)$-dimensional Lorentzian manifold and $\\phi$ a real-valued function on $\\calM$. The spacetime metric $\\bfg_{\\mu \\nu}$ and the scalar field $\\phi$ satisfy the Einstein-scalar-field system:\n\\begin{equation} \\label{eq:ES}\n\\left\\{\n\\begin{aligned}\n\t\\bfR_{\\mu \\nu} - \\frac{1}{2} \\bfg_{\\mu \\nu} R =& 2 \\bfT_{\\mu \\nu}, \\\\\n\t\\nb^{\\mu} \\partial_{\\mu} \\phi =& 0.\n\\end{aligned}\n\\right.\n\\end{equation}\nwhere $\\bfR_{\\mu \\nu}$ is the Ricci curvature of $\\bfg_{\\mu \\nu}$, $R$ is the scalar curvature, and $\\nb_{\\mu}$ is the covariant derivative given by the Levi-Civita connection on $(\\calM, \\bfg)$. The energy-momentum tensor $\\bfT_{\\mu \\nu}$ is given by the scalar field $\\phi$, i.e.\n\\begin{equation}\\label{eq:T} \n\t\\bfT_{\\mu\\nu} = \\partial_{\\mu} \\phi \\partial_{\\nu} \\phi - \\frac{1}{2} \\bfg_{\\mu \\nu} \\partial^{\\lambda} \\phi \\partial_{\\lambda} \\phi.\n\\end{equation}\n\nAssume that the solution $(\\calM, \\bfg_{\\mu \\nu}, \\phi)$ is spherically symmetric, i.e., the group $\\mathrm{SO}(3)$ of three dimensional rotations acts smoothly and isometrically on $(\\calM, \\bfg)$, where each orbit is either a point or is isometric to $\\mathbb S^{2}$ with a round metric. The scalar field $\\phi$ is required to be constant on each of the orbits. These assumptions are propagated by \\eqref{eq:ES}; hence, if $(\\calM, \\bfg_{\\mu \\nu}, \\phi)$ is a Cauchy development, then it suffices to assume spherical symmetry only on the initial data.\n\nThe quotient $\\mathcal M \/ \\mathrm{SO}(3)$ gives rise to a (1+1)-dimensional Lorentzian manifold with boundary, which we denote by $(\\calQ, g_{ab})$. The boundary $\\Gamma$ consists of fixed points of the group action. We define the \\emph{area radius function} $r$ on $\\calQ$ to be\n\\begin{equation*}\n\t r := \\sqrt{\\frac{\\mbox{Area of symmetry sphere}}{4 \\pi}}.\n\\end{equation*}\nand $r=0$ at $\\Gamma$. Note that each component of $\\Gamma$ is a timelike geodesic.\n\nWe assume that $\\Gamma$ is non-empty and connected, and moreover that there exists a \\emph{global double null coordinates} $(u,v)$, i.e. a coordinate system $(u,v)$ covering $\\calQ$ in which the metric takes the form\n\\begin{equation} \\label{eq:defn4Met}\n\tg_{ab} \\mathrm{d} x^{a} \\cdot \\mathrm{d} x^{b} = - \\Omega^{2} \\mathrm{d} u \\cdot \\mathrm{d} v\n\\end{equation}\nfor some $\\Omega > 0$. We remark that both assumptions are easily justified if $(\\mathcal M, {\\bf g})$ is a Cauchy development of a spacelike hypersurface homeomorphic to $\\mathbb R^{3}$. \n\n\nThe metric ${\\bf g}_{\\mu \\nu}$ of $\\mathcal M$ is characterized by $\\Omega$ and $r$ and takes the form\n\\begin{equation}\n\t{\\bf g}_{\\mu \\nu} \\mathrm{d} x^{\\mu} \\cdot \\mathrm{d} x^{\\nu} = - \\Omega^{2} \\mathrm{d} u \\cdot \\mathrm{d} v + r^{2} \\mathrm{d} s^{2}_{\\mathbb S^{2}}\\label{metric}\n\\end{equation}\nwhere $\\mathrm{d} s^{2}_{\\mathbb S^{2}}$ is the standard line element on the unit sphere $\\mathbb S^{2}$. Therefore, we may reformulate the \\emph{spherically symmetric Einstein-scalar-field system} \\eqref{eq:SSESF} in terms of the triplet $(\\phi, r, \\Omega)$ as \n\\begin{equation} \\label{eq:SSESF} \\tag{SSESF}\n\\left\\{\n\\begin{aligned}\n\tr \\partial_{u} \\partial_{v} r =& - \\partial_{u} r \\partial_{v} r - \\frac{1}{4} \\Omega^{2}, \\\\\n\tr^{2} \\partial_{u} \\partial_{v} \\log \\Omega =& \\, \\partial_{u} r\\partial_{v} r + \\frac{1}{4} \\Omega^{2} -  r^{2} \\partial_{u} \\phi \\partial_{v} \\phi, \\\\\n\tr \\partial_{u} \\partial_{v} \\phi =& - \\partial_{u} r \\partial_{v} \\phi - \\partial_{v} r \\partial_{u} \\phi, \\\\\n\t2 \\Omega^{-1} \\partial_{u} r\\,  \\partial_{u} \\Omega =& \\, \\partial^{2}_{u} r + r (\\partial_{u} \\phi)^{2}, \\\\\n\t2 \\Omega^{-1} \\partial_{v} r\\,  \\partial_{v} \\Omega =& \\, \\partial^{2}_{v} r + r (\\partial_{v} \\phi)^{2},\n\\end{aligned}\n\\right.\n\\end{equation}\nwith the boundary condition $r=0$ along $\\Gamma$. \n\n\\subsection{Basic assumptions, notations and conventions}\nIn this subsection, we introduce the basic assumptions on the base manifold $\\calQ$, as well as some notations and conventions that will be used in the rest of the paper.\n\n\\subsubsection*{Definition of $\\calQ$ and $\\calM$}\nDenote by $\\mathbb R^{1+1}$ the (1+1)-dimensional Minkowski space, with the standard double null coordinates $(u,v)$. Let $\\calQ$ be a (1+1)-dimensional Lorentzian manifold which is conformally embedded into $\\mathbb R^{1+1}$ with $\\mathrm{d} s^{2}_{\\calQ} = - \\Omega^{2} \\mathrm{d} u \\cdot \\mathrm{d} v$. Given a non-negative function $r$ on $\\calQ$, we define the set $\\Gamma := \\set{(u,v) \\in \\calQ : r(u,v) = 0}$, called the \\emph{axis of symmetry}. We also define $(\\calM, \\bfg_{\\mu \\nu})$ to be the (1+3)-dimensional Lorentzian manifold with $\\calM = \\calQ \\times \\mathbb S^{2}$ and $\\bfg_{\\mu \\nu}$ given by \\eqref{eq:defn4Met}; this is to be thought of as the full spacetime before the symmetry reduction. (We refer to \\S \\ref{subsec:derivation} for the full interpretation.)\n\n\\subsubsection*{Assumptions on the conformal geometry of $\\calQ$}\nWe assume that $\\Gamma \\subset \\calQ$ is a connected set, which is the image of a future-directed timelike curve emanating from the point $(1,1)$.\nWe also assume that $C_{1} \\subset \\calQ$, where\n\\begin{equation*}\nC_{1} = \\set{(u,v) \\in \\mathbb R^{1+1} : u=1, \\, 1 \\leq v < \\infty}.\n\\end{equation*}\n \nFurthermore, $\\calQ$ is assumed to be the domain of dependence of $\\Gamma$ and $C_{1}$ to the future, in the sense that every causal curve in $\\calQ$\nhas its past endpoint on either $\\Gamma$ or $C_{1}$. \n\n\\subsubsection*{Notations for the conformal geometry of $\\calQ$}\nDenote by $C_{u}$ (resp. $\\underline{C}_{v}$) the constant $u$ (resp. $v$) curve in $\\calQ$. Note that these are null curves in $\\calQ$.\n\nGiven $(u_{0}, v_{0}) \\in \\calQ$, we define the \\emph{domain of dependence} of the line segment $C_{u_{0}} \\cap \\set{v \\leq v_{0}}$, denoted by $\\mathcal D(u_{0}, v_{0})$, to be the set of all points $p \\in \\calQ$ such that all past-directed causal curves passing $p$ intersects $\\Gamma \\cup (C_{u_{0}} \\cap \\set{v \\leq v_{0}})$, plus the line segment $(C_{u_{0}} \\cap \\set{v \\leq v_{0}})$ itself. \n\nAlso, we define the \\emph{future null infinity} $\\mathcal I^{+}$ to be the set of ideal points $(u, +\\infty)$ such that $\\sup_{C_{u}} r = \\infty$. \n\n\\subsubsection*{Integration over null curves}\nWhenever we integrate over a subset of $C_{u}$ (resp. $\\underline{C}_{v}$), we will use the standard line element $\\mathrm{d} v$ ($\\mathrm{d} u$) for integrals over, i.e.,  \n\\begin{align*}\n\t\\int_{\\underline{C}_v\\cap\\{u_1\\leq u\\leq u_2\\}} f = \\int_{u_1}^{u_2} f(u',v) \\, \\mathrm{d} u'  \\\\ \n\t\\int_{C_u\\cap\\{v_1\\leq v\\leq v_2\\}} f = \\int_{v_1}^{v_2} f(u,v') \\, \\mathrm{d} v',\n\\end{align*} \nrespectively.\n\n\\subsubsection*{Functions of bounded variation}\nUnless otherwise specified, functions of bounded variation (BV) considered in this paper will be assumed to be right-continuous. By convention,\n\\begin{equation*}\n\t\\partial_{v} f \\, \\mathrm{d} v  \\hbox{ or } \\partial_{v} f\n\\end{equation*}\nwill refer to the distributional derivative of $f$, which is a finite signed measure, and\n\\begin{equation*}\n\t\\abs{\\partial_{v} f} \\, \\mathrm{d} v  \\hbox{ or } \\abs{\\partial_{v} f}\n\\end{equation*}\nwill denote the total variation measure. Unless otherwise specified, these measures will be evaluated on intervals of the form $(v_{1}, v_{2}]$. Thus, according to our conventions,\n\\begin{align*}\n\t\\int_{v_{1}}^{v_{2}} \\partial_{v} f (v) \\, \\mathrm{d} v =& f(v_{2}) - f(v_{1}), \\\\\n\t\\int_{v_{1}}^{v_{2}} \\abs{\\partial_{v} f (v)} \\, \\mathrm{d} v = & \\mathrm{T.V.}_{(v_{1}, v_{2}]} [f]. \n\\end{align*}\n\n\\subsubsection*{New variables}\nWe introduce the following notation for the directional derivatives of $r$:\n\\begin{equation*}\n\t\\lmb := \\frac{\\partial r}{\\partial v}, \\quad \\nu := \\frac{\\partial r}{\\partial u},\n\\end{equation*}\n\nThe \\emph{Hawking mass} $m(u,v)$ is defined by the relation\n\\begin{equation}\\label{mdef}\n1 - \\frac{2m}{r} = \\partial^{a} r \\partial_{a} r = - 4 \\Omega^{-2} \\partial_{u} r \\partial_{v} r.\n\\end{equation}\n\nFor a solution to \\eqref{eq:SSESF}, the quantity $m$ possesses useful monotonicity properties (see Lemma \\ref{lem:mntn4m}), which will be one of the key ingredients of our analysis. We define the \\emph{mass ratio} to be\n\\begin{equation*}\n\t\\mu := \\frac{2m}{r}.\n\\end{equation*}\n\nWe also define the \\emph{Bondi mass} on $C_{u}$ by $M(u) := \\lim_{v \\to \\infty} m(u, v)$, provided the limit exists. The Bondi mass $M_{i} := M(1) = \\lim_{v \\to \\infty} m(1, v)$ on the initial curve $C_{1}$ is called the \\emph{initial Bondi mass}.\n\n\\subsection{Refomulation in terms of the Hawking mass}\nThe Hawking mass as defined in \\eqref{mdef} turns out to obey useful monotonicity (See \\S \\ref{subsec:monotonicity}). We therefore reformulate \\eqref{eq:SSESF} in terms of $m$ and eliminate $\\Omega$. Notice that by \\eqref{metric} and \\eqref{mdef}, the metric is determined by $r$ and $m$.\n\nWe say that \\emph{$(\\phi, r, m)$ on $\\calQ$ is a solution to }(SSESF) if the following equations hold:\n\\begin{equation} \\label{eq:SSESF:dr}\n\\left\\{\n\\begin{aligned}\n\\partial_{u} \\lmb = & \\frac{\\mu}{(1-\\mu) r} \\lmb \\nu, \\\\\n\\partial_{v} \\nu = & \\frac{\\mu}{(1-\\mu) r} \\lmb \\nu,\n\\end{aligned}\n\\right.\n\\end{equation}\n\\begin{equation} \\label{eq:SSESF:dm}\n\\left\\{\n\\begin{aligned}\n2 \\nu \\partial_{u} m = &  (1-\\mu) r^{2} (\\partial_{u} \\phi)^{2}, \\\\\n2 \\lmb \\partial_{v} m = &  (1-\\mu) r^{2} (\\partial_{v} \\phi)^{2},\n\\end{aligned}\n\\right.\n\\end{equation}\n\\begin{equation} \\label{eq:SSESF:dphi}\n\\partial_{u} \\partial_{v} (r \\phi) = \\frac{\\mu \\lmb \\nu}{(1-\\mu)r} \\phi,\n\\end{equation}\nand moreover, the following boundary conditions hold:\n\\begin{equation*}\n\tr = 0 \\hbox{ and } m = 0 \\hbox{ along } \\Gamma.\n\\end{equation*}\n\nWe remark that using \\eqref{eq:SSESF:dr}, the wave equation \\eqref{eq:SSESF:dphi} for $\\phi$ may be rewritten in either of the following two equivalent forms:\n\\begin{align} \n\\partial_{u} (\\partial_{v} (r \\phi)) = (\\partial_{u} \\lmb) \\phi, \\label{eq:SSESF:dphi'} \\tag{\\ref{eq:SSESF:dphi}$'$} \\\\\n\\partial_{v} (\\partial_{u} (r \\phi)) = (\\partial_{v} \\nu) \\phi. \\label{eq:SSESF:dphi''} \\tag{\\ref{eq:SSESF:dphi}$''$}\n\\end{align}\n\n\\subsection{Choice of coordinates} \\label{subsec:coordSys}\nNote that $\\calQ$ is ruled by the family of null curves $C_{u}$. Since a null curve $C_{u}$ with $u \\neq 1$ cannot intersect $C_{1}$, its past endpoint must be on $\\Gamma$. Therefore, our assumptions so far impose the following conditions on the double null coordinates $(u,v)$ on $\\calQ$: $u$ is constant on each future-directed null curve emanating from $\\Gamma$ and $v$ is constant on each conjugate null curve. However, these conditions are insufficient to give a unique choice of a coordinate system, as the system \\eqref{eq:SSESF} and assumptions so far are invariant under the change of coordinates\n\\begin{equation*}\n\tu \\mapsto U(u), \\quad v \\mapsto V(v), \\quad U(1) = V(1) = 1,\n\\end{equation*}\nfor any strictly increasing functions $U$ and $V$. To remove this ambiguity, we fix the choice of the coordinate system, once and for all, as follows.\n\nWe first fix $v$ on $C_{1}$ relating it with the function $r$. Specifically, we will require that $v = 2r + 1$ on $C_1$, which in particular implies that \n\\begin{equation} \\label{eq:id4dvr}\n\\lmb(1, v) = \\frac{1}{2}.\n\\end{equation}\n\nNext, in order to fix $u$, we prescribe $u$ such that $\\Gamma = \\set{(u,v) : u = v}$. To do so, for every outgoing null curve $\\underline{C}$ in $\\calQ$, follow the incoming null curve to the past starting from $\\underline{C}\\cap \\Gamma$ until the point $p_*$ where it intersects the initial curve $C_1$. We then define the $u$-coordinate value for $\\underline{C}$ to be the $v$-coordinate value for $p_*$. \n\nUnder such coordinate choice, $\\mathcal D(u_{0}, v_{0})$ may be expressed as\n\\begin{equation*}\n\t\\mathcal D(u_{0}, v_{0}) = \\set{(u,v) \\in \\calQ : u \\in [u_{0}, v_{0}], v \\in [u, v_{0}]}.\n\\end{equation*}\n\nMoreover, if $r$ and $\\phi$ are sufficiently regular functions on $\\calQ$, then our coordinate choice leads to $\\lim_{v \\to u+}(\\lmb + \\nu) (u,v) = \\lim_{u \\to v-}(\\lmb + \\nu) (u,v) = 0$ and $\\lim_{v \\to u+} (\\partial_{v} + \\partial_{u}) (r \\phi)(u,v) = \\lim_{u \\to v-} (\\partial_{v} + \\partial_{u}) (r \\phi)(u,v) = 0$. These conditions will be incorporated into precise formulations of solutions to \\eqref{eq:SSESF} with limited regularity in the following subsection.\n\n\\subsection{Characteristic initial value problem}\nIn this paper, we study the characteristic initial value problem for \\eqref{eq:SSESF} with data prescribed on $C_{1}$, under quite general assumptions on the regularity. In this subsection, we give precise formulations of initial data and solutions to \\eqref{eq:SSESF} to be considered in this paper.\n\nWe begin with a discussion on the constraint imposed by \\eqref{eq:SSESF} (more precisely, \\eqref{eq:SSESF:dr}--\\eqref{eq:SSESF:dphi}) on initial data for $(\\phi, r, m)$. In fact, the constraint is very simple, thanks to the fact that they are prescribed on a characteristic (i.e., null) curve $C_{1}$. Once we prescribe $\\phi$ on $C_{1}$, the coordinate condition \\eqref{eq:id4dvr} dictates the initial values of $r$, and the initial values of $m$ are then determined by the constraint \\eqref{eq:SSESF:dm} along the $v$ direction, as well as the boundary condition $m(1, 1) = 0$.  In other words, initial data for $(\\phi, r, m)$ possess only one degree of freedom, namely the prescription of a single real-valued function $\\phi(1, v)$, or equivalently, $\\partial_{v} (r \\phi)(1, v)$.\n\nFollowing Christodoulou \\cite{Christodoulou:1993bt}, we say that an initial data set for $(\\phi, r, m)$ is of \\emph{bounded variation} (BV) if $\\partial_{v}(r \\phi)(1, \\cdot)$ is a (right-continuous) BV function on $[1, \\infty)$ with finite total variation on $(1, \\infty)$. We also define the notion of \\emph{solution of bounded variation} to \\eqref{eq:SSESF} as follows.\n\n\\begin{definition}[Bounded variation solutions to \\eqref{eq:SSESF}] \\label{def:BVsolution}\nA solution $(\\phi, r, m)$ to \\eqref{eq:SSESF} is called a \\emph{solution of bounded variation} on $\\calQ$ if on every compact domain of dependence $\\mathcal D(u_{0}, v_{0})$, the following conditions hold:\n\\begin{enumerate}\n\\item $\\sup_{\\mathcal D(u_{0}, v_{0})} (-\\nu) < \\infty$ and $\\sup_{\\mathcal D(u_{0}, v_{0})} \\lmb^{-1} < \\infty$.\n\\item $\\lmb$ is BV on each $C_{u} \\cap \\mathcal D(u_{0}, v_{0})$ uniformly in $u$, and $\\nu$ is BV on each $\\underline{C}_{v} \\cap \\mathcal D(u_{0}, v_{0})$ uniformly in $v$.\n\\item For each $a$ with $(a, a) \\in \\Gamma$,\n\\begin{equation*}\n\t\\lim_{\\epsilon \\to 0+} (\\nu + \\lmb)(a, a+\\epsilon) = 0.\n\\end{equation*}\n\\item $\\phi$ is an absolutely continuous function on each $C_{u} \\cap \\mathcal D(u_{0}, v_{0})$ with total variation bounded uniformly in $u$, and also an absolutely continuous function on each $\\underline{C}_{v} \\cap \\mathcal D(u_{0}, v_{0})$ with total variation bounded uniformly in $v$.\n\\item For each $a$ with $(a, a) \\in \\Gamma$,\n\\begin{align*}\n\\lim_{\\epsilon \\to 0} \\sup_{0 < \\delta \\leq \\epsilon} \\mathrm{T.V.}_{\\set{a-\\delta} \\times (a-\\delta, a)} [\\phi] =0, \n& \\qquad \\lim_{\\epsilon \\to 0} \\sup_{0 < \\delta \\leq \\epsilon} \\mathrm{T.V.}_{(a-\\epsilon, a-\\delta) \\times \\set{a-\\delta}} [\\phi] =0,  \\\\\n\\lim_{\\epsilon \\to 0} \\sup_{0 < \\delta \\leq \\epsilon} \\mathrm{T.V.}_{(a, a+\\delta) \\times \\set{a+\\delta}} [\\phi] =0, \n& \\qquad \\lim_{\\epsilon \\to 0} \\sup_{0 < \\delta \\leq \\epsilon} \\mathrm{T.V.}_{\\set{a+\\delta} \\times (a+\\delta, a+\\epsilon)} [\\phi] =0.\n\\end{align*}\n\\item $\\partial_{v}(r \\phi)$ is BV on each $C_{u} \\cap \\mathcal D(u_{0}, v_{0})$ uniformly in $u$, and $\\partial_{u}(r \\phi)$ is BV on each $\\underline{C}_{v} \\cap \\mathcal D(u_{0}, v_{0})$ uniformly in $v$.\n\\item For each $a$ with $(a, a) \\in \\Gamma$,\n\\begin{equation*}\n\t\\lim_{\\epsilon \\to 0+} \\big( \\partial_{v}(r \\phi) + \\partial_{u}(r \\phi) \\big) (a, a+\\epsilon) = 0.\n\\end{equation*}\n\\end{enumerate}\n\\end{definition}\n\nWe also consider more regular data and solutions, as follows. We say that an initial data set for $(\\phi, r, m)$ is $C^{1}$ if $\\partial_{v}(r \\phi)(1, \\cdot)$ is $C^{1}$ on $[1, \\infty)$ with $\\sup_{C_{1}} \\abs{\\partial_{v}^{2}(r \\phi)} < \\infty$. In the following definition, we define the corresponding notion of a \\emph{$C^{1}$ solution} to \\eqref{eq:SSESF}.\n\n\\begin{definition}[$C^{1}$ solutions to \\eqref{eq:SSESF}] \\label{def:C1solution}\nA solution $(\\phi, r, m)$ to \\eqref{eq:SSESF} is called a \\emph{$C^{1}$ solution} on $\\calQ$ if the following conditions hold on every compact domain of dependence $\\mathcal D(u_{0}, v_{0})$:\n\\begin{enumerate}\n\\item $\\sup_{\\mathcal D(u_{0}, v_{0})} (-\\nu) < \\infty$ and $\\sup_{\\mathcal D(u_{0}, v_{0})} \\lmb^{-1} < \\infty$.\n\\item $\\lmb$, $\\nu$ are $C^{1}$ on $\\mathcal D(u_{0}, v_{0})$.\n\\item For each $a$ with $(a, a) \\in \\Gamma$,\n\\begin{equation*}\n\t\\lim_{\\epsilon \\to 0+} (\\nu + \\lmb)(a, a+\\epsilon) = \\lim_{\\epsilon \\to 0+} (\\nu + \\lmb)(a-\\epsilon, a) = 0.\n\\end{equation*}\n\\item $\\partial_{v}(r \\phi)$ and $\\partial_{u} (r \\phi)$ are $C^{1}$ on $\\mathcal D(u_{0}, v_{0})$.\n\\item For each $a$ with $(a, a) \\in \\Gamma$,\n\\begin{equation*}\n\t\\lim_{\\epsilon \\to 0+} \\big( \\partial_{v}(r \\phi) + \\partial_{u}(r \\phi) \\big) (a, a+\\epsilon) \n\t= \\lim_{\\epsilon \\to 0+} \\big( \\partial_{v} (r \\phi) + \\partial_{v}(r \\phi) \\big) (a - \\epsilon, a)\n\t= 0.\n\\end{equation*}\n\\end{enumerate}\n\\end{definition}\n\n\\begin{remark} \\label{rem:wp}\nBy \\cite[Theorem 6.3]{Christodoulou:1993bt}, a BV initial data set leads to a unique BV solution to \\eqref{eq:SSESF} on $\\set{(u,v) : 1 \\leq u \\leq 1+\\delta, v \\geq u}$ for some $\\delta > 0$.\nIf the initial data set is furthermore $C^{1}$, then it is not difficult to see that the corresponding solution is also $C^{1}$ (persistence of regularity). In fact, this statement follows from the arguments in Section \\ref{sec.decay2} of this paper; see, in particular, the proof of Lemma \\ref{lem:decay2:key4nullStr}.\n\\end{remark}\n\n\n\\subsection{Local scattering in BV and asymptotic flatness}\nWe are now ready to formulate the precise notion of \\emph{locally BV scattering solutions} to \\eqref{eq:SSESF}, which is the class of solutions that we consider in this paper. In particular, for this class of solutions, we make a priori assumptions on its global geometry. \n\n\\begin{definition}[Local scattering in BV] \\label{def:locBVScat}\nWe say that a BV solution $(\\phi, r, m)$ to \\eqref{eq:SSESF} is \\emph{locally scattering in the bounded variation norm (BV)}, or a \\emph{locally BV scattering solution}, if the following conditions hold:\n\\begin{enumerate}\n\\item \\emph{Future completeness of radial null geodesics}: Every incoming null geodesic in $\\calQ$ has its future endpoint on $\\Gamma$, and every outgoing null geodesic in $\\calQ$ is infinite towards the future  in the affine parameter. \nMoreover, there exists a global system of null coordinates $(u,v)$ and $\\calQ$ is given by\n\\begin{equation} \\label{eq:globalCoords}\n\t\\calQ = \\set{(u,v) : u \\in [1, \\infty), v \\in [u, \\infty)}.\n\\end{equation}\n\n\\item \\emph{Vanishing final Bondi mass}: The final Bondi mass vanishes, i.e.,\n\\begin{equation} \\label{eq:zeroMf}\nM_{f} := \\lim_{u \\to \\infty} M(u) = 0.\n\\end{equation}\n\n\\item \\emph{Scattering in BV in a compact $r$-region}: There exists $R > 0$ such that for $\\PD_{\\mathrm{cpt}}$ defined to be the region $\\set{(u,v) \\in \\calQ : r(u,v) \\leq R}$, we have \n\\begin{equation} \\label{eq:locBVScat}\n\t\\int_{C_{u} \\cap \\PD_{\\mathrm{cpt}}} \\abs{\\partial_{v}^{2} (r \\phi)} \\to 0, \\quad\n\t\\int_{C_{u} \\cap \\PD_{\\mathrm{cpt}}} \\abs{\\partial_{v} \\log \\lmb} \\to 0\n\\end{equation}\nas $u \\to \\infty$.\n\\end{enumerate}\n\\end{definition}\n\nSeveral remarks concerning Definition~\\ref{def:locBVScat} are in order.\n\\begin{remark} \nIn fact, the condition \\eqref{eq:globalCoords} is a consequence of future completeness of radial null geodesics and the preceding assumptions. To see this, first recall our assumption that $C_{1} = \\set{(u,v) : u = 1, v \\in [1, \\infty)}$. Hence from our choice of the coordinate $u$ and future completeness of incoming radial null geodesics, it follows that the range of $u$ must be $[1, \\infty)$. Furthermore, for each $u \\in [1, \\infty)$, the range of $v$ on $C_{u}$ is $[u, \\infty)$ by future completeness of outgoing radial null geodesics and Definition~\\ref{def:BVsolution}. More precisely, future completeness of $C_{u}$ implies that it can be continued past $\\set{u} \\times [u, v_{0}]$ as long as $\\int_{u}^{v_{0}} \\Omega^{2} \\, \\mathrm{d} v < \\infty$, and Definition~\\ref{def:BVsolution} implies\\footnote{We refer to the proof of Proposition~\\ref{prop:geomLocBVScat} below for details of estimating $\\frac{-\\nu}{1-\\mu}$ in terms of assumptions on $\\phi$, $\\partial_{v}(r \\phi)$ and $\\lmb$.} that $\\Omega^{2} = - \\frac{4 \\lambda \\nu}{1-\\mu}$ indeed remains bounded on $\\set{u} \\times [u, v_{0}]$ for every finite $v_{0}$.\n\\end{remark}\n\n\\begin{remark}  \\label{rem:FCGC}\nFor more regular (e.g., $C^{1}$) asymptotically flat solutions, the conditions $(1)$ and $(2)$ in Definition \\ref{def:locBVScat} may be replaced by a single equivalent condition, namely requiring the full spacetime $(\\calM, \\bfg)$ to be \\emph{future casually geodesically complete} as a (1+3)-dimensional Lorentzian manifold. In particular, (2) follows from the deep work \\cite{Christodoulou:1987ta} of Christodoulou, in which it was proved that if $M_{f} > 0$ then the space-time necessarily contains a black-hole and thus is not future causally geodesically complete.\n\\end{remark}\n\n\\begin{remark}\\label{rmk.unif.int}\nAs we will see in the proof, there exists a universal $\\tilde{\\epsilon}_0$ such that (3) in Definition \\ref{def:locBVScat} can be replaced by the weaker requirement that there exists $R>0$ and $U>0$ such that\n\\begin{equation*}\n\t\\int_{C_{u} \\cap \\PD_{\\mathrm{cpt}}} \\abs{\\partial_{v}^{2} (r \\phi)} \\leq \\tilde{\\epsilon}_0, \\quad\n\t\\int_{C_{u} \\cap \\PD_{\\mathrm{cpt}}} \\abs{\\partial_{v} \\log \\lmb} \\leq \\tilde{\\epsilon}_0\n\\end{equation*}\nfor $u\\geq U$. To simplify the exposition, we will omit the proof of this improvement.\n\\end{remark}\n\n\\begin{remark} \nFor a sufficiently regular, asymptotically flat solution to \\eqref{eq:SSESF}, Condition $(1)$ in Definition \\ref{def:locBVScat} is equivalent to requiring that the conformal compactification of $\\calQ$ is depicted by a Penrose diagram as in Figure \\ref{fig.disp} (in the introduction). For more discussion on Penrose diagrams, we refer the reader to \\cite[Appendix C]{DR} and \\cite{Kommemi}. In fact, this equivalence follows easily from the classification of all possible Penrose diagrams for the system \\eqref{eq:SSESF} given in the latter reference.\n\\end{remark}\n\n\nWe also define the precise notion of \\emph{asymptotic flatness} for initial data with BV or $C^{1}$ regularity. As we shall see soon in the main theorems, the rate of decay for the initial data, measured in $r$, is directly related to the rate of decay of the corresponding solution, in both $u$ and $r$.\n\n\\begin{definition}[Asymptotic flatness of order $\\omega'$ in BV or $C^{1}$] \\label{def:AF}\nFor $\\omega' >1$, we make the following definition.\n\n\\begin{enumerate}\n\\item We say that an initial data set is \\emph{asymptotically flat of order $\\omega'$ in BV} if $\\partial_{v} (r \\phi)(1, \\cdot) \\in \\mathrm{BV}[1, \\infty)$ and there exists $\\mathcal I_{1} > 0$ such that \n\\begin{equation}\n\t\\sup_{C_{1}} (1+r)^{\\omega'} \\abs{\\partial_{v}(r \\phi)} \\leq \\mathcal I_{1} < \\infty.\n\\end{equation}\n\n\\item We say that an initial data set is \\emph{asymptotically flat of order $\\omega'$ in $C^{1}$} if $\\partial_{v} (r \\phi)(1, \\cdot) \\in C^{1}[1, \\infty)$ and there exist $\\mathcal I_{2} > 0$ such that \n\\begin{equation}\n\t\\sup_{C_{1}} (1+r)^{\\omega'} \\abs{\\partial_{v}(r \\phi)} + \\sup_{C_{1}} (1+r)^{\\omega'+1} \\abs{\\partial_{v}^{2}(r \\phi)} \\leq \\mathcal I_{2} < \\infty.\n\\end{equation}\n\\end{enumerate}\n\\end{definition}\n\n\\begin{remark} \nThe initial Bondi mass $M_{i} := \\lim_{v \\to \\infty} m(1, v)$ can be easily bounded by $\\leq C \\mathcal I_{1}^{2}$; see Lemma \\ref{lem:bnd4Mi}. \n\\end{remark}\n\n\\begin{remark} \\label{rem:PhiIsZero}\nObserve that both conditions imply that $(r \\phi)(1,v)$ tends to a finite limit as $v \\to \\infty$; in particular, $\\lim_{v \\to \\infty} \\phi(1, v) = 0.$\nThis serves to fix the gauge freedom $(\\phi, r, m) \\mapsto (\\phi + c, r, m)$ for solutions to \\eqref{eq:SSESF}.\n\\end{remark}\n\n\\section{Main results}\\label{sec.main.thm}\n\\subsection{Main theorems}\nWith the definitions of locally BV scattering solutions and asymptotically flat initial data, we now have the necessary means to state the main theorems of this paper. Roughly speaking, these theorems say that locally BV scattering solutions with asymptotically flat initial data exhibits quantitative decay rates, which can be read off from the rate $\\omega'$ in Definition \\ref{def:AF}. The first theorem is for initial data and solutions in BV.\n\n\\begin{theorem}[Main Theorem in BV] \\label{main.thm.1}\nLet $(\\phi, r, m)$ be a locally BV scattering solution to \\eqref{eq:SSESF} with asymptotically flat initial data of order $\\omega'$ in BV. Then for $\\omega := \\min \\set{\\omega', 3}$, there exists a constant $A_{1} > 0$ such that \n\\begin{align} \n\t\\abs{\\phi} \\leq & A_{1} \\min \\set{u^{-\\omega}, r^{-1} u^{-(\\omega-1)}}, \\label{eq:decay1:1} \\\\\n\t\\abs{\\partial_{v}(r \\phi)} \\leq & A_{1} \\min \\set{u^{-\\omega}, r^{-\\omega}}, \\label{eq:decay1:2} \\\\\n\t\\abs{\\partial_{u} (r \\phi)} \\leq & A_{1} u^{-\\omega}. \\label{eq:decay1:3}\n\\end{align}\n\\end{theorem}\n\nThe second theorem is for initial data and solutions in $C^{1}$. \n\n\\begin{theorem}[Main Theorem in $C^{1}$] \\label{main.thm.2}\nLet $(\\phi, r, m)$ be a locally BV scattering solution to \\eqref{eq:SSESF} with asymptotically flat initial data of order $\\omega'$ in $C^{1}$. Then, in addition to the bounds \\eqref{eq:decay1:1}-\\eqref{eq:decay1:3}, there exists a constant $A_{2} > 0$ such that \n\\begin{align} \n\t\\abs{\\partial_{v}^{2} (r \\phi)} \\leq & A_{2} \\min \\set{u^{-(\\omega+1)}, r^{-(\\omega+1)}}, \\label{eq:decay2:1} \\\\\n\t\\abs{\\partial_{u}^{2} (r \\phi)} \\leq & A_{2} u^{-(\\omega+1)}, \\label{eq:decay2:2} \\\\\n\t\\abs{\\partial_{v} \\lmb} \\leq & A_{2} \\min \\set{u^{-3}, r^{-3}}, \\label{eq:decay2:3} \\\\\n\t\\abs{\\partial_{u} \\nu} \\leq & A_{2} u^{-3}. \\label{eq:decay2:4}\n\\end{align}\nfor $\\omega := \\min \\set{\\omega', 3}$.\n\\end{theorem}\n\nSome remarks regarding the main theorems are in order.\n\n\\begin{remark}\nNotice that in Theorem \\ref{main.thm.2}, the decay rates for $\\partial_v\\lambda$ and $\\partial_u\\nu$ are independent of the order $\\omega'$ of asymptotic flatness of the initial data. This is because the scalar field terms enter the equations for $\\partial_u\\partial_v\\log\\lambda$ and $\\partial_v\\partial_u\\log\\nu$ quadratically (see equations \\eqref{eq:eq4dvdvr:normal} and \\eqref{eq:eq4dudur:normal}) and thus as long as $\\omega'>1$, their contributions to the decay rates of $\\partial_v\\lambda$ and $\\partial_u\\nu$ are lower order compared to the term involving the Hawking mass.\n\\end{remark}\n\n\\begin{remark}\nBy Remark \\ref{rem:wp}, a $C^{1}$ initial data set gives rise to a $C^{1}$ solution. Hence Remark \\ref{rem:FCGC} applies, and the conditions (1)--(2) of Definition \\ref{def:locBVScat} may be replaced by a single equivalent condition of \\emph{future causal geodesic completeness} of $(\\calM, {\\bf g})$ in the case of Theorem \\ref{main.thm.2}. \n\\end{remark}\n\n\\begin{remark}\nIn general, the constants $A_1$ and $A_2$ depend not only on the size of the initial data (e.g., $\\mathcal I_{1}$, $\\mathcal I_{2}$), but rather on the full profile of the solution. Nevertheless, for the special case of small initial total variation of $\\partial_{v}(r \\phi)$, $A_{1}$ and $A_{2}$ \\emph{do} depend only on the size of the initial data; see \\S \\ref{sec:mainThm:smallData}.\n\\end{remark}\n\n\\begin{remark}\nIf the initial data also verify higher derivative estimates, then the techniques in proving Theorems \\ref{main.thm.1} and \\ref{main.thm.2} also allow us to derive decay bounds for higher order derivatives. The proof of the higher derivative decay estimates is in fact easier than the proofs of the first and second derivative decay bounds since we have already obtained sufficiently strong control of the scalar field and the geometry of the spacetime. We will omit the details.\n\\end{remark}\n\n\\begin{remark}\nThe decay rates that we obtain in these variables imply as immediate corollaries decay rates for $\\partial_v \\phi$, $\\partial_u \\phi$, etc. See Corollaries \\ref{cor:decay1} and \\ref{cor:decay2}.\n\\end{remark}\n\n\\begin{remark} \\label{rem:coord}\nThe decay rates in the main theorems are measured with respect to the double null coordinates $(u,v)$ normalized at the initial curve and the axis $\\Gamma$ as in \\S \\ref{subsec:coordSys}. To measure the decay rate along null infinity, one can alternatively normalize the $u$ coordinate\\footnote{In particular, this normalization is used in \\cite{DR} for the black hole case. By changing the null coordinates, we can thus more easily compare the decay rates in our setting and that in \\cite{DR}.} by requiring $\\partial_{u} r=-\\frac 12$ at future null infinity. As we will show in \\S \\ref{sec.coord}, for the class of spacetimes considered in this paper, the decay rates with respect to this new system of null coordinates are the same up to a constant multiplicative factor.\n\\end{remark}\n\n\\begin{remark}\nIn view of Remark \\ref{rmk.unif.int}, the assumption of local BV scattering can be replaced by the \\emph{boundedness} of the \\emph{subcritical} quantities\n$$\\int_{C_{u} \\cap \\PD_{\\mathrm{cpt}}} \\abs{\\partial_{v}^{2} (r \\phi)}^p \\leq C, \\quad\n\t\\int_{C_{u} \\cap \\PD_{\\mathrm{cpt}}} \\abs{\\partial_{v} \\log \\lmb}^p \\leq C,\\quad\\mbox{for }p>1.$$\nThis is because for fixed $\\tilde{\\epsilon}_0$, one can choose $R$ to be sufficiently small (depending on $C$) and apply H\\\"older's inequality to ensure\tthat\n$$\\int_{C_{u} \\cap \\PD_{\\mathrm{cpt}}} \\abs{\\partial_{v}^{2} (r \\phi)} \\leq \\tilde{\\epsilon}_0, \\quad\n\t\\int_{C_{u} \\cap \\PD_{\\mathrm{cpt}}} \\abs{\\partial_{v} \\log \\lmb} \\leq \\tilde{\\epsilon}_0.$$\n\\end{remark}\n\n\\begin{remark}\nWe also notice that the proof of our main theorem can be carried out in an identical manner for locally BV scattering solutions arising from asymptotically flat \\emph{Cauchy data}. More precisely, we can consider initial data given on a Cauchy hypersurface\n$$v=f(u),\\quad\\mbox{with }C^{-1}\\leq -f'(u)\\leq C$$\nand satisfying the constraint equation together with the following bounds on the initial data:\n$$(1+r)^{\\omega'}(|\\phi|+|\\partial_v(r\\phi)|+|\\partial_u(r\\phi)|+|\\lambda-\\frac 12|+|\\nu+\\frac 12|)\\leq \\tilde{\\mathcal I}_1,$$\nand\n$$(1+r)^{\\omega'+1}(|\\partial_v^2(r\\phi)|+|\\partial_u^2(r\\phi)|+|\\partial_v \\log\\lambda|+|\\partial_u\\log\\nu|)\\leq \\tilde{\\mathcal I}_2.$$ \nThen, if we assume in addition that the spacetime is locally BV scattering to the future, the conclusions of Theorems \\ref{main.thm.1} and \\ref{main.thm.2} hold.\n\\end{remark}\n\n\\begin{remark}\nOur main theorems can also be viewed as results on upgrading qualitative decay to quantitative decay estimates. Such problems have been widely studied in the \\emph{linear} setting (without the assumption on spherical symmetry) on nontrapping asymptotically flat Lorentzian manifolds \\cite{DR2}, \\cite{Ta}, \\cite{MTT}, as well as for the obstacle problem on Minkowski space \\cite{Morawetz}, \\cite{Strauss}. In the \\emph{nonlinear} setting, we mention the work of Christodoulou-Tahvildar--Zadeh \\cite{CTZ}, who studied the energy critical 2-dimensional spherically symmetric wave map system and proved asymptotic decay for the solution and its derivatives.\n\\end{remark}\n\n\\subsection{BV scattering and the blow-up at infinity scenerio}\n\nThe condition of local BV scattering in the main theorems follows if one rules out, a priori, a blow-up at infinity scenario. More precisely, we say that a solution blows up at infinity if some scale-invariant spacetime norms are infinite as follows:\n\n\\begin{definition}\\label{def.blow.up.infty}\nLet $(\\phi, r, m)$ be a BV solution to \\eqref{eq:SSESF} such that the condition $(1)$ of Definition \\ref{def:locBVScat} (future completeness of radial null geodesics) holds. We say that the solution \\emph{blows up at infinity} if at least one of the following holds:\n\\begin{enumerate}\n\\item $\\displaystyle{\\sup \\lambda_{\\Gamma}^{-1} = \\infty}$, where $\\displaystyle{\\lambda_{\\Gamma}(u) := \\lim_{v \\to u+} \\lmb(u,v)}$.\n\\item $\\displaystyle{\\int_1^{\\infty}\\int_u^{\\infty}|\\partial_v\\lambda\\partial_u\\phi-\\partial_u\\lambda\\partial_v\\phi| \\mathrm{d} v \\mathrm{d} u =\\infty}$,\n\\item $\\displaystyle{\\int_1^{\\infty}\\int_u^{\\infty}|\\partial_u\\phi\\partial_v(\\nu^{-1}\\partial_u(r\\phi))-\\partial_v\\phi\\partial_u(\\nu^{-1}\\partial_u(r\\phi))| \\mathrm{d} v \\mathrm{d} u =\\infty}$.\n\\end{enumerate}\n\\end{definition} \n\n\\begin{remark}\nWe do not prove in the paper the existence or non-existence of solutions that blow up at infinity. We remark that this is analogous to the blow-up at infinity scenarios which have recently been constructed in some simpler \\emph{semilinear}, \\emph{critical} wave equations \\cite{DK}.\n\\end{remark}\n\nIt follows from our main theorem that if a solution does not blow up at infinity, it obeys quantitative decay estimates. More precisely, we have\n\\begin{theorem}[Dichotomy between blow-up at infinity and BV scattering] \\label{thm.dichotomy}\nLet $(\\phi, r, m)$ be a BV solution to \\eqref{eq:SSESF} such that the condition $(1)$ of Definition \\ref{def:locBVScat} (future completeness of radial null geodesics) holds. Assume furthermore that the initial data for $(\\phi, r, m)$ obey the condition\\footnote{By Remark \\ref{rem:PhiIsZero}, note that this is the only condition on $\\lim_{v \\to \\infty} \\phi(1, v)$ which is consistent with asymptotic flatness.} $\\lim_{v \\to \\infty} \\phi(1,v) = 0$ and\n\\begin{equation} \\label{dichotomy.hyp}\n\\int_{C_{1}} \\abs{\\partial_{v}^{2} (r \\phi)} \\, \\mathrm{d} v + \\sup_{C_{1}} \\abs{\\partial_{v}(r \\phi)} < \\infty.\n\\end{equation}\n\nThen either\n\\begin{enumerate}\n\\item the solution blows up at infinity; or\n\\item the solution is globally BV scattering, in the sense that the conditions $(2)$ and $(3)$ of Definition \\ref{def:locBVScat} hold with $R =\\infty$.\n\\end{enumerate}\n\\end{theorem}\n\nThis theorem is established in Section \\ref{sec.dichotomy}. It then follows from our main theorems (Theorems \\ref{main.thm.1} and \\ref{main.thm.2}) that if a BV solution does not blow up at infinity and possesses asymptotically flat initial data, then it obeys quantitative decay estimates.\n\n\\subsection{Refinement in the small data in BV case} \\label{sec:mainThm:smallData}\nBy a theorem of Christodoulou \\cite{Christodoulou:1993bt}, the maximal development of data with small BV norms does not blow up at infinity. The previous theorem applies, and thus the corresponding solution is globally BV scattering, in the sense described in Theorem \\ref{thm.dichotomy}. Moreover, a closer inspection of the proof of the main theorems reveals that the following stronger conclusion holds in this case.\n\n\\begin{theorem} [Sharp decay for data with small BV norm] \\label{thm:smallData}\nThere exists a universal $\\epsilon_{0} > 0$ such that for $0 < \\epsilon \\leq \\epsilon_{0}$, the following statements hold.\n\n\\begin{enumerate}\n\\item If the initial data set is asymptotically flat of order $\\omega'$ in BV and\n\\begin{equation*}\n\t\\int_{C_{1}} \\abs{\\partial_{v}^{2} (r \\phi)} < \\epsilon,\n\\end{equation*}\nthen the maximal development $(\\phi, r, m)$ is globally BV scattering, in the sense that Definition \\ref{def:locBVScat} holds with arbitrarily large $R > 0$. Moreover, it satisfies the estimates \\eqref{eq:decay1:1}--\\eqref{eq:decay1:3} with $A_{1} \\leq C_{\\mathcal I_{1}} (\\mathcal I_{1} + \\epsilon)$.\n\n\\noindent Here (and similarly in (2)), we use the convention that $C_{\\mathcal I_{1}}$ depends on $\\mathcal I_1$ in a non-decreasing fashion.\\footnote{In particular, for $\\mathcal I_1$ sufficiently small, we have the estimate $A_1\\leq C(\\mathcal I_1+\\epsilon)$ for some absolute constant $C$.}\n\n\\item If, in addition, the initial data set is asymptotically flat of order $\\omega'$ in $C^{1}$, then the maximal development also satisfies \\eqref{eq:decay2:1}--\\eqref{eq:decay2:4} with $A_{2} \\leq C_{\\mathcal I_{2}} (\\mathcal I_{2} + \\epsilon)$.\n\\end{enumerate}\n\\end{theorem}\n\nThe point of this theorem is that we only need to know that the initial total variation to be small in order to conclude pointwise decay rates; in particular, $\\mathcal I_{1}$, $\\mathcal I_{2}$ can be arbitrarily large. In this sense, Theorem \\ref{thm:smallData} generalizes both the small BV global well-posedness theorem \\cite[Theorem 6.2]{Christodoulou:1993bt} and the earlier small data scattering theorem \\cite{Christodoulou:1986ue} for data that are small in a weighted $C^1$ norm. A proof of this theorem will be sketched in Section \\ref{sec:smallData}.\n\n\\subsection{Optimality of the decay rates}\n\nOur main theorems show upper bounds for the decay rates of the scalar field $\\phi$ and its derivatives both towards null infinity (i.e., in $r$) and along null infinity (i.e., in $u$). For $\\omega' = \\omega<3$, if the decay rate of the initial data towards null infinity satisfies also a lower bound, then we can show that both the $r$ and $u$ decay rates in Theorem \\ref{main.thm.1} are saturated. More precisely,\n\n\\begin{theorem} [Sharpness of $t^{-\\omega}$ tail for $1 < \\omega < 3$] \\label{thm.opt.1}\nLet $1 < \\omega < 3$. Suppose, in addition to the assumptions of Theorem \\ref{main.thm.1}, that there exists $V\\geq 1$ such that the initial data set satisfies the lower bound\n\\begin{equation*}\n\tr^{\\omega} \\partial_{v}(r \\phi) (1, v) \\geq L > 0,\n\\end{equation*}\nfor $v\\geq V$.\n\nThen there exists a constant $L_{\\omega} >0$ such that\n\\begin{align*}\n\t\\partial_{v} (r \\phi)(u, v) \\geq & L_{\\omega} \\min\\set{r^{-\\omega}, u^{-\\omega}}, \\\\\n\t- \\partial_{u} (r \\phi)(u, v) \\geq & L_{\\omega} u^{-\\omega},\n\\end{align*}\nfor $u$ sufficiently large.\n\\end{theorem}\n\\begin{remark}\nOne can also infer the sharpness of the decay of $\\phi$ from that of its derivatives. We will omit the details.\n\\end{remark}\n\n\nThis theorem will be proved in \\S \\ref{subsec.opt.1}. In fact, the proof of this theorem is similar to the proof of the upper bounds in the first main theorem (Theorem \\ref{main.thm.1}). We will show that after restricting to $u$ sufficiently large, the initial lower bound propagates and the nonlinear terms only give lower order contributions. Notice also that the analogous statement is false for $\\omega'\\geq 3$, since the nonlinear terms may dominate the contribution of the initial data.\n\nFor $\\omega' \\geq 3$, we can show that the decay rates in Theorem \\ref{main.thm.1} are sharp in the following sense:\n\\begin{theorem} [Sharpness of $t^{-3}$ tail] \\label{thm.opt.2}\nFor arbitrarily small $\\epsilon > 0$, there exists a locally BV scattering solution $(\\phi, r, m)$ to \\eqref{eq:SSESF} which satisfies the following properties:\n\\begin{enumerate}\n\\item $\\partial_{v} (r \\phi)(1, v)$ is smooth, compactly supported in the $v$-variable and has total variation less than $\\epsilon$, i.e.,\n\\begin{equation*}\n\t\\int_{C_{1}} \\abs{\\partial_{v}^{2} (r \\phi)} < \\epsilon.\n\\end{equation*}\n\\item There exists a constant $L_{3} > 0$ such that\n\\begin{align*}\n\t\\partial_{v} (r \\phi) (u, v) \\geq & L_{3} \\min \\set{r^{-3}, u^{-3}}, \\\\\n\t- \\partial_{u} (r \\phi) (u, v) \\geq & L_{3} u^{-3},\n\\end{align*}\nfor $u$ sufficiently large.\n\\end{enumerate}\n\\end{theorem}\n\n\nTo prove Theorem \\ref{thm.opt.2}, we will first establish a sufficient condition for the desired lower bounds in terms of (non-vanishing of) a single real number $\\mathfrak{L}$, which is computed from information at the null infinity. This result (Lemma \\ref{lem:LB}) is proved using the decay rates proved in the main theorems, and we believe it might be of independent interest. In \\S \\ref{subsec.opt.2}, we will complete the proof of Theorem \\ref{thm.opt.2} by constructing an initial data set for which $\\mathfrak{L}$ can be bounded away from zero. This can be achieved by showing that the solution is close to that of a corresponding linear problem and controlling the error terms after taking $\\epsilon > 0$ to be sufficiently small and using Theorem \\ref{thm:smallData}.\n\n\\subsection{Strategy of the proof of the main theorems}\n\nRoughly speaking, the proof of decay of $\\phi$ and its derivatives can be split into three steps. In the first two steps, we control the incoming part\\footnote{We call these variables `incoming' because they obey a transport equation in the $\\partial_u$ direction.} of the derivatives of the scalar field and metric components, i.e., $\\partial_v(r\\phi)$, $\\partial_v^2(r\\phi)$ and $\\partial_v\\lmb$. To this end, we split the spacetime into the exterior region $\\PD_{\\mathrm{ext}}:=\\{(u,v)\\in \\mathcal Q: v\\geq 3u\\}$ and the interior region $\\PD_{\\mathrm{int}}:=\\{(u,v)\\in \\mathcal Q: v\\leq 3u\\}$. In the first step, we control the incoming part of the solution in the exterior region. In this region, we have $r \\gtrsim v, u$, thus the negative $r$ weights in the equations give the required decay of $\\phi$ and its derivatives. We then prove bounds in the interior region in the second step. Here, we exploit certain (non-quantitative) smallness in the spacetimes quantities as $u \\to \\infty$ given by the assumption of local BV scattering to propagate the decay estimates from the exterior region to the interior region all the way up to the axis. Finally, in the third step, we control the outgoing part of the solution, i.e., $\\partial_u(r\\phi)$, $\\partial_u^2(r\\phi)$ and $\\partial_u\\nu$, by showing that the decay bounds that we have proved along the axis can be propagated in the outgoing direction.\n\nWe remind the readers that the above sketch is only a heuristic argument and is not true if taken literally. In particular, in order to carry out this procedue, we need to first show that the local BV scattering assumption provides some control over the spacetime geometry. As we will show below, the estimates are derived in slightly different fashions for the first and the second derivatives of $r\\phi$. We note in particular that carrying out this general scheme relies heavily on the analytic structure of the Einstein-scalar field equations, including the montonicity properties as well as the null structure of the (renormalized) equations.\n\n\\subsubsection{Estimates for first derivatives of $r\\phi$}\n\nTo obtain decay bounds for the first derivatives of $r\\phi$, we will rely on the wave equation\n$$\\partial_u \\partial_v(r\\phi)=\\frac{2 m \\lambda \\nu}{(1-\\mu)r^2}\\phi.$$\nNotice that when we solve for the incoming radiation $\\partial_v(r\\phi)$ using this as a transport equation in $u$, the right hand side does not depend explicitly on the outgoing radiation $\\partial_u(r\\phi)$. Instead, the right hand side consists of terms that are either lower order (in terms of derivatives) or satisfy a certain monotonicity property.\n\nIn particular, this equations shows that as long as $\\phi$ can be controlled, we can estimate $\\partial_v(r\\phi)$ by integrating along the incoming $u$ direction. On the other hand, we can also control $\\phi$ once a bound on $\\partial_v(r\\phi)$ is known by integrating along the outgoing $v$ direction.\n\nTo achieve the desired decay rates for $\\phi$, $\\partial_v(r\\phi)$ and $\\partial_u(r\\phi)$, we follow the three steps outlined above:\n\n\\begin{enumerate}\n\\item [(1)] Bounds\\footnote{The estimates in this region are similar to the corresponding bounds for the black hole case in \\cite{DR}. There, it was observed that the quantity $\\partial_v(r \\phi)$, which Dafermos-Rodnianski called an almost Riemann invariant, verifies an equation such that the right hand side has useful weights in $r$ and give the desired decay rates.} for $\\partial_v(r\\phi)$ and $\\phi$ in $v\\geq 3u$: In the exterior region, we have $r\\gtrsim u,v$, it is therefore sufficient to prove the decay in $r$. First, we  prove that $\\sup_{C_u}(1+r)\\phi$ is bounded. This is achieved in a compact region by continuity of the solution\\footnote{In particular, since we are simply using compactness, the constants in Theorem \\ref{main.thm.1} depend not only on the size of the initial data.} and in the region of large $r$ by integrating $\\partial_v(r\\phi)$ in the outgoing direction from the compact region. Since $\\partial_v(r\\phi)$ can in turn be controlled by $\\phi$, we get the desired bound. To improve over this bound we define\n\\begin{equation*}\n\\mathcal B_{1}(U) := \\sup_{u \\in [1, U]} \\sup_{C_{u}} \\Big( u^{\\omega} \\abs{\\phi} + r u^{\\omega-1} \\abs{\\phi} \\Big)\n\\end{equation*}\nand show via the wave equation that\n$$r^\\omega|\\partial_v(r\\phi)|\\leq C(u_1)+\\epsilon(u_1)\\mathcal{B}_1(U),$$\nwhere $\\epsilon\\to 0$ as $u_1\\to \\infty$. This gives the optimal decay rate for $\\partial_v(r\\phi)$ in the exterior region up to an arbitrarily small loss, which can be estimated once $\\mathcal{B}_1(U)$ can be controlled.\n\n\\item [(2)] Bounds for $\\partial_v(r\\phi)$ and $\\phi$ in $v\\leq 3u$: For the decay of the first derivatives, the interior region $\\{v\\leq 3u\\}$ is further divided into the intermediate region $\\{r\\geq R\\}$ and the compact region $\\{r\\leq R\\}$. In these two regions, the $r$ weight in the equation is not sufficient to give the sharp decay rate. Instead, we start from the decay rate $\\partial_v(r\\phi)$ obtained in the first step in the exterior region and propagate this decay estimate inwards. To achieve this, we need to show that $\\int \\frac{2 m \\lambda \\nu}{(1-\\mu)r^2}$ is small when $u$ is sufficiently large. \n\n\\item [(2a)] $r\\geq R$ and $v\\leq 3u$: In the intermediate region where we still have a lower bound on $r$, the required smallness is given by the \\emph{qualitative} information that the Hawking mass approaches $0$. Thus, from some large time onwards, $\\int \\frac{2 m \\lambda \\nu}{(1-\\mu)r^2}$ becomes sufficiently small and we can integrate the wave equation directly to obtain the desired decay bounds.\n\n\\item [(2b)] $r\\leq R$ and $v\\leq 3u$: In this region, we use the local BV scattering assumption to show that $\\int_{\\{r\\leq R\\}} \\frac{2 m \\lambda \\nu}{(1-\\mu)r^2}\\to 0$ as $u\\to\\infty$. This smallness allows us to propagate the decay estimates from the curve $r=R$ to the region $r< R$. At this point, we can also recover the control for $\\mathcal{B}_1(U)$ and close the estimates in step 1. This allows us to derive all the optimal decay rates for $\\phi$ and $\\partial_v(r\\phi)$\n\n\\item [(3)] Bounds for $\\partial_u(r\\phi)$: To achieve the bounds for $\\partial_u(r\\phi)$, first note that along the axis we have $\\partial_u(r\\phi)=-\\partial_v(r\\phi)$. Thus, by the previous derived control for $\\partial_v(r\\phi)$, we also have the decay of $\\partial_u(r\\phi)$ along the axis. We then consider the wave equation as a transport equation in the outgoing direction for $\\partial_u(r\\phi)$ to obtain the sharp decay for $\\partial_u(r\\phi)$ in the whole spacetime. \n\n\\end{enumerate}\n\n\\subsubsection{Estimates for second derivatives of $r\\phi$}\n\nAs for the first derivatives, we control the second derivatives by first integrating the equation in the exterior region up to a curve $v=3u$. We then propagate the decay bounds from the exterior region to the interior region using the estimates already derived for the first derivative of $\\phi$, as well as the local BV scattering assumption. However, at this level of derivatives, some new difficulties arise as we now describe.\n\\\\\n\\fparagraph{Renormalization and the null structure}\nThe assumption of local BV scattering implies that\n\\begin{eqnarray}\n\\int_{C_u\\cap\\{r\\leq R\\}} (|\\partial_v\\phi|+|\\partial_v^2(r\\phi)|)\\to 0 \\label{BV.small.1}\n\\end{eqnarray}\nas $u\\to \\infty$. When combined with Christodoulou's BV theory, this also implies that as $v\\to \\infty$, we have\n\\begin{eqnarray}\n\\int_{\\underline{C}_v\\cap\\{r\\leq R\\}} (|\\partial_u\\phi|+|\\partial_u^2(r\\phi)|) \\to 0.  \\label{BV.small.2}\n\\end{eqnarray}\nNotice that on $C_u$ (resp. $\\underline{C}_v$), we only control the integral of $\\partial_v^2(r\\phi)$ and $\\partial_v\\phi$ (resp. $\\partial_u^2(r\\phi)$ and $\\partial_u\\phi$).\n\nSuppose when integrating along the incoming direction to control $\\partial_v^2(r\\phi)$ and $\\partial_v\\lmb$, we need to estimate terms of the form\n$$\\int_{\\underline{C}_v\\cap\\{r\\leq R\\}} |\\partial_u\\phi \\partial_v\\phi|.$$\nWe can apply the BV theory to show that for $v$ sufficiently large,\n$$\\int_{\\underline{C}_v\\cap\\{r\\leq R\\}} |\\partial_u\\phi| \\leq \\epsilon.$$\nOn the other hand, one can show that\n$$\\sup_{\\underline{C}_v\\cap\\{r\\leq R\\}} |\\partial_v\\phi|\\leq C \\sup_{J^-(\\underline{C}_v\\cap\\PD_{\\mathrm{cpt}})} |\\partial_v^2(r\\phi)|$$\nwhich can be controlled by the quantity that we are estimating.\n\nHowever, in equation \\eqref{eq:eq4dvdvrphi:normal} for $\\partial_v^2(r\\phi)$ derived by differentiating \\eqref{eq:SSESF:dphi}, there are terms of the form\n$$\\partial_v\\phi \\partial_v\\phi$$\nsuch that neither of the factors can be controlled a priori in $L^1$ by the local BV scattering assumption. In other words, the equation does not obey any null condition.\n\nTo deal with this problem, we follow \\cite{Christodoulou:1993bt} and introduce the renormalized variables\n$ \\partial_{v}^{2} (r \\phi) - (\\partial_{v} \\lmb) \\phi, $\n\t$ \\partial_{u}^{2} (r \\phi) - (\\partial_{u} \\nu) \\phi, $\n\t$ \\partial_{v} \\log \\lmb - \\frac{\\lmb}{(1-\\mu)} \\frac{\\mu}{r} + \\partial_{v} \\phi \\Big( \\lmb^{-1} \\partial_{v} (r \\phi) - \\nu^{-1} \\partial_{u} ( r \\phi) \\Big), $\n\t$ \\partial_{u} \\log (-\\nu) - \\frac{\\nu}{(1-\\mu)} \\frac{\\mu}{r} + \\partial_{u} \\phi \\Big( \\lmb^{-1} \\partial_{v} (r \\phi) - \\nu^{-1} \\partial_{u} (r \\phi) \\Big)$\nwhich have the property that the nonlinear terms arising in the equations for these variables in fact have a null structure. In particular, we can apply the above heuristic procedure to obtain decay estimates in the compact region $r\\leq R$.\n\\\\\n\\fparagraph{Non-renormalized variables and decay towards null infinity}\n\nWhile the renormalization allows us to apply the BV theory in the interior region, it does not give the optimal $r$ decay rates in the exterior region. For example, the renormalized quantity\n$$\\partial_{v} \\log \\lmb - \\frac{\\mu}{(1-\\mu)} \\frac{\\lmb}{r} + \\partial_{v} \\phi \\Big( \\lmb^{-1} \\partial_{v} (r \\phi)  - \\nu^{-1} \\partial_{u} (r \\phi) \\Big)$$\ndecays only as $r^{-2}$ towards null infinity due to the contribution of $\\frac{\\mu}{(1-\\mu)} \\frac{\\lmb}{r}$, which is weaker than the desired $r^{-3}$ decay for $\\partial_v\\log \\lmb$. Therefore, in order to obtain the optimal estimates everywhere in the spacetime, we need to use the variables $\\partial_v^2(r\\phi)$, $\\partial_u^2(r\\phi)$, $\\partial_v\\lmb$ and $\\partial_u\\nu$ together with their renormalized versions.\n\\\\\n\\fparagraph{Coupling of the incoming and outgoing parts}\n\nFinally, an additional challenge is that unlike the estimates for the first derivatives of the scalar field, the bounds for the incoming part of the solution $\\partial_v^2(r\\phi)$ and $\\partial_v\\lmb$ are coupled to that for the outgoing part $\\partial_u^2(r\\phi)$ and $\\partial_u\\nu$. Likewise, to control $\\partial_u^2(r\\phi)$, we need estimates for $\\partial_v^2(r\\phi)$ and $\\partial_v\\lmb$. For example, in the equation for $\\partial_{v} \\log \\lmb - \\frac{\\mu}{(1-\\mu)} \\frac{\\lmb}{r} + \\partial_{v} \\phi \\Big( \\lmb^{-1} \\partial_{v} (r \\phi)  - \\nu^{-1} \\partial_{u} (r \\phi) \\Big)$, there is a term involving $\\partial_u^2(r\\phi)$ on the right hand side. In particular, in order to obtain the desired decay for $\\partial_v \\lmb$, we need to at the same time prove the decay for $\\partial_u^2(r\\phi)$.\n\\\\\n\\fparagraph{Strategy for obtaining the decay estimates}\nWith the above difficulties in mind, we can now give a very rough sketch of the strategy of the proof.\n\n\\begin{enumerate}\n\\item [(1)] Bounds for $\\partial_v^2(r\\phi)$ and $\\partial_v\\lmb$ for large $r$: As in the case for the first derivatives, we first prove the optimal $r$ decay for $\\partial_v^2(r\\phi)$ and $\\partial_v\\lmb$ in the exterior region. To this end, we integrate the equations satisfied by the \\emph{non-renormalized} variables. We note that the error terms can all be bounded using the local BV scattering assumption and the decay estimates already proved for the first derivatives.\n\n\\item [(2)] Bounds for all second derivatives: Steps 2 and 3 for the decay bounds for the first derivatives are now coupled. Define \\begin{align*}\n\t\\mathcal B_{2}(U) := \\sup_{u \\in [1, U]} \\sup_{C_{u}} \\Big( & u^{\\omega} \\abs{\\partial_{v}^{2} (r \\phi)} +u^{\\omega} \\abs{\\partial_{u}^{2} (r \\phi)} \n\t\t\t\t\t+ u^{\\omega} \\abs{\\partial_{v} \\lmb} + u^{\\omega} \\abs{\\partial_{u} \\nu} \\Big).\n\\end{align*}\nWe then show that $\\mathcal B_{2}(U)$ can control the error terms arising from integrating the \\emph{renormalized} equations in the sense that we can obtain an inequality of the form\n$$\\abs{\\mbox{weighted renormalized variables}}\\leq C(u_2)+\\epsilon(u_2)\\mathcal B_{2}(U),$$\nwhere $\\epsilon(u_2)\\to 0$ as $u_2\\to \\infty$. We then prove that the renormalized variables in fact control all the weighted second derivatives in $\\mathcal B_2$. After choosing $u_2$ to be sufficiently large, we show that $\\mathcal B_{2}(U)$ is bounded independent of $U$ and thus all the second derivatives have $u^{-\\omega}$ decay.\n\\item [(3)] Optimal bounds in terms of $u$ decay: While we have obtained $u^{-\\omega}$ decay for the second derivatives, the decay rates are not the sharp rates claimed in the main theorem. To finally obtained the desired bounds, we integrate the equations of the \\emph{non-renormalized} variables and use the preliminary estimates obtained in (1) and (2) above. Here, we make use of the fact that the estimates obtained in step (2) above are sufficiently strong (both in terms of regularity and decay) to control the error terms in the non-renormalized equations.\n\\end{enumerate}\n\n\\section{Analytic properties of \\eqref{eq:SSESF}}\\label{sec.anal.prop}\nIn this section, we discuss the analytic properties of \\eqref{eq:SSESF}. These include scaling, monotonicity and the null structure of the system. All these features will play crucial roles in the analysis.\n\\subsection{Scaling}\nFor $a>0$, \\eqref{eq:SSESF} is invariant under the scaling of the coordinate system\n$$ u \\mapsto au,\\quad v\\mapsto av$$\ntogether with the scaling of the functions\n$$r \\mapsto ar,\\quad m\\mapsto am,\\quad \\Omega\\mapsto \\Omega,\\quad\\phi\\mapsto\\phi.$$\nThis in particular implies that the BV norms\n\\begin{equation*}\n\\int_u^{\\infty} |\\partial_v^2(r\\phi)(u,v')| \\mathrm{d} v'\n\\hbox{ and }\n\\int_u^{\\infty} |\\partial_v \\lambda(u,v')| \\mathrm{d} v'\n\\end{equation*}\nare scale invariant. Thus the a priori assumptions \\eqref{eq:locBVScat} are taken with respect to localized versions of scale invariant norms.\n\n\\subsection{Monotonicity properties} \\label{subsec:monotonicity}\nWe first begin with basic monotonicity properties of $r$.\n\\begin{lemma}[Monotonicity of $r$] \\label{lem:mntn4r}\nLet $(\\phi, r, m)$ be a BV solution to \\eqref{eq:SSESF}. Then we have \n\\begin{equation*}\n\t\\nu < 0 \\hbox{ in } \\calQ,\n\\end{equation*}\nand\n\\begin{equation*}\n\t\\left\\{\n\t\\begin{aligned}\n\t\\lmb > 0 & \\hbox{ when } 1-\\mu > 0, \\\\\n\t\\lmb = 0 & \\hbox{ when } 1-\\mu = 0, \\\\\n\t\\lmb < 0 & \\hbox{ when } 1-\\mu < 0.\n\t\\end{aligned}\t\n\t\\right.\n\\end{equation*}\n\\end{lemma}\n\\begin{proof} \nThis was proved in \\cite[Propositions 1.1 and 1.2]{Christodoulou:1993bt}; we reproduce the proof for the reader's convenience. Note the equation\n\\begin{equation*}\n\t\\partial_{u} \\partial_{v} (r^{2}) = - \\frac{1}{2} \\Omega^{2}.\n \\end{equation*}\nwhich easily follows from \\eqref{eq:SSESF}. As $\\partial_{u} r^{2} = 2 r \\partial_{u} r= 0$ on $\\Gamma$ and $r > 0$ on $\\calQ$, we easily see that $\\nu < 0$. Then from the definition of $1-\\mu$, the second conclusion also follows. \\qedhere\n\\end{proof}\n\nAccording to the sign of $\\lmb$, a general Penrose diagram $\\calQ$ is divided into three subregions as follows:\n\\begin{align*}\n\t\\calT := \\set{(u,v) \\in \\calQ : \\lmb < 0}, \\quad \\calA := \\set{(u,v) \\in \\calQ : \\lmb = 0}, \\quad \\calR := \\set{(u,v) \\in \\calQ : \\lmb > 0}.\n\\end{align*}\n\nThese are called the \\emph{trapped region}, \\emph{apparent horizon}, and \\emph{regular region}, respectively. The next lemma, which we borrow from \\cite{Christodoulou:1993bt}, shows that the solutions to \\eqref{eq:SSESF} considered in this paper consist only of the regular region $\\calR$. Therefore, extensive discussion of $\\calT$ and $\\calA$ will be suppressed.\n\n\\begin{lemma}[{\\cite[Proposition 1.4]{Christodoulou:1993bt}}] \\label{lem:regR}\nLet $(\\phi, r, m)$ be a BV solution to \\eqref{eq:SSESF}. Then the causal past of $\\Gamma$ in $\\calQ$ is contained in $\\calR$.\nIn particular, $\\calQ = \\calR$ if $(\\phi, r, m)$ satisfies the condition $(1)$ in Definition \\ref{def:locBVScat} (future completeness of radial null geodesics).\n\\end{lemma}\n\n\nNext, we turn to monotonicity properties of the Hawking mass $m$, which will play an important role in our paper. The following lemma is an obvious consequence of \\eqref{eq:SSESF:dm}.\n\n\\begin{lemma}[Monotonicity of $m$] \\label{lem:mntn4m}\nFor a BV solution $(\\phi, r, m)$ to \\eqref{eq:SSESF}, we have\n\\begin{equation*}\n\t\\partial_{v} m \\geq 0, \\quad  \\partial_{u} m \\leq 0 \\hbox{ in } \\calR.\n\\end{equation*}\n\\end{lemma}\n\nBy the monotonicity $\\partial_{v} m \\geq 0$, the limit $M(u):=\\lim_{v \\to \\infty} m(u,v)$ exists (possibly $+\\infty$ at this point) for each $u$. This is called the \\emph{Bondi mass} at retarded time $u$. The following statement is an easy corollary of the preceding lemma.\n\n\\begin{corollary} [Monotonicity of the Bondi mass] \\label{cor:mntn4Bondi}\nLet $(\\phi, r, m)$ be a BV solution to \\eqref{eq:SSESF}, and suppose that $C_{u} \\subset \\calR$ for $u \\in [u_{1}, u_{2}]$. Then the Bondi mass $M(u)$ is a non-increasing function on $[u_{1}, u_{2}]$.\n\\end{corollary}\n\nThe following lemma shows that $M_{i} < \\infty$ for initial data sets considered in this paper. \n\\begin{lemma} \\label{lem:bnd4Mi}\nSuppose that $\\partial_{v}(r \\phi)(1, \\cdot)$ is asymptotically flat or order $\\omega' > 1$ in the sense of Definition \\ref{def:AF}. Then we have\n\\begin{equation} \\label{eq:bnd4Mi}\n\tM_{i} := \\lim_{v \\to \\infty} m(1, v) \\leq C \\mathcal I_{1}^{2}.\n\\end{equation}\n\\end{lemma}\n\nThis is an easy consequence of \\eqref{eq:SSESF:dm} and Lemma \\ref{lem:mntn4r}; we omit its proof.\nBy the preceding corollary, it follows that $M(u) < \\infty$ for each $u$.\n\nWe conclude this subsection with additional monotonicity properties of solutions to \\eqref{eq:SSESF}, useful for controlling the geometry of locally BV scattering solutions to \\eqref{eq:SSESF}.\n\n\\begin{lemma} \\label{lem:mntn4kpp}\n\tLet $(\\phi, r, m)$ be a BV solution to \\eqref{eq:SSESF}. For $(u,v) \\in \\calR$, we have\n\t\\begin{equation*}\n\t\t\\frac{\\lmb}{1-\\mu}(u,v) \\leq \\frac{\\lmb}{1-\\mu}(1, v),\n\t\\end{equation*}\n\t\n\t$$\\partial_u\\lmb =\\partial_v\\nu \\leq 0.$$\n\t\\end{lemma}\n\n\\begin{proof} \nThe lemma follows from the formula\n\\begin{equation*}\n\t\\partial_{u} \\log \\abs{\\frac{\\lmb}{1-\\mu}} = - (- \\nu)^{-1} r (\\partial_{u} \\phi)^{2}\n\\end{equation*} \nand \\eqref{eq:SSESF:dr}. \\qedhere\n\\end{proof}\n\n\\subsection{Null structure of the evolution equations} \\label{subsec:nullStr}\nIn this subsection, we follow \\cite{Christodoulou:1993bt} and demonstrate that the evolution equations verify a form of null structure. In particular, the null structure occurs in the equations for the second derivatives of the scalar field and the metric. However, it is not apparent if we simply take the derivatives of the equations \\eqref{eq:SSESF:dr} and \\eqref{eq:SSESF:dphi}. Instead, we rewrite the equations in renormalized variables for which the null structure is manifest. We will perform this renormalization separately for the wave equations for $\\phi$ and for the equations for $\\lambda$ and $\\nu$.\n\n\\vspace{.1in}\n{\\it - The wave equation for $\\phi$.}\nTaking $\\partial_{v}$ of the equation \\eqref{eq:SSESF:dphi}, we obtain\n\\begin{equation*}\n\t\\partial_{u} (\\partial_{v}^{2} (r \\phi)) = \\partial_{v} (\\partial_{u} \\lmb \\, \\phi) = \\partial_{u} \\lmb \\, \\partial_{v} \\phi + (\\partial_{v} \\partial_{u} \\lmb) \\phi,\n\\end{equation*}\nor equivalently, after substituting in the first equation in \\eqref{eq:SSESF:dr},\n\\begin{equation} \\label{eq:eq4dvdvrphi:normal}\n\\partial_{u} (\\partial_{v}^{2} (r \\phi)) = \n\\frac{2m \\lmb \\nu}{(1-\\mu) r^{2}} \\, \\partial_{v} \\phi + \\frac{ \\nu}{(1-\\mu) }  (\\partial_{v} \\phi)^{2} \\phi \n + \\frac{2m \\nu}{(1-\\mu) r^{2}} (\\partial_{v} \\lmb) \\phi - \\frac{4m}{(1-\\mu) r^{3}} \\lmb^{2} \\nu \\phi.\n\\end{equation}\n\nSome terms on the right hand side, such as $(1-\\mu)^{-1} \\nu (\\partial_{v} \\phi)^{2} \\phi$, do not exhibit null structure and are dangerous near $\\Gamma$. To tackle this, we rewrite\n\\begin{equation*}\n\t(\\partial_{v} \\partial_{u} \\lmb) \\phi = \\partial_{u} [(\\partial_{v} \\lmb) \\phi ] - \\partial_{v} \\lmb \\, \\partial_{u} \\phi.\n\\end{equation*}\n\nThus, from the first equation, we derive\n\\begin{equation} \\label{eq:eq4dvdvrphi}\n\t\\partial_{u} [\\partial_{v}^{2} (r \\phi) - (\\partial_{v} \\lmb) \\phi] = \\partial_{u} \\lmb \\, \\partial_{v} \\phi - \\partial_{v} \\lmb \\, \\partial_{u} \\phi.\n\\end{equation}\nBy switching $u$ and $v$, we obtain the following analogous equations in the conjugate direction.\n\\begin{equation} \\label{eq:eq4dudurphi:normal}\n\\partial_{v} (\\partial_{u}^{2} (r \\phi)) = \n\\frac{2m \\lmb \\nu}{(1-\\mu) r^{2}} \\, \\partial_{u} \\phi + \\frac{\\lmb }{(1-\\mu) }  (\\partial_{u} \\phi)^{2} \\phi \n + \\frac{2m \\lmb}{(1-\\mu) r^{2}} (\\partial_{u} \\nu) \\phi - \\frac{4m}{(1-\\mu) r^{3}} \\lmb \\nu^{2} \\phi.\n\\end{equation}\n\n\\begin{equation} \\label{eq:eq4dudurphi}\n\t\\partial_{v} [\\partial_{u}^{2} (r \\phi) - (\\partial_{u} \\nu) \\phi] = \\partial_{v} \\nu \\, \\partial_{u} \\phi - \\partial_{u} \\nu \\, \\partial_{v} \\phi.\n\\end{equation}\n\n\n\n\\vspace{.1in}\n{\\it - The equations for $\\lmb$ and $\\nu$.}\nFrom \\eqref{eq:SSESF:dr}, we have\n\\begin{equation*}\n\t\\partial_{u} \\log \\lmb = \\frac{\\mu}{(1-\\mu) r} \\nu, \\quad \\partial_{v} \\log (-\\nu) = \\frac{\\mu}{(1-\\mu) r} \\lmb.\n\\end{equation*}\n\nTake $\\partial_{v}$, $\\partial_{u}$ of the first and second equations respectively. Using \\eqref{eq:SSESF:dr}, it is not difficult to verify that\n\\begin{align}\n\\partial_{u} \\partial_{v} \\log \\lmb\n=& \\frac{1}{(1-\\mu) }  \\lmb^{-1} \\nu (\\partial_{v} \\phi)^{2} - \\frac{4m}{(1-\\mu) r^{3}} \\lmb \\nu, \\label{eq:eq4dvdvr:normal} \\\\\n\\partial_{v} \\partial_{u} \\log (-\\nu)\n=& \\frac{1}{(1-\\mu) }  \\nu^{-1} \\lmb (\\partial_{u} \\phi)^{2} - \\frac{4m}{(1-\\mu) r^{3}} \\lmb \\nu. \\label{eq:eq4dudur:normal}\n\\end{align}\n\nTo reveal the null structure, we must carry out the renormalization as we have done for \\eqref{eq:eq4dvdvrphi}, \\eqref{eq:eq4dudurphi}. Following Christodoulou \\cite{Christodoulou:1993bt}, it is easy to check that the above two equations are equivalent to\n\\begin{equation} \\label{eq:eq4dvdvr}\n\\begin{aligned}\n& \\partial_{u} \\Big[ \\partial_{v} \\log \\lmb - \\frac{\\mu}{(1-\\mu)} \\frac{\\lmb}{r} + \\partial_{v} \\phi \\Big( \\lmb^{-1} \\partial_{v} (r \\phi)  - \\nu^{-1} \\partial_{u} (r \\phi) \\Big) \\Big] \\\\\n& \\qquad = \\partial_{u} \\phi \\, \\partial_{v}\\Big( \\nu^{-1} \\partial_{u} (r \\phi) \\Big)- \\partial_{v} \\phi \\, \\partial_{u} \\Big( \\nu^{-1} \\partial_{u} (r \\phi) \\Big),\n\\end{aligned}\n\\end{equation}\nand the conjugate equation\n\\begin{equation} \\label{eq:eq4dudur}\n\\begin{aligned}\n& \\partial_{v} \\Big[ \\partial_{u} \\log (-\\nu) - \\frac{\\mu}{(1-\\mu)} \\frac{\\nu}{r} + \\partial_{u} \\phi \\Big( \\lmb^{-1} \\partial_{v} (r \\phi) - \\nu^{-1} \\partial_{u} (r \\phi) \\Big) \\Big]  \\\\\n&\\qquad = - \\partial_{u} \\phi \\, \\partial_{v}\\Big( \\lmb^{-1} \\partial_{v} (r \\phi) \\Big) + \\partial_{v} \\phi \\, \\partial_{u} \\Big( \\lmb^{-1} \\partial_{v} (r \\phi) \\Big).\n\\end{aligned}\n\\end{equation}\n\n\n\\section{Basic estimates for locally BV scattering solutions} \\label{sec.geom}\nIn this section, we gather some basic estimates concerning locally BV scattering solutions. These estimates will apply, in particular, to solutions satisfying the hypotheses of Theorem \\ref{main.thm.1}.\n\n\\subsection{Integration lemmas for $\\phi$} \\label{subsec:est4phi}\nWe first derive some basic inequalities for $\\phi$, $\\lmb^{-1} \\partial_v(r\\phi)$ and $\\partial_{v} \\phi$. We remark that these are functional inequalities which hold under very general assumptions, and in particular does not rely on the locally BV scattering assumption.\n\n\\begin{lemma} \\label{lem:est4phi}\nLet $\\phi(u, \\cdot)$ and $r(u, \\cdot)$ be Lipschitz functions on $[u,v]$ with $\\lmb>0$ and $r(u, u) = 0$. \nThen the following inequality holds.\n\\begin{equation} \\label{eq:intEst4phi:1}\n\\abs{\\phi(u,v)} \\leq  \\sup_{v' \\in [u, v]} \\Big\\vert \\frac{\\partial_{v}(r \\phi)}{\\lmb}(u, v') \\Big\\vert.\n\\end{equation}\n\nMore generally, for $u \\leq v_{1} \\leq v_{2}$, we have\n\\begin{equation} \\label{eq:intEst4phi:2}\n\\abs{r \\phi(u,v_{1}) - r \\phi(u, v_{2})} \\leq  \\Big( r(u, v_{2}) - r(u, v_{1}) \\Big) \\sup_{v' \\in [v_{1}, v_{2}]} \\Big\\vert \\frac{\\partial_{v}(r \\phi)}{\\lmb}(u, v') \\Big\\vert.\n\\end{equation}\n\n\n\\end{lemma}\n\n\\begin{proof} \nWe shall prove \\eqref{eq:intEst4phi:2}, since \\eqref{eq:intEst4phi:1} then follows as a special case. Integrating $\\partial_{v} (r \\phi)(u, v')$ over $v' \\in [v_{1}, v_{2}]$, we get\n\\begin{align*}\n\t\\abs{r \\phi(u,v_{1}) - r \\phi (u, v_{2})} \n\t\\leq & \\int_{v_{1}}^{v_{2}} \\abs{\\partial_{v} (r \\phi)(u, v')} \\, \\mathrm{d} v' \\\\\n\t\\leq & \\sup_{v' \\in [v_{1}, v_{2}]} \\Big\\vert \\frac{\\partial_{v}(r \\phi)}{\\lmb} (u, v') \\Big\\vert \\, \\times \\int_{v_{1}}^{v_{2}} \\lmb(u, v') \\, \\mathrm{d} v' \\\\\n\t=& \\Big( r(u, v_{2}) - r(u, v_{1}) \\Big) \\sup_{v' \\in [v_{1}, v_{2}]} \\Big\\vert \\frac{\\partial_{v}(r \\phi)}{\\lmb} (u, v') \\Big\\vert . \\qedhere\n\\end{align*}\n\\end{proof}\n\n\\begin{lemma} \\label{lem:est4dvphi}\nLet $\\phi(u, \\cdot)$ and $r(u, \\cdot)$ be functions on $[u, v]$ such that $\\partial_v\\phi$ is integrable, $r$ is Lipschitz with $\\lambda>0$ and $r(u, u) = 0$. \nSuppose furthermore that $\\lmb^{-1} \\partial_{v} (r \\phi)(u, \\cdot)$ is BV on $[u, v]$. Then the following statements hold.\n\\begin{enumerate}\n\\item We have\n\\begin{equation} \\label{eq:est4dvphi:2}\n\\int_{u}^{v} \\abs{\\partial_{v} \\phi(u,v')} \\, \\mathrm{d} v' \\leq \\int_{u}^{v}\\abs{\\partial_{v}(\\lmb^{-1} \\partial_{v} ( r \\phi))(u,v')} \\, \\mathrm{d} v'.\n\\end{equation}\n\n\\item Suppose, in addition, that $\\lmb^{-1} \\partial_{v}(r \\phi)(u, \\cdot)$ is Lipschitz on $[u,v]$. Then we have\n\\begin{equation} \\label{eq:est4dvphi:1}\n\\abs{\\partial_{v} \\phi(u,v)} \\leq \\frac{1}{2} \\frac{\\sup_{v' \\in [u,v]} \\lmb(u, v')}{\\inf_{v' \\in [u,v]} \\lmb(u, v')}  \\sup_{v' \\in [u, v]} \\abs{\\partial_{v} (\\lmb^{-1} \\partial_{v} ( r \\phi))(u, v')}.\n\\end{equation}\n\\end{enumerate}\n\\end{lemma}\n\n\\begin{proof} \nWe proceed formally to compute\n\\begin{align*}\n\t\\partial_{v} \\phi(u,v) \n\t=& \\frac{\\lmb}{r} \\Big( \\lmb^{-1} \\partial_{v} ( r \\phi) - \\phi\\Big) (u,v) \\\\\n\t=& \\frac{\\lmb}{r^{2}} (u, v) \\int_{u}^{v} \\Big( \\int_{v'}^{v} \\partial_{v} (\\lmb^{-1} \\partial_{v} ( r \\phi)) (u, v'') \\, \\mathrm{d} v'' \\Big) \\lmb(u, v')\\, \\mathrm{d} v' \\\\\n\t=& \\frac{\\lmb}{r^{2}} (u,v) \\int_{u}^{v} r(u, v'') \\partial_{v} (\\lmb^{-1} \\partial_{v} ( r \\phi)) (u, v'') \\, \\mathrm{d} v''.\n\\end{align*}\n\nThe above computation is justified thanks to the hypotheses, where we interpret \n\\begin{equation*}\n\t\\partial_{v} (\\lmb^{-1} \\partial_{v} ( r \\phi)) (u, v'') \\, \\mathrm{d} v''\n\\end{equation*}\nto be the the weak derivative of $\\lmb^{-1} \\partial_{v} ( r \\phi)$, which is a finite signed measure. For a fixed $(u, v)$, observe that\n\\begin{equation*}\n\t\\sup_{v'' \\in [u, v]} r(u, v'') \\int_{v''}^{v} \\frac{\\lmb(u,v')}{r^{2}(u,v')} \\, \\mathrm{d} v' \\leq 1.\n\\end{equation*}\n\nThis proves \\eqref{eq:est4dvphi:2}. For \\eqref{eq:est4dvphi:1}, note that the function $\\lmb^{-1} \\partial_{v} (r \\phi)$ is absolutely continuous on $[u,v]$, so $\\partial_{v}(\\lmb^{-1} \\partial_{v} ( r \\phi)(u, \\cdot))$ exists almost everywhere on $[u, v]$; moreover, it belongs to $L^{\\infty}$ by the Lipschitz assumption. Noting that\n\\begin{equation*}\n\t\\sup_{v' \\in [u, v]} \\frac{\\lmb(u,v')}{r^{2}(u,v')} \\int_{u}^{v'} r(u, v'') \\, \\mathrm{d} v'' \\leq \\frac{1}{2} \\frac{\\sup_{v' \\in [u,v]} \\lmb(u, v')}{\\inf_{v' \\in [u,v]} \\lmb(u, v')}\n\\end{equation*}\nwe obtain \\eqref{eq:est4dvphi:1}.\n\\end{proof}\n\n\\subsection{Geometry of locally BV scattering solutions}\nThe goal of this subsection is to prove the following proposition.\n\\begin{proposition} \\label{prop:geomLocBVScat}\nLet $(\\phi, r, m)$ be a locally BV scattering solution to \\eqref{eq:SSESF} as in Definition \\ref{def:locBVScat}. Assume furthermore that on the initial slice $C_{1}$, we have $\\lmb(1, \\cdot) = \\frac{1}{2}$ and\n\\begin{equation*}\n\t\\sup_{C_{1}} \\abs{\\partial_{v}(r \\phi)} + M_{i} < \\infty.\n\\end{equation*}\n\t\nThen there exist finite constants $K, \\Lambda > 0$ such that the following bounds hold for all $(u, v) \\in \\calQ$:\n\\begin{gather}\n\t\\Lambda^{-1} \\leq \\lmb(u,v) \\leq \\frac{1}{2} \\label{eq:bnd4dvr} \\\\\n\t\\Lambda^{-1} \\leq - \\nu(u,v) \\leq K \\label{eq:bnd4dur} \\\\\n\t1 \\leq (1-\\mu(u,v))^{-1} \\leq K \\Lambda. \\label{eq:bnd4mu}\\\\\n\t0 < \\frac{- \\nu}{1-\\mu(u,v)} \\leq K. \\label{eq:bnd4conjKpp}\n\\end{gather}\n\nMoreover, there exists a finite constant $\\Psi > 0$ such that for all $(u,v) \\in \\calQ$, we have\n\\begin{gather}\n\t\\abs{\\partial_{v}(r \\phi)(u,v)} \\leq \\Psi, \\label{eq:bnd4dvrphi} \\\\\n\t\\abs{\\phi(u,v)} \\leq \\Lambda \\Psi. \\label{eq:bnd4phi}\n\\end{gather}\n\\end{proposition}\n\n{\\bf Once we have this proposition, we will denote by $\\Lambda$, $K$ and $\\Psi$ the best constants such that \\eqref{eq:bnd4dvr}-\\eqref{eq:bnd4phi} hold.}\n\nBy Lemma \\ref{lem:regR}, we already know that $\\lmb > 0$, $- \\nu > 0$ and $(1-\\mu)^{-1} < \\infty$. The first three bounds, namely \\eqref{eq:bnd4dvr}--\\eqref{eq:bnd4mu}, ensure that these bounds concerning the geometry of the spacetime does not degenerate anywhere, in particular along the axis $\\Gamma$. They will be very useful in the analysis in the later section of the paper. \n\nThe proof of Proposition \\ref{prop:geomLocBVScat} will consist of several steps. We begin with elementary bounds for $\\lmb$ and $\\nu$.\n\n\\begin{lemma} \\label{lem:basicEst4dr}\nLet $(\\phi, r, m)$ be a BV solution to \\eqref{eq:SSESF} with $\\calQ = \\calR$. Then for every $(u,v) \\in \\calQ$, we have\n\\begin{align}\n\t\\lmb(u,v) \\leq& \\, \\lmb(1, v), \\label{eq:basicEst4dr:1} \\\\\n\t\\lmb^{-1}(u,v) \\leq& \\, \\lim_{u' \\to v-} \\lmb^{-1}(u',v), \\label{eq:basicEst4dr:2} \\\\\n\t\\nu(u,v) \\leq& - \\lim_{v' \\to u+} \\lmb(u, v'). \\label{eq:basicEst4dr:3}\n\\end{align}\n\\end{lemma}\n\n\\begin{proof}\n By \\eqref{eq:SSESF:dr}, we have\n\\begin{equation*}\n\\begin{aligned}\n\t\\lmb(u,v) =& \\lmb(1, v) \\exp \\Big( \\int_{1}^{u} \\Big( \\frac{2m}{(1-\\mu)r^{2}} \\nu \\Big) (u', v) \\, \\mathrm{d} u' \\Big), \\\\\n\t\\lmb^{-1}(u,v) =& \\lim_{u' \\to v-} \\lmb(u', v)^{-1} \\exp \\Big( \\int_{u}^{v} \\Big( \\frac{2m}{(1-\\mu)r^{2}} \\nu \\Big) (u', v) \\, \\mathrm{d} u' \\Big), \\\\\n\t\\nu(u,v) =& \\lim_{v' \\to u+} \\nu(u, v') \\exp \\Big( \\int_{u}^{v} \\Big( \\frac{2m}{(1-\\mu)r^{2}} \\lmb \\Big) (u, v') \\, \\mathrm{d} v' \\Big).\n\\end{aligned}\n\\end{equation*}\n\nSince $-\\nu, (1-\\mu) > 0$ everywhere, \\eqref{eq:basicEst4dr:1} and \\eqref{eq:basicEst4dr:2} follow. Moreover, since \n\\begin{equation*}\n\\lim_{v' \\to u+} \\nu(u,v') = - \\lim_{v' \\to u+}\\lmb(u,v'),\n\\end{equation*}\nand $\\lmb > 0$ on $\\calQ$, \\eqref{eq:basicEst4dr:3} follows as well. \\qedhere\n\\end{proof}\n\nBy Lemma \\ref{lem:regR}, $\\calQ = \\calR$ holds for a solution \\eqref{eq:SSESF} satisfying the hypotheses of Proposition \\ref{prop:geomLocBVScat}. As an immediate corollary, we have the following easy upper bound for $\\lmb$.\n\\begin{corollary}  \\label{cor:est4dr}\nLet $(\\phi, r, m)$ be a solution to \\eqref{eq:SSESF} satisfying the hypotheses of Proposition \\ref{prop:geomLocBVScat}. Then by the coordinate condition $\\lmb(1, v) = \\frac{1}{2}$ and \\eqref{eq:basicEst4dr:1}, we have\n\\begin{equation*}\n\t\\sup_{\\calQ} \\lmb \\leq \\frac{1}{2} \\, .\n\\end{equation*}\n\\end{corollary}\n\nNext, we proceed to prove the lower bounds of \\eqref{eq:bnd4dvr} and \\eqref{eq:bnd4dur}. We begin with a technical lemma concerning a large-$r$ region, which will also be useful in our proof of \\eqref{eq:bnd4dvrphi} and \\eqref{eq:bnd4phi}.\n\n\\begin{lemma} \\label{lem:babySmllPtnl:1}\nLet $(\\phi, r, m)$ be a solution to \\eqref{eq:SSESF} satisfying the hypotheses of Proposition \\ref{prop:geomLocBVScat}. Then for arbitrarily small $\\epsilon > 0$, there exists $r_{0} > 1$ such that\n\\begin{align} \n\t\\sup_{(u,v) \\in \\set{r \\geq r_{0}}} \\int_{1}^{u} \\abs{\\frac{\\mu}{(1-\\mu)} \\frac{\\nu}{r} (u', v)} \\, \\mathrm{d} u' <&  \\epsilon \\, . \\label{eq:babySmllPtnl:1}\n\\end{align}\n\\end{lemma}\n\\begin{proof} \nFor $(u,v) \\in \\set{r \\geq r_{0}}$, we begin by simply estimating as follows:\n\\begin{equation*}\n\t\\abs{\\frac{\\mu}{(1-\\mu)} \\frac{\\nu}{r}} \\leq \\frac{2 M_{i}}{(1-\\frac{2M_{i}}{r_{0}})} \\frac{(- \\nu)}{r^{2}}\n\\end{equation*}\n\nThe above inequality holds as long as\\footnote{Indeed, it suffices to choose $r_0>2M_i$ here. The condition $r_0> R$ will be used in the proof of Lemma \\ref{lem:babySmllPtnl:2}.} we choose $r_{0} > \\max \\set{2 M_{i}, R}$. Note that if $(u, v) \\in \\set{r \\geq r_{0}}$, then the null curve $\\set{(u', v) : u' \\in [1, u]}$ from the initial slice $C_{1}$ to $(u,v)$ lies entirely in $\\set{r \\geq r_{0}}$. Integrating along this curve, we obtain for $(u, v) \\in \\set{r \\geq r_{0}}$\n\\begin{equation*}\n\t\\int_{1}^{u} \\abs{\\frac{\\mu}{(1-\\mu)} \\frac{\\nu}{r}(u',v)} \\, \\mathrm{d} u' < \\frac{2 M_{i}}{(1-\\frac{2M_{i}}{r_{0}})} \\frac{1}{r_{0}}\n\\end{equation*}\n\nTaking $r_{0}$ sufficiently large, \\eqref{eq:babySmllPtnl:1} follows. \\qedhere\n\n\\end{proof}\n\n\nNext, we prove an analogous result in a large $u$ region. Key to its proof will be the identity \\eqref{eq:babySmllPtnl:pf:0} below, which will also be used to relate \\eqref{eq:babySmllPtnl:1} and \\eqref{eq:babySmllPtnl:2} to the desired lower bounds of $\\lmb$ and $-\\nu$.\n\n\\begin{lemma} \\label{lem:babySmllPtnl:2}\nLet $(\\phi, r, m)$ be a solution to \\eqref{eq:SSESF} satisfying the hypotheses of Proposition \\ref{prop:geomLocBVScat}. Then for arbitrarily small $\\epsilon > 0$, there exists $U > 1$ such that\n\\begin{align} \n\t\\sup_{v \\geq U} \\int_{U}^{v} \\abs{\\frac{\\mu}{1-\\mu} \\frac{\\nu}{r} (u', v)} \\, \\mathrm{d} u' <&  \\epsilon \\, . \\label{eq:babySmllPtnl:2}\n\\end{align}\n\\end{lemma}\n\n\\begin{proof} \nLet $\\epsilon > 0$ be an arbitrary positive number. Using \\eqref{eq:SSESF:dr} and the fact that $1-\\mu > 0, -\\nu > 0$ on $\\calQ$, we have for any $1 \\leq u_{1} \\leq u_{2} < v$,\n\\begin{equation} \\label{eq:babySmllPtnl:pf:0}\n\t\\int_{u_{1}}^{u_{2}} \\abs{\\frac{\\mu}{1-\\mu} \\frac{\\nu}{r} (u', v)} \\, \\mathrm{d} u' = \\log \\lmb(u_{1}, v) - \\log \\lmb(u_{2}, v).\n\\end{equation}\n\nIn order to prove \\eqref{eq:babySmllPtnl:2}, it therefore suffices to exhibit $U > 1$ such that \n\\begin{equation} \\label{eq:babySmllPtnl:pf:1}\n\t\\sup_{(u, v), (u', v') \\in \\set{u \\geq U}} \\abs{\\log \\lmb(u, v) - \\log \\lmb(u', v')} < \\epsilon.\n\\end{equation} \n\nIn order to proceed, we divide $\\calQ$ into three regions: $\\PD_{\\mathrm{cpt}} := \\set{r \\leq R}$, $\\calQ_{[R, r_{0}]} := \\set{R \\leq r \\leq r_{0}}$ and $\\calQ_{[r_{0}, \\infty)} := \\set{r \\geq r_{0}}$, where $r_{0} > \\max\\{2M_i,R\\}$ is chosen via Lemma \\ref{lem:babySmllPtnl:1} so that\n\\begin{equation*} \n\t\\sup_{(u,v) \\in \\calQ_{[r_{0}, \\infty)}}\\int_{1}^{u} \\abs{\\frac{\\mu}{1-\\mu} \\frac{\\nu}{r}(u',v)} \\, \\mathrm{d} u' < \\frac{\\epsilon}{8}.\n\\end{equation*}\n\nUsing \\eqref{eq:babySmllPtnl:pf:0} and the fact that $\\log \\lmb(1, v) = \\frac{1}{2}$, the preceding inequality is equivalent to\n\\begin{equation} \\label{eq:babySmllPtnl:pf:2}\n\t\\sup_{(u,v) \\in \\calQ_{[r_{0}, \\infty)} } \\abs{\\log \\lmb(u,v) - \\frac{1}{2}} < \\frac{\\epsilon}{8}.\n\\end{equation}\n\nNext, we turn to the region $\\calQ_{[R, r_{0}]}$; here we exploit the vanishing of the final Bondi mass. Indeed, taking $U_{1}$ large enough so that $2 M(U_{1}) < R$, we may estimate\n\\begin{equation*}\n\t\\abs{\\frac{\\mu}{1-\\mu} \\frac{\\nu}{r}} \\leq \\frac{2 M(U_{1})}{(1-\\frac{2 M(U_{1})}{R}) R^{2}} (-\\nu)\\quad\\mbox{for }u\\geq U_1.\n\\end{equation*}\n\nConsider now the time-like curve given by $\\gamma_{0} := \\set{(u',v') : r(u', v') = r_{0}}$. On $\\gamma_{0} \\cap \\set{(u,v) : u \\geq U_{1}}$, note that \\eqref{eq:babySmllPtnl:pf:2} holds. Integrating the preceding inequality along incoming null curves emanating from $\\gamma_{0} \\cap \\set{(u,v) : u \\geq U_{1}}$, we obtain for $(u, v) \\in \\calQ_{[R, r_{0}]} \\cap \\set{(u,v) : u \\geq U_{2}}$\n\\begin{equation*} \n\t\\abs{\\log \\lmb(u,v) - \\frac{1}{2}} < \\frac{\\epsilon}{8} + \\frac{2 M(U_{1}) (r_{0} - R)}{(1- \\frac{2 M(U_{1})}{R}) R^{2}} .\n\\end{equation*}\nwhere $U_{2} = U_{2}(U_{1}, r_{0})$ is the future endpoint of the incoming null curve in $\\calQ_{[R, r_{0}]}$ from the past endpoint of $\\gamma_{0} \\cap \\set{(u,v) : u \\geq U_{1}}$; more precisely, $U_{2} = \\sup \\set{u : r(u, V_{1}) \\geq R}$, where $V_{1}$ is the defined by $r(U_{1}, V_{1}) = r_{0}$. Choosing $U_{1}$ sufficiently large, we then obtain\n\\begin{equation} \\label{eq:babySmllPtnl:pf:3}\n\t\\sup_{(u, v) \\in \\calQ_{[R, r_{0}]} \\cap \\set{u \\geq U_{2}}} \\abs{\\log \\lmb(u,v) - \\frac{1}{2}} < \\frac{\\epsilon}{4}.\n\\end{equation}\n\nFinally, in $\\PD_{\\mathrm{cpt}}$, we use the local BV scattering condition \\eqref{eq:locBVScat} to choose $U \\geq U_{2}$ large enough so that we have\n\\begin{equation} \\label{eq:babySmllPtnl:pf:4}\n\t\\sup_{(u, v), (u, v') \\in \\PD_{\\mathrm{cpt}} \\cap \\set{u \\geq U}}\\abs{\\log \\lmb (u, v) - \\log \\lmb(u, v')} < \\frac{\\epsilon}{4}.\n\\end{equation}\n\nTo compare $\\log \\lmb(u, v)$ and $\\log \\lmb(u', v')$ with $u \\neq u'$, we use \\eqref{eq:babySmllPtnl:pf:3}, \\eqref{eq:babySmllPtnl:pf:4} and the triangle inequality. Thus, the desired conclusion \\eqref{eq:babySmllPtnl:pf:1} follows. \\qedhere\n\\end{proof}\n\nAs a corollary of the preceding lemmas and \\eqref{eq:babySmllPtnl:pf:0} (or, more directly, \\eqref{eq:babySmllPtnl:pf:1} and \\eqref{eq:babySmllPtnl:pf:2}), we immediately see that $\\lmb$ and $-\\nu$ is uniformly bounded away from zero.\n\\begin{corollary} \\label{cor:lowerBnd4dvr}\nLet $(\\phi, r, m)$ be a solution to \\eqref{eq:SSESF} satisfying the hypotheses of Proposition \\ref{prop:geomLocBVScat}. Then there exists $0 < \\Lambda < \\infty$ such that for all $(u, v) \\in \\calQ$, we have\n\\begin{equation*}\n\t\\Lambda^{-1} \\leq \\lmb(u, v), \\quad\n\t\\Lambda^{-1} \\leq - \\nu(u,v).\n\\end{equation*}\n\\end{corollary}\n\nTogether with Corollary \\ref{cor:est4dr}, this concludes the proof of \\eqref{eq:bnd4dvr}. Next, using Lemmas \\ref{lem:est4phi}, \\ref{lem:babySmllPtnl:1}, \\ref{lem:babySmllPtnl:2} and the wave equation \\eqref{eq:SSESF:dphi} for $\\phi$, we prove \\eqref{eq:bnd4dvrphi}, \\eqref{eq:bnd4phi} in the following lemma.\n\n\\begin{lemma}  \\label{lem:bnd4dvrphiphi}\nLet $(\\phi, r, m)$ be a solution to \\eqref{eq:SSESF} satisfying the hypotheses of Proposition \\ref{prop:geomLocBVScat}. Then there exists a constant $0 < \\Psi < \\infty$ such that\n\\begin{equation} \\label{eq:bnd4dvrphiphi}\n\t\\sup_{\\calQ} \\abs{\\partial_{v}(r \\phi)} \\leq \\Psi, \\quad \n\t\\sup_{\\calQ} \\abs{\\phi} \\leq \\Lambda \\Psi,\n\\end{equation}\nwhere $\\Lambda$ is the best constant such that Corollary \\ref{cor:lowerBnd4dvr} holds.\n\\end{lemma}\n\\begin{proof} \nNote that the second inequality of \\eqref{eq:bnd4dvrphiphi} is an immediate consequence of the first inequality, Lemma \\ref{lem:est4phi} and Corollary \\ref{cor:lowerBnd4dvr}. The proof of the first inequality will proceed in two steps: First, we shall show that $\\partial_{v}(r \\phi)$ is uniformly bounded on the large $r$ region, essentially via Lemma \\ref{lem:babySmllPtnl:1}. By compactness, it immediately follows that $\\partial_{v}(r \\phi)$ is uniformly bounded on the finite $u$ region. Then in the second step, we shall show that $\\partial_{v}(r \\phi)$ is uniformly bounded on a large $u$ region as well using Lemma \\ref{lem:babySmllPtnl:2}.\n\nBy Lemma \\ref{lem:babySmllPtnl:1}, choose $r_{0} > 0$ so that \n\\begin{equation} \\label{eq:bdd4dvrphiphi:pf:1}\n\t\\sup_{(u,v) \\in \\set{r \\geq r_{0}}} \\int_{1}^{u} \\abs{\\frac{\\mu}{1-\\mu} \\frac{\\nu}{r} (u', v)} \\, \\mathrm{d} u' < \\frac{1}{10 \\Lambda}.\n\\end{equation}\n\nWe also borrow the notation $\\calQ_{[r_{0}, \\infty)} := \\set{(u,v) : r(u,v) \\geq r_{0}}$ from the proof of Lemma \\ref{lem:babySmllPtnl:2}. Given $U \\geq 1$, define $\\Psi_{[r_{0}, \\infty)}(U)$ to be\n\\begin{equation*}\n\t\\Psi_{[r_{0}, \\infty)}(U) := \\sup_{(u, v) \\in \\calQ_{[r_{0}, \\infty)} \\cap \\set{1 \\leq u \\leq U}} \\abs{\\partial_{v} (r \\phi)(u, v)}.  \n\\end{equation*}\n\nLet $(u,v) \\in \\calQ_{[r_{0}, \\infty)}$. Using \\eqref{eq:SSESF:dphi}, we then write\n\\begin{align*}\n\t\\partial_{u} \\partial_{v} ( r \\phi)\n\t= & \\frac{\\mu}{1-\\mu} \\frac{\\nu}{r} \\Big( \\frac{\\lmb}{r} (r \\phi - r_{0} \\phi_{r_{0}}) + \\frac{\\lmb}{r} r_{0} \\phi_{r_{0}} \\Big).\n\\end{align*} \n\nHere, $\\phi_{r_{0}}(u,v) := \\phi(u, v^{\\star}_{0}(u))$, where $v^{\\star}_{0}(u)$ is the unique $v$-value for which $r(u, v^{\\star}_{0}(u)) = r_{0}$. Note that the outgoing null curve from $(u, v^{\\star}_{0}(u))$ to $(u,v) \\in \\calQ_{[r_{0}, \\infty)}$ lies entirely in $\\calQ_{[r_{0}, \\infty)}$. Thus, by Lemma \\ref{lem:est4phi} and \\eqref{eq:bnd4dvr}, we see that for $(u, v) \\in \\calQ_{[r_{0}, \\infty)}$ with $1 \\leq u \\leq U$, \n\\begin{align*}\n\t\\abs{\\partial_{u} \\partial_{v} ( r \\phi)} \n\t\\leq & \\abs{\\frac{\\mu}{1-\\mu} \\frac{\\nu}{r}} \\Big( \\frac{(r - r_{0})}{2 r} \\Lambda \\Psi_{[r_{0}, \\infty)}(U) + \\frac{r_{0}}{2 r} \\abs{\\phi_{r_{0}}} \\Big) \\\\\n\t\\leq & \\abs{\\frac{\\mu}{1-\\mu} \\frac{\\nu}{r}} \\Big( \\Lambda \\Psi_{[r_{0}, \\infty)}(U) + \\abs{\\phi_{r_{0}}} \\Big).\n\\end{align*}\n\nIntegrating this equation over the incoming null curve from $(1, v)$ to $(u, v)$ (which lies in $\\calQ_{[r_{0}, \\infty)} \\cap \\set{1 \\leq u \\leq U}$) and using Lemma \\ref{lem:babySmllPtnl:1}, we then obtain\n\\begin{align*}\n\t\\Psi_{[r_{0}, \\infty)}(U)\t\\leq \\sup_{C_{1} \\cap \\calQ_{[r_{0}, \\infty)}} \\abs{\\partial_{v} (r \\phi)} + \\frac{1}{10} \\Psi_{[r_{0}, \\infty)}(U) + \\frac{1}{10 \\Lambda} \\sup_{\\gamma_{0} \\cap \\set{1 \\leq u \\leq U}} \\abs{\\phi}\n\\end{align*}\nwhere $\\gamma_{0}$ is the time-like curve $\\set{(u,v) : r(u,v) = r_{0}}$. Note that the first term on the right-hand side is finite by the assumptions on the initial data, whereas the last term is finite for every $1 \\leq U < \\infty$ by compactness of $\\gamma_{0} \\cap \\set{(u,v) : 1 \\leq u \\leq U}$ and continuity of $\\phi$. Then, by a simple continuity argument, it follows that $\\Psi_{[r_{0}, \\infty)}(U) < \\infty$ for every $1 \\leq U < \\infty$. Moreover, by compactness of $\\set{(u, v) : r(u,v) \\leq r_{0}, \\, 1 \\leq u \\leq U}$, as well as the uniform BV assumption on $\\partial_{v}(r \\phi)$, we also have\n\\begin{equation*}\n\t\\Psi_{[0, \\infty)}(U) := \\sup_{(u,v) \\in \\set{1 \\leq u \\leq U}} \\abs{\\partial_{v}(r \\phi)(u,v)} < \\infty.\n\\end{equation*}\n\nWe now proceed to deal with the large-$u$ region, namely $\\set{(u,v) : u \\geq U}$. Using Lemma \\ref{lem:babySmllPtnl:2}, we choose $U_{0} \\geq 1$ sufficiently large so that\n\\begin{equation}\n\t\\sup_{v \\geq U_{0}} \\int_{U_{0}}^{v} \\abs{\\frac{\\mu}{1-\\mu} \\frac{\\nu}{r} (u', v)} \\, \\mathrm{d} u' < \\frac{1}{10 \\Lambda}.\n\\end{equation}\n\nProceeding as before via Lemma \\ref{lem:est4phi}, we estimate for $(u,v) \\in \\set{(u,v) : u \\geq U_{0}}$ \n\\begin{align*}\n\t\\abs{\\partial_{u} \\partial_{v} (r \\phi)(u,v)} \\leq \\abs{\\frac{\\mu}{1-\\mu} \\frac{\\nu}{r}} \\, \\Lambda \\sup_{v' \\in [u, v]} \\abs{\\partial_{v}(r \\phi)(u, v')} .\n\\end{align*}\n\nIntegrating along incoming null curves from $C_{U_{0}}$, we see that\n\\begin{equation*}\n\t\\Psi_{[0, \\infty)}(U) \\leq \\Psi_{[0, \\infty)}(U_{0}) + \\frac{1}{10} \\Psi_{[0, \\infty)}(U)\n\\end{equation*}\nfor any $U \\geq U_{0}$. Absorbing the second term on the right-hand side into the left-hand side and taking $U \\to \\infty$, we obtain \\eqref{eq:bnd4dvrphiphi} with $\\Psi \\leq \\frac{10}{9} \\Psi_{[0, \\infty)}(U_{0}) < \\infty$. \\qedhere\n\\end{proof}\n\nWe are finally ready to conclude the proof of Proposition \\ref{prop:geomLocBVScat}, by proving \\eqref{eq:bnd4conjKpp}. Indeed, the upper bounds in \\eqref{eq:bnd4dur} and \\eqref{eq:bnd4mu} would then follow immediately. Moreover, the lower bound in \\eqref{eq:bnd4mu} is trivial, as $\\mu = \\frac{2m}{r} \\geq 0$.\n\n\\begin{lemma} \\label{lem:est4dur}\nLet $(\\phi, r, m)$ be a solution to \\eqref{eq:SSESF} satisfying the hypotheses of Proposition \\ref{prop:geomLocBVScat}. Then there exists a finite constant $K > 0$ such that for all $(u,v) \\in \\calQ$,\n\\begin{equation} \\label{eq:est4dur:key}\t\n \\frac{- \\nu}{1-\\mu} (u,v) \\leq K.\n\\end{equation}\n\\end{lemma}\n\n\\begin{proof} \nTo prove \\eqref{eq:est4dur:key}, we shall rely on the equation\n\\begin{equation} \\label{eq:mntn4durOver1-mu}\n\t\\partial_{v} \\log \\Big( \\frac{-\\nu}{1-\\mu} \\Big) = \\lmb^{-1} r (\\partial_{v} \\phi)^{2},\n\\end{equation}\nwhich may be easily derived from \\eqref{eq:SSESF:dr} and \\eqref{eq:SSESF:dm}.\n\nFor $(u, v) \\in \\calQ$, we begin by integrating \\eqref{eq:mntn4durOver1-mu} on the outgoing null curve from $(u,u) \\in \\Gamma$ to $(u,v)$, which gives\n\\begin{equation*}\n\t\\Big( \\frac{-\\nu}{1-\\mu} \\Big)(u,v) \\leq \\Big( \\lim_{v' \\to u+} \\Big( \\frac{-\\nu}{1-\\mu} \\Big) (u, v') \\Big)  \\exp \\Big( \\int_{u}^{v} \\lmb^{-1} r (\\partial_{v} \\phi)^{2} (u, v') \\, \\mathrm{d} v' \\Big).\n\\end{equation*}\n\nWe claim that $\\lim_{v' \\to u+} (- \\nu) (u, v') = \\lim_{v' \\to u+} \\lmb (u, v')\\leq \\frac 12$ and $\\lim_{v' \\to u+} \\mu(u,v') = 0$. The first assertion is obvious. To prove the second one, we first use \\eqref{eq:SSESF:dm} to write\n\\begin{equation*}\nm(u,v) \\leq \\tfrac{1}{2} \\Big( \\sup_{v' \\in [u, v]} \\abs{r^{2} \\partial_{v} \\phi}(u, v') \\Big) \\int_{u}^{v} \\abs{\\partial_{v} \\phi(u, v')} \\, \\mathrm{d} v'.\n\\end{equation*}\nNow observe that $\\sup_{v' \\in [u, v]} \\abs{r^{2} \\partial_{v} \\phi}(u, v') \\leq C r(u,v) \\sup_{v' \\in [u,v]} \\abs{\\partial_{v}(r \\phi)}$, and the remaining integral goes to $0$ as $v \\to u+$, since $\\phi$ is assumed to be absolutely continuous on $C_{u}$ near the axis by Definition \\ref{def:BVsolution}. \n\nBy the above claim, we have\n\\begin{equation*}\n\t\\Big( \\frac{-\\nu}{1-\\mu} \\Big)(u,v) \\leq \\frac{1}{2} \\exp \\Big( \\int_{u}^{v} \\lmb^{-1} r (\\partial_{v} \\phi)^{2} (u, v') \\, \\mathrm{d} v' \\Big).\n\\end{equation*}\n\nThe lemma would therefore follow if we could prove\n\\begin{equation*}\n\\sup_{(u,v) \\in \\calQ} \\int_{u}^{v} \\lmb^{-1} r (\\partial_{v} \\phi)^{2} (u, v') \\, \\mathrm{d} v' < \\infty.\n\\end{equation*}\n\nTo achieve this, we shall divide the integral into two parts, one in $\\PD_{\\mathrm{cpt}}$ and the other in its complement $\\PD_{\\mathrm{cpt}}^c$. Indeed, defining $v^{\\star}(u)$ to be the unique $v$ value such that $r(u, v^{\\star}(u)) = R$, we divide the integral into $\\int_{u}^{v^{\\star}(u)}$ and $\\int_{v^{\\star}(u)}^{v}$. If $v < v^{\\star}(u)$, the latter integral will be taken to be zero.\n\nFor the first integral, let us begin by pulling out $\\lmb^{-1} r \\partial_{v} \\phi$ from the integral. Using the identity $\\lmb^{-1} r \\partial_{v} \\phi = \\lmb^{-1} \\partial_{v} (r \\phi) - \\phi$ we have\n\\begin{align*}\n&\\int_{u}^{v^{\\star}(u)} \\lmb^{-1} r (\\partial_{v} \\phi)^{2} (u, v') \\, \\mathrm{d} v'\\\\\n& \\quad \\leq \\sup_{v' \\in [u, v^{\\star}(u)]} \\Big( \\lmb^{-1} \\abs{\\partial_{v}(r\\phi)}(u, v') + \\abs{\\phi}(u, v') \\Big) \\int_{u}^{v^{\\star}(u)} \\abs{\\partial_{v} \\phi(u,v')} \\, \\mathrm{d} v'.\n\\end{align*}\n\nThen by Lemmas \\ref{lem:est4dvphi}, \\ref{lem:bnd4dvrphiphi} and the local BV scattering assumption, the right-hand side is uniformly bounded in $u$ from above, as desired. For the second integral, note that, by Lemma \\ref{lem:mntn4kpp} and Corollary \\ref{cor:lowerBnd4dvr}, we have\n\\begin{equation*}\n(1-\\mu)^{-1}(u,v) \\leq \\Lambda \\frac{\\lmb}{1-\\mu}(u,v) \\leq \\frac{\\Lambda}{2} \\sup_{C_{1}} (1-\\mu)^{-1}. \n\\end{equation*}\n\nNotice that the quantity $\\sup_{C_{1}}(1-\\mu)^{-1}$ for the initial data is finite, since $1-\\mu > 0$ everywhere and $1-\\mu(1, v) \\to 1$ as $v \\to \\infty$.\nMoreover, for $v \\geq v^{\\star}(u)$, we have $r(u, v) \\geq R$. Therefore, in view of \\eqref{eq:SSESF:dm}, we may estimate\n\\begin{align*}\n\t\\int_{v^{\\star}(u)}^{v} \\lmb^{-1} r (\\partial_{v} \\phi)^{2} \\, \\mathrm{d} v'\n\t\\leq & \\frac{\\Lambda}{R} \\sup_{C_{1}} (1-\\mu)^{-1} \\int_{v^{\\star}(u)}^{v} \\frac{1}{2} \\lmb^{-1} (1-\\mu) r^{2} (\\partial_{v} \\phi)^{2} (u, v') \\, \\mathrm{d} v' \\\\\n\t\\leq & \\frac{\\Lambda}{R} \\sup_{C_{1}} (1-\\mu)^{-1} (m(u, v) - m(u, v^{\\star}(u))) \\\\\n\t\\leq & C_{\\Lambda, R, M_{i}, \\sup_{C_{1}} (1-\\mu)^{-1}} < \\infty,\n\\end{align*}\nfrom which the lemma follows. \\qedhere\n\\end{proof}\n\nWe conclude this subsection with a pair of identities which are useful for estimating $\\int\\abs{\\partial_{u} \\lmb} \\, \\mathrm{d} u$ and $\\int \\abs{\\partial_{v} \\nu} \\, \\mathrm{d} v$ in terms of information on $\\phi$.\n\\begin{lemma} \\label{lem:auxEqs}\nFrom \\eqref{eq:SSESF}, the following identities hold:\n\\begin{align}\n\t\\int_{u}^{v} \\frac{\\mu}{1-\\mu} \\frac{\\lmb}{r} (u,v') \\, \\mathrm{d} v' = & \\log(1-\\mu)(u,v) + \\int_{u}^{v} \\lmb^{-1} r (\\partial_{v} \\phi)^{2} (u, v') \\, \\mathrm{d} v', \\label{eq:auxEqs:1} \\\\\n\t\\int_{u}^{v} \\frac{\\mu}{1-\\mu} \\frac{(-\\nu)}{r} (u',v) \\, \\mathrm{d} u' = & \\log(1-\\mu)(u,v) + \\int_{u}^{v} (-\\nu)^{-1} r (\\partial_{u} \\phi)^{2} (u', v) \\, \\mathrm{d} u'. \\label{eq:auxEqs:2}\n\\end{align}\n\\end{lemma}\n\n\\begin{proof} \n\tWe shall prove \\eqref{eq:auxEqs:1}, leaving the similar proof of \\eqref{eq:auxEqs:2} to the reader. \n\tFrom the proof of Lemma \\ref{lem:basicEst4dr}, we have\n\t\\begin{equation*}\n\t\t\\int_{u}^{v} \\frac{\\mu}{1-\\mu} \\frac{\\lmb}{r}  (u,v') = \\log \\frac{\\nu(u, v)}{\\lim_{v' \\to u+} \\nu(u,v')}.\n\t\\end{equation*}\n\t\n\tComparing with the integral of \\eqref{eq:mntn4durOver1-mu}, along with the fact that $\\lim_{v' \\to u+} (1-\\mu)(u, v') = 1$, we arrive at \\eqref{eq:auxEqs:1}. \\qedhere\n\\end{proof}\n\n\\subsection{Normalization of the coordinate system}\\label{sec.coord}\n\nIn \\S \\ref{subsec:coordSys}, the coordinates are normalized such that $\\lmb$ is constant on the initial hypersurface $\\{u=1\\}$. Alternatively, one can introduce a new coordinate system $(u_{\\infty},v_{\\infty})$ which is normalized at future null infinity by requiring that $\\nu_{\\infty}\\to-\\frac 12$ along each outgoing null curve towards null infinity and require, as before, that $\\Gamma=\\{(u,v):u=v\\}$. We will show that the coordinate functions $u$ and $u_{\\infty}$ are comparable and thus the main theorem on the decay rates can also be stated in this alternatively normalized coordinate system.\n\nWe can compute explicitly the coordinate change, which is given by\n\\begin{equation*}\n\\frac{du_{\\infty}}{du}(u)=-2\\lim_{v\\to\\infty}\\nu(u,v),\\quad u_\\infty(1)=1\n\\hbox{ and }\nv_{\\infty}(v)=u_{\\infty}(v).\n\\end{equation*}\n\nNotice that the limit $\\displaystyle\\lim_{v\\to\\infty}\\nu(u,v)$ is well-defined due to the monotonicity of $\\nu$.\n\n\\begin{equation*}\n\tu_{\\infty}(u) = - 2\\int_{1}^{u} \\Big(\\lim_{v\\to\\infty}\\nu(u',v)\\Big) \\, \\mathrm{d} u' + 1,\n\\end{equation*}\n\nBy Proposition \\ref{prop:geomLocBVScat}, the following estimate holds:\n\\begin{equation*}\n\t2(\\Lambda)^{-1} (u-1) \\leq u_{\\infty}-1 \\leq 2 K (u-1).\n\\end{equation*}\n\n\\subsection{Consequence of local BV scattering}\nIn this subsection, we give some estimates for $\\partial_u^2(r\\phi)$, $\\partial_u\\phi$ and $\\partial_u \\nu$ that follow from from the local BV scattering assumption. To this end, we will need the analysis for solutions to \\eqref{eq:SSESF} with small bounded variation norm by Christodoulou in \\cite{Christodoulou:1993bt}. In particular, Christodoulou proved\n\\begin{theorem}[{Christodoulou \\cite[Theorem 6.2]{Christodoulou:1993bt}}]\\label{Chr.BV.Thm}\nThere exists universal constants $\\epsilon_0$ and $C_0$ such that for $\\epsilon<\\epsilon_0$, if $\\lmb(1, \\cdot) = \\frac{1}{2}$ and $\\partial_{v} (r \\phi)(1, \\cdot)$ is of bounded variation with\n\\begin{equation} \\label{Chr.BV.Thm.hyp}\n\\int_{C_1} |\\partial_v^2(r\\phi)| <\\epsilon,\n\\end{equation}\nthen its maximal development $(\\phi, r, m)$ satisfies condition $(1)$ in Definition \\ref{def:locBVScat} (future completeness of radial null geodesics) and obeys\n\\begin{gather}\n\t\\frac{1}{3} \\leq \\lmb \\leq \\frac{1}{2}, \\quad \n\t\\frac{1}{3} \\leq - \\nu \\leq \\frac{2}{3}, \\quad\n\t\\frac{2}{3} \\leq (1-\\mu) \\leq 1, \\label{Chr.BV.Thm.geom} \\\\\n\t\\sup_{u \\geq 1} \\int_{C_{u}} \\Big( \\abs{\\partial_{v} (\\lmb^{-1} \\partial_{v} (r \\phi))} + \\abs{\\partial_{v} \\phi} + \\abs{\\partial_{v} \\log \\lmb} \\Big) < C_{0} \\epsilon, \\label{Chr.BV.Thm.dv} \\\\\n\t\\sup_{v \\geq 1} \\int_{\\underline{C}_{v}} \\Big( \\abs{\\partial_{u} (\\nu^{-1} \\partial_{u} (r \\phi))} + \\abs{\\partial_{u} \\phi} + \\abs{\\partial_{u} \\log \\nu} \\Big) < C_{0} \\epsilon. \\label{Chr.BV.Thm.du}\n\\end{gather}\n\\end{theorem}\n\n\\begin{remark} \nWe remark that in \\cite[Theorem 6.2]{Christodoulou:1993bt}, it is implicitly assumed that $\\phi(1, 1) = 0$; see \\cite[Section 4]{Christodoulou:1993bt}. \nNote, however, that the bounds in the above theorem are stated in such a way that they are invariant under the transform $(\\phi, r, m) \\mapsto (\\phi + c, r, m)$, under which \\eqref{eq:SSESF} is also invariant. Any solution may be then transformed to satisfy $\\phi(1, 1) = 0$. As a consequence, we do not need to check $\\phi(1,1) = 0$ in order to apply the theorem.\n\\end{remark}\nUsing Theorem \\ref{Chr.BV.Thm}, we prove the following bound for locally BV scattering solution to \\eqref{eq:SSESF}.\n\n\\begin{theorem} \\label{thm:decayInCpt}\nLet $(\\phi, r, m)$ be a locally BV scattering solution to \\eqref{eq:SSESF}. For every $\\epsilon > 0$, there exists $u_{0} > 1$ such that the following estimate holds.\n\\begin{align*}\n\t\\sup_{v \\in [u_{0}, \\infty)} \\Big( \\int_{\\underline{C}_{v} \\cap \\set{u \\geq u_{0}}\\cap \\PD_{\\mathrm{cpt}}} \\abs{\\partial_{u}^{2} (r \\phi)} \n\t+ \\int_{\\underline{C}_{v} \\cap \\set{u \\geq u_{0}} \\cap \\PD_{\\mathrm{cpt}}} \\abs{\\partial_{u} \\phi} \n\t+ \\int_{\\underline{C}_{v} \\cap \\set{u \\geq u_{0}} \\cap \\PD_{\\mathrm{cpt}}} \\abs{\\partial_{u} \\log \\nu} \\Big) < \\epsilon.\n\\end{align*}\n\nMoreover, we also have\n\\begin{equation} \\label{eq:bnd4durphi}\n\t\\sup_{\\calQ} \\abs{\\partial_{u} (r \\phi)} \\leq C_{K, \\Lambda} \\Psi.\n\\end{equation}\n\\end{theorem}\n\n\\begin{proof}\nWe first show that for a locally BV scattering solution to \\eqref{eq:SSESF},\n\\begin{equation*}\n\t\\int_{C_{u} \\cap \\PD_{\\mathrm{cpt}}} \\abs{\\partial_{v} (\\lmb^{-1} \\partial_{v} (r \\phi))} \\to 0 \\hbox{ as } u \\to \\infty,\n\\end{equation*}\nExpanding this expression, we have\n\\begin{eqnarray*}\n\\int_{C_{u} \\cap \\PD_{\\mathrm{cpt}}} \\abs{\\partial_{v} (\\lmb^{-1} \\partial_{v} (r \\phi))}\n\\leq \\int_{C_{u} \\cap \\PD_{\\mathrm{cpt}}} \\lmb^{-1} (\\abs{ \\partial_{v}^2 (r \\phi)}+\\abs{ (\\partial_{v} \\log \\lambda) \\partial_v(r \\phi)})\n\\end{eqnarray*}\nBy \\eqref{eq:bnd4dvr} and \\eqref{eq:bnd4dvrphi}, we have\n$$\\int_{C_{u} \\cap \\PD_{\\mathrm{cpt}}} \\abs{\\partial_{v} (\\lmb^{-1} \\partial_{v} (r \\phi))}\\leq \nC_{\\Lambda, \\Psi}\\int_{C_{u} \\cap \\PD_{\\mathrm{cpt}}} \\abs{ \\partial_{v}^2 (r \\phi)}+\\abs{ \\partial_{v} \\log \\lambda },$$\nwhich by \\eqref{eq:locBVScat} in Definition \\ref{def:locBVScat} (Scattering in BV in a compact $r$-region) tends to 0 as $u\\to \\infty$.\nNotice that the quantity $\\int_{C_{u} \\cap \\PD_{\\mathrm{cpt}}} \\abs{\\partial_{v} (\\lmb^{-1} \\partial_{v} (r \\phi))}$ which we have controlled is invariant under any rescaling of the coordinate $v$, and also under the transform $(\\phi, r, m) \\to (\\phi + c, r, m)$.\n\nWe now proceed to the proof of the theorem. Let $v_0$ be sufficiently large and $u^{\\star}(v_0)$ be the unique $r(u^{\\star}(v_0),v_0)=R$. By the finite speed of propagation of the equations, the solution on $\\underline{C}_{v_0}\\cap\\PD_{\\mathrm{cpt}}$ depends only on the data on $C_{u^{\\star}(v_0)} \\cap\\PD_{\\mathrm{cpt}}$.\n\nIn order to apply Theorem \\ref{Chr.BV.Thm}, we change coordinates $(u,v)\\mapsto (U(u),V(v))$ in the region bounded by $C_{u^{\\star}(v_0)}$ and $\\underline{C}_{v_0}$ to a new double null coordinate $(U,V)$ such that for $U^{\\star}=U(u^{\\star}(v_0))$, we have $\\lambda(U^{\\star},V)=\\frac 12$. To this end, define $V(v)$ by\n$$\\frac{dV}{dv}=2\\lmb(u^{\\star}(v_0),v),\\quad V(v_0)=v_0.$$\nNotice that this is acceptable since $\\lmb>0$. In order for the condition $U = V$ to hold on $\\Gamma$, we require\n$U(u)=V(u).$\nThen with respect to the coordinate $V$\n$$\\partial_V r(U^{\\star},V)=\\frac 12.$$\nBy \\eqref{eq:bnd4dvr}, we have\n$$\\Lambda^{-1}\\leq \\frac{dV}{dv}, \\frac{dU}{du}\\leq \\frac 12.$$\nMoreover,\n$$\\int_{u^{\\star}(v_0)}^{v_0} |\\frac{d^2V}{dv^2}(v')|dv'\\leq 2\\int_{u^{\\star}(v_0)}^{v_0} |\\partial_v\\lambda(u^{\\star}(v_0),v')| \\mathrm{d} v',$$\nwhich tends to $0$ as $v_0\\to \\infty$ by the assumption of local BV scattering. For $v_0$ sufficiently large, in the $(U,V)$ coordinate system, $\\int_{C_{u^{\\star}(v_0)} \\cap \\PD_{\\mathrm{cpt}}} \\abs{\\partial_{V} ((\\partial_{V} r)^{-1} \\partial_{V} (r \\phi))} dV$ is small and $\\partial_V r=\\frac 12$. The data satisfy the assumptions of Theorem \\ref{Chr.BV.Thm} and therefore\\footnote{More precisely, we apply Theorem \\ref{Chr.BV.Thm} to the truncated initial data $$ \\partial_{V} (r \\widetilde{\\phi})(U^{\\star}, V) = \\left\\{ \\begin{array}{cc} \\partial_{V}(r \\phi)(U^{\\star}, V) & \\hbox{ for } V < v_{0} \\\\ \\partial_{V}(r \\phi)(U^{\\star}, v_{0}) & \\hbox{ for } V \\geq v_{0} \\end{array} \\right.$$}\n$$\\int_{\\underline{C}_{v_0}\\cap\\PD_{\\mathrm{cpt}}}(\\abs{\\partial_{U} ((\\partial_{U} r)^{-1} \\partial_{U} (r \\phi))} + \\abs{\\partial_{U} \\phi} + \\abs{\\partial_{U} \\log \\partial_{U} r} ) dU \\to 0$$\nas $v_{0} \\to \\infty$. Returning to the original coordinate system $(u,v)$, the first statement easily follows.\n\nFinally, for the $L^\\infty$ estimate for $\\partial_{u}(r\\phi)$, notice that $\\abs{\\partial_{u}(r\\phi)} \\leq \\Psi$ at the axis by \\eqref{eq:bnd4dvrphi} and $(7)$ of Definition \\ref{def:BVsolution} (BV solutions to \\eqref{eq:SSESF}). Using \\eqref{eq:SSESF:dphi''}, \\eqref{eq:SSESF:dr} (in particular, the fact that $\\partial_{v} \\nu \\leq 0$), \\eqref{eq:bnd4phi} and \\eqref{eq:bnd4dur}, we have\n\\begin{equation*}\n\\abs{\\partial_{u}(r\\phi)(u,v)} \\leq \\Psi+ \\Lambda \\Psi \\int_{u}^{v} (-\\partial_{v} \\nu) \\, \\mathrm{d} v' \\leq C_{K, \\Lambda} \\Psi. \\qedhere\n\\end{equation*}\n\\end{proof}\n\n\n\\section{Decay of $\\phi$ and its first derivatives}\\label{sec.decay1}\nIn this section, we prove the first main theorem (Theorem \\ref{main.thm.1}). Throughout this section, we assume that $(\\phi, r, m)$ is a locally BV scattering solution to \\eqref{eq:SSESF} with asymptotically flat initial data of order $\\omega'$ in BV, as in Definitions \\ref{def:locBVScat} and \\ref{def:AF}, respectively. Let $\\omega = \\min \\set{\\omega', 3}$.\n\n\\subsection{Preparatory lemmas}\nThe following lemma will play a key role in the proof of both Theorems \\ref{main.thm.1} and \\ref{main.thm.2}. It is a consequence of the scattering assumption \\eqref{eq:locBVScat} and vanishing of the final Bondi mass.\n\n\\begin{lemma} \\label{lem:smallPtnl}\nLet $\\epsilon > 0$ be an arbitrary positive number. For $u_{1} > 1$ sufficiently large, we have\n\\begin{align} \n\t\\sup_{v \\in [u_{1}, \\infty)} \\int_{\\underline{C}_{v} \\cap \\set{u \\geq u_{1}}} \\abs{\\frac{2m \\nu}{(1-\\mu) r^{2}}}   < \\epsilon, \\label{eq:smallPtnl:u} \\\\ \n\t\\sup_{u \\in [u_{1}, \\infty)} \\int_{C_{u}} \\abs{\\frac{2m \\lmb}{(1-\\mu) r^{2}}} < \\epsilon. \\label{eq:smallPtnl:v}\n\\end{align}\n\\end{lemma}\n\n\\begin{proof} \nThe first statement \\eqref{eq:smallPtnl:u} was proved in Lemma \\ref{lem:babySmllPtnl:2}; thus it only remains to prove \\eqref{eq:smallPtnl:v}.\n\nDivide $\\calQ$ into $\\PD_{\\mathrm{cpt}} =\\calQ\\cap\\set{r \\leq R}$ and $\\PD_{\\mathrm{cpt}}^{c} := \\calQ \\setminus \\PD_{\\mathrm{cpt}}$. First, note that by \\eqref{eq:locBVScat} we have\n\\begin{align*}\n\t\\sup_{u \\in [u_{1}, \\infty)} \\int_{C_{u} \\cap \\PD_{\\mathrm{cpt}}} \\abs{\\frac{2m \\lmb}{(1-\\mu) r^{2}}} < \\epsilon\/2\n\\end{align*}\nfor $u_{1}$ sufficiently large. \nNext, we consider $\\PD_{\\mathrm{cpt}}^{c}$. Define $v^{\\star}(u) := \\sup \\set{v \\in [u, \\infty) : r(u,v) \\geq R}$; note that $r(u, v^{\\star}(u)) = R$ by continuity. We now compute\n\\begin{align*}\n\t\\int_{C_{u} \\cap \\PD_{\\mathrm{cpt}}^{c}} \\abs{\\frac{2 m \\lmb}{(1-\\mu) r^{2}}}  \n\t= & \\int_{v^{\\star}(u)}^{\\infty} \\abs{\\frac{2m \\lmb}{(1-\\mu) r^{2}}(u,v')} \\, \\mathrm{d} v' \\\\\n\t\\leq & 2 K \\Lambda M(u_{1}) \\int_{v^{\\star}(u)}^{\\infty} \\frac{\\lmb}{r^{2}}(u, v') \\, \\mathrm{d} v' \\\\\n\t\\leq & 2 R^{-1} K \\Lambda M(u_{1}).\n\\end{align*}\nuniformly in $u \\geq u_{1}$. As $\\lim_{u_{1} \\to \\infty} M(u_{1}) = 0$ by \\eqref{eq:zeroMf} (vanishing final Bondi mass), the last line can be made arbitrarily small by taking $u_{1}$ sufficiently large. This proves \\eqref{eq:smallPtnl:v}. \\qedhere\n\\end{proof}\n\nThe following lemma allows us to estimate $\\phi$ in terms of $\\abs{\\partial_{v} (r \\phi)}$.\n\\begin{lemma} \\label{lem:intEst4phi}\nThe following estimates hold.\n\\begin{align*} \n\t\\abs{\\phi}(u,v) \\leq & \\Lambda \\sup_{C_{u} } \\abs{\\partial_{v} (r \\phi)} , \\\\\n\tr u^{\\omega-1} \\abs{\\phi}(u,v) \\leq & C \\Lambda \\Big( \\sup_{C_{u}} u^{\\omega} \\abs{\\partial_{v} (r \\phi)} + \\sup_{C_{u}} r^{\\omega} \\abs{\\partial_{v} (r \\phi)} \\Big). \n\\end{align*}\n\\end{lemma}\n\n\\begin{proof} \nThe first estimate follows from Lemma \\ref{lem:est4phi} and Proposition \\ref{prop:geomLocBVScat}. The second estimate is a consequence of the first when $r(u,v) \\leq u$, so it suffices to assume $r(u,v) \\geq u$. Introducing a parameter $v_{1} \\in [u, v]$, we estimate\n\\begin{align*}\n\tr u^{\\omega-1} \\abs{\\phi}(u,v) \n\t\\leq & u^{\\omega-1} \\int_{u}^{v} \\abs{\\partial_{v} (r \\phi)(u, v')} \\, \\mathrm{d} v' \\\\\n\t\\leq & \\Lambda u^{\\omega-1}(\\sup_{C_u}|\\partial_v(r\\phi)|)\\int_u^{v_1}\\lmb(u,v')\\mathrm{d} v'+\\Lambda u^{\\omega-1}(\\sup_{C_u}r^{\\omega}|\\partial_v(r\\phi)|)\\int_{v_1}^{v}\\frac{\\lmb}{r^{\\omega}}(u,v')\\mathrm{d} v'\\\\\n\t\\leq & \\Lambda (r(u, v_{1})\/u) \\sup_{C_{u}} u^{\\omega} \\abs{\\partial_{v} (r \\phi)} + \\frac{\\Lambda}{\\omega-1} (u^{\\omega-1}\/r(u, v_{1})^{\\omega-1}) \\sup_{C_{u}} r^{\\omega} \\abs{\\partial_{v} (r \\phi)}.\n\\end{align*}\n\nChoosing $v_{1}$ so that $r(u, v_{1}) = u$ (which is possible since $r(u,v) \\geq u$), the desired estimate follows.\n\\end{proof}\n\n\\subsection{Preliminary $r$-decay for $\\phi$} \\label{subsec:decay1:rDecay}\nIn this subsection, we derive bounds for $\\phi$ which are sharp in terms of $r$-weights. As a consequence, they give sharp decay rates towards null infinity.\n\n\\begin{lemma} \\label{lem:decay1:cptu:0}\n\tThere exists a constant $0 < H_{1} < \\infty$ such that the following estimate holds.\n\t\\begin{equation} \\label{eq:decay1:cptu:0}\n\t\t\\sup_{\\calQ} (1+r) \\abs{\\phi}   \\leq H_{1}.\n\t\\end{equation}\n\\end{lemma}\n\n\\begin{proof} \nLet $r_{1} > 0$ be a large number to be chosen below. Different arguments will be used in $\\set{r \\geq r_{1}}$ and $\\set{r \\leq r_{1}}$. For each $u \\geq 1$ let $v^{\\star}_{1}(u)$ be the unique $v$-value for which $r(u, v_{1}^{\\star}(u)) = r_{1}$. By the fundamental theorem of calculus, we have\n\\begin{equation} \\label{eq:decay1:cptu:0:pf:1}\n\tr \\phi = r_{1} \\phi(u, v^{\\star}_{1} (u)) + \\int_{v^{\\star}_{1}(u)}^{v} \\partial_{v} (r \\phi) (u, v') \\, \\mathrm{d} v'.\n\\end{equation} \n\nIntegrate \\eqref{eq:SSESF:dphi} along the incoming direction from $(1,v)$ to $(u,v)$. By Corollary \\ref{cor:mntn4Bondi} and Proposition \\ref{prop:geomLocBVScat}, we have\n\\begin{align*}\n\t\\abs{\\partial_{v} (r \\phi)(u,v)}\n\t\\leq& \\abs{\\partial_{v} (r \\phi)(1, v)} + \\abs{\\int_{1}^{u} \\frac{2m \\lmb \\nu}{(1-\\mu) r^{3}} (r\\phi) (u', v) \\, \\mathrm{d} u'} \\\\\n\t\\leq &\\abs{\\partial_{v} (r \\phi)(1, v)} + \\frac{K \\Lambda M_{i}}{2} \\frac{1}{r^{2}(u,v)}   \\sup_{u' \\in [1, u]} \\abs{r \\phi(u', v)}.\n\\end{align*}\n\nSubstituting the preceding bound into \\eqref{eq:decay1:cptu:0:pf:1}, we obtain\n\\begin{equation} \\label{eq:decay1:cptu:0:pf:2}\n\\begin{aligned}\n\t\\sup_{C_{u} \\cap \\set{r \\geq r_{1}}} \\abs{r \\phi} \n\t\\leq & \\abs{r_{1} \\phi(u, v^{\\star}_{1} (u))} + \\int_{v_{1}^{\\star}(u)}^{v} \\abs{\\partial_{v} (r \\phi)(1, v')} \\, \\mathrm{d} v' \\\\\n\t& + \\frac{K \\Lambda^{2} M_{i}}{2 r_{1}}  \\sup_{u' \\in [1, u]} \\sup_{C_{u'} \\cap \\set{r \\geq r_{1}}} \\abs{r \\phi}.\n\\end{aligned}\n\\end{equation}\n\nThe first term on the right-hand side is bounded by $r_{1} \\Lambda \\Psi$ by \\eqref{eq:bnd4phi}, whereas the second term depends only on the initial data and can be estimated in terms of $\\mathcal I_{1}$ as follows:\n\\begin{equation*}\n\t\\int_{v_{1}^{\\star}(u)}^{v} \\abs{\\partial_{v} (r \\phi)(1, v')} \\, \\mathrm{d} v' \\leq \\Lambda \\mathcal I_{1} \\int_{1}^{\\infty}  (1+r(1, v'))^{-\\omega'} \\lmb(1,v') \\, \\mathrm{d} v' \\leq \\frac{\\Lambda}{\\omega'-1} \\mathcal I_{1}.\n\\end{equation*}\n\nMoreover, choosing $r_{1}$ to be large enough so that \n\\begin{equation*}\n\\frac{K \\Lambda^{2}  M_{i}}{2 r_{1}} \\leq \\frac{1}{2},\n\\end{equation*}\nthe last term of \\eqref{eq:decay1:cptu:0:pf:2} can be absorbed in to the left-hand side and we conclude\n\\begin{equation*}\n\t\\sup_{\\set{r \\geq r_{1}}} \\abs{r \\phi} \\leq 2 r_{1} \\Lambda \\Psi + \\frac{2}{\\omega'-1}\\Lambda \\mathcal I_{1}.\n\\end{equation*}\n\nOn the other hand, in $\\set{r \\leq r_{1}}$ we have\n\\begin{equation*}\n\t\\sup_{\\set{r \\leq r_{1}}} \\abs{r \\phi} \\leq r_{1} \\Lambda \\Psi\n\\end{equation*}\nby \\eqref{eq:bnd4phi}. Combining the bounds in $\\{r\\geq r_1\\}$ and $\\{r\\leq r_1\\}$, the lemma follows. \\qedhere\n\\end{proof}\n\n\\begin{remark} \nThe preceding argument shows that Lemma \\ref{lem:decay1:cptu:0} holds with\\footnote{Notice that while the constant $C_{\\mathcal I_{1}, K, \\Lambda}$ depends on $\\mathcal I_1$, the preceding argument moreover allows us to choose $C_{\\mathcal I_{1}, K, \\Lambda}$ to be non-decreasing in $\\mathcal I_1$. In particular, \\emph{for $\\mathcal I_1$ sufficiently small}, we have $H_{1} \\leq C_{K, \\Lambda} \\, (\\mathcal I_{1} + \\Psi)$. It is for this reason that we prefer to write the expression $C_{\\mathcal I_{1}, K, \\Lambda} \\, (\\mathcal I_{1} + \\Psi)$ instead of the more general $C_{\\mathcal I_{1}, K, \\Lambda, \\Psi}$.}\n\\begin{equation} \\label{eq:decay1:H1}\n\tH_{1} \\leq C_{\\mathcal I_{1}, K, \\Lambda} \\, (\\mathcal I_{1} + \\Psi).\n\\end{equation}\n\\end{remark}\n\n\n\\subsection{Propagation of $u$-decay for $\\partial_{u} (r \\phi)$}\nHere, we show that $u$-decay estimates proved for $\\partial_{v} (r \\phi)$ and $\\phi$ may be `transferred' to $\\partial_{u} (r \\phi)$; this reduces the proof of Theorem \\ref{main.thm.1} to showing only \\eqref{eq:decay1:1} and \\eqref{eq:decay1:2}. To this end, we integrate $\\partial_{v} \\partial_{u} (r \\phi)$ from the axis $\\Gamma$, along which $\\partial_{u} (r \\phi) = - \\partial_{v} (r \\phi)$. \n\n\\begin{lemma} \\label{lem:decay1:uDecay4durphi}\nSuppose that there exists a finite positive constant $A$ such that \n\\begin{equation*}\n\t\\sup_{\\calQ} \\abs{\\phi} \\leq A u^{-\\omega}, \\qquad \n\t\\sup_{\\calQ} \\abs{\\partial_{v} (r \\phi)} \\leq A u^{-\\omega}.\n\\end{equation*}\n\nThen the following estimate holds.\n\\begin{equation*}\n\t\\sup_{\\calQ} \\abs{\\partial_{u} (r \\phi)} \\leq (1+K)A u^{-\\omega}.\n\\end{equation*}\n\\end{lemma}\n\n\\begin{proof}\nFix $u \\geq 1$ and $v \\geq u$. Integrate \\eqref{eq:SSESF:dphi''} along the outgoing direction from $(u, u)$ to $(u, v)$ and take the absolute value. Using $(7)$ of Definition \\ref{def:BVsolution} (BV solutions to \\eqref{eq:SSESF}), \\eqref{eq:SSESF:dr} (in particular, $\\partial_{v} \\nu\\leq 0$), \\eqref{eq:bnd4dur} and the hypotheses, we have\n\\begin{align*}\n\t\\abs{\\partial_{u}(r\\phi)(u,v)} \n\t\\leq & \\lim_{v' \\to u+} \\abs{\\partial_{v} (r \\phi)(u, v')} + \\sup_{u \\leq v' \\leq v} \\abs{\\phi(u,v')} \\int_{u}^{v} (-\\partial_{v} \\nu) \\, \\mathrm{d} v' \\\\\n\t\\leq & A u^{-\\omega} + K A u^{-\\omega}. \\qedhere\n\\end{align*}\n\\end{proof}\n\n\\subsection{Full decay for $\\phi$ and $\\partial_{v} (r \\phi)$}\\label{sec.full.decay.1}\nIn this subsection, we finish the proof of Theorem \\ref{main.thm.1}. By Lemma \\ref{lem:decay1:uDecay4durphi}, it suffices to establish the full decay of $\\phi$ and $\\partial_{v} ( r \\phi)$, i.e., \\eqref{eq:decay1:1} and \\eqref{eq:decay1:2}. For the convenience of the reader, we recall these estimates below:\n\\begin{align*}\n\t\t\\abs{\\phi} \\leq & A \\min \\set{u^{-\\omega}, r^{-1} u^{-(\\omega-1)}}, \\tag{\\ref{eq:decay1:1}} \\\\\n\t\t\\abs{\\partial_{v} (r \\phi)} \\leq & A \\min \\set{u^{-\\omega}, r^{-\\omega}}. \\tag{\\ref{eq:decay1:2}}\n\\end{align*}\n\nFor $U > 1$, let\n\\begin{equation*}\n\\mathcal B_{1}(U) := \\sup_{u \\in [1, U]} \\sup_{C_{u}} \\Big( u^{\\omega} \\abs{\\phi} + r u^{\\omega-1} \\abs{\\phi}  \\Big).\n\\end{equation*}\n\nNotice that this is finite for every fixed $U$ by Lemma \\ref{lem:decay1:cptu:0}. To establish the decay estimate \\eqref{eq:decay1:1}, it suffices to prove that $\\mathcal B_{1}(U)$ is bounded by a finite constant which is \\emph{independent of $U$}. We will show that this implies also \\eqref{eq:decay1:2}. Divide $\\calQ$ into $\\PD_{\\mathrm{ext}} \\cup \\PD_{\\mathrm{int}}$, defined by\n\\begin{equation*}\n\t\\PD_{\\mathrm{ext}} := \\set{(u,v) \\in \\calQ : v \\geq 3u}, \\quad \\PD_{\\mathrm{int}} := \\set{(u,v) \\in \\calQ : v \\leq 3u}.\n\\end{equation*}\n\nWe first establish a bound for $\\partial_{v} (r \\phi)$ with the sharp $r$-weight, which thus gives the sharp decay rate in $\\PD_{\\mathrm{ext}}$. \n\\begin{lemma} \\label{lem:decay1:extr}\nLet $u_{1} > 1$. Then for $u_{1}\\leq u\\leq U$, the following estimate holds.\n\\begin{equation} \\label{eq:decay1:extr}\n\t\\sup_{C_{u}} r^{\\omega} \\abs{\\partial_{v} (r \\phi)} \\leq \\mathcal I_{1} +   C_{K, M_{i}} \\, u_{1} H_{1} + C u_{1}^{-1} K M_{i} \\, \\mathcal B_{1}(U).\n\\end{equation}\n\\end{lemma}\n\n\\begin{proof} \nWe separate the proof into cases $\\omega \\geq 2 $ and $1<\\omega\\leq 2$.\n\n\\noindent{\\bf Case 1: $\\omega\\geq 2$}\n\nFirst, notice that\n$$|\\phi|\\leq \\mathcal B_1(U) (r^{-1}u^{-(\\omega-1)})^{\\omega-2} (u^{-\\omega})^{1-(\\omega-2)}\\leq \\mathcal B_1(U) r^{-(\\omega-2)}u^{-2}.$$\nApplying Lemma \\ref{lem:decay1:cptu:0}, we also have\n$$|\\phi|\\leq (1+r)^{-1}H_1.$$\nBy Corollary \\ref{cor:mntn4Bondi} and Proposition \\ref{prop:geomLocBVScat}, we have the following pointwise bounds:\n\\begin{equation*}\n\\sup_{u'\\in [1,u_1]} |\\frac{m\\lmb\\nu}{1-\\mu}|\\leq \\frac{K M_i}{2}\\, , \\quad\n\\sup_{u'\\in [u_{1},\\infty)} |\\frac{m\\lmb\\nu}{1-\\mu}|\\leq \\frac{K M(u_{1})}{2}\\, .\n\\end{equation*}\nTherefore, integrating \\eqref{eq:SSESF:dphi} along the incoming direction from $(1, v)$ to $(u, v)$, we have\n\\begin{align*}\n\t&\\abs{\\partial_{v} (r \\phi)(u,v)}\\\\\n\t& \\quad \\leq \\abs{\\partial_{v} (r \\phi)(1, v)} + \\abs{\\int_{1}^{u} \\frac{2m \\lmb \\nu\\phi}{(1-\\mu) r^{2}}  (u', v) \\, \\mathrm{d} u'} \\\\\n\t& \\quad \\leq \\abs{\\partial_{v} (r \\phi)(1, v)} + \\frac{K M_{i}}{r^2(u,v)(1+r(u,v))} H_{1} \\int_{1}^{u_{1}}  \\, \\mathrm{d} u' + \\frac{K M(u_1)}{r^{\\omega}(u,v)} \\mathcal B_{1}(U) \\int_{u_{1}}^{u} (u')^{-2} \\, \\mathrm{d} u' \\\\\n\t& \\quad \\leq \\abs{\\partial_{v} (r \\phi)(1, v)} + \\frac{u_{1} K M_{i}}{r^2(u,v)(1+r(u,v))} H_{1} + \\frac{K M(u_{1})}{u_{1} r^{\\omega}(u,v)} \\mathcal B_{1}(U).\n\\end{align*}\n\nMultiplying both sides by $r^{\\omega}(u,v)$ and using the fact that $r(u,v) \\leq r(1, v)$, we conclude\n\\begin{align*}\n\tr^{\\omega} \\abs{\\partial_{v} (r \\phi)}(u,v) \n\t\\leq & r^{\\omega}\\abs{\\partial_{v} (r \\phi)}(1,v) + u_{1} \\frac{r^{\\omega-2}}{(1+r)} K M_{i} \\, H_{1}+ u_{1}^{-1} K M(u_{1}) \\, \\mathcal B_{1}(U) \\\\\n\t\\leq & \\mathcal I_{1}+ C_{u_{1}, K, M_{i}}  H_{1} + u_{1}^{-1} K M_{i} \\, \\mathcal B_{1}(U).\n\\end{align*}\n\n\\noindent{\\bf Case 2: $1<\\omega\\leq 2$}\n\nWe will use the following bounds for $\\phi$. First, \n$$|\\phi|\\leq \\mathcal B_1(U) (r^{-1}u^{-(\\omega-1)})^{\\omega-1} (u^{-\\omega})^{(2-\\omega)}\\leq \\mathcal B_1(U) r^{-(\\omega-1)}u^{-1}.$$\n\nAlso, Lemma \\ref{lem:decay1:cptu:0} implies\n$$|\\phi|\\leq (1+r)^{-1}H_1.$$\n\nAs in Case 1 we integrate \\eqref{eq:SSESF:dphi} along the incoming direction from $(1, v)$ to $(u, v)$:\n\\begin{align*}\n\t&\\abs{\\partial_{v} (r \\phi)(u,v)}\\\\\n\t& \\quad \\leq \\abs{\\partial_{v} (r \\phi)(1, v)} + \\abs{\\int_{1}^{u} \\frac{2m \\lmb \\nu\\phi}{(1-\\mu) r^{2}}  (u', v) \\, \\mathrm{d} u'} \\\\\n\t& \\quad \\leq \\abs{\\partial_{v} (r \\phi)(1, v)} + \\frac{K \\Lambda M_{i} H_{1}}{(1+r)} \\int_{1}^{u_{1}} \\frac{-\\nu}{r^2}  \\, \\mathrm{d} u' \n\t\t\t\t\t\t\t\t+ \\frac{K \\Lambda M(u_1)}{u_1} \\mathcal B_{1}(U) \\int_{u_{1}}^{u}  \\frac{-\\nu}{r^{\\omega+1}} \\, \\mathrm{d} u' \\\\\n\t& \\quad \\leq \\abs{\\partial_{v} (r \\phi)(1, v)} + \\frac{\\omega K \\Lambda M_{i}}{r(u,v)(1+r(u,v))} H_{1} + \\frac{\\omega K \\Lambda M(u_{1})}{u_{1} r^{\\omega}(u,v)} \\mathcal B_{1}(U).\n\\end{align*}\n\nMultiply both sides by $r^{\\omega}$ to arrive at the conclusion as in Case 1. In this case, note that the second term is a bit better than what is claimed, as there is no dependence on $u_{1} \\geq 1$. \\qedhere\n\\end{proof}\n\n\\begin{remark} \nNote that the proof of this lemma limits $\\omega$ to be $\\leq 3$. More precisely, this limitation comes from the contribution of the right-hand side of \\eqref{eq:SSESF:dphi}\n\\end{remark}\n\n\nWe are now ready to prove bounds \\eqref{eq:decay1:1} and \\eqref{eq:decay1:2}. The idea is  to `propagate' the exterior decay estimate \\eqref{eq:decay1:extr} into $\\PD_{\\mathrm{int}}$ to obtain decay in $u$, using the smallness coming from Lemma \\ref{lem:smallPtnl} in the region where $u$ is sufficiently large. On the other hand, the preliminary $r$-decay estimates proved in \\S \\ref{subsec:decay1:rDecay} will give the desired $r$-decay rates in rest of the space-time.\n\n\\begin{proof}[Proof of \\eqref{eq:decay1:1} and \\eqref{eq:decay1:2}] \nLet $1 \\leq u_{1} \\leq U$. For $(u, v) \\in \\calQ$ with $u \\in [3 u_{1}, U]$, integrate \\eqref{eq:SSESF:dphi} along the incoming direction from $(u\/3, v)$ to $(u, v)$. Then\n\\begin{equation} \\label{eq:decay1:intr:pf:1}\n\\begin{aligned}\n\t\\abs{\\partial_{v} (r \\phi) (u,v)} \\leq \n\t& \\abs{\\partial_{v} (r \\phi) (u\/3,v)} \\\\\n\t& + \\frac{1}{2} (\\sup_{u' \\in [u\/3, u]} \\sup_{C_{u'}} \\abs{\\phi}) \\int_{u\/3}^{u} \\abs{\\frac{2m \\nu}{(1-\\mu) r^{2}} (u', v)} \\, \\mathrm{d} u'.\n\\end{aligned}\n\\end{equation} \n\nMultiply both sides by $u^{\\omega}$ and estimate each term on the right-hand side.  For the first term, the key observation is the following: For $v \\geq u$, the point $(u\/3, v)$ lies in $\\PD_{\\mathrm{ext}}$, where \\eqref{eq:decay1:extr} is effective. Indeed, note that\n\\begin{equation*}\n\t(2\/3\\Lambda)  u \\leq \\Lambda^{-1} ( v- (u\/3) ) \\leq r(u\/3, v).\n\\end{equation*}\n\nThus, by \\eqref{eq:decay1:extr},\n\\begin{align*}\n\tu^{\\omega} \\abs{\\partial_{v} (r \\phi) (u\/3,v)} \n\t\\leq & (3 \\Lambda \/2)^{\\omega} \\Big( r^{\\omega}(u\/3, v) \\abs{\\partial_{v} (r \\phi) (u\/3,v)} \\Big) \\\\\n\t\\leq & (3 \\Lambda\/2)^{\\omega} \\Big( \\mathcal I_{1} +   C_{u_{1}, K, M_{i}}  H_{1} + C u_{1}^{-1} K M_{i} \\, \\mathcal B_{1}(U) \\Big) \\\\\n\t\\leq & C_{u_{1}, K, \\Lambda, M_{i}} (\\mathcal I_{1} + H_{1}) + C_{K, \\Lambda} M_{i} u_{1}^{-1} \\, \\mathcal B_{1}(U).\n\\end{align*}\n\nFor the second term on the right-hand side of \\eqref{eq:decay1:intr:pf:1}, we have\n\\begin{align*}\n\\frac{u^{\\omega}}{2} (\\sup_{u' \\in [u\/3, u]} \\sup_{C_{u'}} \\abs{\\phi}) \\int_{u\/3}^{u} \\abs{\\frac{2m \\nu}{(1-\\mu) r^{2}} (u', v)} \\, \\mathrm{d} u'\n\\leq & \\frac{3^{\\omega}}{2} \\Big( \\int_{u\/3}^{u} \\abs{\\frac{2m \\nu}{(1-\\mu) r^{2}} (u', v)} \\, \\mathrm{d} u' \\Big) \\mathcal B_{1}(U).\n\\end{align*}\n\nCombining these estimates, we deduce\n\\begin{equation} \\label{eq:decay1:intr:pf:2}\n\\begin{aligned}\n\t\\sup_{C_{u}} u^{\\omega} \\abs{\\partial_{v}(r \\phi)(u, v)} \\leq & C_{u_{1}, K, \\Lambda, M_{i}} (\\mathcal I_{1} + H_{1}) \\\\ \n\t&+ \\Big( C_{K, \\Lambda} M_{i} u_{1}^{-1} + C \\int_{u\/3}^{u}  \\abs{\\frac{2m \\nu}{(1-\\mu) r^{2}} (u', v)} \\, \\mathrm{d} u' \\Big) \\, \\mathcal B_{1}(U).\n\\end{aligned}\n\\end{equation}\n\nRecalling the bounds of $\\phi$ in terms of $\\partial_v(r\\phi)$ in Lemmas \\ref{lem:intEst4phi}, we have\n\\begin{align*}\n\t\\mathcal B_{1}(U) \n\t\\leq & (1+2\\Lambda) \\sup_{u \\in [1, U]} \\sup_{C_{u}} \\Big( u^{\\omega} \\abs{\\partial_{v}(r \\phi)} + r^{\\omega} \\abs{\\partial_{v} (r \\phi)} \\Big).\n\\end{align*}\nThe right-hand side can be controlled by \\eqref{eq:decay1:intr:pf:2} and \\eqref{eq:decay1:extr}, from which we conclude\n\\begin{equation} \\label{eq:decay1:intr:pf:key}\n\t\\mathcal B_{1}(U) \\leq C_{u_{1}, K, \\Lambda, M_{i}} (\\mathcal I_{1} + H_{1}) + \\Big( C_{K, \\Lambda} M_{i} u_{1}^{-1} + C \\int_{u\/3}^{u}  \\abs{\\frac{2m \\nu}{(1-\\mu) r^{2}} (u', v)} \\, \\mathrm{d} u'\\Big) \\mathcal B_{1}(U).\n\\end{equation}\n\nAs a consequence of Lemma \\ref{lem:smallPtnl}, the entire coefficient in front of $\\mathcal B_{1}(U)$ can made to be smaller than (say) $1\/2$ by taking $u_{1}$ sufficiently large. Since $\\mathcal B_{1}(U) < \\infty$, we can then absorb this term into the left-hand side. Observing that this bound is independent of $U > 1$, we have thus obtained \\eqref{eq:decay1:1}.\n\nTo prove \\eqref{eq:decay1:2}, simply apply \\eqref{eq:decay1:intr:pf:2} and \\eqref{eq:decay1:extr}, which shows that\n\\begin{align*}\n&\\sup_{u \\in [1, U]} \\sup_{C_{u}} \\Big( u^{\\omega} \\abs{\\partial_{v}(r \\phi)} + r^{\\omega} \\abs{\\partial_{v} (r \\phi)} \\Big)\\\\\n& \\quad \\leq  C_{u_{1}, K, \\Lambda, M_{i}} (\\mathcal I_{1} + H_{1}) + \\Big( C_{K, \\Lambda} M_{i} u_{1}^{-1} + C \\int_{u\/3}^{u}  \\abs{\\frac{2m \\nu}{(1-\\mu) r^{2}} (u', v)} \\, \\mathrm{d} u'\\Big) \\mathcal B_{1}(U).\n\\end{align*}\nThis boundedness of $\\mathcal B_1(U)$ that we just proved thus implies \\eqref{eq:decay1:2}.\n\\qedhere\n\\end{proof}\n\n\\begin{remark} \nAccording to the proof that we have just given, the constant $A_{1} > 0$ depends on our choice of $u_{1} > 1$, which in turn depends on how fast the coefficient in front of $\\mathcal B_{1}(U)$ in \\eqref{eq:decay1:intr:pf:key} vanishes as $u_{1} \\to \\infty$. This explains why $A_{1} > 0$ does not depend only on the size of the initial data, as remarked in Section \\ref{sec.main.thm}. Controlling the size of $u_{1} > 1$ under an additional small data assumption will be key to proving Statement (1) of Theorem \\ref{thm:smallData} in Section \\ref{sec:smallData}.\n\\end{remark}\n\n\\subsection{Additional decay estimates}\nWe end this section with the following decay estimates for $\\partial_{v} \\phi$, $\\partial_{u} \\phi$ and $m$.\n\\begin{corollary} \\label{cor:decay1}\nLet $(\\phi, r, m)$ be a locally BV scattering solution to \\eqref{eq:SSESF} with asymptotically flat initial data of order $\\omega'$ in BV, and define $\\omega = \\min \\set{\\omega', 3}$.\nLet $A_{1}$ be the constant in Theorem \\ref{main.thm.1}. Then the following decay estimates hold.\n\\begin{align} \n\t\\abs{\\partial_{v} \\phi} \\leq & C A_{1} \\min \\set{r^{-1} u^{-\\omega}, r^{-2} u^{-(\\omega-1)}}, \\label{eq:decay1:4} \\\\\n\t\\abs{\\partial_{u} \\phi} \\leq & C_{K} A_{1} \\, r^{-1} u^{-\\omega}, \\label{eq:decay1:5} \\\\\n\tm \\leq & C_{\\Lambda} A_{1}^{2} \\min \\set{r u^{-2\\omega}, u^{-(2\\omega-1)}}.  \\label{eq:decay1:6}\n\\end{align}\n\\end{corollary}\n\\begin{proof} \n\tLet $u \\geq 1$ and $v \\geq u$. Since\n\t\\begin{equation*}\n\t\tr \\partial_{v} \\phi = \\partial_{v} ( r \\phi) - \\lmb \\phi, \\qquad\n\t\tr \\partial_{u} \\phi = \\partial_{u} ( r \\phi) - \\nu \\phi,\n\t\\end{equation*}\n\tthe estimates \\eqref{eq:decay1:4}, \\eqref{eq:decay1:5} follow from \\eqref{eq:decay1:1}--\\eqref{eq:decay1:3} and the fact that $\\sup_{\\calQ} \\abs{\\lmb} \\leq 1\/2$, $\\sup_{\\calQ} \\abs{\\nu} \\leq K$. \n\t\n\tOn the other hand, by \\eqref{eq:SSESF:dm}, we have \n\t\\begin{equation} \\label{eq:decay1:6:pf:1}\n\t\tm(u,v) = \\frac{1}{2} \\int_{u}^{v} \\lmb^{-1} (1-\\mu) r^{2} (\\partial_{v} \\phi)^{2} (u, v')\\, \\mathrm{d} v'.\n\t\\end{equation}\n\t\n\tUsing $\\abs{\\partial_{v} \\phi (u,v) } \\leq C A_{1} r^{-1} u^{-\\omega}$ (which has just been established), we obtain\n\t\\begin{equation*}\n\t\tm(u,v) \\leq C_{\\Lambda} A_{1}^{2} \\, r u^{-2\\omega},\n\t\\end{equation*}\n\twhich proves a `half' of \\eqref{eq:decay1:6}. \n\tTo prove the other `half', let us introduce a parameter $r_{1} > 0$ (to be determined later) and define $v_{1}^{\\star}(u)$ to be the unique $v$-value such that $r(u, v^{\\star}_{1}(u)) = r_{1}$. For $v \\geq v^{\\star}_{1}(u)$, divide the $v'$-integral in \\eqref{eq:decay1:6:pf:1} into $\\int_{u}^{v^{\\star}_{1}(u)} + \\int_{v^{\\star}_{1}(u)}^{v}$ and use $\\abs{\\partial_{v} \\phi (u,v) } \\leq C A_{1} \\, r^{-1} u^{-\\omega}$ for the former and $\\abs{\\partial_{v} \\phi (u,v) } \\leq C A_{1} \\, r^{-2} u^{-(\\omega-1)}$ for the latter. As $m(u,v)$ is non-decreasing in $v$, we then arrive at the estimate\n\t\\begin{equation*}\n\t\t\\sup_{C_{u}} m \\leq C_{\\Lambda} A_{1}^{2} \\, r_{1} u^{-2\\omega} + C_{\\Lambda} A_{1}^{2} \\, r_{1}^{-1} u^{-2(\\omega-1)}.\n\t\\end{equation*}\n\t\nChoosing $r_{1} = u$, we obtain \\eqref{eq:decay1:6}. \\qedhere\n\\end{proof}\n\n\\section{Decay of second derivatives}\\label{sec.decay2}\nIn this section, we establish our second main theorem (Theorem \\ref{main.thm.2}). Throughout the section, we assume that $(\\phi, r, m)$ is a locally BV scattering solution to \\eqref{eq:SSESF} with asymptotically flat initial data of order $\\omega'$ in $C^{1}$, as in Definitions \\ref{def:locBVScat} and \\ref{def:AF}. As discussed in Remark \\ref{rem:wp}, $(\\phi, r, m)$ is then a $C^{1}$ solution to \\eqref{eq:SSESF}. As before, let $\\omega = \\min\\set{\\omega', 3}$. \n\n\\subsection{Preparatory lemmas}\nThe following lemma, along with Lemma \\ref{lem:smallPtnl}, provides the crucial smallness for our proof of Theorem \\ref{main.thm.2}.\n\\begin{lemma} \\label{lem:smallDphi}\nFor every $\\epsilon > 0$, there exists $u_{2} > 1$ such that\n\\begin{align}\n\t\\sup_{v \\in [u_{2}, \\infty)} \\int_{\\underline{C}_{v} \\cap \\set{u \\geq u_{2}}} \\abs{\\partial_{u} \\phi}  < \\epsilon, \\label{eq:smallDphi:u} \\\\ \n\t\\sup_{u \\in [u_{2}, \\infty)} \\int_{C_{u}} \\abs{\\partial_{v} \\phi} < \\epsilon. \\label{eq:smallDphi:v}\n\\end{align}\n\\end{lemma}\n\\begin{proof} \nWe will only prove \\eqref{eq:smallDphi:u}, leaving the similar proof of \\eqref{eq:smallDphi:v} to the reader. As in the proof of Lemma \\ref{lem:smallPtnl}, we divide $\\calQ$ into $\\PD_{\\mathrm{cpt}}:=\\calQ\\cap\\{r\\leq R\\}$ and $\\PD_{\\mathrm{cpt}}^{c} := \\calQ \\setminus \\PD_{\\mathrm{cpt}}$, and argue separately. First, by Theorem \\ref{thm:decayInCpt}, we have\n\\begin{equation*}\n\t\\sup_{v \\in [u_{2}, \\infty)} \\int_{\\underline{C}_{v} \\cap \\set{u \\geq u_{2}} \\cap \\PD_{\\mathrm{cpt}}} \\abs{\\partial_{u} \\phi}  < \\epsilon\/2, \n\\end{equation*}\nfor $u_{2}$ sufficiently large. Next, to derive \\eqref{eq:smallDphi:u} in $\\PD_{\\mathrm{cpt}}^{c}$, we define $u^{\\star}(v) := \\sup \\set{u \\in [u_{2}, v] : r(u,v) \\geq R}$, where we use the convention $u^{\\star}(v) = u_{2}$ when the set is empty. Then using Proposition \\ref{prop:geomLocBVScat} and Schwarz, we compute\n\\begin{align*}\n\t\\int_{\\underline{C}_{v} \\cap \\set{u \\geq u_{2}} \\cap \\PD_{\\mathrm{cpt}}^{c}} \\abs{\\partial_{u} \\phi} \n\t= & \\int_{u_{2}}^{u^{\\star}(v)} \\abs{\\partial_{u} \\phi(u',v)} \\, \\mathrm{d} u' \\\\\n\t\\leq & \\sqrt{\\frac{2 K \\Lambda}{R}} \\sqrt{\\int_{u_{2}}^{u^{\\star}(v)} \\frac{1}{2}(-\\nu)^{-1} (1-\\mu) r^{2} (\\partial_{u} \\phi)^{2} (u', v) \\, \\mathrm{d} u'} \\\\\n\t\\leq & \\sqrt{\\frac{2 K \\Lambda}{R} m(u_{2}, v)} \\leq \\sqrt{\\frac{2 K \\Lambda}{R} M(u_{2})}.\n\\end{align*}\n\nBy \\eqref{eq:zeroMf} (Vanishing final Bondi mass), $\\lim_{u_{2} \\to \\infty} M(u_{2}) = 0$. \\eqref{eq:smallDphi:u} thus follows. \\qedhere\n\\end{proof}\n\nThe next lemma allows us to estimate the first derivative of $\\phi$ at $(u,v)$ in terms of information on $C_{u} \\cap \\set{(u, v'): u \\leq v' \\leq v}$.\n\\begin{lemma} \\label{lem:dphi}\nFor every $(u,v) \\in \\calQ$, the following inequalities hold.\n\\begin{align*}\n\t& \\abs{\\partial_{v} \\phi(u,v)} \\leq \t\t\t\\frac{\\Lambda^{2}}{4} \\sup_{u \\leq v' \\leq v} \\abs{\\partial_{v}^{2}(r\\phi)(u, v')} \\\\\n\t& \\phantom{\\abs{\\partial_{v} \\phi(u,v)} \\leq} \t+ \\frac{\\Lambda^{3}}{4} \\sup_{u \\leq v' \\leq v} \\abs{\\partial_{v} (r \\phi)(u, v')} \\sup_{u \\leq v' \\leq v} \\abs{\\partial_{v} \\lmb (u, v')}, \\\\\n\t& \\abs{\\partial_{u} \\phi(u,v)} \\leq \\Lambda  \\sup_{u \\leq v' \\leq v} (- \\nu)(u, v') \\abs{\\partial_{v} \\phi(u,v')}.\n\\end{align*}\n\\end{lemma}\n\n\\begin{proof} \nThe first is an easy consequence of \\eqref{eq:est4dvphi:1} in \\S \\ref{subsec:est4phi}. To prove the second inequality, we start from the equation\n\\begin{equation*}\n\t\\partial_{v} (r \\partial_{u} \\phi) = - \\nu \\partial_{v} \\phi,\n\\end{equation*}\nwhich follows from \\eqref{eq:SSESF:dr} and \\eqref{eq:SSESF:dphi}. Therefore, we have\n\\begin{align*}\n\t\\abs{\\partial_{u} \\phi (u,v)} \n\t\\leq & \\frac{1}{r(u,v)} \\int_{u}^{v} (-\\nu) \\abs{\\partial_{v} \\phi} (u, v') \\, \\mathrm{d} v',\n\\end{align*}\nfrom which the second inequality easily follows. \\qedhere\n\\end{proof}\n\nIn the next lemma, we show that improved estimates for $m$ near $\\Gamma$ hold if we assume an $L^{\\infty}$ control of ${\\partial_{v} \\phi}$.\n\\begin{lemma} \\label{lem:muOverR}\nFor every $(u,v) \\in \\calQ$, the following inequalities hold:\n\t\\begin{align} \n\t\\frac{\\mu}{r}(u,v) \\leq & \\Lambda^{2} \\sup_{u \\leq v' \\leq v} \\abs{\\partial_{v} (r \\phi)(u, v')} \\sup_{u \\leq v' \\leq v} \\abs{\\partial_{v} \\phi (u, v')}, \\label{eq:muOverR:1} \\\\\n\t\\frac{\\mu}{r^{2}}(u,v) \\leq & \\frac{\\Lambda^{2}}{3} \\sup_{u \\leq v' \\leq v} \\abs{\\partial_{v} \\phi(u, v')}^{2}. \\label{eq:muOverR:2}\n\t\\end{align}\n\\end{lemma}\n\\begin{proof} \n\tRecall $\\mu = 2m\/r$. By \\eqref{eq:SSESF:dm}, we have\n\t\\begin{equation*}\n\t\t2 m(u,v) = \\int_{u}^{v} (1-\\mu) \\lmb^{-1} r^{2} (\\partial_{v} \\phi)^{2} (u, v') \\, \\mathrm{d} v'.\n\t\\end{equation*}\n\t\n\tPulling everything except $r^{2} \\lmb$ outside the integral and using $\\int_{u}^{v} r^{2} \\lmb(u, v') \\, \\mathrm{d} v' = (1\/3) r^{3}(u,v)$, we obtain \\eqref{eq:muOverR:2}. \n\tOn the other hand, using $\\lmb^{-1} r \\partial_{v} \\phi = \\lmb^{-1} \\partial_{v} (r \\phi) - \\phi$ and $\\int_{u}^{v} r \\lmb(u, v') \\, \\mathrm{d} v' = (1\/2) r^{2}(u,v)$, we easily deduce\n\\begin{equation*}\n\t\\frac{\\mu}{r}(u,v) \\leq \\frac{1}{2} \\sup_{u \\leq v' \\leq v} \\Big( \\Lambda^{2} \\abs{\\partial_{v} (r \\phi)(u, v')} + \\Lambda \\abs{\\phi(u, v')} \\Big) \\abs{\\partial_{v} \\phi(u, v')}.\n\\end{equation*}\n\t\n\tFrom the fact that $\\abs{\\phi(u,v)} \\leq \\Lambda \\sup_{u \\leq v' \\leq v} \\abs{\\partial_{v} (r \\phi)(u, v')}$, \\eqref{eq:muOverR:1} easily follows.\\qedhere\n\\end{proof}\n\n\\subsection{Preliminary $r$-decay for $\\partial_{v}^{2} (r \\phi)$ and $\\partial_{v} \\lmb$}\nIn this subsection, we establish decay estimates for $\\partial_{v}^{2} (r \\phi)$ and $\\partial_{v} \\lmb$ which are sharp in terms of $r$-weights in the region $\\PD_{\\mathrm{ext}}$. We remind the reader the decomposition $\\calQ = \\PD_{\\mathrm{ext}} \\cup \\PD_{\\mathrm{int}}$, where\n\\begin{equation*}\n\t\\PD_{\\mathrm{ext}} = \\set{(u,v) \\in \\calQ : v \\geq 3u}, \\quad \\PD_{\\mathrm{int}} = \\set{(u,v) \\in \\calQ : v \\leq 3u}.\n\\end{equation*}\n\nIn particular, note that $r \\geq 2 \\Lambda^{-1} u > 0$ in $\\PD_{\\mathrm{ext}}$.\n\n\\begin{lemma} \\label{lem:decay2:rDecay}\nThe following estimates hold.\n\\begin{align}\n\t \\sup_{\\PD_{\\mathrm{ext}}} r^{3} \\abs{\\partial_{v} \\lmb} \\leq & C_{K, \\Lambda} A_{1}^{2}, \t \\label{eq:decay2:rDecay:1} \\\\\n\t\\sup_{\\PD_{\\mathrm{ext}}} r^{\\omega+1} \\abs{\\partial_{v}^{2} (r \\phi)} \\leq & C \\mathcal I_{2} + C_{K, \\Lambda, M_{i}} A_{1}^{3}. \\label{eq:decay2:rDecay:2}\n\\end{align}\n\\end{lemma}\n\n\\begin{proof} \nWe begin by proving \\eqref{eq:decay2:rDecay:1}. Recall \\eqref{eq:eq4dvdvr:normal}:\n\\begin{equation*} \\tag{\\ref{eq:eq4dvdvr:normal}}\n\\partial_{u} \\partial_{v} \\log \\lmb\n= \\frac{1}{(1-\\mu)}  \\lmb^{-1} \\nu (\\partial_{v} \\phi)^{2} - \\frac{4 m}{(1-\\mu) r^{3}} \\lmb \\nu.\n\\end{equation*}\n\nNote that $\\partial_{v} \\log \\lmb= 0$ on $C_{1}$ by our choice of coordinates. Therefore, integrating the preceding equation along the incoming direction from $(1,v)$ to $(u,v)$, we have\n\\begin{equation*}\n\t\\abs{\\partial_{v} \\log \\lmb(u, v)} \\leq \\int_{1}^{u} \\abs{\\frac{1}{(1-\\mu)}  \\lmb^{-1} \\nu (\\partial_{v} \\phi)^{2} (u', v)} \\, \\mathrm{d} u' + \\int_{1}^{u} \\abs{\\frac{4 m}{(1-\\mu) r^{3}} \\lmb \\nu (u', v)} \\, \\mathrm{d} u'.\n\\end{equation*}\n\nThen \\eqref{eq:decay2:rDecay:1} follows using Proposition \\ref{prop:geomLocBVScat}, \\eqref{eq:decay1:4} and \\eqref{eq:decay1:6}. We remark that the power of $r$ is dictated by the second integral.\n\nThe proof of \\eqref{eq:decay2:rDecay:2} is very similar. We start by recalling \\eqref{eq:eq4dvdvrphi:normal}:\n\\begin{equation*} \\tag{\\ref{eq:eq4dvdvrphi:normal}}\n\\partial_{u} (\\partial_{v}^{2} (r \\phi)) = \n\\frac{2m \\lmb \\nu}{(1-\\mu) r^{2}} \\, \\partial_{v} \\phi + \\frac{ \\nu}{(1-\\mu) }  (\\partial_{v} \\phi)^{2} \\phi \n + \\frac{2m \\nu}{(1-\\mu) r^{2}} (\\partial_{v} \\lmb) \\phi - \\frac{4m}{(1-\\mu) r^{3}} \\lmb^{2} \\nu \\phi.\n\\end{equation*}\n\nFor $u \\geq 1$, we have $r(u,v) \\leq r(1,v)$; moreover, by hypothesis, we have the estimate for the initial data term \n$$(1+r(1,v))^{\\omega'+1} \\abs{\\partial_{v}^{2}(r \\phi)(1,v)} \\leq \\mathcal I_{2} \\, .$$\nTherefore, by the fundamental theorem of calculus, it suffices to bound\n\\begin{align*}\n& \\int_{1}^{u} \\abs{\\frac{2m \\lmb \\nu}{(1-\\mu) r^{2}} \\, \\partial_{v} \\phi (u', v)} \\, \\mathrm{d} u' + \\int_{1}^{u} \\abs{\\frac{ \\nu}{(1-\\mu) }  (\\partial_{v} \\phi)^{2} \\phi (u', v)} \\, \\mathrm{d} u' \\\\\n& \\quad + \\int_{1}^{u} \\abs{\\frac{2m \\nu}{(1-\\mu) r^{2}} (\\partial_{v} \\lmb) \\phi (u', v) } \\, \\mathrm{d} u'  + \\int_{1}^{u} \\abs{\\frac{4m}{(1-\\mu) r^{3}} \\lmb^{2} \\nu \\phi (u', v)} \\, \\mathrm{d} u'\n\\end{align*}\nby $C_{K, \\Lambda, M_{i}} A_{1}^{3} r^{-(\\omega+1)}$. This is an easy consequence of Proposition \\ref{prop:geomLocBVScat}, \\eqref{eq:decay1:1}, \\eqref{eq:decay1:4}, \\eqref{eq:decay1:6} and also \\eqref{eq:decay2:rDecay:1} that has just been established. Note that the last term is what limits $\\omega \\leq 3$. \\qedhere\n\\end{proof}\n\n\\subsection{Propagation of $u$-decay for $\\partial_{u}^{2} (r \\phi)$ and $\\partial_{u} \\nu$}\nHere, we show that certain $u$-decay for $\\partial_{u}^{2} (r \\phi)$ and $\\partial_{u} \\nu$ proved in $\\PD_{\\mathrm{int}}$ can be propagated to $\\calQ$. The technique employed is very similar to that in the previous subsection.\n\\begin{lemma} \\label{lem:decay2:uDecayInExtr}\nFor $U \\geq 1$, suppose that there exists a finite positive constant $A, k_{1}, k_{2}$ such that\n\\begin{equation*}\n\t0 \\leq k_{1} \\leq 2\\omega+1, \\quad\n\t0 \\leq k_{2} \\leq 3\\omega+1, \n\\end{equation*}\nand for $u \\in [1, U]$, we have\n\\begin{equation*}\n\t\\sup_{C_{u} \\cap \\PD_{\\mathrm{int}}} u^{k_{1}} \\abs{\\partial_{u} \\nu} \\leq A, \\quad\n\t\\sup_{C_{u} \\cap \\PD_{\\mathrm{int}}} u^{k_{2}} \\abs{\\partial_{u}^{2}(r\\phi)} \\leq A.\n\\end{equation*}\n\nThen for $u \\in [1, U]$, the following estimates hold.\n\\begin{align}\n\t\\sup_{C_{u}} u^{k_{1}} \\abs{\\partial_{u} \\nu} \\leq & C_{K, \\Lambda} A + C_{K, \\Lambda} A_{1}^{2}, \\label{eq:decay2:uDecayInExtr:1} \\\\\n\t\\sup_{C_{u}} u^{k_{2}} \\abs{\\partial_{u}^{2}(r\\phi)} \\leq & A + C_{K, \\Lambda} A_{1}^{3} + C_{K, \\Lambda} A_{1}^{3} \\, \\sup_{C_{u}} u \\abs{\\partial_{u} \\nu} . \\label{eq:decay2:uDecayInExtr:2}\n\\end{align}\n\nFurthermore, the following alternative to \\eqref{eq:decay2:uDecayInExtr:2} also holds.\n\\begin{equation} \\label{eq:decay2:uDecayInExtr:3}\n\\sup_{C_{u}} u^{k_{2}} \\abs{\\partial_{u}^{2}(r\\phi)} \n\t\\leq A + C_{K, \\Lambda} A_{1}^{3} + \n\tC_{K, \\Lambda} \\Psi \\int_{3u}^{\\infty} \\abs{\\frac{2 m \\lmb}{(1-\\mu) r^{2}}}(u, v') \\, \\mathrm{d} v' \\cdot \\sup_{C_{u}} u^{k_{2}} \\abs{\\partial_{u} \\nu}. \n\\end{equation}\n\\end{lemma}\n\\begin{proof}\nLet us begin with \\eqref{eq:decay2:uDecayInExtr:1}. Recall \\eqref{eq:eq4dudur:normal}\n\\begin{equation*} \\tag{\\ref{eq:eq4dudur:normal}}\n\\partial_{v} \\partial_{u} \\log \\nu\n= \\frac{1}{(1-\\mu)}  \\lmb \\nu^{-1} (\\partial_{u} \\phi)^{2} - \\frac{4m}{(1-\\mu) r^{3}} \\lmb \\nu.\n\\end{equation*}\n\nGiven $(u,v) \\in \\PD_{\\mathrm{ext}}$ (with $u \\in [1,U]$), let us integrate this equation along the outgoing direction from $(u, 3u)$ to $(u,v)$, take the absolute value and multiply by $u^{k_{1}}$. Using the hypothesis\n\\begin{equation*}\n\t\\sup_{\\PD_{\\mathrm{int}} \\cap \\set{(u,v) \\in \\calQ : u \\in [1,U]}} u^{k_{1}} \\abs{\\partial_{u} \\nu} \\leq A,\n\\end{equation*}\n\\eqref{eq:decay2:uDecayInExtr:1} is reduced to showing\n\\begin{align} \n\tu^{k_{1}} \\int_{3u}^{\\infty} \\abs{\\frac{1}{(1-\\mu)} \\lmb \\nu^{-1} (\\partial_{u} \\phi)^{2} (u, v)} \\, \\mathrm{d} v \\leq & C_{K, \\Lambda} A_{1}^{2}, \\label{eq:decay2:uDecayInExtr:pf:1} \\\\ \n\tu^{k_{1}} \\int_{3u}^{\\infty} \\abs{\\frac{4m}{(1-\\mu) r^{3}} \\lmb \\nu (u, v)} \\, \\mathrm{d} v \\leq & C_{K, \\Lambda} A_{1}^{2}, \\label{eq:decay2:uDecayInExtr:pf:2}\n\\end{align}\nfor $u \\in [1, U]$.\n\nUsing Proposition \\ref{prop:geomLocBVScat} and \\eqref{eq:decay1:5}, the left-hand side of \\eqref{eq:decay2:uDecayInExtr:pf:1} is bounded by\n\\begin{equation*}\n\tC_{K, \\Lambda} A_{1}^{2} \\, u^{k_{1} - 2\\omega} \\int_{3u}^{\\infty} \\frac{1}{r^{2}} \\lmb \\, \\mathrm{d} v \n\t\\leq C_{K, \\Lambda} A_{1}^{2} \\, u^{k_{1}- 2\\omega} r^{-1}(u, 3u).\n\\end{equation*}\n\nAs $u \\geq 1$ and $r(u, 3u) \\geq 2 \\Lambda^{-1} u$, \\eqref{eq:decay2:uDecayInExtr:pf:1} follows. Similarly, by \\eqref{eq:bnd4mu} and \\eqref{eq:decay1:6}, the left-hand side of \\eqref{eq:decay2:uDecayInExtr:pf:2} is also bounded by  $C_{K, \\Lambda} A_{1}^{2} \\, u^{k_{1}- 2\\omega} r^{-1}(u, 3u)$, from which \\eqref{eq:decay2:uDecayInExtr:pf:2} immediately follows.\n\nNext, we turn to \\eqref{eq:decay2:uDecayInExtr:2} and \\eqref{eq:decay2:uDecayInExtr:3}; as they are proved similarly as before, we will only outline the main points. Recall \\eqref{eq:eq4dudurphi:normal}:\n\\begin{equation*} \\tag{\\ref{eq:eq4dudurphi:normal}}\n\\partial_{v} (\\partial_{u}^{2} (r \\phi)) = \n\\frac{2m \\lmb \\nu}{(1-\\mu) r^{2}} \\, \\partial_{u} \\phi + \\frac{\\lmb }{(1-\\mu) }  (\\partial_{u} \\phi)^{2} \\phi \n + \\frac{2m \\lmb}{(1-\\mu) r^{2}} (\\partial_{u} \\nu) \\phi - \\frac{4m}{(1-\\mu) r^{3}} \\lmb \\nu^{2} \\phi.\n\\end{equation*}\n\nFix $(u,v) \\in \\PD_{\\mathrm{ext}}$ with $u \\in [1,U]$. We then integrate the preceding equation along the outgoing direction from $(u, 3u)$ to $(u, v)$, take the absolute value and multiply by $u^{k_{2}}$. In order to prove \\eqref{eq:decay2:uDecayInExtr:2}, in view of the hypothesis\n\\begin{equation*}\n\t\\sup_{\\PD_{\\mathrm{int}} \\cap \\set{(u, v) \\in \\calQ : u \\in [1, U]}} u^{k_{2}} \\abs{\\partial_{u}^{2} (r \\phi)} \\leq A,\n\\end{equation*}\nit suffices to establish the following estimates for $u \\in [1, U]$:\n\\begin{align*}\n\tu^{k_{2}} \\int_{3u}^{\\infty} \\abs{\\frac{2m \\lmb \\nu}{(1-\\mu) r^{2}} \\, \\partial_{u} \\phi (u,v)}\\, \\mathrm{d} v \n\t\\leq & C_{K, \\Lambda} A_{1}^{3}, \\\\\n\tu^{k_{2}} \\int_{3u}^{\\infty} \\abs{\\frac{\\lmb }{(1-\\mu) }  (\\partial_{u} \\phi)^{2} \\phi  (u,v)}\\, \\mathrm{d} v \n\t\\leq & C_{K, \\Lambda} A_{1}^{3}, \\\\\n\tu^{k_{2}} \\int_{3u}^{\\infty} \\abs{\\frac{2m \\lmb}{(1-\\mu) r^{2}} (\\partial_{u} \\nu) \\phi  (u,v)}\\, \\mathrm{d} v \n\t\\leq & C_{K, \\Lambda} A_{1}^{3} \\, \\sup_{\\calQ} u \\abs{\\partial_{u} \\nu}, \\\\\n\tu^{k_{2}} \\int_{3u}^{\\infty} \\abs{\\frac{4m}{(1-\\mu) r^{3}} \\lmb \\nu^{2} \\phi (u,v)} \\, \\mathrm{d} v\n\t\\leq & C_{K, \\Lambda} A_{1}^{3}.\n\\end{align*}\n\nThe proof of these estimates are similar to that of \\eqref{eq:decay2:uDecayInExtr:pf:1}, \\eqref{eq:decay2:uDecayInExtr:pf:2}; we omit the details. To prove \\eqref{eq:decay2:uDecayInExtr:3}, we replace the third estimate by\n\\begin{equation*}\n\tu^{k_{2}} \\int_{3u}^{\\infty} \\abs{\\frac{2m \\lmb}{(1-\\mu) r^{2}} (\\partial_{u} \\nu) \\phi  (u,v)}\\, \\mathrm{d} v \n\t\\leq C_{K, \\Lambda} \\Psi \\int_{3u}^{\\infty} \\abs{\\frac{2 m \\lmb}{(1-\\mu) r^{2}} }(u, v') \\, \\mathrm{d} v' \\cdot \\sup_{C_{u}} u^{k_{2}} \\abs{\\partial_{u} \\nu},\n\\end{equation*}\nwhich is an easy consequence of Proposition \\ref{prop:geomLocBVScat}.\n\\qedhere\n\\end{proof}\n\n\n\\subsection{Full decay for $\\partial_{v}^{2} (r \\phi)$, $\\partial_{u}^{2} (r \\phi)$, $\\partial_{v} \\lmb$ and $\\partial_{u} \\nu$} \\label{sec.full.decay.2}\nWith all the preparations so far, we are finally ready to prove Theorem \\ref{main.thm.2}. Our proof consists of two steps.\nThe first step is use the local BV scattering assumption to prove a preliminary decay rate of $u^{-\\omega}$ for $\\partial_{v}^{2} (r \\phi)$, $\\partial_{u}^{2} (r \\phi)$, $\\partial_{v} \\lmb$ and $\\partial_{u} \\nu$. In this step, it is crucial to pass to the \\emph{renormalized variables} and exploit the null structure of \\eqref{eq:SSESF}, in order to utilize the a priori bounds in the local BV scattering assumption. The second step to upgrade these decay rates to those that are claimed in Theorem \\ref{main.thm.2}. Thanks to the preliminary $u^{-\\omega}$ decay from the first step, the null structure is not necessary at this point.\n\nWe now begin with the first step. The null structure of \\eqref{eq:SSESF} as demonstrated in \\S \\ref{subsec:nullStr} is used in an essential way.\n\n\\begin{proposition} \\label{prop:decay2:nullStr}\nThere exists a finite constant $A_{2}' > 0$ such that the following estimates hold.\n\\begin{align*}\n\t& \\abs{\\partial_{v}^{2} (r \\phi)} \\leq A_{2}' u^{-\\omega}, \\quad\n\t\\abs{\\partial_{u}^{2} (r \\phi)} \\leq A_{2}' u^{-\\omega},  \\\\\n\t& \\abs{\\partial_{v} \\lmb} \\leq A_{2}' u^{-\\omega},  \\qquad\n\t\\abs{\\partial_{u} \\nu} \\leq A_{2}' u^{-\\omega}.\n\\end{align*}\n\\end{proposition}\n\n\\begin{proof} \nFor $U > 1$, we define \n\\begin{equation} \\label{eq:decay2:def4B2}\n\t\\mathcal B_{2}(U) := \\sup_{u \\in [1, U]} \\sup_{C_{u}} \\Big(  u^{\\omega} \\abs{\\partial_{v}^{2} (r \\phi)} +u^{\\omega} \\abs{\\partial_{u}^{2} (r \\phi)} \n\t\t\t\t\t+ u^{\\omega} \\abs{\\partial_{v} \\lmb} + u^{\\omega} \\abs{\\partial_{u} \\nu} \\Big).\n\\end{equation}\n \n\n\nNotice that the above is finite for every fixed $U$ due to Lemmas \\ref{lem:decay2:rDecay} and \\ref{lem:decay2:uDecayInExtr}. As indicated earlier, we need to use the null structure of the system \\eqref{eq:SSESF} as in \\S \\ref{subsec:nullStr}. For convenience, we define the shorthands\n\\begin{align*}\n\tF_{1} := & \\partial_{v}^{2} (r \\phi) - (\\partial_{v} \\lmb) \\phi, \\\\\n\tG_{1} := & \\partial_{u}^{2} (r \\phi) - (\\partial_{u} \\nu) \\phi, \n\\end{align*}\nand\n\\begin{align*}\n\tF_{2} := & \\partial_{v} \\log \\lmb - \\frac{\\lmb}{(1-\\mu)} \\frac{\\mu}{r} + \\partial_{v} \\phi \\Big( \\lmb^{-1} \\partial_{v} (r \\phi) - \\nu^{-1} \\partial_{u} ( r \\phi) \\Big), \\\\\n\tG_{2} := & \\partial_{u} \\log (-\\nu) - \\frac{\\nu}{(1-\\mu)} \\frac{\\mu}{r} + \\partial_{u} \\phi \\Big( \\lmb^{-1} \\partial_{v} (r \\phi) - \\nu^{-1} \\partial_{u} (r \\phi) \\Big).\n\\end{align*}\n\nThen \\eqref{eq:eq4dvdvrphi}, \\eqref{eq:eq4dudurphi}, \\eqref{eq:eq4dvdvr} and \\eqref{eq:eq4dudur} may be rewritten in the following fashion.\n\\begin{align} \n& \\partial_{u} F_{1} = \\partial_{u} \\lmb \\, \\partial_{v} \\phi - \\partial_{v} \\lmb \\, \\partial_{u} \\phi, \\label{eq:decay2:nullStr:pf:1} \\\\\n& \\partial_{u} F_{2} = \\partial_{u} \\phi \\, \\partial_{v}\\Big( \\nu^{-1} \\partial_{u} (r \\phi) \\Big)- \\partial_{v} \\phi \\, \\partial_{u} \\Big( \\nu^{-1} \\partial_{u} (r \\phi) \\Big),  \\label{eq:decay2:nullStr:pf:2}  \\\\\n& \\partial_{v} G_{1} = \\partial_{v} \\nu \\, \\partial_{u} \\phi - \\partial_{u} \\nu \\, \\partial_{v} \\phi,\\label{eq:decay2:nullStr:pf:3}  \\\\\n& \\partial_{v} G_{2} = - \\partial_{u} \\phi \\, \\partial_{v} \\Big( \\lmb^{-1} \\partial_{v} (r \\phi) \\Big) + \\partial_{v} \\phi \\, \\partial_{u} \\Big( \\lmb^{-1} \\partial_{v} (r \\phi) \\Big). \\label{eq:decay2:nullStr:pf:4}\n\\end{align}\n\nThe following lemma is the key technical component of the proof.\n\n\\begin{lemma} \\label{lem:decay2:key4nullStr}\nThere exists a finite positive constant $C = C_{A_{1}, \\mathcal I_{2}, K, \\Lambda}$ and positive function $\\epsilon(u)$ satisfying\n\\begin{equation*}\n\t\\epsilon(u) \\to 0 \\hbox{ as } u \\to \\infty\n\\end{equation*}\nsuch that the following inequalities holds for $1 \\leq u_{2} \\leq U$:\n\\begin{align}\n\t\\sup_{\\PD_{\\mathrm{int}} \\cap \\set{(u,v) \\in \\calQ : u \\in [3 u_{2}, U]}} \\Big( u^{\\omega} \\abs{F_{1}} + u^{\\omega} \\abs{G_{1}} \\Big)\n\t \\leq & C_{\\Lambda} \\mathcal I_{2} + C_{K, \\Lambda, M_{i}} A_{1}^{3} + \\epsilon(u_{2}) \\mathcal B_{2}(U), \\label{eq:decay2:key4nullStr:1} \\\\\n\t \\sup_{\\PD_{\\mathrm{int}} \\cap \\set{(u,v) \\in \\calQ : u \\in [3 u_{2}, U]}} \\Big( u^{\\omega} \\abs{F_{2}} + u^{\\omega} \\abs{G_{2}} \\Big)\n\t  \\leq & C_{K, \\Lambda} A_{1}^{2} + \\epsilon(u_{2}) \\mathcal B_{2}(U). \\label{eq:decay2:key4nullStr:2} \n\\end{align}\n\\end{lemma}\n\nWe defer the proof of this lemma until later. Instead, we first finish the proof of Proposition \\ref{prop:decay2:nullStr}, assuming Lemma \\ref{lem:decay2:key4nullStr}.\n\n\\noindent\\emph{Proof of Proposition \\ref{prop:decay2:nullStr}.}\nFirst, we claim that \\eqref{eq:decay2:key4nullStr:1} and \\eqref{eq:decay2:key4nullStr:2} imply\n\\begin{equation} \\label{eq:decay2:nullStr:pf:5}\n\t\\sup_{\\PD_{\\mathrm{int}} \\cap \\set{(u,v) \\in \\calQ : u \\in [3 u_{2}, U]}} u^{\\omega} \\Big( \\abs{\\partial_{v}^{2} (r \\phi)} + \\abs{\\partial_{u}^{2} (r \\phi)} + \\abs{\\partial_{v} \\lmb} + \\abs{\\partial_{u} \\nu} \\Big)\n\t\\leq H_{2} + (\\epsilon + \\epsilon')(u_{2}) \\mathcal B_{2}(U).\n\\end{equation}\nfor some constant $0 < H_{2} < \\infty$ and some positive function $\\epsilon'(u_{2})$ which tends to zero as $u_{2} \\to \\infty$.\n\nThe point is that $F_{1}, F_{2}, G_{1}, G_{2}$ controls $\\partial_{v}^{2} (r \\phi)$, $\\partial_{u}^{2} (r \\phi)$, $\\partial_{v} \\lmb$, $\\partial_{u} \\nu$, respectively, up to higher order terms, which may be absorbed into the second term on the right-hand side. Indeed, consider $u \\in [3 u_{2}, U]$. For $F_{1}$ and $G_{1}$, we estimate\n\\begin{align*}\n\t& u^{\\omega} \\abs{\\partial_{v}^{2} (r \\phi)(u,v)} = u^{\\omega} \\abs{F_{1} +  (\\partial_{v} \\lmb) \\phi}(u,v) \\leq u^{\\omega} \\abs{F_{1}(u,v)} + \\sup_{C_{u}} \\abs{\\phi} \\cdot \\mathcal B_{2}(U), \\\\\n\t& u^{\\omega} \\abs{\\partial_{u}^{2} (r \\phi)(u,v)} = u^{\\omega} \\abs{G_{1} + (\\partial_{u} \\nu) \\phi}(u,v) \\leq u^{\\omega} \\abs{G_{1}(u,v)} + \\sup_{C_{u}} \\abs{\\phi} \\cdot \\mathcal B_{2}(U),\n\\end{align*}\nwhich are acceptable, as $\\sup_{C_{u}} \\abs{\\phi} \\to 0$ as $u \\geq 3 u_{2} \\to \\infty$ by Theorem \\ref{main.thm.1}. \nFor $F_{2}$, we use Proposition \\ref{prop:geomLocBVScat} to estimate\n\\begin{align*}\n\tu^{\\omega} \\abs{\\partial_{v} \\lmb} = & u^{\\omega} \\lmb \\Big\\vert F_{2} + \\frac{\\lmb}{1-\\mu} \\frac{\\mu}{r} + \\partial_{v} \\phi ( \\lmb^{-1} \\partial_{v} (r \\phi) - \\nu^{-1} \\partial_{u} (r \\phi) ) \\Big\\vert \\\\\n\t\\leq &\\frac{1}{2} u^{\\omega} \\abs{F_{2}} + \\frac{K \\Lambda}{4} u^{\\omega} \\abs{\\frac{\\mu}{r}} + \\frac{\\Lambda}{2} u^{\\omega} \\abs{\\partial_{v} \\phi} \\Big( \\abs{\\partial_{v} (r \\phi)} + \\abs{\\partial_{u} (r \\phi)} \\Big).\n\\end{align*}\n\nApplying \\eqref{eq:muOverR:1} (from Lemma \\ref{lem:muOverR}) to the second term on the last line, and then using Lemma \\ref{lem:dphi} to control $u^{\\omega} \\abs{\\partial_{v} \\phi}$, we arrive at \n\\begin{equation*}\nu^{\\omega} \\abs{\\partial_{v} \\lmb(u,v)} \\leq \\frac{1}{2} u^{\\omega} \\abs{F_{2}(u,v)} + C_{K, \\Lambda} \\, \\Psi \\sup_{C_{u}} \\Big( \\abs{\\partial_{v} (r \\phi)} + \\abs{\\partial_{u} (r \\phi)} \\Big) \\cdot \\mathcal B_{2}(U),\n\\end{equation*}\nwhich is acceptable in view of Theorem \\ref{main.thm.1}. Proceeding similarly, but also using the second inequality of Lemma \\ref{lem:dphi} to control $\\abs{\\partial_{u} \\phi}$, we obtain\n\\begin{equation*}\nu^{\\omega} \\abs{\\partial_{u} \\nu(u,v)} \\leq K u^{\\omega} \\abs{G_{2}(u,v)} + C_{K, \\Lambda} \\, \\Psi \\sup_{C_{u}} \\Big( \\abs{\\partial_{v} (r \\phi)} + \\abs{\\partial_{u} (r \\phi)} \\Big) \\cdot \\mathcal B_{2}(U).\n\\end{equation*}\n\nCombining these estimates with \\eqref{eq:decay2:key4nullStr:1} and \\eqref{eq:decay2:key4nullStr:2}, we conclude \\eqref{eq:decay2:nullStr:pf:5} with\n\\begin{align} \n\tH_{2} =& C_{\\Lambda} \\mathcal I_{2} + C_{K, \\Lambda, M_{i}} A_{1}^{3} + C_{K, \\Lambda} A_{1}^{2}, \n\t\\label{eq:decay2:H2} \\\\\n\t\\epsilon'(u_{2}) =& C \\sup_{u \\geq 3u_{2}} \\abs{\\phi} + C_{K, \\Lambda} \\Psi \\sup_{u \\geq 3u_{2}} \\Big( \\abs{\\partial_{v}(r \\phi)} + \\abs{\\partial_{u}(r \\phi)} \\Big).\n\t\\label{eq:decay2:eps'}\n\\end{align}\n\nNext, note that the (non-decreasing) function\n\\begin{equation} \\label{eq:decay2:def4H'2}\n\tH'_{2}(u_{2}) := \\sup_{\\PD_{\\mathrm{int}} \\cap \\set{(u,v) \\in \\calQ : u \\in [1, 3u_{2}]}} u^{\\omega} \\Big( \\abs{\\partial_{v}^{2} (r \\phi)} + \\abs{\\partial_{u}^{2} (r \\phi)} + \\abs{\\partial_{v} \\lmb} + \\abs{\\partial_{u} \\nu} \\Big) \\geq 0\n\\end{equation}\nis always \\emph{finite} for any fixed $u_{2} \\geq 1$, as the region $\\PD_{\\mathrm{int}} \\cap \\set{(u,v) \\in \\calQ : u \\in [1, 3u_{2}]}$ is compact and each of these terms is a continuous function, since $(\\phi, r, m)$ is a $C^{1}$ solution (see Definition \\ref{def:C1solution}). Combining with \\eqref{eq:decay2:nullStr:pf:5}, we obtain\n\\begin{equation*} \n\t\\sup_{\\PD_{\\mathrm{int}} \\cap \\set{(u,v) \\in \\calQ : u \\in [1, U]}} u^{\\omega} \\Big( \\abs{\\partial_{v}^{2} (r \\phi)} + \\abs{\\partial_{u}^{2} (r \\phi)} + \\abs{\\partial_{v} \\lmb} + \\abs{\\partial_{u} \\nu} \\Big)\n\t\\leq H_{2} + H'_{2}(u_{2}) + (\\epsilon + \\epsilon')(u_{2}) \\mathcal B_{2}(U),\n\\end{equation*}\nfor every $u_2\\in [1,U]$.\n\nNow apply \\eqref{eq:decay2:uDecayInExtr:1}, \\eqref{eq:decay2:uDecayInExtr:3} in Lemma \\ref{lem:decay2:uDecayInExtr} to $\\partial_{u}^{2}(r \\phi)$, $\\partial_{u} \\nu$. Apply also Lemma \\ref{lem:decay2:rDecay} (along with the fact that $r(u,v) \\geq 2 \\Lambda^{-1} u$ in $\\PD_{\\mathrm{ext}}$ and $\\omega \\leq 3$) to $\\partial_{v}^{2} (r \\phi)$, $\\partial_{v} \\lmb$ in $\\PD_{\\mathrm{ext}}$. Then we see that there exist a non-negative and non-decreasing function $H''_{2}(u_2)$ and a positive function $\\epsilon''(u_{2})$ such that\n\\begin{equation*} \n\t\\mathcal B_{2}(U) \\leq H_{2}''(u_2) +  \\epsilon''(u_{2}) \\mathcal B_{2}(U),\n\\end{equation*}\nand $\\epsilon''(u_{2}) \\to 0$ as $u_{2} \\to \\infty$. Taking $u_{2}$ sufficiently large, the second term on the right-hand side can be absorbed into the left-hand side; then we conclude that $\\mathcal B_{2}(U) \\leq C_{A_{1}, K, \\Lambda} H''_{2}(u_{2})$. As this bound is independent of $U$, Proposition \\ref{prop:decay2:nullStr} then follows. \\qedhere\n\\end{proof}\n\n\\begin{remark} \nUsing \\eqref{eq:decay2:uDecayInExtr:1}, \\eqref{eq:decay2:uDecayInExtr:3} in Lemma \\ref{lem:decay2:uDecayInExtr} and \\eqref{eq:decay2:rDecay:1}, \\eqref{eq:decay2:rDecay:2} in Lemma \\ref{lem:decay2:rDecay}, the functions $H_{2}''(u_{2})$ and $\\epsilon''(u_{2})$ can be explicitly bounded from the above as follows:\n\\begin{align}\n\tH_{2}''(u_2) \\leq & C_{K, \\Lambda} \\Big( 1 + \\Psi \\int_{3}^{\\infty} \\abs{\\frac{2 m \\lmb}{(1-\\mu) r^{2}}}(u, v') \\, \\mathrm{d} v' \\Big) \\cdot (H_{2} + H'(u_{2}) + A_{1}^{2} + A_{1}^{3}) \t\n\t\\label{eq:decay2:H2''} \\\\\n\t\t\t& + C \\mathcal I_{2} + C_{K, \\Lambda} A_{1}^{2} + C_{K, \\Lambda, M_{i}} A_{1}^{3} \n\t\\notag \\\\\n\t\\epsilon''(u_{2}) \\leq & C_{K, \\Lambda} \\Big( 1 + \\Psi \\int_{3}^{\\infty} \\abs{\\frac{2 m \\lmb}{(1-\\mu) r^{2}}}(u, v') \\, \\mathrm{d} v' \\Big) \\cdot (\\epsilon + \\epsilon')(u_{2}).\n\t\\label{eq:decay2:eps''}\n\\end{align}\n\nThese bounds will be useful in our proof of Theorem \\ref{thm:smallData} in Section \\ref{sec:smallData}.\n\\end{remark}\n\nAt this point, in order to complete the proof of Proposition \\ref{prop:decay2:nullStr}, we are only left to prove Lemma \\ref{lem:decay2:key4nullStr}.\n\\begin{proof}[Proof of Lemma \\ref{lem:decay2:key4nullStr}]\nLet $(u, v) \\in \\PD_{\\mathrm{int}}$ (i.e., $v \\in [u, 3u]$) with $u \\in [3 u_{2}, U]$. In this proof, we will use the notation $\\epsilon(u_{2})$ to refer to a positive quantity which may be made arbitrarily small by choosing $u_{2}$ large enough, which may vary from line to line.\n\nWe first estimate $F_1$ and $F_2$. Integrating \\eqref{eq:decay2:nullStr:pf:1} and \\eqref{eq:decay2:nullStr:pf:2} along the incoming direction from $(u\/3, v)$ to $(u,v)$, we obtain\n\\begin{align*}\n\t\\abs{F_{1}(u,v)} \\leq & \\abs{F_{1}(u\/3, v)} + \\int_{u\/3}^{u} \\abs{\\partial_{u} \\lmb \\, \\partial_{v} \\phi(u', v)} + \\abs{\\partial_{v} \\lmb \\, \\partial_{u} \\phi (u', v)} \\, \\mathrm{d} u',  \\\\\n\t\\abs{F_{2}(u,v)} \\leq & \\abs{F_{2}(u\/3, v)} + \\int_{u\/3}^{u} \\abs{\\partial_{u} \\phi \\, \\partial_{v}( \\nu^{-1} \\partial_{u} (r \\phi) )(u', v)} + \\abs{\\partial_{v} \\phi \\, \\partial_{u} ( \\nu^{-1} \\partial_{u} (r \\phi) )(u', v)}  \\, \\mathrm{d} u'.\n\\end{align*}\n\nMultiply both sides of these inequalities by $u^{\\omega}$. For $v \\in [u, 3u]$, note that $(u\/3, v) \\in \\PD_{\\mathrm{ext}}$ and $u \\leq (3 \\Lambda\/2) r(u\/3, v)$. \nTherefore, using Theorem \\ref{main.thm.1} for $\\phi$, $\\partial_v(r\\phi)$, Corollary \\ref{cor:decay1} for $\\partial_{v} \\phi$, Lemma \\ref{lem:muOverR} for $\\mu\/r$ and Lemma \\ref{lem:decay2:rDecay} for $\\partial_{v}^{2}(r\\phi)$, $\\partial_{v} \\lmb$, we have\n\\begin{align*}\n\tu^{\\omega} \\abs{F_{1} (u\/3, v)} \n\t\t\\leq & C_{\\Lambda} r^{\\omega} \\Big( \\abs{\\partial_{v}^{2} (r \\phi)} + \\abs{(\\partial_{v} \\lmb) \\phi} \\Big) (u\/3, v)  \\\\\n\t\t\\leq & C_{\\Lambda} \\mathcal I_{2} + C_{K, \\Lambda, M_{i}} A_{1}^{3}, \\\\\n\tu^{\\omega} \\abs{F_{2} (u\/3, v)} \n\t\t\\leq & C_{\\Lambda} r^{\\omega} \\Big( \\abs{\\lmb^{-1} \\partial_{v} \\lmb} + \\frac{\\mu}{(1-\\mu)} \\frac{\\lmb}{r} + \\abs{\\partial_{v} \\phi ( \\lmb^{-1} \\partial_{v} (r \\phi) - \\nu^{-1} \\partial_{u} ( r \\phi) )} \\Big) (u\/3, v)\\\\\n\t\t\\leq & C_{K, \\Lambda} A_{1}^{2}.\n\\end{align*}\n\nTherefore, we only need to deal with the $u'$-integrals. For $u \\in [3 u_{2}, U]$, we claim that\n\\begin{align}\n\tu^{\\omega} \\int_{u\/3}^{u} \\abs{\\partial_{u} \\lmb(u', v)} \\abs{\\partial_{v} \\phi(u',v)} \\, \\mathrm{d} u' \\leq & \\epsilon(u_{2}) \\mathcal B_{2}(U), \\label{eq:decay2:key4nullStr:pf:1} \\\\\n\tu^{\\omega} \\int_{u\/3}^{u} \\abs{\\partial_{v} \\lmb(u', v)} \\abs{\\partial_{u} \\phi(u',v)}\\, \\mathrm{d} u' \\leq & \\epsilon(u_{2}) \\mathcal B_{2}(U), \\label{eq:decay2:key4nullStr:pf:2} \\\\\n\tu^{\\omega} \\int_{u\/3}^{u} \\abs{\\partial_{u} \\phi(u', v)} \\abs{\\partial_{v} (\\nu^{-1} \\partial_{u} (r \\phi))(u', v)} \\, \\mathrm{d} u' \\leq & \\epsilon (u_{2}) \\mathcal B_{2}(U), \\label{eq:decay2:key4nullStr:pf:3} \\\\\n\tu^{\\omega} \\int_{u\/3}^{u} \\abs{\\partial_{v} \\phi(u', v)} \\abs{\\partial_{u} (\\nu^{-1} \\partial_{u} (r \\phi))(u', v)} \\, \\mathrm{d} u' \\leq &  \\epsilon (u_{2}) \\mathcal B_{2}(U). \\label{eq:decay2:key4nullStr:pf:4}\n\\end{align}\n\n\\pfstep{Proof of \\eqref{eq:decay2:key4nullStr:pf:1}}\nWe proceed similarly as in the proof of Theorem \\ref{main.thm.1}. By \\eqref{eq:SSESF:dr}, \\eqref{eq:bnd4dvrphi} and Lemma \\ref{lem:dphi}, we estimate\n\\begin{align*}\n& u^{\\omega} \\int_{u\/3}^{u} \\abs{\\partial_{u} \\lmb(u', v)} \\abs{\\partial_{v} \\phi(u',v)} \\, \\mathrm{d} u' \\\\\n& \\quad \\leq C_{\\Lambda} \\Big( \\int_{u_{2}}^{v} \\abs{\\frac{2m \\nu}{(1-\\mu) r^{2}}(u', v)} \\, \\mathrm{d} u' \\Big) \\sup_{u' \\in [u\/3, u]} \\sup_{C_{u'}} (u')^{\\omega} (\\abs{\\partial^{2}_{v}(r \\phi)} + \\Psi \\abs{\\partial_{v} \\lmb} ) \\\\\n& \\quad \\leq C_{\\Lambda, \\Psi} \\Big( \\int_{u_{2}}^{v} \\abs{\\frac{2m \\nu}{(1-\\mu) r^{2}}(u', v)} \\, \\mathrm{d} u' \\Big) \\mathcal B_{2}(U).\n\\end{align*}\n\nThus \\eqref{eq:decay2:key4nullStr:pf:1} follows by Lemma \\ref{lem:smallPtnl}.\n\n\\pfstep{Proof of \\eqref{eq:decay2:key4nullStr:pf:2}}\nWe have\n\\begin{align*}\nu^{\\omega} \\int_{u\/3}^{u} \\abs{\\partial_{v} \\lmb(u', v)} \\abs{\\partial_{u} \\phi(u',v)}\\, \\mathrm{d} u'\n\\leq & C \\Big( \\int_{u_{2}}^{v} \\abs{\\partial_{u} \\phi(u', v)} \\, \\mathrm{d} u' \\Big) \\sup_{u' \\in [u\/3, u]} \\sup_{C_{u'}} (u')^{\\omega} \\abs{\\partial_{v} \\lmb} \\\\\n\\leq & C \\Big( \\int_{u_{2}}^{v} \\abs{\\partial_{u} \\phi(u', v)} \\, \\mathrm{d} u' \\Big) \\mathcal B_{2}(U).\n\\end{align*}\n\nThus \\eqref{eq:decay2:key4nullStr:pf:2} follows by Lemma \\ref{lem:smallDphi}.\n\n\\pfstep{Proof of \\eqref{eq:decay2:key4nullStr:pf:3}}\nWe start with the identity\n\\begin{equation*}\n\t\\partial_{v} (\\nu^{-1} \\partial_{u} (r\\phi)) = - \\frac{2m}{(1-\\mu)r^{2}} \\lmb ( \\nu^{-1} \\partial_{u}(r \\phi) - \\phi).\n\\end{equation*}\nwhich is readily verifiable using \\eqref{eq:SSESF:dr} and \\eqref{eq:SSESF:dphi}. By \\eqref{eq:bnd4phi} and \\eqref{eq:bnd4durphi}, we estimate\n\\begin{align*}\n& u^{\\omega} \\int_{u\/3}^{u} \\abs{\\partial_{u} \\phi(u', v)} \\abs{\\partial_{v} (\\nu^{-1} \\partial_{u} (r \\phi))(u', v)} \\, \\mathrm{d} u' \\\\\n& \\quad \\leq C_{K, \\Lambda} \\Psi \\Big( \\int_{u_{2}}^{v} \\abs{\\frac{2m \\nu}{(1-\\mu) r^{2}} (u', v)} \\, \\mathrm{d} u' \\Big) \\,  \\sup_{u' \\in [u\/3, u]} \\sup_{C_{u'}} (u')^{\\omega}\\abs{\\partial_{u} \\phi}.\n\\end{align*}\n\nThe $u'$-integral vanishes as $u_{2} \\to \\infty$ by Lemma \\ref{lem:smallPtnl}. On the other hand, by Lemma \\ref{lem:dphi} and Proposition \\ref{prop:geomLocBVScat}, we have\n\\begin{equation} \\label{eq:decay2:key4nullStr:pf:3:1}\n\\sup_{C_{u'}} (u')^{\\omega}\\abs{\\partial_{u} \\phi}\n\\leq C_{K, \\Lambda} \\sup_{C_{u'}} (u')^{\\omega}\\abs{\\partial_{v} \\phi} \n\\leq C_{K, \\Lambda, \\Psi} \\mathcal B_{2}(U),\n\\end{equation}\nfor any $u' \\in [1, U]$.\nTherefore, \\eqref{eq:decay2:key4nullStr:pf:3} follows.\n\n\\pfstep{Proof of \\eqref{eq:decay2:key4nullStr:pf:4}}\nHere we divide the integral into two, one in $\\PD_{\\mathrm{cpt}}$ and the other outside. Recall the notation $u^{\\star}(v) = \\sup \\set{u \\in [1, v] : r (u,v) \\geq R}$. Below, we will consider the case $u^{\\star}(v) \\in [u\/3, u]$, i.e., when the line segment $\\set{(u', v) \\in \\calQ : u' \\in [u\/3, u]}$ crosses $\\set{r = R}$; the other case is easier, and can be handled with a minor modification.\n\nWe first deal with the integral over the portion in $\\PD_{\\mathrm{cpt}}$. We claim that\n\\begin{equation*}\n\tu^{\\omega} \\int_{u^{\\star}(v)}^{u} \\abs{\\partial_{v} \\phi(u', v)} \\abs{\\partial_{u} (\\nu^{-1} \\partial_{u} (r \\phi))(u', v)} \\, \\mathrm{d} u' \\leq  \\epsilon (u_{2}) \\mathcal B_{2}(U). \n\\end{equation*}\n\nThis is an easy consequence of the bound for $|\\partial_v\\phi|$ in Lemma \\ref{lem:dphi}, the fact that $u$, $u'$ are comparable over the domain of integration and\n\\begin{equation*}\n\t\\sup_{v \\in [u_{2}, \\infty)} \\int_{\\underline{C}_{v} \\cap \\set{u \\geq u_{2}} \\cap \\PD_{\\mathrm{cpt}}} \\abs{\\partial_{u} (\\nu^{-1} \\partial_{u} (r \\phi))}   \\to 0 \\hbox{ as } u_{2} \\to \\infty\n\\end{equation*}\nwhich follows from \\eqref{eq:decay1:3}, \\eqref{eq:bnd4dur} and Theorem \\ref{thm:decayInCpt}.\n\nWe now consider the remaining contribution to the integral. We begin as follows.\n\\begin{align*}\n\t& u^{\\omega} \\int_{u\/3}^{u^{\\star}(v)} \\abs{\\partial_{v} \\phi(u', v)} \\abs{\\partial_{u} (\\nu^{-1} \\partial_{u} (r \\phi))(u', v)} \\, \\mathrm{d} u'  \\\\\n\t& \\quad \\leq C_{K, \\Lambda} \\Big( \\int_{u\/3}^{u^{\\star}(v)} \\abs{\\partial_{v} \\phi(u', v)} \\, \\mathrm{d} u' \\Big) \\sup_{u' \\in [u\/3, u^{\\star}(v)]} \\sup_{C_{u'}} (u')^{\\omega} (\\abs{\\partial_{u}^{2}(r \\phi)} + \\Psi \\abs{\\partial_{u} \\nu}) \\\\\n\t& \\quad \\leq C_{K, \\Lambda, \\Psi} \\Big( \\int_{u\/3}^{u^{\\star}(v)} \\abs{\\partial_{v} \\phi(u', v)} \\, \\mathrm{d} u' \\Big) \\mathcal B_{2}(U).\n\\end{align*}\n\nFor $u' \\in [u\/3, u^{\\star}(v)]$, we have $r (u', v) \\geq R$. Thus, by \\eqref{eq:decay1:4}, we have\n\\begin{equation*}\n\t\\int_{u\/3}^{u^{\\star}(v)} \\abs{\\partial_{v} \\phi(u', v)} \\, \\mathrm{d} u' \\leq \\frac{C_{K} A_{1}}{R} \\int_{u_{2}}^{\\infty} (u')^{-\\omega} \\, \\mathrm{d} u'  \\leq \\frac{C_{K} A_{1}}{R} u_{2}^{-(\\omega-1)},\n\\end{equation*}\nwhich vanishes as $u_{2} \\to \\infty$. Therefore, in the case under consideration, \\eqref{eq:decay2:key4nullStr:pf:4} follows.\n\n\\vspace{.5em}\n\nWe have therefore obtained the desired bounds for $F_1$ and $F_2$. Next, estimate $G_1$ and $G_2$. Let us integrate \\eqref{eq:decay2:nullStr:pf:3} and \\eqref{eq:decay2:nullStr:pf:4} along the outgoing direction from $(u, u)$ on the axis to $(u,v)$. Then we obtain\n\\begin{align*}\n\t\\abs{G_{1}(u,v)} \\leq & \\lim_{v' \\to u+} \\abs{G_{1}(u, v')} + \\int_{u}^{v} \\abs{\\partial_{v} \\nu \\, \\partial_{u} \\phi (u, v')} + \\abs{\\partial_{u} \\nu \\, \\partial_{v} \\phi (u, v')} \\, \\mathrm{d} v',  \\\\\n\t\\abs{G_{2}(u,v)} \\leq & \\lim_{v' \\to u+} \\abs{G_{2}(u, v')} + \\int_{u}^{v} \\abs{\\partial_{v} \\phi \\, \\partial_{u} ( \\lmb^{-1} \\partial_{v} (r \\phi) ) (u, v')} + \\abs{\\partial_{u} \\phi \\, \\partial_{v} ( \\lmb^{-1} \\partial_{v} (r \\phi) ) (u, v')} \\, \\mathrm{d} v'.\n\\end{align*}\n\nNote that\n\\begin{equation*}\n\t\\lim_{v \\to u+} \\frac{\\mu}{r} (u, v) = 0, \\quad\n\t\\lim_{v \\to u+} \\Big( \\lmb^{-1} \\partial_{v} (r \\phi)(u,v) - \\nu^{-1} \\partial_{u} ( r \\phi) (u,v) \\Big) = 0,\n\\end{equation*}\nsince $(\\phi, r, m)$ is a $C^{1}$ solution. It follows that $\\lim_{v \\to u+} \\partial_{u} \\partial_{v} (r \\phi)(u, v) = 0$ and $\\lim_{v \\to u+} \\partial_{u} \\partial_{v} r (u,v) = 0$. Moreover, we also have\n\\begin{align*}\n\t& \\lim_{v \\to u+} \\partial_{v}^{2} (r \\phi) (u,v) = - \\lim_{v \\to u+} \\partial_{u}^{2} (r \\phi) (u,v), \\quad\n\t\\lim_{v \\to u+} \\partial_{v} \\lmb (u, v)= - \\lim_{v \\to u+} \\partial_{u} \\nu (u,v).\n\\end{align*}\n\nAs a consequence, \n\\begin{equation*}\n\t\\lim_{v' \\to u+} G_{1}(u, v') = - \\lim_{v' \\to u+} F_{1}(u, v'), \\quad\n\t\\lim_{v' \\to u+} G_{2}(u, v') = \\lim_{v' \\to u+} F_{2}(u, v').\n\\end{equation*}\n\nTherefore, by the previous estimates for $F_{1}, F_{2}$, we have\n\\begin{align*}\n\t& u^{\\omega} \\lim_{v' \\to u+} \\abs{G_{1}(u, v')} \\leq C_{\\Lambda} \\mathcal I_{2} + C_{K, \\Lambda, M_{i}} A_{1}^{3} + \\epsilon(u_{2}) \\mathcal B_{2}(U), \\\\\n\t& u^{\\omega} \\lim_{v' \\to u+} \\abs{G_{2}(u, v')} \\leq C_{K, \\Lambda} A_{1}^{2} + \\epsilon(u_{2}) \\mathcal B_{2}(U),\n\\end{align*}\nwhich are acceptable. Recalling that we are considering $(u,v) \\in \\PD_{\\mathrm{int}}$, hence $v \\in [u, 3u]$, we are now left to establish the following estimates: \n\\begin{align}\n\tu^{\\omega} \\int_{u}^{3u} \\abs{\\partial_{v} \\nu(u, v')} \\abs{\\partial_{u} \\phi(u,v')} \\, \\mathrm{d} v' \\leq & \\epsilon(u_{2}) \\mathcal B_{2}(U), \\label{eq:decay2:key4nullStr:pf:5} \\\\\n\tu^{\\omega} \\int_{u}^{3u} \\abs{\\partial_{u} \\nu(u, v')} \\abs{\\partial_{v} \\phi(u,v')}\\, \\mathrm{d} v' \\leq & \\epsilon(u_{2}) \\mathcal B_{2}(U), \\label{eq:decay2:key4nullStr:pf:6} \\\\\n\tu^{\\omega} \\int_{u}^{3u} \\abs{\\partial_{v} \\phi(u, v')} \\abs{\\partial_{u} (\\lmb^{-1} \\partial_{v} (r \\phi))(u, v')} \\, \\mathrm{d} v' \\leq & \\epsilon (u_{2}) \\mathcal B_{2}(U), \\label{eq:decay2:key4nullStr:pf:7} \\\\\n\tu^{\\omega} \\int_{u}^{3u} \\abs{\\partial_{u} \\phi(u, v')} \\abs{\\partial_{v} (\\lmb^{-1} \\partial_{v} (r \\phi))(u, v')} \\, \\mathrm{d} v' \\leq &  \\epsilon (u_{2}) \\mathcal B_{2}(U). \\label{eq:decay2:key4nullStr:pf:8}\n\\end{align}\n\n\\pfstep{Proof of \\eqref{eq:decay2:key4nullStr:pf:5}}\nSubstituting $\\partial_{v} \\nu$ by \\eqref{eq:SSESF:dr} and using \\eqref{eq:decay2:key4nullStr:pf:3:1}, we have\n\\begin{align*}\nu^{\\omega} \\int_{u}^{3u} \\abs{\\partial_{v} \\nu(u, v')} \\abs{\\partial_{u} \\phi(u,v')} \\, \\mathrm{d} v'\n\\leq & K \\Big( \\int_{u}^{\\infty} \\abs{\\frac{2m \\lmb}{(1-\\mu) r^{2}}(u, v')} \\, \\mathrm{d} v' \\Big) \\sup_{v' \\in [u, 3u]} u^{\\omega} \\abs{\\partial_{u} \\phi(u,v')}  \\\\\n\\leq & C_{K, \\Lambda, \\Psi} \\Big( \\sup_{u \\geq 3u_{2}} \\int_{u}^{\\infty} \\abs{\\frac{2m \\lmb}{(1-\\mu) r^{2}}(u, v')} \\, \\mathrm{d} v' \\Big) \\mathcal B_{2}(U).\n\\end{align*}\n\nThus \\eqref{eq:decay2:key4nullStr:pf:5} follows by Lemma \\ref{lem:smallPtnl}.\n\n\\pfstep{Proof of \\eqref{eq:decay2:key4nullStr:pf:6}}\nWe have\n\\begin{align*}\nu^{\\omega} \\int_{u}^{3u} \\abs{\\partial_{u} \\nu(u, v')} \\abs{\\partial_{v} \\phi(u,v')}\\, \\mathrm{d} v'\n\\leq & \\int_{u}^{\\infty} \\abs{\\partial_{v} \\phi(u, v')} \\, \\mathrm{d} v'  \\sup_{v' \\in [u, 3u]} u^{\\omega} \\abs{\\partial_{u} \\nu(u, v')}  \\\\\n\\leq & \\Big( \\sup_{u \\geq 3u_{2}} \\int_{u}^{\\infty} \\abs{\\partial_{v} \\phi(u, v')} \\, \\mathrm{d} v'  \\Big) \\mathcal B_{2}(U).\n\\end{align*}\n\nThus \\eqref{eq:decay2:key4nullStr:pf:6} follows by Lemma \\ref{lem:smallDphi}.\n\n\\pfstep{Proof of \\eqref{eq:decay2:key4nullStr:pf:7}}\nBy \\eqref{eq:SSESF:dr} and \\eqref{eq:SSESF:dphi}, we have the identity\n\\begin{equation*}\n\t\\partial_{u} (\\lmb^{-1} \\partial_{v} (r\\phi)) = - \\frac{2m}{(1-\\mu)r^{2}} \\nu ( \\lmb^{-1} \\partial_{v}(r \\phi) - \\phi).\n\\end{equation*}\n\nThen by Proposition \\ref{prop:geomLocBVScat}, we have\n\\begin{align*}\n&u^{\\omega} \\int_{u}^{3u} \\abs{\\partial_{v} \\phi(u, v')} \\abs{\\partial_{u} (\\lmb^{-1} \\partial_{v} (r \\phi))(u, v')} \\, \\mathrm{d} v' \\\\\n& \\quad \\leq C_{K, \\Lambda} \\Psi \\Big( \\int_{u}^{\\infty} \\abs{\\frac{2m \\lmb}{(1-\\mu) r^{2}} (u, v')} \\, \\mathrm{d} v' \\Big)  \\sup_{v' \\in [u, 3u]} u^{\\omega} \\abs{\\partial_{v} \\phi(u,v')}.\n\\end{align*}\n\nNow \\eqref{eq:decay2:key4nullStr:pf:7} follows by Lemmas \\ref{lem:smallPtnl} and \\ref{lem:dphi} and \\eqref{eq:bnd4dvrphi}.\n\n\\pfstep{Proof of \\eqref{eq:decay2:key4nullStr:pf:8}}\nAs in the proof of \\eqref{eq:decay2:key4nullStr:pf:4}, we will divide the integral into two pieces. More precisely, let us define $v^{\\star}(u)$ to be the unique $v$-value such that $r(u, v^{\\star}(u)) = R$. Assuming $v^{\\star}(u) \\in [u, 3u]$, the integral $\\int_{u}^{3u}$ will be divided into $\\int_{u}^{v^{\\star}(u)}$ and $\\int_{v^{\\star}(u)}^{3u}$. The remaining case $v^{\\star}(u) > 3u$ can be dealt with by adapting the argument for the first integral.\n\nFor the first integral, we claim that\n\\begin{equation*}\n\tu^{\\omega} \\int_{u}^{v^{\\star}(u)} \\abs{\\partial_{u} \\phi(u, v')} \\abs{\\partial_{v} (\\lmb^{-1} \\partial_{v} (r \\phi))(u, v')} \\, \\mathrm{d} v' \\leq \\epsilon(u_{2}) \\mathcal B_{2}(U).\n\\end{equation*}\n\nFrom the locally BV scattering assumption \\eqref{eq:locBVScat}, we have\n\\begin{equation*}\n\t\\sup_{u \\in [3 u_{2}, \\infty)} \\int_{C_{u} \\cap \\PD_{\\mathrm{cpt}}} \\abs{\\partial_{v} (\\lmb^{-1} \\partial_{v} (r \\phi))}  \\to 0 \\hbox{ as } u_{2} \\to \\infty.\n\\end{equation*}\n\nCombined with \\eqref{eq:decay2:key4nullStr:pf:3:1}, the claim follows.\n\nNext, we turn to the second integral. By \\eqref{eq:bnd4dvr} and \\eqref{eq:bnd4dvrphi}, we estimate\n\\begin{align*}\n& u^{\\omega} \\int_{v^{\\star}(u)}^{3u} \\abs{\\partial_{u} \\phi(u, v')} \\abs{\\partial_{v} (\\lmb^{-1} \\partial_{v} (r \\phi))(u, v')} \\, \\mathrm{d} v'  \\\\\n& \\quad \\leq \\sup_{v' \\in [v^{\\star}(u), 3u]} u^{\\omega} \\abs{\\partial_{v} (\\lmb^{-1} \\partial_{v} (r \\phi))(u, v')} \\int_{v^{\\star}(u)}^{3u} \\abs{\\partial_{u} \\phi(u, v')} \\, \\mathrm{d} v' \\\\\n& \\quad \\leq C_{\\Lambda, \\Psi} \\mathcal B_{2}(U) \\int_{v^{\\star}(u)}^{3u} \\abs{\\partial_{u} \\phi(u, v')} \\, \\mathrm{d} v'.\n\\end{align*}\n\nFor $v' \\in [v^{\\star}(u), 3u]$, we have $r (u, v) \\geq R$. Thus, by \\eqref{eq:decay1:5}, we have\n\\begin{equation*}\n\t\\int_{v^{\\star}(u)}^{3u} \\abs{\\partial_{u} \\phi(u, v')} \\, \\mathrm{d} v' \\leq \\frac{C_{K} A_{1}}{R} \\int_{u}^{3u} u^{-\\omega} \\, \\mathrm{d} v'  \\leq \\frac{C_{K} A_{1}}{R} u_{2}^{-(\\omega-1)},\n\\end{equation*}\nwhich vanishes as $u_{2} \\to \\infty$, and therefore finishes the proof of \\eqref{eq:decay2:key4nullStr:pf:8}. We remark that the fact that we are in $\\PD_{\\mathrm{int}}$ is used crucially here, as otherwise the integral would not be convergent. \n\\end{proof}\n\n\\begin{remark} \nIn the case where we have global BV scattering (i.e., conditions $(2)$ and $(3)$ of Definition \\ref{def:locBVScat} are satisfied with $R=\\infty$), we can take $R = \\infty$ in the preceding argument to obtain the following explicit upper bound on $\\epsilon(u_{2})$:\n\\begin{equation} \\label{eq:decay2:eps}\n\\begin{aligned}\n\t\\epsilon(u_{2}) \n\t\\leq & C_{K, \\Lambda, \\Psi} \\sup_{v \\in [u_{2}, \\infty)} \\int_{\\underline{C}_{v} \\cap \\set{u \\geq u_{2}}} \\abs{\\frac{2m \\nu}{(1-\\mu) r^{2}}}\n\t\t+C \\sup_{v \\in [u_{2}, \\infty)} \\int_{\\underline{C}_{v} \\cap \\set{u \\geq u_{2}}} \\abs{\\partial_{u} \\phi} \\\\\n\t\t& + C_{K, \\Lambda, \\Psi} \\sup_{v \\in [u_{2}, \\infty)} \\int_{\\underline{C}_{v} \\cap \\set{u \\geq u_{2}}} \\abs{\\partial_{u} (\\nu^{-1} \\partial_{u} (r \\phi))} \\\\\n\t\t& +C_{K, \\Lambda, \\Psi}  \\sup_{u \\geq 3u_{2}} \\int_{C_{u}} \\abs{\\frac{2m \\lmb}{(1-\\mu) r^{2}}}\n\t\t+C  \\sup_{u \\geq 3u_{2}} \\int_{C_{u}} \\abs{\\partial_{v} \\phi}  \\\\\n\t\t& +C_{K, \\Lambda, \\Psi} \\sup_{u \\in [3 u_{2}, \\infty)} \\int_{C_{u}} \\abs{\\partial_{v} (\\lmb^{-1} \\partial_{v} (r \\phi))}.\n\\end{aligned}\n\\end{equation}\n\nThis will be useful in our proof of the sharp decay rate in the case of small BV norm (Theorem \\ref{thm:smallData}) in Section \\ref{sec:smallData}.\n\\end{remark}\n\nIn the second step of our proof of Theorem \\ref{main.thm.2}, we use the preliminary $u^{-\\omega}$ decay proved in Proposition \\ref{prop:decay2:nullStr} to obtain the optimal the $u$-decay. Key to this step is the following proposition, which claims optimal $u$-decay in $\\PD_{\\mathrm{int}}$.\n\n\\begin{proposition} \\label{prop:decay2:final}\nThere exists a constant $0 < A_{2}'' < \\infty$ such that the following estimates hold.\n\\begin{align} \n\\sup_{\\PD_{\\mathrm{int}}} u^{\\omega+1} \\abs{\\partial_{v}^{2} (r \\phi)} \\leq & A_{2}'',\n\t\\label{eq:decay2:final:1} \\\\\n\\sup_{\\PD_{\\mathrm{int}}} u^{\\omega+1} \\abs{\\partial_{u}^{2} (r \\phi)} \\leq & A_{2}'', \n\t\\label{eq:decay2:final:2} \\\\\n\\sup_{\\PD_{\\mathrm{int}}} u^{3} \\abs{\\partial_{v} \\lmb} \\leq & A_{2}'', \n\t\\label{eq:decay2:final:3} \\\\\n\\sup_{\\PD_{\\mathrm{int}}} u^{3} \\abs{\\partial_{u} \\nu} \\leq & A_{2}''. \n\t\\label{eq:decay2:final:4}\n\\end{align}\n\\end{proposition}\n\nOnce we establish Proposition \\ref{prop:decay2:final}, the desired decay for $\\partial_{v}^{2}(r \\phi)$ and $\\partial_{v} \\lmb$ follow from Lemma \\ref{lem:decay2:rDecay} and the fact that $r \\geq 2 \\Lambda^{-1} u$ in $\\PD_{\\mathrm{ext}}$. Furthermore, the desired decay for $\\partial_{u}^{2}(r \\phi)$ and $\\partial_{u} \\nu$ follow from Lemma \\ref{lem:decay2:uDecayInExtr}.\n\n\\begin{proof} \nThanks to the fact that we have pointwise bounds for sufficient number of derivatives (albeit with sub-optimal decay) near $\\Gamma$ at this point, it suffices to work with the `non-renormalized' equations \\eqref{eq:eq4dvdvrphi:normal}, \\eqref{eq:eq4dudurphi:normal}, \\eqref{eq:eq4dvdvr:normal} and \\eqref{eq:eq4dudur:normal}. In particular, we need not utilize the null structure of \\eqref{eq:SSESF}. \n\nLet $(u,v) \\in \\PD_{\\mathrm{int}}$ (i.e., $v \\in [u, 3u]$) with $u \\geq 3$. We begin with \\eqref{eq:decay2:final:1}. Integrating $\\partial_{u} \\partial_{v}^{2}(r\\phi)$ in the $u$-direction from $u\/3$ to $u$, multiplying by $u^{\\omega+1}$ and using $r(u\/3, v) \\geq (2\/3) \\Lambda^{-1} u$, we obtain\n\\begin{equation} \\label{eq:decay2:final:1:pf}\n\tu^{\\omega+1}\\abs{\\partial_{v}^{2}(r \\phi)}(u,v)\n\t\\leq C_{\\Lambda} r^{\\omega+1} \\abs{\\partial_{v}^{2}(r \\phi)}(u\/3, v) + u^{\\omega+1} \\int_{u\/3}^{u} \\abs{\\partial_{u} \\partial_{v}^{2}(r \\phi)}(u', v) \\, \\mathrm{d} u'.\n\\end{equation}\n\nSince $(u\/3, v) \\in \\PD_{\\mathrm{ext}}$, the first term on the right-hand side is bounded by $\\leq C_{\\Lambda} \\mathcal I_{2} + C_{K, \\Lambda, M_{i}} A_{1}^{3}$, thanks to Lemma \\ref{lem:decay2:rDecay}. To estimate the $u'$-integral, we substitute $\\partial_{u} \\partial_{v}^{2} (r \\phi)$ by \\eqref{eq:eq4dvdvrphi:normal}. Then applying Proposition \\ref{prop:geomLocBVScat}, Lemma \\ref{lem:dphi}, Lemma \\ref{lem:muOverR}, Theorem \\ref{main.thm.1} and Proposition \\ref{prop:decay2:nullStr}, we obtain\n\\begin{equation*}\n\t\\abs{\\partial_{u} \\partial_{v}^{2}(r \\phi)}(u', v)\n\t\\leq C_{A_{1}, K, \\Lambda} (u')^{-3\\omega} A_{1} (A_{2}')^{2}.\n\\end{equation*}\n\nThus we have\n\\begin{equation} \\label{eq:decay2:final:1:explicit}\n\tu^{\\omega+1}\\abs{\\partial_{v}^{2}(r \\phi)}(u,v)\n\t\\leq C_{\\Lambda} \\mathcal I_{2} + C_{K, \\Lambda, M_{i}} A_{1}^{3} + C_{A_{1}, K, \\Lambda} A_{1} (A_{2}')^{2},\n\\end{equation}\nwhere we have used the fact that $\\omega > 1$, and thus $3 \\omega - 1 > \\omega + 1$ to throw away the $u$-weight in the last term. This proves \\eqref{eq:decay2:final:1}.\n\nNext, we prove \\eqref{eq:decay2:final:2}. Integrating $\\partial_{v} \\partial_{u}^{2} (r \\phi)$ in the $v$-direction from $u+$ to $v$ and multiplying by $u^{\\omega+1}$, we have\n\\begin{equation} \\label{eq:decay2:final:2:pf}\n\tu^{\\omega+1} \\abs{\\partial_{u}^{2}(r \\phi)}(u,v)\n\t\\leq \\lim_{v' \\to u+} u^{\\omega+1} \\abs{\\partial_{u}^{2} (r \\phi)}(u, v') + u^{\\omega+1} \\int_{u}^{3u} \\abs{\\partial_{v} \\partial_{u}^{2} (r \\phi)}(u, v') \\, \\mathrm{d} v'.\n\\end{equation}\n\nRecall that $\\lim_{v' \\to u+} \\partial_{u}^{2} (r \\phi)(u, v') = \\lim_{v' \\to u+} \\partial_{v}^{2} (r \\phi)(u, v')$, as $(\\phi, r, m)$ is a $C^{1}$ solution. Thus the first term on the right-hand side can be estimated via \\eqref{eq:decay2:final:1:explicit}. Substitute $\\partial_{v} \\partial_{u}^{2} (r \\phi)$ by \\eqref{eq:eq4dudurphi:normal} and apply, as before, Proposition \\ref{prop:geomLocBVScat}, Lemma \\ref{lem:dphi}, Lemma \\ref{lem:muOverR}, Theorem \\ref{main.thm.1} and Proposition \\ref{prop:decay2:nullStr}. Then we have\n\\begin{equation*}\n\t\\abs{\\partial_{v} \\partial_{u}^{2} (r \\phi)}(u, v')\n\t\\leq C_{A_{1}, K, \\Lambda} u^{-3\\omega} A_{1} (A_{2}')^{2}.\n\\end{equation*}\n\nIt now follows that \n\\begin{equation} \\label{eq:decay2:final:2:explicit}\n\tu^{\\omega+1}\\abs{\\partial_{u}^{2}(r \\phi)}(u,v)\n\t\\leq C_{\\Lambda} \\mathcal I_{2} + C_{K, \\Lambda, M_{i}} A_{1}^{3} + C_{A_{1}, K, \\Lambda} A_{1} (A_{2}')^{2},\n\\end{equation}\nwhich proves \\eqref{eq:decay2:final:2}.\n\nAt this point, combining Lemma \\ref{lem:dphi}, Theorem \\ref{main.thm.1}, Lemma \\ref{lem:decay2:rDecay} and \\eqref{eq:decay2:final:1:explicit}, note that we have the following improved $u$-decay for $\\partial_{v} \\phi$:\n\\begin{equation} \\label{eq:decay2:final:impDvphi} \n\\begin{aligned}\n\t\\sup_{\\calQ} u^{\\omega+1} \\abs{\\partial_{v}\\phi} \n\t\\leq C_{\\Lambda} \\sup_{\\calQ} (u^{\\omega+1} \\abs{\\partial_{v}^{2}(r \\phi)} + u A_{1} \\abs{\\partial_{v} \\lmb})\n\t\\leq B\n\\end{aligned}\n\\end{equation}\nwhere \n\\begin{equation} \\label{eq:decay2:final:aux}\nB := C_{\\Lambda} \\mathcal I_{2} + C_{K, \\Lambda, M_{i}} A_{1}^{3} + C_{A_{1}, K, \\Lambda} A_{1} (A_{2}')^{2} + C_{\\Lambda} A_{1} A_{2}'.\n\\end{equation}\n\nWe now turn to \\eqref{eq:decay2:final:3}. Integrating $\\partial_{u} \\partial_{v} \\log \\lmb$ in the $u$-direction from $u\/3$ to $u$, multiplying by $u^{3}$ and using $r(u\/3, v) \\geq (2\/3) \\Lambda^{-1} u$, we obtain\n\\begin{equation} \\label{eq:decay2:final:3:pf}\n\tu^{3} \\abs{\\partial_{v} \\log \\lmb }(u, v) \\leq \n\tC r^{3} \\abs{\\partial_{v} \\log \\lmb}(u\/3, v) + u^{3} \\int_{u\/3}^{u} \\abs{\\partial_{u} \\partial_{v} \\log \\lmb}(u', v) \\, \\mathrm{d} u'\n\\end{equation}\n\nSince $(u\/3, v) \\in \\PD_{\\mathrm{ext}}$, the first term on the right-hand side is estimated $\\leq C_{K, \\Lambda} A_{1}^{2}$ by Lemma \\ref{lem:decay2:rDecay} and the fact that $\\lmb^{-1} \\leq \\Lambda$. Next, substituting $\\partial_{u} \\log \\lmb$ by \\eqref{eq:eq4dvdvr:normal}, applying Proposition \\ref{prop:geomLocBVScat}, Lemma \\ref{lem:muOverR}, Lemma \\ref{lem:dphi} and using the improved bound \\eqref{eq:decay2:final:impDvphi}, we have \n\\begin{equation*}\n\t\\abs{\\partial_{u} \\partial_{v} \\log \\lmb}(u',v) \\leq C_{K, \\Lambda} B^{2} (u')^{-2(\\omega+1)}.\n\\end{equation*}\n\nTherefore\n\\begin{equation} \\label{eq:decay2:final:3:explicit}\n\tu^{3} \\abs{\\partial_{v} \\lmb}(u,v) \\leq C_{K, \\Lambda} A_{1}^{2} + C_{K, \\Lambda} B^{2},\n\\end{equation}\nwhere we used $2(\\omega + 1) - 1 > 3$ to throw away the $u$-weight in the last term. This proves \\eqref{eq:decay2:final:3}.\n\nFinally, we prove \\eqref{eq:decay2:final:4}. Integrating $\\partial_{v} \\partial_{u} \\log \\nu$ in the $v$-direction from $u+$ to $v$ and multiplying by $u^{3}$, we have\n\\begin{equation} \\label{eq:decay2:final:4:pf}\n\tu^{3} \\abs{\\partial_{u} \\log \\nu}(u,v) \\leq \\lim_{v' \\to u+} u^{3} \\abs{\\partial_{u} \\log \\nu}(u, v') + u^{3} \\int_{u}^{3u} \\abs{\\partial_{v} \\partial_{u} \\log \\nu}(u, v') \\, \\mathrm{d} v'.\n\\end{equation}\n\nSince $\\lim_{v' \\to u+} \\partial_{u} \\nu(u, v') = - \\lim_{v' \\to u+} \\partial_{v} \\lmb(u, v')$, the first term is bounded by \\eqref{eq:decay2:final:3:explicit}. Furthermore, substituting $\\partial_{v} \\partial_{u} \\log \\nu$ by \\eqref{eq:eq4dudur:normal} and applying Proposition \\ref{prop:geomLocBVScat}, Lemma \\ref{lem:muOverR}, Lemma \\ref{lem:dphi} and using the improved bound \\eqref{eq:decay2:final:impDvphi}, we have \n\\begin{equation*}\n\t\\abs{\\partial_{v} \\partial_{u} \\log \\nu}(u, v') \\leq C_{K, \\Lambda} B^{2} u^{-2(\\omega+1)}.\n\\end{equation*}\n\nAs before, it follows that\n\\begin{equation} \\label{eq:decay2:final:4:explicit}\n\tu^{3} \\abs{\\partial_{u} \\nu}(u,v) \\leq C_{K, \\Lambda} A_{1}^{2} + C_{K, \\Lambda} B^{2},\n\\end{equation}\nwhich proves \\eqref{eq:decay2:final:4}.\n\\end{proof}\n\n\\begin{remark} \nCombining \\eqref{eq:decay2:final:1:explicit}, \\eqref{eq:decay2:final:2:explicit}, \\eqref{eq:decay2:final:3:explicit} and \\eqref{eq:decay2:final:4:explicit}, we see that Proposition \\ref{prop:decay2:final} holds with\n\\begin{equation} \\label{eq:decay2:A2''}\n\tA_{2}'' \\leq C_{\\Lambda} \\mathcal I_{2} + C_{K, \\Lambda, M_{i}} A_{1}^{3} + C_{A_{1}, K, \\Lambda} A_{1} (A_{2}')^{2}\n\t\t+ C_{K, \\Lambda} A_{1}^{2} + C_{K, \\Lambda} B^{2}\n\\end{equation}\nwhere $B$ is as in \\eqref{eq:decay2:final:aux}.\n\\end{remark}\n\n\\begin{remark} \nAccording to the argument of this subsection, note that the size of $A_{2}'$ in Proposition \\ref{prop:decay2:nullStr} depends on the choice of $u_{2}$ through the term $H_{2}'(u_{2})$, where the size of $u_{2}$ depends on the rate of convergence of $\\epsilon''(u_{2}) \\to 0$ as $u_{2} \\to \\infty$. This explains why $A_{2}$ does not depend only on the size of the initial data, as remarked in Section \\ref{sec.main.thm}. On the other hand, as stated in Statement (2) of Theorem \\ref{thm:smallData}, we shall show that in the case of small BV initial data, $A_2$ depends only on the size of the initial data. To achieve this, we show in Section \\ref{sec:smallData} that we may take $u_{2} = 1$ under this small data assumption. \n\\end{remark}\n\n\n\\subsection{Additional decay estimates}\nAs in the previous section, we conclude this section by providing additional decay rates concerning second derivatives of $\\phi$, $r$ and improved decay for $m$ near $\\Gamma$.\n\n\\begin{corollary} \\label{cor:decay2}\nLet $(\\phi, r, m)$ be a locally BV scattering solution to \\eqref{eq:SSESF} with asymptotically flat $C^{1}$ initial data of order $\\omega'$.\nLet $A_{1}$ and $A_{2}$ be the constants in Theorems \\ref{main.thm.1} and \\ref{main.thm.2}, respectively. Then \nthe following bounds hold.\n\\begin{align} \n\t\\abs{\\partial_{v} \\phi} \\leq & C_{\\Lambda} (A_{1} + A_{2} + A_{1} A_{2}) \\min \\set{u^{-(\\omega+1)}, r^{-2} u^{-(\\omega-1)}} \\label{eq:decay2:5} \\\\\n\t\\abs{\\partial_{u} \\phi} \\leq & C_{K, \\Lambda} (A_{1} + A_{2} + A_{1} A_{2}) \\min \\set{u^{-(\\omega+1)}, r^{-1} u^{-\\omega}} \\label{eq:decay2:6} \\\\\n\t\\abs{\\partial_{v}^{2} \\phi} \\leq & C_{\\Lambda} (A_{1} + A_{2} + A_{1} A_{2}) \\min \\set{r^{-1} u^{-(\\omega+1)}, r^{-3} u^{-(\\omega-1)}}, \\label{eq:decay2:7}\\\\\n\t\\abs{\\partial_{u} \\partial_{v} \\phi} \\leq & C_{K, \\Lambda} (A_{1} + A_{2} + A_{1} A_{2}) \\min \\set{r^{-1} u^{-(\\omega+1)}, r^{-2} u^{-\\omega}}, \\label{eq:decay2:8}\\\\\n\t\\abs{\\partial_{u}^{2} \\phi} \\leq & C_{K, \\Lambda} (A_{1} + A_{2} + A_{1} A_{2}) \\, r^{-1} u^{-(\\omega+1)}, \\label{eq:decay2:9}\\\\\n\t\\abs{\\partial_{u} \\partial_{v} r} \\leq & C_{K, \\Lambda} (A_{1} + A_{2} + A_{1} A_{2})^{2} \\min \\set{r u^{-(2\\omega+2)}, r^{-2} u^{-(2\\omega-1)}}, \\label{eq:decay2:10} \\\\\n\tm \\leq & C_{K, \\Lambda} (A_{1} + A_{2} + A_{1} A_{2})^{2} \\min \\set{r^{3} u^{-(2\\omega+2)}, u^{-(2\\omega-1)}}. \\label{eq:decay2:11}\n\\end{align}\n\\end{corollary}\n\nThis corollary follows immediately from the estimates derived in Theorem \\ref{main.thm.2}. We sketch the proof below.\n\n\n\\begin{proof} \nFirst, note that \\eqref{eq:decay2:5} and \\eqref{eq:decay2:6} follows from Corollary \\ref{cor:decay1}, Theorem \\ref{main.thm.2} and Lemma \\ref{lem:dphi}. \nNext, \\eqref{eq:decay2:7} and \\eqref{eq:decay2:9} are easy consequences of the preceding estimates, Theorems \\ref{main.thm.1}, \\ref{main.thm.2} and the identities\n\\begin{equation*}\n\tr \\partial_{v}^{2} \\phi = \\partial_{v}^{2} ( r\\phi) - (\\partial_{v} \\lmb) \\phi - 2 \\lmb \\partial_{v} \\phi, \\quad\n\tr \\partial_{u}^{2} \\phi = \\partial_{u}^{2} ( r\\phi) - (\\partial_{u} \\nu) \\phi - 2 \\nu \\partial_{u} \\phi.\n\\end{equation*}\n\nOn the other hand, for \\eqref{eq:decay2:8}, we use the identity\n\\begin{equation*}\n\tr \\partial_{u} \\partial_{v} \\phi = - \\lmb \\partial_{u} \\phi - \\nu \\partial_{v} \\phi,\n\\end{equation*}\nwhich may be verified from \\eqref{eq:SSESF:dr} and \\eqref{eq:SSESF:dphi}.\n\nNext, \\eqref{eq:decay2:11} follows from Corollary \\ref{cor:decay1}, Lemma \\ref{lem:muOverR} and \\eqref{eq:decay2:5}. Finally, using Corollary \\ref{cor:est4dr}, Lemma \\ref{lem:est4dur}, \\eqref{eq:decay2:11} and the equation \\eqref{eq:SSESF:dr}, we conclude \\eqref{eq:decay2:10}. \\qedhere\n\\end{proof}\n\n\\section{Decay and blow up at infinity}\\label{sec.dichotomy}\n\nIn this section, we prove Theorem \\ref{thm.dichotomy}, i.e., unless the solution blows up at infinity, a `future causally geodesically complete' solution scatters in BV.\n\nTake a BV solution to \\eqref{eq:SSESF} satisfying the hypotheses of Theorem \\ref{thm.dichotomy}, which does not blow up at infinity. Note, in particular, that $\\calQ = \\calR$ by $(1)$ of Definition \\ref{def:locBVScat} and Lemma \\ref{lem:regR}. In order to prove Theorem \\ref{thm.dichotomy}, our goal is to show that such a spacetime is in fact BV scattering, i.e., (1), (2) and (3) in Definition \\ref{def:locBVScat} hold and moreover (3) holds with $R=\\infty$.\n\nThe main step will be to show that there exists a constant $C_{\\Lambda}$ such that for every $\\epsilon > 0$, there exists $U$ such that for every $u\\geq U$, we have\n\\begin{equation}\\label{dichotomy.goal}\n\\int_{C_u}|\\partial_v^2(r\\phi)| + \\int_{C_u}|\\partial_v \\lambda| \\leq C_{\\Lambda} \\epsilon.\n\\end{equation}\nThis will be achieved in a sequence of Lemmas and Propositions below.\n\nBefore we proceed, we first prove a preliminary bound on $\\lmb$:\n\\begin{proposition}\nThere exists $0<\\Lambda<\\infty$ such that\n\\begin{equation} \\label{dic.bnd4dvr}\n\t\\Lambda^{-1} \\leq \\lmb(u,v) \\leq \\frac{1}{2}.\n\\end{equation}\n\\end{proposition}\n\\begin{proof}\nBy $(1)$ in Definition \\ref{def.blow.up.infty}, there exists $0 < \\Lambda < \\infty$ such that $\\sup \\lmb_{\\Gamma}^{-1} \\leq \\Lambda$. As $\\lim_{u \\to v-} \\lmb_{\\Gamma}(u) = \\lim_{u \\to v-} \\lmb(u', v)$ (see \\cite[Section 7]{Christodoulou:1993bt}), it follows from Lemma \\ref{lem:basicEst4dr} that for every $(u,v) \\in \\calQ$, we have the estimate \\eqref{dic.bnd4dvr}.\n\\end{proof}\n\nWe now proceed to show \\eqref{dichotomy.goal}. The first step is to show that for $u$ sufficiently large, the integrals along $C_u$ of $|F_1|$ and $|F_2|$ are small. Here, we recall the notation in the proof of Proposition \\ref{prop:decay2:nullStr}, i.e.,\n\\begin{align*}\nF_{1} := & \\partial_{v}^{2} (r \\phi) - (\\partial_{v} \\lmb) \\phi, \\\\\n\tF_{2} := & \\partial_{v} \\log \\lmb - \\frac{\\lmb}{(1-\\mu)} \\frac{\\mu}{r} + \\partial_{v} \\phi \\Big( \\lmb^{-1} \\partial_{v} (r \\phi) - \\nu^{-1} \\partial_{u} ( r \\phi) \\Big).\n\\end{align*}\nOnce we obtain the desired bounds for $F_1$ and $F_2$, we then derive \\eqref{dichotomy.goal} from these bounds. This will be the most technical part (see discussions in Remark \\ref{technical.rmk.dichotomy}).\n\nFirst, we bound the integrals of $F_1$ and $F_2$ in the following proposition:\n\\begin{proposition}\\label{8.1.1}\nFor every $\\epsilon>0$, there exists $V$ sufficiently large such that the following bound holds for $u\\geq V$:\n\\begin{equation} \\label{dichotomy.beginning}\n\\int_{C_u} (|F_1|+|F_2|)(u,v)  \\leq 3\\epsilon.\n\\end{equation}\n\\end{proposition}\n\\begin{proof}\nBy $(2)$ and $(3)$ in Definition \\ref{def.blow.up.infty}, we have\n$$\\int_1^{\\infty}\\int_u^{\\infty}|\\partial_v\\lambda\\partial_u\\phi-  \\partial_u\\lambda\\partial_v\\phi  | \\mathrm{d} v \\mathrm{d} u <\\infty,$$\nand\n$$\\int_1^{\\infty}\\int_u^{\\infty}|\\partial_u\\phi\\partial_v(\\nu^{-1}\\partial_u(r\\phi))-\\partial_v\\phi\\partial_u(\\nu^{-1}\\partial_u(r\\phi))| \\mathrm{d} v \\mathrm{d} u <\\infty.$$\n\nThus, by choosing $V$ sufficiently large, we have\n\\begin{equation}\n\\int_V^{\\infty}\\int_u^{\\infty}|\\partial_v\\lambda\\partial_u\\phi-\\partial_v\\lambda\\partial_v\\phi| \\mathrm{d} v \\mathrm{d} u <\\epsilon,\\label{non.blowup.1}\n\\end{equation}\nand\n\\begin{equation}\n\\int_V^{\\infty}\\int_u^{\\infty}|\\partial_u\\phi\\partial_v(\\nu^{-1}\\partial_u(r\\phi))-\\partial_v\\phi\\partial_u(\\nu^{-1}\\partial_u(r\\phi))| \\mathrm{d} v \\mathrm{d} u <\\epsilon.\\label{non.blowup.2}\n\\end{equation}\n\n\n\nFrom the initial conditions, we easily see that $F_{1}(1, \\cdot)$, $F_{2}(1, \\cdot)$ obey $\\int_{C_{1}} \\abs{F_{1}} + \\abs{F_{2}} < \\infty$. \nThus, by choosing $V$ larger if necessary, we have\n\\begin{equation}\n\\int_V^{\\infty} (|F_1|+|F_2|)(1,v) \\mathrm{d} v\\leq \\epsilon.\\label{data.BV.bd}\n\\end{equation}\nNotice that by equations \\eqref{eq:decay2:nullStr:pf:1} and \\eqref{eq:decay2:nullStr:pf:2}, the estimates \\eqref{non.blowup.1} and \\eqref{non.blowup.2} control $\\iint |\\partial_u F_1| \\mathrm{d} u \\mathrm{d} v$ and $\\iint |\\partial_u F_2| \\mathrm{d} u \\mathrm{d} v$. Thus, we have\n$$\\int_{\\max\\set{u, V}}^{\\infty} (|F_1|+|F_2|)(u,v) \\mathrm{d} v\\leq 3\\epsilon.$$\nfor every $u\\geq 1$. In particular, for $u\\geq V$, we have\n\\begin{equation*}\n\\int_{C_u} (|F_1|+|F_2|)(u,v)  \\leq 3\\epsilon,\n\\end{equation*}\nas desired.\n\\end{proof}\n\nThe inequality \\eqref{dichotomy.beginning} is the starting point for our proof of \\eqref{dichotomy.goal}. More precisely, our basic strategy is to use a continuous induction on $v$, beginning from the axis, to remove the quadratic and higher terms from \\eqref{dichotomy.beginning} and infer \\eqref{dichotomy.goal}. \n\n\\begin{remark} \\label{technical.rmk.dichotomy}\nBefore beginning the proof in earnest, we would like to point out two technical nuisances that we confront: \nFirst, in order to estimate the scalar field $\\phi$ itself from $F_{1}$ and $F_{2}$, we need to integrate essentially from null infinity\\footnote{More precisely, $\\phi$ is determined from $\\partial_{v} (r \\phi)$, which in turn can be determined from $\\int \\abs{\\partial_{v}^{2}(r \\phi)}$ by integrating from $v = \\infty$. Another conceptual reason why information near $v = \\infty$ is relevant for estimating $\\phi$ is that the initial condition $\\lim_{v \\to \\infty} \\phi(1, v) = 0$ implies that $\\lim_{v \\to \\infty} \\phi(u, v)= 0$ for every $u \\geq 1$. See the discussion before \\eqref{dic.0.2}.}, which is opposite to the direction of our method of continuity. Second, as $\\partial_{v}(r \\phi)$ is only assumed to be BV, the left-hand side of \\eqref{dichotomy.goal} is not continuous in $v$ in general. To overcome the first, we make use of the invariance of \\eqref{eq:SSESF} and $F_{1}$, $F_{2}$ under the change $\\phi \\mapsto \\phi + c$. To take care of the second, we carefully keep track of the evolution of discontinuities of $\\partial_{v}(r \\phi)$.\n\\end{remark}\n\nNotice that in order to obtain \\eqref{dichotomy.goal} from \\eqref{dichotomy.beginning}, we only need to integrate on a \\emph{fixed} hypersurface $C_u$. We now fix $u_0\\geq V$ and define a new function $\\overline{\\phi}_{u_{0}}$ by\n\\begin{equation} \\label{dic.0.0}\n\t\\overline{\\phi}_{u_{0}}(u,v) := \\phi(u,v) - \\lim_{v' \\to u_{0} +} \\phi(u_{0}, v').\n\\end{equation}\n\nAs remarked before, note that \\eqref{eq:SSESF} is invariant under the change $(\\phi, r, m) \\mapsto (\\overline{\\phi}_{u_{0}}, r, m)$, i.e., $(\\overline{\\phi}_{u_{0}}, r, m)$ is still a solution to \\eqref{eq:SSESF}. Moreover, it is easy to check that $F_{1}$ and $F_{2}$ are also invariant under this change, i.e.,\n\\begin{equation} \\label{eq:inv4F12}\n\\begin{aligned}\n\tF_{1} =& \\partial_{v}^{2} (r \\overline{\\phi}_{u_{0}}) - (\\partial_{v} \\lmb) \\overline{\\phi}_{u_{0}} \\\\\n\tF_{2} =& \\partial_{v} \\log \\lmb - \\frac{\\lmb}{(1-\\mu)} \\frac{\\mu}{r} +  \\partial_{v} \\overline{\\phi}_{u_{0}} \\Big( \\lmb^{-1} \\partial_{v} (r \\overline{\\phi}_{u_{0}}) - \\nu^{-1} \\partial_{u} ( r \\overline{\\phi}_{u_{0}}) \\Big).\n\\end{aligned}\n\\end{equation}\n\nThe new scalar field has been chosen so that $\\overline{\\phi}_{u_{0}}(u_{0}, \\cdot)$ and $\\partial_{v}(r \\overline{\\phi}_{u_{0}})(u_{0}, \\cdot)$ vanish at the axis, i.e.,\n\\begin{equation} \\label{dic.0.1}\n\\lim_{v \\to u_{0}+}\\overline{\\phi}_{u_{0}}(u_{0},v) = \\lim_{v \\to u_{0}+}\\partial_{v}(r \\overline{\\phi}_{u_{0}}) (u_{0},v) = \\lim_{v \\to u_{0}+} \\partial_{u}(r \\overline{\\phi}_{u_{0}})(u_{0},v) = 0.\n\\end{equation}\n\nWe claim that the original scalar field $\\phi(u, v)$ obeys the condition \n\\begin{equation}\\label{claim.vanishing}\n\\lim_{v \\to \\infty} \\phi(u_{0}, v) = 0\n\\end{equation} \nfor every $u_{0} \\geq 1$.\nTherefore, by the definition given in \\eqref{dic.0.0}, we see that $\\phi$ and $\\overline{\\phi}_{u_0}$ are also related by\n\\begin{equation} \\label{dic.0.2}\n\t\\phi(u,v) = \\overline{\\phi}_{u_{0}}(u,v) - \\lim_{v' \\to \\infty} \\overline{\\phi}_{u_{0}}(u, v').\n\\end{equation}\n\n\nTo establish the claim \\eqref{claim.vanishing}, we proceed as in the proof of Lemma~\\ref{lem:decay1:cptu:0}, but work with $\\phi$ rather than $r \\phi$. \nFix $u_{0} > 1$ and let $r_{1} > 0$ be a large number to be determined. For each $u \\geq 1$, let $v^{\\star}_{1}(u)$ be the unique $v$-value such that $r(u, v^{\\star}_{1}(u)) = r_{1}$. Consider $(u, v) \\in \\set{1 \\leq u \\leq u_{0}} \\cap \\set{r \\geq r_{1}}$. Using the uniform bound of $m$ and $\\frac{\\lmb}{1-\\mu}$ in terms of the data at $u= 1$ (which holds thanks to monotonicity), we may integrate \\eqref{eq:SSESF:dphi} along the incoming direction to estimate\n\\begin{equation*}\n\t\\abs{\\partial_{v} (r \\phi) (u, v) - \\partial_{v} (r \\phi)(1, v)} \n\t\\leq \\frac{C_{0}}{r(u, v)} \\sup_{u' \\in [1, u]} \\abs{\\phi(u', v)},\n\\end{equation*}\nwhere $C_{0}$ depends only on the data at $u=1$. Integrating both sides in the outgoing direction from $v_{1}^{\\star}(u)$ to $v$ (using Lemma~\\ref{dic.bnd4dvr} for the right-hand side) and dividing by $r = r(u, v)$, we obtain\n\\begin{equation} \\label{claim.vanishing.key}\n\\begin{aligned}\n\t\\abs{\\phi(u, v)} \\leq & \\frac{r_{1}}{r} \\abs{\\phi(u, v^{\\star}_{1}(u))} + \\frac{r(1, v^{\\star}_{1}(u))}{r} \\abs{\\phi(1, v^{\\star}_{1}(u))} \\\\\n\t& + \\frac{r(1, v)}{r} \\abs{\\phi(1, v)}\t\t+ \\frac{C_{0} \\Lambda}{r} \\log \\Big( \\frac{r}{r_{1}}\\Big) \\sup_{1 \\leq u' \\leq u, \\, v \\geq v^{\\ast}_{1}(u)} \\abs{\\phi}.\n\\end{aligned}\n\\end{equation}\nNow the idea is to use \\eqref{claim.vanishing.key} to first show that $\\phi$ is bounded on the region $\\set{1 \\leq u \\leq u_{0}}$, and then use \\eqref{claim.vanishing.key} again with the additional boundedness of $\\phi$ to conclude that \\eqref{claim.vanishing} holds.\nTo begin with, observe that $\\phi$ is bounded on each set compact subset of $\\calQ$, since it is a BV solution in the sense of Definition~\\ref{def:BVsolution}. Combined with the hypothesis that $\\phi(1, v) \\to 0$ as $v \\to \\infty$, we see that the first three terms are bounded by a constant that depends on $r_{1}$. On the other hand, by taking $r_{1}$ sufficiently large, the coefficient $(C_{0} \\Lambda \/ r) \\log ( r \/ r_{1} )$ of the last term can be made arbitrarily small for $r \\geq r_{1}$. This smallness allows us to absorb the last term to the left-hand side, and conclude the desired boundedness of $\\phi$ on the region $\\set{1 \\leq u \\leq u_{0}}$.\nThen plugging in $u = u_{0}$ and the uniform bound for $\\phi$ into \\eqref{claim.vanishing.key}, the claim \\eqref{claim.vanishing} follows from the hypothesis $\\lim_{v \\to \\infty} \\phi(1, v)  = 0$.\n\n\n\n\n\n\nLet\n\\begin{eqnarray*}\nI_1(u, v)&:=&\\int_{u}^v |\\partial_v^2(r \\overline\\phi_{u_{0}})| (u,v')dv'  ,\\quad I_2(u, v):=\\int_{u}^v |\\partial_v\\lambda | (u,v')dv' .\n\\end{eqnarray*}\n\nIn the following two lemmas, we will show that\n\\begin{align} \nI_1(u_0, v) \\leq & 3\\epsilon+ C_{\\Lambda} I_{1}(u_0,v) I_2(u_0,v) , \\label{main.ineq.dichotomy.1} \\\\\nI_2(u_0, v)\\leq & 3\\epsilon+ C_{\\Lambda} I_{1}(u_0,v)^{2} (1+I_{1}(u_0,v))^{2} (1+I_{2}(u_0,v))^{2} e^{C_{\\Lambda} I_{1}(u_0,v)^{2} (1+I_{2}(u_0,v))} \\label{main.ineq.dichotomy.2}\n\\end{align}\nfor every $V \\leq u_0 \\leq v$, with $C_{\\Lambda}$ independent of $u_0$ and $v$.\n\n\\begin{lemma} \\label{lem.dic.1}\nThere exists a constant $C_{\\Lambda} > 0$ such that for every $V \\leq u_0 \\leq v$,\n\\begin{equation*}\nI_1(u_0,v) \\leq 3\\epsilon+ C_{\\Lambda} I_{1}(u_0,v) I_2(u_0,v).\n\\end{equation*}\n\\end{lemma}\n\\begin{proof}\nIn this proof, we fix $u_0 \\geq V$ and use the abbreviations\n\\begin{equation} \\label{eq:abbrev-overline-phi}\n\\overline{\\phi} := \\overline{\\phi}_{u_{0}} , \\quad\n\\partial_{v}(r \\overline{\\phi}) := \\partial_{v}(r \\overline{\\phi}_{u_{0}}) \n\\ \\hbox{ and } \\ \n\\partial_{v}^{2}(r \\overline{\\phi}) := \\partial_{v}^{2}(r \\overline{\\phi}_{u_{0}}).\n\\end{equation}\nBy Lemma \\ref{lem:est4phi}, we have\n\\begin{equation} \\label{phi.est}\n\t\\abs{\\overline{\\phi}(u_0,v)} \n\t\\leq \\frac{1}{r} \\int_{u_0}^{v} \\partial_{v}(r \\overline{\\phi}) (u_0, v') \\, \\mathrm{d} v' \n\t\\leq \\Lambda \\sup_{u_0 \\leq v' \\leq v} \\abs{\\partial_{v}(r \\overline{\\phi})(u_0,v')}.\n\\end{equation}\n\nBy the fundamental theorem of calculus and \\eqref{dic.0.1}, note that\n\\begin{equation} \\label{dic.1.0}\n\\sup_{u_0 \\leq v' \\leq v} |\\partial_{v}(r \\overline{\\phi})(u_0,v')|\\leq I_1(u_0,v).\n\\end{equation}\n\nThus, recalling the definition of $F_1$ in \\eqref{eq:inv4F12}, we have\n\\begin{equation*}\n\\begin{split}\nI_1(u_0,v) \\leq &\\int_{u_0}^v |F_1(u_0,v')|dv'+\\int_{u_0}^v|\\partial_v\\lmb||\\overline{\\phi}|(u_0,v')dv' \\\\\n\\leq &\\int_{u_0}^v |F_1(u_0,v')|dv'+ \\Lambda\\,  I_1(u_0,v) I_2(u_0,v) \n\t\t\\leq  3\\epsilon+C_{\\Lambda} I_{1}(u_0,v) I_2(u_0,v). \\qedhere\n\\end{split}\n\\end{equation*}\n\\end{proof}\n\nWe now move on to estimate $I_2(u_0,v)$.\n\\begin{lemma} \\label{lem.dic.2}\nThere exists a constant $C_{\\Lambda} > 0$ such that for every $V \\leq u_0 \\leq v$,\n\\begin{equation*} \nI_2(u_0,v)\\leq 3\\epsilon+ C_{\\Lambda} I_{1}(u_0,v)^{2} (1+I_{1}(u_0,v))^{2} (1+I_{2}(u_0,v))^{2} e^{C_{\\Lambda} I_{1}(u_0,v)^{2} (1+I_{2}(u_0,v))}. \n\\end{equation*}\n\\end{lemma}\n\\begin{proof}\nAgain, we fix $u_0 \\geq V$ and use the abbreviation \\eqref{eq:abbrev-overline-phi}, as well as\n\\begin{equation} \\label{eq:abbrev-du-overline-phi}\n\t\\partial_{u} (r \\overline{\\phi}) (u,v) := \\partial_{u} (r \\overline{\\phi}_{u_{0}})(u,v) .\n\\end{equation}\nRecalling the equation for $F_2$ in \\eqref{eq:inv4F12}, in order to control $I_2(u_0,v)$ from $F_2$, we need to estimate $\\int_{u_0}^{v} (\\frac{\\lambda}{1-\\mu} \\frac{\\mu}{r})(u_0,v') \\mathrm{d} v'$ and $\\int_{u_0}^v \\partial_{v} \\overline{\\phi} ( \\lmb^{-1} \\partial_{v} (r \\overline{\\phi}) - \\nu^{-1} \\partial_{u} ( r \\overline{\\phi})) (u_0,v') \\mathrm{d} v'.$\nBy Lemma \\ref{lem:auxEqs},\n$$\\int_{u_0}^{v} (\\frac{\\lambda}{1-\\mu} \\frac{\\mu}{r})(u_0,v') \\mathrm{d} v' = \\log (1-\\mu(u_0,v))+\\int_{u_0}^v \\frac{r(\\partial_v \\overline{\\phi})^2}{\\lambda}(u_0,v') \\mathrm{d} v'.$$\nSince $\\calQ =\\calR$, the integrand on the left-hand side is non-negative. Notice furthermore that since $\\mu \\geq 0$, $\\log (1-\\mu(u_0,v)) <0$. Thus, \n\\begin{eqnarray*}\n\\int_{u_0}^{v} (\\frac{\\lambda}{1-\\mu} \\frac{\\mu}{r})(u_0,v') \\mathrm{d} v'\n&\\leq & |\\int_{u_0}^v \\frac{r(\\partial_v \\overline{\\phi})^2}{\\lambda}({u_0},v') \\mathrm{d} v'|\\\\\n&\\leq & \\int_{u_0}^v |\\partial_v \\overline{\\phi}({u_0},v')||(\\frac{\\partial_v(r\\overline{\\phi})}{\\lambda}- \\overline{\\phi})({u_0},v')| \\mathrm{d} v'\\\\\n&\\leq & 2 \\Lambda I_{1}({u_0},v)  \\int_{u_0}^v |\\partial_v \\overline{\\phi}({u_0},v')| \\, \\mathrm{d} v' ,\n\\end{eqnarray*}\nwhere we have used \\eqref{phi.est} and \\eqref{dic.1.0} on the last line. Using Lemma \\ref{lem:est4dvphi}, we estimate the integral on the last line by\n\\begin{equation} \\label{dic.2.0}\n\\int_{u_0}^v |\\partial_v \\overline{\\phi}({u_0},v')|dv' \\leq\n\\int_{u_0}^v|\\partial_v(\\lambda^{-1}\\partial_v(r \\overline{\\phi}))({u_0},v')| \\mathrm{d} v' ,\n\\end{equation}\nand the right hand side can in turn be estimated using \\eqref{dic.1.0} by\n\\begin{eqnarray}\n\\int_{u_0}^v|\\partial_v(\\lambda^{-1}\\partial_v(r \\overline{\\phi}))({u_0},v')| \\mathrm{d} v' \n&\\leq &\\int_{u_0}^v \\lambda^{-1}|\\partial_v^2(r \\overline{\\phi})({u_0},v')| \\mathrm{d} v'+\\int_{u_0}^v \\lambda^{-2}|\\partial_v\\lambda\\partial_v(r \\overline{\\phi})({u_0},v')| \\mathrm{d} v'\\notag\\\\\n&\\leq &\\Lambda I_1({u_0},v)+ \\Lambda^2 I_{1}({u_0},v) I_2({u_0},v). \\notag\n\\end{eqnarray}\n\nTherefore, we have\n\\begin{equation}\\label{dic.2.1}\n\\int_{u_0}^{v} (\\frac{\\lambda}{1-\\mu} \\frac{\\mu}{r})({u_0},v') \\mathrm{d} v'\\leq  C_{\\Lambda} I_1({u_0},v)^{2} (1+ I_{2}({u_0},v)).\n\\end{equation}\n\n\n\nWe now move on to bound $\\int_{u_0}^v \\partial_{v} \\overline{\\phi} \\, \\lmb^{-1} \\partial_{v} (r \\overline{\\phi}) ({u_0}, v') \\mathrm{d} v'$. Using \\eqref{dic.1.0} and \\eqref{dic.2.0}, we easily estimate\n\\begin{align}\n\\int_{u_0}^v \\abs{\\partial_{v} \\overline{\\phi} \\, \\lmb^{-1} \\partial_{v} ( r \\overline{\\phi}) ({u_0},v')} \\mathrm{d} v' \\notag\n\\leq & \\Lambda \\int_{{u_0}}^{v} \\abs{\\partial_{v} \\overline{\\phi}}({u_0}, v') \\, \\mathrm{d} v' \\sup_{{u_0} \\leq v' \\leq v} \\abs{\\partial_{v} ( r \\overline{\\phi}) ({u_0},v')} \\\\\n\\leq & C_{\\Lambda} I_{1}({u_0},v)^{2} (1+I_{2}({u_0},v)). \\label{dic.2.2}\n\\end{align}\n\nFinally, we are only left to bound $- \\int_{u_0}^v \\partial_{v} \\overline{\\phi} \\, \\nu^{-1} \\partial_{u} ( r \\overline{\\phi}) ({u_0},v') \\mathrm{d} v'$. As before, we begin by estimating\n\\begin{align}\n\\int_{u_0}^v \\abs{\\partial_{v} \\overline{\\phi} \\, \\nu^{-1} \\partial_{u} ( r \\overline{\\phi}) ({u_0},v')} \\mathrm{d} v'\n\\leq & \\int_{{u_0}}^{v} \\abs{\\partial_{v} \\overline{\\phi}}({u_0}, v') \\, \\mathrm{d} v' \\sup_{{u_0} \\leq v' \\leq v} \\abs{\\nu^{-1} \\partial_{u} ( r \\overline{\\phi}) ({u_0},v')} \\notag \\\\\n\\leq & C_{\\Lambda} I_{1}({u_0},v) (1+I_{2}({u_0},v)) \\sup_{{u_0} \\leq v' \\leq v} \\abs{\\nu^{-1} \\partial_{u} ( r \\overline{\\phi}) ({u_0},v')}. \\label{dic.2.3}\n\\end{align}\n\nIn this case, we do not wish to pull out $\\nu$ as we have not assumed any bound on it. Instead, we consider $\\nu^{-1} \\partial_{u} ( r \\overline{\\phi})$ as a whole and note that\n\\begin{equation}\\label{inv.wave.eqn}\n\\partial_{v}(\\nu^{-1} \\partial_{u} ( r \\overline{\\phi}) ) \n= \t- \\Big( \\frac{\\lmb}{1-\\mu} \\frac{\\mu}{r} \\Big) \\nu^{-1} \\partial_{u} ( r \\overline{\\phi}) \n\t+ \\Big( \\frac{\\lmb}{1-\\mu} \\frac{\\mu}{r} \\Big) \\overline{\\phi}.\n\\end{equation}\nThe equation \\eqref{inv.wave.eqn} holds since by \\eqref{eq:SSESF:dr} and \\eqref{eq:SSESF:dphi}, we have \n\\begin{equation*}\n\\partial_{v}(\\nu^{-1} \\partial_{u} ( r {\\phi}) ) \n= \t- \\Big( \\frac{\\lmb}{1-\\mu} \\frac{\\mu}{r} \\Big) \\nu^{-1} \\partial_{u} ( r {\\phi}) \n\t+ \\Big( \\frac{\\lmb}{1-\\mu} \\frac{\\mu}{r} \\Big) {\\phi}\n\\end{equation*}\nand moreover both the left hand side and the right hand side of the equation are invariant under the transformation $\\phi\\to \\phi+c$.\n\nTherefore, by the variation of constants formula and \\eqref{dic.0.1}, we have\n\\begin{equation*}\n\t\\nu^{-1} \\partial_{u} (r \\overline{\\phi})(u_0, v)\n\t= e^{-J({u_0},v)} \\int_{{u_0}}^{v} e^{J({u_0},v')}  \\frac{\\lmb}{1-\\mu} \\frac{\\mu}{r} \\,  \\overline{\\phi} ({u_0}, v') \\, \\mathrm{d} v',\n\\end{equation*}\nwhere\n\\begin{equation*}\n\tJ({u_0},v) := \\int_{{u_0}}^{v} \\frac{\\lmb}{1-\\mu} \\frac{\\mu}{r} ({u_0}, v') \\, \\mathrm{d} v'.\n\\end{equation*}\n\nBy \\eqref{dic.1.0} and \\eqref{dic.2.1}, we have\n\\begin{equation*}\n\t\\sup_{{u_0} \\leq v' \\leq v}\\abs{\\nu^{-1} \\partial_{u} (r \\overline{\\phi}) ({u_0},v')} \n\t\\leq C_{\\Lambda} I_{1}({u_0},v)^{3} (1+I_{2}({u_0},v)) e^{C_{\\Lambda} I_{1}({u_0},v)^{2} (1+I_{2}({u_0},v))}.\n\\end{equation*}\n\nThen by \\eqref{dic.2.3}, we conclude that \n\\begin{equation} \\label{dic.2.4}\n\\int_{u_0}^v \\abs{\\partial_{v} \\overline{\\phi} \\, \\nu^{-1} \\partial_{u} ( r \\overline{\\phi}) ({u_0},v')} \\mathrm{d} v'\n\t\\leq C_{\\Lambda} I_{1}({u_0},v)^{4} (1+I_{2}({u_0},v))^{2} e^{C_{\\Lambda} I_{1}({u_0},v)^{2} (1+I_{2}({u_0},v))}.\n\\end{equation}\n\nCombining \\eqref{dic.2.1}, \\eqref{dic.2.2} and \\eqref{dic.2.4}, we conclude that \\eqref{main.ineq.dichotomy.2} holds. \\qedhere\n\\end{proof}\n\nNext, we apply \\eqref{main.ineq.dichotomy.1} and \\eqref{main.ineq.dichotomy.2} to show that \n\\begin{proposition} \\label{dic.main.prop}\nFor ${u_0}$ sufficiently large and $v\\geq {u_0}$, we have\n$$I_1({u_0},v)+I_2({u_0},v)\\leq C_{\\Lambda} \\epsilon.$$\n\\end{proposition}\n\\begin{remark}\nIf it is the case that $\\int_{u_0}^v(|\\partial_v^2(r\\overline{\\phi})| + |\\partial_v \\lambda|)({u_0},v')dv'$ is continuous in $v$ for each fixed ${u_0}$, then the desired conclusion follows from \\eqref{main.ineq.dichotomy.1} and \\eqref{main.ineq.dichotomy.2} via a simple continuity argument in $v$. In particular, the conclusion follows in the case where the initial data of $\\partial_v(r \\phi)$ are in $W^{1,1}$ or $C^{1}$. The only remaining difficulty is therefore to control the size of the delta function singularities in $\\partial_v^2(r\\overline{\\phi})$ in the general case where we only have a BV solution.\n\\end{remark}\n\\begin{proof}\nWe begin by studying the propagation of discontinuities for a BV solution to \\eqref{eq:SSESF}. In the general case where $\\partial_v(r \\phi)(1, \\cdot)$ is only in BV and contains jump discontinuities (at which $\\partial_{v}(r \\phi)(1, \\cdot)$ is assumed to be right-continuous), notice that the jump discontinuities for a BV function are discrete, i.e., they occur only at a (possibly infinite) sequence of points $V<v_1<v_2< v_3 < \\cdots$. On the other hand, note that by the initial condition $r = 2v$ on $C_{1}$, we have $\\lmb(1,v) = \\frac{1}{2}$; in particular, $\\lmb$ is continuous initially.\n\nThanks to the initial condition, it follows that $\\lmb$ does not possess any discontinuities outside $\\Gamma$. Indeed, from the definition of a BV solution, $m$ and $r$ are continuous. Then, by \\eqref{eq:SSESF:dr}, we see that that $\\nu$ is Lipschitz in the $v$ direction outside of $\\Gamma$, with bounded Lipschitz constant on each compact interval of $u$. Looking back at \\eqref{eq:SSESF:dr} and recalling that $\\lmb(1,v) = \\frac{1}{2}$, we then see that $\\lmb$ does not possess any discontinuities outside $\\Gamma$ as desired. Since $\\lmb$ is a priori in BV, it follows that $\\int_{u_0}^{v} \\abs{\\partial_{v} \\lmb({u_0},v')} \\, \\mathrm{d} v'$ is continuous in $v \\in ({u_0}, \\infty)$ with $\\int_{{u_0}}^{v} \\abs{\\partial_{v} \\lmb({u_0},v')} \\, \\mathrm{d} v' \\to 0$ as $v \\to {u_0}+$. \n\nBy the above regularity statements and equation \\eqref{eq:SSESF:dphi}, as well as the fact that $\\phi$ is continuous outside $\\Gamma$ by the definition of a BV solution, it now follows that the jump discontinuities of $\\partial_{v}(r \\phi)$ are propagated along constant $v_{i}$ curves. Therefore, for ${u_0} \\geq V$, we see that $\\partial_v (r \\phi)({u_0},v)$ is a right-continuous BV function on $({u_0}, \\infty)$ with jump discontinuities at ${u_0} < v_1 < v_2 < v_3 < \\cdots$ with the same sizes as $\\partial_{v} (r \\phi)(1,v)$. \nFrom \\eqref{dic.0.0}, notice that, using the abbreviations in \\eqref{eq:abbrev-overline-phi} and \\eqref{eq:abbrev-du-overline-phi},\n\\begin{equation*}\n\t\\int_{{u_0}}^{v} \\abs{\\partial_{v}^{2} (r \\overline{\\phi})({u_0},v')} \\, \\mathrm{d} v' = \\int_{{u_0}}^{v} \\abs{(\\partial_{v}^{2} (r \\phi) - c \\partial_{v} \\lmb)({u_0},v')} \\, \\mathrm{d} v',\n\\end{equation*}\nfor the constant $c = \\lim_{v \\to {u_0}+} \\phi({u_0},v)$, which is independent of $v$. Thanks to the continuity property of $\\lmb({u_0}, \\cdot)$, we see that the integral of $\\abs{\\partial_{v}^{2}(r \\overline{\\phi})({u_0}, \\cdot)}$ has the same jump discontinuities as $\\abs{\\partial_{v}^{2} (r \\phi)({u_0}, \\cdot)}$. In particular, by \\eqref{data.BV.bd}, each jump of $I_{1}({u_0},v)$ is at most of size $\\epsilon$.\n\nFix $C_{\\Lambda} > 1$ to be larger than the maximum of the constants from \\eqref{main.ineq.dichotomy.1} and \\eqref{main.ineq.dichotomy.2}. First, a standard continuity argument using \\eqref{main.ineq.dichotomy.1} and \\eqref{main.ineq.dichotomy.2} implies that if \n$$\\lim_{v\\to v_i+}\\int_{u_0}^v |\\partial_v^2(r\\overline{\\phi}) ({u_0},v')| \\mathrm{d} v'\\leq 5 C_{\\Lambda}\\epsilon\n\\hbox{ and }\n\\lim_{v\\to v_i+}\\int_{u_0}^v |\\partial_v \\lambda({u_0},v')| \\mathrm{d} v'\\leq 5\\epsilon,$$\n(with the convention $v_{0} := {u_0}$) then\n$$\\int_{u_0}^v |\\partial_v^2(r\\overline{\\phi}) ({u_0},v')| \\mathrm{d} v'\\leq 4 C_{\\Lambda}\\epsilon\n\\hbox{ and }\n\\int_{u_0}^v |\\partial_v \\lambda({u_0},v')| \\mathrm{d} v'\\leq 4 \\epsilon,$$\nfor $v_i< v< v_{i+1}$.\n\nAssume, for the sake of contradiction, that the conclusion of the proposition is not satisfied. Recall that the integral of $\\abs{\\partial_v\\lambda}$ is continuous. Thus, we have that for some $v_i$ with $i > 0$\n$$\\lim_{v\\to v_i-}\\int_{u_0}^v |\\partial_v^2(r\\overline{\\phi}) ({u_0},v')| \\mathrm{d} v'\\leq 4 C_{\\Lambda}\\epsilon,$$\nholds, but at the same time\n$$\\lim_{v\\to v_i+}\\int_{u_0}^v |\\partial_v^2(r\\overline{\\phi}) ({u_0},v')| \\mathrm{d} v' > 5 C_{\\Lambda}\\epsilon.$$\n\nHowever, we have seen that the size of the jump in $I_1({u_0},v)$ is bounded by $\\epsilon$, which is smaller than $C_{\\Lambda} \\epsilon$ if $C_{\\Lambda}>1$. This leads to a contradiction and thus the conclusion of the proposition holds.\n\\end{proof}\n\nWe are now ready to conclude the proof of Theorem \\ref{thm.dichotomy}.\n\\begin{proof} [Conclusion of Proof of Theorem \\ref{thm.dichotomy}]\nWe first establish \\eqref{dichotomy.goal}. In what follows, we use the abbreviations in \\eqref{eq:abbrev-overline-phi} and \\eqref{eq:abbrev-du-overline-phi}, such as $\\overline{\\phi} = \\overline{\\phi}_{u_{0}}$ etc. The idea is to transform back to $(\\overline{\\phi}, r, m) \\mapsto (\\phi, r, m)$ using \\eqref{dic.0.2}. Note that $\\abs{\\partial_{v} \\lmb}$ remains the same under this change, so it suffices to estimate $\\abs{\\partial_{v}^{2} (r \\phi)}$. By \\eqref{dic.2.0} and Proposition \\ref{dic.main.prop}, for sufficiently large ${u_0}$, the limit $\\overline{\\phi}({u_0}, \\infty) := \\lim_{v \\to \\infty} \\overline{\\phi}({u_0},v)$ exists and satisfies\n\\begin{equation*}\n\t\\abs{\\overline{\\phi}({u_0}, \\infty)} \\leq C_{\\Lambda} \\epsilon,\n\\end{equation*}\nwhere we note that $C_{\\Lambda}$ is independent of ${u_0}$.\n\nBy \\eqref{dic.0.2}, we have $\\phi(u,v) = \\overline{\\phi}(u,v) - \\overline{\\phi}(u, \\infty)$ for all $u$. Thus, using Proposition \\ref{dic.main.prop}, we estimate\n\\begin{align*}\n\t\\int_{{u_0}}^{\\infty} \\abs{\\partial_{v}^{2} ( r\\phi)({u_0},v)} \\, \\mathrm{d} v\n\t= & \\int_{{u_0}}^{\\infty} \\abs{\\partial_{v}^{2} ( r \\overline{\\phi}({u_0},v) - r \\overline{\\phi}({u_0}, \\infty))} \\, \\mathrm{d} v \\\\\n\t\\leq & \\int_{{u_0}}^{\\infty} \\abs{\\partial_{v}^{2} (r \\overline{\\phi})({u_0},v)} \\, \\mathrm{d} v \n\t\t+ \\abs{\\overline{\\phi}({u_0},\\infty)} \\int_{{u_0}}^{\\infty} \\abs{\\partial_{v} \\lmb ({u_0},v)} \\, \\mathrm{d} v \\\\\n\t\\leq & C_{\\Lambda} (\\epsilon + \\epsilon^{2}).\n\\end{align*}\nSince $u_0\\geq V$ is arbitrary, this proves \\eqref{dichotomy.goal}.\n\nFinally, we prove that the conditions $(2)$ and $(3)$ of Definition \\ref{def:locBVScat} hold.\nIndeed, since $\\partial_{v} \\log \\lmb = \\lmb^{-1} \\partial_{v} \\lmb$, $(3)$ in Definition \\ref{def:locBVScat} follows from \\eqref{dichotomy.goal} and \\eqref{dic.bnd4dvr}. In fact, $(3)$ in Definition \\ref{def:locBVScat} holds with arbitrarily large $R > 0$. Next, by \\eqref{eq:SSESF:dm}, non-negativity of $1-\\mu$ and $\\mu$ (by Lemma \\ref{lem:mntn4r}) and the fact that $m$ is invariant under $\\phi \\mapsto \\overline{\\phi}$,\n\\begin{equation*}\n\tm(u_0, v) \n\t\\leq \\frac{1}{2} \\sup_{u_0 \\leq v' \\leq v} \\abs{(\\lmb^{-1} \\partial_{v}(r \\overline{\\phi}) - \\overline{\\phi})(u_0, v')} \\int_{u_0}^{v} \\abs{\\partial_{v} \\overline{\\phi}(u_0, v')} \\, \\mathrm{d} v'\n\\end{equation*}\nwhere the right-hand side is $\\leq C_{\\epsilon, \\Lambda} \\epsilon$ (with $C_{\\epsilon, \\Lambda}$ non-decreasing in $\\epsilon$) by the estimates proved so far. Therefore, $(2)$ of Definition \\ref{def:locBVScat} follows.\nThis concludes the proof of Theorem \\ref{thm.dichotomy}. \\qedhere\n\\end{proof}\n\n\\section{Refinement in the small data case} \\label{sec:smallData}\nIn this section, we sketch a proof of Theorem \\ref{thm:smallData}. The idea is to revisit the proofs of the main theorems (Theorems \\ref{main.thm.1} and \\ref{main.thm.2}), and notice that all the required smallness can be obtained by taking initial total variation of $\\partial_{v}(r \\phi)$ small. Key to this idea is the following lemma.\n\n\\begin{lemma} \\label{lem:smallData}\nThere exist universal constants $\\epsilon_{0}$ and $C_{0}$ such that for $\\epsilon < \\epsilon_{0}$, the following holds. Suppose that $\\lmb(1, \\cdot) = \\frac{1}{2}$ and $\\partial_{v}(r \\phi)(1, \\cdot)$ is of bounded variation with\n\\begin{equation} \\label{eq:smallData:hyp}\n\t\\int_{C_{1}} \\abs{\\partial_{v}^{2} (r \\phi)} < \\epsilon.\n\\end{equation}\n\nSuppose furthermore that $\\lim_{v \\to \\infty} \\phi(1, v) = 0$. Then the maximal development  $(\\phi, r, m)$ satisfies condition $(1)$ of Definition \\ref{def:locBVScat} (future completeness of radial null geodesics) and obeys\n\\begin{gather}\n\t\\sup_{v \\in [1, \\infty)} \\int_{\\underline{C}_{v}} \\abs{\\frac{\\mu }{(1-\\mu)} \\frac{\\nu}{r}} \n\t+ \\sup_{u \\in [1, \\infty)} \\int_{C_{u}} \\abs{\\frac{\\mu }{(1-\\mu)} \\frac{\\lmb}{r}} \\leq C_{0} \\epsilon^{2},\t\\label{eq:smallData:smallPtnl} \\\\\n\t\\sup_{v \\in [1, \\infty)} \\int_{\\underline{C}_{v}} \\abs{\\partial_{u} \\phi}\n\t+ \\sup_{u \\in [1, \\infty)} \\int_{C_{u}} \\abs{\\partial_{v} \\phi} \\leq C_{0} \\epsilon,\t\t\t\t \\label{eq:smallData:smallDphi} \\\\\n\t\\sup_{v \\in [1, \\infty)} \\int_{\\underline{C}_{v}} (\\abs{\\partial_{u}^{2} (r \\phi)} + \\partial_{u} \\log \\nu)\n\t+ \\sup_{u \\in [1, \\infty)} \\int_{C_{u}} (\\abs{\\partial_{v}^{2} (r \\phi)} + \\partial_{v} \\log \\lmb) \\leq C_{0} \\epsilon.\t\t\t\t \\label{eq:smallData:smallTV} \n\\end{gather}\n\nMoreover, the bounds in Proposition \\ref{prop:geomLocBVScat} hold with\n\\begin{equation}  \\label{eq:smallData:geom}\n\tK + \\Lambda \\leq C_{0}, \\quad\n\t\\Psi \\leq C_{0} \\epsilon.\n\\end{equation}\n\\end{lemma}\n\n\\begin{proof} \nThis lemma is an easy consequence of Theorem \\ref{Chr.BV.Thm} and Lemma \\ref{lem:auxEqs} once we show \n\\begin{equation*}\n\\sup_{\\calQ} \\abs{\\partial_{v} (r \\phi)} \\leq C_{0} \\epsilon,\n\\end{equation*}\nusing the additional condition $\\lim_{v \\to \\infty} \\phi(1, v) = 0$. By Lemma \\ref{lem:est4dvphi}, note that $\\int_{C_{1}} \\abs{\\partial_{v} \\phi} \\leq C \\epsilon$; therefore, integrating from $v = \\infty$, we have $\\lim_{v \\to 1+} \\abs{\\phi(1, v)} \\leq C \\epsilon$. Then using \\eqref{eq:smallData:hyp} to integrate from $v = 1$, where we note that $\\lim_{v \\to 1+}\\phi(1, v) = \\lim_{v \\to 1+} \\partial_{v}(r \\phi)(1, v)$, we obtain\n\\begin{equation*}\n\t\\sup_{C_{1}} \\abs{\\partial_{v} (r \\phi)} \\leq C \\epsilon.\n\\end{equation*}\n\nUsing \\eqref{eq:SSESF:dphi'}, $\\partial_{u} \\lmb \\leq 0$, Lemma \\ref{lem:est4phi} (to control $\\abs{\\phi}$ from $\\abs{\\partial_{v}(r \\phi)}$) and $\\frac{1}{3} \\leq \\lmb \\leq \\frac{1}{2}$ (by Theorem \\ref{Chr.BV.Thm}), it follows that\n\\begin{align*}\n\t\\sup_{\\mathcal D(1, v)}\\abs{\\partial_{v}(r \\phi)} \n\t\\leq & \\sup_{1 \\leq v' \\leq v} \\abs{\\partial_{v} (r \\phi)(1, v')} + \\sup_{(u,v) \\in \\mathcal D(1,v)} \\sup_{1 \\leq u' \\leq u} \\abs{\\phi(u', v)} \\int_{1}^{u} (- \\partial_{u} \\lmb)(u', v) \\, \\mathrm{d} u' \\\\\n\t\\leq & C \\epsilon + \\frac{1}{2} \\sup_{\\mathcal D(1,v)} \\abs{\\partial_{v}(r \\phi)},\n\\end{align*} \nwhich proves $\\sup_{\\calQ} \\abs{\\partial_{v} (r \\phi)} \\leq C_{0} \\epsilon$, as desired. \\qedhere\n\\end{proof}\n\nEquipped with Lemma \\ref{lem:smallData}, we now proceed to outline the proof of Theorem \\ref{thm:smallData}.\n\\begin{proof} [Proof of $(1)$ in Theorem \\ref{thm:smallData}]\nThat $(\\phi, r, m)$ is globally BV scattering follows from Theorem \\ref{thm.dichotomy} and the fact that initial data with small total variation cannot lead to a development which blows up at infinity; the latter fact follows from the results proved by Christodoulou \\cite[Section 4, Theorem 6.2]{Christodoulou:1993bt}. \n\nIt remains to prove that \\eqref{eq:decay1:1}--\\eqref{eq:decay1:3} hold with $A_{1} \\leq C_{\\mathcal I_{1}} (\\mathcal I_{1} + \\epsilon)$ if $\\epsilon > 0$ is sufficiently small. By \\eqref{eq:decay1:H1}, it follows Lemma \\ref{lem:decay1:cptu:0} holds with $H_{1} \\leq C_{\\mathcal I_{1}} (\\mathcal I_{1} + \\epsilon)$, and \\eqref{eq:decay1:extr} in Lemma \\ref{lem:decay1:extr} becomes\ufffd\n\\begin{equation*} \\tag{\\ref{eq:decay1:extr}$'$}\n\t\\sup_{C_{u}} r^{\\omega} \\abs{\\partial_{v} (r \\phi)} \\leq C_{\\mathcal I_{1}} u_{1} (\\mathcal I_{1} + \\epsilon) + C M_{i} u_{1}^{-1} \\mathcal B_{1}(U),\n\\end{equation*}\n\nNote that $M_{i} \\leq C \\mathcal I_{1}^{2}$. Then repeating the arguments in \\S \\ref{sec.full.decay.1}, we see that \\eqref{eq:decay1:intr:pf:key} becomes\n\\begin{equation*} \\tag{\\ref{eq:decay1:intr:pf:key}$'$}\n\t\\mathcal B_{1}(U) \\leq C_{\\mathcal I_{1}} u_{1} (\\mathcal I_{1} + \\epsilon) + C (\\mathcal I_{1}^{2} u_{1}^{-1} + \\epsilon^{2}) \\mathcal B_{1}(U). \n\\end{equation*}\n\nIt is important to note that the constant $C$ in the last term does not depend on $\\mathcal I_{1}$. Take $u_{1} = 1000 C (1+\\mathcal I_{15})^{2}$. Then for $\\epsilon > 0$ sufficiently small (independent of $\\mathcal I_{1}$), we derive\n\\begin{equation*}\n\t\\mathcal B_{1}(U) \\leq C_{\\mathcal I_{1}} (\\mathcal I_{1} + \\epsilon)\n\\end{equation*}\n\nIt then follows that \\eqref{eq:decay1:1} and \\eqref{eq:decay1:2} hold with $A_{1} \\leq C_{\\mathcal I_{1}} (\\mathcal I_{1} + \\epsilon)$. Applying Lemma \\ref{lem:decay1:uDecay4durphi}, we conclude that \\eqref{eq:decay1:3} holds with $A_{1} \\leq C_{\\mathcal I_{1}} (\\mathcal I_{1} + \\epsilon)$ as well. \\qedhere\n\\end{proof}\n\n\\begin{proof} [Proof of $(2)$ in Theorem \\ref{thm:smallData}]\nWe need to prove that \\eqref{eq:decay2:1}--\\eqref{eq:decay2:4} hold with $A_{2} \\leq C_{\\mathcal I_{2}} (\\mathcal I_{2} + \\epsilon)$. The key is to show that Proposition \\ref{prop:decay2:nullStr} holds with\n\\begin{equation} \\label{eq:smallData:pf2:key}\n\tA_{2}' \\leq C_{\\mathcal I_{2}} (\\mathcal I_{2} + \\epsilon)\n\\end{equation}\n\nIndeed, by the explicit bounds on the constants (in particular, \\eqref{eq:decay2:rDecay:1}, \\eqref{eq:decay2:rDecay:2}, \\eqref{eq:decay2:uDecayInExtr:1}, \\eqref{eq:decay2:uDecayInExtr:2}, \\eqref{eq:decay2:final:aux} and \\eqref{eq:decay2:A2''}), the desired conclusion easily follows once \\eqref{eq:smallData:pf2:key} is established.\n\nNote that $\\mathcal I_{1} \\leq \\mathcal I_{2}$ by definition, and thus $A_{1} \\leq C_{\\mathcal I_{2}} (\\mathcal I_{2} + \\epsilon)$ by the preceding proof. We furthermore claim that the following statements hold:\n\\begin{itemize}\n\\item Lemma \\ref{lem:decay2:key4nullStr} holds with\n\\begin{equation} \\label{eq:smallData:pf2:1}\n\t \\epsilon(u_{2}) \\leq C \\epsilon,\n\\end{equation}\nfor every $u_{2} \\geq 1$.\n\n\\item We have\n\\begin{equation} \\label{eq:smallData:pf2:2}\n\t H_{2}'(1) \\leq C_{\\mathcal I_{2}} (\\mathcal I_{2} + \\epsilon),\n\\end{equation}\nwhere we remind the reader that\n\\begin{equation*}\nH_{2}'(1) = \\sup_{\\set{(u,v) : u \\in [1, 3] \\, v \\in [u, 3u]}} u^{\\omega} \\Big( \\abs{\\partial_{v}^{2}(r \\phi)} + \\abs{\\partial_{u}^{2}(r \\phi)} + \\abs{\\partial_{v} \\lmb} + \\abs{\\partial_{u} \\nu}\\Big)\n\\end{equation*}\naccording to \\eqref{eq:decay2:def4H'2}.\n\\end{itemize}\n\nThe first claim follows easily from Lemma \\ref{lem:smallData} and \\eqref{eq:decay2:eps}. For the second claim, since $1 \\leq u \\leq 3$, it suffices to prove\\footnote{Recall that $\\mathcal D(1,9) = \\set{(u,v) : u \\in [1, 3] \\, v \\in [u, 3u]}$ is the domain of dependence of $C_{1} \\cap \\set{1 \\leq v \\leq 9}$.}\n\\begin{equation*}\n\t \\sup_{\\mathcal D(1,9)} \\Big( \\abs{\\partial_{v}^{2}(r \\phi)} + \\abs{\\partial_{u}^{2}(r \\phi)} + \\abs{\\partial_{v} \\lmb} + \\abs{\\partial_{u} \\nu}\\Big)\\leq C (\\mathcal I_{2} + \\epsilon),\n\\end{equation*}\nwhich follows from a persistence of regularity argument, similar to our proof of Lemma \\ref{lem:decay2:key4nullStr}. \n\nTo conclude the proof, recall that we had\n\\begin{equation*}\n\t\\mathcal B_{2}(U) \\leq H_{2}''(u_2) + \\epsilon''(u_{2}) \\mathcal B_{2}(U)\n\\end{equation*}\nwhere $\\mathcal B_{2}(U)$ was defined in \\eqref{eq:decay2:def4B2}, and $H_{2}''(u_2)$, $\\epsilon''(u_{2})$ obey the bounds in \\eqref{eq:decay2:H2''} and \\eqref{eq:decay2:eps''} respectively. Thanks to \\eqref{eq:smallData:pf2:1}, it follows that we may take $u_{2} = 1$ and $\\epsilon''(1) \\leq C\\epsilon$, where $C$ does not depend on $\\mathcal I_{2}$. Next, since $u_{2} = 1$, we see that $H_{2}''(1) \\leq C_{\\mathcal I_{2}} (\\mathcal I_{2} + \\epsilon)$ by \\eqref{eq:smallData:pf2:2}. Therefore, for $\\epsilon > 0$ sufficiently small (independent of $\\mathcal I_{2}$), we conclude that\n\\begin{equation*}\n\t\\mathcal B_{2}(U) \\leq C_{\\mathcal I_{2}} (\\mathcal I_{2} + \\epsilon),\n\\end{equation*}\nwhich proves that Proposition \\ref{prop:decay2:nullStr} holds with \\eqref{eq:smallData:pf2:key}, as desired.  \\qedhere\n\\end{proof}\n\n\n\\section{Optimality of the decay rates} \\label{sec.opt}\nIn this section, we show the optimality of the decay rates obtained above, i.e., we prove Theorems \\ref{thm.opt.1} and \\ref{thm.opt.2}. \n\n\\subsection{Optimality of the decay rates, in the case $1 < \\omega' < 3$}  \\label{subsec.opt.1}\nIn this subsection, we prove Theorem \\ref{thm.opt.1}. More precisely, we will demonstrate that the proof of the upper bounds for $\\phi$ and its derivatives can in fact be sharpened to give also lower bounds for $\\partial_v(r\\phi)$ and $\\partial_u(r\\phi)$ if the initial data satisfy appropriate lower bounds for $\\omega<3$.\n\n\\begin{proof}[Proof of Theorem \\ref{thm.opt.1}]\n\nWe first prove the lower bound for $\\partial_v(r\\phi)$. We split the spacetime into the exterior region $\\PD_{\\mathrm{ext}}$ and interior region $\\PD_{\\mathrm{int}}$ as before. Notice that in the exterior region, $u\\lesssim r$ and it suffices to prove a lower bound for $r^\\omega\\partial_v(r\\phi)$. Similarly, in the interior region, $r\\lesssim u$ and it suffices to prove a lower bound for $u^\\omega\\partial_v(r\\phi)$.\n\nRevisiting the proof of Lemma \\ref{lem:decay1:extr}, we note that instead of controlling $\\partial_v(r\\phi)$ by the initial data and error terms, we can bound the difference between $\\partial_v(r\\phi)(u,v)$ and the corresponding initial value of $\\partial_v(r\\phi)(1,v)$. More precisely, from the proof of Lemma \\ref{lem:decay1:extr}, we have\n\n\\begin{align*}\n\t\\abs{\\partial_{v} (r \\phi)(u,v)-\\partial_{v} (r \\phi)(1, v)}\n\t\\leq  \\frac{u_{1} K M_{i}}{r^2(u,v)(1+r(u,v))} H_{1} + \\frac{K M(u_{1})}{u_{1} r^{\\omega}(u,v)} \\mathcal B_{1}(U)\n\\end{align*}\nin the case $2<\\omega<3$ and\n\\begin{align*}\n\t\\abs{\\partial_{v} (r \\phi)(u,v)-\\partial_{v} (r \\phi)(1, v)}\n\t\t\\leq \\frac{\\omega K M_{i}}{r(u,v)(1+r(u,v))} H_{1} + \\frac{\\omega K M(u_{1})}{u_{1} r^{\\omega}(u,v)} \\mathcal B_{1}(U).\n\\end{align*}\nin the case $1<\\omega\\leq 2$. By the decay results proved in \\S \\ref{sec.full.decay.1}, we have\n$$\\sup_u(H_{1}+ \\mathcal B_{1}(u))\\leq A$$\nfor some constant $A$. Therefore, by choosing $u_1$ sufficiently large, we have in the region $3u\\leq v$,\n$$r^{\\omega}\\abs{\\partial_{v} (r \\phi)(u,v)-\\partial_{v} (r \\phi)(1, v)}\\leq \\frac{L}{4},$$\nas long as $u\\geq u_1$.\nWe now apply the assumption on the lower bound for the initial data $r^{\\omega} \\partial_{v} (r \\phi)(1, v)\\geq L$ for $v\\geq V$. Choosing $u$ larger if necessary, we can assume that $u\\geq V$. Then, we derive that in $3u\\leq v$,\n$$r^{\\omega}\\partial_{v} (r \\phi)(u,v)\\geq \\frac{L}{2}.$$\n\nWe now move to the interior region where $3u \\geq v$. To this end, we improve the bounds in \\eqref{eq:decay1:intr:pf:1}. First, notice that the lower bound in the exterior region implies that there exists $L'$ such that \n\\begin{eqnarray}\nu^{\\omega}\\partial_{v} (r \\phi)(u,v)\\geq L'\\label{lower.bd.L1}\n\\end{eqnarray}\nfor $3u \\leq v$. Then, integrating \\eqref{eq:SSESF:dphi} along the incoming direction from $(u\/3, v)$ to $(u, v)$, we get\n\\begin{equation*}\n\\begin{aligned}\n\t\\abs{\\partial_{v} (r \\phi) (u,v)-\\partial_{v} (r \\phi) (u\/3,v)} \n\t\\leq  \\frac{1}{2} (\\sup_{u' \\in [u\/3, u]} \\sup_{C_{u'}} \\abs{\\phi}) \\int_{u\/3}^{u} \\abs{\\frac{2m \\nu}{(1-\\mu) r^{2}} (u', v)} \\, \\mathrm{d} u'.\n\\end{aligned}\n\\end{equation*} \nBy Theorem \\ref{main.thm.1}, we have \n$$\\sup_{C_u}|\\phi|\\leq A_{1} u^{-\\omega}$$\nfor some $A_{1} > 0$. Lemma \\ref{lem:smallPtnl} implies that \n$$\\int_{u\/3}^{u} \\abs{\\frac{2m \\nu}{(1-\\mu) r^{2}} (u', v)} \\, \\mathrm{d} u'\\to 0$$\nas $u\\to \\infty$. Thus the right hand side can be bounded by $\\frac{L' u^{\\omega}}{2}$ after choosing $u$ to be sufficiently large.\nCombining this with the lower bound \\eqref{lower.bd.L1}, we have\n$$u^{\\omega}\\partial_{v} (r \\phi) (u,v) \\geq \\frac{L'}{2}$$\nfor $3u \\leq v$ and $u$ sufficiently large.\n\nWe now proceed to obtain the lower bound for $\\partial_u(r\\phi)$ by revisiting the proof of Lemma \\ref{lem:decay1:uDecay4durphi}. Integrating \\eqref{eq:SSESF:dphi} along the outgoing direction from $(u, u)$ to $(u, v)$, we have\n\\begin{align}\\label{est.durphi.diff}\n\t\\abs{\\partial_{u} (r \\phi)(u,v) - \\lim_{v' \\to u+} \\partial_{u} (r \\phi)(u, v')}\n\t\\leq & \\int_{C_{u}} \\abs{\\frac{\\mu\\lmb\\nu}{(1-\\mu)r}\\phi}.\n\\end{align}\nAs before, we Theorem \\ref{main.thm.1}, i.e.,\n$$\\sup_{C_u}|\\phi|\\leq A_{1} u^{-\\omega}$$\nfor some $A_{1} > 0$. By Lemma \\ref{lem:smallPtnl} and the upper bound \\eqref{eq:bnd4dur} for $|\\nu|$, we have\n$$\\int_{C_{u}} \\abs{\\frac{\\mu\\lmb\\nu}{(1-\\mu)r}} \\to 0, \\quad\\mbox{as }u\\to\\infty.$$\nTherefore, we can choose $u$ sufficiently large such that\n$$u^{\\omega} \\int_{C_{u}} \\abs{\\frac{\\mu\\lmb\\nu}{(1-\\mu)r}\\phi}\\leq \\frac{L'}{4}.$$\nReturning to \\eqref{est.durphi.diff} and recalling that for $u$ large,\n$$-\\lim_{v' \\to u+} \\partial_{u} (r \\phi)(u, v')=\\lim_{v' \\to u+} \\partial_{v} (r \\phi)(u, v')\\geq \\frac{L'}{2}u^{-\\omega},$$\nwe have\n$$-\\partial_{u} (r \\phi)(u,v) \\geq \\frac{L'}{4}u^{-\\omega}$$\nfor $u$ sufficiently large, as desired. \\qedhere\n\n\\end{proof}\n\n\\subsection{Key lower bound lemma}\nThe goal of the remainder of this section is to prove Theorem \\ref{thm.opt.2}. In this subsection we establish the following result, which provides a sufficient condition for the desired lower bounds on the decay of $\\phi$ in terms of a number (called $\\mathfrak{L}$) computed on $\\mathcal I^{+}$. This will be an important ingredient for our proof of Theorem \\ref{thm.opt.2} in the next subsection.\n\n\\begin{lemma}[Key lower bound lemma] \\label{lem:LB}\nLet $(\\phi, r, m)$ be a $C^{1}$ solution to \\eqref{eq:SSESF} which is locally BV scattering and asymptotically flat initial data of order $\\omega = 3$ in $C^{1}$. \nSuppose furthermore that \n\\begin{equation*}\n\t\\mathfrak{L} := \\lim_{v \\to \\infty} r^{3} \\partial_{v} (r \\phi)(1, v) + \\int_{1}^{\\infty} (M \\nu_{\\infty} \\Phi)(u)  \\, \\mathrm{d} u \\neq 0,\n\\end{equation*}\nwhere $M(u) := \\lim_{v \\to \\infty} m(u, v)$, $\\nu_{\\infty}(u) := \\lim_{v \\to \\infty} \\nu(u,v)$ and $\\Phi(u) := \\lim_{v \\to \\infty} r \\phi(u,v)$. \nThen there exist constants $U, L_{3} > 0$ such that the following lower bounds for the decay of $\\partial_{v}(r \\phi)$, $\\partial_{u} (r \\phi)$ hold on $\\set{(u, v)  : u \\geq U}$.\n\\begin{align} \n\t\\abs{\\partial_{v}(r \\phi)(u, v)} \\geq & L_{3} \\min \\set{r(u,v)^{-3}, u^{-3}}, \\label{eq:LB:1} \\\\\n\t\\abs{\\partial_{u}(r \\phi)(u, v)} \\geq & L_{3} u^{-3}. \\label{eq:LB:2}\n\\end{align}\n\\end{lemma}\n\n\\begin{remark} \nNote that $\\partial_{v}(r \\phi)$ and $\\partial_{u}(r \\phi)$ have definite signs by \\eqref{eq:LB:1}, \\eqref{eq:LB:2}. In fact, the proof below shows that the signs of $\\partial_{v} (r \\phi)$ and $-\\partial_{u} (r \\phi)$ agree with that of $\\mathfrak{L}$.\n\\end{remark}\n\n\n\\begin{proof} \nWithout loss of generality, assume that $\\mathfrak{L} > 0$. For $0 < \\eta \\leq 1$, define the \\emph{$\\eta$-exterior region} by\n\\begin{equation*}\n\t\\PD_{\\mathrm{ext}}^{\\eta} := \\set{(u,v) \\in \\calQ : u \\leq \\eta v}.\n\\end{equation*}\n\n\\pfstep{Step 1} In the first step, we make precise the relation between $r$ and $v$ in $\\PD_{\\mathrm{ext}}^{\\eta}$ for small $\\eta$. We claim that $r \\sim v\/2$ in this region; more precisely,\n\\begin{equation} \\label{eq:LB:pf:1}\n\\abs{\\frac{r(u,v)}{v} - \\frac{1}{2}} \\leq \\eta C_{A_{1}, A_{2}, K, \\Lambda}.\n\\end{equation}\n\nIntegrating by parts, we have\n\\begin{align*}\nr(u,v) \n= \\int_{u}^{v} \\lmb(u, v') \\, \\mathrm{d} v' \n= - \\int_{u}^{v} \\partial_{v} \\lmb(u, v') v' \\, \\mathrm{d} v' + v \\lmb(u, v) - u \\lmb(u, u).\n\\end{align*}\n\nTo make the leading term $v \\lmb(u,v)$ and small number $\\frac{u}{v}$ explicit, we rewrite the last expression as follows:\n\\begin{equation*}\n\tr(u,v) = v \\Big[ \\lmb(u,v)  - \\frac{u}{v} \\Big( \\lmb(u,u) + \\int_{u}^{v} \\partial_{v} \\lmb(u, v') \\frac{v'}{u} \\, \\mathrm{d} v' \\Big)  \\Big].\n\\end{equation*}\n\nRecall that $\\lmb$ is uniformly bounded from the above and below on $\\calQ$, i.e., $\\Lambda^{-1} \\leq \\lmb \\leq 1\/2$. Moreover, by the decay estimates for $\\partial_{v} \\lmb$ proved in Theorem \\ref{main.thm.2}, we have\n\\begin{equation*}\n\\sup_{(u,v) \\in \\calQ} \\int_{u}^{v} \\abs{\\partial_{v} \\lmb(u, v')} \\frac{v'}{u} \\, \\mathrm{d} v' \\leq C_{A_{2}}.\n\\end{equation*}\n\nAs a consequence,\n\\begin{equation*}\n\\abs{\\frac{r(u,v)}{v} - \\lmb(u,v)} \\leq \\eta C_{A_{2}, \\Lambda}.\n\\end{equation*}\n\nThus \\eqref{eq:LB:pf:1} will follow once we establish\n\\begin{equation} \\label{eq:LB:pf:1:1}\n\t\\abs{\\lmb(u,v) - \\frac{1}{2}} \\leq \\eta^{2} C_{A_{1}, A_{2}, K, \\Lambda}.\n\\end{equation}\n\nThis inequality is proved by integrating the decay estimate \\eqref{eq:decay2:10} for $\\partial_{u} \\lmb = \\partial_{u} \\partial_{v} r$ along the incoming direction, starting from the normalization $\\lmb(1, v) = 1\/2$. \nHere, we use the easy geometric fact that if $(u,v)$ lies in $\\PD_{\\mathrm{ext}}^{\\eta}$, then so does the incoming null curve from $(1,v)$ to $(u,v)$. \n\n\\pfstep{Step 2} We claim that for $U_{1} \\geq 1$ sufficiently large and $0 < \\eta \\leq 1$ suitably small, we have\n\\begin{equation} \\label{eq:LB:pf:2}\n\t\\partial_{v} (r \\phi)(u,v) \\geq \\frac{\\mathfrak{L}}{2} \\Big( \\frac{v}{2} \\Big)^{-3}\n\\end{equation}\nfor $(u, v) \\in \\PD_{\\mathrm{ext}}^{\\eta} \\cap \\set{u \\geq U_{1}}$.\n\nWe begin with\n\\begin{equation} \\label{eq:LB:pf:2:1}\n\t\\Big( \\frac{v}{2} \\Big)^{3} \\partial_{v}(r \\phi)(u,v) = \\Big( \\frac{v}{2} \\Big)^{3} \\partial_{v}(r \\phi)(1, v) + \\Big( \\frac{v}{2} \\Big)^{3} \\int_{1}^{u} \\frac{2m \\lmb \\nu}{(1-\\mu) r^{3}} r \\phi(u', v) \\, \\mathrm{d} u',\n\\end{equation}\nobtained by integrating the $\\partial_{u} \\partial_{v} (r \\phi)$ equation and multiplying by $(v\/2)^{3}$. To prove \\eqref{eq:LB:pf:2}, it suffices to show that the right-hand side of \\eqref{eq:LB:pf:2:1} is bounded from below by $\\mathfrak{L}\/2$ for $(u, v) \\in \\PD_{\\mathrm{ext}}^{\\eta} \\cap \\set{u \\geq U_{1}}$ with sufficiently large $U_{1} \\geq 1$ and small $0 < \\eta \\leq 1$.\n\nNote that $r = \\frac{v-1}{2}$ on $C_{1}$, and $v \\geq \\eta^{-1}$ if $(u,v) \\in \\PD_{\\mathrm{ext}}^{\\eta}$. Thus for $(u,v) \\in \\PD_{\\mathrm{ext}}^{\\eta}$ and $0 < \\eta \\leq 1$ sufficiently small, we have\n\\begin{equation*}\n\\abs{\\Big( \\frac{v}{2} \\Big)^{3} \\partial_{v} (r \\phi)(1, v) - \\lim_{v \\to \\infty} r^{3} \\partial_{v} (r \\phi)(1, v)} < \\frac{\\mathfrak{L}}{8}.\n\\end{equation*}\n\nIn order to proceed, it is useful to keep in mind the following technical point: For $U_{1} \\geq 1$, by the decay estimates \\eqref{eq:decay1:1} and \\eqref{eq:decay1:6}, we have\n\\begin{equation} \\label{eq:LB:pf:2:2}\n\t\\sup_{v \\geq U_{1}} \\int_{U_{1}}^{v} \\abs{\\frac{2m \\lmb \\nu}{1-\\mu} r \\phi (u', v)} \\, \\mathrm{d} u' \\leq U_{1}^{-6} C_{A_{1}, \\Lambda}.\n\\end{equation}\n\nIn what follows, let $(u, v) \\in \\PD_{\\mathrm{ext}}^{\\eta} \\cap \\set{u \\geq U_{1}}$. Using \\eqref{eq:LB:pf:1}, \\eqref{eq:LB:pf:2:2} and the fact that the null segment from $(1, v)$ to $(u, v)$ lies in $\\PD_{\\mathrm{ext}}^{\\eta}$, we get\n\\begin{equation*}\n\\Big\\vert \\Big( \\frac{v}{2} \\Big)^{3} \\int_{1}^{u} \\frac{2m \\lmb \\nu}{(1-\\mu) r^{3}} r \\phi(u', v) \\, \\mathrm{d} u' - \\int_{1}^{u} \\frac{2m \\lmb \\nu}{1-\\mu} r \\phi(u', v) \\, \\mathrm{d} u'\\Big\\vert\n\\leq \\eta C_{A_{1}, A_{2}, K, \\Lambda}.\n\\end{equation*}\n\nTaking $U_{1} \\geq 1$ large enough and using \\eqref{eq:LB:pf:2:2}, we may arrange \n\\begin{equation*}\n\t\\sup_{v \\geq U_{1}} \\int_{U_{1}}^{v} \\abs{\\frac{2m \\lmb \\nu}{1-\\mu} r \\phi (u', v)} \\, \\mathrm{d} u' + \\int_{U_{1}}^{\\infty} \\abs{M \\nu_{\\infty} \\Phi (u')} \\, \\mathrm{d} u' < \\frac{\\mathfrak{L}}{8}.\n\\end{equation*} \n\nOn the other hand, note that $2m \\lmb \\nu (1-\\mu)^{-1} r \\phi (u, v) \\to M \\nu_{\\infty} \\Phi(u)$ for each $u \\geq 1$ as $v \\to \\infty$. Therefore, by the dominated convergence theorem, for $0 < \\eta \\leq 1$ sufficiently small (so that $v$ is large), we have\n\\begin{align*}\n\t\\abs{\\int_{1}^{U_{1}} \\frac{2m \\lmb \\nu}{1-\\mu} r \\phi (u', v) \\, \\mathrm{d} u' - \\int_{1}^{U_{1}} M \\nu_{\\infty} \\Phi(u')  \\, \\mathrm{d} u'} < \\frac{\\mathfrak{L}}{8}.\n\\end{align*}\n\nPutting these together and taking $0 < \\eta \\leq 1$ sufficiently small, we conclude \\eqref{eq:LB:pf:2}.\n\n\n\\pfstep{Step 3} Next, we claim that there exists $U_{2} = U_{2}(U_{1}, A_{2}, \\Lambda, K, \\eta) \\geq 1$ such that $U_{2} \\geq U_{1}$ and for $(u,v) \\in (\\calQ \\setminus \\PD_{\\mathrm{ext}}^{\\eta}) \\cap \\set{u \\geq U_{2}}$, we have\n\\begin{align} \n\t\\partial_{v}(r \\phi)(u, v) \\geq 2 \\eta^{3} \\mathfrak{L} \\, u^{-3}.\t\t\\label{eq:LB:pf:3}\n\\end{align}\n\nCombined with \\eqref{eq:LB:pf:2} (keeping in mind that $r \\sim v\/2$ in $\\PD_{\\mathrm{ext}}^{\\eta}$ by \\eqref{eq:LB:pf:1}), this would establish \\eqref{eq:LB:1}.\n\nTake $U_{2} \\geq \\eta^{-1} U_{1}$, and consider $(u, v) \\in (\\calQ \\setminus \\PD_{\\mathrm{ext}}^{\\eta}) \\cap \\set{u \\geq U_{2}}$. Integrating \\eqref{eq:SSESF:dphi}, we have\n\\begin{equation*}\n\t\\partial_{v} (r \\phi)(u,v) = \\partial_{v} (r \\phi)(\\eta u, v) + \\int_{\\eta u}^{u} \\frac{2m \\lmb \\nu}{r^{2}} \\phi(u', v) \\, \\mathrm{d} u'.\n\\end{equation*}\n \nNote that $(\\eta u, v) \\in \\PD_{\\mathrm{ext}}^{\\eta} \\cap \\set{u \\geq U_{1}}$ since $v \\geq u$ and $\\eta u \\geq \\eta U_{2} \\geq  U_{1}$. Therefore, by \\eqref{eq:LB:pf:2} and the fact that $\\eta^{-1} u > v$ (as $(u,v) \\in \\calQ \\setminus \\PD_{\\mathrm{ext}}^{\\eta}$), the first term on the right-hand side obeys the lower bound\n\\begin{equation*}\n\t\\partial_{v} (r \\phi)(\\eta u, v) \\geq \\Big( \\frac{\\mathfrak{L}}{2} \\Big) \\Big( \\frac{v}{2} \\Big)^{-3} > 4 \\eta^{3} \\mathfrak{L} \\, u^{-3}.\n\\end{equation*}\n\nOn the other hand, using \\eqref{eq:decay1:1} and \\eqref{eq:decay2:11}, we have\n\\begin{equation*}\n\t\\abs{\\int_{\\eta u}^{u} \\frac{2m \\lmb \\nu}{r^{2}} \\phi(u', v) \\, \\mathrm{d} u'}\n\t\\leq C_{A_{1}, A_{2}, \\Lambda, K} \\int_{\\eta u}^{u} \\frac{1}{(u')^{10}} \\, \\mathrm{d} u' \n\t\\leq C_{A_{1}, A_{2}, \\Lambda, K, \\eta} \\, u^{-9}.\n\\end{equation*}\n\nTaking $U_{2}$ large enough, we conclude that \\eqref{eq:LB:pf:3} holds.\n\n\\pfstep{Step 4} Finally, we claim that there exists $U = U(U_{2}, A_{2}, \\Lambda, K, \\eta) \\geq 1$ such that $U \\geq U_{2} \\geq U_{1}$ and for $(u,v) \\in \\set{u \\geq U}$, we have\n\\begin{align} \n\t- \\partial_{u}(r \\phi)(u, v) \\geq \\eta^{3} \\mathfrak{L} \\, u^{-3}.\t\t\\label{eq:LB:pf:4}\n\\end{align}\n\nThis would prove \\eqref{eq:LB:2}, thereby completing the proof of Lemma \\ref{lem:LB}.\n\nOur argument will be very similar to the previous step. Take $U \\geq U_{2}$ and consider $(u,v) \\in \\set{u \\geq U}$. Integrating \\eqref{eq:SSESF:dphi} along the outgoing direction, we have\n\\begin{equation*}\n\t- \\partial_{u}(r \\phi)(u,v) = - \\partial_{u}(r \\phi)(u, u) - \\int_{u}^{v} \\frac{2 m \\lmb \\nu}{r^{2}} \\phi(u, v') \\, \\mathrm{d} v'.\n\\end{equation*}\n\nRecall that $\\lim_{v \\to u+} \\partial_{u} (r \\phi)(u, v) = - \\lim_{v \\to u+} \\partial_{v}(r \\phi)(u,v)$. By \\eqref{eq:LB:pf:3} and the fact that $u \\geq U \\geq U_{2}$, we see that the first term on the right-hand side obeys the lower bound\n\\begin{equation*}\n\t- \\partial_{u} (r \\phi)(u,u) \\geq 2 \\eta^{3} \\mathfrak{L} \\, u^{-3}.\n\\end{equation*}\n\nOn the other hand, using \\eqref{eq:decay1:1} and \\eqref{eq:decay2:11}, we have\n\\begin{equation*}\n\t\\abs{\\int_{u}^{v} \\frac{2 m \\lmb \\nu}{r^{2}} \\phi(u, v') \\, \\mathrm{d} v'}\n\t\\leq C_{A_{1}, A_{2}, K} \\int_{u}^{v} \\min \\set{u^{-10}, r^{-2} u^{-8}} \\, \\lmb \\, \\mathrm{d} v' \n\t\\leq C_{A_{1}, A_{2}, K} \\, u^{-9}.\n\\end{equation*}\n\nTaking $U$ sufficiently large, we conclude that \\eqref{eq:LB:pf:4} holds. \\qedhere\n\\end{proof}\n\n\\subsection{Optimality of the decay rates, in the case $\\omega' \\geq 3$}  \\label{subsec.opt.2}\nIn this subsection, we prove Theorem \\ref{thm.opt.2} by studying the solution to \\eqref{eq:SSESF} arising from the initial value\n\\begin{equation*}\n\t\\partial_{v}(r \\phi)(1, v) = \\epsilon \\widetilde{\\chi}\\Big( \\frac{v - v_{0}}{N} \\Big),\n\\end{equation*}\nwhere $\\widetilde{\\chi} : (-\\infty, \\infty) \\to [0, \\infty)$ is a smooth function such that\n\\begin{equation*}\n\\mathrm{supp} \\, \\widetilde{\\chi} \\subset (-1\/2, 1\/2), \\quad\n\\int_{\\mathbb R} \\widetilde{\\chi} = 1.\n\\end{equation*}\n\nWe also require that $v_{0} \\geq 2$ and $N \\leq v_{0}$. With such data, the initial total variation is of size $\\leq C\\epsilon$, i.e.,\n\\begin{equation*}\n\t\\int_{1}^{\\infty} \\abs{\\partial_{v}^{2} (r \\phi) (1, v)} \\, \\mathrm{d} v \\leq \\epsilon \\int_{-\\infty}^{\\infty} \\abs{ \\widetilde{\\chi}\\,' \\Big( \\frac{v - v_{0}}{N} \\Big) } \\, \\frac{\\mathrm{d} v}{N} \\leq C \\epsilon.\n\\end{equation*}\n\nWe also see that $\\mathcal I_{1} \\leq C \\epsilon v_{0}^{3}$ and $\\mathcal I_{2} \\leq C \\epsilon v_{0}^{4} \/ N$ with $\\omega' = 3$, as\n\\begin{equation*}\n\t\\sup_{v \\in [1, \\infty)} (1+r)^{3} \\abs{\\partial_{v} (r \\phi)}(1,v) \\leq C \\epsilon v_{0}^{3}, \\quad\n\t\\sup_{v \\in [1, \\infty)} (1+r)^{4} \\abs{\\partial_{v}^{2} (r \\phi)}(1,v) \\leq C \\epsilon \\frac{v_{0}^{4}}{N}.\n\\end{equation*}\n\nWe are now ready to give a proof of Theorem \\ref{thm.opt.2}. The idea is to compute $\\mathfrak{L}$ to the leading order (which turns out to be $- c\\epsilon^{3}$ for some $c > 0$), and then control the lower order terms by taking $\\epsilon > 0$ sufficiently small and applying Theorem \\ref{thm:smallData}.\n\n\\begin{proof} [Proof of Theorem \\ref{thm.opt.2}]\nFor this proof, we fix $v_{0} = 4$ and $N = 1$. We use the shorthand\n\\begin{equation*}\n\t\\chi(v) := \\widetilde{\\chi}(v-4).\n\\end{equation*}\n\nBy the preceding discussion on the size of initial data, we see that Theorem \\ref{thm:smallData} applies when $\\epsilon > 0$ is sufficiently small. Therefore, there exists a constant $C > 0$ independent of $\\epsilon > 0$ such that Theorems \\ref{main.thm.1}, \\ref{main.thm.2} and Proposition \\ref{prop:geomLocBVScat} hold with\n\\begin{equation} \\label{eq:opt2:eps}\n\tA_{1}, A_{2} \\leq C \\epsilon, \\quad K, \\Lambda \\leq C.\n\\end{equation}\n\nWe begin by showing\n\\begin{equation} \\label{eq:opt2:dvrphi}\n\t\\partial_{v}(r \\phi)(u,v) = \\epsilon \\chi(v) + \\mathrm{Err}_{1}(u,v),\n\\end{equation}\nwhere\n\\begin{equation} \\label{eq:opt2:Err1}\n\t\\abs{\\mathrm{Err}_{1}(u,v)} \\leq C \\epsilon^{3} \\min \\set{u^{-3}, r(u,v)^{-3}}.\n\\end{equation}\n\nThe argument is similar to the proof of Theorem \\ref{thm.opt.1}, but this time we rely on Theorem \\ref{thm:smallData} to make the dependence of $\\mathrm{Err}_{1}$ on $\\epsilon$ explicit. Indeed, by \\eqref{eq:SSESF:dphi}, we have\n\\begin{equation*}\n\t\\abs{\\mathrm{Err}_{1}(u,v)}\n\t\\leq \\int_{1}^{u} \\abs{\\frac{\\mu \\lmb \\nu}{(1-\\mu)r}  \\phi} (u', v) \\, \\mathrm{d} u'.\n\\end{equation*}\n\nThen estimating the right-hand side using Theorem \\ref{main.thm.1}, Proposition \\ref{prop:geomLocBVScat} and Corollary \\ref{cor:decay2}, and using \\eqref{eq:opt2:eps} make the $\\epsilon$-dependence explicit, \\eqref{eq:opt2:Err1} follows.\n\nIntegrating \\eqref{eq:opt2:dvrphi}, we also have\n\\begin{align*} \n\tr\\phi(u,v) \t= & \\int_{u}^{v} \\partial_{v}(r \\phi)(u, v') \\, \\mathrm{d} v' \\\\\n\t\t\t= & \\epsilon \\int_{u}^{v} \\chi(v') \\, \\mathrm{d} v' + \\int_{u}^{v} \\mathrm{Err}_{1}(u, v') \\, \\mathrm{d} v' \\\\\n\t\t\t= & \\epsilon X(u,v) + \\mathrm{Err}_{2}(u,v)\n\\end{align*}\nwhere $X(u, v) := \\int_{u}^{v} \\chi(v') \\, \\mathrm{d} v'$ and $\\mathrm{Err}_{2}(u,v) := \\int_{u}^{v} \\mathrm{Err}_{1}(u, v') \\, \\mathrm{d} v'$. Integrating \\eqref{eq:opt2:Err1}, and using the bound $C^{-1} \\leq \\lmb \\leq 1\/2$, we easily obtain\n\\begin{equation} \\label{eq:opt2:Err2}\n\t\\abs{\\mathrm{Err}_{2}(u,v)} \\leq C \\epsilon^{3} \\min \\set{r u^{-3}, u^{-2}}.\t\n\\end{equation}\n\nIn particular, taking $v \\to \\infty$, we see that\n\\begin{equation} \\label{eq:opt2:Phi}\n\t\\abs{\\Phi(u) - \\epsilon X(u, \\infty)} \\leq C \\epsilon^{3} u^{-2}.\n\\end{equation}\n\nWe now proceed to estimate $M(u)$. We begin with the easy observation\n\\begin{equation} \\label{eq:opt2:UBforM}\n\tM(u) \\leq C \\epsilon^{2} u^{-5},\n\\end{equation}\nwhich follows from Corollary \\ref{cor:decay2} and \\eqref{eq:opt2:eps}. On the other hand, recalling the definition of $M(u)$ from \\eqref{eq:SSESF:dm} and using the elementary inequality $(a+b)^{2} \\geq \\frac{1}{2} a^{2} - b^{2}$,\n\\begin{align*}\n\tM(u)\t= & \\frac{1}{2} \\int_{u}^{\\infty} \\frac{1-\\mu}{\\lmb} [ \\partial_{v}(r\\phi) - \\frac{\\lmb}{r} (r \\phi) ]^{2} (u,v) \\, \\mathrm{d} v  \\\\\n\t\t\\geq & \\frac{\\epsilon^{2}}{4} \\int_{u}^{\\infty} \\frac{1-\\mu}{\\lmb}(u,v) [\\chi (v) - \\frac{\\lmb}{r} X (u,v) ]^{2} \\, \\mathrm{d} v \n\t\t\t- \\frac{1}{2}\\int_{u}^{\\infty} \\frac{1-\\mu}{\\lmb} [\\mathrm{Err}_{1} - \\frac{\\lmb}{r} \\mathrm{Err}_{2}]^{2} (u,v) \\, \\mathrm{d} v .\n\\end{align*}\n\nBy \\eqref{eq:opt2:eps}, \\eqref{eq:opt2:Err1} and \\eqref{eq:opt2:Err2}, we have\n\\begin{equation*}\n\\abs{\\frac{1}{2} \\int_{u}^{\\infty} \\frac{1-\\mu}{\\lmb} [\\mathrm{Err}_{1} - \\frac{\\lmb}{r} \\mathrm{Err}_{2}]^{2} (u,v) \\, \\mathrm{d} v} \\leq C \\epsilon^{6}.\n\\end{equation*}\n\nFurthermore, note that $(1-\\mu) \\geq (K \\Lambda)^{-1} \\geq C^{-1} > 0$ by Proposition \\ref{prop:geomLocBVScat} and \\eqref{eq:opt2:eps}. Also, for $(u,v) \\in [1,2] \\times [8, \\infty)$, note that $\\chi(v) = 0$ and $X(u,v) = 1$. Therefore, for $1 \\leq u \\leq 2$, there exists $c > 0$ (independent of $\\epsilon > 0$) such that\n\\begin{align*}\n\\frac{1}{4}\\int_{u}^{\\infty} \\frac{1-\\mu}{\\lmb}(u,v) [\\chi - \\frac{\\lmb}{r} X]^{2} (u,v) \\, \\mathrm{d} v\n\\geq & (4 C)^{-1} \\int_{u}^{\\infty}  [\\chi - \\frac{\\lmb}{r} X]^{2} (u,v) \\, \\lmb^{-1} (u,v)  \\mathrm{d} v \\\\\n\\geq & (4 C)^{-1} \\int_{8}^{\\infty} \\frac{\\lmb}{r^{2}} (u,v) \\, \\mathrm{d} v \\\\\n\\geq & c\\,.\n\\end{align*}\n\nTherefore, we conclude that\n\\begin{equation} \\label{eq:opt2:LBforM}\n\tM(u) \\geq c \\epsilon^{2} - C \\epsilon^{6} \\qquad \\hbox{ for } 1 \\leq u \\leq 2.\n\\end{equation}\n\nWe are now ready to compute $\\mathfrak{L}$ and complete the proof. We begin by observing that \n\\begin{equation*}\n\\lim_{v \\to \\infty} r^{3} \\abs{\\partial_{v}(r \\phi)(1,v)} = 0\n\\end{equation*}\nby our choice of data. Therefore,\n\\begin{align*}\n\t- \\mathfrak{L}\n\t\t = &\\int_{1}^{\\infty} M (-\\nu_{\\infty}) \\Phi(u) \\, \\mathrm{d} u \\\\\n\t\t= & \\epsilon \\int_{1}^{\\infty} M (u) (-\\nu_{\\infty}) (u) X(u,\\infty) \\, \\mathrm{d} u \n\t\t\t+ \\int_{1}^{\\infty} M(u) (-\\nu_{\\infty})(u) \\mathrm{Err}_{2}(u, \\infty) \\, \\mathrm{d} u. \n\\end{align*}\n\nBy Proposition \\ref{prop:geomLocBVScat}, \\eqref{eq:opt2:eps}, \\eqref{eq:opt2:Err2} and \\eqref{eq:opt2:UBforM}, we have\n\\begin{equation*}\n\\abs{\\int_{1}^{\\infty} M(u) (-\\nu_{\\infty})(u) \\mathrm{Err}_{2}(u, \\infty) \\, \\mathrm{d} u } \\leq C \\epsilon^{5}.\n\\end{equation*}\n\nOn the other hand, by Proposition \\ref{prop:geomLocBVScat}, \\eqref{eq:opt2:eps} and \\eqref{eq:opt2:LBforM}, we have (taking $c > 0$ smaller if necessary)\n\\begin{align*}\n\t\t\\epsilon \\int_{1}^{\\infty} M(u) (-\\nu_{\\infty})(u) X(u, \\infty) \\, \\mathrm{d} u \n\t\t\\geq & \\epsilon \\int_{1}^{2} M(u) (-\\nu_{\\infty})(u) X(u, \\infty) \\, \\mathrm{d} u \\\\\n\t\t\\geq & \\Lambda^{-1} \\epsilon \\int_{1}^{2} M(u) \\, \\mathrm{d} u\n\t\t\\geq c \\epsilon^{3} - C \\epsilon^{7}.\n\\end{align*}\n\nTherefore, taking $\\epsilon > 0$ sufficiently small, we see that $- \\mathfrak{L} > \\frac{c}{2} \\epsilon^{3} > 0$. \\qedhere\n\\end{proof}\n\n\\bibliographystyle{amsplain}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{\\label{sec:level1}Introduction}\n\nProcessive motor proteins or molecular motors~\\cite{howard,bray}\n(such as kinesin, cytoplasmic dynein, and myosin V) ``walk'' along\nmolecular tracks (microtubules and actin filaments) taking {\\it\nobserved mechanical steps} of well defined (mean) spacing $d$ , each\nstep being  of {\\it ``negligibly short''} ($\\lesssim 100\\,\\mu s$)\nduration relative to the mean time(s) between steps that are of\norder $1$ to $20\\,\\textrm{ms}$\n~\\cite{howard,bray,mehta99,rief00,nishiyama02,block03,nishiyama03,yildiz03,snyder04,uemura04,baker04,oiwa05,carter05,taniguchi05,clemen05,guydosh06,toba06}.\nSteps may be taken {\\it forwards} ($+$) or {\\it backwards} ($-$).\nThe mean velocity (observed over 10's to 100's ) of steps, $V$, is a\n{\\it function} of the load $\\textbf{F}=(F_x,F_y,F_z)$ exerted on the\nmotor and the fuel concentration [ATP] (for most\ncases)~\\cite{fisher99,kolomeisky03,fisher05} and, in general, of\nother features of the aqueous solution including the pH, ionic\nstrength, temperature $T$, and other reagents\/reactants such as\n[ADP], [Pi], [AMP-PNP], [BeF$_2$],\netc.)\\cite{howard,block03,carter05,guydosh06}.\n\nMotor proteins  are enzymatic {\\it catalysts} that, following\nbiochemical knowledge and principles, turn over one ``fuel molecule\"\n(usually ATP) for each full step via (in the simplest case) a {\\it\nlinear sequence of reversible kinetic transitions} (or {\\it\nreactions}) embodying $N$ (bio)chemical states per\nturnover~\\cite{fisher99,kolomeisky03,fisher05,fisher99a,kolomeisky00,kim05,bustamante01}.\nThis situation is embodied in the following {\\it basic sequential\nmodel}\n\\begin{align}\n&\\cdots\\begin{array}{c}\n    u_{N-1}\\\\\n    \\xleftarrow{\\quad\\quad\\quad}\\\\\n    \\vspace{-10 mm}\\\\\n    \\xrightarrow{\\quad\\quad\\quad}\\\\\n    w_{0\\equiv N}\n\\end{array}\n[0]_l\\begin{array}{c}\n    u_0\\\\\n    \\xleftarrow{\\quad\\quad\\quad}\\\\\n    \\vspace{-10 mm}\\\\\n    \\xrightarrow{\\quad\\quad\\quad}\\\\\n    w_1\n\\end{array}\n(1)_l\\begin{array}{c}\n    u_1\\\\\n    \\xleftarrow{\\quad\\quad\\quad}\\\\\n    \\vspace{-10 mm}\\\\\n    \\xrightarrow{\\quad\\quad\\quad}\\\\\n    w_2\n\\end{array}\n(2)_l\\begin{array}{c}\n    u_2\\\\\n    \\xleftarrow{\\quad\\quad\\quad}\\\\\n    \\vspace{-10 mm}\\\\\n    \\xrightarrow{\\quad\\quad\\quad}\\\\\n    w_3\n\\end{array}\n\\cdots\\nonumber\\\\\n&\\cdots\\begin{array}{c}\n    u_{N-2}\\\\\n    \\xleftarrow{\\quad\\quad\\quad}\\\\\n    \\vspace{-10 mm}\\\\\n    \\xrightarrow{\\quad\\quad\\quad}\\\\\n    w_{N-1}\n\\end{array}\n(N-1)_l\\begin{array}{c}\n    u_{N-1}\\\\\n    \\xleftarrow{\\quad\\quad\\quad}\\\\\n    \\vspace{-10 mm}\\\\\n    \\xrightarrow{\\quad\\quad\\quad}\\\\\n    w_N\n\\end{array}\n[N]_l\\equiv[0]_{l+1}\\begin{array}{c}\n    u_0\\\\\n    \\xleftarrow{\\quad\\quad}\\\\\n    \\vspace{-10 mm}\\\\\n    \\xrightarrow{\\quad\\quad}\\\\\n    w_1\n\\end{array}\n\\cdots,\\label{BasicModel}\n\\end{align}\n\nwhich is understood to repeat periodically as the motor moves\nprocessively along its track. The subscript $l=1,2,\\cdots$ labeling\nthe $N$ basic states $[0]$, $(1)$, $\\cdots(N-1)$, denotes the sites\non the linear track spaced at distance $d$ apart.\n\nBy convention the state $(i)=[0]$ is ``bound'' or\n``nucleotide-free''  so that the transition $[0]\\rightarrow(1)$\nrepresents the {\\it binding} of one fuel molecule to the awaiting\nmotor. Thus we write\n\\begin{equation}\nu_0=k_0[\\mbox{ATP}],\\label{ATPdepend}\n\\end{equation}\nwhere the pseudo-first-order rate constant $k_0$ and all the\nremaining rate constants $u_i$, $w_j$ depend also on $\\textbf{F}$.\nBut under fixed conditions ($\\textbf{F}$, [ATP], $\\cdots$), the\nrates do not change. Note that this formulation embodies the {\\it\ntight coupling principle} of one fuel molecule being consumed per\n(forward) step~\\cite{howard,fisher99,kolomeisky03,bustamante01}.\nThis is {\\it assumed} in the basic model, which also neglects\nirreversible {\\it detachments} from the track (which, however, can\nbe included readily in\nprinciple~\\cite{fisher99,fisher99a,kolomeisky00}).\n\nWhen convenient we will allow the state labels $i$, $j$, $\\cdots$ to\ntake values outside the basic range $[0,N-1]$; for that reason we\nadopt the periodicity convention\n\\begin{equation}\nu_{i+lN}\\equiv u_i,\\;\\;w_{j+lN}\\equiv\nw_j,\\;\\;l=0,\\pm1,\\pm2,\\cdots.\\label{Convention}\n\\end{equation}\n\nNow, in the simplest experimental situation, as observed for\nkinesin, no {\\it mechanical substeps} are\ndetected~\\cite{nishiyama02,carter05} to within the noise level\n(which amounts to $\\Delta x\\lesssim 1\\,\\textrm{nm}$). Furthermore to\nwithin the resolution time ($\\lesssim 100\\,\\mu s$), successive steps\noccur at times, say, $\\cdots, t_{k-1}, \\;t_k, \\;t_{k+1}, \\cdots$.\nThus, between the identifiable mechanical steps of (mean) magnitude\n$d$, the motor {\\it dwells} in a {\\it mechanical state} that, within\nthe noise level $\\Delta x$, appears well defined with no\nsystematically detectable {\\it substeps}, forwards or backwards.\nThen, individual {\\it dwell times} in the mechanical states, namely,\n\\begin{equation}\n\\tau_k=t_k-t_{k-1},\\label{DefDwellTimes}\n\\end{equation}\ncan be measured to reasonable precision and averages may be\ncomputed, over ``many'' observations encompassing, say, $n$ steps,\nto yield an {\\it overall mean dwell time}\n\\begin{equation}\n\\tau=\\left<\\tau_k\\right>\\approx\\frac{1}{n}\\sum_{k=1}^n\\tau_k.\\label{OverallMeanDwellTimes}\n\\end{equation}\nHere and below we use the ``asymptotically equals'' symbol $\\approx$\nto indicate an approximate equality that becomes exact in a long run\nunder steady-state conditions.\n\nGiven a (sufficiently long) sequence of $n$ observed steps with\n$n_+$ forward steps and $n_-$ backward steps, we can also define the\n(steady-state) {\\it step splitting probabilities} or back-step and\nforward-step fractions\n\\begin{equation}\n\\pi_+\\approx n_+\/n,\\qquad\\pi_-\\approx n_-\/n,\\label{StepSplProb}\n\\end{equation}\nwhere, since $n=n_++n_-$, one has\n\\begin{equation}\n\\pi_++\\pi_-=1.\\label{SumOfProbIsOne}\n\\end{equation}\n\nFurthermore, dwell times before a $+$ or $-$ step can be (and have\nbeen~\\cite{carter05}) measured separately leading to distinct {\\it\nprior dwell times}\n\\begin{equation}\n\\tau_+=\\frac{1}{n_+}\\mbox{$\\sum^{+}$}\\tau_k\\qquad\\mbox{and}\\qquad\\tau_-=\\frac{1}{n_-}\\mbox{$\\sum^{-}$}\\tau_k,\\label{PriorDwellTimes}\n\\end{equation}\nthe restricted sums including just $+$ or $-$ steps, respectively.\n\nTo the degree that the runs are long so that $\\pi_+$ and $\\pi_-$ may\nbe accurately considered as probabilities one must evidently also\nhave\n\\begin{equation}\n\\pi_+\\tau_++\\pi_-\\tau_-=\\tau.\\label{TotalDwellasSum}\n\\end{equation}\n\nAs discussed recently in some detail~\\cite{fisher05,kim05}, each\nindividual biochemical state $(i)$, may be characterized by a\ndefinite (mean) {\\it longitudinal location} in physical space, i.e.,\n{\\it along} the track, which we supposed aligned with the $x$\ncoordinate, and, possibly, {\\it transverse} to the track, the\n$y$-coordinate, or {\\it normal} to the track, the $z$-coordinate.\nHence the basic model implies the existence of {\\it substeps}, say,\nof magnitude\n\\begin{equation}\nd_j=x_{j+1}-x_j,\\label{StepMagnitude}\n\\end{equation}\nbetween successive mechanochemical states~\\cite{foot0}. However, the\ngreat majority of these mechanical displacements will be {\\it\nhidden} by noise and so {\\it unobservable}. This is the crucial\nissue.\n\nThe evidence (in particular for kinesin~\\cite{nishiyama02,carter05})\nreveals the existence of one principal or {\\bf major mechanical\nsubstep} of magnitude\n\\begin{equation}\nd_M=x_{M+1}-x_M\\simeq d,\\label{MajorMechStep}\n\\end{equation}\nthat corresponds to a specific transition\n$(M)\\!\\!\\rightarrow\\!\\!(M+1)$ for a {\\it forward} or $+$ {\\it step}.\nSuch a unique forward-step is sometimes called a ``power stroke''.\nThen, clearly, within the basic model a back-step ($-$) corresponds\nto  the specific transition $(M+1)\\!\\rightarrow\\!(M)$.\n\nFor simplicity we will initially suppose that there is {\\it only\none} such single, well defined and observable principal mechanical\ntransition in the processive reaction cycle: it will be referred to\nas a {\\bf major transition} while all other smaller, unobservable\ndisplacements, presumed ``hidden,'' will be termed {\\it substeps}.\n\nIt is of interest, all the same, to analyze situations in which,\nwithin the full cycle, there are a number of {\\it visible} (or\nobservable) {\\it substeps}. Indeed, an initial substep large enough\nto be readily observable was predicted for myosin V by Kolomeisky\nand Fisher~\\cite{kolomeisky03} on the basis of dwell-time data\nobtained at different [ATP] and force levels~\\cite{mehta99}; it was\nlater observed unambiguously by Uemura et al~\\cite{uemura04}. Thus,\nin Sec.~\\ref{sec:level5} below, the case of $K$ ($<N$) distinct\nmajor substeps is considered explicitly~\\cite{foot0A}.\n\nNevertheless, since most of the forward and reverse transitions,\n$(i)\\rightarrow(i-1)$ and $(i-1)\\rightarrow(i)$, are {\\it not}\nobservable, one does {\\it not} know (and {\\it cannot} tell) the\n(bio)chemical (sub)state of the motor {\\it during an observed dwell\ntime}, $\\tau_k$, between steps $k-1$ and $k$:  see\n(\\ref{DefDwellTimes}).  Indeed, the biochemical state will change as\ntime progresses and not necessarily in a uniform sense, e.g., ATP\nmight bind and then be released (or unbind) without undergoing a\nhydrolysis step. For this {\\it crucial reason} once the basic model\nhas $N=2$ or more states the expressions for $\\tau_+$ and $\\tau_-$\nin terms of the basic rates $u_i$ and $w_j$ {\\it cannot be trivial}\n--- and the same goes for the splitting or backwards and forwards probabilities\n$\\pi_+$ and $\\pi_-$.\n\nThe basic theoretical problem is thus to find {\\it explicit\nexpressions} for $\\pi_+$, $\\pi_-$, and for the {\\it partial dwell\ntimes} as well as for {\\it conditional} or {\\it pairwise} stepping\nfractions, $\\pi_{++}$, $\\pi_{-+}$, $\\cdots$, and dwell times,\n$\\tau_{++}$, $\\tau_{-+}$, $\\cdots$, that it is natural to introduce\n(as seen below). Indeed, it is clear that these general statistical\nconcepts are not restricted to linear or translocational motors, on\nwhich we have focussed; in fact, they apply equally to {\\it rotary\nstepping motors} like\nF$_1$-ATPase~\\cite{yasuda98,shimabukuro03,nishizaka04,shimabukuro05},\nF$_1$F$_0$-ATPase~\\cite{diez04,ueno05}, and bacterial flagellar\nmotors~\\cite{bray,sowa05}. However, since, in these respects,\nkinesin and myosin V have been studied more extensively, we will\nretain the language appropriate for processive motor proteins\nwalking on linear tracks.\n\nIn previous theoretical\nstudies~\\cite{kolomeisky03,kolomeisky05,qian06,wang06} splitting\nprobabilities and conditional mean dwell times have been introduced\nin the context of molecular motors {\\it via} the following\ndefinitions (which are distinguished from those used in the\ndiscussion above by haceks) specifically:\n$\\check{\\pi}_+\\equiv\\pi_{0,N}$ and $\\check{\\pi}_-\\equiv\\pi_{0,-N}$\nare the probabilities that a motor starting at a well defined\nphysical site $l$ along the track in the binding state $(j)=[0]$\nwill arrive at the next site $l+1$ or back at the previous site\n$l-1$ (in both cases in state $(j)=[0]$) without having undergone\nthe opposite transition to complete a full backward or forward\ncycle, respectively. Then, similarly,\n$\\check{\\tau}_{\\pm}\\equiv\\tau_{0,\\pm N}$ are the average times a\nmotor spends at site $l$ (starting at $j=0$) before completing a\nfull forward or backward cycle to site $l\\pm 1$ (in state\n$(j)=[0]$). Exact results for such statistics can be derived by\nmapping into a Markov renewal process~\\cite{qian06,wang06}.\n\nWith these definitions the explicit formulae\nobtained~\\cite{kolomeisky03,kolomeisky05} for\n$\\;\\check{\\pi}_{\\pm}\\;$ and $\\;\\check{\\tau}_{\\pm}\\;$ correspond to a\n{\\it full mechanochemical cycle} during which a complete forward or\nbackward step is certainly taken. However, the resulting expressions\ncan be applied to the analysis of experimental stepping data {\\it\nonly if} these data allow one to identify each full cycle. If,\ninstead, the observed noise hides one or more of the $N>1$\nbiochemical or mechanochemical substeps, while only major mechanical\ntransitions are detectible, one {\\it cannot} in general decide\nunambiguously whether a motor executed the detectable steps (or\npower strokes) with or without completing a full cycle. In such\ncases, the previous expressions cannot be applied to account for the\nobserved step fractions and dwell times $\\pi_{\\pm}$ and\n$\\tau_{\\pm}$. Instead, the results must be modified to allow for the\nambiguity arising from the hidden substeps. It transpires, as we\nshow below, that this rather subtle and at first-sight\ninconsequential small difference actually leads to significant\nchanges in the load dependence especially (but not exclusively) when\nthe fractions of back-steps and forward-steps are similar in\nfrequency, i.e., on approaching stall conditions when the velocity,\n$V$, becomes small relative to its load-free\nvalue~\\cite{fisher99,kolomeisky03,fisher05}.\n\n\\begin{figure*}\n\\includegraphics[angle=0, scale=0.70]{PapeFig1Bsm}\n\\caption{\\label{fig:wide} Schematic graphs illustrating why the\nfull-cycle interpretation is not adequate for describing stepping\ndata: Plot {\\bf (a)} depicts a hypothetical time series of forward\nsteps for an ($N=4$) cycle with $M=2$. The (0,1)'s indicate that the\nstates $[0]_l$ and $(1)_l$ are both certain to occur at least once\nbetween the pairs of states $(3)_{l-1}$ and $(2)_l$. (See the text.)\nThen {\\bf (b)} represents a similar time series but with one major\nback-step. The question-marks between the forward-back and\nback-forward steps indicate that one does not know if the motor has\never passed through the states $[0]_l$ and\/or $(1)_l$ in these\nintervals. Finally, {\\bf (c)} and {\\bf (d)} depict two equivalent\ntime series for a motor with $N=3$ and $M=1$ but with different\nnoise levels. The first plot allows one to identify only major\nforward and backward transitions at times $t_k$ (marked on the axis)\nwhile the states $[0]_l$ and corresponding substeps are hidden in\nthe noise. However, the second plot reveals all the transitions and\nsubsteps, so that the $t^0_k$, marking the beginning (or end) of\neach full cycle, can be determined. These schematic examples\ndemonstrate that detectable transitions do not necessarily\ncorrespond to a full biochemical cycle so that a proper statistical\nanalysis must take account of substeps  hidden in the noise. Notice,\nindeed, that the three major transitions identified as forward and\nback steps at times $t_3$, $t_4$ and $t_5$ (on the left) are\nassociated with only a single complete cycle from $t^0_2$ to\n$t^0_3$.}\n\\end{figure*}\n\nTo clarify the issues involved, suppose the cycle has four states\n($N=4$) and the major transition occurs between states $(2)_l$ and\n$(3)_l$ (i.e., $M=2$). Then in stepping time series such as\nillustrated in Figs.~1(a) and (b), one can identify all the moments\nof time at which the motor leaves state $(2)_l$ and reaches state\n$(3)_l$ on moving forwards or when it leaves state $(3)_l$ for state\n$(2)_l$ on moving backwards. When successive forward steps are\nrealized, as illustrated in~Fig.~1(a), one knows that the motor must\npass through the remaining two states, $[0]$ and $(1)$, at some\npoints between the major transitions [see Fig.~1(a)] because state\n$(2)$ cannot otherwise be reached {\\it following} state $(3)$ at the\nsame site $l$. Thus in a sequence of three successive $+$ steps one\ncan conclude that the middle step is associated with a complete\n(forward) cycle. The corresponding observations are equally true for\nsuccessive back steps. On the other hand, when the overall stepping\nsequence encompasses {\\it both} back-steps {\\it and} forward-steps,\nwhich is the interesting (and usual) situation [see Fig.~1(b)], it\nis {\\it impossible}, for example, to be sure that the motor has\ncompleted a full forward cycle when a (detectable) forward step is\nfollowed by a back step; likewise, one cannot tell if a full back\ncycle was completed. In such cases, the full-cycle assumption is not\nvalid.\n\nThe full-cycle assumption can be inadequate even when only a run of\nforward steps is seen, as in Fig.~1(a), in that back steps may\nmerely be infrequent. This is somewhat counterintuitive since one\nmight well argue that each $+$ step does then correspond to a full\ncycle. However, if the enzymatic cycle is reversible there is always\nthe possibility of a completed back step; thus explicit expressions\nin terms of the basic rates $\\{u_j,\\;w_j\\}$ will differ when\npotentially hidden substeps are allowed for. Never-the-less, as we\ndemonstrate in Sec.~\\ref{sec:level4}, there are various cases in\nwhich the quantitative differences may be small.\n\nTo demonstrate further the consequences of different conceivable\ninterpretations, consider an ($N=3$)-state motor with two possible\nsubsteps. Fig.~1(c) illustrates a stepping series with a relatively\nhigh noise level so that only the major transitions, say\n$(1)_l\\rightleftharpoons(2)_l$ for $M=1$, at times $t_k$ (with the\ncorresponding dwell times $\\tau_k=t_k-t_{k-1}$) can be measured. On\nthe other hand, Fig.1(d) shows {\\it exactly the same} series of\nsteps, but with a much lower noise level revealing the previously\nobscured small substeps, $[0]_l\\rightleftharpoons(1)_l$ and\n$(2)_l\\rightleftharpoons[0]_{l+1}$. In the latter case, one can\ndetermine the times $t^0_k$ when the motor reaches the bound state\n$[0]_l$ for the first time (i.e., when a cycle is completed). And\nthen one can reliably determine the number $\\check{n}_+$ of full\nforward, and $\\check{n}_-$ of full backward {\\it cycles}. In\ngeneral, when both forward- and back-steps are present the mean\nvalues of the cycle times $\\check{\\tau}_i=t^0_i-t^0_{i-1}$ (and so\n$\\check{\\tau}$, $\\check{\\tau}_+$, and $\\check{\\tau}_-$) are quite\ndifferent from the mean step-to-step dwell times $\\tau$, $\\tau_+$\nand $\\tau_-$, that one can obtain from the noisy stepping series in\nFig.1(c). The difference between the splitting probabilities,\n$\\check{\\pi}_{\\pm}$ and $\\pi_{\\pm}$, is even more obvious. For\nexample in Fig.~1(d) one has {\\it only} one back cycle since one\nmust not consider the major transitions at times $t_3$ and $t_4$ as\nindicating full stepping cycles because the motor never actually\nreached the next bound state $[0]$: hence from this sequence one\nshould estimate $\\check{\\pi}_+\\simeq5\/6$ and\n$\\check{\\pi}_-\\simeq1\/6$ and $\\check{\\pi}_+\/\\check{\\pi}_-\\simeq5$.\nConversely in Fig.~1(c) one would count 6 forward and 2 back steps\n(or to be more precise, major transitions) leading to the estimates\n$\\pi_+\\simeq3\/4$ and $\\pi_-\\simeq1\/4$ so that $\\pi_+\/\\pi_-\\simeq3$.\n\nFrom a mathematical viewpoint, although most of the transitions and\nbiomechanochemical states remain unseen, there is one bright spot!\nSpecifically, in light of the basic feature or model assumption\nembodied in Eq.~(\\ref{BasicModel}), at the instant of time before\nthe moment, say $t_k$, at which a $+$ step occurs, one can be {\\it\nsure} the motor was in state ($M$) while in the {\\it instant} just\n{\\it after} $t_k$ the motor is in state ($M+1$); and, likewise, just\n{\\it before a backward} ($-$) {\\it step} the state ($M+1$) is\noccupied, while just {\\it after} a back-step the state ($M$) is\ndefinitely occupied. Together with the standard Markovian premise of\nchemical kinetics, namely, that once in a well defined chemical\nstate the subsequent departures are independent of the mode of\narrival, this crucial observation enables the systematic calculation\nof splitting probabilities and conditional dwell times for general\n$N>1$ via the {\\it Theory of First Passage Times:} specifically, as\nwe now explain, we may use the analysis as formulated by van\nKampen~\\cite{kampen}.\n\n\n\n\\section{\\label{sec:level2}Conditional splitting probabilities and dwell times}\n\nBefore undertaking explicit calculations to obtain expressions for\n$\\pi_+$, $\\pi_-$, and $\\tau_+$, $\\tau_-$, in terms of the $u_i$ and\n$w_j$ for general $N$ and $M$, we introduce some further statistical\nproperties that are straightforward to observe experimentally and\nmight prove mechanistically informative. At the same time, they\nenter naturally in to the first-passage analysis that is presented\nin Sec.~\\ref{sec:level3}.\n\nIn addition to the prior dwell times defined\nin~(\\ref{PriorDwellTimes}) one may {\\it separately} observe {\\it\npost dwell times} by measuring intervals {\\it following after} $+$\nor $-$ major steps: we will label the corresponding mean dwell times\n$\\tau_{+\\diamond}$ and $\\tau_{-\\diamond}$, where the subscript\n$\\diamond$ is read as `diamond' and denotes, here and below, a $+$\n{\\it or} a $-$ step. However, such dwell times may be {\\it\ntruncated} by {\\it detachments} (or dissociations or disconnections)\nin which the motor leaves the track (essentially irreversibly) so\nending a run.  The {\\it rates of detachment from states} $(i)$, say\n$\\delta_i$, can certainly be included in the basic sequential\nkinetic model~\\cite{fisher99,fisher99a,kolomeisky00}; but in the\n{\\it first instance} they may be {\\it neglected} provided, as we\nwill suppose, only time intervals between observed $+$ or $-$\nmechanical steps are considered. (Their effects, however, would be\nsignificant if dwell times {\\it prior to detachments or immediately\nfollowing attachments} were considered which might, indeed, prove\ninformative.)\n\nNeglecting such ``initial'' and ``final'' dwell times (although the\nformer have been examined by Veigel {\\it et al.} for myosin V in\nseeking {\\it observable} mechanical substeps~\\cite{veigel05}) one\nmay still observe the {\\it four distinct conditional mean dwell\ntimes}:\n\\begin{equation}\n\\begin{array}{ll}\n\\!\\!\\tau_{++}: & \\mbox{between two successive forward (+) steps},\\\\\n\\!\\!\\tau_{+-}: & \\mbox{between a + step followed by a back-step},\\\\\n\\!\\!\\tau_{-+}: & \\mbox{between a back-step followed by a + step},\\\\\n\\!\\!\\tau_{--}: & \\mbox{between two successive back-steps},\n\\end{array}\\label{DefPairwiseDwells}\n\\end{equation}\ndefined, as in~(\\ref{PriorDwellTimes}), in terms of the observed\nintervals $\\tau^{++}_k=t^{(+)}_k-t^{(+)}_{k-1}$, averaged over\n$n_{++}$ pairs of successive $+$ steps, and likewise for $n_{+-}$\npairs of $+$ steps followed by $-$ steps, etc.\n\nAnother aspect is to note that for realistic runs of limited length,\ndeviations of order $1\/n$ will arise. Thus, for example, for a run\nof length $n=n_++n_-$ {\\it starting} with a $+$ step the overall\nmean dwell time is given by\n\\begin{equation}\n\\tau=[(n_+-1)\\tau_++n_-\\tau_-]\/(n-1),\\label{TauCorrection1}\n\\end{equation}\nthere being only $(n-1)$ measurable (prior) dwell times\n$\\tau_l=t_l-t_{l-1}$. Using the definitions (\\ref{StepSplProb}) then\nyields\n\\begin{equation}\n\\tau=\\pi_+\\tau_++\\pi_-\\tau_--\\frac{\\pi_-(\\tau_+-\\tau_-)}{n-1}.\\label{TauCorrection2}\n\\end{equation}\nIn fitting asymptotic ($n\\gg 1$) expressions to real data from\nfinite runs such systematic deviations should be recognized. Here,\nhowever, we will neglect such finite-$n$ or end-effects.\n\nTo proceed further it is also helpful to introduce the {\\it pairwise\nstep probabilities} $\\;\\pi_{++}$ and $\\pi_{+-}$ defined as the\nprobability that a $+$ step is followed by a $+$ or, respectively,\nby a $-$ step, and likewise, $\\pi_{-+}$ and $\\pi_{--}$. These then\nsatisfy\n\\begin{equation}\n\\pi_{++}+\\pi_{+-}=1\\qquad\\mbox{and}\\qquad\\pi_{-+}+\\pi_{--}=1.\\label{PairwiseSplNorm}\n\\end{equation}\n\nAgain, in a finite run of $n$ steps one can divide the $n-1$\nsuccessive pairs into $n_{++}$ of $+$ steps followed by a $+$ step,\nand so on, and use $\\pi_{++}\\approx n_{++}\/(n_{++}+n_{+-})$,\n$\\pi_{+-}\\approx n_{+-}\/(n_{+-}+n_{++})$, etc. Noting that in a\ngiven run one must have $|n_{+-}-n_{-+}|\\leq 1$, and neglecting\nfinite-$n$ corrections, leads to the valuable relation\n\\begin{equation}\n\\pi_{+}\\pi_{+-}=\\pi_{-}\\pi_{-+}.\\label{ValuableRelation}\n\\end{equation}\nFrom this follows the connections\n\\begin{equation}\n\\pi_+=1-\\pi_-=\\pi_{-+}\/(\\pi_{+-}+\\pi_{-+}),\\label{StepSplInPairSplPl}\n\\end{equation}\n\\begin{equation}\n\\frac{1}{\\pi_+}=1+\\frac{\\pi_{+-}}{\\pi_{-+}}\\qquad\\mbox{and}\\qquad\\frac{1}{\\pi_-}=1+\\frac{\\pi_{-+}}{\\pi_{+-}}.\\label{StepSplInPairSplMi}\n\\end{equation}\nTogether with~(\\ref{PairwiseSplNorm}) these relations show that the\npair $\\pi_{+-}$ and $\\pi_{-+}$ or, equivalently, $\\pi_{++}$ and\n$\\pi_{--}$ serve to determine all the back\/forward or splitting\nprobabilities.\n\nIt is worthwhile to carry these considerations a stage further by\nrecognizing the Markovian character of the basic $N$-state\nmodel~(\\ref{BasicModel}). Thus, neglecting detachments, the four\ndivision or splitting probabilities $\\pi_{++}$, $\\pi_{+-}$,\n$\\pi_{-+}$, and $\\pi_{--}$ satisfying~(\\ref{PairwiseSplNorm}) can be\nregarded as the elements of a $2\\!\\times\\!2$ stepping matrix,\n$[\\pi_{\\alpha\\beta}]$, that stochastically determines the\ntransitions from one (major) step, $+$ or $-$, to the next. By\nvirtue of the conservation of probability, the largest eigenvalue is\n$\\;\\lambda_0=1;\\;$ but the second eigenvalue, which determines the\ndecay per step of step-step correlations, is just\n\\begin{eqnarray}\n\\lambda_1&=&1-\\pi_{+-}-\\pi_{-+}\\nonumber\\\\\n&=&\\pi_{++}-\\pi_{-+}=\\pi_{++}+\\pi_{--}-1.\\label{Eigenvalue2}\n\\end{eqnarray}\nThis vanishes when $\\pi_{+-}=\\pi_{-+}=\\frac{1}{2}$ which corresponds\nto $\\pi_+=\\pi_-$ and hence, to {\\it stall conditions} in which the\nmean velocity, $V$, vanishes.\n\nCounting arguments similar to those\nyielding~(\\ref{TauCorrection1})-(\\ref{ValuableRelation}) also lead\nto relations for the conditional mean dwell times. For completeness\nand consistency with later expressions, we utilize the\n$+\/\\!\\!-`diamond$' notation introduced before. For the prior dwell\ntimes, we thus find\n\\begin{eqnarray}\n&&\\tau_+\\equiv\\tau_{\\diamond+}=\\pi_{++}\\tau_{++}+\\pi_{+-}\\tau_{-+},\\label{StepDwlInPairDwlPl}\\\\\n&&\\tau_-\\equiv\\tau_{\\diamond-}=\\pi_{-+}\\tau_{+-}+\\pi_{--}\\tau_{--},\\label{StepDwlInPairDwlMi}\n\\end{eqnarray}\nwhich, in turn, are fully consistent with the relation\n(\\ref{TotalDwellasSum}) for $\\tau$ in terms of $\\tau_+$ and $\\tau_-$\nwhere we should note\n\\begin{equation}\\label{PiPMDiamond}\n    \\pi_+\\equiv\\pi_{\\diamond+}\\equiv\\pi_{+\\diamond}\\quad\\mbox{and}\\quad\\pi_-\\equiv\\pi_{\\diamond-}\\equiv\\pi_{-\\diamond}.\n\\end{equation}\nThen the {\\it post dwell times} likewise satisfy\n\\begin{eqnarray}\n&&\\tau_{+\\diamond}=\\pi_{++}\\tau_{++}+\\pi_{+-}\\tau_{+-},\\label{PostDwlInPairDwlPl}\\\\\n&&\\tau_{-\\diamond}=\\pi_{-+}\\tau_{-+}+\\pi_{--}\\tau_{--},\\label{PostDwlInPairDwlMi}\n\\end{eqnarray}\nwhile the overall mean dwell time is given by\n\\begin{equation}\\label{OverallDwlTime}\n    \\tau\\equiv\\tau_{\\diamond\\diamond}=\\pi_+\\tau_{+\\diamond}+\\pi_-\\tau_{-\\diamond}.\n\\end{equation}\n\nEach of these pairwise fractions and dwell times can be obtained\nfrom the same experimental data (i.e., stepping time series) that\nhave been used experimentally to obtain the step splitting\nprobabilities and the prior dwell times in the course of studying\nthe dynamics of a motor as a function of load and [ATP], etc. But by\nobserving such further independent statistical parameters one can\ntest the basic theory more completely and hope to obtain more\nreliable and constrained fitting values for the rates determining\nthe full mechanochemical cycle.\n\nAt a more detailed level it is also useful to define\n$\\;\\;n_{i,+}^{\\rho\\sigma}\\;\\;$ and $\\;\\;n_{j,-}^{\\rho\\sigma}\\;\\;$\nwith $\\;\\;\\rho,\\sigma=\\diamond,+,\\;\\mbox{or}\\;-$, as the mean number\nof hidden forward and backward transitions, possibly hidden, from\nstates $(i)$ and $(j)$, respectively, in the intervals between\n(major) steps {\\it subject} to the conditions specified by the pair\n$(\\rho,\\sigma)$. If these transitions prove to be detectable, they\ncan be counted and used in fitting parameters; but if they pertain\nto hidden transitions (e.g., the hydrolysis of ATP, etc.), it is of\ninterest to estimate how often they occur given specific rates. The\nappropriate calculations on the basis of the\nmodel~(\\ref{BasicModel}) are developed below in\nSec.~\\ref{sec:level3e}.\n\nIt is appropriate here to mention various hidden-Markov methods,\netc.~\\cite{smith01,milescu06,mckinney06,milescu06a}, that have been\nderived and employed to locate steps in the presence of noise (and\nto fit their amplitudes, or kinetic parameters, etc.). These\napproaches require an input stochastic\nmodel~\\cite{smith01,milescu06a,foot1}; we believe the present\napproach could provide a valuable complement in the extraction of\nkinetic parameters from such experimental data since, as we will\nsee, it reveals the kinds of behavior different models can generate.\n\n\\section{\\label{sec:level3}Explicit calculations}\n\\subsection{\\label{sec:level3a}Formulation and Notation}\nThe various stepping fractions, dwell times, etc., introduced in\nSec.~\\ref{sec:level1} and \\ref{sec:level2} can be derived explicitly\nin terms of the basic kinetic rates by using van Kampen's analysis\nfor one-dimensional, nearest-neighbor first-passage\nprocesses~\\cite{kampen}. Accordingly, following the basic sequential\nmodel~(\\ref{BasicModel}), with the $N$-periodicity\nconventions~(\\ref{Convention}) for the sequential forward and\nbackward rates, $u_i$ and $w_j$, we envisage a random walker on a\none-dimensional lattice with sites labeled $i,j=0,\\pm1,\\pm2,\\cdots$,\ncorresponding, in turn, to the motor states $(i)$, $(j)$, etc.\n(again subject to the periodicity convention). If the single major\nor observable step per cycle corresponds to the transitions\n$(M)\\rightleftharpoons(M+1)$ with $M\\in[0,N-1]$ we introduce\n(following~\\cite{kampen}) absorbing boundaries on the left and the\nright via\n\\begin{equation}\\label{SpecifyLR}\n    L\\equiv M\\qquad \\mbox{and}\\qquad R\\equiv M+1+N.\n\\end{equation}\n\nIf, for given initial conditions at time $t=0$ (to be selected\nbelow), $q_i(t)$ is the probability that the motor\/walker is in\nstate $(i)$ at time $t$ we may construct the $N\\!\\times\\!N$\ntransition matrix ${\\bf A}=[A_{ij}]$ with elements\n\\begin{equation}\nA_{i,j}=u_j\\delta_{i,j+1}+w_j\\delta_{i+1,j}-(u_j+w_j)\\delta_{i,j},\\label{ElemA}\n\\end{equation}\nwhere $i,j\\in[L+1,R-1]=[M+1,M+N]$. Then if\n$\\textbf{q}^{\\textrm{T}}=[q_{M+1},q_{M+2},\\cdots,q_{M+N}]$ is the\nstate vector, the governing rate equations are\n\\begin{equation}\n\\frac{d\\textbf{q}(t)}{dt}\\;=\\;\\textbf{A}\\textbf{q}(t),\\label{RateEqnForq}\n\\end{equation}\n\nThis completes the first-passage formulation~\\cite{kampen}. Before\nproceeding, however, we record some convenient notation for the\nvarious products and sums of the rate constants that enter the\nanalysis. To that end, our first definition~\\cite{foot2}\nis of the $(m\\!\\geqslant\\!1)$-term product\n\\begin{equation}\n\\Gamma_{l,m}=\\prod_{j=1}^{m}\\frac{w_{l+j}}{u_{l+j}},\\label{DefGamma}\n\\end{equation}\nwhich, by periodicity, is invariant under $l\\Rightarrow l\\pm N$.\nLikewise, the $N$-term product $\\Gamma_{l,N}$ is independent of $l$\nyielding, specifically,\n\\begin{equation}\n\\Gamma_N=\\Gamma_{l,N}=\\prod_{j=0}^{N-1}\\frac{w_j}{u_j},\\label{GammaN}\n\\end{equation}\n\\cite{foot2}. Then for all $l=0,\\pm1,\\pm2,\\ldots$ a central role\nwill be played by the $(n-1)$-term sum\n\\begin{equation}\n\\Delta_{l,n}=\\sum_{m=1}^{n-1}\\Gamma_{l,m}\\qquad(n\\geqslant1),\\label{DefDelta}\n\\end{equation}\nwhere, for the empty sum, we set $\\Delta_{l,1}\\equiv0$. Indeed,\nthese sums appear in previous\nanalyses~\\cite{fisher99,fisher99a,kolomeisky00,foot2} via\n``renormalized'' inverse forward rates (or transition times)\n\\begin{equation}\nr_l=u_l^{-1}(1+\\Delta_{l,N}).\\label{TransitionTimes}\n\\end{equation}\nSpecifically, these enter into the\nexpression~\\cite{fisher99,fisher99a,kolomeisky00}\n\\begin{equation}\nV\/d=(1-\\Gamma_N)\\left\/\\mbox{$\\sum_{l=0}^{N-1}r_l$}\\right.\\label{Velocity}\n\\end{equation}\nfor the {\\it mean velocity} $V$, which we recall here for\nconvenience of reference. (Note that $d$ is the mean spacing of\nsites along the track.) One sees directly from this that {\\it stall\nconditions}, i.e., $V=0$, are determined by $\\Gamma_N({u_i,w_j})=1$.\nThe situation near stall will be a major focus for our discussions\nin Sec.~\\ref{sec:level4}.\n\nThe analysis of van Kampen~\\cite{kampen} may now be put to work.\nReaders uninterested in the details may skip to the next section or\nperuse the main results, namely,\n(\\ref{SplProbPlPl})-(\\ref{StepSplProbMi}) for $\\pi_{++}$, etc.,\n(\\ref{DwelPlPl})-(\\ref{DwelMiMi}) for $\\tau_{++}$, etc., and\n(\\ref{DwelPl})-(\\ref{DwelTot}) for $\\tau_{+}$, $\\tau_{-}$ and\n$\\tau$.\n\n\n\\subsection{\\label{sec:level3b}Pairwise step splitting probabilities}\nTo proceed, let $\\pi_k^L$ be the total probability that a motor\nstarting at $t=0$ in state $(k)$ with $L<k<R$, so that\n$q_i(0)=\\delta_{ik}$, eventually reaches the left absorbing state\n$(L)$ for the first time, i.e., without having been absorbed at\nsites $L$ {\\it or} $R$. A moment's thought confirms\n\\begin{equation}\n    \\pi_k^L=w_{L+1}\\int_0^{\\infty}q_{L+1}(t)dt;\\label{SplGenB}\n\\end{equation}\nlikewise, for reaching the boundary state $(R)$ for the first time,\none has\n\\begin{equation}\n    \\pi_k^R=u_{R-1}\\int_0^{\\infty}q_{R-1}(t)dt.\\label{SplGenF}\n\\end{equation}\nThen we may appeal to Ref.~\\cite{kampen}, Ch.XII, Eq.(2.8) which,\nusing the notation (\\ref{DefGamma}), states\n\\begin{equation}\n\\pi_k^L=1-\\pi_k^R=\\frac{\\sum_{m=k-L}^{R-L-1}\\Gamma_{L,m}}{1+\\sum_{m=1}^{R-L-1}\\Gamma_{L,m}}.\\label{vanKampen28}\n\\end{equation}\n\nFor $L$ and $R$ we have (\\ref{SpecifyLR}). If the motor starts just\nafter a (major) forward or $+$ step it is in the initial state\n$(k)=(M+1)$; then $\\pi_k^L$ and $\\pi_k^R$ correspond, respectively,\nto $\\pi_{+-}$ and $\\pi_{++}$. Conversely, just after a (major) back\nor $-$ step the motor is in state $(M)$ which, by periodicity is\nequivalent to starting $k=M+N$; then we may identify $\\pi_k^L$ and\n$\\pi_k^R$ with $\\pi_{--}$ and $\\pi_{-+}$, respectively. On using the\nnotation (\\ref{GammaN}) and (\\ref{DefDelta}) we thus obtain\n\\begin{eqnarray}\n&&\\pi_{++}(M,N)=1-\\pi_{+-}=\\frac{1}{1+\\Gamma_N+\\Delta_{M,N}},\\label{SplProbPlPl}\\\\\n&&\\pi_{-+}(M,N)=1-\\pi_{--}=\\frac{1+\\Delta_{M,N}}{1+\\Gamma_N+\\Delta_{M,N}}.\\label{SplProbMiPl}\n\\end{eqnarray}\n\nIt is interesting to note the cross-relation\n\\begin{equation}\\label{CrossRelation}\n    \\pi_{--}=\\Gamma_N\\pi_{++}.\n\\end{equation}\nAt stall this implies $\\pi_{++}=\\pi_{--}$ and $\\pi_{+-}=\\pi_{-+}$\nwhich reflects the expected $+\\!\/\\!-$ symmetry. The results\n(\\ref{SplProbPlPl}) and (\\ref{SplProbMiPl}) also yield, via the\nidentities (\\ref{StepSplInPairSplPl}) and\n(\\ref{StepSplInPairSplMi}), the explicit step fraction expressions\n\\begin{eqnarray}\n&&\\pi_{+}=1-\\pi_{-}=\\frac{1+\\Delta_{M,N}}{1+\\Gamma_N+2\\,\\Delta_{M,N}},\\label{StepSplProbPl}\\\\\n&&\\pi_{-}=\\frac{\\Gamma_N+\\Delta_{M,N}}{1+\\Gamma_N+2\\,\\Delta_{M,N}}.\\label{StepSplProbMi}\n\\end{eqnarray}\n\n\n\\subsection{\\label{sec:level3c}Pairwise and prior dwell times}\n\nNow, following van Kampen~\\cite{kampen}, Ch.XII, Eqs.(1.7)-(1.8),\nthe {\\it conditional mean first-passage times} for arriving either\nat the left or right absorbing boundaries starting from a state\npoint $k\\in[L+1,R-1]=[M+1,M+N]$ as before, are\n\\begin{equation}\n\\tau_k^L=\\frac{\\int_0^{\\infty}tq_{L+1}(t)dt}{\\int_0^{\\infty}q_{L+1}(t)dt},\n\\qquad\n\\tau_k^R=\\frac{\\int_0^{\\infty}tq_{R-1}(t)dt}{\\int_0^{\\infty}q_{R-1}(t)dt},\\label{DefDwelGen}\n\\end{equation}\nwhere the $q_i(t)$ are the solutions of (\\ref{RateEqnForq}) subject,\nfor our purposes, to the two, alternative initial conditions,\n\\begin{equation}\\label{InitialConditions}\n    q_i^+(0)=\\delta_{i,M+1}\\quad\\mbox{and}\\quad\n    q_i^-(0)=\\delta_{i,M+N},\n\\end{equation}\nas discussed in deriving (\\ref{SplProbPlPl}) and\n(\\ref{SplProbMiPl}).\n\nTo obtain the pairwise conditional mean dwell times we proceed in\ntwo steps. First, we integrate the kinetic equations\n(\\ref{RateEqnForq}) over all times recognizing that the $q_j(t)$\napproach zero exponentially fast for all $j\\in[M+1,M+N]$, since the\nwalker must eventually be absorbed at either $L$ or $R$. This yields\n\\begin{equation}\n-\\textbf{q}^{\\pm}(0)=\\textbf{A}\\textbf{T}^{\\pm},\\label{RateEqnForQ}\n\\end{equation}\nwhere the superscripts identify the alternative initial conditions\n(\\ref{InitialConditions}). The elements of the vector\n$\\textbf{T}^{\\pm}$ have the dimensions of time and are given simply\nby\n\\begin{equation}\nT_{k}^{\\pm}=\\int_0^{\\infty}q^{\\pm}_k(t)dt.\\label{DefQ}\n\\end{equation}\nBy the definitions (\\ref{SplGenB}) and (\\ref{SplGenF}) with\n(\\ref{SpecifyLR}), we also have the relations\n\\begin{eqnarray}\nT_{M+1}^{+}=\\pi_{+-}\/w_{M+1},&\\;& T_{M+1}^{-}=\\pi_{--}\/w_{M+1},\\label{TatBoundL}\\\\\nT_{M+N}^{+}=\\pi_{++}\/u_{M+N},&\\;&\nT_{M+N}^{-}=\\pi_{-+}\/u_{M+N}.\\label{TatBoundR}\n\\end{eqnarray}\n\nSince $\\textbf{A}$ is a tridiagonal matrix the equations\n(\\ref{RateEqnForQ}) can be inverted recursively to obtain\n\\begin{eqnarray}\nT_j^+&=&\\pi_{++}\\frac{\\Gamma_N+\\Delta_{M,N}-\\Delta_{M,j-M}}{u_j\\Gamma_{M,j-M}},\\label{SolutionQPl}\\\\\nT_j^-&=&\\pi_{--}\\frac{1+\\Delta_{M,j-M}}{u_j\\Gamma_{M,j-M}}.\\label{SolutionQMi}\n\\end{eqnarray}\nOne general approach to this inversion can be found in\nRef.~\\cite{kampen}, Ch.XII, Sec.2. On recalling\n$\\Delta_{l,1}\\equiv0$, one may check that these solutions verify the\nrelations (\\ref{TatBoundL}) and (\\ref{TatBoundR}).\n\nThe next step is to multiply (\\ref{RateEqnForq}) by $t$ and again\nintegrate over all time which yields\n\\begin{equation}\n-\\textbf{T}^{\\pm}=\\textbf{A}\\textbf{S}^{\\pm},\\label{RateEqnForT}\n\\end{equation}\nwhere, with the same superscript conventions, etc., we have\n\\begin{equation}\nS_{k}^{\\pm}=\\int_0^{\\infty}tq^{\\pm}_k(t)dt.\\label{DefT}\n\\end{equation}\nFrom  (\\ref{DefDwelGen}) and following the arguments above\n(\\ref{SplProbPlPl}) and (\\ref{SplProbMiPl}), we can now make the\nidentifications\n\\begin{eqnarray}\n  \\tau_k^{L+} &=& \\tau_{+-}=S_{M+1}^{+}\/T_{M+1}^{+},\\label{TAUboundL} \\\\\n  \\tau_k^{R-} &=& \\tau_{-+}=S_{M+N}^{-}\/T_{M+N}^{-},\\label{TAUboundR}\n\\end{eqnarray}\nand similarly for $\\tau_{--}$ and $\\tau_{++}$. Inverting\n(\\ref{RateEqnForQ}) finally leads to the basic pairwise dwell time\nexpressions\n\\begin{eqnarray}\n\\tau_{++}&=&\\sum_{j=M+1}^{M+N}T_j^+(1+\\Delta_{M,j-M}),\\label{DwelPlPl}\\\\\n\\tau_{-+}&=&\\sum_{j=M+1}^{M+N}T_j^-\\frac{1+\\Delta_{M,j-M}}{1+\\Delta_{M,N}},\\label{DwelMiPl}\\\\\n\\tau_{+-}&=&\\sum_{j=M+1}^{M+N}T_j^+\\frac{\\Gamma_N+\\Delta_{M,N}-\\Delta_{M,j-M}}{\\Gamma_N+\\Delta_{M,N}},\\label{DwelPlMi}\\\\\n\\tau_{--}&=&\\sum_{j=M+1}^{M+N}T_j^-\\frac{\\Gamma_N+\\Delta_{M,N}-\\Delta_{M,j-M}}{\\Gamma_N}.\\label{DwelMiMi}\n\\end{eqnarray}\n\nBy using (\\ref{SplProbPlPl}) and (\\ref{SplProbMiPl}) for $\\pi_{++}$\nand $\\pi_{--}$ in (\\ref{SolutionQPl}) and (\\ref{SolutionQMi}) we may\nestablish the unanticipated, general equality\n\\begin{equation}\\label{TauIdentity}\n    \\tau_{++}=\\tau_{--},\\qquad\\mbox{all}\\;M,\\;N.\n\\end{equation}\n\nEn route to the prior dwell times $\\tau_+$ and $\\tau_-$ it is\nconvenient to introduce\n\\begin{equation}\nT_j=\\pi_+T_j^++\\pi_-T_j^-=\\frac{(1+\\Delta_{j,N})\/u_j}{1+\\Gamma_N+2\\;\\Delta_{M,N}},\\label{ReDefQ}\n\\end{equation}\nwhere we have used (\\ref{StepSplProbPl}) and (\\ref{StepSplProbMi})\nand may recall that the numerator is the inverse rate defined in\n(\\ref{TransitionTimes}) and used in the past. Note also that by\nvirtue of the periodicity we have $T_{j\\pm N}\\equiv T_j$ and,\nlikewise, for the $T_j^{\\pm}$. Then, by utilizing\n(\\ref{StepDwlInPairDwlPl}), (\\ref{StepDwlInPairDwlMi}),\n(\\ref{SplProbPlPl}), and (\\ref{SplProbMiPl}) we obtain the prior\ndwell times in the form\n\\begin{align}\n  &\\tau_{+}\\equiv\\tau_{\\diamond+}=\\frac{1}{\\pi_+}\\sum_{j=M+1}^{M+N}T_j\\frac{1+\\Delta_{M,j-M}}{1+\\Gamma_N+\\Delta_{M,N}},\\label{DwelPl}\\\\\n  &\\tau_{-}\\equiv\\tau_{\\diamond-}=\\frac{1}{\\pi_-}\\sum_{j=M+1}^{M+N}T_j\\frac{\\Gamma_N+\\Delta_{M,N}-\\Delta_{M,j-M}}{1+\\Gamma_N+\\Delta_{M,N}}.\\label{DwelMi}\n\\end{align}\nFinally, with the aid of (\\ref{TotalDwellasSum}), the overall mean\ndwell time is simply\n\\begin{equation}\n    \\tau=\\sum_{j=M+1}^{M+N}T_j\\equiv\\sum_{i=0}^{N-1}T_i.\\label{DwelTot}\n\\end{equation}\n\nAlthough, we have introduced the various step fractions via the\npairwise fractions $\\pi_{++}$, etc., this was not a necessary move\nfrom the mathematical point of view. Indeed, one can the\nresults~(\\ref{StepSplProbPl}) and (\\ref{StepSplProbMi}) for $\\pi_+$\nand $\\pi_-$, and the present expressions for $\\tau_+$, $\\tau_-$ and\n$\\tau$, directly by solving the basic rate\nequations~(\\ref{RateEqnForq}) with the initial conditions\n\\begin{equation}\nq_k(0)=\\pi_+\\delta_{k,M+1}+(1-\\pi_+)\\delta_{k,M+N},\n\\end{equation}\ntogether with the relations~(\\ref{RateEqnForQ}) and\n(\\ref{RateEqnForT}) and appropriate boundary conditions. By this\nroute one need not mention the pairwise splitting probabilities or\npairwise dwell times. Nevertheless, the pairwise stepping fractions\nand dwell times can be useful in data analysis and to test theory,\nsince they represent additional force and [ATP] dependent parameters\nthat can be measured without significant extra experimental effort.\n\n\n\\subsection{\\label{sec:level3d}Individual and post dwell times}\n\nNow the form~(\\ref{DwelTot}) for $\\tau$ strongly suggests that\n$T_j\\equiv T_j^{\\diamond\\diamond}$ is actually the overall {\\it\nindividual mean dwell time} spent in state $(j)$ irrespective, as\nindicated by the use of the $+\\!\/\\!-$ or diamond notation [see\n(\\ref{StepDwlInPairDwlPl})-(\\ref{OverallDwlTime})] of the stepping\nsequence. This conclusion is, indeed, justified since it follows\nfrom~(\\ref{DefQ}) that we may identify $T_j^+=T_j^{+\\diamond}$ and\n$T_j^-=T_j^{-\\diamond}$ as {\\it individual post} $+$ and $-$ step\n{\\it mean dwell times} in state $(j)$, respectively. Consequently,\nthe mean overall {\\it post dwell times}, are given by\n\\begin{equation}\n\\tau_{+\\diamond}=\\sum_{i=0}^{N-1}T_i^{+\\diamond}\\quad\\mbox{and}\\quad\\tau_{-\\diamond}=\\sum_{i=0}^{N-1}T_i^{-\\diamond}.\\label{PostDwelAsSumT}\n\\end{equation}\nFrom these results one may verify that~(\\ref{OverallDwlTime}) is\nsatisfied.\n\nLikewise, we anticipate relations like\n\\begin{equation}\n\\tau_{++}=\\sum_{i=0}^{N-1}T_i^{++},\\label{TauPlPlAsSum}\n\\end{equation}\n etc., and, hence, from\n(\\ref{DwelPlPl}) and (\\ref{DwelMiMi}) we surmise that the {\\it\nconditional individual state dwell times} are\n\\begin{align}\n T_j^{++}&=T_j^{--} \\nonumber\\\\\n &=\\frac{(1+\\Delta_{M,j-M})(\\Gamma_N+\\Delta_{M,N}-\\Delta_{M,j-M})}{u_j\\;\\Gamma_{M,j-M}\\;(1+\\Gamma_N+\\Delta_{M,N})},\\label{DefTPlPl}\n\\end{align}\nwhile (\\ref{DwelPlMi}) and (\\ref{DwelMiPl}) yield\n\\begin{align}\n   &\\!\\!T_j^{-+}=\\frac{\\Gamma_N(1+\\Delta_{M,j-M})^2}{u_j\\Gamma_{M,j-M}(1+\\Delta_{M,N})(1+\\Gamma_N+\\Delta_{M,N})},\\label{DefTMiPl}\\\\\n   &\\!\\!T_j^{+-}=\\frac{(\\Gamma_N+\\Delta_{M,N}-\\Delta_{M,j-M})^2}{u_j\\Gamma_{M,j-M}(\\Gamma_N+\\Delta_{M,N})(1+\\Gamma_N+\\Delta_{M,N})}.\\label{DefTPlMi}\n\\end{align}\nIn terms of these we can define the {\\it prior individual} (or {\\it\npartial}) {\\it dwell times} via\n\\begin{eqnarray}\n   T_j^{\\diamond+}&=&\\pi_{++}T_j^{++}+\\pi_{+-}T_j^{-+},\\nonumber\\\\\n   &=&\\frac{r_j(1+\\Delta_{M,j-M})}{(1+\\Delta_{M,N})(1+\\Gamma_N+\\Delta_{M,N})},\\label{DefPriorTPl}\n\\end{eqnarray}\n\\begin{eqnarray}\n   T_j^{\\diamond-}&=&\\pi_{-+}T_j^{+-}+\\pi_{--}T_j^{--},\\nonumber\\\\\n   &=&\\frac{r_j(\\Gamma_N+\\Delta_{M,N}-\\Delta_{M,j-M})}{(\\Gamma_N+\\Delta_{M,N})(1+\\Gamma_N+\\Delta_{M,N})},\\label{DefPriorTMi}\n\\end{eqnarray}\nwhere the $r_j$ are defined in~(\\ref{TransitionTimes}). Hence one\ncan check the expressions for the mean overall {\\it prior} dwell\ntimes, $\\tau_{\\diamond+}$ and $\\tau_{\\diamond-}$, given\nin~(\\ref{DwelPl}) and~(\\ref{DwelMi}).\n\nEvidently, the analysis presented does not fully justify the\ninferences regarding (\\ref{DefTPlPl})-(\\ref{DefPriorTMi}). However,\nthese expressions have been checked by direct computation for $N=2$\n(as recorded in Appendix A) and the various cross-checks for general\n$M$ and $N$ also serve as validation. However, a complete\njustification requires a more elaborate calculation that we hope to\npresent in the future. By the same route one can derive the mean\nconditional counts, $n_{j,+}^{\\rho\\sigma}$ and\n$n_{j,-}^{\\rho\\sigma}$ [see after\nEqs.~(\\ref{PostDwlInPairDwlPl})-(\\ref{OverallDwlTime})] of the {\\it\nhidden substeps} as we now proceed to demonstrate. Corresponding\nresults for $N=2$ are also presented in Appendix A.\n\n\n\\subsection{\\label{sec:level3e}Counting the hidden substeps}\n\nAs touched on briefly in the penultimate paragraph of\nSec.~\\ref{sec:level2}, it is surely of interest in light of our\nbasic premise to estimate for a particular model how many hidden\nsubsteps actually arise on average in the typical intervals\n$\\tau\\equiv\\tau_{\\diamond\\diamond}$, $\\tau_{+\\diamond}$,\n$\\tau_{++}$, etc., between specified successive observable steps,\ni.e., major transitions between states $(M)$ and $(M+1)$. Granted\nthe results obtained in the previous subsection for the\n$T_j^{\\rho\\sigma}$, where henceforth, $(\\rho,\\sigma)$ runs through\nthe nine combinations\n\\begin{equation}\\label{9combinations}\n    \\{\\rho,\\sigma\\}=\\{\\diamond\\diamond;+\\diamond,-\\diamond,\\diamond+,\\diamond-;++,+-,-+,--\\},\n\\end{equation}\nthis is a reasonably straightforward exercise.\n\nNotice, first, that the mean dwell time in a given state $(j)$, say\n$\\bar{\\tau}_j$, is, by virtue of the Markovian character of the\nrelevant biochemical reactions {\\it independent} of whether the\nstate was reached from state $(j-1)$ or $(j+1)$ {\\it and} of whether\nthe motor departs to states $(j+1)$ or $(j-1)$: formally, we may\nwrite\n\\begin{eqnarray}\n    \\bar{\\tau}_j=(u_j+w_j)^{-1}&=&\\tau_j^{(-)\\rightarrow(+)}=\\tau_j^{(-)\\rightarrow(-)}\\nonumber\\\\\n                               &=&\\tau_j^{(+)\\rightarrow(+)}=\\tau_j^{(+)\\rightarrow(-)},\\label{tauBAR}\n\\end{eqnarray}\nwhere the superscripts have an obvious interpretation.\n\nThen, if $n_{j\\rightarrow j+1}$ and $n_{j\\rightarrow j-1}$ are the\nnumber of transitions, i.e., substeps, forwards or backwards,\nrespectively, from state $(j)$ in an interval between observable\nsteps we desire the conditional mean values\n\\begin{equation}\\label{DefSubnumber}\n    n_{j,+}^{\\rho\\sigma}=\\left\\langle n_{j\\rightarrow j+1}\\right\\rangle_{\\rho\\sigma}\\quad\\mbox{and}\\quad n_{j,-}^{\\rho\\sigma}=\\left\\langle n_{j\\rightarrow j-1}\\right\\rangle_{\\rho\\sigma}.\n\\end{equation}\nIn terms of these we will have for the overall mean numbers of $+$\nor $-$ hidden transitions per step-to-step interval\n\\begin{eqnarray}\n    \\bar{n}_{+}^{\\rho\\sigma}&=&\\frac{1}{N-1}\\sum_{j=M+1}^{M+N-1}n_{j,+}^{\\rho\\sigma},\\label{TotSubnumberPl}\\\\\n    \\bar{n}_{-}^{\\rho\\sigma}&=&\\frac{1}{N-1}\\sum_{j=M+2}^{M+N}n_{j,-}^{\\rho\\sigma},\\label{TotSubnumberMi}\n\\end{eqnarray}\nfor all nine pairs $(\\rho,\\sigma)$. As a moments thought reveals,\nthe limits on the summations here must be carefully set: thus, after\na forward step to state $(M+1)$ any putative forward hidden substep\ntransformation from state $(M+N)$ would represent a full {\\it\nobservable} $+$ step (i.e.,  a major transition) and so is to be\nexcluded from the sum of {\\it substeps}; equally, any back\ntransition from state $(M+1)$ would represent an observable\n(back)step whereas back {\\it sub}steps  from state $(M+N)$, prior to\nthe final forward step, {\\it are} to be counted: See, for example,\nFig.~\\ref{fig2:wide} which can be regarded as a noise-free version\nof Fig.~\\ref{fig:wide}(d) [but for an $(N=4)$-state model].\n\nNow consider $T_j^{\\rho\\sigma}$, the mean time spent during a\n$(\\rho,\\sigma)$ step interval in a state $(j)$ that is {\\it neither}\nan initial {\\it nor} a final state of a major transition, so that\n$j\\neq M,\\;M+1$. On average this state will be visited on\n$T_j^{\\rho\\sigma}\\left\/\\bar{\\tau}_j\\right.=(u_j+w_j)T_j^{\\rho\\sigma}$\nseparate occasions during the interval; thus it entails\n$n_{j,+}^{\\rho\\sigma}$ forward substep {\\it departures} and\n$n_{j,-}^{\\rho\\sigma}$ back substep departures. Equally, it entails,\non average, $n_{j-1,+}^{\\rho\\sigma}$ forward substep {\\it arrivals}\nand $n_{j+1,-}^{\\rho\\sigma}$ back substep arrivals. As a result we\nhave the {\\it frequency relations}\n\\begin{equation}\\label{DefFrequencyJ1}\n    \\omega_j^{\\rho\\sigma}\\equiv(u_j+w_j)T_j^{\\rho\\sigma}=n_{j,+}^{\\rho\\sigma}+n_{j,-}^{\\rho\\sigma},\n\\end{equation}\nfor $j\\in[M+2,M+N-1]$ and, similarly,\n\\begin{equation}\\label{DefFrequencyJ2}\n    \\omega_j^{\\rho\\sigma}=n_{j-1,+}^{\\rho\\sigma}+n_{j+1,-}^{\\rho\\sigma},\n\\end{equation}\nwhere the units of $\\omega_j$ may be regarded as substeps per\ninterval.\n\nFor the reasons explained after~(\\ref{TotSubnumberMi}) and\nillustrated in Fig.~\\ref{fig2:wide} the `boundary states' $j=M+1$\nand $j=M+N$, require special consideration. As noted, one cannot\nhave {\\it sub}steps that go backwards from state $(M+1)$ and, in the\ncase of a prior forward step, the certain incoming arrival must be\nincluded in the individual mean dwell time. Thus we are led to the\nfour boundary frequency relations\n\\begin{eqnarray}\n  \\omega_{M+1}^{++} &=& n_{M+1,+}^{++} = n_{M+2,-}^{++}+1,\\label{FrequencyLPlPl}\\\\\n  \\omega_{M+1}^{+-} &=& n_{M+1,+}^{+-}+1 = n_{M+2,-}^{+-}+1,\\label{FrequencyLPlMi}\\\\\n  \\omega_{M+1}^{-+} &=& n_{M+1,+}^{-+} = n_{M+2,-}^{+-},\\label{FrequencyLMiPl}\\\\\n  \\omega_{M+1}^{--} &=& n_{M+1,+}^{--}+1 = n_{M+2,-}^{--}.\\label{FrequencyLMiMi}\n\\end{eqnarray}\nComplementary arguments apply for the opposite boundary state\n$(j)\\equiv(M+N)$, yielding\n\\begin{eqnarray}\n  \\omega_{M+N}^{++} &=& n_{M+N,-}^{++}+1 = n_{M+N-1,+}^{++},\\label{FrequencyRPlPl}\\\\\n  \\omega_{M+N}^{+-} &=& n_{M+N,-}^{+-} = n_{M+N-1,+}^{+-},\\label{FrequencyRPlMi}\\\\\n  \\omega_{M+N}^{-+} &=& n_{M+N,-}^{-+}+1 = n_{M+N-1,+}^{-+}+1,\\label{FrequencyRMiPl}\\\\\n  \\omega_{M+N}^{--} &=& n_{M+N,-}^{--} = n_{M+N-1,+}^{--}+1.\\label{FrequencyRMiMi}\n\\end{eqnarray}\n\n\\begin{figure*}\n\\includegraphics[angle=0, scale=0.70]{PapeFig2Csm}\n\\caption{\\label{fig2:wide}Schematic plots illustrating substeps and\ntheir counting rules (see text) in the $N=4$ case with $M=3$ for\n$(++)$, $(+-)$, $(-+)$, and $(--)$ stepping intervals. Thus, for\nexample, from the first plot one confirms\n$n^{++}_{0,+}=n^{++}_{1,-}+1$ and, correspondingly, that the motor\nvisited the state $[0]$ three times in this realization; and,\nlikewise, $n^{++}_{1,+}+n^{++}_{1,-}=n^{++}_{0,+}+n^{++}_{2,-}$ is,\nhere, equal to $4$ so that, correspondingly, the motor spent time in\nthe substate $(1)$ on four occasions before undergoing the forward\nmajor transition $(3)\\!\\rightarrow\\!(4\\equiv0)$. Similarly, the\nother plots serve to establish the frequency\nrelations~(\\ref{DefFrequencyJ1})-(\\ref{FrequencyRMiMi}). }\n\\end{figure*}\n\nThe frequency relations~(\\ref{DefFrequencyJ1})\nand~(\\ref{DefFrequencyJ2}) with the boundary\nrelations~(\\ref{FrequencyLPlPl})-(\\ref{FrequencyRMiMi}) constitute,\ntogether with the definition\n$\\omega_j^{\\rho\\sigma}=(u_j+w_j)T_j^{\\rho\\sigma}$\nin~(\\ref{DefFrequencyJ1}), a complete set from which we may derive\nexplicit general expressions for all the $n_{j,+}^{\\rho\\sigma}$ and\n$n_{j,-}^{\\rho\\sigma}$. To proceed, we note first that the purely\ncounting arguments involving $\\pi_+$, $\\pi_-$, and $\\pi_{++}$,\n$\\pi_{--}$, etc., that led to the `reduced'\nrelations~(\\ref{StepDwlInPairDwlPl},\\ref{StepDwlInPairDwlMi})\nand~(\\ref{PostDwlInPairDwlPl},\\ref{PostDwlInPairDwlMi}), apply\nequally to $\\omega_j^{++}$, $\\omega_j^{+-}$, etc. Accordingly, we\nobtain the {\\it post} boundary frequency relations\n\\begin{eqnarray}\n  \\omega_{M+1}^{+\\diamond} &=& n_{M+2,-}^{+\\diamond}+1 = n_{M+1,+}^{+\\diamond}+\\pi_{+-},\\label{FrequencyLPlD}\\\\\n  \\omega_{M+N}^{+\\diamond} &=& n_{M+N-1,+}^{+\\diamond} = n_{M+N,-}^{+\\diamond}+\\pi_{++},\\label{FrequencyRPlD}\\\\\n  \\omega_{M+1}^{-\\diamond} &=& n_{M+2,-}^{-\\diamond} = n_{M+1,+}^{-\\diamond}+\\pi_{--},\\label{FrequencyLMiD}\\\\\n  \\omega_{M+N}^{-\\diamond} &=& n_{M+N-1,+}^{-\\diamond}+1 = n_{M+N,-}^{-\\diamond}+\\pi_{-+},\\label{FrequencyRMiD}\n\\end{eqnarray}\nwhile the {\\it prior} relations are\n\\begin{eqnarray}\n  \\omega_{M+1}^{\\diamond+} &=& n_{M+1,+}^{\\diamond+} = n_{M+2,-}^{\\diamond+}+\\pi_{++},\\label{FrequencyLDPl}\\\\\n  \\omega_{M+N}^{\\diamond+} &=& n_{M+N,-}^{\\diamond+}+1 = n_{M+N-1,+}^{\\diamond+}+\\pi_{+-},\\label{FrequencyRDPl}\\\\\n  \\omega_{M+1}^{\\diamond-} &=& n_{M+1,+}^{\\diamond-}+1 = n_{M+2,-}^{\\diamond-}+\\pi_{-+},\\label{FrequencyLDMi}\\\\\n  \\omega_{M+N}^{\\diamond-} &=& n_{M+N,-}^{\\diamond-} = n_{M+N-1,+}^{\\diamond-}+\\pi_{--},\\label{FrequencyRDMi}\n\\end{eqnarray}\nFinally, in analogy to~(\\ref{OverallDwlTime}), we obtain\n\\begin{eqnarray}\n  \\omega_{M+1}^{\\diamond\\diamond} &=& n_{M+2,-}^{\\diamond\\diamond}+\\pi_+ = n_{M+1,+}^{\\diamond\\diamond}+\\pi_-,\\label{FrequencyLDD}\\\\\n  \\omega_{M+N}^{\\diamond\\diamond} &=& n_{M+N-1,+}^{\\diamond\\diamond}+\\pi_- = n_{M+N,-}^{\\diamond\\diamond}+\\pi_+.\\label{FrequencyRDD}\n\\end{eqnarray}\n\nNow~(\\ref{FrequencyLDD}) clearly implies the result\n\\begin{equation}\\label{nM1DD}\n    n_{M+1,+}^{\\diamond\\diamond}=\\omega_{M+1}^{\\diamond\\diamond}-\\pi_-=(u_j+w_j)T_{M+1}^{\\diamond\\diamond}-\\pi_-,\n\\end{equation}\nwhere $\\pi_-$ and $T_{M+1}^{\\diamond\\diamond}\\equiv\nT_{M+1}^{\\diamond\\diamond}\\!\\left(\\{u_j,w_j\\};M,N\\right)$ are given\nexplicitly in~(\\ref{StepSplProbMi}) and~(\\ref{ReDefQ}). A similar\nresult follows from~(\\ref{FrequencyLDD}) for\n$n_{M+2,-}^{\\diamond\\diamond}$. But by appealing\nto~(\\ref{DefFrequencyJ2}) for $j=M+2$ we also obtain\n\\begin{equation}\\label{nM3DD}\n    n_{M+3,-}^{\\diamond\\diamond}=\\omega_{M+2}^{\\diamond\\diamond}-n_{M+1,+}^{\\diamond\\diamond},\n\\end{equation}\nwhich, with the aid of~(\\ref{nM1DD}), yields an explicit expression\nin terms of $T_{M+1}$ and $T_{M+2}$. Substituting $j=M+3$\nin~(\\ref{DefFrequencyJ1}) then leads to an expression for\n$n_{M+3,+}^{\\diamond\\diamond}$. By proceeding recursively in this\nfashion we eventually obtain, for $k=1,2,\\cdots$, the general\nexpressions\n\\begin{align}\n  &n_{M+2k,-}^{\\diamond\\diamond}=\\sum_{j=M+1}^{M+2k-1}(-)^{j-M-1}(u_j+w_j)T_j^{\\diamond\\diamond}-\\pi_+,\\label{nDDeven}\\\\\n  &n_{M+2k+1,-}^{\\diamond\\diamond}=\\sum_{j=M+1}^{M+2k}(-)^{j-M}(u_j+w_j)T_j^{\\diamond\\diamond}+\\pi_-,\\label{nDDodd}\n\\end{align}\nfor the average number of hidden {\\it back-substeps} from the $N-2$\nhidden intermediate states $(M+2)$ to $(M+N-1)$ and from the\npre-step state $(M+N)\\equiv(M)$. We may also note the sum rule\n\\begin{equation}\\label{SumRule}\n    \\sum_{l=0}^{N-1}(-)^l(u_l+w_l)T^{\\diamond\\diamond}_l=\\left\\{\\begin{array}{cccc}\n        0 & \\mbox{for} & N & \\mbox{even},\\\\\n        1 & \\mbox{for} & N & \\mbox{odd},\n        \\end{array}\\right.\n\\end{equation}\nand the special relations $\\pi_+=u_{M+N}T_{M+N}^{\\diamond\\diamond}$\nand $\\pi_-=w_{M+1}T_{M+1}^{\\diamond\\diamond}$, so that\n$n_{M+1,+}^{\\diamond\\diamond}=u_{M+1}T_{M+1}^{\\diamond\\diamond}$ and\n$n_{M+N,-}^{\\diamond\\diamond}=w_{M+N}T_{M+N}^{\\diamond\\diamond}$.\n\nMore generally~(\\ref{nDDeven}) and~(\\ref{nDDodd}) can be extended to\narbitrary $(\\rho,\\sigma)$ if, following the\nsequence~(\\ref{9combinations}), $\\pi_+$ and $\\pi_-$ are replaced by\n\\begin{eqnarray}\n  &&\\tilde{\\pi}_+=\\{\\pi_+;\\;1,0,\\pi_{++},\\pi_{-+};\\;1,1,0,0\\},\\label{PiSetPl}\\\\\n  &&\\tilde{\\pi}_-=\\{\\pi_-;\\;\\pi_{+-},\\pi_{--},0,1;\\;0,1,0,1\\},\\label{PiSetMi}\n\\end{eqnarray}\nrespectively. Finally, one finds that the mean number of {\\it\nforward substeps} from the $N-2$ hidden intermediate states and the\npost-step state $(M+1)$ in specified intervals can be written\n\\begin{align}\n  &\\!n_{M+2k,+}^{\\rho\\sigma}=\\!\\!\\!\\sum_{j=M+1}^{M+2k}\\!\\!\\!(-)^{j-M}(u_j+w_j)T_j^{\\rho\\sigma}+\\tilde{\\pi}_+,\\label{nRSeven}\\\\\n  &\\!n_{M+2k-1,+}^{\\rho\\sigma}=\\!\\!\\!\\sum_{j=M+1}^{M+2k-1}\\!\\!\\!(-)^{j-M-1}(u_j+w_j)T_j^{\\rho\\sigma}-\\tilde{\\pi}_-,\\label{nRSodd}\n\\end{align}\nfor $k=1,2,\\cdots$. Some examples of these various expressions for\nsmall $N$ are listed in Appendix~A.\n\n\\section{\\label{sec:level4}Discussion and illustrations}\nFirst, it is appropriate to look more closely at the difference\nbetween our present results\n[see~(\\ref{StepSplProbPl}),(\\ref{StepSplProbMi}), and\n(\\ref{DwelPl})-(\\ref{DwelTot})] and those of the previously\npublished analysis~\\cite{kolomeisky03,kolomeisky05}. As in\nSec.~\\ref{sec:level1} [after~(\\ref{MajorMechStep})] we use a\nha$\\check{\\textrm{c}}$ek to distinguish the results that presuppose\nthe completion of a full enzymatic cycle between all pairs of\nsuccessive major, i.e., observable steps. In the present notation\none then has~\\cite{kolomeisky05}\n\\begin{equation}\n   \\check{\\pi}_+=\\frac{1}{1+\\Gamma_N},\\qquad\\check{\\pi}_-=\\frac{\\Gamma_N}{1+\\Gamma_N},\\label{SplOld}\n\\end{equation}\nfor the forward\/back fractions while the mean, prior, and post dwell\ntimes are all given by\n\\begin{equation}\n   \\check{\\tau}=\\check{\\tau}_+=\\check{\\tau}_-=\\sum_{n=0}^{N-1}r_n\\left\/(1+\\Gamma_N)\\right.,\\label{DwelOld}\n\\end{equation}\nwhere $r_n$ is defined in~(\\ref{TransitionTimes}). On the other\nhand, when allowing for hidden substeps, these three times are all\ndistinct: see~(\\ref{DwelPl})-(\\ref{DwelTot}).\n\nThen, if one considers the average velocity defined by\n\\begin{equation}\n   V=d(\\pi_+-\\pi_-)\/\\tau,\\label{DefVelocity}\n\\end{equation}\none finds that the result is the same in both cases, namely,\n\\begin{equation}\n   V=\\check{V}=d(1-\\Gamma_N)\\!\\!\\left\/\\mbox{$\\sum_{n=0}^{N-1}$}r_n\\right.,\\label{VelNew}\n\\end{equation}\nas\npreviously~\\cite{fisher99,kolomeisky03,fisher05,fisher99a,kolomeisky00}.\nThis is not really surprising, because the asymptotic ratio of the\ntotal distance to the total time should not depend on the way one\ntakes into account (or ignores) substeps. Note that $\\Gamma_N=1$ is\nthe condition for stall, when $\\pi_+=\\pi_-$, in both analyses.\n\nNow let us take a closer look at the ratio of splitting fractions,\nbecause this is a quantity which one can readily obtain from\nstepping observations to test theory. The effect of an external\nforce, $\\vec{F}=(F_x,F_y,F_z)$, on the rates can be expressed in\nleading order~\\cite{fisher99,kim05} as\n\\begin{eqnarray}\n&&u_l=u^0_l\\exp(+d\\mbox{\\boldmath $\\theta$}^{+}_{l}\\mbox{\\boldmath $\\cdot$}\\,{\\bf F}\/k_BT),\\label{UexpF}\\\\\n&&w_l=w^0_l\\exp(-d\\mbox{\\boldmath $\\theta$}^{-}_{l}\\mbox{\\boldmath\n$\\cdot$}\\,{\\bf F}\/k_BT),\\label{WexpF}\n\\end{eqnarray}\nwhere $u^0_l,w^0_l$ are the load-free rates while the\nload-distribution vectors $\\mbox{\\boldmath $\\theta$}^+_l$ and\n$\\mbox{\\boldmath $\\theta$}^-_l$ serve to specify how the $2N$\ndistinct rates respond to the stress. The periodicity of the\nstepping along the track (which we suppose is in the $x$ direction)\nimplies\n\\begin{equation}\n\\sum_{n=0}^{N-1}\\left(\\mbox{\\boldmath\n$\\theta$}^{+}_{n}+\\mbox{\\boldmath $\\theta$}^{-}_{n}\\right)=\\hat{\\bf\ne}_x=(1,0,0).\\label{ThetaPerCond}\n\\end{equation}\nRearranging these relations leads to\n\\begin{eqnarray}\n\\Gamma_N&=&\\prod_{n=0}^{N-1}\\frac{w_n}{u_n}=\\exp\\left(-\\frac{dF_x}{k_BT}\\right)\\prod_{n=0}^{N-1}\\frac{w^0_n}{u^0_n}\\nonumber\\\\\n&=&\\exp[-d(F_x-F_{S})\/k_BT],\\label{GammaFstall}\n\\end{eqnarray}\nwhere the stall force is given by~\\cite{fisher99,kim05}\n\\begin{equation}\nF_{S}=\\frac{k_BT}{d}\\ln\\left(\\prod_{n=0}^{N-1}\\frac{w^0_n}{u^0_n}\\right).\\label{DefFstall}\n\\end{equation}\nFrom (\\ref{SplOld}) we thus conclude\n\\begin{equation}\n\\ln(\\check{\\pi}_+\/\\check{\\pi}_-)=-\\ln\\Gamma_N=d(F_x-F_{S})\/k_BT.\\label{LogGamma}\n\\end{equation}\n\nThis is a strong prediction since it asserts that the logarithm of\nthe stepping ratio $(\\check{\\pi}_+\/\\check{\\pi}_-)$ depends linearly\non $F_x$ with a slope, $d\/k_BT$, determined solely by the step size\n$d$. However, this result depends crucially on the assumption that\nthe forward and back steps identified correspond to full enzymatic\ncycles (as the haceks indicate). In a typical experiment, however,\n the hidden substeps result in a violation of\nthis assumption as we have explained.\n\nAccordingly, let us, instead, compute the stepping fraction\n$(\\pi_+\/\\pi_-)$ for which~(\\ref{StepSplProbPl})\nand~(\\ref{StepSplProbMi}) yield\n\\begin{equation}\n\\ln(\\pi_+\/\\pi_-)=d^{*}(F_x-F_S)\/k_BT\\label{LogStepSplProb},\n\\end{equation}\nwhere, for the sake of comparison with~(\\ref{LogGamma}), we have\nintroduced an {\\it effective step size}\n\\begin{equation}\nd^{*}=\\frac{-k_BT}{F_x-F_{S}}\\ln\\left(\\frac{\\Gamma_N+\\Delta_{M,N}}{1+\\Delta_{M,N}}\\right).\\label{DefdEffect}\n\\end{equation}\nIn general, this clearly depends on all the rates $\\{u_j,w_j\\}$ and\nhence on the force $F_x$. In the special case $\\Delta_{M,N}=0$,\nhowever, $d^*$ reduces [via~(\\ref{LogGamma})] to $d$. This condition\nis, in fact, realized when $N=1$ since $\\Delta_{M,1}$ vanishes\nidentically. But an $N=1$ model is unlikely to be adequate. Thus in\nreal systems neither the full linearity vs. $F_x$ nor the equality\n$d^*=d$ are to be expected. In the vicinity of $F_S$, however, we\ncan estimate $d^*$ by expanding in powers of $\\delta F_x\\equiv\nF_x-F_S$. This yields\n\\begin{eqnarray}\n&&\\Gamma_N=1-d\\,\\delta F_x\/k_BT+O(\\delta F_x^2),\\label{GammaExpand}\\\\\n&&\\Delta_{M,N}=\\Delta_S+\\Delta'_S\\delta F_x+O(\\delta\nF_x^2),\\label{DeltaExpand}\n\\end{eqnarray}\nwhere $\\Delta_S=\\Delta_{M,N}(F_x\\!\\!\\,=\\!F_S)$ while $\\Delta'_S$ is\nthe corresponding derivative. Thence we find\n\\begin{equation}\nd^{*}=\\frac{d}{1+\\Delta_S}+O(\\delta F_x).\\label{dEffectExpand}\n\\end{equation}\nIt follows from the definition~(\\ref{DefDelta}) that $\\Delta_S$\ncannot be negative so that, quite generally, one has $d^*\\leqslant\nd$.\n\nOn the other hand, to be concrete, consider an ($N=2$, $M=1$) model\nfor which we have\n\\begin{align}\n\\!\\!\\Delta_S&=(w_0\/u_0)_S=(u_1\/w_1)_S\\nonumber\\\\\n&=(u^0_1\/w^0_1)\\exp\\{d[F_{S}-(\\mbox{\\boldmath\n$\\theta$}^{+}_0+\\mbox{\\boldmath $\\theta$}^{-}_0)\\,\\mbox{\\boldmath\n$\\cdot$}\\,{\\bf F}_S]\/k_BT\\},\\label{dDelta0forN2}\n\\end{align}\nwhere the subscript $S$ denotes evaluation at ${\\bf\nF=F}_S=(F_x\\!\\!=\\!F_S,F_y,F_z)$ in which $F_y$ and $F_z$ need not\nvanish at stall~\\cite{kim05}. It is evident, that $\\Delta_S$ is not\nbounded above so that $d^*\/d$ is not bounded below. Indeed, the\nexperiments on kinesin of Nishiyama {\\it et al.}~\\cite{nishiyama03}\nand of Carter and Cross on kinesin~\\cite{carter05} lead to the\nestimates $d^{*}\\simeq 3.2\\;\\mathrm{nm}$ and $d^{*}\\simeq\n4.0\\;\\mathrm{nm}$, respectively, whereas the step size is $d\\simeq\n8.2\\;\\mathrm{nm}$. This clearly indicates the importance of the\nhidden substeps in understanding the stepping fractions near stall.\n\nAs another concrete example we quote\n\\begin{eqnarray}\n\\Delta_S&=&\\left(\\frac{w_0}{u_0}+\\frac{w_0w_1}{u_0u_1}+\\frac{w_0w_1w_2}{u_0u_1u_2}\\right)_S\\nonumber\\\\\n&=&\\left(\\frac{u_1u_2u_3}{w_1w_2w_3}+\\frac{u_2u_3}{w_2w_3}+\\frac{u_3}{w_3}\\right)_S,\n\\end{eqnarray}\nfor an ($N=4$, $M=3$) model. From this, however, one sees that $d^*$\nwill be close to $d$ if $(u_3\/w_3)_S$ and\/or $(w_0\/u_0)_S$ are\nsufficiently small.\n\n\\begin{figure*}\n\\includegraphics[angle=0, scale=0.70]{PapeFig3Csm}\n\\caption{\\label{fig3:wide}Plots illustrating different dependencies\non the normalized force, $F\/|F_S|$, of {\\bf (a)} the normalized\nvelocity $V(F)\/V(0)$; {\\bf (b)} the forward splitting fraction\n$\\pi_+$; {\\bf (c)} the logarithm of the splitting fraction ratio\n$(\\pi_+\/\\pi_-)$; and {\\bf (d)} the ratio of mean prior backward to\nmean total dwell time, $\\tau_-(F)\/\\tau(F)$. The individual plots,\nlabeled (1) to (5), correspond to ($N=2$, $M=1$) models under scalar\nloading, $\\textbf{F}\\!=\\!(F,0,0)$, with the selected parameter\nvalues presented in Table~I.  }\n\\end{figure*}\n\nMore generally it seems likely that both the full cycle assumption\nand the hidden substep analysis should produce similar results when\nthe rates, $u_M$ and $w_{M+1}$, for the major transition are slow\nrelative to the substep rates. To demonstrate this explicitly let us\nsuppose that the latter rates satisfy\n\\begin{equation}\nu_M=\\eta\\,\\tilde{u}_M\\quad\\mbox{and}\\quad w_{M+1}=\\eta\\,\\tilde{w}_{M+1},\\label{DefTildes}\\\\\n\\end{equation}\nwhere $\\eta$ is small while $\\tilde{u}_M$ and $\\tilde{w}_{M+1}$ and\nall the substep rates, $u_l\\equiv\\tilde{u}_l$ $(l\\neq M)$ and\n$w_l\\equiv\\tilde{w}_l$ $(l\\neq M+1)$, are held fixed. Then one finds\n\\begin{eqnarray}\n\\Gamma_{M,s}=\\prod_{n=1}^{s}\\frac{w_{M+n}}{u_{M+n}}&=&\\eta\\,\\tilde{\\Gamma}_{M,s}\\quad\\mbox{for}\\quad\ns\\in[1,N-1],\\nonumber\\\\\n&=&\\tilde{\\Gamma}_N=\\Gamma_N\\quad\\mbox{for}\\quad s=N,\\label{TildeGs}\n\\end{eqnarray}\nwhere\n$\\tilde{\\Gamma}_{M,s}=\\prod_{n=1}^{s}(\\tilde{w}_{M+n}\/\\tilde{u}_{M+n})$,\nand similarly, via~(\\ref{DefDelta}),\n\\begin{equation}\n\\Delta_{M,N}=\\eta\\sum_{p=1}^{N-1}\\tilde{\\Gamma}_{M,p}\\equiv\\eta\\tilde{\\Delta}_{M,N}.\\label{TildeDelta}\n\\end{equation}\nWith the aid of these expressions we can rewrite the previous\nresults~(\\ref{StepSplProbPl},\\ref{StepSplProbMi})\nand~(\\ref{DwelTot}) as\n\\begin{eqnarray}\n&&\\pi_+=\\frac{1+\\eta\\tilde{\\Delta}_{M,N}}{1+\\tilde{\\Gamma}_N+2\\eta\\tilde{\\Delta}_{M,N}}=1-\\pi_-,\\\\\n&&\\tau=\\check{\\tau}\\frac{1+\\tilde{\\Gamma}_N}{1+\\tilde{\\Gamma}_N+2\\eta\\tilde{\\Delta}_{M,N}}.\n\\end{eqnarray}\nEvidently to zero order in $\\eta$, the analyses are equivalent as\nanticipated.\n\nAs seen in earlier investigations that were confined to the velocity\nvs. force relation~\\cite{fisher99,fisher99a}, a surprisingly wide\nrange of behavior under varying loads is displayed even by the basic\n$(N=2)$-state models when the rates are subject to the exponential\nforce distribution laws embodied\nin~(\\ref{UexpF}-\\ref{ThetaPerCond})~\\cite{fisher99,kim05}. This is\nillustrated in Fig.3(a) which depicts the velocity $V$, normalized\nby its value $V(0)$ under zero load, as a function of the imposed\nload $F$ (supposed parallel to the $x$-axis) normalized by the stall\nforce magnitude $|F_S|$. The labeled plots (1) to (5) in\nFig.~\\ref{fig3:wide} correspond to the selected parameter values\nlisted in Table~I. Note that {\\it superstall loads} are included\n($F\/|F_S|<-1$) which for the parameter sets (1) and (5) results in\nonly a relatively small negative velocity (as uncovered in the\nkinesin experiments of Carter and Cross~\\cite{carter05}). Similarly,\n{\\it assisting loads} ($F\/|F_S|>0$) are also covered and for sets\n(2), (4), and (5) result in a saturating or, even, {\\it decreasing}\nvelocity under increasing load. In the {\\it substall} resistively\nloaded region ($0>F\/|F_S|>-1$) convex, concave, inflected, and even\nnonmonotonic [see parameter set (5)] behavior is realized.\n\nFor these different cases the corresponding forward stepping\nfractions $\\pi_+(F)$ and the logarithmic ratios of forward\/back\nsteps, ($\\pi_+\/\\pi_-$), are depicted in Figs.~3(b) and (c). [Note\nthat in these figures only the {\\it resisting} range of force,\n$F<0$, is displayed.] Although the variation is always monotonically\nincreasing with $F$ (and $\\pi_+=\\pi_-=\\frac{1}{2}$ when\n$F\\!=\\!F_S$), a wide range of forms is evident. In the logarithmic\nplot, Fig.3(c), one sees linear, concave, and inflected variation\nclose to stall. Furthermore, the value of the effective step size\n$d^*$ varies markedly: see the last column in Table~I.\n\nBy using the explicit expressions in the Appendix the nature of\nother statistical observables such as $\\pi_{+-}(F)$, etc., is\nreadily explored. Experiments often measure the overall mean dwell\ntime, $\\tau(F)$, between steps. Under assisting loads $(F>0)$, when\n$\\pi_-$ is negligible, $\\tau(F)$ directly mirrors the reciprocal of\nthe velocity $V(F)$; but, in view of the factor $(\\pi_+-\\pi_-)$\nin~(\\ref{DefVelocity}), it varies somewhat differently under\nresisting loads. More interesting is the behavior of the {\\it\npartial dwell time} $\\tau_-(F)$ observed prior to back steps. This\nis shown in Fig.~3(d) normalized by the overall dwell time\n$\\tau(F)=\\pi_+\\tau_++\\pi_-\\tau_-$ [see~(\\ref{TotalDwellasSum})].\nEven though, the ratio $\\tau_-\/\\tau$ is confined to the range\n$(0,2)$ beyond superstall (since $\\tau_-\/\\tau<1\/\\pi_-$ and\n$\\pi_->\\frac{1}{2}$ for $V<0$), striking nonmonotonic and inflected\nvariation arises.\n\nNeedless-to-say, many more plots exhibiting unexpected and\nsurprising behavior can be generated; but further exploration seems\nmost useful in connection with specific experimental data. Such\napplications are planned.\n\n\\begin{table}\n\\caption{Parameter values for ($N=2$, $M=1$) models under scalar\nloading, $F\\equiv F_x$, employed in Fig.~\\ref{fig3:wide} to\nillustrate different force-dependencies. The last column presents\nthe computed values of the effective step size $d^*$.\n[See~(\\ref{DefdEffect}).]}\n\\begin{tabular}{|c|c|c|c|c|c|c|}\n  \\hline\n \n    & $d|F_S|\/k_BT$ & $w^0_0\/u^0_0$ & $w^0_1\/u^0_0$ & $\\theta^+_1$ & $\\theta^-_0=\\theta^-_1$ & $d^*\/d$ \\\\\\hline\n  (1) & 9.2 & 10$^{-2}$ & 10$^{-2}$ & 0.5 & 0 & 0.501 \\\\\n  (2) & 2.5 & 10$^{-2}$ & 10$^{-2}$ & 0 & 0.5 & 0.966 \\\\\n  (3) & 23 & 10$^{-5}$ & 10$^{-5}$ & 0.07 & 0.43 & 0.503 \\\\\n  (4) & 23 & 10$^{-5}$ & 10$^{-5}$ & $-$0.07 & 0.48 & 0.113 \\\\\n  (5) & 10 & 3.4$\\times$10$^{-4}$ & 2.5$\\times$10$^{-3}$ & $-$0.1 & 0.1 & 0.118 \\\\\n  \\hline\n\\end{tabular}\n\\end{table}\n\n\n\\section{\\label{sec:level5} Multiple observed transitions}\n\nIn the previous sections we have derived expressions only for the\ncase of a {\\it single} major transition in each enzymatic cycle;\nthat, indeed, is the typical situation for experiments on\nconventional\nkinesin~\\cite{nishiyama02,block03,nishiyama03,carter05}. However,\nour results can be generalized to the case in which several substeps\nare sufficiently large to be clearly detected, while others remain\nhidden in the noise. Suppose there are $K$ visible substeps of\n(average) magnitudes $d_A$, $d_B$, $\\cdots$, together totalling\n\\begin{equation}\nd_A+d_B+\\dots+d_K\\simeq d,\\label{SumVisibleStep}\n\\end{equation}\nthat occur between states ($M_J$) and ($M_{J}+1$) with, in sequence,\n\\begin{equation}\n0\\leq M_A<M_B<\\dots<M_K\\leq N-1.\\label{SequenceOfSteps}\n\\end{equation}\nThen between states ($M_{J-1}+1$) and ($M_{J}$) there are\n$M_{J}-M_{J-1}$ hidden states and, consequently, all these states\nbelong (within the noise) to what we may call the same {\\it\nmechanical level}, $J$: see Fig.~\\ref{fig4:wide}.\n\n\\begin{figure*}\n\\includegraphics[angle=0, scale=0.9]{PapeFig4A}\n\\caption{\\label{fig4:wide} Schematic depiction of individual\nbiochemical states, $(i)$, organized into mechanical levels, $A$,\n$B$, $\\cdots$, $J-1$, $J$, $\\cdots$ when there are $K>1$ detectable\nsubsteps or major transitions with substep $J$ going forward from\nstate $(M_J)$ to state $(M_J+1)$.}\n\\end{figure*}\n\nNow one can count all forward and backward detectable transitions in\na long run. Accordingly, let $n^J_{++}$ be the number of pairs of\nobserved substeps that enter the mechanical level $J$ via a $+$\ntransition, i.e., a step ($J-1$), and leave via a $+$ transition\ntaking a step $J$, and similarly for $n^J_{+-}$, $n^J_{-+}$, and\n$n^J_{--}$. Then, as previously, we can estimate pairwise splitting\nfractions for level $J$ via\n\\begin{eqnarray}\n\\pi^J_{++}&=&\\frac{n^J_{++}}{n^J_{++}+n^J_{+-}}=1-\\pi^J_{+-},\\\\\n\\pi^J_{-+}&=&\\frac{n^J_{-+}}{n^J_{-+}+n^J_{--}}=1-\\pi^J_{--}.\n\\end{eqnarray}\nLikewise, we can introduce pair-wise dwell times,\n$\\tau^J_{\\rho\\sigma}$, for each mechanical level $J$ as the mean\ntimes spent between a $\\rho$ and $\\sigma$ substep into and out of\nthat level.\n\nWith these definitions we may adopt the same approach used in\nSec.~\\ref{sec:level3} by setting absorbing boundaries at states\n$(L)=(M_{J-1})$ and $(R)=(M_J+1)$ around each level $J$ and studying\nthe appropriate first-passage processes. If we set\n\\begin{align}\nN_J=\\;&|M_A\\!-\\!M_B|,\\;|M_B\\!-\\!M_C|,\\;\\cdots,\\nonumber\\\\\n&|M_K\\!-\\!M_{K-1}|,\\;|N\\!-\\!M_K\\!+\\!M_A|,\\nonumber\\\\\n&\\mbox{for}\\quad J=A,\\;B,\\;\\cdots,\\;K-1,\\;K,\n\\end{align}\nwe can then conclude, using the previous results\n(\\ref{SplProbPlPl},\\ref{SplProbMiPl}), that\n\\begin{equation}\n\\pi^J_{\\rho\\sigma}=\\pi_{\\rho\\sigma}(M_{J-1},N_{J-1}).\n\\end{equation}\nSimilarly, recalling the definitions\n(\\ref{SolutionQPl},\\ref{SolutionQMi}) for $T^+_j(M,N)$ and\n$T^-_j(M,N)$ and the results (\\ref{DwelPlPl}-\\ref{DwelMiMi}), we\nobtain\n\\begin{equation}\n\\tau^J_{\\rho\\sigma}=\\tau_{\\rho\\sigma}(M_{J-1},N_{J-1}).\n\\end{equation}\n\nThe counting of individual hidden substeps developed in\nSec.~\\ref{sec:level3e} can be carried forward to obtain the mean\nnumber of substep transitions between two specified major\ntransitions. The results (\\ref{nDDeven}-\\ref{nRSodd}) essentially\napply directly with $M\\Rightarrow M_{J-1}$ and $N\\Rightarrow\nN_{J-1}$.\n\nIn terms of the {\\it conditional individual state dwell times}\nintroduced in (\\ref{DefTPlPl}-\\ref{DefTPlMi}) we also have\n\\begin{equation}\n\\tau^J_{\\rho\\sigma}=\\sum_{k=M_{J-1}+1}^{M_J}T_k^{\\rho\\sigma}(M_{J-1},N_{J-1}).\n\\end{equation}\nas a measure of the observable mean overall time spent in the\nmechanical level $J$ subject to the ($\\rho,\\sigma$) conditions.\n\nAs regards potential applications of these results, the case $K=2$\nmay be reasonable for a first analysis of data for myosin V where,\nas mentioned, a significant observable substep was originally\npredicted~\\cite{kolomeisky03} and later observed~\\cite{uemura04};\nhowever, the experiments also suggest~\\cite{uemura04,baker04} that\nstepping may proceed through two (or more) alternative pathways so\nthat a purely sequential model (to which our attention has been\nrestricted) may be inadequate~\\cite{foot0A}. For the F$_1$-ATPase\nmotor~\\cite{nishizaka04,shimabukuro03,ueno05} substeps have also\nbeen reported and occasional back steps have been observed. Thus our\nresults should be applicable.\n\n\\section{\\label{sec:level6}Summary and conclusion}\n\nAs explained in the Introduction, the need to develop the hidden\nsubstep analysis we have presented arises from the fact that\nexperimentally detectable steps in the motion of a motor protein\nalong its track do not necessarily delineate the completion of full\nbiochemical enzymatic cycles. As a consequence, previous analyses\nthat addressed such observable statistics as back-stepping\nfractions, $\\pi_-$, and mean dwell times, $\\tau_+$ and $\\tau_-$,\nmeasured prior to forward (or $+$) and back (or $-$) steps, were not\nadequate to relate the underlying rates in a biomechanochemical\nmodel, say $u_i$ and $w_j$ to the experimental data.\n\nWe have considered the basic $N$-state sequential kinetic model set\nout in~(\\ref{BasicModel}) and specified by $N$ forward rates $u_i$\nfrom biochemical state $(i)$ and $N$ reverse rates $w_j$ from state\n$(j)$. In general, the rates depend on the concentration of various\nreagents [see, e.g.,~(\\ref{ATPdepend})] and, in particular, vary\nexperimentally with the load force ${\\bf F}=(F_x,F_y,F_z)$:\nsee~(\\ref{UexpF},\\ref{WexpF}).\n\nThe basic problem may then be set up by supposing that as the motor\nprogresses (or retrogresses) along its molecular track only a single\n``major transition'' from state $(M)$ to $(M+1)$ $\\:(0\\leq M<N)$, or\nits reverse, corresponds to a ``visible'' or detectable ``step'' in\nthe $N$-state cycle. All the other transitions are ``hidden'': see\nFig.~\\ref{fig:wide}. [Cases in which more than one major transition\nor observable (sub)step occur in each full enzymatic cycle are\nanalyzed in Sec.~\\ref{sec:level5}.] It then transpires that two\ncrucial combinations of the rates $\\{u_i,w_j\\}$  play a central\nrole, namely, $\\Gamma_N$ and $\\Delta_{M,N}$ as defined in\n(\\ref{DefGamma})-(\\ref{DefDelta}).\n\nIndeed, explicit expressions for the forward and backward stepping\nfractions, $\\pi_+$ and $\\pi_-$, are derived in terms of $\\Gamma_N$\nand $\\Delta_{M,N}$ in Sec.~\\ref{sec:level3b} and presented\nin~(\\ref{StepSplProbPl}) and (\\ref{StepSplProbMi}). It proves\nhelpful, furthermore, to relate $\\pi_+$ and $\\pi_-$ to the {\\it\nconditional} or {\\it pairwise step probabilities}, $\\pi_{++}$,\n$\\pi_{+-}$, etc., for $+$ steps followed by a $+$ step, or by a $-$\nstep, etc., that can be defined via counting observations, as\nexplained in~(\\ref{PairwiseSplNorm})-(\\ref{StepSplInPairSplMi}) and\nthe associated text. These pairwise probabilities are likewise\nexpressible in terms of $\\Gamma_N$ and $\\Delta_{M,N}$:\nsee~(\\ref{SplProbPlPl}) and (\\ref{SplProbMiPl}).\n\nFrom these results one can see -- as explained in further detail in\nSec.~\\ref{sec:level4} -- that only when\n$\\;\\Delta_{M,N}\\ll\\min\\{1,\\Gamma_N\\}\\;$ can one neglect the hidden\nsubsteps without risk of serious error. Particularly instructive is\nthe variation of $\\;\\ln(\\pi_+\/\\pi_-)\\;$ as the load $F_x$ passes\nthrough stall (at which $\\;\\pi_+=\\pi_-=\\frac{1}{2}\\;$ so that the\nmean velocity vanishes). One may then define an {\\it effective step\nlength}, $d^*$, via~(\\ref{LogStepSplProb}) and (\\ref{DefdEffect}).\nWhen $\\Delta_{M,N}\\rightarrow 0$ (or if hidden substeps are ignored)\none has the simple {\\it equality} $\\;d^*=d$, where $d$ is the full\nstep size of the motor (per cycle). But, in fact, $d^*\/d$ must\nalways be {\\it less} than unity and, as seen in experiment and\nillustrated in Table~I, this ratio is typically of magnitude only\n0.3 to 0.5.\n\nGoing beyond simple counts of backward and forward steps, one may\nalso define {\\it conditional mean dwell times}, $\\tau_{++}$,\n$\\tau_{+-}$, etc., for the time spent between a pair of successive\n$+$ steps, or a $+$ step followed by a $-$ step, etc.:\nsee~(\\ref{DefPairwiseDwells}). These pairwise mean times can\nlikewise be calculated [see~(\\ref{DwelPlPl}) to (\\ref{DwelMiMi})] in\nterms of {\\it individual-state post dwell times}, $T_j^+(\\equiv\nT_j^{+\\diamond})$ and $T_j^-(\\equiv T_j^{-\\diamond})$, that\nrepresent the mean time spent in a state $(j)$ following a $+$ or\n$-$ step, respectively: see Sec.~\\ref{sec:level3d} for a fuller\nexplanation of the notation, etc. The corresponding explicit\nexpressions, (\\ref{SolutionQPl}) and (\\ref{SolutionQMi}), involve\nthe basic rate products $\\Gamma_{l,m}$ and their sums,\n$\\Delta_{l,n}$, as again defined in (\\ref{DefGamma}) and\n(\\ref{DefDelta}). The final results for $\\tau_+$, $\\tau_-$, and for\nthe overall mean dwell time $\\tau$ (between $+$ or $-$ steps) for\ngeneral $M$ and $N$ entail slightly simpler sums:\nsee~(\\ref{ReDefQ})-(\\ref{DwelTot}).\n\nMore transparent formulae for the stepping fractions and dwell times\nfor $N=2$ models (involving only the rates $u_0$, $u_1$, $w_0$, and\n$w_1$,) and for selected $N=4$ models, are presented in Appendix~A.\nIn addition, the parts of Fig.~\\ref{fig3:wide} and the associated\ndiscussion in Sec.~\\ref{sec:level4}, illustrate that a wide range of\ndifferent types of behavior of $\\pi_+(F)$, $\\ln[\\pi_+(F)\/\\pi_-(F)]$,\nand $\\tau_-(F)$ as functions of load can be realized even within\nsimple $N=2$ models.\n\nAt a higher level of detail, conditional individual-state dwell\ntimes, $T_j^{++}$, $T_j^{+-}$, $\\cdots$, can be derived\n[see~(\\ref{DefTPlPl})-(\\ref{DefTPlMi})] and, likewise, {\\it post}\n(as against the previously mentioned {\\it prior}) dwell times,\n$\\tau_{+\\diamond}$ and $\\tau_{-\\diamond}$:\nsee~(\\ref{PostDwelAsSumT}). Finally, one can obtain the\nexpressions~(\\ref{nDDeven}), (\\ref{nDDodd}), (\\ref{nRSeven}), and\n(\\ref{nRSodd}), for the mean number, $n_{j,+}^{\\rho\\sigma}$ and\n$n_{j,-}^{\\rho\\sigma}$ of hidden, forwards and backwards, substeps\nfrom an individual state $(j)$ that occur in a time interval between\ndetectable steps, i.e., major transitions specified by\n$\\{\\rho,\\sigma\\}=\\{\\cdots,+-,-+,\\cdots\\}$:\nsee~(\\ref{9combinations}). These results provide quantitative\nestimates for the number of ``lost'' or ``hidden'' transitions\noccurring in an enzymatic cycle. Such information could be of\nparticular interest for real motor proteins since, when operating in\ncells to achieve mitosis, etc., they may frequently be in\nclose-to-stall conditions where reverse substeps are likely to be\nmost frequent~\\cite{grill03,pecreaux06}.\n\nIn conclusion, we have provided a detailed analysis of the\nstatistics of mechanochemical transitions that must be hidden in the\nexperimental noise when a motor protein on its track moves\nprocessively via distinct steps, or reaches stall. As experimental\nresolution at the microsecond and nanometer scales improves, we can\nexpect that such analyses will be increasingly valuable for\nextracting reliable inferences about motor mechanisms from\nobservational data.\n\n\\begin{acknowledgments}\nThe authors much appreciate correspondence and interactions with\nDr.~R.~A.~Cross and Dr.~N.J.~Carter. Support from the National\nScience Foundation under Grant No. CHE 03-01101 is acknowledged.\nM.~L. is grateful for the financial support of the Royal Institute\nof Technology and the Wallenberg Foundation and for the hospitality\nof the Institute for Physical Science and Technology in Fall 2005.\n\\end{acknowledgments}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\\label{s-1}\nThe symmetric groups play two different roles in the representation\ntheory of general linear groups.  On one side, let us call it the Weyl\nside, the symmetric group $S_N$ of permutations of $N$ letters is the\nsubgroup of $\\mathit{GL}_N$ of permutation matrices and thus acts on\nany representation of $\\mathit{GL}_N$.  On the other side, say the\nSchur side, $S_M$ is the group of permutation of factors of the $M$th\ntensor power $U^{\\otimes M}=U\\otimes\\cdots\\otimes U$ of a\nrepresentation $U$ of $\\mathit{GL}_N$ and the action of $S_M$ commutes\nwith the natural action of $\\mathit{GL}_N$ on $U^{\\otimes M}$. We will\nfocus on the case where $U$ is an exterior power of the vector\nrepresentation, but let us be general for the time being.\n\nBoth roles admit a $q$-deformed version but the symmetric group needs\nto be replaced by the braid group.  On the Weyl side one has an action\nof the braid group $B_N$ on finite dimensional representations of the\nquantum enveloping algebras $U_q\\mathfrak{gl}_N$ deforming the above\naction of the symmetric group. This is a special case of the action on\nintegrable representations of the Artin braid group corresponding to\nthe Weyl group of any semisimple Lie algebra \\cite{Lusztig1993,\n  LevendorskiiSoibelman1990}.  On the Schur side, the universal $R$-matrix\nof $U_q\\mathfrak gl_N$ defines a representation of the braid group\n$B_M$ on the tensor power $U^{\\otimes M}$ of a finite dimensional\nrepresentation $U$ of $U_q{\\mathfrak{gl}}_N$. The action of the generator $s_i$,\nbraiding the $i$th with the $i+1$st strand is given by the action of\nthe $R$-matrix on the $i$th and $i+1$st factor.\n\nBoth sides are degenerate cases of versions with parameters, called\nspectral (or equivariant) parameters on the Schur side and dynamical\n(or K\u00e4hler) parameters on the Weyl side. In terms of representation\ntheory, this generalization amounts to replacing $U_q\\mathfrak{gl}_N$\nby its affine version $U_q\\widehat{\\mathfrak{gl}}_N$. A feature of\nthe versions with parameters, whose clarification is a goal of this\npaper, is that the symmetric group plays the leading role, rather than\nthe braid group that is relevant in the degenerate limit. The\n$R$-matrices of quantum groups first appeared in statistical mechanics\nand quantum field theory as solutions\n$R(z)\\in\\operatorname{End}(U\\otimes U)$ of the Yang--Baxter equation,\ndepending meromorphically on a complex spectral parameter $z$. The\nYang--Baxter equation is equivalent to the braiding relation\n\\eqref{e-3} for the braiding matrix $\\check R(z)=P\\circ R(z)$\nobtained by composing the $R$-matrix with the permutation\n$P(u\\otimes v)=v\\otimes u$ of factors. Moreover one has the inversion\n(or unitarity) relation $\\check R(z^{-1})\\check R(z)=\\mathrm{id}$.\nOne obtains, on the Schur side, a representation of the symmetric\ngroup $S_M$ on $U^{\\otimes M}$-valued meromorphic functions\n$f(z_1,\\dots, z_M)$ of $M$ variables, such that the generator\n$s_i=(i,i+1)$ is mapped to\n\\[\ns_if(z_1,\\dots,z_N) = \\mathrm{id}^{\\otimes (i-1)}\\otimes \\check\n R(z_i\/z_{i+1})\\otimes \\mathrm{id}^{\\otimes (N-i-1)}\n f(z_1,\\dots,z_{i+1},z_i,\\dots,z_N).\n\\]\nRepresentations of the braid group on $U^{\\otimes M}$ appear in\nthe limits $z_i\/z_{i+1}\\to 0$, $i=1,\\dots,M-1$ or $\\infty$. Indeed $R(z)$\nconverges in the limit $z\\to\\infty$ to the solution of the Yang--Baxter equation\nwithout spectral parameter corresponding to $U_q{\\mathfrak{gl}}_N$. The relation\n$s_i^2=1$ is lost in this limit and one is left with a representation of the\nbraid group.\n\nOn the Weyl side, the discovery of a version of the quantum Weyl group\naction with parameters is more recent\n\\cite{TarasovVarchenko2000,EtingofVarchenko2002} and is called dynamical Weyl\ngroup. Let $T\\subset SL_M(\\mathbb C)$ be the Cartan torus of diagonal\nmatrices, $P_M=\\operatorname{Hom}(T,\\mathbb C^\\times)$ the weight\nlattice. Then finite dimensional representations $U$ of\n$U_q\\mathfrak{sl}_M$ (of type I with $q$ not a root of unity) have a\nweight decomposition $U=\\oplus_{\\mu\\in P_M}U[\\mu]$. The Weyl group\n$S_M$ acts on $T$ and $P_M$, defining a action of the braid group via\nthe canonical homomorphism $B_M\\to S_M$. A dynamical action of the\nbraid group $B_M$ on an $\\mathfrak{sl}_M$-module $U$ is a\nrepresentation $\\rho$ of the braid group on rational $U$-valued\nfunction on $T$ of the form\n\\[\n  (\\rho(g)f)(t)=A_{g}(g^{-1}t)f(g^{-1}t),\n\\]\nfor some rational functions $t\\mapsto A_g(t)\\in\\operatorname{End}(U)$,\ncompatible with the $T$-action on $U$ in the sense that\n$A_g(t)(U[\\mu])\\subset U[g\\mu]$.  The theory of intertwining operators\ndeveloped in \\cite{TarasovVarchenko2000, EtingofVarchenko2002}, which works\nfor arbitrary simple Lie algebras, gives rise to a dynamical action of\nthe braid group with many interesting properties, such as universal\nformulas for $A_g(t)|_{U[\\mu]}$ as the action of an element of\n$U_q\\mathfrak{sl}_M$ independent of $U$ and a compatibility with\ninclusions of Lie subalgebras, implying that $A_{s_i}(t)$ for\ngenerators $s_i$ is obtained as the image of the corresponding element\nof $U_q(\\mathfrak{sl}_2)$ by the inclusion associated with the $i$th\nsimple root.\n\nIn this paper we take $U=\\bigwedge V$, the direct sum of exterior\npowers of the vector representation $V=\\mathbb C^N$ of $U_q{\\mathfrak{gl}}_N$, and\nrelate the two actions of the symmetric group via quantum (skew) Howe\nduality. The classical Howe duality \\cite{Howe1992} is based on the\nobservation that on $\\bigwedge(\\mathbb C^N\\otimes\\mathbb C^M)$ we have\nnatural commuting actions of $\\mathit{GL}_N$ and $\\mathit{GL}_M$ whose\nimages span each other's commutant in the endomorphism ring. This\nimplies that endomorphisms of\n$(\\bigwedge V)^{\\otimes M}=\\bigwedge V\\otimes\\cdots\\otimes\\bigwedge V$\ncommuting with the $\\mathit{GL_N}$ action on $V=\\mathbb C^N$ are\nspanned by a commuting action of $\\mathit{GL}_M$. There is a also a\nsymmetric version of Howe duality, where exterior powers are replaced\nby symmetric powers.  Quantum version of Howe dualities are known. The\nskew case was first considered in \\cite{CautisKamnitzerLicata2010} and \\cite{LehrerZhangZhang2011}.\nSee \\cite{Stroppel2022} for recent developments of this story. \nIn Section \\ref{s-3} we present an approach based on the action of the\nClifford algebra which is more elementary than the categorification\napproach of \\cite{CautisKamnitzerMorrison2014}, see Proposition \\ref{p-5}.\n\nIn particular one obtains a $U_q{\\mathfrak{gl}}_M$-action on\n$(\\bigwedge V)^{\\otimes M}$ commuting with the diagonal\n$U_q{\\mathfrak{gl}}_N$-action.  The upshot is that the action of the symmetric\ngroup on $(\\bigwedge V)^{\\otimes M}$-valued functions by braiding\nmatrices with spectral parameter is realized by a dynamical Weyl group\naction for the $U_q{\\mathfrak{gl}}_M$-action, after a suitable identification of\nspectral with dynamical parameters.  Thus we connect the Schur with\nthe Weyl side. In the limit of infinite spectral\/dynamical parameter\nwe recover the result of \\cite{CautisKamnitzerLicata2010} relating\nbraid group action with quantum Weyl group action.\n\nFor $M=2$ we obtain an explicit formula for the $R$-matrix for pairs\nof exterior powers of the vector representation in terms of\n$U_q{\\mathfrak{gl}}_2$, see Theorem \\ref{t-1}. These $R$-matrices were calculated\nin \\cite{DateOkado1994} by the fusion procedure and are expressed in\nterms of projections onto irreducible subrepresentations of the tensor\nproduct. Our alternative formula does not rely on the decomposition into irreducible\nsubrepresentations. It is a sum of terms, each with a simple\npole in the spectral parameters, and gives an interpretation of the\nresidues at the poles.  When written in terms of Clifford algebra\ngenerators, our formula for the $R$-matrix of\n$\\bigwedge^kV\\otimes\\bigwedge^kV$ converges in the limit $q\\to1,z\\to 1$ to\nthe formula proposed by Smirnov \\cite{Smirnov2016} in the case of the\nYangian. As in \\cite{Smirnov2016}, it has the property that it also\napplies to the limit $N\\to\\infty$, see Theorem \\ref{t-3}.  Finding a conceptual proof of\nSmirnov's formula was an initial motivation of this work.\n\nAs an application of our result, we obtain in Theorem \\ref{t-2} and\nCorollary \\ref{c-1} a dynamical action of the {\\em symmetric group} on\nintegrable representations of $U_q{\\mathfrak{gl}}_M$, rather than of the braid\ngroup as in \\cite{EtingofVarchenko2002}. In fact our dynamical action\ndiffers from the dynamical action of the braid group defined in\n\\cite{EtingofVarchenko2002} by a transformation\n$A_{g}(t)|_{U[\\mu]}\\mapsto A_{g}(t)|_{U[\\mu]}f_{g,\\mu}(t)$ for some\nscalar meromorphic functions $f_{g,\\mu}(t)$ obeying the cocycle\ncondition $f_{gh,\\mu}(t)=f_{g,h\\mu}(ht)f_{h,\\mu}(t)$ for any braids\n$g,h$, see Proposition \\ref{p-3}. While in \\cite{EtingofVarchenko2002}\nthe dynamical action of the braid group factors through the symmetric\ngroup only when restricted to the zero weight space $U[0]$, our\nmodified action defines a representation of the symmetric group on the\nwhole space of $U$-valued functions. We expect that the same\ntransformation gives rise to a dynamical action of the Weyl group on\nintegrable representations of arbitrary reductive Lie algebras.\n\nA summary of the results of this paper was announced in the Oberwolfach report\n\\cite{Felder2022}.\n\n\\subsection{Acknowledgments} This research was supported in part by\nthe National Centre of Competence in Research SwissMAP (grant numbers\n182902, 205607) of the Swiss National Science Foundation. The second\nauthor is also supported by the grant 196892 of the Swiss National\nScience Foundation.  This article was written in part while the second\nauthor was at IHES, which he thanks for hospitality.\n\nWe thank Tommaso Botta for various useful discussions and suggestions.\nThe second author thanks the organizers of the mini-workshop on\n``Recent developments in representation theory and mathematical physics'' at the\nMathematisches Forschungsinstitut Oberwolfach, where these results were presented,\nand the participants, expecially Evgeny Mukhin and Catharina Stroppel for discussions.\nWe thank Andrey Smirnov and Huafeng Zhang for useful correspondence.\n\n\\section{Braiding matrices for exterior powers of the vector representation}\\label{s-2}\n\\subsection{$q$-number notation}\nWe adopt the standard $q$-number notation: for $n\\in\\mathbb Z_{\\geq0}$,\n$[n]_q=\\frac{q^n-q^{-n}}{q-q^{-1}}$. The $q$-factorial is\ndefined by $[0]_q!=1$, $[n]_q!=\\prod_{j=1}^n[j]_q$, $n\\geq 1$\nand the $q$-binomial coefficient (or Gauss polynomial) is\n\\[\n  \\qbinomial{n}{j}=\\frac{[n]_q!}{[j]_q![n-j]_q!}.\n\\]\nAll these objects belong to $\\mathbb Z[q,q^{-1}]$.\n\n\\subsection{Quantum enveloping algebras of type $A_{N-1}$ and\n  $A_{N-1}^{(1)}$}\nDrinfeld--Jimbo quantum enveloping algebras are the subject of several textbooks,\nfor example \\cite{Lusztig1993, ChariPressley1994}. Here we collect our\nconventions.\nLet $q$ be a non-zero complex number which is not a root of unity.\nLet $(a_{ij})_{i,j=0}^{N-1}$ be the Cartan matrix of $A_{N-1}^{(1)}$:\n$a_{ij}=2$ if $i=j$, $a_{ij}=-1$ if $j=i\\pm 1\\mod N$, and $a_{ij}=0$\notherwise.  It is convenient to identify the index set $\\{0,\\dots,N-1\\}$\nwith $\\mathbb Z\/N\\mathbb Z$ in view of the symmetry $a_{i+1,j+1}=a_{ij}$.\nThe Hopf algebra $U'_q(\\widehat{\\mathfrak{sl}}_N)$ is the\nunital algebra over $\\mathbb C$ generated by $e_i,f_i,k_i^{\\pm1}$,\n$(i=0,\\dots, N-1)$ and relations\n\\begin{align*}\n  k_ik_j&=k_jk_i,\\quad k_ik_i^{-1}=k_i^{-1}k_i=1\n  \\\\\n  k_ie_jk_i^{-1}\n        &=q^{a_{ij}}e_j,\\quad k_if_jk_i^{-1}=q^{-a_{ij}}f_j,\n\\\\\n  e_if_j-f_je_i\n        &=\\delta_{ij}\\frac{k_i-k_i^{-1}}{q-q^{-1}}\n  \\\\\n  \\sum_{r=0}^{1-a_{ij}} &\\qbinomial {1-a_{ij}}r e_i^re_je_i^{1-a_{ij}-r}\n  =0,\\quad \\text{if $i\\neq j$,}\n  \\\\\n    \\sum_{r=0}^{1-a_{ij}}& \\qbinomial {1-a_{ij}}r f_i^rf_jf_i^{1-a_{ij}-r}\n  =0,\\quad \\text{if $i\\neq j$.}\n\\end{align*}\nThe coproduct is defined on generators as\n$\\Delta(e_i)=e_i\\otimes 1+k_i\\otimes e_i$,\n$\\Delta(f_i)=f_i\\otimes k_i^{-1}+ 1\\otimes f_i$ and\n$\\Delta(k^{\\pm1}_i)=k^{\\pm1}_i\\otimes k^{\\pm1}_i$.  The element\n$\\prod_{i=0}^{N-1}k_i$ is central and group-like. For finite\ndimensional representations of $U_q'(\\widehat{\\mathfrak{sl}}_N)$ this\ncentral element acts by 1. The quantum loop algebra\n$U_qL{\\mathfrak{sl}}_N$ is the quotient of $U'_q(\\widehat{\\mathfrak{sl}}_N)$\nby the relation $\\prod_{i=0}^{N-1}k_i=1$. The Hopf algebra structure\ndescends to this quotient. It is a deformation of the universal\nenveloping algebra of the Lie algebra $L\\mathfrak{sl}_N={\\mathfrak{sl}}_N[t,t^{-1}]$ of\nloops in ${\\mathfrak{sl}}_N$.  The Hopf algebra\n$U_qL{\\mathfrak{sl}}_N$ contains the Hopf subalgebra\n$U_q {\\mathfrak{sl}}_N$ generated by $e_i,f_i,k_i$ with\n$1\\leq i\\leq N-1$.\n\nFor $k\\in\\mathbb Z_{\\geq0}$ the $k$th divided power of generators is\ndefined as\n\\[\n  e_i^{(k)}=\\frac{e_i^k}{[k]_q!},\\quad f_i^{(k)}=\\frac{f_i^k}{[k]_q!},\n\\]\nwith the convention that $e_i^{(0)}=f_i^{(0)}=1$.\n\n\n\n\nLet $\\mathbb Z^N=\\oplus_{i\\in\\mathbb Z\/N\\mathbb Z}\\mathbb Z\\epsilon_i$\nbe the (co)weight lattice of ${\\mathfrak{gl}}_N$ with standard inner product\n$(\\ |\\ )$, which identifies the Cartan subalgebra of diagonal matrix with its dual.\nThe root lattice of vectors of zero coordinate sum is\nspanned by the simple roots $\\alpha_i=\\epsilon_i-\\epsilon_{i+1}$,\n$(i=1,\\dots, N-1)$. We also set $\\alpha_0=\\epsilon_N-\\epsilon_1$, the\nhighest root.  The Hopf algebra $U_qL{\\mathfrak{gl}}_N$ is obtained by adjoining\ngenerators $t_\\lambda$ for $\\lambda\\in \\mathbb Z^N$ such that\n$t_{\\alpha_i}=k_i$ ($i=1,\\dots,N-1$) with the relations\n\\[\n  t_\\lambda t_\\mu=t_{\\lambda+\\mu},\\quad t_0=1,\\quad t_\\lambda\n  e_it_{-\\lambda}=q^{(\\lambda|\\alpha_i)}e_i, \\quad t_\\lambda\n  f_it_{-\\lambda}=q^{-(\\lambda|\\alpha_i)}e_i.\n\\]\nIt is a Hopf algebra with\n$\\epsilon(t_\\lambda)=1,\nS(t_\\lambda)=t_{-\\lambda},\\Delta(t_\\lambda)=t_\\lambda\\otimes\nt_\\lambda$.  We set $t_i=t_{\\epsilon_i}$ so that $k_i=t_it_{i+1}^{-1}$\nfor $i\\in\\mathbb Z\/N\\mathbb Z$. The Hopf subalgebra $U_q{\\mathfrak{gl}}_N$ is\ngenerated by  $e_i,f_i$ with $1\\leq i\\leq N-1$ and all $t_i^{\\pm1}$.\n\nThe multiplicative group $\\mathbb C^\\times $ acts on $U_qL{\\mathfrak{gl}}_N$ by\nalgebra automorphisms, deforming the rescaling action of the loop\nparameter: the automorphism $\\varphi_z$ associated with\n$z\\in\\mathbb C^\\times$ acts as the identity on all generators except\n$e_0$, $f_0$, for which\n\\[\n  \\varphi_z(e_0)=ze_0,\\quad \\varphi_z(f_0)=z^{-1}f_0.\n\\]\nThe weight space of a $U_q{\\mathfrak{gl}}_N$-module of weight $\\mu\\in \\mathbb Z^N$\nis the simultaneous eigen\\-space of $t_\\lambda$ with eigenvalue\n$q^{(\\mu|\\lambda)}$.  A $U_q{\\mathfrak{gl}}_N$-module is of {\\em type I} if it is the direct\nsum of its finite dimensional weight spaces.  All modules considered\nin this paper are of type I.\n\n\n\\subsection{Exterior powers of the vector representation}\n\\label{ss-2.3}\nLet $V=\\mathbb C^N$ with basis $v_1,\\dots,v_N$ and\n$C_N$ be the Clifford algebra of $V\\oplus V^*$ with quadratic form\n$Q(v\\oplus \\alpha)=\\alpha(v)$. It is the unital algebra  generated by\nthe images of the standard basis  and dual basis vectors\n$\\psi^*_i,\\psi_i$, labeled by $i\\in\\{1,\\dots,N\\}$, which we often identify\nwith $\\mathbb Z\/N\\mathbb Z$,  subject to the  relations\n\\begin{equation}\\label{e-1}\n  \\psi_i\\psi_j=-\\psi_j\\psi_i,\\quad\\psi_i^*\\psi_j^*=-\\psi_j^*\\psi_i^*,\\quad\n  \\psi_i\\psi_j^*+\\psi_j^*\\psi_i=\\delta_{ij}1.\n\\end{equation}\nThe Fock representation of $C_N$ is $\\bigwedge V$ with $\\psi^*_i$\nacting as $v\\mapsto v_i\\wedge v$ and $\\psi_i$ by contraction. It is\nirreducible and generated by $|0\\rangle=1$ such that\n$\\psi_i|0\\rangle=0$ for all $i$. The commuting ``fermion number''\noperators $n_i=\\psi_i^*\\psi_i$ have a common eigenbasis\n$v_{i_1}\\wedge\\cdots\\wedge v_{i_k}$ ($i_1<\\dots<i_k$) and eigenvalue\n$1$ if $i=i_j$ for some $j$ and $0$ otherwise. Thus\n$t^{\\pm1}_i:=q^{\\pm n_i}=q^{\\pm1}\\psi_i^*\\psi_i+\\psi_i\\psi_i^*$ obeys the\nrelations\n\\begin{equation}\\label{e-2}\nt_it_i^{-1}=1,\\quad t_it_j=t_jt_i,\\quad  t_i\\psi_j^*t_i^{-1}=q^{\\delta_{ij}}\\psi_j^*,\\quad\n  t_i\\psi_jt_i^{-1}=q^{-\\delta_{ij}}\\psi_j.\n\\end{equation}\nfor all $i,j$. Let $C_N(q)$ be the algebra with generators $t_i,\\psi_i,\\psi_i^*$,\n($i=1,\\dots,N$) and relations \\eqref{e-1}, \\eqref{e-2}. It is a $\\mathbb Z\/2\\mathbb Z$-graded\nalgebra with $\\psi,\\psi_i^*$ odd and $t_i$ even, and $\\bigwedge V$ is a $\\mathbb Z\/2\\mathbb Z$-graded\nmodule, with the convention that $|0\\rangle$ is even.\n\\begin{prop}[Hayashi \\cite{Hayashi1990}]\\label{p-1}\n  There is a homomorphism $U_qL\\mathfrak{gl}_N\n  \\to C_N(q)$ such that\n  \\[\n    e_i\\mapsto \\psi_i^*\\psi_{i+1},\\quad\n    f_i\\mapsto\\psi_{i+1}^*\\psi_i,\\quad\n    t_i^{\\pm1}\\mapsto t_i^{\\pm 1},\n  \\]\n  for $i\\in\\mathbb Z\/N\\mathbb Z$.\n\\end{prop}\nThus we have a representation of $U_qL{\\mathfrak{gl}}_N$ on $\\bigwedge V$. It is\nthe direct sum of the irreducible eigenspaces $\\bigwedge^kV$ of the\ncentral element $\\prod_{i=1}^N t_i$. These irreducible representations\nare generated by the highest weight vectors\n\\begin{equation}\\label{e-hw}\nv^k=\\psi_1^*\\cdots\\psi_k^*|0\\rangle\n\\end{equation}\nof weight $\\sum_{i=1}^k\\epsilon_i$.  In fact they are obtained from\nthe representation of the classical Lie algebra $L{\\mathfrak{gl}}_N$ on the same\nrepresentation space and with the same action of the generators\n$e_i,f_i$ by setting $t_\\lambda=q^{\\lambda}$ for Cartan elements\n$\\lambda$.  The point is that, since the eigenvalues of $\\epsilon_{i}$\nare $1$ or $0$, the right-hand side $(q^{h_i}-q^{-h_i})\/(q-q^{-1})$ of\nthe commutation relation of $e_i,f_i$ coincides with the action of its\nclassical counterpart $h_i=\\epsilon_i-\\epsilon_{i+1}$.\n\n\\begin{definition}\\label{def-ext}\n  The {\\em $j$-th exterior power of the vector representation} of $U_qL{\\mathfrak{gl}}_N$ with spectral\n  parameter  $z$ is the twist $\\bigwedge^jV(z)$\n  of $\\bigwedge^jV$ by the automorphism $\\varphi_z$. The vector representation is\n  $V(z)=\\bigwedge^1V(z)$.\n\\end{definition}\nThus the action of the generators is given by the formulae of\nProposition \\ref{p-1} except that $e_0,f_0$ are mapped to\n$z\\psi_N^*\\psi_{1}$ and $z^{-1}\\psi_1^*\\psi_N$, respectively.\n\\begin{remark} Our definition of the exterior power  representation\n  has a slightly non-standard choice of signs. A more common convention is to\n  let $e_0$ and $f_0$ act as $(-1)^{j-1}z\\psi_N^*\\psi_1$, $(-1)^{j-1}z^{-1}\\psi_1^*\\psi_N$, respectively,\n  which in the limit $q\\to 1$ correspond to the action of $zE_{N,1},z^{-1}E_{1,N}$, in terms of the\n  standard basis $(E_{i,j})$ of ${\\mathfrak{gl}}_N$, so that $z$ has the\n  interpretation of an ``evaluation point.'' Our convention is better suited to the fermionic representation\n  and somewhat reduces the number of signs in the formulas. The relation of our definition of\n  the exterior power $\\bigwedge^jV(z)$ to the definition $V^{(j)}(z)$ in \\cite{DateOkado1994}\n  is $\\bigwedge^jV(z)=V^{(j)}((-1)^{j-1}z)$,\n\\end{remark}\n\nAnother interpretation of the exterior powers, useful to compute $R$-matrices by the fusion procedure,\nis in terms of $q$-deformed exterior products:\n\\begin{prop}[Jimbo \\cite{Jimbo1986}]\n  Let $k\\geq2$ and $S$ be the subspace of $V^{\\otimes k}$ spanned by\n  $\\alpha\\otimes v_i\\otimes v_i\\otimes\\beta$,\n  $\\alpha\\otimes (v_i\\otimes v_j+q v_j\\otimes v_i)\\otimes \\beta$ for\n  $i>j$, $\\alpha\\in V^{\\otimes r},\\beta\\in V^{\\otimes k-r-2}$,\n  $r=0,\\dots,k-2$. Then $S$ is a $U_qL{\\mathfrak{gl}}_N$ submodule of\n  \\[\n    V^k(z):=V(q^{-k+1}z)\\otimes\\cdots\\otimes V(q^{k-3}z)\\otimes V(q^{k-1}z)\n  \\]\n  and we have the isomorphism of $U_qL{\\mathfrak{gl}}_N$-modules\n  $V^k(z)\/S\\to \\bigwedge^k V(z)$ such that\n  $v_{i_1}\\otimes \\cdots \\otimes v_{i_k}+S\\mapsto\n  v_{i_1}\\wedge\\cdots\\wedge v_{i_k}$ for $1\\leq i_1<\\cdots<i_k\\leq N$.\n\\end{prop}\n\nFor generic $z_1,z_2$ the tensor products module\n$\\bigwedge^{k} V(z_1)\\otimes \\bigwedge^{k'} V(z_2)$ with action of\n$U_q{\\mathfrak{gl}}_N$ given by the coproduct are irreducible and do not depend\non the order of factors up to isomorphism. Thus there is an\nisomorphism of $U_q\\widehat{{\\mathfrak{gl}}}_N$-modules\n\\[\n  \\check R_{k,k'}(z_1\/z_2)\\colon\\textstyle{\\bigwedge^{k} V(z_1)\\otimes\n  \\bigwedge^{k'} V(z_2)\\to \\bigwedge^{k'} V(z_2)\\otimes\n  \\bigwedge^{k} V(z_1)},\n\\]\nwhich, viewed as a linear map between the underlying vector spaces,\ndepends on the ratio of spectral parameters.  We find it convenient to\nnormalize it in such a way that $v^k\\otimes v^{k'}$ is mapped to\n$(-1)^{kk'}v^{k'}\\otimes v^k$. Then the $R$-matrix\n\\[\n  R_{k,k'}(z_1\/z_2)=P\\check R_{k,k'}(z_1\/z_2)\\in\\operatorname{End}_{\\mathbb C}(\n  \\textstyle{\\bigwedge^{k} V(z_1)\\otimes\n  \\bigwedge^{k'} V(z_2)}),\n\\]\nobtained by the composition with the graded flip\n$P\\colon v\\otimes w\\to (-1)^{kk'}w\\otimes v$, restricts to the identity\non the product of highest weight spaces.\n\n  Then by the irreducibility of tensor products\nat generic values of the spectral parameter we have the braiding relation\n\\begin{align}\\label{e-3}\n  (\\check R_{k_2k_3}(z_2\/z_3)\n  &\\otimes\\mathrm{id})\n    (\\mathrm{id}\\otimes\\check R_{k_1k_3}(z_1\/z_3))\n    (\\check R_{k_1k_2}(z_1\/z_2)\\otimes\\mathrm{id})\\notag\n  \\\\\n  &=\n    (\\mathrm{id}\\otimes\\check R_{k_1k_2}(z_1\/z_2))\n    (\\check R_{k_1k_3}(z_1\/z_3)\\otimes\\mathrm{id})\n     (\\mathrm{id}\\otimes\\check R_{k_2k_3}(z_2\/z_3)),\n\\end{align}\non $\n\\bigwedge^{k_1}V(z_1)\\otimes \\bigwedge^{k_2}V(z_2)\\otimes \\bigwedge^{k_3}V(z_3)$,\nequivalent to the Yang--Baxter equation for the $R$-matrices.\nMoreover we have the inversion (or unitarity) relation\n\\begin{equation}\\label{e-4}\n  \\check R_{k'k}(z^{-1})\\check R_{kk'}(z)=\\mathrm{id}.\n\\end{equation}\nThese $R$-matrices were computed\nby Date and Okado \\cite{DateOkado1994} in terms of projections onto irreducible\nsubmodules for the action of $U_q{\\mathfrak{gl}}_N$. The coefficients were computed using\nthe fusion procedure \\cite{KulishReshetikhinSklyanin1981}.\nHere we give an alternative fermionic formula.\n\nLet $\\bigwedge V\\otimes\\bigwedge V$ be the tensor product of\n$\\mathbb Z\/2\\mathbb Z$-graded vector spaces.  The action of tensor\nproducts of linear maps is given by the sign rule:\n$(f\\otimes g)(v\\otimes w)=(-1)^{|g||v|}fv\\otimes gw$ for $g$ of degree\n$|g|$ and $v$ of degree $|v|$. We introduce the following even\nendomorphisms of $\\bigwedge V\\otimes\\bigwedge V$:\n\\begin{equation}\\label{e-EFK}\n      E=\\sum_{i=1}^N\n       (\\psi^{*}_i\\otimes {\\psi}_i)\\prod_{j>i}{K}_j\n  \\qquad\n    F=-\\sum_{i=1}^N\n       \\prod_{j<i}{K}^{{-1}}_j({\\psi}_i\\otimes\\psi^{*}_i)\n  \\qquad\n  {K}=\\prod_{i=1}^N{{K}}_i,\n\\end{equation}\nwith  $K_i=t_i\\otimes t_i^{-1}$ ($i=1,\\dots,N$).\n\\begin{lemma}\\label{l-1}\n  The endomorphisms $E,F,K$ commute with the action of $U_q{\\mathfrak{gl}}_N$ and\n  obey the commutation relations of $U_q\\mathfrak{sl}_2$:\n  \\[\n    [E,F]=\\frac{K-K^{-1}}{q-q^{-1}},\\quad KEK^{-1}=q^2E,\\quad\n    KFK^{-1}=q^{-2}F.\n  \\]\n\\end{lemma}\nThis is an instance of Howe duality and is discussed in the next\nsection. It is convenient to extend this action to an action of\n$U_q{\\mathfrak{gl}}_2$ by setting $T_1=\\prod_{i=1}^N t_i\\otimes 1$,\n$T_2=\\prod_{i=1}^N1\\otimes t_i$ so that\n$\\bigwedge^{k}V\\otimes \\bigwedge^{k'}V$ is the weight space of weight\n$(k,k')$.\n\n\n\\begin{theorem}\\label{t-1}\n  The braiding matrix\n  \\[\n    \\check\n  R_{k,k'}(z_1\/z_2)\\colon\\textstyle{\\bigwedge^kV(z_1)\\otimes\\bigwedge^{k'}V(z_2)\\to\n  \\bigwedge^{k'}V(z_1)\\otimes\\bigwedge^{k}V(z_1)},\n\\]\nnormalized so that\n  $\\check R_{k,k'}(z_1\/z_2)\\,v^k\\otimes v^{k'}=(-1)^{kk'}v^{k'}\\otimes v^{k}$, is\n  given by\n  \\[\n    \\check R_{k,k'}(z_1\/z_2)=(-1)^{\\min(k,k')}A_{k-k'}(z_1\/z_2)\n  \\]\n  where\n  $A_{k-k'}(z_1\/z_2)\\in U_q\\mathfrak{gl}_2$ is defined by\n  \\[\n    A_m(z)=\\sum_{j=0}^n\n    (-q)^{j}\\frac{1-q^{|m|}z}{1-q^{2j+|m|}z}\n    \\begin{cases}\n      E^{(j)}F^{(j+m)},& \\text{if $m\\geq0$,}\n      \\\\\n      E^{(j-m)}F^{(j)},& \\text{if $m\\leq 0$,}\n     \\end{cases}\n  \\]\n  with the divided power notation $E^{(l)}=E^l\/[l]_q!$, $F^{(l)}=F^l\/[l]_q!$ and\n  \\[\n    n=\\min(k,k',N-k,N-k').\n  \\]\n\\end{theorem}\nThe proof relies on Howe duality, which we proceed to explain, and the computation\n\\cite{DateOkado1994} of the $R$-matrices by the fusion procedure.\n\\section{Quantum skew Howe duality}\\label{s-3}\nThe classical skew-symmetric version of Howe duality \\cite[Section 4]{Howe1992}\nstates that the natural actions of $\\mathit{GL}_N$ and\n$\\mathit{GL}_M$ on $\\bigwedge(\\mathbb C^N\\otimes\\mathbb C^M)$ generate\neach other's commutant and one has a decomposition into irreducibles\nof $\\mathit{GL}_N\\times \\mathit{GL}_M$\n\\[\n  \\bigwedge (\\mathbb C^N\\otimes\\mathbb C^M)\\cong\n  \\bigoplus_{\\lambda}V^N_\\lambda\\otimes V^M_{\\lambda^t}.\n\\]\nThe sum is over all Young diagrams $\\lambda$ with at most $N$ rows\nsuch that the transposed diagrams $\\lambda^t$ has at most $M$ rows\n(equivalently, the length of the first row of $\\lambda$ is at most\n$M$) and $V^N_\\lambda$ denotes the irreducible representation of\n$\\mathit{GL}_N$ of highest weight $\\lambda$. The following nice\npictorial description of the corresponding highest weight vectors is\ndue to Howe in {\\it loc.~cit.}: write the tensor products of basis vectors\n$v_{i,r}=v_i\\otimes w_r$ $(i=1,\\dots,N, r=1,\\dots, M)$ in the boxes of an\n$N\\times M$ grid (the tensors $v_{1,r}$ go in the first row,\n$v_{2,r}$ in the second row, etc.).   The Young diagrams\noccuring in the decomposition fit in this rectangle. The highest\nweight vector corresponding to $\\lambda$ is obtained by taking the\nwedge product of the basis vectors in the boxes of the Young diagram\n$\\lambda$. For example, if $\\lambda=(3,1)$,\nwe take $v_{1,1}\\wedge v_{1,2}\\wedge v_{1,2}\\wedge v_{2,1}$.\nIt is indeed of weight $\\lambda$ for $\\mathit{GL}_N$ and of\nweight $\\lambda^t$ for $\\mathit{GL}_M$.\n\nThe quantum group version of Howe duality is known: in the analogous\ncase of the symmetric algebra it is discussed in \\cite{Baumann1999},\n\\cite{ToledanoLaredo2002} and \\cite{Zhang2002}.\nThe skew-symmetric case we use here is\nexplained in \\cite{CautisKamnitzerLicata2010} (for $M=2$), \\cite{LehrerZhangZhang2011} and\n\\cite{CautisKamnitzerMorrison2014}.  \nHere we present\nan equivalent variant based on the Clifford algebra action.\n\nLet $V=\\mathbb C^N$ with basis $v_1,\\dots,v_N$ be the vector\nrepresentation of $U_q\\mathfrak {gl}_N$ and $W=\\mathbb C^M$ with basis\n$w_1,\\dots,w_M$ be the vector representation of\n$U_q\\mathfrak{gl}_M$. Then we have an isomorphism of graded\ncommutative algebras\n\\[\n  \\bigwedge(V\\otimes W)\\cong\\bigwedge(V\\oplus\\cdots\\oplus V) \\to\n  \\left(\\bigwedge V\\right)^{\\otimes M} =\\bigwedge\n  V\\otimes\\cdots\\otimes \\bigwedge V\n\\]\nsending $v_i\\otimes w_j$ to\n$1\\otimes\\cdots\\otimes 1\\otimes v_i\\otimes 1\\otimes \\cdots\\otimes 1$\nwith $v_i$ placed in the $j$-th factor. The right-hand side is a\ntensor product of $U_q\\mathfrak{gl}_N$-modules and thus\n$U_q\\mathfrak{gl}_N$ acts on it by the iterated coproduct. Similarly\nwe have an isomorphism\n$\\bigwedge(V\\otimes W)\\cong \\bigwedge W\\otimes\\cdots \\otimes\\bigwedge\nW$ sending $v_i\\otimes w_j$ to\n$1\\otimes \\cdots\\otimes 1\\otimes w_j\\otimes 1\\otimes \\cdots \\otimes 1$\nwith $i$-th factor $w_j$.  Here we let $U_q\\mathfrak{gl}_M$ act via\nthe {\\em opposite} coproduct $\\Delta'=\\sigma\\circ\\Delta$, with\n$\\sigma(a\\otimes b)=b\\otimes a$.\n\n\\begin{prop}\\label{p-5}\n  The actions of $(U_q\\mathfrak{gl}_N,\\Delta)$ and\n  $(U_q\\mathfrak{gl}_M,\\Delta')$ on $\\bigwedge(V\\otimes W)$ pulled\n  back by the isomorphisms\n  \\[\n    \\left(\\bigwedge W\\right)^{\\otimes N}\\leftarrow\\bigwedge(V\\otimes W)\n    \\rightarrow\\left(\\bigwedge V\\right)^{\\otimes M}\n  \\]\n  commute. If $q$ is not a root of unity the images of\n  $U_q\\mathfrak{gl}_N$ and $U_q\\mathfrak {gl}_M$ in\n  $\\operatorname{End}\\bigwedge(V\\otimes W)$ are commutants of each\n  other and we have a decomposition\n  \\begin{equation}\\label{e-dec}\n    \\bigwedge(V\\otimes W)\\cong\\bigoplus_\\lambda V^N_\\lambda\\otimes V^M_{\\lambda^t}\n  \\end{equation}\n  into simple $U_q\\mathfrak{gl}_N\\otimes U_q\\mathfrak{gl}_M$-modules,\n  where $\\lambda$ runs over Young diagrams with at most $N$ rows and\n  at most $M$ columns.\n\\end{prop}\n\nFor the proof we view $\\bigwedge(V\\otimes W)$ as the Fock module of\nthe Clifford algebra of $V\\otimes W\\oplus (V\\otimes W)^*$. It has\ngenerators $\\psi^*_{i,r},\\psi_{i,r}$ corresponding to the tensor basis\n$v_i\\otimes w_r$ and its dual basis. As in Section \\ref{ss-2.3} we introduce the\noperators\n$t_{i,r}^{\\pm1}=q^{\\pm1}\\psi_{i,r}^*\\psi_{i,r}+\\psi_{i,r}\\psi_{i,r}^*$.\nVia the isomorphism to $(\\bigwedge V)^{\\otimes M}$, $\\psi_{i,r}^*$\nacts as\n$\\mathrm{id}^{\\otimes(r-1)}\\otimes \\psi_i^*\\otimes\\mathrm{id}^{\\otimes\n  (N-r)}$ and $\\psi_{i,r}$ acts as\n$\\mathrm{id}^{\\otimes(r-1)}\\otimes \\psi_i\\otimes\\mathrm{id}^{\\otimes\n  (N-r)}$. Thus the pull-back of the action of $U_q\\mathfrak{gl}_N$ on\n$\\bigwedge(V\\otimes W)$ is given on generators by\n\\begin{align*}\ne_i&\\mapsto\\sum_{r=1}^M\\prod_{s=1}^{r-1} k_{i,s}\n\\psi^*_{i,r}\\psi_{i+1,r}, \\quad k_{i,s}=t_{i,s}t_{i+1,s}^{-1}\\\\\nf_i&\\mapsto\\sum_{r=1}^M\n     \\psi^*_{i+1,r}\\psi_{i,r}\\prod_{s={r+1}}^M k_{i,s}^{-1}\n  \\\\\nt_i^{\\pm1}&\\mapsto \\prod_{r=1}^Mt_{i,r}^{\\pm1}.\n\\end{align*}\nSimilary we can pull-back the action of $U_q\\mathfrak{gl}_M$ on\n$(\\bigwedge W)^{\\otimes N}$ with the opposite coproduct. The action\nof generators (denoted by capital letters to distinguish them)\n\\begin{align*}\n  E_r&\\mapsto\\sum_{i=1}^N\\psi^*_{i,r}\\psi_{i,r+1}\n       \\prod_{j=i+1}^{N} K_{j,r},\\quad K_{j,r}=t_{j,r}t_{j,r+1}^{-1},\n  \\\\\nF_r&\\mapsto\\sum_{i=1}^N\n\\prod_{j=1}^{i-1} K_{j,r}^{-1}\\psi^*_{i,r+1}\\psi_{i,r},\n\\\\\nT_r^{\\pm1}&\\mapsto \\prod_{i=1}^Nt_{i,r}^{\\pm1}.\n\\end{align*}\nIn the case $M=2$ this reduces to the formulae of the previous section, by remembering the sign rule\nin the definition of $F$. By construction, we obtain actions of $U_q{\\mathfrak{gl}}_N$ and $U_q{\\mathfrak{gl}}_M$ and we need\nto check that they commute:\n\\begin{lemma}\\label{l-2}\n  The actions of $U_q{\\mathfrak{gl}}_N$ and $U_q{\\mathfrak{gl}}_M$ on $\\bigwedge(V\\otimes W)$\n  commute.\n\\end{lemma}\n\\begin{proof} It is clear that $t_i^{\\pm1}$ commutes with the action\n  of $U_q{\\mathfrak{gl}}_M$ and that $T_r^{\\pm1}$ commutes with the action of\n  $U_q{\\mathfrak{gl}}_N$.  Let us check that $[e_i,E_r]=0$. The cases of the other\n  generators are dealt with in the same way.\n  The only terms in the sum\n  defining $e_i$ and $E_r$ that contribute non-trivially to the commutator  are\n  \\begin{align*}\n    [e_i,E_r]&=\\prod_{s<r}k_{i,s}  [\n               \\psi_{i,r}^*\\psi_{i+1,r}+k_{i,r}\\psi_{i,r+1}^*\\psi_{i+1,r+1},\n               \\\\\n    &\\quad\n      \\psi_{i,r}^*\\psi_{i,r+1}K_{i+1,r}+\n      \\psi_{i+1,r}^*\\psi_{i+1,r+1}] \\prod_{j>i+1}K_{j,r}.\n  \\end{align*}\n  The bracket on the right-hand side is\n  \\begin{align*}\n    &[\\psi_{i,r}^*\\psi_{i+1,r},      \\psi_{i+1,r}^*\\psi_{i+1,r+1}]\n    +[k_{i,r}\\psi_{i,r+1}^*\\psi_{i+1,r+1},      \\psi_{i,r}^*\\psi_{i,r+1}K_{i+1,r}]\n    \\\\\n    &=\\psi_{i,r}^*\\psi_{i+1,r+1}\n      +k_{i,r}\\psi_{i+1,r+1}\\psi_{i,r}^*K_{i+1,r}\n    \\\\\n    &=\\psi_{i,r}^*\\psi_{i+1,r+1}+t_{i,r}\\psi_{i+1,r+1}\\psi_{i,r}^*t_{i+1,r+1}^{-1}\n  \\end{align*}\n  We have the relations $t_{i,r}\\psi_{i,r}^*=q\\psi_{i,r}^*$ and\n  $\\psi_{i+1,r+1}t_{i+1,r+1}^{-1}=q^{-1}\\psi_{i+1,r+1}$ and we are left\n  with $\\psi_{i,r}^*\\psi_{i+1,r+1}+\\psi_{i+1,r+1}\\psi_{i,r}^*=0$.\n\\end{proof}\nTo complete the proof of Proposition \\ref{p-5}, we observe that for\n$q$ not a root of unity the decomposition into irreducible\nrepresentations in the classical case deforms to the quantum group\ncase.  In fact the highest weight vectors generating irreducible\n$U_q{\\mathfrak{gl}}_N\\otimes U_q{\\mathfrak{gl}}_M$-modules are still given by Howe's\nconstruction.\n\n\n\\begin{proof}[Proof of Theorem \\ref{t-1}]\n  We use the formula of Date and Okado \\cite{DateOkado1994} for the action of the $R$-matrix on highest weight vectors\n  of irreducible $U_q{\\mathfrak{gl}}_N$-subrepresentations in\n  $\\bigwedge^kV\\otimes \\bigwedge^{k'}V$. The decomposition into\n  irreducibles is best understood in terms of Howe duality: we consider\n  the case $M=2$ of the decomposition \\eqref{e-dec}\n  of Proposition \\ref{p-5}. The left-hand side is\n  \\[\n    \\textstyle{\n    \\bigwedge(V\\otimes\\mathbb C^2)=\\bigwedge(V\\oplus V)=\n    \\bigwedge V\\otimes\\bigwedge V=\\oplus_{k,k'=0}^N\\bigwedge^kV\\otimes\n    \\bigwedge^{k'} V\n    }\n  \\]\n  and the summand labeled by $(k,k')\\in\\{0,\\dots,N\\}^2$ is the weight\n  space of weight $(k,k')$ for the $U_q{\\mathfrak{gl}}_2$-action. The right-hand\n  side is a sum over Young diagram $\\lambda$ with at most two columns\n  and at most $N$ rows.  Thus\n  $\\lambda=(2,\\dots,2,1,\\dots,1)=2^{\\ell'}1^{\\ell-\\ell'}$ ($\\ell'$\n  rows of length 2, $\\ell-\\ell'$ rows of length 1) and\n  $\\lambda^t=(\\ell,\\ell')$ with $N\\geq \\ell\\geq\\ell'\\geq0$.  The\n  irreducible $U_q{\\mathfrak{gl}}_2$-module $V^2_{(\\ell,\\ell')}$ has a non-trivial\n  weight space of weight $(k,k')$ if and only if $k+k'=\\ell+\\ell'$\n  for some $s\\geq0$ and $\\ell'-\\ell\\leq k-k'\\leq \\ell-\\ell'$.\n  In this case the weight space is one-dimensional.\n  If $k\\geq k'$ the corresponding highest weights are\n  \\[\n    (\\ell,\\ell')=(k,k'),(k+1,k'-1),\\dots,(k+n,k'-n),\n  \\]\n  where $n=\\min(k',N-k)$, whereas if $k\\leq k'$ they are\n  \\[\n    (\\ell,\\ell')=(k',k),(k'+1,k-1),\\ldots,(k'+n,k-n),\n  \\]\n  where $n=\\min(k,N-k')$. In both cases we can write\n  \\[\n    n=\\min(k,k',N-k,N-k').\n  \\]\n  With this notation, the weight $(k,k')$ subspace in the decomposition \\eqref{e-dec}\n  gives the decomposition\n  \\[\n  {\\textstyle  \\bigwedge^{k}V\\otimes\\bigwedge^{k'}V}\\cong\\oplus_{s=0}^{n}\n    V^N_{(\\overline k+s,\\underline k-s)^t},\\quad\n    \\text{with $\\overline k=\\max(k,k')$ and\n      $\\underline k=\\min(k,k')$},\n  \\]\n  as an $U_q{\\mathfrak{gl}}_N$-module. The $s=0$ component is generated by the\n  highest weight vector $v^{(\\overline{k},\\underline{k})}_0=v^{\\overline{k}}\n  \\otimes v^{\\underline{k}}$, see \\eqref{e-hw},  and\n  the highest weight vector $v^{(k,k')}_s$ of $V^N_{(\\overline k+s,\\underline k-s)^t}$\n  is proportional to $F^{(s)}v_0^{(k+s,k'-s)}$ if $k\\geq k'$ and to\n  $F^{(k'-k+s)}v_0^{(k'+s,k-s)}$ if $k\\leq k'$.\n  We normalize it as in \\cite{DateOkado1994}\n by fixing the coefficient of\n  the basis vector $\\psi^*_k\\cdots\\psi^*_1|0\\rangle$ in the first factor. Namely,\n  \\[\n    v_s^{(k,k')}=\\psi_k^*\\cdots\\psi_1^*|0\\rangle\\otimes\\psi^*_{\\overline\n      k+s}\\cdots \\psi^*_{\\overline k+1}\\psi^*_{\\underline\n      k-s}\\cdots\\psi_1^*|0\\rangle+\\cdots.\n  \\]\n  Let\n  \\[\n    \\textstyle Q_s\\colon \\bigwedge^k\n    V\\otimes\\bigwedge^{k'}V\\to\\bigwedge^{k'} V\\otimes\\bigwedge^{k}V\n  \\]\n  be the unique $U_q{\\mathfrak{gl}}_N$-linear map sending $v^{(k,k')}_s$ to the\n  vector of the same weight $v^{(k',k)}_{s'}$, $s'=k-k'+s$ and vanishing on the\n  other irreducible components. In our\n  notation and with the normalization of Theorem \\ref{t-1}, the formula of\n  \\cite{DateOkado1994}, obtained by the fusion procedure, is\n  \\[\n    \\check R_{k,k'}(z)\n    =\\sum_{s=0}^{n}(-1)^{kk'+(k-k')s}\\prod_{j=1}^s\\frac{z-q^{|k-k'|+2j}}{1-z\n      q^{|k-k'|+2j}}Q_s.\n  \\]\n  To deduce a formula in terms of the $U_q{\\mathfrak{gl}}_2$-action we need\n  to compare the normalizations of the highest weight vectors.\n  If $k\\geq k'$ a straightforward computation using the formula for $F$\n  in \\eqref{e-EFK}  shows that\n  \\[\n    v^{(k,k')}_s=(-1)^{sk}q^{ss'}F^{(s)}v^{(k+s,k'-s)}_0,\n\\quad\n    v^{(k',k)}_{s'}=(-1)^{s'k'}q^{ss'}F^{(s')}v^{(k+s,k'-s)}_0.\n  \\]\n  It follows that for $k\\geq k'$\n  \\[\n    \\check R_{k,k'}(z)F^{(s)}v_0^{(k+s,k'-s)}\n    =(-1)^{k'}\\prod_{j=1}^s\\frac{z-q^{|k-k'|+2j}}{1-z\n      q^{|k-k'|+2j}}F^{(s')}v_0^{(k+s,k'-s)}\n  \\]\n  To prove the formula for $\\check R$ given in the Theorem\n  we need to compute the action of $A_{k-k'}(z)$ on the irreducible\n  representation generated by the highest weight vector\n  $v^{(k+s,k'-s)}_0$ of $\\mathfrak{sl}_2$-weight\n  $\\ell=k-k'+2s\\in\\mathbb Z_{\\geq0}$.  Comparing with the formula for\n  $A_m(z)$ from the Appendix (see Lemma \\ref{l-A1}) we see that\n  $\\check R_{k,k'}(z)$ acts by the same factor on $F^{(s)}v_0^{(k+s,k'-s)}$\n  up to the sign $(-1)^{k'}=(-1)^{\\min(k,k')}$.\n\n  If $k\\leq k'$ the reasoning is similar: in this case the\n  highest weight vector $v^{(k,k')}_s$ is proportional\n  to $F^{(s')}v_0^{(k'+s,k-s)}$ with $s'=k'-k+s$ and\n  \\[\n    v^{(k,k')}_s=(-1)^{sk'}q^{ss'}F^{(s')}v^{(k'+s,k-s)}_0, \\quad\n    v^{(k',k)}_{s'}=(-1)^{s'k}q^{ss'}F^{(s)}v^{(k'+s,k-s)}_0.\n  \\]\n  From this we obtain\n  \\[\n    \\check R_{k,k'}(z)F^{(s')}v_0^{(k'+s,k-s)}\n    =(-1)^{k}\\prod_{j=1}^s\\frac{z-q^{|k-k'|+2j}}{1-z\n      q^{|k-k'|+2j}}F^{(s)}v_0^{(k'+s,k-s)},\n  \\]\n  which is the same as the action of $A_{k-k'}(z)$ up to the sign\n  $(-1)^{k}=(-1)^{\\min(k,k')}$.\n\\end{proof}\n\n\\section{Dynamical Weyl group}\\label{s-4}\nIn Theorem \\ref{t-1} the braiding matrix for the tensor product of two\nrepresentations of $U_qL\\mathfrak{gl}_N$ obtained from exterior powers\nof the vector representation of $U_q\\mathfrak{gl}_N$ are expressed\nin terms of the action of $U_q\\mathfrak{gl}_2$ via Howe duality.\nIn this section we consider braiding matrices on the tensor product of $M$\nexterior powers and notice that they are correspondingly expressed\nin terms of the action of $U_q\\mathfrak{gl}_M$ via the simple root embeddings of\n$U_q{\\mathfrak{gl}}_2$.\nThe braid and inversion relations of the $\\check R$-matrices acting on\na $M$-fold tensor product  is then translated by Howe duality to braid and inversion\nrelations in $U_q{\\mathfrak{gl}}_M$.\nThe result is then that these are relations of the\ndynamical Weyl group.\n\nRecall that the (ordinary) Weyl group of $\\mathit{GL}_M$ is\nthe symmetric group $S_M$. It acts on $\\mathbb C^M$ and\nthe weight lattice $\\mathbb Z^M$ by permutations. It is generated by the transpositions $s_i\\in S_M$ of\n$i$ and $i+1$ ($i=1,\\dots,M-1$) with the relations $s_i^2=1$, $s_is_j=s_js_i$ for\n$|i-j|\\geq 2$ and $s_is_{i+1}s_i=s_{i+1}s_is_{i+1}$ for\n$1\\leq i\\leq M-2$; the dynamical Weyl group relations are a version\nwith parameters.\n\nLet us define formal series $A_{i,\\mu}(z)$ of rational functions of\n$M$-variables $z=(z_1,\\dots,z_M)$ with values in\n$U_q{\\mathfrak{sl}}_M\\subset U_q{\\mathfrak{gl}}_M$ for $1\\leq i\\leq M-1$ and\n$\\mu\\in P_M=\\mathbb Z^M$:\n\\[\n  A_{i,\\mu}(z)=\\sum_{j=0}^{\\infty}\n  (-q)^{j}\n  \\frac{z_{i+1}-q^{|\\mu_i-\\mu_{i+1}|}z_i}\n  {z_{i+1}-q^{2j+|\\mu_i-\\mu_{i+1}|}z_i}\n  E_i^{(j+(\\mu_{i+1}-\\mu_i)_+)}F_i^{(j+(\\mu_i-\\mu_{i+1})_+)}.\n\\]\nHere we set $(x)_+=\\max(x,0)$ and, as before, we denote the generators\nby capital letters to distinguish them from the generators of\n$U_q{\\mathfrak{gl}}_N$. For any finite dimensional $U_q{\\mathfrak{gl}}_N$-module $U$ of type I\nonly finitely many terms contribute non-trivially to the sum and\n$A_{i,\\mu}(z)$ maps the weight space $U[\\mu]$ of weight $\\mu$ to\n$U[s_i\\mu]$.  Notice that $A_{i,\\mu}(z)$ depends on $\\mu$ through its class\nin the weight lattice $P_M=\\mathbb Z^M\/\\mathbb Z(1,\\dots,1)$ of $\\mathfrak{sl}_M$.\nFor $M=2$, the map $\\mathbb Z^2\\to\\mathbb Z$, $\\mu\\mapsto \\mu_1-\\mu_2$ induces\na bijection $P_2\\cong \\mathbb Z$. With this identification, $A_{1,m}(z_1,z_2)=A_m(z_1\/z_2)$ where\n\\begin{equation}\\label{e-5}\n  A_m(z)= \\sum_{j\\geq0} (-q)^{j}\\frac{1-q^{|m|}z}{1-q^{2j+|m|}z}\n  \\begin{cases}\n    E^{(j)}F^{(j+m)},& \\text{if $m\\geq 0$,}\n    \\\\\n    E^{(j+|m|)}F^{(j)},& \\text{if $m\\leq 0$.}  \\end{cases}\n\\end{equation}\nThe general series $A_{i,\\mu}(z)$ reduces to this case via the\nembedding $j_i\\colon U_q\\mathfrak{sl}_2\\hookrightarrow\nU_q\\mathfrak{sl}_M$ sending $E,F,K^{\\pm1}$ to $E_i,F_i,K_i^{\\pm1}$:\n\\[ A_{i,\\mu}(z)=j_i\\left(A_{\\mu(h_i)}(z_i\/z_{i+1})\\right).\n\\] By construction, see Theorem \\ref{t-1}, the action of\n$A_{i,\\mu}(z)$ on the tensor product of exterior powers\n$\\bigwedge^{\\mu_i}V(z_i)$ via Howe duality is the action of the\nbraiding matrix $\\check R_{\\mu_i,\\mu_{i+1}}(z_i\/z_{i+1})$ on the $i$th\nand $i+1$st factors (up to an irrelevant sign). Note that this tensor\nproduct is the weight space of weight $\\mu$ for the action of\n$U_q{\\mathfrak{gl}}_M$ on the tensor product of $M$ copies of $\\bigwedge V$.\n\n\n\n\\begin{theorem}\\label{t-2} Let $U$ be any finite dimensional\n$U_q{\\mathfrak{gl}}_M$-module of type I. Then the restriction of $A_{i,\\mu}(z)$ to\nthe weight space $U[\\mu]$ of weight $\\mu$ obeys the following\nrelations.\n  \\begin{enumerate}\n   \\item[(i)] For $1\\leq i,j\\leq M-1$ such that $|i-j|\\geq 2$,\n     \\[ A_{i,s_i\\mu}(s_jz)A_{j,\\mu}(z) =\nA_{j,s_j\\mu}(s_iz)A_{i,\\mu}(z),\n     \\] in $\\operatorname{Hom}(U[\\mu],U[s_is_j\\mu])$.\n    \\item[(ii)] For all $1\\leq i\\leq M-1$,\n      \\[ A_{i,s_i\\mu}(s_iz)A_{i,\\mu}(z) =\\mathrm{id}\n      \\] in $\\operatorname{End}(U[\\mu])$.\n   \\item[(iii)] For $1\\leq i\\leq M-2$,\n     \\begin{align*} A_{i,s_{i+1}s_i\\mu}&(s_{i+1}s_iz)\nA_{i+1,s_i\\mu}(s_iz)A_{i,\\mu}(z) \\\\ &=\nA_{i+1,s_is_{i+1}\\mu}(s_is_{i+1}z)\nA_{i,s_{i+1}\\mu}(s_{i+1}z)A_{i+1,\\mu}(z),\n     \\end{align*} in $\\operatorname{Hom}(U[\\mu],U[s_is_{i+1}s_i\\mu])$.\n   \\end{enumerate}\n \\end{theorem}\n\n \\begin{proof} By complete reducibility we may assume that $U$ is\nirreducible.  The relations hold for the action of $A_{i,\\mu}$ on\n$\\oplus_{i=1}^M\\bigwedge^{\\mu_i}V(z_i)$ via Howe duality: (i) follows\nfrom the fact that the corresponding $\\check R$-matrices act on\ndifferent factors of the tensor product, (ii) is the inversion\nrelation \\eqref{e-4} and (iii) is the braiding relation \\eqref{e-3}.\nThus the relations hold for any irreducible $U_q{\\mathfrak{gl}}_M$-module\noccurring in the decomposition of the tensor product. But from\nProposition \\ref{p-5} we see that any  partition does occurs as a highest weight if we\ntake $N$ sufficiently large. This covers all type I irreducible\nrepresentations of $U_q\\mathfrak{sl}_M$. For $U_q{\\mathfrak{gl}}_M$ we can obtain\nall representations by tensoring these with a power of the one\ndimensional representations on which $U_q\\mathfrak{sl}_M$ acts\ntrivially (i.e. by the counit) and $t_i$ act by $q^{-1}$ for all\n$i$. This has the effect of shifting the $\\mu_i$s by a common amount\nand does not affect the validity of the claim, since $A_{i,\\mu}(z)$\ndepends on differences of $\\mu_i$s.\n \\end{proof} This result can be reformulated as the construction of a\nrepresentation of the Weyl group on $U$-valued functions. Let $U$ be a\nfinite-dimensional type I $U_q{\\mathfrak{gl}}_M$-module with weight decomposition\n$U=\\oplus_\\mu U[\\mu]$. Let $A_{i,U}(z)$ be the\n$\\operatorname{End}U$-valued rational function such that\n\\begin{equation}\\label{e-6}\n  A_{i,U}(z)|_{U[\\mu]}=A_{i,\\mu}(z)\\colon U[\\mu]\\to U[s_i\\mu]\n\\end{equation}\nfor all weights $\\mu$.\n\n\\begin{cor}\\label{c-1} Let $U$ be a finite-dimensional $U_q{\\mathfrak{gl}}_M$-module of type I.\nThe endomorphisms $s_1,\\dots, s_{M-1}$ of the vector space $U(z)$ of\n$U$-valued rational functions in $M$ variables $z$ given by\n  \\[ (s_if)(z)=A_{i,U}(s_iz)f(s_iz),\\quad i=1,\\dots,M-1,\n  \\] define a representation of the symmetric group $S_M$.\n\\end{cor}\n\\begin{proof} The proof is a straightforward verification that the\nrelations of the symmetric group are equivalent to the identities\n(i)--(iii) of Theorem \\ref{t-2}. For example, to check the\nrelation $s_is_{i+1}s_i=s_{i+1}s_is_{i+1}$ we write\n  \\begin{align*} (s_is_{i+1}s_if)(z) &=A_{i,U}(s_iz)(s_{i+1}s_if)(s_iz)\n\\\\ &=A_{i,U}(s_iz)A_{i+1,U}(s_{i+1}s_iz)(s_if)(s_{i+1}s_iz) \\\\\n&=A_{i,U}(s_iz)A_{i+1,U}(s_{i+1}s_iz)A_{i,U}(s_is_{i+1}s_iz)f(s_{i}s_{i+1}s_iz).\n  \\end{align*}\n  We get the left-hand side of (iii) if we replace $z$ by $ s_is_{i+1}s_iz$ and\n  restrict to the weight space $U[\\mu]$.\n\\end{proof}\n\nWe notice that the operators $A_{i,U}(z)$ have similar properties as\nthe operators of the dynamical Weyl group of $U$ introduced and\nstudied in \\cite{TarasovVarchenko2000,EtingofVarchenko2002} in the\nframework of the theory of intertwiners. As in the case of the\ndynamical Weyl group, their restriction to weight spaces are the image\nof elements of $U_q\\mathfrak{sl}_2$ by embeddings\n$U_q\\mathfrak{sl}_2\\hookrightarrow U_q\\mathfrak{sl}_M$ corresponding\nto simple roots \\footnote{In \\cite{EtingofVarchenko2002} the dynamical\n  Weyl group is defined for any semisimple Lie algebra.}. They obey\nrelations (i) and (iii) of Theorem \\ref{t-2} but not (ii). In\nother words they define a representation of the braid group $B_M$ on\nrational $U$-valued functions. In this representation the squares\n$s_i^2$ of the generators act by multiplication by  a rational function on each weight\nspace.\n\nLet us compare our operators $A_{i,U}(z)$ with the operators\n$\\mathscr A_{s_i,U}(\\lambda)$ of \\cite[Section\n4.2]{EtingofVarchenko2002} defining the dynamical action of generators,\nin the case of $U_q\\mathfrak{sl}_M$. The\nlatter operators are initially defined as functions of sufficiently\nlarge antidominant weights $\\lambda$. They extend (upon choosing a\nlogarithm of $q$) to meromorphic functions of\n$\\lambda\\in \\mathfrak h^*$ in the dual of the Cartan subalgebra\n$\\mathfrak h=\\{x\\in\\mathbb C^M\\,|\\,\\sum_i x_i=0\\}$ of\n$\\mathfrak{sl}_M$.  Let $h_i=(0,\\dots,0,1,-1,0,\\dots,0)$,\nwith $1$ in the position $i=1,\\dots,M-1$, be the basis of coroots. To relate $z$ to $\\lambda$\nwe identify $\\mathfrak h^*$ with $\\mathbb C^M\/\\mathbb C(1,\\dots, 1)$\nand write $z=q^{2\\lambda}=(q^{2\\lambda_1},\\dots,q^{2\\lambda_M})$. This\nis defined up to simultaneous scaling of the variables\n$z_i$. Since $A_{i,U}(z)$ is a function of the ratios of $z_j$\n$A_{i,U}(q^{2\\lambda})$ is well-defined.\n\\begin{prop}\\label{p-3}\n  \\[\n    A_{i,U}(q^{2\\lambda})|_{U[\\mu]}=\n    \\begin{cases}\n      q^{-\\mu(h_i)}\\mathscr{A}_{s_i,U}(\\lambda), &\\text{if $\\mu(h_i)\\geq0$,}\n      \\\\\n\\displaystyle   (-1)^{\\mu(h_i)}   q^{-\\mu(h_i)}\\frac{[\\lambda(h_i)-\\mu(h_i)\/2]_q}{[\\lambda(h_i)+\\mu(h_i)\/2]_q}\n      \\mathscr{A}_{s_i,U}(\\lambda),&\\text{if\n        $\\mu(h_i)\\leq 0$}.\n    \\end{cases}\n  \\]\n\\end{prop}\n\\begin{proof}\n  Since $A_{i,\\mu}$ are images of elements of $U_q\\mathfrak{sl}_2$ by\n  embeddings into $U_q\\mathfrak{sl}_M$, it is sufficient to consider\n  the case $M=2$ and we set $\\lambda=\\lambda_1-\\lambda_2$ and $z=q^{2\\lambda}=z_1\/z_2$.\n  By complete reducibility we may also assume that\n  $U=L_\\ell$ is the irreducible representation of $U_q\\mathfrak{sl}_2$\n  of highest weight $\\ell$ and dimension $\\ell+1$. It has\n  one-dimensional weight spaces $L_\\ell[\\ell-2k]$ with basis\n  $v_{\\ell-2k}=F^{(\\ell)}v_\\ell$, ($k=0,\\dots,\\ell$), in terms of the\n  highest weight vector $v_\\ell$.  In this case we have a unique\n  generator $s_1$ sending $x\\in\\mathfrak h^*$ to $-x$.  The following\n  formula for $\\mathscr A_{s_1,L_\\ell}(\\lambda)$ can be extracted from\n  \\cite{EtingofVarchenko2002}, see Corollary 8 (iii), Proposition 12\n  and the definition of the dynamical action of the braid group at the\n  end of Section 4.2:\n  \\[\n    \\mathscr A_{s_1,L_{\\ell}}(\\lambda)v_{m}= (-1)^kq^{m}\\prod_{j=1}^{k}\n    \\frac{[\\lambda+k-j+\\ell\/2]_q} {[\\lambda-k+j-\\ell\/2]_q}v_{-m},\\quad\n    m=\\ell-2k.\n  \\]\n  By setting $z=q^{2\\lambda}$, this can be rewritten as\n  \\[\n    \\mathscr A_{s_1,L_{\\ell}}(\\lambda)v_{\\ell-2k}=\n    (-1)^kq^{m+k(\\ell-k+1)}\\prod_{j=0}^{k-1}\\frac{1-z\n      q^{-\\ell+2j}}{1-zq^{\\ell-2j}} v_{2k-\\ell}.\n  \\]\n  On the other hand, $A_{1,L_\\lambda}(z)$ acts on the weight $m$\n  subspace of any representation as $A_m(z)$, see \\eqref{e-5}.\n\n  It follows from Lemma \\ref{l-A1} that\n  $A_m(z)=\\varphi_m(z)\\mathscr A_{s,L_\\ell}(q^{2\\lambda})$ with\n  $\\varphi_m(z)=q^{-m}$ for $m\\geq0$. If $m\\leq0$, \\dontprint{\n    \\begin{align*}\n      \\varphi_m(z)\n      &=\n        q^{-m+\\frac14(\\ell-|m|)(\\ell+|m|+2)-\\frac14(\\ell+|m|)(\\ell-|m|+2)}\n        \\frac{\n        \\prod_{j=0}^{\\frac{\\ell-|m|}2-1}\\frac{1-zq^{-\\ell+2j}}{1-zq^{\\ell-2j}}\n        }\n        {\n        \\prod_{j=0}^{\\frac{\\ell+|m|}2-1}\\frac{1-zq^{-\\ell+2j}}{1-zq^{\\ell-2j}}\n        }\n      \\\\\n      &=\n        q^{-m+\\frac14(\\ell-|m|)(\\ell+|m|+2)-\\frac14(\\ell+|m|)(\\ell-|m|+2)}\n        \\prod_{j=\\frac{\\ell-|m|}2}^{\\frac{\\ell+|m|}2-1}\\frac{1-zq^{\\ell-2j}}{1-zq^{-\\ell+2j}}\n      \\\\\n      &=\\frac{1-zq^{-m}}{1-zq^{m}}.\n    \\end{align*}\n  }\n  \\[\n    \\varphi_m(z) =(-1)^m\n    \\prod_{j=\\frac{\\ell-|m|}2}^{\\frac{\\ell+|m|}2-1}\\frac{1-zq^{\\ell-2j}}{1-zq^{-\\ell+2j}}\n    =(-1)^m\\frac{1-zq^{-m}}{1-zq^{m}}, \\quad m\\leq 0.\n  \\]\n  so that\n  $\\varphi_m(q^{2\\lambda})=(-1)^mq^{-m}\\frac{[\\lambda-m\/2]_q}\n  {[\\lambda+m\/2]_q}$.\n\\end{proof}\n\\subsection{$B$-operators, quantum Weyl group, Yangian limit}\nThe limiting values of $R$-matrices as the spectral parameter tends to $0$ or $\\infty$ are\nsolutions of the Yang--Baxter equations without spectral parameter. They are $R$-matrices\nfor quantum enveloping algebras of finite dimensional Lie algebras.\nBy dividing the $A$-operators by their limiting values we obtain weight zero ``$B$-operators''\n\\cite{EtingofVarchenko2002}.\nDefine a formal series $B_m(z)$ by the formula\n\\begin{equation}\\label{e-7}\n  A_m(z)=A_{m}(0)B_m(z).\n\\end{equation}\nThe series $B_m(z)$ is well-defined as an endomorphism of the\n$m$-weight space of any finite dimensional representation and one has\na universal formula:\n\\begin{prop}\\label{p-6}\\\n  \\begin{enumerate}\n    \\item[(i)]\n  For $m\\geq0$,\n  \\[\n    B_m(z)=\\sum_{j=0}^\\infty\n    q^{\\frac{j(j-3)}2}\\frac{(-z)^j}{[j]_q!}F^jE^j\\prod_{i=1}^j\\frac{1-q^2}{1-z\n      q^{m+2i}}\n  \\]\n    \\item[(ii)]\n  For $m\\leq0$,\n  \\[\n    B_m(z)=\\sum_{j=0}^\\infty\n    q^{\\frac{j(j-3)}2}\\frac{(-z)^j}{[j]_q!}E^jF^j\\prod_{i=1}^j\\frac{1-q^2}{1-z\n      q^{-m+2i}}\n  \\]\n\\end{enumerate}\n\\end{prop}\n\\begin{proof}\n  The first formula can be checked directly or deduced from the remark\n  after Corollary 40 in \\cite{EtingofVarchenko2002}, by using\n  Proposition \\ref{p-3}. There is a similar formula in {\\it\n    loc.~cit.}, Proposition 14, but probably the powers of $q$ need to\n  be adjusted there. The second formula follows by applying the Cartan involution, see\n  Corollary \\ref{c-A1}\n\\end{proof}\nFor $m=k-k'$, these operators define via Howe duality\n$U_q{\\mathfrak{gl}}_N$-linear endomorphisms of\n$\\bigwedge^{k} V(z_1)\\otimes\\bigwedge^{k'}V(z_2)$. Their\nrepresentation theoretical meaning in terms of $U_qL{\\mathfrak{gl}}_N$ is not\nclear but in the limit $q\\to 1$, $A_{m}(0)$ tends up to signs to the\npermutation $P$ and in a suitable $z\\to 1$ limit, $B_m(z)$ converges\nto a rational $R$ matrix, intertwining the coproduct and the opposite\ncoproduct of the Yangian. More precisely we let\n$q=\\exp(\\epsilon\\hbar)$, $z=\\exp(2\\epsilon u)$ and consider the limit\n$\\epsilon\\to0$:\n\\begin{theorem}\\label{t-3}\n  Let $\\check R_{k,k'}(z,q)$ be the family of braiding matrices of\n  Theorem \\ref{t-1}, viewed as a linear map on the underlying vector\n  space $\\bigwedge^kV\\otimes \\bigwedge^{k'}V$.  Let\n  \\[\n    R_{k,k'}(z,q)=P\\check R_{k,k'}(z,q)\\in\\textstyle{\\operatorname{End}_{\\mathbb\n    C}(\\bigwedge^kV\\otimes \\bigwedge^{k'}V)},\\text{ with\n    $Pv'\\otimes v=(-1)^{kk'}v\\otimes v'$,}\n\\]\nbe the corresponding\n  $R$-matrix, normalized to be the identity on the product of highest\n  weight spaces.  Then the limit\n  \\[ R^{\\mathrm{rat}}_{k,k'}(u,\\hbar)=\\lim_{\\epsilon\\to\n0}R_{k,k'}(e^{\\hbar \\epsilon},e^{2\\epsilon u})\n  \\] exists and is a ${\\mathfrak{gl}}_N$-invariant solution of the Yang--Baxter\nequation. Moreover\n  \\[ R^{\\mathrm{rat}}_{k,k'}(u,\\hbar)=\n\\sum_{j\\geq0}\\frac{(-\\hbar)^j}{j!}\\prod_{i=1}^j\\frac1{u+\\hbar(i+|k-k'|\/2)}\n    \\begin{cases} F^jE^j&\\text{if $k\\geq k'$,} \\\\ E^jF^j&\\text{if\n$k\\leq k'$.}\n    \\end{cases}\n  \\] Here $E=\\sum_{i=1}^N\\psi^*_i\\otimes\\psi_i$,\n$F=-\\sum_{i=1}^N\\psi_i\\otimes\\psi_i^*$ are the $q\\to 1$ limits of the\n$U_q\\mathfrak{sl}_2$-generators of Section \\ref{s-2}.\n\\end{theorem}\n\\begin{proof} The limit exists since both $E, F$ and the coefficients\nin the formula for $\\check R_{k,k'}$ have a limit as $\\epsilon\\to\n0$. The Yang--Baxter equation then follows from the braiding relation \\eqref{e-3}\nof $\\check R_{k,k'}$.  The action of the subalgebra $U_q{\\mathfrak{gl}}_N$ does\nnot involve the spectral parameter and becomes an action of $U{\\mathfrak{gl}}_N$,\nso that the limit is ${\\mathfrak{gl}}_N$-invariant.\n\nIn a neighbourhood of $q=1$, $\\check R_{k,k'}(z,q)$ is holomorphic in\nthe disk $|z|<1\/2$. Its value at $q=1$ is independent of $z$ and is a\nhomomorphism of $L{\\mathfrak{gl}}_N$-modules, which are irreducible for\n$z=z_1\/ z_2\\neq1$. By Schur's lemma\n$\\check R_{k,k'}(z,1)=\\check R_{k,k'}(0,1)$ must be proportional to\nthe homomorphism $P\\colon u\\otimes v\\mapsto (-1)^{kk'}v\\otimes\nu$. With the normalization of Theorem \\ref{t-1}, the factor of proportionality is 1.\nIt  follows that\n\\[\n  R^{\\mathrm{rat}}_{k,k'}(u,\\hbar) = \\lim_{\\epsilon\\to 0}\\check\n  R_{k,k'}(0,e^{\\epsilon \\hbar})^{-1} \\check R_{k,k'}(e^{2\\epsilon\n    u},e^{\\epsilon\\hbar}).\n\\]\nBy Theorem \\ref{t-1} this can be expressed in terms of\n$A_m(z)=A_m(z,q)$ for $m=k-k'$:\n\\[\n  R^{\\mathrm{rat}}_{k,k'}(u,\\hbar) = \\lim_{\\epsilon\\to\n    0}A_m(0,e^{\\epsilon\\hbar})^{-1}A_m(e^{2\\epsilon\n    u},e^{\\epsilon\\hbar}) = \\lim_{\\epsilon\\to 0}B_m(e^{2\\epsilon\n    u},e^{\\epsilon\\hbar}).\n\\]\nThen, in the limit $\\epsilon\\to 0$, $E=\\sum\\psi_i\\otimes\\psi_i^*$ and\n$F=-\\sum\\psi_i^*\\otimes\\psi_i$ obey $\\mathfrak{sl}_2$ commutation\nrelations and, by Proposition \\ref{p-6}, $B_m$ tends to\n\\[\n  \\sum_{j=0}^\\infty\\frac{(-\\hbar)^j}{j!}F^jE^j\\prod_{i=1}^j\\frac1{u+\\hbar(i+m\/2)}\n\\]\nfor $m\\geq0$ and\n\\[\n  \\sum_{j=0}^\\infty\\frac{(-\\hbar)^j}{j!}E^jF^j\\prod_{i=1}^j\\frac1{u+\\hbar(i-m\/2)}\n\\]\nfor $m\\leq 0$.\n\\end{proof}\n\\begin{remark} If $k=k'$ we recover the formula proposed by Smirnov, see \\cite{Smirnov2016}, eq.~(115).\n  Understanding this formula was part of the motivation of the present work. For general $k,k'$ our formulas seem\n  to be related to the formula in \\cite{Smirnov2016} by a shift of spectral parameter which could be partly explained by\n  a redefinition of exterior powers.\n\\end{remark}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Basic concepts}\t\n\\label{sec:Preliminaries}\n\n We start by introducing the notation used throughout the manuscript along with some basic concepts and definitions regarding symmetric states, entanglement witnesses and copositive matrices.\n\n\\subsection{Symmetric states}\nHenceforth, we focus on bipartite systems. \nLet $\\mathcal H=\\mathbb{C}^{d}\\otimes \\mathbb{C}^{d}$, be the finite dimensional Hilbert space of two qudits, and $\\mathcal{B}(\\mathcal{H})$ the set of its bounded operators. The symmetric subspace $\\mathcal {S} \\subset\\mathcal{H}$, is the convex set formed by the (normalized) pure states $\\ket{\\Psi_S}\\in \\mathcal H$ that are invariant under the exchange of parties. Symmetric states can be mapped to spin systems and, moreover, they span the subspace of maximum spin in the Schur-Weyl representation. The so-called Dicke states form a convenient basis of the symmetric subpace, i.e.,  $\\mathcal{S}\\equiv \\text{span} \\{\\ket{D_{ij}}\\}$, where $\\ket{D_{ij}} =\\ket{D_{ji}}\\equiv(\\ket{ij} + \\ket{ji})\/\\sqrt{2}$ for $i \\neq j,\\; \\ket{D _{ii}} \\equiv \\ket{ii}$ and $\\{\\ket{i}\\}_{i=0}^{d-1}$ is an orthonormal basis of $\\mathbb{C}^{d}$. \nNote the reduced dimensionality of the symmetric subspace, i.e., $\\dim(\\mathcal{S})=d(d+1)\/2 <d^2$. In an abuse of language we denote symmetric quantum states, $\\rho_S\\in \\mathcal{B}(\\mathcal{S})$, as the convex hull of projectors onto pure symmetric normalized states, i.e.,\n$\\rho_{S} =\\sum_{k} p^{(k)}_{S} \\ket{\\Psi^{(k)}_S}\\bra{\\Psi^{(k)}_S}$, with $p^{(k)}_{S} \\geq 0$, $\\sum_{k} p^{(k)}_{S} =1$ and \n$\\ket{\\Psi^{(k)}_{S}}=\\sum_{ij} c^{(k)}_{ij} \\ket{D_{ij}}$, $c^{(k)}_{ij}\\in \\mathbb{C} $.\\\\ Thus, any $\\rho_S\\in \\mathcal{B}(\\mathcal{S})$ is a positive semidefinite operator ($\\rho_S\\succeq 0$) with unit trace ($\\mathrm{Tr} (\\rho_S) = 1$), fulfilling $\\Pi_{S} \\rho_{S} \\Pi_{S} = \\rho_{S} \\Pi_{S} = \\Pi_{S} \\rho_{S}= \\rho_{S}$, where $\\Pi_{S} = \\frac{1}{2}(\\mathds{1} + F)$ is the projector onto the symmetric subspace and $F = \\sum_{i,j = 0}^{d-1} \\ket{ij}\\!\\bra{ji}$ is the flip operator. Using the Dicke basis, symmetric quantum states can be compactly \nexpressed as follows: \n {\\definition{\n \\label{def:SS}\nAny bipartite symmetric state,  $\\rho_{S}\\in \\mathcal{B}(\\mathcal{S})$, can be written as \n\\begin{equation}\n\\rho_{S} = \\sum_{\\substack{0\\leq i\\leq j<d\\\\\n                  0\\leq k\\leq l<d}}\n               \\left(\\rho_{ij}^{kl} \\ket{D_{ij}}\\bra{D_{kl}} + \\emph{h.c.} \\right)~,\n\\end{equation}\nwith $ \\rho_{ij}^{kl} \\in \\mathbb{C}$. Notice that, due to the symmetry of the Dicke states, it holds that $ \\rho_{ij}^{kl} = \\rho_{ji}^{kl}= \\rho_{ij}^{lk}= \\rho_{ji}^{lk} ~\\forall i,j,k,l$.\\\\\n}}\n\n\\noindent Convex mixtures of projectors onto Dicke states are denoted as diagonal symmetric (DS) states, since they are diagonal in the Dicke basis. They form a convex subset of $\\mathcal{S}$ and are particularly relevant for our analysis.\n{ \\definition{\n \\label{def:DSS}\nAny DS state, $\\rho_{\\small{DS}} \\in \\mathcal{B}(\\mathcal{S})$, is of  the form\n \\begin{equation}\n  \\rho_{DS} = \\sum_{0 \\leq i \\leq j < d} p_{ij}\\ket{D_{ij}}\\bra{D_{ij}},\n  \\label{eq:DSS}\n \\end{equation}\nwith $p_{ij} \\geq 0,\\; \\forall\\; i,j$ and $\\sum_{ij} p_{ij}=1$.}}\n\n \\begin{lemma}\n Every symmetric state, $\\rho_{S} \\in \\mathcal{B}(\\mathcal{S})$, can be written as the sum of a DS state, $\\rho_{DS}$, and a traceless symmetric contribution, $\\sigma_{CS}$, which contains all coherences between Dicke states, i.e.,\n\\begin{equation}\n\\label{sym}\n\\rho_{S} = \\rho_{DS}+\\sigma_{CS}= \n\\sum_{0\\le i \\le j < d} p_{ij} \\ket{D_{ij}} \\bra{D_{ij}} + \\underset{{(i,j)\\neq(k,l)}}{\\sum_{ij}\\sum_{kl}} \\left( {\\alpha_{ij}^{kl}} \\ket{D_{ij}} \\bra{D_{kl}} + \\emph{h.c.} \\right),\n\\end{equation}\n\\noindent with  ${\\alpha_{ij}^{kl}} \\in \\mathbb{C}$ and ${\\alpha_{ij}^{kl}} = ({\\alpha_{kl}^{ij}})^*$.\n\\end{lemma}\n\n\n\n\\subsection{ Separability, EWs and copositivity} \n\n\n{\\definition{A bipartite symmetric state $\\rho_{S} \\in \\mathcal{B}(\\mathcal{S})$ is separable (not entangled) if it can be written as a convex combination of projectors onto pure symmetric product states, \\textit{i.e.,}}\n\\begin{equation}\n\\rho_{S}= \\sum_{i} p_i \\ket{e_{i}e_{i}}\\bra{e_{i}e_{i}},\n\\label{eq:sep}\n\\end{equation}\nwith $p_i\\geq 0$, $\\sum_i p_i=1$ and $\\ket{e_{i}}=\\sum_{i} e_{i}^{(k)}\\ket{k}$, where $e_{i}^{(k)} \\in \\mathbb{C}$ and $\\{\\ket{k}\\}_{k=0}^{d-1}$ is an orthonormal basis in $\\mathbb{C}^d$. If a decomposition of this form does not exist, then $\\rho_{S}$ is \\textit{entangled}.}\n\n\\begin{figure}[H]\n\t\\centering\n\t\\includegraphics[width=4cm]{FIGURESEPARABLE2.pdf}\n\t\\caption{Pictorial representation of the set of bipartite symmetric separable states $\\mathcal{D}_S$ (Sym) embedded into the set of bipartite separable states. The cylinder represents the separable set $\\mathcal{D}$. The discontinuous (red) line corresponds to the extremal points (of the form $\\ket{e,f}$) generating the set and the continuous (blue and green) lines corresponds to the respective boundaries (necessarily requiring description as density matrices with rank $>1$ but not maximal). Both the separable and the symmetric separable sets share extremal points of the form $\\ket{e,e}$, here represented by the black dots.}\n\t\\label{fig:separable}\n\\end{figure}\n\n\nWe denote by ${\\mathcal D}$, the compact set of separable quantum states and by $\\mathcal D_{S}$, its analogous symmetric counterpart, which is also compact (see Fig.\\ref{fig:separable}). As a consequence of the Hahn-Banach theorem, the set $\\mathcal D_{S}$ admits also a dual description in terms of its dual cone, $\\mathcal{P_{S}}$, defined as the set of the operators $W$ fulfilling\n\\be\n\\mathcal {P}_{S}= \\{ W=W^{\\dagger}\\;\\;\\mbox{s.t}\\; \\langle W, \\rho \\rangle  \\geq 0 ~, \\forall \\rho_{S} \\in \\mathcal{D}_{S}\\}~,\n\\end{equation}\nwhere $\\langle W, \\rho \\rangle \\equiv \\mbox{Tr} (W^{\\dagger}\\rho)$ is the Hilbert-Schmidt scalar product. \n\n\n{\\definition{A Hermitian operator, $W \\in \\mathcal P_{S}$, is an entanglement witness (EW) of symmetric states if, and only if, it satisfies the following properties: \n\\begin{itemize}\n\t\\item[1.]  $\\emph{Tr}(W\\rho_{S})\\geq 0, ~ \\forall   \\;\\rho_{S}\\in\\mathcal D_{S}$ ,~\n\t\\item[2.] There exists at least one symmetric state $\\rho_{S}$ such that $\\emph{Tr}(W \\rho_{S}) < 0$~. \n\\end{itemize}}}\n\n\\noindent Notice that, by definition, the set of separable symmetric states, $\\mathcal D_{S}$, satisfies the inclusion $\\mathcal D_{S}\\subset \\mathcal D$, but $\\mathcal P \\subset \\mathcal P_S$, where $\\mathcal {P}$ is the dual cone of the convex set $\\mathcal D$, i.e., \n\\begin{equation}\n    \\mathcal {P}= \\{ W=W^{\\dagger}\\;\\; \\mbox{s.t}. \\; \\langle W, \\rho \\rangle \\geq 0~, \\forall \\rho \\in \\mathcal{D}\\}~.\n\\end{equation}\n\nIn other words, any EW acting on $\\mathcal{H}$ that detects an entangled state belongs to $\\mathcal P_S$, but the converse is not necessarily true (see Fig.\\ref{fig:planetary}).\n\\begin{figure}[H]\n\t\\centering\n\t\\includegraphics[width=7cm]{planetary.pdf}\n\t\\caption{Pictorial structure of the quantum states in $\\mathbb{C}^d\\otimes\\mathbb{C}^d$ for $d>5$. Each set contains the sets displayed inside. The colored region (green) represents the set of symmetric states (SYM). Note that, while for $d>5$ there exist diagonal symmetric (DS) states that are PPT-entangled, as represented in the figure, for $d<5$ all PPT-entangled DS state are necessarily separable (SEP) (see the text for details).}\n\t\\label{fig:planetary}\n\\end{figure}\n\\noindent EWs are either decomposable or non-decomposable.\n\\begin{definition}\n\\label{dec}\nAn EW, $W$, is said to be decomposable (non-decomposable) if it can (cannot) be written as\n\\begin{equation}\nW = P + Q^{T_{B}},\n\\end{equation}\nwith $P,Q \\succeq 0$. Here $T_{B} \\equiv \\mathds{1}_{A} \\otimes T$ denotes the partial transposition w.r.t. subsystem $B$, where $T$ stands for the usual matrix tranposition.\\\\\n\\end{definition}\n\\noindent It is easy to show that non-decomposable EWs are the only candidates to detect PPT entanglement. In fact, given a PPTES $\\rho$, for any decomposable EW $W$, it is\n\\begin{equation}\n\\mbox{Tr}(W \\rho) = \\mbox{Tr}(P \\rho) + \\mbox{Tr}(Q^{T_{B}}\\rho) = \\mbox{Tr}(P \\rho) + \\mbox{Tr}(Q \\rho^{T_{B}}) \\geq 0~,\n\\end{equation}\n\\noindent where we have used the properties of the trace and the positive semidefiniteness of the operators $P$ and $Q$.\\\\\n\\noindent In particular from Def.(\\ref{dec}) it follows that a EW is non-decomposable iff it detects at least one PPTES.\\\\\n\n\nRemarkably, despite the apparent simplicity of the symmetric subspace due to its reduced dimensionality ($d(d+1)\/2$ instead of $d^{2}$), entanglement characterization remains, in general, an open problem. For generic symmetric states, sparsity is preserved when the state is expressed in the computational basis but it is lost when the partial transposition is performed. However, for DS states, the corresponding partial transpose remains highly sparse and can be reduced to an associated matrix, $M_d(\\rho_{DS})$, of dimensions $d \\times d$, while generically $\\rho_{S}^{T_{B}}$ is a matrix of dimension $d^2 \\times d^2$.\n{\\definition {The partial transpose of every $\\rho_{DS} \\in \\mathcal{B}(\\mathcal{S})$ has the form\n\\begin{equation}\n\\label{transco}\n\\rho_{DS}^{T_{B}} = M_{d}(\\rho_{DS}) \\bigoplus_{0\\le i \\neq j < d} \\left(p_{ij}\/2\\right)~,\n\\end{equation}\nwhere  $M_{d}(\\rho_{DS})$, which arises from the partially transposed matrix of a DS state in the computational basis, is defined as the $d\\times d$ matrix with (non-negative) entries\n\\begin{equation}\n\\label{mr}\nM_{d}(\\rho_{DS}) :=\n\\left(\\begin{matrix}\np_{00} \t& p_{0 1}\/2  & \\cdots & p_{0, d-1}\/2 \\\\\np_{0 1}\/2  & p_{11} & \\cdots & p_{1, d-1}\/2 \\\\\n\\vdots \t& \\vdots & \\ddots & \\vdots \\\\\np_{0, d-1}\/2 & p_{1, d-1}\/2 & \\cdots & p_{d-1, d-1} \n\\end{matrix}\\right) ~.\n\\end{equation}}}\n\n\nAs shown in previous works \\cite{yu2016separability,tura2018separability}, \ndeciding if a DS state $\\rho_{DS}$ is separable, is equivalent to check the membership of $M_{d}( \\rho_{DS})$ to the cone of completely positive matrices $\\mathcal {CP}_{d}$, i.e., the cone formed by those $d \\times d$ matrices $A_{d}$ that admit a decomposition  of the type $A _{d}= B B^{T}$, where $B$ is a $d \\times k$ matrix, for some $k>1$, with $B_{ij}\\geq 0, B_{ij} \\in \\mathbb{R}$. Thus, if $\\rho_{DS}$ is separable, then its associated matrix of Eq.(\\ref{mr}) must satisfy $M_{d}(\\rho_{DS})=B B^{T}$ \\cite{yu2016separability}. This correspondence can be recast, equivalently, in the dual cone of $\\mathcal {CP}_{d}$, i.e., in the cone  $\\mathcal {COP}_{d}$ of copositive matrices.  As a result, copositive matrices act effectively as EWs for DS states. \nBelow we provide the definition of a copositive matrix together with some properties that will be useful in the following.  \n\n{\\definition{ A real symmetric matrix, $H$, is copositive if, and only if, $\\vec{x}^{T} H \\vec{x} \\geq 0,~ \\forall \\vec{x} \\geq 0$ component-wise}}.\\\\\n\nIt is easy to see that the diagonal elements of a copositive matrix must be non negative, i.e., $H_{ii} \\ge 0$, while negative elements $H_{ij}$ must fulfill $\\sqrt{H_{ii} H_{jj}} \\ge -H_{ij}$. Clearly, every positive semidefinite matrix is also copositive but the converse is, generically, not true. In fact, testing membership to the cone of copositive matrices is known to be a co-NP-hard problem \\cite{murty1985some}\\footnote{The co-NP problems are the complementary of the decision problems in NP.}, and only for $d\\leq 4$, copositivity can be assessed analytically \\cite{ping1993criteria,hiriart2010variational}. \n\nFinally, among copositive matrices, we distinguish extreme and exceptional copositive matrices that stand out for their impossibility to be decomposed.\n\n{\\definition{ A $d\\times d$ copositive matrix $H$ is said to be extreme if $H = H_{1} + H_{2}$ with $H_{1}, H_{2}$ copositive, implies $H_{1} = a H, H_{2} = (1-a)H$ for all $a \\in [0,1]$}}. \n\n{\\definition{ A $d\\times d$ copositive matrix $H$ is said to be \\textit{exceptional} if, and only if, $H$ cannot be decomposed as the sum of a positive semidefinite matrix ($\\mathcal{PSD}_{d}$), and a symmetric entry-wise non-negative matrix ($\\mathcal{N}_{d}$),  i.e., $H \\in \\mathcal {COP}_{d} \\setminus (\\mathcal{PSD}_{d}+ \\mathcal{N}_{d})$}}\\\\\n \n \n\\noindent Remarkably, it has been shown that for $d < 5$ there are no exceptional copositive matrices, \nmeaning that, in this case, $ \\mathcal {COP}_{d} = \\mathcal{PSD}_{d}+ \\mathcal{N}_{d}$ \\cite{diananda1962non}. In\nFig.\\ref{fig:cop} we illustrate, schematically, the relation between the aforementioned classes of copositive matrices.\n\n\\begin{figure}[H]\n\t\\centering\n\t\\includegraphics[width=6.5cm]{Exceptional.pdf}\n\t\\caption{Pictorial representation of the cone of copositive $\\mathcal{COP}_{d}$ and the cones $\\mathcal{PSD}_{d}$ and $\\mathcal{N}_{d}$. The striped region has been overmagnified for clarity and represents the convex hull of the cones $\\mathcal{PSD}_{d}$ and $\\mathcal{N}_{d}$, denoted as $\\mathcal{PSD}_{d}+\\mathcal{N}_{d}$. Note that exceptional copositive matrices exist only for $d>5$ (green). Extremal copositive matrices lie at the border of the cone $\\mathcal{COP}_{d}$.}\n\t\\label{fig:cop}\n\\end{figure}\n \n\\section {Copositive matrices as EWs}\n\\label{sec:EWcopositive}\n\n\nUsing the above definitions we can now show which copositive matrices lead to EWs. We prove explicitly how to construct a decomposable (non-decomposable) EW from a non-exceptional (exceptional) copositive matrix. Since decomposable EWs cannot detect bound entanglement, one is tempted to believe that separability in the symmetric subspace is equivalent to the analysis of exceptional copositive matrices. However, as we shall see later, this is not necessarily the case, and non-exceptional copositive matrices also play a relevant role in detecting bound entanglement.   \nOur findings are summarized in the following theorems.\n\n\\begin{theorem}\n\\label{thext}\nEach copositive matrix $H =\\sum_{i,j=0}^{d-1}  H_{ij} \\ket{i}\\!\\bra{j}$, \nwith at least one negative entry $H_{mn}=H_{nm}<0$ ($m\\neq n$), leads to an EW on $\\mathcal{S}$ of the form $W=(H^{ext})^{T_{B}} =\\sum_{{i,j=0}}^{d-1} H_{ij}\\ket{ij}\\!\\bra{ji}$.\n\\end{theorem} \n\n\\begin{proof} \n\n(i) We extend $H$ to the symmetric subspace as $H^{ext} = \\sum_{i,j= 0}^{d-1} H_{ij} \\ket{i}\\bra{j} \\otimes \\ket{i}\\bra{j}$, and denote $W=(H^{ext} )^{T_B}$. It is straightforward to show that $\\mbox{Tr}(W \\rho_{S}) \\ge 0$ for all $ \\rho_{S} \\in \\mathcal{D}_{S}$ since  $\\bra{e e} W \\ket{e e} = \\braket{e e^{*}| H^{ext}| e e^{*}}=\\sum_{ij}  |c_{i}|^{2} H_{ij} |c_{j}|^{2}= \\vec{x}^{T} H \\vec{x} \\geq 0$, for every state $\\ket{e} = \\sum_{i=0}^{d-1} c_{i} \\ket{i}$, where $c_{i}\\in \\mathbb C$, and $\\{\\ket{i} \\}_{i=0}^{d-1}$ is orthonormal basis of $\\mathbb{C}^{d}$.\n\n(ii) The diagonalization of $W$  shows that its eigenvectors are $\\{ \\ket{ii}, \\ket{\\psi_{ij}^{\\pm}} = (\\ket{ij}\\pm \\ket{ji})\/\\sqrt{2}\\}$, with corresponding eigenvalues $\\{H_{ii}, \\pm H_{ij}\\}$, i.e.,\n\\begin{equation}\n\\label{wtotal}\n\\begin{split}\n\tW &= (H^{ext})^{T_{B}}= \\sum_{i=0}^{d-1} H_{ii} \\ket{ii}\\bra{ii} \n\t+\\\\\n\t&+\\sum_{i < j }^{d-1} H_{ij} \\ket{\\psi_{ij}^{+}}\\bra{\\psi_{ij}^{+}} - \\sum_{i < j }^{d-1} H_{ij} \\ket{\\psi_{ij}^{-}}\\bra{\\psi_{ij}^{-}},\n\\end{split}\n\\end{equation}\nwhere $\\ket{\\psi_{ij}^{+}}=\\ket{D_{ij}}$ and  $\\ket{ii}=\\ket{D_{ii}}$.\nNotice that the $d(d-1)\/2$ eigenvectors corresponding to the projectors $\\ket{\\psi_{ij}^{-}}\\bra{\\psi_{ij}^{-}}$, are orthogonal to the symmetric subspace and, therefore, can be discarded by projecting on the symmetric subspace ($\\mathcal{S}$). \n\\begin{equation}\n\\label{wsym}\n\tW _{S}= \\Pi_{S}W\\Pi_{S}=\\sum_{i=0}^{d-1} H_{ii} \\ket{D_{ii}}\\bra{D_{ii}} \n\t+ \\sum_{i < j}^{d-1} H_{ij} \\ket{D_{ij}}\\bra{D_{ij}}. \n\n\\end{equation}\nFinally, since copositivity requires that $H_{ii}\\geq 0$ $\\forall i$,  $W_{S}$ is an EW iff at least one of the remaining eigenvalues is negative, i.e., if $H$ has at least one negative element $H_{mn}=H_{nm}<0$ for some $m\\neq n$. It is now trivial to see that  $W_{S}$ indeed detects, at least, the entangled state $\\ket{\\psi_{mn}^{+}}$ since $\\mbox{Tr} (W_{S}\\ket{\\psi_{mn}^{+}}\\bra{\\psi_{mn}^{+}} ) = H_{mn} <0$. To conclude, if $W_{S}$ is an EW in the symmetric subspace, so it is $W$ given by \\Cref{wtotal}.\n\\end{proof}\n\n\\begin{theorem}\n\t\\label{the2}\n\t{If $H=H_{\\mathcal{N}} + H_{\\mathcal{PSD}}$ (i.e., H is non-exceptional) with at least one negative element, then  $W=\\Pi_{S} \\left( H^{ext}_{\\mathcal{N}} \\right) ^{T_{B}}\\Pi_{S} + \\Pi_{S}\\left( H^{ext}_{\\mathcal{PSD}} \\right) ^{T_{B}}\\Pi_{S}=\\Pi_{S} \\left( H^{ext}_{\\mathcal{N}} \\right) ^{T_{B}}\\Pi_{S} +\\left( H^{ext}_{\\mathcal{PSD}} \\right) ^{T_{B}}$ is a decomposable EW. The converse is also true, that is, if $W_S=\\Pi_S(H^{ext})^{T_{B}}\\Pi_{S}=P+Q^{T_{B}}$ with $P,Q\\succeq 0$, then $H = H_{\\mathcal{N}} + H_{\\mathcal{PSD}}$}.\n\t\\end{theorem}\nNotice that it is always possible to decompose a copositive matrix in such a way that the semidefinite positive part $H^{ext}_{\\mathcal{PSD}}$ is symmetric.\n\\begin{proof}\n$\\Rightarrow$. Assume that $H =\nH_{\\mathcal{N}}+ H_{\\mathcal{PSD}}$, with $H_{\\mathcal{N}} \\in \\mathcal{N}, ~H_{\\mathcal{PSD}} \\in \\mathcal{PSD}$ and $H_{mn}=H_{nm}<0$. Then\n$(H_{\\mathcal{PSD}})_{mn} =(H_{\\mathcal{PSD}})_{nm}<0$\nby construction and $(H^{ext}_{\\mathcal{PSD}})^{T_{B}} \\nsucceq 0$, but $\\Pi_{S} \\left( H^{ext}_{\\mathcal{N}} \\right) ^{T_{B}}\\Pi_{S}\\succeq 0$. \nThe operator $W=\\Pi_{S} \\left( H^{ext}_{\\mathcal{N}} \\right) ^{T_{B}}\\Pi_{S} + \\left( H^{ext}_{\\mathcal{PSD}} \\right) ^{T_{B}}$ is a decomposable EW: (i) $\\braket{e,e|W| e,e} \\ge 0$ because $H$ is copositive, (ii) $W$ has at least one negative eigenvalue associated to the negative element $(H_{\\mathcal{PSD}})_{mn}$, (iii) $W$ is decomposable since it can be written as $W= P + Q^{T_{B}}$ with $P =\\Pi_{S} \\left( H^{ext}_{\\mathcal{N}} \\right) ^{T_{B}}\\Pi_{S} \\succeq 0 $ and $Q = \\Pi_{S} H^{ext}_{\\mathcal{PSD}}   \\Pi_{S} = H^{ext}_{\\mathcal{PSD}}  \\succeq 0$.\n\n$\\Leftarrow$ Since $W=\\Pi_{S} \\left( H^{ext}\\right)^{T_{B}}\\Pi_{S}$ is an EW, there exists at least one negative element of $H$, $H_{mn}=H_{nm}<0$.  We construct \n$Q=H_{mm}|mm\\rangle\\langle mm| +H_{nn}|nn\\rangle\\langle nn|+H_{mn}(|mn\\rangle\\langle mn| +|nm\\rangle\\langle nm|)$. By construction $Q$ is symmetric and is $Q\\notin\\mathcal{N}$, $Q^{T_B}\\nsucceq 0$ and $Q\\succeq 0$, where the last inequality holds because copositivity of $H$ implies  $\\sqrt{H_{mm} H_{nn}} \\ge -H_{mn}$. Hence, we can identify $Q=H^{ext}_{\\mathcal{PSD}}$. Now, define $P=\\Pi_S (\\sum_{i,j\\neq m,n} H_{ij}|ij\\rangle\\langle ji|)\\Pi_S$, which can be expressed as $P=\\sum_{i,j\\neq m,n} H_{ii}|ii\\rangle\\langle ii|+\\sum_{i,j\\neq m,n}\n(H_{ij}\/2) (|ij\\rangle\\langle ji|+ |ij\\rangle\\langle ij|+|ji\\rangle\\langle ji|+|ji\\rangle\\langle ij|)$. Clearly  $P\\succeq 0, P\\in\\mathcal{N}$, so that \n$P= \\Pi_S (H^{ext}_{\\mathcal{N}})^{T_B}\\Pi_S$ and\n$H = H_{\\mathcal{N}} + H_{\\mathcal{PSD}}$.\n\\end{proof}\n\nLet us illustrate \\Cref{the2} by considering  \nthe following copositive matrix in $d=3$ \n\\begin{equation}\n\tH = \n\t\\begin{pmatrix*}[r]\n\t1 & 1 & 1 \\\\\n\t1 & 1 & -1 \\\\\n\t1 & -1 & 1 \n\t\\end{pmatrix*}~.\n\\end{equation}\nA possible  decomposition of $H = H_{\\mathcal{PSD}}  + H_{\\mathcal{N}}$, with $H_{\\mathcal{PSD}} \\in \\mathcal{PSD}_{3}$ and $ H_{\\mathcal{N}} \\in\\mathcal{N}_{3}$, is given by:\n\\begin{equation}\nH_{\\mathcal {PSD}} = \n\\begin{pmatrix*}[r]\n0 & 0 & 0 \\\\\n0 & 1 & -1 \\\\\n0 & -1 & 1 \n\\end{pmatrix*}, \\quad\nH_{\\mathcal{N}} = \n\\begin{pmatrix*}[r]\n1 & 1 & 1 \\\\\n1 & 0 & 0 \\\\\n1 & 0 & 0 \n\\end{pmatrix*}~.\n\\end{equation}\n\\noindent  The associated EW $W= P + Q^{T_{B}}$, with $P = \\Pi_{S}(H_{\\mathcal{N}}^{ext})^{T_{B}}\\Pi_{S}$, and $Q =H_{\\mathcal{PSD}}^{ext}$ reads\n\n\\begin{equation*}\nP = \\frac{1}{2}\n\\begin{pmatrix}\n\\begin{matrix}\n2 & 0 & 0 \\\\\n0 & 1 & 0 \\\\\n0 & 0 & 1 \n\\end{matrix}\n& \\hspace*{-\\arraycolsep}\\vline\\hspace*{-\\arraycolsep} & \n\\begin{matrix}\n0 & 0 & 0 \\\\\n1 & 0 & 0 \\\\\n0 & 0 & 0 \n\\end{matrix} \n& \\hspace*{-\\arraycolsep}\\vline\\hspace*{-\\arraycolsep} & \n\\begin{matrix}\n0 & 0 & 0 \\\\\n0 & 0 & 0 \\\\\n1 & 0 & 0  \n\\end{matrix} \\\\\n\\hline\n\\begin{matrix}\n0 & 1 & 0 \\\\\n0 & 0 & 0 \\\\\n0 & 0 & 0 \n\\end{matrix} \n& \\hspace*{-\\arraycolsep}\\vline\\hspace*{-\\arraycolsep} &\n\\begin{matrix}\n1 & 0 & 0 \\\\\n0 & 0 & 0 \\\\\n0 & 0 & 0 \n\\end{matrix}\n& \\hspace*{-\\arraycolsep}\\vline\\hspace*{-\\arraycolsep} & \n\\begin{matrix}\n0 & 0 & 0 \\\\\n0 & 0 & 0 \\\\\n0 & 0 & 0  \n\\end{matrix} \\\\\n\\hline\n\\begin{matrix}\n0 & 0 & 1 \\\\\n0 & 0 & 0 \\\\\n0 & 0 & 0 \n\\end{matrix} \n& \\hspace*{-\\arraycolsep}\\vline\\hspace*{-\\arraycolsep} &\n\\begin{matrix}\n0 & 0 & 0 \\\\\n0 & 0 & 0 \\\\\n0 & 0 & 0 \n\\end{matrix}\n& \\hspace*{-\\arraycolsep}\\vline\\hspace*{-\\arraycolsep} & \n\\begin{matrix}\n1 & 0 & 0 \\\\\n0 & 0 & 0 \\\\\n0 & 0 & 0  \n\\end{matrix} \n\\end{pmatrix}~,\\quad Q =\n\\begin{pmatrix}\n\\begin{matrix*}[c]\n0 & 0 & 0 \\\\\n0 & 0 & 0 \\\\\n0 & 0 & 0 \n\\end{matrix*}\n& \\hspace*{-\\arraycolsep}\\vline\\hspace*{-\\arraycolsep} & \n\\begin{matrix*}[c]\n0 & 0 & 0 \\\\\n0 & 0 & 0 \\\\\n0 & 0 & 0 \n\\end{matrix*} \n& \\hspace*{-\\arraycolsep}\\vline\\hspace*{-\\arraycolsep} & \n\\begin{matrix*}[c]\n0 & 0 & 0 \\\\\n0 & 0 & 0 \\\\\n0 & 0 & 0  \n\\end{matrix*} \\\\\n\\hline\n\\begin{matrix}\n0 & 0 & 0 \\\\\n0 & 0 & 0 \\\\\n0 & 0 & 0 \n\\end{matrix} \n& \\hspace*{-\\arraycolsep}\\vline\\hspace*{-\\arraycolsep} &\n\\begin{matrix}\n0 & 0 & 0 \\\\\n0 & 1 & 0 \\\\\n0 & 0 & 0 \n\\end{matrix}\n& \\hspace*{-\\arraycolsep}\\vline\\hspace*{-\\arraycolsep} & \n\\begin{matrix}\n~0 & 0 & 0 \\\\\n~0 & 0 & \\matminus1 \\\\\n~0 & 0 & 0  \n\\end{matrix} \\\\\n\\hline\n\\begin{matrix}\n0 & 0 & 0 \\\\\n0 & 0 & 0 \\\\\n0 & 0 & 0 \n\\end{matrix} \n& \\hspace*{-\\arraycolsep}\\vline\\hspace*{-\\arraycolsep} &\n\\begin{matrix}\n0 & 0 & 0 \\\\\n0 & 0 & 0 \\\\\n0 & \\matminus1 & 0 \n\\end{matrix}\n& \\hspace*{-\\arraycolsep}\\vline\\hspace*{-\\arraycolsep} & \n\\begin{matrix}\n0 & 0 & 0 \\\\\n0 & 0 & 0 \\\\\n0 & 0 & 1  \n\\end{matrix} \n\\end{pmatrix}~,\n\\end{equation*}\n\\noindent but the resulting EW, $W'=P'+Q'^{T_{B}}$, detects exactly the same states in the symmetric subspace.\n\n\nThe link between non-exceptional copositive matrices and decomposable EWs extends also to exceptional copositive matrices and non-decomposable EWs in the symmetric subspace.\n\\begin{theorem}\n\\label{nondeco}\n Associated to each exceptional copositive matrix $H$ (i.e., $H \\in \\mathcal {COP} \\setminus (\\mathcal{PSD}+ \\mathcal{N})$) with at least one negative entry, there is a non-decomposable EW, $W=(H^{ext})^{T_{B}}$, able to detect symmetric PPTES.\n \\end{theorem}\n\n\\begin{proof}\nFor any $H \\in \\mathcal {COP} \\setminus (\\mathcal{PSD}+ \\mathcal{N})$, $H$ always admits a decomposition of the form  $H= H_{\\mathcal{N}}+ H_{\\star}$, where $H_{\\mathcal{N}}$ is a non-negative symmetric matrix and $H_{\\star}$ has at least one negative eigenvalue but is not positive semidefinite. The associated EW $W=P+Q^{T_{B}}$ \nwith $P=\\Pi_{S}( H^{ext}_{\\mathcal{N}})^{T_{B}}\\Pi_{S}$ and $Q= H^{ext}_{\\star}$, is a non-decomposable EW since $P\\succeq 0$ but $Q \\nsucceq 0$. The operator $W=(H^{ext})^{T_{B}}$ is also a non-decomposable EW. \n\\end{proof}\n\n\\begin{corollary}[From \\cite{tura2018separability}]\n\t\\label{d4}\n\tSince for $d<5$ every copositive matrix is not exceptional (i.e., $H = H_{\\mathcal{PSD}}+ H_{\\mathcal{N}}$), all EWs of DS states in $d=3$ and $d=4$ are decomposable.\n\\end{corollary}\n\nThe above corollary rephrases the fact that PPT criterion is necessary and sufficient to assess separability for bipartite DS states $\\rho_{DS} \\in \\mathcal{B}(\\mathbb C^d\\otimes \\mathbb C^d)$ for $d<5$. \n\n\n\\section {Symmetric PPTES}\n\\label{sec:symBoundEntanglement}\n\\noindent \nLet us briefly summarize what we have seen so far. The fact that each DS state, $\\rho_{DS}\\in \\mathcal{B}(\\mathbb C^d\\otimes \\mathbb C^d)$, is associated to a matrix $M_d(\\rho_{DS})$ (see  Eq.\\eqref{mr}), allows to reformulate the problem of entanglement characterization as the equivalent problem of checking the membership of $M_d(\\rho_{DS})$ to the cone of completely positive matrices $\\mathcal{CP}_d$. Equivalently, according to the dual formulation, any entangled state $\\rho_{DS}$ is detected by an EW $W$ which can be constructed from a copositive matrix $H$. PPT entangled diagonal symmetric states (PPTEDS) can only be detected by non-decomposable EWs, which correspond to exceptional copositive matrices. Since for $d<5$, all copositive matrices $H$ are of the form $H = H_{\\mathcal{PSD}}+ H_{\\mathcal{N}}$, all EWs defined as $W=(H^{ext})^{T_B}$ are necessarily of the form $W=P+Q^{T_B}$, with $P,Q\\succeq 0$, meaning that for $d<5$ there are not PPTEDS.\n\n However, for $d>5$, this is not the case, since there exist exceptional copositive matrices, i.e., $H \\notin \\mathcal{PSD}_{d}+ \\mathcal{N}_{d}$. Thus, detecting entanglement of $\\rho_{DS}$ in $d\\geq 5$, is equivalent to checking membership of the corresponding copositive matrix $H \\in \\mathcal {COP}_{d} \\setminus (\\mathcal{PSD}_{d}+ \\mathcal{N}_{d})$, which is, in general, a co-NP-hard problem \\cite{murty1985some}. \n \nWhat can we say about symmetric PPTES $\\rho_S$ that are not DS? In this section, we tackle the problem of entanglement detection for generic states  $\\rho_{S} \\in \\mathcal{B}(\\mathbb{C}^{d}\\otimes \\mathbb{C}^{d})$ in arbitrary dimension $d$. \nFollowing the argument given above, we split our analysis in two different scenarios, namely when $d\\ge 5$ and $d<5$. Remarkably, even outside of the DS paradigm, we find that copositive matrices lie at the core of non-decomposable EWs for symmetric PPTES in arbitrary dimensions. \n\n\\subsection {Symmetric PPTES in $d\\ge 5$}\n\n\\noindent \nThe fact that for $d\\geq 5$ there exist exceptional copositive matrices with at least one negative entry which lead to non-decomposable EWs in $\\mathcal{S}$, implies that (i) such EW can detect a PPTEDS, and (ii) the same EW is able to detect other symmetric, but not DS, PPTES \"around\" it.    \n\n\n\\begin{theorem}\n\t\\label{th}\n\tLet $\\rho_{DS}$ be a PPTEDS. Then any symmetric state $\\rho_{S}  = \\rho_{DS} + \\sigma_{CS}$, such that \n\t$\\rho_{S}^{T_B} \\geq 0$, is PPT entangled.\n\\end{theorem}\n\\begin{proof} \nSince $\\rho_{DS}$ is a PPTEDS state there exists an exceptional copositive matrix $H$ and an associated non\ndecomposable EW $W$ such that $\\mbox{Tr}(W \\rho_{DS})<0$. It follows that $\\mbox{Tr} (W\\rho_{S})= \\mbox{Tr}(W(\\rho_{DS}+\\sigma_{CS}))=\\mbox{Tr}(W \\rho_{DS})=\\mbox{Tr}(H M_{d}(\\rho_{DS}) ) <0$, so that $\\rho_{S}$ is PPT entangled.\n\\end{proof}\n\n\\noindent The paradigmatic example of an exceptional copositive matrix in $d=5$, is the so-called Horn matrix \\cite{hall1963copositive} which is the matrix associated to the quadratic form $\\vec{x}^{T} H \\vec{x} = (x_{1}+x_{2}+x_{3}+x_{4}+x_{5})^2 - 4x_{1}x_{2}- 4x_{2}x_{3}- 4x_{3}x_{4} - 4x_{4}x_{5}- 4x_{5}x_{1}$~. \nExceptional copositive matrices of the Horn type,  $H_{\\mathcal{H}}$, can be generated for any odd $d\\ge 5$ \\cite{johnson2008constructing}, and are of the form\n\\begin{equation}\n\\label{eq:hornodd}\nH_{\\mathcal{H}} = \n\\begin{pmatrix*}[r]\n1  & -1 & 1 & 1 & \\cdots & \\cdots & 1 & 1 & 1 & -1 \\\\\n-1 & 1  & -1 & 1 & \\ddots & & \\ddots & 1 & 1 & 1 \\\\\n1 & -1 & 1 & -1 & 1 &  &  & \\ddots &  1 & 1 \\\\\n1 & 1 & -1 & 1 & -1 & \\ddots & & & \\ddots &  1 \\\\\n\\vdots & \\ddots & 1 & -1 & 1 & \\ddots & & & & \\vdots \\\\\n\\vdots & & & \\ddots & \\ddots & \\ddots & \\ddots & \\ddots & & \\vdots\\\\\n1 & \\ddots & & & & \\ddots & 1 & -1 & 1 & 1 \\\\\n1 & 1 & \\ddots & & & \\ddots & -1 & 1 & -1 & 1 \\\\\n1 & 1 & 1 & \\ddots & & & 1 & -1 & 1 & -1 \\\\\n-1 & 1 & 1 & 1 & \\cdots & \\cdots & 1 & 1 & -1 & 1\n\\end{pmatrix*}~.\n\\end{equation}\n\\noindent Since the $H_{\\mathcal{H}}$ is exceptional and has negative entries, it leads to a non-decomposable EW $W= (H^{ext})^{T_{B}}$, that can be used to detect PPTEDS in any odd dimension $d \\ge 5$. Moreover, due to Th.(\\ref{th}), by adding suitable coherences to such states, the same EW can be used to certify PPT-entanglement also in whole families of symmetric states. Below we provide one of these families.\n\n\\begin{corollary}\n\\label{ijppt}\nGiven a PPTEDS state, $\\rho_{DS}$, any symmetric state of the form $\\rho_{S}= \\rho_{DS} + \\sigma_{CS}$, with  $\\sigma_{CS}=\\sum_{i<j} (\\alpha_{ij}\\ket{D_{ii} }\\bra{D_{jj}}+\\emph{h.c.})$ and $|\\alpha_{ij}| \\leq \\frac{p_{ij}}{2}$ is PPT-entangled.\n\\end{corollary}\n\\begin{proof}\n\n\\noindent The state, $\\rho_{S}$, and its partial transpose, $\\rho_{S}^{T_{B}}$, can be cast  as\n\\begin{align}\n    \\label{rhoco}\n&\\rho_{S\\;} = \\tilde{M}_{d}(\\rho_{S}) \\bigoplus_{i<j} \\frac{p_{ij}}{2} \\begin{pmatrix*}\n1 & 1 \\\\\n1 & 1  \n\\end{pmatrix*}~,\\\\\n&\\rho_{S}^{T_{B}} = M_{d}(\\rho_{DS}) \\bigoplus_{0\\le i < j < d} \n\\begin{pmatrix*}\np_{ij}\/2 & \\alpha_{ij}\\\\\n\\alpha_{ij}^* & p_{ij}\/2\n\\end{pmatrix*}~,\n\\end{align}\nwith \n\\begin{equation*}\n\\tilde{M}_{d}(\\rho_{S}) =\n\\begin{pmatrix*}\np_{00} & \\alpha_{01} &  \\cdots & \\alpha_{0,d-1} \\\\\n\\alpha_{01}^* & p_{11} &  \\cdots & \\alpha_{1,d-1}\\\\\n\\vdots & \\vdots & \\vdots & \\ddots &  \\\\\n\\alpha_{0,d-1}^* & \\alpha_{1,d-1}^* & \\cdots &p_{d-1,d-1}  \n\\end{pmatrix*}~,\n\\end{equation*}\n\\begin{equation*}\nM_{d}(\\rho_{DS}) =\n\\begin{pmatrix*}\np_{00} & p_{01}\/2 &  \\cdots & p_{0,d-1}\/2 \\\\\np_{01}\/2 & p_{11} &  \\cdots & p_{1,d-1}\/2\\\\\n\\vdots & \\vdots & \\vdots & \\ddots &  \\\\\np_{0,d-1}\/2 & p_{1,d-1}\/2 & \\cdots & p_{d-1,d-1}  \n\\end{pmatrix*}~.\n\\end{equation*}\n\n\\noindent Positive semidefiniteness of ${\\rho_{S}}^{T_{B}}$ implies\n$|\\alpha_{ij}|\\le \\frac{p_{ij}}{2}$, so that the state $\\rho_{S}$, generated from a PPTEDS state, remains PPT-entangled -- since it is detected by the same non-decomposable EW -- as long as the coherences respect the condition $|\\alpha_{ij}| \\leq \\frac{p_{ij}}{2}$.\n\\end{proof}\n\nA further connection between copositive matrices and EWs appears when considering {\\it {extreme}} copositive matrices. For instance, let us consider the (generalized) Horn matrix $H_{\\mathcal{H}}$ of Eq.\\eqref{eq:hornodd}, and the so-called Hoffmann-Pereira matrix $H_{\\mathcal{HP}}$ \\cite{johnson2008constructing,hoffman1973copositive}, which, besides of being exceptional, is also extreme. For $d=7$, such copositive matrices take the form\n\n\\begin{equation}\nH_{\\mathcal{H}} =\n\\begin{pmatrix*}[r]\n1 \t&-1 & 1  &  1  & 1  & 1 & -1\\\\\n-1 \t& 1 &-1 &  1  & 1  &1  & 1 \\\\\n1   &-1 & 1  & -1 & 1  &1  & 1\\\\\n1 \t& 1  &-1 &  1  & -1 &1  & 1 \\\\\n1 \t& 1  & 1  & -1 & 1  &-1  & 1\\\\\n1 \t& 1  & 1  & 1 & -1  &1  & -1\\\\\n-1 \t& 1  & 1  & 1 & 1  &-1  & 1\n\\end{pmatrix*}~,\n\\end{equation}\n\\begin{equation}\nH_{\\mathcal{HP}}=\n\\begin{pmatrix*}[r]\n1 \t&-1 & 1  &  0  & 0  & 1 & -1\\\\\n-1 \t& 1 &-1 &  1  & 0  &0  & 1 \\\\\n1   &-1 & 1  & -1 & 1  &0  & 0\\\\\n0 \t& 1  &-1 &  1  & -1 &1  & 0 \\\\\n0 \t& 0  & 1  & -1 & 1  &-1  & 1\\\\\n1 \t& 0  & 0  & 1 & -1  &1  & -1\\\\\n-1 \t& 1  & 0  & 0 & 1  &-1  & 1\n\\end{pmatrix*}~.\n\\end{equation}\nLet us inspect the action of both matrices, $H_\\mathcal{H}$ and \n$H_\\mathcal{HP}$, on a DS state $\\rho_{DS}\\in\\mathcal{B}( \\mathbb C^{7}\\otimes \\mathbb{C}^{7})$, described by its associated $M_7(\\rho_{DS})$ (see Eq.\\eqref{mr}):\n\\begin{equation}\nM_{d}(\\rho_{DS})=\n\\label{PPTEDS7}\n\\begin{pmatrix*}[c]\n1 \t& 1  & 0  & 0 & 0  &0   & 1\/8\\\\\n1 \t& 2  &  1 &  0  & 0  &0   & 0\\\\\n0  & 1  &  2  & 1  &0   &0   & 1\/4\\\\\n0 \t& 0  & 1 &  2  & 1  &0   & 0\\\\\n0 \t& 0  & 0  & 1  & 2  &1   & 0\\\\\n0 \t& 0  & 0  & 0  & 1  &2   & 1\\\\\n1\/8\t & 0  & 1\/4   & 0  &0   & 1  & 1\n\\end{pmatrix*}~.\n\\end{equation}\nIt can be easily checked that \n$\\mbox{Tr}( H_\\mathcal{HP}M_{d}(\\rho_{DS}))= \\mbox{Tr}(\\left(H^{ext}_{\\mathcal{HP}}\\right)^{T_{B}} \\rho_{DS})= -\\frac{1}{4}$. Since both $H_{\\mathcal{H}}$ and $H_\\mathcal{HP}$ are exceptional copositive matrices,  $W_{\\mathcal{HP}}=\\left(H^{ext}_{\\mathcal{HP}}\\right)^{T_{B}}$  and $W_{\\mathcal{H}}=\\left(H^{ext}_{\\mathcal{H}}\\right)^{T_{B}}$ are non-decomposable EWs, so that $\\rho_{DS}$ is a PPT entangled.\n\n In contrast,  $\\mbox{Tr}( H_\\mathcal{H}M_{d}(\\rho_{DS}))=\\mbox{Tr}( W_{\\mathcal{H}} \\rho_{S})=0$, indicating that $H_{\\mathcal{H}}$ fails to detect this state. Moreover, as stated by Th.\\eqref{th}, $W_\\mathcal{HP}= (H^{ext}_{\\mathcal{HP}} )^{T_{B}}$, detects, as well, many other states around the state given by Eq.\\eqref{PPTEDS7}. \nGiven the relationship between non-decomposable EWs and exceptional matrices, we conjecture that extremality in the copositive cone correspond to optimality in the set of EWs. In other words, copositive matrices that are both extreme and exceptional lead to optimal non-decomposable EWs in the sense of \\cite{lewenstein2000optimization}. We complete this subsection with the following theorem regarding extreme copositive matrices (see Fig.\\ref{fig:cop})\n\n\\begin{theorem}\n\\label{extreme}\nLet $H$ be an extreme copositive matrix with at least one negative eigenvalue, and at least one negative element $H_{ij}<0$. Then $H$ must be exceptional.\n\\end{theorem}\n\\begin{proof}\n$H$ cannot belong to neither $ H_{PSD}$ nor to $H_N$ and, while it is extremal, it cannot be a combination of their elements as well. A more detailed proof can be found in \\Cref{sec:appendixB}.\n\\end{proof}\n\n\\subsection {Symmetric PPTES in $d<5$}\n\n In what remains, we are interested in symmetric PPTES of the form $\\rho_{S}=\\rho_{DS}+ \\sigma_{CS}$ where $\\rho_{DS}$ is separable, so that $\\mbox{Tr}( H M_{d}(\\rho_{DS}))\\geq 0$ for all copositive matrices $H$. Moreover, since for $d < 5$, every copositive matrix is non-exceptional, i.e., $H=H_{\\mathcal{N}} + H_{\\mathcal{PSD}}$, the corresponding witness $W= (H^{ext})^{T_{B}}$ will always be decomposable.  For this reason, coherences are needed to create PPTES in low dimensional systems. Here we show that such states symmetric PPTES can nevertheless be detected by EWs which are of the form $W_{S}= W + W_{CS}$, that is by adding to the decomposable EW, $W$, a convenient off-diagonal, symmetric contribution $W_{CS}$ which reads the coherences of $\\rho_S$.\n \nFor the sake of simplicity, we hereby consider symmetric states of the form\n\\begin{equation}\n\\label{coho}\n\\rho_{S}= \\rho_{DS} + \\sigma_{CS} = \\sum_{ij} p_{ij}\\ket{D_{ij}}\\bra{D_{ij}} + \\sum_{i\\neq j\\neq k} (\\alpha_{ijk} \\ket{D_{ii}}\\bra{D_{jk}})+ \\mbox{h.c.}~,\n \\end{equation}\n with $p_{ij}\\geq 0$ $\\forall i,j \\;$, $\\sum p_{ij}=1$ and $\\alpha_{ijk}\\in \\mathbb C$.\n \n Indeed, in this case, both $\\rho_{S}$ and $\\rho_{S}^{T_{B}}$ can be cast as a direct sum of matrices, which highly simplifies our analysis. For instance, for $d=3$, $\\rho_{S}$ and $\\rho_{S}^{T_{B}}$ are of the form\n\\begin{equation}\n\\label{rankrho}\n\\rho_{S} =\n\\begin{pmatrix*}[c]\n\\frac{p_{02}}{2} & \\alpha & \\frac{p_{02}}{2} \\\\\n\\alpha^{*} & p_{11} & \\alpha \\\\\n\\frac{p_{02}}{2} & \\alpha^{*} & \\frac{p_{02}}{2}\n\\end{pmatrix*}\n\\oplus \n\\begin{pmatrix*}[c]\np_{00} & \\beta & \\beta\\\\\n\\beta^{*} &  \\frac{p_{12}}{2} & \\frac{p_{12}}{2} \\\\\n\\beta^{*} &  \\frac{p_{12}}{2}  & \\frac{p_{12}}{2} \n\\end{pmatrix*}\n\\oplus\n\\begin{pmatrix*}[c]\n\\frac{p_{01}}{2} & \\frac{p_{01}}{2} & \\gamma \\\\\n\\frac{p_{01}}{2} &  \\frac{p_{01}}{2} & \\gamma \\\\\n\\gamma^{*} &  \\gamma^{*}  & p_{22} \n\\end{pmatrix*}~,\n\\end{equation}\n\n\\begin{equation}\n\\label{rankrhotb}\n\t\\rho_{S}^{T_{B}} =\n\tM_{d}(\\rho_{DS}) \n\t\\oplus \n\t\\begin{pmatrix*}[c]\n\t\\frac{p_{01}}{2} & \\alpha &\\beta \\\\\n \\alpha^{*} &  \\frac{p_{12}}{2} &  \\gamma \\\\\n\t\\beta^{*} &   \\gamma^{*}  & \\frac{p_{02}}{2} \n\t\\end{pmatrix*} \n\t\\oplus\n\t\\begin{pmatrix*}[c]\n\t\\frac{p_{02}}{2} & \\beta & \\gamma \\\\\n\t\\beta^{*}  &  \\frac{p_{01}}{2} & \\alpha \\\\\n\t\\gamma^{*} &  \\alpha^{*} & \\frac{p_{12}}{2} \n\t\\end{pmatrix*}~,\n\\end{equation}\n\\noindent where we have defined, for easiness of reading, $\\alpha \\equiv \\alpha_{120} = \\alpha_{102}$, $\\beta \\equiv \\alpha_{012}= \\alpha_{021}$ and $\\gamma \\equiv \\alpha_{201} = \\alpha_{210}$. Such structure, which corresponds to a particular direct sum decomposition of the total Hilbert space, bears similitude with the so-called circulant states \\cite{PhysRevA.76.032308}.\n\nIn order to investigate the existence of PPTES we focus on edge states, since their low-dimensional ranks allow for a simpler analysis. \nBy using a notation common in the literature, we say that an edge state $\\rho_{S}$ is of type $(p,q)$ if $p=r(\\rho_{S})$ and $q=r(\\rho_{S}^{T_{B}})$ are the ranks of $\\rho_{S}$ and $\\rho_{S}^{T_{B}}$, respectively.\n While symmetric states in $d=3$ are, generically, of type $(6,9)$, PPT-entangled edge states must have lower ranks. When dealing with states of the form of Eq.(\\ref{coho}), we have found numerically that at least two coherences must be considered.\n For instance, we can set $\\gamma = 0$ and choose $\\alpha$ and $\\beta$ in such a way to lower the value of $(p,q)$. \nIndeed, a direct inspection of Eqs.(\\ref{rankrho})-(\\ref{rankrhotb}), shows that, by setting $|\\alpha|^2= p_{11}p_{02}\/2$ and $|\\beta|^2= p_{02}(p_{01}p_{12}-2p_{02}p_{11})\/4 p_{12}$, it is possible to attain a state of type $(5,7)$. Now, starting from a copositve matrix $H$, we can construct a non-decomposable EW of the form\n\\begin{equation}\n\\label{wits}\nW_{S}= \\Pi_{S}(H^{ext})^{T_{B}}\\Pi_{S} + \\sum_{i\\neq j \\neq k} W^{i}_{jk}\\ket{D_{ii}} \\bra{D_{jk}}+ \\mbox{h.c.}~,\n\\end{equation}\nwhere the coefficients $W^{i}_{jk}$ can be chosen to be real. \\\\\n\nLet us illustrate the above results by providing some explicit examples. We first consider the symmetric edge PPTES provided in \\cite{toth2010separability}. The state is of the form in Eq.(\\ref{coho}) for $d=3$ (\\textit{i.e.,} of the form in Eq.(\\ref{rankrho})), obtained from the DS state $\\rho_{DS}$ with parameters $p_{00}=0.22, p_{11}=0.234\/3, p_{22}=0.183, p_{01}=0.176, p_{02}=0.167\/3, p_{12}=0.254$ and coherences $\\alpha = 0.167\/3, \\beta = -0.059\/\\sqrt{2}, \\gamma = 0 $.\nIts entanglement was previously certified in \\cite{toth2010separability} by means of the PPT symmetric extension proposed in \\cite{doherty2004complete}.\nWith the aid of the same method \\cite{doherty2002distinguishing}, we can derive as well the corresponding non-decomposable EW $W$ via semidefinite programming. The associated copositive matrix, $H$, is easily obtained from the properly symmetrized EW, i.e.,\n$W_S=\\Pi_S W \\Pi_S$, and has the form\n\\begin{equation}\n\\label{with1}\nH \\approx\n\\left(\n\\begin{matrix}\n0.003 &  10.39 & 100.57 \\\\\n10.39 & 59.31 & -21.02 \\\\\n 100.57 &-21.02 & 14.22\n\\end{matrix}\n\\right)~,\n\\end{equation}\nwhile the coefficients $W^{i}_{jk}$ are given by $W^{1}_{02} =23.20$ and $W^{0}_{12} = -37.40$. \nIf we restrict to the DS part of the state $\\rho_S$, it is trivial to check that $\\mbox{Tr}(H M_{d}(\\rho_{DS}))\\geq 0$. This is by no means a surprise, since for DS states, in $d<5$, the PPT condition implies separability.\nFor this reason, the coherences provided by the term $\\sigma_{CS}$ are necessary to induce the PPT entanglement. \nRemarkably, one can vary the value of the coherences $\\alpha$ and $\\beta$ to obtain other symmetric PPTES as certified by the EW, i.e., $\\mbox{Tr}(W_{S}\\rho_{S})<0$. In fact, the EW $W_{S}$ can be used to derive families of PPT entangled states obtained by adding to the state $\\rho_{S}$ any coherent contribution $\\sigma_{CS}$ of the form of Eq.(\\ref{rankrho}) that preserves the positivity of both the state and its partial transpose. Indeed, also in the case $\\gamma \\neq 0$, as long as the conditions $\\rho_{S} \\ge 0$ and $\\rho_{S}^{T_{B}} \\ge 0$ hold, the same non-decomposable EW $W_{S}$, is able to detect, for suitable values of its entries $W^{i}_{jk}$, a whole family of PPTES of the form of Eq.(\\ref{rankrho}), as depicted in Fig.\\ref{fig:wgamma}. \n\n\n\\begin{figure}[H]\n\t\\centering\n\t\\includegraphics[width=10cm]{GUHNE.pdf}\n\t\\caption{\n\tThe PPT entangled states detected by $W_{S}$ of Eq.(\\ref{wits}) with coefficients $|W^{1}_{02}| =23.20, |W^{0}_{12}| = -37.40$ (dark orange) as compared to the whole family of PPT states $\\rho_{S}$ of Eq.(\\ref{rankrho}) with $p_{00}=0.22, p_{11}=0.234\/3, p_{22}=0.183, p_{01}=0.176, p_{02}=0.167\/3, p_{12}=0.254$ (light orange). }\n\t\\label{fig:wgamma}\n\\end{figure}\n\n\\begin{figure}[H]\n\t\\centering\n\t\\includegraphics[width=10cm]{ALBERT.pdf}\n\t\\caption{The PPT entangled states detected by $W_{S}$ of Eq.(\\ref{wits}) with coefficients $|W^{1}_{02}| =\\frac{4595}{191}, |W^{0}_{12}| = \\frac{6114}{113}$ (dark orange) as compared to the whole family of PPT states $\\rho_{S}$ of Eq.(\\ref{rankrho}) with  $p_{00}= p_{11}= p_{12} = \\frac{1848}{7625}$, $p_{22}= \\frac{464}{7625}$, $p_{01}=\\frac{231}{1525}$, $p_{02}=\\frac{462}{7625}$ (light orange). }\n\t\\label{fig:wgamma2}\n\\end{figure}\nIn Fig.\\ref{fig:wgamma2}, we display a new example of a symmetric PPTES $\\rho_{S}$ of the form of Eq.(\\ref{rankrho}), found by semidefinite programming. Also in this case, we have found a non-decomposable EW $W_{S}$ of the form of Eq.(\\ref{wits}), with coefficients $W^{1}_{02} =\\frac{4595}{191}$ and $W^{0}_{12} = -\\frac{6114}{113}$ and whose associated copositive matrix is given by\n\n\\begin{equation}\n\\label{with2}\nH =\n\\left(\n\\begin{matrix}\n 1\/172 & 1009\/151 & 11025\/68 \\\\\n 1009\/151 & 1803\/22 & -5829\/65 \\\\\n 11025\/68 & -5829\/65 & 1224\/7\n\\end{matrix}\n\\right)~.\n\\end{equation}\n\nSimilarly, we can use the same procedure to derive families of PPT entangled symmetric states for $d=4$. In this case, we have found, numerically, that at least three different coherences of the form of Eq.(\\ref{coho}) are needed in order to get a low-dimensional PPT entangled edge state. To the best of our knowledge, there are no explicit examples of symmetric PPT entangled states in $d=4$.\nWe provide an example in \\Cref{sec:appendixA} together with the non-decomposable EW that detects it and its associated copositive matrix.\n\n\\section{Conclusions}\n\\label{sec:Conclusions}\nIn this work we have studied the connection between EWs and copositive matrices, showing how this class of matrices can be effectively used to detect entanglement in bipartite symmetric states.  First, we have proved that non-decomposable (decomposable) EWs can be derived from exceptional (non-exceptional) copositive matrices. Second, we have tackled the problem of the entanglement detection for two symmetric qudits. In this context, our analysis has shown that, on the one hand, for dimension $d \\ge 5$, it is possible to certify bound entanglement in any family of symmetric states constructed by adding coherences to a PPTEDS state. Indeed, we have provided the explicit expression of a symmetric PPTES in $d=7$, along with the exceptional matrix that detects it. On the other hand, for dimension $d<5$, every copositive matrix is not exceptional, so that a different approach is needed in order to construct a non-decomposable EW. Nevertheless, we have shown that, also in this case, an EW of this type can be derived from a not exceptional copositive matrix by adding suitable coherent-like contributions. Indeed, we conjecture that $\\textit{any}$ symmetric PPTES of two qudits can be detected by a non-decomposable EW of the form of Eq.(\\ref{wits}). Our conjecture seems to be well grounded as pointed out by the several examples we have found of symmetric PPTES detected with this method, both for $d=3$ and $d=4$ (see Appendix \\ref{sec:appendixA} for further details).\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{introduction}\n\\label{sec:intro}\nIn the conventional {Private Information Retrieval (PIR) problem}, a user wants to privately download a message belonging to a database with copies stored on a single or multiple remote servers (see~\\cite{yekhanin2010private}). \nThe multiple-server PIR problem has been predominantly studied in the PIR literature, with  breakthrough results for the information-theoretic privacy model in the past few years (see e.g., \\cite{Sun2017, JafarPIR3new, tajeddine2017private1,BU18}, and references therein).\nThe multi-message extension of the PIR problem enables a user to privately download multiple messages from the server(s)~\\cite{Banawan2017,Maddah2018}. \nThere have been a number of recent works on the PIR problem when some side information is present at the user~\\cite{Tandon2017,Wei2017CacheAidedPI,Wei2017FundamentalLO,Chen2017side,Maddah2018}.\n\n\n\nRecently, in~\\cite{Kadhe2017,KGHERS2017}, the authors considered the single-server PIR with Side Information (PIR-SI) problem, wherein the user knows a random subset of messages that is unknown to the server.\nIt was shown that the side information enables the user to substantially reduce the download cost and still achieve information-theoretic privacy for the requested message. The multi-message version of PIR-SI is considered in~\\cite{HKGRS:2018,LG:2018}, and the case of coded side information is considered in~\\cite{HKS2018}. Single-server multi-user PIR-SI problem wherein all users have the same demand but different side-information sets was considered in~\\cite{LiG:18:MU}.\n\nIn this work, we consider the following scenario. Suppose there are $D$ users, each of which wants to privately retrieve a distinct message from a server. The users send their demands to a {\\it trusted agent}. The agent has a subset of $M$ messages, unknown to the server. This side information could have been obtained from the users or from previous interactions with the server. Followed by aggregating the users' requests, the agent then collectively retrieves information from the server. \n\nOne natural solution for the agent to achieve privacy during the retrieval is to successively use the PIR-SI protocol in~\\cite{Kadhe2017} for each request. However, the agent can achieve much higher download rate while preserving the privacy collectively for all the users by using the multi-message PIR protocol in~\\cite{HKGRS:2018,LG:2018}. In this work, we introduce the notion of \\emph{individual-privacy} where the agent is required to protect the privacy only for each individual user, and we refer to this problem as \\emph{Individually-Private Information Retrieval with Side Information (IPIR-SI)}. We seek to answer the following questions: is it possible to further increase the download rate when individual-privacy is required? Moreover, what are the fundamental limits on the download rate for the IPIR-SI problem? We answer the first question affirmatively and take the first steps towards answering the second question.\n\n\\subsection{Main Contributions}\n\nWe first establish a lower bound on the capacity of the IPIR-SI problem (where the capacity is defined as the supremum of all achievable download rates) by presenting a new protocol which builds up on the Generalized Partition and Code (GPC) protocol in~\\cite{HKGRS:2018}. Next, we characterize the capacity of IPIR-SI problem for $M = 1$ and ${D = 2}$. In the process of characterizing the capacity for arbitrary $M$ and $D$ we present a novel combinatorial conjecture, that may be of independent interest. \n\nFor $M=1$ and arbitrary $D$, our conjecture relates the size of an external mother vertex-set (i.e., a minimal subset of nodes from which any other node with nonzero our-degree can be reached via a directed path) of any directed graph $G$ with certain bounds on the in-degree and out-degree of the nodes, to the size of an internal mother vertex-set (i.e., a minimal subset of nodes from which any other node with nonzero in-degree can be reached via a directed path) of the transpose of $G$ which is obtained by reversing the direction of all edges in $G$.\n\n\n\\section{Problem Formulation}\\label{sec:SN}\nLet $\\mathbb{F}_q$ be a finite field of size $q$, and let $\\mathbb{F}_{q^m}$ be an extension field of $\\mathbb{F}_q$ for some integer $m\\geq 1$. Let $L \\triangleq m\\log_2 q$, and let ${\\mathbb{F}_q^{\\times} \\triangleq \\mathbb{F}_q\\setminus \\{0\\}}$. For a positive integer $i$, we denote $\\{1,\\dots,i\\}$ by $[i]$. Also, let $K\\geq 1$, $M\\geq 1$, and $D\\geq 1$ be arbitrary integers such that $D+M\\leq K$. \n\n\nSuppose that there is a server storing a set of $K$ messages $X_1,\\dots,X_K$, with each message $X_i$ being independently and uniformly distributed over $\\mathbb{F}_{q^m}$, i.e., ${H(X_1) = \\dots = H(X_K) = L}$ and $H(X_1,\\dots,X_K) = KL$. Also, suppose that there are $D$ users, each of which demands one distinct message $X_j$. Let $W$ be the index set of the users' demanded messages. The users send the indices of their demanded messages to a trusted agent, called \\emph{aggregator}, who knows $M$ messages $X_S\\triangleq \\{X_j\\}_{j\\in S}$ for some $S\\subset [K]$, $|S|=M$, $S\\cap W=\\emptyset$. Then, the aggregator retrieves the $D$ messages $X_W\\triangleq \\{X_j\\}_{j\\in W}$ from the server. We refer to $W$ as the \\emph{demand index set}, $X_W$ as the \\emph{demand}, $D$ as the \\emph{demand size}, $S$ as the \\emph{side information index set}, $X_S$ as the \\emph{side information}, and $M$ as the \\emph{side information size}. \n\nDenote by $\\mathcal{W}$ and $\\mathcal{S}$ the set of all subsets of $\\mathcal{K}\\triangleq [K]$ of size $D$ and $M$, respectively. Also, let $\\boldsymbol{S}$ and $\\boldsymbol{W}$ be two random variables representing $S$ and $W$, respectively. Denote the probability mass function (PMF) of $\\boldsymbol{S}$ by $p_{\\boldsymbol{S}}(\\cdot)$ and the conditional PMF of $\\boldsymbol{W}$ given $\\boldsymbol{S}$ by $p_{\\boldsymbol{W}|\\boldsymbol{S}}(\\cdot|\\cdot)$. We assume that $\\boldsymbol{S}$ is uniformly distributed over $\\mathcal{S}$, i.e., $p_{\\boldsymbol{S}}(S) = \\binom{K}{M}^{-1}$ for all $S\\in \\mathcal{S}$, and $\\boldsymbol{W}$ (given $\\boldsymbol{S}=S$) is uniformly distributed over ${\\{W\\in \\mathcal{W}:W\\cap S=\\emptyset\\}}$, i.e., \n\\begin{equation*}\np_{\\boldsymbol{W}|\\boldsymbol{S}}(W|S) = \n\\left\\{\\begin{array}{ll}\n\\binom{K-M}{D}^{-1}, & W\\in \\mathcal{W}, W\\cap S=\\emptyset,\\\\\t\n0, & \\text{otherwise}.\n\\end{array}\\right.\t\n\\end{equation*}\n \nWe assume that the server knows the size of $W$ (i.e., $D$) and the size of $S$ (i.e., $M$), as well as the PMF $p_{\\boldsymbol{S}}(\\cdot)$ and the conditional PMF $p_{\\boldsymbol{W}|\\boldsymbol{S}}(\\cdot|\\cdot)$, whereas the realizations $S$ and $W$ are unknown to the server a priori. \n\nFor any $S$ and $W$, in order to retrieve $X_W$, the aggregator sends to the server a query $Q^{[W,S]}$, which is a (potentially stochastic) function of $W$, $S$, and $X_S$. The query $Q^{[W,S]}$ must protect from the privacy of the demand index of every user individually from the server, i.e., \\[\\mathbb{P}(j\\in \\boldsymbol{W}| Q^{[W,S]})= \\mathbb{P}(j\\in \\boldsymbol{W})=\\frac{D}{K}\\] for all $j \\in \\mathcal{K}$. We refer to this condition as the \\emph{individual-privacy condition}. Note that the individual-privacy condition is weaker than the \\emph{joint-privacy condition}, also known as the \\emph{$W$-privacy condition}, being studied in~\\cite{HKGRS:2018}, where the privacy of all indices in the demand index set must be protected jointly.\nThe notions of individual privacy and joint privacy coincide for $D=1$, which was previously settled in~\\cite{Kadhe2017}, and hence, in this work, we focus on $D\\geq 2$.\n\nUpon receiving $Q^{[W,S]}$, the server sends to the aggregator an answer $A^{[W,S]}$, which is a (deterministic) function of the query $Q^{[W,S]}$ and the messages in $X$, i.e., \\[H(A^{[W,S]}| Q^{[W,S]},\\{X_j\\}_{j\\in \\mathcal{K}}) = 0.\\] The answer $A^{[W,S]}$ along with the side information $X_S$ must enable the aggregator to retrieve the demand $X_W$, i.e., \\[H(X_W| A^{[W,S]}, Q^{[W,S]}, X_S)=0.\\] This condition is referred to as the \\emph{recoverability condition}. \n\nThe problem is to design a query $Q^{[W,S]}$ and an answer $A^{[W,S]}$ (for any $W$ and $S$) that satisfy the individual-privacy and recoverability conditions. We refer to this problem as \\emph{single-server multi-message Individually-Private Information Retrieval with Side Information (IPIR-SI)}. \n\nA collection of $Q^{[W,S]}$ and $A^{[W,S]}$ (for all $W$ and $S$) which satisfy the individual-privacy and recoverability conditions, is referred to as an \\emph{IPIR-SI protocol}. We define the \\emph{rate} of an IPIR-SI protocol as the ratio of the entropy of the demand messages, i.e., $DL$, to the average entropy of the answer, i.e., $H(A^{[\\boldsymbol{W},\\boldsymbol{S}]})=\\sum H(A^{[W,S]})p_{\\boldsymbol{W}|\\boldsymbol{S}}(W|S)p_{\\boldsymbol{S}}(S)$, where the average is taken over all $W$ and $S$. The \\emph{capacity} of the IPIR-SI problem is also defined as the supremum of rates over all IPIR-SI protocols. \n\nIn this work, our goal is to characterize the capacity of the IPIR-SI problem, and to design an IPIR-SI protocol that achieves the capacity. \n\n\n\\section{Main Results}\nIn this section, we present our main results. Theorem~\\ref{thm:IPIRSI-Gen} provides a lower bound on the capacity of IPIR-SI problem for $M\\geq 1$ and $D\\geq 2$, and Theorem~\\ref{thm:IPIRSI-Spc} characterizes the capacity of IPIR-SI problem for the special case of $M=1$ and $D=2$. The proofs of Theorems~\\ref{thm:IPIRSI-Gen} and~\\ref{thm:IPIRSI-Spc} are given in Sections~\\ref{sec:IPIRSI-Gen} and~\\ref{sec:IPIRSI-Spc}, respectively.  \n\n\\begin{theorem}\\label{thm:IPIRSI-Gen}\nThe capacity of IPIR-SI problem with $K$ messages, side information size $M\\geq 1$, and demand size $D\\geq 2$ is lower bounded by $D(K-M\\lfloor\\frac{K}{M+D}\\rfloor)^{-1}$ if $\\frac{K-D}{M+D}\\leq \\lfloor \\frac{K}{M+D}\\rfloor$, and by $\\lceil \\frac{K}{M+D}\\rceil^{-1}$ otherwise.\n\\end{theorem}\n\nThe proof is based on constructing an IPIR-SI protocol that achieves the rate ${D(K-M\\lfloor K\/(M+D)\\rfloor)^{-1}}$ or ${\\lceil K\/(M+D)\\rceil^{-1}}$, depending on $K$, $M$, and $D$ (see, for details, Section~\\ref{sec:IPIRSI-Gen}). This protocol, which is a variation of the Generalized Partition and Code (GPC) protocol previously proposed in~\\cite{HKGRS:2018} for single-server multi-message PIR-SI where joint-privacy is required, is referred to as \\emph{GPC for Individual Privacy}, or \\emph{GPC-IP} for short. \n\n\n\\begin{remark}\n\\emph{A lower bound on the capacity of single-server multi-message PIR with side information, when the privacy of the demand indices must be protected jointly, was previously presented in~\\cite[Theorem~1]{HKGRS:2018}. Surprisingly, this lower bound reduces to the lower bound of Theorem~\\ref{thm:IPIRSI-Gen} where $M$ (in ~\\cite[Theorem~1]{HKGRS:2018}) is replaced by $MD$. This correspondence implies that each message in the side information, when achieving individual-privacy, can be as effective as $D$ side information messages when joint-privacy is required. This also suggests that, as one would expect, relaxing the privacy condition (from joint to individual) can increase the capacity.}  \n\\end{remark}\n\n\\begin{theorem}\\label{thm:IPIRSI-Spc}\nThe capacity lower bound given in Theorem~\\ref{thm:IPIRSI-Gen} is tight for $M=1$ and $D=2$.\n\\end{theorem}\n\n\nThe proof of converse is based on a mixture of new combinatorial and information-theoretic arguments relying on two necessary conditions imposed by the individual-privacy and recoverability conditions (see Lemmas~\\ref{lem:2} and~\\ref{lem:3}).\n\n\n\\begin{remark}\n\\emph{As we will show later, the tightness of the result of Theorem~\\ref{thm:IPIRSI-Gen} for arbitrary $M$ and $D$, which remains open in general, is conditional upon the correctness of a novel conjecture in combinatorics, formally stated in Section~\\ref{sec:IPIRSI-Spc}, which may be of independent interest. Interestingly, for ${M=1}$ and ${D\\geq 2}$, our conjecture relates the size of an external mother vertex-set of any directed graph $G$, whose nodes have in-degree at least one and out-degree either zero or at least $D$, to the size of an internal mother vertex-set of the transpose of $G$ (which is the graph obtained by reversing the direction of all edges in $G$). (The notions of external and internal mother vertex-sets, formally defined in Section~\\ref{sec:IPIRSI-Spc}, are two generalizations of the notion of the mother vertex in graph theory.) In this work, we prove the simplest non-trivial case of this conjecture for $M=1$ and $D=2$, and leave the complete proof for the future work.}\t\n\\end{remark}\n\n\n\\section{Proof of Theorem~\\ref{thm:IPIRSI-Gen}}\\label{sec:IPIRSI-Gen}\n\nIn this section, we propose an IPIR-SI protocol, referred to as \\emph{Generalized Partition and Code for Individual Privacy (GPC-IP)}, achieving the rate lower bound of Theorem~\\ref{thm:IPIRSI-Gen}. \n\nDefine $\\alpha\\triangleq M+D$, $\\beta\\triangleq \\lfloor K\/\\alpha\\rfloor$, and $\\rho\\triangleq K-\\alpha\\beta$. (Note that $0\\leq \\rho<\\alpha$.) Also, define $\\gamma\\triangleq \\min\\{\\rho,D\\}$. Assume that $q\\geq \\alpha$, and let $\\omega_1,\\dots,\\omega_{\\alpha}$ be $\\alpha$ distinct elements from $\\mathbb{F}_q$. \n\n\\textit{\\bf GPC-IP Protocol:} This protocol consists of four steps as follows: \n\n\\textbf{\\it Step 1:} First, the aggregator constructs a set $Q_{0}$ of size $\\rho$ from the indices in $\\mathcal{K}$, and $\\beta$ disjoint sets $Q_1,\\dots,Q_{\\beta}$ (also disjoint from $Q_0$), each of size $\\alpha$, from the indices in $\\mathcal{K}$, where the construction procedure is described below. \n\nDefine \\[\\theta_1\\triangleq \\frac{\\binom{\\alpha-1}{M}}{\\prod_{i=1}^{\\beta-1}\\binom{K-i\\alpha}{\\alpha}},\\] \\[\\theta_2 \\triangleq \\frac{\\binom{\\alpha-1}{M+\\rho}\\binom{M+\\rho}{M}(\\frac{\\alpha\\beta}{D-\\rho}-1)}{\\binom{D}{\\rho}\\binom{K-\\alpha}{\\rho}\\prod_{i=1}^{\\beta-1}\\binom{K-i\\alpha-\\rho}{\\alpha}},\\] and \\[\\theta_3\\triangleq \\frac{\\beta\\binom{\\rho}{D}\\binom{K-\\rho}{\\alpha-\\rho}}{\\binom{M}{\\rho-D}\\prod_{i=0}^{\\beta-1}\\binom{K-i\\alpha-\\rho}{\\alpha}}.\\] \n\nThere are two cases based on $\\rho$: (i) ${\\rho<D}$, and (ii) ${\\rho\\geq D}$. \n\n\\emph{Case (i):} With probability $\\frac{\\theta_1}{\\theta_1+\\theta_2}$, the aggregator places $\\rho$ randomly chosen elements (demand indices) from $W$ into $Q_0$ and the remaining elements in $W$ along with all elements in $S$ (side information indices) into $Q_1$. Then the aggregator randomly places all other elements in $\\mathcal{K}$ into $Q_2,\\dots,Q_{\\beta}$ and the remaining positions in $Q_1$; otherwise, with probability $\\frac{\\theta_2}{\\theta_1+\\theta_2}$, the aggregator places all elements in $S\\cup W$ into $Q_1$, and randomly places all other elements in $\\mathcal{K}$ into $Q_0,Q_2,\\dots,Q_{\\beta}$. \n\n\\emph{Case (ii):} With probability $\\frac{\\theta_1}{\\theta_1+\\theta_3}$, the aggregator places all elements in $W$ along with $\\rho-D$ randomly chosen elements from $S$ into $Q_0$, and places the remaining elements of $S$ together with all other elements in $\\mathcal{K}$ into $Q_1,\\dots,Q_{\\beta}$ at random; otherwise, with probability $\\frac{\\theta_3}{\\theta_1+\\theta_3}$, the aggregator places all elements in $S\\cup W$ into $Q_1$, and randomly places all other elements in $\\mathcal{K}$ into $Q_0,Q_2,\\dots,Q_{\\beta}$.\n\nNext, the aggregator creates a collection $Q'$ of $\\gamma$ sequences $Q'_{1},\\dots,Q'_{\\gamma}$, each of length $\\rho$, such that $Q'_{i}  = \\{\\omega^{i-1}_1,\\dots,\\omega^{i-1}_{\\rho}\\}$ for $i\\in [\\gamma]$, and a collection $Q''$ of $D$ sequences $Q''_{1},\\dots,Q''_{D}$, each of length $\\alpha$, such that $Q''_{i}  = \\{\\omega^{i-1}_1,\\dots,\\omega^{i-1}_{\\alpha}\\}$ for $i\\in [D]$.\n\n\\textbf{\\it Step 2:} The aggregator constructs ${Q^{*}_0 = (Q_0,Q')}$ and ${Q^{*}_i = (Q_i,Q'')}$ for $i\\in [\\beta]$, and sends to the server the query ${Q^{[W,S]} = \\{Q^{*}_{0},Q^{*}_{\\sigma^{-1}(1)},\\dots,Q^{*}_{\\sigma^{-1}(\\beta)}\\}}$ for a randomly chosen permutation $\\sigma: [\\beta]\\rightarrow [\\beta]$.\n\n\\textbf{\\it Step 3:} By using ${Q^{*}_0 = (Q_0,Q')}$ and ${Q^{*}_i = (Q_i,Q'')}$ for $i\\in [\\beta]$, the server computes ${A_0 = \\{A^1_0,\\dots,A^{\\gamma}_0\\}}$ by ${A^j_0 = \\sum_{l=1}^{\\rho} \\omega_l^{j-1} X_{i_l}}$ for ${j\\in [\\gamma]}$ where ${Q_0 = \\{i_1,\\dots,i_{\\rho}\\}}$, and computes ${A_i = \\{A^1_i,\\dots,A^D_i\\}}$ for ${i\\in [\\beta]}$ by ${A^{j}_{i} = \\sum_{l=1}^{\\alpha} \\omega^{j-1}_{l} X_{i_j}}$ for $j\\in [D]$ where ${Q_i = \\{i_1,\\dots,i_{\\alpha}\\}}$. The server then sends to the aggregator the answer ${A^{[W,S]}=\\{A_{0},A_{\\sigma^{-1}(1)}\\dots,A_{\\sigma^{-1}(\\beta)}\\}}$.\n\n\\textbf{\\it Step 4:} Upon receiving the answer from the server, the aggregator retrieves $X_j$ for any $j\\in W\\cap Q_0$ (or any $j\\in W\\cap Q_i$ for some $i\\in [\\beta]$) by subtracting off the contribution of the side information messages $\\{X_i\\}_{i\\in S}$ from the $\\gamma$ (or $D$) equations in $A_{0}$ (or $A_i$), and solving the resulting system of $\\gamma$ (or $D$) linear equations with $\\gamma$ (or $D$) unknowns. \n\n\n\n\n\n\n\\begin{lemma}\\label{lem:Ach}\nThe GPC-IP protocol is an IPIR-SI protocol, and achieves the rate ${D(K-M\\lfloor \\frac{K}{M+D}\\rfloor)^{-1}}$ if $\\frac{K-D}{M+D}\\leq \\lfloor \\frac{K}{M+D}\\rfloor$, and the rate ${\\lceil \\frac{K}{M+D}\\rceil^{-1}}$ otherwise.\n\\end{lemma}\n\n\\begin{proof}\nIf $\\frac{K-D}{M+D}\\leq \\lfloor \\frac{K}{M+D}\\rfloor$, then $\\rho< D$. Thus, $\\gamma = \\rho$. In this case, $H(A_0) = \\rho L$ and $H(A_i) = DL$ for $i\\in [\\beta]$, where $L=H(X_i)$ for all $i$. Thus, for any $W\\in \\mathcal{W}$ and $S\\in \\mathcal{S}$ such that $S\\cap W=\\emptyset$, we have $H(A^{[W,S]})=H(A_0,\\dots,A_{\\gamma})=\\sum_{i=0}^{\\beta} H(A_i)=(\\rho+\\beta D)L$. Thus, in this case, the rate is $DL\/H(A^{[\\boldsymbol{W},\\boldsymbol{S}]}) = DL\/H(A^{[W,S]})=D\/(\\rho+\\beta D)$, or equivalently, $D(K-M\\lfloor\\frac{K}{M+D}\\rfloor)^{-1}$. If $\\frac{K-D}{M+D}> \\lfloor \\frac{K}{M+D}\\rfloor$, then $\\rho\\geq D$. Thus, ${\\gamma = D}$. In this case, $H(A_0) = H(A_i) = DL$ for $i\\in [\\beta]$, and thus, ${H(A^{[W,S]})=(\\beta+1)DL}$. In this case, the rate is $D\/(\\beta+1)D$, or equivalently, ${\\lceil \\frac{K}{M+D}\\rceil^{-1}}$. \n\nNext, we prove that the GPC-IP protocol is an IPIR-SI protocol. It should be easy to see that the recoverability condition is satisfied. We only need to prove that the GPC-IP protocol satisfies the individual-privacy condition. \n\nConsider an arbitrary ${Q \\triangleq \\{Q_0,\\dots,Q_{\\beta}\\}}$. We need to show that ${\\mathbb{P}(j\\in \\boldsymbol{W}|Q)=\\mathbb{P}(j\\in \\boldsymbol{W})}$ for all $j\\in \\mathcal{K}$. Equivalently, it suffices to show $\\mathbb{P}(j\\in \\boldsymbol{W}|Q)$ is the same for all $j\\in \\mathcal{K}$.\n\nFirst, suppose that $\\rho<D$. It is easy to see that $\\mathbb{P}(j\\in \\boldsymbol{W}|Q)$ is given by\n\\begin{dmath}\\label{Eq:1}\n\\sum_{i=1}^{\\beta} \\sum_{\\substack{W\\subset Q_i: \\\\ |W|=D-\\rho}} \\sum_{\\substack{S\\subset Q_i\\setminus W:\\\\ |S|=M}}\\mathbb{P}(\\boldsymbol{W}=Q_0\\cup W, \\boldsymbol{S}=S|Q)  \n\\end{dmath} for all $j\\in Q_0$, and \n\\begin{dmath}\\label{Eq:2}\n \\sum_{\\substack{W\\subset Q_i:\\\\ |W|=D, j\\in W}} \\mathbb{P}(\\boldsymbol{W}=W, \\boldsymbol{S}=Q_i\\setminus W|Q) + {\\sum_{\\substack{W\\subset Q_i:\\\\ |W|=D-\\rho, j\\in W}} \\sum_{\\substack{S\\subset Q_i\\setminus W:\\\\ |S|=M}} \\mathbb{P}(\\boldsymbol{W}=Q_0\\cup W, \\boldsymbol{S}=S|Q)}  \n\\end{dmath} for all $j\\in Q_i$, $i\\in [\\beta]$. From~\\eqref{Eq:1} and~\\eqref{Eq:2}, one can see that $\\mathbb{P}(j\\in \\boldsymbol{W}|Q)$ is the same for all $j\\in Q_0$, say equal to $p_0$, and is the same for all $j\\in Q_i$ and all $i\\in [\\beta]$, say equal to $p_1$. We need to show that $p_0$ and $p_1$ are equal. It is easy to show that $p_0$ and $p_1$ are equal if the two quantities\n\\begin{dmath}\\label{Eq:3}\n\\sum_{i=1}^{\\beta} \\sum_{\\substack{W\\subset Q_i: \\\\ |W|=D-\\rho}} \\sum_{\\substack{S\\subset Q_i\\setminus W:\\\\ |S|=M}} \\mathbb{P}(Q|\\boldsymbol{W}=Q_0\\cup W, \\boldsymbol{S}=S) \n\\end{dmath} and \n\\begin{dmath}\\label{Eq:4}\n \\sum_{\\substack{W\\subset Q_i:\\\\ |W|=D, j\\in W}} \\mathbb{P}(Q|\\boldsymbol{W}=W, \\boldsymbol{S}=Q_i\\setminus W) + {\\sum_{\\substack{W\\subset Q_i:\\\\ |W|=D-\\rho, j\\in W}} \\sum_{\\substack{S\\subset Q_i\\setminus W:\\\\ |S|=M}} \\mathbb{P}(Q|\\boldsymbol{W}=Q_0\\cup W, \\boldsymbol{S}=S)}  \n\\end{dmath} are equal. Fix an $i\\in [\\beta]$. For any $W\\subset Q_i$, $|W|=D-\\rho$, and any $S\\subset Q_i\\setminus W$, $|S|=M$, a simple counting yields\n\\begin{align*}\n& \\mathbb{P}(Q|\\boldsymbol{W}=Q_0\\cup W, \\boldsymbol{S}=S) = \\left(\\frac{\\theta_1}{\\theta_1+\\theta_2}\\right) (\\beta-1)!\\left(\\binom{D}{\\rho}\\binom{K-\\alpha}{\\rho}\\prod_{i=1}^{\\beta-1}\\binom{K-i\\alpha-\\rho}{\\alpha}\\right)^{-1}.\t\n\\end{align*} and accordingly,~\\eqref{Eq:3} is equal to \n\\begin{align*}\n& \\left(\\frac{\\theta_1}{\\theta_1+\\theta_2}\\right) \\binom{\\alpha}{M+\\rho}\\binom{M+\\rho}{M} \\beta!\\left(\\binom{D}{\\rho}\\binom{K-\\alpha}{\\rho}\\prod_{i=1}^{\\beta-1}\\binom{K-i\\alpha-\\rho}{\\alpha}\\right)^{-1}.\t\n\\end{align*} For any $W\\subset Q_i$, $|W|=D$ such that $j\\in W$, we have \n\\begin{align*}\n& \\mathbb{P}(Q|\\boldsymbol{W}=W, \\boldsymbol{S}=Q_i\\setminus W) = \\left(\\frac{\\theta_2}{\\theta_1+\\theta_2}\\right) (\\beta-1)!\\left(\\prod_{i=1}^{\\beta-1}\\binom{K-i\\alpha}{\\alpha}\\right)^{-1}, \n\\end{align*} and for any $W\\subset Q_i$, $|W|=D-\\rho$ such that $j\\in W$, and any $S\\subset Q_i\\setminus W$, $|S|=M$, we have \n\\begin{align*}\n& \\mathbb{P}(Q|\\boldsymbol{W}=Q_0 \\cup W, \\boldsymbol{S}=S) = \\left(\\frac{\\theta_1}{\\theta_1+\\theta_2}\\right) (\\beta-1)!\\left(\\binom{D}{\\rho}\\binom{K-\\alpha}{\\rho}\\prod_{i=1}^{\\beta-1}\\binom{K-i\\alpha-\\rho}{\\alpha}\\right)^{-1}. \n\\end{align*} Accordingly,~\\eqref{Eq:4} is equal to \n\\begin{align*}\n& \\binom{\\alpha-1}{M} (\\beta-1)! \\left(\\left(\\frac{\\theta_2}{\\theta_1+\\theta_2}\\right)\\left(\\prod_{i=1}^{\\beta-1}\\binom{K-i\\alpha}{\\alpha}\\right)^{-1}\\right.\\\\\n& \\quad + \\left(\\frac{\\theta_1}{\\theta_1+\\theta_2}\\right)\\left(\\frac{D-\\rho}{D}\\right)\\left.\\left(\\binom{K-\\alpha}{\\rho}\\prod_{i=1}^{\\beta-1}\\binom{K-i\\alpha-\\rho}{\\alpha}\\right)^{-1}\\right).\t\n\\end{align*} It is easy to verify that~\\eqref{Eq:3} and~\\eqref{Eq:4} are equal for the choice of $\\theta_1$ and $\\theta_2$ defined as in the protocol. \n\nNext, consider the case of $\\rho\\geq D$. It is easy to see that $\\mathbb{P}(j\\in \\boldsymbol{W}|Q)$ is given by\n\\begin{dmath}\\label{Eq:5}\n\\sum_{\\substack{W\\subset Q_0: \\\\ |W|=D, j\\in W}} \\sum_{\\substack{S\\subset \\mathcal{K}\\setminus Q_0:\\\\ |S|=\\alpha-\\rho}}\\mathbb{P}(\\boldsymbol{W}=W, \\boldsymbol{S}=S\\cup Q_0\\setminus W|Q)  \n\\end{dmath} for all $j\\in Q_0$, and \n\\begin{dmath}\\label{Eq:6}\n \\hspace{-0.5cm}\\sum_{\\substack{W\\subset Q_i:\\\\ |W|=D, j\\in W}} \\mathbb{P}(\\boldsymbol{W}=W, \\boldsymbol{S}=Q_i\\setminus W|Q)\n\\end{dmath} for all $j\\in Q_i$, $i\\in [\\beta]$. Similarly as before, it can be seen that~\\eqref{Eq:5} and~\\eqref{Eq:6} are equal if the two quantities \n\\begin{dmath}\\label{Eq:7}\n\\sum_{\\substack{W\\subset Q_0: \\\\ |W|=D, j\\in W}} \\sum_{\\substack{S\\subset \\mathcal{K}\\setminus Q_0:\\\\ |S|=\\alpha-\\rho}}\\mathbb{P}(Q|\\boldsymbol{W}=W, \\boldsymbol{S}=S\\cup Q_0\\setminus W)  \n\\end{dmath} and \n\\begin{dmath}\\label{Eq:8}\n \\hspace{-0.5cm}\\sum_{\\substack{W\\subset Q_i:\\\\ |W|=D, j\\in W}} \\mathbb{P}(Q|\\boldsymbol{W}=W, \\boldsymbol{S}=Q_i\\setminus W)\n\\end{dmath} are equal. For any $W\\subset Q_0$, $|W|=D$ such that $j\\in W$, and any $S\\subset \\mathcal{K}\\setminus Q_0$, $|S|=\\alpha-\\rho$, we have \n\\begin{align*}\n& \\mathbb{P}(Q|\\boldsymbol{W}=W, \\boldsymbol{S}=S\\cup Q_0\\setminus W) = \\left(\\frac{\\theta_1}{\\theta_1+\\theta_3}\\right) \\beta!\\left(\\binom{M}{\\rho-D}\\prod_{i=0}^{\\beta-1}\\binom{K-i\\alpha-\\rho}{\\alpha}\\right)^{-1}.\t\n\\end{align*} and accordingly,~\\eqref{Eq:7} is equal to \n\\begin{align*}\n& \\left(\\frac{\\theta_1}{\\theta_1+\\theta_3}\\right) \\binom{\\rho}{D}\\binom{K-\\rho}{\\alpha-\\rho}\\beta!\\left(\\binom{M}{\\rho-D}\\prod_{i=0}^{\\beta-1}\\binom{K-i\\alpha-\\rho}{\\alpha}\\right)^{-1}.\t\n\\end{align*} Fix an $i\\in [\\beta]$. For any $W\\subset Q_i$, $|W|=D$ such that $j\\in W$, we have \n\\begin{align*}\n& \\mathbb{P}(Q|\\boldsymbol{W}=W, \\boldsymbol{S}=Q_i\\setminus W) = \\left(\\frac{\\theta_3}{\\theta_1+\\theta_3}\\right) (\\beta-1)!\\left(\\prod_{i=1}^{\\beta-1}\\binom{K-i\\alpha}{\\alpha}\\right)^{-1}.\t\n\\end{align*} and accordingly,~\\eqref{Eq:8} is equal to \n\\begin{align*}\n& \\left(\\frac{\\theta_3}{\\theta_1+\\theta_3}\\right) \\binom{\\alpha-1}{M}(\\beta-1)!\\left(\\prod_{i=1}^{\\beta-1}\\binom{K-i\\alpha}{\\alpha}\\right)^{-1}.\t\n\\end{align*} Again, for the choice of $\\theta_1$ and $\\theta_3$ as in the protocol, it is easy to verify that~\\eqref{Eq:7} and~\\eqref{Eq:8} are equal.\n\\end{proof}\n\n\\section{Proof of Theorem~\\ref{thm:IPIRSI-Spc}}\\label{sec:IPIRSI-Spc}\nIn this section, we first present a new combinatorial conjecture which, if holds, proves the tightness of the result of Theorem~\\ref{thm:IPIRSI-Gen}. Next, we prove the simplest non-trivial case of this conjecture, yielding the tightness of the capacity lower bound in Theorem~\\ref{thm:IPIRSI-Gen} for $M=1$ and $D=2$. \n\nBefore stating the conjecture, we give two necessary conditions, due to individual-privacy and recoverability, which are essential to relate the IPIR-SI problem to our conjecture. \n\n\\begin{lemma}\\label{lem:2}\nFor any $W\\in \\mathcal{W}$ and $S\\in \\mathcal{S}$ where $S\\cap W=\\emptyset$, and any ${j\\in \\mathcal{K}}$, there must exist $W^{*}\\in \\mathcal{W}$, $j\\in W^{*}$ and ${S^{*}\\in \\mathcal{S}}$ where $S^{*}\\cap W^{*}=\\emptyset$, such that \\[H(X_{W^{*}}| A^{[W,S]}, Q^{[W,S]}, X_{S^{*}}) = 0.\\] \t\n\\end{lemma}\n\n\\begin{proof}\nThe proof is by the way of contradiction, and is omitted for brevity. \t\n\\end{proof}\n\n\\begin{lemma}\\label{lem:3}\nFor any $W\\in \\mathcal{W}$ and $S\\in \\mathcal{S}$ where $S\\cap W=\\emptyset$, and any $J\\subseteq \\mathcal{K}$, if $\\mathbb{P}(\\cup_{j\\in J} E_j|Q^{[W,S]})=1$, then $|J|\\geq \\frac{K}{D}$, where $E_j$ for $j\\in J$ is the event that $j\\in \\boldsymbol{W}$. \n\\end{lemma}\n\n\\begin{proof}\nTake any $J\\subseteq \\mathcal{K}$ such that $\\mathbb{P}(\\cup_{j\\in J} E_j|Q^{[W,S]})=1$. By the union bound, $\\mathbb{P}(\\cup_{j\\in J} E_j|Q^{[W,S]})$ is bounded from above by $\\sum_{j\\in J} \\mathbb{P} (E_j|Q^{[W,S]})$, or equivalently, $\\frac{|J|D}{K}$, noting that $\\mathbb{P}(E_j|Q^{[W,S]})=\\frac{D}{K}$ for all $j\\in \\mathcal{K}$ (by the individual-privacy condition). Since $\\frac{|J|D}{K}\\geq 1$, then $|J|\\geq \\lceil \\frac{K}{D}\\rceil$. \n\\end{proof}\n\n\n \n\nWe would like to show that $H(A^{[W,S]})$, or particularly $H(A^{[W,S]}|Q^{[W,S]})$, for any protocol $(Q^{[W,S]},A^{[W,S]})$ that satisfies the conditions in Lemmas~\\ref{lem:2} and~\\ref{lem:3}, is bounded from below by ${\\min\\{K-M\\lfloor \\frac{K}{M+D}\\rfloor, D\\lceil\\frac{K}{M+D}\\rceil\\}}$. Any such a protocol can be represented by an oracle as follows.  \n\n\n Let $K\\geq 1$, $M\\geq 1$, and $D\\geq 2$ be arbitrary integers such that $D+M\\leq K$. Let $\\mathcal{I}$ and $\\mathcal{J}$ be the set of all subsets $I$ and $J$ of $\\mathcal{K}\\triangleq [K]$ such that $0\\leq |I|\\leq M$ and $|J|\\geq D$, respectively. Let $f: \\mathcal{I}\\rightarrow \\mathcal{J}$ be an arbitrary set relation (mapping). A relation $f$ is called \\emph{good} if the following conditions hold: \n \\begin{itemize}\n \\item[(i)] \t$I\\subseteq f(I)$ for any $I\\in \\mathcal{I}$; \n \\item[(ii)] For any $j\\in \\mathcal{K}$, there exist $I\\in \\mathcal{I}$ and $J\\in \\mathcal{J}$, $|J|=D$, $j\\in J$ where $I\\cap J=\\emptyset$ such that $J\\subseteq f(I)$; \n \\item[(iii)] For any $I_1,I_2\\in \\mathcal{I}$, if $I_2\\subseteq f(I_1)$, then $f(I_2)\\subseteq f(I_1)$;\n \\item[(iv)] For any $J^{*}\\subseteq \\mathcal{K}$, $|J^{*}|< \\lceil \\frac{K}{D}\\rceil$ there exists (non-empty) $I\\in \\mathcal{I}$ such that $f(I)\\cap J^{*}= \\emptyset$. \n \\end{itemize}\n\n\nThinking of the $l$-subsets (for $0\\leq l\\leq M$) in $\\mathcal{I}$ as the potential side information index sets $S^{*}$ and the $D$-subsets in $\\mathcal{J}$ as the possible demand index sets $W^{*}$, one can observe that a good relation $f$, satisfying the conditions (i)-(iv), represents an arbitrary protocol that satisfies the conditions in Lemmas~\\ref{lem:2} and~\\ref{lem:3}. Then, it holds that for any IPIR-SI protocol, $H(A^{[W,S]}|Q^{[W,S]})\\geq K-\\theta$ (for any integer $\\theta\\geq 0$) so long as for any good relation $f$ (defined earlier) there exists a subset $I^{*}\\subseteq \\mathcal{K}$ of size at most $\\theta$ such that the union of $f(I)$ for all $I\\subseteq I^{*}$ is equal to $\\mathcal{K}$. This is because, thinking of $f$ (or in turn, $(Q^{[W,S]}, A^{[W,S]})$) as an oracle, given the messages $\\{X_j\\}_{j\\in I^{*}}$, all other messages $\\{X_j\\}_{j\\in \\mathcal{K}\\setminus I^{*}}$ are recoverable from $A^{[W,S]}$ and $Q^{[W,S]}$; and hence, $H(A^{[W,S]}|Q^{[W,S]})\\geq K-|I^{*}|\\geq K-\\theta$, as desired. \n\n\n\n\n\\begin{conjecture}\\label{conj:1}\nFor any good relation $f$, there exists $I^{*}\\subset \\mathcal{K}$, $|I^{*}|\\leq \\max\\{K-D\\lceil \\frac{K}{M+D}\\rceil, M \\lfloor \\frac{K}{M+D}\\rfloor\\}$ such that $\\cup_{I\\subseteq I^{*}} f(I)=\\mathcal{K}$.\n\\end{conjecture}\n\n\nFor $M=1$ and $D\\geq 2$, the statement of Conjecture~\\ref{conj:1} can be rephrased in the language of graph theory as follows. Let $G = (V,E)$ be an arbitrary directed graph (without parallel edges), where $V$ and $E$ are the set of nodes and edges of $G$, respectively. Denote by $d_{\\mathrm{in}}(v)$ and $d_{\\mathrm{out}}(v)$ the in-degree and out-degree of node $v\\in V$, respectively, over $G$. We define an \\emph{external} (or respectively, \\emph{internal}) \\emph{mother vertex-set} of $G$ as a minimal subset $I^{*}$ of nodes in $V$ from which all other nodes $u$ in $V\\setminus I^{*}$ such that $d_{\\mathrm{out}}(u)\\neq 0$ (or respectively, $d_{\\mathrm{in}}(u)\\neq 0$) can be reached (i.e., for any $u\\in V\\setminus I^{*}$, $d_{\\mathrm{out}}(u)\\neq 0$ (or respectively, $d_{\\mathrm{in}}(u)\\neq 0$), there exists $v\\in I^{*}$ such that there is a directed path from $v$ to $u$ in $G$), and denote the size of an external (or respectively, internal) mother vertex-set $I^{*}$ of $G$ by $\\mu_{\\mathrm{ext}}(G)$ (or respectively, $\\mu_{\\mathrm{int}}(G)$). Also, let $G^{\\mathsf{T}}$ be the transpose of $G$, which is formed by reversing the direction of all edges in $G$ (i.e., $G^{\\mathsf{T}}=(V,E^{\\mathsf{T}})$, where $E^{\\mathsf{T}} = \\{(u,v): (v,u)\\in E\\}$). We call $G$ a \\emph{$D$-graph} if the following conditions hold: \n\\begin{itemize}\n\\item[(i)] For any $v\\in V$, $d_{\\mathrm{in}}(v)\\geq 1$; \n\\item[(ii)] For any $v\\in V$, either $d_{\\mathrm{out}}(v)=0$, or $d_{\\mathrm{out}}(v)\\geq D$; \n\\item[(iii)] $\\mu_{\\mathrm{int}}(G^{\\mathsf{T}})\\geq \\lceil\\frac{K}{D}\\rceil$. \n\\end{itemize}\n\n\\begin{conjecture}\\label{conj:2}\nFor any $D$-graph $G$ on $K$ nodes, ${\\mu_{\\mathrm{ext}}(G)\\leq \\lfloor \\frac{K}{D+1}\\rfloor}$. \t\n\\end{conjecture}\n\n\n\n\nNote that Conjecture~\\ref{conj:2} is equivalent to Conjecture~\\ref{conj:1} for $M=1$. (Since $K-D\\lceil \\frac{K}{D+1}\\rceil\\leq \\lfloor \\frac{K}{D+1}\\rfloor$ for any $D\\leq K-1$, then the upper bound on $|I^{*}|$ in Conjecture~\\ref{conj:1} for $M=1$ reduces to $\\lfloor\\frac{K}{D+1}\\rfloor$.) For any $D$-graph $G = (V,E)$ on $K$ nodes, we can define $f(v)$ for any $v\\in V$ as the set of all nodes (including $v$) that can be reached from node $v$ (via a directed path in $G$). Then, it is easy to verify that $f$ satisfies the conditions (i)-(iv) for a good relation. Note also that $\\mu_{\\mathrm{ext}}(G)$ represents the size of a (minimal) subset $I^{*}\\subseteq V$ such that $\\cup_{v\\in I^{*}} f(v) = V$. This indeed shows the equivalence between the two conjectures for $M=1$.  \n\nIn the following, we prove Conjecture~\\ref{conj:2} for $M=1$ and $D=2$, and hence the proof of Theorem~\\ref{thm:IPIRSI-Spc}.  \n\n\n\\begin{lemma}\\label{lem:4}\nFor any $2$-graph $G$ on $K$ nodes, $\\mu_{\\mathrm{ext}}(G)\\leq \\lfloor \\frac{K}{3}\\rfloor$. \t\n\\end{lemma}\n\n\\begin{proof}\nLet $G$ be an arbitrary $2$-graph on $K$ nodes. Suppose that $\\mu_{\\mathrm{ext}}(G)>\\lfloor \\frac{K}{3}\\rfloor$. We need to show a contradiction. Let $n\\triangleq \\mu_{\\mathrm{ext}}(G)$. Consider an arbitrary partition of the nodes in $G$ into $n$ parts, $V_1,\\dots,V_n$, such that each part $V_j$ contains a node $v_j$ from which all other nodes in $V_j$ can be reached. (Note that a node in a part can potentially reach some other nodes in other parts.) Obviously, $I^{*}\\triangleq \\{v_1,\\dots,v_n\\}$ is an external mother vertex-set of $G$. \n\nBy the minimality of $I^{*}$, it follows that no node $v_j$ can be reached from any node out of the part $V_j$. (Otherwise, from the nodes in $I^{*}\\setminus \\{v_j\\}$ all other nodes can be reached, and this contradicts the minimality of $I^{*}$.) Since $d_{\\mathrm{in}}(v_j)\\geq 1$ (by definition), then there must exist another node $u_j$ in $V_j$ that reaches $v_j$. Also, no part $V_j$ can contain only a single node $v_j$, simply because $d_{\\mathrm{in}}(v_j)\\geq 1$, and the node $v_j$ can be reached from some other node(s) in some other part(s), which again contradicts the minimality of $I^{*}$. \n\nTake an arbitrary part $V_j = \\{v_j,u_j\\}$ of size $2$ (if exists). Since $v_j$ reaches $u_j$ (over $G$), then ${d_{\\mathrm{out}}(v_j)\\geq 1}$, and particularly, ${d_{\\mathrm{out}}(v_j)\\geq 2}$ (noting that $G$ is a $2$-graph). Thus, the node $v_j$ reaches some other node(s), say $w$, in some other part(s) over $G$. Equivalently, the node $w$ reaches both nodes $v_j$ and $u_j$ over $G^{\\mathsf{T}}$. For any other part $V_j$ of size $i\\geq 3$, the nodes $v_j$ and $u_j$ can be reached from each node in $V_j\\setminus \\{v_j,u_j\\}$ over $G^{\\mathsf{T}}$. \n\nPutting these arguments together, it follows that each node in $\\{v_j,u_j\\}_{j\\in [n]}$ can be reached from some node(s) in $J^{*}\\triangleq V\\setminus \\{v_j,u_j\\}_{j\\in [n]}$ via a directed path in $G^{\\mathsf{T}}$. Then, $\\mu_{\\mathrm{int}}(G^{\\mathsf{T}})\\leq |J^{*}|= K-2n$. By assumption, $\\mu_{\\mathrm{ext}}(G)=n>\\lfloor\\frac{K}{3}\\rfloor$. Thus, $|J^{*}|<K-2\\lfloor \\frac{K}{3}\\rfloor$, and consequently, $\\mu_{\\mathrm{int}}(G^{\\mathsf{T}}) <K-2\\lfloor \\frac{K}{3}\\rfloor$. Since ${K-2\\lfloor \\frac{K}{3}\\rfloor\\leq \\lceil\\frac{K}{2}\\rceil}$, then $\\mu_{\\mathrm{int}}(G^{\\mathsf{T}}) <\\lceil\\frac{K}{2}\\rceil$. This is a contradiction because $\\mu_{\\mathrm{int}}(G^{\\mathsf{T}})\\geq \\lceil \\frac{K}{2}\\rceil$ for any $2$-graph $G$ on $K$ nodes. \n\\end{proof}\n\n\\bibliographystyle{IEEEtran}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Optimal transport} \\label{sec1}\n\nLet us state the Kantorovich transport problem.\nWe take $(X, d)$ to be a compact metric space. Our\n``before'' and ``after'' dirtpiles are measures\n$\\mu_0, \\mu_1 \\in P(X)$. They both have mass one.\nWe want to move the total amount of dirt from $\\mu_0$ to $\\mu_1$\nmost efficiently. A moving scheme, maybe not optimal, will be \ncalled a {\\em transference plan}. Intuitively, it amounts to specifying how\nmuch dirt is moved from a point $x_0$ to a point $x_1$. That is, we have a\nprobability measure\n$\\pi \\in P(X\\times X)$, which we informally write as $\\pi(x_0,x_1)$.\nThe statement\nthat $\\pi$ does indeed transport $\\mu_0$ to $\\mu_1$ translates to\nthe condition that \n\\begin{equation} \\label{eqn1}\n(p_0)_* \\pi \\: = \\: \\mu_0,\\qquad (p_1)_* \\pi \\: = \\: \\mu_1,\n\\end{equation}\nwhere $p_0, p_1 \\: : \\: X \\times X \\rightarrow X$ are projections\nonto the first and second factors, respectively.\n\nWe will use optimal transport with quadratic cost function\n(square of the distance). The total cost of the transference plan\n$\\pi$ is given by adding the contributions of $d(x_0, x_1)^2$ with\nrespect to $\\pi$. \nTaking the infimum of this with respect to $\\pi$ \ngives the square of the {\\em Wasserstein distance} \n$W_2(\\mu_0, \\mu_1)$\nbetween $\\mu_0$ and $\\mu_1$, i.e.\n\\begin{equation} \\label{var}\nW_2(\\mu_0, \\mu_1)^2 \\: = \\:\n\\inf_{\\pi}  \\: \\int_{X \\times X} d(x_0, x_1)^2 \\: d\\pi(x_0, x_1),\n\\end{equation}\nwhere $\\pi$ ranges over the set of all transference plans between\n$\\mu_0$ and $\\mu_1$. Any minimizer $\\pi$ for this variational problem\nis called an {\\em optimal transference plan}.\n\nIn \\eqref{var}, one can replace the infimum by the minimum\n\\cite[Proposition 2.1]{TOT}, i.e. there always exists\n(at least) one optimal transference plan.\nIt turns out that $W_2$ is a metric on $P(X)$. \nThe topology that it induces on $P(X)$ is the\nweak-$*$ topology~\\cite[Theorems 7.3 and 7.12]{TOT}.\nWhen equipped with the metric $W_2$, $P(X)$ is a compact metric space.\nIn this way, to each compact metric space $X$ we have assigned\nanother compact metric space $P(X)$.\nThe {\\em Wasserstein space} $(P(X), W_2)$ seems to be a very natural \nobject in mathematics. It generally has infinite topological or\nHausdorff dimension.  (If $X$ is a finite set then $P(X)$ is a\nsimplex, with a certain metric.) It is always contractible, as can\nbe seen by fixing a measure $\\mu_0 \\in P(X)$ and \nlinearly contracting\nother measures $\\mu \\in P(X)$ to $\\mu_0$ by\n$t \\rightarrow t\\mu_0 \\: + \\: (1-t) \\mu$.\n\n\\begin{proposition} \\cite[Corollary 4.3]{LV} \\label{prop1}\nIf $\\lim_{i \\rightarrow \\infty} (X_i, d_i) \\: = \\: \n(X,d)$ in the Gromov-Hausdorff topology \nthen $\\lim_{i \\rightarrow \\infty} (P(X_i), W_2) \\: = \\: \n(P(X), W_2)$ in the Gromov-Hausdorff topology.\n\\end{proposition}\n\n\nA {\\em Monge transport} is a transference plan coming from a map\n$F \\: : \\: X \\rightarrow X$ with $F_* \\mu_0 \\: = \\: \\mu_1$, given\nby $\\pi \\: = \\: ({\\rm Id}\\,, F)_* \\mu_0$. In general an optimal transference\nplan does not have to be a Monge transport, although this may\nbe true under some assumptions.\n\nWhat does optimal transport look like in Euclidean space $\\mathbb R^n$?\nSuppose that $\\mu_0$ and $\\mu_1$ are compactly supported and\nabsolutely continuous with respect to Lebesgue measure.  Brenier \n\\cite{Brenier} and\nRachev-R\\\"uschendorf \\cite{RR} showed that\nthere is a unique optimal transference plan between $\\mu_0$ and\n$\\mu_1$, which is a Monge\ntransport.  Furthermore, there is a convex function $V$ on \n$\\mathbb R^n$ so that for almost all $\\vec{x}$, the Monge transport is given by\n$F(\\vec{x}) \\: = \\: \\vec{\\nabla}_{\\vec{x}} V$. So to find the optimal\ntransport, one finds a convex function $V$ such that\nthe pushforward, under the map\n$\\nabla V \\: : \\: \\mathbb R^n \\rightarrow \\mathbb R^n$, sends $\\mu_0$ to\n$\\mu_1$. This solves the Monge problem for such measures,\nunder our assumption of quadratic cost function.  The solution to\nthe original problem of Monge, with linear cost function, is\nmore difficult; see \\cite{Evans-Gangbo}.\n\nThe statement of the Brenier-Rachev-R\\\"uschendorf theorem may sound like\nanathema to a geometer.  One is identifying the gradient\nof $V$ (at $\\vec{x}$), which is a vector, with the image of \n$\\vec{x}$ under a map, which is a\npoint. Because of this, it is not evident how to extend even the\nstatement of the\ntheorem if one wants to do optimal transport on \na Riemannian manifold. The extension was done by McCann \\cite{McCann (2001)}.\nThe key point is that on $\\mathbb R^n$, we can write\n$\\vec{\\nabla}_{\\vec{x}} V \\: = \\: \\vec{x} - \\vec{\\nabla}_{\\vec{x}} \\phi$,\nwhere $\\phi(\\vec{x}) \\: = \\: \\frac{|\\vec{x}|^2}{2} \\: - \\: V(\\vec{x})$.\nTo understand the relation between $V$ and $\\phi$, we note that if the\nconvex function $V$ were smooth then $\\phi$ would have Hessian\nbounded above by the identity.  On a Riemannian manifold $(M, g)$,\nMcCann's theorem says that\nan optimal transference plan between two compactly supported\nabsolutely continuous\nmeasures is a Monge\ntransport $F$ that satisfies $F(m) \\: = \\: \\exp_m (- \\nabla_m \\phi)$\nfor almost all $m$,\nwhere $\\phi$ is a function on $M$ with Hessian bounded above by $g$\nin a generalized sense. More precisely, $\\phi$\nis $\\frac{d^2}{2}$-concave in the sense that it can\nbe written in the form\n\\begin{equation} \\label{eqn2}\n\\phi(m) \\: = \\: \\inf_{m^\\prime \\in M} \n\\left( \\frac{d(m, m^\\prime)^2}{2} \\: - \\: \\widetilde{\\phi}(m^\\prime) \\right) \n\\end{equation}\nfor some function $\\widetilde{\\phi} \\: : \\: M \\rightarrow [-\\infty, \\infty)$.\n\nReturning to the metric space setting, if $(X, d)$ is a compact\nlength space and one has an optimal transference plan\n$\\pi$ then one would physically perform the transport\nby picking up pieces of dirt in $X$ and moving them along minimal\ngeodesics to other points in $X$, in a way consistent with \nthe transference plan\n$\\pi$. The transference plan $\\pi$ tells us how much dirt has to go\nfrom $x_0$ to $x_1$, but does not say anything about which\nminimal geodesics from $x_0$ to $x_1$ we should actually use. After making\nsuch a choice of minimizing geodesics, we obtain a $1$-parameter family of\nmeasures $\\{\\mu_t\\}_{t \\in [0,1]}$ by stopping the\nphysical transport procedure at time $t$ and looking\nat where the dirt is.\nThis suggests looking at $(P(X), W_2)$ as a length space.\n\n\\begin{proposition} \n\\cite[Corollary 2.7]{LV},\\cite[Proposition 2.10(iii)]{Sturm2} \\label{prop1.5}\nIf $(X, d)$ is a compact length space then $(P(X), W_2)$ is a compact\nlength space.\n\\end{proposition}\n\nHereafter we assume that $(X,d)$ is a compact length space.\nBy definition, a {\\em Wasserstein geodesic} is a minimizing geodesic in the\nlength space $(P(X), W_2)$. (We will always parametrize minimizing\ngeodesics in length spaces to have constant speed.)\nThe length space $(P(X), W_2)$ has some interesting features;\neven for simple $X$, there may be an uncountable number of\nWasserstein geodesics between two measures $\\mu_0, \\mu_1 \\in P(X)$\n\\cite[Example 2.9]{LV}.\n\nAs mentioned above, there is a relation between\nminimizing geodesics in $(P(X), W_2)$  and minimizing geodesics in $X$.\nLet $\\Gamma$ be the set of minimizing\ngeodesics $\\gamma \\: : \\: [0, 1] \\rightarrow X$. \nIt is compact in the uniform topology. \nFor any $t\\in [0,1]$, the {\\em evaluation map} \n$e_t \\: : \\: \\Gamma \\rightarrow X$ defined by \n\\begin{equation} \\label{eqn3}\ne_t(\\gamma) \\: = \\: \\gamma(t) \\end{equation} \nis continuous. Let $E \\: : \\: \\Gamma \\rightarrow X \\times X$ be the\n``endpoints'' map given by $E(\\gamma) \\: = \\: (e_0(\\gamma),e_1(\\gamma))$. \nA {\\em dynamical transference plan} consists of a transference plan $\\pi$ \nand a Borel measure $\\Pi$ on $\\Gamma$ such that $E_* \\Pi \\: = \\: \\pi$;\nit is said to be optimal if $\\pi$ itself is. \nIn words, the transference plan $\\pi$ tells us how much mass goes from\na point $x_0$ to another point $x_1$, but does not tell us about the actual\npath that the mass has to follow. Intuitively, mass should flow\nalong geodesics, but there may be several possible choices of geodesics\nbetween two given points and the transport may be divided among\nthese geodesics; this is the information provided\nby $\\Pi$. \n\nIf $\\Pi$ is an optimal dynamical transference plan then for $t \\in [0,1]$, \nwe put\n\\begin{equation} \\label{eqn4}\n\\mu_t \\: = \\: (e_t)_* \\Pi.\n\\end{equation} \nThe one-parameter family of measures $\\{\\mu_t\\}_{t \\in [0,1]}$ \nis called a {\\em displacement interpolation}.\nIn words, $\\mu_t$ is what has become of the mass of $\\mu_0$ after\nit has travelled from time~0 to time~$t$ according to the dynamical\ntransference plan $\\Pi$.\n\n\\begin{proposition} \\cite[Lemma 2.4 and Proposition 2.10]{LV} \\label{prop2}\nAny displacement interpolation is a Wasserstein geodesic.\nConversely, any Wasserstein geodesic arises as a displacement\ninterpolation from some \noptimal dynamical transference plan.\n\\end{proposition}\n\nIn the Riemannian case, if $\\mu_0, \\mu_1$ are absolutely\ncontinuous with respect to $\\operatorname{dvol}_M$, and $F(m) \\: = \\:\n\\exp_m (- \\: \\nabla_m \\phi)$ is the Monge transport \nbetween them, then there is a unique\nWasserstein geodesic between $\\mu_0$ and $\\mu_1$\ngiven by $\\mu_t \\: = \\: (F_t)_* \\mu_0$, where\n$F_t(m) \\: = \\: \\exp_m (- \\: t \\: \\nabla_m \\phi)$.\nHere $\\mu_t$ is also absolutely continuous\nwith respect to $\\operatorname{dvol}_M$. On the other hand, if $\\mu_0 = \\delta_{m_0}$ and \n$\\mu_1 = \\delta_{m_1}$ then some Wasserstein geodesics from\n$\\mu_0$ to $\\mu_1$ are of the form\n$\\mu_t \\: = \\: \\delta_{c(t)}$, where $c$ is a \nminimizing geodesic from $m_0$ to $m_1$. In particular,\nthe Wasserstein geodesic need not be unique.\n\nIn a remarkable paper \\cite{Otto (2001)}, motivated by PDE problems,\nOtto constructed a {\\em formal}\ninfinite-dimensional\nRiemannian metric $g_{H^{-1}}$ on $P(\\mathbb R^n)$. To describe $g_{H^{-1}}$,\nfor simplicity we work with a compact\nRiemannian manifold $M$ instead of $\\mathbb R^n$. Suppose\nthat $\\mu \\in P(M)$ can be written as $\\mu \\: = \\: \\rho \\:\n\\operatorname{dvol}_M$, with $\\rho$ a smooth positive function. We formally think\nof a tangent vector $\\delta \\mu \\in T_\\mu P(M)$ as being a\nvariation of $\\mu$, which we take to be\n$(\\delta \\rho) \\: \\operatorname{dvol}_M$ with $\\delta \\rho \\in C^\\infty(M)$.\nThere is a $\\Phi \\in C^\\infty(M)$, unique up to constants,\nso that $\\delta \\rho \\: = \\: d^* (\\rho d \\Phi)$. Then by\ndefinition,\n\\begin{equation} \\label{eqn5}\ng_{H^{-1}}(\\delta \\mu, \\delta \\mu) \\: = \\:\n\\int_M |d \\Phi|^2 \\: d\\mu.\n\\end{equation}\nOne sees that in terms of $\\delta \\rho \\in C^\\infty(M)$, $g_{H^{-1}}$\ncorresponds to a weighted $H^{-1}$-inner product.\n\nOtto showed that the\ncorresponding distance function on $P(M)$ is {\\em formally} $W_2$, and that\nthe ``infinite-dimensional Riemannian manifold'' $(P(\\mathbb R^n),\ng_{H^{-1}})$ {\\em formally} has nonnegative sectional curvature.\nOne can make rigorous sense of these statements in terms of Alexandrov\ngeometry.\n\n\\begin{proposition} \n\\cite[Theorem A.8]{LV},\\cite[Proposition 2.10(iv)]{Sturm2} \\label{prop3}\n$(P(M), W_2)$ has nonnegative Alexandrov curvature if and only\nif $M$ has nonnegative sectional curvature.\n\\end{proposition}\n\n\\begin{proposition} \\cite[Proposition A.33]{LV} \\label{prop4}\nIf $M$ has nonnegative sectional curvature then for each\nabsolutely continuous measure $\\mu \\: = \\: \\rho \\: \\operatorname{dvol}_M \\in P(M)$, the\ntangent cone $T_\\mu P(M)$ is an inner product space. If $\\rho$ is smooth\nand positive\nthen the inner product on $T_\\mu P(M)$ equals $g_{H^{-1}}$.\n\\end{proposition}\n\nAn open question is whether there is any good\nsense in which $(P(M), W_2)$, or a large part thereof,\ncarries an infinite-dimensional Riemannian structure.\nThe analogous question for finite-dimensional Alexandrov\nspaces has been much studied.\n\n\\begin{remark} \\label{rmk2}\nIn Sturm's work he\nuses the following interesting metric $D$ on the set of compact \nmetric-measure spaces \\cite[Definition 3.2]{Sturm2}. \nGiven ${\\mathcal X}_1 = (X_1, d_1, \\nu_1)$ \nand ${\\mathcal X}_2 = (X_2, d_2, \\nu_2)$, let $\\widehat{d}$ denote a\nmetric on the disjoint union $X_1 \\coprod X_2$ such that\n$\\widehat{d} \\big|_{X_1 \\times X_1} \\: = \\: d_1$ and \n$\\widehat{d} \\big|_{X_2 \\times X_2} \\: = \\: d_2$. Then\n\\begin{equation} \\label{eqn6}\nD({\\mathcal X}_1, {\\mathcal X}_2 )^2 \\: = \\:\n\\inf_{\\widehat{d}, q} \\int_{X_1 \\times X_2}\n\\widehat{d}(x_1, x_2)^2 \\: dq(x_1, x_2), \n\\end{equation}\nwhere $q$ runs over probability measures on $X_1 \\times X_2$ whose\npushforwards onto $X_1$ and $X_2$ are $\\nu_1$ and $\\nu_2$, respectively.\nIf one restricts to metric-measure spaces with an upper diameter bound\nwhose measures have full\nsupport and satisfy a uniform doubling condition\n(which will be the case with a lower Ricci curvature bound)\nthen the topology coming from $D$ coincides with the MGH topology\nof Definition \\ref{def2} \\cite[Lemma 3.18]{Sturm2},\\cite{TOT2}.\n\\end{remark}\n\n\\section{Motivation for displacement convexity} \\label{sec2}\n\nTo say a bit more about the PDE motivation, we recall that\nthe heat equation $\\frac{\\partial f}{\\partial t} \\: = \\: \\nabla^2 f$\ncan be considered to be the formal gradient flow of the\nDirichlet energy $E(f) \\: = \\: \\frac12 \\: \\int_M |df|^2 \\: \\operatorname{dvol}_M$ on\n$L^2(M, \\operatorname{dvol}_M)$. (Our conventions are that a function decreases\nalong the flowlines of \nits gradient flow, so on a finite-dimensional Riemannian\nmanifold $Y$ the gradient\nflow of a function $F \\in C^\\infty(Y)$ is $\\frac{dc}{dt} \\: = \\:\n- \\: \\nabla F$.) \nJordan-Kinderlehrer-Otto showed that the heat equation on measures\ncan also be formally written as a gradient flow\n\\cite{Jordan-Kinderlehrer-Otto (1998)}.  Namely, for\na smooth\nprobability measure $\\mu \\: = \\: \\rho \\: \\frac{\\operatorname{dvol}_M}{\\operatorname{vol}(M)}$,\nlet us put $H_\\infty(\\mu) \\: = \\: \\int_M \\rho \\: \\log \\rho \\:\n\\frac{\\operatorname{dvol}_M}{\\operatorname{vol}(M)}$. Then the heat equation\n$\\frac{\\partial }{\\partial t} \n\\left( \\rho \\: \\frac{\\operatorname{dvol}_M}{\\operatorname{vol}(M)} \\right) \\: = \\:\n\\nabla^2 \\rho \\: \\frac{\\operatorname{dvol}_M}{\\operatorname{vol}(M)}$ \nis formally the gradient flow of $H_\\infty$\non $P(M)$, where $P(M)$ has Otto's formal Riemannian metric.\nIdentifying a.c. measures and measurable functions using $\\frac{\\operatorname{dvol}_M}{\\operatorname{vol}(M)}$, this\ngave a new way to realize the heat equation as a gradient flow.\n\nAlthough this approach may not give much new\ninformation about the heat equation,\nit has more relevance if one considers other functions $H$ on $P(M)$,\nwhose gradient flows can give rise to interesting nonlinear\nPDE's such as the porous medium equation. Again formally,\nif one has positive lower bounds on the Hessian of $H$ then one can \ndraw conclusions about uniqueness of critical points and \nrates of convergence of the gradient flow to the critical point,\nwhich one can then hope to make rigorous.\nThis reasoning motivated McCann's notion of {\\em displacement convexity},\ni.e. convexity of a function $H$ along Wasserstein geodesics \n\\cite{McCann (1997)}.\n(We recall that on a smooth manifold, a smooth function\nhas a nonnegative Hessian if and only\nif it is convex when restricted to each geodesic.)\n\nIn a related direction, Otto and Villani \\cite{Otto-Villani (2000)} saw that\nconvexity properties on $P(M)$ could be used to give heuristic arguments for\nfunctional inequalities on $M$, such as the log Sobolev inequality.  They\ncould then give rigorous proofs based on these heuristic arguments.\nGiven a smooth background probability measure\n$\\nu \\: = \\: e^{- \\Psi} \\: \\operatorname{dvol}_M$ and an absolutely \ncontinuous probability measure\n$\\mu \\: = \\: \\rho \\: \\nu$, let us now put\n$H_\\infty(\\mu) \\: = \\: \\int_M \\rho \\: (\\log \\rho) \\: d\\nu$. \nAs part of their work,\nOtto and Villani computed the formal Hessian of the\nfunction $H_\\infty$ on $P(M)$\nand found that it is bounded below\nby $K g_{H^{-1}}$ provided that the Bakry-\\'Emery tensor\n$\\operatorname{Ric}_\\infty \\: = \\: \\operatorname{Ric} \\: + \\: \\mathop{{\\rm Hess}\\,}(\\Psi)$ satisfies\n$\\operatorname{Ric}_\\infty \\: \\ge \\: K g$ on $M$. \nThis was perhaps the first indication that Ricci curvature\nis related to convexity properties on Wasserstein space.\n\nAround the same time,\nCordero-Erausquin-McCann-Schmuckenschl\\\"ager \\cite{CEMS01}\ngave a rigorous proof of the convexity of certain functions\non $P(M)$ when $M$ has dimension $n$ and nonnegative Ricci\ncurvature.  Suppose that $A \\: : \\: [0, \\infty) \\rightarrow\n\\mathbb R$ is a continuous convex function\nwith $A(0) = 0$ such that $\\lambda \\rightarrow \n\\lambda^n A(\\lambda^{-n})$ is a convex  function on $\\mathbb R^+$.\nIf $\\mu \\: = \\: \\rho \\: \\frac{\\operatorname{dvol}_M}{\\operatorname{vol}(M)}$ is  an\nabsolutely continuous probability measure then \nput $H_A(\\mu) \\: = \\: \\int_M A(\\rho) \\: \\frac{\\operatorname{dvol}_M}{\\operatorname{vol}(M)}$. The\nstatement is that if \n$\\mu_0, \\mu_1 \\in P(M)$ are absolutely continuous, and\n$\\{\\mu_t\\}_{t \\in [0,1]}$ is the (unique) Wasserstein \ngeodesic between them, then $H_A(\\mu_t)$ is convex in $t$,\nagain under the assumption of nonnegative Ricci curvature.\n\nFinally, von Renesse and Sturm \\cite{Sturm-von Renesse} extended the work of\nCordero-Erausquin-McCann-Schmuckenschl\\\"ager to show that\nthe function $H_\\infty$, defined by\n$H_\\infty \\left(\\rho \\: \\frac{\\operatorname{dvol}_M}{\\operatorname{vol}(M)} \\right) \\: = \\: \\int_M \\rho \\:\n\\log \\rho \\: \\frac{\\operatorname{dvol}_M}{\\operatorname{vol}(M)}$, \nis $K$-convex along Wasserstein geodesics \nbetween  absolutely-continuous measures if and only if $\\operatorname{Ric} \\: \\ge \\: Kg$.\n(The relation with the Otto-Villani result is that\n$\\Psi$ is taken to be constant, so $\\nu \\: = \\: \\frac{\\operatorname{dvol}_M}{\\operatorname{vol}(M)}$.)\nThe ``if'' implication is along the lines of the\nCordero-Erausquin-McCann-Schmuckenschl\\\"ager result and the\n``only if'' implication involves some local arguments.\n\nAlthough these results indicate a formal relation between \nRicci curvature and\ndisplacement convexity, one can ask for a more intuitive\nunderstanding. Here is one example.\n\\begin{example}\nConsider the functional\n$H_\\infty \\left(\n\\rho \\: \\frac{\\operatorname{dvol}_M}{\\operatorname{vol}(M)} \\right) \\: = \\: \\int_M \\rho \\:\n\\log \\rho \\: \\frac{\\operatorname{dvol}_M}{\\operatorname{vol}(M)}$. It is minimized, among\nabsolutely continuous probability measures on $M$, when\n$\\rho \\: = \\: 1$, i.e. when the measure \n$\\mu \\: = \\: \\rho \\: \\frac{\\operatorname{dvol}_M}{\\operatorname{vol}(M)}$ is the\nuniform measure $\\frac{\\operatorname{dvol}_M}{\\operatorname{vol}(M)}$. In this sense, $H_\\infty$ measures\nthe {\\em nonuniformity} of $\\mu$ with respect to\n$\\frac{\\operatorname{dvol}_M}{\\operatorname{vol}(M)}$. Now take $M = S^2$. Let $\\mu_0$ and \n$\\mu_1$ be two small congruent rotationally symmetric\nblobs, centered at the\nnorth and south poles respectively.  Clearly\n$U_\\infty(\\mu_0) \\: = \\: U_\\infty(\\mu_1)$. Consider the\nWasserstein geodesic from $\\mu_0$ to $\\mu_1$. It takes\nthe blob $\\mu_0$ and pushes it down in a certain way \nalong the lattitudes until it becomes $\\mu_1$. At an\nintermediate time, say around $t = \\frac12$, the blob\nhas spread out to form a ring. When it spreads, it becomes\n{\\em more} uniform with respect to $\\frac{\\operatorname{dvol}_M}{\\operatorname{vol}(M)}$. Thus the\n{\\em nonuniformity} at an intermediate time is {\\em at most}\nthat at times $t =0$ or $t = 1$. This can be seen as a\nconsequence of the convexity of $H_\\infty(\\mu_t)$ in $t$,\ni.e. for $t \\in [0,1]$ we have\n$H_\\infty(\\mu_t) \\: \\le H_\\infty(\\mu_0) \\: = \\:\nH_\\infty(\\mu_1)$. In this way the displacement convexity\nof $H_\\infty$ can be seen as an averaged form of the focusing\nproperty of positive curvature.  Of course this\nexample does not indicate why the relevant curvature is\nRicci curvature, as opposed to some other curvature, but\nperhaps gives some indication of why curvature is related\nto displacement convexity.\n\\end{example}\n\n\\section{Entropy functions and displacement convexity} \\label{sec3}\n\nIn this section we give the definition of nonnegative\n$N$-Ricci curvature. We then outline the proof that it is\npreserved under measured Gromov-Hausdorff limits.  In the\nnext section we relate the definition to the classical notion\nof Ricci curvature, in the case of a smooth metric-measure space.\n\n\\subsection{Definitions}\n\nWe first define the relevant ``entropy'' functionals.\n\nLet $X$ be a compact Hausdorff space.\nLet $U \\: : \\: [0, \\infty) \\rightarrow \\mathbb R$ be a continuous\nconvex function with $U(0) \\: = \\: 0$.\nGiven a reference probability measure $\\nu \\in P(X)$,\ndefine the entropy function\n$U_\\nu \\: : \\: P(X) \\rightarrow \\mathbb R \\cup \\{\\infty\\}$ by\n\\begin{equation} \\label{eqn6.5}\nU_\\nu(\\mu) = \n\\int_X U(\\rho(x))\\,d\\nu(x) + U'(\\infty)\\,\\mu_s(X), \n\\end{equation}\nwhere \n\\begin{equation} \\label{eqn7}\n\\mu = \\rho \\nu + \\mu_s \n\\end{equation}\nis the Lebesgue decomposition of $\\mu$ with respect to $\\nu$ \ninto an absolutely continuous part $\\rho\\nu$ and a singular part $\\mu_s$, and\n\\begin{equation} \\label{eqn8}\nU^\\prime(\\infty) \\: = \\: \\lim_{r \\rightarrow \\infty} \\frac{U(r)}{r}.\n\\end{equation}\n\n\\begin{example}\nGiven $N \\in (1, \\infty]$, take the function \n$U_N$ on $[0, \\infty)$ to be\n\\begin{equation} \\label{eqn9}\nU_N(r) \\: = \\: \n\\begin{cases}\nNr (1 - r^{-1\/N})& \\text{ if $1 < N < \\infty$}, \\\\\nr \\log{r} & \\text{ if $N = \\infty$}.\n\\end{cases}\n\\end{equation}\nLet\n$H_{N,\\nu} \\: : \\: P(X) \\rightarrow \\mathbb R \\cup \\{\\infty \\}$ be\nthe corresponding entropy function. If\n$N \\in (1, \\infty)$ then\n\\begin{equation} \\label{eqn10}\nH_{N,\\nu} = N - N \\int_X \\rho^{1-\\frac{1}{N}} \\,d\\nu, \n\\end{equation} \nwhile if $N = \\infty$ then\n\\begin{equation} \\label{eqn11}\nH_{\\infty,\\nu}(\\mu) = \\int_X \\rho\\log\\rho \\,d\\nu\n\\end{equation}\nif $\\mu$ is absolutely continuous with\nrespect to $\\nu$ and $H_{\\infty,\\nu}(\\mu) = \\infty$ otherwise.\n\\end{example}\n\nOne can show that as a function of $\\mu \\in P(X)$, $U_\\nu(\\mu)$ is\nminimized when $\\mu \\: = \\: \\nu$. It would be better to\ncall $U_\\nu$ a ``negative entropy'', but we will be sloppy.\nHere are the technical properties of $U_\\nu$ that we need.\n\n\\begin{proposition} \\cite{Liese-Vajda (1987)},\\cite[Theorem B.33]{LV} \n\\label{prop5}\n(i) $U_\\nu(\\mu)$ is a lower semicontinuous function of $(\\mu,\\nu)\n\\in P(X) \\times P(X)$.\nThat is, if $\\{\\mu_k\\}_{k=1}^\\infty$ and \n$\\{\\nu_k\\}_{k=1}^\\infty$ are sequences in $P(X)$\nwith $\\lim_{k \\rightarrow \\infty} \\mu_k \\: = \\: \\mu$ and\n$\\lim_{k \\rightarrow \\infty} \\nu_k \\: = \\: \\nu$ in the weak-$*$ topology then\n\\begin{equation} \\label{eqn12}\nU_\\nu(\\mu) \\leq \\liminf_{k\\to\\infty} U_{\\nu_k}(\\mu_k).\n\\end{equation}\n\n\\quad (ii) $U_\\nu(\\mu)$ is nonincreasing under pushforward.\nThat is, if $Y$ is a compact Hausdorff space and\n$f \\: : \\: X \\rightarrow Y$ is a Borel map then\n\\begin{equation} \\label{eqn13}\nU_{f_* \\nu} (f_*\\mu) \\leq U_\\nu(\\mu).\n\\end{equation}\n\\end{proposition}\n\nIn fact, the $U^\\prime(\\infty) \\: \\mu_s(X)$ term in (\\ref{eqn6.5}) is \ndictated by the fact that we want\n$U_\\nu$ to be lower semicontinuous on $P(X)$.\n\nWe now pass to the setting of a compact measured length space $(X, d, \\nu)$.\nThe definition of nonnegative $N$-Ricci curvature will be\nin terms of the convexity of certain\nentropy functions on $P(X)$, where the entropy is relative to\nthe background measure $\\nu$. By ``convexity'' we mean\nconvexity along Wasserstein geodesics, i.e. displacement\nconvexity. We first describe the\nrelevant class of entropy functions.\n \nIf $N\\in [1,\\infty)$ then \nwe define ${\\mathcal D}{\\mathcal C}_N$ to be the set of such functions $U$\nso that the function\n\\begin{equation} \\label{eqn14}\n\\psi(\\lambda) = \\lambda^N \\:U(\\lambda^{-N}) \n\\end{equation}\nis convex on $(0, \\infty)$.\nWe further define ${\\mathcal D}{\\mathcal C}_\\infty$ \nto be the set of such functions $U$ so that the function\n\\begin{equation} \\label{eqn15}\n\\psi(\\lambda) = e^\\lambda \\: U(e^{-\\lambda})\n\\end{equation} \nis convex on $(- \\infty, \\infty)$.\nA relevant example of an element of ${\\mathcal D}{\\mathcal C}_N$ is given by\nthe function $U_N$ of (\\ref{eqn9}).\n\n\\begin{definition} \\cite[Definition 5.12]{LV} \\label{def3}\nGiven $N \\in [1, \\infty]$, we say that a compact measured length space\n$(X,d,\\nu)$ has nonnegative $N$-Ricci curvature if for all\n$\\mu_0, \\mu_1 \\in P(X)$ with\n$\\operatorname{supp}(\\mu_0) \\subset \\operatorname{supp}(\\nu)$ and $\\operatorname{supp}(\\mu_1) \\subset \\operatorname{supp}(\\nu)$, \nthere is {\\em some} Wasserstein geodesic\n$\\{\\mu_t\\}_{t \\in [0,1]}$ from $\\mu_0$ to $\\mu_1$ so that\nfor all $U \\in {\\mathcal D}{\\mathcal C}_N$ and all $t \\in [0,1]$,\n\\begin{equation} \\label{ineq1}\nU_\\nu(\\mu_t) \\: \\le \\: t \\: U_\\nu(\\mu_1) \\: + \\: (1-t) \\: \nU_\\nu(\\mu_0).\n\\end{equation}\n\\end{definition}\n\nWe make some remarks about the definition.\n\\begin{remark} \\label{rmk3}\nA similar definition in the case $N = \\infty$, but in terms\nof $U = U_\\infty$ instead of $U \\in {\\mathcal D}{\\mathcal C}_\\infty$, was used in\n\\cite[Definition 4.5]{Sturm2}; see also Remark \\ref{rmk9}.\n\\end{remark}\n\\begin{remark} \\label{rmk4}\nIt is not hard to show that if $(X,d,\\nu)$ has nonnegative\n$N$-Ricci curvature and $N^\\prime \\ge N$ then\n$(X,d,\\nu)$ has nonnegative $N^\\prime$-Ricci curvature. \n\\end{remark}\n\\begin{remark} \\label{rmk5}\nNote that for $t \\in (0,1)$,\nthe intermediate measures $\\mu_t$ are not required to have support\nin $\\operatorname{supp}(\\nu)$. If $(X,d,\\nu)$ has nonnegative $N$-Ricci curvature\nthen $\\operatorname{supp}(\\nu)$ is a convex subset of $X$ and\n$(\\operatorname{supp}(\\nu), d \\big|_{\\operatorname{supp}(\\nu)}, \\nu)$ has nonnegative $N$-Ricci\ncurvature \\cite[Theorem 5.53]{LV}.\n(We recall that a subset $A \\subset X$ is {\\em convex} if for \nany $x_0, x_1 \\in A$ there is a minimizing geodesic from $x_0$ to $x_1$ \nthat lies entirely in $A$. It is {\\em totally convex} if for any\n$x_0, x_1 \\in A$, any minimizing geodesic in $X$ from $x_0$ to\n$x_1$ lies in $A$.)\nSo we don't lose much by assuming that\n$\\operatorname{supp}(\\nu) \\: = \\: X$.\n\\end{remark}\n\\begin{remark} \\label{rmk6}\nThere is supposed to be a single Wasserstein geodesic\n$\\{\\mu_t\\}_{t \\in [0,1]}$ from $\\mu_0$ to $\\mu_1$ so that\n(\\ref{ineq1}) holds along $\\{\\mu_t\\}_{t \\in [0,1]}$ for all\n$U \\in {\\mathcal D}{\\mathcal C}_N$ simultaneously. However,\n(\\ref{ineq1}) is only assumed to\nhold along {\\em some} Wasserstein geodesic from $\\mu_0$ to $\\mu_1$, \nand not necessarily along all such Wasserstein geodesics.  This is what we call\n{\\em weak displacement convexity}. It may be more conventional\nto define convexity on a length space in terms of convexity along\n{\\em all} geodesics. However, the definition\nwith weak displacement convexity turns out to work better under MGH limits, \nand has most of the same implications as if we required\nconvexity along all Wasserstein geodesics from $\\mu_0$ to $\\mu_1$.\n\\end{remark}\n\\begin{remark} \\label{rmk7}\nInstead of requiring that (\\ref{ineq1}) holds for all $U \\in {\\mathcal D}{\\mathcal C}_N$,\nit would be consistent to make a definition in which it is only\nrequired to hold for the function $U = U_N$ of (\\ref{eqn9}).\nFor technical reasons, we\nprefer to require that (\\ref{ineq1}) holds for all $U \\in {\\mathcal D}{\\mathcal C}_N$;\nsee Remark \\ref{rmk13}. Also,\nthe class ${\\mathcal D}{\\mathcal C}_N$ is the natural class of\nfunctions for which the proof of Theorem \\ref{thm2} works.\n\\end{remark}\n\n\\subsection{MGH invariance}\n\nThe next result says that Definition \\ref{def3} satisfies\nCondition 1. of Wishlist \\ref{wishlist}. It shows that for each $N$,\nthere is a self-contained world of measured length spaces with\nnonnegative $N$-Ricci curvature.\n\n\\begin{theorem} \n\\cite[Theorem 5.19]{LV},\\cite[Theorem 4.20]{Sturm2},\n\\cite[Theorem 3.1]{Sturm3} \n\\label{thm1}\nLet $\\{(X_i, d_i, \\nu_i)\\}_{i=1}^\\infty$ be a sequence of\ncompact measured length spaces with $\\lim_{i \\rightarrow \\infty} \n(X_i,d_i,\\nu_i) \\: = \\: (X,d,\\nu)$ in the\nmeasured Gromov-Hausdorff topology.\nFor any $N \\in [1, \\infty]$, if \neach $(X_i, d_i, \\nu_i)$ has nonnegative $N$-Ricci curvature then\n$(X, d, \\nu)$ has nonnegative $N$-Ricci curvature.\n\\end{theorem}\n\\begin{proof}\nWe give an outline of the proof.  For simplicity, we just\nconsider a single $U \\in {\\mathcal D}{\\mathcal C}_N$; the same argument will allow one\nto handle all $U \\in {\\mathcal D}{\\mathcal C}_N$ simultaneously.\n\nSuppose first that $\\mu_0$ and $\\mu_1$ are absolutely continuous\nwith respect to $\\nu$, with continuous densities $\\rho_0, \\rho_1\n\\in C(X)$. \nLet $\\{f_i\\}_{i=1}^\\infty$ be a sequence of $\\epsilon_i$-approximations\nas in Definition \\ref{def2}.\nWe first approximately-lift the measures $\\mu_0$ and $\\mu_1$ to $X_i$. \nThat is, we use $f_i$ to pullback the densities to $X_i$, then\nmultiply by $\\nu_i$ and\nthen normalize to get probability measures.\nMore precisely, we put\n$\\mu_{i,0} \\: = \\: \\frac{f_i^* \\rho_0 \\: \\nu_i}{\n\\int_{X_i} f_i^* \\rho_0 \\: d\\nu_i} \\in P(X_i)$ and \n$\\mu_{i,1} \\: = \\: \\frac{f_i^* \\rho_1 \\: \\nu_i}{\n\\int_{X_i} f_i^* \\rho_0 \\: d\\nu_i} \\in P(X_i)$. One shows that\n$\\lim_{i \\rightarrow \\infty} (f_i)_* \\mu_{i,0} \\: = \\: \\mu_{0}$ and\n$\\lim_{i \\rightarrow \\infty} (f_i)_* \\mu_{i,1} \\: = \\: \\mu_{1}$ in\nthe weak-$*$ topology on $P(X)$. In addition, one shows that\n\\begin{equation} \\label{eqn16}\n\\lim_{i \\rightarrow \\infty} U_{\\nu_i}(\\mu_{i,0}) \\: = \\: U_\\nu(\\mu_0)\n\\end{equation} \nand\n\\begin{equation} \\label{eqn17}\n\\lim_{i \\rightarrow \\infty} U_{\\nu_i}(\\mu_{i,1}) \\: = \\: U_\\nu(\\mu_1).\n\\end{equation}\n\nUp on $X_i$, we are OK in the sense that\nby hypothesis, there is a Wasserstein geodesic\n$\\{\\mu_{i,t}\\}_{t \\in [0,1]}$ from \n$\\mu_{i,0}$ to $\\mu_{i,1}$ in $P(X_i)$ so that for all $t \\in [0,1]$, \n\\begin{equation} \\label{eqn18}\nU_{\\nu_i}(\\mu_{i,t}) \\: \\le \\: t \\: U_{\\nu_i}(\\mu_{i,1}) \\: + \\: (1-t) \\: \nU_{\\nu_i}(\\mu_{i,0}).\n\\end{equation}\nWe now want to take a convergent subsequence\nof these Wasserstein geodesics in an appropriate sense \nto get a Wasserstein geodesic in $P(X)$. This can be done\nusing Proposition \\ref{prop1} and an Arzela-Ascoli-type result.\nThe conclusion is that after\npassing to a subsequence of the $i$'s, there is a Wasserstein geodesic  \n$\\{\\mu_{t}\\}_{t \\in [0,1]}$ from $\\mu_{0}$ to $\\mu_{1}$ in $P(X)$\nso that for each $t \\in [0,1]$, we have $\\lim_{i \\rightarrow \\infty}\n(f_i)_* \\mu_{i,t} \\: = \\: \\mu_t$. \n\nFinally, we want to see what (\\ref{eqn18}) becomes as $i \\rightarrow \\infty$.\nAt the endpoints we have good limits from (\\ref{eqn16}) and (\\ref{eqn17}), so\nthis handles the right-hand-side of (\\ref{eqn18}) as $i \\rightarrow \\infty$.\nWe do not have such a good limit for the left-hand-side. However, this\nis where the lower semicontinuity comes in.\nApplying parts (i) and (ii) of Proposition \\ref{prop5}, we do know that\n\\begin{equation} \\label{eqn19}\nU_{\\nu}(\\mu_{t}) \\: \\le \\: \\liminf_{i \\rightarrow \\infty} \nU_{(f_i)_* \\nu_i}((f_i)_* \\mu_{i,t})\n \\: \\le \\: \\liminf_{i \\rightarrow \\infty} \nU_{\\nu_i}(\\mu_{i,t}).\n\\end{equation}\nThis is enough to give the desired inequality (\\ref{ineq1}) along\nthe Wasserstein geodesic $\\{\\mu_{t}\\}_{t \\in [0,1]}$.\n\nThis handles the case when $\\mu_0$ and $\\mu_1$ have continuous densities.\nFor general $\\mu_0, \\mu_1 \\in P(X)$, using mollifiers we can construct\nsequences $\\{ \\mu_{j,0} \\}_{j=1}^\\infty$\nand $\\{ \\mu_{j,1} \\}_{j=1}^\\infty$ \nof absolutely continuous measures with continuous densities so that\n$\\lim_{j \\rightarrow \\infty} \\mu_{j,0} \\: = \\: \\mu_0$ and\n$\\lim_{j \\rightarrow \\infty} \\mu_{j,1} \\: = \\: \\mu_1$ in the\nweak-$*$ topology.  In addition, one can do the mollifying in such a way\nthat\n$\\lim_{j \\rightarrow \\infty} U_\\nu(\\mu_{j,0}) \\: = \\: U_\\nu(\\mu_0)$ and\n$\\lim_{j \\rightarrow \\infty} U_\\nu(\\mu_{j,1}) \\: = \\: U_\\nu(\\mu_1)$.\nFrom what has already been shown, for each $j$ there is\na Wasserstein geodesic $\\{\\mu_{j,t}\\}_{t \\in [0,1]}$ in\n$P(X)$ from $\\mu_{j,0}$ to $\\mu_{j,1}$ so that\nfor all $t \\in [0,1]$, \n\\begin{equation} \\label{eqn20}\nU_{\\nu}(\\mu_{j,t}) \\: \\le \\: t \\: U_{\\nu}(\\mu_{j,1}) \\: + \\: (1-t) \\: \nU_{\\nu}(\\mu_{j,0}).\n\\end{equation}\nAfter passing to a subsequence, we can assume that the\nWasserstein geodesics $\\{\\mu_{j,t}\\}_{t \\in [0,1]}$ converge\nuniformly as $j \\rightarrow \\infty$ to a Wasserstein geodesic \n$\\{\\mu_{t}\\}_{t \\in [0,1]}$ from $\\mu_0$ to $\\mu_1$. From the\nlower semicontinuity of $U_\\nu$, we have\n$U_{\\nu}(\\mu_{t}) \\: \\le \\: \\liminf_{j \\rightarrow \\infty} \nU_{\\nu}(\\mu_{j,t})$. Equation (\\ref{ineq1}) follows.\n\\end{proof}\n\n\\subsection{Basic properties}\n\nWe now give some basic properties of measured length spaces \n$(X,d,\\nu)$ with\nnonnegative $N$-Ricci curvature.\n\n\\begin{proposition} \\cite[Proposition 5.20]{LV},\n\\cite[Theorem 2.3]{Sturm3} \\label{prop6}\nFor $N \\in (1, \\infty]$, if $(X,d,\\nu)$  has\nnonnegative $N$-Ricci curvature then\nthe measure $\\nu$ is either a delta function or is nonatomic.\nThe support of $\\nu$ is a convex subset of $X$.\n\\end{proposition}\n\nThe next result is an analog of the Bishop-Gromov theorem.\n\n\\begin{proposition} \\cite[Proposition 5.27]{LV},\n\\cite[Theorem 2.3]{Sturm3} \\label{prop7}\nSuppose that $(X,d,\\nu)$ has nonnegative $N$-Ricci curvature, with\n$N \\in [1,  \\infty)$. Then for all \n$x\\in \\operatorname{supp}(\\nu)$ and all $0<r_1\\leq r_2$,\n\\begin{equation} \\label{eqn21}\n\\nu(B_{r_2}(x)) \\leq \\left ( \\frac{r_2}{r_1} \\right )^N \\nu(B_{r_1}(x)).\n\\end{equation}\n\\end{proposition}\n\\begin{proof}\nWe give an outline of the proof.\nThere is a Wasserstein\ngeodesic $\\{ \\mu_t\\}_{t \\in [0,1]}$ between \n$\\mu_0 \\: = \\: \\delta_{x}$ and the restricted measure\n$\\mu_1 \\: = \\: \\frac{1_{B_{r_2}(x)}}{\\nu(B_{r_2}(x))} \\: \\nu$,\nalong which (\\ref{ineq1}) holds. Such a Wasserstein geodesic comes\nfrom a fan of geodesics (the support of $\\Pi$)\nthat go from $x$ to points in $B_{r_2}(x)$.\nThe actual transport, going backwards from $t=1$ to $t=0$, amounts to\nsliding the mass of $\\mu_1$ along these geodesics towards\n$x$. In particular, the support of $\\mu_t$ is contained in\n$B_{tr_2}(x)$. Applying (\\ref{ineq1}) with $U = U_N$ and\n$t \\: = \\: \\frac{r_1}{r_2}$, along with Holder's inequality,\ngives the desired result.\n\\end{proof}\n\nWe give a technical result which will be used in\nderiving functional inequalities.\n\n\\begin{proposition} \\cite[Theorem 5.52]{LV} \\label{prop8}\nSuppose that $(X, d, \\nu)$ has nonnegative $N$-Ricci curvature.\nIf $\\mu_0$ and $\\mu_1$ are absolutely continuous with respect to\n$\\nu$ then the measures in the Wasserstein geodesic \n$\\{\\mu_t\\}_{t \\in [0,1]}$ of Definition \\ref{def3} are all\nabsolutely continuous with respect to $\\nu$.\n\\end{proposition}\n\nFinally, we mention that for {\\em nonbranching} measured\nlength spaces, there is a local-to-global principle which \nsays that having nonnegative $N$-Ricci curvature in a local\nsense implies nonnegative $N$-Ricci curvature in a global\nsense \\cite[Theorem 4.17]{Sturm2},\\cite{TOT2}.\nWe do not know if this holds in the branching case.\n\n\\section{Smooth metric-measure spaces} \\label{sec4}\n\nWe now address Condition 2. of Wishlist \\ref{wishlist}.\nWe want to know what our abstract definition of\n``nonnegative $N$-Ricci curvature'' boils down to in the\nclassical Riemannian case.  To be a bit more general, we\nallow Riemannian manifolds with weights. Let\nus say that a {\\em smooth} measured length space consists\nof a smooth $n$-dimensional Riemannian manifold $M$ along with a smooth\nprobability measure $\\nu \\:  = \\: e^{- \\: \\Psi} \\: \\operatorname{dvol}_M$.\nWe write $(M,g,\\nu)$ for the corresponding measured length\nspace. We are taking $M$ to be compact.\n\nLet us discuss possible Ricci tensors for smooth measured\nlength spaces.\nIf $\\Psi$ is constant, i.e. if $\\nu \\: = \\: \\frac{\\operatorname{dvol}_M}{\\operatorname{vol}(M)}$, then\nthe right notion of a Ricci tensor for $M$ is clearly just the usual\n$\\operatorname{Ric}$. \n\nFor general $\\Psi$, a modified Ricci tensor \n\\begin{equation} \\label{eqn22}\n\\operatorname{Ric}_\\infty \\: = \\: \\operatorname{Ric} \\: + \\: \\mathop{{\\rm Hess}\\,}(\\Psi)\n\\end{equation} \nwas introduced\nby Bakry and \\'Emery \\cite{Bakry-Emery (1985)}. \n(Note that the standard $\\mathbb R^n$ with the Gaussian measure\n$(2\\pi)^{-\\frac{n}{2}} \\: e^{- \\frac{|x|^2}{2}} \\: d^nx$ has a\nconstant Bakry-\\'Emery tensor given by\n$(\\operatorname{Ric}_\\infty)_{ij} \\: = \\: \\delta_{ij}$.) \nTheir motivation came from a desire\nto generalize the Lichnerowicz inequality for the lower positive\neigenvalue $\\lambda_1(\\triangle)$ \nof the Laplacian.  We recall the Lichnerowicz result\nthat if an $n$-dimensional Riemannian manifold has \n$\\operatorname{Ric} \\: \\ge \\: K \\: g$ with $K > 0$  then\n$\\lambda_1(\\triangle) \\: \\ge \\: \\frac{n}{n-1} \\: K$\n\\cite{Lichnerowicz}. \n\nIn the case of a \nRiemannian manifold with a smooth probability measure $\\nu \\: = \\: \ne^{- \\: \\Psi} \\: \\operatorname{dvol}_M$, there is a natural self-adjoint Laplacian\n$\\widetilde{\\triangle}$ acting on the weighted $L^2$-space\n$L^2(M, e^{- \\: \\Psi} \\: \\operatorname{dvol}_M)$, given by\n\\begin{equation} \\label{eqn23}\n\\int_M f_1 (\\widetilde{\\triangle} f_2) \\: e^{- \\: \\Psi} \\: \\operatorname{dvol}_M \\: = \\:\n\\int_M  \\langle \\nabla f_1, \\nabla f_2 \\rangle \\: e^{- \\: \\Psi} \\: \\operatorname{dvol}_M\n\\end{equation}\nfor $f_1, f_2 \\in C^\\infty(M)$. Here $\\langle \\nabla f_1, \\nabla f_2 \\rangle$\nis the usual local inner product\ncomputed using the Riemannian metric $g$.\nBakry and \\'Emery showed that if $\\operatorname{Ric}_\\infty \\: \\ge \\: K g$ then\n$\\lambda_1 (\\widetilde{\\triangle}) \\: \\ge \\: K$. Although this\nstatement is missing the $\\frac{n}{n-1}$ factor of the\nLichnerowicz inequality, it holds independently of $n$ and so\ncan be considered to be a version of the Lichnerowicz inequality\nwhere one allows weights and takes $n \\rightarrow \\infty$. We\nrefer to \\cite{Toulouse (2000)} \nfor more information on the Bakry-\\'Emery tensor\n$\\operatorname{Ric}_\\infty$,\nincluding its relationship to log Sobolev inequalities. \nSome geometric properties of $\\operatorname{Ric}_\\infty$ were studied in\n\\cite{Lott (2003)}. More\nrecently, the Bakry-\\'Emery tensor has appeared as the \nright-hand-side of Perelman's modified Ricci flow equation\n\\cite{Perelman}.\n\nWe have seen that $\\operatorname{Ric}_\\infty$ is a sort of Ricci tensor\nfor the smooth measured length space $(M,g,\\nu)$ when we\nconsider $(M,g,\\nu)$ to have ``effective dimension'' infinity.\nThere is a similar tensor for other effective dimensions.\nNamely, if $N \\in (n, \\infty)$ then we put\n\\begin{equation} \\label{eqn24}\n\\operatorname{Ric}_N \\: = \\: \\operatorname{Ric} \\: + \\: \\mathop{{\\rm Hess}\\,}(\\Psi) \\: - \\: \n\\frac{1}{N-n} \\: d\\Psi \\otimes d\\Psi,\n\\end{equation}\nwhere $\\dim(M) = n$. The intuition is that $(M,g,\\nu)$ has\nconventional dimension $n$ but is pretending to have\ndimension $N$, and $\\operatorname{Ric}_N$ is its effective Ricci tensor\nunder this pretence. There is now a sharp analog of the\nLichnerowicz inequality : if $\\operatorname{Ric}_N \\: \\ge \\: K g$ with\n$K > 0$ then $\\lambda_1(\\widetilde{\\triangle}) \\: \\ge \\:\n\\frac{N}{N-1} \\: K$ \\cite{Bakry}. Geometric properties of\n$\\operatorname{Ric}_N$ were studied in \\cite{Lott (2003)} and \n\\cite{Qian (1997)}.\n\nFinally, if $N < n$, or if $N = n$ and $\\Psi$ is not\nlocally constant, then we take the effective Ricci\ntensor $\\operatorname{Ric}_N$ to be $- \\infty$. To summarize,\n\\begin{definition} \\label{def4}\nFor $N \\in [1, \\infty]$, define the $N$-Ricci tensor $\\operatorname{Ric}_N$ of $(M, g, \\nu)$\nby\n\\begin{equation} \\label{eqn25}\n\\operatorname{Ric}_N \\: = \n\\begin{cases}\n\\operatorname{Ric} \\: + \\: \\mathop{{\\rm Hess}\\,}(\\Psi)  & \\text{ if $N = \\infty$}, \\\\\n\\operatorname{Ric} \\: + \\: \\mathop{{\\rm Hess}\\,}(\\Psi) \\: - \\: \\frac{1}{N-n} \\: d \\Psi \\otimes\nd \\Psi & \\text{ if $n \\: < \\: N \\: < \\: \\infty$}, \\\\\n\\operatorname{Ric} \\: + \\: \\mathop{{\\rm Hess}\\,}(\\Psi) \\: - \\: \\infty \\: (d \\Psi \\otimes\nd \\Psi) & \\text{ if $N = n$}, \\\\\n- \\infty & \\text{ if $N < n$,}\n\\end{cases}\n\\end{equation}\nwhere by convention $\\infty \\cdot 0 \\: = \\: 0$.\n\\end{definition}\n\nWe can now state what the abstract notion of \nnonnegative $N$-Ricci curvature boils down to in the\nsmooth case.\n\n\\begin{theorem} \n\\cite[Theorems 7.3 and 7.42]{LV},\\cite[Theorem 4.9]{Sturm2},\n\\cite[Theorem 1.7]{Sturm3} \\label{thm2}\nGiven $N \\in [1, \\infty]$, the measured length space \n$(M, g, \\nu)$ has nonnegative $N$-Ricci curvature \nin the sense of Definition \\ref{def3} if and\nonly if $\\operatorname{Ric}_N \\: \\ge \\: 0$.\n\\end{theorem}\n\nThe proof of Theorem \\ref{thm2} uses the explicit description of\noptimal transport on Riemannian manifolds.\n\nIn the special case when $\\Psi$ is constant, and so\n$\\nu \\: = \\: \\frac{\\operatorname{dvol}_M}{\\operatorname{vol}(M)}$,\nTheorem \\ref{thm2} shows that we recover the usual notion\nof nonnegative Ricci curvature from our length space definition\nas soon as $N\\geq n$.\n\n\\subsection{Ricci limit spaces}\n\nWe give an application of Theorems \\ref{thm1} and \\ref{thm2} \nto Ricci limit spaces. \nFrom Gromov precompactness, given $N \\in \\mathbb Z^+$ and $D > 0$, the\nRiemannian manifolds with\nnonnegative Ricci curvature, dimension at most $N$ and\ndiameter at most $D$ form a precompact subset of the set of\nmeasured length spaces, with respect to the MGH topology. \nThe problem is to characterize the limit points.  In general\nthe limit points can be very singular, so this is a hard problem.\nHowever, let us ask a simpler question : what are the \nlimit points that happen to be {\\em smooth} measured length\nspaces?  That is, we are trying to characterize the smooth\nlimit points.\n\n\\begin{corollary} \\cite[Corollary 7.45]{LV} \\label{maincor}\nIf $(B, g_B, e^{-  \\Psi} \\operatorname{dvol}_B)$ is a measured \nGromov-Hausdorff limit of Riemannian manifolds with nonnegative\nRicci curvature and dimension at most $N$ then $\\operatorname{Ric}_N(B) \\: \\ge \\:0$. \n(Here $B$ has dimension $n$, which is less than or equal to $N$.)\n\\end{corollary}\n\\begin{proof}\nSuppose that $\\{(M_i, g_i)\\}_{i=1}^\\infty$ is a sequence of\nRiemannian manifolds with nonnegative Ricci curvature and\ndimension at most $N$, with\n$\\lim_{i \\rightarrow \\infty} \\left( M_i, g_i, \n\\frac{\\operatorname{dvol}_{M_i}}{\\operatorname{vol}(M_i)} \\right)\n\\: = \\: (B, g_B, e^{-  \\Psi} \\operatorname{dvol}_B)$. From Theorem \\ref{thm2}, the measured\nlength space $\\left( M_i, g_i, \n\\frac{\\operatorname{dvol}_{M_i}}{\\operatorname{vol}(M_i)} \\right)$ has nonnegative $N$-Ricci\ncurvature.  From Theorem \\ref{thm1}, $(B, g_B, e^{-  \\Psi} \\operatorname{dvol}_B)$ has\nnonnegative $N$-Ricci curvature.  From Theorem \\ref{thm2} again,\n$\\operatorname{Ric}_N(B) \\: \\ge \\: 0$.\n\\end{proof}\n\nThere is a partial converse to Corollary \\ref{maincor}.\n\\begin{proposition} \\cite[Corollary 7.45]{LV} \\label{prop9}\n(i) Suppose that $N$ is an integer. If $(B, g_B, e^{-  \\Psi} \\operatorname{dvol}_B)$ \nhas $\\operatorname{Ric}_N(B) \\: \\ge \\: 0$ with $N \\: \\ge \\: \\dim(B) \\: + \\: 2$ then \n$(B,  g_B, e^{-  \\Psi} \\operatorname{dvol}_B)$ is a measured \nGromov-Hausdorff limit of Riemannian manifolds with nonnegative\nRicci curvature and dimension $N$. \\\\\n(ii) Suppose that $N = \\infty$. If $(B, g_B, e^{-  \\Psi} \\operatorname{dvol}_B)$ has \n$\\operatorname{Ric}_\\infty(B) \\: \\ge \\: 0$\nthen $(B, g_B, e^{-  \\Psi} \\operatorname{dvol}_B)$ is a measured \nGromov-Hausdorff limit of Riemannian manifolds $M_i$ with \n$\\operatorname{Ric}(M_i) \\: \\ge \\: - \\: \\frac{1}{i} \\: g_{M_i}$.\n\\end{proposition}\n\\begin{proof}\nLet us consider part (i). The proof uses the warped\nproduct construction of \\cite{Lott (2003)}. Let $g_{S^{N-\\dim(B)}}$ be the\nstandard metric on the sphere $S^{N-\\dim(B)}$. Let $M_i$ be\n$B \\times S^{N-\\dim(B)}$ with the warped product metric\n$g_i \\: = \\: g_B \\: + \\: i^{-2} e^{- \\: \\frac{\\Psi}{N-\\dim(B)}} \\:\ng_{S^{N-\\dim(B)}}$. The metric $g_i$ is constructed so that\nif $p \\: : \\: B \\times S^{N-\\dim(B)} \\rightarrow B$ is projection\nonto the first factor then $p_* \\operatorname{dvol}_{M_i}$ is a constant\ntimes $e^{-  \\Psi} \\operatorname{dvol}_B$. \nIn terms of the fibration $p$, the Ricci tensor of \n$M_i$ splits into\nhorizontal and vertical components, with the horizontal\ncomponent being exactly $\\operatorname{Ric}_N$.\nAs $i$ increases, the fibers shrink and the vertical\nRicci curvature of $M_i$ becomes dominated by the\nRicci curvature of the small fiber $S^{N-\\dim(B)}$, which is positive\nas we are assuming that $N-\\dim(B) \\ge 2$. Then for large $i$,\n$(M_i, g_i)$ has nonnegative Ricci curvature.  Taking\n$f_i = p$, we see that $\\lim_{i \\rightarrow \\infty} \\left( M_i, g_i, \n\\frac{\\operatorname{dvol}_{M_i}}{\\operatorname{vol}(M_i)} \\right)\n\\: = \\: (B, g_B, e^{-  \\Psi} \\operatorname{dvol}_B)$.\n\nThe proof of (ii) is similar, except that we also allow\nthe dimensions of the fibers to go to infinity.\n\\end{proof}\n\nExamples of singular spaces with nonnegative $N$-Ricci curvature\ncome from group actions.\nSuppose that a compact Lie group $G$ acts isometrically on a \n$N$-dimensional Riemannian manifold $M$ that has\nnonnegative Ricci curvature. Put $X = M\/G$, let\n$p \\: : \\: M \\rightarrow X$ be the quotient map, let\n$d$ be the quotient metric and\nput $\\nu \\: = \\: p_* \\left( \n\\frac{\\operatorname{dvol}_M}{\\operatorname{vol}(M)} \\right)$. Then\n$(X, d, \\nu)$ has nonnegative $N$-Ricci curvature\n\\cite[Corollary 7.51]{LV}.\n\nFinally, we recall the theorem of O'Neill that sectional\ncurvature is nondecreasing under pushforward by a Riemannian\nsubmersion.  There is a Ricci analog of the O'Neill\ntheorem, expressed in terms of the modified Ricci tensor $\\operatorname{Ric}_N$\n\\cite{Lott (2003)}.\nThe proof of this in \n\\cite{Lott (2003)} was by explicit tensor calculations.\nUsing optimal transport, one can give a ``synthetic'' proof\nof this Ricci O'Neill theorem \n\\cite[Corollary 7.52]{LV}.  (This is what first convinced\nthe author that optimal transport is the right approach.)\n\n\\begin{remark} \\label{rmk8}\nWe return to the question of whether one can give a good definition of\n``nonnegative $N$-Ricci curvature'' by just taking the conclusion\nof the Bishop-Gromov theorem and turning it into a definition.\nTo be a bit more reasonable, we consider taking an angular Bishop-Gromov \ninequality as the definition. Such an\ninequality, with parameter $n$, does indeed characterize when\nan $n$-dimensional Riemannian manifold has nonnegative Ricci curvature.\nNamely, from comparison geometry, nonnegative Ricci curvature implies an\nangular Bishop-Gromov inequality. To go the other way, suppose that\nthe angular Bishop-Gromov inequality holds.\nWe use polar coordinates around a point $m \\in M$ and\nrecall that the volume of a infinitesimally small angular sector \ncentered in the direction of a unit vector \n$v \\in T_mM$, and going up to radius $r$, has the\nTaylor expansion \n\\begin{equation} \\label{eqn26}\nV(v, r) \\: = \\: \\operatorname{const.} \\: r^n \\: \\left( 1 \\: - \\: \n\\frac{n}{6(n+2)} \\: \\operatorname{Ric}(v,v) \\: r^2 \\: + \\: \\ldots \\right).\n\\end{equation}\nIf $r^{-n} \\: V(v,r)$ is to be nonincreasing in $r$ then we must have\n$\\operatorname{Ric}(v,v) \\: \\ge \\: 0$. As $m$ and $v$ were arbitrary, we conclude\nthat $\\operatorname{Ric} \\ge 0$. \n\nThere is a version of the angular Bishop-Gromov inequality for\nmeasured length spaces,\ncalled the ``measure contracting property'' (MCP) \\cite{Ohta,Sturm3}.\nIt satisfies Condition 1. of Wishlist \\ref{wishlist}. \n\nThe reason that the MCP notion is not entirely satisfactory can be\nseen by asking what it takes\nfor a smooth measured length space $(M, g, e^{-\\Psi}\n\\: \\operatorname{dvol}_M)$ to satisfy the $N$-dimensional angular Bishop-Gromov inequality.\n(Here $\\dim(M) = n$.)\nThere is a Riccati-type inequality\n\\begin{equation} \\label{eqn27}\n\\frac{\\partial}{\\partial r} \\left( \\operatorname{Tr} \\Pi \\: - \\: \\frac{\\partial \n\\Psi}{\\partial r} \\right) \\: \\le \\: - \\: \\operatorname{Ric}_N(\\partial_r, \\partial_r) \n\\: - \\: \\frac{1}{N-1} \\: \\left( \\operatorname{Tr} \\Pi \\: - \\: \\frac{\\partial \n\\Psi}{\\partial r} \\right)^2,\n\\end{equation}\nwhich looks good.\nAgain there is an expansion for\nthe measure of the infinitesimally small angular sector considered above,\nof the form \n$\\widehat{V}(r) \\: = \\: r^{n} \\left( a_0 \\: + a_1 \\: r \\: + \\: a_2 r^2 \\: + \\:\n\\ldots \\right)$, where the coefficents $a_i$ can be expressed in terms\nof curvature derivatives and the derivatives of $\\Psi$.\nHowever, if $N > n$ then \nsaying that $r^{-N} \\widehat{V}(r)$ is nonincreasing in $r$ does not\nimply anything about the coefficients. Thus having the $N$-dimensional\nangular Bishop-Gromov inequality does not imply that\n$\\operatorname{Ric}_N \\ge 0$.\nIn particular, it does not seem that one can\nprove Corollary \\ref{maincor} using MCP. \n\nHaving nonnegative $N$-Ricci\ncurvature does imply MCP \\cite{Sturm3}.\n\\end{remark}\n\n\\section{$N$-Ricci curvature bounded below by $K$} \\label{sec5}\n\nIn Section \\ref{sec3} we gave the definition of nonnegative $N$-Ricci\ncurvature.  In this section we discuss how to extend this to\na notion of a measured length space having $N$-Ricci curvature\nbounded below by some real number $K$.\n\nWe start with the case $N=\\infty$. As mentioned in Section \\ref{sec2},\nformal computations indicate that in the case of a smooth measured length\nspace $(M,g,e^{-\\Psi} \\: \\operatorname{dvol}_M)$, \nhaving $\\operatorname{Ric}_\\infty \\: \\ge \\: K g$ should imply that\n$H_\\infty$ has Hessian bounded below by $K g_{H^{-1}}$ on $P(M)$.\nIn particular, if $\\{\\mu_t\\}_{t \\in [0,1]}$ is a geodesic in \n$P(M)$ then we would\nexpect that $H_\\infty(\\mu_t) \\: - \\: \\frac{K}{2} \\:\nW_2(\\mu_0, \\mu_1)^2 \\: t^2 $ is\nconvex in $t$. This motivates an adaption of Definition \\ref{def3}.\n\nIn order to handle all $U \\in {\\mathcal D}{\\mathcal C}_\\infty$,  we first make the\nfollowing definition. Given a continuous convex function $U \\:\n: \\: [0, \\infty) \\rightarrow \\mathbb R$, we define its ``pressure'' by\n\\begin{equation} \\label{pdef}\np(r) = r U_+^\\prime(r) - U(r),\n\\end{equation}\nwhere $U_+^\\prime(r)$ is the right-derivative.  Then given \n$K \\in \\mathbb R$, we define \n$\\lambda \\: : \\: {\\mathcal D}{\\mathcal C}_\\infty \\rightarrow \\mathbb R \\cup \\{-\\infty\\}$ by\n\\begin{equation}\\label{lambdainfty} \n\\lambda(U) = \\inf_{r > 0} K \\: \\frac{p(r)}r = \n\\begin{cases}\nK \\lim_{r \\rightarrow 0^+} \\frac{p(r)}r & \\text{if $K > 0$}, \\\\\n0 & \\text{if $K = 0$}, \\\\\nK \\lim_{r \\rightarrow \\infty} \\frac{p(r)}r & \\text{if $K < 0$}.\n\\end{cases}\n\\end{equation}\nNote that if $U = U_\\infty$ (recall that $U_\\infty(r) \\: = \\:\nr \\: \\log r$) then $p(r) = r$ and so\n$\\lambda(U_\\infty) \\: = \\: K$.\n\n\\begin{definition} \\cite[Definition 5.13]{LV} \\label{def5}\nGiven $K \\in \\mathbb R$, we say that $(X,d,\\nu)$ has $\\infty$-Ricci curvature\nbounded below by $K$ if for all\n$\\mu_0, \\mu_1 \\in P(X)$ with\n$\\operatorname{supp}(\\mu_0) \\subset \\operatorname{supp}(\\nu)$ and $\\operatorname{supp}(\\mu_1) \\subset \\operatorname{supp}(\\nu)$,\nthere is {\\em some} Wasserstein geodesic\n$\\{\\mu_t\\}_{t \\in [0,1]}$ from $\\mu_0$ to $\\mu_1$ so that\nfor all $U \\in {\\mathcal D}{\\mathcal C}_\\infty$ and all $t \\in [0,1]$,\n\\begin{equation} \\label{ineq2}\nU_\\nu(\\mu_t) \\: \\le \\: t \\: U_\\nu(\\mu_1) \\: + \\: (1-t) \\: \nU_\\nu(\\mu_0)\n \\: - \\:\n\\frac12 \\: \\lambda(U) \\: t(1-t) W_2(\\mu_0, \\mu_1)^2.\n\\end{equation}\n\\end{definition}\n\n\\begin{remark} \\label{rmk9}\nA similar definition, but in terms\nof $U = U_\\infty$ instead of $U \\in {\\mathcal D}{\\mathcal C}_\\infty$, was used in\n\\cite[Definition 4.5]{Sturm2}.\n\\end{remark}\n\nClearly if $K = 0$ then we recover the notion of\nnonnegative $\\infty$-Ricci curvature in the sense of Definition \\ref{def3}.\nThe $N=\\infty$ results of Sections\n\\ref{sec3} and \\ref{sec4} \ncan be extended to the present case where $K$ may be nonzero.\n\nA good notion of $(X,d,\\nu)$ having $N$-Ricci curvature bounded below\nby $K \\in \\mathbb R$, where $N$ can be finite, is less clear and is essentially\ndue to Sturm \\cite{Sturm3}. The following definition is a variation\nof Sturm's definition and appears in \\cite{LV2}.\n\nGiven $K\\in\\mathbb R$ and $N \\in (1, \\infty]$, define\n\\begin{equation} \\label{beta}\n\\beta_t(x_0,x_1) \\: = \\: \n\\begin{cases} \ne^{\\frac16 K\\: (1-t^2) \\: d(x_0,x_1)^2} \\qquad\\qquad\\qquad & \\text{if \n$N = \\infty$},\\\\\n\\infty \\qquad\\qquad\\qquad & \\text{if $N < \\infty$, $K>0$ and $\\alpha>\\pi$},\\\\\n\\left(\\frac{\\sin(t\\alpha)}{t\\sin\\alpha}\\right)^{N-1}\\qquad\n& \\text{if $N < \\infty$, $K>0$ and $\\alpha\\in[0,\\pi]$}, \\\\\n1 \\qquad \\qquad\\qquad & \\text{if $N < \\infty$ and $K=0$},\\\\\n\\left(\\frac{\\sinh(t\\alpha)}{t\\sinh\\alpha}\\right)^{N-1}\\qquad\n& \\text{if $N < \\infty$ and $K<0$},\n\\end{cases}\n\\end{equation}\nwhere\n\\begin{equation}\n\\alpha \\: = \\: \\sqrt{\\frac{|K|}{N-1}}\\, d(x_0,x_1).\n\\end{equation}\nWhen $N = 1$, define\n\\begin{equation}\n\\beta_t(x_0,x_1) \\: = \\: \n\\begin{cases} \n\\infty \\qquad\\qquad\\qquad & \\text{if \n$K>0$},\\\\\n1\\qquad\n& \\text{if $K \\le 0$},\n\\end{cases}\n\\end{equation}\nAlthough we may not write it explicitly, $\\alpha$ and $\\beta$ depend\non $K$ and $N$.\n\nWe can disintegrate a transference plan $\\pi$ with respect to its\nfirst marginal $\\mu_0$ or its second marginal $\\mu_1$. We write\nthis in a slightly informal way:\n\\begin{equation}\nd\\pi(x_0, x_1) = d\\pi(x_1|x_0) d\\mu_0(x_0) = d\\pi(x_0|x_1) d\\mu_1(x_1).\n\\end{equation}\n\n\\begin{definition} \\cite{LV2} \\label{def6}\nWe say that $(X, d, \\nu)$ has $N$-Ricci curvature bounded below by $K$\nif the following condition is satisfied. Given $\\mu_0, \\mu_1 \\in P(X)$\nwith support in $\\operatorname{supp}(\\nu)$, write\ntheir Lebesgue decompositions with respect to $\\nu$ as\n$\\mu_0=\\rho_0\\,\\nu + \\mu_{0,s}$ and $\\mu_1=\\rho_1\\,\\nu + \\mu_{1,s}$,\nrespectively.\nThen there is {\\em some} \noptimal dynamical transference plan $\\Pi$ from $\\mu_0$ to $\\mu_1$,\nwith corresponding Wasserstein geodesic\n$\\mu_t \\: = \\: (e_t)_*\\Pi$, so that for all $U\\in{\\mathcal D}{\\mathcal C}_N$ and all $t\\in [0,1]$,\nwe have\n\\begin{align} \\label{ineqDCN}\nU_\\nu(\\mu_t) \\: \\leq \\: & (1-t) \\int_{X \\times X} \\beta_{1-t}(x_0,x_1) \\,\nU \\left( \\frac{\\rho_0(x_0)}{\\beta_{1-t}(x_0,x_1)} \\right)\n\\, d\\pi(x_1|x_0)\\,d\\nu(x_0) \\: +  \\\\\n& t \\int_{X \\times X} \\beta_t(x_0,x_1)\\, U \\left( \\frac{\\rho_1(x_1)}\n{\\beta_{t}(x_0,x_1)} \\right) \\, d\\pi(x_0|x_1)\\,d\\nu(x_1) \\: + \\notag \\\\\n& U'(\\infty) \\bigl[ (1-t) \\mu_{0,s}[X] \\: + \\: t \\mu_{1,s}[X] \\bigr]. \\notag\n\\end{align}\n\\end{definition}\n\nHere if $\\beta_t(x_0, x_1) = \\infty$ then we interpret\n$\\beta_t(x_0,x_1)\\, U \\left( \\frac{\\rho_1(x_1)}\n{\\beta_{t}(x_0,x_1)} \\right)$ as\n$U^\\prime(0) \\: \\rho_1(x_1)$, and similarly for\n$\\beta_{1-t}(x_0,x_1) \\,\nU \\left( \\frac{\\rho_0(x_0)}{\\beta_{1-t}(x_0,x_1)} \\right)$.\n\n\\begin{remark} \\label{rmk10}\nIf $\\mu_0$ and $\\mu_1$ are absolutely continuous\nwith respect to $\\nu$ then\nthe inequality can be rewritten in the more symmetric\nform\n\\begin{align} \\label{symform}\nU_\\nu(\\mu_t) \\: \\leq \\: & (1-t) \\int_{X \\times X} \n\\frac{\\beta_{1-t}(x_0,x_1)}{\\rho_0(x_0)} \\,\nU \\left(\\frac{\\rho_0(x_0)}{\\beta_{1-t}(x_0,x_1)} \\right)\\,d\\pi(x_0, x_1)\n\\: + \\\\\n& t \\int_{X \\times X} \\frac{\\beta_t(x_0,x_1)}{\\rho_1(x_1)}\\, \nU \\left( \\frac{\\rho_1(x_1)}\n{\\beta_{t}(x_0,x_1)} \\right) \\, d\\pi(x_0, x_1). \\notag\n\\end{align}\n\\end{remark}\n\n\\begin{remark} \\label{rmk11}\nGiven $K \\ge K^\\prime$ and $N \\le N^\\prime$, if\n$(X, d, \\nu)$ has $N$-Ricci curvature bounded below by $K$ then\nit also has $N^\\prime$-Ricci curvature bounded below by $K^\\prime$.\n\\end{remark}\n\n\\begin{remark} \\label{rmk12}\nThe case $N = \\infty$ of Definition \\ref{def6} is not quite\nthe same as what we gave in Definition \\ref{def5}!  However, it\nis true that having $\\infty$-Ricci curvature bounded below by \n$K$ in the sense of Definition \\ref{def6} implies that one has\n$\\infty$-Ricci curvature bounded below by \n$K$ in the sense of Definition \\ref{def5} \\cite{LV2}. Hence\nany $N =\\infty$ consequences of Definition \\ref{def5} are also\nconsequences of Definition \\ref{def6}.\nWe include the $N=\\infty$ case in Definition \\ref{def6}\nin order to present a unified treatment, but this example shows\nthat there may be some flexibility in the precise definitions.\n\\end{remark}\n\nThe results of Sections \\ref{sec3} and \\ref{sec4} now have extensions\nto the case $K \\neq 0$. However, the proofs of some of the extensions, \nsuch as that of Theorem \\ref{thm1}, may become much more involved\n\\cite[Theorem 3.1]{Sturm3},\\cite{TOT2}. \n\nUsing the extension of Proposition \\ref{prop7}, one obtains a\ngeneralized Bonnet-Myers theorem.\n\n\\begin{proposition} \\cite[Corollary 2.6]{Sturm3} \\label{prop11}\nIf $(X,d,\\nu)$ has $N$-Ricci curvature bounded below by $K > 0$ then\n$\\operatorname{supp}(\\nu)$ has diameter bounded above by\n$\\sqrt{\\frac{N-1}{K}} \\: \\pi$.\n\\end{proposition}\n\n\\section{Analytic consequences} \\label{sec6}\n\nLower Ricci curvature bounds on Riemannian manifolds have\nvarious analytic implications, such as eigenvalue \ninequalities, Sobolev inequalities and local Poincar\\'e\ninequalities. It turns out that these inequalities pass\nto our generalized setting.\n\n\\subsection{Log Sobolev and Poincar\\'e inequalities}\n\nLet us first discuss the so-called log Sobolev inequality.\nIf a smooth measured length space $(M, g, \ne^{- \\Psi} \\: \\operatorname{dvol}_M)$ has $\\operatorname{Ric}_\\infty \\: \\ge \\:\nKg$, with $K > 0$, then for all \n$f \\in C^\\infty(M)$ with $\\int_M f^2 \\: e^{- \\Psi} \\: \\operatorname{dvol}_M\n\\: = \\: 1$, it was shown in \\cite{Bakry-Emery (1985)} that\n\\begin{equation} \\label{eqn28}\n\\int_M f^2 \\: \\log(f^2)  \\: e^{- \\Psi} \\: \\operatorname{dvol}_M \\: \\le \\:\n\\frac{2}{K} \\int_M |\\nabla f|^2 \\: e^{- \\Psi} \\: \\operatorname{dvol}_M.\n\\end{equation}\nThe standard log Sobolev inequality on $\\mathbb R^n$ comes from taking\n$d\\nu \\: = \\: (4\\pi)^{- \\frac{n}{2}} \\: e^{- |x|^2} \\: d^nx$,\ngiving\n\\begin{equation} \\label{eqn29}\n\\int_{\\mathbb R^n} f^2 \\: \\log(f^2)  \\: e^{- |x|^2} \\: d^n x \\: \\le \\:\n\\int_{\\mathbb R^n} |\\nabla f|^2 \\: e^{- |x|^2} \\: d^n x\n\\end{equation}\nwhenever $(4\\pi)^{- \\frac{n}{2}} \\: \\int_{\\mathbb R^n} f^2 \\:\ne^{- |x|^2} \\: d^nx \\: = \\: 1$.\n\nThe log Sobolev inequality for $(M, g, \ne^{- \\Psi} \\: \\operatorname{dvol}_M)$ was given both heuristic and\nrigorous optimal transport proofs by Otto and Villani\n\\cite{Otto-Villani (2000)}. We describe the heuristic proof here.\nFrom Section \\ref{sec2}, having $\\operatorname{Ric}_\\infty \\: \\ge \\: Kg$\nformally implies that $\\mathop{{\\rm Hess}\\,}(H_\\infty) \\: \\ge \\: K\ng_{H^{-1}}$ on $P(M)$.\nTake $\\mu_0 \\: = \\: e^{- \\Psi} \\: \\operatorname{dvol}_M$ and \n$\\mu_1 \\: = \\: f^2 \\: e^{- \\Psi} \\: \\operatorname{dvol}_M$. Let\n$\\{ \\mu_t \\}_{t \\in [0,1]}$ be a Wasserstein geodesic from\n$\\mu_0$ to $\\mu_1$ along which \n\\begin{equation} \\label{eqn30}\nF(t) \\: = \\: H_\\infty(\\mu_t) \\: - \\:\n\\frac{K}{2} \\: W_2(\\mu_0, \\mu_1)^2 \\: t^2\n\\end{equation}\nis convex in $t$. As $F(0) \\: = \\: 0$, we have\n$F(1) \\: \\le \\: F^\\prime(1)$, or\n\\begin{align}\nH_\\infty(\\mu_1) \\: - \\: \\frac{K}{2} \\: W_2(\\mu_0, \\mu_1)^2 \\: & \\le \\:\n\\left\\langle \\frac{d\\mu_t}{dt} \\Big|_{t=1}, (\\grad H_\\infty)(\\mu_1)\n\\right\\rangle_{g_{H^{-1}}} \\: - \\: K \\: W_2(\\mu_0, \\mu_1)^2 \\\\\n& \\le \\:\n\\Bigg| \\frac{d\\mu_t}{dt} \\Big|_{t=1} \\Bigg| \\cdot\n\\Bigg| (\\grad H_\\infty)(\\mu_1) \\Bigg| \\: \n- \\: K \\: W_2(\\mu_0, \\mu_1)^2. \\notag\n\\end{align}\nHere $\\grad H_\\infty$ is the formal gradient of $H_\\infty$ on\n$P(M)$ and the last norms denote lengths with respect to $g_{H^{-1}}$. \nAs $\\{ \\mu_t \\}_{t \\in [0,1]}$ is a minimizing geodesic from\n$\\mu_0$ to $\\mu_1$, we should have\n\\begin{equation} \\label{eqn31}\n\\Bigg| \\frac{d\\mu_t}{dt} \\Big|_{t=1} \\Bigg| \\: = \\:\nW_2(\\mu_0, \\mu_1).\n\\end{equation} \nA formal computation gives\n\\begin{equation} \\label{eqn32}\n\\Bigg| (\\grad H_\\infty)(\\mu_1) \\Bigg|^2 \\: = \\: 4 \\: \n\\int_M |\\nabla f|^2 \\: e^{- \\Psi} \\: \\operatorname{dvol}_M.\n\\end{equation}\nThen\n\\begin{align}\n\\int_M f^2 \\: \\log(f^2)  \\: e^{- \\Psi} \\: \\operatorname{dvol}_M \\: & \\le \\:\n2 \\: W_2(\\mu_0, \\mu_1) \\: \n\\sqrt{\\int_M |\\nabla f|^2 \\: e^{- \\Psi} \\: \\operatorname{dvol}_M} \\: - \\:\n\\frac{K}{2} \\: W_2(\\mu_0, \\mu_1)^2 \\\\\n& \\le \\:\n\\sup_{w \\in \\mathbb R} \\left( \n2 \\: w \\: \n\\sqrt{\\int_M |\\nabla f|^2 \\: e^{- \\Psi} \\: \\operatorname{dvol}_M} \\: - \\:\n\\frac{K}{2} \\: w^2 \\right) \\notag \\\\\n&  = \\: \n\\frac{2}{K} \\int_M |\\nabla f|^2 \\: e^{- \\Psi} \\: \\operatorname{dvol}_M \\notag\n\\end{align}\nwhich is the log Sobolev inequality.\n\nThe rigorous optimal transport proof in \\cite{Otto-Villani (2000)} \nextends to\nmeasured length spaces. To give the statement,\nwe first must say what we mean by\n$|\\nabla f|$.\nWe define the {\\em local gradient norm} of a Lipschitz function \n$f \\in \\operatorname{Lip}(X)$ by the formula\n\\begin{equation} \\label{eqn33}\n|\\nabla f| (x) \\: = \\:\n\\limsup_{y\\to x} \\frac{|f(y)-f(x)|}{d(x,y)}.\n\\end{equation}\nWe don't claim to know the meaning of the gradient $\\nabla f$ on $X$ in\nthis generality, but we can talk about its norm anyway!\nThen we have the following log Sobolev inequality for\nmeasured length spaces.\n\n\\begin{theorem} \\cite[Corollary 6.12]{LV} \\label{thm3}\nSuppose that a compact measured length space\n$(X,d,\\nu)$ has $\\infty$-Ricci curvature bounded below by\n$K > 0$, in the sense of Definition \\ref{def5}. Suppose that\n$f \\in \\operatorname{Lip}(X)$ satisfies $\\int_X f^2 \\: d\\nu \\: = \\: 1$.\nThen\n\\begin{equation} \\label{introeqn}\n\\int_X f^2 \\: \\log(f^2) \\: d\\nu \\: \\le \\: \\frac{2}{K} \\:\n\\int_X |\\nabla f|^2 \\: d\\nu.\n\\end{equation}\n\\end{theorem}\n\nIn the case of Riemannian manifolds, one recovers from \n(\\ref{introeqn}) the log Sobolev inequality (\\ref{eqn28})\nof Bakry and \\'Emery.\n\n\\begin{remark} \\label{rmk13} \nThe proof of Theorem \\ref{thm3}, along with the other\ninequalities in this section, uses the $K>0$ analog of Proposition \\ref{prop8}.\nIn turn,\nthe proof of Proposition \\ref{prop8} uses the fact that (\\ref{ineq1})\nholds for all $U \\in {\\mathcal D}{\\mathcal C}_N$, as opposed to just $U_N$.\n\\end{remark}\n\nAs is well-known, one can obtain a Poincar\\'e inequality\nfrom (\\ref{introeqn}). Take $h \\in \\operatorname{Lip}(X)$ with $\\int_X h\\,d\\nu = 0$\nand put $f^2 \\: = \\: 1 + \\epsilon h$. Taking\n$\\epsilon$ small and expanding the two sides of\n(\\ref{introeqn}) in $\\epsilon$ gives the following result.\n\n\\begin{corollary} \\cite[Theorem 6.18]{LV} \\label{corr}\nSuppose that a compact measured length space\n$(X,d,\\nu)$ has $\\infty$-Ricci curvature bounded below by\n$K > 0$. Then for all\n$h \\in \\operatorname{Lip}(X)$ with $\\int_X h\\,d\\nu = 0$, we have\n\\begin{equation} \\label{eqn34}\n\\int_X h^2\\,d\\nu \\leq \n\\frac1{K} \\int_X |\\nabla h|^2\\,d\\nu. \n\\end{equation}\n\\end{corollary}\n\nIn case of a smooth measured length space\n$(M, g, e^{-\\Psi} \\: \\operatorname{dvol}_M)$, the inequality (\\ref{eqn34}) \ncoincides with\nthe Bakry-\\'Emery extension of the Lichnerowicz inequality,\nnamely $\\lambda_1 (\\widetilde{\\triangle}) \\: \\ge \\: K$.\nFor a general measured length space as in the hypotheses of\nCorollary \\ref{corr}, we do not know if there is a well-defined\nLaplacian.  The Poincar\\'e inequality of Corollary \\ref{corr}\ncan be seen as a generalized eigenvalue inequality that\navoids this issue.  To say a bit more about when one does\nhave a Laplacian,\nif $Q(h) \\: = \\: \\int_X |\\nabla h|^2 \\: d\\nu$ defines a quadratic form on\n$\\operatorname{Lip}(X)$, which in addition is closable in $L^2(X, \\nu)$, then\nthere is a self-adjoint Laplacian $\\triangle_\\nu$\nassociated to $Q$. In this case, Corollary \\ref{corr} implies that \n$\\triangle_\\nu \\: \\ge \\: K$ on the orthogonal complement of the constant\nfunctions.\n\nIn the case of a Ricci limit space, Cheeger and Colding used \nadditional structure in order to show\nthe Laplacian does exist \\cite{Cheeger-Colding (2000)}.\n\n\\subsection{Sobolev inequality}\n\nThe log Sobolev inequality can be viewed as an \ninfinite-dimensional version of an ordinary Sobolev inequality.\nAs such, it is interesting because it is a dimension-independent\nresult.  However, if one has $N$-Ricci curvature \nbounded below by $K > 0$ with $N$ finite then one gets an\nordinary Sobolev inequality, which is a sharper result.\n\n\\begin{proposition} \\cite{LV2} \\label{prop12}\nGiven $N \\in (1, \\infty)$ and $K > 0$, suppose that $(X,d,\\nu)$ has $N$-Ricci\ncurvature bounded below by $K$.\nThen for any nonnegative Lipschitz function $\\rho_0 \\in \\operatorname{Lip}(X)$ with\n$\\int_X \\rho_0 \\: d\\nu \\: = \\: 1$, one has\n\\begin{equation} \\label{Sobolev}\nN - N \\int_X \\rho_0^{1-\\frac{1}{N}} \\: d\\nu \\: \\le \\:\n\\frac{1}{2K} \\left( \\frac{N-1}{N} \\right)^2 \\int_X\n\\frac{\\rho_0^{-1-\\frac{2}{N}}}{\\frac13 + \\frac23 \\rho_0^{-\\frac{1}{N}}} \\:\n|\\nabla \\rho_0|^2 \\: d\\nu.\n\\end{equation}\n\\end{proposition}\n\nTo put Proposition \\ref{prop12} into a more conventional form,\nwe give a slightly weaker inequality.\n\n\\begin{proposition} \\cite{LV2} \\label{prop13}\nGiven $N \\in (2, \\infty)$ and $K > 0$, suppose that $(X,d,\\nu)$ has $N$-Ricci\ncurvature bounded below by $K$.\nThen for any nonnegative Lipschitz function $f \\in \\operatorname{Lip}(X)$ with\n$\\int_X f^{\\frac{2N}{N-2}} \\: d\\nu \\: = \\: 1$, one has\n\\begin{equation} \\label{Sobolev2}\n1 -  \\left( \\int_X f \\: d\\nu \\right)^{\\frac{2}{N+2}} \\: \\le \\:\n\\frac{6}{KN} \\left( \\frac{N-1}{N-2} \\right)^2 \\int_X\n|\\nabla f|^2 \\: d\\nu.\n\\end{equation}\n\\end{proposition}\n\nPutting (\\ref{Sobolev2}) into a homogeneous form, \nthe content of Proposition \\ref{prop13} is that there\nis a bound of the form\n$\\parallel f \\parallel_{\\frac{2N}{N-2}} \\: \\le\nF \\left( \\parallel f \\parallel_1,\n\\parallel \\nabla f \\parallel_2 \\right)$ for some\nappropriate function $F$. This is\nan example of Sobolev embedding. The inequality\n(\\ref{Sobolev2}) is not sharp, due to the\nmany approximations made in its derivation.\n\nOne can use Proposition \\ref{prop12} to prove \na sharp Poincar\\'e inequality.\n\n\\begin{proposition} \\cite{LV2} \\label{prop14}\nGiven $N \\in  (1, \\infty)$ and $K > 0$, \nsuppose that $(X,d, \\nu)$ has $N$-Ricci curvature\nbounded below by $K$. Suppose that $h \\in \\operatorname{Lip}(X)$ has\n$\\int_X h \\: d\\nu \\: = \\: 0$. Then\n\\begin{equation} \\label{eqn35}\n\\int_X h^2\\,d\\nu \\leq \n\\frac{N-1}{KN} \\int_X |\\nabla h|^2\\,d\\nu. \n\\end{equation}\n\\end{proposition}\n\nIn the case of an $N$-dimensional Riemannian manifold with \n$\\operatorname{Ric} \\: \\ge \\: K \\: g$,\none recovers the Lichnerowicz inequality for the lowest positive\neigenvalue of the Laplacian\n\\cite{Lichnerowicz}. It is sharp on round spheres.\n\n\n\\subsection{Local Poincar\\'e inequality}\n\nWhen doing analysis on metric-measure spaces, a useful analytic property\nis a ``local'' Poincar\\'e inequality.\nA metric-measure space $(X, d, \\nu)$ admits a local Poincar\\'e inequality \nif, roughly speaking, for each function\n$f$ and each ball $B$ in $X$, the mean deviation (on $B$) of $f$\nfrom its average value on $B$ is quantitatively controlled by the \ngradient of $f$ on a larger ball.\n\nTo make this precise, if $B = B_r (x)$ is a ball in $X$ then we write\n$\\lambda B$ for $B_{\\lambda r}(x)$.\nThe measure $\\nu$ is said to be {\\em doubling} if\nthere is some $D > 0$ so that for all balls $B$,\n$\\nu(2B) \\: \\le \\: D \\: \\nu(B)$. \nAn {\\em upper gradient} for a function\n$u \\in C(X)$ is a Borel function $g \\: : \\: X \\rightarrow\n[0, \\infty]$ such that for each curve\n$\\gamma \\: : \\: [0,1] \\rightarrow X$ with finite length \n$L(\\gamma)$ \nand constant speed,\n\\begin{equation} \\label{eqn36}\n\\bigl| u(\\gamma(1)) \\: - \\: u(\\gamma(0)) \\bigr| \\: \\le \\:\nL(\\gamma) \\: \\int_0^1 \\: g(\\gamma(t)) \\: dt.\n\\end{equation}\nIf $u$ is Lipschitz then $|\\nabla u|$ is an example of an upper gradient.\n\nThere are many forms of local Poincar\\'e inequalities. \nThe strongest one, in a certain sense, is as follows :\n\\begin{definition} \\label{def7}\nA metric-measure space $(X, d, \\nu)$ admits a local Poincar\\'e\ninequality if\nthere are constants $\\lambda\\geq 1$ and \n$P<\\infty$ such that for all $u\\in C(X)$ and $B=B_r(x)$ with\n$\\nu(B) > 0$,\neach upper gradient $g$ of $u$ satisfies\n\\begin{equation}\\label{PI2}\n\\mathop{-\\mkern-19.5mu\\int}_B |u-\\av{u}_B|\\,d\\nu \\leq P r\\, \\mathop{-\\mkern-19.5mu\\int}_{\\lambda B}\ng \\,d\\nu.\n\\end{equation}\n\\end{definition}\n\nHere the barred integral is the average (with respect to $\\nu$),\ne.g. $\\mathop{-\\mkern-19.5mu\\int}_{\\lambda B}\ng \\,d\\nu \\: = \\: \\frac{\\int_{\\lambda B}\ng \\,d\\nu}{\\nu(\\lambda B)}$,  and\n$\\av{u}_B$ is the average of $u$ over the ball $B$. In the case of\na length space, the local\nPoincar\\'e inequality as formulated in Definition \\ref{def7}\nactually implies stronger inequalities, for which we refer to\n\\cite[Chapters 4 and 9]{Heinonen}.  It is known that the property\nof admitting a local Poincar\\'e inequality is preserved under\nmeasured Gromov-Hausdorff limits \\cite{Keith,Koskela}.\n(This was also shown by Cheeger in unpublished work.)\nCheeger showed that if a metric-measure space\nhas a doubling measure and admits a local Poincar\\'e inequality then\nit has remarkable extra local structure \n\\cite{Cheeger}.\n\nCheeger and Colding showed that local Poincar\\'e inequalities exist for\nRicci limit spaces \\cite{Cheeger-Colding (2000)}.  The method of proof\nwas to show that Riemannian manifolds with lower Ricci curvature\nbounds satisfy a certain ``segment\ninequality''\n\\cite[Theorem 2.11]{Cheeger-Colding}\nand then to show that the property of satisfying\nthe segment inequality is preserved under measured Gromov-Hausdorff \nlimits \\cite[Theorem 2.6]{Cheeger-Colding (2000)}. \nThe segment inequality then implies\nthe local Poincar\\'e inequality.\n\nIt turns out that the argument\nusing the segment inequality can be abstracted and\napplied to certain measured length spaces. For simplicity, we\nrestrict to the case of nonnegative $N$-Ricci curvature.\nWe say that $(X, d, \\nu)$ has\nalmost-everywhere unique geodesics if\nfor $\\nu\\otimes\\nu$-almost all $(x_0,x_1)\\in X\\times X$,\nthere is a unique minimizing geodesic\n$\\gamma \\in \\Gamma$ with $\\gamma(0)=x_0$ and\n$\\gamma(1)=x_1$.\n\n\\begin{theorem} \\cite{LV2,vR,Sturm3} \\label{thm4}\nIf a compact measured length space \n$(X,d,\\nu)$ has nonnegative $N$-Ricci curvature\nand almost-everywhere unique geodesics then it satisfies\nthe local Poincar\\'e inequality of Definition~\\ref{def7} with\n$\\lambda\\: = \\: 2$ and $P\\: = \\: 2^{2N+1}$.\n\\end{theorem}\n\nAs is well-known, a Riemannian manifold has almost-everywhere\nunique geodesics. A sufficient condition for $(X,d, \\nu)$ to\nhave almost-everywhere unique geodesics is that \nalmost every $x \\in X$ is nonbranching in a certain sense\n\\cite{vR,Sturm3}.\n\nThe result of Theorem \\ref{thm4} holds in greater generality.  What one needs\nis a way of joining up points by geodesics, called a ``democratic\ncoupling'' in \\cite{LV2}, and a doubling condition on the measure.\n\nWe do not know whether the condition of \nnonnegative $N$-Ricci curvature is sufficient in itself to imply a\nlocal Poincar\\'e inequality. Having nonnegative $N$-Ricci\ncurvature does not imply almost-everywhere unique geodesics.\nFor a noncompact example, the finite-dimensional Banach space $\\mathbb R^n$ with\nthe $l_1$ norm and the Lebesgue measure has nonnegative\n$n$-Ricci curvature, but certainly does not have almost-everywhere\nunique geodesics.\n\n\\section{Final remarks} \\label{sec7}\nIn this survey we have concentrated on compact spaces.\nThere is also a notion of Ricci curvature bounded below\nfor noncompact measured length spaces $(X,d,\\nu)$\n\\cite[Appendix E]{LV}. Here we want $X$ to be a complete pointed locally\ncompact length space and $\\nu$ to be a nonnegative nonzero \nRadon measure on $X$. We do not require $\\nu$ to be a\nprobability measure.  There is a Wasserstein space $P_2(X)$\nof probability measures on $X$ with finite second moment,\ni.e. \n\\begin{equation} \\label{eqn37}\nP_2(X) \\: = \\: \\left\\{ \\mu \\in P(X) \\: : \\: \\int_X d(\\bp,x)^2 \\: d\\mu(x) \n\\: < \\: \\infty \\right\\},\n\\end{equation}\nwhere $\\bp$ is the basepoint in $X$. Many of the results described\nin this survey extend from compact spaces to such noncompact spaces,\nalthough interesting technical points arise.\n\nIn particular, if $(X, d, \\nu)$ is a compact or\nnoncompact space with nonnegative $N$-Ricci\ncurvature and $\\operatorname{supp}(\\nu) = X$, and if $x$ is a point in $X$, then\na tangent cone at $x$ has nonnegative $N$-Ricci curvature\n\\cite[Corollary E.44]{LV}.\n\nThere are many directions for future research. Any\nspecific problems that we write here may become obsolete, but let us just\nmention two general directions.  One direction is to\nsee whether known results about Riemannian manifolds with\nlower Ricci curvature bounds extend to measured length spaces\nwith lower Ricci curvature bounds.  As a caution, this is not\nalways the case. For example, the Cheeger-Gromoll splitting\ntheorem says that if there is a line in a complete Riemannian manifold \n$M$ with nonnegative Ricci curvature then there is an \nisometric splitting $M = \\mathbb R \\times Y$. This is not true for\nmeasured length spaces with nonnegative $N$-Ricci curvature.\nCounterexamples are given by nonEuclidean $n$-dimensional normed\nlinear spaces,\nequipped with Lebesgue measure, which all have nonnegative\n$n$-Ricci curvature \\cite{TOT2}. However, it is possible that\nthere is some vestige of the splitting theorem left.\n\nThe splitting theorem does hold for a pointed Gromov-Hausdorff limit\nof a sequence $\\{(M_i, g_i)\\}_{i=1}^\\infty$ of complete Riemannian\nmanifolds with Ricci curvature bounded below by $- \\: \\frac{1}{i}$\n\\cite{Cheeger-Colding}, so not every finite-dimensional\n$(X,d,\\nu)$ with nonnegative\n$N$-Ricci curvature arises as a limit in this way.\n(The analogous statement is not known for finite-dimensional\nAlexandrov spaces,\nbut there are candidate Alexandrov spaces that may not\nbe Gromov-Hausdorff limits of Riemannian manifolds with\nsectional curvature uniformly bounded below \\cite{Kapovitch}.)\nOne's attitude towards this fact may depend on whether one intuitively\nfeels that finite-dimensional normed linear spaces should \nor should not have nonnegative Ricci curvature.\n\nAnother direction of research is to find classes of measured length\nspaces $(X,d,\\nu)$ which do or do not have lower Ricci curvature bounds.\nThis usually amounts to understanding optimal transport on\nsuch spaces.\n\n\\bibliographystyle{acm}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section*{Acknowledgments}\n\nWe have benefited from discussions with Dave Morrison, Duco van Straten, and Noriko Yui.\nWe would like to thank the hospitality of SCGP where this work was completed during the 2018 Simons Summer workshop.\n\nThe research of CV is supported by NSF grant PHY-1067976.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nA typical resource allocation problem involves dividing a set of resources among interested agents. We are often concerned with the \\emph{efficiency} of the allocation, e.g., achieving high social welfare or ensuring that there is no other allocation that would make every agent better off than in the current allocation. Another important issue is the \\emph{fairness} of the allocation. For example, we might want the resulting allocation to be \\emph{envy-free}, meaning that every agent regards her bundle as the best among the bundles in the allocation \\cite{Foley67,Varian74}, or \\emph{proportional}, meaning that every agent obtains at least her proportionally fair share \\cite{Steinhaus48}. A common feature of such problems is that one agent's gain is another agent's loss: the setting inherently puts the agents in conflict with one another, and our task is to try to resolve this conflict as best we can according to our objectives.\n\nWe consider a variant of the resource allocation problem where instead of the agents being pitted against one another, they belong to one and the same group. We will collectively allocate a subset of items to this group, and our goal is to make this subset ``agreeable'' to all agents. Agreeability can be thought of as a minimal desirability condition: While an agent may be able to find other subsets of items that she personally prefers, the current set is still acceptable for her and she can agree with the allocation of the set to the group. Without further constraints, the problem described so far would be trivial, since we could simply allocate the whole set of items to the agents. We therefore impose a constraint that the allocated subset should be small. This constraint on size is reasonable in a variety of settings. For instance, the agents could be going together on a trip and there is limited space in the luggage. Alternatively, they could be receiving some items in a resource allocation setting where the preferences of the other groups are not known or are given lower priority, perhaps because the groups have not arrived or are placed lower in a team competition, so we want the subset to be agreeable to the first group while leaving as many items as possible to the remaining groups. \n\nThe problem of allocating a small agreeable subset of items was first studied by Suksompong, who defined the notion of agreeability based on the fairness notion of envy-freeness \\cite{Suksompong16}. A subset of items is said to be \\emph{agreeable} to an agent if the agent likes it at least as much as the complement set. Agreeability, or minor variants thereof, has been considered in the context of fair division, where each group consists of a single agent \\cite{BouveretEnLa10,BramsKiKl12,AzizGaMa15}. In the example of agents going together on a trip, a subset of items that they take is agreeable if they weakly prefer it to the complement subset of items left at home. In the resource allocation example, assuming that we allocate resources between two groups, a subset is agreeable to the first group if no agent in the group would rather switch to the second group. Suksompong showed a tight upper bound on the number of items that may need to be included in the set in order for it to be agreeable to all agents. In particular, for every additional agent, the worst-case bound increases by approximately half an item. This result is rather surprising, since agents can have very different preferences on the items, and yet we only need to pay a relatively mild cost for each extra agent to keep agreeability satisfied for the whole group. When there are two or three agents, Suksompong also gave polynomial-time algorithms that compute an agreeable subset whose size matches the worst-case bound.\n\nWhile Suksompong's results are quite intriguing, some important issues were left unaddressed by his work. Firstly, in many instances, an agreeable subset of smallest size is much smaller than the worst-case bound over all instances with that number of agents and items. Indeed, an extreme example is when there is a single item that every agent likes better than all of the remaining items combined. In this case, it suffices to allocate that item. This results in a much smaller set than the worst-case bound, which is at least half of the items for any number of agents. Secondly, even if we were content with finding a subset that matches the worst-case bound, we might not be able to compute it efficiently, thus rendering the existence result impractical when the number of agents or items is large. A related issue is that of eliciting the preferences on subsets of items from the agents. Since there are an exponential number of subsets, the burden on the agents to determine their preferences and the amount of information that they need to submit to our algorithm is potentially huge. This issue can be circumvented by relying on preferences over single items or allowing the algorithm to query the agents' preferences on a need-to-know basis, as is the case for Suksompong's polynomial time algorithm for two and three agents, respectively.\\footnote{Note that if the algorithm elicits the whole preference relations from the agents, this elicitation step alone already prevents the algorithm from running in time polynomial in the number of items.}\n\nIn this paper, we address all of these issues and investigate the problem of computing an agreeable subset of approximately optimal size for a given instance, as opposed to one whose size is close to the worst-case bound over all instances with that number of agents and items. We tackle the problem using several models that are well-studied in the literature and present computationally efficient algorithms for computing an agreeable subset of approximately optimal size in each of them. Moreover, in all of the models we show that our approximation bounds are virtually tight.\n\nIn Section \\ref{sec:ordinal}, we assume that we only have access to the agents' ordinal preferences on single items rather than subsets of items. Models of this type offer the advantage that the associated algorithms are often simple to implement and the agents do not need to give away or even determine their entire utility functions; such models have therefore received widespread attention \\cite{KohlerCh71,BouveretEnLa10,AzizGaMa15}. With only the ordinal preferences on single items in hand, however, most of the time we cannot tell whether a certain subset is agreeable to an agent or not. Nevertheless, by assuming that preferences are responsive, we can extend preferences on single items to partial preferences on subsets. We show that for any constant number of agents, there exists a subset of $\\frac{m}{2}+o(m)$ items that is always agreeable as long as the full responsive preferences are consistent with the rankings over single items. Since a necessarily agreeable subset always consists of at least $\\frac{m}{2}$ items even for one agent, this bound is essentially tight. We also present a simple randomized algorithm and a deterministic algorithm, both running in polynomial time, to compute such a subset.\n\nNext, in Section \\ref{sec:general}, we consider general preferences using the value oracle model \\cite{FeigeMiVo11}, where the preferences of the agents are represented by utility functions and we are allowed to query the utility of an agent for any subset. We exhibit an efficient approximation algorithm with approximation ratio $O(m\\ln\\ln m\/\\ln m)$ in this model. While this may not seem impressive since the trivial algorithm that always outputs the whole set of items already achieves approximation ratio $O(m)$, we also show that our ratio is essentially the best we can hope for. In particular, there does not exist a polynomial time algorithm with approximation ratio $o(m\/\\ln m)$. \n\nFinally, in Section \\ref{sec:additive}, we assume that the agents are endowed with additive utility functions. Additivity provides a reasonable tradeoff between simplicity and expressiveness; it is commonly assumed in the literature, especially recently \\cite{AmanatidisMaNi15,BouveretLe16,CaragiannisKuMo16}. We show that under additive valuations, it is NP-hard to decide whether there exists an agreeable set containing exactly half of the items even where there are only two agents. On the other hand, using results on covering integer programs, we demonstrate the existence of an $O(\\ln n)$-approximation algorithm for computing a minimum size agreeable set. Moreover, we show that this is tight: It is NP-hard to approximate the problem to within a factor of $(1-\\delta)\\ln n$ for any $\\delta>0$.\n\n\\section{Preliminaries}\n\nLet $N=\\{1,2,\\ldots,n\\}$ denote the set of agents, and $S=\\{x_1,x_2,\\dots,x_m\\}$ the set of items. The agents in $N$ will be collectively allocated a subset of items in $S$. Denote by $\\mathcal{S}$ the set of all subsets of $S$. Each agent $i$ is endowed with a preference relation $\\succeq_i$, a reflexive, complete, and transitive ordering over $\\mathcal{S}$. Let $\\succ_i$ denote the strict part and $\\sim_i$ the indifference part of the relation $\\succeq_i$. For items $x$ and $y$, we will sometimes  write $x\\succeq y$ to mean $\\{x\\}\\succeq\\{y\\}$. We assume throughout the paper that preferences are \\emph{monotonic}, i.e., $T\\cup\\{x\\}\\succeq T$ for all $T\\subseteq S$ and $x\\in S$.\n\nWe are interested in when a set of items is agreeable to an agent. To this end, we must precisely define what agreeability means. The notion of agreeability, defined by Suksompong \\cite{Suksompong16}, is based on the fairness concept of envy-freeness. A subset is considered to be agreeable to an agent if the agent likes it at least as much as the complement set. Put differently, if the complement set is allocated to another agent, then the former agent does not envy the latter.\n\n\\begin{definition}\n\\label{def:agreeable}\nA subset $T\\subseteq S$ of items is said to be \\emph{agreeable} to agent $i$ if $T\\succeq_i S\\backslash T$.\n\\end{definition}\n\nNext, we define a property of preferences called \\emph{responsiveness}, which says that an agent cannot be worse off whenever an item is added to her set or replaced by another item that she weakly prefers to the original item. Responsiveness is a reasonable assumption in many settings and has been widely studied in the literature \\cite{BramsFi00,BramsKiKl12}.\n\n\\begin{definition}\n\\label{def:responsive}\nA preference $\\succeq$ on $\\mathcal{S}$ is called \\emph{responsive} if it satisfies the following two conditions:\n\\begin{itemize}\n\\item $T\\cup\\{x\\}\\succeq T$ for all $T\\subseteq S$ and $x\\in S$ (monotonicity);\n\\item $T\\backslash \\{y\\}\\cup \\{x\\}\\succeq T$ for all $T\\subseteq S$ and $x,y\\in S$ such that $x\\succeq y$, $x\\not\\in T$ and $y\\in T$.\n\\end{itemize}\n\\end{definition}\n\nWhen preferences are responsive, it sometimes suffices to know an agent's preference on single items (i.e., the restriction of $\\succeq_i$ to subsets consisting of single items) in order to deduce that the agent regards a subset as agreeable. This intuition is formalized in the next definition.\n\n\\begin{definition}\n\\label{def:necessaryEF}\nFix a preference $\\succeq^{sing}$ on single items in $S$. A subset $T\\subseteq S$ is \\emph{necessarily agreeable} with respect to $\\succeq^{sing}$ if $T\\succeq S\\backslash T$ for any responsive preference $\\succeq$ on $\\mathcal{S}$ consistent with $\\succeq^{sing}$.\n\\end{definition}\n\nThe following characterization of necessary agreeability will be useful for our results in Section \\ref{sec:ordinal}.\n\n\\begin{lemma}[\\cite{Suksompong16}]\n\\label{lem:necessaryEF}\nFix a preference $\\succeq^{sing}$ on single items in $S$ with \\[x_1\\succeq^{sing}x_2\\succeq^{sing}\\cdots\\succeq^{sing}x_m.\\] \nLet $T\\subseteq S$, and define $I_k=\\{x_1,x_2,\\ldots,x_k\\}$ for all $k\\in\\{1,2,\\ldots,m\\}$. If \\[|I_k\\cap T|\\geq\\frac{k}{2}\\] for all $k\\in\\{1,2,\\ldots,m\\}$, then $T$ is necessarily agreeable with respect to $\\succeq^{sing}$. The converse also holds if the preference $\\succeq^{sing}$ is strict.\n\\end{lemma}\n\n\\section{Ordinal Preferences on Single Items}\n\\label{sec:ordinal}\n\nIn this section, we assume that we only have at our disposal the agents' ordinal preferences on single items. We are interested in computing a small necessarily agreeable subset that is consistent with these preferences. If we had access to the agents' preferences over all subsets of items, it is known that we could always find a subset of size $\\left\\lfloor\\frac{m+n}{2}\\right\\rfloor$ that is agreeable to all $n$ agents \\cite{Suksompong16}. It is not clear, however, how much extra ``penalty'' we have to pay for the information restriction that we are imposing. It could be, for example, that with three agents there exist preferences for which we have to include up to $\\frac{2m}{3}$ or $\\frac{3m}{4}$ items in a necessarily agreeable subset. We show that this is in fact not the case---there always exists a necessarily agreeable subset of size $\\frac{m}{2}+O(\\sqrt{m\\log\\log m})$ as long as the number of agents is constant. \n\nFor the proof of our first result, we will require the law of the iterated logarithm, which gives a bound for the fluctuations of a random walk.\n\n\\begin{lemma}[Law of the iterated logarithm~\\cite{Khintchine24,Kolmogoroff29}]\n\\label{lem:iteratedlog}\nLet $X_1,X_2,\\ldots$ be independent and identically distributed random variables with mean $0$ and variance $1$. Let $S_n:=X_1+\\dots+X_n$. Then\n\\[\n\\limsup_{n\\rightarrow\\infty}\\frac{S_n}{\\sqrt{n\\log\\log n}}=\\sqrt{2}\n\\]\nalmost surely.\n\\end{lemma}\n\n\\begin{theorem}\n\\label{thm:necessaryrandomized}\nIf the number of agents is constant, then there exists a subset of size $\\frac{m}{2}+O(\\sqrt{m\\log\\log m})$ that is necessarily agreeable with respect to the preferences on single items of all agents.\n\\end{theorem}\n\nWe remark that when there are two agents, Suksompong's algorithm computes a necessarily agreeable subset of size $\\left\\lfloor\\frac{m}{2}\\right\\rfloor+1$, which is also the optimal bound for an agreeable subset for two agents even if we know their full preferences.\n\n\\begin{proof}\nLet $X_1,X_2,\\ldots,X_m$ be independent random variables taking values $1$ or $-1$. We will take $X_i=1$ to mean that item $x_i$ is included in our subset, and $X_i=-1$ to mean that it is excluded from the subset. \n\nConsider an arbitrary agent $j$. Suppose that she ranks the single items as \\[x_{\\sigma_j(1)}\\succeq_j x_{\\sigma_j(2)}\\succeq_j\\dots\\succeq_j x_{\\sigma_j(m)}.\\] Using Lemma \\ref{lem:necessaryEF}, we find that our subset is necessarily agreeable for the agent if \\[X_{\\sigma_j(1)}+\\dots+X_{\\sigma_j(i)}\\geq 0\\] for all $1\\leq i\\leq m$.\n\nLet $S_i^j:=X_{\\sigma_j(1)}+\\dots+X_{\\sigma_j(i)}$, and independently set each $X_i$ to be $1$ or $-1$ with probability $\\frac{1}{2}$ each. Lemma \\ref{lem:iteratedlog} implies that for large enough $k$ (and $m$), we have $S_i^j\\leq 2\\sqrt{i\\log\\log i}$ (and by symmetry, $-S_i^j\\geq-2\\sqrt{i\\log\\log i}$) for all $i\\geq k$ with high probability. This means that for large enough $m$, $|S_i^j|\\leq 2\\sqrt{m\\log\\log m}$ for all $1\\leq i\\leq m$ with high probability as well. \n\nSince $n$ is constant, by the union bound we find that for sufficiently large $m$, we can initialize the random variables $X_1,\\dots,X_m$ so that $|S_i^j|\\leq2\\sqrt{m\\log\\log m}$ for all $1\\leq i\\leq m$ and all $j$. We will modify the choice of $X_i$'s slightly. For each agent, include her most preferred $\\sqrt{m\\log\\log m}$ items that have so far been excluded. Thus, we have $S_i^j\\geq 0$ for all $1\\leq i\\leq m$ and all $j$, and our selected subset includes at most \n\\[\\frac{m}{2}+(n+1)\\cdot\\sqrt{m\\log\\log m}=\\frac{m}{2}+O(\\sqrt{m\\log\\log m})\\] \nitems, as desired.\n\\end{proof}\n\nTheorem \\ref{thm:necessaryrandomized} yields a simple  randomized polynomial time algorithm that finds a necessarily agreeable subset for multiple agents by first choosing independently and uniformly at random whether to include each item, and then modifying the selection by including the most preferred $\\sqrt{m\\log\\log m}$ items of each agent that have been excluded.\n\nNext, we present a deterministic polynomial time algorithm that finds a necessarily agreeable subset of size $m\/2+o(m)$ for all agents. We will need the following classical result in combinatorics.\n\n\\begin{lemma}[\\cite{ErdosSz35}]\n\\label{lem:longestsubsequence}\nAny sequence of at least $(r-1)(s-1)+1$ distinct real numbers contains either an increasing subsequence of length $r$ or a decreasing subsequence of length $s$.\n\\end{lemma}\n\n\\begin{theorem}\n\\label{thm:necessarydeterministic}\nIf the number of agents is constant, then there exists a deterministic algorithm, running in time polynomial in the number of items, that computes a subset of size $\\frac{m}{2}+o(m)$ that is necessarily agreeable with respect to the preferences on single items of all agents.\n\\end{theorem}\n\n\\begin{proof}\nWe first observe that when there are $m$ items and two agents whose preferences on these items are the opposite of each other, we can choose a subset of size at most $\\frac{m}{2}+1$ that is necessarily agreeable to both agents. To see this, suppose that the preferences on single items of the two agents are \\[x_1\\succeq_1 x_2\\succeq_1\\dots\\succeq_1 x_m\\] and \\[x_m\\succeq_2 x_{m-1}\\succeq_2\\dots\\succeq_2 x_1.\\] If $m$ is odd, it suffices to choose items $x_1,x_3,x_5,\\dots,x_m$, while if $m$ is even, it suffices to choose items $x_1,x_3,x_5,\\dots,x_{m-1}$ along with item $x_m$.\n\nNow, consider our group of $n$ agents. Assume without loss of generality that the preference on single items of agent 1 is \\[x_1\\succeq_1 x_2\\succeq_1\\dots\\succeq_1 x_m.\\] Using Lemma \\ref{lem:longestsubsequence}, from the preference on single items of agent 2, we can find either an increasing subsequence or a decreasing subsequence of length $\\sqrt{m}$ on the indices of the items. Applying Lemma \\ref{lem:longestsubsequence} again, we find a subsequence of this subsequence of length $m^{1\/4}$ that is either increasing or decreasing in the preference on single items of agent 3. Proceeding in this manner, we find a subsequence of $m^{1\/2^{n-1}}$ items whose indices appear either in that order or in the reverse order in every agent's preference on single items. By our observation above, we can choose a subset of at most $\\frac{1}{2} m^{1\/2^{n-1}}+1$ items that is necessarily agreeable for every agent with respect to this set of $m^{1\/2^{n-1}}$ items.\n\nLet $t:=2^{n-1}$. Note that if a set $A$ of items is necessarily agreeable when the universe of items is taken to be $B\\supseteq A$, and another set $C$ of items is necessarily agreeable with respect to the same preference on single items when the universe of items is $D\\supseteq C$ disjoint from $B$, then $A\\cup C$ is necessarily agreeable with respect to that preference when the universe of items is $B\\cup D$. To obtain our necessarily agreeable subset for all agents, we proceed as follows. When there are $k$ items left, we choose a subset of $k^{\\frac{1}{t}}$ items as above. Within that subset, we choose a subset of at most $\\frac{1}{2} k^{\\frac{1}{t}}+1$ items that is necessarily agreeable for all agents with respect to the $k^{\\frac{1}{t}}$ items, and we remove the $k^{\\frac{1}{t}}$ items from consideration. In the first step, we decrease the number of items by a factor of $1-1\/m^{(t-1)\/t}$, and this factor only decreases in subsequent steps. Hence the number of steps is at most\n\\[\\frac{\\log\\frac{1}{m}}{\\log\\left(1-\\frac{1}{m^{(t-1)\/t}}\\right)}\\approx\\frac{\\log m}{\\frac{1}{m^{(t-1)\/t}}}=m^{\\frac{t-1}{t}}\\log m,\\]\nwhich implies that our chosen subset exceeds half of the items by $o(m)$ items. Moreover, since each step involves finding a longest increasing or decreasing subsequence from a list of length at most $m$ for a constant number of agents, the algorithm runs in polynomial time.\n\\end{proof}\n\nAlthough the algorithm in Theorem~\\ref{thm:necessarydeterministic} has the advantage of being deterministic, the expected number of repetitions of the randomized algorithm in Theorem~\\ref{thm:necessaryrandomized} is very low; in fact, this value is roughly 1 since the algorithm succeeds with high probability. Therefore, we think that the algorithm from Theorem 1 should be preferred in general due to its speed and ease of implementation as well as its superior guarantee.\n\n\\section{General Preferences}\n\\label{sec:general}\n\nWhile our algorithms from the previous section always find an agreeable set of size at most $\\frac{m}{2} + o(m)$, it is unclear how small this set is compared to the optimal if we have information beyond preferences on single items. In other words, our results so far do not yield any guarantee on the approximation ratio beyond the obvious $O(m)$ upper bound for arbitrary preferences over subsets of items. The goal of this section is to explore the possibilities and limitations of achieving better approximation ratios in this general setting.\n\nBefore we move on to our results, let us be more precise about the model we are working with. First, since each agent's preference is reflexive, complete and transitive, there is a utility function $u_i: \\mathcal{S} \\rightarrow [0, 1]$ such that $T \\succeq_i T'$ if and only if $u_i(T) \\geq u_i(T')$; for convenience, we work with these utility functions instead of working directly with the preferences themselves. Since the number of subsets in $\\mathcal{S}$ is exponentially large, the utility functions take exponential space to write down. Hence, it is undesirable to include them as part of the input. Instead, we will work with the \\emph{value oracle model}~\\cite{FeigeMiVo11}, in which the algorithm can query $u_1(T), \\dots, u_n(T)$ for each subset $T \\subseteq S$.\\footnote{When a polynomial number of queries are allowed, it is not hard to see that the value oracle model can be simulated with the preference oracle model (e.g., \\cite{Suksompong16}), in which the algorithm is allowed to query the relative preference of an agent between any two subsets, and vice-versa.} Finally, we note that we do not assume responsiveness of the agents' preferences in this section.\n\nOur first result is a simple polynomial time approximation algorithm with approximation ratio $O(m \\ln\\ln m \/ \\ln m)$. Even though this approximation guarantee is only $\\Omega(\\ln m \/ \\ln\\ln m)$ better than the obvious $O(m)$ bound, we will see later that this is almost the best one can hope for in polynomial time.\n\n\\begin{theorem} \\label{thm:approx-general}\nFor any constant $\\varepsilon > 0$, there exists a deterministic $(\\varepsilon m \\ln\\ln m \/ \\ln m)$-approximation algorithm for finding a minimum size agreeable set that runs in time polynomial in the number of agents and items.\n\\end{theorem}\n\nOur algorithm works as follows. First, we query the agents' utilities on $(\\ln m \/ \\ln\\ln m)$-size subsets $T_1, \\dots, T_{\\poly(m)}$, which are to be specified. If any of these subsets are agreeable, then we output one such subset. Otherwise, we output the whole set $S$.\n\nTo argue about the approximation ratio of the algorithm, we need to show that, if there is a small agreeable subset $T^* \\subseteq S$ (of size $O(\\ln m \/ \\ln\\ln m)$), then at least one of $T_1, \\dots, T_{\\poly(m)}$ is agreeable. A sufficient condition for this is that every subset $T \\subseteq S$ of small size is contained in at least one $T_i$. A similar question has been studied before in combinatorics under the name \\emph{covering design}~\\cite{GordonPaKu96}. Here we will use a construction by Rees et al., stated formally below.\n\n\\begin{lemma}[\\cite{ReesStWe99}] \\label{lem:covering-design}\nFor any set $S$ with $|S|=m$ and any positive integers $p, q$ such that $pq \\leq m$, there exists a deterministic algorithm that outputs subsets $T_1, \\dots, T_{\\binom{\\lceil m \/ p \\rceil}{q}} \\subseteq S$ of size $pq$ such that, for any subset $T \\subseteq S$ of size at most $q$, $T$ is contained in $T_i$ for some $i$. Moreover, the running time of the algorithm is polynomial in $m$ and $\\binom{\\lceil m \/ p \\rceil}{q}$.\n\\end{lemma}\n\n\nWe give a short proof of Lemma~\\ref{lem:covering-design}, which is taken from Rees et al.'s work but modified with the desired range of the parameters.\n\n\\begin{proof}[Proof of Lemma~\\ref{lem:covering-design}]\nFirst, partition $S$ into $\\lceil m\/p \\rceil$ parts $S_1, \\dots, S_{\\lceil m \/ p \\rceil}$ where each part is of size at most $p$. Each set $T_i$ is simply a union of $q$ different $S_j$'s. Note that if the union is of size smaller than $pq$, we can simply add arbitrary elements of $S$ into it to make its size exactly $pq$. Clearly, there are $\\binom{\\lceil m \/ p \\rceil}{q}$ such unions and $T_1, \\dots, T_{\\binom{\\lceil m \/ p \\rceil}{q}}$ satisfy the properties required in the theorem.\n\\end{proof}\n\nWith this lemma in place, we are ready to prove Theorem~\\ref{thm:approx-general}.\n\n\\begin{proof}[Proof of Theorem~\\ref{thm:approx-general}]\nOur algorithm starts by evoking the algorithm from Lemma~\\ref{lem:covering-design} with $q = \\lfloor \\ln m \/ (\\varepsilon \\ln\\ln m) \\rfloor$ and $p = \\lfloor \\varepsilon m \\ln\\ln m \/ (q\\ln m) \\rfloor$ to produce subsets $T_1, \\dots, T_\\ell \\subseteq S$ of size $pq$ where $\\ell = \\binom{\\lceil m \/ p \\rceil}{q}$. For each $i = 1, \\dots, \\ell$, we query $u_1(T_i), \\dots, u_n(T_i), u_1(S \\setminus T_i), \\dots, u_n(S \\setminus T_i)$ to check whether $T_i$ is agreeable. If any $T_i$ is agreeable, we output it. Otherwise, output the whole set $S$. Clearly, the output set is always agreeable.\n\nMoreover, observe that, from our choice of $p, q$, we have \n\\begin{align*}\n\\ell = \\binom{\\lceil m \/ p \\rceil}{q} \n     &\\leq \\left(e \\lceil m \/ p \\rceil \/ q\\right)^q \\\\\n     &= (O(\\ln m))^q \\\\\n     &= \\exp(O(q \\ln\\ln m)) \\\\\n     &= \\exp(O(\\ln m)), \n\\end{align*}\nwhich is polynomial in $m$. Hence, the running time of our algorithm is polynomial in $m$ and $n$.\n\nTo prove the algorithm's approximation guarantee, let us consider two cases. First, if the optimal agreeable set has size more than $q$, then since we output a set of size at most $m$, we obtain an approximation ratio of at most $m \/ (q + 1) \\leq \\varepsilon m \\ln\\ln m \/ \\ln m$.\n\nOn the other hand, if the optimal agreeable set $T^*$ has size at most $q$, then by construction, there exists $i$ such that $T^* \\subseteq T_i$. Since $T^*$ is agreeable, $T_i$ is also agreeable. Hence, our output set has size $pq \\leq \\varepsilon m \\ln\\ln m \/ \\ln m$, which implies an approximation ratio of $\\varepsilon m \\ln\\ln m \/ \\ln m$ as well.\n\\end{proof}\n\nWhile our algorithm may seem rather naive, we will show next that its approximation guarantee is, up to $O(\\ln\\ln m)$ factor, essentially the best one can hope for, even when there is only one agent:\n\n\\begin{theorem} \\label{thm:inapprox-general}\nFor every constant $c> 0$ and every sufficiently large $m$ (depending on $c$), there is no (possibly randomized and adaptive) algorithm that makes at most $m^{c\/8}$ queries and always outputs an agreeable set with expected size at most $m \/ (c \\ln m)$ times the optimum, even when there is only one agent.\n\\end{theorem}\n\nIn other words, the above theorem implies that there is no polynomial time algorithm with approximation ratio $o(m \/ \\ln m)$. We note here that our lower bound is information-theoretic and is not based on any computational complexity assumptions. Moreover, it rules out any algorithm that makes a polynomial number of queries, not only those that run in polynomial time.\n\n\\begin{proof}[Proof of Theorem~\\ref{thm:inapprox-general}]\nLet $g: \\mathcal{S} \\rightarrow [0, 1]$ be defined by\n\\begin{align*}\ng(T) =\n\\begin{cases}\n1 & \\text{ if } |T| \\geq m \/ 2, \\\\\n0 & \\text{ otherwise.}\n\\end{cases}\n\\end{align*}\nMoreover, for each subset $T^* \\subseteq S$, let $f_{T^*}: \\mathcal{S} \\rightarrow [0, 1]$ denote the function\n\\begin{align*}\nf_{T^*}(T) =\n\\begin{cases}\n1 & \\text{ if } |T| \\geq m \/ 2 \\text{ or } T^* \\subseteq T, \\\\\n0 & \\text{ otherwise.}\n\\end{cases}\n\\end{align*}\n\nThat is, $f_{T^*}$ is $g$ together with a planted solution $T^*$.\n\nConsider any algorithm $\\mathcal{A}$ that makes at most $m^{c\/8}$ queries. Assume for the moment that $\\mathcal{A}$ is deterministic. Let us examine a run of $\\mathcal{A}$ when the agent's utility is $g$. Suppose that $\\mathcal{A}$'s queries to $g$ are $T_1, \\dots, T_{\\lfloor m^{c\/8}\\rfloor} \\subseteq S$.\n\nLet $T^*$ be a random subset of $S$ of size $\\lfloor c \\ln m \/ 4 \\rfloor$. Let us now consider the queries $\\mathcal{A}$ made when the agent's utility is $f_{T^*}$; suppose that the queries made are $T'_1, \\dots, T'_{\\lfloor m^{c\/8}\\rfloor} \\subseteq S$. For every $j = 1, \\dots, \\lfloor m^{c\/8}\\rfloor$, if $T_1 = T'_1, \\dots, T_{j - 1} = T'_{j - 1}$ and $g(T_1) = f_{T^*}(T'_1), \\dots, g(T_{j - 1}) = f_{T^*}(T'_{j - 1})$, then $\\mathcal{A}$ goes through the same computation route for both $g$ and $f_{T^*}$, and hence $T_j = T'_j$. Moreover, when both runs share the same computational route so far and $T_j = T'_j$, we can bound the probability that $g(T_j) \\ne f_{T^*}(T'_j)$ as follows. First, if $|T_j| \\geq m \/ 2$, then $g(T_j)$ is always equal to $f_{T^*}(T'_j)$. Otherwise, since $T_j$ is independent of $T^*$, we have\n\\begin{align*}\n\\Pr[g(T_j) \\neq f_{T^*}(T'_j)] &= \\Pr[g(T_j) \\neq f_{T^*}(T_j)] \\\\\n&= \\Pr[T^* \\subseteq T_j] \\\\\n&= \\frac{\\binom{|T_j|}{\\lfloor c \\ln m \/ 4 \\rfloor}}{\\binom{n}{\\lfloor c \\ln m \/ 4 \\rfloor}} \\\\\n&\\leq \\left(\\frac{|T_j|}{n}\\right)^{\\lfloor c \\ln m \/ 4 \\rfloor} \\\\\n&\\leq 2^{-\\lfloor c \\ln m \/ 4 \\rfloor} \\\\\n&\\leq 2 m^{-c \/ 6}.\n\\end{align*}\n\nHence, by union bound, the probability that the two sequences of queries are not identical is at most $(2 m^{-c \/ 6}) \\cdot m^{c \/ 8} = 2 m^{- c \/ 24}$, which is less than $1\/2$ when $m$ is sufficiently large. Furthermore, observe that when the two sequences are identical, $\\mathcal{A}$ must output an agreeable subset for $g$, which is of size at least $m \/ 2$. Thus, the expected size of the output of $\\mathcal{A}$ on $f_{T^*}$ is more than $m \/ 2 \\cdot (1 \/ 2) = m \/ 4$. However, the optimal agreeable set for $f_{T^*}$ has size only $\\lfloor c \\ln m \/ 4 \\rfloor$. As a result, the expected size of the output of $\\mathcal{A}$ is more than $m \/ (c \\ln m)$ times the optimum as desired.\n\nFinally, note that if $\\mathcal{A}$ is randomized, we can use the above argument on each choice of randomness and average over all the choices, which gives a similar conclusion.\n\\end{proof}\n\n\\section{Additive Utilities}\n\\label{sec:additive}\n\nIn this section, we assume that the agents' preferences are represented by additive utility functions. Each agent $i$ has some nonnegative utility $u_i(x_j)$ for item $x_j$, and $u_i(T)=\\sum_{x\\in T}u_i(x)$ for any $i\\in N$ and any subset of items $T\\subseteq S$. \n\nClearly, the problem of deciding whether there exists an agreeable set of a certain size is in NP. Moreover, the following theorem shows that it is indeed NP-complete, even when there are two agents with additive utility functions.\n\n\\begin{theorem}\n\\label{thm:twoplayershardness}\nEven when there are two agents with additive utility functions, it is NP-hard to decide whether there is an agreeable set of size exactly $\\frac{m}{2}$.\n\\end{theorem}\n\n\\begin{proof}\nWe will reduce from the following problem called {\\normalfont \\scshape Balanced 2-Partition}: given a multiset $A$ of non-negative integers, decide whether there exists a subset $B \\subseteq A$ such that $|B| = |A \\setminus B| = |A|\/2$ and $\\sum_{a \\in B} a = \\sum_{a \\in A \\setminus B} a = \\sum_{a \\in A} a \/ 2$. \n\nLike the well-known {\\normalfont \\scshape 2-Partition} where the cardinality constraint is not included, {\\normalfont \\scshape Balanced 2-Partition} is NP-hard.\nFor completeness, we give a simple proof of NP-hardness of {\\normalfont \\scshape Balanced 2-Partition} in Appendix~\\ref{app:np-hard-part}.\n\nThe reduction from {\\normalfont \\scshape Balanced 2-Partition} proceeds as follows. Let $a_1, \\dots, a_{|A|}$ be the elements of $A$. The set $S$ contains $|A|$ items $x_1, \\dots, x_{|A|}$, each associated with an element of $A$. The utility function is then defined by $u_1(x_i) = a_i$ and $u_2(x_i) = M - a_i$ where $M = \\sum_{a \\in A} a$. We will next show that this reduction is indeed a valid reduction.\n\n(YES Case) Suppose that there exists $B \\subseteq A$ such that $|B| = |A|\/2$ and $\\sum_{a \\in B} a = \\sum_{a \\in A} a \/ 2$. We can simply pick $T$ to be all the items corresponding to the elements in $B$. It is obvious that $T$ has size $|A| \/ 2 = m \/ 2$ and that $T$ is agreeable.\n\n(NO Case) We prove the contrapositive; suppose that there is an agreeable subset $T \\subseteq S$ of size $m\/2$. Let $B$ be the corresponding elements in $A$ of all the items in $T$. Since $T$ is agreeable, $\\sum_{x \\in T} u_i(x) \\geq \\sum_{x \\in S \\setminus T} u_i(x)$ for $i \\in \\{1, 2\\}$. When $i = 1$, this implies that $\\sum_{a \\in B} a \\geq \\sum_{a \\in A} a \/ 2$. Moreover, when $i = 2$, using the fact that $|T| = m \/ 2$, we have $\\sum_{a \\in B} a \\leq \\sum_{a \\in A} a \/ 2$. Thus, $\\sum_{a \\in B} a = \\sum_{a \\in A} a \/ 2$. Finally, note that $|B| = m \/ 2 = |A| \/ 2$. Hence, $A$ is a YES instance for {\\normalfont \\scshape Balanced 2-Partition}.\n\\end{proof}\n\nTheorem~\\ref{thm:twoplayershardness} shows that the problem is weakly NP-hard even when there are two agents. Nevertheless, when the number of agents is constant, there exists a pseudo-polynomial time dynamic programming algorithm for computing an optimal agreeable set. In particular, the problem is not strongly NP-hard for a constant number of agents.\n\n\\begin{theorem}\nIf the number of agents is constant, then there exists a pseudo-polynomial time algorithm that computes an agreeable set of minimum size.\n\\end{theorem}\n\n\\begin{proof}\nThe algorithm uses dynamic programming. Assume that the utilities of agent $i$ for the items are integers with sum $\\sigma_i$. We construct a table $\\Sigma$ of size $(m+1)(\\sigma_1+1)\\dots(\\sigma_n+1)$, where for each $0\\leq m'\\leq m$ and each tuple $(y_1,\\dots,y_n)$ with $0\\leq y_i\\leq \\sigma_i$, the entry $\\Sigma(m',y_1,\\dots,y_n)$ of the table corresponds to the minimum number of items from among the first $m'$ items that we need to include so that agent $i$ has utility exactly $y_i$ for all $i$ (if this is achievable). Initially we have $\\Sigma(0,0,\\dots,0)=0$ and $\\Sigma(m',y_1,\\dots,y_n)=\\infty$ otherwise. We then iterate through the values of $m'$ in increasing order. For each $m'\\geq 1$, we update the entries of the table as follows:\n\\begin{itemize}\n\\item If $u_i(x_{m'})\\leq y_i$ for all $i$ and $$1+\\Sigma\\left(m'-1,y_1-u_1(x_{m'})\\dots,y_n-u_n(x_{m'})\\right)<\\Sigma(m'-1,y_1,\\dots,y_n),$$ let $\\Sigma(m',y_1,\\dots,y_n)=1+\\Sigma\\left(m'-1,y_1-u_1(x_{m'})\\dots,y_n-u_n(x_{m'})\\right)$. \n\\item Else, let $\\Sigma(m',y_1,\\dots,y_n)=\\Sigma(m'-1,y_1,\\dots,y_n)$.\n\\end{itemize}\n\nFinally, we look up the entries $\\Sigma(m,y_1,\\dots,y_n)$ for which $y_i\\geq \\frac{\\sigma_i}{2}$ holds for all $i$ and return the minimum value over all such entries. This algorithm takes time $O(m\\sigma_1\\dots \\sigma_n)$. Note that if we also want to return an agreeable set (rather than just the minimum size), we can also keep track of the sets of items along with the values in our table. \n\\end{proof}\n\nWhile there is a pseudo-polynomial time algorithm for the problem when the number of agents is fixed, we show below that when the number of agents is not fixed, the problem becomes strongly NP-hard. In other words, there is no pseudo-polynomial time algorithm for this variation unless P=NP. \n\n\\begin{theorem}\nWhen the number of agents is given as part of the input, it is strongly NP-hard to decide whether there is an agreeable set of size exactly $\\frac{m+1}{2}$.\n\\end{theorem}\n\n\\begin{proof}\nWe will reduce from {\\normalfont \\scshape 3SAT}. Given a 3SAT formula $\\phi$ with $m'$ clauses $C_1, \\dots, C_{m'}$ on $n'$ variables $y_1, \\dots, y_{n'}$, let there be $n=m' + n'$ agents $C_1, \\dots, C_{m'}, y_1, \\dots, y_{n'}$ and $m=2n' + 1$ items where $2n'$ items correspond to all the literals $y_1, \\neg y_1, \\dots, y_{n'}, \\neg y_{n'}$ and the remaining item is called $a$. The utility function is defined by\n\\begin{align*}\nu_{C_i}(b) =\n\\begin{cases}\n1 & \\text{ if } b = a, \\\\\n1 & \\text{ if the literal } b \\text{ is present in } C_i, \\\\\n0 & \\text{ otherwise,}\n\\end{cases}\n\\end{align*}\nand\n\\begin{align*}\nu_{y_i}(b) =\n\\begin{cases}\n1 & \\text{ if } b = a, \\\\\n1 & \\text{ if } b = y_i \\text{ or } b=\\neg y_i \\\\\n0 & \\text{ otherwise.}\n\\end{cases}\n\\end{align*}\nWe will next show that this is a valid reduction. First, note that all the integer parameters are now polynomial in the size of the input. Hence, we are left to show that YES and NO instances of 3SAT map to YES and NO instances of our problem respectively.\n\n(YES Case) Suppose that there exists an assignment that satisfies $\\phi$. For each $y_i$, let $b_i$ be the literal of $y_i$ that is true according to this assignment. Let the set $T$ be $\\{a, b_1, \\dots, b_{n'}\\}$. Since each $C_j$ is satisfied by the assignment, we have $\\sum_{i=1}^{n'} u_{C_j}(b_i) \\geq 1$. Hence, we have $\\sum_{x \\in T} u_{C_j}(x) \\geq 2$, which implies that $T \\succeq_{C_j} S \\setminus T$. \n\nMoreover, for each variable $y_i$, $\\sum_{x \\in T} u_{y_i}(x) = 2$, which also implies that $T \\succeq_{y_i} S \\setminus T$. As a result, $T$ is an agreeable set of size $n' + 1 = \\frac{m+1}{2}$ as desired.\n\n(NO Case) We again prove the contrapositive; suppose that there exists an agreeable set $T \\subseteq S$ of size $\\frac{m+1}{2} = n' + 1$. We can assume without loss of generality that $a \\in T$. Indeed, since the utility of any agent for $a$ is at least as much as the utility of the agent for any other item, if $a \\notin T$, we can remove an arbitrary item from $T$ and add $a$ in while maintaining the agreeability of $T$. Moreover, we also assume without loss of generality that each clause of $\\phi$ has at least two variables---it is obvious that every 3SAT formula can be transformed to this form in polynomial time.\n\nSince $T \\succeq_{y_i} S \\setminus T$, at least one literal corresponding to $y_i$ is included in $T$. Moreover, since the size of $T$ is $n' + 1$ and $a \\in T$, exactly one literal of each $y_i$ is in $T$; let $b_i$ be this literal. Consider the assignment to the variables such that all the $b_i$'s are satisfied. Since $T \\succeq_{C_j} S \\setminus T$ for every $C_j$ and $C_j$ contains at least two literals, at least one literal in $C_j$ is satisfied by this assignment. Hence, this assignment satisfies the formula $\\phi$.\n\\end{proof}\n\nGiven that finding an agreeable set of minimum size is NP-hard, it is natural to attempt to find an approximation algorithm for the problem. When the utilities are additive, this turns out to be closely related to approximating the classical problem {\\normalfont \\scshape Set Cover}. In {\\normalfont \\scshape Set Cover}, we are given a ground set $U$ and a collection $\\mathcal{C}$ of subsets of $U$. The goal is to select a minimum number of subsets whose union is the entire set $U$. \n\n{\\normalfont \\scshape Set Cover} was one of the first problems shown to be NP-hard in Karp's seminal paper~\\cite{Karp72}. Since then, its approximability has been intensely studied and has, by now, been well understood. A simple greedy algorithm is known to yield a $(\\ln |U| + 1)$-approximation for the problem~\\cite{Johnson74,Lovasz75}. On the other hand, a long line of work in hardness of approximation~\\cite{LundYa94,RazSa97,Feige98,AlonMoSa06,Moshkovitz15} culminates in Dinur and Steurer's work, in which a $(1 - \\varepsilon)\\ln |U|$ ratio NP-hardness of approximation for {\\normalfont \\scshape Set Cover} was proved for every constant $\\varepsilon > 0$ \\cite{DinurSt14}.\n\nThe first connection we will make between {\\normalfont \\scshape Set Cover} and approximating minimum size agreeable set is on the negative side---we will show that any inapproximability result of {\\normalfont \\scshape Set Cover} can be translated to that of approximating minimum size agreeable set as well. To do so, we will first state Dinur and Steurer's result more precisely.\n\n\\begin{lemma}[\\cite{DinurSt14}]\nFor every constant $\\varepsilon > 0$, there is a polynomial time reduction from any {\\normalfont \\scshape 3SAT} formula $\\phi$ to a {\\normalfont \\scshape Set Cover} instance $(U, \\mathcal{C})$ and $f(U) = \\poly(|U|)$ such that\n\\begin{itemize}\n\\item (Completeness) if $\\phi$ is satisfiable, the optimum of $(U, \\mathcal{C})$ is at most $f(U)$.\n\\item (Soundness) if $\\phi$ is unsatisfiable, the optimum of $(U, \\mathcal{C})$ is at least $((1 - \\varepsilon) \\ln |U|) f(U)$.\n\\end{itemize}\n\\end{lemma}\n\nWe are now ready to prove hardness of approximation for minimum size agreeable set.\n\n\\begin{theorem}\nFor any constant $\\delta>0$, it is NP-hard to approximate minimum size agreeable set to within a factor $(1 - \\delta) \\ln n$ of the optimum.\n\\end{theorem}\n\n\\begin{proof}\nLet $\\varepsilon = \\delta \/ 2$. Given a 3SAT formula $\\phi$, we first use Dinur-Steurer reduction to produce a {\\normalfont \\scshape Set Cover} instance $(U, \\mathcal{C})$. Let there be $|U|$ agents, each of whom is associated with each element of $U$; it is convenient to think of the set of agents as simply $N = U$. As for the items, let there be one item for each subset $C \\in \\mathcal{C}$ and additionally let there be one special item called $t$; in other words, $S = \\mathcal{C} \\cup \\{t\\}$.\n\nThe utility for each $a \\in U$ is then defined by\n\\begin{align*}\nu_a(s) =\n\\begin{cases}\n|\\{C \\in \\mathcal{C} \\mid a \\in C\\}| - 1 & \\text{ if } s = t, \\\\\n1 & \\text{ if } s \\in \\mathcal{C} \\text{ and } a \\in s, \\\\\n0 & \\text{ otherwise}.\n\\end{cases}\n\\end{align*}\nWe show next that this reduction indeed gives the desired inapproximability result.\n\n(Completeness) If $\\phi$ is satisfiable, then there are $f(|U|)$ subsets from $\\mathcal{C}$ that together cover $U$. We can take $T$ to contain all these subsets and the special item $t$. Clearly, $T$ has size $f(|U|) + 1$ and is agreeable.\n\n(Soundness) If $\\phi$ is unsatisfiable, then any set cover of $(S, \\mathcal{C})$ contains at least $((1 - \\varepsilon) \\ln |U|)f(|U|)$ subsets. Consider any agreeable set $T$. For each $a \\in U$, from our choice of $u_a(t)$, $T$ must include at least one subset that contains $a$. In other words, $T \\setminus \\{t\\}$ is a set cover of $(S, \\mathcal{C})$. Hence, $|T|$ must also be at least $((1 - \\varepsilon) \\ln |U|)f(|U|)$. \n\nThus, it is NP-hard to approximate minimum size agreeable set to within $\\frac{((1 - \\varepsilon) \\ln |U|)f(|U|)}{f(|U|) + 1}$ of the optimum. This ratio is at least $(1 - \\delta) \\ln n$ when the number of agents, $n = |U|$, is sufficiently large.\n\\end{proof}\n\nUnlike the above inapproximability result, it is unclear how algorithms for {\\normalfont \\scshape Set Cover} can be used to approximate minimum size agreeable set.  Fortunately, our problem is in fact a special case of a generalization of {\\normalfont \\scshape Set Cover} called {\\normalfont \\scshape Covering Integer Program} ({\\normalfont \\scshape CIP}), which can be written as follows:\n\\begin{align*}\n\\text{minimize } & c^T x \\\\\n\\text{subject to } & Ax \\geq 1, \\\\\n& 0 \\leq x \\leq u \\\\\n& x \\in \\mathbb{Z}^m\n\\end{align*}\nwhere $c, u \\in \\mathbb{R}^m$ and $A \\in \\mathbb{R}^{n \\times m}$ are given as input.\n\nThe problem of finding a minimum size agreeable set can then be formulated in this form simply by setting $c, u$ and $A$ as follows.\n\\begin{align*}\nc_s = 1 & & \\forall s \\in S \\\\\nu_s = 1 & & \\forall s \\in S \\\\\nA_{i, s} = \\frac{2 u_i(s)}{\\sum_{s' \\in S} u_i(s')} & & \\forall i \\in N, s \\in S\n\\end{align*}\n\nSimilarly to {\\normalfont \\scshape Set Cover}, approximability of {\\normalfont \\scshape CIP} has been well studied; specifically, the problem is known to be approximable to within $O(\\ln n)$ of the optimum in polynomial time~\\cite{KolliopoulosYo05}. This immediately implies an $O(\\ln n)$-approximation algorithm for finding a minimum size agreeable set as well:\n\\begin{theorem}\nWhen the agents' utility functions are additive, there is a polynomial time $O(\\ln n)$-approximation algorithm for finding a minimum size agreeable set.\n\\end{theorem}\n\n\\section{Conclusion}\n\nIn this paper, we study the problem of efficiently computing an approximately optimal agreeable set of items. We consider three well-known models for representing the preferences of the agents and derive essentially tight bounds on the approximation ratio that can be achieved in polynomial time in each model. When we only have access to agents' rankings on single items and the number of agents is constant, we can efficiently compute a necessarily agreeable subset of size $\\frac{m}{2}+O(\\sqrt{m\\log\\log m})$; this is almost tight since any such subset must contain at least $\\frac{m}{2}$ items. Next, for general preferences, we exhibit an efficient algorithm with approximation ratio $O(m\\ln\\ln m\/\\ln m)$ under the value oracle model, and we show that no efficient algorithm can achieve approximation ratio $o(m\/\\ln m)$. On the other hand, for additive valuations, an algorithm with approximation ratio $O(\\ln n)$ exists for computing a minimum size agreeable set; this is complemented by the result that it is NP-hard to approximate the problem to within a factor of $(1-\\delta)\\ln n$ for any $\\delta>0$.\n\nWe conclude the paper by listing some directions for future work. Firstly, in the ranking model, it would be interesting to close the gap between our upper bound of $\\frac{m}{2}+O(\\sqrt{m\\log\\log m})$ and the lower bound of $\\frac{m}{2}+O(1)$ by Suksompong \\cite{Suksompong16} on the minimum size necessarily agreeable set. Secondly, one can consider a probabilistic setting, where the utilities of the agents are drawn from some distributions, and determine the size of the smallest agreeable set that is likely to exist in such a setting. Thirdly, the question of truthfulness has not been addressed in this or previous work. One can ask whether similar approximation ratios can be obtained or whether additional limitations arise in the presence of strategic agents. Finally, a related line of research, which has recently received attention, is to design algorithms that fairly allocate items among groups of agents \\cite{ManurangsiSu17,Suksompong17}.\n\n\\bibliographystyle{abbrv}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\vspace{-0.2cm}\nFor many decades, various methodologies have been proposed for safe or efficient automated vehicle (AV) control under different situations. Rule-based, supervised-learning, and reinforcement-learning are widely studied and applied to control AVs.\n\nAs a rule-based approach, the Finite State Machines (FSMs) which consist of the finite number of predefined states and transition-conditions has been implemented for decision-making under known situations. In the FSMs, either actions or control-strategies are described in each state, and the transitions occur when pre-determined conditions are met. So, the FSMs based controller returns the appropriate action based on the system's conditions. \\cite{kurt2008hybrid} and \\cite{redmill2008ohio} implement the Hierarchical FSMs as a high-level decision-maker within the Hybrid State System (HSS) to control automated vehicles (AVs) under the given several scenarios in the 2007 DARPA Urban Challenge. Although the AV (called OSU-ACT) is controlled via appropriate decision-making under complex situations, the capability of the rule-based controller is fully subordinate to the initially defined rules and conditions. For example, if unexpected situations are encountered, the controller can neither recognize the situations nor make the right decision due to no existing proper rules.\n\nThe Supervised Learning (SL) is focused on seeking the best coefficients or weights of the predefined control models by using a given cost-function and ground truth data. The data consists of pairs of inputs (situations or conditions) and outputs (optimal actions), and the optimal coefficients or weights are obtained by exploring the configuration which minimizes differences between the ground truth and the model's output. In \\cite{gadepally2013framework, kuge2000driver}, the Hidden Markov Model (HMM) is implemented for obtaining a general driver model that enables AV controllers to predict other vehicles' behaviors, which is helpful for safer driving control. The HMM-based driver model is designed by the fixed number of hidden states and observations, and the transition and emission probabilities are identified via the Viterbi algorithm. In addition, various types of Deep Neural Network (DNN) are implemented to create AV control or driver model for predicting future traffic situations or choosing optimal actions respectively. \\cite{han2019driving} proposes to train DNN by using raw observations and estimated general driving characteristics to predict other vehicle's lane-change behavior on the highway. Also, the end-to-end learning-based driving controller, which maps raw camera-images with optimal controls via the Convolutional Neural Network (CNN), is derived by \\cite{bojarski2016end}. \\cite{al2017deep} implements the GoogLeNet to obtain accurate affordance parameters which are essential to take the optimal driving actions. Even though the SL approach has been widely used to get optimal models, a large amount of labeled data and off-line training procedure are required.\n\nThe Reinforcement Learning (RL) approach has been suggested to get an optimal controller which can improve its performance by updating its policy iteratively based on experiences. Unlike the SL, the ground truth data and pre-training process are not necessary, but a reward-function needs to be pre-defined appropriately. While the RL based controller explores or exploits actions at different states, its policy is updated repeatedly in a way to get the highest value (sum of future discounted rewards) at every state. The policy is represented by matrices or DNN. Since the DNN can well approximate value-function, various types of multilayer perceptron (MLP) have been proposed to represent the RL's policy. \\cite{mnih2015human} proposes Deep Q Network (DQN), and \\cite{van2016deep} suggests Double Deep Q-Network (DDQN) which can prevent the over-optimistic value estimation problem observed in the DQN. Since DQN and DDQN can only handle the discrete type of actions, Deep Deterministic Policy Gradient (DDPG) is proposed by \\cite{lillicrap2015continuous} for the continuous type of actions. \\cite{nageshrao2019autonomous} proposed the DRL agent control architecture, which implements DDQN for safe and adaptive AV control.\n\nDue to the model-free attribution, the RL with MLP controller can find the optimal actions under unexpected situations, unlike the rule-based and SL approaches. However, a well-designed reward-function is still required. In this paper, instead of paying efforts for designing a perfect model or reward-function by hand, we propose an evolving methodology (called evolving Finite State Machine), which can derive a stochastic model from scratch by observing variations of system-conditions. Also, the derived stochastic model is used within the online evolving framework to check and revise the action which is chosen by the controller.\n\n\nThis paper is organized as follows: Section \\ref{section:online evolving framework} describes the details of three modules within an on-line evolving framework. Section \\ref{section:Target Scenario and Controller} introduces the DDPG AV controller and the car-following scenario settings. Section \\ref{section:DDPG_controller_eFSM} explains the on-line evolving framework settings for the DDPG controller. Section \\ref{section:experimental results} analyzes car-following performances of the DDPG controller w\/ and w\/o the suggested framework. Lastly, our contributions and further discussion are summarized in section \\ref{section:conclusion}.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{An On-line Evolving Framework}\n\\label{section:online evolving framework}\n\\vspace{-0.2cm}\nAn on-line evolving framework is proposed to inspect or revise an action selected by a controller. In order to inspect the action, it needs to predict future states depending on the action. In the framework shown in  Fig.~\\ref{fig:online_evolving_framework}, there are a controller, evolving Finite State Machine (e-FSM), and action-reviser modules for choosing an optimal action, creating a stochastic model, and inspecting \\& revising the controller's actions respectively.\n\n\\begin{figure}[ht]\n\\begin{center}\n\\includegraphics[width=7.5cm]{figures\/online_evolving_framework.png}   \n\\vspace{-0.2cm}\n\\caption{An on-line evolving framework for controllers} \n\\label{fig:online_evolving_framework}\n\\end{center}\n\\end{figure}\n\nA set of observations $z_t$ at time-step $t$ is fed into the controller and e-FSM modules. The control module makes a decision to choose the best action; the e-FSM module derives a stochastic model which consists of states and state-transition dynamics based on the applied actions; the action-reviser module checks validity of the chosen action $a_t$ and revises the action if necessary. Note that $a_t$, $Prob(S_t)$, $\\mathbf{P}_{t}^{a(r)}\\forall r$, and $\\mathcal{L}(S_t)$ refer to an action chosen by the controller at time $t$, probability distributions over states (e.g., driving situations) which are determined by time $t$, identified state-transition probabilities with respect to actions, and a set of state attributions. In the following sub-sections, details of the e-FSM and the action-reviser modules are described.\n\\vspace{-0.2cm}\n\\subsection{evolving Finite State Machine(e-FSM)}\n\\label{subsec:e-FSM}\n\\vspace{-0.2cm}\nThe e-FSM, an on-line evolving method designed for supporting optimal decision-making, is a hybrid Markov model with states representing specific situations. The states are identified as clusters in the state space, and state dynamics is determined through a set of transition probability matrices associated with the inputs (actions). The framework of e-FSM is shown in Fig.~\\ref{fig:e_FSM_framework}, and some of the key features (online state-determination, state flagging and recognition, and online transition-identification) are described in this paper; see \\cite{han2019online} for e-FSM's further specific properties and descriptions. \n\\vspace{-0.2cm}\n\\begin{figure}[ht]\n\\begin{center}\n\\includegraphics[width=8cm]{figures\/e_FSM_framework.png}   \n\\vspace{-0.2cm}\n\\caption{The framework of an evolving Finite State Machine} \n\\label{fig:e_FSM_framework}\n\\end{center}\n\\end{figure}\n\\vspace{-0.1cm}\n\\subsubsection{- Online State Determination }\n\\vspace{-0.3cm}\nIn the e-FSM, states represent situations that a controller could be encountered. Due to the difficulty of pre-determining all possible situations, the e-FSM aims to determine a state whenever a new situation is observed over time. This feature enables the controller to have opportunities to find the best action by recognizing initially unexpected situations. \n\nTo realize the online state determination, a set of observations $z_t=[o_t(1), ..., o_t(k)]^T$ at every time $t$ is clustered, and each cluster-center is considered as a state, representing individual situation. The evolving Takagi-Sugeno (eTS) online clustering method proposed by \\cite{filev2010markov} is implemented so that infinite number of unknown states can be determined; the eTS can have infinite number of clusters. A state-set can be written by $\\mathcal{S}_{t\\geq1}=\\{s_t(1), ..., s_t(n_t)\\}$ where the states $s_t(i), 1\\leq i\\leq n_t$, are cluster-centers, and $n_t$ is the total number of the states determined by time $t$. This feature enables the structure of the stochastic model to evolve.\n\\vspace{-0.25cm}\n\\subsubsection{- Flagging and Recognition of State}\n\\label{subsubsec:state labeling and recognition}\nThe e-FSM determines states, in which each represents a distinguished situation. Instead of labeling all states, the states are flagged by either unfavorable states or not, where the unfavorable states are defined based on the given criteria. For example, if a given criterion of AV control is safety, the recognized state at the moment of collision is flagged as the unfavorable state. A set of the state attributions at time $t$, $\\mathcal{L}(S_t)$, is transferred to the action-reviser as shown in Fig. \\ref{fig:online_evolving_framework}, where $\\mathcal{L}(s_t(i))\\in\\{none, unfavorable_j\\}, 1\\leq i \\leq n_t$, $1\\leq j\\leq c_n$; $c_n$ is the total number of given criteria.\n\nGiven $z_t$, the current situation is represented by probability distributions over the states determined by time $t$, denoted as $Prob(S_t)=[Prob(s_t(1)), ..., Prob(s_t(n_t))]^T$, where each probability of state $Prob(s_t(\\cdot))$ is obtained by calculating a similarity, $\\lambda_t^i(z_t)$, between $z_t$ and $s_t(i), 1\\leq i \\leq n_t$ as defined in Eq.(\\ref{eq:similarity}).\n\\begin{equation} \\label{eq:similarity}\n\\begin{array}{ll}\n&\\lambda_t^i(z_t)=\\frac{\\eta_t^i(z_t)}{\\sum_{j}\\eta^{j}(z_{t})}  \\\\\n& \\textrm{where}\\,\\,\\,  \\eta_t^i(z_t)=exp\\left(-\\frac{(z_t-s_t^i)^T(z_t-s_t^i)}{var(s_t^i)}\\right)\n\\end{array}\n\\end{equation}\n\\vspace{-0.45cm}\n\\subsubsection{- Online State-Transitions Identification}\nThe identification of state-transitions w.r.t. actions is important to predict future states precisely. For example, if an AV controller knows in advance what the future state would be based on the chosen action, it can get more chances to make a better decision. In the e-FSM, the state-transitions are represented by transition probability matrices (TPMs), of which each matrix is correlated to individual action in the action-set.\n\nThe action-set written by $\\mathcal{A}_d=\\{a(1), ..., a(q)\\}$ is initially defined to have the fixed $q$ number of elements unlike the state-set $\\mathcal{S}_t$ which is a variant set. The e-FSM uses discrete action-set but is compatible to controllers that have continuous action-set ($\\mathcal{A}_c$) by encoding to discrete action-set ($\\mathcal{A}_d$) with an arbitrary interval $\\delta$. For example, the continuous action-set $\\mathcal{A}_c=[-2.0\\,\\, 2.0]$ can be encoded with the interval $\\delta=1.0$ to the discrete action-set such as $\\mathcal{A}_d=\\left\\{a(1), a(2), a(3), a(4)\\right\\}$ where $a(1)=[-2.0\\,\\,-1.0)$, $a(2)=[-1.0\\,\\, 0.0)$, $a(3)=[0.0\\,\\, 1.0)$, $a(4)=[1.0\\,\\, 2.0]$.  \n\nThe state-transitions w.r.t. actions are represented by the $q$ number of TPMs. And, each TPM $\\mathbf{P}_t^{a(r)}\\forall r\\in R^{{n_t}\\times{n_t}}$ is correlated to each discrete action $a(r)\\in \\mathcal{A}_d$, where $1\\leq r \\leq q$ and $q$ is the total number of discrete actions in $\\mathcal{A}_d$. The TPMs are defined by:\n\\begin{equation}\\hspace{-0.3cm}\n\\begin{array}{ll}\n    &\\mathbf{P}_t^{a(r)}=\\{P_t^{a(r)}(i,j)\\}, 1\\leq i,j \\leq n_t\\\\\n    &\\textrm{where}\\,\\,\\,P_t^{a(r)}(i,j)=Prob\\left(s_{t+1}=s(j)|s_t=s(i), a_t=a(r)\\right)\n\\end{array}\n\\end{equation}\nWhen a new state is determined, the dimension of all TPMs is expanded by adding a column and a row, following the proposed method; see detail steps and an example given by \\cite{han2019online}. Otherwise one of TPMs which is correlated to a chosen action is identified via Eq.(\\ref{eq:transition_matrix_efsm}) and (\\ref{eq:transition_matrix_efsm_where}) in a recursive way, designed by \\cite{filev2010markov}. Note $\\tau(t)$ and $\\gamma(t)$ are probability distributions over states at time $t-1$ and $t$, $Prob(S_{t-1})$ and $Prob(S_t)$; $\\varphi$ and $1_{n_t}$ are a learning rate and the $n_t$-dimensional vector of ones respectively. In the beginning, $F^{a(r)}(0)$ and $F_o^{a(r)}(0)$ are initialized by $\\bar{\\epsilon}$ which is a small non-negative constant. \n\\begin{equation} \\label{eq:transition_matrix_efsm}\n        \\mathbf{P}_t^{a(r)}=diag(F_0^{a(r)}(t))^{-1}F^{a(r)}(t)\\\\\n\\end{equation}\n\\begin{equation} \\label{eq:transition_matrix_efsm_where}\n\\begin{array}{ll}\n        &F^{a(r)}(t)=F^{a(r)}(t-1)+\\varphi(\\tau(t)\\gamma(t)^T-F^{a(r)}(t-1))\\\\\n        &F_o^{a(r)}(t)=F_o^{a(r)}(t-1)+\\varphi(\\tau(t)\\gamma(t)^T1_{n_t}-F_o^{a(r)}(t-1))\n\\end{array}\n\\end{equation}\nBy using identified TPMs, the future states can be predictable by calculating probability distributions over states at time $t+1$. Given probability distributions over states at $t$ and an action selected by the controller at $t$, probability distributions over state at $t+1$ or $t+k$ can be obtained by Eq.(\\ref{eq:predict_state}) and (\\ref{eq:predict_state_more}). As shown in Eq.(\\ref{eq:predict_state_more_sub}), $\\mathbf{P_t^*}$ is a marginal TPM, where $Prob(a_t=a(r))$ is set by uniform distribution such as $1\/q\\,\\,\\forall r$.\n\\begin{equation} \\label{eq:predict_state}\n        Prob_{pred}(S_{t+1})=\\mathbf{P}_t^{a(r)}\\cdot Prob(S_t)\n\\end{equation}\n\\begin{equation} \\label{eq:predict_state_more}\n        Prob_{pred}(S_{t+k})=\\mathbf{P}_t^{*}\\cdot Prob(S_{t+1})\n\\end{equation}\n\\begin{equation} \\label{eq:predict_state_more_sub}\n\\begin{array}{ll}\n        &\\mathbf{P}_t^{*}=\\{P_t^{*}(i.j)\\}, 1\\leq i,j \\leq n_t, \\textrm{where}\\\\\n        &P_t^{*}(i,j)=\\sum_{r=1}^{q}Prob\\left(s_{t+1}=s(j)|s_t=s(i), a_t=a(r)\\right)\\\\ \n        & \\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\; \\cdot \\;Prob(a_t=a(r))\n\\end{array}\n\\end{equation}\n\\vspace{-0.5cm}\n\\vspace{-0.2cm}\n\\subsection{Action-Reviser Module} \\label{subsec:action-reviser}\n\\vspace{-0.2cm}\n\nThis module inspects and revises an action $a_t$ chosen by a controller if required. The revision of chosen action is decided based on predicted probabilities of unfavorable states. Given $\\mathcal{L}(S_t)$ and $a_t$, if predicted probabilities of the unfavorable states are higher than a threshold $\\varrho_t$, $a_t$ will be revised. Otherwise, $a_t$ is applied.\n\nThe predicted probability distributions of states at $t+1$, $Prob_{pred}(S_{t+1})$, can be obtained by using Eq. (\\ref{eq:predict_state}), where $\\mathcal{L}(S_t)$, $Prob(S_t)$ and $P_t^{a(\\cdot)}$ are provided from the e-FSM module, and the action $a(r)$ is chosen by a controller. The variant threshold $\\varrho_t$ is set such that if a variance of $Prob_{pred}(S_{t+1})$ is larger, then $\\varrho_t$ is smaller and vice versa as shown in Alg. \\ref{alg:variantThreshold}.\n\n\\begin{algorithm}[h]\n    \\caption{$getVariantThreshold\\,\\,(Prob_{pred}(S_{t+1}))$}\n    \\label{alg:variantThreshold}\n    \\begin{algorithmic}[1]\n    \\Procedure{ $\\varrho_t=f_{threshold}$}{$Prob_{pred}(S_{t+1})$\n        \\State $X \\gets Prob_{pred}(S_{t+1})$\n        \\State $X \\gets$ $Sort$ $X$ by $descending$ $order$\n        \\State $E[X]=\\sum_{j=1}^{n_t}j\\cdot X(j)$\n        \\State $\\varrho_t \\gets X\\left(floor\\left(E[X]\\right)\\right)$\n    \\EndProcedure\n\\end{algorithmic}\n\\end{algorithm}\n\n\nFor revising the chosen action, the action-reviser inspects other possible actions. For example, assuming that a given criterion for AV control is safety in a car-following scenario, an action $a_t$ is chosen by $1.0$, and the action-set (longitudinal acceleration) is defined by $\\{-2.0,-1.0,0.0,1.0,2.0\\} [m\/s^2]$, the action-reviser inspects $a_t$ via calculating $Prob_{pred}(S_{t+1})$ and $\\varrho_t$. If the predicted probability of the unfavorable states (e.g. collision states) are larger than $\\varrho_t$, $a_t$ will be considered as an inappropriate action and replaced by one of other actions in the action-set which is slower than 1.0.\n\n\\section{A Target Scenario and the DDPG Controller Settings}\n\\label{section:Target Scenario and Controller}\n\\vspace{-0.2cm}\nThe DDPG based controller is applied within the on-line evolving framework. In this section, a target car-following scenario and the DDPG based controller settings are discussed. The control performances w\/ and w\/o the on-line evolving framework under the scenario will be compared in Section \\ref{section:experimental results}.\n\n\\vspace{-0.2cm}\n\\subsection{Target Car-Following Scenario}\n\\vspace{-0.2cm}\nA target scenario is car-following on a single lane road as shown in Fig. \\ref{fig:car_following_scenario}. The ego vehicle ($veh_{ego}$) and lead vehicle ($veh_{lead}$) are initialized by randomly assigning speeds and positions as far as the initial distance between two vehicles ($headway_{dist}$) is less than $100[m]$. The ranges of acceleration and velocity for the both vehicles are set by $[-2.0\\,\\,2.0] [m\/s^2]$ and $[0.0\\,\\,\\,32.0] [m\/s]$ respectively. The ego vehicle's acceleration is chosen by the DDPG controller at every simulation time-step $t$=$0.25secs$, while the lead vehicle's acceleration is assigned by randomly selecting 200 secs (800 steps) of velocity profile from the 11 hours of realistic driving profile.\nThe vehicle model in the simulation is a point-mass model defined as Eq.(\\ref{eq:point_mass_vehicle_model}) where $x_t(veh_i), \\dot{x}_t(veh_i),$ and $\\ddot{x}_t(veh_i)$ are position, velocity, and acceleration of $veh_i$ at time $t$.\n\\vspace{0.1cm}\n\\begin{equation} \n    \\label{eq:point_mass_vehicle_model}\n    \\begin{array}{ll}\n            &x_{t+1}(veh_i)=x_t(veh_i)+\\dot{x_t}(veh_i)\\cdot \\Delta t \\\\\n            &\\dot{x}_{t+1}(veh_i)=\\dot{x}_t(veh_i)+\\ddot{x}_t(veh_i)\\cdot \\Delta t\n    \\end{array}\n\\end{equation}\n\nThe goal of the scenario is controlling the ego vehicle to follow the lead vehicle as $fast$ and $safe$ as possible. The episode is terminated in the following three cases: simulation step reaches the maximum step $800$, the $headway_{dist}$ is longer than $200[m]$, and the ego vehicle collides to the lead vehicle.\n\n\\begin{figure}[ht]\n\\begin{center}\n\\includegraphics[width=7.5cm]{figures\/car_following_scenario.png}   \n\\vspace{-0.3cm}\n\\caption{A target car-following scenario} \n\\label{fig:car_following_scenario}\n\\end{center}\n\\end{figure}\n\\vspace{-0.15cm}\n\\subsection{DDPG Controller}\n\\vspace{-0.2cm}\nThe DDPG is implemented as the ego vehicle's controller. It is one of the RL with MLP controllers, of which policy is represented by multilayered neural network. The policy is trained in a way to get the maximum value at every state given a pre-defined reward-function. As proposed by \\cite{van2016deep}, the fixed size $\\omega$ of histories are stored and implemented to train a policy of the RL with MLP controller; the stored histories can be represented by $\\mathcal{H}_t=\\{h_{t-\\omega+1}, ..., h_t\\}$ where $h_t=[\\bar{z}_t, a_t, \\bar{z}_{t+1}, \\eta_t]$; $\\bar{z}_t, \\bar{z}_{t+1}, a_t,$ and $\\eta_t$ are current and next states, an action selected by the controller, and a reward. Especially, the DDPG controller consists of the actor $\\mu(\\bar{z}|\\theta^{\\mu})$ and critic  $Q(\\bar{z},a|\\theta^Q)$ networks and implements ``soft-update\" instead of directly copying weights of the networks in updating target-networks; see descriptions of \\cite{lillicrap2015continuous} for details about the DDPG controller.\n\nIn a similar way in \\cite{nageshrao2019interpretable}, the DDPG controller is configured for the given car-following scenario as the following: the ego vehicle's acceleration is set by $accel_t(veh_{ego})=\\mu(\\bar{z}_t|\\theta^{\\mu})+\\mathcal{N}$, where $\\bar{z}_t$ is a set of observations defined as Eq.(\\ref{eq:state of ddpg1}), and $\\mathcal{N}$ is the temporally correlated exploration noise generated by using the Ornstein-Uhlenbeck noise process proposed by \\cite{uhlenbeck1930theory}. Also, the Adam method is applied for training the networks, as suggested by \\cite{kingma2014adam}. Learning rates for the actor and critic networks are set by $10^{-4}$ and $10^{-3}$ respectively, and a discount factor is set by 0.95. \n\n\\vspace{-0.2cm}\n\\begin{equation} \\label{eq:state of ddpg1}\n\\begin{array}{ll}\n        \\bar{z}_t=[v_t(veh_{ego}), headway_{dist}, v_t(veh_{lead}), \\\\ \\;\\;\\;\\;\\;\\;\\;\\;\\;accel_{t-1}(veh_{ego})]^T\n\\end{array}\n\\end{equation}\n\nSince the given control criteria in the scenario is $speed$ and $safety$, \\cite{nageshrao2019interpretable} designs a reward function to minimize velocity difference and distance between two vehicles, and variation of ego vehicle's acceleration as shown in Eq.(\\ref{eq:ddpg_reward_function1}-\\ref{eq:ddpg_reward_function3}), where $headway_{const}$ and $d_{safe}$ are a constant headway-time and a minimum safe distance. The reward $\\eta_t$ is defined by sum of $\\eta_t^{vel}, \\eta_t^{dist},$ and $\\eta_t^{accel}$. \n\\begin{equation} \\label{eq:ddpg_reward_function1}\n       \\eta_t^{vel}=e^{-\\frac{(v_t(veh_{ego})-v_t(veh_{lead}))^2}{v_{max}}}-1\n\\end{equation}\n\\begin{equation}\\label{eq:ddpg_reward_function2}\n\\eta_t^{dist}=e^{-\\frac{(headway_{dist}-headway_{const}\\cdot d_{safe})^2}{2\\cdot headway_{const}\\cdot d_{safe}}}-1\\\\\n\\end{equation}\n\\begin{equation}\\label{eq:ddpg_reward_function3}\n       \\eta_t^{accel}=e^{-\\frac{(\\Delta accel(veh_{ego})^2}{2\\cdot accel_{max}}}-1\n\\end{equation}\n\n\n\\section{The On-line Evolving Framework Settings with DDPG controller}\n\\label{section:DDPG_controller_eFSM}\nThe on-line evolving framework with the RL with MLP controller can be represented by Fig. \\ref{fig:RL_eFSM_framework}. In this section, the configurations of e-FSM and action-reviser modules are discussed for the given car-following scenario.\n\\begin{figure}[ht]\n\\begin{center}\n\\includegraphics[width=7.5cm]{figures\/RL_eFSM_framework.png}   \n\\vspace{-0.2cm}\n\\caption{An on-line evolving framework with DDPG} \n\\label{fig:RL_eFSM_framework}\n\\end{center}\n\\end{figure}\n\\vspace{-0.35cm}\n\\subsection{e-FSM Configurations}\n\\vspace{-0.2cm}\nAs we discussed, the AV control criteria in the given scenario are $safety$ and $speed$. In simulating the scenario, the DDPG controller chooses longitudinal accelerations via its policy at every time $t$, and the e-FSM determines states and identifies state-transitions independently. Among the states determined by the e-FSM, some will be flagged as the unfavorable states, which represent either unsafe (collision) or slow-speed driving situation ($headwawy_{dist}$ is longer than $200[m]$). At every time-step, $s_t(i)$, where $\\{s_t(i)\\in S_t| i=argmax_i(Prob\\left(s_t(i)\\right)\\}$, is flagged by:\n\\vspace{0.1cm}\n\\[\\mathcal{L}(s_t(i)) = \\left\\{\n  \\begin{array}{ll}\n    &l_n (none): neither\\,\\,crash\\,\\,nor\\,\\,large\\textrm{-}distance\\\\\n    &l_s\\,\\,(unfavorable_{safety}): crash\\\\\n    &l_{ld} (unfavorable_{speed})\\,\\,\\,: large\\textrm{-}distance\n  \\end{array}\n\\right.\n\\]\n\nAs shown in Eq.(\\ref{eq:state_eFSM}), three observations are selected to represent states in the e-FSM. The coefficients of eTS within the e-FSM module, $\\rho$ and $\\epsilon$, are set by 0.7 and 0.3 respectively. Since the DDPG controller uses the continuous action-space, the continuous action-set is encoded to the discrete action-set by using an interval $\\delta=0.2 [m\/s^2]$. The continuous action set $A_c=[-2.0\\,\\,2.0][m\/s^2]$ is given for the both vehicles in the scenario, and encoded to the $q=20$ number of discrete actions.\n\\vspace{0.1cm}\n\\begin{equation} \\label{eq:state_eFSM}\n    z_t=\\{v_t(veh_{ego}), headway_{dist}, v_t(veh_{lead})\\}\n\\end{equation}\n\n\\subsection{Action-Reviser Configurations}\n\\vspace{-0.2cm}\nIf it is predicted that the selected action $a_t$ causes a transition to one of the unfavorable states, the action-reviser explores and selects another action as discussed in section \\ref{subsec:action-reviser}. \n\nThe DDPG controller chooses an action $a_t \\in A_c$ at every time-step, and $a_t$ is encoded to a discrete action $a(r) \\in A_d$. Then, future probability distributions over states are calculated via Eq.(\\ref{eq:predict_state}) to inspect $a(r)$. In the predicted probability distributions, if any of the unfavorable states ($l_s$ or $l_{ld}$) have a higher probability than the dynamic threshold $\\varrho_t$, the action-reviser revises $a(r)$. Otherwise, $a_t$ is returned and applied. The action-reviser continues exploring a discrete action space $(A_d)$ until an appropriate action is found. Specific steps are shown in Alg. \\ref{alg:actionreviser}.\n\n\\begin{algorithm}[b]\n    \\caption{$actionsReviser$}\n    \\label{alg:actionreviser}\n    \\begin{algorithmic}[1]\n    \\Procedure{$\\hat{a}_t=f_{AR}$}{$Prob(S_{t}), a_t, A_d, \\textbf{P}_t^{a(\\cdot)}, \\mathcal{L}(S_t)$\n        \\State{$r \\gets encode(a_t, A_d)$}\n        \\State $Prob_{pred}(S_{t+1}) \\gets \\mathbf{P}_t^{a(r)}\\cdot Prob(S_t)$\n        \\State {$I \\gets f_{IA}\\left(Prob_{pred}(S_{t+1}), \\mathcal{L}(S_t)\\right)$} \\algorithmiccomment{Alg. 3}\n        \\If{$I=0$}  \\algorithmiccomment{$l_s$ is expected}\n            \\While{$I=0 \\And r>0$}\n                \\State{$r \\gets r-1$}\n                \\State{$Prob_{pred}(S_{t+1}) \\gets \\mathbf{P}_t^{a(r)}\\cdot Prob(S_t)$}\n                \\State{$I \\gets f_{IA}\\left(Prob_{pred}(S_{t+1}), \\mathcal{L}(S_t)\\right)$}\n            \\EndWhile\n            \\State{$\\bar{a}_t \\gets decode(a(r),A_d)+\\hat{\\mathcal{N}}(0,V)$}\n            \\State{\\Return $\\hat{a}_t \\gets min(max(\\bar{a}_t,a_{min}),a_{max})$}\n        \\ElsIf{$I=1$}   \\algorithmiccomment{$l_{ld}$ is expected}\n            \\While{$I=1 \\And r<q$}\n                \\State{$r \\gets r+1$}\n                \\State{$Prob_{pred}(S_{t+1}) \\gets \\mathbf{P}_t^{a(r)}\\cdot Prob(S_t)$}\n                \\State{$I \\gets f_{IA}\\left(Prob_{pred}(S_{t+1}), \\mathcal{L}(S_t)\\right)$}\n            \\EndWhile\n            \\State{$\\bar{a}_t \\gets decode(a(r),A_d)+\\hat{\\mathcal{N}}(0,V)$}\n            \\State{\\Return $\\hat{a}_t \\gets min(max(\\bar{a}_t,a_{min}),a_{max})$}\n        \\Else   \\algorithmiccomment{$l_{n}$ is expected}\n            \\State{\\Return $\\hat{a}_t \\gets a_t$}\n        \\EndIf\n    \\EndProcedure\n\\end{algorithmic}\n\\end{algorithm}\n\n\\begin{algorithm}\n    \\caption{$inspectAction$}\n    \\label{alg:inspectAct}\n    \\begin{algorithmic}[1]\n    \\Procedure{$I=f_{IA}$}{$Prob_{pred}(S_{t+1}), \\mathcal{L}(S_t)$\n        \\State $\\varrho_t \\gets f_{threshold}\\left(Prob_{pred}(S_{t+1})\\right)$ \\algorithmiccomment{Alg. 1}\n        \\For{$i=1,2, ..., n_t$}\n            \\If{$Prob_{pred}(s_{t+1}(i))\\geq\\varrho$}\n                \\If{$\\mathcal{L}(s_{t+1}(i))=l_s$}\n                    \\State{\\Return $I\\gets0$}\n                \\ElsIf{$\\mathcal{L}(s_{t+1}(i))=l_{ld}$}\n                    \\State{\\Return $I\\gets1$}\n                \\Else\n                    \\State{\\Return $I\\gets2$}\n                \\EndIf\n            \\EndIf\n        \\EndFor\n    \\EndProcedure\n\\end{algorithmic}\n\\end{algorithm}\n\nAfter action-revising process, $a(r)$ is decoded to a continuous action $\\hat{a}_t$ by calculating a mean of corresponded range since each discrete action represents an individual partial range of the continuous action-set. For example, if the selected discrete action is $a(r)=[0.2\\,\\, 0.4)$, $0.3$ is returned as a revised action $\\hat{a}_t$. In addition, the Gaussian noise with a decreasing variance $\\hat{\\mathcal{N}}(0,V)$ is defined as shown in Eq.(\\ref{eq:revised_action}) and implemented to explore the action space for identifying the e-FSM's TPMs. \n\\vspace{0.1cm}\n\\begin{equation} \\label{eq:revised_action}\n    \\begin{array}{ll}\n    \\textrm{where}\\,\\,\\,\\,V=\\frac{a_{max}}{max\\left(1, (K*Iter_{episode})\\right)}\n    \\end{array}\n\\end{equation}\n\nInitially, $V$ is set by the maximum acceleration of the vehicle, but it is decreased over the number of the episode iterations. Note $K, a_{max}, a_{min},$ and $Iter_{episode}$ are an invariant constant, the maximum\/minimum accelerations of the vehicle, and the number of episode iterations.\nSince the e-FSM does not have any states in the beginning, appropriate inspection and revision of the chosen action are not possible. Thus, the action-reviser module is activated after the $\\xi$ number of episode iterations; $K$ and $\\xi$ are set by 0.001 and 50 in this study.\n\n\\vspace{-0.1cm}\n\\section{Experimental Results}\n\\label{section:experimental results}\n\\vspace{-0.2cm}\nThe car-following control performances of the DDPG w\/ and w\/o the on-line evolving framework are compared in terms of $safety$ and $speed$ criteria. In the beginning of every first episode, the DDPG and e-FSM modules are initialized such that weights of critic and actor networks in the DDPG are randomly assigned, the state-set of e-FSM is set by empty-set $S_0=\\{\\}$, and $F_o^{a(r)}(0)$ and $F^{a(r)}(0)$ are set by $\\bar{\\epsilon}$ for all $r$. While running the scenario repeatedly, the critic and actor networks are updated, and the e-FSM derives a stochastic model by determining states and identifying state-transitions over time.\n\nSince every episode is terminated when the controller fails to avoid collisions or maintain $headway_{dist}$ less than $200[m]$, the controller's performance can be analyzed by comparing the total number of simulated steps in each episode, where the maximum step is 800. The Fig. \\ref{fig:car_following_result_steps} plots the evolution of the car-following performances for DDPG w\/ and w\/o the on-line evolving framework averaged over 10 different runs; 1500 episodes are iterated in each run. The different number of succeeded and failed episodes (in total 15000 episodes) are observed depending on the implementation of the proposed framework as below table.  \n\\begin{table}[h]\n    \\centering\n    \\begin{tabular}{||c | c | c | c||}\n        \\hline\n        Type & Success & Large-distance & Collision \\\\ [0.5ex] \n        \\hline\\hline\n        DDPG & 12249 (81.6\\%) & 1251 & 1500 \\\\ \n        \\hline\n        Our Approach & 14719 (98.1\\%) & 76 & 205 \\\\\n        \\hline\n    \\end{tabular}\n\\end{table}\n\\vspace{-0.3cm}\n\\begin{figure}[h]\n    \\begin{center}\n        \\includegraphics[width=9.0cm]{figures\/ifac_both_intv_new_eval.pdf}   \n        \\vspace{-0.8cm}\n        \\caption{The control performance evolution in car-following by DDPG w\/o (left) and w\/ on-line evolving framework (center). The averaged number of interventions for revising actions chosen by DDPG (right).} \n    \\label{fig:car_following_result_steps}\n    \\end{center}\n\\end{figure}\n\nAs shown in Fig. \\ref{fig:car_following_result_steps}, the control performances of the both approaches are enhanced over iterations. But, the DDPG w\/ the on-line evolving framework (proposed approach) achieves the given control-goal (driving safe and fast) with fewer iterations. Also, our approach does not fail after 68 iterations, but the DDPG w\/o the framework does. it is observed the number of interventions for revising actions is decreasing as DDPG policy is more optimized; there are no interventions during the first 50 episodes because the action-reviser is inactivated. These results imply that the DDPG controller's wrong decision-making is detected and revised appropriately within our proposed framework. Besides, it is noticed that a stochastic model is properly created by the e-FSM module, enabling precise prediction of future states based on the choice of actions.\n\nSince the available maximum speed of ego vehicle depends on the lead vehicle's speed in the car-following, the best way to satisfy the speed criterion is driving as fast as the lead vehicle. Thus, the speed difference between ego and lead vehicles among succeeded episodes is quantified. The overall mean and variance of the velocity difference are 2.0460 and 1.7800 for the DDPG with the on-line evolving framework, while 1.9497 and 1.8057 for the DDPG only. \n\nAlthough the DDPG w\/ the on-line evolving framework controls the ego vehicle slightly slower than the DDPG only, it achieves both criteria, safety and speed, more successfully by trading off priority of the two criteria. \n\n\\vspace{-0.2cm}\n\\section{Discussion and Conclusion}\n\\label{section:conclusion}\n\\vspace{-0.2cm}\nIn this work, the on-line evolving framework is proposed to advance the decision-making capability of controllers. Since the e-FSM module can create a stochastic model by evolving the model's structure and identifying state-transition dynamics through experiences, it becomes possible to predict future states based on the choice of actions. The action-reviser module checks if an action chosen by the controller causes unfavorable situations in the future. And, if one of the unfavorable situations is highly expected, it explores the given action-space to choose another action. \n\nIn the given car-following scenario, DDPG w\/ and w\/o the on-line evolving framework are implemented to control the ego vehicle as $fast$ and $safe$ as possible. As shown in the experimental results, the crashes and large-distances does not occur with our proposed framework after few episode iterations whereas the control failures are continuously occurred w\/o the framework. It shows that the e-FSM generates a stochastic model precisely which enables to detect incorrect decisions effectively and revise them appropriately within the on-line evolving framework. \n\\vspace{-0.3cm}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Appendix}\n\nIn this Appendix we present the basic defect configurations of spin-1\nBose-Einstein condensate (BEC) corresponding to the different broken\nsymmetries of the ferromagnetic (FM) and polar phases. \nUsing these solutions we construct prototype spinor wave functions for\nthe vortex connections across the interface between the polar-FM\nmanifolds (such as those shown \nin Fig. 1 of the main text). The constructed prototype defect\nconfigurations are used as initial states for the numerical studies of\nthe defect stability in the main text. \n\n\\subsection*{Broken symmetries and vortices in the spin-1\n  Bose-Einstein Condensate}\n\nThe wave function of a spin-1 BEC is written in terms of the density\n$n(\\bvec{r}) = |\\Psi(\\bvec{r})|^2$ and a normalized three-component spinor in\nthe basis of spin projection onto the $z$ axis\n\\begin{equation}\n  \\label{eq:spinor_suppl}\n  \\Psi({\\bf r}) = \\sqrt{n({\\bf r})}\\zeta({\\bf r})\n  = \\sqrt{n({\\bf r})}\\threevec{\\zeta_+({\\bf r})}\n                              {\\zeta_0({\\bf r})}\n                              {\\zeta_{-}({\\bf r})}.\n\\end{equation}\nThe Hamiltonian density of a BEC of spin-1 atoms\nis~\\cite{pethick-smith}\n\\begin{equation}\n  \\label{eq:hamiltonian-density-app}\n  \\begin{split}\n    {\\cal H} &=  \\frac{\\hbar^2}{2m}\\abs{\\nabla\\Psi}^2 + V(\\bvec{r})n\n    + \\frac{c_0}{2}n^2\n    + \\frac{c_2}{2}n^2\\abs{\\bvec{\\eva{\\hat{F}}}}^2\\\\\n    &+ g_1n\\eva{\\bvec{B}\\cdot\\bvec{\\hat{F}}}\n    + g_2n\\eva{\\left(\\bvec{B}\\cdot\\bvec{\\hat{F}}\\right)^2},\n  \\end{split}\n\\end{equation}\nwhere $V(\\bvec{r})$ denotes the trapping potential, $m$ is the atomic\nmass, $\\bvec{\\hat{F}}$\nis the spin operator, given by\na vector of spin-1 Pauli matrices, and\n$\\bvec{\\inleva{\\hat{F}}}=\\zeta_\\alpha^\\dagger\\bvec{\\hat{F}}_{\\alpha\\beta}\\zeta_\\beta$.\nIf a weak, external magnetic\nfield is present, this will lead to linear and quadratic Zeeman shifts\ndescribed by the last two terms.\n\nThere are two\ninteraction terms with strengths\n$c_0=4\\pi\\hbar^2(2a_2+a_0)\/3m$ and $c_2=4\\pi\\hbar^2(a_2-a_0)\/3m$,\nwhere $a_{\\!f}$ is the s-wave scattering length in the scattering channel\nwith relative angular momentum $f$.  In $^{23}$Na,\n$c_0\/c_2\\simeq 31$~\\cite{crubellier_epjd_1999} and in $^{87}$Rb \n$c_0\/c_2\\simeq -216$~\\cite{van-kempen_prl_2002}.  The two\ncorresponding healing lengths \nin the spin-1 BEC, $\\xi_n=(8\\pi c_0 n)^{-1\/2}$ and\n$\\xi_F=(8\\pi |c_2| n)^{-1\/2}$,  define the length scales over\nwhich $n$ and $|\\langle\\bvec{\\hat{F}}\\rangle|$ heal when locally\nperturbed.\n\nThe spin-dependent interaction in spin-1 BEC gives rise to two phases, the FM and polar~\\cite{pethick-smith,ho_prl_1998,ohmi_jpsj_1998,leonhardt_jetplett_2000,zhou_ijmpb_2003,ruostekoski_monopole_2003,savage_dirac_2003}.\nBoth phases exhibit distinct sets of possible defects.\nWhen a defect core is formed by a depletion of\nthe density, its size is determined by the density healing length\n$\\xi_n$.  A defect core in the polar (FM) phase may instead be filled\nwith atoms in the FM (polar) phase. The size of the core is then\ndetermined by the spin healing length $\\xi_F$~\\cite{ruostekoski_monopole_2003}.\nIn the following we present the basic defect structures in the FM and polar\nphases of a spin-1 BEC that are used in constructing the defect solutions\ncrossing the coherent topological interface between the FM and polar phases.\n\n{\\em FM phase:}\nIf $c_2<0$, the spin-dependent interaction\nwill favor a state that maximizes the magnitude of the spin, i.e.,\n$\\absF=1$.  The general spinor may then be written\nas spin-rotations\nof $\\zeta = (1,0,0)^T$, together with a condensate phase $\\phi$\n\\begin{equation}\n  \\label{eq:ferro}\n  \\zeta^{\\rm f} =\n  e^{i\\phi}U(\\alpha,\\beta,\\gamma)\\threevec{1}{0}{0}\n  = \\frac{e^{-i\\gamma^\\prime}}{\\sqrt{2}}\n    \\threevec{\\sqrt{2}e^{-i\\alpha}\\cos^2\\frac{\\beta}{2}}\n             {\\sin\\beta}\n             {\\sqrt{2}e^{i\\alpha}\\sin^2\\frac{\\beta}{2}},\n\\end{equation}\nwhere $(\\alpha,\\beta,\\gamma)$ are spin-space Euler angles, and\n\\mbox{$\\gamma^\\prime=\\gamma-\\phi$} absorbs the condensate phase. The\nbroken symmetry of the ground-state manifold therefore corresponds to the\ngroup of three-dimensional rotations \\SO(3). The spin vector is given\nby the Euler angles as\n\\mbox{$\\inleva{\\bvec{\\hat{F}}}=(\\cos\\alpha\\sin\\beta,\\sin\\alpha\\sin\\beta,\\cos\\beta)$.}\n\nTopological stability of line defects can be characterized by studying\nclosed contours around the defect line and the mapping of these contours to the\norder parameter space~\\cite{mermin_rmp_1979}. If the image of a closed\nloop encircling a line defect\nin the order parameter space can be contracted to a point, the defect\nis not topologically\nstable.  $\\SO(3)$ can be represented geometrically as $S^3$ (the unit\nsphere in four dimensions) with diametrically opposite points\nidentified. The only closed contours that cannot be contracted\nto a point are then the ones connecting these diametrically opposite\npoints, and we have only two distinct classes of vortices: singly\nquantized, singular vortices that correspond to noncontractible loops\nand nonsingular vortices representing contractible loops. All other\nvortices can be transformed to either one of these by local\ndeformations of the order parameter. Examples of this first class are\na singular line defect constructed as a winding of the condensate\nphase,\n\\begin{equation}\n  \\label{eq:fmsingular}\n  \\zeta^{\\rm s} =\n  \\frac{e^{i\\varphi}}{\\sqrt{2}}\\threevec{\\sqrt{2}\\cos^2\\frac{\\beta}{2}}\n                                       {\\sin\\beta}\n                                       {\\sqrt{2}\\sin^2\\frac{\\beta}{2}},\n\\end{equation}\nwhere $\\varphi$ is the azimuthal angle, and the singular spin\nvortex constructed as a $2\\pi$ rotation of the spin\nvector~\\cite{ho_prl_1998,ohmi_jpsj_1998} [Fig.~\\ref{fig:fmVortices}(a) and (b)]\n\\begin{equation}\n  \\label{eq:spinvortex}\n  \\zeta^{\\rm sv} =\n  \\frac{1}{\\sqrt{2}}\\threevec{\\sqrt{2}e^{-i\\varphi}\\cos^2\\frac{\\beta}{2}}\n                             {\\sin\\beta}\n                             {\\sqrt{2}e^{i\\varphi}\\sin^2\\frac{\\beta}{2}}.\n\\end{equation}\nA nonsingular, coreless\nvortex is constructed as a combined rotation of the spin vector and\nthe condensate phase [Fig.~\\ref{fig:fmVortices}(c) and (d)]\n\\begin{equation}\n  \\label{eq:cl}\n  \\zeta^{\\rm cl} =\n  \\frac{1}{\\sqrt{2}}\\threevec{\\sqrt{2}\\cos^2\\frac{\\beta}{2}}\n                             {e^{i\\varphi}\\sin\\beta}\n                             {\\sqrt{2}e^{2i\\varphi}\\sin^2\\frac{\\beta}{2}}.\n\\end{equation}\n\\begin{figure}[tb]\n  \\centering\n  \\includegraphics[scale=1]{fmVorticesFig.eps}\n\\caption{Nontrivial vortices in the FM phase.\n  (a) The singular spin vortex is formed as a radial disgyration of\n  the spin vector (red cones) around the singular core.\n  (b) Singular spin vortex with cross-disgyration of the spin vector.\n  (c) The coreless vortex is\n  formed as a combined disgyration of the spin vector and a winding of\n  the condensate phase, corresponding to spin rotations about the\n  local spin vector (indicated by the orthogonal green and blue\n  vectors). The core is\n  nonsingular and filled by the vortex-free spinor component.\n  The direction of the spin vector at the edge depends on\n  boundary conditions.\n  (d) A different nonsingular vortex configuration.\n}\n\\label{fig:fmVortices}\n\\end{figure}\n\n\n$\\SO(3)$ does not strictly speaking support point defects.\nHowever, it is possible to form a spinor with a\nmonopole structure of the spin vector (a radial hedgehog) as the\ntermination of a doubly quantized vortex~\\cite{savage_dirac_2003}\n[Fig.~\\ref{fig:dirac}]\nThis is the analogue of the Dirac monopole in quantum field theory.\n\\begin{equation}\n  \\label{eq:dirac}\n  \\zeta^{\\rm D} = \\frac{1}{\\sqrt{2}}\n                  \\threevec{\\sqrt{2}e^{-2i\\varphi}\\cos^2\\frac{\\theta}{2}}\n                   {e^{-i\\varphi}\\sin\\theta}\n                   {\\sqrt{2}\\sin^2\\frac{\\theta}{2}}.\n\\end{equation}\n\\begin{figure}[tb]\n  \\centering\n  \\includegraphics[scale=1]{diracMonopoleSV.eps}\n\\caption{Dirac monopole; the analogue of the Dirac monopole is\n  formed as the termination of a doubly quantized, singular vortex\n  line (blue) resulting in a hedgehog configuration of the spin vector.\n}\n\\label{fig:dirac}\n\\end{figure}\n\n{\\em Polar phase:} $c_2>0$ favors a state with $\\absF=0$. The general\nspinor is given by spin rotations of $\\zeta=(0,1,0)^T$ together with\na condensate phase\n\\begin{equation}\n  \\label{eq:polar}\n  \\zeta^{\\rm p} =\n  e^{i\\phi}U(\\alpha,\\beta,\\gamma)\\threevec{0}{1}{0}\n  = \\frac{e^{i\\phi}}{\\sqrt{2}}\\threevec{-e^{-i\\alpha}\\sin\\beta}\n                     {\\sqrt{2}\\cos\\beta}\n                     {e^{i\\alpha}\\sin\\beta}.\n\\end{equation}\nThe unit vector\n$\\nematic = (\\cos\\alpha\\sin\\beta,\\sin\\alpha\\sin\\beta,\\cos\\beta)$\ndefines the local direction of macroscopic condensate spin\nquantization.  We may then write the spinor as\n\\begin{equation}\n  \\zeta^{\\rm p} = \\frac{e^{i\\phi}}{\\sqrt{2}}\n                 \\threevec{-d_x+id_y}{\\sqrt{2}d_z}{d_x+id_y}.\n\\end{equation}\nNote, however, that\n$\\zeta^{\\rm p}(\\phi,\\nematic) = \\zeta^{\\rm p}(\\phi+\\pi,-\\nematic)$.\nThese two states must be identified in order to avoid double counting.\nThe order parameter space, which might appear to be $\\U(1)\\times S^2$,\nfrom the condensate phase and the rotations of $\\nematic$, actually becomes\n$(\\U(1) \\times S^2)\/\\mathbb{Z}_2$, where $\\mathbb{Z}_2$ denotes a\ndiscrete two-element group.\nThe vector $\\nematic$ is taken to be {\\em unoriented\\\/} and defines\nthe {\\em nematic axis,} which together with $\\phi$ fully specifies the\nbroken symmetry.\n\nA singly quantized vortex can again be constructed as a $2\\pi$ winding\nof the condensate phase\n\\begin{equation}\n  \\label{eq:psingular}\n  \\zeta^{\\rm 1} =\n  \\frac{e^{i\\varphi}}{\\sqrt{2}}\\threevec{-e^{-i\\alpha}\\sin\\beta}\n                     {\\sqrt{2}\\cos\\beta}\n                     {e^{i\\alpha}\\sin\\beta}.\n\\end{equation}\nThe phase winding may be combined with a $2\\pi$ winding of the nematic\naxis by choosing $\\alpha=\\varphi$\n\\begin{equation}\n  \\label{eq:p012}\n  \\zeta^{\\rm 1^\\prime} =\n  \\frac{1}{\\sqrt{2}}\\threevec{-\\sin\\beta}\n                             {\\sqrt{2}e^{i\\varphi}\\cos\\beta}\n                             {e^{2i\\varphi}\\sin\\beta}.\n\\end{equation}\nBecause of the property that a $\\pi$ change in $\\phi$ can be exactly\ncompensated by a $\\pi$ rotation of $\\nematic$, it is possible to form\nvortices carrying half a quantum of circulation by letting both $\\phi$\nand $\\nematic$ wind by $\\pi$ as the vortex is\nencircled~\\cite{leonhardt_jetplett_2000}, leading to a\ndisclination plane where $\\nematic\\rightarrow-\\nematic$.  If $\\nematic$\nis in the $(x,y)$-plane, a half-quantum vortex can be\nwritten\n\\begin{equation}\n  \\label{eq:hq}\n  \\zeta^{\\rm hq} = \\frac{e^{i\\varphi\/2}}{\\sqrt{2}}\n                   \\threevec{-e^{-i\\varphi\/2}}\n                            {0}\n                            {e^{i\\varphi\/2}}\n\t\t = \\frac{1}{\\sqrt{2}}\n                   \\threevec{-1}\n                            {0}\n                            {e^{i\\varphi}}.\n\\end{equation}\nIn general, the axis about which $\\nematic$ winds need not coincide\nwith the vortex core.  The $\\pi$ winding of the nematic axis still\nallows us to identify the vortex [Fig.~\\ref{fig:hq}].\n\\begin{figure}[tb]\n  \\centering\n  \\includegraphics[scale=1]{halfquantumFig.eps}\n\\caption{Half-quantum vortex.\n  (a) In the half-quantum vortex described by Eq.~(\\ref{eq:hq})\n  the nematic axis (red cones) winds by $\\pi$ as the core is\n  encircled. This is accompanied by a $\\pi$ winding of the condensate\n  phase. The disclination plane is indicated by the dashed line.\n  (b) Another possible $\\pi$ winding of the nematic axis in a\n  half-quantum vortex.\n}\n\\label{fig:hq}\n\\end{figure}\n\nThe polar phase supports also point defects~\\cite{stoof_monopoles_2001,ruostekoski_monopole_2003}.\nWe can form a radial hedgehog of the nematic axis by choosing\n$\\nematic = \\bvec{\\hat{r}} =\n(\\sin\\theta\\cos\\varphi,\\sin\\theta\\sin\\varphi,\\cos\\theta)$,\nwhich represents a spherically symmetric monopole,\n\\begin{equation}\n  \\label{eq:polar_monopole}\n  \\zeta^{\\rm pm} = \\frac{1}{\\sqrt{2}}\\threevec{-e^{-i\\varphi}\\sin\\theta}\n                            {\\sqrt{2}\\cos\\theta}\n\t                    {e^{i\\varphi}\\sin\\theta}.\n\\end{equation}\nHere the origin corresponds to a singular point defect. The two spinor\nwave function components $\\zeta_\\pm$\nform overlapping, singly quantized vortex lines with opposite\ncirculation. The vortex lines are oriented along the $z$ axis, normal\nto a dark-soliton plane in the component $\\zeta_0$. The structure of\nEq.~(\\ref{eq:polar_monopole}) is the analogue of the t'Hooft-Polyakov\nmonopole.\n\n\n\\subsection*{Construction of prototype interface spinors}\n\nWe can construct an approximation of a desired defect configuration that crosses the FM-polar\nboundary by starting from a given FM (polar) defect state and inserting\ndark-soliton planes in some of the spinor wave function components at the interface. For suitable\nchoices of spinor parameters, the resulting configuration will then be sufficiently close to\n a local energetic minimum so that it quickly relaxes to the targeted defect structure. We will now\nexplicitly construct the prototype spinors considered in the main text.\n\n{\\em Singular vortex in both phases:} To illustrate the method\noutlined above, we create a state with a singly quantized vortex on both\nsides of the interface [Fig.~1(a)] by starting from the FM singular vortex,\nEq.~(\\ref{eq:fmsingular}), and inserting a soliton plane in $\\zeta_-$\n\\begin{equation}\n  \\label{eq:1qs}\n  \\zeta^{\\rm 1\\leftrightarrow s} =\n  \\frac{e^{i\\varphi}}{\\sqrt{2}}\\threevec{\\sqrt{2}\\cos^2\\frac{\\beta}{2}}\n                                       {\\sin\\beta}\n                                       {\\pm\\sqrt{2}\\sin^2\\frac{\\beta}{2}},\n\\end{equation}\nwhere the negative sign is used on the polar side of the\ninterface.  Note that for a general $\\beta$ the state created above\nthe interface does not necessarily have $|\\inleva{\\bvec{\\hat{F}}}|=0$,\nand therefore may not strictly be a polar state. However, the state\ncreated has the appropriate vortex structure and very quickly relaxes\nto the singly quantized polar vortex.  The choice $\\beta=\\pi\/2$ yields\na spinor which exactly changes from FM to polar.  Analogous\nconsiderations apply to other prototype spinors.\n\n{\\em Singular to coreless vortex:} Inserting a soliton plane in\n$\\zeta_-$ of the singular vortex $\\zeta^{\\rm 1^\\prime}$ in\nEq.~(\\ref{eq:p012}), results in the interface spinor\n\\begin{equation}\n  \\label{eq:s-cl}\n  \\zeta^{{\\rm 1}\\leftrightarrow{\\rm cl}} =\n  \\frac{1}{\\sqrt{2}}\\threevec{-\\sin\\beta}\n                             {\\sqrt{2}e^{i\\varphi}\\cos\\beta}\n                             {\\pm e^{2i\\varphi}\\sin\\beta},\n\\end{equation}\nwhere the negative sign is used on the FM side of the\nboundary. Choosing $\\beta=\\pi\/4$ or\n$\\beta=3\\pi\/4$ yields $|\\inleva{\\bvec{\\hat{F}}}|=1$ on the FM\nside. The case $\\beta=3\\pi\/4$\ncorresponds to the coreless vortex $\\zeta^{\\rm cl}$ [Eq.~\\ref{eq:cl}]\nwith fountain-like spin profile.\n\n{\\em Dirac monopole on the interface:} A Dirac monopole may be placed\non the interface by inserting a soliton plane into $\\zeta^{\\rm D}$ of\nEq.~(\\ref{eq:dirac})\n\\begin{equation}\n  \\zeta^{\\rm 1 \\leftrightarrow D} = \\frac{1}{\\sqrt{2}}\n                  \\threevec{\\sqrt{2}e^{-2i\\varphi}\\cos^2\\frac{\\theta}{2}}\n                           {e^{-i\\varphi}\\sin\\theta}\n                           {\\pm\\sqrt{2}\\sin^2\\frac{\\theta}{2}}.\n\\end{equation}\nThe spinor after the change of sign has a general structure similar to\nEq.~(\\ref{eq:p012}). The constructed\nspinor therefore approximates a singular vortex in the polar part,\nterminating on the Dirac monopole.  Depending on which side of the\nmonopole is placed on the polar side of the interface, the singular\nDirac string either terminates on the monopole from the FM side, or\nbecomes the singular polar vortex. The structure is equivalent to\n$\\zeta^{{\\rm 1}\\leftrightarrow{\\rm cl}}$, and both spinors are\nrepresented by Fig.~1(b).\n\n{\\em Polar monopole on the interface:}\nWe consider a defect structure with overlapping, singly quantized\nvortex lines in $\\zeta_\\pm$, both oriented normal to the interface\nand of opposite circulation, together with $\\pi$-phase kinks in $\\zeta_+$\nand $\\zeta_0$. This can be parametrized as\n\\begin{equation}\n  \\zeta^{\\rm sv\\leftrightarrow pm} =\n  \\frac{1}{\\sqrt{2}}\\threevec{\\mp e^{-i\\varphi}\\sin\\theta}\n                             {\\sqrt{2}\\cos\\theta}\n\t                     {e^{i\\varphi}\\sin\\theta},\n\\end{equation}\nusing the positive sign on the FM side. The resulting structure on the\npolar side\nrepresents a radial hedgehog of Eq.~(\\ref{eq:polar_monopole}). The\ncontinuation of this on the\nFM side is similar to the singular spin vortex $\\zeta^{\\rm sv}$ in\nEq.~(\\ref{eq:spinvortex}). Hence we have constructed\nan approximation to a spin vortex on the FM side that terminates on\nthe polar monopole on the polar side [Fig.~1(c)].\n\n{\\em Terminating vortices:} Vortices can also be made to terminate at\nthe interface.\nIn Eq.~(\\ref{eq:s-cl}), and in Fig.~1(b) of the main text, a singular\npolar vortex\nperforates across the interface to a coreless FM vortex when we insert\na $\\pi$-phase\nkink in $\\zeta_-$. The resulting defect can be cut in half\nwhile preserving the coherent interface with a continuous\norder-parameter field across the interface.\nWe achieve this by inserting an additional phase kink in $\\zeta_0$\nthat allows the vortices\non either side of the interface to move apart\n\\begin{subequations}\n\\begin{align}\n    \\zeta^{\\rm cut} &=\n    \\frac{1}{\\sqrt{2}}\\threevec{-\\sin\\beta}\n                               {\\sqrt{2}e^{i\\varphi}\\cos\\beta}\n                               {e^{2i\\varphi}\\sin\\beta},\n    \\quad {\\rm for}\\,\\,\\, z>0 \\\\\n    \\zeta^{\\rm cut} & =\n    \\frac{-1}{\\sqrt{2}}\\threevec{\\sin\\beta}\n                               {\\sqrt{2}e^{i\\varphi}\\cos\\beta}\n\t  \t\t       {e^{2i\\varphi}\\sin\\beta}, \\quad {\\rm for}\\,\\,\\, z<0,\n\\end{align}\n\\end{subequations}\nwhere we may choose $\\beta=3\\pi\/4$ as in Eq.~(\\ref{eq:s-cl}).\nThis can yield the configuration in Fig.~1(e), where the singular\npolar vortex and the FM coreless vortex are spatially separated and\nboth terminate on the interface.  Since the vortex lines in the\nindividual spinor components terminate on the soliton planes, it is\nalso possible\nto consider a state where a vortex exists only on one side of the\ninterface, for instance,\n\\begin{subequations}\n\\begin{align}\n    \\zeta^{\\rm pv}  & =\n    \\frac{1}{\\sqrt{2}}\\threevec{-\\sin\\beta}\n                               {\\sqrt{2}e^{i\\varphi}\\cos\\beta}\n                               {e^{2i\\varphi}\\sin\\beta},\n    \\quad {\\rm for}\\,\\,\\, z>0 \\\\\n    \\zeta^{\\rm pv} & =\n    -\\frac{1}{\\sqrt{2}}\\threevec{\\sin\\beta}\n                               {\\sqrt{2}\\cos\\beta}\n\t\t\t       {\\sin\\beta}, \\quad {\\rm for}\\,\\,\\, z<0.\n\\end{align}\n\\end{subequations}\n\n\n\\begin{figure}[t]\n  \\centering\n  \\includegraphics[scale=1]{continuity.eps}\n\\caption{Continuity of the order parameter across the interface.\n  (a)--(c) Density in the individual spinor components\n  $\\zeta_+$,$\\zeta_0$ and $\\zeta_-$, respectively. These do not vanish\n  simultaneously on the interface.  The position of the interface,\n  where $c_2$ changes sign, is\n  indicated by the dashed line.\n  (d) The total atom density is continuous across the interface.\n  (e) The magnitude (color map; dark red at $\\absF=1$) and direction\n  (white arrows) of $\\inleva{\\mathbf{\\hat{F}}}$\n  vary continuously across the interface (indicated by the dashed\n  line; the interface extends across the system, but is indicated only\n  on the sides for clarity.)\n}\n\\label{fig:continuity}\n\\end{figure}\n\n\\subsection*{Coherent stable interface}\n\nThe stable topological interface between the different ground-state\nmanifolds of spin-1 BEC \nconnects the defects and textures continuously across the interface\nboundary. \nAlthough the \npolar and the FM regions exhibit different broken symmetries and\nsupport different defects \nand textures, the order parameter varies smoothly across the\ninterface. This is demonstrated \nin Fig.~\\ref{fig:continuity} for the energetically stable\nconfiguration of Fig.~2 of the main \ntext. \nNote that the total atom density is nonvanishing at the\ninterface and that both the density \nand the spin varies smoothly across the interface (the position where\n$c_2$ changes sign is indicated by a dashed line). The size of the\ninterface region of the relaxed configuration is given by $\\xi_F$ \nrepresenting the healing length of the spin.\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n  Traditionally, integral equations are the\n  subject of functional analysis and operator theory. In the\n  contrast we show that methods of complex\n  geometry and combinatorics are efficient for the study of\n  the following singular integral Poincare-Steklov (briefly, PS) equations\n\n\\begin{equation}\n\\label{PSE}\n\\lambda~\nV.p.\\int\\limits_I\n\\frac{u(t)}{t-x}\\,dt -\nV.p.\\int\\limits_I\n\\frac{u(t)\\,dR(t)}{R(t)-R(x)}\n=const,\\quad x\\in I:=(-1,~1),\n\\end{equation}\nwhere $\\lambda$ is the spectral parameter; $u(t)$ is the unknown function;\n$const$ is independent of $x$. The functional parameter $R(t)$ of the equation\nis a given smooth {\\it nondegenerate} change of variable on the interval $I$:\n\\begin{equation}\n0<\\left|\n\\frac d{dt}\nR(t)\n\\right|<\\infty,\n\\qquad {\\rm when}~ t\\in [-1,1].\n\\label{ND}\n\\end{equation}\nUnder the assumption that $R(t)=:R_3(t)$ is a rational degree three function\nwith separate real critical values different from the endpoints of the interval $I$,\nwe give the constructive representation for the eigenvalues $\\lambda$ and eigenfunctions\n$u(x)$ of equation (\\ref{PSE}). First we say a few words about the origin of PS \nintegral equations and the related background.\n\n\n\\paragraph{Spectral Boundary Value Problem.}\nLet a domain in the plane be subdivided into two simply connected\ndomains $\\Omega_1$ and $\\Omega_2$ by a smooth simple arc $\\Gamma$.\nWe are looking for the values of the spectral parameter $\\lambda$\nwhen the following problem has nonzero solution:\n\n{\\it Find a harmonic function $U_s$ in the domain $\\Omega_s$, $s=1,2$,\nvanishing on the outer portion of the boundary:\n$\\partial\\Omega_s\\setminus\\Gamma$. On the interface\n$\\Gamma$ the functions $U_1$ and $U_2$\ncoincide while their normal derivatives differ by the factor of\n$-\\lambda$:}\n\\begin{equation}\n\\label{InterfaceJump}\n-\\lambda\\frac{\\partial U_1}{\\partial n}=\n\\frac{\\partial U_2}{\\partial n}.\n\\end{equation}\n\n\\paragraph{Applications.}\n\n  Boundary value problems for the Laplace equation with spectral\n  parameter in the boundary conditions were first considered by\n  H.Poincare (1896) and V.A.Steklov (1901). The problems of this \n  kind  arise e.g. in the diffraction, (thermo-) conductivity of \n  composite materials and analysis of 2D models of  oil\n  extraction. \n  \n  \n  This particular\n  problem (\\ref{InterfaceJump}) arises in justification and\n  optimization of {\\it domain decomposition method} for the\n  solution of boundary values problems for elliptic PDE. The\n  eigenvalues $\\lambda$ of the spectral problem and the traces of\n  eigenfunctions $U_1=U_2$ on the interface $\\Gamma$ are\n  respectively the critical values and critical points of the\n  following functional, the ratio of two Dirichlet integrals\n\n\\begin{equation}\nF(U)=\\frac{\\int_{\\Omega_2}|\\nabla U_2|^2 ~d\\Omega_2}\n{\\int_{\\Omega_1}|\\nabla U_1|^2 ~d\\Omega_1},\n \\qquad U\\in H_{oo}^{1\/2}(\\Gamma),\n\\label{Rayleigh}\n\\end{equation}\n  where $U_s$ is the harmonic continuation of the function $U$\n  from interface $\\Gamma$ to the domain $\\Omega_s$, $s=1,2$,\n  vanishing at the outer boundary of the domain.\n\n  \n\n\n\n\n\\paragraph{Integral Equation.}\nThe reduction of the stated above boundary value problem to the interface\nbrings to the equation (\\ref{PSE}). Let $V_s$ be the harmonic function conjugate\nto $U_s$, $s=1,2$. From Cauchy-Riemann equations and (\\ref{InterfaceJump}) it follows\nthat the tangent to the interface derivatives of $V_1$ and $V_2$ differ by the same factor\n$-\\lambda$. Integrating along $\\Gamma$ we get\n\n\\begin{equation}\n\\lambda V_1(y)+V_2(y)=const, \\quad y\\in\\Gamma. \\label{HarmConj}\n\\end{equation}\n\nFor the half-plane the boundary values of conjugate harmonic\nfunctions are related via Hilbert transform. To take advantage of\nthis transformation we consider a conformal mapping $\\omega_s(y)$\nfrom $\\Omega_s$ to the open upper halfplane $\\mbox{\\sets H}$ with\nnormalization $\\omega_s(\\Gamma)=I$, $s=1,2$. Now equation\n(\\ref{HarmConj}) may be rewritten as\n\n$$\n-\\frac{\\lambda}{\\pi}~\nV.p.\\int\\limits_I\n\\frac{U_1(\\omega_1^{-1}(t))}{t-\\omega_1(y)}\\,dt -\n\\frac1{\\pi}~\nV.p.\\int\\limits_I\n\\frac{U_2(\\omega_2^{-1}(t'))}{t'-\\omega_2(y)}\\,dt'\n=const,\n\\quad y\\in\\Gamma.\n$$\nIntroducing new notation\n$x:=\\omega_1(y)\\in I$;\n$R:=\\omega_2\\circ\\omega_1^{-1}:~~I\\to\\Gamma\\to I$;\n$u(t):=U_1(\\omega_1^{-1}(t))$ and the change of variable\n$t'=R(t)$ in the second integral, we arrive at the Poincare-Steklov equation\n(\\ref{PSE}). Note that here $R(t)$ is the decreasing function on $I$.\n\n\n  Operator analysis of equivalent spectral problems, boundary\n  value problem (\\ref{InterfaceJump}) and Poincare-Steklov\n  equation (\\ref{PSE}), may be found e.g.in \\cite{Bog0}. Here\n  we only mention that the spectrum is discreet if (\\ref{ND}) holds, the eigen values\n  are positive and converge to $\\lambda=1$.\n\n\n\\paragraph{Philosophy of the Research.}\n  The aim of our study is to give explicit expressions for the eigen pairs\n  $(\\lambda,u)$ of the PS integral equation. For the rational degree two\n  functions $R(x)=R_2(x)$ the eigen pairs  were expressed in terms of elliptic\n  functions \\cite{Bog3}. Next natural step is to consider degree three\n  rational functions.\n\n  Here the notion of {\\it explicit solution} should be specified. Usually this\n  term means an {\\it elementary function} of parameters or a {\\it quadrature} of\n  it or application of other {\\it permissible operations} (e.g. the concept of\n  Umemura classical functions). The history of mathematics however knows many\n  disappointing results when the solution of the prescribed form does not exist.\n  The nature always forces us to introduce new types of transcendent objects to\n  enlarge the scope of search. Cf.: \"Mais cette \\'etude intime de la nature des\n  fonctions integrales ne peut se faire que par l'introduction de transcendantes\n  nouvelles\" \\cite{Poi}.\n\n  From the\n  philosophical point of view our goal is to study the nature of\n  the solutions of integral equation (\\ref{PSE}) and the means for their \n  constructive representation.\n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n\n\n\n\n\n\n  \\paragraph{Brief Description of the Result.}\n   Given rational degree three function $R_3(t)$, we explicitly associate\n  it to a {\\it pair of pants} in Sect. \\ref{PantsOfR3}. On the\n  other hand, given spectral parameter $\\lambda$ and two\n  auxiliary real parameters, we explicitly construct in Sect.\n  \\ref{Result} another pair of pants which additionally depend on\n  one or two integers. When the above two pants are\n  conformally equivalent, $\\lambda$ is the eigenvalue of the\n  PS integral equation with parameter $R_3(x)$. Essentially,\n  this means that to find the spectrum of the given integral\n  equation (\\ref{PSE}) one has to solve three transcendential\n  equations involving three {\\it moduli of pants}.\n\n\n  Whether this representation of the solutions may be considered as constructive \n  or not is a matter of discussion. On the one hand, today it is possible to \n  numerically evaluate the conformal structure of surfaces (e.g. via circle \n  packing). On the other, this representation allows us to obtain valuable \n  features of the solution: to find the number of zeroes of eigenfunction \n  $u(t)$, to localize the spectrum and to show the discrete mechanism of \n  generating the eigenvalues.\n\n\\section{Space of PS-3 Equations}\n\nIn what follows we consider integral equations (\\ref{PSE}) with rational degree tree real\nfunctional parameter $R(x)=R_3(x)$ and call such equations PS-3. We restrict ourselves to the case when\n$R_3(x)$ has {\\it four distinct real critical values different from} $\\pm1$.\n  The details of our further constructions depend on the\n  topological properties of functional parameter of\n  the integral equation. One may encounter one of five described in\n  section \\ref{Classification} typical situations ${\\cal A}$, ${\\cal B}1$, ${\\cal B}21$,\n  ${\\cal B}22$, ${\\cal B}23$ corresponding to the components in the space of\n  admissible functions $R_3(x)$.\n\n\n\n\n\\subsection{Topology of the Branched Covering}\n\\label{TopBrCov}\n  Degree three rational function $R_3(x)$ defines the three-\n  sheeted branched covering of a Riemann sphere by another Riemann\n  sphere. The Riemann-Hurwitz formula suggests that $R_3(x)$\n  typically has four separate branch points $a_s$, $s=1,4$. This\n  means that every value $a_s$ is covered by a critical point\n  $b_s$, and an ordinary  point $c_s$. We have assumed that\n  all four branch points $a_s$ are distinct, real and differ\n  from $\\pm 1$. Other possible configurations are discussed in\n  \\cite{Bog1}.\n\n\n  Every point $y\\neq a_s$ of the extended real axis\n  $\\hat{\\mbox{\\sets R}}:=\\mbox{\\sets R}\\cup\\{\\infty\\}$ belongs to\n  exactly  one of two types. For the type (3:0) the pre-image\n  $R_3^{-1}(y)$ consists of three distinct real points. For the\n  type (1:2) the pre-image consists of a real and two complex\n  conjugate points. The type of the point is locally constant on\n  the extended real axis and changes when we step over the branch\n  point. Let the branch points $a_s$ be enumerated in the\n  natural cyclic order of $\\hat{\\mbox{\\sets R}}$ so that the\n  intervals $(a_1,a_2)$ and $(a_3,a_4)$ are filled with the\n  points of the type (1:2). We specify the way to\n  exclude the relabeling $a_1\\leftrightarrow a_3$,\n  $a_2\\leftrightarrow a_4$ of branch points in Sect. \\ref{Classification}.\n\n    \\label{LabelBranchPoint}\n\n  The total pre-image $R_3^{-1}(\\hat{\\mbox{\\sets R}})$ consists of\n  the extended real axis  and two pairs of complex conjugate arcs\n  intersecting $\\hat{\\mbox{\\sets R}}$ at points $b_1$, $b_2$,\n  $b_3$, $b_4$ as shown at the left side of Fig.\n  \\ref{CoverTopology}. The compliment of this pre-image on the\n  Riemann sphere has six components, each of them is mapped 1-1\n  onto upper or lower half plane. Note that the points $b_1$,\n  $c_4$, $c_3$, $b_2$... on the left picture of Fig. \\ref{CoverTopology}\n  may follow in inverse order as well.\n\n\n \\begin{figure}\n\\begin{picture}(170,20)\n\\put(110,0){\n\\begin{picture}(70,20)\n\n\\thicklines\n\\put(10,10){\\circle*{1}}\n\\put(20,10){\\circle*{1}}\n\\put(10,10){\\line(1,0){10}}\n\\put(8,12){$a_1$}\n\\put(18,12){$a_2$}\n\\put(30,10){\\circle*{1}}\n\\put(40,10){\\circle*{1}}\n\\put(30,10){\\line(1,0){10}}\n\\put(28,12){$a_3$}\n\\put(38,12){$a_4$}\n\n\\thinlines\n\\multiput(0,10)(2,0){25}{\\line(1,0){1}}\n\\put(51,9){$\\hat{\\mbox{\\sets R}}$}\n\\put(10,25){\\circle{4}}\n\\put(9,24.2){$y$}\n\n\\end{picture}\n}\n\\put(5,0){\n\\begin{picture}(70,20)\n\\thicklines\n\\put(10,10){\\circle*{1}}\n\\put(20,10){\\circle*{1}}\n\\put(10,10){\\line(1,0){10}}\n\\put(8,12){$c_4$}\n\\put(18,12){$c_3$}\n\\put(50,10){\\circle*{1}}\n\\put(60,10){\\circle*{1}}\n\\put(50,10){\\line(1,0){10}}\n\\put(48,12){$c_2$}\n\\put(58,12){$c_1$}\n\\put(15,10){\\oval(30,20)}\n\\put(0,10){\\circle*{1}}\n\\put(30,10){\\circle*{1}}\n\\put(1,12){$b_1$}\n\\put(26,12){$b_2$}\n\\put(55,10){\\oval(30,20)}\n\\put(40,10){\\circle*{1}}\n\\put(70,10){\\circle*{1}}\n\\put(41,12){$b_3$}\n\\put(66,12){$b_4$}\n\\thinlines\n\\multiput(-5,10)(2,0){40}{\\line(1,0){1}}\n\\put(76,9){$\\hat{\\mbox{\\sets R}}$}\n\\put(90,10){$\\longmapsto$}\n\\put(90,15){$R_3(x)$}\n\\put(75,25){\\circle{4}}\n\\put(74,24){$x$}\n\n\\end{picture}}\n\\end{picture}\n\\caption[]\n{\\small The topology of the covering $R_3$ with real branch points}\n\\label{CoverTopology}\n\\end{figure}\n\n\n\n\n\n\\subsection{Classification of Parameters $R_3$}\n\\label{Classification}\n\n  The functional parameter $R_3(x)$ is a\n  nondegenerate change of variable on the segment $[-1,1]$. This\n  in particular means that no critical point $b_s$ belongs to this\n  segment. So exactly one of two cases is realized:\\footnote{\n  Two points on a circle (extended real axis) define two segments.\n  It should be clear which segment we mean: e.g. $b_1,b_2\\not\\in$ $[b_3,b_4]$;~~\n  $b_1,b_4\\not\\in$ $[b_2,b_3]$, etc.}\n\n\\begin{equation}\n\\begin{array}{cl}\nCase~~{\\cal A}: & [-1,1]\\subset[b_2,b_3],\\\\\nCase~~{\\cal B}: & [-1,1]\\subset[b_3,b_4].\n\\end{array}\n\\end{equation}\n  Remaining possibilities (like $[-1,1]\\subset[b_1,b_2]$) are\n  reduced either to ${\\cal A}$ or ${\\cal B}$ by the clever choice of labeling the branch points\n  $a_s$ -- see section \\ref{LabelBranchPoint}. For the case ${\\cal B}$ it is important whether\n  $[-1,1]$ intersects $[c_2,c_1]$ or not. So we consider two subcases:\n\\begin{equation}\n\\begin{array}{cl}\nCase~~{\\cal B}1: & [-1,1]\\cap[c_2,c_1]=\\emptyset,\\\\\nCase~~{\\cal B}2: & [-1,1]\\cap[c_2,c_1]\\neq\\emptyset.\\\\\n\\end{array}\n\\end{equation}\nThe case ${\\cal B}2$ in turn is subdivided into three subcases:\n\\begin{equation}\n\\begin{array}{cl}\nCase~~{\\cal B}21: & [-1,1]\\subset[c_2,c_1],\\\\\nCase~~{\\cal B}22: & [-1,1]\\supset[c_2,c_1],\\\\\nCase~~{\\cal B}23: & all~ the~ rest.\\\\\n\\end{array}\n\\end{equation}\n\n\n\\subsection{Pair of Pants associated to $R_3$}\n\\label{PantsOfR3}\n  For the obvious reason, {\\it a pair of pants} is the name for the\n  Riemann sphere with three holes in it. Pair of pants may\n  be conformally mapped to $\\hat{\\mbox{\\sets C}}$ with  three\n  nonintersecting {\\it real} slots. This mapping is unique up to real\n  linear-fractional mappings. The conformal\n  class of pants with labeled boundary components depends on three\n  real parameters varying in a cell.\n\n\n{\\bf D~e~f~i~n~i~t~i~o~n:}\nTo every PS-3 equation we associate the pair of pants:\n\\begin{equation}\n\\label{R3Pants}\n{\\cal P}(R_3):=Cl\n\\left(\n\\hat{\\mbox{\\sets C}}\n\\setminus\n\\{\n([-1,1]\\triangle [a_1,a_2])\\cup [a_3,a_4]\n\\}\n\\right)\n\\end{equation}\n  where $\\triangle$ is the symmetric difference (union of two\n  sets minus their intersection). {\\it Closure} here and everywhere below\n  is taken with respect to the intrinsic spherical metric\n  when every slot acquires two banks. Boundary components of\n  pants are colored in accordance with the palette:\n\\begin{center}\n\\begin{tabular}{ll}\n$[-1,1]\\setminus [a_1,a_2]$& -- red\\\\\n$[a_1,a_2]\\setminus [-1,1]$& -- blue\\\\\n$[a_3,a_4]$& -- green\n\\end{tabular}\n\\end{center}\n\nThus obtained pair of pants will have boundary ovals of all three colors,\nbut in cases ${\\cal B}21$ (green and two blue ovals) and ${\\cal B}22$\n(green and two red ovals). In case ${\\cal A}$ the red, green and blue slots\nalways follow in the natural cyclic order of the extended real axis.\n\n\\subsection{Gauge Transformations}\nLet us recall that parameter $R(x)$ is not uniquely determined by two\ndomains $\\Omega_1$ and $\\Omega_2$. Composition with linear-fractional transformations\npreserving the segment $[-1,1]$ is admissible. The general appearance of such a mapping is\n\n\\begin{equation}\nL^\\pm_\\alpha(t):=\\pm\\frac{t+\\alpha}{\\alpha t+1},\n\\quad \\alpha\\in (-1,1).\n\\label{LFI}\n\\end{equation}\n\n\n\\begin{lmm}\n1. The gauge transformation $R\\to L_\\alpha^\\pm\\circ R$\ndoes not change neither eigenvalues $\\lambda$ nor the eigenfunctions $u(t)$\nof any PS integral equation.\n\n2. The gauge transformation $R\\to R\\circ L_\\alpha^\\pm$\ndoes not change the eigenvalues $\\lambda$ and slightly changes the eigenfunctions: $u(t)\\to u(L_\\alpha^\\pm(t))$.\n\\end{lmm}\nP~r~o~o~f. To simplify the notations we put $L(t):=L_\\alpha^\\pm(t)$.\n\n1. The gauge transformation just adds a constant term to the right hand side of equation.\n$$\n\\int_{-1}^1\n\\frac{u(t)dL(R(t))}{L(R(t))-L(R(x))}=\n\\int_{-1}^1\n\\frac{u(t)L'(R(t))dR(t)}{[L'(R(t))L'(R(x))]^{1\/2}(R(t)-R(x))}=\n$$\n$$\n\\int_{-1}^1\n\\frac{u(t)dR(t)}{R(t)-R(x)}-\n\\int_{-1}^1\n\\frac{u(t)dR(t)}{R(t)-L^{-1}(\\infty)}.\n$$\n\n2. We define the new variable $x_*:=L(x)$ and the new function $u_*(x_*):=u(x)$.\n$$\n\\int_{-1}^1\n\\frac{u(t)dR(L(t))}{R(L(t))-R(L(x))}=\n\\pm\\int_{-1}^1\n\\frac{u_*(t_*)dR(t_*)}{R(t_*)-R(x_*)},\n$$\n$$\n\\int_{-1}^1\n\\frac{u(t)dt}{t-x}=\n\\pm\\int_{-1}^1\n\\frac{u_*(t_*)dL^{-1}(t_*)}{L^{-1}(t_*)-L^{-1}(x_*)}=\n\\pm\\int_{-1}^1\n\\frac{u_*(t_*)dt_*}{t_*-x_*}\n\\mp\\int_{-1}^1\n\\frac{u_*(t)dt}{t-L(\\infty)}. ~~\\rule[-1mm]{2.4mm}{2.4mm}\n$$\n\nWe see that essentially the space of $PS-3$ equations has real\n  dimension 3, the same as the moduli space of pants. It is easy to check the following:\n\n\\begin{itemize}\n\\item\nAny gauge transformation of the parameter $R_3(x)$ does\nnot change the type (${\\cal A}$, ${\\cal B}1$, \\dots) of integral equation.\n\\item\nThe transformation $R_3\\to R_3\\circ L_\\alpha^\\pm$ does not change the associated\npants and preserves the colors of the boundary ovals.\n\\item\nAssociated to functional parameter $L_\\alpha^\\pm\\circ R_3$ are the pants\n$L_\\alpha^\\pm{\\cal P}(R_3)$. The colors of the boundary ovals are transferred\nby  $L_\\alpha^\\pm$, but in one case. When the type of integral equation is $\\cal A$,\nthe transformation $L_\\alpha^-$ interchanges blue and green colours on the boundaries.\n\\end{itemize}\n\n\n\n\n\n\n\n\n\\subsection{Reconstruction of $R_3(x)$ from the Pants}\n\\label{R3fromPants}\nThe parameter $R_3(x)$ of integral equation may be reconstructed,\ngiven the pants ${\\cal P}(R_3)$ and the type ${\\cal A}$, ${\\cal B}1\\dots$ of the equation.\nOne has to follow the routine described below.\n\n\\paragraph{Restore the Labeling of the Branch Points.} In case ${\\cal B}2$\nwe temporarily paint the real segment separating two non-green slots\nin blue. The (extended) blue segment is set to be $[a_1,a_2]$; the\ngreen segment is $[a_3,a_4]$. The relabeling $a_1\\leftrightarrow\na_2$ and $a_3\\leftrightarrow a_4$ is eliminated by the natural\ncyclic order of the points $a_1,a_2,a_3,a_4$ on the extended real\naxis.\n\n\n\n\\begin{figure}\n\\centerline{\\psfig{figure=r3.eps}}\n\\caption[Two graphs]\n{\\small The graph of $\\tilde {R}_3(x)$ when $c\\in(\\frac13,\\frac12)$.}\n\\label{R3pm}\n\\end{figure}\n\n \\paragraph{Normalized Covering.} Let $L_a$ be the unique linear-fractional map sending\n  the points $a_1$, $a_2$, $a_3$, $a_4$ to respectively $0$,\n  $1$, $a>1$, $\\infty$. The conformal motion $L_b$ of the\n  covering Riemann sphere sends the critical points $b_1$, $b_2$,\n  $b_3$, $b_4$ of $R_3(x)$ (unknown at the moment) to respectively $0$, $1$,\n  $b>1$, $\\infty$. The function $L_a\\circ R_3\\circ\n  L_b^{-1}$ with the normalized critical points and critical values\n  takes a simple form:\n\n  $$\n  \\widetilde{R_3}(x)=x^2L(x),\n  $$\n  with real linear fractional function $L(x)$ satisfying the\n  restrictions:\n$$\n\\begin{array}{ll}\nL(1)=1,&L'(1)=-2,\\\\\nL(b)=a\/b^2,&L'(b)=-2a\/b^3.\n\\end{array}\n$$\nWe got four equations for three parameters of $L(x)$ and the unknown $b$.\nThe first two equations suggest the following expression for the\nlinear-fractional function:\n$$\nL(x)=1+2\\frac{(c-1)(x-1)}{x-c}.\n$$\nThe other two equations are solved parametrically in terms of $c$:\n$$\nb(c)=c\\frac{3c-2}{2c-1};\n\\qquad\na(c)=c\\frac{(3c-2)^3}{2c-1}.\n$$\nGiven $a>1$, there are exactly two real solutions of the equation\n$a(c)=a$. One of the solutions $c$ lies in the segment $(1\/3,1\/2)$, the other\nlies in $(1,\\infty)$. In the case $c\\in(1\/3,1\/2)$ the segments $L_a(a_1,a_2)=(0,1)$\nand $L_a(a_3,a_3)=(a,\\infty)$ are filled with the points of the type (1:2),\nwhich corresponds to our choice of labelling the branchpoints in section \\ref{TopBrCov}.\nThe solution $c\\in(1,\\infty)$ is a fake as the same segments bear the points\nof the type (3:0). Both functions $b(c)$ and $a(c)$ increase from $1$ to\n$\\infty$ when the argument $c$ runs from $1\/3$ to $1\/2$.\n\n\\paragraph{Reconstruction of the Mapping $L_b$.}\n    In the case ${\\cal A}$ the red, green and blue slots follow in the natural\n    cyclic order. Hence, the segment $L_a[-1,1]$ is a subset of $[1,a]$. We\n    choose the unique component of the pre-image $\\widetilde{R_3}^{-1}$ of the\n    segment $L_a[-1,1]$ belonging to the segment  $[1,b]$ -- see Fig.\n    \\ref{R3pm}. For the case ${\\cal B}$ the segment $L_a[-1,1]$ is a subset of\n    $(-\\infty,a]$ and we choose  the pre-image of this segment which lie in\n    $[b,\\infty]$. The requirement: $L_b$ {\\it maps $[-1,1]$ to the chosen\n    segment} determines this map up to a pre-composition with the function\n    (\\ref{LFI}).\\\\[1mm]\n\nWe see that given the pants ${\\cal P}(R_3)$, the functional parameter is\nrecovered up to a gauge transformation $R_3\\to R_3\\circ L_\\alpha^\\pm$. It is not\ndifficult to check, that the described above procedure applied to the pair of\npants $L_\\beta^\\pm{\\cal P}(R_3)$ (in case $\\cal A$ and the mapping $L_\\beta^-$\nreversing the orientation of real axis we additionally have to exchange the blue\nand green colors of the slots) returns the covering map $L_\\beta^\\pm\\circ\nR_3\\circ L_\\alpha^\\pm$. Roughly speaking, the classes of gauge transformation\nof $R_3(x)$ correspond to the conformal classes of pants with suitably colored\nboundary components and each conformal class of pants corresponds to a class of\ncertain functional parameter $R_3(x)$.\n\n\n\n\n\n\n\n\\section{Reduction to Projective Structures}\nPS integral equations possess rich geometrical content\n\\cite{Bog1,Bog2} which we disclose in this section.\nWe describe a three-step reduction\n  of the integral equation to a certain problem \\cite{Bog1} for  projective\n  structures on a riemann surface which has essentially combinatorial solution.\n\n\n\n\\subsection{Step 1: Functional Equation}\n\n  Let us expand the kernel of the second integral in (\\ref{PSE})\n  into a sum of elementary fractions:\n\n\\begin{equation}\n\\label{kernel}\n\\frac{ R_3'(t)}{R_3(t)-R_3(x)}=\n\\frac{d}{dt}\\log(R_3(t)-R_3(x))=\n\\sum_{k=1}^3\\frac{1}{t-z_k(x)}-\\frac{Q'}{Q}(t),\n\\end{equation}\nwhere $Q(t)$ is the denominator in noncancellable representation of $R(t)$\nas the ratio of two polynomials; $z_1(x)=x,~z_2(x),~z_3(x)$ --\nare all solutions (including multiple and infinite) of the equation\n$R_3(z)=R_3(x)$. This expansion suggests to rewrite the original equation (\\ref{PSE})\nas certain relationship for the Cauchy-type integral\n\n\n\\begin{equation}\n\\Phi(x):=\\int\\limits_I\\frac{u(t)}{t-x}dt+const^*,\n\\quad x\\in\\hat{\\mbox{\\sets C}}\\setminus[-1,1].\n\\label{Cauchy}\n\\end{equation}\nKnown $\\Phi(x)$, the solution $u(t)$ may be recovered by the Sokhotskii-Plemelj\nformula:\n\\begin{equation}\n\\label{SP}\nu(t)=(2\\pi i)^{-1}\n\\left[\n\\Phi(t+i0)-\\Phi(t-i0)\n\\right],\n\\quad t\\in I.\n\\end{equation}\nThe constant term $const^*$ in (\\ref{Cauchy}), which we assume to be\n\\begin{equation}\nconst^*:=\\frac{1}{\\lambda - 3}\n\\left[\n\\int\\limits_I\\frac{u(t)\nQ'(t)}{Q(t)}dt - const\n\\right]\n\\label{Const}\n\\end{equation}\nis introduced to cancel the constant terms arising after\nsubstitution of expression (\\ref{Cauchy}) to the equation (\\ref{PSE}).\nIn this way the following result was proven \\cite{Bog2}:\n\n\\begin{lmm}\nFor $\\lambda\\neq 1,3$ the transformations (\\ref{Cauchy}) and\n(\\ref{SP}) bring about a one-to-one correspondence between the\nH\\\"older eigenfunctions $u(t)$ of PS-3 integral equation and the\nholomorphic on the Riemann sphere outside the slot $[-1,1]$\nnontrivial solutions $\\Phi(x)$ of the functional equation\n\\begin{equation}\n\\label{FE}\n\\Phi(x+i0)+\\Phi(x-i0)=\\delta\n\\biggl(\n\\Phi (z_2(x))+\\Phi(z_3(x))\n\\biggl)~,\n\\quad x\\in I,\n\\end{equation}\n\n\\begin{equation}\n\\label{delta}\n\\delta =2\/(\\lambda -1),\n\\end{equation}\nwith H\\\"older boundary values $\\Phi(x\\pm i0)$ at the banks of the slot $[-1,1]$.\n\\end{lmm}\n\n\n\n\n\n\n\\subsection{Step 2: Riemann Monodromy Problem}\n\\label{RHP}\n  In this section we reduce our functional (and\n  therefore integral) equation to the Riemann monodromy\n  problem in the following form. {\\it Find a holomorphic vector\n  $W(y)=(W_1,W_2,W_3)^t$ on the slit sphere\n  ${\\cal P}(R_3)\\setminus[-1,1]$\n  whose boundary values on the opposite sides of\n  every slot are related by the constant matrix specified for each slot.}\n\n\\subsubsection{Monodromy Generators}\nTo formulate the Riemann monodromy problem we introduce $3\\times 3$ permutation matrices\n\\begin{equation}\n\\label{permutations}\n{\\bf D_1}:=\n\\begin{array}{||lll||}\n1&0&0\\\\0&0&1\\\\0&1&0\\\\\n\\end{array}~;\n\\qquad\n{\\bf D_2}:=\n\\begin{array}{||lll||}\n0&0&1\\\\0&1&0\\\\1&0&0\\\\\n\\end{array}~;\n\\qquad\n{\\bf D_3}:=\n\\begin{array}{||lll||}\n0&1&0\\\\1&0&0\\\\0&0&1\\\\\n\\end{array}\n\\qquad\n\\end{equation}\nand a matrix depending on the spectral parameter $\\lambda$:\n\\begin{equation}\n\\label{D}\n{\\bf D}:=\n\\begin{array}{||ccc||}\n-1       & \\delta & \\delta \\\\\n 0       & 1      & 0  \\\\\n 0       & 0      & 1  \\\\\n\\end{array}\n~,\n\\qquad \n\\delta =2\/(\\lambda -1).\n\\end{equation}\n\n\\begin{lmm}\n  Matrices ${\\bf D_1}$, ${\\bf D_2}$, ${\\bf D_3}$,\n  ${\\bf D}$, ${\\bf D_1D}=$ ${\\bf DD_1}$ have order\n  two as $GL_3$ group elements.\n  \\label{DsProperty}\n\\end{lmm}\n\n\\subsubsection{Separating Branches of $R_3^{-1}$}\n\n  Let domain $\\cal O$ be the compliment to the segments\n  $[a_1,a_2]$ and $[a_3,a_4]$ on the Riemann sphere.\n  The pre-image $R_3^{-1}{\\cal O}$\n  consists of three components ${\\cal O}_j$, $j=1,2,3$, mapped one-one to\n  $\\cal O$ -- see Fig. \\ref{CoverTopology}.\n  Two of the components are (topological) discs with\n  a slot and the third is an annulus. The enumeration of domains\n  ${\\cal O}_j$ is determined by the following rule:\n  the segment $[-1,1]$ lies in the closure of ${\\cal O}_1$,\nthe segment $[c_3,c_4]$ lies on the border of ${\\cal O}_2$.\n\n\n\\subsubsection{}\n  Let $u(x)$ be the solution of integral equation (\\ref{PSE}) in the case ${\\cal A}$.\n  We consider the vector\n\\begin{equation}\n\\label{Wofy}\nW(y)=(\\Phi(x_1),\\Phi(x_2),\\Phi(x_3))^t,\n\\qquad y\\in{\\cal O}\\setminus[-1,1],\n\\end{equation}\n  where $\\Phi(x)$ is from (\\ref{Cauchy}) and $x_s$ is the unique\n  solution of the equation $R_3(x_s)=y$ in ${\\cal O}_s$. Vector\n  $W(y)$ will be holomorphic and bounded in the domain ${\\cal\n  O}\\setminus[-1,1]$  as all three points $x_s$, $s=1,2,3$,\n  remain in the holomorphy domain of the function $\\Phi(x)$. We claim\n  that\n\n\\begin{center}\n  \\begin{tabular}{cc}\n  $W(y+i0)={\\bf D}W(y-i0)$,& when $y\\in [-1,1]$,\\\\\n  $W(y+i0)={\\bf D_3}W(y-i0)$,& when $y\\in [a_1,a_2],$\\\\\n  $W(y+i0)={\\bf D_2}W(y-i0)$,& when $y\\in [a_3,a_4].$\n  \\end{tabular}\n\\end{center}\n\n  Indeed, let $y^+:=y+i0$ and $y^-:=y-i0$ be two points on the opposite\n  banks of $[a_1,a_2]$. Their inverse images $x_3^+=x_3^-$, $x_1^\\pm=x_2^\\mp$\n  lie outside the cut $[-1,1]$. Hence $W(y^+)=$\n  ${\\bf D_3}W(y^-)$. For two points $y^\\pm$ lying on the opposite\n  banks of the slot $[a_3,a_4]$, their inverse images satisfy\n  the relations $x_2^+=x_2^-$, $x_1^\\pm=x_3^\\mp$, which means\n  $W(y^+)=$ ${\\bf D_2}W(y^-)$. Finally, let $y^\\pm$ lie on the  banks\n  of $[-1,1]$. Now two points $x_2^+=x_2^-$ and $x_3^+=x_3^-$ lie\n  in the holomorphy domain of $\\Phi(x)$ while $x_1^+$ and $x_1^-$\n  appear on the opposite sides of the cut $[-1,1]$. According to\n  the functional equation (\\ref{FE}),\n$$\n\\Phi(x_1^+)=\n-\\Phi(x_1^-)+\\delta(\\Phi (x_2^-)+\\Phi (x_3^-)),\n$$\ntherefore $W(y^+)=$ ${\\bf D}W(y^-)$ holds on the slot $[-1,1]$.\n\n\\subsubsection{}\n  Conversely, let $W(y)=(W_1,W_2,W_3)^t$ be the bounded solution of the Riemann monodromy\n  problem stated above. We define a piecewise holomorphic\n  function  on the Riemann sphere:\n  \\begin{equation}\n  \\Phi(x):=W_s(R_3(x)),\n  \\qquad {\\rm when}~~x\\in {\\cal O}_s\\setminus R_3^{-1}[-1,1],\n  \\qquad s=1,2,3.\n  \\label{PhifromW}\n  \\end{equation}\n  From the boundary relations for the vector $W(y)$ it\n  immediately follows that the function $\\Phi(x)$ has no jumps on\n  the lifted cuts $[a_1,a_2]$, $[a_3,a_4]$, $[-1,1]$\n  apart from the cut $[-1,1]$ from the upper sphere. Say, if the two points\n  $y^\\pm$ lie on the opposite sides of the cut $[a_1,a_2]$, then\n  $W_3(y^+)=W_3(y^-)$ and  $W_1(y^\\pm)=W_2(y^\\mp)$ which means\n  that the function $\\Phi(x)$ has no jump on the components of\n  $R_3^{-1}[a_1,a_2]$. From the boundary relation on the cut $[-\n  1,1]$ it follows that $\\Phi(x)$ is the solution for the functional\n  equation (\\ref{FE}). Therefore it gives a solution of Poincare-\n  Steklov integral equation with parameter $R_3(x)$. Combining\n  formulae (\\ref{SP}) with (\\ref{PhifromW}) we get the\n  reconstruction rule\n\\begin{equation}\n\\label{uofx}\nu(x)=(2\\pi i)^{-1}\\biggl(W_1(R_3(x)+i0)-W_1(R_3(x)-i0)\\biggr),\n\\qquad x\\in [-1,1].\n\\end{equation}\n\n\n  \\subsubsection{}\nWe have just proved for the case ${\\cal A}$ the following\n\\begin{thrm} \\cite{Bog1}\\label{PSeqRMP}\n~~If $\\lambda\\neq1,3$ then two formulas (\\ref{Wofy}) and (\\ref{uofx})\nimplement the one-to-one  correspondence between the solutions $u(x)$ of\nthe integral equation (\\ref{PSE}) and the bounded solutions $W(y)$ of the\nRiemann monodromy problem in the slit sphere $\\hat{\\mbox{\\sets C}}\\setminus\\{\n[a_1,a_2]\\cup[a_3,a_4]\\cup[-1,1]\\}$ with the following matrices assigned to the slots:\\\\\n\\begin{equation}\n\\label{Rbvp}\n\\begin{array}{c|c|c|c|c|}\n&[-1,1]\\setminus[a_1,a_2]&[a_1,a_2]\\setminus[-1,1]&[a_3,a_4]&[-1,1]\\cap[a_1,a_2]\\\\\n\\hline\n{\\rm Case~{\\cal A}:}&{\\bf D}& {\\bf D_3}& {\\bf D_2}& \\\\\n{\\rm Case~{\\cal B}1:}&{\\bf D}& {\\bf D_1}& {\\bf D_2}& \\\\\n{\\rm Case~{\\cal B}2:}&{\\bf D}& {\\bf D_1}& {\\bf D_2}& {\\bf D_1D}={\\bf DD_1}\\\\\n\\end{array}\n\\end{equation}\n\\end{thrm}\n\n\n\n\\subsubsection{Monodromy Invariant}\nIt may be checked that the matrices $\\bf D$, ${\\bf D_1}$, ${\\bf D_2}$,\n${\\bf D_3}$ generating the monodromy group for the solution $W(y)$\n are pseudo-orthogonal, that is preserve the same quadratic form\n \\begin{equation}\n\\label{J}\nJ(W):=\\sum\\limits_{k=1}^3 W_k^2-\n\\delta\\sum\\limits_{j<s}^3 W_jW_s.\n\\end{equation}\n\n  This form is not degenerate unless $-2\\ne\\delta\\ne 1$, or\n  equivalently $0\\ne\\lambda\\ne 3$. Since the solution $W(y)$ of\n  our monodromy problem is bounded near the cuts, the\n  value of the form $J(W)$ is independent of the variable $y$.\n  Therefore the solution ranges either in the smooth\n  quadric $\\{W\\in\\mbox{\\sets C}^3:~~J(W)=J_0\\neq0\\}$, or the cone\n  $\\{W\\in\\mbox{\\sets C}^3:~~J(W)=0\\}$.\n\n\n\n\\subsubsection{Geometry of Quadric Surface}\nThe nondegenerate projective quadric $\\{J(W)=J_0\\}$ contains two families of\nline elements which for convenience we denote by the signs $'+'$ and $'-'$.\nTwo different lines from the same family are disjoint while two lines from\ndifferent families  intersect. The intersection of those lines with\nthe 'infinitely distant' secant plane gives points on the conic\n\\begin{equation}\n\\label{conica}\n\\{(W_1:W_2:W_3)^t\\in\\mbox{\\sets CP}^2:\\quad J(W)=0\\}\n\\end{equation}\nwhich by means of stereographic projection $p$ may be identified with the Riemann sphere.\nTherefore we have introduced two global coordinates $p^\\pm(W)$ on the quadric,\n'infinite part' of which (i.e. conic (\\ref{conica})) corresponds to coinciding coordinates:\n$p^+=p^-$ (see Fig. \\ref{Quadric}).\n\n\\begin{figure}\n\\centerline{\\psfig{figure=quadric.eps}}\n\\caption[Quadric]\n{\\small Global coordinates $p^+$ and $p^-$ on quadric }\n\\label{Quadric}\n\\end{figure}\n\n\nTo obtain explicit expressions for the coordinate change $W\\leftrightarrow p^\\pm$ on quadric\nwe bring the quadratic form $J(W)$ to the simpler form\n$J_\\bullet(V):=$ $V_1V_3-V_2^2$ by means of the linear coordinate change\n\\begin{equation}\nW={\\bf K}V\n\\label{V}\n\\end{equation}\nwhere\n\\begin{equation}\n\\label{K}\n{\\bf K}:=\n(3\\delta+6)^{-1\/2}\n\\begin{array}{||lll||}\n1&1&1\\\\\n1&\\varepsilon^2&\\varepsilon\\\\\n1&\\varepsilon&\\varepsilon^2\n\\end{array}\n\\cdot\n\\begin{array}{||lll||}\n0&\\mu^{-1}&0\\\\\n0&0&1\\\\\n1&0&0\n\\end{array}\n~,\n\\end{equation}\n$$\n\\varepsilon:=\\exp(2\\pi i\/3),\n\\qquad\n\\mu:=\\sqrt{\\frac{\\delta-1}{\\delta+2}}=\n\\sqrt{\\frac{3-\\lambda}{2\\lambda}}.\n$$\nTranslating the first paragraph of the current section\ninto the language of formulae we get\n\\begin{equation}\n\\label{V2Ppm}\np^\\pm(W):=\\frac{V_2\\pm i\\sqrt{J_0}}{V_1}\n=\\frac{V_3}{V_2\\mp i\\sqrt{J_0}};\n\\end{equation}\nand inverting this dependence,\n\\begin{equation}\n\\label{Ppm2W}\nW(p^+,p^-)=\\frac{2i\\sqrt{J_0}}{p^+-p^-}{\\bf K}\n\\left(\n\\begin{array}{c}\n1\\\\\n(p^++p^-)\/2\\\\\np^+p^-\n\\end{array}\n\\right)~.\n\\end{equation}\nThe point $W(p^+,p^-)$ with coordinate $p^+$ (resp. $p^-$) being fixed moves along the straight\nline with the directing vector ${\\bf K}(1:p^+:(p^+)^2)$ (resp. ${\\bf K}(1:p^-:(p^-)^2)$)\nbelonging to the conic (\\ref{conica}).\n\n\n\\begin{lmm}\nThere exists a (spinor) representation $\\chi:\\quad O_3(J)\\to PSL_2(\\mbox{\\sets C})$ such that:\\\\\n1) The restriction of $\\chi(\\cdot)$ to $SO_3(J)$ is an isomorphism to $PSL_2(\\mbox{\\sets C})$.\\\\\n2) For coordinates $p^\\pm$ on the quadric the following transformation rule holds:\\\\\n\\begin{equation}\n\\label{Ptrans}\n\\begin{array}{ll}\np^\\pm({\\bf T}W)=\\chi({\\bf T})p^\\pm(W),&\n\\qquad{\\bf T}\\in SO_3(J),\\\\\np^\\pm({\\bf T}W)=\\chi({\\bf T})p^\\mp(W),&\n\\qquad{\\bf T}\\in O_3(J)\\setminus SO_3(J).\\\\\n\\end{array}\n\\end{equation}\n3) The linear-fractional mapping $\\chi p:= (ap+b)\/(cp+d)$ is the image of the matrix:\n\\begin{equation}\n\\label{chiT}\n{\\bf T}:=\\frac1{ad-bc}~{\\bf K}~\n\\begin{array}{||ccc||}\nd^2&2cd&c^2\\\\\nbd&ad+bc &ac\\\\\nb^2&2ab&a^2\n\\end{array}\n~{\\bf K}^{-1}\n\\in SO_3(J).\n\\end{equation}\n4) The generators of the monodromy group are mapped to the following elements:\n\\begin{equation}\n\\label{chiD}\n\\begin{array}{ll}\n\\chi({\\bf D}_s)p=\\varepsilon^{1-s}\/p,&\\qquad s=1,2,3;\\\\\n\\chi({\\bf D})p=\n\\displaystyle{\\frac{\\mu p-1}{p-\\mu}}.&\n\\end{array}\n\\end{equation}\n\\end{lmm}\nP~r~o~o~f: We define the action of matrix ${\\bf A}\\in\nSL_2(\\mbox{\\sets C})$ on the vector $V\\in\\mbox{\\sets C}^3$ by the\nformula:\n\n\\begin{equation}\n\\label{SL2action}\n{\\bf A}:=\\begin{array}{||cc||}a&b\\\\c&d\\end{array}:\n\\qquad\n\\begin{array}{||cc||}\nV_3&V_2\\\\\nV_2&V_1\n\\end{array}\n\\quad\n\\longrightarrow\n\\quad\n{\\bf A}~\n\\begin{array}{||cc||}\nV_3&V_2\\\\\nV_2&V_1\n\\end{array}\n~{\\bf A}^t.\n\\end{equation}\n  It is easy to check that (\\ref{SL2action}) gives the faithful\n  representation of connected 3-dimensional group\n  $PSL_2(\\mbox{\\sets C}):=SL_2(\\mbox{\\sets C})\/\\{\\pm {\\bf 1}\\}$ into\n  $SO_3(J_\\bullet)$ and therefore, an isomorphism. Let us  denote\n  $\\chi_\\bullet$ the inverse isomorphism  $SO_3(J_\\bullet)\\to$\n  $PSL_2(\\mbox{\\sets C})$ and let $\\chi(\\pm {\\bf\n  T}):=\\chi_\\bullet({\\bf K}^{-1}{\\bf TK})$ for ${\\bf T}\\in\n  SO_3(J)$. The obtained homomorphism $\\chi:~~$ $O_3(J)\\to$\n  $PSL_2(\\mbox{\\sets C})$ will satisfy statement 1) of the lemma.\n\n  To prove 2) we replace vector $V$ components in the right-hand\n  side of (\\ref{SL2action}) with their representation in terms of\n  the stereographic coordinates $p^\\pm$:\n\n$$\n\\frac{i\\sqrt{J_0}}{p^+-p^-}\n{\\bf A}~\n\\left[\n(p^+,1)^t\\cdot(p^-,1)+\n(p^-,1)^t\\cdot(p^+,1)\n\\right]\n~{\\bf A}^t=\n$$\n$$\ni\\sqrt{J_0}\n\\frac{(cp^++d)(cp^-+d)}{p^+-p^-}\n\\left[\n(\\chi p^+,1)^t\\cdot(\\chi p^-,1)+\n(\\chi p^-,1)^t\\cdot(\\chi p^+,1)\n\\right]\n=\n$$\n$$\n\\frac{i\\sqrt{J_0}}{\\chi p^+-\\chi p^-}\n\\left[\n(\\chi p^+,1)^t\\cdot(\\chi p^-,1)+\n(\\chi p^-,1)^t\\cdot(\\chi p^+,1)\n\\right]\n=\n$$\n$$\n\\begin{array}{||cc||}\nV_3(\\chi p^+,\\chi p^-)&V_2(\\chi p^+,\\chi p^-)\\\\\nV_2(\\chi p^+,\\chi p^-)&V_1(\\chi p^+,\\chi p^-)\n\\end{array}\n~~,\n$$\n  where we set $\\chi p:=(ap+b)\/(cp+d)$. Now (\\ref{Ptrans})\n  follows immediately for ${\\bf T}\\in SO_3(J)$. It remains to\n  check the transformation rule for any matrix ${\\bf T}$ from\n  the other component of the group $O_3(J)$, say ${\\bf T}=-{\\bf 1}$.\n\n  Writing the action (\\ref{SL2action}) component-wise we arrive\n  at conclusion 3) of the lemma.\n\n  An finally, expressions 4) for the generators of monodromy\n  group may be obtained either from analyzing formula\n  (\\ref{chiT}) or finding the eigenvectors of the matrices ${\\bf\n  D}_s,{\\bf D}$ which correspond to the fixed points of linear-\n  fractional transformations. ~~~\\rule[-1mm]{2.4mm}{2.4mm}\n\n\n\\subsection{Step 3: Projective Structures}\nSpeaking informally, complex projective structure\n  \\cite{G,Tu,GKM,Hej,Man} on the Riemann surface $M$ is a\n  meromorphic function $p$ on the universal cover $\\widetilde{M}$ of\n  the surface that transforms linear fractionally under the cover\n  transformations. The appropriate representation $\\pi_1(M)\\to $\n  $PSL_2(\\mbox{\\sets C})$ is called the {\\it monodromy} of the\n  structure $p$. The projective structure is called branched when\n  $p$ has critical points. The set of\n  all critical points of $p(t)$ with their multiplicities survives\n  under the cover transformations of $\\widetilde{\\cal M}$.\n  The projection of this set to the Riemann surface ${\\cal M}$ is known as\n  the  {\\it branching divisor} ${\\sf D}(p)$ of projective structure\n  and the branching number of the structure $p(t)$ is $\\deg{\\sf D}(p)$.\n  The classical examples of unbranched projective structures\n  arise in Fuchsian or Schottky uniformization of Riemann\n  surfaces.\n\n\\subsubsection{Projective Structures Generated by Eigenfunction}\nStereographic coordinates $p^\\pm(y):=p^\\pm(W(y))$ for the solution\nof the Riemann monodromy problem (\\ref{Rbvp}) will give two nowhere\ncoinciding  meromorphic functions in the sphere with three possibly\noverlapping slots. As it follows from the transformation rules\n(\\ref{Ptrans}), the boundary values of two functions $p^\\pm(y)$ on\nthe slot $D_*$, one of $[a_1,a_2]\\setminus [-1,~1]$,\n$[-1,~1]\\setminus [a_1,a_2]$, $[a_1,a_2]\\cap[-1,1]$ or $[a_3,a_4]$,\nare related by the formulas \\begin{equation} \\label{BVp}\n\\begin{array}{ll}\np^\\pm(y+i0)=\\chi({\\bf D}_*)p^\\mp(y-i0),&\n\\qquad y\\in D_*\\neq[a_1,a_2]\\cap[-1,~1],\\\\\np^\\pm(y+i0)=\\chi({\\bf DD_1})p^\\pm(y-i0),&\n\\qquad y\\in [a_1,a_2]\\cap[-1,~1],\n\\end{array}\n\\end{equation}\nwhere ${\\bf D}_*$ is the matrix assigned to the slot $D_*$ in (\\ref{Rbvp}).\n\n\n\nRelations (\\ref{BVp}) allow us to analytically continue both\nfunctions $p^+(y)$ and $p^-(y)$ through any slot to locally single\nvalued functions on the genus 2 Riemann surface \n\\begin{equation}\nM:=\\{w^2=(y^2-1)\\prod\\limits_{s=1}^4(y-a_s)\\}, \n\\label{M} \n\\end{equation} \\\nsince all matrices ${\\bf D}_*$ are involutive -- see Lemma \\ref{DsProperty}. \nFurther continuation gives single valued functions $p^\\pm(\\cdot)$ on the universal \ncovering $\\tilde{M}$. Traveling of the argument $y$ along the handle of the \nsurface $M$ may result in the linear- fractional transformation of the value \n$p^\\pm(y)$. Say, the continuations of $p^+(y)$ from the pants through the red \nand green slots will give two different functions on the second sheet related by \nthe linear-fractional mapping $\\chi({\\bf DD_2})$.\n\n\n\\subsubsection{Branching of Structures $p^\\pm$}\nThe way we have carried out the continuation of functions $p^\\pm(y)$ suggests\nthat the branching divisors of the arising projective structures are related via\nformula:\n\\begin{equation}\n\\label{BranchSymm}\n{\\sf D}(p^+)=H{\\sf D}(p^-)\n\\end{equation}\nwhere  $H(y,w):=(y,-w)$ is the hyperelliptic involution of the surface $M$.\nWe determine the branching numbers of the structures in the proof of\n\n\n\\begin{thrm} \\cite{Bog1} \\label{J0neq0}\n  When $\\lambda\\not\\in\\{0,1,3\\}$ the solutions $u(x)$ of the PS-3 integral equation\n  are in one-one correspondence with the\n  couples of meromorphic in the  slit sphere\n  $\\hat{\\mbox{\\sets C}}\\setminus\\{[a_1,a_2]\\cup[a_3,a_4]\\cup[-1,1]\\}$ functions\n  $p^\\pm(y)$ with boundary values satisfying (\\ref{BVp}) and either non or two\n  critical points in common. The correspondence $u(x)\\to$ $p^\\pm(y)$ is\n  implemented by the sequence of formulae (\\ref{Cauchy}), (\\ref{Wofy}) and\n  (\\ref{V2Ppm}). The inverse dependence is given up to proportionality \n  by the formula\n\\begin{equation}\n\\label{UniU}\nu(x)=\\sqrt{\\frac{\\Omega(y)}{dp^+(y)dp^-(y)}}\n(p^+(y)p^-(y) -\\mu(p^+(y)+p^-(y))+1),\n\\end{equation}\n  where $x\\in[-1,1]$ and $y:=R_3(x)+i0$,\n  $\\displaystyle{\\Omega(y)=(y-y_1)(y-y_2)\\frac{(dy)^2}{w^2(y)}}$\n  is the holomorphic quadratic differential on\n  the Riemann surface $M$ with zeroes at the critical points of\n  the (possibly coinciding) functions $p^+$ and $p^-$, or with\n  double zeroes $y_1=y_2$ (otherwise arbitrary) when $p^+=p^-$ is unbranched.\n\\end{thrm}\n\n{\\bf Remark}:  The number of the critical points of the structures\nin the slit sphere is counted with their {\\it weight and\nmultiplicity}: ~~1) the branching number of $p^\\pm(y)$ at the branch\npoint  $a\\in \\{\\pm1, a_1,\\dots,a_4\\}$ of $M$ is computed with\nrespect to the local parameter $z=\\sqrt{y-a}$, ~~2) every branch\npoint of projective structure on the boundary of the pants should be\nconsidered as a half-point.\n\n{P~r~o~o~f}: {\\bf 1.~~} Let $u(x)$ be an eigenfunction of\nintegral equation PS-3. The stereographic coordinates $p^\\pm(y)$ of\nthe solution $W(y)$ of the associated Riemann monodromy problem are nowhere equal \nmeromorphic functions when the invariant $J_0\\neq0$, or two identically equal\nfunctions when $J_0=0$. In any case they inherit the\nboundary relationship (\\ref{BVp}). \n\nWhat remains is to find the\nbranching numbers of the entangled structures $p^\\pm(y)$. To this\nend we consider the $O_3(J)$-invariant quadratic differential form \n$J(dW)=J_\\bullet(dV)$ transferred to the slit sphere.\n\nIn the general case $J_0\\neq0$ we get (up to proportionality) the \nKleinian quadratic differential:\n\\begin{equation} \n\\label{Klein} \n\\Omega(y)=\\frac{dp^+(y)dp^-(y)}\n{(p^+(y)-p^-(y))^2}, \\qquad y\\in\\hat{\\mbox{\\sets C}}. \n\\end{equation} \nThis\nexpression is the infinitesimal form of the cross ratio, hence it\nremains unchanged after the same linear-fractional transformations\nof the functions $p^+$ and $p^-$. Therefore, (\\ref{Klein}) is a well\ndefined quadratic differential on the entire sphere. Lifting\n$\\Omega(y)$ to the surface $M$ we get a holomorphic differential.\nIndeed, $p^+\\neq p^-$ everywhere and applying suitable\nlinear-fractional transformation we assume that\n$p^+=1+z^{m_+}+\\{terms~of~higher~order\\}$ and $p^-=cz^{m_-}+...$ in\nterms of local parameter $z$ of the surface, $m_\\pm\\ge1$, $c\\neq0$.\nThen $\\Omega=cm_+m_-z^{m_++m_--2}(dz)^2+\\{terms~of~higher~order\\}$.\nTherefore\n$$\n{\\sf D}(p^+)+{\\sf D}(p^-)=(\\Omega).\n$$\nAny holomorphic quadratic differential on genus 2 surface has 4 zeroes. \nThe curve $M$ consists of two copies of the slit sphere interchanged by the \nhyperelliptic involution $H$. Taking\ninto account the symmetry (\\ref{BranchSymm}) of branching divisors,\nwe see that the structures $p^\\pm$ together have two critical points in the \nslit sphere.\n\nIn the special case $J_0=0$ two structures merge: $p^\\pm(y)=:p(y)$ and the \nsame quadratic differential $J(dW)=J_\\bullet(dV)$ on the curve $M$ has the appearance:\n\\begin{equation}\n\\label{OmegaCone}\n\\Omega(y)= [V_1(y)dp(y)]^2 ,\n\\end{equation}\nhere $V_1(y)$ is the first component in the vector $V(y)$ defined by formula (\\ref{V}).\nThe analysis of this representation in local coordinates suggests that\n\\begin{equation}\n2{\\sf D}(p)+2(W)=(\\Omega),\n\\end{equation}\nwhere $(W)$ is the divisor of zeroes of the locally holomorphic (but globally multivalued)\non $M$ vector $W(y)$. To characterize $(W)$ we need the following lemma, which we prove \nat the end of the current section. \n\n\\begin{lmm}\n\\label{ZeroDivW}\nThe vector $W(y)$ cannot have simple zeroes at the fixed points of the hyperelliptic involution\nof $M$ when $J_0=0$ and $\\lambda\\neq0,3$.\n\\end{lmm}\n\nThe divisor $(W)$ is obviously invariant under the hyperelliptic involution $H$. \nFrom this Lemma it follows that either $(W)=0$ (therefore $\\deg {\\sf D}(p)=2$) \nor $(W)$ consists of two points interchanged by $H$ (therefore the structure $p$ \nis unbranched). In other words, $p(y)$ has the branching number 0 or 2 on \nthe slit sphere and the quadratic differential $\\Omega$ is a square of a holomorphic \nlinear differential. \n\n\n\n{\\bf 2.~~}\nConversely, let $p^+(y)$ and $p^-(y)$ be two not identically equal meromorphic \nfunctions on the slit sphere, with boundary conditions (\\ref{BVp}) and the total \nbranching number either zero or two (see remark above). For the meromorphic \nquadratic differential (\\ref{Klein}) on the Riemann surface $M$ we establish \n(using local coordinate on the surface) the inequality:\n\\begin{equation}\n\\label{DivIneq}\n{\\sf D}(p^+)+{\\sf D}(p^-)\\ge(\\Omega)\n\\end{equation}\nwhere the deviation from equality means that there is a point where $p^+=p^-$.\nBut the degree of the divisor on the left of (\\ref{DivIneq}) is zero or four and \n $\\deg(\\Omega)=4$. Therefore, $p^+\\neq p^-$ everywhere (and the total branching \n of this pair of functions in the slit sphere is two). \n \n The holomorphic vector $W(p^+(y),p^-(y))$ in the slit sphere solves the \n Riemann monodromy problem specified in theorem \\ref{PSeqRMP}. We already know \n how to convert the latter vector to the eigenfunction of integral equation PS-3. \nCareful computation gives the restoration formula \n\\begin{equation}\n\\label{UfromP}\n2\\pi u(x)=\\sqrt{\\frac{(\\delta+2)J_0}3}\n\\frac{p^+(y)p^-(y) -\\mu(p^+(y)+p^-(y))+1}\n{p^+(y)-p^-(y)},\n\\end{equation}\nwhere $x\\in[-1,1]$ and $y:=R_3(x)+i0$. Formula (\\ref{UniU}) appears after \nthe substitution of (\\ref{Klein}) to the latter formula.\n\nFinally, suppose that two functions $p^\\pm(y)$ with necessary branching and \nboundary behaviour are identical. For the solution on the cone,\n$V=V_1~(1,p,p^2)^t$ and the first component $V_1$ may be taken from  \n(\\ref{OmegaCone}). Therefore we consider the vector on the slit sphere:\n\\begin{equation}\n\\label{UniW}\nW(y):=\\frac{(y-y_1)}{w(y)p'(y)}\n{\\bf K}(1, p(y), p^2(y))^t,\n\\end{equation}\n  where $y_1$ is the critical point of $p(y)$ or arbitrary point when $p(y)$ is unbranched.\n  One immediately checks that it is holomorphic and solves the Riemann monodromy problem \n  specified in theorem \\ref{PSeqRMP}. Now to find the corresponding eigenfunction is a routine task.\n~~~\\rule[-1mm]{2.4mm}{2.4mm}\n\nP~r~o~o~f~~o~f~~L~e~m~m~a~ \\ref{ZeroDivW}.\nLet $z$ be local coordinate on $M$ in the vicinity of the fixed point $z=0$ \nof the hyperelliptic involution $z\\to -z$. \nBoundary relationship of the vector $W$ on the slots implies the symmetry:\n\\begin{equation}\n\\label{WSymm}\nW(-z)={\\bf D_*}W(z)\n\\end{equation}\nwhere ${\\bf D_*}$ is one of the matrices ${\\bf D_1}$, ${\\bf D_2}$, ${\\bf D_3}$ or \n${\\bf D}$. The matrix ${\\bf D_*}$ has eigenvalues $+1$, $+1$, $-1$ and the \nintersection of the cone $\\{J(W)=0\\}$ with the invariant plane corresponding to \nto the eigenvalue $+1$ contains eigenvector $W^+$ of the matrix. Suppose that \n$W(z)=W^-z+\\{terms~of~higher~order\\}$. \nSubstituting the last ansatz to (\\ref{WSymm}) we see that $W^-$ \nis the eigenvector of the matrix corresponding to eigenvalue $-1$. Let $J(\\cdot,\\cdot)$\nbe the bilinear form polar to quadratic form $J(\\cdot)$, then \n$$\nJ(W^+,W^-)=\nJ({\\bf D_*}W^+,{\\bf D_*}W^-)=\n-J(W^+,W^-)=0.\n$$\nOnce vector $W(z)$ varies in the cone, $J(W^-)=0$. Now we see that the cone contains \nthe entire plane generated by the vectors $W^+$ and $W^-$. Therefore the cone is degenerate \nwhich only happens when $\\lambda=0,3$. ~~~\\rule[-1mm]{2.4mm}{2.4mm}\n\n\n\n\\subsection{Mirror Symmetry of Solution}\n\\label{MirrorSym}\n  In what follows we are looking for {\\it real} solutions $u(x)$\n  of the integral equation (\\ref{PSE}). There is no loss of the\n  generality. Indeed, the restrictions on the monodromy of projective\n  structures \\cite{Bog1} imply that the spectrum\n  of any PS-3 integral equation belongs to the segment $[0,3]$.\n  Now both  real and imaginary parts of any complex eigenfunction\n  $u(x)$ are the solutions of the integral equation.\n\n\n\n  Real solutions $u(x)$ of the integral equation correspond to\n  the solutions of Riemann monodromy problem with mirror symmetry:\n  $W(\\bar{y})=\\overline{W}(y)$. This symmetry for the vector $V(y):={\\bf\n  K}^{-1}W(y)$  takes the form\n$\nV(\\bar{y})=\n(\\overline{V_3}(y)~{\\rm sign}(\\delta+2),\n(\\overline{V_2}(y)~{\\rm sign}(\\delta-1),\n(\\overline{V_1}(y)~{\\rm sign}(\\delta+2))\n$.\n  The values $\\delta+2$ and $\\delta-1$ have the same sign as it follows\n  from the range of spectral parameter $\\lambda\\in[0,3]$. Therefore, real\n  solutions are split into two classes depending on the sign\n  of $(\\delta+2)J_0$:\n  $$\n\\begin{array}{ll}\nSymmetric~~ ((\\delta+ 2)J_0>0),& p^\\pm(\\bar{y})=1\/\\overline{p^\\pm}(y)\\\\\nAntisymmetric~~ ((\\delta+ 2)J_0\\le0),& p^\\pm(\\bar{y})=1\/\\overline{p^\\mp}(y) \\\\\n\\end{array},\n\\qquad y\\in {\\cal P}(R_3)\\setminus[-1,~1],\n$$\n\n\n\n  In the remaining part of the article we give explicit parametrization of all\n  antisymmetric solutions for the integral PS-3 equations of\n  the considered type -- when six points $\\pm1, a_1,\\dots,a_4$ are\n  real and pairwise distinct.\n\nRestricting ourselves to the search of antisymmetric solutions we\nhave to find only one function in the pants, say $p(y):=p^+(y)$ while the remaining function may be\nrecovered from the mirror antisymmetry:\n\\begin{equation}\n\\label{Antisymm}\np^-(y)=1\/\\overline{p^+(\\bar{y})}.\n\\end{equation}\nOn the boundary components \nof the slit sphere this function obeys the rule:\n$$\np^+(y\\pm i0)=\n\\chi({\\bf D}_*)p^-(y\\mp i0)=\n\\chi({\\bf D}_*{\\bf D}_1)\\overline{p^+(y\\pm i0)},\n\\qquad y\\in D_*\\neq\n[-1,1]\\cap[a_1,a_2].\n$$\nTherefore\n\n\\centerline{\n\\begin{tabular}{cc}\n$p\\in\\hat{\\mbox{\\sets R}}$,& when ${\\bf D}_*={\\bf D}_1$;\\\\\n$p\\in\\varepsilon\\hat{\\mbox{\\sets R}}$,& when ${\\bf D}_*={\\bf D}_2$;\\\\\n$p\\in\\varepsilon^2\\hat{\\mbox{\\sets R}}$,& when ${\\bf D}_*={\\bf D}_3$;\\\\\n\\end{tabular}}\n\n\\noindent\nand finally when ${\\bf D}_*={\\bf D}$, the value of $p$ lies on the circle\n\\begin{equation}\n\\label{circle}\nC:=\n\\{~p\\in\\mbox{\\sets C}:\n\\quad |p-\\mu^{-1}|^2=\\mu^{-2}-1~~   \\},\n\\qquad \\mu:=\\sqrt{\\frac{3-\\lambda}{2\\lambda}}\n\\end{equation}\n\nAs an immediate consequence of this observation we give a universal restriction for the\nspectrum of our integral equation\n\\begin{lmm}\n Antisymmetric eigenfunctions correspond to eigenvalues\n$\\lambda\\in [1,3]$.\n\\label{SpLocus}\n\\end{lmm}\nP~r~o~o~f: For the cases ${\\cal A}$, ${\\cal B}1$, ${\\cal B}22$, ${\\cal B}23$ the\nslot $[-1,1]\\setminus[a_1,a_2]$ is not empty and the boundary value of  $p(y)$\non this slot belongs to the circle $C$. This circle is an empty set for $\\mu>1$, or equivalently\n$\\lambda\\in(0,1)$. The proof for the remaining case ${\\cal B}21$ requires special machinery and\n will be given in Sect. \\ref{B2Proof}. ~~\\rule[-1mm]{2.4mm}{2.4mm}\\\\[5mm]\n\n\nThe critical points of two functions $p^+(y)$ and $p^-(y)$ in the considered antisymmetric case\nare complex conjugate as it follows from  (\\ref{Antisymm}). Taking this fact into account we \nreformulate Theorem  \\ref{J0neq0} for the antisymmetric solutions:\n\n\\begin{thrm}\nWhen $\\lambda\\not\\in\\{0,1,3\\}$, the antisymmetric solutions $u(x)$ of\nintegral equation {\\rm PS-3} are in one-two correspondence with the meromorphic \nin the slit sphere $\\hat{\\mbox{\\sets C}}\\setminus\\{[a_1,a_2]\\cup[a_3,a_4]\\cup[-1,1]\\}$\nfunctions $p(y)$ that have either none or one critical point\nin the domain and the following values on the boundary components:\\\\[1mm]\n\n\\begin{tabular}{c||c|c|c}\n$y\\in$ & $[-1,1]\\setminus[a_1,a_2]$ (red)& $[a_1,a_2]\\setminus[-1,1]$ (blue)& $[a_3,a_4]$ (green)\\\\\n\\hline\n$p(y\\pm i0)\\in$& $C$ &\n\\begin{tabular}{c}\n$\\varepsilon^2\\hat{\\mbox{\\sets R}}$ (Case ${\\cal A}$)\\\\\n$\\hat{\\mbox{\\sets R}}$  (Case ${\\cal B}$)\n\\end{tabular}\n& $\\varepsilon\\hat{\\mbox{\\sets R}}$\\\\\n\\end{tabular}\n\n\\vspace{1mm}\n\\noindent\nIn case ${\\cal B}2$ the function $p(y)$ has the jump on the remaining part of the boundary:\n\\begin{equation}\n\\label{Jump}\np(y+i0)=\\chi({\\bf DD}_1)p(y-i0),\n\\quad y\\in [-1,1]\\cap[a_1,a_2].\n\\end{equation}\n\\label{SemiMainTh}\n\\end{thrm}\n\n{\\bf Remark~}\nBy 'one-two' correspondence we mean the following: \ngiven any function $p(y)$ satisfying the conditions of this theorem,\nit's easy to check that its {\\it antisymmetrization} $1\/\\overline{p(\\bar{y})}$\nalso satisfies all the conditions. Therefore, we have a correspondence of an eigenfunction \n$u(x)$ to a couple:  function $p(y)$ and its antisymmetrization, only one of them being independent.\n\n\n\\section{Statement of the main result}\n\\label{Result}\n\n  From the Sect. \\ref{MirrorSym} it follows that every\n  antisymmetric eigenfunction $u(x)$ of PS-3 integral equation\n  induces a mapping $p(y)$ of the pants ${\\cal P}(R_3)$ to a\n  multivalent domain spread possibly with branching over the\n  Riemann sphere. Such surface is known as Kleinian membrane \n  or \\\"Uberlagerungsfl\\\"ache and the complete list of\n  them is given in this section.\n\n\\subsection{Tailoring the Pants}\n\nWe define pants ${\\cal PQ}(\\lambda, h_1, h_2|m_1, \\dots)$ of several\nfashions $\\cal Q$ which parametrically depend on spectral parameter\n$\\lambda$, two other reals $h_1,~h_2$ and one or two integers\n$m_1,\\dots$. Each boundary oval of our pair of pants covers a circle\nand acquires its color in the following way:\n\n\\begin{center}\n\\begin{tabular}{ll}\n$C$&     -- red,\\\\\n$\\varepsilon\\hat{\\mbox{\\sets R}}$ or\n$\\chi({\\bf DD}_1)\\varepsilon\\hat{\\mbox{\\sets R}}$ & -- green,\\\\\n$\\hat{\\mbox{\\sets R}}$ or $\\varepsilon^2\\hat{\\mbox{\\sets R}}$& -- blue.\\\\\n\\end{tabular}\n\\end{center}\n\nAny constructed pair of pants may be obtained from the \"basic\" pants ${\\cal\nPQ}(\\lambda, h_1, h_2|\\dots)$ with lowest possible integer parameters by a\nsurgery procedure known as \"grafting\" and introduced independently by B.Maskit,\nD.Hejhal and D.Sullivan--W.Thurston in 1969-1983.\n\n\n\\subsubsection{Cases ${\\cal A,~B}1$}\n\nFor real $\\lambda\\in(1,2)$ we consider (depending on $\\lambda$) open annulus\n$\\alpha$ bounded by two circles: $C$ defined in (\\ref{circle}) and\n$\\varepsilon\\hat{\\mbox{\\sets R}}$. Another annulus bounded by $C$ and\n$\\varepsilon^2\\hat{\\mbox{\\sets R}}$ we denote $\\bar{\\alpha}$. Note that for\nthe considered values of $\\lambda$ the circle $C$ does not intersect the lines\n$\\varepsilon^{\\pm1}\\mbox{\\sets R}$. The $m-$ sheeted unbranched covering of the\nannuli, $m=1,2,\\dots$, we denote as $m\\cdot\\alpha$ or  $m\\cdot\\bar{\\alpha}$\ncorrespondingly.\n\nThe pants of four fashions ${\\cal PA}_1$, ${\\cal PA}_2$, ${\\cal PA}_3$, ${\\cal PB}1$\nare sewn together of the annuli we have introduced in the way specified in Tab. \\ref{PantsAB1}. \n\n\n\n\\begin{table}[h!]\n\\centering\n\\begin{tabular}\n{l|l|l}\nFashion of Pants  & Range of $h_1,~h_2$ and $m_1,~m_2$ & {\\bf Definition}\\\\\n\\hline\n${\\cal PA}_1(\\lambda,h_1,h_2|~m_1,m_2)$&\n\\begin{tabular}{l}\n$h:=h_1+ih_2\\in\\alpha\\cap\\bar{\\alpha}$, $|h|\\ge1$;\n\\\\\n$m_1,m_2=1,2,\\dots$\n\\end{tabular}&\n\\begin{tabular}{l}\n$Cl\\{(m_1\\cdot\\alpha)\\setminus[\\mu^{-1}, h]\\}+$\\\\\n$Cl\\{(m_2\\cdot\\bar{\\alpha})\\setminus[\\mu^{-1}, h]\\}$\n\\end{tabular}\\\\\n\\hline\n${\\cal PA}_2(\\lambda,h_1,h_2|~m_1,m_2)$&\n\\begin{tabular}{l}\n$0<h_1<h_2$, $h_1h_2\\ge1$;\\\\\n$m_1=1,2,\\dots$, $m_2=0,1,2,\\dots$\n\\end{tabular}&\n\\begin{tabular}{l}\n$Cl\\{(m_1\\cdot\\alpha)\\setminus-\\varepsilon^2[h_1,h_2]\\}+$\\\\\n$Cl\\{m_2\\cdot\\bar{\\alpha}\\}$\n\\end{tabular}\\\\\n\\hline\n${\\cal PA}_3(\\lambda,h_1,h_2|~m_1,m_2)$&\n\\begin{tabular}{l}\n$0<h_1<h_2$, $h_1h_2\\ge1$; \\\\\n$m_1=0,1,2,\\dots$, $m_2=1,2,3,\\dots$\n\\end{tabular}&\n\\begin{tabular}{l}\n$Cl\\{(m_2\\cdot\\bar{\\alpha})\\setminus-\\varepsilon[h_1,h_2]\\}+$\\\\\n$Cl\\{m_1\\cdot{\\alpha}\\}$\n\\end{tabular}\\\\\n\\hline\n${\\cal PB}1(\\lambda,h_1,h_2|~m)$&\n\\begin{tabular}{l}\n$\\mu^{-1}+\\sqrt{\\mu^{-2}-1} <h_1<h_2$;\\\\\n$m=1,2,3,\\dots$\n\\end{tabular}&\n$Cl\\{(m\\cdot\\alpha)\\setminus[h_1,h_2]\\}$\\\\\n\\hline\n\\end{tabular}\n\\caption{Three-parametric families of pairs of pants for the cases $\\cal A$, ${\\cal B}1$;\nparameter $1<\\lambda<2$}\n\\label{PantsAB1}\n\\end{table}\n\n\\vspace{2mm}\nSign '+' in the definitions of Tab. \\ref{PantsAB1} means certain surgery explained below.\n\n\n\\begin{figure}\n\\centerline{\\psfig{figure=fa1.eps}}\n\\caption[]\n{\\small The pair of pants  ${\\cal PA}_1(\\lambda,h_1,h_2|~m_1,m_2)$ is sewn down of two annuli}\n\\label{FA1}\n\\end{figure}\n\n\\begin{figure}\n\\centerline{\\psfig{figure=fa2.eps}}\n\\caption[]\n{\\small The pair of pants ${\\cal PA}_2(\\lambda,h_1,h_2|~m_1,m_2)$ is sewn of\nsimpler pants and the annulus}\n\\label{FA2}\n\\end{figure}\n\n\n\\paragraph{Instructions on sewing annular patches together. }\n\n{\\bf 1.} ${\\cal PA}_1(\\lambda,h_1,h_2|~m_1,m_2)$.\n  Take two annuli $m_1\\cdot\\alpha$ and  $m_2\\cdot\\overline{\\alpha}$.  Cut the\n  top sheet of each annulus along the same segment (dashed red line in the\n  Fig. \\ref{FA1}) starting at the point $h:=h_1+ih_2$ and ending at the circle\n  $C$. Now sew the left bank of one cut on the right bank of the other. The\n  emerging surface is the pair of pants.\n\n{\\bf 2.} ${\\cal PA}_2(\\lambda,h_1,h_2|~m_1,m_2)$.\n    The annulus $m_1\\cdot\\alpha$ with the segment $-\\varepsilon^2[h_1,h_2]$ removed\n  from the top sheet is a pair of pants ${\\cal PA}_2(\\lambda,h_1,h_2|m_1,0)$. Cut\n  the obtained pair of pants along the segment joining the circle $C$ to the\n  slot (dashed blue line in the Fig. \\ref{FA2}). Also cut top sheet of the annulus\n  $m_2\\cdot\\overline{\\alpha}$ along the same segment and sew the left bank of\n  one cut on the right bank of the other. The arising surface is the\n  pair of pants.\n\n{\\bf 3.} ${\\cal PA}_3(\\lambda,h_1,h_2|~m_1,m_2)$.\nThe annulus $m_2\\cdot\\bar{\\alpha}$ with the segment $-\\varepsilon[h_1,h_2]$ removed\nfrom the top sheet, is a pair of pants ${\\cal PA}_3(\\lambda,h_1,h_2|0,m_2)$. As\nin the previous passage, we may sew in the annulus $m_1\\cdot{\\alpha}$ to the\nobtained pants to get the result.\n\n{\\bf 4.} ${\\cal PB}1(\\lambda,h_1,h_2|~m)$ is just the annulus $m\\cdot\\alpha$\nwith the segment $[h_1,h_2]$ removed  from its top sheet.\n\n\n\n\n\n  The limit case of the pants ${\\cal PA}_1$, when the branch point\n  $h_1+ih_2$ tends to $\\varepsilon^{\\pm1}\\mbox{\\sets R}$, coincides with the\n  limit cases of pants ${\\cal PA}_2$ or ${\\cal PA}_3$, when $h_1=h_2>0$.\nThe corresponding unstable two-parametric families of pants \n ${\\cal PA}_{12}$ and  ${\\cal PA}_{13}$ are defined in Tab. \\ref{InterMediate}.\n\n\\begin{table}[h!]\n\\centering\n\\begin{tabular}{l|l}\nFashion of Pants  & {\\bf Definition}\\\\\n\\hline\n${\\cal PA}_{12}(\\lambda,h|~m_1,m_2)$&\n \\begin{tabular}{l}\n  ${\\cal PA}_1(\\lambda, -Re(\\varepsilon^2h),-Im(\\varepsilon^2h)|m_1,m_2)=$\\\\\n  ${\\cal PA}_2(\\lambda, h,h|m_1,m_2)$\n \\end{tabular}\\\\\n \\hline\n${\\cal PA}_{13}(\\lambda,h|~m_1,m_2)$&\n \\begin{tabular}{l}\n  ${\\cal PA}_1(\\lambda, -Re(\\varepsilon h),-Im(\\varepsilon h)|m_1,m_2)=$\\\\\n  ${\\cal PA}_3(\\lambda, h,h|m_1,m_2)$\\\\\n  \\end{tabular}\n\\end{tabular}\n\\caption{Unstable two-parametric families of pants.\nThe range of parameters: $1<\\lambda<2$, $h>0$, $m_1$ and $m_2=1,2,3,\\dots$.}\n\\label{InterMediate}\n\\end{table}\n\n\\subsubsection{Case ${\\cal B}2$}\nTwo circles: $\\varepsilon\\hat{\\mbox{\\sets R}}$ and $\\chi({\\bf\nDD}_1)\\varepsilon\\hat{\\mbox{\\sets R}}$ do not intersect when $\\lambda\\in(1,3)$.\nThey bound the open annulus $\\beta$ depending on $\\lambda$. The $m-$ sheeted\nunbranched covering of the annulus  we denote as $m\\cdot\\beta$, $m=1,2,3,\\dots$\nThe points of the annulus $m\\cdot\\beta$ may be described in the form\n$$\np=\\mu^{-1}+\\rho\\exp(i\\phi),\n$$\nwhere $\\rho>0$ and the argument $\\phi\\in$ {\\sets R} $mod~2\\pi m$.\nThe action of $\\chi({\\bf DD}_1)$ on the sphere (i.e. ~consecutive\nreflections in circles $C$ and $\\hat{\\mbox{\\sets R}}$) lifts to the\ninvolution of the multi-sheeted annulus $m\\cdot\\beta$ in the\nfollowing way:\n\\begin{equation}\n\\Xi: \\mu^{-1}+\\rho\\exp(i\\phi)\\to\\mu^{-1}+\\frac{r^2}\\rho\\exp(-i\\phi)\n\\label{mBetaInv} \n\\end{equation}\nwhere $r:=\\sqrt{\\mu^{-2}-1}$ is the radius of the circle $C$.\n\n{\\bf Definition.} We introduce three pairs of pants ${\\cal PB}21$, ${\\cal PB}22$\nand ${\\cal PB}23$, each of them depend on three reals $\\lambda$, $h_1$, $h_2$ and \nan integer $m$:\n$$\n{\\cal PB}2s(\\lambda,h_1,h_2|~m):=Cl\\{\n(m\\cdot\\beta)\\setminus(E_s^1(h_1)\\cup E_s^2(h_2))\\}\/\\Xi,\n\\quad s=1,2,3,\n$$\nwhere the slots $E_s^1,E_s^2$ are defined in the Tab. \\ref{SlotsDef}. The slots\nare invariant with respect to the involution $\\Xi$ and do not intersect each other as \nwell as the boundary of the annulus $m\\cdot\\beta$.\n\n\\begin{table}[h!]\n\\centering\n\\begin{tabular}{l|l}\n{\\bf Definition} of slots & Range of $h_1,~h_2$\\\\\n\\hline\n\\begin{tabular}{l}\n$E_1^1(h_1):=\\mu^{-1}+r\\exp[-h_1,h_1]$,\\\\\n$E_1^2(h_2):=\\mu^{-1}+r\\exp[-h_2,h_2]\\exp(i\\pi m)$\n\\end{tabular}\n&\n\\begin{tabular}{l}\n$h_1\\ge h_2>0$,\\\\ when $m$ is even;\\\\\n$(\\mu^{-1}+r\\exp~h_1)\\cdot$\\\\\n$(\\mu^{-1}-r\\exp~h_2)\\ge1$,\\\\ when $m$ is odd\n\\end{tabular}\\\\\n\\hline\n\\begin{tabular}{l}\n$E_2^1(h_1):=\\mu^{-1}+r\\exp[-ih_1,ih_1]$,\\\\\n$E_2^2(h_2):=\\mu^{-1}+r\\exp[-ih_2,ih_2]\\exp(i\\pi m)$\n\\end{tabular}\n&\n\\begin{tabular}{l}\n$h_1\\ge h_2$, when $m$ is even;\\\\\n${\\rm Arg}(\\exp(ih_1)+\\mu r)\\ge$\\\\\n${\\rm Arg}(\\exp(ih_2)-\\mu r)$\\\\\nwhen $m$ is odd;\\\\\n$h_1+h_2<m\\pi$, $h_2>0$,\\\\\nfor any $m$\n\\end{tabular}\\\\\n\\hline\n\\begin{tabular}{l}\n$E_3^1(h_1):=\\mu^{-1}+r\\exp[-h_1,h_1]$,\\\\\n$E_3^2(h_2):=\\mu^{-1}+r\\exp[-ih_2,ih_2]\\exp(i\\pi m)$\n\\end{tabular}\n& $h_1>0$, $m\\pi>h_2>0$\\\\\n\\hline\n\\end{tabular}\n\\caption{Slots for the subcases of ${\\cal B}2$, parameter $1<\\lambda<3$.}\n\\label{SlotsDef}\n\\end{table}\n\nTo understand the interrelation of introduced constructions it is very useful \nto imagine how the pants ${\\cal PB}1$\nare transformed to the pants of fashion ${\\cal Q=B}23$ and the latter --\nto the pair of pants  ${\\cal PB}21$ or ${\\cal PB}22$.\n\n\n\n\n\n\n\n\n\n\\subsection{The main theorem}\n\\begin{thrm}\n\n  When $\\lambda\\neq 1,3$ the antisymmetric eigenfunctions of {\\rm PS-3} integral\n  equation for the case ${\\cal Q=A}$, ${\\cal B}1$, ${\\cal B}21$, ${\\cal B}22$,\n  ${\\cal B}23$ are in one to one correspondence with the pairs of pants\n  \\footnote{For the case ${\\cal A}$ there are three stable and two\n  unstable pants fashions ${\\cal PA}_*(\\dots)$ } ${\\cal\n  PQ}(\\lambda, h_1,h_2|m_1..)$ conformally equivalent to the  pants\n  (\\ref{R3Pants}) associated to the functional parameter of integral equation.\n\nLet the function $p(y)$ conformally maps the pair of pants ${\\cal P}(R_3)$  to the pants\n${\\cal PQ}(\\lambda, h_1,h_2|m_1..)$, then up to proportionality\n\n\\begin{equation}\nu(x)=\n\\left\\{\n\\begin{array}{ll}\n\\displaystyle{\n\\sqrt{\\frac{(y-y_1)(y-y_2)}{p'(y^+)p'(y^-)}}\n\\frac{p(y^+)-p(y^-)}{w(y)}},&\\quad x\\in[-1,1]\\setminus[a_1,a_2],\\\\\n\\displaystyle{\n\\sqrt{(y-y_1)(y-y_2)}\n\\frac{{\\rm Im}~p(y^+)}{w(y)|p'(y^+)|}},&\\quad x\\in[-1,1]\\cap[a_1,a_2].\n\\end{array}\n\\right.\n\\label{UviaP}\n\\end{equation}\n\nHere $y:=R_3(x)$, $y^\\pm:=y\\pm i0$. For the fashion ${\\cal Q=A}_1$,\n$y_1=\\overline{y_2}$ is the inner critical point of the function $p(y)$;\nfor other fashions $\\cal Q$ real $y_1$ and $y_2$ are boundary critical points\nof the function $p(y)$.\n\\end{thrm}\n\nThe proof of the main theorem for the cases ${\\cal A,B}1$ is given in Sect. \\ref{AB1Proof}\nand for the case ${\\cal B}2$ -- in Sect. \\ref{B2Proof}.\n\n\n\n\\subsection{Corollaries}\nThe representation (\\ref{UviaP}) cannot be called explicit in the usual sense,\nsince it comprises a transcendent function $p(y)$. We show that nevertheless\nthe representation allows us to understand the following\nproperties of the solutions.\n\n{\\bf 1.} The \"antisymmetric\" part of the spectrum is always a subset of $[1,3]$;\nfor the equations of types ${\\cal A,B}1$ this part of the spectrum always lies in\n$[1,2]\\cup\\{3\\}$.\n\n{\\bf 2.} Every $\\lambda\\in (1,3)$ is the eigenvalue for infinitely many equations PS-3.\n\nP~r~o~o~f.~ Any of the constructed pants may be conformally mapped to the standard\nform: the sphere with three real slots. Now we can apply the procedure of the Sect.\n\\ref{R3fromPants} and get the functional parameter $R_3(x)$ such that the\nassociated pair of pants is conformally equivalent to the pants we started from.\n\n{\\bf 3.} Eigenfunction $u(x)$ related to the pants ${\\cal PQ}(\\dots|m_1,m_2)$\nhas exactly $m_1+m_2+1$ zeroes on the segment $[-1,1]$ when ${\\cal Q=A}$, ${\\cal B}1$.\n\n  P~r~o~o~f.~ According to the formula\n  (\\ref{UviaP}), the number of zeroes of eigenfunction  $u(x)$ is equal to\n  the number of points $y\\in[-1,1]$ where $p(y^+)=p(y^-)$. This\n  number in turn is equal to the number of solutions of the\n  inclusion\n\n\\begin{equation}\nS(y):=\nArg [p(y^-)-\\mu^{-1}] -\nArg [p(y^+)-\\mu^{-1}]\n\\quad\\in 2\\pi\\mbox{\\sets Z},\n\\qquad y\\in[-1,1].\n\\label{Inclus}\n\\end{equation}\n\n  Let the point $p(y)$ goes $m$ times around the circle $C$  when\n  its argument $y$ travels along the banks of $[-1,1]$. Integer $m$\n  is naturally related to the integer parameters of pants ${\\cal\n  PQ}(\\dots)$. The function $S(y)$ strictly increases from $0$\n  to $2\\pi m$ on the segment $[-1,1]$, therefore the inclusion\n  (\\ref{Inclus}) has exactly $m+1$ solutions on the mentioned\n  segment. ~~~\\rule[-1mm]{2.4mm}{2.4mm}\n\n  {\\bf 4.} The mechanism for arising the discrete spectrum of the integral\n  equation is explained. Sewing annuli down to the pants ${\\cal\n  PQ}(\\lambda,h_1,h_2|\\dots)$ one changes the conformal structure of\n  the latter. To return to the conformal structure specified by ${\\cal P}(R_3)$ we have to\n  change the real parameters of the pants, one of them is the\n  spectral parameter $\\lambda$.\n\n  In a sense, the eigenvalue problem (\\ref{PSE}) is reduced to\n  the solution of three equations for three unknown numbers.\n  These equations relate moduli of given pants ${\\cal P}(R_3)$ to\n  the moduli of membrane with real parameters $\\lambda$, $h_1$,\n  $h_2$ and extra discreet parameters.\n\n\n\n\n\n\n\n\\section{Auxiliary constructions}\n\\label{Construct}\nThe combinatorial analysis of the arising projective structure $p(y)$ is based on\ntwo constructions we describe below.\n\n  Let $p(y)$ be a holomorphic map from a Riemann surface $\\cal M$\n  with a boundary  to the sphere and the selected boundary\n  component $(\\partial {\\cal M})_*$ is mapped to a circle. The\n  reflection principle allows us to holomorphically continue\n  $p(y)$ through this selected component to the double of $\\cal\n  M$. Therefore we can talk of the critical points of $p(y)$ on\n  $(\\partial {\\cal M})_*$. When the argument $y$ passes through a\n  simple critical point the value $p(y)$ reverses the direction\n  of its movement on the circle. So there should be even number\n  of critical points (counted with multiplicities) on the\n  selected boundary component.\n\n\\subsection{Construction 1 (no boundary critical points)}\n\\label{Construct1} Using otherwise a composition with suitable\nlinear-fractional map, we suppose that the circle $p((\\partial {\\cal\nM})_*)$ is the boundary of the unitary disc \\begin{equation} \\label{Udisc}\n\\mbox{\\sets U}:=\\{p\\in\\mbox{\\sets C}:\\quad |p|\\le 1\\}, \\end{equation} and a\nsmall annular vicinity of the selected boundary component is mapped\nto the exterior of the unit disc. We define the mapping of a\ndisjoint union ${\\cal M}\\sqcup\\mbox{\\sets U}$ to a sphere:\n\\begin{equation}\n\\tilde{p}(y):=\\left\\{\n\\begin{array}{ll}\np(y),& y\\in {\\cal M},\\\\\nL(y^d),& y\\in \\mbox{\\sets U},\n\\end{array}\n\\right.\n\\end{equation}\nwhere integer $d>0$ is the degree of the mapping\n$p:$  $(\\partial {\\cal M})_*\\to$ $\\partial${\\sets U}\nand $L(y)$ is (at the moment arbitrary) linear fractional mapping\nkeeping the unitary disc (\\ref{Udisc}). The choice of $L(\\cdot)$\nwill be done later to simplify the arising combinatorial analysis.\n\\index{winding number}\n\n  Now we fill in the hole in ${\\cal M}$ by the unit disc,\n  identifying the points of $(\\partial {\\cal M})_*$ and the\n  points of $\\partial${\\sets U} with the same value of\n  $\\tilde{p}$ (there are $d$ ways to do so). The holomorphic\n  mapping $\\tilde{p}(y)$ of the new Riemann surface ${\\cal\n  M}\\cup${\\sets U} to the sphere will have exactly one additional\n  critical point of multiplicity $d-1$ at the center of the glued\n  disc.\n\n\n\n\\subsection{Construction 2 (two boundary critical points)}\\label{Construct2}\n  Let again $p(y)$ be a holomorphic mapping of a bounded Riemann surface $\\cal\n  M$ to the sphere with  selected boundary component $(\\partial {\\cal M})_*$\n  being mapped to the boundary of the unit disc {\\sets U}. Now the mapping\n  $p(y)$  has two simple critical points on the selected boundary component (the\n  case of coinciding critical values is not excluded). Those two points divide\n  the oval $(\\partial {\\cal M})_*$ into two segments: $(\\partial {\\cal M})_*^+$\n  and $(\\partial {\\cal M})_*^-$. We define two positive integers, {\\it partial winding numbers}\n  $d^\\pm$ as follows. As the point $y$ moves round the selected oval in the\n  positive direction, the increment of $\\arg ~p(y)$ on the segment $(\\partial\n  {\\cal M})_*^+$ is $2\\pi d^+-\\phi$, $0<\\phi\\le2\\pi$, and the decrement on the\n  segment $(\\partial {\\cal M})_*^-$ is $2\\pi d^-- \\phi$. We are going to modify\n  the Riemann surface $\\cal M$, sewing down one segment of $(\\partial {\\cal\n  M})_*$ to the other and filling the remaining hole (if any) with the patch\n  $\\mbox{\\sets U}$.\n\n\n\\begin{figure}[h]\n\\begin{picture}(170,50)\n\\thicklines\n\\put(30,40){\\oval(40,10)}\n\\put(5,40){\\oval(10,60)[br]}\n\\put(55,40){\\oval(10,60)[bl]}\n\\thinlines\n\\put(30,10){\\oval(50,13)[b]}\n\\put(20,35){\\vector(-1,0){3}}\n\\put(40,45){\\vector(1,0){3}}\n\\put(30,34){$*$}\n\\put(30,44){$*$}\n\\put(30,10){$\\cal M$}\n\\put(10,48){$(\\partial{\\cal M})_*$}\n\\put(75,23){\\vector(1,0){20}}\n\\put(80,25){$p(y)$}\n\n\\put(120,5){\\psfig{figure=windnum2.eps}}\n\\end{picture}\n\n\\caption[] {\\normalsize Mapping of the boundary component\n$(\\partial{\\cal M})_*$ with two simple branch points $*$ on it and\npartial winding numbers $d^+=1, d^-=2$.} \\label{BdryCritPts}\n\\end{figure}\n\n\n\nWe define the mapping from the disjoint union ${\\cal M}\\sqcup\\mbox{\\sets U}$\nto the sphere:\n\\begin{equation}\n\\tilde{p}(y):=\\left\\{\n\\begin{array}{ll}\np(y),& y\\in {\\cal M},\\\\\nL(y^{d^--d^+}),& y\\in \\mbox{\\sets U},\\\\\n\\end{array}\n\\right.\n\\end{equation}\nwhere $L(\\cdot)$ is a linear fractional mapping\nkeeping the unitary disc (\\ref{Udisc}) invariant.\n\n Without loss of the generality we suppose that $d^-\\ge d^+$. We  sew\n $(\\partial {\\cal M})_*^+$ to a part of $(\\partial {\\cal M})_*^-$, starting from\n one of the boundary critical points and consecutively identifying the points\n of the boundary oval with the same value of $p(y)$. If $d^-=d^+$ the hole disappears,\n otherwise we identify the remnant of $(\\partial {\\cal M})_*^-$ with the boundary of\n {\\sets U} gluing points with the same value of $\\tilde{p}(y)$ as shown in the Fig.\n \\ref{DiscNCut}(a).\n\n The holomorphic mapping $\\tilde{p}(y)$ of the modified Riemann surface\nto the sphere will have an additional critical point of multiplicity $d^--d^+$\nat the center of the artificially attached disc.\nWhen $d^->d^+$ a simple critical point in the place of one of the old boundary critical\npoints arises. When $d^-=d^+$ no additional critical points arises.\n\n\\begin{figure}\n\\begin{picture}(170,50)\n\\thicklines\n\\put(30,25){\\oval(30,30)[t]}\n\\put(30,25){\\oval(30,30)[bl]}\n\\put(30,10){\\line(1,0){40}}\n\\put(45,11){\\line(1,0){25}}\n\\put(45,25){\\line(0,-1){14}}\n\\thinlines\n\\put(69,9.5){$*$}\n\\put(44,8.5){$*$}\n\n\\put(0,25){(a)}\n\\put(80,25){(b)}\n\n\\put(28,23){{\\sets U}}\n\\put(55,23){$\\cal M$}\n\n\\put(25,42){$(\\partial{\\cal M})_*^-$}\n\\put(50,5){$(\\partial{\\cal M})_*^+$}\n\\put(80,-5){\n\\begin{picture}(70,40)\n\\put(10,40){${\\cal P}(R_3)$}\n\\put(30,40){$\\stackrel{inclusion}\\longrightarrow$}\n\\put(50,40){$\\mbox{\\sets C}P^1$}\n\n\\put(15,38){\\vector(2,-1){30}}\n\\put(23,26){$p(y)$}\n\\put(53,38){\\vector(0,-1){15}}\n\\put(58,30){$\\tilde{p}$ is branched}\n\\put(62,26){covering.}\n\\put(50,18){$\\mbox{\\sets C}P^1$}\n\\end{picture}}\n\\end{picture}\n\n\\caption[] {\\normalsize (a) Filling in the hole bounded by $(\\partial{\\cal M})_*$\\hfill\n~~~~~~~~(b) Splitting the mapping $p(y)$}\n\\label{DiscNCut}\n\\end{figure}\n\n\n\\section{Proof for the Cases ${\\cal A,B}1$}\n\\label{AB1Proof}\n\\subsection{Eigenfunction gives Pair of Pants}\n\nWe already know that every antisymmetric eigenfunction of integral equation PS-3\ngenerates the mapping $p(y)$ from the pants ${\\cal P}(R_3)$ to the sphere. The\nboundary ovals of the pants are mapped to three circles specified in Theorem\n\\ref{SemiMainTh} and the function $p(y)$ may have either ({\\bf a}) no critical points,\n({\\bf b}) one simple critical point inside the pants, ({\\bf c}) two boundary simple critical\npoints or ({\\bf d}) one double critical point on the boundary. The first two\npossibilities will be considered in Sect. \\ref{InnerBranch} and the other two --- in the\nSect. \\ref{BoundaryBranch}\n\n\\subsubsection{No critical Points on the Boundary of Pants}\\label{InnerBranch}\n\\paragraph{Branched Covering of a Sphere.}\n\n  Suppose that the point $p(y)$ winds around the corresponding circle $d_r$, $d_g$ and\n  $d_b$ times when the argument $y$ runs around the 'red','green' and 'blue'\n  boundary component of ${\\cal P}(R_3)$ respectively. We can apply the\n  construction of Sect. \\ref{Construct1} and glue three discs, $\\mbox{\\sets U}_r$,\n  $\\mbox{\\sets U}_g$, $\\mbox{\\sets U}_b$ to the holes of the pants.\n  Essentially, we have split our mapping $p(y)$ -- see the commutative\n  diagram on the Fig. \\ref{DiscNCut}(b). The holomorphic mapping\n  $\\tilde{p}$ has three or four ramification points, three of them are in\n  the artificially glued discs and the fourth (if any) is inherited from the\n  projective structure.\n\n  Applying the Riemann--Hurwitz formula we get:\n\n\\begin{equation}\n\\label{RHformula}\n\\begin{array}{ll}\nd_r+d_g+d_b=2N, & p ~ {\\rm is~branched},    \\\\\nd_r+d_g+d_b=2N+1, & p ~ {\\rm is~unbranched},    \\\\\n\\end{array}\n\\quad N:=\\deg\\tilde{p}.\n\\end{equation}\n\n\\paragraph{Intersection of Circles.}\n\n\\begin{lmm}\nIn case ${\\cal A}$ the required projective structure $p(y)$ with a critical point\ninside the pants may exist only if the spectral parameter\n$1<\\lambda<2$ (i.e. when the circle $C$ does not intersect two other circles\n$\\varepsilon^{\\pm1}\\hat{\\mbox{\\sets R}}$).\nThe structure without branching does not exist for any\n$\\lambda$.\n\\end{lmm}\nP~r~o~o~f:~~~\n  {\\bf 1.} We know that the point $0$ lies in the intersection of two\n  circles: $\\varepsilon\\hat{\\mbox{\\sets R}}$ and\n  $\\varepsilon^2\\hat{\\mbox{\\sets R}}$. The total number\n  $\\sharp\\{\\tilde{p}^{-1}(0)\\}$ of the pre-images of this points\n  (counting the multiplicities)  is $N$  and cannot be less than\n  $d_b+d_g$ -- the number of pre-images on the blue and green\n  boundary oval of the pants. Comparing this to\n  (\\ref{RHformula}) we get $d_r\\ge N$ which is only possible when\n\n\\begin{equation}\n\\label{k1k2kRelation}\nd_r=d_g+d_b=N.\n\\end{equation}\n\n  Assuming that the circle $C$ intersects any of the circles\n  $\\varepsilon^{\\pm1}\\hat{\\mbox{\\sets R}}$ we repeat the above\n  argument for the intersection point and arrive at the\n  conclusion $d_b=d_r+d_g=N$ or $d_g=d_r+d_b=N$ incompatible with\n  already established (\\ref{k1k2kRelation}).\n\n  {\\bf 2.} For the unbranched structure $p(y)$ the established inequalities\n$d_b+d_g\\le N$ and $d_r\\le N$ contradict the  Riemann-Hurwitz\nformula (\\ref{RHformula}). ~~~\\rule[-1mm]{2.4mm}{2.4mm}\n\nThe above arguments may be applied to the case ${\\cal B}1$ as well.\nTaking into account that the circles $C$ and $\\hat{\\mbox{\\sets R}}$ always intersect\nwe arrive at\n\\begin{lmm}\nIn case ${\\cal B}1$ the mapping $p(y)$ (if any) will have a\nboundary critical point.\n\\end{lmm}\n\n\n\n\\paragraph{Image of Pants.}\n  Let us investigate where the artificially glued discs in case $\\cal A$ are\n  mapped to. Suppose for instance that the disc $\\mbox{\\sets\n  U}_r$ is mapped to the exterior of the circle $C$. The point\n  $0$ will be covered then at least $d_r+d_g+d_b=2N$ times which\n  is impossible. The discs $\\mbox{\\sets U}_g$ and $\\mbox{\\sets\n  U}_b$  are mapped to the left of the lines\n  $\\varepsilon{\\mbox{\\sets R}}$ and $\\varepsilon^2{\\mbox{\\sets\n  R}}$ respectively, otherwise points from the interior of the\n  circle $C$ will be covered more that $N$ times.  The image of\n  the pair of pants ${\\cal P}(R_3)$ is shown on the\n  Fig. \\ref{BelyiMap}.\n\n  We use the ambiguity in the construction of gluing the discs to the pants and\n  require  that the critical values of $\\tilde{p}$ in the discs $\\mbox{\\sets\n  U}_g$, $\\mbox{\\sets U}_b$ coincide. Now the branched covering $\\tilde{p}$ has\n  only three different branch points shown as $\\bullet$, $\\circ$ and $*$ on the\n  Fig. \\ref{BelyiMap}. The branching type at  $\\bullet$ is the cycle of\n  length $N$; at the point $\\circ$ there are cycles of lengths $d_g$ and $d_b$;\n  and the branch point $*$ is simple. The coverings with three branch points are\n  called Belyi maps and are described by certain graphs known as Grothendieck's\n  {\\it \"Dessins d'Enfants\"}. In our case the {\\it dessin} is the lifting of the\n  segment connecting white and black branch points: $\\Gamma:=\\tilde{p}^{-\n  1}[\\bullet,\\circ]$.\n\n\\begin{figure}[ht!]\n\\begin{picture}(170,55)\n\\put(0,0){\\psfig{figure=imagepants.eps}}\n\\end{picture}\n\\caption[] {\\normalsize Shaded area is the image of pants in case $\\cal A$, index $d_b>0$.}\n\\label{BelyiMap}\n\\end{figure}\n\n\n\\paragraph{Combinatorial Analysis of Dessins.}\n  There is exactly one critical point of $\\tilde{p}$ over the branch point $*$.\n  Hence, the compliment to the graph $\\Gamma$\n  on the upper sphere of the diagram on the Fig.\n  \\ref{DiscNCut}(b) contains exactly one cell mapped $2-1$ to the lower sphere.\n  All the rest components of the compliment are mapped $1-1$.\n  Two types of cells are shown\n  in the Fig.  \\ref{TwoCells} (a) and (b), the\n  lifting of the red circle is not shown to simplify the\n  pictures. The branch point $*=:h_1+ih_2$ should lie in the intersection of\n  two annuli $\\alpha$ and $\\overline{\\alpha}$ otherwise\n  the discs $\\mbox{\\sets U}_g$, $\\mbox{\\sets U}_b$ glued to different boundary components of\n  our pants will intersect: the hypothetical case when the\n  branch point of $p(y)$ belongs to one annulus but does not\n  belong to the other is shown in the Fig. \\ref{TwoCells} (c).\n\n\\begin{figure}[h!]\n\\psfig{figure=graphcell1.eps}\n\\caption[] {\\normalsize (a) Simple cell ($N-2$ copies)~~ (b) Double cell (1 copy)~~ (c) Impossible double cell}\n\\label{TwoCells}\n\\end{figure}\n\n  The cells from the Fig. \\ref{TwoCells} (a), (b) may be assembled in\n  a unique way shown in the Fig. \\ref{Assembly1}. The pants are\n  colored in  white, three artificially sewn discs are\n  shaded. Essentially this picture shows us how to sew together\n  the patches bounded by our three circles $C$,\n  $\\varepsilon^{\\pm1}\\hat{\\mbox{\\sets R}}$ to get the pants\n  conformally equivalent to ${\\cal P}(R_3)$. As a result of the surgery procedure we get\n  the pants ${\\cal PA}_1(\\lambda, h_1, h_2|d_g,d_b)$. Changing the superscript\n  of the projective structure $p^\\pm(y)$ gives us the change of sign for the eigenfunction\n  $u(x)$ and the reflection of the pants ${\\cal PA}_1(\\dots)$ in the unit circle\n  $\\partial${\\sets U}. This is why we consider only the pants with $|h_1+ih_2|\\ge1$.\n\n\n\n\\begin{figure}[ht!]\n\\psfig{figure=assembly1.eps} \\caption[] {\\normalsize Dessin for\n$d_g=3,d_b=2$; the pre-image of the branch point $*$ is at infinity}\n\\label{Assembly1}\n\\end{figure}\n\n\\subsubsection{Boundary critical Points}\n\\label{BoundaryBranch}\nFirst of all we consider the stable case of two simple critical points on the boundary\noval. At the moment we do not know the color of this oval and we use the\n'nicknames' $\\{1,2,3\\}$  for the set of colours $\\{r,g,b\\}$ so that the critical\npoints will be on the oval 3.\n\n\n\\paragraph{Branched Covering of a Sphere.}\n The usage of both constructions from Sect.  \\ref{Construct} allows us to\n include the pants ${\\cal P}(R_3)$ to the sphere attaching two\n discs $\\mbox{\\sets U}_1$ and $\\mbox{\\sets U}_2$ to the first two ovals and\n collapsing the boundary of the third oval and sewing in the third disc\n $\\mbox{\\sets U}_3$ if necessary. Positive integers arising in those\n constructions we denote as $d_1$, $d_2$, $d_3=|d_3^--d_3^+|$ respectively.\n\n  We arrive at the branched covering $\\tilde{p}$ of the diagram\n  on the Fig \\ref{DiscNCut}(b). This mapping has either two or four\n  critical points. Two of them are in the centers of the discs\n  $\\mbox{\\sets U}_1$ and $\\mbox{\\sets U}_2$, another two\n  arise only when $d_3>0$ -- the center of $\\mbox{\\sets U}_3$\n  and one of the boundary critical points of the mapping $p(y)$.\n  The multiplicities of those critical points are respectively\n  $d_1-1$, $d_2-1$, $d_3-1$, 1. Riemann-Hurwitz formula for this covering reads\n \\begin{equation}\n \\label{RH2}\n  d_1+d_2+d_3=2N,\n  \\qquad N:=\\deg\\tilde{p}.\n \\end{equation}\n\n\\begin{lmm}\nThe images of the ovals 1 and 2 do not intersect.\n\\label{ovals12}\n\\end{lmm}\n\nP~r~o~o~f. Suppose the inverse is true and a point $Pt$ lies in the\nintersection of images of the first two ovals. Then\n$N\\ge\\sharp\\tilde{p}^{-1}(Pt)\\ge d_1+d_2$. Any of the critical\npoints from the third oval has at least $d_3+1\\le N$ pre-images counting\nmultiplicities. The last two inequalities contradict (\\ref{RH2}).\n~~\\rule[-1mm]{2.4mm}{2.4mm}\n\n{\\bf Corollaries.}\n\n{\\bf 1.} In case $\\cal A$ the critical points of $p(y)$\nlie either on the blue or on the green boundary of pants.\n(Two circles $\\varepsilon^{\\pm1}\\hat{\\mbox{\\sets R}}$ intersect)\n\n{\\bf 2.} In case ${\\cal B}1$ the critical points of $p(y)$ lie on the\nblue boundary of pants. (Two circles $C$ and $\\hat{\\mbox{\\sets R}}$\nintersect)\n\n{\\bf 3.} In both cases the required function may only exist when\n$\\mu\\in(\\frac12,1)$, or equivalently $\\lambda\\in(1,2)$.\n(Otherwise the circles  $C$ and $\\varepsilon^{\\pm1}\\hat{\\mbox{\\sets R}}$ intersect)\n\n\nTo save space, further proof will be given for the case $\\cal A$ only when both\ncritical points lie on the blue oval. The omitted cases require no extra technique.\nNow the notations $\\mbox{\\sets U}_r$, $\\mbox{\\sets U}_g$, $\\mbox{\\sets U}_b$,\n$d_r$, $d_g$, $d_b$ have the obvious meaning.\n\n\\paragraph{Image of Pants.}\n\\begin{lmm}\nThe image $p({\\cal P}(R_3))$ of the pants is the union $\\alpha\\cup\\bar\\alpha$\nwhen $d_b>0$ or the annulus $\\alpha$ when $d_b=0$.\n\\end{lmm}\nP~r~o~o~f:\nEssentially, we have established the equalities\n$$d_g+d_b=d_r=N$$ setting $Pt=0$ in the proof of Lemma \\ref{ovals12}.\nRepeating the arguments of the same title paragraph of the\nSect. \\ref{InnerBranch} we conclude that the disc $\\tilde{p}(\\mbox{\\sets U}_r)$ fills\nthe interior of $C$, the disc $\\mbox{\\sets U}_g$ is mapped to the left of\nthe line $\\varepsilon{\\mbox{\\sets R}}$ and the disc $\\mbox{\\sets U}_b$ (if any) is\nmapped to the left of $\\varepsilon^2{\\mbox{\\sets R}}$.\nSo the sector $\\{\\frac{2\\pi}3\\le\\arg p\\le\\frac{4\\pi}3 \\}$\nis covered $d_g+d_b=N$ times by the artificially inserted discs.\nThe disc $\\mbox{\\sets U}_g$ covers the half-plane to the left of\nthe line $\\varepsilon{\\mbox{\\sets R}}$\nexactly $d_g$ times, the latter number is $N$ when $d_b=0$.\n\n{\\bf Corollary} ~~ Both critical values of $p(y)$ lie on the ray\n$-\\varepsilon^2(0,\\infty)$.\n\n\\paragraph{Dessin d'Enfants.}\n  Again, we put the critical values of $\\tilde{p}$ in the discs $\\mbox{\\sets\n  U}_g$, $\\mbox{\\sets U}_b$ to the same point $\\circ$ (see Fig.\n  \\ref{BelyiMap}). The only difference from the Sect. \\ref{InnerBranch}:\n  now the inherited from the pants branch point $*$ (if $d_b>0$) lies on the ray $-\n  \\varepsilon^2(0,\\infty)$. We introduce the Grothendieck's {\\it Dessin} as the\n  lifting of the segment connecting white and black branch points:\n  $\\Gamma:=\\tilde{p}^{- 1}[\\bullet,\\circ]$. The compliment to $\\Gamma$\n  is composed of cells shown in the Fig. \\ref{GraphCells2}. In the assembly the\n  double cell may be used only once and only when $d_b>0$.\n\n\\begin{figure}[h!]\n\\psfig{figure=graphcells2.eps}\n\\caption[] {\\normalsize (a) Simple cell ($N-2$ copies)~~ (b) Double cell (1 copy)}\n\\label{GraphCells2}\n\\end{figure}\n\n\n  Given the winding numbers $d_g,d_b$, the cells from the Fig. \\ref{GraphCells2} \n  (a), (b) may be attached to each other in a unique way. When $d_b>0$ the \n  {\\it Dessin} $\\Gamma$ has the same combinatorial structure as in Fig. \n  \\ref{Assembly1}. Of course one  has to replace the old cells by those shown in \n  Fig. \\ref{GraphCells2}. Shown in the Fig. \\ref{Assembly2} is the assembly of \n  cells for $d_b=0$, $d_g=5$. The pants are colored in  white, two artificially \n  inserted discs are shaded.  As a result of the surgery procedure we get the \n  pants ${\\cal PA}_2(\\lambda, h_1, h_2|d_g,d_b)$ with positive reals $h_1,h_2$ \n  determined by the critical values of $p(y)$. To discern the pair of pants \n  ${\\cal PA}_2(\\dots)$ from its reflection in the unit circle we consider the \n  restriction $h_1h_2\\ge1$.\n\n\\begin{figure}[ht!]\n\\psfig{figure=assembly2.eps} \n\\caption[] {\\normalsize Degenerate\nDessin for $d_g=5,d_b=0$; the pre-image of the branch point $\\circ$\nis at infinity} \\label{Assembly2}\n\\end{figure}\n\n\\paragraph{Junction of critical points.}\nTo study the remaining case when the boundary critical points of projective \nstrucure merge, one has to apply the limit case of the Construction 2. In this \nway one arrives at the unstable membranes ${\\cal PA}_{12}$ and ${\\cal PA}_{13}$. \nTo save space we omit the details.\n\n\n\\subsection{Pair of Pants corresponds to Eigenfunction}\nLet the pair of pants (\\ref{R3Pants}) associated to the functional\nparameter of the type ${\\cal Q=A,B}1$ integral equation is\nconformally equivalent to the pants ${\\cal PQ}(\\lambda, \\dots)$.\nThis exactly means that there exists a conformal mapping $p(y)$\nfrom ${\\cal P}(R_3)$ to ${\\cal PQ}(\\lambda, \\dots)$\nrespecting the colors of the boundary ovals. This mapping\nis unique since the conformal self-mapping of pants keeping all boundary ovals invariant \nis trivial.  The mapping $p(y)$ has one simple critical point inside the pants \n(for the membrane ${\\cal PA}_1(\\dots)$) or two simple boundary points (for  ${\\cal PA}_2(\\dots)$,\n${\\cal PA}_3(\\dots)$, ${\\cal PB}1(\\dots)$) or a double boundary\ncritical point (for ${\\cal PA}_{12}(\\dots)$, ${\\cal PA}_{13}(\\dots)$). \nMoreover, $p(y)$ maps the boundary components of ${\\cal P}(R_3)$ to the \ncircles specified by Theorem \\ref{SemiMainTh}. Hence, given $p(y)$ one can consecutively\nrestore: two projective structures $p^\\pm(y)$, the solution $W(y)$\nof Riemann monodromy problem and the eigenfunction $u(x)$. Combining\nthe formulae (\\ref{Antisymm}), (\\ref{UniU}) we obtain the top of the\nreconstruction formulae in (\\ref{UviaP}).\n\n\n\\section{Proof for the Case ${\\cal B}2$}\n\\label{B2Proof}\n\\subsection{Eigenfunction gives Pair of Pants}\nAny antisymmetric eigenfunction of the integral equation PS-3 \ngenerates the mapping $p(y)$ from the pants ${\\cal P}(R_3)$ to the sphere.\nThe principal difference of this case from the one considered in Sect. \n\\ref{AB1Proof} lies in the two-valuedness of the function $p(y)$ in the pants. \nTo reflect this phenomenon we consider the two sheeted \nunbranched cover ${\\cal P}_2\\to$ ${\\cal P}(R_3)$ with trivial monodromy around \nthe green boundary oval. This new surface is a sphere with four holes, each \nboundary inherits the color of the oval it covers -- see Fig. \\ref{AuxSurface}(a). The mapping $p(y)$ is lifted \nto the single-valued mapping $p_2:~~$ ${\\cal P}_2\\to$ $\\mbox{\\sets C}P^1$ \nsatisfying the equivariance condition:\n\n\\begin{equation}\n\\label{EqVar}\np_2\\Xi=\\chi({\\bf DD}_1)p_2,\n\\end{equation}\nwhere $\\Xi$ is the cover transformation (change of sheets) of\n${\\cal P}_2$.\n\n\\begin{figure}[ht!]\n\\psfig{figure=p2p4.eps} \n\\caption[] {\\normalsize (a) The surface ${\\cal P}_2$ is the double cover of pants ${\\cal P}$;\\\\ \n(b) ${\\cal P}_4$ is the reflection of ${\\cal P}_2$ in the blue boundary oval }\n\\label{AuxSurface}\n\\end{figure}\n\n\\vspace{1mm}\n\nNow we can complete the p~r~o~o~f~ o~f~ t~h~e~ L~e~m~m~a\n\\ref{SpLocus}.\\\\ Suppose that there exists the required function\n$p_2(y)$ in the case ${\\cal B}21$ and $\\mu>1$. We show that all\npossible locations of the critical points of this function lead to\nthe contradiction: ({\\bf a}) $p_2$ has no boundary critical points;\n({\\bf b}) all critical points lie on the green ovals of ${\\cal P}_2$;\n({\\bf c}) $p_2$ has at least one critical point on a blue oval.\n\n\n({\\bf a}) We use the Construction 1 and attach four discs to the surface\n${\\cal P}_2$. For the arising ramified covering $\\tilde{p}_2$ the Riemann-Hurwitz\nformula reads\n$$\n\\begin{array}{ll}\n2d_g+d_b+d_b'=2N, & p ~ {\\rm is~branched},  \\\\\n2d_g+d_b+d_b'=2N+2, & p ~ {\\rm is~unbranched},  \\\\\n\\end{array}\n\\quad N:=\\deg\\tilde{p}_2,\n$$\nwhere $d_g$ is the winding number of $p_2(y)$ for each of green ovals; $d_b$ and\n$d_b'$ are the winding numbers for two blue ovals of the surface ${\\cal P}_2$. The point\n$0\\in$ $\\hat{\\mbox{\\sets R}}\\cap$ $\\hat{\\varepsilon\\mbox{\\sets R}}$ is covered\nat least $d_g+d_b+d_b'\\le N$ times. This agrees the previous formula only if\n$d_g\\ge N$. But now $d_b=d_b'=0$ which is impossible.\n\n({\\bf b}) Now each of two green ovals has two boundary critical points of $p_2(y)$. We\nuse both Constructions and eliminate all holes in ${\\cal P}_2$ attaching two\ndiscs to the blue ovals and possibly two more  discs to the green ovals of the\nsurface. The Riemann-Hurwitz formula for the arising ramified covering\n$\\tilde{p}_2$ reads\n$$\n2d_g+d_b+d_b'=2N,\n\\quad N:=\\deg\\tilde{p}_2,\n$$\nwhere $d_g\\ge0$ is the difference of the partial winding numbers for each of the \ngreen ovals of ${\\cal P}_2$. Further argument is exactly as in the previous \nparagraph.\n\n({\\bf c})\nWe claim that in this case there are exactly four critical points of $p_2$ on a \nblue oval of the surface ${\\cal P}_2$. Indeed, given a critical point $Pt$, $\\Xi \nPt$ is also a critical point because of the equivariance (\\ref{EqVar}). When \n$\\mu>1$, the mapping $\\chi({\\bf DD}_1)$ conserves the orientation of the real axis. \nThis means that those two critical points are of the same type (say, local \nmaxima of the real value $p_2$). Hence, $Pt$ and $\\Xi Pt$ are separated by the \ncritical points of the opposite type (local minima in our case). There cannot be \nmore that four critical points of the function $p_2$ on the double cover of \npants ${\\cal P}(R_3)$, so we have listed them all.\n\nLet us consider the {\\it double} of the surface ${\\cal P}_2$\nand cut it along all boundary ovals of  ${\\cal P}_2$, but\nthe blue\noval containing all critical point of $p_2(y)$. This new surface, ${\\cal P}_4$, is a\nsphere with six holes shown in the Fig \\ref{AuxSurface}(b), four boundary ovals are \ngreen and two are blue.  The\nreflection principle allows to continue analytically $p_2(y)$ to the mapping\n$p_4(y)$ of the entire surface ${\\cal P}_4$ to the sphere. This continuation has\nfour inner critical points and no boundary critical points. It maps both blue\novals to $\\hat{\\mbox{\\sets R}}$ and four green ovals to the circles\n $\\varepsilon^{\\pm1}\\hat{\\mbox{\\sets R}}$,  \n $\\chi({\\bf DD_1)})\\varepsilon^{\\pm1}\\hat{\\mbox{\\sets R}}$. The\nusage of Construction 1 allows to fill in all the holes of ${\\cal P}_4$. The\nRiemann-Hurwitz formula for the arising ramified covering $\\tilde{p}_4$\nreads\n$$\n2d_g+d_b=N,\n\\quad N:=\\deg\\tilde{p}_4,\n$$\nwhere the numbers $d_g$, $d_b$ have the obvious meaning. The point $0$ is covered at least\n$2d_g+2d_b>N$ times which is impossible.\n~~~\\rule[-1mm]{2.4mm}{2.4mm}\n\n\\paragraph{The location of the critical points} \nof the mapping $p_2(y)$ is given by the following lemma.\n\n\\begin{lmm}\nThe mapping $p_2(y)$ has exactly two boundary critical points  \non each of the non-green ovals of the surface ${\\cal P}_2$.\n\\label{p2Branch}\n\\end{lmm}\n\nP~r~o~o~f. The mapping $\\chi({\\bf DD}_1)$ changes the orientation of the circles\n$C$ and $\\hat{\\mbox{\\sets R}}$, when $\\mu\\in(0,1)$. When the point $y$ runs along\nthe blue or red oval of ${\\cal P}_2$, the value $p_2(y)$ changes the orientation\nof its motion at least twice due to (\\ref{EqVar}). This means that the argument\n$y$ comes through at least two boundary critical points. Since the mapping\n$p(y)$ has at most two boundary critical points in pants, the lifted mapping $p_2(y)$ has\nat most four in the double cover of pants. ~~~\\rule[-1mm]{2.4mm}{2.4mm}\n\n\\paragraph{Branched covering of the sphere.}\nThe mapping $p_2(y)$ from ${\\cal P}_2$ to the sphere has equal\nwinding numbers $d=d_g$ defined in Sect. \\ref{Construct1} on both green ovals.\nOn each of non-green ovals, $p_2(y)$ has zero index $d:=d^--d^+=0$ introduced in \nSect. \/ref{Construct1}.\nBoth statements are simple consequences of the equivariance condition (\\ref{EqVar}).\nApplying Constructions 1 and 2 to the mapping $p_2$ defined on ${\\cal P}_2$,\nwe get a ramified covering $\\tilde{p}_2$ with two critical points, both of multiplicity\n$d_g-1$.\n\n\\paragraph{Image of the Surface.}\nThe Riemann-Hurwitz formula for the ramified covering $\\tilde{p}_2$ reads\n$d_g=N:=\\deg \\tilde{p}_2$. It is easily seen that two discs attached to\nthe green ovals of ${\\cal P}_2$ are mapped to the left of the line $\\varepsilon$~{\\sets R}\nand to the interior of the circle $\\chi({\\bf DD}_1)$ $\\varepsilon\\hat{\\mbox{\\sets R}}$.\nTherefore, the surface ${\\cal P}_2$ is conformally equivalent to the\nclosure of the annulus $d_g\\cdot\\beta$ with two slots in it.\n\nThe involution $\\Xi$ of ${\\cal P}_2$ (the interchange of sheets) induces the involution \nof the multisheeted annulus. The latter involution is the lifting of $\\chi({\\bf \nDD}_1)$ to $d_g\\cdot\\beta$ and is given by the formula (\\ref{mBetaInv}). The \nslots of $d_g\\cdot\\beta$ are invariant with respect to $\\Xi$ and therefore pass \nthrough the fixed points $\\mu^{-1}+r$ and $\\mu^{-1}+r\\exp{i\\pi m}$ of the \ninvolution. The red slots are projected to the circle $C$, the blue slots are \nprojected to the real line. Given in Tab. \\ref{SlotsDef} inequalities for the \nparameters $h_1,h_2$ specifying the endpoints of the slots allow us to relate any \ngiven antisymmetric eigenfunction to exactly one picture.\n\nA by-product of the explicit description of the image of the pants is the following  \n\\begin{lmm}\n\\label{PpmDifferent}\nIn antisymmetric case ${\\cal B}2$ two structures $p^\\pm(y)$ are different.\n\\end{lmm}\n\nP~r~o~o~f.~ Suppose the opposite, that is \n\\begin{equation}\np(y)\\overline{p(\\bar{y})}\\equiv 1\n\\label{StructEqual}\n\\end{equation}\nfor $p(y)$ satisfying the conditions of theorem \\ref{SemiMainTh}.\n\nIn case ${\\cal B}21$ the value $p(a)\\in${\\sets R} when  $a\\in \\{a_1,a_2\\}$\nis the endpoint of the blue slot. From (\\ref{StructEqual})\nit immediately follows that $p(a)=\\pm1$. But the image of pants\n$p({\\cal P})=\\beta$ avoids both points $\\pm1$. \n\nIn case ${\\cal B}22$ the value $p(a\\pm i0)\\in C=\\{p=\\chi({\\bf DD_1})\\bar{p}\\}$  \nwhen  $a\\in \\{a_1,a_2\\}$ is the endpoint of the red slot. From (\\ref{StructEqual})\nand the jump relationship (\\ref{Jump}) on $[a_1,a_2]$\nit follows that $p(a\\pm i0)=\\pm1$. Again, the image of pants\n$p({\\cal P})$ avoids both points $\\pm1$. \n\nIn case ${\\cal B}23$ any of the above two arguments is applicable. ~~~\\rule[-1mm]{2.4mm}{2.4mm}\n \n{\\bf Corollary~ }\n{\\it In case ${\\cal B}2$ any eigenvalue corresponds to no more than one \nantisymmetric eigenfunction.}\n\nP~r~o~o~f. Suppose that three meromorphic functions $p_s(y)$, $s=1,2,3$, in the pants satisfy \nthe conditions of theorem \\ref{SemiMainTh} and no two of them are identical.\nFrom the second part of the proof of the theorem \\ref{J0neq0} we know\nthat all three values $p_s(y)$ are different at any point $y$.\nWe consider the following differential form on the Riemann surface $M$:\n$$\n\\omega:=dp_1\n\\left(\n\\frac1{p_1-p_2}-\\frac1{p_1-p_3}\n\\right).\n$$\n\nThis form $\\omega$ is the infinitesimal form of cross ratio and it is invariant \nunder the same linear-fractional transformations of all three functions $p_s$. \nTherefore $\\omega$ is well defined on the entire Riemann surface $M$. Using \nlocal coordinates on $M$, it's easy to check that the form is holomorphic and \n$(\\omega)={\\sf D}(p_1)$. Any holomorphic differential on the genus 2 surface has \ntwo zeroes which are interchanged by the hyperelliptic involution of $M$. \nAccording to Lemma \\ref{p2Branch}, the branching divisor of $p_1(y)$ is \ndifferent as it has a branchpoint on each of  non-green ovals of the pants $\\cal \nP$. \n\nTherefore, two of our functions $p_s(y)$ coincide. \nMoreover,   from Lemma \\ref{PpmDifferent} it follows that either all three functions \nare identical, or one of them is the antisymmetrization of the other: \n$p_2(y)=1\/\\overline{p_1(\\bar{y})}$  and $p_3=p_1$. ~~~\\rule[-1mm]{2.4mm}{2.4mm}\n\n\n\n\n\\subsection{Pair of Pants corresponds to Eigenfunction}\nLet the pair of pants ${\\cal PB}2s(\\lambda,h_1,h_2|m)$ be conformally equivalent to the \npair of pants ${\\cal P}(R_3)$ associated to type ${\\cal B}2s$, $s=1,2,3$, integral equation.\nThis exactly means that there exists respecting the colors of the boundary ovals\nequivariant conformal mapping \n$p_2(y)$ from the double cover ${\\cal P}_2(R_3)$ of pants to the closure of the \nmultisheeted annulus $m\\cdot\\beta$ with two slots $E_s^1(h_1)$ and $E_s^2(h_2)$ in it. \nWe represent the double cover ${\\cal P}_2(R_3)$ as two copies of pants ${\\cal P}(R_3)$\ncut along the segment $[-1,1]\\cap[a_1,a_2]$ and attached one to the other.\nThe restriction of $p_2(y)$ to one of such copies gives the function $p(y)$ satisfying \nall the assumptions of the theorem \\ref{SemiMainTh}. The antisymmetric eigenfunction of \nthe integral equation now may be reconstructed via the known procedure\nwhich gives the formulae (\\ref{UviaP}). Moreover, from Lemma \\ref{PpmDifferent}\nwe have learned that the invariant $J_0\\neq0$ for antisymmetric solutions in case\n${\\cal B}2$. So we can use the alternative formula (\\ref{UfromP})\nto reconstruct the eigenfunction $u(x)$ when $x\\in[-1,1]\\setminus[a_1,a_2]$.\nFor $x\\in[-1,1]\\cap[a_1,a_2]$ we can also use the following formula:\n$$\n\\frac{{\\rm Im}~p(y^+)}{|p(y^+)-\\mu|^2+1-\\mu^2},\n\\quad y^+:=R_3(x+i0),\n\\quad x\\in[-1,1]\\cap[a_1,a_2].\n$$\n\n\nThe only nuisance here consists in possible non-uniqueness of the mapping $p(y)$.\nIndeed, when two of the boundary ovals of pants\nhave the same color (blue in case ${\\cal B}21$ of red in case ${\\cal B}22$),\nthe pants may admit conformal involution interchanging the ovals of the same color.\nSuch pants fill in a codimension one manifold in the corresponding moduli space.  \nThe Corollary to Lemma \\ref{PpmDifferent} nevertheless guarantees the uniqueness of \nthe antisymmetric eigenfunction for the given membrane ${\\cal PB}2s$: the composition of \n$p(y)$ with the conformal automorphism of pants coincides with either $p(y)$ \nor its antsymmetrization $1\/\\overline{p(\\bar{y})}$.\n\n\n\n\\section{Conclusion}\nSimilar analysis based on the geometry and combinatorics may be applied to obtain the\nrepresentations of the solutions of PS-3 integral equation in all the dropped cases.\nMuch of the techniques used is helpful for the study of other integral equations with\nrational low degree kernels.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{}\nThe possibility of extracting statistical-mechanical \ninformation from a pure quantum state has been intensively discussed in the context of the foundation of statistical mechanics \n \\cite{Popescu,Lebowitz,SugitaJ,Reimann}.\nAs we shall demonstrate here, it also has a potential significance for \na new formulation of \nstatistical mechanics, \nand for a novel calculation technique.\n\nAs an illustration, let us \nconsider a closed quantum system composed of $N$ spins, \nwhich is enclosed by adiabatic walls.\nIn the ensemble formulation,\nits equilibrium properties are described by the microcanonical \nensemble, which is specified by $E$ (energy), $N$, and so on. \nThe corresponding subspace (energy shell) \nin the Hilbert space $\\mathcal{H}_N$ is denoted\nby $\\mathcal{E}_{E,N}$.\nLet us consider a random vector \n$\n| \\psi \\rangle = \n{\\sum_\\nu}^{\\!\\!\\! \\prime} \\ c_\\nu |\\nu \\rangle\n$\nin $\\mathcal{E}_{E,N}$, \nwhere\n$\\{ |\\nu \\rangle \\}_\\nu$ is an arbitrary orthonormal basis set\nof $\\mathcal{E}_{E,N}$, \n${\\sum_\\nu}^{\\!\\!\\! \\prime}~$ denotes the sum over this basis,\nand $\\{ c_\\nu \\}_\\nu$ is \na set of random complex numbers drawn uniformly\nfrom the unit sphere ${\\sum_\\nu}^{\\!\\!\\! \\prime} \\ |c_\\nu|^2 =1$ in the \ncomplex space of dimension $\\dim \\mathcal{E}_{E,N}$.\nIt was shown in Refs.~\\cite{Popescu,Lebowitz,SugitaJ,Reimann} \nthat almost every such vector \ngives the correct equilibrium values of \na certain class of observables $\\hat{A}$\nby $\\langle \\psi | \\hat{A} | \\psi \\rangle$.\nThis property was proved in Refs.~\\cite{Popescu,Lebowitz}\nfor observables of a subsystem, which is much smaller than the whole system.\nThe case of general observables, including\nobservables of the whole system (such as the total magnetic moment and its fluctuation), \nwas analyzed in Refs.~\\cite{SugitaJ,Reimann}.\nIt was shown that the above property holds {\\em not} for all observables\nbut for observables that are low-degree polynomials \n(i.e., their degree $\\ll N$) of local operators \\cite{SugitaJ}.\nWe here call such observables {\\em mechanical variables}.\nWe \nassume that all mechanical variables are \nnormalized in such a way that \nthey are dimensionless.\n\nFor conceptual clarity, we call {\\em generally} a pure quantum state that \nrepresents an equilibrium state a \n{\\em thermal pure quantum state} (TPQ state).\nStating more precisely for the case where \na state $|\\psi \\rangle$ has random variables \n(such as the random vector \ndiscussed above), \nwe call $|\\psi \\rangle$ a TPQ state \nif for an arbitrary positive number $\\epsilon$\n\\begin{equation}\n{\\rm P}( \n| \n\\langle \\psi | \\hat{A}| \\psi \\rangle\n-\n\\langle \\hat{A} \\rangle^{\\rm eq}_{E, N}\n|\n\\geq \\epsilon\n) \n\\leq\n\\eta_\\epsilon (N)\n\\label{eq:Pto0}\n\\end{equation}\nfor every mechanical variable $\\hat{A}$.\nHere, \n${\\rm P}(x)$ denotes the probability of event $x$,\n$\\langle \\cdot \\rangle^{\\rm eq}_{E, N}$ denotes the ensemble average,\nand \n$\\eta_\\epsilon (N)$\nis a function (of $N$ and $\\epsilon$) which vanishes as $N \\to \\infty$.\nThe above inequality means that for large $N$ \ngetting a {\\em single} realization of \na TPQ state is sufficient, with high probability, \nfor evaluating \nequilibrium values\nof mechanical variables. \nThe vector\n$\n{\\sum_\\nu}^{\\!\\!\\! \\prime} \\ c_\\nu |\\nu \\rangle\n$\nof Refs.~\\cite{Popescu,Lebowitz,SugitaJ,Reimann} \nis a TPQ state.\nHowever, important problems remain to be solved.\nMost crucially,\n{\\em genuine thermodynamic variables}, \nsuch as the entropy and temperature,\ncannot be calculated as $ \\langle \\psi | \\hat{A} | \\psi \\rangle$\nbecause they are not mechanical variables \\cite{vNS}.\nMoreover, \none needs to prepare a basis $\\{ |\\nu \\rangle \\}_\\nu$ of $\\mathcal{E}_{E,N}$ \nto construct ${\\sum_\\nu}^{\\!\\!\\! \\prime} \\ c_\\nu |\\nu \\rangle$.\nSince this is a hard task, such a TPQ state \nis hard to obtain.\n\nIn this Letter, we resolve these problems\nby proposing a new TPQ state, \na novel method of constructing it,\nand new formulas for obtaining genuine thermodynamic variables.\nThis novel formulation of statistical mechanics\nenables one to calculate\n{\\em all} variables of statistical-mechanical interest\nat {\\em finite temperature},\nfrom only a {\\em single} realization of the TPQ state.\nWe also show that \nthis formulation is very useful for practical calculations.\n\n{\\em New TPQ state -- }\nWe consider a discrete quantum system composed of $N$ sites, \nwhich is described by a Hilbert space $\\mathcal{H}_N$ \nof dimension $D=\\lambda^N$,\nwhere $\\lambda$ is a constant of $O(1)$.\n[For a spin-1\/2 system, $\\lambda=2$.]\nOur primary purpose is to obtain results in the thermodynamic limit: \n$N \\rightarrow \\infty$ while $E\/N$ is fixed. \nTherefore, we hereafter use quantities per site,\n$\\hat{h} \\equiv \\hat{H}\/N$ (where $\\hat{H}$ denotes the Hamiltonian), \n$u \\equiv E\/N$, and $(u; N)$ instead of $(E,N)$.\n[We do not write explicitly \nvariables other than $u$ and $N$, such as a magnetic field.]\nWe assume that the system is consistent with thermodynamics\nin the sense that \nthe density of states $g(u;N)$ behaves as \\cite{en:Boltzmann}\n\\begin{equation}\ng(u;N)=  \n\\exp[Ns(u;N)], \n\\\n\\beta'(u;N)\n\\leq 0.\n\\label{eq:TDconsistency}\n\\end{equation}\nHere, $s(u;N)$ is the entropy density,\nwhich converges to the $N$-independent \none $s(u;\\infty)$ as $N\\rightarrow \\infty$,\n$\\beta(u;N) \\equiv \\partial s(u;N)\/\\partial u$\nis the inverse temperature,\nand\n$\\beta'\n\\equiv\n\\partial \\beta \/\\partial u\n$.\nThese conditions are satisfied, for example,\nby spin models and the Hubbard model.\nSince $D$ is finite,\n$\\beta$ may be positive and negative in  \nlower- and higher-energy regions,\nrespectively.\nWe here consider the former region.\n\n\nWe propose the following TPQ state and \nthe procedure for constructing it. \nFirst, take a random vector \n$\n| \\psi_{0} \\rangle \n\\equiv\n\\sum_i c_i | i \\rangle\n$\nfrom the {\\em whole} Hilbert space $\\mathcal{H}_N$.\nHere, $\\{|i\\rangle\\}_i$ is an arbitrary orthonormal basis of $\\mathcal{H}_N$, \nand $\\{c_i\\}_i$ is a set of random complex numbers drawn uniformly\nfrom the unit sphere $\\sum|c_i|^2=1$ of the $D$-dimensional complex space. \nNote that this construction of random vectors is independent of the choice of the orthonormal basis $\\{|i\\rangle\\}_i$.\nOne can therefore use a trivial basis such as a set of product states.\nHence, $| \\psi_{0} \\rangle$ can be generated easily.\nOn ther whole, \nthe amplitude is almost equally distributed over {\\em all} the energy eigenstates \nin this state\n(as is easily seen by choosing the eigenstates of $\\hat{h}$ as the basis $\\{|i\\rangle\\}_i$).\nThus, the distribution of energy in $|\\psi_0\\rangle$ is proportional to $g(u;N)$.\nWe wish to modify this distribution into another distribution $r_k(u;N)$ \nwhich has a peak at an desired energy.\nThis is easily done by operating a suitable polynomial of $\\hat{h}$\nonto $|\\psi_0 \\rangle$ as we shall see below. \n[Operating  $\\hat{h}$ onto a vector is much easier \nthan diagonalizing $\\hat{h}$.]\nWe denote the minimum and the maximum eigenvalues of $\\hat{h}$\nby $e_{\\rm{min}}$ and $e_{\\rm{max}}$, respectively.\nTake a constant $l$ of $O(1)$ \nsuch that $l \\ge e_{\\rm max}$. \nStarting from $| \\psi_{0} \\rangle$, calculate\n\\begin{eqnarray}\nu_{k} &\\equiv& \\langle \\psi_{k} | \\hat{h} | \\psi_{k} \\rangle,\n\\label{eq:uk}\n\\\\\n| \\psi_{k+1} \\rangle \n&\\equiv&\n( l - \\hat{h} ) | \\psi_{k} \\rangle\n\/\n\\| ( l - \\hat{h} ) | \\psi_{k} \\rangle \\|\n\\end{eqnarray}\niteratively for $k=0, 1, 2, \\cdots$. \nFrom Eq.~(\\ref{Nb=k\/L}) below, \n$u_{0}$ corresponds to $\\beta=0$, \ni.e., $g(u;N)$ takes the maximum at $u=u_{0}$.\nWe will \nalso show that $u_{k}$ decreases gradually down to $e_{\\rm min}$ as $k$ is increased, \ni.e., \n$\nu_{0} > u_{1} > \\cdots \\geq e_{\\rm{min}} \n$.\nOne may terminate the iteration \nwhen $u_{k}$ gets low enough for one's purpose.\nWe denote $k$ at this point by $k_{\\rm term}$. \nWe will show that\n$k_{\\rm term} = O(N)$ at finite temperature,\nand that \nthe states\n$\n\t| \\psi_{0} \\rangle, | \\psi_{1} \\rangle, \n\t\\cdots,\n\t| \\psi_{k_{\\rm term}} \\rangle\n$ \nbecome a series of TPQ states corresponding to various energy densities,\n$u_{0}, u_{1}, \\cdots, u_{k_{\\rm term}}$.\nHence, the equilibrium value \nof an arbitrary mechanical variable $\\hat{A}$ is obtained as\n$\\langle \\psi_{k} | \\hat{A} | \\psi_{k} \\rangle$, \nas a function of $u_{k}$.\nFor each realization of $\\{ c_i \\}_i$, \na series of realizations of TPQ states is obtained.\nWe will show that the dependence of \n$\\langle \\psi_{k} | \\hat{A} | \\psi_{k} \\rangle$ on $\\{ c_i \\}_i$ is\nexponentially small in size $N$ as $N$ increases.\nTherefore, \n{\\em only a single realization\nsuffices for getting\na fairly accurate value. }\nWhen better accuracy is required, \none can take the average over many realizations.\n\n\nWe now show that the states obtained with the above procedure are TPQ states.\nSince $| \\psi_{0} \\rangle$ is independent of the choice of the basis, \nwe take \nthe set of energy eigenstates $\\{ | n \\rangle \\}_n$\nas \n$\\{ | i \\rangle \\}_i$ \nin order to see properties of $| \\psi_k \\rangle$\n(although we never use such a basis in practical calculations).\nAfter $k$-times multiplication of $l-\\hat{h}$, \n$ | \\psi_{0} \\rangle\n=\n\\sum_n c_n | n \\rangle\n$\nturns into \n\\begin{equation}\n| \\psi_{k} \\rangle  \n\\propto \n(l-\\hat{h})^k | \\psi_{0} \\rangle\n=  \n\\sum_{n} c_{n} (l-e_{n})^k |n \\rangle,\n\\label{psi=Sdn}\n\\end{equation}\nwhere $\\hat{h} |n \\rangle = e_n |n \\rangle$. \nLet us examine how the energy density $u$ distributes in this state.\nThe (unnormalized) distribution function of $u$ is given by\n$\nr_k(u;N)\n\\equiv\n\\delta_r^{-1} {\\sum_n}^{\\!\\! \\prime \\prime} |c_{n}|^2 (l-e_{n})^{2k}\n$,\nwhere $\\delta_r = o(1)$ and\nthe sum is taken over $n$ such that $e_n$ lies in \na small interval $[u-\\delta_r \/2, u+\\delta_r \/2)$.\nSince the density of states $g(u;N)$ is exponentially large in size $N$, \n$r_k(u;N)$ converges (in probability) exponentially fast\nto its average. Hence, \n\\begin{equation}\nr_k(u;N)\n= D^{-1}  \\exp[N \\xi_{\\kappa}(u;N)],\n\\label{Sd=e^xi}\n\\end{equation}\nwhere $\\xi_{\\kappa}(u;N) \\equiv s(u;N) + 2\\kappa \\ln (l-u)$ with $\\kappa \\equiv k\/N$.\nHereafter we often denote $k$ dependence by $\\kappa$, \ne.g., we express $u_k$ as $u_{\\kappa}$.\nNote that $\\xi_{\\kappa}(u;N)$ does not depend on $\\{ c_i \\}_i$,\nbecause the dependence vanishes\nwhen we have dropped negligible terms in Eq.~(\\ref{Sd=e^xi}). \n$\\xi_{\\kappa}(u;N)$ takes the maximum at $u^{\\ast}_{\\kappa}$ which satisfies\n\\begin{eqnarray}\n\\beta(u^{\\ast}_{\\kappa};N) = 2 \\kappa \/ (l-u^{\\ast}_{\\kappa}).\n\\label{Nb=k\/L}\n\\end{eqnarray}\nSince $\\beta(u^*_\\kappa; N)$ and $l-u^{\\ast}_{\\kappa}$ are $O(1)$, \nwe find $\\kappa = O(1)$, and hence $k = O(N)$.\nExpanding $\\xi_{\\kappa}(u;N)$ around $u^{\\ast}_{\\kappa}$, \nand noticing \n\\begin{eqnarray}\n\\xi''_{\\kappa} \n\\equiv \\partial^2 \\xi_{\\kappa}\/\\partial u^2 \n= \\beta'(u^{\\ast}_{\\kappa};N) - 2\\kappa\/(l-u^{\\ast}_{\\kappa})^2 < 0\n\\nonumber\n\\end{eqnarray}\nfrom Eq.~(\\ref{eq:TDconsistency}), \nwe get \n$\n\\xi_{\\kappa}(u;N) \n=\n\\xi_{\\kappa}(u^{\\ast}_{\\kappa};N) \n- |\\xi''_{\\kappa}| (u-u^{\\ast}_{\\kappa})^2 \/ 2\n+ \\xi'''_{\\kappa} (u-u^{\\ast}_{\\kappa})^3 \/ 6\n+ \\cdots\n$.\nHere, \n$\\xi'''_{\\kappa} \\equiv \\partial^3 \\xi_{\\kappa}\/\\partial u^3\n=\\beta''(u^{\\ast}_{\\kappa};N) - 4\\kappa\/(l-u^{\\ast}_{\\kappa})^3$.\nHence, $r_k(u;N)$ behaves almost as the Gaussian distribution, peaking at \n$u=u^*_\\kappa$, \nwith the vanishingly small variance $1\/N |\\xi''_{\\kappa}|$.\nLet us introduce the density operator \n$\n\\hat{\\rho}_k \\equiv \n(l-\\hat{h})^{2k}\n\/{\\rm Tr}(l-\\hat{h})^{2k}\n$, \nwhich has the same energy distribution $r_k(u;N)$.\nIn the ensemble formulation, \n$\\hat{\\rho}_k$ represents the equilibrium state specified by \n$(u_\\kappa; N)$\nbecause $r_k(u;N)$ has a sharp peak.\nWe call the ensemble corresponding to $\\hat{\\rho}_k$\nthe {\\em smooth microcanonical ensemble} (because\nthe energy distribution is smooth).\nIn a way similar to those of Refs.~\\cite{SugitaJ,Reimann}, \nwe can show that for an arbitrary positive number $\\epsilon$\n\\begin{eqnarray}\n{\\rm P}\\Big( \n\\Big| \\langle \\psi_{k} | \\hat{A} | \\psi_{k} \\rangle\n&-&\n{\\rm Tr}[\\hat{\\rho}_k \\hat{A}] \\Big|\n\\geq \\epsilon\n\\Big) \n\\leq \n{ \\| \\hat{A} \\|^2 r_k(e_{\\rm min};N) \\over \\epsilon^2  r_k(u_\\kappa^*;N)},\n\\quad\n\\label{DeltaA}\n\\\\\n\\overline{\\langle \\psi_{k} | \\hat{A} | \\psi_{k} \\rangle}\n&=&\n{\\rm Tr}[\\hat{\\rho}_k \\hat{A}]\n\\label{<A>}\n\\end{eqnarray}\nfor every mechanical variable $\\hat{A}$.\nHere, \n$\\| \\cdot \\|$ denotes the operator norm \\cite{norm}, and\nthe overline represents the random average.\nWith increasing $N$, \n$\\|\\hat{A} \\|^2$ grows at most as a low-degree polynomial of $N$,\nwhereas \n$r_k(e_{\\rm min};N)\/r_k(u_\\kappa^*;N)$ \ndecreases exponentially at finite temperature \n(i.e., for $u_\\kappa^* > e_{\\rm min}$).\nTherefore, {\\em $| \\psi_{k} \\rangle$ is a TPQ state\nfor the smooth microcanonical ensemble}.\n\n{\\em Genuine thermodynamic variables -- }\nOne might think it impossible to obtain genuine thermodynamic \nvariables \nlike the temperature and entropy by only manipulating pure quantum states.\nHowever, \nour new TPQ state makes it possible.\nIn fact, \nby substituting $u_{\\kappa}$ for $u^{\\ast}_{\\kappa}$ in Eq.~(\\ref{Nb=k\/L}),\nand using Eq.~(\\ref{eq:u'}) below, \nwe obtain \n\\begin{eqnarray}\n\\beta(u_{\\kappa};N) = 2 \\kappa \/ (l-u_{\\kappa}) + O(1\/N).\n\\label{Nb=k\/L+O(1\/N)}\n\\end{eqnarray}\nThis gives $\\beta(u_{\\kappa};N)$, with an error of $O(1\/N)$, \nas a function of $u_{\\kappa}$\n[because\n$\\kappa$ and $l$ are known parameters]. \nThat is, one obtains the temperature of the equilibrium state\nspecified by $(u_{\\kappa};N)$ just by calculating \n $u_{\\kappa}$ with Eq.~(\\ref{eq:uk}).\n\nWe can also obtain formulas with less errors.\nFor example, \nusing Eq.~(\\ref{Sd=e^xi}) and the expansion of $\\xi_\\kappa(u;N)$, we have\n\\begin{equation}\nu^{\\ast}_{\\kappa} = u^{\\bullet}_{\\kappa} + O(1\/N^2),\n\\\nu^{\\bullet}_{\\kappa} \\equiv \nu_{\\kappa} - \\xi'''_{\\kappa}\/2N{\\xi''_{\\kappa}}^2.\n\\label{eq:u'}\n\\end{equation}\nSubstituting \n$u^{\\bullet}_{\\kappa}$ for $u^{\\ast}_{\\kappa}$ in Eq.~(\\ref{Nb=k\/L}),\nwe get a better formula\n\\begin{equation}\n\\beta(u^{\\bullet}_{\\kappa}; N)\n=\n2 \\kappa \/ (l - u^{\\bullet}_{\\kappa})\n+ O(1\/N^2).\n\\label{eq:betaN_better}\\end{equation}\nOne can evaluate \n$\\xi''_{\\kappa}$ and $\\xi'''_{\\kappa}$ \neasily by calculating \n$\n\\langle \\psi_{k} | \n(\\hat{h}  - u_{\\kappa} )^2\n| \\psi_{k} \\rangle\n= 1\/N|\\xi''_{\\kappa}| + O(1\/N^2)\n$\nand\n$\n\\langle \\psi_{k} | \n(\\hat{h}  - u_{\\kappa} )^3\n| \\psi_{k} \\rangle\n=\n\\xi'''_{\\kappa}\/N^2 |\\xi''_{\\kappa}|^3 + O(1\/N^3)\n$.\nHence, \nusing formula (\\ref{eq:betaN_better}), \none obtains  \n$\\beta(u; N)$ (for $u = u^{\\bullet}_0, u^{\\bullet}_1,\\cdots$) \nwith an error of $O(1\/N^2)$.\nIn a similar manner, we can obtain formulas \nwhose errors are of even higher order of $1\/N$.\n\n\n\n\n\nHowever, $\\beta(u; N)$ is the inverse temperature of a {\\em finite} system,\nwhereas we are most interested in its thermodynamic limit $\\beta(u; \\infty)$.\nIn general, the difference $|\\beta(u; N) - \\beta(u; \\infty)|$ decays \nnot so quickly as $O(1\/N^2)$.\nTo obtain an even better formula for $\\beta(u; \\infty)$,\nwe consider $C$ identical copies of the $N$-site system.\nWe denote quantities of this $CN$-site system by tilde,\nsuch as \n$|\\tilde{\\psi}_{0} \\rangle \n\\equiv |\\psi_{0} \\rangle^{\\otimes C}$.\nThe state $| \\tilde{\\psi}_{\\tilde{k}} \\rangle$ \nis given by\n$\n| \\tilde{\\psi}_{\\tilde{k}} \\rangle \n\\propto (\\tilde{l}-\\tilde{h})^{C \\tilde{k}} | \\tilde{\\psi}_{0} \\rangle,\n$ \nwhere \n$\n\\tilde{h} \n\\equiv \n(\n\\hat{H} \\otimes \\hat{1}^{\\otimes (C-1)} \n+ \\hat{1} \\otimes \\hat{H} \\otimes \\hat{1}^{\\otimes (C-2)} \n+ \\cdots + \n\\hat{1}^{\\otimes (C-1)} \\otimes \\hat{H}\n)\/CN$. \nIn the limit of $C \\rightarrow \\infty$, \n$\\tilde{u}_{\\tilde{\\kappa}}$ approaches \nthe canonical average of $u$ in a single copy \nwith inverse temperature $\\tilde{\\beta}(\\tilde{u}_{\\tilde{\\kappa}};\\infty)$.\nAt the point where \n$\\tilde{\\beta}(\\tilde{u}_{\\tilde{\\kappa}};\\infty)\n=\\beta(u^{*}_{\\kappa}; N)$\nis satisfied,\nwe can estimate this canonical average,\nwhich is denoted by $\\tilde{u}^{\\rm c}_\\kappa$, \nin the same manner as Eq.~(11).\nThen, we get \n$\\tilde{u}^{\\rm c}_\\kappa = \\tilde{u}^{\\bullet}_\\kappa +O(1\/N^2)$,\nwhere\n\\begin{equation}\n\\tilde{u}^{\\bullet}_\\kappa\n\\equiv\nu^{\\bullet}_{\\kappa}\n+\\frac{\\xi'''_{\\kappa}+ 4\\kappa\/(l-u^{\\bullet}_\\kappa)^3}\n{2N [\\xi''_{\\kappa} + 2\\kappa\/(l-u^{\\bullet}_{\\kappa})^2]^2}.\n\\label{u=u-z\/g+b''\/b'}\n\\end{equation}\nWe thus find\n\\begin{equation}\n\\tilde{\\beta}(\\tilde{u}^{\\bullet}_\\kappa;\\infty) \n= \n2\\kappa\/(l-u^{\\bullet}_\\kappa)\n+O(1\/N^2),\n\\label{eq:beta_infty}\\end{equation}\nwhich gives the inverse temperature \n$\\tilde{\\beta}(u;\\infty)$ (for $u=\\tilde{u}^{\\bullet}_0,\\tilde{u}^{\\bullet}_1,\\cdots$)\nof an infinite system\ncomposed of an infinite number of $N$-site systems.\nWe expect that $\\tilde{\\beta}(u;\\infty)$ is much closer to $\\beta(u;\\infty)$\nthan $\\beta(u;N)$,\nbecause\ninformation of $\\xi(u;N)$ in the whole spectrum range of $u$\nis included in \n$\\tilde{\\beta}(u;\\infty)$.\n[By contrast, only the information at the peak of \n$\\xi(u;N)$ is included in $\\beta(u;N)$.]\nThis will be confirmed later by numerical computation.\n\nWe can also obtain the entropy density $s$ \nas a function of $u$ and $h_z$,  \nby integrating $\\beta$ over $u$\nand $\\beta m_z$ over $h_z$.\nFor example, for an arbitrarily fixed value of $h_z$, we have\n\\begin{equation}\ns(u_{2p}) - s(u_{2q})\n=\n\\sum_{\\ell = p}^{q-1} \nv(u^{\\bullet}_{2\\ell},u^{\\bullet}_{2\\ell+1},\nu^{\\bullet}_{2\\ell+2})\n+O({1 \\over N^2}).\n\\label{s=Sum beta}\n\\end{equation}\nby generalizing Simpson's rule.\nHere, $u$ stands for \n$(u;N)$ or $(u;\\infty)$,\n$p$ and $q$ are integers, \nand\n$v(x,y,z)\n\\equiv\n(x-z) \\{ \\beta(x)+\\beta(z) \\}\/2\n-\n(x-z)^2\n[x \\{ \\beta(z)-\\beta(y) \\}+y \\{ \\beta(x)-\\beta(z) \\}\n+z \\{ \\beta(y)-\\beta(x) \\}]\n\/6(x-y)(y-z)\n$.\nWe have also developed another method of obtaining $s$, \nin which $g(u;N)$ \nis directly evaluated from the \ninner products\namong different realizations of a TPQ state \\cite{gakkai}.\n\nTo sum up,\none can obtain a series of TPQ states and values of all variables of statistical-mechanical \ninterest, by preparing a random vector \nand simply applying $(l - \\hat{h})$ iteratively.\t\nThat is, \nwe have established a new formulation of statistical mechanics,\nwhose fundamental formulas are \nEqs.~(\\ref{psi=Sdn}) and (\\ref{Nb=k\/L+O(1\/N)}).\n\t\n\n{\\em Numerical results -- }\nOur formulation is easily implemented as a method of numerical \ncomputation.\nWe apply it to the one-dimensional Heisenberg model\nin order to confirm the validity of the formulation.\nWe take\n$\n\\hat{H} \n= \\frac{J}{4} \\sum_{i=1}^{N} \n[\\hat{\\bm{\\sigma}}(i) \\cdot \\hat{\\bm{\\sigma}}(i+1) \n- h_z\\hat{\\sigma}_z(i)],\n$ \nwhere $J=-1$ (ferromagnetic) or $+1$ (antiferromagnetic).\nFor $N \\to \\infty$, \nthe exact results at finite temperature (i.e., $u > e_{\\rm min}$)\nhave been \nderived for \nmagnetization \n$\nm_z\n\\equiv\nN^{-1} \\sum_{i=1}^{N} \n\\langle \\sigma_z(i) \\rangle^{\\rm eq}_{u; N}\n$ \nat all values of $u$ and $h_z$ \\cite{Takahashi}, \nand for the correlation function \n$\n\\phi(j) \\equiv\nN^{-1} \\sum_{i=1}^{N} \\langle \\sigma_z(i) \\sigma_z(i+j) \\rangle^{\\rm eq}_{u; N}\n$ \nat all values of $u$ with $h_z=0$ \\cite{Sato}.\nThey are plotted in Figs.~\\ref{fig:hM} and \\ref{fig:Cor}\nby solid lines, where \ndifferent colors correspond to different values of $u$.\nWe calculate the corresponding results using our formulation,\nby performing numerical computation.\nThe results for $N=24$ are\nplotted by circles, where each circle is obtained \nfrom a {\\em single} realization of TPQ state.\nAccording to Eq.~(\\ref{DeltaA}), \nchoice of the initial random numbers $\\{ c_i \\}_i$ \nhas only an exponentially small effect on the results\nat finite temperature. \nWe have confirmed this fact \nby observing that \nthe standard deviation, \ncomputed from ten realizations of a TPQ state for each data point,\nis smaller than the radius of the circles of\nthese figures.\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=0.9\\linewidth]{hM_hz.eps}\n\\end{center}\n\\vspace{-6mm}\n\\caption{\nMagnetization plotted against a magnetic field for $J=-1$.\nSolid lines represent exact results for $N \\to \\infty$, \nfor various values of the energy density $u$ \\cite{Takahashi}. \nCircles denote results obtained with our formulation for $N=24$.\nResults for $N=4$-$20$ are also shown for $u=-0.3J$.\n}\n\\label{fig:hM}\n\\end{figure}\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=0.9\\linewidth]{Cor_0209.eps}\n\\end{center}\n\\vspace{-6mm}\n\\caption{\nCorrelation function $\\phi(j)$ plotted against $j$ for $J=+1$ and $h_z=0$.\nSolid lines represent exact results for $N \\to \\infty$, \nfor various values of $u$ \\cite{Sato}. \nCircles denote results of our formulation for $N=24$.\n(Left Inset) Results for $N=16$-$24$ at $j=2$ for $u=-0.36J$.\n(Right Inset) $\\phi(j)$ at finite $h_z$, \nobtained from a single realization of the TPQ state at $T \\simeq 0.45J$.\n}\n\\label{fig:Cor}\n\\end{figure}\n\nResults for other values of $N$ \nare plotted in Fig.~\\ref{fig:hM} for $u=-0.3J$, \nand in the left insets of Fig.~\\ref{fig:Cor} for $u=-0.36J$ at $j=2$.\nIt is seen that the $N$-dependence becomes fairly weak for $N \\gtrsim 20$,\nand that the results for $N=24$ agree well with the exact results.\nAs illustrated by this example, \n$N$ should be increased in our method until the \nvariation of the results with increasing $N$ \nbecomes less than the required accuracy.\n\nWe have also computed  \n$\\phi(j)$ at finite $h_z$ and $T$, \nfor which {\\em exact results are unknown}.\nThe results at $T \\simeq 0.45J$ are plotted \nin the right inset of Fig.~\\ref{fig:Cor}.\n\nFor genuine thermodynamic variables, \nthe exact result for $1\/\\beta(u; \\infty)$ \\cite{AF_ET} is \nplotted  by solid lines in Fig.~\\ref{fig:ET}.\nCorresponding results for \n$1\/\\beta(u; N)$ and $1\/\\tilde{\\beta}(u; \\infty)$, \nobtained with our method with $N=24$, are \nplotted by triangles and squares respectively, \nwhere each point is obtained \nfrom a {\\em single} realization of the TPQ state.\n[We have confirmed again that\ndependence on the choice of $\\{ c_i \\}_i$\nis negligibly small.]\nNot only \n$\\beta(u; N)$ but also $\\tilde{\\beta}(u; \\infty)$ \ndepend on $N$.\nHowever, \nthe dependence of $\\tilde{\\beta}(u; \\infty)$ becomes fairly weak\nfor $N \\gtrsim 20$, as shown in the inset.\n$\\tilde{\\beta}(u; \\infty)$ for $N=24$ \nagrees well with\nthe exact result, \nwhereas $\\beta(u; N)$ differs significantly from them for this value of $N$.\nWe have thus confirmed that \n$\\tilde{\\beta}(u; \\infty)$ is much closer \nto $\\beta(u; \\infty)$ than $\\beta(u; N)$, for finite $N$.\nNote however that $\\beta(u;N)$ gives almost correct result for $\\beta$\nof a {\\em finite} system, as seen from Eq.~(\\ref{eq:betaN_better}).\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=0.8\\linewidth]{T_0529.eps}\n\\end{center}\n\\vspace{-6mm}\n\\caption{\nTemperature $T$ plotted against $u$ for $J=+1$.\nSolid lines represent exact results for $N \\to \\infty$, \nfor various values of $h_z$ \\cite{AF_ET}. \nTriangles and squares denote \n$1\/\\beta(u; N)$ (triangles) and \n$1\/\\tilde{\\beta}(u; \\infty)$ (squares)\nfor $N=24$, obtained with our  formulation. \n(Inset) \n$1\/\\tilde{\\beta}(u; \\infty)$ for $N=8$-$24$.\n}\n\\label{fig:ET}\n\\end{figure}\n\nWe have obtained a series of TPQ states \nat the discrete points $u_0, u_1, u_2, \\cdots, u_{\\rm term}$.\nThe discrete points are dense enough because their intervals \nare $O(1\/N)$, vanishing as $N \\rightarrow \\infty$.\nThe intervals also depend on the parameter $l$.\nWhen smaller $l$ is taken, $k$ gets smaller to reach \nthe same $u$ and temperature $T$, \nas seen from Eq.~(\\ref{Nb=k\/L}).\nHence, to obtain results at low $T$, \n$l\\simeq e_{\\rm max}$ is appropriate to reduce the amount of \ncomputation.\nAt high $T$, however, $u_{k}$ moves quickly as $k$ increases, \nfor such a small $l$. \nHence, to obtain results at many values of $u$ at high $T$, \n$l$ should be taken larger. \nWhen computing the data for Figs.~\\ref{fig:hM} and \\ref{fig:Cor},\nwe have taken $l \\simeq e_{\\rm max}$.\nSince the values of $u$ which are specified in these figures \nare not necessarily found among $u_k$'s,\nwe have slightly tuned $l$ in such a way that \n$u_k$ can be found within $0.001J$ of the specified values.\nIn these figures, \n$m_z$ and $\\phi(j)$ at such $u_k$'s are plotted.\nWhen computing the data for Fig.~\\ref{fig:ET}, \nwe have performed computations with two values of $l$; \n$l=e_{\\rm max}$ and $5J$.\nBoth results agree well with each other.\nFor better visualization, \nwe have plotted the results with $l=5J$ (orange and purple) \nand those with $l=e_{\\rm max}$ (red and blue) \nin the high- and low-$T$ regions, respectively.\n\n{\\em Advantages -- }\nWe now discuss advantages of our formulation \nwhen used as a method of numerical calculation.\nAt finite $T$, \nan exponentially large number of states are included in $\\mathcal{E}_{E,N}$.\nThis makes computation of eigenstates in $\\mathcal{E}_{E,N}$ pretty hard.\nIn contrast, \nour method takes full advantage of such a huge number of states,\nas seen, e.g., in the derivation of Eq.~(\\ref{Sd=e^xi}).\nAs a result, \nusing just a single realization of TPQ state, \none can calculate all quantities of statistical-mechanical \ninterest at finite $T$, on the solid theoretical basis that \nis developed in this Letter.\nMoreover, our method is applicable to \nsystems of any spatial dimensions, and to \nfrustrated or fermion systems as well.\nFurthermore, \nour method costs much less computational resources than \nthe numerical diagonalization.\nFor example, \nthe number of non-vanishing elements of \n$\\hat{H}$ of the Heisenberg model is\n$O(N 2^N)$. \nSince $k=O(N)$, \nthe computational time is $O(N^2 2^N)$ in our method, \nwhich is exponentially shorter than that of diagonalization.\nIn fact, \nit took only two hours to compute all data in \nFig.~\\ref{fig:ET} \non a PC.\nComputations can be made even faster by parallelizing \nthe algorithm, which is quite easy and efficient because our method  \nconsists only of matrix multiplications.\n\nFurthermore, \nour method is effective over a wide range of \n$T$ \nbecause the rhs of Eq.~(\\ref{DeltaA}) is exponentially small \nas long as $s$ (and hence $T$) is finite of $O(1)$.\nIn fact, \nFigs.~\\ref{fig:hM}-\\ref{fig:ET} show that \nour results agree well with the rigorous results in a wide range of $T$, \nfrom $T \\ll J$ to $T \\gg J$.\nIn practical computations with finite $N$,\n$T$ ($=1\/\\tilde{\\beta}(u_\\kappa^{\\bullet};\\infty)$) \ncan be lowered as long as $r_k(e_{\\rm min};N)\/r_k(u_\\kappa^*;N) \\ll 1$.\n\n\n\nWe note that the quantum Monte Carlo method may be much faster.\nHowever, it suffers from the sign problem \nin frustrated systems and fermion systems.\nThe density-matrix renormalization group method\nhas been extended to finite temperature,\nand the state obtained in Ref.~\\cite{White}\nmight be close to TPQ states. \nHowever, its effectiveness in two- or more-dimensional systems \nis not clear yet.\nThe states obtained with the microcanonical Lanczos method \\cite{Long},\nwhich tried to obtain not TPQ states but eigenstates,\nmight also be close to TPQ states.\nHowever, the method \ncosts more computational time in Ref.~\\cite{Long}\nthan ours, and \na method of computing $T$ or $s$\nseems more difficult than ours.\nWe therefore expect that our method will \nmake it possible \nto analyze systems which could not be analyzed with other \nmethods.\n\n\n{\\em Concluding remarks -- }\nWe conclude this Letter by making several remarks.\nFirst,\none can evaluate the magnetic susceptibility \n$(\\partial m_z \/ \\partial h_z)_u$ from\nFig.~\\ref{fig:hM} or \\ref{fig:Cor}.\nOne can also obtain \n$(\\partial m_z \/ \\partial h_z)_T$ with the help of \nFig.~\\ref{fig:ET}. \n\nSecond, \n$| \\psi_{k} \\rangle$ remains to be a TPQ state \nafter time evolution, \nsince Eq.~(\\ref{psi=Sdn}) shows that \n$e^{\\hat{H}t\/{i\\hbar}} | \\psi_{k} \\rangle\n\\propto\n\\sum_{n} e^{- ie_n t\/\\hbar} c_{n} (l-e_{n})^k |n \\rangle\n$, \nwhich is just another realization of $| \\psi_{k} \\rangle$.\n\nThird, \nour formulation is advantageous to analyses of phase transitions.\nAs an example, consider the case where the energy density \n$u(T;N)$ for $N \\to \\infty$ is discontinuous at \nthe transition temperature $T_{\\rm tr}$ of a \nfirst-order transition.\nThen, the specific heat $c = {\\partial u \/ \\partial T}$\ndiverges at $T=T_{\\rm tr}$.\nIf one used the canonical formalism, \nwhere $T$ is an independent variable, \ncalculation of \n$u(T;\\infty)$ would be hard around $T = T_{\\rm tr}$.\nIn our formulation, by contrast, \n$u$ is taken as an independent variable, \nand $c$ is obtained as\n$c \n=-\\beta^2\/({\\partial \\beta \/ \\partial u})_{\\scriptscriptstyle N}$\nfrom $\\beta(u;\\infty)$. \nThe function $\\beta(u;\\infty)$\nis {\\em continuous even at the transition point},\nwhere it takes a constant value $1\/T_{\\rm tr}$\nin a finite interval of $u$  \ncorresponding to the phase coexistent region \\cite{AS:phaserule}.\nHence, $\\beta(u;\\infty)$ can be calculated more easily than $u(T;\\infty)$. \nIn fact, one can identify a first-order transition \nby simply observing that \nthe rhs of Eq.~(\\ref{eq:beta_infty}) takes a constant value,\napart from small deviation of $O(1\/N^2)$,\nfor multiple values of $\\tilde{u}^{\\bullet}_\\kappa$ and $\\kappa$.\nRegarding a continuous transition,\nit can be identified from a singularity in $c$, or\nan order parameter $m$, and so on.\nOne can calculate $m$ by adding  \na symmetry-breaking field $f$ to the Hamiltonian,\nand thereby computing $m$ at $f= \\pm |f|$ for small $|f|$.\nOr alternatively, without introducing $f$, one can perform the \n`pure-state decomposition' (i.e., decomposition into \nmacroscopically definite states) by applying\nthe variance-covariance matrix method of Ref.~\\cite{VCM} to a TPQ state. \n\nFourth, \nEq.~(\\ref{psi=Sdn}) can be generalized as\n$\n|\\psi \\rangle \\propto Q(\\hat{h}) |\\psi_{0} \\rangle,\n$ \nwhich defines other new TPQ states.\nHere, $Q(u)$ is any differentiable real \nfunction such that $Q(u)^2 g(u;N)$ has a sharp peak,\nwhose width vanishes as $N \\to \\infty$,\nand $Q(u)^2 g(u;N)$ outside the peak decays quickly. \nUsing this $|\\psi \\rangle$,\none can calculate various quantities as we have done using $|\\psi_k \\rangle$.\nFor instance, the formula corresponding to \nEq.~(\\ref{Nb=k\/L}) is given by\n$\n\\beta(u^{\\ast};N) \n+ 2Q'(u^{\\ast}) \/ N Q(u^{\\ast}) = 0\n$.\n\nFinally,\nalthough a TPQ state (such as $| \\psi_k \\rangle$) and \nthe mixed state (such as $\\hat{\\rho}_k$) of the corresponding ensemble\nare identical with respect to mechanical variables, \n{\\em they are completely different with respect to entanglement}.\nAt $T \\gg J$, for example, $\\hat{\\rho}_k$ has only small \nentanglement (because it is close to the completely mixed state \n$(1\/D) \\hat{1}$, \nwhich has no entanglement), whereas we can show \nthat $| \\psi_k \\rangle$ has exponentially \nlarge entanglement (as previously shown for $T \\to \\infty$ in Ref.~\\cite{SugitaShimizu}). \nIt is thus seen that an equilibrium state can be represented \neither by a TPQ state with huge\nentanglement or by a mixed state with much less entanglement.\nTheir difference can be detected only by high-order polynomials of\nlocal operators, which are \n{\\em not} of statistical-mechanical interest \\cite{SugitaJ, SugitaShimizu}.\n\n\t\n\n\n\n\n\n\n\n\n\\begin{acknowledgments}\nWe thank\nJ. Sato, F. G\\\"{o}hmann, C. Trippe and \nK. Sakai for providing us with numerical\ndata of exact solutions.\nWe also thank H. Tasaki, A. Sugita, Y. Kato, Y. Oono, H. Katsura, \nK. Hukushima, S. Sasa and T. Yuge\nfor helpful discussions.\nThis work was supported by KAKENHI Nos.~22540407 and 23104707.\n\\end{acknowledgments}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nFunctions with values in a (nonlinear) subset of a vector space appear in several applications of imaging and in inverse problems, e.g.\n\n\n\\begin{itemize}\n \\item \\emph{Interferometric Synthetic Aperture Radar (InSAR)} \n       is a technique used in remote sensing and geodesy to generate for example digital elevation \n       maps of the earth's surface. \n       InSAR images represent phase differences of waves between two or more SAR images, cf. \\cite{LiuMas16,RocPraFer97}. \n       Therefore InSAR data are functions $f:\\Omega \\to \\sphere \\subseteq \\mathds{R}^2$.\n       The pointwise function values are on the $\\sphere$, which is considered embedded into $\\mathds{R}^2$.\n \\item A \\emph{color image} can be represented as a function in \\emph{HSV}-space (hue, saturation, value) (see e.g. \\cite{PlaVen00}).\n       Color images are then described as functions $f:\\Omega \\to K \\subseteq \\mathds{R}^3$. Here $\\Omega$ is a plane in $\\mathds{R}^2$, the image domain,\n       and $K$ (representing the HSV-space) is a cone in 3-dimensional space $\\mathds{R}^3$. \n \\item Estimation of the \\emph{foliage angle distribution} has been considered for instance in\n       \\cite{HelAndRobFin15,PutBriManWiePfeZliPfe16}.\n       Thereby the imaging function is from $\\Omega \\subset \\mathds{R}^2$, a part of the Earth's surface, \n       into $\\mathbb{S}^2 \\subseteq \\mathds{R}^3$, representing foliage angle orientation.\n \\item Estimation of functions with values in $SO(3) \\subseteq \\mathds{R}^{3 \\times 3}$. Such problems appear in \\emph{Cryo-Electron Microscopy} \n       (see for instance \\cite{HadSin11,SinShk12,WanSinWen13}).\n\\end{itemize}\nWe emphasize that we are analyzing \\emph{vector}, \\emph{matrix}, \\emph{tensor}-valued functions, where pointwise function \nevaluations belong to some given (sub)set, but are always \\emph{elements} of the underlying vector space. This should not be confused with set-valued functions, where every function \nevaluation can be a set.\n\nInverse problems and imaging tasks, such as the ones mentioned above, might be unstable, or even worse, the solution could be \nambiguous. Therefore, numerical algorithms for imaging need to be \\emph{regularizing} to obtain approximations of the \ndesired solution in a stable manner. \nConsider the operator equation \n\\begin{equation}\\label{eq:basic_problem}\n\\op[w] = v^0,\n\\end{equation} \nwhere we assume that only (noisy) measurement data $v^\\delta$ of $v^0$ become available. \nIn this paper the method of choice is \\emph{variational regularization} which consists \nin calculating a minimizer of the variational regularization functional\n\\begin{equation}\n \\label{eq:energy}\n \\mathcal{F}(w) \\vcentcolon= \\mathcal{D}(\\op[w],v^\\delta) + \\alpha \\mathcal{R}(w).\n\\end{equation}\nHere\n\\begin{description}\n \\item{$w$} is an element of the \\emph{set} of admissible functions.\n \\item{$\\op$} is an operator modeling the image formation process (except the noise).\n \\item{$\\mathcal{D}$} is called the \\textit{data} or \\textit{fidelity term}, which is used to compare a pair \n       of data in the image domain, that is to quantify the difference of the two data sets. \n \\item{$\\mathcal{R}$} is called \\textit{regularization functional}, which is used to impose certain \n       properties onto a minimizer of the regularization functional $\\mathcal{F}$.\n \\item{$\\alpha > 0$} is called \\textit{regularization parameter} and provides a trade off between stability \n       and approximation properties of the minimizer of the regularization functional $\\mathcal{F}$.  \n \\item{$v^\\delta$} denotes measurement data, which we consider noisy. \n \\item{$v^0$} denotes the exact data, which we assume to be not necessarily available.\n\\end{description}\n\nThe main objective of this paper is to introduce a general class of regularization functionals for functions with values in a set of vectors. \nIn order to motivate our proposed class of regularization functionals we review a class of regularization functionals \nappropriate for analyzing \\emph{intensity data}.\n\n\n\\subsection*{Variational regularization for reconstruction of intensity data}\nOpposite to what we consider in the present paper, most commonly, imaging data $v$ and admissible functions $w$, respectively, are \nconsidered to be representable as \\emph{intensity functions}. That is, they are functions from some subset $\\Omega$ of an Euclidean \nspace with \\emph{real values}.\n\nIn such a situation the most widely used regularization functionals use regularization terms \nconsisting of powers of Sobolev (see \\cite{BouSau93,ChaLio97,CimMel12}) or total variation semi-norms \\cite{RudOshFat92}. \nIt is common to speak about \\emph{Tikhonov regularization} (see for instance \\cite{TikArs77}) when the data term and the \nregularization functional are squared Hilbert space norms, respectively. \nFor the \\emph{Rudin, Osher, Fatemi (ROF)} regularization \\cite{RudOshFat92}, also known as total variation \nregularization, the data term is the squared $L^2$-norm and $\\mathcal{R}(w) = |w|_{TV}$ is the total variation semi-norm. \nNonlocal regularization operators based on the generalized nonlocal gradient is used in \\cite{GilOsh08}. \\\\  \nOther widely used regularization functionals are \\emph{sparsity promoting} \\cite{DauDefDem04,KolLasNiiSil12},\n\\emph{Besov space norms} \\cite{LorTre08,LasSak09} and anisotropic regularization norms \\cite{OshEse04,SchWei00}.\nAside from various regularization terms there also have been proposed different fidelity terms other than quadratic norm\nfidelities, like the $p$-th powers of $\\ell^p$ and $L^p$-norms of the differences of \n$F(w)$ and $v$ , \\cite{SchGraGroHalLen09,SchuKalHofKaz12}, Maximum Entropy \\cite{Egg93,EngLan93} and Kullback-Leibler divergence \n\\cite{ResAnd07} (see \\cite{Poe08} for some reference work).\n\nOur work utilizes results from the seminal paper of \\citeauthor{BouBreMir01} \\cite{BouBreMir01}, which \nprovides an equivalent \\emph{derivative-free} characterization of Sobolev spaces and the space $\\BV$, \nthe space of functions of bounded total variation, which consequently, in this context, was analyzed in \\citeauthor{Dav02} \nand \\citeauthor{Pon04b} \\cite{Dav02,Pon04b}, respectively. It is shown in \n\\cite[Theorems 2 \\& 3']{BouBreMir01} and \\cite[Theorem 1]{Dav02} that when $(\\rho_\\varepsilon)_{\\varepsilon > 0}$ \nis a suitable sequence of non-negative, radially symmetric, radially decreasing mollifiers, then\n\\begin{equation}\n\\label{eq:double_integral}\n\\begin{aligned}\n\\lim_{\\varepsilon \\searrow 0} \\tilde{\\mathcal{R}}_\\varepsilon(w) & \n\\vcentcolon= \\lim_{\\varepsilon \\searrow 0}  \\int\\limits_{\\Omega\\times \\Omega} \\frac{\\|w(x)- w(y)\\|_{\\mathds{R}}^p}{\\normN[x-y]^{p}} \\rho_\\varepsilon(x-y) \\,\\mathrm{d}(x,y)\\\\\n&= \n\\begin{cases}\nC_{p,N}|w|^p_{W^{1,p}} & \\mbox{if } w \\in W^{1,p}(\\Omega, \\mathds{R}), \\ 1 < p < \\infty, \\\\\nC_{1,N}|w|_{TV} & \\mbox{if } w \\in \\BV[\\mathds{R}], \\ p = 1, \\\\\n\\infty & \\mbox{otherwise},\n\\end{cases}\n\\end{aligned}\n\\end{equation}\nHence $\\tilde{\\mathcal{R}}_\\varepsilon$ approximates powers of Sobolev semi-norms and the total variation semi-norm, respectively. Variational \nimaging, consisting in minimization of $\\mathcal{F}$ from \\autoref{eq:energy} with $\\mathcal{R}$ replaced by \n$\\tilde{\\mathcal{R}}_\\varepsilon$,  has been considered in \\cite{AubKor09,BouElbPonSch11}. \n\n\n\\subsection*{Regularization of functions with values in a set of vectors}\nIn this paper we generalize the derivative-free characterization of Sobolev spaces and functions of bounded variation to \nfunctions, $u:\\Omega \\to K$, where $K$ is some set of vectors, and use these functionals for variational \nregularization. The applications we have in mind contain that $K$ is a closed subset of $\\mathds{R}^M$ (for instance HSV-data) \nwith non-zero measure, or that $K$ is a sub-manifold (such as for instance InSAR-data).\n\nThe reconstruction of manifold--valued data with variational regularization methods has already been subject \nto intensive research (see for instance \\cite{KimSoc02,CreStr11,CreStr13,CreKoeLelStr13,BacBerSteWei16,WeiDemSto14}).\nThe variational approaches mentioned above use regularization and fidelity functionals based on \nSobolev and TV semi-norms: a total variation regularizer \nfor cyclic data on $\\sphere$ was introduced in \\cite{CreStr13,CreStr11}, see also \\cite{BerLauSteWei14,BerWei16,BerWei15}.\nIn \\cite{BacBerSteWei16,BerFitPerSte17} combined first and second order \ndifferences and derivatives were used for regularization to restore manifold--valued data. The later mentioned papers, however, \nare formulated in a finite dimensional setting, opposed to ours, which is considered in an infinite dimensional setting. \nAlgorithms for total variation minimization problems, including half-quadratic minimization and non-local patch based methods, \nare given for example in \\cite{BacBerSteWei16,BerChaHiePerSte16,BerPerSte16} as well as in \\cite{GroSpr14,LauNikPerSte17}.\nOn the theoretical side the total variation of functions with values in a manifold was investigated by \\citeauthor{GiaMuc07} \nusing the theory of Cartesian currents in \\cite{GiaMuc07,GiaMuc06}, and earlier \\cite{GiaModSou93} if the manifold is a \n$\\sphere$. \n\n\n\\subsection*{The contents and the particular achievements of the paper are as follows}\nThe contribution of this paper is to introduce and analytically analyze double integral \nregularization functionals for reconstructing functions with values in a set of vectors, generalizing functionals \nof the form \\autoref{eq:double_integral}. \nMoreover, we develop and analyze fidelity terms for comparing manifold--valued data. \nSumming these two terms provides a new class of regularization functionals of the form \n\\autoref{eq:energy} for reconstructing manifold--valued data. \n\nWhen analyzing our functionals we encounter several differences to existing \nregularization theory (compare \\autoref{sec: Setting}):\n\\begin{enumerate}\n\\item \n      The \\emph{admissible functions}, where we minimize the regularization functional on, \n      do form only a \\emph{set} but \\emph{not} a \\emph{linear} space.\n      As a consequence, well--posedness \n      of the variational method (that is, existence of a minimizer of the energy functional) cannot directly be proven \n      by applying standard \n      direct methods in the Calculus of Variations \\cite{Dac82,Dac89}. \n\\item The regularization functionals are defined via metrics and not norms, see \\autoref{sec: Existence}. \n\\item In general, the fidelity terms are \\emph{non-convex}.\n      Stability and convergence results are proven in \\autoref{sec: Stability_and_Convergence}. \n\\end{enumerate}      \n\nThe model is validated in \\autoref{sec:Numerical_results} where we present numerical results for denoising \nand inpainting of data of InSAR type.\n\n\n\\section{Setting} \\label{sec: Setting}\nIn the following we introduce the basic notation and the set of admissible functions which we are regularizing \non.\n\n\\begin{assumption}\n \\label{ass:1}\n All along this paper we assume that\n \\begin{itemize}\n  \\item $p_1, p_2 \\in [1, +\\infty)$, $s \\in (0,1]$,\n  \\item $\\Omega_1, \\Omega_2 \\subseteq \\mathds{R}^N$ are nonempty, bounded, and connected open sets with Lipschitz boundary, respectively,\n  \\item $k \\in [0,N]$,\n  \\item $K_1 \\subseteq \\mathds{R}^{M_1}, K_2 \\subseteq \\mathds{R}^{M_2}$ are nonempty and closed subsets of $\\mathds{R}^{M_1}$ and $\\mathds{R}^{M_2}$, respectively.       \n \\end{itemize}\n Moreover, \n \\begin{itemize}\n  \\item $\\normN$ and $\\|\\cdot\\|_{\\mathds{R}^{M_i}}, \\ i=1,2,$ are the Euclidean norms on $\\mathds{R}^N$ and $\\mathds{R}^{M_i}$, respectively.\n  \\item $\\dRMi: \\mathds{R}^{M_i} \\times \\mathds{R}^{M_i} \\rarr [0, +\\infty)$ denotes the Euclidean distance on $\\mathds{R}^{M_i}$ for $i=1,2$ and \n  \\item $\\d_i \\vcentcolon= \\mathrm{d}_{K_i}: K_i \\times K_i \\rarr [0, +\\infty)$ \n          denote arbitrary metrics on $K_i$, which fulfill for $i=1$ and $i=2$\n          \\begin{itemize}\n          \t\\item $\\dRMi|_{K_i \\times K_i} \\leq d_i$,\n          \t\\item $\\d_i$ is continuous with respect to $\\dRMi|_{K_i \\times K_i}$, meaning that for a sequence $\\seq{a}$ in $K_i \\subseteq \\mathds{R}^{M_i}$  converging to some $a \\in K_i$ we also have $\\d_i(a_n,a) \\rarr 0$. \n          \\end{itemize} \n          In particular, this assumption is valid if the metric $d_i$ is equivalent to $\\dRMi|_{K_i \\times K_i}$.\n          When the set $K_i, \\ i=1,2$, is a suitable complete submanifold of $\\mathds{R}^{M_i}$,\n          it seems natural to choose $d_i$ as the geodesic distance on the respective submanifolds.\n  \\item  $(\\rho_{\\varepsilon})_{\\varepsilon > 0}$ is a Dirac family of non-negative, radially symmetric mollifiers, i.e. for every $\\varepsilon > 0$ we have\n    \\begin{enumerate}\n      \\item $\\rho_\\varepsilon \\in \\mathcal{C}^{\\infty}_{c}(\\mathds{R}^N, \\mathds{R})$ is radially symmetric,\n      \\item $\\rho_\\varepsilon \\geq 0$,\n      \\item $\\int \\limits_{\\mathds{R}^N} \\rho_\\varepsilon (x) \\,\\mathrm{d}x = 1$, and \n      \\item for all $\\delta > 0$, $\\lim_{\\varepsilon \\searrow 0}\\limits \\int_{\\set{\\normN[y] > \\delta}} \\rho_\\varepsilon(y) \\,\\mathrm{d}y = 0$.\n    \\end{enumerate} \n    We demand further that, for every $\\varepsilon > 0$, \n    \\begin{enumerate}[resume]\n      \\item there exists a  $\\tau > 0$ and $\\eta_{\\tau} > 0$ such that\n        $\\{ z \\in \\mathds{R}^N : \\rho_{\\varepsilon}(z) \\geq \\tau \\}= \\{z \\in \\mathds{R}^N : \\normN[z] \\leq \\eta_{\\tau} \\}$.\n    \\end{enumerate}\n    This condition holds, e.g., if $\\rho_{\\varepsilon}$ is a radially decreasing continuous function with $\\rho_{\\varepsilon}(0) > 0$.\n  \\item When we write $p$, $\\Omega$, $K$, $M$, then we mean $p_i$, $\\Omega_i$, $K_i$, $M_i$, for either \n        $i=1,2$. In the following we will often omit the subscript indices whenever possible.\n  \\end{itemize}\n\\end{assumption}\n\n\\begin{example}\\label{ex:mol}\nLet $\\hat{\\rho} \\in C_c^\\infty(\\mathds{R},\\mathds{R}_+)$ be symmetric at $0$, monotonically decreasing on $[0, \\infty)$ and satisfy\n\\begin{equation*}\n \\abs{\\mathbb{S}^{N-1}}\\int_0^\\infty \\hat{t}^{N-1} \\hat{\\rho}\\left(\\hat{t}\\right)d \\hat{t} = 1\\;.\n\\end{equation*}\nDefining mappings $\\rho_\\varepsilon: \\mathds{R}^N \\to \\mathds{R}$ by\n\\begin{equation*}\n  \\rho_\\varepsilon(x)\\vcentcolon= \\frac{1}{\\varepsilon^N} \\hat{\\rho}\\left(\\frac{\\normN[x]}{\\varepsilon}\\right)\n\\end{equation*}\nconstitutes then a family $(\\rho_\\varepsilon)_{\\varepsilon > 0}$ which fulfills the above properties \\it{(i) -- (v)}.\nNote here that\n\\begin{itemize}\n \\item by substitution $x = t \\theta$ with $t > 0, \\theta \\in \\mathbb{S}^{N-1}$ and $\\hat{t}=\\frac{t}{\\varepsilon}$, \n\\begin{equation}\n \\label{eq:molII}\n \\begin{aligned}\n  \\int_{\\mathds{R}^N} \\rho_\\varepsilon(x)\\, \\,\\mathrm{d}x &= \\frac{1}{\\varepsilon^N} \\int_{\\mathds{R}^N} \\hat{\\rho}\\left(\\frac{\\normN[x]}{\\varepsilon}\\right) \\,\\mathrm{d}x \\\\\n  &= \\frac{1}{\\varepsilon^N} \\int_0^\\infty t^{N-1} \\hat{\\rho}\\left(\\frac{t}{\\varepsilon}\\right)\\d t \\int_{\\mathbb{S}^{N-1}} \\d\\theta  \\\\ \n  &= \\abs{\\mathbb{S}^{N-1}}\\int_0^\\infty \\hat{t}^{N-1} \\hat{\\rho}\\left(\\hat{t}\\right)\\d \\hat{t} = 1\\;.\n \\end{aligned}\n\\end{equation}\nHere, $d\\theta$ refers to the canonical spherical measure.\n \\item Again by the same substitutions, taking into account that $\\hat{\\rho}$ has compact support, it follows for \n       $\\varepsilon > 0$ sufficiently small that\n\\begin{equation}\n \\label{eq:molIIa}\n \\begin{aligned}\n  \\int_{\\set{y:\\normN[y]>\\delta}} \\rho_\\varepsilon(x)\\, \\,\\mathrm{d}x \n  &= \\frac{1}{\\varepsilon^N} \\int_{\\set{y:\\normN[y]> \\delta}} \\hat{\\rho}\\left(\\frac{\\normN[x]}{\\varepsilon}\\right) \\,\\mathrm{d}x \\\\\n  &= \\frac{1}{\\varepsilon^N} \\int_\\delta^\\infty t^{N-1} \\hat{\\rho}\\left(\\frac{t}{\\varepsilon}\\right)\\d t \\int_{\\mathbb{S}^{N-1}} \\d\\theta  \\\\ \n  &= \\abs{\\mathbb{S}^{N-1}}\\int_{\\delta\/\\varepsilon}^\\infty \\hat{t}^{N-1} \\hat{\\rho}\\left(\\hat{t}\\right)\\d \\hat{t} =0 \\;.\n \\end{aligned}\n\\end{equation}\n\\end{itemize}\n\\end{example}\nIn the following we write down the basic spaces and sets, which will be used in the course of the paper.\n\\begin{definition}\n\\begin{itemize}\n \\item The \\emph{Lebesgue--Bochner space} of $\\mathds{R}^M$--valued functions on $\\Omega$ consists of the set \n  \\begin{equation*}\n  \\begin{aligned}\n    \\Lp \\vcentcolon= \\{ \\phi : \\Omega \\to \\mathds{R}^M : {} & \\phi \\text{ is Lebesgue-Borel measurable and }  \\\\ \n                                           &\\normM[\\phi(\\cdot)]^p: \\Omega \\to \\mathds{R} \\text{ is Lebesgue--integrable on } \\Omega \\},\n   \\end{aligned}\n  \\end{equation*}\n  which is associated with the norm $\\|\\cdot\\|_{\\Lp}$, given by\n  \\begin{gather*}\n    \\norm{\\phi}_{\\Lp} \\vcentcolon= \\Big( \\int_{\\Omega}\\limits \\normM[\\phi(x)]^p \\,\\mathrm{d}x \\Big)^{1\/p} \\; .\n  \\end{gather*}\n \\item Let $0 < s < 1$. Then the \\emph{fractional Sobolev space} of order $s$ can be defined (cf. \\cite{Ada75}) as the set\n\\begin{gather*}\n  \\Wsp \\vcentcolon= \\left\\{ w \\in \\Lp : \\frac{\\normM[w(x) - w(y)]}{\\normN[x-y]^{\\frac{N}{p}+s}} \\in \n  L^p (\\Omega \\times \\Omega, \\mathds{R})  \\right\\} \\\\\n  = \\{w \\in \\Lp : \\abs{w}_{\\Wsp} < \\infty \\},\n\\end{gather*}\nequipped with the norm\n\\begin{equation}\\label{eq:sobolev_norm}\n \\|\\cdot\\|_{\\Wsp} \\vcentcolon= \\big(\\|\\cdot\\|_{\\Lp}^{p} + \\abs{\\cdot}_{\\Wsp}^p \\big)^{1\/p},\n\\end{equation}\nwhere $\\abs{\\cdot}_{\\Wsp}$ is the semi-norm for $\\Wsp$, given by\n\\begin{equation}\\label{eq:sobolev_semi_norm}\n  \\abs{w}_{\\Wsp} \\vcentcolon=  \\Big(\\int\\limits_{\\Omega\\times \\Omega} \\frac{\\normM[w(x) - w(y)]^p}{\\normN[x-y]^{N+ps}} \\,\\mathrm{d}(x,y) \\Big)^{1\/p},\n  \\quad w \\in \\Wsp\\;.\n\\end{equation}\n\\item \nFor $s = 1$ the Sobolev space $W^{1,p}(\\Omega, \\mathds{R}^M)$ consists of all weakly differentiable\nfunctions in $L^1(\\Omega,\\mathds{R}^M)$ for which \n\\begin{equation*}\n\\norm{w}_{W^{1,p}(\\Omega, \\mathds{R}^M)} \n  \\vcentcolon= \\Big( \\norm{w}_{\\Lp}^p +  \\int_{\\Omega}\\limits \\|\\nabla w(x)\\|^p_{\\mathbb{R}^{M\\times N}} \\,\\mathrm{d}x \\Big)^{1\/p} < \\infty\\;,\n\\end{equation*}\nwhere $\\nabla w$ is the weak Jacobian of $w$.\n\\item Moreover, we recall one possible definition of the space $\\BV$ from \\cite{AmbFusPal00}, which consists of all \n      Lebesgue--Borel measurable functions $w:\\Omega \\to \\mathds{R}^M$ for which \n\\begin{gather*}\n\\norm{w}_{\\BV} \\vcentcolon= \\norm{w}_{L^1(\\Omega, \\mathds{R}^M)} + \\abs{w}_{\\BV} < \\infty,\n\\end{gather*}\nwhere \n\\begin{equation*}\n\\begin{aligned}\n~ & \\abs{w}_{\\BV} \\\\\n\\vcentcolon= {}& \\sup \\left\\{ \\int \\limits_\\Omega w(x) \\cdot \\mathrm{Div} \\varphi(x) \\,\\mathrm{d}x : \\ \n\\varphi \\in C_c^1(\\Omega, \\mathds{R}^{M \\times N}) ~ \\mathrm{ such~that }  \\norm{\\varphi}_\\infty \\vcentcolon= \\mathop{\\mathrm{ess~sup}}_{x \\in \\Omega} \\norm{\\varphi(x)}_F \\leq 1 \\right\\},\n\\end{aligned}\n\\end{equation*}\nwhere $\\norm{\\varphi(x)}_F$ is the Frobenius-norm of the matrix $\\varphi(x)$ and \n$\\mathrm{Div}\\varphi = (\\mathrm{div} \\varphi_1, \\dots, \\mathrm{div} \\varphi_M)^\\mathrm{T}$ denotes the row--wise formed divergence of $\\varphi$.\n\\end{itemize}\n\\end{definition}\n\n\\begin{lemma}\n\\label{le:inclusion}\n Let $0 < s \\leq 1$ and $p \\in [1,\\infty)$, then $\\Wsp \\hookrightarrow \\Lp$ and the embedding is compact. Moreover, the embedding\n $\\BV \\hookrightarrow \\Lp$ is compact for all \n   \\begin{gather*}\n    1 \\leq p < 1^* \\vcentcolon=\n    \\begin{cases}\n      +\\infty &\\mbox{if } N = 1 \\\\\n      \\frac{N}{N-1} &\\mbox{otherwise }\n    \\end{cases} \\,.\n  \\end{gather*} \n\\end{lemma}\n\\begin{proof}\n The first result can be found in \\cite{DemDem07} for $0 < s < 1$ and in \\cite{Eva10} for $s = 1$. The second assertion \n is stated in \\cite{AmbFusPal00}.\n\\end{proof}\n\n\n\\begin{remark} \n\\label{re:notes_basic}\nLet \\autoref{ass:1} hold.\n  We recall some basic properties of weak convergence in $\\Wsp$, $W^{1,p}(\\Omega, \\mathds{R}^M)$ and weak* convergence in $\\BV$ (see for instance \\cite{Ada75,AmbFusPal00}) :\n  \\begin{itemize}\n    \\item Let $p > 1$, $s\\in(0,1]$ and assume that $(w_n)_{n \\in \\mathds{N}}$ is bounded in $\\Wsp$. \n          Then there exists a subsequence $(w_{n_k})_{k \\in \\mathds{N}}$ which converges weakly in $\\Wsp$. \n    \\item Assume that $(w_n)_{n \\in \\mathds{N}}$ is bounded in $\\BV$. Then there exists a subsequence $(w_{n_k})_{k \\in \\mathds{N}}$ \n          which converges weakly* in $\\BV$. \n  \\end{itemize}\n\\end{remark}\n\n\nBefore introducing the regularization functional, which we investigate theoretically and numerically, we give \nthe definition of some sets of (equivalence classes of) admissible functions.\n\\begin{definition} \\label{def:spaces_etc}\n\\label{de:basic}\n    For $0 < s \\leq 1$, $p \\geq 1$ and a nonempty closed subset $K \\subseteq \\mathds{R}^M$ we define \n    \\begin{equation}\n    \\label{eq:spacess_K}\n    \\begin{aligned}\n      \\Lp[K]  \\vcentcolon= {} & \\{\\phi \\in \\Lp : \\phi(x) \\in K \\text{ for a.e. } x \\in \\Omega\\}; \\\\\n      \\Wsp[K] \\vcentcolon= {} & \\{w \\in \\Wsp: w(x) \\in K \\text{ for a.e. } x \\in \\Omega  \\}, \\\\\n      \\BV[K]  \\vcentcolon= {} & \\{w \\in \\BV: w(x) \\in K \\text{ for a.e. } x \\in \\Omega  \\}.\n    \\end{aligned}\n    \\end{equation}\n    and equip each of these (in general nonlinear) sets with some subspace topology:\n    \\begin{itemize}\n    \\item $\\Lp[K] \\subseteq \\Lp$ is associated with the strong $\\Lp$-topology,\n    \\item $\\Wsp[K] \\subseteq \\Wsp$ is associated with the weak $\\Wsp$-topology, and \n    \\item $\\BV[K] \\subseteq \\BV$ is associated with the weak* $\\BV$-topology.\n    \\end{itemize}\n\n    Moreover, we define \n      \\begin{equation} \\label{eq:ChooseW}\n      W(\\Omega,K) \\vcentcolon= \n        \\begin{cases}\n          \\Wsp[K] & \\text{ for } p \\in (1, \\infty) \\text { and } s \\in (0,1], \\\\\n          \\BV[K]  & \\text{ for } p = 1             \\text { and } s = 1\\;.\n        \\end{cases}\n      \\end{equation}\n      Consistently, $W(\\Omega,K)$ \n      \\begin{itemize}\n       \\item is associated with the weak $\\Wsp$-topology in the case $p \\in (1, \\infty)$ and $s \\in (0,1]$ and\n       \\item with the weak* $\\BV$-topology when $p=1$ and $s=1$.\n      \\end{itemize}\n      When we speak about \n      \\begin{equation*}\n      \\text{ convergence on } W(\\Omega,K) \\text{ we write } \\overset{W(\\Omega, K)}{\\longrightarrow} \\text{ or simply} \\rarr<W>\n      \\end{equation*}\n      and mean weak convergence on $W^{s,p}(\\Omega,K)$ and weak* convergence on $\\BV[K]$, respectively.\n\\end{definition}\n\\begin{remark} \n\\label{re:notes_choose_w}\n~\\nopagebreak\n\\begin{itemize}[topsep=0pt]\n\\item In general $\\Lp[K], \\Wsp[K]$ and $\\BV[K]$ are sets which do not form a linear space.\n\\item If $K = \\sphere$, then $\\Wsp[K] = \\Wsp[\\sphere]$ as occurred in \\cite{BouBreMir00b}.\n\\item For an embedded manifold $K$ the dimension of the manifold is not necessarily identical with the space dimension of $\\mathds{R}^M$. \n      For instance if $K = \\sphere \\subseteq \\mathds{R}^2$, then the dimension of $\\sphere$ is $1$ and $M=2$.\n\\end{itemize}\n\\end{remark}\nThe following lemma shows that $W(\\Omega,K)$ is a sequentially closed subset of $\\W$.\n\\begin{lemma}[Sequential closedness of $W(\\Omega,K)$ and {$\\Lp[K]$}] \\label{lem:Wsp_weakly_seq_closed_etc} \n  ~\\nopagebreak \n  \\vspace{-0.5\\baselineskip}\n  \\begin{enumerate}[topsep=0pt]\n    \\item \n      Let $w_* \\in \\W$ and $(w_n)_{n\\in \\mathds{N}}$ be a sequence in $\\W[K] \\subseteq \\W$ with \n      $w_n \\overset{W(\\Omega, \\mathds{R}^M)}{\\longrightarrow} w_*$ as $n \\to \\infty$. \n      Then $w_* \\in \\W[K]$ and $w_n \\rarr w_*$ in $\\Lp[K]$.\n    \\item \n        Let $v_* \\in \\Lp$ and $(v_n)_{n \\in \\mathds{N}}$ be a sequence in $\\Lp[K] \\subseteq \\Lp$ with $v_n \\to v_*$ in $\\Lp$ as $n \\to \\infty$.\n        Then $v_* \\in \\Lp[K]$  and there is some subsequence $(v_{n_k})_{k \\in \\mathds{N}}$ which converges to $v_*$ pointwise almost everywhere, i.e. $v_{n_k}(x) \\to v_*(x)$\n        as $k \\to \\infty$ for almost every $x \\in \\Omega$.\n  \\end{enumerate}\n\\end{lemma}\n\\begin{proof}\n  For the proof of the second part, cf. \\cite{Els02}, Chapter VI, Corollary 2.7\n  and take into account the closedness of $K \\subseteq \\mathds{R}^M$.\n  The proof of the first part follows from standard convergence arguments in $\\Wsp$, $\\BV$ and $\\Lp$, respectively, \n  using the embeddings from \\autoref{le:inclusion}, an argument on subsequences and part two.\n\\end{proof}\n\n\\begin{remark}\n  \\autoref{le:inclusion} along with \\autoref{lem:Wsp_weakly_seq_closed_etc} imply that $\\W[K]$ is compactly embedded in $\\Lp[K]$, \n  where these sets are equipped with the bornology inherited from $\\W$ and the topology inherited from $\\Lp$, respectively.\n\\end{remark}\n\n\n\nIn the following we postulate the assumptions on the operator $\\op$ which will be used throughout the paper:\n\\begin{assumption}\n\\label{ass:2}\nLet $\\W<1>$ be as in \\autoref{eq:ChooseW} and assume that $\\op$ is an operator from $\\W<1>$ to $\\Lp<2>$. \n\\end{assumption}\n\nWe continue with the definition of our regularization functionals:\n\\begin{definition} \\label{def:functional}\nLet \\autoref{ass:1} and \\autoref{ass:2} hold. Moreover, let $\\varepsilon > 0$ be fixed and let \n$\\rho \\vcentcolon= \\rho_\\varepsilon$ be a mollifier.\n\nThe regularization functional  \n    $\\F<v><\\alpha>[\\dKt, \\dKo] : \\W<1> \\rarr [0, \\infty]$ is defined as follows\n    \\begin{equation}\\label{eq: functional_with_some_metric}\n    \\boxed{\n      \\F<v><\\alpha>[\\dKt, \\dKo] (w) \\vcentcolon= \\int\\limits_{\\Omega_2} \\dKt^{p_2}(\\op[w](x), v(x)) \\,\\mathrm{d}x + \\alpha \\int\\limits_{\\Omega_1\\times \\Omega_1} \\frac{\\dKo^{p_1}(w(x), w(y))}{\\normN[x-y]^{k+p_1 s}} \\rho^l(x-y) \\,\\mathrm{d}(x,y),}\n        \\end{equation}\n    where \n    \\begin{enumerate}\n      \\item  $v \\in \\Lp<2>$,\n      \\item  $s \\in (0,1]$, \n      \\item  $\\alpha \\in (0, +\\infty)$ is the regularization parameter, \n      \\item  $l \\in \\set{0, 1}$ is an indicator and\n      \\item  \\label{itm:k}\n      $\\begin{cases}\n      k \\leq N &\\mbox{if } \\W<1> = W^{s,p_1}(\\Omega_1, K_1), \\ 0<s<1, \\\\\n      k=0 & \\mbox{if } \\W<1> = W^{1,p_1}(\\Omega_1, K_1) \\text{ or if } \\W<1> = BV(\\Omega_1, K_1), \\text{ respectively.}\n      \\end{cases}$\n    \\end{enumerate}\n    Setting\n    \\begin{equation}\\label{eq:d2}\n    \\boxed{\n     \\mebr[\\phi][\\nu]_{[\\dKt]} \\vcentcolon= \\left( \\int\\limits_{\\Omega_2} \\dKt^{p_2}(\\phi(x),\\nu(x)) \\,\\mathrm{d}x \\right)^{\\frac{1}{p_2}},}\n    \\end{equation}\n    and \n    \\begin{equation}\\label{eq:d3}\n    \\boxed{\n     \\mathcal{R}_{[\\dKo]}(w) \\vcentcolon= \\int\\limits_{\\Omega_1\\times \\Omega_1} \\frac{\\dKo^{p_1}(w(x), w(y))}{\\normN[x-y]^{k+p_1 s}} \\rho^l(x-y) \\,\\mathrm{d}(x,y),}\n    \\end{equation}\n    \\autoref{eq: functional_with_some_metric} can be expressed in compact form\n    \\begin{equation}\n      \\label{eq:functional}\n      \\boxed{\n        \\F<v><\\alpha>[\\dKt,\\dKo](w) = \\mebr[\\op[w]][v]^{p_2}_{[\\dKt]} + \\alpha \\mathcal{R}_{[\\dKo]}(w).}\n    \\end{equation}\n    For convenience we will often skip some of the super- or subscript, and use compact notations like e.g. \n    \\begin{equation*}\n     \\F<v>, \\F[\\dKt, \\dKo] \\text{ or } \\F(w) = \\mebr[\\op[w]][v]^{p_2} + \\alpha \\mathcal{R}(w).\n    \\end{equation*}\n\\end{definition}\n\n\n\\begin{remark}\n~\n  \\begin{enumerate}[topsep=0pt]\n    \\item $l = \\set{0,1}$ is an indicator which allows to consider approximations of Sobolev semi-norms and double integral \n        representations \n        of the type of \\citeauthor{BouBreMir01} \\cite{BouBreMir01} in a uniform manner. \n        \\begin{itemize}\n         \\item when $k=0$, $s=1$, $l=1$ and when $d_1$ is the Euclidean distance, we get the double integrals of the \n         \\citeauthor{BouBreMir01}-form \\cite{BouBreMir01}. Compare with \\autoref{eq:double_integral}.\n         \\item When $d_1$ is the Euclidean distance, $k=N$ and $l=0$, we get Sobolev semi-norms.\n        \\end{itemize}\n        We expect a relation between the two classes of functionals for $l=0$ and $l=1$ as stated in \\autoref{ss:conjecture}.\n    \\item \n      When $d_1$ is the Euclidean distance then the second term in \\autoref{eq: functional_with_some_metric} is similar to the ones used in \n      \\cite{AubKor09,BouElbPonSch11} and \\cite{BouBreMir01, Pon04b, Dav02}. \n\n  \\end{enumerate}\n\\end{remark}\n\nIn the following we state basic properties of $\\mebr_{[\\dKt]}$ and the functional \n$\\F$.\n\n\\begin{proposition} \\label{pr:ExprIsOp}\nLet \\autoref{ass:1} hold. \n\n\\begin{enumerate}\n\\item Then the mapping $ \\mebr_{[\\dKt]} : \\Lp<2> \\times \\Lp<2> \\rarr [0, +\\infty]$ \nsatisfies the metric axioms.\n\n\\item \\label{itm: ExpIsOp} Let, in addition, \\autoref{ass:2} hold, assume that $v \\in \\Lp<2>$ and that both metrics $d_i$, $i=1,2$, \nare equivalent to $\\dRMi|_{K_i \\times K_i}$, respectively.\nThen the functional $\\F<v><\\alpha>[\\dKt, \\dKo]$ does not \nattain the value $+\\infty$ on its domain $\\W<1> \\neq \\emptyset$. \n\\end{enumerate}\n\\end{proposition}\n\n\\begin{proof}\n\\begin{enumerate}\n\\item The axioms of non-negativity, identity of indiscernibles and symmetry are fulfilled by $\\mebr_{[\\dKt]}$ \n      since $\\dKt$ is a metric. To prove the triangle inequality let $\\phi,\\xi,\\nu \\in L^{p_2}(\\Omega_2, K_2)$. \n      In the main case $\\mebr[\\phi][\\nu]_{[\\dKt]}^{p_2} \\in (0, \\infty)$ H\u00f6lder's inequality yields\n\t\\begin{align*}\n\t\\mebr[\\phi][\\nu]_{[\\dKt]}^{p_2} ={}&\n\t\\int\\limits_{\\Omega_2} \\dKt\\big(\\phi(x),\\nu(x) \\big)  \\dKt^{p_2-1}\\big( \\phi(x),\\nu(x) \\big)  \\,\\mathrm{d}x \\\\ \n\t\\leq{}&  \\int\\limits_{\\Omega_2} \\dKt\\big( \\phi(x),\\xi(x) \\big) \\dKt^{p_2-1}\\big( \\phi(x),\\nu(x) \\big) \\,\\mathrm{d}x  \n\t      + \\int\\limits_{\\Omega_2} \\dKt\\big( \\xi(x),\\nu(x) \\big) \\dKt^{p_2-1} \\big( \\phi(x),\\nu(x) \\big) \\,\\mathrm{d}x \\\\\n  \\leq{}&  \n  \\left( \\int\\limits_{\\Omega_2} \\dKt^{p_2} \\big(\\phi(x),\\xi(x) \\big) \\,\\mathrm{d}x  \\right)^{\\frac{1}{p_2}} \n       \\left( \\int\\limits_{\\Omega_2} \\dKt^{p_2}\\big( \\phi(x),\\nu(x) \\big) \\,\\mathrm{d}x  \\right)^{\\frac{p_2-1}{p_2}}\\\\\n  &+ \\left( \\int\\limits_{\\Omega_2} \\dKt^{p_2} \\big(\\xi(x),\\nu(x) \\big) \\,\\mathrm{d}x  \\right)^{\\frac{1}{p_2}} \n       \\left( \\int\\limits_{\\Omega_2} \\dKt^{p_2}\\big( \\phi(x),\\nu(x) \\big) \\,\\mathrm{d}x  \\right)^{\\frac{p_2-1}{p_2}}\\\\\n\t={}& \\left( \\mebr[\\phi][\\xi]_{[\\dKt]} + \\mebr[\\xi][\\nu]_{[\\dKt]} \\right)\n\t\\mebr[\\phi][\\nu]_{[\\dKt]}^{p_2-1},\n\t\\end{align*} \n  meaning \n  \\begin{gather*}\n    \\mebr[\\phi][\\nu]_{[\\dKt]} \\leq \\mebr[\\phi][\\xi]_{[\\dKt]} + \\mebr[\\xi][\\nu]_{[\\dKt]}.\n  \\end{gather*}\n  If $\\mebr[\\phi][\\nu]_{[\\dKt]} = 0$ the triangle inequality is trivially fulfilled. \n  \n  In the remaining case $\\mebr[\\phi][\\nu]_{[\\dKt]} = \\infty$ applying the estimate $(a+b)^p \\leq 2^{p-1} (a^p + b^p)$,\n  see e.g. \\cite[Lemma 3.20]{SchGraGroHalLen09},\n  to $a = \\dKt(\\phi(x), \\xi(x)) \\geq 0$ and $b = \\dKt(\\xi(x), \\nu(x)) \\geq 0$ yields \n  \\begin{gather*}\n    \\mebr[\\phi][\\nu]_{[\\dKt]}^{p_2} \\leq 2^{p_2-1} \\big( \\mebr[\\phi][\\xi]_{[\\dKt]}^{p_2} + \\mebr[\\xi][\\nu]_{[\\dKt]}^{p_2} \\big),\n  \\end{gather*}\n  implying the desired result.\n\\item We emphasize that $\\W<1> \\neq \\emptyset$ because every constant function $w(\\cdot) = a \\in K_1$ belongs to $\\Wsp<1>$ \n      for $p_1 \\in (1, \\infty)$ and $s \\in (0,1]$ as well as to $\\BV<1>$ for $p_1 = 1$ and $s = 1$.\n      Assume now that the metrics $d_i$ are equivalent to $\\dRMi|_{K_i \\times K_i}$ for $i=1$ and $i=2$, respectively,\n      so that we have an upper bound $d_i \\leq C \\dRMi|_{K_i \\times K_i}$.\n      We need to prove that $\\F<v><\\alpha>[\\dKt, \\dKo](w) < \\infty$ for every $w \\in \\W<1>$.\n      Due to $\\mebr[\\phi][\\nu]^{p_2}_{[\\dKt]} \\leq C^{p_2} \\norm{\\phi - \\nu}^{p_2}_{\\Lp<2>[][\\mathds{R}^{M_2}]} < \\infty$ \n      for all $\\phi, \\nu \\in \\Lp<2> \\subseteq \\Lp<2>[][\\mathds{R}^{M_2}]$ it is sufficient to show\n      $\\mathcal{R}_{[\\dKo]}(w) < +\\infty$ for all $w \\in \\W<1>$.\n\\begin{itemize}\n  \\item For $\\W<1> = \\BV<1>$ this is guaranteed by \\cite[Theorem 1.2]{Pon04b}.\n  \\item For $\\W<1> = W^{1,p_1}(\\Omega_1, K_1)$ by \\cite[Theorem 1]{BouBreMir01}.\n  \\item For $\\W<1> = \\Wsp<1>$, $s \\in (0,1)$,  we distinguish between two cases. \\\\\n      If $\\normN[x-y]< 1$ we have that $\\frac{1}{\\normN[x-y]^{k+p_1 s}} \\leq \\frac{1}{\\normN[x-y]^{N+p_1 s}}$ for $k \\leq N$ and hence\n      \\begin{gather*}\\int\\limits_{\\begin{smallmatrix}\n      \t(x,y) \\in \\Omega_1 \\times \\Omega_1 \\\\ \\normN[x-y]< 1\n      \t\\end{smallmatrix}} \\frac{\\dKo^{p_1}(w(x), w(y))}{\\normN[x-y]^{k+p_1 s}} \\rho^l(x-y) \\,\\mathrm{d}(x,y) \n        \\leq C^{p_1} \\norm{\\rho^l}_{\\infty} \\abs{w}_{W^{s,p_1}(\\Omega_1, \\mathds{R}^{M_1})}^{p_1} < \\infty\\;.\n      \\end{gather*}\n      If $\\normN[x-y]\\geq 1$ we can estimate \n      \\begin{gather*}\\int\\limits_{\\begin{smallmatrix}\n      \t(x,y) \\in \\Omega_1 \\times \\Omega_1 \\\\ \\normN[x-y]\\geq 1\n      \t\\end{smallmatrix}} \\frac{\\dKo^{p_1}(w(x), w(y))}{\\normN[x-y]^{k+p_1 s}} \\rho^l(x-y) \\,\\mathrm{d}(x,y) \n        \\leq C^{p_1} \\norm{\\rho^l}_{\\infty} 2^{p_1} |\\Omega_1| \\norm{w}^{p_1}_{L^{p_1}(\\Omega_1, \\mathds{R}^{M_1})} < \\infty\\;.\n     \\end{gather*}\n     In summary adding yields $\\mathcal{R}_{[\\dKo]}(w) < +\\infty$.\n     \\end{itemize}\n\\end{enumerate}\n\\end{proof}\n\n\n\n\\section{Existence} \\label{sec: Existence}\n\nIn order to prove existence of a minimizer of the functional $\\F$ we apply the Direct Method in the Calculus of Variations\n(see e.g. \\cite{Dac82,Dac89}). \nTo this end we verify continuity properties of $\\mebr_{[\\dKt]}$ and $\\mathcal{R}_{[\\dKo]}$, resp. $\\F[\\dKt, \\dKo]$ and apply them along with\nthe sequential closedness of $\\W<1>$, already proven in \\autoref{lem:Wsp_weakly_seq_closed_etc}.\n\n\nIn this context we point out some setting assumptions and their consequences on $\\F$, resp. $\\mebr$ and $\\mathcal{R}$ in the following remark. \nFor simplicity we assume \n$p \\vcentcolon= p_1 = p_2 \\in (1, \\infty)$, $\\Omega \\vcentcolon= \\Omega_1 = \\Omega_2$\nand $(K, \\dK) \\vcentcolon= (K_1, \\dKo) = (K_2, \\dKt)$.\n\\begin{remark} \\label{re:tricks}\n~\n\t\\begin{itemize}[topsep=0pt]\n\t\t\\item\n\t\tThe continuity of $\\dK$ with respect to $\\dRM|_{K \\times K}$ guarantees lower semicontinuity of $\\mebr_{[\\dK]}$ and $\\mathcal{R}_{[\\dK]}$.\n\t\t\\item\n\t\tThe inequality $\\dRM|_{K \\times K} \\leq \\dK$ carries over to the inequalities \n\t\t$\\norm{\\widetilde v - v}_{\\Lp} \\leq \\mebr[\\widetilde v][v]_{[\\dK]}$ for all $\\widetilde v, v \\in \\Lp[K]$,\n\t\tand $|w|_{\\W} \\leq \\mathcal{R}_{[\\dK]}(w)$ for all $w \\in \\W[K]$, allowing to transfer\n\t\tproperties like coercivity from $\\F[\\dRM,\\dRM]$ to $\\F[\\dK,\\dK]$.\n\t\tMoreover, the extended real-valued metric space $(\\Lp[K], \\mebr_{[\\dK]})$ stays \n\t\trelated to the linear space $(\\Lp, \\norm{\\cdot}_{\\Lp})$ in terms of the topology and bornology induced by $\\mebr$, \n\t\tresp. those inherited by\n\t\t$\\norm{\\cdot}_{\\Lp}$.\n\t\t\\item\n\t\tThe closedness of $K \\subseteq \\mathds{R}^M$ is crucial in showing that $\\W[K]$ is a sequentially closed subset of the linear space $\\W$.\n\t\tThis closedness property acts as a kind of replacement for the, a priori not available, notion of completeness with respect to the \n\t\t``space'' $(\\W[K], \\mebr, \\mathcal{R})$.\n\t\\end{itemize}\n\tFor $l=0$, $k=N$ note in the latter item that\n\tequipping $\\W[K]$ with $\\mebr_{[\\dKt]}$ and $\\mathcal{R}_{[\\dKo]}$ does not even lead to an (extended real-valued) metric space,\n\tin contrast to the classical case $(K,\\dK) = (\\mathds{R}^M, \\dRM)$.\n\\end{remark}\n\n\nWe will use the following assumption:\n\n\\begin{assumption} \\label{as:Setting}  \n  Let \\autoref{ass:1} hold, $v^0 \\in \\Lp<2>$ and let $\\W<1>$ and the associated topology be as defined in \\autoref{eq:ChooseW}.\n  \n  In addition we assume: \n  \\begin{itemize}\n   \\item $\\op: \\W<1> \\to \\Lp<2>$ is well--defined and sequentially continuous with respect to the specified topology on $\\W<1>$ and \n   \\item For every $t > 0$ and $\\alpha > 0$ the level sets \n      \\begin{equation}\\label{itm: A} \n      \\text{level}_t(\\F<v^0><\\alpha>[\\dKt, \\dKo]) \\vcentcolon= \\{ w \\in \\W<1> \\ : \\  \\F<v^0><\\alpha>[\\dKt,\\dKo] \\leq t \\}  \n      \\end{equation}\n    are sequentially pre-compact subsets of $W(\\Omega_1, \\mathds{R}^{M_1})$.\n   \\item There exists a $\\bar{t} > 0$ such that $\\text{level}_{\\bar{t}}(\\F<v^0><\\alpha>[\\dKt, \\dKo])$ is nonempty.\n   \\item Only those $v \\in \\Lp<2>$ are considered which additionally fulfill $\\mebr[v][v^0]_{[\\dKt]} < \\infty$. \n  \\end{itemize}\n\\end{assumption}\n\n\\begin{remark}\n  The third condition is sufficient to guarantee $\\F<v^0><\\alpha>[\\dKt, \\dKo]) \\not \\equiv \\infty$. In contrast the condition $v^0 \\in \\Lp<2>$, cf. \n  \\autoref{def:functional}, might not be sufficient if $d_2$ is not equivalent to $\\dRMt|_{K_2 \\times K_2}$.\n\\end{remark}\n\n\n\n\\begin{lemma} \\label{thm:F_and_its_summands_are_seq_weakly_closed} \n   Let \\autoref{as:Setting} hold. \n   Then the  mappings $\\mebr_{[\\dKt]}$, $\\mathcal{R}_{[\\dKo]}$ and $\\F[\\dKt, \\dKo]$ have the following continuity properties:\n   \\begin{enumerate}\n     \\item \\label{enu:continuity_of_mebr}\n       The mapping $\\mebr_{[\\dKt]}: \\Lp<2> \\times \\Lp<2> \\rarr [0, +\\infty]$ is sequentially lower semi-continuous,\n       i.e. whenever sequences $\\seq{\\phi}$, $\\seq{\\nu}$ in $\\Lp<2>$ converge to $\\phi_* \\in \\Lp<2>$ and \n       $\\nu_*\\in \\Lp<2>$, respectively, we have $\\mebr[\\phi_*][\\nu_*]_{[\\dKt]} \\leq \\limi \\limits \\mebr[\\phi_n][\\nu_n]_{[\\dKt]}$.\n     \\item \\label{enu:seq_lscty_of_R} \n       The functional $\\mathcal{R}_{[\\dKo]}: \\W<1> \\rarr [0,\\infty]$ is sequentially lower semi-continuous, i.e. whenever a \n       sequence $(w_n)_{n \\in \\mathds{N}}$ in $\\W<1>$ converges to some $w_* \\in \\W<1>$ we have\n       \\begin{equation*}\n          \\mathcal{R}_{[\\dKo]}(w_*)  \\leq \\limi \\mathcal{R}_{[\\dKo]}(w_n).\n       \\end{equation*}\n     \\item \\label{enu:seq_lscty_of_F} \n       The functional $\\F[\\dKt,\\dKo]: W(\\Omega_1, K_1) \\rarr [0,\\infty]$ is sequentially lower semi-continuous.\n   \\end{enumerate}\n\\end{lemma}\n\n\n\\begin{proof}\n\t\\begin{enumerate}\n\t\\item \\label{enu:lsc_of_mebr}\n\t\tIt is sufficient to show that for \\emph{every} pair of sequences $\\seq{\\phi}$, $\\seq{\\nu}$ in $\\Lp<2>$ which \n\t\tconverge to previously \\emph{fixed} elements  $\\phi_* \\in \\Lp<2>$ and $\\nu_*\\in \\Lp<2>$, respectively, we can extract subsequences \n\t\t$(\\phi_{n_j})_{j \\in \\mathds{N}}$ and $(\\nu_{n_j})_{j \\in \\mathds{N}}$, respectively, with \n\t\t\\begin{gather*}\n\t\t\\mebr[\\phi_*][\\nu_*]_{[\\dKt]} \\leq \\liminf_{j \\rarr \\infty} \\mebr[\\phi_{n_j}][\\nu_{n_j}]_{[\\dKt]}.\n\t\t\\end{gather*}\n\t\tTo this end let $(\\phi_n)_{n \\in \\mathds{N}},(\\nu_n)_{n \\in \\mathds{N}}$ be some sequences in $\\Lp<2>$ with $\\phi_n \\rarr \\phi_*$ and $\\nu_n \\rarr \\nu_*$ in $\\Lp<2>$. \n\t\t\\autoref{lem:Wsp_weakly_seq_closed_etc} ensures that there exist subsequences $(\\phi_{n_j})_{j \\in \\mathds{N}}, (\\nu_{n_j})_{j \\in \\mathds{N}}$ \n\t\tconverging to $\\phi_*$ and $\\nu_*$ pointwise almost everywhere, \n\t\twhich in turn implies $\\big(\\phi_{n_j}(\\cdot), \\nu_{n_j}(\\cdot) \\big) \\to \\big( \\phi_*(\\cdot), \\nu_*(\\cdot) \\big)$ \n\t\tpointwise almost everywhere. Therefrom, together with the continuity  of $\\dKt: K_2 \\times K_2 \\to [0, \\infty)$ \n\t\twith respect to $\\dRMt$, cf. \\autoref{sec: Setting}, we obtain\t\t\n\t\tby using the quadrangle inequality\n\t\tthat\n\t\t\\begin{gather*}\n\t\t   | \\dKt(\\phi_{n_j}(x), \\nu_{n_j}(x)) - \\dKt(\\phi_*(x), \\nu_*(x)) |  \\leq  \\dKt(\\phi_{n_j}(x), \\phi_*(x))  +  \\dKt(\\nu_{n_j}(x), \\nu_*(x)) \\rarr 0,\n\t     \\end{gather*}\n\t\tand hence\t\t\n\t\t\\begin{gather*}\n\t\t\\dKt^{p_2}\\big( \\phi_{n_j}(x), \\nu_{n_j}(x) \\big) \\to \\dKt^{p_2} \\big( \\phi_*(x),\\nu_*(x) \\big) \\text{ for almost every }\n\t\tx \\in \\Omega_2.\n\t\t\\end{gather*}\n\t\tApplying Fatou's lemma we obtain\n\t\t\\begin{gather*}\n\t\t\\mebr[\\phi_*][\\nu_*]_{[\\dKt]} = \\int_{\\Omega_2 }\\limits \\dKt^{p_2}( \\phi_*(x),\\nu_*(x) ) \\,\\mathrm{d}x\n\t\t\\leq \\liminf_{j \\rarr \\infty} \\int_{\\Omega_2}\\limits \\dKt^{p_2} ( \\phi_{n_j}(x), \\nu_{n_j}(x) ) \\,\\mathrm{d}x\n\t\t= \\liminf_{j \\rarr \\infty} \\mebr[\\phi_{n_j}][\\nu_{n_j}]_{[\\dKt]}.\n\t\t\\end{gather*}\n\t\t\\item \n\t\tLet $(w_n)_{n \\in \\mathds{N}}$ be a sequence in $\\W<1>$ with $w_n \\rarr<W> w_*$ as $n \\rarr \\infty$.\n\t\tBy \\autoref{lem:Wsp_weakly_seq_closed_etc} there is a subsequence \n\t\t$(w_{n_j})_{j \\in \\mathds{N}}$ which converges to $w_*$ both in $\\Lp<1>$ and pointwise almost everywhere.\n\t\tThis further implies that\n\t\t\\begin{gather*}\n\t\t\\dKo^{p_1}\\big( w_{n_j}(x), w_{n_j}(y) \\big) \\to \\dKo^{p_1} \\big( w_*(x),w_*(y) \\big)\n\t\t\\end{gather*}\n\t\tfor almost every \n\t\t\\begin{equation}\n\t\t \\label{eq:A} \n\t\t (x,y) \\in \\Omega_1 \\times \\Omega_1 \\supseteq \\{(x,y) \\in \\Omega_1 \\times \\Omega_1 : x \\neq y \\} =\\vcentcolon A.\n\t\t\\end{equation}\n\t\tDefining\n\t\t\\begin{equation*}\n\t\tf_j(x,y) \\vcentcolon= \\left\\{ \\begin{array}{rcl}\n\t\t\\frac{\\dKo^{p_1}(w_{n_j}(x), w_{n_j}(y)) }{\\normN[x-y]^{k+ps}} \\rho^l(x-y) & \\text{ for } & \n\t\t(x,y) \\in A,\\\\\n\t\t0 & \\text{ for } & \n\t\t(x,y) \\in (\\Omega_1 \\times \\Omega_1) \\setminus A,\\\\\n\t\t\\end{array} \\right. \\quad \\text{ for all } j \\in \\mathds{N}.\n\t\t\\end{equation*}\n\t\tand \n\t\t\\begin{equation*}\n\t\tf_*(x,y) \\vcentcolon= \\left\\{ \\begin{array}{ccl}\n\t\t\\frac{\\dKo^{p_1}(w_*(x), w_*(y)) }{\\normN[x-y]^{k+ps}}   \\rho^l(x-y) & \\text{ for } & \n\t\t(x,y) \\in A,\\\\\n\t\t0 & \\text{ for } & \n\t\t(x,y) \\in (\\Omega_1 \\times \\Omega_1) \\setminus A\\\\\n\t\t\\end{array} \\right. \n\t\t\\end{equation*}\n\t\twe thus have $f_*(x,y) = \\lim_{j \\rarr \\infty} f_j(x,y)$\n\t\tfor almost every $(x,y) \\in \\Omega_1 \\times \\Omega_1$.\n\t\tApplying Fatou's lemma to the functions $f_j$ yields the assertion, due to the same reduction as in the proof of the first part.\n\t\t\n\t\t\\item \n\t\tIt is sufficient to prove that the components $\\mathcal{G}(\\cdot) = \\mebr[F(\\cdot)][v]_{[\\dKt]}$ and $\\mathcal{R} = \\mathcal{R}_{[\\dKo]}$ of \n\t\t$\\F[\\dKo,\\dKt] = \\mathcal{G} + \\alpha \\mathcal{R}$ are sequentially lower semi-continuous.\n\t\tTo prove that $\\mathcal{G}$ is sequentially lower semi-continuous in every $w_* \\in W(\\Omega_1, K_1)$\n\t\tlet $(w_n)_{n \\in \\mathds{N}}$ be a sequence in $W(\\Omega_1, K_1)$ with $w_n \\rarr<W> w_*$ as $n \\rarr \\infty$.\n\t\t\\autoref{as:Setting}, ensuring the sequential continuity of $\\op: \\W<1> \\to \\Lp<2>$, implies hence $\\op[w_n] \\rarr \\op[w_*]$ in $\\Lp<2>$ \n\t\tas $n \\rarr \\infty$.\n\t\tBy \\autoref{enu:continuity_of_mebr} we thus obtain $\\mathcal{G}(w_*) = \\mebr[\\op[w_*]][v] \\leq \\limi \\mebr[\\op[w_n]][v] = \\limi \\mathcal{G}(w_n)$. \\\\\n\t\t$\\mathcal{R}$ is sequentially lower semi-continuous by \\autoref{enu:seq_lscty_of_R}.\n\t\\end{enumerate}\n\\end{proof}\n\n\n\\subsection{Existence of minimizers}\n\nThe proof of \n  the existence of a \n  minimizer of $\\F[\\dKt, \\dKo]$ is along the lines of the proof \nin \\cite{SchGraGroHalLen09}, taking into account \\autoref{re:tricks}\nWe will need the following useful lemma, cf. \\cite{SchGraGroHalLen09}, which links $\\text{level}_t(\\F<v^0><\\alpha>)$ and \n$\\text{level}_t(\\F<v><\\alpha>)$ for $\\mebr[v][v^0] < \\infty$.\n\n\n\\begin{lemma}\\label{lem:Ineq_F_parameterchange_to_other_v} \n\tIt holds \n\t\\begin{align*}\n\t\t\\F<v_\\star><>[\\dKt, \\dKo](w)  & \\leq   2^{p_2-1} \\F<v_\\diamond><>[\\dKt, \\dKo](w) + 2^{p_2 -1} \\mebr[v_\\diamond][v_\\star]_{[\\dKt]}^{p_2}\n\t\\end{align*}\n\tfor every \n\t$w \\in W(\\Omega_1, K_1)$ and $v_\\star, v_\\diamond \\in L^{p_2}(\\Omega_2, K_2)$.\n\\end{lemma}\n\n\\begin{proof}\n\tUsing the fact that for $p \\geq 1$ we have that $|a+b|^p \\leq 2^{p-1}(|a|^p + |b|^p), \\ a,b \\in \\mathds{R} \\cup \\{\\infty\\}$ and that $\\mebr_{[\\dKt]}$ fulfills the triangle inequality we obtain\n\t\\begin{align*}\n\t\t\\F<v_\\star><>[\\dKt, \\dKo](w) \n\t\t& = \\mebr[\\op[w]][v_\\star]_{[\\dKt]}^{p_2} + \\alpha \\mathcal{R}_{[\\dKo]}(w) \\\\\n\t\t& \\leq 2^{p_2-1} \\big( \\mebr[\\op[w]][v_\\diamond]_{[\\dKt]}^{p_2} + \\mebr[v_\\diamond][v_\\star]_{[\\dKt]}^{p_2} \\big) + \\alpha \\mathcal{R}_{[\\dKo]}(w) \\\\\n\t\t& \\leq 2^{p_2-1}\\big( \\F<v_\\diamond><>[\\dKt, \\dKo](w) + \\mebr[v_\\diamond][v_\\star]_{[\\dKt]}^{p_2} \\big).\n\t\\end{align*}\n\\end{proof}\n\n\\begin{thm} \\label{thm:F_dK_has_a_minimizer} \nLet \\autoref{as:Setting} hold. Then the functional $\\F<v><\\alpha>[\\dKt, \\dKo]: W(\\Omega_1, K_1) \\rarr [0, \\infty]$ \nattains a minimizer.\n\\end{thm}\n\n\\begin{proof}\n  We prove the existence of a minimizer via the Direct Method.\n  We shortly write $\\F^v$ for $\\F<v><\\alpha>[\\dKt, \\dKo]$.\n  Let $(w_n)_{n \\in \\mathds{N}}$ be a sequence in $W(\\Omega_1, K_1)$ with\n  \\begin{gather}\\label{eq:w_n_is_minimizing_seq}\n    \\lim_{n \\rarr \\infty }\\F^v(w_n) = \\inf_{w \\in W(\\Omega_1, K_1)} \\F^v(w).\n  \\end{gather}\n  The latter infimum is not $+\\infty$, because $\\F^v \\equiv +\\infty$ would imply also $\\F^{v^0} \\equiv +\\infty$ due to \n  \\autoref{lem:Ineq_F_parameterchange_to_other_v}, violating \\autoref{as:Setting}.\n  In particular there is some $c \\in \\mathds{R}$ such that \n  $\\F^v(w_n) \\leq c$ for every $n \\in \\mathds{N}$. \n  Applying \\autoref{lem:Ineq_F_parameterchange_to_other_v} yields\n  $\\F^{v^0}(w_n) \\leq 2^{p_2-1} \\big( \\F^{v}(w_n) + \\mebr[v][v^0] \\big)\\leq 2^{p_2-1} \\big( c + \\mebr[v][v^0] \\big) =\\vcentcolon\\tilde{c} < \\infty$ due to \\autoref{as:Setting}.\n  Since the level set $\\text{level}_{\\tilde{c}}(\\F^{v^0})$ is sequentially pre-compact with respect to the topology given to $W(\\Omega_1, \\mathds{R}^{M_1})$ we get the existence of a\n  subsequence $(w_{n_k})_{k \\in \\mathds{N}}$ which converges to some $w_* \\in W(\\Omega_1, \\mathds{R}^{M_1})$, \n  where actually $w_* \\in W(\\Omega_1, K_1)$ due to \\autoref{lem:Wsp_weakly_seq_closed_etc}.\n  Because $\\F^v$ is sequentially lower semi-continuous, see \\autoref{thm:F_and_its_summands_are_seq_weakly_closed}, we have \n  $\\F^v(w_*) \\leq \\liminf_{k \\rarr \\infty} \\F^v(w_{n_k})$. Combining this with \n  \\autoref{eq:w_n_is_minimizing_seq} we obtain \n  \\begin{gather*}\n    \\inf_{w \\in W(\\Omega_1, K_1)} \\F^v(w) \\leq \\F^v(w_*) \\leq \\liminf_{k \\rarr \\infty} \\F^v(w_{n_k}) \n    = \\lim_{n\\rarr \\infty} \\F^v(w_n) = \\inf_{w \\in W(\\Omega_1, K_1)} \\F^v(w).\n  \\end{gather*}\n  In particular $\\F^v(w_*) = \\inf_{w \\in W(\\Omega_1, K_1)}\\limits \\F^v(w)$, meaning that $w_*$ is a minimizer of $\\F^v$.\n\\end{proof}\n\nIn the following we investigate two examples, which are relevant for the numerical examples in \\autoref{sec:Numerical_results}.\n\\begin{example}\n\\label{ex:coercive}\nWe consider that \n$W(\\Omega_1,K_1) = W^{s, p_1}(\\Omega_1, K_1)$  \nwith $p_1>1, \\ 0 < s < 1$ and fix $k = N$. \n\nIf the operator $\\op$ is norm-coercive in the sense that the implication\n    \\begin{equation} \\label{eq:coercive_F}\n    \\norm{w_n}_{L^{p_1}(\\Omega_1, \\mathds{R}^{M_1})} \\rarr +\\infty \\Rightarrow \\norm{\\op[w_n]}_{L^{p_2}(\\Omega_2, \\mathds{R}^{M_2})} \\rarr +\\infty \n    \\end{equation}\n    holds true for every sequence $(w_n)_{n \\in \\mathds{N}}$ in $W^{s,p_1}(\\Omega_1, K_1)\\subseteq W^{s,p_1}(\\Omega_1, \\mathds{R}^{M_1})$, then the functional \n    \\begin{equation*}\n    \\F[\\dKt, \\dKo] = \\mebr[\\op[w]][v]^{p_2}_{[\\dKt]} + \\alpha \\mathcal{R}_{[\\dKo]}(w): W^{s,p_1}(\\Omega_1, K_1) \\rarr [0, \\infty]\n    \\end{equation*}\n    is coercive. This can be seen as follows: \n    \n    The inequality between $\\dKo$ and $\\dRMo|_{K_1 \\times K_1}$ resp. $\\dKt$ and $\\dRMt|_{K_2 \\times K_2}$, see \\autoref{ass:1}, carries over to \n    $\\F[\\dKt,\\dKo]$ and $\\F[\\dRMt|_{K_2 \\times K_2},\\dRMo|_{K_1 \\times K_1}]$, i.e.\n    \\begin{gather*}\n     \\F[\\dKt, \\dKo] (w) \\geq \\F\\big[\\dRMt|_{K_2 \\times K_2},\\dRMo|_{K_1 \\times K_1}\\big](w) \\text{ for all } w \\in W^{s,p_1}(\\Omega_1, K_1).\n    \\end{gather*} \n    Thus it is sufficient to show that $\\F[\\dRMt|_{K_2 \\times K_2},\\dRMo|_{K_1 \\times K_1}]: W^{s,p_1}(\\Omega_1, K_1) \\rarr [0, \\infty]$ is coercive:    \n    To prove this we write shortly $\\F$ instead of $\\F[\\dRMt|_{K_2 \\times K_2},\\dRMo|_{K_1 \\times K_1}]$ and consider sequences\n    $(w_n)_{n \\in \\mathds{N}}$ in $W^{s,p_1}(\\Omega_1, K_1)$ with $\\norm{w_n}_{W^{s,p_1}(\\Omega_1, \\mathds{R}^{M_1})} \\rarr +\\infty$ as $n \\rarr \\infty$. We \n    show that $\\F(w_n) \\rarr +\\infty$, as $n \\rarr \\infty$.\n    Since  \n    \\begin{equation*} \n     \\norm{w_n}_{W^{s,p_1}(\\Omega_1, \\mathds{R}^{M_1})} =  \\big( \\norm{w_n}_{L^{p_1}(\\Omega_1, \\mathds{R}^{M_1})}^{p_1} + \\abs{w_n}_{W^{s,p_1}(\\Omega_1, \\mathds{R}^{M_1})}^{p_1} \\big)^{\\frac{1}{p_1}}\n    \\end{equation*}\n    the two main cases to be considered are $\\norm{w_n}_{L^{p_1}(\\Omega_1, \\mathds{R}^{M_1})} \\rarr +\\infty$ and \n    $\\abs{w_n}_{W^{s,p_1}(\\Omega_1, \\mathds{R}^{M_1})} \\rarr +\\infty$.\n   \n    \\begin{enumerate}[label=\\textbf{Case \\arabic*}]\n    \\item \\label{ex: coecivity_case1}\n    $\\norm{w_n}_{L^{p_1}(\\Omega_1, \\mathds{R}^{M_1})} \\rarr +\\infty$. \\\\\n    The inverse triangle inequality and the norm-coercivity of $\\op$, \\autoref{eq:coercive_F}, give\n    $\\norm{\\op[w_n] - v}_{L^{p_2}(\\Omega_2, \\mathds{R}^{M_2})} \\geq \\norm{\\op[w_n]}_{L^{p_2}(\\Omega_2, \\mathds{R}^{M_2})} - \\norm{v}_{L^{p_2}(\\Omega_2, \\mathds{R}^{M_2})} \\rarr +\\infty$.\n    Therefore also \n    \\begin{equation*}\n     \\F(w_n) = \\norm{\\op[w_n] - v}^{p_2}_{L^{p_2}(\\Omega_2, \\mathds{R}^{M_2})} \n      + \\alpha \\int\\limits_{\\Omega_1\\times \\Omega_1} \\frac{\\normMo[w_n(x) - w_n(y)]^{p_1}}{\\normN[x-y]^{N+p_1 s}} \\rho^l(x-y) \\,\\mathrm{d}(x,y)\n    \\rarr +\\infty.\n    \\end{equation*}\n\n  \\noindent\n  \\item \\label{ex: coecivity_case2}\n  $\\abs{w_n}_{W^{s,p_1}(\\Omega_1, \\mathds{R}^{M_1})} \\rarr +\\infty$. \\\\\n  If $l=0$, then $\\mathcal{R}_{[\\dKo]}$ is exactly the $W^{s,p_1}(\\Omega_1, \\mathds{R}^{M_1})$-semi-norm $|w|_{W^{s,p_1}(\\Omega_1, \\mathds{R}^{M_1})}$ and we trivially \n  get the desired result. \n  \n  Hence we assume from now on that $l = 1$. \n  The assumptions on $\\rho$ ensure that there exists a  $\\tau > 0$ and $\\eta_{\\tau} > 0$ such that\n  \\begin{align*}\n    \\mathcal{S}_{\\tau} \\vcentcolon= {}&  \\{(x,y) \\in \\Omega_1 \\times \\Omega_1 : \\rho(x-y) \\geq \\tau \\}\n  \\\\\n    = {}& \\{(x,y) \\in \\Omega_1 \\times \\Omega_1 : \\normN[x-y] \\leq \\eta_{\\tau} \\},\n  \\end{align*}\n  cf. \\autoref{fig:stripe}.\n\n  Splitting $\\Omega_1 \\times \\Omega_1$ into $\\mathcal{S}_{\\tau} =\\vcentcolon \\mathcal{S}$ and its complement \n  $(\\Omega_1 \\times \\Omega_1) \\setminus \\mathcal{S}_{\\tau} =\\vcentcolon \\mathcal{S}^c$\n  we accordingly split the integrals \n  $\\abs{w_n}_{W^{s,p_1}(\\Omega_1, \\mathds{R}^{M_1})} = \\int\\limits_{\\Omega_1 \\times \\Omega_1} \\frac{\\normMo[w_n(x) - w_n(y)]^{p_1}}{\\normN[x-y]^{N+p_1 s}} \\,\\mathrm{d}(x,y)$ \n  and consider again two cases \n  $\\int\\limits_{\\mathcal{S}} \\frac{\\normMo[w_n(x) - w_n(y)]^{p_1}}{\\normN[x-y]^{N+p_1 s}} \\,\\mathrm{d}(x,y)  \\rarr +\\infty$ and \n  $\\int\\limits_{\\mathcal{S}^c} \\frac{\\normMo[w_n(x) - w_n(y)]^{p_1}}{\\normN[x-y]^{N+p_1 s}} \\,\\mathrm{d}(x,y) \\rarr +\\infty$, respectively.\n\n\n\\begin{figure}[h!]\n\t\\centering\n\t\\begin{minipage}[t]{0.35\\linewidth}\n\t\t\\vspace*{-110pt}\n\t\t\\begin{tikzpicture}\n\t\t\\draw (0,0) -- (3,0);\n\t\t\\draw (3,0) -- (3,3);\n\t\t\\draw (3,3) -- (0,3);\n\t\t\\draw (0,3) -- (0,0);\n\t\t\\draw [ fill = magenta] (0.25,0) -- (3, 2.75) --(3,3)--(2.75, 3) -- (0,0.25) --(0,0);\n\t\t\\draw [dashed] (0,0) -- (3,3);\n\t\t\\node at (1.35,1.3) {$\\mathcal{S}$};\n\t\t\\node at (2.0,0.5) {$\\mathcal{S}^c$};\n\t\t\\node at (4,3) {$\\Omega_1 \\times \\Omega_1$};\n\t\t\\draw[->] (-0.5,1.5) -- (3.5,1.5);\n\t\t\\node at (3.7,1.5){$x$};\n\t\t\\node at (-1.1,1.5){$y=y_0$};\n\t\t\\end{tikzpicture}\n\t\\end{minipage}\n\t\\hspace{1cm}\n\t\\begin{minipage}[t]{0.35\\linewidth}\n\t\t\\begin{tikzpicture}\n\t\t\\draw[->] (-2,0) -- (2,0);\n\t\t\\node[right] at (2,0) {$x$};\n\t\t\\draw[->](0,0) -- (0,3);\n\t\t\\node[above] at (0,3) {$\\rho(x-y_0)$};\n\t\t\\draw[ thick, blue, out=0, in=180] (0,2.5) to (1.5,0);\n\t\t\\draw[ thick, blue, out=180, in=0] (0,2.5) to (-1.5,0);\n\t\t\\draw[thick, blue] (1.5,0) -- (2,0);\n\t\t\\draw[thick, blue] (-1.5,0) -- (-2,0);\t\n\t\t\\draw[dashed] (-0.6,2) -- (0.6,2);\t\t\n\t\t\\node[right] at (0.6,2) {$\\tau$};\n\t\t\\draw[thick, magenta] (-0.6,0) -- (0.6,0);\t\t\n\t\t\\draw[dashed, magenta] (-0.6,0) to (-0.6,2);\n\t\t\\draw[dashed, magenta] (0.6,0) to (0.6,2);\n\t\t\\node at (0,-0.3){$y_0$};\n\t\t\\end{tikzpicture}\n\t\\end{minipage}\n\t\\caption{The stripe $\\mathcal{S} = \\mathcal{S}_{\\tau}$ if $\\Omega_1$ is an open interval and its connection to the radial mollifier $\\rho$ for fixed $y \\in \\Omega_1$.}\n\t\\label{fig:stripe}\n\\end{figure}\n  \n  \\begin{enumerate}[label=\\textbf{\\ref{ex: coecivity_case2}.\\arabic*}]\n  \\item\t\n  $\\int\\limits_{\\mathcal{S}} \\frac{\\normMo[w_n(x) - w_n(y)]^{p_1}}{\\normN[x-y]^{N+p_1 s}} \\,\\mathrm{d}(x,y) \\rarr + \\infty$. \\\\\n  By definition of $\\mathcal{S}$ we have $\\rho(x-y) \\geq \\tau > 0$ for all $(x,y) \\in \\mathcal{S}$.\n  Therefore\n  \\begin{gather*}\n    \\int\\limits_{\\mathcal{S}} \\frac{\\normMo[w_n(x) - w_n(y)]^{p_1}}{\\normN[x-y]^{N+p_1 s}} \\rho(x-y) \\,\\mathrm{d}(x,y) \n    \\geq \\tau \\int\\limits_{\\mathcal{S}} \\frac{\\normMo[w_n(x) - w_n(y)]^{p_1}}{\\normN[x-y]^{N+p_1 s}} \\,\\mathrm{d}(x,y)\n    \\rarr +\\infty.\n  \\end{gather*}\n  Since $\\alpha > 0$, it follows \n  \\begin{align*}\n    \\F(w_n) &= \\norm{\\op(w_n) - v}^{p_2}_{L^{p_2}(\\Omega_2, \\mathds{R}^{M_2})}\n      + \\underbrace{\\alpha \\int\\limits_{\\mathcal{S}} \\frac{\\normMo[w_n(x) - w_n(y)]^{p_1}}{\\normN[x-y]^{N+p_1 s}} \\rho(x-y) \\,\\mathrm{d}(x,y) \n         }_{\\rarr +\\infty} \\\\\n    & +  \\underbrace{ \\alpha \\int\\limits_{\\mathcal{S}^c} \\frac{\\normMo[w_n(x) - w_n(y)]^{p_1}}{\\normN[x-y]^{N+p_1 s}} \\rho(x-y) \\,\\mathrm{d}(x,y) \n         }_{\\geq 0}\n    \\rarr +\\infty.\n  \\end{align*}\n\n  \\noindent\n  \\item\t\\label{ex: coecivity_case22}\n  $\\int\\limits_{\\mathcal{S}^c} \\frac{\\normMo[w_n(x) - w_n(y)]^{p_1}}{\\normN[x-y]^{N+p_1 s}} \\,\\mathrm{d}(x,y) \\rarr + \\infty$.\\\\\n  For $(x, y) \\in \\mathcal{S}^c$ it might happen that $\\rho(x-y) = 0$, and thus instead of proving \n  $\\F(w_n) \\geq \\int\\limits_{\\mathcal{S}^c} \\frac{\\normMo[w_n(x) - w_n(y)]^{p_1}}{\\normN[x-y]^{N+p_1 s}} \\rho(x-y) \\,\\mathrm{d}(x,y) \\rarr +\\infty$,\n  as in Case 2.1, we rather show that $\\F(w_n) \\geq \\norm{\\op[w_n] - v}^{p_2}_{L^{p_2}(\\Omega_2, \\mathds{R}^{M_2})} \\rarr +\\infty$.\n  For this it is sufficient to show that for every $c > 0$ there is some $C \\in \\mathds{R}$ such that the implication\n  \\begin{gather*}\n     \\norm{\\op[w]-v}^{p_2}_{L^{p_2}(\\Omega_2, \\mathds{R}^{M_2})}\\leq c\n     \\implies\n      \\int\\limits_{\\mathcal{S}^c} \\frac{\\normMo[w(x) - w(y)]^{p_1}}{\\normN[x-y]^{N+p_1 s}}  \\,\\mathrm{d}(x,y) \\leq C,\n  \\end{gather*}\n  holds true for all $w \\in W^{s,p_1}(\\Omega_1, K_1) \\subseteq W^{s,p_1}(\\Omega_1, \\mathds{R}^{M_1})$.\n  To this end let $c > 0$ be given and consider an arbitrarily chosen $w \\in W^{s,p_1}(\\Omega_1, K_1)$ \n  fulfilling $\\norm{\\op[w] - v}^{p_2}_{L^{p_2}(\\Omega_2, \\mathds{R}^{M_2})} \\leq c$.\n  \n  Then $\\norm{\\op[w] - v}_{L^{p_2}(\\Omega_2, \\mathds{R}^{M_2})} \\leq \\sqrt[p_2]{c}$. Using the triangle inequality and the monotonicity\n  of the function $h: t \\mapsto t^{p_2}$ on $[0, +\\infty)$ we get further\n  \\begin{align}\\label{eq: Norm_estimate}\n    \\norm{\\op[w]}^{p_2}_{L^{p_2}(\\Omega_2, \\mathds{R}^{M_2})} \n    &= \\norm{\\op[w] - v + v}^{p_2}_{L^{p_2}(\\Omega_2, \\mathds{R}^{M_2})} \\nonumber\\\\\n    &\\leq \\left( \\norm{\\op[w] - v}_{L^{p_2}(\\Omega_2, \\mathds{R}^{M_2})} + \\norm{v}_{L^{p_2}(\\Omega_2, \\mathds{R}^{M_2})} \\right)^{p_2} \\nonumber\\\\\n    & \\leq \\big( \\sqrt[p_2]{c} + \\norm{v}_{L^{p_2}(\\Omega_2, \\mathds{R}^{M_2})} \\big)^{p_2} =\\vcentcolon \\tilde{c}. \n  \\end{align}\n  \\noindent\n  Due to the norm-coercivity, it thus follows \n  that $\\norm{w}_{L^{p_1}(\\Omega_1, \\mathds{R}^{M_1})} \\leq \\bar{c}$, $\\bar{c}$ some constant. \n  Using \\cite[Lemma 3.20]{SchGraGroHalLen09} it then follows that \n  \\begin{gather}\\label{eq: Convexity_Inequality}\n    \\normMo[w(x) - w(y)]^{p_1} \\leq 2^{p_1-1} \\normMo[w(x)]^{p_1} + 2^{p_1-1} \\normMo[w(y)]^{p_1} \n  \\end{gather}\n  for all $(x,y) \\in \\Omega_1 \\times \\Omega_1$.\n  Using \\autoref{eq: Convexity_Inequality}, Fubini's Theorem and \\autoref{eq: Norm_estimate} we obtain\n  \\begin{align*}\n    \\int\\limits_{\\Omega_1 \\times \\Omega_1} \\normMo[w(x) - w(y)]^{p_1} \\,\\mathrm{d}(x,y)\n    & \\leq \\int\\limits_{\\Omega_1 \\times \\Omega_1}  2^{p_1-1} \\normMo[w(x)]^{p_1} + 2^{p_1-1}\\normMo[w(y)]^{p_1} \\,\\mathrm{d}(x,y) \\\\\n    &= \\abs{\\Omega_1} \\int\\limits_{\\Omega_1}  2^{p_1-1} \\normMo[w(x)]^{p_1} \\,\\mathrm{d}x + \\abs{\\Omega_1} \\int\\limits_{\\Omega_1} 2^{p_1-1}\\normMo[w(y)]^ {p_1} \\,\\mathrm{d}y \\\\\n    &= 2\\abs{\\Omega_1} \\int\\limits_{\\Omega_1}  2^{p_1-1} \\normMo[w(x)]^{p_1}  \\,\\mathrm{d}x \\\\\n    & = 2^{p_1} \\abs{\\Omega_1} \\; \\norm{w}^{p_1}_{L^{p_1}(\\Omega_1, \\mathds{R}^{M_1})} \\leq 2^{p_1} \\abs{\\Omega_1} \\bar{c}^{p_1}.\n  \\end{align*}\n\n  Combining $\\normN[x-y] \\geq \\eta_{\\tau} > 0$ for all $(x,y) \\in \\mathcal{S}^c$ with the previous inequality we obtain the needed estimate\n  \\begin{align*}\n    \\int\\limits_{\\mathcal{S}^c} \\frac{\\normMo[w(x) - w(y)]^{p_1}}{\\normN[x-y]^{N+p_1 s}}  \\,\\mathrm{d}(x,y)\n    & \\leq \\frac{1}{\\eta_{\\tau}^{N+p_1 s}} \\int\\limits_{\\mathcal{S}^c} \\normMo[w(x) - w(y)]^{p_1}  \\,\\mathrm{d}(x,y) \n  \\\\\n    & \\leq \\frac{1}{\\eta_{\\tau}^{N+p_1 s}} \\int\\limits_{\\Omega_1 \\times \\Omega_1} \\normMo[w(x) - w(y)]^{p_1}  \\,\\mathrm{d}(x,y)\n  \\\\\n    & \\leq \\frac{2^{p_1} \\abs{\\Omega_1} \\bar{c}^{p_1}}{\\eta_{\\tau}^{N+p_1 s}} =\\vcentcolon C. \n  \\end{align*}\n  \\end{enumerate}\n  \\end{enumerate}\n  \\end{example}\n  \n  The second example concerns the coercivity of $\\F[\\dKt,\\dKo]$, defined in \\autoref{eq:functional}, when \n  $\\op$ denotes the \\emph{masking operator} occurring in image inpainting. To prove this result we require the following auxiliary lemma:\n  \\begin{lemma}\\label{lem. auxLemma}\n     There exists a constant $C \\in \\mathds{R} $ such that for all $w \\in W^{s,p_1}(\\Omega_1, \\mathds{R}^{M_1}), \\ 0<s < 1, \\ l \\in \\{0,1\\}, \\ 1 < p_1 < \\infty$ and\n     $D \\subsetneq \\Omega_1$ nonempty such that\n  \\begin{equation} \\label{eq: coercivity_inpainting}\n  \\norm{w}_{L^{p_1}(D, \\mathds{R}^{M_1})}^{p_1} \\leq C \\left( \\norm{w}_{L^{p_1}(\\Omega_1 \\setminus D, \\mathds{R}^{M_1})}^{p_1} + \n  \\int\\limits_{\\Omega_1 \\times \\Omega_1} \\frac{\\normMo[w(x) - w(y)]^{p_1}}{\\normN[x-y]^{N+p_1 s}} \\rho^l(x-y) \\,\\mathrm{d}(x,y)  \\right).\n  \\end{equation}\n  \\end{lemma}\n  \\begin{proof}\n   The proof is inspired by the proof of Poincar\u00e9's inequality in \\cite{Eva10}. It is included here for the sake of completeness.\n\n   Assume first that $l=1$. Let $\\mathcal{S}$ be as above, \n   \\begin{align*}\n   \\mathcal{S} \\vcentcolon= {}&  \\{(x,y) \\in \\Omega_1 \\times \\Omega_1 : \\rho(x-y) \\geq \\tau \\}\n   \\\\\n   = {}& \\{(x,y) \\in \\Omega_1 \\times \\Omega_1 : \\normN[x-y] \\leq \\eta \\}.\n   \\end{align*} \n   If the stated inequality \\autoref{eq: coercivity_inpainting} would be false, then for every $n \\in \\mathds{N}$ there would exists a \n   function $w_n \\in W^{s,p_1}(\\Omega_1, \\mathds{R}^{M_1})$ satisfying\n   \\begin{equation}\\label{eq: inpaintingContraEquation}\n   \\norm{w_n}_{L^{p_1}(D, \\mathds{R}^{M_1})}^{p_1} \\geq n \\big( \\norm{w_n}_{L^{p_1}(\\Omega_1 \\setminus D, \\mathds{R}^{M_1})}^{p_1} + \n   \\int\\limits_{\\Omega_1 \\times \\Omega_1} \\frac{\\normMo[w_n(x) - w_n(y)]^{p_1}}{\\normN[x-y]^{N+p_1 s}} \\rho(x-y) \\,\\mathrm{d}(x,y)  \\big).\n   \\end{equation}\n   By normalizing we can assume without loss of generality \n   \\begin{enumerate}\n   \t\\item $\\norm{w_n}_{L^{p_1}(D, \\mathds{R}^{M_1})}^{p_1} = 1$. \\label{itm: inpainting1}\n   \\end{enumerate}\n   Moreover, by \\autoref{eq: inpaintingContraEquation}\n   \\begin{enumerate}\n   \t\\setcounter{enumi}{1}\n   \t\\item  $\\norm{w_n}_{L^{p_1}(\\Omega_1 \\setminus D, \\mathds{R}^{M_1})}^{p_1} < \\frac{1}{n}$, \\label{itm: inpainting2}\n   \t\\item $\\int\\limits_{\\Omega_1 \\times \\Omega_1} \\frac{\\normMo[w_n(x) - w_n(y)]^{p_1}}{\\normN[x-y]^{N+p_1 s}} \\rho(x-y) \\,\\mathrm{d}(x,y) < \\frac{1}{n}$. \\label{itm: inpainting3}\n   \\end{enumerate}\n   \n   By \\autoref{itm: inpainting1} and \\autoref{itm: inpainting2} we get that \n   $ \\norm{w_n}_{L^{p_1}(\\Omega_1, \\mathds{R}^{M_1})}^{p_1} = \\norm{w_n}_{L^{p_1}(D, \\mathds{R}^{M_1})}^{p_1} + \\norm{w_n}_{L^{p_1}(\\Omega_1 \\setminus D, \\mathds{R}^{M_1})}^{p_1} < 1 + \\frac{1}{n} < 2 $\n   is bounded. Moreover\n   \\begin{align*}\n   \\abs{w_n}^{p_1}_{W^{s,p_1}(\\Omega_1, \\mathds{R}^{M_1})} \n   & = \\int\\limits_{\\mathcal{S}} \\frac{\\normMo[w_n(x) - w_n(y)]^{p_1}}{\\normN[x-y]^{N+p_1 s}} \\,\\mathrm{d}(x,y)\n   + \\int\\limits_{\\mathcal{S}^c} \\frac{\\normMo[w_n(x) - w_n(y)]^{p_1}}{\\normN[x-y]^{N+p_1 s}} \\,\\mathrm{d}(x,y) \\\\\n   & \\leq \\frac{1}{\\tau} \\int\\limits_{\\mathcal{S}} \\frac{\\normMo[w_n(x) - w_n(y)]^{p_1}}{\\normN[x-y]^{N+p_1 s}} \\rho(x-y) \\,\\mathrm{d}(x,y)\n   + \\frac{2^p_1 \\abs{\\Omega_1}}{\\eta^{N+p_1 s}}\\norm{w_n}^{p_1}_{L^{p_1}(\\Omega_1, \\mathds{R}^{M_1})} \\\\\n   & < \\frac{1}{\\tau n} + \\frac{2^{p_1+1} \\abs{\\Omega_1}}{\\eta^{N+p_1 s}}\n   \\leq \\frac{1}{\\tau} + \\frac{2^{p_1+1} \\abs{\\Omega_1}}{\\eta^{N+p_1 s}} =\\vcentcolon c < \\infty,\n   \\end{align*}\n   where $c$ is independent of $n$.\n   This yields that the sequence  \n   $(w_n)_{n \\in \\mathds{N}}$ is bounded in $W^{s,p_1}(\\Omega_1, \\mathds{R}^{M_1})$ by $(2 + c)^{\\frac{1}{p_1}}$. \n   By the reflexivity of $\\Wsp<1>[][\\mathds{R}^{M_1}]$ for $p_1 \\in (1, \\infty)$ and \\autoref{lem:Wsp_weakly_seq_closed_etc} \n   there exists a subsequence $(w_{n_k})_{k \\in \\mathds{N}}$ of $(w_n)_{n \\in \\mathds{N}}$  \n   and $w_* \\in W^{s,p_1}(\\Omega_1, \\mathds{R}^{M_1})$\n   such that $w_{n_k} \\rarr w^*$ strongly in $L^{p_1}(\\Omega_1, \\mathds{R}^{M_1})$ and \n   pointwise almost everywhere.\n\n   Using the continuity of the norm and dominated convergence we obtain\n   \\begin{enumerate}\n   \t\\item $\\norm{w^*}_{L^{p_1}(D, \\mathds{R}^{M_1})}^{p_1} = 1$, in particular $w^*$ is not the null-function on D,\n   \t\\item $\\norm{w^*}_{L^{p_1}(\\Omega_1 \\setminus D, \\mathds{R}^{M_1})}^{p_1} = 0$ since $n \\in \\mathds{N}$ is arbitrary and hence $w^* \\equiv 0$ on $\\Omega_1 \\setminus D$.\n   \t\\item \\begin{equation*}\n   \t\\liminf_{n \\rarr \\infty} \\frac{1}{n} \n   \t> \\liminf_{n \\rarr \\infty} \\int\\limits_{\\mathcal{S}} \\frac{\\normMo[w_n(x) - w_n(y)]^{p_1}}{\\normN[x-y]^{N+p_1 s}} \\rho(x-y) \\,\\mathrm{d}(x,y) \n   \t\\geq \\frac{\\tau}{\\eta^{N+p_1 s}} \\int\\limits_{\\mathcal{S}} \\normMo[w^*(x) - w^*(y)]^{p_1},\n   \t\\end{equation*}\n   \ti.e. $w^*(x) = w^*(y) $ for $(x,y) \\in \\mathcal{S}$ yielding that $w^*$ locally constant and hence even constant since $\\Omega_1$ is connected,\n   \\end{enumerate}\n   which gives the contradiction. \n   \n   In the case $l=0$ we use similar arguments, where the distance $\\normN[x-y]$ in the last inequality can be estimated by $\\mathrm{diam}|\\Omega_1|$ (instead of $\\eta$) since $\\Omega_1$ is bounded.\n  \\end{proof}\n  \n  \\begin{remark}\n  \tIn case $l=1$ it follows that the sharper inequality holds true: \n  \tThere exists a constant $C \\in \\mathds{R} $ such that for all $w \\in W^{s,p_1}(\\Omega_1, \\mathds{R}^{M_1}), \\ 0<s < 1, \\ 1 < p_1 < \\infty$ and $D \\subsetneq \\Omega_1$ nonempty such that\n  \t\\begin{equation}\\label{eq: inpaintingIneq2}\n  \t\\norm{w}_{L^{p_1}(D, \\mathds{R}^{M_1})}^{p_1} \\leq C \\left( \\norm{w}_{L^{p_1}(\\Omega_1 \\setminus D, \\mathds{R}^{M_1})}^{p_1} + \n  \t\\int\\limits_{\\mathcal{S}} \\frac{\\normMo[w(x) - w(y)]^{p_1}}{\\normN[x-y]^{N+p_1 s}} \\rho^l(x-y) \\,\\mathrm{d}(x,y)  \\right).\n  \t\\end{equation} \n  \\end{remark}\n  \n  \\begin{example} \n  \\label{ex:in}\n  As in \\autoref{ex:coercive} we consider that \n  $W(\\Omega_1,K_1) = W^{s, p_1}(\\Omega_1, K_1)$  \n  with $p_1>1, \\ 0 < s < 1$ and fix $k = N$. \n  \n  Assume that $\\op$ is the inpainting operator, i.e. \n  \\begin{equation*}\n    \\op(w) = \\chi_{\\Omega_1 \\backslash D} (w),\n  \\end{equation*} \n  where $D \\subseteq \\Omega_1, \\ w \\in W^{s,p_1}(\\Omega_1, K_1)$. Since the dimension of the data $w$ and the image data $\\op(w)$ have the same dimension at every point $x \\in \\Omega_1$, we write $M \\vcentcolon= M_1 = M_2$. \\\\\n  Then the functional \n  \\begin{equation*}\n  \\F[\\dKt, \\dKo] = \\mebr[\\op[w]][v]^{p_2}_{[\\dKt]} + \\alpha \\mathcal{R}_{[\\dKo]}(w): W^{s,p_1}(\\Omega_1, K_1) \\rarr [0, \\infty]\n  \\end{equation*}\n  is coercive for $p_2 \\geq p_1$: \\\\\n  The fact that $p_2 \\geq p_1$ and that $\\Omega_1$ is bounded ensures that\n  \\begin{equation}\\label{eq: LpEmbedding}\n  L^{p_2}(\\Omega_1 \\backslash D, \\mathds{R}^M) \\subseteq L^{p_1}(\\Omega_1 \\backslash D, \\mathds{R}^M).\n  \\end{equation} \n  The proof is done\n  using the same arguments as in the proof of \\autoref{ex:coercive}, where we additionally split \\ref{ex: coecivity_case1} into the two sub-cases \n  \\begin{enumerate}[label=\\textbf{\\ref{ex: coecivity_case1}.\\arabic*}]\n  \t\\item \\label{ex: coecivity_case11}\n  \t$\\norm{w_n}_{L^{p_1}(D, \\mathds{R}^M)} \\rarr +\\infty$\n  \t\\item \\label{ex: coecivity_case12} \n  \t$\\norm{w_n}_{L^{p_1}(\\Omega_1 \\setminus D, \\mathds{R}^M)} \\rarr +\\infty$\n  \\end{enumerate}\n   and using additionally \\autoref{lem. auxLemma}, \\autoref{eq: inpaintingIneq2} and \\autoref{eq: LpEmbedding}.    \n\\end{example}\n\n\n\\section{Stability and Convergence} \\label{sec: Stability_and_Convergence}\nIn this section we will first show a stability and afterwards a convergence result. We use the notation introduced in \n\\autoref{sec:  Setting}. In particular $W(\\Omega_1, K_1)$ is as defined in \\autoref{eq:ChooseW}. \nWe also stress that we use \nnotationally simplified versions $\\F<v>$ of $\\F<v><\\alpha>[\\dKt, \\dKo]$ and $\\mathcal{R}$ of $\\mathcal{R}_{[\\dKo]}$ whenever possible.\nSee \\autoref{eq: functional_with_some_metric}, \\autoref{eq:d2} and \\autoref{eq:d3}.\n\n\\begin{thm} \\label{thm:Stability}\nLet \\autoref{as:Setting} be satisfied.\nLet $v^\\delta \\in L^{p_2}(\\Omega_2, K_2)$ and let $(v_n)_{n \\in \\mathds{N}}$ be a sequence in $L^{p_2}(\\Omega_2, K_2)$ such that\n$\\mebr[v_n][v^\\delta]_{[\\dKt]} \\rarr 0$.\nThen every sequence $\\seq{w}$ with\n  \\begin{equation*}\n    w_n \\in \\arg \\min \\{ \\F<v_n><\\alpha>[\\dKt, \\dKo](w) \\ : \\ w \\in W(\\Omega_1, K_1) \\}\n  \\end{equation*}\t\n  has a converging subsequence w.r.t. the topology of $W(\\Omega_1, K_1)$.\n  The limit $\\tilde{w}$ of any such converging subsequence $(w_{n_k})_{k \\in \\mathds{N}}$ is a minimizer of \n  $\\F^{v^\\delta}[\\dKt, \\dKo]$.\n  Moreover, $(\\mathcal{R}(w_{n_k}))_{k \\in \\mathds{N}}$ converges to $\\mathcal{R}(\\tilde{w})$.\n\\end{thm}\n\nThe subsequent proof of \\autoref{thm:Stability} is similar to the proof of \\cite[Theorem 3.23]{SchGraGroHalLen09}.\n\n\\begin{proof}\nFor the ease of notation we simply write $\\F<v^\\delta>$ instead of $\\F<v^\\delta><\\alpha>[\\dKt, \\dKo]$ and \n$\\mebr[v][\\tilde{v}] = \\mebr[v][\\tilde{v}]_{[\\dKt]}$ \n\nBy assumption the sequence $(\\mebr[v_n][v^\\delta])_{n\\in \\mathds{N}}$ converges to $0$ and thus is bounded, i.e., \nthere exists $B \\in (0, +\\infty)$ such that \n\\begin{gather} \\label{eq:seq_bounded_IN_stability_proof}\n  \\mebr[v_n][v^\\delta] \\leq B \\text{ for all } n \\in \\mathds{N}.\n\\end{gather}\nBecause $w_n \\in \\arg \\min \\{ \\F<v_n>(w)  :  w \\in W(\\Omega_1, K_1) \\}$ it follows that \n\\begin{equation}\\label{eq: w_n Minimizer}\n  \\F<v_n>(w_n) \\leq \\F<v_n>(w) \\text{ for all } w \\in W(\\Omega_1, K_1).\n\\end{equation}\nBy \\autoref{as:Setting} there is a $\\overline{w} \\in W(\\Omega_1, K_1)$ such that $\\F^{v^0}(\\overline{w}) <\\infty$. Set $c \\vcentcolon= 2^{p_2-1}$.  \nUsing \\autoref{as:Setting} and applying \\autoref{lem:Ineq_F_parameterchange_to_other_v}, \n\\autoref{eq: w_n Minimizer} and \\autoref{eq:seq_bounded_IN_stability_proof} implies that for all $n \\in \\mathds{N}$\n\\begin{align*}\n  \\F<v^\\delta> (w_n)  \n  & \\leq c \\F<v_n>(w_n) + c \\mebr[v_n][v^\\delta]^{p_2}   \n\\\\\n  & \\leq c \\F<v_n>(\\overline{w}) + c B^{p_2}\n\\\\\n  & \\leq c \\big[c \\F<v^\\delta>(\\overline{w}) + c \\mebr[v^\\delta][v_n]^{p_2}  \\big] + cB^{p_2}\n\\\\\n  & \\leq c^2 \\F<v^\\delta>(\\overline{w}) + (c^2 + c)B^{p_2}\n\\\\\n  & \\leq c^3 \\big( \\F<v^0>(\\overline{w}) + \\mebr[v^0][v^\\delta] \\big) + (c^2 + c)B^{p_2} =\\vcentcolon m < \\infty.\n\\end{align*}\nApplying again \\autoref{lem:Ineq_F_parameterchange_to_other_v} we obtain\n$\\F<v^0> (w_n) \\leq c \\F<v^\\delta> (w_n) + c \\mebr[v^\\delta][v^0]^{p_2} \\leq m + c \\mebr[v^\\delta][v^0]^{p_2} =\\vcentcolon \\widetilde m < \\infty$.\nHence, from item \\eqref{itm: A} it follows that the sequence $\\seq{w}$\ncontains a converging subsequence.\n\nLet now $(w_{n_k})_{k \\in \\mathds{N}}$ be an arbitrary subsequence of $\\seq{w}$ which converges in $W(\\Omega_1, K_1)$ to some \n$\\tilde w \\in W(\\Omega_1, \\mathds{R}^{M_1})$. Then, from \\autoref{lem:Wsp_weakly_seq_closed_etc} and the continuity properties of $\\op$ \nit follows that $\\tilde w \\in W(\\Omega_1, K_1)$ and $(\\op[w_{n_k}], v_{n_k}) \\rarr (\\op[\\tilde w], v^\\delta)$ in \n$L^{p_2}(\\Omega_2, K_2) \\times L^{p_2}(\\Omega_2, K_2)$. Moreover, using \\autoref{thm:F_and_its_summands_are_seq_weakly_closed},\n\\autoref{eq: w_n Minimizer} \nand the triangle inequality \nit follows that for every $w \\in W(\\Omega_1, K_1)$ the following estimate holds true\n\\begin{align*}\n  \\F<v^\\delta>(\\tilde w) \n  & = \\mebr[\\op(\\tilde w)][v^\\delta]^{p_2} + \\alpha \\mathcal{R}(\\tilde w)\n    \\leq \\mebr[\\op(\\tilde w)][v^\\delta]^{p_2} + \\alpha \\liminf_{k \\rarr \\infty} \\mathcal{R}(w_{n_k})\n    \\leq \\mebr[\\op(\\tilde w)][v^\\delta]^{p_2} + \\alpha \\limsup_{k \\rarr \\infty} \\mathcal{R}(w_{n_k})\n\\\\& \n    \\leq \\liminf_{k \\rarr \\infty}  \\mebr[\\op(w_{n_k})][v_{n_k}]^{p_2} + \\alpha \\limsup_{k \\rarr \\infty} \\mathcal{R}(w_{n_k})    \n    \\leq \n    \\limsup_{k \\rarr \\infty} \\F<v_{n_k}>(w_{n_k})\n    \\leq \\limsup_{k \\rarr \\infty} \\F<v_{n_k}>(w)\n\\\\& \n= \\left( \\limsup_{k \\to \\infty} \\mebr[F(w)][v_{n_k}] \\right)^{p_2} + \\alpha \\mathcal{R}(w)\n    \\leq \\left( \\limsup_{k \\to \\infty} \\big(\\mebr[F(w)][v^\\delta] + \\mebr[v^\\delta][v_{n_k}] \\big) \\right)^{p_2} + \\alpha \\mathcal{R}(w)\n\\\\& = \\F<v^\\delta>(w).\n\\end{align*}\n\n\nThis shows that $\\tilde w$ is a minimizer of $\\F<v^\\delta>$.\nChoosing $w = \\tilde w$ in the previous estimate we obtain the equality\n\\begin{gather*}\n  \\mebr[\\op(\\tilde w)][v^\\delta]^{p_2}  + \\alpha \\mathcal{R}(\\tilde w) \n  = \\mebr[\\op(\\tilde w)][v^\\delta]^{p_2} + \\alpha \\liminf_{k \\rarr \\infty} \\mathcal{R}(w_{n_k}) \n  = \\mebr[\\op(\\tilde w)][v^\\delta]^{p_2} + \\alpha \\limsup_{k \\rarr \\infty} \\mathcal{R}(w_{n_k}) \\,.\n\\end{gather*}\nDue to $\\mebr[\\op(\\tilde w)][v^\\delta]^{p_2} \\leq \\F<v^\\delta>(\\tilde w) \\leq m < \\infty$ this gives\n\\begin{gather*}\n  \\mathcal{R}(\\tilde w) = \\lim_{k \\rarr \\infty} \\mathcal{R}(w_{n_k}).\n\\end{gather*}\n\\end{proof}\n\n\nBefore proving the next theorem we need the following definition, cf. \\cite{SchGraGroHalLen09}.\n\\begin{definition}\n  Let $v^0 \\in \\Lp<2>$.\n  Every element $w^* \\in W(\\Omega_1, K_1)$ fulfilling\n  \\begin{align} \\label{eq: R_minimizing_solution}\n  \\begin{split}\t\n  &\\op[w^*] = v^0 \\\\ \n  &\\mathcal{R}(w^*) = \\min \\{ \\mathcal{R}(w) \\ : \\ w \\in W(\\Omega_1, K_1), \\ \\op[w] = v^0 \\}.\n  \\end{split}\n  \\end{align}\n  is called an \n  \\emph{$\\mathcal{R}$-minimizing solution} of the equation $\\op[w] = v^0$ or shorter just\n  \\emph{$\\mathcal{R}$-minimizing solution}.\n\\end{definition}\n\nThe following theorem and its proof are inspired by \\cite[Theorem 3.26]{SchGraGroHalLen09}. \n\n\\begin{thm} \\label{thm: convergence}\n  Let \\autoref{as:Setting} be satisfied.\t  \n  Let there exist an $\\mathcal{R}$-minimizing solution $w^\\dagger \\in W(\\Omega_1, K_1)$ and \n  let $\\alpha: (0, \\infty) \\rarr (0,\\infty)$ be a function satisfying\n  \\begin{equation}\\label{eq: assumptions_on_alpha}\n    \\alpha(\\delta) \\rarr 0 \\text{ and } \\frac{\\delta^{p_2}}{\\alpha(\\delta)} \\rarr 0\n\t\\text{ for } \\delta \\to 0.\n  \\end{equation} \n  Let $(\\delta_n)_{n \\in \\mathds{N}}$ be a sequence of positive real numbers converging to $0$. Moreover, let \n  $(v_n)_{n \\in \\mathds{N}}$ be a sequence in $L^{p_2}(\\Omega_2, K_2)$ with $\\mebr[v^0][v_n]_{[\\dKt]} \\leq \\delta_n$ and \n  set $\\alpha_n \\vcentcolon= \\alpha(\\delta_n)$.\n  \n  Then every sequence $\\seq{w}$ of minimizers \n  \\begin{equation*}\n   w_n \\in \\arg \\min \\{ \\F^{v_n}_{\\alpha_n}[\\dKt, \\dKo](w) \\ : \\ w \\in W(\\Omega_1, K_1) \\}\n  \\end{equation*}\n  has a converging subsequence $w_{n_k} \\rarr<W> \\tilde{w}$ as $k \\to \\infty$, and the limit $\\tilde{w}$ is always an $\\mathcal{R}$-minimizing solution. \n  In addition, $\\mathcal{R}(w_{n_k}) \\rarr \\mathcal{R}(\\tilde{w})$.\n\t\n  Moreover, if $w^\\dagger$ is unique it follows that $w_n \\rarr<W> w^\\dagger$ and $\\mathcal{R}(w_{n}) \\rarr \\mathcal{R}(w^\\dagger)$.\n\\end{thm}\n\n\\begin{proof}\n  We write shortly $\\mebr$ for $\\mebr_{[\\dKt]}$.\n  Taking into account that $w_n \\in \\argmin \\{ \\F^{v_n}_{\\alpha_n}[\\dKt, \\dKo](w) \\ : \\ w \\in W(\\Omega_1, K_1) \\}$ it follows that\n  \\begin{gather*}\n    \\mebr[\\op[w_n]][v_n]^{p_2} \\leq \\F<v_n><\\alpha_n>(w_n) \\leq \\F<v_n><\\alpha_n>(w^\\dagger) = \n    \\mebr[v^0][v_n]^{p_2} + \\alpha_n \\mathcal{R}(w^\\dagger) \\leq \\delta_n^{p_2} + \\alpha_n \\mathcal{R}(w^\\dagger) \\rarr 0,\n  \\end{gather*}\n  yielding $\\mebr[\\op[w_n]][v_n] \\rarr 0$ as $n \\rarr \\infty$.\n  The triangle inequality gives $\\mebr[\\op[w_n]][v^0] \\leq \\mebr[\\op[w_n]][v_n] + \\mebr[v_n][v^0] \\rarr 0$ as \n  $n \\rarr \\infty$ and\n  \\autoref{re:tricks} ensures \n  $\\norm{F(w_n) - v^0}_{\\Lp<2>[][\\mathds{R}^{M_2}]} \\leq \\mebr[\\op[w_n]][v^0] \\rarr 0$ as \n  $n \\rarr \\infty$, so that  \n  \\begin{gather}\\label{eq:Convergence_of_operator}\n    \\op[w_n] \\rarr v^0 \\text{ in } L^{p_2}(\\Omega_2, \\mathds{R}^{M_2}).\n  \\end{gather} \n  Since\n  \\begin{gather*}\n    \\mathcal{R}(w_n)  \\leq \\frac{1}{\\alpha_n} \\F<v_n><\\alpha_n>(w_n) \\leq \\frac{1}{\\alpha_n} \\F<v_n><\\alpha_n>(w^\\dagger) \n    = \\frac{1}{\\alpha_n}\\big( \\mebr[v^0][v_n]^{p_2} + \\alpha_n \\mathcal{R}(w^\\dagger)  \\big) \\leq \\frac{\\delta_n^{p_2}}{\\alpha_n} + \\mathcal{R}(w^\\dagger),\n  \\end{gather*}\n  we also get \n  \\begin{gather}\\label{eq:Regularizer_values_bounded}\n    \\limsup_{n \\rarr \\infty} \\mathcal{R}(w_n) \\leq \\mathcal{R}(w^\\dagger).\n  \\end{gather}\n  Set $\\alpha_{\\mathrm{max}} \\vcentcolon= \\max\\{\\alpha_n : n \\in \\mathds{N}\\}$. \n  Since \n  \\begin{gather*}\n    \\limsup_{ n \\rarr \\infty } \\F<v^0><\\alpha_n>(w_{n}) \\leq \n    \\limsup_{n \\rarr \\infty } \\big( \\mebr[\\op[w_{n}]][v^0]^{p_2} + \\alpha_{\\mathrm{max}} \\mathcal{R}(w_{n})  \\big) \\leq \\alpha_{\\mathrm{max}} \\mathcal{R}(w^\\dagger)\n  \\end{gather*}\n the sequence $\\F<v^0><\\alpha_{\\mathrm{max}}>(w_{n})$ is bounded. From \\autoref{as:Setting}, item \\eqref{itm: A} it follows that there exists \n a converging subsequence $(w_{n_k})_{k \\in \\mathds{N}}$ of $\\seq{w}$. The limit of $(w_{n_k})_{k \\in \\mathds{N}}$ is denoted by \n $\\tilde{w}$. Then, from \\autoref{lem:Wsp_weakly_seq_closed_etc} it follows that $\\tilde{w} \\in W(\\Omega_1, K_1)$.\n Since the operator $\\op$ is sequentially continuous it follows that $\\op[w_{n_k}] \\rarr \\op[\\tilde{w}]$ in $L^{p_2}(\\Omega_2, K_2)$.\t\n This shows that actually $\\op[\\tilde{w}] = v^0$ since \\autoref{eq:Convergence_of_operator} is valid.\n Then, from \\autoref{thm:F_and_its_summands_are_seq_weakly_closed} it follows that the functional \n $\\mathcal{R}: W(\\Omega_1, K_1) \\rarr [0, +\\infty]$ is sequentially lower semi-continuous,\n\tso that $\\mathcal{R}(\\tilde{w}) \\leq \\liminf_{k \\rarr \\infty} \\mathcal{R}(w_{n_k})$.\n\tCombining this with \\autoref{eq:Regularizer_values_bounded} we also obtain \n\t$$ \\mathcal{R}(\\tilde{w}) \\leq \\liminf_{k \\rarr \\infty} \\mathcal{R}(w_{n_k}) \\leq \\limsup_{k \\rarr \\infty} \\mathcal{R}(w_{n_k}) \\leq \\mathcal{R}(w^\\dagger) \\leq \\mathcal{R}(\\tilde{w}),$$\n\tusing the definition of $w^\\dagger$. \n\tThis, together with the fact that $\\op[\\tilde{w}] = v^0$ we see that $\\tilde{w}$ is an $\\mathcal{R}$-minimizing solution and that \n\t$\\lim_{k \\rarr \\infty} \\mathcal{R}(w_{n_k})= \\mathcal{R}(\\tilde{w})$. \n\t\n\tNow assume that the solution fulfilling \\autoref{eq: R_minimizing_solution} is unique; we call it $w^\\dagger$.  \n\tIn order to prove that $w_n \\rarr<W> w^\\dagger$ it is sufficient to show that any subsequence\n\thas a further subsequence converging to $w^\\dagger$, cf.\n\t\\cite[Lemma 8.2]{SchGraGroHalLen09}.\n\tHence, denote by $(w_{n_k})_{k \\in \\mathds{N}}$ an arbitrary subsequence of $(w_n)$, the sequence of minimizers.\n\tLike before we can show that $\\F<v^0><\\alpha>(w_{n_k})$ is bounded and we can extract a converging subsequence \n\t$(w_{n_{k_l}})_{l \\in \\mathds{N}}$. The limit of this subsequence is $w^\\dagger$ since it is the unique solution fulfilling \n\t\\autoref{eq: R_minimizing_solution}, showing that $w_n \\rarr<W> w^\\dagger$. Moreover, $w^\\dagger \\in W(\\Omega_1, K_1)$. \n\tFollowing the arguments above we obtain as well $\\lim_{n \\rarr \\infty} \\mathcal{R}(w_{n})= \\mathcal{R}(w^\\dagger).$ \n\\end{proof}\n\\begin{remark}\n\\autoref{thm:Stability} guarantees that the minimizers of $\\F<v_n><\\alpha>[\\dKt, \\dKo]$ depend continuously on $v^\\delta$ while \n\\autoref{thm: convergence} ensures that they converge to a solution of $\\op(w) = v^0$, $v^0$ the exact data, while $\\alpha$ tends to zero. \n\\end{remark}\n\n\\section{Discussion of the Results and Conjectures}\nIn this section we summarize some open problems related to double integral \nexpressions\nof functions with values on manifolds.\n\n\\subsection{Relation to single integral representations} \nIn the following we show for one particular case of functions that have values in a manifold, that the double \nintegral formulation $\\mathcal{R}_{[\\dKo]}$, defined in \\autoref{eq:d3}, approximates a single energy integral. The basic \ningredient for this derivation is the exponential map related to the metric $d_1$ on the manifold.\nIn the following we investigate manifold--valued functions $w \\in W^{1,2}(\\Omega, \\mathcal{M})$, where we \nconsider $\\mathcal{M} \\subseteq \\mathds{R}^{M \\times 1}$ to be a connected, complete Riemannian manifold. \nIn this case some of the regularization functionals $\\mathcal{R}_{[\\dKo]}$, defined in \\autoref{eq:d3}, can be \nconsidered as approximations of \\emph{single} integrals. In particular we aim to generalize \\autoref{eq:double_integral} \nin the case $p=2$. \n\nWe have that \n\\begin{equation*}\n \\nabla w = \\begin{bmatrix} \n              \\frac{\\partial w_1}{\\partial x_1} & \\cdots & \\frac{\\partial w_1}{\\partial x_N} \\\\\n              \\vdots & \\ddots & \\vdots\\\\\n              \\frac{\\partial w_M}{\\partial x_1} & \\cdots & \\frac{\\partial w_M}{\\partial x_N}\n\\end{bmatrix} \\in \\mathds{R}^{M \\times N}.\n\\end{equation*}\nIn the following we will write $\\mathcal{R}_{[\\dKo],\\varepsilon}$ instead of $\\tfrac12 \\mathcal{R}_{\\dKo}$ to stress the dependence on $\\varepsilon$ in contrast to above;\nthe factor $\\frac{1}{2}$ was added due to reasons of calculation.\nMoreover, let \n$\\hat{\\rho} : \\mathds{R}_+ \\to \\mathds{R}_+$ be in $C_c^\\infty(\\mathds{R}_+, \\mathds{R}_+)$ and satisfy\n\\begin{equation*}\n\\abs{\\mathbb{S}^{N-1}}\\int_0^\\infty \\hat{t}^{N-1} \\hat{\\rho}\\left(\\hat{t}\\right)d \\hat{t} = 1\\;.\n\\end{equation*}\nThen for every $\\varepsilon > 0$\n\\begin{equation*}\nx \\in \\mathds{R}^n \\mapsto \\rho_\\varepsilon(x)\\vcentcolon= \\frac{1}{\\varepsilon^N} \\hat{\\rho}\\left(\\frac{\\normN[x]}{\\varepsilon}\\right)\n\\end{equation*}\nis a mollifier, cf. \\autoref{ex:mol}. \\\\\n$\\mathcal{R}_{[\\dKo],\\varepsilon}$ \n(with $p_1=2$) then reads as follows: \n\\begin{equation}\n\\label{eq:di_II}\n\n \\mathcal{R}_{[\\dKo],\\varepsilon}(w) \n \\vcentcolon= \n\n \\frac{1}{2}\\int\\limits_{\\Omega\\times \\Omega} \\frac{d_1^2(w(x),w(y))}{\\normN[x-y]^2} \\rho_\\varepsilon(x-y) \\,\\mathrm{d}(x,y)\\,.\n\\end{equation}\nSubstitution with spherical coordinates $y = x - t \\theta \\in \\mathds{R}^{N \\times 1}$ with  \n$\\theta \\in \\mathbb{S}^{N-1} \\subseteq \\mathds{R}^{N \\times 1}$, $t \\geq 0$ gives\n\\begin{equation}\n\\label{eq:reg2}\n\\begin{aligned}\n\\lim_{\\varepsilon \\searrow 0}\n\\mathcal{R}_{[\\dKo],\\varepsilon}(w) &= \n\\lim_{\\varepsilon \\searrow 0}\n\\frac{1}{\\varepsilon^N}\n \\int\\limits_{\\Omega} \\int\\limits_{\\mathbb{S}^{N-1}} \n \\int\\limits_0^\\infty \\frac{1}{2} d_1^2(w(x),w(x-t \\theta)) t^{N-3} \\hat{\\rho}\\left(\\frac{t}{\\varepsilon}\\right) \\mathrm{d}t \\,\\mathrm{d}\\theta \\,\\mathrm{d}x\\;.\n\\end{aligned}\n\\end{equation}\nNow, using that for $m_1 \\in \\mathcal{M}$ fixed and $m_2 \\in \\mathcal{M}$ such that $m_1$ and $m_2$ are joined by a unique minimizing geodesic (see for instance \\cite{FigVil11} where the concept of exponential mappings is explained) \n\\begin{equation}\\label{eq:partial_II}\n \\frac{1}{2} \\partial_2 d_1^2(m_1,m_2) = - (\\exp_{m_2})^{-1}(m_1) \\in \\mathds{R}^{M \\times 1},\n\\end{equation}\nwhere $\\partial_2$ denotes the derivative of $d_1^2$ with respect to the second component. \nBy application of the chain rule we get\n\\begin{equation*}\n \\begin{aligned}\n  - \\frac{1}{2} \\nabla_y d_1^2(w(x),w(y)) &= \n    \\underbrace{(\\nabla w(y))^T}_{\\in \\mathds{R}^{N \\times M}} \\underbrace{(\\exp_{w(y)})^{-1}(w(x))}_{\\in \\mathds{R}^{M \\times 1}}\\in \\mathds{R}^{N \\times 1}\\;,\n \\end{aligned}\n\\end{equation*}\nwhere $w(x)$ and $w(y)$ are joined by a unique minimizing geodesic. This assumption seems reasonable due to the fact \nthat we consider the case $\\varepsilon \\searrow 0$. \nLet $\\cdot$ denote the scalar multiplication of two vectors in $\\mathds{R}^{N \\times 1}$, then the last equality shows that\n\\begin{equation*}\n \\begin{aligned}\n\\frac{1}{2} d_1^2(w(x),w(x-t \\theta)) \n&= - \\frac{1}{2}  \\left[  d_1^2\\big(w(x),w( (x-t\\theta)  + t \\theta )\\big) - d_1^2\\big(w(x),w(x-t \\theta)\\big)  \\right]  \\\\ \n&\\approx \\left( \\left(\\nabla w(x-t \\theta)\\right)^T (\\exp_{w(x-t \\theta)})^{-1}(w(x)) \\right)  \\cdot t\\theta\\;.   \n\\end{aligned}\n\\end{equation*}\nThus from \\autoref{eq:reg2} it follows that \n\\begin{equation}\n\\label{eq:reg3}\n\\begin{aligned}\n~ & \\lim_{\\varepsilon \\searrow 0} \n\\mathcal{R}_{[\\dKo],\\varepsilon}(w) \\\\\n\\approx & \n\\lim_{\\varepsilon \\searrow 0}\n \\frac{1}{\\varepsilon^N}\n \\int\\limits_{\\Omega} \\int\\limits_{\\mathbb{S}^{N-1}} \n \\int\\limits_0^\\infty \\left( \\left( \\nabla w(x-t \\theta)\\right)^T (\\exp_{w(x-t \\theta)})^{-1}(w(x)) \n                              \\right) \\cdot  \n                      \\theta \\left(t^{N-2} \\hat{\\rho}\\left(\\frac{t}{\\varepsilon}\\right)\\right) \\mathrm{d}t \\,\\mathrm{d}\\theta \\,\\mathrm{d}x\\;.\n\\end{aligned}\n\\end{equation}\nNow we will use a Taylor series of power 0 for $ t\\mapsto \\nabla w(x-t \\theta)$ and of power 1 for $t \\mapsto (\\exp_{w(x-t \\theta)})^{-1}(w(x))$ to rewrite \\autoref{eq:reg3}.\nWe write \n\\begin{equation}\n F(w;x,t,\\theta) \\vcentcolon= (\\exp_{w(x-t \\theta)})^{-1}(w(x)) \\in \\mathds{R}^{M \\times 1}\n\\end{equation}\nand define \n\\begin{equation}\n \\dot{F}(w;x,\\theta) \\vcentcolon= \\lim_{t \\searrow 0} \\frac{1}{t} \\left((\\exp_{w(x-t \\theta)})^{-1}(w(x)) - \n \\underbrace{(\\exp_{w(x)})^{-1}(w(x))}_{=0} \n \\right) \\in \\mathds{R}^{M \\times 1}.\n\\end{equation}\nNote that because $(\\exp_{w(x)})^{-1}(w(x))$ vanishes, $\\dot{F}(w(x);\\theta)$ is the leading order term of the expansion of \n$(\\exp_{w(x-t \\theta)})^{-1}(w(x))$ with respect to $t$.\nMoreover, in the case that $\\nabla w(x) \\neq 0$ this is the leading order approximation of $\\nabla w(x-t \\theta)$. In summary we are calculating the leading order term of the expansion \nwith respect to $t$.\n\nThen from \\autoref{eq:reg3} it follows that\n\\begin{equation}\n\\label{eq:reg3a}\n\\lim_{\\varepsilon \\searrow 0}\n\\mathcal{R}_{[\\dKo],\\varepsilon}(w) \n\\approx\n\\lim_{\\varepsilon \\searrow 0}\n\\underbrace{\\frac{1}{\\varepsilon^N} \\int\\limits_0^\\infty t^{N-1} \\hat{\\rho}\\left(\\frac{t}{\\varepsilon}\\right) \\mathrm{d}t}_{= \\abs{\\mathbb{S}^{N-1}}^{-1}}\n \\int\\limits_{\\Omega} \\int\\limits_{\\mathbb{S}^{N-1}} \\left((\\nabla w(x))^T \\dot{F}(w;x,\\theta) \\right) \\cdot \\theta \\;\n  \\mathrm{d}\\theta \\,\\mathrm{d}x\\;.\n\\end{equation}\nThe previous calculations show that the double integral simplifies to a double integral where the inner integration domain \nhas one dimension less than the original integral. Under certain assumption the integration domain can be further simplified:\n\n\\begin{example}\n If $d_1(x,y)=\\normM[x-y]$, $p_1=2$, then \n \\begin{equation*}\n  \\dot{F}(w;x,\\theta) = \\lim_{t \\searrow 0} \\frac{1}{t} \\left(w(x) - w(x-t\\theta)\\right) = \\nabla w(x)\\theta \\in \\mathds{R}^{M \\times 1}.\n \\end{equation*}\n Thus from \\eqref{eq:reg3a} it follows that \n\\begin{equation}\n\\label{eq:reg3b}\n\\lim_{\\varepsilon \\searrow 0}\n\\mathcal{R}_{[\\dKo],\\varepsilon}(w) \n\\approx \\int\\limits_{\\Omega} \\underbrace{(\\nabla w(x))^T \\nabla w(x)}_{\\norm{\\nabla w(x)}^2_{\\mathds{R}^M}} \\,\\mathrm{d}x\\;.\n\\end{equation}\nThis is exactly the identity derived in \\citeauthor{BouBreMir01} \\cite{BouBreMir01}.\n\\end{example}\nFrom these considerations we can view $\\lim_{\\varepsilon \\searrow 0} \\mathcal{R}_{[\\dKo],\\varepsilon}$ as functionals, which generalize \nSobolev and $\\mathrm{BV}$ semi-norms to functions with values on manifolds.\n\n\\subsection{A conjecture on Sobolev semi-norms}\n\\label{ss:conjecture}\nStarting point for this conjecture is \\autoref{eq:d3}. We will write $\\Omega,M$ and $p$ instead of $\\Omega_1, M_1$ and $p_1$.\n\\begin{itemize}\n \\item In the case $l=0$, $k=N$, $0<s<1$ and $\\dKo(w(x), w(y))= \\normM[w(x)-w(y)]$ \n       the functional $\\mathcal{R}_{[\\dKo]}$ from \n       \\autoref{eq:d3} simplifies to the $p$-th \n       power of the Sobolev semi-norm and reads\n       \\begin{equation}\\label{eq:d3a} \n        \\int\\limits_{\\Omega\\times \\Omega} \\frac{\\normM[w(x)-w(y)]^{p}}{\\normN[x-y]^{N+p s}} \\,\\mathrm{d}(x,y).\n       \\end{equation}\n       For a recent survey on fractional Sobolev Spaces see \\cite{DiNPalVal12}.\n \\item On the other hand, when we choose $k=0$, $l=1$ and $\\dKo(w(x), w(y))= \\normM[w(x)-w(y)]$, then \n       $\\mathcal{R}_{[\\dKo]}$ from \\autoref{eq:d3} reads\n       (note $\\rho=\\rho_\\varepsilon$ by simplification of notation):\n       \\begin{equation}\\label{eq:d3b} \n        \\int\\limits_{\\Omega\\times \\Omega} \\frac{\\normM[w(x)-w(y)]^{p}}{\\normN[x-y]^{p s}} \n        \\rho_\\varepsilon(x-y) \n        \\,\\mathrm{d}(x,y).\n    \\end{equation}\n \\item Therefore, in analogy to what we know for $s=1$ from \\cite{BouBreMir01}, we conjecture that\n        \\begin{equation}\\label{eq:d3c}\n        \\lim_{\\varepsilon \\to 0} \\int\\limits_{\\Omega\\times \\Omega} \\frac{\\normM[w(x)-w(y)]^{p}}{\\normN[x-y]^{p s}} \n        \\rho_\\varepsilon(x-y) \n          \\,\\mathrm{d}(x,y) = C \\int\\limits_{\\Omega\\times \\Omega} \\frac{\\normM[w(x)-w(y)]^{p}}{\\normN[x-y]^{N+p s}}\\,\\mathrm{d}(x,y).\n    \\end{equation}\n    The form \\autoref{eq:d3c} is numerically preferable to the standard Sobolev semi-norm \\autoref{eq:d3a}, because \n    $\\rho=\\rho_\\varepsilon$ and thus the integral kernel has compact support.\n\\end{itemize}\n\n\\section{Numerical Examples} \\label{sec:Numerical_results}\n\nIn this section we present some numerical examples for denoising and inpainting of functions with values on the circle \n$\\sphere$. Functions with values on a sphere have already been investigated very diligently (see for instance \n\\cite{BouBreMir00b} out of series of publications of these authors). \nTherefore we review some of their results first. \n\n\\subsection{$\\sphere$-Valued Data}\n\\label{ss: spheredata}\nLet $\\emptyset \\neq \\Omega \\subset \\mathds{R}$ or $\\mathds{R}^2$ be a bounded and simply connected open set with Lipschitz boundary. In \\cite{BouBreMir00b}\nthe question was considered when $w \\in \\Wsp[\\sphere]$ can be represented by some function $u \\in \\Wsp[\\mathds{R}]$ \nsatisfying \n\\begin{equation}\\label{eq:id_sphere_w}\n \\Phi(u) \\vcentcolon= \\mathrm e^{\\i u} = w.\n\\end{equation}\nThat is, the function $u$ is a \\emph{lifting} of $w$.\n\n\\begin{lemma}[\\cite{BouBreMir00b}] \\label{lem: lifting}\n\\begin{itemize}\n \\item Let $\\Omega \\subset \\mathds{R}$, $0 < s < \\infty$, $1 < p < \\infty$. Then for all $w \\in \\Wsp[\\sphere]$ there exists \n       $u \\in \\Wsp[\\mathds{R}]$ satisfying \\autoref{eq:id_sphere_w}.\n \\item Let $\\Omega \\subset \\mathds{R}^N$, $N \\geq 2$, $0 < s < 1$, $1 < p < \\infty$. Moreover, let \n       $sp < 1$ or $sp \\geq N$, then for all $w \\in \\Wsp[\\sphere]$ there exists \n       $u \\in \\Wsp[\\mathds{R}]$ satisfying \\autoref{eq:id_sphere_w}. \n       \n       If  $sp \\in [1,N)$, then there exist functions $w \\in \\Wsp[\\sphere]$ such that \\autoref{eq:id_sphere_w} does not hold with any function\n       $u \\in \\Wsp[\\mathds{R}]$. \n\\end{itemize}\n\n\\end{lemma}\nFor \n\\begin{equation}\n \\label{eq:arccos}\n \\dS(a,b) \\vcentcolon= \\arccos(a^T b) \\, , \\quad a, b \\in \\sphere,\n\\end{equation}\nwe consider the functional (note that by simplification of notation below $\\rho=\\rho_\\varepsilon$ denotes a mollifier)\n\\begin{equation}\n\\label{eq:considered}\n\\mathcal{R}_{[\\dS]}(w) = \\int\\limits_{\\Omega\\times \\Omega} \\frac{\\dS^p(w(x), w(y))}{\\normN[x-y]^{k+ps}} \\rho^l(x-y) \\,\\mathrm{d}(x,y),\n\\end{equation}\non $w \\in \\Wsp[\\sphere]$, \nin accordance to \\autoref{eq:d3}.\n\nWriting $w = \\Phi(u)$ as in \\autoref{eq:id_sphere_w}\nwe get the \nlifted functional\n\\begin{equation}\n\\label{eq:sobolev_alternative}\n\\mathcal{R}_{[\\dS]}^{\\Phi}(u) \\vcentcolon=\n \\int\\limits_{\\Omega\\times \\Omega} \\frac{\\dS^p(\\Phi(u)(x), \\Phi(u)(y))}{\\normN[x-y]^{k+ps}} \\rho^l(x-y) \\,\\mathrm{d}(x,y),\n\\end{equation}\nover the space $\\Wsp[\\mathds{R}]$.\n\\begin{remark}\n\\begin{itemize}\n \\item We note that in the case $k=0$, $s=1$ and $l=1$ these integrals correspond with the ones considered \n in \\citeauthor{BouBreMir01} \\cite{BouBreMir01} for functions with values on $\\sphere$.\n \\item \n If we choose $k=N$, $s=1$ and $l=0$, then this corresponds with Sobolev semi-norms on manifolds. \n \\item Let $\\varepsilon > 0$ fixed (that is, we consider neither a standard Sobolev regularization nor the limiting \n       case $\\varepsilon \\to 0$ as in \\cite{BouBreMir01}). In this case we have proven coercivity of the functional \n       $\\F: \\Wsp[\\sphere] \\rarr [0,\\infty), \\ 0<s<1,$ only with the following regularization functional, \n       cf. \\autoref{ex:coercive} and \\autoref{ex:in}:   \n       $$\\int\\limits_{\\Omega\\times \\Omega} \\frac{\\dS^p(w(x),w(y))}{\\normN[x-y]^{N + p s}} \\rho_\\varepsilon(x-y) \\,\\mathrm{d}(x,y).$$\n \\end{itemize}\n\\end{remark}\n\nWe summarize a few results: The first lemma follows from elementary calculations:\n\\begin{lemma}\n\\label{le:1}\n $\\dS$ and $\\d_{\\mathds{R}^2}\\big|_{\\sphere \\times \\sphere}$ are equivalent.\n\\end{lemma}\n\n\\begin{lemma} \\label{lem:2}\n\tLet $u \\in \\Wsp[\\mathds{R}]$. Then $\\Phi(u) \\in \\Wsp[\\sphere]$.\n\\end{lemma}\n\n\\begin{proof}\n\tThis follows directly from the inequality $\\|\\mathrm e^{ia}-\\mathrm e^{ib}\\| \\leq \\|a-b\\|$ for all $a,b \\in \\mathds{R}$.\n\\end{proof}\nBelow we show that $\\mathcal{R}_{[\\dS]}^{\\Phi}$ is finite on $\\Wsp[\\mathds{R}]$.\n\\begin{lemma} \\label{lem: liftedRegularizer}\n\t$\\mathcal{R}_{[\\dS]}^{\\Phi}$ maps $\\Wsp[\\mathds{R}]$ into $[0,\\infty)$ (i.e. does not attain the value $+\\infty$).\n\\end{lemma}\n\\begin{proof}\n\tLet $u \\in \\Wsp[\\mathds{R}]$. Then by Lemma \\ref{lem:2} we have that $\\Phi(u) \\in \\Wsp[\\sphere]$. Therefore, from Lemma \\ref{le:1} and \\autoref{pr:ExprIsOp} \\autoref{itm: ExpIsOp} it follws that $\\mathcal{R}_{[\\dS]}(\\Phi(u))< \\infty$. Hence, by definition, $\\mathcal{R}_{[\\dS]}^{\\Phi}(u) < \\infty$.  \n\\end{proof}\t\n\t\n\n\n\n\\subsection{Setting of numerical examples}\nIn all numerical examples presented we use a simplified setting with\n\\begin{equation*}\n M_1 = M_2 =\\vcentcolon M,\\;K_1 = K_2 =\\vcentcolon \\sphere,\\;p_1 = p_2 =\\vcentcolon p,\\;k = N,\\;l = 1,\n\\end{equation*}\n$\\Omega_1 = \\Omega_2 =\\vcentcolon \\Omega$ when considering image denoising,\n$\\Omega_1 = \\Omega$, $\\Omega_2 = \\Omega \\setminus D$ when considering image inpainting,\nand\n\\begin{equation*}\nW(\\Omega,\\sphere) = \\Wsp[\\sphere].\n\\end{equation*}\nAs particular mollifier we use $\\rho_\\varepsilon$ (see \\autoref{ex:mol}), which is defined via the one-dimensional \nnormal-distribution $ \\hat{\\rho}(x) = \\frac{1}{\\sqrt{\\pi}} \\mathrm e^{-x^2}.$\n\n\\subsection*{Regularization functionals}\nLet $\\mathcal{R}_{[\\dS]}$ and $\\mathcal{R}_{[\\dS]}^{\\Phi}$ be as defined in \\autoref{eq:considered} and \\autoref{eq:sobolev_alternative}, respectively.\nIn what follows we consider the following regularization functional \n\\begin{equation}\n\\label{eq:reg_numerics}\n\\F<v^\\delta><\\alpha>[\\dS](w) \\vcentcolon= \\int\\limits_\\Omega \\dS^p(\\op[w](x), v^\\delta(x)) \\,\\mathrm{d}x + \\alpha \\mathcal{R}_{[\\dS]}(w),\n\\end{equation}\non $\\Wsp[\\sphere]$ and the lifted variant\n\\begin{equation} \\label{eq: functionalAlternative}\n\\FT<v^\\delta><\\alpha>[\\dS](u) \\vcentcolon= \n\\int\\limits_\\Omega \\dS^p(\\op[\\Phi(u)](x), v^\\delta(x)) \\,\\mathrm{d}x + \\alpha \\mathcal{R}_{[\\dS]}^{\\Phi}(u) \n\\end{equation}\nover the space $\\Wsp[\\mathds{R}]$ (as in \\autoref{ss: spheredata}), where $\\Phi$ is defined as in \\eqref{eq:id_sphere_w}.\nNote that $\\FT = \\F \\circ \\Phi$.\n\n\n\\begin{lemma}\\label{lem: liftedFunctional}\nLet $\\emptyset \\neq \\Omega \\subset \\mathds{R}$ or $\\mathds{R}^2$ be a bounded and simply connected open set with Lipschitz boundary. \nLet $1 < p < \\infty$ and $s \\in (0,1)$. If $N=2$ assume that $sp < 1$ or $sp \\geq 2$. Moreover, let \\autoref{as:Setting} and \\autoref{ass:2} be satisfied. Then the mapping \n$\\FT<v^\\delta><\\alpha>[\\dS]: W^{s,p}(\\Omega, \\mathds{R}) \\rarr [0,\\infty)$ attains a minimizer.\n\\end{lemma}\n\\begin{proof}\n\tLet $u \\in \\Wsp[\\mathds{R}]$. Then by Lemma \\ref{lem:2} we have that $w \\vcentcolon= \\Phi(u) \\in \\Wsp[\\sphere]$.\n\tAs arguing as in the proof of Lemma \\ref{lem: liftedRegularizer} we see that $\\FT<v^\\delta><\\alpha>[\\dS](u) < \\infty$. \\\\\n\tSince we assume that \\autoref{as:Setting} is satisfied we get that $\\F<v^\\delta><\\alpha>[\\dS](w)$ attains a minimizer $w^* \\in \\Wsp[\\sphere]$.\n\tIt follows from \\autoref{lem: lifting} that there exists a function $u^* \\in W^{s,p}(\\Omega, \\mathds{R})$ that can be lifted to $w^*$, i.e. $w^* = \\Phi(u^*)$.\n\tThen $u^*$ is a minimizer of \\eqref{eq: functionalAlternative} \n\tby definition of $\\FT$ and $\\Phi$.\n\\end{proof}\t\n\n\n\n\n\\subsection{Numerical minimization}\nIn our concrete examples we will consider two different operators $\\op$. \nFor numerical minimization we consider the functional from \\autoref{eq: functionalAlternative} \nin a discretized setting. For this purpose we approximate the functions $u \\in W^{s, p}(\\Omega,\\mathds{R})$, $0<s<1,1<p<\\infty$ by quadratic B-Spline \nfunctions and optimize with respect to the coefficients. \nWe remark that this approximation \nis continuous and thus that sharp edges correspond to very steep slopes. \\\\ \nThe noisy data $u^\\delta$ is obtained by adding Gaussian white noise with variance $\\sigma^2$ to the approximation \nor the discretized approximation of $u$. \n\nWe apply a simple Gradient Descent scheme with fixed step length implemented in $\\mathrm{MATLAB}$.\n\n\\subsection{Denoising of $\\sphere$-valued functions - The InSAR problem} \\label{ss: denoising}\nIn this case the operator $\\op: \\Wsp[\\sphere] \\rarr \\Lp[\\sphere]$ is the inclusion operator. It is norm-coercive in the sense of \\autoref{eq:coercive_F} and hence \\autoref{as:Setting} is fulfilled. \nFor \n$\\emptyset \\neq \\Omega \\subset \\mathds{R}$ or $\\mathds{R}^2$ a bounded and simply connected open set, $1 < p < \\infty$ and $s \\in (0,1)$ such that additionally $sp < 1$ or $sp \\geq 2$ if $N=2$ we can apply \\autoref{lem: liftedFunctional} which ensures that the lifted functional \n$\\FT<u^\\delta><\\alpha>[\\dS]: \\Wsp[\\mathds{R}] \\rarr [0,\\infty)$ \nattains a minimizer $u \\in W^{s, p}(\\Omega,\\mathds{R})$. \n\nIn the examples we will just consider the continuous approximation again denoted by $u$.\n\n\n\\subsection*{One dimensional test case}\nLet $\\Omega = (0,1)$ and consider the signal \n$u:\\Omega \\rarr [0,2\\pi)$ representing the angle of a cyclic signal. \\\\\nFor the discrete approximation shown in \\autoref{sfig:signal1-a} the domain $\\Omega$ is sampled equally at 100 points.    \n$u$ is affected by an additive white Gaussian noise with $\\sigma = 0.1$ to obtain the noisy signal which is colored in blue in \\autoref{sfig:signal1-a}.\n \nIn this experiment we show the influence of the parameters $s$ and $p$.\nIn all cases the choice of the regularization parameter $\\alpha$ is 0.19 and $\\varepsilon = 0.01$.\\\\\nThe red signal in \\autoref{sfig:signal1-b} is obtained by choosing \n$s = 0.1$ and $p = 1.1$. \nWe see that the periodicity of the signal is handled correctly and that there is nearly no staircasing.\nIn \\autoref{sfig:signal1-c} the parameter $s$ is changed from $0.1$ to $0.6$. The value of the parameter $p$ stays fixed. \nIncreasing of $s$ leads the signal to be more smooth. We can observe an even stronger similar effect when increasing $p$ \n(here from $1.1$ to $2$) and letting $s$ fixed, see \\autoref{sfig:signal1-d}. This fits the expectation since $s$ only \nappears once in the denominator of the regularizer. At a jump increasing of $s$ leads thus to an increasing of the \nregularization term. The parameter $p$ appears twice in the regularizer. Huge jumps are hence weighted even more.\n\n\\begin{figure}[!h]\n\t\\centering\n\t\\begin{subfigure}[h]{0.35\\linewidth}\n\t\t\\includegraphics[width=1\\linewidth]{Images\/Signal_origAndNoisy_s01_reg019_p11_alpha001_eps001.jpg}\n\t\t\\caption{Original and noisy data}\n\t\t\\label{sfig:signal1-a}\n\t\\end{subfigure}\n    \\begin{subfigure}[h]{0.35\\linewidth}\n    \t\\includegraphics[width=1\\linewidth]{Images\/Signal_denoised_s01_reg019_p11_alpha001_eps001.jpg}\n    \t\\caption{Denoised data}\n    \t\\label{sfig:signal1-b}\n    \\end{subfigure}\t\n\n\t\\begin{subfigure}[h]{0.35\\linewidth}\n\t\t\\includegraphics[width=1\\linewidth]{Images\/Signal_denoised_s06_reg019_p11_alpha001_eps001.jpg}\n\t\t\\caption{Increasing of $s$}\n\t\t\\label{sfig:signal1-c}\n\t\\end{subfigure}\n    \\begin{subfigure}[h]{0.35\\linewidth}\n    \t\\includegraphics[width=1\\linewidth]{Images\/Signal_denoised_s01_reg019_p2_alpha001_eps001.jpg}\n    \t\\caption{Increasing of $p$}\n    \t\\label{sfig:signal1-d}\n    \\end{subfigure}\n\t\n\\caption{Function on $\\sphere$ represented in $[0,2\\pi)$: Left to right, top to bottom: Original data (black) and noisy data (blue) \nwith 100 data points. Denoised data (red) where we chose $s=0.1, p=1.1, \\alpha = 0.19$. Denoised data with \n$s=0.6, p=1.1, \\alpha = 0.19$ resp. $s=0.1, p=2, \\alpha=0.19$. }\n\\label{fig:signal1}\n\\end{figure}\n\nIn \\autoref{sfig:signal2-a} we considered a simple signal with a single huge jump. Again it is described by the angular value. \nWe proceeded as above to obtain the approximated discrete original data (black) and noisy signal with $\\sigma = 0.1$ (blue). We \nchose again $\\varepsilon = 0.01$. \\\\ \nAs we have seen above increasing of $s$ leads to a more smooth signal. \nThis effect can be compensated by choosing a rather small value of $p$, i.e. $p \\approx 1$. In \\autoref{sfig:signal2-b} the value of $s$ is $0.9$. We see that it is still possible to reconstruct jumps by choosing e.g. $p=1.01$. \\\\\nMoreover, we have seen that increasing of $p$ leads to an even more smooth signal. In \\autoref{sfig:signal2-c} we choose a quite large value of $p$,  \n$p=2$ and a \nrather small value of $s$, $s = 0.001$. Even for this very simple signal is was not possible to get sharp edges. This is due to the fact that the parameter $p$ (but not $s$) additionally weights the height of jumps in the regularizing term. \n\n\n\\begin{figure}[!h]\n\t\\centering\n\t\\begin{subfigure}[h]{0.3\\linewidth}\n\t\\includegraphics[width=1\\linewidth]{Images\/Signal_origAndNoisy_s01_reg045_p101_alpha001.jpg}\n\t\\caption{Original and noisy data}\n\t\\label{sfig:signal2-a}\n\t\\end{subfigure}\n\t\\begin{subfigure}[h]{0.3\\linewidth}\n\t\t\\includegraphics[width=1\\linewidth]{Images\/Signal_denoised_s09_reg003_p101_alpha001.jpg}\n\t\t\\caption{$s = 0.9, \\ p = 1.01$}\n\t\t\\label{sfig:signal2-b}\n\t\\end{subfigure}\n    \\begin{subfigure}[h]{0.3\\linewidth}\n    \t\\includegraphics[width=1\\linewidth]{Images\/Signal_denoised_s0001_reg09_p2_alpha001.jpg}\n    \t\\caption{$s = 0.001, \\ p = 2$}\n    \t\\label{sfig:signal2-c}\n    \\end{subfigure}\n    \\caption{Left to right: Original data (black) and noisy data (blue) sampled at 100 data points. Denoised data (red) where we chose $s=0.9, p=1.01, \\alpha = 0.03$. Denoised data with $s=0.001, p=2, \\alpha = 0.9$.}\n\t\\label{fig:signal2}\n\\end{figure} \n\n\n\\subsection*{Denoising of a $\\sphere$-Valued Image}\nOur next example concerned a two-dimensional $\\sphere$-valued image represented by the corresponding angular values.\nWe remark that in this case where $N=2$ the existence of such a representation is always guaranteed in the cases where \n$sp < 1$ or $sp \\geq 2$, see \\autoref{lem: lifting}. \n\nThe domain $\\Omega$ is sampled into $60 \\times 60$ data points and can be considered as discrete grid, \n$\\{1, \\dots,60\\} \\times \\{1, \\dots,60\\} $. \nThe B-Spline approximation evaluated at that grid is given by\n\\begin{equation*}\nu(i,j) = u(i,0) \\vcentcolon= 4\\pi \\frac{i}{60} \\bmod 2\\pi, \\quad i,j \\in \\{1, \\dots,60\\}.\n\\end{equation*}\n\n\\begin{figure} \n\t\\centering\n\t{\\includegraphics[width=6cm]{Images\/rainbow_3dPlot.jpg}}\n\t\\caption{The function $u$ evaluated on the discrete grid.}\n\t\\label{fig: Rainbow_function}\n\\end{figure}\nThe function $u$ is shown in \\autoref{fig: Rainbow_function}. We used the $\\mathrm{hsv}$ colormap \nprovided in $\\mathrm{MATLAB}$ transferred to the interval $[0, 2\\pi]$. \n\nThis experiment shows the difference of our regularizer respecting the periodicity of the data in contrast to the classical \nTotal Variation regularizer. The classical TV-minimization is solved using a fixed point iteration (\\cite{LoeMag}); for the method see also \\cite{VogOma96}.\n\nIn \\autoref{sfig:rainbow-a} the function $u$ can be seen from the top, i.e. the axes correspond to the \n$i$ resp. $j$ axis in \\autoref{fig: Rainbow_function}.\nThe noisy data is obtained by adding white Gaussian noise with $\\sigma = \\sqrt{0.001}$ using the built-in function \n$\\mathtt{imnoise}$ in $\\mathrm{MATLAB}$. It is shown in \\autoref{sfig:rainbow-b}.\nWe choose as parameters $s=0.9, \\ p=1.1, \\ \\alpha = 1,$ and $\\varepsilon = 0.01$.  \nWe observe significant noise reduction in both cases. However, only in \\autoref{sfig:rainbow-d} the color transitions are handled \ncorrectly. This is due to the fact, that our regularizer respects the periodicity, i.e. for the functional there is no jump in \n\\autoref{fig: Rainbow_function} since 0 and $2\\pi$ are identified. Using the classical TV regularizer the values 0 and $2\\pi$ are not \nidentified and have a distance of $2\\pi$. Hence, in the TV-denoised image there is a sharp edge in the middle of the image, see \n\\autoref{sfig:rainbow-c}. \\\\\n\n\n\\begin{figure}[!h]\n\t\\centering\n\t\\begin{subfigure}[h]{0.35\\linewidth}\n\t\t\\includegraphics[width=1\\linewidth]{Images\/Image_Original.jpg}\n\t\t\\caption{Original data}\n\t\t\\label{sfig:rainbow-a}\n\t\\end{subfigure}\n\t\\begin{subfigure}[h]{0.35\\linewidth}\n\t\t\\includegraphics[width=1\\linewidth]{Images\/rainbow_noisy_s09_p11_reg1_alpha02_eps001_400steps.jpg}\n\t\t\\caption{Noisy data}\n\t\t\\label{sfig:rainbow-b}\n\t\\end{subfigure}\t\n\t\n\t\\begin{subfigure}[h]{0.35\\linewidth}\n\t\t\\includegraphics[width=1\\linewidth]{Images\/rainbow_TVdenoised_s09_p11_reg1_alpha02_eps001_400steps.jpg}\n\t\t\\caption{TV-denoised data}\n\t\t\\label{sfig:rainbow-c}\n\t\\end{subfigure}\n\t\\begin{subfigure}[h]{0.35\\linewidth}\n\t\t\\includegraphics[width=1\\linewidth]{Images\/rainbow_denoised_s09_p11_reg1_alpha02_eps001_400steps.jpg}\n\t\t\\caption{Denoised data}\n\t\t\\label{sfig:rainbow-d}\n\t\\end{subfigure}\n\t\n\t\\caption{Left to right, top to bottom: Original and noisy data of an $60 \\times 60$ image. TV-denoised data using a fixed point iteration method. Denoised data where we chose $s=0.9, p=1.1, \\alpha = 1$, 400 steps. }\n\t\\label{fig:rainbow}\n\\end{figure}\n\n\n\\subsection*{Hue Denoising}\n\nThe $\\mathrm{HSV}$ color space is shorthand for Hue, Saturation, Value (of brightness). \nThe hue value of a color image is $\\sphere$-valued, while saturation and value of brightness are real-valued. \nRepresenting colors in this space better match the human perception than representing colors in \nthe RGB space. \n\nIn \\autoref{sfig:fruits-a} we see a part of size $70 \\times 70$ of the RGB image ``fruits''  \n(\\url{https:\/\/homepages.cae.wisc.edu\/~ece533\/images\/}).\n     \nThe corresponding hue data is shown in \\autoref{sfig:fruits-b}, where we used again the colormap hsv, cf. \\autoref{fig: Rainbow_function}. Each pixel-value lies, after transformation, in the interval $[0, 2\\pi)$ and represents the angular value. Gaussian white noise with $\\sigma = \\sqrt{0.001}$ is added to obtain a noisy image, see \\autoref{sfig:fruits-c}.\\\\\nTo obtain the denoised image \\autoref{sfig:fruits-d} we again used the same fixed point iteration, cf. \\cite{LoeMag}, as before. \n\nWe see that the denoised image suffers from artifacts due to the non-consideration of periodicity. The pixel-values in the middle of the apple (the red object in the original image) are close to $2\\pi$ while those close to the border are nearly 0, meaning they have a distance of around $2\\pi$. \\\\\nWe use this TV-denoised image as starting image to perform the minimization of our energy functional. As parameters we choose \n$s = 0.49, \\ p = 2, \\ \\alpha = 2, \\ \\varepsilon = 0.006$. \n\nSince the cyclic structure is respected the disturbing artifacts in image \\autoref{sfig:fruits-d} are removed correctly. The edges are smoothed due to the high value of $p$, see \\autoref{sfig:fruits-e}.\n\n\n\n\\begin{figure}[!h]\n\t\\centering\n\t\\begin{subfigure}[h]{0.3\\linewidth}\n\t\t\\includegraphics[width=1\\linewidth]{Images\/Fruits_OrigRGB.jpg}\n\t\t\\caption{Original RGB image \\newline \\qquad \\newline}\n\t\t\\label{sfig:fruits-a}\n\t\\end{subfigure}\n    \\hspace{1em}\n\t\\begin{subfigure}[h]{0.3\\linewidth}\n\t\t\\includegraphics[width=1\\linewidth]{Images\/Fruits_Orig.jpg}\n\t\t\\caption{Hue component represented in color, which represent function values on $\\sphere$}\n\t\t\\label{sfig:fruits-b}\n\t\\end{subfigure}\t\n    \\hspace{1em}\t\n\t\\begin{subfigure}[h]{0.3\\linewidth}\n\t\t\\includegraphics[width=1\\linewidth]{Images\/Fruits_Noisy_s049.jpg}\n\t\t\\caption{Noisy hue value - again representing function values on $\\sphere$ \\newline}\n\t\t\\label{sfig:fruits-c}\n\t\\end{subfigure}\n\n\t\\begin{subfigure}[h]{0.3\\linewidth}\n\t\t\\includegraphics[width=1\\linewidth]{Images\/Fruits_TVstart_s049.jpg}\n\t\t\\caption{TV-denoised data}\n\t\t\\label{sfig:fruits-d}\n\t\\end{subfigure}\n    \\hspace{1em}\n    \\begin{subfigure}[h]{0.3\\linewidth}\n    \t\\includegraphics[width=1\\linewidth]{Images\/Fruits_s49_reg2_p2_500steps_tau02_eps_0006_TVstart.jpg}\n    \t\\caption{Denoised data}\n    \t\\label{sfig:fruits-e}\n    \\end{subfigure}\n\t\n\t\\caption{Left to right, top to bottom: Original RGB image and its Hue component. Noisy Hue data with $\\sigma^2 = 0.001$. TV minimization is done using an iterative approach. It is serving as starting point for the GD minimization. Denoised data with $s=0.49, p=2, \\alpha = 2$, 500 steps. }\n\t\\label{fig:fruits}\n\\end{figure}\n\n\\subsection{$\\sphere$-Valued Image Inpainting}\nIn this case the operator $\\op: \\Wsp[\\sphere] \\rarr \\Lp[\\sphere]$ is the inpainting operator, i.e. \n  \\begin{equation*}\n    \\op(w) = \\chi_{\\Omega \\backslash D} (w),\n  \\end{equation*}  \nwhere $D \\subseteq \\Omega$ is the area to be inpainted. \n\nWe consider the functional \n\\begin{equation*}\n\\F<v^\\delta><\\alpha>[\\dS](w) \\vcentcolon= \\int\\limits_{\\Omega \\setminus D} \\dS^p(w(x), v^\\delta(x)) \\,\\mathrm{d}x + \n\\alpha \\int\\limits_{\\Omega\\times \\Omega} \\frac{\\dS^p(w(x), w(y))}{\\norm{x-y}_{\\mathds{R}^2}^{2+ps}} \\rho_{\\varepsilon}(x-y) \\,\\mathrm{d}(x,y),\n\\end{equation*}\non $\\Wsp[\\sphere]$.\n\nAccording to \\autoref{ex:in} the functional $\\F$ is coercive and \n\\autoref{as:Setting} is satisfied.\nFor $\\emptyset \\neq \\Omega \\subset \\mathds{R}$ or $\\mathds{R}^2$ a bounded and simply connected open set, $1 < p < \\infty$ and $s \\in (0,1)$ such that additionally $sp < 1$ or $sp \\geq 2$ if $N=2$ \\autoref{lem: liftedFunctional} applies which ensures that there exists a minimizer $u \\in W^{s, p}(\\Omega,\\mathds{R})$ of the lifted functional \n$\\FT<u^\\delta><\\alpha>[\\dS]: \\Wsp[\\mathds{R}] \\rarr [0,\\infty)$ \n$u \\in W^{s, p}(\\Omega,\\mathds{R})$\n\n\\subsection*{Inpainting of a $\\sphere$-Valued Image}\n\nAs a first inpainting test-example we consider two $\\sphere$-valued images of size $28 \\times 28$, \nsee \\autoref{fig:blocks}, represented by its angular values. \nIn both cases the ground truth can be seen in \\autoref{sfig:blocks-a} and \\autoref{sfig:blocks2-a}. \nWe added Gaussian white noise with $\\sigma = \\sqrt{0.001}$ using the $\\mathrm{MATLAB}$ build-in function \n$\\mathtt{imnoise}$. The noisy images can be seen in \\autoref{sfig:blocks-b} and \\autoref{sfig:blocks2-b}. \nThe region $D$ consists of the nine red squares in \\autoref{sfig:blocks-c} and \\autoref{sfig:blocks2-c}.\n\nThe reconstructed data are shown in \\autoref{sfig:blocks-d} and \\autoref{sfig:blocks2-d}. \\\\\nFor the two-colored image we used as parameters $\\alpha = s = 0.3$, $p = 1.01$ and $\\varepsilon = 0.05$.\nWe see that the reconstructed edge appears sharp. \nThe unknown squares, which are completely surrounded by one color are inpainted perfectly.\nThe blue and green color changed slightly.\n\nAs parameters for the three-colored image we used $\\alpha = s = 0.4$, $p=1.01$ and $\\varepsilon = 0.05$. \nHere again the unknown regions lying entirely in one color are inpainted perfectly. The edges are preserved. \nJust the corner in the middle of the image is slightly smoothed. \\\\\nIn \\autoref{sfig:blocks-e} and \\autoref{sfig:blocks2-e} the TV-reconstructed data  \nis shown. The underlying algorithm (\\cite{Get}) uses the split Bregman method (see \\cite{GolSta09}).\n\nIn \\autoref{sfig:blocks-e} the edge is not completely sharp. There are some lighter parts on the blue side. \nThis can be caused by the fact that the unknown domain in this area is not exactly symmetric with respect to the edge. \nThis is also the case in \\autoref{sfig:blocks2-e} where we observe the same effect. Unknown squares lying entirely \nin one color are perfectly inpainted.  \n\n\n\\begin{figure}[!h]\n\t\\centering\n\t\\begin{subfigure}[h]{0.3\\linewidth}\n\t\t\\includegraphics[width=1\\linewidth]{Images\/2blocks_Orig.jpg}\n\t\t\\caption{Original image}\n\t\t\\label{sfig:blocks-a}\n\t\\end{subfigure}\n\t\\begin{subfigure}[h]{0.3\\linewidth}\n\t\t\\includegraphics[width=1\\linewidth]{Images\/2blocks_Noise.jpg}\n\t\t\\caption{Noisy image}\n\t\t\\label{sfig:blocks-b}\n\t\\end{subfigure}\t\t\n\t\\begin{subfigure}[h]{0.3\\linewidth}\n\t\t\\includegraphics[width=1\\linewidth]{Images\/2blocks_NoiseMask.jpg}\n\t\t\\caption{Noisy masked image}\n\t\t\\label{sfig:blocks-c}\n\t\\end{subfigure}\n\t\n\t\\begin{subfigure}[h]{0.3\\linewidth}\n\t\t\\includegraphics[width=1\\linewidth]{Images\/2blocks_Denoised_s0001_regParam07_p101_tau03_eps005_1200steps.jpg}\n\t\t\\caption{Reconstructed image}\n\t\t\\label{sfig:blocks-d}\n\t\\end{subfigure}\n\t\\begin{subfigure}[h]{0.3\\linewidth}\n\t\t\\includegraphics[width=1\\linewidth]{Images\/2blocks_TV_param8.jpg}\n\t\t\\caption{TV-reconstructed image}\n\t\t\\label{sfig:blocks-e}\n\t\\end{subfigure}\n\n    \\begin{subfigure}[h]{0.3\\linewidth}\n    \t\\includegraphics[width=1\\linewidth]{Images\/3blocks_Orig.jpg}\n    \t\\caption{Original image}\n    \t\\label{sfig:blocks2-a}\n    \\end{subfigure}\n    \\begin{subfigure}[h]{0.3\\linewidth}\n    \t\\includegraphics[width=1\\linewidth]{Images\/3blocks_Noise.jpg}\n    \t\\caption{Noisy image}\n    \t\\label{sfig:blocks2-b}\n    \\end{subfigure}\t\t\n    \\begin{subfigure}[h]{0.3\\linewidth}\n    \t\\includegraphics[width=1\\linewidth]{Images\/3blocks_NoiseMask.jpg}\n    \t\\caption{Noisy masked image}\n    \t\\label{sfig:blocks2-c}\n    \\end{subfigure}\n    \n    \\begin{subfigure}[h]{0.3\\linewidth}\n    \t\\includegraphics[width=1\\linewidth]{Images\/3blocks_Denoised_s001_regParam08_p1001_tau04_eps005_1200steps.jpg}\n    \t\\caption{Reconstructed image}\n    \t\\label{sfig:blocks2-d}\n    \\end{subfigure}\n    \\begin{subfigure}[h]{0.3\\linewidth}\n    \t\\includegraphics[width=1\\linewidth]{Images\/3blocks_TV_param8.jpg}\n    \t\\caption{TV-reconstructed image}\n    \t\\label{sfig:blocks2-e}\n    \\end{subfigure}\n\t\n\t\\caption{Left to right. Top to bottom: Original image and the noisy data with $\\sigma^2 = 0.001$. Noisy image with masking filter and denoised data with $s=0.3, p=1.01, \\alpha = 0.3$, 6000 steps. TV denoised data.\\\\\n     Original image and the noisy data with $\\sigma^2 = 0.001$. Noisy image with masking filter and denoised data with $s=0.4, p=1.01, \\alpha = 0.4$, 10000 steps. TV denoised image.}\n\t\\label{fig:blocks}\n\\end{figure}\n\n\\subsection*{Hue Inpainting}\n\nAs a last example we consider again the Hue-component of the image ``fruits'', see \\autoref{sfig:fruits2-a}. \nThe unknown region $D$ is the string $\\mathit{01.01}$ which is shown in \\autoref{sfig:fruits2-b}. \nAs parameters we choose $p=1.1$, $s=0.1$, $\\alpha= 2$ and $\\varepsilon = 0.006$. We get the reconstructed image shown in \n\\autoref{sfig:fruits2-c}. The edges are preserved and the unknown area is restored quite well. This can be also observed in the \nTV reconstructed image, \\autoref{sfig:fruits2-d}, using again the split Bregman method as before, cf. \\cite{Get}.\n\n\\begin{figure}[!h]\n\t\\centering\n\t\\begin{subfigure}[h]{0.35\\linewidth}\n\t\t\\includegraphics[width=1\\linewidth]{Images\/Fruits_Orig.jpg}\n\t\t\\caption{Hue component}\n\t\t\\label{sfig:fruits2-a}\n\t\\end{subfigure}\t\t\n\t\\begin{subfigure}[h]{0.35\\linewidth}\n\t\t\\includegraphics[width=1\\linewidth]{Images\/Inpainting_fruits_date_Orig.jpg}\n\t\t\\caption{Image with masked region}\n\t\t\\label{sfig:fruits2-b}\n\t\\end{subfigure}\n\t\n\t\\begin{subfigure}[h]{0.35\\linewidth}\n\t\t\\includegraphics[width=1\\linewidth]{Images\/fruits_inpainting_1234_p11_regParam2_s01_tau02_eps0006_2000steps.jpg}\n\t\t\\caption{TV-reconstructed image}\n\t\t\\label{sfig:fruits2-c}\n\t\\end{subfigure}\n\t\\begin{subfigure}[h]{0.35\\linewidth}\n\t\t\\includegraphics[width=1\\linewidth]{Images\/fruits_TVinpainting_lambda100.jpg}\n\t\t\\caption{Reconstructed image}\n\t\t\\label{sfig:fruits2-d}\n\t\\end{subfigure}\n\t\n\t\\caption{Left to right, top to bottom: Original image and image with masked region. Reconstructed image with parameters $p=1.1, \\ \n\ts=0.1, \\ \\alpha= 2$ and $\\varepsilon = 0.006$, 2000 steps. TV-reconstructed image.}\n\t\\label{fig:fruits2}\n\\end{figure}\n\n\\subsection{Conclusion}\nIn this paper we developed a functional for regularization of functions with values in a set of vectors. The regularization functional  \nis a derivative-free, nonlocal term, which is based on a characterization of Sobolev spaces of \n\\emph{intensity data} derived by Bourgain, Br\u00e9zis, Mironescu \\& D\u00e1vila. Our objective has been \nto extend their double integral functionals in a natural way to functions with values in a set of vectors, in particular functions with \nvalues on an embedded manifold. These new integral representations are used for regularization on a subset \nof the (fractional) Sobolev space $W^{s,p}(\\Omega, \\mathds{R}^M)$ and the space $BV(\\Omega, \\mathds{R}^M)$, respectively.  \nWe presented numerical results for denoising of artificial InSAR data as well as an example of inpainting. \nMoreover, several conjectures are at hand on relations between double metric integral regularization \nfunctionals and single integral representations.\n\n\\subsection*{Acknowledgements}\nWe thank Peter Elbau for very helpful discussions and comments. MH and OS acknowledge support from \nthe Austrian Science Fund (FWF) within the national research network Geometry and Simulation, project S11704 \n(Variational Methods for Imaging on Manifolds). Moreover, OS is supported by the Austrian Science Fund (FWF), \nwith SFB F68, project F6807-N36 (Tomography with Uncertainties) and I3661-N27 (Novel Error Measures and Source \nConditions of Regularization Methods for Inverse Problems).\n\n\n\n\\section*{References}\n\\renewcommand{\\i}{\\ii}\n\\printbibliography[heading=none]\n\n\n\n\n\\end{document}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\n\\section{Overview}\nAbout DCDP:\n\\begin{enumerate}\n    \\item DCDP works well with a relatively large parameter $\\gamma = C_{\\gamma}K { \\mathcal B_n^{-1}   \\Delta } \\kappa^2$.\n    \\item Local refinement works well both in theory and in practice, though it is not quite ``novel\". Is it good to go? Or should we think about something else?\n    \\item Next step: consider generalizing to nonlinear models like logistic regression.\n    \\item Piece-wise linear signals?\n\\end{enumerate}\n\nAbout CPD for logistic regression and the Bradley-Terry model:\n\\begin{enumerate}\n    \\item Given that the WBS based on loglikelihood performs well in practice, should we try proving some theory for it?\n\\end{enumerate}\n\n\nimprovement in the computation\n\ncompare with others\n\n\\section{Introduction}\n\\label{sec:introduction}\nChange point analysis is a well-established topic in statistics that is concerned with identifying abrupt changes in the data, typically observed as a time series, that are due to structural changes in the underlying distribution. Initially introduced in the 1940s \\citep{wald1945, page1954}, change point analysis has been the subject of a rich statistical literature and has produced a host of well-established methods for statistical inference. Despite their popularity, most existing change point methods available to practitioners are ill-suited or computationally too costly to handle high-dimensional or complex data.  In this paper, we develop a general and flexible framework for high-dimensional change point analysis that enjoys very favorable statistical and computational properties.\n\nWe adopt a standard offline change point analysis set-up, whereby we observe a sequence $\\{ \\bm Z _i\\}_{i\\in [n]}$ of independent data points, where $[n]:=\\{1,\\ldots, n\\}$. We assume that each $ \\bm Z _i$ follows a high-dimensional parametric distribution $\\mathbb{P}_{ { \\bm \\theta } ^*_i}$ specified by an unknown parameter $ { \\bm \\theta } ^*_i$, and that sequence of parameters $\\{ { \\bm \\theta } _i\\}_{i \\in [n]}$ is piece-wise constant over time. For example, in the mean change point model (see ~\\Cref{sec: result mean} below), $ { \\mathbb{E} } ( \\bm Z _i)   =  { \\bm \\theta } ^*_i   \\in \\mathbb R^p$, where $ { \\bm \\theta } ^*_i$ is a vector in $\\mathbb R^p$. In the regression change point model (see ~\\Cref{sec: result regression}),   $ \\bm Z _i = ( \\bm Z _i, y_i) \\in \\mathbb R^p\\times \\mathbb R$ satisfying  $ { \\mathbb{E} } (y_i| \\bm Z _i) =     \\bm Z _i^\\top  { \\bm \\theta }  ^*_i $  where $ { \\bm \\theta } ^*_i$  is a vector of regression parameters.\n\nWe postulate that there exists an unknown sub-sequence \nof {\\it change points} $ 1=\\eta_0 <  \\eta_1  < \\eta_2 < \\ldots <\\eta_K < \\eta _{K+1}= n+1$ such that $ { \\bm \\theta } ^*_{i}\\neq  { \\bm \\theta } ^*_{i - 1}$ if and only if $i \\in \\{\\eta_{k}\\}_{k\\in [K]}$. For each $k\\in [K]=\\{ 1,\\ldots,K\\}$, define the local spacing parameter  and local jump size parameter as\n\\begin{equation}\n    \\Delta_k = \\eta_{k} -\\eta_{k-1} \\quad  \\text{and} \\quad   \\kappa_k \n\t  : =   \\| { \\bm \\theta } ^*_{\\eta_{k}} -  { \\bm \\theta } ^*_{\\eta_k - 1}\\| \n\\end{equation}\nrespectively, where $\\|\\cdot \\|$ is some appropriate  norm that is problem specific.  Throughout the paper, we will allow the parameters of the data generating distributions, the spacing and jump sizes to change with $n$, though we will require $K$ to be bounded.\nOur goal is to estimate the number and locations of the change points sequence $\\{ \\eta_k\\}_{k\\in[K]}$. We will deem any estimator $\\{ \\widehat \\eta_k\\}_{k\\in[ \\widehat{K}]}$ of the change point sequence {\\it consistent} if, with probability tending to $1$ as $n\\rightarrow \\infty$, \n\\begin{equation} \n\\label{eq: consistency general}\n     \\widehat{K} = K \\quad \\text{and} \\quad   \\max_{k\\in [K]}|\\widehat{\\eta}_k - \\eta_k|   = o(\\Delta_{\\min}),\n\\end{equation}\nwhere $\\Delta_{\\min} = \\min_{ k\\in[K] } \\Delta_k$.\n\nRecent years have witnessed significant advances in the fields of high-dimensional change point analysis,  both in terms of methodological developments and theoretical advances. Most change point estimators for high-dimensional problems can be divided into two main categories: those based on variants of the binary segmentation algorithm and those relying on the penalized likelihood. See below for a brief summary of the relevant literature.\n\nIn this paper, we aim to develop a comprehensive framework for estimating change points in high-dimensional models using an $\\ell_0$-penalized likelihood approach. \nWhile $\\ell_0$-based change point algorithms have demonstrated excellent -- in fact, often optimal -- localization rates, their computational costs remain a significant challenge. Indeed, optimizing the $\\ell_0$-penalized objective function using a dynamic programming (DP) approach requires quadratic time complexity \\citep{friedrich2008complexitypenalizedmestimation} and, therefore, is often impractical. \n\nTo overcome this computational bottleneck, we propose a novel class of algorithms for high-dimensional multiple change point estimation problems called \\textit{divide and conquer dynamic programming} (DCDP) -  see \\Cref{algorithm:DCDP}.\nThe DCDP framework is very versatile and can be applied to a wide range of high-dimensional change point problems. At the same time, it yields a substantial reduction in computational complexity compared to the vanilla DP. In particular, when the minimal spacing $\\Delta_{\\min}$ between consecutive change points is of order $n$, DCDP exhibits almost linear time complexity.\n\nMoreover, the DCDP algorithm retains a high degree of statistical accuracy. Indeed, we show that DCDP delivers minimax optimal localization error rates for the important problems of change point localization in sparse high-dimensional mean models, Gaussian graphical models and sparse linear regression problems. \nTo the best of our knowledge, DCDP is the first near-linear time procedure that can provide optimal statistical guarantees in these three different models. See \\Cref{remark:optimal mean} and \\Cref{remark:optimal regression} for more detailed discussions on optimality.\n \n\n    \n\n{\\bf Structure of the paper.}\nBelow we provide a selective review of the recent relevant literature on high-dimensional change point analysis. In \\Cref{sec:method}, we describe the DCDP framework. In \\Cref{section:main}, we provide detailed theoretical studies to demonstrate that DCDP    achieves minimax optimal localization errors in the three models.  In \\Cref{sec:experiment}, we conduct extensive numerical experiments on synthetic and real data to illustrate the superior numerical performance of DCDP compared to existing procedures.\n\n\n\n{\\bf Relevant litearture.} \nBinary Segmentation(BS) is a greedy iterative approach that breaks the multiple-change-point problem down into a sequence of single change-point sub-problems. Originally introduced by \\cite{Scott1974} to handle the case of one change point, the BS algorithm was later shown by \\cite{Venkatraman1992} to be effective also in change point problems involving multiple change points.   \nModern variants of the original BS algorithms include wild-binary segmentation of \\cite{Fryzlewicz2014} and the more computationally efficient  \\textit{Seeded Binary Segmentation} (SBS) algorithm of \\citep{kovacs2020seeded}.\nBS-inspired change point procedures have been designed for various change point problems, including high-dimensional mean models \\citep{eichinger2018mosum,wang_samworth2018}, graphical models \\cite{londschien2021change} and covariance models {\\citep{wang_yi_rinaldo_2021_cov} network models \\cite{wang2021network_cpd}, functional models \\cite{oscar2022nonpara} and many more. \n \n \n\nPenalized likelihood-based approaches are also popular in the change point literature. Broadly, these approaches \n segment the time series by maximizing a likelihood function with an appropriate penalty to avoid over-segmentation.\n \\cite{yao_au1989} \n showed that  $\\ell_0$-penalized  likelihood-based methods yield consistent estimators of change points.  \n\n Relaxing the $\\ell_0$-penalty to the  $\\ell_1$-penalty results in the  Fused Lasso algorithm, whose theoretical and computational properties have been analyzed by, e.g.,  \\citep{lin2017fused} for the mean setting and by \\citep{qian2016fused} for the linear regression setting. More recently, \\cite{unify_sinica2022} proposed a unified framework to analyze Fused-Lasso-based change point estimators in linear models.\n\n\n\nFew recent notable contributions in the literature have focused on designing \\textit{unified methodological frameworks} for offline change point analysis. \n\\cite{verzelen2020hd_mean} developed  a general approach based on local two-sample tests to detect changes in means, but their approach can only consistently estimate the number of change points and the localization accuracy of the estimators is unspecified. \n\\citep{changeforest2022} proposed\n a novel multivariate nonparametric multiple change point detection method based on the likelihood ratio tests.\n\\cite{unify_sinica2022} studied a general framework based on the Fused Lasso to deal with change points in mean and linear regression models,  but their detection boundary is sub-optimal and it is computationally demanding to numerically optimize the  Fused Lasso objective function for high-dimensional time series. Until now, a unified framework for offline change point localization with optimal statistical guarantees and low computational complexity is still missing in the literature.\n\n\n\n\n\n{\\bf Notation.} For   $n \\in \\mathbb Z^+$, denote $[n]:=\\{1,\\cdots, n\\}$. For a vector $ \\bm v \\in \\mathbb{R}^p$,   denote the $i$-th entry as $v_i$, and similarly, for a matrxi $ \\bm A \\in \\mathbb{R}^{m\\times n}$, we use $A_{ij}$ to denote its element at the $i$-th row and $j$-th column. We use $\\mathbb{S}^p_+$ to denote the cone of positive semidefinite matrices in $\\mathbb{R}^{p\\times p}$. For two real numbers $a,b$, we denote $a\\vee b:=\\max\\{a,b\\}$.\n\n$\\|\\cdot\\|_1,\\|\\cdot\\|_2$ refer to the $\\ell_1$ and $\\ell_2$ norm of vectors, i.e., $\\| \\bm v \\|_1 = \\sum_{i\\in [p]}|v_i|$ and $\\| \\bm v \\|_2 = (\\sum_{i\\in [p]}v_i^2)^{1\/2}$. For a square matrix $A\\in \\mathbb{R}^{n\\times n}$, we use $\\| \\bm A \\|_F$ to denote its Frobenius norm, ${\\rm Tr}( \\bm A ) = \\sum_{i\\in [n]}A_{ii}$ to denote its trace, and $| \\bm A |$ to denote its determinant. For a random variable $X\\in \\mathbb{R}$, we denote $\\|X\\|_{\\psi_2}$ as the subgaussian norm \\citep{vershynin2018high}: $\\|X\\|_{\\psi_2}:=\\inf \\{t>0: \\mathbb{E} \\psi_2(|X| \/ t) \\leq 1\\}$ where $\\psi_2(t) = e^{t^2} - 1$.\n\nFor asymptotics, we denote $x_n\\lesssim y_n$\nor $x_n = O(y_n)$ if $\\forall n$,\n$x_n \\leq c_1 y_n$ for some universal constant $c_1>0$. $a_n = o(b_n)$ means $a_n\/b_n\\rightarrow 0$ as $n\\rightarrow \\infty$, and $X_n = o_p(Y_n)$ if $X_n\/Y_n\\rightarrow 0$ in probability. We call a positive sequence $\\{a_n\\}_{n\\in \\mathbb{Z}^+}$ a diverging sequence if $a_n\\rightarrow \\infty$ as $n\\rightarrow \\infty$.\n\n\\section{Methodology}\n\\label{sec:method}\nIn this section, we introduce the DCDP framework and analyze its computational complexity.\nRecall that in our framework,   we observe the time series of independent data  $\\{ \\bm Z _i\\}_{i\\in [n]}$ sampled from the unknown sequence of distributions $ \\{\\mathbb P_{ { \\bm \\theta } _i^*}\\}_{i\\in [n]}$. For a time interval $ \\mclI \\subset [1, n]$  comprised of integers, let $\\mclF( { \\bm \\theta } ,\\mclI) $ denote the value of an appropriately chosen goodness-of-fit function of the subset $\\{ \\bm Z _i\\}_{i\\in  \\mclI }$, and for a fixed and common value of the parameter $ { \\bm \\theta } $. The choice of the goodness-of-fit function is problem dependent.\n\nNext, we use  $\\widehat{ { \\bm \\theta } }_{\\mclI}$ to denote the penalized or unpenalized maximum likelihood estimator of $ { \\bm \\theta } ^*$ within the interval $\\mclI$. Intuitively, \n  $  \\mathcal F( \\widehat  { \\bm \\theta } _\\mathcal I ,\\mathcal I) $\n   can be considered a  local statistic to test for the existence of one or more change points in $\\mathcal I$.\n  \n \n  \n \n  \n  \n\n\n\nDCDP is a two-stage algorithm that entails a divide step and an optional conquer step; see \\Cref{algorithm:DCDP} for details. In the divide step, described in \\Cref{algorithm:DP},  DCDP first computes preliminary estimates of the change point locations by running DDP, a dynamic programming algorithm over a uniformly-spaced grid of time points $\\{s_i=\\lfloor i\\cdot n\/(\\mclQ + 1)\\rfloor\\}_{i\\in [\\mclQ]}$. (DDP can also take as input a random collection of time points, but there are no computational or statistical advantages in randomizing this choice). In the subsequent conquer step, detailed in \\Cref{algo:local_refine_general},  the localization accuracy of the initial estimates is improved using a penalized local refinement (PLR) methodology.\n\n{\\bf Computational complexity of DCDP.} \nThe DCDP procedure achieves substantial computational gains by using a coarse, regular grid of time points $\\{s_i\\}_{i \\in \\mclQ} \\subset [n]$ during the divide step. Additionally, the PLR procedure in the conquer step is a local algorithm and is easily parallelizable. The number of grid points $\\mclQ$ to be given as input to DDP in the divide step should be chosen to be of smaller order than the length of the time course $n$, but large enough to identify the number and the approximate positions of the true change points, with high probability.\n\n\n\\begin{algorithm}[h]\n \\caption{Divide and Conquer Dynamic Programming. DCDP $(\\{ \\bm Z _i\\}_{i \\in [n]}, \\gamma, \\zeta ,  \\mclQ)$}\n\\label{algorithm:DCDP}\n\t\\textbf{Input:} Data $\\{  \\bm Z _i \\}_{i \\in [n ]  }$, tuning parameters $\\gamma,\\zeta,  \\mclQ  > 0$.\n\n  Set grid points  $s_i=\\lfloor\\frac{i\\cdot n}{\\mclQ+1}\\rfloor$ for $i\\in [\\mclQ]$.\n\\vskip 0.2cm \t  \n(Divide Step) Compute the proxy estimators  $\\{\\widehat{\\eta}_k\\}_{k\\in [\\widehat {K}]}$  using GDP $(\\{ \\bm Z _i\\}_{i \\in [n]},\\{s_i\\}_{i\\in [\\mclQ]},\\gamma )$ in \\Cref{algorithm:DP}.\n\\vskip 0.2cm \n(Conquer Step) Compute the final estimators $\\{\\widetilde {\\eta}_k\\}_{k\\in [\\widehat {K}]}$ using $ {\\rm PLR}(\\{\\widehat {\\eta}_k\\}_{k\\in [\\widehat{K}]},\\zeta )$ in \\Cref{algo:local_refine_general}. \n\\vskip 0.2cm \t\n\\textbf{Output:}  The   change point estimators $\\{\\widetilde {\\eta}_k\\}_{k\\in [\\widehat{K}]}$.\n\\end{algorithm} \n\n\n\\begin{algorithm}[h]\n\n\\caption{  Divided Dynamic Programming \n DDP $(\\{ \\bm Z _i\\}_{i \\in [n]},\\{s_i\\}_{i\\in [\\mclQ]},\\gamma )$: \n  the divide step. }\n\\label{algorithm:DP}\n\t\\textbf{Input:} Data $\\{  \\bm Z _i \\}_{i  \\in [n ]}$, ordered integers $\\{s_i\\}_{i\\in [ { \\mathcal Q} ]}\\subset (0,n)$, tuning parameter $\\gamma>0$.\n\n \t Set $ \\widehat {\\mathcal P}= \\emptyset$,  $\\mathfrak{ p} =\\underbrace{(-1,\\ldots, -1)}_{ n }  $,  ${\\bm B} =\\underbrace{( \\gamma, \\infty , \\ldots,  \\infty)}_{n } $.  \n   \\\\\n \t\\For {$r  $ in $\\{s_i\\}_{i\\in [ { \\mathcal Q} ]}$} {\n \t \\For {$l  $ in $\\{s_i\\}_{i\\in [ { \\mathcal Q} ]}$, $ l<r  $} {\n   \\vskip 0.2cm \t\n         $\\mclI \\leftarrow[l , r] \\cap \\{1, \\ldots, n\\}$;\n \t  \\\\\n \t  compute $\\widehat{ { \\bm \\theta } }_{\\mclI} $ and $\\mathcal F (\\widehat{ { \\bm \\theta } }_\\mclI, \\mclI) $ based on $\\{ \\bm Z _i\\}_{i \\in \\I}$  ;\n    \n \t $ b \\leftarrow B_{l}+ \\gamma + \\mathcal F (\\widehat{ { \\bm \\theta } }_\\mclI, \\mclI) ;$\n   \\vskip 0.2cm \t\n \t \\If {$b <  B_r $} {\n \t $B_r \\leftarrow  b $;\n   \\\\\n \t $\\mathfrak{p}_r  \\leftarrow l  $.\n \t } \n \t}\n \t}\n To compute $  \\mathfrak{p}  \\in \\mathbb N^n $, \tset \n $ k \\leftarrow  n  $.\n\n \t  \\While {$k >1$} {\n  $ h  \\leftarrow \\mathfrak{p}_ k  $;\n  \\\\\n   $\\widehat {\\mathcal P} \\leftarrow   \\widehat {\\mathcal P} \\  \\cup \\   \\{h\\}$;\n\\\\ \n  $ k  \\leftarrow h $.\n \t  }\n\t\\textbf{Output:}  The set of estimated change points \n\t$\\widehat {\\mathcal P}$. \n\\end{algorithm} \n\n\\begin{algorithm}[h] \n\\caption{Penalized Local Refinement  ${\\rm PLR}(\\{\\widehat{\\eta}_k\\}_{k\\in [\\widehat{K}]},\\zeta)$: the conquer step.  \n}\n\\label{algo:local_refine_general}\n\t\\textbf{Input:} Data $\\{  \\bm Z _i \\}_{i\\in[n]}$,  estimated change points $\\{\\widehat{\\eta}_k\\}_{k \\in[\\widehat{K}]}$ from \\Cref{algorithm:DP}, tuning parameter  $\\zeta > 0$.\n\n\tLet $(\\widehat{\\eta}_0, \\widehat{\\eta}_{\\widehat{K} + 1}) \\leftarrow (0, n)$.\n\t\n\t\\For{$k = 1, \\ldots, \\widehat{K}$} { \n\t$(s_k, e_k) \\leftarrow (\\frac{2}{3} \\widehat{\\eta}_{k-1} + \\frac{1}{3}\\widehat{\\eta}_{k} , \\ \\  \\frac{1}{3}\\widehat{\\eta}_{k}  \n + \\frac{2}{3}\\widehat{\\eta}_{k+1} )$\n\\begin{align*}\\bigg(\\check{\\eta}_k,\\widehat{ { \\bm \\theta } }^{(1)},&\\widehat{ { \\bm \\theta } }^{(2)}\\bigg) \\leftarrow \\argmin_{\\eta,  { \\bm \\theta } ^{(1)},  { \\bm \\theta } ^{(2)}} \\{\\mclF( { \\bm \\theta } ^{(1)},[s_k,\\eta)) + \n\\\\\n  &\\mclF( { \\bm \\theta } ^{(2)}, [\\eta,e_k)) + \\zeta R( { \\bm \\theta } ^{(1)},  { \\bm \\theta } ^{(2)},\\eta; s_k, e_k )\\}\n\\end{align*}\n\t$$\\widetilde{\\eta}_k \\leftarrow \\argmin_{\\substack{\\eta}} \\left\\{\\mclF(\\widehat  { \\bm \\theta } ^{(1)},[s_k,\\eta)) + \\mclF(\\widehat  { \\bm \\theta } ^{(2)}, [\\eta,e_k))  \\right\\}$$\n\t}\n\t\\textbf{Output:} The refined estimators $\\{\\widetilde{\\eta}_k\\}_{k \\in [\\widehat{K}]}$.\n\\end{algorithm} \n\nIn detail, let   $\\mclC_1(|\\mclI|, p)$ denote the time complexity of  computing the goodness-of-fit function  $ \\mathcal F (\\widehat{\\theta}_\\mclI, \\mclI)  $. Naively, the time  complexity of \\Cref{algorithm:DP} is $O(\\mclQ^2 \\cdot \\mclC_1(n, p))$, where $ { \\mathcal Q} $ is the size of the grid $\\{ s_i\\}_{i \\in [\\mclQ]}$ in \\Cref{algorithm:DP}. \nWith the memorization technique proposed in \\citep{yu2022temporal}, we show in \\Cref{lem: complexity divide step} that the complexity of the divide step can be reduced to $O(n\\mclQ\\cdot \\mclC_2(p))$, and in \\Cref{lem: complexity local refine} that the conquer step can be computed with time complexity $O(n\\cdot \\mclC_2(p))$, where $\\mclC_2(p)$ is independent of $n$. Furthermore, as shown later in \\Cref{section:main} and \\Cref{sec: fundamental lemma}, setting $\\mclQ = \\frac{4n}{\\Delta_{\\min}}\\log^2(n)$ ensures consistency of \\Cref{algorithm:DP}. Therefore, the complexity of DCDP is\n$$O\\left( \\frac{n^2}{\\Delta_{\\min}}\\cdot\\log^2(n)\\cdot \\mclC_2(p) \\right).$$\nWhen $\\Delta_{\\min}$ is of the same order as $n$, the complexity of DCDP becomes $O(n\\log^2(n)\\cdot \\mclC_2(p))$. To the best of our knowledge, DCDP is the first multiple-change-point detection algorithm that can provably achieve near-linear time complexity in the three models presented in \\Cref{section:main}.\n\n{\\bf Statistical accuracy.} As we will show below, though the DDP procedure in the divide step may already be sufficiently accurate to deliver consistent estimates as defined in \\eqref{eq: consistency general}, its error rate is suboptimal. Sharper, even optimal, localization errors can be achieved through the PLR algorithm in the conquer step (see \\Cref{algo:local_refine_general}). The PLR  procedure takes as input the preliminary change points estimates from the divide step\\footnote{More generally, it can be shown that the PLR procedure remains effective as long as it is given as input any change point estimates whose Hausdorff distance from the true change points is bounded by $\\Delta_{\\min}$. Thus, the preliminary estimates need not even be consistent.}, and provably reduces their localization errors --   for some of the models considered in the next section, down to the minimax optimal rates. The effectiveness of local refinement methods to enhance the precision of initial change point estimates has been well-documented in the recent literature on change point analysis \\citep{rinaldo2021cpd_reg_aistats, li2022cpd_btl}.\nIn \\Cref{algo:local_refine_general}, \nthe additional penalty function $R( { \\bm \\theta } ^{(1)},  { \\bm \\theta } ^{(2)},\\eta; s, e)$ in \\Cref{algo:local_refine_general} is introduced to ensure numerical stability of the parameter estimates in high dimensions and, possibly, to reproduce desired structural properties, such as sparsity. Its choice is, therefore, problem specific. For example, in the sparse mean and linear change point model in \\Cref{sec: result mean},  $ { \\bm \\theta } ^{(1)},  { \\bm \\theta } ^{(2)} \\in \\mathbb{R}^{p}$ and we consider the group lasso penalty function\n\\begin{equation}\n\\label{eq:group lasso penalty}\n   \n    R(\\cdot) =   \\sum_{i \\in [p]} \\sqrt{(\\eta - s)( { \\bm \\theta } ^{(1)} )_{i}^2 + (e - \\eta)( { \\bm \\theta }  ^{(2)})_{i} ^2}.\n\\end{equation}\n\\bnrmk[Penalization] In     \\Cref{algorithm:DP}, $\\gamma$  is a tuning parameter to control the number of selected change points and to avoid false discoveries. In \\Cref{algo:local_refine_general}, the tuning parameter $\\zeta$ is used to modulate the impact of the penalty function  $R$. We derive theoretically valid choices of tuning parameters in \\Cref{section:main}, and provide practical guidance on how to select them in a data-driven way in  \\Cref{sec:experiment}.\n \\enrmk\n\n\n\n\n\\section{Main results }\n\\label{section:main}\n\nIn this section, we investigate the theoretical performance of DCDP in three different high-dimensional change point problems. For each of the models examined, we first derive localization rates for the DDP algorithm in the divide step and find that, though they imply consistency, they are worse than the corresponding rates afforded by the computationally costly vanilla DP algorithm \\citep{wang2018univariate, rinaldo2021cpd_reg_aistats}. This suboptimal performance reflects the trade-off between computation efficiency and statistical accuracy and should not come as a surprise. \n  Next, we demonstrate that, by using the PLR algorithm in the conquer step, the estimation accuracy increases and the final localization rates become comparable to the (often minimax) optimal rates.\n\nThroughout the section, we will consider the following high-dimensional offline change point analysis framework of reference.\n\n\\bnassum[]\n\\label{assp: DCDP_general}\nWe observe independent data points $\\{ \\bm Z _{i}\\}_{i\\in[n]}$ such that, for each $i$, $ \\bm Z _{i}$ is a draw from a parametric distribution $\\mathbb{P}_{ { \\bm \\theta } ^*_i}$ specified by an unknown parameter vector $ { \\bm \\theta } ^*_i$.  There exists an unknown  collection of change points $ 1=\\eta_0 <  \\eta_1  < \\eta_2 < \\ldots <\\eta_K < \\eta _{K+1}= n+1$ such that $ { \\bm \\theta } ^*_{i} \\neq  { \\bm \\theta } ^*_{i-1}$ if and only if $i \\in \\{\\eta_k\\}_{k\\in[K]}$. For each change point $\\eta_k$, we will let $\\kappa_k = \\|  { \\bm \\theta } ^*_{\\eta_{k}} -  { \\bm \\theta } _{\\eta_k - 1}^* \\|$ be the size of the corresponding change, where $\\|\\cdot\\|$ is an appropriate norm (to be specified, depending on the model). For simplicity, we further assume that the magnitudes of the changes are of the same order: there exists a $\\kappa>0$ such that \n\t $  \\kappa _k \\asymp  \\kappa $ for all $k\\in [K]$. We denote the spacing between $\\eta_{k}$ and $\\eta_{k-1}$ with \n  $\\Delta_k =  \\eta_{k} -\\eta_{k-1} $ and let  $ \\Delta_{\\min} = \\min_{k\\in[K]}\\Delta_k$ denote the minimal spacing. All the model parameters are allowed to change with $n$, with the exception of $K$.\n\\enassum\n\n   \n\n\n \n\n\n\n\n\n\n\n\\subsection{ Changes in  means}\n\\label{sec: result mean}\n\nChange point detection and localization of a piece-wise constant mean signal is arguably the most traditional and well-studied change point model.  Initially developed in the 1940s for univariate data, the model  has recently been generalized under various high-dimensional settings and thoroughly investigated: see, e.g., \n\\cite{wang_samworth2018, Gao2019, verzelen2020hd_mean, unify_sinica2022}. Below, we show that, for this model,  DCDP achieves the sharp detection boundary and delivers the minimax optimal localization error rate.\n\\bnassum[Mean model]\n\\label{assp: DCDP_mean main}\n   Suppose that for each $i\\in [n]$, $ \\bm Z _i=  \\bm Z _{i }$ satisfies the mean model $ \\bm Z _{i} =     { \\bm \\mu }  _i^*    +  { \\bm \\epsilon } _i\\in \\mathbb{R}^p$ and \\Cref{assp: DCDP_general} holds with $ { \\bm \\theta } ^*_i =  { \\bm \\mu } ^*_i$ and $\\|\\cdot\\| = \\|\\cdot\\|_2$.   \n\n{\\bf (a)} The measurement errors $\\{ { \\bm \\epsilon } _i\\}_{i\\in[n]}$ are independent mean-zero random vectors with independent subgaussian entries such that $0<\\sigma_{\\epsilon} =\\sup_{i\\in[n]}\\sup_{j\\in[p]} \\|( { \\bm \\epsilon } _i)_j\\|_{\\psi_2}<\\infty$.\n   \n\n{\\bf (b)} For each $i \\in [n]$,    there exists a collection of subsets  $  S_i \\subset [p]$,  such that \n\t  $  ( { \\bm \\mu } ^*_{i})_j =0 \\text{ if } j \\not \\in S_i.$\n\t  In addition, the cardinality of the support satisfies $|S_i|\\leq \\mathfrak{s} $. \n\\enassum\n \n \n \n Conditions {\\bf (a)} and {\\bf (b)} above are standard assumptions for the high-dimensional linear regression time series models \\citep{Basu2015,unify_sinica2022}.  In our first result, we establish consistency of the divide step. The proof of the following theorem is in   \\Cref{sec: main proof mean}.\n \n\\bnthm \\label{thm:DCDP mean}\nSuppose  that \\Cref{assp: DCDP_mean main} holds and that \n  \\begin{align}\\label{eq:dp mean snr 1} \\Delta_{\\min} \\kappa^2  \\ge  \\mathcal{B}_n \\sigma_{\\epsilon}^2  { \\mathfrak{s} }  \\log(p\\vee n),\n  \\end{align}\n  for some slowly diverging sequence $\\{ \\mclB_n\\}_{n\\in \\mathbb{Z}^+}$.\n  For sufficiently large constants $C_\\gamma$ and $C_\\mclF$, let \n $\\{\\widehat { \\eta }_k\\}_{k\\in [\\widehat K]}  $ denote the output of  \\Cref{algorithm:DP} with $\\mclQ = \\frac{4n}{\\Delta_{\\min}}\\log^2(n)$,\n \\begin{equation*}\n    \\mathcal F  (\\widehat{ { \\bm \\mu } }_\\I,\\I) := \\begin{cases}\n    \\sum_{i \\in \\I } \\| \\bm Z _i - \\widehat  { \\bm \\mu } _\\I \\|_2 ^2  &\\text{if } |\\I|\\geq C_\\mclF  { \\mathfrak{s} }  \\log(p\\vee n),\\\\\n    0  &\\text{otherwise},\n    \\end{cases}\n\\end{equation*}\n and   $\\gamma = C_\\gamma \\mathcal B_n^{-1\/2} \\Delta_{\\min}\\kappa^2$. Here \n\\begin{equation}\n    \\widehat{ { \\bm \\mu } }_{\\I} = \\argmin_{ { \\bm \\mu } \\in \\mathbb{R}^p}\\| \\bm Z _i -  { \\bm \\mu } \\|_2^2 + \\lambda \\sqrt{|\\I|}\\| { \\bm \\mu } \\|_1,\n\\end{equation}\nwith $\\lambda = C_{\\lambda}\\sqrt{\\log(p\\vee n)}$ and $C_\\lambda $ a\n  sufficiently large constant. \n \n Then, \n with probability  $1 - n^{-3}$, $\\widehat K = K$ and \n \\begin{equation*}\n     \\max_{ k \\in [K] } |\\eta_k-\\widehat \\eta_k|   \\lesssim   \\frac{  \\sigma_{\\epsilon}^2\\log(p\\vee n) +\\gamma }{\\kappa^2 } + \\mathcal{B}_n^{-1\/2} \\Delta_{\\min}.\n \\end{equation*}\n \\enthm\n\nThe \\textit{signal-to-noise-ratio} (SNR) condition \\eqref{eq:dp mean snr 1} assumed in \\Cref{thm:DCDP mean} is frequently used in the change point detection literature \\citep{unify_sinica2022,wang_samworth2018}. Recently,  \\cite{verzelen2020hd_mean} showed that, if $ { \\mathfrak{s} } \\le \\sqrt p$, condition \\eqref{eq:dp mean snr 1} is indeed necessary, in the sense that if \n$$ \\frac{\\Delta_{\\min} \\kappa^2 }{  { \\mathfrak{s} }  \\sigma_{\\epsilon}^2 \\log(p\\vee n)} =o(1) ,$$\nthen there exists a setting for which no change point estimator is consistent.    \nThe localization error of DCDP estimator $\\{ \\widehat \\eta_k\\}_{k\\in[\\widehat K] } $ returned by   \\Cref{algorithm:DP} satisfies \n$$\n \\frac{\\max_{k\\in [K]} |   \\eta_k-\\widehat{\\eta}_k|}{\\Delta_{\\min}} \\lesssim    \\frac{ \\sigma_{\\epsilon}^2 \\log(p\\vee n)}{\\Delta_{\\min}\\kappa^2} +  \\frac{\\gamma}{\\Delta_{\\min}\\kappa^2} +   \\mclB_n^{-1\/2},\n$$\nwith high probability.\n   Thus, using  \\eqref{eq:dp mean snr 1} and given the choice of $\\gamma$, \nit follows  that the resulting estimator is consistent:\n$$ \\frac{\\max_{k\\in [K]}|   \\eta_k-\\widehat{\\eta}_k|}{\\Delta_{\\min}} \\lesssim   \n\\mclB_n^{-1} + \\mclB_n^{-1\/2}\n    =o_p(1). $$ \n\n\\bnrmk[Grid size]\nIn \\Cref{thm:DCDP mean} and in all the results of this section, we choose a  value for the grid size $\\mclQ$ that, while coarse, ensures consistency. Any finer grid can yield the same error rate, at an additional computational cost. \n\\enrmk\n\nCompared to the localization error of the vanilla DP, the localization error of Divided DP \\Cref{algorithm:DP} picks up an additional term $ { \\mathcal B_n^{-1\/2}   \\Delta } $. As remarked above, this is to be expected, as \\Cref{algorithm:DP} only deploys a subset of the data indices.    Starting with the coarse (but still consistent)   preliminary estimators from the divide step  \\Cref{algorithm:DP}, the local refinement algorithm   \\Cref{algo:local_refine_general}  further improves its accuracy and, in fact, yields an optimal error rate.\n \n\\bnthm\n\\label{cor:mean local refinement}  Let $\\{ \\mclB_n\\}_{n\\in \\mathbb{Z}^+}$ be any slowly diverging sequence and suppose  that $\\Delta_{\\min} \\kappa ^2 \\geq  \\mathcal{B}_n \\sigma_{\\epsilon}^2 \\mathfrak{s}^2\\log^3(p\\vee n) $.\n\t Let \n $\\{ \\widetilde \\eta_k\\}_{k\\in[\\widehat K]}$ be the output of \n \\Cref{algo:local_refine_general} with $\\zeta = C_{\\zeta} \\sqrt{\\log(p\\vee n)}$ for sufficiently large constant $C_{\\zeta}$ and $R(\\theta^{(1)}, \\theta^{(2)},\\eta; s, e)$ specified in \\eqref{eq:group lasso penalty}. Then under \\Cref{assp: DCDP_mean main}, for any $\\alpha \\in (0,1) $, with probability at least $1-  (  \\alpha \\vee n^{-1} ) $  it holds that $\\widehat{K} = K$ and\n    \\begin{equation}\n    \\label{eq: mean rate op1}\n        \\max_{k  \\in [K]} |\\eta_k-\\widetilde \\eta_k|\\cdot \\kappa^2   \\lesssim  {  \\sigma_{\\epsilon}^2 \\log ( {1}\/{\\alpha} )  }.\n    \\end{equation}\n\\enthm\n  The proof of \\Cref{cor:mean local refinement} can be found in \\Cref{sec:mean op1}.\n\\bnrmk    \\label{remark:optimal mean}\n The localization error bound \\eqref{eq: mean rate op1} is the tightest in the literature. It improves the existing bounds by \\cite{wang_samworth2018}  and   \\cite{unify_sinica2022}\n by a   factor of $ { \\mathfrak{s} }  \\log(p) $.  It also matches the lower bound established in \\cite{wang_samworth2018}, showing that    $O_p(1\/\\kappa^2)$ is the optimal error order and can not be further improved. \n\\enrmk\n\n\n\n\\subsection{Changes in  regression coefficients}\n\\label{sec: result regression}\nWe now consider the more complex high-dimensional regression change point model in which the regression coefficients are sparse and change in a piecewise constant manner. Recently, various approaches and methods have been proposed to address this challenging scenario; see, in particular, \\cite{rinaldo2021cpd_reg_aistats,wang2021_jmlr,unify_sinica2022,yu2022temporal}. Below, we will show that DCDP yields optimal localization errors also for this class of change point models. \n\n\\bnassum[High-dimensional linear model]\\label{assp:dcdp_linear_reg main}\n   Let  the  observed data   $\\{ \\bm Z _{i }, y_i  \\}_{i\\in [n] }  \\subset \\mathbb R^p \\times \\mathbb R $ be such that   \n\t $y_{i  } =    \\bm Z _i^\\top  { \\bm \\beta }  ^* _i  +\\epsilon_i   $ and let \\Cref{assp: DCDP_general} hold  with $ { \\bm \\theta } ^*_i =  { \\bm \\beta } ^*_i\\in\\mathbb{R}^p$ and $\\|\\cdot\\| = \\|\\cdot\\|_2$. In addition,\n\n{\\bf (a)} Suppose that $\\{  \\bm Z _i\\}_{i\\in [n]} \\overset{i.i.d.}{\\sim} N_p(0,  { \\bm \\Sigma }  )$  and that the minimal and the maximal eigenvalues of  $ { \\bm \\Sigma } $ satisfy \n\t  $\\Lambda_{\\min}   ( { \\bm \\Sigma } )\\geq c_X$ and $\\Lambda_{\\max} ( { \\bm \\Sigma } ) \\le C_X$, with universal constants $c_X, C_X\\in(0,\\infty)$.\n   In addition, suppose that $ \\{ \\epsilon_i \\}_{i\\in [n]}  \\overset{i.i.d.}{\\sim}  N(0, \\sigma^2_ \\epsilon)$  and is  independent of $\\{  \\bm Z _i\\}_{i\\in [n]} $.\n\n{\\bf (b)}  For each $i \\in [n]$,    there exists a collection of indices $  S_i \\subset [p]$,  such that \n\t  $  ( { \\bm \\beta } ^* _ { i})_j =0 \\text{ if } j \\not \\in S_i.$\n\t  In addition, the cardinality of the support satisfies $|S_i|\\leq \\mathfrak{s} $. \n\\enassum\n  \nWe note that   \\Cref{assp:dcdp_linear_reg main}  {\\bf (a)} and {\\bf (b)}   are standard assumptions for Lasso estimators. Similarly to the case of the mean change point model, we first analyze the performance of the divide step of DCDP and find it to be consistent, albeit at a sub-optimal rate.\n\n \n \\bnthm \\label{thm:DCDP regression}   Suppose  \\Cref{assp:dcdp_linear_reg main} holds  and  that \n \\begin{align}\\label{eq:snr divide linear regression}\n \\Delta_{\\min }  \\kappa ^2 \\geq  \\mathcal{B}_n  \\sigma_{\\epsilon}^2 \\mathfrak{s}\\log(p\\vee n) \n \\end{align}for some diverging sequence $\\{ \\mclB_n\\}_{n\\in \\mathbb{Z}^+}$.\nLet  \n $\\{\\widehat { \\eta }_k\\} _{k\\in [\\widehat K]}  $ be the  output of  \\Cref{algorithm:DP}  with $\\mclQ = \\frac{4n}{\\Delta_{\\min}}\\log^2(n)$,    $\\gamma = C_\\gamma  \\mathcal B_n^{-1\/2} \\Delta_{\\min}\\kappa^2$ and\n\\begin{equation*}\n     \\mathcal F  (\\widehat{\\beta}_{\\I},\\mclI) :  = \\begin{cases} \n 0 \\quad \\quad\\quad \\  \\text{if } |\\mclI|< C_\\mclF  { \\mathfrak{s} }  \\log(p\\vee n);\n \\\\ \n \\sum_{i \\in \\mclI } (y_i -  \\bm Z _i^\\top \\widehat  { \\bm \\beta }  _\\mclI ) ^2 \\quad \\text{otherwise,}  \n \\end{cases} \n\\end{equation*}\n for sufficiently large constants $ C_\\gamma $ and $C_\\mclF$ and $\\widehat{ { \\bm \\beta } }_{\\I}$ given by\n\\begin{equation}\n    \\widehat{ { \\bm \\beta } }_{\\I} = \\argmin_{ { \\bm \\beta } \\in \\mathbb{R}^p}(y_i -  \\bm Z _i^\\top  { \\bm \\beta } )^2 + \\lambda \\sqrt{|\\I|}\\| { \\bm \\beta } \\|_1,\n\\end{equation}\nwith $\\lambda = C_{\\lambda}\\sqrt{\\log(p\\vee n)}$, for  $C_{\\lambda}$  a sufficiently large constant. \n Then,\n with probability $1 - n ^{-3}$, $\\widehat K = K$ and that\n $$ \\max_{ k\\in [K]} |\\eta_k-\\widehat \\eta_k|   \\lesssim   \\sigma_{\\epsilon}^2 \\big( \\frac{  \\mathfrak{s} \\log(p\\vee n) +\\gamma }{\\kappa^2 }\\big) +   { \\mathcal B_n^{-1\/2}   \\Delta } .$$\n \\enthm\n\nThe proof of \\Cref{thm:DCDP regression} is deferred to \\Cref{sec: main proof linear}. It is immediate to verify that, under the SNR condition \\eqref{eq:snr divide linear regression} and given the choice of $\\gamma$, estimators satisfy that $ \\max_{k\\in [K]}  |   \\eta_k-\\widehat{\\eta}_k|  \n    =o_p(\\Delta_{\\min}) $ and are therefore consistent.  \n\n \n \nWith a slightly stronger SNR condition than \\eqref{eq:snr divide linear regression},  \nstatistically optimal change point estimators can be obtained  in the conquer step.   \n \n\\bnthm\n \\label{cor:regression local refinement}\n Let $\\{ \\mclB_n\\}_{n\\in \\mathbb{Z}^+}$ be any slowly diverging sequence and suppose  that $\\Delta_{\\min} \\kappa ^2 \\geq  \\mathcal{B}_n  \\sigma_{\\epsilon}^2 \\mathfrak{s}^2\\log^3(p\\vee n) $.\n\n  Let \n $\\{ \\widetilde \\eta_k\\}_{k\\in[\\widehat K]}$ be the output of \n \\Cref{algo:local_refine_general} with $\\zeta = C_{\\zeta} \\sqrt{\\log(p\\vee n)}$ for sufficiently large constant \n $C_\\zeta $ and $R( { \\bm \\theta } ^{(1)},  { \\bm \\theta } ^{(2)},\\eta)$ specified in \\eqref{eq:group lasso penalty}\n Then under \\Cref{assp:dcdp_linear_reg main}, for any $\\alpha\\in (0,1)$, with probability at least  $1 - (\\alpha\\vee n^{-1})$, it holds that $\\widehat {K} = K$ and\n\t\\begin{equation}\n\t   \n      \\max_{k  \\in [K]} | \\eta_k-\\widetilde \\eta_k | \\cdot \\kappa ^2   \\lesssim     {\\sigma_\\epsilon^2 \\log^2({1}\/{\\alpha})}.\n\t \\label{eq: linear reg rate op1}\n\t\\end{equation}\n\\enthm\nThe proof of \\Cref{cor:regression local refinement} can be found in \\Cref{sec:regression op1}.\n\\bnrmk \\label{remark:optimal regression}\nThe localization  error \\eqref{eq: linear reg rate op1} matches the existing lower bound \n established in \\cite{rinaldo2021cpd_reg_aistats}  and, therefore, it is rate minimax optimal. To the best of our knowledge, the only other existing change point algorithm that can achieve optimal localization errors in the high-dimensional linear regression setting is the one developed in  \\cite{yu2022temporal}, which allows for dependent observations. However, the approach by \\cite{yu2022temporal} requires quadratic time complexity.\n It is worth mentioning that both \\cite{rinaldo2021cpd_reg_aistats} and \\cite{yu2022temporal} also assume the SNR condition we use in \\Cref{thm:DCDP regression} and \\Cref{cor:regression local refinement}.\n\\enrmk\n\n\n\n\\subsection{Changes in precision matrices}\n\\label{sec: result covariance}\n\n\nFor our third and final example, we specialize the general change point framework of  \\Cref{assp: DCDP_general} to the case of Gaussian graphical models, in which the distributional changes are induced by a sequence of temporally piece-wise constant precision matrices, with the magnitude of the changes measured in Frobenius norm.\n\n\\bnassum[Gaussian graphical model]\n\\label{assp:DCDP_covariance main} \n  Suppose  for each $i\\in [n]$, $ \\bm Z  _{i }$ is a mean-zero Gaussian vector in $\\mathbb{R}^p$ with covariance matrix $ { \\bm \\Sigma } ^*_i = \\mathbb{E}[ \\bm Z _i \\bm Z _i^\\top]$, and \\Cref{assp: DCDP_general} holds with $ { \\bm \\theta } ^*_i = ( { \\bm \\Sigma } _i^*)^{-1}$  with $\\|\\cdot\\| = \\|\\cdot\\|_F$. Assume that for each $i\\in [n]$, the minimal and maximal eigenvalues of  $ { \\bm \\Sigma } _i^*$ satisfy \n\t  $\\Lambda_{\\min}   ( { \\bm \\Sigma } _i^*)\\geq c_X$ and $\\Lambda_{\\max} ( { \\bm \\Sigma }  _i^*) \\le C_X$, with universal constants $c_X, C_X\\in(0,\\infty)$.\n\\enassum \n\nThe recent literature contains several contributions addressing the problem of detecting change points in precision matrices; see, e.g., \\cite{Gibberd2017ggm,Gibberd2017fusedGlasso,  Bybee2018jmlr,Keshavarz2020jmlr,londschien2021change,liu2021jmlr, unify_sinica2022}. Most of these studies focus on estimating a {\\it single} change point. To the best of our knowledge, only \\cite{unify_sinica2022} has provided theoretical guarantees for the multiple-change-point setting assuming sparse changes in the precision matrices. Below, we show that the divide step of the DCDP procedure is able to detect multiple change points in the precision matrices in the dense regime. \n \n\\bnthm\n\\label{thm:DCDP covariance main} Suppose \\Cref{assp:DCDP_covariance main} holds and that \n\t  \\begin{align}\\label{eq:snr precision 1} \\Delta_{\\min} \\kappa^2  \\geq  \\mathcal{B}_n   p^2\\log(n\\vee p)\n   \\end{align}\n   for some slowly diverging sequence $\\{ \\mclB_n\\}_{n\\in \\mathbb{Z}^+}$. Let  $ \\{ \\widehat \\eta_k\\}_{k\\in [\\widehat K]}$ be the output of \\Cref{algorithm:DP} with $\\mclQ = \\frac{4n}{\\Delta_{\\min}}\\log^2(n)$,  $\\gamma = C_\\gamma  \\mclB_n^{-1\/2}\\Delta_{\\min}\\kappa^2$ and\n\\begin{equation*}\n    \\mathcal F  (\\widehat{ { \\bm \\Omega } }_\\I,\\I) \n     = \\begin{cases}  0  \n\\quad \\quad\\quad \\quad \\quad\\quad \\quad  \\text{ if } |\\I|< C_\\mclF p\\log(p\\vee n);\\\\\n       \\sum_{i \\in \\I } {\\rm Tr}[\\widehat{ { \\bm \\Omega } }_{\\mclI}^\\top  \\bm Z _i \\bm Z _i^\\top] - |\\I|\\log|\\widehat{ { \\bm \\Omega } }_{\\mclI}|   \n       \\ \\text{otherwise}.\n    \\end{cases}\n\\end{equation*}\nfor sufficiently large constants $ C_\\gamma $ and $C_\\mathcal F$. Here $\\widehat{ { \\bm \\Omega } }_{\\I}$ is\n\\begin{equation}\n\\label{eq:goodness-of-fit.precision}\n    \\widehat{ { \\bm \\Omega } }_{\\I} = \\argmin_{ { \\bm \\Omega } \\in \\mathbb{S}^p_+} \\sum_{i \\in \\I } {\\rm Tr}[{ { \\bm \\Omega } }^\\top  \\bm Z _i \\bm Z _i^\\top] - |\\I|\\log| { \\bm \\Omega } |. \n\\end{equation}\n Then\n with probability  at least $1 - n ^{-3}$, $\\widehat  K = K$ and that\n \\begin{equation} \n \\label{eq:loc.rate.precison}\n \\max_{ k\\in [K] } |\\eta_k-\\widehat \\eta_k|   \\lesssim    \\frac{ p^2 \\log(p\\vee n) +\\gamma }{\\kappa^2 } + \\mclB_n^{-\\frac{1}{2}}{\\Delta_{\\min}}.\n \\end{equation}\n\\enthm\nThe proof of \\Cref{thm:DCDP covariance main} is deferred to  \\Cref{sec: main proof covariance}.\n\n\nThe goodness-of-fit function in \\eqref{eq:goodness-of-fit.precision} corresponds to a penalized Gaussian likelihood for the precision matrix and is a natural choice for our model settings.\n\nUnder the assumption of the theorem, the localization rate \\eqref{eq:loc.rate.precison} implies consistency, as defined in \\eqref{eq: consistency general}; indeed, it is easy to see that $ \\max_{k\\in [K]}  |   \\eta_k-\\widehat{\\eta}_k|  \n    =o_p( \\Delta_{\\min}   ). $ \n \nAn analogous condition to Condition \\eqref{eq:snr precision 1} is used in \\cite{unify_sinica2022} under the slightly different settings of sparse changes. More precisely, the authors requires that    $\\Delta_{\\min}\\kappa^2\\geq \\mclB_n d\\log(n\\vee p)$, where $d $   is the maximal number of nonzero entries in the precision matrices. When applied to our dense settings, their SNR condition matches  \\eqref{eq:snr precision 1}.\n\nUnder a slightly stronger SNR condition, we further obtain that the local refinement algorithm in the conquer step improves the localization rate to match the sharpest rate known for this problem.\n \n\n\n\n\n\\bnthm[]\n\\label{cor:covariance local refinement main}Let $ \\mathcal{B}_n$ be an arbitrary slowly diverging sequence and suppose $ \\Delta_{\\min} \\kappa^2  \\geq  \\mathcal{B}_n   p^4\\log^2(n\\vee p)$.\n Let \n $\\{ \\widetilde \\eta_k\\}_{k\\in[\\widehat K]}$ be the output of \n \\Cref{algo:local_refine_general} with $R(\\theta^{(1)}, \\theta^{(2)},\\eta) =0$. \nThen under \\Cref{assp:DCDP_covariance main},  it holds that  with probability at least $1 - n^{-1}$ \n\\begin{equation}\\label{eq:covariance final rate}\n        \\max_{k \\in {[K]}} |\\eta_k-\\widetilde \\eta_k|\\cdot \\kappa^2\\lesssim \\log(n).\n    \\end{equation}\n\\enthm\nThe proof of \\Cref{cor:covariance local refinement main} is in \\Cref{sec:cov op1}.\n The localization error bound obtained for DCDP in\n\\Cref{cor:covariance local refinement main} matches the sharpest error bounds obtained for the precision matrices change point model \\cite{liu2021jmlr, unify_sinica2022} and does not require the precision matrices to be sparse. To the best of our knowledge, DCDP is the first linear time algorithm that can optimally estimate multiple change points in the precision matrices in high dimensions.\n\n\n\n \n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Numerical experiments}\n\\label{sec:experiment}\nWe evaluate the numerical performance of DCDP through examples of synthetic and real data. The tuning parameters  $ \\gamma$ and  $\\zeta$ of DCDP are chosen using cross-validation. The implementations of our numerical experiments   are available online\n\\footnote{\\url{https:\/\/github.com\/MountLee\/DCDP}}.\nMore details, including the implementation for cross-validation and additional numerical  results, can be found in   \\Cref{sec:detail experiment}  due to space constraints.\n\n \n\n\\subsection{Time complexity  and accuracy of DCDP}\n\\label{sec:experiment_time_error_divide}\nWe generate  i.i.d.   Gaussian random variables $\\{y_i\\}_{i\\in[n]} \\subset  \\mathbb R $ with $y_i = \\mu^*_i + \\epsilon_i$ and  $\\sigma_\\epsilon   =1$. We set $n=4\\Delta$ where $\\Delta $ will be specified in each setting.   The three population change points of $\\{\\mu^*_i\\}_{i\\in[n]}$ are set to be $\\mu_{\\eta_0}^* = 0$, $\\mu_{\\eta_1}^* = 5$, $\\mu_{\\eta_2}^* = 0$, $\\mu_{\\eta_3}^* = 5$, where $\\eta_{k} = k\\Delta + \\delta_k$ with $\\delta_k\\sim{\\rm Unif[-\\frac{3}{10}\\Delta,\\frac{3}{10}\\Delta]}$ for $ k = 1,2,3$. We use the Hausdorff distance $H(\\{\\widehat{\\eta}_k\\}_{k\\in [\\widehat{K}]},\\{\\eta_k\\}_{k\\in [K]})$ to quantify the difference between the estimators and the true change points. \n\n\n\\begin{figure}[ht]\n    \\centering\n    \\includegraphics[width = 0.48\n    \\textwidth]{Figure\/Delta5000_time_error_vs_grid_num.pdf}\n    \\caption{Average localization error   and average run time  versus  the number of grid points $\\mclQ$ over 100 trials. The shaded area indicates  the upper and lower 0.1 quantiles of the corresponding quantities. }\n\\label{fig:loc_runtime_vs_n_grid}\n\\end{figure}\nIn the first set of experiments, we set $ \\Delta =5000$ and vary $\\mathcal Q$  from $25$ to $200$, and summarize results in \\Cref{fig:loc_runtime_vs_n_grid}. The left plot of the figure shows that while the localization errors of the divide step are sensitive to the choice of $\\mathcal Q$, the additional conquer step (Algorithm 3) greatly improves the numerical accuracy of the final estimators of DCDP. The right plot of the figure demonstrates that the time complexity of DCDP is quadratic in  $\\mathcal Q$, which is in line with the complexity analysis presented in \\Cref{sec:method}.\n\n\nIn the second set of experiments, we fix $\\mathcal Q = 100 $ and let $\\Delta$ range from 1000 to 6000. The results are summarized in \\Cref{fig:error_runtime_on_n}. The left plot of the figure shows that while the localization errors of the dive step change with $\\Delta$, the accuracy of DCDP is consistently small for all the different values of $\\Delta$. The right plot of the figure shows that the time complexity is linear in $n$, and this observation matches the findings presented in \\Cref{sec:method}.\n \n \n \n\\begin{figure}[ht]\n    \\centering\n    \\includegraphics[width = 0.48 \\textwidth]{Figure\/n_time_error_fine_gamma_Q100.pdf}\n    \\caption{\n    Average localization error and average run time v.s. $\\Delta$ over 100 trials. }\n    \\label{fig:error_runtime_on_n}\n\\end{figure}\n\n\n\n\\subsection{Numerical performance of DCDP}\n\\label{sec: experiment comparison}\n\n\nBelow we report  the outcome of various simulation studies in which we compare the numerical performance of DCDP with that of several other state-of-the-art methods,  for each  of the three models presented in \\Cref{section:main}. \n \n\n \n\nIn the following experiments,\nfor each specific  $\\Delta$  we set the total number of observations $n = (K + 1)\\Delta$ and the locations of true change points $\\eta_k = k\\Delta + \\delta _k$, where $\\delta_k$ is a random variable sampled from  the uniform distribution ${\\rm Unif}[-\\frac{3}{10}\\Delta,\\frac{3}{10}\\Delta]$.\nIn each setting, we conduct  \n100 trials and report the average execution time,   the average  Hausdorff distance between true and estimated change points,  and the frequency of  cases in which $\\widehat{K}=K$, for each method\n\n{\\bf The mean model}\n\\\\\nWe set $K = 3$  and, for $k=0,\\cdots, K$ and $\\delta \\in \\{1, 5 \\}$, we assume  a population mean vector of the form \n\\begin{align*}\n { \\bm \\mu } ^*_{\\eta_k} = (  \\underbrace{ 0, \\ldots, 0}_{5k},  \\underbrace{ \\delta ,  \\ldots, \\delta }_{5} ,   \\underbrace{ 0,\\ldots,0}_{p-5k-5})^\\top   \\in \\mathbb R^{p}.\n\\end{align*}\n\nWe compare DCDP with Change-Forest\n(CF) \\cite{changeforest2022}, Block-wise Fused Lasso (BFL) \\cite{unify_sinica2022}, and Inspect \\citep{wang_samworth2018}. The results are summarized in \\Cref{tab:mean compare_methods}. On average, DCDP outputs the most accurate change point estimators while remaining computationally efficient.  \n \n\n\\begin{table}[h]\n \\centering\n \\begin{tabular}{l l l c}\n \\hline\n Method & $H(\\hat{ { \\bm \\eta } }, { \\bm \\eta } )$ & Time & $\\widehat{\\mathbb{P}}[\\widehat{K}=K]$  \\\\\n \\hline\n   \\multicolumn{4}{c}{$n = 200, p=100, K =3, \\delta = 5$}\n   \\\\\n  DCDP & 0.00 (0.00) & 0.6s (0.0) & 1.00 \\\\\n Inspect & 0.40 (3.50) &  0.0s (0.0) & 0.91 \\\\\n\n  CF & 1.84 (6.27) &  0.8s (0.2) & 0.90  \\\\\n \n  BFL &  47.84 (6.69) &  1.4s (0.2) & 0.00 \\\\\n \\hline\n \\multicolumn{4}{c}{$n = 200, p=100, K =3, \\delta = 1$}  \n  \\\\\n  DCDP & 0.83 (0.87) & 0.8s (0.2) & 1.00 \\\\\n Inspect  &  2.65 (5.16) &   0.0s (0.0) & 0.86  \\\\\n\n  CF &  6.29 (9.57) &  1.1s (0.3) & 0.78 \\\\\n \n  BFL &  47.19 (6.48) &  1.1s (0.2) & 0.00 \\\\\n \\hline\n\n\n\n\n\n\n\n\n \\end{tabular}\n  \n \\caption{Numerical comparison of different methods in the high-dimensional mean shift models. The numbers in the cells indicate the averages over 100 trials and the numbers in the brackets  indicate the corresponding standard errors.}\n \\label{tab:mean compare_methods}\n\\end{table}\n\n{\\bf The linear regression model}\n\nWe set $K = 3$  and, for $k=0,\\cdots, K$,  assume population regression coefficients of the form\n\\begin{align*}\n { \\bm \\beta } ^*_{\\eta_k} = (  \\underbrace{ 0, \\ldots, 0}_{5k},  \\underbrace{ \\delta ,  \\ldots, \\delta }_{5} ,  \\underbrace{ 0,\\ldots,0}_{p-5k-5})^\\top \\in \\mathbb R^{p},\n\\end{align*}\nwhere $\\delta \\in \\{1, 5 \\}$.  \n\n\n\nWe compare the numerical performance of DCDP with   Variance-Projected Wild Binary Segmentation (VPBS) \\citep{wang2021_jmlr} and vanilla Dynamic Programming (DP) \\citep{rinaldo2021cpd_reg_aistats}. The results are summarized in \\Cref{tab:linear compare_methods}. On average, DCDP is the most efficient algorithm with compelling numerical accuracy.  \n\n\n\n\n\\begin{table}[h]\n \\centering\n\\begin{tabular}{l l l c}\n  \\hline\nMethod & $H(\\hat{ { \\bm \\eta } }, { \\bm \\eta } )$ & Time & $\\widehat{\\mathbb{P}}[\\widehat{K}=K]$ \n \\\\\n \\hline\n \\multicolumn{4}{c}{$n = 200, p=100, K =3, \\delta = 5$} \\\\\n  DCDP & 0.13 (0.39) & 18.4s (1.1) & 1.00 \\\\\n  DP & 0.01 (0.10) & 220.3s (16.8) & 0.98 \\\\\n  VPWBS & 15.44 (17.99) &  120.1s (13.1) & 0.70 \\\\\n\n\n\n \\hline\n \\multicolumn{4}{c}{$n = 200, p=100, K =3, \\delta = 1$} \\\\\n  DCDP & 1.45 (8.59) & 8.8s (0.7) & 0.98 \\\\\n  DP & 0.22 (2.00) & 84.4s (5.7) & 0.99 \\\\\n  VPWBS & 11.54 (11.23) &  120.4s (14.5) & 0.65 \\\\\n\n\n\n \\hline\n \\end{tabular}\n\n\\caption{Numerical comparison of different methods in the high-dimensional regression coefficient shift models.}\n \\label{tab:linear compare_methods}\n\\end{table}\n\n{\\bf The Gaussian graphical model}\n\\\\\nWe set $K=3$ and the population  covariance matrix matrices as    $ { \\bm \\Sigma } ^*_{\\eta_0}=  { \\bm \\Sigma } ^*_{\\eta_2} = {\\bm I}_p$ and \n$ { \\bm \\Sigma } ^*_{\\eta_1}=  { \\bm \\Sigma } ^*_{\\eta_3}$ where\n\\begin{equation*}\n( { \\bm \\Sigma } ^*_{\\eta_1})_{ij} =( { \\bm \\Sigma } ^*_{\\eta_3})_{ij} =\\begin{cases}\n    \\delta_1, & i=j;\\\\\n    \\delta_2, & |i-j|=1;\\\\\n    0, & \\text{otherwise},\n\\end{cases}\n\\end{equation*}\nwith $\\delta_1 = 5, \\delta_2 = 0.3$.\n\nWe compare the numerical performance of DCDP with Change-Forest\n(CF) \\cite{changeforest2022} and  Block-wise Fused Lasso (BFL) \\cite{unify_sinica2022}.  \nNote that the BFL algorithm produces empty set in all trials, so we only report DCDP and CF in \\Cref{tab:covariance compare_methods}.\nIt can be seen that on average DCDP outputs  the most accurate change point estimates and  is highly computationally efficient. \n \n\n\n\\begin{table}[h]\n \\centering\n\\begin{tabular}{l l l c}\n  \\hline\n Method & $H(\\hat{ { \\bm \\eta } }, { \\bm \\eta } )$ & Time & $\\widehat{\\mathbb{P}}[\\widehat{K}=K]$ \n \\\\\n \\hline\n\n\n\n\n\n\n\n \\multicolumn{4}{c}{$n = 400, p=10, K =3, \\delta_1 = 5,\\delta_2 = 0.3$} \\\\\n  DCDP & 0.42 (0.64) & 0.5s (0.0) & 1.00  \\\\\n\n\n  CF &  5.54 (14.71) &  0.6s (0.1) & 0.88  \\\\\n\n \\hline\n \\multicolumn{4}{c}{$n = 400, p=20, K =3, \\delta_1 = 5,\\delta_2 = 0.3$} \\\\\n  DCDP & 0.66 (4.37) & 0.9s (0.3) & 1.00 \\\\\n\n\n  CF &  7.37 (18.76) &  1.0s (0.0) & 0.85  \\\\\n\n \\hline\n \\end{tabular}\n\n \\caption{Numerical comparison of different methods in the precision matrix shift models. }\n \\label{tab:covariance compare_methods}\n\\end{table}\n\n\n\n\n\n\n\n\n\\subsection{Real data analysis}\n\\label{sec: application}\nWe also applied DCDP to three popular real data examples and compared it with state-of-the-art methods. The results are deferred to \\Cref{sec:detail application} due to space constraints.\n\n \n\n\n\n\\section{Discussion}\n\\label{sec:discussion}\nIn this paper, we propose  a novel framework called DCDP   for offline change point detection that can efficiently localize multiple change points for a broad range of high-dimensional models. DCDP improves the computational efficiency of vanilla dynamic programming while preserving the accuracy of change point estimation. DCDP serves as a unified methodology for a large family of change point models and theoretical guarantees for the localization errors of DCDP under three specific models are established. Extensive numerical experiments are conducted to compare the performance of DCDP with other popular methods to support our theoretical findings.\n\\subsection{Results with random grid points}\n\n\n\n\\begin{table}[h]\n \\centering\n \\begin{tabular}{l l l l l l l}\n  \\hline\n Setting & Method & $H(\\hat{\\eta},\\eta)$ & Time & $\\hat{K}<K$ &$\\hat{K}=K$ &$\\hat{K}>K$\n \\\\\n \\hline\n \\multirow{1}{5cm}{$n = 200, p=20, K =3, \\delta = 5$} \n & DCDP & 0.00 (0.00) & 0.7s (0.1) & 0 & 100 & 0 \\\\\n \\hline\n \\multirow{1}{5cm}{$n = 200, p=20, K =3, \\delta = 1$} \n & DCDP & 0.51 (0.77) & 0.7s (0.1) & 0 & 100 & 0 \\\\\n \\hline\n \\multirow{1}{5cm}{$n = 200, p=20, K =3, \\delta = 0.5$} \n & DCDP & 16.54 (10.33) & 0.2s (0.0) & 10 & 87 & 3 \\\\\n \\hline\n \\multirow{1}{5cm}{$n = 200, p=100, K =3, \\delta = 5$} \n & DCDP & 0.0 (0.0) & 0.6s (0.1) & 0 & 100 & 0 \\\\\n \\hline\n \\multirow{1}{5cm}{$n = 200, p=100, K =3, \\delta = 1$} \n & DCDP & 0.82 (0.88) & 0.9s (0.1) & 0 & 100 & 0 \\\\\n \\hline\n \\multirow{1}{5cm}{$n = 800, p=100, K =3, \\delta = 0.5$} \n & DCDP &  60.78 (31.08) &  2.0s (0.1) & 5  &  95 & 0 \\\\\n \\hline\n \\end{tabular}\n \\caption{Comparison of DCDP and other methods under the mean model with different simulation settings. 100 trials are conducted in each setting. }\n\n\\end{table}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\begin{table}[H]\n \\centering\n\\begin{tabular}{l l l l l l l}\n  \\hline\n Setting & Method & $H(\\hat{\\eta},\\eta)$ & Time & $\\hat{K}<K$ &$\\hat{K}=K$ &$\\hat{K}>K$\n \\\\\n \\hline\n \\multirow{1}{5cm}{$n = 200, p=20, K =3, \\delta = 5$} \n & DCDP & 0.03 (0.17) & 2.4s (0.1) & 0 & 100 & 0 \\\\\n \\hline\n \\multirow{1}{5cm}{$n = 200, p=20, K =3, \\delta = 1$} \n & DCDP & 1.57 (7.90) & 2.2s (0.1) & 3 & 97 & 0 \\\\\n \\hline\n \\multirow{1}{5cm}{$n = 200, p=100, K =3, \\delta = 5$} \n & DCDP & 0.12 (0.38) & 14.1s (0.5) & 0 & 100 & 0 \\\\\n \\hline\n \\multirow{1}{5cm}{$n = 200, p=100, K =3, \\delta = 1$} \n & DCDP & 2.20 (10.10) & 5.7s (0.2) & 4 & 96 & 0 \\\\\n \\hline\n \\end{tabular}\n \\caption{Comparison of DCDP and other methods under the linear model with different simulation settings. 100 trials are conducted in each setting. }\n\n\\end{table}\n\n\n\n\n\n\n\n\\begin{table}[h]\n \\centering\n\\begin{tabular}{l l l l l l l}\n  \\hline\n Setting & Method & $H(\\hat{\\eta},\\eta)$ & Time & $\\hat{K}<K$ &$\\hat{K}=K$ &$\\hat{K}>K$\n \\\\\n \\hline\n \\multirow{1}{7cm}{$n = 2000, p=5, K =3, \\delta_1 = 2,\\delta_2 = 0.3$} \n & DCDP & 5.10 (6.22) &  0.9s (0.2) & 0 & 100 & 0 \\\\\n \\hline\n \\multirow{1}{7cm}{$n = 2000, p=10, K =3, \\delta_1 = 5,\\delta_2 = 0.3$} \n & DCDP & 0.27 (0.49) & 1.1s (0.1) & 0 & 100 & 0 \\\\\n \\hline\n \\multirow{1}{7cm}{$n = 2000, p=20, K =3, \\delta_1 = 5,\\delta_2 = 0.3$} \n & DCDP & 0.03 (0.17) & 2.5s (0.3) & 0 & 100 & 0 \\\\\n \\hline\n \\multirow{1}{7cm}{$n = 400, p=10, K =3, \\delta_1 = 5,\\delta_2 = 0.3$} \n & DCDP & 0.38 (0.60) & 0.6s (0.1) & 0& 100& 0 \\\\\n \\hline\n \\multirow{1}{7cm}{$n = 400, p=20, K =3, \\delta_1 = 5,\\delta_2 = 0.3$} \n & DCDP & 0.68 (5.07) & 0.9s (0.1) & 0 & 100 & 0 \\\\\n \\hline\n \\end{tabular}\n \\caption{Comparison of DCDP and other methods under the covariance model with different simulation settings. 100 trials are conducted in each setting. }\n\n\\end{table}\n\n\n\\section{Detail of algorithms}\n\n\n \n \n\n\n\n\n\n\n\n\\section{Fundamental lemma}\n\\label{sec: fundamental lemma}\n\n\nIn the proof of localization error of the vanilla dynamic programming, we frequently compare the goodness-of-fit function $\\mclF( \\widehat \\theta_\\mclI , \\mclI)$ over an interval $\\mclI = (s,e]$ with\n\\[\n\\mclF(\\widehat \\theta_{(s,\\eta_{i + 1}] },   (s,\\eta_{i + 1}]) + \\cdots + \\mclF(\\widehat \\theta_{  (\\eta_{i + m},e]},(\\eta_{i + m},e]) + m\\gamma\n\\]\nwhere $\\{\\eta_{i + j}\\}_{j\\in [m]} = \\{\\eta_{\\ell}\\}_{\\ell\\in [K]}\\cap \\mclI$ is the collection of true change points within interval $\\mclI$ and $\\gamma$ is the penalty tuning parameter of the DP. \n\nHowever, for DCDP, we only search over the rough grid $\\{s_{i} = \\lfloor \\frac{i\\cdot n}{\\mclQ + 1}\\rfloor \\}_{i\\in [\\mclQ]}$ that may or may not  contain any true change points. Therefore, we need to   {\\bf a)}  guarantee the existence of some reference points (contained in   $\\{s_{i}\\}_{i\\in [\\mclQ]}$)  that are close enough to true change points, and {\\bf b)} quantify  the deviation of the goodness-of-fit function  evaluated at   the  reference points    compared to that evaluated at the true change points.\n\n\n\n\\paragraph{Reference points.} The grid is given by points $s_q = \\lfloor\\frac{q\\cdot n}{\\mclQ + 1}\\rfloor$ for $q\\in[ { \\mathcal Q} ]$.\n Let \n$ \\{ \\eta_k\\}_{k\\in[K]}$ be the collection of change points and denote \n\\begin{align*} \n  \\mathcal L _k (\\delta)  : = \\bigg  \\{    \\{ s_q\\}_{q\\in[ { \\mathcal Q} ]} \\bigcap   [\\eta_k -\\delta ,\\eta_k  ]  \\not  =\\emptyset \\bigg  \\} , \\quad \\text{and} \\quad    \\mathcal R  _k (\\delta)  : = \\bigg  \\{    \\{ s_q\\}_{q\\in[ { \\mathcal Q} ]} \\bigcap   [\\eta_k ,\\eta_k + \\delta   ]  \\not  =\\emptyset \\bigg  \\} .\n\\end{align*} \nIntuitively, if $ s_q \\in [\\eta_k -\\delta ,\\eta_k  ]$ and $ s_{q'} \\in [\\eta_k ,\\eta_k + \\delta   ]$ , then $s_q, s_{q'}$ can serve as reference points of  the true change point $\\eta_k$.  \nDenote \n\\begin{align} \n\\label{eq:left and right approximation of change points}  \\mathcal L(\\delta)  : = \\bigcap_{k=1}^{K}   \\mathcal L _k  \\big  (   \\delta   \\big  )  \n\\quad \\text{and} \\quad \n\\mathcal R (\\delta)  : = \\bigcap_{k=1}^{K}   \\mathcal R _k  \\big  (  \\delta  \\big  ) .   \n\\end{align}  \n\nThen it is straightforward to see that both events $\\mathcal L(\\delta)$ and $\\mathcal R(\\delta)$ will hold as long as $\\min_{q\\in [\\mclQ + 1]}|s_q - s_{q-1}|<\\frac{\\delta}{2}$, which is guaranteed if $\\mclQ > 3\\frac{n}{\\delta}$. For the proofs in \\Cref{sec: main proof mean}, \\Cref{sec: main proof linear}, and \\Cref{sec: main proof covariance}, we require that $\\mathcal L(\\mclB_n^{-1}\\Delta_{\\min})$ and $\\mathcal R(\\mclB_n^{-1}\\Delta_{\\min})$ hold. Therefore, for the theoretical results in \\Cref{section:main} to hold, $\\mclQ$ should satisfy that\n\\begin{equation*}\n    \\mclQ > \\frac{3n}{\\Delta_{\\min}}\\mclB_n.\n\\end{equation*}\nSince in our paper, $\\{\\mclB_n\\}_{n\\in \\mclZ^+}$ is a slowly diverging sequence, we can take it as $\\mclB_n = \\log(n)$ and then it suffices to take $\\mclQ = \\frac{4n}{\\Delta_{\\min}}\\log^2(n)$.\n\nUnder the fixed-$K$ setting of paper and when $\\{\\Delta_k\\}_{k\\in [K]}$ are of the same order, the existence of reference points will be guaranteed as long as $\\mclQ > 4\\log^2(n)$.\n\n\n\\paragraph{Goodness-of-fit.} The deviation of goodness-of-fit functions at reference points are different from the one that occurs in the proof of the vanilla DP, because the fitted parameters would have some bias since reference points may not locate at true change points. For different models, the deviation of the goodness-of-fit has different orders. We need to analyze each model separately. The deviations are described in \\Cref{lem:mean one change deviation bound}, \\Cref{lem:regression one change deviation bound}, and \\Cref{lem:cov one change deviation bound}.\n\n\\paragraph{Complexity analysis.} In \\Cref{lem: complexity divide step} we analyze the complexity of the divide step.\n\n\\bnlem[Complexity of the divide step]\n\\label{lem: complexity divide step}\nUnder all three models in \\Cref{section:main}, with a memorization technique, the computation complexity of \\Cref{algorithm:DP} would be $O(n\\mclQ\\cdot \\mclC_2(p))$.\n\\enlem\n\\bprf\nFor generality, suppose $\\{s_i\\}_{i\\in [\\mclQ]}$ is an arbitrary grid of integers over $(0,n)$, i.e., $0<s_1<s_2<\\cdots<s_{\\mclQ}<n$, and denote $s_0 = 0$, $s_{\\mclQ + 1} = n$, $\\delta_i = s_{i} - s_{i - 1}$ for $i\\in [\\mclQ + 1]$. \n\nUnder the three models in \\Cref{section:main}, calculating $\\widehat{\\theta}_{\\mclI}$ only involves summations like $\\sum_{i\\in \\mclI}X_i$, $\\sum_{i\\in \\mclI}X_iX_i^\\top$, $\\sum_{i\\in \\mclI}X_iy_i$. In $l$-th step ($l\\geq 1$) of the inner loop of \\Cref{algorithm:DP} at the right end point $s_r$, it suffices to remove $\\delta_l$ terms from the summation. Thus, the complexity for the inner loop at $s_r$ would be $O(s_r\\cdot \\mclC_2(p))$, and the total complexity would be\n$$\n\\sum_{r\\in [\\mclQ]} O(s_r\\cdot \\mclC_2(p)) = O(n\\mclQ\\cdot \\mclC_2(p)).\n$$\n\\eprf\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Mean model} \n\\label{sec: main proof mean}\n\n\nIn this section we show the proof of \\Cref{thm:DCDP mean}. Throughout this section, for any generic interval $\\I\\subset [1,n]$, denote $\\mu^*_{\\I} = \\frac{1}{|\\I|}\\sum_{i\\in \\I}\\mu^*_i$ and\n$$ \\widehat \\mu_\\I = \\argmin_{\\mu\\in \\mathbb{R}^p}\\frac{1}{|\\mclI|}\\sum_{i\\in \\mclI}\\|X_i - \\mu\\|_2^2 + \\frac{\\lambda}{\\sqrt{|\\mclI|}}\\|\\mu\\|_1. $$\nAlso, unless specially mentioned, in this section, we set the goodness-of-fit function $\\mathcal F  (\\I)$ in \\Cref{algorithm:DCDP} to be\n\\begin{equation}\n    \\mathcal F  (\\I) := \\begin{cases}\n    \\sum_{i \\in \\I } \\|X_i - \\widehat \\mu_\\I \\|_2 ^2, &\\text{ when } |\\I|\\geq C_\\mclF\\sigma_{\\epsilon}^2 { \\mathfrak{s} } \\log(n\\vee p),\\\\\n    0, &\\text{otherwise},\n    \\end{cases}.\n\\end{equation}\nwhere $C_\\mclF$ is a universal constant.\n \n\n\n\n\\paragraph{Assumptions.} For the ease of presentation, we combine the SNR condition we will use throughout this section and \\Cref{assp: DCDP_mean main} into a single assumption.\n\n\n\\bnassum[Mean model] \n\\label{assp: DCDP_mean}\nSuppose that \\Cref{assp: DCDP_mean main} holds. In addition, suppose that $ \\Delta_{\\min} \\kappa^2  \\geq  \\mathcal{B}_n   \\frak{s}\\log(n\\vee p)$ as is assumed in \\Cref{thm:DCDP mean}.\n\\enassum\n\n\n\n\n\n\\begin{proof}[Proof of \\Cref{thm:DCDP mean}] \nBy  \\Cref{prop:mean local consistency}, $K \\leq |\\widehat{\\mathcal{P}}| \\leq 3K$.  This combined with \\Cref{prop:mean change points partition size consistency} completes the proof.\n\\end{proof}\n\n\n\n \\bnprop\\label{prop:mean local consistency}\n Suppose  \\Cref{assp: DCDP_mean} holds. Let \n $\\widehat { \\mathcal P }  $ denote the output of  \\Cref{algorithm:DCDP}.\n Then with probability  at least $1 - C n ^{-3}$, the following conditions hold.\n\t\\begin{itemize}\n\t\t\\item [(i)] For each interval  $ \\I = (s, e] \\in \\widehat{\\mathcal{P}}$    containing one and only one true \n\t\t   change point $ \\eta_k $, it must be the case that\n $$\\min\\{ \\eta_k -s ,e-\\eta_k \\}  \\lesssim  \\sigma_{\\epsilon}^2\\bigg( \\frac{   { \\mathfrak{s} } \\log(n\\vee p) +\\gamma }{\\kappa_k^2 }\\bigg) +   { \\mathcal B_n^{-1}   \\Delta }       .$$\n\t\t\\item [(ii)] For each interval  $  \\I = (s, e] \\in \\widehat{\\mathcal{P}}$ containing exactly two true change points, say  $\\eta_ k  < \\eta_ {k+1} $, it must be the case that\n\t\t\t\\begin{align*} \n \\eta_k -s    \\lesssim       { \\mathcal B_n^{-1\/2}   \\Delta }     \\quad \\text{and} \\quad \n e-\\eta_{k+1}  \\lesssim    { \\mathcal B_n^{-1\/2}   \\Delta }      . \n \\end{align*} \n\t\t \n\\item [(iii)] No interval $\\I \\in \\widehat{\\mathcal{P}}$ contains strictly more than two true change points.\n\n\n\t\\item [(iv)] For all consecutive intervals $ \\I_1 $ and $ \\I_2 $ in $\\widehat{ \\mathcal P}$, the interval \n\t\t$ \\I_1 \\cup  \\I_2 $ contains at least one true change point.\n\t\t\t\t\n\t \\end{itemize}\n\\enprop \n\n\\bprf\nThe four cases are proved in \\Cref{lem:mean single cp}, \\Cref{lem:mean two change points}, \\Cref{lem:mean three or more cp}, and \\Cref{lem:mean two intervals}, respectively.\n\\eprf\n\n\n\n\n\n\\bnprop\\label{prop:mean change points partition size consistency}\nSuppose  \\Cref{assp: DCDP_mean} holds. Let \n $\\widehat { \\mathcal P }  $ be the output of  \\Cref{algorithm:DCDP}.\n Suppose \n$ \\gamma \\ge C_\\gamma K { \\mathcal B_n^{-1}   \\Delta }  \\kappa^2 $  for sufficiently large constant $C_\\gamma$. Then\n with probability  at least $1 - C n ^{-3}$,  $| \\widehat { \\mathcal P} | =K $.\n\\enprop\n\n\n\n\\begin{proof}[Proof of   \\Cref{prop:mean change points partition size consistency}] \nDenote $\\mathfrak{G} ^*_n = \\sum_{ i =1}^n \\|X_i  -  \\mu^*_i\\|_2^2$.  Given any collection $\\{t_1, \\ldots, t_m\\}$, where $t_1 < \\cdots < t_m$, and $t_0 = 0$, $t_{m+1} = n$, let \n\\begin{equation}\\label{eq: mean sn-def}\n\t { \\mathfrak{G} }  _n(t_1, \\ldots, t_{m}) = \\sum_{k=1}^{m} \\mclF(\\widehat \\mu_  {(t_{k}, t_{k+1}]}, (t_{k}, t_{k+1}]).\n\\end{equation}\nFor any collection of time points, when defining \\eqref{eq: mean sn-def}, the time points are sorted in an increasing order.\n\nLet $\\{ \\widehat \\eta_{k}\\}_{k=1}^{\\widehat K}$ denote the change points induced by $\\widehat {\\mathcal P}$.  Suppose we can justify that \n\t\\begin{align}\n\t\t { \\mathfrak{G} } ^*_n + K\\gamma  \\ge  & { \\mathfrak{G} }  _n(s_1,\\ldots,s_K)   + K\\gamma - C_1 ( K +1)\\sigma_{\\epsilon}^2  { \\mathfrak{s} }   \\log(n\\vee p) - C_1\\sum_{k\\in [K]}\\kappa_k^2 { \\mathcal B_n^{-1}   \\Delta }   \\label{eq:K consistency step 1} \\\\ \n\t\t\\ge &  { \\mathfrak{G} } _n (\\widehat \\eta_{1},\\ldots, \\widehat \\eta_{\\widehat K } ) +\\widehat K \\gamma - C_1 ( K +1)  \\sigma_{\\epsilon}^2  { \\mathfrak{s} }   \\log(n\\vee p) - C_1\\sum_{k\\in [K]}\\kappa_k^2 { \\mathcal B_n^{-1}   \\Delta }   \\label{eq:K consistency step 2} \\\\ \n\t\t\\ge &   { \\mathfrak{G} }  _n ( \\widehat \\eta_{1},\\ldots, \\widehat \\eta_{\\widehat K } , \\eta_1,\\ldots,\\eta_K ) + \\widehat K \\gamma    - 2C_1 ( K +1)  \\sigma_{\\epsilon}^2  { \\mathfrak{s} }   \\log(n\\vee p)- C_1\\sum_{k\\in [K]}\\kappa_k^2 { \\mathcal B_n^{-1}   \\Delta }   \\label{eq:K consistency step 3}\n\t\\end{align}\n\tand that \n\t\\begin{align}\\label{eq:K consistency step 4}\n\t\t { \\mathfrak{G} }  ^*_n   - { \\mathfrak{G} }  _n ( \\widehat \\eta_{1},\\ldots, \\widehat \\eta_{\\widehat K } , \\eta_1,\\ldots,\\eta_K ) \\le C_2 (K + \\widehat{K} + 2) \\sigma_{\\epsilon}^2  { \\mathfrak{s} }   \\log(n\\vee p).\n\t\\end{align}\n\tThen it must hold that $| \\mathcal{P} | = K$, as otherwise if $\\widehat K \\ge K+1 $, then  \n\t\\begin{align*}\n\t\tC _2 (K + \\widehat{K} + 2)  \\sigma_{\\epsilon}^2  { \\mathfrak{s} }   \\log(n\\vee p)  & \\ge    { \\mathfrak{G} }  ^*_n  - { \\mathfrak{G} }  _n ( \\widehat \\eta_{1},\\ldots, \\widehat \\eta_{\\widehat K } , \\eta_1,\\ldots,\\eta_K ) \\\\\n\t\t& \\ge      (\\widehat K - K)\\gamma  -2C_1 ( K +1)  \\sigma_{\\epsilon}^2  { \\mathfrak{s} }   \\log(n\\vee p) - 2C_1\\sum_{k\\in [K]}\\kappa_k^2 { \\mathcal B_n^{-1}   \\Delta } .\n\t\\end{align*} \n\tTherefore due to the assumption that $| \\widehat p|  =\\widehat K\\le 3K $, it holds that \n\t\\begin{align} \\label{eq:Khat=K} \n\t\t[C_2 (4K + 2) + 2C_1(K+1)] \\sigma_{\\epsilon}^2  { \\mathfrak{s} }   \\log(n\\vee p) +2C_1\\sum_{k\\in [K]}\\kappa_k^2 { \\mathcal B_n^{-1}   \\Delta }  \\ge  (\\widehat K - K)\\gamma \\geq \\gamma,\n\t\\end{align}\n\tNote that \\eqref{eq:Khat=K} contradicts the choice of $\\gamma$.\n\n\n\\\n\\\\\n{\\bf Step 1.} Note that \\eqref{eq:K consistency step 1}  is implied by \n\t\\begin{align}\\label{eq:step 1 K consistency}  \n\t\t\\left| \t { \\mathfrak{G} }  ^*_n -  { \\mathfrak{G} } _n(s_1,\\ldots,s_K)    \\right| \\le  C_3(K+1) \\sigma_{\\epsilon}^2  { \\mathfrak{s} }   \\log(n\\vee p) + C_3\\sum_{k\\in [K]}\\kappa_k^2 { \\mathcal B_n^{-1}   \\Delta } ,\n\t\\end{align}\n\twhich is  an  immediate consequence  of \\Cref{lem:mean one change deviation bound}. \n\t\\\n\t\\\\\n\t\\\\\n\t{\\bf Step 2.} Since $\\{ \\widehat \\eta_{k}\\}_{k=1}^{\\widehat K}$ are the change points induced by $\\widehat {\\mathcal P}$, \\eqref{eq:K consistency step 2} holds because $\\widehat {\\mathcal P}$ is a minimizer.\n\\\\\n{\\bf Step 3.} For every $ \\I  =(s,e]\\in \\widehat p$, by \\Cref{prop:mean local consistency}, we know that $\\I$ contains at most two change points. We only show the proof for the two-change-points case as the other case is easier. Denote\n\t\\[\n\t\t \\I =  (s ,\\eta_{q}]\\cup (\\eta_{q},\\eta_{q+1}] \\cup  (\\eta_{q+1} ,e]  = {  \\mathcal J } _1 \\cup {  \\mathcal J } _2 \\cup {  \\mathcal J } _{3},\n\t\\]\nwhere $\\{ \\eta_{q},\\eta_{q+1}\\} =\\I \\, \\cap \\, \\{\\eta_k\\}_{k=1}^K$. \n\nFor each $ m=1,2,3$, if $|{  \\mathcal J } _m|\\geq C_\\mclF\\sigma_{\\epsilon}^2 { \\mathfrak{s} } \\log(n\\vee p)$, then by \\Cref{lem:mean one change deviation bound}, it holds that\n$$\n\\sum_{ i \\in {  \\mathcal J } _m }\\|y_ i - \\widehat\\mu_{{  \\mathcal J } _m}\\|_2^2   \\leq \\sum_{ i \\in {  \\mathcal J } _ m } \\|y_ i - \\mu^*_i\\|_2^2 + C\\sigma_{\\epsilon}^2 { \\mathfrak{s} } \\log(n\\vee p). \n$$\nThus, we have\n\\begin{equation}\n    \\mclF(\\widehat\\mu_{{  \\mathcal J } _m},{  \\mathcal J } _m)\\leq \\mclF(\\mu^*_{{  \\mathcal J } _m},{  \\mathcal J } _m) + C\\sigma_{\\epsilon}^2 { \\mathfrak{s} } \\log(n\\vee p).\n\\end{equation}\nOn the other hand, by \\Cref{lem:mean loss deviation no change point}, we have\n$$\n\\mclF(\\widehat\\mu_{\\I},{  \\mathcal J } _m)   \\geq \\mclF(\\mu^*_{{  \\mathcal J } _m},{  \\mathcal J } _m) - C\\sigma_{\\epsilon}^2 { \\mathfrak{s} } \\log(n\\vee p). \n$$\nTherefore the last two inequalities above imply that \n\\begin{align}   \n\\mclF(\\widehat \\mu_\\I,\\I) \\geq & \\sum_{m=1}^{3} \\mclF(\\widehat \\mu_\\I,{  \\mathcal J } _m) \\nonumber \\\\\n\\geq & \\sum_{m=1}^{3}   \\mclF(\\widehat{\\mu}_{{  \\mathcal J } _m},{  \\mathcal J } _m) - 6C\\sigma_{\\epsilon}^2 { \\mathfrak{s} } \\log(n\\vee p).\n\\label{eq:K consistency step 3 inequality 3}\n\\end{align}\nNote that \\eqref{eq:K consistency step 3} is an immediate consequence of   \\eqref{eq:K consistency step 3 inequality 3}.\n\n{\\bf Step 4.}\nFinally, to show \\eqref{eq:K consistency step 4},   let  $\\widetilde { \\mathcal P}$ denote the partition induced by $\\{\\widehat \\eta_{1},\\ldots, \\widehat \\eta_{\\widehat K } , \\eta_1,\\ldots,\\eta_K\\}$. Then \n$| \\widetilde { \\mathcal P} | \\le K + \\widehat K+2   $ and that $\\mu^*_i$ is unchanged in every interval $\\I \\in \\widetilde { \\mathcal P}$. \nSo  \\Cref{eq:K consistency step 4} is an immediate consequence of   \\Cref{lem:mean one change deviation bound}.\n\\end{proof} \n\n\n\n\n\n\n\n\n\\subsection{Fundamental lemmas}\n\n\\bnlem[Deviation, mean model]\n\\label{lem:mean one change deviation bound}\nLet $\\mathcal I =(s,e] \\subset (0, n] $ be any generic interval and $$\\widehat \\mu_\\I = \\argmin_{\\mu}  \\frac{1}{|\\mclI|}\\sum_{i\\in \\mclI}\\|X_i - \\mu\\|_2^2 + \\frac{\\lambda}{\\sqrt{|\\mclI|}}\\|\\mu\\|_1   .$$  \n\n{\\bf a.} If $\\I$ contains no change points, then it holds that \n$$\\mathbb P \\bigg( \\bigg|   \\sum_{ i  \\in \\I } \\|X_i   -\\widehat \\mu_\\I   \\|_2^2  - \\sum_{ i   \\in \\I } \\|X_i      - \\mu^*_i   \\|_2^2  \\bigg|  \\ge C \\sigma_{\\epsilon}^2\\mathfrak{s}\\log(n\\vee p)  \\bigg)  \\le  (n\\vee p)^{-3}. $$\n\n{\\bf b.}  Suppose that the interval $ \\I $ contains one and only one change point $ \\eta_k $. Denote\n$$  \\mathcal J = (s,\\eta_k] \\quad \\text{and} \\quad  \\mathcal J' =  (\\eta_k, e]  .$$   Then it holds that \n$$\\mathbb P \\bigg( \\bigg|   \\sum_{ i  \\in \\I } \\|X_i   -\\widehat \\mu_\\I   \\|^2  - \\sum_{ i   \\in \\I } \\|X_i      - \\mu^*_i    \\|^2  \\bigg|  \\geq 2\\frac{ | {  \\mathcal J } ||{  \\mathcal J } '| }{ |\\I| }  \\kappa_k ^2  +C \\sigma_{\\epsilon}^2\\mathfrak{s}\\log(n\\vee p)  \\bigg)   \\le  (n\\vee p)^{-3}. $$\n\\enlem\n\\begin{proof} \nWe show {\\bf b} as {\\bf a} immediately follows from {\\bf b} with $ |{  \\mathcal J } '| =0$. Denote \n$$  \\mathcal J = (s,\\eta_k] \\quad \\text{and} \\quad  \\mathcal J' =  (\\eta_k, e]  .$$ \nDenote  $\\mu_\\I = \\frac{1}{|\\I | } \\sum_{i\\in \\I } \\mu^*_i $.\nThe it holds that \n\\begin{equation*}\n    \\begin{split}\n        &\\sum_{i\\in \\I} \\|X_i -\\widehat \\mu_\\I\\|_2^2  - \\sum_{i\\in \\I} \\|X_i - \\mu^*_i\\|_2^2  \\\\\n        =& \\sum_{i \\in \\I } \\|\\widehat \\mu_\\I-\\mu^*_i \\|_2^2  -2 \\sum_{i \\in \\I } \\epsilon_i^\\top  (\\widehat \\mu_\\I  - \\mu^*_i)\\\\\n        \\leq& 2\\sum_{i \\in \\I } \\|\\widehat \\mu_\\I-\\mu^*_\\I \\|_2^2 + 2\\sum_{i \\in \\I } \\|\\mu^*_\\I-\\mu^*_i \\|_2^2  -2 \\sum_{i \\in \\I } \\epsilon_i^\\top  (\\widehat \\mu_\\I  - \\mu^*_\\I) - 2 \\sum_{i \\in \\I } \\epsilon_i^\\top  (\\mu^*_\\I  - \\mu^*_i).\n    \\end{split}\n\\end{equation*} \nObserve that \n\\begin{equation*}\n    \\mathbb P \\bigg(\\|\\sum_{i\\in \\I } \\epsilon_i \\|_{\\infty}  \\geq C\\sigma_{\\epsilon}\\sqrt {\\log(n\\vee p)|\\I |   }\\bigg) \\leq (n\\vee p)^{-5 } \n\\end{equation*}\nSuppose this good  event holds.\n\n{\\bf Step 1.} By the event and \\Cref{lem:estimation_high_dim_mean}, we have\n\\begin{equation*}\n    \\begin{split}\n        \\sum_{i \\in \\I } \\|\\widehat \\mu_\\I-\\mu^*_\\I \\|_2^2 &\\leq C\\sigma_{\\epsilon}^2 { \\mathfrak{s} } \\log(n\\vee p),\\\\\n        |2 \\sum_{i \\in \\I } \\epsilon_i^\\top  (\\widehat \\mu_\\I  - \\mu^*_\\I)| &\\leq |\\I|\\|\\sum_{i\\in \\I}\\epsilon_i\\|_{\\infty}\\|\\widehat \\mu_\\I-\\mu^*_\\I\\|_1\\leq C\\sigma_{\\epsilon}^2 { \\mathfrak{s} } \\log(n\\vee p).\n    \\end{split}\n\\end{equation*}\n{\\bf Step 2.} Notice that \n\\begin{align*} \n    \\sum_{   i \\in \\I } \\|\\mu^*_\\I-\\mu^*_i\\| ^2  =  \n        & \\sum_{   i \\in \\I }  \\| \\frac{ | {  \\mathcal J }  | \\mu^*_{  \\mathcal J }  + | {  \\mathcal J } '| \\mu^*_{ {  \\mathcal J }  ' }}{|\\I| } -\\mu^*_i  \\|_2 ^2\\\\\n =  &     \\sum_{i\\in {  \\mathcal J }  }   \\| \\frac{ |{  \\mathcal J }  '|  (\\mu^*_{  \\mathcal J }   - \\mu^*_{{  \\mathcal J } '} ) }{|\\I| }  \\|_2^2 + \n     \\sum_{i\\in {  \\mathcal J } ' }   \\| \\frac{ |{  \\mathcal J }   |  (\\mu^*_{  \\mathcal J }   - \\mu^*_{{  \\mathcal J } '} ) }{|\\I| }  \\|_2^2\n  \\\\ = &  \\frac{|{  \\mathcal J }  | | {  \\mathcal J } '| }{|\\I|  }\\|\\mu^*_{  \\mathcal J }   - \\mu^*_{{  \\mathcal J } '}\\|_2 ^2 = \\frac{|{  \\mathcal J }  | | {  \\mathcal J } '| }{|\\I|  }    \\kappa_k  ^2.\n    \\end{align*}\nMeanwhile, it holds that \n\\begin{align*} \n\\sum_{  i     \\in \\I } \\epsilon_i ^\\top (\\mu^*_\\I  - \\mu^*_i  ) =\n& \\sum_{   i \\in \\I }  \\epsilon_i^\\top  \\bigg ( \\frac{ | {  \\mathcal J }  | \\mu^*_{  \\mathcal J }  + | {  \\mathcal J } '| \\mu^*_{ {  \\mathcal J }  ' }}{|\\I| }-\\mu^*_i  \\bigg )  \n\\\\\n=&  \\frac{|{  \\mathcal J } '|}{|\\I|}\\sum_{i\\in {  \\mathcal J }  }   \\epsilon_i^\\top(    \\mu^*_{{  \\mathcal J } '}-\\mu^*_{  \\mathcal J }   ) + \n   \\frac{|{  \\mathcal J } |}{|\\I|}\\sum_{i\\in {  \\mathcal J } ' }   \\epsilon_i^\\top(    \\mu^*_{{  \\mathcal J } }-\\mu^*_{{  \\mathcal J } '}  )\n   \\\\\n   \\le &    C_2\\sigma_{\\epsilon}\\sqrt {   \\frac{|{  \\mathcal J }  | | {  \\mathcal J } '| }{|\\I|  }     \\kappa_k  ^2  \\log(n\\vee p) }\n   \\le \\frac{|{  \\mathcal J }  | | {  \\mathcal J } '| }{|\\I|}   \\kappa_k  ^2    +  C\\sigma_{\\epsilon}^2\\log(n\\vee p)  ,\n\\end{align*}\nwhere the first inequality follows from the fact that the variance is upper bounded by\n$$\\sum_{i\\in {  \\mathcal J }  }  \\sigma_{\\epsilon}^2 \\frac{ |{  \\mathcal J }  '|^2 }{|\\I|^2 } \\|\\mu^*_{  \\mathcal J }   - \\mu^*_{{  \\mathcal J } '} \\|_2^2 + \\sum_{i\\in {  \\mathcal J } ' }  \\sigma_{\\epsilon}^2 \\frac{ |{  \\mathcal J } |^2 }{|\\I|^2 } \\|\\mu^*_{  \\mathcal J }   - \\mu^*_{{  \\mathcal J } '} \\|_2^2 =    \\frac{|{  \\mathcal J }  | | {  \\mathcal J } '| }{|\\I|  }\\sigma_{\\epsilon}^2 \\kappa_k^2 . $$\n\\end{proof} \n\n\n\n\n\\bnlem\n\\label{lem:estimation_high_dim_mean}\nFor any interval $\\mclI\\subset [1, n]$ with $|\\mclI|\\geq C_0\\mathfrak{s}\\log(n\\vee p)$ that   contains finitely many  change points. Let\n\\begin{equation*}\n    \\widehat {\\mu}_{\\mclI}\\defined \\argmin_{\\mu\\in \\mathbb{R}^p}\\frac{1}{|\\mclI|} \\sum_{i\\in \\I }\\|X_i - \\mu\\|_2^2 + \\frac{\\lambda}{\\sqrt{|\\mclI|}}\\|\\mu\\|_1,\n\\end{equation*}\nfor $\\lambda = C_{\\lambda}\\sigma_{\\epsilon}\\sqrt{\\log(n\\vee p)}$ for sufficiently large constant $C_{\\lambda}$. Then it holds with probability at least $1 - (n\\vee p)^{-5}$ that\n\\begin{equation}\n    \\begin{split}\n    \\|\\widehat{\\mu}_{\\I}-\\mu^*_{\\I}\\|_{2}^{2} &\\leq \\frac{C\\sigma_{\\epsilon}^2  { \\mathfrak{s} }  \\log (n\\vee p)}{\\I} \\\\\n\\left\\|\\widehat{\\mu}_{\\I}-\\mu^*_{\\I}\\right\\|_{1} &\\leq C \\sigma_{\\epsilon} { \\mathfrak{s} }  \\sqrt{\\frac{\\log (n\\vee p)}{|\\I|}} \\\\\n\\|\\left(\\widehat{\\mu}_{\\I}-\\mu^*_{\\I}\\right)_{S^{c}}\\|_{1} &\\leq 3\\|\\left(\\widehat{\\mu}_{\\I}-\\mu^{*}_{\\I}\\right)_{S}\\|_{1},\n    \\end{split}\n\\end{equation}\nwhere $\\mu^*_\\mclI = \\frac{1}{|\\I|}\\sum_{i\\in \\I}\\mu^*_i$.\n\\enlem\n\n\n\\begin{proof}\nBy definition, we have $L(\\widehat{\\mu}_{\\I},\\I)\\leq L({\\mu}_{\\I},\\I)$, that is\n\\begin{align}\n\\nonumber\n    &\\sum_{i\\in \\I}\\|Y_i - \\widehat{\\mu}_{\\I}\\|_2^2 + \\lambda\\sqrt{|\\I|}\\|\\widehat{\\mu}_{\\I}\\|_1\\leq \\sum_{i\\in \\I}\\|Y_i - {\\mu}^*_{\\I}\\|_2^2 + \\lambda\\sqrt{|\\I|}\\|{\\mu}^*_{\\I}\\|_1\\\\\n\\nonumber\n    \\Rightarrow & \\sum_{i\\in \\I}(\\widehat{\\mu}_{\\I} - \\mu^*_{\\I})^\\top (2Y_i - \\mu^*_{\\I} - \\widehat{\\mu}_{\\I}) + \\lambda\\sqrt{|\\I|}[\\|{\\mu}^*_{\\I}\\|_1 - \\|\\widehat{\\mu}_{\\I}\\|_1]\\geq 0 \\\\\n    \\nonumber\n    \\Rightarrow & (\\widehat{\\mu}_{\\I} - \\mu^*_{\\I})^\\top (\\sum_{i\\in \\I}\\epsilon_i) + 2(\\widehat{\\mu}_{\\I} - \\mu^*_{\\I})^\\top\\sum_{i\\in \\I}(\\mu^*_i - \\mu^*_{\\I}) - |\\I|\\sum_{i\\in \\I}\\|\\widehat{\\mu}_{\\I} - \\mu^*_{\\I}\\|_2^2 + \\lambda\\sqrt{|\\I|}[\\|{\\mu}^*_{\\I}\\|_1 - \\|\\widehat{\\mu}_{\\I}\\|_1]\\geq 0\\\\\n    \\Rightarrow & \\|\\widehat{\\mu}_{\\I} - \\mu^*_{\\I}\\|_1 \\|2\\sum_{i\\in \\I}\\epsilon_i\\|_{\\infty} + \\lambda\\sqrt{|\\I|}[\\|{\\mu}^*_{\\I}\\|_1 - \\|\\widehat{\\mu}_{\\I}\\|_1]\\geq |\\I|\\sum_{i\\in \\I}\\|\\widehat{\\mu}_{\\I} - \\mu^*_{\\I}\\|_2^2.\n    \\label{tmp_eq:mean_concentration}\n\\end{align}\nBy a union bound, we know that for some universal constant $C>0$, with probability at least $1 - (n\\vee p)^{-5}$,\n$$\n\\|\\sum_{i\\in \\I}\\epsilon_i\\|_{\\infty}\\leq C\\sigma_{\\epsilon}\\sqrt{|\\I|\\log(n\\vee p)}\\leq \\frac{\\lambda}{4}\\sqrt{|\\I|},\n$$\nas long as $C_{\\lambda}$ is sufficiently large. Therefore, based on the sparsity assumption in \\Cref{assp: DCDP_mean}, it holds that\n\\begin{align*}\n    &\\frac{\\lambda}{2}\\|\\widehat{\\mu}_{\\I} - \\mu^*_{\\I}\\|_1 + \\lambda[\\|{\\mu}^*_{\\I}\\|_1 - \\|\\widehat{\\mu}_{\\I}\\|_1]\\geq 0 \\\\\n    \\Rightarrow & \\frac{\\lambda}{2}\\|\\widehat{\\mu}_{\\I} - \\mu^*_{\\I}\\|_1 + \\lambda[\\|({\\mu}^*_{\\I})_S\\|_1 - \\|(\\widehat{\\mu}_{\\I})_S\\|_1]\\geq \\lambda \\|(\\widehat{\\mu}_{\\I})_{S^c}\\|_1\\\\\n    \\Rightarrow & \\frac{\\lambda}{2}\\|\\widehat{\\mu}_{\\I} - \\mu^*_{\\I}\\|_1 + \\lambda\\|({\\mu}^*_{\\I}-\\widehat{\\mu}_{\\I})_S\\|_1 \\geq \\lambda \\|(\\mu^*_{\\I} - \\widehat{\\mu}_{\\I})_{S^c}\\|_1\\\\\n    \\Rightarrow & 3\\|({\\mu}^*_{\\I}-\\widehat{\\mu}_{\\I})_S\\|_1\\geq \\|({\\mu}^*_{\\I}-\\widehat{\\mu}_{\\I})_{S^c}\\|_1.\n\\end{align*}\nNow from \\Cref{tmp_eq:mean_concentration} we can get\n\\begin{align*}\n    |\\I|\\|\\widehat{\\mu}_{\\I} - \\mu^*_{\\I}\\|_2^2\\leq &\\frac{3\\lambda}{2}\\sqrt{|\\I|}\\|\\widehat{\\mu}_{\\I} - \\mu^*_{\\I}\\|_1\\\\\n    \\leq & \\frac{12\\lambda}{2}\\sqrt{|\\I|}\\|(\\widehat{\\mu}_{\\I} - \\mu^*_{\\I})_S\\|_1\\\\\n    \\leq & 6\\lambda \\sqrt{ { \\mathfrak{s} } } \\sqrt{|\\I|}\\|(\\widehat{\\mu}_{\\I} - \\mu^*_{\\I})_S\\|_2\\\\\n    \\leq & 6\\lambda \\sqrt{ { \\mathfrak{s} } } \\sqrt{|\\I|}\\|\\widehat{\\mu}_{\\I} - \\mu^*_{\\I}\\|_2,\n\\end{align*}\nwhich implies that\n$$\n\\|\\widehat{\\mu}_{\\I} - \\mu^*_{\\I}\\|_2\\leq 6C_{\\lambda}\\sigma_{\\epsilon}\\sqrt{\\frac{ { \\mathfrak{s} } \\log(n\\vee p)}{|\\I|}}.\n$$\nThe other inequality follows accordingly.\n\\end{proof}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\subsection{Technical lemmas}\nThroughout this section, let $\\widehat { \\mathcal P }  $ denote the output of  \\Cref{algorithm:DCDP}.\n \n\n\n\\bnlem[No change point]\n\\label{lem:mean loss deviation no change point}\nLet $\\I\\subset [1,\\ldots, n]$ be any interval that contains no change point. Then for any interval ${  \\mathcal J }  \\supset \\I$, it holds with probability at least $1 - (n\\vee p)^{-5}$ that\n\\begin{equation*}\n    \\mclF(\\mu^*_{\\I},\\I)\\leq \\mclF(\\widehat{\\mu}_{  \\mathcal J } ,\\I) + C\\sigma_{\\epsilon}^2 { \\mathfrak{s} } \\log(n\\vee p).\n\\end{equation*}\n\\enlem\n\\begin{proof}\n{\\bf Case 1}. If $|\\I|< C_\\mclF\\sigma_{\\epsilon} { \\mathfrak{s} } \\log(n\\vee p)$, then by definition, we have $\\mclF(\\mu^*_{\\I},\\I) = \\mclF(\\widehat \\mu^*_{{  \\mathcal J } },\\I) = 0$ and the inequality holds.\n\n{\\bf Case 2}. If $|\\I|\\geq  C_\\mclF\\sigma_{\\epsilon} { \\mathfrak{s} } \\log(n\\vee p)$, then take difference and we can get\n\\begin{align*}\n    &\\sum_{i\\in \\I}\\|X_i - \\mu^*_i\\|_2^2 - \\sum_{i\\in \\I}\\|X_i - \\widehat{\\mu}_{  \\mathcal J } \\|_2^2 \\\\\n    =& 2(\\widehat \\mu_{{  \\mathcal J } } - \\mu^*_{\\I})^\\top\\sum_{i\\in \\I}\\epsilon_i -|\\I|\\|\\mu^*_{\\I} - \\widehat \\mu_{{  \\mathcal J } }\\|_2^2\\\\\n    \\leq & 2(\\|(\\widehat \\mu_{{  \\mathcal J } } - \\mu^*_{\\I})_S\\|_1 + \\|(\\widehat \\mu_{{  \\mathcal J } } - \\mu^*_{\\I})_{S^c}\\|_1)\\|\\sum_{i\\in \\I}\\epsilon_i\\|_{\\infty} -|\\I|\\|\\mu^*_{\\I} - \\widehat \\mu_{{  \\mathcal J } }\\|_2^2\\\\\n    \\leq& c_1\\|\\widehat \\mu_{{  \\mathcal J } } - \\mu^*_{\\I}\\|_2\\sigma_{\\epsilon}\\sqrt{ { \\mathfrak{s} } |\\I|\\log(n\\vee p)} + c_2\\sigma_{\\epsilon} { \\mathfrak{s} } \\sqrt{\\frac{\\log(n\\vee p)}{|\\I|}}\\cdot c_1\\sigma_{\\epsilon}\\sqrt{|\\I|\\log(n\\vee p)} - |\\I|\\|\\mu^*_{\\I} - \\widehat \\mu_{{  \\mathcal J } }\\|_2^2 \\\\\n    \\leq & \\frac{1}{2}|\\I|\\|\\mu^*_{\\I} - \\widehat \\mu_{{  \\mathcal J } }\\|_2^2 + 2c_1^2\\sigma_{\\epsilon}^2 { \\mathfrak{s} } \\log(n\\vee p) + c_2\\sigma_{\\epsilon}^2 { \\mathfrak{s} } \\log(n\\vee p) - |\\I|\\|\\mu^*_{\\I} - \\widehat \\mu_{{  \\mathcal J } }\\|_2^2\\\\\n    \\leq & C\\sigma_{\\epsilon}^2 { \\mathfrak{s} } \\log(n\\vee p),\n\\end{align*}\nwhere in the second inequality we use the definition of the index set $S$ and \\Cref{lem:estimation_high_dim_mean}.\n\\end{proof}\n\n\n\n\n\n\n \n\\bnlem[Single change point]\n\\label{lem:mean single cp}\nSuppose the good events \n$\\mathcal L ( { \\mathcal B_n^{-1}   \\Delta }  ) $ and  $\\mathcal R ( { \\mathcal B_n^{-1}   \\Delta } ) $ defined  in \\Cref{eq:left and right approximation of change points} hold. \nLet   $ \\I=(s,e] \\in \\mathcal {\\widehat P}  $  be such that $\\I$ contains exactly  one    change point $ \\eta_k $. \n Then  with probability  at least $1-(n\\vee p)^{-3}$,   it holds that \n $$\\min\\{ \\eta_k -s ,e-\\eta_k \\}  \\lesssim    \\sigma_{\\epsilon}^2\\bigg( \\frac{  { \\mathfrak{s} } \\log(n\\vee p) +\\gamma }{\\kappa_k^2 }\\bigg)  +   { \\mathcal B_n^{-1}   \\Delta }     .$$\n  \\enlem\n\\begin{proof}   \nIf either $ \\eta_k -s \\le  { \\mathcal B_n^{-1}   \\Delta }   $ or $e-\\eta_k\\le  { \\mathcal B_n^{-1}   \\Delta } $, then there is nothing to show. So assume that \n$$  \\eta_k -s  >  { \\mathcal B_n^{-1}   \\Delta }  \\quad \\text{and} \\quad   e-\\eta_k  >  { \\mathcal B_n^{-1}   \\Delta }  . $$\nBy event $\\mathcal R ( { \\mathcal B_n^{-1}   \\Delta }  )$, there exists  $ s_ u  \\in \\{ s_q\\}_{q=1}^ { \\mathcal Q} $ such that \n$$0\\le s_u - \\eta_k \\le  { \\mathcal B_n^{-1}   \\Delta } . $$\n So \n$$ \\eta_k \\le s_ u \\le e .$$\nDenote \n$$ \\I_ 1 = (s,s_u] \\quad \\text{and} \\quad \\I_2 = (s_u, e] .$$\nSince \n$s, e,  s_u \\in  \\{ s_q\\}_{q=1}^ { \\mathcal Q}   $,   it follows that \n\\begin{align}\\nonumber\n\\sum_{i \\in \\I }\\|X_i - \\widehat \\mu_\\I \\|_2 ^2 \\le &  \\sum_{i \\in \\I_1 }\\|X_i - \\widehat \\mu_{\\I_1}  \\|_2 ^2  + \\sum_{i \\in \\I_2 }\\|X_i - \\widehat \\mu_{\\I_2}  \\|_2 ^2   + \\gamma \n\\\\\\nonumber \n\\le &  \\sum_{i \\in \\I_1 }\\|X_i -  \\mu^*_i \\|_2 ^2 + C_1 \\big (  \\sigma_{\\epsilon}^2 { \\mathfrak{s} } \\log(n\\vee p) + (s_u -\\eta_k) \\kappa_k^2 \\big )  \n\\\\ \\nonumber\n& + \\sum_{i \\in \\I_1 }\\|X_i -  \\mu^*_i \\|_2 ^2 + C_1\\sigma_{\\epsilon}^2 { \\mathfrak{s} } \\log(n\\vee p)  + \\gamma\n\\\\ \\nonumber \n=&  \\sum_{i \\in \\I }\\|X_i -  \\mu^*_i  \\|_2 ^2  +C_2 \\big (  \\sigma_{\\epsilon}^2 { \\mathfrak{s} } \\log(n\\vee p) + (s_u -\\eta_k)   \\kappa_k^2 \\big )+ \\gamma \n\\\\\n\\le &  \\sum_{i \\in \\I }\\|X_i -  \\mu^*_i \\|_2 ^2  +C_2 \\big (  \\sigma_{\\epsilon}^2 { \\mathfrak{s} } \\log(n\\vee p) +   { \\mathcal B_n^{-1}   \\Delta }     \\kappa_k^2   \\big ) + \\gamma  , \\label{eq:one change point step 1}\n\\end{align}\nwhere the first inequality follows from the fact that $ \\I =(s ,e ] \\in \\widehat { \\mathcal P} $ and so it is the local minimizer, the second inequality follows from \\Cref{lem:mean one change deviation bound} {\\bf a} and {\\bf b} and the observation that \n$$  \\eta_k -s >  { \\mathcal B_n^{-1}   \\Delta }   \\ge s_u -\\eta_k $$ \nDenote \n$$ {  \\mathcal J } _1  = (s,\\eta_k ] \\quad \\text{and} \\quad {  \\mathcal J } _2 = (\\eta_k , e] .$$\n\\Cref{eq:one change point step 1}  gives \n\\begin{align*}\n\\sum_{i \\in {  \\mathcal J } _1 }\\|X_i - \\widehat \\mu_\\I \\|_2 ^2 +\\sum_{i \\in {  \\mathcal J } _2 }\\|X_i - \\widehat \\mu_\\I \\|_2 ^2  \n \\le \\sum_{i \\in {  \\mathcal J } _1  }\\|X_i -  \\mu^*_{ {  \\mathcal J } _1 } \\|_2 ^2   +\\sum_{i \\in {  \\mathcal J } _2  }\\|X_i -  \\mu^*_{ {  \\mathcal J } _2  } \\|_2 ^2  +C_2  \\big (\\sigma_{\\epsilon}^2 { \\mathfrak{s} } \\log(n\\vee p) +    { \\mathcal B_n^{-1}   \\Delta }  \\kappa_k^2  \\big )  + \\gamma ,\n\\end{align*}\nwhich leads to \n\\begin{align*}\n&\\sum_{i \\in {  \\mathcal J } _1 }\\|\\widehat \\mu_\\I  - \\mu^*_{{  \\mathcal J } _1 }\\|_2^2 +\\sum_{i \\in {  \\mathcal J } _2 }\\|\\widehat \\mu_\\I  - \\mu^*_{{  \\mathcal J } _2  } \\|_2^2  \n\\\\\n \\leq  & 2 \\sum_{i \\in {  \\mathcal J } _1  } \\epsilon_i^\\top (\\widehat \\mu_\\I  -  \\mu^*_{ {  \\mathcal J } _1 } )     +2 \\sum_{i \\in {  \\mathcal J } _2  } \\epsilon_i^\\top (\\widehat \\mu_\\I  -  \\mu^*_{ {  \\mathcal J } _2 } )      +C_2  \\big (\\sigma_{\\epsilon}^2 { \\mathfrak{s} } \\log(n\\vee p) +  \\kappa_k^2   { \\mathcal B_n^{-1}   \\Delta }     \\big )  + \\gamma \n \\\\\n \\leq & 2\\sigma_{\\epsilon}\\sum_{j = 1,2}\\|\\widehat \\mu_\\I  -  \\mu^*_{ {  \\mathcal J } _j }\\|_2 \\sqrt{|{  \\mathcal J } _j|\\log(n\\vee p)}+   C_2  \\big( \\sigma_{\\epsilon}^2 { \\mathfrak{s} } \\log(n\\vee p) +  \\kappa_k^2 { \\mathcal B_n^{-1}   \\Delta } \\big ) + \\gamma  \\\\\n \\leq & \\frac{1}{2} \\sum_{j = 1,2}|{  \\mathcal J } _j|\\|\\widehat \\mu_\\I  -  \\mu^*_{ {  \\mathcal J } _j }\\|_2^2 + C_3  \\big (\\sigma_{\\epsilon}^2 { \\mathfrak{s} } \\log(n\\vee p) +  \\kappa_k^2   { \\mathcal B_n^{-1}   \\Delta }     \\big )  + \\gamma,\n\\end{align*} \nwhere the second inequality holds because the Orlicz norm $\\|\\cdot\\|_{\\psi_2}$ of $\\sum_{i \\in {  \\mathcal J } _1  } \\epsilon_i^\\top (\\mu_\\I  -  \\mu^*_{ {  \\mathcal J } _1 } )$ is upper bounded by $|{  \\mathcal J } _1|\\sigma^2_{\\epsilon}\\|\\mu_\\I  -  \\mu^*_{ {  \\mathcal J } _1 }\\|_2^2$.\n\nIt follows that \n$$ |{  \\mathcal J } _1 |\\|  \\widehat \\mu_\\I  - \\mu^*_{{  \\mathcal J } _1 }\\|_2 ^2 + |{  \\mathcal J } _2 |\\|  \\widehat \\mu_\\I  - \\mu^*_{{  \\mathcal J } _2 }\\|_2 ^2  =  \\sum_{i \\in {  \\mathcal J } _1 }\\|  \\widehat \\mu_\\I  - \\mu^*_{{  \\mathcal J } _1 }\\|_2 ^2 +\\sum_{i \\in {  \\mathcal J } _2 }\\| \\widehat \\mu_\\I  - \\mu^*_{{  \\mathcal J } _2  }  \\|_2^2    \\le C_4\\big ( \\sigma_{\\epsilon}^2 { \\mathfrak{s} } \\log(n\\vee p) + { \\mathcal B_n^{-1}   \\Delta }  \\kappa_k^2    \\big )+ 2\\gamma  .$$\nNote that \n$$ \\inf_{ a \\in \\mathbb R  }  |{  \\mathcal J } _1 |\\|  a - \\mu^*_{{  \\mathcal J } _1 }\\|_2 ^2 + |{  \\mathcal J } _2 |\\| a  - \\mu^*_{{  \\mathcal J } _2 }\\| ^2  = \\kappa_k ^2  \\frac{|{  \\mathcal J } _1| |{  \\mathcal J } _2|}{| \\I| }   \\ge \\frac{ \\kappa_k^2 }{2} \\min\\{ |{  \\mathcal J } _1| ,|{  \\mathcal J } _2| \\}    . $$ \nThis leads to \n$$  \\frac{ \\kappa_k^2 }{2}\\min\\{ |{  \\mathcal J } _1| ,|{  \\mathcal J } _2| \\}    \\le  C_4\\big ( \\sigma_{\\epsilon}^2 { \\mathfrak{s} } \\log(n\\vee p) + { \\mathcal B_n^{-1}   \\Delta }   \\kappa_k^2   + \\gamma  \\big )  ,$$\nwhich is \n$$ \\min\\{ |{  \\mathcal J } _1| ,|{  \\mathcal J } _2| \\}   \\le C_5 \\bigg( \\frac{\\sigma_{\\epsilon}^2 { \\mathfrak{s} } \\log(n\\vee p) + \\gamma  }{\\kappa_k^2 } +   { \\mathcal B_n^{-1}   \\Delta }   \\bigg) .$$\n \\end{proof} \n \n \n \\bnlem [Two change points]\n \\label{lem:mean two change points}\nSuppose the good events \n$\\mathcal L ( { \\mathcal B_n^{-1}   \\Delta }  ) $ and  $\\mathcal R (  { \\mathcal B_n^{-1}   \\Delta }  ) $ defined  in  \\Cref{eq:left and right approximation of change points} hold. \nLet   $ \\I=(s,e] \\in \\mathcal {\\widehat P}  $  be an interval that contains  exactly two  change points $ \\eta_k,\\eta_{k+1}  $.  Suppose in addition that \n\\begin{align} \\label{eq:1D two change points snr}\n\\Delta_{\\min} \\kappa^2 \\ge C    { \\mathcal B_n} ^{1\/2}  \\big(\\sigma_{\\epsilon}^2 { \\mathfrak{s} } \\log(n\\vee p) + \\gamma)    \n\\end{align} \nfor sufficiently large constant $C $. \n Then  with probability  at least $1-(n\\vee p)^{-3}$,   it holds that \n\\begin{align*} \n \\eta_k -s    \\lesssim       { \\mathcal B_n^{-1\/2}   \\Delta }     \\quad \\text{and} \\quad \n e-\\eta_{k+1}  \\lesssim    { \\mathcal B_n^{-1\/2}   \\Delta }      . \n \\end{align*} \n \\enlem \n\\begin{proof} \nSince the events $\\mathcal L (  { \\mathcal B_n^{-1}   \\Delta }   ) $ and  $\\mathcal R (  { \\mathcal B_n^{-1}   \\Delta }  ) $ hold, let $ s_u, s_v$ be such that \n $\\eta_k \\le s_u \\le s_v \\le \\eta_{k+1} $ and that \n $$ 0 \\le s_u-\\eta_k \\le    { \\mathcal B_n^{-1}   \\Delta }  ,  \\quad  0\\le  \\eta_{k+1} - s_v \\le  { \\mathcal B_n^{-1}   \\Delta }    .    $$\n\n \\begin{center}\n \\begin{tikzpicture} \n\\draw[ - ] (-10,0)--(1,0);\n \\node[color=black] at (-8,-0.3) {\\small s};\n \\draw[  (-, ultra thick, black] (-8,0) -- (-7.99,0);\n  \\draw[ { -]}, ultra thick, black] (-1.0002,0) -- (-1,0);\n\n   \\node[color=black] at (-7,-0.3) {\\small $\\eta_k$};\n\\draw(-7 ,0)circle [radius=2pt] ;\n\n  \\node[color=black] at (-6.5,-0.3) {\\small $s_u$};\n\\draw plot[mark=x, mark options={color=black, scale=1.5}] coordinates {(-6.5,0)  }; \n \n\n  \\node[color=black] at (-2.3,-0.3) {\\small $\\eta_{k+1}$};\n\\draw(-2.3 ,0)circle [radius=2pt] ;\n  \\node[color=black] at (-3 ,-0.3) {\\small $s_v$};\n\\draw plot[mark=x, mark options={color=black, scale=1.5}] coordinates {(-3,0)  }; \n  \n   \\node[color=black] at (-1,-0.3) {\\small $e$};\n\n \n\\end{tikzpicture}\n\\end{center} \n\nDenote \n$$ \\mathcal I_ 1= (  s, s _u], \\quad \\I_2  =(s_u, s_v] \\quad \\text{and} \\quad \\I_3  = (s_v,e]. $$\nIn addition, denote \n$$ {  \\mathcal J } _1  = (s,\\eta_k ], \\quad {  \\mathcal J } _2=(\\eta_k, \\eta_{k} + \\frac{ \\eta_{k+1}   -\\eta_k  }{2}], \\quad {  \\mathcal J } _3 =  ( \\eta_k+ \\frac{ \\eta_{k+1}   -\\eta_k  }{2},\\eta_{k+1 } ]  \\quad \\text{and} \\quad {  \\mathcal J } _4 = (\\eta_{k+1}  , e] .$$\nSince \n$s, e,  s_u ,s_v  \\in  \\{ s_q\\}_{q=1}^ { \\mathcal Q}   $, then it follows from the definition of $\\widehat {\\mclP}$ that \n\\begin{align}\\nonumber\n& \\sum_{i \\in \\I }\\|X_i - \\widehat \\mu_\\I \\|_2^2\n\\\\  \\nonumber \n\\le &  \\sum_{i \\in \\I_1 }\\|X_i - \\widehat \\mu_{\\I_1} \\|_2 ^2  + \\sum_{i \\in \\I_2 }\\|X_i - \\widehat \\mu_{\\I_2}\\|_2^2  + \\sum_{i \\in \\I_3 }\\|X_i - \\widehat \\mu_{\\I_3}\\|_2^2  +2 \\gamma \n\\\\\\nonumber \n\\le &  \\sum_{i \\in \\I_1 }\\|X_i -  \\mu^*_i\\|_2^2 + C_1 \\bigg ( \\sigma_{\\epsilon}^2  { \\mathfrak{s} } \\log(n\\vee p) +  \\frac{|{  \\mathcal J } _1| (s_u -\\eta_k ) }{ |{  \\mathcal J } _1| + (s_u -\\eta_k )  }\\kappa_k^2 \\bigg )  +\n \\sum_{i \\in \\I_2 }\\|X_i -  \\mu^*_i\\|_2^2    +C_1   \\sigma_{\\epsilon}^2  { \\mathfrak{s} } \\log(n\\vee p)\n \\\\ \\nonumber \n  + &  \\sum_{i \\in \\I_3 }\\|X_i -  \\mu^*_i\\|_2^2 + C_1 \\bigg( \\sigma_{\\epsilon}^2  { \\mathfrak{s} } \\log(n\\vee p)+  \\frac{|{  \\mathcal J } _4 | (\\eta_{k+1} -s_v   )  }{ |{  \\mathcal J } _4 | + (\\eta_{k+1} -s_v   )   } \\kappa_{k+1} ^2   \\bigg)   +2  \\gamma\n\\\\ \n\\le  &  \\sum_{i \\in \\I }\\|X_i -  \\mu^*_i\\|_2^2  +C_1' \\bigg (  \\sigma_{\\epsilon}^2  { \\mathfrak{s} } \\log(n\\vee p) +  \\frac{|{  \\mathcal J } _1| (s_u -\\eta_k ) }{ |{  \\mathcal J } _1| + (s_u -\\eta_k )  }\\kappa_k^2     +  \\frac{|{  \\mathcal J } _4 | (\\eta_{k+1} -s_v   )  }{ |{  \\mathcal J } _4 | + (\\eta_{k+1} -s_v   )   }  \\kappa_{k+1} ^2        \\bigg )+2  \\gamma \n  \\label{eq:two change points step 1}\n\\end{align}\nwhere the first inequality follows from the fact that $ \\I =(s ,e ] \\in \\widehat { \\mathcal P} $, the second inequality follows from \\Cref{lem:mean one change deviation bound} {\\bf a} and {\\bf b}. \n\\Cref{eq:two change points step 1}  gives \n\\begin{align} \\nonumber \n& \\sum_{i \\in {  \\mathcal J } _1 }\\|X_i - \\widehat \\mu_\\I\\|_2^2 +\\sum_{i \\in {  \\mathcal J } _2 }\\|X_i - \\widehat \\mu_\\I\\|_2^2  +\\sum_{i \\in {  \\mathcal J } _3 }\\|X_i - \\widehat \\mu_\\I\\|_2^2 +\\sum_{i \\in {  \\mathcal J } _4 }\\|X_i - \\widehat \\mu_\\I\\|_2^2   \n\\\\ \\nonumber  \n \\le& \\sum_{i \\in {  \\mathcal J } _1  }\\|X_i -  \\mu^*_{ {  \\mathcal J } _1 }\\|_2^2   +\\sum_{i \\in {  \\mathcal J } _2  }\\|X_i -  \\mu^*_{ {  \\mathcal J } _2  }\\|_2^2 +\\sum_{i \\in {  \\mathcal J } _3  }\\|X_i -  \\mu^*_{ {  \\mathcal J } _3 }\\|_2^2   +\\sum_{i \\in {  \\mathcal J } _4  }\\|X_i -  \\mu^*_{ {  \\mathcal J } _4  }\\|_2^2  \\\\\n + & C_1' \\bigg (\\sigma_{\\epsilon}^2  { \\mathfrak{s} } \\log(n\\vee p) +   \\frac{|{  \\mathcal J } _1| (s_u -\\eta_k ) }{ |{  \\mathcal J } _1| + (s_u -\\eta_k )  }\\kappa_k^2     + \\frac{|{  \\mathcal J } _4 | (\\eta_{k+1} -s_v   )  }{ |{  \\mathcal J } _4 | + (\\eta_{k+1} -s_v   )   }  \\kappa_{k+1}    ^2  \\bigg )  + 2\\gamma.  \\label{eq:1D two change points first}\n\\end{align}\nNote that for $\\ell\\in \\{1,2,3,4 \\}$,\n\\begin{align}\\nonumber  \n  &\\sum_{i \\in {  \\mathcal J } _\\ell }\\|X_i - \\widehat \\mu_\\I\\|_2^2  \n -   \\sum_{i \\in {  \\mathcal J } _\\ell   }\\|X_i -  \\mu^*_{ {  \\mathcal J } _ \\ell }\\|_2^2  -\\sum_{i \\in {  \\mathcal J } _ \\ell   }\\|\\widehat \\mu_\\I  - \\mu^*_{{  \\mathcal J } _ \\ell }\\|_2^2 \n\\\\\n=& \\nonumber \n 2 \\sum_{i \\in {  \\mathcal J } _ \\ell   } \\epsilon_i^\\top ( \\mu^*_{ {  \\mathcal J } _\\ell  }-\\widehat \\mu_\\I    )      \n\\\\\\nonumber   \n\\ge  & -  C\\sigma_{\\epsilon} \\|\\widehat \\mu_\\I  -  \\mu^*_{ {  \\mathcal J } _\\ell }\\|_2 \\sqrt{|{  \\mathcal J } _\\ell|\\log(n\\vee p)}\n\\\\\n\\ge  &  -\\frac{1}{2}|{  \\mathcal J } _\\ell| \\|\\widehat \\mu_\\I  -  \\mu^*_{ {  \\mathcal J } _\\ell }\\|_2^2 - C'\\sigma_{\\epsilon}^2  { \\mathfrak{s} } \\log(n\\vee p). \\nonumber  \n\\end{align} \nwhich gives \n\\begin{align} \\label{eq:1D two change points second}\n  &\\sum_{i \\in {  \\mathcal J } _\\ell }\\|X_i - \\widehat \\mu_\\I \\|_2^2  \n -   \\sum_{i \\in {  \\mathcal J } _\\ell   }\\|X_i -  \\mu^*_{ {  \\mathcal J } _ \\ell }\\|_2^2  \\ge \\frac{1}{2}\\sum_{i \\in {  \\mathcal J } _ \\ell   }\\| \\widehat \\mu_\\I  - \\mu^*_{{  \\mathcal J } _ \\ell }\\|_2^2\n-   C_2\\sigma_{\\epsilon}^2  { \\mathfrak{s} } \\log(n\\vee p) .\n\\end{align} \n\\Cref{eq:1D two change points first} and \n\\Cref{eq:1D two change points second} together implies that\n \n\\begin{align} \\label{eq:1D two change points third}   \\sum_{l=1}^ 4 \n   |{  \\mathcal J } _l | (\\widehat \\mu_\\I  -  \\mu^*_{ {  \\mathcal J } _\\ell } ) ^2        \\le C_3 \\bigg ( \\sigma_{\\epsilon}^2  { \\mathfrak{s} } \\log(n\\vee p) +\\frac{|{  \\mathcal J } _1| (s_u -\\eta_k ) }{ |{  \\mathcal J } _1| + (s_u -\\eta_k )  }  \\kappa_k^2    +\\frac{|{  \\mathcal J } _4 | (\\eta_{k+1} -s_v   )  }{ |{  \\mathcal J } _4 | + (\\eta_{k+1} -s_v   )   } \\kappa_{k+1} ^2    \\bigg ) +4 \\gamma  .  \n   \\end{align}\nNote that \n\\begin{align}  \\label{eq:1D two change points signal lower bound} \\inf_{ a \\in \\mathbb R  }  |{  \\mathcal J } _1 |(  a - \\mu^*_{{  \\mathcal J } _1 }) ^2 + |{  \\mathcal J } _2 |( a  - \\mu^*_{{  \\mathcal J } _2 }) ^2  =&     \\frac{|{  \\mathcal J } _1| |{  \\mathcal J } _2|}{ |{  \\mathcal J } _1| +  |{  \\mathcal J } _2| } \\kappa_k ^2.\n\\end{align} \nSimilarly\n\\begin{align} \\label{eq:1D two change points signal lower bound 2} \\inf_{ a \\in \\mathbb R  }  |{  \\mathcal J } _3 |(  a - \\mu^*_{{  \\mathcal J } _3 }) ^2 + |{  \\mathcal J } _4 |( a  - \\mu^*_{{  \\mathcal J } _4 }) ^2   =     \\frac{|{  \\mathcal J } _3| |{  \\mathcal J } _4|}{|{  \\mathcal J } _3| + |{  \\mathcal J } _4| }       \\kappa_{k+1}  ^2  ,\\end{align} \n\\Cref{eq:1D two change points third} together with \n\\Cref{eq:1D two change points signal lower bound} and \\Cref{eq:1D two change points signal lower bound 2} leads to \n\\begin{align}\n\\label{eq:conclusion of two change points} \n  \\frac{|{  \\mathcal J } _1| |{  \\mathcal J } _2|}{ |{  \\mathcal J } _1| +  |{  \\mathcal J } _2| }  \\kappa_k ^2   +   \\frac{|{  \\mathcal J } _3| |{  \\mathcal J } _4|}{|{  \\mathcal J } _3| + |{  \\mathcal J } _4| } \\kappa_{k+1}  ^2  \\le C_3 \\bigg (\\sigma_{\\epsilon}^2  { \\mathfrak{s} } \\log(n\\vee p) +\\frac{|{  \\mathcal J } _1| (s_u -\\eta_k ) }{ |{  \\mathcal J } _1| + (s_u -\\eta_k )  }  \\kappa_k^2    +\\frac{|{  \\mathcal J } _4 | (\\eta_{k+1} -s_v   )  }{ |{  \\mathcal J } _4 | + (\\eta_{k+1} -s_v   )   } \\kappa_{k+1} ^2    \\bigg ) +4 \\gamma  .\n  \\end{align}\n Note that \n$$ 0\\le s_u -\\eta_k \\le  { \\mathcal B_n^{-1}   \\Delta }  \\quad \\text{and} \\quad \n0 \\le \\eta_{k+1} -s_v \\le  { \\mathcal B_n^{-1}   \\Delta } ,\n $$ \nand so there are four possible cases.  \n\n{\\bf case a.} \nIf \n$$|{  \\mathcal J } _1| \\le  { \\mathcal B_n^{-1\/2}   \\Delta }   \\quad \\text{and} \\quad  \n |{  \\mathcal J } _4|   \\le  { \\mathcal B_n^{-1\/2}   \\Delta }  , $$ \n then the desired result  follows immediately. \n\n{\\bf case b.}   $|{  \\mathcal J } _1|   >   { \\mathcal B_n^{-1\/2}   \\Delta }     $ and  $|{  \\mathcal J } _4|     \\le  { \\mathcal B_n^{-1\/2}   \\Delta }      $. Then since $|{  \\mathcal J } _2| \\ge \\Delta_{\\min} \/2  $, it holds that \n$$   \\frac{|{  \\mathcal J } _1| |{  \\mathcal J } _2|}{ |{  \\mathcal J } _1| +  |{  \\mathcal J } _2| }  \\ge  \\frac{1}{2 }\\min\\{|{  \\mathcal J } _1| , |{  \\mathcal J } _2|  \\} \\ge  \\frac{1}{2 } { \\mathcal B_n^{-1\/2}   \\Delta } .  $$\nIn addition,\n$$\\frac{|{  \\mathcal J } _1| (s_u -\\eta_k ) }{ |{  \\mathcal J } _1| + (s_u -\\eta_k )  }  \\le s_u -\\eta_k \\le  { \\mathcal B_n^{-1}   \\Delta }   \\quad   \\text{and} \\quad  \\frac{|{  \\mathcal J } _4 | (\\eta_{k+1} -s_v   )  }{ |{  \\mathcal J } _4 | + (\\eta_{k+1} -s_v   )   }  \\le \\eta_{k+1} -s_v     \\le  { \\mathcal B_n^{-1}   \\Delta }  . \n$$\nSo  \\Cref{eq:conclusion of two change points}  leads to \n\\begin{align}\\label{eq:conclusion of two change points case two} \n\\frac{1}{2 } { \\mathcal B_n^{-1\/2}   \\Delta }    \\kappa_k ^2   +   \\frac{|{  \\mathcal J } _3| |{  \\mathcal J } _4|}{|{  \\mathcal J } _3| + |{  \\mathcal J } _4| } \\kappa_{k+1}  ^2  \\le C_3 \\bigg (\\sigma_{\\epsilon}^2  { \\mathfrak{s} } \\log(n\\vee p) +  { \\mathcal B_n^{-1}   \\Delta }  \\kappa_k^2    +  { \\mathcal B_n^{-1}   \\Delta }  \\kappa_{k+1} ^2    \\bigg ) +4 \\gamma  .\n  \\end{align} \nSince $ \\kappa_k \\asymp \\kappa$ and $ \\kappa_{k+1}  \\asymp \\kappa$, \\Cref{eq:conclusion of two change points case two} gives \n$$ \\frac{1}{2 } { \\mathcal B_n^{-1\/2}   \\Delta }    \\kappa  ^2  \\le C_4 \\bigg ( \\sigma_{\\epsilon}^2  { \\mathfrak{s} } \\log(n\\vee p) +  { \\mathcal B_n^{-1}   \\Delta }  \\kappa ^2    +  { \\mathcal B_n^{-1}   \\Delta }  \\kappa ^2    \\bigg ) +4 \\gamma   . $$\nSince $ { \\mathcal B_n}  $ is a diverging sequence, the above display gives \n$$ \\Delta_{\\min} \\kappa ^2  \\le C_5 { \\mathcal B_n} ^{1\/2 } (\\log(n\\vee p) + \\gamma ).$$\nThis contradicts \\Cref{eq:1D two change points snr}.\n\n{\\bf case c.}  $|{  \\mathcal J } _1|    \\le    { \\mathcal B_n^{-1\/2}   \\Delta }     $ and \n$|{  \\mathcal J } _4|     >   { \\mathcal B_n^{-1\/2}   \\Delta }      $. Then the same argument  as that in  {\\bf case b} leads to the same contradiction.\n\n {\\bf case d.}     $|{  \\mathcal J } _1|   >   { \\mathcal B_n^{-1\/2}   \\Delta }     $ and  $|{  \\mathcal J } _4|     >   { \\mathcal B_n^{-1\/2}   \\Delta }      $. Then since $|{  \\mathcal J } _2|\\ge \\Delta_{\\min} \/2 , |{  \\mathcal J } _4| \\ge \\Delta_{\\min} \/2  $, it holds that \n$$   \\frac{|{  \\mathcal J } _1| |{  \\mathcal J } _2|}{ |{  \\mathcal J } _1| +  |{  \\mathcal J } _2| }  \\ge  \\frac{1}{2 }\\min\\{|{  \\mathcal J } _1| , |{  \\mathcal J } _2|  \\} \\ge  \\frac{1}{2 } { \\mathcal B_n^{-1\/2}   \\Delta } \\quad \\text{and} \\quad \n \\frac{|{  \\mathcal J } _3| |{  \\mathcal J } _4|}{ |{  \\mathcal J } _3| +  |{  \\mathcal J } _4| }  \\ge  \\frac{1}{2 }\\min\\{|{  \\mathcal J } _3| , |{  \\mathcal J } _4|  \\} \\ge  \\frac{1}{2 } { \\mathcal B_n^{-1\/2}   \\Delta }    $$\nIn addition, \n$$\\frac{|{  \\mathcal J } _4 | (\\eta_{k+1} -s_v   )  }{ |{  \\mathcal J } _4 | + (\\eta_{k+1} -s_v   )   }  \\le \\eta_{k+1} -s_v     \\le  { \\mathcal B_n^{-1}   \\Delta }   \\quad  \n\\frac{|{  \\mathcal J } _1| (s_u -\\eta_k ) }{ |{  \\mathcal J } _1| + (s_u -\\eta_k )  }  \\le s_u -\\eta_k \\le  { \\mathcal B_n^{-1}   \\Delta }  . \n$$\nSo  \\Cref{eq:conclusion of two change points}  leads to \n\\begin{align}\\label{eq:conclusion of two change points case four} \n\\frac{1}{2 } { \\mathcal B_n^{-1\/2}   \\Delta }    \\kappa_k ^2   +   \\frac{1}{2 } { \\mathcal B_n^{-1\/2}   \\Delta }      \\kappa_{k+1}  ^2  \\le C_6 \\bigg (\\sigma_{\\epsilon}^2  { \\mathfrak{s} } \\log(n\\vee p) +  { \\mathcal B_n^{-1}   \\Delta }  \\kappa_k^2    +  { \\mathcal B_n^{-1}   \\Delta }  \\kappa_{k+1} ^2    \\bigg ) +4 \\gamma  .\n  \\end{align} \nNote that $ { \\mathcal B_n}  $ is a diverging sequence. So the above display gives \n$$ \\Delta_{\\min} \\big  ( \\kappa_{k}   ^2+ \\kappa_{k+1}^2  \\big)   \\le C_ 7  { \\mathcal B_n} ^{1\/2 } (\\sigma_{\\epsilon}^2  { \\mathfrak{s} } \\log(n\\vee p) + \\gamma ) $$\nSince   $ \\kappa_k \\asymp \\kappa$ and  $ \\kappa_{k+1}  \\asymp \\kappa$. This contradicts \\Cref{eq:1D two change points snr}. \n \\end{proof}\n\n\n\n \\bnlem[Three or more change points]\n \\label{lem:mean three or more cp}\nSuppose the good events \n$\\mathcal L ( { \\mathcal B_n^{-1}   \\Delta }  ) $ and  $\\mathcal R (  { \\mathcal B_n^{-1}   \\Delta }  ) $ defined  in  \\Cref{eq:left and right approximation of change points} hold. \n Suppose in addition that \n\\begin{align} \\label{eq:1D three change points snr}\n\\Delta \\kappa^2 \\ge C   \\big(\\sigma_{\\epsilon}^2  { \\mathfrak{s} } \\log(n\\vee p) + \\gamma)    \n\\end{align} \nfor sufficiently large constant $C $. \n Then  with probability  at least $1-(n\\vee p)^{-3}$,    there is no interval  $  \\widehat { \\mathcal P}  $  containing three or more true change points.   \n \\enlem  \n \n\n\n\\begin{proof} \nFor  contradiction,  suppose    $ \\I=(s,e] \\in \\mathcal {\\widehat P}  $  be such that $ \\{ \\eta_1, \\ldots, \\eta_M\\} \\subset \\I $ with $M\\ge 3$. Throughout the proof, $M$ is assumed to be a parameter that can potentially change with $n$. \nSince the events $\\mathcal L (  { \\mathcal B_n^{-1}   \\Delta }   ) $ and  $\\mathcal R (  { \\mathcal B_n^{-1}   \\Delta }  ) $ hold, by relabeling $\\{ s_q\\}_{q=1}^ { \\mathcal Q}   $ if necessary,   let $ \\{ s_m\\}_{m=1}^M $ be such that \n $$ 0 \\le s_m  -\\eta_m  \\le  { \\mathcal B_n^{-1}   \\Delta }     \\quad \\text{for} \\quad 1 \\le m \\le M-1  $$ and that\n $$ 0\\le \\eta_M  - s_M \\le  { \\mathcal B_n^{-1}   \\Delta }  .$$ \n Note that these choices ensure that  $ \\{ s_m\\}_{m=1}^M  \\subset \\I . $\n \n \\begin{center}\n \\begin{tikzpicture} \n\\draw[ - ] (-10,0)--(1,0);\n \\node[color=black] at (-8,-0.3) {\\small s};\n \\draw[  (-, ultra thick, black] (-8,0) -- (-7.99,0);\n  \\draw[ { -]}, ultra thick, black] (-1.0002,0) -- (-1,0);\n\n   \\node[color=black] at (-7,-0.3) {\\small $\\eta_1$};\n\\draw(-7 ,0)circle [radius=2pt] ;\n\n  \\node[color=black] at (-6.5,-0.3) {\\small $s_1$};\n\\draw plot[mark=x, mark options={color=black, scale=1.5}] coordinates {(-6.5,0)  }; \n\n  \\node[color=black] at (-5,-0.3) {\\small $\\eta_2$};\n\\draw(-5 ,0)circle [radius=2pt] ;\n  \\node[color=black] at (-4.5,-0.3) {\\small $s_2$};\n\\draw plot[mark=x, mark options={color=black, scale=1.5}] coordinates {(-4.5,0)  }; \n \n\n  \\node[color=black] at (-2.5,-0.3) {\\small $\\eta_3$};\n\\draw(-2.5 ,0)circle [radius=2pt] ;\n  \\node[color=black] at (-3 ,-0.3) {\\small $s_3$};\n\\draw plot[mark=x, mark options={color=black, scale=1.5}] coordinates {(-3,0)  }; \n  \n   \\node[color=black] at (-1,-0.3) {\\small $e$};\n\n \n\\end{tikzpicture}\n\\end{center}\n\n{\\bf Step 1.}\n Denote \n $$ \\mathcal I_ 1= (  s, s _1], \\quad \\I_m   =(s_{m-1} , s_m] \\text{ for }  2 \\le m \\le M     \\quad \\text{and} \\quad \\I_{M+1}   = (s_M,e]. $$\nThen since \n$ s , e,  \\{ s_m \\}_{m=1}^M  \\subset   \\{ s_q\\}_{q=1}^ { \\mathcal Q}   $,   it follows that \n\\begin{align}\\nonumber\n& \\sum_{i \\in \\I }\\|X_i - \\widehat \\mu_\\I\\|_2^2\n\\\\  \\nonumber \n\\le &  \\sum _{m=1}^{M+1 } \\sum_{i \\in \\I_m}\\|X_i - \\widehat y_{\\I_m }  \\|_2^2   + M  \\gamma \n\\\\  \\label{eq: 1D three change points deviation term 1}\n\\le &  \\sum_{i \\in \\I_1 }\\|X_i -  \\mu^*_i\\|_2^2 + C_1 \\bigg ( \\sigma_{\\epsilon}^2  { \\mathfrak{s} } \\log(n\\vee p) +  \\frac{ (\\eta_1 -s ) ( s_1-\\eta_ 1)  }{  s_1-s  }\\kappa_1^2 \\bigg )\n\\\\   \\label{eq: 1D three change points deviation term 2} \n+ & \n \\sum_{m=2}^{M-1} \\sum_{i \\in \\I_m  }\\|X_i -  \\mu^*_i\\|_2^2    +C_1  \\bigg(\\sigma_{\\epsilon}^2  { \\mathfrak{s} } \\log(n\\vee p)+     \\frac{(\\eta_m -s_{m-1} )(s_m-\\eta_{m } ) }{ s _{m}-s _{m-1}  }  \\kappa_m^2 \\bigg ) \n \\\\  \\label{eq: 1D three change points deviation term 3}\n  + &    C_1 \\sigma_{\\epsilon}^2  { \\mathfrak{s} } \\log(n\\vee p)\n\\\\ \\label{eq: 1D three change points deviation term 4}\n+ & \\sum_{i\\in \\I_{M+1}}\\|X_i-\\mu^*_i\\|_2^2+ C_1  \\bigg( \\sigma_{\\epsilon}^2  { \\mathfrak{s} } \\log(n\\vee p) + \\frac{(\\eta_M -s_{M} )(e-\\eta_{M} ) }{ e-s _{M}  }  \\kappa_{M} ^2 \\bigg ) + M\\gamma, \n \\end{align}  \n where Equations \\eqref{eq: 1D three change points deviation term 1}, \\eqref{eq: 1D three change points deviation term 2}\n \\eqref{eq: 1D three change points deviation term 3} and \\eqref{eq: 1D three change points deviation term 4} follow from \\Cref{lem:mean one change deviation bound} and in particular,   \\Cref{eq: 1D three change points deviation term 3} corresponds to the interval $\\I _M = (s_{M-1},s_M] $ which by assumption  containing no change points. \n Note that \n\\begin{align*} & \\frac{ (\\eta_1 -s ) ( s_1-\\eta_ 1)  }{  s_1-s  } \\le s_1-\\eta_1  \\le  { \\mathcal B_n^{-1}   \\Delta }  ,\n\\\\ \n&   \\frac{(\\eta_m -s_{m-1} )(s_m-\\eta_{m } ) }{ s _{m}-s _{m-1}  } \\le  s_m-\\eta_m  \\le  { \\mathcal B_n^{-1}   \\Delta }  ,   \\ \\text{ and } \n\\\\\n&   \\frac{(\\eta_M -s_{M} )(e-\\eta_{M} ) }{ e-s _{M}  } \\le \\eta_M-s_m  \\le  { \\mathcal B_n^{-1}   \\Delta }   \n \\end{align*} \n and  that \n $ \\kappa_k \\asymp \\kappa $ for all $ 1\\le k \\le K$. Therefore \n\\begin{equation}\n    \\label{eq:1D three change points step 1}\n  \\sum_{i \\in \\I }\\|X_i - \\widehat \\mu_\\I\\|_2^2 \\le  \\sum _{ i \\in \\I } \\|X_i - \\mu^*_i\\|_2^2 +  C_2  \\bigg( M \\sigma_{\\epsilon}^2  { \\mathfrak{s} } \\log(n\\vee p)+   M  { \\mathcal B_n^{-1}   \\Delta }  \\kappa^2\n+  M\\gamma \\bigg),\n\\end{equation}\nwhere $ C_2$ is some large constant independent of $M$.\n\n{\\bf Step 2.}  Let \n$$  {  \\mathcal J } _1 =(s,  \\eta_1], \\ {  \\mathcal J } _m = (\\eta_{m-1}, \\eta_m]  \\text{ for } 2 \\le m \\le M , \\ {  \\mathcal J } _{M+1}  =(\\eta_M, e]. $$\nNote that $\\mu^*_i$ is unchanged in any of $\\{ {  \\mathcal J } _m\\}_{m=0}^{M+1}$.  \nSo  for $ 1 \\le m \\le M+1  $,\n\\begin{align}\\nonumber  \n  &\\sum_{i \\in  {  \\mathcal J } _m  }\\|X_i - \\widehat \\mu_\\I\\|_2^2  \n -   \\sum_{i \\in {  \\mathcal J } _  m    }\\|X_i -  \\mu^*_{ {  \\mathcal J } _  m }\\|_2^2  -\\sum_{i \\in {  \\mathcal J } _  m   }\\|\\widehat \\mu_\\I  - \\mu^*_{{  \\mathcal J } _  m }\\|_2^2 \n\\\\\n=& \\nonumber \n 2 \\sum_{i \\in {  \\mathcal J } _  m} \\epsilon_i^\\top (\\mu^*_{{  \\mathcal J } _ m}-\\widehat \\mu_\\I)   \n\\\\\\nonumber   \n\\ge  & - C\\sigma_{\\epsilon}\\|\\widehat \\mu_\\I  -  \\mu^*_{ {  \\mathcal J } _ m  }\\|_2\\sqrt{|{  \\mathcal J } _m|\\log(n\\vee p)}  \n\\\\\n\\ge &   -  C_3  \\sigma_{\\epsilon}^2  { \\mathfrak{s} } \\log(n\\vee p) - \\frac{1 }{2 } |{  \\mathcal J } _  m |\\|\\widehat \\mu_\\I  -  \\mu^*_{ {  \\mathcal J } _ m  }\\|_2^2   \\nonumber  \n\\end{align} \nwhich gives \n\\begin{align} \\label{eq:1D three change points second}\n  &\\sum_{i \\in {  \\mathcal J } _m  }\\|X_i - \\widehat \\mu_\\I\\|_2^2 \n -   \\sum_{i \\in {  \\mathcal J } _ m   }\\|X_i -  \\mu^*_{ {  \\mathcal J } _  m }\\|_2^2  \\geq \\frac{1}{2}\\sum_{i \\in {  \\mathcal J } _  m   }\\|\\widehat \\mu_\\I  - \\mu^*_{{  \\mathcal J } _  m  }\\|_2 ^2 - C_3\\sigma_{\\epsilon}^2  { \\mathfrak{s} } \\log(n\\vee p).\n\\end{align} \nTherefore\n\\begin{align} \\label{eq:1D three change points third} \\sum_{ m=1}^{M+1}   |{  \\mathcal J } _m |\\|\\widehat \\mu_\\I  - \\mu^*_{{  \\mathcal J } _  m  }\\|_2^2 = \\sum_{ m=1 }^{M+1}  \\sum_{i \\in {  \\mathcal J } _  m   }\\|\\widehat \\mu_\\I  - \\mu^*_{{  \\mathcal J } _  m  }\\|_2^2 \\le  C_4  M\\bigg(\\sigma_{\\epsilon}^2  { \\mathfrak{s} } \\log(n\\vee p)  +     { \\mathcal B_n^{-1}   \\Delta }  \\kappa^2\n+   \\gamma \\bigg),\n\\end{align}\nwhere the equality follows from the fact that  $\\mu^*_i$ is unchanged in any of $\\{ {  \\mathcal J } _m\\}_{m=0}^{M+1}$, and\n the inequality follows from \\Cref{eq:1D three change points step 1} and \n\\Cref{eq:1D three change points second}.\n\\\\\n\\\\\n{\\bf Step 3.}\nFor any $  m \\in\\{2, \\ldots, M\\}$, it holds that\n\\begin{align}  \\label{eq:1D three change points signal lower bound} \\inf_{ a \\in \\mathbb R  }  |{  \\mathcal J } _{m-1} |\\|a - \\mu^*_{{  \\mathcal J } _{m-1} }\\|_2^2 + |{  \\mathcal J } _{m} |\\|a  - \\mu^*_{{  \\mathcal J } _{m} }\\|_2^2  =&     \\frac{|{  \\mathcal J } _{m-1}| |{  \\mathcal J } _m|}{ |{  \\mathcal J } _{m-1}| +  |{  \\mathcal J } _m| } \\kappa_m ^2  \\ge \\frac{1}{2} \\Delta_{\\min} \\kappa^2,\n\\end{align}  \nwhere the last inequality follows from the assumptions that $\\eta_k - \\eta_{k-1}\\ge \\Delta_{\\min}  $ and $ \\kappa_k \\asymp \\kappa$  for all $1\\le k \\le K$. So\n\\begin{align}   \\nonumber &2 \\sum_{ m=1}^{M }   |{  \\mathcal J } _m |\\|\\widehat \\mu_\\I  - \\mu^*_{{  \\mathcal J } _  m  }\\|_2^2   \n\\\\ \n\\ge &  \\nonumber    \\sum_{m=2}^M  \\bigg(  |{  \\mathcal J } _{m-1}  | \\|\\widehat \\mu_\\I  - \\mu^*_{{  \\mathcal J } _ { m-1}  }\\|_2^2   + |{  \\mathcal J } _m |\\|\\widehat \\mu_\\I  - \\mu^*_{{  \\mathcal J } _  m  }\\|_2^2   \\bigg) \n\\\\ \\label{eq:1D three change points signal lower bound two} \n\\ge & (M-1)  \\frac{ 1}{2} \\Delta_{\\min} \\kappa^2  \\ge \\frac{M}{4} \\Delta_{\\min} \\kappa^2 ,\n\\end{align} \nwhere the second inequality follows from  \\Cref{eq:1D three change points signal lower bound} and the last inequality follows from $M\\ge 3$. \\Cref{eq:1D three change points third} and  \\Cref{eq:1D three change points signal lower bound two} together imply that \n\\begin{align}\\label{eq:1D three change points signal lower bound three} \n \\frac{M}{4} \\Delta_{\\min} \\kappa^2 \\le 2  C_4  M\\bigg(\\sigma_{\\epsilon}^2  { \\mathfrak{s} } \\log(n\\vee p) +  { \\mathcal B_n^{-1}   \\Delta }  \\kappa^2 + \\gamma \\bigg) .\n\\end{align}\nSince $ { \\mathcal B_n} \\to \\infty $, it follows that for sufficiently large $n$,   \\Cref{eq:1D three change points signal lower bound three} gives \n$$ \\Delta_{\\min}\\kappa^2 \\le C_5 \\big(\\sigma_{\\epsilon}^2  { \\mathfrak{s} } \\log(n\\vee p) +\\gamma),$$  \nwhich contradicts \\Cref{eq:1D three change points snr}.\n \\end{proof} \n \n \n \n \\bnlem[Two consecutive intervals]\n \\label{lem:mean two intervals}\n Suppose $ \\gamma \\ge C_\\gamma K { \\mathcal B_n^{-1}   \\Delta }  \\kappa^2 $ for sufficiently large constant $C_\\gamma $. \nWith probability  at least $1-(n\\vee p)^{-3}$,    there are no  two consecutive intervals $\\I_1= (s,t ] \\in \\widehat {\\mathcal  P}  $,   $ \\I_2=(t, e]  \\in   \\widehat  {\\mathcal  P}  $    such that $\\I_1 \\cup \\I_2$ contains no change points.  \n \\enlem \n \\begin{proof} \n For contradiction, suppose that \n $$ \\I :  =\\I_1\\cup  \\I_2 $$\n contains no change points. \n Since \n $ s,t,e \\in \\{ s_q\\}_{q=1}^ { \\mathcal Q}  $, it follows that\n $$ \\sum_{i\\in \\I_1} \\|X_i - \\widehat \\mu_{\\I_1}\\|^2  +\\sum_{i\\in \\I_2} \\|X_i - \\widehat \\mu_{\\I_2}\\|_2^2 +\\gamma  \\le \\sum_{i\\in \\I } \\|X_i - \\widehat \\mu_{\\I }\\|_2^2. $$\n By \\Cref{lem:mean one change deviation bound}, it follows that \n\\begin{align*} \n& \\sum_{i\\in \\I_1} \\|X_i - \\mu^*_i\\|_2^2 \\le  C_1 \\sigma_{\\epsilon}^2 { \\mathfrak{s} } \\log(n\\vee p) +  \\sum_{i\\in \\I_1} \\|X_i - \\widehat \\mu_{\\I_1}\\|_2^2 ,\n \\\\\n & \\sum_{i\\in \\I_2} \\|X_i - \\mu^*_i\\|_2^2 \\le  C_1 \\sigma_{\\epsilon}^2 { \\mathfrak{s} } \\log(n\\vee p) +  \\sum_{i\\in \\I_2} \\|X_i - \\widehat \\mu_{\\I_2}\\|_2^2\n  \\\\\n  & \\sum_{i\\in \\I} \\|X_i - \\widehat{\\mu}_{\\I}\\|_2^2 \\le  C_1 \\sigma_{\\epsilon}^2 { \\mathfrak{s} } \\log(n\\vee p) + \\sum_{i\\in \\I} \\|X_i - \\mu_{i}^*\\|_2^2.\n \\end{align*} \n So \n $$\\sum_{i\\in \\I_1} \\|X_i - \\mu^*_i\\|_2^2  +\\sum_{i\\in \\I_2}\\|X_i - \\mu^*_i\\|_2^2 -2C_1 \\sigma_{\\epsilon}^2 { \\mathfrak{s} } \\log(n\\vee p) +\\gamma    \\le \\sum_{i\\in \\I }\\|X_i - \\mu^*_i\\|_2^2 +C_1\\sigma_{\\epsilon}^2 { \\mathfrak{s} } \\log(n\\vee p) .  $$\n Since $\\mu^*_i$ is unchanged when $i\\in \\I$, it follows that \n $$ \\gamma \\le 3C_1\\sigma_{\\epsilon}^2 { \\mathfrak{s} } \\log(n\\vee p).$$\n This is a contradiction when $C_\\gamma> 3C_1. $\n \\end{proof} \n \n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Linear model}\n\\label{sec: main proof linear}\n\nIn this section we show the proof of \\Cref{thm:DCDP regression}. Throughout this section, \nfor any generic interval $\\I\\subset [1,n]$, denote $\\beta^*_{\\I} = \\frac{1}{|\\I|}\\sum_{i\\in \\I}\\beta^*_i$ and\n$$ \\widehat \\beta_\\I = \\argmin_{\\beta\\in \\mathbb{R}^p}\\frac{1}{|\\mclI|}\\sum_{i\\in \\mclI}(y_i - X_i^\\top \\beta)^2 + \\frac{\\lambda}{\\sqrt{|\\mclI|}}\\|\\beta\\|_1. $$\nAlso, unless specified otherwise, for the output of \\Cref{algorithm:DCDP}, we always set the goodness-of-fit function $\\mclF(\\cdot, \\cdot)$ to be\n\\begin{equation}\n\\mathcal F  (\\beta, \\I) :  = \\begin{cases} \n \\sum_{i \\in \\I } (y_i - X_i^\\top \\beta) ^2  &\\text{if } |\\I|\\ge C_\\mclF { \\mathfrak{s} }  \\log(n\\vee p) ,\n \\\\ \n 0 &\\text{otherwise,}\n \\end{cases}\n\\end{equation} \nwhere $C_{\\mclF}$ is a universal constant which is larger than $C_s$, the constant in sample size in \\Cref{lemma:interval lasso} and \\Cref{lemma:consistency}.\n\n\n\\paragraph{Assumptions.} For the ease of presentation, we combine the SNR condition we will use throughout this section and \\Cref{assp:dcdp_linear_reg main} into a single assumption.\n\n\n\\bnassum[Linear model] \n\\label{assp:dcdp_linear_reg}\nSuppose that \\Cref{assp:dcdp_linear_reg main} holds. In addition, suppose that $ \\Delta_{\\min} \\kappa^2  \\geq  \\mathcal{B}_n   \\frak{s}\\log(n\\vee p)$ as is assumed in \\Cref{thm:DCDP regression}.\n\\enassum\n\n\n\n\n\\begin{proof}[Proof of \\Cref{thm:DCDP regression}] \nBy  \\Cref{prop:regression local consistency}, $K \\leq |\\widehat{\\mathcal{P}}| \\leq 3K$.  This combined with \\Cref{prop:regression change points partition size consistency} completes the proof.\n\\end{proof}\n\n\n\n\n\n\n\n \\bnprop\\label{prop:regression local consistency}\n Suppose  \\Cref{assp:dcdp_linear_reg} holds. Let \n $\\widehat { \\mathcal P }  $ denote the output of  \\Cref{algorithm:DCDP} with $\\gamma =C_\\gamma  K { \\mathcal B_n^{-1}   \\Delta } \\kappa^2$. Then with probability  at least $1 - n ^{-3}$, the following properties hold.\n\t\\begin{itemize}\n\t\t\\item [(i)] For each interval  $ \\I = (s, e] \\in \\widehat{\\mathcal{P}}$    containing one and only one true \n\t\t   change point $ \\eta_k $, it must be the case that\n $$\\min\\{ \\eta_k -s ,e-\\eta_k \\}  \\lesssim \\frac{\\sigma_{\\epsilon}^2\\vee 1  }{\\kappa^2 }\\bigg(   { \\mathfrak{s} }  \\log(n\\vee p) +\\gamma \\bigg) +   { \\mathcal B_n^{-1}   \\Delta } .$$\n\t\\item [(ii)] For each interval  $  \\I = (s, e] \\in \\widehat{\\mathcal{P}}$ containing exactly two true change points, say  $\\eta_ k  < \\eta_ {k+1} $, it must be the case that\n\\begin{align*} \n \\eta_k -s    \\lesssim   \\frac{\\sigma_{\\epsilon}^2\\vee 1  }{\\kappa^2 }\\bigg(   { \\mathfrak{s} }  \\log(n\\vee p) +\\gamma \\bigg) +   { \\mathcal B_n^{-1}   \\Delta }       \\ \\text{and} \\ \n e-\\eta_{k+1}   \\le  C   \\frac{\\sigma_{\\epsilon}^2\\vee 1  }{\\kappa^2 }\\bigg(   { \\mathfrak{s} }  \\log(n\\vee p) +\\gamma \\bigg) +   { \\mathcal B_n^{-1}   \\Delta } . \n \\end{align*} \n\t\t \n\\item [(iii)] No interval $\\I \\in \\widehat{\\mathcal{P}}$ contains strictly more than two true change points; and \n\n\n\t\\item [(iv)] For all consecutive intervals $ \\I_1 $ and $ \\I_2 $ in $\\widehat{ \\mathcal P}$, the interval \n\t\t$ \\I_1 \\cup  \\I_2 $ contains at least one true change point.\n\t\t\t\t\n\t \\end{itemize}\n\\enprop \n\n\n\n\n\n\n\\bprf\nThe four cases are proved in \\Cref{lemma: regression dcdp one change point}, \\Cref{lemma:regression two change points}, \\Cref{lem:regression three or more cp}, and \\Cref{lem:regression two intervals} respectively.\n\\eprf\n\n\n\n \n\n\\bnprop\\label{prop:regression change points partition size consistency}\nSuppose  \\Cref{assp:dcdp_linear_reg} holds. Let \n $\\widehat { \\mathcal P }  $ denote the output of  \\Cref{algorithm:DCDP}. Suppose \n$\\gamma \\ge  C_\\gamma K  { \\mathcal B_n^{-1}   \\Delta }   \\kappa^2$  for sufficiently large constant $C_\\gamma$. Then\n with probability  at least $1 - C n ^{-3}$,  $| \\widehat { \\mathcal P} | =K $.\n\\enprop\n\n\n\n\\begin{proof}[Proof of   \\Cref{prop:regression change points partition size consistency}] \nDenote $\\mathfrak{G} ^*_n = \\sum_{ i =1}^n (y_i  -  X_i^\\top \\beta^*_i)^2$.  Given any collection $\\{t_1, \\ldots, t_m\\}$, where $t_1 < \\cdots < t_m$, and $t_0 = 0$, $t_{m+1} = n$, let \n\t\\begin{equation}\\label{regression eq-sn-def}\n\t\t { \\mathfrak{G} }  _n(t_1, \\ldots, t_{m}) = \\sum_{k=1}^{m} \\sum_{ i = t_k +1}^{t_{k+1}} \\mclF(\\widehat{\\beta} _  {(t_{k}, t_{k+1}]}, (t_{k}, t_{k+1}]). \n\t\\end{equation}\nFor any collection of time points, when defining \\eqref{regression eq-sn-def}, the time points are sorted in an increasing order.\n\nLet $\\{ \\widehat \\eta_{k}\\}_{k=1}^{\\widehat K}$ denote the change points induced by $\\widehat {\\mathcal P}$.  Suppose we can justify that \n\t\\begin{align}\n\t\t { \\mathfrak{G} } ^*_n + K\\gamma  \\ge  & { \\mathfrak{G} }  _n(s_1,\\ldots,s_K)   + K\\gamma - C_1 ( K +1)   { \\mathfrak{s} }  \\log(n\\vee p) - C_1\\sum_{k\\in [K]}\\kappa_k^2 { \\mathcal B_n^{-1}   \\Delta }   \\label{eq:regression K consistency step 1} \\\\ \n\t\t\\ge &  { \\mathfrak{G} } _n (\\widehat \\eta_{1},\\ldots, \\widehat \\eta_{\\widehat K } ) +\\widehat K \\gamma - C_1 ( K +1)   { \\mathfrak{s} }  \\log(n\\vee p) - C_1\\sum_{k\\in [K]}\\kappa_k^2 { \\mathcal B_n^{-1}   \\Delta }   \\label{eq:regression K consistency step 2} \\\\ \n\t\t\\ge &   { \\mathfrak{G} }  _n ( \\widehat \\eta_{1},\\ldots, \\widehat \\eta_{\\widehat K } , \\eta_1,\\ldots,\\eta_K ) + \\widehat K \\gamma    - C_1 ( K +1)   { \\mathfrak{s} }  \\log(n\\vee p)- C_1\\sum_{k\\in [K]}\\kappa_k^2 { \\mathcal B_n^{-1}   \\Delta }   \\label{eq:regression K consistency step 3}\n\t\\end{align}\n\tand that \n\t\\begin{align}\\label{eq:regression K consistency step 4}\n\t\t { \\mathfrak{G} }  ^*_n   - { \\mathfrak{G} }  _n ( \\widehat \\eta_{1},\\ldots, \\widehat \\eta_{\\widehat K } , \\eta_1,\\ldots,\\eta_K ) \\le C_2 (K + \\widehat{K} + 2)  { \\mathfrak{s} }  \\log(n\\vee p) .\n\t\\end{align}\n\tThen it must hold that $| \\widehat p | = K$, as otherwise if $\\widehat K \\ge K+1 $, then  \n\t\\begin{align*}\n\t\tC _2 (K + \\widehat{K} + 2)   { \\mathfrak{s} }  \\log(n\\vee p)  & \\ge    { \\mathfrak{G} }  ^*_n  - { \\mathfrak{G} }  _n ( \\widehat \\eta_{1},\\ldots, \\widehat \\eta_{\\widehat K } , \\eta_1,\\ldots,\\eta_K ) \\\\\n\t\t& \\ge      (\\widehat K - K)\\gamma  -C_1 ( K +1)   { \\mathfrak{s} }  \\log(n\\vee p) - C_1\\sum_{k\\in [K]}\\kappa_k^2 { \\mathcal B_n^{-1}   \\Delta } .\n\t\\end{align*} \n\tTherefore due to the assumption that $| \\widehat p|  =\\widehat K\\le 3K $, it holds that \n\t\\begin{align} \\label{eq:regression Khat=K} \n\t\tC_2 (4K + 2)  { \\mathfrak{s} }  \\log(n\\vee p)  + C_1(K+1)  { \\mathfrak{s} }  \\log(n\\vee p) +C_1\\sum_{k\\in [K]}\\kappa_k^2 { \\mathcal B_n^{-1}   \\Delta }  \\ge  (\\widehat K - K)\\gamma \\geq \\gamma,\n\t\\end{align}\n\tNote that \\eqref{eq:regression Khat=K} contradicts the choice of $\\gamma$.\n\n\n\\\n\\\\\n{\\bf Step 1.} Note that \\eqref{eq:regression K consistency step 1}  is implied by \n\t\\begin{align}\\label{eq:regression step 1 K consistency}  \n\t\t\\left| \t { \\mathfrak{G} }  ^*_n   -    { \\mathfrak{G} } _n(s_1,\\ldots,s_K)    \\right| \\le  C_3(K+1) \\lambda^2 + C_3\\sum_{k\\in [K]}\\kappa_k^2 { \\mathcal B_n^{-1}   \\Delta }  ,\n\t\\end{align}\n\twhich is  an  immediate consequence  of \\Cref{lem:regression one change deviation bound}. \n\t\\\n\t\\\\\n\t\\\\\n\t{\\bf Step 2.} Since $\\{ \\widehat \\eta_{k}\\}_{k=1}^{\\widehat K}$ are the change points induced by $\\widehat {\\mathcal P}$, \\eqref{eq:regression K consistency step 2} holds because $\\widehat p$ is a minimizer.\n\\\\\n\\\\\n\t{\\bf Step 3.}\nFor every $ \\I  =(s,e]\\in \\widehat p$, by \\Cref{prop:regression local consistency}, we know that with probability at least $1 - (n\\vee p)^{-5}$, $\\I$ contains at most two change points. We only show the proof for the two-change-point case as the other case is easier. Denote\n\t\\[\n\t\t \\I =  (s ,\\eta_{q}]\\cup (\\eta_{q},\\eta_{q+1}] \\cup  (\\eta_{q+1} ,e]  = {  \\mathcal J } _1 \\cup {  \\mathcal J } _2 \\cup {  \\mathcal J } _{3},\n\t\\]\nwhere $\\{ \\eta_{q},\\eta_{q+1}\\} =\\I \\, \\cap \\, \\{\\eta_k\\}_{k=1}^K$. \n\nFor each $m\\in\\{1,2,3\\}$, by \\Cref{lem:regression one change deviation bound}, it holds that\n\\[\n\\sum_{ i \\in {  \\mathcal J } _m }(y_ i - X_i^\\top \\widehat{\\beta} _{{  \\mathcal J } _m}    )^2   \\leq \\sum_{ i \\in {  \\mathcal J } _ m } (y_ i  - X_i^\\top \\beta^*_i   )^2 + C\\sigma_{\\epsilon}^2 { \\mathfrak{s} }  \\log(n\\vee p).\n\\]\nBy \\Cref{lem:regression loss deviation no change point}, we have\n\\[\n\\sum_{ i \\in {  \\mathcal J } _m }(y_ i - X_i^\\top\\widehat {\\beta} _\\I    )^2   \\ge \\sum_{ i \\in {  \\mathcal J } _ m } (y_ i  - X_i^\\top \\beta_i^*   )^2 - C\\sigma_{\\epsilon}^2 { \\mathfrak{s} }  \\log(n\\vee p).\n\\]\n Therefore the above inequality implies that \n\t\\begin{align} \\label{eq:regression K consistency step 3 inequality 3}  \\sum_{i \\in \\I }  (y_ i - X_i^*\\widehat {\\beta} _\\I    )^2   \\ge \\sum_{m=1}^{3} \\sum_{ i \\in {  \\mathcal J } _m }(y_ i - X_i^\\top\\widehat {\\beta} _{{  \\mathcal J } _m}    )^2  -C\\sigma_{\\epsilon}^2 { \\mathfrak{s} } \\log(n\\vee p).  \n\t\\end{align}\nNote that   \\eqref{eq:regression K consistency step 3} is an immediate consequence of   \\eqref{eq:regression K consistency step 3 inequality 3}.\n\n\t{\\bf Step 4.}\nFinally, to show \\eqref{eq:regression K consistency step 4},   let  $\\widetilde { \\mathcal P}$ denote    the partition induced by $\\{\\widehat \\eta_{1},\\ldots, \\widehat \\eta_{\\widehat K } , \\eta_1,\\ldots,\\eta_K\\}$. Then \n$| \\widetilde { \\mathcal P} | \\le K + \\widehat K+2   $ and that $\\beta^*_i$ is unchanged in every interval $\\I \\in \\widetilde { \\mathcal P}$. \n\tSo  \\Cref{eq:regression K consistency step 4} is an immediate consequence of   \\Cref{lem:regression one change deviation bound}.\n\\end{proof} \n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\subsection{Fundamental lemmas}\n\n\\bnlem \\label{lem:regression one change deviation bound}\nLet $\\mathcal I =(s,e] $ be any generic interval.  \n\\\\\n{\\bf a.} If $\\I$ contains no change points and that \n$|\\I| \\ge C_s  { \\mathfrak{s} }  \\log(n\\vee p) $ where $C_s$ is the universal constant in \\Cref{lemma:interval lasso}. Then it holds that \n$$\\mathbb P \\bigg( \\bigg|   \\sum_{ i  \\in \\I } (y_i   - X_i ^\\top \\widehat  \\beta _\\I   )^2  - \\sum_{ i   \\in \\I } (y_i      -  X_i^\\top \\beta^*_\\I    )^2  \\bigg|  \\ge C     { \\mathfrak{s} }   \\log(n\\vee p)  \\bigg)  \\le  n^{-4}. $$\n\\\n\\\\\n{\\bf b.}  Suppose that   the   interval $ \\I=(s,e]$ contains one  and only change point $ \\eta_k $ and that \n$|\\I| \\ge C_s  { \\mathfrak{s} }  \\log(n\\vee p) $. Denote  $\\widehat \\mu_\\I = \\frac{1}{|\\I | } \\sum_{i\\in \\I } y_i $ and \n$$  \\mathcal J = (s,\\eta_k] \\quad \\text{and} \\quad  \\mathcal J' =  (\\eta_k, e]  .$$   Then it holds that \n$$\\mathbb P \\bigg( \\bigg|   \\sum_{ i  \\in \\I } (y_i   - X_i^\\top \\widehat \\beta_\\I   )^2  - \\sum_{ i   \\in \\I } (y_i      - X_i^\\top \\beta^* _i    )^2  \\bigg|  \\ge  C\\bigg\\{   \\frac{ | {  \\mathcal J } ||{  \\mathcal J } '| }{ |\\I| }  \\kappa_k ^2  +   { \\mathfrak{s} }  \\log(n\\vee p)\\bigg\\}   \\bigg)   \\le  n^{-4}. $$\n\\enlem\n\\begin{proof} \nWe show {\\bf b} as {\\bf a} immediatelly follows from {\\bf b} with $ |{  \\mathcal J } '| =0$.\n Denote \n$$  \\mathcal J = (s,\\eta_k] \\quad \\text{and} \\quad  \\mathcal J' =  (\\eta_k, e]  .$$ \nDenote  $  \\beta _\\I^*  = \\frac{1}{|\\I | } \\sum_{i\\in \\I } \\beta ^* _i $. Note that \n\\begin{align}  \\nonumber \n   \\bigg|  \\sum_{ i  \\in \\I } (y_i   - X_i^\\top \\widehat \\beta_\\I   )^2  - \\sum_{ i   \\in \\I } (y_i      - X_i^\\top \\beta^* _i    )^2  \\bigg|   \n = &\\bigg| \\sum_{i\\in \\I } \\big\\{  X_i^\\top  ( \\widehat \\beta_\\I  -   \\beta^* _i )   \\big\\} ^2 -  2 \\sum_{i\\in \\I }  \\epsilon_i X_i^\\top ( \\widehat \\beta_\\I   -  \\beta^*_i ) \\bigg| \n \\\\ \\label{eq:regression one change point deviation bound term 1}\n \\le & 2 \\sum_{i\\in \\I } \\big\\{  X_i^\\top  ( \\widehat \\beta_\\I  -    \\beta^*_\\I  )   \\big\\}   ^2 \n \\\\ \\label{eq:regression one change point deviation bound term 2}\n + & 2 \\sum_{i\\in \\I } \\big\\{  X_i^\\top  (    \\beta^*_\\I  -\\beta^*_i  )    \\big\\} ^2  \n \\\\ \\label{eq:regression one change point deviation bound term 3}\n +& 2 \\bigg|    \\sum_{i\\in \\I }  \\epsilon_i X_i^\\top ( \\widehat \\beta_\\I   -  \\beta^*_\\I ) \\bigg|  \n \\\\ \\label{eq:regression one change point deviation bound term 4}\n +&  2 \\bigg|    \\sum_{i\\in \\I }  \\epsilon_i X_i^\\top (   \\beta^*_\\I -\\beta^*_i  ) \\bigg|  .\n\\end{align}\n\\\n\\\\\nSuppose all the good events in \\Cref{lemma:interval lasso} holds. \n\\\\\n\\\\\n{\\bf Step 1.} By \\Cref{lemma:interval lasso},   $\\widehat \\beta_\\I  -    \\beta^*_\\I  $ satisfies the cone condition that \n$$ \\| (\\widehat \\beta_\\I  -    \\beta^*_\\I  )_{S^c}\\|_1 \\le 3 \\| (\\widehat \\beta_\\I  -    \\beta^*_\\I )_S  \\|_1   .$$\nIt follows from \\Cref{corollary:restricted eigenvalues 2} that with probability at least $ 1-n^{-5}$,\n\\begin{align*} \n \\bigg| \\frac{1}{|\\I| } \\sum_{i\\in \\I } \\big\\{  X_i^\\top  ( \\widehat \\beta_\\I  -    \\beta^*_\\I  )   \\big\\} ^2    - ( \\widehat \\beta_\\I  -    \\beta^*_\\I    )^\\top \\Sigma  ( \\widehat \\beta_\\I  -    \\beta^*_\\I  )  \\bigg|  \\le C_1  \\sqrt { \\frac{ { \\mathfrak{s} }  \\log(n\\vee p) }{|\\I| }} \\| \\widehat \\beta_\\I  -    \\beta^*_\\I     \\|_2 ^2  .\n\\end{align*}\nThe above display   gives\n\\begin{align*}    \\bigg| \\frac{1}{|\\I| } \\sum_{i\\in \\I } \\big\\{  X_i^\\top  ( \\widehat \\beta_\\I  -    \\beta^*_\\I  )    \\big\\}^2 \\bigg| \\le &  \\| \\Sigma\\|_{\\text{op}}   \\| \\widehat \\beta_\\I  -    \\beta^*_\\I     \\|_2 ^2   + C_1  \\sqrt { \\frac{ { \\mathfrak{s} }  \\log(n\\vee p) }{|\\I| }} \\| \\widehat \\beta_\\I  -    \\beta^*_\\I     \\|_2 ^2 \n\\\\\n\\le & C_x \\| \\widehat \\beta_\\I  -    \\beta^*_\\I     \\|_2 ^2 + C_1  \\sqrt { \\frac{ { \\mathfrak{s} }  \\log(n\\vee p) }{C_\\zeta  { \\mathfrak{s} }  \\log(n\\vee p)  }} \\| \\widehat \\beta_\\I  -    \\beta^*_\\I     \\|_2 ^2   \n\\\\\n\\le &  \\frac{ C_2 { \\mathfrak{s} } \\log(n\\vee p)}{|\\I| }   \\end{align*} \nwhere the second inequality follows from the assumption that  $|\\I| \\ge C_\\zeta  { \\mathfrak{s} }  \\log(n\\vee p) $ and the last inequality follows from  \\Cref{eq:lemma:interval lasso term 1}  in \\Cref{lemma:interval lasso}.\nThis gives \n$$    \\bigg|   \\sum_{i\\in \\I } \\big\\{  X_i^\\top  ( \\widehat \\beta_\\I  -    \\beta^*_\\I  )    \\big\\}^2 \\bigg| \\le  2 C_2 { \\mathfrak{s} } \\log(n\\vee p) . $$\n\\\n\\\\\n{\\bf Step 2.}  Observe that \n$ X_i^\\top (\\beta_\\I ^* - \\beta^*_i) $ is Gaussian with mean $0$ and variance \n\\begin{align*} \\omega_i ^2 = (\\beta_\\I ^* - \\beta^*_i )^\\top \\Sigma  (\\beta_\\I ^* - \\beta^*_i)  . \n\\end{align*}\nSince \n$$ \\beta_\\I ^* = \\frac{|{  \\mathcal J } | \\beta^*_{  \\mathcal J }  +|{  \\mathcal J } '| \\beta^*_{{  \\mathcal J } '}   }{|\\I | },$$\nit follows that \n$$ \\omega_i ^2 = \\begin{cases} \n\\bigg(  \\frac{ | {  \\mathcal J } '| (\\beta^*_{  \\mathcal J }  -\\beta^*_{{  \\mathcal J } '}   ) }{ | \\I| }  \\bigg)^\\top  \\Sigma \\bigg(  \\frac{ | {  \\mathcal J } '| (\\beta^*_{  \\mathcal J }  -\\beta^*_{{  \\mathcal J } '}   ) }{ | \\I| }  \\bigg)   \\le   \\frac{| {  \\mathcal J } '| ^2  \\kappa_k^2 }{ |\\I|^2 }   &\\text{when }  i \\in {  \\mathcal J }  ,\n\\\\\n\\bigg(  \\frac{ | {  \\mathcal J } | (\\beta^*_{{  \\mathcal J } '} -\\beta^*_{  \\mathcal J }     ) }{ | \\I| }  \\bigg) ^\\top  \\Sigma \\bigg(  \\frac{ | {  \\mathcal J } '| (  \\beta^*_{{  \\mathcal J } '}  -\\beta^*_{  \\mathcal J }    ) }{ | \\I| }  \\bigg)   \\le   \\frac{| {  \\mathcal J } | ^2  \\kappa_k^2 }{ |\\I|^2 }   &\\text{when }  i \\in {  \\mathcal J } '.\n\\end{cases} $$\n Consequently,\n$   \\{ X_i^\\top (\\beta_\\I ^* - \\beta^*_i)\\} ^2 $ is sub-Exponential with parameter $  \\omega_i ^2$.  \nBy standard sub-Exponential tail bounds, it follows that \n\\begin{align*}\n &\\mathbb P \\bigg(  \\bigg|   \\sum_{ i \\in \\I }\\{ X_i^\\top (\\beta_\\I ^* - \\beta^*_i)\\} ^2 - { \\mathbb{E} } {\\sum_{ i \\in \\I }\\{ X_i^\\top (\\beta_\\I ^* - \\beta^*_i)\\} ^2}    \\bigg| \\ge C_3 \\tau    \\bigg)  \n \\\\\n \\le &  \\exp\\bigg (-c \\min\\bigg \\{ \\frac{  \\tau^2}{\\sum_{i\\in \\I } \\omega_i ^4  }  ,  \\frac{ \\tau }{ \\max_{i\\in \\I }   \\omega_i ^2 } \\bigg\\}  \\bigg)\n \\\\\n \\le &  \\exp\\bigg (-c' \\min\\bigg \\{ \\frac{  \\tau^2}{\\sum_{i\\in \\I } \\omega_i ^2  }  ,  \\frac{ \\tau }{ \\max_{i\\in \\I }   |\\omega_i | } \\bigg\\}  \\bigg)\n \\\\\n \\le & \\exp\\bigg (-c ''\\min\\bigg \\{  \\tau^2  \\bigg( \\frac{|\\I |   }{| {  \\mathcal J } '| |{  \\mathcal J } |  } \\kappa_k^{-2}  \\bigg)    ,    \\tau   \\frac{|\\I| }{ \\max \\{|{  \\mathcal J } |, |{  \\mathcal J } '|  \\}}   \\kappa_k ^{-1} \\bigg\\}  \\bigg) ,\n \\end{align*}\nwhere the second inequality follows from the observation that \n$$  \\omega_i^2     \\le \\kappa_k |\\omega_i| \\le  C_\\kappa  |\\omega_i| \\text{ for all } i \\in \\I , $$\n and the last inequality follows from the observation that \n\\begin{align*}\n \\sum_{i\\in \\I} \\omega_i ^2    \n \\le  C_x | {  \\mathcal J } | \\frac{| {  \\mathcal J } '| ^2  \\kappa_k^2 }{ |\\I|^2 } + C_x | {  \\mathcal J }  ' | \\frac{| {  \\mathcal J } | ^2  \\kappa_k^2 }{ |\\I|^2 }  \n=   C_x \\frac{| {  \\mathcal J } '| |{  \\mathcal J } | }{|\\I | } \\kappa_k^2 .\n \\end{align*} \n So there exists a sufficiently large constant $C_4$ such that with probability at  least $1- n^{-5}$, \n\\begin{align*}\n &  \\bigg|   \\sum_{ i \\in \\I }\\{ X_i^\\top (\\beta_\\I ^* - \\beta^*_i)\\} ^2 - { \\mathbb{E} } {\\sum_{ i \\in \\I }\\{ X_i^\\top (\\beta_\\I ^* - \\beta^*_i)\\} ^2}    \\bigg|\n  \\\\ \n   \\le &  C_ 4   \\bigg\\{\\sqrt { \\frac{| {  \\mathcal J } '| |{  \\mathcal J } | }{|\\I | }  \\log(n)  \\kappa _k ^2  } + \\log(n)    \\frac{\\max \\{|{  \\mathcal J } |, |{  \\mathcal J } '|  \\}  }{|\\I| } \\kappa _k  \\bigg\\}     \n \\\\\n \\le &  C_ 4'  \\bigg\\{      \\frac{| {  \\mathcal J } '| |{  \\mathcal J } | }{|\\I | }  \\kappa_k^2  + \\log(n)   +  \\log(n)    \\frac{\\max \\{|{  \\mathcal J } |, |{  \\mathcal J } '|  \\}  }{|\\I| } \\kappa _k  \\bigg\\}      \n \\\\\n \\le &C_5  \\bigg\\{      \\frac{| {  \\mathcal J } '| |{  \\mathcal J } | }{|\\I | }  \\kappa_k^2  + \\log(n)     \\bigg\\}      \n \\end{align*}\n where $ \\kappa_k \\asymp \\kappa \\le C_\\kappa $ is used in the last inequality.\nSince $  { \\mathbb{E} } {\\sum_{ i \\in \\I }\\{ X_i^\\top (\\beta_\\I ^* - \\beta^*_i)\\} ^2} = \\sum_{i\\in \\I }\\omega_i ^2 \\le C_x \\frac{| {  \\mathcal J } '| |{  \\mathcal J } | }{|\\I | } \\kappa_k^2$, it follows that \n$$ \\mathbb P \\bigg (  \\bigg|   \\sum_{ i \\in \\I }\\{ X_i^\\top (\\beta_\\I ^* - \\beta^*_i)\\} ^2    \\bigg|  \\le ( C_5 +C_x )  \\frac{| {  \\mathcal J } '| |{  \\mathcal J } | }{|\\I | }  \\kappa_k^2  + C_5 \\log(n) \\bigg ) \\ge 1-n^{-5}.$$\n\\\n\\\\\n{\\bf Step 3.} For \\Cref{eq:regression one change point deviation bound term 3},  it follows that with probability at least $1-2n^{-4}$\n\\begin{align*} \n   \\frac{1}{|\\I| } \\sum_{i\\in \\I }  \\epsilon_i X_i^\\top ( \\widehat \\beta_\\I   -  \\beta^*_\\I )\n  \\le  C_6 \\sqrt { \\frac{\\log(n\\vee p) }{ |\\I| } } \\|  \\widehat \\beta_\\I   -  \\beta^*_\\I \\|_1 \\le C_7  \\frac{ { \\mathfrak{s} }  \\log(n\\vee p) }{ |\\I| } \n\\end{align*}\nwhere the first inequality is a consequence of \\Cref{eq:independent condition 1c 1}, the second inequality follows from  \\Cref{eq:lemma:interval lasso term 3} in \\Cref{lemma:interval lasso}. \n\\\n\\\\\n\\\\\n{\\bf Step 4.}  From {\\bf Step 2}, we have  that\n$ X_i^\\top (\\beta_\\I ^* - \\beta^*_i) $ is Gaussian with mean $0$ and variance \n$$ \\omega_i ^2 = \\begin{cases} \n\\bigg(  \\frac{ | {  \\mathcal J } '| (\\beta^*_{  \\mathcal J }  -\\beta^*_{{  \\mathcal J } '}   ) }{ | \\I| }  \\bigg)^\\top  \\Sigma \\bigg(  \\frac{ | {  \\mathcal J } '| (\\beta^*_{  \\mathcal J }  -\\beta^*_{{  \\mathcal J } '}   ) }{ | \\I| }  \\bigg)   \\le   \\frac{| {  \\mathcal J } '| ^2  \\kappa_k^2 }{ |\\I|^2 }   &\\text{when }  i \\in {  \\mathcal J }  ,\n\\\\\n\\bigg(  \\frac{ | {  \\mathcal J } | (\\beta^*_{{  \\mathcal J } '} -\\beta^*_{  \\mathcal J }     ) }{ | \\I| }  \\bigg) ^\\top  \\Sigma \\bigg(  \\frac{ | {  \\mathcal J } '| (  \\beta^*_{{  \\mathcal J } '}  -\\beta^*_{  \\mathcal J }    ) }{ | \\I| }  \\bigg)   \\le   \\frac{| {  \\mathcal J } | ^2  \\kappa_k^2 }{ |\\I|^2 }   &\\text{when }  i \\in {  \\mathcal J } '.\n\\end{cases} $$\n Consequently,\n$   \\epsilon_i  X_i^\\top (\\beta_\\I ^* - \\beta^*_i)  $ is   centered  sub-Exponential with parameter $  \\omega_i \\sigma_\\epsilon  $.  \nBy standard sub-Exponential tail bounds, it follows that \n\\begin{align*}\n &\\mathbb P \\bigg(  \\bigg|   \\sum_{ i \\in \\I } \\epsilon_i  X_i^\\top (\\beta_\\I ^* - \\beta^*_i)       \\bigg| \\ge C_8 \\tau    \\bigg)  \n \\\\\n \\le &  \\exp\\bigg (-c \\min\\bigg \\{ \\frac{  \\tau^2}{\\sum_{i\\in \\I } \\omega_i ^2  }  ,  \\frac{ \\tau }{ \\max_{i\\in \\I }  |\\omega_i| } \\bigg\\}  \\bigg)\n \\\\\n \\le & \\exp\\bigg (-c '\\min\\bigg \\{  \\tau^2  \\bigg( \\frac{|\\I |   }{| {  \\mathcal J } '| |{  \\mathcal J } |  } \\kappa_k^{-2}  \\bigg)    ,    \\tau   \\frac{|\\I| }{ \\max \\{|{  \\mathcal J } |, |{  \\mathcal J } '|  \\}}   \\kappa_k ^{-1} \\bigg\\}  \\bigg) ,\n \\end{align*}\nwhere the last inequality follows from the observation that \n\\begin{align*}\n \\sum_{i\\in \\I} \\omega_i ^2    \n \\le  C_x | {  \\mathcal J } | \\frac{| {  \\mathcal J } '| ^2  \\kappa_k^2 }{ |\\I|^2 } + C_x | {  \\mathcal J }  ' | \\frac{| {  \\mathcal J } | ^2  \\kappa_k^2 }{ |\\I|^2 }  \n=   C_x \\frac{| {  \\mathcal J } '| |{  \\mathcal J } | }{|\\I | } \\kappa_k^2 .\n \\end{align*} \n So there exists a sufficiently large constant $C_9$ such that with probability at  least $1- n^{-5}$, \n\\begin{align*}\n &  \\bigg|   \\sum_{ i \\in \\I } \\epsilon_i X_i^\\top (\\beta_\\I ^* - \\beta^*_i) \\bigg|\n  \\\\ \n   \\le &  C_ 9   \\bigg\\{\\sqrt { \\frac{| {  \\mathcal J } '| |{  \\mathcal J } | }{|\\I | }  \\log(n)  \\kappa _k ^2  } + \\log(n)    \\frac{\\max \\{|{  \\mathcal J } |, |{  \\mathcal J } '|  \\}  }{|\\I| } \\kappa _k  \\bigg\\}     \n \\\\\n \\le &  C_ 9'  \\bigg\\{      \\frac{| {  \\mathcal J } '| |{  \\mathcal J } | }{|\\I | }  \\kappa_k^2  + \\log(n)   +  \\log(n)    \\frac{\\max \\{|{  \\mathcal J } |, |{  \\mathcal J } '|  \\}  }{|\\I| } \\kappa _k  \\bigg\\}      \n \\\\\n \\le &C_9  \\bigg\\{      \\frac{| {  \\mathcal J } '| |{  \\mathcal J } | }{|\\I | }  \\kappa_k^2  + \\log(n)     \\bigg\\}      \n \\end{align*}\n where $ \\kappa_k \\asymp \\kappa \\le C_\\kappa $ is used in the last inequality.\n\\end{proof} \n\n \n\\bnlem \\label{lemma:interval lasso} Suppose  \\Cref{assp:dcdp_linear_reg} holds. Let $$ \\widehat \\beta _{ \\mathcal I } = \\arg\\min_{\\beta \\in \\mathbb R^p }  \\frac{1}{ |\\mathcal I |  }\\sum_{i \\in \\mathcal I}   (y_i -X_i^\\top \\beta) ^2 + \\lambda \\|\\beta\\|_1$$ with $\\lambda = C_\\lambda (\\sigma_{\\epsilon} \\vee 1)\\sqrt { \\log(n\\vee p) }$ for some sufficiently large constant $C _ \\lambda $. There exists a sufficiently large constant $ C_s$ such that for all   $ \\mathcal I  \\subset (0, n] $   such that $| \\mathcal I | \\ge C_s { \\mathfrak{s} }  \\log(n\\vee p)$, it holds with probability at least $ 1-(n\\vee p)^{-3}$ that \n\\begin{align} \\label{eq:lemma:interval lasso term 1}\n&   \\| \\widehat \\beta_\\I -\\beta^*_\\I \\| _2^2 \\le \\frac{C (\\sigma_{\\epsilon}^2\\vee 1) { \\mathfrak{s} } \\log(n\\vee p)}{ |\\I  |  } ;\n \\\\ \\label{eq:lemma:interval lasso term 2}\n &     \\| \\widehat \\beta_\\I  -\\beta^*_\\I \\| _1  \\le     C(\\sigma_{\\epsilon}\\vee 1)  { \\mathfrak{s} } \\sqrt {  \\frac{\\log(n\\vee p)} {|\\I  |  } }  ;\n \\\\ \\label{eq:lemma:interval lasso term 3}\n &    \\| (\\widehat \\beta_\\I -\\beta^*_\\I)_{S^c} \\| _1 \\le  3 \\| (\\widehat \\beta _\\I  -\\beta^*_\\I )_{S } \\| _1   .\n\\end{align} \nwhere $ \\beta^*_\\I  = \\frac{1}{|\\I|} \\sum_{i\\in \\I } \\beta^*_i $.\n \\enlem \n\\begin{proof}  Denote $S=\\bigcup_{k=1}^K S_{\\eta_k+1} $. Since $K< \\infty$, \n$|S| \\asymp  { \\mathfrak{s} } .$ \n  It follows from the definition of $\\widehat{\\beta}_\\I$ that \n\t\\[\n\t \\frac{1}{|\\I | }\t\\sum_{ i \\in I} (y_i - X_i^{\\top}\\widehat{\\beta}_\\I )^2 + \\frac{ \\lambda}{ \\sqrt{  |\\I|  }}    \\|\\widehat{\\beta} _\\I\\|_1 \\leq  \n\t \\frac{1}{|\\I | } \\sum_{t \\in \\I} (y_i - X_i^{\\top}\\beta^*_\\I )^2 + \\frac{ \\lambda}{ \\sqrt{  |\\I|  }}  \\|\\beta^*_\\I  \\|_1.\n\t\\]\n\tThis gives \n\t\\begin{align*}\n\t\t \\frac{1}{|\\I | } \\sum_{i \\in \\I} \\bigl\\{X_i ^{\\top}(\\widehat{\\beta}_\\I - \\beta^*_\\I )\\bigr\\}^2 +    \\frac{ 2 }{|\\I | }   \\sum_{ i \\in \\I}(y_i - X_i^{\\top}\\beta^*_\\I)X_i^{\\top}(\\beta^*_\\I  - \\widehat{\\beta}_\\I )  +  \\frac{ \\lambda}{ \\sqrt{  |\\I|  }}   \\bigl\\|\\widehat{\\beta}_\\I \\bigr\\|_1 \n\t\t\\leq  \\frac{ \\lambda}{ \\sqrt{  |\\I|  }}    \\bigl\\|\\beta^*_\\I \\bigr\\|_1,\n\t\\end{align*}\n\tand therefore\n\t\\begin{align}\n\t\t& \\frac{1}{|\\I | }\t  \\sum_{i  \\in \\I} \\bigl\\{X_i ^{\\top}(\\widehat{\\beta} _\\I  - \\beta^*_\\I )\\bigr\\}^2 + \\frac{ \\lambda}{ \\sqrt{  |\\I|  }} \\bigl\\|\\widehat{\\beta}_\\I \\bigr\\|_1  \\nonumber \\\\\n\t\t \\leq  &  \\frac{ 2}{|\\I | }\t \\sum_{i \\in  \\I } \\epsilon_i  X_i ^{\\top}(\\widehat{\\beta}_\\I  - \\beta^*_\\I ) \n+ 2   (\\widehat{\\beta}_\\I  - \\beta^*_\\I )^{\\top}\\frac{  1 }{|\\I | }\t \\sum_{i\\in   \\I} X_ i  X_i^{\\top}(    \\beta^*_i -\\beta^*_\\I   )\t\t \n\t\t+  \\frac{ \\lambda}{ \\sqrt{  |\\I|  }}    \\bigl\\|\\beta^*_\\I \\bigr\\|_1   .\t \\label{eq-lem10-pf-2}\n\t\\end{align}\nTo bound\t\n\t\t$\\left\\|\\sum_{ i  \\in \\I} X_ i  X_i ^\\top (\\beta^*  _\\I  -\\beta^*_i ) \\right\\|_{\\infty},  \n$\n\tnote  that  for any $j \\in  \\{ 1, \\ldots, p\\}$, the $j$-th entry of \n\t\\\\$\\sum_{i \\in \\I} X _ i X _i^\\top (\\beta^*_\\I   -\\beta_i )$ satisfies  \n\t\\begin{align*}\n\t\t  E \\left\\{\\sum_{ i  \\in \\I}   X_i (j) X _ i ^\\top (\\beta^*_\\I   - \\beta^*_ i  )\\right\\} = \\sum_{ i \\in \\I}  E \\{X _ i (j  ) X_i ^\\top \\}\\{\\beta^*_\\I  - \\beta^*_ i \\}  \n\t\t=     \\mathbb{E}\\{X _1( j ) X _1 ^\\top \\} \\sum_{ i \\in \\I}\\{\\beta^*_\\I    - \\beta^*_ i  \\} = 0.\n\t\\end{align*}\n\tSo   $  E\\{ \\sum_{ i  \\in \\I} X_ i  X_i ^\\top (\\beta^*  _\\I  -\\beta^*_i )\\} =0 \\in \\mathbb R^p.$ \nBy \\Cref{lemma:consistency}{\\bf b},\n\t\\begin{align*}\n\t  \\bigg|  (    \\beta^*_i -\\beta^*_\\I   ) ^\\top  \\frac{  1 }{|\\I | }\t \\sum_{i\\in   \\I} X_t X_t^{\\top} (\\widehat{\\beta}_\\I  - \\beta^*_\\I )\t \t   \\bigg| \\le & C_1 \\big(\\max_{1\\le i \\le n }  \\|\\beta^*_i -\\beta^*_\\I \\|_2 \\big) \\sqrt { \\frac{\\log(n\\vee p) }{ | \\I|  }} \\| \\widehat{\\beta}_\\I  - \\beta^*_\\I  \\|_1   \n\t  \\\\\n\t  \\le & C_2 \\sqrt { \\frac{\\log(n\\vee p) }{ | \\I| }} \\| \\widehat{\\beta}_\\I  - \\beta^*_\\I  \\|_1  \n\t  \\\\\n\t  \\le &  \\frac{\\lambda}{8\\sqrt { |\\I| } }   \\|\\widehat{\\beta}_\\I  - \\beta^*_\\I \\|_1 \n \t\\end{align*}\n \twhere the second inequality follows from \\Cref{lemma:beta bounded 1} and the last inequality follows from $\\lambda = C_\\lambda\\sigma_{\\epsilon} \\sqrt { \\log(n\\vee p) }$ with sufficiently large constant $C_\\lambda$.\n\tIn addition  by \\Cref{lemma:consistency}{\\bf a},\n\t\\begin{equation*}\n\t    \\frac{ 2}{|\\I | }\t \\sum_{i \\in  \\I } \\epsilon_i  X_i ^{\\top}(\\widehat{\\beta}_\\I  - \\beta^*_\\I )  \\le C  \\sigma_{\\epsilon}\\sqrt { \\frac{ \\log(n\\vee p) }{ |\\I| } }\\|\\widehat{\\beta}_\\I  - \\beta^*_\\I \\|_1  \\le  \\frac{\\lambda}{8\\sqrt { |\\I| } }   \\|\\widehat{\\beta}_\\I  - \\beta^*_\\I \\|_1 .\n\t\\end{equation*}\n\tSo   \\eqref{eq-lem10-pf-2}  gives \n\t\\begin{align*}\n\t\t  \\frac{1}{|\\I | }\t  \\sum_{i  \\in \\I} \\bigl\\{X_i ^{\\top}(\\widehat{\\beta} _\\I  - \\beta^*_\\I )\\bigr\\}^2 + \\frac{ \\lambda}{ \\sqrt{  |\\I|  }} \\bigl\\|\\widehat{\\beta}_\\I \\bigr\\|_1  \n\t\t \\leq  \\frac{\\lambda}{4\\sqrt { |\\I| } }   \\|\\widehat{\\beta}_\\I  - \\beta^*_\\I \\|_1  \n\t\t+  \\frac{ \\lambda}{ \\sqrt{  |\\I|  }}    \\bigl\\|\\beta^*_\\I \\bigr\\|_1   .\t \n\t\\end{align*} \nLet $\\Theta = \\widehat  \\beta _\\I   - \\beta^*_\\I   $. The above inequality implies\n\\begin{align}\n\\label{eq:two sample lasso deviation 1} \\frac{1}{|\\I|} \\sum_{i \\in \\I } \\left( X_i^\\top  \\Theta \\right)^2 + \\frac{ \\lambda}{2\\sqrt{  |\\I|  } }\\|  (\\widehat \\beta _\\I)   _{ S ^c}\\|_1  \n \\le & \\frac{3\\lambda}{2\\sqrt{  |\\I|  }  } \\| ( \\widehat  \\beta _\\I  -  \\beta^*_\\I  )    _{S} \\| _1  ,\n \\end{align}  \nwhich also implies that \n$$ \\frac{\\lambda }{2}\\| \\Theta _{S^c}  \\|_1 = \\frac{ \\lambda}{2 }\\|   (\\widehat \\beta_\\I)  _{ S ^c}\\|_1    \\le  \n\\frac{3\\lambda}{2 } \\|  ( \\widehat  \\beta _\\I   - \\beta^* _\\I  ) _{S} \\| _1  = \\frac{3\\lambda}{2 } \\|  \\Theta  _{S} \\| _1 . $$\nThe above inequality and  \\Cref{corollary:restricted eigenvalues 2} imply that with probability at least $1-n^{-5}$,\n$$ \\frac{1}{|\\I| } \\sum_{i \\in \\I }  \\left( X_i^\\top  \\Theta \\right)^2  = \\Theta^\\top \\widehat \\Sigma _\\I  \\Theta  \n\\ge \n \\Theta^\\top \\Sigma \\Theta - C_3\\sqrt{ \\frac{  { \\mathfrak{s} } \\log(n\\vee p)}{ |\\I| }}  \\| \\Theta\\|_2^2   \\ge  \\frac{c_x}{2} \\|\\Theta\\|_2 ^2   ,$$\n where the last inequality follows from the assumption that $| \\mathcal I | \\ge C_s  { \\mathfrak{s} }  \\log(n\\vee p)   $ for sufficiently large $C_s$.\nTherefore \\Cref{eq:two sample lasso deviation 1} gives\n\\begin{align}\n\\label{eq:two sample lasso deviation 2}  c'\\|\\Theta\\|_2 ^2  + \\frac{ \\lambda}{2\\sqrt{  |\\I|  }  }\\| ( \\widehat \\beta_\\I  - \\beta^*_\\I ) _{ S ^c}\\|_1  \n \\le   \\frac{3\\lambda}{2\\sqrt{  |\\I|  }  } \\| \\Theta_{S} \\| _1  \\le \\frac{3\\lambda \\sqrt  { \\mathfrak{s} }  }{2 \\sqrt{  |\\I|  } } \\| \\Theta  \\| _2 \n \\end{align}  \n and so \n $$ \\|\\Theta\\|_2  \\le   \\frac{C \\lambda \\sqrt   { \\mathfrak{s} } }{\\sqrt{| \\I |} } . $$ \n The above display   gives  \n $$  \\| \\Theta_{S} \\| _1 \\le  \\sqrt { { \\mathfrak{s} } }  \\| \\Theta_{S} \\| _2 \\le   \\frac{C\\lambda  { \\mathfrak{s} } }{\\sqrt{|\\I| }}. $$ \n Since \n  $ \\| \\Theta_{S^c } \\| _1 \\le 3 \\| \\Theta_{S} \\| _1 ,$ \nit also holds that \n$$\\| \\Theta  \\| _1 = \\| \\Theta_{S  } \\| _1 +\\| \\Theta_{S^c } \\| _1  \\le 4 \\| \\Theta_{S} \\| _1 \\le  \\frac{4C\\lambda  { \\mathfrak{s} } }{\\sqrt{|\\I|} } .$$\n \\end{proof}\n\n \n \n\n \n \n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\subsection{Technical lemmas}\n\nThroughout this section, let $\\widehat { \\mathcal P }  $ denote the output of  \\Cref{algorithm:DCDP}.\n \n \n \n\n\n\n\n \\bnlem[No change point]\n \\label{lem:regression loss deviation no change point}\nLet $\\I\\subset [1,T]$ be any interval that contains no change point. Then under \\Cref{assp:dcdp_linear_reg}, for any interval ${  \\mathcal J }  \\supset \\I$, it holds with probability at least $1 - (n\\vee p)^{-5}$ that\n\\begin{equation*}\n    \\mclF(\\beta^*_{\\I},\\I)\\leq \\mclF(\\widehat{\\beta}_{  \\mathcal J } ,\\I) + C(\\sigma_{\\epsilon}^2\\vee 1) { \\mathfrak{s} } \\log(n\\vee p).\n\\end{equation*}\n \\enlem\n\\begin{proof} \n\\noindent \\textbf{Case 1.} If $ |\\I| < C_{\\mclF}  { \\mathfrak{s} }  \\log(n p)$, then by the definition of $\\mclF(\\beta, \\mclI)$, we have $\\mclF(\\beta^*_{\\I},\\I)= \\mclF(\\widehat{\\beta}_{  \\mathcal J } ,\\I) =0$ and the inequality holds automatically.\n \n\\noindent \\textbf{Case 2.} If \n\t\\begin{equation}\\label{eq-lem16-i-cond} \n\t\t|\\I| \\geq  C_{\\mclF}  { \\mathfrak{s} }  \\log(n p),\n\t\\end{equation}\n\tthen letting $\\delta_\\I = \\beta^*_\\I  -\\widehat \\beta_{  \\mathcal J } $ and consider the high-probability event given in \\Cref{lem:restricted eigenvalue}, we have\n\t\\begin{align}\n\t\t& \\sqrt{\\sum_{t \\in \\I} (X_t^{\\top} \\delta_\\I)^2} \\geq {c_1'\\sqrt{|\\I|}} \\|\\delta_\\I\\|_2 - c_2'\\sqrt{\\log(p)} \\|\\delta_\\I\\|_1 \\nonumber \\\\\n\t\t= & {c_1'\\sqrt{|\\I|}} \\|\\delta_\\I\\|_2 - c_2'\\sqrt{\\log(p)} \\|(\\delta_I)_{S}\\|_1 - c_2'\\sqrt{\\log(p)} \\|(\\delta_\\I)_{S^c}\\|_1 \\nonumber \\\\\n\t\t\\geq & c_1'\\sqrt{|\\I|} \\|\\delta_\\I\\|_2 - c_2'\\sqrt{ { \\mathfrak{s} } \\log(p)} \\|\\delta_\\I\\|_2 - c_2'\\sqrt{\\log(p)} \\|(\\delta_\\I)_{S^c}\\|_1 \\nonumber \\\\\n\t\t\\geq & \\frac{c_1'}{2}\\sqrt{|\\I|} \\|\\delta_\\I\\|_2 - c_2'\\sqrt{\\log(p)} \\|(\\widehat{\\beta}_{  \\mathcal J } )_{S^c}\\|_1 \\geq c_1\\sqrt{|\\I|} \\|\\delta_\\I\\|_2 - c_2\\sqrt{\\log(p)} \\frac{ { \\mathfrak{s} }  \\lambda}{\\sqrt{|\\I|}}, \\label{eq-lem16-pf-1}\n\t\\end{align} \n\twhere the last inequality follows from \\Cref{lemma:interval lasso} and the assumption that $(\\beta^*_t)_i = 0$, for all $t\\in [T]$ and $i\\in S^c$. Then by the fact that $(a - b)^2\\geq \\frac{1}{2}a^2 - b^2$ for all $a,b\\in \\mathbb{R}$, it holds that\n\t\\begin{equation}\n\t    \\sum_{t \\in \\I} (X_t^{\\top} \\delta_\\I)^2\\geq \\frac{c_1^2}{2}{|\\I|} \\|\\delta_\\I\\|_2^2 - \\frac{c_2^2  \\lambda^2 { \\mathfrak{s} } ^2\\log(p)}{|\\I|}.\n\t\\end{equation}\n\tNotice that\n\t\\begin{align*}\n\t    & \\sum_{t \\in \\I} (y_t - X_t^{\\top} \\beta^*_\\I)^2 - \\sum_{t \\in \\I} (y_t - X_t^{\\top} \\widehat{\\beta}_{  \\mathcal J } )^2  = 2 \\sum_{t \\in \\I} \\epsilon_t X_t^{\\top}\\delta_\\I  - \\sum_{t \\in \\I} (X_t^{\\top}\\delta_\\I)^2 \\\\\n\t\\leq & 2\\|\\sum_{t \\in \\I }  X_t \\epsilon_t \\|_{\\infty }  \\left( \\sqrt { { \\mathfrak{s} } } \\| (\\delta_\\I)_{S} \\|_2  + \\| (\\widehat \\beta_{  \\mathcal J } )_{S^c} \\|_1  \\right)- \\sum_{t \\in \\I} (X_t^{\\top}\\delta_\\I)^2.\n\t\\end{align*}\n\tSince for each $t$, $\\epsilon_t$ is subgaussian with $\\|\\epsilon_t\\|_{\\psi_2}\\leq \\sigma_{\\epsilon}$ and for each $i\\in [p]$, $(X_t)_i$ is subgaussian with $\\|(X_t)_i\\|_{\\psi_2}\\leq C_x$, we know that $(X_t)_i\\epsilon_t$ is subexponential with $\\|(X_t)_i\\epsilon_t\\|_{\\psi_1}\\leq C_x\\sigma_{\\epsilon}$. Therefore, by Bernstein's inequality (see, e.g., Theorem 2.8.1 in \\cite{vershynin2018high}) and a union bound, for $\\forall u\\geq 0$ it holds that\n\t\\begin{equation*}\n\t    \\mathbb{P}(\\|\\sum_{t \\in \\I }  X_t \\epsilon_t \\|_{\\infty }> u)\\leq 2p\\exp(-c\\min\\{\\frac{u^2}{|\\I|C_x^2\\sigma_{\\epsilon}^2}, \\frac{u}{C_x\\sigma_{\\epsilon}}\\}).\n\t\\end{equation*}\n\tTake $u = cC_x\\sigma_{\\epsilon}\\sqrt{|\\I|\\log(n\\vee p)}$, then by the fact that $|\\I|\\geq C_{\\mclF} { \\mathfrak{s} } \\log(n\\vee p)$, it follows that with probability at least $1 - (n\\vee p)^{-7}$,\n\t\\begin{equation*}\n\t    \\|\\sum_{t \\in \\I }  X_t \\epsilon_t \\|_{\\infty } \\leq CC_x\\sigma_{\\epsilon}\\sqrt{|\\I|\\log(n\\vee p)}\\leq \\lambda \\sqrt{|\\I|},\n\t\\end{equation*}\n\twhere we use the fact that $\\lambda=C_{\\lambda}(\\sigma_{\\epsilon}\\vee 1)\\sqrt{\\log(n\\vee p)}$. Therefore, we have\n\t\\begin{align*}\n\t& \\sum_{t \\in \\I} (y_t - X_t^{\\top} \\beta^*_\\I)^2 - \\sum_{t \\in \\I} (y_t - X_t^{\\top} \\widehat{\\beta}_{  \\mathcal J } )^2 \\\\\n\t\\leq & 2\\lambda\\sqrt{|\\I| { \\mathfrak{s} } }\\|\\delta_\\I\\|_2 + 2\\lambda\\sqrt{|\\I|}\\cdot \\frac{\\lambda  { \\mathfrak{s} } }{ \\sqrt{|\\I|}} - \\frac{c_1^2 |\\I|}{2}\\|\\delta_\\I\\|_2^2 + \\frac{c_2^2 \\lambda^2  { \\mathfrak{s} } ^2\\log(p)}{|\\I|}\\\\  \n\t\\leq & 2\\lambda\\sqrt{|\\I| { \\mathfrak{s} } }\\|\\delta_\\I\\|_2 + {2\\lambda^2  { \\mathfrak{s} } } - \\frac{c_1^2 |\\I|}{2}\\|\\delta_\\I\\|_2^2 + \\frac{c_2^2 \\lambda^2  { \\mathfrak{s} } ^2\\log(p)}{|\\I|} \\\\\n\t\\leq & \\frac{4}{c_1^2}\\lambda^2 { \\mathfrak{s} }  + \\frac{c_1^2}{4}|\\I| \\|\\delta_\\I\\|^2_2 + {2\\lambda^2  { \\mathfrak{s} } } - \\frac{c_1^2}{2}|\\I|\\|\\delta_\\I\\|^2_2 + \\frac{c_2^2 \\lambda^2  { \\mathfrak{s} } ^2\\log(p)}{C_{\\mclF} { \\mathfrak{s} } \\log(n\\vee p)} \\\\\n\t\\leq & {c_3\\lambda^2 { \\mathfrak{s} } } + {2\\lambda^2  { \\mathfrak{s} } } + \\frac{c_2^2 \\lambda^2  { \\mathfrak{s} } ^2\\log(p)}{C_{\\mclF} { \\mathfrak{s} } \\log(n\\vee p)}\\\\\n\t\\leq & c_4\\lambda^2  { \\mathfrak{s} } .\n\t\\end{align*}\n\twhere the third inequality follows from $2ab \\leq a^2 + b^2$. \n\n\\end{proof} \n\n\n\n\n\n\n\n\n\n\n \n \n \n \\bnlem[Single change point]\n \\label{lemma: regression dcdp one change point}\nSuppose the good events \n$\\mathcal L ( { \\mathcal B_n^{-1}   \\Delta }  ) $ and  $\\mathcal R ( { \\mathcal B_n^{-1}   \\Delta } ) $ defined  in \\Cref{eq:left and right approximation of change points} hold. \nLet   $ \\I=(s,e] \\in \\mathcal {\\widehat P}  $  be such that $\\I$ contains exactly  one  true  change point $ \\eta_k $. \nSuppose $\\gamma \\geq C_\\gamma K { \\mathcal B_n^{-1}   \\Delta }  \\kappa^2   $.  Then  with probability  at least $1- n^{-3}$,   it holds that \n\\begin{equation*}\n    \\min\\{ \\eta_k -s ,e-\\eta_k \\}   \\lesssim \\frac{\\sigma_{\\epsilon}^2\\vee 1  }{\\kappa^2 }\\bigg(   { \\mathfrak{s} }  \\log(n\\vee p) +\\gamma \\bigg)  +   { \\mathcal B_n^{-1}   \\Delta } .\n\\end{equation*}\n\\enlem\n\\begin{proof}    \nIf either $ \\eta_k -s \\le  C_\\mclF  { \\mathfrak{s} }  \\log(n\\vee p)   $ or $e-\\eta_k\\le C_\\mclF  { \\mathfrak{s} }  \\log(n\\vee p)  $, then \n$$ \\min\\{ \\eta_k -s ,e-\\eta_k \\}   \\le C_\\mclF  { \\mathfrak{s} }  \\log(n\\vee p)  $$\nand there is nothing to show. So assume that \n$$  \\eta_k -s  > C_\\mclF  { \\mathfrak{s} }  \\log(n\\vee p)  \\quad \\text{and} \\quad   e-\\eta_k  >C_\\mclF  { \\mathfrak{s} }  \\log(n\\vee p)  . $$\nBy event $\\mathcal R ( { \\mathcal B_n^{-1}   \\Delta }  )$, there exists  $ s_ u  \\in \\{ s_q\\}_{q=1}^ { \\mathcal Q} $ such that \n$$0\\le s_u - \\eta_k \\le  { \\mathcal B_n^{-1}   \\Delta } . $$\n\n\\begin{center}\n \\begin{tikzpicture} \n\\draw[ - ] (-10,0)--(-1,0);\n \\node[color=black] at (-8,-0.3) {\\small s};\n \\draw[  (-, ultra thick, black] (-8,0) -- (-7.99,0);\n  \\draw[ { -]}, ultra thick, black] (-3.0002,0) -- (-3,0);\n\n   \\node[color=black] at (-7,-0.3) {\\small $\\eta_k$};\n\\draw(-7 ,0)circle [radius=2pt] ;\n\n  \\node[color=black] at (-6.5,-0.3) {\\small $s_u$};\n\\draw plot[mark=x, mark options={color=black, scale=1.5}] coordinates {(-6.5,0)  }; \n  \n  \\node[color=black] at (-3 ,-0.3) {\\small $e$};\n\\end{tikzpicture}\n\\end{center} \n\n{\\bf Step 1.} Denote \n$$ \\I_ 1 = (s,s_u] \\quad \\text{and} \\quad \\I_2 = (s_u, e] .$$\nSince \n$ \\eta_k-s \\ge C_\\mclF  { \\mathfrak{s} }  \\log(n\\vee p)    , $\nit follows that \n   $ |\\I| \\ge C_\\mclF  { \\mathfrak{s} }  \\log(n\\vee p)   $ and $ |\\I_1| \\ge C_\\mclF  { \\mathfrak{s} }  \\log(n\\vee p)   $. Thus \n$$ \\mathcal F  (\\I)  = \\sum_{ i \\in \\I } (y_i - X_i^\\top \\widehat \\beta_\\I )^2 \\quad \\text{and} \\quad \\mathcal F  (\\I_1)  = \\sum_{ i \\in {\\I_1}  } (y_i - X_i^\\top \\widehat \\beta_{\\I_1}  )^2  .$$\nSince $\\I\\in \\widehat { \\mathcal P}$, it holds that \n\\begin{align}\\label{eq:regression one change basic inequality step 1} \\mathcal F  (\\I)  \\le  \\mathcal F  (\\I_1)  +  \\mathcal F  (\\I_2)  + \\gamma. \n\\end{align}\n\n{\\bf Case a.} Suppose $|\\I_2| <C_\\mclF  { \\mathfrak{s} }  \\log(n\\vee p)      $.\nIt follows from \\Cref{eq:regression one change basic inequality step 1} that \n\\begin{align}\\nonumber\n\\sum_{i \\in \\I }(y_i - X_i^\\top \\widehat \\beta _\\I ) ^2 \\le &  \\sum_{i \\in \\I_1 }(y_i -X_i^\\top  \\widehat \\beta _{\\I_1}  ) ^2 +  0  + \\gamma \n\\\\ \\nonumber \n\\le &  \\sum_{i \\in \\I_1 }(y_i -  X_i^\\top  \\beta^*  _i  ) ^2 + C_1 \\big ( (s_u -\\eta_k) \\kappa_k^2   +  { \\mathfrak{s} }   \\log(n\\vee p)  \\big )   +    \\gamma\n\\\\ \\nonumber \n\\le &  \\sum_{i \\in \\I _1}(y_i -  X_i^\\top  \\beta^*  _i ) ^2  +C_1 \\big (   { \\mathcal B_n^{-1}   \\Delta }   \\kappa_k^2   +   { \\mathfrak{s} }  \\log(n\\vee p )    \\big ) +  \\gamma\n\\\\\n\\le &   \\sum_{i \\in \\I  }(y_i -  X_i^\\top  \\beta^*  _i  ) ^2  +C_1 \\big (   { \\mathcal B_n^{-1}   \\Delta }   \\kappa_k^2   +   { \\mathfrak{s} }  \\log(n\\vee p )    \\big ) +  \\gamma  , \\label{eq:regression one change point step 1  case a}\n\\end{align}\nwhere the first inequality follows from the fact that $  \\mathcal F  (\\I_2)  = 0  $ when $|\\I_2| < C_\\mclF  { \\mathfrak{s} }  \\log(n\\vee p)   $,    the second inequality follows from \\Cref{lem:regression one change deviation bound}   {\\bf b}, the third inequality follows from  the assumption  that \n$   (s_u-\\eta_k) \\le   { \\mathcal B_n^{-1}   \\Delta }    $,   and the inequality holds because $ \\sum_{i \\in \\I _2 }(y_i -  X_i^\\top  \\beta^*  _i  ) ^2\\ge 0 $.\n\n{\\bf Case b.} Suppose $|\\I_2|  \\ge C_\\mclF  { \\mathfrak{s} }  \\log(n\\vee p)   $.\nIt follows from \\Cref{eq:regression one change basic inequality step 1} that  \n\\begin{align}\\nonumber\n\\sum_{i \\in \\I }(y_i - X_i^\\top \\widehat \\beta _\\I ) ^2 \\le &  \\sum_{i \\in \\I_1 }(y_i -X_i^\\top  \\widehat \\beta _{\\I_1}  ) ^2 +  \\sum_{i \\in \\I_2  }(y_i -X_i^\\top  \\widehat \\beta _{\\I_2 }  ) ^2   + \\gamma \n\\\\ \\nonumber \n\\le &  \\sum_{i \\in \\I_1 }(y_i -  X_i^\\top  \\beta^*  _i  ) ^2 + C_1 \\big ( (s_u -\\eta_k) \\kappa_k^2   +  { \\mathfrak{s} }   \\log(n\\vee p)  \\big )  \n\\\\  \\nonumber \n+ & \\sum_{i \\in \\I_2 }(y_i -  X_i^\\top  \\beta^*  _i  ) ^2 + C_1      { \\mathfrak{s} }   \\log(n\\vee p)     + 2  \\gamma\n\\\\ \n\\le &  \\sum_{i \\in \\I }(y_i -  X_i^\\top  \\beta^*  _i ) ^2  +C_2 \\big (   { \\mathcal B_n^{-1}   \\Delta }   \\kappa_k^2   +   { \\mathfrak{s} }  \\log(n\\vee p )    \\big ) +  \\gamma, \\label{eq:regression one change point step 1 case b}\n\\end{align}\nwhere    the second inequality follows from \\Cref{lem:regression one change deviation bound}  {\\bf a} and {\\bf b}, and the third inequality follows from  the assumption  that \n$   (s_u-\\eta_k) \\le   { \\mathcal B_n^{-1}   \\Delta }    $. Combing two cases leads to \n\\begin{align} \n\\sum_{i \\in \\I }(y_i - X_i^\\top \\widehat \\beta _\\I ) ^2  \n\\le     \\sum_{i \\in \\I  }(y_i -  X_i^\\top  \\beta^*  _i  ) ^2  +C_2 \\big (   { \\mathcal B_n^{-1}   \\Delta }   \\kappa_k^2   +   { \\mathfrak{s} }  \\log(n\\vee p )    \\big ) +  \\gamma . \\label{eq:two change point step 1}\n\\end{align}\n\\\\\n\\\\\n{\\bf Step 2.}\nDenote \n$$ {  \\mathcal J } _1  = (s,\\eta_k ] \\quad \\text{and} \\quad {  \\mathcal J } _2 = (\\eta_k , e] .$$\n\\Cref{eq:one change point step 1}  gives \n\\begin{align}  \\nonumber \n& \\sum_{i \\in {  \\mathcal J } _1 }(y_i  -X_i^\\top  \\widehat \\beta _\\I ) ^2 +\\sum_{i \\in {  \\mathcal J } _2 }(y_i - X_i^\\top  \\widehat \\beta _\\I  ) ^2  \n\\\\ \n \\le \\label{eq:regression step 2 first term} & \\sum_{i \\in {  \\mathcal J } _1  }(y_i -  X_i^\\top    \\beta _{{  \\mathcal J }  _1}  ^*    ) ^2   +\\sum_{i \\in {  \\mathcal J } _2  }(y_i - X_i^\\top    \\beta _{{  \\mathcal J }  _1}  ^*  ) ^2  +C_2  \\big (      { \\mathcal B_n^{-1}   \\Delta }  \\kappa_k^2  + { \\mathfrak{s} }  \\log(n\\vee p)  \\big )  +   \\gamma \n\\end{align}\nThe above display  leads to \n\\begin{align} \\nonumber\n&\\sum_{i \\in {  \\mathcal J } _1 } \\big\\{ X_i^\\top  (  \\widehat \\beta _\\I  -  \\beta^* _{{  \\mathcal J } _1 }) \\big\\}  ^2 +\\sum_{i \\in {  \\mathcal J } _2  } \\big\\{ X_i^\\top  (  \\widehat \\beta _\\I  -  \\beta^* _{{  \\mathcal J } _2 }) \\big\\}  ^2  \n\\\\ \\nonumber \n \\le  & 2 \\sum_{i \\in {  \\mathcal J } _1  } \\epsilon_i  X_i^\\top (  \\widehat \\beta _\\I  -  \\beta^* _{{  \\mathcal J } _1 })     +2 \\sum_{i \\in {  \\mathcal J } _2  } \\epsilon_i  X_i^\\top (  \\widehat \\beta _\\I  -  \\beta^* _{{  \\mathcal J } _2 })      +C_2  \\big (      { \\mathcal B_n^{-1}   \\Delta }    \\kappa_k^2   + { \\mathfrak{s} } \\log(n\\vee p)  \\big )  +  \\gamma \n \\\\ \\label{eq:regression one change step 2 term 2}\n \\le  & C_3  \\sqrt { \\log(n\\vee p)    |{  \\mathcal J } _1| }  \\|  \\widehat \\beta _\\I  -  \\beta^* _{{  \\mathcal J } _1 }\\| _1  + C_3  \\sqrt { \\log(n\\vee p)  |{  \\mathcal J } _2| }  \\|  \\widehat \\beta _\\I  -  \\beta^* _{{  \\mathcal J } _2  }\\| _1+ C_2  \\big (      { \\mathcal B_n^{-1}   \\Delta }  \\kappa_k^2  + { \\mathfrak{s} }  \\log(n\\vee p)  \\big ) +    \\gamma \n\\end{align} \nwhere the last inequality follows from \\Cref{lemma:consistency} and  that  $|{  \\mathcal J } _1|   \\ge  C_\\mclF  { \\mathfrak{s} }  \\log(n\\vee p)   $ and $|{  \\mathcal J } _2|  \\ge C_\\mclF  { \\mathfrak{s} }  \\log(n\\vee p)   $.\nNote that \n\\begin{align*} \n\\|  ( \\widehat \\beta _\\I  -  \\beta _{{  \\mathcal J } _1}^* )_{S^c } \\|_1 = \\|  ( \\widehat \\beta _\\I )_{S^c} \\|_{1} \n=   \\|  ( \\widehat \\beta _\\I -\\beta _\\I ^* )_{S^c} \\|_{1}  \\le 3 \\|  ( \\widehat \\beta _\\I -\\beta _\\I ^* )_{S } \\|_{1}   \\le C_5  { \\mathfrak{s} }  \\sqrt { \\frac{ \\log(n\\vee p)}{| \\I|} }    , \n\\end{align*} \nwhere the last two inequalities follows from \\Cref{lemma:interval lasso}.  So\n\\begin{align}  \\label{eq:regression one change step 2 term 1}\n\\|   \\widehat \\beta _\\I  -  \\beta _{{  \\mathcal J } _1}^*   \\|_1 = \\|  ( \\widehat \\beta _\\I  -  \\beta _{{  \\mathcal J } _1}^* )_{S  } \\|_1 +\\|  ( \\widehat \\beta _\\I  -  \\beta _{{  \\mathcal J } _1}^* )_{S^c } \\|_1  \\le \\sqrt { { \\mathfrak{s} }  }  \\|  \\widehat \\beta _\\I  -  \\beta _{{  \\mathcal J } _1}^*   \\|_2 + C_5  { \\mathfrak{s} }  \\sqrt { \\frac{ \\log(n\\vee p)}{| \\I|} } .\n\\end{align} \nTherefore  \\Cref{eq:regression one change step 2 term 2} gives \n\\begin{align} \\nonumber \n& \\sum_{i \\in {  \\mathcal J } _1 } \\big\\{ X_i^\\top  (  \\widehat \\beta _\\I  -  \\beta^* _{{  \\mathcal J } _1 }) \\big\\}  ^2 +\\sum_{i \\in {  \\mathcal J } _2  } \\big\\{ X_i^\\top  (  \\widehat \\beta _\\I  -  \\beta^* _{{  \\mathcal J } _2 }) \\big\\}  ^2   \n\\\\  \\nonumber \n\\le & C_3  \\sqrt { \\log(n\\vee p)    |{  \\mathcal J } _1| }   \\bigg(  \\sqrt { { \\mathfrak{s} }  }  \\|  \\widehat \\beta _\\I  -  \\beta _{{  \\mathcal J } _1}^*   \\|_2 + C_5  { \\mathfrak{s} }  \\sqrt { \\frac{ \\log(n\\vee p)}{| \\I|} } \\bigg) \n\\\\ \\nonumber  \n+ & C_3  \\sqrt { \\log(n\\vee p)    |{  \\mathcal J } _2| }   \\bigg(  \\sqrt { { \\mathfrak{s} }  }  \\|  \\widehat \\beta _\\I  -  \\beta _{{  \\mathcal J } _2}^*   \\|_2 + C_5  { \\mathfrak{s} }  \\sqrt { \\frac{ \\log(n\\vee p)}{| \\I|} } \\bigg)  +2\\gamma \n\\\\ \\nonumber  \n\\le & \\frac{ c_x | {  \\mathcal J } _1|}{64} \\|  \\widehat \\beta _\\I  -  \\beta _{{  \\mathcal J } _1}^*   \\|_2  ^2 + \\frac{ c_x | {  \\mathcal J } _2|}{64} \\|  \\widehat \\beta _\\I  -  \\beta _{{  \\mathcal J } _2}^*   \\|_2  ^2   + C_5'    { \\mathfrak{s} }  \\log(n\\vee p)\n\\\\ +&C_2  \\big (      { \\mathcal B_n^{-1}   \\Delta }  \\kappa_k^2  + { \\mathfrak{s} }  \\log(n\\vee p)  \\big )   +2\\gamma,\n \\label{eq:regression one change step 2 term 3}\n\\end{align}\n   $ |{  \\mathcal J } _1| \\le |\\I| , |{  \\mathcal J } _2| \\le |\\I|  $ are used in the last inequality. \n\\\\\n\\\\\n{\\bf Step 3.}  Since     $|{  \\mathcal J } _1|   \\ge  C_\\mclF   { \\mathfrak{s} }  \\log(n\\vee p)   $ and $|{  \\mathcal J } _2|  \\ge C_\\mclF  { \\mathfrak{s} }  \\log(n\\vee p)   $, for $\\ell=1, 2$,\n  it holds that\n \\begin{align}\\nonumber \n\t\t&  \\sum_{i \\in {  \\mathcal J } _\\ell  } \\big\\{ X_i^\\top  (  \\widehat \\beta _\\I  -  \\beta^* _{{  \\mathcal J } _ \\ell  }) \\big\\}  ^2 \n\t\t \\\\  \\nonumber  \n\t\t  \\ge&  \\frac{c_x|{  \\mathcal J } _\\ell | }{16} \\|\\widehat \\beta _\\I  -  \\beta^* _{{  \\mathcal J } _ \\ell  } \\|_2^2 - C_6  \\log(p) \\|\\widehat \\beta _\\I  -  \\beta^* _{{  \\mathcal J } _ \\ell  } \\|_1  ^2 \n\t\t \\\\\\nonumber  \n\t\t \\ge & \\frac{c_x|{  \\mathcal J } _\\ell | }{16} \\|\\widehat \\beta _\\I  -  \\beta^* _{{  \\mathcal J } _ \\ell  } \\|_2^2 - C_6'  { \\mathfrak{s} }    \\log(p) \\|\\widehat \\beta _\\I  -  \\beta^* _{{  \\mathcal J } _ \\ell  } \\|_ 2   ^2  - C_6'  \\frac{   { \\mathfrak{s} } ^2 \\log(p) \\log(n\\vee p)  }{|\\I| }\n\t\t \\\\ \\label{eq:regression one change step 3 term 1}\n\t\t \\ge &  \\frac{c_x|{  \\mathcal J } _\\ell | }{32} \\|\\widehat \\beta _\\I  -  \\beta^* _{{  \\mathcal J } _ \\ell  } \\|_2^2 - C_7   { \\mathfrak{s} }  \\log(n\\vee p),\n\t\\end{align} \n\twhere the first inequality follows from   \n \\Cref{lem:restricted eigenvalue},  the second inequality follows from \\Cref{eq:regression one change step 2 term 1}   and the last inequality follows from the observation that \n $$ |\\I| \\ge |{  \\mathcal J } _\\ell|  \\ge   C_\\mclF  { \\mathfrak{s} }  \\log(n\\vee p). $$\n \\Cref{eq:regression one change step 2 term 3} and \\Cref{eq:regression one change step 3 term 1} together lead to \n $$   |{  \\mathcal J } _1  |   \\|\\widehat \\beta _\\I  -  \\beta^* _{{  \\mathcal J } _ 1  } \\|_2^2  + |{  \\mathcal J } _2  |   \\|\\widehat \\beta _\\I  -  \\beta^* _{{  \\mathcal J } _ 2  } \\|_2^2  \\le C_8( { \\mathfrak{s} }  \\log(n\\vee p) +  { \\mathcal B_n^{-1}   \\Delta }  \\kappa_k^2 + \\gamma) .$$\nObserve  that \n$$ \\inf_{  \\beta  \\in \\mathbb R ^p   }  |{  \\mathcal J } _1 |  \\| \\beta    -  \\beta^* _{{  \\mathcal J } _ 1  } \\|_2^2  + |{  \\mathcal J } _2 | \\|  \\beta    -  \\beta^* _{{  \\mathcal J } _ 2  } \\|_2^2    = \\kappa_k ^2  \\frac{|{  \\mathcal J } _1| |{  \\mathcal J } _2|}{| \\I| }   \\ge \\frac{ \\kappa_k^2 }{2} \\min\\{ |{  \\mathcal J } _1| ,|{  \\mathcal J } _2| \\}    . $$ \nThis leads to \n$$  \\frac{ \\kappa_k^2 }{2}\\min\\{ |{  \\mathcal J } _1| ,|{  \\mathcal J } _2| \\}    \\le  C_8( { \\mathfrak{s} }  \\log(n\\vee p)+ { \\mathcal B_n^{-1}   \\Delta }   + \\gamma)  .$$\nSince $\\kappa_k \\asymp \\kappa $, it follows that \n$$ \\min\\{ |{  \\mathcal J } _1| ,|{  \\mathcal J } _2| \\}   \\le C_9 \\bigg( \\frac{ { \\mathfrak{s} }  \\log(n\\vee p) + \\gamma  }{\\kappa \t^2 } +   { \\mathcal B_n^{-1}   \\Delta }   \\bigg) .$$\n \\end{proof} \n  \n\n\n \\bnlem[Two change points] \n \\label{lemma:regression two change points}\nSuppose the good events \n$\\mathcal L ( { \\mathcal B_n^{-1}   \\Delta }  ) $ and  $\\mathcal R (  { \\mathcal B_n^{-1}   \\Delta }  ) $  in  \\Cref{eq:left and right approximation of change points} hold. \nLet   $ \\I=(s,e] \\in \\mathcal {\\widehat P}  $  be such that $\\I$ contains  exactly two  change points $ \\eta_k,\\eta_{k+1}  $.  Suppose in addition that \n\\begin{align} \\label{eq:regression two change points snr}\n\\Delta_{\\min} \\kappa^2 \\ge  C     \\big( \\sigma_{\\epsilon}^2 { \\mathfrak{s} }  \\log(n\\vee p) + \\gamma)    \n\\end{align} \nfor sufficiently large constant $C   $ and that \n$\\gamma \\ge  C_\\gamma \\mclB_n^{-1}\\Delta_{\\min}\\kappa^2   $.\n Then  with probability  at least $1- n^{-3}$,   it holds that \n\\begin{align*} \n \\eta_k -s     \\lesssim  \\frac{\\sigma_{\\epsilon}^2\\vee 1  }{\\kappa^2 }\\bigg(   { \\mathfrak{s} }  \\log(n\\vee p) +\\gamma \\bigg) +   { \\mathcal B_n^{-1}   \\Delta }    \\quad \\text{and} \\quad \n e-\\eta_{k+1}   \\lesssim \\frac{\\sigma_{\\epsilon}^2\\vee 1  }{\\kappa^2 }\\bigg(   { \\mathfrak{s} }  \\log(n\\vee p) +\\gamma \\bigg) +   { \\mathcal B_n^{-1}   \\Delta }   ,\n \\end{align*} \n where $C_0\\ge 1 $ is some sufficiently large constant.\n \\enlem \n\\begin{proof} \nBy symmetry, it suffices to show that $\\eta_k-s\\lesssim \\frac{\\sigma_{\\epsilon}^2\\vee 1  }{\\kappa^2 }\\bigg(   { \\mathfrak{s} }  \\log(n\\vee p) +\\gamma \\bigg) +   { \\mathcal B_n^{-1}   \\Delta } $.  If \n$$\\eta_k-s  \\le  C_\\mclF  { \\mathfrak{s} }  \\log(n\\vee p) , $$\nthen the desired result follows immediately.  \nSo it suffices to assume that \n$$\\eta_k-s > C_\\mclF  { \\mathfrak{s} }  \\log(n\\vee p) .  $$\nSince the events $\\mathcal L (  { \\mathcal B_n^{-1}   \\Delta }   ) $ and  $\\mathcal R (  { \\mathcal B_n^{-1}   \\Delta }  ) $ hold, let $ s_u, s_v$ be such that \n $\\eta_k \\le s_u \\le s_v \\le \\eta_{k+1} $ and that \n $$ 0 \\le s_u-\\eta_k \\le    { \\mathcal B_n^{-1}   \\Delta }  ,  \\quad  0\\le  \\eta_{k+1} - s_v \\le  { \\mathcal B_n^{-1}   \\Delta }    .    $$\n \\\n \\\\\n \\begin{center}\n \\begin{tikzpicture} \n\\draw[ - ] (-10,0)--(1,0);\n \\node[color=black] at (-8,-0.3) {\\small s};\n \\draw[  (-, ultra thick, black] (-8,0) -- (-7.99,0);\n  \\draw[ { -]}, ultra thick, black] (-1.0002,0) -- (-1,0);\n\n   \\node[color=black] at (-7,-0.3) {\\small $\\eta_k$};\n\\draw(-7 ,0)circle [radius=2pt] ;\n\n  \\node[color=black] at (-6.5,-0.3) {\\small $s_u$};\n\\draw plot[mark=x, mark options={color=black, scale=1.5}] coordinates {(-6.5,0)  }; \n \n\n  \\node[color=black] at (-2.3,-0.3) {\\small $\\eta_{k+1}$};\n\\draw(-2.3 ,0)circle [radius=2pt] ;\n  \\node[color=black] at (-3 ,-0.3) {\\small $s_v$};\n\\draw plot[mark=x, mark options={color=black, scale=1.5}] coordinates {(-3,0)  }; \n  \n   \\node[color=black] at (-1,-0.3) {\\small $e$};\n\n \n\\end{tikzpicture}\n\\end{center} \n\\\n\\\\\n{\\bf Step 1.} Denote \n $$ \\mathcal I_ 1= (  s, s _u], \\quad \\I_2  =(s_u, s_v] \\quad \\text{and} \\quad \\I_3  = (s_v,e]. $$\n \\\n \\\\\nSince  $ |\\I| \\ge \\eta_{k+1} -\\eta_k \\ge C_\\mclF  { \\mathfrak{s} }  \\log(n\\vee p)   $, \n$$ \\mathcal F(\\I ) = \\sum_{i \\in \\I }(y_i - X_i^\\top \\widehat \\beta _\\I ) ^2  . $$\nSince \n $ |\\I_1|  \\ge  \\eta_k-s      \\ge C_\\mclF  { \\mathfrak{s} }  \\log(n\\vee p)   , $ \nit follows that \n $$ \\mathcal F(\\I_1 ) = \\sum_{i \\in \\I_1 }(y_i - X_i^\\top \\widehat \\beta _{\\I_1}  ) ^2  . $$\nIn addition   since  $ |\\I_1| \\ge C_\\mclF  { \\mathfrak{s} }  \\log(n\\vee p)  $, then \n \\begin{align*}\\mathcal F(\\I_1)  = & \\sum_{i \\in \\I_1 }(y_i - X_i^\\top \\widehat \\beta _{\\I_1}  ) ^2 \n \\\\\n \\le & \\sum_{i \\in \\I_1 }(y_i - X_i^\\top   \\beta ^*  _i   ) ^2  + C_1  \\bigg\\{ \\frac{( \\eta_k-s) (s_u -\\eta_k ) }{ ( \\eta_k-s)  + (s_u -\\eta_k )  } \\kappa ^2 +  { \\mathfrak{s} } \\log(n\\vee p)  \\bigg\\} \n \\\\\n  \\le & \\sum_{i \\in \\I_1 }(y_i - X_i^\\top   \\beta ^*  _i   ) ^2  +  C_1 \\bigg\\{ (s_u -\\eta_k )   \\kappa ^2 +   { \\mathfrak{s} } \\log(n\\vee p)  \\bigg\\} \n  \\\\\n  \\le & \\sum_{i \\in \\I_1 }(y_i - X_i^\\top   \\beta ^*  _ i   ) ^2  + C_1 \\bigg\\{   { \\mathcal B_n^{-1}   \\Delta }    \\kappa ^2 +   { \\mathfrak{s} } \\log(n\\vee p)  \\bigg\\} ,\n \\end{align*}\nwhere the first  inequality follows  from \\Cref{lem:regression one change deviation bound}  and that $ \\kappa_{k} \\asymp \\kappa$.\n Similarly, since \n $ |\\I_2|  \\ge \\Delta_{\\min} \/2 \\ge   C_\\mclF  { \\mathfrak{s} }  \\log(n\\vee p)   , $ \nit follows that \n $$ \\mathcal F(\\I_2 ) = \\sum_{i \\in \\I_2 }(y_i - X_i^\\top \\widehat \\beta _{\\I_2}  ) ^2  . $$\n  Since $|\\I_2| \\ge C_\\mclF  { \\mathfrak{s} }  \\log(n\\vee p)   $ and  $\\I_2$ contains no change points, by \\Cref{lem:regression one change deviation bound},\n  \\begin{align*}\\mathcal F(\\I_2)  \\le \\sum_{i \\in \\I_2 }(y_i - X_i^\\top   \\beta ^*  _ i  ) ^2   + C_1 { \\mathfrak{s} } \\log(n\\vee p).\n  \\end{align*}  \n  \\\\\n  \\\\\n {\\bf Step 2.} If $ |\\I_3 | \\ge C_\\mclF  { \\mathfrak{s} }  \\log(n\\vee p)   $, then \n  \\begin{align*}\\mathcal F(\\I_3)  = & \\sum_{i \\in \\I_3 }(y_i - X_i^\\top \\widehat \\beta _{\\I_3}  ) ^2 \n \\\\\n \\le & \\sum_{i \\in \\I_3 }(y_i - X_i^\\top   \\beta ^*  _i   ) ^2  + C_1  \\bigg\\{ \\frac{( \\eta_{k+1}-s_v) (e -\\eta_{k+1} ) }{ ( \\eta_{k+1}-s_v)+  (e -\\eta_{k+1} )  } \\kappa ^2 +  { \\mathfrak{s} } \\log(n\\vee p)  \\bigg\\} \n \\\\\n  \\le & \\sum_{i \\in \\I_3 }(y_i - X_i^\\top   \\beta ^*  _i   ) ^2  +  C_1 \\bigg\\{ ( \\eta_{k+1}-s_v)    \\kappa ^2 +   { \\mathfrak{s} } \\log(n\\vee p)  \\bigg\\} \n  \\\\\n  \\le & \\sum_{i \\in \\I_3 }(y_i - X_i^\\top   \\beta ^*  _ i   ) ^2  + C_1 \\bigg\\{   { \\mathcal B_n^{-1}   \\Delta }    \\kappa ^2 +   { \\mathfrak{s} } \\log(n\\vee p)  \\bigg\\} ,\n \\end{align*}\n where the first inequality follows from \\Cref{lem:regression one change deviation bound}{\\bf b} and that $ \\kappa_{k+1} \\asymp \\kappa$. \n If $|\\I_3| < C_\\mclF  { \\mathfrak{s} }  \\log(n\\vee p)   $, then \n$\\mathcal F(\\I_3)  =  0 $.\n So \nboth cases imply that  \n $$\\mathcal F(\\I_3)  \\le \\sum_{i \\in \\I_3 }(y_i - X_i^\\top   \\beta ^*  _ i   ) ^2  + C_1 \\bigg\\{   { \\mathcal B_n^{-1}   \\Delta }    \\kappa_k^2 +   { \\mathfrak{s} } \\log(n\\vee p)  \\bigg\\}  .$$\n \\\n \\\\\n {\\bf Step 3.} Since $\\I \\in \\widehat{\\mathcal P} $, we have\n \\begin{align}\\label{eq:regression two change points local min} \\mathcal F(\\I ) \\le \\mathcal F(\\I_1  ) +\\mathcal F(\\I_2 )+\\mathcal F(\\I_3 ) + 2\\gamma. \n \\end{align} The above display and the calculations in  {\\bf Step 1} and  {\\bf Step 2} implies that\n\\begin{align} \n \\sum_{i \\in \\I }(y_i -  X_i^\\top \\widehat \\beta _\\I ) ^2  \n\\le   \\sum_{i \\in \\I  }(y_i -   X_i^\\top \\beta_i^*   ) ^2  \n +   3  C_1 \\bigg\\{   { \\mathcal B_n^{-1}   \\Delta }    \\kappa ^2 +   { \\mathfrak{s} } \\log(n\\vee p) \\bigg\\}\n+2\\gamma  .\n  \\label{eq:regression two change points step 2 term 1}\n\\end{align}\nDenote \n$$ {  \\mathcal J } _1  = (s,\\eta_k ], \\quad {  \\mathcal J } _2=(\\eta_k, \\eta_{k} +   \\eta_{k+1}    ]  \\quad \\text{and} \\quad {  \\mathcal J } _3 = (\\eta_{k+1}  , e] .$$\n\\Cref{eq:regression two change points step 2 term 1}  gives \n\\begin{align} \n \\sum_{\\ell=1}^3 \\sum_{i \\in {  \\mathcal J } _{\\ell}  }(y_i - X_i^\\top  \\widehat \\beta _\\I ) ^2 \n \\le& \\sum_{\\ell=1}^3 \\sum_{i \\in {  \\mathcal J } _ \\ell  }(y_i -  X_i^\\top    \\beta^*  _{{  \\mathcal J } _\\ell }  ) ^2    \n +  3  C_1 \\bigg\\{   { \\mathcal B_n^{-1}   \\Delta }    \\kappa ^2 +   { \\mathfrak{s} } \\log(n\\vee p) \\bigg\\} \n+2\\gamma     \\label{eq:regression two change points first}\n\\end{align} \n\\\n\\\\\n{\\bf Step 4.} \nNote that for $\\ell\\in \\{1,2,3\\}$, \n\\begin{align*} \n\\|  ( \\widehat \\beta _\\I  -  \\beta _{{  \\mathcal J } _\\ell }^* )_{S^c } \\|_1 = \\|  ( \\widehat \\beta _\\I )_{S^c} \\|_{1} \n=   \\|  ( \\widehat \\beta _\\I -\\beta _\\I ^* )_{S^c} \\|_{1}  \\le 3 \\|  ( \\widehat \\beta _\\I -\\beta _\\I ^* )_{S } \\|_{1}   \\le C_2  { \\mathfrak{s} }  \\sqrt { \\frac{ \\log(n\\vee p)}{| \\I|} }    , \n\\end{align*} \nwhere the last two inequalities follows from \\Cref{lemma:interval lasso}.  So\n\\begin{align}  \\label{eq:regression two change step 3 term 1}\n\\|   \\widehat \\beta _\\I  -  \\beta _{{  \\mathcal J } _\\ell}^*   \\|_1 = \\|  ( \\widehat \\beta _\\I  -  \\beta _{{  \\mathcal J } _\\ell }^* )_{S  } \\|_1 +\\|  ( \\widehat \\beta _\\I  -  \\beta _{{  \\mathcal J } _\\ell}^* )_{S^c } \\|_1  \\le \\sqrt { { \\mathfrak{s} }  }  \\|  \\widehat \\beta _\\I  -  \\beta _{{  \\mathcal J } _\\ell}^*   \\|_2 + C_2  { \\mathfrak{s} }  \\sqrt { \\frac{ \\log(n\\vee p)}{| \\I|} } .\n\\end{align} \nNote that by assumptions,\n$$|{  \\mathcal J } _1|\\ge C_\\mclF  { \\mathfrak{s} }  \\log(n\\vee p)    \\quad \\text{and} \\quad |{  \\mathcal J } _2 |\\ge C_\\mclF  { \\mathfrak{s} }  \\log(n\\vee p) .$$\n So for $\\ell\\in \\{ 1, 2\\} $,\n  it holds that\n \\begin{align}\\nonumber \n\t\t&  \\sum_{i \\in {  \\mathcal J } _\\ell  } \\big\\{ X_i^\\top  (  \\widehat \\beta _\\I  -  \\beta^* _{{  \\mathcal J } _ \\ell  }) \\big\\}  ^2 \n\t\t \\\\  \\nonumber  \n\t\t  \\ge&  \\frac{c_x|{  \\mathcal J } _\\ell | }{16} \\|\\widehat \\beta _\\I  -  \\beta^* _{{  \\mathcal J } _ \\ell  } \\|_2^2 - C_3 \\log(p) \\|\\widehat \\beta _\\I  -  \\beta^* _{{  \\mathcal J } _ \\ell  } \\|_1  ^2 \n\t\t \\\\\\nonumber  \n\t\t \\ge & \\frac{c_x|{  \\mathcal J } _\\ell | }{16} \\|\\widehat \\beta _\\I  -  \\beta^* _{{  \\mathcal J } _ \\ell  } \\|_2^2 - C_3'  { \\mathfrak{s} }    \\log(p) \\|\\widehat \\beta _\\I  -  \\beta^* _{{  \\mathcal J } _ \\ell  } \\|_ 2   ^2  - C_3'  \\frac{   { \\mathfrak{s} } ^2 \\log(p) \\log(n\\vee p)  }{|\\I| }\n\t\t \\\\ \\label{eq:regression two change step 3 term 2}\n\t\t \\ge &  \\frac{c_x|{  \\mathcal J } _\\ell | }{32} \\|\\widehat \\beta _\\I  -  \\beta^* _{{  \\mathcal J } _ \\ell  } \\|_2^2 - C_4   { \\mathfrak{s} }  \\log(n\\vee p),\n\t\\end{align} \n\twhere the first inequality follows from \\Cref{lem:restricted eigenvalue}, the second inequality follows from \\Cref{eq:regression two change step 3 term 1}   and the last inequality follows from the observation that \n $$  |\\I|\\ge  |{  \\mathcal J } _\\ell|   \\ge C_\\gamma  { \\mathfrak{s} }  \\log(n\\vee p). \n $$ \n So for $\\ell\\in \\{ 1,2\\}$, \n\\begin{align*} &\\sum_{i \\in {  \\mathcal J } _\\ell  }(y_i - X_i^\\top  \\widehat \\beta _\\I ) ^2  - \\sum_{i \\in {  \\mathcal J } _\\ell   }(y_i - X_i^\\top    \\beta^*  _{{  \\mathcal J } _\\ell }  ) ^2\n=  \\sum_{i \\in {  \\mathcal J } _\\ell    }\\big\\{ X_i^\\top (\\widehat \\beta _\\I -\\beta^*  _{{  \\mathcal J } _4} )\\big\\}^2 -2 \\sum_{i \\in {  \\mathcal J } _\\ell   } \\epsilon_i X_i^\\top (\\widehat \\beta _\\I -\\beta^*  _{{  \\mathcal J } _\\ell  }  )\n\\\\\n\\ge  & \\sum_{i \\in {  \\mathcal J } _\\ell   }\\big\\{ X_i^\\top (\\widehat \\beta _\\I -\\beta^*  _{{  \\mathcal J } _\\ell  } )\\big\\}^2  - 2 \\| \\sum_{i \\in {  \\mathcal J } _\\ell   }  \\epsilon_i X_i^\\top \\|_\\infty \\| \\widehat \\beta _\\I -\\beta^*  _{{  \\mathcal J } _\\ell  }  \\|_1\n\\\\\n\\ge &\\sum_{i \\in {  \\mathcal J } _\\ell   }\\big\\{ X_i^\\top (\\widehat \\beta _\\I -\\beta^*  _{{  \\mathcal J } _\\ell  } )\\big\\}^2  - C_5 \\sqrt{ \\log(n\\vee p) |{  \\mathcal J } _\\ell|   }\\bigg(\\sqrt { { \\mathfrak{s} }  }  \\|  \\widehat \\beta _\\I  -  \\beta _{{  \\mathcal J } _1}^*   \\|_2 + C_2  { \\mathfrak{s} }  \\sqrt { \\frac{ \\log(n\\vee p)}{| \\I|} } \\bigg) \n\\\\\n\\ge &  \\frac{c_x|{  \\mathcal J } _\\ell | }{32} \\|\\widehat \\beta _\\I  -  \\beta^* _{{  \\mathcal J } _ \\ell  } \\|_2^2 - C_4   { \\mathfrak{s} }  \\log(n\\vee p)  - C_5 \\sqrt{ \\log(n\\vee p) |{  \\mathcal J } _\\ell|  }\\bigg(\\sqrt { { \\mathfrak{s} }  }  \\|  \\widehat \\beta _\\I  -  \\beta _{{  \\mathcal J } _1}^*   \\|_2 + C_2  { \\mathfrak{s} }  \\sqrt { \\frac{ \\log(n\\vee p)}{| \\I|} } \\bigg)  \n\\\\\n\\ge & \\frac{c_x|{  \\mathcal J } _\\ell | }{32} \\|\\widehat \\beta _\\I  -  \\beta^* _{{  \\mathcal J } _ \\ell  } \\|_2^2 -\\frac{c_x|{  \\mathcal J } _\\ell | }{64} \\|\\widehat \\beta _\\I  -  \\beta^* _{{  \\mathcal J } _ \\ell  } \\|_2^2 -C_6  { \\mathfrak{s} }  \\log(n\\vee p)  \n= \\frac{c_x|{  \\mathcal J } _\\ell | }{64} \\|\\widehat \\beta _\\I  -  \\beta^* _{{  \\mathcal J } _ \\ell  } \\|_2^2     -C_6  { \\mathfrak{s} }  \\log(n\\vee p)  ,\n\\end{align*}\nwhere the second inequality follows from the standard sub-Exponential tail bound and \\Cref{eq:regression two change step 3 term 1}, the third inequality follows from \\Cref{eq:regression two change step 3 term 2}, and the fourth inequality follows from  $ {  \\mathcal J } _\\ell \\subset \\I $ and so $ |\\I | \\ge |{  \\mathcal J } _\\ell|$.\n\\\\\n\\\\\nSo for $\\ell \\in \\{1,2 \\}$, \n\\begin{align} \\label{eq:regression change point step 3 last item}\\sum_{i \\in {  \\mathcal J } _\\ell  }(y_i - X_i^\\top  \\widehat \\beta _\\I ) ^2  - \\sum_{i \\in {  \\mathcal J } _\\ell   }(y_i - X_i^\\top    \\beta^*  _{{  \\mathcal J } _\\ell }  ) ^2 \n\\ge \\frac{c_x|{  \\mathcal J } _\\ell | }{64} \\|\\widehat \\beta _\\I  -  \\beta^* _{{  \\mathcal J } _ \\ell  } \\|_2^2     -C_6  { \\mathfrak{s} }  \\log(n\\vee p).\n\\end{align} \n\\\n\\\\\n\\\\\n{\\bf Step 5.} For ${  \\mathcal J } _3$, if $|{  \\mathcal J } _3| \\ge C_\\mclF  { \\mathfrak{s} }  \\log(n\\vee p)  $,  following the same calculations as in {\\bf Step 4}, \n$$\\sum_{i \\in {  \\mathcal J } _ 3 }(y_i - X_i^\\top  \\widehat \\beta _\\I ) ^2  - \\sum_{i \\in {  \\mathcal J } _ 3 }(y_i - X_i^\\top    \\beta^*  _{{  \\mathcal J } _3   }  ) ^2 \\ge \\frac{c_x|{  \\mathcal J } _3   | }{64} \\|\\widehat \\beta _\\I  -  \\beta^* _{{  \\mathcal J } _ 3  } \\|_2^2     -C_6  { \\mathfrak{s} }  \\log(n\\vee p)  \\ge -C_6  { \\mathfrak{s} } \\log(n\\vee p). \n$$\nIf $|{  \\mathcal J } _3 | < C_\\mclF  { \\mathfrak{s} }  \\log(n\\vee p) $, then\n\\begin{align} & \\sum_{i \\in {  \\mathcal J } _3   }(y_i - X_i^\\top  \\widehat \\beta _\\I ) ^2  - \\sum_{i \\in {  \\mathcal J } _3   }(y_i - X_i^\\top    \\beta^*  _{{  \\mathcal J } _ 3}  ) ^2 \\nonumber \n=   \\sum_{i \\in {  \\mathcal J } _ 3   }\\big\\{ X_i^\\top (\\widehat \\beta _\\I -\\beta^*  _{{  \\mathcal J } _3} )\\big\\}^2 -2 \\sum_{i \\in {  \\mathcal J } _3   } \\epsilon_i X_i^\\top (\\widehat \\beta _\\I -\\beta^*  _{{  \\mathcal J } _3 }  )\n\\\\  \\nonumber \n\\ge  & \\sum_{i \\in {  \\mathcal J } _3   }\\big\\{ X_i^\\top (\\widehat \\beta _\\I -\\beta^*  _{{  \\mathcal J } _3 } )\\big\\}^2  -\\frac{1}{2 }\\sum_{i \\in {  \\mathcal J } _3   }\\big\\{ X_i^\\top (\\widehat \\beta _\\I -\\beta^*  _{{  \\mathcal J } _3 } )\\big\\}^2  - 4   \\sum_{i \\in {  \\mathcal J } _3   }\\epsilon_i^2 \n\\\\  \\nonumber \n\\ge & \\frac{1}{2 }\\sum_{i \\in {  \\mathcal J } _3   }\\big\\{ X_i^\\top (\\widehat \\beta _\\I -\\beta^*  _{{  \\mathcal J } _3 } )\\big\\}^2  -C_7\\bigg(  \\sqrt { \\gamma   \\log(n)} + \\log(n)+ \\gamma  \\bigg) \n\\\\  \\nonumber  \n\\ge & \\frac{1}{2 }\\sum_{i \\in {  \\mathcal J } _3   }\\big\\{ X_i^\\top (\\widehat \\beta _\\I -\\beta^*  _{{  \\mathcal J } _3 } )\\big\\}^2  -C_7' \\bigg(     \\log(n)+ \\gamma   \\bigg)  \n\\\\ \\ge \n& \\frac{1}{2 }\\sum_{i \\in {  \\mathcal J } _3   }\\big\\{ X_i^\\top (\\widehat \\beta _\\I -\\beta^*  _{{  \\mathcal J } _3 } )\\big\\}^2  -C_8    (  { \\mathfrak{s} } \\log(n\\vee p) + \\gamma) \n\\ge -C_8    (   { \\mathfrak{s} } \\log(n\\vee p) +\\gamma)   \\label{eq:regression change point step 4 last item}\n\\end{align}\nwhere the second inequality follows from the standard sub-exponential deviation bound.\n\\\n\\\\\n\\\\\n{\\bf Step 6.} Putting \\Cref{eq:regression two change points first}, \\eqref{eq:regression change point step 3 last item} and \\eqref{eq:regression change point step 4 last item} together, it follows that \n $$  \\sum_{\\ell =1}^2 \\frac{c_x|{  \\mathcal J } _\\ell | }{64} \\|\\widehat \\beta _\\I  -  \\beta^* _{{  \\mathcal J } _ \\ell  } \\|_2^2      \\le C_9( { \\mathfrak{s} }  \\log(n\\vee p) +  { \\mathcal B_n^{-1}   \\Delta }  \\kappa ^2 + \\gamma) .$$\n This leads to \n  $$     |{  \\mathcal J } _1 |   \\|\\widehat \\beta _\\I  -  \\beta^* _{{  \\mathcal J } _ 1  } \\|_2^2     + |{  \\mathcal J } _2 |   \\|\\widehat \\beta _\\I  -  \\beta^* _{{  \\mathcal J } _ 2  } \\|_2^2       \\le C_9( { \\mathfrak{s} }  \\log(n\\vee p) +  { \\mathcal B_n^{-1}   \\Delta }  \\kappa ^2 + \\gamma) .$$ \nObserve  that \n$$ \\inf_{  \\beta  \\in \\mathbb R ^p   }  |{  \\mathcal J } _1 |  \\| \\beta    -  \\beta^* _{{  \\mathcal J } _ 1  } \\|_2^2  + |{  \\mathcal J } _2 | \\|  \\beta    -  \\beta^* _{{  \\mathcal J } _ 2  } \\|_2^2    = \\kappa_k   ^2  \\frac{|{  \\mathcal J } _1| |{  \\mathcal J } _2|}{| {  \\mathcal J } _1| +|{  \\mathcal J } _2|  }   \\ge \\frac{ \\kappa_k  ^2 }{2} \\min\\{ |{  \\mathcal J } _1| ,|{  \\mathcal J } _2| \\}   \\ge   \\frac{ c \\kappa   ^2 }{2} \\min\\{ |{  \\mathcal J } _1| ,|{  \\mathcal J } _2| \\}   . $$ \nThus\n$$    \\kappa ^2 \\min\\{ |{  \\mathcal J } _1| ,|{  \\mathcal J } _2| \\}    \\le  C_{10} ( { \\mathfrak{s} }  \\log(n\\vee p)+ { \\mathcal B_n^{-1}   \\Delta }  \\kappa^2  + \\gamma)  ,$$\nwhich is \n$$ \\min\\{ |{  \\mathcal J } _1| ,|{  \\mathcal J } _2| \\}   \\le C_5 \\bigg( \\frac{ { \\mathfrak{s} }  \\log(n\\vee p) + \\gamma  }{\\kappa ^2 } +   { \\mathcal B_n^{-1}   \\Delta }   +\\frac{ \\gamma}{\\kappa ^2} \\bigg) .$$ \nSince  $ |{  \\mathcal J } _2  | \\ge \\Delta_{\\min} \\ge     \\frac{  C     (  { \\mathfrak{s} }  \\log(n\\vee p) + \\gamma) }{   \\kappa^{2}}   $\n for sufficiently large constant $C$, \n it follows that\n $$ |{  \\mathcal J } _2| \\ge \\Delta_{\\min}>  C_5 \\bigg( \\frac{ { \\mathfrak{s} }  \\log(n\\vee p) + \\gamma  }{\\kappa ^2 } +   { \\mathcal B_n^{-1}   \\Delta }   +\\frac{ \\gamma}{\\kappa ^2} \\bigg) .$$  So it holds that \n $$   |{  \\mathcal J } _1|    \\le C_5 \\bigg( \\frac{ { \\mathfrak{s} }  \\log(n\\vee p) + \\gamma  }{\\kappa ^2 } +   { \\mathcal B_n^{-1}   \\Delta }   \\bigg) .$$ \n \\end{proof} \n\n \\bnlem[Three or more change points]\n\\label{lem:regression three or more cp}\nSuppose the good events \n$\\mathcal L ( { \\mathcal B_n^{-1}   \\Delta }  ) $ and  $\\mathcal R (  { \\mathcal B_n^{-1}   \\Delta }  ) $ defined  in  \\Cref{eq:left and right approximation of change points} hold. \n Suppose in addition that \n\\begin{align} \\label{eq:regression three change points snr}\n\\Delta_{\\min} \\kappa^2 \\ge C      \\big(  { \\mathfrak{s} }  \\log(n\\vee p) + \\gamma)    \n\\end{align} \nfor sufficiently large constant $C $. \n Then  with probability  at least $1- n^{-3}$,    there is no intervals in   $  \\widehat { \\mathcal P}  $  containing three or more true change points.   \n \\enlem  \n \n\n\n\\begin{proof} For  contradiction,  suppose    $ \\I=(s,e] \\in \\mathcal {\\widehat P}  $  be such that $ \\{ \\eta_1, \\ldots, \\eta_M\\} \\subset \\I $ with $M\\ge 3$.  \n\\\\\n\\\\ \nSince the events $\\mathcal L (  { \\mathcal B_n^{-1}   \\Delta }   ) $ and  $\\mathcal R (  { \\mathcal B_n^{-1}   \\Delta }  ) $ hold, by relabeling $\\{ s_q\\}_{q=1}^ { \\mathcal Q}   $ if necessary,   let $ \\{ s_m\\}_{m=1}^M $ be such that \n $$ 0 \\le s_m  -\\eta_m  \\le  { \\mathcal B_n^{-1}   \\Delta }     \\quad \\text{for} \\quad 1 \\le m \\le M-1  $$ and that\n $$ 0\\le \\eta_M  - s_M \\le  { \\mathcal B_n^{-1}   \\Delta }  .$$ \n Note that these choices ensure that  $ \\{ s_m\\}_{m=1}^M  \\subset \\I . $\n \\\n \\\\\n \\begin{center}\n \\begin{tikzpicture} \n\\draw[ - ] (-10,0)--(1,0);\n \\node[color=black] at (-8,-0.3) {\\small s};\n \\draw[  (-, ultra thick, black] (-8,0) -- (-7.99,0);\n  \\draw[ { -]}, ultra thick, black] (-1.0002,0) -- (-1,0);\n\n   \\node[color=black] at (-7,-0.3) {\\small $\\eta_1$};\n\\draw(-7 ,0)circle [radius=2pt] ;\n\n  \\node[color=black] at (-6.5,-0.3) {\\small $s_1$};\n\\draw plot[mark=x, mark options={color=black, scale=1.5}] coordinates {(-6.5,0)  }; \n\n  \\node[color=black] at (-5,-0.3) {\\small $\\eta_2$};\n\\draw(-5 ,0)circle [radius=2pt] ;\n  \\node[color=black] at (-4.5,-0.3) {\\small $s_2$};\n\\draw plot[mark=x, mark options={color=black, scale=1.5}] coordinates {(-4.5,0)  }; \n \n\n  \\node[color=black] at (-2.5,-0.3) {\\small $\\eta_M$};\n\\draw(-2.5 ,0)circle [radius=2pt] ;\n  \\node[color=black] at (-3 ,-0.3) {\\small $s_M$};\n\\draw plot[mark=x, mark options={color=black, scale=1.5}] coordinates {(-3,0)  }; \n  \n   \\node[color=black] at (-1,-0.3) {\\small $e$};\n\n \n\\end{tikzpicture}\n\\end{center}\n\\\n\\\\\n{\\bf Step 1.}\n Denote \n $$ \\mathcal I_ 1= (  s, s _1], \\quad \\I_m   =(s_{m-1} , s_m] \\text{ for }  2 \\le m \\le M     \\quad \\text{and} \\quad \\I_{M+1}   = (s_M,e]. $$\n Then since $|\\I| \\ge \\Delta_{\\min} \\ge  C_s  { \\mathfrak{s} }  \\log(n\\vee p)  $, it follows that \n Since  $ |\\I| \\ge \\eta_{k+1} -\\eta_k \\ge  C_s  { \\mathfrak{s} }  \\log(n\\vee p) $, \n$$ \\mathcal F(\\I ) = \\sum_{i \\in \\I }(y_i - X_i^\\top \\widehat \\beta _\\I ) ^2  . $$\nSince  $ | \\I_m | \\ge \\Delta_{\\min} \/2 \\ge   C_s  { \\mathfrak{s} }  \\log(n\\vee p) $ for all $ 2 \\le m \\le M $,  it follows  from the same argument as   {\\bf Step 1} in the proof  of  \\Cref{lemma:regression two change points} that \n \\begin{align*}\\mathcal F(\\I_m)  = & \\sum_{i \\in \\I_m }(y_i - X_i^\\top \\widehat \\beta _{\\I_m}  ) ^2\n  \\le   \\sum_{i \\in \\I_m }(y_i - X_i^\\top   \\beta ^*  _ i   ) ^2  + C_1 \\bigg\\{   { \\mathcal B_n^{-1}   \\Delta }    \\kappa ^2 +   { \\mathfrak{s} } \\log(n\\vee p)  \\bigg\\} \\quad \\text{for all } 2 \n  \\le m \\le M.\n \\end{align*}\n \\\n \\\\\n {\\bf Step 2.} It follows  from the same argument as   {\\bf Step 2} in the proof  of  \\Cref{lemma:regression two change points} that    \n  \\begin{align*}\n  &\\mathcal F(\\I_1)  \n  \\le   \\sum_{i \\in \\I_1 }(y_i - X_i^\\top   \\beta ^*  _ i   ) ^2  + C_1 \\bigg\\{   { \\mathcal B_n^{-1}   \\Delta }    \\kappa ^2 +   { \\mathfrak{s} } \\log(n\\vee p)  \\bigg\\} , \\text{ and}\n  \\\\\n  &\\mathcal F(\\I_{M+1} )  \n  \\le   \\sum_{i \\in \\I_{M+1} }(y_i - X_i^\\top   \\beta ^*  _ i   ) ^2  + C_1 \\bigg\\{   { \\mathcal B_n^{-1}   \\Delta }    \\kappa ^2 +   { \\mathfrak{s} } \\log(n\\vee p)  \\bigg\\}  \n \\end{align*}\n \\\n \\\\\n {\\bf Step 3.} Since $\\I \\in \\widehat{\\mathcal P} $, we have\n \\begin{align}\\label{eq:regression three change points local min} \\mathcal F(\\I ) \\le \\sum_{m=1}^{M+1}\\mathcal F(\\I_m  )  +  M\\gamma. \n \\end{align} The above display and the calculations in  {\\bf Step 1} and  {\\bf Step 2} implies that\n\\begin{align} \n \\sum_{i \\in \\I }(y_i -  X_i^\\top \\widehat \\beta _\\I ) ^2  \n\\le   \\sum_{i \\in \\I  }(y_i -   X_i^\\top \\beta_i^*   ) ^2  \n +   (M +1) C_1 \\bigg\\{   { \\mathcal B_n^{-1}   \\Delta }    \\kappa ^2 +   { \\mathfrak{s} } \\log(n\\vee p) \\bigg\\}\n+M\\gamma  .\n  \\label{eq:regression three change points step 2 term 1}\n\\end{align}\nDenote \n   $$  {  \\mathcal J } _1 =(s,  \\eta_1], \\ {  \\mathcal J } _m = (\\eta_{m-1}, \\eta_m]  \\quad \\text{for}\\quad  2 \\le m \\le M , \\ {  \\mathcal J } _{M+1}  =(\\eta_M, e]. $$  \n\\Cref{eq:regression three change points step 2 term 1}  gives \n\\begin{align} \n \\sum_{m=1}^{M+1}  \\sum_{i \\in {  \\mathcal J } _m   }(y_i - X_i^\\top  \\widehat \\beta _\\I ) ^2 \n \\le& \\sum_{m=1}^{M+1} \\sum_{i \\in {  \\mathcal J } _  m   }(y_i -  X_i^\\top    \\beta^*  _{{  \\mathcal J } _m  }  ) ^2    \n +  (M+1)   C_1 \\bigg\\{   { \\mathcal B_n^{-1}   \\Delta }    \\kappa ^2 +   { \\mathfrak{s} } \\log(n\\vee p) \\bigg\\} \n+M\\gamma     \\label{eq:regression three change points first}\n\\end{align}  \n \\\n \\\\\n {\\bf Step 4.} Using the same argument as in the {\\bf Step 4} in the proof  of  \\Cref{lemma:regression two change points},\nit follows that \n\\begin{align} \\label{eq:regression three change point step 3 last term}\\sum_{i \\in {  \\mathcal J } _m   }(y_i - X_i^\\top  \\widehat \\beta _\\I ) ^2  - \\sum_{i \\in {  \\mathcal J } _m   }(y_i - X_i^\\top    \\beta^*  _{{  \\mathcal J } _m }  ) ^2 \n\\ge \\frac{c_x|{  \\mathcal J } _m | }{64} \\|\\widehat \\beta _\\I  -  \\beta^* _{{  \\mathcal J } _ m  } \\|_2^2     -C_2  { \\mathfrak{s} }  \\log(n\\vee p) \\quad \\text{for all}  \\   2 \\le m \\le M. \n\\end{align} \n \\\n \\\\\n {\\bf Step 5.}\n  Using the same argument as in the {\\bf Step 4} in the proof  of  \\Cref{lemma:regression two change points},  it follows that \n\\begin{align} \\label{eq:regression three change point step 5 first term}& \\sum_{i \\in {  \\mathcal J } _ 1 }(y_i - X_i^\\top  \\widehat \\beta _\\I ) ^2  - \\sum_{i \\in {  \\mathcal J } _ 1 }(y_i - X_i^\\top    \\beta^*  _{{  \\mathcal J } _1   }  ) ^2  \\ge -C_3  (  { \\mathfrak{s} } \\log(n\\vee p)  +\\gamma) \\text{ \n and }\n\\\\\\label{eq:regression three change point step 5 second term}\n& \\sum_{i \\in {  \\mathcal J } _ {M+ 1} }(y_i - X_i^\\top  \\widehat \\beta _\\I ) ^2  - \\sum_{i \\in {  \\mathcal J } _  {M+ 1}  }(y_i - X_i^\\top    \\beta^*  _{{  \\mathcal J } _ {M+ 1}   }  ) ^2  \\ge -C_3 (  { \\mathfrak{s} } \\log(n\\vee p)  +\\gamma) \n \\end{align}\n\\\n\\\\\n{\\bf  Step 6.} Putting \\Cref{eq:regression three change points first}, \\eqref{eq:regression three change point step 3 last term}, \\eqref{eq:regression three change point step 5 first term} and  \\eqref{eq:regression three change point step 5 second term}, \n it follows that \n\\begin{align} \\label{eq:regression three change points step six} \\sum_{ m  =2}^M \\frac{c_x|{  \\mathcal J } _m  | }{64} \\|\\widehat \\beta _\\I  -  \\beta^* _{{  \\mathcal J } _  m  } \\|_2^2      \\le C_4M ( { \\mathfrak{s} }  \\log(n\\vee p) +  { \\mathcal B_n^{-1}   \\Delta }  \\kappa ^2 + \\gamma) .\n\\end{align} \n For any $  m \\in\\{2, \\ldots, M\\}$, it holds that\n\\begin{align}  \\label{eq:regression three change points signal lower bound} \\inf_{  \\beta  \\in \\mathbb R^p   }  |{  \\mathcal J } _{m-1} |  \\|   \\beta - \\beta^* _{{  \\mathcal J } _ {m-1}   }  \\|  ^2 + |{  \\mathcal J } _{m} | \\|   \\beta - \\beta^* _{{  \\mathcal J } _ m  }  \\|  ^2   =&     \\frac{|{  \\mathcal J } _{m-1}| |{  \\mathcal J } _m|}{ |{  \\mathcal J } _{m-1}| +  |{  \\mathcal J } _m| } \\kappa_m ^2  \\ge \\frac{1}{2} \\Delta_{\\min} \\kappa^2,\n\\end{align}  \nwhere the last inequality follows from the assumptions that $\\eta_k - \\eta_{k-1}\\ge \\Delta_{\\min}  $ and $ \\kappa_k \\asymp \\kappa$  for all $1\\le k \\le K$. So\n\\begin{align}   \\nonumber &2 \\sum_{ m=1}^{M }   |{  \\mathcal J } _m |\\|\\widehat \\beta _\\I  -  \\beta^* _{{  \\mathcal J } _ m  } \\|_2^2   \n\\\\ \n\\ge &  \\nonumber    \\sum_{m=2}^M  \\bigg(  |{  \\mathcal J } _{m-1}  \\|\\widehat \\beta _\\I  -  \\beta^* _{{  \\mathcal J } _ {m-1}   } \\|_2^2      + |{  \\mathcal J } _m |\\|\\widehat \\beta _\\I  -  \\beta^* _{{  \\mathcal J } _ m  } \\|_2^2    \\bigg) \n\\\\ \\label{eq:regression three change points signal lower bound two} \n\\ge & (M-1)  \\frac{ 1}{2} \\Delta_{\\min} \\kappa^2  \\ge \\frac{M}{4} \\Delta_{\\min} \\kappa^2 ,\n\\end{align} \nwhere the second inequality follows from  \\Cref{eq:regression three change points signal lower bound} and the last inequality follows from $M\\ge 3$. \\Cref{eq:regression three change points step six} and  \\Cref{eq:regression three change points signal lower bound two} together imply that \n\\begin{align}\\label{eq:regression three change points signal lower bound three} \n \\frac{M}{4} \\Delta_{\\min} \\kappa^2 \\le 2  C_5  M\\bigg(   { \\mathfrak{s} }    \\log(n\\vee p ) +     { \\mathcal B_n^{-1}   \\Delta }  \\kappa^2\n+   \\gamma \\bigg) .\n\\end{align}\nSince $ { \\mathcal B_n} \\to \\infty $, it follows that for sufficiently large $n$,   \\Cref{eq:regression three change points signal lower bound three} gives \n$$ \\Delta_{\\min}\\kappa^2 \\le C_5 \\big(   { \\mathfrak{s} }  \\log(n\\vee p) +\\gamma),$$  \nwhich contradicts \\Cref{eq:regression three change points snr}. \n \n  \n \\end{proof} \n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n \n \\bnlem[Two consecutive intervals]\n \\label{lem:regression two intervals}\n Suppose $ \\gamma \\ge C_\\gamma  { \\mathfrak{s} }   \\log(n\\vee p) $ for sufficiently large constant $C_\\gamma $. \nWith probability  at least $1- n^{-3}$,    there are no  two consecutive intervals $\\I_1= (s,t ] \\in \\widehat {\\mathcal  P}  $,   $ \\I_2=(t, e]  \\in   \\widehat  {\\mathcal  P}  $    such that $\\I_1 \\cup \\I_2$ contains no change points.  \n \\enlem \n \\begin{proof} \n For contradiction, suppose that \n $$ \\I :  =\\I_1\\cup  \\I_2 $$\n contains no change points. \nFor $\\I_1$, note that if $|\\I_1| \\ge  C_\\zeta  { \\mathfrak{s} }  \\log(n\\vee p) $, then \n by \\Cref{lem:regression one change deviation bound} {\\bf a}, it follows that \n \\begin{align*} \n \\bigg|   \\mathcal F(\\I_1)   -  \\sum_{i\\in \\I_1} (y_i - X_i^\\top  \\beta^*  _{i} )^2 \\bigg|  = \\bigg|   \\sum_{i\\in \\I_1} (y_i - X_i^\\top \\widehat \\beta _{\\I_1} )^2- \\sum_{i\\in \\I_1} (y_i - X_i^\\top \\beta_i^*   )^2    \\bigg| \\le  C_1  { \\mathfrak{s} }  \\log(n\\vee p)  .\n \\end{align*}\n If $|\\I_1| <  C_\\zeta  { \\mathfrak{s} }  \\log(n\\vee p)  $, then\n\\begin{align*} \n& \\bigg|   \\mathcal F(\\I_1)   -  \\sum_{i\\in \\I_1} (y_i - X_i^\\top  \\beta^*  _{i} )^2 \\bigg|   =   \\bigg|     \\sum_{i\\in \\I_1} (y_i - X_i^\\top  \\beta^*  _{i} )^2  \\bigg|  \n=\\sum_{i\\in \\I_1}   \\epsilon_i^2 \n \\\\ \n \\le & |\\I_1| E(\\epsilon^2_1) + C_2 \\sqrt {  |\\I_1|  \\log(n)  } + \\log(n)  \\le C_2'  { \\mathfrak{s} }  \\log(n\\vee p) .\n \\end{align*} \n So\n \\begin{align*} \n \\bigg|   \\mathcal F(\\I_1)   -  \\sum_{i\\in \\I_1} (y_i - X_i^\\top  \\beta^*  _{i} )^2 \\bigg| \\le   C_3    { \\mathfrak{s} }  \\log(n\\vee p)  .\n \\end{align*} \n Similarly, \n\\begin{align*} \n&\\bigg|   \\mathcal F(\\I_2)   -  \\sum_{i\\in \\I_2} (y_i - X_i^\\top  \\beta^*  _{i} )^2 \\bigg| \\le   C_3    { \\mathfrak{s} }  \\log(n\\vee p), \\quad \\text{and}\n\\\\\n&\\bigg|   \\mathcal F(\\I )   -  \\sum_{i\\in \\I } (y_i - X_i^\\top  \\beta^*  _{i} )^2 \\bigg| \\le   C_3    { \\mathfrak{s} }  \\log(n\\vee p)  .\n \\end{align*} \n So \n $$\\sum_{i\\in \\I_1} (y_i - X_i^\\top \\beta_i^*   )^2  +\\sum_{i\\in \\I_2} (y_i - X_i^\\top \\beta_i^*    )^2 -2C_1   { \\mathfrak{s} }  \\log(n\\vee p ) +\\gamma    \\le \\sum_{i\\in \\I } (y_i - X_i^\\top \\beta_i^*   )^2 +C_1  { \\mathfrak{s} }   \\log(n\\vee p) .  $$\n Since $\\beta^* _i$ is unchanged when $i\\in \\I$, it follows that \n $$ \\gamma \\le 3C_1  { \\mathfrak{s} }  \\log(n\\vee p).$$\n This is a contradiction when $C_\\gamma> 3C_1. $\n \\end{proof} \n  \n  \n  \n  \n   \n\n\n\n\n\n\n\n\\bnlem\nLet $\\mathcal{S}$ be any linear subspace in $\\mathbb{R}^n$ and $\\mathcal{N}_{1\/4}$\tbe a $1\/4$-net of $\\mathcal{S} \\cap B(0, 1)$, where $B(0, 1)$ is the unit ball in $\\mathbb{R}^n$.  For any $u \\in \\mathbb{R}^n$, it holds that\n\t\\[\n\t\t\\sup_{v \\in \\mathcal{S} \\cap B(0, 1)} \\langle v, u \\rangle \\leq 2 \\sup_{v \\in \\mathcal{N}_{1\/4}} \\langle v, u \\rangle,\n\t\\]\n\twhere $\\langle \\cdot, \\cdot \\rangle$ denotes the inner product in $\\mathbb{R}^n$.\n\\enlem\n\n\\begin{proof}\nDue to the definition of $\\mathcal{N}_{1\/4}$, it holds that for any $v \\in \\mathcal{S} \\cap B(0, 1)$, there exists a $v_k \\in \\mathcal{N}_{1\/4}$, such that $\\|v - v_k\\|_2 < 1\/4$.  Therefore,\n\t\\begin{align*}\n\t\t\\langle v, u \\rangle = \\langle v - v_k + v_k, u \\rangle = \\langle x_k, u \\rangle + \\langle v_k, u \\rangle \\leq \\frac{1}{4} \\langle v, u \\rangle + \\frac{1}{4} \\langle v^{\\perp}, u \\rangle + \\langle v_k, u \\rangle,\n\t\\end{align*}\n\twhere the inequality follows from  $x_k = v - v_k = \\langle x_k, v \\rangle v + \\langle x_k, v^{\\perp} \\rangle v^{\\perp}$.  Then we have\n\t\\[\n\t\t\\frac{3}{4}\\langle v, u \\rangle \\leq \\frac{1}{4} \\langle v^{\\perp}, u \\rangle + \\langle v_k, u \\rangle.\n\t\\]\n\tIt follows from the same argument that \n\t\\[\n\t\t\\frac{3}{4}\\langle v^{\\perp}, u \\rangle \\leq \\frac{1}{4} \\langle v, u \\rangle + \\langle v_l, u \\rangle,\n\t\\]\n\twhere $v_l \\in \\mathcal{N}_{1\/4}$ satisfies $\\|v^{\\perp} - v_l\\|_2 < 1\/4$.  Combining the previous two equation displays yields\n\t\\[\n\t\t\\langle v, u \\rangle \\leq 2 \\sup_{v \\in \\mathcal{N}_{1\/4}} \\langle v, u \\rangle,\n\t\\]\n\tand the final claims holds.\n\\end{proof}\n\n \n\n\\Cref{lem:deviation piecewise constant} is an adaptation of Lemma 3 in \\cite{wang2021_jmlr}.\n\n\\bnlem\\label{lem:deviation piecewise constant}\n\tGiven any interval $I = (s, e] \\subset \\{1, \\ldots, n\\}$. Let $\\mclR_m := \\{v \\in \\mathbb{R}^{(e-s)}| \\|v\\|_2 = 1, \\sum_{t = 1}^{e-s-1} \\mathbf{1}\\{v_i \\neq v_{i+1}\\} = m\\}$. Then for data generated from \\Cref{assp:dcdp_linear_reg}, it holds that for any $\\delta > 0$, $i \\in \\{1, \\ldots, p\\}$, \n\t\\[\n\t\t\\mathbb{P}\\left\\{\\sup_{v\\in \\mclR_m} \\left|\\sum_{t = s+1}^e v_t \\epsilon_t (X_t)_i\\right| > \\Delta_{\\min} \\right\\} \\leq C(e-s-1)^m 9^{m+1} \\exp \\left\\{-c \\min\\left\\{\\frac{\\delta^2}{4C_x^2}, \\, \\frac{\\delta}{2C_x \\|v\\|_{\\infty}}\\right\\}\\right\\}.\n\t\\]\n\\enlem\n\n\\begin{proof}\nFor any $v \\in \\mathbb{R}^{(e-s)}$ satisfying $\\sum_{t = 1}^{e-s-1}\\mathbbm{1}\\{v_i \\neq v_{i+1}\\} = m$, it is determined by a vector in $\\mathbb{R}^{m+1}$ and a choice of $m$ out of $(e-s-1)$ points.  Therefore we have,\n\t\\begin{align*}\n\t\t& \\mathbb{P}\\left\\{\\sup_{\\substack{v \\in \\mathbb{R}^{(e-s)}, \\, \\|v\\|_2 = 1\\\\ \\sum_{t = 1}^{e-s-1} \\mathbf{1}\\{v_i \\neq v_{i+1}\\} = m}} \\left|\\sum_{t = s+1}^e v_t \\epsilon_t (X_t)_i\\right| > \\Delta_{\\min} \\right\\} \\\\\n\t\t\\leq & {(e-s-1) \\choose m} 9^{m+1} \\sup_{v \\in \\mathcal{N}_{1\/4}}\t\\mathbb{P}\\left\\{\\left|\\sum_{t = s+1}^e v_t \\epsilon_t (X_t)_i\\right| > \\delta\/2 \\right\\} \\\\\n\t\t\\leq & {(e-s-1) \\choose m} 9^{m+1} C\\exp \\left\\{-c \\min\\left\\{\\frac{\\delta^2}{4C_x^2}, \\, \\frac{\\delta}{2C_x \\|v\\|_{\\infty}}\\right\\}\\right\\} \\\\\n\t\t\\leq & C(e-s-1)^m 9^{m+1} \\exp \\left\\{-c \\min\\left\\{\\frac{\\delta^2}{4C_x^2}, \\, \\frac{\\delta}{2C_x \\|v\\|_{\\infty}}\\right\\}\\right\\}.\n\t\\end{align*}\n\n\\end{proof}\n\n\n\n\n\n\n \n\n\n\n\n\n\n\n\n\n\n \n\n\n\\subsection{Additional Technical Results} \n\\bnlem\n\\label{theorem:restricted eigenvalues 2}\nSuppose  $\\{X_{i } \\}_{1\\le i \\le n } \\overset{i.i.d.} {\\sim}  N_p (0, \\Sigma )  $. \nDenote $\\mathcal C_S  := \\{ v : \\mathbb R^p : \\| v_{S^c }\\|_1 \\le 3\\| v_{S}\\|_1 \\}  $, where $ |S| \\le  { \\mathfrak{s} }  $.\nThen there exists constants $c$ and $C$ such that for all $\\eta\\le 1$, \n\\begin{align} \nP \\left( \\sup_{v \\in \\mathcal C _S , \\|v\\|_2=1   } \\left | v^\\top ( \\widehat \\Sigma -\\Sigma )  v \\right| \\ge C\\eta   \\Lambda_{\\max} (\\Sigma)   \\right)\n\\le 2\\exp( -c n\\eta ^2 + 2 { \\mathfrak{s} }  \\log(p) ) .\n\\end{align}\n\\enlem\n\\begin{proof}\nThis is a well known restricted eigenvalue property for Gaussian design. The proof can be found  in \\cite{Basu2015} or \\cite{Loh2012}.\n\\end{proof}  \n\n\\bnlem\n\\label{corollary:restricted eigenvalues 2} Suppose  $\\{X_{i } \\}_{1\\le i \\le n } \\overset{i.i.d.} {\\sim}  N_p (0, \\Sigma )  $. \n Denote $\\mathcal C_S  := \\{ v : \\mathbb R^p : \\| v_{S^c }\\|_1 \\le 3\\| v_{S}\\|_1 \\}  $, where $ |S| \\le  { \\mathfrak{s} }  $.  With probability at least $1- n ^{-5}$, it holds that\n$$  \\left | v^\\top ( \\widehat \\Sigma_\\I  -\\Sigma )  v \\right| \\le C \\sqrt { \\frac{ { \\mathfrak{s} } \\log(n\\vee p) }{|\\I| } }     \\|v\\|_2^2   \n $$\n   for all $v \\in \\mathcal C _S $ and all $\\I \\subset  (0, n]$ such that $| \\I |\\ge C_s  { \\mathfrak{s} } \\log(n\\vee p)  $, where $ C_s$\nis the constant in \\Cref{lemma:consistency} which is independent of $n, p$.   \n\\enlem\n\\begin{proof} \nFor any $\\I \\subset  (0, n]$ such that $| \\I |\\ge C_s { \\mathfrak{s} }  \\log(n\\vee p)   $, by \\Cref{theorem:restricted eigenvalues 2}, it holds that \n\\begin{align*} \nP \\left( \\sup_{v \\in \\mathcal C _S , \\|v\\|_2=1   } \\left | v^\\top ( \\widehat \\Sigma _\\I -\\Sigma )  v \\right| \\ge C\\eta   \\Lambda_{\\max} (\\Sigma)   \\right)\n\\le 2\\exp( -c   |\\I| \\eta ^2 + 2 { \\mathfrak{s} }  \\log(p) ) .\n\\end{align*}\nLet  $\\eta= C_1 \\sqrt { \\frac{ { \\mathfrak{s} } \\log(n\\vee p) }{|\\I | } } $ for sufficiently large constant $C_1$. Note that $\\eta< 1$ if $ |\\I| > C_1^2  { \\mathfrak{s} } \\log(n\\vee p) $. Then   with probability at least \n$(n\\vee p)^{-7} $, \n$$\\sup_{v \\in \\mathcal C _S , \\|v\\|_2=1   } \\left | v^\\top ( \\widehat \\Sigma _\\I -\\Sigma )  v \\right| \\ge C_2 \\sqrt { \\frac{s\\log(n\\vee p) }{|\\I |  } } .$$\nSince there are at most $n^2$ many different  choices of $ \\I \\subset (0,n]$, the desired result follows from a union bound argument. \n\\end{proof}  \n\n\n\\bnlem\n\\label{lem:restricted eigenvalue}\nUnder \\Cref{assp:dcdp_linear_reg}, it holds that\n \\begin{align*}\n\t\t \\mathbb P \\bigg( \\sum_{t\\in \\I } ( X_t^\\top v )^2 \\ge \\frac{c_x|\\I| }{4} \\|v\\|_2^2 - C_2 \\log(n\\vee p) \\|v\\|_1 ^2 \\ \\forall v \\in \\mathbb R^p  \\text{ and } \\forall |\\I|\\ge C_s  { \\mathfrak{s} }  \\log(n\\vee p)  \\bigg)  \\le  n^{-5} \n\t\\end{align*} \n\twhere $  C_2 > 0$ is an  absolute constant  only depending on $C_x$.\n\\enlem\n\\begin{proof}\nBy the well known restricted eigenvalue condition, for any $\\I$, it holds that \n \\begin{align*}\n\t\t \\mathbb P \\bigg( \\sum_{t\\in \\I } ( X_t^\\top v )^2 \\ge \\frac{c_x |\\I| }{4} \\|v\\|_2^2 - C_2 \\log(n\\vee p) \\|v\\|_1 ^2  \\ \\  \\forall v \\in \\mathbb R^p    \\bigg)  \\le  C_3 \\exp(-c_3|\\I| ).\n\t\\end{align*} \n  Since $|\\I|  \\ge  C_s  { \\mathfrak{s} }  \\log(n\\vee p) $, \n  \\begin{align*}\n\t\t \\mathbb P \\bigg( \\sum_{t\\in \\I } ( X_t^\\top v )^2 \\ge \\frac{c_ x |\\I| }{4} \\|v\\|_2^2 - C_2 \\log(n\\vee p) \\|v\\|_1^2 \\ \\ \\forall v \\in \\mathbb R^p    \\bigg)  \\le  n^{-4}.\n\t\\end{align*} \n\tSince there are at most $n^2$ many subinterval $\\I \\subset (0,n]$, it follows from a union bound argument  that \n\t \\begin{align*}\n\t\t \\mathbb P \\bigg( \\sum_{t\\in \\I } ( X_t^\\top v )^2 \\ge \\frac{c_x |\\I| }{4} \\|v\\|_2^2 - C_2 \\log(n\\vee p) \\|v\\|_1^2  \\ \\ \\forall v \\in \\mathbb R^p  \\text{ and } \\forall |\\I|\\ge C_s  { \\mathfrak{s} }  \\log(n\\vee p)  \\bigg)  \\le  n^{-2} .\n\t\\end{align*} \n\\end{proof} \n\n\n\n\n\\bnlem \\label{lemma:consistency}\nSuppose  \\Cref{assp:dcdp_linear_reg} holds. There exists  a sufficient large constant $ C_s$ such that the following conditions holds.    \n\\\\\n\\\\\n{\\bf a.} With probability at least $1-n^{-3} $, it holds that\n\\begin{align}\\label{eq:independent condition 1c 1}\n   \\left |  \\frac{1}{|\\I | } \\sum_{i \\in \\mathcal I }  \\epsilon_i  X_i^\\top \\beta \\right| \\le C\\sigma_{\\epsilon}\\sqrt { \\frac{\\log(n\\vee p)}{|\\I |   } } \\|\\beta\\|_1      \n \\end{align}\nuniformly  for all $ \\beta \\in \\mathbb R^p$ and all $\\mathcal I \\subset (0, n]$ such that $|\\I|\\ge C_s  { \\mathfrak{s} }  \\log(n\\vee p) $, \n\\\\\n\\\\\n {\\bf b.} Let $\\{ u_i\\}_{i=1}^n \\subset  \\mathbb R^p$  be a collection of  deterministic vectors. Then with probability at least $1-n^{-3} $, it holds that   \n\\begin{align}\\label{eq:independent condition 1c}\n       \\left |  \\frac{1}{|\\I |   } \\sum_{i \\in \\I }    u^\\top_i   X_i   X_i^\\top \\beta  - \\frac{1}{ |\\I |  } \\sum_{i \\in \\mathcal I }   u^\\top  _i  \\Sigma \\beta  \\right| \\le C \\left(  \\max _{1\\le i \\le n }\\|u_i\\|_2  \\right) \\sqrt {  \\frac{\\log(n\\vee p)}{|\\I |   } } \\|\\beta\\|_1    \n \\end{align}\n uniformly for all $ \\beta \\in \\mathbb R^p$ and all $\\mathcal I \\subset (0, n]$ such that $|\\I|\\ge  C_s  { \\mathfrak{s} }  \\log(n\\vee p) $.    \n\\enlem \n \\begin{proof}\n   The justification   of the  \\eqref{eq:independent condition 1c 1}  is   similar and simpler  than the justification  of \\eqref{eq:independent condition 1c}. For conciseness, only the justification  of \\eqref{eq:independent condition 1c} is presented.\n   \nFor any $\\mathcal I \\subset (0, n]$  such  that $|\\I|\\ge  C_s  { \\mathfrak{s} }  \\log(n\\vee p) $, it holds  that\n \\begin{align*}\n &\\left|  \\frac{1}{ |\\I |  } \\sum_{i\\in \\I }  u^\\top  _i  X_i   X_i^\\top \\beta   - \\frac{1}{ |\\I |  }  \\sum_{i\\in \\I }   u^\\top_i  \\Sigma  \\beta \\right| \n \\\\\n =\n & \\left|  \\left(  \\frac{1}{|\\I |  } \\sum_{i\\in \\I }   u^\\top _i    X_i   X_i^\\top     - \\frac{1}{|\\I |  }    \\sum_{i\\in \\I }   u^\\top_i  \\Sigma  \\right) \\beta  \\right| \n \\\\\n \\le \n & \\max_{1\\le j \\le p }  \\left |  \\frac{1}{|\\I |  } \\sum_{i\\in \\I }   u^\\top  _i   X_i   X_{i,j}      - \\frac{1}{|\\I |  }    \\sum_{i\\in \\I }   u^\\top_i  \\Sigma  (, j)  \\right| \\|\\beta\\|_1.\n \\end{align*}\n Note that \n $ E(u^\\top  _i  X_i   X_{i,j} ) =u^\\top _i \\Sigma  (, j)    $ and in addition, \n $$u^\\top _ i   X_i   \\sim N(0, u^\\top _i  \\Sigma u _i )  \\quad \\text{ and } \\quad X_{i,j} \\sim N(0, \\Sigma({j,j}))  .$$\n So $u^\\top _i   X_i X_{i,j} $  is a sub-exponential random variable such that \n $$ u^\\top _i   X_i X_{i,j} \\sim SE( u^\\top_i  \\Sigma u _i  \\Sigma ({j,j})).$$\n As a result, for $\\gamma<1$ and every $j$,\n$$ P \\left( \\left |  \\frac{1}{|\\I |  } \\sum_{i \\in \\I }  u^\\top _ i     X_i   X_{i,j}      - u^\\top \\Sigma  (, j)  \\right|  \\ge \\gamma \\sqrt {  \\max_{1\\le i\\le n } ( u^\\top_i  \\Sigma u_i)  \\Sigma ( {j,j} ) }\\right) \\le \\exp(-c \\gamma^2 |\\I |  ). $$\nSince \n$$  \\sqrt { u^\\top _i  \\Sigma u _i \\Sigma ({j,j} ) } \\le C_x  \\|u_i \\|_2, $$\nby union bound,\n$$ \\mathbb P \\left(  \\max_{1\\le j \\le p }\\left |  \\frac{1}{|\\I |  } \\sum_{i \\in \\I}  u^\\top _ i     X_i   X_{i,j}      - \\frac{1}{|\\I |  } \\sum_{i \\in\\I}  u^\\top _i  \\Sigma  (, j)  \\right|  \\ge \\gamma \nC_x  \\left(  \\max _{1\\le i \\le n }\\|u_i\\|_2  \\right)  \\right) \\le p\\exp(-c \\gamma^2  |\\I| ). $$\nLet $\\gamma =  3\\sqrt { \\frac{\\log(n\\vee p) }{c  |\\I| }  }$. Note that   $ \\gamma<1$ if  $|\\I|\\ge C_s  { \\mathfrak{s} }  \\log(n\\vee p) $ for sufficiently large $C_s$.  Therefore \n$$ \\mathbb P \\left(  \\max_{1\\le j \\le p }\\left |  \\frac{1}{|\\I |  } \\sum_{i \\in \\I}  u^\\top _ i     X_i   X_{i,j}      - \\frac{1}{|\\I |  } \\sum_{i \\in\\I}  u^\\top _i  \\Sigma  (, j)  \\right|  \\ge  \nC_1\\sqrt{ \\frac{ \\log(n\\vee p)}{|\\I| } } \\left(  \\max _{1\\le i \\le n }\\|u_i\\|_2  \\right)  \\right) \\le p\\exp(-9\\log(n\\vee p)  ). $$\nSince there are at most $n^2$ many intervals $\\I \\subset(0,n]$, it follows that \n\\begin{align*}\n &\\mathbb P \\left(  \\max_{1\\le j \\le p }\\left |  \\frac{1}{|\\I |  } \\sum_{i \\in \\I}  u^\\top _ i     X_i   X_{i,j}      - \\frac{1}{|\\I |  } \\sum_{i \\in\\I}  u^\\top _i  \\Sigma  (, j)  \\right|  \\ge  \nC_1\\sqrt{ \\frac{ \\log(n\\vee p)}{|\\I| } } \\left(  \\max _{1\\le i \\le n }\\|u_i\\|_2   \\right) \\ \\forall    |\\I| \\ge C_s  { \\mathfrak{s} }  \\log(n\\vee p)  \\right)\n\\\\\n \\le & p n^2 \\exp(-9\\log(n\\vee p)  ) \\le n^{-3}. \n\\end{align*}\nThis immediately  gives \\eqref{eq:independent condition 1c}.\n \\end{proof} \n \n \n \n \n \\bnlem\\label{lem-x-bound}\nUder \\Cref{assp:dcdp_linear_reg}, for any interval $\\I \\subset (0, n]$, for any \n\t\\[\n\t\t\\lambda \\geq \\lambda_1 := C_{\\lambda}\\sigma_{\\epsilon} \\sqrt{\\log(n p)},\n\t\\]\n\twhere $C_{\\lambda} > 0$ is a large enough absolute constant, it holds with probability at least $1 - n^{-5}$ that\n\t\\[\n    \\|\\sum_{i \\in \\I} \\epsilon_i X_i \\|_{\\infty} \\leq \\lambda \\sqrt{\\max\\{|\\I|, \\, \\log(n p)\\}}\/8,\n\t\\]\n\twhere $c_3 > 0$ is an absolute constant depending only on the distributions of covariants $\\{X_i\\}$ and $\\{\\epsilon_i\\}$.\n\\enlem\n\n\\begin{proof}\n\tSince $\\epsilon_i$'s are sub-Gaussian random variables and $X_i$'s are sub-Gaussian random vectors, we have that $\\epsilon_i X_i$'s are sub-Exponential random vectors with $\\|\\epsilon_iX_i\\|_{\\psi_1}\\leq C_x \\sigma_{\\epsilon}$ \\citep[see e.g.~Lemma~2.7.7 in][]{vershynin2018high}. It then follows from Bernstein's inequality \\citep[see e.g.~Theorem~2.8.1 in][]{vershynin2018high} that for any $t > 0$,\n\t\t\\[\n\t\\mathbb{P}\\left\\{\\|\\sum_{i \\in \\I} \\epsilon_i X_i \\|_{\\infty} > t\\right\\} \\leq 2p \\exp\\left\\{-c \\min\\left\\{\\frac{t^2}{|\\I|C_x^2\\sigma^2_{\\epsilon}}, \\, \\frac{t}{C_x \\sigma_{\\epsilon}}\\right\\}\\right\\}.\n\t\t\\]\n\t\tTaking \n\t\t\\[\n\t\t\tt = C_{\\lambda}C_x\/4 \\sigma_{\\epsilon} \\sqrt{\\log(n p)} \\sqrt{\\max\\{|\\I|, \\, \\log(n p)\\}}\n\t\t\\] \n\t\tyields the conclusion.\n\\end{proof}\n \n \n \n \n\n\\bnlem \\label{lemma:beta bounded 1}\nSuppose   \\Cref{assp:dcdp_linear_reg} holds. Let $\\I \\subset [1,n] $. Denote \n $ \\kappa = \\min_{k\\in \\{1,\\ldots, K\\} } \\kappa_k,$ \nwhere $ \\{ \\kappa_k\\}_{k=1}^K$ are defined in  \\Cref{assp:dcdp_linear_reg}. Then for any $i\\in [T]$,\n   $$\\|\\beta^*_\\I - \\beta^*_i\\|_2  \\le C\\kappa  \\le CC_\\kappa,$$\n  for some absolute constant $C$ independent of $n$.\n\\enlem \n \\begin{proof}\n It suffices to consider $\\I =[1,n]$  and  $\\beta_i^*= \\beta_1^*$ as the general case is similar. Observe that  \n \\begin{align*}\n  \\| \\beta^*_{[1,n]}  - \\beta_ 1^*  \\|_2  =& \n  \\|  \\frac{1}{n} \\sum_{i=1}^n \\beta_i ^*  - \\beta_1^*  \n   \\|_2 \n = \\|  \\frac{1}{n} \\sum_{k=0}^{K} \\Delta_k \\beta_{\\eta_k+ 1} ^*   -    \\frac{1}{n}\\sum_{k=0}^{K} \\Delta_k  \\beta_1 ^*\n  \\|_2  \n  \\\\\n \\le \n &    \\frac{1}{n} \\sum_{k=0}^{K} \\left\\| \\Delta_k  (\\beta_{\\eta_k+ 1} ^* -\\beta_1 ^* ) \\right\\|_2    \\le \n \\frac{1}{n}\\sum_{k=0}^{K} \\Delta_k  (K+1)  \\kappa   \\le  (K+1)\\kappa . \n  \\end{align*}\n  By \\Cref{assp:dcdp_linear_reg}, both $\\kappa $ and $K $ bounded above. \n \\end{proof}\n \n \\bnlem\\label{eq:cumsum upper bound}\n Let $t\\in \\I =(s,e] \\subset [1,n]$. Denote \n $ \\kappa_{\\max}= \\max_{k\\in \\{1,\\ldots, K\\} } \\kappa_k,$ \nwhere $ \\{ \\kappa_k\\}_{k=1}^K$ are defined in  \\Cref{assp:dcdp_linear_reg}.  Then   \n $$ \\sup_{0< s < t<e\\le n  }\\| \\beta_{ (s, t]}  ^* - \\beta_{(t,e] }  ^* \\|_2  \\le C\\kappa  \\le CC_\\kappa. $$ \nfor some absolute constant $C$ independent of $n$. \n \\enlem\n \\begin{proof}\n It suffices to consider $(s,e]=(0,n]$, as the general case is similar.  \n Suppose that $\\eta_{q} <t \\le \\eta_{q+1} $.\n Observe that \n \\begin{align*}\n&  \\| \\beta^*_{(1,t]}  - \\beta ^* _{(t,n]}  \\|_2  \n  \\\\\n  =& \n  \\left\\|  \\frac{1}{t } \\sum_{i=1}^t \\beta_i ^*  - \\frac{1}{ n- t } \\sum_{i=t+1}^n \\beta_i ^*\n  \\right \\|_2 \n \\\\\n = & \\left\\|  \\frac{1}{t}  \\left( \\sum_{k=0}^{q-1} \\Delta_k \\beta_{\\eta_k+ 1}  ^*  + (t-\\eta_{q} )\\beta^*_{\\eta_q + 1} \\right) -    \\frac{1}{n-t } \\left( \\sum_{k=q+1}^{K}  \\Delta_k \\beta_{\\eta_k+ 1} ^*   + ( \\eta_{q+1}-t  )\\beta^*_{\\eta_q + 1} \\right) \n  \\right \\|_2  \n  \\\\\n=\n & \\left\\|  \\frac{1}{t}  \\left( \\sum_{k=0}^{q-1} \\Delta_k (\\beta ^* _{\\eta_k+ 1}-\\beta^*_{\\eta_q+1} ) \\right)  +  \\beta^*_{\\eta_q + 1} -    \\frac{1}{n-t } \\left( \\sum_{k=q+1}^{K}  \\Delta_k (  \\beta ^* _{\\eta_k+ 1}  -\\beta ^* _{\\eta_q+1} ) \\right)   -  \\beta^*_{\\eta_q + 1} \n  \\right \\|_2 \n  \\\\\n  =\n  &\\left\\|  \\frac{1}{t}  \\left( \\sum_{k=0}^{q-1} \\Delta_k (\\beta_{\\eta_k+ 1} ^* -\\beta^*_{\\eta_q+1} ) \\right)    -    \\frac{1}{n-t } \\left( \\sum_{k=q+1}^{K}  \\Delta_k (  \\beta_{\\eta_k+ 1} ^*   -\\beta_{\\eta_q+1} ^*  ) \\right)   \n  \\right \\|_2 \n  \\\\\n  \\le \n  &\\frac{1}{t} \\sum_{k=0}^{q-1} \\Delta_k    K \\kappa      +\\frac{1}{n-t} \\sum_{k=q+1}^{K} \\Delta_k    K  \\kappa \n  \\le 2K\\kappa  . \n  \\end{align*}\n  \n \\end{proof}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Gaussian graphical model}\n\\label{sec: main proof covariance}\n\nIn this section, we will present the proof of \\Cref{thm:DCDP covariance main}. Throughout this section, we use $\\Sigma$ for covariance matrices and $\\Omega$ for precision matrices. For any generic interval $\\I\\subset [1,n]$, denote $\\Omega^*_{\\I} = \\frac{1}{|\\I|}\\sum_{i\\in \\I}\\Omega^*_i$ and\n$$ \\widehat{\\Omega}_{\\I} = \\argmin_{\\Omega\\in \\mathbb{S}^p_+} \\sum_{i \\in \\I } {\\rm Tr}[{\\Omega}^\\top X_iX_i^\\top] - |\\I|\\log|\\Omega|. $$\nAlso, unless specially mentioned, in this section, we set the goodness-of-fit function $\\mathcal F  (\\I)$ in \\Cref{algorithm:DCDP} to be\n\\begin{equation}\n    \\mathcal F  (\\widehat{\\Omega}_\\I,\\I) \n     = \\begin{cases}  0  \n & \\text{ if } |\\I|< C_\\mclF p\\log(p\\vee n);\\\\\n       \\sum_{i \\in \\I } {\\rm Tr}[\\widehat{\\Omega}^\\top X_iX_i^\\top] - |\\I|\\log|\\widehat{\\Omega}|   \n       &\\text{otherwise}.\n    \\end{cases}\n\\end{equation}\nwhere $C_\\mclF$ is a universal constant.\n\n\\paragraph{Additional notations.} Before presenting more details on Gaussian graphical model, we introduce some additional notations while reviewing some notations we used in the main text. We use $\\mathbb{S}^p_+$ to denote the cone of positive semidefinite matrices in $\\mathbb{R}^{p\\times p}$. For a matrix $A\\in \\mathbb{R}^{m\\times n}$, we use $\\|A\\|_F:=\\sqrt{\\sum_{i\\in [m]}\\sum_{j\\in [n]}A_{ij}^2}$ to denote its Forbenius norm, $\\|A\\|_{op} = \\sup_{v\\in \\mathbb{R}^p}\\|Av\\|_2\/\\|v\\|_2$ as its operator norm, and ${\\rm Tr}(A) = \\sum_{i\\in [m\\wedge n]}A_{ii}$ to denote its trace. For a square matrix $A\\in \\mathbb{R}^{n\\times n}$, denote its determinant by $|A|$. For two matrices $A, B\\in\\mathbb{R}^{p\\times p}$, $A\\preceq B$ means that $B - A\\in \\mathbb{S}_+^p$. For a random vector $X\\in \\mathbb{R}^p$, we denote $g_X$ as the subgaussian norm \\citep{vershynin2018high}: $g_X:=\\sup \\{\\|v^\\top X\\|_{\\psi_2}: v\\in \\mathbb{R}^p, \\|v\\|_2=1\\}$.\n\n\n\\paragraph{Assumptions.} For the ease of presentation, we combine the SNR condition we will use throughout this section and \\Cref{assp:DCDP_covariance main} into a single assumption. Besides, we would like to point out that although we assume that $\\{X_i\\}_{i\\in[n]}$ are Gaussian vectors in \\Cref{assp:DCDP_covariance main}, it is actually only compulsory for the proof of the conquer step. Throughout this section for the divide step, it suffices to assume that $\\{X_i\\}_{i\\in[n]}$ are subgaussian vectors with bounded Orlicz norm $\\sup_{i\\in[n]}\\|X_i\\|_{\\psi_2}\\leq g_X<\\infty$ where $g_X$ is some absolute constant. Thus, we keep $g_X$ in all results in this section, although when $\\{X_i\\}_{i\\in[n]}$ are Gaussian it holds that $g_X = C_X$.\n\n\n\\bnassum[Gaussian graphical model] \n\\label{assp:DCDP_covariance}\nSuppose that \\Cref{assp:DCDP_covariance} holds. In addition, suppose that $ \\Delta_{\\min} \\kappa^2  \\geq \\mclB_n   p^2\\log^2(n\\vee p)$ as is assumed in \\Cref{thm:DCDP covariance main}.\n\\enassum\n\n\\bnprop\\label{prop:covariance local consistency}\n Suppose  \\Cref{assp:DCDP_covariance} holds. Let \n $\\widehat { \\mathcal P }  $ denote the output of \\Cref{algorithm:DP}. \n Then with probability  at least $1 - C n ^{-3}$, the following conditions hold.\n    \\begin{itemize}\n        \\item [(i)] For each interval  $ \\I = (s, e] \\in \\widehat{\\mathcal{P}}$    containing one and only one true \n           change point $ \\eta_k $, it must be the case that\n $$\\min\\{ \\eta_k -s ,e-\\eta_k \\}  \\lesssim  C_{\\gamma} g_X^4\\frac{C_X^2}{c_X^6}\\frac{p^2\\log(n\\vee p)}{\\kappa_k^2} +g_X^4\\frac{C_X^6}{c_X^6} { \\mathcal B_n^{-1}   \\Delta }  .$$\n        \\item [(ii)] For each interval  $  \\I = (s, e] \\in \\widehat{\\mathcal{P}}$ containing exactly two true change points, say  $\\eta_ k  < \\eta_ {k+1} $, it must be the case that\n            \\begin{align*} \n \\eta_k -s    \\lesssim  { \\mathcal B_n^{-1\/2}   \\Delta }     \\quad \\text{and} \\quad \n e-\\eta_{k+1}  \\lesssim   { \\mathcal B_n^{-1\/2}   \\Delta } . \n \\end{align*} \n         \n\\item [(iii)] No interval $\\I \\in \\widehat{\\mathcal{P}}$ contains strictly more than two true change points; and \n\n\n    \\item [(iv)] For all consecutive intervals $ \\I_1 $ and $ \\I_2 $ in $\\widehat{ \\mathcal P}$, the interval \n        $ \\I_1 \\cup  \\I_2 $ contains at least one true change point.\n                \n     \\end{itemize}\n\\enprop \n\\bprf\nThe four cases are proved in \\Cref{lem:cov single cp}, \\Cref{lem:cov two cp}, \\Cref{lem:cov three or more cp}, and \\Cref{lem:cov two consecutive interval}, respectively.\n\\eprf\n\n\n\n\n\n\n\n\n\\bnprop\\label{prop:cov change points partition size consistency}\nSuppose  \\Cref{assp:DCDP_covariance} holds. Let \n $\\widehat { \\mathcal P }  $ be the output of \\Cref{algorithm:DP}. Suppose \n$ \\gamma \\ge C_\\gamma K { \\mathcal B_n^{-1}   \\Delta }  \\kappa^2 $  for sufficiently large constant $C_\\gamma$. Then\n with probability  at least $1 - C n ^{-3}$,  $| \\widehat { \\mathcal P} | =K $.\n\\enprop\n\n\n\n\\begin{proof}[Proof of   \\Cref{prop:cov change points partition size consistency}] \nDenote $\\mathfrak{G} ^*_n = \\sum_{ i =1}^n [{\\rm Tr}[(\\Omega_i^*)^\\top X_iX_i^\\top] - \\log|\\Omega_i^*|]$. Given any collection $\\{t_1, \\ldots, t_m\\}$, where $t_1 < \\cdots < t_m$, and $t_0 = 0$, $t_{m+1} = n$, let \n\\begin{equation}\\label{eq: cov sn-def}\n     { \\mathfrak{G} }  _n(t_1, \\ldots, t_{m}) = \\sum_{k=1}^{m} \\sum_{ i = t_k +1}^{t_{k+1}} \\mclF(\\widehat {\\Omega}_{(t_{k}, t_{k+1}]}, (t_{k}, t_{k+1}]).\n\\end{equation}\nFor any collection of time points, when defining \\eqref{eq: cov sn-def}, the time points are sorted in an increasing order.\n\nLet $\\{ \\widehat \\eta_{k}\\}_{k=1}^{\\widehat K}$ denote the change points induced by $\\widehat {\\mathcal P}$.  Suppose we can justify that \n    \\begin{align}\n         { \\mathfrak{G} } ^*_n + K\\gamma  \\ge  & { \\mathfrak{G} }  _n(s_1,\\ldots,s_K)   + K\\gamma - C(K + 1)\\frac{g_X^2}{c_{X}^4}\\frac{ p^2\\log(n\\vee p)}{\\kappa^2} - C\\sum_{k\\in [K]}\\kappa_k^2\\mclB_n^{-1}\\Delta_{\\min} \\label{eq:cov K consistency step 1} \\\\ \n        \\ge &  { \\mathfrak{G} } _n (\\widehat \\eta_{1},\\ldots, \\widehat \\eta_{\\widehat K } ) +\\widehat K \\gamma - C_1'(K + 1)\\frac{g_X^2}{c_{X}^4}\\frac{ p^2\\log(n\\vee p)}{\\kappa^2} - C_1'\\sum_{k\\in [K]}\\kappa_k^2\\mclB_n^{-1}\\Delta_{\\min}  \\label{eq:cov K consistency step 2} \\\\ \n        \\ge &   { \\mathfrak{G} }  _n ( \\widehat \\eta_{1},\\ldots, \\widehat \\eta_{\\widehat K } , \\eta_1,\\ldots,\\eta_K ) + \\widehat K \\gamma    - C_1(K + 1)\\frac{g_X^2}{c_{X}^4}\\frac{ p^2\\log(n\\vee p)}{\\kappa^2} - C_1\\sum_{k\\in [K]}\\kappa_k^2\\mclB_n^{-1}\\Delta_{\\min} \\label{eq:cov K consistency step 3}\n    \\end{align}\n    and that \n    \\begin{align}\\label{eq:cov K consistency step 4}\n         { \\mathfrak{G} }  ^*_n   - { \\mathfrak{G} }  _n ( \\widehat \\eta_{1},\\ldots, \\widehat \\eta_{\\widehat K } , \\eta_1,\\ldots,\\eta_K ) \\le C_2 (K + \\widehat{K} + 2) \\frac{g_X^4}{c_X^2} p^2\\log(n\\vee p).\n    \\end{align}\n    Then it must hold that $| \\widehat{\\mclP} | = K$, as otherwise if $\\widehat K \\geq K+1 $, then  \n    \\begin{align*}\n        C _2 (K + \\widehat{K} + 2)  \\frac{g_X^4}{c_X^2} p^2\\log(n\\vee p)  & \\ge    { \\mathfrak{G} }  ^*_n  - { \\mathfrak{G} }  _n ( \\widehat \\eta_{1},\\ldots, \\widehat \\eta_{\\widehat K } , \\eta_1,\\ldots,\\eta_K ) \\\\\n        & \\ge      (\\widehat K - K)\\gamma  -C_1 ( K +1)  \\frac{g_X^4}{c_X^2} p^2\\log(n\\vee p).\n    \\end{align*} \n    Therefore due to the assumption that $| \\widehat{\\mclP}|=\\widehat K\\leq 3K $, it holds that \n    \\begin{align} \\label{eq:cov Khat=K} \n        [C_2 (4K + 2) + C_1(K+1)] \\frac{g_X^4}{c_X^2} p^2\\log(n\\vee p) \\geq  (\\widehat K - K)\\gamma \\geq \\gamma.\n    \\end{align}\n    Note that \\eqref{eq:cov Khat=K} contradicts the choice of $\\gamma$. Therefore, it remains to show \\Cref{eq:cov K consistency step 1} to \\Cref{eq:cov K consistency step 4}.\n\n{\\bf Step 1.} \\Cref{eq:cov K consistency step 1} holds because $\\widehat{\\Omega}_{\\I}$ is (one of) the minimizer of $\\mclF(\\Omega, \\I)$ for any interval $\\I$.\n\n{\\bf Step 2.} \\Cref{eq:cov K consistency step 2} is guaranteed by the definition of $\\widehat{\\mclP}$.\n\n{\\bf Step 3.} For every $ \\I  =(s,e]\\in \\widehat{\\mclP}$, by \\Cref{prop:covariance local consistency}, we know that $\\I$ contains at most two change points. We only show the proof for the two-change-points case as the other case is easier. Denote\n    \\[\n         \\I =  (s ,\\eta_{q}]\\cup (\\eta_{q},\\eta_{q+1}] \\cup  (\\eta_{q+1} ,e]  = {  \\mathcal J } _1 \\cup {  \\mathcal J } _2 \\cup {  \\mathcal J } _{3},\n    \\]\nwhere $\\{ \\eta_{q},\\eta_{q+1}\\} =\\I \\, \\cap \\, \\{\\eta_k\\}_{k=1}^K$. \n\nFor each $ m=1,2,3$, by definition it holds that\n\\begin{equation}\n    \\mclF(\\widehat\\Omega_{{  \\mathcal J } _m},{  \\mathcal J } _m)\\leq \\mclF(\\Omega^*_{{  \\mathcal J } _m},{  \\mathcal J } _m).\n\\end{equation}\nOn the other hand, by \\Cref{lem:cov loss deviation no change point}, we have\n$$\n\\mclF(\\widehat\\Omega_{\\I},{  \\mathcal J } _m)   \\geq \\mclF(\\Omega^*_{{  \\mathcal J } _m},{  \\mathcal J } _m) - C\\frac{g_X^4}{c_X^2}p^2\\log(n\\vee p). \n$$\nTherefore the last two inequalities above imply that \n\\begin{align} \n\\sum_{i \\in \\I }  \\mclF(\\widehat \\Omega_\\I,\\I) \\geq & \\sum_{m=1}^{3} \\sum_{ i \\in {  \\mathcal J } _m }\\mclF(\\widehat \\Omega_\\I,{  \\mathcal J } _m) \\nonumber \\\\\n\\geq & \\sum_{m=1}^{3}   \\sum_{ i \\in {  \\mathcal J } _m}\\mclF(\\widehat \\Omega_{{  \\mathcal J } _m},{  \\mathcal J } _m) - C\\frac{g_X^4}{c_X^2}p^2\\log(n\\vee p). \\label{eq:cov K consistency step 3 inequality 3}\n\\end{align}\nThen \\eqref{eq:cov K consistency step 3} is an immediate consequence of \\eqref{eq:cov K consistency step 3 inequality 3}.\n\n{\\bf Step 4.}\nFinally, to show \\eqref{eq:cov K consistency step 4},   let  $\\widetilde { \\mathcal P}$ denote    the partition induced by $\\{\\widehat \\eta_{1},\\ldots, \\widehat \\eta_{\\widehat K } , \\eta_1,\\ldots,\\eta_K\\}$. Then \n$| \\widetilde { \\mathcal P} | \\le K + \\widehat K+2   $ and that $\\Omega^*_i$ is unchanged in every interval $\\I \\in \\widetilde { \\mathcal P}$. So \\Cref{eq:cov K consistency step 4} is an immediate consequence of \\Cref{lem:cov no cp}.\n\\end{proof} \n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\subsection{Fundamental lemmas}\n\n\n\n\n\\bnlem[Deviation, Gaussian graphical model]\n\\label{lem:cov one change deviation bound}\nLet $\\mathcal I =(s,e] $ be any generic interval, and define the loss function $\\mclF(\\Omega, \\I)  = \\sum_{i\\in \\I}{\\rm Tr}[\\Omega^\\top (X_iX_i^\\top)] - |\\I|\\log|\\Omega|$.  Define $\\widehat{\\Omega}_{\\I} = \\argmin_{\\Omega\\in \\mathbb{S}_+}L(\\Omega,\\I)$ and $\\mclF^*(\\Omega^*, \\I) = \\sum_{i\\in \\I}[{\\rm Tr}((\\Omega_i^*)^\\top (X_iX_i^\\top)) - \\log|\\Omega_i^*|]$.\n\n{\\bf a.} If $\\I$ contains no change points. Then it holds that \n$$\\mathbb{P} \\bigg( |   \\mclF(\\widehat{\\Omega}_{\\I},\\I)  - \\mclF^*(\\Omega^*, \\I) |  \\ge C \\frac{g_X^4}{c_X^2}p^2\\log(n\\vee p)  \\bigg)  \\le  (n\\vee p)^{-3}. $$\n\n{\\bf b.}  Suppose that the interval $ \\I=(s,e]$ contains one and only one change point $ \\eta_k $. Denote\n$$  \\mathcal J = (s,\\eta_k] \\quad \\text{and} \\quad  \\mathcal J' =  (\\eta_k, e].$$ \nThen it holds that \n$$\\mathbb{P} \\bigg( | \\mclF(\\widehat{\\Omega}_{\\I},\\I) - \\mclF^*(\\I)  | \\geq \\frac{C_X^2p}{c_X^8}\\frac{|{  \\mathcal J } ||{  \\mathcal J }  '| }{|\\I|}\\kappa_k^2 +  C\\frac{g_X^4 C_X^2}{c_X^4}p^2\\log(n\\vee p) \\bigg)   \\le  (n\\vee p)^{-3}. $$\n\\enlem\n\\begin{proof} \nWe show {\\bf b} as {\\bf a} immediately follows from {\\bf b} with $ |{  \\mathcal J } '| =0$. Denote \n$$  \\mathcal J = (s,\\eta_k] \\quad \\text{and} \\quad  \\mathcal J' =  (\\eta_k, e]  .$$ \nLet $\\widetilde{\\Omega}_{\\I} = (\\frac{1}{|\\I|}\\sum_{i\\in\\I}\\Sigma_i^*)^{-1}$.\nThen by Taylor expansion and \\Cref{lem:estimation covariance}, we have\n\\begin{align}\n    & |\\mclF(\\widehat{\\Omega}_{\\I},\\I)-\\mclF(\\widetilde{\\Omega}_{\\I},\\I)| \\nonumber \\\\\n    \\leq & |{\\rm Tr}[(\\widehat{\\Omega}_{\\I} - \\widetilde{\\Omega}_{\\I})^\\top (\\sum_{i\\in\\I} X_iX_i^\\top - |\\I|\\widetilde{\\Omega}_{\\I}^{-1})]| + \\frac{C_X^2}{2}|\\I|\\|\\widehat{\\Omega}_{\\I} - \\widetilde{\\Omega}_{\\I}\\|_F^2 \\nonumber \\\\\n    \\leq & |\\I|\\|\\widehat{\\Omega}_{\\I} - \\widetilde{\\Omega}_{\\I}\\|_F \\|\\widehat{\\Sigma}_{\\I} - \\widetilde{\\Omega}_{\\I}^{-1}\\|_F + \\frac{C_X^2}{2}|\\I|\\|\\widehat{\\Omega}_{\\I} - \\widetilde{\\Omega}_{\\I}\\|_F^2 \\nonumber\\\\\n    \\leq & C\\frac{g_X^4}{c_X^2}p^2\\log(n\\vee p) + C\\frac{g_X^4 C_X^2}{c_X^4}p^2\\log(n\\vee p)\\leq C\\frac{g_X^4 C_X^2}{c_X^4}p^2\\log(n\\vee p). \\label{eq:cov mixed deviation 1}\n\\end{align}\nOn the other hand, it holds that\n\\begin{align}\n    & |\\mclF(\\widetilde{\\Omega}_{\\I},\\I)-\\mclF^*(\\Omega^*, \\I)| \\nonumber \\\\\n    \\leq & |{\\rm Tr}[(\\widetilde{\\Omega}_{\\I} - {\\Omega}_{{  \\mathcal J } })^\\top (\\sum_{i\\in{  \\mathcal J } } X_iX_i^\\top - |{  \\mathcal J } |\\Omega_{{  \\mathcal J } }^{-1})]| + \\frac{C_X^2}{2}|{  \\mathcal J } |\\|\\widetilde{\\Omega}_{\\I} - {\\Omega}_{{  \\mathcal J } }\\|_F^2 \\nonumber\\\\\n     & + |{\\rm Tr}[(\\widetilde{\\Omega}_{\\I} - {\\Omega}_{{  \\mathcal J }  '})^\\top (\\sum_{i\\in{  \\mathcal J } '} X_iX_i^\\top - |{  \\mathcal J }  '|\\Omega_{{  \\mathcal J }  '}^{-1})]| + \\frac{C_X^2}{2}|{  \\mathcal J }  '|\\|\\widetilde{\\Omega}_{\\I} - {\\Omega}_{{  \\mathcal J }  '}\\|_F^2.\\label{eq:cov mixed deviation 2}\n\\end{align}\nTo bound $\\|\\widetilde{\\Omega}_{\\I} - {\\Omega}_{{  \\mathcal J } }\\|_F$ and $\\|\\widetilde{\\Omega}_{\\I} - {\\Omega}_{{  \\mathcal J }  '}\\|_F$, notice that for two positive definite matrices $\\Sigma_1,\\Sigma_2\\in \\mathbb{S}_+$ and two positive numbers $w_1,w_2$ such that $w_1 + w_2 = 1$, we have\n\\begin{align*}\n    & \\|(w_1\\Sigma_1 + w_2\\Sigma_2)^{-1} - \\Sigma_1^{-1}\\|_F\\\\\n    \\leq &\\sqrt{p}\\|(w_1\\Sigma_1 + w_2\\Sigma_2)^{-1} - \\Sigma_1^{-1}\\|_{op}\\\\\n    = &\\sqrt{p}\\|(w_1\\Sigma_1 + w_2\\Sigma_2)^{-1}[\\Sigma_1 - (w_1\\Sigma_1 + w_2\\Sigma_2)]\\Sigma_1^{-1}\\|_{op}\\\\\n    \\leq & \\sqrt{p}\\|(w_1\\Sigma_1 + w_2\\Sigma_2)^{-1}\\|_{op}\\|\\Sigma_1 - (w_1\\Sigma_1 + w_2\\Sigma_2)\\|_{op}\\|\\Sigma_1^{-1}\\|_{op}\\\\\n    \\leq & \\|(w_1\\Sigma_1 + w_2\\Sigma_2)^{-1}\\|_{op}\\|\\Sigma_1^{-1}\\|_{op}\\cdot\\sqrt{p}w_2\\|\\Sigma_1 - \\Sigma_2\\|_{op} \\\\\n    \\leq & \\|(w_1\\Sigma_1 + w_2\\Sigma_2)^{-1}\\|_{op}\\|\\Sigma_1^{-1}\\|_{op}\\|\\Sigma_1\\|_{op}\\|\\Sigma_2\\|_{op}\\cdot\\sqrt{p}w_2\\|\\Sigma_1^{-1} - \\Sigma_2^{-1}\\|_{op}.\n\\end{align*}\nTherefore, under \\Cref{assp:DCDP_covariance}, it holds that\n\\begin{equation}\n    \\|\\widetilde{\\Omega}_{\\I} - {\\Omega}_{{  \\mathcal J } }\\|_F\\leq \\frac{C_X^2}{c_X^2}\\frac{|{  \\mathcal J }  '|}{|\\I|}\\sqrt{p}\\kappa_k, \\|\\widetilde{\\Omega}_{\\I} - {\\Omega}_{{  \\mathcal J }  '}\\|_F\\leq \\frac{C_X^2}{c_X^2}\\frac{|{  \\mathcal J } |}{|\\I|}\\sqrt{p}\\kappa_k,\n\\end{equation}\nwhere in the second inequality we use the fact that $2ab\\leq a^2+ b^2$. As a consequence, \\Cref{eq:cov mixed deviation 2} can be bounded as\n\\begin{align}\n     & |\\mclF(\\widetilde{\\Omega}_{\\I},\\I)-\\mclF^*(\\Omega^*, \\I)| \\nonumber \\\\\n    \\leq & C\\frac{g_X^2p\\sqrt{\\log(n\\vee p)}}{c_X^2C_X^{-2}}\\kappa_k(\\frac{|{  \\mathcal J }  '||{  \\mathcal J } |^{\\frac{1}{2}}}{|\\I|} + \\frac{|{  \\mathcal J }  '|^{\\frac{1}{2}}|{  \\mathcal J } |}{|\\I|}) + \\frac{C_X^6 p}{2c_X^4}\\kappa_k^2(\\frac{|{  \\mathcal J } ||{  \\mathcal J }  '|^2 }{|\\I|^2} + \\frac{|{  \\mathcal J }  '||{  \\mathcal J } |^2 }{|\\I|^2}) \\nonumber\\\\\n    \\leq & C\\frac{g_X^4p^2{\\log(n\\vee p)}}{C_X^2} + \\frac{C_X^6p}{c_X^4}\\frac{|{  \\mathcal J } ||{  \\mathcal J }  '| }{|\\I|}\\kappa_k^2.\\label{eq:cov mixed deviation 3}\n\\end{align}\nCombine \\Cref{eq:cov mixed deviation 1} and \\Cref{eq:cov mixed deviation 3} and we can get\n\\begin{equation*}\n    \\begin{split}\n        |\\mclF(\\widehat{\\Omega}_{\\I},\\I) - \\mclF^*(\\Omega^*,\\I)|\n        \\leq & |\\mclF(\\widehat{\\Omega}_{\\I},\\I)-\\mclF({\\Omega}_{\\I},\\I)| +|\\mclF({\\Omega}_{\\I},\\I) - \\mclF^*(\\I)|\\\\\n        \\leq& C\\frac{g_X^4 C_X^2}{c_X^4}p^2\\log(n\\vee p) + \\frac{C_X^6 p}{c_X^4}\\frac{|{  \\mathcal J } ||{  \\mathcal J }  '| }{|\\I|}\\kappa_k^2.\n    \\end{split}\n\\end{equation*}\n\\end{proof} \n\n\\bnrmk\nIt can be seen later that the $p$ factor in the signal term $\\frac{C_X^6 p}{c_X^4}\\frac{|{  \\mathcal J } ||{  \\mathcal J }  '| }{|\\I|}\\kappa_k^2$ will require an additional $p$ factor in the number of points in the grid for DCDP, leading to an additional $p^2$ factor in the computation time.\n\nThis factor is hard to remove because it is rooted in the approximation error\n$$\n\\|(w_1\\Sigma_1 + w_2\\Sigma_2)^{-1} - \\Sigma_1^{-1}\\|_F.\n$$\nWe can try another slightly neater way of bounding this term. As is mentioned in \\cite{kereta2021ejs}, for two matrices $\\mathbf{G}, \\mathbf{H}\\in \\mathbb{R}^{d_1\\times d_2}$, it holds that\n$$\n\\left\\|\\mathbf{H}^{\\dagger}-\\mathbf{G}^{\\dagger}\\right\\|_F \\leq \\min \\left\\{\\left\\|\\mathbf{H}^{\\dagger}\\right\\|_{op}\\left\\|\\mathbf{G}^{\\dagger} (\\mathbf{H}-\\mathbf{G})\\right\\|_F,\\left\\|\\mathbf{G}^{\\dagger}\\right\\|_{op}\\left\\|\\mathbf{H}^{\\dagger} (\\mathbf{H}-\\mathbf{G})\\right\\|_{op}\\right\\},\n$$\nif $\\operatorname{rank}(\\mathbf{G})=\\operatorname{rank}(\\mathbf{H})=\\min \\left\\{d_1, d_2\\right\\}$. Therefore, we have\n\\begin{align*}\n    \\|(w_1\\Sigma_1 + w_2\\Sigma_2)^{-1} - \\Sigma_1^{-1}\\|_F\\leq & \\|(w_1\\Sigma_1 + w_2\\Sigma_2)^{-1}\\|_{op}\\|\\Sigma_{1}^{-1}(w_1\\Sigma_1 + w_2\\Sigma_2 - \\Sigma_1)\\|_F\\\\\n    \\leq & \\|(w_1\\Sigma_1 + w_2\\Sigma_2)^{-1}\\|_{op}\\|\\Sigma_{1}^{-1}\\|w_2\\|\\Sigma_2 - \\Sigma_1\\|_F.\n\\end{align*}\nHowever, to relate $\\|\\Sigma_2 - \\Sigma_1\\|_F$ to $\\|\\Sigma_2^{-1} - \\Sigma_1^{-1}\\|_F$, we need to proceed in the following way:\n\\begin{align*}\n    \\|\\Sigma_2 - \\Sigma_1\\|_F\\leq & \\sqrt{p}\\|\\Sigma_2 - \\Sigma_1\\|_{op} \\\\\n    \\leq & \\|\\Sigma_1\\|_{op}\\|\\Sigma_2\\|_{op} \\cdot \\sqrt{p}\\|\\Sigma_2^{-1} - \\Sigma_1^{-1}\\|_{op}\\\\\n    \\leq & \\|\\Sigma_1\\|_{op}\\|\\Sigma_2\\|_{op} \\cdot \\sqrt{p}\\|\\Sigma_2^{-1} - \\Sigma_1^{-1}\\|_F,\n\\end{align*}\nwhich leads to the same bound in \\Cref{lem:cov one change deviation bound}.\n\\enrmk\n\n\n\n\n\n\n\n\\bnlem\n\\label{lem:estimation covariance}\nLet $\\{X_i\\}_{i\\in [n]}$ be a sequence of subgaussian vectors in $\\mathbb{R}^{p}$ with Orlicz norm upper bounded by $g_X<\\infty$. Suppose $\\mathbb{E}[X_i] = 0$ and $\\mathbb{E}[X_iX_i^\\top] = \\Sigma_i$ for $i\\in [n]$. Consider the change point setting in \\Cref{assp:DCDP_covariance} and consider a generic interval $\\I\\subset [1,n]$. Let $\\widehat{\\Sigma}_{\\I} = \\frac{1}{|\\I|}\\sum_{i \\in \\I} X_i X_i^\\top$ and $\\Sigma_{\\I} = \\frac{1}{|\\I|}\\sum_{i\\in \\I}\\Sigma^*_i$. Then for any $u>0$, it holds with probability at least $1 - \\exp(-u)$ that\n\\begin{equation}\n    \\|\\widehat{\\Sigma}_{\\I} - \\Sigma_{\\I}\\|_{op}\\lesssim g_X^2(\\sqrt{\\frac{p + u}{|\\mclI|}}\\vee \\frac{p + u}{|\\mclI|}).\n    \\label{eq: cov concentration 1}\n\\end{equation}\nAs a result, when $n\\geq C_s p\\log(n\\vee p)$ for some universal constant $C_s>0$, it holds with probability at least $1 - (n\\vee p)^{-7}$ that\n\\begin{equation}\n    \\|\\widehat{\\Sigma}_{\\I} - \\Sigma_{\\I}\\|_{op}\\leq Cg_X^2\\sqrt{\\frac{p\\log(n\\vee p)}{|\\I|}},\n    \\label{eq: cov concentration 2}\n\\end{equation}\nwhere $C$ is some universal constant that does not depend on $n, p, g_X$, and $C_s$. In addition let $\\widehat{\\Omega}_{\\I} = \\argmin_{\\Omega\\in \\mathbb{S}_+}L(\\Omega,\\I)$ and $\\widetilde{\\Omega}_{\\I} = (\\frac{1}{|\\I|}\\sum_{i\\in\\I}\\Sigma^*_i)^{-1}$. if $|\\I|\\geq C_s p\\log(n\\vee p)g_X^4 \/ c_X^2$ for sufficiently large constant $C_s>0$, then it holds with probability at least $1 - (n\\vee p)^{-7}$ that\n\\begin{equation}\n    \\|\\widehat{\\Omega}_{\\I} -\\widetilde{\\Omega}_{\\I}\\|_{op}\\leq C \\frac{g_X^2}{c_X^2}\\sqrt{\\frac{p\\log(n\\vee p)}{|\\I|}}.\n    \\label{eq: precision concentration}\n\\end{equation}\n\\enlem\n\\bprf\nIf there is no change point in $\\mclI$, then the two inequalities \\eqref{eq: cov concentration 1} and \\eqref{eq: cov concentration 2} are well-known results in the literature, see, e.g., \\cite{kereta2021ejs}. Otherwise, suppose $\\mclI$ is split by change points into $q$ subintervals $\\mclI_1,\\cdots, \\mclI_q$. By \\Cref{assp:DCDP_covariance}, we know that $q\\leq C$ for some constant $C<\\infty$. Thus with probability at least $1 - \\exp(-u)$,\n\\begin{align*}\n    \\|\\widehat{\\Sigma}_{\\I} - \\Sigma_{\\I}\\|_{op} &\\leq \\|\\frac{1}{|\\mclI|}\\sum_{k\\in [q]}|\\mclI_k|(\\widehat{\\Sigma}_{\\I_k} - \\Sigma_{\\I_k})\\|_{op}\\\\\n    & \\leq C_1g_X^2\\frac{\\sqrt{p + u}}{|\\mclI|}\\sum_{k\\in [q]}\\sqrt{|\\mclI_k|\\vee (p + u)} \\\\\n    &\\leq C_2g_X^2\\frac{\\sqrt{p + u}}{|\\mclI|}\\max_{k\\in [q]}{|\\mclI_k|\\vee (p + u)}\\\\\n    &\\leq C_2g_X^2\\sqrt{\\frac{{p + u}}{|\\mclI|}}(\\sqrt{\\frac{\\max_{k\\in [q]}{|\\mclI_k|}}{|\\mclI|}}\\vee \\sqrt{\\frac{{p + u}}{|\\mclI|}}) \\leq C_2 g_X^2(\\sqrt{\\frac{{p + u}}{|\\mclI|}}\\vee \\frac{{p + u}}{|\\mclI|}).\n    \\end{align*}\nIt is then straightforward to see that \\Cref{eq: cov concentration 2} holds with probability at least $1 - (n\\vee p)^{-7}$ when $n\\geq C_sp\\log(n\\vee p)$ for some sufficiently large constant $C_s>0$.\n\nFor \\Cref{eq: precision concentration}, first vanish the gradient of the loss function $L(\\Omega, \\I)$ and we get\n\\begin{equation*}\n    \\widehat{\\Omega}_{\\I} = (\\widehat{\\Sigma}_{\\I})^{\\dagger}.\n\\end{equation*}\nThen \\Cref{eq: precision concentration} is implied by \\Cref{eq: cov concentration 2} and the well-known property that\n$$\n\\left\\|\\mathbf{H}^{\\dagger}-\\mathbf{G}^{\\dagger}\\right\\|_{op} \\leq C\\max \\left\\{\\|\\mathbf{G}^{\\dagger}\\|^2_{op},\\|\\mathbf{H}^{\\dagger}\\|^2_{op}\\right\\}\\left\\|\\mathbf{H}-\\mathbf{G}\\right\\|_{op},\n$$\nfor two matrices $G,H\\in \\mathbb{R}^{p\\times p}$.\n\\eprf\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\subsection{Technical lemmas}\n\n\n\n\n\n\\bnlem[No change point]\n\\label{lem:cov no cp}\nFor interval $\\mclI$ containing no change point, it holds with probability at least $1 - n^{-5}$ that\n\\begin{equation}\n    \\mclF(\\widehat{\\Omega}_{\\I}, \\mclI) -\\mclF({\\Omega}^*, \\mclI) \\geq -g_X^4 p^2\\log(n\\vee p)\\max_{k\\in[K + 1]}\\|\\Omega^*_{\\eta_k}\\|_{op}^2.\n\\end{equation}\n\\enlem\n\\bprf\nIf $\\I< C_s\\frac{g_X^4}{c_X^2}p\\log(n\\vee p)$, then $\\mclF (\\widehat{\\Omega}_{\\I}, \\mclI) = \\mclF ({\\Omega}^*, \\mclI)=0$ and the conclusion holds automatically. If $\\I\\geq C_s\\frac{g_X^4}{c_X^2}p\\log(n\\vee p)$, then by \\Cref{lem:estimation covariance}, it holds with probability at least $1 - n^{-7}$ that\n\\begin{align}\n\\mclF(\\widehat{\\Omega}_{\\I}, \\mclI) -\\mclF({\\Omega}^*, \\mclI) \\geq & |\\mclI|{\\rm Tr}[(\\widehat{\\Omega}_{\\I} - \\Omega^*)^\\top (\\widehat{\\Sigma}_{\\I} - \\Sigma^*)] + \\frac{c|\\mclI|}{2\\|\\Omega^*\\|_{op}^2}\\|\\widehat{\\Omega}_{\\I} - \\Omega^*\\|_F^2.    \\\\\n\\geq & -|\\mclI|\\|\\widehat{\\Omega}_{\\I} - \\Omega^*\\|_F \\|\\widehat{\\Sigma}_{\\I} - \\Sigma^*\\|_F \\\\\n\\geq & -|\\mclI|p \\|\\widehat{\\Omega}_{\\I} - \\Omega^*\\|_{op} \\|\\widehat{\\Sigma}_{\\I} - \\Sigma^*\\|_{op}\\\\\n\\geq & -g_X^4 p^2\\log(n\\vee p)\\|\\Omega^*\\|_{op}^2.\n\\end{align}\n\\eprf\n\n\n\n\n\\bnlem\n\\label{lem:cov loss deviation no change point}\nLet $\\I\\subset [1,T]$ be any interval that contains no change point. Then for any interval ${  \\mathcal J }  \\supset \\I$, it holds with probability at least $1 - (n\\vee p)^{-5}$ that\n\\begin{equation*}\n    \\mclF(\\Omega^*_{\\I},\\I)\\leq \\mclF(\\widehat{\\Omega}_{  \\mathcal J } ,\\I) + C\\frac{g_X^4}{c_X^2}p^2\\log(n\\vee p).\n\\end{equation*}\n\\enlem\n\\begin{proof}\nThe conclusion is guaranteed by \\Cref{lem:cov no cp}   $\\widehat{\\Omega}_{\\I}$ is   the minimizer of $\\mclF(\\Omega,\\I)$.\n\\end{proof}\n\n\n\n\n\n\n\n\n\n\\bnlem[Single change point]\n\\label{lem:cov single cp}\nSuppose the good events \n$\\mathcal L ( { \\mathcal B_n^{-1}   \\Delta }  ) $ and  $\\mathcal R ( { \\mathcal B_n^{-1}   \\Delta } ) $ defined  in \\Cref{eq:left and right approximation of change points} hold. \nLet   $ \\I=(s,e] \\in \\mathcal {\\widehat P}  $  be such that $\\I$ contains exactly  one    change point $ \\eta_k $. \n Then  with probability  at least $1-(n\\vee p)^{-3}$,   it holds that \n\\begin{equation}\n    \\min\\{ \\eta_k -s ,e-\\eta_k \\} \\leq CC_{\\gamma} g_X^4\\frac{C_X^2}{c_X^6}\\frac{p^2\\log(n\\vee p)}{\\kappa_k^2} + Cg_X^4\\frac{C_X^6}{c_X^6} { \\mathcal B_n^{-1}   \\Delta } .\n\\end{equation}\n\\enlem\n\\bprf\nIf either $ \\eta_k -s \\le  { \\mathcal B_n^{-1}   \\Delta }   $ or $e-\\eta_k\\le  { \\mathcal B_n^{-1}   \\Delta } $, then there is nothing to show. So assume that \n$$  \\eta_k -s  >  { \\mathcal B_n^{-1}   \\Delta }  \\quad \\text{and} \\quad   e-\\eta_k  >  { \\mathcal B_n^{-1}   \\Delta }  . $$\nBy event $\\mathcal R (p^{-1} { \\mathcal B_n^{-1}   \\Delta }  )$, there exists  $ s_ u  \\in \\{ s_q\\}_{q=1}^ { \\mathcal Q} $ such that \n$$0\\le s_u - \\eta_k \\le p^{-1} { \\mathcal B_n^{-1}   \\Delta } . $$\n So \n$$ \\eta_k \\le s_ u \\le e .$$\nDenote \n$$ \\I_ 1 = (s,s_u] \\quad \\text{and} \\quad \\I_2 = (s_u, e],$$\nand $\\mclF^*({  \\mathcal J } ) = \\sum_{i\\in {  \\mathcal J } }[{\\rm Tr}((\\Omega^*_i)^\\top X_iX_i^\\top) -\\log|\\Omega_i^*|]$. Since\n$s, e,  s_u \\in  \\{ s_q\\}_{q=1}^ { \\mathcal Q}   $, by the definition of $\\widehat{P}$ and $\\widehat{\\Omega}$, and \\Cref{lem:cov one change deviation bound}, it holds that\n\\begin{align}\n    \\mclF(\\widehat{\\Omega}_{\\I},\\I)\\leq& \\mclF(\\widehat{\\Omega}_{\\I_1},\\I_1) + \\mclF(\\widehat{\\Omega}_{\\I_2},\\I_2) + \\gamma \\nonumber \\\\\n    \\leq & \\mclF^*(\\I_1) + \\frac{C_X^6 p}{c_X^4}(s_u - \\eta_k)\\kappa_k^2 +  C\\frac{g_X^4 C_X^2}{c_X^4}p^2\\log(n\\vee p) + \\mclF^*(\\I_2) + \\gamma \\nonumber \\\\\n    \\leq & \\mclF^*(\\I) + \\frac{C_X^6}{c_X^4}\\mclB_n^{-1}\\Delta\\kappa_k^2 +  C\\frac{g_X^4 C_X^2}{c_X^4}p^2\\log(n\\vee p) +  \\gamma,\n\\end{align}\nwhere the last inequality is due to $s_u - \\eta_k\\leq \\mclB_n^{-1}\\Delta$. Let \n$$\\widetilde{\\gamma} = \\frac{C_X^6}{c_X^4}\\mclB_n^{-1}\\Delta\\kappa_k^2 +  C\\frac{g_X^4 C_X^2}{c_X^4}p^2\\log(n\\vee p) +  \\gamma.$$ \nThen by Taylor expansion and \\Cref{lem:estimation covariance} we have\n\\begin{align}\n    \\frac{c}{\\max_{k\\in [K]}\\|\\Omega_{\\I_k}^*\\|_{op}^2}\\sum_{t\\in \\mclI}\\|\\widehat{\\Omega}_{\\I} - \\Omega^*_t\\|_F^2 \n    &\\leq \\widetilde{\\gamma} + \\sum_{i=k-1}^k|\\I_i|{\\rm Tr}[(\\Omega_{\\I_i}^* - \\widehat{\\Omega}_{\\I})^\\top (\\widehat{\\Sigma}_{\\I_i}  - \\Sigma^*_{\\I_i})] \\nonumber \\\\\n    &\\leq \\widetilde{\\gamma} + \\sum_{i=k-1}^k {|\\mclI_{i}|}\\|\\widehat{\\Sigma}_{\\I_i} - \\Sigma^*_{\\I_i}\\|_F\\|\\widehat{\\Omega}_{\\I} - \\Omega^*_{\\I_i}\\|_F \\nonumber \\\\\n    &\\leq \\widetilde{\\gamma} + C_1g_X^2 p\\log^{\\frac{1}{2}}(n\\vee p)\\left[ \\sum_{i=k-1}^k \\sqrt{|\\mclI_{i}|}\\|\\widehat{\\Omega}_{\\I} - \\Omega^*_{\\I_i}\\|_F \\right] \\nonumber \\\\\n    &\\leq \\widetilde{\\gamma} + C_1g_X^2 p\\log^{\\frac{1}{2}}(n\\vee p) \\sqrt{\\sum_{t\\in \\mclI}\\|\\widehat{\\Omega}_{\\I} - \\Omega^*_t\\|_F^2}.\n\\end{align}\nThe inequality above implies that\n\\begin{equation}\n    \\sum_{t\\in \\mclI}\\|\\widehat{\\Omega}_{\\I} - \\Omega^*_t\\|_F^2 \\leq \\frac{2}{c}\\max\\|\\Omega_{\\I_k}^*\\|_{op}^2\\left[\\widetilde{\\gamma} + \\max\\|\\Omega_{\\I_k}^*\\|_{op}^2\\|X\\|^4_{\\psi_2} p^{2}\\log(n\\vee p)\\right].\n\\end{equation}\nOn the other hand,\n\\begin{equation}\n    \\sum_{t\\in \\mclI}\\|\\widehat{\\Omega}_{\\I} - \\Omega^*_t\\|_F^2\\geq \\frac{|\\mclI_{k-1}||\\mclI_{k}|}{|\\mclI|}\\|\\Omega^*_{\\I_{k-1}} - \\Omega^*_{\\I_k}\\|_F^2,\n\\end{equation}\nwhich implies that\n\\begin{equation}\n    \\min\\{|\\mclI_{k-1}|,|\\mclI_{k}|\\}\\leq C_2C_{\\gamma} g_X^4\\frac{C_X^2}{c_X^6}\\frac{p^2\\log(n\\vee p)}{\\kappa_k^2} + C_2g_X^4\\frac{C_X^6}{c_X^6} { \\mathcal B_n^{-1}   \\Delta } .\n\\end{equation}\n\\eprf\n\n\nRecall that we assume for $i\\in [n]$, $c_XI_p \\preceq \\Sigma_i \\preceq C_X I_p$ for some universal constants $c_X>0, C_X<\\infty$.\n\n\\bnlem[Two change points]\n\\label{lem:cov two cp}\nSuppose the good events \n$\\mathcal L ( { \\mathcal B_n^{-1}   \\Delta }  ) $ and  $\\mathcal R (  { \\mathcal B_n^{-1}   \\Delta }  ) $ defined  in  \\Cref{eq:left and right approximation of change points} hold. Let   $ \\I=(s,e] \\in \\mathcal {\\widehat P}  $  be an interval that contains exactly two change points $ \\eta_k,\\eta_{k+1}$. Suppose in addition that $\\gamma \\geq C_{\\gamma}\\frac{g_X^4}{c_X^2} p^2\\log(n\\vee p)$, and\n\n\\begin{equation}\n    \\Delta_{\\min} \\kappa^2 \\geq \\mathcal{B}_n \\frac{g_X^4}{c_X^4}p^2\\log(n\\vee p),\n\\end{equation}\nthen   with probability at least $1 - n^{-5}$ it holds  that\n\\begin{equation}\n    \\max\\{\\eta_k - s, e - \\eta_{k + 1}\\}\\leq  { \\mathcal B_n^{-1\/2}   \\Delta } .\n\\end{equation}\n\\enlem\n\\bprf\nSince the events $\\mathcal L (  { \\mathcal B_n^{-1}   \\Delta }   ) $ and  $\\mathcal R (  { \\mathcal B_n^{-1}   \\Delta }  ) $ hold, let $ s_u, s_v$ be such that \n $\\eta_k \\le s_u \\le s_v \\le \\eta_{k+1} $ and that \n $$ 0 \\le s_u-\\eta_k \\le    { \\mathcal B_n^{-1}   \\Delta }  ,  \\quad  0\\le  \\eta_{k+1} - s_v \\le  { \\mathcal B_n^{-1}   \\Delta }    .    $$\n\n \\begin{center}\n \\begin{tikzpicture} \n\\draw[ - ] (-10,0)--(1,0);\n \\node[color=black] at (-8,-0.3) {\\small s};\n \\draw[  (-, ultra thick, black] (-8,0) -- (-7.99,0);\n  \\draw[ { -]}, ultra thick, black] (-1.0002,0) -- (-1,0);\n\n   \\node[color=black] at (-7,-0.3) {\\small $\\eta_k$};\n\\draw(-7 ,0)circle [radius=2pt] ;\n\n  \\node[color=black] at (-6.5,-0.3) {\\small $s_u$};\n\\draw plot[mark=x, mark options={color=black, scale=1.5}] coordinates {(-6.5,0)  }; \n \n\n  \\node[color=black] at (-2.3,-0.3) {\\small $\\eta_{k+1}$};\n\\draw(-2.3 ,0)circle [radius=2pt] ;\n  \\node[color=black] at (-3 ,-0.3) {\\small $s_v$};\n\\draw plot[mark=x, mark options={color=black, scale=1.5}] coordinates {(-3,0)  }; \n  \n   \\node[color=black] at (-1,-0.3) {\\small $e$};\n\n \n\\end{tikzpicture}\n\\end{center} \n\nDenote \n$$ \\mathcal I_ 1= (  s, s _u], \\quad \\I_2  =(s_u, s_v] \\quad \\text{and} \\quad \\I_3  = (s_v,e]. $$\nIn addition, denote \n$$ {  \\mathcal J } _1  = (s,\\eta_k ], \\quad {  \\mathcal J } _2=(\\eta_k, \\eta_{k} + \\frac{ \\eta_{k+1}   -\\eta_k  }{2}], \\quad {  \\mathcal J } _3 =  ( \\eta_k+ \\frac{ \\eta_{k+1}   -\\eta_k  }{2},\\eta_{k+1 } ]  \\quad \\text{and} \\quad {  \\mathcal J } _4 = (\\eta_{k+1}  , e] .$$\nSince \n$s, e,  s_u ,s_v  \\in  \\{ s_q\\}_{q=1}^ { \\mathcal Q}   $, by the event $\\mclL(p^{-1} { \\mathcal B_n^{-1}   \\Delta } )$ and $\\mclR(p^{-1} { \\mathcal B_n^{-1}   \\Delta } )$, it holds with probability at least $1 - n^{-3}$ that\n$$\n0\\leq s_u - \\eta_k\\leq p^{-1} { \\mathcal B_n^{-1}   \\Delta } ,\\ 0\\leq \\eta_{k+1} - s_v\\leq p^{-1} { \\mathcal B_n^{-1}   \\Delta } .\n$$\nDenote\n$$\\I_1 = (s ,\\eta_k], \\I_2 = (\\eta_k, \\eta_{k + 1}], \\I_3 = (\\eta_{k + 1}, e].$$\nBy the definition of DP and $\\widehat{\\Omega}_{\\I}$, it holds that\n\\begin{align}\n        \\mclF(\\widehat {\\Omega}_{\\I},\\I)\n        \\leq & \\sum_{i = 1}^3 \\mclF(\\widehat {\\Omega}_{\\I_i},\\I_i) + 2\\gamma \\\\\n        \\leq & \\sum_{i = 1}^3 \\mclF({\\Omega}_{\\I_i}^*,\\I_i) +  2\\gamma + \\frac{C_X^6 p}{c_X^4} \\frac{|{  \\mathcal J } _1|(s_u - \\eta_k)}{|{  \\mathcal J } _1|+s_u - \\eta_k} \\kappa_k^2 + \\frac{C_X^6 p}{c_X^4} \\frac{|{  \\mathcal J } _4|(\\eta_{k + 1} - s_v)}{|{  \\mathcal J } _4|+\\eta_{k + 1} - s_v} \\kappa_k^2 + C\\frac{g_X^4 C_X^2}{c_X^4}p^2\\log(n\\vee p) \\\\\n        \\leq & \\sum_{i = 1}^3 \\mclF({\\Omega}_{\\I_i}^*,\\I_i) +  2\\gamma + \\frac{C_X^6}{c_X^4}  { \\mathcal B_n^{-1}   \\Delta }  \\kappa_k^2 + C\\frac{g_X^4 C_X^2}{c_X^4}p^2\\log(n\\vee p).\n\\end{align}\nLet\n$$\n\\widetilde{\\gamma} = 2\\frac{C_X^6}{c_X^4}\\mclB_n^{-1}\\Delta\\kappa_k^2 +  C\\frac{g_X^4 C_X^2}{c_X^4}p^2\\log(n\\vee p) +  2\\gamma.\n$$\nThen by Taylor expansion and \\Cref{lem:estimation covariance} we have\n\\begin{align}\n    cc_X^2\\sum_{t\\in \\mclI}\\|\\widehat{\\Omega}_{\\I} - \\Omega^*_t\\|_F^2 &\\leq \\widetilde{\\gamma} + \\sum_{i=1}^3|\\I_i|{\\rm Tr}[(\\Omega_{\\I_i}^* - \\widehat{\\Omega}_{\\I})^\\top (\\widehat{\\Sigma}_{\\I_i}  - \\Sigma^*_{\\I_i})] \\nonumber \\\\\n    &\\leq \\widetilde{\\gamma} + Cg_X^2 p\\log^{\\frac{1}{2}}(n\\vee p)\\left[ \\sum_{i=1}^3 \\sqrt{|\\mclI_{i}|}\\|\\widehat{\\Omega}_{\\I} - \\Omega^*_{i}\\|_F \\right]  \\nonumber \\\\\n    &\\leq \\widetilde{\\gamma} + Cg_X^2 p\\log^{\\frac{1}{2}}(n\\vee p) \\sqrt{\\sum_{t\\in \\mclI}\\|\\widehat{\\Omega}_{\\I} - \\Omega^*_t\\|_F^2}.\n\\end{align}\nThe inequality above implies that\n\\begin{equation}\n    \\sum_{t\\in \\mclI}\\|\\widehat{\\Omega}_{\\I} - \\Omega^*_t\\|_F^2 \\leq \\frac{C_1}{c_X^2}\\left[\\widetilde{\\gamma} + \\frac{1}{c_X^2}\\|X\\|^4_{\\psi_2} p^{2}\\log(n\\vee p)\\right].\n\\end{equation}\nBy the choice of $\\gamma$, it holds that\n\\begin{equation}\n    \\sum_{t\\in \\I_1\\cup\\I_2}\\|\\widehat{\\Omega}_{\\I} - \\Omega^*_t\\|_F^2 \\leq \\frac{C_1C_{\\gamma}}{c_X^4} \\|X\\|^4_{\\psi_2} p^{2}\\log(n\\vee p).\n\\end{equation}\nOn the other hand,\n\\begin{equation}\n    \\sum_{t\\in \\I_1\\cup\\I_2}\\|\\widehat{\\Omega}_{\\I} - \\Omega^*_t\\|_F^2\\geq \\frac{|\\mclI_{1}||\\mclI_{2}|}{|\\mclI|}\\|\\Omega^*_{k-1} - \\Omega^*_k\\|_F^2\\geq \\frac{1}{2}\\min\\{|\\I_1|,|\\I_2|\\}\\|\\Omega^*_{k-1} - \\Omega^*_k\\|_F^2.\n\\end{equation}\nSuppose $|\\I_1|\\geq |\\I_2|$, then the inequality above leads to\n\\begin{equation*}\n    \\Delta_{\\min} \\kappa^2\\leq \\frac{C_1C_{\\gamma}}{c_X^4} \\|X\\|^4_{\\psi_2} p^{2}\\log(n\\vee p),\n\\end{equation*}\nwhich is contradictory to the assumption on $\\Delta$. Therefore, $|\\I_1|<|\\I_2|$ and we have \n\\begin{equation}\n    s-\\eta_k = |\\I_1|\\leq CC_{\\gamma} \\frac{g_X^4}{c_X^4\\|\\Omega_k^* - \\Omega_{k-1}^*\\|_F^2}p^2\\log(n\\vee p).\n\\end{equation}\nThe bound for $e - \\eta_{k+1}$ can be proved similarly.\n\\eprf\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\bnlem[Three or more change points]\n\\label{lem:cov three or more cp}\nSuppose the assumptions in \\Cref{assp:DCDP_covariance} hold. Then with probability at least $1-(n\\vee p)^{-3}$, there is no intervals in $\\widehat { \\mathcal P}$ containing three or more true change points.   \n \\enlem \n\\bprf\nWe prove by contradiction. Suppose    $ \\I=(s,e] \\in \\mathcal {\\widehat P}  $  be such that $ \\{ \\eta_1, \\ldots, \\eta_M\\} \\subset \\I $ with $M\\ge 3$. Throughout the proof, $M$ is assumed to be a parameter that can potentially change with $n$. \nSince the events $\\mathcal L (  { \\mathcal B_n^{-1}   \\Delta }   ) $ and  $\\mathcal R (  { \\mathcal B_n^{-1}   \\Delta }  ) $ hold, by relabeling $\\{ s_q\\}_{q=1}^ { \\mathcal Q}   $ if necessary,   let $ \\{ s_m\\}_{m=1}^M $ be such that \n $$ 0 \\le s_m  -\\eta_m  \\le  { \\mathcal B_n^{-1}   \\Delta }     \\quad \\text{for} \\quad 1 \\le m \\le M-1  $$ and that\n $$ 0\\le \\eta_M  - s_M \\le  { \\mathcal B_n^{-1}   \\Delta }  .$$ \n Note that these choices ensure that  $ \\{ s_m\\}_{m=1}^M  \\subset \\I . $\n \n \\begin{center}\n \\begin{tikzpicture} \n\\draw[ - ] (-10,0)--(1,0);\n \\node[color=black] at (-8,-0.3) {\\small s};\n \\draw[  (-, ultra thick, black] (-8,0) -- (-7.99,0);\n  \\draw[ { -]}, ultra thick, black] (-1.0002,0) -- (-1,0);\n\n   \\node[color=black] at (-7,-0.3) {\\small $\\eta_1$};\n\\draw(-7 ,0)circle [radius=2pt] ;\n\n  \\node[color=black] at (-6.5,-0.3) {\\small $s_1$};\n\\draw plot[mark=x, mark options={color=black, scale=1.5}] coordinates {(-6.5,0)  }; \n\n  \\node[color=black] at (-5,-0.3) {\\small $\\eta_2$};\n\\draw(-5 ,0)circle [radius=2pt] ;\n  \\node[color=black] at (-4.5,-0.3) {\\small $s_2$};\n\\draw plot[mark=x, mark options={color=black, scale=1.5}] coordinates {(-4.5,0)  }; \n \n\n  \\node[color=black] at (-2.5,-0.3) {\\small $\\eta_3$};\n\\draw(-2.5 ,0)circle [radius=2pt] ;\n  \\node[color=black] at (-3 ,-0.3) {\\small $s_3$};\n\\draw plot[mark=x, mark options={color=black, scale=1.5}] coordinates {(-3,0)  }; \n  \n   \\node[color=black] at (-1,-0.3) {\\small $e$};\n\n \n\\end{tikzpicture}\n\\end{center}\n\n{\\bf Step 1.}\n Denote \n $$ \\mathcal I_ 1= (  s, s _1], \\quad \\I_m   =(s_{m-1} , s_m] \\text{ for }  2 \\le m \\le M     \\quad \\text{and} \\quad \\I_{M+1}   = (s_M,e]. $$\nThen since \n$ s , e,  \\{ s_m \\}_{m=1}^M  \\subset   \\{ s_q\\}_{q=1}^ { \\mathcal Q}   $,   it follows that \n\n\n\nSuppose $\\I = (s,e]\\in \\widehat{\\mclP}$ and there are $M\\geq 3$ true change points $\\{\\eta_{q + i}\\}_{i\\in [M]}$ inside $\\I$, and denote\n\\begin{equation*}\n    \\I_1 = (s, \\eta_{q + 1}],\\ \\I_m = (\\eta_{q + m - 1},\\eta_{q + m}],\\ \\I_{M+1} = (\\eta_{q + M},e].\n\\end{equation*}\nThen by the definition of $\\widehat{\\mclP}$ and $\\widehat{\\Omega}_{\\I_m}$, it holds that\n\\begin{equation*}\n    \\mclF(\\widehat {\\Omega}_{\\I},\\I)\\leq \\sum_{i = 1}^{M+1} \\mclF (\\widehat {\\Omega}_{\\I_i},\\I_i) + M\\gamma\\leq \\sum_{i = 1}^{M+1} \\mclF ({\\Omega}_{\\I_i}^*,\\I_i) + M\\gamma,\n\\end{equation*}\nwhich implies that\n\\begin{align}\n    & \\sum_{t\\in \\I}{\\rm Tr}(\\widehat{\\Omega}_{\\I}^\\top (X_tX_t^\\top)) - |\\I|\\log|\\widehat{\\Omega}_{\\I}| \\nonumber  \\\\\n    \\leq & \\sum_{i=1}^{M+1} \\sum_{t\\in \\I_i}{\\rm Tr}(({\\Omega}_{\\I_i}^*)^\\top (X_tX_t^\\top)) - \\sum_{i=1}^{M+1}|\\I_i|\\log|{\\Omega}_{\\I_i}^*| + M\\gamma.\n\\end{align}\nBy Taylor expansion and \\Cref{lem:estimation covariance} we have\n\\begin{align}\n    cc_X^2\\sum_{t\\in \\mclI}\\|\\widehat{\\Omega}_{\\I} - \\Omega^*_t\\|_F^2 &\\leq M\\gamma + \\sum_{i=1}^{M+1}|\\I_i|{\\rm Tr}[(\\Omega_{\\I_i}^* - \\widehat{\\Omega}_{\\I})^\\top (\\widehat{\\Sigma}_{\\I_i}  - \\Sigma^*_{\\I_i})] \\nonumber \\\\\n    &\\leq M\\gamma + Cg_X^2 p\\log^{\\frac{1}{2}}(n\\vee p)\\left[ \\sum_{i=1}^{M+1} \\sqrt{|\\mclI_{i}|}\\|\\widehat{\\Omega}_{\\I} - \\Omega^*_{i}\\|_F \\right]  \\nonumber \\\\\n    &\\leq M\\gamma + Cg_X^2 p\\log^{\\frac{1}{2}}(n\\vee p) \\sqrt{\\sum_{t\\in \\mclI}\\|\\widehat{\\Omega}_{\\I} - \\Omega^*_t\\|_F^2}.\n\\end{align}\nThe inequality above implies that\n\\begin{equation}\n    \\sum_{t\\in \\mclI}\\|\\widehat{\\Omega}_{\\I} - \\Omega^*_t\\|_F^2 \\leq \\frac{C_1}{c_X^2}\\left[M\\gamma + \\frac{1}{c_X^2}\\|X\\|^4_{\\psi_2} p^{2}\\log(n\\vee p)\\right].\n\\end{equation}\nOn the other hand, for each $i\\in [M]$, we have\n\\begin{equation}\n    \\sum_{t\\in \\I_i\\cup\\I_{i +1}}\\|\\widehat{\\Omega}_{\\I} - \\Omega^*_t\\|_F^2\\geq \\frac{|\\mclI_{i}||\\mclI_{i + 1}|}{|\\mclI|}\\|\\Omega^*_{\\eta_{q + i + 1}} - \\Omega^*_{\\eta_{q + i}}\\|_F^2.\n\\end{equation}\nIn addition, for each $i\\in \\{2,\\cdots,M\\}$, by definition, it holds that $\\min\\{|\\I_i|,|\\I_{i + 1}|\\}\\geq \\Delta_{\\min}$. Therefore, we have\n\\begin{align*}\n    (M-2)\\Delta_{\\min} \\kappa^2\\leq \\frac{C_1}{c_X^2}\\left[M\\gamma + \\frac{1}{c_X^2}\\|X\\|^4_{\\psi_2} p^{2}\\log(n\\vee p)\\right].\n\\end{align*}\nSince $M\/(M-2)\\leq 3$ for any $M\\geq 3$, it holds that\n\\begin{equation}\n    \\Delta_{\\min} \\leq CC_{\\gamma} \\frac{g_X^4}{c_X^4\\|\\Omega_k^* - \\Omega_{k-1}^*\\|_F^2}p^2\\log(n\\vee p),\n\\end{equation}\nwhich is contradictory to the assumption on $\\Delta$, and the proof is complete.\n\\eprf\n\n\n\n\n\n\n \\bnlem[Two consecutive intervals]\n \\label{lem:cov two consecutive interval}\n Under \\Cref{assp:DCDP_covariance} and the choice that $$\\gamma\\geq C_{\\gamma}\\frac{g_X^4}{c_X^2} p^2\\log(n\\vee p),$$ with probability at least $1-(n\\vee p)^{-3}$, there are no  two consecutive intervals $\\I_1= (s,t ] \\in \\widehat{\\mathcal P}$, $\\I_2=(t, e] \\in   \\widehat{\\mathcal P}$ such that $\\I_1 \\cup \\I_2$ contains no change points.  \n \\enlem \n \\begin{proof} \n We prove by contradiction. Suppose that $\\I_1,\\I_2\\in \\widehat{\\mclP}$ and\n $$ \\I :  =\\I_1\\cup  \\I_2 $$\n contains no change points. \n By the definition of $\\widehat{\\mclP}$ and $\\widehat{\\Omega}_{\\I}$, it holds that\n $$ \\mclF (\\widehat{\\Omega}_{\\I_1},\\I_1)  + \\mclF(\\widehat{\\Omega}_{\\I_2},\\I_2) + \\gamma \\leq  \\mclF (\\widehat{\\Omega}_{\\I},\\I)\\leq \\mclF ({\\Omega}_{\\I}^*,\\I)$$\n By \\Cref{lem:cov no cp}, it follows that \n\\begin{align*} \n\\mclF({\\Omega}^*_{\\I_1}, \\mclI_1)\\leq &  \\mclF(\\widehat{\\Omega}_{\\I_1}, \\mclI_1) + C\\frac{g_X^4}{c_X^2} p^2\\log(n\\vee p),\n \\\\\n\\mclF({\\Omega}^*_{\\I_2}, \\mclI_2)\\leq &  \\mclF(\\widehat{\\Omega}_{\\I_2}, \\mclI_2) +  C\\frac{g_X^4}{c_X^2} p^2\\log(n\\vee p) \n \\end{align*}\n So\n $$\\mclF({\\Omega}^*_{\\I_1}, \\mclI_1) +\\mclF({\\Omega}^*_{\\I_2}, \\mclI_2) -2C\\frac{g_X^4}{c_X^2} p^2\\log(n\\vee p)  +\\gamma    \\leq  \\mclF ({\\Omega}_{\\I}^*,\\I).  $$\n Since $\\I$ does not contain any change points, ${\\Omega}^*_{\\I_1} = {\\Omega}^*_{\\I_2} = {\\Omega}^*_{\\I}$, and it follows that \n $$ \\gamma \\leq 2C\\frac{g_X^4}{c_X^2} p^2\\log(n\\vee p).$$\n This is a contradiction when $C_\\gamma$ is sufficiently large.\n \\end{proof} \n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Penalized local refinement}\n\\label{sec: proof local refinement}\nIn this section, we prove consistency results in \\Cref{section:main} for penalized local refinement, or the conquer step. We also provide more details on the computational complexity of local refinement using memorization technique which is summarized in \\Cref{sec:method}. In particular,\n\\begin{enumerate}\n    \\item In \\Cref{sec:complexity local refine}, we analyze the complexity of the local refinement step and show that it is linear in terms of $n$, as is mentioned in \\Cref{sec:method}.\n    \\item \\Cref{sec:op1 fundations} presents some fundamental lemmas to prove other results.\n    \\item \\Cref{sec:mean op1} prove results for the mean model, i.e., \\Cref{cor:mean local refinement}.\n    \\item \\Cref{sec:regression op1} prove results for the linear regression model, i.e., \\Cref{cor:regression local refinement}.\n    \\item \\Cref{sec:cov op1} prove results for the Gaussian graphical model, i.e., \\Cref{cor:covariance local refinement main}.\n\\end{enumerate}\n\n\n\n\\subsection{Complexity analysis}\n\\label{sec:complexity local refine}\nWe show in \\Cref{lem: complexity local refine} that the complexity of the conquer step (\\Cref{algo:local_refine_general}) can be as low as $O(n\\cdot \\mathcal{C}_2(p))$.\n\n\\bnlem[Complexity of the conquer step]\n\\label{lem: complexity local refine}\nFor all three models we discussed in \\Cref{section:main}, with a memorization technique, the complexity of \\Cref{algo:local_refine_general} would be $O(n\\cdot \\mathcal{C}_2(p))$.\n\\enlem\n\\bprf\nIn \\Cref{algo:local_refine_general}, for each $k\\in [\\widehat{K}]$, we search over the interval of length $\\frac{2}{3}(\\widehat{\\Delta}_{k-1} + \\widehat{\\Delta}_k)$ where $\\widehat{\\Delta}_k:=\\widehat{\\eta}_{k+1} - \\widehat{\\eta}_{k}$. Without \nany algorithmic optimization, the complexity would be $O((\\widehat{\\Delta}_{k-1} + \\widehat{\\Delta}_k)\\mathcal{C}_1(\\widehat{\\Delta}_{k-1} + \\widehat{\\Delta}_k, p))$ where $\\mathcal{C}_1(m, p)$ is the complexity of calculating $\\widehat{\\theta}_{\\mclI}$ and $\\mclF(\\widehat{\\theta}_{\\mclI}, \\mclI)$ for an interval of length $m$,\n\nUnder the three models in \\Cref{section:main}, calculating $\\widehat{\\theta}_{\\mclI}$ involves the calculation of some sufficient statistics or gradients and a gradient descent or coordinate descent procedure which is independent of $|\\mclI|$. Therefore, $\\mathcal{C}_1(|\\mclI|, p) = O(|\\mclI|) + O(\\mclC_2(p))$. For instance, solving Lasso only takes $O(p)$ time once $\\sum_{i\\in[n]}X_iX_i^\\top$ and $\\sum_{{i\\in[n]}}X_iy_i$ are known. In the conquer step, each time we only update the two summations $(\\sum_{i\\in[n]}X_iX_i^\\top,\\sum_{{i\\in[n]}}X_iy_i)$ by one term, so we can use memorization trick to reduce $\\mathcal{C}_1(\\widehat{\\Delta}_{k-1} + \\widehat{\\Delta}_k, p)$ to $O(1) + O(\\mclC_2(p))$. Consequently, the complexity at the $k-th$ step of \\Cref{algo:local_refine_general} can be reduced to $O((\\widehat{\\Delta}_{k-1} + \\widehat{\\Delta}_k)\\mclC_2(p))$. Taking summation over $k\\in [\\widehat{K}]$ and considering the fact that $\\widehat{\\mclP}$ is a segmentation of $[1,n]$, the total complexity of the conquer step would be\n$$\n\\sum_{k\\in [\\widehat{K}]}O((\\widehat{\\Delta}_{k-1} + \\widehat{\\Delta}_k)\\cdot \\mclC_2(p)) = O(n\\cdot \\mclC_2(p)).\n$$\n\\eprf\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\subsection{Fundamental lemma}\n\\label{sec:op1 fundations}\n As is introduced in \\Cref{sec:introduction}, the sub-gaussian norm of a random variable is defined as \\citep{vershynin2018high}: $\\|X\\|_{\\psi_2}:=\\inf \\{t>0: \\mathbb{E} \\psi_2(|X| \/ t) \\leq 1\\}$ where $\\psi_2(t) = e^{t^2} - 1$.\n\n Similarly, for sub-exponential random variables, one can define its Orlicz norm as $\\|X\\|_{\\psi_1}:=\\inf \\{t>0: \\mathbb{E} \\psi_1(|X| \/ t) \\leq 2\\}$ where $\\psi_1(t) = e^{t}$.\n\n\\bnlem\n\\label{lem:subexp deviation martingale}\nSuppose $\\left\\{z_i\\right\\}_{i=1}^{\\infty}$ is a collection of independent centered sub-exponential random variables with $0<\\sup _{1 \\leq i<\\infty}\\left\\|z_i\\right\\|_{\\psi_1} \\leq 1$. Then for any integer $d>0, \\alpha>0$ and any $x>0$\n$$\n\\mathbb{P}\\left(\\max _{k \\in[d,(1+\\alpha) d]} \\frac{\\sum_{i=1}^k z_i}{\\sqrt{k}} \\geq x\\right) \\leq \\exp \\left\\{-\\frac{x^2}{2(1+\\alpha)}\\right\\}+\\exp \\left\\{-\\frac{\\sqrt{d} x}{2}\\right\\} .\n$$\n\\enlem\n\\bprf\nDenote $S_n=\\sum_{i=1}^n z_i$. Let $\\zeta=\\sup _{1 \\leq i \\leq \\infty}\\left\\|z_i\\right\\|_{\\psi_1}$. For any two integers $m<n$ and any $t \\leq \\frac{C_1}{\\zeta}$\n$$\n\\mathbb{E}\\left(\\exp \\left(t\\left(S_n-S_m\\right)\\right)\\right) \\leq \\prod_{i=m+1}^n \\mathbb{E}\\left(\\exp \\left(t z_i\\right)\\right) \\leq \\prod_{i=m+1}^n \\mathbb{E}\\left(C_1^2 t^2 \/ 2\\right)=\\mathbb{E}\\left[(n-m) C_1^2 t^2 \/ 2\\right].\n$$\nLet $\\mathcal{F}_k$ denote the sigma-algebra generated by $\\left(z_1, \\ldots, z_k\\right)$. Without loss of generality, assume that $C_1=1$. Since $S_k$ is independent of $S_n-S_k$, this implies that when $t \\geq 1 \/ \\zeta$\n$$\n\\mathbb{E}\\left(\\exp \\left\\{t S_n-\\frac{t^2 n}{2}\\right\\} \\mid \\mathcal{F}_k\\right)=\\exp \\left\\{t S_k-\\frac{t^2 k}{2}\\right\\} \\mathbb{E}\\left(\\exp \\left\\{t\\left(S_n-S_k\\right)\\right\\}-\\frac{t^2(n-k)}{2}\\right) \\leq \\exp \\left\\{t S_k-\\frac{t^2 k}{2}\\right\\}\n$$\nTherefore $\\exp \\left\\{t S_n-\\frac{t^2 n}{2}\\right\\}$ is a super-Martingale. Let $x$ be given and\n$$\nA=\\inf \\left\\{n \\geq d, S_n \\geq \\sqrt{n} x\\right\\} .\n$$\nThen $S_A \\geq \\sqrt{A} x \\geq \\sqrt{d} x$. Thus for $t \\geq 1 \/ \\zeta$,\n$$\n\\mathbb{E}\\left(\\exp \\left\\{t \\sqrt{d} x-\\frac{t^2 A}{2}\\right\\}\\right) \\leq \\mathbb{E}\\left(\\exp \\left\\{t S_A-\\frac{t^2 A}{2}\\right\\}\\right) \\leq \\mathbb{E}\\left(\\exp \\left\\{t S_1-\\frac{t^2}{2}\\right\\}\\right) \\leq 1 .\n$$\nBy definition of $A$,\n$$\n\\mathbb{P}\\left(\\max _{k \\in[d,(1+\\alpha) d]} \\frac{\\sum_{i=1}^k z_i}{\\sqrt{k}} \\geq x\\right) \\leq \\mathbb{P}(A \\leq(1+\\alpha) d) .\n$$\nSince $u \\rightarrow \\exp \\left(s \\sqrt{d} x-\\frac{t^2 u}{2}\\right)$ is decreasing, it follows that\n$$\n\\mathbb{P}\\left(\\max _{k \\in[d,(1+\\alpha) d]} \\frac{\\sum_{i=1}^k z_i}{\\sqrt{k}} \\geq x\\right) \\leq \\mathbb{P}\\left(\\exp \\left\\{t \\sqrt{d} x-t^2 A \/ 2\\right\\} \\geq \\exp \\left\\{t \\sqrt{d} x-t^2(1+\\alpha) d \/ 2\\right\\}\\right) .\n$$\nMarkov's inequality implies that when $t \\geq \\frac{1}{\\zeta}$,\n$$\n\\mathbb{P}\\left(\\max _{k \\in[d,(1+\\alpha) d]} \\frac{\\sum_{i=1}^k z_i}{\\sqrt{k}} \\geq x\\right) \\leq \\exp \\left\\{-t \\sqrt{d} x+t^2(1+\\alpha) d \/ 2\\right\\}\n$$\nSet\n$$\nt=\\min \\left\\{\\frac{1}{\\zeta}, \\frac{x}{(1+\\alpha) \\sqrt{d}}\\right\\}\n$$\nIf $\\frac{1}{\\zeta} \\geq \\frac{x}{(1+\\alpha) \\sqrt{d}}$. Then $t=\\frac{x}{(1+\\alpha) \\sqrt{d}}$ and therefore\n$$\n-t \\sqrt{d} x+t^2(1+\\alpha) d \/ 2=-\\frac{x^2}{2(1+\\alpha)}\n$$\nSo\n$$\n\\mathbb{P}\\left(\\max _{k \\in[d,(1+\\alpha) d]} \\frac{\\sum_{i=1}^k z_i}{\\sqrt{k}} \\geq x\\right) \\leq \\exp \\left\\{-\\frac{x^2}{2(1+\\alpha)}\\right\\} .\n$$\nIf $\\frac{1}{\\zeta} \\leq \\frac{x}{(1+\\alpha) \\sqrt{d}}$. Then $t=\\frac{1}{\\zeta} \\leq \\frac{x}{(1+\\alpha) \\sqrt{d}}$ and so\n$$\n-t \\sqrt{d} x+t^2(1+\\alpha) d \/ 2 \\leq \\frac{-\\sqrt{d} x}{\\zeta}+\\frac{1}{\\zeta} \\frac{x}{(1+\\alpha) \\sqrt{d}} \\frac{(1+\\alpha) d}{2}=\\frac{-\\sqrt{d} x}{2 \\zeta} .\n$$\nSo\n$$\n\\mathbb{P}\\left(\\max _{k \\in[d,(1+\\alpha) d]} \\frac{\\sum_{i=1}^k z_i}{\\sqrt{k}} \\geq x\\right) \\leq \\exp \\left\\{-\\frac{\\sqrt{d} x}{2 \\zeta}\\right\\} \\leq \\exp \\left\\{-\\frac{\\sqrt{d} x}{2}\\right\\},\n$$\nwhere $\\zeta \\leq 1$ is used in the last inequality. Putting the two cases together leads the desired result.\n\\eprf\n\n\n\n\\bnlem\n\\label{lem:subexp uniform loglog law}\nSuppose $\\left\\{z_i\\right\\}_{i=1}^{\\infty}$ is a collection of independent centered sub-exponential random variable with $0<\\sup _{1 \\leq i<\\infty}\\left\\|z_i\\right\\|_{\\psi_1} \\leq 1$. Let $\\nu>0$ be given. For any $x>0$, it holds that\n$$\n\\mathbb{P}\\left(\\sum_{i=1}^r z_i \\leq 4 \\sqrt{r\\{\\log \\log (4 \\nu r)+x+1\\}}+4 \\sqrt{r \\nu}\\{\\log \\log (4 \\nu r)+x+1\\} \\text { for all } r \\geq 1 \/ \\nu\\right) \\geq 1-2 \\exp (-x) \\text {. }\n$$\n\\enlem\n\\bprf\nLet $s \\in \\mathbb{Z}^{+}$and $\\mathcal{T}_s=\\left[2^s \/ \\nu, 2^{s+1} \/ \\nu\\right]$. By \\Cref{lem:subexp deviation martingale}, for all $x>0$,\n$$\n\\mathbb{P}\\left(\\sup _{r \\in \\mathcal{T}_s} \\frac{\\sum_{i=1}^r z_i}{\\sqrt{r}} \\geq x\\right) \\leq \\exp \\left\\{-\\frac{x^2}{4}\\right\\}+\\exp \\left\\{-\\frac{\\sqrt{2^s \/ \\nu} x}{2}\\right\\} \\leq \\exp \\left\\{-\\frac{x^2}{4}\\right\\}+\\exp \\left\\{-\\frac{x}{2 \\sqrt{\\nu}}\\right\\} .\n$$\nTherefore by a union bound,\n\\begin{align}\n    & \\mathbb{P}\\left(\\exists s \\in \\mathbb{Z}^{+}: \\sup _{r \\in \\mathcal{T}_s} \\frac{\\sum_{i=1}^r z_i}{\\sqrt{r}} \\geq 2 \\sqrt{\\log \\log ((s+1)(s+2))+x}+2 \\sqrt{\\nu}\\{\\log \\log ((s+1)(s+2))+x\\}\\right) \\nonumber \\\\\n\\leq & \\sum_{s=0}^{\\infty} 2 \\frac{\\exp (-x)}{(s+1)(s+2)}=2 \\exp (-x) \\label{tmp_eq: sub-gauss iterated log}.\n\\end{align}\nFor any $r \\geq 2^s \/ \\nu, s \\leq \\log (r \\nu) \/ \\log (2)$, and therefore\n$$\n(s+1)(s+2) \\leq \\frac{\\log (2 r \\nu) \\log (4 r \\nu)}{\\log ^2(2)} \\leq\\left(\\frac{\\log (4 r \\nu)}{\\log (2)}\\right)^2 .\n$$\nThus\n$$\n\\log ((s+1)(s+2)) \\leq 2 \\log \\left(\\frac{\\log (4 r \\nu)}{\\log (2)}\\right) \\leq 2 \\log \\log (4 r \\nu)+1 .\n$$\nThe above display together with \\eqref{tmp_eq: sub-gauss iterated log} gives\n$$\n\\mathbb{P}\\left(\\sup _{r \\geq 1 \/ \\nu} \\frac{\\sum_{i=1}^r z_i}{\\sqrt{r}} \\geq 2 \\sqrt{2 r \\log \\log (4 r \\nu)+x+1}+2 \\sqrt{r \\nu}\\{\\log \\log (4 r \\nu)+x+1\\}\\right) \\leq 2 \\exp (-x) .\n$$\n\\eprf\n\n\n\n\n\n\nNext we present two analogous lemmas for sub-gaussian random variables.\n\n\n\n\n\\bnlem\n\\label{lem:subgaussian deviation martingale}\nSuppose $\\left\\{z_i\\right\\}_{i=1}^{\\infty}$ is a collection of independent centered sub-gaussian random variables with $0<\\sup _{1 \\leq i<\\infty}\\left\\|z_i\\right\\|_{\\psi_2} \\leq \\sigma$. Then for any integer $d>0, \\alpha>0$ and any $x>0$\n$$\n\\mathbb{P}\\left(\\max _{k \\in[d,(1+\\alpha) d]} \\frac{\\sum_{i=1}^k z_i}{\\sqrt{k}} \\geq x\\right) \\leq \\exp \\left\\{-\\frac{x^2}{2(1+\\alpha)\\sigma^2}\\right\\} .\n$$\n\\enlem\n\\bprf\nDenote $S_n=\\sum_{i=1}^n z_i$. Let $\\zeta=\\sup _{1 \\leq i \\leq \\infty}\\left\\|z_i\\right\\|_{\\psi_2}$. For any two integers $m<n$,\n$$\n\\mathbb{E}\\left(\\exp \\left(t\\left(S_n-S_m\\right)\\right)\\right) \\leq \\prod_{i=m+1}^n \\mathbb{E}\\left(\\exp \\left(t z_i\\right)\\right) \\leq \\prod_{i=m+1}^n \\mathbb{E}\\left(\\zeta^2 t^2 \/ 2\\right)=\\mathbb{E}\\left[(n-m) \\zeta^2 t^2 \/ 2\\right].\n$$\nLet $\\mathcal{F}_k$ denote the sigma-algebra generated by $\\left(z_1, \\ldots, z_k\\right)$. Since $S_k$ is independent of $S_n-S_k$, this implies that\n\\begin{align*}\n    \\mathbb{E}\\left(\\exp \\left\\{t S_n-\\frac{\\zeta^2t^2 n}{2}\\right\\} \\mid \\mathcal{F}_k\\right)\n    &=\\exp \\left\\{t S_k-\\frac{\\zeta^2t^2 k}{2}\\right\\} \\mathbb{E}\\left(\\exp \\left\\{t\\left(S_n-S_k\\right)\\right\\}-\\frac{\\zeta^2 t^2(n-k)}{2}\\right) \\\\\n    &\\leq \\exp \\left\\{t S_k-\\frac{\\zeta^2 t^2 k}{2}\\right\\}\n\\end{align*}\nTherefore $\\exp \\left\\{t S_n-\\frac{\\zeta^2t^2 n}{2}\\right\\}$ is a super-martingale. Let $x$ be given and\n$$\nA=\\inf \\left\\{n \\geq d, S_n \\geq \\sqrt{n} x\\right\\} .\n$$\nThen $S_A \\geq \\sqrt{A} x \\geq \\sqrt{d} x$. Thus for $t >0$,\n$$\n\\mathbb{E}\\left(\\exp \\left\\{t \\sqrt{d} x-\\frac{\\zeta^2 t^2 A}{2}\\right\\}\\right) \\leq \\mathbb{E}\\left(\\exp \\left\\{t S_A-\\frac{\\zeta^2 t^2 A}{2}\\right\\}\\right) \\leq \\mathbb{E}\\left(\\exp \\left\\{t S_1-\\frac{\\zeta^2 t^2}{2}\\right\\}\\right) \\leq 1 .\n$$\nBy definition of $A$,\n$$\n\\mathbb{P}\\left(\\max _{k \\in[d,(1+\\alpha) d]} \\frac{\\sum_{i=1}^k z_i}{\\sqrt{k}} \\geq x\\right) \\leq \\mathbb{P}(A \\leq(1+\\alpha) d) .\n$$\nSince $u \\rightarrow \\exp \\left(s \\sqrt{d} x-\\frac{\\zeta^2 t^2 u}{2}\\right)$ is decreasing, it follows that\n$$\n\\mathbb{P}\\left(\\max _{k \\in[d,(1+\\alpha) d]} \\frac{\\sum_{i=1}^k z_i}{\\sqrt{k}} \\geq x\\right) \\leq \\mathbb{P}\\left(\\exp \\left\\{t \\sqrt{d} x - \\zeta^2 t^2 A \/ 2\\right\\} \\geq \\exp \\left\\{t \\sqrt{d} x-\\zeta^2 t^2(1+\\alpha) d \/ 2\\right\\}\\right) .\n$$\nMarkov's inequality implies that,\n$$\n\\mathbb{P}\\left(\\max _{k \\in[d,(1+\\alpha) d]} \\frac{\\sum_{i=1}^k z_i}{\\sqrt{k}} \\geq x\\right) \\leq \\exp \\left\\{-t \\sqrt{d} x+\\zeta^2 t^2(1+\\alpha) d \/ 2\\right\\}\n$$\nSet $t= \\frac{x}{\\zeta^2(1+\\alpha) \\sqrt{d}}$, then\n$$\n-t \\sqrt{d} x+\\zeta^2 t^2(1+\\alpha) d \/ 2=-\\frac{x^2}{2(1+\\alpha)\\zeta^2}\n$$\nSo\n$$\n\\mathbb{P}\\left(\\max _{k \\in[d,(1+\\alpha) d]} \\frac{\\sum_{i=1}^k z_i}{\\sqrt{k}} \\geq x\\right) \\leq \\exp \\left\\{-\\frac{x^2}{2(1+\\alpha)\\zeta^2}\\right\\} .\n$$\n\\eprf\n\n\n\n\\bnlem\n\\label{lem:subgaussian uniform loglog law}\nSuppose $\\left\\{z_i\\right\\}_{i=1}^{\\infty}$ is a collection of independent centered sub-gaussian random variable with $0<\\sup _{1 \\leq i<\\infty}\\left\\|z_i\\right\\|_{\\psi_2} \\leq \\sigma$. Let $\\nu>0$ be given. For any $x>0$, it holds that\n$$\n\\mathbb{P}\\left(\\sum_{i=1}^r z_i \\leq 4\\sigma \\sqrt{r\\{\\log \\log (4 \\nu r)+x+1\\}} \\text { for all } r \\geq 1 \/ \\nu\\right) \\geq 1-2 \\exp (-x) \\text {. }\n$$\n\\enlem\n\\bprf\nLet $s \\in \\mathbb{Z}^{+}$and $\\mathcal{T}_s=\\left[2^s \/ \\nu, 2^{s+1} \/ \\nu\\right]$. By \\Cref{lem:subgaussian deviation martingale}, for all $x>0$,\n$$\n\\mathbb{P}\\left(\\sup _{r \\in \\mathcal{T}_s} \\frac{\\sum_{i=1}^r z_i}{\\sqrt{r}} \\geq x\\right) \\leq \\exp \\left\\{-\\frac{x^2}{4\\sigma^2}\\right\\}.\n$$\nTherefore by a union bound,\n\\begin{align}\n    & \\mathbb{P}\\left(\\exists s \\in \\mathbb{Z}^{+}: \\sup _{r \\in \\mathcal{T}_s} \\frac{\\sum_{i=1}^r z_i}{\\sqrt{r}} \\geq 2\\sigma \\sqrt{\\log \\log ((s+1)(s+2))+x}\\right) \\nonumber \\\\\n\\leq & \\sum_{s=0}^{\\infty} 2 \\frac{\\exp (-x)}{(s+1)(s+2)}=2 \\exp (-x) \\label{tmp_eq: iterated log}.\n\\end{align}\nFor any $r \\geq 2^s \/ \\nu, s \\leq \\log (r \\nu) \/ \\log (2)$, and therefore\n$$\n(s+1)(s+2) \\leq \\frac{\\log (2 r \\nu) \\log (4 r \\nu)}{\\log ^2(2)} \\leq\\left(\\frac{\\log (4 r \\nu)}{\\log (2)}\\right)^2 .\n$$\nThus\n$$\n\\log ((s+1)(s+2)) \\leq 2 \\log \\left(\\frac{\\log (4 r \\nu)}{\\log (2)}\\right) \\leq 2 \\log \\log (4 r \\nu)+1 .\n$$\nThe above display together with \\eqref{tmp_eq: iterated log} gives\n$$\n\\mathbb{P}\\left(\\sup _{r \\geq 1 \/ \\nu} \\frac{\\sum_{i=1}^r z_i}{\\sqrt{r}} \\geq 2\\sigma \\sqrt{2 r \\log \\log (4 r \\nu)+x+1}\\right) \\leq 2 \\exp (-x) .\n$$\n\\eprf\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\clearpage\n\\subsection{Local Refinement in the mean model }\n\\label{sec:mean op1}\nFor the ease of notations, we re-index the observations in the $k$-th interval by $[n_0]:\\{1,\\cdots, n_0\\}$ (though the sample size of the problem is still $n$), and denote the $k$-th jump size as $\\kappa$ and the minimal spacing between consecutive change points as $\\Delta$ (instead of $\\Delta_{\\min}$ in the main text).\n\nBy \\Cref{assp: DCDP_mean} and the setting of the local refinement algorithm, we have for some $\\alpha^*,\\beta^*\\in \\mathbb{R}^p$ that\n$$\ny_i=\\begin{cases}\n\\alpha^*+\\epsilon_i & \\text { when } i \\in(0, \\eta] \\\\\n\\beta^*+\\epsilon_i & \\text { when } i \\in(\\eta, n_0]\n\\end{cases}\n$$\nwhere $\\{\\epsilon_i\\}$ is an i.i.d sequence of subgaussian variables such that $\\|\\epsilon_i\\|_{\\psi_2}=\\sigma_\\epsilon<\\infty$. In addition, there exists $\\theta \\in(0,1)$ such that $\\eta=\\lfloor n \\theta\\rfloor$ and that $\\|\\alpha^*-\\beta^*\\|_2=\\kappa<\\infty$. By \\Cref{assp: DCDP_mean}, it holds that $\\|\\alpha^*\\|_0 \\leq \\mathfrak{s},\\|\\beta^*\\|_0 \\leq \\mathfrak{s}$ and\n\\begin{equation}\n    \\frac{\\mathfrak{s}^2 \\log^2 (n\\vee p)}{\\Delta \\kappa^2} \\rightarrow 0.\n    \\label{op1_eq:mean snr}\n\\end{equation}\nBy \\Cref{lem:mean refinement step 1}, with probability at least $1 - n^{-2}$, there exist $\\widehat{\\alpha}$ and $\\widehat{\\beta}$ such that\n\\begin{align*}\n  & \\|\\widehat{\\alpha}-\\alpha^*\\|_2^2\\leq C\\frac{\\mathfrak{s} \\log (n\\vee p)}{\\Delta} \\text{ and } \\|\\widehat{\\alpha}-\\alpha^*\\|_1\\leq C\\mathfrak{s} \\sqrt{\\frac{\\log (n\\vee p)}{\\Delta}}; \\\\\n& \\|\\widehat{\\beta}-\\beta^*\\|_2^2\\leq C\\frac{\\mathfrak{s} \\log (n\\vee p)}{\\Delta} \\text{ and } \\|\\widehat{\\beta}-\\beta^*\\|_1\\leq C \\mathfrak{s} \\sqrt{\\frac{\\log (n\\vee p)}{\\Delta}}.\n\\end{align*}\n\nLet\n$$\n\\widehat{\\mathcal{Q}}(k)=\\sum_{i=1}^k\\|y_i- \\widehat{\\alpha}\\|_2^2+\\sum_{i=k+1}^{n_0}\\|y_i-\\widehat{\\beta}\\|_2^2 \\quad \\text { and } \\quad \\mathcal{Q}^*(k)=\\sum_{i=1}^k\\|y_i- \\alpha^*\\|_2^2+\\sum_{i=k+1}^{n_0}\\|y_i- \\beta^*\\|_2^2 .\n$$\n\n\\bnlem[Refinement for the mean model]\nLet\n$$\n\\eta+r=\\underset{k \\in(0, n_0]}{\\arg \\max } \\widehat{\\mathcal{Q}}(k) .\n$$\nThen under the assumptions above, for any given $\\alpha\\in (0,1)$, it holds with probability $1 - (\\alpha\\vee n^{-1})$ that\n$$\n\\kappa^2 r \\leq C \\log\\frac{1}{\\alpha}.\n$$\n\\enlem\n\\bprf\nWithout loss of generality, suppose $r \\geq 0$. Since $\\eta+r$ is the minimizer, it follows that\n$$\n\\widehat{\\mathcal{Q}}(\\eta+r) \\leq \\widehat{\\mathcal{Q}}(\\eta) .\n$$\nIf $r \\leq \\frac{1}{\\kappa^2}$, then there is nothing to show. So for the rest of the argument, for contradiction, assume that\n$$\nr \\geq \\frac{1}{\\kappa^2}\n$$\nObserve that\n$$\n\\begin{aligned}\n\\widehat{\\mathcal{Q}}(\\eta + r)-\\widehat{\\mathcal{Q}}(\\eta) &=\\sum_{i=\\eta+1}^{\\eta+r}\\|y_i- \\widehat{\\alpha}\\|_2^2-\\sum_{i=\\eta+1}^{\\eta+r}\\|y_i-\\widehat{\\beta}\\|_2^2 \\\\\n\\mathcal{Q}^*(\\eta + r)-\\mathcal{Q}^*(\\eta) &=\\sum_{i=\\eta+1}^{\\eta+r}\\|y_i- \\alpha^*\\|_2^2-\\sum_{i=\\eta+1}^{\\eta+r}\\|y_i- \\beta^*\\|_2^2\n\\end{aligned}\n$$\n\\textbf{Step 1}. It follows that\n$$\n\\begin{aligned}\n& \\sum_{i=\\eta+1}^{\\eta+r}\\|y_i- \\widehat{\\alpha}\\|_2^2-\\sum_{i=\\eta+1}^{\\eta+r}\\|y_i-\\alpha^*\\|_2^2 \\\\\n=& \\sum_{i=\\eta+1}^{\\eta+r}\\|\\widehat{\\alpha}- \\alpha^*\\|_2^2+2\\left(\\widehat{\\alpha}-\\alpha^*\\right)^{\\top} \\sum_{i=\\eta+1}^{\\eta+r}\\left(y_i-\\alpha^*\\right) \\\\\n=& \\sum_{i=\\eta+1}^{\\eta+r}\\|\\widehat{\\alpha}- \\alpha^*\\|_2^2+2r\\left(\\widehat{\\alpha}-\\alpha^*\\right)^{\\top} \\left(\\beta^*-\\alpha^*\\right) +2\\left(\\widehat{\\alpha}-\\alpha^*\\right)^{\\top} \\sum_{i=\\eta+1}^{\\eta+r}\\epsilon_i\n\\end{aligned}\n$$\nBy assumptions, we have\n$$\n\\sum_{i=\\eta+1}^{\\eta+r}\\|\\widehat{\\alpha}- \\alpha^*\\|_2^2 \\leq C_1 r \\frac{\\mathfrak{s} \\log (p)}{\\Delta}.\n$$\nSimilarly\n$$\nr\\left(\\widehat{\\alpha}-\\alpha^*\\right)^{\\top}\\left(\\beta^*-\\alpha^*\\right)\\leq r\\|\\widehat{\\alpha}-\\alpha^*\\|_2\\|\\beta^*-\\alpha^*\\|_2 \\leq\nC_1 r\\kappa \\sqrt{\\frac{\\mathfrak{s} \\log (p)}{\\Delta} }\n$$\nwhere the second equality follows from $\\|\\beta^*-\\alpha^*\\|_2=\\kappa$, and the last equality follows from \\eqref{op1_eq:mean snr}. In addition,\n$$\n\\begin{aligned}\n&\\left(\\widehat{\\alpha}-\\alpha^*\\right)^{\\top} \\sum_{i=\\eta+1}^{\\eta+r} \\epsilon_i \\leq\\|\\widehat{\\alpha}-\\alpha^*\\|_1\\|\\sum_{i=\\eta+1}^{\\eta+r}\\epsilon_i\\|_{\\infty} \\\\\n=& C_2\\mathfrak{s} \\sqrt{\\frac{\\log (p)}{\\Delta}} \\sqrt{r \\log (p)}=C_2\\mathfrak{s}\\log(p)\\sqrt{\\frac{r}{\\Delta}}.\n\\end{aligned}\n$$\nTherefore\n\\begin{align}\n    \\sum_{i=\\eta+1}^{\\eta+r}\\|y_i- \\widehat{\\alpha}\\|_2^2-\\sum_{i=\\eta+1}^{\\eta+r}\\|y_i-\\alpha^*\\|_2^2 &\\leq C_1 r \\frac{\\mathfrak{s} \\log (p)}{\\Delta} + C_1 r\\kappa \\sqrt{\\frac{\\mathfrak{s} \\log (p)}{\\Delta} } + C_2\\mathfrak{s}\\log(p)\\sqrt{\\frac{r}{\\Delta}}\\nonumber \\\\\n    &\\leq C_1 r\\kappa^2 \\frac{\\mathfrak{s} \\log (p)}{\\Delta\\kappa^2} + C_1 r\\kappa^2 \\sqrt{\\frac{\\mathfrak{s} \\log (p)}{\\Delta\\kappa^2} } + C_2\\mathfrak{s}\\log(p)\\sqrt{\\frac{r\\kappa^2}{\\Delta\\kappa^2}}\\nonumber \\\\\n    &\\leq C_3 r \\kappa^2 \\frac{\\mathfrak{s} \\log (p)}{\\sqrt{\\Delta\\kappa^2}}.\n\\end{align}\n\\textbf{Step 2}. Using the same argument as in the previous step, it follows that\n$$\n\\sum_{i=\\eta+1}^{\\eta+r}\\|y_i- \\widehat{\\beta}\\|_2^2-\\sum_{i=\\eta+1}^{\\eta+r}\\|y_i- \\beta^*\\|_2^2\\leq C_3 r \\kappa^2 \\frac{\\mathfrak{s} \\log (p)}{\\sqrt{\\Delta\\kappa^2}}.\n$$\nTherefore\n\\begin{equation}\n    \\left|\\widehat{\\mathcal{Q}}(\\eta+r)-\\widehat{\\mathcal{Q}}(\\eta)-\\left\\{\\mathcal{Q}^*(\\eta+r)-\\mathcal{Q}^*(\\eta)\\right\\}\\right|\\leq C_3 r \\kappa^2 \\frac{\\mathfrak{s} \\log (p)}{\\sqrt{\\Delta\\kappa^2}}\n    \\label{tmp_eq:op1 mean main prop eq 1}\n\\end{equation}\nNotice that $\\widehat{\\mathcal{Q}}(\\eta+r)-\\widehat{\\mathcal{Q}}(\\eta)\\leq 0$, so our goal is to find a regime where $\\mathcal{Q}^*(\\eta+r)-\\mathcal{Q}^*(\\eta)\\geq 0$, in order to get rid of the $|\\cdot|$.\n\n\\textbf{Step 3}. Observe that\n$$\n\\begin{aligned}\n\\mathcal{Q}^*(\\eta+r)-\\mathcal{Q}^*(\\eta) \n=& \\sum_{i=\\eta+1}^{\\eta+r}\\|y_i- \\alpha^*\\|_2^2-\\sum_{i=\\eta+1}^{\\eta+r}\\|y_i- \\beta^*\\|_2^2 \\\\\n=& r\\| \\alpha^*- \\beta^*\\|_2^2-2 \\sum_{i=\\eta+1}^{\\eta+r}(y_i- \\beta^*)( \\alpha^*- \\beta^*) \\\\\n=& r\\| \\alpha^*- \\beta^*\\|_2^2 -2 (\\alpha^*- \\beta^*)^\\top\\sum_{i=\\eta+1}^{\\eta+r} \\epsilon_i\n\\end{aligned}\n$$\nLet\n$$\nw_i=\\frac{1}{\\kappa} \\epsilon_i^\\top\\left( \\alpha^*-\\beta^*\\right)\n$$\nThen $\\left\\{w_i\\right\\}_{i=1}^{\\infty}$ are subgaussian random variables with bounded $\\psi_2$ norm. Therefore by \\Cref{lem:subgaussian uniform loglog law}, uniformly for all $r \\geq 1 \/ \\kappa^2$, with probability at least $1 -\\alpha\/2$,\n$$\n\\sum_{i=1}^r w_i\\leq4\\sqrt{r\\left\\{\\log \\log \\left(\\kappa^2 r\\right) +\\log\\frac{4}{\\alpha}+1\\right\\}}\n$$\nIt follows that\n$$\n\\sum_{i=\\eta+1}^{\\eta+r} \\epsilon_i^\\top \\left( \\alpha^*-\\beta^*\\right)\\leq 4\\sqrt{r \\kappa^2\\left\\{\\log \\log \\left(\\kappa^2 r\\right)+\\log\\frac{4}{\\alpha}+1\\right\\}}.\n$$\nTherefore\n\\begin{equation}\n    \\begin{split}\n        \\mathcal{Q}^*(\\eta+r)-\\mathcal{Q}^*(\\eta) &\\geq r\\kappa^2 - 4\\sqrt{r \\kappa^2\\left\\{\\log \\log \\left(\\kappa^2 r\\right)+\\log\\frac{4}{\\alpha}+1\\right\\}}\\\\\n        &\\geq \n        r\\kappa^2 - 4\\sqrt{r \\kappa^2\\{1\\vee \\log \\log \\left(\\kappa^2 r\\right)\\}}-4\\sqrt{r \\kappa^2 \\log\\frac{4}{\\alpha}}-4\\sqrt{r \\kappa^2}\n    \\end{split}\n\\label{tmp_eq:op1 mean main prop eq 2}\n\\end{equation}\nSince $\\frac{x}{144} - \\log\\log (x)\\geq 0$ for all $x>0$, when $r\\kappa^2\\geq \\max\\{144\\log\\frac{4}{\\alpha},144\\}$, we have $\\mathcal{Q}^*(\\eta+r)-\\mathcal{Q}^*(\\eta)\\geq 0$.\n\n\\textbf{Step 4}. \\Cref{tmp_eq:op1 mean main prop eq 1} and \\Cref{tmp_eq:op1 mean main prop eq 2} together give, uniformly for all $r$ such that $r \\kappa^2 \\geq 144 (1\\vee \\log\\frac{4}{\\alpha})$,\n$$\n 0\\leq r\\kappa^2 - 4\\sqrt{r \\kappa^2\\left\\{\\log \\log \\left(\\kappa^2 r\\right)+\\log\\frac{4}{\\alpha}+1\\right\\}} \\leq C_3 r \\kappa^2 \\frac{\\mathfrak{s} \\log (p)}{\\sqrt{\\Delta\\kappa^2}}.\n$$\nSince we assume that $\\frac{\\mathfrak{s}^2\\log^2(p)}{\\Delta \\kappa^2}\\rightarrow 0$, this either leads to a contradiction or implies that $r\\kappa^2\\leq C_4(1\\vee \\log\\frac{1}{\\alpha})$. \n\\eprf\n\n\n\\bnlem[Local refinement step 1]\n\\label{lem:mean refinement step 1}\nThe output $\\check{\\eta}$ of step 1 of the local refinement satisfies that with probability at least $1 - n^{-3}$,\n\\begin{equation}\n    \\max_{k\\in [K]}|\\check {\\eta}_k - \\eta_k|\\leq \\frac{C \\sigma_{\\epsilon}^2 { \\mathfrak{s} } \\log(p)}{\\kappa^2}.\n\\end{equation}\n\\enlem\n\n\n\n\n\n\n\n\n\n\n\n\n\\begin{proof}[Proof of \\Cref{lem:mean refinement step 1}]\nFor each $k\\in [K]$, let $\\widehat {\\mu}_t = \\widehat {\\mu}^{(1)}$ if $s_k<t< \\check {\\eta}_k$ and $\\widehat {\\mu}_t =\\widehat {\\mu}^{(2)}$ otherwise, and ${\\mu}^*_t$ be the true parameter at time point $t$. First we show that under conditions $\\tilde{K} = K$ and $\\max_{k\\in [K]}|\\tilde{\\eta}_k - \\eta_k|\\leq \\Delta \/ 5$, there is only one true change point $\\eta_k$ in $(s_k, e_k)$. It suffices to show that\n\\begin{equation}\n    |\\tilde{\\eta}_k - \\eta_k|\\leq \\frac{2}{3}(\\tilde{\\eta}_{k+1} - \\tilde{\\eta}_{k}),\\ \\text{and}\\ |\\tilde{\\eta}_{k+1} - \\eta_{k+1}|\\leq \\frac{1}{3}(\\tilde{\\eta}_{k+1} - \\tilde{\\eta}_{k}).\n    \\label{tmp_eq: mean goal_lr}\n\\end{equation}\nDenote $R = \\max_{k\\in [K]}|\\tilde{\\eta}_k - \\eta_k|$, then\n\\begin{align*}\n    \\tilde{\\eta}_{k+1} - \\tilde{\\eta}_{k} &= \\tilde{\\eta}_{k+1} - {\\eta}_{k+1} + {\\eta}_{k+1} - {\\eta}_{k} + {\\eta}_{k} - \\tilde{\\eta}_{k} \\\\\n    &= ({\\eta}_{k+1} - {\\eta}_{k}) + (\\tilde{\\eta}_{k+1} - {\\eta}_{k+1}) + ({\\eta}_{k} - \\tilde{\\eta}_{k})\\in [{\\eta}_{k+1} - {\\eta}_{k} - 2R,{\\eta}_{k+1} - {\\eta}_{k} + 2R].\n\\end{align*}\nTherefore, \\Cref{tmp_eq: mean goal_lr} is guaranteed as long as\n\\begin{equation*}\n    R\\leq \\frac{1}{3}(\\Delta - 2R),\n\\end{equation*}\nwhich is equivalent to $R\\leq \\Delta\/5$.\n\nNow without loss of generality, assume that $s_k<\\eta_k<\\check {\\eta}_k<e_k$. Denote $\\mclI_k = \\{s_k + 1, \\cdots, e_k\\}$. Consider two cases:\n\n\\textbf{Case 1} If\n\\begin{equation*}\n    \\check {\\eta}_k - \\eta_k < \\max\\{C\\sigma_{\\epsilon}^2 { \\mathfrak{s} } \\log(n\\vee p), C\\sigma_{\\epsilon}^2 { \\mathfrak{s} } \\log(n\\vee p)\/\\kappa^2\\},\n\\end{equation*}\nthen the proof is done.\n\n\\textbf{Case 2} If\n\\begin{equation*}\n    \\check {\\eta}_k - \\eta_k \\geq \\max\\{C\\sigma_{\\epsilon}^2 { \\mathfrak{s} } \\log(n\\vee p), C\\sigma_{\\epsilon}^2 { \\mathfrak{s} } \\log(n\\vee p)\/\\kappa^2\\},\n\\end{equation*}\nthen we proceed to prove that $|\\check {\\eta}_k - \\eta_k|\\leq C\\sigma_{\\epsilon}^2 { \\mathfrak{s} } \\log(n\\vee p)\/\\kappa^2$ with probability at least $1 - (Tn)^{-3}$. Then we either prove the result or get an contradiction, and complete the proof in either case.\n\nBy definition, we have\n\\begin{equation*}\n    \\sum_{t\\in \\I}\\|y_t - \\widehat {\\mu}_t\\|_2^2 + \\zeta\\sum_{i = 1}^p \\sqrt{\\sum_{i\\in \\I}(\\widehat {\\mu}_t)_i^2}\\leq \\sum_{t\\in \\I}\\|y_t - {\\mu}^*_t\\|_2^2 + \\zeta\\sum_{i = 1}^p \\sqrt{\\sum_{i\\in \\I}({\\mu}^*_t)_i^2},\n\\end{equation*}\nwhich implies that\n\\begin{equation*}\n    \\sum_{t\\in \\I}\\|\\mu^*_t - \\widehat {\\mu}_t\\|_2^2 + \\zeta\\sum_{i = 1}^p \\sqrt{\\sum_{i\\in \\I}(\\widehat {\\mu}_t)_i^2}\\leq 2\\sum_{t\\in \\I}(y_t - \\mu^*_t)^\\top (\\widehat {\\mu}_t - \\mu^*_t) + \\zeta\\sum_{i = 1}^p \\sqrt{\\sum_{i\\in \\I}({\\mu}^*_t)_i^2}.\n\\end{equation*}\nDenote $\\delta_t = \\widehat {\\mu}_t - \\mu^*_t$. Notice that\n\\begin{align*}\n    & \\sum_{i  \\in [p]} \\sqrt{\\sum_{i\\in \\I}({\\mu}^*_t)_i^2} - \\sum_{i  \\in [p]} \\sqrt{\\sum_{i\\in \\I}(\\widehat {\\mu}_t)_i^2} \\\\ \n    =& \\sum_{i \\in [p]} \\sqrt{\\sum_{i\\in \\I}({\\mu}^*_t)_i^2} - \\sum_{i \\in S} \\sqrt{\\sum_{i\\in \\I}(\\widehat {\\mu}_t)_i^2} - \\sum_{i \\in S^c} \\sqrt{\\sum_{i\\in \\I}(\\widehat {\\mu}_t)_i^2} \\\\\n    \\leq & \\sum_{i \\in S} \\sqrt{\\sum_{i\\in \\I}(\\delta_t)_i^2} - \\sum_{i \\in S^c} \\sqrt{\\sum_{i\\in \\I}(\\delta_t)_i^2}.\n\\end{align*}\nNow we check the cross term. Notice that the variance of $\\sum_{t\\in \\I}(\\epsilon_t)_i(\\delta_t)_i$ is $\\sum_{t\\in \\I}(\\delta_t)_i^2$, so with probability at least $1 - (n\\vee p)^{-5}$,\n\\begin{equation*}\n    \\sum_{t\\in \\I}(y_t - \\mu^*_t)^\\top (\\widehat {\\mu}_t - \\mu^*_t)\\leq C\\sigma_{\\epsilon}\\sqrt{\\log(n\\vee p)}\\sum_{i \\in [p]}\\sqrt{\\sum_{t\\in \\I}(\\delta_t)_i^2}\\leq \\frac{\\zeta}{4}\\sum_{i \\in [p]}\\sqrt{\\sum_{t\\in \\I}(\\delta_t)_i^2},\n\\end{equation*}\nsince $\\zeta = C_{\\zeta}\\sigma_{\\epsilon}\\sqrt{\\log(n\\vee p)}$ with sufficiently large constant $C_{\\zeta}$. Combining inequalities above, we can get\n\\begin{align*}\n    \\sum_{t\\in \\I}\\|\\delta_t\\|_2^2 +\\frac{\\zeta}{2}\\sum_{i\\in S^c}\\sqrt{\\sum_{t\\in \\I}(\\delta_t)_i^2}\n    \\leq & \\frac{3\\zeta}{2}\\sum_{i\\in S}\\sqrt{\\sum_{t\\in \\I}(\\delta_t)_i^2} \\\\\n    \\leq & \\frac{3\\zeta}{2}\\sqrt{ { \\mathfrak{s} } }\\sqrt{\\sum_{t\\in \\I}\\|(\\delta_t)_S\\|_2^2} \\\\\n    \\leq & \\frac{3\\zeta}{2}\\sqrt{ { \\mathfrak{s} } }\\sqrt{\\sum_{t\\in \\I}\\|\\delta_t\\|_2^2},\n\\end{align*}\nwhich implies that\n\\begin{equation}\n    \\sum_{t\\in \\I}\\|\\delta_t\\|_2^2\\leq \\frac{9}{4} { \\mathfrak{s} } \\zeta^2\\leq C { \\mathfrak{s} } \\sigma_{\\epsilon}^2\\log(n\\vee p).\n    \\label{tmp_eq:upper_bound_delta_local_refine}\n\\end{equation}\nWithout loss of generality, assume that $\\check {\\eta}>\\eta_k$ and denote\n$$\n{  \\mathcal J } _1 = [s_k, \\eta_k), {  \\mathcal J } _2 = [\\eta_k, \\check {\\eta}_k), {  \\mathcal J } _3 = [\\check {\\eta}_k, e_k),\n$$\nand $\\mu^{(1)} = \\mu^*_{\\eta_k - 1}$, $\\mu^{(2)} = \\mu^*_{\\eta_k}$. Then \\Cref{tmp_eq:upper_bound_delta_local_refine} is equivalent to\n\\begin{equation*}\n    {  \\mathcal J } _1\\|\\widehat {\\mu}^{(1)} - \\mu^{(1)}\\|_2^2 + {  \\mathcal J } _2\\|\\widehat {\\mu}^{(1)} - \\mu^{(2)}\\|_2^2 + {  \\mathcal J } _3\\|\\widehat {\\mu}^{(2)} - \\mu^{(2)}\\|_2^2 \\leq C { \\mathfrak{s} } \\sigma_{\\epsilon}^2\\log(n\\vee p).\n\\end{equation*}\nSince $|\\mclJ_1|=\\eta_k - s_k\\geq c_0\\Delta$ with some constant $c_0$ under \\Cref{assp: DCDP_mean}, we have\n\\begin{equation}\n    \\Delta \\|\\widehat {\\mu}^{(1)} - \\mu^{(1)}\\|_2^2\\leq c_0|\\mclJ_1|\\|\\widehat {\\mu}^{(1)} - \\mu^{(1)}\\|_2^2 \\leq C \\sigma_{\\epsilon}^2 { \\mathfrak{s} } \\log(n\\vee p)\\leq c_2\\Delta \\kappa^2,\n\\end{equation}\nwith some constant $c_2\\in (0,1\/4)$, where the last inequality is due to the fact that $ { \\mathcal B_n}  \\rightarrow \\infty$. Thus we have\n\\begin{equation*}\n    \\|\\widehat {\\mu}^{(1)} - \\mu^{(1)}\\|_2^2\\leq c_2\\kappa^2.\n\\end{equation*}\nTriangle inequality gives\n\\begin{equation*}\n    \\|\\widehat {\\mu}^{(1)} - \\mu^{(2)}\\|_2\\geq \\|\\mu^{(1)} - \\mu^{(2)}\\|_2 - \\|\\widehat {\\mu}^{(1)} - \\mu^{(1)}\\|_2 \\geq \\kappa\/2.\n\\end{equation*}\nTherefore, $\\kappa^2|\\mclJ_2|\/4\\leq |\\mclJ_2|\\|\\widehat {\\mu}^{(1)} - \\mu^{(2)}\\|_2^2\\leq C \\sigma_{\\epsilon}^2 { \\mathfrak{s} } \\log(n\\vee p)$ and\n\\begin{equation*}\n    |\\check {\\eta}_k - \\eta_k| = |\\mclJ_2|\\leq \\frac{C \\sigma_{\\epsilon}^2 { \\mathfrak{s} } \\log(n\\vee p)}{\\kappa^2}.\n\\end{equation*}\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\\clearpage\n\\subsection{Local refinement in the regression model}\n\\label{sec:regression op1}\nFor the ease of notations, we re-index the observations in the $k$-th interval by $[n_0]:\\{1,\\cdots, n_0\\}$ (though the sample size of the problem is still $n$), and denote the $k$-th jump size as $\\kappa$ and the minimal spacing between consecutive change points as $\\Delta$ (instead of $\\Delta_{\\min}$ in the main text).\n\nBy \\Cref{assp:dcdp_linear_reg} and the setting of the local refinement algorithm, we have\n$$\ny_i=\\left\\{\\begin{array}{ll}\nX_i^{\\top} \\alpha^*+\\epsilon_i & \\text { when } i \\in(0, \\eta] \\\\\nX_i^{\\top} \\beta^*+\\epsilon_i & \\text { when } i \\in(\\eta, n_0]\n\\end{array} .\\right.\n$$\nIn addition, there exists $\\theta \\in(0,1)$ such that $\\eta=\\lfloor n_0 \\theta\\rfloor$ and that $\\|\\alpha^*-\\beta^*\\|_2=\\kappa<\\infty$. By \\Cref{assp:dcdp_linear_reg}, it holds that $\\left\\|\\alpha^*\\right\\|_0 \\leq \\mathfrak{s},\\left\\|\\beta^*\\right\\|_0 \\leq \\mathfrak{s}$, and\n\\begin{equation}\n    \\frac{\\mathfrak{s}^2 \\log ^{3}(n\\vee p)}{\\Delta \\kappa^2} \\rightarrow 0.\n    \\label{op1_eq:regression snr}\n\\end{equation}\nBy \\Cref{lem: regression local refinement step 1}, with probability at least $1 - n^{-2}$, the output of the first step of the PLR algorithm (\\Cref{algo:local_refine_general}) $\\widehat{\\alpha}$ and $\\widehat{\\beta}$ satisfies that\n\\begin{equation}\n    \\begin{split}\n          & \\|\\widehat{\\alpha}-\\alpha^*\\|_2^2\\leq C\\frac{\\mathfrak{s} \\log (n\\vee p)}{\\Delta} \\text{ and } \\|\\widehat{\\alpha}-\\alpha^*\\|_1\\leq C\\mathfrak{s} \\sqrt{\\frac{\\log (n\\vee p)}{\\Delta}}; \\\\\n& \\|\\widehat{\\beta}-\\beta^*\\|_2^2\\leq C\\frac{\\mathfrak{s} \\log (n\\vee p)}{\\Delta} \\text{ and } \\|\\widehat{\\beta}-\\beta^*\\|_1\\leq C \\mathfrak{s} \\sqrt{\\frac{\\log (n\\vee p)}{\\Delta}}.\n    \\end{split}\n    \\label{eq: regression op1 condition}\n\\end{equation}\nLet\n$$\n\\widehat{\\mathcal{Q}}(k)=\\sum_{i=1}^k\\left(y_i-X_i^{\\top} \\widehat{\\alpha}\\right)^2+\\sum_{i=k+1}^{n_0}\\left(y_i-X_i^{\\top} \\widehat{\\beta}\\right)^2 \\quad \\text { and } \\quad \\mathcal{Q}^*(k)=\\sum_{i=1}^k\\left(y_i-X_i^{\\top} \\alpha^*\\right)^2+\\sum_{i=k+1}^{n_0}\\left(y_i-X_i^{\\top} \\beta^*\\right)^2 .\n$$\n\n\\bnlem[Refinement for regression]\nLet\n$$\n\\eta+r=\\underset{k \\in(0, n_0]}{\\arg \\max } \\widehat{\\mathcal{Q}}(k) .\n$$\nThen under the assumptions above, it holds with probability at least $1 - (\\alpha\\vee n^{-1})$ that\n$$\n r\\kappa^2\\leq C \\log^2\\frac{1}{\\alpha}.\n$$\nwhere $C$ is a universal constant that only depends on $C_{\\kappa}$, $\\Lambda_{\\min}$, $\\sigma_{\\epsilon}$.\n\\enlem\n\\bprf\nFor the brevity of notations, we denote $p_n:=n\\vee p$ throughout the proof. Without loss of generality, suppose $r \\geq 0$. Since $\\eta+r$ is the minimizer, it follows that\n$$\n\\widehat{\\mathcal{Q}}(\\eta+r) \\leq \\widehat{\\mathcal{Q}}(\\eta) .\n$$\nIf $r \\leq \\frac{1}{\\kappa^2}$, then there is nothing to show. So for the rest of the argument, for contradiction, assume that\n$$\nr \\geq \\frac{1}{\\kappa^2}\n$$\nObserve that\n$$\n\\begin{aligned}\n\\widehat{\\mathcal{Q}}(t)-\\widehat{\\mathcal{Q}}(\\eta) &=\\sum_{i=\\eta+1}^{\\eta+r}\\left(y_i-X_i^{\\top} \\widehat{\\alpha}\\right)^2-\\sum_{i=\\eta+1}^{\\eta+r}\\left(y_i-X_i^{\\top} \\widehat{\\beta}\\right)^2 \\\\\n\\mathcal{Q}^*(t)-\\mathcal{Q}^*(\\eta) &=\\sum_{i=\\eta+1}^{\\eta+r}\\left(y_i-X_i^{\\top} \\alpha^*\\right)^2-\\sum_{i=\\eta+1}^{\\eta+r}\\left(y_i-X_i^{\\top} \\beta^*\\right)^2\n\\end{aligned}\n$$\n\\textbf{Step 1}. It follows that\n$$\n\\begin{aligned}\n& \\sum_{i=\\eta+1}^{\\eta+r}\\left(y_i-X_i^{\\top} \\widehat{\\alpha}\\right)^2-\\sum_{i=\\eta+1}^{\\eta+r}\\left(y_i-X_i^{\\top} \\alpha^*\\right)^2 \\\\\n=& \\sum_{i=\\eta+1}^{\\eta+r}\\left(X_i^{\\top} \\widehat{\\alpha}-X_i^{\\top} \\alpha^*\\right)^2+2\\left(\\widehat{\\alpha}-\\alpha^*\\right)^{\\top} X_i \\sum_{i=\\eta+1}^{\\eta+r}\\left(y_i-X_i^{\\top} \\alpha^*\\right) \\\\\n=& \\sum_{i=\\eta+1}^{\\eta+r}\\left(X_i^{\\top} \\widehat{\\alpha}-X_i^{\\top} \\alpha^*\\right)^2+2\\left(\\widehat{\\alpha}-\\alpha^*\\right)^{\\top} \\sum_{i=1}^r X_i X_i^{\\top}\\left(\\beta^*-\\alpha^*\\right)+2\\left(\\widehat{\\alpha}-\\alpha^*\\right)^{\\top} \\sum_{i=\\eta+1}^{\\eta+r} X_i \\epsilon_i\n\\end{aligned}\n$$\nBy \\Cref{lem:regression op1 base lemma 1}, uniformly for all $r$,\n$$\n\\left\\|\\frac{1}{r} \\sum_{i=1}^r X_i X_i^{\\top}-\\Sigma\\right\\|_{\\infty} \\leq C\\left(\\sqrt{\\frac{\\log (p_n)}{r}}+\\frac{\\log (p_n)}{r}\\right) .\n$$\nTherefore\n$$\n\\begin{aligned}\n\\sum_{i=\\eta+1}^{\\eta+r}\\left(X_i^{\\top} \\widehat{\\alpha}-X_i^{\\top} \\alpha^*\\right)^2 &=\\sum_{i=\\eta+1}^{\\eta+r}\\left(\\widehat{\\alpha}-\\alpha^*\\right)^{\\top} \\sum_{i=1}^r\\left\\{X_i X_i^{\\top}-\\Sigma\\right\\}\\left(\\widehat{\\alpha}-\\alpha^*\\right)+r\\left(\\widehat{\\alpha}-\\alpha^*\\right)^{\\top} \\Sigma\\left(\\widehat{\\alpha}-\\alpha^*\\right) \\\\\n& \\leq\\|\\widehat{\\alpha}-\\alpha^*\\|_1^2\\left\\|\\sum_{i=1}^r X_i X_i^{\\top}-\\Sigma\\right\\|_{\\infty}+\\Lambda_{\\max } r\\|\\widehat{\\alpha}-\\alpha^*\\|_2^2\\\\\n&\\leq C_1\\frac{\\mathfrak{s}^2 \\log (p_n)}{\\Delta}(\\sqrt{r \\log (p_n)}+\\log (p_n))+C_1 r \\frac{\\mathfrak{s} \\log (p)}{\\Delta} \\\\\n&\\leq  C_1\\sqrt{r}\\frac{\\mathfrak{s}^2 \\log^{3\/2} (p_n)}{\\Delta}+C_1\\frac{\\mathfrak{s}^2 \\log^2 (p_n)}{\\Delta}+C_1 r \\frac{\\mathfrak{s} \\log (p_n)}{\\Delta}\n\\end{aligned}\n$$\nwhere the second inequality follows from \\Cref{lem:regression op1 base lemma 1}. Similarly\n$$\n\\begin{aligned}\n\\left(\\widehat{\\alpha}-\\alpha^*\\right)^{\\top} \\sum_{i=1}^r X_i X_i^{\\top}\\left(\\beta^*-\\alpha^*\\right) &=\\left(\\widehat{\\alpha}-\\alpha^*\\right)^{\\top} \\sum_{i=1}^r\\left\\{X_i X_i^{\\top}-\\Sigma\\right\\}\\left(\\beta^*-\\alpha^*\\right)+r\\left(\\widehat{\\alpha}-\\alpha^*\\right)^{\\top} \\Sigma\\left(\\beta^*-\\alpha^*\\right) \\\\\n& \\leq \\|\\widehat{\\alpha}-\\alpha^*\\|_1\\|\\left(\\beta^*-\\alpha^*\\right)^{\\top}\\left\\{\\sum_{i=1}^r X_i X_i^{\\top}-\\Sigma\\right\\}\\|_{\\infty}+\\Lambda_{\\max } r\\|\\widehat{\\alpha}-\\alpha^*\\|_2\\|\\beta^*-\\alpha^*\\|_2 \\\\\n&\\leq C_2\\mathfrak{s} \\sqrt{\\frac{\\log (p_n)}{\\Delta}} (\\kappa\\sqrt{r \\log (p_n)}+\\kappa\\log (p_n))+C_2 r\\kappa \\sqrt{\\frac{\\mathfrak{s} \\log (p_n)}{\\Delta} } .\\\\\n&\\leq C_2\\mathfrak{s}\\kappa\\log (p_n)\\sqrt{\\frac{r}{\\Delta}} +C_2\\mathfrak{s}\\kappa\\sqrt{\\frac{\\log^3 (p_n)}{\\Delta}} + C_2 r\\kappa \\sqrt{\\frac{\\mathfrak{s} \\log (p_n)}{\\Delta} }.\n\\end{aligned}\n$$\nwhere the second equality follows from $\\|\\beta^*-\\alpha^*\\|_2=\\kappa$ and \\Cref{lem:regression op1 base lemma 2}. In addition,\n$$\n\\begin{aligned}\n&\\left(\\widehat{\\alpha}-\\alpha^*\\right)^{\\top} \\sum_{i=\\eta+1}^{\\eta+r} X_i \\epsilon_i \\leq\\|\\widehat{\\alpha}-\\alpha^*\\|_1\\|\\sum_{i=\\eta+1}^{\\eta+r} X_i \\epsilon_i\\|_{\\infty} \\\\\n\\leq & C_3\\mathfrak{s} \\sqrt{\\frac{\\log (p_n)}{\\Delta}}(\\sqrt{r \\log (p_n)}+\\log (p_n))\\leq C_3\\mathfrak{s}\\log (p_n)\\sqrt{\\frac{r}{\\Delta}} +C_3 { \\mathfrak{s} }  \\sqrt{\\frac{\\log^3 (p_n)}{\\Delta}}.\n\\end{aligned}\n$$\nwhere the second equality follows from \\Cref{lem:regression op1 base lemma 1}. Therefore\n\\begin{align*}\n    &\\sum_{i=\\eta+1}^{\\eta+r}\\left(y_i-X_i^{\\top} \\widehat{\\alpha}\\right)^2-\\sum_{i=\\eta+1}^{\\eta+r}\\left(y_i-X_i^{\\top} \\alpha^*\\right)^2\\\\\n    \\leq & C_4(\\kappa + 1)\\mathfrak{s}\\log (p_n)\\sqrt{\\frac{r}{\\Delta}} + C_4(\\kappa + 1) { \\mathfrak{s} }  \\sqrt{\\frac{\\log^3 (p_n)}{\\Delta}} + C_4 r\\kappa^2 \\sqrt{\\frac{ { \\mathfrak{s} } \\log (p_n)}{\\Delta\\kappa^2}}\\\\\n    & \\quad + C_1\\frac{ { \\mathfrak{s} } ^2\\log^2(p_n)}{\\Delta} + C_1\\sqrt{r}\\frac{ { \\mathfrak{s} } ^2\\log^{3\/2}(p_n)}{\\delta}\\\\\n    \\leq & C_4({\\kappa} + 1)r\\kappa^2\\sqrt{\\frac{\\mathfrak{s}^2\\log^2 (p_n)}{\\Delta\\kappa^2}} + C_4(\\kappa^2 + \\kappa) \\sqrt{\\frac{ { \\mathfrak{s} } ^2\\log^3 (p_n)}{\\Delta\\kappa^2}}  + C_4 {\\kappa}\\sqrt{r\\kappa^2} \\frac{ { \\mathfrak{s} } ^2\\log^{3\/2}(p_n)}{\\Delta \\kappa^2}\\\\\n    \\leq & C_4({\\kappa} + 1)r\\kappa^2\\sqrt{\\frac{\\mathfrak{s}^2\\log^2 (p_n)}{\\Delta\\kappa^2}} + C_4\\kappa(\\kappa + 1 + \\sqrt{r\\kappa^2}) \\sqrt{\\frac{ { \\mathfrak{s} } ^2\\log^3 (p_n)}{\\Delta\\kappa^2}}\\leq C_5(C_\\kappa^2 + 1){r\\kappa^2} \\sqrt{\\frac{ { \\mathfrak{s} } ^2\\log^3 (p_n)}{\\Delta\\kappa^2}}.\n\\end{align*}\nwhere we use the assumption that $\\Delta \\kappa^2 \\geq \\mathcal{B}_n { \\mathfrak{s} } ^2\\log^2(p_n)$, $\\kappa\\leq C_{\\kappa}$, and $r\\kappa^2 \\geq 1$.\n\n\\textbf{Step 2}. Using the same argument as in the previous step, it follows that\n$$\n\\sum_{i=\\eta+1}^{\\eta+r}\\left(y_i-X_i^{\\top} \\widehat{\\beta}\\right)^2-\\sum_{i=\\eta+1}^{\\eta+r}\\left(y_i-X_i^{\\top} \\beta^*\\right)^2\\leq C_5(C_\\kappa^2 + 1){r\\kappa^2} \\sqrt{\\frac{ { \\mathfrak{s} } ^2\\log^3 (p_n)}{\\Delta\\kappa^2}}.\n$$\nTherefore\n\\begin{equation}\n    \\left|\\widehat{\\mathcal{Q}}(\\eta+r)-\\widehat{\\mathcal{Q}}(\\eta)-\\left\\{\\mathcal{Q}^*(\\eta+r)-\\mathcal{Q}^*(\\eta)\\right\\}\\right|\\leq C_5(C_\\kappa^2 + 1){r\\kappa^2} \\sqrt{\\frac{ { \\mathfrak{s} } ^2\\log^3 (p_n)}{\\Delta\\kappa^2}}.\n    \\label{tmp_eq:op1 regression main prop eq 1}\n\\end{equation}\n\\textbf{Step 3}. Observe that\n$$\n\\begin{aligned}\n& \\mathcal{Q}^*(\\eta+r)-\\mathcal{Q}^*(\\eta) \\\\\n=& \\sum_{i=\\eta+1}^{\\eta+r}\\left(y_i-X_i^{\\top} \\alpha^*\\right)^2-\\sum_{i=\\eta+1}^{\\eta+r}\\left(y_i-X_i^{\\top} \\beta^*\\right)^2 \\\\\n=& \\sum_{i=\\eta+1}^{\\eta+r}\\left(X_i^{\\top} \\alpha^*-X_i^{\\top} \\beta^*\\right)^2-2 \\sum_{i=\\eta+1}^{\\eta+r}\\left(y_i-X_i^{\\top} \\beta^*\\right)\\left(X_i^{\\top} \\alpha^*-X_i^{\\top} \\beta^*\\right) \\\\\n=& \\sum_{i=\\eta+1}^{\\eta+r}\\left(\\alpha^*-\\beta^*\\right)^{\\top}\\left\\{X_i^{\\top} X_i-\\Sigma\\right\\}\\left(\\alpha^*-\\beta^*\\right)+r\\left(\\alpha^*-\\beta^*\\right)^{\\top} \\Sigma\\left(\\alpha^*-\\beta^*\\right)-2 \\sum_{i=\\eta+1}^{\\eta+r} \\epsilon_i\\left(X_i^{\\top} \\alpha^*-X_i^{\\top} \\beta^*\\right)\n\\end{aligned}\n$$\nNote that\n$$\nz_i=\\frac{1}{\\kappa^2}\\left(\\alpha^*-\\beta^*\\right)^{\\top}\\left\\{X_i^{\\top} X_i-\\Sigma\\right\\}\\left(\\alpha^*-\\beta^*\\right)\n$$\nis a sub-exponential random variable with bounded $\\psi_1$ norm. Therefore by \\Cref{lem:subexp uniform loglog law}, uniformly for all $r \\geq 1 \/ \\kappa^2$, with probability at least $1 - \\alpha \/ 2$,\n$$\n\\sum_{i=1}^r z_i\\leq 4\\left(\\sqrt{r\\left\\{\\log \\log \\left(\\kappa^2 r\\right) + \\log\\frac{4}{\\alpha}+1\\right\\}}+\\sqrt{r \\kappa^2}\\left\\{\\log \\log \\left(\\kappa^2 r\\right)+\\log\\frac{4}{\\alpha}+1\\right\\}\\right).\n$$\nIt follows that\n\\begin{align*}\n    \\sum_{i=\\eta+1}^{\\eta+r}\\left(\\alpha^*-\\beta^*\\right)^{\\top} &\\left\\{X_i^{\\top} X_i-\\Sigma\\right\\}\\left(\\alpha^*-\\beta^*\\right) \\\\\n    &\\leq 4\\left(\\kappa^2\\sqrt{r\\left\\{\\log \\log \\left(\\kappa^2 r\\right) + \\log\\frac{4}{\\alpha}+1\\right\\}}+\\kappa^3\\sqrt{r}\\left\\{\\log \\log \\left(\\kappa^2 r\\right)+\\log\\frac{4}{\\alpha}+1\\right\\}\\right).\n\\end{align*}\nSimilarly, let\n$$\nw_i=\\frac{1}{\\kappa} \\epsilon_i\\left(X_i^{\\top} \\alpha^*-X_i^{\\top} \\beta^*\\right)\n$$\nThen $\\left\\{w_i\\right\\}_{i=1}^{\\infty}$ are sub-exponential random variables with bounded $\\psi_1$ norm. Therefore by \\Cref{lem:subexp uniform loglog law}, uniformly for all $r \\geq 1 \/ \\kappa^2$,\n$$\n\\sum_{i=1}^r w_i\\leq 4\\left(\\sqrt{r\\left\\{\\log \\log \\left(\\kappa^2 r\\right) +\\log\\frac{4}{\\alpha}+1\\right\\}}+\\sqrt{r \\kappa^2}\\left\\{\\log \\log \\left(\\kappa^2 r\\right)+\\log\\frac{4}{\\alpha}+1\\right\\}\\right)\n$$\nIt follows that\n\\begin{align*}\n    \\sum_{i=\\eta+1}^{\\eta+r} \\epsilon_i &\\left(X_i^{\\top} \\alpha^*-X_i^{\\top} \\beta^*\\right)\\\\\n    &\\leq 4\\left(\\sqrt{r \\kappa^2\\left\\{\\log \\log \\left(\\kappa^2 r\\right)+\\log\\frac{4}{\\alpha}+1\\right\\}}+\\kappa^2\\sqrt{r}\\left\\{\\log \\log \\left(\\kappa^2 r\\right)+\\log\\frac{4}{\\alpha}+1\\right\\}\\right).\n\\end{align*}\nTherefore\n\\begin{equation}\n    \\begin{split}\n        \\mathcal{Q}^*(\\eta+r)-\\mathcal{Q}^*(\\eta) \\geq & \\Lambda_{\\min } r \\kappa^2 - 4(\\kappa + 1)\\sqrt{r \\kappa^2\\left\\{\\log \\log \\left(\\kappa^2 r\\right)+\\log\\frac{4}{\\alpha}+1\\right\\}}\\\\\n        & \\quad - 4(\\kappa^2 + \\kappa)\\sqrt{r \\kappa^2}\\left\\{\\log \\log \\left(\\kappa^2 r\\right)+\\log\\frac{4}{\\alpha}+1\\right\\}\\\\\n        \\geq & \\Lambda_{\\min } r \\kappa^2 - 16\\sqrt{r\\kappa^2}(\\kappa^2\\vee 1)(1 + \\log\\frac{4}{\\alpha} + \\{1\\vee \\log\\log (r\\kappa^2)\\})\\\\\n        \\geq & \\Lambda_{\\min } r\\kappa^2  - 16\\sqrt{r\\kappa^2}(C_{\\kappa}^2\\vee 1)(1 + \\log\\frac{4}{\\alpha} + \\{1\\vee \\log\\log (r\\kappa^2)\\}).\n    \\end{split}\n\\label{tmp_eq:op1 regression main prop eq 2}\n\\end{equation}\nwhere $\\Lambda_{\\min}$ is the minimal eigenvalue of $\\Sigma$. By \\Cref{lem: x - c log^2log x}, for $r\\kappa^2 \\geq \\frac{48^2(C_{\\kappa}^2\\vee 1)^2}{\\Lambda_{\\min}^2} \\vee e^{2e}$, $\\frac{\\Lambda_{\\min}}{3}r\\kappa^2\\geq \\sqrt{r\\kappa^2}\\log\\log (r\\kappa^2)$. Thus, when $r\\kappa^2\\geq (\\frac{48^2(C_{\\kappa}^2\\vee 1)^2}{\\Lambda_{\\min}^2}\\log^2\\frac{4}{\\alpha}) \\vee e^{2e}$, we have $\\mathcal{Q}^*(\\eta+r)-\\mathcal{Q}^*(\\eta)\\geq 0$.\n\n\\textbf{Step 4}. \\Cref{tmp_eq:op1 regression main prop eq 1} and \\Cref{tmp_eq:op1 regression main prop eq 1} together give that, uniformly for all $r$ such that $r\\kappa^2 \\geq (\\frac{48^2(C_{\\kappa}^2\\vee 1)^2}{\\Lambda_{\\min}^2}\\log^2\\frac{4}{\\alpha}) \\vee e^{2e}$, with probability at least $1 - (\\alpha\\vee n^{-1})$\n$$\n\\Lambda_{\\min } r \\kappa^2 - 16\\sqrt{r\\kappa^2}(C_{\\kappa}^2\\vee 1)(1 + \\log\\frac{4}{\\alpha} + \\{1\\vee \\log\\log (r\\kappa^2)\\}) \\leq C_5(C_\\kappa^2 + 1){r\\kappa^2} \\sqrt{\\frac{ { \\mathfrak{s} } ^2\\log^3 (p_n)}{\\Delta\\kappa^2}},\n$$\nwhich either leads to a contradiction or implies the conclusion.\n\\eprf\n\nIn what follows, we first show that the first step in the local refinement gives estimators $\\hat{\\alpha}, \\hat{\\beta}$ that satisfies \\Cref{eq: regression op1 condition}, and then prove some relevant lemmas.\n\n\n\n\n\n\n\n\n\n\n\n\\bnlem[Local refinement step 1]\n\\label{lem: regression local refinement step 1}\nFor each $k\\in [K]$, let $\\check{\\eta}_k, \\widehat{\\beta}^{(1)}, \\widehat{\\beta}^{(2)}$ be the output of step 1 of the local refinement algorithm for linear regression, with\n\\begin{equation*}\n    R(\\theta^{(1)}, \\theta^{(2)},\\eta; s, e) = \\zeta  \\sum_{i \\in [p]} \\sqrt{(\\eta - s)(\\theta^{(1)}_{i} )^2 + (e - \\eta)(\\theta_{i} ^{(2)}) ^2}.\n\\end{equation*}\nand $\\zeta = C_{\\zeta}\\sqrt{\\log (n\\vee p)}$. Then with probability at least $1 - n^{-3}$, it holds that\n\\begin{equation}\n    \\max_{k\\in [\\hat{K}]} |\\check{\\eta}_k - \\eta_k|\\leq C\\frac{ { \\mathfrak{s} }  \\log(n\\vee p)}{\\kappa^2}.\n\\end{equation}\n\\enlem\n\n\n\\begin{proof}[Proof of \\Cref{lem: regression local refinement step 1}]\nFor each $k\\in [K]$, let $\\widehat {\\beta}_t = \\widehat {\\beta}^{(1)}$ if $s_k<t< \\widehat {\\eta}_k$ and $\\widehat {\\beta}_t =\\widehat {\\beta}^{(2)}$ otherwise. Let ${\\beta}^*_t$ be the true parameter at time point $t$, and $\\beta^{(1)} = \\beta^*_{\\eta_k}$ and $\\beta^{(2)} = \\beta^*_{\\eta_k + 1}$. First we show that under conditions $\\tilde{K} = K$ and $\\max_{k\\in [K]}|\\tilde{\\eta}_k - \\eta_k|\\leq \\Delta \/ 5$, there is only one true change point $\\eta_k$ in $(s_k, e_k)$. It suffices to show that\n\\begin{equation}\n    |\\tilde{\\eta}_k - \\eta_k|\\leq \\frac{2}{3}(\\tilde{\\eta}_{k+1} - \\tilde{\\eta}_{k}),\\ \\text{and}\\ |\\tilde{\\eta}_{k+1} - \\eta_{k+1}|\\leq \\frac{1}{3}(\\tilde{\\eta}_{k+1} - \\tilde{\\eta}_{k}).\n    \\label{tmp_eq:goal_lr}\n\\end{equation}\nDenote $R = \\max_{k\\in [K]}|\\tilde{\\eta}_k - \\eta_k|$, then\n\\begin{align*}\n    \\tilde{\\eta}_{k+1} - \\tilde{\\eta}_{k} &= \\tilde{\\eta}_{k+1} - {\\eta}_{k+1} + {\\eta}_{k+1} - {\\eta}_{k} + {\\eta}_{k} - \\tilde{\\eta}_{k} \\\\\n    &= ({\\eta}_{k+1} - {\\eta}_{k}) + (\\tilde{\\eta}_{k+1} - {\\eta}_{k+1}) + ({\\eta}_{k} - \\tilde{\\eta}_{k})\\in [{\\eta}_{k+1} - {\\eta}_{k} - 2R,{\\eta}_{k+1} - {\\eta}_{k} + 2R].\n\\end{align*}\nTherefore, \\Cref{tmp_eq:goal_lr} is guaranteed as long as\n\\begin{equation*}\n    R\\leq \\frac{1}{3}(\\Delta - 2R),\n\\end{equation*}\nwhich is equivalent to $R\\leq \\Delta\/5$.\n\nNow without loss of generality, assume that $s_k<\\eta_k<\\widehat {\\eta}_k<e_k$. Denote $\\mclI_k = \\{s_k + 1, \\cdots, e_k\\}$. Consider two cases:\n\n\\textbf{Case 1.} If\n\\begin{equation*}\n    \\widehat {\\eta}_k - \\eta_k < \\max\\{C_s(\\sigma_{\\epsilon}^2\\vee 1) { \\mathfrak{s} } \\log(n\\vee p), C_s(\\sigma_{\\epsilon}^2\\vee 1) { \\mathfrak{s} } \\log(n\\vee p)\/\\kappa^2\\},\n\\end{equation*}\nthen the proof is done.\n\n\\textbf{Case 2.} If\n\\begin{equation*}\n    \\widehat {\\eta}_k - \\eta_k \\geq \\max\\{C_s(\\sigma_{\\epsilon}^2\\vee 1) { \\mathfrak{s} } \\log(n\\vee p), C_s(\\sigma_{\\epsilon}^2\\vee 1) { \\mathfrak{s} } \\log(n\\vee p)\/\\kappa^2\\},\n\\end{equation*}\nthen we proceed to prove that $|\\widehat {\\eta}_k - \\eta_k|\\leq C(\\sigma_{\\epsilon}^2\\vee 1) { \\mathfrak{s} } \\log(n\\vee p)\/\\kappa^2$ with probability at least $1 - (n\\vee p)^{-5}$. Then we either prove the result or get an contradiction, and complete the proof in either case. The first step is to prove that with probability at least $1 - (n\\vee p)^{-5}$,\n\t\\[\n\t\t\\sum_{t = s_k + 1}^{e_k}\\|\\widehat{\\beta}_t - \\beta^*_t\\|_2^2 \\leq C_1 { \\mathfrak{s} }  \\zeta^2.\n\t\\]\nBy definition, it holds that\n\t\\begin{align}\\label{eq:reg local refine 1}\n\t\t& \\sum_{t = s_k + 1}^{e_k} (y_t - X_t^{\\top}\\widehat{\\beta}_t)^2 + \\zeta \\sum_{i = 1}^p \\sqrt{\\sum_{t = s_k + 1}^{e_k} \\bigl(\\widehat{\\beta}_t\\bigr)_i^2} \\leq \\sum_{t = s_k + 1}^{e_k} (y_{t} - X_t^{\\top}\\beta^*_t )^2 + \\zeta \\sum_{i = 1}^p \\sqrt{\\sum_{t = s_k + 1}^{e_k} \\bigl(\\beta^*_t\\bigr)_{i}^2}.\n\t\\end{align}\n\tLet $\\delta_t = \\widehat{\\beta}_t - \\beta^*_t$. It holds that $\\sum_{t = s_k + 1}^{e_k - 1}\\mathbbm{1}\\left\\{\\delta_t \\neq \\delta_{t+1}\\right\\} = 2$.\tThen \\Cref{eq:reg local refine 1} implies that\n\t\\begin{align}\\label{eq:reg local refine 2}\n\t\t\\sum_{t = s_k + 1}^{e_k} (\\delta_t^{\\top} X_t)^2 + \\zeta  \\sum_{i = 1}^p \\sqrt{\\sum_{t = s_k + 1}^{e_k} \\bigl(\\widehat{\\beta}_t\\bigr)_{i}^2}\\leq 2\\sum_{t = s_k + 1}^{e_k} (y_t - X_t^{\\top}\\beta^*_t)\\delta_t^{\\top} X_t + \\zeta \\sum_{i = 1}^p \\sqrt{\\sum_{t = s_k + 1}^{e_k} \\bigl(\\beta^*_t\\bigr)_{i}^2}.\n\t\\end{align}\nNote that\n\t\\begin{align}\n\t\t& \\sum_{i = 1}^p \\sqrt{\\sum_{t = s_k + 1}^{e_k} \\bigl(\\beta^*_t\\bigr)_{i}^2} - \\sum_{i = 1}^p \\sqrt{\\sum_{t = s_k + 1}^{e_k} \\bigl(\\widehat{\\beta}_t\\bigr)_{i}^2} \\nonumber \\\\\n\t\t=& \\sum_{i \\in S} \\sqrt{\\sum_{t = s_k + 1}^{e_k} \\bigl(\\beta^*_t\\bigr)_{i}^2} - \\sum_{i \\in S} \\sqrt{\\sum_{t = s_k + 1}^{e_k} \\bigl(\\widehat{\\beta}_t\\bigr)_{i}^2}  - \\sum_{i \\in S^c} \\sqrt{\\sum_{t = s_k + 1}^{e_k} \\bigl(\\widehat{\\beta}_t\\bigr)_{i}^2} \\nonumber \\\\\n\t\t\\leq & \\sum_{i \\in S} \\sqrt{\\sum_{t = s_k + 1}^{e_k} \\bigl(\\delta_t\\bigr)_{i}^2}  - \\sum_{i \\in S^c} \\sqrt{\\sum_{t = s_k + 1}^{e_k} \\bigl(\\delta_t\\bigr)_{i}^2}.\\label{eq:reg local refine 3}\n\t\\end{align}\nWe then examine the cross term, with probability at least $1 - (n\\vee p)^{-5}$, which satisfies the following\n\t\\begin{align}\n\t\t& \\left|\\sum_{t = s_k + 1}^{e_k} (y_t - X_t^{\\top}\\beta^*_t)\\delta_t^{\\top} X_t\\right| = \\left|\\sum_{t = s_k + 1}^{e_k} \\epsilon_{t} \\delta_t^{\\top} X_t\\right| = \\sum_{i = 1}^p \\left\\{\\left|\\frac{\\sum_{t = s_k + 1}^{e_k} \\epsilon_{t} (\\delta_t)_i (X_t)_i}{\\sqrt{\\sum_{t = s_k + 1}^{e_k} (\\delta_t)^2_i}}\\right| \\sqrt{\\sum_{t = s_k + 1}^{e_k} (\\delta_t)^2_i}\\right\\} \\nonumber \\\\\n\t\t\\leq & \\sup_{i = 1, \\ldots, p} \\left|\\frac{\\sum_{t = s_k + 1}^{e_k} \\epsilon_{t} (\\delta_t)_i (X_t)_i}{\\sqrt{\\sum_{t = s_k + 1}^{e_k} (\\delta_t)^2_i}}\\right| \\sum_{i = 1}^p \\sqrt{\\sum_{t = s_k + 1}^{e_k} (\\delta_t)^2_i} \\leq (\\zeta\/4) \\sum_{i = 1}^p \\sqrt{\\sum_{t = s_k + 1}^{e_k} (\\delta_t)^2_i}, \\label{eq:reg local refine 4}\n\t\\end{align}\n\twhere the second inequality follows from \\Cref{lem:deviation piecewise constant}.\n\nCombining \\eqref{eq:reg local refine 1}, \\eqref{eq:reg local refine 2}, \\eqref{eq:reg local refine 3} and \\eqref{eq:reg local refine 4} yields\n\t\\begin{equation}\\label{eq:reg local refine 5}\n\t\t\\sum_{t = s_k + 1}^{e_k} (\\delta_t^{\\top} X_t)^2\t + \\frac{\\zeta}{2}\\sum_{i \\in S^c} \\sqrt{\\sum_{t = s_k + 1}^{e_k} \\bigl(\\delta_t\\bigr)_{i}^2} \\leq \\frac{3\\zeta}{2}\\sum_{i \\in S} \\sqrt{\\sum_{t = s_k + 1}^{e_k} \\bigl(\\delta_t\\bigr)_{i}^2}.\n\t\\end{equation}\nNow we are to explore the restricted eigenvalue inequality. Let\n\t\\[\n\t\t\\I_1 = (s_k, \\eta_k], \\quad \\I_2 = (\\eta_k, \\widehat{\\eta}_k], \\quad \\I_3 = (\\widehat{\\eta}_k, e_k].\n\t\\]\nThen for $\\I_1$, it holds that\n\t\\begin{align}\n\t\t& \\eta_k - s_k = \\eta_k - \\frac{2}{3}\\widetilde{\\eta}_k - \\frac{1}{3}\\widetilde{\\eta}_k \\nonumber \\\\\n\t\t= & \\frac{2}{3}(\\eta_k - \\eta_{k-1}) + \\frac{2}{3}(\\widetilde{\\eta}_k - \\eta_k) -  \\frac{2}{3}(\\widetilde{\\eta}_{k-1} - \\eta_{k-1}) + (\\eta_k - \\widetilde{\\eta}_k) \\nonumber \\\\\n\t\t\\geq & \\frac{2}{3}\\Delta - \\frac{1}{3}\\Delta = \\frac{1}{3}\\Delta, \\label{tmp_eq:regression I_1}\n\t\\end{align}\t\nwhere the inequality follows from \\Cref{assp:dcdp_linear_reg} and \\Cref{tmp_eq:goal_lr}. \n\nFor $\\I_3$, by the design of the local refinement algorithm in \\Cref{algo:local_refine_general}, we have $|\\I_3|\\geq C_s  { \\mathfrak{s} } \\log(n\\vee p)$. Since $\\min\\{|\\I_1|,|\\I_3|\\}\\geq C_s { \\mathfrak{s} } \\log(n\\vee p)$, by \\Cref{lem:restricted eigenvalue}, it holds with probability at least $1 - (n\\vee p)^{-5}$ that,\n\t\\begin{align*}\n\t\t& \\sum_{i = 1, 3}\\sum_{t \\in \\I_i}\\|\\delta_{\\I_i}^{\\top}X_t\\|_2^2 \\\\\n\t\t\\geq & \\sum_{i = 1, 3} \\left(c'_1\\sqrt{|\\I_i|} \\|\\delta_{\\I_i}\\|_2 - c_2 \\sqrt{\\log(p)} \\|\\delta_{\\I_i}\\|_1\\right)^2 \\\\\n\t\t\\geq & \\sum_{i = 1, 3} \\left(c_1\\sqrt{|\\I_i|} \\|\\delta_{\\I_i}\\|_2 - c_2 \\sqrt{\\log(p)} \\|(\\delta_{\\I_i})_{S^c}\\|_1\\right)^2,\n\t\\end{align*}\n\twhere the last inequality follows from $\\|(\\delta_{\\I})_S\\|_1\\leq \\sqrt{ { \\mathfrak{s} } }\\|\\delta_{\\I}\\|_2$ and the fact that $\\min\\{|\\I_1|,|\\I_3|\\}>C_s  { \\mathfrak{s} }  \\log(n\\vee p)$. Similarly, since $|\\I_2| > \\Delta> C_s  { \\mathfrak{s} }  \\log(n\\vee p)$, we have\n\t\\[\n\t\t\\sqrt{\\sum_{t \\in \\I_2}(\\delta_{\\I_2}^{\\top}X_t)^2} \\geq {c_1\\sqrt{|\\I_2|}} \\|\\delta_{\\I_2}\\|_2 - c_2 \\sqrt{\\log(p)} \\|\\delta_{\\I_2}(S^c)\\|_1.\n\t\\]\nDenote $n_0 = C_s { \\mathfrak{s} } \\log(n\\vee p)$. We first bound the terms with $\\|\\cdot\\|_1$. Note that\n\t\\begin{align*}\n\t&\\sum_{i = 1}^3\\sum_{j\\in S^c}|(\\delta_{\\I_i})_j|\n\t\t\\leq \\sqrt{3}\\sqrt{\\sum_{i = 1}^3 (\\sum_{j \\in S^c} |(\\delta_{\\I_i})_j|)^2 } \\\\ \n\t\t\\leq & \\sqrt{3}\\sqrt{\\sum_{i = 1}^3{\\frac{|\\I_i|}{n_0}} (\\sum_{j \\in S^c}|(\\delta_{\\I_i})_j| )^2}\t\n\t\t\\leq  \\sqrt{\\frac{3}{n_0}} \\sum_{i = 1}^3   \\sqrt{|\\I_i|}(\\sum_{j \\in S^c}|(\\delta_{\\I_i})_j| ) \\\\\n\t\t\\leq &\n\t\t\\sqrt{\\frac{3}{n_0}} \\sum_{j \\in S^c}  \\sqrt{\\sum_{t = s_k + 1}^{e_k} (\\delta_t)_j^2}\n\t\t\\leq \\frac{3\\sqrt{3}}{\\sqrt{n_0}}\\sum_{j \\in S}  \\sqrt{\\sum_{t = s_k + 1}^{e_k} (\\delta_t)_j^2}\n\t\t\\\\\n\t\t\\leq & \\frac{3\\sqrt{3}}{\\sqrt{n_0}} \\sqrt{ { \\mathfrak{s} }  \\sum_{j \\in S} \\sum_{t = s_k + 1}^{e_k} (\\delta_t)_j^2} \\leq \\frac{c}{ \\sqrt{\\log(n\\vee p)}} \\sqrt{\\sum_{t = s_k + 1}^{e_k} \\|\\delta_t\\|_2^2}.\n\t\\end{align*}\nTherefore,\n\t\\begin{align*}\n\t\t& c_1\\sqrt{\\sum_{t = s_k + 1}^{e_k} \\|\\delta_t\\|_2^2} - \\frac{c_2}{ \\sqrt{\\log(n\\vee p)}} \\sqrt{\\sum_{t = s_k + 1}^{e_k} \\|\\delta_t\\|_2^2} \\\\\n\t\t\\leq & \\sum_{i = 1}^3 c_1 \\|\\delta_{I_i}\\|_2 - \\frac{c_2}{ \\sqrt{\\log(n\\vee p)}} \\sqrt{\\sum_{t = s_k + 1}^{e_k} \\|\\delta_t\\|_2^2} \\leq \\sqrt{3} \\sqrt{ \\sum_{t = s_k + 1}^{e_k} (\\delta_t^{\\top} X_t)^2 } \\\\\n\t\t\\leq & \\frac{3\\sqrt{\\zeta}}{\\sqrt{2}} { \\mathfrak{s} } ^{1\/4} \\left(\\sum_{t = s_k + 1}^{e_k} \\|\\delta_t\\|_2^2\\right)^{1\/4} \\leq \\frac{9\\zeta  { \\mathfrak{s} } ^{1\/2}}{4c_1} + \\frac{c_1}{2}\\sqrt{\\sum_{t = s_k + 1}^{e_k} \\|\\delta_t\\|_2^2}\n\t\\end{align*}\n\twhere the third inequality follows from \\eqref{eq:reg local refine 5} and the fact that $\\sum_{i\\in S}\\sqrt{\\sum_{t=s_k + 1}^{e_k}(\\delta_t)_i^2}\\leq \\sqrt{s}\\sqrt{\\sum_{t=s_k + 1}^{e_k}\\|\\delta_t\\|_2^2}$. The inequality above implies that\n\t\\[\n\t\t\\frac{c_1}{4}\\sqrt{\\sum_{t = s_k + 1}^{e_k} \\|\\delta_t\\|_2^2} \\leq \\frac{9\\zeta  { \\mathfrak{s} } ^{1\/2}}{4c_1}\n\t\\]\t\n\tTherefore,\n\t\\[\n\t\t\\sum_{t = s_k + 1}^{e_k}\\|\\widehat{\\beta}_t - \\beta^*_t\\|_2^2 \\leq 81\\zeta^2 { \\mathfrak{s} } \/c_1^4.\n\t\\]\n  Recall that $\\beta^{(1)} = \\beta^*_{\\eta_k}$ and $\\beta^{(2)} = \\beta^*_{\\eta_k + 1}$. We have that\n\t\\[\n\t\t\\sum_{t = s_k + 1}^{e_k}\\|\\widehat{\\beta}_t - \\beta^*_t\\|_2^2 = |I_1| \\|\\beta^{(1)} - \\widehat{\\beta}^{(1)}\\|_2^2 + |I_2| \\|\\beta^{(2)} - \\widehat{\\beta}^{(1)}\\|_2^2 + |I_3| \\|\\beta^{(2)} - \\widehat{\\beta}^{(2)}\\|_2^2.\n\t\\]\nSince $\\eta_k - s_k \\geq\\frac{1}{3}\\Delta$ as is shown in \\Cref{tmp_eq:regression I_1}. we have that\n\t\\[\n\t\t\\Delta\\|\\beta^{(1)} - \\widehat{\\beta}^{(1)}\\|_2^2\/3 \\leq |I_1| \\|\\beta^{(1)} - \\widehat{\\beta}^{(1)}\\|_2^2 \\leq \\frac{C_1 C_{\\zeta}^2\\Delta \\kappa^2} { { \\mathfrak{s} }  K \\sigma^2_{\\epsilon}  { \\mathcal B_n}  } \\leq c_3 \\Delta \\kappa^2,\n\t\\]\n\twhere $1\/4 > c_3 > 0$ is an arbitrarily small positive constant.  Therefore we have\n\t\\[\n\t\t\\|\\beta^{(2)} - \\widehat{\\beta}_1\\|_2^2 \\leq 3c_3 \\kappa^2.\n\t\\]\nIn addition we have\n\t\\[\n\t\t\\|\\beta^{(2)} - \\widehat{\\beta}^{(1)}\\|_2 \\geq \\|\\beta^{(2)} - \\beta^{(1)}\\|_2 - \\|\\beta^{(1)} - \\widehat{\\beta}^{(1)}\\|_2 \\geq \\kappa\/2.\n\t\\]\t\n\tTherefore, it holds that \t\n\t\\[\n\t\t\\kappa^2 |I_2|\/4 \\leq |I_2| \\|\\beta^{(2)} - \\widehat{\\beta}^{(1)}\\|_2^2 \\leq C_2 { \\mathfrak{s} } \\zeta^2,\n\t\\]\n\twhich implies that \n\t\\[\n\t\t|\\widehat{\\eta}_k - \\eta_k| \\leq \\frac{4C_2 { \\mathfrak{s} }  \\zeta^2}{\\kappa^2},\n\t\\]\nwhich gives the bound we want. \n\\end{proof}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\bnlem\n\\label{lem:regression op1 base lemma 1}\nSuppose $\\left\\{X_i\\right\\}_{i=1}^n \\stackrel{i . i . d}{\\sim} N_p(0, \\Sigma)$ and $\\left\\{\\epsilon_i\\right\\}_{i=1}^n \\stackrel{i i . d}{\\sim} N\\left(0, \\sigma^2\\right)$. Then it holds that\n$$\n\\begin{aligned}\n&\\mathbb{P}\\left(\\|\\frac{1}{r} \\sum_{i=1}^r X_i X_i^{\\top}-\\Sigma\\|_{\\infty} \\geq C_1\\left(\\sqrt{\\frac{\\log (p\\vee n)}{r}}+\\frac{\\log (p\\vee n)}{r}\\right) \\text { for all } 1 \\leq r \\leq n\\right) \\leq (n\\vee p)^{-2}, \\\\\n&\\mathbb{P}\\left(\\|\\frac{1}{r} \\sum_{i=1}^r X_i \\epsilon_i\\|_{\\infty} \\geq C_2\\left(\\sqrt{\\frac{\\log (p\\vee n)}{r}}+\\frac{\\log (p\\vee n)}{r}\\right) \\text { for all } 1 \\leq r \\leq n\\right) \\leq (n\\vee p)^{-2}\n\\end{aligned}\n$$\n\\enlem\n\\bprf\nProof. For the first probability bound, observe that for any $j, k \\in[1, \\ldots, p], X_j X_k-\\Sigma_{j k}$ is subexponential random variable. Therefore for any $r>0$,\n$$\n\\mathbb{P}\\left\\{\\left|\\frac{1}{r} \\sum_{i=1}^r X_{i j} X_{i k}-\\Sigma_{j k}\\right| \\geq x\\right\\} \\leq \\exp \\left(-r c_1 x^2\\right)+\\exp \\left(-r c_2 x\\right)\n$$\nSo\n$$\n\\mathbb{P}\\left\\{\\left\\|\\frac{1}{r} \\sum_{i=1}^r X_{i j} X_{i k}-\\Sigma_{j k}\\right\\|_{\\infty} \\geq x\\right\\} \\leq p \\exp \\left(-r c_1 x^2\\right)+p \\exp \\left(-r c_2 x\\right) .\n$$\nThis gives, for sufficiently large $C_1>0$,\n$$\n\\mathbb{P}\\left\\{\\left\\|\\frac{1}{r} \\sum_{i=1}^r X_{i j} X_{i k}-\\Sigma_{j k}\\right\\|_{\\infty} \\geq C_1\\left(\\sqrt{\\frac{\\log (p\\vee n)}{r}}+\\frac{\\log (p\\vee n)}{r}\\right)\\right\\} \\leq (n\\vee p)^{-3} .\n$$\nBy a union bound,\n$$\n\\mathbb{P}\\left\\{\\left\\|\\frac{1}{r} \\sum_{i=1}^r X_{i j} X_{i k}-\\Sigma_{j k}\\right\\|_{\\infty} \\geq C_1\\left(\\sqrt{\\frac{\\log (p\\vee n)}{r}}+\\frac{\\log (p\\vee n)}{r}\\right) \\text { for all } 1 \\leq r \\leq n\\right\\} \\leq (n\\vee p)^{-2}\n$$\nThe desired result follows from the assumption that $p \\geq n^\\alpha$. The second probability bound follows from the same argument and therefore is omitted for brevity.\n\\eprf\n\n\n\n\n\\bnlem\n\\label{lem:regression op1 base lemma 2}\n Suppose $\\left\\{X_i\\right\\}_{i=1}^n \\stackrel{i.i.d.}{\\sim} N_p(0, \\Sigma)$ and $u \\in \\mathbb{R}^p$ is a deterministic vector such that $|u|_2=1$. Then it holds that\n$$\n\\mathbb{P}\\left(\\|u^{\\top}\\left\\{\\frac{1}{r} \\sum_{i=1}^r X_i X_i^{\\top}-\\Sigma\\right\\}\\|_{\\infty} \\geq C_1\\left(\\sqrt{\\frac{\\log (p\\vee n)}{r}}+\\frac{\\log (p\\vee n)}{r}\\right) \\text { for all } 1 \\leq r \\leq n\\right) \\leq (n\\vee p)^{-2} .\n$$\n\\enlem\n\\bprf\nFor fixed $j \\in[1, \\ldots, p]$, let\n$$\nz_i=u^{\\top} X_i X_{i j}-u^{\\top} \\Sigma_{\\cdot j},\n$$\nwhere $\\Sigma_{\\cdot j}$ denote the $j$-th column of $\\Sigma$. Note that $z_i$ is a sub-exponential random variable with bounded $\\psi_1$ norm. The desired result follows from the same argument as \\Cref{lem:regression op1 base lemma 1}.\n\\eprf\n\n\n\n\\bnlem\n\\label{lem: x - c log^2log x}\nGiven a fixed constant $c>0$, for $x\\geq c^2\\vee e^{2e}$, it holds that\n$$\nx\\geq c(\\log\\log x)^2.\n$$\n\\enlem\n\\bprf\nLet $f(x)  =x- c(\\log\\log x)^2$ for $x>1$. We have $f'(x) = 1 - \\frac{2c\\log\\log x}{x\\log x}$. Therefore, when $x\\geq (2c)\\vee e^e$, $f'(x)> 0$. Let $x_0= c^2 \\vee e^{2e}$, and then\n$$\nf(x_0)\\geq ce^e - c\\log\\log e^{2e}= c[e^e - \\log 2 - 1]>0,\n$$\nand thus $f(x)>0$ for $x\\geq x_0 = c^2 \\vee e^{2e}$.\n\\eprf\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\clearpage\n\\subsection{Local refinement in the Gaussian graphical model}\n\\label{sec:cov op1}\nFor the ease of notations, we re-index the observations in the $k$-th interval by $[n_0]:\\{1,\\cdots, n_0\\}$ (though the sample size of the problem is still $n$), and denote the $k$-th jump size as $\\kappa$ and the minimal spacing between consecutive change points as $\\Delta$ (instead of $\\Delta_{\\min}$ in the main text).\n\nBy \\Cref{assp:DCDP_covariance main} and the setting of the local refinement algorithm, we have for some $G^*,H^*\\in \\mathbb{S}_+^{p}$ that\n$$\n\\mathbb{E}[X_iX_i^\\top]=\\left\\{\\begin{array}{ll}\nG^* & \\text { when } i \\in(0, \\eta] \\\\\nH^* & \\text { when } i \\in(\\eta, n_0]\n\\end{array} .\\right.\n$$\nIn addition, there exists $\\theta \\in(0,1)$ such that $\\eta=\\lfloor n_0 \\theta\\rfloor$ and that $\\|G^*-H^*\\|_F=\\kappa_{F}<\\infty$. By \\Cref{assp:DCDP_covariance main}, it holds that $c_XI_d \\preceq G^* \\preceq C_X I_d$, $c_XI_d \\preceq H^* \\preceq C_X I_d$, and\n\\begin{equation}\n    \\frac{p^4\\log^2(p_n)}{\\Delta\\kappa_F^2}\\rightarrow 0, \\ \\frac{p^5\\log^3(p_n)}{\\Delta}\\rightarrow 0\n    \\label{op1_eq:cov snr}\n\\end{equation}\nBy \\Cref{lem:estimation covariance 2}, there exist $\\widehat{G},\\widehat{H}$ such that\n\\begin{equation}\n    \\begin{split}\n    & \\|\\widehat{G}-G^*\\|_{op}\\leq C\\sqrt{\\frac{p \\log (n\\vee p)}{\\Delta}} \\text{ and } \\|\\widehat{G}-G^*\\|_{F}\\leq C p \\sqrt{\\frac{\\log (n\\vee p)}{\\Delta}}; \\\\\n& \\|\\widehat{H}-H^*\\|_{op}\\leq C\\sqrt{\\frac{p \\log (n\\vee p)}{\\Delta}} \\text{ and } \\|\\widehat{H}-H^*\\|_F\\leq C p \\sqrt{\\frac{\\log (n\\vee p)}{\\Delta}}.\n    \\end{split}\n\\end{equation}\nLet\n$$\n\\widehat{\\mathcal{Q}}(k)=\\sum_{i=1}^k\\|X_iX_i^\\top- \\widehat{G}\\|_F^2+\\sum_{i=k+1}^{n_0}\\|X_iX_i^\\top-\\widehat{H}\\|_F^2 \\quad \\text { and } \\quad \\mathcal{Q}^*(k)=\\sum_{i=1}^k\\|X_iX_i^\\top- G^*\\|_F^2+\\sum_{i=k+1}^{n_0}\\|X_iX_i^\\top- H^*\\|_F^2 .\n$$\nThrough out this section, we use $\\kappa_{F} = \\|G^*-H^*\\|_F$ to measure the signal.\n\\bnlem[Refinement for covariance model]\nLet\n$$\n\\eta+r=\\underset{k \\in(0, n_0]}{\\arg \\max } \\widehat{\\mathcal{Q}}(k) .\n$$\nThen under the assumptions above, it holds that\n$$\n\\kappa_{F}^2 r=O_P(\\log(n)) .\n$$\n\\enlem\n\\bprf\nFor the brevity of notations, we denote $n\\vee p$ as $p_n$ throughout the proof. Without loss of generality, suppose $r \\geq 0$. Since $\\eta+r$ is the minimizer, it follows that\n$$\n\\widehat{\\mathcal{Q}}(\\eta+r) \\leq \\widehat{\\mathcal{Q}}(\\eta) .\n$$\nIf $r \\leq C\\frac{\\log(n)}{\\kappa_{F}^2}$, then there is nothing to show. So for the rest of the argument, for contradiction, assume that\n$$\nr \\geq C\\frac{\\log(n)}{\\kappa_{F}^2}\n$$\nObserve that\n$$\n\\begin{aligned}\n\\widehat{\\mathcal{Q}}(t)-\\widehat{\\mathcal{Q}}(\\eta) &=\\sum_{i=\\eta+1}^{\\eta+r}\\|X_iX_i^\\top- \\widehat{G}\\|_F^2-\\sum_{i=\\eta+1}^{\\eta+r}\\|X_iX_i^\\top- \\widehat{H}\\|_F^2 \\\\\n\\mathcal{Q}^*(t)-\\mathcal{Q}^*(\\eta) &=\\sum_{i=\\eta+1}^{\\eta+r}\\|X_iX_i^\\top- {G}^*\\|_F^2-\\sum_{i=\\eta+1}^{\\eta+r}\\|X_iX_i^\\top-{H}^*\\|_F^2\n\\end{aligned}\n$$\n\\textbf{Step 1}. It follows that\n$$\n\\begin{aligned}\n& \\sum_{i=\\eta+1}^{\\eta+r}\\|X_iX_i^\\top- \\widehat{G}\\|_F^2-\\sum_{i=\\eta+1}^{\\eta+r}\\|X_iX_i^\\top- {G}^*\\|_F^2 \\\\\n=& \\sum_{i=\\eta+1}^{\\eta+r}\\|\\widehat{G}- G^*\\|_F^2+2\\left\\langle G^*-\\widehat{G}, \\sum_{i=\\eta+1}^{\\eta+r}(X_iX_i^\\top-G^*)\\right\\rangle \\\\\n=& r\\|\\widehat{G}- G^*\\|_F^2+2r \\left\\langle G^* -\\widehat{G}, H^*-G^*\\right\\rangle +2\\left\\langle G^*-\\widehat{G}, \\sum_{i=\\eta+1}^{\\eta+r}(X_iX_i^\\top -H^*)\\right\\rangle\n\\end{aligned}\n$$\nBy assumptions, we have\n$$\nr\\|\\widehat{G}- G^*\\|_F^2 \\leq C_1 r \\frac{p^2 \\log (p_n)}{\\Delta}.\n$$\nSimilarly\n$$\nr\\left\\langle G^* -\\widehat{G}, H^*-G^*\\right\\rangle\\leq r\\|G^* -\\widehat{G}\\|_F\\| H^*-G^* \\|_F \\leq C_2 r\\kappa_{F} p\\sqrt{\\frac{ \\log (p_n)}{\\Delta} },\n$$\nwhere the second equality follows from $\\|G^*-H^*\\|_F=\\kappa_{F}$, and the last equality follows from \\eqref{op1_eq:cov snr}. In addition,\n$$\n\\begin{aligned}\n&\\left\\langle G^*-\\widehat{G}, \\sum_{i=\\eta+1}^{\\eta+r}(X_iX_i^\\top -H^*)\\right\\rangle \\leq\\| G^*-\\widehat{G}\\|_F\\|\\sum_{i=\\eta+1}^{\\eta+r}(X_iX_i^\\top -H^*)\\|_{F} \\\\\n\\leq & C_3 p \\sqrt{\\frac{\\log (p_n)}{\\Delta}} (p\\sqrt{r\\log (p_n)}+ p^{3\/2}\\log (p_n)) \\leq C_3 p^2 \\log (p_n) \\sqrt{\\frac{r}{\\Delta}} + C_3 p^2  \\sqrt{\\frac{p \\log^3 (p_n)}{\\Delta}}.\n\\end{aligned}\n$$\nTherefore\n\\begin{align*}\n    & \\sum_{i=\\eta+1}^{\\eta+r}\\|X_iX_i^\\top- \\widehat{G}\\|_F^2-\\sum_{i=\\eta+1}^{\\eta+r}\\|X_iX_i^\\top- {G}^*\\|_F^2\\\\\n    \\leq & C_1 p^2\\log(p_n) \\frac{r}{\\Delta} + C_2r\\kappa_F p\\sqrt{\\frac{\\log(p_n)}{\\Delta}} + C_3 p^2 \\log (p_n) \\sqrt{\\frac{r}{\\Delta}} + C_3 p^2  \\sqrt{\\frac{p \\log^3 (p_n)}{\\Delta}}\\\\\n    \\leq & C_4 r\\kappa_F^2\\sqrt{\\frac{p^4\\log^2(p_n)}{\\Delta\\kappa_F^2}} + C_4 \\sqrt{\\frac{p^5\\log^3(p_n)}{\\Delta}}.\n\\end{align*}\n\\textbf{Step 2}. Using the same argument as in the previous step, it follows that\n$$\n\\sum_{i=\\eta+1}^{\\eta+r}\\|X_iX_i^\\top- \\widehat{H}\\|_F^2-\\sum_{i=\\eta+1}^{\\eta+r}\\|X_iX_i^\\top- {H}^*\\|_F^2\\leq C_4 r\\kappa_F^2\\sqrt{\\frac{p^4\\log^2(p_n)}{\\Delta\\kappa_F^2}} + C_4 \\sqrt{\\frac{p^5\\log^3(p_n)}{\\Delta}}.\n$$\nTherefore\n\\begin{equation}\n    \\left|\\widehat{\\mathcal{Q}}(\\eta+r)-\\widehat{\\mathcal{Q}}(\\eta)-\\left\\{\\mathcal{Q}^*(\\eta+r)-\\mathcal{Q}^*(\\eta)\\right\\}\\right|\\leq C_4 r\\kappa_F^2\\sqrt{\\frac{p^4\\log^2(p_n)}{\\Delta\\kappa_F^2}} + C_4 \\sqrt{\\frac{p^5\\log^3(p_n)}{\\Delta}}.\n    \\label{tmp_eq:op1 cov main prop eq 1}\n\\end{equation}\n\\textbf{Step 3}. Observe that\n$$\n\\begin{aligned}\n\\mathcal{Q}^*(\\eta+r)-\\mathcal{Q}^*(\\eta) \n=&\\sum_{i=\\eta+1}^{\\eta+r}\\|X_iX_i^\\top- {G}^*\\|_F^2-\\sum_{i=\\eta+1}^{\\eta+r}\\|X_iX_i^\\top- {H}^*\\|_F^2 \\\\\n=& r\\| G^*- H^*\\|_F^2-2 \\left\\langle H^*- G^*, \\sum_{i=\\eta+1}^{\\eta+r}(X_iX_i^\\top- H^*) \\right\\rangle\n\\end{aligned}\n$$\nDenote $D^* = H^*- G^*$, then we can write the noise term as\n$$\n\\left\\langle H^*- G^*, X_iX_i^\\top- H^* \\right\\rangle = X_i^\\top D^* X_i - \\mathbb{E}[X_i^\\top D^* X_i].\n$$\nSince $X_i$'s are Gaussian, denote $\\Sigma_{i} = \\mathbb{E}[X_iX_i^\\top] =U_i^\\top \\Lambda_i U_i$, then\n$$\n\\left\\langle H^*- G^*, \\sum_{i=\\eta+1}^{\\eta+r}(X_iX_i^\\top- H^*) \\right\\rangle = Z^\\top \\tilde{D} Z^\\top - \\mathbb{E}[Z^\\top \\tilde{D} Z^\\top],\n$$\nwhere $Z\\in \\mathbb{R}^{rd}$ is a standard Gaussian vector and \n$$\n\\widetilde{D} = {\\rm diag}\\{U_1D^*U_1^\\top,U_2D^*U_2^\\top,\\cdots, U_r D^*U_r^\\top\\}.\n$$\nSince $\\|\\widetilde{D}\\|_F = r\\kappa_F^2$, by Hanson-Wright inequality, with probability at least $1 - n^{-3}$, it holds uniformly for all $r\\geq C\\frac{\\log(n)}{\\kappa_{F}^2}$ that\n$$\n|\\left\\langle H^*- G^*, \\sum_{i=\\eta+1}^{\\eta+r}(X_iX_i^\\top- H^*) \\right\\rangle| \\leq C_5 \\|X\\|_{\\psi_2}^2\\sqrt{r}\\kappa_{F}\\log(r\\kappa_{F}^2).\n$$\nTherefore, by Hanson-Wright inequality, uniformly for all $r\\geq C \\frac{\\log(n)}{\\kappa_{F}^2}$ it holds that\n\\begin{equation}\n    \\mathcal{Q}^*(\\eta+r)-\\mathcal{Q}^*(\\eta) \\geq r \\kappa_{F}^2 - C_5 \\|X\\|_{\\psi_2}^2\\sqrt{r}\\kappa_{F}\\log(r\\kappa_{F}^2),\n\\label{tmp_eq:op1 cov main prop eq 2}\n\\end{equation}\nand thus when $r\\geq C (\\|X\\|_{\\psi_2}^4\\vee 1)\\frac{\\log(n)}{\\kappa_{F}^2}$, $\\mathcal{Q}^*(\\eta+r)-\\mathcal{Q}^*(\\eta)\\geq 0$.\n\n\\textbf{Step 4}. \\Cref{tmp_eq:op1 cov main prop eq 1} and \\Cref{tmp_eq:op1 cov main prop eq 2} together give, uniformly for all $r \\geq C\\log(n) \/ \\kappa_{F}^2$,\n$$\n  r \\kappa_{F}^2 - C_5 \\|X\\|_{\\psi_2}^2\\sqrt{r}\\kappa_{F}\\log(r\\kappa_{F}^2) \\leq C_4 r\\kappa_F^2\\sqrt{\\frac{p^4\\log^2(p_n)}{\\Delta\\kappa_F^2}} + C_4 \\sqrt{\\frac{p^5\\log^3(p_n)}{\\Delta}},\n$$\nwhich either leads to a contradiction or proves the conclusion since we assume that $\\frac{p^4\\log^2(p_n)}{\\Delta\\kappa_F^2}\\rightarrow 0$ and $\\frac{p^5\\log^3(p_n)}{\\Delta}\\rightarrow 0$.\n\\eprf\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\bnlem\n\\label{lem:estimation covariance 2}\nLet $\\{X_i\\}_{i\\in [n]}$ be a sequence of subgaussian vectors in $\\mathbb{R}^{d}$ with orlitz norm upper bounded $\\|X\\|_{\\psi_2}<\\infty$. Suppose $\\mathbb{E}[X_i] = 0$ and $\\mathbb{E}[X_iX_i^\\top] = \\Sigma$ for $i\\in [n]$. Let $\\widehat{\\Sigma}_{n} = \\frac{1}{n}\\sum_{i \\in [n]} X_i X_i^\\top$. Then for any $u>0$, it holds with probability at least $1 - \\exp(-u)$ that\n\\begin{equation}\n    \\|\\widehat{\\Sigma}_{n} - \\Sigma\\|_{op}\\lesssim \\|X\\|_{\\psi_2}^2(\\sqrt{\\frac{d + u}{n}}\\vee \\frac{d + u}{n}).\n\\end{equation}\n\\enlem\n\\bprf\nThis is the same as \\Cref{lem:estimation covariance}.\n\\eprf\n\n\n\n\\bnlem[Hanson-Wright inequality]\nLet $X=\\left(X_1, \\ldots, X_n\\right) \\in \\mathbb{R}^n$ be a random vector with independent, mean zero, sub-gaussian coordinates. Let $A$ be an $n \\times n$ matrix. Then, for every $t \\geq 0$, we have\n$$\n\\mathbb{P}\\left\\{\\left|X^{\\top} A X-\\mathbb{E} X^{\\top} A X\\right| \\geq t\\right\\} \\leq 2 \\exp \\left[-c \\min \\left(\\frac{t^2}{K^4\\|A\\|_F^2}, \\frac{t}{K^2\\|A\\|_{op}}\\right)\\right],\n$$\nwhere $K=\\max _i\\left\\|X_i\\right\\|_{\\psi_2}$\n\\enlem\n\\bprf\nSee \\cite{vershynin2018high} for a proof and \\cite{hanson-wright-general} for a generalization to random vectors with dependence.\n\\eprf\n\n\n\n\n\n\n\\bncor[Local refinement step 1]\n\\label{lem: covariance local refinement step 1}\nUnder \\Cref{assp:DCDP_covariance main}, let $\\{\\widetilde{\\eta}_k\\}_{k \\in [\\widetilde {K}]}$ be a set of time points satisfying\n\t\\begin{equation}\n\t\t\\max_{k \\in [K]} |\\widetilde{\\eta}_k - \\eta_k| \\leq \\Delta\/5.\n\t\\end{equation}\n\tLet $\\{\\check{\\eta}_k\\}_{k\\in [\\widehat{K}]}$ be the change point estimators generated from step 1 of the local refinement algorithm with $\\{\\widetilde{\\eta}_k\\}_{k\\in [\\widehat{K}]}$ as inputs and the penalty function $R(\\cdot) = 0$. Then\n with probability at least $1 - C n ^{-3}$, $\\widehat K = K$ and that\n \\[ \n \\max_{ k\\in [K] } |\\check {\\eta}_k - \\eta_k|   \\lesssim   \\frac{\\|X\\|_{\\psi_2}^4}{c_{X}^4}\\frac{ p^2\\log(n\\vee p)}{\\kappa^2}.\n \\]\n\\encor\n\\bprf\nDenote $\\I = (s_k, e_k)$ as the input interval in the local refinement algorithm. Without loss of generality, assume that\n$$\n\\I = \\I_1\\cup \\I_2\\cup \\I_3 = [s,\\eta_k)\\cup [\\eta_k,\\check {\\eta}_k)\\cup [\\check {\\eta}_k, \\eta_{k + 1}).\n$$\nFor $\\I_2$, there are two cases.\n\n\\textbf{Case 1.} If\n$$|\\I_2| < \\max\\{C_s p\\log(n\\vee p), C_s p\\log(n\\vee p)\/\\kappa^2\\},$$\nthen the proof is complete.\n\n\\textbf{Case 2.} If\n$$|\\I_2| \\geq \\max\\{C_s p\\log(n\\vee p), C_s p\\log(n\\vee p)\/\\kappa^2\\},$$\nThen we proceed to prove that $|\\check {\\eta}_k - \\eta_k|\\leq C\\frac{\\|X\\|_{\\psi_2}^4}{c_X^4} p^2\\log(n\\vee p)\/\\kappa^2$ for some universal constant $C>0$ with probability at least $1 - (n\\vee p)^{-5}$.\n\nFor $t\\in \\I$, let $\\widehat{\\Omega}_t$ be the estimator at index $t$. By definition, we have\n\\begin{equation}\n    \\sum_{t \\in \\I}{\\rm Tr}(\\widehat{\\Omega}_t^\\top X_tX_t^\\top) - \\sum_{t\\in \\I}\\log|\\widehat{\\Omega}_t|\\leq \\sum_{t \\in \\I}{\\rm Tr}(({\\Omega}_t^*)^\\top X_tX_t^\\top) - \\sum_{t\\in \\I}\\log|{\\Omega}_t^*|\n    \\label{dp_tmp_eq:cov local refine 1}\n\\end{equation}\nDue to the property that\n\\begin{equation}\n    \\ell_t(\\widehat{\\Omega}) - \\ell_t({\\Omega}^*)\\geq {\\rm Tr}[(\\widehat{\\Omega} - \\Omega^*)^\\top (X_tX_t^\\top - \\Sigma^*)] + \\frac{c}{2}\\frac{1}{\\|\\Omega^*\\|_{op}^2}\\|\\widehat{\\Omega} - \\Omega^*\\|_F^2,\n\\end{equation}\nequation \\eqref{dp_tmp_eq:cov local refine 1} implies that\n\\begin{align}\n    & \\sum_{i=1}^3 \\frac{|\\I_i|}{\\|\\widehat{\\Omega}_{\\I_i}^*\\|_{op}^2}\\|\\widehat{\\Omega}_{\\I_i} - {\\Omega}_{\\I_i}^*\\|_F^2 \\nonumber \\\\\n    \\leq & c_1\\sum_{i = 1}^3 |\\I_i|{\\rm Tr}[({\\Omega}^*_{\\I_i} - \\widehat{\\Omega}_{\\I_i})^\\top (\\widehat{\\Sigma}_{\\I_i} - {\\Sigma}^*_{\\I_i})] \\nonumber \\\\\n    \\leq & c_1\\sum_{i = 1}^3 |\\I_i|\\|{\\Omega}^*_{\\I_i} - \\widehat{\\Omega}_{\\I_i}\\|_F \\|\\widehat{\\Sigma}_{\\I_i} - {\\Sigma}^*_{\\I_i}\\|_F \\nonumber \\\\\n    \\leq & \\sum_{i=1}^3 \\frac{|\\I_i|}{2\\|\\widehat{\\Omega}_{\\I_i}^*\\|_{op}^2}\\|\\widehat{\\Omega}_{\\I_i} - {\\Omega}_{\\I_i}^*\\|_F^2 + c_2\\sum_{i = 1}^3 |\\I_i|\\|\\widehat{\\Sigma}_{\\I_i} - {\\Sigma}^*_{\\I_i}\\|_F^2\\|\\Omega_{\\I_i}^*\\|_{op}^2,\\label{dp_tmp_eq:cov local refine 2}\n\\end{align}\nwhere we denote $\\widehat{\\Omega}_{\\I_1} = \\widehat{\\Omega}_{\\I_2} = \\widehat{\\Omega}_{[s_k,\\check {\\eta}_k)}$, $\\widehat{\\Omega}_{\\I_3} = \\widehat{\\Omega}_{[\\check {\\eta}_k,e_k)}$, ${\\Omega}_{\\I_1}^* = \\Omega^*_{\\eta_k - 1}$, and ${\\Omega}_{\\I_2}^* = {\\Omega}_{\\I_3}^* = \\Omega^*_{\\eta_k}$.\n\nBy the setting of local refinement, we have $\\min\\{|\\I_1|,|\\I_3|\\}\\geq C_s p\\log(n\\vee p)$. Therefore, by \\Cref{lem:estimation covariance 2}, for $i=1,2,3$, it holds with probability at least $1 - (n\\vee p)^{-7}$ that\n$$\n\\|\\widehat{\\Sigma}_{\\I_i} - {\\Sigma}^*_{\\I_i}\\|_F^2\\leq p\\|\\widehat{\\Sigma}_{\\I_i} - {\\Sigma}^*_{\\I_i}\\|_{op}^2\\leq C \\|X\\|_{\\psi_2}^4 \\frac{p^2\\log(n\\vee p)}{|\\I_i|}.\n$$\nConsequently, we have\n\\begin{equation}\n    \\sum_{i=1}^3 \\frac{|\\I_i|}{\\|\\widehat{\\Omega}_{\\I_i}^*\\|_{op}^2}\\|\\widehat{\\Omega}_{\\I_i} - {\\Omega}_{\\I_i}^*\\|_F^2\\leq c_2\\sum_{i = 1}^3 \\|\\Omega_{\\I_i}^*\\|_{op}^2 \\|X\\|_{\\psi_2}^4 {p^2\\log(n\\vee p)}.\n    \\label{dp_tmp_eq:cov local refine 3}\n\\end{equation}\nIn particular, $\\Delta\\kappa^2>\\mclB_n \\frac{\\|X\\|_{\\psi_2}^4}{c_X^4} p^2\\log(n\\vee p)$, we have\n\\begin{align*}\n    |\\I_1|\\|\\widehat{\\Omega}_{\\I_1} - {\\Omega}_{\\I_1}^*\\|_F^2\\leq& c_2\\|\\widehat{\\Omega}_{\\I_1}^*\\|_{op}^2\\sum_{i = 1}^3 \\|\\Omega_{\\I_i}^*\\|_{op}^2 \\|X\\|_{\\psi_2}^4 {p^2\\log(n\\vee p)}\\\\\n    \\leq & 3c_2\\frac{\\|X\\|_{\\psi}^4}{c_X^4} p^2\\log(n\\vee p)\\leq \\frac{1}{12}\\Delta\\kappa^2,\n\\end{align*}\nfor sufficiently large $n$ because $\\mclB_n\\rightarrow \\infty$ as $n\\rightarrow \\infty$. Since $|\\I_1|\\geq \\frac{1}{3}\\Delta$, it follows from the inequality above that $\\|\\widehat{\\Omega}_{\\I_1} - {\\Omega}_{\\I_1}^*\\|_F\\leq \\frac{\\kappa}{2}$ and thus,\n\\begin{equation*}\n    \\|\\widehat{\\Omega}_{\\I_2} - {\\Omega}_{\\I_2}^*\\|_F\\geq \\|\\widehat{\\Omega}_{\\I_2} - {\\Omega}_{\\I_1}^*\\|_F + \\|{\\Omega}^*_{\\I_1} - {\\Omega}_{\\I_2}^*\\|_F\\geq \\frac{\\kappa}{2}.\n\\end{equation*}\nPlug this back into \\Cref{dp_tmp_eq:cov local refine 3} and we can get\n\\begin{equation}\n    \\frac{\\kappa^2}{4}|\\I_2|\\leq c_4\\frac{\\|X\\|_{\\psi_2}^4}{c_X^4}p^2\\log(n\\vee p),\n\\end{equation}\nwhich completes the proof.\n\\eprf\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Detail of application}\n\\label{sec:detail application}\nIn this section, we show the detail of numerical experiments on real data mentioned in \\Cref{sec: application}. We present results on three datasets that have been previously studied in the change point literature. All the three datasets we used in this section are publicly available.\n\n\\subsection{Bladder tumor micro-array data}\nThis dataset contains the micro-array records of 43 patients with bladder tumor, collected and studied by \\cite{acgh2006}. The result is visualized in \\Cref{fig:acgh}, where we only show the data of 10 patients are shown for the ease of presentation and reading. While there is no accurate ground truth of locations of change points, the 37 change points spotted by DCDP align well with previous research \\citep{ecp2015, wang_samworth2018}. \\Cref{fig:acgh} provides virtual support for the findings by DCDP.\n\n\n\\begin{figure}[ht]\n    \\centering\n    \\includegraphics[width = 0.8 \\textwidth]{Figure\/acgh.pdf}\n    \\caption{Estimated change points in the micro-array data. The result is based on the data of all 43 patients, while only the data of 10 patients is presented. The estimated change points are indicated by dashed vertical lines.}\n    \\label{fig:acgh}\n\\end{figure}\n\n\n\n\n\n \n\n\n\\subsection{Dow Jones industrial average index}\nWe apply DCDP to the weekly log return of the 29 companies composing the \\textit{Dow Jones Industrial Average(DJIA)} from April, 1990 to January, 2012, to detect changes in the covariance structure. We use the version of the data provided in \\cite{ecp2015}. Two change points at September 22, 2008 and May 4, 2009 are detected, which corresponds to the months during which the market is impacted by the financial crisis in 2008. The estimates by DCDP match well with previous research \\cite{ecp2015} on this data.\n\nTo give a virtual evaluation on estimated change points, in \\Cref{fig:djia} we show the estimated precision matrices on the three segments of the data split at estimated change points.\n\n\n\\begin{figure}[ht]\n    \\centering\n    \\includegraphics[width = 0.8 \\textwidth]{Figure\/djia_precision.pdf}\n    \\caption{Estimated change points in the FRED data.}\n    \\label{fig:djia}\n\\end{figure}\n\n\n\n\\subsection{FRED data}\nWe also apply DCDP to \\textit{Federal Reserve Economic Database (FRED)} data.\\footnote{The dataset is publicly available at \\url{https:\/\/research.stlouisfed.org\/econ\/mccracken\/fred-databases}.} We use the subset of monthly data spanning from January 2000 to December 2019, which consists of $n = 240$ samples. The original data has 128 features, including the date. We use the R package \\texttt{fbi}\\citep{fbi} to transform the raw data to be stationary and remove outliers, as is suggested by the data collector \\citep{fred2016}. After preprocessing, there are 118 features, including the date.\n\nWe use logarithm of the monthly growth rate of the US industrial production index (INDPRO named in the dataset) as the response variable, i.e., let $y_t = \\log({\\rm INDPRO}_t\/{\\rm INDPRO}_{t - 1})$, and other 116 macroeconomic variables as predictors. Previous research \\citep{wang2022testing_linear, yu2022temporal} suggests that there exist change points in the association between $y_t$ and predictors.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\nA change point at January 2008 is spotted, which is consistent with previous research on this data \\citep{wang2022testing_linear,yu2022temporal}.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Additional Experiments}\n\\label{sec:detail experiment}\nThis section serves as a supplement to \\Cref{sec:experiment}. In \\Cref{sec:add_experiment}, we discuss the selection of $\\gamma$. In \\Cref{sec:experiment supp results}, we present full results of numerical experiments in \\Cref{sec: experiment comparison}.\n\n\n\\subsection{Selection of $\\gamma$}\n\\label{sec:add_experiment}\nIn the theory of DCDP, we need $\\gamma = C_{\\gamma}\\mclB_n^{-1\/2}\\Delta_{\\min}\\kappa^2$, which involves unknown population parameter $\\Delta_{\\min}$ and $\\kappa^2$. It is common in the change point literature and even broader literature that theoretically best tuning parameters involve unknown quantities, and a typical practical solution is to perform cross validation to select the best tuning parameter from a list of candidates.\n\nSuppose we have data $\\{ \\bm Z _i\\}_{i\\in [n]}$ with $ \\bm Z _i\\sim \\mathbb{P}_{ { \\bm \\theta } _i}$. Without loss of generality, suppose $n = 2m$ for some $m\\in\\mathbb{Z}^+$. We split the data by indices, such that data with odd indices $\\{ \\bm Z _{2i - 1}\\}_{i\\in [m]}$ is the training set and data with even indices $\\{ \\bm Z _{2i}\\}_{i\\in [m]}$ is the test set. This is a common way to conduct cross validation in the change point literature. Given a set of candidate parameters $G = \\{(\\gamma_i,\\zeta_i)\\}_{i\\in [l]}$, for each $i\\in [l]$, the CV has three steps: \n\\begin{enumerate}\n    \\item Run DCDP on $\\{ \\bm Z _{2i - 1}\\}_{i\\in \\mclI_k}$ with $(\\gamma_i,\\zeta_i)$ to get a segmentation $\\widetilde{P} = \\{\\mclI_k\\}_{k\\in[\\widehat{K}+1]}$ of $[1,m]$ where $\\mclI_k = [\\widetilde{\\eta}_{k-1}, \\widetilde{\\eta}_{k})$.\n    \\item Calculate $\\{\\widehat{ { \\bm \\theta } }_k\\}_{k\\in [\\widehat K+1]}$ from $\\{\\{ \\bm Z _{2i - 1}\\}_{i\\in \\mclI_k}\\}_{k\\in [\\widehat K+1]}$ and $$R_i = \\sum_{k\\in [\\widehat{K} + 1]}\\mclF(\\widehat{ { \\bm \\theta } }_k,\\mclI_k)$$ from $\\{\\{ \\bm Z _{2i}\\}_{i\\in \\mclI_k}\\}_{k\\in [\\widehat K + 1]}$.\n    \\item Select $(\\gamma_{i_{cv}},\\zeta_{i_{cv}})$ with the index $i_{cv}=\\argmin_{i\\in[l]}R_i$.\n\\end{enumerate}\n\n\n\n\n\n\n\n\\clearpage\n\\subsection{More results on comparisons}\n\\label{sec:experiment supp results}\nIn this section we present full results of comparisons between DCDP and other methods in \\Cref{tab:mean compare_methods full}, \\Cref{tab:linear compare_methods full}, and \\Cref{tab:covariance compare_methods full}, as a supplement to \\Cref{sec: experiment comparison}. Among all involved methods, DCDP is implemented in Python, ChangeForest is implemented in Rust and provides Python API, Inspect, Variance-Projected WBS, vanilla DP, and Block-Fused-Lasso are implemented in R based on \\texttt{Rcpp}. For fair comparison, we first generate data in Python and then load the data in R for R-based methods. All experiments for DCDP and ChangeForest are run on a virtual machine of Google Colab with Intel(R) Xeon(R) CPU of 2 cores 2.30\nGHz and 12GB RAM (one setting at a time). All other experiments are run on a personal computer with Intel Core i7 8850H CPU of 6 cores 2.60GHz and 64GB RAM (one setting at a time). Notice that programs implemented by \\texttt{Rcpp} is usually faster than Python, and the machine to run \\texttt{Rcpp}-based methods has better parameters than the virtual machine to run DCDP and ChangeForest, the comparison of execution time would not be unfair against \\texttt{Rcpp}-based methods.\n\n\n\\Cref{tab:mean compare_methods full} shows full results of the comparison under the mean shift model.\n\n\\begin{table}[H]\n \\centering\n \\begin{tabular}{l l l l l l l}\n  \\hline\n Setting & Method & $H(\\hat{\\eta},\\eta)$ & Time & $\\hat{K}<K$ &$\\hat{K}=K$ &$\\hat{K}>K$\n \\\\\n \\hline\n \\multirow{4}{3.6cm}{$n = 200, p=20, K =3, \\delta = 5$} \n & DCDP & 0.00 (0.00) & 0.7s (0.2) & 0 & 100 & 0 \\\\\n & Inspect & 0.54 (4.46) &  0.0s (0.0) & 0 & 96 & 4 \\\\\n\n & CF &  3.59 (10.10) &  0.3s (0.0) & 0 & 84 & 16 \\\\\n\n & BFL &  42.56 (6.95) &  3.5s (0.6) & 100 & 0 & 0 \\\\\n \\hline\n \\multirow{4}{3.6cm}{$n = 200, p=20, K =3, \\delta = 1$} \n & DCDP & 0.51 (0.77) & 0.7s (0.2) & 0 & 100 & 0 \\\\\n & Inspect &  3.13 (5.50) &  0.0s (0.0) & 0 & 67 & 33 \\\\\n\n & CF &  4.38 (10.13) &  0.4s (0.1) & 0 & 81 & 19 \\\\\n\n & BFL &  43.30 (8.25) &  2.9s (0.6) & 100 & 0 & 0 \\\\\n \\hline\n \\multirow{4}{3.6cm}{$n = 200, p=20, K =3, \\delta = 0.5$} \n & DCDP & 8.30 (12.90) & 0.4s (0.0) & 8 & 90 & 2 \\\\\n & Inspect & 6.85 (7.53) &  0.0s (0.0) & 0 & 78 & 22 \\\\\n\n & CF &  7.15 (9.57) &  0.4s (0.1) & 1 & 78 & 21 \\\\\n\n & BFL &  54.48 (20.98) &  2.8s (1.1) & 100 & 0 & 0 \\\\\n \\hline\n \\multirow{4}{3.6cm}{$n = 200, p=100, K =3, \\delta = 5$} \n & DCDP & 0.0 (0.0) & 0.6s (0.0) & 0 & 100 & 0 \\\\\n & Inspect & 0.40 (3.50) &  0.0s (0.0) & 0 & 91 & 9 \\\\\n\n & CF & 2.85 (7.50) &  0.8s (0.2) & 0 & 85 & 15 \\\\\n\n & BFL &  47.80 (6.66) &  1.5s (0.3) & 100 & 0 & 0 \\\\\n \\hline\n \\multirow{4}{3.6cm}{$n = 200, p=100, K =3, \\delta = 1$} \n & DCDP & 0.83 (0.87) & 0.8s (0.2) & 0 & 100 & 0 \\\\\n & Inspect &  2.65 (5.16) &  0.0s (0.0) & 0 & 86 & 14 \\\\\n\n & CF &  3.28 (7.01) &  1.3s (0.1) & 0 & 85 & 15 \\\\\n\n & BFL &  47.59 (6.08) &  1.1s (0.2) & 100 & 0 & 0 \\\\\n \\hline\n \\multirow{4}{3.6cm}{$n = 800, p=100, K =3, \\delta = 0.5$} \n & DCDP &  9.36 (29.96) &  2.1s (0.3) & 3  &  97 & 0 \\\\\n & Inspect & 12.55 (22.14) &  0.1s (0.0) & 0 & 77 & 23 \\\\\n\n & CF &  14.73 (30.50) &  5.5s (0.3) & 0 & 82 & 18 \\\\\n\n & BFL &  80.10 (137.33) &  15.7s (3.8) & 28 & 71 & 1 \\\\\n \\hline\n \\end{tabular}\n \\caption{Comparison of DCDP and other methods under the mean model with different simulation settings. 100 trials are conducted in each setting. For the localization error and running time (in seconds), the average over 100 trials is shown with standard error in the bracket. The three columns on the right record the number of trials in which $\\hat{K}<K$, $\\hat{K}=K$, and $\\hat{K}>K$ respectively.}\n \\label{tab:mean compare_methods full}\n\\end{table}\n\n\n\n\\Cref{tab:linear compare_methods full} shows full results of the comparison under the linear regression coefficient shift model.\n\n\n\n\\begin{table}[H]\n \\centering\n\\begin{tabular}{l l l l l l l}\n  \\hline\n Setting & Method & $H(\\hat{\\eta},\\eta)$ & Time & $\\hat{K}<K$ &$\\hat{K}=K$ &$\\hat{K}>K$\n \\\\\n \\hline\n \\multirow{4}{3.6cm}{$n = 200, p=20, K =3, \\delta = 5$} \n & DCDP & 0.03 (0.17) & 5.1s (0.3) & 0 & 100 & 0 \\\\\n & DP & 0.04 (0.20) & 17.0s (0.5) & 0 & 100 & 0 \\\\\n & VPWBS & 7.69 (15.53) &  28.4s (3.5) & 1 & 71 & 28 \\\\\n\n\n & BFL &  84.45 (15.33) &  4.2s (0.7) & 100 & 0 & 0 \\\\\n \\hline\n \\multirow{4}{3.6cm}{$n = 200, p=20, K =3, \\delta = 1$} \n & DCDP & 0.94 (5.17) & 2.3s (0.2) & 2 & 98 & 0 \\\\\n & DP & 0.05 (0.22) & 12.8s (0.5) & 0 & 100 & 0 \\\\\n & VPWBS & 11.71 (19.82) &  30.4s (2.2) & 21 & 73 & 6 \\\\\n\n\n & BFL &  43.31 (8.82) &  3.1s (0.8) & 100 & 0 & 0 \\\\\n \\hline\n \\multirow{4}{3.6cm}{$n = 200, p=100, K =3, \\delta = 5$} \n & DCDP & 0.13 (0.39) & 18.4s (1.1) & 0 & 100 & 0 \\\\\n & DP & 0.01 (0.10) & 220.3s (16.8) & 0 & 98 & 2 \\\\\n & VPWBS & 15.44 (17.99) &  120.1s (13.1) & 18 & 70 & 12 \\\\\n\n\n & BFL &  47.84 (6.69) &  1.4s (0.2) & 100 & 0 & 0 \\\\\n \\hline\n \\multirow{4}{3.6cm}{$n = 200, p=100, K =3, \\delta = 1$} \n & DCDP & 1.45 (8.59) & 8.8s (0.7) & 2 & 98 & 0 \\\\\n & DP & 0.22 (2.00) & 84.4s (5.7) & 0 & 99 & 1 \\\\\n & VPWBS & 11.54 (11.23) &  120.4s (14.5) & 3 & 65 & 32 \\\\\n\n\n & BFL &  47.19 (6.48) &  1.1s (0.2) & 100 & 0 & 0 \\\\\n \\hline\n \\end{tabular}\n \\caption{Comparison of DCDP and other methods under the linear model with different simulation settings. 100 trials are conducted in each setting. For the localization error and running time (in seconds), the average over 100 trials is shown with standard error in the bracket. The three columns on the right record the number of trials in which $\\hat{K}<K$, $\\hat{K}=K$, and $\\hat{K}>K$ respectively.}\n \\label{tab:linear compare_methods full}\n\\end{table}\n\n\n\n\n\n\\Cref{tab:covariance compare_methods full} shows full results of the comparison under the precision shift model. In \\Cref{tab:covariance compare_methods full}, we didn't present the results of BFL because it produces empty set in all trials, for some unknown reason. We tried to fine tune the parameters in BFL, but didn't manage to produce nonempty sets, probably because the precision matrices under our setting are not sparse enough for BFL to perform well.\n\n\\begin{table}[H]\n \\centering\n\\begin{tabular}{l l l l l l l}\n  \\hline\n Setting & Method & $H(\\hat{\\eta},\\eta)$ & Time & $\\hat{K}<K$ &$\\hat{K}=K$ &$\\hat{K}>K$\n \\\\\n \\hline\n \\multirow{2}{3.6cm}{$n = 2000, p=5, K =3, \\delta_1 = 2,\\delta_2 = 0.3$} \n & DCDP & 5.16 (6.52) & 0.7s (0.3) & 0 & 100 & 0 \\\\\n\n\n & CF &  58.25 (151.74) &  1.8s (0.3) & 2 & 69 & 29 \\\\\n\n \\hline\n \\multirow{2}{3.6cm}{$n = 2000, p=10, K =3, \\delta_1 = 5,\\delta_2 = 0.3$} \n & DCDP & 0.27 (0.49) & 0.7s (0.1) & 0 & 100 & 0 \\\\\n\n\n & CF &  42.5 (137.92) &  2.9s (0.2) & 0 & 84 & 16 \\\\\n\n \\hline\n \\multirow{2}{3.6cm}{$n = 2000, p=20, K =3, \\delta_1 = 5,\\delta_2 = 0.3$} \n & DCDP & 0.03 (0.17) & 1.2s (0.2) & 0 & 100 & 0 \\\\\n\n\n & CF &  27.68 (97.20) &  4.8s (0.4) & 0 & 86 & 14 \\\\\n\n \\hline\n \\multirow{2}{3.6cm}{$n = 400, p=10, K =3, \\delta_1 = 5,\\delta_2 = 0.3$} \n & DCDP & 0.42 (0.64) & 0.5s (0.0) & 0& 100& 0 \\\\\n\n\n & CF &  5.54 (14.71) &  0.6s (0.1) & 0 & 88 & 12 \\\\\n\n \\hline\n \\multirow{2}{3.6cm}{$n = 400, p=20, K =3, \\delta_1 = 5,\\delta_2 = 0.3$} \n & DCDP & 0.66 (4.37) & 0.9s (0.3) & 0 & 100 & 0 \\\\\n\n\n & CF &  7.37 (18.76) &  1.0s (0.0) & 0 & 85 & 15 \\\\\n\n \\hline\n \\end{tabular}\n \\caption{Comparison of DCDP and other methods under the covariance model with different simulation settings. 100 trials are conducted in each setting. For the localization error and running time (in seconds), the average over 100 trials is shown with standard error in the bracket. The three columns on the right record the number of trials in which $\\hat{K}<K$, $\\hat{K}=K$, and $\\hat{K}>K$ respectively.}\n \\label{tab:covariance compare_methods full}\n\\end{table}\n\n\\section{Overview}\nAbout DCDP:\n\\begin{enumerate}\n    \\item DCDP works well with a relatively large parameter $\\gamma = C_{\\gamma}K { \\mathcal B_n^{-1}   \\Delta } \\kappa^2$.\n    \\item Local refinement works well both in theory and in practice, though it is not quite ``novel\". Is it good to go? Or should we think about something else?\n    \\item Next step: consider generalizaing to nonlinear models like logistic regression.\n    \\item Piece-wise linear signals?\n\\end{enumerate}\n\nAbout CPD for logistic regression and Bradley-Terry model:\n\\begin{enumerate}\n    \\item Given that the WBS based on loglikelihood performs well in practice, should we try proving some theory for it?\n\\end{enumerate}\n\n\nimprovement in the computation\n\ncompare with others\n\n\\section{Numerical experiments}\n\\subsection{Run time and localization error of the divide step}\n\\label{sec:experiment_time_error_divide}\nIn the first experiemnt, we generate $T=4\\Delta$ samples from the univariate Gaussian mean model $y_t = \\mu^*_t + \\epsilon_t$ with 3 change points at $\\eta_{k} = k\\Delta + 1$ for $k = 1,2,3$ and $\\mu_{\\eta_0} = 0$, $\\mu_{\\eta_1} = 5$, $\\mu_{\\eta_2} = 0$, $\\mu_{\\eta_3} = 5$. In \\Cref{fig:loc_vs_n_grid} and \\Cref{fig:runtime_n_grid}, we present the localization error rate and run time of DCDP with $\\mclR = 5$ as $\\mclQ$, the number of random samples in the grid changes from 25 to 200. The localization error rate is defined as $H(\\{\\hat{\\eta}\\},\\{\\eta\\})\/T$ where $H(\\cdot,\\cdot)$ is the Hausdorff distance between two sets.\n\n\n\n\nWe show results for $\\Delta = 500$ and $\\Delta = 5000$. For the tuning parameter $\\gamma$, we search over a list from $0.2\\Delta$ to $0.4\\Delta$. To tune $\\gamma$, we run DCDP on each value of $\\gamma$ on the training data and get estimates of mean values on induced intervals, then check the goodness-of-fit of the testing data on the estimated mean values, and pick the $\\gamma$ with the best goodness-of-fit value. Here the training and testing data are generated from the same model above. In practice, one can do a odd-even split on the data. In experiments, $60$ trials are run for each value of $\\mclQ$.\n\\begin{figure}[H]\n    \\centering\n    \\includegraphics[width = 0.46 \\textwidth]{Figure\/Delta500_loc_error_grid_num.png}\n    \\includegraphics[width = 0.47\n    \\textwidth]{Figure\/Delta5000_loc_error_grid_num.png}\n    \\caption{Localization error vs. number of points $\\mclQ$ in the grid. Left: $\\Delta = 500$; right: $\\Delta = 5000$. The line is the average and the shaded area shows the upper and lower 0.1 quantiles from 60 trials. Blue: only divide step; orange: dvide step followed by a local refinement.}\n    \\label{fig:loc_vs_n_grid}\n\\end{figure}\n\n\\Cref{fig:runtime_n_grid} illustrates the efficiency of DCDP in the sense that the original DP with the grid set to be $[T]$ corresponds to $\\mclQ = T$, which will take much longer time to run. As a reference, the WBS method implemented by our own takes 0.14 and 10.79 seconds with average error $5.6$ and $0.0$ for $\\Delta = 500, 5000$.\n\\begin{figure}[H]\n    \\centering\n    \\includegraphics[width = 0.46 \\textwidth]{Figure\/Delta500_runtime_grid_num.png}\n    \\includegraphics[width = 0.47\n    \\textwidth]{Figure\/Delta5000_runtime_grid_num.png}\n    \\caption{Run time vs. number of points $\\mclQ$ in the grid. Left: $\\Delta = 500$; right: $\\Delta = 5000$. The line is the average and the shaded area shows the upper and lower 0.1 quantiles from 60 trials. Blue: only divide step; orange: dvide step followed by a local refinement.}\n    \\label{fig:runtime_n_grid}\n\\end{figure}\n\n\n\\subsection{Comparison with other methods}\nIn this section, we will compare the performance of DCDP with several competitors in the literature. Apart from the standard Wild Binary Segmentation \\citep{Fryzlewicz2014} and the standard Dynamic Programming \\citep{pelt2012, rinaldo2021cpd_reg_aistats}, we also consider serveral competitors that were developed recently, including the Seeded WBS \\citep{kovacs2020seeded}, the Block Fused Lasso \\citep{unify_sinica2022}, and the Change Forest \\citep{changeforest2022}.\n\\section{Proof}\n\\label{sec:proof}\n\\subsection{Fundamental lemma}\nIn the proof of localization error of the standard dynamic programming, we frequently compare the goodness-of-fit $\\mclF(\\mclI)$ over an interval $\\mclI = (s,e]$ with\n\\[\n\\mclF((s,\\eta_{i + 1}]) + \\cdots + \\mclF((\\eta_{i + m},e]) + m\\gamma\n\\]\nwhere $\\{\\eta_{i + j}\\}_{j\\in [m]} = \\{\\eta_{\\ell}\\}_{\\ell\\in [K]}\\cap \\mclI$ is the collection of true change points within interval $\\mclI$ and $\\gamma$ is the penalty coefficient in the DP. \n\nHowever, for DCDP, we only search over the random grid $\\{s_{\\ell}\\}_{\\ell\\in [\\mclQ]}$ that does not contain any true change points in general. Therefore, we need two fundamental lemmas that guarantee the existence of some reference points that are close to true change points, and the deviation of goodness-of-fit evaluated at the output of DCDP compared to those reference points.\n\n\\paragraph{Reference points.} Suppose  the sample points $\\{ s_q\\}_{q=1}^ { \\mathcal Q}  $ are uniformly and independently  sampled from $(0,n)$. \n Let \n$ \\{ \\eta_k\\}_{k=1}^{K}$ be the collection of change points and denote \n\\begin{align*} \n  \\mathcal L _k (\\delta)  : = \\bigg  \\{    \\{ s_q\\}_{q=1}^ { \\mathcal Q}  \\cap   [\\eta_k -\\delta ,\\eta_k  ]  \\not  =\\emptyset \\bigg  \\} , \\quad \\text{and} \\quad    \\mathcal R  _k (\\delta)  : = \\bigg  \\{    \\{ s_q\\}_{q=1}^ { \\mathcal Q}  \\cap   [\\eta_k ,\\eta_k + \\delta   ]  \\not  =\\emptyset \\bigg  \\}  .\n\\end{align*} \nDenote \n\\begin{align} \n\\label{eq:left and right approximation of change points}  \\mathcal L(\\delta)  : = \\bigcap_{k=1}^{K}   \\mathcal L _k  \\big  (   \\delta   \\big  )  \n\\quad \\text{and} \\quad \n\\mathcal R (\\delta)  : = \\bigcap_{k=1}^{K}   \\mathcal R _k  \\big  (  \\delta  \\big  ) .   \n\\end{align}  \n\\bnlem Suppose  the sample points $\\{ s_q\\}_{q=1}^ { \\mathcal Q}  $ are uniformly and independently  sampled from $(0,n)$. \nIt holds that \n\\[\n\\mathbb P \\bigg (    \\bigcap_{k=1}^{K}   \\mathcal  L _k (\\delta)  \\bigg ) \\ge 1 -  \\exp( - 2\\delta  { \\mathcal Q}  \/ n+ \\log(K)  ) \n\\quad \\text{and} \\quad \n \\mathbb P \\bigg (    \\bigcap_{k=1}^{K}   \\mathcal  R  _k (\\delta)  \\bigg ) \\ge 1 -  \\exp( - 2\\delta  { \\mathcal Q}  \/ n+ \\log(K)  ).\n\\]\nIf in addition that   $ { \\mathcal Q} \\ge 4 (  n\/\\Delta ) \\log(n)   { \\mathcal B_n}    $ for some slowly diverging sequence $ { \\mathcal B_n} $, then \n\\[\n\\mathbb P \\bigg (        \\mathcal L   (   { \\mathcal B_n^{-1}   \\Delta }    \\big  )  \\bigg )     \\ge 1 - \\frac{1}{n} \\quad \\text{and}  \\quad  \\mathbb P \\bigg (      \\mathcal R   \\big  (   { \\mathcal B_n^{-1}   \\Delta }    \\big  )  \\bigg )     \\ge 1 - \\frac{1}{n}  .\n\\]\n\\enlem\n\n\\begin{proof} \nNote that \n\\[\n\\mathbb P( \\mathcal L _k (\\delta) ^c  ) =  \\bigg( 1-  \\frac{ \\delta }{ n } \\bigg)^  { \\mathcal Q}  \\le \\exp( -  \\delta  { \\mathcal Q}  \/ n ).\n\\]\nConsequently,\n\\[\n\\mathbb P \\bigg (    \\bigg\\{  \\bigcap_{k=1}^{K}   \\mathcal  L _k (\\delta)  \\bigg\\}^c  \\bigg ) \\le  \\sum_{k=1}^K \\mathbb P ( \\mathcal  L _k (\\delta) ^c  )  \n\\le K \\exp( - \\delta  { \\mathcal Q}  \/ n ) =  \\exp( -  \\delta  { \\mathcal Q}  \/ n+ \\log(K)  ) . \n\\]\nSetting $\\delta =   { \\mathcal B_n^{-1}   \\Delta }   $ and noting that $K\\le n\/\\Delta$, it follows that if $ { \\mathcal Q} \\ge 4 (  n\/\\Delta )  { \\mathcal B_n}  \\log(n)   $, then \n\\[\n\\mathbb P \\bigg (     \\bigcap_{k=1}^{K}   \\mathcal  L _k  \\big  ( { \\mathcal B_n^{-1}   \\Delta }    \\big  )  \\bigg )  \\ge 1- \\exp \\bigg( -  4 \\log (n )  + \\log( n\/\\Delta )\\bigg)  \\ge 1 - \\frac{1}{n^3} .\n\\]\n\\end{proof} \n\n\n\n\n\n\n\n\n\n\n\n\n\\paragraph{Goodness-of-fit.} For different models, the deviation of the goodness-of-fit has different orders. We need to analyze each model separately.\n\n\\bnlem[Mean model]\n\\label{lem:mean one change deviation bound}\nLet $\\mathcal I =(s,e] $ be any generic interval and $\\widehat \\mu_\\I = \\argmin_{\\mu}[\\frac{1}{|\\mclI|}\\sum_{i\\in \\mclI}\\|X_i - \\mu\\|_2^2 + \\frac{\\lambda}{\\sqrt{|\\mclI|}}\\|\\mu\\|_1]$. \n\n{\\bf a.} If $\\I$ contains no change points. Then it holds that \n$$\\mathbb P \\bigg( \\bigg|   \\sum_{ i  \\in \\I } \\|X_i   -\\widehat \\mu_\\I   \\|_2^2  - \\sum_{ i   \\in \\I } \\|X_i      - \\mu^*_i   \\|_2^2  \\bigg|  \\ge C \\sigma_{\\epsilon}^2\\mathfrak{s}\\log(np)  \\bigg)  \\le  (np)^{-3}. $$\n\n{\\bf b.}  Suppose that the interval $ \\I=(s,e]$ contains one and only one change point $ \\eta_k $. Denote\n$$  \\mathcal J = (s,\\eta_k] \\quad \\text{and} \\quad  \\mathcal J' =  (\\eta_k, e]  .$$   Then it holds that \n$$\\mathbb P \\bigg( \\bigg|   \\sum_{ i  \\in \\I } \\|X_i   -\\widehat \\mu_\\I   \\|^2  - \\sum_{ i   \\in \\I } \\|X_i      - \\mu^*_i    \\|^2  \\bigg|  \\geq \\frac{ | {  \\mathcal J } ||{  \\mathcal J } '| }{ |\\I| }  \\kappa_k ^2  +C \\sigma_{\\epsilon}^2\\mathfrak{s}\\log(np)  \\bigg)   \\le  (np)^{-3}. $$\n\\enlem\n\\begin{proof} \nWe show {\\bf b} as {\\bf a} immediately follows from {\\bf b} with $ |{  \\mathcal J } '| =0$. Denote \n$$  \\mathcal J = (s,\\eta_k] \\quad \\text{and} \\quad  \\mathcal J' =  (\\eta_k, e]  .$$ \nDenote  $\\mu_\\I = \\frac{1}{|\\I | } \\sum_{i\\in \\I } \\mu^*_i $.\nThe it holds that \n\\begin{equation*}\n    \\begin{split}\n        &\\sum_{i\\in \\I} \\|X_i -\\widehat \\mu_\\I\\|_2^2  - \\sum_{i\\in \\I} \\|X_i - \\mu^*_i\\|_2^2  \\\\\n        =& \\sum_{i \\in \\I } \\|\\widehat \\mu_\\I-\\mu^*_i \\|_2^2  -2 \\sum_{i \\in \\I } \\epsilon_i^\\top  (\\widehat \\mu_\\I  - \\mu^*_i)\\\\\n        \\leq& \\sum_{i \\in \\I } \\|\\widehat \\mu_\\I-\\mu^*_\\I \\|_2^2 + \\sum_{i \\in \\I } \\|\\mu^*_\\I-\\mu^*_i \\|_2^2  -2 \\sum_{i \\in \\I } \\epsilon_i^\\top  (\\widehat \\mu_\\I  - \\mu^*_\\I) - 2 \\sum_{i \\in \\I } \\epsilon_i^\\top  (\\mu^*_\\I  - \\mu^*_i).\n    \\end{split}\n\\end{equation*} \nObserve that \n\\begin{equation*}\n    \\mathbb P \\bigg(\\|\\sum_{i\\in \\I } \\epsilon_i \\|_{\\infty}  \\geq C\\sigma_{\\epsilon}\\sqrt {\\log(np)|\\I |   }\\bigg) \\leq (np)^{-5 } \n\\end{equation*}\nSuppose this event holds.\n\n{\\bf Step 1.} By the event and \\Cref{lem:estimation_high_dim_mean}, we have\n\\begin{equation*}\n    \\begin{split}\n        \\sum_{i \\in \\I } \\|\\widehat \\mu_\\I-\\mu^*_\\I \\|_2^2 &\\leq C\\sigma_{\\epsilon}^2 { \\mathfrak{s} } \\log(np),\\\\\n        |2 \\sum_{i \\in \\I } \\epsilon_i^\\top  (\\widehat \\mu_\\I  - \\mu^*_\\I)| &\\leq |\\I|\\|\\sum_{i\\in \\I}\\epsilon_i\\|_{\\infty}\\|\\widehat \\mu_\\I-\\mu^*_\\I\\|_1\\leq C\\sigma_{\\epsilon}^2 { \\mathfrak{s} } \\log(np).\n    \\end{split}\n\\end{equation*}\n{\\bf Step 2.} Notice that \n\\begin{align*} \n    \\sum_{   i \\in \\I } \\|\\mu^*_\\I-\\mu^*_i\\| ^2  =  \n        & \\sum_{   i \\in \\I }  \\| \\frac{ | {  \\mathcal J }  | \\mu^*_{  \\mathcal J }  + | {  \\mathcal J } '| \\mu^*_{ {  \\mathcal J }  ' }}{|\\I| } -\\mu^*_i  \\|_2 ^2\\\\\n =  &     \\sum_{i\\in {  \\mathcal J }  }   \\| \\frac{ |{  \\mathcal J }  '|  (\\mu^*_{  \\mathcal J }   - \\mu^*_{{  \\mathcal J } '} ) }{|\\I| }  \\|_2^2 + \n     \\sum_{i\\in {  \\mathcal J } ' }   \\| \\frac{ |{  \\mathcal J }   |  (\\mu^*_{  \\mathcal J }   - \\mu^*_{{  \\mathcal J } '} ) }{|\\I| }  \\|_2^2\n  \\\\ = &  \\frac{|{  \\mathcal J }  | | {  \\mathcal J } '| }{|\\I|  }(\\mu^*_{  \\mathcal J }   - \\mu^*_{{  \\mathcal J } '} ) ^2 = \\frac{|{  \\mathcal J }  | | {  \\mathcal J } '| }{|\\I|  }    \\kappa_k  ^2.\n    \\end{align*}\nMeanwhile, it holds that \n\\begin{align*} \n\\sum_{  i     \\in \\I } \\epsilon_i ^\\top (\\mu^*_\\I  - \\mu^*_i  ) =\n& \\sum_{   i \\in \\I }  \\epsilon_i^\\top  \\bigg ( \\frac{ | {  \\mathcal J }  | \\mu^*_{  \\mathcal J }  + | {  \\mathcal J } '| \\mu^*_{ {  \\mathcal J }  ' }}{|\\I| }-\\mu^*_i  \\bigg )  \n\\\\\n=&  \\frac{|{  \\mathcal J } '|}{|\\I|}\\sum_{i\\in {  \\mathcal J }  }   \\epsilon_i^\\top(    \\mu^*_{{  \\mathcal J } '}-\\mu^*_{  \\mathcal J }   ) + \n   \\frac{|{  \\mathcal J } |}{|\\I|}\\sum_{i\\in {  \\mathcal J } ' }   \\epsilon_i^\\top(    \\mu^*_{{  \\mathcal J } }-\\mu^*_{{  \\mathcal J } '}  )\n   \\\\\n   \\le &    C_2\\sigma_{\\epsilon}\\sqrt {   \\frac{|{  \\mathcal J }  | | {  \\mathcal J } '| }{|\\I|  }     \\kappa_k  ^2  \\log(np) }\n   \\le \\frac{|{  \\mathcal J }  | | {  \\mathcal J } '| }{|\\I|}   \\kappa_k  ^2    +  C\\sigma_{\\epsilon}^2\\log(np)  ,\n\\end{align*}\nwhere the first inequality follows from the fact that the variance is upper bounded by\n$$\\sum_{i\\in {  \\mathcal J }  }  \\sigma_{\\epsilon}^2 \\frac{ |{  \\mathcal J }  '|^2 }{|\\I|^2 } \\|\\mu^*_{  \\mathcal J }   - \\mu^*_{{  \\mathcal J } '} \\|_2^2 + \\sum_{i\\in {  \\mathcal J } ' }  \\sigma_{\\epsilon}^2 \\frac{ |{  \\mathcal J } |^2 }{|\\I|^2 } \\|\\mu^*_{  \\mathcal J }   - \\mu^*_{{  \\mathcal J } '} \\|_2^2 =    \\frac{|{  \\mathcal J }  | | {  \\mathcal J } '| }{|\\I|  }\\sigma_{\\epsilon}^2 \\kappa_k^2 . $$\n\\end{proof} \n\n\n\n\n\\bnlem\n\\label{lem:estimation_high_dim_mean}\nFor any interval $\\mclI\\subset [1, n]$ with $|\\mclI|\\geq C_0\\mathfrak{s}\\log(np)$ that might contain change points. Let\n\\begin{equation*}\n    \\hat{\\mu}_{\\mclI}\\defined \\argmin_{\\mu\\in \\mathbb{R}^p}\\frac{1}{|\\mclI|}\\|X_i - \\mu\\|_2^2 + \\frac{\\lambda}{\\sqrt{|\\mclI|}}\\|\\mu\\|_1,\n\\end{equation*}\nfor $\\lambda = C_{\\lambda}\\sigma_{\\epsilon}\\sqrt{\\log(np)}$ for sufficiently large constant $C_{\\lambda}$. Then it holds with probability at least $1 - (np)^{-5}$ that\n\\begin{equation}\n    \\begin{split}\n    \\|\\widehat{\\mu}_{\\I}-\\mu^*_{\\I}\\|_{2}^{2} &\\leq \\frac{C\\sigma_{\\epsilon}^2  { \\mathfrak{s} }  \\log (np)}{\\I} \\\\\n\\left\\|\\widehat{\\mu}_{\\I}-\\mu^*_{\\I}\\right\\|_{1} &\\leq C \\sigma_{\\epsilon} { \\mathfrak{s} }  \\sqrt{\\frac{\\log (np)}{|\\I|}} \\\\\n\\|\\left(\\widehat{\\mu}_{\\I}-\\mu^*_{\\I}\\right)_{S^{c}}\\|_{1} &\\leq 3\\|\\left(\\widehat{\\mu}_{\\I}-\\mu^{*}_{\\I}\\right)_{S}\\|_{1},\n    \\end{split}\n\\end{equation}\nwhere $\\mu_\\mclI = \\frac{1}{|\\I|}\\sum_{i\\in \\I}\\mu^*_i$.\n\\enlem\n\n\n\\begin{proof}\nBy definition, we have $L(\\widehat{\\mu}_{\\I},\\I)\\leq L({\\mu}_{\\I},\\I)$, that is\n\\begin{align}\n\\nonumber\n    &\\sum_{i\\in \\I}\\|Y_i - \\widehat{\\mu}_{\\I}\\|_2^2 + \\lambda\\sqrt{|\\I|}\\|\\widehat{\\mu}_{\\I}\\|_1\\leq \\sum_{i\\in \\I}\\|Y_i - {\\mu}^*_{\\I}\\|_2^2 + \\lambda\\sqrt{|\\I|}\\|{\\mu}^*_{\\I}\\|_1\\\\\n\\nonumber\n    \\Rightarrow & \\sum_{i\\in \\I}(\\widehat{\\mu}_{\\I} - \\mu^*_{\\I})^\\top (2Y_i - \\mu^*_{\\I} - \\widehat{\\mu}_{\\I}) + \\lambda\\sqrt{|\\I|}[\\|{\\mu}^*_{\\I}\\|_1 - \\|\\widehat{\\mu}_{\\I}\\|_1]\\geq 0 \\\\\n    \\nonumber\n    \\Rightarrow & (\\widehat{\\mu}_{\\I} - \\mu^*_{\\I})^\\top (\\sum_{i\\in \\I}\\epsilon_i) + 2(\\widehat{\\mu}_{\\I} - \\mu^*_{\\I})^\\top\\sum_{i\\in \\I}(\\mu^*_i - \\mu^*_{\\I}) - |\\I|\\sum_{i\\in \\I}\\|\\widehat{\\mu}_{\\I} - \\mu^*_{\\I}\\|_2^2 + \\lambda\\sqrt{|\\I|}[\\|{\\mu}^*_{\\I}\\|_1 - \\|\\widehat{\\mu}_{\\I}\\|_1]\\geq 0\\\\\n    \\Rightarrow & \\|\\widehat{\\mu}_{\\I} - \\mu^*_{\\I}\\|_1 \\|\\sum_{i\\in \\I}\\epsilon_i\\|_{\\infty} + \\lambda\\sqrt{|\\I|}[\\|{\\mu}^*_{\\I}\\|_1 - \\|\\widehat{\\mu}_{\\I}\\|_1]\\geq |\\I|\\sum_{i\\in \\I}\\|\\widehat{\\mu}_{\\I} - \\mu^*_{\\I}\\|_2^2.\n    \\label{tmp_eq:mean_concentration}\n\\end{align}\nBy a union bound, we know that for some universal constant $C>0$, with probability at least $1 - (np)^{-5}$,\n$$\n\\|\\sum_{i\\in \\I}\\epsilon_i\\|_{\\infty}\\leq C\\sigma_{\\epsilon}\\sqrt{|\\I|\\log(np)}\\leq \\frac{\\lambda}{2}\\sqrt{|\\I|},\n$$\nas long as $C_{\\lambda}$ is sufficiently large. Therefore, based on the sparsity assumption in \\Cref{assp: DCDP_mean}, it holds that\n\\begin{align*}\n    &\\frac{\\lambda}{2}\\|\\widehat{\\mu}_{\\I} - \\mu^*_{\\I}\\|_1 + \\lambda[\\|{\\mu}^*_{\\I}\\|_1 - \\|\\widehat{\\mu}_{\\I}\\|_1]\\geq 0 \\\\\n    \\Rightarrow & \\frac{\\lambda}{2}\\|\\widehat{\\mu}_{\\I} - \\mu^*_{\\I}\\|_1 + \\lambda[\\|({\\mu}^*_{\\I})_S\\|_1 - \\|(\\widehat{\\mu}_{\\I})_S\\|_1]\\geq \\lambda \\|(\\widehat{\\mu}_{\\I})_{S^c}\\|_1\\\\\n    \\Rightarrow & \\frac{\\lambda}{2}\\|\\widehat{\\mu}_{\\I} - \\mu^*_{\\I}\\|_1 + \\lambda\\|({\\mu}^*_{\\I}-\\widehat{\\mu}_{\\I})_S\\|_1 \\geq \\lambda \\|(\\mu^*_{\\I} - \\widehat{\\mu}_{\\I})_{S^c}\\|_1\\\\\n    \\Rightarrow & 3\\|({\\mu}^*_{\\I}-\\widehat{\\mu}_{\\I})_S\\|_1\\geq \\|({\\mu}^*_{\\I}-\\widehat{\\mu}_{\\I})_{S^c}\\|_1.\n\\end{align*}\nNow from \\Cref{tmp_eq:mean_concentration} we can get\n\\begin{align*}\n    |\\I|\\|\\widehat{\\mu}_{\\I} - \\mu^*_{\\I}\\|_2^2\\leq &\\frac{3\\lambda}{2}\\sqrt{|\\I|}\\|\\widehat{\\mu}_{\\I} - \\mu^*_{\\I}\\|_1\\\\\n    \\leq & \\frac{12\\lambda}{2}\\sqrt{|\\I|}\\|(\\widehat{\\mu}_{\\I} - \\mu^*_{\\I})_S\\|_1\\\\\n    \\leq & 6\\lambda \\sqrt{ { \\mathfrak{s} } } \\sqrt{|\\I|}\\|(\\widehat{\\mu}_{\\I} - \\mu^*_{\\I})_S\\|_2\\\\\n    \\leq & 6\\lambda \\sqrt{ { \\mathfrak{s} } } \\sqrt{|\\I|}\\|\\widehat{\\mu}_{\\I} - \\mu^*_{\\I}\\|_2,\n\\end{align*}\nwhich implies that\n$$\n\\|\\widehat{\\mu}_{\\I} - \\mu^*_{\\I}\\|_2\\leq 6C_{\\lambda}\\sigma_{\\epsilon}\\sqrt{\\frac{ { \\mathfrak{s} } \\log(np)}{|\\I|}}.\n$$\nThe other inequality follows accordingly.\n\\end{proof}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\subsection{Mean model}\nIn this section we show the proof of \\Cref{thm:DCDP mean} and \\Cref{cor:mean local refinement}. Throughout this section, for any generic interval $\\I\\subset [1,n]$, denote $\\mu^*_{\\I} = \\frac{1}{|\\I|}\\sum_{i\\in \\I}\\mu^*_i$ and\n$$ \\widehat \\mu_\\I = \\argmin_{\\mu}\\frac{1}{|\\mclI|}\\sum_{i\\in \\mclI}\\|X_i - \\mu\\|_2^2 + \\frac{\\lambda}{\\sqrt{|\\mclI|}}\\|\\mu\\|_1. $$\nAlso, unless specially mentioned, in this section, we set the goodness-of-fit function $\\mathcal F  (\\I)$ in \\Cref{algorithm:DCDP_divide} to be\n\\begin{equation}\n    \\mathcal F  (\\I) := \\begin{cases}\n    \\sum_{i \\in \\I } \\|X_i - \\widehat \\mu_\\I \\|_2 ^2, &\\text{ when } |\\I|\\geq C_\\mclF\\sigma_{\\epsilon}^2 { \\mathfrak{s} } \\log(np),\\\\\n    0, &\\text{otherwise},\n    \\end{cases}.\n\\end{equation}\nwhere $C_\\mclF$ is a universal constant.\n \n \\begin{proof}[Proof of \\Cref{thm:DCDP mean}]\n By  \\Cref{prop:mean local consistency}, $K \\leq |\\widehat{\\mathcal{P}}| \\leq 3K$.  This combined with \\Cref{prop:change points partition size consistency} completes the proof.\n \\end{proof}\n \n\n \\bnprop\\label{prop:mean local consistency}\n Suppose  \\Cref{assp: DCDP_mean} holds. Let \n $\\widehat { \\mathcal P }  $ denote the output of  \\Cref{algorithm:DCDP_divide}.\n Then with probability  at least $1 - C n ^{-3}$, the following conditions hold.\n\t\\begin{itemize}\n\t\t\\item [(i)] For each interval  $ \\I = (s, e] \\in \\widehat{\\mathcal{P}}$    containing one and only one true \n\t\t   change point $ \\eta_k $, it must be the case that\n $$\\min\\{ \\eta_k -s ,e-\\eta_k \\}  \\lesssim  \\sigma_{\\epsilon}^2\\bigg( \\frac{   { \\mathfrak{s} } \\log(np) +\\gamma }{\\kappa_k^2 }\\bigg) +   { \\mathcal B_n^{-1}   \\Delta }       .$$\n\t\t\\item [(ii)] For each interval  $  \\I = (s, e] \\in \\widehat{\\mathcal{P}}$ containing exactly two true change points, say  $\\eta_ k  < \\eta_ {k+1} $, it must be the case that\n\t\t\t\\begin{align*} \n \\eta_k -s    \\lesssim       { \\mathcal B_n^{-1\/2}   \\Delta }     \\quad \\text{and} \\quad \n e-\\eta_{k+1}  \\lesssim    { \\mathcal B_n^{-1\/2}   \\Delta }      . \n \\end{align*} \n\t\t \n\\item [(iii)] No interval $\\I \\in \\widehat{\\mathcal{P}}$ contains strictly more than two true change points; and \n\n\n\t\\item [(iv)] For all consecutive intervals $ \\I_1 $ and $ \\I_2 $ in $\\widehat{ \\mathcal P}$, the interval \n\t\t$ \\I_1 \\cup  \\I_2 $ contains at least one true change point.\n\t\t\t\t\n\t \\end{itemize}\n\\enprop \n\n\n\\bnprop\\label{prop:change points partition size consistency}\nSuppose  \\Cref{assp: DCDP_mean} holds. Let \n $\\widehat { \\mathcal P }  $ be the output of  \\Cref{algorithm:DCDP_divide}.\n Suppose \n$ \\gamma \\ge C_\\gamma K { \\mathcal B_n^{-1}   \\Delta }  \\kappa^2 $  for sufficiently large constant $C_\\gamma$. Then\n with probability  at least $1 - C n ^{-3}$,  $| \\widehat { \\mathcal P} | =K $.\n\\enprop\n\n\n\n\\begin{proof}[Proof of   \\Cref{prop:change points partition size consistency}] \nDenote $\\mathfrak{G} ^*_n = \\sum_{ i =1}^n \\|X_i  -  \\mu^*_i\\|_2^2$.  Given any collection $\\{t_1, \\ldots, t_m\\}$, where $t_1 < \\cdots < t_m$, and $t_0 = 0$, $t_{m+1} = n$, let \n\\begin{equation}\\label{eq-sn-def}\n\t { \\mathfrak{G} }  _n(t_1, \\ldots, t_{m}) = \\sum_{k=1}^{m} \\sum_{ i = t_k +1}^{t_{k+1}} \\mclF(\\widehat \\mu_  {(t_{k}, t_{k+1}]}, (t_{k}, t_{k+1}]).\n\\end{equation}\nFor any collection of time points, when defining \\eqref{eq-sn-def}, the time points are sorted in an increasing order.\n\nLet $\\{ \\widehat \\eta_{k}\\}_{k=1}^{\\widehat K}$ denote the change points induced by $\\widehat {\\mathcal P}$.  Suppose we can justify that \n\t\\begin{align}\n\t\t { \\mathfrak{G} } ^*_n + K\\gamma  \\ge  & { \\mathfrak{G} }  _n(s_1,\\ldots,s_K)   + K\\gamma - C_1 ( K +1)\\sigma_{\\epsilon}^2  { \\mathfrak{s} }   \\log(np) - C_1\\sum_{k\\in [K]}\\kappa_k^2 { \\mathcal B_n^{-1}   \\Delta }   \\label{eq:K consistency step 1} \\\\ \n\t\t\\ge &  { \\mathfrak{G} } _n (\\widehat \\eta_{1},\\ldots, \\widehat \\eta_{\\widehat K } ) +\\widehat K \\gamma - C_1 ( K +1)  \\sigma_{\\epsilon}^2  { \\mathfrak{s} }   \\log(np) - C_1\\sum_{k\\in [K]}\\kappa_k^2 { \\mathcal B_n^{-1}   \\Delta }   \\label{eq:K consistency step 2} \\\\ \n\t\t\\ge &   { \\mathfrak{G} }  _n ( \\widehat \\eta_{1},\\ldots, \\widehat \\eta_{\\widehat K } , \\eta_1,\\ldots,\\eta_K ) + \\widehat K \\gamma    - C_1' ( K +1)  \\sigma_{\\epsilon}^2  { \\mathfrak{s} }   \\log(np)- C_1'\\sum_{k\\in [K]}\\kappa_k^2 { \\mathcal B_n^{-1}   \\Delta }   \\label{eq:K consistency step 3}\n\t\\end{align}\n\tand that \n\t\\begin{align}\\label{eq:K consistency step 4}\n\t\t { \\mathfrak{G} }  ^*_n   - { \\mathfrak{G} }  _n ( \\widehat \\eta_{1},\\ldots, \\widehat \\eta_{\\widehat K } , \\eta_1,\\ldots,\\eta_K ) \\le C_2 (K + \\widehat{K} + 2) \\sigma_{\\epsilon}^2  { \\mathfrak{s} }   \\log(np).\n\t\\end{align}\n\tThen it must hold that $| \\hatp | = K$, as otherwise if $\\widehat K \\ge K+1 $, then  \n\t\\begin{align*}\n\t\tC _2 (K + \\widehat{K} + 2)  \\sigma_{\\epsilon}^2  { \\mathfrak{s} }   \\log(np)  & \\ge    { \\mathfrak{G} }  ^*_n  - { \\mathfrak{G} }  _n ( \\widehat \\eta_{1},\\ldots, \\widehat \\eta_{\\widehat K } , \\eta_1,\\ldots,\\eta_K ) \\\\\n\t\t& \\ge      (\\widehat K - K)\\gamma  -C_1 ( K +1)  \\sigma_{\\epsilon}^2  { \\mathfrak{s} }   \\log(np) - C_1\\sum_{k\\in [K]}\\kappa_k^2 { \\mathcal B_n^{-1}   \\Delta } .\n\t\\end{align*} \n\tTherefore due to the assumption that $| \\hatp|  =\\widehat K\\le 3K $, it holds that \n\t\\begin{align} \\label{eq:Khat=K} \n\t\t[C_2 (4K + 2) + C_1(K+1)] \\sigma_{\\epsilon}^2  { \\mathfrak{s} }   \\log(np) +C_1\\sum_{k\\in [K]}\\kappa_k^2 { \\mathcal B_n^{-1}   \\Delta }  \\ge  (\\widehat K - K)\\gamma \\geq \\gamma,\n\t\\end{align}\n\tNote that \\eqref{eq:Khat=K} contradicts the choice of $\\gamma$.\n\n\n\\\n\\\\\n{\\bf Step 1.} Note that \\eqref{eq:K consistency step 1}  is implied by \n\t\\begin{align}\\label{eq:step 1 K consistency}  \n\t\t\\left| \t { \\mathfrak{G} }  ^*_n -  { \\mathfrak{G} } _n(s_1,\\ldots,s_K)    \\right| \\le  C_3(K+1) \\sigma_{\\epsilon}^2  { \\mathfrak{s} }   \\log(np) + C_3\\sum_{k\\in [K]}\\kappa_k^2 { \\mathcal B_n^{-1}   \\Delta } ,\n\t\\end{align}\n\twhich is  an  immediate consequence  of \\Cref{lem:mean one change deviation bound}. \n\t\\\n\t\\\\\n\t\\\\\n\t{\\bf Step 2.} Since $\\{ \\widehat \\eta_{k}\\}_{k=1}^{\\widehat K}$ are the change points induced by $\\widehat {\\mathcal P}$, \\eqref{eq:K consistency step 2} holds because $\\hatp$ is a minimizer.\n\\\\\n\\\\\n\t{\\bf Step 3.}\nFor every $ \\I  =(s,e]\\in \\hatp$, by \\Cref{prop:mean local consistency}, we know that $\\I$ contains at most two change points. We only show the proof for the two-change-points case as the other case is easier. Denote\n\t\\[\n\t\t \\I =  (s ,\\eta_{q}]\\cup (\\eta_{q},\\eta_{q+1}] \\cup  (\\eta_{q+1} ,e]  = {  \\mathcal J } _1 \\cup {  \\mathcal J } _2 \\cup {  \\mathcal J } _{3},\n\t\\]\nwhere $\\{ \\eta_{q},\\eta_{q+1}\\} =\\I \\, \\cap \\, \\{\\eta_k\\}_{k=1}^K$. \n\nFor each $ m=1,2,3$, if $|{  \\mathcal J } _m|\\geq C_\\mclF\\sigma_{\\epsilon}^2 { \\mathfrak{s} } \\log(np)$, then by \\Cref{lem:mean one change deviation bound}, it holds that\n$$\n\\sum_{ i \\in {  \\mathcal J } _m }\\|y_ i - \\widehat\\mu_{{  \\mathcal J } _m}\\|_2^2   \\leq \\sum_{ i \\in {  \\mathcal J } _ m } \\|y_ i - \\mu^*_i\\|_2^2 + C\\sigma_{\\epsilon}^2 { \\mathfrak{s} } \\log(np). \n$$\nThus, we have\n\\begin{equation}\n    \\mclF(\\widehat\\mu_{{  \\mathcal J } _m},{  \\mathcal J } _m)\\leq \\mclF(\\mu^*_{{  \\mathcal J } _m},{  \\mathcal J } _m) + C\\sigma_{\\epsilon}^2 { \\mathfrak{s} } \\log(np).\n\\end{equation}\nOn the other hand, by \\Cref{lem:mean loss deviation no change point}, we have\n$$\n\\mclF(\\widehat\\mu_{\\I},{  \\mathcal J } _m)   \\geq \\mclF(\\mu^*_{{  \\mathcal J } _m},{  \\mathcal J } _m) - C\\sigma_{\\epsilon}^2 { \\mathfrak{s} } \\log(np). \n$$\nTherefore the last two inequalities above imply that \n\\begin{align} \n\\label{eq:K consistency step 3 inequality 3}  \n\\sum_{i \\in \\I }  \\mclF(\\widehat \\mu_\\I,\\I) \\geq & \\sum_{m=1}^{3} \\sum_{ i \\in {  \\mathcal J } _m }\\mclF(\\widehat \\mu_\\I,{  \\mathcal J } _m) \\\\\n\\geq & \\sum_{m=1}^{3}   \\sum_{ i \\in {  \\mathcal J } _m}\\mclF(\\widehat \\mu_{{  \\mathcal J } _m},{  \\mathcal J } _m) - C\\sigma_{\\epsilon}^2 { \\mathfrak{s} } \\log(np).\n\\end{align}\nNote that \\eqref{eq:K consistency step 3} is an immediate consequence of   \\eqref{eq:K consistency step 3 inequality 3}.\n\n{\\bf Step 4.}\nFinally, to show \\eqref{eq:K consistency step 4},   let  $\\widetilde { \\mathcal P}$ denote    the partition induced by $\\{\\widehat \\eta_{1},\\ldots, \\widehat \\eta_{\\widehat K } , \\eta_1,\\ldots,\\eta_K\\}$. Then \n$| \\widetilde { \\mathcal P} | \\le K + \\widehat K+2   $ and that $\\mu^*_i$ is unchanged in every interval $\\I \\in \\widetilde { \\mathcal P}$. \n\tSo  \\Cref{eq:K consistency step 4} is an immediate consequence of   \\Cref{lem:mean one change deviation bound}.\n\\end{proof} \n\n\n\n\n\n\n\\begin{proof}[Proof of \\Cref{cor:mean local refinement}]\nFor each $k\\in [K]$, let $\\hat{\\mu}_t = \\hat{\\mu}^{(1)}$ if $s_k<t< \\hat{\\eta}_k$ and $\\hat{\\mu}_t =\\hat{\\mu}^{(2)}$ otherwise, and ${\\mu}^*_t$ be the true parameter at time point $t$. First we show that under conditions $\\tilde{K} = K$ and $\\max_{k\\in [K]}|\\tilde{\\eta}_k - \\eta_k|\\leq \\Delta \/ 5$, there is only one true change point $\\eta_k$ in $(s_k, e_k)$. It suffices to show that\n\\begin{equation}\n    |\\tilde{\\eta}_k - \\eta_k|\\leq \\frac{2}{3}(\\tilde{\\eta}_{k+1} - \\tilde{\\eta}_{k}),\\ \\text{and}\\ |\\tilde{\\eta}_{k+1} - \\eta_{k+1}|\\leq \\frac{1}{3}(\\tilde{\\eta}_{k+1} - \\tilde{\\eta}_{k}).\n    \\label{tmp_eq:goal_lr}\n\\end{equation}\nDenote $R = \\max_{k\\in [K]}|\\tilde{\\eta}_k - \\eta_k|$, then\n\\begin{align*}\n    \\tilde{\\eta}_{k+1} - \\tilde{\\eta}_{k} &= \\tilde{\\eta}_{k+1} - {\\eta}_{k+1} + {\\eta}_{k+1} - {\\eta}_{k} + {\\eta}_{k} - \\tilde{\\eta}_{k} \\\\\n    &= ({\\eta}_{k+1} - {\\eta}_{k}) + (\\tilde{\\eta}_{k+1} - {\\eta}_{k+1}) + ({\\eta}_{k} - \\tilde{\\eta}_{k})\\in [{\\eta}_{k+1} - {\\eta}_{k} - 2R,{\\eta}_{k+1} - {\\eta}_{k} + 2R].\n\\end{align*}\nTherefore, \\Cref{tmp_eq:goal_lr} is guaranteed as long as\n\\begin{equation*}\n    R\\leq \\frac{1}{3}(\\Delta - 2R),\n\\end{equation*}\nwhich is equivalent to $R\\leq \\Delta\/5$.\n\nNow without loss of generality, assume that $s_k<\\eta_k<\\hat{\\eta}_k<e_k$. Denote $\\mclI_k = \\{s_k + 1, \\cdots, e_k\\}$. Consider two cases:\n\n\\textbf{Case 1} If\n\\begin{equation*}\n    \\hat{\\eta}_k - \\eta_k < \\max\\{C\\sigma_{\\epsilon}^2 { \\mathfrak{s} } \\log(np), C\\sigma_{\\epsilon}^2 { \\mathfrak{s} } \\log(np)\/\\kappa^2\\},\n\\end{equation*}\nthen the proof is done.\n\n\\textbf{Case 2} If\n\\begin{equation*}\n    \\hat{\\eta}_k - \\eta_k \\geq \\max\\{C\\sigma_{\\epsilon}^2 { \\mathfrak{s} } \\log(np), C\\sigma_{\\epsilon}^2 { \\mathfrak{s} } \\log(np)\/\\kappa^2\\},\n\\end{equation*}\nthen we proceed to prove that $|\\hat{\\eta}_k - \\eta_k|\\leq C\\sigma_{\\epsilon}^2 { \\mathfrak{s} } \\log(np)\/\\kappa^2$ with probability at least $1 - (Tn)^{-3}$. Then we either prove the result or get an contradiction, and complete the proof in either case.\n\nBy definition, we have\n\\begin{equation*}\n    \\sum_{t\\in \\I}\\|y_t - \\hat{\\mu}_t\\|_2^2 + \\zeta\\sum_{i = 1}^p \\sqrt{\\sum_{i\\in \\I}(\\hat{\\mu}_t)_i^2}\\leq \\sum_{t\\in \\I}\\|y_t - {\\mu}^*_t\\|_2^2 + \\zeta\\sum_{i = 1}^p \\sqrt{\\sum_{i\\in \\I}({\\mu}^*_t)_i^2},\n\\end{equation*}\nwhich implies that\n\\begin{equation*}\n    \\sum_{t\\in \\I}\\|\\mu^*_t - \\hat{\\mu}_t\\|_2^2 + \\zeta\\sum_{i = 1}^p \\sqrt{\\sum_{i\\in \\I}(\\hat{\\mu}_t)_i^2}\\leq 2\\sum_{t\\in \\I}(y_t - \\mu^*_t)^\\top (\\hat{\\mu}_t - \\mu^*_t) + \\zeta\\sum_{i = 1}^p \\sqrt{\\sum_{i\\in \\I}({\\mu}^*_t)_i^2}.\n\\end{equation*}\nDenote $\\delta_t = \\hat{\\mu}_t - \\mu^*_t$. Notice that\n\\begin{align*}\n    & \\sum_{i  \\in [p]} \\sqrt{\\sum_{i\\in \\I}({\\mu}^*_t)_i^2} - \\sum_{i  \\in [p]} \\sqrt{\\sum_{i\\in \\I}(\\hat{\\mu}_t)_i^2} \\\\ \n    =& \\sum_{i \\in [p]} \\sqrt{\\sum_{i\\in \\I}({\\mu}^*_t)_i^2} - \\sum_{i \\in S} \\sqrt{\\sum_{i\\in \\I}(\\hat{\\mu}_t)_i^2} - \\sum_{i \\in S^c} \\sqrt{\\sum_{i\\in \\I}(\\hat{\\mu}_t)_i^2} \\\\\n    \\leq & \\sum_{i \\in S} \\sqrt{\\sum_{i\\in \\I}(\\delta_t)_i^2} - \\sum_{i \\in S^c} \\sqrt{\\sum_{i\\in \\I}(\\delta_t)_i^2}.\n\\end{align*}\nNow we check the cross term. Notice that the variance of $\\sum_{t\\in \\I}(\\epsilon_t)_i(\\delta_t)_i$ is $\\sum_{t\\in \\I}(\\delta_t)_i^2$, so with probability at least $1 - (np)^{-5}$,\n\\begin{equation*}\n    \\sum_{t\\in \\I}(y_t - \\mu^*_t)^\\top (\\hat{\\mu}_t - \\mu^*_t)\\leq C\\sigma_{\\epsilon}\\sqrt{\\log(np)}\\sum_{i \\in [p]}\\sqrt{\\sum_{t\\in \\I}(\\delta_t)_i^2}\\leq \\frac{\\zeta}{4}\\sum_{i \\in [p]}\\sqrt{\\sum_{t\\in \\I}(\\delta_t)_i^2},\n\\end{equation*}\nsince $\\zeta = C_{\\zeta}\\sigma_{\\epsilon}\\sqrt{\\log(np)}$ with sufficiently large constant $C_{\\zeta}$. Combining inequalities above, we can get\n\\begin{align*}\n    \\sum_{t\\in \\I}\\|\\delta_t\\|_2^2 +\\frac{\\zeta}{2}\\sum_{i\\in S^c}\\sqrt{\\sum_{t\\in \\I}(\\delta_t)_i^2}\n    \\leq & \\frac{3\\zeta}{2}\\sum_{i\\in S}\\sqrt{\\sum_{t\\in \\I}(\\delta_t)_i^2} \\\\\n    \\leq & \\frac{3\\zeta}{2}\\sqrt{ { \\mathfrak{s} } }\\sqrt{\\sum_{t\\in \\I}\\|(\\delta_t)_S\\|_2^2} \\\\\n    \\leq & \\frac{3\\zeta}{2}\\sqrt{ { \\mathfrak{s} } }\\sqrt{\\sum_{t\\in \\I}\\|\\delta_t\\|_2^2},\n\\end{align*}\nwhich implies that\n\\begin{equation}\n    \\sum_{t\\in \\I}\\|\\delta_t\\|_2^2\\leq \\frac{9}{4} { \\mathfrak{s} } \\zeta^2\\leq C { \\mathfrak{s} } \\sigma_{\\epsilon}^2\\log(np).\n    \\label{tmp_eq:upper_bound_delta_local_refine}\n\\end{equation}\nWithout loss of generality, assume that $\\hat{\\eta}>\\eta_k$ and denote\n$$\n{  \\mathcal J } _1 = [s_k, \\eta_k), {  \\mathcal J } _2 = [\\eta_k, \\hat{\\eta}_k), {  \\mathcal J } _3 = [\\hat{\\eta}_k, e_k),\n$$\nand $\\mu^{(1)} = \\mu^*_{\\eta_k - 1}$, $\\mu^{(2)} = \\mu^*_{\\eta_k}$. Then \\Cref{tmp_eq:upper_bound_delta_local_refine} is equivalent to\n\\begin{equation*}\n    {  \\mathcal J } _1\\|\\hat{\\mu}^{(1)} - \\mu^{(1)}\\|_2^2 + {  \\mathcal J } _2\\|\\hat{\\mu}^{(1)} - \\mu^{(2)}\\|_2^2 + {  \\mathcal J } _3\\|\\hat{\\mu}^{(2)} - \\mu^{(2)}\\|_2^2 \\leq C { \\mathfrak{s} } \\sigma_{\\epsilon}^2\\log(np).\n\\end{equation*}\nSince $|\\mclJ_1|=\\eta_k - s_k\\geq c_0\\Delta$ with some constant $c_0$ under \\Cref{assp: DCDP_mean}, we have\n\\begin{equation}\n    \\Delta \\|\\hat{\\mu}^{(1)} - \\mu^{(1)}\\|_2^2\\leq c_0|\\mclJ_1|\\|\\hat{\\mu}^{(1)} - \\mu^{(1)}\\|_2^2 \\leq C \\sigma_{\\epsilon}^2 { \\mathfrak{s} } \\log(np)\\leq c_2\\Delta \\kappa^2,\n\\end{equation}\nwith some constant $c_2\\in (0,1\/4)$, where the last inequality is due to the fact that $ { \\mathcal B_n}  \\rightarrow \\infty$. Thus we have\n\\begin{equation*}\n    \\|\\hat{\\mu}^{(1)} - \\mu^{(1)}\\|_2^2\\leq c_2\\kappa^2.\n\\end{equation*}\nTriangle inequality gives\n\\begin{equation*}\n    \\|\\hat{\\mu}^{(1)} - \\mu^{(2)}\\|_2\\geq \\|\\mu^{(1)} - \\mu^{(2)}\\|_2 - \\|\\hat{\\mu}^{(1)} - \\mu^{(1)}\\|_2 \\geq \\kappa\/2.\n\\end{equation*}\nTherefore, $\\kappa^2|\\mclJ_2|\/4\\leq |\\mclJ_2|\\|\\hat{\\mu}^{(1)} - \\mu^{(2)}\\|_2^2\\leq C \\sigma_{\\epsilon}^2 { \\mathfrak{s} } \\log(np)$ and\n\\begin{equation*}\n    |\\hat{\\eta}_k - \\eta_k| = |\\mclJ_2|\\leq \\frac{C \\sigma_{\\epsilon}^2 { \\mathfrak{s} } \\log(np)}{\\kappa^2}.\n\\end{equation*}\n\\end{proof}\n\n\n\n\n\n\n\\subsubsection{Technical lemmas and proofs}\nThroughout this section, let $\\widehat { \\mathcal P }  $ denote the output of  \\Cref{algorithm:DCDP_divide}.\n \n \n \n\n\n\n\n\n\\bnlem\n\\label{lem:mean loss deviation no change point}\nLet $\\I\\subset [1,T]$ be any interval that contains no change point. Then for any interval ${  \\mathcal J }  \\supset \\I$, it holds with probability at least $1 - (np)^{-5}$ that\n\\begin{equation*}\n    \\mclF(\\mu^*_{\\I},\\I)\\leq \\mclF(\\widehat{\\mu}_{  \\mathcal J } ,\\I) + C\\sigma_{\\epsilon}^2 { \\mathfrak{s} } \\log(np).\n\\end{equation*}\n\\enlem\n\\begin{proof}\n{\\bf Case 1}. If $|\\I|< C_\\mclF\\sigma_{\\epsilon} { \\mathfrak{s} } \\log(np)$, then by definition, we have $\\mclF(\\mu^*_{\\I},\\I) = \\mclF(\\widehat \\mu^*_{{  \\mathcal J } },\\I) = 0$ and the inequality holds.\n\n{\\bf Case 2}. If $|\\I|\\geq  C_\\mclF\\sigma_{\\epsilon} { \\mathfrak{s} } \\log(np)$, then take difference and we can get\n\\begin{align*}\n    &\\sum_{i\\in \\I}\\|X_i - \\mu^*_i\\|_2^2 - \\sum_{i\\in \\I}\\|X_i - \\widehat{\\mu}_{  \\mathcal J } \\|_2^2 \\\\\n    =& 2(\\widehat \\mu_{{  \\mathcal J } } - \\mu^*_{\\I})^\\top\\sum_{i\\in \\I}\\epsilon_i -|\\I|\\|\\mu^*_{\\I} - \\widehat \\mu_{{  \\mathcal J } }\\|_2^2\\\\\n    \\leq & 2(\\|(\\widehat \\mu^*_{{  \\mathcal J } } - \\mu^*_{\\I})_S\\|_1 + \\|(\\widehat \\mu_{{  \\mathcal J } } - \\mu^*_{\\I})_{S^c}\\|_1)\\|\\sum_{i\\in \\I}\\epsilon_i\\|_{\\infty} -|\\I|\\|\\mu^*_{\\I} - \\widehat \\mu^*_{{  \\mathcal J } }\\|_2^2\\\\\n    \\leq& c_1\\|\\widehat \\mu_{{  \\mathcal J } } - \\mu^*_{\\I}\\|_2\\sigma_{\\epsilon}\\sqrt{ { \\mathfrak{s} } |\\I|\\log(np)} + c_2\\sigma_{\\epsilon}\\sqrt{\\frac{\\log(np)}{|\\I|}}\\cdot c_1\\sigma_{\\epsilon}\\sqrt{|\\I|\\log(np)} - |\\I|\\|\\mu^*_{\\I} - \\widehat \\mu_{{  \\mathcal J } }\\|_2^2 \\\\\n    \\leq & \\frac{1}{2}|\\I|\\|\\mu^*_{\\I} - \\widehat \\mu_{{  \\mathcal J } }\\|_2^2 + 2c_1^2\\sigma_{\\epsilon}^2 { \\mathfrak{s} } \\log(np) + c_2\\sigma_{\\epsilon}^2 { \\mathfrak{s} } \\log(np) - |\\I|\\|\\mu^*_{\\I} - \\widehat \\mu_{{  \\mathcal J } }\\|_2^2\\\\\n    \\leq & C\\sigma_{\\epsilon}^2 { \\mathfrak{s} } \\log(np),\n\\end{align*}\nwhere in the second inequality we use the definition of the index set $S$ and \\Cref{lem:estimation_high_dim_mean}.\n\\end{proof}\n\n\n\n\n\n\n \n\\bnlem\n\\label{lem:mean single cp}\nSuppose the good events \n$\\mathcal L ( { \\mathcal B_n^{-1}   \\Delta }  ) $ and  $\\mathcal R ( { \\mathcal B_n^{-1}   \\Delta } ) $ defined  in \\Cref{eq:left and right approximation of change points} hold. \nLet   $ \\I=(s,e] \\in \\mathcal {\\widehat P}  $  be such that $\\I$ contains exactly  one    change point $ \\eta_k $. \n Then  with probability  at least $1-(np)^{-3}$,   it holds that \n $$\\min\\{ \\eta_k -s ,e-\\eta_k \\}  \\lesssim    \\sigma_{\\epsilon}^2\\bigg( \\frac{  { \\mathfrak{s} } \\log(np) +\\gamma }{\\kappa_k^2 }\\bigg)  +   { \\mathcal B_n^{-1}   \\Delta }     .$$\n  \\enlem\n\\begin{proof}   \nIf either $ \\eta_k -s \\le  { \\mathcal B_n^{-1}   \\Delta }   $ or $e-\\eta_k\\le  { \\mathcal B_n^{-1}   \\Delta } $, then there is nothing to show. So assume that \n$$  \\eta_k -s  >  { \\mathcal B_n^{-1}   \\Delta }  \\quad \\text{and} \\quad   e-\\eta_k  >  { \\mathcal B_n^{-1}   \\Delta }  . $$\nBy event $\\mathcal R ( { \\mathcal B_n^{-1}   \\Delta }  )$, there exists  $ s_ u  \\in \\{ s_q\\}_{q=1}^ { \\mathcal Q} $ such that \n$$0\\le s_u - \\eta_k \\le  { \\mathcal B_n^{-1}   \\Delta } . $$\n So \n$$ \\eta_k \\le s_ u \\le e .$$\nDenote \n$$ \\I_ 1 = (s,s_u] \\quad \\text{and} \\quad \\I_2 = (s_u, e] .$$\nSince \n$s, e,  s_u \\in  \\{ s_q\\}_{q=1}^ { \\mathcal Q}   $,   it follows that \n\\begin{align}\\nonumber\n\\sum_{i \\in \\I }\\|X_i - \\widehat \\mu_\\I \\|_2 ^2 \\le &  \\sum_{i \\in \\I_1 }\\|X_i - \\widehat \\mu_{\\I_1}  \\|_2 ^2  + \\sum_{i \\in \\I_2 }\\|X_i - \\widehat \\mu_{\\I_2}  \\|_2 ^2   + \\gamma \n\\\\\\nonumber \n\\le &  \\sum_{i \\in \\I_1 }\\|X_i -  \\mu^*_i \\|_2 ^2 + C_1 \\big (  \\sigma_{\\epsilon}^2 { \\mathfrak{s} } \\log(np) + (s_u -\\eta_k) \\kappa_k^2 \\big )  \n\\\\ \\nonumber\n& + \\sum_{i \\in \\I_1 }\\|X_i -  \\mu^*_i \\|_2 ^2 + C_1\\sigma_{\\epsilon}^2 { \\mathfrak{s} } \\log(np)  + \\gamma\n\\\\ \\nonumber \n=&  \\sum_{i \\in \\I }\\|X_i -  \\mu^*_i  \\|_2 ^2  +C_2 \\big (  \\sigma_{\\epsilon}^2 { \\mathfrak{s} } \\log(np) + (s_u -\\eta_k)   \\kappa_k^2 \\big )+ \\gamma \n\\\\\n\\le &  \\sum_{i \\in \\I }\\|X_i -  \\mu^*_i \\|_2 ^2  +C_2 \\big (  \\sigma_{\\epsilon}^2 { \\mathfrak{s} } \\log(np) +   { \\mathcal B_n^{-1}   \\Delta }     \\kappa_k^2   \\big ) + \\gamma  , \\label{eq:one change point step 1}\n\\end{align}\nwhere the first inequality follows from the fact that $ \\I =(s ,e ] \\in \\widehat { \\mathcal P} $ and so it is the local minimizer, the second inequality follows from \\Cref{lem:mean one change deviation bound} {\\bf a} and {\\bf b} and the observation that \n$$  \\eta_k -s >  { \\mathcal B_n^{-1}   \\Delta }   \\ge s_u -\\eta_k $$ \nDenote \n$$ {  \\mathcal J } _1  = (s,\\eta_k ] \\quad \\text{and} \\quad {  \\mathcal J } _2 = (\\eta_k , e] .$$\n\\Cref{eq:one change point step 1}  gives \n\\begin{align*}\n\\sum_{i \\in {  \\mathcal J } _1 }\\|X_i - \\widehat \\mu_\\I \\|_2 ^2 +\\sum_{i \\in {  \\mathcal J } _2 }\\|X_i - \\widehat \\mu_\\I \\|_2 ^2  \n \\le \\sum_{i \\in {  \\mathcal J } _1  }\\|X_i -  \\mu^*_{ {  \\mathcal J } _1 } \\|_2 ^2   +\\sum_{i \\in {  \\mathcal J } _2  }\\|X_i -  \\mu^*_{ {  \\mathcal J } _2  } \\|_2 ^2  +C_2  \\big (\\sigma_{\\epsilon}^2 { \\mathfrak{s} } \\log(np) +    { \\mathcal B_n^{-1}   \\Delta }  \\kappa_k^2  \\big )  + \\gamma \n\\end{align*}\nwhich leads to \n\\begin{align*}\n&\\sum_{i \\in {  \\mathcal J } _1 }\\|\\widehat \\mu_\\I  - \\mu^*_{{  \\mathcal J } _1 }\\|_2^2 +\\sum_{i \\in {  \\mathcal J } _2 }\\|\\widehat \\mu_\\I  - \\mu^*_{{  \\mathcal J } _2  } \\|_2^2  \n\\\\\n \\leq  & 2 \\sum_{i \\in {  \\mathcal J } _1  } \\epsilon_i^\\top (\\widehat \\mu_\\I  -  \\mu^*_{ {  \\mathcal J } _1 } )     +2 \\sum_{i \\in {  \\mathcal J } _2  } \\epsilon_i^\\top (\\widehat \\mu_\\I  -  \\mu^*_{ {  \\mathcal J } _2 } )      +C_2  \\big (\\sigma_{\\epsilon}^2 { \\mathfrak{s} } \\log(np) +  \\kappa_k^2   { \\mathcal B_n^{-1}   \\Delta }     \\big )  + \\gamma \n \\\\\n \\leq & 2\\sigma_{\\epsilon}\\sum_{j = 1,2}\\|\\widehat \\mu_\\I  -  \\mu^*_{ {  \\mathcal J } _j }\\|_2 \\sqrt{|{  \\mathcal J } _j|\\log(np)}+   C_2  \\big( \\sigma_{\\epsilon}^2 { \\mathfrak{s} } \\log(np) +  \\kappa_k^2 { \\mathcal B_n^{-1}   \\Delta } \\big ) + \\gamma  \\\\\n \\leq & \\frac{1}{2} \\sum_{j = 1,2}|{  \\mathcal J } _j|\\|\\widehat \\mu_\\I  -  \\mu^*_{ {  \\mathcal J } _j }\\|_2^2 + C_3  \\big (\\sigma_{\\epsilon}^2 { \\mathfrak{s} } \\log(np) +  \\kappa_k^2   { \\mathcal B_n^{-1}   \\Delta }     \\big )  + \\gamma,\n\\end{align*} \nwhere the second inequality holds because the variance of $\\sum_{i \\in {  \\mathcal J } _1  } \\epsilon_i^\\top (\\mu_\\I  -  \\mu^*_{ {  \\mathcal J } _1 } )$ is upper bounded by $|{  \\mathcal J } _1|\\sigma^2_{\\epsilon}\\|\\mu_\\I  -  \\mu^*_{ {  \\mathcal J } _1 }\\|_2^2$.\n\nIt follows that \n$$ |{  \\mathcal J } _1 |\\|  \\widehat \\mu_\\I  - \\mu^*_{{  \\mathcal J } _1 }\\|_2 ^2 + |{  \\mathcal J } _2 |\\|  \\widehat \\mu_\\I  - \\mu^*_{{  \\mathcal J } _2 }\\|_2 ^2  =  \\sum_{i \\in {  \\mathcal J } _1 }\\|  \\widehat \\mu_\\I  - \\mu^*_{{  \\mathcal J } _1 }\\|_2 ^2 +\\sum_{i \\in {  \\mathcal J } _2 }\\| \\widehat \\mu_\\I  - \\mu^*_{{  \\mathcal J } _2  }  \\|_2^2    \\le C_4\\big ( \\sigma_{\\epsilon}^2 { \\mathfrak{s} } \\log(np) + { \\mathcal B_n^{-1}   \\Delta }  \\kappa_k^2    \\big )+ 2\\gamma  .$$\nNote that \n$$ \\inf_{ a \\in \\mathbb R  }  |{  \\mathcal J } _1 |\\|  a - \\mu^*_{{  \\mathcal J } _1 }\\|_2 ^2 + |{  \\mathcal J } _2 |\\| a  - \\mu^*_{{  \\mathcal J } _2 }\\| ^2  = \\kappa_k ^2  \\frac{|{  \\mathcal J } _1| |{  \\mathcal J } _2|}{| \\I| }   \\ge \\frac{ \\kappa_k^2 }{2} \\min\\{ |{  \\mathcal J } _1| ,|{  \\mathcal J } _2| \\}    . $$ \nThis leads to \n$$  \\frac{ \\kappa_k^2 }{2}\\min\\{ |{  \\mathcal J } _1| ,|{  \\mathcal J } _2| \\}    \\le  C_4\\big ( \\sigma_{\\epsilon}^2 { \\mathfrak{s} } \\log(np) + { \\mathcal B_n^{-1}   \\Delta }   \\kappa_k^2   + \\gamma  \\big )  ,$$\nwhich is \n$$ \\min\\{ |{  \\mathcal J } _1| ,|{  \\mathcal J } _2| \\}   \\le C_5 \\bigg( \\frac{\\sigma_{\\epsilon}^2 { \\mathfrak{s} } \\log(np) + \\gamma  }{\\kappa_k^2 } +   { \\mathcal B_n^{-1}   \\Delta }   \\bigg) .$$\n \\end{proof} \n \n \n \\bnlem \nSuppose the good events \n$\\mathcal L ( { \\mathcal B_n^{-1}   \\Delta }  ) $ and  $\\mathcal R (  { \\mathcal B_n^{-1}   \\Delta }  ) $ defined  in  \\Cref{eq:left and right approximation of change points} hold. \nLet   $ \\I=(s,e] \\in \\mathcal {\\widehat P}  $  be an interval that contains  exactly two  change points $ \\eta_k,\\eta_{k+1}  $.  Suppose in addition that \n\\begin{align} \\label{eq:1D two change points snr}\n\\Delta \\kappa^2 \\ge C    { \\mathcal B_n} ^{1\/2}  \\big(\\sigma_{\\epsilon}^2 { \\mathfrak{s} } \\log(np) + \\gamma)    \n\\end{align} \nfor sufficiently large constant $C $. \n Then  with probability  at least $1-(np)^{-3}$,   it holds that \n\\begin{align*} \n \\eta_k -s    \\lesssim       { \\mathcal B_n^{-1\/2}   \\Delta }     \\quad \\text{and} \\quad \n e-\\eta_{k+1}  \\lesssim    { \\mathcal B_n^{-1\/2}   \\Delta }      . \n \\end{align*} \n \\enlem \n\\begin{proof} \nSince the events $\\mathcal L (  { \\mathcal B_n^{-1}   \\Delta }   ) $ and  $\\mathcal R (  { \\mathcal B_n^{-1}   \\Delta }  ) $ hold, let $ s_u, s_v$ be such that \n $\\eta_k \\le s_u \\le s_v \\le \\eta_{k+1} $ and that \n $$ 0 \\le s_u-\\eta_k \\le    { \\mathcal B_n^{-1}   \\Delta }  ,  \\quad  0\\le  \\eta_{k+1} - s_v \\le  { \\mathcal B_n^{-1}   \\Delta }    .    $$\n\n \\begin{center}\n \\begin{tikzpicture} \n\\draw[ - ] (-10,0)--(1,0);\n \\node[color=black] at (-8,-0.3) {\\small s};\n \\draw[  (-, ultra thick, black] (-8,0) -- (-7.99,0);\n  \\draw[ { -]}, ultra thick, black] (-1.0002,0) -- (-1,0);\n\n   \\node[color=black] at (-7,-0.3) {\\small $\\eta_k$};\n\\draw(-7 ,0)circle [radius=2pt] ;\n\n  \\node[color=black] at (-6.5,-0.3) {\\small $s_u$};\n\\draw plot[mark=x, mark options={color=black, scale=1.5}] coordinates {(-6.5,0)  }; \n \n\n  \\node[color=black] at (-2.3,-0.3) {\\small $\\eta_{k+1}$};\n\\draw(-2.3 ,0)circle [radius=2pt] ;\n  \\node[color=black] at (-3 ,-0.3) {\\small $s_v$};\n\\draw plot[mark=x, mark options={color=black, scale=1.5}] coordinates {(-3,0)  }; \n  \n   \\node[color=black] at (-1,-0.3) {\\small $e$};\n\n \n\\end{tikzpicture}\n\\end{center} \n\nDenote \n$$ \\mathcal I_ 1= (  s, s _u], \\quad \\I_2  =(s_u, s_v] \\quad \\text{and} \\quad \\I_3  = (s_v,e]. $$\nIn addition, denote \n$$ {  \\mathcal J } _1  = (s,\\eta_k ], \\quad {  \\mathcal J } _2=(\\eta_k, \\eta_{k} + \\frac{ \\eta_{k+1}   -\\eta_k  }{2}], \\quad {  \\mathcal J } _3 =  ( \\eta_k+ \\frac{ \\eta_{k+1}   -\\eta_k  }{2},\\eta_{k+1 } ]  \\quad \\text{and} \\quad {  \\mathcal J } _4 = (\\eta_{k+1}  , e] .$$\nSince \n$s, e,  s_u ,s_v  \\in  \\{ s_q\\}_{q=1}^ { \\mathcal Q}   $, then it follows from the definition of $\\hat{\\mclP}$ that \n\\begin{align}\\nonumber\n& \\sum_{i \\in \\I }\\|X_i - \\widehat \\mu_\\I \\|_2^2\n\\\\  \\nonumber \n\\le &  \\sum_{i \\in \\I_1 }\\|X_i - \\widehat \\mu_{\\I_1} \\|_2 ^2  + \\sum_{i \\in \\I_2 }\\|X_i - \\widehat \\mu_{\\I_2}\\|_2^2  + \\sum_{i \\in \\I_3 }\\|X_i - \\widehat \\mu_{\\I_3}\\|_2^2  +2 \\gamma \n\\\\\\nonumber \n\\le &  \\sum_{i \\in \\I_1 }\\|X_i -  \\mu^*_i\\|_2^2 + C_1 \\bigg ( \\sigma_{\\epsilon}^2  { \\mathfrak{s} } \\log(np) +  \\frac{|{  \\mathcal J } _1| (s_u -\\eta_k ) }{ |{  \\mathcal J } _1| + (s_u -\\eta_k )  }\\kappa_k^2 \\bigg )  +\n \\sum_{i \\in \\I_2 }\\|X_i -  \\mu^*_i\\|_2^2    +C_1   \\sigma_{\\epsilon}^2  { \\mathfrak{s} } \\log(np)\n \\\\ \\nonumber \n  + &  \\sum_{i \\in \\I_3 }\\|X_i -  \\mu^*_i\\|_2^2 + C_1 \\bigg( \\sigma_{\\epsilon}^2  { \\mathfrak{s} } \\log(np)+  \\frac{|{  \\mathcal J } _4 | (\\eta_{k+1} -s_v   )  }{ |{  \\mathcal J } _4 | + (\\eta_{k+1} -s_v   )   } \\kappa_{k+1} ^2   \\bigg)   +2  \\gamma\n\\\\ \n\\le  &  \\sum_{i \\in \\I }\\|X_i -  \\mu^*_i\\|_2^2  +C_1' \\bigg (  \\sigma_{\\epsilon}^2  { \\mathfrak{s} } \\log(np) +  \\frac{|{  \\mathcal J } _1| (s_u -\\eta_k ) }{ |{  \\mathcal J } _1| + (s_u -\\eta_k )  }\\kappa_k^2     +  \\frac{|{  \\mathcal J } _4 | (\\eta_{k+1} -s_v   )  }{ |{  \\mathcal J } _4 | + (\\eta_{k+1} -s_v   )   }  \\kappa_{k+1} ^2        \\bigg )+2  \\gamma \n  \\label{eq:two change points step 1}\n\\end{align}\nwhere the first inequality follows from the fact that $ \\I =(s ,e ] \\in \\widehat { \\mathcal P} $, the second inequality follows from \\Cref{lem:mean one change deviation bound} {\\bf a} and {\\bf b}. \n\\Cref{eq:two change points step 1}  gives \n\\begin{align} \\nonumber \n& \\sum_{i \\in {  \\mathcal J } _1 }\\|X_i - \\widehat \\mu_\\I\\|_2^2 +\\sum_{i \\in {  \\mathcal J } _2 }\\|X_i - \\widehat \\mu_\\I\\|_2^2  +\\sum_{i \\in {  \\mathcal J } _3 }\\|X_i - \\widehat \\mu_\\I\\|_2^2 +\\sum_{i \\in {  \\mathcal J } _4 }\\|X_i - \\widehat \\mu_\\I\\|_2^2   \n\\\\ \\nonumber  \n \\le& \\sum_{i \\in {  \\mathcal J } _1  }\\|X_i -  \\mu^*_{ {  \\mathcal J } _1 }\\|_2^2   +\\sum_{i \\in {  \\mathcal J } _2  }\\|X_i -  \\mu^*_{ {  \\mathcal J } _2  }\\|_2^2 +\\sum_{i \\in {  \\mathcal J } _3  }\\|X_i -  \\mu^*_{ {  \\mathcal J } _3 }\\|_2^2   +\\sum_{i \\in {  \\mathcal J } _4  }\\|X_i -  \\mu^*_{ {  \\mathcal J } _4  }\\|_2^2  \\\\\n + & C_1' \\bigg (\\sigma_{\\epsilon}^2  { \\mathfrak{s} } \\log(np) +   \\frac{|{  \\mathcal J } _1| (s_u -\\eta_k ) }{ |{  \\mathcal J } _1| + (s_u -\\eta_k )  }\\kappa_k^2     + \\frac{|{  \\mathcal J } _4 | (\\eta_{k+1} -s_v   )  }{ |{  \\mathcal J } _4 | + (\\eta_{k+1} -s_v   )   }  \\kappa_{k+1}    ^2  \\bigg )  + 2\\gamma.  \\label{eq:1D two change points first}\n\\end{align}\nNote that for $\\ell\\in \\{1,2,3,4 \\}$,\n\\begin{align}\\nonumber  \n  &\\sum_{i \\in {  \\mathcal J } _\\ell }\\|X_i - \\widehat \\mu_\\I\\|_2^2  \n -   \\sum_{i \\in {  \\mathcal J } _\\ell   }\\|X_i -  \\mu^*_{ {  \\mathcal J } _ \\ell }\\|_2^2  -\\sum_{i \\in {  \\mathcal J } _ \\ell   }\\|\\widehat \\mu_\\I  - \\mu^*_{{  \\mathcal J } _ \\ell }\\|_2^2 \n\\\\\n=& \\nonumber \n 2 \\sum_{i \\in {  \\mathcal J } _ \\ell   } \\epsilon_i^\\top ( \\mu^*_{ {  \\mathcal J } _\\ell  }-\\widehat \\mu_\\I    )      \n\\\\\\nonumber   \n\\ge  & -  C\\sigma_{\\epsilon} \\|\\widehat \\mu_\\I  -  \\mu^*_{ {  \\mathcal J } _\\ell }\\|_2 \\sqrt{|{  \\mathcal J } _\\ell|\\log(np)}\n\\\\\n\\ge  &  -\\frac{1}{2}|{  \\mathcal J } _\\ell| \\|\\widehat \\mu_\\I  -  \\mu^*_{ {  \\mathcal J } _\\ell }\\|_2^2 - C'\\sigma_{\\epsilon}^2  { \\mathfrak{s} } \\log(np). \\nonumber  \n\\end{align} \nwhich gives \n\\begin{align} \\label{eq:1D two change points second}\n  &\\sum_{i \\in {  \\mathcal J } _\\ell }\\|X_i - \\widehat \\mu_\\I \\|^2  \n -   \\sum_{i \\in {  \\mathcal J } _\\ell   }\\|X_i -  \\mu^*_{ {  \\mathcal J } _ \\ell }\\|^2  \\ge \\frac{1}{2}\\sum_{i \\in {  \\mathcal J } _ \\ell   }\\| \\widehat \\mu_\\I  - \\mu^*_{{  \\mathcal J } _ \\ell }\\| ^2\n-   C_2\\sigma_{\\epsilon}^2  { \\mathfrak{s} } \\log(np) .\n\\end{align} \n\\Cref{eq:1D two change points first} and \n\\Cref{eq:1D two change points second} together implies that\n \n\\begin{align} \\label{eq:1D two change points third}   \\sum_{l=1}^ 4 \n   |{  \\mathcal J } _l | (\\widehat \\mu_\\I  -  \\mu^*_{ {  \\mathcal J } _\\ell } ) ^2        \\le C_3 \\bigg ( \\sigma_{\\epsilon}^2  { \\mathfrak{s} } \\log(np) +\\frac{|{  \\mathcal J } _1| (s_u -\\eta_k ) }{ |{  \\mathcal J } _1| + (s_u -\\eta_k )  }  \\kappa_k^2    +\\frac{|{  \\mathcal J } _4 | (\\eta_{k+1} -s_v   )  }{ |{  \\mathcal J } _4 | + (\\eta_{k+1} -s_v   )   } \\kappa_{k+1} ^2    \\bigg ) +4 \\gamma  .  \n   \\end{align}\nNote that \n\\begin{align}  \\label{eq:1D two change points signal lower bound} \\inf_{ a \\in \\mathbb R  }  |{  \\mathcal J } _1 |(  a - \\mu^*_{{  \\mathcal J } _1 }) ^2 + |{  \\mathcal J } _2 |( a  - \\mu^*_{{  \\mathcal J } _2 }) ^2  =&     \\frac{|{  \\mathcal J } _1| |{  \\mathcal J } _2|}{ |{  \\mathcal J } _1| +  |{  \\mathcal J } _2| } \\kappa_k ^2.\n\\end{align} \nSimilarly\n\\begin{align} \\label{eq:1D two change points signal lower bound 2} \\inf_{ a \\in \\mathbb R  }  |{  \\mathcal J } _3 |(  a - \\mu^*_{{  \\mathcal J } _3 }) ^2 + |{  \\mathcal J } _4 |( a  - \\mu^*_{{  \\mathcal J } _4 }) ^2   =     \\frac{|{  \\mathcal J } _3| |{  \\mathcal J } _4|}{|{  \\mathcal J } _3| + |{  \\mathcal J } _4| }       \\kappa_{k+1}  ^2  ,\\end{align} \n\\Cref{eq:1D two change points third} together with \n\\Cref{eq:1D two change points signal lower bound} and \\Cref{eq:1D two change points signal lower bound 2} leads to \n\\begin{align}\n\\label{eq:conclusion of two change points} \n  \\frac{|{  \\mathcal J } _1| |{  \\mathcal J } _2|}{ |{  \\mathcal J } _1| +  |{  \\mathcal J } _2| }  \\kappa_k ^2   +   \\frac{|{  \\mathcal J } _3| |{  \\mathcal J } _4|}{|{  \\mathcal J } _3| + |{  \\mathcal J } _4| } \\kappa_{k+1}  ^2  \\le C_3 \\bigg (\\sigma_{\\epsilon}^2  { \\mathfrak{s} } \\log(np) +\\frac{|{  \\mathcal J } _1| (s_u -\\eta_k ) }{ |{  \\mathcal J } _1| + (s_u -\\eta_k )  }  \\kappa_k^2    +\\frac{|{  \\mathcal J } _4 | (\\eta_{k+1} -s_v   )  }{ |{  \\mathcal J } _4 | + (\\eta_{k+1} -s_v   )   } \\kappa_{k+1} ^2    \\bigg ) +4 \\gamma  .\n  \\end{align}\n Note that \n$$ 0\\le s_u -\\eta_k \\le  { \\mathcal B_n^{-1}   \\Delta }  \\quad \\text{and} \\quad \n0 \\le \\eta_{k+1} -s_v \\le  { \\mathcal B_n^{-1}   \\Delta } ,\n $$ \nand so there are four possible cases.  \n\n{\\bf case a.} \nIf \n$$|{  \\mathcal J } _1| \\le  { \\mathcal B_n^{-1\/2}   \\Delta }   \\quad \\text{and} \\quad  \n |{  \\mathcal J } _4|   \\le  { \\mathcal B_n^{-1\/2}   \\Delta }  , $$ \n then the desired result  follows immediately. \n\n{\\bf case b.}   $|{  \\mathcal J } _1|   >   { \\mathcal B_n^{-1\/2}   \\Delta }     $ and  $|{  \\mathcal J } _4|     \\le  { \\mathcal B_n^{-1\/2}   \\Delta }      $. Then since $|{  \\mathcal J } _2| \\ge \\Delta \/2  $, it holds that \n$$   \\frac{|{  \\mathcal J } _1| |{  \\mathcal J } _2|}{ |{  \\mathcal J } _1| +  |{  \\mathcal J } _2| }  \\ge  \\frac{1}{2 }\\min\\{|{  \\mathcal J } _1| , |{  \\mathcal J } _2|  \\} \\ge  \\frac{1}{2 } { \\mathcal B_n^{-1\/2}   \\Delta } .  $$\nIn addition,\n$$\\frac{|{  \\mathcal J } _1| (s_u -\\eta_k ) }{ |{  \\mathcal J } _1| + (s_u -\\eta_k )  }  \\le s_u -\\eta_k \\le  { \\mathcal B_n^{-1}   \\Delta }   \\quad   \\text{and} \\quad  \\frac{|{  \\mathcal J } _4 | (\\eta_{k+1} -s_v   )  }{ |{  \\mathcal J } _4 | + (\\eta_{k+1} -s_v   )   }  \\le \\eta_{k+1} -s_v     \\le  { \\mathcal B_n^{-1}   \\Delta }  . \n$$\nSo  \\Cref{eq:conclusion of two change points}  leads to \n\\begin{align}\\label{eq:conclusion of two change points case two} \n\\frac{1}{2 } { \\mathcal B_n^{-1\/2}   \\Delta }    \\kappa_k ^2   +   \\frac{|{  \\mathcal J } _3| |{  \\mathcal J } _4|}{|{  \\mathcal J } _3| + |{  \\mathcal J } _4| } \\kappa_{k+1}  ^2  \\le C_3 \\bigg (\\sigma_{\\epsilon}^2  { \\mathfrak{s} } \\log(np) +  { \\mathcal B_n^{-1}   \\Delta }  \\kappa_k^2    +  { \\mathcal B_n^{-1}   \\Delta }  \\kappa_{k+1} ^2    \\bigg ) +4 \\gamma  .\n  \\end{align} \nSince $ \\kappa_k \\asymp \\kappa$ and $ \\kappa_{k+1}  \\asymp \\kappa$, \\Cref{eq:conclusion of two change points case two} gives \n$$ \\frac{1}{2 } { \\mathcal B_n^{-1\/2}   \\Delta }    \\kappa  ^2  \\le C_4 \\bigg ( \\sigma_{\\epsilon}^2  { \\mathfrak{s} } \\log(np) +  { \\mathcal B_n^{-1}   \\Delta }  \\kappa ^2    +  { \\mathcal B_n^{-1}   \\Delta }  \\kappa ^2    \\bigg ) +4 \\gamma   . $$\nSince $ { \\mathcal B_n}  $ is a diverging sequence, the above display gives \n$$ \\Delta \\kappa ^2  \\le C_5 { \\mathcal B_n} ^{1\/2 } (\\log(n) + \\gamma ).$$\nThis contradicts \\Cref{eq:1D two change points snr}.\n\n{\\bf case c.}  $|{  \\mathcal J } _1|    \\le    { \\mathcal B_n^{-1\/2}   \\Delta }     $ and \n$|{  \\mathcal J } _4|     >   { \\mathcal B_n^{-1\/2}   \\Delta }      $. Then the same argument  as that in  {\\bf case b} leads to the same contradiction.\n\n {\\bf case d.}     $|{  \\mathcal J } _1|   >   { \\mathcal B_n^{-1\/2}   \\Delta }     $ and  $|{  \\mathcal J } _4|     >   { \\mathcal B_n^{-1\/2}   \\Delta }      $. Then since $|{  \\mathcal J } _2|\\ge \\Delta \/2 , |{  \\mathcal J } _4| \\ge \\Delta \/2  $, it holds that \n$$   \\frac{|{  \\mathcal J } _1| |{  \\mathcal J } _2|}{ |{  \\mathcal J } _1| +  |{  \\mathcal J } _2| }  \\ge  \\frac{1}{2 }\\min\\{|{  \\mathcal J } _1| , |{  \\mathcal J } _2|  \\} \\ge  \\frac{1}{2 } { \\mathcal B_n^{-1\/2}   \\Delta } \\quad \\text{and} \\quad \n \\frac{|{  \\mathcal J } _3| |{  \\mathcal J } _4|}{ |{  \\mathcal J } _3| +  |{  \\mathcal J } _4| }  \\ge  \\frac{1}{2 }\\min\\{|{  \\mathcal J } _3| , |{  \\mathcal J } _4|  \\} \\ge  \\frac{1}{2 } { \\mathcal B_n^{-1\/2}   \\Delta }    $$\nIn addition, \n$$\\frac{|{  \\mathcal J } _4 | (\\eta_{k+1} -s_v   )  }{ |{  \\mathcal J } _4 | + (\\eta_{k+1} -s_v   )   }  \\le \\eta_{k+1} -s_v     \\le  { \\mathcal B_n^{-1}   \\Delta }   \\quad  \n\\frac{|{  \\mathcal J } _1| (s_u -\\eta_k ) }{ |{  \\mathcal J } _1| + (s_u -\\eta_k )  }  \\le s_u -\\eta_k \\le  { \\mathcal B_n^{-1}   \\Delta }  . \n$$\nSo  \\Cref{eq:conclusion of two change points}  leads to \n\\begin{align}\\label{eq:conclusion of two change points case four} \n\\frac{1}{2 } { \\mathcal B_n^{-1\/2}   \\Delta }    \\kappa_k ^2   +   \\frac{1}{2 } { \\mathcal B_n^{-1\/2}   \\Delta }      \\kappa_{k+1}  ^2  \\le C_6 \\bigg (\\sigma_{\\epsilon}^2  { \\mathfrak{s} } \\log(np) +  { \\mathcal B_n^{-1}   \\Delta }  \\kappa_k^2    +  { \\mathcal B_n^{-1}   \\Delta }  \\kappa_{k+1} ^2    \\bigg ) +4 \\gamma  .\n  \\end{align} \nNote that $ { \\mathcal B_n}  $ is a diverging sequence. So the above display gives \n$$ \\Delta \\big  ( \\kappa_{k}   ^2+ \\kappa_{k+1}^2  \\big)   \\le C_ 7  { \\mathcal B_n} ^{1\/2 } (\\sigma_{\\epsilon}^2  { \\mathfrak{s} } \\log(np) + \\gamma ) $$\nSince   $ \\kappa_k \\asymp \\kappa$ and  $ \\kappa_{k+1}  \\asymp \\kappa$. This contradicts \\Cref{eq:1D two change points snr}. \n \\end{proof}\n\n\n\n \\bnlem \nSuppose the good events \n$\\mathcal L ( { \\mathcal B_n^{-1}   \\Delta }  ) $ and  $\\mathcal R (  { \\mathcal B_n^{-1}   \\Delta }  ) $ defined  in  \\Cref{eq:left and right approximation of change points} hold. \n Suppose in addition that \n\\begin{align} \\label{eq:1D three change points snr}\n\\Delta \\kappa^2 \\ge C    { \\mathcal B_n}   \\big(\\sigma_{\\epsilon}^2  { \\mathfrak{s} } \\log(np) + \\gamma)    \n\\end{align} \nfor sufficiently large constant $C $. \n Then  with probability  at least $1-(np)^{-3}$,    there is no interval  $  \\widehat { \\mathcal P}  $  containing three or more true change points.   \n \\enlem  \n \n\n\n\\begin{proof} \nFor  contradiction,  suppose    $ \\I=(s,e] \\in \\mathcal {\\widehat P}  $  be such that $ \\{ \\eta_1, \\ldots, \\eta_M\\} \\subset \\I $ with $M\\ge 3$. Throughout the proof, $M$ is assumed to be a parameter that can potentially change with $n$. \nSince the events $\\mathcal L (  { \\mathcal B_n^{-1}   \\Delta }   ) $ and  $\\mathcal R (  { \\mathcal B_n^{-1}   \\Delta }  ) $ hold, by relabeling $\\{ s_q\\}_{q=1}^ { \\mathcal Q}   $ if necessary,   let $ \\{ s_m\\}_{m=1}^M $ be such that \n $$ 0 \\le s_m  -\\eta_m  \\le  { \\mathcal B_n^{-1}   \\Delta }     \\quad \\text{for} \\quad 1 \\le m \\le M-1  $$ and that\n $$ 0\\le \\eta_M  - s_M \\le  { \\mathcal B_n^{-1}   \\Delta }  .$$ \n Note that these choices ensure that  $ \\{ s_m\\}_{m=1}^M  \\subset \\I . $\n \n \\begin{center}\n \\begin{tikzpicture} \n\\draw[ - ] (-10,0)--(1,0);\n \\node[color=black] at (-8,-0.3) {\\small s};\n \\draw[  (-, ultra thick, black] (-8,0) -- (-7.99,0);\n  \\draw[ { -]}, ultra thick, black] (-1.0002,0) -- (-1,0);\n\n   \\node[color=black] at (-7,-0.3) {\\small $\\eta_1$};\n\\draw(-7 ,0)circle [radius=2pt] ;\n\n  \\node[color=black] at (-6.5,-0.3) {\\small $s_1$};\n\\draw plot[mark=x, mark options={color=black, scale=1.5}] coordinates {(-6.5,0)  }; \n\n  \\node[color=black] at (-5,-0.3) {\\small $\\eta_2$};\n\\draw(-5 ,0)circle [radius=2pt] ;\n  \\node[color=black] at (-4.5,-0.3) {\\small $s_2$};\n\\draw plot[mark=x, mark options={color=black, scale=1.5}] coordinates {(-4.5,0)  }; \n \n\n  \\node[color=black] at (-2.5,-0.3) {\\small $\\eta_3$};\n\\draw(-2.5 ,0)circle [radius=2pt] ;\n  \\node[color=black] at (-3 ,-0.3) {\\small $s_3$};\n\\draw plot[mark=x, mark options={color=black, scale=1.5}] coordinates {(-3,0)  }; \n  \n   \\node[color=black] at (-1,-0.3) {\\small $e$};\n\n \n\\end{tikzpicture}\n\\end{center}\n\n{\\bf Step 1.}\n Denote \n $$ \\mathcal I_ 1= (  s, s _1], \\quad \\I_m   =(s_{m-1} , s_m] \\text{ for }  2 \\le m \\le M     \\quad \\text{and} \\quad \\I_{M+1}   = (s_M,e]. $$\nThen since \n$ s , e,  \\{ s_m \\}_{m=1}^M  \\subset   \\{ s_q\\}_{q=1}^ { \\mathcal Q}   $,   it follows that \n\\begin{align}\\nonumber\n& \\sum_{i \\in \\I }\\|X_i - \\widehat \\mu_\\I\\|_2^2\n\\\\  \\nonumber \n\\le &  \\sum _{m=1}^{M+1 } \\sum_{i \\in \\I_m}\\|X_i - \\widehat y_{\\I_m }  \\|_2^2   + M  \\gamma \n\\\\  \\label{eq: 1D three change points deviation term 1}\n\\le &  \\sum_{i \\in \\I_1 }\\|X_i -  \\mu^*_i\\|_2^2 + C_1 \\bigg ( \\sigma_{\\epsilon}^2  { \\mathfrak{s} } \\log(np) +  \\frac{ (\\eta_1 -s ) ( s_1-\\eta_ 1)  }{  s_1-s  }\\kappa_1^2 \\bigg )\n\\\\   \\label{eq: 1D three change points deviation term 2} \n+ & \n \\sum_{m=2}^{M-1} \\sum_{i \\in \\I_m  }\\|X_i -  \\mu^*_i\\|_2^2    +C_1  \\bigg(\\sigma_{\\epsilon}^2  { \\mathfrak{s} } \\log(np)+     \\frac{(\\eta_m -s_{m-1} )(s_m-\\eta_{m } ) }{ s _{m}-s _{m-1}  }  \\kappa_m^2 \\bigg ) \n \\\\  \\label{eq: 1D three change points deviation term 3}\n  + &    C_1 \\sigma_{\\epsilon}^2  { \\mathfrak{s} } \\log(np)\n\\\\ \\label{eq: 1D three change points deviation term 4}\n+ & \\sum_{i\\in \\I_{M+1}}\\|X_i-\\mu^*_i\\|_2^2+ C_1  \\bigg( \\sigma_{\\epsilon}^2  { \\mathfrak{s} } \\log(np) + \\frac{(\\eta_M -s_{M} )(e-\\eta_{M} ) }{ e-s _{M}  }  \\kappa_{M} ^2 \\bigg ) + M\\gamma, \n \\end{align}  \n where Equations \\eqref{eq: 1D three change points deviation term 1}, \\eqref{eq: 1D three change points deviation term 2}\n \\eqref{eq: 1D three change points deviation term 3} and \\eqref{eq: 1D three change points deviation term 4} follow from \\Cref{lem:mean one change deviation bound} and in particular,   \\Cref{eq: 1D three change points deviation term 3} corresponds to the interval $\\I _M = (s_{M-1},s_M] $ which by assumption  containing no change points. \n Note that \n\\begin{align*} & \\frac{ (\\eta_1 -s ) ( s_1-\\eta_ 1)  }{  s_1-s  } \\le s_1-\\eta_1  \\le  { \\mathcal B_n^{-1}   \\Delta }  ,\n\\\\ \n&   \\frac{(\\eta_m -s_{m-1} )(s_m-\\eta_{m } ) }{ s _{m}-s _{m-1}  } \\le  s_m-\\eta_m  \\le  { \\mathcal B_n^{-1}   \\Delta }  ,   \\ \\text{ and } \n\\\\\n&   \\frac{(\\eta_M -s_{M} )(e-\\eta_{M} ) }{ e-s _{M}  } \\le \\eta_M-s_m  \\le  { \\mathcal B_n^{-1}   \\Delta }   \n \\end{align*} \n and  that \n $ \\kappa_k \\asymp \\kappa $ for all $ 1\\le k \\le K$. Therefore \n\\begin{equation}\n    \\label{eq:1D three change points step 1}\n  \\sum_{i \\in \\I }\\|X_i - \\widehat \\mu_\\I\\|_2^2 \\le  \\sum _{ i \\in \\I } \\|X_i - \\mu^*_i\\|_2^2 +  C_2  \\bigg( M \\sigma_{\\epsilon}^2  { \\mathfrak{s} } \\log(np)+   M  { \\mathcal B_n^{-1}   \\Delta }  \\kappa^2\n+  M\\gamma \\bigg),\n\\end{equation}\nwhere $ C_2$ is some large constant independent of $M$.\n\n{\\bf Step 2.}  Let \n$$  {  \\mathcal J } _1 =(s,  \\eta_1], \\ {  \\mathcal J } _m = (\\eta_{m-1}, \\eta_m]  \\text{ for } 2 \\le m \\le M , \\ {  \\mathcal J } _{M+1}  =(\\eta_M, e]. $$\nNote that $\\mu^*_i$ is unchanged in any of $\\{ {  \\mathcal J } _m\\}_{m=0}^{M+1}$.  \nSo  for $ 1 \\le m \\le M+1  $,\n\\begin{align}\\nonumber  \n  &\\sum_{i \\in  {  \\mathcal J } _m  }\\|X_i - \\widehat \\mu_\\I\\|_2^2  \n -   \\sum_{i \\in {  \\mathcal J } _  m    }\\|X_i -  \\mu^*_{ {  \\mathcal J } _  m }\\|_2^2  -\\sum_{i \\in {  \\mathcal J } _  m   }\\|\\widehat \\mu_\\I  - \\mu^*_{{  \\mathcal J } _  m }\\|_2^2 \n\\\\\n=& \\nonumber \n 2 \\sum_{i \\in {  \\mathcal J } _  m} \\epsilon_i^\\top (\\mu^*_{{  \\mathcal J } _ m}-\\widehat \\mu_\\I)   \n\\\\\\nonumber   \n\\ge  & - C\\sigma_{\\epsilon}\\|\\widehat \\mu_\\I  -  \\mu^*_{ {  \\mathcal J } _ m  }\\|_2\\sqrt{|{  \\mathcal J } _m|\\log(np)}  \n\\\\\n\\ge &   -  C_3  \\sigma_{\\epsilon}^2  { \\mathfrak{s} } \\log(np) - \\frac{1 }{2 } |{  \\mathcal J } _  m |\\|\\widehat \\mu_\\I  -  \\mu^*_{ {  \\mathcal J } _ m  }\\|_2^2   \\nonumber  \n\\end{align} \nwhich gives \n\\begin{align} \\label{eq:1D three change points second}\n  &\\sum_{i \\in {  \\mathcal J } _m  }\\|X_i - \\widehat \\mu_\\I\\|_2^2 \n -   \\sum_{i \\in {  \\mathcal J } _ m   }\\|X_i -  \\mu^*_{ {  \\mathcal J } _  m }\\|_2^2  \\geq \\frac{1}{2}\\sum_{i \\in {  \\mathcal J } _  m   }\\|\\widehat \\mu_\\I  - \\mu^*_{{  \\mathcal J } _  m  }\\|_2 ^2 - C_3\\sigma_{\\epsilon}^2  { \\mathfrak{s} } \\log(np).\n\\end{align} \nTherefore\n\\begin{align} \\label{eq:1D three change points third} \\sum_{ m=1}^{M+1}   |{  \\mathcal J } _m |\\|\\widehat \\mu_\\I  - \\mu^*_{{  \\mathcal J } _  m  }\\|_2^2 = \\sum_{ m=1 }^{M+1}  \\sum_{i \\in {  \\mathcal J } _  m   }\\|\\widehat \\mu_\\I  - \\mu^*_{{  \\mathcal J } _  m  }\\|_2^2 \\le  C_4  M\\bigg(\\sigma_{\\epsilon}^2  { \\mathfrak{s} } \\log(np)  +     { \\mathcal B_n^{-1}   \\Delta }  \\kappa^2\n+   \\gamma \\bigg),\n\\end{align}\nwhere the equality follows from the fact that  $\\mu^*_i$ is unchanged in any of $\\{ {  \\mathcal J } _m\\}_{m=0}^{M+1}$, and\n the inequality follows from \\Cref{eq:1D three change points step 1} and \n\\Cref{eq:1D three change points second}.\n\\\\\n\\\\\n{\\bf Step 3.}\nFor any $  m \\in\\{2, \\ldots, M\\}$, it holds that\n\\begin{align}  \\label{eq:1D three change points signal lower bound} \\inf_{ a \\in \\mathbb R  }  |{  \\mathcal J } _{m-1} |\\|a - \\mu^*_{{  \\mathcal J } _{m-1} }\\|_2^2 + |{  \\mathcal J } _{m} |\\|a  - \\mu^*_{{  \\mathcal J } _{m} }\\|_2^2  =&     \\frac{|{  \\mathcal J } _{m-1}| |{  \\mathcal J } _m|}{ |{  \\mathcal J } _{m-1}| +  |{  \\mathcal J } _m| } \\kappa_m ^2  \\ge \\frac{1}{2} \\Delta \\kappa^2,\n\\end{align}  \nwhere the last inequality follows from the assumptions that $\\eta_k - \\eta_{k-1}\\ge \\Delta  $ and $ \\kappa_k \\asymp \\kappa$  for all $1\\le k \\le K$. So\n\\begin{align}   \\nonumber &2 \\sum_{ m=1}^{M }   |{  \\mathcal J } _m |\\|\\widehat \\mu_\\I  - \\mu^*_{{  \\mathcal J } _  m  }\\|_2^2   \n\\\\ \n\\ge &  \\nonumber    \\sum_{m=2}^M  \\bigg(  |{  \\mathcal J } _{m-1}  | \\|\\widehat \\mu_\\I  - \\mu^*_{{  \\mathcal J } _ { m-1}  }\\|_2^2   + |{  \\mathcal J } _m |\\|\\widehat \\mu_\\I  - \\mu^*_{{  \\mathcal J } _  m  }\\|_2^2   \\bigg) \n\\\\ \\label{eq:1D three change points signal lower bound two} \n\\ge & (M-1)  \\frac{ 1}{2} \\Delta \\kappa^2  \\ge \\frac{M}{4} \\Delta \\kappa^2 ,\n\\end{align} \nwhere the second inequality follows from  \\Cref{eq:1D three change points signal lower bound} and the last inequality follows from $M\\ge 3$. \\Cref{eq:1D three change points third} and  \\Cref{eq:1D three change points signal lower bound two} together imply that \n\\begin{align}\\label{eq:1D three change points signal lower bound three} \n \\frac{M}{4} \\Delta \\kappa^2 \\le 2  C_4  M\\bigg(\\sigma_{\\epsilon}^2  { \\mathfrak{s} } \\log(np) +  { \\mathcal B_n^{-1}   \\Delta }  \\kappa^2 + \\gamma \\bigg) .\n\\end{align}\nSince $ { \\mathcal B_n} \\to \\infty $, it follows that for sufficiently large $n$,   \\Cref{eq:1D three change points signal lower bound three} gives \n$$ \\Delta\\kappa^2 \\le C_5 \\big(\\sigma_{\\epsilon}^2  { \\mathfrak{s} } \\log(np) +\\gamma),$$  \nwhich contradicts \\Cref{eq:1D three change points snr}.\n \\end{proof} \n \n \n \n \\bnlem Suppose $ \\gamma \\ge C_\\gamma K { \\mathcal B_n^{-1}   \\Delta }  \\kappa^2 $ for sufficiently large constant $C_\\gamma $. \nWith probability  at least $1-(np)^{-3}$,    there are no  two consecutive intervals $\\I_1= (s,t ] \\in \\widehat {\\mathcal  P}  $,   $ \\I_2=(t, e]  \\in   \\widehat  {\\mathcal  P}  $    such that $\\I_1 \\cup \\I_2$ contains no change points.  \n \\enlem \n \\begin{proof} \n For contradiction, suppose that \n $$ \\I :  =\\I_1\\cup  \\I_2 $$\n contains no change points. \n Since \n $ s,t,e \\in \\{ s_q\\}_{q=1}^ { \\mathcal Q}  $, it follows that\n $$ \\sum_{i\\in \\I_1} \\|X_i - \\widehat \\mu_{\\I_1}\\|^2  +\\sum_{i\\in \\I_2} \\|X_i - \\widehat \\mu_{\\I_2}\\|_2^2 +\\gamma  \\le \\sum_{i\\in \\I } \\|X_i - \\widehat \\mu_{\\I }\\|_2^2. $$\n By \\Cref{lem:mean one change deviation bound}, it follows that \n\\begin{align*} \n& \\sum_{i\\in \\I_1} \\|X_i - \\mu^*_i\\|_2^2 \\le  C_1 \\sigma_{\\epsilon}^2 { \\mathfrak{s} } \\log(np) +  \\sum_{i\\in \\I_1} \\|X_i - \\widehat \\mu_{\\I_1}\\|_2^2 ,\n \\\\\n & \\sum_{i\\in \\I_2} \\|X_i - \\mu^*_i\\|_2^2 \\le  C_1 \\sigma_{\\epsilon}^2 { \\mathfrak{s} } \\log(np) +  \\sum_{i\\in \\I_2} \\|X_i - \\widehat \\mu_{\\I_2}\\|_2^2\n  \\\\\n  & \\sum_{i\\in \\I} \\|X_i - \\widehat{\\mu}_{\\I}\\|_2^2 \\le  C_1 \\sigma_{\\epsilon}^2 { \\mathfrak{s} } \\log(np) + \\sum_{i\\in \\I} \\|X_i - \\mu_{i}^*\\|_2^2.\n \\end{align*} \n So \n $$\\sum_{i\\in \\I_1} \\|X_i - \\mu^*_i\\|_2^2  +\\sum_{i\\in \\I_2}\\|X_i - \\mu^*_i\\|_2^2 -2C_1 \\sigma_{\\epsilon}^2 { \\mathfrak{s} } \\log(np) +\\gamma    \\le \\sum_{i\\in \\I }\\|X_i - \\mu^*_i\\|_2^2 +C_1\\sigma_{\\epsilon}^2 { \\mathfrak{s} } \\log(np) .  $$\n Since $\\mu^*_i$ is unchanged when $i\\in \\I$, it follows that \n $$ \\gamma \\le 3C_1\\sigma_{\\epsilon}^2 { \\mathfrak{s} } \\log(np).$$\n This is a contradiction when $C_\\gamma> 3C_1. $\n \\end{proof} \n \n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\clearpage\n\\subsection{Covariance model}\n\n\n\\bnprop\\label{prop:covariance local consistency}\n Suppose  \\Cref{assp:DCDP_covariance} holds. Let \n $\\widehat { \\mathcal P }  $ denote the output of DP.\n Then with probability  at least $1 - C n ^{-3}$, the following conditions hold.\n\t\\begin{itemize}\n\t\t\\item [(i)] For each interval  $ \\I = (s, e] \\in \\widehat{\\mathcal{P}}$    containing one and only one true \n\t\t   change point $ \\eta_k $, it must be the case that\n $$\\min\\{ \\eta_k -s ,e-\\eta_k \\}  \\lesssim  C_{\\gamma} \\|X\\|_{\\psi_2}^4\\frac{C_X^2}{c_X^6}\\frac{p^2\\log(np)}{\\kappa_k^2} +\\|X\\|_{\\psi_2}^4\\frac{C_X^6}{c_X^6} { \\mathcal B_n^{-1}   \\Delta }  .$$\n\t\t\\item [(ii)] For each interval  $  \\I = (s, e] \\in \\widehat{\\mathcal{P}}$ containing exactly two true change points, say  $\\eta_ k  < \\eta_ {k+1} $, it must be the case that\n\t\t\t\\begin{align*} \n \\eta_k -s    \\lesssim  { \\mathcal B_n^{-1\/2}   \\Delta }     \\quad \\text{and} \\quad \n e-\\eta_{k+1}  \\lesssim   { \\mathcal B_n^{-1\/2}   \\Delta } . \n \\end{align*} \n\t\t \n\\item [(iii)] No interval $\\I \\in \\widehat{\\mathcal{P}}$ contains strictly more than two true change points; and \n\n\n\t\\item [(iv)] For all consecutive intervals $ \\I_1 $ and $ \\I_2 $ in $\\widehat{ \\mathcal P}$, the interval \n\t\t$ \\I_1 \\cup  \\I_2 $ contains at least one true change point.\n\t\t\t\t\n\t \\end{itemize}\n\\enprop \n\\bprf\nThe four cases are proved in \\Cref{lem:cov no cp}, \\Cref{lem:cov single cp}, \\Cref{lem:cov two cp}, and \\Cref{lem:cov two consecutive interval}.\n\\eprf\n\n\n\n\n\n\n\n\n\\bnprop\\label{prop:change points partition size consistency}\nSuppose  \\Cref{assp:DCDP_covariance} holds. Let \n $\\widehat { \\mathcal P }  $ be the output of DP. Suppose \n$ \\gamma \\ge C_\\gamma K { \\mathcal B_n^{-1}   \\Delta }  \\kappa^2 $  for sufficiently large constant $C_\\gamma$. Then\n with probability  at least $1 - C n ^{-3}$,  $| \\widehat { \\mathcal P} | =K $.\n\\enprop\n\n\n\n\\begin{proof}[Proof of   \\Cref{prop:change points partition size consistency}] \nDenote $\\mathfrak{G} ^*_n = \\sum_{ i =1}^n [{\\rm Tr}[(\\Omega_i^*)^\\top X_iX_i^\\top] - \\log|\\Omega_i^*|]$. Given any collection $\\{t_1, \\ldots, t_m\\}$, where $t_1 < \\cdots < t_m$, and $t_0 = 0$, $t_{m+1} = n$, let \n\\begin{equation}\\label{eq-sn-def}\n\t { \\mathfrak{G} }  _n(t_1, \\ldots, t_{m}) = \\sum_{k=1}^{m} \\sum_{ i = t_k +1}^{t_{k+1}} \\mclF(\\widehat {\\Omega}_{(t_{k}, t_{k+1}]}, (t_{k}, t_{k+1}]).\n\\end{equation}\nFor any collection of time points, when defining \\eqref{eq-sn-def}, the time points are sorted in an increasing order.\n\nLet $\\{ \\widehat \\eta_{k}\\}_{k=1}^{\\widehat K}$ denote the change points induced by $\\widehat {\\mathcal P}$.  Suppose we can justify that \n\t\\begin{align}\n\t\t { \\mathfrak{G} } ^*_n + K\\gamma  \\ge  & { \\mathfrak{G} }  _n(s_1,\\ldots,s_K)   + K\\gamma - C(K + 1)\\frac{\\|X\\|_{\\psi_2}^2}{c_{X}^4}\\frac{ p^2\\log(np)}{\\kappa^2} - C\\sum{k\\in [K]}\\kappa_k^2\\mclB_n^{-1}\\Delta \\label{eq:cov K consistency step 1} \\\\ \n\t\t\\ge &  { \\mathfrak{G} } _n (\\widehat \\eta_{1},\\ldots, \\widehat \\eta_{\\widehat K } ) +\\widehat K \\gamma - C_1'(K + 1)\\frac{\\|X\\|_{\\psi_2}^2}{c_{X}^4}\\frac{ p^2\\log(np)}{\\kappa^2} - C_1'\\sum{k\\in [K]}\\kappa_k^2\\mclB_n^{-1}\\Delta  \\label{eq:cov K consistency step 2} \\\\ \n\t\t\\ge &   { \\mathfrak{G} }  _n ( \\widehat \\eta_{1},\\ldots, \\widehat \\eta_{\\widehat K } , \\eta_1,\\ldots,\\eta_K ) + \\widehat K \\gamma    - C_1(K + 1)\\frac{\\|X\\|_{\\psi_2}^2}{c_{X}^4}\\frac{ p^2\\log(np)}{\\kappa^2} - C_1\\sum{k\\in [K]}\\kappa_k^2\\mclB_n^{-1}\\Delta \\label{eq:cov K consistency step 3}\n\t\\end{align}\n\tand that \n\t\\begin{align}\\label{eq:cov K consistency step 4}\n\t\t { \\mathfrak{G} }  ^*_n   - { \\mathfrak{G} }  _n ( \\widehat \\eta_{1},\\ldots, \\widehat \\eta_{\\widehat K } , \\eta_1,\\ldots,\\eta_K ) \\le C_2 (K + \\widehat{K} + 2) \\frac{\\|X\\|_{\\psi_2}^4}{c_X^2} p^2\\log(np).\n\t\\end{align}\n\tThen it must hold that $| \\widehat{\\mclP} | = K$, as otherwise if $\\widehat K \\geq K+1 $, then  \n\t\\begin{align*}\n\t\tC _2 (K + \\widehat{K} + 2)  \\frac{\\|X\\|_{\\psi_2}^4}{c_X^2} p^2\\log(np)  & \\ge    { \\mathfrak{G} }  ^*_n  - { \\mathfrak{G} }  _n ( \\widehat \\eta_{1},\\ldots, \\widehat \\eta_{\\widehat K } , \\eta_1,\\ldots,\\eta_K ) \\\\\n\t\t& \\ge      (\\widehat K - K)\\gamma  -C_1 ( K +1)  \\frac{\\|X\\|_{\\psi_2}^4}{c_X^2} p^2\\log(np).\n\t\\end{align*} \n\tTherefore due to the assumption that $| \\widehat{\\mclP}|=\\widehat K\\leq 3K $, it holds that \n\t\\begin{align} \\label{eq:cov Khat=K} \n\t\t[C_2 (4K + 2) + C_1(K+1)] \\frac{\\|X\\|_{\\psi_2}^4}{c_X^2} p^2\\log(np) \\geq  (\\widehat K - K)\\gamma \\geq \\gamma.\n\t\\end{align}\n\tNote that \\eqref{eq:cov Khat=K} contradicts the choice of $\\gamma$. Therefore, it remains to show \\Cref{eq:cov K consistency step 1} to \\Cref{eq:cov K consistency step 4}.\n\n{\\bf Step 1.} \\Cref{eq:cov K consistency step 1} holds because $\\widehat{\\Omega}_{\\I}$ is (one of) the minimizer of $\\F(\\Omega, \\I)$ for any interval $\\I$.\n\n{\\bf Step 2.} \\Cref{eq:cov K consistency step 2} is guaranteed by the definition of $\\widehat{\\mclP}$.\n\n{\\bf Step 3.} For every $ \\I  =(s,e]\\in \\widehat{\\mclP}$, by \\Cref{prop:covariance local consistency}, we know that $\\I$ contains at most two change points. We only show the proof for the two-change-points case as the other case is easier. Denote\n\t\\[\n\t\t \\I =  (s ,\\eta_{q}]\\cup (\\eta_{q},\\eta_{q+1}] \\cup  (\\eta_{q+1} ,e]  = {  \\mathcal J } _1 \\cup {  \\mathcal J } _2 \\cup {  \\mathcal J } _{3},\n\t\\]\nwhere $\\{ \\eta_{q},\\eta_{q+1}\\} =\\I \\, \\cap \\, \\{\\eta_k\\}_{k=1}^K$. \n\nFor each $ m=1,2,3$, by definition it holds that\n\\begin{equation}\n    \\mclF(\\widehat\\Omega_{{  \\mathcal J } _m},{  \\mathcal J } _m)\\leq \\mclF(\\Omega^*_{{  \\mathcal J } _m},{  \\mathcal J } _m).\n\\end{equation}\nOn the other hand, by \\Cref{lem:cov loss deviation no change point}, we have\n$$\n\\mclF(\\widehat\\Omega_{\\I},{  \\mathcal J } _m)   \\geq \\mclF(\\Omega^*_{{  \\mathcal J } _m},{  \\mathcal J } _m) - C\\frac{\\|X\\|_{\\psi_2}^4}{c_X^2}p^2\\log(np). \n$$\nTherefore the last two inequalities above imply that \n\\begin{align} \n\\sum_{i \\in \\I }  \\mclF(\\widehat \\Omega_\\I,\\I) \\geq & \\sum_{m=1}^{3} \\sum_{ i \\in {  \\mathcal J } _m }\\mclF(\\widehat \\Omega_\\I,{  \\mathcal J } _m) \\nonumber \\\\\n\\geq & \\sum_{m=1}^{3}   \\sum_{ i \\in {  \\mathcal J } _m}\\mclF(\\widehat \\Omega_{{  \\mathcal J } _m},{  \\mathcal J } _m) - C\\frac{\\|X\\|_{\\psi_2}^4}{c_X^2}p^2\\log(np). \\label{eq:cov K consistency step 3 inequality 3}\n\\end{align}\nThen \\eqref{eq:cov K consistency step 3} is an immediate consequence of \\eqref{eq:cov K consistency step 3 inequality 3}.\n\n{\\bf Step 4.}\nFinally, to show \\eqref{eq:cov K consistency step 4},   let  $\\widetilde { \\mathcal P}$ denote    the partition induced by $\\{\\widehat \\eta_{1},\\ldots, \\widehat \\eta_{\\widehat K } , \\eta_1,\\ldots,\\eta_K\\}$. Then \n$| \\widetilde { \\mathcal P} | \\le K + \\widehat K+2   $ and that $\\Omega^*_i$ is unchanged in every interval $\\I \\in \\widetilde { \\mathcal P}$. So \\Cref{eq:cov K consistency step 4} is an immediate consequence of \\Cref{lem:cov no cp}.\n\\end{proof} \n\n\n\n\n\n\n\n\n\\subsubsection{Technical lemmas}\n\n\n\n\n\n\\bnlem\n\\label{lem:estimation covariance}\nLet $\\{X_i\\}_{i\\in [n]}$ be a sequence of subgaussian vectors in $\\mathbb{R}^{p}$ with orlitz norm upper bounded $\\|X\\|_{\\psi_2}<\\infty$. Suppose $\\mathbb{E}[X_i] = 0$ and $\\mathbb{E}[X_iX_i^\\top] = \\Sigma_i$ for $i\\in [n]$. Consider the change point setting in \\Cref{assp:DCDP_covariance} and consider a generic interval $\\I\\subset [1,n]$. Let $\\widehat{\\Sigma}_{\\I} = \\frac{1}{|\\I|}\\sum_{i \\in \\I} X_i X_i^\\top$ and $\\Sigma_{\\I} = \\frac{1}{|\\I|}\\Sigma^*_i$. Then for any $u>0$, it holds with probability at least $1 - \\exp(-u)$ that\n\\begin{equation}\n    \\|\\widehat{\\Sigma}_{\\I} - \\Sigma_{\\I}\\|_2\\lesssim \\|X\\|_{\\psi_2}^2(\\sqrt{\\frac{p + u}{n}}\\vee \\frac{p + u}{n}).\n\\end{equation}\nAs a corollary, when $n\\geq C_s p\\log(np)$ for some universal constant $C_s>0$, it holds with probability at least $1 - (np)^{-7}$ that\n\\begin{equation}\n    \\|\\widehat{\\Sigma}_{\\I} - \\Sigma_{\\I}\\|_2\\leq C\\|X\\|_{\\psi_2}^2\\sqrt{\\frac{p\\log(np)}{|\\I|}},\n\\end{equation}\nwhere $C$ is some universal constant that does not depend on $n, p, \\|X\\|_{\\psi_2}$, and $C_s$. In addition let $\\widehat{\\Omega}_{\\I} = \\argmin_{\\Omega\\in \\mathbb{S}_+}L(\\Omega,\\I)$ and $\\widetilde{\\Omega}_{\\I} = (\\frac{1}{|\\I|}\\sum_{i\\in\\I}\\Sigma^*_i)^{-1}$. if $|\\I|\\geq C_s p\\log(np)\\|X\\|_{\\psi_2}^4 \/ c_X^2$ for sufficiently large constant $C_s>0$, then it holds with probability at least $1 - (np)^{-7}$ that\n\\begin{equation}\n    \\|\\widehat{\\Omega}_{\\I} -\\widetilde{\\Omega}_{\\I}\\|_2\\leq C \\frac{\\|X\\|_{\\psi_2}^2}{c_X^2}\\sqrt{\\frac{p\\log(np)}{|\\I|}}.\n\\end{equation}\n\\enlem\n\\bprf\nVanishing the gradient of the loss function $L(\\Omega, \\I)$ gives\n\\begin{equation*}\n    \\widehat{\\Omega}_{\\I} = (\\widehat{\\Sigma}_{\\I})^{\\dagger}.\n\\end{equation*}\n\\eprf\n\n\n\n\n\n\n\n\n\n\\bnlem[Covariance model]\n\\label{lem:cov one change deviation bound}\nLet $\\mathcal I =(s,e] $ be any generic interval, and define the loss function $L(\\Omega, \\I) = \\sum_{i\\in \\I}{\\rm Tr}[\\Omega^\\top (X_iX_i^\\top)] - |\\I|\\log|\\Omega|$ and goodness-of-fit function $\\mclF(\\Omega, \\I) = L(\\Omega,\\I)$. Define $\\widehat{\\Omega}_{\\I} = \\argmin_{\\Omega\\in \\mathbb{S}_+}L(\\Omega,\\I)$ and $\\mclF^*(\\I) = \\sum_{i\\in \\I}[{\\rm Tr}((\\Omega_i^*)^\\top (X_iX_i^\\top)) - \\log|\\Omega_i^*|]$.\n\n{\\bf a.} If $\\I$ contains no change points. Then it holds that \n$$\\mathbb{P} \\bigg( |   \\mclF(\\widehat{\\Omega}_{\\I},\\I)  - \\mclF^*(\\I) |  \\ge C \\frac{\\|X\\|_{\\psi_2}^4}{c_X^2}p^2\\log(np)  \\bigg)  \\le  (np)^{-3}. $$\n\n{\\bf b.}  Suppose that the interval $ \\I=(s,e]$ contains one and only one change point $ \\eta_k $. Denote\n$$  \\mathcal J = (s,\\eta_k] \\quad \\text{and} \\quad  \\mathcal J' =  (\\eta_k, e].$$ \nThen it holds that \n$$\\mathbb{P} \\bigg( | \\mclF(\\widehat{\\Omega}_{\\I},\\I) - \\mclF^*(\\I)  | \\geq \\frac{C_X^2p}{c_X^8}\\frac{|{  \\mathcal J } ||{  \\mathcal J }  '| }{|\\I|}\\kappa_k^2 +  C\\frac{\\|X\\|_{\\psi_2}^4 C_X^2}{c_X^4}p^2\\log(np) \\bigg)   \\le  (np)^{-3}. $$\n\\enlem\n\\begin{proof} \nWe show {\\bf b} as {\\bf a} immediately follows from {\\bf b} with $ |{  \\mathcal J } '| =0$. Denote \n$$  \\mathcal J = (s,\\eta_k] \\quad \\text{and} \\quad  \\mathcal J' =  (\\eta_k, e]  .$$ \nLet $\\widetilde{\\Omega}_{\\I} = (\\frac{1}{|\\I|}\\sum_{i\\in\\I}\\Sigma_i^*)^{-1}$.\nThen by Taylor expansion and \\Cref{lem:estimation covariance}, we have\n\\begin{align}\n    & |\\mclF(\\widehat{\\Omega}_{\\I},\\I)-\\mclF(\\widetilde{\\Omega}_{\\I},\\I)| \\nonumber \\\\\n    \\leq & |{\\rm Tr}[(\\widehat{\\Omega}_{\\I} - \\widetilde{\\Omega}_{\\I})^\\top (\\sum_{i\\in\\I} X_iX_i^\\top - |\\I|\\widetilde{\\Omega}_{\\I}^{-1})]| + \\frac{C_X^2}{2}|\\I|\\|\\widehat{\\Omega}_{\\I} - \\widetilde{\\Omega}_{\\I}\\|_F^2 \\nonumber \\\\\n    \\leq & |\\I|\\|\\widehat{\\Omega}_{\\I} - \\widetilde{\\Omega}_{\\I}\\|_F \\|\\widehat{\\Sigma}_{\\I} - \\widetilde{\\Omega}_{\\I}^{-1}\\|_F + \\frac{C_X^2}{2}|\\I|\\|\\widehat{\\Omega}_{\\I} - \\widetilde{\\Omega}_{\\I}\\|_F^2 \\nonumber\\\\\n    \\leq & C\\frac{\\|X\\|_{\\psi_2}^4}{c_X^2}p^2\\log(np) + C\\frac{\\|X\\|_{\\psi_2}^4 C_X^2}{c_X^4}p^2\\log(np)\\leq C\\frac{\\|X\\|_{\\psi_2}^4 C_X^2}{c_X^4}p^2\\log(np). \\label{eq:cov mixed deviation 1}\n\\end{align}\nOn the other hand, it holds that\n\\begin{align}\n    & |\\mclF(\\widetilde{\\Omega}_{\\I},\\I)-\\mclF^*(\\I)| \\nonumber \\\\\n    \\leq & |{\\rm Tr}[(\\widetilde{\\Omega}_{\\I} - {\\Omega}_{{  \\mathcal J } })^\\top (\\sum_{i\\in{  \\mathcal J } } X_iX_i^\\top - |{  \\mathcal J } |\\Omega_{{  \\mathcal J } }^{-1})]| + \\frac{C_X^2}{2}|{  \\mathcal J } |\\|\\widetilde{\\Omega}_{\\I} - {\\Omega}_{{  \\mathcal J } }\\|_F^2 \\nonumber\\\\\n     & + |{\\rm Tr}[(\\widetilde{\\Omega}_{\\I} - {\\Omega}_{{  \\mathcal J }  '})^\\top (\\sum_{i\\in{  \\mathcal J } '} X_iX_i^\\top - |{  \\mathcal J }  '|\\Omega_{{  \\mathcal J }  '}^{-1})]| + \\frac{C_X^2}{2}|{  \\mathcal J }  '|\\|\\widetilde{\\Omega}_{\\I} - {\\Omega}_{{  \\mathcal J }  '}\\|_F^2.\\label{eq:cov mixed deviation 2}\n\\end{align}\nTo bound $\\|\\widetilde{\\Omega}_{\\I} - {\\Omega}_{{  \\mathcal J } }\\|_F$ and $\\|\\widetilde{\\Omega}_{\\I} - {\\Omega}_{{  \\mathcal J }  '}\\|_F$, notice that for two positive definite matrices $\\Sigma_1,\\Sigma_2\\in \\mathbb{S}_+$ and two positive numbers $w_1,w_2$ such that $w_1 + w_2 = 1$, we have\n\\begin{align*}\n    & \\|(w_1\\Sigma_1 + w_2\\Sigma_2)^{-1} - \\Sigma_1^{-1}\\|_F\\\\\n    \\leq &\\sqrt{p}\\|(w_1\\Sigma_1 + w_2\\Sigma_2)^{-1} - \\Sigma_1^{-1}\\|_2\\\\\n    = &\\sqrt{p}\\|(w_1\\Sigma_1 + w_2\\Sigma_2)^{-1}[\\Sigma_1 - (w_1\\Sigma_1 + w_2\\Sigma_2)]\\Sigma_1^{-1}\\|_2\\\\\n    \\leq & \\sqrt{p}\\|(w_1\\Sigma_1 + w_2\\Sigma_2)^{-1}\\|_2\\|\\Sigma_1 - (w_1\\Sigma_1 + w_2\\Sigma_2)\\|_2\\|\\Sigma_1^{-1}\\|_2\\\\\n    \\leq & \\|(w_1\\Sigma_1 + w_2\\Sigma_2)^{-1}\\|_2\\|\\Sigma_1^{-1}\\|_2\\cdot\\sqrt{p}w_2\\|\\Sigma_1 - \\Sigma_2\\|_2 \\\\\n    \\leq & \\|(w_1\\Sigma_1 + w_2\\Sigma_2)^{-1}\\|_2\\|\\Sigma_1^{-1}\\|_2\\|\\Sigma_1\\|_2\\|\\Sigma_2\\|_2\\cdot\\sqrt{p}w_2\\|\\Sigma_1^{-1} - \\Sigma_2^{-1}\\|_2.\n\\end{align*}\nTherefore, under \\Cref{assp:DCDP_covariance}, it holds that\n\\begin{equation}\n    \\|\\widetilde{\\Omega}_{\\I} - {\\Omega}_{{  \\mathcal J } }\\|_F\\leq \\frac{C_X^2}{c_X^2}\\frac{|{  \\mathcal J }  '|}{|\\I|}\\sqrt{p}\\kappa_k, \\|\\widetilde{\\Omega}_{\\I} - {\\Omega}_{{  \\mathcal J }  '}\\|_F\\leq \\frac{C_X^2}{c_X^2}\\frac{|{  \\mathcal J } |}{|\\I|}\\sqrt{p}\\kappa_k,\n\\end{equation}\nwhere in the second inequality we use the fact that $2ab\\leq a^2+ b^2$. As a consequence, \\Cref{eq:cov mixed deviation 2} can be bounded as\n\\begin{align}\n     & |\\mclF(\\widetilde{\\Omega}_{\\I},\\I)-\\mclF^*(\\I)| \\nonumber \\\\\n    \\leq & C\\frac{\\|X\\|_{\\psi_2}^2p\\sqrt{\\log(np)}}{c_X^2C_X^{-2}}\\kappa_k(\\frac{|{  \\mathcal J }  '||{  \\mathcal J } |^{\\frac{1}{2}}}{|\\I|} + \\frac{|{  \\mathcal J }  '|^{\\frac{1}{2}}|{  \\mathcal J } |}{|\\I|}) + \\frac{C_X^6 p}{2c_X^4}\\kappa_k^2(\\frac{|{  \\mathcal J } ||{  \\mathcal J }  '|^2 }{|\\I|^2} + \\frac{|{  \\mathcal J }  '||{  \\mathcal J } |^2 }{|\\I|^2}) \\nonumber\\\\\n    \\leq & C\\frac{\\|X\\|_{\\psi_2}^4p^2{\\log(np)}}{C_X^2} + \\frac{C_X^6p}{c_X^4}\\frac{|{  \\mathcal J } ||{  \\mathcal J }  '| }{|\\I|}\\kappa_k^2.\\label{eq:cov mixed deviation 3}\n\\end{align}\nCombine \\Cref{eq:cov mixed deviation 1} and \\Cref{eq:cov mixed deviation 3} and we can get\n\\begin{equation*}\n    \\begin{split}\n        |\\mclF(\\widehat{\\Omega}_{\\I},\\I) - \\mclF^*(\\I)|\n        \\leq & |\\mclF(\\widehat{\\Omega}_{\\I},\\I)-\\mclF({\\Omega}_{\\I},\\I)| +|\\mclF({\\Omega}_{\\I},\\I) - \\mclF^*(\\I)|\\\\\n        \\leq& C\\frac{\\|X\\|_{\\psi_2}^4 C_X^2}{c_X^4}p^2\\log(np) + \\frac{C_X^6 p}{c_X^4}\\frac{|{  \\mathcal J } ||{  \\mathcal J }  '| }{|\\I|}\\kappa_k^2.\n    \\end{split}\n\\end{equation*}\n\\end{proof} \n\n\\bnrmk\nIt can be seen later that the $p$ factor in the signal term $\\frac{C_X^6 p}{c_X^4}\\frac{|{  \\mathcal J } ||{  \\mathcal J }  '| }{|\\I|}\\kappa_k^2$ will require an additional $p$ factor in the number of points in the grid for DCDP, leading to an additional $p^2$ factor in the computation time.\n\nThis factor is hard to remove because it is rooted in the approximation error\n$$\n\\|(w_1\\Sigma_1 + w_2\\Sigma_2)^{-1} - \\Sigma_1^{-1}\\|_F.\n$$\nWe can try another slightly neater way of bounding this term. As is mentioned in \\cite{kereta2021ejs}, for two matrices $\\mathbf{G}, \\mathbf{H}\\in \\mathbb{R}^{d_1\\times d_2}$, it holds that\n$$\n\\left\\|\\mathbf{H}^{\\dagger}-\\mathbf{G}^{\\dagger}\\right\\|_F \\leq \\min \\left\\{\\left\\|\\mathbf{H}^{\\dagger}\\right\\|_2\\left\\|\\mathbf{G}^{\\dagger} (\\mathbf{H}-\\mathbf{G})\\right\\|_F,\\left\\|\\mathbf{G}^{\\dagger}\\right\\|_2\\left\\|\\mathbf{H}^{\\dagger} (\\mathbf{H}-\\mathbf{G})\\right\\|_2\\right\\},\n$$\nif $\\operatorname{rank}(\\mathbf{G})=\\operatorname{rank}(\\mathbf{H})=\\min \\left\\{d_1, d_2\\right\\}$. Therefore, we have\n\\begin{align*}\n    \\|(w_1\\Sigma_1 + w_2\\Sigma_2)^{-1} - \\Sigma_1^{-1}\\|_F\\leq & \\|(w_1\\Sigma_1 + w_2\\Sigma_2)^{-1}\\|_2\\|\\Sigma_{1}^{-1}(w_1\\Sigma_1 + w_2\\Sigma_2 - \\Sigma_1)\\|_F\\\\\n    \\leq & \\|(w_1\\Sigma_1 + w_2\\Sigma_2)^{-1}\\|_2\\|\\Sigma_{1}^{-1}\\|w_2\\|\\Sigma_2 - \\Sigma_1\\|_F.\n\\end{align*}\nHowever, to relate $\\|\\Sigma_2 - \\Sigma_1\\|_F$ to $\\|\\Sigma_2^{-1} - \\Sigma_1^{-1}\\|_F$, we need to proceed in the following way:\n\\begin{align*}\n    \\|\\Sigma_2 - \\Sigma_1\\|_F\\leq & \\sqrt{d}\\|\\Sigma_2 - \\Sigma_1\\|_2 \\\\\n    \\leq & \\|\\Sigma_1\\|_2\\|\\Sigma_2\\|_2 \\cdot \\sqrt{d}\\|\\Sigma_2^{-1} - \\Sigma_1^{-1}\\|_2\\\\\n    \\leq & \\|\\Sigma_1\\|_2\\|\\Sigma_2\\|_2 \\cdot \\sqrt{d}\\|\\Sigma_2^{-1} - \\Sigma_1^{-1}\\|_F,\n\\end{align*}\nwhich leads to the same bound in \\Cref{lem:cov one change deviation bound}.\n\\enrmk\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\bnlem[No change point]\n\\label{lem:cov no cp}\nFor interval $\\mclI$ containing no change point, it holds with probability at least $1 - n^{-5}$ that\n\\begin{equation}\n    \\F(\\widehat{\\Omega}_{\\I}, \\mclI) -\\F({\\Omega}^*, \\mclI) \\geq -\\|X\\|_{\\psi_2}^4 p^2\\log(np)\\max_{k\\in[K + 1]}\\|\\Omega^*_{\\eta_k}\\|_2^2.\n\\end{equation}\n\\enlem\n\\bprf\nIf $\\I< C_s\\frac{\\|X\\|_{\\psi_2}^4}{c_X^2}p\\log(np)$, then $\\F(\\widehat{\\Omega}_{\\I}, \\mclI) =\\F({\\Omega}^*, \\mclI)=0$ and the conclusion holds automatically. If $\\I\\geq C_s\\frac{\\|X\\|_{\\psi_2}^4}{c_X^2}p\\log(np)$, then by \\Cref{lem:estimation covariance}, it holds with probability at least $1 - n^{-7}$ that\n\\begin{align}\n\\F(\\widehat{\\Omega}_{\\I}, \\mclI) -\\F({\\Omega}^*, \\mclI) \\geq & |\\mclI|{\\rm Tr}[(\\widehat{\\Omega}_{\\I} - \\Omega^*)^\\top (\\widehat{\\Sigma}_{\\I} - \\Sigma^*)] + \\frac{c|\\mclI|}{2\\|\\Omega^*\\|_2^2}\\|\\widehat{\\Omega}_{\\I} - \\Omega^*\\|_F^2.    \\\\\n\\geq & -|\\mclI|\\|\\widehat{\\Omega}_{\\I} - \\Omega^*\\|_F \\|\\widehat{\\Sigma}_{\\I} - \\Sigma^*\\|_F \\\\\n\\geq & -|\\mclI|p \\|\\widehat{\\Omega}_{\\I} - \\Omega^*\\|_2 \\|\\widehat{\\Sigma}_{\\I} - \\Sigma^*\\|_2\\\\\n\\geq & -\\|X\\|_{\\psi_2}^4 p^2\\log(np)\\|\\Omega^*\\|_2^2.\n\\end{align}\n\\eprf\n\n\n\n\n\\bnlem\n\\label{lem:cov loss deviation no change point}\nLet $\\I\\subset [1,T]$ be any interval that contains no change point. Then for any interval ${  \\mathcal J }  \\supset \\I$, it holds with probability at least $1 - (np)^{-5}$ that\n\\begin{equation*}\n    \\mclF(\\Omega^*_{\\I},\\I)\\leq \\mclF(\\widehat{\\Omega}_{  \\mathcal J } ,\\I) + C\\frac{\\|X\\|_{\\psi_2}^4}{c_X^2}p^2\\log(np).\n\\end{equation*}\n\\enlem\n\\begin{proof}\nThe conclusion is guaranteed by \\Cref{lem:cov no cp} and the fact that $\\F(\\Omega,\\I) = L(\\Omega,\\I)$ and $\\widehat{\\Omega}_{\\I}$ is (one of) the minimizer of $L(\\Omega,\\I)$.\n\\end{proof}\n\n\n\n\n\n\n\n\n\n\\bnlem[Single change point]\n\\label{lem:cov single cp}\nSuppose the good events \n$\\mathcal L ( { \\mathcal B_n^{-1}   \\Delta }  ) $ and  $\\mathcal R ( { \\mathcal B_n^{-1}   \\Delta } ) $ defined  in \\Cref{eq:left and right approximation of change points} hold. \nLet   $ \\I=(s,e] \\in \\mathcal {\\widehat P}  $  be such that $\\I$ contains exactly  one    change point $ \\eta_k $. \n Then  with probability  at least $1-(np)^{-3}$,   it holds that \n\\begin{equation}\n    \\min\\{ \\eta_k -s ,e-\\eta_k \\} \\leq CC_{\\gamma} \\|X\\|_{\\psi_2}^4\\frac{C_X^2}{c_X^6}\\frac{p^2\\log(np)}{\\kappa_k^2} + C\\|X\\|_{\\psi_2}^4\\frac{C_X^6}{c_X^6} { \\mathcal B_n^{-1}   \\Delta } .\n\\end{equation}\n\\enlem\n\\bprf\nIf either $ \\eta_k -s \\le  { \\mathcal B_n^{-1}   \\Delta }   $ or $e-\\eta_k\\le  { \\mathcal B_n^{-1}   \\Delta } $, then there is nothing to show. So assume that \n$$  \\eta_k -s  >  { \\mathcal B_n^{-1}   \\Delta }  \\quad \\text{and} \\quad   e-\\eta_k  >  { \\mathcal B_n^{-1}   \\Delta }  . $$\nBy event $\\mathcal R (p^{-1} { \\mathcal B_n^{-1}   \\Delta }  )$, there exists  $ s_ u  \\in \\{ s_q\\}_{q=1}^ { \\mathcal Q} $ such that \n$$0\\le s_u - \\eta_k \\le p^{-1} { \\mathcal B_n^{-1}   \\Delta } . $$\n So \n$$ \\eta_k \\le s_ u \\le e .$$\nDenote \n$$ \\I_ 1 = (s,s_u] \\quad \\text{and} \\quad \\I_2 = (s_u, e],$$\nand $\\mclF^*({  \\mathcal J } ) = \\sum_{i\\in {  \\mathcal J } }[{\\rm Tr}((\\Omega^*_i)^\\top X_iX_i^\\top) -\\log|\\Omega_i^*|]$. Since\n$s, e,  s_u \\in  \\{ s_q\\}_{q=1}^ { \\mathcal Q}   $, by the definition of $\\widehat{P}$ and $\\widehat{\\Omega}$, and \\Cref{lem:cov one change deviation bound}, it holds that\n\\begin{align}\n    \\mclF(\\widehat{\\Omega}_{\\I},\\I)\\leq& \\mclF(\\widehat{\\Omega}_{\\I_1},\\I_1) + \\mclF(\\widehat{\\Omega}_{\\I_2},\\I_2) + \\gamma \\nonumber \\\\\n    \\leq & \\mclF^*(\\I_1) + \\frac{C_X^6 p}{c_X^4}(s_u - \\eta_k)\\kappa_k^2 +  C\\frac{\\|X\\|_{\\psi_2}^4 C_X^2}{c_X^4}p^2\\log(np) + \\mclF^*(\\I_2) + \\gamma \\nonumber \\\\\n    \\leq & \\mclF^*(\\I) + \\frac{C_X^6}{c_X^4}\\mclB_n^{-1}\\Delta\\kappa_k^2 +  C\\frac{\\|X\\|_{\\psi_2}^4 C_X^2}{c_X^4}p^2\\log(np) +  \\gamma,\n\\end{align}\nwhere the last inequality is due to $s_u - \\eta_k\\leq \\mclB_n^{-1}\\Delta$. Let \n$$\\widetilde{\\gamma} = \\frac{C_X^6}{c_X^4}\\mclB_n^{-1}\\Delta\\kappa_k^2 +  C\\frac{\\|X\\|_{\\psi_2}^4 C_X^2}{c_X^4}p^2\\log(np) +  \\gamma.$$ \nThen by Taylor expansion and \\Cref{lem:estimation covariance} we have\n\\begin{align}\n    \\frac{c}{\\max_{k\\in [K]}\\|\\Omega_{\\I_k}^*\\|_2^2}\\sum_{t\\in \\mclI}\\|\\widehat{\\Omega}_{\\I} - \\Omega^*_t\\|_F^2 \n    &\\leq \\widetilde{\\gamma} + \\sum_{i=k-1}^k|\\I_i|{\\rm Tr}[(\\Omega_{\\I_i}^* - \\widehat{\\Omega}_{\\I})^\\top (\\widehat{\\Sigma}_{\\I_i}  - \\Sigma^*_{\\I_i})] \\nonumber \\\\\n    &\\leq \\widetilde{\\gamma} + \\sum_{i=k-1}^k {|\\mclI_{i}|}\\|\\widehat{\\Sigma}_{\\I_i} - \\Sigma^*_{\\I_i}\\|_F\\|\\widehat{\\Omega}_{\\I} - \\Omega^*_{\\I_i}\\|_F \\nonumber \\\\\n    &\\leq \\widetilde{\\gamma} + C_1\\|X\\|_{\\psi_2}^2 p\\log^{\\frac{1}{2}}(np)\\left[ \\sum_{i=k-1}^k \\sqrt{|\\mclI_{i}|}\\|\\widehat{\\Omega}_{\\I} - \\Omega^*_{\\I_i}\\|_F \\right] \\nonumber \\\\\n    &\\leq \\widetilde{\\gamma} + C_1\\|X\\|_{\\psi_2}^2 p\\log^{\\frac{1}{2}}(np) \\sqrt{\\sum_{t\\in \\mclI}\\|\\widehat{\\Omega}_{\\I} - \\Omega^*_t\\|_F^2}.\n\\end{align}\nThe inequality above implies that\n\\begin{equation}\n    \\sum_{t\\in \\mclI}\\|\\widehat{\\Omega}_{\\I} - \\Omega^*_t\\|_F^2 \\leq \\frac{2}{c}\\max\\|\\Omega_{\\I_k}^*\\|_2^2\\left[\\widetilde{\\gamma} + \\max\\|\\Omega_{\\I_k}^*\\|_2^2\\|X\\|^4_{\\psi_2} p^{2}\\log(np)\\right].\n\\end{equation}\nOn the other hand,\n\\begin{equation}\n    \\sum_{t\\in \\mclI}\\|\\widehat{\\Omega}_{\\I} - \\Omega^*_t\\|_F^2\\geq \\frac{|\\mclI_{k-1}||\\mclI_{k}|}{|\\mclI|}\\|\\Omega^*_{\\I_{k-1}} - \\Omega^*_{\\I_k}\\|_F^2,\n\\end{equation}\nwhich implies that\n\\begin{equation}\n    \\min\\{|\\mclI_{k-1}|,|\\mclI_{k}|\\}\\leq C_2C_{\\gamma} \\|X\\|_{\\psi_2}^4\\frac{C_X^2}{c_X^6}\\frac{p^2\\log(np)}{\\kappa_k^2} + C_2\\|X\\|_{\\psi_2}^4\\frac{C_X^6}{c_X^6} { \\mathcal B_n^{-1}   \\Delta } .\n\\end{equation}\n\\eprf\n\n\n\n\\bnlem[Two change point]\n\\label{lem:cov two cp}\nSuppose the good events \n$\\mathcal L ( { \\mathcal B_n^{-1}   \\Delta }  ) $ and  $\\mathcal R (  { \\mathcal B_n^{-1}   \\Delta }  ) $ defined  in  \\Cref{eq:left and right approximation of change points} hold. Let   $ \\I=(s,e] \\in \\mathcal {\\widehat P}  $  be an interval that contains exactly two change points $ \\eta_k,\\eta_{k+1}$. Suppose in addition that $\\gamma \\geq C_{\\gamma}\\|X\\|_{\\psi_2}^4\\max_{k\\in [K]}\\|\\Omega_{\\I_k}^*\\|_2^2p^2\\log(np)$, and\n\\begin{equation}\n    \\Delta \\kappa^2 \\geq \\mathcal{B}_n \\frac{\\|X\\|_{\\psi_2}^4}{c_X^4}p^2\\log(np),\n\\end{equation}\nthen it holds with probability at least $1 - n^{-5}$ that\n\\begin{equation}\n    \\max\\{\\eta_k - s, e - \\eta_{k + 1}\\}\\leq  { \\mathcal B_n^{-1\/2}   \\Delta } .\n\\end{equation}\n\\enlem\n\\bprf\nSince the events $\\mathcal L (  { \\mathcal B_n^{-1}   \\Delta }   ) $ and  $\\mathcal R (  { \\mathcal B_n^{-1}   \\Delta }  ) $ hold, let $ s_u, s_v$ be such that \n $\\eta_k \\le s_u \\le s_v \\le \\eta_{k+1} $ and that \n $$ 0 \\le s_u-\\eta_k \\le    { \\mathcal B_n^{-1}   \\Delta }  ,  \\quad  0\\le  \\eta_{k+1} - s_v \\le  { \\mathcal B_n^{-1}   \\Delta }    .    $$\n\n \\begin{center}\n \\begin{tikzpicture} \n\\draw[ - ] (-10,0)--(1,0);\n \\node[color=black] at (-8,-0.3) {\\small s};\n \\draw[  (-, ultra thick, black] (-8,0) -- (-7.99,0);\n  \\draw[ { -]}, ultra thick, black] (-1.0002,0) -- (-1,0);\n\n   \\node[color=black] at (-7,-0.3) {\\small $\\eta_k$};\n\\draw(-7 ,0)circle [radius=2pt] ;\n\n  \\node[color=black] at (-6.5,-0.3) {\\small $s_u$};\n\\draw plot[mark=x, mark options={color=black, scale=1.5}] coordinates {(-6.5,0)  }; \n \n\n  \\node[color=black] at (-2.3,-0.3) {\\small $\\eta_{k+1}$};\n\\draw(-2.3 ,0)circle [radius=2pt] ;\n  \\node[color=black] at (-3 ,-0.3) {\\small $s_v$};\n\\draw plot[mark=x, mark options={color=black, scale=1.5}] coordinates {(-3,0)  }; \n  \n   \\node[color=black] at (-1,-0.3) {\\small $e$};\n\n \n\\end{tikzpicture}\n\\end{center} \n\nDenote \n$$ \\mathcal I_ 1= (  s, s _u], \\quad \\I_2  =(s_u, s_v] \\quad \\text{and} \\quad \\I_3  = (s_v,e]. $$\nIn addition, denote \n$$ {  \\mathcal J } _1  = (s,\\eta_k ], \\quad {  \\mathcal J } _2=(\\eta_k, \\eta_{k} + \\frac{ \\eta_{k+1}   -\\eta_k  }{2}], \\quad {  \\mathcal J } _3 =  ( \\eta_k+ \\frac{ \\eta_{k+1}   -\\eta_k  }{2},\\eta_{k+1 } ]  \\quad \\text{and} \\quad {  \\mathcal J } _4 = (\\eta_{k+1}  , e] .$$\nSince \n$s, e,  s_u ,s_v  \\in  \\{ s_q\\}_{q=1}^ { \\mathcal Q}   $, by the event $\\mclL(p^{-1} { \\mathcal B_n^{-1}   \\Delta } )$ and $\\mclR(p^{-1} { \\mathcal B_n^{-1}   \\Delta } )$, it holds with probability at least $1 - n^{-3}$ that\n$$\n0\\leq s_u - \\eta_k\\leq p^{-1} { \\mathcal B_n^{-1}   \\Delta } ,\\ 0\\leq \\eta_{k+1} - s_v\\leq p^{-1} { \\mathcal B_n^{-1}   \\Delta } .\n$$\nDenote\n$$\\I_1 = (s ,\\eta_k], \\I_2 = (\\eta_k, \\eta_{k + 1}], \\I_3 = (\\eta_{k + 1}, e].$$\nBy the definition of DP and $\\widehat{\\Omega}_{\\I}$, it holds that\n\\begin{align}\n        \\F(\\hat{\\Omega}_{\\I},\\I)\n        \\leq & \\sum_{i = 1}^3 \\F(\\hat{\\Omega}_{\\I_i},\\I_i) + 2\\gamma \\\\\n        \\leq & \\sum_{i = 1}^3 \\F({\\Omega}_{\\I_i}^*,\\I_i) +  2\\gamma + \\frac{C_X^6 p}{c_X^4} \\frac{|{  \\mathcal J } _1|(s_u - \\eta_k)}{|{  \\mathcal J } _1|+s_u - \\eta_k} \\kappa_k^2 + \\frac{C_X^6 p}{c_X^4} \\frac{|{  \\mathcal J } _4|(\\eta_{k + 1} - s_v)}{|{  \\mathcal J } _4|+\\eta_{k + 1} - s_v} \\kappa_k^2 + C\\frac{\\|X\\|_{\\psi_2}^4 C_X^2}{c_X^4}p^2\\log(np) \\\\\n        \\leq & \\sum_{i = 1}^3 \\F({\\Omega}_{\\I_i}^*,\\I_i) +  2\\gamma + \\frac{C_X^6}{c_X^4}  { \\mathcal B_n^{-1}   \\Delta }  \\kappa_k^2 + C\\frac{\\|X\\|_{\\psi_2}^4 C_X^2}{c_X^4}p^2\\log(np).\n\\end{align}\nLet\n$$\n\\widetilde{\\gamma} = 2\\frac{C_X^6}{c_X^4}\\mclB_n^{-1}\\Delta\\kappa_k^2 +  C\\frac{\\|X\\|_{\\psi_2}^4 C_X^2}{c_X^4}p^2\\log(np) +  2\\gamma.\n$$,\nThen by Taylor expansion and \\Cref{lem:estimation covariance} we have\n\\begin{align}\n    cc_X^2\\sum_{t\\in \\mclI}\\|\\widehat{\\Omega}_{\\I} - \\Omega^*_t\\|_F^2 &\\leq \\widetilde{\\gamma} + \\sum_{i=1}^3|\\I_i|{\\rm Tr}[(\\Omega_{\\I_i}^* - \\widehat{\\Omega}_{\\I})^\\top (\\widehat{\\Sigma}_{\\I_i}  - \\Sigma^*_{\\I_i})] \\nonumber \\\\\n    &\\leq \\widetilde{\\gamma} + C\\|X\\|_{\\psi_2}^2 p\\log^{\\frac{1}{2}}(np)\\left[ \\sum_{i=1}^3 \\sqrt{|\\mclI_{i}|}\\|\\widehat{\\Omega}_{\\I} - \\Omega^*_{i}\\|_F \\right]  \\nonumber \\\\\n    &\\leq \\widetilde{\\gamma} + C\\|X\\|_{\\psi_2}^2 p\\log^{\\frac{1}{2}}(np) \\sqrt{\\sum_{t\\in \\mclI}\\|\\widehat{\\Omega}_{\\I} - \\Omega^*_t\\|_F^2}.\n\\end{align}\nThe inequality above implies that\n\\begin{equation}\n    \\sum_{t\\in \\mclI}\\|\\widehat{\\Omega}_{\\I} - \\Omega^*_t\\|_F^2 \\leq \\frac{C_1}{c_X^2}\\left[\\widetilde{\\gamma} + \\frac{1}{c_X^2}\\|X\\|^4_{\\psi_2} p^{2}\\log(np)\\right].\n\\end{equation}\nBy the choice of $\\gamma$, it holds that\n\\begin{equation}\n    \\sum_{t\\in \\I_1\\cup\\I_2}\\|\\widehat{\\Omega}_{\\I} - \\Omega^*_t\\|_F^2 \\leq \\frac{C_1C_{\\gamma}}{c_X^4} \\|X\\|^4_{\\psi_2} p^{2}\\log(np).\n\\end{equation}\nOn the other hand,\n\\begin{equation}\n    \\sum_{t\\in \\I_1\\cup\\I_2}\\|\\widehat{\\Omega}_{\\I} - \\Omega^*_t\\|_F^2\\geq \\frac{|\\mclI_{1}||\\mclI_{2}|}{|\\mclI|}\\|\\Omega^*_{k-1} - \\Omega^*_k\\|_F^2\\geq \\frac{1}{2}\\min\\{|\\I_1|,|\\I_2|\\}\\|\\Omega^*_{k-1} - \\Omega^*_k\\|_F^2.\n\\end{equation}\nSuppose $|\\I_1|\\geq |\\I_2|$, then the inequality above leads to\n\\begin{equation*}\n    \\Delta \\kappa^2\\leq \\frac{C_1C_{\\gamma}}{c_X^4} \\|X\\|^4_{\\psi_2} p^{2}\\log(np),\n\\end{equation*}\nwhich is contradictory to the assumption on $\\Delta$. Therefore, $|\\I_1|<|\\I_2|$ and we have \n\\begin{equation}\n    s-\\eta_k = |\\I_1|\\leq CC_{\\gamma} \\frac{\\|X\\|_{\\psi_2}^4}{c_X^4\\|\\Omega_k^* - \\Omega_{k-1}^*\\|_F^2}p^2\\log(np).\n\\end{equation}\nThe bound for $e - \\eta_{k+1}$ can be proved similarly.\n\\eprf\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\bnlem[Three or more change points]\nSuppose the assumptions in \\Cref{assp:DCDP_covariance} hold. Then with probability at least $1-(np)^{-3}$, there is no interval $\\widehat { \\mathcal P}$ containing three or more true change points.   \n \\enlem \n\\bprf\nWe prove by contradiction. Suppose    $ \\I=(s,e] \\in \\mathcal {\\widehat P}  $  be such that $ \\{ \\eta_1, \\ldots, \\eta_M\\} \\subset \\I $ with $M\\ge 3$. Throughout the proof, $M$ is assumed to be a parameter that can potentially change with $n$. \nSince the events $\\mathcal L (  { \\mathcal B_n^{-1}   \\Delta }   ) $ and  $\\mathcal R (  { \\mathcal B_n^{-1}   \\Delta }  ) $ hold, by relabeling $\\{ s_q\\}_{q=1}^ { \\mathcal Q}   $ if necessary,   let $ \\{ s_m\\}_{m=1}^M $ be such that \n $$ 0 \\le s_m  -\\eta_m  \\le  { \\mathcal B_n^{-1}   \\Delta }     \\quad \\text{for} \\quad 1 \\le m \\le M-1  $$ and that\n $$ 0\\le \\eta_M  - s_M \\le  { \\mathcal B_n^{-1}   \\Delta }  .$$ \n Note that these choices ensure that  $ \\{ s_m\\}_{m=1}^M  \\subset \\I . $\n \n \\begin{center}\n \\begin{tikzpicture} \n\\draw[ - ] (-10,0)--(1,0);\n \\node[color=black] at (-8,-0.3) {\\small s};\n \\draw[  (-, ultra thick, black] (-8,0) -- (-7.99,0);\n  \\draw[ { -]}, ultra thick, black] (-1.0002,0) -- (-1,0);\n\n   \\node[color=black] at (-7,-0.3) {\\small $\\eta_1$};\n\\draw(-7 ,0)circle [radius=2pt] ;\n\n  \\node[color=black] at (-6.5,-0.3) {\\small $s_1$};\n\\draw plot[mark=x, mark options={color=black, scale=1.5}] coordinates {(-6.5,0)  }; \n\n  \\node[color=black] at (-5,-0.3) {\\small $\\eta_2$};\n\\draw(-5 ,0)circle [radius=2pt] ;\n  \\node[color=black] at (-4.5,-0.3) {\\small $s_2$};\n\\draw plot[mark=x, mark options={color=black, scale=1.5}] coordinates {(-4.5,0)  }; \n \n\n  \\node[color=black] at (-2.5,-0.3) {\\small $\\eta_3$};\n\\draw(-2.5 ,0)circle [radius=2pt] ;\n  \\node[color=black] at (-3 ,-0.3) {\\small $s_3$};\n\\draw plot[mark=x, mark options={color=black, scale=1.5}] coordinates {(-3,0)  }; \n  \n   \\node[color=black] at (-1,-0.3) {\\small $e$};\n\n \n\\end{tikzpicture}\n\\end{center}\n\n{\\bf Step 1.}\n Denote \n $$ \\mathcal I_ 1= (  s, s _1], \\quad \\I_m   =(s_{m-1} , s_m] \\text{ for }  2 \\le m \\le M     \\quad \\text{and} \\quad \\I_{M+1}   = (s_M,e]. $$\nThen since \n$ s , e,  \\{ s_m \\}_{m=1}^M  \\subset   \\{ s_q\\}_{q=1}^ { \\mathcal Q}   $,   it follows that \n\n\n\nSuppose $\\I = (s,e]\\in \\widehat{\\mclP}$ and there are $M\\geq 3$ true change points $\\{\\eta_{q + i}\\}_{i\\in [M]}$ inside $\\I$, and denote\n\\begin{equation*}\n    \\I_1 = (s, \\eta_{q + 1}],\\ \\I_m = (\\eta_{q + m - 1},\\eta_{q + m}],\\ \\I_{M+1} = (\\eta_{q + M},e].\n\\end{equation*}\nThen by the definition of $\\widehat{\\mclP}$ and $\\widehat{\\Omega}_{\\I_m}$, it holds that\n\\begin{equation*}\n    \\F(\\hat{\\Omega}_{\\I},\\I)\\leq \\sum_{i = 1}^{M+1} \\F(\\hat{\\Omega}_{\\I_i},\\I_i) + M\\gamma\\leq \\sum_{i = 1}^{M+1} \\F({\\Omega}_{\\I_i}^*,\\I_i) + M\\gamma,\n\\end{equation*}\nwhich implies that\n\\begin{align}\n    & \\sum_{t\\in \\I}{\\rm Tr}(\\widehat{\\Omega}_{\\I}^\\top (X_tX_t^\\top)) - |\\I|\\log|\\widehat{\\Omega}_{\\I}| \\nonumber  \\\\\n    \\leq & \\sum_{i=1}^{M+1} \\sum_{t\\in \\I_i}{\\rm Tr}(({\\Omega}_{\\I_i}^*)^\\top (X_tX_t^\\top)) - \\sum_{i=1}^{M+1}|\\I_i|\\log|{\\Omega}_{\\I_i}^*| + M\\gamma.\n\\end{align}\nBy Taylor expansion and \\Cref{lem:estimation covariance} we have\n\\begin{align}\n    cc_X^2\\sum_{t\\in \\mclI}\\|\\widehat{\\Omega}_{\\I} - \\Omega^*_t\\|_F^2 &\\leq M\\gamma + \\sum_{i=1}^{M+1}|\\I_i|{\\rm Tr}[(\\Omega_{\\I_i}^* - \\widehat{\\Omega}_{\\I})^\\top (\\widehat{\\Sigma}_{\\I_i}  - \\Sigma^*_{\\I_i})] \\nonumber \\\\\n    &\\leq M\\gamma + C\\|X\\|_{\\psi_2}^2 p\\log^{\\frac{1}{2}}(np)\\left[ \\sum_{i=1}^{M+1} \\sqrt{|\\mclI_{i}|}\\|\\widehat{\\Omega}_{\\I} - \\Omega^*_{i}\\|_F \\right]  \\nonumber \\\\\n    &\\leq M\\gamma + C\\|X\\|_{\\psi_2}^2 p\\log^{\\frac{1}{2}}(np) \\sqrt{\\sum_{t\\in \\mclI}\\|\\widehat{\\Omega}_{\\I} - \\Omega^*_t\\|_F^2}.\n\\end{align}\nThe inequality above implies that\n\\begin{equation}\n    \\sum_{t\\in \\mclI}\\|\\widehat{\\Omega}_{\\I} - \\Omega^*_t\\|_F^2 \\leq \\frac{C_1}{c_X^2}\\left[M\\gamma + \\frac{1}{c_X^2}\\|X\\|^4_{\\psi_2} p^{2}\\log(np)\\right].\n\\end{equation}\nOn the other hand, for each $i\\in [M]$, we have\n\\begin{equation}\n    \\sum_{t\\in \\I_i\\cup\\I_{i +1}}\\|\\widehat{\\Omega}_{\\I} - \\Omega^*_t\\|_F^2\\geq \\frac{|\\mclI_{i}||\\mclI_{i + 1}|}{|\\mclI|}\\|\\Omega^*_{\\eta_{q + i + 1}} - \\Omega^*_{\\eta_{q + i}}\\|_F^2.\n\\end{equation}\nIn addition, for each $i\\in \\{2,\\cdots,M\\}$, by definition, it holds that $\\min\\{|\\I_i|,|\\I_{i + 1}|\\}\\geq \\Delta$. Therefore, we have\n\\begin{align*}\n    (M-2)\\Delta \\kappa^2\\leq \\frac{C_1}{c_X^2}\\left[M\\gamma + \\frac{1}{c_X^2}\\|X\\|^4_{\\psi_2} p^{2}\\log(np)\\right].\n\\end{align*}\nSince $M\/(M-2)\\leq 3$ for any $M\\geq 3$, it holds that\n\\begin{equation}\n    \\Delta \\leq CC_{\\gamma} \\frac{\\|X\\|_{\\psi_2}^4}{c_X^4\\|\\Omega_k^* - \\Omega_{k-1}^*\\|_F^2}p^2\\log(np),\n\\end{equation}\nwhich is contradictory to the assumption on $\\Delta$, and the proof is complete.\n\\eprf\n\n\n\n\n\n\n \\bnlem[Two consecutive intervals]\n \\label{lem:cov two consecutive interval}\n Under \\Cref{assp:DCDP_covariance} and the choice that $$\\gamma\\geq C_{\\gamma}\\|X\\|_{\\psi_2}^4\\max_{k\\in [K+1]}\\|\\Omega_{\\eta_k}^*\\|_2^2 p^2\\log(np),$$ with probability at least $1-(np)^{-3}$, there are no  two consecutive intervals $\\I_1= (s,t ] \\in \\widehat{\\mathcal P}$, $\\I_2=(t, e] \\in   \\widehat{\\mathcal P}$ such that $\\I_1 \\cup \\I_2$ contains no change points.  \n \\enlem \n \\begin{proof} \n We prove by contradiction. Suppose that $\\I_1,\\I_2\\in \\widehat{\\mclP}$ and\n $$ \\I :  =\\I_1\\cup  \\I_2 $$\n contains no change points. \n By the definition of $\\widehat{\\mclP}$ and $\\widehat{\\Omega}_{\\I}$, it holds that\n $$ \\F(\\widehat{\\Omega}_{\\I_1},\\I_1)  + \\F(\\widehat{\\Omega}_{\\I_2},\\I_2) + \\gamma \\leq  \\F(\\widehat{\\Omega}_{\\I},\\I)\\leq \\F({\\Omega}_{\\I}^*,\\I)$$\n By \\Cref{lem:cov no cp}, it follows that \n\\begin{align*} \n\\F({\\Omega}^*_{\\I_1}, \\mclI_1)\\leq &  \\F(\\widehat{\\Omega}_{\\I_1}, \\mclI_1) + C\\frac{\\|X\\|_{\\psi_2}^4}{c_X^2} p^2\\log(np),\n \\\\\n\\F({\\Omega}^*_{\\I_2}, \\mclI_2)\\leq &  \\F(\\widehat{\\Omega}_{\\I_2}, \\mclI_2) +  C\\frac{\\|X\\|_{\\psi_2}^4}{c_X^2} p^2\\log(np) \n \\end{align*}\n So\n $$\\F({\\Omega}^*_{\\I_1}, \\mclI_1) +\\F({\\Omega}^*_{\\I_2}, \\mclI_2) -2C\\frac{\\|X\\|_{\\psi_2}^4}{c_X^2} p^2\\log(np)  +\\gamma    \\leq  \\F({\\Omega}_{\\I}^*,\\I).  $$\n Since $\\I$ does not contain any change points, ${\\Omega}^*_{\\I_1} = {\\Omega}^*_{\\I_2} = {\\Omega}^*_{\\I}$, and it follows that \n $$ \\gamma \\leq 2C\\frac{\\|X\\|_{\\psi_2}^4}{c_X^2} p^2\\log(np).$$\n This is a contradiction when $C_\\gamma> 2C$.\n \\end{proof} \n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\clearpage\n\\subsection{High-dimensional regression}\n\n\nThroughout this section, unless specified otherwise, for the output of \\Cref{algorithm:DCDP_divide}, we always set the loss function $\\mclF(\\cdot, \\cdot)$ to be\n\\begin{equation}\n\\mathcal F  (\\beta, \\I) :  = \\begin{cases} \n \\sum_{i \\in \\I } (y_i - X_i^\\top \\beta) ^2  &\\text{if } |\\I|\\ge C_\\mclF { \\mathfrak{s} }  \\log(np) ,\n \\\\ \n 0 &\\text{otherwise,}\n \\end{cases}\n\\end{equation} \nwhere $C_{\\mclF}$ is a universal constant which is larger than $C_s$, the constant in sample size in \\Cref{lemma:interval lasso} and \\Cref{lemma:consistency}.\n\n\n\n \\bnprop\\label{prop:regression local consistency}\n Suppose  \\Cref{assp:dcdp_linear_reg} holds. Let \n $\\widehat { \\mathcal P }  $ denote the output of  \\Cref{algorithm:DCDP_divide} with\n\nSuppose that $\\gamma \\ge C_\\gamma  K { \\mathcal B_n^{-1}   \\Delta } \\kappa^2$. Uniformly with probability  at least $1 - C n ^{-3}$, the following conditions hold.\n\t\\begin{itemize}\n\t\t\\item [(i)] For each interval  $ \\I = (s, e] \\in \\widehat{\\mathcal{P}}$    containing one and only one true \n\t\t   change point $ \\eta_k $, it must be the case that\n $$\\min\\{ \\eta_k -s ,e-\\eta_k \\}  \\lesssim \\frac{\\sigma_{\\epsilon}^2\\vee 1  }{\\kappa^2 }\\bigg(   { \\mathfrak{s} }  \\log(np) +\\gamma \\bigg) +   { \\mathcal B_n^{-1}   \\Delta } .$$\n\t\\item [(ii)] For each interval  $  \\I = (s, e] \\in \\widehat{\\mathcal{P}}$ containing exactly two true change points, say  $\\eta_ k  < \\eta_ {k+1} $, it must be the case that\n\\begin{align*} \n \\eta_k -s    \\lesssim   \\frac{\\sigma_{\\epsilon}^2\\vee 1  }{\\kappa^2 }\\bigg(   { \\mathfrak{s} }  \\log(np) +\\gamma \\bigg) +   { \\mathcal B_n^{-1}   \\Delta }       \\ \\text{and} \\ \n e-\\eta_{k+1}   \\le  C   \\frac{\\sigma_{\\epsilon}^2\\vee 1  }{\\kappa^2 }\\bigg(   { \\mathfrak{s} }  \\log(np) +\\gamma \\bigg) +   { \\mathcal B_n^{-1}   \\Delta } . \n \\end{align*} \n\t\t \n\\item [(iii)] No interval $\\I \\in \\widehat{\\mathcal{P}}$ contains strictly more than two true change points; and \n\n\n\t\\item [(iv)] For all consecutive intervals $ \\I_1 $ and $ \\I_2 $ in $\\widehat{ \\mathcal P}$, the interval \n\t\t$ \\I_1 \\cup  \\I_2 $ contains at least one true change point.\n\t\t\t\t\n\t \\end{itemize}\n\\enprop \n\n \n\n\\bnprop\\label{prop:regression change points partition size consistency}\nSuppose  \\Cref{assp:dcdp_linear_reg} holds. Let \n $\\widehat { \\mathcal P }  $ denote the output of  \\Cref{algorithm:DCDP_divide}. Suppose \n$\\gamma \\ge  C_\\gamma K  { \\mathcal B_n^{-1}   \\Delta }   \\kappa^2$  for sufficiently large constant $C_\\gamma$. Then\n with probability  at least $1 - C n ^{-3}$,  $| \\widehat { \\mathcal P} | =K $.\n\\enprop\n\n\n\n\\begin{proof}[Proof of   \\Cref{prop:regression change points partition size consistency}] \nDenote $\\mathfrak{G} ^*_n = \\sum_{ i =1}^n (y_i  -  X_i^\\top \\beta^*_i)^2$.  Given any collection $\\{t_1, \\ldots, t_m\\}$, where $t_1 < \\cdots < t_m$, and $t_0 = 0$, $t_{m+1} = n$, let \n\t\\begin{equation}\\label{regression eq-sn-def}\n\t\t { \\mathfrak{G} }  _n(t_1, \\ldots, t_{m}) = \\sum_{k=1}^{m} \\sum_{ i = t_k +1}^{t_{k+1}} \\mclF(\\widehat{\\beta} _  {(t_{k}, t_{k+1}]}, (t_{k}, t_{k+1}]). \n\t\\end{equation}\nFor any collection of time points, when defining \\eqref{regression eq-sn-def}, the time points are sorted in an increasing order.\n\nLet $\\{ \\widehat \\eta_{k}\\}_{k=1}^{\\widehat K}$ denote the change points induced by $\\widehat {\\mathcal P}$.  Suppose we can justify that \n\t\\begin{align}\n\t\t { \\mathfrak{G} } ^*_n + K\\gamma  \\ge  & { \\mathfrak{G} }  _n(s_1,\\ldots,s_K)   + K\\gamma - C_1 ( K +1)   { \\mathfrak{s} }  \\log(np) - C_1\\sum_{k\\in [K]}\\kappa_k^2 { \\mathcal B_n^{-1}   \\Delta }   \\label{eq:regression K consistency step 1} \\\\ \n\t\t\\ge &  { \\mathfrak{G} } _n (\\widehat \\eta_{1},\\ldots, \\widehat \\eta_{\\widehat K } ) +\\widehat K \\gamma - C_1 ( K +1)   { \\mathfrak{s} }  \\log(np) - C_1\\sum_{k\\in [K]}\\kappa_k^2 { \\mathcal B_n^{-1}   \\Delta }   \\label{eq:regression K consistency step 2} \\\\ \n\t\t\\ge &   { \\mathfrak{G} }  _n ( \\widehat \\eta_{1},\\ldots, \\widehat \\eta_{\\widehat K } , \\eta_1,\\ldots,\\eta_K ) + \\widehat K \\gamma    - C_1 ( K +1)   { \\mathfrak{s} }  \\log(np)- C_1\\sum_{k\\in [K]}\\kappa_k^2 { \\mathcal B_n^{-1}   \\Delta }   \\label{eq:regression K consistency step 3}\n\t\\end{align}\n\tand that \n\t\\begin{align}\\label{eq:regression K consistency step 4}\n\t\t { \\mathfrak{G} }  ^*_n   - { \\mathfrak{G} }  _n ( \\widehat \\eta_{1},\\ldots, \\widehat \\eta_{\\widehat K } , \\eta_1,\\ldots,\\eta_K ) \\le C_2 (K + \\widehat{K} + 2)  { \\mathfrak{s} }  \\log(np) .\n\t\\end{align}\n\tThen it must hold that $| \\hatp | = K$, as otherwise if $\\widehat K \\ge K+1 $, then  \n\t\\begin{align*}\n\t\tC _2 (K + \\widehat{K} + 2)   { \\mathfrak{s} }  \\log(np)  & \\ge    { \\mathfrak{G} }  ^*_n  - { \\mathfrak{G} }  _n ( \\widehat \\eta_{1},\\ldots, \\widehat \\eta_{\\widehat K } , \\eta_1,\\ldots,\\eta_K ) \\\\\n\t\t& \\ge      (\\widehat K - K)\\gamma  -C_1 ( K +1)   { \\mathfrak{s} }  \\log(np) - C_1\\sum_{k\\in [K]}\\kappa_k^2 { \\mathcal B_n^{-1}   \\Delta } .\n\t\\end{align*} \n\tTherefore due to the assumption that $| \\hatp|  =\\widehat K\\le 3K $, it holds that \n\t\\begin{align} \\label{eq:regression Khat=K} \n\t\tC_2 (4K + 2)  { \\mathfrak{s} }  \\log(np)  + C_1(K+1)  { \\mathfrak{s} }  \\log(np) +C_1\\sum_{k\\in [K]}\\kappa_k^2 { \\mathcal B_n^{-1}   \\Delta }  \\ge  (\\widehat K - K)\\gamma \\geq \\gamma,\n\t\\end{align}\n\tNote that \\eqref{eq:regression Khat=K} contradicts the choice of $\\gamma$.\n\n\n\\\n\\\\\n{\\bf Step 1.} Note that \\eqref{eq:regression K consistency step 1}  is implied by \n\t\\begin{align}\\label{eq:regression step 1 K consistency}  \n\t\t\\left| \t { \\mathfrak{G} }  ^*_n   -    { \\mathfrak{G} } _n(s_1,\\ldots,s_K)    \\right| \\le  C_3(K+1) \\lambda^2 + C_3\\sum_{k\\in [K]}\\kappa_k^2 { \\mathcal B_n^{-1}   \\Delta }  ,\n\t\\end{align}\n\twhich is  an  immediate consequence  of \\Cref{lam:regression one change deviation bound}. \n\t\\\n\t\\\\\n\t\\\\\n\t{\\bf Step 2.} Since $\\{ \\widehat \\eta_{k}\\}_{k=1}^{\\widehat K}$ are the change points induced by $\\widehat {\\mathcal P}$, \\eqref{eq:regression K consistency step 2} holds because $\\hatp$ is a minimizer.\n\\\\\n\\\\\n\t{\\bf Step 3.}\nFor every $ \\I  =(s,e]\\in \\hatp$, by \\Cref{prop:regression local consistency}, we know that with probability at least $1 - (np)^{-5}$, $\\I$ contains at most two change points. We only show the proof for the two-change-point case as the other case is easier. Denote\n\t\\[\n\t\t \\I =  (s ,\\eta_{q}]\\cup (\\eta_{q},\\eta_{q+1}] \\cup  (\\eta_{q+1} ,e]  = {  \\mathcal J } _1 \\cup {  \\mathcal J } _2 \\cup {  \\mathcal J } _{3},\n\t\\]\nwhere $\\{ \\eta_{q},\\eta_{q+1}\\} =\\I \\, \\cap \\, \\{\\eta_k\\}_{k=1}^K$. \n\nFor each $m\\in\\{1,2,3\\}$, by \\Cref{lam:regression one change deviation bound}, it holds that\n\\[\n\\sum_{ i \\in {  \\mathcal J } _m }(y_ i - X_i^\\top \\widehat{\\beta} _{{  \\mathcal J } _m}    )^2   \\leq \\sum_{ i \\in {  \\mathcal J } _ m } (y_ i  - X_i^\\top \\beta^*_i   )^2 + C\\sigma_{\\epsilon}^2 { \\mathfrak{s} }  \\log(np).\n\\]\nBy \\Cref{lam:regression goodness of fit J superset of I}, we have\n\\[\n\\sum_{ i \\in {  \\mathcal J } _m }(y_ i - X_i^\\top\\widehat {\\beta} _\\I    )^2   \\ge \\sum_{ i \\in {  \\mathcal J } _ m } (y_ i  - X_i^\\top \\beta_i^*   )^2 - C\\sigma_{\\epsilon}^2 { \\mathfrak{s} }  \\log(np).\n\\]\n Therefore the above inequality implies that \n\t\\begin{align} \\label{eq:regression K consistency step 3 inequality 3}  \\sum_{i \\in \\I }  (y_ i - X_i^*\\widehat {\\beta} _\\I    )^2   \\ge \\sum_{m=1}^{3} \\sum_{ i \\in {  \\mathcal J } _m }(y_ i - X_i^\\top\\widehat {\\beta} _{{  \\mathcal J } _m}    )^2  -C\\sigma_{\\epsilon}^2 { \\mathfrak{s} } \\log(np).  \n\t\\end{align}\nNote that   \\eqref{eq:regression K consistency step 3} is an immediate consequence of   \\eqref{eq:regression K consistency step 3 inequality 3}.\n\n\t{\\bf Step 4.}\nFinally, to show \\eqref{eq:regression K consistency step 4},   let  $\\widetilde { \\mathcal P}$ denote    the partition induced by $\\{\\widehat \\eta_{1},\\ldots, \\widehat \\eta_{\\widehat K } , \\eta_1,\\ldots,\\eta_K\\}$. Then \n$| \\widetilde { \\mathcal P} | \\le K + \\widehat K+2   $ and that $\\beta^*_i$ is unchanged in every interval $\\I \\in \\widetilde { \\mathcal P}$. \n\tSo  \\Cref{eq:regression K consistency step 4} is an immediate consequence of   \\Cref{lam:regression one change deviation bound}.\n\\end{proof} \n\n\n\n\n\n\n\\begin{proof}[Proof of \\Cref{cor:regression local refinement}]\nFor each $k\\in [K]$, let $\\hat{\\beta}_t = \\hat{\\beta}^{(1)}$ if $s_k<t< \\hat{\\eta}_k$ and $\\hat{\\beta}_t =\\hat{\\beta}^{(2)}$ otherwise. Let ${\\beta}^*_t$ be the true parameter at time point $t$, and $\\beta^{(1)} = \\beta^*_{\\eta_k}$ and $\\beta^{(2)} = \\beta^*_{\\eta_k + 1}$. First we show that under conditions $\\tilde{K} = K$ and $\\max_{k\\in [K]}|\\tilde{\\eta}_k - \\eta_k|\\leq \\Delta \/ 5$, there is only one true change point $\\eta_k$ in $(s_k, e_k)$. It suffices to show that\n\\begin{equation}\n    |\\tilde{\\eta}_k - \\eta_k|\\leq \\frac{2}{3}(\\tilde{\\eta}_{k+1} - \\tilde{\\eta}_{k}),\\ \\text{and}\\ |\\tilde{\\eta}_{k+1} - \\eta_{k+1}|\\leq \\frac{1}{3}(\\tilde{\\eta}_{k+1} - \\tilde{\\eta}_{k}).\n    \\label{tmp_eq:goal_lr}\n\\end{equation}\nDenote $R = \\max_{k\\in [K]}|\\tilde{\\eta}_k - \\eta_k|$, then\n\\begin{align*}\n    \\tilde{\\eta}_{k+1} - \\tilde{\\eta}_{k} &= \\tilde{\\eta}_{k+1} - {\\eta}_{k+1} + {\\eta}_{k+1} - {\\eta}_{k} + {\\eta}_{k} - \\tilde{\\eta}_{k} \\\\\n    &= ({\\eta}_{k+1} - {\\eta}_{k}) + (\\tilde{\\eta}_{k+1} - {\\eta}_{k+1}) + ({\\eta}_{k} - \\tilde{\\eta}_{k})\\in [{\\eta}_{k+1} - {\\eta}_{k} - 2R,{\\eta}_{k+1} - {\\eta}_{k} + 2R].\n\\end{align*}\nTherefore, \\Cref{tmp_eq:goal_lr} is guaranteed as long as\n\\begin{equation*}\n    R\\leq \\frac{1}{3}(\\Delta - 2R),\n\\end{equation*}\nwhich is equivalent to $R\\leq \\Delta\/5$.\n\nNow without loss of generality, assume that $s_k<\\eta_k<\\hat{\\eta}_k<e_k$. Denote $\\mclI_k = \\{s_k + 1, \\cdots, e_k\\}$. Consider two cases:\n\n\\textbf{Case 1.} If\n\\begin{equation*}\n    \\hat{\\eta}_k - \\eta_k < \\max\\{C_s(\\sigma_{\\epsilon}^2\\vee 1) { \\mathfrak{s} } \\log(np), C_s(\\sigma_{\\epsilon}^2\\vee 1) { \\mathfrak{s} } \\log(np)\/\\kappa^2\\},\n\\end{equation*}\nthen the proof is done.\n\n\\textbf{Case 2.} If\n\\begin{equation*}\n    \\hat{\\eta}_k - \\eta_k \\geq \\max\\{C_s(\\sigma_{\\epsilon}^2\\vee 1) { \\mathfrak{s} } \\log(np), C_s(\\sigma_{\\epsilon}^2\\vee 1) { \\mathfrak{s} } \\log(np)\/\\kappa^2\\},\n\\end{equation*}\nthen we proceed to prove that $|\\hat{\\eta}_k - \\eta_k|\\leq C(\\sigma_{\\epsilon}^2\\vee 1) { \\mathfrak{s} } \\log(np)\/\\kappa^2$ with probability at least $1 - (np)^{-5}$. Then we either prove the result or get an contradiction, and complete the proof in either case. The first step is to prove that with probability at least $1 - (np)^{-5}$,\n\t\\[\n\t\t\\sum_{t = s_k + 1}^{e_k}\\|\\widehat{\\beta}_t - \\beta^*_t\\|_2^2 \\leq C_1 { \\mathfrak{s} }  \\zeta^2.\n\t\\]\nDue to \\Cref{eq-g-lasso}, it holds that\n\t\\begin{align}\\label{eq:reg local refine 1}\n\t\t& \\sum_{t = s_k + 1}^{e_k} \\|y_t - X_t^{\\top}\\widehat{\\beta}_t\\|_2^2 + \\zeta \\sum_{i = 1}^p \\sqrt{\\sum_{t = s_k + 1}^{e_k} \\bigl(\\widehat{\\beta}_t\\bigr)_i^2} \\leq \\sum_{t = s_k + 1}^{e_k} \\|y_{t} - X_t^{\\top}\\beta^*_t \\|_2^2 + \\zeta \\sum_{i = 1}^p \\sqrt{\\sum_{t = s_k + 1}^{e_k} \\bigl(\\beta^*_t\\bigr)_{i}^2}.\n\t\\end{align}\n\tLet $\\delta_t = \\widehat{\\beta}_t - \\beta^*_t$. It holds that $\\sum_{t = s_k + 1}^{e_k - 1}\\mathbbm{1}\\left\\{\\delta_t \\neq \\delta_{t+1}\\right\\} = 2$.\tThen \\Cref{eq:reg local refine 1} implies that\n\t\\begin{align}\\label{eq:reg local refine 2}\n\t\t\\sum_{t = s_k + 1}^{e_k} \\|\\delta_t^{\\top} X_t\\|_2^2 + \\zeta  \\sum_{i = 1}^p \\sqrt{\\sum_{t = s_k + 1}^{e_k} \\bigl(\\widehat{\\beta}_t\\bigr)_{i}^2}\\leq 2\\sum_{t = s_k + 1}^{e_k} (y_t - X_t^{\\top}\\beta^*_t)\\delta_t^{\\top} X_t + \\zeta \\sum_{i = 1}^p \\sqrt{\\sum_{t = s_k + 1}^{e_k} \\bigl(\\beta^*_t\\bigr)_{i}^2}.\n\t\\end{align}\nNote that\n\t\\begin{align}\n\t\t& \\sum_{i = 1}^p \\sqrt{\\sum_{t = s_k + 1}^{e_k} \\bigl(\\beta^*_t\\bigr)_{i}^2} - \\sum_{i = 1}^p \\sqrt{\\sum_{t = s_k + 1}^{e_k} \\bigl(\\widehat{\\beta}_t\\bigr)_{i}^2} \\nonumber \\\\\n\t\t=& \\sum_{i \\in S} \\sqrt{\\sum_{t = s_k + 1}^{e_k} \\bigl(\\beta^*_t\\bigr)_{i}^2} - \\sum_{i \\in S} \\sqrt{\\sum_{t = s_k + 1}^{e_k} \\bigl(\\widehat{\\beta}_t\\bigr)_{i}^2}  - \\sum_{i \\in S^c} \\sqrt{\\sum_{t = s_k + 1}^{e_k} \\bigl(\\widehat{\\beta}_t\\bigr)_{i}^2} \\nonumber \\\\\n\t\t\\leq & \\sum_{i \\in S} \\sqrt{\\sum_{t = s_k + 1}^{e_k} \\bigl(\\delta_t\\bigr)_{i}^2}  - \\sum_{i \\in S^c} \\sqrt{\\sum_{t = s_k + 1}^{e_k} \\bigl(\\delta_t\\bigr)_{i}^2}.\\label{eq:reg local refine 3}\n\t\\end{align}\nWe then examine the cross term, with probability at least $1 - (np)^{-5}$, which satisfies the following\n\t\\begin{align}\n\t\t& \\left|\\sum_{t = s_k + 1}^{e_k} (y_t - X_t^{\\top}\\beta^*_t)\\delta_t^{\\top} X_t\\right| = \\left|\\sum_{t = s_k + 1}^{e_k} \\epsilon_{t} \\delta_t^{\\top} X_t\\right| = \\sum_{i = 1}^p \\left\\{\\left|\\frac{\\sum_{t = s_k + 1}^{e_k} \\epsilon_{t} (\\delta_t)_i (X_t)_i}{\\sqrt{\\sum_{t = s_k + 1}^{e_k} (\\delta_t)^2_i}}\\right| \\sqrt{\\sum_{t = s_k + 1}^{e_k} (\\delta_t)^2_i}\\right\\} \\nonumber \\\\\n\t\t\\leq & \\sup_{i = 1, \\ldots, p} \\left|\\frac{\\sum_{t = s_k + 1}^{e_k} \\epsilon_{t} (\\delta_t)_i (X_t)_i}{\\sqrt{\\sum_{t = s_k + 1}^{e_k} (\\delta_t)^2_i}}\\right| \\sum_{i = 1}^p \\sqrt{\\sum_{t = s_k + 1}^{e_k} (\\delta_t)^2_i} \\leq (\\zeta\/4) \\sum_{i = 1}^p \\sqrt{\\sum_{t = s_k + 1}^{e_k} (\\delta_t)^2_i}, \\label{eq:reg local refine 4}\n\t\\end{align}\n\twhere the second inequality follows from \\Cref{lem:deviation piecewise constant}.\n\nCombining \\eqref{eq:reg local refine 1}, \\eqref{eq:reg local refine 2}, \\eqref{eq:reg local refine 3} and \\eqref{eq:reg local refine 4} yields\n\t\\begin{equation}\\label{eq:reg local refine 5}\n\t\t\\sum_{t = s_k + 1}^{e_k} \\|\\delta_t^{\\top} X_t\\|_2^2\t + \\frac{\\zeta}{2}\\sum_{i \\in S^c} \\sqrt{\\sum_{t = s_k + 1}^{e_k} \\bigl(\\delta_t\\bigr)_{i}^2} \\leq \\frac{3\\zeta}{2}\\sum_{i \\in S} \\sqrt{\\sum_{t = s_k + 1}^{e_k} \\bigl(\\delta_t\\bigr)_{i}^2}.\n\t\\end{equation}\nNow we are to explore the restricted eigenvalue inequality. Let\n\t\\[\n\t\t\\I_1 = (s_k, \\eta_k], \\quad \\I_2 = (\\eta_k, \\widehat{\\eta}_k], \\quad \\I_3 = (\\widehat{\\eta}_k, e_k].\n\t\\]\nThen for $\\I_1$, it holds that\n\t\\begin{align}\n\t\t& \\eta_k - s_k = \\eta_k - \\frac{2}{3}\\widetilde{\\eta}_k - \\frac{1}{3}\\widetilde{\\eta}_k \\nonumber \\\\\n\t\t= & \\frac{2}{3}(\\eta_k - \\eta_{k-1}) + \\frac{2}{3}(\\widetilde{\\eta}_k - \\eta_k) -  \\frac{2}{3}(\\widetilde{\\eta}_{k-1} - \\eta_{k-1}) + (\\eta_k - \\widetilde{\\eta}_k) \\nonumber \\\\\n\t\t\\geq & \\frac{2}{3}\\Delta - \\frac{1}{3}\\Delta = \\frac{1}{3}\\Delta, \\label{tmp_eq:regression I_1}\n\t\\end{align}\t\nwhere the inequality follows from \\Cref{assp:dcdp_linear_reg} and \\Cref{eq-lr-cond-1}. \n\nFor $\\I_3$, by the design of the local refinement algorithm in \\Cref{algo:local_refine_general}, we have $|\\I_3|\\geq C_s  { \\mathfrak{s} } \\log(np)$. Since $\\min\\{|\\I_1|,|\\I_3|\\}\\geq C_s { \\mathfrak{s} } \\log(np)$, by \\Cref{lem:restricted eigenvalue}, it holds with probability at least $1 - (np)^{-5}$ that,\n\t\\begin{align*}\n\t\t& \\sum_{i = 1, 3}\\sum_{t \\in \\I_i}\\|\\delta_{\\I_i}^{\\top}X_t\\|_2^2 \\\\\n\t\t\\geq & \\sum_{i = 1, 3} \\left(c'_1\\sqrt{|\\I_i|} \\|\\delta_{\\I_i}\\|_2 - c_2 \\sqrt{\\log(p)} \\|\\delta_{\\I_i}\\|_1\\right)^2 \\\\\n\t\t\\geq & \\sum_{i = 1, 3} \\left(c_1\\sqrt{|\\I_i|} \\|\\delta_{\\I_i}\\|_2 - c_2 \\sqrt{\\log(p)} \\|(\\delta_{\\I_i})_{S^c}\\|_1\\right)^2,\n\t\\end{align*}\n\twhere the last inequality follows from $\\|(\\delta_{\\I})_S\\|_1\\leq \\sqrt{ { \\mathfrak{s} } }\\|\\delta_{\\I}\\|_2$ and the fact that $\\min\\{|\\I_1|,|\\I_3|\\}>C_s  { \\mathfrak{s} }  \\log(np)$. Similarly, since $|\\I_2| > \\Delta> C_s  { \\mathfrak{s} }  \\log(np)$, we have\n\t\\[\n\t\t\\sqrt{\\sum_{t \\in \\I_2}\\|\\delta_{\\I_2}^{\\top}X_t\\|_2^2} \\geq {c_1\\sqrt{|\\I_2|}} \\|\\delta_{\\I_2}\\|_2 - c_2 \\sqrt{\\log(p)} \\|\\delta_{\\I_2}(S^c)\\|_1.\n\t\\]\nDenote $n_0 = C_s { \\mathfrak{s} } \\log(np)$. We first bound the terms with $\\|\\cdot\\|_1$. Note that\n\t\\begin{align*}\n\t&\\sum_{i = 1}^3\\sum_{j\\in S^c}|(\\delta_{\\I_i})_j|\n\t\t\\leq \\sqrt{3}\\sqrt{\\sum_{i = 1}^3 (\\sum_{j \\in S^c} |(\\delta_{\\I_i})_j|)^2 } \\\\ \n\t\t\\leq & \\sqrt{3}\\sqrt{\\sum_{i = 1}^3{\\frac{|\\I_i|}{n_0}} (\\sum_{j \\in S^c}|(\\delta_{\\I_i})_j| )^2}\t\n\t\t\\leq  \\sqrt{\\frac{3}{n_0}} \\sum_{i = 1}^3   \\sqrt{|\\I_i|}(\\sum_{j \\in S^c}|(\\delta_{\\I_i})_j| ) \\\\\n\t\t\\leq &\n\t\t\\sqrt{\\frac{3}{n_0}} \\sum_{j \\in S^c}  \\sqrt{\\sum_{t = s_k + 1}^{e_k} (\\delta_t)_j^2}\n\t\t\\leq \\frac{3\\sqrt{3}}{\\sqrt{n_0}}\\sum_{j \\in S}  \\sqrt{\\sum_{t = s_k + 1}^{e_k} (\\delta_t)_j^2}\n\t\t\\\\\n\t\t\\leq & \\frac{3\\sqrt{3}}{\\sqrt{n_0}} \\sqrt{ { \\mathfrak{s} }  \\sum_{j \\in S} \\sum_{t = s_k + 1}^{e_k} (\\delta_t)_j^2} \\leq \\frac{c}{ \\sqrt{\\log(np)}} \\sqrt{\\sum_{t = s_k + 1}^{e_k} \\|\\delta_t\\|_2^2}.\n\t\\end{align*}\nTherefore,\n\t\\begin{align*}\n\t\t& c_1\\sqrt{\\sum_{t = s_k + 1}^{e_k} \\|\\delta_t\\|_2^2} - \\frac{c_2}{ \\sqrt{\\log(np)}} \\sqrt{\\sum_{t = s_k + 1}^{e_k} \\|\\delta_t\\|_2^2} \\\\\n\t\t\\leq & \\sum_{i = 1}^3 c_1 \\|\\delta_{I_i}\\|_2 - \\frac{c_2}{ \\sqrt{\\log(np)}} \\sqrt{\\sum_{t = s_k + 1}^{e_k} \\|\\delta_t\\|_2^2} \\leq \\sqrt{3} \\sqrt{ \\sum_{t = s_k + 1}^{e_k} \\|\\delta_t^{\\top} X_t\\|_2^2 } \\\\\n\t\t\\leq & \\frac{3\\sqrt{\\zeta}}{\\sqrt{2}} { \\mathfrak{s} } ^{1\/4} \\left(\\sum_{t = s_k + 1}^{e_k} \\|\\delta_t\\|_2^2\\right)^{1\/4} \\leq \\frac{9\\zeta  { \\mathfrak{s} } ^{1\/2}}{4c_1} + \\frac{c_1}{2}\\sqrt{\\sum_{t = s_k + 1}^{e_k} \\|\\delta_t\\|_2^2}\n\t\\end{align*}\n\twhere the third inequality follows from \\eqref{eq:reg local refine 5} and the fact that $\\sum_{i\\in S}\\sqrt{\\sum_{t=s_k + 1}^{e_k}(\\delta_t)_i^2}\\leq \\sqrt{s}\\sqrt{\\sum_{t=s_k + 1}^{e_k}\\|\\delta_t\\|_2^2}$. The inequality above implies that\n\t\\[\n\t\t\\frac{c_1}{4}\\sqrt{\\sum_{t = s_k + 1}^{e_k} \\|\\delta_t\\|_2^2} \\leq \\frac{9\\zeta  { \\mathfrak{s} } ^{1\/2}}{4c_1}\n\t\\]\t\n\tTherefore,\n\t\\[\n\t\t\\sum_{t = s_k + 1}^{e_k}\\|\\widehat{\\beta}_t - \\beta^*_t\\|_2^2 \\leq 81\\zeta^2 { \\mathfrak{s} } \/c_1^4.\n\t\\]\n  Recall that $\\beta^{(1)} = \\beta^*_{\\eta_k}$ and $\\beta^{(2)} = \\beta^*_{\\eta_k + 1}$. We have that\n\t\\[\n\t\t\\sum_{t = s_k + 1}^{e_k}\\|\\widehat{\\beta}_t - \\beta^*_t\\|_2^2 = |I_1| \\|\\beta^{(1)} - \\widehat{\\beta}^{(1)}\\|_2^2 + |I_2| \\|\\beta^{(2)} - \\widehat{\\beta}^{(1)}\\|_2^2 + |I_3| \\|\\beta^{(2)} - \\widehat{\\beta}^{(2)}\\|_2^2.\n\t\\]\nSince $\\eta_k - s_k \\geq\\frac{1}{3}\\Delta$ as is shown in \\Cref{tmp_eq:regression I_1}. we have that\n\t\\[\n\t\t\\Delta\\|\\beta^{(1)} - \\widehat{\\beta}^{(1)}\\|_2^2\/3 \\leq |I_1| \\|\\beta^{(1)} - \\widehat{\\beta}^{(1)}\\|_2^2 \\leq \\frac{C_1 C_{\\zeta}^2\\Delta \\kappa^2} { { \\mathfrak{s} }  K \\sigma^2_{\\epsilon}  { \\mathcal B_n}  } \\leq c_3 \\Delta \\kappa^2,\n\t\\]\n\twhere $1\/4 > c_3 > 0$ is an arbitrarily small positive constant.  Therefore we have\n\t\\[\n\t\t\\|\\beta^{(2)} - \\widehat{\\beta}_1\\|_2^2 \\leq 3c_3 \\kappa^2.\n\t\\]\nIn addition we have\n\t\\[\n\t\t\\|\\beta^{(2)} - \\widehat{\\beta}^{(1)}\\|_2 \\geq \\|\\beta^{(2)} - \\beta^{(1)}\\|_2 - \\|\\beta^{(1)} - \\widehat{\\beta}^{(1)}\\|_2 \\geq \\kappa\/2.\n\t\\]\t\n\tTherefore, it holds that \t\n\t\\[\n\t\t\\kappa^2 |I_2|\/4 \\leq |I_2| \\|\\beta^{(2)} - \\widehat{\\beta}^{(1)}\\|_2^2 \\leq C_2 { \\mathfrak{s} } \\zeta^2,\n\t\\]\n\twhich implies that \n\t\\[\n\t\t|\\widehat{\\eta}_k - \\eta_k| \\leq \\frac{4C_2 { \\mathfrak{s} }  \\zeta^2}{\\kappa^2},\n\t\\]\nwhich gives the bound we want. \n\\end{proof}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\subsubsection{Technical lemmas}\n\nThroughout this section, let $\\widehat { \\mathcal P }  $ denote the output of  \\Cref{algorithm:DCDP_divide}.\n \n \n \n\n\\bnlem\nLet $\\mathcal{S}$ be any linear subspace in $\\mathbb{R}^n$ and $\\mathcal{N}_{1\/4}$\tbe a $1\/4$-net of $\\mathcal{S} \\cap B(0, 1)$, where $B(0, 1)$ is the unit ball in $\\mathbb{R}^n$.  For any $u \\in \\mathbb{R}^n$, it holds that\n\t\\[\n\t\t\\sup_{v \\in \\mathcal{S} \\cap B(0, 1)} \\langle v, u \\rangle \\leq 2 \\sup_{v \\in \\mathcal{N}_{1\/4}} \\langle v, u \\rangle,\n\t\\]\n\twhere $\\langle \\cdot, \\cdot \\rangle$ denotes the inner product in $\\mathbb{R}^n$.\n\\enlem\n\n\\begin{proof}\nDue to the definition of $\\mathcal{N}_{1\/4}$, it holds that for any $v \\in \\mathcal{S} \\cap B(0, 1)$, there exists a $v_k \\in \\mathcal{N}_{1\/4}$, such that $\\|v - v_k\\|_2 < 1\/4$.  Therefore,\n\t\\begin{align*}\n\t\t\\langle v, u \\rangle = \\langle v - v_k + v_k, u \\rangle = \\langle x_k, u \\rangle + \\langle v_k, u \\rangle \\leq \\frac{1}{4} \\langle v, u \\rangle + \\frac{1}{4} \\langle v^{\\perp}, u \\rangle + \\langle v_k, u \\rangle,\n\t\\end{align*}\n\twhere the inequality follows from  $x_k = v - v_k = \\langle x_k, v \\rangle v + \\langle x_k, v^{\\perp} \\rangle v^{\\perp}$.  Then we have\n\t\\[\n\t\t\\frac{3}{4}\\langle v, u \\rangle \\leq \\frac{1}{4} \\langle v^{\\perp}, u \\rangle + \\langle v_k, u \\rangle.\n\t\\]\n\tIt follows from the same argument that \n\t\\[\n\t\t\\frac{3}{4}\\langle v^{\\perp}, u \\rangle \\leq \\frac{1}{4} \\langle v, u \\rangle + \\langle v_l, u \\rangle,\n\t\\]\n\twhere $v_l \\in \\mathcal{N}_{1\/4}$ satisfies $\\|v^{\\perp} - v_l\\|_2 < 1\/4$.  Combining the previous two equation displays yields\n\t\\[\n\t\t\\langle v, u \\rangle \\leq 2 \\sup_{v \\in \\mathcal{N}_{1\/4}} \\langle v, u \\rangle,\n\t\\]\n\tand the final claims holds.\n\\end{proof}\n\n \n\n\\Cref{lem:deviation piecewise constant} is an adaptation of Lemma 3 in \\cite{wang2021_jmlr}.\n\n\\bnlem\\label{lem:deviation piecewise constant}\n\tGiven any interval $I = (s, e] \\subset \\{1, \\ldots, n\\}$. Let $\\mclR_m := \\{v \\in \\mathbb{R}^{(e-s)}| \\|v\\|_2 = 1, \\sum_{t = 1}^{e-s-1} \\mathbf{1}\\{v_i \\neq v_{i+1}\\} = m\\}$. Then for data generated from \\Cref{assp:dcdp_linear_reg}, it holds that for any $\\delta > 0$, $i \\in \\{1, \\ldots, p\\}$, \n\t\\[\n\t\t\\mathbb{P}\\left\\{\\sup_{v\\in \\mclR_m} \\left|\\sum_{t = s+1}^e v_t \\epsilon_t (X_t)_i\\right| > \\delta \\right\\} \\leq C(e-s-1)^m 9^{m+1} \\exp \\left\\{-c \\min\\left\\{\\frac{\\delta^2}{4C_x^2}, \\, \\frac{\\delta}{2C_x \\|v\\|_{\\infty}}\\right\\}\\right\\}.\n\t\\]\n\\enlem\n\n\\begin{proof}\nFor any $v \\in \\mathbb{R}^{(e-s)}$ satisfying $\\sum_{t = 1}^{e-s-1}\\mathbbm{1}\\{v_i \\neq v_{i+1}\\} = m$, it is determined by a vector in $\\mathbb{R}^{m+1}$ and a choice of $m$ out of $(e-s-1)$ points.  Therefore we have,\n\t\\begin{align*}\n\t\t& \\mathbb{P}\\left\\{\\sup_{\\substack{v \\in \\mathbb{R}^{(e-s)}, \\, \\|v\\|_2 = 1\\\\ \\sum_{t = 1}^{e-s-1} \\mathbf{1}\\{v_i \\neq v_{i+1}\\} = m}} \\left|\\sum_{t = s+1}^e v_t \\epsilon_t (X_t)_i\\right| > \\delta \\right\\} \\\\\n\t\t\\leq & {(e-s-1) \\choose m} 9^{m+1} \\sup_{v \\in \\mathcal{N}_{1\/4}}\t\\mathbb{P}\\left\\{\\left|\\sum_{t = s+1}^e v_t \\epsilon_t (X_t)_i\\right| > \\delta\/2 \\right\\} \\\\\n\t\t\\leq & {(e-s-1) \\choose m} 9^{m+1} C\\exp \\left\\{-c \\min\\left\\{\\frac{\\delta^2}{4C_x^2}, \\, \\frac{\\delta}{2C_x \\|v\\|_{\\infty}}\\right\\}\\right\\} \\\\\n\t\t\\leq & C(e-s-1)^m 9^{m+1} \\exp \\left\\{-c \\min\\left\\{\\frac{\\delta^2}{4C_x^2}, \\, \\frac{\\delta}{2C_x \\|v\\|_{\\infty}}\\right\\}\\right\\}.\n\t\\end{align*}\n\n\\end{proof}\n\n\n\n\n\n\n \n\n\n\n\n\n\n\n\n\n\n\n\n \\bnlem\\label{lam:regression goodness of fit J superset of I}\nLet $\\I\\subset [1,T]$ be any interval that contains no change point. Then under \\Cref{assp:dcdp_linear_reg}, for any interval ${  \\mathcal J }  \\supset \\I$, it holds with probability at least $1 - (np)^{-5}$ that\n\\begin{equation*}\n    \\mclF(\\mu^*_{\\I},\\I)\\leq \\mclF(\\widehat{\\mu}_{  \\mathcal J } ,\\I) + C(\\sigma_{\\epsilon}^2\\vee 1) { \\mathfrak{s} } \\log(np).\n\\end{equation*}\n \\enlem\n\\begin{proof} \n\\noindent \\textbf{Case 1.} If $ |\\I| < C_{\\mclF}  { \\mathfrak{s} }  \\log(n p)$, then by the definition of $\\mclF(\\beta, \\mclI)$, we have $\\mclF(\\beta^*_{\\I},\\I)= \\mclF(\\widehat{\\beta}_{  \\mathcal J } ,\\I) =0$ and the inequality holds automatically.\n \n\\noindent \\textbf{Case 2.} If \n\t\\begin{equation}\\label{eq-lem16-i-cond} \n\t\t|\\I| \\geq  C_{\\mclF}  { \\mathfrak{s} }  \\log(n p),\n\t\\end{equation}\n\tthen letting $\\delta_\\I = \\beta^*_\\I  -\\widehat \\beta_{  \\mathcal J } $ and consider the high-probability event given in \\Cref{lem:restricted eigenvalue}, we have\n\t\\begin{align}\n\t\t& \\sqrt{\\sum_{t \\in \\I} (X_t^{\\top} \\delta_\\I)^2} \\geq {c_1'\\sqrt{|\\I|}} \\|\\delta_\\I\\|_2 - c_2'\\sqrt{\\log(p)} \\|\\delta_\\I\\|_1 \\nonumber \\\\\n\t\t= & {c_1'\\sqrt{|\\I|}} \\|\\delta_\\I\\|_2 - c_2'\\sqrt{\\log(p)} \\|(\\delta_I)_{S}\\|_1 - c_2'\\sqrt{\\log(p)} \\|(\\delta_\\I)_{S^c}\\|_1 \\nonumber \\\\\n\t\t\\geq & c_1'\\sqrt{|\\I|} \\|\\delta_\\I\\|_2 - c_2'\\sqrt{ { \\mathfrak{s} } \\log(p)} \\|\\delta_\\I\\|_2 - c_2'\\sqrt{\\log(p)} \\|(\\delta_\\I)_{S^c}\\|_1 \\nonumber \\\\\n\t\t\\geq & \\frac{c_1'}{2}\\sqrt{|\\I|} \\|\\delta_\\I\\|_2 - c_2'\\sqrt{\\log(p)} \\|(\\widehat{\\beta}_{  \\mathcal J } )_{S^c}\\|_1 \\geq c_1\\sqrt{|\\I|} \\|\\delta_\\I\\|_2 - c_2\\sqrt{\\log(p)} \\frac{ { \\mathfrak{s} }  \\lambda}{\\sqrt{|\\I|}}, \\label{eq-lem16-pf-1}\n\t\\end{align} \n\twhere the last inequality follows from \\Cref{lemma:interval lasso}t and the assumption that $(\\beta^*_t)_i = 0$, for all $t\\in [T]$ and $i\\in S^c$. Then by the fact that $(a - b)^2\\geq \\frac{1}{2}a^2 - b^2$ for all $a,b\\in \\mathbb{R}$, it holds that\n\t\\begin{equation}\n\t    \\sum_{t \\in \\I} (X_t^{\\top} \\delta_\\I)^2\\geq \\frac{c_1^2}{2}{|\\I|} \\|\\delta_\\I\\|_2^2 - \\frac{c_2^2  \\lambda^2 { \\mathfrak{s} } ^2\\log(p)}{|\\I|}.\n\t\\end{equation}\n\tNotice that\n\t\\begin{align*}\n\t    & \\sum_{t \\in \\I} (y_t - X_t^{\\top} \\beta^*_\\I)^2 - \\sum_{t \\in \\I} (y_t - X_t^{\\top} \\widehat{\\beta}_{  \\mathcal J } )^2  = 2 \\sum_{t \\in \\I} \\epsilon_t X_t^{\\top}\\delta_\\I  - \\sum_{t \\in \\I} (X_t^{\\top}\\delta_\\I)^2 \\\\\n\t\\leq & 2\\|\\sum_{t \\in \\I }  X_t \\epsilon_t \\|_{\\infty }  \\left( \\sqrt { { \\mathfrak{s} } } \\| (\\delta_\\I)_{S} \\|_2  + \\| (\\widehat \\beta_{  \\mathcal J } )_{S^c} \\|_1  \\right)- \\sum_{t \\in \\I} (X_t^{\\top}\\delta_\\I)^2.\n\t\\end{align*}\n\tSince for each $t$, $\\epsilon_t$ is subgaussian with $\\|\\epsilon_t\\|_{\\psi_2}\\leq \\sigma_{\\epsilon}$ and for each $i\\in [p]$, $(X_t)_i$ is subgaussian with $\\|(X_t)_i\\|_{\\psi_2}\\leq C_x$, we know that $(X_t)_i\\epsilon_t$ is subexponential with $\\|(X_t)_i\\epsilon_t\\|_{\\psi_1}\\leq C_x\\sigma_{\\epsilon}$. Therefore, by Bernstein's inequality (see, e.g., Theorem 2.8.1 in \\cite{vershynin2018high}) and a union bound, for $\\forall u\\geq 0$ it holds that\n\t\\begin{equation*}\n\t    \\mathbb{P}(\\|\\sum_{t \\in \\I }  X_t \\epsilon_t \\|_{\\infty }> u)\\leq 2p\\exp(-c\\min\\{\\frac{u^2}{|\\I|C_x^2\\sigma_{\\epsilon}^2}, \\frac{u}{C_x\\sigma_{\\epsilon}}\\}).\n\t\\end{equation*}\n\tTake $u = cC_x\\sigma_{\\epsilon}\\sqrt{|\\I|\\log(np)}$, then by the fact that $|\\I|\\geq C_{\\mclF} { \\mathfrak{s} } \\log(np)$, it follows that with probability at least $1 - (np)^{-7}$,\n\t\\begin{equation*}\n\t    \\|\\sum_{t \\in \\I }  X_t \\epsilon_t \\|_{\\infty } \\leq CC_x\\sigma_{\\epsilon}\\sqrt{|\\I|\\log(np)}\\leq \\lambda \\sqrt{|\\I|},\n\t\\end{equation*}\n\twhere we use the fact that $\\lambda=C_{\\lambda}(\\sigma_{\\epsilon}\\vee 1)\\sqrt{\\log(np)}$. Therefore, we have\n\t\\begin{align*}\n\t& \\sum_{t \\in \\I} (y_t - X_t^{\\top} \\beta^*_\\I)^2 - \\sum_{t \\in \\I} (y_t - X_t^{\\top} \\widehat{\\beta}_{  \\mathcal J } )^2 \\\\\n\t\\leq & 2\\lambda\\sqrt{|\\I| { \\mathfrak{s} } }\\|\\delta_\\I\\|_2 + 2\\lambda\\sqrt{|\\I|}\\cdot \\frac{\\lambda  { \\mathfrak{s} } }{ \\sqrt{|\\I|}} - \\frac{c_1^2 |\\I|}{2}\\|\\delta_\\I\\|_2^2 + \\frac{c_2^2 \\lambda^2  { \\mathfrak{s} } ^2\\log(p)}{|\\I|}\\\\  \n\t\\leq & 2\\lambda\\sqrt{|\\I| { \\mathfrak{s} } }\\|\\delta_\\I\\|_2 + {2\\lambda^2  { \\mathfrak{s} } } - \\frac{c_1^2 |\\I|}{2}\\|\\delta_\\I\\|_2^2 + \\frac{c_2^2 \\lambda^2  { \\mathfrak{s} } ^2\\log(p)}{|\\I|} \\\\\n\t\\leq & \\frac{4}{c_1^2}\\lambda^2 { \\mathfrak{s} }  + \\frac{c_1^2}{4}|\\I| \\|\\delta_\\I\\|^2_2 + {2\\lambda^2  { \\mathfrak{s} } } - \\frac{c_1^2}{2}|\\I|\\|\\delta_\\I\\|^2_2 + \\frac{c_2^2 \\lambda^2  { \\mathfrak{s} } ^2\\log(p)}{C_{\\mclF} { \\mathfrak{s} } \\log(np)} \\\\\n\t\\leq & {c_3\\lambda^2 { \\mathfrak{s} } } + {2\\lambda^2  { \\mathfrak{s} } } + \\frac{c_2^2 \\lambda^2  { \\mathfrak{s} } ^2\\log(p)}{C_{\\mclF} { \\mathfrak{s} } \\log(np)}\\\\\n\t\\leq & c_4\\lambda^2  { \\mathfrak{s} } .\n\t\\end{align*}\n\twhere the third inequality follows from $2ab \\leq a^2 + b^2$. \n\n\\end{proof} \n\n\n\n\n\n\n\n\n\n\n \n \n \n \\bnlem \\label{lemma: regression dcdp one change point}\nSuppose the good events \n$\\mathcal L ( { \\mathcal B_n^{-1}   \\Delta }  ) $ and  $\\mathcal R ( { \\mathcal B_n^{-1}   \\Delta } ) $ defined  in \\Cref{eq:left and right approximation of change points} hold. \nLet   $ \\I=(s,e] \\in \\mathcal {\\widehat P}  $  be such that $\\I$ contains exactly  one  true  change point $ \\eta_k $. \nSuppose $\\gamma \\geq C_\\gamma K { \\mathcal B_n^{-1}   \\Delta }  \\kappa^2   $.  Then  with probability  at least $1- n^{-3}$,   it holds that \n\\begin{equation*}\n    \\min\\{ \\eta_k -s ,e-\\eta_k \\}   \\lesssim \\frac{\\sigma_{\\epsilon}^2\\vee 1  }{\\kappa^2 }\\bigg(   { \\mathfrak{s} }  \\log(np) +\\gamma \\bigg)  +   { \\mathcal B_n^{-1}   \\Delta } .\n\\end{equation*}\n\\enlem\n\\begin{proof}    \nIf either $ \\eta_k -s \\le  C_\\mclF  { \\mathfrak{s} }  \\log(np)   $ or $e-\\eta_k\\le C_\\mclF  { \\mathfrak{s} }  \\log(np)  $, then \n$$ \\min\\{ \\eta_k -s ,e-\\eta_k \\}   \\le C_\\mclF  { \\mathfrak{s} }  \\log(np)  $$\nand there is nothing to show. So assume that \n$$  \\eta_k -s  > C_\\mclF  { \\mathfrak{s} }  \\log(np)  \\quad \\text{and} \\quad   e-\\eta_k  >C_\\mclF  { \\mathfrak{s} }  \\log(np)  . $$\nBy event $\\mathcal R ( { \\mathcal B_n^{-1}   \\Delta }  )$, there exists  $ s_ u  \\in \\{ s_q\\}_{q=1}^ { \\mathcal Q} $ such that \n$$0\\le s_u - \\eta_k \\le  { \\mathcal B_n^{-1}   \\Delta } . $$\n\n\\begin{center}\n \\begin{tikzpicture} \n\\draw[ - ] (-10,0)--(-1,0);\n \\node[color=black] at (-8,-0.3) {\\small s};\n \\draw[  (-, ultra thick, black] (-8,0) -- (-7.99,0);\n  \\draw[ { -]}, ultra thick, black] (-3.0002,0) -- (-3,0);\n\n   \\node[color=black] at (-7,-0.3) {\\small $\\eta_k$};\n\\draw(-7 ,0)circle [radius=2pt] ;\n\n  \\node[color=black] at (-6.5,-0.3) {\\small $s_u$};\n\\draw plot[mark=x, mark options={color=black, scale=1.5}] coordinates {(-6.5,0)  }; \n  \n  \\node[color=black] at (-3 ,-0.3) {\\small $e$};\n\\end{tikzpicture}\n\\end{center} \n\n{\\bf Step 1.} Denote \n$$ \\I_ 1 = (s,s_u] \\quad \\text{and} \\quad \\I_2 = (s_u, e] .$$\nSince \n$ \\eta_k-s \\ge C_\\mclF  { \\mathfrak{s} }  \\log(np)    , $\nit follows that \n   $ |\\I| \\ge C_\\mclF  { \\mathfrak{s} }  \\log(np)   $ and $ |\\I_1| \\ge C_\\mclF  { \\mathfrak{s} }  \\log(np)   $. Thus \n$$ \\mathcal F  (\\I)  = \\sum_{ i \\in \\I } (y_i - X_i^\\top \\widehat \\beta_\\I )^2 \\quad \\text{and} \\quad \\mathcal F  (\\I_1)  = \\sum_{ i \\in {\\I_1}  } (y_i - X_i^\\top \\widehat \\beta_{\\I_1}  )^2  .$$\nSince $\\I\\in \\widehat { \\mathcal P}$, it holds that \n\\begin{align}\\label{eq:regression one change basic inequality step 1} \\mathcal F  (\\I)  \\le  \\mathcal F  (\\I_1)  +  \\mathcal F  (\\I_2)  + \\gamma. \n\\end{align}\n\n{\\bf Case a.} Suppose $|\\I_2| <C_\\mclF  { \\mathfrak{s} }  \\log(np)      $.\nIt follows from \\Cref{eq:regression one change basic inequality step 1} that \n\\begin{align}\\nonumber\n\\sum_{i \\in \\I }(y_i - X_i^\\top \\widehat \\beta _\\I ) ^2 \\le &  \\sum_{i \\in \\I_1 }(y_i -X_i^\\top  \\widehat \\beta _{\\I_1}  ) ^2 +  0  + \\gamma \n\\\\ \\nonumber \n\\le &  \\sum_{i \\in \\I_1 }(y_i -  X_i^\\top  \\beta^*  _i  ) ^2 + C_1 \\big ( (s_u -\\eta_k) \\kappa_k^2   +  { \\mathfrak{s} }   \\log(np)  \\big )   +    \\gamma\n\\\\ \\nonumber \n\\le &  \\sum_{i \\in \\I _1}(y_i -  X_i^\\top  \\beta^*  _i ) ^2  +C_1 \\big (   { \\mathcal B_n^{-1}   \\Delta }   \\kappa_k^2   +   { \\mathfrak{s} }  \\log(np )    \\big ) +  \\gamma\n\\\\\n\\le &   \\sum_{i \\in \\I  }(y_i -  X_i^\\top  \\beta^*  _i  ) ^2  +C_1 \\big (   { \\mathcal B_n^{-1}   \\Delta }   \\kappa_k^2   +   { \\mathfrak{s} }  \\log(np )    \\big ) +  \\gamma  , \\label{eq:regression one change point step 1  case a}\n\\end{align}\nwhere the first inequality follows from the fact that $  \\mathcal F  (\\I_2)  = 0  $ when $|\\I_2| < C_\\mclF  { \\mathfrak{s} }  \\log(np)   $,    the second inequality follows from \\Cref{lam:regression one change deviation bound}   {\\bf b}, the third inequality follows from  the assumption  that \n$   (s_u-\\eta_k) \\le   { \\mathcal B_n^{-1}   \\Delta }    $,   and the inequality holds because $ \\sum_{i \\in \\I _2 }(y_i -  X_i^\\top  \\beta^*  _i  ) ^2\\ge 0 $.\n\n{\\bf Case b.} Suppose $|\\I_2|  \\ge C_\\mclF  { \\mathfrak{s} }  \\log(np)   $.\nIt follows from \\Cref{eq:regression one change basic inequality step 1} that  \n\\begin{align}\\nonumber\n\\sum_{i \\in \\I }(y_i - X_i^\\top \\widehat \\beta _\\I ) ^2 \\le &  \\sum_{i \\in \\I_1 }(y_i -X_i^\\top  \\widehat \\beta _{\\I_1}  ) ^2 +  \\sum_{i \\in \\I_2  }(y_i -X_i^\\top  \\widehat \\beta _{\\I_2 }  ) ^2   + \\gamma \n\\\\ \\nonumber \n\\le &  \\sum_{i \\in \\I_1 }(y_i -  X_i^\\top  \\beta^*  _i  ) ^2 + C_1 \\big ( (s_u -\\eta_k) \\kappa_k^2   +  { \\mathfrak{s} }   \\log(np)  \\big )  \n\\\\  \\nonumber \n+ & \\sum_{i \\in \\I_2 }(y_i -  X_i^\\top  \\beta^*  _i  ) ^2 + C_1      { \\mathfrak{s} }   \\log(np)     + 2  \\gamma\n\\\\ \n\\le &  \\sum_{i \\in \\I }(y_i -  X_i^\\top  \\beta^*  _i ) ^2  +C_2 \\big (   { \\mathcal B_n^{-1}   \\Delta }   \\kappa_k^2   +   { \\mathfrak{s} }  \\log(np )    \\big ) +  \\gamma, \\label{eq:regression one change point step 1 case b}\n\\end{align}\nwhere    the second inequality follows from \\Cref{lam:regression one change deviation bound}  {\\bf a} and {\\bf b}, and the third inequality follows from  the assumption  that \n$   (s_u-\\eta_k) \\le   { \\mathcal B_n^{-1}   \\Delta }    $. Combing two cases leads to \n\\begin{align} \n\\sum_{i \\in \\I }(y_i - X_i^\\top \\widehat \\beta _\\I ) ^2  \n\\le     \\sum_{i \\in \\I  }(y_i -  X_i^\\top  \\beta^*  _i  ) ^2  +C_2 \\big (   { \\mathcal B_n^{-1}   \\Delta }   \\kappa_k^2   +   { \\mathfrak{s} }  \\log(np )    \\big ) +  \\gamma . \\label{eq:two change point step 1}\n\\end{align}\n\\\\\n\\\\\n{\\bf Step 2.}\nDenote \n$$ {  \\mathcal J } _1  = (s,\\eta_k ] \\quad \\text{and} \\quad {  \\mathcal J } _2 = (\\eta_k , e] .$$\n\\Cref{eq:one change point step 1}  gives \n\\begin{align}  \\nonumber \n& \\sum_{i \\in {  \\mathcal J } _1 }(y_i  -X_i^\\top  \\widehat \\beta _\\I ) ^2 +\\sum_{i \\in {  \\mathcal J } _2 }(y_i - X_i^\\top  \\widehat \\beta _\\I  ) ^2  \n\\\\ \n \\le \\label{eq:regression step 2 first term} & \\sum_{i \\in {  \\mathcal J } _1  }(y_i -  X_i^\\top    \\beta _{{  \\mathcal J }  _1}  ^*    ) ^2   +\\sum_{i \\in {  \\mathcal J } _2  }(y_i - X_i^\\top    \\beta _{{  \\mathcal J }  _1}  ^*  ) ^2  +C_2  \\big (      { \\mathcal B_n^{-1}   \\Delta }  \\kappa_k^2  + { \\mathfrak{s} }  \\log(np)  \\big )  +   \\gamma \n\\end{align}\nThe above display  leads to \n\\begin{align} \\nonumber\n&\\sum_{i \\in {  \\mathcal J } _1 } \\big\\{ X_i^\\top  (  \\widehat \\beta _\\I  -  \\beta^* _{{  \\mathcal J } _1 }) \\big\\}  ^2 +\\sum_{i \\in {  \\mathcal J } _2  } \\big\\{ X_i^\\top  (  \\widehat \\beta _\\I  -  \\beta^* _{{  \\mathcal J } _2 }) \\big\\}  ^2  \n\\\\ \\nonumber \n \\le  & 2 \\sum_{i \\in {  \\mathcal J } _1  } \\epsilon_i  X_i^\\top (  \\widehat \\beta _\\I  -  \\beta^* _{{  \\mathcal J } _1 })     +2 \\sum_{i \\in {  \\mathcal J } _2  } \\epsilon_i  X_i^\\top (  \\widehat \\beta _\\I  -  \\beta^* _{{  \\mathcal J } _2 })      +C_2  \\big (      { \\mathcal B_n^{-1}   \\Delta }    \\kappa_k^2   + { \\mathfrak{s} } \\log(np)  \\big )  +  \\gamma \n \\\\ \\label{eq:regression one change step 2 term 2}\n \\le  & C_3  \\sqrt { \\log(np)    |{  \\mathcal J } _1| }  \\|  \\widehat \\beta _\\I  -  \\beta^* _{{  \\mathcal J } _1 }\\| _1  + C_3  \\sqrt { \\log(np)  |{  \\mathcal J } _2| }  \\|  \\widehat \\beta _\\I  -  \\beta^* _{{  \\mathcal J } _2  }\\| _1+ C_2  \\big (      { \\mathcal B_n^{-1}   \\Delta }  \\kappa_k^2  + { \\mathfrak{s} }  \\log(np)  \\big ) +    \\gamma \n\\end{align} \nwhere the last inequality follows from \\Cref{lemma:consistency} and  that  $|{  \\mathcal J } _1|   \\ge  C_\\mclF  { \\mathfrak{s} }  \\log(np)   $ and $|{  \\mathcal J } _2|  \\ge C_\\mclF  { \\mathfrak{s} }  \\log(np)   $.\nNote that \n\\begin{align*} \n\\|  ( \\widehat \\beta _\\I  -  \\beta _{{  \\mathcal J } _1}^* )_{S^c } \\|_1 = \\|  ( \\widehat \\beta _\\I )_{S^c} \\|_{1} \n=   \\|  ( \\widehat \\beta _\\I -\\beta _\\I ^* )_{S^c} \\|_{1}  \\le 3 \\|  ( \\widehat \\beta _\\I -\\beta _\\I ^* )_{S } \\|_{1}   \\le C_5  { \\mathfrak{s} }  \\sqrt { \\frac{ \\log(np)}{| \\I|} }    , \n\\end{align*} \nwhere the last two inequalities follows from \\Cref{lemma:interval lasso}.  So\n\\begin{align}  \\label{eq:regression one change step 2 term 1}\n\\|   \\widehat \\beta _\\I  -  \\beta _{{  \\mathcal J } _1}^*   \\|_1 = \\|  ( \\widehat \\beta _\\I  -  \\beta _{{  \\mathcal J } _1}^* )_{S  } \\|_1 +\\|  ( \\widehat \\beta _\\I  -  \\beta _{{  \\mathcal J } _1}^* )_{S^c } \\|_1  \\le \\sqrt { { \\mathfrak{s} }  }  \\|  \\widehat \\beta _\\I  -  \\beta _{{  \\mathcal J } _1}^*   \\|_2 + C_5  { \\mathfrak{s} }  \\sqrt { \\frac{ \\log(np)}{| \\I|} } .\n\\end{align} \nTherefore  \\Cref{eq:regression one change step 2 term 2} gives \n\\begin{align} \\nonumber \n& \\sum_{i \\in {  \\mathcal J } _1 } \\big\\{ X_i^\\top  (  \\widehat \\beta _\\I  -  \\beta^* _{{  \\mathcal J } _1 }) \\big\\}  ^2 +\\sum_{i \\in {  \\mathcal J } _2  } \\big\\{ X_i^\\top  (  \\widehat \\beta _\\I  -  \\beta^* _{{  \\mathcal J } _2 }) \\big\\}  ^2   \n\\\\  \\nonumber \n\\le & C_3  \\sqrt { \\log(np)    |{  \\mathcal J } _1| }   \\bigg(  \\sqrt { { \\mathfrak{s} }  }  \\|  \\widehat \\beta _\\I  -  \\beta _{{  \\mathcal J } _1}^*   \\|_2 + C_5  { \\mathfrak{s} }  \\sqrt { \\frac{ \\log(np)}{| \\I|} } \\bigg) \n\\\\ \\nonumber  \n+ & C_3  \\sqrt { \\log(np)    |{  \\mathcal J } _2| }   \\bigg(  \\sqrt { { \\mathfrak{s} }  }  \\|  \\widehat \\beta _\\I  -  \\beta _{{  \\mathcal J } _2}^*   \\|_2 + C_5  { \\mathfrak{s} }  \\sqrt { \\frac{ \\log(np)}{| \\I|} } \\bigg)  +2\\gamma \n\\\\ \\nonumber  \n\\le & \\frac{ c_x | {  \\mathcal J } _1|}{64} \\|  \\widehat \\beta _\\I  -  \\beta _{{  \\mathcal J } _1}^*   \\|_2  ^2 + \\frac{ c_x | {  \\mathcal J } _2|}{64} \\|  \\widehat \\beta _\\I  -  \\beta _{{  \\mathcal J } _2}^*   \\|_2  ^2   + C_5'    { \\mathfrak{s} }  \\log(np)\n\\\\ +&C_2  \\big (      { \\mathcal B_n^{-1}   \\Delta }  \\kappa_k^2  + { \\mathfrak{s} }  \\log(np)  \\big )   +2\\gamma,\n \\label{eq:regression one change step 2 term 3}\n\\end{align}\n   $ |{  \\mathcal J } _1| \\le |\\I| , |{  \\mathcal J } _2| \\le |\\I|  $ are used in the last inequality. \n\\\\\n\\\\\n{\\bf Step 3.}  Since     $|{  \\mathcal J } _1|   \\ge  C_\\mclF   { \\mathfrak{s} }  \\log(np)   $ and $|{  \\mathcal J } _2|  \\ge C_\\mclF  { \\mathfrak{s} }  \\log(np)   $, for $\\ell=1, 2$,\n  it holds that\n \\begin{align}\\nonumber \n\t\t&  \\sum_{i \\in {  \\mathcal J } _\\ell  } \\big\\{ X_i^\\top  (  \\widehat \\beta _\\I  -  \\beta^* _{{  \\mathcal J } _ \\ell  }) \\big\\}  ^2 \n\t\t \\\\  \\nonumber  \n\t\t  \\ge&  \\frac{c_x|{  \\mathcal J } _\\ell | }{16} \\|\\widehat \\beta _\\I  -  \\beta^* _{{  \\mathcal J } _ \\ell  } \\|_2^2 - C_6  \\log(p) \\|\\widehat \\beta _\\I  -  \\beta^* _{{  \\mathcal J } _ \\ell  } \\|_1  ^2 \n\t\t \\\\\\nonumber  \n\t\t \\ge & \\frac{c_x|{  \\mathcal J } _\\ell | }{16} \\|\\widehat \\beta _\\I  -  \\beta^* _{{  \\mathcal J } _ \\ell  } \\|_2^2 - C_6'  { \\mathfrak{s} }    \\log(p) \\|\\widehat \\beta _\\I  -  \\beta^* _{{  \\mathcal J } _ \\ell  } \\|_ 2   ^2  - C_6'  \\frac{   { \\mathfrak{s} } ^2 \\log(p) \\log(np)  }{|\\I| }\n\t\t \\\\ \\label{eq:regression one change step 3 term 1}\n\t\t \\ge &  \\frac{c_x|{  \\mathcal J } _\\ell | }{32} \\|\\widehat \\beta _\\I  -  \\beta^* _{{  \\mathcal J } _ \\ell  } \\|_2^2 - C_7   { \\mathfrak{s} }  \\log(np),\n\t\\end{align} \n\twhere the first inequality follows from   \n \\Cref{lem:restricted eigenvalue},  the second inequality follows from \\Cref{eq:regression one change step 2 term 1}   and the last inequality follows from the observation that \n $$ |\\I| \\ge |{  \\mathcal J } _\\ell|  \\ge   C_\\mclF  { \\mathfrak{s} }  \\log(np). $$\n \\Cref{eq:regression one change step 2 term 3} and \\Cref{eq:regression one change step 3 term 1} together lead to \n $$   |{  \\mathcal J } _1  |   \\|\\widehat \\beta _\\I  -  \\beta^* _{{  \\mathcal J } _ 1  } \\|_2^2  + |{  \\mathcal J } _2  |   \\|\\widehat \\beta _\\I  -  \\beta^* _{{  \\mathcal J } _ 2  } \\|_2^2  \\le C_8( { \\mathfrak{s} }  \\log(np) +  { \\mathcal B_n^{-1}   \\Delta }  \\kappa_k^2 + \\gamma) .$$\nObserve  that \n$$ \\inf_{  \\beta  \\in \\mathbb R ^p   }  |{  \\mathcal J } _1 |  \\| \\beta    -  \\beta^* _{{  \\mathcal J } _ 1  } \\|_2^2  + |{  \\mathcal J } _2 | \\|  \\beta    -  \\beta^* _{{  \\mathcal J } _ 2  } \\|_2^2    = \\kappa_k ^2  \\frac{|{  \\mathcal J } _1| |{  \\mathcal J } _2|}{| \\I| }   \\ge \\frac{ \\kappa_k^2 }{2} \\min\\{ |{  \\mathcal J } _1| ,|{  \\mathcal J } _2| \\}    . $$ \nThis leads to \n$$  \\frac{ \\kappa_k^2 }{2}\\min\\{ |{  \\mathcal J } _1| ,|{  \\mathcal J } _2| \\}    \\le  C_8( { \\mathfrak{s} }  \\log(np)+ { \\mathcal B_n^{-1}   \\Delta }   + \\gamma)  .$$\nSince $\\kappa_k \\asymp \\kappa $, it follows that \n$$ \\min\\{ |{  \\mathcal J } _1| ,|{  \\mathcal J } _2| \\}   \\le C_9 \\bigg( \\frac{ { \\mathfrak{s} }  \\log(np) + \\gamma  }{\\kappa \t^2 } +   { \\mathcal B_n^{-1}   \\Delta }   \\bigg) .$$\n \\end{proof} \n  \n\n\n \\bnlem  \\label{lemma:regression two change points}\nSuppose the good events \n$\\mathcal L ( { \\mathcal B_n^{-1}   \\Delta }  ) $ and  $\\mathcal R (  { \\mathcal B_n^{-1}   \\Delta }  ) $  in  \\Cref{eq:left and right approximation of change points} hold. \nLet   $ \\I=(s,e] \\in \\mathcal {\\widehat P}  $  be such that $\\I$ contains  exactly two  change points $ \\eta_k,\\eta_{k+1}  $.  Suppose in addition that \n\\begin{align} \\label{eq:regression two change points snr}\n\\Delta \\kappa^2 \\ge  C     \\big( \\sigma_{\\epsilon}^2 { \\mathfrak{s} }  \\log(np) + \\gamma)    \n\\end{align} \nfor sufficiently large constant $C   $ and that \n$\\gamma \\ge  C_\\gamma  { \\mathcal B_n} ^{-1}\\Delta\\kappa^2   $.\n Then  with probability  at least $1- n^{-3}$,   it holds that \n\\begin{align*} \n \\eta_k -s     \\lesssim  \\frac{\\sigma_{\\epsilon}^2\\vee 1  }{\\kappa^2 }\\bigg(   { \\mathfrak{s} }  \\log(np) +\\gamma \\bigg) +   { \\mathcal B_n^{-1}   \\Delta }    \\quad \\text{and} \\quad \n e-\\eta_{k+1}   \\lesssim \\frac{\\sigma_{\\epsilon}^2\\vee 1  }{\\kappa^2 }\\bigg(   { \\mathfrak{s} }  \\log(np) +\\gamma \\bigg) +   { \\mathcal B_n^{-1}   \\Delta }   ,\n \\end{align*} \n where $C_0\\ge 1 $ is some sufficiently large constant.\n \\enlem \n\\begin{proof} \nBy symmetry, it suffices to show that $\\eta_k-s\\lesssim \\frac{\\sigma_{\\epsilon}^2\\vee 1  }{\\kappa^2 }\\bigg(   { \\mathfrak{s} }  \\log(np) +\\gamma \\bigg) +   { \\mathcal B_n^{-1}   \\Delta } $.  If \n$$\\eta_k-s  \\le  C_\\mclF  { \\mathfrak{s} }  \\log(np) , $$\nthen the desired result follows immediately.  \nSo it suffices to assume that \n$$\\eta_k-s > C_\\mclF  { \\mathfrak{s} }  \\log(np) .  $$\nSince the events $\\mathcal L (  { \\mathcal B_n^{-1}   \\Delta }   ) $ and  $\\mathcal R (  { \\mathcal B_n^{-1}   \\Delta }  ) $ hold, let $ s_u, s_v$ be such that \n $\\eta_k \\le s_u \\le s_v \\le \\eta_{k+1} $ and that \n $$ 0 \\le s_u-\\eta_k \\le    { \\mathcal B_n^{-1}   \\Delta }  ,  \\quad  0\\le  \\eta_{k+1} - s_v \\le  { \\mathcal B_n^{-1}   \\Delta }    .    $$\n \\\n \\\\\n \\begin{center}\n \\begin{tikzpicture} \n\\draw[ - ] (-10,0)--(1,0);\n \\node[color=black] at (-8,-0.3) {\\small s};\n \\draw[  (-, ultra thick, black] (-8,0) -- (-7.99,0);\n  \\draw[ { -]}, ultra thick, black] (-1.0002,0) -- (-1,0);\n\n   \\node[color=black] at (-7,-0.3) {\\small $\\eta_k$};\n\\draw(-7 ,0)circle [radius=2pt] ;\n\n  \\node[color=black] at (-6.5,-0.3) {\\small $s_u$};\n\\draw plot[mark=x, mark options={color=black, scale=1.5}] coordinates {(-6.5,0)  }; \n \n\n  \\node[color=black] at (-2.3,-0.3) {\\small $\\eta_{k+1}$};\n\\draw(-2.3 ,0)circle [radius=2pt] ;\n  \\node[color=black] at (-3 ,-0.3) {\\small $s_v$};\n\\draw plot[mark=x, mark options={color=black, scale=1.5}] coordinates {(-3,0)  }; \n  \n   \\node[color=black] at (-1,-0.3) {\\small $e$};\n\n \n\\end{tikzpicture}\n\\end{center} \n\\\n\\\\\n{\\bf Step 1.} Denote \n $$ \\mathcal I_ 1= (  s, s _u], \\quad \\I_2  =(s_u, s_v] \\quad \\text{and} \\quad \\I_3  = (s_v,e]. $$\n \\\n \\\\\nSince  $ |\\I| \\ge \\eta_{k+1} -\\eta_k \\ge C_\\mclF  { \\mathfrak{s} }  \\log(np)   $, \n$$ \\mathcal F(\\I ) = \\sum_{i \\in \\I }(y_i - X_i^\\top \\widehat \\beta _\\I ) ^2  . $$\nSince \n $ |\\I_1|  \\ge  \\eta_k-s      \\ge C_\\mclF  { \\mathfrak{s} }  \\log(np)   , $ \nit follows that \n $$ \\mathcal F(\\I_1 ) = \\sum_{i \\in \\I_1 }(y_i - X_i^\\top \\widehat \\beta _{\\I_1}  ) ^2  . $$\nIn addition   since  $ |\\I_1| \\ge C_\\mclF  { \\mathfrak{s} }  \\log(np)  $, then \n \\begin{align*}\\mathcal F(\\I_1)  = & \\sum_{i \\in \\I_1 }(y_i - X_i^\\top \\widehat \\beta _{\\I_1}  ) ^2 \n \\\\\n \\le & \\sum_{i \\in \\I_1 }(y_i - X_i^\\top   \\beta ^*  _i   ) ^2  + C_1  \\bigg\\{ \\frac{( \\eta_k-s) (s_u -\\eta_k ) }{ ( \\eta_k-s)  + (s_u -\\eta_k )  } \\kappa ^2 +  { \\mathfrak{s} } \\log(np)  \\bigg\\} \n \\\\\n  \\le & \\sum_{i \\in \\I_1 }(y_i - X_i^\\top   \\beta ^*  _i   ) ^2  +  C_1 \\bigg\\{ (s_u -\\eta_k )   \\kappa ^2 +   { \\mathfrak{s} } \\log(np)  \\bigg\\} \n  \\\\\n  \\le & \\sum_{i \\in \\I_1 }(y_i - X_i^\\top   \\beta ^*  _ i   ) ^2  + C_1 \\bigg\\{   { \\mathcal B_n^{-1}   \\Delta }    \\kappa ^2 +   { \\mathfrak{s} } \\log(np)  \\bigg\\} ,\n \\end{align*}\nwhere the first  inequality follows  from \\Cref{lam:regression one change deviation bound}  and that $ \\kappa_{k} \\asymp \\kappa$.\n Similarly, since \n $ |\\I_2|  \\ge \\Delta \/2 \\ge   C_\\mclF  { \\mathfrak{s} }  \\log(np)   , $ \nit follows that \n $$ \\mathcal F(\\I_2 ) = \\sum_{i \\in \\I_2 }(y_i - X_i^\\top \\widehat \\beta _{\\I_2}  ) ^2  . $$\n  Since $|\\I_2| \\ge C_\\mclF  { \\mathfrak{s} }  \\log(np)   $ and  $\\I_2$ contains no change points, by \\Cref{lam:regression one change deviation bound},\n  \\begin{align*}\\mathcal F(\\I_2)  \\le \\sum_{i \\in \\I_2 }(y_i - X_i^\\top   \\beta ^*  _ i  ) ^2   + C_1 { \\mathfrak{s} } \\log(np).\n  \\end{align*}  \n  \\\\\n  \\\\\n {\\bf Step 2.} If $ |\\I_3 | \\ge C_\\mclF  { \\mathfrak{s} }  \\log(np)   $, then \n  \\begin{align*}\\mathcal F(\\I_3)  = & \\sum_{i \\in \\I_3 }(y_i - X_i^\\top \\widehat \\beta _{\\I_3}  ) ^2 \n \\\\\n \\le & \\sum_{i \\in \\I_3 }(y_i - X_i^\\top   \\beta ^*  _i   ) ^2  + C_1  \\bigg\\{ \\frac{( \\eta_{k+1}-s_v) (e -\\eta_{k+1} ) }{ ( \\eta_{k+1}-s_v)+  (e -\\eta_{k+1} )  } \\kappa ^2 +  { \\mathfrak{s} } \\log(np)  \\bigg\\} \n \\\\\n  \\le & \\sum_{i \\in \\I_3 }(y_i - X_i^\\top   \\beta ^*  _i   ) ^2  +  C_1 \\bigg\\{ ( \\eta_{k+1}-s_v)    \\kappa ^2 +   { \\mathfrak{s} } \\log(np)  \\bigg\\} \n  \\\\\n  \\le & \\sum_{i \\in \\I_3 }(y_i - X_i^\\top   \\beta ^*  _ i   ) ^2  + C_1 \\bigg\\{   { \\mathcal B_n^{-1}   \\Delta }    \\kappa ^2 +   { \\mathfrak{s} } \\log(np)  \\bigg\\} ,\n \\end{align*}\n where the first inequality follows from \\Cref{lam:regression one change deviation bound}{\\bf b} and that $ \\kappa_{k+1} \\asymp \\kappa$. \n If $|\\I_3| < C_\\mclF  { \\mathfrak{s} }  \\log(np)   $, then \n$\\mathcal F(\\I_3)  =  0 $.\n So \nboth cases imply that  \n $$\\mathcal F(\\I_3)  \\le \\sum_{i \\in \\I_3 }(y_i - X_i^\\top   \\beta ^*  _ i   ) ^2  + C_1 \\bigg\\{   { \\mathcal B_n^{-1}   \\Delta }    \\kappa_k^2 +   { \\mathfrak{s} } \\log(np)  \\bigg\\}  .$$\n \\\n \\\\\n {\\bf Step 3.} Since $\\I \\in \\widehat{\\mathcal P} $, we have\n \\begin{align}\\label{eq:regression two change points local min} \\mathcal F(\\I ) \\le \\mathcal F(\\I_1  ) +\\mathcal F(\\I_2 )+\\mathcal F(\\I_3 ) + 2\\gamma. \n \\end{align} The above display and the calculations in  {\\bf Step 1} and  {\\bf Step 2} implies that\n\\begin{align} \n \\sum_{i \\in \\I }(y_i -  X_i^\\top \\widehat \\beta _\\I ) ^2  \n\\le   \\sum_{i \\in \\I  }(y_i -   X_i^\\top \\beta_i^*   ) ^2  \n +   3  C_1 \\bigg\\{   { \\mathcal B_n^{-1}   \\Delta }    \\kappa ^2 +   { \\mathfrak{s} } \\log(np) \\bigg\\}\n+2\\gamma  .\n  \\label{eq:regression two change points step 2 term 1}\n\\end{align}\nDenote \n$$ {  \\mathcal J } _1  = (s,\\eta_k ], \\quad {  \\mathcal J } _2=(\\eta_k, \\eta_{k} +   \\eta_{k+1}    ]  \\quad \\text{and} \\quad {  \\mathcal J } _3 = (\\eta_{k+1}  , e] .$$\n\\Cref{eq:regression two change points step 2 term 1}  gives \n\\begin{align} \n \\sum_{\\ell=1}^3 \\sum_{i \\in {  \\mathcal J } _{\\ell}  }(y_i - X_i^\\top  \\widehat \\beta _\\I ) ^2 \n \\le& \\sum_{\\ell=1}^3 \\sum_{i \\in {  \\mathcal J } _ \\ell  }(y_i -  X_i^\\top    \\beta^*  _{{  \\mathcal J } _\\ell }  ) ^2    \n +  3  C_1 \\bigg\\{   { \\mathcal B_n^{-1}   \\Delta }    \\kappa ^2 +   { \\mathfrak{s} } \\log(np) \\bigg\\} \n+2\\gamma     \\label{eq:regression two change points first}\n\\end{align} \n\\\n\\\\\n{\\bf Step 4.} \nNote that for $\\ell\\in \\{1,2,3\\}$, \n\\begin{align*} \n\\|  ( \\widehat \\beta _\\I  -  \\beta _{{  \\mathcal J } _\\ell }^* )_{S^c } \\|_1 = \\|  ( \\widehat \\beta _\\I )_{S^c} \\|_{1} \n=   \\|  ( \\widehat \\beta _\\I -\\beta _\\I ^* )_{S^c} \\|_{1}  \\le 3 \\|  ( \\widehat \\beta _\\I -\\beta _\\I ^* )_{S } \\|_{1}   \\le C_2  { \\mathfrak{s} }  \\sqrt { \\frac{ \\log(np)}{| \\I|} }    , \n\\end{align*} \nwhere the last two inequalities follows from \\Cref{lemma:interval lasso}.  So\n\\begin{align}  \\label{eq:regression two change step 3 term 1}\n\\|   \\widehat \\beta _\\I  -  \\beta _{{  \\mathcal J } _\\ell}^*   \\|_1 = \\|  ( \\widehat \\beta _\\I  -  \\beta _{{  \\mathcal J } _\\ell }^* )_{S  } \\|_1 +\\|  ( \\widehat \\beta _\\I  -  \\beta _{{  \\mathcal J } _\\ell}^* )_{S^c } \\|_1  \\le \\sqrt { { \\mathfrak{s} }  }  \\|  \\widehat \\beta _\\I  -  \\beta _{{  \\mathcal J } _\\ell}^*   \\|_2 + C_2  { \\mathfrak{s} }  \\sqrt { \\frac{ \\log(np)}{| \\I|} } .\n\\end{align} \nNote that by assumptions,\n$$|{  \\mathcal J } _1|\\ge C_\\mclF  { \\mathfrak{s} }  \\log(np)    \\quad \\text{and} \\quad |{  \\mathcal J } _2 |\\ge C_\\mclF  { \\mathfrak{s} }  \\log(np) .$$\n So for $\\ell\\in \\{ 1, 2\\} $,\n  it holds that\n \\begin{align}\\nonumber \n\t\t&  \\sum_{i \\in {  \\mathcal J } _\\ell  } \\big\\{ X_i^\\top  (  \\widehat \\beta _\\I  -  \\beta^* _{{  \\mathcal J } _ \\ell  }) \\big\\}  ^2 \n\t\t \\\\  \\nonumber  \n\t\t  \\ge&  \\frac{c_x|{  \\mathcal J } _\\ell | }{16} \\|\\widehat \\beta _\\I  -  \\beta^* _{{  \\mathcal J } _ \\ell  } \\|_2^2 - C_3 \\log(p) \\|\\widehat \\beta _\\I  -  \\beta^* _{{  \\mathcal J } _ \\ell  } \\|_1  ^2 \n\t\t \\\\\\nonumber  \n\t\t \\ge & \\frac{c_x|{  \\mathcal J } _\\ell | }{16} \\|\\widehat \\beta _\\I  -  \\beta^* _{{  \\mathcal J } _ \\ell  } \\|_2^2 - C_3'  { \\mathfrak{s} }    \\log(p) \\|\\widehat \\beta _\\I  -  \\beta^* _{{  \\mathcal J } _ \\ell  } \\|_ 2   ^2  - C_3'  \\frac{   { \\mathfrak{s} } ^2 \\log(p) \\log(np)  }{|\\I| }\n\t\t \\\\ \\label{eq:regression two change step 3 term 2}\n\t\t \\ge &  \\frac{c_x|{  \\mathcal J } _\\ell | }{32} \\|\\widehat \\beta _\\I  -  \\beta^* _{{  \\mathcal J } _ \\ell  } \\|_2^2 - C_4   { \\mathfrak{s} }  \\log(np),\n\t\\end{align} \n\twhere the first inequality follows from \\Cref{lem:restricted eigenvalue}, the second inequality follows from \\Cref{eq:regression two change step 3 term 1}   and the last inequality follows from the observation that \n $$  |\\I|\\ge  |{  \\mathcal J } _\\ell|   \\ge C_\\gamma  { \\mathfrak{s} }  \\log(np). \n $$ \n So for $\\ell\\in \\{ 1,2\\}$, \n\\begin{align*} &\\sum_{i \\in {  \\mathcal J } _\\ell  }(y_i - X_i^\\top  \\widehat \\beta _\\I ) ^2  - \\sum_{i \\in {  \\mathcal J } _\\ell   }(y_i - X_i^\\top    \\beta^*  _{{  \\mathcal J } _\\ell }  ) ^2\n=  \\sum_{i \\in {  \\mathcal J } _\\ell    }\\big\\{ X_i^\\top (\\widehat \\beta _\\I -\\beta^*  _{{  \\mathcal J } _4} )\\big\\}^2 -2 \\sum_{i \\in {  \\mathcal J } _\\ell   } \\epsilon_i X_i^\\top (\\widehat \\beta _\\I -\\beta^*  _{{  \\mathcal J } _\\ell  }  )\n\\\\\n\\ge  & \\sum_{i \\in {  \\mathcal J } _\\ell   }\\big\\{ X_i^\\top (\\widehat \\beta _\\I -\\beta^*  _{{  \\mathcal J } _\\ell  } )\\big\\}^2  - 2 \\| \\sum_{i \\in {  \\mathcal J } _\\ell   }  \\epsilon_i X_i^\\top \\|_\\infty \\| \\widehat \\beta _\\I -\\beta^*  _{{  \\mathcal J } _\\ell  }  \\|_1\n\\\\\n\\ge &\\sum_{i \\in {  \\mathcal J } _\\ell   }\\big\\{ X_i^\\top (\\widehat \\beta _\\I -\\beta^*  _{{  \\mathcal J } _\\ell  } )\\big\\}^2  - C_5 \\sqrt{ \\log(np) |{  \\mathcal J } _\\ell|   }\\bigg(\\sqrt { { \\mathfrak{s} }  }  \\|  \\widehat \\beta _\\I  -  \\beta _{{  \\mathcal J } _1}^*   \\|_2 + C_2  { \\mathfrak{s} }  \\sqrt { \\frac{ \\log(np)}{| \\I|} } \\bigg) \n\\\\\n\\ge &  \\frac{c_x|{  \\mathcal J } _\\ell | }{32} \\|\\widehat \\beta _\\I  -  \\beta^* _{{  \\mathcal J } _ \\ell  } \\|_2^2 - C_4   { \\mathfrak{s} }  \\log(np)  - C_5 \\sqrt{ \\log(np) |{  \\mathcal J } _\\ell|  }\\bigg(\\sqrt { { \\mathfrak{s} }  }  \\|  \\widehat \\beta _\\I  -  \\beta _{{  \\mathcal J } _1}^*   \\|_2 + C_2  { \\mathfrak{s} }  \\sqrt { \\frac{ \\log(np)}{| \\I|} } \\bigg)  \n\\\\\n\\ge & \\frac{c_x|{  \\mathcal J } _\\ell | }{32} \\|\\widehat \\beta _\\I  -  \\beta^* _{{  \\mathcal J } _ \\ell  } \\|_2^2 -\\frac{c_x|{  \\mathcal J } _\\ell | }{64} \\|\\widehat \\beta _\\I  -  \\beta^* _{{  \\mathcal J } _ \\ell  } \\|_2^2 -C_6  { \\mathfrak{s} }  \\log(np)  \n= \\frac{c_x|{  \\mathcal J } _\\ell | }{64} \\|\\widehat \\beta _\\I  -  \\beta^* _{{  \\mathcal J } _ \\ell  } \\|_2^2     -C_6  { \\mathfrak{s} }  \\log(np)  ,\n\\end{align*}\nwhere the second inequality follows from the standard sub-Exponential tail bound and \\Cref{eq:regression two change step 3 term 1}, the third inequality follows from \\Cref{eq:regression two change step 3 term 2}, and the fourth inequality follows from  $ {  \\mathcal J } _\\ell \\subset \\I $ and so $ |\\I | \\ge |{  \\mathcal J } _\\ell|$.\n\\\\\n\\\\\nSo for $\\ell \\in \\{1,2 \\}$, \n\\begin{align} \\label{eq:regression change point step 3 last item}\\sum_{i \\in {  \\mathcal J } _\\ell  }(y_i - X_i^\\top  \\widehat \\beta _\\I ) ^2  - \\sum_{i \\in {  \\mathcal J } _\\ell   }(y_i - X_i^\\top    \\beta^*  _{{  \\mathcal J } _\\ell }  ) ^2 \n\\ge \\frac{c_x|{  \\mathcal J } _\\ell | }{64} \\|\\widehat \\beta _\\I  -  \\beta^* _{{  \\mathcal J } _ \\ell  } \\|_2^2     -C_6  { \\mathfrak{s} }  \\log(np).\n\\end{align} \n\\\n\\\\\n\\\\\n{\\bf Step 5.} For ${  \\mathcal J } _3$, if $|{  \\mathcal J } _3| \\ge C_\\mclF  { \\mathfrak{s} }  \\log(np)  $,  following the same calculations as in {\\bf Step 4}, \n$$\\sum_{i \\in {  \\mathcal J } _ 3 }(y_i - X_i^\\top  \\widehat \\beta _\\I ) ^2  - \\sum_{i \\in {  \\mathcal J } _ 3 }(y_i - X_i^\\top    \\beta^*  _{{  \\mathcal J } _3   }  ) ^2 \\ge \\frac{c_x|{  \\mathcal J } _3   | }{64} \\|\\widehat \\beta _\\I  -  \\beta^* _{{  \\mathcal J } _ 3  } \\|_2^2     -C_6  { \\mathfrak{s} }  \\log(np)  \\ge -C_6  { \\mathfrak{s} } \\log(np). \n$$\nIf $|{  \\mathcal J } _3 | < C_\\mclF  { \\mathfrak{s} }  \\log(np) $, then\n\\begin{align} & \\sum_{i \\in {  \\mathcal J } _3   }(y_i - X_i^\\top  \\widehat \\beta _\\I ) ^2  - \\sum_{i \\in {  \\mathcal J } _3   }(y_i - X_i^\\top    \\beta^*  _{{  \\mathcal J } _ 3}  ) ^2 \\nonumber \n=   \\sum_{i \\in {  \\mathcal J } _ 3   }\\big\\{ X_i^\\top (\\widehat \\beta _\\I -\\beta^*  _{{  \\mathcal J } _3} )\\big\\}^2 -2 \\sum_{i \\in {  \\mathcal J } _3   } \\epsilon_i X_i^\\top (\\widehat \\beta _\\I -\\beta^*  _{{  \\mathcal J } _3 }  )\n\\\\  \\nonumber \n\\ge  & \\sum_{i \\in {  \\mathcal J } _3   }\\big\\{ X_i^\\top (\\widehat \\beta _\\I -\\beta^*  _{{  \\mathcal J } _3 } )\\big\\}^2  -\\frac{1}{2 }\\sum_{i \\in {  \\mathcal J } _3   }\\big\\{ X_i^\\top (\\widehat \\beta _\\I -\\beta^*  _{{  \\mathcal J } _3 } )\\big\\}^2  - 4   \\sum_{i \\in {  \\mathcal J } _3   }\\epsilon_i^2 \n\\\\  \\nonumber \n\\ge & \\frac{1}{2 }\\sum_{i \\in {  \\mathcal J } _3   }\\big\\{ X_i^\\top (\\widehat \\beta _\\I -\\beta^*  _{{  \\mathcal J } _3 } )\\big\\}^2  -C_7\\bigg(  \\sqrt { \\gamma   \\log(n)} + \\log(n)+ \\gamma  \\bigg) \n\\\\  \\nonumber  \n\\ge & \\frac{1}{2 }\\sum_{i \\in {  \\mathcal J } _3   }\\big\\{ X_i^\\top (\\widehat \\beta _\\I -\\beta^*  _{{  \\mathcal J } _3 } )\\big\\}^2  -C_7' \\bigg(     \\log(n)+ \\gamma   \\bigg)  \n\\\\ \\ge \n& \\frac{1}{2 }\\sum_{i \\in {  \\mathcal J } _3   }\\big\\{ X_i^\\top (\\widehat \\beta _\\I -\\beta^*  _{{  \\mathcal J } _3 } )\\big\\}^2  -C_8    (  { \\mathfrak{s} } \\log(np) + \\gamma) \n\\ge -C_8    (   { \\mathfrak{s} } \\log(np) +\\gamma)   \\label{eq:regression change point step 4 last item}\n\\end{align}\nwhere the second inequality follows from the standard sub-exponential deviation bound.\n\\\n\\\\\n\\\\\n{\\bf Step 6.} Putting \\Cref{eq:regression two change points first}, \\eqref{eq:regression change point step 3 last item} and \\eqref{eq:regression change point step 4 last item} together, it follows that \n $$  \\sum_{\\ell =1}^2 \\frac{c_x|{  \\mathcal J } _\\ell | }{64} \\|\\widehat \\beta _\\I  -  \\beta^* _{{  \\mathcal J } _ \\ell  } \\|_2^2      \\le C_9( { \\mathfrak{s} }  \\log(np) +  { \\mathcal B_n^{-1}   \\Delta }  \\kappa ^2 + \\gamma) .$$\n This leads to \n  $$     |{  \\mathcal J } _1 |   \\|\\widehat \\beta _\\I  -  \\beta^* _{{  \\mathcal J } _ 1  } \\|_2^2     + |{  \\mathcal J } _2 |   \\|\\widehat \\beta _\\I  -  \\beta^* _{{  \\mathcal J } _ 2  } \\|_2^2       \\le C_9( { \\mathfrak{s} }  \\log(np) +  { \\mathcal B_n^{-1}   \\Delta }  \\kappa ^2 + \\gamma) .$$ \nObserve  that \n$$ \\inf_{  \\beta  \\in \\mathbb R ^p   }  |{  \\mathcal J } _1 |  \\| \\beta    -  \\beta^* _{{  \\mathcal J } _ 1  } \\|_2^2  + |{  \\mathcal J } _2 | \\|  \\beta    -  \\beta^* _{{  \\mathcal J } _ 2  } \\|_2^2    = \\kappa_k   ^2  \\frac{|{  \\mathcal J } _1| |{  \\mathcal J } _2|}{| {  \\mathcal J } _1| +|{  \\mathcal J } _2|  }   \\ge \\frac{ \\kappa_k  ^2 }{2} \\min\\{ |{  \\mathcal J } _1| ,|{  \\mathcal J } _2| \\}   \\ge   \\frac{ c \\kappa   ^2 }{2} \\min\\{ |{  \\mathcal J } _1| ,|{  \\mathcal J } _2| \\}   . $$ \nThus\n$$    \\kappa ^2 \\min\\{ |{  \\mathcal J } _1| ,|{  \\mathcal J } _2| \\}    \\le  C_{10} ( { \\mathfrak{s} }  \\log(np)+ { \\mathcal B_n^{-1}   \\Delta }  \\kappa^2  + \\gamma)  ,$$\nwhich is \n$$ \\min\\{ |{  \\mathcal J } _1| ,|{  \\mathcal J } _2| \\}   \\le C_5 \\bigg( \\frac{ { \\mathfrak{s} }  \\log(np) + \\gamma  }{\\kappa ^2 } +   { \\mathcal B_n^{-1}   \\Delta }   +\\frac{ \\gamma}{\\kappa ^2} \\bigg) .$$ \nSince  $ |{  \\mathcal J } _2  | \\ge \\Delta \\ge     \\frac{  C     (  { \\mathfrak{s} }  \\log(np) + \\gamma) }{   \\kappa^{2}}   $\n for sufficiently large constant $C$, \n it follows that\n $$ |{  \\mathcal J } _2| \\ge \\Delta>  C_5 \\bigg( \\frac{ { \\mathfrak{s} }  \\log(np) + \\gamma  }{\\kappa ^2 } +   { \\mathcal B_n^{-1}   \\Delta }   +\\frac{ \\gamma}{\\kappa ^2} \\bigg) .$$  So it holds that \n $$   |{  \\mathcal J } _1|    \\le C_5 \\bigg( \\frac{ { \\mathfrak{s} }  \\log(np) + \\gamma  }{\\kappa ^2 } +   { \\mathcal B_n^{-1}   \\Delta }   \\bigg) .$$ \n \\end{proof} \n\n \\bnlem \nSuppose the good events \n$\\mathcal L ( { \\mathcal B_n^{-1}   \\Delta }  ) $ and  $\\mathcal R (  { \\mathcal B_n^{-1}   \\Delta }  ) $ defined  in  \\Cref{eq:left and right approximation of change points} hold. \n Suppose in addition that \n\\begin{align} \\label{eq:regression three change points snr}\n\\Delta \\kappa^2 \\ge C      \\big(  { \\mathfrak{s} }  \\log(np) + \\gamma)    \n\\end{align} \nfor sufficiently large constant $C $. \n Then  with probability  at least $1- n^{-3}$,    there is no intervals in   $  \\widehat { \\mathcal P}  $  containing three or more true change points.   \n \\enlem  \n \n\n\n\\begin{proof} For  contradiction,  suppose    $ \\I=(s,e] \\in \\mathcal {\\widehat P}  $  be such that $ \\{ \\eta_1, \\ldots, \\eta_M\\} \\subset \\I $ with $M\\ge 3$.  \n\\\\\n\\\\ \nSince the events $\\mathcal L (  { \\mathcal B_n^{-1}   \\Delta }   ) $ and  $\\mathcal R (  { \\mathcal B_n^{-1}   \\Delta }  ) $ hold, by relabeling $\\{ s_q\\}_{q=1}^ { \\mathcal Q}   $ if necessary,   let $ \\{ s_m\\}_{m=1}^M $ be such that \n $$ 0 \\le s_m  -\\eta_m  \\le  { \\mathcal B_n^{-1}   \\Delta }     \\quad \\text{for} \\quad 1 \\le m \\le M-1  $$ and that\n $$ 0\\le \\eta_M  - s_M \\le  { \\mathcal B_n^{-1}   \\Delta }  .$$ \n Note that these choices ensure that  $ \\{ s_m\\}_{m=1}^M  \\subset \\I . $\n \\\n \\\\\n \\begin{center}\n \\begin{tikzpicture} \n\\draw[ - ] (-10,0)--(1,0);\n \\node[color=black] at (-8,-0.3) {\\small s};\n \\draw[  (-, ultra thick, black] (-8,0) -- (-7.99,0);\n  \\draw[ { -]}, ultra thick, black] (-1.0002,0) -- (-1,0);\n\n   \\node[color=black] at (-7,-0.3) {\\small $\\eta_1$};\n\\draw(-7 ,0)circle [radius=2pt] ;\n\n  \\node[color=black] at (-6.5,-0.3) {\\small $s_1$};\n\\draw plot[mark=x, mark options={color=black, scale=1.5}] coordinates {(-6.5,0)  }; \n\n  \\node[color=black] at (-5,-0.3) {\\small $\\eta_2$};\n\\draw(-5 ,0)circle [radius=2pt] ;\n  \\node[color=black] at (-4.5,-0.3) {\\small $s_2$};\n\\draw plot[mark=x, mark options={color=black, scale=1.5}] coordinates {(-4.5,0)  }; \n \n\n  \\node[color=black] at (-2.5,-0.3) {\\small $\\eta_M$};\n\\draw(-2.5 ,0)circle [radius=2pt] ;\n  \\node[color=black] at (-3 ,-0.3) {\\small $s_M$};\n\\draw plot[mark=x, mark options={color=black, scale=1.5}] coordinates {(-3,0)  }; \n  \n   \\node[color=black] at (-1,-0.3) {\\small $e$};\n\n \n\\end{tikzpicture}\n\\end{center}\n\\\n\\\\\n{\\bf Step 1.}\n Denote \n $$ \\mathcal I_ 1= (  s, s _1], \\quad \\I_m   =(s_{m-1} , s_m] \\text{ for }  2 \\le m \\le M     \\quad \\text{and} \\quad \\I_{M+1}   = (s_M,e]. $$\n Then since $|\\I| \\ge \\Delta \\ge  C_s  { \\mathfrak{s} }  \\log(np)  $, it follows that \n Since  $ |\\I| \\ge \\eta_{k+1} -\\eta_k \\ge  C_s  { \\mathfrak{s} }  \\log(np) $, \n$$ \\mathcal F(\\I ) = \\sum_{i \\in \\I }(y_i - X_i^\\top \\widehat \\beta _\\I ) ^2  . $$\nSince  $ | \\I_m | \\ge \\Delta \/2 \\ge   C_s  { \\mathfrak{s} }  \\log(np) $ for all $ 2 \\le m \\le M $,  it follows  from the same argument as   {\\bf Step 1} in the proof  of  \\Cref{lemma:regression two change points} that \n \\begin{align*}\\mathcal F(\\I_m)  = & \\sum_{i \\in \\I_m }(y_i - X_i^\\top \\widehat \\beta _{\\I_m}  ) ^2\n  \\le   \\sum_{i \\in \\I_m }(y_i - X_i^\\top   \\beta ^*  _ i   ) ^2  + C_1 \\bigg\\{   { \\mathcal B_n^{-1}   \\Delta }    \\kappa ^2 +   { \\mathfrak{s} } \\log(np)  \\bigg\\} \\quad \\text{for all } 2 \n  \\le m \\le M.\n \\end{align*}\n \\\n \\\\\n {\\bf Step 2.} It follows  from the same argument as   {\\bf Step 2} in the proof  of  \\Cref{lemma:regression two change points} that    \n  \\begin{align*}\n  &\\mathcal F(\\I_1)  \n  \\le   \\sum_{i \\in \\I_1 }(y_i - X_i^\\top   \\beta ^*  _ i   ) ^2  + C_1 \\bigg\\{   { \\mathcal B_n^{-1}   \\Delta }    \\kappa ^2 +   { \\mathfrak{s} } \\log(np)  \\bigg\\} , \\text{ and}\n  \\\\\n  &\\mathcal F(\\I_{M+1} )  \n  \\le   \\sum_{i \\in \\I_{M+1} }(y_i - X_i^\\top   \\beta ^*  _ i   ) ^2  + C_1 \\bigg\\{   { \\mathcal B_n^{-1}   \\Delta }    \\kappa ^2 +   { \\mathfrak{s} } \\log(np)  \\bigg\\}  \n \\end{align*}\n \\\n \\\\\n {\\bf Step 3.} Since $\\I \\in \\widehat{\\mathcal P} $, we have\n \\begin{align}\\label{eq:regression three change points local min} \\mathcal F(\\I ) \\le \\sum_{m=1}^{M+1}\\mathcal F(\\I_m  )  +  M\\gamma. \n \\end{align} The above display and the calculations in  {\\bf Step 1} and  {\\bf Step 2} implies that\n\\begin{align} \n \\sum_{i \\in \\I }(y_i -  X_i^\\top \\widehat \\beta _\\I ) ^2  \n\\le   \\sum_{i \\in \\I  }(y_i -   X_i^\\top \\beta_i^*   ) ^2  \n +   (M +1) C_1 \\bigg\\{   { \\mathcal B_n^{-1}   \\Delta }    \\kappa ^2 +   { \\mathfrak{s} } \\log(np) \\bigg\\}\n+M\\gamma  .\n  \\label{eq:regression three change points step 2 term 1}\n\\end{align}\nDenote \n   $$  {  \\mathcal J } _1 =(s,  \\eta_1], \\ {  \\mathcal J } _m = (\\eta_{m-1}, \\eta_m]  \\quad \\text{for}\\quad  2 \\le m \\le M , \\ {  \\mathcal J } _{M+1}  =(\\eta_M, e]. $$  \n\\Cref{eq:regression three change points step 2 term 1}  gives \n\\begin{align} \n \\sum_{m=1}^{M+1}  \\sum_{i \\in {  \\mathcal J } _m   }(y_i - X_i^\\top  \\widehat \\beta _\\I ) ^2 \n \\le& \\sum_{m=1}^{M+1} \\sum_{i \\in {  \\mathcal J } _  m   }(y_i -  X_i^\\top    \\beta^*  _{{  \\mathcal J } _m  }  ) ^2    \n +  (M+1)   C_1 \\bigg\\{   { \\mathcal B_n^{-1}   \\Delta }    \\kappa ^2 +   { \\mathfrak{s} } \\log(np) \\bigg\\} \n+M\\gamma     \\label{eq:regression three change points first}\n\\end{align}  \n \\\n \\\\\n {\\bf Step 4.} Using the same argument as in the {\\bf Step 4} in the proof  of  \\Cref{lemma:regression two change points},\nit follows that \n\\begin{align} \\label{eq:regression three change point step 3 last term}\\sum_{i \\in {  \\mathcal J } _m   }(y_i - X_i^\\top  \\widehat \\beta _\\I ) ^2  - \\sum_{i \\in {  \\mathcal J } _m   }(y_i - X_i^\\top    \\beta^*  _{{  \\mathcal J } _m }  ) ^2 \n\\ge \\frac{c_x|{  \\mathcal J } _m | }{64} \\|\\widehat \\beta _\\I  -  \\beta^* _{{  \\mathcal J } _ m  } \\|_2^2     -C_2  { \\mathfrak{s} }  \\log(np) \\quad \\text{for all}  \\   2 \\le m \\le M. \n\\end{align} \n \\\n \\\\\n {\\bf Step 5.}\n  Using the same argument as in the {\\bf Step 4} in the proof  of  \\Cref{lemma:regression two change points},  it follows that \n\\begin{align} \\label{eq:regression three change point step 5 first term}& \\sum_{i \\in {  \\mathcal J } _ 1 }(y_i - X_i^\\top  \\widehat \\beta _\\I ) ^2  - \\sum_{i \\in {  \\mathcal J } _ 1 }(y_i - X_i^\\top    \\beta^*  _{{  \\mathcal J } _1   }  ) ^2  \\ge -C_3  (  { \\mathfrak{s} } \\log(np)  +\\gamma) \\text{ \n and }\n\\\\\\label{eq:regression three change point step 5 second term}\n& \\sum_{i \\in {  \\mathcal J } _ {M+ 1} }(y_i - X_i^\\top  \\widehat \\beta _\\I ) ^2  - \\sum_{i \\in {  \\mathcal J } _  {M+ 1}  }(y_i - X_i^\\top    \\beta^*  _{{  \\mathcal J } _ {M+ 1}   }  ) ^2  \\ge -C_3 (  { \\mathfrak{s} } \\log(np)  +\\gamma) \n \\end{align}\n\\\n\\\\\n{\\bf  Step 6.} Putting \\Cref{eq:regression three change points first}, \\eqref{eq:regression three change point step 3 last term}, \\eqref{eq:regression three change point step 5 first term} and  \\eqref{eq:regression three change point step 5 second term}, \n it follows that \n\\begin{align} \\label{eq:regression three change points step six} \\sum_{ m  =2}^M \\frac{c_x|{  \\mathcal J } _m  | }{64} \\|\\widehat \\beta _\\I  -  \\beta^* _{{  \\mathcal J } _  m  } \\|_2^2      \\le C_4M ( { \\mathfrak{s} }  \\log(np) +  { \\mathcal B_n^{-1}   \\Delta }  \\kappa ^2 + \\gamma) .\n\\end{align} \n For any $  m \\in\\{2, \\ldots, M\\}$, it holds that\n\\begin{align}  \\label{eq:regression three change points signal lower bound} \\inf_{  \\beta  \\in \\mathbb R^p   }  |{  \\mathcal J } _{m-1} |  \\|   \\beta - \\beta^* _{{  \\mathcal J } _ {m-1}   }  \\|  ^2 + |{  \\mathcal J } _{m} | \\|   \\beta - \\beta^* _{{  \\mathcal J } _ m  }  \\|  ^2   =&     \\frac{|{  \\mathcal J } _{m-1}| |{  \\mathcal J } _m|}{ |{  \\mathcal J } _{m-1}| +  |{  \\mathcal J } _m| } \\kappa_m ^2  \\ge \\frac{1}{2} \\Delta \\kappa^2,\n\\end{align}  \nwhere the last inequality follows from the assumptions that $\\eta_k - \\eta_{k-1}\\ge \\Delta  $ and $ \\kappa_k \\asymp \\kappa$  for all $1\\le k \\le K$. So\n\\begin{align}   \\nonumber &2 \\sum_{ m=1}^{M }   |{  \\mathcal J } _m |\\|\\widehat \\beta _\\I  -  \\beta^* _{{  \\mathcal J } _ m  } \\|_2^2   \n\\\\ \n\\ge &  \\nonumber    \\sum_{m=2}^M  \\bigg(  |{  \\mathcal J } _{m-1}  \\|\\widehat \\beta _\\I  -  \\beta^* _{{  \\mathcal J } _ {m-1}   } \\|_2^2      + |{  \\mathcal J } _m |\\|\\widehat \\beta _\\I  -  \\beta^* _{{  \\mathcal J } _ m  } \\|_2^2    \\bigg) \n\\\\ \\label{eq:regression three change points signal lower bound two} \n\\ge & (M-1)  \\frac{ 1}{2} \\Delta \\kappa^2  \\ge \\frac{M}{4} \\Delta \\kappa^2 ,\n\\end{align} \nwhere the second inequality follows from  \\Cref{eq:regression three change points signal lower bound} and the last inequality follows from $M\\ge 3$. \\Cref{eq:regression three change points step six} and  \\Cref{eq:regression three change points signal lower bound two} together imply that \n\\begin{align}\\label{eq:regression three change points signal lower bound three} \n \\frac{M}{4} \\Delta \\kappa^2 \\le 2  C_5  M\\bigg(   { \\mathfrak{s} }    \\log(np ) +     { \\mathcal B_n^{-1}   \\Delta }  \\kappa^2\n+   \\gamma \\bigg) .\n\\end{align}\nSince $ { \\mathcal B_n} \\to \\infty $, it follows that for sufficiently large $n$,   \\Cref{eq:regression three change points signal lower bound three} gives \n$$ \\Delta\\kappa^2 \\le C_5 \\big(   { \\mathfrak{s} }  \\log(np) +\\gamma),$$  \nwhich contradicts \\Cref{eq:regression three change points snr}. \n \n  \n \\end{proof} \n \n \\bnlem Suppose $ \\gamma \\ge C_\\gamma  { \\mathfrak{s} }   \\log(np) $ for sufficiently large constant $C_\\gamma $. \nWith probability  at least $1- n^{-3}$,    there is no  two consecutive intervals $\\I_1= (s,t ] \\in \\widehat {\\mathcal  P}  $,   $ \\I_2=(t, e]  \\in   \\widehat  {\\mathcal  P}  $    such that $\\I_1 \\cup \\I_2$ contains no change points.  \n \\enlem \n \\begin{proof} \n For contradiction, suppose that \n $$ \\I :  =\\I_1\\cup  \\I_2 $$\n contains no change points. \nFor $\\I_1$, note that if $|\\I_1| \\ge  C_\\zeta  { \\mathfrak{s} }  \\log(np) $, then \n by \\Cref{lam:regression one change deviation bound} {\\bf a}, it follows that \n \\begin{align*} \n \\bigg|   \\mathcal F(\\I_1)   -  \\sum_{i\\in \\I_1} (y_i - X_i^\\top  \\beta^*  _{i} )^2 \\bigg|  = \\bigg|   \\sum_{i\\in \\I_1} (y_i - X_i^\\top \\widehat \\beta _{\\I_1} )^2- \\sum_{i\\in \\I_1} (y_i - X_i^\\top \\beta_i^*   )^2    \\bigg| \\le  C_1  { \\mathfrak{s} }  \\log(np)  .\n \\end{align*}\n If $|\\I_1| <  C_\\zeta  { \\mathfrak{s} }  \\log(np)  $, then\n\\begin{align*} \n& \\bigg|   \\mathcal F(\\I_1)   -  \\sum_{i\\in \\I_1} (y_i - X_i^\\top  \\beta^*  _{i} )^2 \\bigg|   =   \\bigg|     \\sum_{i\\in \\I_1} (y_i - X_i^\\top  \\beta^*  _{i} )^2  \\bigg|  \n=\\sum_{i\\in \\I_1}   \\epsilon_i^2 \n \\\\ \n \\le & |\\I_1| E(\\epsilon^2_1) + C_2 \\sqrt {  |\\I_1|  \\log(n)  } + \\log(n)  \\le C_2'  { \\mathfrak{s} }  \\log(np) .\n \\end{align*} \n So\n \\begin{align*} \n \\bigg|   \\mathcal F(\\I_1)   -  \\sum_{i\\in \\I_1} (y_i - X_i^\\top  \\beta^*  _{i} )^2 \\bigg| \\le   C_3    { \\mathfrak{s} }  \\log(np)  .\n \\end{align*} \n Similarly, \n\\begin{align*} \n&\\bigg|   \\mathcal F(\\I_2)   -  \\sum_{i\\in \\I_2} (y_i - X_i^\\top  \\beta^*  _{i} )^2 \\bigg| \\le   C_3    { \\mathfrak{s} }  \\log(np), \\quad \\text{and}\n\\\\\n&\\bigg|   \\mathcal F(\\I )   -  \\sum_{i\\in \\I } (y_i - X_i^\\top  \\beta^*  _{i} )^2 \\bigg| \\le   C_3    { \\mathfrak{s} }  \\log(np)  .\n \\end{align*} \n So \n $$\\sum_{i\\in \\I_1} (y_i - X_i^\\top \\beta_i^*   )^2  +\\sum_{i\\in \\I_2} (y_i - X_i^\\top \\beta_i^*    )^2 -2C_1   { \\mathfrak{s} }  \\log(np ) +\\gamma    \\le \\sum_{i\\in \\I } (y_i - X_i^\\top \\beta_i^*   )^2 +C_1  { \\mathfrak{s} }   \\log(np) .  $$\n Since $\\beta^* _i$ is unchanged when $i\\in \\I$, it follows that \n $$ \\gamma \\le 3C_1  { \\mathfrak{s} }  \\log(np).$$\n This is a contradiction when $C_\\gamma> 3C_1. $\n \\end{proof} \n  \n  \n  \n  \n   \n \n\n\\subsection{Additional Results Related to Lasso}\n\n\\bnlem \\label{lam:regression one change deviation bound}\nLet $\\mathcal I =(s,e] $ be any generic interval.  \n\\\\\n{\\bf a.} If $\\I$ contains no change points and that \n$|\\I| \\ge C_s  { \\mathfrak{s} }  \\log(np) $ where $C_s$ is the universal constant in \\Cref{lemma:interval lasso}. Then it holds that \n$$\\mathbb P \\bigg( \\bigg|   \\sum_{ i  \\in \\I } (y_i   - X_i ^\\top \\widehat  \\beta _\\I   )^2  - \\sum_{ i   \\in \\I } (y_i      -  X_i^\\top \\beta^*_\\I    )^2  \\bigg|  \\ge C     { \\mathfrak{s} }   \\log(np)  \\bigg)  \\le  n^{-4}. $$\n\\\n\\\\\n{\\bf b.}  Suppose that   the   interval $ \\I=(s,e]$ contains one  and only change point $ \\eta_k $ and that \n$|\\I| \\ge C_s  { \\mathfrak{s} }  \\log(np) $. Denote  $\\widehat \\mu_\\I = \\frac{1}{|\\I | } \\sum_{i\\in \\I } y_i $ and \n$$  \\mathcal J = (s,\\eta_k] \\quad \\text{and} \\quad  \\mathcal J' =  (\\eta_k, e]  .$$   Then it holds that \n$$\\mathbb P \\bigg( \\bigg|   \\sum_{ i  \\in \\I } (y_i   - X_i^\\top \\widehat \\beta_\\I   )^2  - \\sum_{ i   \\in \\I } (y_i      - X_i^\\top \\beta^* _i    )^2  \\bigg|  \\ge  C\\bigg\\{   \\frac{ | {  \\mathcal J } ||{  \\mathcal J } '| }{ |\\I| }  \\kappa_k ^2  +   { \\mathfrak{s} }  \\log(np)\\bigg\\}   \\bigg)   \\le  n^{-4}. $$\n\\enlem\n\\begin{proof} \nWe show {\\bf b} as {\\bf a} immediatelly follows from {\\bf b} with $ |{  \\mathcal J } '| =0$.\n Denote \n$$  \\mathcal J = (s,\\eta_k] \\quad \\text{and} \\quad  \\mathcal J' =  (\\eta_k, e]  .$$ \nDenote  $  \\beta _\\I^*  = \\frac{1}{|\\I | } \\sum_{i\\in \\I } \\beta ^* _i $. Note that \n\\begin{align}  \\nonumber \n   \\bigg|  \\sum_{ i  \\in \\I } (y_i   - X_i^\\top \\widehat \\beta_\\I   )^2  - \\sum_{ i   \\in \\I } (y_i      - X_i^\\top \\beta^* _i    )^2  \\bigg|   \n = &\\bigg| \\sum_{i\\in \\I } \\big\\{  X_i^\\top  ( \\widehat \\beta_\\I  -   \\beta^* _i )   \\big\\} ^2 -  2 \\sum_{i\\in \\I }  \\epsilon_i X_i^\\top ( \\widehat \\beta_\\I   -  \\beta^*_i ) \\bigg| \n \\\\ \\label{eq:regression one change point deviation bound term 1}\n \\le & 2 \\sum_{i\\in \\I } \\big\\{  X_i^\\top  ( \\widehat \\beta_\\I  -    \\beta^*_\\I  )   \\big\\}   ^2 \n \\\\ \\label{eq:regression one change point deviation bound term 2}\n + & 2 \\sum_{i\\in \\I } \\big\\{  X_i^\\top  (    \\beta^*_\\I  -\\beta^*_i  )    \\big\\} ^2  \n \\\\ \\label{eq:regression one change point deviation bound term 3}\n +& 2 \\bigg|    \\sum_{i\\in \\I }  \\epsilon_i X_i^\\top ( \\widehat \\beta_\\I   -  \\beta^*_\\I ) \\bigg|  \n \\\\ \\label{eq:regression one change point deviation bound term 4}\n +&  2 \\bigg|    \\sum_{i\\in \\I }  \\epsilon_i X_i^\\top (   \\beta^*_\\I -\\beta^*_i  ) \\bigg|  .\n\\end{align}\n\\\n\\\\\nSuppose all the good events in \\Cref{lemma:interval lasso} holds. \n\\\\\n\\\\\n{\\bf Step 1.} By \\Cref{lemma:interval lasso},   $\\widehat \\beta_\\I  -    \\beta^*_\\I  $ satisfies the cone condition that \n$$ \\| (\\widehat \\beta_\\I  -    \\beta^*_\\I  )_{S^c}\\|_1 \\le 3 \\| (\\widehat \\beta_\\I  -    \\beta^*_\\I )_S  \\|_1   .$$\nIt follows from \\Cref{corollary:restricted eigenvalues 2} that with probability at least $ 1-n^{-5}$,\n\\begin{align*} \n \\bigg| \\frac{1}{|\\I| } \\sum_{i\\in \\I } \\big\\{  X_i^\\top  ( \\widehat \\beta_\\I  -    \\beta^*_\\I  )   \\big\\} ^2    - ( \\widehat \\beta_\\I  -    \\beta^*_\\I    )^\\top \\Sigma  ( \\widehat \\beta_\\I  -    \\beta^*_\\I  )  \\bigg|  \\le C_1  \\sqrt { \\frac{ { \\mathfrak{s} }  \\log(np) }{|\\I| }} \\| \\widehat \\beta_\\I  -    \\beta^*_\\I     \\|_2 ^2  .\n\\end{align*}\nThe above display   gives\n\\begin{align*}    \\bigg| \\frac{1}{|\\I| } \\sum_{i\\in \\I } \\big\\{  X_i^\\top  ( \\widehat \\beta_\\I  -    \\beta^*_\\I  )    \\big\\}^2 \\bigg| \\le &  \\| \\Sigma\\|_{\\text{op}}   \\| \\widehat \\beta_\\I  -    \\beta^*_\\I     \\|_2 ^2   + C_1  \\sqrt { \\frac{ { \\mathfrak{s} }  \\log(np) }{|\\I| }} \\| \\widehat \\beta_\\I  -    \\beta^*_\\I     \\|_2 ^2 \n\\\\\n\\le & C_x \\| \\widehat \\beta_\\I  -    \\beta^*_\\I     \\|_2 ^2 + C_1  \\sqrt { \\frac{ { \\mathfrak{s} }  \\log(np) }{C_\\zeta  { \\mathfrak{s} }  \\log(np)  }} \\| \\widehat \\beta_\\I  -    \\beta^*_\\I     \\|_2 ^2   \n\\\\\n\\le &  \\frac{ C_2 { \\mathfrak{s} } \\log(np)}{|\\I| }   \\end{align*} \nwhere the second inequality follows from the assumption that  $|\\I| \\ge C_\\zeta  { \\mathfrak{s} }  \\log(np) $ and the last inequality follows from  \\Cref{eq:lemma:interval lasso term 1}  in \\Cref{lemma:interval lasso}.\nThis gives \n$$    \\bigg|   \\sum_{i\\in \\I } \\big\\{  X_i^\\top  ( \\widehat \\beta_\\I  -    \\beta^*_\\I  )    \\big\\}^2 \\bigg| \\le  2 C_2 { \\mathfrak{s} } \\log(np) . $$\n\\\n\\\\\n{\\bf Step 2.}  Observe that \n$ X_i^\\top (\\beta_\\I ^* - \\beta^*_i) $ is Gaussian with mean $0$ and variance \n\\begin{align*} \\omega_i ^2 = (\\beta_\\I ^* - \\beta^*_i )^\\top \\Sigma  (\\beta_\\I ^* - \\beta^*_i)  . \n\\end{align*}\nSince \n$$ \\beta_\\I ^* = \\frac{|{  \\mathcal J } | \\beta^*_{  \\mathcal J }  +|{  \\mathcal J } '| \\beta^*_{{  \\mathcal J } '}   }{|\\I | },$$\nit follows that \n$$ \\omega_i ^2 = \\begin{cases} \n\\bigg(  \\frac{ | {  \\mathcal J } '| (\\beta^*_{  \\mathcal J }  -\\beta^*_{{  \\mathcal J } '}   ) }{ | \\I| }  \\bigg)^\\top  \\Sigma \\bigg(  \\frac{ | {  \\mathcal J } '| (\\beta^*_{  \\mathcal J }  -\\beta^*_{{  \\mathcal J } '}   ) }{ | \\I| }  \\bigg)   \\le   \\frac{| {  \\mathcal J } '| ^2  \\kappa_k^2 }{ |\\I|^2 }   &\\text{when }  i \\in {  \\mathcal J }  ,\n\\\\\n\\bigg(  \\frac{ | {  \\mathcal J } | (\\beta^*_{{  \\mathcal J } '} -\\beta^*_{  \\mathcal J }     ) }{ | \\I| }  \\bigg) ^\\top  \\Sigma \\bigg(  \\frac{ | {  \\mathcal J } '| (  \\beta^*_{{  \\mathcal J } '}  -\\beta^*_{  \\mathcal J }    ) }{ | \\I| }  \\bigg)   \\le   \\frac{| {  \\mathcal J } | ^2  \\kappa_k^2 }{ |\\I|^2 }   &\\text{when }  i \\in {  \\mathcal J } '.\n\\end{cases} $$\n Consequently,\n$   \\{ X_i^\\top (\\beta_\\I ^* - \\beta^*_i)\\} ^2 $ is sub-Exponential with parameter $  \\omega_i ^2$.  \nBy standard sub-Exponential tail bounds, it follows that \n\\begin{align*}\n &\\mathbb P \\bigg(  \\bigg|   \\sum_{ i \\in \\I }\\{ X_i^\\top (\\beta_\\I ^* - \\beta^*_i)\\} ^2 - { \\mathbb{E} } {\\sum_{ i \\in \\I }\\{ X_i^\\top (\\beta_\\I ^* - \\beta^*_i)\\} ^2}    \\bigg| \\ge C_3 \\tau    \\bigg)  \n \\\\\n \\le &  \\exp\\bigg (-c \\min\\bigg \\{ \\frac{  \\tau^2}{\\sum_{i\\in \\I } \\omega_i ^4  }  ,  \\frac{ \\tau }{ \\max_{i\\in \\I }   \\omega_i ^2 } \\bigg\\}  \\bigg)\n \\\\\n \\le &  \\exp\\bigg (-c' \\min\\bigg \\{ \\frac{  \\tau^2}{\\sum_{i\\in \\I } \\omega_i ^2  }  ,  \\frac{ \\tau }{ \\max_{i\\in \\I }   |\\omega_i | } \\bigg\\}  \\bigg)\n \\\\\n \\le & \\exp\\bigg (-c ''\\min\\bigg \\{  \\tau^2  \\bigg( \\frac{|\\I |   }{| {  \\mathcal J } '| |{  \\mathcal J } |  } \\kappa_k^{-2}  \\bigg)    ,    \\tau   \\frac{|\\I| }{ \\max \\{|{  \\mathcal J } |, |{  \\mathcal J } '|  \\}}   \\kappa_k ^{-1} \\bigg\\}  \\bigg) ,\n \\end{align*}\nwhere the second inequality follows from the observation that \n$$  \\omega_i^2     \\le \\kappa_k |\\omega_i| \\le  C_\\kappa  |\\omega_i| \\text{ for all } i \\in \\I , $$\n and the last inequality follows from the observation that \n\\begin{align*}\n \\sum_{i\\in \\I} \\omega_i ^2    \n \\le  C_x | {  \\mathcal J } | \\frac{| {  \\mathcal J } '| ^2  \\kappa_k^2 }{ |\\I|^2 } + C_x | {  \\mathcal J }  ' | \\frac{| {  \\mathcal J } | ^2  \\kappa_k^2 }{ |\\I|^2 }  \n=   C_x \\frac{| {  \\mathcal J } '| |{  \\mathcal J } | }{|\\I | } \\kappa_k^2 .\n \\end{align*} \n So there exists a sufficiently large constant $C_4$ such that with probability at  least $1- n^{-5}$, \n\\begin{align*}\n &  \\bigg|   \\sum_{ i \\in \\I }\\{ X_i^\\top (\\beta_\\I ^* - \\beta^*_i)\\} ^2 - { \\mathbb{E} } {\\sum_{ i \\in \\I }\\{ X_i^\\top (\\beta_\\I ^* - \\beta^*_i)\\} ^2}    \\bigg|\n  \\\\ \n   \\le &  C_ 4   \\bigg\\{\\sqrt { \\frac{| {  \\mathcal J } '| |{  \\mathcal J } | }{|\\I | }  \\log(n)  \\kappa _k ^2  } + \\log(n)    \\frac{\\max \\{|{  \\mathcal J } |, |{  \\mathcal J } '|  \\}  }{|\\I| } \\kappa _k  \\bigg\\}     \n \\\\\n \\le &  C_ 4'  \\bigg\\{      \\frac{| {  \\mathcal J } '| |{  \\mathcal J } | }{|\\I | }  \\kappa_k^2  + \\log(n)   +  \\log(n)    \\frac{\\max \\{|{  \\mathcal J } |, |{  \\mathcal J } '|  \\}  }{|\\I| } \\kappa _k  \\bigg\\}      \n \\\\\n \\le &C_5  \\bigg\\{      \\frac{| {  \\mathcal J } '| |{  \\mathcal J } | }{|\\I | }  \\kappa_k^2  + \\log(n)     \\bigg\\}      \n \\end{align*}\n where $ \\kappa_k \\asymp \\kappa \\le C_\\kappa $ is used in the last inequality.\nSince $  { \\mathbb{E} } {\\sum_{ i \\in \\I }\\{ X_i^\\top (\\beta_\\I ^* - \\beta^*_i)\\} ^2} = \\sum_{i\\in \\I }\\omega_i ^2 \\le C_x \\frac{| {  \\mathcal J } '| |{  \\mathcal J } | }{|\\I | } \\kappa_k^2$, it follows that \n$$ \\mathbb P \\bigg (  \\bigg|   \\sum_{ i \\in \\I }\\{ X_i^\\top (\\beta_\\I ^* - \\beta^*_i)\\} ^2    \\bigg|  \\le ( C_5 +C_x )  \\frac{| {  \\mathcal J } '| |{  \\mathcal J } | }{|\\I | }  \\kappa_k^2  + C_5 \\log(n) \\bigg ) \\ge 1-n^{-5}.$$\n\\\n\\\\\n{\\bf Step 3.} For \\Cref{eq:regression one change point deviation bound term 3},  it follows that with probability at least $1-2n^{-4}$\n\\begin{align*} \n   \\frac{1}{|\\I| } \\sum_{i\\in \\I }  \\epsilon_i X_i^\\top ( \\widehat \\beta_\\I   -  \\beta^*_\\I )\n  \\le  C_6 \\sqrt { \\frac{\\log(np) }{ |\\I| } } \\|  \\widehat \\beta_\\I   -  \\beta^*_\\I \\|_1 \\le C_7  \\frac{ { \\mathfrak{s} }  \\log(np) }{ |\\I| } \n\\end{align*}\nwhere the first inequality is a consequence of \\Cref{eq:independent condition 1c 1}, the second inequality follows from  \\Cref{eq:lemma:interval lasso term 3} in \\Cref{lemma:interval lasso}. \n\\\n\\\\\n\\\\\n{\\bf Step 4.}  From {\\bf Step 2}, we have  that\n$ X_i^\\top (\\beta_\\I ^* - \\beta^*_i) $ is Gaussian with mean $0$ and variance \n$$ \\omega_i ^2 = \\begin{cases} \n\\bigg(  \\frac{ | {  \\mathcal J } '| (\\beta^*_{  \\mathcal J }  -\\beta^*_{{  \\mathcal J } '}   ) }{ | \\I| }  \\bigg)^\\top  \\Sigma \\bigg(  \\frac{ | {  \\mathcal J } '| (\\beta^*_{  \\mathcal J }  -\\beta^*_{{  \\mathcal J } '}   ) }{ | \\I| }  \\bigg)   \\le   \\frac{| {  \\mathcal J } '| ^2  \\kappa_k^2 }{ |\\I|^2 }   &\\text{when }  i \\in {  \\mathcal J }  ,\n\\\\\n\\bigg(  \\frac{ | {  \\mathcal J } | (\\beta^*_{{  \\mathcal J } '} -\\beta^*_{  \\mathcal J }     ) }{ | \\I| }  \\bigg) ^\\top  \\Sigma \\bigg(  \\frac{ | {  \\mathcal J } '| (  \\beta^*_{{  \\mathcal J } '}  -\\beta^*_{  \\mathcal J }    ) }{ | \\I| }  \\bigg)   \\le   \\frac{| {  \\mathcal J } | ^2  \\kappa_k^2 }{ |\\I|^2 }   &\\text{when }  i \\in {  \\mathcal J } '.\n\\end{cases} $$\n Consequently,\n$   \\epsilon_i  X_i^\\top (\\beta_\\I ^* - \\beta^*_i)  $ is   centered  sub-Exponential with parameter $  \\omega_i \\sigma_\\epsilon  $.  \nBy standard sub-Exponential tail bounds, it follows that \n\\begin{align*}\n &\\mathbb P \\bigg(  \\bigg|   \\sum_{ i \\in \\I } \\epsilon_i  X_i^\\top (\\beta_\\I ^* - \\beta^*_i)       \\bigg| \\ge C_8 \\tau    \\bigg)  \n \\\\\n \\le &  \\exp\\bigg (-c \\min\\bigg \\{ \\frac{  \\tau^2}{\\sum_{i\\in \\I } \\omega_i ^2  }  ,  \\frac{ \\tau }{ \\max_{i\\in \\I }  |\\omega_i| } \\bigg\\}  \\bigg)\n \\\\\n \\le & \\exp\\bigg (-c '\\min\\bigg \\{  \\tau^2  \\bigg( \\frac{|\\I |   }{| {  \\mathcal J } '| |{  \\mathcal J } |  } \\kappa_k^{-2}  \\bigg)    ,    \\tau   \\frac{|\\I| }{ \\max \\{|{  \\mathcal J } |, |{  \\mathcal J } '|  \\}}   \\kappa_k ^{-1} \\bigg\\}  \\bigg) ,\n \\end{align*}\nwhere the last inequality follows from the observation that \n\\begin{align*}\n \\sum_{i\\in \\I} \\omega_i ^2    \n \\le  C_x | {  \\mathcal J } | \\frac{| {  \\mathcal J } '| ^2  \\kappa_k^2 }{ |\\I|^2 } + C_x | {  \\mathcal J }  ' | \\frac{| {  \\mathcal J } | ^2  \\kappa_k^2 }{ |\\I|^2 }  \n=   C_x \\frac{| {  \\mathcal J } '| |{  \\mathcal J } | }{|\\I | } \\kappa_k^2 .\n \\end{align*} \n So there exists a sufficiently large constant $C_9$ such that with probability at  least $1- n^{-5}$, \n\\begin{align*}\n &  \\bigg|   \\sum_{ i \\in \\I } \\epsilon_i X_i^\\top (\\beta_\\I ^* - \\beta^*_i) \\bigg|\n  \\\\ \n   \\le &  C_ 9   \\bigg\\{\\sqrt { \\frac{| {  \\mathcal J } '| |{  \\mathcal J } | }{|\\I | }  \\log(n)  \\kappa _k ^2  } + \\log(n)    \\frac{\\max \\{|{  \\mathcal J } |, |{  \\mathcal J } '|  \\}  }{|\\I| } \\kappa _k  \\bigg\\}     \n \\\\\n \\le &  C_ 9'  \\bigg\\{      \\frac{| {  \\mathcal J } '| |{  \\mathcal J } | }{|\\I | }  \\kappa_k^2  + \\log(n)   +  \\log(n)    \\frac{\\max \\{|{  \\mathcal J } |, |{  \\mathcal J } '|  \\}  }{|\\I| } \\kappa _k  \\bigg\\}      \n \\\\\n \\le &C_9  \\bigg\\{      \\frac{| {  \\mathcal J } '| |{  \\mathcal J } | }{|\\I | }  \\kappa_k^2  + \\log(n)     \\bigg\\}      \n \\end{align*}\n where $ \\kappa_k \\asymp \\kappa \\le C_\\kappa $ is used in the last inequality.\n\\end{proof} \n\n \n\\bnlem \\label{lemma:interval lasso} Suppose  \\Cref{assp:dcdp_linear_reg} holds. Let $$ \\widehat \\beta _{ \\mathcal I } = \\arg\\min_{\\beta \\in \\mathbb R^p }  \\frac{1}{ |\\mathcal I |  }\\sum_{i \\in \\mathcal I}   (y_i -X_i^\\top \\beta) ^2 + \\lambda \\|\\beta\\|_1$$ with $\\lambda = C_\\lambda (\\sigma_{\\epsilon} \\vee 1)\\sqrt { \\log(np) }$ for some sufficiently large constant $C _ \\lambda $. There exists a sufficiently large constant $ C_s$ such that for all   $ \\mathcal I  \\subset (0, n] $   such that $| \\mathcal I | \\ge C_s { \\mathfrak{s} }  \\log(np)$, it holds with probability at least $ 1-(np)^{-3}$ that \n\\begin{align} \\label{eq:lemma:interval lasso term 1}\n&   \\| \\widehat \\beta_\\I -\\beta^*_\\I \\| _2^2 \\le \\frac{C (\\sigma_{\\epsilon}^2\\vee 1) { \\mathfrak{s} } \\log(np)}{ |\\I  |  } ;\n \\\\ \\label{eq:lemma:interval lasso term 2}\n &     \\| \\widehat \\beta_\\I  -\\beta^*_\\I \\| _1  \\le     C(\\sigma_{\\epsilon}\\vee 1)  { \\mathfrak{s} } \\sqrt {  \\frac{\\log(np)} {|\\I  |  } }  ;\n \\\\ \\label{eq:lemma:interval lasso term 3}\n &    \\| (\\widehat \\beta_\\I -\\beta^*_\\I)_{S^c} \\| _1 \\le  3 \\| (\\widehat \\beta _\\I  -\\beta^*_\\I )_{S } \\| _1   .\n\\end{align} \nwhere $ \\beta^*_\\I  = \\frac{1}{|\\I|} \\sum_{i\\in \\I } \\beta^*_i $.\n \\enlem \n\\begin{proof}  Denote $S=\\bigcup_{k=1}^K S_{\\eta_k+1} $. Since $K< \\infty$, \n$|S| \\asymp  { \\mathfrak{s} } .$ \n  It follows from the definition of $\\widehat{\\beta}_\\I$ that \n\t\\[\n\t \\frac{1}{|\\I | }\t\\sum_{ i \\in I} (y_i - X_i^{\\top}\\widehat{\\beta}_\\I )^2 + \\frac{ \\lambda}{ \\sqrt{  |\\I|  }}    \\|\\widehat{\\beta} _\\I\\|_1 \\leq  \n\t \\frac{1}{|\\I | } \\sum_{t \\in \\I} (y_i - X_i^{\\top}\\beta^*_\\I )^2 + \\frac{ \\lambda}{ \\sqrt{  |\\I|  }}  \\|\\beta^*_\\I  \\|_1.\n\t\\]\n\tThis gives \n\t\\begin{align*}\n\t\t \\frac{1}{|\\I | } \\sum_{i \\in \\I} \\bigl\\{X_i ^{\\top}(\\widehat{\\beta}_\\I - \\beta^*_\\I )\\bigr\\}^2 +    \\frac{ 2 }{|\\I | }   \\sum_{ i \\in \\I}(y_i - X_i^{\\top}\\beta^*_\\I)X_i^{\\top}(\\beta^*_\\I  - \\widehat{\\beta}_\\I )  +  \\frac{ \\lambda}{ \\sqrt{  |\\I|  }}   \\bigl\\|\\widehat{\\beta}_\\I \\bigr\\|_1 \n\t\t\\leq  \\frac{ \\lambda}{ \\sqrt{  |\\I|  }}    \\bigl\\|\\beta^*_\\I \\bigr\\|_1,\n\t\\end{align*}\n\tand therefore\n\t\\begin{align}\n\t\t& \\frac{1}{|\\I | }\t  \\sum_{i  \\in \\I} \\bigl\\{X_i ^{\\top}(\\widehat{\\beta} _\\I  - \\beta^*_\\I )\\bigr\\}^2 + \\frac{ \\lambda}{ \\sqrt{  |\\I|  }} \\bigl\\|\\widehat{\\beta}_\\I \\bigr\\|_1  \\nonumber \\\\\n\t\t \\leq  &  \\frac{ 2}{|\\I | }\t \\sum_{i \\in  \\I } \\epsilon_i  X_i ^{\\top}(\\widehat{\\beta}_\\I  - \\beta^*_\\I ) \n+ 2   (\\widehat{\\beta}_\\I  - \\beta^*_\\I )^{\\top}\\frac{  1 }{|\\I | }\t \\sum_{i\\in   \\I} X_ i  X_i^{\\top}(    \\beta^*_i -\\beta^*_\\I   )\t\t \n\t\t+  \\frac{ \\lambda}{ \\sqrt{  |\\I|  }}    \\bigl\\|\\beta^*_\\I \\bigr\\|_1   .\t \\label{eq-lem10-pf-2}\n\t\\end{align}\nTo bound\t\n\t\t$\\left\\|\\sum_{ i  \\in \\I} X_ i  X_i ^\\top (\\beta^*  _\\I  -\\beta^*_i ) \\right\\|_{\\infty},  \n$\n\tnote  that  for any $j \\in  \\{ 1, \\ldots, p\\}$, the $j$-th entry of \n\t\\\\$\\sum_{i \\in \\I} X _ i X _i^\\top (\\beta^*_\\I   -\\beta_i )$ satisfies  \n\t\\begin{align*}\n\t\t  E \\left\\{\\sum_{ i  \\in \\I}   X_i (j) X _ i ^\\top (\\beta^*_\\I   - \\beta^*_ i  )\\right\\} = \\sum_{ i \\in \\I}  E \\{X _ i (j  ) X_i ^\\top \\}\\{\\beta^*_\\I  - \\beta^*_ i \\}  \n\t\t=     \\mathbb{E}\\{X _1( j ) X _1 ^\\top \\} \\sum_{ i \\in \\I}\\{\\beta^*_\\I    - \\beta^*_ i  \\} = 0.\n\t\\end{align*}\n\tSo   $  E\\{ \\sum_{ i  \\in \\I} X_ i  X_i ^\\top (\\beta^*  _\\I  -\\beta^*_i )\\} =0 \\in \\mathbb R^p.$ \nBy \\Cref{lemma:consistency}{\\bf b},\n\t\\begin{align*}\n\t  \\bigg|  (    \\beta^*_i -\\beta^*_\\I   ) ^\\top  \\frac{  1 }{|\\I | }\t \\sum_{i\\in   \\I} X_t X_t^{\\top} (\\widehat{\\beta}_\\I  - \\beta^*_\\I )\t \t   \\bigg| \\le & C_1 \\big(\\max_{1\\le i \\le n }  \\|\\beta^*_i -\\beta^*_\\I \\|_2 \\big) \\sqrt { \\frac{\\log(np) }{ | \\I|  }} \\| \\widehat{\\beta}_\\I  - \\beta^*_\\I  \\|_1   \n\t  \\\\\n\t  \\le & C_2 \\sqrt { \\frac{\\log(np) }{ | \\I| }} \\| \\widehat{\\beta}_\\I  - \\beta^*_\\I  \\|_1  \n\t  \\\\\n\t  \\le &  \\frac{\\lambda}{8\\sqrt { |\\I| } }   \\|\\widehat{\\beta}_\\I  - \\beta^*_\\I \\|_1 \n \t\\end{align*}\n \twhere the second inequality follows from \\Cref{lemma:beta bounded 1} and the last inequality follows from $\\lambda = C_\\lambda\\sigma_{\\epsilon} \\sqrt { \\log(np) }$ with sufficiently large constant $C_\\lambda$.\n\tIn addition  by \\Cref{lemma:consistency}{\\bf a},\n\t\\begin{equation*}\n\t    \\frac{ 2}{|\\I | }\t \\sum_{i \\in  \\I } \\epsilon_i  X_i ^{\\top}(\\widehat{\\beta}_\\I  - \\beta^*_\\I )  \\le C  \\sigma_{\\epsilon}\\sqrt { \\frac{ \\log(np) }{ |\\I| } }\\|\\widehat{\\beta}_\\I  - \\beta^*_\\I \\|_1  \\le  \\frac{\\lambda}{8\\sqrt { |\\I| } }   \\|\\widehat{\\beta}_\\I  - \\beta^*_\\I \\|_1 .\n\t\\end{equation*}\n\tSo   \\eqref{eq-lem10-pf-2}  gives \n\t\\begin{align*}\n\t\t  \\frac{1}{|\\I | }\t  \\sum_{i  \\in \\I} \\bigl\\{X_i ^{\\top}(\\widehat{\\beta} _\\I  - \\beta^*_\\I )\\bigr\\}^2 + \\frac{ \\lambda}{ \\sqrt{  |\\I|  }} \\bigl\\|\\widehat{\\beta}_\\I \\bigr\\|_1  \n\t\t \\leq  \\frac{\\lambda}{4\\sqrt { |\\I| } }   \\|\\widehat{\\beta}_\\I  - \\beta^*_\\I \\|_1  \n\t\t+  \\frac{ \\lambda}{ \\sqrt{  |\\I|  }}    \\bigl\\|\\beta^*_\\I \\bigr\\|_1   .\t \n\t\\end{align*} \nLet $\\Theta = \\widehat  \\beta _\\I   - \\beta^*_\\I   $. The above inequality implies\n\\begin{align}\n\\label{eq:two sample lasso deviation 1} \\frac{1}{|\\I|} \\sum_{i \\in \\I } \\left( X_i^\\top  \\Theta \\right)^2 + \\frac{ \\lambda}{2\\sqrt{  |\\I|  } }\\|  (\\widehat \\beta _\\I)   _{ S ^c}\\|_1  \n \\le & \\frac{3\\lambda}{2\\sqrt{  |\\I|  }  } \\| ( \\widehat  \\beta _\\I  -  \\beta^*_\\I  )    _{S} \\| _1  ,\n \\end{align}  \nwhich also implies that \n$$ \\frac{\\lambda }{2}\\| \\Theta _{S^c}  \\|_1 = \\frac{ \\lambda}{2 }\\|   (\\widehat \\beta_\\I)  _{ S ^c}\\|_1    \\le  \n\\frac{3\\lambda}{2 } \\|  ( \\widehat  \\beta _\\I   - \\beta^* _\\I  ) _{S} \\| _1  = \\frac{3\\lambda}{2 } \\|  \\Theta  _{S} \\| _1 . $$\nThe above inequality and  \\Cref{corollary:restricted eigenvalues 2} imply that with probability at least $1-n^{-5}$,\n$$ \\frac{1}{|\\I| } \\sum_{i \\in \\I }  \\left( X_i^\\top  \\Theta \\right)^2  = \\Theta^\\top \\widehat \\Sigma _\\I  \\Theta  \n\\ge \n \\Theta^\\top \\Sigma \\Theta - C_3\\sqrt{ \\frac{  { \\mathfrak{s} } \\log(np)}{ |\\I| }}  \\| \\Theta\\|_2^2   \\ge  \\frac{c_x}{2} \\|\\Theta\\|_2 ^2   ,$$\n where the last inequality follows from the assumption that $| \\mathcal I | \\ge C_s  { \\mathfrak{s} }  \\log(np)   $ for sufficiently large $C_s$.\nTherefore \\Cref{eq:two sample lasso deviation 1} gives\n\\begin{align}\n\\label{eq:two sample lasso deviation 2}  c'\\|\\Theta\\|_2 ^2  + \\frac{ \\lambda}{2\\sqrt{  |\\I|  }  }\\| ( \\widehat \\beta_\\I  - \\beta^*_\\I ) _{ S ^c}\\|_1  \n \\le   \\frac{3\\lambda}{2\\sqrt{  |\\I|  }  } \\| \\Theta_{S} \\| _1  \\le \\frac{3\\lambda \\sqrt  { \\mathfrak{s} }  }{2 \\sqrt{  |\\I|  } } \\| \\Theta  \\| _2 \n \\end{align}  \n and so \n $$ \\|\\Theta\\|_2  \\le   \\frac{C \\lambda \\sqrt   { \\mathfrak{s} } }{\\sqrt{| \\I |} } . $$ \n The above display   gives  \n $$  \\| \\Theta_{S} \\| _1 \\le  \\sqrt { { \\mathfrak{s} } }  \\| \\Theta_{S} \\| _2 \\le   \\frac{C\\lambda  { \\mathfrak{s} } }{\\sqrt{|\\I| }}. $$ \n Since \n  $ \\| \\Theta_{S^c } \\| _1 \\le 3 \\| \\Theta_{S} \\| _1 ,$ \nit also holds that \n$$\\| \\Theta  \\| _1 = \\| \\Theta_{S  } \\| _1 +\\| \\Theta_{S^c } \\| _1  \\le 4 \\| \\Theta_{S} \\| _1 \\le  \\frac{4C\\lambda  { \\mathfrak{s} } }{\\sqrt{|\\I|} } .$$\n \\end{proof}\n\n \n \n\n \n \n\\subsection{Additional Technical Results} \n\\bnlem\n\\label{theorem:restricted eigenvalues 2}\nSuppose  $\\{X_{i } \\}_{1\\le i \\le n } \\overset{i.i.d.} {\\sim}  N_p (0, \\Sigma )  $. \nDenote $\\mathcal C_S  := \\{ v : \\mathbb R^p : \\| v_{S^c }\\|_1 \\le 3\\| v_{S}\\|_1 \\}  $, where $ |S| \\le  { \\mathfrak{s} }  $.\nThen there exists constants $c$ and $C$ such that for all $\\eta\\le 1$, \n\\begin{align} \nP \\left( \\sup_{v \\in \\mathcal C _S , \\|v\\|_2=1   } \\left | v^\\top ( \\widehat \\Sigma -\\Sigma )  v \\right| \\ge C\\eta   \\Lambda_{\\max} (\\Sigma)   \\right)\n\\le 2\\exp( -c n\\eta ^2 + 2 { \\mathfrak{s} }  \\log(p) ) .\n\\end{align}\n\\enlem\n\\begin{proof}\nThis is a well known restricted eigenvalue property for Gaussian design. The proof can be found  in \\cite{Basu2015} or \\cite{Loh2012}.\n\\end{proof}  \n\n\\bnlem\n\\label{corollary:restricted eigenvalues 2} Suppose  $\\{X_{i } \\}_{1\\le i \\le n } \\overset{i.i.d.} {\\sim}  N_p (0, \\Sigma )  $. \n Denote $\\mathcal C_S  := \\{ v : \\mathbb R^p : \\| v_{S^c }\\|_1 \\le 3\\| v_{S}\\|_1 \\}  $, where $ |S| \\le  { \\mathfrak{s} }  $.  With probability at least $1- n ^{-5}$, it holds that\n$$  \\left | v^\\top ( \\widehat \\Sigma_\\I  -\\Sigma )  v \\right| \\le C \\sqrt { \\frac{ { \\mathfrak{s} } \\log(np) }{|\\I| } }     \\|v\\|_2^2   \n $$\n   for all $v \\in \\mathcal C _S $ and all $\\I \\subset  (0, n]$ such that $| \\I |\\ge C_s  { \\mathfrak{s} } \\log(np)  $, where $ C_s$\nis the constant in \\Cref{lemma:consistency} which is independent of $n, p$.   \n\\enlem\n\\begin{proof} \nFor any $\\I \\subset  (0, n]$ such that $| \\I |\\ge C_s { \\mathfrak{s} }  \\log(np)   $, by \\Cref{theorem:restricted eigenvalues 2}, it holds that \n\\begin{align*} \nP \\left( \\sup_{v \\in \\mathcal C _S , \\|v\\|_2=1   } \\left | v^\\top ( \\widehat \\Sigma _\\I -\\Sigma )  v \\right| \\ge C\\eta   \\Lambda_{\\max} (\\Sigma)   \\right)\n\\le 2\\exp( -c   |\\I| \\eta ^2 + 2 { \\mathfrak{s} }  \\log(p) ) .\n\\end{align*}\nLet  $\\eta= C_1 \\sqrt { \\frac{ { \\mathfrak{s} } \\log(np) }{|\\I | } } $ for sufficiently large constant $C_1$. Note that $\\eta< 1$ if $ |\\I| > C_1^2  { \\mathfrak{s} } \\log(np) $. Then   with probability at least \n$(np)^{-7} $, \n$$\\sup_{v \\in \\mathcal C _S , \\|v\\|_2=1   } \\left | v^\\top ( \\widehat \\Sigma _\\I -\\Sigma )  v \\right| \\ge C_2 \\sqrt { \\frac{s\\log(np) }{|\\I |  } } .$$\nSince there are at most $n^2$ many different  choices of $ \\I \\subset (0,n]$, the desired result follows from a union bound argument. \n\\end{proof}  \n\n\n\\bnlem\n\\label{lem:restricted eigenvalue}\nUnder \\Cref{assp:dcdp_linear_reg}, it holds that\n \\begin{align*}\n\t\t \\mathbb P \\bigg( \\sum_{t\\in \\I } ( X_t^\\top v )^2 \\ge \\frac{c_x|\\I| }{4} \\|v\\|_2^2 - C_2 \\log(np) \\|v\\|_1 ^2 \\ \\forall v \\in \\mathbb R^p  \\text{ and } \\forall |\\I|\\ge C_s  { \\mathfrak{s} }  \\log(np)  \\bigg)  \\le  n^{-5} \n\t\\end{align*} \n\twhere $  C_2 > 0$ is an  absolute constant  only depending on $C_x$.\n\\enlem\n\\begin{proof}\nBy the well known restricted eigenvalue condition, for any $\\I$, it holds that \n \\begin{align*}\n\t\t \\mathbb P \\bigg( \\sum_{t\\in \\I } ( X_t^\\top v )^2 \\ge \\frac{c_x |\\I| }{4} \\|v\\|_2^2 - C_2 \\log(np) \\|v\\|_1 ^2  \\ \\  \\forall v \\in \\mathbb R^p    \\bigg)  \\le  C_3 \\exp(-c_3|\\I| ).\n\t\\end{align*} \n  Since $|\\I|  \\ge  C_s  { \\mathfrak{s} }  \\log(np) $, \n  \\begin{align*}\n\t\t \\mathbb P \\bigg( \\sum_{t\\in \\I } ( X_t^\\top v )^2 \\ge \\frac{c_ x |\\I| }{4} \\|v\\|_2^2 - C_2 \\log(np) \\|v\\|_1^2 \\ \\ \\forall v \\in \\mathbb R^p    \\bigg)  \\le  n^{-4}.\n\t\\end{align*} \n\tSince there are at most $n^2$ many subinterval $\\I \\subset (0,n]$, it follows from a union bound argument  that \n\t \\begin{align*}\n\t\t \\mathbb P \\bigg( \\sum_{t\\in \\I } ( X_t^\\top v )^2 \\ge \\frac{c_x |\\I| }{4} \\|v\\|_2^2 - C_2 \\log(np) \\|v\\|_1^2  \\ \\ \\forall v \\in \\mathbb R^p  \\text{ and } \\forall |\\I|\\ge C_s  { \\mathfrak{s} }  \\log(np)  \\bigg)  \\le  n^{-2} .\n\t\\end{align*} \n\\end{proof} \n\n\n\n\n\\bnlem \\label{lemma:consistency}\nSuppose  \\Cref{assp:dcdp_linear_reg} holds. There exists  a sufficient large constant $ C_s$ such that the following conditions holds.    \n\\\\\n\\\\\n{\\bf a.} With probability at least $1-n^{-3} $, it holds that\n\\begin{align}\\label{eq:independent condition 1c 1}\n   \\left |  \\frac{1}{|\\I | } \\sum_{i \\in \\mathcal I }  \\epsilon_i  X_i^\\top \\beta \\right| \\le C\\sigma_{\\epsilon}\\sqrt { \\frac{\\log(np)}{|\\I |   } } \\|\\beta\\|_1      \n \\end{align}\nuniformly  for all $ \\beta \\in \\mathbb R^p$ and all $\\mathcal I \\subset (0, n]$ such that $|\\I|\\ge C_s  { \\mathfrak{s} }  \\log(np) $, \n\\\\\n\\\\\n {\\bf b.} Let $\\{ u_i\\}_{i=1}^n \\subset  \\mathbb R^p$  be a collection of  deterministic vectors. Then with probability at least $1-n^{-3} $, it holds that   \n\\begin{align}\\label{eq:independent condition 1c}\n       \\left |  \\frac{1}{|\\I |   } \\sum_{i \\in \\I }    u^\\top_i   X_i   X_i^\\top \\beta  - \\frac{1}{ |\\I |  } \\sum_{i \\in \\mathcal I }   u^\\top  _i  \\Sigma \\beta  \\right| \\le C \\left(  \\max _{1\\le i \\le n }\\|u_i\\|_2  \\right) \\sqrt {  \\frac{\\log(np)}{|\\I |   } } \\|\\beta\\|_1    \n \\end{align}\n uniformly for all $ \\beta \\in \\mathbb R^p$ and all $\\mathcal I \\subset (0, n]$ such that $|\\I|\\ge  C_s  { \\mathfrak{s} }  \\log(np) $.    \n\\enlem \n \\begin{proof}\n   The justification   of the  \\eqref{eq:independent condition 1c 1}  is   similar and simpler  than the justification  of \\eqref{eq:independent condition 1c}. For conciseness, only the justification  of \\eqref{eq:independent condition 1c} is presented.\n   \nFor any $\\mathcal I \\subset (0, n]$  such  that $|\\I|\\ge  C_s  { \\mathfrak{s} }  \\log(np) $, it holds  that\n \\begin{align*}\n &\\left|  \\frac{1}{ |\\I |  } \\sum_{i\\in \\I }  u^\\top  _i  X_i   X_i^\\top \\beta   - \\frac{1}{ |\\I |  }  \\sum_{i\\in \\I }   u^\\top_i  \\Sigma  \\beta \\right| \n \\\\\n =\n & \\left|  \\left(  \\frac{1}{|\\I |  } \\sum_{i\\in \\I }   u^\\top _i    X_i   X_i^\\top     - \\frac{1}{|\\I |  }    \\sum_{i\\in \\I }   u^\\top_i  \\Sigma  \\right) \\beta  \\right| \n \\\\\n \\le \n & \\max_{1\\le j \\le p }  \\left |  \\frac{1}{|\\I |  } \\sum_{i\\in \\I }   u^\\top  _i   X_i   X_{i,j}      - \\frac{1}{|\\I |  }    \\sum_{i\\in \\I }   u^\\top_i  \\Sigma  (, j)  \\right| \\|\\beta\\|_1.\n \\end{align*}\n Note that \n $ E(u^\\top  _i  X_i   X_{i,j} ) =u^\\top _i \\Sigma  (, j)    $ and in addition, \n $$u^\\top _ i   X_i   \\sim N(0, u^\\top _i  \\Sigma u _i )  \\quad \\text{ and } \\quad X_{i,j} \\sim N(0, \\Sigma({j,j}))  .$$\n So $u^\\top _i   X_i X_{i,j} $  is a sub-exponential random variable such that \n $$ u^\\top _i   X_i X_{i,j} \\sim SE( u^\\top_i  \\Sigma u _i  \\Sigma ({j,j})).$$\n As a result, for $\\gamma<1$ and every $j$,\n$$ P \\left( \\left |  \\frac{1}{|\\I |  } \\sum_{i \\in \\I }  u^\\top _ i     X_i   X_{i,j}      - u^\\top \\Sigma  (, j)  \\right|  \\ge \\gamma \\sqrt {  \\max_{1\\le i\\le n } ( u^\\top_i  \\Sigma u_i)  \\Sigma ( {j,j} ) }\\right) \\le \\exp(-c \\gamma^2 |\\I |  ). $$\nSince \n$$  \\sqrt { u^\\top _i  \\Sigma u _i \\Sigma ({j,j} ) } \\le C_x  \\|u_i \\|_2, $$\nby union bound,\n$$ \\mathbb P \\left(  \\max_{1\\le j \\le p }\\left |  \\frac{1}{|\\I |  } \\sum_{i \\in \\I}  u^\\top _ i     X_i   X_{i,j}      - \\frac{1}{|\\I |  } \\sum_{i \\in\\I}  u^\\top _i  \\Sigma  (, j)  \\right|  \\ge \\gamma \nC_x  \\left(  \\max _{1\\le i \\le n }\\|u_i\\|_2  \\right)  \\right) \\le p\\exp(-c \\gamma^2  |\\I| ). $$\nLet $\\gamma =  3\\sqrt { \\frac{\\log(np) }{c  |\\I| }  }$. Note that   $ \\gamma<1$ if  $|\\I|\\ge C_s  { \\mathfrak{s} }  \\log(np) $ for sufficiently large $C_s$.  Therefore \n$$ \\mathbb P \\left(  \\max_{1\\le j \\le p }\\left |  \\frac{1}{|\\I |  } \\sum_{i \\in \\I}  u^\\top _ i     X_i   X_{i,j}      - \\frac{1}{|\\I |  } \\sum_{i \\in\\I}  u^\\top _i  \\Sigma  (, j)  \\right|  \\ge  \nC_1\\sqrt{ \\frac{ \\log(np)}{|\\I| } } \\left(  \\max _{1\\le i \\le n }\\|u_i\\|_2  \\right)  \\right) \\le p\\exp(-9\\log(np)  ). $$\nSince there are at most $n^2$ many intervals $\\I \\subset(0,n]$, it follows that \n\\begin{align*}\n &\\mathbb P \\left(  \\max_{1\\le j \\le p }\\left |  \\frac{1}{|\\I |  } \\sum_{i \\in \\I}  u^\\top _ i     X_i   X_{i,j}      - \\frac{1}{|\\I |  } \\sum_{i \\in\\I}  u^\\top _i  \\Sigma  (, j)  \\right|  \\ge  \nC_1\\sqrt{ \\frac{ \\log(np)}{|\\I| } } \\left(  \\max _{1\\le i \\le n }\\|u_i\\|_2   \\right) \\ \\forall    |\\I| \\ge C_s  { \\mathfrak{s} }  \\log(np)  \\right)\n\\\\\n \\le & p n^2 \\exp(-9\\log(np)  ) \\le n^{-3}. \n\\end{align*}\nThis immediately  gives \\eqref{eq:independent condition 1c}.\n \\end{proof} \n \n \n \n \n \\bnlem\\label{lem-x-bound}\nUder \\Cref{assp:dcdp_linear_reg}, for any interval $\\I \\subset (0, n]$, for any \n\t\\[\n\t\t\\lambda \\geq \\lambda_1 := C_{\\lambda}\\sigma_{\\epsilon} \\sqrt{\\log(n p)},\n\t\\]\n\twhere $C_{\\lambda} > 0$ is a large enough absolute constant, it holds with probability at least $1 - n^{-5}$ that\n\t\\[\n    \\|\\sum_{i \\in \\I} \\epsilon_i X_i \\|_{\\infty} \\leq \\lambda \\sqrt{\\max\\{|\\I|, \\, \\log(n p)\\}}\/8,\n\t\\]\n\twhere $c_3 > 0$ is an absolute constant depending only on the distributions of covariants $\\{X_i\\}$ and $\\{\\epsilon_i\\}$.\n\\enlem\n\n\\begin{proof}\n\tSince $\\epsilon_i$'s are sub-Gaussian random variables and $X_i$'s are sub-Gaussian random vectors, we have that $\\epsilon_i X_i$'s are sub-Exponential random vectors with $\\|\\epsilon_iX_i\\|_{\\psi_1}\\leq C_x \\sigma_{\\epsilon}$ \\citep[see e.g.~Lemma~2.7.7 in][]{vershynin2018high}. It then follows from Bernstein's inequality \\citep[see e.g.~Theorem~2.8.1 in][]{vershynin2018high} that for any $t > 0$,\n\t\t\\[\n\t\\mathbb{P}\\left\\{\\|\\sum_{i \\in \\I} \\epsilon_i X_i \\|_{\\infty} > t\\right\\} \\leq 2p \\exp\\left\\{-c \\min\\left\\{\\frac{t^2}{|\\I|C_x^2\\sigma^2_{\\epsilon}}, \\, \\frac{t}{C_x \\sigma_{\\epsilon}}\\right\\}\\right\\}.\n\t\t\\]\n\t\tTaking \n\t\t\\[\n\t\t\tt = C_{\\lambda}C_x\/4 \\sigma_{\\epsilon} \\sqrt{\\log(n p)} \\sqrt{\\max\\{|\\I|, \\, \\log(n p)\\}}\n\t\t\\] \n\t\tyields the conclusion.\n\\end{proof}\n \n \n \n \n\n\\bnlem \\label{lemma:beta bounded 1}\nSuppose   \\Cref{assp:dcdp_linear_reg} holds. Let $\\I \\subset [1,n] $. Denote \n $ \\kappa = \\min_{k\\in \\{1,\\ldots, K\\} } \\kappa_k,$ \nwhere $ \\{ \\kappa_k\\}_{k=1}^K$ are defined in  \\Cref{assp:dcdp_linear_reg}. Then for any $i\\in [T]$,\n   $$\\|\\beta^*_\\I - \\beta^*_i\\|_2  \\le C\\kappa  \\le CC_\\kappa,$$\n  for some absolute constant $C$ independent of $n$.\n\\enlem \n \\begin{proof}\n It suffices to consider $\\I =[1,n]$  and  $\\beta_i^*= \\beta_1^*$ as the general case is similar. Observe that  \n \\begin{align*}\n  \\| \\beta^*_{[1,n]}  - \\beta_ 1^*  \\|_2  =& \n  \\|  \\frac{1}{n} \\sum_{i=1}^n \\beta_i ^*  - \\beta_1^*  \n   \\|_2 \n = \\|  \\frac{1}{n} \\sum_{k=0}^{K} \\Delta_k \\beta_{\\eta_k+ 1} ^*   -    \\frac{1}{n}\\sum_{k=0}^{K} \\Delta_k  \\beta_1 ^*\n  \\|_2  \n  \\\\\n \\le \n &    \\frac{1}{n} \\sum_{k=0}^{K} \\left\\| \\Delta_k  (\\beta_{\\eta_k+ 1} ^* -\\beta_1 ^* ) \\right\\|_2    \\le \n \\frac{1}{n}\\sum_{k=0}^{K} \\Delta_k  (K+1)  \\kappa   \\le  (K+1)\\kappa . \n  \\end{align*}\n  By \\Cref{assp:dcdp_linear_reg}, both $\\kappa $ and $K $ bounded above. \n \\end{proof}\n \n \\bnlem\\label{eq:cumsum upper bound}\n Let $t\\in \\I =(s,e] \\subset [1,n]$. Denote \n $ \\kappa_{\\max}= \\max_{k\\in \\{1,\\ldots, K\\} } \\kappa_k,$ \nwhere $ \\{ \\kappa_k\\}_{k=1}^K$ are defined in  \\Cref{assp:dcdp_linear_reg}.  Then   \n $$ \\sup_{0< s < t<e\\le n  }\\| \\beta_{ (s, t]}  ^* - \\beta_{(t,e] }  ^* \\|_2  \\le C\\kappa  \\le CC_\\kappa. $$ \nfor some absolute constant $C$ independent of $n$. \n \\enlem\n \\begin{proof}\n It suffices to consider $(s,e]=(0,n]$, as the general case is similar.  \n Suppose that $\\eta_{q} <t \\le \\eta_{q+1} $.\n Observe that \n \\begin{align*}\n&  \\| \\beta^*_{(1,t]}  - \\beta ^* _{(t,n]}  \\|_2  \n  \\\\\n  =& \n  \\left\\|  \\frac{1}{t } \\sum_{i=1}^t \\beta_i ^*  - \\frac{1}{ n- t } \\sum_{i=t+1}^n \\beta_i ^*\n  \\right \\|_2 \n \\\\\n = & \\left\\|  \\frac{1}{t}  \\left( \\sum_{k=0}^{q-1} \\Delta_k \\beta_{\\eta_k+ 1}  ^*  + (t-\\eta_{q} )\\beta^*_{\\eta_q + 1} \\right) -    \\frac{1}{n-t } \\left( \\sum_{k=q+1}^{K}  \\Delta_k \\beta_{\\eta_k+ 1} ^*   + ( \\eta_{q+1}-t  )\\beta^*_{\\eta_q + 1} \\right) \n  \\right \\|_2  \n  \\\\\n=\n & \\left\\|  \\frac{1}{t}  \\left( \\sum_{k=0}^{q-1} \\Delta_k (\\beta ^* _{\\eta_k+ 1}-\\beta^*_{\\eta_q+1} ) \\right)  +  \\beta^*_{\\eta_q + 1} -    \\frac{1}{n-t } \\left( \\sum_{k=q+1}^{K}  \\Delta_k (  \\beta ^* _{\\eta_k+ 1}  -\\beta ^* _{\\eta_q+1} ) \\right)   -  \\beta^*_{\\eta_q + 1} \n  \\right \\|_2 \n  \\\\\n  =\n  &\\left\\|  \\frac{1}{t}  \\left( \\sum_{k=0}^{q-1} \\Delta_k (\\beta_{\\eta_k+ 1} ^* -\\beta^*_{\\eta_q+1} ) \\right)    -    \\frac{1}{n-t } \\left( \\sum_{k=q+1}^{K}  \\Delta_k (  \\beta_{\\eta_k+ 1} ^*   -\\beta_{\\eta_q+1} ^*  ) \\right)   \n  \\right \\|_2 \n  \\\\\n  \\le \n  &\\frac{1}{t} \\sum_{k=0}^{q-1} \\Delta_k    K \\kappa      +\\frac{1}{n-t} \\sum_{k=q+1}^{K} \\Delta_k    K  \\kappa \n  \\le 2K\\kappa  . \n  \\end{align*}\n  \n \\end{proof}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\clearpage\n\n\n\\subsection{Local refinement}\nIn this section, we prove results related to the $O_p(1)$ rates of localization error.\n\\begin{enumerate}\n    \\item \\Cref{sec:op1 fundations} presents two fundamental lemmas to prove other results.\n    \\item \\Cref{sec:mean op1} prove results for the mean model.\n    \\item \\Cref{sec:cov op1} prove results for the covariance model.\n    \\item \\Cref{sec:regression op1} prove results for the regression model.\n\\end{enumerate}\n\n\n\\subsubsection{Fundamental lemma}\n\\label{sec:op1 fundations}\n\n\n\\bnlem\n\\label{lem:subexp deviation martingale}\nSuppose $\\left\\{z_i\\right\\}_{i=1}^{\\infty}$ is a collection of independent centered sub-exponential random variable with $0<\\sup _{1 \\leq i<\\infty}\\left\\|z_i\\right\\|_{\\psi_1} \\leq 1$. Then for any integer $d>0, \\alpha>0$ and any $x>0$\n$$\n\\mathbb{P}\\left(\\max _{k \\in[d,(1+\\alpha) d]} \\frac{\\sum_{i=1}^k z_i}{\\sqrt{k}} \\geq x\\right) \\leq \\exp \\left\\{-\\frac{x^2}{2(1+\\alpha)}\\right\\}+\\exp \\left\\{-\\frac{\\sqrt{d} x}{2}\\right\\} .\n$$\n\\enlem\n\\bprf\nDenote $S_n=\\sum_{i=1}^n z_i$. Let $\\zeta=\\sup _{1 \\leq i \\leq \\infty}\\left\\|z_i\\right\\|_{\\psi_1}$. For any two integers $m<n$ and any $t \\leq \\frac{C_1}{\\zeta}$\n$$\n\\mathbb{E}\\left(\\exp \\left(t\\left(S_n-S_m\\right)\\right)\\right) \\leq \\prod_{i=m+1}^n \\mathbb{E}\\left(\\exp \\left(t z_i\\right)\\right) \\leq \\prod_{i=m+1}^n \\mathbb{E}\\left(C_1^2 t^2 \/ 2\\right)=\\mathbb{E}\\left[(n-m) C_1^2 t^2 \/ 2\\right].\n$$\nLet $\\mathcal{F}_k$ denote the sigma-algebra generated by $\\left(z_1, \\ldots, z_k\\right)$. Without loss of generality, assume that $C_1=1$. Since $S_k$ is independent of $S_n-S_k$, this implies that when $t \\geq 1 \/ \\zeta$\n$$\n\\mathbb{E}\\left(\\exp \\left\\{t S_n-\\frac{t^2 n}{2}\\right\\} \\mid \\mathcal{F}_k\\right)=\\exp \\left\\{t S_k-\\frac{t^2 k}{2}\\right\\} \\mathbb{E}\\left(\\exp \\left\\{t\\left(S_n-S_k\\right)\\right\\}-\\frac{t^2(n-k)}{2}\\right) \\leq \\exp \\left\\{t S_k-\\frac{t^2 k}{2}\\right\\}\n$$\nTherefore $\\exp \\left\\{t S_n-\\frac{t^2 n}{2}\\right\\}$ is a super-Martingale. Let $x$ be given and\n$$\nA=\\inf \\left\\{n \\geq d, S_n \\geq \\sqrt{n} x\\right\\} .\n$$\nThen $S_A \\geq \\sqrt{A} x \\geq \\sqrt{d} x$. Thus for $t \\geq 1 \/ \\zeta$,\n$$\n\\mathbb{E}\\left(\\exp \\left\\{t \\sqrt{d} x-\\frac{t^2 A}{2}\\right\\}\\right) \\leq \\mathbb{E}\\left(\\exp \\left\\{t S_A-\\frac{t^2 A}{2}\\right\\}\\right) \\leq \\mathbb{E}\\left(\\exp \\left\\{t S_1-\\frac{t^2}{2}\\right\\}\\right) \\leq 1 .\n$$\nBy definition of $A$,\n$$\n\\mathbb{P}\\left(\\max _{k \\in[d,(1+\\alpha) d]} \\frac{\\sum_{i=1}^k z_i}{\\sqrt{k}} \\geq x\\right) \\leq \\mathbb{P}(A \\leq(1+\\alpha) d) .\n$$\nSince $u \\rightarrow \\exp \\left(s \\sqrt{d} x-\\frac{t^2 u}{2}\\right)$ is decreasing, it follows that\n$$\n\\mathbb{P}\\left(\\max _{k \\in[d,(1+\\alpha) d]} \\frac{\\sum_{i=1}^k z_i}{\\sqrt{k}} \\geq x\\right) \\leq \\mathbb{P}\\left(\\exp \\left\\{t \\sqrt{d} x-t^2 A \/ 2\\right\\} \\geq \\exp \\left\\{t \\sqrt{d} x-t^2(1+\\alpha) d \/ 2\\right\\}\\right) .\n$$\nMarkov's inequality implies that when $t \\geq \\frac{1}{\\zeta}$,\n$$\n\\mathbb{P}\\left(\\max _{k \\in[d,(1+\\alpha) d]} \\frac{\\sum_{i=1}^k z_i}{\\sqrt{k}} \\geq x\\right) \\leq \\exp \\left\\{-t \\sqrt{d} x+t^2(1+\\alpha) d \/ 2\\right\\}\n$$\nSet\n$$\nt=\\min \\left\\{\\frac{1}{\\zeta}, \\frac{x}{(1+\\alpha) \\sqrt{d}}\\right\\}\n$$\nIf $\\frac{1}{\\zeta} \\geq \\frac{x}{(1+\\alpha) \\sqrt{d}}$. Then $t=\\frac{x}{(1+\\alpha) \\sqrt{d}}$ and therefore\n$$\n-t \\sqrt{d} x+t^2(1+\\alpha) d \/ 2=-\\frac{x^2}{2(1+\\alpha)}\n$$\nSo\n$$\n\\mathbb{P}\\left(\\max _{k \\in[d,(1+\\alpha) d]} \\frac{\\sum_{i=1}^k z_i}{\\sqrt{k}} \\geq x\\right) \\leq \\exp \\left\\{-\\frac{x^2}{2(1+\\alpha)}\\right\\} .\n$$\nIf $\\frac{1}{\\zeta} \\leq \\frac{x}{(1+\\alpha) \\sqrt{d}}$. Then $t=\\frac{1}{\\zeta} \\leq \\frac{x}{(1+\\alpha) \\sqrt{d}}$ and so\n$$\n-t \\sqrt{d} x+t^2(1+\\alpha) d \/ 2 \\leq \\frac{-\\sqrt{d} x}{\\zeta}+\\frac{1}{\\zeta} \\frac{x}{(1+\\alpha) \\sqrt{d}} \\frac{(1+\\alpha) d}{2}=\\frac{-\\sqrt{d} x}{2 \\zeta} .\n$$\nSo\n$$\n\\mathbb{P}\\left(\\max _{k \\in[d,(1+\\alpha) d]} \\frac{\\sum_{i=1}^k z_i}{\\sqrt{k}} \\geq x\\right) \\leq \\exp \\left\\{-\\frac{\\sqrt{d} x}{2 \\zeta}\\right\\} \\leq \\exp \\left\\{-\\frac{\\sqrt{d} x}{2}\\right\\},\n$$\nwhere $\\zeta \\leq 1$ is used in the last inequality. Putting the two cases together leads the desired result.\n\\eprf\n\n\n\n\\bnlem\n\\label{lem:subexp uniform loglog law}\nSuppose $\\left\\{z_i\\right\\}_{i=1}^{\\infty}$ is a collection of independent centered sub-exponential random variable with $0<\\sup _{1 \\leq i<\\infty}\\left\\|z_i\\right\\|_{\\psi_1} \\leq 1$. Let $\\nu>0$ be given. For any $x>0$, it holds that\n$$\n\\mathbb{P}\\left(\\sum_{i=1}^r z_i \\leq 4 \\sqrt{r\\{\\log \\log (4 \\nu r)+x+1\\}}+4 \\sqrt{r \\nu}\\{\\log \\log (4 \\nu r)+x+1\\} \\text { for all } r \\geq 1 \/ \\nu\\right) \\geq 1-2 \\exp (-x) \\text {. }\n$$\n\\enlem\n\\bprf\nLet $s \\in \\mathbb{Z}^{+}$and $\\mathcal{T}_s=\\left[2^s \/ \\nu, 2^{s+1} \/ \\nu\\right]$. By \\Cref{lem:subexp deviation martingale}, for all $x>0$,\n$$\n\\mathbb{P}\\left(\\sup _{r \\in \\mathcal{T}_s} \\frac{\\sum_{i=1}^r z_i}{\\sqrt{r}} \\geq x\\right) \\leq \\exp \\left\\{-\\frac{x^2}{4}\\right\\}+\\exp \\left\\{-\\frac{\\sqrt{2^s \/ \\nu} x}{2}\\right\\} \\leq \\exp \\left\\{-\\frac{x^2}{4}\\right\\}+\\exp \\left\\{-\\frac{x}{2 \\sqrt{\\nu}}\\right\\} .\n$$\nTherefore by a union bound,\n\\begin{align}\n    & \\mathbb{P}\\left(\\exists s \\in \\mathbb{Z}^{+}: \\sup _{r \\in \\mathcal{T}_s} \\frac{\\sum_{i=1}^r z_i}{\\sqrt{r}} \\geq 2 \\sqrt{\\log \\log ((s+1)(s+2))+x}+2 \\sqrt{\\nu}\\{\\log \\log ((s+1)(s+2))+x\\}\\right) \\nonumber \\\\\n\\leq & \\sum_{s=0}^{\\infty} 2 \\frac{\\exp (-x)}{(s+1)(s+2)}=2 \\exp (-x) \\label{tmp_eq: iterated log}.\n\\end{align}\nFor any $r \\geq 2^s \/ \\nu, s \\leq \\log (r \\nu) \/ \\log (2)$, and therefore\n$$\n(s+1)(s+2) \\leq \\frac{\\log (2 r \\nu) \\log (4 r \\nu)}{\\log ^2(2)} \\leq\\left(\\frac{\\log (4 r \\nu)}{\\log (2)}\\right)^2 .\n$$\nThus\n$$\n\\log ((s+1)(s+2)) \\leq 2 \\log \\left(\\frac{\\log (4 r \\nu)}{\\log (2)}\\right) \\leq 2 \\log \\log (4 r \\nu)+1 .\n$$\nThe above display together with \\eqref{tmp_eq: iterated log} gives\n$$\n\\mathbb{P}\\left(\\sup _{r \\geq 1 \/ \\nu} \\frac{\\sum_{i=1}^r z_i}{\\sqrt{r}} \\geq 2 \\sqrt{2 r \\log \\log (4 r \\nu)+x+1}+2 \\sqrt{r \\nu}\\{\\log \\log (4 r \\nu)+x+1\\}\\right) \\leq 2 \\exp (-x) .\n$$\n\\eprf\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\clearpage\n\\subsubsection{Mean}\n\\label{sec:mean op1}\nSuppose $\\alpha^*,\\beta^*\\in \\mathbb{R}^p$ and\n$$\ny_i=\\begin{cases}\n\\alpha^*+\\epsilon_i & \\text { when } i \\in(0, \\eta] \\\\\n\\beta^*+\\epsilon_i & \\text { when } i \\in(\\eta, n]\n\\end{cases}\n$$\nwhere $\\{\\epsilon_i\\}$ is an i.i.d sequence of subgaussian variables such that $\\|\\epsilon_i\\|_{\\psi_2}=\\sigma_\\epsilon<\\infty$. In addition, suppose that there exists $\\theta \\in(0,1)$ such that $\\eta=\\lfloor n \\theta\\rfloor$ and that $\\|\\alpha^*-\\beta^*\\|_2=\\kappa<\\infty$. Suppose that $\\|\\alpha^*\\|_0 \\leq \\mathfrak{s},\\|\\beta^*\\|_0 \\leq \\mathfrak{s}$ and that\n\\begin{equation}\n    \\frac{\\mathfrak{s} \\log ^{3 \/ 2}(p)}{\\sqrt{n}} \\rightarrow 0, \\quad \\frac{\\mathfrak{s}^2 \\log ^{3 \/ 2}(p)}{n \\kappa} \\rightarrow 0, \\quad \\frac{\\mathfrak{s} \\log ^{3 \/ 2}(p)}{\\sqrt{n} \\kappa} \\rightarrow 0, \\quad \\frac{\\mathfrak{s} \\log (p)}{n \\kappa^2} \\rightarrow 0.\n    \\label{op1_eq:mean snr}\n\\end{equation}\nSuppose there exists $\\widehat{\\alpha}$ and $\\widehat{\\beta}$ such that\n\\begin{align*}\n  & \\|\\widehat{\\alpha}-\\alpha^*\\|_2^2=O_p\\left(\\frac{\\mathfrak{s} \\log (p)}{n}\\right) \\text{ and } \\|\\widehat{\\alpha}-\\alpha^*\\|_1=O_p\\left(\\mathfrak{s} \\sqrt{\\frac{\\log (p)}{n}}\\right); \\\\\n& \\|\\widehat{\\beta}-\\beta^*\\|_2^2=O_p\\left(\\frac{\\mathfrak{s} \\log (p)}{n}\\right) \\text{ and } \\|\\widehat{\\beta}-\\beta^*\\|_1=O_p\\left(\\mathfrak{s} \\sqrt{\\frac{\\log (p)}{n}}\\right).\n\\end{align*}\nLet\n$$\n\\widehat{\\mathcal{Q}}(k)=\\sum_{i=1}^k\\|y_i- \\widehat{\\alpha}\\|_2^2+\\sum_{i=k+1}^n\\|y_i-\\widehat{\\beta}\\|_2^2 \\quad \\text { and } \\quad \\mathcal{Q}^*(k)=\\sum_{i=1}^k\\|y_i- \\alpha^*\\|_2^2+\\sum_{i=k+1}^n\\|y_i- \\beta^*\\|_2^2 .\n$$\n\n\\bnlem[Refinement for the mean model]\nLet\n$$\n\\eta+r=\\underset{k \\in(0, n]}{\\arg \\max } \\widehat{\\mathcal{Q}}(k) .\n$$\nThen under the assumptions above, it holds that\n$$\n\\kappa^2 r=O_p(1) .\n$$\n\\enlem\n\\bprf\nBy assumptions above, we have\n$$\n\\frac{\\mathfrak{s} \\log ^{3 \/ 2}}{\\sqrt{n}} \\rightarrow 0, \\quad \\frac{\\mathfrak{s}^2 \\log ^{3 \/ 2}(p)}{n \\kappa} \\rightarrow 0, \\quad \\frac{\\mathfrak{s} \\log ^{3 \/ 2}(p)}{\\sqrt{n} \\kappa} \\rightarrow 0, \\quad \\frac{\\mathfrak{s} \\log (p)}{n \\kappa^2} \\rightarrow 0 .\n$$\nWithout loss of generality, suppose $r \\geq 0$. Since $\\eta+r$ is the minimizer, it follows that\n$$\n\\widehat{\\mathcal{Q}}(\\eta+r) \\leq \\widehat{\\mathcal{Q}}(\\eta) .\n$$\nIf $r \\leq \\frac{1}{\\kappa^2}$, then there is nothing to show. So for the rest of the argument, for contradiction, assume that\n$$\nr \\geq \\frac{1}{\\kappa^2}\n$$\nObserve that\n$$\n\\begin{aligned}\n\\widehat{\\mathcal{Q}}(\\eta + r)-\\widehat{\\mathcal{Q}}(\\eta) &=\\sum_{i=\\eta+1}^{\\eta+r}\\|y_i- \\widehat{\\alpha}\\|_2^2-\\sum_{i=\\eta+1}^{\\eta+r}\\|y_i-\\widehat{\\beta}\\|_2^2 \\\\\n\\mathcal{Q}^*(\\eta + r)-\\mathcal{Q}^*(\\eta) &=\\sum_{i=\\eta+1}^{\\eta+r}\\|y_i- \\alpha^*\\|_2^2-\\sum_{i=\\eta+1}^{\\eta+r}\\|y_i- \\beta^*\\|_2^2\n\\end{aligned}\n$$\n\\textbf{Step 1}. It follows that\n$$\n\\begin{aligned}\n& \\sum_{i=\\eta+1}^{\\eta+r}\\|y_i- \\widehat{\\alpha}\\|_2^2-\\sum_{i=\\eta+1}^{\\eta+r}\\|y_i-\\alpha^*\\|_2^2 \\\\\n=& \\sum_{i=\\eta+1}^{\\eta+r}\\|\\widehat{\\alpha}- \\alpha^*\\|_2^2+2\\left(\\widehat{\\alpha}-\\alpha^*\\right)^{\\top} \\sum_{i=\\eta+1}^{\\eta+r}\\left(y_i-\\alpha^*\\right) \\\\\n=& \\sum_{i=\\eta+1}^{\\eta+r}\\|\\widehat{\\alpha}- \\alpha^*\\|_2^2+2r\\left(\\widehat{\\alpha}-\\alpha^*\\right)^{\\top} \\left(\\beta^*-\\alpha^*\\right) +2\\left(\\widehat{\\alpha}-\\alpha^*\\right)^{\\top} \\sum_{i=\\eta+1}^{\\eta+r}\\epsilon_i\n\\end{aligned}\n$$\nBy assumptions, we have\n$$\n\\sum_{i=\\eta+1}^{\\eta+r}\\|\\widehat{\\alpha}- \\alpha^*\\|_2^2 \\leq C r \\frac{\\mathfrak{s} \\log (p)}{n} \\\\\n=o_p\\left(r \\sigma^2_{\\epsilon} \\kappa^2\\right).\n$$\nSimilarly\n$$\nr\\left(\\widehat{\\alpha}-\\alpha^*\\right)^{\\top}\\left(\\beta^*-\\alpha^*\\right)\\leq r\\|\\widehat{\\alpha}-\\alpha^*\\|_2\\|\\beta^*-\\alpha^*\\|_2 \\leq\nO_p\\left(r\\kappa \\sqrt{\\frac{\\mathfrak{s} \\log (p)}{n} } \\right)=o_p\\left(r\\kappa^2\\right)\n$$\nwhere the second equality follows from $\\|\\beta^*-\\alpha^*\\|_2=\\kappa$, and the last equality follows from \\eqref{op1_eq:mean snr}. In addition,\n$$\n\\begin{aligned}\n&\\left(\\widehat{\\alpha}-\\alpha^*\\right)^{\\top} \\sum_{i=\\eta+1}^{\\eta+r} \\epsilon_i \\leq\\|\\widehat{\\alpha}-\\alpha^*\\|_1\\|\\sum_{i=\\eta+1}^{\\eta+r}\\epsilon_i\\|_{\\infty} \\\\\n=& O_p\\left(\\mathfrak{s} \\sqrt{\\frac{\\log (p)}{n}}\\right) O_p(\\sqrt{r \\log (p)})=O_p(\\kappa \\sqrt{r})\n\\end{aligned}\n$$\nTherefore\n$$\n\\sum_{i=\\eta+1}^{\\eta+r}\\|y_i- \\widehat{\\alpha}\\|_2^2-\\sum_{i=\\eta+1}^{\\eta+r}\\|y_i-\\alpha^*\\|_2^2=O_p(\\kappa \\sqrt{r})+o_p\\left(r \\kappa^2\\right) .\n$$\n\\textbf{Step 2}. Using the same argument as in the previous step, it follows that\n$$\n\\sum_{i=\\eta+1}^{\\eta+r}\\|y_i- \\widehat{\\beta}\\|_2^2-\\sum_{i=\\eta+1}^{\\eta+r}\\|y_i- \\beta^*\\|_2^2=O_p(\\kappa \\sqrt{r})+o_p(r \\kappa^2) .\n$$\nTherefore\n\\begin{equation}\n    \\left|\\widehat{\\mathcal{Q}}(\\eta+r)-\\widehat{\\mathcal{Q}}(\\eta)-\\left\\{\\mathcal{Q}^*(\\eta+r)-\\mathcal{Q}^*(\\eta)\\right\\}\\right|=O_p(\\kappa \\sqrt{r})+o_p\\left(r \\kappa^2\\right)\n    \\label{tmp_eq:op1 mean main prop eq 1}\n\\end{equation}\n\\textbf{Step 3}. Observe that\n$$\n\\begin{aligned}\n\\mathcal{Q}^*(\\eta+r)-\\mathcal{Q}^*(\\eta) \n=& \\sum_{i=\\eta+1}^{\\eta+r}\\|y_i- \\alpha^*\\|_2^2-\\sum_{i=\\eta+1}^{\\eta+r}\\|y_i- \\beta^*\\|_2^2 \\\\\n=& r\\| \\alpha^*- \\beta^*\\|_2^2-2 \\sum_{i=\\eta+1}^{\\eta+r}(y_i- \\beta^*)( \\alpha^*- \\beta^*) \\\\\n=& r\\| \\alpha^*- \\beta^*\\|_2^2 -2 (\\alpha^*- \\beta^*)^\\top\\sum_{i=\\eta+1}^{\\eta+r} \\epsilon_i\n\\end{aligned}\n$$\nLet\n$$\nw_i=\\frac{1}{\\sigma_{\\epsilon}\\kappa} \\epsilon_i^\\top\\left( \\alpha^*-\\beta^*\\right)\n$$\nThen $\\left\\{w_i\\right\\}_{i=1}^{\\infty}$ are subgaussian random variables with bounded $\\psi_2$ norm. Therefore by \\Cref{lem:subexp uniform loglog law}, uniformly for all $r \\geq 1 \/ \\kappa^2$,\n$$\n\\sum_{i=1}^r w_i=O_p\\left(\\sqrt{r\\left\\{\\log \\log \\left(\\kappa^2 r\\right)+1\\right\\}}+\\sqrt{r \\kappa^2}\\left\\{\\log \\log \\left(\\kappa^2 r\\right)+1\\right\\}\\right)\n$$\nSince $\\kappa<\\infty$, it follows that\n$$\n\\sum_{i=\\eta+1}^{\\eta+r} \\epsilon_i^\\top \\left( \\alpha^*-\\beta^*\\right)=O_p\\left(\\sqrt{r \\kappa^2\\left\\{\\log \\log \\left(\\kappa^2 r\\right)+1\\right\\}}+\\sqrt{r \\kappa^2}\\left\\{\\log \\log \\left(\\kappa^2 r\\right)+1\\right\\}\\right)\n$$\nTherefore\n\\begin{equation}\n    \\mathcal{Q}^*(\\eta+r)-\\mathcal{Q}^*(\\eta) = r \\kappa^2+O_p\\left(\\sqrt{r \\kappa^2\\left\\{\\log \\log \\left(\\kappa^2 r\\right)+1\\right\\}}+\\sqrt{r \\kappa^2}\\left\\{\\log \\log \\left(\\kappa^2 r\\right)+1\\right\\}\\right)\n\\label{tmp_eq:op1 mean main prop eq 2}\n\\end{equation}\n\n\\textbf{Step 4}. \\Cref{tmp_eq:op1 mean main prop eq 1} and \\Cref{tmp_eq:op1 mean main prop eq 2} together give, uniformly for all $r \\geq 1 \/ \\kappa^2$,\n$$\n r \\kappa^2+O_p\\left(\\sqrt{r \\kappa^2\\left\\{\\log \\log \\left(\\kappa^2 r\\right)+1\\right\\}}+\\sqrt{r \\kappa^2}\\left\\{\\log \\log \\left(\\kappa^2 r\\right)+1\\right\\}\\right) \\leq O_p(\\kappa \\sqrt{r})+o_p\\left(r \\kappa^2\\right)\n$$\nThis implies that\n$$\nr \\kappa^2=O_p()\n$$\n\\eprf\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\clearpage\n\\subsubsection{Covariance}\n\\label{sec:cov op1}\nSuppose $G^*,H^*\\in \\mathbb{R}^d$ and\n$$\n\\mathbb{E}[X_iX_i^\\top]=\\left\\{\\begin{array}{ll}\nG^* & \\text { when } i \\in(0, \\eta] \\\\\nH^* & \\text { when } i \\in(\\eta, n]\n\\end{array} .\\right.\n$$\nIn addition, suppose that there exists $\\theta \\in(0,1)$ such that $\\eta=\\lfloor n \\theta\\rfloor$ and that $\\|G^*-H^*\\|_2=\\kappa_{\\Sigma}<\\infty$. Suppose that $c_XI_d \\preceq G^* \\preceq C_X I_d$, $c_XI_d \\preceq H^* \\preceq C_X I_d$, and that\n\\begin{equation}\n    \\frac{d^{5\/2} \\log ^{3 \/ 2}(nd)}{\\sqrt{n}} \\rightarrow 0, \\quad \\frac{d^2 \\log ^{3 \/ 2}(nd)}{n \\kappa_{\\Sigma}} \\rightarrow 0, \\quad \\frac{d \\log ^{3 \/ 2}(nd)}{\\sqrt{n} \\kappa_{\\Sigma}} \\rightarrow 0, \\quad \\frac{d \\log (nd)}{n \\kappa_{\\Sigma}^2} \\rightarrow 0.\n    \\label{op1_eq:cov snr}\n\\end{equation}\nBy \\Cref{lem:estimation covariance}, there exist $\\widehat{G},\\widehat{H}$ such that\n\\begin{align*}\n  & \\|\\widehat{G}-G^*\\|_2=O_p\\left(\\sqrt{\\frac{d \\log (nd)}{n}}\\right) \\text{ and } \\|\\widehat{G}-G^*\\|_F=O_p\\left(d \\sqrt{\\frac{\\log (nd)}{n}}\\right); \\\\\n& \\|\\widehat{H}-H^*\\|_2=O_p\\left(\\sqrt{\\frac{d \\log (nd)}{n}}\\right) \\text{ and } \\|\\widehat{H}-H^*\\|_F=O_p\\left(d \\sqrt{\\frac{\\log (nd)}{n}}\\right).\n\\end{align*}\nLet\n$$\n\\widehat{\\mathcal{Q}}(k)=\\sum_{i=1}^k\\|X_iX_i^\\top- \\widehat{G}\\|_F^2+\\sum_{i=k+1}^n\\|X_iX_i^\\top-\\widehat{H}\\|_F^2 \\quad \\text { and } \\quad \\mathcal{Q}^*(k)=\\sum_{i=1}^k\\|X_iX_i^\\top- G^*\\|_F^2+\\sum_{i=k+1}^n\\|X_iX_i^\\top- H^*\\|_F^2 .\n$$\nThrough out this section, we use $\\kappa_{\\Sigma} = \\|G^*-H^*\\|_F$ to measure the signal.\n\\bnlem[Refinement for regression]\nLet\n$$\n\\eta+r=\\underset{k \\in(0, n]}{\\arg \\max } \\widehat{\\mathcal{Q}}(k) .\n$$\nThen under the assumptions above, it holds that\n$$\n\\kappa_{\\Sigma}^2 r=O_P(\\log(n)) .\n$$\n\\enlem\n\\bprf\nBy assumptions above, we have\n$$\n\\frac{d^{5\/2} \\log ^{3 \/ 2}(nd)}{\\sqrt{n}} \\rightarrow 0, \\quad \\frac{d^2 \\log ^{3 \/ 2}(nd)}{n \\kappa_{\\Sigma}} \\rightarrow 0, \\quad \\frac{d \\log ^{3 \/ 2}(nd)}{\\sqrt{n} \\kappa_{\\Sigma}} \\rightarrow 0, \\quad \\frac{d \\log (nd)}{n \\kappa_{\\Sigma}^2} \\rightarrow 0 .\n$$\nWithout loss of generality, suppose $r \\geq 0$. Since $\\eta+r$ is the minimizer, it follows that\n$$\n\\widehat{\\mathcal{Q}}(\\eta+r) \\leq \\widehat{\\mathcal{Q}}(\\eta) .\n$$\nIf $r \\leq C\\frac{\\log(n)}{\\kappa_{\\Sigma}^2}$, then there is nothing to show. So for the rest of the argument, for contradiction, assume that\n$$\nr \\geq C\\frac{\\log(n)}{\\kappa_{\\Sigma}^2}\n$$\nObserve that\n$$\n\\begin{aligned}\n\\widehat{\\mathcal{Q}}(t)-\\widehat{\\mathcal{Q}}(\\eta) &=\\sum_{i=\\eta+1}^{\\eta+r}\\|X_iX_i^\\top- \\widehat{G}\\|_F^2-\\sum_{i=\\eta+1}^{\\eta+r}\\|X_iX_i^\\top- \\widehat{H}\\|_F^2 \\\\\n\\mathcal{Q}^*(t)-\\mathcal{Q}^*(\\eta) &=\\sum_{i=\\eta+1}^{\\eta+r}\\|X_iX_i^\\top- {G}^*\\|_F^2-\\sum_{i=\\eta+1}^{\\eta+r}\\|X_iX_i^\\top-{H}^*\\|_F^2\n\\end{aligned}\n$$\n\\textbf{Step 1}. It follows that\n$$\n\\begin{aligned}\n& \\sum_{i=\\eta+1}^{\\eta+r}\\|X_iX_i^\\top- \\widehat{G}\\|_F^2-\\sum_{i=\\eta+1}^{\\eta+r}\\|X_iX_i^\\top- {G}^*\\|_F^2 \\\\\n=& \\sum_{i=\\eta+1}^{\\eta+r}\\|\\widehat{G}- G^*\\|_F^2+2\\left\\langle G^*-\\widehat{G}, \\sum_{i=\\eta+1}^{\\eta+r}(X_iX_i^\\top-G^*)\\right\\rangle \\\\\n=& r\\|\\widehat{G}- G^*\\|_F^2+2r \\left\\langle G^* -\\widehat{G}, H^*-G^*\\right\\rangle +2\\left\\langle G^*-\\widehat{G}, \\sum_{i=\\eta+1}^{\\eta+r}(X_iX_i^\\top -H^*)\\right\\rangle\n\\end{aligned}\n$$\nBy assumptions, we have\n$$\nr\\|\\widehat{G}- G^*\\|_F^2 =O_p\\left(r \\frac{d^2 \\log (nd)}{n}\\right) \\\\\n=o_p\\left(r \\kappa_{\\Sigma}^2\\right).\n$$\nSimilarly\n$$\n\\left\\langle G^* -\\widehat{G}, H^*-G^*\\right\\rangle\\leq r\\|G^* -\\widehat{G}\\|_F\\| H^*-G^* \\|_F \\leq\nO_p\\left(r\\kappa_{\\Sigma} d\\sqrt{\\frac{ \\log (nd)}{n} } \\right)=o_p\\left(r \\kappa_{\\Sigma}^2\\right)\n$$\nwhere the second equality follows from $\\|G^*-H^*\\|_2=\\kappa_{\\Sigma}$, and the last equality follows from \\eqref{op1_eq:cov snr}. In addition,\n$$\n\\begin{aligned}\n&\\left\\langle G^*-\\widehat{G}, \\sum_{i=\\eta+1}^{\\eta+r}(X_iX_i^\\top -H^*)\\right\\rangle \\leq\\| G^*-\\widehat{G}\\|_F\\|\\sum_{i=\\eta+1}^{\\eta+r}(X_iX_i^\\top -H^*)\\|_{F} \\\\\n=& O_p\\left(d \\sqrt{\\frac{\\log (nd)}{n}}\\right) O_p(d\\sqrt{r\\log (nd)}+ d^{3\/2}\\log (nd))=O_p(\\kappa_{\\Sigma} \\sqrt{r})+O_p(1)\n\\end{aligned}\n$$\nTherefore\n$$\n\\sum_{i=\\eta+1}^{\\eta+r}\\|X_iX_i^\\top- \\widehat{G}\\|_F^2-\\sum_{i=\\eta+1}^{\\eta+r}\\|X_iX_i^\\top- {G}^*\\|_F^2=O_p(\\kappa_{\\Sigma} \\sqrt{r})+O_p(1)+o_p\\left(r \\kappa_{\\Sigma}^2\\right) .\n$$\n\\textbf{Step 2}. Using the same argument as in the previous step, it follows that\n$$\n\\sum_{i=\\eta+1}^{\\eta+r}\\|X_iX_i^\\top- \\widehat{H}\\|_F^2-\\sum_{i=\\eta+1}^{\\eta+r}\\|X_iX_i^\\top- {H}^*\\|_F^2=O_p(\\kappa_{\\Sigma} \\sqrt{r})+O_p(1)+o_p(r \\kappa_{\\Sigma}^2) .\n$$\nTherefore\n\\begin{equation}\n    \\left|\\widehat{\\mathcal{Q}}(\\eta+r)-\\widehat{\\mathcal{Q}}(\\eta)-\\left\\{\\mathcal{Q}^*(\\eta+r)-\\mathcal{Q}^*(\\eta)\\right\\}\\right|=O_p(\\kappa_{\\Sigma} \\sqrt{r})+O_p(1)+o_p\\left(r \\kappa_{\\Sigma}^2\\right)\n    \\label{tmp_eq:op1 cov main prop eq 1}\n\\end{equation}\n\\textbf{Step 3}. Observe that\n$$\n\\begin{aligned}\n\\mathcal{Q}^*(\\eta+r)-\\mathcal{Q}^*(\\eta) \n=&\\sum_{i=\\eta+1}^{\\eta+r}\\|X_iX_i^\\top- {G}^*\\|_F^2-\\sum_{i=\\eta+1}^{\\eta+r}\\|X_iX_i^\\top- {H}^*\\|_F^2 \\\\\n=& r\\| G^*- H^*\\|_F^2-2 \\left\\langle H^*- G^*, \\sum_{i=\\eta+1}^{\\eta+r}(X_iX_i^\\top- H^*) \\right\\rangle\n\\end{aligned}\n$$\nDenote $D^* = H^*- G^*$, then we can write the noise term as\n$$\n\\left\\langle H^*- G^*, X_iX_i^\\top- H^* \\right\\rangle = X_i^\\top D^* X_i - \\mathbb{E}[X_i^\\top D^* X_i].\n$$\nSince $X_i$'s are Gaussian, denote $\\Sigma_{i} = \\mathbb{E}[X_iX_i^\\top] =U_i^\\top \\Lambda_i U_i$, then\n$$\n\\left\\langle H^*- G^*, \\sum_{i=\\eta+1}^{\\eta+r}(X_iX_i^\\top- H^*) \\right\\rangle = Z^\\top \\tilde{D} Z^\\top - \\mathbb{E}[Z^\\top \\tilde{D} Z^\\top],\n$$\nwhere $Z\\in \\mathbb{R}^{rd}$ is a standard Gaussian vector and \n$$\n\\tilde{D} = {\\rm diag}\\{U_1D^*U_1^\\top,U_2D^*U_2^\\top,\\cdots, U_r D^*U_r^\\top\\}.\n$$\nTherefore, by Hanson-Wright inequality, uniformly for all $r\\geq C\\frac{\\log(n)}{\\kappa_{\\Sigma}^2}$ it holds that\n\\begin{equation}\n    \\mathcal{Q}^*(\\eta+r)-\\mathcal{Q}^*(\\eta) \\leq r \\kappa_{\\Sigma}^2 + O_P(\\|X\\|_{\\psi_2}^2\\sqrt{r}\\kappa_{\\Sigma}\\log(r\\kappa_{\\Sigma}^2)),\n\\label{tmp_eq:op1 cov main prop eq 2}\n\\end{equation}\n\n\\textbf{Step 4}. \\Cref{tmp_eq:op1 cov main prop eq 1} and \\Cref{tmp_eq:op1 cov main prop eq 2} together give, uniformly for all $r \\geq C\\log(n) \/ \\kappa_{\\Sigma}^2$,\n$$\n r \\kappa_{\\Sigma}^2+O_P(\\|X\\|_{\\psi_2}^2\\sqrt{r}\\kappa_{\\Sigma}\\log(r\\kappa_{\\Sigma}^2)) \\leq O_p(\\kappa_{\\Sigma} \\sqrt{r})+O_p(1)+o_p\\left(r \\kappa_{\\Sigma}^2\\right),\n$$\nwhich leads to contradiction and thus proves that\n$$\nr \\kappa_{\\Sigma}^2=O_P(\\log(n)).\n$$\n\\eprf\n\n\n\n\\bnlem\n\\label{lem:estimation covariance}\nLet $\\{X_i\\}_{i\\in [n]}$ be a sequence of subgaussian vectors in $\\mathbb{R}^{d}$ with orlitz norm upper bounded $\\|X\\|_{\\psi_2}<\\infty$. Suppose $\\mathbb{E}[X_i] = 0$ and $\\mathbb{E}[X_iX_i^\\top] = \\Sigma$ for $i\\in [n]$. Let $\\widehat{\\Sigma}_{n} = \\frac{1}{n}\\sum_{i \\in [n]} X_i X_i^\\top$. Then for any $u>0$, it holds with probability at least $1 - \\exp(-u)$ that\n\\begin{equation}\n    \\|\\widehat{\\Sigma}_{n} - \\Sigma\\|_2\\lesssim \\|X\\|_{\\psi_2}^2(\\sqrt{\\frac{d + u}{n}}\\vee \\frac{d + u}{n}).\n\\end{equation}\n\\enlem\n\n\n\n\n\\bnlem[Hanson-Wright inequality]\nLet $X=\\left(X_1, \\ldots, X_n\\right) \\in \\mathbb{R}^n$ be a random vector with independent, mean zero, sub-gaussian coordinates. Let $A$ be an $n \\times n$ matrix. Then, for every $t \\geq 0$, we have\n$$\n\\mathbb{P}\\left\\{\\left|X^{\\top} A X-\\mathbb{E} X^{\\top} A X\\right| \\geq t\\right\\} \\leq 2 \\exp \\left[-c \\min \\left(\\frac{t^2}{K^4\\|A\\|_F^2}, \\frac{t}{K^2\\|A\\|}\\right)\\right],\n$$\nwhere $K=\\max _i\\left\\|X_i\\right\\|_{\\psi_2}$\n\\enlem\n\n\n\n\n\n\n\n\n\\clearpage\n\\subsubsection{Regression}\n\\label{sec:regression op1}\nSuppose\n$$\ny_i=\\left\\{\\begin{array}{ll}\nX_i^{\\top} \\alpha^*+\\epsilon_i & \\text { when } i \\in(0, \\eta] \\\\\nX_i^{\\top} \\beta^*+\\epsilon_i & \\text { when } i \\in(\\eta, n]\n\\end{array} .\\right.\n$$\nIn addition, suppose that there exists $\\theta \\in(0,1)$ such that $\\eta=\\lfloor n \\theta\\rfloor$ and that $\\left|\\alpha^*-\\beta^*\\right|_2=\\kappa<\\infty$. Suppose that $\\left\\|\\alpha^*\\right\\|_0 \\leq \\mathfrak{s},\\left\\|\\beta^*\\right\\|_0 \\leq \\mathfrak{s}$ and that\n\\begin{equation}\n    \\frac{\\mathfrak{s} \\log ^{3 \/ 2}(p)}{\\sqrt{n}} \\rightarrow 0, \\quad \\frac{\\mathfrak{s}^2 \\log ^{3 \/ 2}(p)}{n \\kappa} \\rightarrow 0, \\quad \\frac{\\mathfrak{s} \\log ^{3 \/ 2}(p)}{\\sqrt{n} \\kappa} \\rightarrow 0, \\quad \\frac{\\mathfrak{s} \\log (p)}{n \\kappa^2} \\rightarrow 0.\n    \\label{op1_eq:regression snr}\n\\end{equation}\nSuppose there exists $\\widehat{\\alpha}$ and $\\widehat{\\beta}$ such that\n\\begin{align*}\n  & \\|\\widehat{\\alpha}-\\alpha^*\\|_2^2=O_p\\left(\\frac{\\mathfrak{s} \\log (p)}{n}\\right) \\text{ and } \\|\\widehat{\\alpha}-\\alpha^*\\|_1=O_p\\left(\\mathfrak{s} \\sqrt{\\frac{\\log (p)}{n}}\\right); \\\\\n& \\|\\widehat{\\beta}-\\beta^*\\|_2^2=O_p\\left(\\frac{\\mathfrak{s} \\log (p)}{n}\\right) \\text{ and } \\|\\widehat{\\beta}-\\beta^*\\|_1=O_p\\left(\\mathfrak{s} \\sqrt{\\frac{\\log (p)}{n}}\\right).\n\\end{align*}\nLet\n$$\n\\widehat{\\mathcal{Q}}(k)=\\sum_{i=1}^k\\left(y_i-X_i^{\\top} \\widehat{\\alpha}\\right)^2+\\sum_{i=k+1}^n\\left(y_i-X_i^{\\top} \\widehat{\\beta}\\right)^2 \\quad \\text { and } \\quad \\mathcal{Q}^*(k)=\\sum_{i=1}^k\\left(y_i-X_i^{\\top} \\alpha^*\\right)^2+\\sum_{i=k+1}^n\\left(y_i-X_i^{\\top} \\beta^*\\right)^2 .\n$$\n\n\\bnlem[Refinement for regression]\nLet\n$$\n\\eta+r=\\underset{k \\in(0, n]}{\\arg \\max } \\widehat{\\mathcal{Q}}(k) .\n$$\nThen under the assumptions above, it holds that\n$$\n\\kappa^2 r=O_p(1) .\n$$\n\\enlem\n\\bprf\nBy assumptions above, we have\n$$\n\\frac{\\mathfrak{s} \\log ^{3 \/ 2}(p)}{\\sqrt{n}} \\rightarrow 0, \\quad \\frac{\\mathfrak{s}^2 \\log ^{3 \/ 2}(p)}{n \\kappa} \\rightarrow 0, \\quad \\frac{\\mathfrak{s} \\log ^{3 \/ 2}(p)}{\\sqrt{n} \\kappa} \\rightarrow 0, \\quad \\frac{\\mathfrak{s} \\log (p)}{n \\kappa^2} \\rightarrow 0 .\n$$\nWithout loss of generality, suppose $r \\geq 0$. Since $\\eta+r$ is the minimizer, it follows that\n$$\n\\widehat{\\mathcal{Q}}(\\eta+r) \\leq \\widehat{\\mathcal{Q}}(\\eta) .\n$$\nIf $r \\leq \\frac{1}{\\kappa^2}$, then there is nothing to show. So for the rest of the argument, for contradiction, assume that\n$$\nr \\geq \\frac{1}{\\kappa^2}\n$$\nObserve that\n$$\n\\begin{aligned}\n\\widehat{\\mathcal{Q}}(t)-\\widehat{\\mathcal{Q}}(\\eta) &=\\sum_{i=\\eta+1}^{\\eta+r}\\left(y_i-X_i^{\\top} \\widehat{\\alpha}\\right)^2-\\sum_{i=\\eta+1}^{\\eta+r}\\left(y_i-X_i^{\\top} \\widehat{\\beta}\\right)^2 \\\\\n\\mathcal{Q}^*(t)-\\mathcal{Q}^*(\\eta) &=\\sum_{i=\\eta+1}^{\\eta+r}\\left(y_i-X_i^{\\top} \\alpha^*\\right)^2-\\sum_{i=\\eta+1}^{\\eta+r}\\left(y_i-X_i^{\\top} \\beta^*\\right)^2\n\\end{aligned}\n$$\n\\textbf{Step 1}. It follows that\n$$\n\\begin{aligned}\n& \\sum_{i=\\eta+1}^{\\eta+r}\\left(y_i-X_i^{\\top} \\widehat{\\alpha}\\right)^2-\\sum_{i=\\eta+1}^{\\eta+r}\\left(y_i-X_i^{\\top} \\alpha^*\\right)^2 \\\\\n=& \\sum_{i=\\eta+1}^{\\eta+r}\\left(X_i^{\\top} \\widehat{\\alpha}-X_i^{\\top} \\alpha^*\\right)^2+2\\left(\\widehat{\\alpha}-\\alpha^*\\right)^{\\top} X_i \\sum_{i=\\eta+1}^{\\eta+r}\\left(y_i-X_i^{\\top} \\alpha^*\\right) \\\\\n=& \\sum_{i=\\eta+1}^{\\eta+r}\\left(X_i^{\\top} \\widehat{\\alpha}-X_i^{\\top} \\alpha^*\\right)^2+2\\left(\\widehat{\\alpha}-\\alpha^*\\right)^{\\top} \\sum_{i=1}^r X_i X_i^{\\top}\\left(\\beta^*-\\alpha^*\\right)+2\\left(\\widehat{\\alpha}-\\alpha^*\\right)^{\\top} \\sum_{i=\\eta+1}^{\\eta+r} X_i \\epsilon_i\n\\end{aligned}\n$$\nBy \\Cref{lem:regression op1 base lemma 1}, uniformly for all $r$,\n$$\n\\left\\|\\frac{1}{r} \\sum_{i=1}^r X_i X_i^{\\top}-\\Sigma\\right\\|_{\\infty} \\geq O_p\\left(\\sqrt{\\frac{\\log (p)}{r}}+\\frac{\\log (p)}{r}\\right) .\n$$\nTherefore\n$$\n\\begin{aligned}\n\\sum_{i=\\eta+1}^{\\eta+r}\\left(X_i^{\\top} \\widehat{\\alpha}-X_i^{\\top} \\alpha^*\\right)^2 &=\\sum_{i=\\eta+1}^{\\eta+r}\\left(\\widehat{\\alpha}-\\alpha^*\\right)^{\\top} \\sum_{i=1}^r\\left\\{X_i X_i^{\\top}-\\Sigma\\right\\}\\left(\\widehat{\\alpha}-\\alpha^*\\right)+r\\left(\\widehat{\\alpha}-\\alpha^*\\right)^{\\top} \\Sigma\\left(\\widehat{\\alpha}-\\alpha^*\\right) \\\\\n& \\leq\\|\\widehat{\\alpha}-\\alpha^*\\|_1^2\\left\\|\\sum_{i=1}^r X_i X_i^{\\top}-\\Sigma\\right\\|_{\\infty}+\\Lambda_{\\max } r\\|\\widehat{\\alpha}-\\alpha^*\\|_2^2\\\\\n&=O_p\\left(\\frac{\\mathfrak{s}^2 \\log (p)}{n}\\right)O_p(\\sqrt{r \\log (p)}+\\log (p))+O_p\\left(r \\frac{\\mathfrak{s} \\log (p)}{n}\\right) \\\\\n&=O_p\\left(\\sqrt{r \\kappa^2}\\right)+O_p(1)+o_p\\left(r \\kappa^2\\right)\n\\end{aligned}\n$$\nwhere the inequality follows from \\Cref{lem:regression op1 base lemma 1} and last equality follows from assumption \\eqref{op1_eq:regression snr}. Similarly\n$$\n\\begin{aligned}\n\\left(\\widehat{\\alpha}-\\alpha^*\\right)^{\\top} \\sum_{i=1}^r X_i X_i^{\\top}\\left(\\beta^*-\\alpha^*\\right) &=\\left(\\widehat{\\alpha}-\\alpha^*\\right)^{\\top} \\sum_{i=1}^r\\left\\{X_i X_i^{\\top}-\\Sigma\\right\\}\\left(\\beta^*-\\alpha^*\\right)+r\\left(\\widehat{\\alpha}-\\alpha^*\\right)^{\\top} \\Sigma\\left(\\beta^*-\\alpha^*\\right) \\\\\n& \\leq \\|\\widehat{\\alpha}-\\alpha^*\\|_1\\|\\left(\\beta^*-\\alpha^*\\right)^{\\top}\\left\\{\\sum_{i=1}^r X_i X_i^{\\top}-\\Sigma\\right\\}\\|_{\\infty}+\\Lambda_{\\max } r\\|\\widehat{\\alpha}-\\alpha^*\\|_2\\|\\beta^*-\\alpha^*\\|_2 \\\\\n&=O_p\\left(\\mathfrak{s} \\sqrt{\\frac{\\log (p)}{n}}\\right) O_p(\\kappa\\{\\sqrt{r \\log (p)}+\\log (p)\\})+O_p\\left(r\\kappa \\sqrt{\\frac{\\mathfrak{s} \\log (p)}{n} } \\right).\\\\\n&=O_p(\\kappa \\sqrt{r})+O_p(1)+o_p\\left(r \\kappa^2\\right)\n\\end{aligned}\n$$\nwhere the second equality follows from $\\|\\beta^*-\\alpha^*\\|_2=\\kappa$ and \\Cref{lem:regression op1 base lemma 2}, and the last equality follows from \\eqref{op1_eq:regression snr}. In addition,\n$$\n\\begin{aligned}\n&\\left(\\widehat{\\alpha}-\\alpha^*\\right)^{\\top} \\sum_{i=\\eta+1}^{\\eta+r} X_i \\epsilon_i \\leq\\|\\widehat{\\alpha}-\\alpha^*\\|_1\\|\\sum_{i=\\eta+1}^{\\eta+r} X_i \\epsilon_i\\|_{\\infty} \\\\\n=& O_p\\left(\\mathfrak{s} \\sqrt{\\frac{\\log (p)}{n}}\\right) O_p(\\sqrt{r \\log (p)}+\\log (p))=O_p(\\kappa \\sqrt{r})+O_p(1)\n\\end{aligned}\n$$\nwhere the second equality follows from \\Cref{lem:regression op1 base lemma 1}. Therefore\n$$\n\\sum_{i=\\eta+1}^{\\eta+r}\\left(y_i-X_i^{\\top} \\widehat{\\alpha}\\right)^2-\\sum_{i=\\eta+1}^{\\eta+r}\\left(y_i-X_i^{\\top} \\alpha^*\\right)^2=O_p(\\kappa \\sqrt{r})+O_p(1)+o_p\\left(r \\kappa^2\\right) .\n$$\n\\textbf{Step 2}. Using the same argument as in the previous step, it follows that\n$$\n\\sum_{i=\\eta+1}^{\\eta+r}\\left(y_i-X_i^{\\top} \\widehat{\\beta}\\right)^2-\\sum_{i=\\eta+1}^{\\eta+r}\\left(y_i-X_i^{\\top} \\beta^*\\right)^2=O_p(\\kappa \\sqrt{r})+O_p(1)+o_p\\left(r \\kappa^2\\right) .\n$$\nTherefore\n\\begin{equation}\n    \\left|\\widehat{\\mathcal{Q}}(\\eta+r)-\\widehat{\\mathcal{Q}}(\\eta)-\\left\\{\\mathcal{Q}^*(\\eta+r)-\\mathcal{Q}^*(\\eta)\\right\\}\\right|=O_p(\\kappa \\sqrt{r})+O_p(1)+o_p\\left(r \\kappa^2\\right)\n    \\label{tmp_eq:op1 regression main prop eq 1}\n\\end{equation}\n\\textbf{Step 3}. Observe that\n$$\n\\begin{aligned}\n& \\mathcal{Q}^*(\\eta+r)-\\mathcal{Q}^*(\\eta) \\\\\n=& \\sum_{i=\\eta+1}^{\\eta+r}\\left(y_i-X_i^{\\top} \\alpha^*\\right)^2-\\sum_{i=\\eta+1}^{\\eta+r}\\left(y_i-X_i^{\\top} \\beta^*\\right)^2 \\\\\n=& \\sum_{i=\\eta+1}^{\\eta+r}\\left(X_i^{\\top} \\alpha^*-X_i^{\\top} \\beta^*\\right)^2-2 \\sum_{i=\\eta+1}^{\\eta+r}\\left(y_i-X_i^{\\top} \\beta^*\\right)\\left(X_i^{\\top} \\alpha^*-X_i^{\\top} \\beta^*\\right) \\\\\n=& \\sum_{i=\\eta+1}^{\\eta+r}\\left(\\alpha^*-\\beta^*\\right)^{\\top}\\left\\{X_i^{\\top} X_i-\\Sigma\\right\\}\\left(\\alpha^*-\\beta^*\\right)+r\\left(\\alpha^*-\\beta^*\\right)^{\\top} \\Sigma\\left(\\alpha^*-\\beta^*\\right)-2 \\sum_{i=\\eta+1}^{\\eta+r} \\epsilon_i\\left(X_i^{\\top} \\alpha^*-X_i^{\\top} \\beta^*\\right)\n\\end{aligned}\n$$\nNote that\n$$\nz_i=\\frac{1}{\\kappa^2}\\left(\\alpha^*-\\beta^*\\right)^{\\top}\\left\\{X_i^{\\top} X_i-\\Sigma\\right\\}\\left(\\alpha^*-\\beta^*\\right)\n$$\nis a sub-exponential random variable with bounded $\\psi_1$ norm. Therefore by \\Cref{lem:subexp uniform loglog law}, uniformly for all $r \\geq 1 \/ \\kappa^2$,\n$$\n\\sum_{i=1}^r z_i=O_p\\left(\\sqrt{r\\left\\{\\log \\log \\left(\\kappa^2 r\\right)+1\\right\\}}+\\sqrt{r \\kappa^2}\\left\\{\\log \\log \\left(\\kappa^2 r\\right)+1\\right\\}\\right) .\n$$\nSince $\\kappa<\\infty$, it follows that\n$$\n\\sum_{i=\\eta+1}^{\\eta+r}\\left(\\alpha^*-\\beta^*\\right)^{\\top}\\left\\{X_i^{\\top} X_i-\\Sigma\\right\\}\\left(\\alpha^*-\\beta^*\\right)=O_p\\left(\\sqrt{r \\kappa^2\\left\\{\\log \\log \\left(\\kappa^2 r\\right)+1\\right\\}}+\\sqrt{r \\kappa^2}\\left\\{\\log \\log \\left(\\kappa^2 r\\right)+1\\right\\}\\right)\n$$\nSimilarly, let\n$$\nw_i=\\frac{1}{\\kappa} \\epsilon_i\\left(X_i^{\\top} \\alpha^*-X_i^{\\top} \\beta^*\\right)\n$$\nThen $\\left\\{w_i\\right\\}_{i=1}^{\\infty}$ are sub-exponential random variables with bounded $\\psi_1$ norm. Therefore by \\Cref{lem:subexp uniform loglog law}, uniformly for all $r \\geq 1 \/ \\kappa^2$,\n$$\n\\sum_{i=1}^r w_i=O_p\\left(\\sqrt{r\\left\\{\\log \\log \\left(\\kappa^2 r\\right)+1\\right\\}}+\\sqrt{r \\kappa^2}\\left\\{\\log \\log \\left(\\kappa^2 r\\right)+1\\right\\}\\right)\n$$\nSince $\\kappa<\\infty$, it follows that\n$$\n\\sum_{i=\\eta+1}^{\\eta+r} \\epsilon_i\\left(X_i^{\\top} \\alpha^*-X_i^{\\top} \\beta^*\\right)=O_p\\left(\\sqrt{r \\kappa^2\\left\\{\\log \\log \\left(\\kappa^2 r\\right)+1\\right\\}}+\\sqrt{r \\kappa^2}\\left\\{\\log \\log \\left(\\kappa^2 r\\right)+1\\right\\}\\right)\n$$\nTherefore\n\\begin{equation}\n    \\mathcal{Q}^*(\\eta+r)-\\mathcal{Q}^*(\\eta) \\geq \\Lambda_{\\min } r \\kappa^2+O_p\\left(\\sqrt{r \\kappa^2\\left\\{\\log \\log \\left(\\kappa^2 r\\right)+1\\right\\}}+\\sqrt{r \\kappa^2}\\left\\{\\log \\log \\left(\\kappa^2 r\\right)+1\\right\\}\\right)\n\\label{tmp_eq:op1 regression main prop eq 2}\n\\end{equation}\nwhere $\\Lambda_{\\min }$ is the minimal eigenvalue of $\\Sigma$.\n\n\\textbf{Step 4}. \\Cref{tmp_eq:op1 regression main prop eq 1} and \\Cref{tmp_eq:op1 regression main prop eq 1} together give, uniformly for all $r \\geq 1 \/ \\kappa^2$,\n$$\n\\Lambda_{\\min } r \\kappa^2+O_p\\left(\\sqrt{r \\kappa^2\\left\\{\\log \\log \\left(\\kappa^2 r\\right)+1\\right\\}}+\\sqrt{r \\kappa^2}\\left\\{\\log \\log \\left(\\kappa^2 r\\right)+1\\right\\}\\right) \\leq O_p(\\kappa \\sqrt{r})+O_p(1)+o_p\\left(r \\kappa^2\\right)\n$$\nThis implies that\n$$\nr \\kappa^2=O_p(1)\n$$\n\\eprf\n\n\n\\bnlem\n\\label{lem:regression op1 base lemma 1}\nSuppose $\\left\\{X_i\\right\\}_{i=1}^n \\stackrel{i . i . d}{\\sim} N_p(0, \\Sigma)$ and $\\left\\{\\epsilon_i\\right\\}_{i=1}^n \\stackrel{i i . d}{\\sim} N\\left(0, \\sigma^2\\right)$. Then it holds that\n$$\n\\begin{aligned}\n&\\mathbb{P}\\left(\\|\\frac{1}{r} \\sum_{i=1}^r X_i X_i^{\\top}-\\Sigma\\|_{\\infty} \\geq C_1\\left(\\sqrt{\\frac{\\log (p)}{r}}+\\frac{\\log (p)}{r}\\right) \\text { for all } 1 \\leq r \\leq n\\right) \\leq n^{-2}, \\\\\n&\\mathbb{P}\\left(\\|\\frac{1}{r} \\sum_{i=1}^r X_i \\epsilon_i\\|_{\\infty} \\geq C_2\\left(\\sqrt{\\frac{\\log (p)}{r}}+\\frac{\\log (p)}{r}\\right) \\text { for all } 1 \\leq r \\leq n\\right) \\leq n^{-2}\n\\end{aligned}\n$$\n\\enlem\n\\bprf\nProof. For the first probability bound, observe that for any $j, k \\in[1, \\ldots, p], X_j X_k-\\Sigma_{j k}$ is subexponential random variable. Therefore for any $r>0$,\n$$\n\\mathbb{P}\\left\\{\\left|\\frac{1}{r} \\sum_{i=1}^r X_{i j} X_{i k}-\\Sigma_{j k}\\right| \\geq x\\right\\} \\leq \\exp \\left(-r c_1 x^2\\right)+\\exp \\left(-r c_2 x\\right)\n$$\nSo\n$$\n\\mathbb{P}\\left\\{\\left\\|\\frac{1}{r} \\sum_{i=1}^r X_{i j} X_{i k}-\\Sigma_{j k}\\right\\|_{\\infty} \\geq x\\right\\} \\leq p \\exp \\left(-r c_1 x^2\\right)+p \\exp \\left(-r c_2 x\\right) .\n$$\nThis gives, for sufficiently large $C_1>0$,\n$$\n\\mathbb{P}\\left\\{\\left\\|\\frac{1}{r} \\sum_{i=1}^r X_{i j} X_{i k}-\\Sigma_{j k}\\right\\|_{\\infty} \\geq C_1\\left(\\sqrt{\\frac{\\log (p n)}{r}}+\\frac{\\log (p n)}{r}\\right)\\right\\} \\leq n^{-3} .\n$$\nBy a union bound,\n$$\n\\mathbb{P}\\left\\{\\left\\|\\frac{1}{r} \\sum_{i=1}^r X_{i j} X_{i k}-\\Sigma_{j k}\\right\\|_{\\infty} \\geq C_1\\left(\\sqrt{\\frac{\\log (p n)}{r}}+\\frac{\\log (p n)}{r}\\right) \\text { for all } 1 \\leq r \\leq n\\right\\} \\leq n^{-2}\n$$\nThe desired result follows from the assumption that $p \\geq n^\\alpha$. The second probability bound follows from the same argument and therefore is omitted for brevity.\n\\eprf\n\n\n\n\n\\bnlem\n\\label{lem:regression op1 base lemma 2}\n Suppose $\\left\\{X_i\\right\\}_{i=1}^n \\stackrel{2.2 . d}{\\sim} N_p(0, \\Sigma)$ and $u \\in \\mathbb{R}^p$ is a deterministic vector such that $|u|_2=1$. Then it holds that\n$$\n\\mathbb{P}\\left(\\|u^{\\top}\\left\\{\\frac{1}{r} \\sum_{i=1}^r X_i X_i^{\\top}-\\Sigma\\right\\}\\|_{\\infty} \\geq C_1\\left(\\sqrt{\\frac{\\log (p)}{r}}+\\frac{\\log (p)}{r}\\right) \\text { for all } 1 \\leq r \\leq n\\right) \\leq n^{-2} .\n$$\n\\enlem\n\\bprf\nFor fixed $j \\in[1, \\ldots, p]$, let\n$$\nz_i=u^{\\top} X_i X_{i j}-u^{\\top} \\Sigma_{\\cdot j},\n$$\nwhere $\\Sigma_{\\cdot j}$ denote the $j$-th column of $\\Sigma$. Note that $z_i$ is a sub-exponential random variable with bounded $\\psi_1$ norm. The desired result follows from the same argument as \\Cref{lem:regression op1 base lemma 1}.\n\\eprf\n\n\n\n\n\n\n\n\n\\section{Discussion}\n\\label{sec:discussion}\n\\section{Detail of algorithms}\n\\label{sec:dp}\n\nThe $\\ell_0$ penalization approach considers the joint optimization problem\n\\begin{equation}\n    \\hat{\\mclP} = \\argmin_{\\mclP} \\left\\lbrace \\sum_{I\\in \\mclP} L(\\hat{\\theta}(\\mclI),\\mclI) + \\gamma |\\mclP| \\right\\rbrace, \\quad \\hat{\\theta}(\\mclI) = \\argmin_{\\theta} L(\\theta,\\mclI),\n    \\label{eq:P_hat_high_dim}\n\\end{equation}\nwhere $\\mclP$ is a collections of points in $(0,T]$ and for any interval $\\mclI = (s,e]$, $\\mclI\\in \\mclP$ means $\\{s,e\\}\\in \\mclP$. If we put the constraint $|\\mclP|\\leq K$, then a brute-force search can find the global minimizer $\\hat{\\mclP}$ with $O(T^K)$ time complexity. For a given $\\gamma$, it can be seen that $|\\hat{\\mclP}|\\leq \\frac{1}{\\gamma}L(\\hat{\\theta}((0,T]),(0,T])$. Instead, by the dynamic programming algorithm \\ref{algorithm:DP} with grid $[T]$, we can solve \\eqref{eq:P_hat_high_dim} with $T^2$ complexity, which is free of parameter $\\gamma$.\n\n\n\n\n\n\\section{Additional Experiments}\n\\label{sec:add_experiment}\n\n\n\\subsection{Selection of $\\gamma$}\nIn the theory of DCDP, we need $\\gamma = C_{\\gamma}K { \\mathcal B_n^{-1}   \\Delta } $, which is increasing with $\\Delta$. If $\\gamma$ gets smaller, it's more likely that the DP will select more false positive change points. In experiments, we can observe a similar phenomenon: in \\Cref{fig:small_big_gamma}, the localization error using CV list of $\\gamma$ with bigger values is smaller than that using CV list with smaller values.\n\nThe next question is, if we include both small and big values of $\\gamma$ in the CV list, can the algorithm pick the big values by itself? The answer is NO. \\Cref{fig:pick_gamma} shows the best gamma that selected by the CV step, and there is no clear pattern for selecting bigger $\\gamma$. On the other hand, \\Cref{fig:error_gamma} shows that bigger $\\gamma$ indeed leads to better localization error. Therefore, in conclusion, we do need bigger $\\gamma$ for DCDP, but the algorithm itself cannot pick bigger $\\gamma$ from a wide range: what we can do is to manually set a CV list of $\\gamma$ with big values.\n\n\\begin{figure}[H]\n    \\centering\n    \\includegraphics[width = 0.92 \\textwidth]{Figure\/small_big_gamma.pdf}\n    \\caption{Localization error vs. $\\mclQ$ with $\\Delta = 5000$. Left: the CV list of $\\gamma$ is $\\{10, 20, 40, 80\\}$; right: the CV list of $\\gamma$ is $\\{1000,2000,3000\\}$. The line is the average and the shaded area shows the upper and lower 0.1 quantiles from 60 trials.}\n    \\label{fig:small_big_gamma}\n\\end{figure}\n\n\\begin{figure}[H]\n    \\centering\n    \\includegraphics[width = 0.92 \\textwidth]{Figure\/pick_gamma.pdf}\n    \\caption{$\\Delta = 5000$. Left: Localization error; right: the selected best $\\gamma$ in the CV. The CV list of $\\gamma$ is $\\{10, 20, 40, 80, 200, 500, 1000, 2000, 3000, 4000, 5000\\}$;. The line is the average and the shaded area shows the upper and lower 0.1 quantiles from 100 trials.}\n    \\label{fig:pick_gamma}\n\\end{figure}\n\n\n\\begin{figure}[H]\n    \\centering\n    \\includegraphics[width = 0.6 \\textwidth]{Figure\/scatter_gamma_error.pdf}\n    \\caption{$\\Delta = 5000$. Scatter plot of localization error versus $\\gamma$ picked in the CV step. Each point corresponds to a trial in the experiment of \\Cref{fig:pick_gamma}.}\n    \\label{fig:error_gamma}\n\\end{figure}\n\\section{Introduction}\n\\label{sec:introduction}\nChange point detection is a classical problem in statistics that can date back to 1940s \\citep{wald1945, page1954}. In 1980s people established asymptotic theory for change point detection methods \\citep{vostrikova1981,james1987,yao_au1989}. Most of these classical papers study the univariate mean model. Recently with more advanced theoretical tools developed in modern statistics, more delicate analysis of change point detection came out in univariate mean models \\citep{wang2018univariate}, in high-dimensional mean models \\citep{cho2015,jirak2015,aston2018,wang_samworth2018}, in covariance models \\citep{aue2009, avanesov2018, wang_yi_rinaldo_2021_cov}, in high-dimensional regression models \\citep{rinaldo2021cpd_reg_aistats, wang2021_jmlr}, in network models \\citep{wang2021network_cpd}, and in temporally-correlated times series \\citep{lavielle2000, davis2006, aue2009, wang2020sepp}.\n\nSuppose we observe $Z_t$ at time $t$ for $t\\in [T]$ and all observations are independent. Each $Z_t$ follows the same family of distribution $\\mathbb{P}(\\theta_t)$ where $\\theta_t$ is a parameter that can change over time. We are interested in capturing abrupt changes in the distributions and equivalently in the parameters. Towards that goal, we assume that there is a collection of change points $ 0=\\eta_0 <  \\eta_1  < \\eta_2 < \\ldots <\\eta_K < \\eta _{K+1}= n  $ such that $\\theta_{t}\\neq \\theta_{t + 1}$ only when $t \\in \\{\\eta_{k}\\}_{k\\in [K]}$. Define the spacing and jump size as\n\\begin{equation}\n    \\Delta_k = \\eta_{k+1} -\\eta_{k},\\ \\kappa_k \n\t  : =   \\|\\theta_{\\eta_{k}+1} - \\theta_{\\eta_k}\\|_2,\n\\end{equation}\nwhere $\\|\\cdot \\|$ is some norm in the parameter space $\\Theta$, and let $\\Delta = \\min_{k\\in [K] } \\Delta_k$, $\\kappa = \\min_{k\\in [K]}\\kappa_k$. Change point detection aims to provide \\textit{consistent} estimates $\\{\\hat{\\eta}_k\\}$.\n\nIt is well-known that the dynamic programming algorithm can solve the $\\ell_0$ penalization problem for change point detection with $O(T^2 \\mclC(T, p))$ time complexity, where $\\mclC(T,p)$ is the time complexity for getting the estimator $\\hat{\\theta}(\\mclI)$. \n\nThe main concern upon the dynamic programming approach is that DP is more expensive to execute in practice compared to the wild binary segmentation. This motivates our project on improving the computational efficiency of DP with theoretical guarantee on the localization rate.\n\nImproving the computational efficiency of change point detection algorithms is an important research direction. \\cite{kovacs2020seeded} propose an algorithm called seeded binary segmentation that can reduce the computational complexity of Wild Binary Segmentation from $O(TM\\mclC(T))$ to $O(T\\log(T)\\mclC(T))$ where $M$ is the number of initial intervals in the WBS algorithm and $\\mclC(T)$ is the time complexity of fitting an estimator on samples of order $T$ which depends on the model.\n\nWe propose a new algorithm for retrospective change point detection called divide and conquer dynamic programming (DCDP) that can greatly improve the computational efficiency of the dynamic programming method that are currently used in the literature. We show by theory and numerical experiments that this algorithm works well for a broad range of problems and thus serve as a unified framework for change point detection.\n\nOur contributions:\n\\begin{enumerate}\n    \\item A new efficient algorithm for change point detection with guarantee\n    \\item It's a unified method\tfor change point detection problems. We establish theoretical guarantee for high-dimensional mean model and high-dim linear model.\n\\end{enumerate}\n\n\n\\section{Methodology}\nWe first discuss the general form of DCDP and its time  complexity of it.\n\nSuppose we have $T$ observations and for any interval $\\mclI \\subset (0,T]$, we have some loss function $L(\\theta,\\mclI)$ and some goodness of fit function $\\mclF(\\theta,\\mclI)$ that are defined according to the setting of the problem. For instance, when considering change points in high-dimensional linear models, we have $L(\\beta,\\mclI) = \\sum_{i \\in \\mathcal I}   (y_i -X_i^\\top \\beta) ^2 + \\lambda\\sqrt{|\\I|} \\|\\beta\\|_1$, where\n\\begin{equation}\n    \\widehat \\beta _{ \\mathcal I } = \\arg\\min_{\\beta \\in \\mathbb R^p } \\sum_{i \\in \\mathcal I}   (y_i -X_i^\\top \\beta) ^2 + \\lambda \\sqrt{|\\I|} \\|\\beta\\|_1,\n\\end{equation}\nand\n\\begin{equation}\n    \\mathcal F  (\\beta, \\I) := \\begin{cases}\n    \\sum_{i \\in \\I } (y_i - X_i^\\top \\beta)^2, &\\text{ when } |\\I|\\geq C_\\mclF { \\mathfrak{s} } \\log(np),\\\\\n    0, &\\text{otherwise}.\n    \\end{cases}.\n\\end{equation}\nDCDP contains two steps: divide and conquer. In the divide step, we randomly sample $\\mclQ$ integer points $\\{s_i\\}_{i\\in [\\mclQ]}\\subset [n]$ and find best approximations to the true change points by \\Cref{algorithm:DP}. The conquer step aims to further improve the localization accuracy based on the output of the divide step. It applies the local refinement in \\Cref{algo:local_refine_general} to the output of the divide step.\n\\begin{algorithm}[htbp]\n\t\\textbf{INPUT:} Data $\\{ y_i, X_i \\}_{i =1}^{n }$, tuning parameters $\\lambda, \\gamma,  \\mclQ  > 0$.\n \n Randomly uniformly  sample $\\mclQ $ points $\\{ s_1, \\ldots, s_ \\mclQ \\}$ from $(0, n)$.\n \t  \n(Divide) Get $\\{\\tilde{\\eta}_k\\}_{k\\in [\\tilde{K}]}$ with DP $(\\{y_i, X_i\\}_{i =1}^{n},\\{s_i\\}_{i\\in [\\mclQ]}, \\lambda, \\gamma)$.\n\n(Conquer) $\\{\\hat{\\eta}_k\\}_{k\\in [\\hat{K}]}\\leftarrow {\\rm Refinement}(\\{\\tilde{\\eta}_k\\}_{k\\in [\\tilde{K}]})$. \n\t\n\\textbf{OUTPUT:}  The estimated change points $\\{\\hat{\\eta}_k\\}_{k\\in [\\hat{K}]}$.\n \t\n \\caption{Divide and Conquer Dynamic Programming. DCDP $(\\{y_i, X_i\\}_{i =1}^{n}, \\lambda, \\gamma ,  \\mclQ)$}\n\\label{algorithm:DCDP_divide}\n\\end{algorithm} \n\n\\begin{algorithm}[htbp]\n\t\\textbf{INPUT:} Data $\\{ y_i, X_i \\}_{i =1}^{n }$, tuning parameters $\\lambda, \\gamma$, grid $\\{s_i\\}_{i\\in [m]}$ to search over.\n \t \n \t Set $\\mathcal P= \\emptyset$,  $\\mathfrak{ p} =\\underbrace{(-1,\\ldots, -1)}_{ n }  $,  $B =\\underbrace{( \\gamma, \\infty , \\ldots,  \\infty)}_{n } $.  Denote $B_i$ to be  the $i$-$th$ entry of $B$. \n \t \n \t\\For {$r  $ in $\\{s_i\\}_{i\\in [m]}$} {\n \t \\For {$l  $ in $\\{s_i\\}_{i\\in [m]}$, $ l<r  $} {\n \t $$ b \\leftarrow B_{l}+ \\gamma + \\mathcal F (\\hat{\\theta}_\\mclI, \\mclI)  \\quad \\text{where} \\quad \\mclI=( l, \\ldots, r]  ;$$\n \t \\If {$b <  B_r $} {\n \t $B_r \\leftarrow  b $;\n \t $\\mathfrak{p}_r  \\leftarrow l  $.\n \t } \n \t}\n \t}\n\n To compute  the change point estimates from $  \\mathfrak{p}  \\in \\mathbb N^n $, \t\n $ k \\leftarrow  n  $.\n \n \t  \\While {$k >1$} {\n \n  $ h  \\leftarrow \\mathfrak{p}_ k  $;\n  $\\mathcal P = \\mathcal P \\cup  h $;\n  $ k  \\leftarrow h $.\n \t  }\n\t\\textbf{OUTPUT:}  The estimated change points $\\mathcal P$. \n \n \\caption{Dynamic Programming. DP $(\\{y_i, X_i\\}_{i =1}^{n},\\{s_i\\}_{i\\in [m]}, \\lambda, \\gamma)$}\n\\label{algorithm:DP}\n\\end{algorithm} \n\n\n\\begin{algorithm}[htbp]\n\t\\textbf{INPUT:} Data $\\{(Z_i)\\}_{i=1}^{n}$, $\\{\\widetilde{\\eta}_k\\}_{k = 1}^{\\widetilde{K}}$ ,  $\\zeta > 0$.\n\n\tLet $(\\widetilde{\\eta}_0, \\widetilde{\\eta}_{\\widetilde{K} + 1}) \\leftarrow (0, n)$, $n_0\\leftarrow C_{\\mclF} { \\mathfrak{s} } \\log(np)$.\n\t\n\t\\For{$k = 1, \\ldots, \\widetilde{K}$} { \n\t$(s_k, e_k) \\leftarrow (2\\widetilde{\\eta}_{k-1}\/3 + \\widetilde{\\eta}_{k}\/3, \\widetilde{\\eta}_{k}\/3 + 2\\widetilde{\\eta}_{k+1}\/3)$\n\n\n\t\t\\begin{align}\n\t\t\t\\left(\\hat{\\eta},\\hat{\\theta}^{(1)},\\hat{\\theta}^{(2)}\\right) &\\leftarrow \\argmin_{\\substack{\\eta \\in \\{s_k + n_0, \\ldots, e_k - n_0\\} \\\\ \\theta^{(1)}, \\theta^{(2)} \\in \\mathbb{R}^{p},\\, \\theta^{(1)} \\neq \\theta^{(2)}}}   \\Bigg\\{\\mclF(\\theta^{(1)},(s_k,\\eta)) + \\mclF(\\theta^{(2)}, [\\eta,e_k)) \\nonumber \\\\\n\t\t    & \\hspace{1.5cm} + \\zeta  \\sum_{i = 1}^p \\sqrt{(\\eta - s_k)(\\theta^{(1)})_{i}^2 + (e_k - \\eta)(\\theta^{(2)})_{i}^2}\\Bigg\\} \\label{eq-g-lasso}\n\t\t\\end{align}\n\t}\n\t\\textbf{OUTPUT:} $\\{\\widehat{\\eta}_k\\}_{k = 1}^{\\widetilde{K}}$.\n\\caption{Local Refinement. \n\\label{algo:local_refine_general}\n\\end{algorithm} \n\n\nIn general, the computational complexity of the divide step is $O(\\mclQ^2 \\mclC(n))$ where $\\mclC(n)$ is the computational complexity of getting $\\hat{\\theta}(\\mclI)$ where $|\\mclI|\\asymp n$. In contrast, the standard DP costs $O(n^2 \\mclC(n))$. Thus we want smaller $\\mclQ$ to make the improvement significant. As we will show later, in the current framework of DCDP, when the minimal spacing is $\\Delta\\geq \\mclB_n \\Delta_0$, we need $\\mclQ \\geq \\frac{n}{\\Delta}\\mclB_n \\log(n)$ to guarantee consistency. Here $\\{\\mclB_n\\}$ is any diverging sequence and $\\Delta_0$ is usually the minimal sample size for the estimator of the model to have $o(1)$ estimation error. Therefore, the complexity of the divide step is in general\n\\[\nO(\\left(\\frac{n}{\\Delta}\\right)^2\\log^3(n) \\cdot \\mclC(n)).\n\\]\nThe reduction in complexity gets bigger as $\\Delta$ gets bigger. For models with linear structures such as linear regression and covariance estimation, by some pre-calculation step that costs $O(n)$, we can further reduce $\\mclC(n)$ to be free of $n$, as we will discuss next. However, for nonlinear models like logistic regression, $\\mclC(n)$ is generally at least of order $n$. In either case, when $\\Delta\\asymp n$, the complexity will be of order $O(n\\log^3(n))$, which is a big improvement compared to the standard DP.\n\\section{Theoretical analysis}\nWe present some results on the localization error of DCDP in the case of mean model and high dimensional linear regression. Briefly, with the reduction on the computational complexity, the localization error with a single divide step will be larger than that of the vanilla DP by a term $ { \\mathcal B_n^{-1\/2}   \\Delta } $. As is shown in \\Cref{cor:mean local refinement} and \\Cref{cor:regression local refinement}, this extra term can be removed by applying a local refinement algorithm to the output of the divide step.\n\nConsider a generic interval $\\I = (s,e)$ containing a single change point $\\eta$ and let $\\I_1 = (s,\\eta]$, $\\I_2 = (\\eta,e)$. In the proof of DP, a key property to prove is\n\\begin{equation}\n    L(\\widehat{\\theta}_{\\I},\\I) - L(\\theta_1^*,\\I_1) - L(\\theta_1^*,\\I_1) \\geq {\\rm signal}(\\I_1,\\I_2) + {\\rm noise},\n\\end{equation}\nand when $|\\I_1|,|\\I_2|$ are large enough,\n\\begin{equation}\n    \\gamma < {\\rm signal}(\\I_1,\\I_2) - |{\\rm noise}|,\\ {\\rm signal}(\\I_1,\\I_2) > |{\\rm noise}|.\n\\end{equation}\nFor DCDP, we need to prove that in addition,\n\\begin{equation}\n    L(\\widehat{\\theta}_{\\I},\\I) - L(\\theta_1^*,\\I_1) - L(\\theta_1^*,\\I_1) \\leq {\\rm approx}(\\I_1,\\I_2) + {\\rm noise},\n\\end{equation}\nand when $|\\I_1|,|\\I_2|$ are small enough,\n\\begin{equation}\n    \\gamma > {\\rm approx}(\\I_1,\\I_2) + |{\\rm noise}|.\n\\end{equation}\n\n\n\n\n\\subsection{Change in the mean}\n\n\\bnassum[Mean model]\n\\label{assp: DCDP_mean}\n   Suppose  for $1 \\le i \\le n$, the data  $\\{X_{i } \\}_{1\\le i \\le n }   $ satisfy mean model $X_{i} =    \\mu _i    +\\epsilon_i\\in \\mathbb{R}^p$ with a collection of change points $ 0=\\eta_0 <  \\eta_1  < \\eta_2 < \\ldots <\\eta_K < \\eta _{K+1}= n  $ and that  the following conditions hold.   \n\\begin{enumerate}\n     \\item The measurement errors are independent mean-zero subgaussian random variables with $\\sigma_{\\epsilon} =\\sup_{i} \\|\\epsilon_i\\|_{\\psi_2}<\\infty$.\n   \n     \\item For each $i \\in \\{ 1,\\ldots, n\\}$,    there exists a collection of subsets  $  S_i \\subset\\{1, \\ldots, p \\}$,  such that \n\t  $$  \\mu _ { i,j} =0 \\text{ if } j \\not \\in S_i.$$\n\t  In addition the  cardinality  of the support satisfies $|S_i|\\leq \\mathfrak{s} $. \n\t \\item Denote \n\t $ \\kappa_k \n\t  : =   \\| \\mu_{\\eta_{k}+1}^* - \\mu_{\\eta_k}^* \\|_2 $.   Suppose that \n\t $  \\kappa _k \\asymp  \\kappa$ for all $1\\le k \\le K$.\n\t \\item Denote   \n\t  $\\Delta_k = \\eta_{k+1} -\\eta_{k}. $ \n\t and \n\t $\\Delta = \\min_{1\\le k \\le K } \\Delta_k$. Let $ \\mathcal{B}_n$ be any diverging sequence.  Suppose that\n\t  $ \\Delta \\kappa^2  \\ge  \\mathcal{B}_n \\sigma_{\\epsilon}^2  { \\mathfrak{s} }  \\log(np)  $.\n\\end{enumerate}\n\\enassum\n\n\\bnthm \\label{thm:DCDP mean}\nSuppose  \\Cref{assp: DCDP_mean} holds and $\\gamma\\ge C_\\gamma K { \\mathcal B_n^{-1}   \\Delta } \\kappa^2$  for sufficiently large constant $C_\\gamma$. Let \n $\\widehat { \\mathcal P }  $ denote the output of  \\Cref{algorithm:DP} with \n \\begin{equation}\n    \\mathcal F  (\\widehat{\\mu}_\\I,\\I) := \\begin{cases}\n    \\sum_{i \\in \\I } \\|X_i - \\widehat \\mu_\\I \\|_2 ^2, &\\text{ when } |\\I|\\geq C_\\mclF\\sigma_{\\epsilon}^2 { \\mathfrak{s} } \\log(np),\\\\\n    0, &\\text{otherwise},\n    \\end{cases}.\n\\end{equation}\nwhere $\\widehat{\\mu}_{\\I}$ is given by\n\\begin{equation}\n    \\widehat{\\mu}_{\\I} = \\argmin_{\\mu\\in \\mathbb{R}^p}\\|X_i - \\mu\\|_2^2 + \\lambda \\sqrt{|\\I|}\\|\\mu\\|_1,\n\\end{equation}\nwith $\\lambda = C_{\\lambda}\\sqrt{\\log(np)}$. Here $C_{\\lambda}$ and $C_{\\mclF}$ are sufficiently large universal constants. Let $\\{ \\widehat \\eta_k\\}_{k=1}^{\\widehat K } $ be  the collection of change points induced by $\\widehat { \\mathcal P}$.\n Then\n with probability  at least $1 - C n ^{-3}$, $\\widehat K = K$ and that\n \\[ \n \\max_{ 1\\le k \\le K } |\\eta_k-\\widehat \\eta_k|   \\lesssim   \\sigma_{\\epsilon}^2\\frac{  \\log(n) +\\gamma }{\\kappa^2 } +  { \\mathcal B_n^{-1\/2}   \\Delta }  .\n \\]\n \\enthm\n \n\\bncor\n\\label{cor:mean local refinement}\nUnder the conditions of \\Cref{thm:DCDP mean}, let $\\{\\widehat{\\eta}_k\\}_{k = 1}^{K}$ be a set of time points satisfying $\\widehat K = K$ and\n\t\\begin{equation}\\label{eq-lr-cond-1}\n\t\t\\max_{k = 1, \\ldots, K} |\\widehat{\\eta}_k - \\eta_k| \\leq \\Delta\/5.\n\t\\end{equation}\n\tLet $\\{\\widetilde{\\eta}_k\\}_{k = 1}^{\\widehat{K}}$ be the change point estimators generated from \\Cref{algo:local_refine_general} with $\\{\\widehat{\\eta}_k\\}_{k = 1}^{K}$ and $\\zeta = 0$. Then\n\t\\[ \n \\max_{ 1\\le k \\le K } (\\eta_k-\\widehat \\eta_k)\\kappa^2   = O_P(\\sigma_{\\epsilon}^2).\n \\]\n\\encor\n\n\\subsection{Change in the covariance}\n\n\\bnassum[Covariance model]\n\\label{assp:DCDP_covariance}\n  Suppose  for $i\\in [n]$,    the data  $\\{X _{i } \\}_{i\\in [n]}   $ are mean-zero Gaussian variables in $\\mathbb{R}^p$ with non-singular covariance matrix $\\Sigma_i = \\mathbb{E}[X_iX_i^\\top]$ and Orlicz norm $g_X = \\sup_{i\\geq 1}\\|X_i\\|_{\\psi_2}<\\infty$. Assume there is a collection of change points $ 0=\\eta_0 <  \\eta_1  < \\eta_2 < \\ldots <\\eta_K < \\eta _{K+1}= n  $ and that  the following conditions hold.   \n\\begin{enumerate}\n    \\item For $i\\in [n]$, $c_XI_p \\preceq \\Sigma_i \\preceq C_X I_p$ where $c_X>0, C_X<\\infty$ are universal constants.\n\t \\item Denote \n\t $ \\kappa_k \n\t  : =   \\|\\Sigma^{-1}_{\\eta_{k}+1} - \\Sigma^{-1}_{\\eta_k}\\|_F$.   Suppose that \n\t $  \\kappa _k \\asymp  \\kappa$ for all $1\\le k \\le K$.\n\t \\item Denote   \n\t  $\\Delta_k = \\eta_{k+1} -\\eta_{k}. $ \n\t and \n\t $\\Delta = \\min_{1\\le k \\le K } \\Delta_k$. Let $ \\mathcal{B}_n$ be any diverging sequence.  Suppose that\n\t  $ \\Delta \\kappa^2  \\geq  \\mathcal{B}_n \\frac{g_X^4}{c_X^4} p^2\\log(np)$.\n\\end{enumerate}\n\\enassum\n\n\\bnrmk\nSince $\\{X_i\\}_{i\\in [n]}$ are Gaussian, we can make $g_X = C_X$ since $\\|X_i\\|_{\\psi_2} = \\lambda_{\\max}(\\Sigma_{X})$. I am trying to generalize to subgaussian variables with convex concentration property.\n\\enrmk\n\n\n\\bnthm\n\\label{thm:DCDP covariance}\nSuppose \\Cref{assp:DCDP_covariance} holds and $\\gamma\\ge C_\\gamma K { \\mathcal B_n^{-1}   \\Delta } \\kappa^2$  for sufficiently large constant $C_\\gamma$. Let \n $\\widehat { \\mathcal P }  $ denote the output of  \\Cref{algorithm:DP} with \n \\begin{equation}\n    \\mathcal F  (\\widehat{\\Omega}_\\I,\\I) := \\begin{cases}\n    \\sum_{i \\in \\I } {\\rm Tr}[\\widehat{\\Omega}^\\top X_iX_i^\\top] - |\\I|\\log|\\widehat{\\Omega}| , &\\text{ when } |\\I|\\geq C_\\mclF\\frac{g_X^4}{c_X^2}p\\log(np),\\\\\n    0, &\\text{otherwise},\n    \\end{cases}.\n\\end{equation}\nwhere $\\widehat{\\Omega}_{\\I}$ is given by\n\\begin{equation}\n    \\widehat{\\Omega}_{\\I} = \\argmin_{\\Omega\\in \\mathbb{S}^p_+} \\sum_{i \\in \\I } {\\rm Tr}[{\\Omega}^\\top X_iX_i^\\top] - |\\I|\\log|\\Omega|.\n\\end{equation}\nHere $C_{\\mclF}$ is a sufficiently large universal constant. Let $\\{ \\widehat \\eta_k\\}_{k=1}^{\\widehat K } $ be  the collection of change points induced by $\\widehat { \\mathcal P}$.\n Then\n with probability  at least $1 - C n ^{-3}$, $\\widehat K = K$ and that\n \\[ \n \\max_{ 1\\le k \\le K } |\\eta_k-\\widehat \\eta_k|   \\lesssim   \\frac{g_X^4C_X^2}{c_X^6}\\frac{ p^2 \\log(np) +\\gamma }{\\kappa^2 } +  { \\mathcal B_n^{-1\/2}   \\Delta }  .\n \\]\n\\enthm\n\n\n\n\\bncor\n\\label{cor:covariance local refinement}\nUnder the conditions of \\Cref{thm:DCDP covariance}, let $\\{\\widehat{\\eta}_k\\}_{k = 1}^{K}$ be a set of time points satisfying $\\widehat K = K$ and\n\t\\begin{equation}\\label{eq-lr-cond-1}\n\t\t\\max_{k = 1, \\ldots, K} |\\widehat{\\eta}_k - \\eta_k| \\leq \\Delta\/5.\n\t\\end{equation}\n\tLet $\\{\\widetilde{\\eta}_k\\}_{k = 1}^{\\widehat{K}}$ be the change point estimators generated from \\Cref{algo:local_refine_general} with\n\t$$\n\t\\mclF(\\theta, \\I) = \\|\\theta - \\frac{1}{|\\I|}\\sum_{i\\in \\I}X_iX_i^\\top\\|_F^2,\n\t$$\n\t$\\{\\widehat{\\eta}_k\\}_{k = 1}^{K}$ and $\\zeta = 0$. Then  \n \\[ \n \\max_{ 1\\le k \\le K } (\\eta_k-\\widehat \\eta_k)\\kappa^2  = O_P(\\log(n)).\n \\]\n\\encor\n\n\n\n\n\n\n\\subsection{Change in the linear model}\n\n\n \\bnassum[High-dimensional linear regression]\\label{assp:dcdp_linear_reg}\n   Suppose  for $1 \\le i \\le n$,    $\\{X_{i } \\}_{1\\le i \\le n }  \\subset \\mathbb R^p   $ and $ \\{ y_i\\}_{i=1}^n \\subset \\mathbb R  $ satisfy the regression model\n\t$y_{i  } =   X_i^\\top \\beta ^* _i     +\\epsilon_i  $ with a collection of change points $ 0=\\eta_0 <  \\eta_1  < \\eta_2 < \\ldots <\\eta_K < \\eta _{K+1}=n $\nand that  the following conditions hold.   \n\\begin{enumerate}\n    \\item Suppose that $\\{ X_i\\}_{i=1}^n \\overset{i.i.d.}{\\sim} N_p(0, \\Sigma )$  and that the minimal and the maximal eigenvalues of  $\\Sigma$ satisfy \n\t  $$  0< c_x \\le  \\Lambda_{\\min}   (\\Sigma) \\le    \\Lambda_{\\max} (\\Sigma ) \\le C_x < \\infty .   $$ In addition, suppose that $ \\{ \\epsilon_i \\}_{i=1}^n  \\overset{i.i.d.}{\\sim}  N(0, \\sigma^2_ \\epsilon)$ is a collection of  random  errors independent of $\\{ X_i\\}_{i=1}^n $.\n\t  \\item For each $i \\in \\{ 1,\\ldots, n\\}$,    there exists a collection of subsets  $  S_i \\subset\\{1, \\ldots, p \\}$,  such that \n\t  $$  \\beta^* _ { i,j} =0 \\text{ if } j \\not \\in S_i.$$\n\t  In addition the  cardinality  of the support satisfies $|S_i|\\leq \\mathfrak{s} $. \n\t  \\item There exists a parameter  $\\kappa>0$ such that for each  $ k \\in \\{ 0,\\ldots, K\\}   $,  it holds that\n\t $$  \n\t   \\| \\beta^* _{\\eta_k+1} - \\beta_{\\eta_k} ^* \\|_2 \\asymp \\kappa  .$$\n\t   In addition,  suppose that $ \\kappa  < C_\\kappa$  for some universal constant $C_{\\kappa}<\\infty$.\n\t   \\item Denote the spacing of change points as\n\t  $\\Delta_k = \\eta_{k+1} -\\eta_{k}. $ \n\t and \n\t $\\Delta = \\min_{1\\le k \\le K } \\Delta_k$. It holds that $\\Delta  \\kappa ^2 \\geq  \\mathcal{B}_n (\\sigma_{\\epsilon}^2\\vee 1)\\mathfrak{s}\\log(np) $.\n\\end{enumerate}\n\\enassum\n\n\n \\bnthm \\label{thm:DCDP regression}\nSuppose \\Cref{assp:dcdp_linear_reg} holds and $\\gamma\\ge C_\\gamma K  { \\mathcal B_n^{-1}   \\Delta } \\kappa^2$  for sufficiently large constant $C_\\gamma$. Let \n $\\widehat { \\mathcal P }  $ denote the output of  \\Cref{algorithm:DCDP_divide} and\n \\[\n \\mathcal F  (\\hat{\\beta}_{\\I},\\mclI) :  = \\begin{cases} \n \\sum_{i \\in \\mclI } (y_i - X_i^\\top \\widehat \\beta _\\mclI ) ^2  &\\text{if } |\\mclI|\\ge C_\\mclF  { \\mathfrak{s} } \\log(np),\n \\\\ \n 0 &\\text{otherwise,}  \n \\end{cases}   \n \\]\n where $C_{\\mclF}$ is a universal constant. Let \n$ \\{ \\widehat \\eta_k\\}_{k=1}^{\\widehat K } $ be  the collection of change points induced by $\\widehat { \\mathcal P}$.\n Then\n with probability  at least $1 - C n ^{-3}$, $\\widehat K = K$ and that\n $$ \\max_{ 1\\le k \\le K } |\\eta_k-\\widehat \\eta_k|   \\lesssim  (\\sigma_{\\epsilon}^2\\vee 1)\\bigg( \\frac{  \\mathfrak{s} \\log(np) +\\gamma }{\\kappa^2 }\\bigg) +   { \\mathcal B_n^{-1\/2}   \\Delta } $$\n \\enthm\n \n Once we get the output $\\{\\hat{\\eta}_k\\}_{k\\in [\\hat{K}]}$ from the DP, we can apply the local refinement algorithm and get refined estimates $\\{\\widetilde{\\eta}_k\\}_{k\\in [\\hat{K}]}$ that further reduce the term $ { \\mathcal B_n^{-1\/2}   \\Delta } $ in the localization error.\n \n\\bncor\n \\label{cor:regression local refinement}\nUnder the conditions of \\Cref{thm:DCDP regression}, let $\\{\\widehat{\\eta}_k\\}_{k = 1}^{K}$ be a set of time points satisfying $\\widehat K = K$ and\n\t\\begin{equation}\\label{eq-lr-cond-1}\n\t\t\\max_{k = 1, \\ldots, K} |\\widehat{\\eta}_k - \\eta_k| \\leq \\Delta\/5.\n\t\\end{equation}\n\tLet $\\{\\widetilde{\\eta}_k\\}_{k = 1}^{\\widehat{K}}$ be the change point estimators generated from \\Cref{algo:local_refine_general} with $\\{\\widehat{\\eta}_k\\}_{k = 1}^{K}$  and $\\zeta = 0$. Then\n\t\\begin{align*}\n\t \\max_{k = 1, \\ldots, K}(\\widetilde{\\eta}_k - \\eta_k)\\kappa^2 = O_P(\\sigma_{\\epsilon}^2\\vee 1).\n\t\\end{align*}\n\\encor","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nThe goal of this work is to study the complexity classes {\\sf P} and \n{\\sf NP} via functions, and via semigroups of functions, rather than \njust as sets of languages.\nThis approach is intuitive (in particular, because of the immediate \nconnection with certain one-way functions), and quickly leads to results. \nIt is not clear whether this will contribute to a solution of the famous \n{\\sf P}-vs.-{\\sf NP} problem, but the semigroups considered here, as well \nas the ``inversive reductions'' and the accompanying completeness results \nfor one-way functions, are interesting in their own right. \n \nThe starting point is a certain kind of {\\em one-way functions}, and the \nwell-known fact that one-way functions of this kind exist iff \n${\\sf P} \\neq {\\sf NP}$. \n\nFirst, some notation: \nWe fix an alphabet $A$, which will be $\\{0,1\\}$ unless another alphabet\nis explicitly mentioned. The set of all strings over $A$, including the\nempty string, is denoted by $A^*$.\nFor a partial function $f$, the domain (i.e., the inputs $x$ for which\n$f(x)$ is defined) is denoted by ${\\sf Dom}(f)$. The image set of $f$ is\ndenoted by ${\\sf Im}(f)$ or by $f(A^*)$ or $f({\\sf Dom}(f))$. As a rule we\nwill use partial functions, even when the word ``partial'' is omitted; we\nsay {\\it total function} for functions whose domain is $A^*$.  \nAs usual,\n{\\sf P} and {\\sf NP} are the class of languages accepted by deterministic,\nrespectively nondeterministic, polynomial-time Turing machines\n\\cite{DuKo,Papadim}.\n\n\\medskip\n\n\\noindent {\\bf Definition scheme:} A function $f: A^* \\to A^*$ is called \n{\\bf ``one-way''} \\ iff \\ from $x$ and a description of $f$ it is \n{\\sl ``easy''} to compute $f(x)$, but from $f$ and $y \\in {\\sf Im}(f)$ it \nis {\\sl ``difficult''} to find any $x \\in A^*$ such that $f(x) = y$. \n\n\\medskip\n\nThis is an old idea going back at least to W.S.\\ Jevons in 1873, who also \ncompared the difficulties of multiplication and factorization of integers \n(as pointed out in \\cite{Handbook}). \nThe concept became well-known after the work of Diffie and Hellman\n\\cite{DiffHell}.  \nLevin's paper \\cite{LevinTale} discusses some deeper connections of one-way \nfunctions.\nThe definition scheme can be turned into precise definitions, in many\n(non-equivalent) ways, by defining ``easy'' and ``difficult'' (and, if needed, \n``description'' of $f$).\n \n\n\\begin{defn}\nA partial function $f: A^* \\to A^*$ is {\\em polynomially balanced} iff \nthere exists polynomials $p, q$ such that for all $x \\in {\\sf Dom}(f)$:\n \\, $|f(x)| \\leq p(|x|)$ and $|x| \\leq q(|f(x)|)$.\n\\end{defn}\nWe call the polynomial $q$ above an {\\em input balance} function of $f$.\nThe word ``honest'' is often used in the literature for polynomially \nbalanced.\n\nWe introduce the following set of ``easy'' functions.\n\n\\begin{defn} {\\bf (the semigroup {\\sf fP}).} \n \\ {\\sf fP} is the set of partial functions $f: A^* \\to A^*$ that are \npolynomially balanced, and such that \n \\ $x \\in {\\sf Dom}(f) \\longmapsto f(x)$ \\ is computable by a \ndeterministic polynomial-time Turing machine. \n(It follows from the second condition that ${\\sf Dom}(f)$ is in {\\sf P}.)\n\\end{defn}\nAs a rule, a machine that computes a partial function $f$ can always also\nbe used as an acceptor of ${\\sf Dom}(f)$.  \n\nWhen $A$ is an arbitrary alphabet (as opposed to $\\{0,1\\}$) we write \n${\\sf fP}_A$ or ${\\sf fP}_{|A|}$.\nThe complexity class {\\sf fP} is different from the complexity class \n{\\sf FP}, considered in the literature \\cite{Papadim}; {\\sf FP} is a set \nof relations (viewed as search problems) whereas {\\sf fP} is a set of \npartial functions. \nIt is easy to see that ${\\sf fP}_{|A|}$ is closed under composition, so it \nis a semigroup. \n\n\n\\begin{defn} {\\bf (worst-case deterministic one-way function).}  \nA partial function $f: A^* \\to A^*$ is {\\em one-way} iff $f \\in {\\sf fP}$, \nbut there exists {\\em no} deterministic polynomial-time algorithm which, on \nevery input $y \\in {\\sf Im}(f)$, \noutputs some $x \\in A^*$ such that \\ $f(x) = y$.\nThere is no requirement when $y \\not\\in {\\sf Im}(f)$.\n\\end{defn} \nThis kind of one-way functions is defined in terms of worst-case complexity, \nhence it is not ``cryptographic''. However, it is relevant to the \n{\\sf P}-vs.-{\\sf NP} problem because of the following fact (see e.g., \n\\cite{HemaOgi} p.\\ 33 for a proof and history).  \n\n\\begin{pro} {\\bf (folklore).} \n \\ One-way functions (in the worst-case sense) exist iff \n \\ ${\\sf P} \\neq {\\sf NP}$. \\ \\ \\ $\\Box$\n\\end{pro}\nThe concept of an {\\it inverse} is central to one-way functions.\nThe following lemma is straightforward to prove.\n\n{\\sf Notation:} \nFor a partial function $f$ and a set $S$, the restriction of $f$ to $S$ is \ndenoted by $f|_S$; for the restriction of the identity map to $S$ we simply \nwrite ${\\sf id}_S$.\n\n\n\\begin{lem} \\label{concept} {\\bf (concept of inverse).} \n\n\\noindent\nFor partial functions $f, f': A^* \\to A^*$ the following are equivalent.\n\n\\smallskip\n\n\\noindent $\\bullet$ \\ For all $y \\in {\\sf Im}(f)$, \\ $f'(y)$ is defined\nand $f(f'(y)) = y$.  \n\n\\smallskip\n\n\\noindent $\\bullet$ \\ $f \\circ f'|_{{\\sf Im}(f)} = {\\sf id}_{{\\sf Im}(f)}$ .\n\n\\smallskip\n\n\\noindent $\\bullet$ \\ $f \\circ f' \\circ f = f$. \\ \\ \n\n\\smallskip\n\n\\noindent These properties imply ${\\sf Im}(f) \\subseteq {\\sf Dom}(f')$.  \n \\ \\ \\ \\ \\  $\\Box$\n\\end{lem}\n\n\n\\begin{defn} \\ A function $f'$ such that $f \\circ f' \\circ f = f$ is called \nan {\\em inverse of} $f$.\n\\end{defn}\nThe following recipe gives more intuition about inverses.\n\n\\medskip\n\n\\noindent Pseudo-algorithm: {\\it How any inverse $f'$ of a given $f$ is \nmade.}\n\n\\smallskip\n\n\\noindent (1) \\ Choose ${\\sf Dom}(f')$ such that \n${\\sf Im}(f) \\subseteq {\\sf Dom}(f')$. \n\n\\smallskip\n\n\\noindent (2) \\ For every $y \\in {\\sf Im}(f)$, choose $f'(y)$ to be any\n$x \\in f^{-1}(y)$.\n\n\\smallskip\n\n\\noindent (3) \\ For every $y \\in {\\sf Dom}(f') - {\\sf Im}(f)$, choose \n$f'(y)$ arbitrarily in $A^*$.  \n\n\\bigskip\n\n\\noindent {\\bf Remark.} When $f'$ is an inverse of $f$, the restriction \n$f'|_{ {\\sf Im}(f)}: y \\in {\\sf Im}(f) \\mapsto f'(y)$ is the {\\em choice\nfunction} corresponding to $f'$. For set theory in general, the existence \nof choice functions (and the existence of inverses) for every partial \nfunction is equivalent to the Axiom of Choice. The existence of one-way \nfunctions in our sense amounts to the non-existence of certain inverses; \nthe existence of one-way functions is thus equivalent to the {\\em non-validity \nof the Axiom of Choice} in the (highly restricted) context of deterministic \npolynomial time-complexity. \n\nFrom the definition of polynomially balanced we can see now:\n{\\em If $f$ is polynomially balanced then so is every choice function \ncorresponding to any inverse of $f$. } \n\n\n\\begin{defn} \\ Let $S$ be a semigroup. An element $x \\in S$ is called\n{\\em regular} iff there exists $x' \\in S$ such that $x x' x = x$. \nIn that case, $x'$ is called an {\\em inverse} of $x$. \nA semigroup $S$ is called regular iff every element of $S$ is regular.\n\nLet $S$ be a monoid with identity {\\bf 1}. Then $x'$ is a {\\em left}- (or \n{\\em right}-) inverse of $x$ iff $x'x = {\\bf 1}$ (respectively \n$x x' = {\\bf 1}$). If $x'x = x x' = {\\bf 1}$ then $x'$ is a two-sided inverse\nor group-inverse. {\\rm (See \\cite{CliffPres,Grillet}.)}\n\\end{defn}\n\n\\noindent The following summarizes what we have seen, and gives the\ninitial motivation for studying the class {\\sf NP} via certain functions\nand semigroups. \n\n\\begin{pro} \\  \nThe monoid {\\sf fP} is {\\em regular} iff one-way functions do {\\sl not} \nexist (iff \\ ${\\sf P} = {\\sf NP}$).  \\ \\ \\ \\ \\ $\\Box$\n\\end{pro} \nSome properties of the image set of functions in {\\sf fP}:\n\n\n\\begin{pro}  \\label{ImfP} \\!\\!\\! .\n\n\\noindent (1) \\ For every $f \\in {\\sf fP}$, \\, ${\\sf Im}(f) \\in {\\sf NP}$.\n\n\\noindent (2) \\ If $f \\in {\\sf fP}$ and $f$ is regular then\n${\\sf Im}(f) \\in {\\sf P}$.\n\n\\noindent (3) \\ For every language $L \\in {\\sf NP}$ there exists\n$f_L \\in {\\sf fP}$ such that $L = {\\sf Im}(f_L)$. \n\n \\ Moreover, the set of functions $\\{f_L \\in {\\sf fP} : L \\in {\\sf NP}\\}$ \ncan be chosen so that $f_L$ is regular iff \\ $L \\in {\\sf P}$.\nThe map $L \\in {\\sf NP} \\mapsto f_L \\in {\\sf fP}$ is an embedding of\n{\\sf NP} (as a set) into {\\sf fP}, such that {\\sf P} (and only {\\sf P}) is \nmapped into the regular elements of {\\sf fP}. \n\\end{pro}\n\\noindent {\\bf Proof.} (1) is obvious (polynomial balance is needed).\n\n\\noindent (2)  Let $f' \\in {\\sf fP}$ be an inverse of $f$.\nIf $y \\in {\\sf Im}(f)$ then $ff'(y) = y$.\nIf $y \\not\\in {\\sf Im}(f)$, then either $f'(y)$ is not defined, or\n$ff'(y) \\in {\\sf Im}(f)$, hence $ff'(y) \\neq y$. Thus, $y \\in {\\sf Im}(f)$ \niff $ff'(y) = y$.\nWhen $f, f' \\in {\\sf fP}$ then on input $y$ the properties $ff'(y) = y$,\n$y \\not\\in {\\sf Dom}(f')$, $ff'(y) \\neq y$, can be decided deterministically\nin polynomial time.\n\n\\noindent (3) Let $M$ be a nondeterministic Turing machine accepting $L$, \nsuch that all computations of $M$ are polynomially bounded, and do not halt\nbefore the whole input has been read. We can assume that $M$ has binary \nnondeterminism, i.e., in each transition it has at most two nondeterministic \nchoices.  Define \n\n\\smallskip\n\n \\ \\ \\ \\ \\ \\   \n $f_L(x,s) = x$ \\ \\ iff \\ \\ $M$, with {\\it choice sequence} $s$, \n accepts $x$; \n\n \\ \\ \\ \\ \\ \\    \n $f_L(x,s)$ is undefined otherwise. \n\n\\smallskip \n\n\\noindent Choice sequences are also called guessing sequences, or advice \nsequences.\nThen $L = {\\sf Im}(f_L)$ and $f_L \\in {\\sf fP}$; balancing comes from the \nfact that $s$ has polynomially bounded length.\n\nWe saw that if $f_L$ is regular then ${\\sf Im}(f_L) = L \\in {\\sf P}$.\nMoreover, if $L \\in {\\sf P}$ then the Turing machine $M$ can be chosen to\nbe deterministic, and the choice sequence $s$ is the all-0 word;\nso in that case, $f_L$ is not one-way.\n  \\ \\ \\ $\\Box$ \n\n\n\\begin{cor}  \\ The {\\em image transformation} {\\sf Im}: \n$f \\mapsto {\\sf Im}(f)$ maps {\\sf fP} onto {\\sf NP}, and maps the set of \nregular elements of {\\sf fP} onto {\\sf P}. \nMoreover, {\\sf NP} is embedded into {\\sf fP} (by the transformation \n$L \\mapsto f_L$ above), and {\\sf NP} is a {\\em retract} of {\\sf fP} (by the \ntransformations $L \\mapsto f_L$ and {\\sf Im}).\n \\ \\ \\   \\ \\ \\ $\\Box$\n\\end{cor}\nThe following suggests that ${\\sf Im}(f) \\in {\\sf P}$ is not equivalent \nto the regularity of $f$.\n\n\n\\begin{thm} \n \\ If $\\Pi_2^{\\sf P} \\neq \\Sigma_2^{\\sf P}$ then there exist surjective \none-way functions. {\\rm (Proved in \\cite{JCBcircsizeinv}.)} \n\\end{thm}\n\nIn the next sections we show various properties of {\\sf fP} and of \nclosely related semigroups. The fact that these semigroups have \ninteresting properties, and that the proofs are not difficult, gives \na second motivation for studying these semigroups.\n\n\n\n\\section{The Green relations of polynomial-time function semigroups}\n\nWe give a few properties of the Green relations $\\leq_{\\cal R}, \n \\ \\leq_{\\cal L}, \\ \\leq_{\\cal J}, \\ \\equiv_{\\cal D}$ (see\n\\cite{CliffPres,Grillet}).  First some notation.\n\nLet $F: X \\to Y$ be a partial function; then $F^{-1}: Y \\to X$ denotes \nthe {\\em inverse relation} of $F$, i.e., for all \n$(x,y) \\in X \\times Y$: \\ $x \\in F^{-1}(y)$ iff $F(x) = y$. By\n${\\sf mod}F$ we denote the partition on ${\\sf Dom}(F)$, defined by\n\\ $x_1$ ${\\sf mod}F$ $x_2$ iff $F(x_1) = F(x_2)$.\nThe set of blocks (equivalence classes) of ${\\sf mod}F$ is \n\\ $\\{ F^{-1}F(x) : x \\in {\\sf Dom}(F)\\}$.\nFor two partial functions $F, C: X \\to Y$ we write\n${\\sf mod}C \\leq {\\sf mod}F$ (the partition of $C$ is {\\em coarser} \nthan the partition of $F$, or the partition of $F$ {\\em refines} the \npartition of $C$) iff \\ ${\\sf Dom}(C) \\subseteq {\\sf Dom}(F)$ \\ and\n \\ $F^{-1}F(x) \\subseteq C^{-1}C(x)$ \\ for all $x \\in {\\sf Dom}(C)$;\nequivalently, every ${\\sf mod}C$-class is a union of ${\\sf mod}F$-classes.\n\n\n\\begin{pro}{\\rm (regular ${\\cal L}$- and $\\cal R$-orders).}\n\\label{regLR}   \nIf $f, r \\in {\\sf fP}$ and $r$ is {\\em regular} with an inverse\n$r' \\in {\\sf fP}$ then:\n\n\\smallskip\n\n\\noindent $\\bullet$ \\ $f \\leq_{\\cal R} r$ \\ \\ iff \\ \\ $f = r r' f$\n \\ \\ iff \\ \\ ${\\sf Im}(f) \\subseteq {\\sf Im}(r)$.\n\n\\smallskip\n\n\\noindent $\\bullet$ \\ $f \\leq_{\\cal L} r$ \\ \\ iff \\ \\ $f = fr'r$  \n \\ \\ iff \\ \\ ${\\sf mod}f \\leq {\\sf mod}r$. \n\\end{pro}\n{\\bf Proof.} [$\\cal R$-order]:  \\ \n$f \\leq_{\\cal R} r$ \\ iff \\ for some $u \\in {\\sf fP}: f = ru$. Then \n$f = r r' r u = r r' f$. \nAlso, it is straightforward that $f = ru$ implies \n ${\\sf Im}(f) \\subseteq {\\sf Im}(r)$.\n\nConversely, if ${\\sf Im}(f) \\subseteq {\\sf Im}(r)$ then \n \\ ${\\sf id}_{{\\sf Im}(f)} \\ = \\ $\n${\\sf id}_{{\\sf Im}(r)} \\circ {\\sf id}_{{\\sf Im}(f)} \\ = \\ $\n$r \\circ r'|_{{\\sf Im}(r)} \\circ {\\sf id}_{{\\sf Im}(f)}$. Hence,\n$f \\ = \\ {\\sf id}_{{\\sf Im}(f)} \\circ f \\ = \\ $\n$r \\circ r'|_{{\\sf Im}(r)} \\circ {\\sf id}_{{\\sf Im}(f)} \\circ f \\ = \\ $\n$r \\circ r'|_{{\\sf Im}(r)} \\circ f \\ \\leq_{\\cal R} r$. \n \n\\medskip\n\n[$\\cal L$-order]: \\\n$f \\leq_{\\cal L} r$ \\ iff \\ for some $v \\in {\\sf fP}: f = vr$. Then \n$f = v r r' r = f r' r$. \nAnd it is straightforward that $f = vr$ implies\n ${\\sf mod}f \\leq {\\sf mod}r$.\n\nConversely, if ${\\sf mod}f \\leq {\\sf mod}r$ then for all \n$x \\in {\\sf Dom}(f)$, \\ $r^{-1}r(x) \\subseteq f^{-1}f(x)$.  And for \nevery $x \\in {\\sf Dom}(f)$, \\ $\\{f(x)\\} = f \\circ f^{-1} \\circ f(x)$.\nMoreover,\n$f \\circ r^{-1} \\circ r(x) \\subseteq f \\circ f^{-1} \\circ f(x) = \\{f(x)\\}$, \nand since $r^{-1} \\circ r(x) \\neq \\varnothing$, it follows that \n$f \\circ r^{-1} \\circ r(x) = \\{f(x)\\}$.  So, $f = f \\circ r^{-1} \\circ r$. \nMoreover, $f \\circ r' \\circ r(x) \\in f \\circ r^{-1} \\circ r(x) = \\{f(x)\\}$,\nhence $f \\circ r' \\circ r(x) = f(x)$.  Hence, $f = f r' r \\leq_{\\cal L} r$.  \n \\ \\ \\ $\\Box$\n \n\\bigskip\n\n\\noindent The ${\\cal D}$-relation between elements of {\\sf fP} with infinite\nimage sets seems difficult, even in the case of regular elements.\nA first question (inspired from the Thompson-Higman monoids \n\\cite{JCBmonThH}): Are all regular elements of {\\sf fP} with infinite image \nin the same ${\\cal D}$-class, i.e., the ${\\cal D}$-class of \n${\\sf id}_{A^*}$?\n\n\n\\begin{pro} \\label{D_inj}  Let $f \\in {\\sf fP}$ be regular. \nThen $f \\equiv_{\\cal D} {\\sf id}_{A^*}$ iff there exists $g \\in {\\sf fP}$ \nsuch that $g$ is injective, total, and regular, and such that \n${\\sf Im}(f) = {\\sf Im}(g)$.\n\\end{pro}\n{\\bf Proof.} Assume $f \\equiv_{\\cal D} {\\sf id}_{A^*}$. By Prop.\\ \\ref{regLR},\n$f \\equiv_{\\cal R} {\\sf id}_L$, where $L = {\\sf Im}(f)$.\nAnd ${\\sf id}_{A^*} \\equiv_{\\cal D} {\\sf id}_L$ iff there exists \n$g \\in {\\sf fP}$ such that \n${\\sf id}_{A^*} \\equiv_{\\cal L} g \\equiv_{\\cal R} {\\sf id}_L$.\nHence, ${\\sf id}_{A^*} = g' \\, g$ for some $g' \\in {\\sf fP}$; this equality\nimplies that $g$ is total and injective. The existence of $g' \\in {\\sf fP}$\nimplies that $g$ is regular. Since $g \\equiv_{\\cal R} {\\sf id}_L$, \n${\\sf Im}(g) = L$. Hence, $g$ has the required properties. \n\nTo prove the converse we will use the following:\n\n\\medskip\n\n\\noindent {\\sf Claim.} \\ For every $g \\in {\\sf fP}$ we have:\n \\ $g$ is injective, total, and regular \\ iff \n \\ $(\\exists g' \\in {\\sf fP}) \\, g' g = {\\sf id}_{A^*}$. \n\n\\noindent Proof of the Claim. \nThe right-to-left implication is straightforward. In the other direction, \nif $g$ is regular then there exists $g' \\in {\\sf fP}$ such that $gg'g =g$. \nAnd if $g$ is total and injective, there exists a partial function $h$ such \nthat $hg = {\\sf id}_{A^*}$. Now $gg'g =g$ implies $hgg'g = hg$, hence by \nusing $hg = {\\sf id}_{A^*}$ we obtain: $g' g = {\\sf id}_{A^*}$. \nThis proves the Claim.\n\n\\medskip\n \nFor the converse of the Proposition, assume there exists $g \\in {\\sf fP}$\nwith the required properties. \nIf such a $g$ exists, then $f \\equiv_{\\cal R} g$, by Prop.\\ \\ref{regLR}. \nMoreover, $g \\equiv_{\\cal L} {\\sf id}_{A^*}$; this follows from \n$g' g = {\\sf id}_{A^*}$, which holds by the Claim. Hence\n$f \\equiv_{\\cal R} g \\equiv_{\\cal L} {\\sf id}_{A^*}$.\n \\ \\ \\ $\\Box$\n\n\n\\bigskip\n\n\\noindent However, it is an open problem whether every infinite language $L$ \nin {\\sf P} is the image of an injective, total, polynomial-time computable \nfunction $g$ (and whether $g$ can be taken to be regular or one-way).\nAlso, not much is known about which infinite languages in {\\sf P} can be \nmapped onto each other by maps in {\\sf fP}.\n\n\nWhen ${\\sf Im}(f)$ is a right ideal, more can be said.  \nBy definition, a {\\em right ideal} of $A^*$ is a subset $R \\subseteq A^*$\nsuch that $R \\, A^* = R$ (i.e., $R$ is closed under right-concatenation by \nany string).\nEquivalently, a right ideal is a set of the form $R = L \\, A^*$, for any set \n$L \\subseteq A^*$; in that case we also say that $L$ {\\em generates} $R$\nas a right ideal.\nA {\\em prefix code} in $A^*$ is a set $P \\subseteq A^*$ such that no word \nin $P$ is a prefix of another word in $P$.  It is not hard to prove that \nfor any right ideal $R$ there exists a unique prefix code $P_{_R}$ such that\n$R = P_{_{\\!\\! R}} \\, A^*$; in other words, $P_{_R}$ is the minimum generating set \nof $R$, as a right ideal.\n\nA {\\em right ideal morphism} is a partial function $h: A^* \\to A^*$ such\nthat for all $x \\in {\\sf Dom}(h)$ and all $w \\in A^*$: \\ $h(xw) = h(x) \\, w$.\nOne proves easily that then ${\\sf Dom}(h)$ and ${\\sf Im}(h)$ are right \nideals.  \n\nWe also consider $A^{\\omega}$ (the $\\omega$-sequences over $A$, see e.g.\\\n\\cite{PerrinPin}). For a set $L \\subseteq A^*$ we define ${\\sf ends}(L)$ to \nconsist of all elements of $A^{\\omega}$ that have a prefix in $L$.  \nThe Cantor space topology on $A^{\\omega}$ uses the sets of the form \n${\\sf ends}(L)$ (for $L \\subseteq A^*$) as its open sets; here we can \nassume without loss of generality that $L$ is a prefix code or a right \nideal of $A^*$. \n\n\n\\begin{lem} \\label{codeVSrideal} \\\nIf a right ideal $R \\subseteq A^*$ belongs to {\\sf P} then the corresponding \nprefix code $P$ (such that $R = PA^*$) also belongs to {\\sf P}. Conversely, \nif $L$ is in {\\sf P} then $LA^*$ is in {\\sf P}.\n\\end{lem}\n{\\bf Proof.} The first statement follows immediately from the fact \nthat $x \\in P$ iff $x \\in R$ and every strict prefix of $x$ does not belong\nto $R$. The converse is straightforward.   \\ \\ \\ $\\Box$\n\n\\bigskip\n\n\\noindent {\\bf Notation.} \\ Below, $\\ov{PA^*}$ \\, denotes $A^* - PA^*$ \n(complement).\n\n\\begin{pro} \\label{Pp_0} \\ Let $P \\subseteq A^*$ be a prefix code that \nbelongs to {\\sf P}, and let $p_0 \\in P$. Then all {\\em regular} elements \n$r \\in {\\sf fP}$ whose image is \\ ${\\sf Im}(r) \\, = \\, L_P \\, = \\, $\n$(P - \\{p_0\\}) \\, A^* \\, \\cup \\ p_0 \\, (p_0 A^* \\ \\cup \\ \\ov{PA^*})$ \\ are \nin the ${\\cal D}$-class of \\, ${\\sf id}_{A^*}$.\nWe can view $L_P$ as an ``approximation'' of the right ideal $PA^*$ since \n\n\\medskip\n\n\\hspace{1in}  $(P - \\{p_0\\}) \\, A^* \\ \\subset \\ L_P \\ \\subset \\ P A^*$. \n \n\\end{pro} \n{\\bf Proof.} \\ Let $L = L_P = {\\sf Im}(r)$. By Prop.\\ \\ref{regLR}, \n$r \\equiv_{\\cal R} {\\sf id}_L$, so it suffices to prove that \n \\ ${\\sf id}_L \\equiv_{\\cal D} {\\sf id}_{A^*}$.\nWe define $\\pi, \\pi' \\in {\\sf fP}$ by\n\\[ \\pi(x) = \\left\\{  \n   \\begin{array}{ll}\n       x     & \\mbox{if  \\ $x \\in (P - \\{p_0\\}) A^*$, } \\\\  \n       p_0 x & \\mbox{otherwise (i.e., if $x \\in p_0A^*$ or $x \\not\\in PA^*$);}\n   \\end{array}  \\right.\n\\]\n\\[ \\pi'(x) = \\left\\{ \n   \\begin{array}{ll} \n        x   & \\mbox{if \\ $x \\in (P - \\{p_0\\}) A^*$, } \\\\  \n        z   & \\mbox{if \\ $x \\in p_0A^*$ with $x = p_0z$, } \\\\  \n        {\\rm undefined} & \\mbox{otherwise (i.e., when $x \\not\\in PA^*$). }\n   \\end{array}  \\right.\n\\]\n\\noindent Then $\\pi$ is a total and injective function, and\n${\\sf Im}(\\pi) = L$. Hence, $\\pi \\equiv_{\\cal R} {\\sf id}_L$.\nMoreover, $\\pi' \\circ \\pi = {\\sf id}_{A^*}$, as is easily verified, \nhence $\\pi \\equiv_{\\cal L} {\\sf id}_{A^*}$. In summary,\n \\ $r \\equiv_{\\cal R} {\\sf id}_L \\equiv_{\\cal R} \\pi \\equiv_{\\cal L}$\n${\\sf id}_{A^*}$. \n  \\ \\ \\ \\ \\ $\\Box$\n\n\\bigskip\n\n\\noindent \nFunctions that have right ideals as domain and image are of great interest, \nbecause of the remarkable properties of the Thompson-Higman \ngroups and monoids \\cite{McKTh,Th,Scott,Hig74,CFP,BiThomps,BiCoNP} \nand \\cite{JCBonewPerm, JCBmonThH, JCBrl}.\nProp.\\ \\ref{Pp_0} gives an additional motivation for looking at the special\nrole of right ideals.\nThis motivates the following.\n\n\n\\begin{defn} \\label{RM2P} \n \\ \\ \\ ${\\cal RM}_{_{|A|}}^{\\sf P} \\ = \\ \\{ f \\in {\\sf fP} \\ : \\ $\n$f$ is a right ideal morphism of $A^* \\}$.\n\\end{defn}\nWhen $f$ is a right ideal morphism, ${\\sf Dom}(f)$ and ${\\sf Im}(f)$ are\nright ideals.\n${\\cal RM}_{_{|A|}}^{\\sf P}$ is closed under composition, and\n${\\cal RM}_2^{\\sf P}$ is a submonoid of {\\sf fP}. \n\n\nAn interesting submonoid of ${\\cal RM}_{_{|A|}}^{\\sf P}$ is \n${\\cal RM}_{_{|A|}}^{\\sf fin}$, consisting of all those\n$f \\in {\\cal RM}_{_{|A|}}^{\\sf P}$ for which ${\\sf Dom}(f)$ (and hence also\n${\\sf Im}(f)$) is a {\\it finitely generated} right ideal. The monoid\n${\\cal RM}_{_{|A|}}^{\\sf fin}$ is used to define the Thompson-Higman monoid\n$M_{|A|,1}$ in \\cite{JCBmonThH}. \n\n\\begin{pro} \\label{fPinvinRM} \nIf an element $f \\in {\\cal RM}_2^{\\sf P}$ has an inverse in {\\sf fP} then\n$f$ also has an inverse in ${\\cal RM}_2^{\\sf P}$.\n\\end{pro}\n{\\bf Proof.} Let $f_0' \\in {\\sf fP}$ be an inverse of $f$; we want to \nconstruct an inverse $f'$ of $f$ that belongs to ${\\cal RM}_2^{\\sf P}$.\nSince $f$ is regular in {\\sf fP}, we know from Prop.\\ \\ref{ImfP} that \n${\\sf Im}(f)$ is in {\\sf P}.\nHence we can restrict $f_0'$ to ${\\sf Im}(f)$, i.e., \n${\\sf Dom}(f_0') = {\\sf Im}(f)$.\nWe proceed to define $f'(y)$ for $y \\in {\\sf Im}(f)$.\n\nFirst, we compute the shortest prefix $p$ of $y$ that satisfies \n$p \\in {\\sf Dom}(f_0') = {\\sf Im}(f)$. Since  ${\\sf Im}(f) \\in {\\sf P}$, \nthis can be done in polynomial time. Now, $y = p \\, z$ for some  string $z$. \n\nSecond, we define $f'(y) \\ = \\ f'_0(p) \\ z$, \\ where $p$ and $z$ are as \nabove.  Thus, $f'$ is a right-ideal morphism.\n\n\\smallskip\n\nLet us verify that $f'$ has the claimed properties.\nClearly, $f'$ is polynomial-time computable, and polynomially \nbalanced (the latter following from the fact that $f'$ is an inverse of $f$,\nwhich we prove next).\nTo prove that $f'$ is an inverse of $f$, let $x \\in {\\sf Dom}(f)$.\nThen $f (f' (f(x))) = f (f'(p \\, z))$, where $y = f(x) = p \\, z$, and $p$\nis the shortest prefix of $y$ such that $p \\in {\\sf Im}(f)$.\nThen, $f'(p \\, z) = f_0'(p) \\, z$, by the definition of $f'$.\nThen, since $f$ is a right-ideal morphism, $f(f_0'(p) \\, z)$\n$= f(f_0'(p)) \\, z = p \\, z$ (the latter since $f'_0$ is an inverse of $f$,\nand since $p \\in {\\sf Im}(f)$).\nHence, $ff'|_{{\\sf Im}(f)} = {\\sf id}_{{\\sf Im}(f)}$. Thus, by Prop.\\ \n\\ref{concept}, $f'$ is an inverse of $f$. \n \\ \\ \\ $\\Box$\n\n\\bigskip\n\n\\noindent {\\bf Remark and notation:} \n \\ For $f \\in {\\cal RM}_{_{|A|}}^{\\sf P}$ we saw that ${\\sf Dom}(f)$ and \n${\\sf Im}(f)$ are right ideals. Let ${\\sf domC}(f)$, called the \n{\\em domain code}, be the prefix code that generates ${\\sf Dom}(f)$ as a \nright ideal. Similarly, let ${\\sf imC}(f)$, called the {\\em image code}, \nbe the prefix code that generates ${\\sf Im}(f)$.\n\nIn general, ${\\sf imC}(f) \\subseteq f({\\sf domC}(f))$, and it can happen that\n${\\sf imC}(f) \\neq f({\\sf domC}(f))$. However the last paragraph of proof of\nProp.\\ \\ref{fPinvinRM} shows that in any case: \n \\ {\\em If $f \\in {\\cal RM}_{_{|A|}}^{\\sf P}$ is regular then $f$ has an \ninverse $f' \\in {\\cal RM}_{_{|A|}}^{\\sf P}$ such that \n${\\sf domC}(f') = {\\sf imC}(f)$. }\n\n\\bigskip\n\n\\noindent {\\bf Notation:} \\ For two words $u,v \\in A^*$,\n$(v \\leftarrow u)$ denotes the right ideal morphism $ux \\mapsto vx$\n(for all $x \\in A^*$). In particular,\n$(\\varepsilon \\leftarrow \\varepsilon) = {\\sf id}_{A^*}$, where\n$\\varepsilon$ denotes the empty word.\nThe morphism $(v \\leftarrow u)$ is length-balanced because $|u|, |v|$ are\nconstants for a given morphism.\n\n\n\\begin{pro} \\label{j0} \\ For every alphabet $A$, the monoid \n${\\cal RM}_{_{|A|}}^{\\sf P}$ is ${\\cal J}^0$-simple (i.e., the only ideals\nare $\\{0\\}$ and ${\\cal RM}_{_{|A|}}^{\\sf P}$ itself).\n\\end{pro}\n{\\bf Proof.} For any $f \\in {\\cal RM}_{|A|}^{\\sf P}$ that is not the \nempty map, there exist words $x_0, y_0$ such that $f(x_0) = y_0$. Then \n$(\\varepsilon \\leftarrow \\varepsilon)$ $=$ \n$(\\varepsilon \\leftarrow y_0) \\circ f \\circ (x_0 \\leftarrow \\varepsilon)$.\nHence, ${\\sf id}_{A^*} \\leq_{\\cal J} f$ for every non-empty element \n$f \\in {\\cal RM}_{|A|}^{\\sf P}$.  \n \\ \\ \\ $\\Box$  \n\n\n\\begin{pro} \\ \\ {\\sf fP} is not ${\\cal J}^0$-simple, and it has regular \nelements in different non-0 ${\\cal J}$-classes.\n\\end{pro}\n{\\bf Proof.} The map $\\ell: x \\in \\{0,1\\}^* \\longmapsto 0^{|x|}$\nis in {\\sf fP} and it is an idempotent. \n\nMoreover, $\\ell \\not\\equiv_{\\cal J} {\\sf id}_{A^*}$.  \nIndeed, if there exist functions $\\beta, \\alpha$ such that for all \n$x \\in A^*$, $x = \\beta \\, \\ell \\, \\alpha(x) = \\beta(0^{|\\alpha(x)|})$, then \n$|\\alpha(x)|$ is different for every $x \\in A^*$. But then $\\alpha$ is not\npolynomially balanced, since $|\\alpha(x)|$ would have to range over \n$|A|^{|x|}$ values.  \n \\ \\ \\ $\\Box$\n\n\n\\begin{cor} \\ \\ {\\sf fP} and ${\\cal RM}_{_{|A|}}^{\\sf P}$ are not\nisomorphic. \\ \\ \\  \\ \\ \\ $\\Box$\n\\end{cor}\n\n\n\\noindent  As a consequence of Prop. \\ref{Pp_0} we have:\n\n\\begin{cor} \\ Every {\\em regular} element $r \\in {\\cal RM}_2^{\\sf P}$ is \n``close'' to an element of {\\sf fP} belonging to the ${\\cal D}$-class of\n${\\sf id}_{A^*}$. \nHere, $h_{p_0} \\in {\\sf fP}$ is  called ``close'' to $r$ \\, iff \\  \n${\\sf Im}(r) = PA^*$ for a prefix code $P$, and there exists $p_0 \\in P$ such\nthat:\n\n\\smallskip\n\n\\noindent $\\bullet$ \\ \n$(P - \\{p_0\\}) \\, A^* \\ \\subseteq \\ {\\sf Im}(h_{p_0}) \\ \\subseteq PA^*$, \n \\ and \n\n\\smallskip\n\n\\noindent $\\bullet$ \n \\ $h_{p_0}(x) = r(x)$ whenever $r(x) \\in {\\sf Im}(h_{p_0})$. \n\n\\end{cor}\n{\\bf Proof.} Let $P = {\\sf domC}(r)$, so $PA^* = {\\sf Im}(r)$. \nFor every $p_0 \\in P$, $r$ is close to ${\\sf id}_{L_P} \\circ r$, whose image \nset is $L_P \\ = \\ $ \n$(P - \\{p_0\\}) \\, A^* \\ \\cup \\ p_0 \\, (p_0 A^* \\cup \\ov{PA^*})$, hence\n \\ $(P - \\{p_0\\}) \\, A^* \\subset L_P \\subset PA^*$. \nAnd ${\\sf id}_{L_P} \\circ r \\equiv_{\\cal R} {\\sf id}_{L_P}$ since\n$L_P \\subset PA^*$. \n \\ \\ \\ $\\Box$\n\n\\medskip\n\n\\noindent Recall the notation $(v \\leftarrow u)$ given just before Prop.\\\n\\ref{j0}.\n\n\\begin{lem}  \\label{RM1LR} \\   \nIn ${\\cal RM}_2^{\\sf P}$, the $\\cal L$-class of ${\\sf id}_{A^*}$ is \n$\\{(v \\leftarrow \\varepsilon) \\in {\\cal RM}_2^{\\sf P} : v \\in A^*\\}$.\nThis is the set of elements of ${\\cal RM}_2^{\\sf P}$ that are injective \nand total (i.e., defined for all $x \\in A^*$).\n\nThe $\\cal R$-class of ${\\sf id}_{A^*}$ is \n$\\{ f \\in {\\cal RM}_2^{\\sf P} : \\,  \\varepsilon \\in {\\sf Im}(f)\\}$. \nThis is the set of elements of ${\\cal RM}_2^{\\sf P}$ that are surjective \n(i.e., map onto $A^*$).\n\\end{lem}\n{\\bf Proof.} If $f \\equiv_{\\cal L} {\\sf id}_{A^*}$ then \n$\\varepsilon \\in {\\sf Dom}(f) = A^*$, so there is $v \\in A^*$ such that \n$v = f(\\varepsilon)$. Then $f(x) = vx$ for all $x\\in A^*$. \nConversely, if $f(x) = vx$ for all $x\\in A^*$ then \n$(\\varepsilon \\leftarrow v) \\circ f = {\\sf id}_{A^*}$.\n\nIf $f \\equiv_{\\cal R} {\\sf id}_{A^*}$ then ${\\sf Im}(f) = A^*$. So\n$\\varepsilon \\in {\\sf Im}(f)$. \nConversely, if $f$ satisfies $\\varepsilon \\in {\\sf Im}(f)$, i.e., \n$\\varepsilon = f(x_0)$ for some $x_0 \\in A^*$, then \n$f \\circ (x_0 \\leftarrow \\varepsilon) = (\\varepsilon \\leftarrow \\varepsilon)$\n$= {\\sf id}_{A^*}$.\n \\ \\ \\ $\\Box$\n\n\\bigskip\n\n\\noindent Lemma \\ref{RM1LR} shows that the $\\cal L$-class of\n${\\sf id}_{A^*}$ in ${\\cal RM}_2^{\\sf P}$ is also the $\\cal L$-class of\n${\\sf id}_{A^*}$ in ${\\cal RM}_2^{\\sf fin}$.\n\n\\begin{pro} \\ ${\\cal RM}_2^{\\sf P}$ has trivial group of units, i.e., the \n$\\cal D$-class of the identity ${\\sf id}_{A^*}$ is $\\cal H$-trivial.\n\\end{pro}\n{\\bf Proof.} \nIf $f \\equiv_{\\cal H} {\\sf id}_{A^*}$ then by Lemma \\ref{RM1LR} (for the \n$\\cal L$-class of ${\\sf id}_{A^*}$), $f(x) = vx$ for all $x$. \nAlso by Lemma \\ref{RM1LR} (for the $\\cal R$-class of ${\\sf id}_{A^*}$), \n$f(x_1) = v x_1 = \\varepsilon$, for some $x_1$.\nThis implies $v = \\varepsilon$, hence $f = {\\sf id}_{A^*}$.\n \\ \\ \\ $\\Box$\n\n\\bigskip\n\n\\noindent  As a consequence of Lemma \\ref{RM1LR}, ${\\cal RM}_2^{\\sf P}$ can \nbe injectively mapped (non-homomorphically) into the $\\cal R$-class of \n${\\sf id}_{A^*} \\in {\\cal RM}_2^{\\sf P}$.  Let us define $f \\mapsto \\psi_f$ \nby ${\\sf Dom}(\\psi_f) = \\{0\\} \\, \\cup \\, 1 \\, {\\sf Dom}(f)$, and\n\n\\smallskip\n\n \\ \\ \\ \\ \\ $\\psi_f(0) \\ = \\ \\varepsilon$, \\ \\ and\n\n\\smallskip\n\n \\ \\ \\ \\ \\ $\\psi_f(1x) \\ = \\ 1 \\, f(x)$, \\, for all $x \\in {\\sf Dom}(f)$.\n\n\\smallskip\n\n\\noindent Then for all $1x \\in 1 \\, A^*$, \n \\ $\\psi_{g \\circ f}(1x) = (\\psi_g \\circ \\psi_f)(1x) = 1 \\, (f g)(x)$.\nSo, $f \\mapsto \\psi_f|_{1A^*}$ is a morphism (where $\\psi_f|_{1A^*}$ is\nthe restriction of $\\psi_f$ to $1A^*$). \nBut $\\psi$ is not a morphism; indeed, since ${\\cal RM}_2^{\\sf P}$ \ncontains non-trivial groups, but the $\\cal D$-class of ${\\sf id}_{A^*}$ is \n$\\cal H$-trivial, there cannot be a homomorphic embedding of \n${\\cal RM}_2^{\\sf P}$ into the $\\cal D$-class of ${\\sf id}_{A^*}$. \n\n\n\n\\section{\\bf Embedding {\\sf fP} into ${\\cal RM}_2^{\\sf P}$ }\n\n\n\\noindent {\\bf Transforming any map into a right-ideal morphism:}\n\n\\medskip\n\n\\noindent The semigroup {\\sf fP} uses the alphabet $\\{0,1\\}$; let $\\#$ be \na new letter.  For $f \\in {\\sf fP}$ we define\n$f_{\\#}: \\{0, 1, \\#\\}^* \\to \\{0, 1, \\#\\}^*$ \\, by letting \\,  \n${\\sf Dom}(f_{\\#}) \\ = \\ {\\sf Dom}(f) \\, \\# \\, \\{0, 1, \\#\\}^*$, and\n\n\\smallskip\n\n \\ \\ \\ \\ \\ \\ $f_{\\#}(x \\# w) \\ = \\ f(x) \\ \\# \\, w$, \n\n\\smallskip\n\n\\noindent for all $x \\in {\\sf Dom}(f)$ $( \\subseteq \\{0,1\\}^*)$, and all \n$w \\in \\{0, 1, \\#\\}^*$. So ${\\sf domC}(f_{\\#}) = {\\sf Dom}(f) \\, \\# $. \n\n\n\n\\begin{pro}\\hspace{-.06in}{\\bf .}  \n\n\\noindent (1) \\ For any $L \\subseteq \\{0,1\\}^*$, $L \\#$ is a prefix code \nin $\\{0,1,\\#\\}^*$. \n\n\\smallskip\n\n\\noindent (2) \\ $L$ is in {\\sf P} \\ iff \\ $L \\#$ is in {\\sf P}.\n\n\\smallskip\n\n\\noindent (3) \\ For any partial function $f: \\{0,1\\}^* \\to \\{0,1\\}^*$,\n \\, $f_{\\#}$ is a right ideal morphism of $\\{0,1,\\#\\}^*$.\n\n\\smallskip\n\n\\noindent \n(4) \\ $f \\in {\\sf fP}$ \\ iff \\ \\, $f_{\\#} \\in {\\cal RM}^{\\sf P}_3$.\n\\end{pro}\n{\\bf Proof.} This is straightforward.  \\ \\ \\ $\\Box$\n\n\\medskip\n\n\\bigskip\n\n\\noindent {\\bf Coding from three letters to two letters:}\n  \n\\medskip\n\n\\noindent We consider the following encoding from the 3-letter alphabet \n$\\{0, 1, \\#\\}$ to the 2-letter alphabet $\\{0,1\\}$. \n\n\\smallskip\n\n \\ \\ \\ \\ \\ \\ ${\\sf code}: \\, \\{0,1,\\#\\} \\, \\to \\, \\{00, 01, 11\\}$ \n \\ \\ is defined by\n\n\\smallskip\n\n \\ \\ \\ \\ \\ \\ ${\\sf code}(0) = 00$, \\ \\ \\ ${\\sf code}(1) = 01$, \n \\ \\ \\ ${\\sf code}(\\#) = 11$.\n\n\\smallskip\n\n\\noindent For a word $w \\in \\{0,1,\\#\\}^*$, ${\\sf code}(w)$ is the \nconcatenation of the encodings of the letters in $w$. \n\nThe choice of this code is somewhat arbitrary; e.g., we could \nhave picked the encoding \n$c$ from $\\{0,1,\\#\\}$ onto the maximal prefix code $\\{00, 01, 1\\}$, defined \nby $c(0) = 00$, $c(1) = 01$, $c(\\#) = 1$. \n\n\\begin{defn} \\label{f_encoding} \n \\ We define $f^C: \\{0,1\\}^* \\to \\{0,1\\}^*$ by letting \n \\, ${\\sf Dom}(f^C) \\ = \\ {\\sf code}({\\sf Dom}(f) \\, \\#) \\ \\{0, 1\\}^*$, and \n\n\\smallskip\n\n \\ \\ \\ \\ \\ \\   \n$f^C({\\sf code}(x \\#) \\, v) \\ = \\ {\\sf code}(f(x) \\, \\#) \\ v$,\n\n\\smallskip\n\n\\noindent for all $x \\in {\\sf Dom}(f)$ $( \\subseteq \\{0,1\\}^*)$,\nand all $v \\in \\{0,1\\}^*$.\nWe call $f^C$ the {\\em encoding of $f$}. \n\\end{defn}\n\n\n\\begin{pro}.\n\n\\noindent (1) \\ For any $L \\subseteq \\{0,1\\}^*$, \\, ${\\sf code}(L \\#)$ is a \nprefix code.\n\n\\smallskip\n\n\\noindent (2) \\ $L$ is in {\\sf P}\n \\ iff \\ ${\\sf code}(L \\#)$ is in {\\sf P}.\n\n\\smallskip\n\n\\noindent (3) \\ For any partial function $f: \\{0,1\\}^* \\to \\{0,1\\}^*$, \n \\, $f^C$ is a right ideal morphism of $\\{0,1\\}^*$.\n\n\\smallskip\n\n\\noindent (4) \\ $f \\in {\\sf fP}$ \n \\ iff \\ $f^C \\in {\\cal RM}_2^{\\sf P}$.\n\\end{pro}\n{\\bf Proof.} This is straightforward. \\ \\ \\ $\\Box$\n\n\n\\begin{pro}.\n\n\\noindent (1) \\ The transformations \n$f \\in {\\sf fP} \\longmapsto f_{\\#} \\in {\\cal RM}_3^{\\sf P}$ and  \n$f \\in {\\sf fP} \\longmapsto f^C \\in {\\cal RM}_2^{\\sf P}$ are injective\ntotal homomorphisms from {\\sf fP} into ${\\cal RM}_3^{\\sf P}$, respectively\n${\\cal RM}_2^{\\sf P}$.\n\n\\smallskip\n\n\\noindent (2) \\ $f$ is regular in {\\sf fP} \\ iff \\ $f_{\\#}$ is regular in\n${\\cal RM}^{\\sf P}_3$ \\ iff \\ $f^C$ is regular in ${\\cal RM}_2^{\\sf P}$.\n\n\\smallskip\n\n\\noindent (3) \\ There are one-to-one correspondences between the inverses\nof $f$ in {\\sf fP}, the inverses of $f_{\\#}$ in ${\\cal RM}^{\\sf P}_3$, and\nthe inverses of $f^C$ in ${\\cal RM}_2^{\\sf P}$.\n\\end{pro} \n{\\bf Proof.} \\ (1) is straightforward, and (2) follows from injectiveness \nand from the fact that the homomorphic image of an inverse is an inverse. \n\n(3) Let $G \\in {\\cal RM}^{\\sf P}_3$ be such that \n$f_{\\#} \\circ G \\circ f_{\\#} = f_{\\#}$; i.e., \n$f_{\\#}(G(f(x) \\# w)) = f(x) \\# w$, for all $x \\in \\{0,1\\}^*$ and \n$w \\in \\{0,1,\\#\\}^*$. Since $f_{\\#}(G(f(x) \\# w))$ ($= f(x) \\# w$) contains \n$\\#$, $G(f(x) \\# w)$ is of the form $G(f(x) \\# w) = z \\# v$, for some\n$z \\in \\{0,1\\}^*$ and $v \\in \\{0,1,\\#\\}^*$. Hence $f_{\\#}(G(f(x) \\# w)) = $\n$f_{\\#}(z \\# v) = f(z) \\# v =  f(x) \\# w$, so $f(z) = f(x)$ and $v = w$. So,\n$G(f(x) \\# w) = z \\# w$ for some $z \\in f^{-1}f(x)$.\nThus there exists a function $g: \\{0,1\\}^* \\to \\{0,1\\}^*$ such that\n$G(y\\#) = g(y) \\#$ for all $y \\in {\\sf Im}(f)$; then $G(y\\# w) = g(y) \\# w$,\nfor all $w \\in \\{0,1,\\#\\}^*$. Hence $g$ is an inverse \nof $f$.  Moreover, $g$ is clearly in {\\sf fP} if $G$ is in \n${\\cal RM}^{\\sf P}_3$. \n\nLet $H \\in {\\cal RM}^{\\sf P}_2$ be such that $f^C \\circ H \\circ f^C = f^C$;\ni.e., $f^C(H({\\sf code}(f(x) \\#) \\, v)) = {\\sf code}(f(x) \\#) \\, v$, for all \n$x, v \\in \\{0,1\\}^*$. Since $f^C$ outputs ${\\sf code}(f(x) \\#) \\, v$ \non input $H({\\sf code}(f(x) \\#) \\, v)$, the definition of $f^C$ implies \nthat for all $v \\in \\{0,1\\}^*:$ \\,  $H({\\sf code}(f(x) \\#) \\ v)$ is of the \nform ${\\sf code}(z \\#) \\, v$ for some $z \\in \\{0,1\\}^*$ with $f(z) = f(x)$. \nHence there exists a function $h: \\{0,1\\}^* \\to \\{0,1\\}^*$ such that \n$H({\\sf code}(y \\#)) =  {\\sf code}(h(y) \\#)$ for all $y$; then  \n$H({\\sf code}(y \\#) \\, v) =  {\\sf code}(h(y) \\#) \\, v$, for all $y,v$.\nHence $h$ is an inverse of $f$.  \nMoreover, $h$ is clearly in {\\sf fP} if $H$ is in ${\\cal RM}_2^{\\sf P}$. \n \\ \\ \\ $\\Box$\n\n\\bigskip\n\n\\noindent We will show in Section 5 that the encoding $f \\mapsto f^C$ \ncorresponds to an ``inversive reduction''.\n\n\\bigskip\n\n\\noindent In summary we have the following \n{\\bf relation between {\\sf fP} and ${\\cal RM}_2^{\\sf P}$:}\n\n\n$${\\sf fP} \\ \\stackrel{ \\, C}{\\hookrightarrow} \\ {\\cal RM}_2^{\\sf P} \n \\ \\hookrightarrow \\ [ {\\sf id} ]_{_{{\\cal J}({\\sf fP})}}^0 \\ \\hookrightarrow \n \\ {\\sf fP} .$$\n\n\\noindent Here $[ {\\sf id} ]^0_{_{{\\cal J}({\\sf fP})}}$ is the \n${\\cal J}$-class of the identity {\\sf id} of {\\sf fP}, together with the \nzero element (i.e., it is the {\\it Rees quotient} of the ${\\cal J}$-class\nof {\\sf id} in {\\sf fP}).\nThe embedding into $[ {\\sf id} ]^0_{_{{\\cal J}({\\sf fP})}}$ holds because \n${\\cal RM}_2^{\\sf P}$ is ${\\cal J}^0$-simple. \n\n\n\\bigskip\n\n\\section{\\bf Evaluation maps}\n\n\\bigskip\n\n\\noindent The {\\em Turing machine evaluation function}\n${\\sf eval}_{_{\\sf TM}}$ is the input-output function of a universal Turing\nmachine; it has the form\n \\ ${\\sf eval}_{_{\\sf TM}}(w, x) \\ = \\ \\phi_w(x)$,\nwhere $\\phi_w$ is the input-output function described by the word \n(``program'') $w$. \n(Recall that by ``function'' we always mean partial function.)\nSimilarly, there is an {\\em evaluation function for acyclic circuits}, \n${\\sf eval}_{\\sf circ}(C, x) = f_C(x)$, where $f_C$ is the input-output map \nof the circuit $C$. Here we will only consider length-preserving circuits, \ni.e., $|f_C(x)| = |x|$. \nWe also identify the circuit with a bitstring that describes the circuit.\nThe map ${\\sf eval}_{\\sf circ}$ is polynomial-time computable, but not \npolynomially balanced (since the size of input component $C$ is not bounded \nin terms of the output length $|f_C(x)|$).\n\nLevin \\cite{Levin1W} noted that functions of the form \n\n\\smallskip\n\n\\hspace{1in}  ${\\sf ev}(w, x) \\ = \\ (w, \\phi_w(x))$, \n\n\\smallskip\n\n\\noindent (under some additional assumptions) are polynomially balanced and \npolynomial-time computable; and he observed that {\\sf ev} is a {\\it critical \none-way function} in the following sense: \n\n\\begin{defn}\nA function $e \\in {\\sf fP}$ is {\\em critical} (or {\\sf fP}{\\em -critical}) \niff the following holds:\nOne-way functions exist iff the function $e$ is a one-way function.\nSimilarly, a set $L \\in {\\sf NP}$ is {\\em critical} (or \n{\\sf P}{\\em -critical})  iff  the following holds: ${\\sf P} \\neq {\\sf NP}$ \niff $L \\not\\in {\\sf P}$.\n\\end{defn}\nThe literature calls these functions ``universal'' one-way functions; \nhowever, not all critical one-way functions are universal (in the sense of \nuniversal Turing machines, or other universal computing devices).\nLevin's idea of a ``universal'' (critical) one-way function has also been \nused in probabilistic settings for one-way functions (see e.g., \n\\cite{Goldreich}). \n\nTo make {\\sf ev} polynomial-time computable, additional features have to be\nintroduced. One approach is to simply build a counter into the program of \n{\\sf ev} that stops the computation of ${\\sf ev}(w, x)$ after a polynomial\nnumber of steps (for a fixed polynomial). For example, the computation could \nbe stopped after a quadratic number of steps, i.e., $c \\, (|w| + |x|)^2 + c$ \nsteps (for a fixed constant $c \\geq 1$); we call this function \n${\\sf ev}_{(2)}$. \nThere exist other approaches; see e.g., section 2.4 of \\cite{Goldreich}, \nor p.\\ 178 of \\cite{AroraBarak}, where it is proved that ${\\sf ev}_{(2)}$ \nis {\\sf fP}-critical.\n\nHere is another simple example of a critical one-way function:\n\n\\smallskip\n\n\\hspace{1in} ${\\sf ev}_{\\sf circ}(C, x) \\ = \\ (C, f_C(x))$,\n\n\\smallskip\n\n\\noindent where $C$ ranges over finite acyclic circuits (more precisely, \nstrings that describe finite acyclic circuits), and $f_C$ is the \ninput-output map of a circuit $C$. \nThe function ${\\sf ev}_{\\sf circ}$ is in {\\sf fP}; it is balanced since \n$|x| \\leq |C|$ and $C$ is part of the output. Here we only consider\nlength-preserving circuits, i.e., $|f_C(x)| = |x|$.\nWe will prove later that ${\\sf ev}_{\\sf circ}$ is not only critical, but \nalso {\\em complete} with respect to a reduction that is appropriate for \none-way functions. \n\nA similar example of a critical function is \n$(B, \\tau) \\longmapsto (B, B(\\tau))$, where $B$ ranges over all boolean\nformulas (or over all boolean formulas in 3CNF), $\\tau$ is any truth-value \nassignment for $B$ (i.e., a bitstring whose length is the number of boolean\nvariables in $B$), and $B(\\tau)$ is the truth-value of $B$ for the \ntruth-value assignment $\\tau$. More generally we have:\n\n\\begin{pro} \\label{f_Mcrit} \\,   \nFor a nondeterministic polynomial-time Turing machine $M$, let $f_M$ be \ndefined by  \n\n\\smallskip\n\n \\ \\ \\ $f_M(x,s) = x$ \\ iff \\ $M$, with {\\it choice sequence} $s$,\naccepts $x$ (and undefined otherwise).\n\n\\smallskip\n\n\\noindent Then $f_M$ is {\\sf fP}-critical iff the language (in {\\sf NP})\naccepted by $M$ is {\\sf P}-critical. \n\\end{pro}\n{\\bf Proof.} We studied the functions $f_M$ in Prop.\\ \\ref{ImfP} (where we \nused the notation $f_L$). We saw that $f_M$ is one-way iff \n${\\sf Im}(f_M) \\not\\in {\\sf P}$. Moreover, ${\\sf Im}(f_M)$ is the language \naccepted by $M$, and ${\\sf Im}(f_M) \\in {\\sf NP}$. So, $f_M$ is one-way iff \n${\\sf P} \\neq {\\sf NP}$.  \n \\ \\ \\ $\\Box$ \n\n\\bigskip\n\n\\bigskip\n\n\\bigskip\n\n\n\\noindent {\\bf Machine model for {\\sf fP}}:\n\n\\medskip\n\n\\noindent Every function in {\\sf fP} can be computed by a Turing machine\nwith a built-in polynomial-time counter, that is used for enforcing \ntime-complexity and input balance. \nAs usual, to say that time-complexity or balance functions are \n``polynomial'' means that they have polynomial upper-bounds.\nMore precisely, we will describe every polynomial-time multi-tape Turing \nmachine $M$ by a program $v$ (which consists of the list of transitions of \nthe Turing machine, as well as its start and accept states), and a \npolynomial $p$ such that $p(n)$ is an upper-bound on the time-complexity \nand the input balance of $M$ on all inputs of length $\\le n$. Since we only \nrequire polynomial {\\it upper-bounds}, we can take $p$ of the form \n$p(n) = a \\, n^k + a$, where $k, a$ are positive integers.\nSo $p$ is determined by two integers (stored as bitstrings). \nWe do not need to assume anything about the time-complexity of the Turing \nmachine with program $v$ (and in general it is undecidable whether $v$\nhas polynomial-time complexity); instead, we want to consider pairs $(v,p)$ \nwhere $v$ is a Turing machine program, and $p$ is a polynomial (given by \ntwo integers $k,a$). \nBased on pairs $(v,p)$ we define the following: A partial function $f$ \nis computed by $(v,p)$ iff for all $x \\in A^*$, $f(x)$ is computed by the  \nTuring machine with program $v$ in time $\\le a \\, |x|^k + a = p(|x|)$ and \ninput balance $\\le p(|f(x)|)$; when $f(x)$ is undefined then the program\neither gives no output, or violates the time bound or the input balance\nbound.\nIn this way {\\sf fP} can be recursively enumerated by pairs $(v,p)$. \nFor {\\sf P} and {\\sf NP} this (or a similar idea) goes back to the work of \nHartmanis, Lewis, Stearns, and others in the 1970's; compare with the \ngeneric {\\sf NP}-complete problem in \\cite{Hartmanis}, the proof of the \ncomplexity hierarchy theorems in chapter 12 in \\cite{HU}, and the section \non critical (``universal'') one-way functions in \\cite{Goldreich}.  \n\nHowever, pairs $(v,p)$ do not form a machine model, being hybrids\nconsisting of a machine and two numbers.\nIn order to obtain a machine model for {\\sf fP} we take a Turing machine \nwith program $v$, and add an extra tape that will be used as a counter. \nWe assume that every tape has a left endmarker.  On input $x$, a Turing \nmachine with counter first computes $p(|x|) = a \\, |x|^k + a$, and \nmoves the head of the counter tape $p(|x|)$ positions to the right. \nAfter the counter has been prepared (and the head on the input $x$ has been \nmoved back to the left end), the Turing machine executes program $v$ on the \nother tapes, while in each transition the head on the counter tape moves \nleft by one position. If the counter head gets back to the left endmarker\n(``it triggers the counter''), the Turing machine stops and rejects (and \nproduces no output). If the machine halts before triggering the counter, \nthe counter has no effect on the result of program $v$ on input $x$. After \nthis, if there is an output $y$ the Turing machine with counter checks the \ninput balance: If $|y| \\ge |x|$ the balance condition obviously holds, so\n$y$ is the final output. If $|y| < |x|$, the machine computes $p(|y|)$ \n($< p(|x|)$); if $|x| > p(|y|)$ the machine rejects (and produces no final \noutput); otherwise, $y$ is the final output.    \n\n\\smallskip\n\nIn order to mark off space of length $p(|x|)$ on the counter tape, we need \nan algorithm for computing $p(|x|) = a \\, |x|^k + a$, and we will look at \nthe time-complexity of this algorithm.  Recall that the bitlength of a \npositive integer $n$ is $\\lfloor \\log_2 n \\rfloor + 1$ (in unsigned binary\nrepresentation). \n\n\\smallskip\n\n\\noindent \n(1) First, we compute $|x|$ in binary, by repeatedly dividing $|x|$ by 2, \nusing two tapes: On one tape, we start with a length $n = |x|$, \nthen a mod-2 counter produces $\\lfloor n\/2 \\rfloor$ on the 2nd tape and \nrecords the remainder (0 or 1) on a 3rd tape; then a mod-2 counter computes \nhalf of the 2nd tape and writes it on the 1st tape, and records the \nremainder on tape 3, etc. This takes time \n \\ $\\le \\sum_{i=0}^{\\lfloor \\log_2 |x| \\rfloor} |x|\/2^i$ \n$ \\ = \\ 2^{\\lfloor \\log_2 |x| \\rfloor + 1} - 1 \\ < \\ 2 \\, |x|$.\n\n\\smallskip\n\n\\noindent \n(2) We compute $a \\, |x|^k + a$ in binary, using $k$ multiplications \nand one addition. This takes time \n\n\\smallskip\n\n$\\le \\ c \\, (\\log_2 |x|)^2 + 2 c (\\log_2 |x|)^2 + \\ \\ldots \\ + $\n$ (k-1) c (\\log_2 |x|)^2$ $+$ $c k \\log_2 |x| \\ \\log_2 a$\n\n \\ \\ \\ \\ \\ $+$ $ \\ c\\, \\max\\{k \\log_2 |x|, \\log_2 a\\}$ \n\n\\smallskip\n\n$< \\ c \\, k^2 \\, (\\log_2 |x|)^2 + c \\, k \\ \\log_2 a + c \\, k \\ \\log_2 |x|$ \n \\ \\ \\ \\ \\ (when \\ $k \\log_2 |x| \\ge \\log_2 a$, i.e., $|x| \\ge a^{1\/k}$)\n\n\\smallskip\n\n$< \\ c \\ k^2 \\ \\log_2 a \\ (\\log_2 |x|)^2$,\n\n\\smallskip\n\n\\noindent where $c$ is a positive constant that depends on the details of \nthe multiplication and addition algorithms. Two integers $n_1, n_2$ (in \nbinary) can be multiplied in time $\\le c \\, \\log_2 n_1 \\, \\log_2 n_2$, and \nadded in time $\\le c \\, \\max\\{\\log_2 n_1, \\ \\log_2 n_2\\}$. The bitlength \nof the product $n_1 n_2$ is $\\le$ $\\lfloor \\log_2 n_1 \\rfloor$ $+$ \n$\\lfloor \\log_2 n_2\\rfloor + 2$. \nFor the last expression in the calculation above, we have \n \\ $c \\ k^2 \\cdot \\log_2 a \\cdot (\\log_2 |x|)^2 \\ < \\ |x|$ \n \\ for large enough $|x|$, i.e., when $c_p \\le |x|$ \\, (where $c_p$ is \na positive integer depending on  $p$).  \n\nRemark (concrete upper-bound on $c_p$): We define $c_p$ to be the smallest \nnumber $N$ such that for all $x \\in A^{\\ge N}$, \nthe time to prepare the counter is $\\le |x|$.  We saw that this time is\n$\\le |x|$ when \\ $|x| \\ge a^{1\/k}$ \\ and\n \\ $|x| \\ge c\\, k^2 \\, \\log_2 a \\ (\\log_2 |x|)^2$.\nOne proves easily that $n \\ge (\\log_2 n)^2$ for all $n \\ge 16$.\nSo, $|x|  = |x|^{1\/2} \\cdot |x|^{1\/2}$\n$ \\ge c\\, k^2 \\, \\log_2 a \\cdot (\\log_2 |x|)^2$ \\ is implied by\n \\ $|x|^{1\/2} \\ge c\\, k^2 \\, \\log_2 a$ \\ and \\ $|x|^{1\/2} \\ge 16$. \nThus we have: \\ $c_p \\ \\le \\ $\n$\\max\\{256, \\ \\ c^2 \\, k^4 \\, (\\log_2 a)^2, \\ \\ a^{1\/k} \\}$.\n\n\\smallskip\n\n\\noindent \n(3) We mark off space of length $p(|x|)$ by converting the binary \nrepresentation of $p(|x|)$ to ``unary'': This is done by the Horner scheme \nwith repeated doubling (where the doubling is done by using two tapes, \nand writing two spaces on the 2nd tape for each space on the 1st tape). \nThis takes time \n \\ $\\sum_{i=0}^{\\lfloor \\log_2 p(|x|) \\rfloor + 1} 2^i \\ < \\ 4 \\, p(|x|)$. \n\n\\smallskip\n\nThus, the total time used to prepare the counter on input $x$ is \n$< \\ 2 \\, |x| + |x| + 4 \\, p(|x|) \\, \\le \\, 7 \\, p(|x|)$, when \n$|x| \\ge c_p$. \n\nFor inputs $x$ with $|x| < c_p$, the counter will also receive space \n$p(|x|)$, but the time used for this could be more than $7 \\, p(|x|)$.\nWe can remove the exception of the finitely many inputs of length $< c_p$ \nas follows. For these inputs we let the Turing machine operate as a\nfinite-state machine (without using the work tapes or any counter); for \nsuch an input $x$, the time to set up the counter will then be \n \\ $\\le \\, |x| + p(|x|)$ ($< 7 \\, p(|x|)$). \n\nAfter execution of program $v$ on input $x$ for time $\\le p(|x|)$, if\nthere is an output $y$ so far, the input balance is checked. \nIf $|y| \\ge |x|$ input balance holds automatically; checking whether\n$|y| \\ge |x|$ takes time $\\le |x|$. If $|x| > |y|$,\nwe compute $p(|y|)$ ($< p(|x|)$) in binary, in the same ways as in steps \n(1) and (2) of the counter-tape preparation above. \nThis takes time \\, $\\le \\, 2 \\, |y| + |y| \\, < \\, 3 \\, |x|$, when \n$|y| \\ge c_p$. The time needed to compare the binary representations of \n$|x|$ and $|y|$ (of length $\\le \\lfloor \\log_2 |x| \\rfloor + 1$) is \nabsorbed in the time to calculate $p(|y|)$. \nSo, the time for checking the input balance is $< |x| + 3 \\, |x|$\n$<$ $4 \\, p(|x|)$.\n\n\\smallskip\n\nSo far we have obtained a machine, described by $(v,p)$, whose \ntime-complexity is $\\le 7 \\, p(.) + p(.) + 4\\, p(.) = 12 \\ p(.)$ and whose \ninput balance is $\\le p(.)$.\nBecause of the preparation of the counter, the time-complexity of the new \nmachine is always larger than $p(.)$; therefore we will further modify the \nconstruction, as follows. Let $p(n) = a \\, (n^k + 1)$; we will assume from \nnow on that $a \\ge 12$. Let $p'(n) = (a - a\\%12) \\cdot (n^k + 1)$, where \n$a\\%12$ is the remainder of the division of $a$ by 12. \nSo, $a - a\\%12 \\ge 12$ and $a - a\\%12$ is a multiple of 12.   \n \\ (1) Instead of marking off space of length $p(|x|)$, the new machine marks \noff length $\\frac{1}{12} \\, p'(|x|)$, in time $\\leq \\frac{7}{12} \\, p'(|x|)$.\n \\ (2) It executes the program $v$ for time $\\le \\frac{1}{12} \\, p'(|x|)$, \nusing the marked-off counter.\n(3) It checks the input balance in time $\\le \\frac{4}{12} \\, p'(|x|)$. \n The total time of the modified machine is then \n$\\le p'(|x|) \\le p(|x|)$, and the input balance is $\\le p'(|x|) \\le p(|x|)$.\n\n\nSo for every Turing machine program $v$ and any polynomial $p$ this \nmodified machine computes a function in {\\sf fP}. Conversely, if $v$ is a \nprogram with polynomial time for a function $f \\in {\\sf fP}$, then the \nmodified program with pair $(v,p)$ correctly computes $f$ if \n$\\frac{1}{12} \\, p'(.)$ is larger than the time (and balance) that $v$ uses \non all inputs; if $f$ belongs to {\\sf fP} then a polynomial $p$ such that\n$\\frac{1}{12} \\, p'(.)$ is large enough for bounding the time and the input \nbalance, exists. \nSuch a modified machine will be called {\\em Turing machine with polynomial\ncounter}. A program for such a machine consists of a pair $(v,p)$ and an\nextra program for preparing the counter and checking input balance; let's\ncall that extra program $u_p$ (it only depends on $p$, and not on $v$). \nThe triple $(v, p, u_p)$, or more precisely, the word \n${\\sf code}(v \\# k \\# a \\# u_p)$ with the numbers $k,a$ written in \nbinary, will be called a {\\it polynomial program}.  Thus, Turing machines \nwith polynomial counter are a machine model for {\\sf fP}.\nSo we have proved:\n\n\\begin{pro} \\label{Polyn_TM}\nThere exists a class of modified Turing machines, called {\\em Turing machine \nwith polynomial counter}, with the following properties:\nFor every $f \\in {\\sf fP}$ there exists a Turing machine with polynomial \ncounter that computes $f$; and for every Turing machine with polynomial\ncounter, the input-output function belongs to {\\sf fP}.  \n  \\ \\ \\ $\\Box$\n\\end{pro} \n{\\bf Notation:} \\ A polynomial program \n$w = {\\sf code}(v \\# k \\# a \\# u_p)$, based on a Turing machine program $v$ \nand a polynomial $p$ (with $p(n) = a \\, n^k + a$), will be denoted by \n$\\langle v,p \\rangle$. The polynomial $p$ that appears in $w$ will often be \ndenoted by $p_w$. The function computed by a polynomial program \n$w = \\langle v,p \\rangle$ will be denoted by $\\phi_w$ ($\\in {\\sf fP}$).\n\n\\bigskip\n\n\n\\noindent {\\bf Evaluation maps for {\\sf fP}}:\n\n\\medskip\n\nAt first we consider a function ${\\sf ev}_{\\sf poly}$ defined by \n${\\sf ev}_{\\sf poly}(w, x) = (w, \\phi_w(x))$, \nwhere $w = \\langle v,p \\rangle$ is any polynomial program.\nBut ${\\sf ev}_{\\sf poly}$ is {\\em not} in {\\sf fP}. Indeed, the output \nlength (and hence the time-complexity) of  ${\\sf ev}_{\\sf poly}$ on input \n$(w,x)$ is equal to $p(|x|)$ (in infinitely many cases, when $p$ is a tight \nupper-bound); as $w$ varies, the degree of $p$ is unboundedly large, hence \nthe time-complexity of ${\\sf ev}_{\\sf poly}$ has no polynomial upper-bound. \nWe will nevertheless be able to build {\\sf fP}-critical functions. For a \nfixed polynomial $q$ (of the form $q(n) = a \\, n^k + a$), let\n\n\\medskip\n\n\\hspace{1in}   ${\\sf fP}^{q} \\ = \\ \\{ \\phi_w \\in {\\sf fP} : \\ p_w \\le q\\}$.\n\n\\medskip\n\n\\noindent More explicitly, $p_w \\le q$ means that the polynomial program $w$\nhas time-complexity $\\le p_w(|x|) \\le q(|x|)$ and input-balance \n$|x| \\leq p_w(|\\phi_w(x)|) \\le q(|\\phi_w(x)|)$, for all \n$x \\in {\\sf Dom}(\\phi_w)$.\n\nIn general, for polynomials $q_1, q_2$ we say $q_1 \\le q_2$ iff for all \nnon-negative integers $n$: $q_1(n) \\le q_2(n)$.\nInterestingly, for polynomials of the form $q_i(n) = a_i \\, n^{k_i} + a_i$ \nwe have: \n$q_1 \\le q_2$ \\ iff \\ $k_1 \\le k_2$ and $a_1 \\le a_2$.  \nHence it is easy to check whether $p_w \\le q$, given $w$ and the two numbers\nthat determine $q$.  \n\n\\smallskip\n\nA polynomial program $w$ such that $p_w \\le q$ (for a fixed polynomial $q$) \nis called a {\\em $q$-polynomial program}. We define ${\\sf ev}_{q}$ by \n\n\\smallskip\n\n\\hspace{1in}  ${\\sf ev}_{q}(w, x) = (w, \\phi_w(x))$,\n\n\\smallskip\n\n\\noindent where $w$ is any $q$-polynomial program. \nThe function ${\\sf ev}_{q}$ above has two input and two output strings. \nTo make ${\\sf ev}_{q}$ fit into our framework of functions with one input \nand one output string we encode ${\\sf ev}_{q}$ as \n${\\sf ev}_{q}^C: \\{0,1\\}^* \\to \\{0,1\\}^*$ where for all \n$w, x \\in \\{0,1\\}^*$ such that $w$ is a $q$-polynomial program, \n\n\\medskip\n\n\\hspace{1in}   ${\\sf ev}_{q}^C\\big({\\sf code}(w) \\, 11 \\, x\\big) = $\n        ${\\sf code}(w) \\, 11 \\, \\phi_w(x)$.\n\n\\medskip\n\n\\noindent From now on we will call ${\\sf ev}_{q}^C$ an {\\it ``evaluation \nmap''}.  We observe that in the special case where $\\phi_w$ (for a fixed \n$w$) is a right ideal morphism, the function \n\n\\medskip\n\n\\hspace{1in} ${\\sf ev}_{q}^C\\big({\\sf code}(w) \\, 11 \\, \\cdot\\big)$:\n$ \\ \\ x \\ \\longmapsto \\ \\ {\\sf code}(w) \\, 11 \\, \\phi_w(x)$ \n\n\\medskip\n\n\\noindent is also a right ideal morphism.\n\n\\bigskip\n\n\n\\noindent {\\bf Criticality of ${\\sf ev}_{q}^C$:}\n\n\\medskip\n\n\\noindent For any fixed word $v \\in \\{0,1\\}^*$ we define the prepending \nmap\n\n\\smallskip\n\n\\hspace{1in}  $\\pi_v: x \\in \\{0,1\\}^* \\longmapsto v \\, x$ ;\n\n\\smallskip\n\n\n\\noindent and for any fixed positive integer $k$ we define \n\n\\smallskip\n\n\\hspace{1in} $\\pi_k': z \\, x \\in \\{0,1\\}^* \\longmapsto x$, \\ where $|z| = k$\n \\ (with $\\pi_k'(t)$ undefined when $|t| < k$). \n\n\\smallskip\n\n\\noindent Clearly, $\\pi_v, \\pi_k' \\in {\\cal RM}_2^{\\sf P}$, and we have \n \\ $\\pi'_{|v|} \\circ \\pi_v = {\\sf id}_{A^*}$ \n \\ (i.e., $\\pi'_{|v|}$ is a left inverse of $\\pi_v$).\n\nWe observe that $\\pi_v$ can be written as a composite of the\nmaps $\\pi_0$ and $\\pi_1$, for any $v \\in \\{0,1\\}^*$. Similarly, $\\pi_k'$ \nis the $k$th power of $\\pi_1'$. \n\n\n\\begin{pro} \\label{ev_q} \n \\ Let $q$ be any polynomial such that for all $n \\ge 0$, $q(n) > cn+c$\n(where $c > 1$ is a constant).  \nThen ${\\sf ev}_{q}^C$ belongs to {\\sf fP}, and ${\\sf ev}_{q}^C$ \nis a one-way function if one-way functions exist. \n\\end{pro}\n{\\bf Proof.} We saw that testing whether $p_w \\le q$ is easy for \npolynomials of the form that we consider. \nBy reviewing the workings of a universal Turing machine (e.g., \nin \\cite{HU}) we see that the time-complexity of ${\\sf ev}_{q}(w,x)$ is \\, \n$\\leq \\ c_0 \\ |w| \\cdot p_w(|x|)^2$ (when $p_w$ is at least linear); here, \n$c_0 \\ge 1$ is a constant (independent of $x$ and $w$). \nThe factor $p_w(|x|)^2$ comes the fact that Turing machines can have any \nnumber of tapes, whereas a Turing machine for ${\\sf ev}_{q}$ has a fixed \nnumber of tapes; any number of tapes can be converted to one tape, but the\ncomplexity increases by a square (the more efficient Hennie-Stearns \nconstruction converts any number of tapes to two tapes, with a complexity\nincrease from $T$ to $T \\log T$). The universal Turing machine \nsimulates each transition of program $w$ (modified into a 1-tape Turing \nmachine) using $\\le c_1 \\, |w|$ steps (for a constant $c_1 \\ge 1$). \n\nFor input balance: When $|x| \\le |\\phi_w(x)|$ we also have \n$|w| + |x| \\le |w| + |\\phi_w(x)|$ so balance is automatic.\nWhen $|x| > |\\phi_w(x)|$ then (since $p_w$ bounds the input balance of\n$\\phi_w$), the input-length satisfies \\,  \n$|w| + |x| \\leq |w| + p_w(|\\phi_w(x)|) \\ \\leq \\ |w| + q(|\\phi_w(x)|)$\n$\\le$ $q(|w| + |\\phi_w(x)|)$. \nHence ${\\sf ev}_{q}$, and similarly ${\\sf ev}_{q}^C$, belongs to {\\sf fP}.\n\nCriticality: \nIf the function ${\\sf ev}_{q}$ has an inverse $e'_q \\in {\\sf fP}$, then \n \\, ${\\sf ev}_{q} \\circ e'_q \\circ {\\sf ev}_{q}(w,x) = (w, \\phi_w(x)$.\nHence, for any function $\\phi_w \\in {\\sf fP}^{q}$ with a fixed program $w$ \nwe have:  \n$\\phi_w \\circ \\pi_{|w|}' \\circ e'_q \\circ \\pi_w \\circ \\phi_w = \\phi_w$,\nwhere $\\pi_{|w|}': (w,v) \\mapsto v$, and $\\pi_w: y \\mapsto (w,y)$. \nFor a fixed $w$ we have $\\pi_{|w|}', \\pi_w \\in {\\sf fP}$, so \\, \n$\\pi_{|w|}' \\circ e'_q \\circ \\pi_w$ \\, is an inverse of $\\phi_w$. Hence, if\n${\\sf ev}_{q}$ is not one-way, no function in ${\\sf fP}^{q}$ is one-way. \nBut there are functions (e.g., some $f_M$ as seen in Prop.\\ \\ref{f_Mcrit}) \nthat are {\\sf fP}-critical, even when $q$ is linear (since by padding \narguments one can obtain {\\sf NP}-complete languages with nondeterministic\nlinear time-complexity).  So some {\\sf fP}-critical $f_M$ would have an \ninverse in {\\sf fP}. The same proof is easily adapted to ${\\sf ev}_{q}^C$. \n \\ \\ \\ $\\Box$ \n\n\\bigskip\n\n\n\\noindent {\\bf Finite generation of {\\sf fP}:}\n\n\\medskip\n\n\\noindent We will show that ${\\sf ev}_{q}^C$ can be used to simulate\nuniversal evaluation maps, and to prove that {\\sf fP} is finitely generated. \nThis is based on the universality of ${\\sf ev}_{q_2}^C$ for \n${\\sf fP}^{q_2}$, combined with a padding argument; here $q_2$ is a\npolynomial of degree 2, with $q_2(n) \\ge  c\\, n^2 +c$ for a constant \n$c \\ge 12$.  (We chose 12 in view of the reasoning before Prop.\\ \n\\ref{Polyn_TM}.)\nWe need some auxiliary functions first.\n\n\\medskip\n\n\\noindent We define the {\\em expansion (or padding) map}, first as a \nmulti-variable function for simplicity:\n\n\\smallskip\n\n\\hspace{.6in} ${\\sf expand}(w, x) \\ = \\ $\n    $({\\sf e}(w), \\ (0^{4 \\, |x|^2 + 7 \\, |x| + 2}, \\ x))$\n\n\\smallskip\n\n\\noindent where ${\\sf e}(w)$ is such that \\    \n\n\\smallskip\n\n\\hspace{.6in} \n$\\phi_{{\\sf e}(w)}(0^k, \\ x) \\ = \\ (0^k, \\ \\phi_w(x))$, \\ for all $k \\geq 0$.\n\n\\smallskip\n\n\\noindent The program ${\\sf e}(w)$ is easily obtained from the program \n$w$, since it just processes the padding in front of the input and in front \nof the output of $\\phi_w$, and acts as $\\phi_w$ on $x$; and the time and\nbalance polynomial is decreased due to padding.  \nAs a one-variable function, {\\sf expand} is defined by\n\n\\medskip\n\n\\hspace{.6in}  ${\\sf expand}\\big({\\sf code}(w) \\ 11 \\ x\\big) \\ = \\ \n   {\\sf code}({\\sf ex}(w)) \\ 11 \\ 0^{4 \\, |x|^2 + 7 \\, |x| + 2} \\ 11 \\ x$,\n\n\\medskip\n\n\\noindent where for one-variable functions, the program ${\\sf ex}(w)$ \nis such that \\    \n\n\\medskip\n\n\\hspace{.6in} $\\phi_{{\\sf ex}(w)}(0^k \\ 11 \\ x) \\ = \\ 0^k \\ 11 \\ \\phi_w(x)$.\n\n\\medskip\n\n\\noindent Again, ${\\sf ex}(w)$ is a slight modification of \n$w = {\\sf code}(v \\# k \\# a \\# u_p)$ (following the notation for the\nmachine model for {\\sf fP}), to allow inputs and outputs with padding, and \nto readjust the complexity and balance polynomial. \nMore precisely, ${\\sf ex}(w)$ is of the form \n${\\sf code}(r \\# v \\# \\lceil k\/2 \\rceil \\# a_e \\# u_{p_e})$, where $r$ is \na preprocessing subprogram by which the prefix $0^k 11$ of the input is \nsimply copied to the output; at the end of execution of $r$, the state and \nhead-positions are the start state and start positions of the subprogram \n$w$. The appropriate complexity and balance polynomial stored in \n${\\sf ex}(w)$ is \n\n\\medskip\n\n \\ \\ \\ $p_e(n) = a_e \\, n^{\\lceil k\/2 \\rceil} + a_e$, \\ with \\ \n$a_e \\ = \\ \\max\\{12, \\ \\lceil a\/2^k \\rceil +1\\}$.  \n\n\\medskip\n\n\\noindent Indeed, if $m = |x|$, an input \n$0^{4 \\, |x|^2 + 7 \\, |x| + 2} \\ 11 \\ x$ of $\\phi_{{\\sf ex}(w)}$ has \nlength $i = 4 \\, m^2 + 8 \\, m + 4$. So, $m = \\sqrt{i}\/2 - 1$.  \nLet $a \\, (m^k + 1)$ be the polynomial of program $w$.\nThe complexity of $\\phi_{{\\sf ex}(w)}$ on its input \nis $4 \\, m^2 + 7 \\, m + 4$ (for reading the part $0^* 11$ of the input), \nplus \\, $a \\, (m^k + 1)$ (for using $x$ and computing $\\phi_w(x)$).  \nSo in terms of its input length $i$, the complexity of \n$\\phi_{{\\sf ex}(w)}$ is \\ $< \\ i + a \\, (m^k + 1)$\n$\\le \\ i^{\\lceil k\/2 \\rceil} \\ + \\ a \\, ((\\sqrt{i}\/2 - 1)^k + 1)$ \n$\\le \\ i^{\\lceil k\/2 \\rceil} \\ + \\ a\/2^k \\, i^{k\/2}$; \nthe last step uses the fact that $(z - 1)^m \\le z^m - 1$ for all \n$z \\ge 0, \\ m \\ge 1$.  Hence the complexity of $\\phi_{{\\sf ex}(w)}$ is \n$< \\ (a\/2^k + 1) \\, i^{\\lceil k\/2 \\rceil}$.  \nFor the input balance of $\\phi_{{\\sf ex}(w)}$ we have: The input-length\nis bounded by twice the output-length.\nIndeed, the input length is $i = 4 \\, |x|^2 + 8 \\, |x| + 4$\n$<$ $2 \\cdot |0^{4 \\, |x|^2 + 7 \\, |x| + 2} \\ 11|$\n$<$ $2 \\cdot |0^{4 \\, |x|^2 + 7 \\, |x| + 2} \\, 11 \\, \\phi_w(x)|$ $=$ \n$2 \\cdot |\\phi_{{\\sf ex}(w)}(0^{4 \\, |x|^2 + 7 \\, |x| + 2} \\, 11 \\, x)|$.\nMoreover, we want $a_e$ to stay $\\ge 12$ (in view of the reasoning before \nProp.\\ \\ref{Polyn_TM}). \n\n\\smallskip\n\nIn order to achieve an arbitrarily large polynomial amount of padding we \niterate the quadratic padding operation. Therefore we define a {\\em repeated \nexpansion (or re-padding)} map, first as a two-variable function:\n\n\\medskip\n\n\\hspace{.6in} ${\\sf reexpand}(u, (0^h, x)) \\ = \\ $\n       $({\\sf e}(u), (0^{4 \\, h^2 + 8 \\, h + 2}, x))$, \\ for all $h > 0$, \n\n\\medskip\n\n\\noindent where ${\\sf e}(.)$ is as above. As a one-variable function, \n\n\\medskip\n\n\\hspace{.6in} ${\\sf reexpand}\\big({\\sf code}(u) \\ 11 \\ 0^h \\ 11 \\ x \\big)$ \n$ \\ = \\ $ \n${\\sf code}({\\sf ex}(u)) \\ 11 \\ 0^{4 \\, h^2 + 8 \\, h + 2} \\ 11 \\ x$, \n         \\ for any $h \\geq 0$,\n\n\\medskip\n\n\\noindent with ${\\sf ex}(.)$ as in {\\sf expand} above.\n\n\\medskip\n\nWe also introduce a {\\em contraction (or unpadding) map}, which is a partial\nleft inverse of {\\sf expand}. We define {\\sf contr} first as a multi-variable \nfunction:\n\n\\medskip\n\n\\hspace{.6in} \n$ {\\sf contr}(w, \\ (0^h, \\ y)) \\ = \\ ({\\sf c}(w), \\ y)$, \n  \\ \\ if $h \\leq 4 \\, |y|^2 + 7 \\, |y| + 2$ \\ \\ (undefined otherwise).  \n \n\\medskip\n\n\\noindent As a one-variable function, {\\sf contr} is defined by\n\n\\medskip \n\n\\hspace{.6in} \n${\\sf contr}\\big({\\sf code}(w) \\ 11 \\ 0^h \\ 11 \\ y \\big) \\ = \\  $\n${\\sf code}({\\sf co}(w))\\ 11 \\ y$, \\ \\ if $h \\leq 4\\, |y|^2 + 7\\, |y| + 2$ \n  \n\\hspace{.6in} (undefined otherwise). \n\n\\medskip\n\n\\noindent The program transformations ${\\sf c}(.)$ and ${\\sf co}(.)$ are\ninverses of ${\\sf e}(.)$, respectively ${\\sf ex}(.)$. So, ${\\sf c}(.)$ and\n${\\sf co}(.)$ erase the prefix $r$ in ${\\sf ex}(u)$, and replace the\npolynomial $b \\, n^h + b$, encoded in ${\\sf ex}(u)$, by\n$b_c \\, n^{2h} + b_c$, where  \\, $b_c = (b-1) \\, 2^{2h}$.\n\n\\smallskip\n\nTo invert repeated padding we introduce a {\\em repeated contraction (or \nunpadding)} map, first as a multi-variable function. \nNote that if $h = 4 \\, k^2 + 8 \\, k + 2$ (which is the amount of padding \nintroduced by {\\sf reexpand}), then $k = \\frac{1}{2} \\, \\sqrt{h+2} -1$.\nTherefore, for any $h \\ge 0$ we define\n\n\\medskip\n\n\\hspace{.6in} ${\\sf recontr}(u, \\ (0^h, \\ y)) \\ = \\ $ \n$({\\sf c}(u), \\ (0^{\\max\\{1, \\ \\lfloor \\sqrt{h+2}\/2 \\rfloor - 1\\}}, \\ y))$\n   \\ \\ (undefined on other inputs).\n \n\\medskip\n\n\\noindent As a one-variable function, {\\sf recontr} is defined by\n\n\\medskip\n\n\\hspace{.6in} \n${\\sf recontr}\\big({\\sf code}(u) \\ 11 \\ 0^h \\ 11 \\ y \\big)$\n $ \\ = \\ $ \n${\\sf code}({\\sf co}(u))\\ 11$\n    $0^{\\max\\{1, \\ \\lfloor \\sqrt{h+2}\/2 \\rfloor - 1\\}} \\ 11\\ y$ \n\n\\hspace{.6in} (undefined on other inputs).\n\n\\medskip\n\n\nThe maps {\\sf expand}, {\\sf reexpand}, {\\sf contr}, and {\\sf recontr} belong \nto {\\sf fP}, and they are regular (they have polynomial-time inverses). \n\n\n\n\\begin{pro} \\label{fPfingen} \n \\ \\ {\\sf fP} is finitely generated.\n\\end{pro}\n{\\bf Proof.} \\ We will show that the following is a generating set of \n{\\sf fP}: \n\n\\medskip\n \n\\hspace{1in} \n$\\{{\\sf expand}, \\ {\\sf reexpand}, \\ {\\sf contr}, \\ {\\sf recontr}, $\n$ \\ \\pi_0, \\ \\pi_1, \\ \\pi_1', \\ {\\sf ev}_{q_2}^C \\}$, \n\n\\medskip\n\n\\noindent where $q_2$ is the polynomial $q_2(n) = c \\, n^2 + c$, with\n$c \\geq 12$ (the number 12 comes from the discussion before Prop.\\\n\\ref{Polyn_TM}).\n\n\\smallskip\n\n\\noindent {\\sf Remark:} \\, The functions {\\sf expand}, {\\sf reexpand}, \n{\\sf contr}, and {\\sf recontr} all have quadratic time-complexity and\nbalance functions, so they can be generated by $\\pi_0, \\pi_1, \\pi_1'$, and\n${\\sf ev}_{q_2}^C$ (provided that the constant $c$ in $q_2$ is chosen large \nenough, so that ${\\sf ev}_{q_2}^C$ can execute {\\sf expand}, {\\sf reexpand}, \n{\\sf contr}, and {\\sf recontr}).  Thus \\, \n\n\\medskip\n\n\\hspace{1in} \n$\\{\\pi_0, \\ \\pi_1, \\ \\pi_1', \\ {\\sf ev}_{q_2}^C \\}$ \\, \n\n\\medskip\n\n\\noindent is a generating set of {\\sf fP}.  \nWe use the larger generating set since it yields simpler formulas.\n\n\\medskip\n\nLet $w$ be a program with polynomial counter and let $m$ be an integer \nupper-bound on $\\log_2(a + k)$, where $p_w (n) = a \\, n^k + a$. \nWe also assume that the program $w = \\langle v, p \\rangle$ is such that \nfor any polynomial $P(.) > p_w(.)$, $\\langle v, P \\rangle$ also computes \n$\\phi_w$; indeed, since $\\phi_w \\in {\\sf fP}$, we can choose $v$ and $p$ so \nthat the execution of $v$ by the Turing machine with polynomial counter \n(described by $\\langle v, p \\rangle$) never triggers the counter; in that \ncase, making the counter larger does not change the function.  \nThen for all $x \\in \\{0,1\\}^*$,\n\n\\bigskip\n\n\\noindent $(\\star)$ \n\\hspace{.5in} $\\phi_w(x) \\ = \\ $\n$\\pi_{_{2 \\, |w'| + 2}}' \\circ {\\sf contr} \\circ {\\sf recontr}^{2 \\, m}$ \n$\\circ$ ${\\sf ev}_{q_2}^C$ $\\circ$ \n${\\sf reexpand}^{2m} \\circ {\\sf expand}$  $\\circ$\n$\\pi_{_{{\\sf code}(w) \\, 11}}(x)$ ,\n\n\\bigskip\n\n\\noindent where $w' = {\\sf co}^{2m+1} \\circ {\\sf ex}^{2m+1}(w)$.\nIndeed,\n\n\\medskip\n\n $x \\ \\ \\stackrel{\\pi_{_{{\\sf code}(w) \\, 11}}}{\\longmapsto} \\ \\ $\n${\\sf code}(w) \\ 11 \\ x \\ \\ $\n$\\stackrel{\\sf expand}{\\longmapsto} \\ \\ $\n${\\sf code}({\\sf ex}(w)) \\ 11 \\ 0^{4 \\, |x|^2 + 7\\, |x| +2} \\ 11 \\ x$ \n\n\\medskip\n\n$\\stackrel{{\\sf reexpand}^{2m}}{\\longmapsto} \\ \\ $ \n${\\sf code}({\\sf ex}^{2m+1}(w)) \\ 11 \\ 0^{N_{2m+1}} \\ 11 \\ x$. \n\n\\medskip\n\n\\noindent Here $N_1 = |x|^2 + 7\\, |x| +2$, and \n$|0^{N_1} \\, 11 \\, x| = (2 \\, (|x| + 1))^2$;     inductively, \n$N_i = 4\\, N_{i-1}^2 + 8\\,  N_{i-1} + 2$ \\ for $1 < i \\leq 2m+1$,\nand $|0^{N_i} \\, 11| = (2 \\, (N_{i-1} + 1))^2$.\n\n\\noindent Continuing the calculation, \n\n\\medskip\n\n$\\stackrel{{\\sf ev}_{q_2}^C}{\\longmapsto} \\ \\ $\n${\\sf code}({\\sf ex}^{2m+1}(w)) \\ 11 \\ 0^{N_{2m+1}} \\ 11 \\ \\phi_w(x) $\n\n\\medskip\n\n$\\stackrel{{\\sf recontr}^{2m}}{\\longmapsto} \\ \\ $\n${\\sf code}(w') \\ 11 \\ 0^{\\ell} \\ 11 \\ \\phi_w(x) \\ \\ $\n\n\\medskip\n\n$\\stackrel{\\sf contr}{\\longmapsto} \\ \\ $\n${\\sf code}(w') \\ 11 \\, \\phi_w(x) \\ \\ $\n$\\stackrel{\\pi_{_{2 \\, |w'| + 2}}'}{\\longmapsto} \\ \\ \\phi_w(x)$\n\n\\medskip\n\n\\noindent where $w' = {\\sf co}^{2m+1} \\circ {\\sf ex}^{2m+1}(w)$.\nWe use $2m$ in ${\\sf recontr}^{2 m}$ because $\\phi_w(x)$ could be much \nshorter than $x$ (but by input balance, $|x| \\le p_w(|\\phi_w(x)|)$).\nAs a consequence, we also use $2m$ in ${\\sf reexpand}^{2 m}$ in order to have \nequal numbers of program transformations ${\\sf ex}(.)$ and ${\\sf co}(.)$.   \nNote that doing more input padding than necessary does not do any harm; also,\nif $w$ contains a polynomial $p_w$ larger than needed for computing \n$\\phi_w$, this does not cause a problem (by our assumption on $v$).  \nBy the choice of $2m$, the value of $\\ell$ above is less than \n$4 \\, |\\phi_w(x)|^2 + 7 \\, |\\phi_w(x)| + 2$, so {\\sf contr} can be applied \ncorrectly.  \n\nThe argument of ${\\sf ev}_{q_2}^C$ in the above calculation has length\n$> N_{2m+1} + 2 + |x|$, which is much larger than the time it \ntakes to simulate the machine with program $w$ on input $x$ (that time is \n$< c_0 |w| \\, p_w(|x|)^2$). In fact, by the choice of $m$, the polynomial \nencoded in ${\\sf ex}^{2m+1}(w)$ is the linear polynomial $12 \\, (n + 1)$ \n(which is $< q_2(n)$).  Hence, ${\\sf ev}_{q_2}^C$ works correctly on its \ninput in this context.\n \\ \\ \\ $\\Box$\n\n\\medskip\n\nWe saw that {\\sf fP} does not have an evaluation map in the same sense as\nthe Turing evaluation map. However, formula $(\\star)$ in the proof of  \nProp.\\ \\ref{fPfingen} shows that the map ${\\sf ev}_{q_2}^C$ simulates every\nfunction in {\\sf fP}, in the following sense: \n$f_2$ {\\em simulates} $f_1$ (denoted by $f_1 \\preccurlyeq f_2$) iff there \nexist $\\beta, \\alpha \\in {\\sf fP}$ such that \n$f_1 = \\beta \\circ f_2 \\circ \\alpha$; this is discussed further at the \nbeginning of Section 5.  \nFormula $(\\star)$ in Prop.\\ \\ref{fPfingen} implies:\n\n\\begin{pro} \\label{evalfP} \n \\ Every function $f \\in {\\sf fP}$ is simulated by ${\\sf ev}^C_{q_2}$.\n \\ \\ \\  \\ \\ \\  $\\Box$\n\\end{pro}  \nIt follows from this and the definition of simulation that\n${\\sf ev}^C_{q_2}$ belongs to the $\\cal J$-class of ${\\sf id}_{A^*}$ in \n{\\sf fP}. \n\n\\medskip\n\nSince {\\sf fP} is finitely generated we now have two ways of representing \neach element $g \\in {\\sf fP}$ by a word: \n(1) We have $g = \\phi_w$ for some polynomial program $w \\in A^*$ (as seen \nin Prop.\\ \\ref{ev_q}), and \n(2) $g$ can be represented by a string of generators (considering the finite \nset of generators of {\\sf fP} as an alphabet). The next proposition \ndescribes the translation between these two representations.\n\n\n\\begin{pro} \\label{compilers}\nThere are total computable maps $\\alpha, \\beta$ such that for any word $s$ \nover a finite generating set of {\\sf fP}, $\\alpha(s)$ is a polynomial \nprogram for the function given by $s$; and for any polynomial program $u$, \n$\\beta(u)$ is a word for $\\phi_u$ over the generators of {\\sf fP}.\n\nMore precisely, let $\\Gamma$ be a finite generating set of {\\sf fP}. \nFor any $s \\in \\Gamma^*$, let $\\Pi s \\in {\\sf fP}$ be the element of \n{\\sf fP} obtained by composing the generators in the sequence $s$. \nThere exist total recursive {\\em ``compiler maps''} \n$\\alpha: \\Gamma^* \\to \\{0,1\\}^*$ and $\\beta: \\{0,1\\}^* \\to \\Gamma^*$ such \nthat for all $s \\in \\Gamma^*$ and all $w \\in  \\{0,1\\}^*$: \\, \n$f_{\\alpha(s)} = \\Pi s$, and $\\Pi \\beta(w) = \\phi_w$.  \n\\end{pro}\n\n\\smallskip\n\n\\noindent\n{\\bf Proof.} The map $\\beta$ is given by formula $(\\star)$ in the proof \nof Prop.\\ \\ref{fPfingen}, where a representation over the generators is \nexplicitly constructed. When $u$ is not a well-formed polynomial program\nwe let $\\beta(u)$ be a sequence of generators for the empty function.  \n\nConversely, by composing a sequence of generators, a function in {\\sf fP} \nis obtained (note that every sequence $s$ of generators has a finite \nlength).  More precisely, if $f_1, f_2 \\in {\\sf fP}$ have as complexity \nand balance bounds the polynomials \n$q_i(n) = a_i \\, n^{k_i} + a_i$ ($i = 1, 2$), then \n$f_2 \\circ f_1$ has input balance $\\le q_2 \\circ q_1(n)$ (obviously), and \ntime-complexity $\\le q_1(n) + q_2 \\circ q_1(n)$. \nIndeed, a polynomial-time program for $f_2 \\circ f_1$ is obtained by first \ntaking the program for $f_1$ on input $x$, and then applying the program \nfor $f_2$ to $f_1(x)$ (in time $\\le q_2(|f_1(x)|)$). \nThe corresponding polynomial upper-bound is\n \\, $q_1(n) + q_2 \\circ q_1(n)$ $=$ \n$a_1 \\, (n^{k_1} + 1) + a_2 \\, a_1^{k_2} \\, (n^{k_1} + 1)^{k_2} + a_2$ $<$\n$(a_1 + a_2 \\, a_1^{k_2})\\, (n^{k_1} + 1)^{k_2} + a_2$.\nIn order to obtain a polynomial upper-bound of the form $a \\, n^k + a$, we\nuse the inequality  \n\n\\smallskip\n\n$(n + 1)^j \\ \\le \\ 2^{j-1} \\, (n^j + 1)$, \\ for all $n \\ge 0, \\ j \\ge 1$. \n\n\\smallskip\n\n\\noindent (To prove this inequality apply calculus to the function \n$f(x) = 2^{j-1} (x^j + 1) - (x + 1)^j$.) \nThus for $f_2 \\circ f_1$ we get a complexity and balance upper-bound \n\n\\smallskip\n\n$q(n) = a \\, n^{k_1 k_2} + a$, where\n \\ $a = a_2 + a_1 + a_2 \\, a_1^{k_2} \\, 2^{k_2}$.\n\\smallskip\n\n\\noindent This yields an algorithm for obtaining a polynomial program for \n$f_2 \\circ f_1$ from polynomial programs for $f_1$ and $f_2$. For a sequence\nof generators $s$, this algorithm can be repeated $|s| - 1$ times to yield\na polynomial programs for the sequence $s$ of generators.  \n \\ \\ \\ $\\Box$ \n\n\\medskip\n\n\\noindent A finite generating set $\\Gamma$ for {\\sf fP} can be used to \nconstruct a {\\em generator-based evaluation map} for {\\sf fP}, defined by\n \\ $(s, x) \\in \\Gamma^* \\times A^* \\ \\longmapsto \\ {\\sf ev}_{\\Gamma}(s,x)$\n$=$ $(s, \\, (\\Pi s)(x))$. \n However, ${\\sf ev}_{\\Gamma}$ does not belong to {\\sf fP}, for the \nsame reasons as we saw at the beginning of Sect.\\ 4 for ${\\sf ev}_{\\sf poly}$.\n(But just as for ${\\sf ev}_{\\sf poly}$ we could restrict ${\\sf ev}_{\\Gamma}$ \nto a function that belongs to {\\sf fP} and that simulates every element \nof {\\sf fP}.)\n\n\n\\begin{pro} \\label{notFinPres}\n \\ {\\sf fP} is {\\em not} finitely presented. Its word problem is \nco-r.e.,  but not r.e.\n\\end{pro}\n{\\bf Proof.} The word problem is co-r.e.: Let $U, V \\in \\Gamma^*$; using \nProp.\\ \\ref{compilers} we effectively find programs $u, v \\in A^*$ from \n$U, V$ such that $\\phi_u = \\Pi U$, $\\phi_v = \\Pi V$.\nIf $\\phi_u \\neq \\phi_v$ then by exhaustive search we will find $x$ such that\n$\\phi_u(x) \\neq \\phi_v(x)$, thus showing that $U \\neq V$ in {\\sf fP}.\nWhen $U = V$ in {\\sf fP} then this procedure rejects by not halting.\n\nThe word problem of {\\sf fP} is undecidable, since the equality problem for\nlanguages in {\\sf P} can be reduced to this (reducing $L$ to ${\\sf id}_L$ \nor to ${\\sf id}_{{\\sf code}(L\\#)}$).\nAnd the equality problem for languages in {\\sf P} is undecidable, since the\nuniversality problem of context-free languages can be reduced to the equality\nproblem for languages in {\\sf P}; all context-free languages are in {\\sf P}.\nThe universality problem for context-free language is the question whether\nfor a given context-free grammar $G$ with terminal alphabet $A$ (with\n$|A| \\geq 2$), the language generated by $G$ is $A^*$; this problem is\nundecidable (see \\cite{HU} Thm.\\ 8.11).\n\nSince the word problem is co-r.e.\\ but undecidable, it is not r.e.  Hence\nthese finitely generated monoids are not finitely presented (since the word\nproblem of a finitely presented monoid is r.e.).\n \\ \\ \\ $\\Box$\n\n\n\\begin{pro} \\ \\ {\\sf fP} is finitely generated by {\\em regular} elements.\n\\end{pro}\n{\\bf Proof.} All the listed generators of {\\sf fP} are regular, except \npossibly ${\\sf ev}_{q_2}^C$.\nLet us define a partial function $E_{q_2} \\in {\\sf fP}^{q_2}$ by \\,\n $E_{q_2}(w, x) = (w, \\phi_w(x), x)$, when $\\phi_w \\in {\\sf fP}^{q_2}$. \nObviously, $E_{q_2}$ is not one-way.\nBut ${\\sf ev}_{q_2}$ (as a two-variable function) can be expressed as a\ncomposition of $E_{q_2}$ and the other (regular) generators.\nIn more detail, ${\\sf ev}_{q_2} = \\pi'_{q_2} \\circ E_{q_2}$, where \n$\\pi'_{q_2}(w,z,x) = (w,z)$ if $|z| \\le q_2(|x|)$ and $|x| \\le q_2(|z|)$. \nSo ${\\sf ev}_{q_2}^C$ can be replaced by $E_{q_2}^C$ as a generator.\n \\ \\ \\ $\\Box$\n\n\n\\begin{pro} \\ There are elements of {\\sf fP} and of ${\\cal RM}_2^{\\sf P}$\nthat are critical (i.e., non-regular if ${\\sf P} \\neq {\\sf NP}$), whose \nproduct is a non-zero idempotent.\n\\end{pro}\n{\\bf Proof.} For $i = 0, 1$, let $e_i \\in {\\sf fP}$ be defined, as a\ntwo-variable function, by\n\n\\smallskip\n\n \\ \\ \\ \\ \\ \\\n$e_i(w,x) \\ = \\\n\\left\\{\n\\begin{array}{ll}\n(w, \\phi_w(x)) & \\mbox{if $x \\in i\\, \\{0,1\\}^*$,\\ \n     $\\phi_w(x) \\in i\\, \\{0,1\\}^*$,\n     and $|\\phi_w(x)| = |x|$;} \\\\\n(w, 0^{|x|}) & \\mbox{otherwise.}\n\\end{array}  \\right. $\n\n\\smallskip\n\n\\noindent Then $(e_1 \\circ e_0)(w,x) = (w, 0^{|x|})$ for all $(w, x)$,\nso $e_1 \\circ e_0$ is an idempotent.\n\nTo prove that $e_i$ is critical we reduce the satisfiability problem to\nthe inversion problem of $e_i$.\nThe reduction for $e_i$ maps a boolean formula $B$ with $n$ variables to\n$(b, i^n 1)$, where $b$ is a program such that\n$f_b(i \\, \\tau) = i^n B(\\tau)$; i.e., for a truth-value assignment\n$\\tau \\in \\{0,1\\}^n$, $f_b$ evaluates $B$ on $\\tau$, and outputs the \nresulting truth-value, prefixed with $n$ copies of $i$.\nIf $e_i$ were regular then ${\\sf Im}(e_i)$ would be in {\\sf P}, by\nProp.\\ \\ref{ImfP}.\nThen satisfiability of $B$ could be checked by a {\\sf P}-algorithm,\nsince  $B$ is satisfiable iff $(b, i^n 1) \\in {\\sf Im}(e_i)$.\nTo obtain one-variable functions we can take $e_i^C$.\n\nTo prove the proposition for ${\\cal RM}_2^{\\sf P}$ we define $e_i \\in$\n${\\cal RM}_2^{\\sf P}$ for $i = 0, 1$ as follows, first as two-variable\nfunctions:\n\n\\smallskip\n\n \\ \\ \\ \\ \\ \\\n$e_i(w,x) \\ = \\\n\\left\\{\n\\begin{array}{ll}\n(w, \\phi_w(x)) & \\mbox{if $x \\in 0i \\, \\{0,1\\}^*$,\n \\ $\\phi_w(x) \\in 0i \\, \\{0,1\\}^*$, and $|\\phi_w(x)| = |x|$;} \\\\\n(w, x)      & \\mbox{if $x \\in 1 \\{0,1\\}^*$;} \\\\\n\\mbox{undefined} & \\mbox{otherwise.}\n\\end{array}  \\right. $\n\n\\smallskip\n\n\\noindent Then $(e_1 \\circ e_0)(w,x) = (w,x)$ when $x \\in 1 \\, \\{0,1\\}^*$, \nand $(e_1 \\circ e_0)(w,x)$ is undefined otherwise; so $e_1 \\circ e_0$ is a\npartial identity.\nThe reduction of the satisfiability problem to the inversion problem of $e_i$\nis similar to the case of {\\sf fP}.\n \\ \\ \\ $\\Box$\n\n\n\n\\section{Reductions and completeness}\n\n\\noindent The usual reduction between partial functions \n$f_1, f_2: A^* \\to A^*$ is as follows. \n\n\\begin{defn} \\label{simul} \\   \n \\ $f_1$ {\\em is simulated by} $f_2$ (denoted by\n$f_1 \\preccurlyeq f_2$) \\ iff \\ there exist polynomial-time computable\npartial functions $\\beta, \\alpha$ such that \n$ \\, f_1 = \\beta \\circ f_2 \\circ \\alpha$.\n\\end{defn} \nRecall polynomial-time {\\em many-to-one reduction} that is used for \nlanguages; it is defined by $L_1 \\preccurlyeq_{\\sf m:1} L_2$ iff for some \npolynomial-time computable function $\\alpha$ and for all $x \\in A^*$: \n$x \\in L_1$ iff $\\alpha(x) \\in L_2$.  \nThis is equivalent to $L_1 = \\alpha^{-1}(L_2)$, and also to \n$\\chi_{_{L_1}} = \\chi_{_{L_2}} \\circ \\alpha$  (where $\\chi_{_{L_j}}$ \ndenotes the characteristic function of $L_j$). \nSo $L_1 \\preccurlyeq_{\\sf m:1} L_2$ implies that $\\chi_{_{L_1}}$ is simulated\nby $\\chi_{_{L_2}}$.\n\nMoreover, when we talk about simulations between functions we will always \nuse the following \n\n\\medskip\n\n\\noindent {\\bf Addendum to Definition 5.1.} \\ {\\it We assume that\n$\\beta, \\alpha \\in {\\sf fP}$.\nFor a simulation between two right-ideal morphisms of $A^*$ we assume}\n$\\beta, \\alpha \\in {\\cal RM}_{|A|}^{\\sf P}$.\n\n\\bigskip\n\nWe can define simulation for monoids in general. For monoids $M_0 \\leq M_1$\nand $s, t \\in M_1$, simulation $s \\preccurlyeq t$ is the same thing as  \n$s \\leq_{{\\cal J}(M_0)} t$, i.e., the submonoid ${\\cal J}$-order on $M_1$, \nusing multipliers in the submonoid $M_0$.\n\n\\smallskip\n\nSimulation tells us which functions are harder to compute than \nothers, but it does not say anything about the hardness of inverses of\nfunctions. We want a reduction with the property that if a one-way \nfunction $f_1$ reduces to a function $f_2 \\in {\\sf fP}$ then $f_2$ is \nalso one-way. The intuitive idea is that $f_1$ ``reduces inversively'' to \n$f_2$ iff (1) $f_1$  is simulated by $f_2$, and \n(2) the ``easiest inverses'' of $f_1$ are simulated by the ``easiest \ninverses'' of $f_2$.  But ``easiest inverses'' are difficult to define.\nWe rigorously define inversive reduction as follows.\n\n\n\\begin{defn} {\\bf (inversive reduction).} \n \\ Let $f_1, f_2: A^* \\to A^*$ be any partial functions.\nWe say that $f_1$ {\\em reduces inversively} to $f_2$ (notation, \n$f_1 \\leqslant_{\\sf inv} f_2$) \\ iff \n\n\\smallskip\n\n\\noindent (1) \\ \\ $f_1 \\preccurlyeq f_2$  \\ \\ and\n\n\\smallskip\n\n\\noindent (2) \\ \\ for every inverse $f_2'$ of $f_2$ there exists an\ninverse $f_1'$ of $f_1$ such that $f_1' \\preccurlyeq f_2'$ .\n\\end{defn}\nHere, $f_1, f_2, f_1', f_2'$ range over all partial functions $A^* \\to A^*$.\n\n\\medskip\n\n\\noindent The relation $\\leqslant_{\\sf inv}$ can be defined on monoids \n$M_0 \\leq M_1 \\leq M_2$ in general: We let $f_1, f_2$ range over $M_1$, and \nlet inverses $f_1', f_2'$ range over $M_2$. For simulation $\\preccurlyeq$ \nwe pick $\\leq_{{\\cal J}(M_0)}$ (i.e., multipliers are in $M_0$).  \nWe should assume that $M_1$ is regular within $M_2$ in order to avoid empty\nranges for the quantifiers ``$(\\forall f_2')(\\exists f_1')$''; otherwise,\nwhen $f_2$ has no inverse in $M_2$, $f_1 \\leqslant_{\\sf inv} f_2$ is \ntrivially equivalent to $f_1 \\preccurlyeq f_2$.\n\n\n\\begin{pro} \\ $\\leqslant_{\\sf inv}$ is transitive and reflexive.\n\\end{pro}\n{\\bf Proof.} \\ Simulation is obviously transitive. Moreover, if\n$f_1 \\leqslant_{\\sf inv} f_2$ and $f_2 \\leqslant_{\\sf inv} f_3$, then\nfor each $f_3'$ there exists an inverse\n$f_2' = \\beta_{23} \\circ f_3' \\circ \\alpha_{23}$, and for $f_2'$ there \nis an inverse $f_1' = \\beta_{12} \\circ f_2' \\circ \\alpha_{12}$. \nThen $f_1' = \\beta_{12} \\circ  \\beta_{23} \\circ f_3'$ $\\circ$\n$\\alpha_{23} \\circ \\alpha_{12}$, so $f_3'$ simulates some inverse of $f_1$. \n \\ \\ \\ $\\Box$\n\n\n\\begin{pro} \\ If $f_1 \\leqslant_{\\sf inv} f_2$, \n \\ $f_2 \\in {\\sf fP}$, and $f_2$ is regular, then $f_1 \\in {\\sf fP}$ and\n$f_1$ is regular.\n\n\\noindent Contrapositive: If $f_1, f_2 \\in {\\sf fP}$, \n$f_1 \\leqslant_{\\sf inv} f_2$, and $f_1$ is one-way, then $f_2$ is one-way.\n\\end{pro}\n{\\bf Proof.} \\ The property $f_1 \\in {\\sf fP}$ follows from simulation. \nIf $f_2$ is regular, then it has an inverse $f_2' \\in {\\sf fP}$, and $f_1$ \nhas an inverse $f_1' = \\beta \\circ f_2' \\circ  \\alpha$. All the factors \nare in {\\sf fP}, so $f_1' \\in {\\sf fP}$. \n \\ \\ \\ $\\Box$\n \n\n\\begin{defn} \\ \nA partial function $g$ is {\\em complete} (or {\\sf fP}{\\em -complete}) with \nrespect to $\\leqslant_{\\sf inv}$ \\ iff \\ $g \\in {\\sf fP}$, and for every\n$f \\in {\\sf fP}$ we have $f \\leqslant_{\\sf inv} g$. In a similar way we can\ndefine ${\\cal RM}_2^{\\sf P}${\\em -complete}.  \n\\end{defn}\nObservation:  If $g$ is {\\sf fP}-complete then \n$g \\equiv_{\\cal J} {\\sf id}_{A^*}$. \n\n\n\\begin{pro} \\label{ev_complete}\n \\ The map ${\\sf ev}_{q_2}^C$ is {\\sf fP}-{\\em complete} with respect to \ninversive reduction.\n\\end{pro}\n{\\bf Proof.} \\ Any $\\phi_w \\in {\\sf fP}$ with $q_2$-polynomial program $w$ is\nsimulated by ${\\sf ev}_{q_2}^C$; recall formula $(\\star)$ in the proof of \nProp.\\ \\ref{fPfingen}:\n\n\\smallskip\n\n \\ \\ \\ \\ \\ $\\phi_w \\ = \\ $\n$\\pi_{_{2 \\, |w'| + 2}}' \\circ {\\sf contr} \\circ {\\sf recontr}^{2 \\, m}$\n$\\circ$ ${\\sf ev}_{q_2}^C$ $\\circ$\n${\\sf reexpand}^{2 \\, m} \\circ {\\sf expand}$  $\\circ$\n$\\pi_{_{{\\sf code}(w) \\, 11}}$, \n\n\\smallskip\n\n\\noindent where $w' = {\\sf co}^{2m+1} \\circ {\\sf ex}^{2m+1}(w)$.\n\nTo prove the inversive property, let ${\\sf e}'$ be any inverse of \n${\\sf ev}_{q_2}^C$. We apply ${\\sf e}'$ to a string of the form \\, \n${\\sf code}({\\sf ex}^{2m+1}(w)) \\ 11 \\ 0^{N_{2m+1}} \\ 11 \\ y$,\nwhere $\\phi_{{\\sf ex}^{2m+1}(w)} \\in {\\sf fP}^{q_2}$ and \n$y \\in {\\sf Im}(\\phi_w)$.\nThus, $N_{2m+1}$ is at least as large as the time of the computation that \nled to output $y$. Note that we use $2m$ in $N_{2m+1}$ because the input \nthat led to output $y$ could be polynomially longer than $y$ (by polynomial \n$q_2$).  Then we have: \n\n\\smallskip\n\n \\ \\ \\ \\ \\ \n${\\sf e}'\\big({\\sf code}({\\sf ex}^{2m+1}(w))\\ 11\\ 0^{N_{2m+1}}\\ 11\\ y \\big)$\n$ \\ = \\ $\n${\\sf code}({\\sf ex}^{2m+1}(w)) \\ 11 \\ 0^{N_{2m+1}} \\, 11 \\, x_i$, \n \\ \\ for some $x_i \\in \\phi_w^{-1}(y)$.\n\n\\smallskip\n\n\\noindent We don't care whether and how ${\\sf e}'(Z)$ is defined when the\ninput $Z$ is not of the above form.\nThen ${\\sf e}'$ simulates an inverse $f'$ of $\\phi_w$ defined by \n\n\\smallskip\n\n \\ \\ \\ \\ \\ $f'(y) \\ = \\ $ \n$\\pi_{_{2 \\, |w'| + 2}}' \\circ {\\sf contr} \\circ {\\sf recontr}^{2 \\, m}$\n$\\circ$ ${\\sf e}'$ $\\circ$\n${\\sf reexpand}^{2 \\, m} \\circ {\\sf expand}$  $\\circ$\n$\\pi_{_{{\\sf code}(w) \\, 11}}(y)$\n\n\\smallskip\n\n\\noindent for all $y \\in {\\sf Im}(\\phi_w)$.  Indeed, $f'(y) = x_i$ \n($\\in \\phi_w^{-1}(y)$ as above). \n\nWhen $y \\not\\in {\\sf Im}(\\phi_w)$ the right side of the above formula may\ngive a value to $f'(y)$; but it does not matter whether and how $f'$ is  \ndefined outside of ${\\sf Im}(\\phi_w)$.  Thus, \nevery inverse of ${\\sf ev}_{q_2}^C$ simulates an inverse of $\\phi_w$.\n \\ \\ \\ $\\Box$\n\n\\medskip\n\nIn a similar way one can prove that the generator-based evaluation map \n${\\sf ev}_{\\Gamma,q}$ (for a large enough polynomial $q$) is complete\nin {\\sf fP}.\n\n\\medskip\n\n\\noindent {\\bf Notation:} \\ For a partial function $f: A^* \\to A^*$, the \n{\\em set of all inverses} $f': A^* \\to A^*$ of $f$ is denoted by \n${\\sf Inv}(f)$.  \n\n\n\\begin{defn} {\\bf (uniform inversive reduction).} \n \\ Let $f, g$ be partial functions.  An inversive reduction \n$f \\leqslant_{\\sf inv} g$ is called {\\em uniform} \\ iff\n \\ $f \\preccurlyeq g$, and $(\\exists \\beta, \\alpha \\in {\\sf fP})$\n$(\\forall g' \\in {\\sf Inv}(g))$ $(\\exists f' \\in {\\sf Inv}(f))$ \n$[ \\, f' = \\beta \\circ g' \\circ \\alpha \\, ]$. So $\\beta$ and $\\alpha$ only\ndepend on $f$ and $g$, but not on $g'$ or $f'$.\n\\end{defn}\nWe observe that in the proof of Prop.\\ \\ref{ev_complete} the simulation of\n$f'$ by ${\\sf e'}$ only depends on $\\phi_w$ and ${\\sf e}$, but not on \n$f'$ nor on ${\\sf e'}$. We conclude:\n  \n\\begin{cor} \nThe map ${\\sf ev}_{q_2}^C$ is {\\sf fP}-complete with respect to \n{\\em uniform} inversive reduction.  \\ \\ \\ $\\Box$\n\\end{cor}\n\n\nNext we study the completeness of the circuit evaluation map \n ${\\sf ev}_{\\sf circ}$ (defined at the beginning of Section 4).\nSince it is defined in terms of length-preserving circuits, \n${\\sf ev}_{\\sf circ}$ is itself length-preserving, i.e., it belongs to \nthe submonoid of {\\em length-preserving} partial functions in\n{\\sf fP},\n\n\\smallskip\n\n \\ \\ \\  \\ \\ \\ ${\\sf fP}_{\\sf lp} \\ = \\ $\n$\\{ f \\in {\\sf fP} \\ : \\ |f(x)| = |x|$ \\ for all $x \\in {\\sf Dom}(f) \\}$. \n\n\n\\begin{pro} \\label{LevComplLP}\nThe critical map ${\\sf ev}_{\\sf circ}$ is complete in the submonoid\n${\\sf fP}_{\\sf lp}$ with respect to inversive reduction.\n\\end{pro}\n\n\\noindent {\\bf Proof.} Let $f \\in {\\sf fP}_{\\sf lp}$ be a fixed \nlength-preserving partial function, and let $M$ be a fixed deterministic \npolynomial-time Turing machine that computes $f$.\n\nSimulation of $f$ by ${\\sf ev}_{\\sf circ}$: It is well known that for every \ninput length $n$ (of inputs of $f$) one can construct an acyclic circuit \n$C_n$ such that $C_n(x) = f(x)$ for all $x$ of length $n$. The circuit can \nbe constructed from $M$ and $n$ in polynomial time (as a function of $n$). \nLet $\\alpha(x) = (C_{|x|}, x)$, and let $\\beta(C_n, y) = y$, where \n$|y| = n$.\nThen $f = \\beta \\circ {\\sf ev}_{\\sf circ} \\circ \\alpha$.\n  \nSimulation between inverses: Any inverse $e'$ of ${\\sf ev}_{\\sf circ}$ has \nthe form $e'(C,y) = (C, x_i)$ for some $x_i \\in C^{-1}(y)$, when\n$y \\in {\\sf Im}(C)$. When $y \\not\\in {\\sf Im}(C)$, $e'(C,y)$ could be any \npair of bitstrings.\nThen an inverse $f'$ of $f$ is obtained by defining \n$f'(y) = \\beta \\circ e' \\circ \\alpha(y)$, where $\\alpha, \\beta$ are as in\nthe simulation of $f$ (in the first part of this proof).  Indeed, when \n$y \\in {\\sf Im}(f)$ we have $\\alpha: y \\longmapsto (C_n, y)$, where $|y| = n$.\nNext, $e': (C_n, y) \\longmapsto (C_n, x_i)$ for some \n$x_i \\in C_n^{-1}(y) = f^{-1}(y)$; recall that we only use length-preserving\ncircuits, so $|y| = n = |x_i|$.\nFinally, $\\beta: (C_n, x_i) \\longmapsto x_i \\in f^{-1}(y)$.  So $f'$ is an \ninverse of $f$ on ${\\sf Im}(f)$; outside of ${\\sf Im}(f)$, the values of \n$f'$ do not matter.\n \\ \\ \\ $\\Box$\n\n\\bigskip\n\n\\noindent To show completeness of ${\\sf ev}_{\\sf circ}$ in {\\sf fP} (rather\nthan just in ${\\sf fP}_{\\sf lp}$), a stronger inversive reduction is needed, \nthat overcomes the limitation of length-preservation in ${\\sf ev}_{\\sf circ}$. \n\n\\bigskip\n\n\\noindent {\\bf Remark:} \nCircuits are usually generalized to allow the output length to be different \nfrom the input length. But that would not simplify the completeness proof for\n${\\sf ev}_{\\sf circ}$, \nbecause the main limitation is that all inputs of a circuit have the same \nlength, and all outputs of a circuit have the same length. \n\n\n\\begin{defn} \\label{Tsimulation} {\\bf (polynomial-time Turing simulation).}\nLet $f_1, f_2: A^* \\to A^*$ be two partial functions.\nBy definition, $f_1 \\preccurlyeq_{\\sf T} f_2$ \\ iff \\ $f_1$ is computed by a \ndeterministic polynomial-time Turing machine that can make oracle calls to \n$f_2$; these can include, in particular, calls on the membership problem of \n${\\sf Dom}(f_2)$.\n\\end{defn}  \nIn the next proofs we do not need the full power of Turing reductions. \nThe following, much weaker reduction, will be sufficient.\n\n\\medskip\n\n\\noindent {\\bf Notation:} Let $L \\subseteq A^*$. Then ${\\sf fP}^L$ denotes \nthe set of all polynomially balanced partial functions computed by \ndeterministic polynomial-time Turing machines that can make oracle calls to \nthe membership problem of $L$. In particular we will consider \n${\\sf fP}^{{\\sf Dom}(f)}$ for any partial function $f: A^* \\to A^*$. \n\n\\begin{defn} \\label{weakTsimulation} {\\bf (weak Turing simulation).} \nA {\\em weak Turing simulation} of $f_1$ by $f_2$ consists of two partial \nfunctions $\\beta, \\alpha$ such that $f_1 = \\beta \\circ f_2 \\circ \\alpha$, \nwhere $\\alpha \\in {\\sf fP}^{{\\sf Dom}(f_2)}$ and $\\beta \\in {\\sf fP}$.\nThe existence of a weak Turing simulation of $f_1$ by $f_2$ is denoted\nby $f_1 \\preccurlyeq_{\\sf wT} f_2$.\n\\end{defn}\nInformally we also write \n$f_1 = \\beta \\circ f_2 \\circ \\alpha^{{\\sf Dom}(f_2)}$. \nIn a weak Turing simulation by $f_2$, only the domain of $f_2$ is repeatedly\nqueried; $f_2$ itself is called only once, and this call of $f_2$ takes the \nform of an ordinary (not a Turing) simulation.\n\n\n\\begin{defn} \\label{inversif} {\\bf (inversification of a simulation).}\nFor any simulation relation $\\preccurlyeq_{\\sf X}$ between partial functions,\nthe corresponding {\\em inversive reduction} $\\leqslant_{\\sf inv,X}$ is\ndefined as follows:\n\n \\ \\ $f_1 \\leqslant_{\\sf inv,X} f_2$ \\ \\ iff\n\n\\smallskip\n\n \\ \\   \n$f_1 \\preccurlyeq_{\\sf X} f_2$, and for every inverse $f_2'$ of $f_2$ there \nexists an inverse $f_1'$ of $f_1$ such that $f_1' \\preccurlyeq_{\\sf X} f_2'$.\n\\end{defn}\nOne easily proves:\n\n\\begin{pro}\n \\ If $\\preccurlyeq_{\\sf X}$ is transitive then $\\leqslant_{\\sf inv,X}$ is\ntransitive.  \\ \\ \\ \\ \\ \\ $\\Box$\n\\end{pro}\nBased on this general definition we can introduce {\\em polynomial-time \ninversive Turing reductions}, denoted by $\\leqslant_{\\sf inv,T}$, and \n{\\em polynomial-time inversive weak Turing reductions}, denoted by\n$\\leqslant_{\\sf inv,wT}$.\nThe following is straightforward to prove.\n\n\\begin{pro} \nIf $f_1 \\leqslant_{\\sf inv,T} f_2$ then: \n\n\\noindent $\\bullet$ \\ $f_2 \\in {\\sf fP}$ implies $f_1 \\in {\\sf fP}$;\n\n\\noindent $\\bullet$ \\ $f_2 \\in {\\sf fP}$ and $f_2$ is regular, implies\n$f_1$ is regular.\n \\ \\ \\ \\ \\ $\\Box$\n\\end{pro}\nThe following shows that $\\leqslant_{\\sf inv, wT}$ can overcome the \nlimitations of length-preservation.\n\n\\begin{pro} \\label{lpVSall} \nFor every $f \\in {\\sf fP}$ there exists $\\ell \\in {\\sf fP}_{\\sf lp}$ \nsuch that $f \\leqslant_{\\sf inv,wT} \\ell$.\n\\end{pro}\n{\\bf Proof.} For any $f \\in {\\sf fP}$ we define \n$\\ell_f \\in {\\sf fP}_{\\sf lp}$ by\n\n\\medskip\n\n \\ \\ \\ \\ \\ \\ \\ $\\ell_f(0^n 1 \\, x) \\ = \\ \\left\\{ \n \\begin{array}{ll}\n    0^{|x|} 1 \\, f(x) \\ \\ \\ \\ \\ \\      & \\mbox{if $n = |f(x)|$, } \\\\\n    {\\rm undefined}         & \\mbox{on all other inputs.}\n \\end{array}  \\right.\n$\n\n\\medskip\n\n\\noindent Let $p_f(.)$ be a polynomial upper-bound on the time-complexity and \non the balance of $f$.\n\n\\smallskip\n\n\\noindent 1. \\ Proof that $f \\preccurlyeq \\ell_f$ (simulation):\n \\ We have $f = \\beta \\circ \\ell_f \\circ \\alpha$, where \n \\ $\\alpha(x) = 0^{|f(x)|} 1 \\, x$ \\ for all $x \\in A^*$; and  \n \\ $\\beta(0^m 1 \\, y) = y$ \\ for all $y \\in A^*$ and all \n $m \\leq p_f(|y|)$ \\ ($\\beta$ is undefined otherwise).\n\n\\smallskip\n\n\\noindent  2. \\ Proof that for every inverse $\\ell'$ of $\\ell_f$ there is\nan inverse $f'$ of $f$ such that $f' \\preccurlyeq_{\\sf wT} \\ell_f'$ :\n\n\\smallskip\n\nEvery element of ${\\sf Im}(\\ell_f)$ has the form $0^m 1 \\, y$\nwhere $y \\in {\\sf Im}(f)$, for some appropriate $m$. More precisely,\n$\\ell_f^{-1}(0^m \\, 1 \\, y)$ $ = $ \n$\\{ 0^{|y|} \\, 1 \\, x : x \\in f^{-1}(y) \\cap A^m\\}$.  Hence, \n$0^m \\, 1 \\, y \\in {\\sf Im}(\\ell_f)$ iff \n$f^{-1}(y) \\cap A^m \\neq \\varnothing$.  Therefore, any inverse $\\ell'$ \nsatisfies $\\ell'(0^m 1 \\, y) = 0^{|y|} 1 \\, x_i$ for some choice of \n$x_i \\in f^{-1}(y) \\cap A^m$; we do not care about the values of $\\ell'$ \nwhen its inputs are not in ${\\sf Im}(\\ell_f)$.\nThus we can define an inverse of $f$ on each $y \\in {\\sf Im}(f)$ by \n\n\\smallskip\n\n$f'(y) = x_i$, \\, for $x_i$ chosen in $f^{-1}(y) \\cap A^m$ \n\n\\hspace{.5in} where $m$ is the minimum integer such that \n$f^{-1}(y) \\cap A^m \\neq \\varnothing$. \n\n\\smallskip\n\n\\noindent We don't care what $f'(y)$ is when $y \\not\\in {\\sf Im}(f)$.\n \nTo obtain an inversive weak Turing reduction we need to compute \n$x_i = f'(y)$ from $y$, based on oracle calls to ${\\sf Dom}(\\ell')$ and\none simulation of $\\ell'$.  This computation of $x_i$ is done in two steps:  \n First we compute the minimum $m$ ($= |x_i|$) such that \n$f^{-1}(y) \\cap A^m \\neq \\varnothing$ (see Step 1 below for details). \n Second, we apply $\\ell'$ to compute\n$\\ell'(0^{|x_i|} 1 \\, y) = 0^{|y|} 1 \\, x_i$. From this we obtain $x_i$ by \napplying the map $\\beta$ defined above (in part 1 of this proof). \nThe first step is a Turing reduction to the domain of $\\ell'$. \nThe second step is a simulation by $\\ell'$.  In more detail:\n\nStep 1: \nBy input balance we have $|x_i| \\leq p_f(|y|)$ when $x_i \\in f^{-1}(y)$.\nFor each $m \\in \\{0, 1, \\ldots, p_f(|y|)\\}$, in increasing order, we make \nan oracle call to the membership problem in ${\\sf Dom}(\\ell')$ with query \ninput $0^m 1 \\, y$.\nIf $y \\in {\\sf Im}(f)$ then the first of these queries with a positive\nanswer determines $m$, and $0^m 1 \\, y$ is returned.\n\nStep 2: To the result $0^m 1 \\, y$ of step 1 we apply the functions\n$\\ell'$ and $\\beta$.  This yields $x_i$, which is $f'(y)$. \nThus, step 2 is just a simulation. \n\nTogetherm, steps 1 and 2 form a weak polynomial Turing simulation of $f'$ \nby $\\ell'$.  \n \\ \\ \\ $\\Box$\n\n\n\\begin{cor}\nThe critical map ${\\sf ev}_{\\sf circ}$ is {\\sf fP}-complete with respect to \ncomposites of polynomial inversive weak Turing reductions and polynomial\ninversive simulation reductions ($\\leqslant_{{\\sf inv,wT}}$ and \n$\\leqslant_{\\sf inv}$).\n\\end{cor}  \n{\\bf Proof.} For every $f \\in {\\sf fP}$ we first reduce $f$ to a\nlength-preserving function $\\ell_f$, by Prop.\\ \\ref{lpVSall}. Then we reduce \n$\\ell_f$ to ${\\sf ev}_{\\sf circ}$ by Prop. \\ref {LevComplLP}.\n \\ \\ \\ $\\Box$\n\n\\bigskip\n\n\n\\noindent {\\bf Reduction and completeness in ${\\cal RM}_2^{\\sf P}$}\n\n\\medskip\n\n\\noindent The following shows that the encoding that embeds {\\sf fP} into\n${\\cal RM}_2^{\\sf P}$ does not make inversion easier.\n\n\\begin{pro} \\label{code_invred} \n \\ For all $f \\in {\\sf fP}$ we have $f \\leqslant_{\\sf inv} f^C$, \nwhere $\\leqslant_{\\sf inv}$ is based on simulation in {\\sf fP}. \n\\end{pro}\n{\\bf Proof.} Recall the encoding maps\n $(.)_{\\#}: x \\in \\{0,1\\}^* \\longmapsto x \\# \\in \\{0,1, \\#\\}^*$, and \n${\\sf code}$ which replaces $0, 1, \\#$ by respectively $00, 01, 11$, defined\nin Section 3; and recall $f^C$ from Def.\\ \\ref{f_encoding}.  \nWe now introduce inverses of these maps.\nLet ${\\sf dec}: {\\sf code}(x) \\in \\{00, 01, 11\\}^* $\n$\\longmapsto x \\in \\{0,1\\}^*$ (undefined outside of $\\{00, 01, 11\\}^*$), \nand $r: x \\# \\longmapsto x \\in \\{0,1\\}^*$ (undefined outside of \n$\\{0,1\\}^*\\#$).  Then \n\n\\smallskip\n\n \\ \\ \\  \\ \\ \\ $f \\ = \\ $\n$r \\circ {\\sf dec} \\circ f^C \\circ {\\sf code} \\circ (.)_{\\#}$ .\n\n\\smallskip\n\n\\noindent Clearly, $(.)_{\\#}, \\, {\\sf code}, \\, {\\sf dec}, \\, r$\n$ \\in {\\sf fP}$.  Hence  $f^C$ simulates $f$. \n\nFor the inversive part of the reduction, let $\\varphi'$ be any inverse of\n$f^C$; we want to find an inverse $f'$ of $f$ such that \n$f' \\preccurlyeq \\varphi'$, where $\\preccurlyeq$ denotes simulation in \n{\\sf fP}. \nAny element of ${\\sf Im}(f^C)$ has the form ${\\sf code}(s) \\, 11 \\, t$, \nwith $s, t \\in \\{0,1\\}^*$, and $s \\in {\\sf Im}(f)$. Moreover, if \n${\\sf code}(s) \\, 11 \\, t \\in {\\sf Im}(f^C)$ then\n${\\sf code}(s) \\, 11 \\in {\\sf Im}(f^C)$. Let us define $f'$ for any\n$s \\in {\\sf Im}(f)$ by $f'(s) = x_1$ where $x_1$ is such that \n$\\varphi'({\\sf code}(s) \\, 11) = {\\sf code}(x_1) \\, 11$ \n$ \\in (f^C)^{-1}({\\sf code}(s) \\, 11)$. Then $x_1 \\in f^{-1}(s)$. \nIn general, finally, we define $f'$ by \n\n\\smallskip \n\n \\ \\ \\ \\ \\ \\ $f' \\ = \\ $\n    $r \\circ {\\sf dec} \\circ \\varphi' \\circ {\\sf code} \\circ (.)_{\\#}$ .\n\n\\smallskip\n\n\\noindent For $s \\in {\\sf Im}(f)$ we indeed have then:\n$r \\circ {\\sf dec} \\circ \\varphi' \\circ {\\sf code} \\circ (.)_{\\#}(s) \\ = \\ $\n$r \\circ {\\sf dec} \\circ \\varphi'({\\sf code}(s) \\, 11) \\ = \\ $\n$r \\circ {\\sf dec}({\\sf code}(x_1) \\, 11) \\ = \\ x_1$, \\,  where \n$x_1 \\in f^{-1}(s)$, as above. So this definition makes $f'$ an inverse of \n$f$ on ${\\sf Im}(f)$; hence $f'$ is an inverse of $f$. \nThe above formula for $f'$ explicitly shows that $f' \\preccurlyeq \\varphi'$. \n \\ \\ \\ $\\Box$\n\n                \n\\medskip\n\nLet $\\equiv_{\\sf inv}$ denote $\\leqslant_{\\sf inv}$-equivalence (i.e.,\n$f \\equiv_{\\sf inv} g$ iff $f \\leqslant_{\\sf inv} g$ and \n$g \\leqslant_{\\sf inv} f$).\nThe $\\leqslant_{\\sf inv}$-complete functions of {\\sf fP} obviously form an\n$\\equiv_{\\sf inv}$-class, and this is the maximum class for the \n$\\leqslant_{\\sf inv}$-preorder. \nSimilarly, the complete functions of ${\\cal RM}_2^{\\sf P}$ are the maximum \ninversive reducibility class in ${\\cal RM}_2^{\\sf P}$.\nThe non-empty regular elements of ${\\cal RM}_2^{\\sf P}$ also form an \nequivalence class in ${\\cal RM}_2^{\\sf P}$, and this is the minimum class \nof all non-empty functions, as the following shows:\n\n\n\\begin{pro} \\ For every $f, r \\in {\\cal RM}_2^{\\sf P}$ where $r$ is \nregular and $f$ is non-empty, we have $r \\leqslant_{\\sf inv} f$.\n\\end{pro}\n{\\bf Proof.} The simulation $r \\preccurlyeq f$ follows from \n${\\cal J}^0$-simplicity of ${\\cal RM}_2^{\\sf P}$.\nLet $f'$ be any inverse of $f$ (with $f'$ not necessarily in \n${\\cal RM}_2^{\\sf P}$). Since $r$ is regular, there is an inverse \n$r' \\in {\\cal RM}_2^{\\sf P}$ of $r$. Since $f'$ is not the empty map there \nexist $x_0, y_0$ with $f'(y_0) = x_0$. Then $(x_0 \\leftarrow y_0)$ is \nsimulated by $f'$, since\n$(x_0 \\leftarrow y_0) = {\\sf id}_{\\{x_0\\}} \\circ f'$. \nMoreover, $(x_0 \\leftarrow y_0)$ is regular and $(x_0 \\leftarrow y_0)$ \nbelongs to ${\\cal RM}_2^{\\sf P}$, so $(x_0 \\leftarrow y_0)$ simulates $r'$ \n(again by ${\\cal J}^0$-simplicity of ${\\cal RM}_2^{\\sf P}$).  Thus, $f'$ \nsimulates $r'$.\n \\ \\ \\ $\\Box$\n\n\n\\begin{pro} \\ In both {\\sf fP} and ${\\cal RM}_2^{\\sf P}$:  The \n$\\equiv_{\\cal D}$-relation is a refinement of $\\equiv_{\\sf inv}$.\n\\end{pro}\n{\\bf Proof.} Is suffices to prove that both $\\equiv_{\\cal R}$ and \n$\\equiv_{\\cal L}$ refine $\\equiv_{\\sf inv}$.\nWe will prove that $f \\equiv_{\\cal R} g$ implies $f \\equiv_{\\sf inv} g$.\n(The same reasoning works for $\\equiv_{\\cal L}$.) \nWhen $f \\equiv_{\\cal R} g$, there exist $\\alpha, \\beta \\in {\\sf fP}$ (or \n$\\in {\\cal RM}_2^{\\sf P}$) such that $f = g \\, \\alpha$ and $g = f \\, \\beta$. \nSo, $f$ and $g$ simulate each other.\n\nFor any inverse $f'$ of $f$ we have $f = f \\, f' \\, f $\n$ = g \\, \\alpha f' f$. Right-multiplying by $\\beta$ we obtain\n$g = g \\, \\alpha f' \\, g$, hence $\\alpha f'$ is an inverse of $g$, and \n$\\alpha f'$ is obviously simulated by $f'$. So, $g$ inversely reduces to \n$f$. Similarly, $f$ inversely reduces to $g$.\n \\ \\ \\ $\\Box$ \n\n\n\n\\section{\\bf The polynomial hierarchy}\n\n\\noindent The classical polynomial hierarchy for languages is defined by\n$\\Sigma_1^{\\sf P} = {\\sf NP}$, $\\Pi_1^{\\sf P} = {\\sf coNP}$, and \nfor all $k > 0$: $\\Sigma_{k+1}^{\\sf P} = {\\sf NP}^{\\Sigma_k^{\\sf P}}$\n(i.e., all languages accepted by nondeterministic Turing machines with oracle \nin $\\Sigma_k^{\\sf P}$, or equivalently, with oracle in $\\Pi_k^{\\sf P}$); and\n$\\Pi_{k+1}^{\\sf P} = ({\\sf coNP})^{\\Sigma_k^{\\sf P}}$ \n$\\big( = {\\sf co}({\\sf NP}^{\\Sigma_k^{\\sf P}})\\big)$. Moreover,\n${\\sf PH} \\ = \\ \\bigcup_k \\Sigma_k^{\\sf P}$ \\ \\ ($\\subseteq {\\sf PSpace}$).\n\n\\medskip\n\n\\noindent {\\sf Polynomial hierarchy for functions:}\n\n\\smallskip\n\n${\\sf fP}^{\\Sigma_k^{\\sf P}}$ \\, consists of all polynomially balanced \npartial functions $A^* \\to A^*$ computed by deterministic polynomial-time \nTuring machines with oracle in $\\Sigma_k^{\\sf P}$ (or equivalently, with \noracle in $\\Pi_k^{\\sf P}$);\n\n\\smallskip\n\n${\\sf fP}^{\\sf PH}$ \\, consists of all polynomially balanced partial \nfunctions $A^* \\to A^*$ computed by deterministic polynomial-time Turing \nmachines with oracle in {\\sf PH}. Equivalently, \n${\\sf fP}^{\\sf PH} = \\, \\bigcup_k \\, {\\sf fP}^{\\Sigma_k^{\\sf P}}$.\n\n\\smallskip\n\nMoreover, ${\\sf fPSpace}$ \\, consists of all polynomially balanced\npartial functions (on $A^*$) computed by deterministic polynomial-space \nTuring machines.\n\n\\smallskip\n\nWe can also define a polynomial hierarchy over ${\\cal RM}_2^{\\sf P}$.\n\n\n\\begin{pro} \\label{fPNP_inv} \\ Every $f \\in {\\sf fP}$ has an inverse in \n${\\sf fP}^{\\Sigma_1^{\\sf P}}$, and\nevery $f \\in {\\sf fP}^{\\Sigma_k^{\\sf P}}$ has an inverse in\n${\\sf fP}^{\\Sigma_{k+1}^{\\sf P}}$. The monoids \n${\\sf fP}^{\\sf PH}$ and ${\\sf fPSpace}$ are regular.\n\\end{pro}\n{\\bf Proof.} \\ The following is an inverse of $f$:\n\n\\smallskip\n\n \\ \\ \\ \\ \\ $f'_{\\sf min}(y) \\ = \\ \\left\\{\n   \\begin{array}{ll}\n    {\\sf min}(f^{-1}(y))    & \\mbox{if $y \\in {\\sf Im}(f)$, } \\\\\n    {\\rm undefined}         & \\mbox{otherwise,} \n   \\end{array}  \\right.\n$\n\n\\smallskip\n\n\\noindent where ${\\sf min}(S)$ denotes the minimum of a set of strings $S$ \nin dictionary order (or alternatively in length-lexicographic order).\nTo show that $f'_{\\sf min} \\in {\\sf fP}^{\\sf NP}$ when $f \\in {\\sf fP}$ we \nfirst observe that for any fixed $f \\in {\\sf fP}$ the following problems are \nin {\\sf NP}: \n\n\\noindent (1) On input $y \\in A^*$, decide whether $y \\in {\\sf Im}(f)$.\n\n\\noindent (2) Fix $u \\in A^*$; on input  $y \\in A^*$, decide whether \n$y \\in f(u \\, A^*)$ (i.e., decide whether there exists $x \\in u \\, A^*$ such \nthat $f(x) = y$).\n\nWhen $y \\not\\in {\\sf Im}(f)$ then it doesn't matter what value we choose \nfor $f'_{\\sf min}(y)$; we choose $f'_{\\sf min}(y)$ to be undefined then.\n\nHere is an ${\\sf fP}^{\\sf NP}$-algorithm for computing $f'_{\\sf min}(y)$.\nIt is a form of binary search in the sorted list $A^*$, that ends when \na string in $f^{-1}(y)$ has been found. A growing prefix $z$ of \n$x = f'_{\\sf min}(y)$ is constructed; at each step we query whether $z$ can \nbe extended by a 0 or a 1; i.e., we ask whether $y \\in f(z0 A^*)$; we don't \nneed to ask whether $y \\in f(z1 A^*)$ too, since we tested already that \n$y \\in {\\sf Im}(f)$.  Oracle calls are denoted by angular brackets \n$\\langle \\ldots \\rangle$, and $\\varepsilon$ denotes the empty word. \n\n\\smallskip\n\n\\noindent Algorithm for $f'_{\\sf min}$ on input $y:$\n\n\\smallskip\n\n{\\tt if $\\langle y \\in {\\sf Im}(f) \\rangle$ then\n\n \\ \\ \\ $z := \\varepsilon$;\n \n \\ \\ \\ while $\\langle z \\not\\in f^{-1}(y) \\rangle$ do \\hspace{1in}\n    \/\/ assume $y \\in f(z A^*)$\n\n \\ \\ \\  \\ \\ \\ if $\\langle y \\in f(z0 A^*) \\rangle$ then \n    $z := z 0$;   \n\n \\ \\ \\  \\ \\ \\ else $z := z 1$; \n\n \\ \\ \\ output $z$.\n}  \n\n\\smallskip\n\n\\noindent One can prove in a similar way that when \n$f \\in {\\sf fP}^{\\Sigma_k^{\\sf P}}$ then $f'_{\\sf min} \\in$\n${\\sf fP}^{\\Sigma_{k+1}^{\\sf P}}$: In that case the problems (1) and (2)\nabove are in ${\\sf NP}^{ \\Sigma_k^{\\sf P}} = \\Sigma_{k+1}^{\\sf P}$.   \n\nThe regularity of ${\\sf fP}^{\\sf PH}$ follows immediately from the \nfact about ${\\sf fP}^{\\Sigma_k^{\\sf P}}$ for each $k$.\nThe regularity of ${\\sf fPSpace}$ holds because the above algorithm \ncan be carried out in ${\\sf fPSpace}$.  \n \\ \\ \\ $\\Box$\n\n\\bigskip\n\nThe above algorithm is similar to the proofs in the literature  that \n{\\sf P} $\\neq$ {\\sf NP} iff one-way functions exist; see e.g.\\\n\\cite{HemaOgi} p.\\ 33.\n\n\\medskip\n\nIn the proof of Prop.\\ \\ref{fPNP_inv} we used the function $f'_{\\sf min}$.\nIn a similar way, by using ${\\sf max}(f^{-1}(y))$ one can define\n$f'_{\\sf max} \\in {\\sf fP}^{\\Sigma_1^{\\sf P}}$, which is also an inverse of \n$f$. Yet more inverses can be defined:  for any positive integer $i$ let\n\n\\smallskip\n\n \\ \\ \\ \\ \\ $ f'_i(y) \\ = \\ \\left\\{\n\\begin{array}{ll}\ni^{\\rm th} \\ {\\rm word \\ in} \\ f^{-1}(y) \\ \\ \\ \n                        & \\mbox{if $y \\in {\\sf Im}(f)$,} \\\\\n{\\rm undefined}         & \\mbox{otherwise.}\n\\end{array}  \\right.\n$\n\n\\smallskip\n\n\\noindent Here, ``$i^{\\rm th}$ word'' refers to the dictionary order; also, \nwhen $i > |f^{-1}(y)|$, the $i^{\\rm th}$ word is taken to be the maximum \nword in $f^{-1}(y)$. Then $f'_i$ is an inverse of $f$ and \n$f'_i \\in {\\sf fP}^{\\Sigma_1^{\\sf P}}$; note that $i$ is fixed for each \nfunction $f'_i$. \n\n  \n\\begin{pro} \\ For any {\\em {\\sf fP}-critical} partial function \n$f \\in {\\sf fP}$ we have:\n \\ $f$ is one-way \\, iff \\, $f'_{\\sf min} \\not\\in {\\sf fP}$.\nSimilarly,  \\,  $f$ is one-way \\, iff \\, $f'_{\\sf max} \\not\\in {\\sf fP}$\n\\, iff \\, $(\\exists i > 0)[ \\, f'_i \\not\\in {\\sf fP} \\, ]$.\n\\end{pro}\n{\\bf Proof.}  Since $f'_{\\sf min}$ is an inverse of $f$, the direction \n``$\\Rightarrow$'' is obvious by the definition of one-way function. \nConversely, we saw that if $f \\in {\\sf fP}$ then \n$f'_{\\sf min} \\in {\\sf fP}^{\\Sigma_1^{\\sf P}}$. \nIf $f'_{\\sf min} \\not\\in {\\sf fP}$ then \n${\\sf fP} \\neq {\\sf fP}^{\\Sigma_1^{\\sf P}}$, hence ${\\sf P} \\neq {\\sf NP}$, \nhence one-way functions exist.\nThen any {\\sf fP}-critical function $f$ is one-way.\n \\ \\ \\ $\\Box$\n\n\\medskip\n\n\\noindent Recall that a partial function $f$ is called {\\sf fP}-critical iff\n$f \\in {\\sf fP}$ and the existence of one-way functions implies that $f$ is\none-way. In particular, {\\sf fP}-complete functions (with respect to inversive\nreduction) are {\\sf fP}-critical.\nAn interesting consequence of the above Proposition is that now we do not \nonly have critical functions, but these functions also have {\\em critical \ninverses}.\n\n\\begin{defn} \\ Let $f$ be an {\\sf fP}-critical function.  We say that an \ninverse $f'$ of $f$ is a {\\em critical inverse} of $f$ \\, iff \\,    \n$f' \\not\\in {\\sf fP}$ implies that $f$ is one-way.\n\\end{defn}\n\n\\begin{cor}\nFor the {\\sf fP}-critical function ${\\sf ev}_{\\sf circ}$, the inverses \n$({\\sf ev}_{\\sf circ})'_{\\sf min}$, $({\\sf ev}_{\\sf circ})'_{\\sf max}$\nand $({\\sf ev}_{\\sf circ})'_i$ are critical inverses.\n \\ \\ \\ $\\Box$ \n\\end{cor}\nThus, to decide whether ${\\sf P} \\neq {\\sf NP}$ it suffices to consider \none function, and one of its inverses.\n\n\n\\begin{pro} \\label{polyHierFinGen}  \\ For each $k \\geq 1$ the monoid \n${\\sf fP}^{\\Sigma_k^{\\sf P}}$ is finitely generated, but not finitely \npresented. The monoid ${\\sf fPSpace}$ is also finitely generated, \nbut not finitely presented.\n\nThe monoid ${\\sf fP}^{\\sf PH}$ is not finitely generated, unless the\npolynomial hierarchy collapses.\n\\end{pro}\n{\\bf Proof.} The proof for ${\\sf fPSpace}$ is similar to the proof that we \ngave for {\\sf fP} in Prop.\\ \\ref{notFinPres}. \n\nFor ${\\sf fP}^{\\Sigma_k^{\\sf P}}$, let $Q_k$ be any \n$\\Sigma_k^{\\sf P}$-complete problem; we can assume that all oracle calls\nare calls to $Q_k$. Then every $f \\in {\\sf fP}^{\\Sigma_k^{\\sf P}}$ has a\nprogram which is like an {\\sf fP}-program, but with oracle calls to $Q_k$ \nadded. \nFor every polynomial $q \\geq q_2$, an evaluation function ${\\sf ev}_q^{Q_k}$ \nfor ${\\sf fP}^{\\Sigma_k^{\\sf P}}$ can then be designed; in the computation of \n${\\sf ev}_q^{Q_k}(w, x)$, oracle calls to $Q_k$ are made whenever the program \n$w$ being executed makes calls to $Q_k$. Then, \n${\\sf ev}_q^{Q_k}(w, x) = (w, \\phi_w(x))$.\nBy using ${\\sf ev}_q^{Q_k}$ the proof for ${\\sf fP}^{\\Sigma_k^{\\sf P}}$\nis similar to the proof of Prop.\\ \\ref{notFinPres}.  \n\nIf ${\\sf fP}^{\\sf PH}$ were finitely generated then let $m$ be the lowest\nlevel in the hierarchy that contains a finite generating set. Then \n${\\sf fP}^{\\sf PH} \\subseteq {\\sf fP}^{\\Sigma_m^{\\sf P}}$.\n \\ \\ \\ $\\Box$\n\n\\bigskip\n\nInstead of using all of ${\\sf fP}^{\\sf NP}$ to obtain inverses for the \nelements of {\\sf fP}, we could simply adjoin inverses to {\\sf fP} (within\n${\\sf fP}^{\\sf NP}$). \nIt turns out that it suffices to adjoin just one inverse \n$e' \\in {\\sf fP}^{\\sf NP}$ of a function $e$ that is {\\sf fP}-complete for \n$\\leqslant_{\\sf inv}$.\n\n\\medskip\n\n\\noindent \n{\\bf Notation:} For a semigroup $S$ and a subset $W \\subseteq S$, the \nsubsemigroup of $S$ generated by $W$ is denoted by $\\langle W \\rangle_S$.\nFor any $h \\in {\\sf fP}^{\\sf NP}$, we denote \n$\\langle {\\sf fP} \\cup \\{h\\} \\rangle_{{\\sf fP}^{\\sf NP}}$ by \n${\\sf fP}[h]$ (called {\\em ``{\\sf fP} with $h$ adjoined''}).\nSo, \\ \\   \n${\\sf fP} \\ \\subseteq \\ {\\sf fP}[h] \\ \\subseteq \\ {\\sf fP}^{\\sf NP}$.\n\n\n\n\\begin{pro}\nLet $g \\in {\\sf fP}$ be any function that is {\\sf fP}-complete with respect\nto $\\leqslant_{\\sf inv}$, and let $g'$ be any inverse of $g$ such that \n$g' \\in {\\sf fP}^{\\sf NP}$.\nThen the subsemigroup ${\\sf fP}[g']$ of ${\\sf fP}^{\\sf NP}$ contains at least\none inverse of each element of {\\sf fP}. \n\\end{pro}\n{\\bf Proof.} From the assumption that $g$ is complete it follows that\n\n\\smallskip\n\n \\ \\ \\   \\ \\ \\   \\ \\ \\   $(\\forall f \\in {\\sf fP})$\n$(\\forall g' \\in {\\sf Inv}(g) \\cap {\\sf fP}^{\\sf NP})$\n$(\\exists f' \\in {\\sf Inv}(f))$ $(\\exists \\beta, \\alpha \\in {\\sf fP})$ \n$[ \\, f' = \\beta \\, g' \\, \\alpha \\, ]$. \n\n\\smallskip\n\n\\noindent So for any fixed $g' \\in {\\sf Inv}(g) \\cap {\\sf fP}^{\\sf NP}$, \nevery $f \\in{\\sf fP}$ has an inverse of the form \n$f' = \\beta \\, g' \\, \\alpha$, for some $\\beta, \\alpha \\in {\\sf fP}$ (that\ndepend on $f'$).  Hence $f' \\in {\\sf fP}[g']$.  \n\\ \\ \\ $\\Box$\n\n\\bigskip\n\n\n\n\\noindent {\\bf Observations:} \n\n\\smallskip\n\n\\noindent 1.\nWe saw in the proof of Prop.\\ \\ref{polyHierFinGen} that ${\\sf fP}^{\\sf NP}$ \nhas complete elements with respect to simulation. For any \n${\\sf fP}^{\\sf NP}$-complete element $h$ we have\n${\\sf fP}^{\\sf NP} = {\\sf fP}[h]$. This raises the question: \nIs ${\\sf fP}^{\\sf NP} \\neq {\\sf fP}[g']$, when $g' \\in {\\sf fP}^{\\sf NP}$ and\n$g'$ is an inverse of an element $g$ that is {\\sf fP}-complete \n(for $\\leqslant_{\\sf inv}$)? \nIn one direction we have:\n\n\\smallskip\n\n{\\it If there exists $g$ which is {\\sf fP}-complete with respect \nto $\\leqslant_{\\sf inv}$, and an inverse $g' \\in {\\sf fP}^{\\sf NP}$ such that \n${\\sf fP}^{\\sf NP} \\neq {\\sf fP}[g']$, then ${\\sf P} \\neq {\\sf NP}$.\n}\n\nIndeed, if ${\\sf fP}^{\\sf NP} \\neq {\\sf fP}[g']$ then\n${\\sf fP} \\subseteq {\\sf fP}[g'] \\varsubsetneq {\\sf fP}^{\\sf NP}$, hence\n${\\sf fP} \\neq {\\sf fP}^{\\sf NP}$, hence ${\\sf P} \\neq {\\sf NP}$.\n\n\\medskip\n\n\\noindent 2. {\\it If there exist $g_1, g_2$ (not necessarily different) that\nare {\\sf fP}-complete with respect to $\\leqslant_{\\sf inv}$, and inverses\n$g'_1, g'_2 \\in {\\sf fP}^{\\sf NP}$ of $g_1$, respectively $g_2$, such that\n${\\sf fP}[g'_1] \\neq {\\sf fP}[g'_2]$, then ${\\sf P} \\neq {\\sf NP}$.\n}\n\nIndeed, by contraposition, if ${\\sf P} = {\\sf NP}$ then \n${\\sf fP} = {\\sf fP}^{\\sf NP}$, hence $g'_1, g'_2 \\in {\\sf fP}$. Then\n${\\sf fP}[g'_1] = {\\sf fP} = {\\sf fP}[g'_2]$.\n\n\\medskip\n\n\\noindent 3. {\\it The following two statements are equivalent: \n \\ (1) \\ ${\\sf P} \\neq {\\sf NP}$; \n \\ (2) \\ there exist $g$ which is {\\sf fP}-complete with respect to\n$\\leqslant_{\\sf inv}$, and an inverse $g' \\in {\\sf fP}^{\\sf NP}$ such that\n${\\sf fP} \\neq {\\sf fP}[g']$.\n}\n\nIndeed, if such a $g$ and $g'$ exist then $g$ is a one-way function,\nhence ${\\sf P} \\neq {\\sf NP}$.\nIf for such a $g$ and $g'$ we have ${\\sf fP} = {\\sf fP}[g']$, then $g$ is\nan {\\sf fP}-complete function which is not one-way, hence one-way functions\ndo not exist.  \n\n\n\\bigskip\n\n\\noindent {\\bf Other monoids:} \n\n\\smallskip\n\n\\noindent (1) We have: \\, ${\\sf fP}^{\\sf PSpace} = {\\sf fPSpace}$. \n\nIndeed, the monoid ${\\sf fP}^{\\sf PSpace}$ consists of polynomially \nbalanced functions that are polynomial-time computable, with calls to \n{\\sf PSpace} oracles. \nThe monoid {\\sf fPSpace} consists of polynomially balanced functions that \nare polynomial-space computable (hence they might use exponential time).\nObviously, ${\\sf fP}^{\\sf PSpace} \\subseteq {\\sf fPSpace}$. But the \nconverse holds too, since the polynomially many output bits of a function\nin {\\sf fPSpace} can be found one by one, by a polynomial number of calls\nto {\\sf PSpace} oracles.\n\n\\medskip\n\n\\noindent (2) We define {\\sf fLog} (``functions in log-space'') to \nconsist of the polynomially balanced partial functions that are computable \nin deterministic log space.  {\\sf fLog} is closed under composition (see \n\\cite{HU}), and ${\\sf fLog} \\subseteq {\\sf fP}$. \n\n\nIf ${\\sf P} \\neq {\\sf NP}$ then {\\sf fLog} is non-regular;\nmore strongly, in that case {\\sf fLog} contains one-way functions (with \nno inverse in {\\sf fP}). Indeed, the 3{\\sc cnf} \nsatisfiability problem reduces to the inversion of the map\n$(B, \\alpha) \\mapsto (B, B(\\alpha))$, where $B$ is any boolean formula in \n3{\\sc cnf}, and $\\alpha$ is a truth-value assignment for $B$. It is not\ndifficult to prove that this map is in {\\sf fLog} when $B$ is in 3{\\sc cnf}. \nOne of the referees observed that {\\sf fLog} is regular iff {\\sf NP} $=$\n{\\sf L}, i.e., the class of languages accepted in deterministic log-space. \n\n\\medskip\n\n\\noindent (3) We define {\\sf fLin} (``functions in linear time'') to \nconsist of the linearly balanced partial functions that are computable \nin deterministic linear time. {\\sf fLin} is closed under composition, and\nit is non-regular \\ iff \\ ${\\sf P} \\neq {\\sf NP}$. More strongly, if ${\\sf P}\n\\neq {\\sf NP}$ then {\\sf fLin} contains one-way functions (with no inverse \nin {\\sf fP}); this is proved by padding arguments.\n\n\\bigskip\n\n\\bigskip\n\n\\noindent {\\bf Acknowledgement:}  This paper benefitted greatly form \ncorrections offered by the referees. \n\n\n\n\n{\\small\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nTiming noise is low-frequency fluctuation in the rotation-rate of\npulsars and is evident in the timing residuals of all young and\nmiddle-aged pulsars. The basic properties of timing noise were reviewed\nrecently by \\cite[Hobbs et al. (2010)]{hlk10} who presented the results of an\nanalysis of the rotation of 366 pulsars.  In summary, timing noise is\nseen as smooth changes in the timing residuals (and rotation\nfrequency). The timing residuals are often asymmetric, with peaks and\ntroughs having different radii of curvature; the variations are\noften quasi-periodic with timescales which are typically 1-10\nyears. For long, it was thought to arise from the fluid interiors of\nthe neutron stars.\n\nA breakthrough was made in 2006, when \\cite[Kramer et al.]{klo+06}\nshowed that the timing noise in the long-term intermittent pulsar\nB1931+24 could be explained in detail by switching in the magnitude of\nthe slowdown rate $\\dot{\\nu}$ betwen the \"ON\" and \"OFF\" emission\nstates of the pulsar, indicating that changes in the current flow from\nthe pulsar resulted in changes of both the radio emission and of the\nbraking torque.  The implication that changes in magnetospheric\ncurrents could alter pulsar emission properties as well as slowdown\nrate led \\cite[Lyne et al. (2010)]{lhk+10} (hereafter LHKSS) to study\nthe detailed pulse profiles of some of those pulsars having the\nlargest amounts of timing noise.  They demonstrated that six of these\npulsars exhibited the well-known phenomenon of mode-changing, in which\na pulsar switches abruptly between two stable profiles.  Moreover, in\nall six pulsars, there was a high degree of correlation between the\npulse shape and slowdown rate, the pulsars switching rapidly between\nlow- and high-spindown rates.\n\nTwo years after the publication of LHKSS, we review the observational\nevidence for switched changes in magnetospheric states, both in\nintermittent pulsars and in mode-changing pulsars, and discuss the\nrelationship between the two phenomena.\n\n\\section{Intermittent pulsars}\nMany pulsars show intermittency in their radio emission, although\nusually the durations of the \"ON\" and \"OFF\" states are measured in\nseconds to hundreds of seconds, timescales which are far too short to\npermit the determination of any change in slowdown rate between the\nstates.  This is the phenomenon of pulse nulling which has been known\nsince shortly after the discovery of pulsars (\\cite[Backer 1970]{bac70}).\n\n \\begin{figure}[t]\n \\begin{center}\n  \\includegraphics[scale=0.31, angle=-90]{fig1.eps} \n  \\caption{The rotational frequency evolution of PSR~J1832+0029\n (\\cite{llm+12}).} \n    \\label{fig1}\n \\end{center}\n \\end{figure}\n\n \\begin{figure}[t]\n \\begin{center}\n  \\includegraphics[scale=0.31, angle=-90]{fig2.eps} \n  \\caption{The rotational frequency evolution of PSR~J1841$-$0500\n (\\cite[Camilo et al. 2012]{crc+12} and Lyne, priv. comm.).} \n    \\label{fig2}\n \\end{center}\n \\end{figure}\n\nHowever, the intermittent pulsar B1931+24 is typically ON for 1 week\nand OFF for about 1 month, permitting \\cite[Kramer et\nal. (2006)]{klo+06} to show that the ratio of ON- and OFF- slowdown\nvalues $\\dot{\\nu}_{\\rm ON}\/\\dot{\\nu}_{\\rm OFF}=1.5\\pm0.1$, roughly\nconsistent with an absence of all magnetospheric currents during\nthe OFF phase, in accordance with the calculations of the braking\neffects of magnetospheric currents by \\cite[Goldreich \\& Julian\n(1969)]{gj69}.\n\nShortly after that publication, a second long-term intermittent\npulsar was discovered (PSR~J1832+0029) and reported to show similar\nlarge changes in in slowdown rate ($\\dot{\\nu}_{\\rm ON}\/\\dot{\\nu}_{\\rm\nOFF}=1.7\\pm0.1$; \\cite[Kramer 2008]{kra08}, \\cite[Lyne 2009]{lyn09},\n\\cite[Lorimer et al. 2012]{llm+12}).  Fig.~1 shows the measured values\nof rotation rate during the 10 years since its discovery.  With rather\npoor statistics, the lengths of the ON and OFF states are typically\nmany hundreds of days, compared with tens of days for B1931+24.\n\nMore recently, a third long-term intermittent object\n(PSR~J1841$-$0500), also with timescales measured in hundreds of days,\nhas been reported by \\cite[Camilo et al. (2012)]{crc+12}.  Even though\nthe statistics are also poor for this pulsar, it is clear that this\nhas an even greater slowdown rate ratio ($\\dot{\\nu}_{\\rm\nON}\/\\dot{\\nu}_{\\rm OFF}=2.5\\pm0.2$; see Fig.~2).\n\n\n\\section{Profile-switching pulsars}\nLHKSS studied those pulsars in the Jodrell Bank timing database which\nshowed the largest amounts of timing noise, measured as the ratio of\nmaximum to minimum values of slowdown rate.  Seventeen examples are\nshown in Fig.~3. Pulsars typically have peak-to-peak values of about\n1\\% of the mean, over a 4-orders-of-magnitude range of slowdown rates.\nIndividual pulsars may have a factor of 10 times more or less than\nthis.  LHKSS found that six of the 17 pulsars showed pulse-shape\nchanges which were correlated with these slowdown rate variations.\nSubsequent studies (Lyne et al., in prep) have now shown that a\nfurther four of these 17 also have significant pulse-shape variations\nthat are correlated with slowdown rate (PSRs B0919+06, B1642$-$03,\nB1826$-$17 and B1903+07) as well as two others (PSRs B1740$-$03 and\nB0105+65).\n\n \\begin{figure}[hbt]\n \\begin{center}\n  \\includegraphics[width=0.4\\textwidth]{fig3new.eps} \n  \\caption{The slowdown rate ($\\dot{\\nu}$) of 17 pulsars (from Lyne et\n al. 2010).} \n    \\label{fig3}\n \\end{center}\n \\end{figure}\n\n \\begin{figure}[hbt]\n \\begin{center}\n  \\includegraphics[scale=0.35, angle=0]{fig4.eps} \n  \\caption{The profiles of the ``normal'' (top) and ``abnormal''\n(bottom) states of PSR~J2047+5029 (Janssen et al., in prep.).} \n    \\label{fig4}\n \\end{center}\n \\end{figure}\n\nA further striking example of correlated changes in pulse shape and\nslowdown is displayed by PSR~J2047+5029, discovered at Westerbork in\nthe 8gr8 survey (\\cite[Janssen et al. 2009]{jsb+09}).  At discovery,\nthe observed profile showed a main pulse and an interpulse with an\nassociated precursor having about 1\/3 of the flux density of the main\npulse (Fig.~4 top).  However, when monitoring commenced at Jodrell\nBank, the main pulse had reduced in intensity by a factor of about 10,\nso that it was then much weaker than the interpulse (Fig.~4 bottom).\nThe pulsar has since changed from this ``abnormal mode'' back to the\nearlier ``normal'' mode.  These changes were accompanied by changes in\nslowdown rate, which was larger when in the normal mode than in the\nabnormal mode, again consistent with the notion that the particles\nresponsible for much of the normal-mode main-pulse radio flux density\nwere also responsible for the increase in braking.\n\nIn total there are now 16 pulsars with established synchronised\nchanges in the radio emission and the slowdown rate.  The properties\nof these pulsars are summarised in Table~1, in decreasing order of the\nmagnitude of the timing noise, measured as the ratio of the maximum and\nminimum slowdown rates.\n\n\\begin{table}\n  \\begin{center}\n  \\caption{Timing noise slowdown rate ratios ($\\dot{\\nu_1}\/\\dot{\\nu_2}$) and emission changes in 16 pulsars.}\n  \\label{tab1}\n  \\scriptsize\n  \\begin{tabular}{|l|l|l|l|}\\hline \n{\\bf PULSAR} & {\\bf $\\dot{\\nu}_1\/\\dot{\\nu}_2$}~~ & {\\bf Emission Change} & {\\bf Reference} \\\\ \\hline\nJ1841$-$0500~~ & 2.5  & Deep null & Camilo et al. (2012) \\\\\nJ1832+0029   & 1.7  & Deep null & Lorimer et al. (2012) \\\\\nB1931+24     & 1.5  & Deep null & Kramer et al. (2012) \\\\ \nB2035+36     & 1.13 & 28\\% change in W$_{\\rm eq}$ & Lyne et al. (2010) \\\\\nB1740$-$03   & 1.13 & 70\\% change in component ratio & This paper \\\\\nB0105+65     & 1.11 & 30\\% change in W$_{\\rm eq}$ & This paper \\\\\nB1903+07     & 1.07 & 10\\% change in W$_{\\rm 10}$ & This paper \\\\\nJ2043+2740   & 1.06 & 100\\% change in W$_{\\rm 50}$ & Lyne et al. (2010) \\\\\nB1822$-$09   & 1.033& 100\\% change in precursor\/interpulse~~ & Lyne et al. (2010) \\\\\nJ2047+5029   & 1.030& 90\\% change in main pulse & Janssen et al. (in prep)~~ \\\\\nB1642$-$03   & 1.025& 20\\% change in cone\/core & This paper \\\\\nB1540$-$06   & 1.017& 12\\% change in W$_{\\rm 10}$ & Lyne et al. (2010) \\\\\nB1828$-$11   & 1.007& 100\\% change in W$_{\\rm 10}$ & Lyne et al. (2010) \\\\\nB1826$-$17   & 1.007& 10\\% change in cone\/core & This paper \\\\\nB0919+06     & 1.007& 30\\% change in component ratio & This paper \\\\\nB0740$-$28   & 1.007& 20\\% change in W$_{\\rm 75}$ & Lyne et al. (2010) \\\\\n\\hline\n  \\end{tabular}\n \\end{center}\n\\vspace{1mm}\n \\scriptsize\n\\end{table}\n\n\\section{The nature of the switching}\n\nThe time sequences of slowdown rates shown in Fig.~2 are usually\nbounded by well-defined maximum and minimum levels, each extreme level\nbeing identifiable with a characteristic emission profile or flux\ndensity.  As reported by LHKSS, each pulsar is usually seen to switch\nabruptly between these extreme states.  The fact that the patterns in\nFig.~2 are generally smooth and do not display abrupt switching\nbehaviour was demonstrated to arise from changing statistical\nproperties of the mode-changing phenomenon, and the observed profile\nshape parameter is determined by the proportion of time spent in the\ntwo modes.  This is most clearly illustrated in Fig.~5 which shows a\nnumber of 8-hour observations of PSR~B1828$-$11 chosen at different\nphases of the 500-day oscillating pattern displayed by this pulsar in\nFig.~3.  We note that although pulse nulling and profile mode-changing\nwas first observed in 1970 (\\cite[Backer 1970]{bac70}; \\cite[Backer\n1970a]{bac70a}), the following four decades have seen no study of the\nstability of the statistics of nulling or mode-changing.\n\n\n \\begin{figure}[bt]\n \\begin{center}\n  \\includegraphics[scale=0.38, angle=-90]{fig5.eps} \n  \\caption{Four 8-hour time-sequences of the profile state of\n PSR~B1828$-$11, taken at four different phases of the 500-day oscillation\n seen in Fig.~3 for this pulsar. Although the switching timescale may\n be short, the fraction of time spent in one state or the other\n changes slowly (Stairs et al. in prep.)} \n    \\label{fig5}\n \\end{center}\n \\end{figure}\n\n\\section{The relationship between nulling and mode-changing}\nThe processes of nulling and mode-changing are similar in many\nways. Both are switched phenomena between (usually) two discrete\nemission states.  They both have similar large ranges of timescales\nand both have a major synchronisation with the spindown rate of the pulsar.\nThey are both understood in terms of changes in magnetospheric particle\ncurrents.  The natural conclusion is that nulling is probably an extreme form of\nmode-changing.  This view is supported by a few cases in which an\napparently nulling pulsar has been found to have low-level emission in\nthe null state.  One such example is PSR~B0826$-$34 shown in Fig.~6,\nin which integration of the data during apparently ``null'' episodes\nshows pulsed emission at a level of about 2\\% of the un-nulled pulses.\n \nPerhaps one telescope's nulling pulsar is a larger telescope's\nmode-changing pulsar ! \n\n\n \\begin{figure}[hbt]\n \\begin{center}\n  \\includegraphics[scale=0.35, angle=-90]{fig6new.eps} \n  \\caption{The integrated ``normal'' (top) and ``abnormal'' (bottom) \nprofiles of PSR~B0826$-$34 at 1374~MHz (from \\cite[Esamdin et al. 2005]{elg+05}).} \n    \\label{fig6}\n \\end{center}\n \\end{figure}\n\n\\section{Conclusions}\nPulsar magnetospheres switch between a small number of discrete\nstates, usually two, each of which corresponds to an apparently\nquasi-stable magnetospheric configuration.  It seems that changes in\nmagnetospheric current flows between these states cause variations in\nboth the emission beam snd the slow-down rate. This is supported by\nthe general observation that the larger slowdown-rate is nearly\nalways associated with enhanced emission, particularly of the pulsar\n``core'' emission.  I have no understanding of why there are these\ndiscrete states, or of the origin of the multi-year quasi-periodicities that\nmodulate the statistical properties of the states.  Free-precession of\nthe neutron star and orbiting asteroids have been proposed, but any\nlinks are obscure.\n\nFinally, it must be emphasised that these phenomena are\nwidespread. The majority of pulsars of young and intermediate\ncharacteristic age display detectable timing noise.  The studies of\nprofiles described here have mostly been of those pulsars which have\nthe largest fractional changes in slowdown rates and hence may be\nexpected to suffer the greatest magnetospheric changes and\ncorresponding variation in emission properties.  In fact, 12 of the 19\npulsars with largest timing noise show correlated emission\nvariations. Most of the remaining 7 have much poorer signal-to-noise\nratio, making the precise determination of pulse shape changes\nchallenging.  The changes expected in less timing-noisy pulsars are\nlikely to be much more subtle and a challenge to detect. At present,\nthere is no reason to doubt that all timing noise has its origin in\nswitched magnetospheric states.\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\\label{introduction}\nIn \\cite{SpeyerSturmfels} the authors associated Gr\\\"obner toric\ndegenerations $\\mathrm{Gr}_2(\\mathbb{C}^n)^{\\mathcal{T}}$ of $\\mathrm{Gr}_2(\\mathbb{C}^n)$\nto each\ntrivalent tree $\\mathcal{T}$ with $n$ leaves. These degenerations\ninduce toric\ndegenerations  $M_{\\mathbf{r}}^{\\mathcal{T}}$ of $M_{\\mathbf{r}}$, the\nspace of $n$ ordered, weighted (by $\\mathbf{r}$) points on the projective line.  \nWe denote the  corresponding toric fibers by\n$\\mathrm{Gr}_2(\\mathbb{C}^n)_0^{\\mathcal{T}}$ and $(M_{\\mathbf{r}})_0^{\\mathcal{T}}$ respectively.\nOur goal in this paper is to give a\ngeometric (Euclidean polygon) description of the toric fibers as\nstratified symplectic spaces (see \\cite{SjamaarLerman} for this\nnotion) and describe the action of the compact part of the torus\nas ``bendings of polygons.''\n\n\n\\subsection{The Grassmannian and imploded spin-framed polygons}\nWe start by identifying the Grassmannian $\\mathrm{Gr}_2(\\mathbb{C}^n)$\nwith the moduli space of\n``imploded spin-framed'' $n$-gons in $\\mathbb{R}^3$.\nWe define the space of \\emph{imploded framed vectors}, which is\ntopologically the cone $C \\mathrm{SO}(3,\\mathbb{R})$ of $\\mathrm{SO}(3,\\mathbb{R})$, as the space\n$$\\Big\\{(F,e) \\in \\mathrm{SO}(3,\\mathbb{R}) \\times \\mathbb{R}^3 \\mid\n\\text{ $e = tF(\\epsilon_1)$ for some $t \\in \\mathbb{R}_{\\geq 0}$}\\Big\\}$$\nmodulo the equivalence relation\n$(F_1,0) \\sim (F_2,0)$ for all $F_1,F_2 \\in \\mathrm{SO}(3,\\mathbb{R})$,\nwhere $\\epsilon_1 = (1,0,0)$ is the first standard basis vector of\n$\\mathbb{R}^3$.\nWe will see later that this equivalence relation is ``implosion'' in\nthe sense of \\cite{GuilleminJeffreySjamaar}.\nWe call the equivalence class of $(F,e)$ an ``imploded framed vector''.  Note that the isotropy $T_{\\mathrm{SO}(3,\\mathbb{R})}$ of $\\epsilon_1$ in $\\mathrm{SO}(3,\\mathbb{R})$ has a natural right action on the space of imploded framed vectors. \n\nNow fix a covering homomorphism $\\pi : \\mathrm{SU}(2) \\to \\mathrm{SO}(3,\\mathbb{R})$ such that the Cartan subgroup of diagonal matrices $T_{\\mathrm{SU}(2)}$ in $\\mathrm{SU}(2)$\nmaps onto $T_{\\mathrm{SO}(3,\\mathbb{R})}$.\nWe define the space of\nimploded \\emph{spin}-framed vectors, which is topologically\nthe cone $C \\mathrm{SU}(2) \\cong \\mathbb{C}^2$, as the space\n$$\\Big\\{(F,e) \\in \\mathrm{SU}(2) \\times \\mathbb{R}^3 \\mid\n\\text{ $e = t \\pi(F)(\\epsilon_1)$ for some $t \\in \\mathbb{R}_{\\geq 0}$}\\Big\\}$$\nmodulo the equivalence relation\n$(F_1,0) \\sim (F_2,0)$ for all\n$F_1,F_2 \\in \\mathrm{SU}(2)$.\nWe call the equivalence class of $(F,e)$ an ``imploded spin-framed vector''.\nAn \\emph{imploded spin-framed $n$-gon} is an $n$-tuple\n $((F_1,e_1),\\ldots,(F_n,e_n))$ of imploded spin-framed vectors\n such that $e_1 + e_2 + \\cdots + e_n = 0$.\nLet $\\widetilde{P}_n(\\mathrm{SU}(2))$\ndenote the space of imploded spin-framed $n$-gons.\nThere is an\naction of $\\mathrm{SU}(2)$ on $\\widetilde{P}_n(\\mathrm{SU}(2))$ given by\n$$F \\cdot ((F_1,e_1),\\ldots,(F_n,e_n)) =\n((F F_1, \\pi(F)(e_1)),\\ldots,(F F_n, \\pi(F)(e_n))).$$\nWe let $P_n(\\mathrm{SU}(2))$ denote the quotient\nspace. Note that since we may scale the edges of an $n$-gon the\nspace $P_n(\\mathrm{SU}(2))$ is a cone with vertex the zero $n$-gon (all the\nedges are the zero vector in $\\mathbb{R}^3$).  Finally, note that \nthere is a natural right action of $T_{\\mathrm{SU}(2)^n}$ on \n$\\widetilde{P}_n(\\mathrm{SU}(2))$, which rotates frames but fixes \nthe vectors:\n$$(t_1,\\ldots,t_n) \\cdot ((F_1,e_1),\\ldots,(F_n,e_n)) = \n((F_1 t_1, e_1),\\ldots,(F_n t_n, e_n)).$$\n\n\\begin{remark}\nWe will see later that $P_n(\\mathrm{SU}(2))$\noccurs naturally in equivariant symplectic\ngeometry, \\cite{GuilleminJeffreySjamaar}: it is the symplectic\nquotient by the left diagonal action of $\\mathrm{SU}(2)$ on the right\nimploded cotangent bundle of $\\mathrm{SU}(2)^n$. The space\n$\\widetilde{P}_n(\\mathrm{SU}(2))$ is the zero level set of the momentum map\nfor the left diagonal action on the right imploded product of\ncotangent bundles. Hence we find that the space $P_n(\\mathrm{SU}(2))$ has a\n(residual) right action of an $n$--torus $T_{\\mathrm{SU}(2)^n}$, the\nmaximal torus in $\\mathrm{SU}(2)^n$ which rotates the imploded spin-frames.\n\\end{remark}\n\nLet $Q_n(\\mathrm{SU}(2))$ be the quotient of the subspace of imploded spin-framed $n$-gons\nof perimeter $1$ by the action of the diagonal embedded circle in\n$T_{\\mathrm{SU}(2)^n}$. In what follows $\\mathrm{Aff} \\mathrm{Gr}_2(\\mathbb{C}^n)$ denotes the affine cone over\nthe Grassmannian $\\mathrm{Gr}_2(\\mathbb{C}^n)$ for the Pl\\\"ucker embedding. Thus $\\mathrm{Aff} \\mathrm{Gr}_2(\\mathbb{C}^n)$ is the\nsubcone of $\\bigwedge^2(\\mathbb{C}^n)$ consisting of the decomposable\nbivectors (the zero locus of the Pl\\\"ucker equations). The reason\nwhy we consider $Q_n(\\mathrm{SU}(2))$ here is the following theorem (proved in \\S \\ref{HJconstruction})\nwhich gives a polygonal interpretation of $\\mathrm{Gr}_2(\\mathbb{C}^n)$. It is the\nstarting point of our work.\n\n\\begin{theorem}\\label{polygonGrass}\\hfill\n\\begin{enumerate}\n\\item $P_n(\\mathrm{SU}(2))$ and\n$\\mathrm{Aff} \\mathrm{Gr}_2(\\mathbb{C}^n)$ are homeomorphic.\n\\item This homeomorphism induces a homeomorphism between  $Q_n(\\mathrm{SU}(2))$ and $\\mathrm{Gr}_2(\\mathbb{C}^n).$\n\\end{enumerate}\n\\end{theorem}\n\n\n\\subsection{Triangulations, trivalent trees, and the construction of Kamiyama-Yoshida}\nLet $P$ denote a fixed convex\nplanar $n$-gon.\nThroughout the paper we will use the\nsymbol $\\mathcal{T}$ to denote either a triangulation of $P$ or its dual\ntrivalent tree. Accordingly we fix a triangulation $\\mathcal{T}$ of $P$.\nThe points of\n$\\mathrm{Gr}_2(\\mathbb{C}^n)_0^{\\mathcal{T}}$ are ``imploded spin-framed $n$--gons'' with a\nfixed perimeter in $\\mathbb{R}^3$\nmodulo an equivalence relation  called $\\mathcal{T}$-congruence and denoted\n$\\sim_{\\mathcal{T}}$ that depends on the triangulation $\\mathcal{T}$.\n\n\\begin{figure}\n\\centering\n\\includegraphics[scale = 0.2]{newpoly.eps}\n\\caption{A framed spatial polygon with chosen triangulation.}\\label{fig:TriangulatedPolygon}\n\\end{figure}\n\n\nNow that we have a polygonal interpretation of $\\mathrm{Gr}_2(\\mathbb{C}^n)$  we\nwill impose the equivalence relation of $\\mathcal{T}$-congruence (to be\ndescribed below) on our space of framed polygons and obtain a polygonal\ninterpretation of $\\mathrm{Gr}_2(\\mathbb{C}^n)_0^{\\mathcal{T}}$\ncorresponding to the triangulation (tree) $\\mathcal{T}$. We now describe\nthe equivalence relation of $\\mathcal{T}$-congruence.\nHere we will discuss only the case\nof the {\\it the standard triangulation} $\\mathcal{T}_0$, that is the\ntriangulation of $P$ given by drawing the diagonals from the first\nvertex to the remaining nonadjacent vertices. The dual tree to the\nstandard triangulation will be called the caterpillar or fan\n(see Figure \\ref{fig:CaterpillarTree7Leaves}). However we will state \nour theorems in the generality in which they are proved in the paper \nnamely for all trivalent trees $\\mathcal{T}$ with $n$ leaves.\n\n\n\n\n\n\\begin{figure}\n\\centering\n\\includegraphics[scale = 0.2]{NP3.eps}\n\\caption{The standard triangulation of the model $7$-gon with dual\ncaterpillar tree.}\\label{fig:CaterpillarTree7Leaves}\n\\end{figure}\n\n\n\n\n\n\n\n\n\nThe reader is urged to refer to the pictures below to\nunderstand the following description. The  equivalence relation\nfor the standard triangulation (in the case of $n$-gon linkages)\ndescribed below was first introduced in \\cite{KamiyamaYoshida}. We\nhave extended their definition to all triangulations and to\n$n$-gons equipped with imploded spin-frames. Label the diagonals\nof the triangulation counterclockwise by $1$ through $n-3$. For each $S\n\\subset \\{1,2,\\cdots,n-3\\}$ let $\\widetilde{P}_n(\\mathrm{SU}(2))^{[S]}$ denote the\nsubspace of $\\widetilde{P}_n(\\mathrm{SU}(2))$ where the diagonals\ncorresponding to the elements in $S$ are zero and all other diagonals are nonzero.\nLet $\\mathbf{F}=\n((F_1,e_1),(F_2,e_2),\\cdots,(F_n,e_n))$ be a point\nin $\\widetilde{P}_n(\\mathrm{SU}(2))^{[S]}$.  Suppose that $|S| = i$.\nSince $i$ diagonals are zero there will be $i+1$ sums of the form\n$e_j + e_{j+1}+ \\cdots + e_{j+k_j}$ that are zero, and the\n$n$-gon underlying $\\mathbf{F}$ will be the wedge of $i+1$ closed\nsubpolygons corresponding to the $i+1$ closed subpolygons in the\ncollapsed reference polygon. Thus we can divide $\\mathbf{F}$ into\n$i+1$ imploded spin-framed closed subpolygons ( in terms of\nformulas we can break up the above $n$-tuple into $i+1$ sub\n$k_j$-tuples of edges for $1 \\leq j \\leq i+1$). We may act on\neach imploded spin-framed subpolygon ($k_j$-tuple) by a copy of\n$\\mathrm{SU}(2)$. We pass to the quotient $\\widetilde{P}_n(\\mathrm{SU}(2))\/\\sim_{\\mathcal{T} _0}$\nby dividing out $\\widetilde{P}_n(\\mathrm{SU}(2))^{[S]}$ by the resulting action\nof $\\mathrm{SU}(2)^{i+1}$. By definition $\\mathcal{T}$-congruence is\nthe equivalence relation induced by the above quotient operation,\nsee Figure \\ref{fig:TreeEquivalence}.\n\n\\begin{figure}[htp]\n\\centering\n\\includegraphics[scale = 0.2]{newpoly3.eps}\n\\caption{A picture of equivalent polygons in $V_n^\\mathcal{T}$ which\nare not equivalent in $P_n(\\mathrm{SU}(2))$.}\\label{fig:TreeEquivalence}\n\\end{figure}\n\nWe let $V_n^{\\mathcal{T}_0}$ denote the quotient space\n$\\widetilde{P}_n(\\mathrm{SU}(2))\/\\sim_{\\mathcal{T}_0}$. Thus $V_n^{\\mathcal{T}_0}$ is\ndecomposed into the pieces $(V_n^{\\mathcal{T}_0})^{[S]} =\n\\widetilde{P}_n(\\mathrm{SU}(2))^{[S]}\/\\sim_{\\mathcal{T}_0}$. We will call the\nresulting decomposition in the special case of $\\mathcal{T}_0$\nthe Kamiyama-Yoshida decomposition (or\nKY-decomposition).  In this paper we use a\nweakened definition of the term {\\it decomposition}; we will refer\nto decompositions of spaces where the pieces are products\nof spaces with isolated singularities.\n\n\n\n\\begin{remark}\nThe equivalence relation of $\\mathcal{T}$-congruence can be defined \nanalogously for\nany triangulation of $P$ (equivalently any trivalent tree with\n$n$-leaves) and induces an equivalence relation on $Q_n(\\mathrm{SU}(2))$\n(and many other spaces associated to spaces of $n$-gons in $\\mathbb{R}^3$,\nfor example, the space of $n$-gons itself or the space of\n$n$-gon linkages). We will use the symbol $\\sim_{\\mathcal{T}}$ to denote\nall such equivalence relations.\n\\end{remark}\n\n\nWe will use $W_n^{\\mathcal{T}}$ to denote the quotient of\n$Q_n(\\mathrm{SU}(2))$ by\nthe equivalence relation $\\sim_{\\mathcal{T}}$. We can now state our\nfirst main result (this is proved in \\S \\ref{homeomorphismVtoP}).\n\n\\begin{theorem} \\label{firstmaintheorem}\\hfill\n\\begin{enumerate}\n\\item The toric fiber of the toric degeneration of\n$\\mathrm{Aff} \\mathrm{Gr}_2(\\mathbb{C}^n)$ corresponding to\nthe trivalent tree $\\mathcal{T}$ is homeomorphic to  $V_n^{\\mathcal{T}}$.\n\\item The toric fiber of the toric degeneration of\n$\\mathrm{Gr}_2(\\mathbb{C}^n)$\ncorresponding to the trivalent tree $\\mathcal{T}$ is homeomorphic to\n$W_n^{\\mathcal{T}}$.\n\\end{enumerate}\n\\end{theorem}\n\n\\begin{remark}\nThe quotient map from\n$Q_n(\\mathrm{SU}(2))$ to\n$W_n^{\\mathcal{T}}$ given by passing to $\\mathcal{T}$-congruence classes\nmaps the generic fiber of the toric degeneration onto the special\n(toric) fiber.\n\\end{remark}\n\nWe next describe the space $V_{\\mathbf{r}}^{\\mathcal{T}_0}$ that will be proved\nlater to be homeomorphic to the toric fiber $(M_{\\mathbf{r}})_0^{\\mathcal{T}_0}$.\nAgain we will restrict ourselves to the standard triangulation\nin our description. Starting with the space $\\widetilde{M}_{\\mathbf{r}} =\n\\{(e_1,\\ldots,e_n) \\in (\\mathbb{R}^3)^n \\mid \\sum_i e_i = 0, \\; \\|e_i\\| = r_i\\}$\nof closed $n$-gon linkages with side-lengths $\\mathbf{r}$, we define\n$$V_{\\mathbf{r}}^{\\mathcal{T}_0} = \\widetilde{M}_{\\mathbf{r}}\/\\sim_{\\mathcal{T}_0}.$$\nThis is the construction of \\cite{KamiyamaYoshida} - see below\nfor some pictures.\n\n\n\\begin{theorem} \\label{secondmaintheorem}\nThe toric fiber $(M_{\\mathbf{r}})^{\\mathcal{T}}_0$ of the toric degeneration\nof $M_{\\mathbf{r}}$ corresponding to\nthe trivalent tree $\\mathcal{T}$ is homeomorphic to\n$V_{\\mathbf{r}}^{\\mathcal{T}}$.\n\\end{theorem}\n\n\\begin{remark} The quotient map from $M_{\\mathbf{r}}$ to\n$V_{\\mathbf{r}}^{\\mathcal{T}}$ maps the generic fiber of the toric degeneration onto the special\n(toric) fiber.\n\\end{remark}\n\nThe result in the previous theorem was\nconjectured by Philip Foth and Yi Hu in \\cite{FothHu}.\n\n\\subsection{Bending flows, edge rotations, and the toric structure of $W_{n}^{\\mathcal{T}_0}$}\nThe motivation for the above construction becomes more clear once we introduce the bending flows of\n\\cite{KapovichMillson} and \\cite{Klyachko}. The lengths of the $n-3$ diagonals created are continuous\nfunctions on $M_{\\mathbf{r}}$ and are smooth where they are not zero.\nThey give rise to Hamiltonian flows which were called {\\em bending\nflows} in \\cite{KapovichMillson}. The bending flow associated to a\ngiven diagonal has the following description. The part of the\n$n$-gon to one side of the diagonal does not move, while the other\npart rotates around the diagonal at constant speed. The lengths of\nthe diagonals are action variables which generate the bending\nflows, the conjugate angle variables are the dihedral angles\nbetween the fixed and moving parts. However {\\em the bending flow\nalong the $i$-th diagonal is not defined at those $n$-gons where\nthe $i$-th diagonal is zero}. If the bending flows are everywhere\ndefined (for example if one of the side-lengths is much larger\nthan the rest) then we may apply the theorem of Delzant,\n\\cite{Delzant} to conclude that $M_{\\mathbf{r}}$ is toric. However for\nmany $\\mathbf{r}$ (including the case of regular $n$\n-gons) the bending\nflows are not everywhere defined. The point of\n\\cite{KamiyamaYoshida} was to make the bending flows well-defined\nby dividing out the subspaces of $\\widetilde{M}_{\\mathbf{r}}$ where a\ncollection of $i$ diagonals vanish by $\\mathrm{SO}(3,\\mathbb{R})^{i+1}$. We\nillustrate their construction with two examples.\\vspace {1 mm}\n\n\nFirst, let $\\mathbf{r} = (1,1,1,1,1,1)$ so $M_{\\mathbf{r}}$ is the space of\nregular hexagons with side-lengths all equal to $1$. Let\n$M_{\\mathbf{r}}^{(2)}$\nbe the subspace of $M_{\\mathbf{r}}$\nwhere the middle (second) diagonal vanishes. Thus $M_{\\mathbf{r}}^{(2)}$\nis the space of ``bowties'' (see Figure \\ref{fig:Bowties}) modulo the diagonal action of $\\mathrm{SO}(3,\\mathbb{R})$\non the two equilateral triangles. We can no longer bend on the\nsecond diagonal because we have no axis of rotation. Passing to\n$\\mathcal{T}$-congruence classes\ncollapses the\nspace $M_{\\mathbf{r}}^{(2)}$ to a point by dividing by the action\nof $\\mathrm{SO}(3,\\mathbb{R}) \\times \\mathrm{SO}(3,\\mathbb{R})$. Bending along the second diagonal fixes \nthis point by definition.\n\n\n\n\\begin{figure}[htbp]\n\\centering\n\\includegraphics[scale = 0.2]{NP4.eps}\n\\caption{The bowties are all $\\mathcal{T}_0$-congruent\nand so define a single point in $V_\\mathbf{r}^{\\mathcal{T}_0}$.\nHowever in $M_\\mathbf{r}$ the subspace of bowties is homeomorphic to $\\mathrm{SO}(3,\\mathbb{R})$.}\\label{fig:Bowties}\n\\end{figure}\n\n\nFor our second example,  we consider the space of regular octagons\nwith all side-lengths equal to $1$ and  the subspace $M_{\\mathbf{r}}^{(3)}$ where\nthe middle diagonal vanishes. Thus $M_{\\mathbf{r}}^{(3)}$ is the space of\nwedges of rhombi modulo the diagonal action of $\\mathrm{SO}(3,\\mathbb{R})$ on the two\nrhombi, see Figure \\ref{fig:TwoRhombi}.  Passing to $\\mathcal{T}$-congruence classes\namounts to dividing by the action of $\\mathrm{SO}(3,\\mathbb{R}) \\times \\mathrm{SO}(3,\\mathbb{R})$ on the\ntwo rhombi.\n\n\n\n\n\n\\begin{figure}[htbp]\n\\centering\n\\includegraphics[scale = 0.2]{NP5.eps}\n\\caption{Here\n$n=8$ and $S= \\{3 \\}$.\nThe middle three components\nof $T^5$ act trivially, and the\nquotient $2$-torus acts by\nbending along the first and fifth diagonals.}\\label{fig:TwoRhombi}\n\\end{figure}\n\n\n\n\n\n\n We obtain the action of the bending flows on $V_n^{\\mathcal{T}_0}$\nas follows. In this case we will be given a lift of the\none-parameter group bending along a diagonal to $\\mathrm{SU}(2)$. The one\nparameter group acts through its quotient in $\\mathrm{SO}(3,\\mathbb{R})$ by bending\nalong the diagonal. If an edge moves under this bending then the\nimploded spin-frame is moved by the one-parameter group in $\\mathrm{SU}(2)$\nin the same way. Hence, one part of the imploded spin-framed\npolygon is fixed and the other moves by a ``rigid motion'' --i.e. all\nthe spin-framed edges of the second part are moved by the same\none-parameter group in $\\mathrm{SU}(2)$. The bendings give rise to an\naction of an $n-3$ torus $T_{bend}$ on $V_n^{\\mathcal{T}}$. There are also ''edge-rotations''\nthat apply a one-parameter group to the imploded spin-frame but do\nnot move the edge. This action on frames is the action of the torus\n$T_{\\mathrm{SU}(2)^n}$ coming from the theory of the imploded cotangent bundle\nof $\\mathrm{SU}(2)^n$. This action is not faithful, the diagonal subtorus acts\ntrivially. The bendings together with the edge rotations give rise to\nan action of a compact $2n-4$ torus $T = T_{bend} \\times T_{\\mathrm{SU}(2)^n}$.\n\n\\subsection{A sketch of the proofs.}\n\nThe main step in proving Theorems \\ref{firstmaintheorem} and \\ref{secondmaintheorem} is to\nproduce a space $P_n^{\\mathcal{T}_0}(\\mathrm{SU}(2))$ that ``interpolates''\nbetween $V_n^{\\mathcal{T}_0}$ and $\\mathrm{Gr}_2(\\mathbb{C}^n)^{\\mathcal{T}_0}_0$.\nOur construction  of $P_n^{\\mathcal{T}_0}(\\mathrm{SU}(2))$\nwas motivated by the construction of the toric\ndegeneration of $\\mathrm{SU}(2)$-character varieties of fundamental groups\nof surfaces given by Hurtubise and Jeffrey in\n\\cite{HurtubiseJeffrey}. The connection is that the  space\n$M_{\\mathbf{r}}$ can be interpreted as the (relative) character variety\nof the fundamental group of the $n$-punctured two-sphere with\nvalues in the translation subgroup of the Euclidean group\n$\\mathbb{E}_3$ - a small loop around the $i$-th puncture maps to\ntranslations by the $i$-th edge of the polygon (considered as a\nvector in $\\mathbb{R}^3$).\n\nTake the triangulated model (convex planar) $n$-gon $P$ and break\nit apart into $n-2$ triangles $T_1,\\cdots,T_{n-2}$.\nEquivalently we break apart the dual tree $\\mathcal{T}$ into a forest\n$\\mathcal{T}^D$ consisting of $n-2$ tripods.\n\n\\begin{figure}[htbp]\n\\centering\n\\includegraphics[scale = 0.2]{newpoly2.eps}\n\\caption{The decomposed polygon and the decomposed dual tree $\\mathcal{T}^D$.}\\label{fig:decomposed_ngon}\n\\end{figure}\n\n\n\nAttach to each of the $3(n-2)$ edges  of the $n-2$ triangles\n(or each edge of the forest $\\mathcal{T}^D$) a copy\nof $T^*(\\mathrm{SU}(2))$. Now right-implode each copy of $T^*(\\mathrm{SU}(2))$ so that  a\ncopy of $\\mathcal{E} T^\\ast(\\mathrm{SU}(2))) \\cong \\mathbb{C}^2$, the imploded cotangent bundle of\n$\\mathrm{SU}(2)$, is attached to each edge. Since\n$\\mathcal{E} T^\\ast(\\mathrm{SU}(2))$ admits\nan action of the circle (from the right) and $\\mathrm{SU}(2)$ from the left\nthe resulting space admits an action of a torus $\\mathbb{T}$ of\ndimension $3(n-2)$ and a\ncommuting action of $\\mathrm{SU}(2)^{3(n-2)}$. The torus has a product\ndecomposition $\\mathbb{T} = \\mathbb{T}_e \\times\\mathbb{T}_d$\nwhere $\\mathbb {T}_e$ is the product of factors corresponding to\nthe $n$ edges of the polygon $P$ and $\\mathbb{T}_d$ is the\nproduct of factors corresponding to the $n-3$ diagonals of $P$.\nNote that each diagonal of $P$ occurs in two triangles so corresponds to\ntwo edges of $\\mathcal{T}^D$. Hence each diagonal gives\nrise to a two-torus $S^1 \\times S^1$ in $\n\\mathbb{T}$ which we will\nrefer to as the two-torus corresponding to that diagonal. Define a\nsubtorus $\\mathbb{T}_d^-$ of $\\mathbb{T}_d$ of dimension $n-3$ by\ntaking an antidiagonal embedding of $S^1$ in each two-torus\ncorresponding to a diagonal.\n\n\nNow take the symplectic quotient (at level zero) of (the product\nof) the three copies of $\\mathbb{C}^2$ associated to the three sides of\neach triangle by $\\mathrm{SU}(2)$ acting diagonally. For each triangle\n(or each tripod) we\nobtain a resulting copy of $\\bigwedge^2(\\mathbb{C}^3)$.  The resulting\nproduct $(\\bigwedge^2(\\mathbb{C}^3))^{n-2}$ has an induced action of the\ntorus $\\mathbb{T}$. Glue the copies $\\bigwedge^2(\\mathbb{C} ^3)$\nassociated to the triangles together along the edges of the\ntriangles associated to diagonals by taking the symplectic quotient at level zero by the\ntorus $\\mathbb{T}_d^- \\subset \\mathbb{T}_d$ described above. Each of\nthe two previous symplectic quotients has a corresponding GIT quotient.\n Taking both GIT quotients we obtain\na space which is an affine torus quotient of affine space. Hence the combined\nsymplectic quotient is the space underlying the\naffine toric variety\n$$P_n^{\\mathcal{T}_0}(\\mathrm{SU}(2)) =(\\mathrm{\\bigwedge}\n\\space^2(\\mathbb{C}^3))^{n-2}\/\\!\/\n_0 \\mathbb{T}_d^-.$$\nLet $\\mathbf{t}_e(\\lambda)$ be the element in complexified torus $\\underline{\\mathbb{T}} \\cong (\\mathbb{C}^\\ast)^{3n-6}$ of $\\mathbb{T}$ \nsuch that all the edge\ncomponents coincide with $\\lambda \\in \\mathbb{C}^{\\ast}$ and\nall the components corresponding to\ndiagonals are $1$. Then Theorem \\ref{firstmaintheorem} follows by\nputting together items 2 and 4 in the next theorem, which is proved in\n\\S \\ref{homeomorphismVtoP}.\n\n\\begin{theorem} \\label{firstauxiliarytheorem}\n\\hfill\n\\begin{enumerate}\n\\item The toric varieties $\\mathrm{Aff} \\mathrm{Gr}_2(\\mathbb{C}^n)_0^{\\mathcal{T}}$ \nand $P_n^{\\mathcal{T}}(\\mathrm{SU}(2))$ are isomorphic as\naffine toric varieties.\n\\item The grading action of $\\lambda \\in\n\\mathbb{C}^*$ on $\\mathrm{Aff} \\mathrm{Gr}_2(\\mathbb{C}^n)_0^{\\mathcal{T}}$ corresponds to\nthe action of $\\mathbf{t}_e(\\sqrt{\\lambda}^{-1})$ on\n$P_n^{\\mathcal{T} }(\\mathrm{SU}(2))$ (this is well-defined).\nConsequently $ \\mathrm{Gr}_2(\\mathbb{C}^n)_0^{\\mathcal{T}}$\nis projectively isomorphic to the quotient of\n$P_n^{\\mathcal{T}}(\\mathrm{SU}(2))$ by this $\\mathbb{C}^{\\ast }$ action - we will\ndenote this quotient by $Q_n^{\\mathcal{T}}(\\mathrm{SU}(2))$.\n\\item There is a\nhomeomorphism (that creates imploded spin-frames along the\ndiagonals of the triangulation)\n$$\\Psi_n^{\\mathcal{T}}:V_n^{\\mathcal{T}} \\to P_n^{\\mathcal{T}}(\\mathrm{SU}(2)).$$\n\\item The homeomorphism\n$\\Psi_n^{\\mathcal{T}}$ induces a\nhomeomorphism from $W_n^{\\mathcal{T}}$ to the projective toric variety\n$Q_n^{\\mathcal{T}}(\\mathrm{SU}(2))$.\n\\end{enumerate}\n\\end{theorem}\nThe quotient\n$P_n^{\\mathcal{T}_0}(\\mathrm{SU}(2))=\\bigwedge^2(\\mathbb{C}^3)^{n-2} \/\\!\/ \\mathbb{T}_d^-$\nadmits a residual\naction by the quotient torus $\\mathbb{T}\/ \\mathbb{T}_d^-$. This\nquotient torus contains a factor that can be identified with\n$\\mathbb{T}_e$. The toric fiber $(M_{\\mathbf{r}})_0$ is obtained from\n$P_n^{\\mathcal{T}_0}(\\mathrm{SU}(2))$ by taking the symplectic quotient by\n$\\mathbb{T}_e$ at level $\\mathbf{r}$. Theorem \\ref{secondmaintheorem}\nfollows from the two statements of the following theorem.\n\n\\begin{theorem}\\label{secondauxiliarytheorem}\\hfill\n\\begin{enumerate}\n\\item The toric variety $P_n^{\\mathcal{T}}(\\mathrm{SU}(2))\/\\!\/_{\\mathbf{r}} \\mathbb{T}_e $\nis isomorphic to the toric variety\n$(M_{\\mathbf{r}})_0^{\\mathcal{T}}$.\n\\item For each $\\mathbf{r}$ the homeomorphism $\\Psi_n^{\\mathcal{T}}$ induces a homeomorphism\n$\\Psi_{\\mathbf{r}}:V^{\\mathcal{T}}_{\\mathbf{r}} \\to P_n^{\\mathcal{T}}(\\mathrm{SU}(2))\/\\!\/_{\\mathbf{r}} \\mathbb{T}_e .$\n\\end{enumerate}\\end{theorem}\n\n\nNote that the toric varieties\n$(\\bigwedge^2(\\mathbb{C}^3))^{n-2} \/\\!\/ \\mathbb{T}_d^-$\nresp. $\\bigwedge^2(\\mathbb{C}^3)^{n-2}\n\/\\!\/_{\\mathbf{r},0} (\\mathbb{T}_e \\times\\mathbb{T}_d^-)$ mediate between the\nKamiyama-Yoshida spaces $V_n^{\\mathcal{T}}$ resp. $V_{\n\\mathbf{r} }^{\\mathcal{T}}$ and the toric\nvarieties $\\mathrm{Gr}_2(\\mathbb{C}^n)_0^{\\mathcal{T}}$ resp. $(M_{ \\mathbf{r}})_0^{\\mathcal{T}}$.\n\n\\subsection{The Hamiltonian nature of the edge rotations and the bending flows}\nIt remains to place the edge rotations and bending flows in their proper context (in\nterms of  symplectic and algebraic geometry). We do this with the following theorems. \nWe note that since the spaces\n$P_n^{\\mathcal{T}}(\\mathrm{SU}(2))$ and $P_{\\mathbf{r}}^{\\mathcal{T}}(\\mathrm{SU}(2))$ are quotients of\nthe affine space $(\\bigwedge^2(\\mathbb{C}^3))^{n-2}$ by tori they inherit symplectic\nstratifications from the orbit type stratification of\n$(\\bigwedge^2(\\mathbb{C}^3))^{n-2}$ according to \\cite{SjamaarLerman}.  \nWe identify the edge flows and bending flows with the action of\nthe maximal compact subgroups (compact torus) of the complex tori\nthat act holomorphically with open orbits on the toric\nvarieties $\\mathrm{Gr}_2(\\mathbb{C}^n)_0^{\\mathcal{T}}$ and\n$(M_{ \\mathbf{r}})_0^{\\mathcal{T}}$. This is accomplished\nby the following theorem, which is proved\nby Theorems \\ref{noncompact} and \\ref{noncompact2}.\n\n\\begin{theorem}\n\\hfill\n\\begin{enumerate}\n\\item The action of the edge rotations corresponds under\n$\\Psi_n^{\\mathcal{T}}$ to the residual action of $\\mathbb{T} \/ \\mathbb{T}_d^-$.\n\\item The action of the bending flows corresponds under\n$\\Psi_n^{\\mathcal{T}}$ (resp. $\\Psi_{\\mathbf{r}}^{\\mathcal{T}}$) to the residual action\nof $\\mathbb{T} \/ \\mathbb{T}_e \\times \\mathbb{T}_d^-$. \n\\end{enumerate}\n\n\\end{theorem}\nWe may now give the edge rotations and the \nbending flows on $V_n^{\\mathcal{T}}$ and $V_{\\mathbf{r}}^{\\mathcal{T}}$ \na natural Hamiltonian interpretation.  This is\nproved by the last theorem along with Proposition \\ref{pullback}\nand Theorem \\ref{diagonalpullback}.\n\n\\begin{theorem}\\label{stratification}\n\\hfill\n\\begin{enumerate}\n\\item The edge rotations on $V_n^{\\mathcal{T}}$ are the stratified\nsymplectic Hamiltonian flows (in the sense of \\cite{SjamaarLerman})\nassociated to the lengths of edges in $\\mathcal{T}$-congruence\nclasses of spin-framed polygons. \n\\item The bending flows on $V_n^{\\mathcal{T}}$ (resp. $V_{\\mathbf{r}}^{\\mathcal{T}}$) are the\nstratified Hamiltonian flows associated to the lengths of the diagonals \nof $\\mathcal{T}$-congruence classes of spin-framed polygons (resp. polygonal linkages).  \n\\end{enumerate}\n\n\\end{theorem}\n\n\n\n\n\\medskip\n\n{\\bf Acknowledgements.} \nWe thank Bill Goldman, Henry King and Reyer Sjamaar for useful\nconversations. The  authors would also like to thank\nAndrew Snowden and Ravi Vakil, a number of ideas from the collaboration\n \\cite{HowardMillsonSnowdenVakil} have reappeared in this paper.\n We thank Philip Foth and Yi Hu for posing the\nproblem solved in this paper. They first observed that the toric\ndegenerations of flag varieties constructed by \\cite{AlexeevBrion}\ncould be descended to the associated weight varieties. We\nespecially thank Philip Foth for pointing out that the\nconstruction of \\cite{AlexeevBrion} gives one toric degeneration\nof $M_{\\mathbf{r}}$ for each triangulation of the model convex\n$n$-gon.\nThis led us to consider triangulations other than the standard\none. We thank Bernd Sturmfels for telling us\nabout \\cite{SpeyerSturmfels} and \\cite{BuczynskaWisniewski} which\nled us to understand from the point of view of tree metrics why\nthere was one toric degeneration for each triangulation.\nWe thank Allen Knutson for pointing out that  toric degenerations\nof Grassmannians were first constructed by Sturmfels in\n\\cite{Sturmfels}. Finally we should emphasize that\nthe notion of $\\mathcal{T}$-congruence\nis based on the work of Kamiyama and Yoshida \\cite{KamiyamaYoshida}\nand that the notion of bending flows is based on \\cite{KapovichMillson}\nand \\cite{Klyachko}.\n\n\n\n\n\n\n\\section{The moduli spaces  of $n$-gons and $n$-gon linkages in $\\mathbb{R}^3$}\nThroughout this paper the term $n$-gon will mean a closed $n$-gon\nin $\\mathbb{R}^3$ modulo translations. More precisely an $n$-gon $\\mathbf{e}$\nwill be an $n$-tuple $\\mathbf{e} = (e_1,e_2,\\cdots,e_n)$ of vectors\nin $\\mathbb{R}^3$ satisfying the closing condition\n$$e_1 + e_2 + \\cdots + e_n =0.$$\nWe will say the $e_i$ is the $i$-th edge of $\\mathbf{e}$.\nWe will say two $n$-gons $\\mathbf{e}$ and $\\mathbf{e}^{\\prime}$ are congruent\nif there exists a rotation $ g \\in \\mathrm{SO}(3,\\mathbb{R})$ such that\n$$ e_i^{\\prime} = g e_i , 1 \\leq i \\leq n.$$\nWe will let ${\\mathrm{Pol}_ n}$ denote the space of closed $n$-gons in $\\mathbb{R}^3$\nand $\\overline{\\mathrm{Pol}}_ n$ denote the quotient space of $n$-gons modulo congruence.\n\nNow let $\\mathbf{r} = (r_1,r_2,\\cdots,r_n)$ be an $n$-tuple of nonegative\nreal numbers. We will say an $n$-gon $\\mathbf{e}$ is an $n$-gon { \\em linkage}\nwith side-lengths $\\mathbf{r}$ if the $i$-th edge of $\\mathbf{e}$ has length $r_i,\n1 \\leq i \\leq n$. We will say an $n$-gon or $n$-gon linkage is\n{\\it degenerate} if it is contained in a line.\n\n\n\nWe define the configuration space $\\widetilde{M_{\\mathbf{r}}}$ to be\nthe set of $n$--gon linkages with side-lengths $\\mathbf{r}$.  We will define the moduli space $M_{\\mathbf{r}}$ of $n$-gon linkages to be the quotient of\nthe configuration space by $\\mathrm{SO}(3,\\mathbb{R})$. The space $M_{\\mathbf{r}}$ is a complex analytic space\n(see \\cite{KapovichMillson} with isolated singularities at the degenerate $n$-gon linkages.\n\nRecall that we have defined a reference convex planar $n$-gon $P$.\nLet $u_i, u_j$ be an ordered pair of nonconsecutive vertices of $P$. For any $n$-gon $\\mathbf{e}\n\\in \\mathbb{R}^3$ we have corresponding vertices $v_i$ and $v_j$ defined up to\nsimultaneous translation.  The vector in $\\mathbb{R}^3$ pointing from $v_i$ to $v_j$\nwill be called a {\\em diagonal} of $\\mathbf{e}$.  We let $d_{ij}(\\mathbf{e})$\nbe the length of this diagonal.\nIn  \\cite{KapovichMillson} and \\cite{Klyachko} the authors described\nthe Hamiltonian flow corresponding to $d_{ij}(\\mathbf{e})$ - see the Introduction.\nIn \\cite{KapovichMillson} these flows were called {\\em bending flows}.\nFurthermore it was proved in \\cite{KapovichMillson} and \\cite{Klyachko} if two such diagonals do not\nintersect then the corresponding bending flows commute.\nSince each triangulation $\\mathcal{T}$ of $P$ contains $n-3 = \\frac{1}{2} \\dim (M_{\\mathbf{r}})$\nnonintersecting diagonals it follows that each one has an integrable\nsystem on $M_{\\mathbf{r}}$ for each such $\\mathcal{T}$.  Unfortunately these flows\nare not everywhere defined. The bending flow\ncorresponding to $d_{ij}$ is not well-defined for those $\\mathbf{e}$ where $d_{ij}(\\mathbf{e})$ is zero and the Hamiltonian $d_{ij}$ is not differentiable\nat such $\\mathbf{e}$.\n\n\n\n\n\n\n\n\n\\section{The space of imploded spin-framed Euclidean $n$--gons and the\nGrassmannian of two planes in complex $n$ space} \\label{HJconstruction}\n\nIn this section we will construct the space $P_n(\\mathrm{SU}(2))$ of imploded spin-framed $n$--gons in $\\mathbb{R}^3$ modulo $\\mathrm{SU}(2)$ and prove that this space is\nisomorphic\nto $\\mathrm{Aff} \\mathrm{Gr}_2(\\mathbb{C}^n)$ as a symplectic manifold\nand as a complex projective variety. We will first\nconstruct $P_n(\\mathrm{SU}(2))$ using the extension and implosion technique\nof \\cite{HurtubiseJeffrey} without reference to $n$--gons in $\\mathbb{R}^3$,\nthen\nrelate the\nresult to $\\mathrm{Gr}_2(\\mathbb{C}^n)$. Then we will show that a point in $P_n(\\mathrm{SU}(2))$\ncan be interpreted as a Euclidean $n$-gon equipped with an\nimploded spin-frame.\n\n\\subsection{The imploded extended moduli spaces $P_n(G)$ of $n$-gons\nand $n$-gon linkages.}\nIn this subsection we will define the imploded extended moduli\nspace $P_n(G)$ of $n$-gons in $\\mathfrak{g}^*$ for a\ngeneral semisimple Lie group $G$.\nWe have included this subsection to make the connection\nwith \\cite{HurtubiseJeffrey}.\nThroughout we assume that $G$ is semisimple.\\subsubsection{The moduli\nspaces of $n$-gons and $n$-gon linkages}\nTo motivate the definition of the next subsection we briefly recall\ntwo definitions.\\begin{definition}\nAn $n$-gon in $\\mathfrak{g}^*$ is an $n$-tuple of vectors $e_i, 1 \\leq\ni \\leq n,  \\in\\mathfrak{g}^*$ satisfying the closing condition$$e_1 +\n\\cdots + e_n = 0.$$ We define the moduli space of $n$-gons to\nbe the set of all $n$-gons modulo the diagonal coadjoint action of\n$G$.\n\\end{definition}\nNow choose $n$ coadjoint orbits $\\mathcal{O}_i, 1 \\leq i \\leq\nn$.\n\n\\begin{definition}\nWe define an $n$-gon {\\em linkage} to be an $n$-gon such that $e_i \\in\n\\mathcal{O}_i, 1 \\leq i \\leq n$. We define the moduli space\nof $n$-gon linkages to be the set of all $n$-gon linkages modulo\nthe diagonal coadjoint action of $G$.\n\\end{definition}\nWe leave the proof of the following lemma to the reader.\n\n\\begin{lemma}\nThe moduli space of $n$-gon linkages is the symplectic quotient$$G \\backslash \\! \\backslash\n(\\prod_{i=1}^n \\mathcal{O}_i).$$\n\\end{lemma}\n\n\n\\begin{remark}\\label{moduliofconnections}\nThe moduli space of $n$-gons is in fact\na moduli space of flat connections modulo gauge\ntransformations (equivalently\na character variety).  The moduli space\nof $n$-gon linkages is a moduli space of\nflat connections with the conjugacy classes\nof holonomies fixed in advance (a relative character\nvariety). In this case the\nflat connections are over an $n$-fold punctured $2$-sphere and the\nstructure group is the cotangent bundle $T^*(G) = G \\ltimes \\mathfrak{g}^*$.\nThe holonomy around each puncture is a ``translation'', i.e. a group\nelement of the form $(1,v), v \\in \\mathfrak{g}^*$.\nTo see this connection in more detail the reader is referred to\n\\S 5 of\n\\cite{KapovichMillson}. We will not need this connection in\nwhat follows.\n\\end{remark}\n\n\\subsubsection{The extended moduli spaces of $n$-gons and $n$-gon\nlinkages}\nThe following definition is motivated by the definition of the extended\nmoduli spaces of flat connections of \\cite{Jeffrey}.\n\n\\begin{definition}\nWe define the extended moduli space $M_n(G)$ to be\nthe symplectic quotient of  $T^*(G)^n$ by the left diagonal action\nof $G$: $$M_n(G) = G \\backslash \\! \\backslash T^*(G)^n.$$\n\\end{definition}\nThe space $M_n(G)$ has a $G^n$ action coming from right multiplication\non $T^*(G)$.\nWe take each $T^*(G)$ to be identified with $G\\times\\mathfrak{g^*}$ using\nthe\nleft-invariant trivialization.\n\n\\begin{lemma}$M_n(G) = G\\backslash\\{((g_1,\\alpha_1),\n\\ldots,(g_n,\\alpha_n)): \\sum_{i=1}^n Ad_{g_i}(\\alpha_i) = 0\n\\}.$\n\\end{lemma}\n\n\\begin{proof}\nThe momentum mapping associated to the diagonal left action\non $T^*(G)^n$ (identified with $(G \\times \\mathfrak{g}^*)^n$ using the\nleft-invariant trivialization) is\n$$\\mu_L((g_1,\\alpha_1), \\ldots, (g_n,\\alpha_n)) =- \\sum_{i=1}^n\nAd_{g_i}(\\alpha_i).$$\nThe expression on the right above\nclearly corresponds to the 0-momentum level of $\\mu_L$ which\nis $M_n(G)$.\n\\end{proof}\n\n\\begin{remark}\nThe  reason for the term {\\em extended} moduli space of $n$-gons is that\nthis space is obtained\nfrom the moduli space of $n$-gons  by adding the frames $g_1,g_2,\\cdots,g_n$.\nNote that the moduli space of $n$-gons is embedded in the extended\nmoduli space as the subspace corresponding to $g_1 = g_2 = \\cdots = g_n = e$.\nIf we wish to fix the conjugacy classes of the second components we\nwill call the above the extended moduli space of $n$-gon {\\em\nlinkages}.\n\\end{remark}\n\n\\subsubsection{The imploded extended moduli spaces of $n$-gons and $n$-gon\nlinkages}\nChoose a maximal torus $T_G\\subset G$ and a (closed) Weyl chamber $\\Delta$\ncontained\nin the\nLie algebra $\\mathfrak{t}$ of $T_G$. Note that the action of $G^n$ \non $M_n(G)$ by right multiplication induces an action\nby the torus $T_G^n$ on $M_n(G)$.\nWe now obtain the imploded extended moduli space $P_n(G)$ by {\\it imploding},\nfollowing \\cite{GuilleminJeffreySjamaar}, the extended moduli space $M_n(G)$.\n\n\\begin{definition}\\label{Definitionofimplodedextended moduli space $P_n(G)$} $$P_n(G) = M_n(G)_{impl}.$$\n\\end{definition}\nFollowing \\cite{HurtubiseJeffrey} we will use $\\mathcal{E}T^*(G)$\nto denote the imploded cotangent bundle\n$$\\mathcal{E}T^*(G) = T^*(G)_{impl}.$$\nFor the benefit of the reader we will recall the definition\nof $\\mathcal{E}T^*(G)$. We have\n$$\\mathcal{E}T^*(G) = \\mu_G^{-1}(\\Delta)\/\\sim $$\nHere $\\mu_G$ is the momentum map for the action of $G$\nby right translation and we have identified $\\mathfrak{t}$\n(resp. $\\Delta$) with $\\mathfrak{t}^*$ (resp. a dual Weyl chamber) using the\nKilling form.\nThe equivalence relation $\\sim$ is described as\nfollows.\nLet $F$ be a open face of $\\Delta$ and let $h_F$ be a generic element of $F$.\nLet $G_F$ be the subgroup of $G$ which is the derived subgroup of the\nstabilizer of $h_F$ under the\nadjoint representation. We define $x$ and $y$ in $\\mu_T^{-1}(\\Delta)$\nto be equivalent if $\\mu_T(x)$ and $\\mu(y)$ lie in the same face $F$\nof $\\Delta$ and $x$ and $y$ are in the same orbit under $G_F$. Thus we\ndivide\nout the inverse images of  faces by different subgroups of $G$.\nIn what follows we will use the symbol $\\sim$ to denote\nthis equivalence relation (assuming the group $G$, the torus $T$\nand the chamber $\\Delta$ are understood).\n\nWe define the space $E_n(G)$ by\n$$E_n(G) = \\mathcal{E}T^*(G)^n.$$\nNoting that right implosion commutes\nwith left symplectic quotient we have\n$$P_n(G) = G \\backslash E_n(G) \n=  G\\backslash \\Big\\{ ([g_1,\\alpha_1], \\ldots, [g_n,\\alpha_n]) \\mid \\alpha_1,\n\\ldots, \\alpha_n \\in \\Delta,\\sum_{i=1}^n Ad_\\mathfrak{g_i}(\\alpha_i) =\n0\\Big\\}.$$\nIn the above $[g_i, \\alpha_i]$ denotes the equivalence class\nin $T^*(G)$ relative to the equivalence relation $\\sim$ above.\nWe will sometimes use $E_n(G)$ to denote the product\n$\\mathcal{E}T^*(G)^n = \\mathcal{E}T^*(G^n)$. In what follows we let\n$\\mathbf{t}(\\lambda)$ denote the element of the complexification maximal torus\n$\\underline{T}_{\\mathrm{SU}(2)^n}$.\n\n\\subsection{The isomorphism of  $Q_n(\\mathrm{SU}(2))$ and $\\mathrm{Gr}_2(\\mathbb{C}^n)$}\nIn this subsection we will prove Theorem \\ref{polygonGrass}\nby calculating $P_n(\\mathrm{SU}(2))$, the imploded extended\nmoduli space for the group $\\mathrm{SU}(2)$ and its quotient $Q_n(\\mathrm{SU}(2))$.\nWe will in fact need a slightly more precise version than that\nstated in the Introduction.\n\n\\begin{theorem}\\label{polygonGrasspreciseversion}\n\\hfill\n\\begin{enumerate}\n\\item There exists a homeomorphism\n$\\psi:\\mathrm{Aff} \\mathrm{Gr}_2(\\mathbb{C}^n) \\to P_n(\\mathrm{SU}(2))$.\n\\item The homeomorphism $\\psi$ intertwines the natural action of the maximal torus $T_{\\mathrm{U}(n)}$ of \\ $\\mathrm{U}(n)$ with  (the inverse of) that of $T_{\\mathrm{SU}(2)}^n$ acting on imploded frames.\n\\item The homeomorphism $\\psi$ intertwines the grading circle (resp. $\\mathbb{C}^{\\ast}$)\nactions on $\\mathrm{Aff} \\mathrm{Gr}_2(\\mathbb{C}^n)$ with the actions of $\\mathbf{t}((\\exp{i\\theta})^{-1\/2})$ (resp. $\\mathbf{t}((\\lambda)^{-1\/2})$)\n\\item The homeomorphism $\\psi$ induces a homeomorphism between $\\mathrm{Gr}_2(\\mathbb{C}^n)$\nand $Q_n(\\mathrm{SU}(2)).$\n\\end{enumerate}\n\n\\end{theorem}\nIn order to prove the theorem we first\nneed to compute $\\mathcal{E}T^*(\\mathrm{SU}(2))$.\n\n\\subsubsection{The imploded cotangent bundle of $\\mathrm{SU}(2)$ }\n\nIn this section we will review the formula\nof \\cite{GuilleminJeffreySjamaar} for the (right) imploded\ncotangent bundle $\\mathcal{E}T^*(\\mathrm{SU}(2))$. Let $T_{\\mathrm{SU}(2)}$ be the maximal\ntorus of $\\mathrm{SU}(2)$ consisting of the diagonal matrices.\nLet $\\mathfrak{t}$ be the Lie algebra of $T_{\\mathrm{SU}(2)}$ and let $\\Delta$ be\nthe\npositive Weyl chamber in $\\mathfrak{t}$ (so $\\Delta$ is a ray in\nthe one-dimensional vector space $\\mathfrak{t}$).\n\n\nWe will take as basis for $\\mathfrak{t}$ the coroot $\\alpha^{\\vee}$\n(multiplied by $i$), that is\n$$\\alpha^{\\vee}= \\begin{pmatrix}  i & 0 \\\\  0 & -i \\\\  \\end{pmatrix}$$\nThen $\\alpha^{\\vee}$ may be\n  identified  with a\n  basis vector over $\\mathbb{Z}$ for the cocharacter lattice $X_*(T_{\\mathrm{SU}(2)})$.\nIt is tangent at the identity to a unique cocharacter.\n   Let $\\varpi_1$ be the fundamental weight\nof $\\mathrm{SL}(2,\\mathbb{C})$ thus\n$$ \\varpi_1(\\alpha^{\\vee}) = 1.$$\nThen $\\varpi_1$ may be identifed with a character of $T$ (it\nis the derivative at the identity of a unique character) which is a basis\nfor\nthe character lattice of $T_{\\mathrm{SU}(2)}$.\n\n\n\n\\smallskip\nLet $\\mu_R$ be the momentum map for the action of $\\mathrm{SU}(2)$ on $T^*\\mathrm{SU}(2)$\ninduced by right multiplication. Specializing the\ndefinition of the imploded cotangent bundle to $G = \\mathrm{SU}(2)$ we find that\nthe (right)\nimplosion of $T^*\\mathrm{SU}(2)$ is  the set of equivalence\nclasses\n$$\\mathcal{E} T^\\ast (\\mathrm{SU}(2)) = \\mu_R^{-1}(\\Delta)\/\\sim$$\nwhere two points $x,y \\in \\mu_R^{-1}(\\Delta)$ are\nequivalent if\n$$\\mu_R(x) = \\mu_R(y) = 0 \\ \\text{and} \\ x = y \\cdot g \\ \\text{for\nsome}\\ g \\in \\mathrm{SU}(2).$$\nIt follows from the general theory of  \\cite{GuilleminJeffreySjamaar}\nthat $\\mathcal{E}T^*(\\mathrm{SU}(2))$ has an induced\nstructure of a stratified symplectic space in the sense of\n\\cite{SjamaarLerman} with induced isometric actions\nof $\\mathrm{SU}(2)$  induced by the left action of $\\mathrm{SU}(2)$ on $T^*\\mathrm{SU}(2)$\nand $T_{\\mathrm{SU}(2)}$ induced by the right action of $T_{\\mathrm{SU}(2)}$ on $T^*(\\mathrm{SU}(2))$.\nHowever the exceptional\nfeature of this special case is that $\\mathcal{E}T^*(\\mathrm{SU}(2))$ is a\n{\\em manifold}. The multiplicative group $\\mathbb{R}_+$ acts on $\\mathcal{E}T^*(\\mathrm{SU}(2))$\nby the formula\n$$\\mu \\cdot [g,\\lambda \\varpi_1] = [g, \\mu \\lambda \\varpi_1]$$\nwhereas the actions of $g_0 \\in \\mathrm{SU}(2)$ and $t \\in T_{\\mathrm{SU}(2)}$ described\nabove are given by\n$$g_0 \\cdot [g, \\lambda \\varpi_1] = [g_0 g, \\lambda\n\\varpi_1] \\ \\text{ and } t \\cdot [g, \\lambda \\varpi_1] = [gt, \\lambda\n\\varpi_1].$$\n\nThe product group $\\mathrm{SU}(2) \\times \\mathbb{R}_+$ fixes the point $[e,0]$\nand acts transitively on the complement of $[e,0]$ in $\\mathcal{E}T^*(\\mathrm{SU}(2))$.\nThe reader will verify, see Example 4.7 of \\cite{GuilleminJeffreySjamaar},\nthat the symplectic form $\\omega$ on $\\mathcal{E}T^*(\\mathrm{SU}(2))$\nis homogeneous of degree one under $\\mathbb{R}_+$.\n\n\nThe following lemma is extracted from Example 4.7\nof \\cite{GuilleminJeffreySjamaar}. Since this lemma is central to what\nfollows\nwe will go into more detail for its proof than is\nin \\cite{GuilleminJeffreySjamaar}.\n\n\\begin{lemma}\\label{implodedcotangentbundle}\nThe map $\\phi: \\mathcal{E} T^\\ast (\\mathrm{SU}(2)) \\to \\mathbb{C}^2$\ngiven by\n$$\\phi([g,\\lambda \\varpi_1]) = \\sqrt{2\\lambda} g\n\\begin{pmatrix}1\\\\0\\end{pmatrix} \\ \\text{and} \\ \\phi([e,0]) =\n\\begin{pmatrix}0\\\\0\\end{pmatrix}$$\ninduces a symplectomorphism of stratified symplectic\nspaces onto $\\mathbb{C}^2$\nwhere $\\mathbb{C}^2$ is given its standard symplectic structure.\nUnder this isomorphism the action of\n$\\mathrm{SU}(2)$ on $\\mathcal{E}T^*(\\mathrm{SU}(2))$\ngoes to the standard action on $\\mathbb{C}^2$ and\nthe action of $T$ on $\\mathcal{E}T^*(\\mathrm{SU}(2))$ goes to the action of $T$\ngiven by $t\\cdot (z,w) = (t^{-1}z, t^{-1}w)$.\n\\end{lemma}\n\n\n\\begin{proof}\nThe inverse to $\\phi$ is given (on nonzero elements of $\\mathbb{C}^2$)\nby $$\\psi((z,w)) = \\Big[g, \\frac{|z|^2 + |w|^2}{2} \\varpi_1\\Big],$$\nwhere $$g= \\frac{1}{\\sqrt{|z|^2 + |w|^2}} \\begin{pmatrix}z &\n-\\overline{w} \\\\w & \\overline{z}\\end{pmatrix}.$$\nIt follows that $\\phi$ is a homeomorphism. It remains to check\nthat it is a symplectomorphism.\nNoting that $\\phi$ is homogeneous under $\\mathbb{R}_+$ of degree one-half\nand the standard symplectic form $\\omega_{\\mathbb{C}^2}$ is homogeneous of\ndegree two it\nfollows (from the transitivity of the action of $\\mathrm{SU}(2) \\times \\mathbb{R}_+$) that\nthe  symplectic form $\\phi^{\\ast} \\omega_{\\mathbb{C}^2}$ is a constant\nmultiple of $\\omega$.\nThus is suffices to prove that the two above forms both take value $1$ on\nthe ordered\npair of tangent vectors $(\\partial\/\\partial \\lambda, \\alpha^{\\vee})$. We\nleave\nthis to the reader.\n\\end{proof}\n\n\\begin{remark}\nThe fact that we have inverted the usual action of the circle on the\ncomplex plane will have\nthe effect\nof changing the signs of the torus Hamiltonians that occur in the rest\nof this paper (from {\\it minus} one-half a sum of squares of norms of complex\nnumbers to {\\it plus} one-half of the corresponding sum).\n\\end{remark}\nTaking into account the above remark we obtain a formula for the\nmomentum map for the action of $T_{\\mathrm{SU}(2)}$ on $\\mathcal{E}T^*(\\mathrm{SU}(2))$\nwhich will be\ncritical in what follows.\nNow we have\n\\begin{lemma}\\label{momentummap}\n  The momentum map $\\mu_T$ for the action of $T_{\\mathrm{SU}(2)}$ on\n$\\mathcal{E} T^\\ast (\\mathrm{SU}(2))=\\mathbb{C}^2$\n  is given by\n  $$\\mu_T((z,w)) = (1\/2)(|z|^2 + |w|^2) \\varpi_1.$$\n  Consequently, the Hamiltonian $f_{\\alpha^{\\vee}}(z,w)$ for the\nfundamental\n  vector field on $\\mathcal{E}T^\\ast (\\mathrm{SU}(2))$ induced by $\\alpha^{\\vee}$\n  is given by\n  $$f_{\\alpha^{\\vee}}(z,w) = (1\/2)(|z|^2 + |w|^2).$$\n\\end{lemma}\n\n\\subsubsection{The computation of $P_n(\\mathrm{SU}(2))$ and $Q_n(\\mathrm{SU}(2))$}\nWe can now prove Theorem \\ref{polygonGrasspreciseversion}. Recall we have introduced\nthe abbreviation $E_n(\\mathrm{SU}(2))$ for the product $\\mathcal{E} T^\\ast (\\mathrm{SU}(2))^n$.\nWe will use $\\mathbf{E}$ to denote an element of $E_n(\\mathrm{SU}(2))$.\nIt will be convenient to regard  $\\mathbf{E}$ as a function from\nthe edges of the reference polygon $P$ into $\\mathcal{E} T^\\ast (\\mathrm{SU}(2))$.\nWe note that there is a map from $E_n(\\mathrm{SU}(2))$ to not necessarily closed\n$n$-gons in $\\mathbb{R}^3$,\nthe map that scales the first element of the frame by $\\lambda$ and forgets\nthe other two elements of the frame this sends\neach equivalence class $[g_i, \\lambda \\varpi_1]$ to $Ad^{\\ast} g_i (\\lambda_i \\varpi_1)$ for  $1 \\leq i \\leq n$.\n\n\nBy the previous lemmas we may represent an element  of $E_n(\\mathrm{SU}(2))$\nby the $2$ by $n$ matrix\n\\begin{center}$A = \\begin{pmatrix}z_1 & z_2 & \\ldots & z_{n-1} & z_n\n\\\\w_1 & w_2 & \\ldots & w_{n-1} & w_n\\end{pmatrix}$.\\end{center}\nThe left diagonal action of the group $\\mathrm{SU}(2)$ on $E_n(\\mathrm{SU}(2))$\nis then represented\nby the diagonal action of $\\mathrm{SU}(2)$ on the columns of $A$.\nWe let $[A]$ denote the orbit equivalence class of $A$ for this action.\n  We let $T_{\\mathrm{U}(n)}$ be the compact $n$ torus\nof diagonal elements in\nin $\\mathrm{U}(n))$. Then $T_{\\mathrm{U}(n)}$ acts by scaling the columns of the matrices $A$.\nThis action coincides with the inverse of the action of the $n$-torus $T_{\\mathrm{SU}(2)^n}$\ncoming from the theory of the imploded cotangent bundle. By this we mean\nthat if an element in $T_{\\mathrm{U}(n)}$ acts by scaling a column by $\\lambda$\nthen the corresponding element acts by scaling the same column by\n$\\lambda^{-1}$.\n\nThe first statement of\nTheorem \\ref{polygonGrasspreciseversion} will be a consequence of the following two lemmas.\nLet $M_{2,n}(\\mathbb{C})$ be the space of $2$ by $n$ complex\nmatrices $A$ as above. In what follows we will let $Z$ denote the first\nrow\nand $W$ the second row of the above matrix $A$. We give $M_{2,n}(\\mathbb{C})$\nthe Hermitian structure given by\n$$( A,B) = Tr (AB^*).$$\nWe will use $\\mu_G$ to denote the momentum map\nfor the action of $\\mathrm{SU}(2)$ on the left of $A$.\n\n\n\\begin{lemma}\\label{levelzero}$$\\mu_G(A) = (1\/2)\\begin{pmatrix}\n(\\|Z\\|^2 - \\|W\\|^2)\/2 & Z \\cdot W \\\\W \\cdot Z & (\\|W\\|^2 -\n\\|Z\\|^2)\/2\\end{pmatrix}$$\nHence $\\mu_G(A) = 0$ $\\iff$ the two rows of $A$ have the same length\nand are orthogonal.\n\\end{lemma}\n\n\\begin{proof}\nFirst note that the momentum map $\\mu$ for the action of $\\mathrm{SU}(2)$\non $\\mathbb{C}^2$ is given by$$\\mu(z,w) = (1\/2)\\begin{pmatrix} (|z|^2 -\n|w|^2)\/2 & z\\overline{w} \\\\w\\overline{z}  & (|w|^2 -\n|z|^2)\/2\\end{pmatrix}.$$\nNow add the momentum maps for each of the columns to get the lemma.\n\\end{proof}\nWe find that $$ \\mathrm{SU}(2) \\backslash \\! \\backslash M_{2,n}(\\mathbb{C}) = \\mathrm{SU}(2) \\backslash \\{ (Z,W):\n\\|Z\\|^2 = \\|W\\|^2, Z \\cdot W = 0 \\}.$$\n\n\\begin{lemma}\\label{affinegrass}\n $$\\mathrm{SU}(2) \\backslash \\! \\backslash M_{n,2}(\\mathbb{C}) \\cong \\mathrm{Aff} \\mathrm{Gr}_2(\\mathbb{C}^n).$$\n\\end{lemma}\n\n\\begin{proof}\nThe map $\\varphi:\\mu^{-1}(0) \\to \\bigwedge^2(\\mathbb{C}^n)$ given by $F(A) = Z\\wedge\nW$  maps onto the decomposable vectors, descends to the quotient\nby $\\mathrm{SU}(2)$, and induces the required isomorphism.\n\\end{proof}\n\nIn what follows it will be important to understand the above result\nin terms of the GIT quotient of $M_{2,n}(\\mathbb{C})$ by $\\mathrm{SL}(2,\\mathbb{C})$\nacting on the\nleft. Since we are taking a quotient of affine space\nby a reductive group  the Geometric Invariant Theory\nquotient of $M_{2,n}(\\mathbb{C})$ by $\\mathrm{SL}(2, \\mathbb{C})$\ncoincides with the symplectic quotient by $\\mathrm{SU}(2)$ and consequently is\n$\\mathrm{Aff} \\mathrm{Gr}_2(\\mathbb{C}^n)$, \\cite{KempfNess}, see also \\cite{Schwarz}, Theorem\n4.2. Note that since $\\mathrm{SU}(2)$ is simple the question of normalizing the\nmomentum map does not arise.  It will be\nimportant to\nunderstand this in terms of the\nsubring $\\mathbb{C}[M_{2,n}(\\mathbb{C})]^{\\mathrm{SL}(2, \\mathbb{C})}$ of\ninvariant\npolynomials\non $M_{2,n}(\\mathbb{C})$. In what follows let $Z_{ij} = Z_{ij}(A)$ or $[i,j]$ be the\ndeterminant\nof the $2$ by $2$ submatrix of $A$ given by taking columns $i$ and $j$ of $A$\n( the $ij$-th Pl\\\"ucker coordinate or bracket). The following result on one of\nthe basic results in invariant theory, see \\cite{Dolgachev},Chapter 2.\n\n\\begin{proposition}\\label{PnGGIT}\n$\\mathbb{C}[M_{2,n}(\\mathbb{C})]^{\\mathrm{SL}(2,\n\\mathbb{C})}$ is generated by\nthe $Z_{ij}$ subject to the Pl\\\"ucker relations.\n\\end{proposition}\nBut the above ring is the homogeneous coordinate ring of $\\mathrm{Gr}_2(\\mathbb{C}^n)$.\nThis gives the required invariant-theoretic proof of Theorem \\ref{polygonGrass}.\n\n\\begin{remark}\nIn fact we could do {\\em all} of the previous analysis in terms\nof invariant theory by using Example 6.12 of\n\\cite{GuilleminJeffreySjamaar} to replace the\nimploded cotangent bundle $\\mathcal{E}T^{\\ast}\\mathrm{SU}(2)\\cong \\mathbb{C}^2$ by the\nquotient $\\mathrm{SL}(2,\\mathbb{C})\/\\!\/ N \\cong \\mathbb{C}^2$ where $N$ is the subgroup of $\\mathrm{SL}(2,\\mathbb{C})$ of\nstrictly\nupper-triangular matrices. We leave the details to the reader.\n\\end{remark}\nWe have now proved the first statement in Theorem \\ref{polygonGrasspreciseversion}\nfor both symplectic and GIT quotients. We note the consequence\n(since every element of $\\bigwedge^2(\\mathbb{C}^3)$ is decomposable)\n\n\\begin{equation}\\label{P3}\nP_3(\\mathrm{SU}(2)) \\cong \\mathrm{\\bigwedge} \\space^2(\\mathbb{C}^3).\n\\end{equation}\nIt is clear that $\\psi$ is equivariant as claimed and the second statement\nfollows.\n\nSince the grading action of $\\lambda \\in \\mathbb{C}^{\\ast}$ on \n$\\mathrm{Aff}\\mathrm{Gr}_2(\\mathbb{C}^n)$ is the action that\nscales each Pl\\\"ucker coordinate by $\\lambda$ the third statement is also clear.\nIt remains to prove the fourth statement of Theorem \\ref{polygonGrasspreciseversion}.\nFirst recall  the standard $\\mathrm{U}(n)$ invariant positive definite\nHermitian form $( \\ , \\ )$ on $\\bigwedge^2(\\mathbb{C}^n)$ is given on the\nbivector $a \\wedge b$ by\n$$ (a\\wedge b, a \\wedge b ) = det \\begin{pmatrix} (a,a) & (a,b) \\\\\n(b,a) & (b,b)\n\\end{pmatrix}.$$\nWe can obtain $\\mathrm{Gr}_2(\\mathbb{C}^n)$ as a symplectic manifold  the cone $\\mathrm{Aff} \\mathrm{Gr}_2(\\mathbb{C}^n)$ of decomposable\nbivectors by taking the subset of decomposable bivectors of norm squared $R$\nthen dividing by the action of the circle by scalar multiplication\n(symplectic quotient by $S^1$ of level $\\frac{R}{2}$). Thus we can find\na manifold diffeomorphic to $\\mathrm{Gr}_2(\\mathbb{C}^n)$ by computing the pull-backs of\nthe function $f(a\\wedge b) = (a \\wedge b, a \\wedge b)$ and scalar multiplication\nof bivectors by the circle under the map $\\varphi$. We represent elements of $P_n(\\mathrm{SU}(2))$\nby left $\\mathrm{SU}(2)$ orbit-equivalence classes $[A]$ of $2$ by $n$ matrices $A$ such that the rows\n$Z$ and $W$ of $A$ are orthogonal and of the same length.\n\n\\begin{lemma}\n$$\\varphi^{\\ast} f (A) = \\Bigg(\\frac{||Z||^2 + ||W||^2}{2}\\Bigg)^2.$$\n\\end{lemma}\n\n\\begin{proof}\nWe have\n$$\\varphi^{\\ast} f (A) = det \\begin{pmatrix} (Z,Z) & (Z,W) \\\\\n(W,Z) & (W,W)\n\\end{pmatrix}.$$\nBut $(Z,Z) = (W,W)$ and $(Z,W) = (W,Z) = 0$.\nHence\n$$\\varphi^{\\ast} f (A) = ||Z||^2 ||W||^2 = \\Bigg(\\frac{||Z||^2 + ||W||^2}{2}\\Bigg)\\Bigg(\\frac{||Z||^2 + ||W||^2}{2}\\Bigg).$$\n\\end{proof}\nNow we ask the reader to look ahead and see that in subsection \\ref{frommatricestopolygons}\nwe define a map $F_n$ from $P_n(\\mathrm{SU}(2))$ to the space of $n$-gons in $\\mathbb{R}^3$.\nFurthermore $F_n([A])$ is the $n$-gon underlying the imploded spin-framed $n$-gon\nrepresented by $[A]$. Accordingly we obtain the following corollary of the\nabove lemma.\n\n\\begin{corollary}\n$\\varphi^{\\ast} f (A)$ is four times the square of the perimeter of the $n$-gon\nin $\\mathbb{R}^3$ underlying the imploded spin-framed $n$-gon represented by $[A]$.\n\\end{corollary}\nNow  observe that scaling the bivector $Z \\wedge W$ by $\\exp{i\\theta}$\ncorresponds to scaling the class of the matrix $[A]$ by the square-root of $\\exp{i\\theta}$,\nNote that the choice square-root is irrelevant since\n$$[-A] = [A].$$\nBut scaling the columns of $A$ corresponds to rotating the corresponding imploded\nspin-frames.\nThus fixing the perimeter of the underlying $n$-gon to be one corresponds to\ntaking the symplectic quotient of the cone of decomposable bivectors by $S^1$ at level $2$.\nand we have now completed the proof of the second part of Theorem \\ref{polygonGrass}.\n\n\n\n\n\\subsection{The momentum polytope for the $T_{\\mathrm{U}(n)}=T_{\\mathrm{SU}(2)^n}$ action on $P_n(\\mathrm{SU}(2))$}\nWe have identified the momentum map for the action of $\\mathrm{SU}(2)$\non $M_{2,n}(\\mathbb{C})$. The torus $T_{\\mathrm{U}(n)}$ acts on $M_{2,n}(\\mathbb{C})$ by scaling\nthe\ncolumns.\nWe leave the proof of the following lemma to the reader.\nLet $\\mu_{T_n}$ be the momentum map for the above action of $T_{\\mathrm{U}(n)}$.\n\n\\begin{lemma}$$\\mu_{T_n}(A) = \\Bigg(\\frac{||C_1||^2}{2},\\cdots ,\n\\frac{||C_n||^2}{2}\\Bigg).$$\n\\end{lemma}\nThe action of $T_{\\mathrm{U}(n)}$ descends to the quotient $P_n(\\mathrm{SU}(2))$ with the\nsame momentum map (only now $A$ must satisfy $\\mu_G(A) =0$). Thus we have\na formula for the momentum map for the action of $T_{\\mathrm{U}(n)}$ on the\nGrassmannian\nin terms of special representative matrices $A$ (of $\\mathrm{SU}(2)$-momentum\nlevel zero). It will be important to extend this formula to\nall $A \\in M_{2,n}(\\mathbb{C})$ of rank two. We do this by relating the\nabove  formula to\nthe usual formula for momentum map of the action of $T_{\\mathrm{U}(n)}$ on a general\ntwo\nplane in $\\mathbb{C}^n$ represented by a general rank 2 matrix $A \\in M_{2,n}(\\mathbb{C})$.\nThis momentum map\nis described in Proposition 2.1 of \\cite{GGMS}.\nLet  $A$ be a $2$ by $n$ matrix of rank two and $[A]$\ndenote the two plane in $\\mathbb{C}^n$ spanned by its columns.\nThen the $i$-th component of the momentum map of $T_{\\mathrm{U}(n)}$ is given\nby$$\\mu_i([A]) = \\frac{1}{\\|A\\|^2}\\sum_{j,j \\neq i} \\|Z_{ij}(A)\n\\|^2.$$\nHere $\\|A\\|^2$ denotes the sum of the squares of the norms of the\nPl\\\"ucker coordinates of $[A]$.\nThe required extension formula is them implied by\n\n\\begin{lemma}\\label{equalityofHamiltonians}\nSuppose that $\\mu_G(A) = 0$. Let $C_i$ be the $i$-th column of $A$.\nThen\n\n$$\\frac{\\|C_i\\|^2 }{2}= \\frac{1}{\\|A\\|^2}\\sum_{j,j\\neq i}\n\\|Z_{ij}(A) \\|^2.$$\n\n\\end{lemma}\n\n\\begin{proof}\nIt suffices to prove the formula in the special case that $i=1$.\nIf the first row of $A$ is zero then both sides of the equation are zero.\nSo assume the first row is not zero. Apply $g$ to $A$ so that the first\nrow of $Ag$ is of the form $(r,0)$ where $r = \\|R_1\\|$. Let $z_i', w_i'$\nbe the $i$-th row of $Ag$. Note\nthat $\\mu_G(Ag)$ is still zero, hence the length of the second column of $Ag$\nis equal to $(1\/2)\\|Ag\\|^2 = (1\/2)\\|A\\|^2$.  Also $Z_{ij}(Ag) =\nZ_{ij}(A)$. Now compute\nthe left-hand side of the equation.\nWe have $ Z_{1j}(A) = Z_{1j}(Ag) = r w_j', 2 \\leq j \\leq n$ and hence (noting\nthat $w_1' =0$) we have$$\\sum_{j,j >1} \\|Z_{1j}(A) \\|^2 = r^2\n\\sum_{j=1}^n |w_j'|^2.$$\nNote that the sum on the right-hand side is the length squared of the\nsecond column of $Ag$. The lemma follows.\n\\end{proof}\nLet $D_n \\subset \\Delta^n$ be the cone defined by$$D_n = \\{ \\mathbf{r}\n\\in \\Delta^n : 2 r_i \\leq \\sum_{j=1}^n r_j, 1 \\leq i \\leq n\\}.$$\nThen $D_n$ is the cone on the hypersimplex $\\Delta^2_{n-2}$ of\n\\cite{GGMS} \\S 2.2.\nNote that $D_3$ is the set of nonnegative real numbers satisfying the\nusual\ntriangle inequalities.\nWe can now apply Lemma \\ref{equalityofHamiltonians} and the results of\n\\S 2.2 of \\cite{GGMS} to\ndeduce\n\n\\begin{proposition}\\label{triangleinequalities}$$\\mu_{T_n}(P_n(\\mathrm{SU}(2))\n= D_n.$$\n\\end{proposition}\nWe give another proof of this theorem in terms of the side-lengths\ninequalities\nfor Euclidean $n$-gons in the next subsection.\n\n\n\\subsection{The relation between points of $P_n(\\mathrm{SU}(2))$ and\nframed Euclidean $n$--gons}\\label{frommatricestopolygons}\n\nThe point of this subsection is to show that a point of $P_n(\\mathrm{SU}(2))$\nmay be interpreted as a Euclidean $n$--gon equipped with an imploded\nspin-frame. The critical role in this interpretation is played by\na map $F:\\mathbb{C}^2 \\to \\mathbb{R}^3$.  The map $F$ is  the Hopf map.\n\n\\subsubsection{The equivariant map $F$ from $\\mathcal{E} T^\\ast (\\mathrm{SU}(2))$ to\n$\\mathbb{R}^3$}\\label{maptoR3}\n\nIn this section we define a map $F$ from\nthe imploded cotangent bundle $\\mathcal{E} T^\\ast (\\mathrm{SU}(2)) \\cong \\mathbb{C}^2$\nto $\\mathbb{R}^3$. This map will give the critical connection between\nthe previous constructions and polygonal linkages in $\\mathbb{R}^3$.\n\nWe first define $\\pi: \\mathcal{E} T^\\ast (\\mathrm{SU}(2)) \\to \\mathfrak{su}^\\ast(2)$\nby$$\\pi([g, \\lambda \\varpi_1]) = Ad^\\ast(g)(\\lambda \\varpi_1),$$\nwhere $\\varpi_1$\ndenotes the fundamental weight of $\\mathrm{SU}(2)$.\nWe leave the following lemma to the reader.\n\n\\begin{lemma}\\label{circlebundle}\n$\\pi$ factors through the action of\n$T_{\\mathrm{SU}(2)}$. The map $\\pi$ restricted\nto\nthe complement of the point $[e,0]$ is a principal $T_{\\mathrm{SU}(2)}$ bundle\n(the Hopf bundle).\n\\end{lemma}\nDefine the normal frame bundle of $\\mathbb{R}^3-{0}$ to be the set\nof triples $\\mathbf{F} = (v,u_1,u_2)$ where $v$ is a nonzero vector in $\\mathbb{R}^3$\nand the pair $u_1,u_2$\nis a properly oriented frame for the normal plane to $v$. The projection\nto $v$ is a principal circle bundle. The normal frame\nbundle is homotopy equivalent to $\\mathrm{SO}(3,\\mathbb{R})$ and admits a unique nontrivial\ntwo-fold cover which again is a principal circle bundle over $\\mathbb{R}^3 -0$\nwhich we call the bundle of normal spin-frames for $\\mathbb{R}^3 -0$.\n\nIn order to prove Proposition \\ref{framedngons} below we will need the\nfollowing lemma. We leave its proof to the reader. In what follows we\nidentify the coadjoint action of $\\mathrm{SU}(2)$ with its action (through its\nquotient $\\mathrm{SO}(3,\\mathbb{R})$) on $\\mathbb{R}^3$.\n\n\\begin{lemma}\\label{framedvectors}\\hfill\n\\begin{enumerate}\\item  The map\n$\\pi$ restricted to the subset of $\\mathcal{E}T^\\ast (\\mathrm{SU}(2))$ with\n$\\lambda \\neq 0$ is the spin-frame bundle of $\\mathbb{R}^3 -0$. \\item The\naction of $T_{\\mathrm{SU}(2)}$ on $\\mathcal{E}T^\\ast (\\mathrm{SU}(2))$ preserves the\nset where $\\lambda \\neq 0$ and induces the circle\naction on the normal spin-frames.\n\\end{enumerate}\n\n\\end{lemma}\nWe have that $\\mathcal{E}T^\\ast (\\mathrm{SU}(2))$ is symplectomorphic to $\\mathbb{C}^2$\nand $\\mathfrak{su}^\\ast(2)$ is isomorphic as a Lie algebra to $\\mathbb{R}^3$,\nand so\nthe map $\\pi$ induces a map from $\\mathbb{C}^2$ to $\\mathbb{R}^3$.  The induced map\nis the map $F$. We now give details.\n\n\n\nIdentify $\\mathfrak{su}^\\ast(2)$ with the traceless Hermitian matrices\n$\\mathcal{H}_2^0$\nvia the pairing $\\mathcal{H}_2^0 \\times \\mathfrak{su}(2) \\to \\mathbb{R}$ given\nby\n$$\\langle X, Y \\rangle = \\Im (\\mathrm{tr}(XY)).$$\nIn this\nidentification\nwe have\n$$\\varpi_1 = \\begin{pmatrix}\\frac{1}{2} & 0 \\\\0 & -\\frac{1}{2}\n\\\\\\end{pmatrix}.$$\nWe recall there is a Lie algebra isomomorphism $g$ from $\\mathbb{R}^3$ to\nthe traceless skew-Hermitian matrices given by  given\nby\n$$g(x_1,x_2,x_3) = \\frac{1}{2}\\begin{pmatrix}ix_1 & ix_2 - x_3\n\\\\ix_2 + x_3 & -ix_1\\end{pmatrix}.$$\nThe isomorphism $f$ dual to $g$ from\nthe traceless Hermitian matrices $\\mathcal{H}_2^0$ to $\\mathbb{R}^3$ is given\nby\n$$ f\\begin{pmatrix}x_1 & x_2 + \\sqrt{-1} x_3 \\\\x_2 - \\sqrt{-1}x_3 &\n-x_1 \\\\\\end{pmatrix} = (x_1,x_2,x_3).$$\nNote that in particular, $f(\\varpi_1) = (1\/2,0,0) $.\n\n\nWe  now define the map $F$.\nDefine $h:\\mathbb{C}^2 \\to \\mathcal{H}_2$ as follows. Recall\n$$\\mu_{\\mathrm{U}(2)}(z,w)\n= 1\/2 \\begin{pmatrix}z \\bar{z} & w \\bar{z} \\\\z \\bar{w} & w \\bar{w}\n\\\\\\end{pmatrix}.$$\nThen define $h(z,w) = \\mu_{\\mathrm{U}(2)}(z,w)^0$ where the superscript zero\ndenotes traceless\nprojection. Hence$$ h(z,w) = (1\/4)\\begin{pmatrix}z\\bar{z} - w \\bar{w}\n& 2w\\bar{z} \\\\  2z \\bar{w} & w\\bar{w} - z \\bar{z} \\\\\\end{pmatrix}.$$\nWe define $F$ by$$F = f \\circ h.$$\n\n\\begin{remark}\nThe map $h$ is the momentum map for the action of $\\mathrm{SU}(2)$ on $\\mathbb{C}^2$\nafter we identify $\\mathcal{H}_2^0$ with the dual of the Lie algebra\n  of $\\mathrm{SU}(2)$ using the imaginary part of the trace. Accordingly\n  we will use the notations $h$ and $\\mu_{\\mathrm{SU}(2)}$ interchangeably.\n\\end{remark}\nIn the next lemma we note an important equivariance property of $F$.\nLet $\\rho:\\mathrm{SU}(2) \\to \\mathrm{SO}(3,\\mathbb{R})$ be the double cover. We leave its\nproof to the reader.\n\n\\begin{lemma}\\label{equivariance}\nLet $g \\in \\mathrm{SU}(2)$. Then$$F \\circ g = \\rho(g) \\circ F.$$\n\\end{lemma}\nWe next have\n\n\\begin{lemma} We have a commutative diagram\n\\[ \\begin{CD}\\mathcal{E} T^\\ast (\\mathrm{SU}(2)) @>\\pi>> \\mathfrak{su}(2)^{\\ast}\n\\\\@V\\phi VV  @VVfV \\\\\\mathbb{C}^2 @>F>> \\mathbb{R}^3\\end{CD}\\]\n\\end{lemma}\n\n\\begin{proof}\nFirst observe that $f\\circ \\pi$ and $F\\circ \\phi$ are homogeneous of\ndegree one with respect to the $\\mathbb{R}_+$ actions on their domain and range\n(note that $\\phi$ is homogeneous of degree $1\/2$ and $F$ is homogeneous\nof degree two). Also both intertwine the $\\mathrm{SU}(2)$ action on $\\mathcal{E}\nT^\\ast (\\mathrm{SU}(2))$\nwith its action through the double cover on $\\mathbb{R}^3$. Since the action\nof $\\mathrm{SU}(2) \\times \\mathbb{R}_+$ on $\\mathcal{E} T^\\ast (\\mathrm{SU}(2))$ is transitive away\nfrom $[e,0]$ it suffices to prove the two above maps coincide at the\npoint $[e,\\varpi_1]$. But it is immediate that both maps take the\nvalue $(1\/2,0,0)$ at this point.\n\\end{proof}\n\n\\begin{remark}\\label{SpinF}\nWe will need the following calculation.\n\n$$F \\circ \\phi([g, \\lambda\\varpi_1]) = \\lambda f( Ad_g^*(\\varpi_1))$$\nIn particular this implies that $\\|F \\circ \\phi([g, \\lambda\\varpi_1])\\| = \\frac{1}{2}\\lambda$.\n\\end{remark}\nThe following lemma is a direct calculation.\n\n\\begin{lemma}\\label{factoroftwo}\nThe formula for $F: \\mathbb{C}^2 \\to \\mathbb{R}^3$ in the usual coordinates is$$F(z,\nw) = \\frac{1}{4}\\Big(z \\bar{z} - w \\bar{w}, \\; 2\\Re(w \\bar{z}), \\;\n2\\Im(w\\bar{z})\\Big).$$\nConsequently, the Euclidean length of the vector $F(z,w) \\in \\mathbb{R}^3$ is\ngiven by$$ \\|F(z,w)\\| = \\frac{1}{4}(|z|^2 + |w|^2) .$$\n\\end{lemma}\n\n\\begin{corollary}\\label{SpinLength}\nNote that the length of $F$ is related to the Hamiltonians\n$f_{\\alpha^{\\vee}}(z,w)$\nfor the infinitesimal action of $\\mathfrak{t}$ by$$||F(z,w)|| = (1\/2)\nf_{\\alpha^{\\vee}}(z,w) .$$\nAlso, by the above remark if $\\phi([g, \\lambda\\varpi_1]) = (z, w)$ then\n$$\\lambda = f_{\\alpha^{\\vee}}(z,w).$$\n\\end{corollary}\nLater we will need the following determination of the fibers of $F$.\n\n\\begin{lemma}\\label{phasefactor}\n$$F(z_1,w_1) = F(z_2,w_2) \\iff z_1 = c\nz_2 \\ \\text{and} \\ w_1 = c w_2 \\ \\text{with} \\ |c|\n=1.$$\n\\end{lemma}\n\n\\begin{proof}\nThe implication $\\Leftarrow$ is immediate. We prove the reverse\nimplication.\nThus we are assuming the equations\n\\begin{align*}\n|z_1|^2 - |w_1|^2 &= |z_2|^2 - |w_2|^2  \\\\\nz_1 \\bar{w_1} &= z_2 \\bar{w_2}\\end{align*}\nSquare each side of the first equation.\nTake four times the norm squared of each side of the second equation and\nadding the\nresulting equation to the new first equation to obtain$$ (|z_1|^2 +\n|w_1|^2)^2 = (|z_2|^2 + |w_2|^2)^2.$$\nHence $|z_1|^2 + |w_1|^2=|z_2|^2 + |w_2|^2$ and consequently$$|z_1|^2\n= |z_2|^2 \\ \\text{and} \\ |w_1|^2 = |w_2|^2.$$\nNow it is an elementary version of the first fundamental theorem\nof invariant theory that if we are given two ordered pairs of vectors\nin the plane so that the lengths of corresponding vectors\nare equal and the symplectic inner products between the two\nvectors in each pair coincide then there is an element in $\\mathrm{SO}(2)$\nthat carries one ordered pair to the other (this is one definition\nof the oriented angle between two vectors).\n\\end{proof}\n\nWe now construct the required map to Euclidean $n$-gons.\nWe define a map $F_n:P_n(\\mathrm{SU}(2)) \\to (\\mathbb{R}^3)^n\/\\mathrm{SO}(3,\\mathbb{R})$ by defining\n$F_n(A)$ to be the orbit of  $(F(C_1),\\ldots,F(C_n))$ under the diagonal\naction of $\\mathrm{SO}(3,\\mathbb{R})$. Here $C_i$ is the $i$-th column of $A$.\nLet $\\mathrm{Pol}_ n(\\mathbb{R}^3)$ denote the space of closed $n$-gons in\n$\\mathbb{R}^3$.\n\n\\begin{theorem}\\label{framedngons}\\hfill\n\\begin{enumerate}\n\\item $F_n$ induces a homeomorphism from $P_n(\\mathrm{SU}(2))\/T_{\\mathrm{SU}(2)^n}$ onto\n$\\mathrm{Pol}_ n(\\mathbb{R}^3)\/\\mathrm{SO}(3,\\mathbb{R})$.\n\\item The fiber of $F_n$ over a Euclidean $n$-gon\n is\nnaturally homeomorphic to the set\nof imploded spin framings of the edges of that\n$n$-gon.\n\\item The side-lengths of $F_n(A)$ are related to the norm squared of the columns\nof $A$ by\n$$||e_i(F_n(A))|| = \\frac{||C_i||^2}{4} = \\frac{|z_i|^2 + |w_i|^2}{4}.$$\nHere $e_i(F_n(A)) = F_n(C_i)$ is the $i$-th edge of any $n$-gon in the\ncongruence class represented by $F_n(A)$.\n\n\\end{enumerate}\n\\end{theorem}\n\n\\begin{proof}\nWe first prove that if $A$ is in the zero level set of $\\mu_G$\nthen $F_n(A)$ is a {\\it closed} $n$-gon.\nRecall $C_i,1 \\leq i \\leq n$ be the $i$-th column of $A$. Then we\nhave$$F_n(A) = (f((C_1C_1^*)^0),\\cdots,f((C_nC_n^*)^0)).$$\nHence the sum $s$ of the edges $F_n(A)$ is given by$$s = f((C_1C_1^* +\n\\cdots + C_nC_n^*)^0).$$\nNow it is a formula in elementary matrix multiplication that we\nhave$$AA^* = C_1C_1^* + \\cdots + C_nC_n^*.$$\nBut by Lemma \\ref{levelzero} we find that $AA^*$ is scalar\nwhence$(AA^*)^0 =0$ and$$s = f((AA^*)^0) = f(0) = 0.$$\nWe next prove that $F_n$ is onto.\nIt is clear that $F_n$ maps $M_{n,2}(\\mathbb{C})$ onto $(\\mathbb{R}^3)^n$ (this may\nbe proved one column at a time). Let $\\mathbf{e}=(e_1,\\cdots,e_n)$\nbe the edges of a closed $n$-gon in $\\mathbb{R}^3$. Choose $A \\in M_{2,n}(\\mathbb{C})$\nsuch that $F_n(A) = \\mathbf{e}$. But by the above$$f((AA^*)^0) = e_1 +\n\\cdots + e_n = 0.$$\nHence $(AA^*)^0 =0$ whence $AA^*$ is scalar and $\\mu_G(A) = 0$.\nWe now prove that $F_n$ is injective.\nSuppose there exists $g \\in \\mathrm{SU}(2)$ such that $F_n(A) = \\rho(g) F_n(A')$.\nThen $F(A) = F(gA')$. Hence by Lemma \\ref{phasefactor} we have $A = g A' t$\nfor some $t \\in T_{\\mathrm{U}(n)}$.\n\nThe second statement follows because the action of $T_{\\mathrm{U}(n)}$ corresponds to\na transitive action on the set\nof imploded spin framings.  Indeed using the identification above between\nbetween $\\mathbb{R}^3$ and $\\mathfrak{su}(2)^{\\ast}$ we may replace $F_n$ by the\nmap $\\pi_n :\\mathcal{E}T^\\ast(\\mathrm{SU}(2))^n \\to (\\mathfrak{su}(2)^{\\ast})^n$ given\nby$$\\pi_n([g_1,\\lambda_1 \\varpi_1],\\cdots,[g_n,\\lambda_n \\varpi_i])=\n(Ad^{\\ast}g_1(\\lambda_1 \\varpi_1), \\cdots, Ad^{\\ast}g_n(\\lambda_n\n\\varpi_1)).$$ If no $\\lambda_i$ is zero  by Lemma \\ref{framedvectors}\nthe (classes of ) the $n$-tuple $[g_1, \\lambda_1 \\varpi_1],\\cdots,\n[g_n,\\lambda_n \\varpi_1]$ represent the imploded\nspin-frames over the $n$ vectors in the image.\nThen by Lemma \\ref{circlebundle} and Lemma \\ref{framedvectors} two\nsuch $n$--tuples correspond to\nframes over the same image $n$-gon if and only the two $n$--tuples are\nrelated\nby right multiplication by $t_1,\\cdots, t_n$ with $t_i$\nfixing $\\varpi_1$ for all $i$. Hence  if no $\\lambda_i$ is zero then\nthe two $n$--tuples are related as above if\nand only if they are in the same $T_{\\mathrm{U}(n)}$ orbit by definition of\nthe $T_{\\mathrm{U}(n)}$ action. Since $T_{\\mathrm{U}(n)}$ acts transitively\non the spin-frames over a given $n$-gon we see that in this case the\nfiber of $\\pi_n$ is the set of spin-frames over the image $n$-gon and the second statement\nis proved.  If some subset  of the $\\lambda_i$'s is zero then\nwe replace the corresponding $g_i$'s by the identity (this does not\nchange the imploded frame) and procede as above with the remaining\ncomponents.\n\nThe last statement in the theorem is Lemma \\ref{factoroftwo}.\n\n\\end{proof}\n\n\n\n\nLet $\\sigma:\\mathrm{Pol}_ n(\\mathbb{R}^3) \\to \\mathbb{R}_+^n$ be the map that assigns\nto a closed $n$-gon the lengths of its sides. It is standard (see for\nexample the Introduction of \\cite{KapovichMillson}) that\nthe image of $\\sigma$ is the polyhedral cone $D_n$.\nWe can now give another proof of Proposition \\ref{triangleinequalities}\nbased on Euclidean geometry.\nWe restate it for the convenience of the reader.\n\n\\begin{proposition}$$\\mu_{T_n}(P_n(\\mathrm{SU}(2))) = D_n.$$\\end{proposition}\\begin{proof}\nThe proposition follows from the commutative diagram\\[\n\\begin{CD}P_n(\\mathrm{SU}(2)) @>\\mu_{T_n}>> \\mathbb{R}_+^n \\\\@VF_n VV  @VVIV\n\\\\ \\mathrm{Pol}_n(\\mathbb{R}^3) @>\\sigma>>\\mathbb{R}_+^n\\end{CD}\\]\\end{proof}\n\n\\subsection{The space $P_n(\\mathrm{SO}(3,\\mathbb{R})$}\n\nWe now compute $P_n(\\mathrm{SO}(3,\\mathbb{R}))$ using the fact that $\\mathrm{SO}(3,\\mathbb{R})$ is covered by $\\mathrm{SU}(2)$.\nLet the semi simple Lie Group $\\bar{G}$ be a quotient of the semi simple Lie\nGroup $G$\nby a finite group $\\Gamma$\n\n$$\\Gamma \\rightarrow G \\xrightarrow{\\pi} \\bar{G}$$\n\nIn Example 4.7 of \\cite{GuilleminJeffreySjamaar}, Guillemin-Jeffrey-Sjamaar, show that\nthis quotient gives a description nomeomorphism $\\phi:\\mathcal{E}(T^*G)\/\\Gamma \\to  \\mathcal{E}(T^* \\overline{G}))$. The following lemma is left to\nthe reader to prove.\n\n\\begin{lemma}\\label{quotientimplosion}\nThe homeomorphism $\\phi$ induces a homeomorphism from $\\Gamma \\backslash \\! \\backslash P_n(G)$\nto $P_n(\\overline{G})$.\n\\end{lemma}\n\n\n\\begin{theorem}\\label{compare} The double cover $\\phi:\\mathrm{SU}(2) \\to \\mathrm{SO}(3,\\mathbb{R})$\ninduces a homeomorphism from $H \\backslash \\! \\backslash P_n(\\mathrm{SU}(2))$ to $P_n(\\mathrm{SO}(3,\\mathbb{R}))$.\nHere  H is the finite 2-group $Z(\\mathrm{SU}(2))^n$ and $Z(\\mathrm{SU}(2)) \\cong \\mathbb{Z}\/2$\nis the center of $\\mathrm{SU}(2)$.\n\\end{theorem}\n\nWe  have the following analogue of Theorem \\ref{framedngons} above.\nWe leave its proof to the reader. We now have orthogonal (imploded) framings\ninstead of imploded spin framings.  We note that the map\n$F_n$ above descends to a map $\\overline{F}_n: P_n(\\mathrm{SO}(3,\\mathbb{R})) \\to\n\\mathrm{Pol}_ n(\\mathbb{R}^3)\/\\mathrm{SO}(3,\\mathbb{R})$.\n\n\\begin{theorem}\\label{orthogonalframedngons}\\hfill\n\\begin{enumerate}\n\\item $\\overline{F}_n$ induces a homeomorphism from $P_n(\\mathrm{SO}(3,\\mathbb{R}))\/T_{\\mathrm{SO}(3,\\mathbb{R})^n}$ onto the polygon space\n$\\mathrm{Pol}_ n(\\mathbb{R}^3)\/\\mathrm{SO}(3,\\mathbb{R})$.\n\\item The fiber of $\\overline{F}_n$ over a Euclidean $n$-gon\nof the induced map from $P_n(\\mathrm{SU}(2))$ onto $\\mathrm{Pol}_ n(\\mathbb{R}^3)\/\\mathrm{SO}(3,\\mathbb{R})$ is\nnaturally homeomorphic to the set\nof imploded orthogonal framings of the edges of that\n$n$-gon.\n\\end{enumerate}\n\\end{theorem}\n\n\n\n\n\n\n\n\n\n\\section{Toric degenerations associated to trivalent trees}\n\nSuppose one has a planar regular $n$-gon subdivided into\ntriangles. We call this a triangulation of the $n$-gon.  The dual\ngraph is a tree with $n$ leaves and $n-2$ internal trivalent nodes\n(see Figure \\ref{fig:triangulated_hexagon}).  We will see below\nhow the tree $\\mathcal{T}$ determines a Gr\\\"obner degeneration of the\nGrassmannian $\\mathrm{Gr}_2(\\mathbb{C}^n)$ to a toric variety. These toric\ndegenerations first appeared in \\cite{SpeyerSturmfels}.\n\n\\begin{figure}[htbp]\n\\centering\n\\includegraphics[scale = 0.2]{NP6.eps}\n\\caption{ A triangulated hexagon with associated dual tree.}\\label{fig:triangulated_hexagon}\n\\end{figure}\n\n\n\\subsection{Toric degenerations of $\\mathrm{Gr}_2(\\mathbb{C}^n)$}\n\nRecall that the standard coordinate ring $R = \\mathbb{C}[\\mathrm{Gr}_2(\\mathbb{C}^n)]$ (for\nthe Pl\\\"ucker embedding) of the Grassmannian is generated by\n$Z_{i,j}$, for $1 \\leq i < j \\leq n$, subject to the\nquadric relations, $Z_{i,j}Z_{k,l} - Z_{i,k}Z_{j,l} + Z_{i,l}Z_{j,k} = 0$ for\n$1 \\leq i < j < k < l \\leq n$. These relations generate the\nPl\\\"ucker ideal $I_{2,n}$.  For each pair of indices $i,j$ let\n$w_{i,j}^\\mathcal{T}$ denote the length of the unique path in $\\mathcal{T}$\njoining leaf $i$ to leaf $j$. For example, in Figure\n\\ref{fig:triangulated_hexagon} we have $w_{1,4}^\\mathcal{T} = 5$. To any\nmonomial $m = \\prod_k Z_{i_k, j_k}$ we assign a weight $w^\\mathcal{T}(m) =\n\\sum_k w_{i_k,j_k}^\\mathcal{T}.$  Let $I_{2,n}^\\mathcal{T}$ denote the initial\nideal with respect to the weighting $w^\\mathcal{T}$.\n\nIt is a standard result in the theory of Gr\\\"obner degenerations\nthat one has a flat degeneration of $\\mathbb{C}[\\mathrm{Gr}_2(\\mathbb{C}^n)] =\n\\mathbb{C}[\\{Z_{i,j}\\}_{i < j}] \/ I_{2,n}$ to $\\mathbb{C}[\\{Z_{i,j}\\}_{i<j}] \/\nI_{2,n}^\\mathcal{T}$. Below we outline how this works using the Rees\nalgebra.\n\n\nThe weight $w^\\mathcal{T}$ induces an (increasing) filtration on the\nring of the Grassmannian.  Let $F_m^\\mathcal{T}$ be the vector subspace\nof $R$ spanned by monomials of weight at most $m$. Then, for any\nelements $x \\in F_{m_1}^\\mathcal{T}$ and $y \\in F_{m_2}^\\mathcal{T}$, the\nproduct $xy$ belongs to $F_{m_1 + m_2}^\\mathcal{T}$. Let $R^\\mathcal{T}$\ndenote the associated graded ring $R^\\mathcal{T} = \\bigoplus_{m=0}^\\infty F_m^\\mathcal{T} \/ F_{m-1}^\\mathcal{T}$,\nwhere $F_{-1}^\\mathcal{T} := 0$.\nThe Rees algebra $\\mathcal{R}^\\mathcal{T}$ is given by,\n$\\mathcal{R}^\\mathcal{T} = \\bigoplus_{m=0}^\\infty t^m F_m^\\mathcal{T}$,\nwhere $t$ is an indeterminant. Then (cf. \\cite{AlexeevBrion}) \n$\\mathcal{R}^\\mathcal{T}$ is flat over $\\mathbb{C}[t]$, \n$\\mathcal{R}^\\mathcal{T} \\otimes_{\\mathbb{C}[t]} \\mathbb{C}[t,t^{-1}] \\cong\nR[t,t^{-1}]$, and $\\mathcal{R}^\\mathcal{T} \\otimes_{\\mathbb{C}[t]} (\\mathbb{C}[t]\/(t))\n\\cong R^\\mathcal{T}$.\n\nSince the ring $R$ is already graded, there is a natural grading\nof the Rees algebra $\\mathcal{R}^\\mathcal{T}$, where the degree of the\nindeterminant variable $t$ is defined to be zero, and in general,\nthe degree of a product $t^m x$, where $x \\in R^{(k)} \\cap\nF_m^\\mathcal{T}$, is equal to $\\deg(x) = k$. Now, the specializations\n$\\mathcal{R}^\\mathcal{T} \\otimes_{\\mathbb{C}[t]} \\mathbb{C}[t]\/(t-a)$ (at $t = a$) are\nall graded algebras, where $\\deg(y \\otimes \\overline{p(t)}) =\n\\deg(y)$ for $y \\in \\mathcal{R}^\\mathcal{T}$. In particular the\nassociated graded algebra $R^\\mathcal{T}$ is bigraded; one grading comes\nfrom the grading of $R$, and the other from the filtration of $R$.\nWe have that $\\mathrm{Proj} (\\mathcal{R}^\\mathcal{T})$ has a flat\nmorphism to the affine line $\\mathbb{A}^1 = \\mathrm{Spec} (\\mathbb{C}[t])$\nwith all fibers  projective varieties, with general fiber\nisomorphic to $\\mathrm{Gr}_2(\\mathbb{C}^n)$ and special fiber $\\mathrm{Gr}_2(\\mathbb{C}^n)_0^\\mathcal{T}$ at\n$t=0$.\n\n\\begin{definition}\nWe shall reserve a special name for the case that the\ntriangulation is such that all diagonals contain the initial\nvertex, forming a fan.  The associated tree has the appearance of\na caterpillar and was called such in \\cite{SpeyerSturmfels}. We\nshall call this triangulation the ``fan'' and the associated\ndegeneration the LG--degeneration, since the special fiber in this\ncase agrees with the special fiber of the degeneration of\n$\\mathrm{Gr}_2(\\mathbb{C}^n)$ given by Lakshmibai and Gonciulea in\n\\cite{LakshmibaiGonciulea}.\n\\end{definition}\n\nNow we will see why the special fiber $\\mathcal{R} \\otimes_{\\mathbb{C}[t]}\n\\mathbb{C}[t]\/(t)$ is toric.  This was also proved in\n\\cite{SpeyerSturmfels} where these degenerations were first\ndiscovered. First we establish a basis for $\\mathbb{C}[\\mathrm{Gr}_2(\\mathbb{C}^n)]$ as a\ncomplex vector space. Suppose that the vertices $1,2,\\ldots,n$ of\na multigraph (multiple edges allowed between two vertices) are\ndrawn on the unit circle, in cyclic clockwise order, and all edges\nare drawn as straight line segments (a chord of the circle). Then,\nif no two edges cross (it is okay for an edge to have multiplicity\ngreater than one) we call the graph a Kempe graph, in honor of the\nwork of A. Kempe in 1894 (see \\cite{Kempe}). For an example see\nFigure \\ref{fig:Kempe_graph}.  In fact two edges $ij$ and $kl$\ncross exactly when $i < k < j < l$, assuming that $i < k$.\n\n\\begin{figure}[htbp]\n\\centering\n\\includegraphics[scale = 0.3]{Kempe_graph.eps}\n\\caption{ A Kempe graph for $n=6$.}\\label{fig:Kempe_graph}\n\\end{figure}\n\n\nWe assign a monomial $m_G$ to each graph $G$, given by\n$$m_G = \\prod_{ij \\in E(G)} Z_{i,j},$$\nwhere $E(G)$ is the multi-set of edges of $G$. For a graph $G$,\nlet $\\deg(G)$ be the number of edges in $G$, which is one half of\nthe total sum of valencies of the all the vertices.  Note that\n$\\deg(G) = \\deg(m_G)$. The proof of the proposition and theorem\nbelow appears in \\cite{HowardMillsonSnowdenVakil}.\n\\begin{proposition}\nThe monomials $m_G$ as $G$ runs over the set of all Kempe graphs\nwith vertex set $V(G) = \\{1,2,\\ldots,n\\}$ form a $\\mathbb{C}$-basis for\n$\\mathbb{C}[\\mathrm{Gr}_2(\\mathbb{C}^n)]$. Furthermore, the $m_G$ for which $\\deg(G) = k$\nand $G$ is a Kempe graph form a basis for the $k$-th graded piece\nof $R$.\n\\end{proposition}\n\n\\begin{theorem}\\label{leadingterm}\nSuppose that $G_1$ and $G_2$ are Kempe graphs.  Suppose that\n$$m_{G_1} m_{G_2} = \\sum_{\\text{$G$ is Kempe}} c_G m_G.$$\nThen, there is a unique Kempe graph $G^\\ast$ such that\n\\begin{itemize}\n\\item $c_{G^\\ast} = 1$, \\item $w^\\mathcal{T}(m_{G^\\ast}) =\nw^\\mathcal{T}(m_{G_1}) + w^\\mathcal{T}(m_{G_2})$, \\item for all $G$, if $c_G\n\\neq 0$ and $G \\neq G^\\ast$ then $w^\\mathcal{T}(m_G) <\nw^\\mathcal{T}(m_{G^\\ast})$.\n\\end{itemize}\n\\end{theorem}\n\n\\begin{proof}\n\nSuppose that $i_1 j_1$ is an edge of $G_1$ and $i_2 j_2$ is an\nedge of $G_2$ such that $i_1 j_1$ and $i_2 j_2$ cross.  Without\nloss of generality we can assume that $i_1 < i_2 < j_1 < j_2$. We\nhave the Pl\\\"ucker relation $Z_{i_1,j_1}Z_{i_2,j_2} =\nZ_{i_1,i_2}Z_{j_1,j_2} + Z_{i_1,j_2}Z_{i_2,j_1}$, and the latter two terms\n(viewed as graphs having two edges) are Kempe graphs since neither\npair of edges cross. Denote $G_1 \\cdot G_2$ as the graph with edge\nset $E(G_1) \\coprod E(G_2)$. Let $G_0 = (G_1 \\cdot G_2)$ with the\ntwo edges $i_1 j_1$ and $i_2 j_2$ removed. We have,\n$$m_{G_1} m_{G_2} = m_{G_0 \\cdot i_1 j_1 \\cdot i_2 j_2} =\nm_{G_0 \\cdot i_1 i_2 \\cdot j_1 j_2} + m_{G_0 \\cdot i_1j_2 \\cdot\ni_2j_1}.$$ We will show that one of the two monomials on the right\nhand side of the above equation has the same $\\mathcal{T}$--weight\n($w^\\mathcal{T}$) as does $m_{G_1} m_{G_2}$ and the other term has\nstrictly smaller weight.  After sufficiently many Pl\\\"ucker\nrelations as above are applied we get $m_{G_1} m_{G_2}$ expressed\nas a unique integral combination of Kempe graphs with a unique\nterm of maximal weight equal to the weight of $m_{G_1} m_{G_2}$.\n\nLet $\\gamma(i,j)$ denote the shortest path in $\\mathcal{T}$ joining $i$\nto $j$.  Note that the paths $\\gamma(i_1,j_1)$ and\n$\\gamma(i_2,j_2)$ must intersect one another since they cross when\ndrawn as straight line segments in the graph $G_1 \\cdot G_2$. Now\nconsider the two pairs of paths in $\\mathcal{T}$:\n\n\\begin{itemize}\n\\item $\\gamma(i_1,i_2)$ and $\\gamma(j_1,j_2)$. \\item\n$\\gamma(i_1,j_2)$ and $\\gamma(i_2,j_1)$.\n\\end{itemize}\nExactly one of the above two above pairs of paths cover precisely\nthe same set of edges in $\\mathcal{T}$ as does the crossing pair\n($\\gamma(i_1,j_1)$,$\\gamma(i_2,j_2)$.   Say for example it is the\npair ($\\gamma(i_1,i_2)$, $\\gamma(j_1,j_2)$). Then\n$w^\\mathcal{T}(Z_{i_1,i_2}Z_{j_1,j_2}) = w^\\mathcal{T}(Z_{i_1,j_1}Z_{i_2,j_2})$.\nFurthermore the two paths $\\gamma(i_1,i_2)$ and $\\gamma(j_1,j_2)$\nmust meet one another within the tree $\\mathcal{T}$ along some internal\nedges $e_1,\\ldots,e_k$ (although they are non-crossing when drawn\nas straight line segments).  The edges $e_1,\\ldots,e_k$ are also\nthe edges common to the two paths $\\gamma(i_1,j_1)$ and\n$\\gamma(i_2,j_2)$. The third pair of paths ($\\gamma(i_1,j_2)$,\n$\\gamma(i_2,j_1)$) covers the same set of edges as do the first\ntwo pairs of paths, excepting the edges $e_1,\\ldots,e_k$ above. In\nparticular we have that\n$$w^\\mathcal{T}(Z_{i_1,i_2}Z_{j_1,j_2}) = w^\\mathcal{T}(Z_{i_1,j_1}Z_{i_2,j_2}) =\nw^\\mathcal{T}(Z_{i_1,j_2}Z_{i_2,j_1}) + 2k.$$\n\\end{proof}\n\n\\begin{definition}\nLet $G_1 \\ast_\\mathcal{T} G_2$ denote the Kempe graph $G^\\ast$ from the\nabove theorem.   Let $\\mathcal{S}_n^\\mathcal{T}$ denote the graded commutative\nsemigroup of Kempe graphs $G$ with binary operation $(G_1,G_2)\n\\mapsto G_1 \\ast_\\mathcal{T} G_2$.  The grading is given by $\\deg(G) =\n\\deg(m_G)$.\n\\end{definition}\n\n\n\\begin{corollary} (to Theorem \\ref{leadingterm})\nThe special fiber at $t=0$ of the degeneration, $\\mathcal{R}\n\\otimes_{\\mathbb{C}[t]} \\mathbb{C}[t]\/(t)$, is isomorphic to the semigroup algebra\n$\\mathbb{C}[\\mathcal{S}_n^\\mathcal{T}]$.\n\\end{corollary}\nGiven any graph $G$, there is an associated weighting $w_G$ of the\ntree $\\mathcal{T}$ where the weight assigned to an edge $e$ of $\\mathcal{T}$\nis equal to the number of edges $ij$ in $G$ such that the path\n$\\gamma(i,j)$ in $\\mathcal{T}$ joining $i$ to $j$ passes through $e$. We\ndenote this weight by $w_G(e)$.  We now determine the image of the\nmap $G \\mapsto w_G$.\n\n\\begin{proposition}\\label{admissibleweighting}\nGiven a weighting $w$ of nonnegative integers to the edges of\n$\\mathcal{T}$, then $w = w_G$ for some graph $G$ iff for each triple\n$e_1, e_2, e_3$ of edges meeting at a common (internal) vertex of\n$\\mathcal{T}$ the sum $w(e_1) + w(e_2) + w(e_3)$ is even, and\n$w(e_1),w(e_2),w(e_3)$ satisfy the triangle inequalities.\n\\end{proposition}\n\n\\begin{proof}\nLet a $G$--path mean a path in the tree $\\mathcal{T}$ joining vertices\n$i$ to $j$ where $ij$ is an edge of $G$.  Thus $G$-paths are in\nbijection with edges of $G$, and are meant to be counted with\nmultiplicity.  Now fix an internal vertex $v_0$ of $\\mathcal{T}$ with\nneighboring vertices $v_1, v_2, v_3$ with connecting respective\nedges $e_1, e_2, e_3$.  For each $i,j$ with $1 \\leq i < j \\leq 3$,\nlet $x_{ij}$ be the number of $G$-paths passing through each of\n$v_i$ and $v_j$.  Thus \\begin{itemize} \\item $w_G(e_1) = x_{12} +\nx_{13}$. \\item $w_G(e_2) = x_{12} + x_{23}$. \\item $w_G(e_3) =\nx_{13} + x_{23}$.\n\\end{itemize}  Now solving for the $x_{ij}$ we have:\n\\begin{itemize} \\item $x_{12} = \\frac{1}{2}(w_G(e_1) + w_G(e_2) - w_G(e_3))$.\n\\item $x_{13} = \\frac{1}{2}(w_G(e_1) + w_G(e_3) - w_G(e_2))$.\n\\item $x_{23} = \\frac{1}{2}(w_G(e_2) + w_G(e_3) - w_G(e_1))$.\n\\end{itemize} Since the $x_{ij}$ are non-negative we have that\n$w_G(e_1)$, $w_G(e_2)$, and $w_G(e_3)$ satisfy the triangle\ninequalities. Since the $x_{ij}$ are integers we have that\n$w_G(e_1) + w_G(e_2) + w_G(e_3)$ is even.\n\nConversely, given a weighting of $\\mathcal{T}$ satisfying the above\nconditions, we will get non-negative integers $x_{12}(v),\nx_{13}(v), x_{23}(v)$ at each internal vertex $v$.  We may thus\ndefine a graph $G_v$ on the three neighboring vertices\n$v_1,v_2,v_3$ by setting the multiplicity of edge $ij$ to be\n$x_{ij}(v)$. If $v'$ is a neighboring internal vertex, say for\nexample sharing edge $e_1$ with $v$, then we have $x_{12}(v) +\nx_{13}(v) = x_{12}(v') + x_{13}(v') = w(e_1)$.  Hence we may glue\ntogether $G_{v}$ and $G_{v'}$ to form a Kempe graph $G_{v,v'}$ on\nthe neighboring vertices of $v$ and $v'$ by adjoining the edges\nwhose respective paths go through the edge $vv'$ of $\\mathcal{T}$.  Note\nthat the weighting of the edges of the tree pertaining to\n$G_{v,v'}$ is equal to $w$. Continuing in this way, we may form a\nKempe graph $G$ on the leaves of $\\mathcal{T}$ such that $w = w_G$, by\ngluing together all the $G_v$ for $v$ and internal vertex of\n$\\mathcal{T}$.\n\\end{proof}\n\n\\begin{definition}\nWe shall call a weighting $w$ or $\\mathcal{T}$ an admissible weighting\nif $w = w_G$ for some graph $G$.\n\\end{definition}\n\n\\begin{proposition}\nFor each admissible weighting $w$ of $\\mathcal{T}$, there exists a\nunique Kempe graph $G$ such that $w = w_G$.  Furthermore, if $G_1$\nand $G_2$ are Kempe graphs, then $w_{G_1} + w_{G_2} = w_{G_1\n\\ast_\\mathcal{T} G_2}$.\n\\end{proposition}\n\n\\begin{proof}\nCertainly the paths in the tripods of the proof of Proposition\n\\ref{admissibleweighting} may be drawn so that they are\nnon-crossing.  Such a non-crossing tripod graph is unique.\nIn the process of gluing all these tripod graphs together\nto form a graph $G$ such that $w_G = w$, it is clear there is only\none way to join the paths so that no two paths are non-crossing.\nNow by straightening these non-crossing paths into line segments,\nwe see that they remain non-crossing and so form a Kempe graph.\n\nA quick inspection of the proof of Theorem \\ref{leadingterm}\nreveals that the graph $G_1 \\ast_\\mathcal{T} G_2$ has weight equal to\nthe sum $w_{G_1} + w_{G_2}$.\n\\end{proof}\nNote that $w^\\mathcal{T}(G) = \\sum_{ij \\in E(G)} w_{i,j}^\\mathcal{T} = \\sum_{e\n\\in E(\\mathcal{T})} w_G(e)$. We define the degree of an admissible weighting\n$w^{\\mathcal{T}}$ to be one -half the sum of the weights of the leaf edges (a leaf edge\nis an edge incident to a leaf).  Since each edge of the graph $G$ contributes\na weight of one to two leaf edges we have\n$$deg(m_G) = deg(G) = deg(w^{\\mathcal{T}}).$$\n\n\\begin{definition}\nDefine $\\mathcal{W}_{n}^{\\mathcal{T}}$ to be the graded semigroup of\nadmissible weightings of the edges of $\\mathcal{T}$.\n\\end{definition}\n\n\n\n\\begin{proposition}\\label{gradedsemigroupiso}\nThe map that associates to an element of $\\mathcal{S}^\\mathcal{T}_n$ the induced\nweighting of the edges of $\\mathcal{T}$ induces an isomorphism\n$\\Omega_n:\\mathcal{S}_{n}^{\\mathcal{T}} \\to \\mathcal{W}_{n}^{\\mathcal{T}}$.\n\\end{proposition}\n\n\n\\subsection{Induced toric degeneration of the space $M_\\mathbf{r}$ of polygonal linkages}\n\nHere we briefly review the construction of the GIT quotients of\nthe Grassmannian by the maximal torus $T$ in $\\mathrm{SL}(n,\\mathbb{C})$. The GIT quotient \n(or technically, the subspace of closed points in the GIT quotient) \nis homeomorphic to a \nconfiguration space $M_\\mathbf{r}$ of polygonal\nlinkages in $\\mathbb{R}^3$ with prescribed integral side lengths $\\mathbf{r} =\n(r_1,\\ldots,r_n)$. See \\cite{KapovichMillson}, Theorem 2.3, or \n\\cite{Klyachko} for more\ndetails.\n\nLet $T$ be the torus of diagonal matrices in $\\mathrm{SL}(n,\\mathbb{C})$.  There\nis a natural action of $T$ on the Grassmannian $\\mathrm{Gr}_2(\\mathbb{C}^n)$.  The\nGIT quotient $\\mathrm{Gr}_2(\\mathbb{C}^n) \/\\!\/ T$ depends upon the choice of a\n$T$-linearized line bundle of $\\mathrm{Gr}_2(\\mathbb{C}^n)$.  Let $L$ be the line\nbundle associated to the Pl\\\"ucker embedding of $\\mathrm{Gr}_2(\\mathbb{C}^n)$.\nLet $P \\subset \\mathrm{SL}(n,\\mathbb{C})$ be the parabolic subgroup fixing the point\n$e_1 \\wedge e_2 \\in \\mathrm{Gr}_2(\\mathbb{C}^n)$.  It is the subgroup of matrices\n$[a_{ij}]_{1 \\leq i,j \\leq n}$ of determinant one, such that\n$a_{ij} = 0$ for $i=1,2$ and $j > 2$. We identify $\\mathrm{Gr}_2(\\mathbb{C}^n)$\nwith the homogeneous space $\\mathrm{SL}(n,\\mathbb{C})\/P$,\nby identifying $gP \\in \\mathrm{SL}(n,\\mathbb{C})\/P$ with\n$g \\cdot (e_1 \\wedge e_2) \\in \\mathrm{Gr}_2(\\mathbb{C}^n)$.  Let $\\varpi_2 =\n(1,1,0,\\ldots,0) \\in \\mathbb{Z}^n$ be the second fundamental weight of\n$\\mathrm{SL}(n,C)$.  Associated to $\\varpi_2$ is a character\n$\\chi : P \\to \\mathbb{C}^\\ast$ given by $\\chi([a_{ij}]_{1 \\leq i,j \\leq n})\n= \\det\n\\begin{pmatrix}\na_{11} & a_{12} \\\\\na_{21} & a_{22}\n\\end{pmatrix}$.\nFollowing the Borel-Weil construction, we may take the total space\nof $L$ to be the product $\\mathrm{SL}(n,\\mathbb{C}) \\times \\mathbb{C}$ modulo\nthe equivalence relation $(g,z) \\sim (gp, \\chi(p)z)$, for all\n$g \\in \\mathrm{SL}(n,\\mathbb{C})$, $p \\in P$, and $z \\in \\mathbb{C}$. We denote the equivalence\nclass of $(g,z)$ as above by $[g,z]$.\nThe bundle map $\\pi : L \\to \\mathrm{Gr}_2(\\mathbb{C}^n)$ is given by\n$[g,z] \\mapsto gP$.  The Plucker coordinate ring of $\\mathrm{Gr}_2(\\mathbb{C}^n)$\nis now $\\bigoplus_{N=0}^\\infty \\Gamma(\\mathrm{Gr}_2(\\mathbb{C}^n),L^{\\otimes N})$, and it is\ngenerated in degree one by the brackets $[i,j]$\nwhich are associated to global sections $s_{ij}$ of $L$, by\n$s_{ij}(gP) := [g, g_{i1}g_{j2} - g_{i2}g_{j1}]$, where\n$g = [g_{kl}]_{1 \\leq k,l \\leq n}$.\n\n\n\nWe suppose that $|\\mathbf{r}| = r_1 + \\cdots + r_n$ is an even integer.\nThe line bundle $L^{\\otimes |\\mathbf{r}|\/2}$ of $L$ may be identified\nwith the product $\\mathrm{SL}(n,\\mathbb{C}) \\times \\mathbb{C}$ modulo the relation\n$(g,z) \\sim (gp, \\chi(p)^{\\otimes |\\mathbf{r}|\/2} z)$.\nWe define an action of $T$ on $L^{\\otimes |\\mathbf{r}|\/2}$ via the character\n$\\chi_\\mathbf{r}$ of $T$, given by\n$\\chi_\\mathbf{r}(t_1,\\ldots,t_n) := \\prod_{i=1}^n t_i^{r_i}$.\nWe define the action of $T$ on $L^{\\otimes |\\mathbf{r}|\/2}$ by\n$$(t_1,\\ldots,t_n) \\cdot [g,z] = [\\mathbf{t} \\cdot g,\n\\chi_\\mathbf{r}(\\mathbf{t})z],$$\nwhere $\\mathbf{t} = (t_1,\\ldots,t_n) \\in T$.  We call this the\n$\\mathbf{r}$--linearization of $L^{\\otimes |\\mathbf{r}|\/2}$.\n\nThe space $M_\\mathbf{r}$ is homeomorphic to \n$$\\mathrm{Gr}_2(\\mathbb{C}^n) \/\\!\/_\\mathbf{r} T = \\mathrm{Projm} \\Big(\n\\bigoplus_{N=0}^\\infty \\Gamma(\\mathrm{Gr}_2(\\mathbb{C}^n), L^{\\otimes N |\\mathbf{r}|\/2})^T\n\\Big),$$\nwhere $\\Gamma(\\mathrm{Gr}_2(\\mathbb{C}^n), L^{\\otimes N |\\mathbf{r}|\/2})^T$ indicates\nthe $T$-invariant global sections of $L^{\\otimes N |\\mathbf{r}|\/2}$.\nThe ring \n$$R_\\mathbf{r} = \n\\bigoplus_{N=0}^\\infty \\Gamma(\\mathrm{Gr}_2(\\mathbb{C}^n), L^{\\otimes N |\\mathbf{r}|\/2})^T$$\nis naturally graded by $N$.\nThe invariant sections of $L^{\\otimes N|\\mathbf{r}|\/2}$ are spanned by\nmonomials $m = m_G$ over all graphs $G$ having multi-degree $N\\mathbf{r}$,\ni.e. the valency of vertex $i$ is $r_i$ for each $i$.\nThe degree of $m_G$ is then $N$, if $G$ has multidegree $N \\mathbf{r}$.\n\nRestricting to torus invariants is an exact functor so the flat\ndegenerations of the Grassmannian described above restrict to flat\ndegenerations of $\\mathrm{Gr}_2(\\mathbb{C}^n) \/\\!\/_\\mathbf{r} T$.  Furthermore, the\nspecial fiber of this restricted degeneration is toric, since it\nis the $T$-quotient of a toric variety.\nTherefore we obtain a flat degeneration of $\\mathrm{Gr}_2(\\mathbb{C}^n) \/\\!\/_\\mathbf{r} T$ to a\ntoric variety $(\\mathrm{Gr}_2(\\mathbb{C}^n) \/\\!\/_\\mathbf{r} T)^\\mathcal{T}$ for each triangulation $\\mathcal{T}$ of the\nmodel $n$-gon.\nThe associated semigroup\n$\\mathcal{S}^\\mathcal{T}_\\mathbf{r}$ is the set of Kempe graphs having valency a multiple\nof $\\mathbf{r}$; it is a sub-semigroup of $\\mathcal{S}_n^\\mathcal{T}$, however the grading\nof $\\mathcal{S}^\\mathcal{T}_\\mathbf{r}$ is not the same as that of $\\mathcal{S}_n^\\mathcal{T}$.  Instead\nthe degree of a Kempe graph $G \\in \\mathcal{S}^\\mathcal{T}_\\mathbf{r}$ of multidegree \n$N \\mathbf{r}$ is $N$, rather than $N |\\mathbf{r}|\/2$ as it would have in $\\mathcal{S}_n^\\mathcal{T}$.\nWe have that\n$$(\\mathrm{Gr}_2(\\mathbb{C}^n) \/\\!\/_\\mathbf{r} T)^\\mathcal{T} = \\mathrm{Projm}(\\mathbb{C}[S^\\mathcal{T}_\\mathbf{r}]),$$\nwith $\\mathcal{S}^\\mathcal{T}_\\mathbf{r}$ graded as mentioned above.\n\n\\begin{definition}\nDefine the  graded semigroup $\\mathcal{W}_{\\mathbf{r}}^{\\mathcal{T}}$ to be the graded subsemigroup of\n$\\mathcal{W}^{\\mathcal{T}}_n$ with leaf-edge weights that are integral multiples of $\\mathbf{r}$.\n\\end{definition}\nThe admissible weightings of the tree $\\mathcal{T}$ relating\nto the sub-semigroup $\\mathcal{S}^\\mathcal{T}_\\mathbf{r}$ must satisfy that the weighting of\nthe outer edges $e_1,\\ldots,e_n$ (the edge\n$e_i$ is adjacent to leaf $i$) is some multiple $N$ of $\\mathbf{r}$,\nmeaning that $w(e_i) = Nr_i$ for each $i$,\nand the multiple $N$ is the degree of the weighting.\nThus we have\n\\begin{proposition}\\label{gradedsubsemigroupiso}\nThe isomorphism $\\Omega_n$  of Proposition \\ref{gradedsemigroupiso} induces an isomorphism\n$\\Omega_{\\mathbf{r}}:\\mathcal{S}^{\\mathcal{T}}_{\\mathbf{r}} \\to \\mathcal{W}_{\\mathbf{r}}^{\\mathcal{T}}$.\n\\end{proposition}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{$\\mathcal{T}$-congruence of polygons and polygonal linkages}\\label{KYspace}\n\nIn this section we will define an equivalence relation on polygons and polygonal linkages \nwhich depends on the choice of a trivalent tree $\\mathcal{T}$.  First we will collect some results \nabout trivalent trees which will be useful in what follows.\n\n\\subsection{Trivalent trees and their decompositions into forests}\nLet $\\mathcal{T}$ be a trivalent tree with $n$ leaves which we assume is dual\nto a triangulation of $P$.  We will\nsay a vertex is {\\it internal} if it is not a leaf. The triangles in the triangulation of $P$ correspond to the internal vertices of $\\mathcal{T}$.\nWe will say an edge is a {\\it leaf edge} or an {\\it outer edge} if it is incident to a leaf. Thus the leaf edges are dual to the edges of $P$.\nAn edge of $\\mathcal{T}$ which does not border a leaf will be called an\n{\\it inner edge}. Thus the inner edges of $\\mathcal{T}$ are dual to the\ndiagonals of the triangulation of $P$.\n\n\\begin{definition}\nWe say two leaves of $\\mathcal{T}$ are a matched pair of leaves if they have a common neighbor.\n\\end{definition}\nThe following technical lemma will be very useful for giving inductive proofs.\n\n\\begin{lemma}\\label{trivalentinduction}\nFor any trivalent tree $\\mathcal{T}$ it is possible\nto find a sequence of subtrees\n\n$$\\mathcal{T}_0 \\subset \\ldots \\subset \\mathcal{T}_{n-3} = \\mathcal{T}$$\nsuch that\n\\begin{itemize}\n\\item The tree $\\mathcal{T}_0$ is a tripod.\n\\item The tree $\\mathcal{T}_i$ can be identified with $\\mathcal{T}_{i-1}$ joined with a tripod along some $e_i$.\n\\item Each internal edge of $\\mathcal{T}$ appears as some $e_i$.\n\\end{itemize}\n\\end{lemma}\n\n\\begin{proof}\nLet $\\mathcal{T}$ be a trivalent tree.  let $\\mathcal{T'}$ be the (not necessarily trivalent) subtree of $\\mathcal{T}$ with vertex set the internal vertices of $\\mathcal{T}$ and edge set equal to all edges of $\\mathcal{T}$ not connected to leaves.  It is easy to see that $\\mathcal{T'}$ is also a tree and therefore has a leaf.  Let $n \\in \\mathcal{T}$ be the vertex corresponding to this leaf, then $n$ is trivalent by definition and therefore must be connected to two leaves in $\\mathcal{T}$.  This shows that $\\mathcal{T}$ must have a vertex $v$ connected to two leaf edges, $v$ is connected to a third edge, $e$,\nby splitting $\\mathcal{T}$ along $e$ we obtain a tripod and a new trivalent tree.  The three items above now follow by induction.\n\\end{proof}\nFor each tree $\\mathcal{T}$ we choose once and for all an ordering on the\ntrivalent vertices of $\\mathcal{T}$ as follows.  Pick any matched pair\nof leaves and label their trivalent vertex $\\tau_{n-2}$\nwith $n$ the number of leaves of $\\mathcal{T}$. Then,\nrepeat this proceedure for the tree given by removing $\\tau_{n-2}$\nand its matched pair. Note that this ordering is not unique.\n\n\nWe now discuss the decompositions  of $\\mathcal{T}$ obtained\nby splitting certain internal edges.\nWe first describe  the decompositions of $P$ which will induce the required\ndecompositions of $\\mathcal{T}$. For a diagonal $d$ in the triangulation $\\mathcal{T}$, \nwe define the operation of splitting $P$ along $d$ by removing a small\ntubular neighborhood of $d$ in $P$.  This results in a disjoint union\nof two triangulated polytongs $P' \\cup P'' = P^{\\{d\\}}$. \nWe may generalize this operation by performing the same operation for any subset $S$ \nof the set of diagonals of $P$ to obtain a union of triangulated polygons $P^S$. \nThe dual graph to $P^S$ is a forest $\\mathcal{T}^S$ that may be obtained by removing\na small open interval from each edge corresponding to a diagonal in $S$. \nIf we choose $S$ to  be the entire set of diagonals we obtain the decomposition \n$P^D$ of $P$ into triangles and $\\mathcal{T}$ into the forest $\\mathcal{T}^D$ of trinodes.\n\nIt will be important in what follows that we have the quotient map\n$$\\pi^{S}: \\mathcal{T}^S \\to \\mathcal{T}$$\nthat glues together the pairs of vertices that are the boundaries of\nthe open intervals removed from $\\mathcal{T}$.\nWe will say two vertices of $\\mathcal{T}^S$ are equivalent if they have the same\nimage under $\\pi^{S}$ and two edges of $\\mathcal{T}^S$ are equivalent\nif they contain equivalent vertices.\n\nThere are two types of leaf edges of $\\mathcal{T}^S$.\nThe first type correspond to edges of $P$ (leaf edges of $\\mathcal{T}$) and the second correspond to (split)\ndiagonals of $P$.  The $n$ edges of $\\mathcal{T}^S$ corresponding to leaf edges of $\\mathcal{T}$ will from now on be referred to as distinguished\nedges, we denote this set $dist(\\mathcal{T}^S)$.  For a connected component $C$ of $\\mathcal{T}^S$ we let $dist(C) = dist(\\mathcal{T}^S)\\cap C$.\nNote that these components uniquely partition the set $dist(\\mathcal{T}^S) = \\bigcup_{C_i \\subset \\mathcal{T}^S} dist(C_i)$.\n\n\\subsection{$\\mathcal{T}$-congruence of polygonal linkages}\n\nFix a triangulation of the model n-gon with dual tree $\\mathcal{T}$.\nRecall $\\mathrm{Pol}_ n(\\mathbb{R}^3)$, the space of n-gons  in $\\mathbb{R}^3$.\nA polygon $\\mathbf{e} \\in \\mathrm{Pol}_ n(\\mathbb{R}^3)$ comes with a fixed ordering on its edges,\nthese ordered edges are in bijection with the leaves of the tree $\\mathcal{T}$.\nA set of diagonals $S$ corresponds to a set of internal edges of\n$\\mathcal{T}$. We know that such a set defines a\nunique partition of the edges of $\\mathbf{e} $, $E(\\mathbf{e} ) = E_1(\\mathbf{e} ) \\cup \\ldots \\cup E_m(\\mathbf{e} )$\ngiven by grouping together the distinguished edges of $\\mathcal{T}^S$ that lie in a component tree of the  forest $\\mathcal{T}^S$.\n\n\\begin{proposition}\nFor $\\mathbf{e}  \\in \\mathrm{Pol}_ n(\\mathbb{R}^3)$, if all diagonals in the set $S$ are zero then\nthe edge sets $E_i(\\mathbf{e} )$ define closed polygons.\n\\end{proposition}\n\n\\begin{proof}\nSplit the polygon $P$ along $S$ to obtain a union of polygons.\nChoose a  polygon $P_i$ in the union. Then $P_i$ is dual to a component $C_i$ of the forest $\\mathcal{T}^S$.\nThe edges of the  polygon $P_i$  are either edges of the\norginal original polygon (so distinguished leaves of the tree $C_i$)\nand hence edges of the Euclidean polygon $\\mathbf{e}$\nor diagonals of $S$ and hence zero diagonals of the Euclidean\npolygon $\\mathbf{e}$. We assume that  we have chosen $i$ consistently\nwith the division of the distinguished edges above whence\nthe set of distinguished edges of $C_i$ equals\nthe set $E_i(\\mathbf{e})$. Now since $P_i$ closes up the sum of\nall the vectors in $\\mathbb{R}^3$ associated to the edges of $P_i$\nis zero. But this sum is the sum of the vectors in $\\mathbb{R}^3$ associated\nto the distinguished edges of $C_i$ (that is the elements\nin $E_i(\\mathbf{e})$) and a set of vectors all of which are zero.\n\\end{proof}\nThe following groups will be useful in defining structures on\nspaces of polygons.\n\n\\begin{definition}\nLet $G^{dist(\\mathcal{T}^S)}$ be the group of maps from\nthe set $dist(\\mathcal{T}^S)$ into a group $G$.  Define\n$G^{D(S)}$ to be the subgroup of $G^{dist(\\mathcal{T}^S)}$\nof maps which are constant along the distinguished edges\nof each component $C$ of $\\mathcal{T}^S$\n\\end{definition}\nNotice that $G^{D(S)}$ naturally splits as a\nproduct of $G$ over the components of $\\mathcal{T}^S$.\nLet $G^{D(C)}$ be the component corresponding\nto the component $C$. The sets $S$ of zero diagonals define a filtration of the space $\\mathrm{Pol}_ n(\\mathbb{R}^3)$\nwhere the subspace $\\mathrm{Pol}_ n(\\mathbb{R}^3)^S$ in the filtration is defined to be the\nset of all points $\\mathbf{e} $ such that the diagonals in $S$ have zero length.\nThis in turn defines a decomposition of $\\mathrm{Pol}_ n(\\mathbb{R}^3)$ into subspaces.\n$$\\mathrm{Pol}_ n(\\mathbb{R}^3)^{[S]} = \\mathrm{Pol}_ n(\\mathbb{R}^3)^S \\setminus \\cup_{S \\subset J} \\mathrm{Pol}_ n(\\mathbb{R}^3)^J$$\nThe subspace $\\mathrm{Pol}_ n(\\mathbb{R}^3)^{[S]}$ is the collection of all points in $\\mathrm{Pol}_ n(\\mathbb{R}^3)$\nsuch that exactly the diagonals in $S$ have zero length.  This also\ninduces a filtration on $M_{\\mathbf{r}}$.\n\n\\begin{definition}\nDefine an action of $\\mathrm{SO}(3,\\mathbb{R})^{D(S)}$ on $\\mathrm{Pol}_ n(\\mathbb{R}^3)^{[S]}$ by\nletting $\\mathrm{SO}(3, \\mathbb{R})^{D(C_i)}$ act diagonally on the edges in $E_i(\\mathbf{e} )$.\nThe equivalence relation given by $\\mathrm{SO}(3,\\mathbb{R})^{D(S)}$ orbit type on $\\mathrm{Pol}_ n(\\mathbb{R}^3)^{[S]}$\nfit together to give an equivalence relation $\\sim_{\\mathcal{T}}$,\nwhich we call $\\mathcal{T}$-congruence, on $\\mathrm{Pol}_ n$.\n\\end{definition}\nWe may describe the above equivalence relation geometrically as follows.\nA polygon $\\mathbf{e} \\in \\mathrm{Pol}_ n(\\mathbb{R}^3)^{[S]}$ is a wedge of $|S| +1$ closed polygons $\\mathbf{e}_i$\nwedged together at certain vertices of the polygon $\\mathbf{e}$. Although each $\\mathbf{e}_i$\nmay contain several wedge points, since the vertices of $\\mathbf{e}$ are ordered there\nwill be a first wedge point $v_i$. Then we apply\na rotation $g_i$ about $v_i$ to each $\\mathbf{e}_i$ for $1 \\leq i \\leq |S|+1$ and identify\npoints in the resulting orbit of $\\mathrm{SO}(3,\\mathbb{R})^{D(S)}$.\n\n$\\mathcal{T}$-congruence for $\\mathrm{Pol}_ n(\\mathbb{R}^3)$ induces an equivalence relation on the space of $n$-gon linkages  $\\widetilde{M}_{\\mathbf{r}} \\subset \\mathrm{Pol}_ n(\\mathbb{R}^3)$,\nwhich we also call $\\mathcal{T}$-congruence.\n\n\\begin{definition}\nDefine $$V_{\\mathbf{r}}^{\\mathcal{T}} = \\widetilde{M}_{\\mathbf{r}} \/ \\sim_{\\mathcal{T}}$$\n\\end{definition}\nKamiyama and Yoshida studied the space $V_{\\mathbf{r}}^{\\mathcal{T}}$ for the special case\nwhen $\\mathcal{T}$ was the {\\it fan} tree.\nNote that $V_{\\mathbf{r}}^{\\mathcal{T}}$ inherits a filtration by the subspaces $(V_{\\mathbf{r}}^{\\mathcal{T}})^{S}$.\nLet $(V_{\\mathbf{r}}^{\\mathcal{T}})^{[S]} = (V_{\\mathbf{r}}^{\\mathcal{T}})^{S} \\setminus \\cup_{S \\subset J} (V_{\\mathbf{r}}^{\\mathcal{T}})^J$.\nThe spaces  $(V_{\\mathbf{r}}^{\\mathcal{T}})^{[S]}$ define a decomposition of $V_{\\mathbf{r}}^{\\mathcal{T}}$.\nWe can say more about the pieces of this decomposition.\n\n\\begin{theorem}\n$(V_{\\mathbf{r}}^{\\mathcal{T}})^{[S]}$ is canonically homeomorphic to $\\prod_{C_i \\subset \\mathcal{T}^S} M_{\\mathbf{r}_i}^o$\nwhere $\\mathbf{r}_i$ is the subvector of linkage lengths corresponding to the elements in $dist(C_i)$,\nand $M_{\\mathbf{r}_i}^o$ is the dense open subset of $M_{\\mathbf{r}_i}$ corresponding to polygons\nwith no zero diagonals.\n\\end{theorem}\n\n\\begin{proof}\nLet us first describe a map $\\tilde{F}: M_{\\mathbf{r}}^{S} \\to V_{\\mathbf{r}_1} \\times \\ldots \\times V_{\\mathbf{r}_k}$.\nThe diagonals $S$ define a partition of edges sets $E_i(\\mathbf{e})$ for each $\\mathbf{e} \\in \\widetilde{M}_{\\mathbf{r}}^{S}$.  By the above proposition, $E_i(\\mathbf{e})$ corresponds to a closed polygon.\nSo we may send a member of the equivalence class of $\\mathbf{e}$ to the product of the equivalence classes\ndefining these closed polygons in $V_{\\mathbf{r}_1} \\times \\ldots \\times V_{\\mathbf{r}_k}$, with the appropriate\n$\\mathbf{r}_i$.  This map is certainly onto, and by the definition of $V_{\\mathbf{r}_i}$, it factors through\nthe relation $\\sim{\\mathcal{T}}$. Hence, we get a 1-1 and onto continuous function\n $F: (V_{\\mathbf{r}}^{\\mathcal{T}})^{S} \\to V_{\\mathbf{r}_1} \\times \\ldots \\times V_{\\mathbf{r}_k}$, which is a homeomorphism\nby the compactness of $V_{\\mathbf{r}_1} \\times \\ldots \\times V_{\\mathbf{r}_k}$.\n\n\nWe may restrict this map to the spaces $(V_{\\mathbf{r}}^{\\mathcal{T}})^{[S]}$ which define the induced\ndecomposition of $V_{\\mathbf{r}}^{\\mathcal{T}}$.  Recall that these are the polygons with exactly the $S$\ndiagonals zero.  We therefore obtain that $(V_{\\mathbf{r}}^{\\mathcal{T}})^{[S]}$ is homeomorphic to $M_{\\mathbf{r}_1}^o \\times \\ldots \\times M_{\\mathbf{r}_k}^o$,\n\\end{proof}\n\n\\begin{remark}\nSince the fibers of $\\pi : M_\\mathbf{r} \\to V^\\mathcal{T}_\\mathbf{r}$ are sometimes\nodd-dimensional, the quotient map $\\pi$ cannot be algebraic\neven when $\\mathbf{r}$ is integral.\n\\end{remark}\n\n\n\n\\subsection{$\\mathcal{T}$-congruence of imploded spin-framed polygons}\n\nIn this section we introduce a generalization of the $\\mathcal{T}$-congruence\nrelation by lifting $\\mathcal{T}$-congruence from $\\mathrm{Pol}_ n(\\mathbb{R}^3)$ to $P_n(\\mathrm{SU}(2))$.\nIn section 3 we constructed a map $F_n: P_n(\\mathrm{SU}(2)) \\rightarrow \\mathrm{Pol}_ n(\\mathbb{R}^3)\/\\mathrm{SO}(3,\\mathbb{R})$. \nPulling back the decomposition of $\\mathrm{Pol}_ n(\\mathbb{R}^3)\/\\mathrm{SO}(3,\\mathbb{R})$ into $\\mathcal{T}$-\ncongruence classes\n by $F_n$ produces a decomposition of $P_n(\\mathrm{SU}(2))$ by the spaces\nwe will denote $P_n(\\mathrm{SU}(2))^{[S]} =F_n^{-1}(\\mathrm{Pol}_ n(\\mathbb{R}^3)^{[S]})$.\n\n\\begin{lemma}\\label{P_nStrat}\nElements of $P_n(\\mathrm{SU}(2))^{[S]}$ are the spin-framed polygons\nin $P_n(\\mathrm{SU}(2))$ such that the following equation holds,\n\n$$\\sum \\lambda_iAd^*_{g_i}(\\varpi_1) = 0$$\nwhere the sum is over all $[g_i, \\lambda_i\\varpi]\\in \\mathbf{E}_j(\\mathbf{e})$, the\nedges in the $j$-th partition defined by $S$, for each $j$.\n\\end{lemma}\n\n\\begin{proof}\nThis follows from Theorem \\ref{framedngons} and Remark \\ref{SpinF}.\n\\end{proof}\nBy the previous lemma we have a decomposition of $\\mathbf{e} \\in P_n(\\mathrm{SU}(2))^{[S]}$ into imploded spin-framed\npolygons\n\n$$\\mathbf{E} = \\mathbf{E}_1(\\mathbf{e} ) \\cup \\ldots \\cup \\mathbf{E}_{|S|+1}(\\mathbf{e} ).$$\nWe define an action of $\\mathrm{SU}(2)^{D(S)}$ on $P_n(\\mathrm{SU}(2))^{[S]}$ by\nletting $\\mathrm{SU}(2)^{D(C)}$ act diagonally on the framed edges associated\nto the component $C$ of $\\mathcal{T}^S$.\n\n\n\\begin{definition}\nThe   $\\mathrm{SU}(2)^{D(S)}$-orbits  on $P_n(\\mathrm{SU}(2))^{[S]}$\nfit together to give an equivalence relation $\\sim_{\\mathcal{T}}$,\nwhich we again call $\\mathcal{T}$-congruence, on $P_n(\\mathrm{SU}(2))$.\n\\end{definition}\n\n\\begin{definition}\n$$V_n^{\\mathcal{T}} = P_n(\\mathrm{SU}(2))\/ \\sim_{\\mathcal{T}}$$\n\\end{definition}\nNote that this defines a decomposition on $V_n^{\\mathcal{T}}$ into\nsubspaces $(V_n^{\\mathcal{T}})^{[S]} = P_n(\\mathrm{SU}(2))^{[S]}\/\\mathrm{SU}(2)^{D(S)}$.\nWe have seen that  $P_n(\\mathrm{SU}(2))$ carries an action of $T_{\\mathrm{SU}(2)^n}$ given\nby the following formula. Let $\\mathbf{t} = (t_1, \\cdots, t_n) \\in T_{\\mathrm{SU}(2)^n}$,\n\n$$ \\mathbf{t} \\circ ([g_1, \\lambda_1\\varpi_1, \\ldots, [g_n, \\lambda_n \\varpi_1)   =\n([g_1t_1, \\lambda_1\\varpi_1], \\ldots, [g_nt_n, \\lambda_n\\varpi_1).$$\nIn particular, note that the action of the diagonal $(-1)$ element in $T_{\\mathrm{SU}(2)^n}$ is trivial, because\nthis corresponds to acting on the left by the nontrivial central $\\mathrm{SU}(2)$ element.\nSince $t_i$ fixes $\\varpi_1$ for $1 \\leq i \\leq n$  this action does not change the image of a point under\n$F_n$, hence it fixes each piece of the decomposition.  Also this action commutes\nwith the $\\mathrm{SU}(2)^{D(S)}$-action on the piece $P_n(\\mathrm{SU}(2))^{[S]}$, so\nthe action of $T_{\\mathrm{SU}(2)^n}$ must descend to $V_n^{\\mathcal{T}}$.\n\n\n\n\n\n\n\n\\section{The toric varieties $P^{\\mathcal{T}}_{n}(\\mathrm{SU}(2))$ and $Q^{\\mathcal{T}}_{n}(\\mathrm{SU}(2))$}\n\nIn this section we will construct the affine toric variety $P^{\\mathcal{T}}_{n}(\\mathrm{SU}(2))$ of imploded triangulated $\\mathrm{SU}(2)$--framed $n$--gons (without imposing the condition that the side-lengths are $\\mathbf{r}$) and its projective quotient\n$Q^{\\mathcal{T}}_{n}(\\mathrm{SU}(2))$.\n\nWe have tried to follow the notation of \\cite{HurtubiseJeffrey} when possible. In \\cite{HurtubiseJeffrey} the superscript $D$ on $P^D$ refers to a ``pair of pants'' decomposition $D$ of the surface. For us superscript $\\mathcal{T}$ on $P_n^{\\mathcal{T}}(\\mathrm{SU}(2))$ refers to the triangulation of the $n$-gon. The connection is the following. Under the correspondence between moduli spaces of $n$-gons and character varieties briefly explained in Remark \\ref{moduliofconnections}\n(and explained in detail in \\S 5 of \\cite{KapovichMillson}) the standard triangulation corresponds to the following ``pair of pants'' decomposition of the $n$ times punctured two-sphere. Represent the sphere as the complex plane with a point at infinity. Take the punctures to be the points $1,2,\\cdots,n$ on the real line. Draw small circles around the punctures. Now draw n-3 more circles with centers on the $x$-axis, so that the first circle contains the small circles around $1$ and $2$, the next circle contains the circle just drawn and the small circle around 3 and the last circle contains all the previous circles except the small circles around $n-1$ and $n$. The graph dual to the pair of pants decomposition is the tree $\\mathcal{T}$ that is dual to the triangulation.\nFurthermore the decomposed tree $\\mathcal{T}^D$ is dual to the pair-of-pants\ndecomposition of the $n$-punctured sphere obtained by cutting the sphere\napart along the above $2n-3$ circles - we might say that $\\mathcal{T}^D$ is the\npair-of-pants decomposition of $\\mathcal{T}$.\nUsing this correspondence the reader should be able to relate what follows with \\cite{HurtubiseJeffrey} for the case of the $n$-fold punctured sphere.\n\n\n\nIt will be important in what follows that we have earlier defined the quotient map\n$$\\pi^{D}: \\mathcal{T}^D \\to \\mathcal{T}$$\nthat glues together the pairs of vertices that are the boundaries of\nthe open intervals removed from $\\mathcal{T}$.\n\n\n\\subsection{The space $E^{\\mathcal{T}}_n(\\mathrm{SU}(2))$ of imploded framed edges}\n\nWe define $E^{\\mathcal{T}}_n(\\mathrm{SU}(2))$ to be the product of $\\mathcal{E}T^*(\\mathrm{SU}(2))$\nover the edges of $\\mathcal{T}^D$, hence an element $\\mathbf{T} \\in\nE^{\\mathcal{T}}_n(\\mathrm{SU}(2))$  is a map\nfrom the $3(n-2)$ edges of $\\mathcal{T}^D$ into $\\mathcal{E}T^*(\\mathrm{SU}(2))$ or equivalently\nan assignment of an element of $\\mathcal{E}T^*(\\mathrm{SU}(2))$ to each\neach of the $3(n-2)$ edges of the triangles $\\tau_i,1 \\leq n-2$ in the triangulation\nof $P$.  It will be important later to note that there is a forgetful\nmap that restricts $\\mathbf{T}$ to the distinguished edges of $\\mathcal{T}^D$ to obtain\nan element $\\mathbf{E}$ of $E_n(\\mathrm{SU}(2))$.\n\n\nIt will be convenient to represent\nthe resulting product $\\mathcal{E}T^*\\mathrm{SU}(2)^{3n-6} \\cong (\\mathbb{C}^2)^{3n-6}$ by a\n$2$ by $3n-6$ matrix. To do this we will linearly order the $3n-6$ edges by\nfirst ordering the $n-2$ triangles (tripods) and then ordering the $3$ edges for\neach triangle (tripod). Now use the lexicographic ordering on pairs $(triangle,edge)$\n($(tripod,edge)$). The point is that in the matrix $A^{\\mathcal{T}}$ below the $(\\mathbb{C}^2)$'s belonging\nto the same triangle (tripod) appear consecutively. Let $(\\tau_i, j)$ denote the\n$j$-th edge of the $i$-th tripod.\n\nWe will use the following notation for the entries in the matrix $A^{\\mathcal{T}}$ to indicate\nthat each successive group of $3$ columns of $A^{\\mathcal{T}}$ belongs to a triangle in the\ntriangulation. \n$$A^{\\mathcal{T}} =\n\\begin{pmatrix}\nz_1(\\tau_1) & z_2(\\tau_1) & z_3(\\tau_1) & z_1(\\tau_2) & z_2(\\tau_2) & z_3(\\tau_2) & \\cdots & z_1(\\tau_{n-2}) & z_2(\\tau_{n-2}) & z_3(\\tau_{n-2}) \\\\\nw_1(\\tau_1) & w_2(\\tau_1) & w_3(\\tau_1) & w_1(\\tau_2) & w_2(\\tau_2) & w_3(\\tau_2) & \\cdots & w_1(\\tau_{n-2}) & w_2(\\tau_{n-2}) & w_3(\\tau_{n-2})\n\\end{pmatrix}$$\nIn what follows we will abbreviate $\\mathcal{E}T^*(\\mathrm{SU}(2))^{3n-6}$\nto $E^{\\mathcal{T}}_n(\\mathrm{SU}(2))$. \n\n\n\n\n\n\n\\subsection{The space $X^{\\mathcal{T}}_n(\\mathrm{SU}(2))$ of imploded framed triangles}\n\nThe action of the group $\\mathrm{SU}(2)^{3n-6}$ on $\\mathcal{E}T^*(\\mathrm{SU}(2))^{3n-6}$ is then\nrepresented by acting on the columns of $A^{\\mathcal{T}}$. We will represent elements $\\mathbf{g}$ of\n$\\mathrm{SU}(2)^{3n-6}$ as $3n-6$-tuples\n\n$$\\mathbf{g} = (f_1(\\tau_1),f_2(\\tau_1),f_3(\\tau_1)|\\cdots|f_1(\\tau_{n-2}),,f_2(\\tau_{n-2}),\nf_3(\\tau_{n-2})).$$\nWe let $\\mathrm{SU}(2)^{n-2}$ denote the ``diagonal'' subgroup of $\\mathrm{SU}(2)^{3n-6}$ defined\nby the condition that for each triangle $T_i$ (tripod $\\tau_i$) we have\n\n$$f_1(\\tau_i) = f_2(\\tau_i) = f_3(\\tau_i).$$\nWe will regard $\\mathrm{SU}(2)^{n-2}$ as the space of mappings $\\mathbf{f}$ from the tripods in $\\mathcal{T}^D$ to $\\mathrm{SU}(2)$.\n\n\n\\begin{definition}\nWe then define the space $X^{\\mathcal{T}}_n(\\mathrm{SU}(2))$ as the symplectic quotient\n$$ X^{\\mathcal{T}}_n(\\mathrm{SU}(2)) = \\mathrm{SU}(2)^{n-2}\\backslash \\! \\backslash (\\mathcal{E}T^*\\mathrm{SU}(2))^{3n-6}.$$\n\\end{definition}\n\nThus $X^{\\mathcal{T}}_n(\\mathrm{SU}(2))$ is obtained by taking the symplectic quotient of each\ntriple of edges belonging to one of the triangles $T_i, 1 \\leq i \\leq n-2$, by\nthe group $\\mathrm{SU}(2)^{n-2}$. The resulting space $X_n(\\mathrm{SU}(2))$ is clearly the product of $n-2$\ncopies of $(P_3(\\mathrm{SU}(2)))^{n-2} \\cong (\\bigwedge^{2}(\\mathbb{C}^3))^{n-2}$.\\\\\n\nWe will often denote an element of $X^{\\mathcal{T}}_n(\\mathrm{SU}(2))$ by $\\mathbf{T}=([T_1],[T_2],\\cdots,[T_{n-2}])$\nwhere $[T_i]$, the $i$-th triangle denotes the equivalence class of the matrix\n\n$A_{\\tau_i} =\n\\begin{pmatrix}\nz_1(\\tau_i) & z_2(\\tau_i) & z_3(\\tau_i)\\\\\nw_1(\\tau_i) & w_2(\\tau_i) & w_3(\\tau_i)\n\\end{pmatrix}$\nsuch that the momentum image of $A_i$ under the momentum map for $\\mathrm{SU}(2)$ is zero\n(equivalently the rows are orthogonal with the same length).\n\n\n\n\\subsection{The affine toric variety $P_n^{\\mathcal{T}}(\\mathrm{SU}(2))$}\\label{mastertoricspace}\n\nIn equation (\\ref{P3}) we have seen that\n$$P_3(\\mathrm{SU}(2)) \\cong \\mathrm{\\bigwedge} \\space^2(\\mathbb{C}^3).$$\nSince the space of imploded triangles is the product of\n$n-2$ copies of $P_3(\\mathrm{SU}(2))$ we see that\n$X^{\\mathcal{T}}_n(\\mathrm{SU}(2))$ is the affine space obtained by taking the product of $n-2$ copies\nof $\\bigwedge^2(\\mathbb{C}^3)$. It has an action of a $3n-6$ torus $\\mathbb{T}$ induced\nby the right actions of the torus $\\mathbb{T} = T _{\\mathrm{SU}(2)}^{3n-6}$ on the $3n-6$ copies\nof $\\mathcal{E}T^*(\\mathrm{SU}(2))$. In terms of our matrices $A^{\\mathcal{T}}$ this \namounts to scaling the\ncolumns of $A^{\\mathcal{T}}$ (right multiplication of $A^{\\mathcal{T}}$ by $\\mathbb{T}$).\nHowever note that the  entry of an element of $\\mathbb{T}$ corresponding to\nthe edge $e$ of $\\mathcal{T}^D$\nscales the column of $A^{\\mathcal{T}}$ corresponding to $e$ by its {\\it inverse}.\nWe will use the notation $\\underline{\\mathbb{T}}$ to indicate the complexification of \n$\\mathbb{T}$.  Similarly for any compact group $G$ that appears in this paper, \nthe notation $\\underline{G}$ indicates the complexification of $G$.\n\nLet $\\mathbb{T}_e$, $\\mathbb{T}_d$, $\\mathbb{T}_d^-$ be as in the introduction.\nNow we glue the diagonals of $P$ back together by taking the\nsymplectic quotient by $\\mathbb{T}_d^-$ at level $0$.\n\n\n\\begin{definition}\n$$P_n^{\\mathcal{T}}(\\mathrm{SU}(2)) = X_n^{\\mathcal{T}}(\\mathrm{SU}(2)\/\/\\mathbb{T}_d^- = (\\mathrm{\\bigwedge} \\space^2(\\mathbb{C}^3))^{n-2}\/\/\\mathbb{T}_d^-.$$\n\\end{definition}\n\n\n\\subsection{The space $P_n^{\\mathcal{T}}(\\mathrm{SO}(3,\\mathbb{R}))$}\\label{orthogonalanalogues}\nThe spaces $E_n^{\\mathcal{T}}(\\mathrm{SU}(2)), X_n^{\\mathcal{T}}(\\mathrm{SU}(2))$ and $P_n^{\\mathcal{T}}(\\mathrm{SU}(2))$ all have analogues when $\\mathrm{SU}(2))$ is replaced by\n$\\mathrm{SO}(3,\\mathbb{R})$ which are quotients\nby a finite group of the corresponding spaces for $\\mathrm{SU}(2)$. There\nalso analogues of the tori $\\mathbb{T},\\mathbb{T}_e$ and $\\mathbb{T}_d^-$\nfor $\\mathrm{SO}(3,\\mathbb{R})$ which we will denote $\\mathbb{T}(\\mathrm{SO}(3,\\mathbb{R})),\\mathbb{T}_e(\\mathrm{SO}(3,\\mathbb{R}))$ and $\\mathbb{T}_d^-(\\mathrm{SO}(3,\\mathbb{R}))$ respectively which are quotients of the\ncorresponding tori for $\\mathrm{SU}(2)$.  We leave the details to the reader.\n\n\\subsection{$P_n^{\\mathcal{T}}(\\mathrm{SU}(2))$ as a GIT quotient}\\label{firsttorusquotient}\nSince affine GIT quotients coincide with symplectic quotients, see  \\cite{KempfNess} and  \\cite{Schwarz},\nTheorem 4.2,  we may also obtain $P_n^{\\mathcal{T}}(\\mathrm{SU}(2))$ as the GIT quotient of $(\\bigwedge^2(\\mathbb{C}^3))^{n-2}$\nby the complex torus $\\underline{\\mathbb{T}}^d$. For each triangle $T_k$ (tripod $\\tau_k$) we have a corresponding\n$\\bigwedge^2(\\mathbb{C}^3)$ with Pl\\\"ucker coordinates $Z_{ij}(\\tau_k), 1 \\leq i,j \\leq 3$. The coordinate\nring of the affine variety $P_n^{\\mathcal{T}}(\\mathrm{SU}(2))$ will be the ring of invariants\n$\\mathbb{C}[(Z_{ij}(\\tau_k))]^{\\underline{\\mathbb{T}}^-_d}$. This ring of invariants will be spanned by the ring of\ninvariant {\\em monomials} which we now determine. There is a technical problem\nhere. We need to know that we have chosen the correctly normalized\nmomentum map for the action $\\mathbb{T}^-_d$. But by Theorem \\ref{twistandshift}\nthe correct normalization is the one that is homogeneous linear in the\nsquares of the norms of the coordinates which is the one we have used here.\n\n\\subsection{The semigroup $\\mathcal{P}^{\\mathcal{T}}_n$}\n\nAs a geometric quotient of affine space by a torus the space\n$P_n^{\\mathcal{T}}(\\mathrm{SU}(2))$ is an affine toric variety. We now analyze the\naffine coordinate ring of $P_n^{\\mathcal{T}}(\\mathrm{SU}(2))$.   In what follows we label the leaf of the tripod\n$\\tau_i$ incident to the edge $(\\tau_i,k)$ by $k$ - we will do this only when the tripod of which\n$k$ is a vertex is clearly indicated. We leave the proof of the following lemma to the reader.\n\n\n\\begin{lemma}\nThe monomial $f(Z)=\n\\prod_{i=1}^{n-2}Z_{12}(\\tau_i)^{x_{12}(\\tau_i)}Z_{13}(\\tau_i)^{x_{13}(\\tau_i)}\nZ_{23}(\\tau_i)^{x_{23}(\\tau_i)}$ is $\\underline{\\mathbb{T}}_d^-$--invariant if and only if the\nexponents $\\mathbf{x} = (x_{jk}(\\tau_i))$ satisfy the system of equations\n\n$$x_{k,m}(\\tau_i) + x_{k,l}(\\tau_i) = x_{l,k}(\\tau_j) + x_{l,m}(\\tau_j), \\quad \\text{for } \\ (\\tau_i, k) \\ \\text{identified to } (\\tau_j, l) \\ \\text{in $\\mathcal{T}$}.$$\n\\end{lemma}\nBefore stating a corollary of the lemma we need a definition.\n\n\\begin{definition}\nLet $\\mathcal{P}^{\\mathcal{T}}_n$ be the subset of $\\mathbf{x} \\in (\\mathbb{N}^3)^{(n-2)}$ satisfying the equations\nin the above lemma.  Clearly $\\mathcal{P}^{\\mathcal{T}}_n$ is a semigroup under addition.\nLet $\\mathbb{C}[\\mathcal{P}_n^{\\mathcal{T}}]$ denote the associated semigroup algebra.\n\\end{definition}\n\n\\begin{corollary}\nThe affine coordinate ring of\n$P_n^{\\mathcal{T}}(\\mathrm{SU}(2))$ is isomorphic to the semigroup ring $\\mathbb{C}[\\mathcal{P}_n^{\\mathcal{T}}]$.\n\\end{corollary}\n\n\nNow we will relate the semigroup $\\mathcal{P}^{\\mathcal{T}}_n$ and the monomials of the lemma to graphs on vertices of the decomposed tree $\\mathcal{T}^D$. Monomials in the Pl\\\"ucker coordinates $Z_{ij}(\\tau)$ for $\\tau$ fixed  correspond to graphs on the vertices $i,j,k$  of the tripod $\\tau$ as follows. We associate to  the exponent $x_{ij}(\\tau)$ of $Z_{ij}(\\tau)$    the graph consisting of $x_{ij}(\\tau)$ arcs joining the vertex  $i$ of $\\tau$ to the vertex $j$. Each triple $\\{x_{ij}(\\tau)\\}$ for $\\tau$ fixed determines a graph on the leaves of the tripod $\\tau$.\nThus each element $\\mathbf{x} \\in \\mathcal{P}_n^{\\mathcal{T}}$ corresponds to a collection of $n-2$ graphs,\neach on 3 vertices, one for each tripod in  $\\mathcal{T}^D$. Also each such element\n$\\mathbf{x}$  corresponds to a product of monomials attached to tripods. This leads to a bijective correspondence\nbetween monomials and graphs.\n\n\n\n\\begin{figure}[htbp]\n\\centering\n\\includegraphics[scale = 0.4]{tripod.eps}\n\\caption{A single tripod $\\tau$ with vertex $v$, with $x_{ef}(\\tau) = 3$, $x_{eg}(\\tau)=2$, $x_{fg}(\\tau)=1$.  Hence $x_{ef}(\\tau)+x_{eg}(\\tau)=5$, $x_{ef}(\\tau)+x_{fg}(\\tau)=4$, $x_{eg}(\\tau)+x_{fg}(\\tau)=3$. }\n\\label{fig:tripod}\n\\end{figure}\nThe condition\n\n$$x_{km}(\\tau_i) + x_{k\\ell}(\\tau_i) = x_{\\ell,k}(\\tau_j) + x_{\\ell,m}(\\tau_j)$$\nspecifies that the number of arcs associated to the identified  edges $(\\tau_i, k)$ of\nthe tripod $\\tau_i$, and $(\\tau_j,\\ell)$ of the tripod $\\tau_j$ must agree.  Later, this will allow\nus to glue these ``local'' graphs to obtain a ``global'' graph\non $n$ vertices.  In the next section we will associate a weighting of $\\mathcal{T}$ to the $\\{x_{ij}(\\tau_k)\\}$.\n\n\\subsection{The projective toric variety $Q_n^{\\mathcal{T}}(\\mathrm{SU}(2))$}\n\nWe define $Q_n^{\\mathcal{T}}(\\mathrm{SU}(2))$, a projectivization of $P_n^{\\mathcal{T}}(\\mathrm{SU}(2))$,\nwhich will be isomorphic to $\\mathrm{Gr}_2(\\mathbb{C}^n)^\\mathcal{T}_0$. The coordinate ring of the affine\ntoric variety $P_n^{\\mathcal{T}}(\\mathrm{SU}(2))$ is the semigroup algebra $\\mathbb{C}[\\mathcal{P}_n^{\\mathcal{T}}]$.\nHence to define the  $\\mathbb{C}^*$ action required\nfor projectivization, it suffices to define a grading on $\\mathcal{P}^{\\mathcal{T}}_n$.\nFirst, for $\\mathbf{x} \\in P^{\\mathcal{T}}$ we define the associated weight of the\nedge $(\\tau,i)$  of the tripod $\\tau$ in\n$\\mathcal{T}^D$.\nlet $w_{(\\tau,i)}(\\mathbf{x} ) = x_{ij}(\\tau) + x_{ik}(\\tau)$.\nWe define the degree of an element $\\mathbf{x} \\in \\mathcal{P}_n^{\\mathcal{T}}$  by\n\n$$degree(\\mathbf{x}) = \\frac{1}{2} \\sum_{(\\tau,i) \\in dist(\\mathcal{T}^D)} w_{(\\tau, i)}(\\mathbf{x})$$\nwhere the sum is over all $(\\tau, i)$ which represent leaf edges of $\\mathcal{T}$.\n\n\\begin{remark}\nWe will see below that the under the isomorphism from the semigroup $\\mathcal{P}_n^{\\mathcal{T}}$\nto the semigroups of admissible weightings $\\mathcal{W}_n^{\\mathcal{T}}$  the number $w_{(\\tau,i)}(\\mathbf{x})$ will be the weight assigned to the edge $(\\tau,i)$\nof the tree $\\mathcal{T}$. Thus the degree of $\\mathbf{x}$ is then half the sum of the weights of the leaf edges.\n\\end{remark}\n\n\n\\begin{definition}\nGive $\\mathcal{P}^{\\mathcal{T}}_n$ the grading defined above, then we define\n$$Q_n^{\\mathcal{T}}(\\mathrm{SU}(2)) = \\mathrm{Projm}(\\mathbb{C}[\\mathcal{P}_n^{\\mathcal{T}}])$$\n\\end{definition}\nNote that as defined $Q_n^{\\mathcal{T}}(\\mathrm{SU}(2))$ is a GIT $\\mathbb{C}^*$-quotient of $P_n^{\\mathcal{T}}(\\mathrm{SU}(2))$.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{The toric varieties $Q^{\\mathcal{T}}_n(\\mathrm{SU}(2))$ and $\\mathrm{Gr}_2(\\mathbb{C}^n)^{\\mathcal{T}}_0$ are isomorphic}\n\n\nIn this section we will prove the first statement of Theorem \\ref{firstauxiliarytheorem} namely that the toric variety $Q_n^{\\mathcal{T}}(\\mathrm{SU}(2))$ constructed in\nthe previous section is the isomorphic to $\\mathrm{Gr}_2(\\mathbb{C}^n)^{\\mathcal{T}}_0$ by proving\nProposition \\label{threeisomorphicsemigroups} below.\n\nRecall that in Section 4 it was shown that the two semigroups $\\mathcal{S}^{\\mathcal{T}}_n$ and $\\mathcal{W}^{\\mathcal{T}}_n$ are isomorphic.\n$\\mathcal{S}^{\\mathcal{T}}_n$ is the semigroup of Kempe Graphs on $n = |edge(\\mathcal{T})|$ vertices, with the product $\\ast_{\\mathcal{T}}$.  $\\mathcal{W}^{\\mathcal{T}}_n$ is the semigroup of admissible weightings on $\\mathcal{T}$ under addition.  Recall that an admissible weighting is an integer weight satisfying the triangle inequalities about each internal vertex of $\\mathcal{T}$, along with the condition that the sum of the weights about each internal node be even. It was also shown in section 4 that the semigroup algebra of $\\mathcal{S}_n^{\\mathcal{T}} \\cong \\mathcal{W}_n^{\\mathcal{T}}$ is isomorphic to the coordinate ring of $\\mathrm{Gr}_2(\\mathbb{C}^n)_0^{\\mathcal{T}}$.\n\n\\begin{proposition}\\label{twoisomorphicsemigroups}\nThe graded semigroups $\\mathcal{W}^{\\mathcal{T}}_n$ and $\\mathcal{P}^{\\mathcal{T}}_n$\nare isomorphic.\n\\end{proposition}\n\n\\begin{proof}\nIn all that follows we deal with tripods, so we give the unique leaf edge incident to the leaf labelled $i$ the label $i$, and vice-versa.\nWith this in mind, $X_{ij}$ can be thought of as giving the number of arcs in a graph on the\nleaves of a tripod $Y$ between the $i$-th and $j$-th leaves.  The number $N_i$ is a natural number\nassigned to the $i$-th edge of a tripod $Y$ obtained by counting the number of arcs which have unique\npath in $Y$ containing the $i$-th edge.\nThe elements of both  $\\mathcal{P}^{\\mathcal{T}}_n$ and $\\mathcal{W}^{\\mathcal{T}}_n$ both associate triples of\nintegers to each tripod of ${\\mathcal{T}^D}$ with certain ``gluing conditions''. Let us consider a single tripod $Y \\in \\mathcal{T}^D$.  A triple of numbers $N_1, N_2, N_3$ is an admissible weighting of $Y$ if and only if there are integers $X_{ij}$, $i, j \\in \\{1, 2, 3\\}$ such that $X_{ij} + X_{ik} = N_i$.  To see this simply note that the equations\n$$X_{ij} = \\frac{N_i + N_j - N_k}{2}$$\nhave natural solutions if and only if $(N_1, N_2, N_3)$ is admissible.\nTherefore we may define a map from $\\mathcal{W}^{\\mathcal{T}}_n$ to $\\mathcal{P}^{\\mathcal{T}}_n$ by solving for $x_{ij}(\\tau)$, with an obvious inverse given by solving for the weighting on the edge $(\\tau, i)$.  To see that these maps are well defined, note that the gluing condition\n$$x_{km}(\\tau_i) + x_{k\\ell}(\\tau_i) = x_{\\ell,k}(\\tau_j) + x_{\\ell,m}(\\tau_j), \\quad \\text{for } (\\tau_i, k) \\text{ identified to } (\\tau_j, \\ell) \\text{$\\in \\mathcal{T}$}.$$\nis exactly equivalent to the weights on $(\\tau_i, n)$ and $(\\tau_j, l)$ being equal when these tuples represent the same diagonal in $\\mathcal{T}$.  Since both semigroup operations are defined by adding integers, and since both $\\Phi$ and its inverse are linear functions over each trinode, these maps are semigroup isomorphisms.\nFinally, note that the grading on $\\mathcal{P}^{\\mathcal{T}}_n$ was chosen specifically to match the grading on $\\mathcal{S}^{\\mathcal{T}}_n$, we leave\ndirect verification of this to the reader.\n\\end{proof}\n\n\n\n\\subsubsection{An explicit description of the ring isomorphism}\n The semigroup $\\mathcal{S}^{\\mathcal{T}}_n$ is generated as a graded semigroup by the elements corresponding to\ngraphs with exactly one edge.  The element corresponding to the graph with one edge, between the $i$ and $j$ vertices corresponds to the Pl\\\"ucker coordinate $Z_{ij}$.  Recall that the unique path in the tree $\\mathcal{T}$ joining\nthe leafs $i$ and $j$ has been denoted $\\gamma(ij)$. The path $\\gamma(ij)$\ngives rise to a sequence of edges of the forest $\\mathcal{T}^D$ (where we pass from\none tripod to the next by passing from an edge $(\\tau_i, k)$ to an equivalent\nedge $(\\tau_j,\\ell)$. We let $Z_{\\gamma(ij)}$ be the corresponding\nproduct of Pl\\\"ucker coordinates $Z_{st}(\\tau_k)$.  Thus\n$$Z_{\\gamma(ij)} = \\prod Z_{st}(\\tau)$$\nwhere the $(\\tau, s)$ and $(\\tau, t)$ are the edges in the unique path\ndefined by the path $\\gamma(i,j)$ corresponding to $Z_{ij}$ in $\\mathcal{T}$.\nNote that $Z_{\\gamma(ij)}$ is a $\\underline{\\mathbb{T}}_d^-$-invariant monomial\nin the homogeneous coordinate ring of $(\\bigwedge^2(\\mathbb{C}^3))^{n-2}$ -\nmoreover it is a generator of the ring of invariants.\nThe isomorphism of toric rings $\\Phi$ from the homogeneous\ncoordinate ring of $\\mathrm{Gr}_2(\\mathbb{C}^n)^{\\mathcal{T}}_0$ to $P^{\\mathcal{T}}_n(\\mathrm{SU}(2)$ is then given on generators\nby\n$$\\Phi(Z_{ij}) = Z_{\\gamma(ij)}.$$\n\\begin{remark}\nIt is important to see that the degree assigned to $Z_{\\gamma(ij)} = \\prod Z_{st}(\\tau)$ by the isomorphism $\\Phi$ (namely {\\it one}) is\ndifferent from that given by counting the $Z_{st}(\\tau)$ in the product formula\nfor $Z_{\\gamma(i,j)}$. The latter\ncount is in fact the Speyer-Sturmfels weight $w^{\\mathcal{T}}_{i,j}$ of the Pl\\\"ucker coordinate $Z_{ij}$.\n\\end{remark}\n\n\\begin{figure}[htbp]\n\\centering\n\\includegraphics[scale = 0.3]{invar.eps}\n\\caption{This illustrates $\\Phi(Z_{14}) = Z_{\\gamma(14)} = Z_{13}(\\tau_1)Z_{12}(\\tau_2)Z_{13}(\\tau_4)$ in \nthe case $n=6$ with the symmetric tree.}\\label{fig:tripod2}\n\\end{figure}\n\n\\subsubsection{The grading circle action on $P_n^{\\mathcal{T}}(\\mathrm{SU}(2))$}\n\nWe verify the second statement of Theorem \\ref{firstauxiliarytheorem}.\nWe have seen that the action of $\\lambda \\in \\mathbb{C}^*$ that gives the grading\nscales each $Z_{\\gamma(i,j)}$ by $\\lambda$.\nClearly this action is induced by the action on the matrix $A^{\\mathcal{T}}$\nthat scales each row corresponding to a leaf edge by $\\sqrt{\\lambda}$,\nin other words by the action of the element $\\mathbf{t}_e((\\sqrt{\\lambda})^{-1})$ as claimed in\nthe second statement of Theorem \\ref{firstauxiliarytheorem}.\nWe conclude by explaining why the actions of $\\mathbf{t}_e(\\sqrt{\\lambda})$\nand $\\mathbf{t}_e(-\\sqrt{\\lambda})$ coincide. It suffices to prove that\n$\\mathbf{t}_e(-1)$ acts trivially. But since the operation\nthat scales all rows of $A^{\\mathcal{T}}$ by $-1$ is induced by\nan element of $\\mathrm{SU}(2)^{n-2}$ it acts trivially on $X_n^{\\mathcal{T}}(\\mathrm{SU}(2))$\nand hence on $P_n^{\\mathcal{T}}(\\mathrm{SU}(2))$. But also the operation of scaling all\nthe rows of $A^{\\mathcal{T}}$ that belong to nonleaf edges of $\\mathcal{T}^D$\nis induced by an element of $\\underline{\\mathbb{T}}_d^{-}$ hence  this\noperation too is trivial on $P_n^{\\mathcal{T}}(\\mathrm{SU}(2))$. But $\\mathbf{t}_e(-1)$\nis the composition of the two operations just proved to be trivial.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{The spaces $W^{\\mathcal{T}}_n$ and $Q_n^{\\mathcal{T}}(\\mathrm{SU}(2))$ are homeomorphic.}\\label{homeomorphismVtoP}\n\nWe now give the proof of Theorem \\ref{firstmaintheorem}.\nWe first note that we have identified an ordered subset of edges of $\\mathcal{T}^D$\nwith the leaf edges of $\\mathcal{T}$ (equivalently the edges of $P$). This\nidentification gives an isomorphism   $\\rho:T_{\\mathrm{SU}(2)^n} \\to \\mathbb{T}_e$.\n\n\\subsection{A homeomorphism of affine varieties}\\label{secondtorusquotient}\nWe will  prove the following.\n\n\\begin{theorem}\\label{noncompact}\nThe spaces $V^{\\mathcal{T}}_n$ and $P^{\\mathcal{T}}_n(\\mathrm{SU}(2))$ are equivariantly homeomorphic with respect to $\\rho$.\n\\end{theorem}\nWe will use the symbol\n$\\Psi^{\\mathcal{T}}_n$ to denote the above equivariant homeomorphism\nand $\\Phi^{\\mathcal{T}}_n$ to denote its inverse.\nBy the results of subsection \\ref{firsttorusquotient} it suffices to\nconstruct a $\\rho$-equivariant homeomorphism (and its inverse) from $V^{\\mathcal{T}}_n$\nto the above symplectic quotient which we will continue to denote\n$P^{\\mathcal{T}}_n(\\mathrm{SU}(2))$.  \nWe first construct the  $\\rho$-equivariant map \n$\\Phi^{\\mathcal{T}}_n : P_n^{\\mathcal{T}}(\\mathrm{SU}(2)) \\to V^{\\mathcal{T}}_n$.\nThere is a simple idea behind this map. We have an inclusion of\nthe leaf edges of $\\mathcal{T}$ into the edges of $\\mathcal{T}^D$.\nNow an element of $E_n^{\\mathcal{T}}(\\mathrm{SU}(2))$ is a map $\\mathbf{T}$ from the edges\nof $\\mathcal{T}^D$ into $\\mathbb{C}^2$. The map $\\Phi^{\\mathcal{T}}_n$ is induced by the map\n$\\widetilde{\\Phi}^{\\mathcal{T}}_n:E_n^{\\mathcal{T}}(SU(2)) \\to E_n(\\mathrm{SU}(2))$  that\nrestricts $\\mathbf{T}$ to the leaf edges of $\\mathcal{T}$.  However we need\nto verify that the image of an element of momentum level zero\nfor $\\mathrm{SU}(2)^{n-2} \\times \\mathbb{T}_d^-$ has $\\mathrm{SU}(2)$--momentum level zero,\nand that the induced map of zero momentum levels descends  to the required quotients.\nFor the rest of this discussion, let $\\textbf{T} = (F_1(\\tau_1), \\dots, F_{n-2}(\\tau_{n-2}))$ be an \nelement of $E_n^{\\mathcal{T}}(\\mathrm{SU}(2))$.  Each $F_i$ is an\nimploded spin framing of the edges of the triangle $\\tau_i$.\nThis means that\n$$F_i = [(g_1(i), \\lambda_1(i)\\varpi_1), (g_2(i), \\lambda_2(i)\\varpi_1), (g_3(i), \\lambda_3(i)\\varpi_1)]$$\nsuch that\n$$\\lambda_1(i)g_1(i)\\varpi_1 + \\lambda_2(i)g_2(i)\\varpi_1 + \\lambda_3(i)g_3(i)\\varpi_1 = 0$$\nWe will henceforth denote $T_i = F_i(\\tau_i)$, thus the symbol\n$T_i$ stands for a triangle together with an imploded spin-frame on its\nedges.\n\n\n\n\\subsubsection{$\\varrho$ flips and normalized framings}\\label{normalizedframings}\n\nLet $\\varrho = \\begin{pmatrix} 0 & 1 \\\\ -1 & 0\\end{pmatrix}$.  Note that if\n$t \\in \\mathrm{SU}(2)$ fixes $\\varpi_1$ by conjugation, then $t$ is diagonal and\n$\\varrho t \\varrho^{-1} = t^{-1}$. Furthermore, $\\varrho \\varpi_1 \\varrho^{-1} = - \\varpi_1$.\nLet $[[T_1],[T_2],\\ldots,[T_{n-2}]$ be an element of $P^{\\mathcal{T}}_n(\\mathrm{SU}(2))$.\n\n\\begin{lemma}\\label{rhoflip}\nSuppose that the $k$-th edge of $\\tau_i$ is identified with the $\\ell$-th edge\nof $\\tau_j$ in $\\mathcal{T}$ and that the edge $(\\tau_j, \\ell)$ comes after\nthe edge $(\\tau_i, k)$ in the above ordering of edges of $\\mathcal{T}^D$. Then using the left actions of $\\mathrm{SU}(2)$ we may arrange\nthat\n\n$$g_{(\\tau_j,\\ell)} = g_{(\\tau_i,k)}\\varrho.$$\n\\end{lemma}\n\n\\begin{proof}\nSuppose we have\n\n$$[T_i] = [(a_1, \\ell_1 \\varpi_1),(a_2, \\ell_2\\varpi_1), (h, d \\varpi_1)],$$\nIf the diagonal defined by $(\\tau_i, \\ell)$ and $(\\tau_j, k)$ has length $0$\nthere is nothing to prove since any two frames are equivalent. Suppose then\nthat this diagonal is not $0$. Then by applying $g =h \\varrho (h')^{-1}$ to $T_{j}$,\nwe get\n\n$$g \\cdot T_{j} = [(h \\varrho, d \\varpi_1),(g a_3,\\ell_3 \\varpi_1),(g a_4, \\ell_4 \\varpi_1)],$$\nwhich is equivalent to $T_{j}$.  Hence the consecutive diagonal frames (the third\nframe of $T_{i}$ and the first frame of $g \\cdot T_{i+1}$) are now related\nby right multiplication by $\\varrho$.\n\\end{proof}\nWe shall henceforth choose representatives in $E_n^{\\mathcal{T}}(\\mathrm{SU}(2))$ for\nelements of $P^{\\mathcal{T}}_n(\\mathrm{SU}(2))$ so that nonzero frames associated to equivalent edges\nsatisfy this ``$\\varrho$ flip condition''.  If  the frame associated to\none edge of a pair of equivalent edges is zero then we require  that\nthe frame associated to the other edge of the pair is also zero.  We say \nsuch elements of $P_n^{\\mathcal{T}}(\\mathrm{SU}(2))$ are {\\it normalized}.\nNote that the definition of normalized depends on the ordering\nof the triangles $T_i$.\n\n\nIt is important to note that if an element $A=([T_1],[T_2],\\cdots,[T_{n-2}])$ is normalized\nthen so is $t\\cdot A$ for any $t \\in \\mathbb{T}_d^-$. This is because $h\\varrho t^{-1} = ht\\varrho$. We leave the details\nto the reader. Thus we may speak of a $\\mathbb{T}_d^-$-equivalence class as being $normalized$.\nWe will define $\\Phi_n^{\\mathcal{T}}$ on such normalized elements.\n\n\n\\begin{lemma}\\label{norm}\nFor any $\\textbf{T} \\in  E_n^{\\mathcal{T}}(\\mathrm{SU}(2))$ there exist $\\mathbf{f} \\in \\mathrm{SU}(2)^{n-2}$ and  a normalized $\\textbf{T}' \\in  E_n^{\\mathcal{T}}(\\mathrm{SU}(2))$ such that\n$$\\mathbf{f} \\mathbf{T} = \\mathbf{T}'.$$\n\\end{lemma}\n\n\n\\begin{proof}\nLet $\\mathcal{T}' \\subset \\mathcal{T}$ be connected, and let $Y \\subset \\mathcal{T}$\nbe a tripod which shares exactly one edge, $d$ with $\\mathcal{T}'$.  Let $(\\tau, i)$\nand $(\\tau', j)$ be the edges of $\\mathcal{T}^D$ which map to $d$.\nSuppose $\\textbf{T} \\in E_n^{\\mathcal{T}}(\\mathrm{SU}(2))$ is normalized\nat all diagonals corresponding to internal edges of $\\mathcal{T}'$.\nLet $\\mathbf{f} \\in \\mathrm{SU}(2)^{n-2}$ be the element\nwhich is $g_{(\\tau, i)}\\varrho g_{(\\tau', j)}^{-1}$ on\nthe $\\tau'$-th factor, and the identity elsewhere.\nThen $\\textbf{T}' = \\mathbf{f}\\textbf{T}$ is normalized\nat all diagonals corresponding to internal edges of $\\mathcal{T}' \\cup Y$.\nBy lemma \\ref{trivalentinduction} we can always find a sequence\nof connected trees $\\mathcal{T}_i \\subset \\mathcal{T}_{i+1}$\nwith $Y_i = \\mathcal{T}_{i+1}^D \\setminus \\mathcal{T}_i^D$ a tripod\nsuch that each internal edge of $\\mathcal{T}$ appears\nas an edge shared by the images of $Y_i$ and $\\mathcal{T}_i$\nin $\\mathcal{T}$ for some $i$.  The lemma now follows by induction.\n\\end{proof}\n\n\n\\subsubsection{Definition of the maps $\\Phi_n^{\\mathcal{T}}$ and $\\Psi_n^{\\mathcal{T}}$}\n\nGiven a normalized element $\\textbf{T} \\in E^{\\mathcal{T}}_n(\\mathrm{SU}(2))$\ndefine $\\widetilde{\\Phi}_n^{\\mathcal{T}}(\\textbf{T})$ to be the element in $E_n(\\mathrm{SU}(2))$ given by projecting on the components\n$(g_{(\\tau_i, j)}, \\lambda_{(\\tau_i, j)}\\varpi_1) \\in \\textbf{T}$ such that $(\\tau_i, j) \\in \\mathcal{T}^D$\nmaps to a leaf edge in $\\mathcal{T}$ under $\\pi_{\\mathcal{T}}$.\n\n\\begin{lemma}\nIf $\\textbf{T} \\in \\mu_{\\mathrm{SU}(2)^{n-2}}^{-1}(0) \\cap \\mu_{\\mathbb{T}_d^-}^{-1}(0)$\nand is normalized then $\\widetilde{\\Phi}_n^{\\mathcal{T}}(\\textbf{T}) \\in\n\\widetilde{P}_n(\\mathrm{SU}(2))$, that is the polygon in $\\mathbb{R}^3$ associated to $\\widetilde{\\Phi}_n^{\\mathcal{T}}(\\textbf{T})$ closes up.\n\\end{lemma}\n\\begin{proof}\nFirst  observe   that $\\lambda g\\varpi_1 = - \\lambda g\\varrho\\varpi_1$, Now\nbecause each triangle closes up the sum of $g \\lambda \\varpi_1$ over all edges of $\\mathcal{T}^D$ is zero. But by the observation just above the sum over pairs of equivalent edges of $\\mathcal{T}^D$\ncancel. Hence the sum over leaf edges of $\\mathcal{T}$\nis zero as required.\n\\end{proof}\n\n\\begin{lemma}\\label{WDPhi}\n$\\widetilde{\\Phi}_n^{\\mathcal{T}} $ induces a well-defined map on the quotient\n$$\\widehat{\\Phi}_n^{\\mathcal{T}}: \\mathrm{SU}(2)^{n-2} \\backslash \\! \\backslash (\\mu_{\\mathrm{SU}(2)^{n-2}}^{-1}(0) \\cap \\mu_{\\mathbb{T}_d^-}^{-1}(0)) \\to V_n^{\\mathcal{T}}$$\nand a well-defined map\n$$\\Phi_n^{\\mathcal{T}}:P_n^{\\mathcal{T}}(\\mathrm{SU}(2)) \\to V_n^{\\mathcal{T}}.$$\n\\end{lemma}\n\n\\begin{proof}\nTo define $\\widehat{\\Phi}_n^{\\mathcal{T}}$ we need only show that for any pair of  normalized $\\textbf{T}$ and $\\textbf{T}'$ in $\\mu_{\\mathrm{SU}(2)^{n-2}}^{-1}(0) \\cap \\mu_{\\mathbb{T}_d^-}^{-1}(0)$ and $\\mathbf{f} = (f_1,\\cdots, f_{n-2}) \\in G$  such that $\\mathbf{f} \\textbf{T}=  \\textbf{T}'$\nwe have\n$$\\widetilde{\\Phi}^{\\mathcal{T}}_n(\\textbf{T}) =  \\widetilde{\\Phi}_n^{\\mathcal{T}}(\\textbf{T}')$$\nin $V_n^{\\mathcal{T}}$.\n\nThe assumption $\\mathbf{f} \\textbf{T}=  \\textbf{T}'$ is equivalent to\n\n$$(f_ig_{(\\tau_i, j)}, \\lambda_{(\\tau_i, j)}\\varpi_1) = (g'_{(\\tau_i, j)}, \\lambda'_{(\\tau_i, j)}\\varpi_1)$$\nfor each $j \\in \\{1, 2, 3\\}$ and $1 \\leq i \\leq n-2$.  Hence $\\lambda_{(\\tau_i, j)} = \\lambda'_{(\\tau_i, j)}$, and in particular a (pair of) diagonals of $\\textbf{T}$ vanishes if and only if the same diagonals vanish for $\\textbf{T}'$\nthat is\n\n$$S(\\mathbf{T}) = S(\\mathbf{T}').$$\nHence the forests $\\mathcal{T}^{S(\\mathbf{T})}$ and $\\mathcal{T}^{S(\\mathbf{T}')}$ coincide.\nWe must show that the images of $\\widetilde{\\Phi}_n^{\\mathcal{T}}(\\mathbf{T})$ and $\\widetilde{\\Phi}_n^{\\mathcal{T}}(\\mathbf{T}')$ in $V_n^{\\mathcal{T}}$ coincide, which amounts to proving that the $\\mathbf{f}$ is constant on any connected component in the\nforest $\\mathcal{T}^{S(\\mathbf{T})}$ (recall that we think of $\\mathbf{f}$ as a function from\nthe trivalent vertices of $\\mathcal{T}$ to $\\mathrm{SU}(2)$).\nClearly it suffices to prove that if $(\\tau_i, j)$ and $(\\tau_h, k)$\nare equivalent  then\n\n$$f_h = f_i.$$\nSuppose the imploded spin framings on the two edges are\nrespectively\n$(g_{(\\tau_i, j)}, \\lambda_{(\\tau_i, j)}\\varpi_1)$ and\n$(g_{(\\tau_h, k)}, \\lambda_{(\\tau_h, k)}\\varpi_1)$.\nWe have equations\n\n$$f_ig_{(\\tau_i,j)} = g_{(\\tau_i,j)}'$$\n$$f_hg_{(\\tau_h,k)} = g_{(\\tau_h,k)}'$$\nand because we have assumed both $\\textbf{T}$ and $\\textbf{T}'$ are normalized we have\n\n$$g_{(\\tau_h,k)} = g_{(\\tau_i,j)}\\varrho,$$\n$$g_{(\\tau_h,k)}' = g_{(\\tau_i,j)}'\\varrho.$$\nWe can rearrange these equations to get\n\n$$f_ig_{(\\tau_h,k)}\\varrho^{-1} = g_{(\\tau_h,k)}'\\varrho^{-1}.$$\nWe cancel $\\varrho^{-1}$ to obtain $f_ig_{(\\tau_h,k)} = g_{(\\tau_h,k)}'$.\nBut from above $f_hg_{(\\tau_h,k)} = g_{(\\tau_h,k)}'$\nThis proves the first statement of the lemma.\nIt is clear that $\\widehat{\\Phi}_n^{\\mathcal{T}}$ is constant on $\\mathbb{T}_d^-$ orbits, so it descends to give $\\Phi_n^{\\mathcal{T}}$.\n\\end{proof}\nThis shows that $\\Phi_n^{\\mathcal{T}}$ is well-defined and continuous\nbecause of the continuity of $\\widetilde{\\Phi}_n^{\\mathcal{T}}.$ \n\n\\begin{lemma}\n$\\Phi_n^{\\mathcal{T}} $ is proper (hence closed).\n\\end{lemma}\n\n\\begin{proof}\nSuppose $K$ is a compact subset of $P_n(\\mathrm{SU}(2))$. Then the lengths of\nthe columns of any matrix $A$ (so the edge-lengths of the corresponding $n$-gon) representing an element of $K$ are uniformly\nbounded by a constant $C$. We may reinterpret the above uniform bound as a bound\non all edge lengths  of all triangles in the given triangulation of $P$ that are\nalso edges of $P$.\nBut we may assume all our matrices $A$ are\nin the zero level sets of  the momentum maps for $\\mathrm{SU}(2)^{n-2}$ and\n$\\mathbb{T}_d^-$.  It follows by Lemma \\ref{trivalentinduction} and the triangle inequalities that the lengths of all columns of any matrix\n$A^{\\mathcal{T}}$ representing any element in $P_n^{\\mathcal{T}}(\\mathrm{SU}(2))$ in the inverse image\nof $K$ are bounded by $N(\\mathcal{T})C$ where $N(\\mathcal{T})$ is a positive integer\ndepending only on the tree $\\mathcal{T}$. Then $(\\Phi_n^{\\mathcal{T}})^{-1}(K)$\nis contained in the image of a subset of  $E_n^{\\mathcal{T}}(\\mathrm{SU}(2))$ homeomorphic to the product of $3(n-2)$ copies of the ball\nof radius $N(\\mathcal{T})C$ in $\\mathbb{C}^2$\n\\end{proof}\nWe now construct the  map $\\Psi_n^{\\mathcal{T}} : V_n^{\\mathcal{T}} \\to P_n^{\\mathcal{T}}(\\mathrm{SU}(2))$ inverse to $\\Phi_n^{\\mathcal{T}}$.\nWe first define $\\widetilde{\\Psi}_n^{\\mathcal{T}}$ on $E_n(\\mathrm{SU}(2))$ then verify that the resulting map descends to $V_n^{\\mathcal{T}}$.\nWe will need to be able to add a diagonal frame on a triangle with two already framed edges.\n\n\n\\begin{lemma}\\label{framing}\nSuppose that two sides of an imploded spin-framed triangle $(g_1, \\lambda_1 \\varpi_1)$, $(g_2, \\lambda_2 \\varpi_1)$ are given.  Then we can find $g_3 \\in \\mathrm{SU}(2)$ and $\\lambda_3 \\in \\mathbb{R}$ so that\n$(g_1, \\lambda_1 \\varpi_1)$, $(g_2, \\lambda_2 \\varpi_1)$ $(g_3, \\lambda_3 \\varpi_1)$ is an imploded spin-framed triangle.  Precisely we may solve\n\\begin{equation}\\label{closingequation}\n\\lambda_1g_1\\varpi_1 + \\lambda_2g_2\\varpi_1 + \\lambda_3g_3\\varpi_1 = 0.\n\\end{equation}\nMoreover $\\lambda_3$ is uniquely determined by $\\lambda_1$ and $\\lambda_2$ and if\n$\\lambda_3 \\neq 0$ any two choices of $g_3$ for given $g_1$ and $g_2$ are related by right multiplication of $T_{\\mathrm{SU}(2)}$.\n\\end{lemma}\n\n\\begin{proof}\nIf both $\\lambda_1$ and $\\lambda_2$ are zero then we take $\\lambda_3$ to be zero and $g_3 = 1$. Suppose then that exactly one of $\\lambda_1$ and $\\lambda_2$ is nonzero.  Without loss of generality, assume $\\lambda_1$ is nonzero. Put $\\lambda_3 = \\lambda_1$ and choose $g_3 = g_1\\varrho$.  Hence we have\n$$\\lambda_1g_1\\varpi_1 = -\\lambda_3g_3\\varpi_1.$$\nNote that the map $S^3 \\to S^2$ given by $g \\to g\\varpi_1$ is the Hopf fibration and consequently we may solve the equation $g\\varpi_1 = -g_1\\varpi_1$ locally in a neighborhood of $g_1$, in particular we can solve this equation in a neighborhood of $(g_1, 1)$ of $\\mathrm{SU}(2) \\times \\mathrm{SU}(2)$.\nThe number $\\lambda_3 \\geq 0$ is uniquely determined  but $g_3$ is defined only up to right multiplication by an element of $\\mathbb{T}$.\nNow assume that both $\\lambda_1$ and $\\lambda_2$ are nonzero.  First assume that the sum $\\lambda_1g_1\\varpi_1 +\\lambda_2g_2\\varpi_1$ is zero, then we are forced to take $\\lambda_3 = 0$ and we may choose any framing data we wish, so we choose $g_3 = 1$. Assume then that $\\lambda_1g_1\\varpi_1 + \\lambda_2g_2\\varpi_1$ is nonzero. We choose $g_3 \\in \\mathrm{SU}(2)$ and $\\lambda_3$ so that $$\\lambda_1g_1\\varpi_1 + \\lambda_2g_2\\varpi_1 = -\\lambda_3g_3\\varpi_1.$$\nAgain, $\\lambda_3 \\neq 0$ is uniquely determined, and $g_3$ is defined up to right multiplication by an element of $\\mathbb{T}$. We define the required framed triangle $T$ by\n$$T = (g_1, \\lambda_1 \\varpi_1), (g_2, \\lambda_2 \\varpi_1), (g_3, \\lambda_3 \\varpi_1)$$\n\\end{proof}\nWe make use of Lemma \\ref{framing} to extend any framing of\nthe edges of $P$ to an element of $E_n^{\\mathcal{T}}(\\mathrm{SU}(2))$. The resulting object will be a normalized element\n$\\mathbf{T} \\in \\mu_{\\mathrm{SU}(2)^{n-2}}^{-1}(0) \\cap \\mu_{\\mathbb{T}_d^-}^{-1}(0) \\subset  E_n^{\\mathcal{T}}(\\mathrm{SU}(2))$.\n\n\\begin{lemma}\\label{extend}\nSuppose we are given an imploded spin framing $\\mathbf{E}$ of the edges of the model convex $n$-gon $P$ which is of momentum level zero for the action\nof $\\mathrm{SU}(2)$. Then we may extend the framing by choosing\nimploded spin-frames on the diagonals and an enumeration of the triangles\nof $P$ so that the resulting element $\\mathbf{T} \\in E_n^{\\mathcal{T}}(\\mathrm{SU}(2))$\nis\n\\begin{enumerate}\n\\item normalized\n\\item of momentum level zero for $\\mathrm{SU}(2)^{n-2}$\n\\item of momentum level zero for $\\mathbb{T}_d ^-$.\n\\end{enumerate}\n\nMoreover any two such extensions are equivalent under the action of\nthe torus $\\mathbb{T}_d^-$.\n\n\\end{lemma}\n\n\\begin{proof}\nWe prove the existence of the framing by induction on $n$. We take as base\ncase $n=3$, here there is nothing to prove.\n\nLet $P$ be a model convex $n$-gon  with an imploded spin framing on\nits edges. Define $P_{i_1} =P$.\nChoose a triangle $T$ in the triangulation of $P$ which shares two edges with $P$.\nHence two sides of $T$ are framed. Let $e$ be the remaining side (a diagonal\nof $P$). Suppose $e$ has length $\\lambda$. Define $T_{i_1} = T$.\nApply Lemma \\ref{framing} to frame the third side $e$ such that the resulting\nframing is of momentum zero level for $\\mathrm{SU}(2)$. Split $T$ off from $P$\nto obtain an $n-1$--gon $P_{i_2}$. Suppose the framing on $e$\nis $[g, \\lambda \\varpi_1]$.  Give the edge $e'$ of $P_{i_2}$ which is not yet\nframed the framing $[g \\varrho, \\lambda \\varpi_1]$. We obtain the existence\npart of the lemma by induction.\n\nNow we prove uniqueness. Suppose that $\\mathbf{T}'$ is another extension of $\\mathbf{E}$, and\nthat $\\mathbf{T}'$ assigns the frame $[h, \\lambda \\varpi_1]$ to $e$ with $h \\neq g$. \nThen again by Lemma \\ref{framing} either $\\lambda = 0$ or there exists an element $t \\in T_{\\mathrm{SU}(2)}$ such that\n$h = gt$. In case $\\lambda \\neq 0$ the frame on $e'$ is necessarily \n$[gt \\varrho, \\lambda \\varpi_1] = [ g \\varrho t^{-1}, \\lambda \\varpi_1]$. Thus the new frames\non the pair of equivalent edges of $\\mathcal{T}^D$ are\n$[gt ,\\lambda \\varpi_1]$ followed by $[g\\varrho t^{-1},\\lambda \\varpi_1]$.\nNote that the framing of $P_{i_2}$ obtained by restriction of $\\mathbf{T}'$ \nsatisfies the three properties in the statement above, and also it is an extension of the framing of its boundary\ninduced by $\\mathbf{T}'$. Hence for the case $\\lambda \\neq 0$\nthe induction step of the uniqueness part of the lemma is completed.\nIn case $\\lambda = 0$ then we may represent both frames by\n$[I ,\\lambda \\varpi_1]$. But since $\\lambda = 0$ we have (by definition of implosion) for any $t \\in\nT_{\\mathrm{SU}(2)}$\n$$[I ,\\lambda \\varpi_1] = [t ,\\lambda \\varpi_1] = [t^{-1} ,\\lambda \\varpi_1].$$\nThis completes the induction step in the uniqueness part of the lemma.\n\\end{proof}\n\n\nBy the above lemma we obtain a well-defined map\n\n$$\\widetilde{\\Psi}_n^{\\mathcal{T}}: E_n(\\mathrm{SU}(2)) \\to (\\mu_{\\mathrm{SU}(2)^{n-2}}^{-1}(0) \\cap \\mu_{\\mathbf{T}_d^-}^{-1}(0))\/\\!\/ \\mathbb{T}_d^-.$$\nWe prove this map descends to $V_n^{\\mathcal{T}}$. We let\n$\\widehat{\\Psi}_n^{\\mathcal{T}}$ denote the map obtained from $\\widetilde{\\Psi}_n^{\\mathcal{T}}$\nby postcomposing it with projection to $P_n^{\\mathcal{T}}(\\mathrm{SU}(2))$.\nLet us note the following obvious refinement of Lemma \\ref{extend}.\nSuppose $P_1$ is a subpolygon of $P$ which meets $P$ along\ndiagonals of length zero in the realization of\n$P$ given by the framing of the edges. Then given any extension of\nthe framing of $P$ to all diagonals of $P_1$ we may find an extension of\nthe framing of $P$ to all diagonals of $P$ agreeing with the given one on $P_1$.\n\n\\begin{lemma}\nThe map $\\widehat{\\Psi}_n^{\\mathcal{T}}$ descends to\n$V_n^{\\mathcal{T}}$.\n\\end{lemma}\n\n\n\\begin{proof}\nWe first verify that $\\widehat{\\Psi}_n^{\\mathcal{T}}$ descends to $P_n(\\mathrm{SU}(2))$.\nIn fact we will show it is equivariant under $\\mathrm{SU}(2)$ where the action\nof $\\mathrm{SU}(2)$ on $E_n^{\\mathcal{T}}(\\mathrm{SU}(2))$ is by the diagonal in $\\mathrm{SU}(2)^{n-2}$.\nLet $f \\in \\mathrm{SU}(2)$. Let $\\mathbf{E}$ be a framing of the\nedges of $P$ and $\\mathbf{T}$ be the extension of Lemma \\ref{extend}.\nApply Lemma \\ref{extend} to extend $f \\mathbf{F}$ to an\nelement $\\mathbf{T}'$ of $E^{\\mathcal{T}}_n(\\mathrm{SU}(2))$\nso that this extension also satisfies the three properties in\nthe statement of Lemma \\ref{extend}.  We wish to prove that the resulting element\nof  $E_n^{\\mathcal{T}}(\\mathrm{SU}(2))$ is in the\n$\\mathbb{T}_d^-$-orbit of $f\\mathbf{T}$. But since any two extensions satisfying the three properties of Lemma \\ref{extend} are equivalent under $\\mathbb{T}_d^-$\nit suffices to prove that the image of {\\it any one} of the above extensions is   \nequal to $f \\mathbf{T}$. But there is an\nextension that is obviously equivalent to $f \\mathbf{T}$ namely\n$f \\mathbf{T}$ itself.  \n\nIt remains to check that $\\widehat{\\Psi}_n^{\\mathcal{T}}$ descends to $V_n^{\\mathcal{T}}$.\nLet $\\mathbf{E}$ and $\\mathbf{T}$ be as above. Let $\\mathbf{E}'$ be another framing\nof the edges of $P$ in the same $\\mathcal{T}$-congruence class as\n$\\mathbf{E}$. We have to prove that\n$$\\widehat{\\Psi}_n^{\\mathcal{T}}(\\mathbf{E}) = \\widehat{\\Psi}_n^{\\mathcal{T}}(\\mathbf{E}').$$\nBy transitivity of the $\\mathcal{T}$-congruence relation it suffices to prove this\nfor $\\mathbf{E}'$.\n\nLet $C_1$ be a component of\n$\\mathcal{T}^{S(\\mathbf{T})}$. We may assume that $C_1$ contains leaf edges.  \nApply an element $f \\in \\mathrm{SU}(2)$ to the\nframes of the leaf edges of $\\mathcal{T}_1$ to obtain a new framing $\\mathbf{E}'$\nof the edges of $P$. We now prove\n$$\\widehat{\\Psi}_n^{\\mathcal{T}}(\\mathbf{E}) = \\widehat{\\Psi}_n^{\\mathcal{T}}(\\mathbf{E}')$$\nfor this choice of $\\mathbf{E}'$.\nApply Lemma \\ref{extend} to  $\\mathbf{E}'$ to obtain an element $\\mathbf{T}'$\nof  $E_n^{\\mathcal{T}}(\\mathrm{SU}(2))$ such that the three properties\nin Lemma \\ref{extend} are satisfied.  We must prove that $\\mathbf{T}$\nand $\\mathbf{T}'$  have the same image in $P_n^{\\mathcal{T}}(\\mathrm{SU}(2))$.\nAs before in the paragraph above it suffices to find one extension\n$\\mathbf{T}'$ of  $\\mathbf{E}'$ which has the same image as $\\mathbf{T}$.\nWe now construct such an extension $\\mathbf{T}''$\nThe subtree $C_1$ corresponds to a subpolygon $P'$ of $P$ such that\nthe edges of $P'$ that correspond to diagonals of $P$  have zero length.  Define\n$\\mathbf{T}''$ by defining the frame on an edge $e$ of $\\mathcal{T}^D$ that\nis in $C_1$ to be $f$ applied to the value of $\\mathbf{T}$ on $e$\nand if $e$ is not in  $C_1$ then define the value of $\\mathbf{T}''$\non that edge to be the value of $\\mathbf{T}$ on that edge.\nWe leave to the reader that $\\mathbf{T}''$ satisfies the three properties\nof Lemma \\ref{extend}. Now we construct an element $\\tilde{f}  \\in \\mathrm{SU}(2)^{n-2}$\nsuch that\n$$\\mathbf{T}'' = \\tilde{f}\\mathbf{T}.$$\nRecall that an element of $\\mathrm{SU}(2)^{n-2}$ may be regarded as a function\nfrom the trivalent nodes of $\\mathcal{T}$ to $\\mathrm{SU}(2)$.\nWe define $\\tilde{f}$ to be the function that is $I$ if the trivalent node\nis not in $C_1$, and is $f$ if the trinode is in $C_1$. Since an edge $e$ of $\\mathcal{T}$ is in $C_1$\nif and only if the trivalent node incident to $e$ we do indeed have\n$$\\mathbf{T}'' = \\tilde{f}\\mathbf{T}.$$\n\\end{proof}\n\nBecause $\\Psi_n^{\\mathcal{T}}$ is the inverse of a closed map we have\n\n\\begin{lemma}\n$\\Psi_n^{\\mathcal{T}}$ is continuous.\n\\end{lemma}\n\nFinally we have\n\n\\begin{lemma}\n $\\Phi^{\\mathcal{T}}_n$ and $\\Psi^{\\mathcal{T}}_n$ are inverses to one another\n\\end{lemma}\n\\begin{proof}\nIt is obvious that $\\Phi^{\\mathcal{T}}_n \\circ \\Psi^{\\mathcal{T}}_n = I_{P_n(\\mathrm{SU}(2))}$.\nThis is because neither of the two map changes the framing of the edges of $P$.\nNow let $\\mathbf{T}$ be a normalized element of  $\\mathrm{SU}(2)^{n-2} \\times \\mathbb{T}_d^-$-momentum level zero in $E_n^{\\mathcal{T}}(\\mathrm{SU}(2))$. Let $\\mathbf{F}$ be the restriction\nof $\\mathbf{T}$ to the edges of $P$. Thus $\\mathbf{T}$ is an extension\nof $\\mathbf{F}$ satisfying the three properties of Lemma \\ref{extend}. But\n$\\Psi^{\\mathcal{T}}_n \\circ \\Phi^{\\mathcal{T}}_n(\\mathbf{T})$ also has restriction\nto $P$ given by $\\mathbf{F}$ and also satisfies the three properties of\nLemma \\ref{extend}. Hence $\\mathbf{T}$ and $\\Psi^{\\mathcal{T}}_n \\circ \\Phi^{\\mathcal{T}}_n(\\mathbf{T})$ are in the same $\\mathbb{T}_d^-$-orbit and\nhence their images in $P_n^{\\mathcal{T}}(\\mathrm{SU}(2))$ coincide.\n\\end{proof}\nIt is clear from  the definition of $\\Phi^{\\mathcal{T}}_n$ that this map intertwines the actions of\n$T_{\\mathrm{SU}(2)^n}$ and $\\mathbb{T}_e$.\nThis proves Theorem \\ref{noncompact}.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\subsection{Pulling back Hamiltonian functions from $P_n^{\\mathcal{T}}(\\mathrm{SU}(2))$ to $V_n^{\\mathcal{T}}$}\n\nIn the previous subsection we constructed a homeomorphism\n$\\Psi_n^{\\mathcal{T}}:V_n^{\\mathcal{T}} \\to P_n^{\\mathcal{T}}(\\mathrm{SU}(2))$ with inverse\n$\\Phi_n^{\\mathcal{T}}$. The torus $\\mathbb{T}$ acts on $P_n^{\\mathcal{T}}(\\mathrm{SU}(2))$\nsuch that the Hamiltonian for the circle factor corresponding to the edge $(\\tau_i,j)$ of $\\mathcal{T}^D$ is given by $f_{\\tau_i,j}(A) = (1\/2)|z_j(\\tau_i)|^2 + |w_j(\\tau_i)|^2$.  In spin-framed coordinates this is $f_{\\tau_i,j}(A) = \\lambda_{(\\tau_i, j)}$.\nWe will compute the pullback of these functions to $V_n^{\\mathcal{T}}$ via $\\Psi_n^{\\mathcal{T}}$.\n\n\\begin{definition}\nLet $v_{(\\tau, i)}^{\\mathcal{T}}: E_n^{\\mathcal{T}}(\\mathrm{SU}(2)) \\to \\mathfrak{su}(2)^*$\nby the composition of the projection on the $(\\tau, i)$-th factor\nof $E_n^{\\mathcal{T}}(\\mathrm{SU}(2))$ with the function $h$ from subsection \\ref{frommatricestopolygons}.\nLet $v_e: E_n(\\mathrm{SU}(2)) \\to \\mathfrak{su}(2)^*$ be the composition of the projection\non the $e$-th factor of $E_n(\\mathrm{SU}(2))$ with the function $h$ from subsection \\ref{frommatricestopolygons}.\n\\end{definition}\n\nFor a distinguished edge $(\\tau, i)$ of $\\mathcal{T}^D$ let $e(\\tau, i)$\nbe associated edge of the model convex planar $n$--gon.  Then\nwe have the following identity\n\n$$(\\tilde{\\Phi}_n^{\\mathcal{T}})^*(v_{e(\\tau, i)}) = v_{(\\tau, i)}^{\\mathcal{T}}.$$\n\n\\subsubsection{Pulling back distinguished edge Hamiltonian functions}\n\n\nIn what follows\nwe will need to compute the pull-back of $f_{(\\tau_i,j)}$ to $V_n^{\\mathcal{T}}$\nfor those distinguished edges  $(\\tau_i,j)$.  Let $f$ be the map from subsection\n\\ref{frommatricestopolygons}, then we have\n\n$$f_{\\tau_i, j}(\\mathbf{E}) = 2 \\| f \\circ v_{(\\tau_i, j)}^{\\mathcal{T}}(\\mathbf{T}) \\|$$\nfor any normalized $\\mathbf{T} \\in \\mu_{\\mathrm{SU}(2)^{n-2} \\times \\mathbb{T}_d^-}^{-1}(0)$ which maps to $\\mathbf{E}$.\n\n\\begin{proposition}\\label{pullback}\nFor $A \\in V_n^{\\mathcal{T}}$ we have\n$$(\\Psi^{\\mathcal{T}}_n)^* f_{\\tau_i,j}(A) = 2 \\|f \\circ v_{e(\\tau_i, j)}(\\tilde{A})\\|.$$\nwhere $\\tilde{A} \\in E_n(\\mathrm{SU}(2))$ maps to $A$.\n\\end{proposition}\n\n\\begin{proof}\nThe proposition follows from\n\n$$(\\Psi^{\\mathcal{T}}_n)^*(f_{\\tau_i, j})(A) = f_{\\tau_i, j}(\\Psi^{\\mathcal{T}}_n(A)) = \n2 \\| f \\circ (\\Psi^{\\mathcal{T}}_n)^*(v_{(\\tau_i, j)}^{\\mathcal{T}})(\\tilde{A}) \\| = \n2 \\| f \\circ v_{e(\\tau_i, j)}(\\tilde{A}) \\|.$$\n\\end{proof}\n\n\\begin{remark}\nNote that $\\| f \\circ v_{e(\\tau_i, j)}(\\tilde{A}) \\|$ is the length\nof the $e(\\tau_i, j)$ edge of $F_n(A)$.\n\\end{remark}\n\n\n\\subsubsection{Pullback of the internal edge Hamiltonian functions}\n\nWe also need to compute the $\\Psi_n^{\\mathcal{T}}$ pullbacks\nof the Hamiltonian functions $f_{(\\tau_j, i)}$ for $(\\tau_j, i)$\nan internal edge of $\\mathcal{T}$, this will be important when we\nwish to indentify the Hamiltonian flows of these pullbacks\nwith the bending flows. Note that $(\\tau_j, i)$ corresponds\nto a diagonal $d(\\tau_j, i)$ in a model $n$-gon $P$.\n\n\\begin{lemma}\nLet $\\mathcal{T}' \\subset \\mathcal{T}$ be a connected subtree\nsuch that every leaf of $\\mathcal{T}'$ except one, say $(\\tau, k)$, is a\nleaf of $\\mathcal{T}$. Then for any element $\\mathbf{E} \\in P_n^{\\mathcal{T}}(\\mathrm{SU}(2))$\nwe can compute\n\n$$f_{\\tau, k}(\\mathbf{E}) = \\| \\sum_{(\\tau_i, j) \\in dist(\\mathcal{T}')} v_{(\\tau_i, j)}^{\\mathcal{T}}(\\mathbf{T}) \\|$$\n\nFor $\\mathbf{T}$ a normalized element of $\\mu_{\\mathrm{SU}(2)^{n-2}\\times\\mathbb{T}_d^-}^{-1}(0)$ which\nmaps to $\\mathbf{E}$.\n\\end{lemma}\n\n\\begin{proof}\nFirst note that we compute $f_{\\tau, k}$ by\ntaking half the length of the $(\\tau, k)$ edge for\nany normalized element $\\mathbf{T} \\in \\mu_{\\mathrm{SU}(2)^{n-2}\\times\\mathbb{T}_d^-}^{-1}(0)$\nwhich maps to $\\mathbf{E}$, this is equal to $\\lambda_{(\\tau, k)}$.\nBy the normalized condition and the closing condition imposed\non $\\mathbf{T}$, the leaf edge of $\\mathcal{T}'$ form a closed polygon,\nwhich implies that $\\lambda_{(\\tau, k)} =  \\| \\sum_{(\\tau_i, j) \\in dist(\\mathcal{T}')} \\lambda_{(\\tau_i, j)}(\\mathbf{T}) Ad^*_{g_{(\\tau_i,j)}(\\mathbf{T})}(\\varpi_1) \\|$.\n\\end{proof}\n\nThe previous proposition allows us to conclude the following\ntheorem.\n\n\\begin{theorem}\\label{diagonalpullback}\nLet $d(\\tau_j, i)(A)$ be the associated diagonal in\n$F_n(A)$.\n\n$$(\\Psi_n^{\\mathcal{T}})^*(f_{(\\tau_j, i)})(A) =  2\\| d(\\tau_j, i)(A)\\|$$\n\n\\end{theorem}\n\n\\begin{proof}\nWe have the following equation\n\n$$(\\Psi_n^{\\mathcal{T}})^*(f_{(\\tau_j, i)})(A) = \\| \\sum_{(\\tau_i, j) \\in dist(\\mathcal{T}')} v_{(\\tau_i, j)}^{\\mathcal{T}}(\\tilde{\\Psi}_n^{\\mathcal{T}}(\\tilde{A})) \\|.$$\nThe right hand side of this equation is equal to\n\n$$2\\| \\sum_{(\\tau_i, j) \\in dist(\\mathcal{T}')} f \\circ v_{e(\\tau_i, j)}(\\tilde{A}) \\|.$$\nThis last expression is in turn equal to $2\\|d(\\tau_i, j)(A)\\|$.\n\\end{proof}\n\nThis, along with Proposition \\ref{pullback}, proves Theorem \\ref{stratification}.\n\n\n\n\\subsection{The homeomorphism of projective varieties}\\label{thirdtorusquotient}\nIn this subsection we will descend the homeomorphisms of affine varieties $\\Phi_n^{\\mathcal{T}}$\nand $\\Psi_n^{\\mathcal{T}}$ to their projective quotients completing the proof\nof Theorem \\ref{firstauxiliarytheorem} and hence the proof of\nTheorem \\ref{firstmaintheorem}.\nRecall that we defined  elements $\\mathbf{t}(\\lambda) \\in T_{\\mathrm{SU}(2)^n}$\nand $\\mathbf{t}_e(\\lambda) \\in \\mathbb{T}_e$ for $\\lambda \\in S^1$. We note\n$$\\rho(\\mathbf{t}(\\lambda)) = \\mathbf{t}_e(\\lambda).$$\nSince $\\Psi^{\\mathcal{T}}_n$\nis equivariant we have\n\\begin{equation}\\label{degreeequivariance}\n\\Psi^{\\mathcal{T}}_n \\circ \\mathbf{t}(\\lambda)  = \\mathbf{t}_e(\\lambda) \\circ\n\\Psi^{\\mathcal{T}}_n.\n\\end{equation}\n\n\\begin{proposition}\nThe map $\\Psi^{\\mathcal{T}}_n$ induces a homeomorphism from $W_n^{\\mathcal{T}}$\nto the symplectic quotient of $P_n^{\\mathcal{T}}(\\mathrm{SU}(2))$ by the grading\ncircle action.\n\\end{proposition}\n\n\\begin{proof}\nIt suffices to prove that  the pull-back by $\\Psi_n^{\\mathcal{T}}$ of the grading circle action on $P_n^{\\mathcal{T}}(\\mathrm{SU}(2))$\nis the grading circle action on $V_n^{\\mathcal{T}}$.\nThis follows\nby replacing $\\lambda$\nby $(\\sqrt{\\lambda})^{-1}$ in equation (\\ref{degreeequivariance}).\n\\end{proof}\n\n\n\\subsection{The symplectic and GIT quotients coincide}\\label{quotientscoincide}\nWe complete the proof of Theorems \\ref{firstauxiliarytheorem}\nand \\ref{firstmaintheorem} by proving\n\n\\begin{proposition}\nThe symplectic quotient of $P_n^{\\mathcal{T}}(\\mathrm{SU}(2)$ by the grading circle\naction coincides with the GIT quotient of $P_n^{\\mathcal{T}}(\\mathrm{SU}(2))$ by the\ngrading $\\mathbb{C}^{\\ast}$-action linearized by acting by the identity on the\nfiber of the trivial complex line bundle over $P_n^{\\mathcal{T}}(\\mathrm{SU}(2))$.\n\\end{proposition}\n\nWe are forced to give an indirect argument because $P_n^{\\mathcal{T}}(\\mathrm{SU}(2)$\nis not smooth and there do not seem to be theorems asserting\nthe isomorphism of symplectic and GIT quotients of nonsmooth spaces.\n\nWe first note that the above GIT quotient is canonically isomorphic  to the GIT quotient of $(\\bigwedge^2(\\mathbb{C}^3))^{n-2}$ by the product group\n$\\mathbb{C}^{\\ast} \\times \\underline{\\mathbb{T}}_d^-$ where the first factor acts by\nthe grading action linearized by the identity action on the fiber $\\mathbb{C}$\nof the trivial complex line bundle over $(\\bigwedge^2(\\mathbb{C}^3))^{n-2}$.\nThis follows because we can take the ring of invariants for a product\ngroup acting on a ring $R$ by first taking the invariants of one factor and then taking the invariants of the resulting ring by the second factor.\nThe corresponding quotient by stages for symplectic quotients is\nalso true but slightly harder. It is proved in Theorem 4.1 of \\cite{SjamaarLerman}.\n\nThus it remains to prove that the symplectic quotient of $(\\bigwedge^2(\\mathbb{C}^3))^{n-2}$ by the product of the grading $S^1$-action\nand $\\mathbb{T}_d^-$ is isomorphic to\nthe GIT quotient of $(\\bigwedge^2(\\mathbb{C}^3))^{n-2}$ by the product of\nthe grading $\\mathbb{C}^{\\ast}$ action and the complexified torus \n$\\underline{\\mathbb{T}}_d^-$ where the first factor\nacts by the grading action and the product is linearized by acting by\nthe identity map applied to the projection on the first factor.\nThis follows immediately from Theorem \\ref{twistandshift}\nonce we establish that the momentum map for the product\nof the grading $S^1$-action\nand $\\mathbb{T}_d^-$ is proper.\nWe now prove that the momentum map for the product of the grading circle\naction and $\\mathbb{T}_d^-$ is proper. In fact we prove a different result\nthat turns out to be equivalent to the one we need here because we will need this different result below.\n\n\\begin{proposition}\\label{propernesstheorem}\nThe momentum mapping $\\mu : (\\bigwedge^2 \\mathbb{C}^3)^{n-2} \\to \\mathbb{R}^{2n-3}$\nfor the action of $\\mathbb{T}_e \\times \\mathbb{T}_d^-$ is proper.\n\\end{proposition}\n\n\\begin{proof}\nEarlier we identified $\\bigwedge^2 \\mathbb{C}^3$ with the space of\nframed triangles $P_3(\\mathrm{SU}(2))$. Let\n$(\\ldots, [g_{(\\tau_i, j)}, \\lambda_{(\\tau_i, j)}\\varpi_1], \\ldots)$\nbe an element of $P_3(\\mathrm{SU}(2))^{n-2}$.\nUnder this identification we have that\n$\\mu^{-1}_{\\mathbb{T}_e}(\\ldots, [g_{(\\tau_i, j)}, \\lambda_{(\\tau_i, j)}\\varpi_1], \\ldots)$\nis the vector of elements $\\lambda_{(\\tau_i, j)}$ where $(\\tau_i, j)$\nis a distinguished edge of $\\mathcal{T}^D$.  Similarly\n$\\mu_{\\mathbb{T}_d^-}(\\ldots, [g_{(\\tau_i, j)}, \\lambda_{(\\tau_i, j)}\\varpi_1], \\ldots)$\nis the vector of elements $\\lambda_{(\\tau_i, j)} - \\lambda_{(\\tau_k, \\ell)}$\nwhere $(\\tau_i, j)$ and $(\\tau_k, \\ell)$ represent the same internal edge of $\\mathcal{T}$.\nIn order to show that $\\mu_{\\mathbb{T}_e \\times \\mathbb{T}_d^-}$ is a proper map, we must show\nthat if all $\\lambda_{(\\tau_i, j)}$ are bounded for $(\\tau_i, j)$ distinguished,\nand all differences $\\lambda_{(\\tau_i, j)} - \\lambda_{(\\tau_k, \\ell)}$ are bounded\nfor all $(\\tau_i, j)$ and $(\\tau_k, \\ell)$ which represent the same internal edge of $\\mathcal{T}$,\nthen all $\\lambda_{(\\tau_i, j)}$ are bounded.\n\nLet $([g_1, \\lambda_1\\varpi_1], [g_2, \\lambda_2\\varpi_1], [g_3, \\lambda_3\\varpi_1])$\nbe an element of $P_3(\\mathrm{SU}(2))$.  Since\n$$\\sum_{i = 1}^3 \\lambda_i Ad_{g_i}^*(\\varpi_1) = 0,$$\nthe $\\lambda_i$ are side-lengths of a closed triangle, hence if two of the three are bounded, so is the third.\nFurthermore, if $(\\tau_i, j)$ and $(\\tau_k, \\ell)$ represent the same internal edge of $\\mathcal{T}$,\nand $\\lambda_{(\\tau_i, j)}$ and the difference $\\lambda_{(\\tau_i, j)} - \\lambda_{(\\tau_k, \\ell)}$ are bounded,\nthen $\\lambda_{(\\tau_k, \\ell)}$ is bounded.  The proposition now follows from\ntrivalent induction, see lemma \\ref{trivalentinduction}.\n\\end{proof}\n\n\\begin{corollary}\nThe momentum mapping $\\mu : (\\bigwedge^2 \\mathbb{C}^3)^{n-2} \\to \\mathbb{R}^{2n-3}$\nfor the action of $S^1\\times \\mathbb{T}_d^-$ is proper.\nHere $S^1$ acts by the grading circle action.\n\\end{corollary}\n\\begin{proof}\nThe Hamiltonian for the grading circle action is the half the sum of the\nHamiltonians for the circle $\\mathbb{T}_e$ factors of $\\mathbb{T}_e$.\nBut note that each of the summands is bounded if and only if the sum\nis (since they have the same sign).\n\\end{proof}\n\nThere is one more technical point. We claim that the GIT\nquotient coincides with the symplectic quotient at momentum level\n$(1,0)$. We will prove first that the level for the\n$\\mathbb{T}_d^{-1}$ is zero. But this follows\nbecause  the torus  $\\mathbb{T}_d^{-1}$ acts linearly on\n$\\bigwedge^{n-2}(\\mathbb{C}^3)$ and trivially on the fiber whereas the grading\ncircle action is twisted by the identity action on the fiber.\n\n\\section{The space $V_{\\mathbf{r}}^{\\mathcal{T}}$ is homeomorphic to the\ntoric variety $(M_{\\mathbf{r}})^{\\mathcal{T}}_0$}\nIn this section we will prove Theorem \\ref{secondmaintheorem}\nby proving Theorem \\ref{secondauxiliarytheorem}.\n\n\\subsection{The toric varieties $(M_{\\mathbf{r}})^{\\mathcal{T}}_0$ and $P^{\\mathcal{T}}_{\\mathbf{r}}(\\mathrm{SU}(2))$ are isomorphic}\nIn this subsection we prove the first statement of Theorem \\ref{secondauxiliarytheorem}.\nBy definition the toric variety $P^{\\mathcal{T}}_{\\mathbf{r}}(\\mathrm{SU}(2))$ is obtained\nas\nthe GIT quotient of $P^{\\mathcal{T}}_{n}(\\mathrm{SU}(2))$ by $\\underline{\\mathbb{T}}_e$\nusing the linearization given by the character\n$$\\chi_{\\mathbf{r}}(\\mathbf{t}(\\mathbf{\\lambda})) = \\lambda_1^{r_1} \\cdots \\lambda_n^{r_n}.$$\nHere $\\mathbf{t}(\\mathbf{\\lambda})$ denotes the element of $\\underline{\\mathbb{T}}_e$\ncorresponding to $\\mathbf{\\lambda} = (\\lambda_1,\\cdots,\\lambda_n) \\in (\\mathbb{C}^{\\ast})^n$.\nWe recall that the graded ring\nof $(M_{\\mathbf{r}})^{\\mathcal{T}}_0$ is the semigroup ring $\\mathbb{C}[\\mathcal{S}^{\\mathcal{T}}_{\\mathbf{r}}]$ where\n$\\mathcal{S}^{\\mathcal{T}}_{\\mathbf{r}}$ is the graded subsemigroup of $\\mathcal{S}^{\\mathcal{T}}_n$ defined by taking\ngraphs with valence $k\\mathbf{r}$ for positive integers $k$.\nDefine the subsemigroup $\\mathcal{P}^{\\mathcal{T}}_{\\mathbf{r}}$ of $\\mathcal{P}^{\\mathcal{T}}_n$\nto be the inverse image of  $\\mathcal{W}_{\\mathbf{r}}^{\\mathcal{T}}$ under the isomorphism\nfrom $\\mathcal{P}^{\\mathcal{T}}_n$ to $\\mathcal{W}^{\\mathcal{T}}_n$ under the\nisomorphism of Proposition \\ref{twoisomorphicsemigroups}.\n\n\\begin{lemma}\n\\\nThe $\\underline{\\mathbb{T}}_d^-$-invariant monomial\n$$f(Z)=\n\\prod_{i=1}^{n-2}Z_{12}(\\tau_i)^{x_{12}(\\tau_i)}Z_{13}(\\tau_i)^{x_{13}(\\tau_i)}\nZ_{23}(\\tau_i)^{x_{23}(\\tau_i)}$$\nis $\\underline{\\mathbb{T}}_e$--invariant for the twist\n$\\chi_{\\mathbf{r}}^p$ if and only if the\nexponents $\\{x_{jk}(\\tau_i)\\}$ satisfy the system of equations\n$$x_{k,m}(\\tau) + x_{k,\\ell}(\\tau_i) = p r_{(\\tau_i,k)}, \\quad \\text{for all leaf edges} \\ (\\tau_i, k) \\ \\text{where} \\ k,\\ell,m = 1,2,3.$$\nBy Proposition \\ref{twoisomorphicsemigroups} we may rewrite the\nabove equation as a condition on the leaf weights\n$$w_{(\\tau_i,k)}^{\\mathcal{T}} = p  r_{(\\tau_i,k)}, \\ (\\tau_i, k) \\in \\mathcal{L}.$$\n\\end{lemma}\n\n\n\\begin{proof}\nThe lemma follows from the formula for $\\mathbf{t}(\\mathbf{\\lambda})$\nacting on $f$, namely\n$$\\mathbf{t}(\\mathbf{\\lambda}) \\circ f(Z) = \\prod_{(\\tau_i, k) \\in \\mathcal{L}}\n\\lambda_{(\\tau_i,k)}^{[p r_{(\\tau_i,k)} - (x_{k,m}(\\tau_i) + x_{k,\\ell}(\\tau_i))]} f(Z).$$\nHere $\\lambda_{(\\tau_i,k)}$ is the coordinate of $\\underline{\\mathbb{T}}_e$ corresponding to the\nleaf edge $(\\tau_i,k)$ and $k,\\ell,m = 1,2,3$.\n\\end{proof}\n\n\\begin{remark}\nThe  set of $\\underline{\\mathbb{T}}_e$-invariant monomials in $Z_{ij}(\\tau)$ {\\it for the twist by $\\chi_{\\mathbf{r}}^p$}\nis the subset of invariants {\\it of degree $p$} (the $p$-th graded piece of the\nassociate semigroup of lattice points).\n\\end{remark}\n\nAs an immediate consequence we have\n\\begin{corollary}\\label{rthsemigroup}\nThe toric variety $P_{\\mathbf{r}}^{\\mathcal{T}}(\\mathrm{SU}(2))$ is the projective toric variety\nassociated to the graded semigroup $\\mathcal{P}^{\\mathcal{T}}_{\\mathbf{r}}$.\n\\end{corollary}\n\n\n\nNow by Proposition \\ref{gradedsubsemigroupiso} we have\nan isomorphism of graded semigroups  \n$\\Omega_{\\mathbf{r}}: \\mathcal{S}_{\\mathbf{r}}^{\\mathcal{T}} \\to \\mathcal{W}_{\\mathbf{r}}^{\\mathcal{T}}$.  \nSince by definition we have an isomorphism\nof graded semigroups $\\mathcal{P}^{\\mathcal{T}}_{\\mathbf{r}} \\cong \\mathcal{W}_{\\mathbf{r}}^{\\mathcal{T}}$\nwe obtain the required isomorphism of graded semigroups\n$\\mathcal{S}^{\\mathcal{T}}_{\\mathbf{r}} \\cong \\mathcal{P}^{\\mathcal{T}}_{\\mathbf{r}}$\nand we have proved the desired isomorphism of projective toric varieties.\n\n\n\n\\subsection{$V_{\\mathbf{r}}^{\\mathcal{T}}$ is homeomorphic to $P^{\\mathcal{T}}_{\\mathbf{r}}(\\mathrm{SU}(2))$}\nIn this section we will prove the second statement of Theorem \\ref{secondauxiliarytheorem}.\n\n\n\\begin{proposition}\n$V_{\\mathbf{r}}^{\\mathcal{T}}$ is homeomorphic to the symplectic quotient\nof $P^{\\mathcal{T}}_n(\\mathrm{SU}(2))$ by $\\mathbb{T}_e$ at level $\\mathbf{r}$.\n\\end{proposition}\n\n\n\n\\begin{proof}\nRecall that in Proposition \\ref{pullback} we proved the following formula.\nLet  $\\mathbf{E} \\in V_n^{\\mathcal{T}}$ then\n\n$$(\\Psi^{\\mathcal{T}}_n)^* f_{\\tau_i,j}(A) =  2\\|f \\circ v_{e(\\tau_i, j)}(\\tilde{A})\\| = \\| v_{e(\\tau_i, j)}(\\tilde{A}) \\|.$$\n\nWhere $e(\\tau_i, j)$ is the leaf edge of $\\mathcal{T}$\ncorresponding to the leaf edge $(\\tau_i,j)$ of the decomposed tree $\\mathcal{T}^D$.\nThus $\\Psi_n^{\\mathcal{T}}$ induces a homeomorphism between the $\\mathcal{T}$-congruence classes\nof imploded spin-framed $n$-gons with side-lengths $\\mathbf{r}$ and the $\\mathbf{r}$-th level set for\nthe momentum map of $\\mathbb{T}_e$. But also $\\Psi_n^{\\mathcal{T}}$ is $\\rho$-equivariant,\nhence the above bijection descends to a homeomorphism from the $T_{\\mathrm{SU}(2)^n}$-quotient of the imploded spin-framed $n$-gons\nto the  $\\mathbb{T}_e$ symplectic quotient at level $\\mathbf{r}$.\n\\end{proof}\n\n\nIt remains to prove that the symplectic quotient of $P_n^{\\mathcal{T}}(\\mathrm{SU}(2))$\nat level $\\mathbf{r}$ coincides with the GIT quotient for the action on the\ntrivial bundle using the twist by $\\chi_{\\mathbf{r}}$.\n\n\\subsubsection{The symplectic quotient coincides with the GIT quotient}\n\\label{fourthtorusquotient}\nLet $L$ be the trivial complex line bundle over $\\bigwedge^{n-2}(\\mathbb{C}^3)$.\nThen $L$ descends to the trivial complex  line bundle $\\overline{L}$\nover $P^{\\mathcal{T}}_n(\\mathrm{SU}(2))$. The torus\n$\\underline{\\mathbb{T}}_e$ acts on $\\overline{L}$ by twisting by\nthe character $\\chi_{\\mathbf{r}}$.\nWe  need to prove that the GIT quotient of\n$P^{\\mathcal{T}}_n(\\mathrm{SU}(2))$ by $\\underline{\\mathbb{T}}_e$ with linearization the above\naction on $\\overline{L}$ is homeomorphic to the symplectic quotient by the maximal compact subgroup of $\\underline{\\mathbb{T}}_e$ at level $\\mathbf{r}$.\nThe argument is almost  the same as  that of subsection \\ref{quotientscoincide}.\nIn particular we use reduction in stages to reduce to the\ncorresponding problem for the quotients of $\\bigwedge^{n-2}(\\mathbb{C}^3)$ by\nthe product torus $\\underline{\\mathbb{T}}_e \\times \\underline{\\mathbb{T}}_d^{-}$.\nHere we twist the action by the character that is $\\chi_{\\mathbf{r}}$ on $\\underline{\\mathbb{T}}_e$ and trivial on\n$\\underline{\\mathbb{T}}_d^{-}$.\nSince the momentum map for the action by $\\underline{\\mathbb{T}}_e \\times \\underline{\\mathbb{T}}_d^{-}$ \nis proper by Proposition \\ref{propernesstheorem}\nthe equality of quotients follows by Theorem \\ref{twistandshift}.\nThe momentum level for the product group is then $(\\mathbf{r},0)$.\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{The residual action of $\\mathbb{T} \/ (\\mathbb{T}_e \\times \\mathbb{T}_d^-)$ and bending flows}\\label{bendingflowsection}\n\nNow we will relate the action of $\\mathbb{T} \/ (\\mathbb{T}_e \\times \\mathbb{T}_d^-)$ on $P_n^{\\mathcal{T}}(\\mathrm{SU}(2))$ and $P_\\mathbf{r}^\\mathcal{T}(\\mathrm{SU}(2))$ to bending flows.\n\n\n\\subsection{A complement to $\\mathbb{T}_d^-$ in $\\mathbb{T}_d$}\nIn this subsection we will define a complement $\\mathbb{T}_d^+$\nto $\\mathbb{T}_d^-$ in $\\mathbb{T}_d$. It will be more convenient\nto work with this complement rather than the quotient\n$\\mathbb{T}_d\/ \\mathbb{T}_d^-$.\n\nWe recall that an element of $\\mathbb{T}$ corresponds to a function\n$f$ from the edges of $\\mathcal{T}^D$ to the circle. The subgroup $\\mathbb{T}_d$\ncorresponds to the functions with value the identity on the distinguished\nedges of $\\mathcal{T}^D$. The nondistinguished edges of $\\mathcal{T}^d$ occur in\nequivalent pairs. The subtorus $\\mathbb{T}_d^-$ of $\\mathbb{T}_d$\nconsists of those $f$ whose values on equivalent pairs of\nnondistinguished edges are inverse to each other.\n\nIn order to construct the complement $\\mathbb{T}_d^+$ we need to choose\nan edge from each distinguished pair. To do this in a systematic way\nwe use the ordering of the trinodes of $\\mathcal{T}^D$ induced by\nthe trivalent induction construction, see Lemma \\ref{trivalentinduction}. Suppose the edges\n$(\\tau, i)$ and $(\\tau', j)$ are equivalent. The two edges occur\nin different trinodes. We relabel the edges by ${\\epsilon}^+$ and\n${\\epsilon}^-$ by labeling the edge that comes in the first trinode\nby (the superscript) plus and the edge that comes in a later trinode\nby minus. Thus every nondistinguished edge is either a plus edge\nor a minus edge. We now define the subtorus $\\mathbb{T}_d^+$ as the subtorus\nof $\\mathbb{T}_d$ consisting of those functions that take value the identity on \nall nondistinguished minus edges. It is clear that $\\mathbb{T}_d^+$ is the required complement.\nIt is also clear that $\\mathbb{T}_d^+$ is a complement to\n$\\mathbb{T}_e \\times \\mathbb{T}_d^-$ in $\\mathbb{T}$. We point out\nhere that the choice of where to place the identity\nin the definition of this complement is irrelevant with respect\nto the Hamiltonian functions - see below. \n\n\n\\subsection{The action of $\\mathbb{T}_d^+$ coincides with bending} \\hfill\n\nWe now prove the following theorem.\n\\begin{theorem}\\label{noncompact2}\n\\hfill\n\\begin{enumerate}\n\\item The homeomorphism $\\Psi^{\\mathcal{T}}_n$ intertwines the bending flows\non $V_n^{\\mathcal{T}}$ with the action of $\\mathbb{T}_d^+$ on $P_n^{\\mathcal{T}}(\\mathrm{SU}(2))$. \\item The homeomorphism $\\Psi^{\\mathcal{T}}_{\\mathbf{r}}$ intertwines the bending flows\non $V_{\\mathbf{r}}^{\\mathcal{T}}(\\mathrm{SU}(2))$ with the action of $\\mathbb{T}_d^+$ on $P_{\\mathbf{r}}^{\\mathcal{T}}(\\mathrm{SU}(2))$.\n\\end{enumerate}\n\\end{theorem}\n\\begin{proof}\n\nLet $\\mathbf{T} \\in P_n^{\\mathcal{T}}(\\mathrm{SU}(2))$ and $\\mathbf{t} \\in \\mathbb{T}_d^+$.\nLet $d$ be a diagonal of the triangulated $n$-gon $P$ and suppose\nthe triangles $T_i$ and $T_j$ share the diagonal $d$. Suppose\n$d$ divides $P$ into two polygons $P'$ and $P''$ with\n$T \\in P'$ and $T' \\in P''$. Let $\\tau$ and $\\tau'$\nbe the trinodes associated to $T$ and $T'$ respectively,\nand let $[g_{(\\tau, i)}, \\lambda_{(\\tau, i)}\\varpi_1]$ and\n$[g_{(\\tau', j)}, \\lambda_{(\\tau', j)}\\varpi_1]$ be spin-framed\nrepresentatives of the edges of $\\mathbf{T}$ corresponding to\nthe diagonal $d$.  Suppose $\\epsilon(d)^- = (\\tau', j)$.\n\nSince the representatives are normalized we have\n$$g_{(\\tau, i)} = g_{(\\tau', j)} \\varrho.$$\nLet $t$ be the $\\epsilon(d)^+$-th component of $\\mathbf{t}$ (the\n$\\epsilon(d)^-$-th component is $1$). Under the action\nof $\\mathbf{t}$ the edge $[g_{(\\tau, i)}, \\lambda_{(\\tau, i)}\\varpi_1]$\nbecomes $[g_{(\\tau, i)}t,\\lambda_{(\\tau, i)}\\varpi_1]$ and all other\ncomponents are unchanged.  Thus the resulting\nelement of $E_n^{\\mathcal{T}}(\\mathrm{SU}(2))$ is no longer normalized.\n\nTo normalize this element we multiply all the frames on the edges and diagonals\nof $P''$ by $g_{(\\tau', j)} t g_{(\\tau', j)}^{-1}$. Note that the image of $h=g_{(\\tau', j)} t g_{(\\tau', j)}^{-1} \\in \\mathrm{SU}(2)$.\nis a rotation about the oriented line in $\\mathbb{R}^3$ corresponding to\n$Ad^{\\ast}h(\\varpi_1) \\in \\mathfrak{su}(2)^{\\ast} \\cong \\mathbb{R}^3$ which is  the diagonal corresponding to $d$ of\nthe Euclidean $n$-gon $\\mathbf{e}$ underlying the imploded\nframed $n$-gon $\\mathbf{T}$. Thus the Euclidean $n$-gon underlying\nthe imploded framed $n$-gon $\\mathbf{T}$ is bent along the\ndiagonal $d$. Furthermore the frames  of all the edges\nand diagonals of $P''$  are transformed by applying the element of the one-parameter group \n$g_{(\\tau', j)} t g_{(\\tau', j)}^{-1}$ in $\\mathrm{SU}(2)$ which covers\nrotation along the diagonal. Applying $\\Phi_n^{\\mathcal{T}}$ we forget the\nframes along the diagonals but the frames on the edges transform\nin the same way as before.  This amounts to bending the framed \n$n$-gon $\\Phi_n^{\\mathcal{T}}(\\mathbf{T})$ along the diagonal $d$.\n\nThe first statement implies the second statement because the\nbending flows on $V_{\\mathbf{r}}^{\\mathcal{T}}$ are descended from\nthose on $V_{n}^{\\mathcal{T}}$, and the action of\n$\\mathbb{T}_d^+$ on $P_{\\mathbf{r}}^{\\mathcal{T}}(\\mathrm{SU}(2))$ is descended from the action\non $P_{n}^{\\mathcal{T}}(\\mathrm{SU}(2))$\n\\end{proof}\n\n\\subsection{The Hamiltonians for the residual action}\nIt remains to prove that the action of $\\mathbb{T}_d^+$ is Hamiltonian\nwith the given Hamiltonians.\n\n\nWe claim that $\\mathbb{T}_d^+$ preserves the orbit-type stratification.\nSince $\\mathbb{T} = (\\mathbb{T}_e \\times \\mathbb{T}_d^-) \\times \\mathbb{T}_d^+$ is abelian,\nthe isotropy subgroup $(\\mathbb{T}_e \\times \\mathbb{T}_d^-)_x \\subset \\mathbb{T}_e \\times \\mathbb{T}_d^-$ of\n$x \\in P_\\mathbf{r}^\\mathcal{T}(\\mathrm{SU}(2))$ is equal to the isotropy\ngroup $(\\mathbb{T}_e \\times \\mathbb{T}_d^-)_{t \\cdot x}$ of $t \\cdot x$, for any $t \\in \\mathbb{T}$; in particular this is true for\n$t \\in \\mathbb{T}_d^+$.  Although the space $P_\\mathbf{r}^\\mathcal{T}(\\mathrm{SU}(2))$ is possibly\nsingular, we can work in a given symplectic stratum.\nThus it makes sense to say that $\\mathbb{T}_d^+$ acts in a Hamiltonian\nfashion (on each individual stratum), and we may identify\nthe torus action on the whole space\nby the Hamiltonians of the $S^1$ factors.\nThese Hamiltonians are smooth in the sense\nof \\cite{SjamaarLerman}, since these functions are obtained by\nrestricting $\\mathbb{T}_e \\times \\mathbb{T}_d^-$ invariant continuous functions on\n$(\\bigwedge^2 \\mathbb{C}^3)^{n-2}$ to individual strata.\n\nEach $\\mathbb{C}^2 \\subset E_n^\\mathcal{T}(\\mathrm{SU}(2))$ is indexed by\nan edge of $\\mathcal{T}^D$.\nGiven an internal edge $\\epsilon$,\nthe factor $(S^1)_{\\epsilon}$ of $\\mathbb{T}_d^+$\nhas Hamiltonian function\n$\\|(z,w)_{\\epsilon^-}\\|^2\/2 = \\|(z,w)_{\\epsilon^+}\\|^2\/2$\non any given orbit-type stratum.\nThe $(z,w)_{\\epsilon^-}$ corresponds to the diagonal\n$d_{\\epsilon}$ of the associated $n$-gon in\n$\\mathbb{R}^3$.\nBy Lemma \\ref{diagonalpullback} and Lemma \\ref{SpinLength}\nwe have $\\|d_{\\epsilon}\\| = \\frac{1}{4} \\|(z,w)_{\\epsilon^-}\\|^2$,\nand so $\\|d_{\\epsilon}\\|$ is\n\\emph{one-half} that of\nthe Hamiltonian for the $\\epsilon$-th factor $(S^1)_\\epsilon$ of\n$\\mathbb{T}_d^+$.\nIn what follows we will need the following elementary lemma.\n\\begin{lemma}\nSuppose $A: S^1 \\times X \\to X$ is a Hamiltonian action of\nthe circle on a stratified symplectic space $X$. Suppose the action\nis generated by the Hamiltonian potential $f$ and that the element\n$-1 \\in S^1$ acts trivially. Let $\\overline{A}$ be the induced action\nof the quotient circle $\\overline{S^1} = S^1\/\\{\\pm 1\\}$.\nThen the Hamiltonian potential for the $\\overline{A}$ action\nis $\\frac{f}{2}$.\n\\end{lemma}\n\n\\begin{proof}\nWe have\n$$\\overline{A}(\\exp{\\sqrt{-1}t},x) = A(\\exp{\\sqrt{-1}t\/2},x).$$\n\\end{proof}\n\nThe bending flow torus $$T_{bend} = \\prod_{\\text{$d$ diagonal}} (S^1)_d,$$\nacting on $P_\\mathbf{r}^\\mathcal{T}(\\mathrm{SU}(2))$, is such that\nthe $d(\\epsilon)$-th factor has Hamiltonian $\\|d_\\epsilon\\|$.\nTherefore, the $\\mathbb{T}_d^+$ action factors through the action of $T_{bend}$ via\na two to one cover on each component of $\\mathbb{T}_d^+$.  Thus we may conclude\nthat $T_{bend}$ and $\\mathbb{T}_d^+(\\mathrm{SO}(3,\\mathbb{R}))$ coincide.\n\n\\begin{theorem}\nLet $\\mathbb{T}_d^+(\\mathrm{SO}(3,\\mathbb{R})) = \\mathbb{T}_d^+ \/ (\\mathbb{Z}\/2\\mathbb{Z})^{n-3}$ be the image of $\\mathbb{T}_d^+$ in $\\mathrm{SO}(3,\\mathbb{R})^{n-3}$ under the\nsurjection $\\mathrm{SU}(2) \\to \\mathrm{SO}(3,\\mathbb{R})$.  The action of $\\mathbb{T}_d^+(\\mathrm{SO}(3,\\mathbb{R}))$ on $P_\\mathbf{r}^\\mathcal{T}(\\mathrm{SU}(2))$\ncoincides with the bending flows. Precisely the action is Hamiltonian\nand the circle factor\ncorresponding to $d(\\epsilon)$ has Hamiltonian potential $\\|d_{\\epsilon}\\|$.\n\\end{theorem}\n\n\\begin{proof}\nLet $\\pi : \\mathbb{T}_d^+ \\to \\mathbb{T}_d^+(\\mathrm{SO}(3,\\mathbb{R}))$ be the quotient map.\nLet $\\epsilon$ be any internal edge of $\\mathcal{T}$. The action of\n$(S^1)_\\epsilon \\subset \\mathbb{T}_d^+$ on $P_\\mathbf{r}^\\mathcal{T}(\\mathrm{SU}(2))$\nfactors through the action of $\\pi((S^1)_\\epsilon) \\in \\mathbb{T}_d^+(\\mathrm{SO}(3,\\mathbb{R}))$,\nthrough the map $t \\mapsto t^2$.  Consequently by the lemma\nabove the Hamiltonian function\nof $(S^1)_\\epsilon \\subset \\mathbb{T}_d^+$ is twice that of the Hamiltonian\nof $\\pi((S^1)_\\epsilon) \\subset \\mathbb{T}_d^+(\\mathrm{SO}(3,\\mathbb{R}))$.\nHence the Hamiltonian for $\\pi((S^1)_\\epsilon) \\subset \\mathbb{T}_d^+(\\mathrm{SO}(3,\\mathbb{R}))$ is\n$\\|d_{\\epsilon}\\|$, as it is for $(S^1)_{d_{\\epsilon}} \\subset T_{bend}$.\nSince the Hamiltonian functions\nof $T_d^+(\\mathrm{SO}(3,\\mathbb{R}))$ and $T_{bend}$ coincide, their actions must coincide on\n$P_\\mathbf{r}^\\mathcal{T}(\\mathrm{SU}(2))$.\n\\end{proof}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Appendix : symplectic and GIT quotients of affine space}\\\nThe results in this appendix were obtained with the aid of W. Goldman.\n\n\\subsection{Fiber twists and normalizing the momentum map}\n\\label{fibertwists}\nThe goal of this appendix is to prove Theorem \\ref{twistandshift} below -\nwe match the level for the symplectic quotient with the\ntwist used to define the linearization in forming the GIT quotient.\nIn four places in the paper,\nnamely subsections \\ref{firsttorusquotient},\\  \n\\ref{secondtorusquotient},\\ \n\\ref{thirdtorusquotient} and subsubsection \\ref{fourthtorusquotient} we applied the results of \\cite{Sjamaar} to deduce\nthat for a torus acting on affine space linearized by acting\non the trivial line bundle by a character $\\chi$ the GIT\nquotient is homeomorphic to the symplectic quotient at level\nthe derivative of $\\chi$ at the identity provided the momentum map was proper.\nHowever there is a technical problem \nabout the normalization  of the momentum\nmap chosen for the\naction of the torus \n(there is an indeterminancy of an  additive\nconstant vector) \nThe correct normalization of\nthe momentum map  must depend on  the action of the torus on the total space of the line\nbundle. It is given in formula (2.3), page 116 of \\cite{Sjamaar} which\nwe now state for the convenience of the reader. \n\nLet  $p:E \\to M$ be a Hermitian line bundle $L$. Let $G$ be a compact group\nacting on $E$ by automorphisms of the Hermitian structure. \nLet $\\xi \\in \\mathfrak{g}, e \\in E$ and $m=p(e)$.   In what follows\n$\\xi_E$ denotes the vector field on $E$ induced by $\\xi$,\n$\\xi_M^{hor}$ denotes the horizontal lift of the vector\nfield $\\xi_M$ induced on $M$ and $\\nu_E$ denotes the canonical vertical\nvector field (induced by the $\\mathrm{U}(1)$ action). The connection and curvature\nforms take values in the Lie algebra $\\mathfrak{u}(1) = \\sqrt{-1} \\mathbb{R}$\nof $\\mathrm{U}(1)$. We can now state the formula from \\cite{Sjamaar}:\n\\begin{equation} \\label{normalization}\n\\xi_E(e)  = \\xi_M^{hor}(e)  + 2\\pi <\\mu(m),\\xi>  \\nu_E.\n\\end{equation}\n\n\\begin{definition}\nWe will say a momentum map satisfying equation (\\ref{normalization})\nis {\\em normalized} relative to the linearization (action of $G$\non the total space of the bundle).\n\\end{definition}\n\n\\begin{remark}\nWe can check the conventions involved in equation (\\ref{normalization})\nby applying the connection form $\\theta$ to both sides to obtain\n$$\\theta(\\xi_E(e)) = 2\\pi \\sqrt{-1}<\\mu(m),\\xi>.$$ \nApplying $d$ to each side and Cartan's formula we find that \n$<\\mu(m),\\xi>$ is a Hamiltonian potential for $\\xi_M $ if and only if\nthe symplectic form $\\omega$ and the connection form $\\theta$ are related\nby\n$$\\omega = - \\frac{1}{2\\pi \\sqrt{-1}} d \\theta.$$\n\\end{remark}\n\nSuppose now we twist the action of the torus $G$ on the total\nspace of the line bundle by scaling each fiber by a fixed character\n$\\chi$. This changes the invariant sections and hence changes the\nGIT quotient. Note that the differential of $\\chi$ at the identity\nof $G$ is an element $\\dot{\\chi}$ of $\\mathfrak{g}^*$. Thus we could\nchange the momentum map $\\mu$  by adding $\\dot{\\chi}$ and obtain a\nnew momentum map. The following lemma is an immediate consequence\nof equation (\\ref{normalization}).\n\n\\begin{lemma}\\label{twistshift}\nSuppose we twist the action of $G$ by a character $\\chi$. Then the\nnormalized momentum map for the new action is obtained by adding\n$\\dot{\\chi}$.\n\\end{lemma}\n\nWe now restrict to the case of a torus $T$ acting linearly on a\nsymplectic vector space $V,\\omega$. We assume we have chosen a\n$T$-invariant complex structure $J$ on $V$ so that $\\omega$ is \nof type $(1,1)$ for $J$ (this means $J$ is an isometry of $\\omega$) and \nthe symmetric form $B$ given by $B(v,v) = \\omega(v,Jv)$\nis positive definite.\nWe let $W$ be the subspace of $V \\otimes \\mathbb{C}$ of type $(1,0)$ vectors,\nthat is $W=\\{ v -\\sqrt{-1} Jv: v \\in V\\}$. We define a \npositive-definite Hermitian form $H$ on $W$ by \n$$H(v_1 - \\sqrt{-1}Jv_1, v_2 - \\sqrt{-1}Jv_2)= B(v_1,v_2) - \\sqrt{-1}\n\\omega(v_1,v_2).$$\nWe will abbreviate $\\sqrt{H(v,v)}$ to $\\|v\\|$ in what follows. We define\na symplectic form $A$ on $W$ by $A(w_1,w_2) = - \\Im H(w_1,w_2)$.\nWe note that the map $w \\to \\Re w$ is a symplectomorphism\nfrom $W,A$ to $V,\\omega$.\n\nSuppose that we have chosen an $H$-orthonormal basis for $W$ so we\nhave identified\n$$W \\cong \\mathbb{C}^n.$$\nWe let $T_0 \\cong (S^1)^n$ be the  compact torus\nconsisting of the diagonal matrices with unit length elements on\nthe diagonal and $\\underline{T}_0$ be the complexification of $T_0$.  We let $\\mathfrak{t}_0^*\n\\cong \\mathbb{R}^n$ be the dual of the Lie algebra of $T_0$.  We define\n$\\mu_0:W \\to \\mathfrak{t}_0 ^*$ by\n$$\\mu_0((z_1,\\cdots,z_n)) = (-\\frac{|z_1|^2}{2},\\cdots,-\\frac{|z_n|^2}{2})$$\nThen $\\mu_0$ is a momentum map for the Hamiltonian action of $T_0$\non $W$.\nWe note that  all possible momentum maps\nare obtained from $\\mu_0$ by adding a vector $\\mathbf{c} = (c_1,\\cdots,c_n)\n\\in \\mathfrak{t}_0^{\\ast}$.  Thus $\\mu_0$ is the unique momentum map\nvanishing at the origin of $W$.  \n\nLet $E= W \\times \\mathbb{C}$ be total space of the trivial line bundle $L$ over $W$. We first describe the Hermitian holomorphic structure on $L$.\nWe give $L$ a holomorphic  structure by requiring that the nowhere vanishing section $s_0$ of $E$ defined by\n$$s_0(w) = (w,1)$$\nis holomorphic. \nThus if $U$ is an open subset of $W$ and $s$ is a local section over $U$\nthen $s$ is holomorphic if and only if the function $f$ on $U$ defined by\n$$s = f s_0|U$$\nis holomorphic.\n\nWe  define a Hermitian  structure on $L$ by defining\n$$||s_0||(w) = \\exp{(-\\frac{\\pi}{2}\\|w\\|^2)}.$$ \nHence the section $\\sigma_0$ given by\n$$\\sigma_0(w) = \\exp{(\\frac{\\pi}{2}\\|w\\|^2)} s_0(w)$$\nhas unit length  at every point.\nWe note that\n$$ - \\frac{1}{2\\pi \\sqrt{-1}}\\  \\overline{\\partial}\\partial \\ \\log ||s_0||^2 = \\sum _{i=1} ^{n} dx_i \\wedge dy_i$$\nin agreement with \\cite{Sjamaar}, pg. 115.\nLet $E_0$ denote the principal circle bundle of unit length vectors in \n$E$. \n\n\nOur goal is to describe the lift of the action of $T_0$ to $E$\nso that the normalized momentum map corresponding to this lifted\naction is $\\mu_0$. We note there is a distinguished lift given by\n$$t \\circ(w,z) = (tw,z), t \\in T_0, w \\in W, z \\in \\mathbb{C}.$$\nBy definition this lift leaves invariant the holomorphic section $s_0$\nbut it also leaves invariant the unit length section $\\sigma_0$\nbecause the function $\\exp{(-\\frac{\\pi}{2}\\|w\\|^2)}$ is invariant\nunder $\\mathrm{U}(n)$ and hence under $T_0$.  Hence $T_0$ leaves fixed the Hermitian structure on $E$\nand  $\\underline{T}_0$ leaves the holomorphic section $s_0$ fixed.\nWe will call such lifts {\\it untwisted} and we will say the linearization\nconsisting of the trivial bundle together with the previous action\nis untwisted. We will now prove\n\\begin{proposition}\\label{untwistedstandard}\nThe normalized momentum map of $T_0$ corresponding to the untwisted linearization\nis $\\mu_0$.\n\\end{proposition}\n\nThe proposition will be a consequence of the next lemma and corollary.\n\n\nWe will compute $\\mu_0$ in the trivialization of $E_0$ given by $\\sigma_0$.\nWe note that since $\\sigma_0$ is invariant under $T_0$ the untwisted lift of the compact torus above (relative\nto the trivialization by $s_0$) remains untwisted in the trivialization\nby $\\sigma_0$.  We let $\\psi$ denote the coordinate in the fiber circle\nof $E_0$ so $\\nu_{E_0} = \\partial \/ \\partial \\psi$. Hence if $z_i = x_i + \\sqrt{-1}y_i, 1 \\leq i \\leq n$, then\n$x_1,y_1,\\cdots,x_n,y_n, \\psi$ are coordinates in $E_0$.\n\n\n\\begin{lemma}\\label{connectionform}\nThe canonical connection on $E$ is given by\n$$\\nabla \\sigma_0 = - \\pi \\sqrt{-1} (\\sum_{i=1}^n (x_i dy_i - y_i dx_i) \\otimes \\sigma_0.$$\nEquivalently in the above coordinates the connection form $\\theta$ of the canonical connection is given by\n$$\\theta = \\sqrt{-1} d\\psi - \\pi \\sqrt{-1} (\\sum_{i=1}^n (x_i dy_i - y_i dx_i).$$\n\\end{lemma}\n\\begin{proof}\nThe reader will verify that $\\nabla$ satisfies\n$$ \\nabla_{\\partial \/ \\partial \\overline{z}_j} s_0 =0, 1 \\leq j \\leq n, $$\nand has curvature $- 2\\pi \\sqrt{-1}\\omega$.  Hence $\\nabla$ is the unique Hermitian connection with\ncurvature $d \\theta = - 2\\pi \\sqrt{-1}\\omega$.\n\\end{proof}\n\nWe then have the  following corollary.\n\n\\begin{corollary}\nThe horizontal lift of the vector field $x_i \\partial\/\\partial y_i -\ny_i \\partial\/\\partial x_i$ is $x_i \\partial\/\\partial y_i - y_i\n\\partial\/\\partial x_i + \\pi (x_i^2 + y_i^2)\\partial \/ \\partial \\psi$.\n\\end{corollary}\n\n\\begin{proof}\nBy the formula for $\\theta$ we see that the horizontal lift of\n$\\partial\/\\partial x_i$ is $\\partial\/\\partial x_i - \\pi y_i \\partial \/ \\partial \\psi\n$ and the horizontal lift of $\\partial\/\\partial y_i$ is\n$\\partial\/\\partial y_i +  \\pi x_i \\partial \/ \\partial \\psi$. Since the operation\nof taking horizontal lifts is linear over the functions the corollary\nfollows.\n\\end{proof}\n\nProposition \\ref{untwistedstandard} follows from the corollary and  equation (\\ref{normalization}).\n\nNow suppose $T$ is a compact torus with complexification $\\underline{T}$ and $T$ acts on $W$ through a\nrepresentation \n$\\rho:T \\to T_0$. Assume further that we linearize the action \nof $\\underline{T}$ on $W$ by the untwisted linearization.\nIt is  clear from equation (\\ref{normalization}) that we obtain the {\\it normalized} momentum\nmap $\\mu_T$ for $T$ by restricting the {\\it normalized} momentum map for $T_0$. More precisely\nlet $\\rho^{\\ast} : \\mathfrak{t}_0^{\\ast} \\to \\mathfrak{t}^{\\ast}$\nbe the induced map on dual spaces.\nThen we have\n$$\\mu_T = \\rho^{\\ast} \\circ \\mu_0.$$\nThus the normalized momentum map corresponding to the untwisted linearization\nof the linear action of $T$ is  homogeneous linear  in the squares of the $|z_i|$'s.\nWe now obtain the desired result in this appendix by\napplying Lemma \\ref{twistshift} and Theorem 2.18 (page 122) of \\cite{Sjamaar}.\n \n\\begin{theorem}\\label{twistandshift}\nLet $\\underline{T}$ be a complex torus with maximal compact  subtorus $T$ and let $\\chi$ be a character of $\\mathbb{T}$.\nThen the  GIT quotient for a linear action of  $\\underline{T}$ on a complex vector space $W$ with  linearization given by $W \\times \\mathbb{C}$ and the action  \n$$t \\circ (w,z) = (tw, \\chi(t) z)$$\nis homeomorphic to the symplectic quotient by \n$T$ obtained using the momentum\nmap $(\\mu_T)_0 + \\dot{\\chi}$ if the momentum map for the action of $T$\nis proper. Here $(\\mu_T)_0$ is the momentum map that vanishes at the origin\nof $W$.\n\\end{theorem}\n\n\n\n\n\n\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section*{Introduction}\n\nThroughout this paper, all spaces are path-connected with homotopy types of $CW$-complexes. \nWe do not distinguish between a map and its homotopy class.\n\nLet $X$ be a connected space, $x_0\\in X$ a base-point and $\\mathbb{S}^1$ the circle. The  \\textit{Gottlieb group} $G(X,x_0)$ of $X$ defined in \\cite{gottlieb1}  is the subgroup of the fundamental group $\\pi_1(X,x_0)$ consisting of all elements which can be represented by a map $\\alpha \\colon \\mathbb{S}^1\\to X$ such that $\\mathrm{id}_X \\vee \\alpha \\colon X\\vee \\mathbb{S}^1\\to X$ extends (up to homotopy) to a map $F \\colon X\\times \\mathbb{S}^1\\to X$.\nFollowing \\cite{gottlieb1}, we recall that $P(X,x_0)$ is the set of elements of $\\pi_1(X,x_0)$ whose Whitehead products\nwith all elements of all homotopy groups $\\pi_m(X,x_0)$  are zero for $m\\geq 1$. It turns out that $P(X,x_0)$\nforms a subgroup of $\\pi_1(X,x_0)$ called the \\textit{Whitehead center group} and, by \\cite[Theorem~I.4]{gottlieb1}, it holds $G(X,x_0)\\subseteq P(X,x_0)$.\n\nNow, given a map $f \\colon X\\to Y$, in view of \\cite{gottlieb} (see also \\cite{kim}),\nthe  \\textit{generalized Gottlieb group} $G^f(Y,f(x_0))$ is defined \nas the subgroup of  $\\pi_1(Y,f(x_0))$ consisting of all elements which can be\nrepresented by a map $\\alpha \\colon \\mathbb{S}^1\\to Y$ such that $f\\vee \\alpha \\colon X\\vee \\mathbb{S}^1\\to Y$ extends (up to homotopy) to a map $F \\colon X\\times \\mathbb{S}^1\\to Y$. \n\nThe  \\textit{generalized Whitehead center group}  $P^f(Y,f(x_0))$ as defined in \\cite{kim} consists of all elements $\\alpha\\in\\pi_1(Y,f(x_0))$ whose Whitehead products $[f\\beta,\\alpha]$ are zero for all  $\\beta\\in\\pi_m(X,x_0)$ with $m\\ge 1$. It turns out that $P^f(Y,f(x_0))$ forms a subgroup of $\\pi_1(Y,f(x_0))$ and $G^f(Y,f(x_0))\\subseteq P^f(Y,f(x_0))\\subseteq \\mathcal{Z}_{\\pi_1(Y,f(x_0))}f_*(\\pi_1(X,x_0))$, the centralizer of $f_*(\\pi_1(X,x_0))$ in $\\pi_1(Y,f(x_0))$. \n\nIf $X=Y$ then the group $G^f(Y,f(x_0))$  is considered in \\cite[Chapter~II, 3.5~Definition]{jiang}, denoted by $J(f,x_0)$ and called the \\textit{Jiang subgroup} of the map $f\\colon Y \\to Y$. The role of the $J(f,x_0)$ played in that theory has been intensively studied in the book \\cite{brown} as well. More precisely, it is observed that the group $J(f,x_0)$  acts on the right on the set of all fixed point classes of $f$, and any two  equivalent fixed point classes under this action have the same index. Further, Bo-Ju Jiang in \\cite[Chapter~II, 3.1~Definition]{jiang} considered also the group $J(\\tilde{f}_0)$ for a fixed lifting $\\tilde{f}_0$ of $f$ to the universal covering of $Y$ and stressed its importance to the Nielsen--Wecken theory of fixed point classes. \n\nIf $f=\\mathrm{id}_X$ then, by \\cite[Theorem~II.1]{gottlieb1}, the groups $J(f,x_0)$ and $J(\\tilde{f}_0)$ are isomorphic and, according to \\cite[Chapter~II, 3.6~Lemma]{jiang}, the groups $J(f,x_0)$ and $J(\\tilde{f}_0)$ are isomorphic for any self-map $f\\colon X\\to X$ but no proof is given.\n\n\nThe aim of this paper is to follow the proof of \\cite[Theorem~II.1]{gottlieb1} and give not only a proof of \\cite[Chapter~II, 3.6~Lemma]{jiang} but also present a proof of its generalized version for any map $f\\colon X \\to Y$.\n\nThe paper is divided into two sections. Section~\\ref{sec.1} follows some results from \\cite{gottlieb1} and  deals with some properties of fibre-preserving maps and deck transformations used in the sequel. In particular, we show the functoriality of the fundamental group via deck transformations.  Section~\\ref{sec.2} takes up the systematic study of the group $G^f(Y,f(x_0))$. If $X=Y$, $f=\\mathrm{id}_X$ and $x_0\\in X$ is a base-point then the group $G^f(X,x_0)=G(X,x_0)$ has been described in  \\cite[Theorem~II.1]{gottlieb1} via the deck transformation group of $X$ and by \\cite[Chapter~II, 3.6~Lemma]{jiang} the groups $G^f(X,x_0)=J(f,x_0)$ and $J(\\tilde{f}_0)$ are isomorphic for a self-map $f\\colon X\\to X$.\n\nDenote by $\\mathcal{D}(Y)$ the group of all deck transformations of a space $Y$. Given a map $f\\colon X \\to Y$, write $\\mathcal{L}^f(Y)$ for the set of all liftings  $\\tilde{f}\\colon \\tilde{X}\\to \\tilde{Y}$ of $f$ with respect to universal coverings of $X$ and $Y$, respectively.  Now, for a fixed $\\tilde{f}_0\\in \\mathcal{L}^f(Y)$, we denote by $\\mathcal{D}^{\\tilde{f}_0}(Y)$ the set (being a group) of all elements $h\\in  \\mathcal{D}(Y)$ such that $\\tilde{f}_0\\simeq_{\\tilde{H}} h\\tilde{f}_0$, where  $\\tilde{H}\\colon \\tilde{X}\\times I \\to \\tilde{Y}$ is a fibre-preserving homotopy with respect to the universal covering maps $p\\colon \\tilde{X}\\to X$ and $q\\colon \\tilde{Y}\\to Y$. Then, the main result, Theorem~\\ref{qq} generalizes \\cite[Theorem~II.1]{gottlieb1} and \\cite[Chapter~II, 3.6~Lemma]{jiang} as follows:\n\n\\textit{Given $f\\colon X \\to Y$,  the canonical isomorphism $\\pi_1(Y,f(x_0))\\xrightarrow{\\approx}\\mathcal{D}(Y)$ restricts to an isomorphism $G^f(Y,f(x_0))\\xrightarrow{\\approx} \\mathcal{D}^{\\tilde{f}_0}(Y)$.}\n\n\\section{Preliminaries}\\label{sec.1}\n\nLet $p\\colon X\\to A$ and $q\\colon Y\\to B$ be maps. We say that $f\\colon X\\to Y$ is a \\textit{fibre-preserving map} with respect to $p,q$ provided $p(x)=p(x')$ implies $qf(x)=qf(x')$ for any $x,x'\\in X$. \n\nWe say that  $H\\colon X\\times I \\to Y$ is a \\textit{fibre-preserving homotopy} with respect to  $p,q$ if $H$ is a fibre-preserving map with respect to $p\\times \\mathrm{id}_I\\colon X\\times I  \\to A\\times I$ and $q\\colon Y \\to B$.\n\nIt is clear that the commutativity of a diagram \\[ \\xymatrix{ X\\ar[r]^{f}  \\ar[d]_p & Y\\ar[d]^q \\\\ A \\ar[r]^g& B }\\] guarantee that $f$ is a fibre-preserving map. \n\n\n\\begin{remark}\n(1) Let $p,q,f$ be maps as above. If $p\\colon X \\to A$ is surjective then there exists a map $g\\colon A\\to B$ such that $qf=g p$. In addition, if $p$ is a quotient map, then $g$ is continuous.\n\n(2) Given discrete groups $H$ and $K$, consider actions $H\\times X \\to X$ and $K\\times Y \\to Y$ and write $p\\colon X \\to X\/H$ and $q\\colon Y \\to Y\/K$ for the quotient maps. If $f\\colon X \\to Y$ is a $\\varphi$-equivariant map for a homomorphism $\\varphi\\colon H \\to K$ then $f$ is a fibre-preserving map with respect to $p$ and $q$.\n\\end{remark}\n\nIf $f\\colon X \\to Y$ is a fibre-preserving map and $g=\\mathrm{id}_A$ then the map $f$ is a fibrewise map in the sense of \\cite[Chapter~1]{james}. But, the reciprocal of that does not hold, as it is shown below:\n\n\n\\begin{example}\nLet $p=q\\colon \\mathbb{S}^1\\times I \\to \\mathbb{S}^1$ be the projection. Fix $1\\neq \\lambda\\in \\mathbb{S}^1$ and define $f\\colon \\mathbb{S}^1\\times I\\to \\mathbb{S}^1\\times I$ by $f(z,t)= (\\lambda z,t)$ for $(z,t)\\in \\mathbb{S}^1\\times I$. Then, $f$ is a fibre-preserving map but clearly $qf\\neq p$.\n\\end{example}\n\nWrite $\\mathcal{D}(X)$ for the group of all deck transformations of $X$ and  recall that there is an isomorphism $\\mathcal{D}(X)\\approx \\pi_1(X,x_0)$. Next, given  a map $f\\colon X\\to Y$, consider the set $\\mathcal{L}^f(Y)$ of all maps $\\tilde{f}\\colon \\tilde{X}\\to \\tilde{Y}$ such that the diagram \\[ \\xymatrix{ \\tilde{X}\\ar[r]^{\\tilde{f}}  \\ar[d]_p & \\tilde{Y}\\ar[d]^q \\\\ X \\ar[r]^f & Y }\\] is commutative, where $p,q$ are universal covering maps. \n\nFixing $\\tilde{f}_0\\in \\mathcal{L}^f(Y)$, we follow \\cite[Chapter~I, 1.2~Proposition]{jiang} to show:\n\n\\begin{proposition}\\label{star}If $f\\colon X\\to Y$ then for any lifting $\\tilde{f}\\colon \\tilde{X}\\to \\tilde{Y}$ of $f$ there is a unique $h\\in \\mathcal{D}(Y)$ such that $\\tilde{f}=h\\tilde{f}_0$. \n\\end{proposition}\n\\begin{proof} First, fix $x_0\\in X$ and $\\tilde{x}_0\\in p^{-1}(x_0)$, and\nwrite $y_0=f(x_0)$, $\\tilde{y}_0=\\tilde{f}_0(\\tilde{x}_0)$, and\n$\\tilde{y}=\\tilde{f}(\\tilde{x}_0)$. Obviously, $\\tilde{y}_0,\\tilde{y}\\in\nq^{-1}(y_0)$. Then, there exists a unique $h\\in \\mathcal{D}(Y)$ with\n$h(\\tilde{y}_0)=\\tilde{y}$, that is, $h\\tilde{f}_0(\\tilde{x}_0)=\n\\tilde{f}(\\tilde{x}_0)$. Since both $\\tilde{f}$ and $h\\tilde{f}_0$ are\nlifts of $fp\\colon \\tilde{X}\\to Y$, the unique lifting property\nguarantees that $\\tilde{f}=h\\tilde{f}_0$. \n\\par Now, suppose that $\\tilde{f}=h\\tilde{f}_0=h'\\tilde{f}_0$ for some $h,h'\\in \\mathcal{D}(Y)$. Then, $h\\tilde{f}_0(\\tilde{x}_0)=h'\\tilde{f}_0(\\tilde{x}_0)$ implies $h(\\tilde{y}_0)=h'(\\tilde{y}_0)$. Consequently, $h=h'$ and the proof is complete.\n\\end{proof}\n\nFor a deck transformation $l\\in \\mathcal{D}(X)$, we notice that $\\tilde{f}_0l$ is also a lifting of $f$. By Proposition~\\ref{star}, there exists a unique $h_l\\in \\mathcal{D}(Y)$ such that $\\tilde{f}_0l=h_l\\tilde{f}_0$. \nThen, we define \\[f_*\\colon \\mathcal{D}(X)\\to \\mathcal{D}(Y)\\] by $f_*(l)=h_l$ for any $l\\in \\mathcal{D}(X)$. Obviously, the map $f_*$ is a homomorphism. Notice that the map $f_*$ has been already defined in \\cite[Chapter~II, 1.1~Definition]{jiang} for any self-map $f\\colon X \\to X$.\n\n\nGiven $\\tilde{f}_1,\\tilde{f}_2\\in \\mathcal{L}^f(Y)$, we define $\\tilde{f}_1\\ast\\tilde{f}_2=h_1h_2\\tilde{f}_0$, where $\\tilde{f}_1=h_1\\tilde{f}_0$ and $\\tilde{f}_2=h_2\\tilde{f}_0$ for   $h_1,h_2\\in \\mathcal{D}(Y)$ as in Proposition~\\ref{star}. This leads to a group structure on $\\mathcal{L}^f(Y)$ with $\\tilde{f}_0$ as the identity element. Notice that the groups $\\mathcal{L}^f(Y)$ and $\\mathcal{D}(Y)$ are isomorphic. In the sequel we identify those two groups, if necessary.\n\nFor a homotopy  $\\tilde{H}\\colon \\tilde{X}\\times I \\to \\tilde{Y}$, we write $\\tilde{H}_t=\\tilde{H}(-,t)$ with $t\\in I$.\n\n\\begin{lemma}\\label{Ht}Let $f\\colon X \\to Y$. A homotopy $\\tilde{H}\\colon \\tilde{X}\\times I \\to \\tilde{Y}$ with $\\tilde{H}_0=\\tilde{f}_0$ is a fibre-preserving homotopy if and only if for any $l\\in \\mathcal{D}(X)$ and $t\\in I$ the following diagram \n\\[ \\xymatrix@C=1.5cm{ \\tilde{X}\\ar[r]^{\\tilde{H}_t}  \\ar[d]_-{l} & \\tilde{Y}\\ar[d]^-{f_*(l)} \\\\ \\tilde{X} \\ar[r]^{\\tilde{H}_t} & \\tilde{Y}  } \\] commutes.\n\\end{lemma}\n\n\\begin{proof}Let $(\\tilde{x},t),(\\tilde{x}',t')\\in \\tilde{X}\\times I$ with  $(p\\times \\mathrm{id}_I)(\\tilde{x},t)=(p\\times \\mathrm{id}_I)(\\tilde{x}',t')$. Then, $p(\\tilde{x})=p(\\tilde{x}')$ and $t=t'$. Next, consider $l\\in \\mathcal{D}(X)$ such that $l(\\tilde{x})=\\tilde{x}'$. Because $\\tilde{H}_t=f_*(l)\\tilde{H}_tl^{-1}$, we conclude that $q\\tilde{H}_t(\\tilde{x})=qf_*(l)\\tilde{H}_tl^{-1}(\\tilde{x})=qf_*(l)\\tilde{H}_t(\\tilde{x}')=q\\tilde{H}_t(\\tilde{x}')$. Hence $\\tilde{H}$ is fibre-preserving.\n\nSuppose $\\tilde{H}$ is fibre-preserving, $l\\in \\mathcal{D}(X)$ and take $\\tilde{x}\\in \\tilde{X}$, $t\\in I$. Then, $\\tilde{x}$ and $ l(\\tilde{x})$ are in the same fibre of $p$. Since $\\tilde{H}_t(\\tilde{x})$ and $\\tilde{H}_t(l(\\tilde{x}))$ are in the same fibre of $q$, there exists a unique $h\\in \\mathcal{D}(Y)$ such that $h\\tilde{H}_t(\\tilde{x})=\\tilde{H}_tl(\\tilde{x})$. If $\\varepsilon>0$ is sufficiently small, $h\\tilde{H}_{t-\\varepsilon}(\\tilde{x})=\\tilde{H}_{t-\\varepsilon}l(\\tilde{x})$. Thus, the greatest lower bound of the set of $t$'s such that $h\\tilde{H}_t(\\tilde{x})=\\tilde{H}_tl(\\tilde{x})$ must occur when $t=0$. Therefore, by continuity, $h\\tilde{H}_0(\\tilde{x})=\\tilde{H}_0l(\\tilde{x})$. But $\\tilde{H}_0=\\tilde{f}_0$, so we get $h\\tilde{f}_0(\\tilde{x})=\\tilde{f}_0l(\\tilde{x})=f_*(l)\\tilde{f}_0(\\tilde{x})$. This can occur only when $h=f_*(l)$. Consequently, $\\tilde{H}_tl=f_*(l)H_t$ and the proof is complete.\n\\end{proof}\n\nNow, fix $\\tilde{f}_0\\in \\mathcal{L}^{f}(Y)$ and consider the subset $\\mathcal{D}^{\\tilde{f}_0}(Y)$ of elements $h\\in  \\mathcal{D}(Y)$  such that $\\tilde{f}_0\\simeq_{\\tilde{H}} h\\tilde{f}_0$, where $\\tilde{H}\\colon \\tilde{X}\\times I \\to \\tilde{Y}$ is a fibre-preserving homotopy with respect to the universal covering maps $p,q$. \nEquivalently, in view of Lemma~\\ref{Ht}, the set $\\mathcal{D}^{\\tilde{f}_0}(Y)$ coincides with the set of all elements $h\\in  \\mathcal{D}(Y)$ for which there is a homotopy $f\\simeq_H f$ which lifts to a homotopy $\\tilde{H}$ with $\\tilde{f}_0\\simeq_{\\tilde{H}}h\\tilde{f}_0$.\n\nNext, write $\\mathcal{Z}_{\\mathcal{D}(Y)} f_*(\\mathcal{D}(X))$ for the centralizer of $ f_*(\\mathcal{D}(X))$ in $\\mathcal{D}(Y)$. Then, the result below generalizes \\cite[Chapter~II, 3.2~Proposition, 3.3~Lemma]{jiang} as follows:\n\n\\begin{proposition}\\label{prop.center}The subset $\\mathcal{D}^{\\tilde{f}_0}(Y)$  is contained in $\\mathcal{Z}_{\\mathcal{D}(Y)} f_*(\\mathcal{D}(X))$ and is a subgroup of $\\mathcal{D}(Y)$.\n\\end{proposition}\n\n\\begin{proof}\nLet $h\\in \\mathcal{D}^{\\tilde{f}_0}(Y)$. Then, there is a fibre-preserving homotopy  $\\tilde{H}\\colon \\tilde{X}\\times I \\to \\tilde{Y}$ with $\\tilde{H}_0=\\tilde{f}_0$ and  $\\tilde{H}_1=h\\tilde{f}_0$. But, by Lemma~\\ref{Ht}, it holds $f_*(l)\\tilde{H}_t=\\tilde{H}_tl$ for any $l\\in \\mathcal{D}(X)$ and $t\\in I$. Hence,  for $t=0,1$ we get  $f_*(l)\\tilde{f}_0=\\tilde{f}_0l$ and $f_*(l)h\\tilde{f}_0=h\\tilde{f}_0l=hf_*(l)\\tilde{f}_0$. Consequently,  $f_*(l)h=hf_*(l)$ and we get that $h\\in \\mathcal{Z}_{\\mathcal{D}(Y)} f_*(\\mathcal{D}(X))$.\n\n\nTo show the second part, take $h,h'\\in \\mathcal{D}^{\\tilde{f}_0}(Y)$. Then, there are fibre-preserving homotopies $\\tilde{H},\\tilde{H}'\\colon \\tilde{X}\\times I \\to \\tilde{Y}$ with $\\tilde{H}_0=\\tilde{H}_0'=\\tilde{f}_0$, $\\tilde{H}_1=h\\tilde{f}_0$ and $\\tilde{H}'_1=h'\\tilde{f}_0$. Next,  consider the map $\\tilde{H}''\\colon \\tilde{X}\\times I \\to \\tilde{Y}$ given by $\\tilde{H}''(\\tilde{x},t)=hh'^{-1}\\tilde{H}'(\\tilde{x},1-t)$ for $(\\tilde{x},t)\\in \\tilde{X}\\times I$ and notice that $\\tilde{H}''$ is a fibre-preserving homotopy with $\\tilde{H}''_0=h\\tilde{f}_0$ and  $\\tilde{H}''_1=hh'^{-1}\\tilde{f}_0$. Finally, the concatenation $\\tilde{H}\\bullet \\tilde{H}'' \\colon \\tilde{X}\\times I \\to \\tilde{Y}$ is a fibre-preserving homotopy with $(\\tilde{H}\\bullet \\tilde{H}'')_0=\\tilde{f}_0$ and  $(\\tilde{H}\\bullet \\tilde{H}'')_1=hh'^{-1}\\tilde{f}_0$. Consequently, $hh'^{-1}\\in \\mathcal{D}^{\\tilde{f}_0}(Y)$ and the proof is complete.\n\\end{proof}\n\nNotice that if $f\\colon X \\to X$ is a self-map then the group $\\mathcal{D}^{\\tilde{f}_0}(X)$ coincides with the group $J(\\tilde{f}_0)$ defined in \\cite[Chapter~II, 3.1~Definition]{jiang}. \n\n\n\\section{Main result}\\label{sec.2}\n\nGiven spaces $X$ and $Y$, write $Y^X$ for the space of continuous maps from $X$ into $Y$ with the compact-open topology. Next, consider the evaluation map $\\mathrm{ev} \\colon Y^X\\to Y$, i.e., $\\mathrm{ev}(f)=f(x_0)$ for $f\\in  Y^X$ and the base-point $x_0\\in X$. \nThen, it holds \\[G^f(Y,f(x_0))=\\operatorname{Im}\\bigl(\\mathrm{ev}_\\ast \\colon \\pi_1(Y^X,f)\\to \\pi_1(Y,f(x_0))\\bigr).\\] \nCertainly, $G^f(X,f(x_0))$ coincides with the group $J(f,x_0)$ defined in \\cite[Chapter~II, 3.5~Definition]{jiang} for a self-map $f\\colon X\\to X$.\n\\par Now, we follow \\textit{mutatis mutandis} the result \\cite[Theorem~II.1]{gottlieb1} to generalize \\cite[Chapter~II, 3.6~Lemma]{jiang} as follows: \n\n\\begin{theorem}\\label{qq}Given $f\\colon X \\to Y$, the canonical isomorphism $\\pi_1(Y,f(x_0))\\xrightarrow{\\approx} \\mathcal{D}(Y)$ restricts to an isomorphism $G^f(Y,f(x_0))\\xrightarrow{\\approx} \\mathcal{D}^{\\tilde{f}_0}(Y)$.\n\\end{theorem}\n\n\\begin{proof}Let $\\alpha\\in G^f(Y,f(x_0))$ and $h\\in \\mathcal{D}(Y)$ be the corresponding deck transformation. Then, there is a homotopy $H\\colon X\\times I \\to Y$ such that $H_0=H_1=f$ and $H(x_0,-)=\\alpha$, where $x_0\\in X$ is a base-point. Next, consider the commutative diagram\n\\[ \\xymatrix@C=1.5cm{\n\\tilde{X} \\ar[d]_{i_0}\\ar[rr]^{\\tilde{f}_0} && \\tilde{Y} \\ar[d]^q \\\\\n\\tilde{X}\\times I \\ar[r]^{p\\times \\mathrm{id}_I} & X\\times I \\ar[r]^H & Y\\rlap{.}\n}  \\] Then, by the lifting homotopy property there is a map $\\tilde{H}\\colon \\tilde{X}\\times I \\to \\tilde{Y}$ such that $\\tilde{H}i_0=\\tilde{H}_0=\\tilde{f}_0$ and $q\\tilde{H}=H(p\\times \\mathrm{id}_I)$. This implies that $\\tilde{H}$ is a fibre-preserving homotopy. Further, because $H_0=H_1=f$, we also derive that $\\tilde{H}_0,\\tilde{H}_1\\in \\mathcal{L}^f(Y)$.\n\nNow, since the path $\\tilde{\\tau}\\colon I\\to \\tilde{Y}$ defined by $\\tilde{\\tau}=\\tilde{H}(\\tilde{x}_0,-)$ runs from $\\tilde{f}_0(\\tilde{x}_0)$ to $\\tilde{H}_1(\\tilde{x}_0)$, we derive that $\\alpha=q\\tilde{\\tau}$. Consequently, by means of Proposition~\\ref{star} we get $\\tilde{H}_1=h\\tilde{f}_0$ and so $\\tilde{H}$ is the required fibre-preserving homotopy with $\\tilde{f}_0\\simeq_{\\tilde{H}} h\\tilde{f}_0$. \n\nConversely, given $h\\in \\mathcal{D}^{\\tilde{f}_0}(Y)$, there is a fibre-preserving homotopy $\\tilde{H}\\colon \\tilde{X}\\times I \\to \\tilde{Y}$ with $\\tilde{f}_0\\simeq_{\\tilde{H}} h\\tilde{f}_0$. This implies a homotopy $H\\colon X\\times I \\to Y$ such that $H_0=H_1=f$ and $q\\tilde{H}=H(p\\times \\mathrm{id}_I)$. Then, the path $\\tau\\colon I\\to Y$ given by $\\tau=H(x_0,-)$  leads to the required loop in $G^f(Y,f(x_0))$.\n\\end{proof}\nNotice that by Theorem~\\ref{qq} the group $\\mathcal{D}^{\\tilde{f}_0}(Y)$ is independent of the lifting $\\tilde{f}_0\\in\\mathcal{L}^f(Y)$. Further, the advantage of $G^f(Y,f(x_0))$ over $\\mathcal{D}^{\\tilde{f}_0}(Y)$ is that it does not involve the covering spaces $\\tilde{X}$ and $\\tilde{Y}$ explicitly, hence it is easier to handle.\n\nNext, Lemma~\\ref{Ht} and Theorem~\\ref{qq} yield:\n\\begin{corollary}\\label{cor2}If $f\\colon X \\to Y$ then $G^f(Y,f(x_0))$ is isomorphic to the subgroup of $\\mathcal{D}(Y)$ given by those deck transformations $h$ for which there are homotopies $\\tilde{H}\\colon \\tilde{X}\\times I \\to \\tilde{Y}$ such that $\\tilde{f}_0\\simeq_{\\tilde{H}} h\\tilde{f}_0$  and the diagrams \n\\[ \\xymatrix@C=1.5cm{ \\tilde{X}\\ar[r]^{\\tilde{H}_t}  \\ar[d]_-{l} & \\tilde{Y}\\ar[d]^-{f_*(l)} \\\\ \\tilde{X} \\ar[r]^{\\tilde{H}_t} & \\tilde{Y}  } \\] commute for any $l\\in \\mathcal{D}(X)$ and $t\\in I$. Equivalently, the homotopies $\\tilde{H}\\colon \\tilde{X}\\times I \\to \\tilde{Y}$ are $f_*$-equivariant.\n\\end{corollary}\n\nLet $\\mathcal{H}^{\\tilde{f}_0}(Y)$ be the subset of all $h\\in\\mathcal{Z}_{\\mathcal{D}(Y)}f_*(\\mathcal{D}(X))$ such that $\\tilde{f}_0\\simeq h\\tilde{f}_0$. By similar arguments as in the proof of Proposition~\\ref{prop.center}, it is easy to verify that  $\\mathcal{H}^{\\tilde{f}_0}(Y)$ is a subgroup of $\\mathcal{Z}_{\\mathcal{D}(Y)}f_*(\\mathcal{D}(X))$. \n\nNow, we process as in the proof of \\cite[Theorem~II.6]{gottlieb1} to show:\n\n\\begin{proposition}\\label{pp}Given $f\\colon X \\to Y$, there are inclusions \\[ G^f(Y,f(x_0))\\subseteq \\mathcal{H}^{\\tilde{f}_0}(Y)\\subseteq P^f(Y,f(x_0)). \\]\n\\end{proposition}\n\n\\begin{proof} Certainly, the inclusion  $G^f(Y,f(x_0))\\subseteq \\mathcal{H}^{\\tilde{f}_0}(Y)$ is a direct consequence of Proposition~\\ref{prop.center} and Theorem~\\ref{qq}.\n\nNow, let $h\\in \\mathcal{H}^{\\tilde{f}_0}(Y)$  and $\\tilde{H} \\colon \\tilde{X}\\times I\\to \\tilde{Y}$ be a  homotopy with   $\\tilde{f}_0\\simeq_{\\tilde{H}} h\\tilde{f}_0$. Next, consider the path $\\tilde{\\phi} \\colon I\\to \\tilde{Y}$ defined by $\\tilde{\\phi}=\\tilde{H}(\\tilde{x}_0,-)$, where $p(\\tilde{x}_0)=x_0$. Then, the loop $\\phi=q\\tilde{\\phi}$ corresponds to $h$.\n\nNotice that $\\phi$ acts trivially on $f_\\ast(\\pi_m(X,x_0))$ for $m>1$ if and only if there is a map $F \\colon \\mathbb{S}^m\\times \\mathbb{S}^1\\to Y$ such that the diagram\n\\[ \\xymatrix@C=2cm{\\mathbb{S}^m\\vee \\mathbb{S}^1\\ar@{^{(}->}[d] \\ar[r]^-{f\\alpha\\vee \\phi} & Y \\\\ \n\\mathbb{S}^m\\times \\mathbb{S}^1 \\ar@{-->}[ru]_F &} \\]\ncommutes (up to homotopy) for any $\\alpha\\in \\pi_m(X,x_0)$.\\\n\nGiven $\\alpha\\in \\pi_m(X,x_0)$ with $m>1$, there exists $\\tilde{\\alpha}\\in\\pi_m(\\tilde{X},\\tilde{x}_0)$ such that $p\\tilde{\\alpha}=\\alpha$. Thus, we define a map \\[F' \\colon \\mathbb{S}^m\\times I\\xrightarrow{\\tilde{\\alpha}\\times \\mathrm{id}_I}\\tilde{X}\\times I\\xrightarrow{\\tilde{H}}\\tilde{Y}\\xrightarrow{q}Y.\\]\nBecause $F'(s,0)=q\\tilde{H}(\\tilde{\\alpha}(s),0)=q\\tilde{f}_0(\\tilde{\\alpha}(s))=fp\\tilde{\\alpha}(s)$ and\n$F'(s,1)=q\\tilde{H}(\\tilde{\\alpha}(s),1)=qh\\tilde{f}_0(\\tilde{\\alpha}(s))=q\\tilde{f}_0(\\tilde{\\alpha}(s))=fp\\tilde{\\alpha}(s)$ for $s\\in \\mathbb{S}^m$, the map $F'$\nimplies the required map $F\\colon  \\mathbb{S}^m\\times \\mathbb{S}^1\\to Y$. \n\nSince  $\\mathcal{H}^{\\tilde{f}_0}(Y)\\subseteq \\mathcal{Z}_{\\mathcal{D}(Y)}f_*(\\mathcal{D}(X))$, we derive that $\\phi$ acts trivially also on $f_\\ast(\\mathcal{D}(X))$. This gives the inclusion $\\mathcal{H}^{\\tilde{f}_0}(Y)\\subseteq P^f(Y,f(x_0))$ and the proof is complete.\n\\end{proof}\n\nLet $H$ be a finite group acting freely on a $(2n+1)$-homotopy sphere $\\Sigma(2n+1)$.\nIf $\\Sigma(2n+1)\/H$ is the corresponding space form then, following \\cite[Chapter~VII, Proposition~10.2]{brown}, the action of $H=\\mathcal{D}(\\Sigma(2n+1)\/H)$  on  $\\pi_m(\\Sigma(2n+1)\/H,y_0)$ is trivial for $m>1$. In particular, $H$ acts trivially on $\\pi_{2n+1}(\\Sigma(2n+1)\/H,y_0)\\approx \\pi_{2n+1}(\\Sigma(2n+1),\\tilde{y}_0)$. This implies that for any $h\\in H$, the induced homeomorphism $h_*\\colon \\Sigma(2n+1)\\to \\Sigma(2n+1)$ is homotopic to $\\mathrm{id}_{\\Sigma(2n+1)}$. Consequently, if $f\\colon X\\to \\Sigma(2n+1)\/H$ is a map then $\\mathcal{H}^{\\tilde{f}_0}(\\Sigma(2n+1)\/H)=\\mathcal{Z}_{H}f_*(\\mathcal{D}(X))$. Because $P^f(\\Sigma(2n+1)\/H,f(x_0))\\subseteq \\mathcal{Z}_{H}f_*(\\mathcal{D}(X))$, Proposition~\\ref{pp} yields \\[\\mathcal{H}^{\\tilde{f}_0}(\\Sigma(2n+1)\/H)=P^f(\\Sigma(2n+1)\/H,f(x_0))=\\mathcal{Z}_{H}f_*(\\mathcal{D}(X)). \\]\n\nFurther, the result \\cite[Theorem~1.17]{gol}, Theorem~\\ref{qq} and Proposition~\\ref{pp} lead to:\n\n\n\\begin{corollary}\nIf $f\\colon X\\to\\Sigma(2n+1)\/H $ is a map as in \\cite[Theorem~1.14]{gol} then $\\mathcal{D}^{\\tilde{f}_0}(\\Sigma(2n+1)\/H)=\\mathcal{H}^{\\tilde{f}_0}(\\Sigma(2n+1)\/H)=\\mathcal{Z}_Hf_*(\\mathcal{D}(X))$. In particular, $J(f,x_0)=\\mathcal{Z}_Hf_*(H)$ for any self-map $f\\colon \\Sigma(2n+1)\/H\\to \\Sigma(2n+1)\/H$.\n\\end{corollary}\n\nGiven a free action  of a finite group $H$ on $\\mathbb{S}^{2n+1}$, Oprea \\cite[\\textsc{Theorem~A}]{oprea} has shown that $G(\\mathbb{S}^{2n+1}\/H,y_0)=\\mathcal{Z}H$, the center of $H$.\nIn the special case of a free linear action of $H$ on $\\mathbb{S}^{2n+1}$, the description of $G(\\mathbb{S}^{2n+1}\/H,y_0)$ via deck transformations presented in \\cite[Theorem~II.1]{gottlieb1} has been applied  in \\cite{bro} to get a very nice representation-theoretic proof of \\cite[\\textsc{Theorem~A}]{oprea}. The result stated in Theorem~\\ref{qq} might be applied to extend the methods from \\cite{bro} to simplify the  proof of  \\cite[Theorem~1.17]{gol} on the case of a free linear action  of $H$ on $\\mathbb{S}^{2n+1}$ as well.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{sec:intro}\nThe solar wind is a plasma flow, electrically neutral, which emanates from the basis of the solar corona, permeating and shaping the whole heliosphere. During solar activity minima, when the meridional branches of the polar coronal holes (CH hereafter) reach the equatorial regions of the sun, an observer located in the ecliptic plane would record a repeated occurrence of fast (700-800 km\/s) and slow (300-400 km\/s) wind samples. The balance between the two classes of wind would change during the evolution of the 11-year solar cycle depending on changes in the topology of the heliomagnetic equator and its inclination on the ecliptic plane. The fast wind originates from unipolar open field line regions, typical of the coronal holes \\citep{Schatten69, Hassler99}. The slow wind origin is much more uncertain, although we know this class of wind is generated within regions mainly characterized by closed field lines configuration that likely inhibit the escape of the wind \\citep{Wang90, Antonucci2005, Bavassano97, Wu2000, Hick99}. Indeed, the different composition and mass flux of the slow wind and the different degree of elemental fractionation  with respect to  the corresponding photospheric regions, strongly suggest that such slow plasma flow could be initially magnetically trapped and then released \\citep{Geiss95a, Geiss95b}. Interchange reconnection process plays a fundamental role in opening up part of the closed field lines linked to the convective cells at photospheric level \\citep{Fisk99, Fisk2001, Schwadron2002, Schwadron2005, Wu2000, Fisk2020}. This process should preferentially develop close to the border of the CH, influencing as well the neighbour regions, i.e. the coronal hole boundary layer (hereafter CHBL), and the closed loop corona. It has been empirically proven that the expansion factor of the magnetic field line configuration is anti-correlated to the wind speed \\citep{Wang90}. A smaller expansion factor, typical of the central region of the coronal hole, would therefore produce a faster wind while larger expansion factor, typical of the border of the coronal hole, would produce instead a slower wind \\citep{Wang90, Wang2006, D'Amicis2015}. Several other contributions to the slow wind come from more specific processes like, e.g. the release of plasmoids by helmet streamers or the continuous leakage of plasma from the highest regions of the large coronal loops where magnetic pressure is no longer largely dominating the thermal pressure \\citep[among others]{Bavassano97, Hick99, Antonucci2005}. \nDuring the wind expansion, magnetic field lines from coronal holes of opposite polarities are stretched into the heliosphere by fast wind streams being separated by the heliospheric current sheet, an ideal plane, magnetically neutral, permeated by slow wind plasma. During the expansion into the interplanetary space, because of solar rotation, the high speed plasma would overtake and impact on any slower plasma ahead of it creating a compression region \\citep{Hundhausen1972}.\n\\begin{figure}[h!]\n\t\\centering\n\t\\includegraphics[width=7cm]{Fig1_Stream_sketc.png}\n\t\\caption{Sketch of a stream structure in the ecliptic plane, adapted from Hundhausen 1972. The spiral structure is a consequence of solar rotation. The spiral inclination changes as the solar wind velocity changes. When the high-speed stream (in blue) compresses the slower ambient solar wind (in black), a compression region downstream (in red) and a rarefaction region upstream (in pink) are formed.}\n\t\\label{fig01}\n\\end{figure}\nAs a consequence of this interaction, a so called co-rotating interaction region (CIR), characterized by a rather rapid increase in the wind speed, from values typical of the slow wind (300-400 km\/s) to values typical of fast wind (700-800 km\/s), forms at the stream-stream interface. The CIR is characterized by strong compressive phenomena (red zone in Figure \\ref{fig01}) affecting both magnetic field and plasma \\citep{Richardson2018}. The CIR is followed by a region where the wind speed persist at its highest values (blue zone in Figure \\ref{fig01}), resembling a sort of plateau, while the other plasma and magnetic field parameters rapidly decrease to remain rather stable across its extension. Beyond this region, highlighted at times by a a sharp knee in the radial velocity, the wind speed decreases monotonically to reach values typical of the following slow wind (black zone in Figure \\ref{fig01}), which permeates the interplanetary current sheet. This decreasing wind speed region is slightly more rarefied than the high speed region which immediately follows the CIR and is commonly called rarefaction region (pink zone in Figure \\ref{fig01}). The three regions described above can be considered the imprint in the interplanetary medium of the coronal structure from which the wind originated \\citep{Hundhausen1972}. The fast wind plateau corresponds to the core of the CH while CIR and rarefaction region correspond to the same CHBL encircling the CH, though within the CIR the CHBL is compressed by the dynamical interaction described earlier between fast and slow wind while, within the rarefaction region the CHBL is stretched by the wind expansion into the interplanetary medium.\nBeyond a few solar radii, the  solar wind (both fast and slow) becomes supersonic and super-Alfv\\'enic. Fluctuations of interplanetary magnetic field and plasma parameters start to develop their turbulent character since the first tens of solar radii \\citep{Kasper2019} and, in particular, turbulence features within fast and slow wind differ dramatically \\citep[and references therein]{BrunoCarbone2013}. Fast wind is rather uncompressive and mostly characterized by strong Alfv\\'enic correlations between velocity and magnetic field fluctuations, though carrying also compressive fast and slow magnetosonic modes  \\citep{Marsch90,Marsch93,Klein2012, Howes2012, Verscharen2019}. The origin of  Alfv\\'enic and compressive fluctuations is rather different. The former appear to be generated at large scales through the shuffling of the magnetic field lines foot-points (by photospheric motion \\citep{Hollweg2006}) and at small scales by magnetic reconnection processes together with fast\/slow magnetosonic modes \\citep{Parker57, Lazarian2015, Kigure2010, Cranmer2018}. On the other hand, following \\cite{Marsch93} and \\cite{Tu94} compressive modes may also originate during the wind expansion from a superposition of pressure balanced structures of solar origin and magnetoacoustic fluctuations generated by the dynamical interaction of contiguous flux-tubes. Uncompressive and compressive fluctuations eventually experience a non-linear cascade to smaller and smaller scales.\nAlfv\\'{e}nicity in the slow wind is in general much lower with compressive effects affecting more the plasma dynamics and reflecting the complex magnetic and plasma structure of its source regions, close to the heliomagnetic equator\n\nThis paper deals with a peculiar phenomenon that we observed in the Alfv\\'{e}nic turbulence when passing from the fast wind plateau to the rarefaction region. The border between these two regions  is often marked by a rather sharp knee in the velocity profile. As detailed in the following, We noticed that, while downstream the knee, fluctuations have large amplitude and are highly Alfv\\'enic, in the upstream region, fluctuations dramatically reduce their amplitude and the Alfv\u00e9nic correlation weakens, beginning to fluctuate between positive and negative values. Interesting enough, the region where the remarkable depletion of these fluctuations takes place can be very short in time, of the order of tens of minutes, and is located around the velocity knee.\nTo our knowledge, no specific analyses, before the present work, have ever been devoted to understand the reason of such a sudden change of the Alfv\\'enic turbulence across the velocity knee. \\par \nAs we will explain in the following section, after analyzing the turbulent character of the stream, we first looked for the presence of a tangential discontinuity between the fast wind region and the rarefaction region. Then, we made a composition analysis of the minor ions of the solar wind. Finally, we suggested a mechanism that we believe may explain the rapid depletion of fluctuations and Alfv\u00e9nicity observed at the beginning of the rarefaction region.\n\\section{Data analysis}\n\\label{sec:data}\n\nWe analyzed 3 second averages of magnetic field and plasma observed at 1AU from the WIND spacecraft and 2 hour averages minor ions parameters from the ACE spacecraft during the last minimum of solar activity. The data refers to the Heliocentric Earth Ecliptic (HEE) system, which has an X axis pointing along the Sun-Earth line, and a Z axis pointing along the ecliptic north pole \\citep{Thompson2006}. In particular, after a visual inspection of several co-rotating high speed streams towards the end of solar cycle 24, we focused on a series of wind streams in which the phenomenon described above appears, though for sake of brevity here we report results related to the high speed stream observed on 3-9 August 2017, representative of those streams showing a similar fluctuations reduction phenomenon. We also show, for comparison, a case in which the reduction in fluctuations and Alfv\u00e9nicity occurs more slowly, without being able to identify a clear velocity knee in this case.\nSince the magnetic field and plasma data collected by WIND are not provided with synchronized time stamps, we re-sampled the time series with a 6 second cadence in order to build up a single merged data-set with a uniform time base to be used to characterize the statistics of the turbulent fluctuations of the solar wind parameters under study. \nFor the minor ions analysis we were forced to employ the 2 hour cadence being the observations with the highest time resolution available during the selected streams.\n\nTo characterize the turbulence in the solar wind we built the Els\\\"asser variables $ \\bf{z^\\pm} = \\bf{v} \\pm \\bf{b} $ \\citep{Elsasser1950}, where $\\bf{v}$ is the velocity vector and $\\bf{b}$ is the magnetic field vector expressed in Alfv\\'en units \\citep{Tu95, BrunoCarbone2013}. \n\nThe second order moments linked to Els\\\"asser variables are \\citep{BrunoCarbone2013} :\n\\begin{subequations}\n\\begin{align}\n   e^{\\pm} &=\\frac{1}{2} \\langle \\left(\\bf{z^{\\pm}} \\right)^2 \\rangle & \\: (energies\\; related\\; to\\; \\mathbf{z^+} \\;and\\; \\mathbf{z^-})\\\\\n   e_{v}&=\\frac{1}{2} \\langle v^2 \\rangle & (kinetic\\; energy)\\\\\n   e_{b}&=\\frac{1}{2} \\langle b^2 \\rangle & (magnetic\\; energy)\\\\\n   e_c &= \\frac{1}{2} \\langle \\bf{v} \\cdot \\bf{b} \\rangle & (cross-helicity)\n\\end{align}\n\\end{subequations}\nwhere angular brackets indicate averaging process over the established time range. In our case $e^{\\pm}$, $e_{v}$, $e_{b}$ and $e_{c}$ represent the variances calculated on hourly moving window for the whole stream.\nIn order to describe the degree of correlation between v and b, it is convenient to use normalized quantities:\\\\\n\\begin{subequations}\n\\begin{align}\n\\sigma_c &= \\frac{e^+ - e^-}{e^+ + e^-}  & \\;\\; (normalized \\;cross-helicity)  \n\\label{def_sigma_c}   \\\\\n\\sigma_r &= \\frac{e_v - e_b}{e_v + e_b}  & \\;\\; (normalized \\;residual \\; energy) \\label{def_sigma_r}\n\\end{align}\n\\end{subequations}\nWhere $-1 \\leq \\sigma_c \\leq 1 $ and $-1 \\leq \\sigma_r \\leq 1 $.\n\n\\vspace{7mm}\n\\begin{figure}\n\t\\centering\n\t\\includegraphics[width=9cm]{Figura2_A17_Stream.png}\n\t\\caption{Temporal profiles of solar wind parameters from $3^{th}$ to $9^{th}$ August 2017, at 6 seconds time resolution. Panels show from top to bottom: proton velocity components in HEE (Heliocentric Earth Ecliptic) reference system, respectively $v_x$ (panel a), $v_y$ (panel b), $v_z$ (panel c); proton number density (d); proton temperature (e); magnetic field magnitude (f); magnetic field components in HEE, respectively $B_x$ (panel g), $B_y$ (panel h), $B_z$ (panel i); angle between the average magnetic field direction and the radial direction (j). In the latest panel the black dashed curve represent the local Parker spiral direction, referred to the high-speed region and rarefaction region. The red dashed line in each panel denotes the velocity knee between fast wind region and rarefaction region. The green dashed line in each panel denotes the location of the Tangential Discontinuity found at almost the end of the rarefaction region (see section \\ref{sec:data}). On the higher part of the figure (outside the panels) are indicated the different solar wind region, vertically interpreted, with color-correspondence to sketch of Figure \\ref{fig01}.}\n\t\\label{fig02}\n\\end{figure}\nThe multiple panels of Figure \\ref{fig02} show some of the plasma and magnetic field features characterizing the particular stream we chose observed by WIND on August 2017. From top to bottom we show the time profiles of the three velocity components $v_x$, $v_y$ and $v_z$ (panels a to c), proton number density (panel d), proton temperature (panel e), magnetic field intensity (panel f), magnetic field components $B_x$, $B_y$ and $B_z$ (panels g to i) and, at the bottom (panel j), the angle $ \\theta = cos^{-1} \\left(|B_{x}| \/ |B| \\right) $ (limited between $0$ and $90^{\\circ}$) between the local field and the radial direction, respectively. This is a canonical corotating high-speed stream in which it can be easily recognized an interaction region, roughly occurring around noon of August the $4^{th}$ and characterized by strong magnetic field intensity and plasma number density enhancements followed by a clear proton temperature increase, as expected for a compressive region, because of the dynamical interaction of this stream with the slow wind ahead of it \\citep{BrunoCarbone2013}. This interaction region, within which the wind speed rapidly increases from about 400 km\/s to about 700 km\/s, is followed by a speed plateau about two days long. The latter is characterized by fluctuations in velocity and magnetic field components of remarkable amplitude and rather constant values of number density \\begin{math} { n \\sim 3 \\; cm^{-3} } \\end{math} and field intensity \\begin{math}{|B| \\sim 5 \\; nT }\\end{math}. In addition, the temperature profile (panel e), although lower than that within the preceding interaction region (CIR), is much higher than in the surrounding low speed regions. The high speed plateau is then followed by a slowly decreasing wind speed profile identifying the rarefaction region of this corotating stream. These features are typical for most of the high speed corotating streams we examined. What is less common is the abrupt depletion of large amplitude fluctuations detected in the high speed plateau, across the velocity knee (defined later, when describing Figure \\ref{figura5}) indicated in Figure \\ref{fig02} by the vertical dashed red line around day 218 (6 August) at noon. This phenomenon is highlighted by the profile of the curve in the bottom panel j related to the angle $\\theta$ between the magnetic field vector and the radial direction. The difference in the behavior of this parameter before and after the velocity knee is dramatic and, as shown later on, this transition is rather fast, of the order of minutes.\nLarge angular fluctuations between the magnetic field vector and the radial direction are due to the presence of large amplitude, uncompressive  Alfv\\'{e}nic fluctuations populating the high speed plateau that force the tip of the magnetic field vector to randomly move on the surface of a hemisphere centered around the background mean field direction \\citep[e.g. see figure 4 in ][]{Bruno2001}.\n\\begin{figure}\n\t\\centering\n\t\\includegraphics[width=9cm]{Figura3_M17.png}\n\t\\caption{Temporal profiles of solar wind parameters from $28^{th}$ February to $15^{th}$ March 2017, at 6 seconds time resolution. Panels show from top to bottom: proton velocity components in HEE (Heliocentric Earth Ecliptic) reference system, respectively $v_x$ (panel a), $v_y$ (panel b), $v_z$ (panel c); proton number density (d); proton temperature (e); magnetic field magnitude (f); magnetic field components in HEE, respectively $B_x$ (panel g), $B_y$ (panel h), $B_z$ (panel i); angle between the average magnetic field direction and the radial direction (j). In the latest panel the black dashed curve represent the local Parker spiral direction referred to the high-speed region and rarefaction region.}\n\t\\label{fig03}\n\\end{figure}\n\nOn the other hand, not all streams have the same abrupt depletion of fluctuations after the fast wind plateau; an example is the stream of March 2017 shown in Figure \\ref{fig03}. In the multiple panels there are: velocity components $v_x$, $v_y$ and $v_z$ (panels a to c), proton density (panel d), proton temperature (panel e), magnetic field magnitude (panel f), magnetic field components $B_x$, $B_y$ and $B_z$ (panels g to i) and the angle $\\theta$ (panel j) between the local field and the radial direction. We can clearly identify the interaction region characterized by a compression (enhancement of proton density in panel d), followed by an increase in magnetic field, proton temperature and velocity of the solar wind, typical of high speed corotating streams.\nImmediately after, there are wide fluctuations in velocity and magnetic field components in correspondence of the fast wind plateau of this stream, that has a longer duration with respect to the previous stream. In this case, unlike the previous one, even though there is a decrease in velocity and magnetic fluctuations, it occurs gradually over time (days). This gradual decreasing trend is also present in the temperature profile (panel e). Moreover, the angular fluctuations between the local field and the radial direction (panel j) change rapidly over time, without having a clear trend in the rarefaction region, unlike the previous stream. Hence the magnetic field vector fluctuates continuously around the radial direction for almost the entire duration of this second stream.\n\nComparing the time profiles of the angle between the average direction of the magnetic field with respect to the radial direction (panels h of Figures \\ref{fig02} and \\ref{fig03}), we note that in correspondence of the fast wind region of both August (Figure \\ref{fig02} h) and March (Figure \\ref{fig03} h) streams, most of the values are above the black solid curve, which indicates the predicted Parker spiral angle based on the wind speed, computed as $\\Psi = \\arctan(\\Omega R \/ |v|)$ where R is the distance from the Sun to the Earth and $\\Omega$ is the sidereal angular velocity of the Sun at the equator, i.e. $2.97 \\cdot 10^{-6} rad\/s$.\nAs a consequence, before the velocity knee, the background magnetic field is over-wound \\citep{BrunoBavassano1997}, most likely as a consequence of the dynamical interaction of the fast stream with the slow wind ahead \\citep{Schwenn90}. Within the rarefaction region, the time profile of this angle for the August stream shows a smaller variability with an average direction closer to the radial one and smaller than the local Parker spiral direction, as expected for a rarefaction region \\citep{SchwadronMcComas2005}. This region maps in the interplanetary space the coronal hole boundary layer (CHBL) encircling the fast wind source region at the sun \\citep{McComas2003}. The solar wind observed in space and emanating from this source region, because of the solar rotation, is characterized by compressive phenomena when detected ahead of the high velocity stream and rarefaction phenomena when it is observed following the high speed plateau.  The existence of this sub-Parker spiral orientation of the background magnetic field depends on the rate of the motion of open magnetic field foot points across the coronal hole boundary at photospheric level \\citep{SchwadronMcComas2005}. Such motion is responsible for connecting and stretching magnetic field lines from fast and slow wind sources across the CHBL, as explained in the model by \\citep{SchwadronMcComas2005}.\n\nIt is interesting to remark how different are the profiles of $\\theta$ for these two streams within the rarefaction regions. For the stream of August, the $B_x$ component is generally positive and the profile of $\\theta$ is largely confined to values below $45 ^{\\circ} $ with sporadic jumps to larger values.  For the March stream, the $B_x$ profile is much more structured and  $\\theta$ continuously fluctuates between $0^{\\circ}$ and $90^{\\circ}$. Moreover, for this last stream, $B_x$ continuously jumps from positive to negative values suggesting that we must be close to the heliomagnetic equator.\n\\begin{figure}\n\t\\centering\n\t\\includegraphics[width=9.2cm]{Fig4_Backprojection_A17_M17.png}\n\t\\caption{Source Surface Synoptic Charts at  $3.25 R_s$ related to July-August 2017 on the left side and February-March 2017 on the right side (Source: Wilcox Solar Observatory).The level curves indicate values of constant magnetic field (at $3.25 R_s$), the black line represents the heliomagnetic equator (current sheet), the two different shades of gray indicate the magnetic field polarity: dark gray and red lines correspond to negative polarity while light gray and blue lines correspond to positive polarity. The orange dashed line represent the Earth trajectory projected to the Solar surface at $3.25 R_s$ . The different purple lines correspond to the periods of the studied streams. In the bottom two panels are shown the velocity profiles of August 2017 stream (on the left) and March 2017 stream (on the right).}\n\t\\label{figura4}\n\\end{figure}\nAs a matter of fact, in our analysis we have observed many cases like the one (August 2017) described, on the other hand we also observed several other cases where the disappearance of the Alfv\\'enic fluctuations is not characterized by a sudden event like the one observed here on August the $6^{th}$. Individual streams can be different from each other and one cannot simply infer their turbulence properties without proper analysis.\nAs observed by \\cite{Shi2021}, structures such as heliospheric current sheets can play an important role in modifying the properties of turbulence in the solar wind, also supported by 2D MHD simulations of Alfv\u00e9nic waves on a current sheet \\citep{Malara1996}, which showed that initially high values of $\\sigma_c$ are quickly destroyed near a current sheet.\nWe observed that the sudden depletion of the Alfv\\'{e}nic fluctuations beyond the velocity knee might be related to the pitch angle with which an observer, in the ecliptic plane, crosses the heliomagnetic equator. \nIn the upper panel of the left-hand-side of Figure \\ref{figura4} we show the Source Surface Synoptic Chart relative to July-August 2017 taken from Wilcox Solar Observatory. The level curves indicate values of constant magnetic field (at 3:25Rs), the black line represents the heliomagnetic equator, which runs in the middle of the heliospheric current sheet. The two different shades of gray indicate the magnetic field polarity: dark gray and red lines correspond to negative polarity while light gray and blue lines correspond to positive polarity. The orange dashed line represent the Earth trajectory projected to the Solar surface at 3:25Rs. The purple lines indicate the beginning and the end of the high speed stream studied in this work, whose speed profile is shown in the lower panel. In this case (4-8 August 2017), the heliospheric current sheet is highly inclined with respect to the trajectory of the observer, i.e. WIND s\/c. In the right-hand-side of this Figure, we show, in the same format as of the left-hand-side, one of those cases for which we do not see a clear and abrupt depletion of the fluctuations. This is a high speed stream observed on March 2017. In this case, the speed profile shows a less clear velocity knee and a slow and progressive disappearance of the fluctuations without any abrupt event like the one observed in the previous stream. The main difference, in this case, is that the heliospheric current sheet is very flat and the observer experiences a sort of surfing along this structure instead of a clear and rapid crossing. \n\\begin{figure}\n\t\\centering\n\t\\includegraphics[width=9cm]{Figura5_completa_A17.png}\n\t\\caption{Fast Wind region and rarefaction region of the August 2017 stream. From top to bottom: solar wind speed at 6 seconds time resolution (a), values, computed on 1 hour moving window, of kinetic energy in semi-logarithmic scale (b), magnetic energy in semi-logarithmic scale (c), density compressibility in semi-logarithmic scale (d), compressibility of the magnetic field magnitude in semi-logarithmic scale (e), normalized cross-helicity  (f) and normalized residual-energy (g).  The bottom panel (h) shows an enlargement of the total pressure (yellow), magnetic pressure(orange) and thermal pressure (blue) trend in a small region of few hours close to the velocity knee, the latter represented by the red dashed line in each panel. The green dashed line, in panels from (a) to (g), denotes the location of the tangential discontinuity (TD) founded at almost the end of the rarefaction region (see section \\ref{sec:data}).}\n\t\\label{figura5}\n\\end{figure}\n\nAs anticipated, the high speed plateau of the stream under study is characterized by large amplitude Alfv\\'{e}nic fluctuations as shown in Figure \\ref{figura5}. The top panel (a) shows part of the speed profile of our corotating high speed stream. The location of the velocity knee, beyond which large amplitude fluctuations seems to turn off, is indicated by the vertical dashed red line and determined as follows. Panels b and c show hourly values of kinetic and magnetic energy, respectively, on a semi-logarithmic vertical scale.\nThe sharp variation in these two parameters, at the end of the speed plateau, identifies what we have defined as velocity knee. They highlight remarkably the different physical situation before and after the velocity knee and the abrupt change across it. We notice that\nkinetic energy is the parameter that experiences the largest decrease, of a factor $\\sim 20$. As a consequence, fluctuations within this region become strongly magnetically dominated. In addition, this decrease happens in correspondence of a strong temperature reduction as shown in the previous Figure \\ref{fig02} by the thermal energy parameter. This strong depletion of proton temperature reflects the thermal pressure decrease (Figure \\ref{figura5} h) that starts at the velocity knee together with a corresponding increase of the magnetic pressure such to maintain the wind in a pressure balanced status, achieved at the edge of the Alfv\\'{e}nic surface during the wind initial expansion \\citep{SchwadronMcComas2005}. Figure \\ref{figura5} (d) shows the density compressibility over time in a semi-logarithmic vertical scale, computed as ${\\sigma_{n}}^2 \/ n^{2}$; its value remains quite low (on average $3 \\cdot 10^{-3}$) both during the fast wind region and the rarefaction region, indicative of a fairly incompressible wind, given that the density remains approximately constant in both regions.\nOn the other hand, the compressibility of magnetic field magnitude, computed as ${\\sigma_{B}}^2 \/ |B|^{2}$ and shown in panel \\ref{figura5} (e) on a semi-logarithmic vertical scale, denotes that magnetic fluctuations, in modulus, are more compressive in correspondence of the fast wind plateau rather than in the rarefaction region; there is a sudden reduction at the velocity knee of about an order of magnitude. The two other panels, f and g, highlight the Alfv\\'{e}nic nature of these fluctuations within the high speed plateau. Panel f shows high values of the normalized cross helicity $\\sigma_C$ while panel g shows that the normalized residual energy, although dominated by magnetic energy, is not far from the equipartition expected for Alfv\\'enic fluctuations. Beyond the velocity knee, these two parameters experience a considerable change towards a much lower Alfv\\'enicity although the transition is not as abrupt as the one relative to the amplitude of these fluctuations, as discussed above. \\\\\nThe behaviour of $\\sigma_C$ is similar to that recently observed in the SW from Parker Solar Probe near perihelions of the orbits $(R \\le 0.3 - 0.4 \\;AU) $, as described by \\cite{Shi2021}, where $\\sigma_c$ is usually close to $1$, implying a status of dominating outward-propagating Alfv\u00e9n waves. There are also periods when $\\sigma_C$ oscillates and becomes negative; these periods correspond to Parker Solar Probe observing heliospheric large-scale inhomogeneous structures, such as the heliospheric current sheet.\\\\\n\\begin{figure}\n\t\\centering\n\t\\includegraphics[width=9.4cm]{Figura6_M17.png}\n\t\\caption{Fast Wind region and rarefaction region of the March 2017 stream. From top to bottom: solar wind speed at 6 seconds time resolution (a), values computed on 1 hour moving window, of kinetic energy in semi-logarithmic scale (b), magnetic energy in semi-logarithmic scale (c), density compressibility in semi-logarithmic scale (d), compressibility of the magnetic field magnitude in semi-logarithmic scale (e), normalized cross-helicity (f) and normalized residual-energy (g).}\n\t\\label{figura6}\n\\end{figure}\nThe same analysis of Figure \\ref{figura5} for the August 2017 stream was also carried out for the March 2017 stream and shown in Figure \\ref{figura6}. In this case the decrease in fluctuations and Alfv\u00e9nicity occurs very gradually over time (few days). The velocity profile (panel a) of the March 2017 stream does not allow us to define a velocity knee in this case. Furthermore, the magnetic energy (panel b) and kinetic energy (panel c) decrease more slowly in the rarefaction region of the stream, with respect to the previous one. The Alfv\u00e9nic character of the fast wind remains evident, as shown by the profile of $\\sigma_c$ in panel e and $\\sigma_r$ in panel f. Even in this case the density compressibility (panel d) oscillates around very low values, of the order of $10^{-3}$, while in this stream the compressibility of magnetic field magnitude (panel e) does not show an effective decrease after the velocity plateau, unlike the previous case (August 2017 stream).\\\\\nThe MHD simulations of an ensemble of Alfv\u00e9nic fluctuations propagating in an expanding solar wind including the presence of fast and slow solar wind streams, made by \\cite{Shi2020}, showed that the decrease in $\\sigma_c$ is more significant in the compression and rarefaction regions of the stream (as we observed in Figure \\ref{figura5}) than within the fast and slow streams. In particular, in the rarefaction region of such simulations $\\sigma_c$ rapidly decreases from 1 to 0.6 at $ R \\approx 80 R_s $ and remains around this value until the end of the simulation. While leaving open the hypothesis that the phenomenon under study may develop during the radial expansion of the solar wind, the very rapid (few minutes) reduction of the fluctuations and Alfv\u00e9nicity between the fast wind plateau and the rarefaction region makes us lean towards the hypothesis that this phenomenon should originate at the source regions of the solar wind. \\\\\nThe sudden changes in some of the parameters described above for the August stream, between the end of the fast wind plateau and the beginning of the rarefaction region, suggested to look for magnetic structures like tangential discontinuities (TD, hereafter), which might represent a barrier for the propagation of the Alfv\\'enic fluctuations. The presence of a TD, where by definition there is no magnetic component normal to the discontinuity surface, could explain the depletion of Alfv\u00e9n waves, that propagate along the magnetic field direction \\citep{Hundhausen1972}. In fact, these fluctuations would not propagate across this kind of discontinuity since the magnetic component perpendicular to the discontinuity plane vanishes. This would prevent their propagation into the CHBL or rarefaction region and would produce the observed sudden depletion of the Alfv\\'enic fluctuations. \n\n\\begin{figure}\n\t\\centering\n\t\\includegraphics[width=9cm]{Figura7.png}\n\t\\caption{Top three panels: time profile of velocity components (blue lines) and magnetic field components in Alfv\u00e9n units (black lines) at 6 seconds (panels a, b, c), in about an 8 hr time interval.  Bottom panel (d): 5 minute values of magnetic $e_b$ (black line) and kinetic $e_v$ (blue line) energies within a narrower time interval of almost 5 hours around the estimated velocity knee (red dashed line).}\n\t\\label{figura7}\n\\end{figure}\n\nTo better determine the starting time of the depletion of magnetic and kinetic fluctuations, we show in Figure \\ref{figura7} the time profile of 6 sec averages of magnetic field components, expressed in Alfv\\'en units, and velocity components within a short time interval, slightly longer than 8 hrs and roughly centered around the previously estimated time location of the velocity knee. There is a clear change in the amplitude of the fluctuations around the location of the dashed line. This phenomenon is highlighted by the time profile of magnetic and kinetic energies, $e_b$ and $e_v$, respectively, computed within 5 min running window, shown in the bottom panel (panel d). The sharp profile of these curves allows to better estimate the time when fluctuations are abrupt reduced, that is on day  $218^d13^h24^m$ (i.e. 6 August).\nHowever, we did not find any relevant TD around the estimated time, which could be associated with the observed phenomenon. Thus TD can not be a valid explanation of the rapid decrease of the Alfv\u00e9nic fluctuations at the velocity knee, leaving open the possibility that this phenomenon could be due to a different evolution of the solar wind during its expansion or to a possible reconnection mechanism at the base of the Solar Corona, as we will explain in details in the following..\nOn the other hand we point out the presence of a TD two days later the velocity knee, on August the $8^{th}$ at $14^h34^m$, easily recognizable in Figure \\ref{fig02} by looking at the switchback in the $B_x$ component and highlighted in Figure \\ref{figura8}. \n\n \\begin{figure}\n\t\\centering\n\t\\includegraphics[width=9cm]{Figura8_minvar.png}\n\t\\caption{Left-hand-side: panels from top to bottom show the three components of magnetic field (panels (a), (b), (c))and velocity (panels (d), (e), (f)), rotated in the magnetic field minimum variance reference system, and the total pressure (panel g). The 3D figures in the right-hand-side panels show magnetic field (panel h) and velocity components (panel i) rotated in the field minimum variance reference system. All these quantities are plotted with data at 6 sec time resolution. The green dashed line, in panels from (a) to (g), denotes the location of the tangential discontinuity (TD) founded at almost the end of the rarefaction region (see section \\ref{sec:data}).}\n\t\\label{figura8}\n\\end{figure}\nIn the latter, panels on the left hand side show (from top to bottom): magnetic field (panels a , b and c) and velocity components (panels d, e and f), and total pressure (panel g).  Magnetic field and velocity components are rotated in the magnetic field minimum variance reference system, as better shown in the 3D plots in the right hand side panels. A TD discontinuity is an MHD structure for which the total pressure is conserved, as shown in Figure \\ref{figura8} panel (g), the component of the magnetic field normal to the discontinuity plane is identically zero on both sides (Figure \\ref{figura8} panel c) and number density, thermal pressure and tangential component of the magnetic field vector can be discontinuous across the layer. In this particular TD, the magnetic field rotates in less than 1 minute in the minimum variance plane shown on the right-hand-side panel and characterized by robust eigenvalues ratios: $\\Lambda_1 \/ \\Lambda_3 = 0.015 $ and  $\\Lambda_2 \/ \\Lambda_3 = 0.10 $.\n\n\n\nTotal pressure remains roughly constant across the discontinuity and there is no relevant mass flow across the latter, because the velocity component normal to the discontinuity plane is quite constant throughout the whole time interval of Figure \\ref{figura8}, indicating that it is at rest in the plasma reference frame. Looking back at Figure \\ref{fig02}, magnetic field and plasma parameters rapidly become more variable upstream the TD, approaching the slow wind.  This TD seems to mark the physical border between the  rarefaction region and the surrounding slow wind. In other words, it could be the coronal discontinuity (CD) observed by \\cite{McComas98} in Ulysses data during March 1995, when Ulysses was rapidly crossing the low-latitudinal region of the solar wind towards the fast polar wind flow and modeled by \\cite{SchwadronMcComas2005}. The relevant difference would be that this is the first time this particular structure is recognized between the rarefaction region of an high speed stream and the following slow solar wind.\n\nMinor ions components of the solar wind carry relevant information about its source region, i.e. the CHBL. This is the source of a sort of transition region between the fast coronal wind and the slow wind. As a matter of fact, minor ions parameters like freezing-in temperature and relevance of low-FIP elements \\citep{Geiss95a, Geiss95b, McComas2003} monotonically increase from fast to slow wind.\nAt this point we searched for possible signatures in plasma composition and charge state assuming the rapid depletion of fluctuations and Alfv\u00e9nicity might have been due to differences in the source regions (at the basis of the solar Corona) of the stream high speed plateau and the subsequent rarefaction region. As a matter of fact, fast and slow wind largely differ for ion composition, charge state and freezing-in temperature (\\cite{Bochsler IoniMinori}) and the region in between, i.e. the rarefaction region, might be a sort of wind largely mixed with the slower wind, and this would be highlighted by the composition analysis. \\cite{Geiss95a} and \\cite{Phillips95}, based on Ulysses observations, found several dramatic differences in ions composition between fast and slow wind. One of the most relevant is the strong bias of the slow wind in favor of low FIP (first ionization potential) elements with respect to the fast wind. \\cite{Geiss95b}, using charge state ratios for C6+\/C5+ and O7+\/O6+ estimated the corresponding freezing-in temperatures for these elements and found that slow wind sources at the sun are quite hotter than fast wind sources. Finally, fast and slow wind greatly differ also for the higher abundance of He++ within fast and hot wind with respect to slow and cold wind \\citep{Kasper2012}. \n\\begin{figure}\n\t\\centering\n\t\\includegraphics[width=9cm]{Figura9_ioni_A17.png}\n\t\\caption{Temporal profiles of ions compositions during the August 2017 stream. In the top panel(a): Alpha particles over protons ratio $ {He^{++}}\/{p+}$ at 6 sec time resolution (in blue) and its hourly moving average (in red). Second panel b: charge state ratio, at 2 hours time resolution, of ${C^{+6}}\/{C^{+5}}$ in blue, ${O^{+7}}\/{O^{+6}}$ in red and ${O^{+8}}\/{O^{+6}}$ in yellow. Third panel c: iron average charge state, at 2 hours time resolution. Bottom panel d: iron over Oxygen abundance ratio, at 2 hours time resolution. The red dashed line denotes the velocity knee between fast wind region and rarefaction region.The green dashed line denotes the location of the tangential discontinuity (TD) founded at almost the end of the rarefaction region (see section \\ref{sec:data}). }\n\t\\label{figura9}\n\\end{figure}\nThe four panels of Figure \\ref{figura9} show, from top to bottom, the relative abundance of He++\/p+ (panel a), some charge state ratios for carbon and oxygen (panel b), charge state for iron (panel c) and relative abundance of iron over oxygen (panel d) respectively. This figure (\\ref{figura9}) refers to the entire stream, i.e. the compressive region, followed by the high-speed plateau, followed by the rarefaction region and finally by the slow wind (for comparison keep the Figure \\ref{fig02} in mind). As expected, the last three panels of Figure \\ref{figura9} show a gradual increase from the fast wind plateau to the rarefaction region, but without a sharp jump across the velocity knee. \n\\begin{figure}\n\t\\centering\n\t\\includegraphics[width=9cm]{Figura10_ioni_M17.png}\n\t\\caption{Temporal profiles of ions compositions during the March 2017 stream. In the top panel(a): Alpha particles over protons ratio $ {He^{++}}\/{p+}$ at 64 sec time resolution (in blue) and its hourly moving average (in red). Second panel(b): charge state ratio, at 2 hours time resolution, of ${C^{+6}}\/{C^{+5}}$ in blue, ${O^{+7}}\/{O^{+6}}$ in red and ${O^{+8}}\/{O^{+6}}$ in yellow. Third panel(c): iron average charge state, at 2 hours time resolution. Bottom panel(d): iron over Oxygen abundance ratio, at 2 hours time resolution.}\n\t\\label{figura10}\n\\end{figure}\nA very similar result can be seen in Figure \\ref{figura10} that is the equivalent result of minor ions analysis of Figure \\ref{figura9}, but referred to entire March 2017 stream. Looking at the minor ions analysis (both Figure \\ref{figura9} and \\ref{figura10}) we can clearly see that there are no jumps in the quantities in correspondence of the end of the fast wind plateau, nor are there large differences between the two streams of August and March.\nAs a matter of fact, although the time resolution of minor ions parameters provided by ACE related to the selected time period is not very high to allow to estimate small scale structures, but it's enough to show that there are not strong discontinuities in ions compositions at the location of the velocity knee such to justify, in terms of differences in the source region, abrupt changes in the fluctuations. The top panel shows the He++\/p+ ratio and, although higher values for this parameter are found within the high speed plateau, the transition across the velocity knee is quite smooth. Thus, the results from this composition analysis point towards a smooth transition from fast to slow wind and did not unravel dramatic changes to be associated with the observed fluctuations abrupt depletion. Therefore, these similarities strongly suggest that the reason for the sudden depletion of turbulence observed within the stream of August 2017 and not observed within the stream of March 2017 can not be hidden in differences in the source region at photospheric level.\nMoreover, the charge state ratio profile of $C_{+6}\/C_{+5}$ (see the blue curve on panel b of both Figures \\ref{figura9} and \\ref{figura10}) is in agreement with the results shown by \\cite{Schwadron2005}. They modeled a solar minimum configuration that gives rise to a co-rotating region considering the differential motion of magnetic field foot points at the Sun across the CHBL. In their model, they started the simulation at $30 R_s$ ($ \\sim 0.14$  AU ) with the presence of a discontinuity at the stream interface (that comes before the fast wind plateau) and another discontinuity at the CHBL. They showed that the discontinuity at the stream interface remains whereas the discontinuity at the CHBL, in the rarefaction region, is eroded as the distance from the Sun increases (they have extended the simulation up to 5 AU). This result is in agreement with what we observe in the carbon charge ratio (panel b of Figure \\ref{figura9} and Figure \\ref{figura10}): we can notice a rapid decrease at the stream interface, corresponding to CH discontinuity before the fast wind plateau and, more important, we also observe a gradual increase of $C_{+6}\/C_{+5}$  in the rarefaction region, corresponding to the CHBL described in model of \\cite{Schwadron2005}.\n\\begin{figure}\n\t\\centering\n\t\\includegraphics[width=9.2cm]{Figura11_Riconnessione_AB.png}\n\t\\caption{Proposed mechanism of interchange reconnection invoked to explain the abrupt depletion of Alfv\u00e9nic fluctuations often observed in co-rotating high velocity streams. More details in the text.}\n\t\\label{figura11}\n\\end{figure}\n\nAt this point we invoke a simple mechanism, as shown in Figure \\ref{figura11}, which might solve this puzzle, taking into account that there is no TD there, nor is there an abrupt change in the composition analysis of the SW. The left-hand panel (Figure \\ref{figura11} A) sketches open magnetic field lines rooted inside a coronal hole, indicated by CH, and coronal loops rooted inside a coronal hole boundary layer, indicated by CHBL, encircling the CH. Field lines inside the CH are populated by Alfv\\'enic fluctuations while coronal loops are not. \nAlthough it is expected that Alfv\u00e9nic fluctuations largely populate the photospheric magnetic carpet \\citep{Title1998} transverse Alfv\u00e9nic fluctuations would experience  a different fate in open or closed magnetic structures since in the latter ones the presence of counter-propagating Alfv\u00e9nic fluctuations would ignite turbulence process with consequent transfer of energy to smaller and smaller scales to eventually dissipate and heat the plasma. Thus, we would expect smaller amplitude and weaker Alfv\u00e9nic character in those fluctuations coming from originally closed field regions and larger amplitude and stronger Alfv\u00e9nic character for those coming from open-field-line regions.\nInterchange reconnection between open and closed field lines plays a major role in the regions bordering the coronal hole \\citep{Fisk99, Fisk2001, Fisk2005, Fisk2020,Schwadron2005}. This mechanism governs the diffusion of open magnetic flux outside the CH. The motion of foot-points of magnetic field lines due to photospheric dynamics is at the basis of interchange reconnection, which tends to mix open field line regions with nearby closed loop regions. Within this framework, an open field line region, populated by propagating Alfv\\'enic fluctuations, might reconnect with a closed loop region and cause a sudden disappearance of these fluctuations as shown in the right-hand panel (Figure \\ref{figura11} B). This phenomenon might happen at the base of the corona during the initial phase of the wind expansion \\citep{Fisk2005}. \n\n\n\\section{Summary}\n\nCo-rotating high speed streams are generally characterized by a high speed plateau followed by a slow velocity decrease, commonly identified as the stream rarefaction region. These two portions of the stream, in some cases, are separated by a clear velocity knee. In these cases, the knee marks also the point beyond which Alfv\\'enic fluctuations, which generally populate the high speed plateau, are dramatically and rapidly depleted.  Interesting enough, this thin border region is not characterized by any relevant TD that could be responsible for such an abrupt depletion of the fluctuations. On the other hand, we could not find either clue in the composition analysis which might have indicated a clear different origin in the source region between the wind observed in the high speed plateau and that observed in the rarefaction region. Thus, the origin of the observed rapid depletion of the fluctuations does not seem to be related neither to differences at the source region nor to a TD separating the high speed plateau from the following rarefaction region. \nNot excluding the hypothesis that this reduction may occur during the expansion of the solar wind, the rapid time scales over which this abrupt decrease in fluctuations  (few minutes) and Alfv\u00e9nicity (less than one hour) is observed make us lean towards the hypothesis that a magnetic reconnection event during the initial phase of the wind expansion, within the lowest layers of the solar atmosphere, might be the mechanism responsible for what we observed and described in this paper. Phenomena of interchange magnetic reconnection favored by the motion of the foot-points of magnetic field lines due to photospheric dynamics would mix together field lines originating from open field line regions, robustly populated by large amplitude Alfv\\'enic fluctuations, with magnetic loops, characteristic of surrounding closed field line regions, populated by smaller amplitude and less Alfv\\'enic fluctuations. \nThis suggested mechanism, which we cannot prove but propose as an hypothesis at the present, will be tested in the near future as soon as the nominal phase of the Solar Orbiter mission will start and it will become possible to directly link in-situ and remote observations.  This sort of behavior could be related to the topology of the heliospheric current sheet at the moment of the crossing by the observer. Highly inclined current sheet crossings seem to result in an abrupt depletion of the fluctuations whereas during flat current sheet crossings we observe neither a clear velocity knee nor an abrupt reduction of the fluctuations.\n\n\\begin{acknowledgements}\nThis work was partially supported by the Italian Space Agency (ASI) under contract ACCORDO ATTUATIVO n. 2018-30-HH.O. Results from the data analysis presented in this paper are directly available from the authors. WIND data have been accessed through the NASA SPDF-Coordinated Data Analysis Web. G.C. acknowledges support from the Laboratoire de M\u00e9canique des Fluides et d'Acoustique at the  \u00c9cole centrale de Lyon, France. G.C. acknowledges SWICo's recognition through the award \"Premio Mariani\" 2021.\n\\end{acknowledgements}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nModification and external control of the electronic structure of two-dimensional materials via the proximity effect is of great experimental and technological interest for designing systems with novel magnetic and spin properties~\\cite{Han2014,Sierra2021,RevModPhys.92.021003}. Of particular importance is the introduction of \nspin interactions in graphene, whose Dirac electrons exhibit only weak spin-orbit coupling. Placing graphene in proximity to transition metal dichalcogenides (TMDCs) is a viable route, resulting in spin-orbit couplings on the meV scale~\\cite{Gmitra2015,Wang2015, Gmitra2016, Gani2020},\nalso tunable by twisting \\cite{Li2019,David2019,naimer2021twistangle,pezo2021manipulation}. While the proximity spin-orbit coupling is in general sublattice dependent \\cite{Frank2016, Kochan2017}, \nTMDC substrates induce valley-Zeeman and Rashba couplings, providing uniform pseudospin spin-orbit fields, opposite at $K$ and $K'$~\\cite{Gmitra2015, Wang2015,PhysRevB.94.241106}. \nSimilar proximity spin-orbit physics and the appearance of the valley-Zeeman coupling has been predicted for graphene on Bi$_2$Se$_3$-family of topological insulators \\cite{Song2018,PhysRevB.100.165141}, but also for \nbilayer Jacutingaite\\cite{Rademaker2021}. There already exists a significant body of experimental evidence demonstrating the presence of valley-Zeeman coupling in proximitized graphene~\\cite{Garcia2018,Island2019,Karpiak2019,Tang2020,Hoque2021,Ghiasi2017,Benitez2018,Ghiasi2019,Safeer2019,Herling2020,Bentez2020,Avsar2014,Banszerus2020,Khokhriakov2020, Wakamura2019, Wakamura2020,amann2021gatetunable,2021boosting,InglaAyns2021}.\n\nIn proximitized graphene with valley-Zeeman coupling, two groups of edge states---pseudohelical (intervelley) and intravalley states---form within the spin-orbit gap. At each edge there are those two pairs of states, conforming to the trivial topology of the system \\cite{Frank2018}. However, for ribbons on the nanoscale, with the widths less than a micron, the intravalley states are gapped out by the confinement-induced hybridization and only the lone pseudohelical pair remains at each edge. This pair is fully protected against backscattering by time-reversal defects, similarly to helical states of the spin quantum Hall effect \\cite{Kane2005}. But, unlike helical states, the pseudohelical states in a flake change the helicity from one sigzag edge to the other, flipping the spin along the armchair edge at which perfect tunneling of the states occurs. In large flakes, where also intravalley states propagate, the backscattering protection by time-reversal symmetry is lifted, as the pseudohelical states reflect at armchair edges to intravalley states at the same edge. \n\nIs there a way to reinstate the lone pseudopotential pair also in larger ribbons and flakes, where there are nominally intravelley states as well. We show in this paper that placing proximitized graphene in a perpendicular magnetic field achieves exactly that. Magnetic effects are typicall twofold: Zeeman-like, providing spin imbalance, and orbital effects, leading to Landau quantization. Zeeman effects in proximitized graphene were investigated earlier, demonstrating the appearance of the quantum anomalous Hall effect \\cite{Hogl2020} and chiral Majorana modes \\cite{Hogl2020a}. Magnetic orbital response of helical edge states to magnetic fields was studied in previous works \\cite{Luo2020,Frank2020,Yang2016,Li2018,DeMartino2011,Lado2015,Delplace2010,Gani2020}. Rather surprisingly, the quantum spin Hall edge states, which are generated by uniform intrinsic (Kane-Mele) spin-orbit coupling, are not necessarily destroyed by the cyclotron effect~\\cite{Shevtsov2012, Goldman2012}, which can theoretically be used to switch between the quantum spin Hall and quantum Hall regimes by gating. However, there can be a crossover between topological and trivial regimes in the presence of a perpendicular magnetic field  ~\\cite{Scharf2015,Bottcher2019}. \n\nIn this paper we study theoretically the response of the pseudohelical and intravalley edge states in proximitized graphene (using realistic paramaters for a graphene\/WSe2 heterostucture) to an external perpendicular magnetic field, as depicted in Fig. 1.(a). We employ a tight-binding model supplemented with Peierl's substitution to study the electronic structure of graphene zigzag nanoribbons and finite flakes. The Landau levels calculated by the tight-binding approach are in excellent agreement with the bulk Landau level predictions \\cite{Frank2020}. The pairs of pseudohelical  edge states, see Fig.\\ref{fig:schema}(b), are preserved even if time-reversal symmetry is broken by the magnetic field. \nThis is \nsimilar to what happens in the quantum spin Hall effect, where helical edge states are preserved in an applied field~\\cite{Shevtsov2012}.\n\nHowever, the  intravalley states (originating from Rashba SOC~\\cite{Frank2018}) disappear once the magnetic field increases beyond some critical value $B_c$. At $B > B_c$, intravalley edge states merge with the conduction and valence bands, opening an intravalley gap. \nInside the gap one finds a lone pair of pseudohelical states, at each zigzag edge. Effectively, the magnetic field gaps out the weakly localized intravalley states, mimicking finite-size confinement. \\cite{Frank2018}. \n\nThe paper is organized as follows. In Sec. II. we introduce a tight-binding model Hamiltonian for graphene with proximity induced spin-orbit coupling, and present a scaling method to obtain the energy spectrum for microscopic structures. In Sec. III. we analyze the electronic structures of zigzag ribbons and finite flakes states in the presence of an external perpendicular magnetic field. We also discuss the crossover at which the intravalley edges states are gapped, referring to bulk Landau level results. Finally, in Sec. IV., we briefly summarize the main results.\n\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=.5\\textwidth]{1.pdf}\n\\end{center}\n\t\\caption{(a) Sketch of a graphene ribbon or flake on a substrate such as a TMDC which yields valley-Zeeman spin-orbit and Rashba couplings. A perpendicular magnetic field is applied to affect the orbital states of the Dirac electrons. (b) Two types of edge states near the Fermi level in graphene with valley-Zeeman spin-orbit interaction: (1-4) spin-polarized pseudohelical (intervalley) states, and (5) intravalley states. (c) Lone pairs of pseudohelical states above the crossover magnetic field.}\n\\label{fig:schema}\n\\end{figure}\n\n\\section{Model and Methods}\n\nWe consider Dirac electrons in proximitized graphene, with sizeable (on the meV scale) spin-orbit interactions of the valley-Zeeman and Rashba types. Such interactions are induced by the proximity effects with TMDCs or TIs, as discussed above. To investigate finite systems, zigzag ribbons and flakes, we implement the following tight-binding  Hamiltonian, \\cite{Gmitra2016,Kochan2017,Gmitra2013}\n\\begin{align}\n    \\hat{\\mathcal{H}}&=\\sum_{\\langle i,j \\rangle}t c_{is}^\\dagger  c_{js}+\\sum_{i}\\Delta\\xi_i c_{is}^\\dagger  c_{is} \\nonumber \\\\\n    &+\\frac{2i}{3}\\sum_{\\langle i,j \\rangle}\\lambda_R c_{is}^\\dagger  c_{js^\\prime}\\left[\\left(\\hat{\\mathbf{s}}\\times \\mathbf{d}_{ij}\\right)_{z}\\right]_{ss^\\prime} \\nonumber \\\\\n    &+\\frac{i}{3}\\sum_{\\langle\\langle i,j \\rangle\\rangle}\\frac{\\lambda_{I}^{i}}{\\sqrt{3}}c_{is}^\\dagger c_{js} \\left[\\nu_{ij}\\hat{\\mathbf{s}}_z\\right]~,  \\label{eq:tb-ham}\n\\end{align}\nwhere $c_{is}^\\dagger$ and $c_{is}$ are the creation and annihilation operators for site $i$ and spin $s$, $\\langle i,j \\rangle$ denotes the nearest, $\\langle\\langle i,j \\rangle\\rangle$ the next nearest neighbors.\nThe Hamiltonian of Eq.~\\eqref{eq:tb-ham} has four terms: First, the nearest-neighbour hopping with amplitude $t$ occurs between sites $i$ and $j$ with spin preservation. Second, the proximity effects induces the staggered potential $\\Delta$ with signs $\\xi_i$ equal to $+1$ or $-1$ for A and B sublattices, respectively. The third term describes\nRashba spin-orbit coupling \\cite{Rashba2009,Tsaran2014} with amplitude $\\lambda_{R}$, which breaks horizontal reflection symmetry and mixes states of opposite spins and sublattices. Symbols $\\mathbf{d}_{ij}$ and $\\hat{\\mathbf{s}}$ denote the unit vector from site $j$ to site $i$ and the vector of spin Pauli matrices, respectively. Finally, the fourth term models the  valley-Zeeman spin-orbit coupling \\cite{Gmitra2015, Wang2015, Yang2016}. This term preserves the spin, but the intra-sublattice hopping is different for  clockwise ($\\nu_{ij} = -1$) and counterclockwise ($\\nu_{ij} = +1$) paths along a hexagonal ring from site $j$ to $i$. The intrinsic spin-orbit coupling $\\lambda_{I}^{i}$ is written here in a more general way, allowing for different strengths at $A$ and $B$ sublattices. The orbital effects of a perpendicular magnetic field are modelled by Peierl's phase \\cite{Peierls1933,PhysRev.115.1460}. We do not consider\nthe Zeeman effects of the field, as they are negligible for the fields we consider, on the millitesla scale. \n\nIn the following, when we present numerical results, we use parameters from first-principles\ncalculations for graphene\/WSe$_2$, see Ref. \\cite{Gmitra2016}: the nearest-neighbor hopping $t=-2.507$~eV, staggered potential $\\Delta=0.56$~meV, Rashba SOC parameter $\\lambda_R=0.54$~meV, and intrinsic SOC parameters $\\lambda_I^A=1.22$~meV and $\\lambda_I^B=-1.16$~meV. \n\nWe wish to investigate micron-size systems, in which magnetic orbital effects are not quenched, and in which the intravalley edge states are not gapped out. Since the spin-orbit parameters are on the meV scales, the considered systems need to be large enough to resolve such energies in the subband structure. To reduce computational efforts and avoid dealing with intractably large structures, we employ a scaling trick ~\\cite{Liu2015}, which allows us to consider smaller structures but with rescaled parameters. \nFinite-size level spacings in graphene ribbons are on the order of $\\Delta E\\approx\\pi\\hbar v_F \/ w$~\\cite{Lin2008}, which can be reduced by changing the Fermi velocity $v_F\\rightarrow v_F\/r= \\sqrt{3}a t \/2\\hbar r$, by rescaling the nearest-neighbor hopping $t$. Such a rescaling affects the interpretation of the lattice constant $a$, which is mapped to $ra$, in order to keep the energy spectrum invariant. The cyclotron energy in graphene is given by $\\hbar\\omega_c = \\sqrt{2e\\hbar B} v_F$. To preserve this energy scale, we rescale the external perpendicular magnetic field by $B\\rightarrow Br^2$. The rescaling does not influence the underlying physics as long as we consider low-energy states for which the linear dispersion holds\\cite{Liu2015}. \n\nThe scaling trick permits us to keep the cyclotron energy scale at the level of SOC parameters, while decreasing the computational burden to simulate large-scale systems. The main criterion for choosing the rescaling parameter is $r\\ll3t\\pi\/E_{\\text{max}}$, where $E_{\\text{max}}$ is the maximal energy of interest~\\cite{Liu2015}. We wish to resolve energies up to $E_{\\text{max}}=2$ meV, which safely covers the spin-orbit gap region in which the edge states form, the scaling parameter  $r\\ll3t\\pi\/E_{\\text{max}}$=11809. In our work we we choose $r=400$. With that, the finite-size effects are decreased to $0.05$~meV for a width of 400 unit cells. The magnetic field strength equivalent to this energy scale is about $10^{-5}$~T, well below the value for the crossover magnetic field (see the next chapter) for gap closing.\n\nTo calculate the band structure of zigzag nanoribbons and the electronic states of a graphene flake with proximity induced spin-orbit interaction, we implement the above model using the Python-based numerical package KWANT~\\cite{Groth2014}.\n\n\\section{Results}\n\n\\subsection{Bulk results}\n\nThe starting point for analyzing the behavior of edge states of proximitized graphene in the presence of an external perpendicular magnetic field is the study of Landau levels in the bulk system. A detailed derivation of the Hamiltonian of the Landau levels for bulk proximitized graphene is presented in Ref.~\\cite{Frank2020}. Graphene with spin-orbit coupling in magnetic fields was studied from different perspectives by other groups as well~\\cite{Cysne2018, DeMartino2011}.\n\nWe construct the Landau fan diagram, shown in Fig.~\\ref{fig:bulk_landau}. Comparing to the Kane-Mele model \\cite{Kane2005,Shevtsov2012} for graphene, a staggered intrinsic SOC does not preserve the gap in the presence of an external magnetic field. The bulk band gap mainly forms between nonzero Landau levels from the $K$ and $K^{\\prime}$ valley, which can be shifted by the magnetic field. On the other hand, in the Kane-Mele model, a bulk band gap forms between the zero Landau levels, which is stable under an external magnetic field. In the systems with valley-Zeeman spin-orbit coupling, the bulk band gap closes and reopens at a crossover value of the external magnetic field equal to \\cite{Frank2020}\n\\begin{align}\n    B_{c}=\n    \\frac{\n    \\left[\\lambda_I^A-\\lambda_I^B-2\\Delta\\right]\n    \\left[\\left(\\Delta+\\lambda_I^A\\right)\\left(\\Delta-\\lambda_I^B\\right)+4\\lambda_R^2\\right]\n    }\n    {\n    2e\\hbar v_F^2\\left[2\\Delta+\\lambda_I^A-\\lambda_I^B\\right]\n    }.\n\\end{align}\nFor the given parameters set from Ref.~\\cite{Gmitra2015,Frank2018} of our Hamiltonian\\eqref{eq:tb-ham}, the crossover value of the magnetic field will be equal to 1.942~mT for the graphene\/WSe$_2$ heterostructure. This crossover point distinguishes two regimes, below the crossover field, a bulk band gap formed by the different nonzero Landau levels, while above the crossover field bounded by the first Landau levels.\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=.5\\textwidth]{2.png}\n\\end{center}\n\\label{fig:bulk_landau}\n\\caption{Calculated evolution of the bulk Landau levels in graphene\/WSe$_2$ with increasing of the external magnetic field. The crossover magnetic field is indicated as a dashed line. The color code corresponds to the $s_z$ expectation value in the left column and $\\kappa_z$ (valley) expectation value in the right column, see Ref.~\\onlinecite{Frank2020}.}\n\\end{figure}\n\n\n\\subsection{Zigzag ribbon results}\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=.5\\textwidth]{3.png}\n\\end{center}\n\\label{fig:band}\n\\caption{Calculated band structures of a proximitized graphene zigzag ribbon in the presence of a perpendicular magnetic field B=0~mT (a), B=1.5~mT, and B=3~mT. The color code denotes spin polarization in the left column and edge polarization in the right column. Thin dashed lines indicate bulk Landau levels.}\n\\end{figure}\n\nThe band structure of a zigzag graphene ribbon is calculated for different values of the external perpendicular magnetic field. We consider zigzag ribbons\nwith the width of 4.1~$\\mu$m. With the scaling factor $r=400$, the simulation size of the ribbon is reduced to  72 carbon-carbon bonds. The results for the zigzag ribbon band structure for magnetic fields of 0~mT, 1.5~mT, and 3.0~mT are shown in Fig.~\\ref{fig:band}. At zero magnetic field, we can distinguish two types of edge states: spin unpolarized intravalley states and pseudohelical intervalley states with strong spin polarization~\\cite{Frank2018, Gmitra2016}. The intravalley edge states are more delocalized towards the bulk than intervalley states, due to their spectral proximity to the bulk states. \n\nTo compare with the bulk~\\cite{Frank2020}, we also plot bulk Landau levels at $K$ and $K^{\\prime}$. For small magnetic fields (1.5~mT), the Landau levels start to form and bands at the $K$ and $K^{\\prime}$ valleys lose their dispersion. In general, the Landau levels are rather bulk-like, indicated by the absence of edge polarization. They coincide with the analytic prediction\\cite{Frank2020}. \n\nAt the borders of the bulk continuum, edge states form, which are responsible for the quantum Hall effect in graphene. With the increase of the external magnetic field, intervalley edge states shift in $k$-space: states with the same spin polarization shift in the same direction, while states with the opposite spin polarization shift in the opposite direction. Generally, intervalley edge states are not much affected by the perpendicular magnetic field. However, the intravalley states get strongly modified,  merging with the bulk bands; this happens already at magnetic fields less than 1.5~mT. This is similar to what is predicted for 2D topological insulators in a magnetic field~\\cite{Bottcher2019}. The bulk gap is openning at the crossover magnetic field $B_c \\approx  1.942$~mT. As \na consequence, the intravalley edge states disappear, leaving behind the lone pairs of pseudohelical edge states, ona pair at each zigzag edge. \n\n\\subsection{Flake results}\nTo analyze the nature of the edge states, we have calculated low energy states of differently sized proximitized graphene flakes. At zero magnetic field the pseudohelical states which form at the zigzag edges reflect back to the intravalley states, which generates standing waves spread out through the flake. In nanoribbons, with sizes less than a micron, the intravalley states are gapped out and the pseudohelical states tunnel through the armchair edges, allowing them to fully propagate through the flake edges. \n\nAs we showed above, for $B > B_c$ the intravalley states in wide ribbons are gapped out as well. Can then the pseudohelical states follow the same scenario as in nanoribbons? The answer is no. While at zero magnetic field the pseudohelical states cannot reflect back at the armchair edges due to time-reversal symmetry (the pseudohelical pair is formed by time-reversal partners), in the presence of a magnetic field such a reflection is possible. In fact, as we demonstrate below, this reflection is perfect, leading to the localization of the pseudohelical pair at zigzag edges and formation of standing waves that carry pure spin current. \nThe qualitative difference between propagating pseudohelical edge states and pure spin current non-propagating states is depicted in Fig.\\ref{fig:states}.\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=.5\\textwidth]{2_.pdf}\n\\end{center}\n\\caption{(a) Schematic representation of pseudohelical states in a proximitized graphene flake. The states form at zigzag edges, propagate towards the armchair edges, along which they tunnel and undergo spin flip due to Rashba coupling. The colors represent the spin which is perpendicular to the sheet. At zero magnetic field the tunneling along the armchair edges is perfect. (b) In large flakes, if $B > B_c$, the counterpropagating pseudohelical edge states are perfectly reflected at the armchair edges. The pair then forms a standing, nonpropagating wave, which carries net spin current, but no charge current. The resulting state is spin unpolarized. See also Fig.~\\ref{fig:schema}(c).}\n\\label{fig:states}\n\\end{figure}\n\nWhile the formation of the pseudohelical states can be explained by perfect tunnelling of the intervalley states across the armchair edge, the formation of the pure spin current states corresponds to perfect reflection off the armchair edge. Such a difference in the mechanisms of the formation of states is possible due to the exponential decay of the probability of tunnelling between zigzag edges as the distance between them increases. It is not possible to continuously go from one limit to another due to the periodic subband gap opening and closing~\\cite{Frank2018}. However, we can clearly distinguish pseudohelical states for a narrow graphene flake and pure spin current states for a wide graphene flake, because these states represent two different physical limits. \nWe estimated the distance between zigzag edges at which perfect tunneling still occurs as the typical spin-flip length of the Rashba spin-orbit interaction\n\\begin{align}\n    l_{R}\\approx3a\\frac{t}{\\lambda_R}.\n\\end{align}\nFor our parameters (graphene\/WSe$_2$) $l_{R}$ is about 3.4~$\\mu$m, which is less than the flake width we use in this work. \n\nPseudohelical states and pure spin current states are formed by combinations of intervalley states. They both consist of states (1) and (2) at the left edge and of (3) and (4) at the right edge, see  Fig.\\ref{fig:states}(c). Both types of states, pseudohelical states and pure spin current states, are formed using the same edge states and the only difference is the mechanism of their combination.\n\nWe have specifically investigated pure spin current (i.e., spin current without any charge current) states for a flake whose size is equal to 16.3584~$\\mu$m $\\times$ 4.0896~$\\mu$m, taking into account the scaling factor $r=400$, which corresponds to an auxiliary 288 $\\times$ 72 carbon-carbon bond sized flake. The width of this flake is greater than the Rashba spin-flip length, so the perfect reflection mechanism is dominant in this case. We trace the evolution of the highest occupied state with incremental steps in the external perpendicular magnetic field. \n\nThe spin and charge currents and densities through a cut in the middle of the flake, orthogonal to the armchair direction are shown in Fig.\\ref{fig:cut}. The data indicates that with the increase of the magnetic field, the charge current and spin densities disappear and complete edge localization of the state occurs. Also, the spatial spin current oscillations vanish and the highest occupied states become pure spin current states with negligible spin density. A similar situation occurs with the lowest unoccupied state. \n\nThe evolution of the spin currents of the highest occupied and the lowest unoccupied states are shown in Fig.\\ref{fig:spin}. Both states exhibit similar behavior of their spin currents, located on different zigzag edges of the flake. Pure spin currents arise in the presence of an external magnetic above the crossover value $B_c$. The evolution of the charge density is similar to the previous case, depicted in Fig.~\\ref{fig:charge}. With an external magnetic field below the crossover value, the states closest to the Fermi level are a mixture of strongly localized intervalley states and weakly localized intravalley states; they form a standing wave at the zigzag boundary, as tunneling through the armchair edge is forbidden (this can explicitly be seen in gapped ribbon armchair band structures~\\cite{Frank2018}).\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=.5\\textwidth]{4.pdf}\n\\end{center}\n\\caption{Calculated spin (a) and charge (b) currents of the highest occupied flake states (see text for flake parameters) through the cut in the middle of the flake and in the presence of a\nperpendicular magnetic field. Spin (c) and charge (d) densities of the highest occupied state at the cut in the middle of the flake and in the presence of a magnetic field.}\n\\label{fig:cut}\n\\end{figure}\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=.5\\textwidth]{5_new.png}\n\\end{center}\n\\caption{Evolution of the spin current of the highest occupied flake state in the left column and of the lowest unoccupied flake state in the right column.}\n\\label{fig:spin}\n\\end{figure}\n\n\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=.5\\textwidth]{6_new.png}\n\\end{center}\n\\caption{Evolution of the charge density of the highest occupied flake state in the left column and of the lowest unoccupied flake state in the right column.}\n\\label{fig:charge}\n\\end{figure}\n\n\nHowever, with a magnetic field above the crossover value $B_c$, we find states with pronounced pure spin current, localized strongly at a single edge. From the zigzag ribbon band-structure analysis, we can conclude that the pure spin current states correspond to the combination of the intervalley edge states on the same edge, which is stable under the application of a magnetic field. Similar to the pseudohelical states, pure spin current states are stable against scattering on defects at the zigzag edge of the graphene flake, see Appendix A.\n\n\\section{Conclusion}\nWe investigated the behavior of the electronic structure of proximitized graphene ribbons and flakes of microscopic sized in the presence of a perpendicular magnetic field. We have chosen \ngraphene\/WSe$_2$ heterostructure for the numerical parameters, as in such structures pseudohelical states were predicted to exist. We found that under the influence of a magnetic field, the pseudohelical (intervalley) edge states are preserved, while the intravalley edge states disappear at the crossover value of the magnetic field $B_c \\approx 1.9~mT$. This value corresponds to the closing\/reopening of the gap between the bulk Landau levels. \n\nWe also studied a finite flake of micron sizes, \nfinding that instead of perfect tunneling of the pseudohelical states at the armchair edges at zero magnetic field, for $B > B_c$ the states perfectly reflect to their counterpropagating partners and form non-propagating, spin unpolarized pure spin current states which are stable under scattering off zigzag edge defects. \n\nPure spin current states should be observable in wide flakes of at least a few microns (for graphene\/TMDC heterostructures) already at rather weak (a few millitesla) fields. Lone pseudohelical pairs at a given edge should be observable in wide ribbons for $B > B_c$ of in nanosized flakes where perfect tunneling through armchair edges is allowed and intravalley states are gapped even at zero magnetic field. \n\n\n\\begin{acknowledgments}\nWe would like to thank D. Kochan for useful discussions. This work was supported by the DFG SFB Grant No. 1277 (A09 and B07), DFG SPP 2244 (Project-ID 443416183), the International Doctorate Program Topological Insulators of the Elite Network of Bavaria. The authors gratefully acknowledge the Gauss Centre for Supercomputing e.V. (www.gauss-centre.eu) for providing computing time on the GCS Supercomputer SuperMUC at Leibniz Supercomputing Centre (LRZ, www.lrz.de).\n\\end{acknowledgments}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{sec:intro} \n\n\nThe Kronecker-Weierstrass theory of $m \\times n$ matrix pencils provides a complete classification in terms of \n$\\GL_{m,n}(\\mathbb{F})$-orbits, which are equivalence classes under the action of $\\GL_{m}(\\mathbb{F}) \\times \\GL_{n}(\\mathbb{F})$:\n\\begin{equation*}\n \\left( P, \\, U \\right) \\cdot \\left( \\mu A + \\lambda B \\right) =   \\mu P A U^\\T + \\lambda P B U^\\T,\n\\end{equation*}\nwhere the pencil $(A,B)$ is expressed in homogeneous coordinates. \nHere, $\\mathbb{F}$ denotes a field, usually $\\mathbb{R}$ or $\\mathbb{C}$.\nThese orbits are represented by Kronecker canonical forms, which are characterized by unique minimal indices describing the singular \npart of the pencil along with elementary divisors associated with its regular part \\cite{Gantmacher1960}. \nIt follows then that these elementary divisors are $\\GL_{m}(\\mathbb{F}) \\times \\GL_{n}(\\mathbb{F})$-invariant.\n\n\nThis theory has been extended by Ja'Ja' and Atkinson \\cite{JaJa1979b, Atkinson1991}, who have characterized the orbits of the larger \ngroup of tensor equivalence transformations $\\GL_{m}(\\mathbb{F}) \\times \\GL_{n}(\\mathbb{F}) \\times \\GL_2 (\\mathbb{F})$, acting on \npencils via\n\\begin{equation*}\n \\left( P, \\, U, \\, T \\right) \\cdot \\left( \\mu A + \\lambda B \\right) \n  =  \\mu P (t_{11} A + t_{12} B) U^\\T + \\lambda P (t_{21} A + t_{22} B) U^\\T,\n\\end{equation*} \nwhere \n\\begin{equation*}\n T = \\begin{pmatrix}\n      t_{11} & t_{12} \\\\ t_{21} & t_{22}\n     \\end{pmatrix}\n     \\in \\GL_2 (\\mathbb{F}).\n\\end{equation*} \nFor simplicity of notation, we use the shorthands $\\GL_{m,n,2}(\\mathbb{F}) \\triangleq \\GL_{m}(\\mathbb{F}) \\times \n\\GL_{n}(\\mathbb{F}) \\times \\GL_2 (\\mathbb{F})$ and $\\GL_{m,n}(\\mathbb{F}) \\triangleq \\GL_{m}(\\mathbb{F}) \\times \\GL_{n}(\\mathbb{F})$. \nJa'Ja' \\cite{JaJa1979b} has shown that the Kronecker minimal indices of a pencil are $\\GL_{m,n,2}(\\mathbb{F})$-invariant, and so the \nsingular part of a pencil is preserved by $\\GL_{m,n,2}(\\mathbb{F})$ as well. \nHowever, the elementary divisors of its regular part are not. Nevertheless, their powers still remain the same, which motivates the \nterminology ``invariant powers,'' used by Ja'Ja' \\cite{JaJa1979b}. Atkinson \\cite{Atkinson1991} went on to prove that, for an \nalgebraically closed field $\\mathbb{F}$, the equivalence classes of regular pencils are characterized by those powers and also by \ncertain ratios which completely describe the elementary divisors. Specifically, recalling that over such a field all elementary \ndivisors are powers of linear factors of the form\n\\begin{equation*}\n  \\phi_i(\\mu, \\lambda) = \\alpha_i \\mu + \\beta_i \\lambda  \\quad \\text{for some } \\alpha_i, \\beta_i \\in \\mathbb{F},\n\\end{equation*}\nthese ratios are defined as $\\gamma_i \\triangleq \\alpha_i \/ \\beta_i \\in \\mathbb{F} \\cup \\{\\infty\\}$.\n\n\nWhen viewed as a tensor, the (tensor) rank\\footnote{This is not to be confused with the \\emph{normal rank} of the pencil \n$\\mu A + \\lambda B$, simply defined as $\\rank(\\mu A + \\lambda B)$ \\cite{Gantmacher1960, Ikramov1993}.} of a pencil $\\mu A + \\lambda B$ \nis defined as the minimal number $r$ of rank-one matrices $U_1, \\ldots, U_r \\in \\mathbb{F}^{m \\times n}$ such that $A, B \\in \\sspan \n\\{U_1, \\ldots, U_r\\}$ \\cite{JaJa1979}. Equivalently, it is given by the \nminimal number $r$ such that one can find vectors $u_i \\in \\mathbb{F}^m$, $v_i \\in \\mathbb{F}^n$ and $w_i \\in \\mathbb{F}^2$ satisfying\n\\begin{equation*}\n A \\otimes e_1 + B \\otimes e_2 = \\sum_{i=1}^r u_i \\otimes v_i \\otimes w_i,\n\\end{equation*} \nwhere $e_i$ denotes the canonical basis vector of $\\mathbb{F}^2$. Under the action of $(P,U,T) \\in \\GL_{m,n,2}(\\mathbb{F})$, this \nexpression is transformed into the tensor\n\\begin{equation*}\n \\left( P, U, T \\right) \\cdot \\left( A \\otimes e_1 + B \\otimes e_2 \\right) \n   = \\sum_{i=1}^r  \\left(  P \\, u_i \\right) \\otimes \\left( U \\, v_i \\right) \\otimes \\left( T \\, w_i \\right),\n\\end{equation*} \nfrom which it is visible that the tensor rank is $\\GL_{m,n,2}(\\mathbb{F})$-invariant. The rank of $m \\times n \\times 2$ tensors \ncan thus be studied by considering $\\GL_{m,n,2}(\\mathbb{F})$-orbits and associated representatives (see, e.g., the classification of \n$\\GL_{2,2,2}(\\mathbb{R})$-orbits undertaken by De Silva and Lim \\cite{deSilva2008}).\n\n\nOne application of the study of $\\GL_{m,n,2}(\\mathbb{F})$-orbits is in algebraic complexity theory, since the tensor rank \nof $\\mu A + \\lambda B$ quantifies the minimal number of multiplications needed to simultaneously evaluate a pair of \nbilinear forms $g_1(x,y) = \\langle x, A y \\rangle$ and $g_2(x,y) = \\langle x, B y \\rangle$ \\cite{JaJa1979}. In the case where $\\mathbb{F}$ is \nalgebraically closed, Ja'Ja' \\cite{JaJa1979} has derived results which allow determining the tensor rank of any pencil based on its \nKronecker canonical form. Sumi et al.~\\cite{Sumi2009} have extended these results to pencils over any field $\\mathbb{F}$.\n\nTensor equivalence transformations  can also be employed to avoid so-called infinite elementary divisors of regular \npencils \\cite{Ikramov1993}. These arise when matrix $B$ is singular (note that both $A$ and $B$ can be singular but still \nsatisfy $\\det(\\mu A + \\lambda B) \\not\\equiv 0$). In non-homogeneous coordinates, the polynomial $\\det (A + \\lambda B)$ of an $n \n\\times n$ pencil $A + \\lambda B$ has degree $s = \\rank B$, and its characteristic polynomial is said to have infinite \nelementary divisors of combined degree $n - s$. In this case,\nthe tensor equivalence transformation $A + \\lambda B \\mapsto B + \\lambda (A + \\alpha B)$ can be employed for any $\\alpha \\in \n\\mathbb{F}\\setminus\\{0\\}$ such that $\\rank (A + \\alpha B) = n$, yielding a pencil having only finite elementary divisors, \nincluding some of the form $\\lambda^q$ induced by the infinite elementary divisors of $A + \\lambda B$.\nThe existence of such an $\\alpha$ is guaranteed by definition, since $A + \\lambda B$ is regular. In other words, every regular pencil \n$\\mu A + \\lambda B$ is $\\GL_{m,n,2}(\\mathbb{F})$-equivalent to another pencil $\\mu A' + \\lambda B'$ such that $B'$ is \nnonsingular. In fact, it is always $\\GL_{m,n,2}(\\mathbb{F})$-equivalent to some $\\mu A' + \\lambda B'$ with nonsingular \nmatrices $A'$ and $B'$.\n\nOn the other hand, not every regular pencil $\\mu A + \\lambda B$ constituted by nonsingular matrices $A$ and $B$ is \n$\\GL_{n,n,2}(\\mathbb{F})$-equivalent to some other pencil $\\mu A' + \\lambda B'$ such that either $A'$ or $B'$ are \nsingular (or both). Take, for instance, $\\mathbb{F} = \\mathbb{R}$ and\n\\begin{equation*}\n A = \\begin{pmatrix} 1 & 0 \\\\ 0 & 1 \\end{pmatrix}, \\quad B = \\begin{pmatrix} 0 & -1 \\\\ 1 & 0 \\end{pmatrix}.\n\\end{equation*} \nNo tensor equivalence transformation in $\\GL_{m,n,2}(\\mathbb{R})$ of this pencil can yield $\\mu A' + \\lambda B'$ such that \neither $A'$ or $B'$ is singular. Obviously, this property is $\\GL_{m,n,2}(\\mathbb{R})$-invariant.\nIt turns out that each $\\GL_{m,n,2}(\\mathbb{F})$-orbit $\\mathcal{O}$ of a matrix pencil space can be classified on the basis \nof its associated \\emph{minimal ranks} $r$ and $s$, with $r \\ge s$, which are such that every pencil $\\mu A + \\lambda B \\in \n\\mathcal{O}$ with $\\rank A \\ge \\rank B$ satisfies $\\rank A \\ge r$ and $\\rank B \\ge s$.\nThis notion of intrinsic complexity of a matrix pencil is complementary to its tensor rank, in the sense that pencils of same \ntensor rank do not necessarily have the same minimal ranks and vice-versa. For simplicity, we will compactly denote the minimal ranks \nof a pencil by $ \\rho(A,B) = (r,s)$.\n\nIt turns out that there is a direct connection between the minimal ranks of a pencil and the decomposition of its associated \nthird-order tensor in block terms  consisting of matrix-vector outer products, as introduced by De Lathauwer \n\\cite{deLathauwer2008b}. Namely, the components of $\\rho(A,B)$ are the minimal numbers $r$ and $s$ satisfying\n\\begin{equation*}\n A \\otimes e_1 + B \\otimes e_2 = \\left( \\sum_{i=1}^{r} u_i \\otimes v_i \\right) \\otimes w +\n                                 \\left( \\sum_{i=1}^{s} x_i \\otimes y_i \\right) \\otimes z,\n\\end{equation*} \nwhere $r \\ge s$ and $\\{w, z\\}$ forms a basis for $\\mathbb{F}^2$. The theoretical properties of such block-term decompositions \n(henceforth abbreviated as BTD) of $m \\times n \\times 2$ tensors are therefore related to properties of matrix pencils via this notion \nof minimal ranks. \n\nIn this paper, we will more generally define the minimal ranks of $m \\times n$ matrix polynomials $\\sum_{k=1}^{d} \\lambda^{k-1} \nA_k$, which include matrix pencils as a special case. This property of matrix polynomials is directly related to the BTD of $m \n\\times n \\times d$ tensors. In particular, similarly to the tensor rank, it induces a hierarchy of matrix polynomials, albeit a more \ninvolved one. \nWe derive results which determine the minimal ranks of any matrix pencil in Kronecker canonical form.\nA classification of $\\GL_{m,n,2}(\\mathbb{R})$-orbits of real $m \\times n$ pencils in terms of their minimal ranks is then\ncarried out for $m, n \\le 4$.\nOn the basis of these results, we proceed to show that:\n\\begin{enumerate}\n \\item The set of real $2k \\times 2k$ pencils which are $\\GL_{2k,2k,2}(\\mathbb{R})$-equivalent to some $\\mu A + \\lambda B$\nwith $\\rank A \\le 2k-1$ and $\\rank B \\le 2k-1$ is not closed in the norm topology for any positive integer $k$.\n \\item No real $2k \\times 2k$ pencil having minimal ranks $(2k,2k)$ admits a best approximation in the set above described. \n\\end{enumerate}\nThe first above result is analogous to the fact that a set of tensors having rank bounded by some number $r>1$ is generally not \nclosed. \nSimilarly, the second one parallels the fact that no element of certain sets of real rank-$r$ tensors admits a best approximation of \na certain rank\\footnote{For instance, no real $2 \\times 2 \\times 2$ tensor of rank $3$ admits a best rank-$2$ approximation.} $r' < r$ \nin the norm topology \\cite{Stegeman2006, deSilva2008}. \nThis second result is of consequence to applications relying on the BTD, since the set of real pencils having minimal ranks $(2k,2k)$ \nis open in the norm topology, thus having positive volume. For complex-valued pencils, the results of \\cite{Qi2017} imply that such a \nnon-existence phenomenon can only happen over sets of zero volume. We shall give a template of examples of (possibly complex) pencils \nhaving no best approximation on a given set of pencils with strictly lower minimal ranks.\n\nIt should be noted that the fact that a tensor might not admit an approximate BTD with a certain prescribed structure (referring \nto the number of blocks and their multilinear ranks \\cite{deLathauwer2008b}) is already known. Specifically, De Lathauwer \n\\cite{deLathauwer2008c} has provided an example relying on a construction similar to that of De Silva and Lim \\cite{deSilva2008} \nconcerning the case of low-rank tensor approximation. Our example given in \\Cref{sec:ill-posed-examples} is in the same spirit. \nNonetheless, to our knowledge ours is the first work showing the existence of a positive-volume set of tensors having no approximate \nBTD of a given structure, a phenomenon which is known to happen for low-rank tensor approximation \\cite{Stegeman2006, deSilva2008}.\n\n\n\n \n\\section{Minimal ranks of pencils}\n\\label{sec:pencils}\n\nFor brevity, we will henceforth express matrix pencils only in non-homogeneous coordinates.\n\n\\subsection{Definition and basic results}\n\\label{sec:min-ranks}\n\nGiven its prominent role in what follows, the $\\GL_{m,n,2}(\\mathbb{F})$-orbit of a pencil deserves a special notation:\n\\begin{equation*}\n \\mathcal{O}(A,B) \\triangleq \n                   \\{(P,U,T) \\cdot (A + \\lambda B) \\ | \\ (P,U,T) \\in \\GL_{m,n,2}(\\mathbb{F}) \\}.\n\\end{equation*}\nIt will also be helpful to introduce the sets\n\\begin{equation}\n\\label{Bset}\n \\mathcal{B}_{r,s} \\triangleq \n                   \\left\\{ A + \\lambda B \\; | \\;\n                   \\exists \\, A' + \\lambda B' \\in \\mathcal{O}(A,B) \n                   \\mbox{ such that }\n                   \\rank A' \\le r, \\, \\rank B' \\le s \\right\\}.\n\\end{equation} \nClearly, $\\mathcal{B}_{r,s} = \\mathcal{B}_{s,r}$, and thus we shall assume that $r \\ge s$ without loss of generality.\nFor simplicity, when $r \\ge s = 0$ we can also write $\\mathcal{B}_r$ instead of $\\mathcal{B}_{r,0}$. According to this convention and  \nto definition \\eqref{Bset}, we have, for instance, $\\mathcal{B}_{r} \\subseteq \\mathcal{B}_{r,s} \\subseteq \\mathcal{B}_{r+1,s}$. \nFurthermore, $\\mathcal{B}_{r,s}$ is by definition $\\GL_{m,n,2}(\\mathbb{F})$-invariant, because the relation $(A,B) \\sim \n(A',B')$ defined as $A' + \\lambda B' \\in \\mathcal{O}(A,B)$ is reflexive and transitive, i.e., it defines an equivalence class. Hence:\n\n\\begin{lem}\n $A + \\lambda B \\in \\mathcal{B}_{r,s}$ if and only if $\\mathcal{O}(A,B) \\subseteq \\mathcal{B}_{r,s}$.\n\\end{lem}\t\n\nAs far as the question of whether $A + \\lambda B$ is in $\\mathcal{B}_{r,s}$ for some $(r,s)$ is concerned, all that matters is the \naction of $\\GL_2(\\mathbb{F})$. Indeed, if $A' + \\lambda B' = (P,U,T) \\cdot (A + \\lambda B)$ are such that $\\rank A' = r$ and $\\rank B' \n= s$, then\\footnote{Note that we denote the identity of $\\GL_m(\\mathbb{F})$ by $E_m$ or, when no ambiguity arises, simply by $E$.} $A'' \n+ \\lambda B'' = (P^{-1},U^{-1},E) \\cdot (A' + \\lambda B') = (E,E,T) \\cdot (A + \\lambda B)$ satisfies $\\rank A'' = \\rank P^{-1} A' \n U^{-\\T} = \\rank A'$ and $\\rank B'' = \\rank P^{-1} B' U^{-\\T} = \\rank B'$. We have shown the following.\n\n\\begin{lem}\n\\label{lem:GL2}\n$A + \\lambda B \\in \\mathcal{B}_{r,s}$ if and only if there exists $T \\in \\GL_2(\\mathbb{F})$ such that $A' + \\lambda B' = (E, E, T) \n\\cdot (A + \\lambda B)$ satisfies $\\rank A' \\le r$ and $\\rank B' \\le s$. In other words, $\\mathcal{B}_{r,s}$ can be equivalently \ndefined as the set of all pencils which are $\\GL_2(\\mathbb{F})$-equivalent to some $A' + \\lambda B'$ satisfying $\\rank A' \\le r$ and \n$\\rank B' \\le s$.\n\\end{lem}\n\n\nLet us now formally define the minimal ranks in terms of the introduced notation.\n\n\\begin{defn}\n\\label{def:min-ranks}\nLet $A + \\lambda B$ be an $m \\times n$ pencil over $\\mathbb{F}$. The \\emph{minimal ranks} of $A + \\lambda B$, denoted as \n$\\rho(A,B) = (r,s)$, are defined as\n\\begin{align}\n \\label{s-def}\ns \\triangleq & \\ \\min_{(t,u) \\neq 0} \\rank (t A + u B),  \\\\\nr \\triangleq & \\  \\min_{(t',u') \\notin \\sspan\\{(t^\\star,u^\\star)\\}} \\rank(t' A + u' B), \n\\label{r-def}\n\\end{align} \nwhere $(t^\\star,u^\\star)$ is any minimizer\\footnote{Observe that such a minimizer is not unique. Besides the obvious family of \nminimizers of the form $(c t^\\star, c u^\\star)$, there may be also multiple non-collinear minimizers. For instance, for the pencil $E_4 \n+ \\lambda (a E_2 \\oplus b E_2)$, with $a \\neq b$, there are two non-collinear minimizers: $(-a,1)$ and $(-b,1)$.} of \\eqref{s-def}. We \nobviously have $r\\ge s$.\nWhen denoting a pencil as $P(\\lambda) = A + \\lambda B$, we shall also use the notation $\\rho(P) = \\rho(A,B)$.\n\\end{defn}\n\nThe first thing to note is that $r$ is well-defined, i.e., its value is always the same regardless of the minimizer $(t^\\star,u^\\star)$\npicked in the definition \\eqref{r-def}. For different collinear minimizers of \\eqref{s-def}, this is immediately clear. Now if two \nnon-collinear minimizers $(t^\\star,u^\\star)$ and $(t^{\\star\\star}, u^{\\star\\star})$ exist for \\eqref{s-def}, then $r = s$ must hold.\nIt is also clear from \\eqref{s-def} and \\eqref{r-def} that the minimal ranks of a pencil  $A + \\lambda B$ are the ranks of \nmatrices $A'$ and $B'$ of some pencil $A' + \\lambda B'$ in the $\\GL_{2}(\\mathbb{F})$-orbit  of $A + \\lambda B$. Indeed, \ntaking $B' = t_{21} A + t_{22} B$ and $A' = t_{11} A + t_{12} B$, where $(t_{21},t_{22})$ and $(t_{11},t_{12})$ are the minimizers of \n\\eqref{s-def} and \\eqref{r-def}, respectively, then  $A + \\lambda B = (E,E,T^{-1}) \\cdot (A' + \\lambda B')$, with $T = (t_{ij})  \\in \n\\GL_2(\\mathbb{F})$.  Moreover, the minima in \\eqref{s-def} and \\eqref{r-def} are unchanged under a transformation from \n$\\GL_{m,n}(\\mathbb{F})$, implying that the value of $\\rho$ is an invariant of $\\GL_{m,n,2}$ action. Summarizing, we have:\n\n\\begin{prop}\n \\label{prop:rho-orbit}\n Every $A' + \\lambda B' \\in \\mathcal{O}(A,B)$ satisfies $\\rho(A,B) = \\rho(A',B')$. In particular, if $A' + \\lambda B' \\in \n\\mathcal{O}(A,B)$ satisfies $(\\rank A', \\rank B') = \\rho(A,B)$, we say that $A' + \\lambda B'$ attains the minimal ranks of $A + \\lambda \nB$.\n\\end{prop}\n\nFrom the above discussion, $\\rho(A,B) = (r,s)$ implies $A + \\lambda B \\in \\mathcal{B}_{r,s}$. However, the converse is not true. For \ninstance, $E_n + \\lambda E_n \\in \\mathcal{B}_{n,n}$ but $\\rho(E_n, E_n) = (n,0)$. In general, if $A = c B$ or $B = c A$ for \nsome $c \\in \\mathbb{F}$ (including the possibility $c = 0$), then $\\rho(A,B) = (r,0)$  with $r = \\max\\{\\rank A, \\rank B\\}$. \nConversely, $s = 0$ only if $A$ and $B$ are proportional. We thus have the following result. \n\n\\begin{lem}\n \\label{lem:rho-r-0}\nA pencil $A + \\lambda B$ satisfies $\\rho(A,B) = (r,0)$ if and only if $A$ and $B$ are proportional. Furthermore, $r = \n\\max\\{\\rank A, \\rank B\\}$.\n\\end{lem}\n\n\nThe terminology ``minimal ranks'' is motivated by the fact that, if $\\rho(A,B) = (r,s)$ and $A + \\lambda B \\in \n\\mathcal{B}_{r',s'}$ for \nsome $r' \\ge s'$, then \\emph{both} $r' \\ge r$ and $s' \\ge s$ must hold. The definitions in \\eqref{s-def} and \\eqref{r-def} clearly \nimply $A + \\lambda B \\notin \\mathcal{B}_{r',s'}$ for any pair $r'$ and $s' < s$ or any pair $r' < r$ and $s' = s$.\nIt remains to show that $A + \\lambda B \\notin \\mathcal{B}_{r',s'}$ also for $r > r' \\ge s' > s$. \nSuppose on the contrary that $A + \\lambda B \\in \\mathcal{B}_{r',s'}$ with $r > r' \\ge s' > s$. \nThis implies there exists $T \\in \\GL_2\\left( \\mathbb{F} \\right)$ such that \n\\begin{equation*}\n   \\rank (t_{11} A + t_{12} B) \\le r' \\quad \\text{and} \\quad \\rank (t_{21} A + t_{22} B) \\le s', \\quad \n   \\text{with} \\quad r > r' \\ge s' > s.\n\\end{equation*}\nBut then, $r' > s$ implies $(t_{11},t_{12}) \\notin \\sspan\\{(t^\\star,u^\\star)\\}$, where $(t^\\star,u^\\star)$ is a minimizer of \n\\eqref{s-def}. This contradicts the definition of $r$ given by \\eqref{r-def}. \n\n\n\\begin{prop}\n\\label{prop:r-s}\nIf $\\rho(A,B) = (r,s)$ and $r' \\ge s'$, then $A + \\lambda B \\in \\mathcal{B}_{r',s'}$ if and only if $r' \\ge r$ and $s' \\ge s$.\n\\end{prop}\n\n\nWe consider now some examples.\n\n\\begin{exmp}\nA regular $n \\times n$ pencil $A + \\lambda B$ can only belong to $\\mathcal{B}_r$ if $r = n$. Indeed, $A + \\lambda B \\in \\mathcal{B}_r$ \nimplies $A$ and $B$ are proportional,  say $B = \\alpha A$, and $\\rank(t A + u B) = \\rank((t + \\alpha u) A) \\le r$ for any \n$(t,u) \\in \\mathbb{F}^2$. As a concrete example, $E + \\lambda E$ is clearly in $\\mathcal{B}_n$ but not in any $\\mathcal{B}_{r'}$ with \n$r' < n$.\n\\end{exmp}\n\n\\begin{exmp}\nRegular $n \\times n$ pencils can also be in $\\mathcal{B}_{r,s}$ for some $n > r \\ge s > 0$. For example, the regular $3 \\times 3$ \npencil\\footnote{The dimensions of the zero blocks in that expression should be clear from the context.} $E_2 \\oplus 0 + \\lambda (0 \n\\oplus E_1)$ is in $\\mathcal{B}_{2,1}$. Yet, the constraint $r + s \\ge n$ must be satisfied. Indeed, $A + \\lambda B \\in \n\\mathcal{B}_{r,s}$ means $A + \\lambda B$ is $\\GL_{m\\times n\\times 2}(\\mathbb{F})$-equivalent to some $A' + \\lambda B'$, with $\\rank A' \n\\le r$ and $\\rank B' \\le s$. If $r + s < n$, then clearly $\\det(A' + \\lambda B') \\equiv 0$, implying neither $A' + \\lambda B'$ nor $A \n+ \\lambda B$ is regular.\n\\end{exmp}\n\nThe next two examples underline how the elementary divisors of a regular pencil determine its minimal ranks. A general result \nestablishing this connection will be presented ahead.\n\n\\begin{exmp}\n The pencil \n \\begin{equation*}\n   Q + \\lambda E =\n             \\begin{pmatrix} a & b  \\\\ -b & a \\end{pmatrix}\n   + \\lambda \\begin{pmatrix} 1 & 0  \\\\ 0 & 1 \\end{pmatrix} \n \\end{equation*} \n with $b \\neq 0$ has minimal ranks $\\rho(Q,E) = (2,2)$ in $\\mathbb{R}$, because\n \\begin{equation*}\n             \\rank \\begin{pmatrix} u + t a & t b  \\\\ -t b & u + t a \\end{pmatrix} = 2\n \\end{equation*}  \nfor any  $(t,u) \\in \\mathbb{R}^2 \\setminus \\{0\\}$. However, this is not true in $\\mathbb{C}$, because the $\\GL_2(\\mathbb{C})$ \ntransformation $(Q,E) \\mapsto ((-a + bi) E + Q, E )$ yields $B + \\lambda E$, where\n\\begin{equation*}\n B = \\begin{pmatrix} bi & b \\\\ -b & bi \\end{pmatrix} = \n      \\begin{pmatrix} bi & bi \\\\ -b & -b \\end{pmatrix} \\begin{pmatrix} 1 & 0 \\\\ 0 & -i \\end{pmatrix} \n\\end{equation*} \nhas rank 1. This difference comes from the fact that $Q + \\lambda E$ has a single elementary divisor over $\\mathbb{R}$, namely \n$\\lambda^2 + 2a\\lambda + (a^2 + b^2)$, which cannot be factored into powers of linear forms since its roots are complex. \nIn fact, $Q$ is diagonalizable over $\\mathbb{C}$, since it is similar to\n\\begin{equation*}\n B' = \\begin{pmatrix} a + bi & 0 \\\\ 0 & a - bi \\end{pmatrix}\n\\end{equation*}\nFrom \\Cref{prop:rho-orbit}, we have $\\rho(Q,E) = \\rho(B',E)$, and it is not hard to see that $\\rho(B',E) = (1,1)$. \n\\end{exmp}\n\n\\begin{exmp}\nDefining\n\\begin{equation*}\n H  = \\begin{pmatrix} a & 1 & 0  \\\\ 0 & a & 1 \\\\ 0 & 0 & a \\end{pmatrix}, \\quad\n H'  = \\begin{pmatrix} a & 1 & 0  \\\\ 0 & a & 0 \\\\ 0 & 0 & a \\end{pmatrix}\n   \\quad \\text{and} \\quad \n H'' = \\begin{pmatrix} a & 1 & 0  \\\\ 0 & a & 0 \\\\ 0 & 0 & b \\end{pmatrix}\n\\end{equation*} \nwith $a \\neq b$, we have $\\rho(H,E) = (3,2)$, $\\rho(H',E) = (3,1)$ and $\\rho(H'',E) = (2,2)$. Note that the three \nconsidered pencils are regular and, in particular, the eigenvalues of the first two are the same but their elementary divisors are not. \nIn fact, their invariant polynomials are $\\{(a+\\lambda)^3,1,1\\}$ for $H + \\lambda E$, $\\{(a+\\lambda)^2,a+\\lambda,1\\}$ for $H' \n+ \\lambda E$, and $\\{(a+\\lambda)^2,b+\\lambda,1\\}$ for $H'' + \\lambda E$. \n\\end{exmp}\n\n\n\\subsection{Induced hierarchy of matrix pencils}\n\\label{sec:hierarchy}\n\nThe tensor rank induces a straightforward hierarchy in any tensor space, namely, $\\mathcal{S}_0 \\subset \\mathcal{S}_1 \\subset \n\\mathcal{S}_2 \\subset \\dots$, where $\\mathcal{S}_r$ contains all tensors of rank up to $r$. Our definition of minimal ranks also \ninduces a hierarchy which can be expressed by using the definition of the sets $\\mathcal{B}_{r,s}$ given in \\eqref{Bset}. However, \nsuch a hierarchy is more intricate, as now we have, for instance, $\\mathcal{B}_{r,s} \\subset \n\\mathcal{B}_{r+1,s}$ and $\\mathcal{B}_{r,s} \\subset \\mathcal{B}_{r,s+1}$ but $\\mathcal{B}_{r+1,s} \\not\\subset \n\\mathcal{B}_{r,s+1}$ and $\\mathcal{B}_{r,s+1} \\not\\subset \\mathcal{B}_{r+1,s}$.\n\n\n\\Cref{fig:hierarchy} contains a diagram depicting the hierarchy of $\\mathcal{B}_{r,s}$ sets in the space of $m \\times n$ real pencils, \ndenoted by $\\mathcal{P}_{m,n}(\\mathbb{R})$. \nWe assume $m \\le n$ without loss of generality, since $\\mathcal{P}_{m,n}(\\mathbb{R})$ and $\\mathcal{P}_{n,m}(\\mathbb{R})$ have \nidentical structures. \nFor even $m$, the set $\\mathcal{B}_{m,m} \\subset \\mathcal{P}_{m,n}(\\mathbb{R})$ is always non-empty, but for odd $m$ the set \n$\\mathcal{B}_{m,m} \\subseteq \\mathcal{P}_{m,n}(\\mathbb{R})$ is non-empty if and only if $m < n$.\nThis is because an $m \\times m$ pencil has full minimal ranks if and only if it is regular and its elementary divisors cannot be \nwritten as powers of linear forms, as we shall prove in the next section. In $\\mathcal{P}_{m,m}(\\mathbb{R} )$, this means that a \npencil $A + \\lambda B$ satisfies $\\rho(A,B) = (m,m)$ if and only if it is $\\GL_{m,m,2}(\\mathbb{R})$-equivalent to some other pencil \n$Q + \\lambda E$ where no eigenvalue of $Q$ is in $\\mathbb{R}$. Since complex-valued eigenvalues of a real matrix necessarily \narise in pairs, this can evidently only happen for even values of $m$.\nFor concreteness, three examples concerning $\\mathcal{P}_{2,2}(\\mathbb{R})$, $\\mathcal{P}_{2,3}(\\mathbb{R})$ and \n$\\mathcal{P}_{3,3}(\\mathbb{R})$ are shown in \\Cref{fig:hierarchy}.\n\n\n\\begin{figure}[h!]\n\\centering\n \\includegraphics[width=0.8\\textwidth]{hierarchy.eps}\n\\caption{Hierarchy of sets of real pencils in $\\mathcal{P}_{m,n}(\\mathbb{R})$ according to their minimal ranks: illustration of \nthe general form (left) and concrete examples for three pencil spaces (right). The notation $\\mathcal{B}_{r,s} \\relbar\\joinrel\\mathrel\\RHD \n\\mathcal{B}_{r',s'}$ stands for $\\mathcal{B}_{r,s} \\subset \\mathcal{B}_{r',s'}$. See \\eqref{Bset} for a definition of the sets \n$\\mathcal{B}_{r,s}$.}\n \\label{fig:hierarchy}\n\\end{figure}\n\n\\subsection{Minimal ranks of Kronecker canonical forms}\n\\label{sec:Kronecker}\n\nWe now show how the minimal ranks of a pencil can be determined from its Kronecker canonical form. The notation $J_m(a) \n\\triangleq a E_m + H_m$, where $H_m \\triangleq \\sum_{l=1}^{m-1} e_l \\otimes e_{l+1}$, will be used for a \nJordan block of size $m$ associated with the finite elementary divisor $(a + \\lambda)^m$. In this definition, the vectors $e_l$ \ndenote as usual the canonical basis vectors of their corresponding spaces. \nA canonical $v \\times v$ block associated with an infinite elementary divisor will be expressed as $N_v(\\lambda) \\triangleq E_m \n+ \\lambda H_v$.\nLet us first consider regular pencils.\n\n\n\\begin{lem}\n\\label{lem:rho-regular} \n Let $A + \\lambda B$ be a regular $n \\times n$ pencil and let \n$$ \n     A' + \\lambda B' =  \\left[ N_{v_1}(\\lambda) \\oplus \\dots \\oplus N_{v_j}(\\lambda) \\right] \\oplus \n     \\left[ (A_1 \\oplus \\dots \\oplus A_k \\oplus Q) + \\lambda E \\right]\n$$ \nbe its Kronecker canonical form, where the elementary divisors of $ Q + \\lambda E_q$ cannot be factored into powers of linear forms\nand $A_l = J_{m_l}(a_l)$ for some $a_l \\in \\mathbb{F}$, $l \\in \\{1,\\ldots,k\\}$. Let $k_s$ be the largest number of blocks $A_l$ whose \nelementary divisors share a common factor $a^\\star + \\lambda$ and $k_r$ be the second largest number of blocks $A_l$ whose elementary \ndivisors share a common factor $a^{\\star\\star} + \\lambda$ (with $a^{\\star\\star} \\neq a^{\\star}$). Then \n\\begin{equation}\n\\label{rho-reg}\n  \\rho(A,B) = \\left( n - k'_r, n - k'_s \\right),\n\\end{equation}\nwhere $k'_s$ and $k'_r$ are the first and second largest components of $(j,k_s,k_r)$, respectively. \n\\end{lem}\n\n\\begin{proof}\nBy virtue of \\Cref{prop:rho-orbit}, we have $\\rho(A, B) = \\rho(A',B')$. It thus suffices to show that $\\rho(A',B') = (n - k'_r, n - \nk'_s)$. The steps are as follows.\n\\begin{enumerate}\n\n\\item First, we claim that $t Q + u E_q$ is nonsingular for any $(t,u) \\neq 0$. \nThis claim is trivially true if $t=0$ and $u \\neq 0$. For $t \\neq 0$ (and $u$ possibly null), the argument is as follows. Suppose for \na contradiction that $t Q + u E_q$ is singular, with $t \\neq 0$. Without loss of generality, we may take $t=1$. Then, there \nexists $P \\in \\GL_q(\\mathbb{F})$ such that \n\\begin{equation*}\n P \\left( Q + u E_q \\right) P^{-1} = P Q P^{-1} + u E_q = F \\oplus J_p(0),\n\\end{equation*} \nwhere $0 < p \\le q$. But then, $P Q P^{-1} = (F - u E_{q-p}) \\oplus J_p(-u)$, implying $Q + \\lambda E_q$ has an elementary \ndivisor of the form $(-u + \\lambda)^p$, which contradicts the hypothesis that the elementary divisors of $Q$ cannot be written as\npowers of linear forms. As a consequence, $\\rho(Q,E_q) = (q,q)$. In particular, if $Q = A'$ (i.e., $n = q$), then \\eqref{rho-reg} \nyields $\\rho(A',B') = (n,n)$ (because $j = k_s = k_r = 0$), as required.\n\n\\item  Now, note that $\\rank\\, (t J_{m}(a) + u E_m) < m$ for $(t,u) \\neq 0$ if and only if $(t,u) = c(1,-a)$ for some $c \\neq 0$, in \nwhich case $\\rank\\, (t J_{m}(a) + u E_m) = \\rank J_{m}(0) = m-1$. Moreover, $\\rank\\, (t E_v + u H_v) < v$ for $(t,u) \\neq 0$ if and \nonly if $(t,u) = (0,c)$ for some $c \\neq 0$, implying $\\rank\\, (t E_v + u H_v) = v-1$.\nHence, since by definition $k_s \\ge k_r$, we have three cases:\n\\begin{enumerate}[(i)]\n\n   \\item If $k_s \\ge j \\ge k_r$, then we can apply the following transformation\n      \\begin{equation*}\n      (A',B') \\mapsto (B', A' -a^\\star B') \n      \\end{equation*}    \nto obtain an $\\GL_{n,n,2}(\\mathbb{F})$-equivalent pencil attaining the minimal ranks of $(A',B')$. Indeed, $\\rank \\, (A' -a^\\star B') \n= n - k_s$ is minimal among all linear combinations $t A' + u B'$ with $(t,u) \\neq 0$. Hence, \\eqref{s-def} must equal $n - k_s = n - \nk'_s$. Furthermore, $\\rank B' = n - j$ is minimal among all linear combinations $t A' + u B'$ with $(t,u) \\neq c (1, -a^\\star)$. So, \n\\eqref{r-def} must equal $r' = n - j = n - k'_r$.\n\n   \\item If $k_s \\ge k_r \\ge j$, then using a similar argument we deduce that the $\\GL_{n,n,2}(\\mathbb{F})$-equivalent pencil $(A' \n-a^{\\star\\star} B') + \\lambda( A' -a^\\star B' )$ attains the minimal ranks of $(A',B')$, showing that $\\rho(A',B') = (n-k_r, n-k_s) \n= (n-k'_r, n-k'_s)$.\n\n  \\item Finally, if $j \\ge k_s \\ge k_r$, then following the same line of thought we have that $(A' -a^{\\star} B') + \\lambda B'$ \nattains the minimal ranks of $(A',B')$, that is, $\\rho(A',B') = (n-k_r, n-j) = (n-k'_r, n-k'_s)$.\n\n\\end{enumerate}\n\n\\end{enumerate}\n\\end{proof}\n\n\nFor a singular $m \\times n$ canonical pencil $A + \\lambda B$ having no regular part, computing the minimal ranks is \nstraightforward, because $a_{i,j}$ and $b_{i,j}$ cannot be both nonzero for any given pair of indices $(i,j)$. Indeed, the \ncanonical block $L_k(\\lambda)$ related to a minimal index $k$ associated with the columns is the $k \\times (k+1)$ pencil of \nthe form\n\\begin{equation*}\nL_k(\\lambda) \\triangleq\n \\begin{pmatrix}\n    \\lambda & 1 & & & \\\\\n     & \\lambda & 1 & & \\\\\n     &  & \\ddots & \\ddots & \\\\\n     & &  & \\lambda & 1\n \\end{pmatrix}.\n\\end{equation*} \nAny $\\GL_2(\\mathbb{F})$ transformation applied to $L_k(\\lambda)$ yields some pencil $A' + \\lambda B'$ such that $\\rank \nA' = \\rank B' = k$. In other words, $L_k(\\lambda)$ has minimal ranks $(k,k)$. By the same argument, the canonical block $R_l(\\lambda)$ \nrelated to a minimal index $l$ associated with the rows, which is an $(l+1) \\times l$ pencil defined as $R_l(\\lambda) \\triangleq \nL^\\T_l(\\lambda)$, has minimal ranks $(l,l)$. The special case $k = 0$ (or $l = 0$) also adheres to that \nrule, as its minimal ranks are $(0,0)$.\nNow, adjoining blocks having these forms yields a singular pencil \n$L_{k_1}(\\lambda) \\oplus \\dots \\oplus L_{k_p}(\\lambda) \\oplus R_{l_1}(\\lambda) \\oplus \\dots \\oplus R_{l_q}(\\lambda)$ whose minimal \nranks are clearly the sum of the minimal ranks of the blocks. Note that this is true even for the zero minimal indices $k_1 = \\dots = \nk_{p'} = l_1 = \\dots = l_{q'} = 0$, since they correspond to $p'$ null columns and $q'$ null rows, and so the minimal ranks must be \nbounded by $\\min\\{m - q', n-p'\\}$. We have arrived at the following result. \n\n\\begin{lem}\n\\label{lem:rho-singular}\nLet $A + \\lambda B$ be a singular $m \\times n$ pencil having the form $A + \\lambda B = L_{k_1}(\\lambda) \\oplus \\dots \\oplus \nL_{k_p}(\\lambda) \\oplus R_{l_1}(\\lambda) \\oplus \\dots \\oplus \nR_{l_q}(\\lambda)$, where $k_1, \\ldots, k_p$ are the minimal indices associated with its columns (henceforth called minimal column \nindices) and $l_1,\\ldots,l_q$ are the minimal indices associated with its rows (minimal row indices). \nThen, \\begin{equation*}\n  \\rho(A,B) = (\\bar{s},\\bar{s}), \\quad \\text{where } \\bar{s} = k_1 + \\dots + k_p + l_1 + \\dots + l_q.\n\\end{equation*}\n\\end{lem}\n\nFor an arbitrary $m \\times n$ pencil $A + \\lambda B$, the block diagonal structure of its Kronecker canonical form allows a direct \ncombination of the previous results, yielding the main theorem of this section.\n\n\\begin{theo}\n\\label{theo:rho-canonical}\nLet $A + \\lambda B$ be an arbitrary $m \\times n$ pencil with Kronecker canonical form $S(\\lambda) \\oplus (A' + \\lambda B')$, where \n$S(\\lambda)$ is its singular part and $A' + \\lambda B'$ is regular. Suppose $S(\\lambda)$ has minimal column indices $k_1, \\ldots, k_p$ \nand minimal row indices $l_1, \\ldots, l_q$. Define $\\bar{s} = k_1 + \\dots + k_p + l_1 + \\dots + l_q$. Then, its minimal ranks are given \nby\n\\begin{equation}\n \\label{rho-canonical}\n  \\rho(A, B) = (r' + \\bar{s}, s' + \\bar{s}), \n\\end{equation} \nwhere $(r', s') = \\rho(A',B')$, whose components are given by \\Cref{lem:rho-regular}.\n\\end{theo}\n\nThe above result implies that both minimal ranks of an $n \\times n$ singular pencil must be strictly smaller than $n$. This is because \nthe sum of the minimal indices of its singular part (which equals $\\bar{s}$ in \\eqref{rho-canonical}) can never attain the largest \ndimension of that part. Hence, if an $n \\times n$ pencil has minimal ranks $(n,n)$ or $(n,n-1)$, then it is necessarily regular. \nIn particular, $n \\times n$ pencils with full minimal ranks can be characterized as follows.\n\n\\begin{corol}\n\\label{cor:full-min-rank}\nAn $n \\times n$ pencil $A + \\lambda B$ satisfies $\\rho(A,B) = (n,n)$ if and only if it is regular and its elementary divisors cannot be \nwritten as powers of linear forms.\n\\end{corol}\n\n\n\n\n\\subsection{Classification of $\\GL_{m,n,2}(\\mathbb{R})$-orbits for $m,n \\le 4$}\n\\label{sec:classif}\n\nUsing \\Cref{theo:rho-canonical}, a complete classification of all Kronecker canonical forms of $m \\times n$ pencils over \n$\\mathbb{R}$ is provided in \\Cref{tab:1,tab:2,tab:3,tab:4} for $m, n \\in \\{1,\\ldots,4\\}$. Each such form is associated with a \nfamily of $\\GL_{m,n,2}(\\mathbb{R})$-orbits. We denote canonical blocks whose elementary divisors are powers of second-order \nirreducible polynomials by\n\\begin{equation*}\n  Q_{2k}(a,b) = E_k \\kron \\begin{pmatrix} a & b \\\\ -b & a \\end{pmatrix} + J_2(0) \\kron E_k \\quad \\text{with} \\quad b \\neq 0,\n\\end{equation*}\nwhere $\\kron$ denotes the Kronecker product. Observe that, because we consider orbits of $\\GL_{m,n,2}(\\mathbb{R})$ action, we \ncan represent each family of orbits by a canonical form having no infinite elementary divisors (which can always be avoided by \nemploying a $\\GL_{2}(\\mathbb{R})$ transformation).\n\nIt can be checked that each described family with $m, n \\le 3$ corresponds to a single orbit,\\footnote{Atkinson \n\\cite{Atkinson1991} had already pointed out that, over an algebraically closed field $\\mathbb{F}$, there are only finitely many \n$\\GL_{3,n,2}(\\mathbb{F})$-orbits for any $n$. Thus, over $\\mathbb{R}$ this must be true of orbits whose elementary divisors are \npowers of linear forms.} except for $\\mathscrc{R}'_{3,2}$, which encompasses an infinite number of non-equivalent orbits. All families \nhaving dimensions $m = 3$ and $n = 4$ (or $m = 4$ and $n = 3$) also contain only one orbit each. These properties can be \nverified by inspecting the equivalent pencils of \\Cref{tab:equiv} shown ahead in \\Cref{sec:equiv}: for $\\min\\{m,n\\} \\le 3$, the only \nfamily whose given canonical form depends on a parameter is that of $\\mathscrc{R}'_{3,2}$.\nFor $m = n = 4$, infinitely many non-equivalent orbits are contained by each family in general. \nTo avoid redundancies, families of orbits having zero minimal indices are omitted in the tables, since each such family corresponds \nto some other one of lower dimensions. For instance for $m=n=4$, if a singular pencil $A + \\lambda B$ has minimal indices \n$k_1 = k_2 = l_1 = l_2 = 0$, then it can be expressed in the form $0 \\oplus (A' + \\lambda B')$, where both blocks in this \ndecomposition have size $2 \\times 2$. So, the canonical form of $A' + \\lambda B'$ can be inspected to determine the properties \nof $A + \\lambda B$. Similarly, not all combinations of canonical blocks are included for the singular part, because the shown \nproperties remain the same if we transpose these blocks. To exemplify, it can be checked that $L_1 \\oplus R_2$ and $R_1 \\oplus L_2$ \nhave the same dimensions, tensor rank, multilinear rank and minimal ranks, because the roles played by column and row minimal indices \nare essentially the same, up to a transposition.\n\nA family is denoted with the letter $\\mathscrc{R}$ or the letter $\\mathscrc{S}$ if it encompasses regular or singular pencils, \nrespectively. The subscript indices of each family indicate its minimal ranks, and primes are used to distinguish among otherwise \nidentically labeled families. The tensor rank of each canonical form was determined using Corollary 2.4.1 of Ja'Ja' \\cite{JaJa1979} \nand Theorem 4.6 of Sumi et al.~\\cite{Sumi2009}, which requires taking into account the minimal indices of the pencil and also its \nelementary divisors. The values given in the column ``multilinear rank'' were determined by inspection; for a definition see \n\\cite{deSilva2008}. Specifically, for an $m \\times n$ pencil $A + \\lambda B$ viewed as an $m \\times n \\times 2$ tensor $A \n\\otimes e_1 + B \\otimes e_2$, the multilinear rank is the triple $(r_1,r_2,r_3)$ satisfying\n\\begin{equation*}\n r_1 = \\rank \\begin{pmatrix} A & B \\end{pmatrix},     \\quad \n r_2 = \\rank \\begin{pmatrix} A^\\T & B^\\T \\end{pmatrix}, \\quad \n r_3 = \\dim \\sspan \\{A, B\\}.\n\\end{equation*} \nIt should be noted that, in the above equation, $\\sspan \\{A, B\\}$ denotes the subspace of $\\mathbb{R}^{m \\times n}$ spanned by \nthe matrices $A$ and $B$, whose dimension is at most two. Finally, in order to determine the minimal ranks (column labeled ``$\\rho$'') \nof each family, \\Cref{theo:rho-canonical} was applied.\n\n\nWe point out that another classification of pencil orbits is given by Pervouchine \n\\cite{Pervouchine2004}, but his study is concerned with closures of orbits and pencil bundles, not with tensor rank or minimal ranks. \nOur list is therefore a complement to the one he provides. Furthermore, the hierarchy of closures of pencil bundles he has \npresented bears no direct connection with the hierarchy of sets $\\mathcal{B}_{r,s}$ we present in \\Cref{sec:hierarchy}, which is \neasily determined by the numbers $r,s$ associated with each such set.\n\n\\begin{table}[t!]\n\\begin{center}\n\\def1.6{1.6}\n\\begin{tabular}{c | c |c|c|c|c}\n\\hline \n\\multirow{2}{*}{Family} & \\multirow{2}{*}{Canonical form} & $m \\times n$ & tensor & multilinear & \\multirow{2}{*}{$\\rho$} \\\\[-8pt]\n& & &   rank   &    rank    &   \\\\\n\\hline\n$\\mathscrc{R}_{1,0}$ & $a + \\lambda\\, E_1$  & $1 \\times 1$ & 1  & $(1,1,1)$ & $(1,0)$ \\\\[2pt] \n\\hline \n$\\mathscrc{S}_{1,1}$ & $L_1(\\lambda)$ & $1 \\times 2$ & 2 & $(1,2,2)$ & $(1,1)$ \\\\[2pt] \n\\hline\n\\end{tabular}\n\\end{center}\n \\caption{Families of canonical forms of $1 \\times 1$ and $1 \\times 2$ real pencils having no zero minimal indices, with $a_i \n\\neq a_j$ for $i \\neq j$.}\n \\label{tab:1}\n\\end{table}\n\n\\begin{table}[t!]\n\\begin{center}\n\\def1.6{1.6}\n\\begin{tabular}{c | c |c|c|c|c}\n\\hline \n\\multirow{2}{*}{Family} & \\multirow{2}{*}{Canonical form} & $m \\times n$ & tensor & multilinear & \\multirow{2}{*}{$\\rho$} \\\\[-8pt] \n& & &   rank   &    rank    &   \\\\\n\\hline\n$\\mathscrc{R}_{1,1}$ & $a_1 \\oplus a_2 + \\lambda\\, E_2$ & $2 \\times 2$ & 2  & $(2,2,2)$ & $(1,1)$ \\\\ \n$\\mathscrc{R}_{2,0}$ & $a \\oplus a + \\lambda\\, E_2$ & $2 \\times 2$  & 2  & $(2,2,1)$ & $(2,0)$ \\\\[2pt]\n\\hline \n$\\mathscrc{R}_{2,1}$ & $J_2(a) + \\lambda\\, E_2$ & $2 \\times 2$  & 3  & $(2,2,2)$ & $(2,1)$  \\\\[2pt] \n\\hline \n$\\mathscrc{R}_{2,2}$ & $Q_2(a,b) + \\lambda\\, E_2$ & $2 \\times 2$  & 3  & $(2,2,2)$ & $(2,2)$  \\\\[2pt] \n\\hline\n$\\mathscrc{S}_{2,2}$ & $L_2(\\lambda)$ & $2 \\times 3$  & 3  & $(2,3,2)$ & $(2,2)$  \\\\[2pt] \n$\\mathscrc{S}_{2,1}$ & $L_1(\\lambda) \\oplus (a + \\lambda\\, E_1)$ & $2 \\times 3$  & 3  & $(2,3,2)$ & $(2,1)$  \\\\[2pt] \n\\hline\n$\\mathscrc{S}'_{2,2}$ & $L_1(\\lambda) \\oplus L_1(\\lambda)$ & $2 \\times 4$  & 4  & $(2,4,2)$ & $(2,2)$  \\\\[2pt] \n\\hline\n\\end{tabular}\n\\end{center}\n \\caption{Families of canonical forms of $2 \\times 2$ and $2 \\times 3$ real pencils having no zero minimal indices, with $a_i \n\\neq a_j$ for $i \\neq j$ and $b \\neq 0$.}\n \\label{tab:2}\n\\end{table}\n\n\n\\begin{table}[t!]\n\\begin{center}\n\\def1.6{1.6}\n\\begin{tabular}{c | c |c|c|c|c}\n\\hline \n\\multirow{2}{*}{Family} & \\multirow{2}{*}{Canonical form} & $m \\times n$ & tensor & multilinear & \\multirow{2}{*}{$\\rho$} \\\\[-8pt] \n& & &   rank   &    rank    &   \\\\\n\\hline \n$\\mathscrc{S}''_{2,2}$ & $L_1(\\lambda) \\oplus R_1(\\lambda)$ & $3 \\times 3$ & 4  & $(3,3,2)$ & $(2,2)$ \\\\[2pt] \n\\hline \n$\\mathscrc{R}'_{2,2}$ & $a_1 \\oplus a_2 \\oplus a_3 + \\lambda\\, E_3$ & $3 \\times 3$  & 3  & $(3,3,2)$ & $(2,2)$ \\\\ \n$\\mathscrc{R}'_{2,1}$ & $a_1 \\oplus a_2 \\oplus a_2 + \\lambda\\, E_3$ & $3 \\times 3$  & 3  & $(3,3,2)$ & $(2,1)$  \\\\ \n$\\mathscrc{R}_{3,0}$ & $a \\oplus a \\oplus a + \\lambda\\, E_3$ & $3 \\times 3$  & 3  & $(3,3,1)$ & $(3,0)$  \\\\[2pt] \n\\hline \n$\\mathscrc{R}_{3,2}$ & $J_3(a) + \\lambda\\, E_3$ & $3 \\times 3$  & 4  & $(3,3,2)$ & $(3,2)$  \\\\ \n$\\mathscrc{R}_{3,1}$ & $a \\oplus J_2(a) + \\lambda\\, E_3$ & $3 \\times 3$  & 4  & $(3,3,2)$ & $(3,1)$  \\\\ \n$\\mathscrc{R}''_{2,2}$ & $a_1 \\oplus J_2(a_2) + \\lambda\\, E_3$ & $3 \\times 3$  & 4  & $(3,3,2)$ & $(2,2)$  \\\\[2pt]\n\\hline \n$\\mathscrc{R}'_{3,2}$ & $a \\oplus Q_2(c,d) + \\lambda\\, E_3$ & $3 \\times 3$  &  4  & $(3,3,2)$ & $(3,2)$ \\\\[2pt] \n\\hline\n$\\mathscrc{S}'''_{2,2}$ & $L_1(\\lambda) \\oplus (a_1 \\oplus a_2 + \\lambda\\, E_2)$ & $3 \\times 4$  &  4  & $(3,4,2)$ & $(2,2)$ \\\\ \n$\\mathscrc{S}_{3,1}$ & $L_1(\\lambda) \\oplus (a \\oplus a + \\lambda\\, E_2)$ & $3 \\times 4$  &  4  & $(3,4,2)$ & $(3,1)$ \\\\ \n$\\mathscrc{S}_{3,2}$ & $L_1(\\lambda) \\oplus (J_2(a) + \\lambda\\, E_2)$ & $3 \\times 4$  &  5  & $(3,4,2)$ & $(3,2)$ \\\\ \n$\\mathscrc{S}_{3,3}$ & $L_1(\\lambda) \\oplus (Q_2(a,b) + \\lambda\\, E_2)$ & $3 \\times 4$  &  5  & $(3,4,2)$ & $(3,3)$ \\\\\n$\\mathscrc{S}'_{3,2}$ & $L_2(\\lambda) \\oplus (a + \\lambda\\, E_1)$ & $3 \\times 4$  &  4  & $(3,4,2)$ & $(3,2)$ \\\\ \n$\\mathscrc{S}'_{3,3}$ & $L_3(\\lambda)$ & $3 \\times 4$  &  4  & $(3,4,2)$ & $(3,3)$ \\\\[2pt] \n\\hline\n\\end{tabular}\n\\end{center}\n \\caption{Families of canonical forms of $3 \\times 3$ and $3 \\times 4$ real pencils having no zero minimal indices, with $a_i \n\\neq a_j$ for $i \\neq j$, $b \\neq 0$ and $d \\neq 0$.}\n \\label{tab:3}\n\\end{table}\n\n\n\\begin{table}[t!]\n\\begin{center}\n\\def1.6{1.6}\n\\begin{tabular}{c | c |c|c|c|c}\n\\hline \n\\multirow{2}{*}{Family} & \\multirow{2}{*}{Canonical form} & $m \\times n$ & tensor & multilinear & \\multirow{2}{*}{$\\rho$} \\\\[-8pt]\n& & &  rank   &    rank    &   \\\\\n\\hline \n$\\mathscrc{S}''_{3,2}$ & $L_1(\\lambda) \\oplus R_1(\\lambda) \\oplus (a + \\lambda\\, E_1)$ & $4 \\times 4$ & 5  & $(4,4,2)$ & $(3,2)$  \n\\\\[2pt] \n$\\mathscrc{S}''_{3,3}$ & $L_2(\\lambda) \\oplus R_1(\\lambda) $ & $4 \\times 4$ & 5  & $(4,4,2)$ & $(3,3)$  \\\\[2pt] \n\\hline\n$\\mathscrc{R}_{4,0}$ & $a \\oplus a \\oplus a \\oplus a + \\lambda\\, E_4$ & $4 \\times 4$ & 4  & $(4,4,1)$ & $(4,0)$  \\\\ \n$\\mathscrc{R}'''_{2,2}$ & $a_1 \\oplus a_1 \\oplus a_2 \\oplus a_2 + \\lambda\\, E_4$ & $4 \\times 4$ & 4 & $(4,4,2)$ & $(2,2)$  \\\\ \n$\\mathscrc{R}'_{3,1}$ & $a_1 \\oplus a_2 \\oplus a_2 \\oplus a_2 + \\lambda\\, E_4$ & $4 \\times 4$ & 4  &  $(4,4,2)$ & $(3,1)$  \\\\ \n$\\mathscrc{R}''_{3,2}$ & $a_1 \\oplus a_2 \\oplus a_3 \\oplus a_3 + \\lambda\\, E_4$ & $4 \\times 4$ & 4 &   $(4,4,2)$ & $(3,2)$  \\\\ \n$\\mathscrc{R}_{3,3}$ & $a_1 \\oplus a_2 \\oplus a_3 \\oplus a_4 + \\lambda\\, E_4$ & $4 \\times 4$ & 4  & $(4,4,2)$ & $(3,3)$  \\\\[2pt] \n\\hline\n$\\mathscrc{R}'''_{3,2}$ & $a_1 \\oplus a_2 \\oplus J_2(a_2) + \\lambda\\, E_4$ & $4 \\times 4$ & 5  & $(4,4,2)$ & $(3,2)$  \\\\ \n$\\mathscrc{R}''''_{3,2}$ & $a_1 \\oplus a_1 \\oplus J_2(a_2) + \\lambda\\, E_4$ & $4 \\times 4$ & 5  & $(4,4,2)$ & $(3,2)$  \\\\\n$\\mathscrc{R}_{4,1}$ & $a \\oplus a \\oplus J_2(a) + \\lambda\\, E_4$ & $4 \\times 4$ & 5  & $(4,4,2)$ & $(4,1)$  \\\\ \n$\\mathscrc{R}'_{3,3}$ & $J_2(a_1) \\oplus J_2(a_2) + \\lambda\\, E_4$ & $4 \\times 4$ & 5  & $(4,4,2)$ & $(3,3)$  \\\\ \n$\\mathscrc{R}''_{3,3}$ & $a_1 \\oplus a_2 \\oplus J_2(a_3) + \\lambda\\, E_4$ & $4 \\times 4$ & 5  & $(4,4,2)$ & $(3,3)$  \\\\ \n$\\mathscrc{R}'''_{3,3}$ & $a_1 \\oplus J_3(a_2) + \\lambda\\, E_4$ & $4 \\times 4$ & 5  & $(4,4,2)$ & $(3,3)$  \\\\ \n$\\mathscrc{R}_{4,2}$ & $a \\oplus J_3(a) + \\lambda\\, E_4$ & $4 \\times 4$ & 5  & $(4,4,2)$ & $(4,2)$  \\\\ \n$\\mathscrc{R}'_{4,2}$ & $J_2(a) \\oplus J_2(a) + \\lambda\\, E_4$ & $4 \\times 4$ & 6  & $(4,4,2)$ & $(4,2)$  \\\\ \n$\\mathscrc{R}_{4,3}$ & $J_4(a) + \\lambda\\, E_4$ & $4 \\times 4$ & 5  & $(4,4,2)$ & $(4,3)$  \\\\[2pt] \n\\hline\n$\\mathscrc{R}''''_{3,3}$ & $a_1 \\oplus a_2 \\oplus Q_2(c,d) + \\lambda\\, E_4$ & $4 \\times 4$ &  5  & $(4,4,2)$ & $(3,3)$  \\\\ \n$\\mathscrc{R}''_{4,2}$ & $a \\oplus a \\oplus Q_2(c,d) + \\lambda\\, E_4$ & $4 \\times 4$ &  5  & $(4,4,2)$ & $(4,2)$  \\\\\n$\\mathscrc{R}'_{4,3}$ & $J_2(a) \\oplus Q_2(c,d) + \\lambda\\, E_4$ & $4 \\times 4$ &  5  & $(4,4,2)$ & $(4,3)$  \\\\[2pt] \n\\hline\n$\\mathscrc{R}_{4,4}$ & $Q_2(a,b) \\oplus Q_2(c,d) + \\lambda\\, E_4$ & $4 \\times 4$ &  5  & $(4,4,2)$ & $(4,4)$ \\\\ \n$\\mathscrc{R}'_{4,4}$ & $Q_4(a,b) + \\lambda\\, E_4$ & $4 \\times 4$ &  5  & $(4,4,2)$ & $(4,4)$ \\\\ \n$\\mathscrc{R}''_{4,4}$ & $Q_2(a,b) \\oplus Q_2(a,b) + \\lambda\\, E_4$ & $4 \\times 4$ &  6  & $(4,4,2)$ & $(4,4)$ \\\\[2pt] \n\\hline\n\\end{tabular}\n\\end{center}\n \\caption{Families of canonical forms of $4 \\times 4$ real pencils having no zero minimal indices, with $a_i \n\\neq a_j$ for $i \\neq j$, $b \\neq 0$ and $d \\neq 0$.}\n \\label{tab:4}\n\\end{table}\n\n\\subsection{Minimal ranks of matrix polynomials}\n\\label{sec:matrix-poly}\n\n \nOne can generalize \\Cref{def:min-ranks} to matrix polynomials of finite degree as follows. The minimal ranks of a \ndegree-$(d-1)$ polynomial $P(\\lambda) = \\sum_{k=1}^{d} \\lambda^{k-1} A_k$ should correspond to the minimal values $r_1 \\ge \\dots \\ge \nr_d$ such that we can find \na transformation $T \\in \\GL_d(\\mathbb{F})$ for which $(E,E,T) \\cdot P(\\lambda) = \\sum_{k=1}^{d} \\lambda^{k-1} A'_k$ where $\\rank A'_k \n= r_k$ for $k=1,\\ldots,d$. One can thus introduce the \\emph{rank-minimizing subspaces} $\\mathcal{T}_1, \\ldots, \\mathcal{T}_l$ of \n$\\mathbb{F}^d$ with respect to $P(\\lambda)$, where $1 \\le l \\le d$, which are inductively defined in the following manner. First, \nlet $\\mathcal{T}_1$ be the subspace of $\\mathbb{F}^d$ spanned by all solutions of \n\\begin{equation*}\n   \\min_{ (t_{1}, \\ldots, t_{d}) \\neq 0 } \\rank \\left(  \\sum_{k=1}^{d} t_{k} A_k \\right) = \\bar{r}_1.\n\\end{equation*} \nIf $\\dim \\mathcal{T}_1 = d$, then $\\mathcal{T}_1$ is the only rank-minimizing subspace, that is, $l=1$. Otherwise, we define next \n$\\mathcal{T}_2$ as the subspace of $\\mathbb{F}^d$ spanned by all solutions of \n\\begin{equation*}\n   \\min_{ (t_{1}, \\ldots, t_{d}) \\notin \\mathcal{T}_1 } \\rank \\left(  \\sum_{k=1}^{d} t_{k} A_k \\right) = \\bar{r}_2.\n\\end{equation*} \nIf $\\dim \\mathcal{T}_1 + \\dim \\mathcal{T}_2 = d$, then we have $\\mathcal{T}_1 \\oplus \\mathcal{T}_2 = \\mathbb{F}^2$ and $l=2$. \nOtherwise, one continues in this fashion until $\\dim \\mathcal{T}_1 + \\dots + \\dim \\mathcal{T}_l = d$ for some $l$, which must happen \nafter finitely many steps. So, each $\\mathcal{T}_p$ is defined as the subspace spanned by the solutions of \n\\begin{equation}\n \\label{ms-Tp}\n   \\min_{ (t_{1}, \\ldots, t_{d}) \\notin \\bar{\\mathcal{T}} } \n     \\rank \\left(  \\sum_{k=1}^{d} t_{k} A_k \\right) = \\bar{r}_p,\n     \\quad \\text{where} \\quad\n     \\bar{\\mathcal{T}} = \n     \\begin{cases}\n       \\{ 0 \\}, & p = 1, \\\\\n       \\mathcal{T}_1 \\oplus \\dots \\oplus \\mathcal{T}_{p-1}, & 1 < p \\le l.\n     \\end{cases}\n\\end{equation} \nIt is clear that $\\mathcal{T}_1 \\oplus \\dots \\oplus \\mathcal{T}_l = \\mathbb{F}^d$. The minimal ranks are then associated with this \ndecomposition, as defined below.\n\n\\begin{defn}\n\\label{def:min-ranks-poly}\nLet $ P(\\lambda) = \\sum_{k=1}^{d} \\lambda^{k-1} A_k$ be an $m \\times n$ matrix polynomial over $\\mathbb{F}$ of degree (at most) $d-1$, \nand let $\\mathcal{T}_1, \\ldots, \\mathcal{T}_l$ be the rank-minimizing subspaces of $\\mathbb{F}^d$ associated with $P(\\lambda)$, with \n$\\dim \\mathcal{T}_p = d_p$.\nThe minimal ranks of $P(\\lambda)$, denoted by $\\rho(P)$, are defined as the components of the $d$-tuple \n\\begin{equation*}\n \\rho(P) = (r_1,\\ldots,r_d) \\triangleq ( \n            \\underbrace{\\bar{r}_l,\\ldots,\\bar{r}_l}_{d_l \\text{ times}},\n            \\underbrace{\\bar{r}_{l-1},\\ldots,\\bar{r}_{l-1}}_{d_{l-1} \\text{ times}},\n            \\ldots,\n            \\underbrace{\\bar{r}_2,\\ldots,\\bar{r}_2}_{d_2 \\text{ times}} \n            \\underbrace{\\bar{r}_1,\\ldots,\\bar{r}_1}_{d_1 \\text{ times}} \n            ),\n\\end{equation*}\nwhere the numbers $\\bar{r}_p$ are given by \\eqref{ms-Tp} and satisfy $\\bar{r}_l > \\bar{r}_{l-1} > \\dots > \\bar{r}_1$.\nWe say that the components of $\\rho(P)$ which equal $\\bar{r}_p$ are associated with $\\mathcal{T}_p$.\n\\end{defn}\n\nFor $d=2$, \\Cref{def:min-ranks-poly} is equivalent to \\Cref{def:min-ranks}. In particular, $\\rho(A,B) = (r,s)$ satisfies $r = s$ if \nand only if $\\mathbb{F}^2$ has a single associated rank-minimizing subspace $\\mathcal{T}_1 = \\mathbb{F}^2$ with respect to $A + \n\\lambda B$. An analogue of \\Cref{lem:GL2} also clearly holds for matrix polynomials.\n\nLet us introduce \n\\begin{equation*}\n\\label{Bset-poly}\n \\mathcal{B}_{r_1,\\ldots,r_d} \\triangleq \n                   \\left\\{ P(\\lambda) \n                   \\; \\Bigg| \\;\n                   \\exists \\, \\sum_{k=1}^{d} \\lambda^{k-1} A'_k \\in \\mathcal{O}(P) \n                   \\mbox{ such that }\n                   \\rank A_k' \\le r_k \\mbox{ for } k=1,\\ldots,d \\right\\},\n\\end{equation*}\nwhere $\\mathcal{O}(P)$ stands for the $\\GL_{m,n,d}$-orbit of $P(\\lambda)$.\n$\\mathcal{B}_{r_1,\\ldots,r_d}$ is clearly invariant with respect to a permutation of $r_1,\\ldots,r_d$, and thus we shall assume \n$r_1 \\ge \\dots \\ge r_d$. \nAssuming $\\rho(P) = (r_1,\\ldots,r_d)$, the construction of the rank-minimizing subspaces shows there is a transformation $T \\in \n\\GL_d(\\mathbb{F})$ yielding $(E,E,T) \\cdot P(\\lambda) = \\sum_{k=1}^{d} \\lambda^{k-1} A'_k$ such that $\\rank A'_k = r_k$,\nand thus by definition $P(\\lambda) \\in \\mathcal{B}_{r_1,\\ldots,r_d}$. \nNow, since the rows of any other $T' \\in \\GL_d(\\mathbb{F})$ must span $\\mathbb{F}^d$, it is easy to see \nthat $P(\\lambda)$ cannot belong to any  $\\mathcal{B}_{r_1,\\ldots,r_{q-1},r'_q,r_{q+1},\\ldots,r_d}$  such that $r_q > r'_q \\ge \nr_{q+1}$. Indeed, this would contradict the construction of the subspaces $\\mathcal{T}_p$ as being spanned by \\emph{all} solutions of \nthe rank minimization problem \\eqref{ms-Tp}.\nIn fact, if $s_1 \\ge \\dots \\ge s_d$ and $P(\\lambda) \\in \\mathcal{B}_{s_1,\\ldots,s_d}$, then we must have $s_q \\ge r_q$ for all $q \\in \n\\{1,\\ldots,d\\}$, because the assumption $r_q > s_q \\ge s_{q+1} \\ge \\dots \\ge s_d$ is inconsistent with the above construction \nof the rank-minimizing subspaces $\\mathcal{T}_1,\\ldots,\\mathcal{T}_l$ of $\\mathbb{F}^d$ with respect to $P(\\lambda)$. This is the \ncentral argument of the following generalization of \\Cref{prop:r-s}.\n\n\n\\begin{prop}\nLet $ P(\\lambda) = \\sum_{k=1}^{d} \\lambda^{k-1} A_k$ be an $m \\times n$ matrix polynomial over $\\mathbb{F}$.\nIf $\\rho(P) = (r_1,\\ldots,r_d)$ and $s_1 \\ge \\dots \\ge s_d$, then $P(\\lambda) \\in \\mathcal{B}_{s_1,\\ldots,s_d}$ if and only if \n$s_k \\ge r_k$ for all $k \\in \\{1,\\ldots,d\\}$.\n\\end{prop}\n\n\n\nIn view of these extensions, it makes sense to consider the hierarchy of $m \\times n$ matrix polynomials of any finite degree \nin terms of their minimal ranks, leading to a diagram such as that of \\Cref{fig:hierarchy}.\n \n\n\\subsection{Decomposing third-order tensors into matrix-vector tensor products of minimal ranks}\n\\label{sec:MVD}\n\n\nThe connection between $m \\times n$ matrix pencils and $m \\times n \\times 2$ tensors has been exploited time and \nagain to derive many results, such as the characterization of $\\GL_{m,n,2}(\\mathbb{R})$-orbits in terms of their tensor ranks carried \nout by Ja'Ja' and Sumi et al.~\\cite{JaJa1979, Sumi2009}. This correspondence is also well-suited to study block-term \ndecompositions of $m \\times n \\times 2$ tensors composed by matrix-vector products, as we can directly associate the ranks of these \nblocks with the ranks of the matrices constituting the pencil.\n\nMore generally, a natural connection exists between a block-term decomposition of an $m \\times n \\times d$ tensor and $m \\times n$ \npolynomial of degree $d-1$. Given any third order tensor $X \\in \\mathbb{F}^{m} \\otimes \\mathbb{F}^{n} \\otimes \\mathbb{F}^{d}$, it can \nalways be written in the form\n\\begin{equation*}\n  X = \\sum_{k=1}^d \\left( \\sum_{j=1}^{r_k} u^{(k)}_j \\otimes v^{(k)}_j \\right) \\otimes w_k,\n\\end{equation*} \nwhere the vectors $w_1, \\ldots, w_d$ are linearly independent. Choosing coordinates for these spaces, this expression can be \nidentified with\n\\begin{equation}\n \\label{BTD}\n  X = \\sum_{k=1}^d A_k \\otimes w_k \\quad \\in \\mathbb{F}^{m \\times n \\times d},\n\\end{equation} \nwhere $A_k$ is a matrix of rank at most $r_k$. By suitably defining an isomorphism $\\mathbb{F}^d \\simeq \n\\mathbb{F}_{d-1}[\\lambda]$, where $\\mathbb{F}_{d-1}[\\lambda]$ denotes the space of degree-$(d-1)$ polynomials, the tensor in \n\\eqref{BTD} can be associated with the matrix polynomial\n\\begin{equation*}\n  P(\\lambda) = \\sum_{k=1}^d \\lambda^{k-1} A_k.\n\\end{equation*}\n\n\nFrom this link, it becomes evident that the minimal ranks of $P(\\lambda)$ characterize the intrinsic complexity of the tensor $X$ in \n\\eqref{BTD} in terms of the ranks of the matrices appearing in the sum. It thus quantifies the dimensions of the smooth manifolds \nwhose join set contains $X$; see the recent work by Breiding,  and Vannieuwenhoven \\cite{Breiding2018} for a discussion on this \ninterpretation of \\eqref{BTD}. In the case of a matrix pencil $P(\\lambda) = A + \\lambda B$ ($d=2$), our results in \n\\Cref{sec:Kronecker} allow one to compute $\\rho(A,B)$ from the Kronecker canonical form of the pencil. Similar characterizations for \nhigher values of $d$ would be surely valuable for the study of the BTD and its properties.\n\n\n\n\\section{A pencil may have no best approximation with strictly lower minimal ranks}\n\\label{sec:ill-posed}\n\nThis section investigates the existence of optimal approximations of a pencil $A + \\lambda B \\in \\mathcal{P}_{m,n}(\\mathbb{F})$ on \n$\\mathcal{B}_{r,s}$, for a given pair $(r,s)$ satisfying $r \\ge s$, for $\\mathbb{F} = \\mathbb{R}$ or $\\mathbb{F} = \n\\mathbb{C}$. \nDefining such approximations requires a topology for $\\mathcal{P}_{m,n}(\\mathbb{F})$. We shall pick the norm topology, which is \nthe same regardless of the chosen norm, since $\\mathcal{P}_{m,n}(\\mathbb{F})$ has a finite dimension. It can be introduced by \nconsidering the inner product\n\\begin{equation}\n \\label{inner}\n   \\langle A + \\lambda B, C + \\lambda D \\rangle \\triangleq \\langle A , C  \\rangle + \\langle B, D \\rangle, \n\\end{equation}\nwhere the inner product appearing in the right-hand side is the standard Euclidian one given by\n\\begin{equation*}\n   \\langle A , C \\rangle \\triangleq \\trace A^* C, \n\\end{equation*}\nwith $A^*$ denoting the adjoint of $A$.  This leads to the Euclidian norm\n\\begin{equation*}\n  \\| A + \\lambda B \\| \\triangleq \\sqrt{ \\langle A + \\lambda B , A + \\lambda B \\rangle  }\n                              = \\sqrt{ \\|A\\|^2 + \\|B\\|^2},\n\\end{equation*}\nwhich induces the topology. In the above expression, $\\|A\\| \\triangleq \\sqrt{\\langle A, A \\rangle}$ is the Frobenius norm of $A$.\nWith this definition, the approximation problem can be formulated as\n\\begin{equation}\n \\label{best-app}\n  \\inf_{ A' + \\lambda B' \\in \\mathcal{B}_{r,s} }  \\| A + \\lambda B - (A' + \\lambda B' ) \\|.\n\\end{equation}\nWe are thus interested in determining whether this infimum is attained for a given pencil $A + \\lambda B$ and some choice of ranks $(r,s)$. \nEvidently, this question is only of interest at all when $A + \\lambda B \\not\\in \\mathcal{B}_{r,s}$. To check whether this holds, we \nshall rely on the concept of minimal ranks and its associated results developed in \\Cref{sec:pencils}.\n\nIn view of the isomorphism between $\\mathcal{P}_{m,n}(\\mathbb{F})$ and $\\mathbb{F}^{m \\times n \\times 2}$ discussed in \\Cref{sec:MVD}, \nthe results of the present section apply also to the approximation of tensors from $\\mathbb{F}^{m \\times n \\times 2}$ by a sum \nof two matrix-vector tensor products, in the tensor norm topology.\nTo define this topology, one can consider the Frobenius norm \n\\begin{equation*}\n  \\| A \\otimes e_1 + B \\otimes e_2 \\| \\triangleq \\sqrt{\\langle A \\otimes e_1 + B \\otimes e_2 , A \\otimes e_1 + B \\otimes e_2 \\rangle  }\n\\end{equation*}\nwhere the scalar product is defined for rank-one tensors as\n\\begin{equation*}\n  \\langle u \\otimes v \\otimes w, \\, x \\otimes y \\otimes z  \\rangle \\triangleq \\langle u, x \\rangle \\cdot \\langle v, y \\rangle \\cdot \\langle w, z \\rangle\n\\end{equation*}\nand extends bi-linearly to tensors of arbitrary rank. Therefore, \n$$ \\| A \\otimes e_1 + B \\otimes e_2 \\| = \\sqrt{\\|A\\|^2 + \\|B\\|^2} = \\| A + \\lambda B \\|.$$\n\n\n\\subsection{A template for ill-posed pencil approximation problems}\n\\label{sec:ill-posed-examples}\n\nExamples of tensors having no best rank-$r$ approximation in the norm topology have been known for quite a while; see \n\\cite{Comon2014, deSilva2008} and references therein. De Lathauwer \\cite{deLathauwer2008c} has employed the same kind of \nconstruction to provide an example of a tensor having no best approximate block-term decomposition constituted by two blocks of \nmultilinear rank $(2,2,2)$. In the next proposition, we resort to a similar expedient to derive a template of ill-posed instances \nof problem \\eqref{best-app} of a certain kind.\n\n\\begin{prop}\n\\label{prop:exist-PL}\n Let $A, B$ be $m \\times s$ matrices and $C, D$ be $n \\times s$ matrices and suppose \n\\begin{equation}\n\\label{cond-ex}\n\\min\\left\\{ \\, \\rank \\begin{pmatrix} A & B \\end{pmatrix}, \\, \n      \\rank \\begin{pmatrix} C & D \\end{pmatrix} \\,\\right\\} > r =  \\frac{3s}{2} .  \n\\end{equation} \n Then, the pencil \n \\begin{equation}\n   P(\\lambda) = \\left( A C^\\T + B D^\\T \\right) + \\lambda \\, B C^\\T \\quad \\in \\mathcal{P}_{m,n}(\\mathbb{F})\n \\end{equation}\nhas no best approximation in $\\mathcal{B}_{s,s}$.\n\\end{prop}\n\n\\begin{proof}\nFirst, let us show that for any transformation $T \\in \\GL_2(\\mathbb{F})$, at least one of the matrices $t_{11} (A C^\\T + B D^\\T) + \nt_{12} B C^\\T$ and $t_{21} (A C^\\T + B D^\\T) + t_{22} B C^\\T$ has rank strictly larger than $s$. \nNote that we can write  $t_{i1} (A C^\\T + B D^\\T) + t_{i2} B C^\\T = F_i \\begin{pmatrix} C & D \\end{pmatrix}^\\T$, where\n$$ F_i = \\begin{pmatrix} A & B \\end{pmatrix} \\begin{pmatrix} t_{i1} E_s & 0 \\\\ t_{i2} E_s & t_{i1} E_s \\end{pmatrix}. $$\nWe have $\\rank F_i = \\rank \\begin{pmatrix} A & B \\end{pmatrix} > r$ for $t_{i1} \\neq 0$, and so Sylvester's inequality implies that \nthe product $F_i \\begin{pmatrix} C & D \\end{pmatrix}^\\T$ has rank strictly greater than $2(r - s) = s$.\nBecause $t_{11}$ and $t_{21}$ obviously cannot be both zero, the statement is true. Hence, we \nconclude that $P(\\lambda) \\notin \\mathcal{B}_{s,s}$.\nNext, let\n \\begin{align}\n \\label{Pn}\nP_n(\\lambda) \\triangleq & \\ n \n  \\left[ \\left( B + \\frac{1}{n} A \\right)\\left( C + \\frac{1}{n} D \\right)^\\T  \\right] \n  \\left( 1 + \\frac{1}{n} \\lambda \\right) - n \\left( B C^\\T \\right)  \\\\\n    = & \\ P(\\lambda) + \\frac{1}{n}\\left[  \\left( A C^\\T \\right) \\lambda +\n                 \\left( B D^\\T \\right) \\lambda +\n                 \\left( A D^\\T \\right) \\right] \n                 + \\frac{1}{n^2} \\left( A D^\\T \\right) \\lambda.  \n \\label{Pn2}\n \\end{align}\nBy construction, $P_n(\\lambda) \\in \\mathcal{B}_{s,s}$, while \\eqref{Pn2} reveals that $P_n(\\lambda) \\rightarrow P(\\lambda) $ as \n$n \\rightarrow \\infty$. Hence, since $P(\\lambda) \\notin \\mathcal{B}_{s,s}$, it holds that \n\\begin{equation*}\n \\inf_{A' + \\lambda B' \\in \\mathcal{B}_{s,s}} \\, \n    \\| P(\\lambda) - (A' + \\lambda B') \\| = 0\n\\end{equation*} \nis not attained.\n\\end{proof}\n\n\\begin{rem}\n The condition \\eqref{cond-ex} is tight in the sense that one can find matrices $A$, $B$, $C$ and $D$ satisfying $\\rank \n\\begin{pmatrix} A & B\\end{pmatrix} \\le r$ and $\\rank \\begin{pmatrix} C & D \\end{pmatrix} \\le r$ such that \n $P(\\lambda) = \\left( A C^\\T + B D^\\T \\right) + \\lambda \\, B C^\\T \\in \\mathcal{B}_{s,s}$. Take, for instance, $m, n \\ge 6$, $s=4$ and\n\\begin{align*}\n   A = \\begin{pmatrix} a_1 & a_2 & a_3 & a_4 \\end{pmatrix}, \\quad & \\quad\n   B = \\begin{pmatrix} a_1 & a_2 & b_3 & b_4 \\end{pmatrix}, \\\\\n   C = \\begin{pmatrix} c_1 & c_2 & c_3 & c_4 \\end{pmatrix}, \\quad & \\quad\n   D = \\begin{pmatrix} d_1 & d_2 & c_3 & c_4 \\end{pmatrix},\n\\end{align*}\nwhere $a_1, a_2, a_3, a_4, b_3, b_4$ are linearly independent, and the same applies to $c_1, c_2, c_3, c_4, d_1, d_2$. We have then\n$\\rank \\begin{pmatrix} A & B \\end{pmatrix} = \\rank \\begin{pmatrix} C & D \\end{pmatrix} = 6 = \\frac{3s}{2} = r$. Choosing $t_{i1} = 1$ \nand $t_{i2} = -1$ yields $F_i \\begin{pmatrix} C & D \\end{pmatrix}^\\T  = (a_3 - b_3) c_3^\\T + (a_4 - b_4) c_4^\\T + B D^\\T $, which \nclearly cannot have rank larger than $s = 4$.\n\\end{rem}\n\n\n\n\\subsection{Ill-posedness over a positive-volume set of real pencils}\n\\label{sec:positive-volume}\n\nIn this section, we will prove that no (regular) $2k \\times 2k$ pencil having only complex-valued eigenvalues admits a best \napproximation in $\\mathcal{B}_{2k-1,2k-1}$ in the norm topology, for any positive integer $k$. The set containing all such pencils is \ndefined as \n\\begin{align}\n \\label{C-set}\n  \\mathcal{C} \\triangleq & \\ \\{ A + \\lambda B \\in \\mathcal{P}_{2k,2k}(\\mathbb{R}) \\ | \\ \\mathcal{O}(A,B) \\text{ contains } Q + \\lambda \nE, \n   \\text{ where } Q  \\text{ has no real eigenvalues }\\} \\\\\n   = & \\ \\{ A + \\lambda B \\in \\mathcal{P}_{2k,2k}(\\mathbb{R}) \\ | \\ \\rho(A,B) = (2k,2k)\\},  \\nonumber\n\\end{align} \nwhere the equality is due to \\Cref{cor:full-min-rank}. \nFor instance, in the case $k=2$ this set is constituted by all orbits of the families $\\mathscrc{R}_{4,4}$, $\\mathscrc{R}'_{4,4}$ and \n$\\mathscrc{R}''_{4,4}$ of \\Cref{tab:4}. We start by showing $\\mathcal{C}$ is open, and therefore has positive volume (since \nit is always nonempty).\n\n\\begin{lem}\n\\label{lem:C-open}\nThe set $\\mathcal{C} \\subset \\mathcal{P}_{2k,2k}(\\mathbb{R})$ defined by \\eqref{C-set} is open in the norm topology.\n\\end{lem}\n\n\\begin{proof}\nTake an arbitrary pencil $A + \\lambda B \\in \\mathcal{C}$. By definition, it can be written as $(P,U,T) \\cdot (Q + \\lambda E)$,\nwhere $(P,U,T) \\in \\GL_{2k,2k,2}(\\mathbb{R})$ and \n$Q = Q_{2 k_1}(a_1,b_1) \\oplus Q_{2 k_2}(a_2,b_2) \\oplus \\dots \\oplus Q_{2 k_l}(a_l,b_l)$, with $k_1 + \\dots + k_l = k$ and \n$b_i \\neq 0$ for $i=1,\\ldots,l$.\nConsider any other pencil $A' + \\lambda B'$ lying in an open ball of radius $\\sqrt{\\epsilon}$ centered on $A + \\lambda B$. We \nhave\n\\begin{equation*}\n  \\left\\| (P,U,T) \\cdot (Q + \\lambda E) - (A' + \\lambda B') \\right\\|^2\n= \\left\\| (P,U,T) \\cdot \\left(Q - A'' + \\lambda(E - B'') \\right) \\right\\|^2 < \\epsilon,\n\\end{equation*} \nwhere $A'' + \\lambda B'' = (P^{-1}, U^{-1}, T^{-1}) \\cdot (A' + \\lambda B') \\not\\equiv Q + \\lambda E$. But, since $(P,U,T) \\in \n\\GL_{2k,2k,2}(\\mathbb{R})$, then we have\n$\n  \\| (P,U,T) \\cdot (C + \\lambda D) \\| \\ge \\sigma \\|C + \\lambda D\\|\n$ \nfor any $C + \\lambda D \\in \\mathcal{P}_{2k,2k}(\\mathbb{R})$, where $\\sigma > 0$ is the smallest singular value of the linear \noperator $(P,U,T) : \\mathcal{P}_{2k,2k}(\\mathbb{R}) \\rightarrow \\mathcal{P}_{2k,2k}(\\mathbb{R})$. So, \n\\begin{equation*}\n    \\left\\| Q - A'' \\right\\|^2 + \\left\\| E - B'' \\right\\|^2 \n\\le \\sigma^{-2} \\left\\| (P,U,T) \\cdot \\left(Q - A'' + \\lambda(E - B'') \\right) \\right\\|^2\n <  \\sigma^{-2} \\, \\epsilon.\n\\end{equation*}\nHence, a sufficiently small $\\epsilon$ can be chosen to guarantee that $\\|Q - A''\\| \\le \\epsilon_1$ and $\\| E - B'' \\| \\le \\epsilon_2$ \nfor any $\\epsilon_1, \\epsilon_2 > 0$. By continuity of the eigenvalues of the pencil $Q + \\lambda E$, it follows that there \nexists $\\epsilon > 0$ such that every such $A'' + \\lambda B''$ can be written as $A'' + \\lambda B'' = \\left( X, Y, Z \\right) \\cdot \n(Q' + \\lambda E)$ for some $(X,Y,Z) \\in \\GL_{2k,2k,2}(\\mathbb{R})$ and $Q'$ having $k$ pairs of complex conjugate eigenvalues \n(with possibly some identical pairs). Therefore, $A' + \\lambda B' = \\left( PX, UY, TZ \\right) \\cdot (Q' + \\lambda E) \\in \n\\mathcal{C}$. Because this applies to every $A' + \\lambda B'$ in the chosen open ball of radius $\\epsilon$, then $A + \\lambda B$ is an \ninterior point of $\\mathcal{C}$.\n\\end{proof}\n\n\\begin{corol}\n\\label{cor:Y-bestapp}\n The set $\\mathcal{B}_{2k,2k-1} \\subset \\mathcal{P}_{2k,2k}(\\mathbb{R})$ is closed in the norm topology. Consequently, \n $$\n  \\inf_{A' + \\lambda B' \\in \\mathcal{B}_{2k,2k-1}} \\| A + \\lambda B - (A' + \\lambda B') \\|\n $$\n is always attained by some $A' + \\lambda B' \\in \\mathcal{B}_{2k,2k-1}$.\n\\end{corol}\n\n\\begin{proof}\nFollows from \\Cref{lem:C-open} by using the fact that $\\mathcal{B}_{2k,2k-1} = \\mathcal{P}_{2k,2k}(\\mathbb{R}) \\setminus \n\\mathcal{C}$.\n\\end{proof}\n\n\nAnother way to define $\\mathcal{C}$ is by stating that it contains all $2k \\times 2k$ pencils $A + \\lambda B$ such that $\\det(u A + t \nB) = 0$ if and only if $t=u=0$. This corresponds to the set of absolutely nonsingular $2k \\times 2k \\times 2$ tensors defined by Sumi \net al.~in their paper \\cite{Sumi2013}. It was shown in \\cite[Theorem 2.5]{Sumi2013} that, for any positive integer $k$ and $d > 1$, \nthe set of absolutely nonsingular $2k \\times 2k \\times d$ tensors is open. \\Cref{lem:C-open} therefore provides an alternative proof \nof that fact for the case $d = 2$.\n\n\n\\begin{theo}\n\\label{theo:closure}\n The set $\\mathcal{B}_{2k-1,2k-1} \\subset \\mathcal{P}_{2k,2k}(\\mathbb{R})$ is not closed in the norm topology. Furthermore, its \nclosure is given by $ \\overline{\\mathcal{B}_{2k-1,2k-1}} =  \\mathcal{B}_{2k,2k-1} \\subset \\mathcal{P}_{2k,2k}(\\mathbb{R})$.\n\\end{theo}\n\n\\begin{proof}\nTo prove this claim, it is sufficient\\footnote{Indeed, if $\\mathcal{X} \\subset \\mathcal{Y}$, $\\mathcal{Y} = \n\\overline{\\mathcal{Y}}$ and $\\mathcal{Y} \\subseteq \\overline{\\mathcal{X}}$, then $\\overline{\\mathcal{X}} \\subseteq \n\\overline{\\mathcal{Y}}$ and thus $\\mathcal{Y} \\subseteq \\overline{\\mathcal{X}} \\subseteq \\overline{\\mathcal{Y}} = \\mathcal{Y}$, \nimplying $\\overline{\\mathcal{X}} = \\mathcal{Y}$.} to show that for every pencil $A + \\lambda B \\in \\mathcal{B}_{2k,2k-1} \\subset \n\\mathcal{P}_{2k,2k}(\\mathbb{R})$ having minimal ranks $\\rho(A,B) = (2k,2k-1)$, we can find a sequence of pencils in \n$\\mathcal{B}_{2k-1,2k-1}$ which converges to $A + \\lambda B$. \nFirst, recall that every pencil $A + \\lambda B$ with $\\rho(A,B) = (2k,2k-1)$ must be regular.\nFrom \\Cref{lem:rho-regular}, $\\rho(A,B) = (2k,2k-1)$ holds precisely when the canonical form of $A + \\lambda B$ comprises \neither a single Jordan block $J_{m_1}(b)$ for some $b \\in \\mathbb{R}$, or a single block $N_{m_1}(\\lambda)$ having an infinite \nelementary divisor, with even dimension $m_1 > 0$. Let us first focus on the case with no infinite elementary divisors, where the \ncanonical form is either of the form $J_{2k}(a) + \\lambda E$ (with $m_1 = 2k$) or of the form $J_{m_1}(a) \\oplus Q + \\lambda \nE$, where the elementary divisors of $Q + \\lambda E$ cannot be factored into powers of linear forms. The latter possibility \nexists of course only when $k > 1$. Let us consider each case separately:\n\\begin{enumerate}[(i)]\n\n \\item In the former case, define $Z_p(\\lambda) \\triangleq W_p + \\lambda E$, where \n \\begin{equation*}\nW_p \\triangleq J_{2k-1}(a) \\oplus \\left(a + \\frac{1}{p}\\right) + e_{2k-1} \\otimes e_{2k} = \n\\begin{pmatrix}\na & 1 &  &  &  &  \\\\\n & a & 1 &  & &  \\\\\n  &   & \\ddots &  \\ddots &   & \\\\\n &  &  & a & 1 &  \\\\\n &  & &  & a & 1 \\\\  \n &  & &  &  & a + 1\/p \\\\\n\\end{pmatrix},\n\\end{equation*}\nwhere $p$ is a positive integer. It is not hard to see that $Z_p(\\lambda)$ is $\\GL_{2k,2k,2}(\\mathbb{R})$-equivalent to \n$J_{2k-1}(a) \\oplus (a + 1\/p) + \\lambda E$, since their elementary divisors coincide.\nHence, by \\Cref{lem:rho-regular}, $Z_p(\\lambda)$ has minimal ranks $(2k-1,2k-1)$, and thus belongs to $\\mathcal{B}_{2k-1,2k-1}$. \nOn the other hand, $A + \\lambda B = \\left( P, U, T  \\right) \\cdot (J_{2k}(a) + \\lambda E)$ for some $\\left( P, U, T  \\right) \n\\in \\GL_{2k,2k,2}(\\mathbb{R})$, and so we have\n\\begin{equation*}\n   \\lim_{p \\rightarrow \\infty}\n    \\left\\| A + \\lambda B - \\left( P, U, T  \\right) \\cdot Z_p(\\lambda) \\right\\|\n = \\lim_{p \\rightarrow \\infty}\n     \\left\\| \\left( P, U, T  \\right) \\cdot \\left( J_{2k}(a) - W_p \\right) \\right\\| = 0.\n\\end{equation*} \nIt follows that the sequence of pencils $\\{ \\left( P, U, T  \\right) \\cdot Z_p(\\lambda)\\}_{p \\in \\mathbb{N}}$ converges to $A + \\lambda \nB$ as $p \\rightarrow \\infty$.\n\n\\item The same argument can be employed in the second case, now with $Z'_p(\\lambda) \\triangleq W'_p + \\lambda E$, where\n$$ W'_p \\triangleq J_{m_1 - 1}(a) \\oplus (a + 1\/p) \\oplus Q + e_{m_1 - 1} \\otimes e_{m_1}.$$\nFrom this definition, $Z_p(\\lambda)$ is $\\GL_{2k,2k,2}(\\mathbb{R})$-equivalent to $J_{m_1-1}(a) \\oplus (a + 1\/p) \\oplus \nQ + \\lambda E$, which is also in $\\mathcal{B}_{2k-1,2k-1}$. Writing again $A + \\lambda B = \\left( P, U, T  \\right) \\cdot (J_{m_1}(a) \n\\oplus Q + \\lambda E)$, it follows that\n\\begin{equation*}\n   \\lim_{p \\rightarrow \\infty}\n     \\left\\| A + \\lambda B - \\left( P, U, T  \\right) \\cdot Z'_p(\\lambda) \\right\\|\n = \\lim_{p \\rightarrow \\infty}\n     \\left\\| \\left( P, U, T  \\right) \\cdot \\left( J_{m_1}(a) \\oplus Q - W'_p \\right) \\right\\| = 0.\n\\end{equation*}\n\\end{enumerate}\nFinally, note that the above argument extends easily to the case where $A + \\lambda B$ has one infinite divisor, because there \nstill exists $(P,U,T) \\in \\GL_{m,n,2}(\\mathbb{R})$ such that either $A + \\lambda B = \\left( P, U, T  \\right) \\cdot (J_{2k}(a) + \\lambda \nE)$ or $A + \\lambda B = \\left( P, U, T  \\right) \\cdot (J_{m_1}(a) \\oplus Q + \\lambda E)$, since infinite elementary divisors can always \nbe avoided with a $\\GL_{2}(\\mathbb{R})$ transformation.\n\\end{proof}\n\nMotivated by the fact that pencils with minimal ranks $(2k,2k-1)$ can be arbitrarily well approximated by pencils having minimal ranks \n$(2k-1,2k-1)$, one may define the ``border minimal ranks'' of a pencil in the same fashion as the border rank is defined for tensors \n(see, e.g., \\cite{deSilva2008}). Specifically, \\Cref{theo:closure} shows that every real $2k \\times 2k$ pencil with minimal ranks \n$(2k,2k-1)$ has ``border minimal ranks'' $(2k-1,2k-1)$.\nAs a consequence, no pencil with minimal ranks $(2k,2k-1)$ has a best approximation $\\mathcal{B}_{2k-1,2k-1}$ in the norm topology, as \nstated next.\n\n\n\\begin{corol}\n\\label{corol:approx-43} \n If $A + \\lambda B \\in \\mathcal{P}_{2k,2k}(\\mathbb{R})$ satisfies $\\rho(A,B) = (2k,2k-1)$, then  \n \\begin{equation*}\n  \\inf_{A' + \\lambda B' \\in \\mathcal{B}_{2k-1,2k-1}} \\| A + \\lambda B - (A' + \\lambda B') \\| = 0\n \\end{equation*}\nis not attained by any $A' + \\lambda B' \\in \\mathcal{B}_{2k-1,2k-1}$.\n\\end{corol}\n\nThe next result establishes that the best approximation of any pencil $A + \\lambda B \\in \\mathcal{C}$ on $\\mathcal{B}_{2k,2k-1}$ \nmust have minimal ranks $(2k,2k-1)$, otherwise it is not optimal. It is thus in the same spirit of Lemma 8.2 of De Silva and Lim \n\\cite{deSilva2008}, which states that for positive integers $r,s$ such that $r \\ge s$, the best approximation of a rank-$r$ tensor \nhaving rank up to $s$ always has rank $s$.  \n\n\\begin{lem}\n\\label{lem:approx-43}\n Let $A + \\lambda B \\in \\mathcal{C}$. Then, in the norm topology we have \n \\begin{equation*}\n  \\arg\\min_{A' + \\lambda B' \\in \\mathcal{B}_{2k,2k-1}} \\| A + \\lambda B - (A' + \\lambda B') \\|\n  \\quad \\subset \\quad \\mathcal{B}_{2k,2k-1} \\setminus  \\left( \\mathcal{B}_{2k,2k-2} \\cup \\mathcal{B}_{2k-1,2k-1} \\right).\n \\end{equation*} \n In other words, every best approximation $A' + \\lambda B'$ of $A + \\lambda B$ on $\\mathcal{B}_{2k,2k-1}$ is such that \n$\\rho(A',B') = (2k,2k-1)$. \n\\end{lem}\n\n\\begin{proof}\nTake any $A' + \\lambda B' \\in \\mathcal{B}_{2k,2k-2} \\cup \\mathcal{B}_{2k-1,2k-1}$. By definition, it can be written as\n$(t_{11} U + t_{12} V) + \\lambda (t_{21} U + t_{22} V)$, with $T = (t_{ij}) \\in \\GL_2(\\mathbb{R})$ and either $(\\rank U, \\rank V) \\le \n(2k-1,2k-1)$ or $(\\rank U, \\rank V) \\le (2k,2k-2)$, where the inequality is meant entry-wise. Let us define now $Z_1(\\lambda) \n\\triangleq \nt_{11}(A- A') + \\lambda t_{21}(B- B')$ and $Z_2(\\lambda) \\triangleq t_{12}(A- A') + \\lambda t_{22}(B- B')$. Since $A' + \\lambda B' \n\\not\\equiv A + \\lambda B$, then $Z_i(\\lambda) \\not\\equiv 0$ for at least one $i \\in \\{1,2\\}$. Thus, there exists a rank-one matrix $W$ \nsuch that $\\langle Z_i(\\lambda), W + \\lambda W \\rangle \\neq 0$ for a certain $i \\in \\{1,2\\}$. Without loss of generality, we can \nassume that $\\|t_{1i} W + \\lambda t_{2i} W \\|^2 = (t_{1i}^2 + t_{2i}^2) \\|W\\|^2 = 1$. For any $c \\in \\mathbb{R}$, we have\n\\begin{multline*}\n \\| A + \\lambda B - (A' + \\lambda B' + c \\, (t_{1i} W + \\lambda t_{2i} W) ) \\|^2 \\\\\n = \\| A + \\lambda B - (A' + \\lambda B') \\|^2 - 2c \\langle Z_i(\\lambda) , W + \\lambda W \\rangle + c^2.\n\\end{multline*} \nNow, if we choose $c = \\langle Z_i(\\lambda) , W + \\lambda W \\rangle  \\neq 0$, then clearly \n\\begin{equation*}\n \\| A + \\lambda B - (A' + \\lambda B' + c \\, (t_{1i} W + \\lambda t_{2i} W) ) \\|^2\n  = \\| A + \\lambda B - (A' + \\lambda B') \\|^2 - c^2  < \\| A + \\lambda B - (A' + \\lambda B') \\|^2.\n\\end{equation*}\nThis means that $A' + \\lambda B' + c \\, (t_{1i} W + \\lambda t_{2i} W)$ is closer to $A + \\lambda B$ than $A' + \\lambda B'$. Using now \nthe expressions given for $A'$ and $B'$, we find that\n\\begin{equation*}\nA'' + \\lambda B'' \\triangleq\nA' + \\lambda B' + c \\, (t_{1i} W + \\lambda t_{2i} W) =\n    (t_{11} U + t_{12} V + c t_{1i} W) + \\lambda (t_{21} U + t_{22} V + c t_{2i} W).\n\\end{equation*}\nIf $i=1$, we have $ (E,E,T^{-1}) \\cdot (A'' + \\lambda B'') = (U + c W) + \\lambda V $, otherwise \n$ (E,E,T^{-1}) \\cdot (A'' + \\lambda B'') = U + \\lambda (V + c  W) $. Either way, since $\\rank W = 1$, then $A'' + \\lambda B'' \n\\in \\mathcal{B}_{2k,2k-1}$. This shows that for any $A' + \\lambda B' \\in \\mathcal{B}_{2k-1,2k-1} \\cup \\mathcal{B}_{2k,2k-2}$, we can \nalways find some other pencil $A'' + \\lambda B''$ in $\\mathcal{B}_{2k,2k-1}$ such that $\\| A + \\lambda B - (A'' + \\lambda B'') \n\\| < \\| A + \\lambda B - (A' + \\lambda B') \\|$. Because a best approximation of $A + \\lambda B \\in \\mathcal{B}_{2k,2k-1}$ must exist \ndue to \\Cref{cor:Y-bestapp}, we conclude that it can only belong to  $\\mathcal{B}_{2k,2k-1} \\setminus \\left( \\mathcal{B}_{2k-1,2k-1} \n\\cup \\mathcal{B}_{2k,2k-2}\\right)$, i.e., it necessarily has minimal ranks $(2k,2k-1)$.\n\\end{proof}\n\nWe are now in a position to prove the main result of this section.\n\n\\begin{theo}\nIn the norm topology, if $A + \\lambda B \\in \\mathcal{C}$ then \n \\begin{equation*}\n  \\inf_{A' + \\lambda B' \\in \\mathcal{B}_{2k-1,2k-1}} \\| A + \\lambda B - (A' + \\lambda B') \\| \n \\end{equation*}\nis not attained by any $A' + \\lambda B' \\in \\mathcal{B}_{2k-1,2k-1}$. In other words, the problem above stated has no solution \nwhen $\\rho(A,B)=(2k,2k)$. \n\\end{theo}\n\n\\begin{proof}\n \\Cref{cor:Y-bestapp} and \\Cref{lem:approx-43} imply that for every pencil $A + \\lambda B \\in \\mathcal{C}$ there exists another \npencil $A' + \\lambda B'$ such that $\\rho(A',B') = (2k,2k-1)$ and\n\\begin{equation}\n\\label{B43-B33}\n \\left\\| A + \\lambda B - (A' + \\lambda B') \\right\\| < \\left\\| A + \\lambda B - (C + \\lambda D) \\right\\|, \n \\quad \\forall \\, C + \\lambda D \\in \\mathcal{B}_{2k-1,2k-1} \\subset \\mathcal{B}_{2k,2k-1}. \n\\end{equation}\nBut, it follows from \\Cref{theo:closure} that any such $A' + \\lambda B'$ can be arbitrarily well approximated by pencils from \n$\\mathcal{B}_{2k-1,2k-1}$, that is, \n\\begin{equation}\n   \\inf_{C + \\lambda D \\in \\mathcal{B}_{2k-1,2k-1}} \\| A' + \\lambda B' - (C + \\lambda D) \\| = 0, \n\\end{equation}\nwhilst no $C + \\lambda D \\in \\mathcal{B}_{2k-1,2k-1}$ can attain that infimum because $A' + \\lambda B' \\notin \\mathcal{B}_{2k-1,2k-1}$.\nCombining the above facts, we conclude that\n\\begin{equation}\n   \\inf_{C + \\lambda D \\in \\mathcal{B}_{2k-1,2k-1}} \\| A + \\lambda B - (C + \\lambda D) \\| = \\| A + \\lambda B - (A' + \\lambda B') \\|\n\\end{equation} \ncannot be attained by any $C + \\lambda D \\in \\mathcal{B}_{2k-1,2k-1}$.\n\\end{proof}\n\n\n \n \\section{Conclusion}\n\\label{sec:concl}\n\nThis work defines and studies a fundamental property of a matrix pencil, which we have called its minimal ranks. \nThe structure of a space of pencils can be better understood on the basis of this notion and its properties.\nIn particular, endowing $\\mathcal{P}_{m,n}( \\mathbb{F} )$ with a norm, we have studied the problem of approximation of a \npencil by another one having strictly lower minimal ranks in the induced norm topology. An optimal approximation may not exist, and \nour results show that this is true for every pencil of the set $\\mathcal{C} \\subset \\mathcal{P}_{2k,2k}( \\mathbb{R} )$ if an \napproximation is sought over $\\mathcal{B}_{2k-1,2k-1}$ for any positive integer $k$. $\\mathcal{C}$ is open, which shows that, \ncontrarily to the complex-valued case, this phenomenon can happen for pencils forming a positive volume set. Translated to a tensor \nviewpoint, our result states that certain $2k \\times 2k \\times 2$ real tensors forming a positive-volume set have no best approximate \nblock-term decomposition with two rank-$(2k-1)$ blocks. \n\nAs we have shown, the definition and essential properties of the minimal ranks can be readily extended to matrix polynomials, which \nare associated with more general $m \\times n \\times d$ tensors. We believe this should provide a useful element for the study of \nthird-order block-term decompositions composed by matrix-vector tensor products. In particular, results enabling the computation \nof the minimal ranks of a general matrix polynomial would certainly be helpful in this regard.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\nThe ejection of material from the surfaces of atmosphereless bodies is a ubiquitous phenomenon in the Solar System. Prominent examples are comets, active asteroids, ejecta clouds from hypervelocity impacts, or plumes from active satellites. For the dynamics of the ejected material, it is in many cases possible to neglect any other forces than the mass point gravity of the source body  to a good degree of approximation. This is for instance the case for impact-generated dust clouds around planetary satellites as were detected around the Galilean moons \\citep{1999Natur.399..558K, 2003Icar..164..170K} or the Moon \\citep{2015Natur.522..324H}, or dust plumes ejected from cryovolcanically active satellites \\citep{2006Sci...311.1416S, 2006Sci...311.1393P, 2015GeoRL..4210541S}. We note that for higher-order gravity terms to be negligible, the source body does not necessarily need to be spherical. For instance, mass-point gravity can be a good approximation to describe dust ejection from a satellite with surface topography.\n\n\nIn this paper we derive a semianalytical model to assess the spatial configuration of the emitted dust.  The model relates the distribution of dust sources on the surface of the atmosphereless body and the parameters of ejection (e.g., source strength or directional and velocity distribution) to observable quantities such as number density, fluxes, or optical depth. The mathematical foundations are described in Sect. \\ref{sec:mathematics}. Expanding on work in the literature \\citep{2003P&SS...51..251K, Sremcevic:2003gf}, our model can handle emission through inclined jets, and we develop a method for carrying out two of three integrations over the velocity distribution analytically. The code that implements the new model, carrying out the one remaining integration numerically, is called DUDI (for \\textquotedblleft dust distribution\\textquotedblright) and is freely available under the GNU General Public License on https:\/\/github.com\/Veyza\/dudi. Aspects of the numerical algorithm for the integration are outlined in Sect. \\ref{sec:numerics}. Examples for an application of the model to current problems in planetary science are given in Sect. \\ref{sec:applications}, including cases of steady and nonsteady dust emission.\n\n\\section{Mathematical formulation}\n\\label{sec:mathematics}\n\\subsection{Phase-space density}\nWe followed the derivations by \\citet{2003P&SS...51..251K}, \\citet{Sremcevic:2003gf}, and \\citet{2011Natur.474..620P} to relate the phase-space density of dust in a certain point of interest in space to the distributions that describe the ejection of the dust from a source on the moon surface. We generalized the existing model to allow emission from a point source in a direction that is not normal to the surface, with an axisymmetric distribution of ejection angles around this direction. Moreover, we allowed for a general coupling of the distribution of ejection velocities and grain size.\n\n\nOur model was developed initially to fit in situ measurements by the Cassini Cosmic Dust Analyzer at the Saturn satellite Enceladus. For convenience, we use the words \\textquotedblleft spacecraft position\\textquotedblright \\ or \\textquotedblleft spacecraft coordinates\\textquotedblright\\   from now on to denote the point in space at which the dust density is calculated. We also use the term \\textquotedblleft density\\textquotedblright,  which at any point can be understood as the number density, mass density, the average radius of dust particles, or the cross section that is covered by the dust at the spacecraft position. The model allows us to obtain any of these quantities with a change of only one parameter.\n\nWe first consider a stationary process. We can equate the differential number of dust particles in a certain point of phase space \n        \\begin{equation}\n                \\label{dninspace}\n                \\mathrm{d}n = n(r, \\alpha, \\beta, v, \\theta, \\lambda,R)r^2\\sin\\alpha \\mathrm{d}r\\mathrm{d}\\alpha \\mathrm{d}\\beta\\  v^2\\sin\\theta \\mathrm{d}v\\mathrm{d}\\theta \\mathrm{d}\\lambda \\mathrm{d}R\n        \\end{equation}\nto the number of particles ejected from the satellite surface\n        \\begin{equation}\n                \\label{dnonsurface}\n                \\mathrm{d}n = \\gamma \\mathrm{d}t\\ f(\\alpha_M, \\beta_M, u, \\psi,\\lambda_M, R)\\sin\\alpha_M \\mathrm{d}\\alpha_M \\mathrm{d}\\beta_M \\mathrm{d}u \\sin\\psi \\mathrm{d}\\psi \\mathrm{d}\\lambda_M \\mathrm{d}R\n        .\\end{equation}\nThe variables used here are defined in Table \\ref{vars}, and Fig. \\ref{sptr} illustrates the geometry of the problem.\nFor the phase-space  density at the spacecraft, we obtain\n\\begin{multline}\n\\label{ndens0}\nn(r, \\alpha, \\beta, v, \\theta, \\lambda, R) v^2 \\sin \\theta = \\frac{\\gamma}{|\\mathrm{d}r\/\\mathrm{d}t|r^2}\\frac{\\sin\\alpha_M\\sin\\psi}{\\sin\\alpha}\\times\\\\\\times f(\\alpha_M, \\beta_M, u, \\psi, \\lambda_M, R) \\abs*{\\frac{\\partial (\\alpha_M, \\beta_M, u, \\psi, \\lambda_M)}{\\partial(\\alpha, \\beta, v, \\theta, \\lambda)}}.\n\\end{multline}\nFor the two-body problem, the Jacobian can be obtained analytically (see \\citet{Sremcevic:2003gf}),\n\\begin{equation}\n\\label{Jalpha}\n\\abs*{\\frac{\\partial (\\alpha_M, \\beta_M, u, \\psi, \\lambda_M)}{\\partial(\\alpha, \\beta, v, \\theta, \\lambda)}} = \\frac{r}{r_M}\\frac{v^2}{u^2}\\frac{|\\cos\\theta|}{\\cos\\psi}\\frac{\\sin\\alpha}{\\sin\\alpha_M}\n.\\end{equation}\n\n\\begin{figure}\n        \\centering\n        \\includegraphics[width=0.5\\textwidth]{sph_triangle.pdf}\n        \\caption{Directions to the north and to the positions of the source and the spacecraft form a spherical triangle on a sphere. This establishes the relations between the angles of the problem. Here $\\Delta\\beta$ is the angle between the projections of vectors $\\mathbf{r}$ and $\\mathbf{r}_M$ on the equatorial plane.}\n        \\label{sptr}\n\\end{figure}\n\n\\begin{table*}[t]\n        \\caption{Definition of variables}\n        \\vskip0.3cm\n        \\begin{tabular}{|c|p{6cm}|c|p{6cm}|}\n                \\hline\n                Variable & Definition & Variable & Definition \\\\\n                \\hline\n                $r$ & Radial distance from the moon center to a point in space where the density is to be calculated (spacecraft position) & $r_M$ & Radial distance of the dust source from the center of the moon (source position)\\\\\n                $\\alpha$ & Colatitude of the spacecraft (measured from the moon north pole) & $\\alpha_M$ & Colatitude of the source  (measured from the moon north pole) \\\\\n                $\\beta$ & Eastern longitude of the spacecraft & $\\beta_M$ & Eastern longitude of the source \\\\\n                $v$ & Particle speed at the spacecraft position & $u$ & Particle speed at the moment of ejection \\\\\n                $\\theta$  & Angle\n                between the particle velocity and position vectors & $\\psi$  & Initial angle\n                between the particle velocity and position vectors\\\\\n                $\\lambda$ & Azimuth angle of the particle velocity, measured clockwise from local north. & $\\lambda_M$ & Azimuth angle of the particle initial velocity, measured clockwise from local north.\\\\\n                $R$ & Radius of the particle & $\\gamma$ & Rate of dust particle production \\\\\n                $n$ & Phase-space density of particles with fixed radius & $f$ & Distribution describing the dust ejection process\\\\\n                $\\zeta$ & Zenith angle of the source symmetry axis & $\\eta$ & Azimuth of the source symmetry axis measured clockwise from local north \\\\\n                 & & $\\eta^*$ & Auxiliary angle used in the derivation of the expression for the ejection direction distribution in case of a tilted symmetry axis\\\\\n                \\hline\n        \\end{tabular}\n        \\label{vars}\n\\end{table*}\nWe assume that the distribution function $f$ factorizes, and that the distributions of the source position ($f_{\\alpha_M,\\beta_M}$), ejection speed ($f_u$), ejection direction ($f_{\\psi,\\lambda_M}$), and size of the ejected dust particles ($f_R$) can be defined separately,\n\\begin{equation}\n\\label{fff}\nf(\\alpha_M, \\beta_M, u, \\psi, \\lambda_M, R) = f_{\\alpha_M,\\beta_M} (\\alpha_M,\\beta_M) f_{\\psi, \\lambda_M}(\\psi, \\lambda_M) f_u(u, R)  f_R(R).\n\\end{equation}\n\nIt is physically plausible that the distribution of the ejection speed of the dust particles depends on the grain size \\citep{2008Natur.451..685S, 2011Natur.474..620P}, which we emphasize with the notation $f_u(u,R)$.  \n\nWe describe the position of the point source located at the coordinates $(\\alpha_M^0,\\beta_M^0)$ on the surface of the spherical moon as the product of two Dirac $\\delta$-functions,\n\\begin{equation}\n\\label{fpos}\nf_{\\alpha_M,\\beta_M} (\\alpha_M,\\beta_M) = \\frac{\\delta(\\alpha_M - \\alpha_M^0)\\,\\delta(\\beta_M - \\beta_M^0)}{\\sin\\alpha_M} \\equiv\\frac{\\delta\\left((\\alpha_M,\\beta_M)-(\\alpha_M^0,\\beta_M^0)\\right)}{\\sin\\alpha_M}\n.\\end{equation}\nThe term $\\sin\\alpha_M$ in the denominator comes from the normalization.\n\nTo formulate the directional distribution of $\\psi$ and $\\lambda_M$ so that it describes the ejection of dust that is axisymmetric around the axis of an inclined jet, we consider two coordinate systems centered at the location of the point source. The \\textit{Z}-axis of system $(X,Y,Z)$ points along the local normal to the surface, and the $X$-axis points toward the local north. The  $\\tilde{Z}$-axis\nof system $(\\tilde{X}, \\tilde{Y}, \\tilde{Z})$ is aligned with the axis of the jet. \n The axis $\\tilde{X}$ lies on the line of nodes, so that the angle $\\eta^*$ measured from $X$ to $\\tilde{X}$ is related to the jet azimuth $\\eta$ as $\\eta^* = \\eta - \\pi \/ 2$. We define azimuth angles always \\emph{\\textup{clockwise}} from the local north, allowing a direct comparison to the derivations in \\citet{2003P&SS...51..251K} and \\citet{Sremcevic:2003gf}. Then, the transformation of the $(X,Y,Z)$ coordinate system to the $(\\tilde{X}, \\tilde{Y}, \\tilde{Z})$ coordinate system may be performed as two subsequent rotations, as shown in Fig. \\ref{2sys}. The first rotation is clockwise around the $Z$-axis with angle $\\eta^*$. The second rotation is counterclockwise around the $\\tilde{X}$-axis with angle $\\zeta$.\n\n\n\\begin{figure}\n        \\centering\n        \\includegraphics[width=0.5\\textwidth]{twosystems_newnew}\n        \\caption{Two coordinate systems centered at the location of the dust source on the surface of a spherical body. The  $Z$-axis of system ($X,Y,Z$) is normal to the surface, and $X$ points to the local north. The $\\tilde{Z}$-axis of the coordinate system ($\\tilde{X},\\tilde{Y}, \\tilde{Z}$) is aligned with a jet that is tilted from the surface normal by an angle $\\zeta$.}\n        \\label{2sys}\n\\end{figure}\n\nLet $(\\psi,\\lambda_M)$ and $(\\tilde{\\psi}, \\tilde{\\lambda}_M)$ be the polar angle and azimuth in the two systems $(X,Y,Z)$ and $(\\tilde{X}, \\tilde{Y}, \\tilde{Z})$, respectively. The distribution of the ejection direction we wish to use can be formulated in a simple manner in terms of the variables $(\\tilde{\\psi}, \\tilde{\\lambda}_M)$ because the distribution is symmetrical with respect to the axis $\\tilde{Z}$. \nHowever, in Eq. (\\ref{ndens0}) and in the formulae that are to be derived later in the course of solving the two-body problem, we have to deal with the angles $(\\psi,\\lambda_M)$ that are defined in the local horizontal coordinate system. Therefore a replacement of the coordinates must be performed in the expression to obtain the distribution of $\\psi$ and $\\lambda_M$ , which can be used in further calculations. The desired function $f_{\\psi, \\lambda_M}(\\psi, \\lambda_M)$ can be obtained by multiplication with the Jacobian,\n\\begin{equation}\n\\label{transpsi}\nf_{\\psi, \\lambda_M}(\\psi, \\lambda_M)\\sin\\psi = f_{\\tilde{\\psi}, \\tilde{\\lambda}_M}(\\tilde{\\psi}, \\tilde{\\lambda}_M) \\sin\\tilde{\\psi} \\abs*{\\frac{\\partial (\\tilde{\\psi}, \\tilde{\\lambda}_M)}{\\partial (\\psi, \\lambda_M)}}\n.\\end{equation}\nTo express $\\tilde{\\psi}$ and $\\tilde{\\lambda}_M$ through $\\psi$ and $\\lambda_M$ , we consider a unit vector $\\mathbf{k}$ pointing in an arbitrary direction (Fig. \\ref{2sys}). In both coordinate systems the vector can be defined by Cartesian coordinates related to the corresponding polar coordinates as (we recall that $\\lambda_M$ and $\\tilde{\\lambda}_M$ are azimuthal angles counted \\emph{\\textup{clockwise}} from the $X$ and $\\tilde X$ axes, respectively)\n\n\\begin{equation}\n\\label{kvec}\n\\mathbf{k} = \\colvec{3}{k_1}{k_2}{k_3} = \\colvec{3}{\\sin\\psi\\cos\\lambda_M}{-\\sin\\psi\\sin\\lambda_M}{\\cos\\psi},\\ \n\\mathbf{k} = \\colvec{3}{\\tilde{k}_1}{\\tilde{k}_2}{\\tilde{k}_3} = \\colvec{3}{\\sin\\tilde{\\psi}\\cos\\tilde{\\lambda}_M}{-\\sin\\tilde{\\psi}\\sin\\tilde{\\lambda}_M}{\\cos\\tilde{\\psi}}\n.\\end{equation}\nThe Cartesian coordinates of $\\mathbf{k}$ in the two systems are related through the rotation matrix, which can be expressed in terms of  $\\eta$,\n\\begin{equation}\n\\label{rotmatr}\n\\colvec{3}{\\tilde{k}_1}{\\tilde{k}_2}{\\tilde{k}_3} = \\begin{pmatrix}\n\\sin\\eta & \\cos\\eta & 0\\\\\n-\\cos\\eta\\cos \\zeta &\\sin \\eta \\cos \\zeta & \\sin \\zeta\\\\\n\\cos\\eta\\sin \\zeta& -\\sin\\eta\\sin \\zeta & \\cos \\zeta\n\\end{pmatrix}\\colvec{3}{k_1}{k_2}{k_3}\n.\\end{equation}\nUsing Eqs. \\ref{kvec} and \\ref{rotmatr}, we obtain\n\\begin{equation}\n\\label{wpsi}\n\\tilde{\\psi} = \\arccos (\\cos \\zeta\\cos\\psi + \\cos(\\eta-\\lambda_M)\\sin \\zeta\\sin\\psi),\n\\end{equation}\n\\begin{equation}\n\\label{wlambdaM}\n \\tilde{\\lambda}_M =\\arctan \\left( \\frac{\\cos\\zeta \\sin\\psi \\cos(\\eta-\\lambda_M) - \\sin\\zeta \\sin\\psi)}{\\sin\\psi \\sin(\\eta-\\lambda_M)} \\right)\n.\\end{equation}\nThis gives the Jacobian\n\\begin{multline}\n\\label{Jpsi}\n\\abs*{\\frac{\\partial (\\tilde{\\psi}, \\tilde{\\lambda}_M)}{\\partial (\\psi, \\lambda_M)}} = 4\\sin\\psi\\ \/ [10-2\\cos 2\\psi-3\\cos 2(\\psi - \\zeta) - 2\\cos 2\\zeta -3\\cos 2(\\psi + \\zeta) \\\\- 8\\cos 2(\\lambda_M-\\eta)\\sin^2\\zeta\\sin^2\\psi -8\\cos(\\lambda_M-\\eta)\\sin 2\\zeta \\sin 2\\psi] ^{1\/2}.\n\\end{multline}\n\n\n\n\\subsection{Integration}\n\\label{integration}\nTo compute the density of dust at the point $(r, \\alpha,\\beta),$ we must integrate Eq. (\\ref{ndens0}) over all possible velocities and over all possible particle sizes,\n\\begin{multline}\n\\label{ndens}\nn(r,\\alpha,\\beta,R_{min} < R < R_{max}) = \\frac{\\gamma}{r r_M} \\int_{v_{min}}^{v_{max}}\\mathrm{d}v\\int_{0}^{\\pi}\\mathrm{d}\\theta\\int_{0}^{2\\pi}\\mathrm{d}\\lambda \\frac{v}{u^2}G^p_u(R_{min}, R_{max})\\times\\\\\\times\\frac{f_{\\psi, \\lambda_M}(\\psi, \\lambda_M)\\sin\\psi }{\\cos\\psi}\\,\\frac{\\delta\\left((\\alpha_M(\\theta,\\lambda),\\beta_M(\\theta,\\lambda))-(\\alpha_M^0,\\beta_M^0)\\right)}{\\sin\\alpha_M}\n.\\end{multline}\nHere,\n\\begin{equation}\n\\label{Gu}\nG^p_u(R_{min},R_{max}) \\equiv \\int_{R_{min}}^{R_{max}}\\mathrm{d}Rf_R(R)f_u(u,R)R^p\n\\end{equation}\nis defined in a similar way as in \\citet{2011Natur.474..620P}. The parameter $p$ defines the moment of the size distribution related to the quantity we are interested in. Using p=0, we obtain the number density of particles in the specified range of sizes. Setting p=1 gives the average radius of the grains per unit volume. For p=2 we obtain the average cross section of the dust particles per volume. This setting is used below to compute the geometrical optical depth of the dust population. Finally, p=3 gives the average volume occupied by dust grains per unit volume. This setting is used to compute the mass density of the dust. For more details of the evaluation of $G^p_u(R_{min},R_{max}),$ see Appendix B.\nWe replace variables in the argument of the $\\delta$ function in equation (\\ref{ndens}) as\n\\begin{equation}\n\\label{repl}\n\\int_{0}^{\\pi}\\mathrm{d}\\theta\\int_{0}^{2\\pi}\\mathrm{d}\\lambda\\, \\delta\\left((\\alpha_M(\\theta,\\lambda),\\beta_M(\\theta,\\lambda))-(\\alpha_M^0,\\beta_M^0)\\right)F(\\theta,\\lambda) =\\sum_{i} \\frac{F(\\theta_i^*,\\lambda_i^*)}{\\abs*{\\frac{\\partial(\\alpha_M,\\beta_M)}{\\partial(\\theta,\\lambda)}}}_{\\theta_i^*,\\lambda_i^*}\\,\n.\\end{equation}\nto integrate over $\\theta$ and $\\lambda$ analytically.  Eq. (\\ref{repl}) is derived in greater detail in Appendix A.\nHere, $F(\\theta,\\lambda)$ represents the integrand of equation \\ref{ndens}, while $\\theta_i$ and $\\lambda_i$ are the roots of the equation\n\\begin{equation}\n\\label{alphaMbetaM}\n\\alpha_M(\\theta_i^*,\\lambda_i^*) = \\alpha_M^0,\\quad \\beta_M(\\theta_i^*,\\lambda_i^*)=\\beta_M^0\n.\\end{equation}\nAll the necessary dependences between the variables in question can be found from spherical trigonometry  \\citep{2003P&SS...51..251K, Sremcevic:2003gf}, for instance,\n\\begin{equation}\n\\label{alphaM}\n\\alpha_M = \\arccos\\left(\\cos\\alpha\\cos\\Delta\\phi(\\theta) - \\sin\\alpha\\sin\\Delta\\phi(\\theta)\\cos\\lambda\\right)\n\\end{equation}\nand\n\\begin{equation}\n\\label{betaM}\n\\beta_M = \\beta \\pm \\arcsin\\left(\\frac{\\sin\\Delta\\phi(\\theta)\\sin\\lambda}{\\sin\\alpha_M(\\theta,\\lambda)}\\right)\n.\\end{equation}\n\nThe spherical triangle used to obtain these relations is shown in Fig. \\ref{sptr}. The angle $\\Delta\\phi$ is the angle between the position vectors of the spacecraft and the source location on the moon.\nBecause $\\theta$ enters expressions \\ref{alphaM} and \\ref{betaM} only through $\\Delta\\phi$, the partial derivatives of $\\alpha_M$ and $\\beta_M$ with respect to $\\theta$ can be computed as partial derivatives with respect to $\\Delta\\phi$ multiplied by $\\partial\\Delta\\phi \/ \\partial\\theta$. The Jacobian reads\n\\begin{equation}\n\\label{JalphaM}\n\\abs*{\\frac{\\partial(\\alpha_M,\\beta_M)}{\\partial(\\theta,\\lambda)}}=\\frac{\\sin\\Delta\\phi}{\\sin\\alpha_M}\\abs*{\\frac{\\partial\\Delta\\phi}{\\partial\\theta}},\n\\end{equation}\nand our final formula is\n\\begin{multline}\n\\label{workformula}\nn(r,\\alpha,\\beta,R_{min}<R<R_{max}) = \\frac{\\gamma}{rr_M\\sin\\Delta\\phi} \\int_{v_{min}}^{v_{max}}\\mathrm{d}v\\frac{v}{u^2}G^p_u(R_{min},R_{max})\\times\\\\ \\times \\sum_{i}\\frac{f_{\\psi,\\lambda_M}(\\psi_i,\\lambda_{Mi})\\sin\\psi_i}{\\cos\\psi_i}\\abs*{\\frac{\\partial\\Delta\\phi}{\\partial\\theta}}_{\\theta^*_i}^{-1}.\n\\end{multline}\n\nThe integration over $v$ must be carried out numerically. The lower integration limit is restricted by the minimum energy, or minimum semimajor axis, of the orbits that pass through the two points $(r_M,\\alpha_M,\\beta_M)$ and $(r,\\alpha,\\beta)$. It is obtained from\n\\begin{equation}\n        \\label{vmin}\n        v_{min} = \\sqrt{GM\\left(\\frac{2}{r}-\\frac{1}{a_{min}}\\right)}\n,\\end{equation}\nwhere\n \\begin{equation}\n\\label{amin}\na_{min} = \\frac{r + r_M}{4} + \\frac{1}{2} \\sqrt{\\frac{r^2 + r_M^2}{4} - \\frac{r r_M \\cos\\Delta\\phi}{2}}\n.\\end{equation}\nThe upper limit is constrained by the maximum ejection speed,\n\\begin{equation}\n        \\label{vmax}\n        v_{max} = \\sqrt{u_{max}^2+ 2GM \\left(\\frac{1}{r}-\\frac{1}{r_M}\\right)}\n.\\end{equation}\n\nThe angle $\\Delta\\phi$  in the expressions above is the angle between the position vector of the dust source, $\\mathbf{r}_M$ , and the position vector of the spacecraft, $\\mathbf{r}$. For a fixed position of the source and spacecraft, the angle $\\Delta\\phi$ is also fixed, but it formally depends on $v$ and $\\theta$. In the integrand of equation \\ref{workformula}, the value $v$ is given and $\\Delta\\phi(\\theta)$ is a function of the variable $\\theta$ alone. From the conservation equations of the two-body problem, we obtain \n\\begin{equation}\n\\label{wp}\n\\tilde{p} = 2\\tilde{r}^2\\tilde{v}^2\\sin^2\\theta,\n\\end{equation}\n\\begin{equation}\ne^2 = 1 + 4 \\tilde{r}^2\\tilde{v}^2\\sin^2\\theta\\left(\\tilde{v}^2-\\frac{1}{\\tilde{r}}\\right),\n\\end{equation}\n\\begin{equation}\n\\label{cosphiM}\n\\cos\\phi_M= \\frac{1}{e}\\left(\\frac{\\tilde{p}}{1}-1\\right),\n\\end{equation}\n\\begin{equation}\n\\label{cosphi}\n\\cos\\phi= \\frac{1}{e}\\left(\\frac{\\tilde{p}}{\\tilde{r}}-1\\right),\n\\end{equation}\n\\begin{equation}\n\\label{dphi}\n\\Delta\\phi = \\phi - \\phi_M,\n\\end{equation}\nwhich give the relation between $\\Delta\\phi$ and $\\theta$. Following the notational convention of \\citet{2003P&SS...51..251K}, we use dimensionless variables $\\tilde{r} = r\/r_M$ and $\\tilde{v} = v\/ v_{escape}$, where $v_{escape}$ is the escape velocity on the satellite surface. The angles $\\phi$ and $\\phi_M$ are the true anomalies at $\\mathbf{r}$ and $\\mathbf{r}_M$, respectively.\n\nEquations (\\ref{wp}) -- (\\ref{dphi}) can be used to calculate the partial derivative $\\partial\\Delta\\phi \/ \\partial\\theta$. However, it is not possible to reverse these expressions to obtain $\\theta$ from a given value of $\\Delta\\phi$ analytically. The desired $\\theta^*_i$ are the values of $\\theta$ that (for a given $v$) lead to a $\\Delta\\phi$ satisfying equations \\ref{alphaM} and \\ref{betaM}. We use a geometrical approach to calculate all possible $\\theta_i^*$. We consider the two-body problem, therefore the motion is restricted to a plane. We define a two-dimensional coordinate system in the plane containing the vectors $\\mathbf{r}$ and $\\mathbf{r_M}$. The origin is located at the moon center and $\\mathbf{r}$ points along the $x$-axis. We know the lengths of $\\mathbf{r}$ and $\\mathbf{r_M}$ as well as the angle $\\Delta\\phi$ between them. Then the coordinates of the vectors $\\mathbf{r}$ and $\\mathbf{r_M}$ in the plane are ($r,0$) and ($r_M\\cos\\Delta\\phi,r_M\\sin\\Delta\\phi$). \n\nWe wish to obtain $\\theta$ , which is the angle between $\\mathbf{r}$ and the grain velocity vector, which is tangential to the particle trajectory at the point $\\mathbf{r}$. This angle can be calculated when we know the equation of the trajectory, which is either an ellipse or a hyperbola. At this step (calculating the integrand of equation \\ref{workformula}), we know the value of the particle speed and the distance to the moon center, which determines the orbital energy, and thus, the semimajor axis $a$. We also know whether the particle moves along an ellipse (negative orbital engery) or along a hyperbola (positive orbital energy).\n\nWe first consider the case of an ellipse (Fig. \\ref{ellipse}). One of the focal points ($F_1$) is located in the origin, which is the center of the moon. To draw the ellipse, we must find the position of the second focus. At each point of the ellipse, the sum of the distances to the focal points is a constant equal to $2a$. Therefore the second focus must be removed from point $\\mathbf{r}$ by $2a-r$ and from point $\\mathbf{r_M}$ by $2a-r_M$. These conditions are met at the intersection points of two circles centered at $\\mathbf{r}$ and $\\mathbf{r_M}$ with radii $2a-r$ and $2a-r_M$, respectively.\n\\begin{figure\n        \\centering\n        \\begin{minipage}{0.45\\textwidth}\n                \\centering\n                \\includegraphics[width=0.9\\textwidth]{ellipse_new.pdf}\n                \\caption{Finding the second focus for an elliptic trajectory and the solutions for $\\theta^*_i$\\\\\n                }\n                \\label{ellipse}\n        \\end{minipage}\\hfill\n        \\begin{minipage}{0.45\\textwidth}\n                \\centering\n                \\includegraphics[width=0.9\\textwidth]{hyperbola_new.pdf}\n                \\caption{Finding the second focus for an hyperbolic trajectory and the solutions for $\\theta^*_i$. Only $\\theta^*_1$ corresponds to a physically possible trajectory.}\n                \\label{hyperbola}\n        \\end{minipage}\\hfill\n\\end{figure}\nThis leads to the conditions\n\\begin{equation}\n\\label{circleel}\n\\begin{array}{rl}\n(x-r_M\\cos\\Delta\\phi)^2 + (y-r_M\\sin\\Delta\\phi)^2 & = (2a-r_M)^2,\\\\\n(x-r)^2 + y^2 &= (2a-r)^2.\n\\end{array}\n\\end{equation}\nBy solving the system of equations \\ref{circleel}, we find two possible positions for the second focal point, $F_2^1$ and $F_2^2$. In either case, we can calculate the eccentricity $e$ of the ellipse, the true anomaly at point $\\mathbf{r,}$ and, finally, the two solutions for $\\theta$ using Eqs. (\\ref{eccentricity}) -- (\\ref{theta}). Knowing $\\cos\\phi_M$ is sufficient to obtain $\\phi_M$ because we know that the ejection point $\\mathbf{r_M}$ cannot have a true anomaly $\\phi_M > \\pi$. We have\n\\begin{equation}\n\\label{eccentricity}\ne = \\frac{|\\overline{F_2F_1}|}{2a},\n\\end{equation} \n\\begin{equation}\n\\label{cosf1}\n        \\cos\\phi_M = \\frac{\\overline{F_2F_1}\\cdot\\mathbf{r}_M}{|\\overline{F_2F_1}|r_M},\n\\end{equation}\n\\begin{equation}\n        \\label{difphi}\n        \\phi = \\phi_M + \\Delta\\phi,\n\\end{equation}\n\\begin{equation}\n\\label{theta}\n\\theta = \\frac{\\pi}{2} - \\arctan\\frac{e\\sin \\phi}{\\sqrt{1+e\\cos\\phi}},\n\\end{equation}\n\\begin{equation}\n        \\label{rad}\n        r = \\frac{a(1-e^2)}{1+e\\cos\\phi}\n.\\end{equation}\nThese equations determine the solutions $\\theta^*_i$ used in equation \\ref{workformula} if the particle travels from $\\mathbf{r_M}$ to $\\mathbf{r}$ along the shorter arc of the ellipse. However, there are cases when a particle reaches $\\mathbf{r}$ over an arc of $2\\pi - \\Delta\\phi$ (Fig. \\ref{bigdphi}), leaving $\\mathbf{r_M}$ in the opposite direction. To distinguish this case, we must recalculate $r$ from the obtained value for $\\phi$ using Eq. (\\ref{rad}) and verify that it matches the starting value for $r$ that we used to obtain $\\phi$. If this is not the case, then $\\Delta\\phi$ in Eq. (\\ref{difphi}) must be replaced by $2\\pi-\\Delta\\phi$, which corresponds to the motion along the same ellipse, but in the opposite direction. This case is relatively rare, and in the examples we explored was only encountered at large distances from the source.\n\n\\begin{figure}\n        \\centering\n        \\includegraphics[width=0.5\\textwidth]{bigdphi_new.pdf}\n        \\caption{The case when $\\Delta\\phi$ should be replaced by $2\\pi-\\Delta\\phi.$}\n        \\label{bigdphi}\n\\end{figure}\n\n\n\nIn case of hyperbolic motion ($a<0$, Fig. \\ref{hyperbola}), we follow almost the same line. For every point on a hyperbola, the difference between the distances to the focal points is the same. Because $r_M$ and $r$ must be distances to the nearest focus, the system of equations for the coordinates of the second focal point reads \n\\begin{equation}\n\\label{circlehy}\n\\begin{array}{rl}\n(x-r_M\\cos\\Delta\\phi)^2 + (y-r_M\\sin\\Delta\\phi)^2 & = (r_M+2|a|)^2,\\\\\n(x-r)^2 + y^2 &= (r+2|a|)^2.\n\\end{array}\n\\end{equation}\n\\vskip0.4cm\nEquations \\ref{eccentricity} -- \\ref{theta} remain the same for the hyperbolic case. However, when the coordinates of the second focus are found, we must make sure that the particle does not pass the pericenter on its way from $\\mathbf{r_M}$ to $\\mathbf{r}$. For this purpose, we verify that the points $\\mathbf{r_M}$ and $\\mathbf{r}$ lie on the same side of the line $F_1F_2$. If this condition is not satisfied, the solution is rejected. In Fig. \\ref{hyperbola} the hyperbola with second focus at the point $F_2^2$  does not meet this condition. Therefore only one hyperbolic trajectory is possible to get from $\\mathbf{r_M}$ to $\\mathbf{r}$. Furthermore, the motion along the hyperbola is possible only in one direction. The value for $\\sin\\phi$ is always positive and $\\Delta\\phi=\\phi-\\phi_M$.\n\nThe values of $\\lambda_i$ can be inferred from spherical trigonometry. The two solutions can be either identical or they differ by $180^\\circ$ because the motion is restricted to a plane,\n\\begin{equation}\n\\label{coslam}\n\\cos\\lambda= \\frac{\\cos\\alpha_M\\cos\\Delta\\phi-\\cos\\alpha}{\\sin\\alpha_M\\sin\\Delta\\phi},\n\\end{equation}\n\\begin{equation}\n\\label{sinlam}\n\\sin\\lambda=\\pm\\frac{\\sin\\alpha_M\\sin(\\beta-\\beta_M)}{\\sin\\Delta\\phi}\n.\\end{equation}\nThe sign in Eq. (\\ref{sinlam}) depends on the specific orientation of $\\mathbf{r}$ and $\\mathbf{r_M}$ relative to the direction of zero-longitude. When the particle travels from $\\mathbf{r_M}$ to $\\mathbf{r}$ over the angle of $2\\pi-\\Delta\\phi$ , the signs of $\\sin\\lambda$ and $\\cos\\lambda$ both change because the sign of the $\\sin\\Delta\\phi$ term in the denominator changes.\n\nAs soon as the values of $(v,\\theta^*_i,\\lambda_i)$ that satisfy the orbital geometry are known, they may be used to calculate the corresponding $(u,\\psi_i,\\lambda_{Mi})$ from Eqs. (\\ref{ufromv}) -- (\\ref{coslambdaM}) and the integrand in Eq. (\\ref{workformula}) is fully determined,\n\\begin{equation}\n\\label{ufromv}\nu = \\sqrt{v^2_{escape}+2\\left(\\frac{v^2}{2}-\\frac{GM}{r}\\right)},\n\\end{equation}\n\\begin{equation}\n\\label{psi}\n\\sin\\psi = \\frac{rv}{r_M u}\\sin\\theta,\n\\end{equation}\n\\begin{equation}\n\\label{sinlambdaM}\n\\sin\\lambda_M=\\frac{\\sin\\alpha\\sin\\lambda}{\\sin\\alpha_M},\n\\end{equation}\n\\begin{equation}\n\\label{coslambdaM}\n\\cos\\lambda_M = \\frac{\\cos\\alpha -\\cos\\alpha_M\\cos\\Delta\\phi}{\\sin\\alpha_M\\sin\\Delta\\phi}\n.\\end{equation}\n\n\\subsection{Nonstationary case}\nNonstationary dust ejection can be modeled by allowing a time-dependent production rate $\\gamma$  in Eq. (\\ref{workformula})\nthat will result in a time-dependent spatial distribution of the dust. When we determine the orbital geometry for a fixed velocity value at the given point in space (Sect. \\ref{integration}), the time $\\Delta t$ required for traveling from $\\mathbf{r_{M}}$ to $\\mathbf{r}$ along the Keplerian orbit can be calculated from Kepler's equation. Thus, we know that the properties of the dust configuration at location $\\mathbf{r}$ and time $t$ are caused by the production of dust at the source location $\\mathbf{r_{M}}$ with the rate $\\gamma(t-\\Delta t)$.\nIn this case, the production rate cannot be put outside the integral and Eq. (\\ref{workformula}) becomes\n\\begin{multline}\n\\label{timedep}\n        n(r,\\alpha,\\beta,R_{min}<R<R_{max},t) = \\frac{1}{rr_M\\sin\\Delta\\phi} \\int_{v_{min}}^{v_{max}}\\mathrm{d}v\\frac{v}{u^2}G^p_u(R_{min},R_{max})\\times\\\\\\times\\sum_{i}\\gamma(t-\\Delta t_i)\\frac{f_{\\psi,\\lambda_M}(\\psi_i,\\lambda_{Mi})}{\\cos\\psi_i}\\abs*{\\frac{\\partial\\Delta\\phi}{\\partial\\theta}}_{\\theta^*_i}^{-1}.\n\\end{multline}\nThe two solutions for $\\Delta t_i$ follow from Kepler's equation using the two solutions for eccentricity from equation \\ref{eccentricity} and computing the eccentric anomaly with the half-angle formula from the true anomaly given by equation \\ref{difphi}.\n\n\n\\subsection{Singularities of the coordinates}\n\\label{spepoints}\nThere are coordinates for which solutions of equations (\\ref{workformula}) or (\\ref{timedep}) cannot be obtained (see Table \\ref{sppoints}). The problem arises from the use of a spherical coordinate system. In practice, it is possible to avoid the singularities by carefully choosing the pole axis of the coordinate system after the source location and the detector position of interest are known. If coordinates close to the singularities need to be evaluated, then the stability and accuracy of the numerical integration becomes a challenge. However, double-precision calculations allow approaching the singular angles as close as $10^{-4}$ radians. This difference from the values listed in Table \\ref{sppoints} can be considered safe, and with a possible moderate loss of accuracy, the model can be applied within an even closer vicinity of the singularities.\n\\begin{table*}\n        \\caption{Coordinate singularities of the model}\n        \\begin{tabular}{|c|c|}\n                \\hline\n                Variable value    &  Physical and geometrical meaning   \\\\    \n                \\hline\n                $\\alpha_M = 0^\\circ$ & Point source of dust is located at the north or south pole,  \\\\\n                $\\alpha_M = 180^\\circ$ & its latitude is not defined, and no spherical triangle from Fig. \\ref{sptr} exists\\\\\n                \\hline\n                $\\alpha = 0^\\circ$ & Spacecraft is located directly above the north or south pole, \\\\\n                $\\alpha = 180^\\circ$ & its latitude  is not defined, no spherical triangle from Fig. \\ref{sptr} exists\\\\\n                \\hline\n                $\\beta = \\beta_M$ & Spacecraft has the same longitude \\\\\n                ($\\Delta\\beta = 0^\\circ$) &as the source. No spherical triangle from Fig. \\ref{sptr} exists\\\\\n                \\hline\n                $\\Delta\\phi = 0^\\circ$& Spacecraft position, source position, and moon center \\\\\n                $\\Delta\\phi = 180^\\circ$ & are on a straight line. No spherical triangle from Fig. \\ref{sptr} exists\\\\\n                &  and the orbit geometry (Figs. \\ref{ellipse} and \\ref{hyperbola}) is undefined\\\\\n                \\hline\n                $\\psi = z$, & A particle is ejected exactly along\\\\\n                $z \\neq 0^\\circ$ &  the jet axis, the azimuth of ejection is undefined, and the Jacobian\\\\\n                &  in Eq. (\\ref{Jpsi}) diverges\\\\\n                \\hline\n        \\end{tabular}\n        \\label{sppoints}\n\\end{table*}\n\n\\section{Numerical integration}\n\\label{sec:numerics}\nIn this section we present and discuss the algorithm for the numerical solution of equations \\ref{workformula} and \\ref{timedep}. We have implemented this algorithm in a code written in Fortran-95, which we call dust distribution (DUDI). The source code with technical documentation and instructions for usage and compilation is freely available under the GNU General Public License on https:\/\/github.com\/Veyza\/dudi. \nThe \\textit{makefile} provided for compilation uses the \\textit{gfortran} compiler. DUDI can be compiled and run without installing additional libraries. The library of \\textit{OpenMP}, which is used to speed the computation up, is included in the compiler. \n\nDUDI allows us to compute the number density of dust or related quantities at given points in space as a mean radius,  average cross section, or mass density of the dust grains ejected from the surface of a spherical body without an atmosphere. The input data are the spacecraft coordinates, the properties of the source (i.e., the\\ location and the distributions of the direction and speed of the ejection) and the three parameters of $G^p_u$. \n\nThe first preliminary step is to calculate $G^p_u$ on a grid of $u$-values. The array of pairs $(u, G^p_u)$ is later used to interpolate $G^p_u(R_{min}, R_{max})$ for the actually required value $u$ under the integrand of equations \\ref{workformula} or \\ref{timedep}. The second preliminary step is to compute the values of $\\Delta\\phi$ and $\\Delta\\beta$ for the given positions of the source and the spacecraft.\n\nThen we proceed directly to the numerical evaluation of the integral over velocity in Eq (\\ref{workformula}) in the stationary case, or Eq. (\\ref{timedep}) if there is a time-dependence. The ejection speed distribution implies a certain lower limit for the possible ejection speed $u_{min}$. At any point $\\mathbf{r,}$ this minimum ejection velocity restricts the corresponding minimum velocity at spacecraft position $v_{min}^0$ , which may be higher than the lower integration limit given by Eqs. (\\ref{amin}) -- (\\ref{vmin}). The actual numerical integration is performed over the interval where $(v_{min},v_{max})$ and $(v_{min}^0,v_{max})$ overlap. The minimum and maximum ejection speed $u_{min}$ and $u_{max}$ (in m\/s) must be explicitly specified as a property of the dust source, along with the expression for the ejection speed distribution.\n\nAt each step of the integration, we find for a given $v$, $r$, and $\\Delta\\phi$ the solutions for the angles $\\theta$ and $\\lambda$ as described in Sect. \\ref{integration}. We control the accuracy of the solution for $\\theta$ by recalculating the value of $\\Delta\\phi$ from Eqs. (\\ref{wp})--(\\ref{dphi}) and comparing it to the starting value of $\\Delta\\phi$ as given by the positions of the dust source and the spacecraft. We require the difference between the two values of $\\Delta\\phi$ to be smaller than $10^{-4}\\Delta\\phi$. In most cases the accuracy is much better, but it may degrade for certain values of $v$ near the singularities of the coordinates (see Sect. \\ref{spepoints}) or when $r \\approx r_M$. Even for poor accuracy for $\\theta$ , this means that the accuracy decreases only for one or two integration steps. The error in the final result is smaller. We consider the relative accuracy of $10^{-4}$ sufficient for the $\\theta$ solutions. If it is worse, a warning is produced by the program.\n\n We divide the integration domain into two regions as $(v_{min},v_{par})$ and $(v_{par},v_{max})$ to obtain the number density of the particles on elliptic and hyperbolic (escaping) trajectories separately.  Here $v_{par}$ (the subscript stands for \"\\emph{\\textup{parabolic\"}}) is the minimum escape velocity at radial distance $r$. \n\n When $v_{min} > v_{min}^0$ , the integrand in Eq. (\\ref{workformula}) has a pole at $v = v_{min}$. We replace $v_{min}$ from Eq. (\\ref{vmin}) with $v_{min} + \\Delta$, where $\\Delta = 10^{-10}$ has turned out to be a good choice to evaluate the integrand near the pole to reasonable accuracy in a stable manner. To better resolve the pole, we additionally subdivide the elliptic part of the integral into two parts that are treated separately. The first integration subinterval is $(v_{min},v_{1})$, where $v_{1} = v_{min} + 10^{-4}(v_{par} - v_{min})$ is the domain that contains the pole. Here integration is performed using the trapezoidal rule with a large number of supports that become denser toward $v_{min}$ as\n \\begin{equation}\n \\label{vsteps}\n v_i = v_{min} + \\left(\\frac{i}{N}\\right)^k (v_{1} - v_{min})\n ,\\end{equation}\n where $N$ is the number of supports. Then we use the Gauss-Legendre quadrature of a moderate order to compute the integral from $v_{1}$ to $v_{par}$. The integration over the hyperbolic velocities is also performed with a Gauss-Legendre quadrature. The choice of the quadrature order depends on the integration domain and ejection speed distribution. The nodes and weights of the Gauss-Legendre formula are tabulated in our code for the following orders: 5, 10, 20, 30, 40, and 50. \n \n Figures \\ref{integrandpole} -- \\ref{integrand3} show examples for the behavior of the integrand in the three domains. Depending on the choice of the ejection speed distribution $f_u(u,R),$ the integrand may decrease (Fig. \\ref{integrand1}) or increase (Fig. \\ref{integrand2}) toward higher velocities. Remarkably, the integrand can jump at $v = v_{par}$ (Fig. \\ref{integrand3}) if a significant part of the dust number density is due to the particles on their way back to the moon after passage of their apocenter,\n\n\n\n\\begin{figure\n        \\centering\n        \\begin{minipage}{0.45\\textwidth}\n                \\centering\n                \\includegraphics[width=0.9\\textwidth]{integrand_vs_velocity_pole.pdf}\n                \\caption{Pole at $v = v_{min.}$\\\\\n                }\n                \\label{integrandpole}\n        \\end{minipage}\\hfill\n        \\begin{minipage}{0.45\\textwidth}\n                \\centering\n                \\includegraphics[width=0.9\\textwidth]{integrand_vs_velocity1.pdf}\n                \\caption{Integrand from Eq. (\\ref{workformula}) for an ejection speed distribution that favors low velocities.}\n                \\label{integrand1}\n        \\end{minipage}\\hfill\\\\\n        \\begin{minipage}{0.45\\textwidth}\n                \\centering\n                \\includegraphics[width=0.9\\textwidth]{integrand_vs_velocity2.pdf} \n                \\caption{Integrand from Eq. (\\ref{workformula}) for an ejection speed distribution that favors high velocities.\\\\\n                }\n                \\label{integrand2}\n        \\end{minipage}\\hfill\n        \\begin{minipage}{0.45\\textwidth}\n                \\centering\n                \\includegraphics[width=0.9\\textwidth]{integrand_vs_velocity3.pdf} \n                \\caption{Integrand from Eq. (\\ref{workformula}). A high abundance of dust falling back causes a jump at the transition from the elliptic to the hyperbolic case.}\n                \\label{integrand3}\n        \\end{minipage}\\hfill\n        \\captionsetup{labelformat=empty}\n\\end{figure}\n\n The sharpness of the pole varies. At each integration step, the given velocity $v$ determines the values of $\\theta$ that enter the integrand of equation \\ref{workformula} through the derivative $\\partial\\Delta\\phi \/ \\partial\\theta$ and $\\psi(\\theta)$. The latter is needed to compute the ejection angle distribution $f_{\\psi,\\lambda_M}(\\psi,\\lambda_M)$. The factor $\\partial\\Delta\\phi \/ \\partial\\theta$ is the reason for the pole. Its value depends on $\\theta$ and on the spacecraft position relative to the source. The pole is less strongly peaked if the value of $\\psi(\\theta)$ corresponds to a very unlikely ejection direction. Thus, the sharpness of the pole depends on the spacecraft position relative to the source position and also on the ejection angle distribution, and so does the number of integration steps required to achieve a given accuracy goal.\n\nThe accuracy can be estimated from Eq. (\\ref{eps}), where $P$ is the value of the pole integral (between $v_{min}$ and $v_1$) obtained with $N$ steps for the integration with the trapezoidal rule, and $N_{max}$ is the maximum reasonable number of steps. $N_{max}$ is limited by accumulated rounding errors, and we determine its value in test integrations. $I(N_{max})$ is the sum of $P(N_{max})$ and the remaining part of the integral between $v_1$ and $v_{max}$. In this way, we can quantify the discrepancy induced by the pole integration in the final result,\n\n\\begin{equation}\n        \\label{eps}\n        \\epsilon(N) = |P(N) - P(N_{max})| \/ I(N_{max})\n.\\end{equation}\n\nWe require $\\epsilon \\le 10^{-3}$ and perform tests to determine the corresponding number of pole integration steps $N$ necessary to achieve this goal. This number we compute for different spacecraft positions relative to the source and to the axis of ejection symmetry (the polar angle in the coordinate system $\\tilde{X} \\tilde{Y} \\tilde{Z}$ in Fig. \\ref{2sys}, in the following denoted by $\\xi$). We adopt \n\\begin{equation}\n\\label{psidistr}\nf_{\\tilde{\\psi}, \\tilde{\\lambda}_M}(\\tilde{\\psi}, \\tilde{\\lambda}_M)\\sin\\tilde{\\psi} = e^{-(\\tilde{\\psi}-\\tilde{\\psi}_{max})^2\/2\\omega^2} \\frac{\\sin\\tilde{\\psi}}{2\\pi}\n.\\end{equation}\nfor the ejection direction distribution. Normalization in this expression does not matter for an evaluation of $\\epsilon$ from Eq. (\\ref{eps}). We vary the parameters $\\psi_{max}$ and $\\omega$, along with the polar angle of the ejection symmetry axis, to investigate the behavior of the pole for different ejection distributions, of which two main classes can be defined. The \\textquotedblleft jets\\textquotedblright\\ are the sources with a preferred direction of ejection, and the \\textquotedblleft diffuse sources\\textquotedblright \\ have no such direction. \n We find that for diffuse ejection ($\\omega = 45^\\circ$ and $\\psi_{max} = 45^\\circ$), N = 15 is a sufficient number of supports to achieve $\\epsilon \\le 10^{-3}$ at any spacecraft position. For a vertical jet ($\\psi_{max} = 0^\\circ$ and $\\omega$ in the range of $3^\\circ$ and $5^\\circ$), the value of $P$ can be neglected for all $\\xi > 40^\\circ$ and $N = 15$ is sufficient for $\\xi < 40^\\circ$.\n \n However, for a narrow and inclined jet, we find that there are points where a large number of supports is required to integrate the pole accurately. The narrower and the more inclined the jet, the greater the number of these points and the greater the required $N$. We focus on the worst-case scenarios generally to constrain an optimal number of steps required for the pole integration.\n\nEmpirically, we find that an accuracy of $10^{-3}$ can be achieved with a minimum number of steps when an exponent of $k = 4$ is used in Eq. (\\ref{vsteps}). Figs. \\ref{accumap1} and \\ref{accumap2} show examples of how the minimum number of steps required to integrate the pole with the given accuracy is distributed over $r$ and $\\xi$ values. $N=0$ means that the pole does not have to be integrated at all because its value is negligible.\n\n\\begin{figure}[H]\n        \\centering\n        \\begin{minipage}{0.45\\textwidth}\n                \\centering\n                \\includegraphics[scale=0.5]{omega3deg_inc20.pdf}\n                \\caption{Minimum number of supports necessary to achieve an accuracy of $10^{-3}$ (equation \\ref{eps}) in the integration of the pole of the integrand (Fig.\\ref{integrandpole})  for the case of a narrow jet ($\\omega = 3^\\circ,\\ \\psi_{max} = 0^\\circ$), tilted by $z = 20^\\circ.$}\n                \\label{accumap1}\n        \\end{minipage}\\hfill\n        \\begin{minipage}{0.45\\textwidth}\n                \\centering\n                \\includegraphics[scale=0.5]{omega3deg_inc30.pdf}\n                \\caption{Minimum number of supports necessary to achieve an accuracy of $10^{-3}$ (equation \\ref{eps}) in the integration of the pole of the integrand (Fig. \\ref{integrandpole})  for the case of a narrow jet ($\\omega = 3^\\circ,\\ \\psi_{max} = 0^\\circ$), tilted by $z = 30^\\circ.$}\n                \\label{accumap2}\n        \\end{minipage}\\hfill\n        \n\\end{figure}\n\nFor jets, we select the number of supports $N$ for the pole integration\nbased on the distributions shown in Figs. \\ref{accumap1} and \\ref{accumap2} as follows: \n\\begin{equation}\n\\label{Nprestep}\nN = \\left\\{ \\begin{array}{cl}\n0,& \\xi > 45^\\circ\\ \\mathrm{or}\\ \\xi < 10^\\circ,\\ r\/r_M > 1.05,\\\\\n80,&\\xi > 45^\\circ\\ \\mathrm{or}\\ \\xi < 10^\\circ,\\ r\/r_M < 1.05,\\\\\n15 + 10z[\\mathrm{deg}],& 10^\\circ < \\xi < 45^\\circ, \\ r\/r_M <2,\\\\\n10 + 5z[\\mathrm{deg}],& 10^\\circ < \\xi < 45^\\circ, \\ r\/r_M >2.\\\\\n\\end{array}\\right.\n\\end{equation}\n\n\\section{Applications}\n\\label{sec:applications}\nIn this section we present three applications of the model to phenomena of scientific interest in the Solar System. The purpose of these examples is to demonstrate the wide range of applicability of the model. We leave a rigorous scientific analysis of these problems with a comprehensive comparison to data for future work.\n\n\\subsection{Density profile of the Enceladus dust plume}\n\\label{enceladusex}\nOn July 14, 2005, the Cassini spacecraft performed a flyby at the Saturnian moon Enceladus (labeled E2). During the flyby, the High Rate Detector (HRD), a subsystem of the Cassini Cosmic Dust Analyzer instrument \\citep{Srama:2004uz}, measured the number density of dust particles in the vicinity of the satellite. The significant increase in dust density near Enceladus (see Fig. \\ref{hrdex}), about one minute prior to the closest approach of the spacecraft to the satellite, was the first in situ measurement of particles in the Enceladus dust plume \\citep{2006Sci...311.1416S}. Dust and vapor are emitted from four fissures called the tiger stripes in the anomalously warm south polar terrain of Enceladus \\citep{2006Sci...311.1401S, 2006Sci...311.1393P}.\nA part of this dust escapes the moon gravity and forms the dusty E ring of Saturn \\citep{2009sfch.book..511H, 2018eims.book..195K}.\n\nFrom an analysis of high phase-angle images, \\citet{2014AJ....148...45P} suggested a list of 100 jets of dust emission for which the coordinates and tilts were derived from images (see also \\citet{2015Natur.521...57S}). To demonstrate an application of our model to the dust emission from Enceladus, we selected one single jet from this list with coordinates ($-80.25^\\circ$\\ N, $ 55.23^\\circ$\\ E), which is tilted by $5^\\circ$ from the surface normal in an azimuthal direction $38^\\circ$ away from local north. The ejection is stationary, so that the production rate $\\gamma(t)$ is constant. The distributions we implemented for particle sizes, ejection speed, and direction are given by \n\n\\begin{equation}\n\\label{lognorm}\nf_R(R) = \\frac{1}{\\sigma \\sqrt{2\\pi}}\\frac{1}{R}\\exp\\left(-\\frac{(\\ln R - \\mu)^2}{2\\sigma ^2}\\right)\n,\\end{equation}\n\n\\begin{equation}\n\\label{fu}\nf_u(u,R) = \\frac{R}{R_c}\\left(1 + \\frac{R}{R_c}\\right)\\frac{u}{u_{gas}^2}\\left(1 - \\frac{u}{u_{gas}}\\right)^{\\frac{R}{R_c}-1}\n,\\end{equation}\nand\n\\begin{equation}\n\\label{unicon}\nf_{\\tilde{\\psi}, \\tilde{\\lambda_M}}(\\tilde{\\psi}, \\tilde{\\lambda}_M)\\sin\\tilde{\\psi} = \\left\\{ \\begin{array}{rl}\n\\frac{\\sin\\tilde{\\psi}}{1-\\cos\\omega}\\frac{1}{2\\pi},\\ \\tilde{\\psi} \\le \\omega,\\\\\n0,\\ \\tilde{\\psi} > \\omega.\n\\end{array}\\right.\n\\end{equation}\n Equation (\\ref{fu}) was derived by \\citet{2008Natur.451..685S} to describe the acceleration of dust grains in the gas flux in the vents that supply the sources. Particles smaller than $R_c$ (measured in the same units as $R$) tend to accelerate up to the gas velocity ($u_{gas}$), while particles larger than $R_c$ are significantly slower. For this distribution, we have $u_{min} = 0,$ while $u_{max}$ is equal to the gas velocity $u_{gas}$. Table \\ref{enceladus_params} lists the parameters of the distributions and other parameters that are necessary to set up the model.\nFigure \\ref{hrdex} shows the result for the number density of dust obtained from the model for the single jet, evaluated along the trajectory of Cassini during the E2 flyby. The model was evaluated for grains with a radius larger that 1.6 micron, which corresponds to the size threshold for the HRD data shown in the plot. We multiplied the model profile by a factor so that the peak matches the measured peak density. To match the HRD profile at a large distance from the plume, we added a constant background of 0.01 particles\/$m^3$ to the model number density. The selection of the grain size in the model was realized by adjusting the function $G^p_u$ appropriately (see Table \\ref{enceladus_params}). We obtained the position of the spacecraft from the reconstructed spice kernels of the mission (https:\/\/naif.jpl.nasa.gov\/pub\/naif\/CASSINI\/kernels\/), using the NAIF Spice toolkit (https:\/\/naif.jpl.nasa.gov\/naif\/toolkit.html). Using these parameters, we recovered the location of the maximum number density on the Cassini trajectory (Fig. \\ref{hrdex}). Our model could now be applied to all the jets identified by \\citep{2014AJ....148...45P}, and the results could be fit to in situ\\emph{} data. Similarly, we could calculate the geometrical optical depths of the dust emitted along a given line of sight and compare this to the brightness distribution in images (see Sect. \\ref{ioex} for an example). For a quantitative comparison to images, we can apply light scattering modeling to the dust configuration that is derived from the dust distribution model.\n\\begin{table*}\n        \\caption{Parameters used to model the number density profile of the E2 flyby}\n        \\begin{tabular}{|c|c|}\n                \\hline\n                Parameter & Comment \\\\\n                \\hline\n        $\\gamma(t) = 1.35 \\cdot 10^{14} \\ s^{-1}$ & The dust ejection is stationary, the rate \\\\\n        & was chosen to fit the data\\\\\n        \\hline\n        Eq. (\\ref{lognorm}) with $\\mu = -1.0$ and $\\sigma = 1.5$  & Small particles dominate the population,\\\\\n        is used as a size distribution & but the distribution is not too steep\\\\\n        \\hline\n         Eq. (\\ref{fu}) with $R_c=0.5\\ \\mu m$  & This expression describes  \\\\\n         and $u_{gas} = 1000$ m\/s is used  & the dust acceleration by the gas flux \\\\\n         as the ejection speed distribution&inside the channels as it is at Enceladus\\\\\n         \\hline\n         Eq. (\\ref{unicon}) with $\\omega=10^\\circ$ is used& The distribution describes uniform emission \\\\\n         as the distribution  & into a cone of a given width\\\\\n         of the ejection direction&\\\\\n         \\hline\n         $R_{min} = 1.6 \\ \\mu m$ & The lower sensitivity threshold of the HRD \\\\\n         \\hline\n          & An arbitrarily chosen but reasonably  \\\\\n           &high value, meaning that we can neglect\\\\\n           & particles with larger sizes.  With our\\\\\n         $R_{max} = 6 \\ \\mu m$ & choice of size distribution and the size  \\\\\n         & -dependent ejection speed distribution, it \\\\\n          &is highly unlikely that the HRD detects \\\\\n          & a particle of this size at the altitude\\\\\n          &  of E2 flyby\\\\\n          \\hline\n         $p=0$ & We are interested in the number density \\\\\n         \\hline\n        \\end{tabular}\n        \\label{enceladus_params}\n\\end{table*}\n\\begin{figure}[H]\n        \\centering\n        \\includegraphics[width=0.5\\textwidth]{hrd_profile}\n        \\caption{Dust number density profile observed by the HRD during the E2 flyby of the Cassini spacecraft at Enceladus and results from modeling the emission from one single jet in the south polar terrain (see text for details).}\n        \\label{hrdex}\n\\end{figure}\n\n\n\n\\subsection{Surface deposition of material from plumes on Europa}\n\\label{europaex}\nThere is evidence that cryovolcanic activity also generates plumes on the Jupiter satellite Europa \\citep{2000Icar..144...54F, 2000JGR...10522579P, 2014Sci...343..171R,Sparks:2016gl,Jia:2018ii}. Owing to the higher gravity of Europa, plumes would be more confined than those on Enceladus, and they may be harder to observe or sample directly \\citep{2015GeoRL..4210541S, Quick:2020kp}. However, surface features from plume deposits may provide evidence for past cryovolcanic activity, and if spectral features exist, allow the remote characterization of material from the interior \\citep{2000Icar..144...54F,2000JGR...10522579P,Quick:2020kp}. \n\n\nWe calculated the radial variation of the mass flux of dust grains falling back onto the surface, considering four different dust sources with identical characteristics except for the particle size distribution and the distribution of ejection directions. We considered two size distributions and two ejection modes: one describing a narrow jet, and the other describing a broader, more diffuse ejection. For the size distributions we employed a power law\n\n\\begin{equation}\n\\label{pow}\nf_R(R) = \\frac{1-q}{R_2^{1-q}-R_1^{1-q}}R^{-q}\n,\\end{equation}\nwith two different values of the exponent $q$. The greater $q$, the more abundant the small dust particles. For the ejection direction, we used a pseudo-Gaussian distribution of the polar angle \n\\begin{equation}\n\\label{Gauss}\nf_{\\tilde{\\psi}, \\tilde{\\lambda}_M}(\\tilde{\\psi}, \\tilde{\\lambda}_M)\\sin\\tilde{\\psi} = C_{norm}e^{-(\\tilde{\\psi}-\\tilde{\\psi}_{max})^2\/2\\omega^2} \\frac{\\sin\\tilde{\\psi}}{2\\pi}\n,\\end{equation}\ngiving a nonzero probability of ejection in any direction. The normalization constant $C_{norm}$ was found numerically for fixed values of $\\psi_{max}$ and $\\omega$. The dust production rate $\\gamma$ was identical for all the sources, meaning that they produced the same number of dust particles per unit time, but we then obtained different mass production rates for the different size distributions. For the ejection speed distribution we again used Eq. (\\ref{fu}). Table \\ref{europa_params} lists the distribution parameters.\n\nWe can compute the rate of dust mass produced per second from the size distribution as\n\\begin{equation}\n        \\label{totmass}\n        \\frac{\\mathrm{d}m}{\\mathrm{d}t} =\\gamma\\rho\\frac{4\\pi}{3}\\int_{R_{min}}^{R_{max}}f_R(R)R^3\\mathrm{d}R,\n\\end{equation}\nwhere $\\rho$ is the density of the ice grains, and the grains were assumed to be spherical.\n\nTo compute flux instead of density, we must modify Eq. (\\ref{workformula}) as\n\\begin{multline}\n        \\label{flux}\n        n(r,\\alpha,\\beta,R_{min}<R<R_{max}) = \\frac{\\gamma}{rr_M\\sin\\Delta\\phi} \\int_{v_{min}}^{v_{max}}\\frac{G^p_u(R_{min},R_{max})}{u^2}\\times\\\\\\times \\sum_{i}\\frac{f_{\\psi,\\lambda_M}(\\psi_i,\\lambda_{Mi})}{\\cos\\psi_i}\\abs*{\\frac{\\partial\\Delta\\phi}{\\partial\\theta}}_{\\theta^*_i}^{-1}v^2|\\cos\\theta_i^*|dv.  \n\\end{multline}\nAs  the vector $\\mathbf{r}$ is normal to the surface of the satellite, the factor $v|\\cos\\theta|$ turns the density into the flux of dust falling back to the moon. Multiplying equation \\ref{flux} by $4\\pi\\rho\/3 $ ($\\rho$ is the density of the particle material) and setting $p=3$ and $r=r_M$ gives the mass flux onto the surface. Because $r = r_M$ on the surface, it is geometrically impossible to obtain $\\theta^*_i < \\pi\/2$ or any particles moving upward.\n\nWith the parameters from Table \\ref{europa_params}, Eq (\\ref{totmass}) gives $0.6$ kg of dust produced each second for the source with the shallow size distribution ($q=3$) and $0.05$ kg for the source with the steep size distribution ($q=5$). Figure \\ref{deposits} shows the distribution of the dust deposition (mass flux onto the surface) with distance from the source on the moon surface.\n \\begin{table*}\n        \\caption{Parameters used to model the radial distribution of dust deposition on the surface of Europa}\n        \\begin{tabular}{|c|c|}\n                \\hline\n                Parameter & Comment \\\\\n                \\hline\n        $\\gamma(t) = 10^{14} \\ s^{-1}$ & The dust ejection is considered stationary\\\\\n                \\hline\n        Eq.( \\ref{pow}) with $q = 3$, $R_1 = 0.2\\ \\mu m$& We paired these size and ejection \\\\\n          and $R_2 = 20\\ \\mu m$ as   & direction distributions\\\\\n           the shallow\\ size distribution & to model four sources: a narrow jet\\\\\n         and with the same boundaries, but $q=5$ & with a shallow size distribution,\\\\\n         as the steep\\ size distribution & a narrow jet with a steep size distribution,\\\\\n         & a diffuse source  with a shallow size\\\\\n         Eq. (\\ref{Gauss}) with $\\omega=5^\\circ$ and $\\psi_{max} = 0^\\circ$&  distribution, and a diffuse source\\\\\n         for the  narrow jet\\ and with $\\omega=45^\\circ$ & with a steep size distribution \\\\\n         and $\\psi_{max}=45^\\circ$ for the  diffuse source& \\\\\n        \\hline\n                Eq. (\\ref{fu}) with $R_c=0.5\\ \\mu m$  & This expression describes  \\\\\n                and $u_{gas} = 700$ m\/s is used&the acceleration of the dust by the gas  \\\\\n                 as the ejection speed distribution &flux inside the vents as it is thought be at Europa \\\\\n                \\hline\n & We wish to compute the total mass\\\\\n                $R_{min} = 0.2 \\ \\mu m$ &  produced by each source, \\\\\n                $R_{max} = 20 \\ \\mu m$ & therefore the interval $(R_{min},R_{max})$ \\\\\n &coincides with $(R_1,R_2)$\\\\\n                \\hline\n                $p=3$ & We are interested in the mass \\\\\n                \\hline\n                $\\rho = 920\\ kg\/m^3$ & Water-ice density\\\\\n                \\hline\n        \\end{tabular}\n        \\label{europa_params}\n \\end{table*}\n \\begin{figure}[H]\n        \\centering\n        \\includegraphics[width=0.5\\textwidth]{mass_deposition}\n        \\caption{Dust flux on the surface of the moon at various distances from the dust source. The production rate of the sources with shallow dust size distribution is 0.6 kg\/s, and the production rate of the sources with steep size distribution is 0.05 kg\/s.}\n        \\label{deposits}\n \\end{figure}\n\nTaking the observational evidence together, we expect the outgassing activity on Europa to be intermittent or episodic, with a potentially complex time dependence. This case could be modeled using a time-dependent function for the dust production rate. We consider a time-dependent dust ejection in our final example.\n\\subsection{Images of a volcanic eruption on Io}\n\\label{ioex}\nThe innermost satellite of Jupiter, Io, is a geologically very active body. It has multiple volcanic centers scattered over its surface \\citep{1979Natur.280..733S,2001JGR...10633025K, 2004Icar..169...29G}. The volcanic plumes vary in many features as shape, size and period of activity and often they are asymmetric. Our model allows to investigate such plumes by using inclined jets with time-dependent eruptions, or even a simultaneous dust ejection from sources with different properties.\n\nThe main flaw of our model in application to the Io volcanic plumes is that the gas emerging from the vents is generally not collisionless. In the dense cores of the plumes, condensation occurs up to a height of several hundred kilometers \\citep{2002JGRE..107.5109C}. Moreover, the dynamics of the dust will be influenced by the surrounding gas. Condensation (i.e.,\\ dust production) in the plume might be taken into account in principle by modeling such a plume with additional point sources placed within the condensation column, with appropriately adjusted time-dependent dust production rates. For this purpose, we implemented in the code the possibility of computing the dust density also at locations below (closer to the moon center) the location of the source. In this case, we still use the assumption that the true anomaly at the location of the source is lower than $\\pi$, that is, downward ejection from a source located above the surface is not possible. \n\nIn our simple example presented here, we considered just one source located on the surface to represent a volcanic plume with a finite duration of activity. We took several images of the expanding dust plume from the same point in space with a favorable geometry when the volcano is located at the limb of the moon. We also assumed that the glow of the surface of Io was already subtracted from the images but a homogeneous background brightness was present, so that the moon disk looks dark against this background. We tilted the volcano slightly by $3^\\circ$ toward local south. The volcano was not exactly in the center of the image for the purpose of reducing the computational difficulties arising when $\\Delta\\phi$ or $\\Delta\\beta \\approx 0$ (see Sect. \\ref{spepoints}). \n\nSynthetic images of size 128x128 pixels were then constructed in the following way. Each pixel corresponds to a line of sight. We placed a grid of points on the line of sight for which we calculated the total cross section covered by the dust grains ($p=2$ in Eq. (\\ref{Gu})). Then we integrated over the lines of sight to obtain the geometrical optical depth (total particle cross section per area covered by the pixel). The color of a pixel then corresponded to the value determined for the geometric optical depth.\n\nWe employed Eq. (\\ref{pow}) for the size distribution, Eq. (\\ref{Gauss}) for the distribution of the ejection direction, and a simple size-independent expression (Eq. (\\ref{fuun})) for a uniform ejection speed distribution,\n\\begin{equation}\n\\label{fuun}\nf_u(u,R) = \\frac{1}{u_{max} - u_{min}}\n.\\end{equation}\n\nWhen the time interval in which the plume is observed is much shorter than the characteristic time of the plume variability, the ejection can be considered steady. However, here we modeled a nonstationary volcanic plume that was active for only 1000 seconds and constructed nine snapshots of this plume to show the evolution of the space distribution of dust with time \n\\begin{equation}\n\\label{gammat}\n\\gamma(t) = \\left\\{ \\begin{array}{rl}\n\\gamma_0\\frac{-t^2+2t_{max}t}{t_{max}^2} ,\\ 0 < t < 2t_{max},\\\\\n0,\\ t < 0\\ \\text{or}\\ t > 2t_{max},\n\\end{array}\\right\n.\\end{equation}\nwhere $t_{max} = 500$ s. Table \\ref{io_params} describes the model setup, and the images are shown in Fig. \\ref{9im}\n\\begin{table*}\n        \\caption{Parameters used to construct the volcanic plume images.}\n        \\begin{tabular}{|c|c|}\n                \\hline\n                Parameter & Comment \\\\\n                \\hline\n                Eq. (\\ref{gammat}) with $t_{max} = 500$ s  & The volcanic plume \\\\\n                and $\\gamma_0 = 10^{14} \\ s^{-1}$ as the time- & ejected dust for 1000 s\\\\\n                dependent dust production rate& \\\\\n                \\hline\n                Eq. (\\ref{pow}) with $q = 3.0$, & \\\\\n                 $R_1 = 0.2\\ \\mu m$, and $R_2 = 20\\ \\mu m$  & \\\\\n                is used as the size distribution & \\\\\n                \\hline\n                The ejection speed is uniformly  & The narrow range of initial  \\\\\n                distributed between &velocities allows us to obtain \\\\\n                700 m\/s and 750 m\/s & the umbrella-shaped plume\\\\\n                \\hline\n                Eq. (\\ref{Gauss}) with $\\psi_{max} = 0^\\circ$ & The jets are very narrow,   \\\\\n                and $\\omega=5^\\circ$ is used as the distribution  &but there is a nonzero probability \\\\\n                of the ejection direction & of ejection in any direction\\\\\n                \\hline\n                $R_{min} = 0.2 \\ \\mu m$ & We observe the particles with the sizes \\\\\n                $R_{max} = 0.4 \\ \\mu m$ &  close to the optical wavelength range, \\\\\n                 & so that the radii are twice smaller\\\\\n                \\hline\n                $p=2$ & We are interested in the area \\\\\n                & covered by the particles \\\\\n                \\hline\n        \\end{tabular}\n        \\label{io_params}\n\\end{table*}\n\n\\begin{figure*}[h!]\n        \\centering\n        \\begin{minipage}{0.33\\textwidth}\n                \\centering\n                \\includegraphics[width=0.95\\textwidth]{5}\n        \\end{minipage}\\hfill\n        \\begin{minipage}{0.33\\textwidth}\n                \\centering\n                \\includegraphics[width=0.95\\textwidth]{10}\n        \\end{minipage}\\hfill\n        \\begin{minipage}{0.33\\textwidth}\n                \\centering\n                \\includegraphics[width=0.95\\textwidth]{15}\n        \\end{minipage}\\hfill\\\\\n        \\vskip0.1cm\n        \\begin{minipage}{0.33\\textwidth}\n                \\centering\n                \\includegraphics[width=0.95\\textwidth]{20}\n        \\end{minipage}\\hfill\n        \\begin{minipage}{0.33\\textwidth}\n                \\centering\n                \\includegraphics[width=0.95\\textwidth]{25}\n        \\end{minipage}\\hfill\n        \\begin{minipage}{0.33\\textwidth}\n                \\centering\n                \\includegraphics[width=0.95\\textwidth]{30}\n        \\end{minipage}\\hfill\n        \\vskip0.1cm\n        \\begin{minipage}{0.33\\textwidth}\n                \\centering\n                \\includegraphics[width=0.95\\textwidth]{35}\n        \\end{minipage}\\hfill\n        \\begin{minipage}{0.33\\textwidth}\n                \\centering\n                \\includegraphics[width=0.95\\textwidth]{40}\n        \\end{minipage}\\hfill\n        \\begin{minipage}{0.33\\textwidth}\n                \\centering\n                \\includegraphics[width=0.95\\textwidth]{45}\n        \\end{minipage}\\hfill\n        \\caption{Images of a fictive volcanic plume on Io taken at different stages of eruption.}\n        \\label{9im}\n\\end{figure*}\n\nCompared to the examples from Sects \\ref{enceladusex} and \\ref{europaex}, the construction of images is computationally expensive. Therefore it makes sense to avoid calculations of the properties in those points where the value would not affect the final result. In the case considered here, the maximum ejection velocity was set to be lower than the escape velocity of Io, so that there is a maximum height that the particles can reach. Moreover, the disk of the moon covers part of the images. Taking these two facts into account, we excluded the points on the lines of sight from the calculations that crossed the moon disk and the points that lie at a greater distance from the surface than the maximum height. \n\n \n\\section{Discussion}\nWe developed a semianalytical model that allows the user to derive the average properties of dust (number or mass density, fluxes, and optical depths) ejected from an atmosphereless body. Physically, our approach is based on the two-body problem, that is,\\ it neglects any forces on the dust particles other than the point-mass gravity of the source body. We show that the model still has a wide range of applications. These are situations where the gravity of other perturbing bodies and higher-order gravity from a nonspherical mass distribution of the central body can be neglected along the entire path that a dust particle takes from its source to its sink. The nongravitational forces must also be negligibly small, for instance, electromagnetic forces acting on charged grains, radiation-induced forces (solar radiation pressure and Poynting-Robertson drag), and drag exerted by ambient gas or plasma. For instance, the model can be applied to the Enceladus dust plume in a region that is sufficiently close to the dust sources on the south polar terrain of this satellite. It cannot be applied to estimate the dust density at higher northern latitudes of Enceladus, however, because this region is too far away from the sources near the south pole, and three-body forces due to Saturn have already affected the shape of the dust configuration. This can be seen in a peculiar pattern of color variation \\citep{2011Icar..211..740S}, that can be matched with two wedge shaped regions extending deep into the northern hemisphere \\citep{2011epsc.conf.1358S,2017LPI....48.2601S}, for which three body models of the plume predict enhanced fall back rates of south polar dust \\citep{2010Icar..206..446K}. The model can be applied to investigate the dust clouds around the Galilean moons \\citep{1999Natur.399..558K} at any longitude and latitude, however, because in this case, the dust emission occurs (nearly) uniformly over the whole surface of the satellites as long as the point of interest is located deeply enough in the Hill sphere of the satellite. \n\nMathematically, our model relates the dust distribution at the site of ejection to the distribution at the point of interest (spacecraft position) by using the conservation laws of energy and momentum provided by the two-body problem \\citep{2003P&SS...51..251K, Sremcevic:2003gf}. We use the fact that the position of the spacecraft, the position of the source, and the center of the moon define the plane to which the movement is restricted. In the evaluation of the dust properties at a given point, this allows us to carry out two of the three integrations over velocity space analytically. Only one remaining integration must then be performed numerically. The distribution-based approach is very flexible, and because it allows employing asymmetric and nonstationary modes of dust emission, it allows modeling quite complex situations. A time dependence could relatively easily be introduced in the distributions of ejection speed, direction, and size as well. Furthermore, the model could also be extended to model dust emission from a comet or an active asteroid, making the Sun the central body and using a rescaled solar mass to account for the radiation pressure. \n\n\nRelying only on one numerical integration, the model becomes computationally very efficient, so that even image reconstruction becomes feasible, although it involves the evaluation of the dust properties in the three-dimensional region that is in the field of view. From the examples discussed in Sect. \\ref{sec:applications}, the image of the volcano (Sect. \\ref{ioex}) is the most computationally demanding. Nevertheless, on a usual four-core PC, each image in Fig. \\ref{9im} required 0.4 s of elapsed time to be obtained.\n\n The model is implemented in Fortran-95, and the package called DUDI is available at \\\\\n https:\/\/github.com\/Veyza\/dudi for free usage under the terms of GNU General Public License. A user can use the  probability density functions and choose from several variants that are already implemented in the described examples, or they can implement new variants.  \n\n\\section*{Acknowledgements}\nThis work was supported by the Academy of Finland.\n\n\n\\section*{Appendix A. Replacement of the variable in the argument of Dirac's $\\delta$-function}\n\\label{replsec}\n\nLet $x\\in S \\subset \\mathbb{R}^n$; $f,g : S \\rightarrow \\mathbb{R}^n$. Then we can perform a replacement of the variable under the integral based on Eqs. (\\ref{dg}) and (\\ref{dint}) from \\citet{GelfShil},\n\\begin{equation}\n\\label{dg}\n\\delta(g(x)) = \\sum_{i}\\frac{\\delta(x-x_i)}{|g'(x_i)|}\n.\\end{equation}\nHere, $g'$ is the Jacobian matrix of the function $g$, $|g'|$ denotes the Jacobi determinant, and $x_i$ are the zeros of $g$,\n\\begin{equation}\n\\label{dint}\n\\int_{\\mathbb{R}^n}\\delta(g(x))f(g(x))|g'(x)|dx = \\int_{g(S)}\\delta(h)f(h)dh\n\\end{equation}\n\n\\begin{eqnarray}\n\\nonumber\n\\int_S \\delta(g(x))f(x)dx &=& \\int_S \\delta(g(x))f(g^{-1}(g(x)))dx\\\\\n\\nonumber\n&=& \\int_S\\delta(g(x))f^*(g(x))dx \\\\\n\\nonumber\n&=&\\int_S \\delta(g(x))\\frac{f^*(g(x))}{| g'(x)|}|g'(x)|dx \\\\\n\\nonumber\n&=&\\int_S \\delta(g(x))\\frac{f^*(g(x))}{| g'(g^{-1}(g(x)))|}|g'(x)|dx \\\\\n\\nonumber\n&=&\\int_S \\delta(g(x))f^*_D(g(x))|g'(x)|dx \\\\\n\\nonumber\n&=& \\int_{g(S)}\\delta(h)f^*_D(h)dh,\n\\end{eqnarray}\nwhere\n\\begin{equation*}\nf^*_D(h) = \\frac{f(x)}{\\abs*{g'(x)}},\\quad h = g(x).\n\\end{equation*}\nUpon integration, we obtain\n\\begin{equation*}\n\\int_{g(S)}\\delta(h)f^*_D(h)dh = \\sum_{i}\\frac{f(x_i)}{\\abs*{g'(x_i)}},\\ \\ g(x_i) = \\textbf{0}.\n\\end{equation*}\n\nIn our case, $n=2,$ the function $g$ performs a transformation from $(\\alpha_M,\\beta_M)$ to $(\\theta,\\lambda),$ and the function $f$ represents all the dependences on $\\theta$ and $\\lambda$ in the integrand in Eq. (\\ref{ndens}).\n\n\\section*{Appendix B. Physical meaning of the function $G_u$}\nFor each source, the first step in the numerical calculations is to obtain values of $G^p_u(R_{min}, R_{max})$ (Eq. (\\ref{Gu})) on a dense grid of $u$. This means that we perform integrations over the particle size $R$ treating $u$ as a parameter. The size distribution $f_R(R)$ can be defined on an interval of grain sizes, its lower and upper boundaries being parameters of the distribution. \n\nThese boundaries need not necessarily be equal to $R_{min}$ and $R_{max}$. Instead, $R_{min}$ and $R_{max}$ should be understood as limits of an observable interval of particle radii. Particles smaller than $R_{min}$ or larger than $R_{max}$ may exist, and they contribute to the normalization of $f_R(R)$, but they would not contribute to the value of $n$ (Eq. (\\ref{workformula})). This concept is illustrated in Figs. \\ref{GRexlognorm} and \\ref{GRexpower}. The quantity plotted in the two figures is $f_R(R)f_u(u,R)$. In Fig. \\ref{GRexlognorm} ,$f_R(R)$ is a lognormal distribution that is formally defined in the interval $(0,+\\infty)$, or in other words, $\\int_{0}^{+\\infty}f_R(R)dR = 1$, and the interval $(R_{min}, R_{max})$ does not cover the whole domain of $f_R(R)$. In Fig. \\ref{GRexpower}, the particle sizes are distributed between certain $R_1$ and $R_2$ as Eq. (\\ref{pow}). In this example, $R_{min} = R_1$ , but $R_{max} < R_2$. If $R_{min}$ were lower than $R_1$ , it would not have changed the result because for $R < R_1$ , we have $f_R(R) = 0$.\n\\begin{figure}\n        \\centering\n        \\begin{minipage}{0.45\\textwidth}\n                \\centering\n                \\includegraphics[width=0.9\\textwidth]{GR_lognorm_example.pdf}\n                \\caption{Integrand in Eq (\\ref{Gu}) with a lognormal size distribution and a fixed value of $u.$}\n                \\label{GRexlognorm}\n        \\end{minipage}\\hfill\n        \\begin{minipage}{0.45\\textwidth}\n                \\centering\n                \\includegraphics[width=0.9\\textwidth]{GR_powerlow_example.pdf}\n                \\caption{Integrand in Eq. (\\ref{Gu}) with a power-law size distribution and a fixed value of $u.$}\n                \\label{GRexpower}\n        \\end{minipage}\\hfill\n        \\captionsetup{labelformat=empty}\n\\end{figure}\n\nAlthough we defined the size distribution outside the interval $(R_{min}, R_{max}),$ its shape there does not affect the final result as long as it remains the same inside $(R_{min}, R_{max})$. However, we can investigate the defined size distribution by applying different values of $R_{min}$ and $R_{max}$. This approach implies that the size distribution and the initial speed distribution are physical characteristics of the dust source, while $R_{min}$ and $R_{max}$ represent the sensitivity range of the instrument with which we perform observations. In the model implementation DUDI, the interval $(R_{min},R_{max})$ can coincide with the $f_R(R)$ domain or can be even larger.\n\nThe units of the particle radius $R$ matter only in the definition of $f_R$ and $f_u$, so that we suggest measuring $R$ in microns to have simpler numbers in the expressions.\nWith different formulae for $f_R$ and $f_u$ , shorter expressions for $G^p_u$ can be obtained that require less cumbersome computations (see, e.g., \\citet{2011Natur.474..620P}). However, we purposefully consider a set of $f_R$ and $f_u$ in our model that cannot be simplified to show the general form of the solution and to allow flexibility in applying the model.\n\\newpage\n\\bibliographystyle{apalike}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{INTRODUCTION}\n\t\nStudies on the topology of band structures in ${\\bf k}$-space reveal various nontrivial phases of insulators and semimetals~\\cite{Hasan10,Qi11,Hirayama18r}.\nIn recent years, higher-order topological phases have attracted high attention,\nwhere higher-order topological states appear due to nontrivial bulk topology.\nThere are two types of higher-order topological phases: those supporting currents~\\cite{Langbehn17,Song17,Schindler18,Schindler18b,Tang18} and those supporting charges~\\cite{Benalcazar17,Benalcazar17b,Peterson18,Benalcazar19,Schindler19}.\nOne of the realistic materials having chiral hinge states is bismuth.\nSuch chiral hinge states have been confirmed by the scanning tunneling microscope (STM)~\\cite{Schindler18b}.\nThe other example is Bi$_4$Br$_4$, where the bulk is completely insulating unlike bismuth~\\cite{Tang18}.\nOn the other hand, there has been no proposal for a realistic three-dimensional material having higher-order topological states supporting quantized charges.\n\nHere, we focus on electride as a system that can realize a higher-order topological phase.\nIn electrides, electrons enter the voids surrounded by cations\nand are responsible for stabilizing the structure as anions~\\cite{Ellaboudy83,Singh93,Sushko03,Matsuishi03}.\nInterstitial electrons are not strongly stabilized by the electric field from positive ions,\nas compared to electrons belonging to atomic orbitals stabilized by the strong electric field from the nuclei.\nTherefore, the work function of an electride is generally small~\\cite{Singh93,Toda07}, and electrides are actively studied in the chemistry field such as catalyst~\\cite{Kitano12}.\nCompared with covalent crystals originating from $p$ orbitals, there is less structural instability in electrides when making the cleavage plane~\\cite{Hirayama18}.\n\nIn recent years, it has been proposed that electrides have a high affinity with topological phases in physics~\\cite{Hirayama18}.\nSince the interstitial electrons lead to a small work function, the bands originating from interstitial electrons appear near the Fermi level.\nTherefore, electrides are likely to have band inversion.\nActually, various topological phases with\/without the spin-orbit interaction (SOI) are proposed in the electrides~\\cite{Hirayama18,Huang18,Zhang18,Park18}.\nFor example, a two-dimensional electride Sc$_2$C is a nontrivial insulator having a quantized Zak phase $\\pi$~\\cite{Hirayama18}.\nBecause anionic electrons exist in the two-dimensional regions between Sc$_2$C layers, floating topological charges protected by the bulk topology appear on the surface.\n\n\nIn this letter, we propose an apatite electride as a higher-order topological crystalline insulator (TCI).\nThe apatite electride is a one-dimensional electride, having anionic electrons in one-dimensional hollows surrounded by cations~\\cite{Zhang15}.\nWe show that quantized topological charges protected by the $C_6$ symmetry appear at the hinge.\n\n\\begin{figure}[htp]\n\t\\centering \n\t\\includegraphics[clip,width=0.45\\textwidth ]{fig1.pdf} \n\n\n\t\\caption{Bulk structure and symmetry of the topological apatite A$_6$B$_4$(SiO$_4$)$_6$.\n\t\t(a) Crystal structure. \n\t\tThe green, brown, black and gray balls represent A, B, Si and O atoms, respectively.\n\t\t(b) One-dimensional hollow in the crystal structure.\n\t\t(c) Symmetry of the unit cell.\n\t}\n\t\\label{str}\n\n\\end{figure} \n\n\n\\begin{figure*}[htp]\n\t\\includegraphics[width=15cm]{fig2.pdf}\n\t\\caption{\n\t\tElectronic band structure.\n\t\t(a) (b) Electronic band structure of the La, Y apatite, respectively. \n\t\tThe energy is measured from the Fermi level.\n\t\t(c) Brillouin zone for bulk and the ($01\\bar{1}0$) surface.\n\t\t(d-f) Eigenfunctions at the $\\Gamma$ point.\n\t\tThe blue and yellow represent the positive and negative components of the wave function, respectively.\n\t\tThese eigenstates are marked in the band structure in (a).\n\t\t(g) Electronic band structure of the Wannier functions in the La apatite.\n\t\tThe four Wannier functions are at Wyckoff position $4e$.\n\t\tThe blue solid line is the Wannier band and the black dotted line is the Kohn-Sham band for comparison.\n\t\tThe band in the inset originates from the twelve Wannier functions at Wyckoff position $12i$.\n\t\tThe energy is measured from the Fermi level.\n\t}\n\t\\label{bulk}\n\\end{figure*} \n\n\n\nWe calculate the band structures within the density functional theory.\nThe electronic structure is calculated using the generalized gradient approximation (GGA).\nWe use VASP for the lattice optimization. \nThe energy cutoff is 50 Ry for the calculation by VASP~\\cite{vasp}.\nWe use the \\textit{ab initio} code OpenMX for the calculation of the electronic structure~\\cite{openmx}. \nand the energy cutoff for the numerical integrations is 150 Ry for the calculation by OpenMX.\nThe $6\\times 8\\times 6$ regular ${\\bf k}$-mesh is employed for the bulk.\nWe construct Wannier functions from the Kohn-Sham bands, using the maximally localized Wannier function~\\cite{Marzari97,Souza01}.\nThe density of states on the ($01\\bar{1}0$) surface is calculated by the recursive Green's function method~\\cite{Turek97}.\nIn the slab (wire) calculations, we tune the on-site potential of the Wannier function at the surface (hinge) to satisfy the charge neutrality condition, respectively.  \nWe confirm the effect of the charge neutrality and the long-ranged Coulomb interaction by first-principles calculations (see Supplementary Material~\\cite{SM6}).\n\t\n\t\n\t\n\t\n\t\n\n\nFigure~\\ref{str}(a) shows the crystal structure of the apatite electride A$_6$B$_4$(SiO$_4$)$_6$.\nCrystals with apatite structure are so stable that they appear in nature. The most well-known one is\nhydroxyapatite Ca$_{10}$(PO$_4$)$_6$(OH)$_2$, \nthat is the main constituent of bones and teeth.\nIt has a hexagonal crystal structure having a cleavage plane of hexagonal shape around the (0001) direction~\\cite{Aizawa05}.\nThere are no atoms on the boundary of the unit cell except for the top and bottom planes.\nSince the apatite has a one-dimensional structure, it is used in the study of ionic conductors.\nExperimentally, the apatite electride A$_6$B$_4$(SiO$_4$)$_6$ can be synthesized from A$_6$B$_4$(SiO$_4$)$_6$O$_2$ by releasing two oxygen atoms located in a region surrounded by cations.\nThe apatite electride has a one-dimensional hollow along the $z$-axis as shown in Fig.~\\ref{str}(b).\nThe positive ions in the hollows are located at $z=1\/4$ and $3\/4$, where the lattice constant \nalong the $z$-direction is set to be unity.\nSince the Wannier orbitals in the interstitial region are $s$-orbital-like and there is no nucleus inside the interstitial region,\nthe SOI of the interstitial electron is very weak.\nIn the following, the apatite refers to the apatite electride, unless otherwise specified.\n\n\n\nFigure~\\ref{str}(c) is the spatial symmetry of the apatite.\nThe space group of the apatite is No. 176 ($P6_3\/m$).\nOne of the space inversion centers lies at $(x, y, z) = (0, 0, 0)$ of the unit cell.\nIt also has $C_6$ screw rotation symmetry with $1\/2$ translation along the $c$-axis with its screw axis at the hollow $(x, y) = (0, 0)$.\nThe unit cell consists of two triangular prisms, which are transformed to each other by space inversion.\nEach triangular prism region has the $C_3$ rotational axis around the $c$-axis at the center.   \nThe system also has mirror symmetry perpendicular to the $c$-axis.\n\n\n\\begin{figure*}[htp]\n\t\\includegraphics[width=15cm]{fig3.pdf}\n\t\\caption{Surface state.\n\t\t(a) ($01\\bar{1}0$) slab. The purple and the red circles represent the hollows along the $c$-axis supporting the bulk and surface bands, respectively, and the numbers in the figure represent\n\t\tthe fillings (i.e. the number of filled bands) in the bulk and the surface. \n\t\t(b) Electronic band structure of the La apatite for the ($01\\bar{1}0$) surface.\n\t\tThe energy is measured from the Fermi level.\n\t\t(c)  Electronic band structure of the La apatite for the ($01\\bar{1}0$) surface subtracted the bulk contribution.\n\t}\n\t\\label{surface}\n\n\\end{figure*}\n\n\nFigures~\\ref{bulk}(a) and (b) show electronic band structures of La and Y apatites, respectively.\nThe corresponding Brillouin zone is shown in Fig.~\\ref{bulk}(c).\nWhile the A site is La$^{3+}$\/Y$^{3+}$,\nthe B site is completely dissolved with La$^{3+}$\/Y$^{3+}$ and another positive ion or void and has $+2.5e$ charge on average.\nIn our numerical calculation, such mixed states are handled by the virtual crystal approximation.\nThe La apatite is an insulator, while the Y apatite has a semimetallic band structure.\nReflecting a quasi one-dimensional crystal structure, they have a large band dispersion along the $k_z$ direction.\nBoth the valence band and the conduction band near the Fermi level originate from the interstitial electron as shown in Figs.~\\ref{bulk} (d)-(f).\nThe corresponding Wannier bands for the interstitial electron are shown in Fig.~\\ref{bulk}(g).\nAs shown in Supplementary Material~\\cite{SM1}, we find that many topological materials such as Na$_3$Bi~\\cite{Wang12} and Ca$_3$P$_2$~\\cite{Xie15,Chan16} are topological electrides in which either the bottom of the conduction band or the top of the valence band originates from the interstitial orbital.\nNo topological material has been known so far in which the band inversion occurs between the interstitial orbitals, and apatite is a first example of such a case.\nWe discuss the irreducible representation (irrep) for the occupied bands in Supplementary Material~\\cite{SM2}.\nThe detail of the Wannier function is also shown in Supplementary Material~\\cite{SM3}.\n\n\n\nWe show our results on surface states of the La apatite. \nWe consider the slab with the $(01\\bar{1}0)$ surfaces as shown in Fig~\\ref{surface}(a),\nwhich preserves space inversion and $C_2$ symmetry.\nFigure~\\ref{surface}(b) is the surface states calculated\nusing the recursive Green's function.\nThe effect of the lattice optimization is shown in Supplementary Material~\\cite{SM2}.\nThe surface states are gapped similarly to the bulk states.\nThis gapped surface state means that the Zak phase of the bulk is zero~\\cite{Hirayama18r,Vanderbilt93,Hirayama17}.\nThe detail of the Zak phase is discussed in Supplementary Material~\\cite{SM4}.\nTwo spinless states are occupied in the hollow in the bulk, and one spinless state is occupied at the surface (Fig.~\\ref{surface}(a)), excluding the double degeneracy due to the spins.\nBecause the filling of the surface bands is an integer, the surface bands are gapped, as seen from Fig.~\\ref{surface}(c).\n\n\n\nNext we show the hinge band structure of the La apatite.\nHere, we consider a hexagonal prism, which preserves the space inversion and the $C_6$ symmetry (Fig.~\\ref{hinge}(a)).\nFigures~\\ref{hinge}(b),(c) shows the one-dimensional band structure of the La apatite prism.\nWhile the band structure is gapped in the bulk and the surface, as is shown in Fig.~\\ref{surface}(b), topological gapless hinge states appear at the Fermi level.  \nSuch hinge states are protected by nontrivial topology of the bulk La apatite, \nand is characterized as a higher-order topological insulator protected by a rotational symmetry, \nproposed in Ref.~\\cite{Benalcazar19}. \nIn Ref.~\\cite{Benalcazar19}, to describe this higher-order topology, integer topological invariants for the high symmetry $\\Pi$ point is defined in terms of \nthe $C_{n'}$ eigenvalues at high-symmetry points $\\Pi$ compared to those at the $\\Gamma$ point\n\\begin{align}\n\t[\\Pi _p^{(n')}]=\\# \\Pi _p^{(n')} -\\# \\Gamma _p ^{(n')},\n\t\\label{eq:pi}\n\\end{align}\nwhere $C_{n'}$ is a local symmetry at the high-symmetry point $\\Pi$, $\\Pi _p^{(n')}$ ($=e^{2\\pi i(p-1)\/n'}$)  is the $C_{n'}$ eigenvalue labeled by $p$ at the high-symmetry point denoted by $\\Pi$, and $\\# \\Pi _p^{(n')}$ is the number of the occupied bands having the eigenvalue $\\Pi _p^{(n')}$.\nIn Ref.~\\cite{Benalcazar19}, it was proposed that \nthe topological class of the two-dimensional TCI having the $C_6$ symmetry is characterized by $\\chi ^{(6)}=([M_1^{(2)}],[K_1^{(3)}])$.\nBy using the connections between the positions of the Wannier orbitals and the irrep at high-symmetry points, the corner charge is given in terms of the topological numbers by \n\\cite{Benalcazar19}\n\\begin{align}\n\tQ_{\\rm corner}^{\\text{H}(6)}=-\\frac{|e|}{4}[M_1^{(2)}]-\\frac{|e|}{6}[K_1^{(3)}]\\ \\text{mod}\\ e,\n\t\\label{eq:Qcorner}\n\\end{align}\nwhere we adopt the convention $e=-|e|<0$, following Ref.~\\cite{Benalcazar19}. \nThe rotational symmetry along the $c$-axis in the apatite is $C_6$ screw rotation $6_3$.\nSince the apatite is insulating, the eigenvalues for the rotational symmetries are same between $k_z=0$ and $k_z=\\pi$ of the high-symmetry points,\nand we can limit our discussion to the irrpdf at $k_z=0$ for characterization of the hinge charge.\nTherefore, the $6_3$ helically symmetric system is essentially the same as the 6-fold symmetric system in terms of the topological charges.\n\n\n\n\n\\begin{figure*}[htp]\n\t\\includegraphics[width=15cm]{fig4.pdf}\n\t\\caption{Topological hinge states and higher-order TCI phase in the apatite electride.\n\t\t(a) Crystal with a shape of a hexagonal prism along the $c$ axis. Along the $a$,$b$ directions, it consists of triangular blocks\n\t\tand electrons are located at the Wyckoff position $a$. \n\t\tThe purple, red and orange circles represent the hollows along the\n\t\t$c$-axis supporting the bulk, surface, and hinge bands, respectively, and the numbers in the figure represent\n\t\tthe fillings (i.e. the number of filled bands) of the hollows in the bulk, surfaces and hinges. \n\t\t(b)(c) Electronic structure of the $120^\\circ $ hinge of the La apatite.\n\t\tThe energy is measured from the Fermi level. The high-symmetry wavenumbers are $\\tilde{\\Gamma}=0$ and $\\tilde{Z}=\\pi$.\t\t\n\t\tIn the magnified figure (c) the bands originate from the surface and hinge are colored in orange and yellow, respectively. \n\t\t(d) Maximal Wyckoff positions in the unit cell consisting of hexagonal blocks.\n\t\t(e) Maximal Wyckoff positions in the unit cell consisting of triangular blocks.\n\t\t(f) Relationship between the hexagonal and triangular blocks.\n\t\t(g) Crystal consisting of hexagonal blocks. In contrast with (a), the Wannier orbitals at the Wyckoff position $a$ do not lead to hinge states.\n\t}\n\t\\label{hinge}\n\\end{figure*}\n\n\nNonetheless, this formula (\\ref{eq:Qcorner}) does not apply to our system. Equation (\\ref{eq:Qcorner}) is derived for a crystal consisting of \nhexagonal blocks (Fig.~\\ref{hinge}(d)), with the sixfold rotation axis at the center of the hexagon, and Eq.~(\\ref{eq:Qcorner})\nrepresents a corner charge for a 2D crystal with a hexagonal shape, composed of the \nhexagonal blocks (Fig.~\\ref{hinge}(g)).\nMeanwhile, in the present case, if we focus on the crystal shape along the $ab$ plane, the fundamental unit is a regular triangle, rather than a hexagon.\nThe whole crystal forming a hexagonal prism is composed of the triangular blocks, and the unit cell consists of \ntwo triangular blocks (Fig.~\\ref{hinge}(e)).\nThis choice of the fundamental units of the crystal makes a\ndifference in the corner charge. In the present case, the Wannier orbital is at the hollow sites, located at the Wyckoff position $1a$.\nIn this case the representations are trivial, and the same between the high-symmetry points, and $[M_1^{(2)}]=0$ and $[K_1^{(3)}]=0$. This leads to an absence of the corner charge if we use Eq.~(\\ref{eq:Qcorner})\nwhich is natural because the Wyckoff position $1a$ is at the center of the\nhexagon. \nNonetheless,\nit is not valid,\nbecause the fundamental blocks are triangular and this Wyckoff position $1a$ is at the corners of the triangle (Fig.~\\ref{hinge}(f)), giving rise to a nonzero quantized corner charge as we show later. \nThus, even for trivial representations, one can have a nonvanishing quantized corner charge. \n\nFor a triangular block,\nwe show maximal Wyckoff positions in Fig.~\\ref{hinge}(e). For each Wyckoff position, \none can calculate the corner charge and irreducible representations at high-symmetry points\nsimilarly as in Ref.~\\cite{Benalcazar19}. \nThrough some calculations whose details are given in Supplementary Material~\\cite{SM5}, \nthe corner charge for $C_6$-symmetric 2D systems is given by  \n\\begin{align}\n\tQ^{\\text{T}(6)}_{\\text{corner}}\n\t=\\frac{|e|}{4}[M^{(2)}_1]\n\t+\\frac{|e|}{3}[K^{(3)}_1]\n\t+\\frac{|e|}{3}\\nu\\ \\text{mod}\\ e,\n\t\\label{eq:Qcorner2}\n\\end{align}\nwhere $\\nu$ is a number of electrons per unit cell, i.e. the number of occupied bands. \nIt means that the filling of the corner states is $Q_{\\text{corner}}\/|e|$, and it is well defined only when the \nbulk and the surface are gapped. In the present case, the surface is gapped only when $\\nu$ is even. \nThis equation is quite different from \n(\\ref{eq:Qcorner}) in that it depends on $\\nu$. \nWe now apply this theory to the apatite. We consider a hexagonal prism, having a structure within the $ab$ plane as shown in Fig.~\\ref{hinge}(a).\nLet us consider the whole system as a two-dimensional system, and consider only the interstitial electrons. \nThen we have\n$[M_1^{(2)}]=0$, $[K_1^{(3)}]=0$, and $\\nu=2$,\ngiving rise to \n$Q_{\\rm corner}^{\\text{T}(6)}=\\frac{2|e|}{3} \\ \\text{mod} \\ e$.\nFrom the charge neutrality condition, the filling of the topological hinge state is $\\frac{2}{3}$.\n\n\n\n\n\t\n\t\n\t\n\nTo summarize, \nwe show that the apatite electrides are higher-order topological insulators, having topological gapless hinge states, while the bulk and the \nsurface are gapped. The hinge states reside at interstitial sites forming  one-dimensional hollows in the electride. \nIn the bulk and the surface, these interstitial sites are integer-filled, but at the 120${}^\\circ$ hinge, they are $2\/3$ filled due to the \ntopological properties of the bulk occupied bands. \nRealization of the topological bands stemming from the band inversion, together with the floating hinge states at the hinges \noriginates from the characteristics as elelctrides.\n\n\t\n\n\n\t\n\t\n\\begin{acknowledgments}\n\t\n\nThis work was supported by JSPS KAKENHI Grant Number 18H03678,  and by the MEXT Elements Strategy Initiative to Form Core Research\nCenter (TIES).\n\\end{acknowledgments}\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nThe primary target of objective image quality assessment (IQA) is to automatically predict the perceptual visual quality, providing a cost-effective alternative for the cumbersome subjective user study~\\cite{athar2019comprehensive}. Full-reference IQA (FR-IQA) feeds the pristine image ${\\bf x}$ and the counterpart distorted image ${\\bf y}$ into different perceptual distance measures. The predicted quality score is used to evaluate the image processing system and optimize various real-world applications, such as image compression, restoration, and rendering. The FR-IQA models can be summarized from a Bayesian perspective~\\cite{duanmu2021quantifying}:\n\\begin{equation}\\label{eq:gauss_assump_fr}\n    p(s | {\\bf x}, {\\bf y}, \\bm{\\theta}, \\beta) = \\mathcal{N}(s | d({\\bf x}, {\\bf y}; \\bm{\\theta}), \\beta),\n\\end{equation}\nwhere $s$ is the subjective quality rating which is assumed to follow a Gaussian distribution with the mean $d({\\bf x}, {\\bf y}; \\bm{\\theta})$ and variance $\\beta$. Herein, we denote the $d({\\bf x}, {\\bf y}; \\bm{\\theta})$ as the perceptual distance measured between ${\\bf x}$ and  ${\\bf y}$, with the $\\theta$ encoding the prior knowledge of  human vision system (HVS). In the last decades, efforts have been mainly devoted to exploring a meaningful and powerful parameter distribution $p\\bm{(\\theta)}$, aiming to achieve a fully perceptual consistent measure. Those models can be classified into knowledge-driven and data-driven approaches.\n\n\n\\begin{figure*}[t]\n \\centering\n \\includegraphics[trim={0cm 0cm 0cm 0cm},clip, width=.99\\textwidth]{.\/figs\/fig_1.png\n \\caption{ Illustration of the contrastive preference by human and existing quality measures.  Left: Human and DID prefer the image (c) from generative adversarial network (GAN)~\\cite{navarrete2018multi} over the JPEG2000 distorted image (a), but the distance-based measure (PSNR, SSIM, LPIPS, and DISTS) prefer image (a) over (c); Right: Human, DISTS, and DID prefer the resampling texture image (f) over the pink noise image (d), but the PSNR, SSIM, and LPIPS prefer image (d) over (f). The better quality scores are highlighted in boldface.\n }\n \\label{fig:sameples}\n\\end{figure*}\n\nThe \\textit{knowledge-driven approaches} have dominated the  FR-IQA models for more than a  half-century. Mean squared error (MSE) is one of the most popular error visibility methods owing to its simplicity, clear physical meaning and desired properties for optimization, but it shows poor correlation with the HVS~\\cite{wang2009mean}. Afterward, methods that correlate better with HVS were developed, such as the structural similarity index (SSIM)~\\cite{wang2004image}, the visual information fidelity (VIF)~\\cite{sheikh2006image}, and the normalized Laplacian pyramid distance (NLPD)~\\cite{laparra2016perceptual}. Recently, the \\textit{data-driven approaches} are prevailing due to the perceptual meaningful characteristic of deep pre-trained convolution neural network (CNN), denoted as $\\tilde{\\bf x} = h(\\bf{x}, \\bm{\\theta}_c)$ and $\\bm{\\theta}_c$ is the parameters of network $h(\\cdot)$. In the deep learning feature domain, various distance measures have been developed, \\textit{i}.\\textit{e}., $d(\\tilde{\\bf x}, \\tilde{\\bf y}; \\bm{\\theta}_c)$. For the element-wise methods, Johnson~\\textit{et al}.~ constructs the perceptual loss by computing the weighted summation of $\\ell_2$-norm distance in both image and feature domains~\\cite{johnson2016perceptual}. The learned perceptual image patch similarity (LPIPS)~\\cite{zhang2018unreasonable} calculates the weighted MSE result with corresponding deep representations, attempting to account for the ``unreasonable'' effectiveness. Ding~\\textit{et al}.~ computes the global or local deep feature statistics~(\\textit{i}.\\textit{e}., mean and variance) to unify the structure and texture similarity~(DISTS)~\\cite{ding2020image,ADISTS}. In addition, the internal feature dependency (\\textit{i}.\\textit{e}., distribution-wise) comparisons also play an essential role in perceptual visual quality predictions. Representative examples include style loss~\\cite{gatys2016image}, deep self-dissimilarity~\\cite{kligvasser2021deep}, and deep Wasserstein distance (DeepWSD)~\\cite{liao2022deepwsd}. \n\n\n\n\n\n\n\n\nThe knowledge-driven and data-driven approaches primarily rely on the deterministic comparisons of the images in various domains. The deterministic comparisons based on distance metrics, though faithfully reflecting the fidelity, may fail when the test images are generated instead of physically acquired. This can be attributed to the rooted view that images with perfect quality can be feasibly modeled as the output of a stochastic source. One example can be observed in Fig.~\\ref{fig:sameples}, where the image (c) with pleasant texture  is disfavored by the distance-based measures, due to the large distance caused by the generated textures. In addition, as shown in the right case of Fig.~\\ref{fig:sameples}, human perception is usually invariant to the texture resampling even the resampling brings distance boosting in both signal space and feature space. \nThose phenomenons inspire us to bring the shift from the traditional distance-based FR-IQA paradigm, to the dependency modeling from statistical perspective. In particular, though statistical models dominate the no-reference IQA methods, much less work has been dedicated to characterizing the statistics for FR-IQA. More importantly, statistical and perceptual modeling of visual signals are broadly recognized as the dual problems~\\cite{sheikh2005information}.\nAs such, the shift is grounded on the mild assumption that the perception of texture variation can be well reflected by a reliable feature dependency measure.\n\nIn this work, we adopt the Brownian Distance Covariance (BDC) as the ideal feature dependency measure for FR-IQA, and propose the Deep Image Dependency (DID)\nbased FR-IQA model. The BDC is defined as the weighted Euclidean distance between the joint characteristic function and the product of the marginal characteristic functions~\\cite{szekely2009brownian}. The proposed DID model presents several desired advantages. First, the model is able to  naturally capture the feature dependency against both linear and non-linear transformations. Second, in the deep feature space, the model does not rely on any trainable parameters, demonstrating promising flexibility and generalization capability. \n{Third, the model presents superior performance on texture perception, no matter the texture is artificially generated, randomly reasmpled and geometrically transformed.}\nExtensive experiments based upon classical IQA datasets,  texture similarity datasets, and geometric transformation  dataset demonstrate that DID achieves state-of-the-art performance according to the correlation with mean opinion scores (MOSs). \nIt is also worth mentioning that though DID obtains competitive performance in quality evaluation tasks, it is independent of the training data (\\textit{i}.\\textit{e}., MOSs) and does not contain any controllable parameters. \n\n\n \n\n\n\n\n\\section{Related Work}\n\\subsection{Full-reference Image Quality Assessment}\nThe early works for FR-IQA capture the distortion relying on signal fidelity measure \\textit{e}.\\textit{g}., MSE, and peak signal-to-noise ratio (PSNR). Although those works enjoy the calculation simplicity and mathematical convenience, the low consistency with human perception has been widely criticized \\cite{lin2003discriminative}. In \\cite{wang2004image}, the SSIM was proposed by introducing three important components, including luminance, contrast, and structural similarity. This work was extended to several more advanced quality measures, such as MS-SSIM \\cite{wang2003multiscale}, IW-SSIM \\cite{wang2010information} and CW-SSIM \\cite{wang2005translation}. In the deep-learning era, pioneering works concentrate on image comparison in the deep learning feature space. For example, in  LPIPS \\cite{zhang2018unreasonable}, the multi-scale features were extracted from the pre-trained VGG  \\cite{simonyan2014very} network and the image quality is estimated by measuring the feature fidelity loss with Euclidean distance. Analogously, the combination of spatial averages and  correlations of the feature maps was adopted in DISTS \\cite{ding2020image}, aiming for the estimation of  texture similarity and structure similarity.  The work was further improved by processing the structure and texture information adaptively in A-DISTS \\cite{ADISTS}. Instead of measuring the feature distance point-by-point, the the Wasserstein distance was utilized in DeepWSD \\cite{liao2022deepwsd} to capture the  quality contamination. Driven by the quality annotated data, the human perception knowledge can also be learned by CNN, such as DeepQA \\cite{kim2017deep}, WaDIQaM \\cite{bosse2017deep}, PieAPP \\cite{prashnani2018pieapp} and JSPL \\cite{cao2022incorporating}. However, compared with the features extracted from pre-trained networks, the learned models usually suffer from the over-fitting problem due to the limited labeled data. \n\n\n\n\\subsection{Data Dependency Measure}\nClassical data dependency is measured only in the linear scenario. For example, the product-moment correlation and covariance are two widely used dependency measures between two random variables. In the case that the joint distribution of two vectors is multivariate normal distribution, the covariance matrix can be used to measure the dependence of different dimensions. This  measure which captures the high-order information is also utilized in computer vision tasks, \\textit{i}.\\textit{e}., the style transferring \\cite{gatys2016image} and deep self-similarity~\\cite{kligvasser2021deep}. For the image stylization, the Gram matrices of the neural activations of different CNN layers are extracted under the view that the channel dependency captured by Gram matrices well represents the artistic style of an image. However, in a more generic scene, the nonlinear or nonmonotone dependence is expected to be effectively captured. In \\cite{szekely2009brownian,szekely2007measuring}, the BDC was proposed which is able to measure the data dependency efficiently even in the nonlinear scenario. The BDC is designed based on the construction of joint characteristics of two variables, such that it could be more effective than only the marginal distribution involved. Benefiting from the properties, the BDC was also introduced for few-shot classification tasks \\cite{xie2022joint}, showing its robustness in different settings. \n\n\n\\section{Methodology}\n\\subsection{Preliminary of Brownian distance covariance}\nLet $X \\in \\mathbb{R}^p$, $Y \\in \\mathbb{R}^q$ be two random vectors, where $p$ and $q$ are their dimensions. The characteristic functions of $X$ and $Y$ are denoted as $f_{X}$ and $f_{Y}$ and their joint characteristic function is  $f_{XY}$. Assuming $X$ and $Y$ have finite first moments, analogous to classical covariance, the BDC measure is defined  as follows,\n\\begin{equation}\n\\begin{aligned}\n\\mathcal{V}^2(X, Y ; w) &=\\left\\|f_{X, Y}(t, s)-f_X(t) f_Y(s)\\right\\|_w^2,\n\\end{aligned}\n\\end{equation}\n where $\\|\\cdot\\|_w^2$-norm is defined by,\n\\begin{equation}\n\\|\\gamma(t, s)\\|_w^2=\\int_{\\mathbb{R}^{p+q}}|\\gamma(t, s)|^2 w(t, s) d t  d s.\n\\end{equation}\nThe $w(t, s)$ is a positive weight function for which the integral above exists. As such,\n\\begin{equation}\n\\begin{aligned}\n\\mathcal{V}^2(X, Y ; w) =\\int_{\\mathbb{R}^{p+q}}\\left|f_{X, Y}(t, s)-f_X(t) f_Y(s)\\right|^2 w(t, s) d t d s.\n\\label{v1}\n\\end{aligned}\n\\end{equation}\nTo endow the $\\mathcal{V}^2(X, Y; w)$ with the capability to capture  the dependence  between  $X$ and $Y$, a suitable  weight function can be found as follows \\cite{szekely2009brownian}, \n\\begin{equation}\nw(t, s)=\\left(c_p c_q|t|_p^{1+p}|s|_q^{1+q}\\right)^{-1},\n\\label{wts}\n\\end{equation}\nwhere\n\\begin{equation}\nc_p=\\frac{\\pi^{(1+p) \/ 2}}{\\Gamma((1+p) \/ 2)},  \\quad  c_q=\\frac{\\pi^{(1+q) \/ 2}}{\\Gamma((1+q) \/ 2)}, \n\\end{equation}\nand $\\Gamma (\\cdot)$ is the complete gamma function. From the Eq. (\\ref{v1}) and Eq. (\\ref{wts}),  the BDC measure can be formed by,\n\\begin{equation}\n \\begin{aligned}\n\\mathcal{V}^{2}(X, Y) &=\\left\\|f_{X, Y}(t, s)-f_{X}(t) f_{Y}(s)\\right\\|^{2} \\\\\n&=\\frac{1}{c_{p} c_{q}} \\int_{\\mathbb{R}^{p+q}} \\frac{\\left|f_{X, Y}(t, s)-f_{X}(t) f_{Y}(s)\\right|^{2}}{|t|_{p}^{1+p}|s|_{q}^{1+q}} d t d s.\n\\end{aligned}\n\\end{equation}\nHerein, we omit the $w$ in $\\mathcal{V}^2(X, Y; w)$ for simplification. In practice, the observations of $X$ and $Y$  are usually discrete. For $n$ i.i.d. observed  random vectors $(\\mathbf{X}, \\mathbf{Y})=\\left\\{\\left(X_k, Y_k\\right): k=1, \\ldots, n\\right\\}$, the  BDC measure $\\mathcal{V}^{2}(X, Y)$ can be efficiently acquired by \\cite{szekely2009brownian,szekely2007measuring},\n\\begin{equation}\n\\mathcal{V}^{2}(\\mathbf{X}, \\mathbf{Y}) = \\frac{1}{n^{2}} \\sum_{k, l = 1}^{n}{\\bf{A}}_{k l} {\\bf{B}}_{k l},\n\\label{pro}\n\\end{equation}\nwhere ${\\bf{A}}_{k l}=a_{k l}-\\bar{a}_{k \\cdot}-\\bar{a}_{\\cdot l}+\\bar{a}_{. .}$ and \n\n\\begin{equation}\na_{k l}=\\left\\|X_k-X_l\\right\\|_z, \\quad \\bar{a}_{k \\cdot}=\\frac{1}{n} \\sum_{l=1}^n a_{k l}, \\quad \\bar{a}_{\\cdot l},=\\frac{1}{n} \\sum_{k=1}^n a_{k l},  \\quad  \\bar{a}_{. .} =\\frac{1}{n^2} \\sum_{k, l=1}^n a_{k l},\n\\label{akl}\n\\end{equation}\nwhere $\\|\\cdot\\|_z$ means the z-norm.  Analogously,  we define ${\\bf{B}}_{k l}=b_{k l}-\\bar{b}_{k \\cdot}-\\bar{b}_{\\cdot l}+\\bar{b}_{. .}$, for $k,l =1, \\ldots, n$. The model enjoys several interesting properties which are summarized as follows,\n\\begin{enumerate}[label=(\\roman*)]\n\\item $\\mathcal{V}^{2}(\\mathbf{X}, \\mathbf{Y}) \\geq 0$.\n\\item $\\mathcal{V}^{2}(\\mathbf{X}, \\mathbf{Y})= 0$, if and only if $\\mathbf{X}$ and  $\\mathbf{Y}$ are independent.\n\\item For all constant vector $t_1$, $t_2$, nonzero real number $s_1$, $s_2$  and orthogonal matrix $\\mathbf{R_1}$,  $\\mathbf{R_2}$, $\\mathcal{V}^{2}(t_1+s_1\\mathbf{X}\\mathbf{R_1}, t_2+s_2\\mathbf{X}\\mathbf{R_2})= \\left|{s_1}{s_2}\\right|\\mathcal{V}^{2}(\\mathbf{X}, \\mathbf{Y})$.\n\\end{enumerate}\n{\nThe above properties reveal that the  BDC is  non-negative and able to  capture the dependency of the signals well under the translations and orthonormal transformations. Those properties exhibit surprising consistency with human  texture perception, which is usually not sensitive to texture resampling and geometric transformations. Incorporating the BDC in our method endows our method with a more powerful quality prediction capability and its effectiveness can be verified by the extensive experiments in Sec. \\ref{exp}.\n}\n\n\n\\subsection{BDC based FR-IQA model}\nGiven the reference image ${\\bf x  \\in \\mathbb{R}^3}$ and  the test image ${\\bf y  \\in \\mathbb{R}^3}$, the aim of FR-IQA is to predict the image quality  ${\\bf q  \\in \\mathbb{R}^1}$ of ${\\bf y}$. Herein, directly calculating the dependency between ${\\bf x}$ and  ${\\bf y}$ in the pixel space may not be adequate, as human perceptual sensitivity  is  usually non-uniform~\\cite{wang2008maximum,berardino2017eigen}. Recently, deep neural networks have shown a surprising power in capturing image distortions, popularly adopted as the quality-aware representation generator \\cite{zhang2018unreasonable,prashnani2018pieapp,ding2020image}. Following the vein, we first adopt the VGG16 network~\\cite{simonyan2014very} to nonlinearly  transform the images (${\\bf x}$ and  ${\\bf y}$) to the deep representations. Then, the BDC is calculated to evaluate the dependency between the representations of ${\\bf x}$ and  ${\\bf y}$, deemed that the higher dependency corresponds to the higher quality. The design details are shown in Fig.~\\ref{fig:frame}. In particular, the VGG16 network contains five stages in total and is pre-trained on the ImageNet~\\cite{deng2009imagenet} dataset. We empirically  abandon the layers after the fourth stage and use the rest layers as the deep-feature extractor. Supposing the extractor is represented by $\\phi(\\cdot)$, the deep-features of ${\\bf x}$ and  ${\\bf y}$ can be obtained by,\n\\begin{equation}\n\\mathbf{X}= \\phi(\\bf {x}),\n\\mathbf{Y}= \\phi (\\bf{y}),\n\\label{vgg}\n\\end{equation}\nwhere $\\mathbf{X,Y} \\in \\mathbb{R}^{h \\times w \\times d}$, $h$ and $w$ are the spatial dimensions and $d$ is the channel number. We reshape the $\\mathbf{X}$ and $\\mathbf{Y}$ into $hw \\times\\ d$, \\textit{i}.\\textit{e}., $\\mathbf{X,Y} \\in \\mathbb{R}^{hw \\times\\ d}$, then treat each column vector of $\\mathbf{X}$ and $\\mathbf{Y}$ (denoted as $X \\in \\mathbb{R}^{hw}$, $Y \\in \\mathbb{R}^{hw}$) as the observations of random vectors sampled from the marginal distributions of  $f_{X}$ and $f_{Y}$, respectively. However, the i.i.d. assumption may not hold for  $X$ and $Y$ due to distinct semantic information lying  in different channels. As such, instead of directly using the inner product in Eq. (\\ref{pro}), suggested by~\\cite{xie2022joint}, the cosine similarity is adopted for dependency estimation,\n\\begin{equation}\n{\\bf q}= \\mathcal{D}(\\mathbf{X,Y})= \\frac{\\bf A_u {\\bf B_u^T}}{\\left\\| \\bf A_u \\right\\|_2 \\left\\| \\bf B_u \\right\\|_2},\n\\label{qua}\n\\end{equation}\nwhere $\\bf  A_u$ and $\\bf  B_u$  are the flattened results of the upper triangular portions of $\\bf  A$ and $\\bf  B$ with ${\\bf{A}}_{k l}=a_{k l}-\\bar{a}_{k \\cdot}-\\bar{a}_{\\cdot l}+\\bar{a}_{. .}$ and ${\\bf{B}}_{k l}=b_{k l}-\\bar{b}_{k \\cdot}-\\bar{b}_{\\cdot l}+\\bar{b}_{. .}$. For $a_{k l}$ and $b_{k l}$, we adopt $z=2$ in Eq. (\\ref{akl}), \n\\begin{equation}\na_{k l}=\\left\\|X_k-X_l\\right\\|_2,  b_{k l}=\\left\\|Y_k-Y_l\\right\\|_2.\n\\label{akl2}\n\\end{equation}\n\n\\begin{figure*}[t]\n \\centering\n \\includegraphics[ width=.95\\textwidth]{.\/figs\/fram.png}\n \\caption{Overall structure of DID. The tailored VGG16 (only the first four stages are reserved) network is adopted as the deep feature extractor. Then the features dependency is measured by the BDC and higher dependency indicates better quality. }\n \\label{fig:frame}\n\\end{figure*}\n\n   Algorithm 1 summarizes the framework of our method.  The $\\mathcal{D}(\\mathbf{X,Y})$ depicts the dependency level of $\\mathbf{Y}$ with $\\mathbf{X}$, ranging from (-1,1). Higher $\\mathcal{D}(\\mathbf{X,Y})$ corresponds to better quality of test image. It should be noted that although our DID-based IQA model is a deep learning-based quality measure, it enjoys the learning-free advantage, avoiding over-fitting to a specific dataset. \n\n\n\n\\begin{algorithm}[t]\n\t\\caption{DID based FR-IQA model.}\n\t\\label{alg:algorithm1}\n\t\\KwIn{ Reference image ${\\bf x}$; test image  ${\\bf y}$. }\n\t\\KwOut{The quality of test image  ${\\bf q}$.}  \n\t\\BlankLine\n        Step 1. Extract the deep representations $\\mathbf{X, Y}$ of ${\\bf x, y}$ by the tailored VGG16; \\\\[2pt]\n        Step 2. Obtain the matrices ${\\bf{A}}_{k l}=a_{k l}-\\bar{a}_{k \\cdot}-\\bar{a}_{\\cdot l}+\\bar{a}_{. .}$ and ${\\bf{B}}_{k l}=b_{k l}-\\bar{b}_{k \\cdot}-\\bar{b}_{\\cdot l}+\\bar{b}_{. .}$ by Eq.~(\\ref{akl}) and Eq. (\\ref{akl2}). \\\\[2pt]\n        Step 3. Select the upper triangular portions of $\\bf  A$ and $\\bf  B$  as $\\bf  A_u$ and $\\bf  B_u$. \\\\[2pt]\n        Step 4. Estimate the test image quality  ${\\bf q} = \\mathcal{V}^{2}(\\mathbf{X}, \\mathbf{Y}) = \\frac{1}{n^{2}} \\sum_{k, l = 1}^{n}{\\bf{A}}_{k l} {\\bf{B}}_{k l}$ by Eq. (\\ref{qua}).\n\\end{algorithm}\n\n\n\\begin{table}[t]\n  \\centering\n  \\small\n    \\begin{tabular}{l|cc|cc|cc|cc|cc}\n    \\toprule\n    \\multicolumn{1}{c|}{\\multirow{2}[4]{*}{Method}} & \\multicolumn{2}{c|}{LIVE} & \\multicolumn{2}{c|}{CSIQ} & \\multicolumn{2}{c|}{TID2013} & \\multicolumn{2}{c|}{KADID-10k} & \\multicolumn{2}{c}{PIPAL} \\\\\n\\cmidrule{2-11}          & SRCC  & PLCC  & SRCC  & PLCC  & SRCC  & PLCC  & SRCC  & PLCC  & SRCC  & PLCC \\\\\n    \\midrule\n    PSNR  & 0.873  & 0.868  & 0.809  & 0.815  & 0.688  & 0.679  & 0.676  & 0.680  & 0.407  & 0.415  \\\\\n    SSIM  & 0.931  & 0.928  & 0.872  & 0.868  & 0.720  & 0.745  & 0.724  & 0.723  & 0.498  & 0.505  \\\\\n    MS-SSIM & 0.931  & 0.931  & 0.908  & 0.896  & 0.798  & 0.810  & 0.802  & 0.801  & 0.552  & 0.590  \\\\\n    VIF   & 0.927  & 0.925  & 0.902  & 0.887  & 0.690  & 0.732  & 0.593  & 0.602  & 0.443  & 0.468  \\\\\n    FSIM  & \\textbf{0.965 } & \\textbf{0.961 } & 0.931  & 0.919  & \\textbf{0.851}  & \\textbf{0.877}  & 0.854  & 0.851  & 0.589  & 0.615  \\\\\n    NLPD  & 0.914  & 0.914  & 0.917  & 0.911  & 0.808  & 0.823  & 0.810  & 0.810  & 0.469  & 0.509  \\\\\n    \\midrule\n    Style & 0.898  & 0.882  & 0.853  & 0.837  & 0.675  & 0.681  & 0.701  & 0.707  & 0.339  & 0.337  \\\\\n    PieAPP & 0.908  & 0.919  & 0.877  & 0.892  & 0.850  & 0.848  & 0.836  & 0.836  & \\textbf{0.700 } & \\textbf{0.712 } \\\\\n    LPIPS & 0.939  & 0.945  & 0.883  & 0.906  & 0.695  & 0.759  & 0.720  & 0.729  & 0.573  & 0.618  \\\\\n    DISTS & \\textbf{0.955}  & \\textbf{0.954}  & 0.939  & 0.941  & 0.848  & 0.870  & \\textbf{0.890 } & \\textbf{0.889 } & 0.624  & 0.644  \\\\\n    DSD   & 0.577  & 0.552  & 0.603  & 0.700  & 0.548  & 0.657  & 0.439  & 0.527  & 0.274  & 0.350  \\\\\n    DeepWSD & 0.896  & 0.890  & \\textbf{0.963 } & \\textbf{0.953 } & \\textbf{0.874 } & \\textbf{0.896 } & 0.888  & 0.888  & 0.514  & 0.517  \\\\\n    \\midrule\n    DID   & 0.949  & 0.943  & \\textbf{0.945 } & \\textbf{0.942 } & 0.850  & 0.871  & \\textbf{0.913 } & \\textbf{0.911 } & \\textbf{0.677 } & \\textbf{0.697 } \\\\\n    \\bottomrule\n    \\end{tabular}%\n  \\caption{Performance comparison of DID against twelve existing FR-IQA models on five standard IQA datasets. The best two results are highlighted in boldface.}    \n  \\label{tab:mainresult}%\n\\end{table}%\n\n\n\\section{Experiment}\n\\label{exp}\nIn this section, we first describe the experimental setup. Next, we conduct comprehensive experiments to verify the effectiveness of the proposed model, including image quality prediction, texture quality assessment, and invariance of geometric transformation. Finally, the  ablation studies are performed. \n\\subsection{Experimental Setups}\nThe VGG16 is tailored at ``ReLU4$\\_$3\" layer and pre-trained on the ImageNet~\\cite{deng2009imagenet}. Inspired by the SSIM~\\cite{wang2004image} and DISTS~\\cite{ding2020image}, we resize the shorter side of the input images to $224$ while keeping the aspect ratio. We apply the Spearman's rank-order correlation coefficient (SRCC) and Pearson linear correlation coefficient (PLCC) to evaluate the monotonicity and linearity. The larger SRCC and PLCC values reflect better quality prediction results. In particular,  a five-parameter nonlinear logistic function is fitted to map the predicted scores to the same scale as MOSs when computing PLCC~\\cite{video2000final}. \n\n\\subsection{Performance on Image Quality Prediction}\nWe compare DID with $12$ FR-IQA models on five standard IQA datasets, including the LIVE~\\cite{sheikh2003image} CSIQ~\\cite{larson2010most}, TID2013~\\cite{ponomarenko2015image}, KADID-10k~\\cite{2019KADID}, and PIPAL~\\cite{jinjin2020pipal}. In particular, LIVE, CSIQ, and TID2013 contain limited image contents and distortion types, and they have been widely used for more than ten years. The KADID-10k and PIPAL are two large-scale IQA datasets with more than ten thousand distorted images. KADID-10k has $81$ pristine images, and $25$ distortion types with $5$ levels are adopted to generate $10,125$ distorted images. PIPAL is so far the largest human-rated IQA dataset with $23,200$ images, which are generated by $200$ reference images with $40$ distortions types. It is worth noting that PIPAL introduces $19$ GAN-based distortions, challenging the existing FR-IQA a lot. In addition, the $12$ FR-IQA models cover various design methodologies: the error visibility methods - PSNR, and NLPD~\\cite{laparra2016perceptual}, the structural similarity methods - SSIM~\\cite{wang2004image}, MS-SSIM~\\cite{wang2003multiscale}, and FSIM~\\cite{zhang2011fsim}; the information-theoretic methods - MAD~\\cite{larson2010most}, and VIF~\\cite{sheikh2006image}; the learning-based methods - PieAPP~\\cite{prashnani2018pieapp}, LPIPS~\\cite{zhang2018unreasonable}, and DISTS~\\cite{ding2020image}; the distribution-based methods - Style~\\cite{gatys2016image}, DSD~\\cite{kligvasser2021deep}, DeepWSD~\\cite{liao2022deepwsd}. \nThe experimental results are reported in the Table~\\ref{tab:mainresult}, from which we can find DID achieves superior performance on both classical (LIVE, CSIQ, and TID2013) and latest (KADID-10k and PIPAL) IQA datasets.\nIt demonstrates that the dependency-based model is well correlated with human ratings, and the learning-free advantage equips the DID with strong generalization capability. \nIn addition, the knowledge-driven methods (\\textit{e}.\\textit{g}., FSIM) generally perform better on small-scale IQA datasets, indicating the potential over-fitting problem because of the extensive parameter tuning. \nMoreover, DeepWSD outperforms most learning-based methods on synthetic distortions, which further reflects the success of the joint distribution based FR-IQA model. \nFinally, though PieAPP obtains the best performance on the PIPAL dataset, it requires plenty of the human-rated images to train the model~\\cite{prashnani2018pieapp}.\n\n\n\\begin{table}[t]\n\t\\begin{minipage}[b]{0.48\\linewidth}\n\t\t\\centering\n\t\t\\small\n\t\t\\resizebox{1.0\\textwidth}{!}{\n\t\t\n    \\begin{tabular}{l|cc}\n    \\toprule\n    \\multicolumn{1}{c|}{\\multirow{2}[4]{*}{Method}} & \\multicolumn{2}{c}{PIPAL (GAN distortion)} \\\\\n\\cmidrule{2-3}          & SRCC  & PLCC  \\\\\n    \\midrule\n    SSIM  & 0.322  & 0.472  \\\\\n    MS-SSIM & 0.387  & 0.615  \\\\\n    VIF   & 0.324  & 0.543  \\\\\n    FSIM  & 0.410  & 0.621  \\\\\n    NLPD  & 0.341  & 0.570  \\\\\n    \\midrule\n    LPIPS & 0.486  & 0.617  \\\\\n    PieAPP & \\textbf{0.553 } & \\textbf{0.632 } \\\\\n    DISTS & 0.549  & 0.607  \\\\\n    DeepWSD & 0.397  & 0.560  \\\\\n    \\midrule\n    \\textit{FID} & 0.413  & 0.496  \\\\\n    \\midrule\n    DID   & \\textbf{0.5742} & \\textbf{0.6403} \\\\\n    \\bottomrule\n    \\end{tabular}%\n\t\t}\n\t\t\\caption{Performance comparison of DID against state-of-the-art methods on the GAN distortion of PIPAL dataset. The measure specifically designed for GAN images is represented in italics.}\n\t\t\\label{tab:gan}\t\t\n\t\\end{minipage}\n\t\\hfill\n\t\\begin{minipage}[b]{0.48\\linewidth}  \n\t\t\\centering\n\t    \\small\n   \n\t\n              \\resizebox{200pt}{102pt}{\n\n    \\begin{tabular}{l|cc|cc}\n    \\toprule\n    \\multicolumn{1}{c|}{\\multirow{2}[4]{*}{Method }} & \\multicolumn{2}{c|}{SynTEX} & \\multicolumn{2}{c}{TQD} \\\\\n\\cmidrule{2-5}          & SRCC  & PLCC  & SRCC  & PLCC \\\\\n    \\midrule\n    SSIM  & 0.579  & 0.598  & 0.352  & 0.418  \\\\\n    VIF   & 0.606  & 0.697  & 0.549  & 0.614  \\\\\n    FSIM  & 0.081  & 0.115  & 0.386  & 0.272  \\\\\n    NLPD  & 0.606  & 0.607  & 0.409  & 0.457  \\\\\n    \\midrule\n    LPIPS & 0.788  & 0.788  & 0.203  & 0.188  \\\\\n    PieAPP & 0.715  & 0.719  & 0.718  & 0.721  \\\\\n    DISTS & \\textbf{0.923 } & \\textbf{0.901 } & \\textbf{0.910 } & \\textbf{0.903 } \\\\\n    DISTS$_\\text{s}$ & 0.877  & 0.868  & 0.795  & 0.780  \\\\\n    \\midrule\n   \\textit{STSIM} & 0.643  & 0.650  & 0.408  & 0.422  \\\\\n    \\textit{NPTSM} & 0.496  & 0.505  & 0.679  & 0.678  \\\\\n    \\textit{ISGTQA} & 0.820  & 0.816  & 0.802  & 0.804  \\\\\n    \\midrule\n    DID   & \\textbf{0.896 } & \\textbf{0.874 } & \\textbf{0.889 } & \\textbf{0.917 } \\\\\n    \\bottomrule\n    \\end{tabular}%\n\t\t}\n\t\t\\caption{Performance comparison of DID against state-of-the-art methods on two texture quality datasets. The texture similarity models are represented in italics.}\n\t\t\\label{tab:texutre}\t\n\t\\end{minipage}\n\\end{table}\n\n\n\\begin{figure*}\n    \\centering\n   \n    \\subfloat[MOS $\\uparrow$ \/ DID $\\uparrow$]{\\includegraphics[width=0.24\\textwidth]{figs\/textures\/13-0.png}}\\hskip.2em\n    \\subfloat[0.873 \/ 0.991 ]{\\includegraphics[width=0.24\\textwidth]{.\/figs\/textures\/13-4.png}}\\hskip.2em\n    \\subfloat[0.727 \/ 0.980]{\\includegraphics[width=0.24\\textwidth]{.\/figs\/textures\/13-2.png}}\\hskip.2em\n    \\subfloat[0.693 \/ 0.931]{\\includegraphics[width=0.24\\textwidth]{.\/figs\/textures\/13-15.png}}\n    \\vspace{-8pt}\n    \\addtocounter{subfigure}{0}\n    \\subfloat[0.528 \/ 0.821]{\\includegraphics[width=0.24\\textwidth]{.\/figs\/textures\/13-9.png}}\\hskip.2em\n    \\subfloat[0.481 \/ 0.758]{\\includegraphics[width=0.24\\textwidth]{.\/figs\/textures\/13-14.png}}\\hskip.2em\n    \\subfloat[0.437 \/ 0.506]{\\includegraphics[width=0.24\\textwidth]{.\/figs\/textures\/13-11.png}}\\hskip.2em\n    \\subfloat[0.401 \/ 0.210]{\\includegraphics[width=0.24\\textwidth]{.\/figs\/textures\/13-10.png}}\n    \\vspace{-5pt}\n    \\caption{Texture  images sampled from TQD~\\cite{ding2020image}. (a) Reference image. (b) Resampling image. (c) Resampling image. (d) Texture synthesis method~\\cite{snelgrove2017high}. (e) Pink noise. (f) Texture synthesis method~\\cite{gatys2015texture}. (g) Color\nquantization. (h) Chromatic aberration. }\\label{fig:texture}\n\n\\end{figure*}\n\nWe further compare DID with the FR-IQA models on the GAN-generated images of the PIPAL dataset. As shown in Table~\\ref{tab:gan}, most distance-based DR-IQA models present poor performance, and the underlying reason may lie in that synthesized textures which not appear in reference images are usually introduced by the generation networks.  Although the  Fr\u00e9chet Inception Distance (FID)~\\cite{heusel2017gans} is designed especially for the quality evaluation of  GAN models,  the poor performance reveals its limitations and the great challenges of GAN-generated IQA.  Compared with those methods, our model achieves the best result, the dependency rather than the distance mitigates the strict requirement of point-by-point alignment  during feature comparison. \n\n\n\n\\subsection{Performance on Texture Similarity}\n\\label{sec:texture}\nTo verify the effectiveness of our method on texture quality prediction,\nwe conduct  experiments on SynTEX~\\cite{golestaneh2015effect} and TQD~\\cite{ding2020image} datasets. In particular, SynTEX consists of $105$ synthesized texture images, which were generated by five texture synthesis methods for $21$ high-quality texture images. TQD contains ten reference texture images, and each of them is degraded by 15 distortion types, including seven synthetic distortions, four texture synthesis methods, and four randomly resampling versions. We show several sampled texture images in Fig.~\\ref{fig:sameples}. For performance comparison, four representative knowledge-driven methods: SSIM~\\cite{wang2004image}, VIF~\\cite{sheikh2006image}, FSIM~\\cite{zhang2011fsim}, and NLPD~\\cite{laparra2016perceptual} and three data-driven methods: LPIPS~\\cite{zhang2018unreasonable}, PieAPP~\\cite{prashnani2018pieapp}, and DISTS~\\cite{ding2020image} are selected. Furthermore, three models: STSIM~\\cite{zujovic2013structural}, NPTSM~\\cite{alfarraj2016content}, and IGSTOA~\\cite{golestaneh2018synthesized}, that designed especially for texture similarity are also included. \n\nThe SRCC and PLCC results are listed in Table~\\ref{tab:texutre}. It is not surprising that DISTS achieves the best performance on two texture quality datasets since a great number of texture images were used to train the weights for measuring texture similarity. However, when the training session for the texture similarity term is absent (denoted as DISTS$_{\\text{s}}$ in Table~\\ref{tab:texutre}), DID outperforms DISTS$_{\\text{s}}$ by a significant margin. Besides, ISGTQA exhibits noteworthy improvements for the texture similarity models but still falls behind DID. Furthermore, as shown in Fig.~\\ref{fig:texture}, we rank the sample texture images based on the DID score and MOS. We can observe that DID is consistent with human perception of texture quality. \\bl{In particular,  the images with visible artifacts have lower quality scores while the resampling images correspond to higher quality scores.} Thus, we may draw the conclusion that the proposed dependency-based model provides a promising texture perception in the scenarios of texture synthesis, resampling, and transformation.\n\n\n\\subsection{Performance on Geometric Transformations}\n\n\\begin{table}[t]\n  \\centering\n  \\small\n    \\begin{tabular}{l|cc|cc|cc|cc|cc}\n    \\toprule\n    \\multicolumn{1}{c|}{\\multirow{2}[4]{*}{Method}} & \\multicolumn{2}{c|}{Translation} & \\multicolumn{2}{c|}{Rotation} & \\multicolumn{2}{c|}{Scaling} & \\multicolumn{2}{c|}{Mixed} & \\multicolumn{2}{c}{Overall} \\\\\n\\cmidrule{2-11}          & SRCC  & PLCC  & SRCC  & PLCC  & SRCC  & PLCC  & SRCC  & PLCC  & SRCC  & PLCC \\\\\n    \\midrule\n    PSNR  & 0.104  & 0.365  & 0.088  & 0.365  & 0.088  & 0.366  & 0.093  & 0.373  & 0.102  & 0.320  \\\\\n    SSIM  & 0.194  & 0.388  & 0.199  & 0.390  & 0.196  & 0.394  & 0.207  & 0.393  & 0.211  & 0.232  \\\\\n    MS-SSIM & 0.183  & 0.381  & 0.191  & 0.373  & 0.202  & 0.387  & 0.213  & 0.389  & 0.191  & 0.202  \\\\\n    VIF   & 0.239  & 0.417  & 0.224  & 0.409  & 0.209  & 0.402  & 0.214  & 0.399  & 0.369  & 0.373  \\\\\n    FSIM  & 0.370  & 0.558  & 0.382  & 0.575  & 0.376  & 0.566  & 0.393  & 0.590  & 0.336  & 0.522  \\\\\n    NLPD  & 0.153  & 0.180  & 0.170  & 0.196  & 0.189  & 0.223  & 0.172  & 0.202  & 0.265  & 0.324  \\\\\n    \\midrule\n    Style & 0.740  & 0.758  & 0.735  & 0.747  & 0.729  & 0.738  & 0.740  & 0.752  & 0.744  & 0.758  \\\\\n   \n    LPIPS & 0.781  & 0.788  & 0.806  & 0.808  & 0.811  & 0.822  & 0.843  & 0.853  & 0.746  & 0.762  \\\\\n    DISTS & \\textbf{0.885 } & \\textbf{0.886 } & \\textbf{0.887 } & \\textbf{0.887 } & \\textbf{0.881 } & \\textbf{0.882 } & \\textbf{0.905 } & \\textbf{0.905 } & \\textbf{0.890 } & \\textbf{0.890 } \\\\\n    DSD   & 0.497  & 0.670  & 0.500  & 0.682  & 0.501  & 0.686  & 0.487  & 0.688  & 0.496  & 0.638  \\\\\n    DeepWSD & 0.163  & 0.189  & 0.187  & 0.207  & 0.215  & 0.194  & 0.280  & 0.302  & 0.234  & 0.288  \\\\\n    \\midrule\n    DID   & \\textbf{0.899 } & \\textbf{0.907 } & \\textbf{0.899 } & \\textbf{0.905 } & \\textbf{0.891 } & \\textbf{0.898 } & \\textbf{0.913 } & \\textbf{0.918 } & \\textbf{0.903 } & \\textbf{0.909 } \\\\\n    \\bottomrule\n    \\end{tabular}%\n        \\caption{Performance comparison of DID against the state-of-the-art FR-IQA models on LIVE-GT  dataset. The best two results are highlighted in boldface.}\n  \\label{tab:geo}%\n\\end{table}%\n\n\n\\bl{Texture image quality usually presents an invariance to mild geometric transformations~\\cite{ding2020image}. To study the performance of  existing FR-IQA models on such a prior, we construct a new IQA dataset denoted as LIVE-GT  with four geometric transformations (translation, rotation, scaling, and their mixed) involved. \n In particular, the four geometric transformations are implemented by randomly shifting $5\\%$ pixels in vertical or horizontal directions, randomly rotating $3^{\\circ}$ in clockwise or anticlockwise directions, scaling the image by a factor of $1.05$, and mixing the above-mentioned transformations.  We impose the four transformations to the reference images in the LIVE dataset and a total of $3,895$ distorted images are finally generated. Herein, we make a mild assumption  that the modest geometric transformations will not change the MOS of each image. Then, the performance comparison on the  LIVE-GT dataset can be conducted. \n\n\n \n\n\n\n\nWe list the SRCC and PLCC results for overall and individual geometric transformation types in Table~\\ref{tab:geo}. We can observe that the performance of knowledge-driven models drops dramatically on the LIVE-GT dataset while  our DID model achieves the best performance on all four transformations.  We believe the invariance property is brought by the dependency-based measure which avoids measuring the feature difference in a deterministic way.\nIn addition, it is worth noting that the DeepWSD  is also a joint distribution-based FR-IQA. However, it is vulnerable to transformations as the feature statistic comparison is performed locally. On the contrary, we construct the feature joint distribution in a global manner with both spatial and channel dimensions involved. In summary, we can conclude that the dependency derived from the global statistic of features contributes  to the robustness of our model.\n\n\n}\n\n\n\n\\subsection{Ablation studies}\n\\begin{table}[t]\n  \\centering\n  \\small \n\\setlength{\\tabcolsep}{4pt}\n    \\begin{tabular}{c|c|c|c|c|c|c|c|c}\n    \\toprule\n    \\multirow{2}[4]{*}{Backbone} & \\multirow{2}[4]{*}{Layer} & \\multicolumn{4}{c|}{Quality prediction} & \\multicolumn{2}{c|}{Texture similarity} & Geo invariance \\\\\n\\cmidrule{3-9}          &       & LIVE  & TID2013 & KADID-10k & PIPAL & SynTEX & TQD   & LIVE-GT  \\\\\n    \\midrule\n    \\multirow{5}[2]{*}{VGG16} & ReLU1\\_2 & 0.825  & 0.637  & 0.709  & 0.573  & 0.699  & 0.733  & 0.679  \\\\\n          & ReLU2\\_2 & 0.892  & 0.771  & 0.854  & 0.601  & 0.837  & 0.806  & 0.774  \\\\\n          & ReLU3\\_3 & 0.938  & 0.845  & \\textbf{0.899 } & 0.652  & 0.869  & 0.860  & 0.892  \\\\\n          & \\underline{ReLU4\\_3} & {0.948}  & {\\textbf{0.858 }} & {\\textbf{0.905 }} & {\\textbf{0.677 }} & {0.896}  & {0.889}  & {0.903}  \\\\\n          & ReLU5\\_3 & 0.946  & 0.855  & 0.881  & 0.656  & 0.893  & 0.867  & 0.893  \\\\\n    \\midrule\n    \\multirow{2}[2]{*}{ResNet50} & Layer\\_3 & \\textbf{0.953 } & \\textbf{0.865 } & 0.877  & 0.612  & 0.903  & 0.897  & 0.889  \\\\\n          & Layer\\_4 & \\textbf{0.950 } & 0.857  & 0.875  & 0.640  & \\textbf{0.925 } & 0.898  & \\textbf{0.905 } \\\\\n    \\midrule\n    \\multirow{2}[2]{*}{DenseNet121} & Block\\_3 & 0.937  & 0.856  & 0.850  & \\textbf{0.658 } & 0.916  & \\textbf{0.920 } & \\textbf{0.919 } \\\\\n          & Block\\_4 & 0.949  & \\textbf{0.865 } & 0.881  & 0.654  & \\textbf{0.918 } & \\textbf{0.921 } & 0.856  \\\\\n    \\bottomrule\n    \\end{tabular\n    \\caption{Ablation experiments of DID with different CNN backbones and tailored at different layers in terms of SRCC results. Geo invariance is the abbreviation of geometric invariance. The best two results are highlighted in boldface, and the default setting of DID is highlighted with an underline.}\n  \\label{tab:ablation}%\n\\end{table}%\n\nIn this subsection, we conduct ablation experiments to investigate the effect of the pre-trained CNN backbone and the tailored layer in the proposed model. Except for the default VGG16~\\cite{simonyan2014very} backbone, we applied the proposed BDC based FR-IQA method to other two widely used ImageNet pre-trained image classification networks - ResNet50~\\cite{he2016deep} and DenseNet121~\\cite{huang2017densely}. We tailored VGG16 at the last ReLU nonlinearity layer of each stage. We take the last two stages of ResNet50 (denoted as Layer$\\_$3 and Layer$\\_$4, respectively) and DenseNet121 (denote as Block$\\_$3 and Block$\\_$4, respectively) as the comparison variants. The results of three kinds of quality assessment tasks, \\textit{i}.\\textit{e}., quality prediction, texture similarity, and geometric invariance, are shown in Table~\\ref{tab:ablation}, \\bl{from which we can have the following observations. \nFirst, the proposed dependency model performs better in the deeper layers, as the deeper layers are able to capture the semantic information.\nSecond, the proposed BDC based model is quite robust to the CNN backbones according to the superior performance. Last, while the ResNet50 and DenseNe121 outperform the VGG16 in some small-scale datasets, we still choose VGG16 as the default backbone due to the satisfactory trade-off between model complexity and performance.}\n\n\n\n\\section{Conclusion}\nIn this paper, we have presented the new design philosophy for FR-IQA method and shown that the feature-dependency is particularly effective for generative images. The paradigm shift brings a fresh new perspective regarding how image quality shall be alternatively defined, given the available reference image. We obtain the conclusion that instead of gauging the feature distance, the dependency which is characterized by BDC could well reflect the image quality. In addition,  the proposed measure can be treated as a plug-and-play module and incorporated into  different backbones, dynamically adapting the application scenarios. We also believe the paradigm shifted from distance to dependency will shed light on more generalized quality measures and inspire more works on the exploration of feature dependency.\n\n\n\n\n\\section*{Acknowledgements} The authors would like to thank Keyan Ding for providing the TQD dataset and Haoming Cai for giving the rule of the name for the distortion images in PIPAL.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{INTRODUCTION.}\n\nEdge magnetoplasmons (EMP's) have atracted considerable interest in the past\nyears \\cite{1}-\\cite{11} and Ref. \\cite{1} contains a discussion of older\nstudies. In \\cite{11} we have treated EMPs, $\\propto A(\\omega,q_{x},y) \\exp[\n-i(\\omega t-q_{x} x)]$, for $\\nu=1,2$ and very low temperatures when the\nunperturbed density profile drops sharply at the edge on a length of the\norder of the magnetic length $\\ell_{0}$. Such a profile is valid for $\nk_{B}T\\ll \\hbar v_{g}\/\\ell_{0}$, where $v_{g}$ is the group velocity of the\nedge states. Then for $\\nu=1,2$ the unperturbed electron density $n_{0}(y)$,\nnormalized to the bulk value $n_{0}$, is calculated as $n_{0}(y)\/n_{0}=\\{1+\n\\Phi[(y_{re}-y)\/\\ell_{0}]\\}\/2$, where $y_{re}$ is the coordinate of the\nright edge and $\\Phi(y)$ the probability integral.\n\nThe main results of Ref. \\cite{11} were the existence of many {\\it symmetric}\nand {\\it antisymmetric} modes with respect to the edge, some of which can\npropagate nearly undamped even when the dissipation is strong and were,\ntherefore, termed {\\it edge helicons}. In this {\\it quasi-microscopic}\ntreatment, however, the contributions to the current density\n$j_{\\mu}(\\omega,q_{x},y)$ were obtained from microscopic expressions valid\nwhen the components of the electric field ${\\bf E}$, associated with the\nwave, are smooth on the $\\ell_{0}$ scale. This is not well justified.\nIn Ref. \\cite{11} we neglected possible nonlocal effects and approximated the\ncontributions to $j_{\\mu}(\\omega,q_{x},y)$ with those obtained when\n$E_{y}(y)$ is smooth on the scale of $\\ell_{0}$.\nThe model \\cite{11} also neglects\nthe screening of the two-dimensional electron gas (2DEG). Here, using a RPA\nframework we study the effect of nonlocality and of edge-states screening on\nEMPs for filling factors $6\\geq\\nu\\geq 1$. For even filling factors $\\nu$ we\nneglect the spin splitting. As we will show, the existence of additional\nmodes for $\\nu=1(2)$, predicted in Ref. \\cite{11}, is confirmed by the\npresent {\\it fully microscopic} treatment and the results for the\nfundamental mode remain valid for weak dissipation despite the neglect of\nscreening and nonlocality in \\cite{11}.\n\nIn Sec. II we present the channel-edge characteristics and derive an\nintegral equation for the inhomogeneous wave charge density within a RPA\nframework. In Sec. III we obtain the fundamental and dipole EMP's for\n$\\nu=1,2$ and in Sec. IV we consider fundamental EMPs for $\\nu=4$\nand $\\nu=6$. Finally, in Sec. V we discuss briefly the results and\nmake concluding remarks.\n\n\\section{BASIC RELATIONS}\n\n\\subsection{Channel-edge characteristics}\n\nWe consider a zero-thickness 2DEG of width $W$ and of length $L_x=L$ in the\npresence of a strong magnetic field $B$ along the $z$ axis. We take the\nconfining potential flat in the interior of the 2DEG ($V_{y}=0$) and parabolic\nat its edges, $V_{y} =m^*\\Omega^2 (y-y_r)^2\/2$, $y\\geq y_r$. $V_{y}$ is\nassumed smooth on the scale of $\\ell_{0}=(\\hbar\/m^{*} \\omega_{c})^{1\/2}$\nsuch that $\\Omega \\ll \\omega_{c}$, where $\\omega_{c}=|e|B\/m^* $ is the\ncyclotron frequency ($e<0$). In the Landau gauge for the vector potential\n${\\bf A}=(-By, 0, 0)$ the one-electron Hamiltonian $\\hat{h}^0$ is given by\n\n\\begin{equation}\n\\hat{h}^0=[(\\hat{p}_x +eBy)^2 +\\hat{p}_{y}^{2}]\/2m^* +V_{y} \\;\\; , \\label{1}\n\\end{equation}\nwhere $\\hat{{\\bf p}}$ is the momentum operator. The eigenvalues and\neigenfunctions corresponding to Eq. (\\ref{1}) near the right edge of the\nchannel, with $y_{0} \\equiv y_{0}(k_{x})=\\ell_{0}^{2} k_{x} \\geq y_{r}$, are\nwell approximated by\n\n\\begin{equation}  \nE_{\\alpha}\\equiv E_{n}(k_{x})= (n+1\/2)\\hbar \\omega_{c}+m^*\\Omega^2\n(y_{0}-y_{r})^2\/2 , \\\\    \\label{2}\n\\end{equation}\nand\n\\begin{equation}  \n|\\alpha> \\equiv \\psi_\\alpha({\\bf r}) =e^{ik_x x}\\Psi_n(y-y_{0})\/\\sqrt{L}, \\\\\n\\label{3}\n\\end{equation}\nrespectively. Here ${\\bf r}=\\{x,y\\}$, $\\alpha\\equiv \\{n,k_x\\}$, $\\Psi_n(y)$\nis a harmonic oscillator function. \nThe energy spectrum (\\ref{2}) of the n-th LL leads to the group velocity of\nthe edge states $v_{gn}=\\partial E_{n}(k_{r}+k_{e}^{(n)})\/\\hbar\\partial\nk_{x}=\\hbar \\Omega^{2}k_{e}^{(n)}\/m^{*}\\omega_{c}^{2}$ with characteristic\nwave vector $k_{e}^{(n)}=(\\omega_{c}\/ \\hbar\\Omega) \\sqrt{2m^{*}\\Delta_{Fn}}$,\n$\\Delta_{Fn}=E_{F}-(n+1\/2)\\hbar \\omega_{c}$, and $E_F$ is the Fermi\nenergy. The edge of the n-th LL is denoted by $y_{rn}=y_{r}+\n\\ell_{0}^{2}k_{e}^{(n)}=\\ell_{0}^{2}k_{rn}$,\nwhere $k_{rn}=k_{r}+k_{e}^{(n)}$,\nand $W=2y_{r0}$. We can also write $v_{gn}=E_{en}\/B$, where $E_{en}=\\Omega\n\\sqrt{2m^{*}\\Delta_{Fn}}\/|e|$ is the electric field associated with the\nconfining potential $V_{y}$ at $y_{rn}$. We also introduce the wave vector\n$k_{r}=y_{r}\/\\ell_{0}^{2}$.\n\nFor definiteness, we take the background dielectric constant $\\epsilon$ to\nbe spatially homogeneous. Assuming $|q_{x}| W \\gg 1$, we can consider an EMP\nalong the right edge of the channel of the form $A(\\omega, q_{x}, y) \\exp[\n-i(\\omega t-q_{x} x)]$ totally independent of the left edge. For $\\nu$ even\nthe spin-splitting is neglected.\n\n\\subsection{ Wave charge density and electric potential at the channel edge}\n\nAs in Refs. \\cite{11} and \\cite{12}, we assume that without interaction the\none-electron density matrix $\\hat{\\rho}^{(0)}$ is diagonal, i.e., $<\\alpha|\n\\hat{\\rho}^{(0)}|\\beta>=f_{\\alpha}\\delta_{\\alpha\\beta}$, where\n$f_{\\alpha}=1\/[1+exp((E_{\\alpha}-E_F)\/k_BT)]$ is the Fermi-Dirac function.\n\nFor the application of the RPA we follow the self-consistent field approach\nof Ref. \\cite{13} and references cited therein. The one-electron Hamiltonian\nin the presence of a self-consistent wave potential $V(x,y,t)=V(%\n\\omega_{0},q_{x},y) \\exp[-i(\\omega_{0} t-q_{x} x)]+ c.c.$, is $\\hat{H}(t)=%\n\\hat{h}^{0}+V(x,y,t)$. The corresponding equation of motion for the\none-electron density matrix $\\hat{\\rho}$ reads\n\n\\begin{equation}\ni\\hbar\\frac{\\partial \\hat{\\rho}}{\\partial t}=[\\hat{H}(t),\\hat{\\rho}]%\n-\\frac{i\\hbar}{\\tau}(\\hat{\\rho}-\\hat{\\rho}^{(0)}),   \\label{5}\n\\end{equation}\nwhere $[,]$ denotes the commutator. On the right hand side (RHS) of Eq.\n(\\ref{5}) we have introduced the phenomenological infinitesimal term\n$\\propto 1\/\\tau$ ($\\tau \\rightarrow \\infty$) that leads to the correct\nrules for contour integration around the singularities, cf. Refs. \\cite{13}\nand \\cite{14}. Notice that $\\tau \\rightarrow \\infty$ corresponds to the\ncollisionless case while using a finite $\\tau$ provides the possibility of\nestimating roughly the influence of collisions. Here the most effective\ncollisions are related to intra-LL and intra-edge transitions,\ncf. Refs. \\cite{11}-\\cite{12}.\n\nWe take the Laplace trasform, with respect to the time t, of Eq. (\\ref{5}),\nwrite $\\hat{R}(\\omega)=\\int_{0}^{\\infty}e^{i\\omega t} \\hat{\\rho} dt$ and\n$R_{\\alpha \\beta}(\\omega)=<\\alpha|\\hat{R}(\\omega)|\\beta>$, and look for\nits solution in a power series in $V$\n\n\\begin{equation}\nR_{\\alpha \\beta}(\\omega)=\\sum_{n=0}^{\\infty} R_{\\alpha \\beta}^{(n)}(\\omega),\n\\label{6}\n\\end{equation}\nwhere $R_{\\alpha \\beta}^{(0)}(\\omega)=(i f_{\\alpha}\/\\omega) \\delta_{\\alpha\n\\beta}$. Because we consider linear EMPs, it is sufficient  to take into\naccount only the first two terms $n=0$ and $n=1$ on the RHS of\nEq. (\\ref{6}).  Then in $V(x,y,t)$ we can consider only the term\n$V(\\omega_{0},q_{x},y) \\exp[-i(\\omega_{0} t-q_{x} x)]$, which leads to\n\n\\begin{equation}\nR_{\\alpha \\beta}^{(1)}(\\omega)=\\frac{i(f_{\\beta}-f_{\\alpha})\n<\\alpha|V(\\omega_{0},q_{x},y) e^{iq_{x} x}|\\beta>} {(\\omega-\\omega_{0})\n[E_{\\beta}-E_{\\alpha}+\\hbar \\omega+i\\hbar\/\\tau]} .  \\label{7}\n\\end{equation}\nTaking the trace of $\\hat{\\rho}$ with the electron charge density operator,\n$e \\delta({\\bf r}-\\hat{{\\bf r}})$, gives the wave charge density as\n\n\\begin{equation}\n\\delta \\rho(t,x,y) \\equiv \\rho(t,x,y)=\\frac{e}{2\\pi} \\int_{-\\infty+i\\eta}^{\n\\infty+i\\eta} d \\omega e^{-i\\omega t} \\sum_{\\alpha \\beta} R_{\\alpha\n\\beta}^{(1)}(\\omega) \\psi_{\\beta}^{*}({\\bf r}) \\psi_\\alpha({\\bf r}) ,\n\\label{8}\n\\end{equation}\nwhere $\\eta>0$. From Eqs. (\\ref{7}) and (\\ref{8}) it follows that\n$\\rho(t,x,y)=\\rho(t,q_{x},y) \\exp(iq_{x} x)$. Moreover, for $t\/\\tau \\gg 1$,\nthe contributions related to transitional processes are already negligible.\nIt follows that\n$\\rho(t,q_{x},y)=\\rho(\\omega_{0},q_{x},y) \\exp(-i\\omega_{0}t)$.\n\nFurther, from Poisson's equation the wave electric potential $\\phi(t,\nq_{x},y)$ induced by $\\rho(t,q_{x},y)$ is given as\n\n\\begin{equation}\n\\phi(t, q_{x},y)=\\frac{2}{\\epsilon}\\int_{-\\infty}^{\\infty}\ndy^{\\prime}K_{0}(|q_{x}||y-y^{\\prime}|)\\rho(t, q_{x},y^{\\prime}), \\label{9}\n\\end{equation}\nwhere $K_{0}(x)$ is the modified Bessel function; $\\phi$ and $\\rho$ pertain\nto the 2D plane.\n\n\\subsection{ Integral equation for EMP's}\n\nFor $t\/\\tau \\gg 1$ the relation $\\rho(t, q_{x},y^{\\prime})= \\rho(\\omega_{0},\nq_{x},y^{\\prime})\\exp(-i\\omega_0 t)$ holds; then from Eq. (\\ref{9}) it\nfollows that $\\phi(t, q_{x},y)=\\phi(\\omega_0, q_{x},y)\\exp(-i\\omega_0 t)$.\nIn the absence of an external potential we have $V(\\omega_{0},q_{x},y)=e\n\\phi(\\omega_0, q_{x},y)$. As a result,  from Eqs. (\\ref{7})-(\\ref{9}), for\n$t\/\\tau \\gg 1$ , we obtain the integral equation for $\\rho(\\omega, q_{x},y)$\nas\n\n\\begin{eqnarray}\n&&\\rho(\\omega, q_{x},y)=\\frac{2e^{2}}{\\epsilon L} \\sum_{n_{\\alpha},n_{\n\\beta}=0}^{\\infty} \\sum_{k_{x\\alpha}} \\frac{f_{\\beta}-f_{\\alpha}}{\nE_{\\beta}-E_{\\alpha}+\\hbar \\omega+ i \\hbar\/\\tau} \\\n\\Pi_{n_{\\alpha}n_{\\beta}}( y, k_{x\\alpha}, k_{x\\beta})  \\nonumber \\\\\n* \\   \\nonumber \\\\\n&&\\times\\int_{-\\infty}^{\\infty} d\\tilde{y} \\int_{-\\infty}^{\\infty}\ndy^{\\prime}\\ \\Pi_{n_{\\alpha}n_{\\beta}}( \\tilde{y}, k_{x\\alpha}, k_{x\\beta})\nK_{0}(|q_{x}||\\tilde{y}-y^{\\prime}|)\\ \\rho(\\omega, q_{x},y^{\\prime}) ,\n\\label{10}\n\\end{eqnarray}\nwhere $\\Pi_{n_{\\alpha}n_{\\beta}}( y, k_{x\\alpha}, k_{x\\beta})=\n\\Psi_{n_{\\alpha}}(y-y_0(k_{x\\alpha})) \\Psi_{n_{\\beta}}(y-y_0(k_{x\\beta}))$,\n$k_{x\\beta}=k_{x\\alpha}-q_{x}$.\nWe dropped the subscript $0$ from $\\omega_{0}$.\n\nFor flat LL's, i.e., for $\\Omega \\rightarrow 0$ and fixed width of the 2DEG,\nwe first apply the Fourier transformation along $y$ to Eq. (\\ref{10}) and\nthen carry out the sum over $k_{x\\alpha}$ as well as the integral over\n$y^{\\prime}$. This leads to $\\rho(\\omega, q_{x},q_{y}) \\epsilon_{l}(\\omega,\nq_{x},q_{y})\/\\epsilon=0$, where $\\epsilon_{l}(\\omega,\nq_{x},q_{y})=\\epsilon_{l}(\\omega, {\\bf q})  \\equiv \\epsilon_{l}(\\omega, q)$\nis the RPA longitudinal dielectric function (${\\bf q}=\\{q_{x},q_{y}\\}$).\nThat is, for $\\rho(\\omega, q_{x},q_{y}) \\neq 0$ we have\n\n\\begin{equation}\n\\frac{\\epsilon_{l}(\\omega, {\\bf q})}{\\epsilon}=1- \\frac{2 \\pi e^{2}}{%\n\\epsilon q L^{2}} \\sum_{\\alpha,\\beta} \\frac{f_{\\beta}-f_{\\alpha}}{%\nE_{\\beta}-E_{\\alpha}+\\hbar \\omega+ i \\hbar\/\\tau} |<\\alpha|e^{i\\vec q \\cdot\n\\vec r}|\\beta>|^{2}=0 .  \\label{12}\n\\end{equation}\nThis equation gives the RPA dispersion relation for \"bulk\" longitudinal wave\nexcitations of a 2DEG in a strong magnetic field. It also shows that the\n\"bulk\" excitations should have $\\omega>\\omega_{c}$. For definitness, we take\n$\\omega>0$.\n\nThe integral equation (\\ref{10}) can be considered as a generalization of\nEq. (10) of Ref. \\cite{11} since it takes into account nonlocal\ncontributions to the current density $\\propto \\int dy^{\\prime}\\sigma_{\\mu\n\\gamma}(y,y^{\\prime}) E_{\\gamma}(y^{\\prime})$ and the screening by the edge\nand bulk states of the 2DEG. As shown in Ref. \\cite{16}, the screening by\nedge states can be strong. We will obtain only solutions of Eq. (\\ref{10})\nthat are low in frequency, i.e., solutions for which $\\omega\/\\omega_{c} \\ll 1\n$. The appearence of any wave branch with $\\omega < \\omega_{c}$ is\nprincipally related to the existence of the edge of the 2DEG. Hence, any\nwave with $\\omega < \\omega_{c}$ should be localized at the edge and can be\ncalled edge magnetoplasmon.\n\nWe consider very low temperatures $T$ satisfying $\\hbar v_{gn}\\gg \\ell_{0}\nk_{B} T$. Further, we will assume the long-wavelength limit $q_{x} \\ell_{0}\n\\ll 1$, which is well satisfied, e.g., for fundamental EMP \\cite{11} in the\nlow-frequency region. Then if we compare the terms $\\propto f_{\\beta^{*}}$,\nfor given $n_{\\beta^{*}}$, on the RHS of Eq. (\\ref{10}), we obtain that the\ncontribution to the sum over $n_{\\alpha}$ with $n_{\\alpha}=n_{\\beta^{*}}$ is\nmuch larger than any other term of this sum or the sum of all terms with\n$n_{\\alpha} \\neq n_{\\beta^{*}}$. The small parameter is $|\\omega-q_{x}v_{g\nn_{\\beta^{*}}}(k_{x \\beta})|\/\\omega_{c} \\ll 1$, where $v_{g\nn_{\\beta^{*}}}(k_{x \\beta})=\\hbar^{-1} \\partial E_{n_{\\beta^{*}}}(k_{x\n\\beta})\/\\partial k_{x \\beta}$ is the group velocity of any occupied state\n$\\{ n_{\\beta^{*}},k_{x \\beta} \\}$ of the $n_{\\beta^{*}}$ LL. We assume that\neach occupied n-th LL has one intersection with the Fermi level at the edge\nof the channel and denote the group velocity of its edge states as $v_{gn}\n\\equiv v_{gn}(k_{rn})$. The small parameter given above implies $q_{x}\nv_{g0}\/\\omega_{c} \\ll 1$, as $v_{g0}$ is typically the largest among\n$v_{gn}$. Similar results follow from an analysis of the terms $\\propto\nf_{\\alpha^{*}}$ in the sum over $n_{\\beta}$ on the RHS of Eq. (\\ref{10}).\nHence, for $\\omega \\ll \\omega_{c}$ and $q_{x} v_{g 0} \\ll \\omega_{c}$ the\nterms with $n_{\\alpha} \\neq n_{\\beta}$ can be neglected. This leads to the\nintegral equation\n\n\\begin{eqnarray}\n&&\\rho(\\omega, q_{x},y)=\\frac{2e^{2}}{\\epsilon L}\n\\sum_{n_{\\alpha}=0}^{\\bar{n}}\n\\sum_{k_{x\\alpha}} \\frac{f_{n_{\\alpha},k_{x\\alpha}-q_x}\n-f_{n_{\\alpha},k_{x\\alpha}}} {E_{n_{\\alpha},k_{x\\alpha}-q_x}\n-E_{n_{\\alpha},k_{x\\alpha}}+ \\hbar \\omega+i \\hbar\/\\tau} \\\n\\Pi_{n_{\\alpha}n_{\\alpha}}( y, k_{x\\alpha}, k_{x\\alpha}-q_x)  \\nonumber \\\\\n* \\   \\nonumber \\\\\n&&\\times\\int_{-\\infty}^{\\infty} d\\tilde{y} \\int_{-\\infty}^{\\infty}\ndy^{\\prime}\\ \\Pi_{n_{\\alpha}n_{\\alpha}}( \\tilde{y}, k_{x\\alpha},\nk_{x\\alpha}-q_x) K_{0}(|q_{x}||\\tilde{y}-y^{\\prime}|)\\ \\rho(\\omega,\nq_{x},y^{\\prime}) ,  \\label{14}\n\\end{eqnarray}\nwhere $\\bar{n}$ denotes the highest occupied LL. For even $\\nu$ the RHS of\nEq. (\\ref{14}) should be multiplied by 2, the spin degeneracy factor. We now\nstudy EMPs following from Eq. (\\ref{14}).\n\n\\section{EMPs FOR $\\protect\\nu=1$($2$)}\n\nWe first consider the case $\\nu=1$ and then explain how the pertinent\nformulas should be  modified for $\\nu=2$. For $\\nu=1$ we have $\\bar{n}=0$\nand Eq. (\\ref{14}) takes the form\n\n\\begin{eqnarray}\n&&\\rho(\\omega, q_{x},y)=\\frac{e^{2}}{\\pi \\hbar \\epsilon} \\int_{-\\infty}^{%\n\\infty} dk_{x\\alpha} \\frac{f_{0,k_{x\\alpha}-q_{x}}-f_{0,k_{x\\alpha}}} {%\n\\tilde{\\omega}-v_{g0}(k_{x \\alpha}) q_{x}} \\ \\Pi_{00}( y, k_{x\\alpha},\nk_{x\\alpha}-q_{x})  \\nonumber \\\\\n* \\   \\nonumber \\\\\n&&\\times\\int_{-\\infty}^{\\infty} d\\tilde{y} \\int_{-\\infty}^{\\infty}\ndy^{\\prime}\\ \\Pi_{00}( \\tilde{y}, k_{x\\alpha}, k_{x\\alpha}-q_{x}) \\\nK_{0}(|q_{x}||\\tilde{y}-y^{\\prime}|)\\ \\rho(\\omega, q_{x},y^{\\prime}) ,\n\\label{15}\n\\end{eqnarray}\nwhere $\\tilde{\\omega}=\\omega+i\/\\tau$. As in Ref. \\cite{11}, we seek a\nsolution of Eq. (\\ref{15}) in the form\n\n\\begin{equation}\n\\rho_{0}(\\omega,q_{x},y)=\\Psi_{0}^{2}(\\bar{y}) \\sum_{n=0}^{\\infty}\n\\rho^{(n)}_{0}(\\omega,q_{x}) H_{n}(\\bar{y}\/\\ell_{0}) = \\sum_{n=0}^{\\infty}\n\\sqrt{2^{n}n!} \\ \\rho^{(n)}_{0}(\\omega,q_{x})\n\\Psi_{n}(\\bar{y})\\Psi_{0}(\\bar{y}).  \\label{16}\n\\end{equation}\nHere $\\bar{y}=y-y_{r0}=y-y_{0}(k_{r0})$ and Eq. (\\ref{16}) is the {\\it exact}\nsolution due to the orthogonality of the Hermite polynomials $H_{n}(x)$. The\nexpansion (\\ref{16}) is specific to $\\nu=1 (2)$ under the assumed\nconditions. This implies that the charge distortion far from the LL edge,\n$\\agt 10 \\ell_{0}$, is much smaller than near this LL edge, $\\alt \\ell_{0}$.\nPhysically this assumption can be well justified. We call the terms $n=0, 1,\n2,...$, the monopole, dipole, quadrupole, etc. terms in the expansion of\n$\\rho_{0}(\\omega,q_{x},y)$ relative to $y=y_{r0}$. Notice that expansion\n(\\ref{16}) can also be understood as an expansion in a complete set of\noscillatory wave functions corresponding to all, occupied ($n=0$) and\nempty ($n \\geq 1$) LLs.\n\nWe now multiply Eq. (\\ref{15}) by $H_{m}(\\bar{y}\/\\ell_{0})$ and integrate\nover $y$ from $y_{r0}-\\Delta y_{0}$ to $y_{r0}+\\Delta y_{0}$, where $\\Delta\ny_{0} \\agt 3 \\ell_{0}$. With the abbreviation $\\rho_{0}^{(m)}(\\omega,\nq_{x})\\equiv \\rho_{0}^{(m)}$, we obtain\n\n\\begin{equation}\n\\rho_{0}^{(m)}=\\frac{e^{2}}{\\pi \\hbar \\epsilon} \\int_{-\\infty}^{\\infty}\ndk_{x\\alpha} \\frac{f_{0,k_{x\\alpha}-q_{x}}-f_{0,k_{x\\alpha}}}{\\tilde{\\omega}\n-v_{g0}(k_{x \\alpha}) q_{x}} \\ d_{m0}(q_{x},\\delta k_{x\\alpha})\n\\sum_{n=0}^{\\infty} \\Big(\\frac{2^{n}n!}{2^{m}m!}\\Big)^{1\/2}\nb_{n0}^{(0)}(q_{x},|q_{x}|,\\delta k_{x\\alpha}) \\rho_{0}^{(n)},  \\label{17}\n\\end{equation}\nwhere $\\delta k_{x\\alpha}=k_{x\\alpha}-k_{r0}$, $\\delta\nk_{x\\alpha^{^{\\prime}}}=\\delta k_{x\\alpha}-q_{x}$,\n\n\\begin{equation}\nd_{m0}(q_{x},\\delta k_{x\\alpha})= \\frac{1}{\\sqrt{2^{m}m!}}\n\\int_{-\\infty}^{\\infty} d\\bar{y} \\ H_{m}(\\bar{y}\/\\ell_{0}) \\ \\Pi_{00}( \\bar{y%\n}, \\delta k_{x\\alpha}, \\delta k_{x\\alpha^{^{\\prime}}})   \\label{18}\n\\end{equation}\nand\n\n\\begin{equation}\nb_{n0}^{(0)}(q_{x},|q_{x}|,\\delta k_{x\\alpha})= \\int_{-\\infty}^{\\infty} d\n\\bar{y}\\ \\int_{-\\infty}^{\\infty} d\\bar{y}^{\\prime}\\ \\Pi_{00}( \\bar{y},\n\\delta k_{x\\alpha}, \\delta k_{x\\alpha^{^{\\prime}}}) K_{0}(|q_{x}||\\bar{y}-\n\\bar{y}^{\\prime}|) \\Psi_{n}(\\bar{y}^{\\prime})\\Psi_{0}(\\bar{y}^{\\prime}) \\ .\n\\label{19}\n\\end{equation}\nIn both $d_{m0}(q_{x},\\delta k_{x\\alpha})$ and\n$b_{n0}^{(0)}(q_{x},|q_{x}|,\\delta k_{x\\alpha})$ the first argument\nrepresents the $q_{x}$-dependent term of the argument of the wave function\n$\\Psi_{0}(\\bar{y}-\\delta k_{x\\alpha^{^{\\prime}}})$ and, if replaced by 0, it\nmeans this term can be neglected. We will also use the coefficients $a_{mn}(\nq_{x})$ given by \\cite{11}\n\n\\begin{equation}\na_{mn}( q_{x})=a_{nm} ( q_{x})=\\int_{-\\infty}^{\\infty} dx\\\n\\Psi_{m}(x)\\Psi_{0}(x) \\int_{-\\infty}^{\\infty} dx^{\\prime}\\\nK_{0}(|q_{x}||x-x^{\\prime}|)\\ \\Psi_{n}(x^{\\prime})\\Psi_{0} (x^{\\prime}),\n\\label{20}\n\\end{equation}\nand satisfying $b_{n0}^{(0)}(0,|q_{x}|,0)=a_{n0}(q_{x})$.\n\n\\subsection{Fundamental EMP}\n\nLet us approximate the numerator on the RHS of Eq. (\\ref{17}) by the first\nnonzero term in the expansion over $q_{x}$: $f_{0,k_{x\\alpha}-q_x\n}-f_{0,k_{x\\alpha}} \\approx -q_{x} (\\partial f_{0,k_{x\\alpha}}\/\\partial\nk_{x\\alpha})= \\delta(k_{x\\alpha}-k_{r0}) q_{x}$. Then Eq. (\\ref{17}), after\nintegration over $k_{x\\alpha}$, gives\n\n\\begin{equation}\n\\rho_{0}^{(m)}=\\frac{e^{2}}{\\pi \\hbar \\epsilon} \\ \\frac{q_{x}}{\\tilde{\\omega}%\n_{0}} \\ d_{m0}(q_{x},0) \\sum_{n=0}^{\\infty} \\Big(\\frac{2^{n}n!}{2^{m}m!}\\Big)%\n^{1\/2} \\ b_{n0}^{(0)}(q_{x},|q_{x}|, 0)\\ \\rho_{0}^{(n)},  \\label{21}\n\\end{equation}\nwhere $\\tilde{\\omega}_{0}=(\\tilde{\\omega}-v_{g0} q_{x})$. Further, if we\nneglect a small term $\\propto q_{x} \\ell_{0}$\nin the argument of the wave function $\\Psi_{0}(\\bar{y}+\\ell_{0}^{2}q_{x})$,\nwe obtain $d_{m0}(q_{x},0) \\approx d_{m0}(0,0)=\\delta_{m,0}$ and\n$b_{n0}^{(0)}(q_{x},|q_{x}|,0) \\approx b_{n0}^{(0)}(0,|q_{x}|,0)=\na_{n0}(q_{x})$. Substituting the former in Eq. (\\ref{21}), we obtain\n$\\rho_{0}^{(m)} \\equiv 0$ for $m \\geq 1$; thus, only $\\rho_{0}^{(0)}$ can be\nfinite. Then Eq. (\\ref{21}) can be rewritten as\n\n\\begin{equation}\n\\rho_{0}^{(0)}=\\frac{e^{2}}{\\pi \\hbar \\epsilon} \\ \\frac{q_{x}}{\\tilde{\\omega}%\n_{0}}\\ a_{00}(q_{x})\\ \\rho_{0}^{(0)}.  \\label{22}\n\\end{equation}\nFor $\\rho_{0}^{(0)} \\neq 0$ Eq. (\\ref{22}) gives the dispersion relation\n\n\\begin{equation}\n\\omega=q_{x} v_{g0}+\\frac{2}{\\epsilon} \\sigma_{yx}^{0} q_{x} [\\ln(\\frac{1}{%\n|q_{x}|\\ell_{0}})+\\frac{3}{4}]-\\frac{i}{\\tau} ,  \\label{23}\n\\end{equation}\nwhere $\\sigma_{yx}^{0}=e^{2}\/2 \\pi \\hbar$ for $\\nu=1$ and $%\n\\sigma_{yx}^{0}=e^{2}\/\\pi \\hbar$ for $\\nu=2$. We have used the result\n$a_{00}(q_{x}) \\approx [\\ln(1\/|q_{x}|\\ell_{0})+3\/4]$ \\cite{11} for\n$|q_{x}|\\ell_{0} \\ll 1$. If we neglect the dissipation, the dispersion\nrelation (\\ref{23}) and the corresponding charge density profile coincide\nwith the corresponding expressions for the fundamental mode obtained in Ref.\n\\cite{11}.\n\nUntill now nonlocal effects and screening by the 2DEG have not changed the\nresults for the fundamental EMP of Ref. \\cite{11}. In Ref. \\cite{11} it is\nshown that the next term that can affect the fundamental mode of the $n=0$\nLL is the quadrupole term. Then, if we neglect dissipation $\\omega-q_{x}\nv_{g0}$ becomes slightly larger, by a factor $[1+0.125\/a_{00}^{2}(q_{x})]\n\\approx 1$. Typically we have $a_{00}(q_{x}) \\agt 10$. In addition, the\nspatial structure of the fundamental EMP acquires a small quadrupole term\n$|\\rho_{0}^{(2)}\/\\rho_{0}^{(0)}| \\approx 1\/8 a_{00}(q_{x}) \\ll 1$ \\cite{11}.\nIt can be shown from Eqs. (\\ref{17})-(\\ref{19}) that here too the\ncorrections to the fundamental EMP due to the quadrupole term are very\nsmall; namely, $a_{00}(q_{x})$ on the RHS of Eq. (\\ref{23}) should be\nchanged to $a_{00}(q_{x})+a_{20}(q_{x})q_{x}^{2} \\ell_{0}^{2}\/2\\sqrt{2}\n\\approx a_{00}(q_{x})-q_{x}^{2} \\ell_{0}^{2}\/8$. Further, the spatial\nstructure of the fundamental EMP acquires a small quadrupole term:\n$\\rho_{0}^{(2)}\/\\rho_{0}^{(0)}=q_{x}^{2} \\ell_{0}^{2}\/8 \\ll 1$. Thus, for\nweak dissipation the results of Ref. \\cite{11} for the fundamental EMP are\nnearly the same as those of the present microscopic treatment.\n\n\\subsection{Dipole EMP}\n\nFor a more acurate calculation of the RHS of Eq. (\\ref{17}) we must take\ninto account higher order terms in the expansion over $q_{x}$. As shown\nbelow, when this is done it leads to additional branches. To obtain the\ndipole branch correctly we must keep the first three nonzero terms in the\nexpansion of the numerator of RHS of Eq. (\\ref{17}) over $q_{x}$ by writing\n\n\\begin{equation}\nf_{0,k_{x\\alpha}-q_x }-f_{0,k_{x\\alpha}} \\approx [q_{x}-\\frac{q_{x}^{2}}{2!}%\n\\frac{\\partial}{\\partial k_{x\\alpha}}+ \\frac{q_{x}^{3}}{3!}\\frac{\\partial^{2}%\n}{\\partial k_{x\\alpha}^{2}}] \\delta(k_{x\\alpha}-k_{r0}).  \\label{24}\n\\end{equation}\nFurther, we will consider only the first two terms, $n=0$ and $n= 1$ in the\nsum of Eq. (\\ref{16}) . Then from Eq. (\\ref{17})  for $m=0$ and $m=1$ we\nobtain the following system of linear equations for $\\rho_{0}^{(0)}$ and\n$\\rho_{0}^{(1)}$\n\n\\begin{equation}\n\\rho_{0}^{(0)}=\\frac{e^{2}}{\\pi \\hbar \\epsilon \\tilde{\\omega}_{0}}\n\\int_{-\\infty}^{\\infty} dk_{x\\alpha} (f_{0, k_{x\\alpha}-q_x\n}-f_{0,k_{x\\alpha}}) \\ d_{00}(q_{x},\\delta k_{x\\alpha}) \\sum_{n=0}^{1} \\Big(%\n2^{n}n!\\Big)^{1\/2} \\ b_{n0}^{(0)}(q_{x},|q_{x}|,\\delta k_{x\\alpha})\n\\rho_{0}^{(n)},  \\label{25}\n\\end{equation}\n\n\\begin{eqnarray}\n\\rho_{0}^{(1)}=\\frac{e^{2}}{\\pi \\hbar \\epsilon \\tilde{\\omega}_{0}}\n\\int_{-\\infty}^{\\infty} dk_{x\\alpha} (f_{0,k_{x\\alpha}-q_x\n}-f_{0,k_{x\\alpha}}) \\ d_{10}(q_{x},\\delta k_{x\\alpha}) \\sum_{n=0}^{1} \\Big(%\n\\frac{2^{n}n!}{2^{1}1!}\\Big)^{1\/2} \\ b_{n0}^{(0)}(q_{x},|q_{x}|,\\delta\nk_{x\\alpha}) \\rho_{0}^{(n)}.  \\label{26}\n\\end{eqnarray}\nWe assumed that, for $T \\rightarrow 0$, $v_{g 0}(k_{x \\alpha})$ is\nindependent of $k_{x \\alpha}$ in the vicinity of $k_{x \\alpha}=k_{r0}$ and\nreplaced it by $v_{g 0}$. This assumption does not impose any important\nrestriction on the analysis. Therefore, taking into account Eq. (\\ref{24})\nwe can replace\n$1\/(\\tilde{\\omega}- v_{g 0}(k_{x \\alpha}) q_{x})$ by\n$1\/(\\tilde{\\omega}-q_{x} v_{g 0})$ and place it in front of the\nintegrals in Eqs. (\\ref{25})-(\\ref{26}).\n\nIn Eq. (\\ref{25}) we can assume\n$d_{00}(q_{x},\\delta k_{x \\alpha})=1$ since corrections to it, varying as\n$\\sim (q_{x} \\ell_{0})^{2}$, are not essential. To evaluate the\ncontribution $\\propto \\rho_{0}^{(0)}$ on the RHS of Eq. (\\ref{25}) it is\nsufficient to keep only the first term, $\\propto q_{x}$, on the RHS\nof Eq. (\\ref{24}). This gives\n\n\\begin{equation}\n\\rho_{0}^{(0)}=\\frac{e^{2}}{\\pi \\hbar \\epsilon \\tilde{\\omega}_{0}} \\{q_{x}\nb_{00}^{(0)}(0,|q_{x}|,0) \\rho_{0}^{(0)}+ \\sqrt{2} I_{01}^{(1)}(q_{x})\n\\rho_{0}^{(1)} \\} ,  \\label{27}\n\\end{equation}\nwhere\n\n\\begin{equation}\nI_{01}^{(1)}(q_{x})=\\int_{-\\infty}^{\\infty} dk_{x\\alpha} (f_{0,\nk_{x\\alpha}-q_x }-f_{0,k_{x\\alpha}}) b_{10}^{(0)}(q_{x},|q_{x}|,\\delta k_{x\n\\alpha}) ,  \\label{28}\n\\end{equation}\nand where $b_{00}^{(0)}(q_{x},|q_{x}|,0)$ has justifiably been approximated\nby $b_{00}^{(0)}(0,|q_{x}|,0)$. Before calculating $I_{01}^{(1)}(q_{x})$ we\nnotice that the approximation for\n$d_{00}(q_{x},\\delta k_{x \\alpha})$ entails the neglect of small corrections\nto $I_{01}^{(1)}(q_{x})$ proportional to $(q_{x} \\ell_{0})^{k}$ if\n$k \\geq 4$. Then Eqs. (\\ref{24}) and Eq. (\\ref{28}) give\n\n\\begin{eqnarray}\nI_{01}^{(1)}(q_{x})&=& -\\frac{1}{\\sqrt{2}\\ell_{0}} (q_{x} \\ell_{0})^{2}\na_{11}(q_{x})+  \\nonumber \\\\\n* && +\\frac{q_{x}^{2}}{2} \\int_{-\\infty}^{\\infty} dk_{x\\alpha}\n\\delta(k_{x\\alpha}-k_{r0}) [1+ \\frac{q_{x}}{3}\\frac{\\partial}{\\partial\nk_{x\\alpha}}] \\frac{\\partial}{\\partial k_{x\\alpha}}\nb_{10}^{(0)}(q_{x},|q_{x}|,\\delta k_{x \\alpha}) .  \\label{29}\n\\end{eqnarray}\nThe corrections $\\propto (q_{x} \\ell_{0})^{k}$, $k \\geq 4$,  we neglected in\nthe first term on the RHS  are related to the expansion\n\n\\begin{equation}\n\\Psi_{0}(\\bar{y}+\\ell_{0}^{2}q_{x})=\\Psi_{0}(\\bar{y})- \\frac{1}{\\sqrt{2}}\n\\Psi_{1}(\\bar{y}) (q_{x} \\ell_{0})+ \\frac{1}{4}[\\sqrt{2} \\Psi_{2}(\\bar{y}%\n)-\\Psi_{0}(\\bar{y})] (q_{x} \\ell_{0})^{2}+...  \\label{30}\n\\end{equation}\nIt can be shown that the term $(q_{x}\/3)\\partial\/\\partial k_{x\\alpha}$ in\nthe square brackets \n[...] of Eq. (\\ref{29}) gives, upon integration over  $k_{x\\alpha}$, a\nvanishing contribution  since the  remaining double integral \nchanges sign upon making the changes $\\bar{y}  \\rightarrow -\\bar{y}$\nand $\\bar{y}^{\\prime}\\rightarrow -\\bar{y}^{\\prime}$. As for the term\ncorresponding to $1$ in [...], its evaluation gives $q_{x}^{2}\\ell_{0}\na_{11}(q_{x})\/\\sqrt{2}$. Hence, the two finite terms on the RHS of\nEq. (\\ref{29}) cancel each other. The final result is $I_{01}^{(1)}(q_{x})=0$\nand Eq. (\\ref{27}) takes the form of Eq. (\\ref{22}).\nThat is,  in this approximation the fundamental mode is purely {\\it monopole}\nand is totally independent of {\\it dipole} excitations.\n\nWe now consider Eq. (\\ref{26}). Here we should consider six different\ncontributions $I_{1n}^{(1)}(q_{x})$, $I_{1n}^{(2)}(q_{x})$, and\n$I_{1n}^{(3)}(q_{x})$ for $n=0,1$ related, respectively, to the first\n($\\propto q_{x}$), second ($\\propto q_{x}^{2}$), and third\n($\\propto q_{x}^{3}$) term on the RHS of Eq. (\\ref{24}).\nThen Eq. (\\ref{26}) can be rewritten as\n\n\\begin{equation}\n\\rho_{0}^{(1)}=\\frac{e^{2}}{\\pi \\hbar \\epsilon \\tilde{\\omega}_{0}}\n\\sum_{n=0}^{1} \\sum_{l=1}^{3} I_{1n}^{(l)}(q_{x}) \\rho_{0}^{(n)} ,\n\\label{31}\n\\end{equation}\nwhere\n\n\\begin{equation}\nI_{1n}^{(1)}(q_{x})=\\Big(\\frac{2^{n}n!}{2^{1}1!}\\Big)^{1\/2} q_{x}\nd_{10}(q_{x},0) b_{n0}^{(0)}(q_{x},|q_{x}|,0) ,  \\label{32}\n\\end{equation}\n\n\\begin{equation}\nI_{1n}^{(2)}(q_{x})=-\\frac{\\sqrt{2^{n-1}n! }}{2!} \\ q_{x}^{2}\n\\int_{-\\infty}^{\\infty} dk_{x\\alpha} d_{10}(q_{x},\\delta k_{x\\alpha})\nb_{n0}^{(0)}(q_{x},|q_{x}|,\\delta k_{x\\alpha}) \\frac{\\partial}{\\partial\nk_{x\\alpha}}\\delta(k_{x\\alpha}-k_{r0}) ,  \\label{33}\n\\end{equation}\n\n\\begin{equation}\nI_{1n}^{(3)}(q_{x})=\\frac{\\sqrt{2^{n-1}n!}}{3!} \\ q_{x}^{3}\n\\int_{-\\infty}^{\\infty} dk_{x\\alpha} d_{10}(q_{x},\\delta k_{x\\alpha})\nb_{n0}^{(0)}(q_{x},|q_{x}|,\\delta k_{x\\alpha}) \\frac{\\partial^{2}}{\\partial\nk_{x\\alpha}^{2}} \\delta(k_{x\\alpha}-k_{r0}).  \\label{34}\n\\end{equation}\n\nAgain we will neglect contributions $\\propto (q_{x} \\ell_{0})^{m}$ for $m\n\\geq 4$. In Eq. (\\ref{32}) we have $d_{10}(q_{x},0)=-(q_{x} \\ell_{0})\/\\sqrt{2%\n}+O((q_{x} \\ell_{0})^{3})$ and $%\nb_{00}^{(0)}(q_{x},|q_{x}|,0)=a_{00}(q_{x})+O((q_{x} \\ell_{0})^{2})$, $%\nb_{10}^{(0)}(q_{x},|q_{x}|,0)=-(q_{x} \\ell_{0}\/\\sqrt{2}) \\\na_{11}(q_{x})+O((q_{x} \\ell_{0})^{3})$. It follows\n\n\\begin{equation}\nI_{10}^{(1)}(q_{x})=-\\frac{(q_{x} \\ell_{0})^{2}}{2 \\ell_{0}} a_{00}(q_{x}) ,\n\\;\\;\\;\\;\\; I_{11}^{(1)}(q_{x})=\\frac{(q_{x} \\ell_{0})^{3}}{2 \\ell_{0}} \\\na_{11}(q_{x}) .  \\label{35}\n\\end{equation}\nEvaluating the integral in Eq. (\\ref{33}) by parts we obtain\n\n\\begin{eqnarray}\nI_{1n}^{(2)}(q_{x})&=&\\frac{(2^{n-1}n!)^{1\/2}}{2!} \\ q_{x}^{2}\\ \\Big\\{ \\frac{%\nd}{d \\delta k_{x\\alpha}} \\ [d_{10}(q_{x},\\delta k_{x\\alpha})]_{|\\delta\nk_{x\\alpha}=0} \\ b_{n0}^{(0)}(q_{x},|q_{x}|,0)+  \\nonumber \\\\\n* &&d_{10}(q_{x},0)\\ \\frac{d}{d \\delta k_{x\\alpha}} \\\n[b_{n0}^{(0)}(q_{x},|q_{x}|,\\delta k_{x\\alpha})]_{|\\delta k_{x\\alpha}=0} %\n\\Big\\}.  \\label{36}\n\\end{eqnarray}\nThis gives the contributions\n\n\\begin{equation}\nI_{10}^{(2)}(q_{x})=\\frac{(q_{x} \\ell_{0})^{2}}{2 \\ell_{0}} \\ a_{00}(q_{x})\n, \\;\\;\\;\\;\\; I_{11}^{(2)}(q_{x})=-\\frac{(q_{x} \\ell_{0})^{3}}{\\ell_{0}} \\\na_{11}(q_{x}) .  \\label{37}\n\\end{equation}\nEvaluating the integral in Eq. (\\ref{34}) by parts we obtain\n\n\\begin{equation}\nI_{10}^{(3)}(q_{x})=0 , \\;\\;\\;\\;\\; I_{11}^{(3)}(q_{x})=\\frac{2(q_{x}\n\\ell_{0})^{3}}{3\\ell_{0}} \\ a_{11}(q_{x}) .  \\label{38}\n\\end{equation}\nUsing Eqs. (\\ref{35}),(\\ref{37}) and (\\ref{38}) we can rewrite\nEq. (\\ref{31}) as\n\n\\begin{equation}\n\\rho_{0}^{(1)}=\\frac{e^{2} \\ell_{0}^{2} a_{11}(q_{x})} {6\\pi \\hbar \\epsilon\n\\tilde{\\omega}_{0}} q_{x}^{3} \\rho_{0}^{(1)} .  \\label{39}\n\\end{equation}\nEq. (\\ref{39}) shows that the {\\it dipole} branch, similar to that in Ref.\n\\cite{11}, is not coupled to {\\it monopole} excitations of the charge\ndensity. Further, Eq. (\\ref{39}), with $a_{11}(q_{x})=1\/2$ for\n$|q_{x}|\\ell_{0} \\ll 1$ and precision\n$\\alt 0.2 \\% $ \\cite{11}, gives the dispersion relation of the dipole\nbranch, $\\rho_{0}^{(1)}(\\omega,q_{x}) \\neq 0$,\n\n\\begin{equation}\n\\omega=v_{g 0}q_{x}+\\frac{\\ell_{0}^{2}}{6 \\epsilon} \\sigma_{yx}^{0}\nq_{x}^{3} -\\frac{i}{\\tau} ,  \\label{40}\n\\end{equation}\nwhere $\\sigma_{yx}^{0}=e^{2}\/2 \\pi \\hbar$ for $\\nu=1$ and\n$\\sigma_{yx}^{0}=e^{2}\/\\pi \\hbar$ for $\\nu=2$.\n\nAn important difference of Eq. (\\ref{40}) from the corresponding result, Eq.\n(40), of Ref. \\cite{11} for the pure dipole EMP is that here we have\n$\\omega-v_{g 0}q_{x} \\propto q_{x}^{3}$ whereas in Ref. \\cite{11} this\nbecomes $\\omega-v_{g 0}q_{x} \\propto q_{x}$. That is, the term caused by the\nCoulomb interaction has a {\\it nonacoustic} behavior if we neglect\ndissipation. The calculations demonstrate clearly that the spatial\ndispersion effects, although they depend on $q_{x}$, they are essentially\nrelated to the wave structure along the $y$-direction. It is known that\nspatial dispersion is directly related to the nonlocality of responses, as\nexpressed by the dielectric function, etc. \\cite{151}. In Eq. (\\ref{40})\nboth nonlocal effects and edge states screening are taken into account.\nWe point out that in Ref. \\cite {17}\na similar dispersion law, $\\omega \\propto a q_{x}^{3}$, was obtained for\nshort-range forces, i.e., for an interaction that is essentially not of the\nCoulomb type.\n\n\\section{Fundamental EMPs for $\\protect\\nu=4$ and $6$}\n\nFor $\\nu=2(\\bar{n}+1)$ the $n=0$,...,$n=\\bar{n}$ LLs have intersections with\nthe Fermi level at $y_{r0}$,..., $y_{r\\bar{n}}$, respectively, and\nEq. (\\ref{14}) can be written as\n\n\\begin{eqnarray}\n\\rho(\\omega, q_{x},y&)&=\\frac{2e^{2}}{\\pi \\hbar \\epsilon} \\sum_{n=0}^{\\bar{n}%\n} \\int_{-\\infty}^{\\infty} d{k_{x\\alpha}} \\frac{f_{n,k_{x\\alpha}-q_x}\n-f_{n,k_{x\\alpha}}} {\\tilde{\\omega}-v_{gn}(k_{x \\alpha}) q_{x}} \\ \\Pi_{nn}(\ny, k_{x\\alpha}, k_{x\\alpha}-q_{x})  \\nonumber \\\\\n* \\   \\nonumber \\\\\n&&\\times\\int_{-\\infty}^{\\infty} d\\tilde{y} \\int_{-\\infty}^{\\infty}\ndy^{\\prime}\\ \\Pi_{nn}( \\tilde{y}, k_{x\\alpha}, k_{x\\alpha}-q_{x})\nK_{0}(|q_{x}||\\tilde{y}-y^{\\prime}|)] \\rho(\\omega, q_{x},y^{\\prime}) .\n\\label{41}\n\\end{eqnarray}\nFor confining potentials smooth on the $\\ell_{0}$ scale we have $\\Delta\ny_{m-1,m}=y_{rm-1}-y_{rm} \\gg \\ell_{0}$, where $m \\leq \\bar{n}$. For $\\nu=4$\nthere is only one inter-LL length $\\Delta y_{0,1}=y_{r0}-y_{r1}$. Further,\nmaking the same  approximations, in the long-wavelength limit $q_{x}\\ell_{0}\n\\ll 1$, as in Sec. IIIA and integrating over $k_{x\\alpha}$ in\nEq. (\\ref{41}), we obtain\n\n\\begin{eqnarray}\n\\rho(\\omega, q_{x},y)&=&\\frac{2e^{2}}{\\pi \\hbar \\epsilon} \\sum_{n=0}^{\\bar{n}%\n} \\frac{q_{x}}{\\tilde{\\omega}_{n}} \\Psi_{n}^{2}(y-y_{rn})  \\nonumber \\\\\n* \\   \\nonumber \\\\\n&&\\times\\int_{-\\infty}^{\\infty} d\\tilde{y} \\int_{-\\infty}^{\\infty}\ndy^{\\prime}[\\Psi_{n}^{2}(\\tilde{y}-y_{rn}) K_{0}(|q_{x}||\\tilde{y}%\n-y^{\\prime}|)] \\rho(\\omega, q_{x},y^{\\prime}) ,  \\label{42}\n\\end{eqnarray}\nwhere $\\tilde{\\omega}_{n} \\equiv \\tilde{\\omega}-v_{gn} q_{x}$. It follows\nthat $\\rho(\\omega, q_{x},y)$ can be represented by a sum of charges\n$\\rho_{n}(\\omega, q_{x},y)= \\rho_{n}^{(0)}(\\omega, q_{x}) \\Psi_{n}^2(y-y_{rn})\n$, localized at the edge of the $n$-th LL, within a region of extent $\\sim\n\\sqrt{2n+1} \\ell_{0}$, $n=0,...,\\bar{n}$. The result is\n\n\\begin{equation}\n\\rho(\\omega, q_{x},y)=\\sum_{n=0}^{\\bar{n}} \\rho_{n}^{(0)}(\\omega, q_{x})\n\\Psi_{n}^{2}(y-y_{rn}) .  \\label{43}\n\\end{equation}\nWe substitute Eq. (\\ref{43}) into Eq. (\\ref{42}) and demand that the\ncoefficients of $\\Psi_{n}^{2}(y-y_{rn})$ on both sides of\nEq. (\\ref{42})  be equal.\nThis leads to $\\bar{n}+1$ linear homogeneous\nequations for $\\rho_{n}^{(0)}(\\omega, q_{x})$.\n\n\\subsection{$\\protect\\nu=4$}\n\nDue to $\\Delta y_{0,1} \\gg \\ell_{0}$ we can neglect the exponentially small\noverlap between $\\rho_{0}(\\omega, q_{x},y)$ and $\\rho_{1}(\\omega, q_{x},y)$.\nWe will assume $q_{x} \\Delta y_{01} \\ll 1$.\nAs shown above, from Eqs. (\\ref{42}) and (\\ref{43}), with $\\bar{n}=1$, we\nobtain the system\n\n\\begin{equation}\n\\rho_{0}^{(0)}=\\frac{2e^{2}}{\\pi\\hbar\\epsilon} \\frac{q_x}{\\tilde{\\omega}_{0}}\n[a_{00}(q_{x})\\rho_0^{(0)}+ a_{00}^{11}( q_{x}, \\Delta y_{01})\\rho_{1}^{(0)}\n] ,  \\label{44}\n\\end{equation}\n\n\\begin{equation}\n\\rho_{1}^{(0)}=\\frac{2e^{2}}{\\pi\\hbar\\epsilon} \\frac{q_x}{\\tilde{\\omega}_{1}}\n[a_{00}^{11}( q_{x}, \\Delta y_{01}) \\rho_{0}^{(0)}+ a_{11}^{11}( q_{x},\n0)\\rho_{1}^{(0)} ] ,  \\label{45}\n\\end{equation}\nwhere $\\rho_{n}^{(0)}(\\omega, q_{x}) \\equiv \\rho_{n}^{(0)}$ and\n\n\\begin{equation}\na_{nn}^{mm}( q_{x}, \\Delta y)=\n\\int_{-\\infty}^{\\infty}\\int_{-\\infty}^{\\infty} dx\\\ndx^{\\prime}\\Psi_{n}^2(x)\\Psi_{m}^2(x^{^{\\prime}})\nK_{0}(|q_{x}||x-x^{\\prime}+\\Delta y|).  \\label{46}\n\\end{equation}\nHere $a_{nn}^{mm}( q_{x}, \\Delta y)=a_{mm}^{nn}( q_{x}, \\Delta y)$,\n$a_{nn}^{mm}( q_{x}, \\Delta y)=a_{nn}^{mm}( q_{x}, -\\Delta y)$ and\n$a_{00}^{00}( q_{x},0)=a_{00}(q_{x})$. For a nontrivial solution of the\nsystem of Eqs. (\\ref{44}) and (\\ref{45}) the determinant of the coefficients\nmust vanish. This gives the dispersion relation of the renormalized\nfundamental EMP of the $n=0$ LL as\n\n\\begin{equation}\n\\omega=q_{x} v_{01} +\\frac{\\sigma_{yx}^{0}}{\\epsilon} \\ q_{x} [2 \\ln(\\frac{1%\n}{|q_{x}|\\ell_{0}})-\\ln(\\frac{\\Delta y_{01}}{\\ell_{0}})+ \\frac{3}{5}]-\\frac{i%\n}{\\tau} ,  \\label{47}\n\\end{equation}\nand that of the $n=1$ LL as\n\n\\begin{equation}\n\\omega=q_{x} v_{01} +\\frac{\\sigma_{yx}^{0}}{\\epsilon} \\ q_{x} [\\ln(\\frac{%\n\\Delta y_{01}}{\\ell_{0}})+\\frac{2}{5}]-\\frac{i}{\\tau} ,  \\label{48}\n\\end{equation}\nwhere $\\sigma_{yx}^{0}=2e^{2}\/\\pi \\hbar$ and $v_{01} =(v_{g0}+v_{g1})\/2$.\nAll double integrals involved in the coefficients of Eqs. (\\ref{44}) and\n(\\ref{45}) can be evaluated within the approximation $K_{0}(x) \\approx\n\\ln(2\/|x|)-\\gamma$, where $\\gamma$ is the Euler constant. We obtain\n$a_{11}^{11}( q_{x}, 0) \\approx \\ln(1\/|q_{x}|\\ell_{0})+1\/4$ and\n$a_{00}^{11}(q_{x}, \\Delta y_{01}) \\approx\n(\\ln(2)-\\gamma)- \\ln(|q_{x}|\\Delta y_{01}) \\approx \\ln(1\/|q_{x}\\Delta\ny_{01}|)+0.1$. Notice that in the absence of dissipation Eqs. (\\ref{47}) and\n(\\ref{48}) coincide with the quasi-microscopic results of Ref. \\cite{18}.\n>From Eqs. (\\ref{44}) and (\\ref{45}) it is easy to see that, if the inter-LL\nCoulomb coupling is neglected by setting $a_{00}^{11}( q_{x}, \\Delta\ny_{01})\\equiv 0$, the dispersion laws of the {\\it decoupled} fundamental\nEMPs of the $n=0$ and $n=1$ LLs are given, respectively, by Eq. (\\ref{23})\nand by\n\n\\begin{equation}\n\\omega=q_{x} v_{g1}+ \\frac{2e^{2}}{\\pi \\hbar \\epsilon} q_{x} [\\ln(\\frac{1}{%\n|q_{x}|\\ell_{0}})+\\frac{1}{4}]-\\frac{i}{\\tau} .  \\label{49}\n\\end{equation}\nSubstituting Eq. (\\ref{47}) in Eq. (\\ref{44}) and Eq. (\\ref{48}) in Eq. (\\ref\n{45}) we obtain, respectively, $\\rho_{1}^{(0)}\/\\rho_{0}^{(0)} \\approx 1$\nand $\\rho_{0}^{(0)}\/\\rho_{1}^{(0)} \\approx -1$. This means that in the\nformer case the wave charges localized at the edges of the $n=0$ LL and $n=1$\nLLs are in phase whereas in the latter they are out of phase. Therefore, the\nformer EMP has an {\\it acoustic} spatial structure along the $y$ axis while\nthe latter EMP has an {\\it optical} spatial structure though its dispersion\nlaw is purely acoustic.\n\n\\subsection{$\\protect\\nu=6$}\n\nFor $\\nu=6$ we have the intersection of the $n=2$ LL with the Fermi level,\nat $y_{r2}$, in addition those of the $n=0$ and $n=1$ LLs; $\\bar{n}=2$.\nCorresponding to Eqs. (\\ref{42}) and Eq. (\\ref{43}) we now obtain\n\n\\begin{equation}\n\\rho_{0}^{(0)}=\\frac{2e^{2}}{\\pi\\hbar\\epsilon} \\frac{q_x}{\\tilde{\\omega}_{0}}\n[a_{00}(q_{x})\\rho_{0}^{(0)}+ a_{00}^{11}( q_{x}, \\Delta\ny_{01})\\rho_{1}^{(0)}+ a_{00}^{22}( q_{x}, \\Delta y_{02})\\rho_{2}^{(0)} ] ,\n\\label{50}\n\\end{equation}\n\n\\begin{equation}\n\\rho_{1}^{(0)}=\\frac{2e^{2}}{\\pi\\hbar\\epsilon} \\frac{q_x}{\\tilde{\\omega}_{1}}\n[a_{00}^{11}( q_{x}, \\Delta y_{01}) \\rho_{0}^{(0)}+ a_{11}^{11}( q_{x}, 0)\n\\rho_{1}^{(0)}+ a_{11}^{22}( q_{x}, \\Delta y_{12}) \\rho_{2}^{(0)} ] ,\n\\label{51}\n\\end{equation}\n\n\\begin{equation}\n\\rho_{2}^{(0)}=\\frac{2e^{2}}{\\pi\\hbar\\epsilon} \\frac{q_x}{\\tilde{\\omega}_{2}}\n[a_{00}^{22}( q_{x}, \\Delta y_{02}) \\rho_{0}^{(0)}+ a_{11}^{22}( q_{x},\n\\Delta y_{12}) \\rho_{1}^{(0)}+ a_{22}^{22}( q_{x}, 0) \\rho_{2}^{(0)} ] .\n\\label{52}\n\\end{equation}\nThe vanishing of the determinant of the coefficients leads to the cubic\nequation\n\n\\begin{equation}\n\\omega^{` 3}+a_{2}(q_{x}) \\omega^{` 2}+ a_{1}(q_{x})\n\\omega^{`}+a_{0}(q_{x})=0 ,  \\label{53}\n\\end{equation}\nwhere $\\omega^{` }=\\tilde{\\omega}\/ (2e^{2}q_{x}\/\\pi \\hbar \\epsilon)$. The\nexpressions for the coefficients $a_{k}(q_{x}), \\ k=0,1,2$, are given in the\nappendix. We will assume that $q_{x} \\Delta y_{ij} \\ll 1$, $i \\neq j \\leq 2$.\n\nIt can be shown that all three roots of Eq. (\\ref{53}) are real and\ndifferent as they correspond to the irreducible case. Assuming $\\Delta\ny_{02}, \\Delta y_{12}, \\Delta y_{01} \\gg \\ell_{0}$,  these roots give the\ndispersion law for the renormalized fundamental EMP of the $n=0$ LL as\n\n\\begin{equation}\n\\omega=q_{x} v_{012} + \\frac{2\\sigma_{yx}^{0}}{3\\epsilon} q_{x} \\{3 \\ln(%\n\\frac{1}{|q_{x}|\\ell_{0}})- \\frac{2}{3}[\\ln(\\frac{\\Delta y_{02}}{\\ell_{0}})+\n\\ln(\\frac{\\Delta y_{01}}{\\ell_{0}})+ \\ln(\\frac{\\Delta y_{12}}{\\ell_{0}})]+\n\\frac{8}{15} \\}-\\frac{i}{\\tau} ,  \\label{57}\n\\end{equation}\nand that for the renormalized fundamental EMP of the $n=1$ LL, $\\omega_{+}$,\nand $n=2$ LL, $\\omega_{-}$, as\n\n\\begin{eqnarray}\n\\omega_{\\pm}=&&q_{x} v_{012} + \\frac{2\\sigma_{yx}^{0}}{9\\epsilon} q_{x}\n\\{\\ln(\\frac{\\Delta y_{02}}{\\ell_{0}})+ \\ln(\\frac{\\Delta y_{01}}{\\ell_{0}})+\n\\ln(\\frac{\\Delta y_{12}}{\\ell_{0}})+\\frac{7}{10} \\} \\pm  \\nonumber \\\\\n* \\   \\nonumber \\\\\n&&\\frac{2 \\sqrt{2}\\sigma_{yx}^{0}}{9\\epsilon} q_{x} \\{\\ln^{2}(\\frac{\\Delta\ny_{02}}{\\Delta y_{01}})+ \\ln^{2}(\\frac{\\Delta y_{02}}{\\Delta y_{12}})+\n\\ln^{2}(\\frac{\\Delta y_{01}}{\\Delta y_{12}})+A \\}^{1\/2}-\\frac{i}{\\tau} .\n\\label{58}\n\\end{eqnarray}\nHere $\\sigma_{yx}^{0}=3e^{2}\/\\pi \\hbar$,\n$v_{012} =(v_{g0}+v_{g1}+v_{g2})\/3$, and\n\n\\begin{eqnarray}\nA=&&(\\tilde{v}_{g0}+3\/4)[ (\\tilde{v}_{g0}- \\tilde{v}_{g2}+ 3\/4)\/2+\\ln(\\Delta\ny_{02}\\Delta y_{01}\/ {\\Delta y_{12}}^{2})]  \\nonumber \\\\\n* &&+(\\tilde{v}_{g1}+1\/4)[(\\tilde{v}_{g1}- \\tilde{v}_{g0}- 1\/2)\/2+\n\\ln(\\Delta y_{01}\\Delta y_{12}\/ {\\Delta y_{02}}^{2})]  \\nonumber \\\\\n* &&+\\tilde{v}_{g2} [ (\\tilde{v}_{g2}- \\tilde{v}_{g1}-1\/4)\/2+\\ln(\\Delta\ny_{02}\\Delta y_{12}\/ {\\Delta y_{01}}^{2})] ,  \\label{59}\n\\end{eqnarray}\nwhere $\\tilde{v}_{gi}=v_{gi}\/(2e^{2}\/\\pi \\hbar \\epsilon)$, $i=0,1,2$. Notice\nthat the fundamental EMPs of the $n=1$ and $n=2$ LLs have purely acoustic\ndispersion laws, cf. Eq. (\\ref{58}) . Only the fundamental EMP of the $n=0$\nLL, Eq. (\\ref{57}), behaves in the previously obtained manner $\\propto q_{x}\n\\ln(q_{x})$, cf. Refs. \\cite{1}-\\cite{10}), for a fundamental EMP. It is\ndifficult to clearly disentangle the contributions of the $n=2$ LL in\nEq. (\\ref{58}). It can be shown though that, if they are neglected by\nsetting $\\tilde{a}_{22}^{22}$, $a_{11}^{22}$, $a_{00}^{22}$, $v_{g2}$\nto zero, then we have $\\omega^{^{\\prime}}_{2}=0$ and Re$\\omega_{-}=0$ as\nwe should in this limit.\n\nThe solutions of the cubic equation, Eqs. (\\ref{57}) and  (\\ref{58}), are\nnot obtained using the standard expressions,  reproduced in the appendix,\nwhich become very unwieldy for the  present case. Instead, the following\nreasoning is used. If $\\omega$ has an acoustic character for some branches,\nthen $\\omega^{`}$ in Eq. (\\ref{53}) is independent of $q_{x}$ and should\nsatisfy the quadratic equation obtained by differentiating Eq. (\\ref{53})\nwith respect to $q_{x}$\n\n\\begin{equation}\n\\frac{\\partial a_{2}(q_{x})}{\\partial q_{x}} \\omega^{`2}+ \\frac{\\partial\na_{1}(q_{x})}{\\partial q_{x}} \\omega^{`}+ \\frac{\\partial a_{0}(q_{x})}{%\n\\partial q_{x}}=0 .  \\label{60}\n\\end{equation}\nThe two roots $\\omega_{2}^{`}$ and $\\omega^{`}_{3}$ of Eq. (\\ref{60}) are\nindeed independent of $q_{x}$ and lead to the solutions $\\omega_{-}$ and\n$\\omega_{+}$, respectively, given by Eq. (\\ref{58}). The other root is given\nby $\\omega^{`}_{1}=-a_{2}(q_{x})-(\\omega^{^{\\prime}}_{2}+\n\\omega^{^{\\prime}}_{3})$ and results in Eq. (\\ref{57}). Of course both\nmethods give the same solutions $\\omega^{^{\\prime}}_{j}$, $j=1,2,3$.\nNotice that on the RHS of Eqs. (\\ref{57}) and (\\ref{58}) we  used\n{\\it approximate} expressions of $a_{ii}^{jj}(q_{x},\\Delta y_{ij})$ which ,\nassuming $\\Delta y_{ij}\/\\ell_{0} \\gg 1$, $i \\neq j$, are obtained with the\napproximation\n\n\\begin{equation}\n\\int_{-\\infty}^{\\infty}\\int_{-\\infty}^{\\infty} dx\\\ndx^{\\prime}\\Psi_{i}^2(x)\\Psi_{j}^2(x^{^{\\prime}}) \\ln(|x-x^{\\prime}+\\Delta\ny_{ij}|\/\\ell_{0}) \\approx \\ln(\\Delta y_{ij}\/\\ell_{0}) .  \\label{61}\n\\end{equation}\nBelow we will demonstrate that the approximation (\\ref{61}) is well\njustified even for not-too-large values $\\Delta y_{ij}\/\\ell_{0}$. For that\nwe will calculate the dispersion laws of the renormalized fundamental EMPs\nusing the {\\it exact} expressions of $a_{ii}^{jj}(q_{x},\\Delta y_{ij})$,\ni.e., with the double integral on the LHS of Eq. (\\ref{61}) evaluated\nnumerically. In the latter case we will calculate $\\omega^{^{\\prime}}_{j}$\nfrom the standard expressions (\\ref{A1}) -(\\ref{A3}) of the appendix.\n\nFor a $GaAs$-based 2DEG and negligible dissipation the dispersion laws\n(\\ref{57}) and (\\ref{58}) for the renormalized, by the inter-LL Coulomb\ncoupling, fundamental EMPs are shown in Fig. 1 by the solid curves. The\nparameters are $m^{*} \\approx 6.1 \\times 10^{-29} g$, $\\epsilon \\approx 12.5$,\nand $\\Omega \\approx 7.8 \\times 10^{11} sec^{-1}$ \\cite{19}. For $\\nu=6$\nand $B=3 $Tesla these parameters lead to $\\hbar \\omega_{c} \\approx 5 meV$,\n$\\omega_{c}\/\\Omega \\approx 10$. Here $\\omega_{*}=2e^{2}\/\\pi \\hbar\n\\epsilon \\ell_{0}$ is a characteristic frequency almost equal, for the\nconditions stated,  to $\\omega_{c}$. We have also assumed\n$\\Delta_{F2}=\\hbar \\omega_{c}\/2$. This gives $v_{g0} \\approx \\sqrt{5}\n\\Omega \\ell_{0}$, $v_{g1} \\approx \\sqrt{3} \\Omega \\ell_{0}$, $v_{g2} \\approx\n\\Omega \\ell_{0}$, $\\Delta y_{02}\/\\ell_{0} \\approx (\\omega_{c}\/\\Omega)\n[\\sqrt{5}-1]$, $\\Delta y_{01}\/\\ell_{0} \\approx (\\omega_{c}\/\\Omega)\n[\\sqrt{5}-\\sqrt{3}]$, and $\\Delta y_{12}\/\\ell_{0} \\approx (\\omega_{c}\/\n\\Omega)[\\sqrt{3}-1]$. The dashed curves show the fundamental EMPs of\nthe totally decoupled $n=0$ (Eq. (\\ref{23}), topmost dashed curve),\n$n=1$, and $n=2$ (lowest dashed curve) LLs, i.e., the modes obtained\nby neglecting the inter-LL Coulomb coupling. In this case all coefficients\n$a_{ii}^{jj}$ on the RHS of Eqs. (\\ref{50})-(\\ref{52}) vanish for $i \\neq j$.\nThis leads directly to the dispersion laws represented by the dashed curves\nin Fig. 1. The dotted curves show the dispersion laws of the renormalized\nfundamental EMPs using the exact expressions for $a_{ii}^{jj}(q_{x},\\Delta\ny_{ij})$, $i \\neq j$. It is seen that each dotted curve is very close to the\ncorresponding solid curve.\n\nFor the same parameters and conditions as in Fig. 1  we show, in Fig. 2 ,\n$\\rho(\\omega,q_{x},y)\/\\rho_{*}$ (solid and dotted curves) and\n$\\rho(\\omega,q_{x},y)\/2\\rho_{*}$ (dashed curve) as a function of\n$Y=(y-y_{r2})\/\\ell_{0}$ with\n$\\rho_{*}=\\rho_{0}^{(0)}(\\omega,q_{x})\/\\sqrt{\\pi} \\ell_{0}$. The solid curve\ncorresponds to the topmost solid curve in Fig. 1, Eq. (\\ref{57}), with\n$\\rho_{1}^{(0)}\/\\rho_{0}^{(0)} \\approx 1.0$ and\n$\\rho_{2}^{(0)}\/\\rho_{0}^{(0)} \\approx 1$.\nAs can be seen the renormalized fundamental EMP of the $n=0$ LL has an\n{\\it acoustic} spatial structure along the $y$ axis. The dashed curve\ncorresponds to the lowest solid curve in Fig. 1, $\\omega_{-}$ of Eq. (\\ref\n{58}), with $\\rho_{1}^{(0)}\/\\rho_{0}^{(0)} \\approx -2.0$ and\n$\\rho_{2}^{(0)}\/\\rho_{0}^{(0)} \\approx 1$. Hence the renormalized fundamental\nEMP of the $n=2$ LL has an {\\it optical} spatial structure though its\ndispersion law is purely acoustic. The dotted curve corresponds to the\nmiddle solid curve in Fig. 1, $\\omega_{+}$ of Eq. (\\ref{58}), with\n$\\rho_{1}^{(0)}\/\\rho_{0}^{(0)} \\approx -0.1$ and $\\rho_{2}^{(0)}\/\n\\rho_{0}^{(0)} \\approx -1.0$. Thus, the renormalized fundamental EMP\nof the $n=1$ LL has an {\\it optical} spatial structure though its\ndispersion law is purely acoustic. Notice that the dependence of\n$\\rho_{j}^{(0)}\/\\rho_{0}^{(0)}$, $j=1,2$, on $q_{x}$ is typically weak.\n\nIn Fig. 3 we use the same values for $m^{*}$, $\\epsilon$, $\\Omega$, and\n$\\Delta_{F2}=\\hbar \\omega_{c}\/2$, $\\nu=6$ as in Fig. 1. However, we take\n$B=1.5$Tesla and this leads to $\\omega_{c}\/\\Omega \\approx 5$. This gives\n$\\sqrt{2}$ times larger values for $v_{g0} \\approx \\sqrt{5} \\Omega \\ell_{0}$, $v_{g1} \\approx\n\\sqrt{3} \\Omega \\ell_{0}$, and $v_{g2} \\approx \\Omega \\ell_{0}$, and twice\nsmaller values for $\\Delta y_{02}\/\\ell_{0} \\approx (\\omega_{c}\/\\Omega)[\n\\sqrt{5}-1]$, $\\Delta y_{01}\/\\ell_{0} \\approx (\\omega_{c}\/\\Omega)[\\sqrt{5}-\n\\sqrt{3}]$, and $\\Delta y_{12}\/\\ell_{0} \\approx (\\omega_{c}\/\\Omega)[\\sqrt{3}\n-1]$ as compared with those in Fig. 1. All  curves are marked as in Fig. 1\nand show the corresponding  dispersion laws.\n\nIn Fig. 4 we plot $\\rho(\\omega,q_{x},y)\/\\rho_{*}$ as a function of\n$Y=(y-y_{r2})\/\\ell_{0}$ for the conditions corresponding to the solid curves\nin Fig. 3. All curves are marked as in Fig. 2. We have, respectively,\n$\\rho_{1}^{(0)}\/\\rho_{0}^{(0)} \\approx 1.0$ and\n$\\rho_{2}^{(0)}\/\\rho_{0}^{(0)} \\approx 1$ (solid  curve),\n$\\rho_{1}^{(0)}\/\\rho_{0}^{(0)} \\approx -2.1$ and\n$\\rho_{2}^{(0)}\/\\rho_{0}^{(0)} \\approx 1.2$ (dashed curve), and\n$\\rho_{1}^{(0)}\/\\rho_{0}^{(0)} \\approx -0.1$ and\n$\\rho_{2}^{(0)}\/\\rho_{0}^{(0)} \\approx -1.0$ (dotted  curve).\nAs in Fg. 2, the renormalized fundamental EMP of the $n=0$ LL has an\n{\\it acoustic} spatial structure along the $y$ axis while those of the\n$n=1$ and $n=2$ LL have an {\\it optical} spatial structure though their\ndispersion laws are purely acoustic. As noted above, the dependence of\n$\\rho_{j}^{(0)}\/\\rho_{0}^{(0)}$, $j=1,2$, on $q_{x}$ is typically weak.\n\n\\section{discussion and concluding remarks}\n\nWe presented a fully {\\it microscopic} model of EMPs in a RPA framework\nvalid for integer $\\nu\\geq 1$ and confining potentials that are smooth on\nthe $\\ell_{0}$ scale but still sufficiently steep that LL flattening can be\nneglected \\cite{20}. The model takes into account LL coupling and treats\nonly very weak dissipation. The main results of the present work are as\nfollows.\n\ni) For moderately steep confining potential we presented a microscopic model\nthat improves the quasi-microscopic approach of Refs. \\cite{11} and\n\\cite{18}. In particular, the model combines features of the following\ndistinct edge wave mechanisms. In mostly classical models, e.g., \\cite{1},\n\\cite{7}, the position of the edge does not vary but the charge density\nprofile at the edge does. In a sence this edge wave mechanism is the analog\nof that for the Kelvin wave \\cite{21} at the edge of a rotating \"shallow\"\nsea with chirality determined by the Coriolis parameter which corresponds to\nthe cyclotron frequency $\\omega_{c}$. Another edge-wave mechanism,\nfully quantum mechanical, is that of Refs. \\cite{10}, \\cite{17}, and\n\\cite{22}-\\cite{23}, in which, for $\\nu=1$, only the edge position of an\nincompressible 2DEG of the lowest LL varies and with respect to which the\ndensity profile is that of the undisturbed 2DEG. The approach of these\nworks is limited to the subspace of the lowest LL wave functions,\nneglects LL mixing and dissipation, and results in a single EMP with\ndispersion law similar to that in Ref. \\cite{1}.\n\nii) We confirmed, for $\\nu=1 (2)$, the existence of EMP modes in addition to\nthe fundamental EMP and single one obtained for vertically steep unperturbed\nelectron density profile \\cite{1}. This is in line with our earlier\nquasi-microscopic results \\cite{11} for moderately steep \\cite{11} confining\npotentials. The additional modes result from an {\\bf exact} solution of\nEq. (\\ref{16}) in terms of the complete set of the Hermite polynomials.\nWith this expansion we can also make contact, for $\\nu=1 (2)$, with other\nmicroscopic theories \\cite{10}, \\cite{17}, and \\cite{22}-\\cite{23} that are\nlimited to the subspace of the lowest LL wave functions. If we retain only\nthe $n=0$ term in Eq. (\\ref{16}), we obtain only the $n=0$ LL fundamental\nmode with the same dispersion relation $\\omega\\sim q_{x}\\ln q_{x}$. The\nadditional modes result from retaining ( correspond to) higher-order terms\nin the expansion Eq. (\\ref{16}). This shows that the limitation to this\nLL subspace is too restrictive and indicates the importance of LL\ncoupling even for $\\nu=1 (2)$.\n\niii) An important additional mode is the {\\it dipole} EMP, presented in Sec.\nIII B, with dispersion relation $\\omega\\sim q_{x}^{3} $. This differs\nmarkedly from the corresponding result $\\omega\\sim q_{x}$ of Ref. \\cite{11}\nand signals {\\it nonlocal} responses that were previously neglected. This\nand other differences between the present results and those of Ref. \\cite{11}\nfurther indicate the importance of having a fully {\\it microscopic} theory\nfor EMPs.\n\niv) As we showed, taking LL coupling into account is an essential ingredient\nespecially for $\\nu>2$. As Fig. 1 and Fig. 3 demonstrate, the coupling\nstrongly renormalizes the {\\it uncoupled} fundamental LL modes and results\nin one mode behaving as the usual one $\\omega\\sim q_{x}\\ln q_{x}$ and the\nothers behaving in an {\\it acoustic} manner $\\omega\\sim q_{x}$. This partly\nreminds the results of Ref. \\cite{7} but the resemblance should not be\noverestimated because the two models are drastically different.\n\nv) We have not treated spin effects beyond the HA or RPA. Our study is\nfocused on important wave effects of non-spin nature and spin splitting is\nassumed negligible for even $\\nu$. Thus, skyrmions and spin textures are not\ndealt with in this work. Though neglecting the spin splitting is a\nreasonable approximation, for even $\\nu \\geq 2$\nin the bulk of the channel, its validity near the edges remains uncertain\nin view of the work of Refs. \\cite{16} and \\cite{24}.\n\n   (vi) Finally, a  few remarks are in order about  the studies\nof Refs.  \\cite{25} and \\cite{26}  that used the RPA. \nThe study of Ref. \\cite{26} is too simplified and in essence  \nrepeats the results of Ref. \\cite{25} and of works cited therein\nfor the \"optical\" EMP. As for the results of Ref. \\cite{25}, \ndue to omitted important contributions,\nthey are essentially different from ours for both the acoustic\nand the optical modes. As Sec. IV A shows, an important logarithmic term,\ncaused by the Coulomb interaction, is missed\nin Eqs.  (54a) and (54b) of Ref. \\cite{25}. \nIn addition,  Refs. \\cite{25} and \\cite{26} did not study a\ndipole mode or   multipole modes pertaining to any occupied LL.\nThese modes of ours are totally different than those of\nSecs. 3.1.6 and  3.1.7 of Ref. \\cite{25}.\n\n\\acknowledgements\nThis work was supported by NSERC Grant No. OGP0121756. In addition, O G B\nacknowledges partial support by the Ukrainian SFFI Grant No. 2.4\/665.\n\\ \\newline\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nA tensor is a multidimensional array of numbers. Formally, an order-$k$ tensor is an element of the tensor product of $k$ vector spaces. A first order tensor is a vector, and a second order tensor is a matrix. The rank of a tensor is the minimal number of terms in its expression as a sum of simple tensors. See Cichocki et al. \\cite{Cichocki} for a recent survey of matrix and tensor decomposition algorithms, as well as applications in areas such as data mining, email surveillance, gene expression classification, and signal processing. See also Kroonenberg \\cite{Kroonenberg}, and Smilde et al. \\cite{Smilde}, and references therein.\n\nA tensor is called symmetric if its elements remain fixed under any permutation of the indices. Then we can define the symmetric rank of a symmetric tensor to be the minimal number of terms in its expression as a sum of simple symmetric tensors. Further, we can associate a homogeneous polynomial in $\\mathbb{F}[x_0, \\dots, x_n]_d$ to any symmetric tensor $X \\in S^d V$, and the determination of the symmetric rank of $X$ is equivalent to the Big Waring Problem: determining the minimum integer $r$ such that a generic form of degree $d$ in $n+1$ variables can be written as a sum of $r$ $d$th powers of linear forms \\cite{Comon1}. See \\cite{B1, B2, Bernardi, B3, B4} and references therein for more information on symmetric tensors, symmetric rank, and decompositions of homogeneous polynomials.\n\nThe determination of canonical forms of tensors is a generalization of the matrix case. For any $m \\times n$ matrix $A$ (with $m \\leq n$) there exist invertible matrices $P \\in GL_m(\\mathbb{F})$ and $Q \\in GL_n(\\mathbb{F})$ such that \n\\[\nPAQ = \n\\begin{bmatrix} I_r & 0 \\\\ 0 & 0 \\end{bmatrix},\n\\]\nwhere $I_r$ is the $r \\times r$ identity and $r \\leq m$ is the rank. The matrix $PAQ$ is called the Smith normal form (canonical form), and is the orbit representative for the action of $GL_m(\\mathbb{F}) \\times GL_n(\\mathbb{F})$, the direct product of general linear groups. When $m, n = 2$, there are three canonical forms,\n\\[\n\\begin{bmatrix} 0 & 0 \\\\ 0 & 0 \\end{bmatrix}, \\qquad\n\\begin{bmatrix} 1 & 0 \\\\ 0 & 0 \\end{bmatrix}, \\qquad\n\\begin{bmatrix} 1 & 0 \\\\ 0 & 1 \\end{bmatrix}.\n\\]\nDetermining the canonical forms of order-$k$ tensors for $k>2$ is a more complicated problem. For the next simplest case, consider $2 \\times 2 \\times 2$ tensors. Le Paige (1881) determined there are 7 canonical forms over $\\mathbb{C}$ \\cite{LePaige}, and later Oldenburger determined there are 8 over $\\mathbb{R}$ \\cite{Oldenburger1}. In previous work, we considered the same problem over the prime fields $\\mathbb{F}_2$ and $\\mathbb{F}_3$ \\cite{BremnerStavrou}. The maximum rank of these tensors over $\\mathbb{R}$, $\\mathbb{C}$, and the prime fields we considered is 3. For larger tensor formats over $\\mathbb{R}$ and $\\mathbb{C}$, the problem is more complicated and, in general, there is not a finite classification \\cite{deSilvaLim}. \n\nIn previous work, we considered the same problem of determining canonical forms over prime fields, but restricted our attention to symmetric tensors. For the $2 \\times 2 \\times 2$ and $2 \\times 2 \\times 2 \\times 2$ symmetric formats over $\\mathbb{R}$ and $\\mathbb{C}$, this problem was analyzed independently by Gurevich \\cite{Gurevich} first, then by Weinberg \\cite{Weinberg}.\nIn \\cite{Stavrou}, we extended these results by determining the canonical forms of $2 \\times 2 \\times 2$ symmetric tensors over the prime fields $\\mathbb{F}_p$ for $p = 2, 3, 5, 7, 11, 13, 17$, as well as $2 \\times 2 \\times 2 \\times 2$ symmetric tensors over $\\mathbb{F}_p$ for $p = 2, 3, 5, 7$. \n\nWe continue in this direction by considering order-$k$ (for $k = 2,3,4$) symmetric tensors of format $3 \\times \\dots \\times 3$ over $\\mathbb{F}_p$, for $p \\in \\{2,3,5\\}$. For $3 \\times 3$ symmetric tensors (i.e. symmetric matrices), we work over $\\mathbb{F}_p$ for $p = 2, 3, 5$. The maximum symmetric rank is 4, which is larger than the maximum rank of 3.  For $3 \\times 3 \\times 3$ symmetric tensors, we work over $\\mathbb{F}_p$ for $p=2,3$. The maximum symmetric rank is 7, which is larger than the maximum rank of 5 over $\\mathbb{C}$ \\cite{PierreComon}. Lastly, for $3 \\times 3 \\times 3 \\times 3$ symmetric tensors, we work over $\\mathbb{F}_p$ for $p=2, 3$. Over $\\mathbb{F}_3$, the number of symmetric tensors was too large to determine, and so the procedure had to be terminated. At termination, there were symmetric tensors with symmetric rank 13, thus we have a lower bound on the symmetric rank.\n\nIn each case, we compute the symmetric tensors of each rank. The general linear group GL$_3(\\mathbb{F}_p)$ partitions the tensors of each rank into a disjoint union of orbits, where the tensors in each orbit are equivalent. The elements are arranged in lexical order and the minimal element is the canonical form of its orbit under this group action. We comment here that due to limited computer memory, we are not able to consider larger prime fields, or other tensor formats. \n\n\n\\section{Preliminaries}\n\nAn order-$k$ {\\bf tensor} $X$ is an element of the {\\bf tensor product} of $k$ vector spaces $V_1 \\otimes \\dots \\otimes V_k$, where the {\\bf order} refers to the number of dimensions. Once we fix a basis in each vector space, we can associate to $X$ a $k$-dimensional array. Then an order-1 tensor is a vector, an order-2 tensor is a matrix, and we denote an order-$k$ tensor of format $d_1 \\times \\dots \\times d_k$ as $[x_{i_1 \\dots i_k}] \\in \\mathbb{F}^{d_1 \\times \\dots \\times d_k}$. We can represent the tensor $X$ in {\\bf vectorized (flattened)} form by writing the columns of $X$ in a vector format, where the entries are in lexical order of the $k$-tuples of the subscripts. \n\nAn order-$k$ tensor $[x_{i_1 \\dots i_k}] \\in \\mathbb{F}^{n \\times \\dots \\times n}$ is called {\\bf symmetric} if \n\\[\nx_{i_{\\pi(1)} \\dots i_{\\pi(k)}} = x_{i_1 \\dots i_k}, \\qquad i_1, \\dots, i_k \\in \\{1, \\dots, n\\},\n\\]\nfor all permutations $\\pi \\in S_k$  \\cite{Comon1}. \n\n\\begin{proposition} [Proposition 3.7 \\cite{Comon1}]\nLet $X = [x_{i_1 \\dots i_k}] \\in \\mathbb{F}^{n \\times \\dots \\times n}$ be an order-$k$ tensor. Then \n\\[\n\\pi(X) = X\n\\]\nfor all permutations $\\pi \\in S_k$ if and only if \n\\[\nx_{i_{\\pi(1)} \\dots i_{\\pi(k)}} = x_{i_1 \\dots i_k}, \\qquad i_1, \\dots, i_k \\in \\{1, \\dots, n\\}\n\\]\nfor all permutations $\\pi \\in S_k$.\n\\end{proposition}\n\nIn this paper we consider $3 \\times \\dots \\times 3$ symmetric tensors of order-$k$ for $k=2,3,4$. For $k = 2$, a symmetric matrix has the form\n\\[\n\\begin{bmatrix}\na & b & c \\\\\nb & d & e \\\\\nc & e & f\n\\end{bmatrix},\n\\]\nwhere the entries are scalars in $\\mathbb{F}_p$. For $k = 3$, \na $3 \\times 3 \\times 3$ symmetric tensor, in terms of its $3 \\times 3$ frontal slices, has the form \n\\[\n\\left[\n\\begin{array}{ccc|ccc|ccc}\na & b & c & b & d & e & c & e & f \\\\\nb & d & e & d & j & g & e & g & h \\\\\nc & e & f & e & g & h & f & h & k\n\\end{array}\n\\right].\n\\]\nLastly, for $k = 4$, a $3 \\times 3 \\times 3 \\times 3$ symmetric tensor, in terms of its $3 \\times 3$ frontal slices, has the form\n\\[\n\\left[\n\\begin{array}{ccc|ccc|ccc}\na & b & f & b & c & g & f & g & j \\\\\nb & c & g & c & d & i & g & i & h \\\\\nf & g & j & g & i & h & j & h & k \\\\ \\hline\nb & c & g & c & d & i & g & i & h \\\\\nc & d & i & d & e & l & i & l & m \\\\\ng & i & h & i & l & m & h & m & n \\\\ \\hline\nf & g & j & g & i & h & j & h & k \\\\\ng & i & h & i & l & m & h & m & n \\\\\nj & h & k & h & m & n & k & n & p\n\\end{array}\n\\right].\n\\]\n\nWe will denote the set of all order-$k$, $n$-dimensional symmetric tensors over the field $\\mathbb{F}$ by $\\mathcal{S}^k(\\mathbb{F}^n) \\subset \\mathbb{F}^{n \\times \\dots \\times n}$. The set of such tensors satisfies the property $\\pi(X) = X$ for all $\\pi \\in S_k$ and $X \\in \\mathcal{S}^k(\\mathbb{F}^n)$. \n\n\\begin{definition}\\cite{Hitchcock1} \nA tensor $X \\in \\mathbb{F}^{d_1 \\times \\dots \\times d_k}$ is {\\bf simple} if it can be written as\n\\[\nX = u^{(1)} \\otimes u^{(2)} \\otimes \\cdots \\otimes u^{(k)},\n\\]\nwith non-zero $u^{(i)} \\in \\mathbb{F}^{d_i}$ for $i = 1, \\dots, k$. The $(i_1, i_2, \\dots, i_k)$th entry of $X$ is \n\\[\nx_{ i_1 i_2 \\cdots i_k } = u^{(1)}_{i_1} u^{(2)}_{i_2} \\cdots u^{(k)}_{i_k}.\n\\]\n\\end{definition}\n\n\\begin{definition}\nA tensor $X \\in \\mathcal{S}^k(\\mathbb{F}^n)$ is called {\\bf simple symmetric} if it can be written as \n\\[\nX = u \\otimes \\dots \\otimes u\n\\]\nwith $k$-many non-zero vectors $u \\in \\mathbb{F}^n$. The $(i_1, \\dots, i_k)$th entry of $X$ is \n\\[\nx_{i_1 \\dots i_k} = u_{i_1} \\dots u_{i_k}.\n\\]\n\\end{definition}\n\n\\begin{definition}\\cite{Hitchcock1}\nA tensor has {\\bf outer product rank} $r$ if it can be written as a sum of $r$ (and no fewer) decomposable tensors, \n\\[\nX = \\sum_{i = 1}^r  u_i^{(1)} \\otimes \\dots \\otimes u_i^{(k)} = u_1^{(1)} \\otimes \\dots \\otimes u_1^{(k)} + \\dots + u_r^{(1)} \\otimes \\dots \\otimes u_r^{(k)}\n\\]\nwhere $u_i^{(1)} \\in \\mathbb{F}^{d_1}, \\dots, u_i^{(k)} \\in \\mathbb{F}^{d_k}, i = 1, \\dots, r$.\nWe write rank$(X)$ to denote the outer product rank of $X$.\n\\end{definition}\n\n\\begin{definition}\nA tensor has {\\bf symmetric outer product rank} $s$ if it can be written as a sum of $s$ (and no fewer) symmetric simple tensors,\n\\[\nX = \\sum_{i=1}^s u_i^{\\otimes k}.\n\\]\nWe write rank$_S(X)$ to denote the symmetric outer product rank of $X$. \n\\end{definition}\nThe only rank-0 symmetric tensor is the zero tensor. We will drop the words outer product and simply say symmetric rank. \n\n\\begin{definition}\nThe {\\bf maximum rank} is defined to be  \n\\[\n\\text{max}\\{ \\text{rank}(X) \\mid X \\in \\mathbb{F}^{d_1 \\times \\dots \\times d_k} \\}. \n\\]\nIf we replace rank$(X)$ with rank$_S(X)$ we get the analogous definition of maximum symmetric rank.\n\n\\end{definition}\n\nIn order to compute the canonical forms of symmetric tensors we use a tensor-matrix multiplication. {\\bf Multilinear matrix multiplication} is a tensor-matrix multiplication that allows us to multiply matrices on each of the modes of the tensor. If $X = [x_{i_1 \\dots i_k}] \\in \\mathbb{F}^{d_1 \\times \\dots \\times d_k}$ and \n\\[\nA_1 = [a^{(1)}_{u_1 i_1}] \\in \\mathbb{F}^{c_1 \\times d_1}, \\dots, A_k = [a^{(k)}_{u_k i_k}] \\in \\mathbb{F}^{c_k \\times d_k},\n\\]\nthen $Y= (A_1, \\dots, A_k)\\cdot X = [y_{u_1 \\dots u_k}] \\in \\mathbb{F}^{c_1 \\times \\dots \\times c_k}$ is the new tensor defined by\n\\[\ny_{u_1 \\dots u_k} = \\sum_{i_1, \\dots, i_k = 1}^{d_1, \\dots, d_k} a^{(1)}_{u_1 i_1}  \\dots  a^{(k)}_{u_k i_k} x_{i_1 \\dots i_k}.\n\\]\nSince we are restricting ourselves to symmetric tensors, we impose the condition that $A_1 = \\dots = A_k$, otherwise, multilinear matrix multiplication can transform a symmetric tensor into a non-symmetric tensor.\n\n\nWe consider the action of the symmetry group GL$_3(\\mathbb{F}_p)$, which does not change the rank of a tensor \\cite{deSilvaLim}. The action of this group decomposes the set of order-$k$ symmetric tensors into a disjoint union of orbits, where the tensors in each orbit are equivalent under the group action. The orbit of $X$ is the set \n\\[\n\\mathcal{O}_X := \\{ (g, \\dots, g) \\cdot X \\mid g \\in \\text{GL}_3(\\mathbb{F}_p)\\}.\n\\] \nWe define the {\\bf canonical form} of $X$ to be the minimal element in its orbit with respect to the lexical ordering. \n\n\n\n\\section{Algorithms}\n\nUsing computer algebra, we generate all the symmetric tensors in each symmetric rank. Then we apply the group action which decomposes the set of tensors within each rank into a disjoint union of orbits. The minimal element of each orbit with respect to the lexical ordering is the canonical form.  \n\nWe determine the symmetric tensors in each symmetric rank by first generating the set $R_1$ of simple symmetric tensors, which is achieved by computing all possible products $v \\otimes \\dots \\otimes v$ of non-zero vectors $v \\in \\mathbb{F}^3_p$, for each prime. Then to determine the elements in $R_i$ we compute all possible sums $(T + S)\\ \\textrm{mod} \\ p$ for $T \\in R_{i-1}$ and $S \\in R_1$, and then subtract the tensors already computed in the previous ranks: $R_i = R_i \\setminus \\cup_{k=1}^{i-1} R_k$. The procedure terminates once no new symmetric tensors are generated when taking sums.  In our previous work \\cite{Stavrou}, when $T+S$ was computed, the procedure searched to see if $T+S$ already existed in the current rank and the lower ranks. If it did not, then it was added to the current rank set. In this paper, we made the necessary modification of subtracting the sets so that the ranks were computed more efficiently. This is what allowed us to consider larger formats than what we considered in \\cite{Stavrou}.\n\n\nNext we apply the action of the general linear group to decompose the tensors in each rank into a disjoint union of orbits. The pseudocode is provided in Table \\ref{groupalgorithmtable}. This algorithm was used in our previous paper \\cite{Stavrou}. In this paper, we made two major modifications to the program. The first modification was removing an index (mode) from the procedures in order to consider second order $3 \\times 3$ symmetric tensors.  The second modification was to index the elements to 3, since each mode has length 3. The pseudocode in Table \\ref{groupalgorithmtable} displays the order-4 case. For the order-3 case, remove the $\\ell$ index, and for the order-2 case, remove the $\\ell$ and $k$ indices.\n\n\n\n\\begin{table}\n\\begin{itemize}\n\\item[]\n\\texttt{unflatten}$( x )$\n\\item[] \\quad\nset $t \\leftarrow 0$\n\\item[] \\quad\nfor $i, j, k, \\ell = 1, 2, 3$ do:\nset $t \\leftarrow t + 1$;\nset $y_{ijk\\ell} \\leftarrow x_t$\n\\item[] \\quad\n$\\texttt{return}( y )$\n\\end{itemize}\n\\medskip\n\\begin{itemize}\n\\item[]\n\\texttt{groupaction}$( g, x, m )$\n\\item[] \\quad\nset $y \\leftarrow \\texttt{unflatten}( x )$\n\\item[] \\quad\nif $m = 1$ then\nfor $j,k,\\ell = 1,2,3$ do:\n\\item[] \\quad \\quad\nset $v \\leftarrow [ \\, y_{1jk\\ell}, \\, y_{2jk\\ell} \\, ]$;\n\\item[] \\quad \\quad\nset $w \\leftarrow [ \\, g_{11} v_1 {+} g_{12} v_2 {+} g_{13}v_3 \\, \\mathrm{mod}\\,p,\n\\dots,\n\\, g_{31} v_1 {+} g_{32} v_2{+} g_{33} v_3 \\, \\mathrm{mod}\\,p \\,]$\n\\item[] \\quad \\quad\nfor $i = 1, 2, 3$ do: set $y_{ijk\\ell} \\leftarrow w_i$\n\\item[] \\quad\nif $m = 2$ then \\dots \\emph{(similar for second subscript)}\n\\item[] \\quad\nif $m = 3$ then \\dots \\emph{(similar for third subscript)}\n\\item[] \\quad\nif $m = 4$ then \\dots \\emph{(similar for fourth subscript)}\n\\item[] \\quad\n\\texttt{return}( \\texttt{flatten}( $y$ ) )\n\\end{itemize}\n\\medskip\n\\begin{itemize}\n\\item[]\n\\texttt{smallorbit}$( x )$\n\\item[] \\quad\nset $\\texttt{result} \\leftarrow \\{\\,\\}$\n\\item[] \\quad\nfor $a \\in GL_3(\\mathbb{F}_p)$ do:\n\\item[] \\quad \\quad\nset $y \\leftarrow \\texttt{groupaction}( a, x, 1 )$\n\\item[] \\quad \\quad\nset $z \\leftarrow \\texttt{groupaction}( a, y, 2 )$\n\\item[] \\quad \\quad \nset $w \\leftarrow \\texttt{groupaction}( a, z, 3 )$\n\\item[] \\quad \\quad \nset $u \\leftarrow \\texttt{groupaction}( a, w, 4 )$\n\\item[] \\quad\nset $\\texttt{result} \\leftarrow \\texttt{result} \\cup \\{ u \\}$\n\\item[] \\quad\n\\texttt{return}( \\texttt{result} )\n\\end{itemize}\n\\medskip\n\\begin{itemize}\n\\item\nfor $r = 0,\\dots,\\texttt{maximumrank}$ do:\n\\item[] \\quad\nset $\\texttt{representatives}[r] \\leftarrow \\{\\,\\}$;\nset $\\texttt{remaining} \\leftarrow \\texttt{arrayset}[r]$\n\\item[] \\quad\nwhile $\\texttt{remaining} \\ne \\{\\,\\}$ do:\n\\item[] \\quad \\quad\nset $x \\leftarrow \\texttt{remaining}[1]$;\nset $\\texttt{xorbit} \\leftarrow \\texttt{largeorbit}( x )$\n\\item[] \\quad \\quad\nappend $\\texttt{xorbit}[1]$ to $\\texttt{representatives}[r]$\n\\item[] \\quad \\quad\nset $\\texttt{remaining} \\leftarrow \\texttt{remaining} \\setminus \\texttt{xorbit}$\n\\end{itemize}\n\\medskip\n\\caption{Algorithm for group action (pseudocode)}\n\\label{groupalgorithmtable}\n\\end{table}\n\n\n\n\n\n\n\\section{Symmetric Tensors of Format $3 \\times 3$}\n\nEvery order-$k$ symmetric tensor of dimension $n$ may be uniquely associated with a homogeneous polynomial (also called a quantic) of degree $k$ in $n$ variables \\cite{Comon1}. The problem of determining symmetric rank is equivalent to the Big Waring Problem: determining the minimal number of $p$th powers of linear terms \\cite{Ehrenborg2} \\cite{Ehrenborg3}. The Alexander-Hirschowitz (AH) Theorem \\cite{AH} completely describes the calculation of the generic rank of symmetric tensors. \n\n\n\n\\subsection{Canonical Forms of $3 \\times 3$ Symmetric Tensors over $F_2$.}\n\nThere are 512 $3 \\times 3$ tensors over $\\mathbb{F}_2$, where 64 are symmetric. Every symmetric tensor has a symmetric decomposition. The maximum symmetric rank is 3, which is equal to the maximum rank. The symmetric ranks, orders of each orbit, and the minimal representatives of each orbit are given in Table \\ref{table33symmetricmod2}. The number of tensors in each symmetric rank is listed below.\n\\[\n\\begin{array}{lrrrrrrrr}\n\\text{rank} & 0 & 1 & 2 & 3 \\\\\n\\text{number} & 1 & 7 & 21 & 35 \\\\\n\\text{$\\approx$ $\\%$} & 1.5625\\% & 10.9375\\% & 32.8125\\% & 54.6875\\%\n\\end{array}\n\\]\n\n  \\begin{table}\n  \\[\n  \\begin{array}{ccc}\n  \\text{symmetric rank} & \\text{orbit size} &  \\text{canonical form} \n    \\\\\n  \\toprule \n0 & 1 & \\begin{bmatrix}0&0&0\\\\0&0&0\\\\0&0&0\\end{bmatrix}\n\\\\\n\\midrule \n1 & 7 & \\begin{bmatrix}0&0&0\\\\0&0&0\\\\0&0&1\\end{bmatrix}\n\\\\\n\\midrule\n2 & 21 &  \\begin{bmatrix}0&0&0\\\\0&0&1\\\\0&1&1\\end{bmatrix}\n\\\\\n\\midrule\n3 & 7 & \\begin{bmatrix}0&0&0\\\\0&0&1\\\\0&1&0\\end{bmatrix}\n\\\\\n\\midrule\n3 & 28 & \\begin{bmatrix}0&0&1\\\\0&1&0\\\\1&0&0\\end{bmatrix}\n  \\\\\n  \\bottomrule\n  \\end{array}\n  \\]\n  \\medskip\n  \\caption{Canonical forms of $3 \\times 3$ symmetric tensors over $\\mathbb{F}_2$}\n  \\label{table33symmetricmod2}\n  \\end{table}\n\n\n\\subsection{Canonical Forms of $3 \\times 3$ Symmetric Tensors over $F_3$.}\n\nThere are 19,683 $3 \\times 3$ tensors over $\\mathbb{F}_3$, where 729 are symmetric. Every symmetric tensor has a symmetric decomposition. The maximum symmetric rank in this case is 4, which is larger than the maximum rank of 3. The symmetric ranks, orders of each orbit, and the minimal representatives of each orbit are given in Table \\ref{table33symmetricmod3}. The number of tensors in each symmetric rank is listed below.\n\\[\n\\begin{array}{lrrrrrrrr}\n\\text{rank} & 0 & 1 & 2 & 3 & 4 \\\\\n\\text{number} & 1 & 13 & 91 & 390 & 234 \\\\\n\\text{$\\approx$ $\\%$} & 0.1372\\% & 1.7833\\% & 12.4829\\% & 53.4979\\% & 32.0988\n\\end{array}\n\\]\n\n  \\begin{table}\n  \\[\n  \\begin{array}{ccc}\n  \\text{symmetric rank} & \\text{orbit size} &  \\text{canonical form} \n    \\\\\n  \\toprule \n0 & 1 & \\begin{bmatrix}0&0&0\\\\0&0&0\\\\0&0&0\\end{bmatrix}\n\\\\\n\\midrule \n1 & 13 & \\begin{bmatrix}0&0&0\\\\0&0&0\\\\0&0&1\\end{bmatrix}\n\\\\\n\\midrule\n2 & 13 &  \\begin{bmatrix}0&0&0\\\\0&0&0\\\\0&0&2\\end{bmatrix}\n\\\\\n\\midrule\n2 & 78 &  \\begin{bmatrix}0&0&0\\\\0&1&0\\\\0&0&1\\end{bmatrix}\n\\\\\n\\midrule\n3 & 156 & \\begin{bmatrix}0&0&0\\\\0&0&1\\\\0&1&0\\end{bmatrix}\n\\\\\n\\midrule\n3 & 234 &  \\begin{bmatrix}0&0&1\\\\0&2&0\\\\1&0&0\\end{bmatrix}\n\\\\\n\\midrule\n4 & 234 & \\begin{bmatrix}0&0&1\\\\0&1&0\\\\1&0&0\\end{bmatrix}\n  \\\\\n  \\bottomrule\n  \\end{array}\n  \\]\n  \\medskip\n  \\caption{Canonical forms of $3 \\times 3$ symmetric tensors over $\\mathbb{F}_3$}\n  \\label{table33symmetricmod3}\n  \\end{table}\n\n\n\n\n\n\n\\subsection{Canonical Forms of $3 \\times 3$ Symmetric Tensors over $F_5$.}\n\nThere are 1,953,125 $3 \\times 3$ tensors over $\\mathbb{F}_5$, where 15,625 are symmetric.  The maximum symmetric rank is again 4. The order of the group GL$(\\mathbb{F}_5)$ is 427,307 which our program cannot handle, and thus we do not determine the orbits under the group action. The symmetric ranks, number of tensors in each rank, and the minimal representative of each rank are given in Table \\ref{table33symmetricmod5}. \n\\[\n\\begin{array}{lrrrrrrrr}\n\\text{rank} & 0 & 1 & 2 & 3 & 4 \\\\\n\\text{number} & 1 & 62 & 1922 & 7440 & 6200 \\\\\n\\text{$\\approx$ $\\%$} & 0.0064\\% & 0.3968\\% & 12.3008\\% & 47.6160\\% & 39.6800\n\\end{array}\n\\]\n\n  \\begin{table}\n  \\[\n  \\begin{array}{ccc}\n  \\text{symmetric rank} & \\text{rank size} &  \\text{minimal element} \n    \\\\\n  \\toprule \n0 & 1 & \\begin{bmatrix}0&0&0\\\\0&0&0\\\\0&0&0\\end{bmatrix}\n\\\\\n\\midrule \n1 & 62 & \\begin{bmatrix}0&0&0\\\\0&0&0\\\\0&0&1\\end{bmatrix}\n\\\\\n\\midrule\n2 & 1922 &  \\begin{bmatrix}0&0&0\\\\0&0&0\\\\0&0&2\\end{bmatrix}\n\\\\\n\\midrule\n3 & 7440 &  \\begin{bmatrix}0&0&0\\\\0&1&0\\\\0&0&2\\end{bmatrix}\n\\\\\n\\midrule\n4 & 6200 & \\begin{bmatrix}0&0&1\\\\0&2&0\\\\1&0&0\\end{bmatrix}\n  \\\\\n  \\bottomrule\n  \\end{array}\n  \\]\n  \\medskip\n  \\caption{Minimal elements of $3 \\times 3$ symmetric tensors over $\\mathbb{F}_5$}\n  \\label{table33symmetricmod5}\n  \\end{table}\n  \n  \n  \n  \n  \n  \n  \n\n\n\n\n\\section{Symmetric Tensors of Format $3 \\times 3 \\times 3$}\n\nIn Dickson's 1908 paper \\cite{Dickson}, he cites that a complete set of canonical forms of ternary cubics was determined over $\\mathbb{C}$. He considers canonical forms by determining the algebraic irrationalities occurring in the reducing linear transformations over $\\mathbb{F}_3$, and finds 11 distinct forms. \n\n\nMore recently, Comon \\cite{PierreComon} summarize the generic ranks of symmetric tensors of some specific formats. In particular, they tabulate the equivalence classes for ternary cubics (i.e. symmetric $3 \\times 3 \\times 3$ tensors). Below are the orbits under the action of the group of invertible three-dimensional changes of coordinates.\n\\[\n\\begin{array}{c|c}\n\\text{Orbit} & \\text{rank} \\\\ \\hline\nx^3 & 1 \\\\\nx^3+y^3 & 2 \\\\\nx^2y & 3 \\\\\nx^3 + 3y^2z & 4 \\\\\nx^3 + y^3 + 6xyz & 4 \\\\\nx^3 + 6xyz & 4 \\\\\na(x^3 + y^3 + z^3) + 6bxyz & 4 \\\\\nx^2y + xz^2 & 5\n\\end{array}\n\\]\n\n\nKogan and Maza \\cite{Kogan} determined the equivalence classes of ternary cubics under general complex linear changes of variables using a computational approach. See in particular Theorem 5 of \\cite{Kogan}. Their contribution provided a method of computing the signature manifolds for each of the equivalence classes, and their algorithm matches a cubic with its canonical form while producing the required linear transformation explicitly.\n\n\n\n\n\\subsection{Canonical Forms of $3 \\times 3 \\times 3$ Symmetric Tensors over $\\mathbb{F}_2$}\n\n\nThere are 1024 $3 \\times 3 \\times 3$ symmetric tensors over $\\mathbb{F}_2$. Only 128 (12.5\\%) of these have a symmetric decomposition. The maximum symmetric rank is 7. The symmetric ranks, orders of each orbit, and the minimal representatives (in flattened form) of each orbit are given in Table \\ref{table333symmetricmod2}. To make the canonical forms more legible we replace the zero entries with $\\cdot$. The number of tensors in each symmetric rank is listed below (percentages add to 12.5\\%).\n\\[\n\\begin{array}{lrrrrrrrr}\n\\text{rank} & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 \\\\\n\\text{number} & 1 & 7 & 21 & 35 & 35 & 21 & 7 & 1 \\\\\n\\text{$\\approx$ $\\%$} & 0.10\\% & 0.68\\% & 2.05\\% & 3.42\\% & 3.42\\% & 2.05\\% & 0.68\\% & 0.10 \\%\n\\end{array}\n\\]\n\n  \\begin{table}\n  \\[\n  \\begin{array}{ccc}\n  \\text{symmetric rank} & \\text{orbit size} &  \\text{canonical form (flattened)} \n    \\\\\n  \\toprule \n0 & 1 & [ \\cdot  \\cdot  \\cdot  \\cdot  \\cdot  \\cdot  \\cdot  \\cdot  \\cdot  \\cdot  \\cdot  \\cdot  \\cdot  \\cdot  \\cdot  \\cdot  \\cdot  \\cdot  \\cdot  \\cdot  \\cdot  \\cdot  \\cdot  \\cdot  \\cdot  \\cdot  \\cdot ]\n\\\\\n\\midrule \n1 & 7 & [\\cdot \\cdot \\cdot \\cdot \\cdot \\cdot \\cdot \\cdot \\cdot \\cdot \\cdot \\cdot \\cdot \\cdot \\cdot \\cdot \\cdot \\cdot \\cdot \\cdot \\cdot \\cdot \\cdot \\cdot \\cdot \\cdot 1] \n\\\\\n\\midrule\n2 & 21 &  [\\cdot \\cdot \\cdot \\cdot \\cdot \\cdot \\cdot \\cdot \\cdot \\cdot \\cdot \\cdot \\cdot \\cdot 1 \\cdot 1 1 \\cdot \\cdot \\cdot \\cdot 1 1 \\cdot 1 1]\n\\\\\n\\midrule\n3 & 7 & [\\cdot \\cdot \\cdot \\cdot \\cdot \\cdot \\cdot \\cdot \\cdot \\cdot \\cdot \\cdot \\cdot \\cdot 1 \\cdot 1 1 \\cdot \\cdot \\cdot \\cdot 1 1 \\cdot 1 \\cdot]\n\\\\\n3 & 18 &  [\\cdot \\cdot 1 \\cdot \\cdot \\cdot 1 \\cdot 1 \\cdot \\cdot \\cdot \\cdot 1 \\cdot \\cdot \\cdot \\cdot 1 \\cdot 1 \\cdot \\cdot \\cdot 1 \\cdot 1]\n\\\\\n\\midrule\n4 & 7 &  [\\cdot \\cdot \\cdot \\cdot \\cdot 1 \\cdot 1 \\cdot \\cdot \\cdot 1 \\cdot \\cdot 1 1 1 1 \\cdot 1 \\cdot 1 1 1 \\cdot 1 \\cdot]  \n\\\\  \n4 & 18 & [\\cdot \\cdot 1 \\cdot \\cdot \\cdot 1 \\cdot 1 \\cdot \\cdot \\cdot \\cdot 1 \\cdot \\cdot \\cdot \\cdot 1 \\cdot 1 \\cdot \\cdot \\cdot 1 \\cdot \\cdot]\n\\\\   \n\\midrule\n5 & 21 & [\\cdot \\cdot \\cdot \\cdot \\cdot 1 \\cdot 1 \\cdot \\cdot \\cdot 1 \\cdot \\cdot 1 1 1 1 \\cdot 1 \\cdot 1 1 1 \\cdot 1 1]    \n\\\\\n\\midrule \n6 & 7 & [\\cdot \\cdot \\cdot \\cdot \\cdot 1 \\cdot 1 \\cdot \\cdot \\cdot 1 \\cdot \\cdot \\cdot 1 \\cdot \\cdot \\cdot 1 \\cdot 1 \\cdot \\cdot \\cdot \\cdot 1]\n\\\\  \n\\midrule\n7 & 1 & [\\cdot \\cdot \\cdot \\cdot \\cdot 1 \\cdot 1 \\cdot \\cdot \\cdot 1 \\cdot \\cdot \\cdot 1 \\cdot \\cdot \\cdot 1 \\cdot 1 \\cdot \\cdot \\cdot \\cdot \\cdot]   \n  \\\\\n  \\bottomrule\n  \\end{array}\n  \\]\n  \\medskip\n  \\caption{Canonical forms of $3 \\times 3 \\times 3$ symmetric tensors over $\\mathbb{F}_2$}\n  \\label{table333symmetricmod2}\n  \\end{table}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\subsection{Canonical Forms of $3 \\times 3 \\times 3$ Symmetric Tensors over $\\mathbb{F}_3$}\n\nThere are 59,049 $3 \\times 3 \\times 3$ symmetric tensors over $\\mathbb{F}_3$, and unlike in the previous case, every symmetric tensor has a symmetric decomposition. The maximum symmetric rank is 7. The symmetric ranks, orders of each orbit, and the minimal representatives of each orbit are given in Table \\ref{table333symmetricmod3}. The number of tensors in each rank is listed in the table below. We mention now that we cannot consider larger primes because of insufficient computer memory. There are over 9.7 million symmetric $3 \\times 3 \\times 3$ tensors over $\\mathbb{F}_5$.\n\n\\[\n\\begin{array}{lrrrrrrrr}\n\\text{rank} & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 \\\\\n\\text{number} & 1 & 26 & 312 & 2288 & 11440 & 30342 & 14352 & 288 \\\\\n\\text{$\\approx$ $\\%$} & 0.00\\% & 0.05\\% & 0.53\\% & 3.87\\% & 19.37\\% & 51.38\\% & 24.31\\% & 0.49 \\%\n\\end{array}\n\\]\n\n\\begin{table}\n \\[\n  \\begin{array}{cccc}\n  \\text{symmetric rank} & \\text{orbit size} &  \\text{canonical form (flattened)} \n    \\\\\n  \\toprule\n0 & 1 &  [\\cdot  \\cdot  \\cdot  \\cdot  \\cdot  \\cdot  \\cdot  \\cdot  \\cdot  \\cdot  \\cdot  \\cdot  \\cdot  \\cdot  \\cdot  \\cdot  \\cdot  \\cdot  \\cdot  \\cdot  \\cdot  \\cdot  \\cdot  \\cdot  \\cdot  \\cdot  \\cdot]   \n\\\\\n\\midrule  \n1 & 26 & [\\cdot \\cdot \\cdot \\cdot \\cdot \\cdot \\cdot \\cdot \\cdot \\cdot \\cdot \\cdot \\cdot \\cdot \\cdot \\cdot \\cdot \\cdot \\cdot \\cdot \\cdot \\cdot \\cdot \\cdot \\cdot \\cdot 1]  \n\\\\\n\\midrule\n2 & 312 &  [\\cdot \\cdot \\cdot \\cdot \\cdot \\cdot \\cdot \\cdot \\cdot \\cdot \\cdot \\cdot \\cdot \\cdot 1 \\cdot 1 \\cdot \\cdot \\cdot \\cdot \\cdot 1 \\cdot \\cdot \\cdot 1]  \n\\\\\n\\midrule  \n3 & 104 &  [\\cdot \\cdot \\cdot \\cdot \\cdot \\cdot \\cdot \\cdot \\cdot \\cdot \\cdot \\cdot \\cdot \\cdot \\cdot \\cdot \\cdot 1 \\cdot \\cdot \\cdot \\cdot \\cdot 1 \\cdot 1 \\cdot]  \n\\\\ \n3 & 312 &  [\\cdot \\cdot \\cdot \\cdot \\cdot \\cdot \\cdot \\cdot \\cdot \\cdot \\cdot \\cdot \\cdot \\cdot 1 \\cdot 1 \\cdot \\cdot \\cdot \\cdot \\cdot 1 \\cdot \\cdot \\cdot 2]    \n\\\\\n3 & 1872 & [\\cdot \\cdot 1 \\cdot \\cdot \\cdot 1 \\cdot \\cdot \\cdot \\cdot \\cdot \\cdot 1 \\cdot \\cdot \\cdot \\cdot 1 \\cdot \\cdot \\cdot \\cdot \\cdot \\cdot \\cdot 1]     \n\\\\\n\\midrule  \n4 & 208 & [\\cdot \\cdot \\cdot \\cdot \\cdot \\cdot \\cdot \\cdot \\cdot \\cdot \\cdot \\cdot \\cdot \\cdot \\cdot \\cdot \\cdot 1 \\cdot \\cdot \\cdot \\cdot \\cdot 1 \\cdot 1 1]  \n\\\\\n4 & 1872 & [\\cdot \\cdot \\cdot \\cdot \\cdot \\cdot \\cdot \\cdot 1 \\cdot \\cdot \\cdot \\cdot 1 \\cdot \\cdot \\cdot \\cdot \\cdot \\cdot 1 \\cdot \\cdot \\cdot 1 \\cdot \\cdot ]    \n\\\\\n4 & 468 & [\\cdot \\cdot \\cdot \\cdot \\cdot 1 \\cdot 1 \\cdot \\cdot \\cdot 1 \\cdot \\cdot \\cdot 1 \\cdot \\cdot \\cdot 1 \\cdot 1 \\cdot \\cdot \\cdot \\cdot \\cdot]    \n\\\\\n4 & 1404 & [\\cdot \\cdot 1 \\cdot \\cdot \\cdot 1 \\cdot \\cdot \\cdot \\cdot \\cdot \\cdot \\cdot 1 \\cdot 1 \\cdot 1 \\cdot \\cdot \\cdot 1 \\cdot \\cdot \\cdot 1]    \n\\\\\n4 & 5616 & [\\cdot \\cdot 1 \\cdot \\cdot \\cdot 1 \\cdot \\cdot \\cdot \\cdot \\cdot \\cdot 1 \\cdot \\cdot \\cdot \\cdot 1 \\cdot \\cdot \\cdot \\cdot \\cdot \\cdot \\cdot 2]    \n\\\\\n4 & 1872 & [\\cdot \\cdot 1 \\cdot \\cdot \\cdot 1 \\cdot \\cdot \\cdot \\cdot \\cdot \\cdot 1 \\cdot \\cdot \\cdot 1 1 \\cdot \\cdot \\cdot \\cdot 1 \\cdot 1 1]   \n\\\\\n\\midrule  \n5 & 624 & [\\cdot \\cdot \\cdot \\cdot \\cdot \\cdot \\cdot \\cdot 1 \\cdot \\cdot \\cdot \\cdot \\cdot 1 \\cdot 1 \\cdot \\cdot \\cdot 1 \\cdot 1 \\cdot 1 \\cdot 1]     \n\\\\\n5 & 3744 & [\\cdot \\cdot \\cdot \\cdot \\cdot \\cdot \\cdot \\cdot 1 \\cdot \\cdot \\cdot \\cdot 1 \\cdot \\cdot \\cdot \\cdot \\cdot \\cdot 1 \\cdot \\cdot \\cdot 1 \\cdot 1]    \n\\\\\n5 & 2808  & [\\cdot \\cdot \\cdot \\cdot \\cdot 1 \\cdot 1 \\cdot \\cdot \\cdot 1 \\cdot \\cdot \\cdot 1 \\cdot \\cdot \\cdot 1 \\cdot 1 \\cdot \\cdot \\cdot \\cdot 1]    \n\\\\\n5 & 5616  &   [\\cdot \\cdot \\cdot \\cdot \\cdot 1 \\cdot 1 \\cdot \\cdot \\cdot 1 \\cdot 1 \\cdot 1 \\cdot \\cdot \\cdot 1 \\cdot 1 \\cdot \\cdot \\cdot \\cdot 1]    \n\\\\\n5 & 702  & [\\cdot \\cdot \\cdot \\cdot 1 \\cdot \\cdot \\cdot 1 \\cdot 1 \\cdot 1 \\cdot \\cdot \\cdot \\cdot \\cdot \\cdot \\cdot 1 \\cdot \\cdot \\cdot 1 \\cdot \\cdot]    \n \\\\\n5 & 5616 & [\\cdot \\cdot 1 \\cdot \\cdot \\cdot 1 \\cdot \\cdot \\cdot \\cdot \\cdot \\cdot 1 \\cdot \\cdot \\cdot 1 1 \\cdot \\cdot \\cdot \\cdot 1 \\cdot 1 \\cdot]    \n\\\\ \n5 & 5616  &  [\\cdot \\cdot 1 \\cdot \\cdot \\cdot 1 \\cdot \\cdot \\cdot \\cdot \\cdot \\cdot 1 \\cdot \\cdot \\cdot 1 1 \\cdot \\cdot \\cdot \\cdot 1 \\cdot 1 2]    \n\\\\\n5 & 5616  & [\\cdot \\cdot 1 \\cdot \\cdot \\cdot 1 \\cdot \\cdot \\cdot \\cdot \\cdot \\cdot 1 \\cdot \\cdot \\cdot 2 1 \\cdot \\cdot \\cdot \\cdot 2 \\cdot 2 \\cdot]    \n\\\\\n\\midrule  \n6 & 624 & [\\cdot \\cdot \\cdot \\cdot \\cdot \\cdot \\cdot \\cdot 1 \\cdot \\cdot \\cdot \\cdot \\cdot 1 \\cdot 1 \\cdot \\cdot \\cdot 1 \\cdot 1 \\cdot 1 \\cdot \\cdot]   \n\\\\\n6 & 624 & [\\cdot \\cdot \\cdot \\cdot \\cdot \\cdot \\cdot \\cdot 1 \\cdot \\cdot \\cdot \\cdot \\cdot 1 \\cdot 1 \\cdot \\cdot \\cdot 1 \\cdot 1 \\cdot 1 \\cdot 2]     \n\\\\ \n6 & 5616 & [\\cdot \\cdot \\cdot \\cdot 1 \\cdot \\cdot \\cdot 1 \\cdot 1 \\cdot 1 \\cdot \\cdot \\cdot \\cdot \\cdot \\cdot \\cdot 1 \\cdot \\cdot \\cdot 1 \\cdot 1]    \n\\\\\n6 & 3744 & [\\cdot \\cdot 1 \\cdot 1 \\cdot 1 \\cdot \\cdot \\cdot 1 \\cdot 1 \\cdot 2 \\cdot 2 1 1 \\cdot \\cdot \\cdot 2 1 \\cdot 1 1]    \n\\\\\n6 & 3744 & [\\cdot \\cdot 1 \\cdot 1 \\cdot 1 \\cdot \\cdot \\cdot 1 \\cdot 1 \\cdot 2 \\cdot 2 1 1 \\cdot \\cdot \\cdot 2 1 \\cdot 1 2]    \n \\\\\n \\midrule \n7 & 288  & [\\cdot \\cdot 1 \\cdot 1 \\cdot 1 \\cdot \\cdot \\cdot 1 \\cdot 1 \\cdot 2 \\cdot 2 1 1 \\cdot \\cdot \\cdot 2 1 \\cdot 1 \\cdot] \n\\\\ \n  \\bottomrule\n \\end{array}\n  \\]\n  \\medskip\n  \\caption{Canonical forms of $3 \\times 3 \\times 3$ symmetric tensors over $\\mathbb{F}_3$}\n  \\label{table333symmetricmod3}\n  \\end{table}\n\n\n\n\n\n\n\\section{Symmetric Tensors of Format $3 \\times 3 \\times 3 \\times 3$}\n\nWe consider $3 \\times 3 \\times 3 \\times 3$ symmetric tensors over $\\mathbb{F}_2$. Since the order of the group is too large for our algorithm, we do not determine the orbits under the group action. Instead, we determine the maximum symmetric rank, the number of symmetric tensors in each rank, and the minimal element.\n\n\n\\subsection{Canonical Forms of $3 \\times 3 \\times 3 \\times 3$ Symmetric Tensors over $\\mathbb{F}_2$}\nThere are $2^{81}$ tensors over $\\mathbb{F}_2$, where 32,768 are symmetric. Only 128 (approximately 0.29\\%) of these symmetric tensors have a symmetric decomposition.\n\\[\n\\begin{array}{lrrrrrrrr}\n\\text{rank} & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 \\\\\n\\text{number} & 1 & 7 & 21 & 35 & 35 & 21 & 7 & 1 \\\\\n\\text{$\\approx$ $\\%$} & 0.0031\\% & 0.0214\\% & 0.0641\\% & 0.1068\\% & 0.1068\\% & 0.0641\\% & 0.0214\\% & 0.0031 \\%\n\\end{array}\n\\]\n\nThe rank-0 representative is the zero tensor. The rank-1 minimal representative is the symmetric tensor with every entry equal to 0 except for the $(3,3,3,3)$th entry, which equals 1. The rank-2,3,4,5,6,7 minimal representatives are given respectively by \n\\begin{align*}\n&\n\\left[\n\\begin{array}{ccc|ccc|ccc}\n0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\ \\hline\n0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n0 & 0 & 0 & 0 & 0 & 1 & 0 & 1 & 1 \\\\\n0 & 0 & 0 & 0 & 1 & 1 & 0 & 1 & 1 \\\\ \\hline\n0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n0 & 0 & 0 & 0 & 1 & 1 & 0 & 1 & 1 \\\\ \n0 & 0 & 0 & 0 & 1 & 1 & 0 & 1 & 1 \\\\ \n\\end{array}\n\\right], \n\\quad\n\\left[\n\\begin{array}{ccc|ccc|ccc}\n0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\ \\hline\n0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n0 & 0 & 0 & 0 & 0 & 1 & 0 & 1 & 1 \\\\\n0 & 0 & 0 & 0 & 1 & 1 & 0 & 1 & 1 \\\\ \\hline\n0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n0 & 0 & 0 & 0 & 1 & 1 & 0 & 1 & 1 \\\\ \n0 & 0 & 0 & 0 & 1 & 1 & 0 & 1 & 0 \\\\ \n\\end{array}\n\\right],\n\\\\\n&\n\\left[\n\\begin{array}{ccc|ccc|ccc}\n0 & 0 & 0 & 0 & 0 & 1 & 0 & 1 & 0 \\\\\n0 & 0 & 1 & 0 & 0 & 1 & 1 & 1 & 1 \\\\\n0 & 1 & 0 & 1 & 1 & 1 & 0 & 1 & 0 \\\\ \\hline\n0 & 0 & 1 & 0 & 0 & 1 & 1 & 1 & 1 \\\\\n0 & 0 & 1 & 0 & 0 & 1 & 1 & 1 & 1 \\\\\n1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\\\ \\hline\n0 & 1 & 0 & 1 & 1 & 1 & 0 & 1 & 0 \\\\\n1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\\\ \n0 & 1 & 0 & 1 & 1 & 1 & 0 & 1 & 0 \\\\\n\\end{array}\n\\right],\n\\quad\n\\left[\n\\begin{array}{ccc|ccc|ccc}\n0 & 0 & 0 & 0 & 0 & 1 & 0 & 1 & 0 \\\\\n0 & 0 & 1 & 0 & 0 & 1 & 1 & 1 & 1 \\\\\n0 & 1 & 0 & 1 & 1 & 1 & 0 & 1 & 0 \\\\ \\hline\n0 & 0 & 1 & 0 & 0 & 1 & 1 & 1 & 1 \\\\\n0 & 0 & 1 & 0 & 0 & 1 & 1 & 1 & 1 \\\\\n1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\\\ \\hline\n0 & 1 & 0 & 1 & 1 & 1 & 0 & 1 & 0 \\\\\n1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\\\ \n0 & 1 & 0 & 1 & 1 & 1 & 0 & 1 & 1 \\\\\n\\end{array}\n\\right],\n\\\\\n&\n\\left[\n\\begin{array}{ccc|ccc|ccc}\n0 & 0 & 0 & 0 & 0 & 1 & 0 & 1 & 0 \\\\\n0 & 0 & 1 & 0 & 0 & 1 & 1 & 1 & 1 \\\\\n0 & 1 & 0 & 1 & 1 & 1 & 0 & 1 & 0 \\\\ \\hline\n0 & 0 & 1 & 0 & 0 & 1 & 1 & 1 & 1 \\\\\n0 & 0 & 1 & 0 & 0 & 0 & 1 & 0 & 0 \\\\\n1 & 1 & 1 & 1 & 0 & 0 & 1 & 0 & 0 \\\\ \\hline\n0 & 1 & 0 & 1 & 1 & 1 & 0 & 1 & 0 \\\\\n1 & 1 & 1 & 1 & 0 & 0 & 1 & 0 & 0 \\\\ \n0 & 1 & 0 & 1 & 0 & 0 & 0 & 0 & 1 \\\\\n\\end{array}\n\\right],\n\\quad\n\\left[\n\\begin{array}{ccc|ccc|ccc}\n0 & 0 & 0 & 0 & 0 & 1 & 0 & 1 & 0 \\\\\n0 & 0 & 1 & 0 & 0 & 1 & 1 & 1 & 1 \\\\\n0 & 1 & 0 & 1 & 1 & 1 & 0 & 1 & 0 \\\\ \\hline\n0 & 0 & 1 & 0 & 0 & 1 & 1 & 1 & 1 \\\\\n0 & 0 & 1 & 0 & 0 & 0 & 1 & 0 & 0 \\\\\n1 & 1 & 1 & 1 & 0 & 0 & 1 & 0 & 0 \\\\ \\hline\n0 & 1 & 0 & 1 & 1 & 1 & 0 & 1 & 0 \\\\\n1 & 1 & 1 & 1 & 0 & 0 & 1 & 0 & 0 \\\\ \n0 & 1 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\\\\n\\end{array}\n\\right].\n\\end{align*}\n\nWe attempted to work over $\\mathbb{F}_3$ but the computations could not be completed. We were able to determine that the maximum symmetric rank is at least 13 (since the program was terminated at this step while many symmetric tensors still had not been found). Thus, larger prime fields and larger tensor formats cannot be dealt with by our computer algorithm in a reasonable amount of time.\n\n\n\\section{Conclusion}\nWe began by summarizing known results, and our own previous work about canonical forms of tensors (symmetric and non-symmetric). For third and fourth order tensors, the simplest cases for the classification of canonical forms is for $2 \\times \\dots \\times 2$ tensor formats. The classification over $\\mathbb{R}$ and $\\mathbb{C}$ has appeared many times throughout the literature. In previous work, we extended the results by determining the canonical forms of $2 \\times 2 \\times 2$ tensors over the prime fields $\\mathbb{F}_p$ for $p = 2, 3, 5$. We also examined the same problem for the larger tensor format $2 \\times 2 \\times 2 \\times 2$ over $\\mathbb{F}_p$ for $p = 2, 3$, where the choice of $p$ is restricted by the memory capabilities of our computer.\n\nNext, we summarized known results about $2 \\times 2 \\times 2$ symmetric and $2 \\times 2 \\times 2 \\times 2$ symmetric tensors. The canonical forms of these tensor formats have been enumerated in the past over $\\mathbb{R}$ and $\\mathbb{C}$ by Weinberg \\cite{Weinberg} and Gurevich \\cite{Gurevich}. In recent work, we extended these results by determining the canonical forms over $\\mathbb{F}_p$. For the third order case, we considered all primes $p \\leq 17$. For the fourth order case, we considered $p \\leq 5$. \n\nIn this paper, we determined the canonical forms of second, third, and fourth order symmetric tensors of format $3 \\times \\dots \\times 3$ over $\\mathbb{F}_p$ for $p \\in \\{2, 3, 5\\}$. We determined that the maximum symmetric rank over finite fields is at least 4 in the second order case, 7 in the third order case, and 13 in the fourth order case. Due to memory limitations we could not consider larger primes or tensor formats. \n\n\n\n\n\\section*{Acknowledgements}\nThe author would like to thank Edoardo Ballico for his insights and suggestions. \n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\n \\label{sec:introduction}\n\nQuantum chromodynamics (QCD)\nshows a variety of phases, and in passing over a phase\nboundary one would encounter either first (discontinuous) or second\norder (continuous) transition, depending on temperature, density, quark\nmasses, the number of flavors, etc.\nChiral phase transition of QCD with two massless quarks at the\nvanishing chemical potential has been studied with various approaches\nfor a long time, since it provides us with a solid basis in the study\nof $2+1$-flavor QCD in the real world.\nNevertheless, the nature of the transition of this relatively simple\nsystem is yet ambiguous, and is counted as one of the longstanding\nproblems.\n\nBased on the universality argument and the results of the leading\norder $\\epsilon$ expansion, Pisarski and Wilczek analyzed the\nrenormalization group (RG) flow of the three-dimensional scalar field\ntheory, which shares the same internal symmetry with massless QCD around\nthe critical temperature ($T_c$), and pointed out that, in the\ntwo-flavor case, the order of the transition could crucially depend on\nthe presence (or the absence) of the flavor singlet axial ($U_A(1)$)\nsymmetry at $T_c$~\\cite{Pisarski:1983ms}.\nIf $U_A(1)$ symmetry is largely violated at $T_c$, the second order\nphase transition with the $O(4)$ scaling becomes possible although not\nmandatory.\nOn the other hand, when the symmetry is effectively and fully restored at\n$T_c$, the leading order calculation of the $\\epsilon$ expansion\nsuggests no infrared fixed point (IRFP), and hence the second order\nphase transition is excluded.\nBut, later, further studies using different advanced techniques\nfound evidence of IRFP, and the confirmation of the presence of the\nIRFP is under active\ninvestigation~\\cite{Butti:2003nu,Aoki:2012yj,Pelissetto:2013hqa,Nakayama:2014sba}.\nThus, the transition in this case again can be either of first or\nsecond order.\nRecently, a novel possibility is pointed out, following the RG flow\nanalysis: in the presence of small but finite $U_A(1)$ symmetry\nbreaking, the system may undergo the second order transition with the\n$O(4)$ scaling but one of the critical exponents related to the\nscaling dimension of the leading irrelevant operator is different from\nthat of the $O(4)$~\\cite{Sato:2014axa}, although again the second order\ntransition is not mandatory.\n\nNumerical simulations of QCD on the lattice can, in principle,\ndetermine the order of the transition as well as the universality class,\nto which massless two-flavor QCD belongs, by performing the scaling\nstudy~\\cite{Karsch:1993tv,Karsch:1994hm,Iwasaki:1996ya,AliKhan:2000iz,\nD'Elia:2005bv,Bonati:2009yg,RBCBi09,Kanaya:2010qj,Bonati:2014kpa}.\nHowever, it is not easy to keep all the systematic and statistical\nuncertainties under control in the chiral limit due to large\ncomputational costs.\nFurthermore, it appears that, in practice, the standard scaling study\nmay not be efficient enough to distinguish the first and the second\norder transitions because the scaling functions are similar between the\n$Z_2$ and $O(4)$ universality classes~\\cite{Brandt:2013mba}.\nWith lattice QCD simulations, one can also study the presence (or\nabsence) of $U_A(1)$ symmetry through the Dirac spectrum.\nFor recent progress, see, for example,\nRefs.~\\cite{Cossu:2013uua,Buchoff:2013nra,Bhattacharya:2014ara,Dick:2015twa}.\n\nClarifying this point is important not only for understanding the QCD phase\ndiagram but also for the scenario of the axion dark matter.\nThe axion abundance is essentially determined by the temperature\ndependence of the topological susceptibility, $\\chi_t(T)$, which\nvanishes when $U_A(1)$ symmetry is fully and effectively restored.\nIf $\\chi_t$ vanishes very rapidly right above $T_c$, too many axions\nwould be produced, and the axion dark matter scenario becomes hard or is\neven excluded~\\cite{Kitano:2015fla}, depending on how rapidly it\nvanishes.\nThe lattice studies to test the axion dark matter scenario has recently\nbegun in the quenched\napproximation~\\cite{Kitano:2015fla,Berkowitz:2015aua,Borsanyi:2015cka}.\n\nIn this paper, we follow the approach proposed in\nRef.~\\cite{Ejiri:2012rr}, in which the phase transition of two-flavor\nQCD is studied by adding many extra heavy quarks.\nWe call this the many flavor approach.\nThe transition of two-flavor QCD at a finite quark mass is known to be\na crossover, but it can turn to a first order transition when adding\nmany extra flavors.\nThis property is used to explore the nature of the phase transition of\nmassless two-flavor QCD.\nThe extra quarks are incorporated in the form of the hopping parameter\nexpansion (HPE) through the reweighting.\nThen the number of the extra flavors ($N_f$) and their mass parameter\n($\\kappa_h$) appear in a single parameter\n\\begin{eqnarray}\n     h\n &=& 2\\,N_f\\,(2\\kappa_h)^{N_t} \\ ,\n  \\label{eq:hcdef}\n\\end{eqnarray}\nwhere $N_t$ denotes the number of lattice sites in the temporal\ndirection.\nWe determine the critical value of the parameter ($h_c$) at which the\nfirst order and the crossover regions are separated, and examine its\ndependence on the two-flavor mass.\nThe order of the transition for a given $h$ is discriminated by the\nshape of the constraint effective potential at $T_c$, which is\nconstructed from the probability distribution function (PDF) for a\ngeneralized plaquette.\nNamely, it is discriminated by whether the potential at $T_c$ is in\nsingle- or double-well shape.\nIt is important to note that, in the determination of $h_c$ in this\napproach, the convergence of the HPE is not the matter since we can\nconsider arbitrary small $\\kappa_h$ by considering arbitrary large $N_f$ as\nseen from Eq.(\\ref{eq:hcdef}).\n\nWe perform an exploratory study on $N_t=4$ and try to see how $h_c$\ndepends on the two-flavor mass.\nThen, $h_c$ is found to stay constant against the change of the\ntwo-flavor mass in the range we have studied and to remain positive and\nfinite in the chiral limit.\nSince the two-flavor system is equivalent to the $2+N_f$-flavor system\nwith $h=0$, our result suggests that massless two-flavor QCD belongs to\nthe region of second order phase transition.\nThis kind of extension of QCD is useful also for the study of the phase\nstructure in the presence of finite chemical\npotential~\\cite{Ejiri:2012rr}.\nThe similar approach but introducing a finite imaginary chemical\npotential is taken in Ref.~\\cite{Bonati:2014kpa}.\n\nThe paper is organized as follows.\nAfter the central idea of many flavor approach is explained in\nSec.~\\ref{sec:approach}, the method is described in detail in\nSec.~\\ref{sec:method}.\nThe lattice setup and the main part of this paper are given in\nSec.~\\ref{sec:numerical_results}.\nIn Sec.~\\ref{sec:consistency-check}, two independent analyses are\nperformed for the consistency check.\nFinally, the conclusion and perspectives are stated in\nSec.~\\ref{sec:summary}.\nThe preliminary result of this work is available in\nRef.~\\cite{Ejiri:2015pva}.\n\n\n\\section{Many flavor approach}\n\\label{sec:approach}\n\nThe central idea of many flavor approach is outlined.\nFigure~\\ref{fig:columbia-plot} (a) shows the so-called Columbia plot for\n$2+1$-flavor QCD~\\cite{Brown:1990ev}, which summarizes the present\nknowledge on the mass dependent nature of the phase transition of\nQCD as a function of $m_{ud}$ and $m_s$.\nThe physical point is believed to be located in the crossover\nregion~\\cite{RBCBi09,Aoki:2006we}.\nThe plot tells us that there are two distinct first order regions lying\naround the quenched limit ($m_{ud}=m_s=\\infty$) and the chiral limit of\nthree-flavor QCD ($m_{ud}=m_s=0$), respectively.\nIn what follows, we focus on the latter.\n\\begin{figure}[tb]\n\\begin{center}\n\\begin{tabular}{cc}\n\\includegraphics*[width=0.5 \\textwidth,clip=true]\n{figs\/columbia-2+1_nopoint_cap_uds_rev20151029.eps}&\n\\includegraphics*[width=0.5 \\textwidth,clip=true]\n{figs\/columbia-2+nf_nopoint_cap_udh_rev20151029.eps}\\\\\n (a)\\ 2+1 flavor QCD&\n (b)\\ 2+$N_f$ flavor QCD\\\\\n\\end{tabular}\n\\vspace{-1ex}\n\\caption{Basic idea of \"many flavor approach\".}\n\\label{fig:columbia-plot}\n\\end{center}\n\\end{figure}\n\nOur interest is in whether the massless two-flavor QCD point ($m_{ud}=0$\nand $m_s=\\infty$) is inside the first order region or not and hence in\nthe shape of the first order region.\nIf it ends at a finite $m_s$, called the tricritical point, massless\ntwo-flavor QCD is on the line of the second order phase transition (the\nsolid curve on the $m_{ud}=0$ line).\nOn the other hand, if the first order region extends to $m_s=\\infty$,\nmassless two-flavor QCD should undergo the first order transition ({\\it\ni.e.} the dotted curve near the $m_{ud}=0$ line).\nIn either case, if we could resolve which of the solid or dotted curves\nis realized, we would be able to answer the question.\nHowever, it is difficult to trace the critical line for the $2+1$-flavor\ncase, because the critical line is located in the small $m_s$ region\nwhen $m_{ud}$ is moderately\nsmall~\\cite{RBCBi09,Ding:2013lfa,Jin:2014hea}.\n\nThe situation will change in $2+N_f$-flavor QCD as shown in\nFig.~\\ref{fig:columbia-plot} (b).\nThe bottom-right corner of Fig.~\\ref{fig:columbia-plot} (b) is of\nfirst order for $N_f \\geq 3$.\nThe critical line is expected to move upward as $N_f$ increases, and for\nsufficiently large $N_f$ it could enter the region where the hopping\nparameter expansion from the static limit (crosses at\n$m_h=\\infty$) works well (above the dashed line).\nThen, one should be able to easily identify how the critical line runs\nas a function of $m_{ud}$.\nIf the critical heavy mass $m_h^c(m_{ud})$ remains finite in the\n$m_{ud}\\to 0$ limit, it immediately means that massless two-flavor QCD\ncorresponding to the point $(m_{ud},m_h)=(0,\\infty)$ is the outside of\nthe first order region.\nAn important remark is that, in the limit of $m_s\\to\\infty$ or\n$m_h\\to\\infty$, both $2+1$- and $2+N_f$-flavor QCD end up with the same\ntheory, which we want to study.\nThus, the original question is simplified to the one whether the\ncritical heavy mass in the chiral limit stays finite or not.\n\nWhile most of the current knowledge on the shape of the first order\nregion are only qualitative, in a context of the Taylor-expanded\nreweighting method~\\cite{BS02} we can derive a solid statement about the\nslope at $m_{ud}=m_h$ if the same quark action is employed for two- and\n$N_f$-flavors.\nSuppose that $m_{ud}=m + \\Delta m_{ud}$ and $m_{h}=m + \\Delta m_h$ and\nexpand the logarithm of the quark determinants in terms of $\\Delta\nm_{ud}$ and $\\Delta m_h$.\nThen, one will be aware that the partition function does not change as\nlong as $\\Delta m_h=- 2\\, \\Delta m_{ud}\/N_f$ and hence that physics is\nidentical along the line of slope $-2\/N_f$ near $m_{ud} = m_h$ line in\nthe $(m_{ud}, m_h)$ plane~\\cite{ejiri04}.\nThis means that the critical line in $2+N_f$-flavor QCD crosses the line\nof $m_{ud}=m_h$ with a slope milder than the $2+1$ flavor case.\n\nIn Ref~\\cite{Ejiri:2012rr}, it is demonstrated that by adding extra\nflavors the end point (or the critical line) indeed enters the region\nreachable by the hopping parameter expansion.\nIn this paper, we examine the light quark mass dependence of the\nend point.\nAs we will explain below, $m_h^c(m_{ud})$ seems to remain finite in\nthe chiral limit of $m_{ud}$, suggesting the phase transition of\nmassless two-flavor QCD is of second order.\n\n\n\\section{Calculational method}\n\\label{sec:method}\n\nWe first generate two-flavor configurations at finite temperatures\nfollowing the standard hybrid Monte Carlo method.\nUsing the hopping parameter expansion and the reweighting method,\nwe incorporate extra $N_f$ flavors of heavy quarks into those\nconfigurations, and measure the probability distribution function\nfor a generalized plaquette to construct the constraint effective\npotential for $2+N_f$-flavor QCD.\n\nAfter the HPE, the hopping parameter for heavy quarks $\\kappa_h$ and\n$N_f$ appear only in a single parameter, $h$ [see Eq.~(\\ref{eq:h})].\nIf the parameter $h$ is in the crossover region, the effective potential\nshould take a single-well shape at the pseudocritical temperature,\n$T_{pc}$.\nOn the other hand, when the parameter $h$ enters the first order region,\na double-well shape should emerge at $T_c$.\nBy scanning $h$, we determine the critical value $h_c$, at which the\nfirst order and the crossover regions are separated.\nThe critical value $h_c$ is determined at four values of two-flavor\nmass to see the light quark mass dependence of $h_c$.\nIn the following, the calculational procedure is described in detail.\n\nThe PDF was introduced in Refs.~\\cite{Bruce:1981,Binder:1981sa} and has\nbeen extensively used in various fields to study the critical\nproperties of various materials~\\cite{Plascak:2013} or the phase\ndiagrams of QCD~\\cite{whot11}.\nIn our study of 2+$N_f$-flavor QCD, the PDF $w$ for a quantity $\\hat X$\nis defined by\n\\begin{eqnarray}\n    w(X; \\beta, \\kappa_l, \\kappa_h,N_f)\n&=& \\int \\!\\! {\\cal D} U\\, \\delta(X- \\hat{X}) \\ \n    \\big[\\det M(\\kappa_h)\\big]^{N_f}\n    e^{-S_{\\rm gauge}(\\beta) - S_{\\rm light}(\\kappa_l)},\n\\label{eq:pdist}\n\\end{eqnarray}\nwhere $S_{\\rm gauge}(\\beta)$ and $S_{\\rm light}(\\kappa_l)$ are the lattice\nactions for the gauge field and two flavors of light quarks,\nrespectively.\n$\\beta=6\/g_0^2$ is the simulation parameter setting the temperature\nthrough the lattice spacing, and $\\kappa_l$ is the light quark mass\nparameter.\nNote that the method  described below works for any kinds of light quark\naction unless it contains $\\beta$ dependent coefficients.\nThe action for the $N_f$ extra flavors is written in the determinant\nform in Eq.~(\\ref{eq:pdist}), where $M(\\kappa_h)$ is the lattice Dirac\noperator for heavy quarks with a mass parameter $\\kappa_h$.\nA quantity to be constrained, $\\hat X$, is basically arbitrary, but the\norder parameter would be the most natural choice if it is available.\nIn this paper, following previous\nworks~\\cite{Ejiri:2007ga,whot11,Ejiri:2012rr,Ejiri:2013lia}, $\\hat X$ is\nchosen to be the generalized plaquette\n\\begin{eqnarray}\n  \\hat P\n= c_0\\, \\hat W_P + 2 c_1\\, \\hat W_R\\ ,\n\\end{eqnarray}\nwhere $\\hat W_P$ and $\\hat W_R$ denote the averaged plaquette and\nrectangle, respectively, and $c_0$ and $c_1$ satisfying $c_0=1-8 c_1$\nare the improvement coefficients for lattice gauge action.\nIn terms of $\\hat P$, the gauge action is written as\n\\begin{eqnarray}\n    S_{\\rm gauge}(\\beta)\n= - 6\\, N_{\\rm site}\\, \\beta\\, \\hat P\\, ,\n\\end{eqnarray}\nwhere $N_{\\rm site}= N_{\\rm s}^3 \\times N_t$ represents the number of\nsites in four-dimensional lattice volume.\nOur main analysis is carried out with $\\hat X=\\hat P$, but the\ncalculation for $\\hat X=\\hat L$ with $\\hat L$ the real part of the\nPolyakov loop averaged over spatial sites is also performed as a\nconsistency check (see Sec.~\\ref{subsec:polyakov}).\nChoosing $\\hat X=\\hat P$ brings a great simplification in the\nnumerical analysis as explained below.\nIn principle, we could choose other quantities, {\\it e.g.} the\nchiral condensate, to be $\\hat X$.\nBut, whenever a quantity other than $\\hat P$ is chosen, we lose not only\nthe advantage for $\\hat X=\\hat P$ but also the accuracy in the results\nas demonstrated in Sec.~\\ref{subsec:polyakov} for $\\hat X=\\hat L$.\n\nWith the PDF thus obtained, the constraint effective potential $V$ is\ncalculated by\n\\begin{eqnarray}\n       V(X;\\beta,\\kappa_l,\\kappa_h,N_f)\n&=&  - \\ln w(X;\\beta,\\kappa_l,\\kappa_h,N_f)\n \\label{eq:effective-potential}\n\\end{eqnarray}\nIn practice, the PDF is not directly accessible, and hence we instead\ncalculate the histogram defined by\n\\begin{eqnarray}\n    H(X;\\beta,\\kappa_l,\\kappa_h,N_f)\n&=& \\frac{w(X;\\beta,\\kappa_l,\\kappa_h,N_f)}{Z(\\beta,\\kappa_l,\\kappa_h,N_f)}\\ ,\n \\label{eq:histogram}\n\\end{eqnarray}\nwhere $Z(\\beta)$ is the partition function.\nNote that $Z(\\beta)$ is not calculable, but its ratio at two\ndifferent $\\beta$ values is calculable~\\cite{Ejiri:2008xt}.\n\nIn order to see the shape of the potential, we need to calculate the\npotential over a certain range of $X$.\nHowever, a simulation at a single $\\beta$ provides the potential only in\na limited range of $X$.\nThus, the potential calculated at a certain $\\beta$ needs to be\ntranslated to that at other $\\beta$.\nWe call the temperature, at which we want to calculate the potential,\nthe reference temperature (or $\\beta_{\\rm ref}$).\nThen, the potential is calculated as\n\\begin{eqnarray}\n&&      V(X;\\beta_{\\rm ref},\\kappa_l,\\kappa_h,N_f) + \\ln Z(\\beta^*,\\kappa_l',\\kappa_h',N_f')\n\\nonumber\\\\\n&=& - \\ln H(X';\\beta',\\kappa_l',\\kappa_h',N_f')\n    - \\ln \\left(\\frac{w(X'';\\beta'',\\kappa_l'' ,\\kappa_h'' ,N_f'')}\n                     {w(X';\\beta'  ,\\kappa_l',\\kappa_h',N_f')}\n          \\right)\n    - \\ln \\left(\\frac{w(X;\\beta_{\\rm ref},\\kappa_l ,\\kappa_h ,N_f)}\n                     {w(X'';\\beta''       ,\\kappa_l'',\\kappa_h'',N_f'')}\n          \\right)\n\\nonumber\\\\\n& & - \\ln \\frac{Z(\\beta',\\kappa_l',\\kappa_h',N_f')}{Z(\\beta^*,\\kappa_l',\\kappa_h',N_f')}\n\\ .\n \\label{eq:effective-potential-2}\n\\end{eqnarray}\nAll the intermediate quantities such as $\\beta'$, $X'$, $\\kappa_l'$, $\\kappa_h'$\nand $N_f'$ are arbitrary as well as $\\beta^*$.\nFor a practical reason, we take\n\\begin{eqnarray}\n X'=X''=X,\\ \\ \\\n \\beta'' = \\beta_{\\rm ref},\\ \\ \\\n \\kappa_l' = \\kappa_l'' = \\kappa_l,\\ \\ \\\n \\kappa_h' = \\kappa_h'' = 0\\,\\ \\ \\\n N_f' = N_f'' = 0\\ ,\n \\label{eq:setup}\n\\end{eqnarray}\nand $\\beta^*$ is chosen to be the vicinity of $\\beta_{\\rm ref}$.\n\nIn the following, we take $\\hat X=\\hat P$, then the effective potential\nis simplified as\n\\begin{eqnarray}\n    V(P; \\beta_{\\rm ref},\\kappa_l,\\kappa_h,N_f) + \\ln Z(\\beta^*,\\kappa_l',\\kappa_h',N_f')\n&=&   V_{\\rm light}(P; \\beta_{\\rm ref},\\kappa_l)\n    - \\ln R(P;\\beta_{\\rm ref},\\kappa_l,\\kappa_h,N_f)\\ .\n \\nonumber\\\\\n\\label{eq:vefftrans}\n\\end{eqnarray}\nThe first term is defined by\n\\begin{eqnarray}\n    V_{\\rm light}(P; \\beta_{\\rm ref},\\kappa_l)\n= - \\ln H(P;\\beta,\\kappa_l,0,0)  \n  - 6\\,N_{\\rm site}\\,(\\beta_{\\rm ref} - \\beta)\\,P\n  - \\ln \\frac{Z(\\beta,\\kappa_l,0,0)}{Z(\\beta^*,\\kappa_l,0,0)}\n\\label{eq:V-two-flavor}\n\\end{eqnarray}\nand represents the constraint effective potential for two flavors alone.\nThe second term of Eq.~(\\ref{eq:vefftrans}) is defined by\n\\begin{eqnarray}\n    R(P;\\beta_{\\rm ref},\\kappa_l,\\kappa_h,N_f)\n&=& \\left\\langle\n    \\displaystyle\n    \\left[ \\det M(\\kappa_h) \\right]^{N_f}\n    \\right\\rangle_{P: {\\rm fixed},(\\beta_{\\rm ref},\\kappa_l)}, \\ \n\\label{eq:lnr}\\\\\n    \\langle \\cdots \\rangle_{P: {\\rm fixed}, (\\beta_{\\rm ref},\\kappa_l)}\n&\\equiv&\n    \\frac{\\langle \\delta(P- \\hat{P}) \\cdots \\rangle_{(\\beta_{\\rm ref},\\kappa_l)}}\n         {\\langle \\delta(P- \\hat{P}) \\rangle_{(\\beta_{\\rm ref},\\kappa_l)}}\\ ,\n\\label{eq:exp-fixedP}\n\\end{eqnarray}\nwhere $\\langle \\cdots \\rangle_{(\\beta,\\kappa_l)}$ denotes the ensemble\naverage over two-flavor configurations generated with $\\beta$ and $\\kappa_l$.\nIt is important to note that, separating the effective potential into\nthe two-flavor part and the extra heavy part as in\nEq.~(\\ref{eq:vefftrans}), the latter becomes independent of\n$\\beta_{\\rm ref}$.\nThe reason is as follows.\nDue to the operator $\\delta(P-\\hat P)$, the factor of\n$\\exp(6 N_{\\rm site}\\beta_{\\rm ref} P)$ comes out of the brackets in\nEq.~(\\ref{eq:exp-fixedP}).\nSince this factor cancels between the numerator and the denominator in\nEq.~(\\ref{eq:exp-fixedP}), $\\beta_{\\rm ref}$ dependence disappears.\nThis simplification takes place only for $\\hat X=\\hat P$.\n\nFor the extra heavy quarks, we employ the unimproved Wilson fermion\nbecause it suffices for the present purpose.\nFor sufficiently small $\\kappa_h$, the determinant in Eq.~(\\ref{eq:lnr}) can\nbe approximated by the leading order of the HPE as\n\\begin{eqnarray}\n   \\ln \\left[ \\det M (\\kappa_h) \\right]^{N_f}\n&\\approx&\n   N_f\n    \\left( 288\\, N_{\\rm site}\\, \\kappa_h^4\\, \\hat W_P\n         + 12 N_s^3 (2 \\kappa_h)^4\\, \\hat{L}\n    \\right)\n\\nonumber\\\\\n&=& 6\\,N_s^3\\,h\\,\\hat Y\n\\label{eq:detmw-1}\\\\\n&=&   9\\,N_{\\rm site}\\,\\frac{h}{c_0}\\, \\hat P\n    + 6\\,N_s^3\\,h\\,\\hat Z \\ ,\n\\label{eq:detmw-2}\n\\end{eqnarray}\nwhere the following quantities have been introduced:\n\\begin{eqnarray}\n    h\n&=& 2\\,N_f\\,(2 \\kappa_h)^4\\ ,\n\\label{eq:h}\\\\\n    \\hat Y\n&=& 6\\,\\hat W_P + \\hat{L}\\ ,\n\\label{eq:Y}\\\\\n    \\hat Z\n&=& - \\frac{12\\,c_1}{c_0}\\,\\hat W_R + \\hat{L}\\ .\n\\label{eq:Z}\n\\end{eqnarray}\nHere and hereafter, $N_t=4$ is assumed because we take that value in\nnumerical simulations, but the extension to other values of $N_t$ is\nstraightforward though the expression becomes complicated.\nThen, $\\ln R$ in Eq.~(\\ref{eq:vefftrans}) can be approximated as\n\\begin{eqnarray}\n    \\ln R(P;\\kappa_l,h)\n&\\approx&\n    \\ln \\left\\langle\n    \\displaystyle\n    \\exp\\left( 6\\,N_s^3\\,h\\,\\hat Y \\right)\n    \\right\\rangle_{P: {\\rm fixed},(\\beta,\\kappa_l)}\n\\label{eq:r-1}\\\\\n&=&  9\\,N_{\\rm site}\\,\\frac{h}{c_0}\\, P\n   + \\ln R'(P;\\kappa_l,h)\n\\label{eq:r-2}\n\\end{eqnarray}\nwhere\n\\begin{eqnarray}\n    \\ln R'(P;\\kappa_l,h)\n&=& \\ln \\left\\langle\n    \\displaystyle\n    \\exp\\left( 6\\,N_s^3\\,h\\,\\hat Z \\right)\n    \\right\\rangle_{P: {\\rm fixed},(\\beta,\\kappa_l)}\\ .\n\\label{eq:rdash}\n\\end{eqnarray}\nAlthough Eqs.~(\\ref{eq:r-1}) and (\\ref{eq:r-2}) are algebraically\nidentical, the equality is not necessarily trivial in numerical data\nbecause the $\\delta$ function is approximated by\n\\begin{eqnarray}\n        \\delta(x)\n \\approx 1\/(\\Delta \\sqrt{\\pi}) \\exp[-(x\/\\Delta)^2]\\ .\n \\label{eq:delta-func}\n\\end{eqnarray}\nThen, the difference can arise when\n\\begin{eqnarray}\n  \\big\\langle e^{6 N_s^3 h \\hat Y}\n          \\exp\\left[-(P-\\hat P)^2\/\\Delta^2\\right]\n  \\big\\rangle_{(\\beta,\\kappa_l)}\n- e^{9 N_{\\rm site} \\frac{h}{c_0}P}\n  \\big\\langle e^{6 N_s^3 h \\hat Z}\n  \\exp\\left[-(P-\\hat P)^2\/\\Delta^2\\right]\n  \\big\\rangle_{(\\beta,\\kappa_l)}\n  \\neq 0  ,\\nonumber\n\\end{eqnarray}\nwhich should vanish for sufficiently small $\\Delta$, but then the\nstatistical error will enlarge.\nIn the following analysis, both expressions are examined to check\nthe consistency.\n\nIt is important to note that, after the HPE, the number of extra heavy\nflavors ($N_f$) and their mass parameter ($\\kappa_h$) appear only in a single\nparameter $h$, Eq.~(\\ref{eq:h}).\nBecause of this, $\\kappa_h$ and $N_f$ have been replaced by $h$ in the\narguments of $R$ and $R'$, and our purpose turns to finding the critical\nvalue of $h$, $h_c$.\nIt should be also noted that the second derivatives of\nEqs.~(\\ref{eq:r-1}) and (\\ref{eq:rdash}) with regard to $P$ are\nidentical because the difference is proportional to $P$.\n\nOne side remark is below.\nThus far, we have restricted the extra heavy quarks to be degenerate.\nBut the extension to the nondegenerate case is straightforward by\ninterpreting $h$ as $h=2 \\sum_{f=1}^{N_f} (2 \\kappa_h)^{N_t}$.\nIn the following, we only consider the degenerate case for simplicity.\n\nChoosing $\\hat X=\\hat P$ significantly simplifies the procedure to find\nthe critical value of $h$~\\cite{Ejiri:2007ga,Ejiri:2012rr} as follows.\nWe are interested in the shape of the potential at the (pseudo)critical\ntemperature, which requires $\\beta_{\\rm ref}$ to be tuned to its\n(pseudo)critical value, $\\beta_{c}$ (or $\\beta_{pc}$).\nThis tuning can be totally skipped if we look at the curvature, {\\it\ni.e.} the second derivative of the potential with respect to $P$,\nbecause it is independent of $\\beta_{\\rm ref}$.\n$R$ is independent of $\\beta_{\\rm ref}$ as stated above.\n$V_{\\rm light}$ depends on $\\beta_{\\rm ref}$, but its second derivative\ndoes not as explained below.\n\nThe finite temperature transition of two-flavor QCD is always a\ncrossover for the two-flavor masses adopted in this paper.\nThen, at any temperatures, the shape of the PDF (or equivalently\nhistogram) for $\\hat P$ in two-flavor QCD can be well approximated,\naround the peak, by Gaussian form,\n\\begin{eqnarray}\n   w(P;\\beta,\\kappa_l,0,0)\\big|_{P\\sim \\bar P(\\beta,\\kappa_l)}\n\\propto\n   \\exp\\left[ - \\frac{6\\,N_{\\rm site}\\,\n   \t              (\\,P - \\bar P(\\beta,\\kappa_l)\\,)^2}\n                     {2\\,\\chi_P(\\beta,\\kappa_l)}\n       \\right]\\ ,\n\\end{eqnarray}\nwhere $\\bar P(\\beta,\\kappa_l)=\\langle \\hat P \\rangle_{\\beta,\\kappa_l}$ is the\naverage of generalized plaquette at $\\beta$ and $\\kappa_l$, and $\\chi_P$\nis the susceptibility of $P$, given by\n\\begin{eqnarray}\n  \\chi_P(\\beta,\\kappa_l)\n= 6 N_{\\rm site}\n  \\langle\\, (\\,\\hat P - \\bar P(\\beta,\\kappa_l)\\, )^2\\,\n  \\rangle_{\\beta,\\kappa_l}\\ .\n\\end{eqnarray}\nSubstituting this into Eq.~(\\ref{eq:V-two-flavor}) yields, up to a\nconstant shift,\n\\begin{eqnarray}\n      V_{\\rm light}(P; \\beta_{\\rm ref},\\kappa_l)\n      \\big|_{P\\sim \\bar P(\\beta,\\kappa_l)}\n&=&   \\frac{6\\,N_{\\rm site}\\,\n          (\\, P - \\bar P(\\beta,\\kappa_l)\\,)^2}\n\t {2\\,\\chi_P(\\beta,\\kappa_l)}\n    - 6(\\beta_{\\rm ref} - \\beta)\\,N_{\\rm site}\\,P\n    \\ .\n\\end{eqnarray}\nThen, the first and second derivatives are given by\n\\begin{eqnarray}\n      \\frac{d V_{\\rm light}(P;\\beta_{\\rm ref},\\kappa_l)}{dP}\n      \\bigg|_{P\\sim \\bar P(\\beta,\\kappa_l)}\n&=&   \\frac{6\\,N_{\\rm site}\\,\n            (\\, P - \\bar P(\\beta,\\kappa_l)\\,)}\n\t   {\\chi_P(\\beta,\\kappa_l)}\n    - 6(\\beta_{\\rm ref} - \\beta)\\,N_{\\rm site}\\ ,\n\\label{eq:d1V0}\n\\\\\n      \\frac{d^2 V_{\\rm light}(P;\\kappa_l)}{dP^2}\n      \\bigg|_{P\\sim \\bar P(\\beta,\\kappa_l)}\n&=&   \\frac{6\\,N_{\\rm site}}{\\chi_P(\\beta,\\kappa_l)}\\ .\n\\label{eq:d2V0}\n\\end{eqnarray}\nThus, we can calculate the curvature of the two-flavor part\nby collecting $\\chi_P(\\beta,\\kappa_l)$ obtained at various $\\beta$.\nImportantly, Eq.~(\\ref{eq:d2V0}) is independent of\n$\\beta_{\\rm ref}$.\nIn summary, the curvature of the total effective potential\n\\begin{eqnarray}\n  \\frac{d^2 V(P;\\beta_{\\rm ref},\\kappa_l,h)}{dP^2}\n= \\frac{d^2 V_{\\rm light}(P;\\beta_{\\rm ref},\\kappa_l)}{dP^2}\n- \\frac{d^2 \\ln R(P; \\kappa_l, h)}{dP^2}\\ ,\n\\label{eq:curvature}\n\\end{eqnarray}\nis independent of $\\beta_{\\rm ref}$.\n\nThe procedure to identify $h_c$ in the chiral limit of two flavors goes\nas follows.\nAt $h=0$, the contribution of the extra heavy flavors is trivially zero,\nand the system is reduced to two-flavor QCD, where the transition is a\ncrossover.\nTherefore, the second derivative of the potential is always positive.\nAs $h$ is increased from zero, the minimum of the curvature takes zero\nat some point, which gives $h_c$.\nIn this procedure, one needs not tune $\\beta_{\\rm ref}$ to\n$\\beta_{\\rm pc}$ or $\\beta_{\\rm c}$, because the curvature is\nindependent of the temperature or $\\beta_{\\rm ref}$.\nThis simplification does not occur in general, and one such example\nis explicitly shown in Sec.~\\ref{subsec:polyakov}.\nBy looking at the light quark mass dependence of $h_c(\\kappa_l)$, we try to\nextract $h_c$ in the chiral limit.\n\n\n\\section{Numerical results}\n\\label{sec:numerical_results}\n\\subsection{Simulation parameters}\n\\label{subsec:lat-para}\n\nFollowing Ref.~\\cite{whot10}, we take the Iwasaki gauge action\n($c_1=-0.331$) and the $O(a)$-improved Wilson fermion action with the\nperturbatively improved $c_{\\rm sw}$ for two flavors of light quarks.\nSimulations are performed on $N_{\\rm site}=16^3\\times 4$ lattices with\n25 to 32 $\\beta$ values at each of four $\\kappa_l$,\nand 10,000 to 40,000 trajectories have been accumulated at each\nsimulation point.\nFour light quark masses are ranging from $\\kappa_l=0.145$ to 0.1505.\nFigure \\ref{fig:histogram-action} shows the histogram of the generalized\nplaquette.\n\\begin{figure}[tb]\n\\begin{center}\n\\begin{tabular}{cc}\n\\includegraphics*[width=0.5 \\textwidth,clip=true]\n{figs\/plqhist_L16x4_nf2-rg-k0.1450-csw1.650-nosmr-apbc-sld.eps}& \n\\includegraphics*[width=0.5 \\textwidth,clip=true]\n{figs\/plqhist_L16x4_nf2-rg-k0.1475-csw1.677-nosmr-apbc-sld.eps}\\\\\n\\includegraphics*[width=0.5 \\textwidth,clip=true]\n{figs\/plqhist_L16x4_nf2-rg-k0.150-csw1.707-nosmr-apbc-sld.eps}&\n\\includegraphics*[width=0.5 \\textwidth,clip=true]\n{figs\/plqhist_L16x4_nf2-rg-k0.1505-csw1.712-nosmr-apbc-sld.eps}\\\\\n\\end{tabular}\n\\vspace{-1ex}\n\\caption{Histogram of the generalized plaquette at four values of\n $\\kappa_l$.}\n\\label{fig:histogram-action}\n\\end{center}\n\\end{figure}\n\n\\begin{table}[t]\n \\centering\n \\begin{tabular}{cc|ccccc}\n $\\kappa_l$ & $c_{\\rm sw}$ & $\\beta_{\\rm pc}$ & $a m_\\pi$ & $a m_\\rho$\n& $m_\\pi\/m_\\rho$ & $a m_{\\rm pcac}$\\\\\n  \\hline\n   0.1450 & 1.650 & 1.778 &0.779(1) & 1.172(\\ 2)& 0.665(33)\n          & 0.0535(1)\\\\\n   0.1475 & 1.677 & 1.737 & 0.651(1) & 1.130(\\ 5)& 0.576(28)\n          & 0.0350(2)\\\\\n   0.1500 & 1.707 & 1.691 & 0.514(2) & 1.099(10) & 0.468(24)\n          & 0.0202(2)\\\\\n   0.1505 & 1.712 & 1.681 & 0.495(2) & 1.082(13) & 0.458(23)\n          & 0.0186(2)\\\\\n \\end{tabular}\n \\caption{Simulation parameters ($\\kappa_l$ and $c_{\\rm sw}$) and the\n pseudocritical $\\beta$ ($\\beta_{\\rm pc}$) in two-flavor QCD,\n determined from the peak of the susceptibility for the generalized\n plaquette.\n $m_\\pi\/m_\\rho$ and other quantities are determined on a $16^3\\times 32$\n lattice at $\\beta_{\\rm pc}$ for each $\\kappa_l$.}\n \\label{tab:spectrum}\n\\end{table}\n\nAt the pseudocritical point $\\beta_{\\rm pc}$ for each $\\kappa_l$, we carried\nout zero temperature simulations on $16^3\\times 32$ lattices to find the\nmass ratio of pseudoscalar and vector mesons, $m_\\pi\/m_\\rho$, and the\nquark mass defined through the partially conserved axialvector current\n(PCAC),\n\\begin{eqnarray}\n    am_{\\rm pcac}\n&=& \\frac{\\langle \\sum_{\\vec x}\n          (A_4(N_t\/2+1,\\vec x)-A_4(N_t\/2-1,\\vec x)) P(0)\\rangle}\n         {4\\, \\langle \\sum_{\\vec x}P(N_t\/2,\\vec x) P(0) \\rangle}\\ ,\n\\end{eqnarray}\nwhere $A_4(x)$ and $P(x)$ are the flavor nonsinglet, local axialvector\nand pseudoscalar operator, respectively, and $N_t=32$.\nThen, the four values of $\\kappa_l$ in this paper turn out to cover\n$0.46 < m_\\pi\/m_\\rho < 0.66$, or $0.019 < am_{\\rm pcac} < 0.054$.\nThese results are tabulated in Table~\\ref{tab:spectrum}.\n\n\n\\subsection{Main results}\n\\label{subsec:main-result}\n\nWe present the numerical results for the two terms in the right-hand\nside of Eq.~(\\ref{eq:curvature}) separately, and focus on the second\nterm first.\nWith the approximated $\\delta$ function (\\ref{eq:delta-func}),\n$\\ln R(P;\\kappa_l,h)$ [Eq.~(\\ref{eq:r-1})] and $\\ln R'(P;\\kappa_l,h)$\n[Eq.~(\\ref{eq:rdash})] are calculated.\nWe take two values of $\\Delta$, 0.0001, 0.00025, to see the\nstability of the results, and the discrepancy arising from a different\nchoice of $\\Delta$ is taken as the systematic uncertainties.\nIn the following plots, the results with $\\Delta=0.0001$ are shown\nunless otherwise stated.\n\nThe $P$ dependence of $\\ln R(P;\\kappa_l,h)$ and $\\ln R'(P;\\kappa_l,h)$ are shown\nin Fig.~\\ref{fig:lnr}.\n\\begin{figure}[t]\n\\begin{center}\n\\begin{tabular}{cc}\n\\includegraphics*[width=0.5 \\textwidth,clip=true]\n{figs\/lnR-k0.145-d0.0005-Xsp1.eps}&\n\\includegraphics*[width=0.5 \\textwidth,clip=true]\n{figs\/lnR-k0.1505-d0.0005-Xsp1.eps}\\\\[-2ex]\n (a)&(b)\\\\[2ex]\n\\includegraphics*[width=0.5 \\textwidth,clip=true]\n{figs\/lnR-k0.145-d0.0005-Xsp2.eps}&\n\\includegraphics*[width=0.5 \\textwidth,clip=true]\n{figs\/lnR-k0.1505-d0.0005-Xsp2.eps}\\\\[-2ex]\n (c)&(d)\\\\\n\\end{tabular}\n\\vspace{-1ex}\n\\caption{$P$ dependence of $\\ln R$ [Eq.~(\\ref{eq:r-1})]\n at $\\kappa_l=0.145$ (a) and 0.1505 (b),\n and $\\ln R'$ [Eq.~(\\ref{eq:rdash})] at $\\kappa_l=0.145$ (c) and 0.1505 (d).\n The results at $h$=0.2 to 0.4 are shown.\n }\n\\label{fig:lnr}\n\\vspace{-1ex}\n\\end{center}\n\\end{figure}\nThe statistical errors are invisible on this scale.\nThe sizes of $\\ln R$ and $\\ln R'$ differ by an order of magnitude, which\nindicates that the difference proportional to $P$ is large and explains\nwhy the curvature in $\\ln R$ is less clear than that in $\\ln R'$.\nThe curvature in $\\ln R$ and $\\ln R'$ originates from the rapid increase\nof the Polyakov loop ($\\hat L$) contained in Eqs.~(\\ref{eq:Y}) or\n(\\ref{eq:Z}) around $\\beta_{\\rm pc}$.\n\nWe then calculate the slope and curvature by fitting the data\nof $\\ln R$ and $\\ln R'$.\nHere let us mention the fits and the selection of the results.\nEach data point shown in Fig.~\\ref{fig:lnr} is obtained in a separate\nsimulation, and is totally independent of other points.\nThe data points are fit to polynomial functions of $P$ as\n\\begin{eqnarray}\n    \\ln {R^{(')}}^{\\rm fit}(P)\n&=& \\sum_{i=0}^{N_{\\rm poly}} c_i\\, P^i \\ ,\n\\label{eq:fit-func}\n\\end{eqnarray}\nover three fit ranges (or three different numbers of the data points),\ntwo different polynomial orders, and two values of $\\Delta$.\nSince not all the fits are successful, we only keep the fit results,\nwhich satisfy $\\chi^2\/{\\rm dof} < 3$, in the following analysis.\n\nFigure~\\ref{fig:chi2} shows the values of $\\chi^2\/{\\rm dof}$ obtained\nfrom the fit of $\\ln R$ and $\\ln R'$ as a function of the number of the\ndata points used, where only the results with $\\chi^2\/{\\rm dof}<3$ are\nplotted.\nIt is seen that $\\chi^2\/{\\rm dof}$ with $\\Delta=0.00025$ is always\nlarger than that with $\\Delta=0.0001$.\nOnce the fit parameters are determined, it is straightforward to\ncalculate the curvature of the potential.\n\\begin{figure}[t]\n\\begin{center}\n\\vspace*{-3ex}\n\\begin{tabular}{cc}\n\\includegraphics*[width=0.5 \\textwidth,clip=true]\n{figs\/scan-chi-k0.145-dall-Xsp1.eps}&\n\\includegraphics*[width=0.5 \\textwidth,clip=true]\n{figs\/scan-chi-k0.1475-dall-Xsp1.eps}\\\\[-2ex]\n\\includegraphics*[width=0.5 \\textwidth,clip=true]\n{figs\/scan-chi-k0.150-dall-Xsp1.eps}&\n\\includegraphics*[width=0.5 \\textwidth,clip=true]\n{figs\/scan-chi-k0.1505-dall-Xsp1.eps}\\\\[-2ex]\n\\includegraphics*[width=0.5 \\textwidth,clip=true]\n{figs\/scan-chi-k0.145-dall-Xsp2.eps}&\n\\includegraphics*[width=0.5 \\textwidth,clip=true]\n{figs\/scan-chi-k0.1475-dall-Xsp2.eps}\\\\[-2ex]\n\\includegraphics*[width=0.5 \\textwidth,clip=true]\n{figs\/scan-chi-k0.150-dall-Xsp2.eps}&\n\\includegraphics*[width=0.5 \\textwidth,clip=true]\n{figs\/scan-chi-k0.1505-dall-Xsp2.eps}\\\\[-2ex]\n\\end{tabular}\n\\vspace{-3ex}\n\\caption{The resulting $\\chi^2$\/dof as a function of the number of data\n points.\n The results for the $n$-th order polynomial fit are shown.}\n\\label{fig:chi2}\n\\end{center}\n\\end{figure}\n\nAlthough we analyze the second derivative to derive the main result,\nwe discuss the first derivative because it is instructive.\nThe first derivative of Eq.~(\\ref{eq:vefftrans}) is obtained as follows.\nFirst, the numerical values of $\\bar P(\\beta,\\kappa_l)$ and\n$\\chi_P(\\beta,\\kappa_l)$ are calculated and substituted in\nEq.~(\\ref{eq:d1V0}) to obtain the two-flavor contribution.\nThen, the $N_f$ flavor's contribution, the first derivative of\n$\\ln R^{(')}$, is determined using the fit results of\nEq.~(\\ref{eq:fit-func}).\nBy adding up, we obtain the first derivative of the full effective\npotential.\nFigure~\\ref{fig:dvdp} shows the typical behavior of the first derivative\nof the potential, where the five curves in each plot represent the\nresults for $h=0.0$, 0.1, $\\dots$, 0.4 from top to bottom and the fit\nresults with $\\Delta=0.00010$ and $n=5$ are used.\nIt is clear that for $h=0$ the curve is monotonically increasing for all\n$\\kappa_l$ while the ``S'' shape is seen for $h=0.4$.\nIn principle, it may be possible to determine the critical value of $h$\nusing these plots, however it is not easy to clearly distinguish an\n``S'' shape from a monotonic increase.\nThus, we use the second derivative to determine $h_c$ as described\nbelow.\n\nThe results for the curvature are plotted in Fig.~\\ref{fig:d2lnr} for\n$\\ln R$ (solid curves) and $\\ln R'$ (dashed curves), where the results\nfor $h=0.2$ and 0.4 are shown as examples.\nAgain, the fit results with $\\Delta=0.00010$ and $n=5$ are used.\nThe difference in the curvature between $\\ln R$ and $\\ln R'$ turns out\nto be reasonably small at all $\\kappa_l$.\n\\begin{figure}[t]\n\\vspace*{-1ex}\n\\begin{center}\n\\begin{tabular}{cc}\n\\includegraphics*[width=0.45 \\textwidth,clip=true]\n{figs\/dvdp-k0.145-Xsp1.eps}&\n\\includegraphics*[width=0.45 \\textwidth,clip=true]\n{figs\/dvdp-k0.1505-Xsp1.eps}\\\\\n\\includegraphics*[width=0.45 \\textwidth,clip=true]\n{figs\/dvdp-k0.145-Xsp2.eps}&\n\\includegraphics*[width=0.45 \\textwidth,clip=true]\n{figs\/dvdp-k0.1505-Xsp2.eps}\\\\\n\\end{tabular}\n\\vspace{-1ex}\n\\caption{The first derivative of the full effective potential is shown\n as a function of $P$.\n $h$ = 0.0, 0.1, 0.2, 0.3, 0.4 from top to bottom.}\n\\label{fig:dvdp}\n \\vspace{-1ex}\n\\end{center}\n\\end{figure}\n\\begin{figure}[tb]\n\\begin{center}\n\\begin{tabular}{cc}\n\\includegraphics*[width=0.45 \\textwidth,clip=true]\n{figs\/d2lnR-k0.145-d0.0001.eps}&\n\\includegraphics*[width=0.45 \\textwidth,clip=true]\n{figs\/d2lnR-k0.1475-d0.0001.eps}\\\\\n\\includegraphics*[width=0.45 \\textwidth,clip=true]\n{figs\/d2lnR-k0.150-d0.0001.eps}&\n\\includegraphics*[width=0.45 \\textwidth,clip=true]\n{figs\/d2lnR-k0.1505-d0.0001.eps}\\\\\n\\end{tabular}\n\\vspace{-2ex}\n\\caption{The second derivative of the first term and the second term of\n Eq.~(\\ref{eq:curvature}) are shown as a function of $P$.\n The second term contribution exceeds that of the first term in a range\n of $P$ when $h=0.4$, which indicates the occurrence of the first order\n transition at such a value of $h$.}\n\\label{fig:d2lnr}\n \\vspace{-2ex}\n\\end{center}\n\\end{figure}\n\nNext, the curvature of the first term in Eq.~(\\ref{eq:curvature}) is\npresented, which can be easily calculated using the averaged value and\nthe susceptibility of $\\hat P$ at each $\\beta$ as in\nEq.~(\\ref{eq:d2V0}).\nThe curvatures thus obtained are shown in Fig.~\\ref{fig:d2lnr} together\nwith the statistical error, where the fit results obtained with a fifth\norder of polynomial are shown by dotted curves.\nIt is seen that, independently of $\\kappa_l$, $d^2V_{\\rm light}\/dP^2$ is\nalways positive as expected.\n\nFigure~\\ref{fig:d2lnr} shows that $d^2 \\ln R\/dP^2$ and $d^2 \\ln R'\/dP^2$\nhave a peak slightly below the $P$ value at which\n$d^2V_{\\rm light}\/dP^2$ takes the minimum.\nThis indicates that, in the many flavor system, the phase transition or\nrapid crossover occurs at $P$ smaller than the two-flavor case.\nFor all $\\kappa_l$, it is observed that the peak of $d^2 \\ln R\/dP^2$ or\n$d^2 \\ln R'\/dP^2$ is almost touching the curve of\n$d^2V_{\\rm light}\/dP^2$ at $h=0.2$, and exceeds $d^2V_{\\rm light}\/dP^2$\nat $h=0.4$ in a certain region of $P$.\n\nThe resulting curvature of the full effective potentials\nEq.~(\\ref{eq:curvature}) for $\\kappa_l=0.1450$ and 0.1505 are shown in\nFig.~\\ref{fig:d2Veff-action}.\nIt is seen that the minimum of the curvature with $h=0.0$ (solid curve)\napproaches to zero towards the chiral limit of two light flavors.\nThus, with this observation alone, one might expect that the potential\nwith $\\kappa_l=0.1505$ requires only a small $h$ to bring it into the double\nwell shape.\nHowever, at $\\kappa_l=0.1505$, it is also true that adding the heavy quarks\ndoes not reduce the minimum of the curvature by much.\nAs a consequence, $h_c$ takes a similar value at $\\kappa_l=0.1450$ and\n0.1505.\n\\begin{figure}[tb]\n\\begin{center}\n\\begin{tabular}{cc}\n\\includegraphics*[width=0.6 \\textwidth,clip=true]\n{figs\/d2veff-n6-rng3-d0.0001-Xsp2.eps}\\\\\n\\end{tabular}\n\\vspace{-2ex}\n\\caption{The curvature of the effective potential at $\\kappa_l=0.1450$ and\n 0.1505 and for $h=0.0$ (solid), 0.2 (dashed) and 0.4 (dotted).}\n\\label{fig:d2Veff-action}\n\\vspace{-2ex}\n\\end{center}\n\\end{figure}\n\n\\begin{figure}[h]\n\\vspace*{-2ex}\n\\begin{center}\n\\begin{tabular}{cc}\n\\includegraphics*[width=0.5 \\textwidth,clip=true]\n{figs\/scan-fit-k0.145-dall-Xsp1.eps}&\n\\includegraphics*[width=0.5 \\textwidth,clip=true]\n{figs\/scan-fit-k0.1475-dall-Xsp1.eps}\\\\[-2ex]\n\\includegraphics*[width=0.5 \\textwidth,clip=true]\n{figs\/scan-fit-k0.150-dall-Xsp1.eps}&\n\\includegraphics*[width=0.5 \\textwidth,clip=true]\n{figs\/scan-fit-k0.1505-dall-Xsp1.eps}\\\\[-2ex]\n\\includegraphics*[width=0.5 \\textwidth,clip=true]\n{figs\/scan-fit-k0.145-dall-Xsp2.eps}&\n\\includegraphics*[width=0.5 \\textwidth,clip=true]\n{figs\/scan-fit-k0.1475-dall-Xsp2.eps}\\\\[-2ex]\n\\includegraphics*[width=0.5 \\textwidth,clip=true]\n{figs\/scan-fit-k0.150-dall-Xsp2.eps}&\n\\includegraphics*[width=0.5 \\textwidth,clip=true]\n{figs\/scan-fit-k0.1505-dall-Xsp2.eps}\\\\[-2ex]\n\\end{tabular}\n\\vspace{-2ex}\n \\caption{The critical value of $h$ as a function of the number of the\n data points used.\n }\n\\label{fig:scan-hc}\n\\end{center}\n\\end{figure}\n\nTo determine $h_c$, we iterate the calculation with $h$ varying in\nsteps of 0.02.\nFigure~\\ref{fig:scan-hc} shows the critical values of $h$\nas a function of the number of data points used in the fit,\ncorresponding to Fig.~\\ref{fig:chi2}.\nSince no reason exists to select the best result from them,\nwe take all the results satisfying $\\chi^2\/$dof $<$ 3 as the final\nresults, and the systematic uncertainty is chosen to cover the whole\naccepted results.\n\n\\begin{figure}[tb]\n\\vspace*{-1ex}\n\\begin{center}\n\\begin{tabular}{c}\n\\includegraphics*[width=0.6 \\textwidth,clip=true]\n{figs\/scan-hc-P-k0.145-0.1505-Xspall.eps}\\\\\n\\end{tabular}\n\\vspace{-1ex}\n\\caption{$h_c$ and $P$ at $h_c$: The results from $\\ln R$ (open symbols)\n and $\\ln R'$ (filled symbols).}\n\\label{fig:hc-P}\n\\end{center}\n\\vspace{-2ex}\n\\end{figure}\n\nWe can also determine the value of $P$ at $h_c$, denoted by $P_c$\n(for the numerical values, see Table~\\ref{tab:hc-result}).\nIn Fig.~\\ref{fig:hc-P}, all the results with $\\chi^2\/{\\rm dof} < 3$ are\nplotted together on the $P_c$-$h_c$ plane.\nWhile $h_c$ is insensitive to the two-flavor mass, $P_c$ is found\nto decrease towards the chiral limit of the two-flavor mass.\nThis qualitative feature is tested in the direct simulations of\n$2+N_f$-flavor QCD in Sec.~\\ref{subsec:direct-sim}.\n\n\\begin{table}[tb]\n \\centering\n \\begin{tabular}{c|cc}\n $\\kappa_l$ & $h_c$ & $P$ at $h_c$\\\\\n  \\hline\n   0.1450 & 0.23( 6) & 1.627(34) \\\\\n   0.1475 & 0.27( 8) & 1.595(27) \\\\\n   0.1500 & 0.26( 5) & 1.561(41) \\\\\n   0.1505 & 0.27(10) & 1.572(24) \\\\\n \\end{tabular}\n \\caption{Numerical results of $h_c$ and $P$ at $h_c$.\n }\n \\label{tab:hc-result}\n\\vspace{-1ex}\n\\end{table}\n\n\\begin{figure}[t]\n\\vspace*{-1ex}\n\\begin{center}\n\\begin{tabular}{cc}\n\\includegraphics*[width=0.5 \\textwidth,clip=true]\n{figs\/hc-mpmr2-0.eps} &\n\\includegraphics*[width=0.5 \\textwidth,clip=true]\n{figs\/hc-pcac-0.eps} \\\\\n\\includegraphics*[width=0.5 \\textwidth,clip=true]\n{figs\/hc-mpmr2.eps} &\n\\includegraphics*[width=0.5 \\textwidth,clip=true]\n{figs\/hc-pcac.eps} \\\\\n\\end{tabular}\n\\vspace{-2ex}\n\\caption{The light quark mass dependence of $h_c$.\n Top: A constant fit.\n Bottom: A linear fit is performed with the horizontal axis\n proportional to $m_l^{2\/5}$, testing the mean field prediction\n Eq.~(\\ref{eq:meanfield-scaling}).\n The band appearing in the top right corner is $h_c$ determined in the\n direct simulation with $\\kappa_l=0$ and $N_f=50$ (for details, see\n Sec.~\\ref{subsec:direct-sim}).\n }\n\\label{fig:lq-dep}\n\\vspace{-2ex}\n\\end{center}\n\\end{figure}\n\nThe light quark mass dependence of $h_c$ is plotted in\nFig.~\\ref{fig:lq-dep} as a function of the $m_\\pi\/m_\\rho$ ratio (left)\nand the PCAC quark mass (right).\nThe error is dominated by the systematic uncertainty associated with\nthe fitting procedure.\nIn the top-right corner of each plot, $h_c$ determined from the direct\n$2+N_f$ flavor simulation with $\\kappa_l=0$ ({\\it i.e.} $m_l=\\infty$) and\n$N_f=50$ is shown with an uncertainty (for the direct simulations, see\nSec.~\\ref{subsec:direct-sim}).\n$h_c$ at $\\kappa_l=0$ is clearly larger than those around\n$0.145 \\le \\kappa_l \\le 0.1505$, which indicates that $h_c$ gradually\ndecreases towards the chiral limit as a global behavior.\nIn the range of $0.145 \\le \\kappa_l \\le 0.1505$, $h_c$ does not show\nsignificant dependence on $m_l$ within the error, and a constant fit\nyields $h_c=0.23(1)$ in the chiral limit.\n\nIf this mild dependence is continued down to the chiral limit and hence\n$h_c$ in the chiral limit remains positive and finite, $h_c$ in the\nchiral limit corresponds to the tricritical point in\nFig.~\\ref{fig:columbia-plot}.\nMean field analysis of an effective theory predicts the tricritical\nscaling~\\cite{Wilczek:1992sf,Rajagopal:1992qz,Ukawa:1995tc,Ejiri:2008nv},\n\\begin{eqnarray}\n h_c \\sim (\\mbox{const.})\\times m_l^{2\/5} + \\mbox{const.}\n \\label{eq:meanfield-scaling}\n\\end{eqnarray}\nin the vicinity of the tricritical point, where the power $2\/5$ is\nindependent of $N_f$.\nIn addition to the constant fit (top), we also fit the data to a linear\nfunction of $m_l^{2\/5}$ in each plot, yielding $h_c=0.35(26)$ and\n$0.32(17)$ in the chiral limit of the two-flavor mass, respectively.\nNote that the slope is undetermined and consistent with zero.\nIn either case, a positive value of $h_c$ is favored in the chiral\nlimit, which suggests the second order transition of massless two-flavor\nQCD.\nFurther checks require more extensive lattice calculations and are\npostponed to future papers.\n\n\n\\section{Consistency check}\n\\label{sec:consistency-check}\n\n\\subsection{Effective potential constraining Polyakov loop}\n\\label{subsec:polyakov}\n\nAs an independent check of the results obtained in\nSec.~\\ref{sec:numerical_results}, we try to estimate $h_c$ with a\ndifferent method.\nThe quantity to be constrained to obtain the PDF is arbitrary as long as\nit has an overlap with the order parameter.\nIn this section, we take the real part of the Polyakov loop, $\\hat L$.\nRecalling Eq.~(\\ref{eq:effective-potential-2}), the constraint effective\npotential for $\\hat L$ is given by\n\\begin{eqnarray}\n      V_L(L;\\beta_{\\rm ref},\\kappa_l,\\kappa_h,N_f)\n&=& - \\ln w_L(L;\\beta_{\\rm ref},\\kappa_l,0,0)\n    - \\ln \\left(\\frac{w_L(L;\\beta_{\\rm ref},\\kappa_l,\\kappa_h,N_f)}\n                     {w_L(L;\\beta_{\\rm ref},\\kappa_l,0  ,0)}\n\t\t   \\right)\\\\\n&=&   V_{L,\\,\\rm light}(L; \\beta_{\\rm ref},\\kappa_l)\n    - \\ln R_L(L;\\beta_{\\rm ref},\\kappa_l,\\kappa_h,N_f)\\ ,\n \\label{eq:effective-potential-poly}\n\\end{eqnarray}\nwhere\n\\begin{eqnarray}\n      V_{L,\\,\\rm light}(L; \\beta_{\\rm ref},\\kappa_l)\n&=& - \\ln w_L(L;\\beta_{\\rm ref},\\kappa_l,0,0) \\\\\n&=& - \\ln \\langle\\,\n             \\delta(L-\\hat L)\\,\n             e^{6\\,(\\beta_{\\rm ref}-\\beta)\\,N_{\\rm site}\\, \\hat P}\\,\n          \\rangle_{(\\beta,\\kappa_l)}\\, ,\\\\\n    R_L(L; \\beta_{\\rm ref},\\kappa_l,\\kappa_h,N_f)\n&=& e^{6\\,N_s^3\\,h\\,L}\n    \\left\\langle \\displaystyle\n       \\exp\\left[\n            36\\,N_s^3\\,h\\,\\hat W_P\n\t  +  6\\,(\\beta_{\\rm ref}-\\beta)\\,N_{\\rm site}\\, \\hat P\n\t   \\right]\n    \\right\\rangle_{L: {\\rm fixed},(\\beta,\\kappa_l)}\\ .\n\\end{eqnarray}\nUnlike the case with $\\hat X=\\hat P$, $\\beta_{\\rm ref}$ dependence\nremains in the second derivative of the potential with respect to $L$,\nwhich means $\\beta_{\\rm ref}$ has to be explicitly tuned to the\n(pseudo)critical temperature for each value of $h$.\nIn the crossover region, the minimum of the potential and the minimum of\nits second derivative are realized at the same value of $L$.\nOn the other hand, in the first order region, $\\beta_{\\rm ref}$ has to\nbe tuned until the two minima in the potential take the same depth.\n\nThe potential, $V_L(L)$, is shown in Fig.~\\ref{fig:Veff-RePoly}\nfor $h=0.0$, 0.2 and 0.4, where $V_L(L)$ at each $h$ is shifted in a\nvertical direction for comparison.\n\\begin{figure}[tb]\n\\begin{center}\n\\begin{tabular}{cc}\n\\includegraphics*[width=0.5 \\textwidth,clip=true]\n{figs\/poly-case-k0.145.eps}&\n\\includegraphics*[width=0.5 \\textwidth,clip=true]\n{figs\/poly-case-k0.1505.eps}\n\\end{tabular}\n\\vspace{-1ex}\n\\caption{The effective potential constructed by constraining $\\hat L$\n near the critical temperatures.\n The absolute values of the potential are shifted for comparison.\n The results at $h=0.0$, 0.2 and 0.4 are shown for $\\kappa_l=0.145$ (left)\n and 0.1505 (right).\n }\n\\label{fig:Veff-RePoly}\n\\end{center}\n\\end{figure}\nFirst of all, the effective potential for the Polyakov loop is not as\nclean as that for the generalized plaquette, especially at nonzero $h$,\nwhich only allows us to extract the qualitative feature.\nIt is seen that the potential has a positive curvature at $h=0$ in our\nlightest and heaviest light quarks.\nWhen $h=0.2$, the potential around the minimum becomes almost flat,\nindicating that it is close to the end point.\nIf $h$ is further increased to 0.4, the double well shape appears to emerge\nthough the statistical error makes it ambiguous.\nThese qualitative features are consistent with the findings in the\nprevious section.\n\n\n\\subsection{Direct simulations of $2+N_f$ flavor QCD}\n\\label{subsec:direct-sim}\n\nAlthough the convergence of hopping parameter expansion is not the\nmatter for the discussion in the previous section, it is interesting to\ninvestigate the convergence of the HPE for future applications.\nWe study this by explicitly performing simulations of $2+N_f$ flavor QCD\nand comparing the results with those based on the HPE.\nHowever, thoroughly precise calculations of the many flavor system\nrequire an extensive scan of simulation parameters ($\\kappa_h$, $\\beta$ and\n$N_f$) even after fixing $\\kappa_l$.\nFurthermore, in general, it is not easy to locate the end point of the\nthe first order transition accurately, because the statistical noise\ngrows as one approaches the end point.\nInstead, we draw a thermal cycle on the $\\beta$-$P$ plane, which is\nobtained by increasing (or decreasing) $\\beta$ until passing its\n(pseudo) critical value and then reversing the direction.\nFor representative values of $\\kappa_h$, $\\kappa_l$ and $N_f$, we accumulated\n400 trajectories at each point of the thermal cycle.\n\nWe take $\\kappa_l=0, 0.145$ and 0.1505, and choose $N_f$ ranging from 4 to\n50, depending on $\\kappa_l$, and monitor each thermal cycle whether the\nhysteresis curve occurs or not.\nIf it occurs, the parameters chosen turn out to be in the first order\nregion.\nBy repeating this, we try to find the critical value of $\\kappa_h$, $\\kappa_{h_c}$,\nfor fixed values of $\\kappa_l$ and $N_f$.\nThe thermal cycles at several simulation parameters are shown in\nFig.~\\ref{fig:hysteresis}.\n\\begin{figure}[tb]\n\\begin{center}\n\\begin{tabular}{cc}\n\\includegraphics*[width=0.5 \\textwidth,clip=true]\n{figs\/plq_L16x4_nf2+16_rg-KL0.0-cswL0.0-nosmr-apbc.eps}&\n\\includegraphics*[width=0.5 \\textwidth,clip=true]\n{figs\/plq_L16x4_nf2+50_rg-KL0.0-cswL0.0-nosmr-apbc.eps}\\\\[-2ex]\n (a)&(b)\\\\[2ex]\n\\includegraphics*[width=0.5 \\textwidth,clip=true]\n{figs\/plq_L16x4_nf2+32_rg-KL0.1505-cswL1.712-nosmr-apbc.eps}&\n\\includegraphics*[width=0.5 \\textwidth,clip=true]\n{figs\/plq_L16x4_nf2+10_rg-KL0.1505-cswL1.712-nosmr-apbc.eps}\\\\[-2ex]\n (c)&(d)\\\\[2ex]\n\\end{tabular}\n\\vspace{-1ex}\n\\caption{The thermal cycle on the ($P$, $\\beta$)-plane in $2+N_f$ flavor\n QCD. The vicinity of the end point, {\\it i.e.} $\\kappa_h\\approx\\kappa_{h_c}$, is\n shown.}\n\\label{fig:hysteresis}\n\\end{center}\n\\end{figure}\n\nFigures~\\ref{fig:hysteresis} (a) and \\ref{fig:hysteresis} (b) show the\nthermal cycle at $\\kappa_l=0$ and $N_f=10$, 16, 32, 50, which tells us that\n$\\kappa_{h_c}$ decreases with $N_f$.\nFigure~\\ref{fig:hysteresis} (c) shows the same but with nonzero $\\kappa_l$.\nComparing the $N_f=16$ data in Figs.~\\ref{fig:hysteresis} (a) and\n\\ref{fig:hysteresis} (c) or the $N_f=32$ data in\nFigs.~\\ref{fig:hysteresis} (b) and \\ref{fig:hysteresis} (c), it is found\nthat $\\kappa_{h_c}$ is clearly different between $\\kappa_l=0$ and 0.1505.\nIn Fig.~\\ref{fig:hc-P} of Sec.~\\ref{subsec:lat-para}, we have discussed\nthat the HPE predicts that $P$ at $h_c$ decreases with $\\kappa_l$.\nFigure~\\ref{fig:hysteresis} (d) seems to show that it is the case, at\nleast, qualitatively.\n\nThe critical values, $\\kappa_{h_c}$, obtained at each $\\kappa_l$ and $N_f$\nare translated into $h_c$ using Eq.~(\\ref{eq:h}), and plotted as a\nfunction of $N_f$ in Fig.~\\ref{fig:direct-hc}.\n\\begin{figure}[tb]\n\\begin{center}\n\\begin{tabular}{cc}\n\\includegraphics*[width=0.7 \\textwidth,clip=true]\n{figs\/nf_vs_h_c.eps}\\\\\n\\end{tabular}\n\\vspace{-1ex}\n\\caption{$N_f$ dependence of $h_c$ through direct simulations of $2+N_f$\n flavor QCD.}\n\\label{fig:direct-hc}\n\\end{center}\n\\end{figure}\nIt is expected that, for a sufficiently large $N_f$,\n$h_c$ approaches an asymptotic value, but the value of $N_f$ we have\nstudied seems not to reach such a region yet although the increasing\nrate looks slowing down.\nImportantly, it is seen that $\\kappa_{h_c}$ does not differ by much between\n$\\kappa_l=0.145$ and 0.1505, which is consistent with the observation in the\nHPE analysis.\nIt is interesting to include the next-to-leading order contribution of\nthe HPE analysis.\n\n\n\\section{Summary and discussion}\n\\label{sec:summary}\n\nWe have studied the finite temperature phase transition of QCD with two\nlight and many heavy quarks at zero chemical potential, where the heavy\nquarks are introduced in the form of the hopping parameter expansion via\nthe reweighting method.\nThe phase structure was scanned on the $\\kappa_l$-$h$ plane to identify the\ncritical line separating the continuous crossover and the first order\nregions.\n\nThe nature of the transition is identified by the shape of the\nconstraint effective potential constructed from the probability\ndistribution function of the generalized plaquette.\nFor $h=0$, the system reduces to two-flavor QCD, which always shows\ncontinuous crossover and hence the potential has a single well.\nAs one increases $h$, at some point the potential takes a double-well\nshape, which defines a critical value, $h_c$.\nWe have determined $h_c$ at four light quark masses, and observed that\n$h_c$ is independent of two-flavor mass in the range we have studied\n($0.46 \\le m_\\pi\/m_\\rho\\le 0.66$).\nThis result indicates that the critical heavy mass remains finite in the\nchiral limit of the two flavors, suggesting the phase\ntransition of massless two-flavor QCD is of second order.\nSome of the qualitative features observed in the main analysis were\nchecked by two independent analyses.\n\nThe approach in this study can be said as follows.\nTwo-flavor QCD with a finite mass is enforced to undergo a first order\nphase transition by adding extra quarks.\nIt is then likely that those extra quarks are necessary to keep the\nfirst order transition down to the chiral limit of two-flavor QCD.\nThis method is applicable for any kinds of lattice fermions unless they\ncontain $\\beta$ dependent coefficients.\nAccording to the definition of $h$, Eq.~(\\ref{eq:h}), $\\kappa_{h_c}$ can be\nconsidered to be arbitrarily small for a given $h_c$ by assuming\narbitrarily large $N_f$, and thus we do not have to care about the\nconvergence of the hopping parameter expansion.\nNevertheless, it is interesting to see the limitation of the HPE for a\nfixed $N_f$ for further applications.\n\nIn order to establish our finding, possible systematic uncertainties,\nwhich are not investigated in the present paper, need to be\nunderstood.\nSince, at the end point, the second order phase transition occurs, a\nsizable finite volume effect is possible in the vicinity of the point.\nHowever, we do not expect it to be significant in the many flavor\napproach.\nIn this approach, the end point is determined through the extrapolation\nof the effective potential with regard to $h$, and the extrapolation is\nperformed in a region free from finite size effects since the two-flavor\nconfigurations are all generated at a parameter region away from the\nsecond order end point.\nNamely, at the price of the uncertainty due to the extrapolation, we\ncould have avoided the finite volume effect associated with the second\norder phase transition.\nNevertheless, it is clearly important to explicitly check that the\neffect is under good control.\nSuch work is ongoing.\n\nAlthough the behavior observed in Fig.~\\ref{fig:lq-dep} seems to suggest\nthat the chiral limit of $h_c$ is finite and positive, it then has to\nshow the tricritical scaling [Eq.~(\\ref{eq:meanfield-scaling})].\nAt the present, our results allow us to fit to any smooth function\nin $m_l$.\nIn order to improve the situation, we need to explore lighter quark\nmasses and reduce the systematic uncertainty associated with the\nfitting procedure.\nHowever, during the preliminary study, we realized that the lightest\nquark mass presented in this paper is the lower limit in our lattice\nsetup.\nTo go beyond the limit, the setup has to be changed.\nTowards the ultimate goal, the discretization effects also have to be\nexamined.\nA systematic study of these uncertainties requires large scale\nsimulations, and we postpone them to future works.\n\nWe can extend the many flavor approach to explore QCD at finite chemical\npotential as initiated in Refs.~\\cite{Ejiri:2012rr,Iwami:2015mqa}.\nIn this case, mean field analysis predicts that the critical line\nruns like ${m_l}^c \\sim |\\mu|^5$~\\cite{Ejiri:2008nv}.\nWe believe that such a study brings valuable information to\nunderstand the rich QCD phase diagram.\n\n\n\\section*{Acknowledgments}\n\nWe would like to thank members of the WHOT-QCD Collaboration for\nuseful discussions.\nWe also thank Ken-Ichi Ishikawa\nfor providing us his simulation codes.\nThis work is in part supported by\nJSPS KAKENHI Grant-in-Aid for Scientific Research (B)\n(No.\\ 15H03669 [N.~Y.],\n       26287040 [S.~E.]\n)\nand (C)\n(No.\\ 26400244 [S.~E.]), and by the Large Scale Simulation Program of High\nEnergy Accelerator Research Organization (KEK) No.\\ 14\/15-23.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{\\label{sec:level1} Introduction}\nCircular dichroism (CD) spectroscopy is a widely used experimental technique which enables the characterization of chiral molecular samples. This technique, which measures the differential absorption of molecules to left- and right-handed circularly polarized light (CPL) excitation, permits, for instance, to characterize the secondary structure of biomolecules and to determine the enantiopurity of pharmaceutical drugs ~\\cite{Barron}. Despite its wide applicability, the sensitivity of CD measurements is quite limited due to the weak nature of chiral light-matter interactions. Nanostructured materials have been instrumental enhancing the sensitivity of other molecular spectroscopic techniques such as Surface-Enhanced Raman Scattering (SERS) and Surface-Enhanced Infrared Absorption (SEIRA) spectroscopy, increasing their performance several orders of magnitude \\cite{SEIRA1, SEIRA2, SERS1, SERS2, SERS3}. Similarly, increasing the sensitivity of CD spectroscopy with the aid of nanostructures would be highly desirable and it is a research objective which has been pursued in the recent past.\n\nSince the seminal contributions of Y. Tang and A. E. Cohen~\\cite{tang_cohen}, several nanophotonic platforms have been proposed to augment the sensitivity of CD experiments through the local and global enhancement of $C(\\mathbf{r})$, the local density of electromagnetic chirality. To mention a few, high refractive index nanoparticles~\\cite{PRB, CDEnh2, OldACSAitzol, Curto, Vladimiro, IvanACS}, plasmonic chiral~\\cite{PRXDesignPples, fCD3000} and nonchiral nanoantennas~\\cite{CDEnh1, MicOrCD, fCD3000}, optical waveguides~\\cite{Vazquez} and metasurfaces~\\cite{ACSDionne, NanophotonicPlatforms, Quidant} have been explored as enablers of surface-enhanced circular dichroism and other chiral spectroscopy techniques. \n\nGlobally enhancing CD spectroscopy around a nanostructure necessitates duality symmetry in the design of the device so that the helicity of the scattered fields is conserved ~\\cite{PRLMolina}. Unfortunately, common nanophotonic platforms can only guarantee this symmetry at particular, non-resonant, excitation wavelengths. This limits the efficiency of light-matter interactions between the scattered fields and the molecules under study, limiting the magnitude of the achievable enhancements in CD spectroscopy. \n\nIn this work, we present a novel approach to design helicity preserving and optically resonant nanostructured devices to enhance the CD signal exploiting the lattice modes of periodic arrays of dual high refractive index nanoparticles. Starting from the well-known example of a silicon nanosphere capable of enhancing CD spectroscopy by a factor of 10~\\cite{PRB}, we first characterize the resonances of an infinite 1D array of such nanoparticles. We observe that these systems support two different types of emerging lattice resonances: far-field diffractive modes and a non-diffractive lattice resonance mediated by mid- and near-field interactions. Secondly, we find that in finite particle arrays finite chain modes can also be observed and exploited. \n\nWe show that all of these helicity preserving resonant modes can globally enhance the sensitivity of CD spectroscopy over two orders of magnitude, providing one additional order of magnitude enhancement over the response of a single, non-resonant nanoparticle ~\\cite{PRB}. Even if the maximum enhancement is given for the far-field diffractive modes, we find that enhancements obtained for the rest of the modes are comparable.\n\n\\section{\\label{sec:level2} Enhanced chiral light-matter interactions and helicity conservation}\n\nFor molecules illuminated by a left ($+$) or right ($-$) handed circularly polarized plane wave in vacuum, the CD signal of a molecule, $\\text{CD}_{cpl}$, can be expressed as \\cite{tang_cohen, PRB} \n\\begin{equation}\n\\text{CD}_{cpl}=-\\frac{4}{\\varepsilon_0}\\text{Im}(G)|C_{cpl}|,\n\\end{equation}\nwhere $G$ is the molecular chiral polarizability, $C_{cpl}=\\pm \\frac{\\varepsilon_0\\omega}{2c}E_0^2$ the local density of chirality of a plane wave in vacuum, $c$ the speed of light, $\\omega$ the angular frequency, $E_0$ the amplitude of the incoming electric field and $\\varepsilon_0$ the permittivity of free space. Ref.~8 concluded that in the presence of nonchiral antennas, the CD signal could be locally expressed as\n\\begin{equation}\n\\text{CD} = -\\frac{4}{\\varepsilon_0}\\text{Im}(G)C(\\mathbf{r}),\n\\label{CD_C}\n\\end{equation}\nwith $C(\\mathbf{r}) = -\\frac{{\\omega}}{2c^2}\\text{Im}\\Big( \\mathbf{E}(\\mathbf{r})^*\\cdot \\mathbf{H}(\\mathbf{r}) \\Big)$  being the local density of optical chirality\\cite{tang_cohen}, where $\\mathbf{E}(\\mathbf{r})$ and $\\mathbf{H}(\\mathbf{r})$ are the local electric and magnetic fields, respectively. Although $G$ is a fixed molecular parameter, $C(\\mathbf{r})$, can be in principle engineered and enhanced in the presence of optical antennas. Importantly, in the presence of nonchiral antennas one can define a local enhancement factor such that $\\text{CD}=f_\\text{CD}\\text{CD}_{cpl}$ where\n\n\\begin{equation} \\label{fCD}\nf_\\text{CD}=\\frac{\\text{CD}}{\\text{CD}_{cpl}}=\\frac{C(\\mathbf{r})}{|C_{cpl}|}=-\\frac{Z_0}{E_0^2}\\text{Im}\\Big(\\mathbf{E}(\\mathbf{r})^*\\cdot \\mathbf{H}(\\mathbf{r})\\Big),\n\\end{equation}\n$Z_0$ being the impedance of vacuum.\n\nContrary to what could be intuitively expected, the local density of \nchirality, $C(\\mathbf{r})$, is not necessarily related to the local handedness of the field. Nevertheless, as shown in Eq. (\\ref{CD_C}), the CD signal is directly proportional to $C(\\mathbf{r})$ in the presence of nonchiral environments. Therefore, an essential element to enhance the CD signal is to engineer the local fields, so that the overall sign of $C(\\mathbf{r})$ (and, thus, also the sign of $f_{\\text{CD}}$) is preserved, while its magnitude is maximized. Our work is focused on finding a nanostructure which, when illuminated by CPL of a given handedness, can enhance the absolute value of $C(\\mathbf{r})$ while locally preserving its sign in space. In such a scenario, molecules in the vicinity of our optical resonator will interact with the local field in a way that always contributes positively to CD signal. This allows for a strict spatial control on the molecular absorption rates and facilitates the practical measurement of the field-enhanced CD signal.\n\nIn the case of monochromatic fields, the electromagnetic density of chirality  is  intimately related to the local density of helicity~\\cite{barnett, cameron, nieto2017chiral, Lisa}. \nThe helicity operator can be expressed as\n\\begin{equation}\n\\Lambda = \\frac{\\mathbf{J} \\cdot \\mathbf{P}}{|\\mathbf{P}|} = \\frac{1}{k}\\nabla \\times\n\\label{helicity}\n\\end{equation}\nwhere $\\mathbf{J}$ is the total angular momentum of light, $\\mathbf{P}$ is the linear momentum and the second equality in Eq. \\eqref{helicity} is only held for monochromatic fields \\cite{Messiah}. The helicity operator is the generator of dual transformations, i.e. rotations of the electric and magnetic fields~\\cite{PRLMolina}. Hence, the helicity is a conserved magnitude in the interaction of light with systems symmetric under the exchange of electric and magnetic fields, the so-called dual systems.\n\nIn single particles with a dominant dipolar response, duality symmetry is fulfilled for wavelengths which meet the first Kerker's condition, in other words, for wavelengths where the electric and magnetic polarizabilities of the nanostructure are equal ($\\alpha_e=\\alpha_m$) \\cite{nieto2011angle,Mole_k,Zambrana, Aitzol_moebius,olmos2019enhanced}. This condition can be achieved for particular wavelengths of light and materials~\\cite{strong_magnetic}. Similarly, in a collection of nanoparticles, duality is a symmetry of the whole system if every individual particle satisfies the first Kerker's condition~\\cite{PRLMolina,MikolajPRL} \n\nHelicity conservation in such scenarios allows us to consider the following expressions for the local electric and magnetic fields\\cite{MikolajPRL}\n\\begin{align}\n\\label{DualE}\n\\Lambda \\mathbf{E}(\\mathbf{r}) &=  \\frac{1}{k}\\nabla \\times \\mathbf{E}(\\mathbf{r}) = p\\mathbf{E}(\\mathbf{r})\\\\\n\\label{DualH}\n\\Lambda \\mathbf{H}(\\mathbf{r}) &=  \\frac{1}{k}\\nabla \\times \\mathbf{H}(\\mathbf{r}) = p\\mathbf{H}(\\mathbf{r}),\n\\end{align}\nwhere the helicity eigenvalue, $p = \\pm 1$. From Eqs. \\eqref{DualE}-\\eqref{DualH} and Maxwell's equations one obtains $\\mathbf{H}(\\mathbf{r}) = \\frac{-i p k}{\\omega\\mu_0}\\mathbf{E}(\\mathbf{r})$. Applying this expression to Eq. \\eqref{fCD}, the local CD enhancement factor for dual systems can be reformulated as:\n\n\\begin{equation}\n\\label{fCDdualE}\nf^{dual}_{\\text{CD}} = \\frac{1}{E_0^2}|\\mathbf{E}(\\mathbf{r})|^2.\n\\end{equation}\n\nOn the one hand, Eq. \\eqref{fCDdualE} confirms that under duality symmetry, $C(\\mathbf{r})$ does not change sign locally upon scattering. Most importantly, this result predicts that we can enhance the local CD experimental signal on a dual structure as far as we are able to enhance the fields close to our optical system. Finally, this expression comes into agreement with other results in the literature, which express the chirality in terms of the Riemann-Silberstein vectors~\\cite{RS, IvanACS}.\n\n\\section{\\label{sec:level3}Lattice resonances}\n\\begin{figure}[ht]\n\t\\centering\n\t\\includegraphics[width=1\\textwidth, scale=0.6]{jon_spheres_v55.png}\n\t\\caption{Sketch of the setup under study: a group of nanoparticles arranged in a 1D periodic arrangement. In the bottom right part, the extinction cross section of a single isolated silicon nanoparticle in our system.}\n\t\\label{Array}\n\\end{figure}\n\nAs we determined in the previous section, dual systems allow for the enhancement of $f_\\text{CD}$ while keeping its local sign unaltered.  Nevertheless, duality symmetry constriction restricts the chirality enhancement to specific wavelengths which fulfill Kerker's first condition. Typically, this wavelength does not coincide with resonant modes of optical cavities and resonators. As a consequence, the electromagnetic fields used in typical surface-enhanced CD devices are not very intense on isolated nanoparticles. Previous results for single high refractive index nanoparticles \\cite{PRB} predict a modest enhancement of one order of magnitude in the molecular CD signal. In order to overcome this limitation we propose to exploit lattice resonances arising in ordered collections of similar dual, non-resonant, individual dielectric nanoparticles. For the sake of clarity, we restrict our study to 1D periodic arrangements of silicon nanoparticles. Nevertheless, ideas presented in this article can be easily generalized for 2D or 3D particle arrays.\n\nLattice resonances are high-quality factor collective excitations of periodically arranged groups of nanoparticles. Equation \\eqref{fCDdualE} indicates that on any dual system $f_\\text{CD}$ scales with the square of the field enhancement that is produced. Thus, our hypothesis is that although the individual nanoparticles are non-resonant at the first Kerker's condition, their collective behaviour at particular separation distances can give rise to resonant, very intense circularly polarized fields capable of uniformly enhancing CD spectroscopic signal of molecules over the entire extension of the array. Thus, in order to guarantee duality symmetry, we will operate at a particular frequency at which Kerker's first condition is fulfilled by every particle in the lattice, while we modify the separation distance between them to induce a resonant behaviour of the periodic system.\n\nUnder certain circumstances, the optical response of each nanoparticle can be described as a combination of a point electric dipole and a point magnetic dipole. Thus, in such situations, the collective behaviour of the system can be analyzed using an approach based on the Coupled Dipole Approximation (CDA). For a finite collection of $N$ particles, the optical response of the coupled electric and magnetic dipoles has a well-known algebraic solution:\n\n\\begin{equation}\n\\vec{\\Psi} = \\left[\\bm{I} - k^2 \\bm{G}_F \\bm{\\alpha}_F\\right]^{-1} \\vec{\\Psi}_{0}.\n\\end{equation}\n$\\vec{\\Psi}$ is a $6N$ dimensional vector which contains the self-consistent solution for each electric and magnetic field components in every site. $\\bm{I}$ is the $6N \\times 6N$ identity matrix and $\\bm{\\alpha}_F$ is a matrix of the same size containing the values of the electric and magnetic polarizabilities. $\\bm{G}_F$ is a matrix that contains the coupling coefficients (built as lattice sums of Green's functions) between all the different dipoles in the system. Finally, $\\vec{\\Psi}_{0}$ is a $6N$ dimensional vector containing the values of the incident electric and magnetic fields at each of the positions of the particles (more detailed information can be found in the Supplementary Material).\n\nCollective resonances emerge when $\\text{det}\\left( \\bm{I} - k^2 \\bm{G}_F \\bm{\\alpha}_F \\right) \\approx 0$ (notice that real poles would signal the existence of bound states). In these situations the scattered fields around the nanoparticles are strongly enhanced and, thus, $f_\\text{CD}$ increases substantially. Note that since the incident wavelength and, therefore, the polarizabilities of the particles are fixed to fulfill duality symmetry, the emergence of this collective modes depends only on geometrical values, namely, the spatial distribution of the nanoparticles and the direction of incidence of the probing CPL.\n\nIt is important to note that there are different interaction mechanisms through which nanoparticles can interact in a periodic structure to give rise to collective resonances. On the one hand, if the photonic elements are placed at relatively long separation distances, they interact through a $r^{-1}$ dependent far-field dipolar interaction giving rise to diffractive modes ~\\cite{hicks}. On the other hand, if the particles are more closely spaced, $r^{-3}$ and $r^{-2}$ dependent near-field interactions can also give rise to hybridized chain modes~\\cite{ross,halas_review}.  \n\nIn what follows, we show that lattice resonances based on both types of interaction mechanisms are, in practice, useful to enhance the local density of optical chirality $C(\\mathbf{r})$ and, thus, enhance the CD signal, setting a useful photonic paradigm for experimental testing. We have chosen our proof-of-concept system to be a finite 1D array of silicon dual nanoparticles showing that both types of lattice resonances give rise to an important enhancement of $f_\\text{CD}$.\n\n\\section{\\label{sec:level4}A finite chain of dual nanospheres}\n\nIn the following, we study a 1D array of $N=2000$ silicon nanospheres of radius $a = 150$ nm located along the $OY$ axis such that the nanospheres are placed at $\\mathbf{r}_n = (n-1)D \\mathbf{\\hat{u}_y}$ where $n = 1,2,..., N$ and $D$ is the distance between two adjacent nanoparticles from center to center. In addition, the wavevector ($\\mathbf{k}$) of the incoming circularly polarized plane wave is contained in the $YZ$ plane and forms an angle $\\theta$ with the $OZ$ axis perpendicular to the chain. A sketch of the parameters of the system can be seen in Fig. \\ref{Array}.\n\nTo characterize the resonant features of the system, we first study the extinction cross section, $C_{ext}$ of the particle chain. In this particular system, Kerker's first condition is exclusively fulfilled at $\\lambda = \\lambda_k = 1208$ nm, so we explore the extinction cross section of the system as a function of two parameters: the incidence angle, $\\theta$, and the interparticle distance, $D$. The results of this calculation are shown in Fig. \\ref{Diego_Extinction}a, where the normalized extinction cross section, $Q_{ext} = C_{ext}\/(N\\cdot C_{ext,1})$, is shown and $C_{ext,1}$ is the extinction cross section of a single silicon particle at $\\lambda_k$. In Fig. \\ref{Diego_Extinction}b, a comparison of our results with the extinction cross section of an infinite chain of particles is shown (details of the theoretical formalism describing the infinite case can also be found in the Supplementary Material). As expected, both calculations give very similar results since collective interactions are known to converge for a given number of elements in a periodic structure~\\cite{plasmon_pol, size_matters}. Interestingly, the number of nanospheres required in our system to reproduce the behaviour of the infinite case is very big comparing to what is found in the literature.\n\n\\begin{figure}[ht]\n\t\\centering\n\t\\includegraphics[width=1\\textwidth]{AfterDiego_ExtMap_a-b_changed_HOT_back.png}\n\t\\caption{Normalized extinction cross section map under Kerker's first condition as a function of interparticle distance for: {\\bf a)} a finite periodic 1D array of 2000 silicon nanospheres, {\\bf b)} an infinitely periodic 1D array of silicon nanospheres.}\n\t\\label{Diego_Extinction}\n\\end{figure}\n\nThe analytic calculations for the infinite chain allow us to identify the narrow peaks at low angles ($\\theta \\sim 0$) as diffractive modes emerging at $D\/\\lambda = m\/(1 \\pm \\sin \\theta)$ where $m$ takes positive integer values. As stated above, these peaks emerge as a consequence of the constructive interference of the far-field interactions. This is also observable in Fig. \\ref{LambdaSweep}b, where, together with the single-particle spectrum, narrow diffraction lines emerge when nanoparticles are placed at $\\lambda=5\\lambda_k$.\n\n\\begin{figure}[ht]\n\t\\centering\n\t\\includegraphics[width=1\\textwidth]{LambdaSweep2.png}\n\t\\caption{Normalized extinction cross section map as a function of the illuminating wavelength for: {\\bf a)} a single silicon nanosphere, {\\bf b)} a chain of 2000 nanospheres with $D = 5\\lambda_k$, {\\bf c)} a chain of 2000 nanospheres with $D = 1.5\\lambda_k$ and {\\bf d)} a chain of 2000 nanospheres with $D = 0.5\\lambda_k$.}\n\t\\label{LambdaSweep}\n\\end{figure}\n\nHowever, close to grazing incidence ($\\theta \\rightarrow \\pi\/2$) a much wider resonance emerges. This mode does not fit with diffractive characteristics and, thus, has to be interpreted in terms of a different interaction mechanism. As we show in Fig. \\ref{LambdaSweep}, this resonance emerges when particles in the chain are placed closer than $D = 1.5\\lambda_k$ and, therefore, near- and mid-field interparticle contributions become important. As particles are placed closer together, the magnetic single-particle resonance of silicon ($\\lambda_{mag} \\sim 1110$ nm) redshifts and it can approach the wavelength of the Kerker condition, $\\lambda_k$. All these considerations point out that this resonance is related to a \\emph{bonding} transversal mode of the chain, a configuration which has been extensively studied in nanoparticle dimers and metamaterials~\\cite{halas_review}.\n\nMoreover, at $\\theta = \\pi\/2$, the complete physical configuration (incident wave included) is rotationally symmetric. Due to the simultaneous duality and axial symmetry, helicity can be expressed in this situation as:\n\n\\begin{equation}\n\\Lambda = \\frac{J_yP_y}{|P_y|},\n\\end{equation}\nwhere both $J_y$ (y-component of the total angular momentum of light) and $\\Lambda$ are conserved quantities. Therefore, $P_y$ (y-component of the linear momentum of light) cannot change sign under scattering and backward scattering is not allowed~\\cite{kerker}. As a consequence, this mode in an exclusively forward propagating one.\n\nOnce we have thoroughly analyzed the optical resonances that can emerge in a dual 1D array of nanospheres, we proceed to show that, as predicted, these resonances give rise to important enhancements of $f_\\text{CD}$. In order to model realistic environments of practical interest, we define two averaged enhancement factors: the average CD value around a particle in position $i$ ($f_{\\text{CD}}^{i}$) and the array-averaged CD enhancement factor, $f_{\\text{CD}}^{avg}$, defined as\n\n\\begin{eqnarray}\n\\label{fCDi}\nf_{\\text{CD}}^{i} &=& \\int_{S_i}\\frac{1}{4\\pi} f_{\\text{CD}}(\\xi,\\eta)\\sin\\xi d\\xi d\\eta, \\\\\n\\label{fCDavg}\nf_{\\text{CD}}^{avg} &=& \\frac{1}{N}\\sum_{i = 1}^N f_{\\text{CD}}^{i}.\n\\end{eqnarray}\nThe integrals are carried out around a spherical surface $S_i$ ($\\xi$ and $\\eta$ are, respectively, the polar and azimuthal spherical coordinates respect to the center of the $i$-th nanoparticle) separated $1~\\text{nm}$ apart from the surface of the $i$-th silicon nanoparticle.  \n\nFig. \\ref{Extinction_fCD}a presents the normalized extinction cross section of the silicon array as a function of $D\/\\lambda$, for $\\theta = 0$. Under this incidence, the extinction shows a resonant behaviour at the diffraction condition $D = m\\lambda_k$, for $m \\in \\mathbb{N}$. However, the strength of the resonances decreases as $m$ is increased, something which is already noticeable in Fig. \\ref{Extinction_fCD}a. In Fig. \\ref{Extinction_fCD}b we can compare the extinction resonances with the CD enhancement averaged over the whole structure, $f_{\\text{CD}}^{avg}$, for $\\theta=0$. We obtain two orders of magnitude CD enhancement (a factor of $\\sim 110$) averaged over the entire structure for the diffractive mode peaking at $D\/\\lambda=1$. Note that previous results for induvidual nanoparticles \\cite{PRB} concluded that individual silicon nanoparticles of similar radius, were capable of inducing an enhancement of one order of magnitude in the molecular CD signal. Thus, this dual lattice design strategy provides an additional order of magnitude enhancement in $f_{\\text{CD}}^{avg}$. \n\nAlso, the CD enhancement variation among the individual spheres of the chain is shown in Fig. \\ref{Extinction_fCD}c. In this figure, we reflect the $f_{\\text{CD}}^i$ enhancement value around every sphere, for the two resonant distances appearing in Fig. \\ref{Extinction_fCD}b. The average values of the resonant enhancement for the whole structure (coinciding with the peak values of Fig. \\ref{Extinction_fCD}b) are presented as black dotted line. We can observe that both resonances have similar symmetric field distributions, whose maximum is located right at the center of the array.\n\n\\begin{figure}[ht]\n\t\\centering\n\t\\includegraphics[width=0.92\\textwidth]{Image_inkscape_theta0_and_thetapi41.png}\n\t\\caption{ {\\bf a)} Normalized extinction for an incidence of $\\theta = 0$. {\\bf b)} $f_\\text{CD}^{avg}$ for $\\theta = 0$. {\\bf c)} Distribution of the local CD enhancement factor for the two resonant conditions in b). {\\bf d)} Normalized extinction for an incidence of $\\theta = \\pi\/2$. {\\bf e)} $f_\\text{CD}^{avg}$ for $\\theta = \\pi\/2$. {\\bf f)} Distribution of the local CD enhancement factor for the two resonant conditions in e). Black dotted line: average enhancement for the whole structure ($f_\\text{CD}^{avg}$) at the resonant conditions in Figs. {\\bf b)} and {\\bf c)}.}\n\t\\label{Extinction_fCD}\n\\end{figure}\n\n\nOn the other hand, under incidence parallel to the array axis, at $\\theta = \\pi\/2$, the normalized extinction cross section in Fig. \\ref{Extinction_fCD}d shows a single broad peak around $D\/\\lambda \\sim 1.1$. Analyzing the array-averaged CD enhancement factor, though, two peaks are evident in Fig. \\ref{Extinction_fCD}e, one at $D\/\\lambda \\sim 1.1$ and another one around $D\/\\lambda \\sim 0.5$, both showing maximum values of $f_\\text{CD}^{avg} \\sim 60$. As pointed out above, the $f_\\text{CD}^{avg}$ peak centered at $D\/\\lambda \\sim 1.1$ is related to the redshifted single-particle magnetic resonance and, thus, it is directly related to the the peak in Fig. \\ref{Extinction_fCD}d.\n\nHowever, the second $f_\\text{CD}^{avg}$ peak at $D\/\\lambda \\sim 0.5$ is not related to an extinction peak, but to the small oscillations that the normalized extinction shows in Fig. \\ref{Extinction_fCD}d for $D\/\\lambda \\le 0.5$. This oscillations do not appear in the infinite case, which allows us to understand it as a finite chain effect. Finite chain modes have been widely studied over the literature and are known to appear for the geometric condition $D\/\\lambda < 0.5$ (concretely for $D\/\\lambda = \\frac{(N-2)n + 1}{2N(N-1)}$, with $n \\in [1,2,..., N]$)~\\cite{finite_chain}, which very well fits the range in which both the extinction oscillations and the $f_\\text{CD}^{avg}$ peak are found. The nature of these modes makes them have nearly $0$ dipole moment, which implies that they can barely contribute to the extinction but can still present very strong near-fields~\\cite{DarkPlasmon}, giving rise to the second peak in Fig. \\ref{Extinction_fCD}e. This happens because finite chain modes are typically characterized by sign changing dipole distributions~\\cite{finite_chain}, whose contributions interfere destructively in the far-field. Figure \\ref{Extinction_fCD}f shows the spatial distribution of $f_\\text{CD}^{avg}$ in the chain for the two resonant conditions in Fig. \\ref{Extinction_fCD}e. Finally, note that the $f_\\text{CD}$ distribution for the finite chain mode in Fig. \\ref{Extinction_fCD}f represents a guided mode of the finite chain, with a considerable propagation length.\n\nIn conclusion, we have presented a systematic procedure to design periodic nanophotonic platforms capable of enhancing molecular CD spectroscopy resonantly. As an example, we applied the method to design a dual 1D periodic nanostructure made of high refractive index nanoparticles. By analyzing the emergent lattice resonances in the system we have found that they can be classified in three different types: diffractive modes, non-diffractive modes and finite chain modes. We have shown that far-field diffractive resonances can increase the array-averaged CD enhancement up to a factor $\\sim 110$, providing an additional order of magnitude enhancement compared to the optical response of a single particle. Moreover, we find that non-diffractive and finite chain modes, which have not been previously considered in the field-enhanced CD literature, also give rise to comparable enhancement magnitudes (over a factor of $\\sim 60$). This method can be easily extended to more complicated platforms such as 2D metasurfaces or 3D photonic crystals, opening venues to new designs and research directions.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nGalaxy formation is a complex process.\nDespite continuing efforts incorporating higher resolution numerical simulations with more and more sophisticated subgrid stellar physics, it has hitherto proven impossible to satisfy observational constraints simultaneously at both dwarf and massive galaxy scales.\nIntroduction of feedback from massive black holes has helped alleviate the problem of excessive production of  massive galaxies~\\citep{2014MNRAS.445..175G}, but  nearby dwarf galaxies provide a well studied and more challenging environment.\nSupernova feedback seems incapable of resolving their  paucity~\\citep{2015ApJ...807..154B}, the too-big-to-fail problem~\\citep{2013MNRAS.433.3539G}, and the ``missing'' baryon fraction issue~\\citep{2015JATIS...1d5003B}.\n\nThis failure has motivated many attempts at modifying the nature of Dark Matter (DM), for example into warm~\\citep{Dodelson:1993je,Dolgov:2000ew,2017MNRAS.464.4520B}, fuzzy~\\citep{Hu:2000ke,2014MNRAS.437.2652M} and strongly self-interacting~\\citep{Spergel:1999mh,2015MNRAS.453...29E} variants.\nAll of these attempts seem to create as many problems as they try to\nresolve~\\citep{2016arXiv161109362S}.\n\n\\smallskip\n\nHere we take a different tack via the mirror DM~\\citep{Berezhiani:2005ek,Foot:2004pa,1983SvA....27..371B,1991SvA....35...21K}.\nWe demonstrate that a subdominant component of dissipative dark matter, containing dark baryons and dark photons identical to ordinary sector particles, naturally produces Intermediate Mass Black Holes (IMBHs), in the mass range $(10^4-10^5) \\, \\rm M_\\odot$, at the epoch of first structure formation.\nThis behavior derives from the suppression of mirror molecular hydrogen, due to a much lower fraction of free mirror electrons, which act as catalyzers.\n\nBy accretion, a few of these black holes can transform into the Supermassive Black Holes (SMBHs) observed at $z \\sim 7$ (see~\\cite{2017arXiv170303808V} and references therein), whose existence is still an unexplained issue in astrophysics.\nThis can happen because we have massive seeds and they can accrete two non-interacting dissipative matter sectors (ordinary and mirror).\n\nThe paper is organized as follows. In Sec.~\\ref{Sec:Mirror} we describe our dark matter model, whose thermal history we study in Sec.~\\ref{Sec:Thermal}.\nSection~\\ref{Sec:Structures} is devoted to the description of the structure formation in the mirror sector and the estimate of the IMBH number density.\nWe discuss the accretion of the BH seeds in~\\ref{Sec:Accretion}, and summarize our results in~\\ref{Sec:Conclusions}.\n\n\n\\section{Mirror World}\n\\label{Sec:Mirror}\nWe assume the existence of a parallel sector of mirror particles which is completely identical, in terms of particle physics properties, to the Standard Model (SM) particle sector.\nMirror particles interact with the SM only via gravitational interactions and all the portals (e.g. photon and Higgs portals) are chosen to be very small.\nWe further assume the existence of a cold DM component, which does not interact appreciably with the baryons (ordinary and mirror).\n\nWe leave the particle physics details to future work.\nFor the purpose of this article, it will suffice to note that, in the simplest scenario, the whole theory is invariant with respect to an unbroken discrete mirror parity that exchanges the fields in the two sectors, although there needs to be a breaking in the very early universe to allow different initial conditions in the two sectors~\\cite{Berezhiani:2000gw}.\nThe DM component can simply be an axion, i.e. the Goldstone of an anomalous, spontaneously broken $U(1)_{\\rm PQ}$ Peccei-Quinn (PQ) symmetry, with the $U(1)_{\\rm PQ}$ charges carried by both the ordinary and mirror Higgses (for an example, see~\\citep{Berezhiani:2000gh}).\nIn summary, from a cosmological point of view, we have:\n\\begin{itemize}\n    \\item[$\\diamond$]\n    A duplicate of the SM matter.\n    The relativistic degrees of freedom of this sector are mirror photons and neutrinos, contributing an energy density $\\Omega_{\\rm r}'$.\n    The non-relativistic degrees of freedom are mirror baryons with energy density $\\Omega_{\\rm b}'$.\nHere and in the following, the symbol ($'$) denotes the physical quantities of the mirror world.\n    \\item[$\\diamond$]\n    A Cold Dark Matter (CDM) candidate, whose energy density is denoted $\\Omega_{\\rm c}$, such that the total matter energy fraction is $\\Omega_{\\rm m} = \\Omega_{\\rm c} + \\Omega_{\\rm b} + \\Omega_{\\rm b}'$.\n\\end{itemize}\n\nAll the differences between the two sectors can be described in terms of two macroscopic parameters which are the only free parameters of the model:\n\\begin{equation}\n    x = T'_\\gamma\/T_\\gamma \\, , \\qquad \\beta = \\Omega_{\\rm b}'\/\\Omega_{\\rm b} \\, ,\n\\end{equation}\n$T_\\gamma$ being the photon temperature.\nFor simplicity, the results showed in the next sections are derived by taking  $\\beta = 1$, i.e.~$\\Omega_{\\rm b}'=\\Omega_{\\rm b}$.\n\nIn order to avoid the CMB (Cosmic Microwave Background) and BBN (Big Bang Nucleosynthesis) bounds on dark radiation, one needs the condition $x\\lesssim 0.3$~\\citep{Berezhiani:2000gw,Foot:2014uba}.\nIf this is the case, we will see in the next section that mirror matter behaves like CDM at the time of CMB last scattering  (mirror baryons are bounded in neutral mirror hydrogen atoms).\n\n\n\\section{A brief thermal history of the mirror universe}\n\\label{Sec:Thermal}\nAs discussed, in our setup the Friedmann equation reads:\n\\begin{multline}\n    H(z) = H_0 \\Bigl[ \\Omega_{\\rm r}(1+x^4)(1+z)^4 + \\\\\n    \\left(\\Omega_{\\rm b}(1+\\beta)\n    + \\Omega_{\\rm c}\\right)(1+z)^3\n    +\\Omega_\\Lambda \\Bigr]^{1\/2} \\ ,\n\\end{multline}\nwhere $H_0$ is today's Hubble constant.\n\nAn important stage for structure formation is the matter-radiation equality epoch, which occurs at the redshift\n\\begin{equation}\n\\begin{split}\n    1+z_{\\rm eq} =& \\frac{\\Omega_{\\rm m}}{\\Omega_{\\rm r}^{\\rm tot}}\n    =\\frac{\\Omega_{\\rm b}(1+\\beta) + \\Omega_{\\rm c}}{\\Omega_{\\rm r}(1+x^4)} \\\\\n    =& \\frac{\\rho_{\\rm c}^0}{T_{\\gamma,0}^4} \\, \\frac{\\Omega_{\\rm m}}{ \\pi^2\/30 \\,  g_*(T_{\\gamma, 0})\\,  (1+x^4)} \\, \\ ,\n\\end{split}\n\\end{equation}\nwhere $g_*$ is the number of relativistic degrees of freedom and $T_{\\gamma,0}$ is the CMB temperature today.\nUsing the best-fit Planck parameters~\\cite{Ade:2015xua}, one gets $1+z_{\\rm eq} \\simeq 3396\/(1+x^4)$.\nSince $x \\ll 1$, the matter-radiation equality is untouched in presence of a colder mirror sector.\n\n\\medskip\nThe evolution of the free electron number fraction $X_e$ and gas temperature $T_g$ as a function of the redshift $z$ for the ordinary and mirror sectors are ruled by the following coupled differential equations~\\cite{Giesen:2012rp}:\n\\begin{align}\n    \\begin{split}\n        \\frac{{\\rm d}X_e}{{\\rm d}z} ={}& \\frac{\\mathcal P_2}{(1+z)H(z)}  \\Big(\\mathcal \\alpha_H(T_g)  n_{\\rm H} X_e^2 \\\\\n        &- \\mathcal \\beta_H(T_g) e^{-E_\\alpha\/T_g} (1-X_e) \\Big) \\, ,\n    \\end{split} \\\\\n    \\frac{{\\rm d}T_{\\rm g}}{{\\rm d}z} ={}& \\frac{1}{1+z} \\[ 2 T_g - \\gamma_{\\rm C} \\(T_\\gamma(z) - T_g \\) \\] \\, ,\n\\label{eq:evolXe&Tg}\n\\end{align}\nwhere $E_{\\alpha}$ is the Ly-$\\alpha$ energy, $\\beta_H$ is the effective photoionization rate from $n = 2$ (per atom in the $2s$ state), and $\\alpha_H$ is the case-B recombination coefficient.\nWe have defined the dimensionless coefficient\n\\begin{equation}\n    \\gamma_{\\rm C} \\equiv \\frac{8 \\sigma_{\\rm T} a_r T_\\gamma^4}{3 H m_e c} \\frac{X_e}{1 +X_{\\rm He} + X_e} \\, ,\n    \\label{eq:compton}\n\\end{equation}\n(and analogous for the mirror sector) with $\\sigma_{\\rm T}$ the Thomson cross-section, $a_r$ the radiation constant, $m_e$ the electron mass and $X_{\\rm He}$ the number fraction of helium.\nThe coefficient $\\mathcal P_2$ represents the probability for an electron in the $n = 2$ state to get to the ground state before being ionized, given by~\\cite{Giesen:2012rp}\n\\begin{equation}\n    \\mathcal{P}_2 = \\frac{1 + K_H \\Lambda_H n_H (1-X_e)}{1 + K_H (\\Lambda_H + \\beta_H) n_H (1-X_e)} \\, ,\n\\end{equation}\n(and analogous expression for the mirror sector) where $\\Lambda_H = 8.22458 \\, {\\rm s^{-1}}$ is the decay rate of the $2s$ level, and $K_H = \\lambda^3_{\\rm Ly\\alpha}\/(8 \\pi H(z))$ accounts for the cosmological redshifting of the Ly-$\\alpha$ photons.\n\nFor the ordinary sector, the boundary conditions are $X_e(z_M)=1$ and $T_g(z_M)=T_{\\gamma, 0}(1+z_M)$~\\footnote{Here $z_M=2500$. We have checked that for $z>z_M$ the solutions are stable.}.\nFor the mirror sector, the equations take the same form, with the substitutions $X_e \\to X_e'$, $T_g \\to T_g'$, $n_H \\to n_{H'}$, $\\gamma_C \\to \\gamma'_C$, $T_{\\gamma}(z) \\to x\\, T_{\\gamma}(z)$.\nThe boundary conditions in the mirror sector are $X'_e(z_M\/x)=1$ and $T'_g(z_M\/x)=T_{\\gamma, 0}(1+z_M\/x)$.\n\n\\medskip\nFrom eq.~\\eqref{eq:compton} in the mirror sector, we notice that the Compton rate is a factor $x^{4}$ smaller than that in the ordinary sector.\nAs a consequence, the recombination in the speculative sector is much faster.\nWe solve numerically Eqs.~\\eqref{eq:evolXe&Tg}, showing the results in fig.~\\ref{fig:recombination}.\nThe left panel shows the electron fraction $X_e$ for the ordinary and mirror sectors as a function of redshift. For the mirror sector we show the results for three benchmark values of $x$: $x=0.3$ (blue dot-dashed line), $x=0.1$ (magenta dot-dashed line) and $x=0.01$ (red dot-dashed line).\n\nAt the time of the ordinary recombination ($z\\simeq 1100$) the mirror hydrogen is fully recombined.\nIndeed, the residual mirror ionization fraction $X_e'$ at $z=1100$ is always less than $10^{-5}$ for the three benchmark models we consider.\nAs a consequence, mirror hydrogen behaves like CDM with respect to the ordinary plasma evolution before the CMB last scattering.\nHence, by $z \\simeq 1100$, the total amount of CDM is exactly the one measured by the Planck satellite: i.e.~$\\Omega_{\\rm DM}=\\Omega_{\\rm m}-\\Omega_{\\rm b} = \\Omega_{\\rm b}'+\\Omega_{\\rm c}$.\nThe right panel of fig.~\\ref{fig:recombination} shows instead the evolution of the gas temperatures $T_g$, $T'_g$ as a function of redshift.\nSince the Compton heating process is not efficient in keeping the mirror baryons and mirror photons in thermal equilibrium, the temperature of the mirror gas at redshifts relevant for the structure formation is much smaller than the ordinary one (not simply by a factor $x$).\n\\begin{figure*}\n\\centering\n\\includegraphics[width=0.4925\\textwidth]{Xe} \\\n\\includegraphics[width=0.4925\\textwidth]{TIGM}\n\\caption{{\\bf Left panel:} Free electron number abundance $X_e$ as a function of redshift $z$ for the ordinary (black solid), and mirror sector for different values of the $x$ parameter (dot-dashed lines). {\\bf Right panel:} Background gas temperature $T_g$ as a function of redshift $z$.}\n\\label{fig:recombination}\n\\end{figure*}\n\n\\section{Structure formation}\n\\label{Sec:Structures}\n\\subsection{Qualitative picture}\nThe different chemical initial conditions of the ordinary and mirror gas are crucial for understanding the differences in the formation of the first structures in the two sectors.\n\nLet us follow a spherical overdensity of mirror baryons.\nWhile $\\delta \\equiv \\delta \\rho\/\\rho \\ll 1$, it will expand with the rest of the universe, but lagging behind as $\\delta \\rho \\sim a^{-2}$.\nAt some point, when $\\delta \\sim 4.55$, the overdensity turns around and starts to collapse~\\citep{2010gfe..book.....M}.\nIf the matter interacts only gravitationally (as the CDM component), the final result of the collapse will be a halo of particles supported by velocity dispersion, corresponding to an effective virial temperature $T_{\\rm vir} = \\mu m_p G_N M \/ (5 k_B R)$, where $\\mu$ is the mean molecular weight, $m_p$ the proton mass, and $M$, $R$ the mass and radius of the overdensity, respectively.\nHowever, unlike CDM, the mirror gas does not undergo shell-crossing, being instead heated by shocks.\nThe end result of the collapse of the  gas is approximately a mirror cloud heated to a temperature $\\sim (\\gamma-1) T_{\\rm vir}$, where $\\gamma \\simeq 5\/3$ is the adiabatic index, at the virial density $\\rho_{\\rm vir} \\simeq 178 \\, \\rho_{cr} \\Omega_{\\rm b}'(1+z)^3$~\\cite{2010gfe..book.....M}.\n\nAt this point, the chemistry of the gas needs to be considered.\nIn particular, we have to ask whether the mirror cloud can cool, losing pressure support and contracting further, or it will remain as a hot dilute halo (resembling a CDM component).\nAt low temperatures, in the absence of heavy elements, there is not enough energy to excite mirror atomic transitions.\nTherefore, as in the ordinary sector, the main cooling mechanism for a hydrogen-helium gas is through $\\mathrm{H}_2$, which at low densities is produced mainly through the reactions $\\mathrm{H} + e^{-} \\to \\mathrm{H}^{-} + \\gamma$, $\\mathrm{H}^{-} + \\mathrm{H} \\to \\mathrm{H}_2 + e^{-}$, in which free electrons act as catalyzers.\n\nSince molecular cooling can bring the temperature down to $T_g \\sim 200 \\, \\mathrm{K}$, ordinary baryons are able to form small structures.\nHowever, in the mirror sector the initial abundance of free electrons, as shown in the left panel of Fig.~\\ref{fig:recombination}, is very suppressed, and therefore the production of mirror molecular hydrogen is slower.\nThis allows some mirror clouds to not undergo catastrophic cooling, as we will show.\n\nFirst of all, it is clear that mirror clouds with very high virial temperatures, $T_{\\rm vir} \\gtrsim 10^4 \\, \\rm K$, will behave essentially as ordinary clouds, as the mirror hydrogen quickly undergoes full ionization, independently on the initial conditions.\nThe same fate happens to massive clouds which are adiabatically heated to high temperatures at the beginning of their evolution.\nOn the other hand, at very low virial temperatures, $T_{\\rm vir} \\lesssim 1000 \\, \\rm K$, a mirror cloud is a dilute cloud of neutral atomic hydrogen, which does not cool efficiently (at least in the absence of metals) and just behaves as cold DM, as shown in Sec.~\\ref{sec:timecol}.\nWe thus expect a range of masses in between these two extrema in which the behavior of mirror clouds can be different from ordinary baryons.\n\n\n\n\n\\subsection{Semi-analytical model}\n\\begin{figure*}\n\\centering\n\\includegraphics[width=0.93\\textwidth]{Plottatutto_z40}\n\n\\caption{Evolution of several physical quantities as a function of the gas number density $n_g$ at $z_{\\rm vir}=40$.\nThe panels in the first column describe the properties of the ordinary sector while the second and third columns refer to the mirror sectors with $x=0.1$ and $x=0.01$ respectively.\nIn all plots, the light gray area on the left denotes the region of the parameter space where the gas is not yet virialized.\n{\\bf First row:} Evolution of the free fall $t_{\\rm ff}$ (grey lines), sound-crossing $t_{\\rm s}$ (red lines) and cooling $t_{\\rm c}$ (blue lines) timescales for three virial temperatures ($T_{\\rm vir}=1000\\, \\rm K$ (solid lines), $T_{\\rm vir}=3500\\, \\rm K$ (dot-dashed lines) and $T_{\\rm vir}=12000\\, \\rm K$ (dotted lines)).\n{\\bf Second row:} Evolution of the gas temperature $T_g$. Three scenarios are possible: efficient gas cooling at $\\sim 200\\, \\rm K$ (red lines); quasi-isothermal collapse in the range $\\sim (500 - 900) \\, \\rm K$ (orange lines, depending on the value of $x$); quasi-isothermal collapse at $\\sim 9000 \\, \\rm K$ (blue lines).\n{\\bf Third row:} Evolution of the number fraction of molecular hydrogen $X_{\\rm H_2}$ for the scenarios discussed before.\n{\\bf Fourth row:} Evolution of the mass accretion rate estimated as the Jeans mass to free fall time $\\dot M = M_{\\rm J}\/t_{\\rm ff}$. The cyan region gives a threshold above which a {\\sc DCBH}\\ is  possible.}\n\\label{fig:All_Results}\n\\end{figure*}\n\nWe can try to understand more in detail the dynamics of a mirror cloud by considering the evolution of averaged temperature, total number density and species abundances, as done for instance in~\\cite{2000ApJ...534..809O,2001ApJ...546..635O}.\nOf course, this is only a first approximation, but it is useful to check the existence of halos in which very massive black holes may be formed by direct collapse.\n\nThe chemistry evolution of the cloud is solved using the \\textsc{krome} package~\\cite{2014MNRAS.439.2386G}.\nFor the details about the model, we refer to the built-in one-zone collapse released with the package.\nWe follow the abundance of 9 mirror species ($\\mathrm{H}$, $\\mathrm{H}^{-}$, $p$, $e$, $\\mathrm{H}_2$, $\\mathrm{H}_2^{+}$, $\\mathrm{He}$, $\\mathrm{He}^{+}$, $\\mathrm{He}^{++}$), tracking 21 reactions.\nFor the ordinary sector we use $\\mu = 1.22$, corresponding to a gas of hydrogen and helium in their standard BBN abundances.\nFor the mirror sector, we study two models, $x=0.1$ and $x=0.01$, for which \\cite{Berezhiani:2000gw} showed that the mirror helium abundance is negligible, therefore we have $\\mu'=1$.\nThe thermodynamic evolution is given by the equation\n\\begin{equation}\\label{eq:evolution}\n\\frac{\\dot{T_g}}{T_g} - (\\gamma-1 ) \\frac{\\dot{n}_g}{n_g} = \\frac{(\\gamma-1)}{k_{\\rm B} T_g n_g} (\\mathcal{H-C}) \\, ,\n\\end{equation}\nwhere $\\mathcal{C}$ and $\\mathcal{H}$ are, respectively, the cooling and heating rates per unit volume.\nThe inverse of the right hand side of Eq.~\\eqref{eq:evolution} is defined as (minus) the cooling time scale $t_{\\rm c}$.\nThe density evolution is approximated by a free-fall or isobaric evolution, depending on the shorter timescale between the sound-crossing time $t_{\\rm s}$ and the free-fall time $t_{\\rm ff}$~\\footnote{The sound-crossing time is $t_{\\rm s} = R\/c_s$, where $R$ is the radius of the cloud and $c_s$ the sound speed. The free-fall time is $t_{\\rm ff} = (3 \\pi \/ (32 G \\rho_{\\rm tot}))^{1\/2}$, $\\rho_{\\rm tot}$ being the total mass density.}.\nIf $t_{\\rm s} > t_{\\rm ff}$, we take the number density evolution to be $\\dot{n}_g = n_g\/t_{\\rm ff}$~\\cite{2000ApJ...534..809O}; if instead $t_{\\rm s} < t_{\\rm ff}$, the number density is inversely proportional to the temperature, $n_g \\propto T_g^{-1}$.\n\nOur results are shown in fig.~\\ref{fig:All_Results}, which illustrates the evolution of several physical quantities as a function of the gas density $n_g$ at a virialization redshift $z_{\\rm vir}=40$.\nFrom top to bottom, the rows show the evolution of: free-fall, sound-crossing and cooling timescales; gas temperature $T_g$; number fraction of molecular hydrogen $X_{\\rm H_2}$; and finally the accretion rate, estimated as  $\\dot{M} = M_{\\rm J}\/t_{\\rm ff}$, where $M_{\\rm J}$ is the Jeans mass.\nFrom left to right, we show the ordinary sector, the mirror with $x=0.1$ and the mirror with $x=0.01$.\nFrom the second to fourth row, the different curves correspond to different halo virial masses.\n\nFocusing on the phase diagram $(T_g-n_g)$, we can see that all the halos of the ordinary sector, after an initial phase of adiabatic contraction (in some cases followed by a short isobaric evolution) produce enough molecular hydrogen to quickly cool down to $\\sim 200 \\, \\rm K$ (red lines), according to the results reported in~\\cite{2000ApJ...534..809O,2001ApJ...546..635O}.\nThe mirror sectors, instead, shows markedly different behavior for moderate virial temperatures.\nFirst, as shown by the red curves, the more massive halos which manage to attain a temperature above the Ly$-\\alpha$ line will ionize the mirror neutral hydrogen, thus behaving as ordinary halos.\nHowever, at lower virial temperatures, the orange curves show that the evolution settles down to a quasi-isothermal collapse at a temperature in the range $\\sim (500 - 900)\\, \\rm K$ (depending on the value of $x$), at least until the density reaches $10^{10}\\,  \\mathrm{cm^{-3}}$, when 3-body reactions become important.\nAs apparent from the evolution of the timescales and of the $\\mathrm{H_2}$ abundance, this behavior is due to a balance between the cooling induced by trace amounts of molecular hydrogen and the compressional heating.\nWe also observe a third qualitative behavior in a narrow range of virial masses, illustrated by the blue curves.\nThese halos follow a trajectory which brings them close to the Ly-$\\alpha$ line, and they start to produce molecular hydrogen.\nHowever, before cooling occurs, the density reaches a critical value for which the cooling function changes behavior in $n_g$~\\cite{1999MNRAS.305..802L}.\nAfter that, there is a collapse in which the temperature decreases very slowly, with negligible amounts of $\\mathrm{H}_2$.\nThe end result of this scenario looks somewhat similar to the case in which molecular hydrogen is destroyed by Lyman-Werner photons~\\cite{2003ApJ...596...34B}, but in our case the time evolution of the halos is very slow ($\\sim \\rm Gyr$).\nTherefore, as we discuss below, these halos cannot collapse fast enough before undergoing merger events.\n\nAt this point, we would like to discuss the endpoint of the collapse of ordinary and mirror halos, whether we produce Population III ({\\sc POPIII}) stars or direct collapse black holes ({\\sc DCBH}).\nWe rely on the results of~\\cite{2014MNRAS.443.2410F,2016PASA...33...51L}, which give a threshold $\\dot{M} \\sim 10^{-2} \\, \\rm M_{\\odot}\/\\mathrm{yr}$ above which the result of the halo collapse is a {\\sc DCBH}.\nThis is shown as the cyan region in the last row of fig.~\\ref{fig:All_Results}.\nIf we choose to evaluate $\\dot{M}$ when the halo reaches a minimum temperature, which presumably means it fragments into Jeans-supported structures, we find that the ordinary sector can only form {\\sc POPIII}\\ stars, while the mirror sector is able to form {\\sc DCBH} s in the cases represented by the orange and blue curves (which however evolve too slowly).\nFrom the dashed black lines depicted in the second row of fig.~\\ref{fig:All_Results} we estimate the mass of such black holes, {\\it at their formation time}, as $(10^4-10^5) \\rm \\, M_{\\odot}$, the Jeans mass at (presumed) fragmentation (for the orange curves).\nThe number of black holes we expect to form can be as low as a few and as high as $(\\Omega'_{\\rm b}\/{\\Omega_{\\rm c}}) \\, M_{\\rm vir}\/M_{\\rm J,min}$, where $M_{\\rm vir}$ is the virial mass of the original halo and $M_{\\rm J,min}$ the Jeans mass at the temperature minimum.\nFor illustration, in the mirror sector with $x=0.1$, the maximum number of DCBHs with a mass $\\simeq 10^4\\, \\rm M_\\odot$, in a single halo, is in the range $(100-500)$ at $z_{\\rm vir}=40$.\n\n\\subsection{Time of collapse}\n\\label{sec:timecol}\nBefore concluding that each halo produces black holes, we need to check that the single-halo evolution discussed above is a reasonable approximation.\nIn particular, we have neglected the fact that structures in a $\\Lambda$CDM universe form by continuous merging of smaller objects into larger halos.\nTherefore, to be consistent, we have to require that the collapse time $t_{\\rm Coll}$ of any halo, defined as the time when the density grows super-exponentially, is shorter that a typical ``merger'' timescale, which we take as the Hubble time at the virialization epoch $t_{\\rm H}$.\nWe show this in fig.~\\ref{fig:Collasso}.\nFrom left to right, we have the ordinary sector, mirror with $x=0.1$, and mirror with $x=0.01$.\nIn each plot, we show the estimated age of the universe when we consider the halo collapsed (and a BH or a star formed), as a function of the virial temperature, for different virialization redshifts (at fixed $T_{\\rm vir}$, the redshift increases from top to bottom).\nThe gray region represents times larger than the age of the Universe today.\nThe stars denote the halos which satisfy our criterion that $t_{\\rm Coll} < t_{\\rm H}$, while the dots denote the ones which do not.\n\nThe evolution of the ordinary sector is always very fast, and we conclude that each halo we consider ends up producing {\\sc POPIII}\\ stars.\nThe evolution in both mirror sectors is instead very different.\nThere are 2 different ranges in virial temperatures which we can describe within the single-halo approximation, which correspond to all of the red curves (higher $T_{\\rm vir}$), which end up forming mirror {\\sc POPIII}\\ stars, and some of the orange curves (lower $T_{\\rm vir}$) in fig.~\\ref{fig:All_Results}, which end up forming IMBH seeds.\nAs anticipated, all the blue curves in fig.~\\ref{fig:All_Results} evolve so slowly we cannot really trust our conclusions.\nWe can say the same about the small halos with low $T_{\\rm vir}$ (dots on the left in the mirror panels of fig.~\\ref{fig:Collasso}).\nThese will evolve very slowly, thus resembling a cold dark matter component before undergoing mergers with other halos.\n\\begin{figure*}\n\\includegraphics[width=0.31\\textwidth]{Collasso_Ordinary} \\\n\\includegraphics[width=0.31\\textwidth]{Collasso_Mirrorp1} \\\n\\includegraphics[width=0.31\\textwidth]{Collasso_Mirrorp01}\n\\caption{Typical collapse timescales as a function of the virial temperature $T_{\\rm vir}$. From the left to the right we show the ordinary and the mirror sectors with $x=0.1$ and $x=0.01$ respectively. For a given $T_{\\rm vir}$, the different points are obtained by assuming several virialization redshifts. From bottom ($z_{\\rm vir} =60$) to top ($z_{\\rm vir} =10$) the step in redshift is $\\Delta z = 5$. The stars satisfy the criterium $t_{\\rm Coll}<t_H$ while the dots do not.}\n\\label{fig:Collasso}\n\\end{figure*}\n\\begin{table*}\n\\centering\n\\begin{tabular}{c|c|c|c|c|c|c|c|c|c|}\n\\cline{2-10}\n& \\multicolumn{3}{c|}{Ordinary}\n& \\multicolumn{3}{c|}{$x=0.1$}\n& \\multicolumn{3}{c|}{$x=0.01$} \\\\\n\\cline{2-10}\n& $z_{\\rm vir}=50$       & $z_{\\rm vir}=40$ &  $z_{\\rm vir}=30$\n& $z_{\\rm vir}=50$      & $z_{\\rm vir}=40$ & $z_{\\rm vir}=30$\n& $z_{\\rm vir}=50$       & $z_{\\rm vir}=40$ & $z_{\\rm vir}=30$  \\\\\n\\hline\n\\hline\n\\hline\n\\multicolumn{1}{|l|}{DCBH seeds}  & $0$      & $0$     & $0$        & $403$        & $1.8\\times 10^5$     & $2.7 \\times 10^7$  & $0.01$ & $0$ &   $0$ \\\\\n\\hline\n\\multicolumn{1}{|l|}{$M_{\\rm vir}^{\\rm min}$ [$M_\\odot$]}  & $\/$      & $\/$   & $\/$   & $3.4 \\times 10^5$  & $6.2 \\times 10^5$    & $1.4 \\times 10^6$  & $1.7 \\times 10^6$  & $\/$ & $\/$ \\\\\n\\multicolumn{1}{|l|}{$M_{\\rm vir}^{\\rm max}$ [$M_\\odot$]}  & $\/$       & $\/$   & $\/$   & $4.7 \\times 10^6$  & $5.0 \\times 10^6$    & $7.7 \\times 10^6$  & $2.4 \\times 10^6$  & $\/$ & $\/$ \\\\ \\hline\n\\hline\n\\multicolumn{1}{|l|}{Star sites} & $1.1\\times 10^5$     & $3.6 \\times 10^7$     & $4.3\\times 10^9$                    & $1.2 \\times 10^{-8}$        & $0.01$  & $505$  & $1.2 \\times 10^{-8}$  & $1.4 \\times 10^{-2}$  &  $258$ \\\\\n\\hline\n\\multicolumn{1}{|l|}{$M_{\\rm vir}^{\\rm min}$ [$M_\\odot$]}  & $9.0 \\times 10^4$      & $1.2 \\times 10^5$  & $1.9 \\times 10^5$  & $8.0 \\times 10^6$  & $1.1 \\times 10^7$    & $1.9 \\times 10^7$  & $8.0 \\times 10^6$  & $1.3 \\times 10^7$ & $1.9 \\times 10^7$ \\\\\n\\multicolumn{1}{|l|}{$M_{\\rm vir}^{\\rm max}$ [$M_\\odot$]}  & $\\infty$     & $\\infty$  & $\\infty$  & $\\infty$  & $\\infty$    & $\\infty$  & $\\infty$  & $\\infty$ & $\\infty$ \\\\    \\hline\n\\end{tabular}\n\\caption{Number density of candidate halos, in $\\mathrm{Gpc}^{-3}$. For each virialization redshift we show the number density of halos able to produce {\\sc DCBH} s seed or {\\sc POPIII}\\ star forming sites,\nas well as the minimum and maximum virial masses, which are the extremes of integration of the Press-Schechter mass function (inferred from Fig.~\\ref{fig:Collasso}).\nFrom left to right, we show results for the ordinary and mirror sectors with $x=0.1$ and $x=0.01$.}\n\\label{tab:Candidates}\n\\end{table*}\n\\subsection{Number of candidate halos}\nFinally, we would like to estimate the number of black hole seeds produced by the mirror sector.\nWe can say that each halo which undergoes quasi-isothermal collapse at $T_g\\simeq (500-900)\\, \\rm K$ (orange lines in the second panel of Fig.~\\ref{fig:All_Results}) could hosts at least one, at most $(\\Omega'_{\\rm b}\/{\\Omega_{\\rm c}}) \\, M_{\\rm vir}\/M_{\\rm J}$ black holes.\nAn approximate answer to our question is the estimate of the number of such halos, which we will do using the Press-Schechter mass function.\nIn detail, we integrate the mass function over the ranges of halo virial masses able to produce {\\sc DCBH} s or stars, as identified in fig.~\\ref{fig:Collasso}.\nOur results, in number of halos per comoving $\\mathrm{Gpc}^3$, are shown in table~\\ref{tab:Candidates}, for 3 representative redshifts of virialization ($z_{\\rm vir}=30, 40, 50$).\n\nAs one can see, the amount of halos containing stars in the mirror sector is much smaller with respect to the ordinary sector, especially at high redshift.\nWe then expect that the mirror sector is, in general, less contaminated by metals, which cause halo cooling and fragmentation.\nThis can be understood by noticing that, in the case of the lowest virialization redshift we use ($z_{\\rm vir} = 30$), in the ordinary sector there is already one star formation site per halo with a mass above $\\simeq 2 \\times 10^5 \\, \\rm M_{\\odot}$ (corresponding to the lowest virial temperature we consider).\nIndeed, our estimate is of the same order ($\\sim 10^{9}\/\\rm Gpc^3$ when the Universe was $t_{\\rm vir}(z=30)+t_{\\rm coll} \\simeq 200$ Myr old) of the (extrapolated) galaxy number density of the Universe~\\cite{2016ApJ...830...83C,2015ApJ...803...34B,2013ApJ...763..129S}, while in the mirror sector we expect the star formation sites to be in negligible number ($\\sim 500$).\nFor this reason, we do not fully trust our calculations at a lower redshift, as we expect that at least the ordinary sector is polluted by metals, thus changing the collapse dynamics of both the ordinary and the mirror sector.\n\nThe star formation history in the mirror sector is different than in the ordinary sector, and we expect that mirror stars are less abundant and bigger than the ordinary ones, at the same epoch.\nFurthermore, for the mirror sector with $x=0.1$ we estimate that there are many {\\sc DCBH}\\ seeds ($\\sim 10^{7}\/\\rm Gpc^3$), more than the mirror star formation sites.\nFrom the numbers we get, we estimate that {\\sc DCBH}\\ seeds are about a factor $10^{-2}$ less abundant than the sites of star formation in the ordinary sector, in other words we expect that about one percent of galaxies will host a black hole.\n\nAs for the sector with $x=0.01$, we see that it forms very few {\\sc DCBH}\\ seeds, formed only above $z=40$, and a few mirror stars.\nIndeed, this sector is very cold and remains mostly dilute during the cosmic evolution.\n\n\n\\section{Accretion of BH seeds}\n\\label{Sec:Accretion}\nOur results so far give an interesting indication that many halos can produce very massive black hole seeds, by direct collapse of mirror baryon clouds.\nThe next question is to understand whether we can reach BH masses of up to $(10^9 - 10^{10}) \\, \\rm M_{\\odot}$ by redshift $\\sim 7$, as observed~\\cite{2017arXiv170303808V}.\n\nAfter formation, we expect that the seed black hole will grow by accretion, of both ordinary and mirror baryons.\nIn first approximation, we assume~\\cite{2004ApJ...614L..25Y} that a BH formed at time $t_0$ with mass $M_0$ grows continuously to reach a mass at time $t$\n\\begin{equation}\\label{eq:accretion}\nM(t) = M_0 \\, e^{(t-t_0)\/t_{\\rm Sal}} \\, .\n\\end{equation}\nHere, $t_{\\rm Sal}$ is the Salpeter time~\\cite{1964ApJ...140..796S}\n\\begin{equation}\nt_{\\rm Sal} = \\frac{\\epsilon M c^2}{(1-\\epsilon) L}\n\\simeq 400 \\, \\mathrm{ Myr} \\frac{\\epsilon}{1-\\epsilon} \\frac{L_{\\rm Edd}}{L} \\, ,\n\\end{equation}\nwhere $\\epsilon$ is the radiative efficiency, $L$ the BH luminosity and $L_{\\rm Edd}$ the Eddington luminosity.\nIn the presence of an ordinary and a mirror sector, which only interact gravitationally with each other, we expect that the Salpeter time roughly halves, assuming that the accretion happens in both sectors at similar luminosity and efficiency.\nTherefore, in a universe with both ordinary and mirror matter, even small seed black holes grow very fast, with an e-folding timescale of $\\sim 22 \\, \\mathrm{Myr}$ if the accretion happens at Eddington luminosity and $\\epsilon \\simeq 0.1$.\nNow, focusing on the mirror sector with $x=0.1$, by $z_{\\rm vir}=30$ we have a large amount of {\\sc DCBH}\\ seeds ($\\sim 10^{7}\/\\rm Gpc^3$) with initial masses $M_0 \\simeq 10^4 \\, \\rm M_\\odot$ formed when the Universe was $t_{\\rm vir}(z=30)+t_{\\rm Coll} \\equiv t_0 \\simeq 200 \\, \\rm Myr$.\nBy substituting these numbers in Eq.~\\eqref{eq:accretion}, we get that, at the time $t(z=7) \\simeq 800 \\, \\rm Myr$ when SMBHs with mass bigger than $10^9\\, \\rm M_\\odot$ are observed~\\cite{2011Natur.474..616M}, in our framework we are able to saturate such high masses even if the {\\sc DCBH}\\ seeds continuously accrete mass with a high radiative efficiency and luminosity less than Eddington in either sector.\nAs far as we know, ours is the only model~\\footnote{More generally, this happens in models with dissipative matter which can form dense structures.} which can accelerate the black hole growth so quickly.\nEven significant super-Eddington growth of SMBH in the ordinary sector requires seed black holes of at least $\\sim  100 \\rm \\, M_\\odot$ present at $z\\sim 20$~\\cite{2016MNRAS.458.3047P}, and in any case observations of the SMBH in quasars point out to accretion at a fraction of the Eddington luminosity~\\cite{2017ApJ...836L...1T}.\n\n\\medskip\nNevertheless, we cannot reliably estimate the final mass reached by redshift $\\sim 7$ because the exponential accretion is too simple a scenario. There are indeed two major issues to be addressed:\n\\begin{itemize}\n\\item\nFirst, there might be radiative feedback effects which shut off the mass accretion above a limiting mass~\\cite{2014MNRAS.443.2410F}.\n\n\\item\nThen, perhaps more importantly, our seed BHs are born in relatively small halos, of up to $\\sim 10^7 \\, \\rm M_{\\odot}$.\nAs a consequence, we should quantify whether the BH, once it has swallowed most of its host environment, can grow further.\nThis will be possible, because of halo accretion.\nIn fact, using the (extrapolation of the) median mass accretion rate derived in~\\cite{2010MNRAS.406.2267F}, we can determine the evolution of a typical halo by solving $\\mathrm{d} M \/ \\mathrm{d} z = 25.3 (1+1.65 z)\/(1+z) \\left[ M \/ (10^{12} {\\rm M_{\\odot}}) \\right]^{1.1} H_0^{-1} {\\rm M_{\\odot}} \\,\\mathrm{yr}^{-1}$.\nFocusing again on the mirror with $x=0.1$, we estimate that the final halo mass at $z = 7$ is $\\sim 10^8 \\, \\rm M_{\\odot}$ and $\\sim 1.6 \\times 10^9 \\, \\rm M_{\\odot}$ for, respectively, initial virial temperatures $T_{\\rm vir} = 1200$ and $T_{\\rm vir} = 3900$, corresponding to the limits of the range of BH-producing halos virialized at $z_{\\rm vir}=30$ (as shown in Fig.~\\ref{fig:Collasso}).\nThe mass in both baryonic components is a fraction $\\Omega_{\\rm b} (1+\\beta)\/\\Omega_{\\rm m}$.\nThe accretion history of a single halo has a large variability in its final mass, and the numbers we quote correspond to a median rate.\nGiven that the largest halos we consider accrete, in the median, a baryonic mass of the order of the few observed SMBHs, we find it likely that these can be produced by our mechanism.\n\n\\end{itemize}\n\n\n\n\\section{Conclusions and Outlook}\n\\label{Sec:Conclusions}\nThe presence of supermassive black holes at redshift as high as 7 is still an open issue in astrophysics.\nMoreover, baryonic feedback by supernovae has difficulty in resolving outstanding problems in dwarf galaxy formation.\nA new ingredient may be needed  that might involve IMBH feedback in dwarfs, but it is far from obvious how such IMBH might form in the early universe.\n\nWe provide a DM solution to this problem by appealing to a subdominant dissipative component of the dark sector, which for simplicity we consider to be a mirror copy of the Standard Model, while the rest of the DM sector is assumed to be a standard cold candidate.\nConcerning the particle physics details, we have in mind a model consisting of a mirror copy of the Standard Model with the same coupling constants and masses, with an axion-like particle shared by both ordinary and mirror sector.\nThe axion can potentially solve the strong-CP problem~\\cite{Berezhiani:2000gh}, but we leave the details to future work.\n\nOf course, this is not the only model which can give the phenomenology we discussed.\nWe expect that similar conclusions can be reached by changing couplings and masses (or even the particle content).\nIn particular, models with larger binding energy of the dark hydrogen atom will have a larger atomic line cooling temperature, and the cooling by molecular hydrogen is hindered because the formation rate is less efficient.\nThe cooling due to a dark hydrogen-like atom has been recently analyzed in~\\cite{Rosenberg:2017qia}.\n\n\\smallskip\n\nComing back to our mirror scenario, using a zero-dimensional approach, we showed that, if the mirror CMB temperature is much lower than the ordinary one, the thermodynamics and chemistry of collapsing mirror clouds follow a different path, leading to the production of IMBH seeds by direct collapse.\nWe have estimated the frequency, mass range and growth by accretion of the seeds, finding that they might in fact be helpful to resolve the many issues that perplex astrophycisists.\nMore in detail, we expect that by $z \\simeq 7$, in the case $x=0.1$, the bulk of the population of black holes has masses around $10^{7} - 10^{8} \\, \\rm M_{\\odot}$, with a tail that extends above $\\sim 10^{9}\\, \\rm M_{\\odot}$, and power the quasars we observe.\nThe mirror sector IMBHs are sufficiently numerous to be at the centers of the dwarf galaxies, such that their feedback could plausibly resolve many of the dwarf galaxy issues in the standard sector.\nOf course, detailed numerical simulations need to be performed in order to transform our estimates into more precise values, and analyze cosmological data in terms of our scenario.\n\nIn addition to the black holes, we will form mirror stars, more massive and much less abundant than in the ordinary sector, as shown in table~\\ref{tab:Candidates}.\nIn star-forming sites, we expect that part of the dissipative dark matter will organize itself in disks, rather than halos.\nThis has been already pointed out in similar models, for instance in~\\cite{Fan:2013yva,Fan:2013tia,Agrawal:2017rvu,Agrawal:2017pnb}.\nThe mirror substructures can be detected by large-scale structure observations, such as lensing, and by small-scale (galactic) probes, such as stellar dynamics.\n\nToday we have a new window to look at the sky: gravitational waves.\nWe expect to have IMBH mergers, and a few mergers of smaller black holes born from the death of mirror stars, at redshifts all the way up to $\\sim 30$.\nWhile IMBH mergers belong to the LISA range of frequencies, smaller black holes can be observed by LIGO.\nIt is therefore very interesting to evaluate the expected rate and mass range of these merger events.\n\n\\smallskip\n\n\n\n\n\\bigskip\n\n\\section*{acknowledgments}\nWe thank Stephane Charlot, Pratika Dayal, Anna Feltre, Andrea Macci\\`o, Mallory Roberts and Dominik Schleicher for useful discussions.\nP.P., J.S. and A.L. acknowledge support by the European Research Council ({\\sc Erc}) under the EU Seventh Framework Programme (FP7\/2007-2013)\/{\\sc Erc} Advanced Grant 267117 (`{\\sc Dark}') hosted by Universit\\'e Pierre \\& Marie Curie - Paris 6.\nA.L. aknowledges support by the European Research Council (ERC), Project no. 614199 (`BLACK').\nS.B. acknowledges funding through the DFG priority program `The Physics of the Interstellar Medium' (projects BO 4113\/1-2).\nP.P. is grateful to the Institute d'Astrophysique de Paris where this work was initiated.\n\n\n\\bibliographystyle{apsrev4-1}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nLet  $G$ be a group and  $n\\geqslant 1$. Recall that an \\emph{elementary Nielsen transformation} on a $n$-tuple  $T=(g_1,\\ldots, g_n)$ of elements of  $G$   is one of the following three types of moves:\n\\begin{enumerate}\n\\item[(T1)] For some $i\\in \\{1,\\ldots, n\\}$ replace $g_i$ by $g_i^{-1}$ in $T$.\n\\item[(T2)] For some $i\\neq j$, $i,j\\in \\{1, \\ldots, n\\}$ interchange $g_i$ and $g_j$ in $T$. \n\\item[(T3)] For some $i\\neq j$, $i,j\\in \\{1, \\ldots, n\\}$ replace $g_i$ by $g_ig_j$ in $T$.\n\\end{enumerate}\nWe  say that two $n$-tuples $T$ and $T'$   are \\emph{Nielsen equivalent}  if there exists a finite sequence of $n$-tuples \n$$T=T_0, T_1, \\ldots, T_{k-1}, T_k=T'$$ \nsuch that $T_i$ is obtained from $T_{i-1}$ by an elementary Nielsen transformation for $1\\leqslant i\\leqslant k$. Nielsen equivalence clearly defines an equivalence relation for tuples of elements of $G$.\n\n\\smallskip\n \nNielsen equivalence in Fuchsian groups  have been extensively studied by many authors, see  \\cite{Fine, Lustig1, Lustig2, Lustig3, Lustig4, P,Rosenberger1,  R, Zieschang} for example. However the techniques deployed so far are mainly algebraic such as normal form and K-theoretic arguments.\tIn \\cite{Dutra} we give a new geometric insight to the problem by showing that  any generating tuple of the fundamental group of a sufficiently large $2$-orbifold is represented by a special almost orbifold covering.  We still do not know if the same holds  for non sufficiently large orbifolds. The interesting case are orbifolds  with three cone points whose underlying surface is a sphere. The purpose of  this note  is to show that   generating pairs of the fundamental group of such orbifolds also have this geometric description.\n\\begin{theorem}{\\label{mainthm}}\nLet  $\\mathcal{O}=S^2(p_1, p_2, p_3)$   and  \n$$G=\\pi_1^o(\\mathcal{O})\\cong \\Delta(p_1, p_2, p_3)=\\langle s_1, s_2, s_3 \\ | \\ s_1^{p_1}=s_2^{p_2}=s_3^{p_3}=s_1s_2s_3=1\\rangle .$$ \n\nThen for any generating pair   $T$  of  $G$  there is a special almost orbifold covering  $\\eta':\\mathcal{O}'\\rightarrow \\mathcal{O}$ and a generating pair  $T' $ of $\\pi_1^o(\\mathcal{O}')$ such that $\\eta_{\\ast}'(T') $ and $T$ are Nielsen   equivalent.   \n\\end{theorem}\n\n\n\nA natural question that arises  is  whether  this holds  for arbitrary generating tuples, that is: \n\\begin{question}\nAre all  generating tuples of a triangle group also  represented by a (special)  almost orbifold coverings?\n\\end{question}\n\n  \n  \nThe proof of Theorem~\\ref{mainthm} relies heavily on the  description of all generating pairs of two-generated Fuchsian groups given  by Fine-Knapp-Matelsky-Purzitsky-Rosenberger \\cite{Fine, Knapp,Matelski, P,R}.  \n  \n  \n  \n\\section{Orbifolds and almost orbifold coverings}{\\label{secorbifold}}\n\nIn this section we give a quick review  about  orbifolds, orbifold fundamental groups and orbifold coverings.  More details about orbifolds can be found in the beautiful  article of P. Scott~\\cite{Scott}.\n\n \nA  $2$-\\emph{orbifold}  $\\mathcal{O}$ is a pair $(F, p)$ where $F$ is a  connected  surface, called the \\emph{underlying surface} of $\\mathcal{O}$, and $p:F\\rightarrow \\mathbb{N}$ is a function such that \n$$\\Sigma(\\mathcal{O}):=\\{x\\in F \\ | \\ p(x)\\geqslant 2\\}$$ \nis finite and contained in the interior   of $F$. The number $p(x)$ is called the \\emph{order} of   $x$ and the points of $\\Sigma(\\mathcal{O})$   are called \\emph{cone points}.  An  orbifold is said to be \\emph{compact} (resp. \\emph{closed}) if its underlying surface is  compact (resp. closed).  \n\nA compact orbifold   $\\mathcal{O}=(F, p)$  with $\\Sigma(\\mathcal{O})=\\{x_1, \\ldots, x_r\\}$  will also be denoted by   $F(p_1, \\ldots, p_r)$   where   $p(x_i)=p_i$ for $i=1,\\ldots, r$. \n\n\n\\smallskip \n\nThe \\emph{fundamental group} $\\pi_1^{o}(\\mathcal{O})$ of  $\\mathcal{O}=(F, p)$  is defined in terms of the fundamental group of the underlying surface $F$  as follows.  Let $x_1, x_2, \\ldots$ be the cone points of $\\mathcal{O}$ and for each $i=1, 2 , \\ldots$ let  $D_i\\subseteq \\text{Int} (F)$ be a disk centered at $x_i$ such that $ {D}_i\\cap {D}_j= \\emptyset$  for $i\\neq j$.   Set   \n$$ S_{\\mathcal{O}} = {F -  \\underset{i=1,2,\\ldots}\\cup \\text{Int}(D_i )}.$$  \nThen $\\pi_1^{o}(\\mathcal{O})$ is the  group obtained from $\\pi_1 (S_{\\mathcal{O}})$ by adding the relations $s_i^{p_i}=1$ for all $i=1,2,\\ldots$,   where $s_i$ is the homotopy class represented by $\\partial D_i$ and $p_i$ is the order of  $x_i$. For example the fundamental group of   $\\mathcal{O}=S^2(p_1, p_2, p_3)$  has presentation   \n$$\\pi_1^o(\\mathcal{O})=\\langle s_1, s_2, s_3 \\ | \\  s_1^{p_1}=s_2^{p_2}=s_3^{p_3}=s_1s_2s_3=1 \\rangle.$$\n\\begin{figure}[h!]\n\\begin{center}\n\\includegraphics[scale=0.8]{sphere_1.pdf}\n\\end{center}\n\\caption{The generator $s_i$ is represented by a simple closed curve that goes once around the cone point $x_i$.}\\label{fig:almost1}\n\\end{figure} \n \n \n\nIn order to define almost orbifold coverings we first  recall the definition of an orbifold covering. Let $\\mathcal{O}=(F,p)$ and $\\mathcal{O}'=(F', p')$ be orbifolds. An \\emph{orbifold covering} $\\eta:\\mathcal{O}'\\rightarrow \\mathcal{O}$ is a continuous surjective map $\\eta:F'\\rightarrow F$ \nwith the following properties:\n\\begin{enumerate}\n\\item For each point $y\\in F'$, the order of $y$ divides the order of $\\eta(y)$.\n\n\\item For each point $x\\in \\text{Int}(F)$ the set $\\eta^{-1}(x)\\subseteq F'$ is discrete and, over a small disk in $F$ centered at $x$,  the map $\\eta$ is equivalent to the map \n$$(z, y)\\in \\mathbb{D}^2 \\times \\eta^{-1}(x) \\mapsto e^{( \\frac{2\\pi p'(y)}{p(x)}i ) } z\\in \\mathbb{D}^2$$ \nand $x$ corresponds to $0$ in $\\mathbb{D}^2$.\n\n\n\\end{enumerate} \nNote that the map $\\eta|_{F'- \\eta^{-1}(\\Sigma(\\mathcal{O}))}:F'- \\eta^{-1}(\\Sigma(\\mathcal{O}))\\rightarrow F-\\Sigma(\\mathcal{O})$ is a genuine covering.  The \\emph{degree}  of  $\\eta:\\mathcal{O}'\\rightarrow\\mathcal{O}$ is defined as  the degree of $\\eta|_{F'- \\eta^{-1}(\\Sigma(\\mathcal{O}))}$. It is not hard to see that an orbifold covering $\\eta:\\mathcal{O}'\\rightarrow \\mathcal{O}$ induces a monomorphism $\\eta_{\\ast}:\\pi_1^o(\\mathcal{O}')\\rightarrow \\pi_1^o(\\mathcal{O}).$ \nConversely, for any subgroup $H\\leq \\pi_1^o(\\mathcal{O})$ there is an orbifold covering $\\eta:\\mathcal{O}_{H}\\rightarrow  \\mathcal{O}$ such that $\\eta_{\\ast}(\\pi_1^o(\\mathcal{O}_{H}))=H$.  \n\n\n  \n\\begin{definition*}[Almost orbifold covering]{\\label{def:almost}}\nLet $\\mathcal{O}'=(F', p')$ and $\\mathcal{O}=(F, p)$ be compact  orbifolds.  An \\emph{almost orbifold covering} $\\eta:\\mathcal{O}'\\rightarrow \\mathcal{O}$ is a (non necessarily surjective) continuous map $\\eta:F'\\rightarrow F$ having the following properties:\n\\begin{enumerate}\n\\item[(C1)] For each $y\\in F'$ the order of $y$ divides the order  of $\\eta(y)$. \n\n\\item[(C2)] There is a point $x\\in\\text{Int}(F)$ of order $p\\geqslant 1$,  called the \\emph{exceptional point}, and a disk ${D}\\subseteq\\text{Int}(F)$ centered at $x\\in F$, called the \\emph{exceptional disk}, with \n$$(D-\\{x\\})\\cap \\Sigma(\\mathcal{O})=\\emptyset$$  \nsuch that $\\eta$ restricted to $F'-\\eta^{-1}(\\text{Int}(D))$ defines  an orbifold covering of finite degree between the compact $2$-orbifolds  \n$$\\mathcal{Q}':=(F'-\\eta^{-1}(\\text{Int}(D)), p'|_{F'-\\eta^{-1}(\\text{Int}(D))})\\subseteq \\mathcal{O}'$$ and $$\\mathcal{Q}:=(F-\\text{Int}(D), p|_{F-\\text{Int}(D)})\\subseteq\\mathcal{O}.$$\n\n\\item[(C3)] $\\eta^{-1}(D)=D_1\\sqcup D_2\\sqcup\\ldots\\sqcup D_t\\sqcup C_1\\sqcup \\ldots \\sqcup C_u$, where $t\\geqslant 0$, $u\\geqslant 1$  and  \n\\begin{enumerate}\n\\item[(C3.a)]  $C_1, \\ldots, C_u\\subseteq \\partial F'$ are boundary components of $\\mathcal{O}'$, called the \\emph{exceptional boundary components} of $\\mathcal{O}'$.\n\n\\item[(C3.b)] Each $D_j\\subseteq \\text{Int}(F')$ ($1\\leqslant j\\leqslant t$) is a disk and $\\eta|_{D_j}:D_j\\rightarrow D$ is equivalent to the map\n$$z\\in \\mathbb{D}^2\\longmapsto e^{\\frac{2\\pi p}{q}i}  z\\in \\mathbb{D}^2$$\nand  $x\\in D$ corresponds to $0$ in $\\mathbb{D}^2$, where  $q$ is the order of the  point $\\eta^{-1}(x)\\cap D_j$.  \n\\end{enumerate}\n\n\\end{enumerate}\nThe degree of $\\eta:\\mathcal{O}'\\rightarrow \\mathcal{O}$ is defined as the degree of the orbifold covering  $$\\eta|_{\\mathcal{Q}'}:\\mathcal{Q}'\\rightarrow \\mathcal{Q}.$$  \nWe further call an almost orbifold covering   $\\eta :\\mathcal{O}'\\rightarrow \\mathcal{O}$   \\emph{special} if $u=1$ (i.e.   $\\mathcal{Q}'$  has a single exceptional boundary component) and  the degree of the map  $\\eta|_{C_1}:C_1\\rightarrow \\partial D$ is at most  the order $p$ of the exceptional point $x$.\n\\end{definition*} \n\n  \n\n\\begin{example}{\\label{ex:almost}}\nLet  $\\mathcal{O}=(F,p)$ be a closed orbifold and  $x$ an arbitrary  point of $F$. Let further   $D\\subseteq F$ be a small disk centered at $x$  such that $D$ contains no cone points except possibly  $x$. Denote by $\\mathcal{O}_0$ the orbifold $( F-\\text{Int}(D), p|_{F-\\text{Int}(D)})$ and by $j$ the inclusion map $ F-\\text{Int}(D) \\hookrightarrow F$. \n\nAny orbifold covering  $\\eta':\\mathcal{O}'\\rightarrow \\mathcal{O}_0$    of finite degree induces an almost orbifold covering  of $\\mathcal{O}$,  namely the map  $\\eta:= j\\circ \\eta':F'\\rightarrow F.$\n Note that the  exceptional point is   $x$,  the exceptional disk is $D$ and \n$$\\eta^{-1}(D)=\\partial \\mathcal{O}'= C_1\\sqcup \\ldots \\sqcup C_u$$  are  the exceptional boundary components of $\\eta$.  When   the degree of $\\eta'$ is one, an hence the degree of $\\eta$ is also   one,  then  $\\mathcal{O}'=\\mathcal{O}_0$.  In this case   we say that $\\eta$   is   trivial.\t\n\\end{example}\n\n \n \n\n\\section{Proof of Theorem~\\ref{mainthm}}\nThroughout  this section $\\mathcal{O}$ will denote the orbifold $S^2(p_1,p_2,p_3)$   and $G$ its fundamental  group. In \\cite{Matelski} (see also Knapp \\cite{Knapp} for the particular case of  generating pairs consisting  of elliptic elements and \\cite{Fine} for a more general discussion)  Matelski  implicitly shows that   if  $T$ is  a generating pair of $G$ (the arguments in \\cite{Matelski} work also  for non-hyperbolic triangle groups)  then  $T$ is equivalent to $T'$ where one of the following holds:\n\\begin{enumerate}\n\\item $T'=(s_{i_1}^{\\nu_1},s_{i_2}^{\\nu_2})$ with  $1\\leqslant i_1<i_2\\leqslant 3$ and $(p_{i_j}, \\nu_j)=1$ for $j=1,2$.\n\n\n\\item $p_1=2$,  $p_3\\geqslant 3$ odd and  $T'=( s_2^{\\nu}, s_1 s_2^{\\nu'}  s_1^{-1})= ( s_2^{\\nu}, s_1  s_2^{\\nu'} s_1 )$  with  $\\nu $ and $\\nu'$ coprime with $p_2$. \n \n\\item  $p_1=2$, $p_2=3$  and $p_3\\geqslant 3$ is odd,   and   $T'=(s_1, s_2^{-1}   \\cdot  s_3^{\\nu}  \\cdot  s_2)$ with $(\\nu, p_3)=1$.\n\n \n\\item $p_1=2$, $p_2=3$ and $p_3\\geqslant 4$  coprime with  $3$, and   $$T'=( s_1s_2s_1 \\cdot  s_3^{\\nu} \\cdot (s_1s_2s_1)^{-1}, s_2) $$   with $\\nu$ coprime with  $p_3$.\n\n\\item $p_1=2$, $p_2=3$ and $p_3\\geqslant 5$  coprime with  $4$, and   \n$$T'=( (s_2s_1)^2 \\cdot  s_3^{\\nu}\\cdot    (s_2s_1)^{-2}, s_3^{\\nu'})$$ \nwith $ \\nu$ and $\\nu'$ coprime with  $p_3$.\n\n\n\\item  $(p_1,p_2,p_3)=(2,3,5)$, and one of the following holds:\n\\begin{enumerate}\n\\item[(6.a)]  $T'=( s_1, (s_2^2s_1)^2\\cdot  s_2 \\cdot  (s_2^2s_1)^{-2})$. \n \n\\item[(6.b)]  $T'=(s_2\\cdot s_1 \\cdot s_2^{-1}, (s_1s_2)^2 s_1\\cdot   s_3^{\\nu}\\cdot   s_1^{-1} (s_1s_2)^{-1})$ with  $\\nu \\in \\{1, 2\\}$.\n \n \n \\item[(6.c)]  $T'=( s_2s_1 \\cdot s_2 \\cdot  (s_2s_1)^{-1}, s_3^3  s_1\\cdot    s_3^{\\nu} \\cdot   (s_3^3s_1)^{-1}) $ with $\\nu\\in \\{1, 2\\}$.\n \n \n \\item[(6.d)] $T'=(s_2  s_1 \\cdot  s_2  \\cdot (s_2s_1)^{-1}, (s_1s_2)^2 s_1\\cdot   s_3^{\\nu}\\cdot    s_1^{-1} (s_1s_2)^{-1})$. \n\\end{enumerate}\n \n\n\\item $(p_1,p_2,p_3)=(2,3,7)$, and  one of the following holds:   \n\\begin{enumerate}\n\\item[(7.a)] $T'=(s_2 ,  s_3^{-3} s_1 \\cdot    s_3^{\\nu} \\cdot  s_1    s_3^3)$ with $ \\nu$ coprime with $7$\n\n\\item[(7.b)] $T'= ((s_1s_2^2s_1s_2)^2s_2s_1s_2, (s_2(s_2s_1)^2)^2 s_2s_1) $.\n\\end{enumerate}\n\n \n\\item $p_1=p_2=3$ and $p_3\\geqslant 4$  coprime with $3$,  and   $$T'= (s_1s_2^2, s_2^2s_1).$$  \n\n \n\n\\item $p_1=2$, $p_2=4$ and $p_3\\geqslant 7$ odd, and \n$$T'=( s_1s_2^2,  s_2^3  s_1  s_2^3).$$ \n\n\n\\item $p_1=2$, $p_2=3$ and $p_3\\geqslant 7$ coprime with  $6$, and  $$T'=(s_1s_2s_1s_2^2, s_2^2s_1s_2s_1).$$\n\\end{enumerate}\nNote that the list (1)-(10) above does not give a classification of generating pairs, that is, pairs from distinct items might be Nielsen equivalent.  For example the   generating  pair $T'=(s_2^{\\nu}, s_1s_2^{\\nu'}s_1^{-1})$ given in (2) with $\\nu=\\nu'=1$   is Nielsen equivalent to the standard pair  $(s_1, s_3^2)$.  For a complete classification of generating pairs of triangle groups see \\cite{Fine}.  \n \nFor  each   generating pair given in  (1)-(10) we will  describe a  special almost orbifold covering that represents $T$.  First note that standard generating tuples, that is, those given in   (1),  are represented by trivial almost orbifold coverings, see Example~\\ref{ex:almost} and  \\cite[Example 1.9]{Dutra}.  Thus  we are left with the generating pairs given in  items  (2)-(10). \n\n\\smallskip \n\n\nLet $D\\subseteq S^2$ be a disk centered at the   cone point  $x_3$  that does not contain $x_1$ and $x_2$ as shown in Fig.~\\ref{fig:case0}. Denote the orbifold obtained from $\\mathcal{O}$ by removing the interior of  $D$  by $\\mathcal{O}_0$. Then $\\mathcal{O}_0$  is a disk  $D_0:=S^2-\\text{Int}(D)$ with two cone points $x_1$ and $x_2$ of order $p_1$ and $p_2$ respectively. Consequently  $\\pi_1^o(\\mathcal{O}_0)$ has the following presentation\n$$ \\langle S_1 , S_2 \\ | \\ S_1^{p_1}=S_2^{p_2}=1\\rangle.$$  \nThe  inclusion map $i:D_0\\rightarrow S^2$  induces a surjective  homomorphism   $$i_{\\ast} :\\pi_1^{o}(\\mathcal{O}_0)\\twoheadrightarrow \\pi_1^o(\\mathcal{O})$$\n that sends $S_i$ onto $s_i$ ($i=1,2$)  and $S_1S_2$ onto $s_3^{-1}$. \n\nLet further  $\\alpha$ and $\\beta$    be arcs in $S^2$  with $\\partial \\alpha =\\{x_1, x_3\\} $ and  $\\partial \\beta= \\{x_2, x_3\\}$  as shown in   Fig.~\\ref{fig:case0}.   Let further $\\alpha_0$ and  $\\beta_0$ be the arcs $D_0\\cap \\alpha $ and $ D_0\\cap \\beta$  in $D_0$. We also  label the cone point   $x_1$ (of order $p_1$) by a small square an the cone point $x_2$ (of order $p_2$) by a small disk. The  arcs $\\alpha$, $\\alpha_0$, $\\beta$ and $\\beta_0$   and  the labels of the points  $x_1$ and $x_2$  will help us to visualize the defined orbifold maps. \n\\begin{figure}[h!]\n\\begin{center}\n\\includegraphics[scale=0.8]{graph_gamma.pdf}\n\\end{center}\n\\caption{ The  arcs  $\\alpha $ and $ \\beta$ in $S^2$  and the orbifold $\\mathcal{O}_0=D_0(p_1,p_2)\\subseteq \\mathcal{O}$.}\\label{fig:case0}\n\\end{figure} \n\n \n\nIn all cases we will describe an almost orbifold covering $\\mathcal{O}'\\rightarrow \\mathcal{O}$  where the   orbifold $\\mathcal{O}'$  is either a disk with two cone points   or a torus with no cone points that has an open disk removed. Moreover, in all cases   $x_3$ will be  the exceptional point and $D$  the exceptional disk.\n\n \\smallskip \n\n\\noindent \\textbf{Case (2).} Recall that $p_1=2$ and that  $p_3\\geqslant 3$ is odd. The pair $T'$ is equal to $(s_2^{\\nu} , s_1s_2^{\\nu'}s_1^{-1})$ with $(\\nu, p_2)=(\\nu', p_2)=1$. \n\nConsider the surjecitve  map $\\eta':D^2\\twoheadrightarrow D_0$ described in Fig.~\\ref{fig:case2}, that is, $\\eta'(\\alpha_1)=\\eta'(\\alpha_2)=\\alpha_0$, $\\eta'(\\beta_1)=\\eta'(\\beta_2)=\\beta_0$ and $\\eta'$  maps each component of $$D^2-\\alpha_1\\cup \\alpha_2\\cup \\beta_1\\cup \\beta_2$$ homeomorphically onto $D_0-\\alpha_0\\cup   \\beta_0$.\n \n Denote by  $\\mathcal{O}'$   the orbifold with underlying surface $ D^2$   and with cone points $y_1$ and $y_2$ both of order $p_2$. Thus $\\mathcal{O}'=D^2(p_2,p_2)$.  Then  it is not hard to see that $\\eta'$ defines an orbifold covering of degree $2$  from $\\mathcal{O}'$ onto $\\mathcal{O}_0\\subseteq \\mathcal{O}$  that corresponds to the subgroup $\\langle S_2 , S_1S_2S_1^{-1}\\rangle$ of $\\pi_1^o(\\mathcal{O}_0)$. As $\\eta_{\\ast}':\\pi_1^o(\\mathcal{O}')\\rightarrow \\pi_1^o(\\mathcal{O}_0)$ is 1-1 we  may assume that $$\\pi_1^o(\\mathcal{O}')=\\langle S_2 , S_1S_2S_1^{-1}\\rangle.$$ \n\nIt follows from Example~\\ref{ex:almost} that the map $\\eta:=i\\circ \\eta':D^2\\rightarrow S^2$ defines an almost orbifold covering from $\\mathcal{O}'$ to $\\mathcal{O}$. As the degree of $\\eta|_C:C\\rightarrow \\partial D$ is equal to two, which is strictly smaller than $p_3$, we conclude that $\\eta$ is   special. Moreover, \n$$ \\eta_{\\ast}(S_2^{\\nu})=i_{\\ast}(S_2^{\\nu})= s_2^{\\nu} \\ \\  \\text{ and } \\ \\   \\eta_{\\ast}(S_1S_2^{\\nu'}S_1^{-1})=i_{\\ast}(S_1S_2^{\\nu'}S_1^{-1})=s_1s_2^{\\nu'}s_1^{-1}$$ \nwhich means that $\\eta$ represents the generating pair $T'$.\n\\begin{figure}[h]\n\\begin{center}\n\\includegraphics[scale=0.8]{sphere_case_2.pdf}\n\\end{center}\n\\caption{ $\\eta=\\eta'\\circ i:D^2\\rightarrow S^2$ defines a special almost orbifold covering of degree $2$. }\\label{fig:case2}\n\\end{figure} \n  \n\\smallskip \n\n \n\\noindent\\textbf{Case  (3).} Recall that in this case we have $p_1=2$, $p_2=3$ and $p_3\\geqslant 3$ odd, and  $T'=(s_1, s_2^{-1}    s_3^{\\nu}   s_2)$ with $(\\nu, p_3)=1$.  \nLet $\\eta:D^2\\rightarrow S^2$ be the map  described in Fig.~\\ref{fig:case3}, that is,   $\\eta(\\alpha_1)=\\eta(\\alpha_2)=\\alpha_0$, $\\eta(\\alpha_3)=\\alpha$, $\\eta (\\beta_1)=\\eta (\\beta_2)=\\beta_0$,  $\\eta (\\beta_3)=\\beta$ and $\\eta $ maps each component of \n$$D^2-\\text{Int}(D_1)\\cup \\alpha_1\\cup\\alpha_2\\alpha_3\\cup\\beta_1\\cup\\beta_2\\cup \\beta_3$$\nhomeomorphically onto $D_0-\\alpha_0\\cup \\beta_0$ and the disk  $D_1$ homeomorphically onto $D$.  \n\n\nDenote by $\\mathcal{O}'$   the orifold  with underlying surface $D^2$ and with two cone pints $y_1$  and $y_2$ of order $p_1=2$ and $p_3$ respectively, i.e. $\\mathcal{O}'=D^2(p_1, p_3)$.   Then  $\\eta$  defines an almost orbifold covering of degree $3$ from $\\mathcal{O}'$ to $\\mathcal{O}$. \n\n\nBy the description of $\\eta$ we see that   $\\eta^{-1}(D)=D_1 \\sqcup  C$.  The restriction of $\\eta$ to the disk $D_1$ defines an orbifold covering of degree $1$ from the orbifold $\\mathcal{Q}'=D_1(p_2)$ (a disk with a single cone point, $y_2$ ,  order $p_3$)   onto  $\\mathcal{Q}=D(p_3)$ (a disk with a single cone point, $x_3$ , of order $p_3$).  Moreover,   the exceptional boundary component $C$ is mapped  by $\\eta$ onto $\\partial D$ with degree $2$ which implies that   $\\eta$ is a special almost orbifold covering. \n\\begin{figure}[h!]\n\\begin{center}\n\\includegraphics[scale=0.8]{sphere_case_3.pdf}\n\\end{center}\n\\caption{ Case \\textbf{(3)}.  $\\eta:D^2\\rightarrow S^2$ defines a special almost orbifold covering of degree $3$.  }\\label{fig:case3}\n\\end{figure} \n\nPut $F:= D^2-\\text{Int}(D_1)$. It is easy to see that  the map \n$$\\eta':=\\eta|_{F}:F:=D^2-\\text{Int}(D_1)\\twoheadrightarrow D_0$$  defines an orbifold covering of degree $3$ from the orbifold $\\mathcal{O}_0'=F(p_1)$ (an annulus  with a single cone point $y_1$ of order $p_1=2$) onto the orbifold $\\mathcal{O}_0\\subseteq \\mathcal{O}$ and that   $\\eta'$ corresponds to the subgroup  $\\langle  S_1 , S_2^{-1}\\cdot  S_1S_2\\cdot  S_2\\rangle$ of $\\pi_1^o(\\mathcal{O}_0)$. Thus we may assume that   \n$$\\pi_1^o(\\mathcal{O}_0')=\\langle S_1 , S_2^{-1}\\cdot S_1S_2\\cdot S_2\\rangle.$$\nThe inclusion map $j:F\\hookrightarrow D^2$  induces a surjective homomorphism $$j_{\\ast}:\\pi_1^o(\\mathcal{O}_0')\\rightarrow \\pi_1^o(\\mathcal{O}').$$ Consequently $\\pi_1^o(\\mathcal{O}')$ is generated by  \n$$t_1:=j_{\\ast}(S_1) \\ \\ \\text{ and } \\ \\ t_2:=j_{\\ast}(S_2^{-1}\\cdot S_1S_2 \\cdot S_2).$$     \nAs  $\\eta_{\\ast}\\circ j_{\\ast}=i_{\\ast}\\circ \\eta_{\\ast}'$ and   $i_{\\ast}(S_1S_2)=s_3$,  we see that $\\eta_{\\ast}$ sends the generating pair $(t_1, t_2^{-\\nu})$  of $\\pi_1^o(\\mathcal{O}')$   onto the pair $T'$, and hence  the special almost orbifold covering $\\eta:\\mathcal{O}'\\rightarrow \\mathcal{O}$ represents the generating pair $T'$. \n\n\n\\smallskip \n\n\\noindent \\textbf{Case (4).}  In this case we have $p_1=2$, $p_2=3$ and $p_3\\geqslant 4$ with $(p_3, 3)=1$, and \n$$T'=( s_1s_2s_1 \\cdot   s_3^{\\nu} \\cdot   (s_1s_2s_1)^{-1}, s_2) $$ \nwith $(\\nu, p_3)=1$. Consider the map   $\\eta:D^2\\rightarrow S^2$   described in Fig.~\\ref{fig:case4} (its precise description is given as in   case (3)) and let $\\mathcal{O}'$ be the orifold  with underlying surface $D^2$ and with two cone points $y_1$  and $y_2$ of order $p_2=3$ and $p_3$ respectively, i.e. $\\mathcal{O}'=D^2(p_2, p_3)$.   Then $\\eta$  defines an almost orbifold covering of degree $4$ from $\\mathcal{O}'$ to $\\mathcal{O}$. \n \\begin{figure}[h!]\n\\begin{center}\n\\includegraphics[scale=0.8]{sphere_case_4.pdf}\n\\end{center}\n\\caption{ Case \\textbf{(4)}.  $\\eta:D^2\\rightarrow S^2$ defines a special almost orbifold covering of degree $4$. }\\label{fig:case4}\n\\end{figure}\n\nThe description of $\\eta$  implies  that   $\\eta^{-1}(D)=D_1  \\sqcup  C$.  The restriction of $\\eta$ to the disk $D_1$ defines an orbifold covering of degree $1$ from the orbifold $\\mathcal{Q}'=D_1(p_2)$ (a disk with a single cone point $y_2$  order $p_3$)   onto  $\\mathcal{Q}=D(p_3)$ (a disk with a single cone point $x_3$  of order $p_3$).    Moreover,   the exceptional boundary component $C$ is mapped  by $\\eta$ onto $\\partial D$ with degree $3$,  and so $\\eta$ is a special almost orbifold covering. \n \n \nThe restriction $\\eta'$  of $\\eta$ to the  surface  $F:=D^2-\\text{Int}(D_1) \\subseteq D^2$ defines an orbifold covering of degree $4$ from the orbifold $\\mathcal{O}_0'=F(p_2)$ (an annulus  with a single cone point $y_1$ of order $p_2=3$) onto the orbifold $\\mathcal{O}_0\\subseteq \\mathcal{O}$ such that \n$$\\eta_{\\ast}'(\\pi_1^o(\\mathcal{O}_0'))= \\langle S_1S_2S_1   \\cdot S_1S_2  \\cdot S_1S_2^2S_1\n, S_2\\rangle. $$ \nThus we may assume that  $\\pi_1^o(\\mathcal{O}_0')=\\langle S_1S_2S_1   \\cdot S_1S_2  \\cdot S_1S_2^2S_1 , S_2\\rangle.$ \n\nNow the inclusion map $j:F\\hookrightarrow D^2$  induces a surjective homomorphism $j_{\\ast}$ from $\\pi_1^o(\\mathcal{O}_0')$ onto $\\pi_1^o(\\mathcal{O}').$ Consequently $\\pi_1^o(\\mathcal{O}')$ is generated by  $$t_1:=j_{\\ast}(S_1S_2S_1   \\cdot S_1S_2  \\cdot S_1S_2^2S_1) \\ \\ \\text{ and } \\ \\  t_2:=j_{\\ast}(S_2). $$ \nSince $\\eta_{\\ast}\\circ j_{\\ast}=i_{\\ast}\\circ \\eta_{\\ast}'$ and since $i_{\\ast}(S_1S_2)=s_3^{-1}$,  we see that $\\eta_{\\ast}$ sends the generating pair $(t_1^{-\\nu}, t_2)$  of $\\pi_1^o(\\mathcal{O}')$   onto the pair $T'=( s_1s_2s_1   s_3^{\\nu}  (s_1s_2s_1)^{-1}, s_2)$. \n \n \n  \n\\smallskip \n\n \n \n\\noindent \\textbf{Case (5).} Recall that $p_1=2$, $p_2=3$ and $p_3\\geqslant 5$ with $(p_3, 4)=1$, and that   $T'$ is equal to $( (s_2s_1)^2\\cdot   s_3^{\\nu} \\cdot    (s_2s_1)^{-2}, s_3^{\\nu'})$ with $\\nu$ and $\\nu'$ coprime with $p_3$. \n\nConsider the  map $\\eta:D^2\\rightarrow S^2$  described in Fig.~\\ref{fig:case5} (again, its precise definition is given as in case (3))  and let $\\mathcal{O}'$ be the orifold  with underlying surface $D^2$ and with two cone pints $y_1$  and $y_2$  both of  order $p_3$, i.e. $\\mathcal{O}'=D^2(p_3, p_3)$.   Then it is easy to see that  $\\eta$  defines an almost orbifold covering of degree $6$ from $\\mathcal{O}'$ to $\\mathcal{O}$. \n\\begin{figure}[h!]\n\\begin{center}\n\\includegraphics[scale=0.8]{sphere_case_5.pdf}\n\\end{center}\n\\caption{ Case \\textbf{(5)} $\\eta:D^2\\rightarrow S^2$ defines a special almost orbifold covering of degree $6$.}\\label{fig:case5}\n\\end{figure} \n  \n  \nPut $F:=D^2-\\text{Int}(D_1)\\cup \\text{Int}(D_2)$.   It is not hard to see that  the map   $$\\eta':=\\eta|_{F}:F  \\rightarrow D_0$$ defines an orbifold covering of degree $6$ from the orbifold $\\mathcal{O}_0'=F$ (a pair of pants without cone points)  onto the orbifold $\\mathcal{O}_0\\subseteq \\mathcal{O}$.  Moreover,  \n$$\\eta_{\\ast}'(\\pi_1^o(\\mathcal{O}_0'))= \\langle    (S_2S_1)^2 \\cdot S_2^2S_1 \\cdot     (S_2S_1)^{-2}, S_2^2S_1 )  \\rangle.  $$ \nThus  we can assume that    $\\pi_1^o(\\mathcal{O}_0')=\\langle    (S_2S_1)^2 \\cdot S_2^2S_1 \\cdot     (S_2S_1)^{-2}, S_2^2S_1 )  \\rangle$. \n \n \nThe inclusion map $ j:F\\hookrightarrow D^2$  induces  a  surjective homomorphism $j_{\\ast}$ from $\\pi_1^o(\\mathcal{O}_0')$ onto $ \\pi_1^o(\\mathcal{O}')$.  Consequently $\\pi_1^o(\\mathcal{O}')$ is generated by  \n$$t_1:=j_{\\ast}((S_2S_1)^2 \\cdot S_2^2S_1 \\cdot     (S_2S_1)^{-2})  \\ \\ \\text{ and } \\ \\  t_2:=j_{\\ast}(S_2^2S_1).$$ \nAs $\\eta_{\\ast}\\circ j_{\\ast}=i_{\\ast}\\circ \\eta_{\\ast}'$ and since $i_{\\ast}$ maps the element $(S_2^2S_1)$ onto $ s_3$,  we see that  the   pair $(t_1^{-\\nu}, t_2)$  is mapped by $\\eta_{\\ast}$    onto the pair $T'$. \n\n \n\\smallskip\n  \n  \n  \n\\noindent \\textbf{Case (6).}  $(p_1,p_2,p_3)=(2,3,5)$. It is not hard to see that    all generating pairs given in (6.a)-(6.d) are equivalent to those given in (1)-(2).  For example, suppose that    (6.b) holds, that is,     $T'=(s_2s_1s_2^{-1}, (s_1s_2)^2s_1 \\cdot s_3^{\\nu} \\cdot s_1(s_1s_2)^{-2})$ with $\\nu \\in \\{1,2\\}$. If  $\\nu =1$  then we have \n$$ T'\\sim_{NE} (s_1 , s_3^2)$$ \n which is a standard generating pair,  and if $ \\nu=2$ then  we have\n$$T'\\sim_{NE} (s_3^2 , s_1 s_3^2  s_1^{-1})$$ \nwhich is case (2.b) with $p_2=3$ and $p_3=5$. We point out that for all cases (6.a)-(6.d)  it is also possible to construct a special almost orbifold covering $\\eta:\\mathcal{O}'\\rightarrow \\mathcal{O}$   and find a   generating pair $T_{\\mathcal{O'}}$    of $\\pi_1^o(\\mathcal{O}')$ that is mapped by $\\eta_{\\ast}$ onto $T'$.  If we take  case (6.b)   for example,  then  the almost orbifold covering  $\\eta$ described in Figure~\\ref{fig:case_6} represents $\\tau'$. \n\\begin{figure}[h!]\n\\begin{center}\n\\includegraphics[scale=1]{sphere_case_6.pdf}\n\\end{center}\n\\caption{ Case \\textbf{(6.b)}.  $\\eta$ has degree $6$ and $\\mathcal{O}'$ is equal to $D^2(p_1, p_3)$.}\\label{fig:case_6}\n\\end{figure} \n \n\n\\smallskip  \n \n\\noindent \\textbf{Case (7).} $(p_1, p_2, p_3)=(2,3,7)$.  We start with the generating par  given in (7.a), that is, we define  a special  almost orbifold covering  that represents  \n$$T'=(s_2 ,  s_3^{-3} s_1 \\cdot    s_3^{\\nu} \\cdot  s_1    s_3^3)$$ \nwhere $(\\nu, 7)=1$. \n\n\nLet $\\eta:D^2\\rightarrow S^2$ be the map described in Fig.~\\ref{fig:case7a} and let $\\mathcal{O}'$ be the orbifold  with underlying surface $D^2$ and with two cone pints $y_1$  and $y_2$ of order $p_2=3$ and $p_3=7$ respectively, that is, $\\mathcal{O}'=D^2(3,  7)$.   Then $\\eta$  defines an almost orbifold covering of degree $10$ from $\\mathcal{O}'$ to $\\mathcal{O}$. \n\nThe description of $\\eta$ implies that  $\\eta^{-1}(D)=D_1\\sqcup D_2 \\sqcup  C$ and that the restriction of $\\eta$ to the disk $D_1$    defines an orbifold covering of degree $7$ from the orbifold $\\mathcal{Q}'=D_1$ (a disk without cone points) onto  $\\mathcal{Q}=D(p_3)$ (a disk with a single cone point $x_3$  of order $p_3=7$). Moreover,  $\\eta|_{ D_2}:D_2\\rightarrow D$  defines a  genuine   covering (of spaces) of degree one.   The exceptional boundary component $C$ is mapped  by $\\eta$ onto $\\partial D$ with degree $3$ which is smaller than $p_e=7$. Thus   $\\eta$ is a special almost orbifold covering. \n\\begin{figure}[h!]\n\\begin{center}\n\\includegraphics[scale=0.8]{sphere_case_7_a.pdf}\n\\end{center}\n\\caption{ Case \\textbf{(7.a)}. $\\eta:D^2\\rightarrow S^2$ defines a special almost orbifold  covering of degree $10$.}\\label{fig:case7a}\n\\end{figure} \n\n\nIt is not hard to see that    the restriction $\\eta'$  of $\\eta$ to the  surface  $$P:=D^2-\\text{Int}(D_1)\\cup \\text{Int}(D_2)\\subseteq D^2$$  \ndefines an orbifold covering of degree $10$ from the orbifold $\\mathcal{O}_0'=P(p_2)$ (a pair of pants with a single cone point $y_1$ of order $p_2=3$) onto the orbifold $\\mathcal{O}_0\\subseteq \\mathcal{O}$.   Moreover,  we have  $\\eta_{\\ast}'(\\pi_1^o(\\mathcal{O}_0'))= \\langle  T_1, T_2, T_1T_2\\rangle $ \nwhere $T_1= S_2$, \n$$T_2= S_1S_2S_1S_2S_1S_2^2 \\cdot S_1S_2 \\cdot S_2S_1 S_2^2S_1S_2^2S_1$$ \nand $T_3=(S_1S_2)^7$. By the injectivity of $\\eta_{\\ast}'$   we can assume that  $\\pi_1^o(\\mathcal{O}_0')=\\langle T_1 , T_2 ,T_3\\rangle.$\nUsing the relations in $G$  we see that  $i_{\\ast} \\circ \\eta_{\\ast}' (T_1)=s_2$, that \n\\begin{eqnarray}\ni_{\\ast} \\circ \\eta_{\\ast}' (T_2) & = &  s_1s_2s_1s_2s_1s_2^2 \\cdot s_1s_2 \\cdot s_2s_1 s_2^2s_1s_2^2s_1 \\nonumber \\\\ \n          & = & s_3^{-3} s_1 \\cdot  s_3^{-1} \\cdot s_1  s_3^3 \\nonumber\n\\end{eqnarray} \nand that  $i_{\\ast} \\circ \\eta_{\\ast}'(T_3)=(s_1s_2)^7=s_3^{-7}=1.$ The inclusion map $j:P\\hookrightarrow D^2$ induces a surjective homomorphism $j_{\\ast}:\\pi_1^o(\\mathcal{O}_0')\\rightarrow \\pi_1^o(\\mathcal{O}')$   whose kernel is normally generated by $T_3$.   Consequently $\\pi_1^o(\\mathcal{O}')$ is generated by  \n$$t_1:=j_{\\ast}(T_1) \\ \\ \\text{ and } \\ \\ t_2:=j_{\\ast}(T_2).$$   \n As we clearly have $\\eta_{\\ast}\\circ j_{\\ast}=i_{\\ast}\\circ \\eta_{\\ast}'$ we see that $\\eta_{\\ast}$ sends the generating pair $(t_1, t_2^{\\nu})$  of $\\pi_1^o(\\mathcal{O}')$   onto the pair $T'$.\n\n\\smallskip \n\n \nFor the reaming of this paper  $F$  will denote  the surface  obtained from a torus by removing an open  open disk   and   $\\mathcal{O}'$  will denote  the orbifold with underlying surface $F$ and without cone points, that is, $\\mathcal{O}'$ is isomorphic to the surface $F$.\n\n\n\\smallskip\n\nWe now turn our attention to case (7.b). Thus we must find a special almost orbifold covering that represents  the pair \n$$T'= ((s_1s_2^2s_1s_2)^2s_2s_1s_2, (s_2(s_2s_1)^2)^2 s_2s_1).$$ \nThe map $\\eta:F\\rightarrow S^2$  described  in Fig.~\\ref{fig:case7b}  defines an almost orbifold covering of degree $17$ from the orbifold  $\\mathcal{O}' $ to $\\mathcal{O}$. The description of $\\eta$ implies that\n$$\\eta^{-1}(D)=D_1\\sqcup D_2 \\sqcup  C$$  and that  \n$$\\eta|_{D_i}:D_i\\rightarrow D$$  defines an orbifold covering of degree $7$ from the orbifold $\\mathcal{Q}_i'=D_i$ (a disk without cone points) onto $\\mathcal{Q}=D(p_3)$ (a disk with a single point of order $p_3=7$) for $i=1,2$. As   $\\eta|_{C}:C\\rightarrow \\partial D$ has degree $4$ we conclude that   $\\eta$ is a special almost  orbifold covering.\n\\begin{figure}[h!]\n\\begin{center}\n\\includegraphics[scale=0.8]{sphere_case_7_b.pdf}\n\\end{center}\n\\caption{ Case \\textbf{(7.b)}. $\\eta:F\\rightarrow S^2$ defines a special  almost orbifold covering of degree $17$.}\\label{fig:case7b}\n\\end{figure} \n \nPut $F':=F-\\text{Int}(D_1)\\cup \\text{Int}(D_2)$.  It is not hard to see that the  map   \n$$\\eta':=\\eta|_{F'}: F'  \\rightarrow D_0$$  \n defines an orbifold covering of degree $17$ from the  orbifold $\\mathcal{O}_0'=F'$  onto the orbifold $\\mathcal{O}_0\\subseteq \\mathcal{O}$.  Moreover, we have \n $$\\eta_{\\ast}'(\\pi_1^o(\\mathcal{O}_0'))= \\langle  T_1, T_2, T_3 , T_4, T_5\\rangle \\leq \\pi_1^o(\\mathcal{O}_0)$$ \n where \n\\begin{eqnarray}\nT_1 & = & (S_1S_2^2S_1S_2)^2S_2S_1S_2  \\nonumber \\\\\nT_2 & = &  (S_2(S_2S_1)^2)^2 S_2S_1) \\nonumber \\\\\nT_3 & = &  S_1S_2^2\\cdot (S_1S_2)^7\\cdot S_1S_2^{2} \\nonumber \\\\\nT_4 & = & (S_1S_2)^{3}S_2S_1\\cdot (S_1S_2)^7 S_1S_2 (S_1S_2)^3 \\nonumber \\\\\nT_5 & = &  (S_1S_2)^{7} \\nonumber \n\\end{eqnarray}\nBy the injectivity of $\\eta_{\\ast}'$   we can assume that   $\\pi_1^o(\\mathcal{O}_0')=\\langle T_1 , T_2 ,T_3, T_4, T_5\\rangle.$\nUsing the relations in $G$  we see that  $i_{\\ast} \\circ \\eta_{\\ast}' (T_1)= (s_1s_2^2s_1s_2)^2s_2s_1s_2$, that  $i_{\\ast} \\circ \\eta_{\\ast}' (T_2)= (s_2(s_2s_1)^2)^2 s_2s_1)$ and that $$i_{\\ast} \\circ \\eta_{\\ast}' (T_3)=i_{\\ast} \\circ \\eta_{\\ast}' (T_4)=i_{\\ast} \\circ \\eta_{\\ast}' (T_5)=1.$$\n\n Now the inclusion map    $j: F'\\hookrightarrow F$  induces a surjective homomorphism  $$j_{\\ast}:\\pi_1^o(\\mathcal{O}_0')\\rightarrow \\pi_1^o(\\mathcal{O}')$$    whose kernel is normally generated by $\\{T_3, T_4, T_5\\}$.   Consequently $\\pi_1^o(\\mathcal{O}')$ is generated by  $$t_1:=j_{\\ast}(T_1) \\ \\ \\text{ and } \\ \\  t_2:=j_{\\ast}(T_2).$$   \nSince $\\eta_{\\ast}\\circ j_{\\ast}=i_{\\ast}\\circ \\eta_{\\ast}'$ we see that $\\eta_{\\ast}$ sends the  pair $(t_1, t_2)$   onto the pair $T$, and hence  the special almost orbifold covering $\\eta:\\mathcal{O}'\\rightarrow \\mathcal{O}$ represents  $T'$. \n \n\\smallskip\n\n\\noindent{\\textbf{Case (8).}}  Recall that  $p_1=p_2=3$ and $p_3\\geqslant 4$ with $(p_3, 3)=1$. We must find a special almost orbifold covering that represents the generating pair $T'= (s_1s_2^2, s_2^2s_1).$  \n\nLet $\\eta':F\\twoheadrightarrow D_0\\subseteq S^2$ be the surjective map described in Fig.~\\ref{fig:case9}, that is, $\\eta'(\\alpha_i)=\\alpha_0$, $\\eta'(\\beta_i)=\\beta$ for $1\\leqslant i\\leqslant 3$ and $\\eta'$ maps each component of $$F- \\alpha_1\\cup\\alpha_2\\cup\\alpha_3\\cup \\beta_1\\cup \\beta_2\\cup\\beta_3$$ \nhomeomorphically onto $D_0-\\alpha_0\\cup\\beta_0$. \n\\begin{figure}[h!]\n\\begin{center}\n\\includegraphics[scale=0.8]{sphere_case_9.pdf}\n\\end{center}\n\\caption{ Case \\textbf{(8)}.   $\\eta =i\\circ \\eta' :F\\rightarrow S^2$ defines an almost orbifold covering of degree $3$.}\\label{fig:case9}\n\\end{figure} \n\n\nIt is not hard to see that  $\\eta'$  defines an  orbifold covering   of degree $3$ from $\\mathcal{O}'$ onto  $\\mathcal{O}_0$. Note further  that  the orbifold covering $\\eta'$ corresponds  to the subgroup \n $\\langle  t_1, t_2\\rangle$ of $\\pi_1^o(\\mathcal{O}_0)$ where $t_1= S_1S_2^2$ and $t_2= S_2^2S_1$. Thus   we  can  assume  that $\\pi_1^o(\\mathcal{O}')= \\langle t_1 , t_2\\rangle$.  Moreover,  $\\eta'$  maps the boundary $C$ of $\\mathcal{O}'$ onto  $\\partial  D$  with degree $3$. \n\nIt follows from Example~\\ref{ex:almost} that $\\eta:=i\\circ \\eta':F\\rightarrow S^2$  defines an almost orbifold covering from $\\mathcal{O}'$ to $\\mathcal{O}$. Since $\\eta|_C$ has degree $3$, which is smaller than $p_3$, we conclude that $\\eta$ is a special almost orbifold covering. Now   the generating pair $(t_1, t_2)$ is clearly  mapped by $\\eta_{\\ast}=i_{\\ast}\\circ \\eta_{\\ast}'$ onto the pair $T'$. \n\n\n\\smallskip\n\n\n\\noindent{\\textbf{Case (9).}} In this case   $p_1=2$, $p_2=4$ and $p_3\\geqslant 7$ odd, and  $T'=( s_1s_2^2,  s_2^3  s_1  s_2^3)$.\n\n\nLet $\\eta':F\\twoheadrightarrow D_0\\subseteq S^2$ be the surjective map described in Fig.~\\ref{fig:case10}. Thus  $\\eta'$ defines an  orbifold covering of degree $4$ from $\\mathcal{O}'$ onto  $\\mathcal{O}_0$.  Moreover,  \n$$(\\eta_n')_{\\ast}\\pi_1^o(\\mathcal{O}')=\\langle T_1, T_2\\rangle$$ \nwhere   $t_1= S_1S_2^2$  and $t_2= S_2^3  S_1  S_2^3$, consequently we may assume that $\\pi_1^o(\\mathcal{O}')$ is generated by the pair  $(t_1, t_2)$.  \n\\begin{figure}[h!]\n\\begin{center}\n\\includegraphics[scale=0.8]{sphere_case_10.pdf}\n\\end{center}\n\\caption{ Case \\textbf{(9)}.  $\\eta =i\\circ \\eta ' :F\\rightarrow S^2$ defines a special almost orbifold covering of degree $4$.}\\label{fig:case10}\n\\end{figure}\n\n\n\nIt follows from Example~\\ref{ex:almost} that $\\eta:=i\\circ \\eta':F\\rightarrow S^2$  defines an almost orbifold covering from $\\mathcal{O}'$ to $\\mathcal{O}$ which is special as  the degree of the map  $\\eta|_C$  is equal to $4$ which  is smaller than $p_3$. The generating pair $(t_1, t_2)$  of $\\pi_1^o(\\mathcal{O}'$)   clearly gets mapped by $\\eta_{\\ast}=i_{\\ast}\\circ \\eta_{\\ast}'$ onto the pair $\\tau'$. \n \n\n\\smallskip \n\n\\noindent{\\textbf{Case (10).}} In this case we have $p_1=2$, $p_2=3$ and $p_3\\geqslant 7$ with $(p_3,6)=1$, and  $T'$ is equal to  $(s_1s_2s_1s_2^2, s_2^2s_1s_2s_1).$\n\n\nThe map $\\eta':F\\twoheadrightarrow D_0$  described   in Fig.~\\ref{fig:case8} defines an orbifold covering of degree $6$ from $\\mathcal{O}'$ onto $\\mathcal{O}_0$ such that \n$$\\eta_{\\ast}'(\\pi_1^o(\\mathcal{O}'))  = \\langle S_1S_2S_1S_2^2 ,  S_2^2S_1S_2S_1  \\rangle.  $$\nBy applying  Example~\\ref{ex:almost} we conclude that $\\eta:=i\\circ \\eta'$ defines a almost orbifold covering from $\\mathcal{O}'$ to $\\mathcal{O}$. Moreover it is easy to see that \n$$ \\eta_{\\ast}(S_1S_2S_1S_2^2 ,  S_2^2S_1S_2S_1 )=T'$$  \nwhich concludes the proof. \n\\begin{figure}[h!]\n\\begin{center}\n\\includegraphics[scale=0.8]{sphere_case_8.pdf}\n\\end{center}\n\\caption{ Case \\textbf{(10)}.   $\\eta =i\\circ \\eta ' :F\\rightarrow S^2$ defines a special almost orbifold covering of degree $6$.}\\label{fig:case8}\n\\end{figure}\n \n  \n \n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{sec:introduction}\n\nQuasi-probability distributions such as the Wigner function have been an important tool in quantum mechanics since the early days of the theory \\cite{wigner:1932}. Indeed, the Wigner function lies at the heart of the phase-space representation of quantum mechanics which is tightly connected to classical statistical mechanics and provides an alternative route to quantization \\cite{qm_phase:book}. In classical statistical mechanics, averages of observables can be described as averages taken with respect to some underlying probability distribution. This probability distribution fully describes the system under consideration and reflects the lack of knowledge of the observer. If one has perfect knowledge of the behavior of all degrees of freedom, there would be no need for probability distributions and measurement outcomes could be predicted in a deterministic manner. {This is not possible in quantum mechanics due to the probabilistic nature of the theory.} Similarly to classical statistical mechanics, the fluctuations in quantum mechanics can be encoded in \\textit{quasi}-probability distributions. These distributions can take on negative values which is ultimately a consequence of the non-commutativity of operators, reflecting the fact that a measurement inevitably disturbs all subsequent measurements of observables which do not commute with the first one. Negative quasi-probability distributions therefore reflect the impossibility of accessing observables without disturbing them, indicating a non-classical behavior of the system which requires to take into account the action of the measurement apparatus to predict its outputs.\nThe concept of negative quasi-probability distributions, or \\textit{negativity} in short, is therefore closely related to the concept of contextuality which deals with the dependence of measurement outcomes on the experimental setup \\cite{spekkens:2008,ferrie:2008,ferrie:2009,ferrie:2011}.\n\nSo far, most quasi-probability distributions describe the quantum state of a system. This holds true for the Wigner function and its generalizations to finite dimensional Hilbert spaces \\cite{wootters:1987,gibbons:2004,ferrie:2011}, as well as a number of other distributions used in quantum optics \\cite{cahill:1969,walls:book}. Negativity in these distributions indicates non-classicality of the quantum state. However, a state that exhibits negativity in some distribution might not do so in another one \\cite{lutkenhaus:1995}, implying that non-classicality is not necessarily a general feature but depends on the way a state is interrogated (see also Refs.~\\cite{revzen:2005,spekkens:2008,ferrie:2008,ferrie:2009}). A well studied quasi-probability distribution which does not describe the quantum state of a system but rather its dynamics is the full counting statistics (FCS) \\cite{levitov:1996,nazarov:book2,nazarov:2003,bednorz:2010,clerk:2011,bednorz:2012,hofer:2016}. Negativity in the FCS has recently been connected to a peculiar interference effect and it was shown that systems which do not show any negativity in their Wigner function can still show negativity in the FCS \\cite{hofer:2016}. This further stresses the fact that systems might exhibit completely classical behavior in some aspects, e.g. their instantaneous states, while exhibiting non-classicality in others, e.g. their dynamics which determine the temporal fluctuations within the system. It would therefore be highly desirable to have access to a quasi-probability distribution which encodes only the aspects of interest of a system.\nThis is in marked contrast to Refs.~\\cite{spekkens:2008,ferrie:2008,ferrie:2009,zhu:2016} which consider a full description of states and measurements in terms of quasi-probabilities.\n\nThe goal of the present paper is to introduce a general quasi-probability distribution that can be tailored to the problem at hand and thus provides a general framework for investigating fluctuations of non-commuting observables. Because this distribution is based on the Keldysh formalism, we call it the Keldysh quasi-probability distribution (KQPD). When the observables of interest are the position and momentum operators, the KQPD reduces to the Wigner function. Similarly, when one is interested in the time-integral of an observable, the FCS of the corresponding operator is obtained. Furthermore, the outcome of von Neumann type measurements \\cite{vonneumann:book} can be predicted by convolving the corresponding KQPD with the Wigner function of the detectors [cf.~Sec.~\\ref{sec:vonneumann}]. This convolution takes into account the imprecision as well as the back-action of the performed measurement. An example is provided by the Husimi $Q$-function which is obtained by convolving the Wigner function with Gaussians and predicts the outcomes of simultaneous but imprecise measurements of position and momentum \\cite{stenholm:1992}.\n\nThe rest of the article is structured as follows. In Sec.~\\ref{sec:keldysh}, we introduce the KQPD, discuss the Wigner function, the FCS, and two subsequent spin measurements as examples, and discuss negativity of the KQPD  as an indicator for non-classical behavior. In Sec.~\\ref{sec:vonneumann}, we provide a more physical motivation for investigating the KQPD by connecting it to measurement outcomes which requires the introduction of measurement imprecision and back-action. Before concluding in Sec.~\\ref{sec:conclusions}, we discuss applications of the KQPD in Sec.~\\ref{sec:applications}, including the distribution of thermodynamic work, (anomalous) weak values, and Leggett-Garg inequalities. Throughout this paper, we set $\\hbar=1$.\n\n\n\\section{The Keldysh quasi-probability distribution}\n\\label{sec:keldysh}\nIn this section we briefly introduce the Keldysh formalism, a powerful framework for tackling out-of-equilibrium problems, where the KQPD arises quite naturally through its moment generating function. Here we only touch upon the ideas in the Keldysh formalism which are relevant for our discussion. For a detailed introduction to the subject we refer the reader to Ref.~\\cite{kamenev:book}. We will start by considering the somewhat trivial example of the statistics of a single observable before we generalize the results to the highly non-trivial scenario of multiple, non-commuting observables at different times.\n\n\\subsection{Single observable}\n\nThe main idea that we will be concerned with is the evolution along the closed time contour illustrated in Fig.~\\ref{fig:ctc}, an idea that goes back to Schwinger \\cite{schwinger:1961}. In this section, we are interested in the statistics of a single observable $\\hat{A}$ at time $\\tau$. Its average value can be written as\n\\begin{equation}\n\\label{eq:avga}\n\\begin{aligned}\n\\langle A \\rangle_\\tau &= {\\rm Tr}\\left\\{\\hat{U}(0,\\tau)\\hat{A}\\hat{U}(\\tau,0)\\hat{\\rho}_0\\right\\} \\\\&=\\frac{1}{2}{\\rm Tr}\\left\\{\\hat{U}(0,\\mathcal{T})\\hat{U}(\\mathcal{T},\\tau)\\hat{A}\\hat{U}(\\tau,0)\\hat{\\rho}_0\\right\\} +\\frac{1}{2}{\\rm Tr}\\left\\{\\hat{U}(0,\\tau)\\hat{A}\\hat{U}(\\tau,\\mathcal{T})\\hat{U}(\\mathcal{T},0)\\hat{\\rho}_0\\right\\},\n\\end{aligned}\n\\end{equation}\nwhere $\\hat{U}(t',t)=\\hat{U}^\\dag(t,t')$ is the time-evolution operator from time $t$ to time $t'$ and $\\hat{\\rho}_0$ denotes the density matrix at time $t=0$. Throughout this article, we consider unitary time evolution. Generalizing the results to non-unitary time evolution is in principle straightforward and discussed in App.~\\ref{app:nonunitary}. Writing the density matrix as a mixture of pure states, the first line of the last expression can be interpreted as time evolving the states up to time $\\tau$, evaluating the operator $\\hat{A}$, and evolving the states backward to the initial time $t=0$ [cf.~the left-hand side of Fig.~\\ref{fig:ctc}]. On the second line, we inserted identities $\\mathbb{1}=\\hat{U}(\\tau,\\mathcal{T})\\hat{U}(\\mathcal{T},\\tau)$, extending the contour along which the state is time evolved. We call the resulting contour, from $t=0$ to $t=\\mathcal{T}$ and back again, the Keldysh contour. We can evaluate the operator $\\hat{A}$ either along the forward-in-time or along the backward-in-time branch of the contour. As discussed below, the most natural choice is to treat both contours on an equal footing as illustrated graphically in Fig.~\\ref{fig:ctc}.\n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=\\textwidth]{ctc}\n\\caption{Illustration of the closed time contour. The average $\\langle\\hat{A}\\rangle_\\tau$ can be obtained by evolving the state (or a mixture of states) up to time $\\tau$, where the operator $\\hat{A}$ is evaluated, and back again. Alternatively, one can extend this closed-time contour to the Keldysh contour (from $t=0$ to $t= \\mathcal{T}$ and back again) and evaluate the operator either on the forward-in-time or on the backward-in-time branch.}\n  \\label{fig:ctc}\n\\end{figure}\n\nMore information on the observable $\\hat{A}$ can be obtained by considering its moment generating function given by\n\\begin{equation}\n\\label{eq:momgena1}\n\\Lambda(\\lambda;\\tau)=\\langle e^{-i\\lambda\\hat{A}}\\rangle_\\tau = {\\rm Tr}\\left\\{\\hat{U}(0,\\tau)e^{-i\\frac{\\lambda}{2}\\hat{A}}\\hat{U}(\\tau,\\mathcal{T})\\hat{U}(\\mathcal{T},\\tau)e^{-i\\frac{\\lambda}{2}\\hat{A}}\\hat{U}(\\tau,0)\\hat{\\rho}_0\\right\\}.\n\\end{equation}\nAll higher moments can be obtained from this generating function by taking derivatives with respect to $\\lambda$\n\\begin{equation}\n\\label{eq:momentsa1}\n\\langle \\hat{A}^k\\rangle_\\tau=\\left.i^k\\partial_\\lambda^k\\Lambda(\\lambda;\\tau)\\right|_{\\lambda=0}.\n\\end{equation}\nIn anticipation of the discussion below, we wrote Eq.~\\eqref{eq:momgena1} symmetrically in terms of the Keldysh contour. Taking the derivative with respect to $\\lambda$ at $\\lambda=0$, we recover Eq.~\\eqref{eq:avga}. Since the moment generating function is the average of the exponential of $\\hat{A}$, it corresponds to the Fourier transform of a probability distribution associated to the fluctuations of $\\hat{A}$\n\\begin{equation}\n\\label{eq:probaa1}\n\\Lambda(\\lambda;\\tau)=\\int_{-\\infty}^\\infty dA \\mathcal{P}(A;\\tau)e^{-i\\lambda A},\\hspace{1cm}\\mathcal{P}(A;\\tau)=\\sum_{A'}\\delta(A-A')\\langle A'|\\hat{U}(\\tau,0)\\hat{\\rho}_0\\hat{U}^\\dag(\\tau,0)|A'\\rangle,\n\\end{equation}\nwhere the sum over $A'$ goes over all eigenstates $\\hat{A}|A'\\rangle=A'|A'\\rangle$ and has to be replaced by an integral if $\\hat{A}$ has a continuous spectrum. In the case of a single observable, the KQPD $\\mathcal{P}(A;\\tau)$ is a positive probability distribution which predicts the outcome of projectively measuring $\\hat{A}$ at time $\\tau$.\n\nThe symmetric treatment with respect to the forward-in-time and the backward-in-time branches of the Keldysh contour might seem like an unnecessary complication at this point. Below, we show how such a treatment for multiple observables leads to a time-ordering prescription which is relevant for von Neumann type measurements. Before discussing multiple observables in detail, we express the moment generating function in terms of the classical and quantum fields obtained by a Keldysh rotation [cf.~Eq.~\\eqref{eq:keldyshrot}]. To this end, we assume that the observable $\\hat{A}$ has a discrete spectrum and that its eigenstates span the whole Hilbert space, i.e. $\\mathbb{1}=\\sum_{A}|A\\rangle\\langle A|.$\nWe then divide the time evolution along the Keldysh contour in Eq.~\\eqref{eq:momgena1} into $2N$ steps ($N$ for the forward-, $N$ for the backward-in-time branch) of infinitesimal length $\\delta t$ and insert identities in between the steps. Writing the trace in terms of the eigenstates of $\\hat{A}$, we then find\n\\begin{equation}\n\\label{eq:momgensuma1}\n\\Lambda(\\lambda;\\tau)=\\sum_{\\boldsymbol{A}_+,\\boldsymbol{A}_-}\\delta_{A_\\mathcal{T}^+,A_\\mathcal{T}^-}e^{-i\\frac{\\lambda}{2}(A_\\tau^++A_\\tau^-)}\\mathcal{A}(\\boldsymbol{A}_+)\\mathcal{A}^*(\\boldsymbol{A}_-)\\langle A_0^+|\\hat{\\rho}|A_0^-\\rangle.\n\\end{equation}\nHere, $A_t^{+(-)}$ denotes the state that was inserted at time $t$ on the forward-in-time (backward-in-time) branch, the vectors $\\boldsymbol{A}_\\pm$ gather all $A_t^\\pm$ and the amplitudes along the branches read\n\\begin{equation}\n\\label{eq:amplitudesa1}\n\\mathcal{A}(\\boldsymbol{A}_\\pm)=\\langle A_\\mathcal{T}^\\pm|\\hat{U}(\\mathcal{T},\\mathcal{T}-\\delta t)|A_{\\mathcal{T}-\\delta t}^\\pm\\rangle\\langle A_{\\mathcal{T}-\\delta t}^\\pm|\\cdots |A^\\pm_{\\delta t}\\rangle\\langle A_{\\delta t}^\\pm|\\hat{U}(\\delta t, 0)|A_0^\\pm\\rangle.\n\\end{equation}\nThe Kronecker delta in Eq.~\\eqref{eq:momgensuma1} is due to the fact that the point $t=\\mathcal{T}$ connects the two branches (cf.~Fig.~\\ref{fig:ctc}). We can write the moment generating function in the form of a path integral\n\\begin{equation}\n\\label{eq:pathintegral}\n\\Lambda(\\lambda;\\tau)=\\int \\mathcal{D}[A_t^+,A_t^-]e^{-i\\frac{\\lambda}{2}(A_\\tau^++A_\\tau^-)}e^{iS[A_t^+,A_t^-]},\n\\end{equation}\nwhere the action $S$ is determined by $\\exp(iS[A_t^+,A_t^-])=\\mathcal{A}(\\boldsymbol{A}_+)\\mathcal{A}^*(\\boldsymbol{A}_-)\\langle A_0^+|\\hat{\\rho}|A_0^-\\rangle$\nand the integration measure reads\n$\\int \\mathcal{D}[A_t^+,A_t^-]=\\sum_{\\boldsymbol{A}_+,\\boldsymbol{A}_-}\\delta_{A_\\mathcal{T}^+,A_\\mathcal{T}^-}$.\nFinally, we perform a Keldysh rotation by introducing the new fields \\cite{kamenev:book}\n\\begin{equation}\n\\label{eq:keldyshrot}\nA^{cl}_t=\\frac{1}{2}(A_t^++A_t^-),\\hspace{2cm} A^q_t = A_t^+-A_t^-,\n\\end{equation}\nwhere the superscript stands for classical and quantum respectively. This nomenclature derives from the fact that the forward- and backward-in-time evolutions are equal for classical processes. Using these new integration variables, the path integral in Eq.~\\eqref{eq:pathintegral} reduces to\n\\begin{equation}\n\\label{eq:pathintegral2}\n\\Lambda(\\lambda;\\tau)=\\int \\mathcal{D}[A_t^{cl},A_t^q]e^{-i\\lambda A_\\tau^{cl}}e^{iS[A_t^{cl},A_t^q]}.\n\\end{equation}\nThe corresponding probability distribution then reads\n\\begin{equation}\n\\label{eq:probapathint}\n\\mathcal{P}(A;\\tau)=\\frac{1}{2\\pi}\\int_{-\\infty}^{\\infty}e^{i\\lambda A}d\\lambda\\Lambda(\\lambda;\\tau)=\\int \\mathcal{D}[A_t^{cl},A_t^q]\\delta(A-A_\\tau^{cl})e^{iS[A_t^{cl},A_t^q]}.\n\\end{equation}\nThe moment generating function thus describes the fluctuations of $A_\\tau^{cl}$ and the probability density $\\mathcal{P}(A;\\tau)$ is given by the sum over all paths for which $A_\\tau^{cl}=A$, weighted by the complex exponential of the action in complete analogy to Feynman path integrals \\cite{altland:book}. In the case of a single observable, the quantum field is zero (i.e. $A_\\tau^+=A_\\tau^-$), making its introduction seem like an unnecessary complication. However, when multiple observables are involved, the quantum fields are no longer bound to be zero. Considering the fluctuations of the classical fields then corresponds to a given time-ordering prescription. We now generalize the results of this section to an arbitrary number of (possibly non-commuting) observables at arbitrary times.\n\n\\subsection{Multiple observables}\n\nThe KQPD, denoted by $\\mathcal{P}(\\boldsymbol{A})$, can be defined via its moment generating function which is obtained by generalizing Eq.~\\eqref{eq:momgena1} and reads\n\\begin{equation}\n\\label{eq:kmomgen}\n\\begin{aligned}\n\\Lambda(\\boldsymbol{\\lambda})&\\equiv\\int d\\boldsymbol{\\lambda}e^{-i\\boldsymbol{\\lambda}\\cdot\\boldsymbol{A}}\\mathcal{P}(\\boldsymbol{A})\\\\\n&\\equiv{\\rm Tr}\\left\\{\\hat{\\mathcal{T}}_+\\exp\\left[-\\frac{i}{2}\\int_0^\\mathcal{T} d\\tau\\sum_lf_l(\\tau)\\lambda_l\\hat{A}_l(\\tau)\\right]\\hat{\\rho}_0\\hat{\\mathcal{T}}_-\\exp\\left[-\\frac{i}{2}\\int_0^\\mathcal{T} d\\tau\\sum_lf_l(\\tau)\\lambda_l\\hat{A}_l(\\tau)\\right]\\right\\}\\\\&\n={\\rm Tr}\\left\\{\\hat{\\mathcal{T}}_K\\exp\\left[-\\frac{i}{2}\\int_K d\\tau\\sum_lf_l(\\tau)\\lambda_l\\hat{A}_l(\\tau)\\right]\\hat{\\rho}_0\\right\\}.\n\\end{aligned}\n\\end{equation}\nHere $\\hat{\\mathcal{T}}_{+(-)}$ is the (anti) time-ordering operator, the variables $\\lambda_l$ and $A_l$ are grouped in vectors $\\boldsymbol{\\lambda}$ and  $\\boldsymbol{A}$, and the operators are given in the Heisenberg picture, i.e.\n\\begin{equation}\n\\label{eq:aheisen}\n\\hat{A}_l(t)=\\hat{U}(0,t)\\hat{A}_l\\hat{U}(t,0),\n\\end{equation}\nwhere $\\hat{A}$ denotes the operator in the Schr\u00f6dinger picture (where all considered operators are time-independent). In the last line of Eq.~\\eqref{eq:kmomgen}, $\\hat{\\mathcal{T}}_K$ denotes time-ordering along the Keldysh contour (from $t=0$ to $t=\\mathcal{T}$ and back again). The functions $f_l(\\tau)$ take into account a possibly time-dependent coupling to the observables $\\hat{A}_l$. When using the Keldysh contour, the functions $f_l$ take on the same values on the forward- and the backward-in-time branch.\nThe case of $N$ observables probed instantaneously at different times $\\tau_l$ is obtained by taking $f_l(\\tau_l)=\\delta(t-\\tau_l)$ (for $l\\leq N$) resulting in the moment generating function\n\\begin{equation}\n\\label{eq:momgenmultsub}\n\\Lambda(\\boldsymbol{\\lambda})={\\rm Tr}\\left\\{e^{-i\\frac{\\lambda_N}{2}\\hat{A}_N(\\tau_N)}\\cdots e^{-i\\frac{\\lambda_1}{2}\\hat{A}_1(\\tau_1)}\\hat{\\rho}_0e^{-i\\frac{\\lambda_1}{2}\\hat{A}_1(\\tau_1)}\\cdots e^{-i\\frac{\\lambda_N}{2}\\hat{A}_N(\\tau_N)} \\right\\}.\n\\end{equation}\nThe apparent symmetry between terms on the left and on the right of the density matrix is equivalent to treating the forward- and the backward-in-time branch of the time evolution symmetrically and determines the time-ordering of operators when evaluating higher moments (see below).\n\n\n\\subsection{Examples}\nWe now introduce two well-known examples of the KQPD: the Wigner function and the Full counting statistics (FCS). To recover the Wigner function from the moment generating function in Eq.~\\eqref{eq:kmomgen}, we set\n\\begin{equation}\n\\label{eq:lamsetwig}\n\\sum_lf_l(\\tau)\\lambda_l\\hat{A}_l(\\tau)=\\delta(\\tau)\\left[\\lambda_x\\hat{x}(\\tau)+\\lambda_p\\hat{p}(\\tau)\\right],\n\\end{equation}\nwhich results in\n\\begin{equation}\n\\label{eq:wigmom}\n\\Lambda(\\lambda_x,\\lambda_p)={\\rm Tr}\\left\\{e^{-\\frac{i}{2}(\\lambda_x\\hat{x}+\\lambda_p\\hat{p})}\\hat{\\rho}_0e^{-\\frac{i}{2}(\\lambda_x\\hat{x}+\\lambda_p\\hat{p})}\\right\\},\n\\end{equation}\nwhere we identified $\\hat{x}=\\hat{x}(0)$ and similarly for $\\hat{p}$. It is straightforward to verify that the Wigner function is indeed given by\n\\begin{equation}\n\\label{eq:wigner}\n\\mathcal{W}(x,p)=\\frac{1}{(2\\pi)^2}\\int d\\lambda_xd\\lambda_p e^{i\\lambda_x x+i\\lambda_p p} \\Lambda(\\lambda_x,\\lambda_p)=\\frac{1}{2\\pi}\\int d\\lambda_pe^{-i\\lambda_p p}\\left\\langle x+\\frac{\\lambda_p}{2}\\right|\\hat{\\rho}_0\\left|x-\\frac{\\lambda_p}{2}\\right\\rangle.\n\\end{equation}\nThe Wigner function is thus recovered as the KQPD corresponding to a simultaneous observation of $\\hat{x}$ and $\\hat{p}$. In the next section, we discuss how the Wigner function can be used to predict a simultaneous (but imprecise) measurement of position and momentum \\cite{arthurs:1964}. The moments obtained from the Wigner function\n\\begin{equation}\n\\label{eq:momwig}\n\\langle x^n p^m\\rangle_W=\\int dx dp \\,x^np^m \\mathcal{W}(x,p),\n\\end{equation}\nare determined by the Weyl order \\cite{carmichael:book} obtained by complete symmetrization, i.e. every possible order is given the same weight, e.g.\n\\begin{equation}\n\\label{eq:weylord}\n\\langle x p\\rangle_W=\\frac{1}{2}{\\rm Tr}\\left\\{(\\hat{x}\\hat{p}+\\hat{p}\\hat{x})\\hat{\\rho}_0\\right\\},\\hspace{1cm}\n\\langle x^2p\\rangle_W=\\frac{1}{3}{\\rm Tr}\\left\\{(\\hat{x}^2\\hat{p}+\\hat{x}\\hat{p}\\hat{x}+\\hat{p}\\hat{x}^2)\\hat{\\rho}_0\\right\\}.\n\\end{equation}\nThis is a general result for the KQPD; when considering multiple operators at equal times, the moments generated by the KQPD are determined by the quantum averages of the completely symmetrized operators.\n\nThe next example we consider in this section is the FCS which is obtained from the moment generating function in Eq.~\\eqref{eq:kmomgen} upon setting\n\\begin{equation}\n\\label{eq:lamsetfcs}\n\\sum_lf_l(\\tau)\\lambda_l\\hat{A}_l(\\tau)=\\lambda \\hat{A}(\\tau),\n\\end{equation}\nwhich results in\n\\begin{equation}\n\\label{eq:momgenfcs}\n\t\\Lambda(\\lambda)={\\rm Tr}\\left\\{\\hat{\\mathcal{T}}_+e^{-i\\frac{\\lambda}{2}\\int_{0}^{\\mathcal{T}}d\\tau\\hat{A}(\\tau)}\\hat{\\rho}_0 \\hat{\\mathcal{T}}_-e^{-i\\frac{\\lambda}{2}\\int_{0}^{\\mathcal{T}}d\\tau\\hat{A}(\\tau)}\\right\\}={\\rm Tr}\\left\\{\\hat{\\mathcal{T}}_Ke^{-i\\frac{\\lambda}{2}\\int_Kd\\tau\\hat{A}(\\tau)}\\hat{\\rho}_0\\right\\}.\n\\end{equation}\nThe FCS is given by the Fourier transform of the last expression\n\\begin{equation}\n\\label{eq:fcs}\n\\mathcal{F}(m)\\equiv\\frac{1}{2\\pi}\\int d\\lambda e^{i\\lambda m}\\Lambda(\\lambda),\n\\end{equation}\nand gives information on the fluctuations of the time integral of $\\hat{A}$. The FCS has been investigated intensively in the context of electron transport \\cite{nazarov:book2}, where $\\hat{A}$ corresponds to the current operator and one is interested in the fluctuations of the charge that is transferred through a conductor during a time interval $[0,\\mathcal{T}]$.\n\nThe moments of the FCS can conveniently be expressed using the Keldysh contour\n\\begin{equation}\n\\label{eq:momentsfcs}\n\\int dm\\, m^k\\mathcal{F}(m)=i^k\\partial_\\lambda^k\\Lambda(\\lambda)\\big|_{\\lambda=0}=\\int_Kd\\tau_1\\cdots d\\tau_k{\\rm Tr}\\left\\{ \\hat{\\mathcal{T}}_K\\hat{A}(\\tau_1)\\cdots \\hat{A}(\\tau_k) \\hat{\\rho}_0\\right\\}.\n\\end{equation}\nWe thus find that the time-ordering corresponding to the KQPD is determined by the \\textit{Keldysh time-ordering}, implying that operators are ordered along the Keldysh contour.\nThe Wigner function as well as the FCS have been discussed extensively in the literature. Explicit evaluations exhibiting negativity can be found in, e.g., Refs.~\\cite{kenfack:2004,hofer:2016}.\n\nTo further illustrate the KQPD, we consider the very simple example of two subsequent spin measurements. The operators of interest are given by the rotated Pauli matrices $\\hat{\\sigma}_1$ and $\\hat{\\sigma}_2$. For simplicity, we consider a qubit in a pure state $|+_0\\rangle$ which is an eigenstate of $\\hat{\\sigma}_0$ with eigenvalue $+1$. The KQPD then reads\n\\begin{equation}\n\\label{eq:kqpdspins}\n\\mathcal{P}(\\Sigma_1,\\Sigma_2)=\\frac{1}{(2\\pi)^2}\\int d\\lambda_1d\\lambda_2e^{i\\lambda_1\\Sigma_1+i\\lambda_2\\Sigma_2}{\\rm Tr}\\left\\{e^{-i\\frac{\\lambda_2}{2}\\hat{\\sigma}_2}e^{-i\\frac{\\lambda_1}{2}\\hat{\\sigma}_1}|+_0\\rangle \\langle +_0|e^{-i\\frac{\\lambda_1}{2}\\hat{\\sigma}_1}e^{-i\\frac{\\lambda_2}{2}\\hat{\\sigma}_2}\\right\\}.\n\\end{equation}\nWriting the eigenstates of $\\hat{\\sigma}_j$ in the eigenbasis of $\\hat{\\sigma}_1$\n\\begin{equation}\n\\label{eq:basis}\n|+_0\\rangle = \\alpha|+_1\\rangle +\\beta|-_1\\rangle\\hspace{2cm}|+_2\\rangle = \\gamma|+_1\\rangle +\\delta|-_1\\rangle,\n\\end{equation}\nwe can evaluate Eq.~\\eqref{eq:kqpdspins}, which can be written as\n\\begin{equation}\n\\mathcal{P}(\\Sigma_1,\\Sigma_2)=\\sum_{\\sigma_1=0,\\pm 1}\\sum_{\\sigma_2=\\pm 1}\\tilde{\\mathcal{P}}(\\sigma_1,\\sigma_2)\\delta(\\Sigma_1-\\sigma_1)\\delta(\\Sigma_2-\\sigma_2),\n\\end{equation}\nwith the discrete probability distribution\n\\begin{equation}\n\\label{eq:kqpdspinsdisc}\n\\begin{aligned}\n&\\tilde{\\mathcal{P}}(+1,+1)=|\\alpha|^2|\\gamma|^2,\\hspace{1cm}\\tilde{\\mathcal{P}}(-1,+1)=|\\beta|^2|\\delta|^2,\\hspace{1cm}\\tilde{\\mathcal{P}}(0,+1)=2{\\rm Re}\\left\\{\\alpha\\beta^*\\gamma^*\\delta\\right\\},\\\\&\n\\tilde{\\mathcal{P}}(+1,-1)=|\\alpha|^2|\\delta|^2,\\hspace{1cm}\\tilde{\\mathcal{P}}(-1,-1)=|\\beta|^2|\\gamma|^2,\\hspace{1cm}\\tilde{\\mathcal{P}}(0,-1)=-2{\\rm Re}\\left\\{\\alpha\\beta^*\\gamma^*\\delta\\right\\}.\n\\end{aligned}\n\\end{equation}\nNote that although the operators $\\hat{\\sigma}_j$ have eigenvalues $\\pm 1$, the KQPD is non-zero also for $\\sigma_1=0$. Furthermore, these terms can become negative. We will discuss the origin of these terms and their negativity in more detail in the next section. For now, we only mention that the negativity reflects the fact that a measurement of $\\hat{\\sigma}_1$ disturbs a subsequent measurement of $\\hat{\\sigma}_2$. The negative terms thus vanish if these two operators are simultaneously measurable, $[\\hat{\\sigma}_1,\\hat{\\sigma}_2]=0$, if the first measurement does not influence the initial state, $[\\hat{\\sigma}_0,\\hat{\\sigma}_1]=0$, or if a measurement of $\\hat{\\sigma}_2$ has a completely random outcome no matter if $\\hat{\\sigma}_1$ is measured or not. The latter is the case if $\\hat{\\sigma}_0=\\hat{\\sigma}_x$, $\\hat{\\sigma}_1=\\hat{\\sigma}_y$, and $\\hat{\\sigma}_2=\\sigma_z$. In all these cases, a measurement of $\\hat{\\sigma}_2$ yields outcomes which are independent of the presence or absence of a previous measurement of $\\hat{\\sigma}_1$. The KQPD for two subsequent spin measurements is illustrated in Fig.~\\ref{fig:barchart}.\n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=.6\\textwidth]{barchart}\n\\caption{Keldysh quasi-probability distribution for two subsequent spin measurements on a pure qubit state. While the second argument is restricted to the eigenvalues of $\\hat{\\sigma}_2$, the first argument can also be zero. (a) $\\hat{\\sigma}_0=\\hat{\\sigma}_2=\\hat{\\sigma}_x$ and $\\hat{\\sigma}_1=\\hat{\\sigma}_z$. In this case, a measurement of $\\hat{\\sigma}_1$ influences a subsequent measurement of $\\hat{\\sigma}_2$ which results in a negative value for $\\tilde{\\mathcal{P}}(0,-1)$. (b) $\\hat{\\sigma}_0=\\hat{\\sigma}_x$, $\\hat{\\sigma}_1=\\hat{\\sigma}_y$, and $\\hat{\\sigma}_2=\\sigma_z$. In this case, the measurement outcome of $\\hat{\\sigma}_2$ is completely random, no matter if the first measurement is performed or not. A measurement of $\\hat{\\sigma}_1$ does therefore not influence a subsequent measurement of $\\hat{\\sigma}_2$ and the KQPD is purely positive.}\n  \\label{fig:barchart}\n\\end{figure}\n\n\n\\subsection{Negativity}\n\\label{sec:negativity}\n{As mentioned above, negativity in the KQPD indicates that measuring the considered observables necessarily influences the outcomes (i.e. measuring all of the operators results in different statistics than measuring each of them in a different realization). Following Ref.~\\cite{hofer:2016}, we corroborate this claim by connecting negative values in the KQPD to coherent superpositions of different eigenstates of the observables. In the presence of such superpositions, any measurement (partly) collapses the state, possibly influencing subsequent measurement outcomes.} For clarity, we consider the case of $N$ observables at subsequent times as described by the moment generating function in Eq.~\\eqref{eq:momgenmultsub}.\nAssuming that the eigenstates of each operator $\\hat{A}_l$ span the whole Hilbert space, we replace\n\\begin{equation}\n\\label{eq:exprepl}\ne^{-i\\frac{\\lambda_l}{2}\\hat{A}_l(\\tau_l)}=\\sum_{A_l^\\alpha}e^{-i\\frac{\\lambda_l}{2}A_{l}^\\alpha}\\hat{U}(0,\\tau_l)|A_{l}^{\\alpha}\\rangle\\langle A_{l}^{\\alpha}|\\hat{U}(\\tau_l,0),\n\\end{equation} \nwhere $\\hat{A}_l|A_{l}^\\alpha\\rangle=A_{l}^\\alpha|A_{l}^\\alpha\\rangle$ and the superscript $\\alpha=L,R$ labels if the replacement happened on the left ($L$) or on the right ($R$) of the density matrix in Eq.~\\eqref{eq:momgenmultsub}, i.e. $\\alpha$ labels the branch (forward- or backward-in-time).\nThis allows us to take the $\\lambda$-dependent factors out of the trace and obtain the KQPD by a Fourier transformation\n\\begin{equation}\n\\label{eq:kqpdtraj}\n\\mathcal{P}(\\boldsymbol{A})=\\sum_{\\boldsymbol{A}_L,\\boldsymbol{A}_R}\\delta_{A_N^L,A_N^R}\\delta\\left(\\boldsymbol{A}-\\frac{\\boldsymbol{A}_L+\\boldsymbol{A}_R}{2}\\right)\\mathcal{A}(\\boldsymbol{A}_L)\\mathcal{A}^*(\\boldsymbol{A}_R)\\langle A^L_1|\\hat{\\rho}_{\\tau_1}|A^R_1\\rangle.\n\\end{equation}\nHere the vectors $\\boldsymbol{A}_{L\/R}$ have as their entries $A_l^{L\/R}$ and thus denote a particular manifestation of eigenvalues that we call a trajectory. The amplitudes associated with such a trajectory read\n\\begin{equation}\n\\label{eq:aampl}\n\\mathcal{A}(\\boldsymbol{A}_\\alpha)=\\langle A_N^\\alpha|\\hat{U}(\\tau_N,\\tau_{N-1})|A_{N-1}^\\alpha\\rangle\\langle A_{N-1}^\\alpha|\\cdots|A_{2}^\\alpha\\rangle\\langle A_{2}^\\alpha| \\hat{U}(\\tau_2,\\tau_1)|A_1^\\alpha\\rangle,\n\\end{equation}\nthe Dirac delta for vectors is defined as\n\\begin{equation}\n\\delta\\left(\\boldsymbol{A}-\\frac{\\boldsymbol{A}_L+\\boldsymbol{A}_R}{2}\\right)=\\prod_l \\delta\\left(A_l-\\frac{A_l^R+A_l^L}{2}\\right),\n\\end{equation}\nand $\\hat{\\rho}_{\\tau_1}=\\hat{U}(\\tau_1,0)\\hat{\\rho}_0\\hat{U}(0,\\tau_1)$.\n\nEquation \\eqref{eq:kqpdtraj} expresses the KQPD as a sum over pairs of trajectories. These pairs, here denoted left ($L$) and right ($R$) trajectory, are equivalent to a single trajectory along the Keldysh contour. A given pair of trajectories contributes with a weight that is given by the interference term $\\mathcal{A}(\\boldsymbol{A}_L)\\mathcal{A}^*(\\boldsymbol{A}_R)$ times a density matrix element $\\langle A^L_1|\\hat{\\rho}_{\\tau_1}|A^R_1\\rangle$ corresponding to the ``starting point'' of the trajectories. Each pair contributes to the KQPD at the argument which corresponds to the \\textit{mean} value of the observable on the two trajectories, again underlining the fact that the KQPD considers the fluctuations of the \\textit{classical} field in the Keldysh language (see below). Finally, the Kronecker delta in Eq.~\\eqref{eq:kqpdtraj} results from the trace and reflects the fact that the disturbance of the last measurement is irrelevant.\n\nFrom Eq.~\\eqref{eq:kqpdtraj}, we see that as long as $\\boldsymbol{A}_L=\\boldsymbol{A}_R$, the KQPD is strictly greater or equal to zero. It is therefore the interference terms of trajectories that differ from each other which allow the KQPD to become negative. This is a genuine quantum effect since these interference terms are a direct result of a coherent superposition between different eigenstates of the observables $\\hat{A}_l$. {Although superposition can be encountered in classical (wave) theories, a coherent superposition between two states which correspond to two different values for an observable does not exist in classical theories (indeed, the presence of such superpositions violates the assumptions made by Leggett and Garg to obtain their inequalities \\cite{leggett:1985}, see Sec.~\\ref{eq:seclg})}.\n\nIn analogy to Eqs.~\\eqref{eq:keldyshrot} and \\eqref{eq:probapathint}, we can introduce a Keldysh rotation\n\\begin{equation}\n\\label{eq:keldyshrot2}\n\\boldsymbol{A}_{cl}=\\frac{1}{2}(\\boldsymbol{A}_L+\\boldsymbol{A}_R),\\hspace{2cm} \\boldsymbol{A}_q = \\boldsymbol{A}_L-\\boldsymbol{A}_R,\n\\end{equation}\nand write the KQPD in Eq.~\\eqref{eq:kqpdtraj} as a path integral\n\\begin{equation}\n\\label{eq:probapathint2}\n\\mathcal{P}(\\boldsymbol{A})=\\int \\mathcal{D}[\\boldsymbol{A}_{cl},\\boldsymbol{A}_q]\\delta(\\boldsymbol{A}-\\boldsymbol{A}_{cl})e^{iS[\\boldsymbol{A}_{cl},\\boldsymbol{A}_q]},\n\\end{equation}\nwith the action\n\\begin{equation}\n\\label{eq:action}\ne^{iS[\\boldsymbol{A}_{cl},\\boldsymbol{A}_q]}=\\mathcal{A}(\\boldsymbol{A}_{cl}+\\boldsymbol{A}_q\/2)\\mathcal{A}^*(\\boldsymbol{A}_{cl}-\\boldsymbol{A}_q\/2)\\langle A^{cl}_1+A^q_1\/2|\\hat{\\rho}_{\\tau_1}| A^{cl}_1-A^q_1\/2\\rangle,\n\\end{equation}\nand the integration measure\n$\\int \\mathcal{D}[\\boldsymbol{A}_{cl},\\boldsymbol{A}_q]=\\sum_{\\boldsymbol{A}_{cl},\\boldsymbol{A}_q}\\delta_{A_N^q,0}$. In this language, the KQPD is given by the sum over all paths for which $\\boldsymbol{A}=\\boldsymbol{A}_{cl}$, weighted by the complex exponential of the action. From Eqs.~\\eqref{eq:probapathint2} and \\eqref{eq:action}, we see that any negativity in the KQPD necessarily implies the presence of a non-zero \\textit{quantum} field $\\boldsymbol{A}_q\\neq0$.\nIn contrast to Eq.~\\eqref{eq:probapathint}, which describes a single observable, the quantum field in Eq.~\\eqref{eq:probapathint2} is not necessarily zero implying that the statistics of $\\boldsymbol{A}_{cl}$, determined by the Keldysh time-ordering, are not necessarily equal to the statistics of $\\boldsymbol{A}_L$ or $\\boldsymbol{A}_R$. \n\nWe now illustrate these ideas with the KQPD for two subsequent qubit measurements introduced in the last section. In terms of trajectories, the discrete KQPD in Eq.~\\eqref{eq:kqpdspinsdisc} can be expressed as\n\\begin{equation}\n\\label{eq:kqpdtwospintraj}\n\\tilde{\\mathcal{P}}(\\sigma_1,\\sigma_2)=\\sum_{\\sigma_1^{L\/R}=\\pm }\\delta_{\\sigma_1,(\\sigma_1^L+\\sigma_1^R)\/2}\\langle\\sigma_2|\\sigma_1^L\\rangle\\langle \\sigma_1^L|+_0\\rangle\\langle +_0|\\sigma_1^R\\rangle\\langle\\sigma_1^R|\\sigma_2\\rangle.\n\\end{equation}\nWe thus see that the terms where $\\sigma_1^L\\neq\\sigma_1^R$ are responsible for the terms $\\tilde{\\mathcal{P}}(0,\\sigma_2)$. Note that these terms are only non-zero if the initial state is in a coherent superposition of the two eigenstates of $\\hat{\\sigma}_1$. One can further show that\n\\begin{equation}\n\\sum_{\\sigma_1,\\sigma_2=\\pm 1}\\tilde{\\mathcal{P}}(\\sigma_1,\\sigma_2)=1,\n\\end{equation}\nwhich directly implies that $\\tilde{\\mathcal{P}}(0,+1)=-\\tilde{\\mathcal{P}}(0,-1)$. Thus any contribution to Eq.~\\eqref{eq:kqpdtwospintraj} from pairs of trajectories with $\\sigma_1^L\\neq\\sigma_1^R$ results in negativity in the KQPD as illustrated in Fig.~\\ref{fig:barchart}.\n\n{We close this section with a brief comparison to other non-classicality indicators based on negative quasi-probabilities. As mentioned above, the Wigner function (at time $t$) is the KQPD for the observables $\\hat{x}(t)$ and $\\hat{p}(t)$. A negative Wigner function implies an impossibility of describing the corresponding state classically in the canonical position-momentum phase space (i.e. as a mixture of states with well defined positions and momenta). The KQPD generalizes this concept to arbitrary observables. Negativity in the KQPD implies that it is impossible to describe the system and its time evolution as a mixture of states with well defined values for the observable outcomes. We note that negative values in a discrete analogue of the Wigner function have recently been connected to the computational speed-up of a quantum computer \\cite{veitch:2012,veitch:2013,howard:2014}.\n\nIn quantum optics, one often considers negativity, or strong singularity, in the Glauber-Sudarshan $P$ function as an indicator of non-classicality \\cite{titulaer:1965,mandel:1986}. This criterion captures all states with non-positive Wigner functions as well as squeezed states and entangled states \\cite{sperling:2009}. A non-classical $P$ function implies that the state can not be interpreted as a mixture of coherent states which indicates that the state can not be fully described by classical optics \\cite{mandel:1986}. We note that the $P$ function has recently been extended to multiple times \\cite{vogel:2008,krumm:2016,krumm:2017}. In contrast to the KQPD, this distribution is relevant for photo-detection and its moments correspond to normal ordering and regular time ordering.\n\nWhile these QPDs only contain information on the states and possibly their time-evolution, quasi-probability \\textit{representations} also describe measurements with QPDs \\cite{spekkens:2008,ferrie:2008,ferrie:2009,ferrie:2011}. It has been shown that a positive representation (i.e. a representation that only uses positive probability distributions) of all states and measurements is impossible. Restricting the states and measurements to a given experimental procedure, a procedure is then said to require non-classical resources if no positive representation can be found \\cite{ferrie:2008}. It is an interesting question if negativity in the KQPD implies the absence of a positive representation for a corresponding experimental procedure.}\n\n\\section{von Neumann measurements}\n\\label{sec:vonneumann}\nIn this section, we consider von Neumann type measurements and we show that the corresponding probability distributions can be obtained by convolving the KQPD with the back-action and imprecision associated with the measurement. We start by considering the instantaneous measurement of a single observable before moving on to subsequent instantaneous measurements, simultaneous measurements, and measurements of finite duration.\n\n\\subsection{Instantaneous measurements}\nFollowing von Neumann \\cite{vonneumann:book}, we model the measurement of an observable $\\hat{A}$ at time $\\tau$ by coupling it to a detector described by the canonically conjugate observables $\\hat{r}$ and $\\hat{\\pi}$ via the interaction Hamiltonian\n\\begin{equation}\n\\label{eq:hvn}\n\\hat{H}_m = \\delta(t-\\tau)\\chi\\hat{A}\\hat{\\pi},\n\\end{equation}\nwhere $\\chi$ denotes the coupling strength. This interaction induces the time-evolution\n$\\hat{U}_m=\\exp(-i\\chi\\hat{A}\\hat{\\pi})$,\nwhich displaces the detector coordinate $\\hat{r}$ by an amount that depends on the state of the system. Projectively measuring the position of the detector then yields the measurement of $\\hat{A}$. Measuring a single observable in this way yields the probability distribution (see App.~\\ref{app:vonneumann})\n\\begin{equation}\n\\label{eq:probaone}\nP(A)=\\int \\chi dA'\\langle  \\chi(A-A')|\\hat{\\rho}_{\\rm det}|\\chi(A-A')\\rangle\\mathcal{P}(A';\\tau),\n\\end{equation}\nwhere $\\hat{\\rho}_{\\rm det}$ is the density matrix of the detector, $|\\chi(A-A')\\rangle$ is an eigenstate of the operator $\\hat{r}$, and $\\mathcal{P}(A';\\tau)$ is the KQPD for a single observable given in Eq.~\\eqref{eq:probaa1}. Note that the position uncertainty in the initial detector state directly translates into a measurement uncertainty. A projective measurement can be obtained by choosing a position eigenstate as the initial state for the detector. In this case, the measured distribution coincides with the KQPD. This is due to the fact that the back-action of the measurement is irrelevant for a single observable since we are not interested in the post-measured state.\n\nFor multiple subsequent (instantaneous) measurements, we couple each observable to a different detector. The measurement Hamiltonian then reads \n\\begin{equation}\n\\label{eq:hvnm}\n\\hat{H}_m = \\sum_{l=1}^N\\delta(t-\\tau_l)\\chi_l\\hat{A}_l\\hat{\\pi}_l.\n\\end{equation}\nIn contrast to a single observable, each measurement can now influence all subsequent measurements. To capture this back-action effect, we evaluate the KQPD with the additional term $\\sum_l\\delta(t-\\tau_l)\\gamma_l\\hat{A}_l$ in the Hamiltonian,\nwhere the $\\gamma_j$ are random variables which are distributed according to the detector states (see below). This yields the modified moment generating function\n\\begin{equation}\n\\label{eq:momgenmultsubba}\n\\Lambda(\\boldsymbol{\\lambda};\\boldsymbol{\\gamma})={\\rm Tr}\\left\\{e^{-i\\left(\\frac{\\lambda_N}{2}+\\gamma_N\\right)\\hat{A}_N(\\tau_N)}\\cdots e^{-i\\left(\\frac{\\lambda_1}{2}+\\gamma_1\\right)\\hat{A}_1(\\tau_1)}\\hat{\\rho}_0e^{-i\\left(\\frac{\\lambda_1}{2}-\\gamma_1\\right)\\hat{A}_1(\\tau_1)}\\cdots e^{-i\\left(\\frac{\\lambda_N}{2}-\\gamma_N\\right)\\hat{A}_N(\\tau_N)} \\right\\},\n\\end{equation}\nwhich can be obtained by replacing $\\lambda_l\\rightarrow \\lambda_l\\pm 2\\gamma_l$ in Eq.~\\eqref{eq:momgenmultsub}, where the $+$ ($-$) sign corresponds to terms on the left (right) side of the density matrix.\n\nThe probability distribution which describes subsequent von Neumann measurements can then be shown to read (for a derivation, see App.~\\ref{app:vonneumann})\n\\begin{equation}\n\\label{eq:probasub}\nP(\\boldsymbol{A})=\\int d\\boldsymbol{A}'d\\boldsymbol{\\gamma}\\left[\\prod_l \\mathcal{W}_l(\\chi_l[A_l-A_l'],\\gamma_l\/\\chi_l)\\right]\\mathcal{P}(\\boldsymbol{A}';\\boldsymbol{\\gamma}),\n\\end{equation}\nwhere $\\mathcal{W}_l(r,\\gamma)$ denotes the Wigner function of detector $l$ and $\\mathcal{P}(\\boldsymbol{A}';\\boldsymbol{\\gamma})$ is the KQPD defined as the Fourier transform of the moment generating function in Eq.~\\eqref{eq:momgenmultsubba}. The measurement outcomes are determined by a convolution of the KQPD and the Wigner functions of the detectors. We can thus interpret the KQPD as describing the \\textit{intrinsic} fluctuations of the observables of interest which is being influenced by the effects of the measurement \\cite{clerk:2011}. The position distributions of the detectors determine the precision of the measurement while the momentum distributions determine the back-action via the $\\gamma_l$-dependent terms in the KQPD. The Heisenberg uncertainty relation implies a trade-off between these two effects, indicating that we can never eliminate both influences.\n\nAs an example, we consider a measurement of position $\\hat{x}$ at $\\tau=0$ followed immediately by a momentum $\\hat{p}$ measurement. The probability distribution describing such a measurement reads\n\\begin{equation}\n\\label{eq:probaxp}\nW_{x\\rightarrow p}(x,p)=\\int dx'dp'd\\gamma_xd\\gamma_p\\mathcal{W}_p(\\chi_p[p-p'],\\gamma_p\/\\chi_p)\\mathcal{W}_x(\\chi_x[x-x'],\\gamma_x\/\\chi_x)\\mathcal{W}(x',p'+\\gamma_x).\n\\end{equation} \nHere $\\mathcal{W}_{x\/p}$ denotes the Wigner function of the detector measuring $x\/p$ and $\\mathcal{W}$ without a subscript denotes the KQPD which in this case is given by the Wigner function of the system. The position measurement disturbs the momentum of the system as reflected by the second argument of the KQPD. Note that it is the momentum distribution of the $x$-detector which determines the effect of this disturbance. This reflects the trade-off between the precision of the position measurement (which requires a peaked position distribution for the $x$-detector) and the disturbance of the subsequent momentum distribution (which is minimized for a peaked momentum distribution for the $x$-detector). Note that the last expression is independent of the momentum distribution of the $p$-detector (this can be seen by performing the integral over $\\gamma_p$), reflecting the fact that the back-action of the momentum measurement is irrelevant since it is the last measurement. Inverting the order of the measurements yields a probability distribution which is given by the last expression upon exchanging $\\mathcal{W}(x',p'+\\gamma_x)\\rightarrow \\mathcal{W}(x'-\\gamma_p,p')$.\n\nReturning to arbitrary observables and choosing Gaussian Wigner functions for the detectors\n\\begin{equation}\n\\mathcal{W}_l(r,\\gamma)=\\frac{1}{2\\pi\\sigma_{r_l}\\sigma_{\\gamma_l}}e^{-\\frac{r^2}{2\\sigma_{r_l}^2}}e^{-\\frac{\\gamma^2}{2\\sigma_{\\gamma_l}^2}},\n\\end{equation}\nwe can write the probability distribution in Eq.~\\eqref{eq:probasub} in terms of trajectories [cf.~Eq.~\\eqref{eq:kqpdtraj}] yielding\n\\begin{equation}\n\\label{eq:probasubtraj}\n\\begin{aligned}\nP(\\boldsymbol{A})=\\sum_{\\boldsymbol{A}_L,\\boldsymbol{A}_R}\\delta_{A_N^L,A_N^R}&\\left[\\prod_l\\frac{1}{\\sqrt{2\\pi}\\sigma_{{\\rm imp},l}}e^{-\\frac{1}{2\\sigma_{{\\rm imp},l}^2}\\left(A_l-\\frac{A_l^L+A_l^R}{2}\\right)^2}e^{-\\frac{\\sigma_{{\\rm BA},l}^2}{2}\\left(A_l^L-A_l^R\\right)^2}\\right]\\\\&\\times\\mathcal{A}(\\boldsymbol{A}_L)\\mathcal{A}^*(\\boldsymbol{A}_R)\\langle A^L_1|\\hat{\\rho}_{\\tau_1}|A^R_1\\rangle.\n\\end{aligned},\n\\end{equation}\nwhere $\\sigma_{{\\rm imp},l}=\\sigma_{r_l}\/\\chi_l$ and $\\sigma_{{\\rm BA},l}=\\sigma_{\\gamma_l}\\chi_l$. Comparing the last expression with the KQPD in Eq.~\\eqref{eq:kqpdtraj}, we can clearly identify the effects of the measurement imprecision and back-action. The first Gaussian in the last expression replaces the Dirac delta in Eq.~\\eqref{eq:kqpdtraj}. It implies that even terms with $A_l\\neq (A_l^L+A_l^R)\/2$ contribute to the probability of measuring $A_l$. This effect captures the imprecision of the measurement which becomes more pronounced as the uncertainty in the position of detector $l$ grows. The second Gaussian in the last expression reduces the weight of terms which have $A_l^L\\neq A_l^R$, i.e. a finite quantum field in the Keldysh language. {This effect is due to the measurement back-action which (partly) collapses the coherent superpositions between eigenstates of $\\hat{A}_l$}. The coupling Hamiltonian in Eq.~\\eqref{eq:hvnm} entangles the system with the detector leading to dephasing in the reduced state of the system. As the uncertainty in the detector momentum grows, this effect becomes more pronounced. From the Heisenberg uncertainty relation, we find\n\\begin{equation}\n\\label{eq:heisenberg}\n\\sigma_{{\\rm imp},l}\\sigma_{{\\rm BA},l}=\\sigma_{r_l}\\sigma_{\\gamma_l}\\geq\\frac{1}{2},\n\\end{equation}\nreflecting the trade-off between measurement imprecision and back-action {(see also Ref.~\\cite{clerk:2010})} which ensures that the measured distribution is positive even if the KQPD exhibits negativities. We note that coupling the system to additional degrees of freedom, such as a thermal reservoir, introduces dephasing analogously to the measurement back-action \\cite{hofer:2016}.\n\nTo illustrate the effect of the measurement, we return to our example of two subsequent spin measurements [cf.~Eq.~\\eqref{eq:kqpdspins}]. Since the back-action of the second measurement is irrelevant, we consider a measurement of intermediate strength of $\\hat{\\sigma}_1$ followed by a projective measurement of $\\hat{\\sigma}_2$.\nIn terms of the discrete KQPD given in Eq.~\\eqref{eq:kqpdspinsdisc}, we find for the distribution describing a von Neumann measurement (with Gaussian detectors)\n\\begin{equation}\n\\label{eq:kqpdspinsmeas}\nP(\\sigma_1,\\sigma_2)=\\frac{1}{\\sqrt{2\\pi}\\sigma_{\\rm imp}}\\sum_{\\sigma'_1=0,\\pm 1}e^{-\\frac{1}{2\\sigma_{\\rm imp}^2}(\\sigma_1-\\sigma_1')^2}e^{-2\\sigma_{\\rm BA}^2\\delta_{\\sigma'_1,0}}\\tilde{\\mathcal{P}}(\\sigma_1',\\sigma_2).\n\\end{equation}\nThe effect of the measurement is illustrated in Fig.~\\ref{fig:measured}. We note that the distribution of measurement outcomes for the first measurement depends very strongly on the outcome of the second measurement. Specifically, we see from Fig.~\\ref{fig:measured} that only considering events where the second (projective) measurement yields $-1$ allows one to distinguish the outcomes of the first measurement much better (for a given measurement strength). This effect, which arises from the negativity of the KQPD, is responsible for the occurrence of anomalous weak values as discussed in Sec.~\\ref{sec:weakvalues}. \n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=.8\\textwidth]{measured}\n\\caption{Effect of measurement imprecision and back-action on the KQPD. Here $\\hat{\\sigma}_0=\\hat{\\sigma}_2=\\hat{\\sigma}_x$ and $\\hat{\\sigma}_1=\\hat{\\sigma}_z$ to maximize the negativity in the KQPD. The measurement imprecision and back-action modify the KQPD resulting in a positive probability distribution. Here $\\sigma_{\\rm imp}\\sigma_{\\rm BA}=1\/2$, minimizing the influence of the measurement. Depending on the measurement outcome of the second (projective) measurement, the distribution for the first measurement results in two clearly distinguishable outcomes or not. This is directly related to the terms $\\tilde{\\mathcal{P}}(0,\\sigma_2)$ which are responsible for the negative values in the KQPD. Note that the peaks are shifted with respect to the eigenvalues of $\\hat{\\sigma}_1$ (dashed lines). This effect allows for the occurrence of anomalous weak values as discussed in Sec.~\\ref{sec:weakvalues}.}\n  \\label{fig:measured}\n\\end{figure}\n\n{Note that subsequent von Neumann measurements can also be used to model continuous measurements, a subject which has received a lot of attention recently, see e.g. \\cite{korotkov:1999,korotkov:2001,chantasri:2013,weber:2014,chantasri:2015}.}\n\n\n\\subsection{Simultaneous measurements of position and momentum}\n\nWe now turn to the simultaneous measurement of position and momentum. In analogy to Eq.~\\eqref{eq:hvn} (and following Arthurs and Kelly \\cite{arthurs:1964}), we model the measurement by the simultaneous coupling of two detectors to the position and momentum of the system respectively\n\\begin{equation}\n\\label{eq:hvnmsim}\n\\hat{H}_m = \\delta(t)\\left(\\chi_x\\hat{x}\\hat{\\pi}_x+\\chi_p\\hat{p}\\hat{\\pi}_p\\right).\n\\end{equation}\nIt can be shown that this measurement is described by the probability distribution \\cite{di_lorenzo:2011}\n\\begin{equation}\n\\label{eq:probaxpsim}\nW(x,p)=\\int dx'dp'd\\gamma_xd\\gamma_p\\mathcal{W}_p(\\chi_p[p-p'],\\gamma_p\/\\chi_p)\\mathcal{W}_x(\\chi_x[x-x'],\\gamma_x\/\\chi_x)\\mathcal{W}(x'-\\gamma_p\/2,p'+\\gamma_x\/2).\n\\end{equation}\nThe arguments of the last Wigner function reflect the fact that in a simultaneous measurement, both position and momentum are being disturbed. Note the close similarity between the last expression and the probability distribution describing subsequent measurements of position and momentum [cf.~Eq.~\\eqref{eq:probaxp}]. Notably, the KQPD is in both cases given by the Wigner function. This is a consequence of the commutation relation $[\\hat{x},\\hat{p}]=i$ and does not hold in general for non-commuting observables.\n\nFor a detailed discussion on the von Neumann type simultaneous measurement of position and momentum, we refer the reader to Ref.~\\cite{stenholm:1992}. Here we only discuss the case of Gaussian states of minimal uncertainty for the detectors ($k=x,p$)\n\\begin{equation}\n\\mathcal{W}_{k}(r,\\gamma)=\\frac{1}{\\pi}e^{-r^2\/\\sigma_{k}^2}e^{-\\gamma^2\\sigma_{k}^2},\n\\end{equation}\nwith the additional constraint $\\sigma_x\\sigma_p\/(\\chi_x\\chi_p)=1\/2$. This constrained minimizes the disturbance of the measurement and yields the Husimi $Q$-function as the measured probability distribution \\cite{husimi:1940,stenholm:1992}\n\\begin{equation}\n\\label{eq:husimiq}\nW(x,p)=\\frac{1}{\\pi}\\int dx' dp' e^{-\\frac{(x-x')^2}{2\\sigma^2}}e^{-2\\sigma^2(p-p')^2}\\mathcal{W}(x',p'),\n\\end{equation}\nwhere $\\sigma=\\sigma_x\/\\chi_x$. In addition to this von Neumann type measurement, the Husimi $Q$-function also describes heterodyne measurements of two field quadratures \\cite{leonhardt:1995}.\n\n\n\\subsection{Measuring the FCS}\n\n\nIn this section, we consider a von Neumann type measurement of the time-integral of an observable \\cite{nazarov:2003}. To this end, we couple a detector to the observable of interest for a finite amount of time\n\\begin{equation}\n\\label{eq:hvnc}\n\\hat{H}_m = \\Theta(t)\\Theta(\\mathcal{T}-t)\\chi\\hat{A}\\hat{\\pi}.\n\\end{equation}\nThis measurement can be shown to be described by the probability distribution \\cite{nazarov:2003}\n\\begin{equation}\n\\label{eq:measfcs}\nF(m) = \\int d\\gamma \\mathcal{W}(\\chi[m-m'],\\gamma\/\\chi)\\mathcal{F}(m';\\gamma),\n\\end{equation}\nwhere $\\mathcal{F}(m;\\gamma)$ is obtained through Eqs.~\\eqref{eq:momgenfcs} and \\eqref{eq:fcs} upon adding the term $\\Theta(t)\\Theta(\\mathcal{T}-t)\\gamma\\hat{A}$ to the Hamiltonian.\nAgain, we find a trade-off between measurement imprecision, depending on the spread in the detector position, and back-action which depends on the spread in the detector momentum. As discussed in Ref.~\\cite{hofer:2016}, the last equation can be expressed in terms of trajectories yielding (for a Gaussian detector)\n\\begin{equation}\n\\label{eq:fcsmeastraj}\nF(m)=\\frac{1}{\\sqrt{2\\pi}\\sigma_{\\rm imp}}\\sum_{\\boldsymbol{A}_L,\\boldsymbol{A}_R}\\delta_{A_\\mathcal{T}^L,A_\\mathcal{T}^R}e^{-\\frac{\\left(m-\\frac{1}{2}m_L-\\frac{1}{2}m_R\\right)^2}{2\\sigma^2_{\\rm imp}}} e^{-\\frac{\\sigma_{\\rm BA}^2}{2}\\left(m_L-m_R\\right)^2}\\langle A_0^L|\\hat{\\rho}|A_0^R\\rangle \\mathcal{A}(\\boldsymbol{A}_L)\\mathcal{A}^*(\\boldsymbol{A}_R),\n\\end{equation}\nwhere the trajectory amplitudes are obtained by dividing the time evolution into $N$ time-slices of infinitesimal duration $\\delta t$ and inserting identities resolved in the eigenstates of $\\hat{A}$ after each slice\n\\begin{equation}\n\\label{eq:aamplfcs}\n\\mathcal{A}(\\boldsymbol{A}_\\alpha)=\\langle A_\\mathcal{T}^\\alpha|\\hat{U}(\\mathcal{T},\\mathcal{T}-\\delta t)|A_{\\mathcal{T}-\\delta t}^\\alpha\\rangle\\langle A_{\\tau-\\delta t}^\\alpha|\\cdots|A_{\\delta t}^\\alpha\\rangle\\langle A_{\\delta t}^\\alpha| \\hat{U}(\\delta t,0)|A_0^\\alpha\\rangle.\n\\end{equation}\nThe discrete time-integral over such a trajectory is denoted by $m_\\alpha =\\sum_l A^\\alpha_l \\delta t$. The Heisenberg uncertainty relation applied to the detector variables again enforces $\\sigma_{\\rm imp}\\sigma_{\\rm BA}\\geq 1\/2$, reflecting the trade-off between smearing out the negativities in the FCS by measurement imprecision and reducing the weight of negative terms (which necessarily have $m_L\\neq m_R$) by measurement back-action in complete analogy to Eq.~\\eqref{eq:probasubtraj}.\n\nThis concludes our treatment on von Neumann measurements. Expressing the measured probability distributions in terms of the KQPD suggests that the latter should be interpreted as describing the intrinsic fluctuations of the observables of interest. In the next section, we will build on this interpretation and demonstrate the utility of studying the KQPD to understand the quantum behavior of dynamic systems.\n\n\n\\section{Applications of the KQPD}\n\\label{sec:applications}\nWe now turn to situations where it is beneficial to study the KQPD in the absence of any measurement device. We are especially interested in situations where the KQPD fails to be positive. As discussed in Sec.~\\ref{sec:negativity}, this indicates non-classical behavior. The most prominent KQPD is by far the Wigner function, which is a representation of the density matrix and is used extensively to illustrate the state of a quantum system and to capture its non-classicality \\cite{deleglise:2008,kenfack:2004}. An example for a KQPD which goes beyond instantaneous states is provided by the FCS which describes the fluctuations of an observable over a certain time-interval. While the FCS has mostly been employed to describe electronic transport in systems where it is a purely positive function, its utility for detecting non-classical behavior has been recognized \\cite{hofer:2016,clerk:2011,bednorz:2010,bednorz:2012,belzig:2001,clerk:2010}. Here we discuss three additional applications of the KQPD, stressing its versatility and its utility in capturing non-classical behavior in the dynamics of quantum systems.\n\n\\subsection{Work fluctuations}\n\nSince work is not a state function, it is in general not sufficient to interrogate a system at a single time in order to determine the work performed by the system. As we have seen above, performing measurements at multiple times generally results in a probability distribution which is influenced by the measurement itself. As a consequence, the observed amount of work will in general depend on the way it is measured \\cite{perarnau:2017}. Here we use the KQPD to describe work as a measurement-independent, fluctuating quantity. We note that the KQPD has been used before to describe work fluctuations \\cite{esposito:2009,solinas:2015,solinas:2016}. \n\nIn classical statistical mechanics, work is defined as the energy change in the macroscopic, accessible degrees of freedom, while heat is the energy change in the microscopic, inaccessible degrees of freedom. In quantum systems, all degrees of freedom are in principle microscopic. We can however still define work as being the energy change mediated by the accessible degrees of freedom. Here we consider two different scenarios: In the first, one has access to the energy stored in a part of the system that we call the work storage device. The energy increase of this device is then interpreted as the work performed by the system. {This is the analog of determining the work performed on a suspended weight by measuring its height.} In the second scenario, one has access to the power produced by the system and the work is defined as the time-integral of the power. This captures the situation in thermoelectric heat engines, where one can access the power through a measurement of the electrical current. {In terms of the suspended weight, this corresponds to the analog of measuring the power by measuring the velocity of the weight.}\n\nFor the first scenario, we write the total Hamiltonian as $\\hat{H}=\\hat{H}_w+\\hat{H}_{\\rm rest}$, where $\\hat{H}_w$ denotes the Hamiltonian of the work storage device. Work fluctuations can then be described by the KQPD corresponding to the difference of two subsequent energy measurements on the work storage device\n\\begin{equation}\n\\label{eq:work}\n\\begin{aligned}\n\\mathcal{P}(w;\\tau)=&\\frac{1}{2\\pi}\\int d\\lambda e^{i\\lambda w}{\\rm Tr}\\left\\{e^{-i\\frac{\\lambda}{2}\\hat{H}_w}\\hat{U}(\\tau,0)e^{i\\frac{\\lambda}{2}\\hat{H}_w}\\hat{\\rho}_0e^{i\\frac{\\lambda}{2}\\hat{H}_w}\\hat{U}(0,\\tau)e^{-i\\frac{\\lambda}{2}\\hat{H}_w}\\right\\}\\\\\n=&\\sum_{E_\\tau,E_L,E_R}\\delta\\left(w-\\left[E_\\tau-\\frac{E_L+E_R}{2}\\right]\\right){\\rm Tr}\\left\\{\\langle E_\\tau|\\hat{U}(\\tau,0)|E_L\\rangle\\langle E_L|\\hat{\\rho}_0|E_R\\rangle \\langle E_R|\\hat{U}(0,\\tau)|E_\\tau\\rangle\\right\\}.\n\\end{aligned}\n\\end{equation}\nHere $\\hat{H}_w|E_j\\rangle=E_j|E_j\\rangle$ and $\\tau$ denotes the time interval between the two measurements. For initial states which commute with $\\hat{H}_w$, we find $E_L=E_R$ and we recover the probability for obtaining $w$ as the difference of two projective energy measurements \\cite{talkner:2007}. Any negativity in the above distribution is thus due to the coherence in the initial state. We note that the probability distribution obtained by performing two Gaussian energy measurements, as discussed in Ref.~\\cite{talkner:2016}, is obtained from Eq.~\\eqref{eq:work} by including the imprecision and back-action of the measurement as discussed in Sec.~\\ref{sec:vonneumann}.\nThe moments generated by the KQPD are given by\n\\begin{equation}\n\\label{eq:momentswork}\n\\langle w^k\\rangle = \\sum_{l=0}^{k}\\frac{1}{2^k}{k \\choose l}{\\rm Tr}\\left\\{ \\hat{\\mathcal{T}}_-\\left(\\hat{H}_w(\\tau)-\\hat{H}_w\\right)^l\\hat{\\mathcal{T}}_+\\left(\\hat{H}_w(\\tau)-\\hat{H}_w\\right)^{(k-l)}\\hat{\\rho}_0\\right\\},\n\\end{equation}\nwhere $\\hat{H}_w(\\tau)=\\hat{U}(0,\\tau)\\hat{H}_w\\hat{U}(\\tau,0)$. For the first two moments, i.e. $k=1,2$ in the last expression, this reduces to \n\\begin{equation}\n\\label{eq:momentswork12}\n\\langle w^k\\rangle={\\rm Tr}\\left\\{ \\left(\\hat{H}_w(\\tau)-\\hat{H}_w\\right)^k\\hat{\\rho}_0\\right\\},\\hspace{1cm}{\\rm for}\\hspace{.2cm} k=1,2.\n\\end{equation}\nThe mean work obtained through the KQPD is thus just given by the difference of the average energy in the work storage device at $t=\\tau$ and $t=0$. This is in contrast to the probability distribution obtained by performing two projective energy measurements, where the mean value only coincides with Eq.~\\eqref{eq:momentswork12} for initial states that commute with $\\hat{H}_w$. For an explicit evaluation of Eq.~\\eqref{eq:work}, we refer to Ref.~\\cite{solinas:2016}. See Ref.~\\cite{batalhao:2014} for an experimental reconstruction of the work distribution by measuring its moment generating function (for an initial state that commutes with the Hamiltonian).\n\nIn case one has access to a power operator $\\hat{P}$, the KQPD describing work fluctuations is given by the FCS of the power operator. The KQPD and its moments are then given by Eqs.~(\\ref{eq:momgenfcs}-\\ref{eq:momentsfcs}) upon replacing $\\hat{A}$ with $\\hat{P}$. {This scenario is relevant for thermoelectric devices, where the power operator is given by the electrical current operator times the voltage across the device $\\hat{P}=\\hat{I}V$. In this case, the work fluctuations reduce to the problem of charge FCS \\cite{nazarov:book2}. We now return to systems described by $\\hat{H}=\\hat{H}_w+\\hat{H}_{\\rm rest}$, where we take the Hamiltonians to be time-independent for simplicity (see Ref.~\\cite{venkatesh:2015} for a discussion on the power operator for a time-dependent Hamiltonian). In this case, the power operator is given by the energy change of the work storage device per unit time, i.e. $\\hat{P}=i[\\hat{H},\\hat{H}_w]$.}\nIf the power operator commutes with the total Hamiltonian, i.e. $[\\hat{P},\\hat{H}]=0$, we find\n\\begin{equation}\n\\label{eq:powerop}\ne^{-i\\frac{\\lambda}{2}\\hat{H}_w}\\hat{U}(\\tau,0)e^{i\\frac{\\lambda}{2}\\hat{H}_w}=\\hat{\\mathcal{T}}_+e^{-i\\frac{\\lambda}{2}\\int_0^\\tau dt \\hat{P}(t)}.\n\\end{equation}\nThis implies that the FCS of the power operator is equal to the KQPD for two energy measurements given in Eq.~\\eqref{eq:work} \\cite{esposito:2009}. The difference of the two scenarios considered here is then only relevant when taking into account the disturbance from an actual measurement. {For a thermoelectric device, the energy in the work storage device corresponds to the electric potential energy stored in an electronic contact, i.e. $\\hat{H}_w=V\\hat{Q}$, where $\\hat{Q}$ denotes the charge operator in the contact with $\\hat{I}=i[\\hat{H},\\hat{Q}]$. The work provided by a thermoelectric device can thus either be determined by measuring the electrical current (determining the power) or by measuring the number of electrons in an electric contact (determining the energy in the work storage device). We note that both approaches have been considered in the theory of charge FCS \\cite{nazarov:book2}.}\n\n\n\\subsection{Weak values}\n\\label{sec:weakvalues}\nThe weak value of an observable $\\hat{A}$ is obtained by preparing the system in some state $|I\\rangle$, performing a weak measurement of $\\hat{A}$, and post-selecting the measurement outcomes on the final state $|F\\rangle$ \\cite{aharonov:1988}. Describing the weak measurement as a von Neumann measurement, the post-measured state of the \\textit{detector} is determined by the weak value\n\\begin{equation}\n\\label{eq:weakvalue}\nA_w=\\frac{\\langle F|\\hat{A}|I\\rangle}{\\langle F|I\\rangle},\n\\end{equation}\nand the average of the measurement outcome is given by the real part of the last expression \\cite{aharonov:1988,dressel:2015}. Any possible time-evolution between pre-selection, measurement and post-selection is absorbed in the definition of the initial and final states. In the KQPD framework, we can describe the weak value measurement using the distribution describing two subsequent measurements, one for the weak measurement and one for the post-selection. For simplicity, we do not treat the pre-selection as an additional measurement but impose it by setting $\\hat{\\rho}_0=|I\\rangle\\langle I|$. The relevant KQPD then reads\n\\begin{equation}\n\\label{eq:kqpdweak1}\n\\begin{aligned}\n\\mathcal{P}(A,B)=&\\frac{1}{(2\\pi)^2}\\int d\\lambda_Ad\\lambda_Be^{i\\lambda_A A}e^{i\\lambda_B B}{\\rm Tr}\\left\\{e^{-i\\frac{\\lambda_B}{2}\\hat{B}}e^{-i\\frac{\\lambda_A}{2}\\hat{A}}|I\\rangle\\langle I|e^{-i\\frac{\\lambda_A}{2}\\hat{A}}e^{-i\\frac{\\lambda_B}{2}\\hat{B}}\\right\\}\\\\\n=&\\sum_{B_F,A_L,A_R}\\delta\\left(B-B_F\\right)\\delta\\left(A-\\frac{A_L+A_R}{2}\\right) \\langle B_F|A_L\\rangle\\langle A_L|I\\rangle\\langle I|A_R\\rangle\\langle A_R|B_F\\rangle,\n\\end{aligned}\n\\end{equation}\nwhere $|B_F\\rangle$ are eigenstates of $\\hat{B}$. The KQPD corresponding to the measurement of the weak value is then obtained by taking into account the post-selection, i.e. fixing the argument corresponding to the second measurement\n\\begin{equation}\n\\label{eq:kqpdweak2}\n\\mathcal{P}_w(A)=\\frac{\\mathcal{P}(A,F)}{\\int dA\\mathcal{P}(A,F)}=\\frac{1}{|\\langle F|I\\rangle|^2}\\sum_{A_L,A_R}\\delta\\left(A-\\frac{A_L+A_R}{2}\\right)\\langle F|A_L\\rangle\\langle A_L|I\\rangle\\langle I|A_R\\rangle\\langle A_R|F\\rangle.\n\\end{equation}\nThe statistics of weak value measurements (with post-selection performed by a projective measurement) can be described using the last expression by including measurement imprecision and back-action as discussed in Sec.~\\ref{sec:vonneumann}. In the weak measurement limit, the disturbance of the measurement does not alter the average value of the KQPD, which is equal to the real part of Eq.~\\eqref{eq:weakvalue}\n\\begin{equation}\n\\label{eq:evgweakvalue}\n\\langle A\\rangle_w =\\int dA\\,A\\mathcal{P}_w(A)={\\rm Re}\\left\\{\\frac{\\langle F|\\hat{A}|I\\rangle}{\\langle F|I\\rangle}\\right\\}.\n\\end{equation}\n\nFrom Eq.~\\eqref{eq:kqpdweak2}, we see that the KQPD vanishes for arguments that lie outside the spectrum of $\\hat{A}$. However Eq.~\\eqref{eq:evgweakvalue} can take on values which lie outside the spectrum of $\\hat{A}$. In such cases, the weak value is called \\textit{anomalous}.\nWe now show that an anomalous weak value implies that the distribution $\\mathcal{P}_w(A)$ takes on negative values. To see this, assume that $\\mathcal{P}_w(A)\\geq 0$. In that case, we find\n\\begin{equation}\n\\label{eq:wvnegkqpd}\n\\langle A\\rangle_w =\\int_{A_{\\rm min}}^{A_{\\rm max}} dA\\,A\\mathcal{P}_w(A)\\leq A_{\\rm max} \\int_{A_{\\rm min}}^{A_{\\rm max}} dA\\mathcal{P}_w(A)=A_{\\rm max},\n\\end{equation}\nwhere $A_{\\rm min \\,(max)}$ is the smallest (largest) eigenvalue of $\\hat{A}$ and a similar inequality restricts $\\langle A\\rangle_w\\geq A_{\\rm min}$. Note that the inequality in Eq.~\\eqref{eq:wvnegkqpd} is only valid for a strictly positive KQPD.\nAn anomalous weak value thus necessarily implies that the KQPD $\\mathcal{P}_w(A)$ [and also $\\mathcal{P}(A,B)$, cf.~Eq.~\\eqref{eq:kqpdweak2}] exhibits negativity indicating non-classical behavior. This is an agreement with Ref.~\\cite{pusey:2014}, where anomalous weak values have been shown to imply contextuality. We note that weak values have been connected to quasi-probability distributions before \\cite{dressel:2015}. In particular, the average momentum at a given position that is obtained from the Wigner function has been shown to be determined by the weak value of the momentum\n\\begin{equation}\n\\label{eq:weakvaluep}\n\\frac{\\int dp\\, p\\mathcal{W}(x,p)}{\\int dp \\mathcal{W}(x,p)}={\\rm Re}\\left\\{\\frac{\\langle x|\\hat{p}|I\\rangle}{\\langle x|I\\rangle}\\right\\},\n\\end{equation}\nwhich follows from our approach when taking $\\hat{A}=\\hat{p}$ and $\\hat{B}=\\hat{x}$.\n\n{Finally, we note that the distribution in Eq.~\\eqref{eq:kqpdweak2} is relevant for a measurement of $\\hat{A}$ with an arbitrary strength. It has the variance}\n\\begin{equation}\n\\label{eq:weakvariance}\n\\langle \\hat{A}^2\\rangle_w-\\left(\\langle \\hat{A}\\rangle_w\\right)^2 = \\frac{1}{2}{\\rm Re}\\left\\{\\frac{\\langle F|\\hat{A}^2|I\\rangle}{\\langle F|I\\rangle}\\right\\}-\\frac{1}{2}\\left({\\rm Re}\\left\\{\\frac{\\langle F|\\hat{A}|I\\rangle}{\\langle F|I\\rangle}\\right\\}\\right)^2.\n\\end{equation}\nFor $\\hat{A}=\\hat{p}$ and $|F\\rangle =|x\\rangle$, the last expression is proportional to the \\textit{quantum potential energy} \\cite{dressel:2015}, a central object in Bohmian mechanics \\cite{bohm:1952}. {See also Ref.~\\cite{dressel:2012} for a connection between weak values and von Neumann measurements of arbitrary strength and post-selection.}\n\nFor our example of two subsequent spin measurements, we find [using Eqs.~\\eqref{eq:kqpdspins}, \\eqref{eq:kqpdspinsdisc}, and \\eqref{eq:kqpdweak2}]\n\\begin{equation}\n\\label{eq:kqpdwvspins}\n\\mathcal{P}_w(\\sigma_1)=\\frac{|\\alpha|^2|\\delta|^2\\delta(\\sigma_1-1)+|\\beta|^2|\\gamma|^2\\delta(\\sigma_1+1)-2{\\rm Re}\\{\\alpha\\beta^*\\gamma^*\\delta\\}\\delta(\\sigma_1)}{|\\alpha\\delta-\\beta\\gamma|^2},\n\\end{equation}\nwhere we chose $|-_2\\rangle$ as the final state $|F\\rangle$.\nWe see that for a sufficiently small denominator, the probability density associated with $\\sigma_1=\\pm 1$ can become very large as long as the presence of the negative term ensures the normalization of $\\mathcal{P}_w(\\sigma_1)$. In this way, the weak value, determined by $\\mathcal{P}_w(1)-\\mathcal{P}_w(-1)$ can be amplified resulting in values far outside the range $[-1,+1]$. This can however only occur for a sufficiently large negative term in the KQPD as can be seen from Eq.~\\eqref{eq:kqpdwvspins} which implies\n\\begin{equation}\n\\label{eq:weakvaluespins}\n\\langle \\sigma_1\\rangle_w>1\\hspace{.5cm}\\Leftrightarrow\\hspace{.5cm} {\\rm Re}\\{\\alpha\\beta^*\\gamma^*\\delta\\}>|\\beta|^2|\\gamma|^2.\n\\end{equation}\n\n\n\\subsection{Leggett-Garg inequalities}\n\\label{eq:seclg}\n\nIn Ref.~\\cite{leggett:1985}, Leggett and Garg derived inequalities which hold under the assumption of \\textit{macroscopic realism} [i.e., a (macroscopic) system will at all times be in one (and only one) of the states available to it] and \\textit{non-invasive measurability}. Clearly quantum systems do not fulfill either of these assumptions and can thus violate Leggett-Garg inequalities. In contrast to Bell inequalities, where multiple spatially separated parties apply local operations, Leggett-Garg inequalities are obtained by probing the same system at multiple times.\nWe note that even if a Leggett-Garg inequality is violated, one can always argue that this happened because the measurement did influence the system, and therefore all subsequent measurements, in an unexpected (but in principle avoidable) way. This is known as the \\textit{clumsiness} loophole. Since it is not possible to certify the non-invasiveness of a measurement, the clumsiness loophole can never be closed \\cite{emary:2014}.\n\nWe will now show that the violation of a particular Leggett-Garg inequality directly implies the occurence of negative values in the corresponding KQPD. The considered inequality reads\n\\begin{equation}\n\\label{eq:leggettgarg}\nK=C_{21}+C_{32}-C_{31}\\leq 1,\n\\end{equation}\nwhere the correlators are given by\n\\begin{equation}\n\\label{eq:lgcorr}\nC_{ij} = \\sum_{Q_i,Q_j=\\pm 1}Q_iQ_jP_{ij}(Q_i,Q_j).\n\\end{equation}\nHere $P_{ij}(Q_i,Q_j)$ is the probability of obtaining the outcome $Q_i$ at time $t_i$ and $Q_j$ at $t_j$ if the system is measured at these two times only. The measurement outcomes are restricted to $Q_i=\\pm 1$. Violation of the inequality in Eq.~\\eqref{eq:leggettgarg} implies that the correlators $C_{ij}$ can not be obtained from a positive probability distribution which describes all three measurements and is independent of the choice of measurements that are performed \\cite{leggett:1985,emary:2014}, i.e. \n\\begin{equation}\n\\label{eq:corrneq}\nK>1\\hspace{.5cm}\\Rightarrow \\hspace{.5cm}C_{ij}\\neq \\sum_{Q_1,Q_2,Q_3=\\pm 1}Q_iQ_jP(Q_3,Q_2,Q_1),\n\\end{equation}\nwith $P(Q_3,Q_2,Q_1)\\geq 0$.\n\nFor projective measurements of a dichotomic observable $\\hat{Q}$, the correlator to be used in Eq.~\\eqref{eq:leggettgarg} is given by \\cite{fritz:2010}\n\\begin{equation}\n\\label{eq:corrproj}\nC_{ij}={\\rm Tr}\\left\\{\\frac{1}{2}\\left\\{\\hat{Q}(t_i),\\hat{Q}(t_j)\\right\\}\\hat{\\rho}_0\\right\\}.\n\\end{equation}\nWe now introduce the KQPD for three subsequent measurements of $\\hat{Q}$\n\\begin{equation}\n\\label{eq:kqpdlg}\n\\begin{aligned}\n\\mathcal{P}(Q_3,Q_2,Q_1)=&\\frac{1}{(2\\pi)^3}\\int d\\lambda_3d\\lambda_2\\lambda_1e^{i\\lambda_3 Q_3+i\\lambda_2 Q_2+i\\lambda_1 Q_1}\\\\&\\times{\\rm Tr}\\left\\{e^{-i\\frac{\\lambda_3}{2}\\hat{Q}(t_3)}e^{-i\\frac{\\lambda_2}{2}\\hat{Q}(t_2)}e^{-i\\frac{\\lambda_1}{2}\\hat{Q}(t_1)}\\hat{\\rho}_0e^{-i\\frac{\\lambda_1}{2}\\hat{Q}(t_1)}e^{-i\\frac{\\lambda_2}{2}\\hat{Q}(t_2)}e^{-i\\frac{\\lambda_3}{2}\\hat{Q}(t_3)}\\right\\}.\n\\end{aligned}\n\\end{equation}\nIt is straightforward to show that the correlators in Eq.~\\eqref{eq:corrproj} can directly be obtained from the KQPD\n\\begin{equation}\n\\label{eq:lgcorrkqpd}\nC_{ij} = \\int dQ_1dQ_2dQ_3\\,Q_iQ_j\\mathcal{P}(Q_3,Q_2,Q_1).\n\\end{equation}\nIf the KQPD was strictly positive, it would thus provide a positive probability distribution which describes all three measurements and is independent of the choice of measurements that are performed. The existence of such a probability distribution prevents the Leggett-Garg inequality to be violated \\cite{emary:2014} (see also App.~\\ref{app:lg}).\nInverting the argument, we conclude that a violation of the Leggett-Garg inequality in Eq.~\\eqref{eq:leggettgarg} implies negative values for the KQPD in Eq.~\\eqref{eq:kqpdlg} and thus non-classical behavior in the time-evolution of $\\hat{Q}$.\n\n{While we discussed projective measurements, the KQPD could also find applications when considering Leggett-Garg inequalities obtained from weak continuous measurements \\cite{ruskov:2006,jordan:2006}.}\n\n\\section{Conclusions}\n\\label{sec:conclusions}\nThe KQPD provides a generalization of the Wigner function and the FCS to describe arbitrary observables within a quasi-probability formalism. This provides a unified framework for the description of fluctuations in dynamic quantum systems. Negative values of the KQPD can be directly linked to an interference between paths through Hilbert space which visit different eigenstates of the observables of interest. This requires coherent superpositions {of states that correspond to different measurement outcomes}, implying that negativity in the KQPD indicates non-classical behavior. As we have shown, this negativity is witnessed by anomalous weak values and violated Leggett-Garg inequalities (see also Refs.~\\cite{williams:2008,groen:2013} for a connection between anomalous weak values and Leggett-Garg inequalities). The versatility of the KQPD allows to describe the problem at hand with a tailor-made quasi-probability distribution which focuses on the observables of interest.\nWhen measuring these observables in von Neumann measurements, the outcome is determined by the KQPD, corrupted by measurement imprecision and back-action. \nHowever, the underlying KQPD can in principle be reconstructed tomographically by coupling the observables of interest to qubits and performing state tomography on the qubits. This has been discussed extensively for the FCS \\cite{levitov:1996,clerk:2011,hofer:2016,dasenbrook:2016,lebedev:2016}.\n\n{While we presented the point of view of the KQPD being a measurement-independent quantity that captures non-classicality of the system, a complementary viewpoint is provided by only considering von Neumann measurement setups. The appearance of negative values in the KQPD then reflects the presence of non-trivial correlations between the system and the detectors during the measurement.}\n\nHere we only discussed a small number of applications for the KQPD. Other scenarios where the KQPD might be of interest include: measurement based quantum computing \\cite{briegel:2009}; could the KQPD in this case capture the non-classical features that allow for outperforming a classical computer? Quantum-to-classical transition; how much noise is necessary for ensuring positivity of the KQPD and in which cases can the back-action in a measurement be neglected? Bell non-locality, entanglement, and steering; are these notions of non-classicality captured by a negative KQPD?\nIn addition to these open questions, we believe that there are many useful applications for the KQPD yet to be discovered.\n\n\\section*{Acknowledgments}\nI acknowledge stimulating discussions with A. Clerk, N. Brunner, C. Flindt, and M. Perarnau-Llobet. I gratefully acknowledge the hospitality of McGill University where part of this work has been done. This work was funded by the Swiss National Science foundation.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\n \nThe AdS\/{\\sc CFT} duality \n\\cite{Maldacena:1997re,Witten:1998qj,Gubser:2002tv} identifies a gravitational\ntheory in asymptotic anti-de-Sitter (AdS) space with a strongly \ncoupled conformal field theory ({\\sc CFT}) on the boundary of the same space. This identification\ncan be used to translate a non-perturbative computation from a\nstrongly coupled condensed matter system to \nsemi-classical gravity, and thereby make it conceptually easier to treat. This\napproach has shown much promise to deal with systems such as the\nquark gluon plasma and strange metals \\cite{Hartnoll:2008kx,Hartnoll:2009sz,Horowitz:2010gk}, whose behavior is very\ndifficult to calculate by other methods. \n \n\nIn a previous paper \\cite{Hossenfelder:2014gwa}, we pointed out the\npossibility to use the AdS\/{\\sc CFT} duality to arrive at a new type of duality\nwhich connects a strongly with a weakly coupled condensed matter system.\nThis can be done by combining AdS\/{\\sc CFT} with analog gravity.\nAnalog\ngravity \\cite{Unruh:1980cg, Barcelo:2005fc} is a way to assign an effective metric to certain types of weakly\ncoupled condensed matter systems. It can be shown that in these systems perturbations\npropagate in the matter background according to an equation of motion \nidentical to that of particles traveling in curved space. Gravitational analogs are known to exist for the Schwarzschild black hole \\cite{Visser:1997ux,Garay:1999sk,Barcelo:2000tg} and expanding Friedmann-Robertson-Walker space-times that mimic\ninflation in the early universe \\cite{Volovik:2000ua,Weinfurtner:2004mu,Jain:2007gg,Lin:2012ft,Bilic:2013qpa}. \nResearch in this area is presently very active, and theory and experiments both are being\nrapidly developed \\cite{Weinfurtner:2010nu,ex1}. \n\n\nThe effective metric \nis obtained from the behavior of the\nbackground field, and completely specified by the degrees of freedom of the\ncondensed matter system. If this effective metric is also a gravitational dual of\na strongly coupled system, then this leads to a relation between the two\ncondensed matter systems that give rise to the same metric. The connection\nbetween analog gravity and the AdS\/{\\sc CFT} duality was also explored in \n\\cite{Bilic:2014dda,Das:2010mk,holo,holo2,Chen:2012uc}.\n\nNot all metrics can be obtained as effective analog metrics from condensed\nmatter systems. It must both be possible to bring the metric into a specific\nform, which amounts to a certain gauge condition, and it must then be\ndemonstrated that the condensed matter system needed to obtain this\nmetric does fulfill the equations of motion. In the previous paper \\cite{Hossenfelder:2014gwa}\nit was demonstrated that the type of metrics used to model strongly coupled systems \nvia the AdS\/{\\sc CFT} duality can indeed also be obtained\nas effective metrics of a weakly coupled condensed matter system. \n\nWhile intriguing evidence, this finding by itself does not suffice to show that there is a new duality between\nweakly and strongly coupled condensed matter systems, because the\nidentification of the metric used in the AdS\/{\\sc CFT} correspondence as an\neffective metric does only demonstrate that the semi-classical limit is\nidentical. It does however show that this necessary condition \nis fulfilled and thus represents a first step on the way to a more general\nproof.\n\n\nIn the present paper, we will look at the next step, which is to demonstrate that there exists a relativistic\ncompletion of the system used in  \\cite{Hossenfelder:2014gwa}. In \\cite{Hossenfelder:2014gwa} it was found that the\nnon-relativistic limit is good towards the boundary of AdS-space, but not close by the horizon. This \nis unfortunate because the near-horizon region is the most interesting part, so we will here derive a \nrelativistic acoustic metric and recover the non-relativistic limit. We will find\nthat this also tells us more about the form of the Lagrangian of the analog\ngravity system than could be extracted from the non-relativistic limit.\n\nThroughout this paper we use units in which the speed of light and $\\hbar=1$. \nThe constant $c$ denotes the speed of sound and {\\sl not} the speed of light.\nThe metric signature is $(-1,1,1,1)$. Small Greek indices run from zero to three. With non-relativistic we refer to the limit\nin which the four-velocity is $\\ll c$. \n\n\n\\section{Framing the question}\n\nThe most instructive way to obtain the effective analog metric of a fluid is to\nuse the Lagrangian approach in a mean-field approximation and then\nderive the equations of motion for perturbations around the mean \\cite{Barcelo:2005fc,Bilic:2013qpa,Hossenfelder:2014gwa}. It can\nbe shown then that the perturbations obey a wave-equation that is identical\nto the wave-equation in a curved background whose metric is the effective\nanalog metric.  The form of the metric one obtains in this way depends on the \nLagrangian, and we will be dealing here specifically with an \neffective metric known as the `acoustic metric' because it determines the\npropagation of sound waves. There are other types of analog metrics\nfor different systems, for example the optic metric \\cite{Barcelo:2005fc,Gordon,optic1,optic2}, but these will not be discussed here.\n\nIt must be emphasized that the key finding of analog gravity is not trivial. While the assumption\nof Lorentz-invariance of the original Lagrangian \nalready tells us that the resulting equation of motion for the perturbation must\nrespect this symmetry too, this alone does not single out a wave-equation in\ncurved space.\nThe important property of the resulting equation is that it separates the \nbackground from the perturbations in just the right way so that the degrees\nof freedom of the background can be collected in something that takes the\nform of a metric tensor, or that the derivatives can be reformulated as\ncovariant derivatives in curved space respectively. Just by looking at all the terms that are possible \nwhen one\nrequires that indices are contracted suitably, there could\nbe combinations between the background field and the perturbations that do\nnot lend themselves to the description in terms of an effective metric.\n\nIndeed, it is interesting to observe that Lorentz-invariance in the equations governing\nthe propagation of the exitations can emerge even if the equations of the background\nthemselves are in the non-relativistic limit \\cite{Fagnocchi:2010sn}. The Lorentz-group that is relevant here\nis that in which the limiting velocity is the speed of sound, and not the speed of\nlight. At high energies, this emergent Lorentz-invariance can be violated, a phenomenon\nthat has been used to study the robustness under UV-corrections of quantum\nfield theory in curved background \\cite{Liberati:2005id}. \n\nConcretely, we take a Lagrangian for a real scalar field $\\theta$ of the form\n\\beqn\n{\\cal L} = {\\cal L}(\\chi (\\partial \\theta) - V )~. \\label{L}\n\\eeqn\nwhich depends on a kinetic energy term\n\\beqn\n\\chi = \\eta^{\\mu\\nu} \\left( \\partial_\\nu \\theta \\right) \\left( \\partial_\\mu \\theta \\right)\n\\eeqn\nand some additional potential $V$ that may include a mass term (more about the potential later).  $\\eta$ is the metric in the laboratory\nthat hosts the analog gravity system. We assume that this space-time in the laboratory is flat, ie that its curvature tensor vanishes. However,\nsince we are free to choose a coordinate system, the metric might not have the Minkowski-form. \n\n\nWe then\nmake a perturbation around a background field that is assumed to fulfill the equations of motion, $\\theta= \\theta_0 + \\varepsilon \\theta_1$. The inverse of the\nanalog metric in terms of derivatives with respect to the field then takes the rather simple form \\cite{Barcelo:2005fc,Bilic:1999sq,Bilic:2013qpa,Hossenfelder:2014gwa}\n\\beqn\n\\sqrt{-g} g^{\\mu \\nu} = - \\sqrt{-\\eta} \\frac{\\partial^2 {\\cal L}}{\\partial (\\partial_\\nu \\theta_0) \\partial (\\partial_\\mu \\theta_0)} ~. \\label{geff}\n\\eeqn\nHere and in the following, the lower index $0$ refers to quantities describing the background field (at zeroth order). \nThis metric then has to be rewritten into the hydrodynamic variables of the background fluid, and be inverted.\n\nOne can identify the pressure $p_0$, the density $\\rho_0$ and the fluid-velocity $u_\\nu$ by comparing the stress-energy\nderived from the Lagrangian (\\ref{L}) to the familiar stress-energy tensor of a fluid, from which one finds \n\\beqn\nu_\\nu = \\frac{\\partial_\\nu \\theta_0}{\\sqrt{\\chi}}~,~p_0 = {\\cal L}~,~\\rho_0 = 2 \\chi \\frac{\\partial {\\cal L}}{\\partial \\chi} - {\\cal L}~. \\label{ident}\n\\eeqn\nTo match this to the notation of \\cite{Visser:2010xv}, it is $\\chi = (\\rho_0+p_0)^2\/n_0^2$, where $n_0$ is the particle-density of the fluid\nand $\\chi$ is the specific enthalpy. The four-velocity is normalized to minus one\n\\beqn\n\\eta^{\\mu\\nu}u_\\mu u_\\nu = - 1~. \\label{norm}\n\\eeqn\nWith these variables one then gets the acoustic metric\n\\beqn\ng_{\\mu \\nu} =  \\left(  \\frac{\\rho_0+p_0}{c \\chi} \\right)  \\left( \\eta_{\\mu\\nu} + \\left(1- c^2 \\right) u_\\mu u_\\nu \\right) ~, \\label{acmet}\n\\eeqn\nwhere $c$ is the speed of sound and defined by\n\\beqn\n\\frac{1}{c^2} = \\frac{\\partial \\rho_0}{\\partial p_0}  = \\frac{2 \\chi \\partial^2 {\\cal L}\/\\partial \\chi^2 + \\partial {\\cal L}\/\\partial \\chi}{\\partial{\\cal L}\/\\partial \\chi} ~. \\label{sos}\n\\eeqn\n\nWhat we aim to show here is that there exists a scalar field Lagrangian of the general form  (\\ref{L}) that gives rise to an acoustic metric which describes the planar black hole \nin asymptotic $4+1$ dimensional AdS that reads\n\\beqn\n{\\rm d}s^2 = - \\frac{L^2}{ z^2}  \\gamma(z)  {\\rm d} {\\tilde t}^2 + \n\\frac{L^2}{z^2} \\gamma( z)^{-1} {\\rm d} z^2 + \\frac{L^2}{z^2}  \\sum_{i=1}^{3} {\\rm d} x^i {\\rm d} x^i  ~. \\label{charged}\n\\eeqn\nwhere \\cite{Hartnoll:2009sz}\n\\beqn\n \\gamma(z) =  1 - \\left( \\frac{ z}{z_0}\\right)^4 . \\label{gammach}\n\\eeqn\nWe have introduced the tilde for the coordinate $\\tilde t$ for later convenience. \nThe length scale $L$ is the AdS radius and inversely related to the cosmological constant. \nThe analog gravitational system will have to reproduce the metric (\\ref{charged}) on a space-like\nslice of dimension $3+1$ perpendicular to the horizon. Since the metric (\\ref{charged}) is translationally\ninvariant into the directions parallel to the horizon, this just means that for the effective metric the sum in the last term runs\nonly over $1$ and $2$. \n\n\n\\section{Gauging the metric}\n\nWe now have to find a transformation that brings the metric (\\ref{charged}) into the form (\\ref{acmet}). As\nnoted in \\cite{Hossenfelder:2014gwa}, this procedure generally isn't unique and one can spend a lot of time changing\ncoordinate systems in AdS space just to then realize that the resulting acoustic metric cannot be derived from any Lagrangian.\nFor this reason we will stay as close as possible to the transformation that was found to work previously and use the coordinate transformation $ \\tilde t \\to  t = \\tilde t - f(z)$ also used earlier, but now\nadd the rescaling $z \\to \\tilde z = g^{-1}(z)$. This transformation does not change the $1\/z^2$ prefactor of the\nAdS metric, except that now $z$ is implicitly a function of $\\tilde z$. We therefore can read off \n\\beqn\n\\frac{\\partial {\\cal L}}{\\partial \\chi} = \\frac{c L^2}{z^2}~\\label{pref}\n\\eeqn\ndirectly by comparing (\\ref{acmet}) with (\\ref{charged}).\n\nFurther comparing the components of the metric in the new coordinates with (\\ref{acmet}) one obtains\n\\beqn\ng_{tt} &:& - \\gamma = -1 + (1-c^2) u_t u_t ~, \\label{trafo1} \\\\\ng_{\\tilde z \\tilde z} &:& \\left(-\\gamma (f')^2 + \\frac{1}{\\gamma}\\right) (g')^2 = 1 + (1-c^2)u_{\\tilde z} u_{\\tilde z}~, \\label{trafo2}\\\\\ng_{t \\tilde z} &:& - \\gamma f' g'  = (1-c^2) u_t u_{\\tilde z} ~, \\label{trafo3}\n\\eeqn\nwhere a dash denotes a derivative with respect to $z$. We have assumed here that like the metric of the planar black hole the condensed\nmatter system too is static and translationally invariant in the directions parallel to the horizon, so that the components of the velocity in the $x_i$-directions\nall vanish. \n\nWe now have three equations (\\ref{trafo1},\\ref{trafo2},\\ref{trafo3}) for four unknown variables $f',g',c$ and one of the components of the four-velocity.\nThis does not look too promising but luckily with simple algebraic manipulations one can solve this system to\n\\beqn\nc = g'~,~u_t^2 = \\frac{z^4}{z_0^4(1- c^2)}~,~u_{\\tilde z}^2 = \\frac{c^2 - \\gamma}{1-c^2}~, \\label{rels}\n\\eeqn\nand $f'$ can then be obtained by integration of\n\\beqn\nf' =  \\frac{\\sqrt{(c^2-\\gamma)(1-\\gamma)}}{\\gamma c}~.\\label{f'}\n\\eeqn\n\nTo get the non-relativistic limit that reproduces the case discussed in \\cite{Hossenfelder:2014gwa}, one needs $u_t^2 \\to 1$, $u_{\\tilde z}^2 \\to z^4$ and\nand $\\tilde z \\to z$. \nThis tells us that the speed of sound must have an expansion of the form \n\\beqn\nc^2 = \\gamma + a \\left( \\frac{z}{z_0} \\right)^8 + {\\cal O}(z^9)~, \\label{cexp}\n\\eeqn \nwhere $a$ is a dimensionless\ncoefficient that we will determine in the next section.\n\n\\section{Checking that the equations of motion are fulfilled}\n\nSo, encouragingly we have seen that there is a straight-forward way to bring the metric of the planar black\nhole into the right form for the acoustic metric. Next we will have to check whether a fluid with the degrees of freedom\nidentified in the previous section fulfills the equations of motion. As was noted in \\cite{Hossenfelder:2014gwa}, the\nEuler-equation doesn't actually give a constraint on the system, it instead determines the potential  necessary to \nobtain the right pressure gradient. The remaining consistency requirement is that the continuity equation be fulfilled,\nwhich most conveniently can be written as the $t$-component of stress-energy conservation:\n\\beqn\n\\partial_\\nu T^{\\nu t} = 0~.\n\\eeqn\n\nSince our system is time-independent and translationally invariant into the directions perpendicular to $z$ and the metric is\ndiagonal, this\nsimplifies to\n\\beqn\n\\partial_{z} \\left( (\\rho_0 + p_0) u^t u^{\\tilde z} \\right) = \\partial_z \\left( \\frac{\\partial {\\cal L}}{\\partial \\chi} \\chi u^t u^{\\tilde z} \\right) = 0~. \\label{cont}\n\\eeqn\nBecause $\\tilde z$ is just a rescaling $z$ and $c \\partial_z = \\partial_{\\tilde z}$, it does not matter whether we take the derivative in this\nequation with respect to $z$ or $\\tilde z$.\n\nThe key to demonstrating that the continuity equation is fulfilled is to look at the equation of motion of the background field $\\theta_0$. We will now, crucially, assume that the field $\\theta_0$ describes the phase of some other field $\\Phi_0 = \\phi_0 \\exp({\\rm i} \\theta_0)$. Then the kinetic term $\\chi$ has only mass dimension two, and\nthere must be a prefactor of mass dimension two in the metric that comes from the amplitude of the field $\\phi_0$. We will denote this prefactor with $m^2$ so that now\n\\beqn\n\\frac{\\partial {\\cal L}}{\\partial \\chi} = \\frac{c L^2}{z^2} m^2~.\\label{prefm}\n\\eeqn\n\nThe assumption that $\\theta_0$ is a phase is sufficient to arrive at a metric that fulfils all requirements, but it might not be the only choice. It is chosen here because in the non-relativistic limit it reduces to the case earlier discussed in \\cite{Hossenfelder:2014gwa}. However, it is quite possible that there are other ways to obtain the correct relativistic metric. The purpose here is merely to demonstrate that it is possible to obtain the relativistic metric, not to show that this procedure is the unique way to obtain it.\n\nIf $\\theta_0$ is a phase, this also means that the potential $V$ will typically be a function of $\\phi_0^2$ (and, possibly, the coordinates), but be independent of $\\theta_0$. The equation of motion of\n$\\theta_0$ then reads\n\\beqn\n\\partial_\\mu \\sqrt{- \\eta} \\eta^{\\mu \\nu} \\frac{\\partial {\\cal L}}{\\partial \\chi} \\partial_\\nu \\theta_0 = 0~. \\label{thetaeom}\n\\eeqn\nIt is then further natural to assume that the time-dependence of $\\theta_0$ is just $mt$.\nUsing (\\ref{pref}), we can then solve (\\ref{thetaeom}) with\n\\beqn\n\\partial_t \\theta_0 = m~,~\\partial_{\\tilde z} \\theta_0 =  m  \\frac{z^2}{L^2c}~.\n\\eeqn\n\nWith this we get from (\\ref{rels})\n\\beqn\nu_0^2 = \\frac{m^2}{\\chi} = \\frac{z^4}{z_0^4(1- c^2)}~, u_{\\tilde z}^2 = \\frac{m^2 z^4}{\\chi c^2 L^4} = \\frac{c^2 -\\gamma}{1-c^2}~, \\label{chi2}\n\\eeqn\nand so\n\\beqn\n\\chi = m^2 \\frac{z_0^4}{z^4} (1-c^2)~, \\label{chi}\n\\eeqn\nand\n\\beqn\nc^2 (c^2 - \\gamma) = \\frac{z^8}{L^4 z_0^4} ~\\Rightarrow~ c^2 = \\frac{\\gamma}{2} + \\sqrt{ \\frac{\\gamma^2}{4} +  \\left( \\frac{z^2}{z_0 L}\\right)^4}~,\\label{c2}\n\\eeqn\nwhere we have picked a sign so that $c^2 \\to 1$ for $z \\to 0$. With this solution for $c$ we could now integrate $f$ and $g$ via Eqs (\\ref{rels}) and (\\ref{f'}) but these\nfunctions are not particularly illuminating. More interesting is that the speed of sound can be expanded as\n\\beqn\nc^2 = 1 - \\left( \\frac{z}{z_0}\\right)^4 + \\left( \\frac{z}{z_0}\\right)^4 \\bigg( \\frac{z}{L}\\bigg)^4 + {\\cal O}(z^{12})~,\n\\eeqn\nand thus has indeed the form anticipated in (\\ref{cexp}).\n\nWe can then finally insert all these expressions into the continuity equation (factors appear in the same order as in (\\ref{cont})):\n\\beqn\n\\partial_z \\left( \\left( m^2 \\frac{c L^2}{z^2} \\right) \\left( m^2 \\frac{z_0^4}{z^4} (1-c^2) \\right) \\sqrt{\\frac{z^4 (c^2 - \\gamma)}{z_0^4 (1-c^2)^2}} \\right) &=& 0~.\n\\eeqn\nand see that with (\\ref{c2}) this equation is fulfilled. \n\nOne convinces oneself easily that the reason\nthis equation is fulfilled has nothing to do with the particular form of the four-velocity or the prefactor of the metric. \nIt is not the scaling of the \ncomponents that is relevant here, but that\nthe $z$-coordinate is\nchosen as the coordinate system in the laboratory in which the metric is of the Minkowski-form. If $z$\nwas a radial coordinate for example then this equation would not be fulfilled.\n\nNow that we have a relativistic analog metric, we have to note however that\nit does not have a non-relativistic limit! \nThe reason is that the limit where $u_z \\ll 1$ is the same limit in\nwhich $c \\to 1$. In the limit $c \\to 1$ the acoustic metric (\\ref{acmet}) seems to become ill-defined. However, one\ncan absorb the factors $(1-c^2)$ from the metric (\\ref{acmet}) into\nthe four-velocity and then take the limit $c\\to 1$. The metric then remains well-defined but the \nfour-velocity becomes a null-vector, rendering the non-relativistic limit meaningless.\n\nTo avoid this, one \ncan rescale the\ntime coordinate $t \\to t \\kappa$ with an arbitrary constant $\\kappa$, which has the effect of\nchanging the limit of $c$ from $1$ to $\\kappa$. The equations (\\ref{trafo1},\\ref{trafo2},\\ref{trafo3}) then\nbecome\n\\beqn\ng_{tt} &:& - \\gamma \\kappa^2 = -1 + (1-c^2) u_t u_t ~, \\label{trafo1a} \\\\\ng_{\\tilde z \\tilde z} &:& \\left(-\\gamma (f')^2 + \\frac{1}{\\gamma}\\right) (g')^2 = 1 + (1-c^2)u_{\\tilde z} u_{\\tilde z}~, \\label{trafo2a}\\\\\ng_{t \\tilde z} &:& - \\gamma f' g' \\kappa  = (1-c^2) u_t u_{\\tilde z} ~, \\label{trafo3a}\n\\eeqn\nand one finds similarly as before (compare to \\ref{rels})\n\\beqn\nc = \\kappa g'~,~u_t^2 = \\frac{1-\\kappa^2 \\gamma}{1- c^2}~,~u_{\\tilde z}^2 =  \\frac{c^2 - \\kappa^2 \\gamma}{1-c^2}~. \\label{rels2}\n\\eeqn\nThe continuity equation is still fulfilled in this case, and the speed of sound now is approximately\n\\beqn\nc^2 = \\kappa^2 \\gamma(z) + \\left( \\frac{1-\\kappa^2}{\\kappa^2}\\right) \\frac{z^4}{L^4} + {\\cal O} (z^8)~.\n\\eeqn\n\nThe above calculation reveals that this finding that the continuity equation is fulfilled is independent of the number of\ndimensions of the AdS-space of which our analog systems describes a spatial slice. This might seem surprising because \nit was found in \\cite{Hossenfelder:2014gwa} that the non-relativistic limit\nonly works if the AdS-space is $4+1$ dimensional. How can this be? The reason can be found in equation\n(\\ref{trafo2}) or (\\ref{trafo2a}) respectively. Once one has identified the\nfour-velocity it turns out the square of the\nvelocity $u_{\\tilde z}$ is comparable to terms that appear also in the metric coefficients. This means that if one\nstarts with the non-relativistic metric, in which the term $u_{\\tilde z}^2$ in Eq (\\ref{trafo2}) or (\\ref{trafo2a}) is just absent, one neglects\na relevant term and obtains a different $f'$ that leads to another coordinate system. If one identifies the degrees\nof freedom in this different coordinate system, the\ncontinuity equation is fulfilled only in $4+1$ dimensions. So the discrepancy comes about because\ntwo different coordinate systems were used.\n\n\\section{Working towards a Lagrangian}\n\nWe have seen that the degrees of freedom of a fluid which can be extracted from the metric of\nthe planar black hole do fulfill the equations of motion. This leads to the question whether we\ncan find a Lagrangian for this system. From Eqs (\\ref{sos}), (\\ref{prefm}) and (\\ref{chi2}) we can infer that this Lagrangian\nmust have the property that\n\\beqn\n \\frac{\\partial {\\cal L}}{\\partial \\chi} = m^2 \\frac{c L^2}{z^2}~,~\\frac{\\partial^2 {\\cal L}}{\\partial \\chi^2} = \\frac{z^2 L^2}{2cz_0^4}~. \\label{req}\n\\eeqn\n\nWe can obtain a Lagrangian for example by making the ansatz\n\\beqn\n{\\cal L} = \\frac{2}{3} m \\left( \\chi - V\/m^2 \\right)^{3\/2}~, \\label{lans}\n\\eeqn\nwhere the constant $m$ is introduced for dimensional reasons.\nLagrangians with fractional powers seem strange from the perspective of particle physics, but they are a common\noccurrence in effective theories for condensed matter systems \\cite{Son:2002zn}. The reason is, roughly speaking, that to obtain the\neffective Lagrangian one uses a limit in which part of the equations of motion can be solved and then substitutes this\nsolution back into the Lagrangian. Since this partial solution itself depends generically on the kinetic \nterm $\\chi$ this can lead to\nthe odd-looking fractional powers. Interestingly enough, \nLagrangians with the particular power $3\/2$ of the\nkinetic term have recently been proved useful in the cosmological context, see for example \\cite{Berezhiani:2015bqa} and references\ntherein.  \n\nThe ansatz (\\ref{lans}), when compared with (\\ref{req}) leads to\n\\beqn\n{\\cal L} = \\frac{2}{3}m \\left( \\chi - m^2 z_0^4\/z^4 \\right)^{3\/2}~. \\label{lagex}\n\\eeqn\nAgain the potential term might appear somewhat unmotivated, but keep in mind that this is\nan effective theory. It might come about for example from a Lagrangian that originally was of the form\n\\beqn\n{\\cal L} = \\frac{2}{3} \\left[ \\left( \\eta^{\\mu \\nu } (\\partial_\\nu \\Phi_0) (\\partial_\\mu \\Phi_0^*)\/(\\Phi \\Phi^*) -  \\tilde{V}  z_0^4\/z^4  \\right) \\right]^{3\/2}~,~ \n\\tilde{V} = 2 \\Phi \\Phi^* - \\frac{(\\Phi \\Phi^*)^2}{m^2} \\label{lagphi}\n\\eeqn\nwhere $\\Phi_0 = \\phi_0 \\exp {\\rm i} \\theta_0$. One can then decompose the kinetic term of $\\Phi_0$ as\n\\beqn\n\\eta^{\\mu \\nu } (\\partial_\\nu \\Phi_0) (\\partial_\\mu \\Phi_0^*)  =  \\eta^{\\mu \\nu } (\\partial_\\nu \\phi_0) (\\partial_\\mu \\phi_0) + \\phi^2 \\chi~,\n\\eeqn\nand derive the equation of motion for $\\phi_0$\n\\beqn\n\\partial_{\\tilde z} \\frac{1}{\\phi_0^2} \\frac{\\partial {\\cal L}}{\\partial \\chi} \\partial_{\\tilde z} \\phi_0 -  \\frac{\\partial {\\cal L}}{\\partial \\chi} \\frac{1}{2} \\frac{(\\partial_{\\tilde z} \\phi_0)^2}{ \\phi^3_0} + \\frac{\\partial {\\cal L}}{\\partial \\chi} \\frac{{\\rm d} \\tilde V}{{\\rm d} \\phi_0} =0~.\n\\eeqn\n\nThis equation of motion has a simple solution in which the field $\\phi_0$ is just constant and sits in the potential minimum where\n\\beqn\n\\phi_0^2 = \\tilde V = m^2~.\n\\eeqn\nIf one inserts this solution back into the Lagrangian (\\ref{lagphi}) one obtains (\\ref{lagex}). This is just to demonstrate\nhow this type of potential can arise. It would be nicer of course\nif one could find a background that creates the $z$-dependence rather than putting this into the potential to begin with.\nAnd besides this, while the Lagrangian (\\ref{lagphi}) looks more plausible than (\\ref{lagex}) it would be interesting to see whether the Lagrangian (\\ref{lagphi}) itself can be obtained as the low-energy\nlimit from some other theory, but we will leave this to a future work. For now it is good to have an effective Lagrangian \nat all, because this allows one to look for other solutions to the field equations.\n\n\\section{Summary}\n\nWe have shown here that a $3+1$ dimensional slice of the metric of the planar black hole in AdS-space of arbitrary dimension can\nbe obtained as an effective acoustic metric, and that a simple effective Lagrangian exists for this system which has a fractional\npower of $3\/2$ in the kinetic term.\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nLet $(\\Omega,{\\cal F},P)$ be a complete probability space,\ncarrying a standard Brownian motion $W=\\{W_t\\}_{t\\ge 0}$ with\nvalues in $R^d$. $\\{{\\cal F}_t\\}_{t\\ge 0}$ is the natural\nfiltration of the Brownian motion $W$ augmented by the $P$-null\nsets of ${\\cal F}$.\n\nConsider two players $I$ and $II$, who use their respective\nswitching control processes $a(\\cdot)$ and $b(\\cdot)$ to control\nthe following BSDE \\begin{equation}\\label{system}\\begin{array}{rcl} U(s)&=&\\displaystyle\n\\xi+\\left(A^{a(\\cdot)}(T)-A^{a(\\cdot)}(s)\\right)-\\left(B^{b(\\cdot)}(T)-B^{b(\\cdot)}(s)\\right)\\\\\n&&\\displaystyle +\\int_s^T\\psi(r, U(r),V(r), a(r),b(r))\\,\ndr-\\int_s^TV(r)\\, dW(r).\\end{array}\\end{equation} Here,  $\\xi$ is an $m$-dimensional\nrandom variable measurable with respect to the past of $W$ up to\ntime $T$. $\\xi$ is called the terminal condition and $\\psi$ is called the\ncoefficient (also called the generator). $A^{a(\\cdot)}(\\cdot)$ and\n$B^{b(\\cdot)}(\\cdot)$ are the cost processes associated with the\nswitching control processes $a(\\cdot)$ and $b(\\cdot)$,\nrespectively; they are c\\`adl\\`ag processes. Under suitable\nconditions, the above BSDE has a unique adapted solution, denoted\nby $(U^{a(\\cdot),b(\\cdot)}, V^{a(\\cdot),b(\\cdot)})$.\n\nPlayer $I$ chooses the switching control $a(\\cdot)$ from a given\nfinite set $\\Lambda$ to minimize the cost \\begin{equation}\\label{cost} J(a(\\cdot),\nb(\\cdot))=U^{a(\\cdot),b(\\cdot)}(0),\\end{equation} and each of his\ninstantaneous switchings from one scheme $i\\in \\Lambda$ to another\ndifferent scheme $i'\\in \\Lambda$ incurs  a positive cost which will be\nspecified by the function $k(i,i'), (i,i')\\in \\Lambda\\times \\Lambda$. While\nPlayer $II$ chooses the switching control $b(\\cdot)$ from a given\nfinite set $\\Pi$ to maximize the cost $J(a(\\cdot), b(\\cdot))$, and\neach of his instantaneous switchings from one scheme $j\\in \\Pi$ to\nanother different scheme $j'\\in \\Pi$ incurs  a positive cost which\nwill be specified by the function $l(j,j'), (j,j')\\in \\Pi\\times\n\\Pi$. We are interested in the existence and the construction of\nthe value process as well as the saddle point.\n\nThe solution of the above-stated switching game will appeal to the\nfollowing new type of reflected backward stochastic differential\nequation (RBSDE for short)  with oblique reflection: for $(i,j)\\in\n\\Lambda\\times\\Pi$ and $t\\in [0,T]$,\n\\begin{equation}\\label{RBSDEi}\\left\\{\\begin{array}{rcl}Y_{ij}(t)&=&\\displaystyle\\xi_{ij}+\\int_t^T\n\\psi(s,Y_{ij}(s),Z_{ij}(s),i,j)\\,\nds\\\\\n& &\\displaystyle-\\int_t^TdK_{ij}(s)+\\int_t^TdL_{ij}(s)-\\int_t^TZ_{ij}(s)\\, dW(s),\\\\\nY_{ij}(t)&\\le&\\displaystyle \\mathop{\\rm min}_{i'\\not=i}\\{Y_{i'j}(t)+k(i,i')\\}, \\\\\nY_{ij}(t)&\\ge&\\displaystyle\n\\mathop{\\rm max}_{j'\\not=j}\\{Y_{ij'}(t)-l(j,j')\\},\\\\\n &&\\displaystyle\n\\int_0^T\\left(Y_{ij}(s)-\\mathop{\\rm min}_{i'\\not=i}\\{Y_{i'j}(s)+k(i,i')\\}\\right)dK_{ij}(s)=0,\\\\\n&\n&\\displaystyle\\int_0^T\\left(Y_{ij}(s)-\\mathop{\\rm max}_{j'\\not=j}\\{Y_{ij'}(s)-l(j,j')\\}\\right)dL_{ij}(s)=0.\n\\end{array}\\right.\\end{equation} Here,  the unknowns are the processes\n$\\{Y(t)\\}_{t\\in [0,T]}$, $\\{Z(t)\\}_{t\\in [0,T]}$, $\\{K(t)\\}_{t\\in\n[0,T]}$, and $\\{L(t)\\}_{t\\in [0,T]}$, which are required to be\nadapted with respect to the natural completed filtration of the\nBrownian motion $W$. Moreover, $K$ and $L$ are componentwisely increasing\nprocesses. The last two relations in (\\ref{RBSDEi}) are called the  upper and lower\nminimal boundary conditions.\n\nOne-dimensional RBSDEs were first studied by El Karoui et al.\n\\cite{ElKPPQ} in the\n case of one obstacle, and then by Cvitanic and Karatzas~\\cite{CvtanicKaratzas}\n in the case of two obstacles. In both papers, it is recognized that one-dimensional reflected\n BSDEs, with one obstacle and with two obstacles, are generalizations of\noptimal stopping and Dynkin games, respecively. Nowadays, the\nliterature on one-dimensional reflected BSDEs is very rich. The\nreader is referred to Hamad\\`ene and Hassani~\\cite{HH} and Peng\nand Xu~\\cite{PengXu}, among others, for the one-dimensional\nreflected BSDEs with two obstacles.\n\n Multidimensional RBSDE was studied by\nGegout-Petit and Pardoux \\cite{Gegout-PetitPardoux}, but their\nBSDE is reflected on the boundary of a convex domain along the\ninward normal direction, and their method depends heavily on the\nproperties of this inward normal reflection (see (1)-(3) in\n\\cite{Gegout-PetitPardoux}). We note that in a very special case\n(e.g., $\\psi$ is independent of $z$), Ramasubramanian\n\\cite{Ramasubramanian} studied a BSDE in an orthant with oblique\nreflection. Multi-dimensional BSDEs reflected along an oblique\ndirection rather than a normal direction, still remains to be open\nin general, even in a convex domain, let alone in a nonconvex\ndomain. Note  that there are some papers dealing with SDEs with\noblique reflection (see, e.g. \\cite{LionsSznitman} and\n\\cite{DupuisIshii}).\n\nRecently, the authors~\\cite{HuTang} studied the optimal switching\nproblem for \\break one-dimensional BSDEs, and the associated following\ntype of obliquely reflected \\break multi-dimensional  BSDEs: for $i\\in\n\\Lambda$,\n\\begin{equation}\\label{RBSDEswitching}\\left\\{\\begin{array}{rcl}Y_i(t)&=&\\displaystyle\\xi_i+\\int_t^T\n\\psi(s,Y_i(s),Z_i(s),i)\\,\nds\\\\\n& &\\displaystyle-\\int_t^TdK_i(s)-\\int_t^TZ_i(s)\\, dW(s),\\\\\nY_i(s)&\\le&\\displaystyle \\mathop{\\rm min}_{i'\\not=i}\\{Y_{i'}(s)+k(i,i')\\}, \\\\\n &&\\displaystyle\n\\int_0^T\\left(Y_i(s)-\\mathop{\\rm min}_{i'\\not=i}\\{Y_{i'}(s)+k(i,i')\\}\\right)dK_i(s)=0.\\\\\n\\end{array}\\right.\\end{equation} It should be added that an incomplete and less\ngeneral form of RBSDE (\\ref{RBSDEswitching}) (where the minimal\ncondition of (\\ref{RBSDEswitching}) is missing and the generator\n$\\psi$ does not depend on $(y,z)$) is suggested by \\cite{CarLud}.\nBut they did not discuss the existence and uniqueness of solution,\nwhich is considered to be difficult. See Remark 3.1 in\n\\cite{CarLud}. Lately, Hamad\\`ene and Zhang~\\cite{HZ} also studied\nRBSDE (\\ref{RBSDEswitching}) in a more general form.\n\nMore recently, Tang and Zhong~\\cite{TangZhong} discussed the mixed\nswitching and stopping problem for one-dimensional BSDEs, and\nobtained the existence and uniqueness result for the the associated\nfollowing type of obliquely reflected multi-dimensional BSDEs: for\n$i\\in \\Lambda$ and $t\\in [0,T]$,\n\\begin{equation}\\label{RBSDEswitchingStopping}\\left\\{\\begin{array}{rcl}Y_i(t)&=&\\displaystyle\\xi_i+\\int_t^T\n\\psi(s,Y_i(s),Z_i(s),i)\\,\nds\\\\\n& &\\displaystyle-\\int_t^TdK_i(s)+\\int_t^T\\, dL_i(s)-\\int_t^TZ_i(s)\\, dW(s),\\\\\nY_i(s)&\\le&\\displaystyle \\mathop{\\rm min}_{i'\\not=i}\\{Y_{i'}(s)+k(i,i')\\}, \\quad Y_i(t)\\ge S(t),\\\\\n &&\\displaystyle\n\\int_0^T\\left(Y_i(s)-\\mathop{\\rm min}_{i'\\not=i}\\{Y_{i'}(s)+k(i,i')\\}\\right)dK_i(s)=0,\\\\\n&&\\displaystyle \\int_0^T(Y_i(t)-S(t))\\, dL_i(t)=0.  \\end{array}\\right.\\end{equation}\nHere, $S$ is a previously given $\\{{\\cal F}_t, 0\\le t\\le T\\}$-adapted\nprocess with some suitable regularity.\n\nRBSDE (\\ref{RBSDEi}) arises from the switching game for BSDEs, and\nits form is more complicated than that of RBSDE\n(\\ref{RBSDEswitching}), which arises from the optimal switching\nproblem for BSDEs. For each fixed $j\\in \\Pi$, if we do not impose\nthe following constraint: \\begin{equation} Y_{ij}(t)\\ge\n\\mathop{\\rm max}_{j'\\not=j}\\{Y_{ij'}(t)-l(j,j')\\}, \\quad t\\in [0,T],\\end{equation} and\nits related boundary condition: \\begin{equation}\n\\int_0^T\\left(Y_{ij}(s)-\\mathop{\\rm max}_{j'\\not=j}\\{Y_{ij'}(s)-l(j,j')\\}\\right)dL_{ij}(s)=0,\\end{equation}\nthen we can take $L\\equiv 0$, and RBSDE (\\ref{RBSDEi}) is reduced\nto RBSDE (\\ref{RBSDEswitching}).\n\n RBSDE (\\ref{RBSDEi}) evolves in the closure $\\overline{Q}$ of domain $Q$:\n\\begin{equation} \\begin{array}{rcl} Q&:=&\\displaystyle \\biggl\\{(y_{ij})\\in\n\\mathrm{R}^{m_1\\times m_2}: y_{ij}< y_{i'j}+k(i,i') \\\\\n&&\\displaystyle\\qquad \\mbox{\\rm for any } i,i'\\in \\Lambda \\mbox{\n\\rm such that }i'\\not=i \\hbox{\n\\rm and }  j\\in \\Pi; \\\\\n&& \\qquad \\qquad \\qquad\\qquad \\displaystyle y_{ij}>y_{ij'}-l(j,j') \\\\\n&&\\displaystyle\\qquad  \\mbox{\\rm for any } j,j'\\in \\Pi \\mbox{ \\rm\nsuch that }j'\\not=j \\hbox{ \\rm and }  i\\in \\Lambda\\biggr\\},\\end{array}\\end{equation}\nwhich is convex and unbounded. The boundary $\\partial Q$ of domain\n$Q$ consists of the boundaries $\\partial D_{ij}^-$ and $\\partial\nD_{ij}^+$, $(i,j)\\in\\Lambda\\times \\Pi,$ with\n$$D_{ij}^-:=\\{(y_{ij})\\in \\mathrm{R}^m:y_{ij}<y_{i'j}+k(i,i'),\n \\mbox{ for any }i'\\in\\Lambda \\mbox{ such that } i'\\not=i\\}$$\n and\n$$D_{ij}^+:=\\{(y_{ij})\\in \\mathrm{R}^m:y_{ij}>y_{ij'}-l(j,j'),\n \\mbox{ for any }j'\\in\\Pi \\mbox{ such that } j'\\not=j\\}$$\n for $(i,j)\\in\\Lambda\\times \\Pi.$\nThat is,\n$$\\partial Q=\\mathop{\\cup}_{i=1}^{m_1}\\mathop{\\cup}_{j=1}^{m_2}\\left(\\partial D_{ij}^-\\cup \\partial D_{ij}^+\\right). $$\nIn the interior of $\\overline {Q}$, each equation in\n(\\ref{RBSDEi}) is independent of others. On the boundary, say\n$\\partial D_{ij}^-$ (resp. $\\partial D_{ij}^+$), the $(i,j)$-th\nequation  is switched to another one $(i',j)$ (resp. $(i,j')$),\nand the solution is reflected along the oblique direction\n$-e_{ij}$ (resp. $e_{ij}$), which is the negative (resp. positive)\ndirection of the $(i,j)$-th coordinate axis.\n\nThe existence and uniqueness of solution for RBSDE (\\ref{RBSDEi})\nconstitutes a main contribution of this paper. We prove the\nexistence by a penalization method, whereas the uniqueness is\nobtained by a verification method: first we introduce an optimal\nswitching problem for multi-dimensional RBSDEs of form (\\ref{RBSDEswitching}) , then we prove that\nthe first component $Y$ of any adapted solution $(Y,Z,K, L)$ of\nRBSDE (\\ref{RBSDEi}) is the (vector) value for the optimal\nswitching problem.\n\nSolution of RBSDE (\\ref{RBSDEi}) presents new difficulties when one\nfollows either our previous work~\\cite{HuTang} using the penalization method,\nor Hamad\\`ene and Zhang~\\cite{HZ} using a Picard\napproximation. In fact, even for the proof of the existence, we\nhave to use the representation for obliquely reflected BSDEs---an\nextended version of our previous representation result proved in\n\\cite{HuTang}. Moreover, we have to impose, for the proof of the\nexistence, the additional technical condition that the generator\n$\\psi$ is uniformly bounded.\n\nThere exist different methods in the literature for the study of\nswitching control and game problems. For the classical method of\nquasi-variational inequalities, the reader is referred to the book\nof Bensoussan and Lions~\\cite{BL}. See Tang and Yong~\\cite{TY} and\nTang and Hou~\\cite{TangHou} and the references therein for the\ntheory of variational inequalities and the dynamic programming for\noptimal stochastic switching control and switching games. But\nthese works are restricted to the Markovian case. Recently, using\nthe method of  Snell envelope (see, e.g. El Karoui \\cite{ElK})\ncombined with the theory of scalar valued RBSDEs, Hamadene and\nJeanblanc \\cite{HaJean} studied the switching problem with two\nmodes (i.e., $m=2$) in the non-Markovian context. Djehiche,\nHamadene and Popier \\cite{DHP} generalized their result to the\nabove switching problem with multi modes. The BSDE approach,\nfirstly developed in Hu and Tang~\\cite{HuTang} for optimal\nstochastic switching and taking the advantage of modern theory and\ntechniques of BSDEs, permits  to state and solve these\nproblems in a general non-Markovian framework. This paper is\ndevoted to the development of the BSDE approach for stochastic\nswitching games.\n\nThe paper is organized as follows: in Section 2, we introduce some\nnotation and formulate the switching game for one-dimensional\nBSDEs. We prove the existence of solution by a penalization method\nin Section 3, whereas in Section 4 we study the uniqueness.\nThe last section is devoted to the proof of the existence of the value process\nand the construction of the saddle point for our switching game.\n\n\n\n\\section{Notations, and Formulation of our switching game}\n\n\\subsection{Notations}\n\nLet us fix a nonnegative real number $T>0$. First of all,\n$W=\\{W_t\\}_{t\\ge 0}$ is a standard Brownian motion with values in\n$R^d$ defined on some complete probability space $(\\Omega,{\\cal\nF},P)$. $\\{{\\cal F}_t\\}_{t\\ge 0}$ is the natural filtration of the\nBrownian motion $W$ augmented by the $P$-null sets of ${\\cal F}$.\nAll the measurability notions will refer to this filtration. In\nparticular, the sigma-field of predictable subsets of $[0,T]\\times\n\\Omega$ is denoted by ${\\cal P}$.\n\nWe denote by $S^2(R^{m_1\\times m_2})$ or simply by $S^2$  the set of\n$R^{m_1\\times m_2}$-valued, adapted and c\\`adl\\`ag processes\n$\\{Y(t)\\}_{t\\in [0,T]}$ such that\n$$||Y||_{S^2}:=E\\left[\\mathop{\\rm sup}_{t\\in [0,T]}|Y(t)|^2\\right]^{1\/2}<+\\infty.$$\n$(S^2,||\\cdot||_{S^2})$ is then a Banach space.\n\nWe denote by $M^2((R^{m_1\\times m_2})^d)$ or simply by $M^2$ the set\nof (equivalent classes of) predictable processes $\\{Z(t)\\}_{t\\in\n[0,T]}$ with values in $(R^{m_1\\times m_2})^d$ such that\n$$||Z||_{M^2}:=E\\left[\\int_0^T |Z(s)|^2ds\\right]^{1\/2}<+\\infty.$$\n$M^2$ is then a Banach space endowed with this norm.\n\nWe define  also\n\\begin{eqnarray*}\nN^2(R^{m_1\\times m_2}):&=&\\{K=(K_{ij})\\in S^2 :\\mbox{ for any }\n(i,j)\\in\n\\Lambda\\times \\Pi, K_{ij}(0)=0, \\\\\n& &\\mbox{ and } K_{ij}(\\cdot) \\mbox{ is increasing }\\},\n\\end{eqnarray*}\nwhich is abbreviated as $N^2$. $(N^2,||\\cdot||_{S^2})$ is then a\nBanach space.\n\n\\subsection{Hypotheses}\n\nConsider now the RBSDE (\\ref{RBSDEi}). The generator $\\psi$ is a\nrandom function $\\psi:[0,T]\\times \\Omega\\times R\\times R^d\\times\n\\Lambda\\times \\Pi \\to R$ whose component $\\psi(\\cdot,i,j)$ is\nmeasurable with respect to ${\\cal P}\\otimes {\\cal B}(R)\\otimes\n{\\cal B}(R^d)$ for each pair $(i,j)\\in \\Lambda\\times \\Pi$,  and the\nterminal condition $\\xi$ is simply an $R^{m_1\\times m_2}$-valued\n${\\cal F}_T$-measurable random variable. The cost functions $k$\nand $l$ for two players are defined on $\\Lambda\\times \\Lambda$ and\n$\\Pi\\times \\Pi$, respectively; their values are both positive.\n\nWe assume the following Lipschiz condition on the generator.\n\\begin{hypothesis}\\label{psi}\n(i) The generator $\\psi(\\cdot,0,0):=(\\psi(\\cdot,0,0,i,j))_{i\\in\n\\Lambda,j\\in \\Pi} \\in M^2$.\n\n(ii) There exists a constant $C> 0$ such that,  for each\n$(t,y,y',z,z',i,j)\\in [0,T]\\times R\\times R\\times R^{d}\\times\nR^{d}\\times\\Lambda\\times \\Pi$,\n$$|\\psi(t,y,z,i,j)-\\psi(t,y',z',i,j)|\\le C (|y-y'|+|z-z'|), \\quad a.s.$$\n\\end{hypothesis}\n\nWe make the following assumption on the cost functions $k$ and $l$\nof both players, which is standard in the literature.\n\n\\begin{hypothesis}\\label{k}\n(i) For $i\\in \\Lambda$, $k(i,i)=0$.  For $(i,i')\\in \\Lambda\\times\n\\Lambda$ such that $i\\not=i'$, $k(i,i')> 0$.\n\n(ii) For $j\\in \\Pi$, $l(j,j)=0$.  For $(j,j')\\in \\Pi\\times \\Pi$\nsuch that $j\\not=j'$, $l(j,j')> 0$.\n\n(iii) For any $(i,i',i'')\\in \\Lambda\\times \\Lambda\\times \\Lambda$\nsuch that $i\\not=i'$ and $i'\\not=i''$,\n$$k(i,i')+k(i',i'')> k(i,i'').$$\n\n(iv) For any $(j,j',j'')\\in \\Lambda\\times \\Lambda\\times \\Lambda$\nsuch that $j\\not=j'$ and $j'\\not=j''$,\n$$l(j,j')+l(j',j'')> l(j,j'').$$\n\n\\end{hypothesis}\n\n\n\\subsection{Statement of our switching game}\n\nLet $\\{\\theta_j\\}_{j=0}^\\infty$ be an increasing sequence of\nstopping times with values in $[0,T]$ and $\\forall j$, $\\alpha_j$\nis an ${\\cal F}_{\\theta_j}$-measurable random variable with values\nin $\\Lambda$, and $\\chi$ is the indicator function. We assume\nmoreover that there exists an integer-valued random variable\n$N(\\cdot)$ such that $\\theta_{N}=T$ $P$- $a.s.$ and $N\\in L^2({\\cal F}_T)$. Then we define the\nadmissible switching strategy for player $I$ as follows:\n\\begin{equation}\\label{admi}\na(s)=\\alpha_0\\chi_{\\{\\theta_0\\}}(s)+\\sum_{j=1}^N\n\\alpha_{j-1}\\chi_{(\\theta_{j-1},\\theta_j]}(s).\n\\end{equation}\n\nWe denote by ${\\cal A}$ the set of all these admissible switching\nstrategies for Player $I$, and by ${\\cal A}^i$ the subset of\n${\\cal A}$ consisting of admissible switching strategies starting\nfrom the scheme $i\\in \\Lambda$. In the same way, we denote by ${\\cal\nA}_t$ the set of all the admissible strategies for Player $I$,\nstarting at the time $t$ (or equivalently $\\theta_0=t$ ),  and by\n${\\cal A}^i_t$ the subset of ${\\cal A}_t$ consisting of admissible\nswitching strategies starting at time $t$ from the scheme $i\\in\n\\Lambda$.\n\nFor any $a(\\cdot)\\in {\\cal A}$, we define the associated (cost)\nprocess $A^{a(\\cdot)}$ as follows:\n\\begin{equation}\\label{UA}\nA^{a(\\cdot)}(s)=\\sum_{j=1}^{N-1}\nk(\\alpha_{j-1},\\alpha_j)\\chi_{[\\theta_{j},T]}(s).\n\\end{equation}\nObviously, $A^{a(\\cdot)}(\\cdot)$ is a c\\`adl\\`ag process.\n\nIn an identical way, we define the admissible switching\nstrategy $b(\\cdot)$ for Player $II$, and introduce the notations\n${\\cal B}, {\\cal B}^j, {\\cal B}_t, {\\cal B}_t^j$  for the scheme\n$j\\in \\Pi$ for Player $II$, as well as ${\\cal B}^{b(\\cdot)}$ for\n$b(\\cdot)\\in {\\cal B}$.\n\n\nNow we are in position to introduce the switched BSDEs for both\nplayers. \\begin{equation} \\label{UV} \\begin{array}{rcl} U(s) &=&\\displaystyle\n\\xi_{a(T)b(T)}\n+\\left(A^{a(\\cdot)}(T)-A^{a(\\cdot)}(s)\\right)-\\left(B^{b(\\cdot)}(T)-B^{(\\cdot)}(s)\\right)\\\\\n&&\\displaystyle+\\int_s^T\\psi(r,U(r),V(r),a(r), b(r))\\,\ndr-\\int_s^TV(r)\\, dW(r).\\end{array} \\end{equation} This is a (slightly) generalized\nBSDE: it is equivalent to the following standard BSDE: \\begin{equation}\n\\label{UV1}\\begin{array}{rcl} {\\bar U}(s) &=&\\displaystyle\\xi_{a(T)b(T)}\n+A^{a(\\cdot)}(T)-B^{b(\\cdot)}(T)\\\\\n&&\\displaystyle+\\int_t^T\\psi(r,\\bar{U}(r)-A^{a(\\cdot)}(r)+B^{b(\\cdot)}(r),\\bar{V}(r),a(r))\\,\ndr\\\\\n&&\\displaystyle-\\int_t^T\\bar{V}(r)\\, dW(r),\\end{array}\\end{equation} via the simple\nchange of variable:\n$$\\bar{U}(s)=U(s)+A^{a(\\cdot)}(s)-B^{b(\\cdot)}(s),\\quad \\bar{V}(s)=V(s).$$\n\nHence, BSDE (\\ref{UV}) has a solution in $S^2\\times M^2$. We\ndenote by $\\left(U^{a(\\cdot), b(\\cdot)},V^{a(\\cdot),\nb(\\cdot)}\\right)$ this solution. We note that $U$ is only a c\\`adl\\`ag process.\n\n\nThe switching game problem with the initial scheme\n$(i,j)\\in\\Lambda\\times \\Pi$ is stated as follows: Player $I$ aims\nto minimize $U^{a(\\cdot), b(\\cdot)}(t)$ over $a(\\cdot)\\in {\\cal\nA}_t^i$, while Player $II$ aims to maximize $U^{a(\\cdot),\nb(\\cdot)}(t)$ over $b(\\cdot)\\in {\\cal B}_t^j$.\n\n\n\\section{Existence}\n\nIn this section, we state and prove our existence result for\nRBSDE (\\ref{RBSDEi}).\n\nWe need the following additional technical assumption.\n\n\\begin{hypothesis}\\label{UB} Assume that the generator $\\psi$ is\nuniformly bounded with respect to all its arguments.\n\\end{hypothesis}\n\nwe shall use $\\|\\psi\\|_\\infty$ to denote the least upper bound of\n$|\\psi|$.\n\n\\begin{definition}  A solution to RBSDE (\\ref{RBSDEi}) is defined to be a set\n$(Y, Z, K, L)=\\{Y(t),Z(t),K(t), L(t)\\}_{t\\in [0,T]}$ of predictable\nprocesses with values in $ \\left(R^{m_1\\times\nm_2}\\right)^{1+d+1+1}$ such that $P$-a.s., $t\\mapsto Y(t)$,\n$t\\mapsto K(t)$, and $t\\mapsto L(t)$ are continuous, $t\\mapsto\nZ(t)$ belongs to $L^2(0,T)$, $t\\mapsto \\psi(t,Y_i(t),Z_i(t),i)$\nbelongs to $L^1(0,T)$ and $P$-a.s., RBSDE (\\ref{RBSDEi})\nholds. \\end{definition}\n\nWe are now in position to state the existence result.\n\\begin{theorem}\\label{existence}\nLet Hypotheses \\ref{psi}, \\ref{k} and \\ref{UB} be satisfied.\nAssume that\n$$\\xi\\in L^2(\\Omega,{\\cal F}_T,P;R^m)$$ takes values in\n$\\bar{Q}$. Then RBSDE (\\ref{RBSDEi}) has a solution $(Y,Z,K,L)$ in\n$S^2\\times M^2\\times (N^2)^2$.\n\\end{theorem}\n\nWe shall use a penalization method to construct a solution to\nRBSDE (\\ref{RBSDEi}). We observe (as mentioned in the introducion)\nthat RBSDE (\\ref{RBSDEi}) consists of the $m_2$ systems of\n$m_1$-dimensional obliquely reflected BSDEs of the form like\n(\\ref{RBSDEswitching}):\n\\begin{equation}\\left\\{\\begin{array}{rcl}Y_{ij}(t)&=&\\displaystyle\\xi_{ij}+\\int_t^T\n\\psi(s,Y_{ij}(s),Z_{ij}(s),i,j)\\,\nds\\\\\n& &\\displaystyle-\\int_t^TdK_{ij}(s)+\\int_t^TdL_{ij}(s)-\\int_t^TZ_{ij}(s)\\, dW(s),\\\\\nY_{ij}(t)&\\le&\\displaystyle \\mathop{\\rm min}_{i'\\not=i}\\{Y_{i'j}(t)+k(i,i')\\}, \\\\\n &&\\displaystyle\n\\int_0^T\\left(Y_{ij}(s)-\\mathop{\\rm min}_{i'\\not=i}\\{Y_{i'j}(s)+k(i,i')\\}\\right)dK_{ij}(s)=0;\n\\quad i\\in \\Lambda, \\end{array}\\right.\\end{equation} with the unknown processes being\n$$(Y_{ij}, Z_{ij},K_{ij}; i=1,2, \\ldots, m_1)$$\n(the process $(L_{1j}, \\cdots, L_{m_1j})$ is taken to be\npreviously given) for $j=1,2,\\ldots,m_2$. These $m_2$ systems have\nbeen well studied by Hu and Tang~\\cite{HuTang}. In RBSDE\n(\\ref{RBSDEi}), they are coupled together by the processes\n$(L_{1j}, \\cdots, L_{m_1j})$ through the constraint \\begin{equation}\nY_{ij}(t)\\ge \\mathop{\\rm max}_{j'\\not=j}\\{Y_{ij'}(t)-l(j,j')\\}, \\quad (i,j)\\in\n\\Lambda\\times \\Pi \\end{equation} and the minimal boundary condition: \\begin{equation}\n\\int_0^T\\left(Y_{ij}(s)-\\mathop{\\rm max}_{j'\\not=j}\\{Y_{ij'}(s)-l(j,j')\\}\\right)dL_{ij}(s)=0,\n\\quad (i,j)\\in \\Lambda\\times \\Pi. \\end{equation} Therefore, it is natural to\nconsider the following penalized system of RBSDEs (the unknown\nprocesses are $(Y_{ij}, Z_{ij},K_{ij}; i\\in \\Lambda, j\\in \\Pi)$):\n\\begin{equation}\\label{penalized}\\left\\{\\begin{array}{rcl}Y_{ij}(t)&=&\\displaystyle\\xi_{ij}+\\int_t^T\n\\psi(s,Y_{ij}(s),Z_{ij}(s),i,j)\\,\nds\\\\\n& &\\displaystyle\n+n\\sum_{j'=1}^{m_2}\\int_t^T\\left(Y_{ij}(s)-Y_{ij'}(s)+l(j,j')\\right)^-\\,\nds \\\\\n& &\\displaystyle-\\int_t^TdK_{ij}(s)-\\int_t^TZ_{ij}(s)\\, dW(s);\\\\\nY_{ij}(t)&\\le&\\displaystyle \\mathop{\\rm min}_{i'\\not=i}\\{Y_{i'j}(t)+k(i,i')\\}; \\\\\n &&\\displaystyle\n\\int_0^T\\left(Y_{ij}(s)-\\mathop{\\rm min}_{i'\\not=i}\\{Y_{i'j}(s)+k(i,i')\\}\\right)dK_{ij}(s)=0;\n\\quad (i,j)\\in \\Lambda\\times \\Pi. \\end{array}\\right.\\end{equation} Note that when $j'=j$, we have, in\nview of Hypothesis \\ref{k} (ii),\n\\begin{equation}(Y_{ij}(s)-Y_{ij'}(s)+l(j,j'))^-=0. \\end{equation}\n\nA striking difference between RBSDE (\\ref{penalized}) and the RBSDE which is studied by Hu and Tang~\\cite{HuTang},\nlies in the fact  that the $i$-th set of unknown variables $$(y_i,z_i,k_i)\\in R\\times R^d\\times R$$ in the latter\nis replaced in the former with\n$$((Y_{ij})_{j\\in \\Pi},(Z_{ij})_{j\\in \\Pi}, (K_{ij})_{j\\in \\Pi})\\in\nR^{m_2}\\times \\left(R^{m_2}\\right)^d\\times R^{m_2}.\n$$\nBy slightly adapting the relevant arguments in our previous work~\\cite{HuTang},\nwe can show the following theorem.\n\n\\begin{theorem}\\label{th2}  For any integer $n$,\nRBSDE (\\ref{penalized}) has a unique solution $(Y^n,Z^n, K^n)$ in\nthe space $S^2\\times M^2\\times N^2$. Furthermore,  we have\nthe following representation:\n\\begin{equation} Y^n_{ij}(t)=\\mathop{\\mbox{\\rm essinf }}_{a(\\cdot)\\in {\\cal A}_t^{i}}\nU_{j}^{a(\\cdot),n}(t), \\quad (t,i,j)\\in [0,T]\\times \\Lambda\\times \\Pi.\n\\end{equation}\nHere, for any $a(\\cdot)\\in {\\cal A}^i_t$,\n$(U^{a(\\cdot),n},V^{a(\\cdot),n})$\nis the unique solution to the following BSDE: \\begin{equation}\\begin{array}{rcl} U_j(s)&=&\n\\displaystyle\\xi_{a(T)j}+[A^{a(\\cdot)}(T)-A^{a(\\cdot)}(s)]\n+\\int_s^T \\psi(r, U_j,V_j,a(r), j)\\, dr\\\\\n&&\\displaystyle +n\\sum_{j'=1}^{m_2}\\int_s^T(U_j-U_{j'}+l(j,j'))^-\\,\ndr-\\int_s^TV_j(r)\\, dW(r),\\\\\n&&\\displaystyle \\qquad\\qquad j\\in \\Pi, \\quad s\\in [t,T].\\end{array} \\end{equation}\n\\end{theorem}\n\nIntuitively, as $n$ tends to $+\\infty$, we expect that the sequence\nof solutions $$\\{(Y^n,Z^n, K^n)\\}_{n=1}^\\infty$$ together with the\npenalty term\n$$\nL^n_{ij}(t):=n\\sum_{j'=1}^{m_2}\\int_0^t\\left(Y_{ij}(s)-Y_{ij'}(s)+l(j,j')\\right)^-\\,\nds, \\quad (t,i,j)\\in [0,T]\\times \\Lambda\\times \\Pi\n$$\nwill have a limit $(Y,Z, K, L)$, which solves RBSDE (\\ref{RBSDEi}).\n\nFor this purpose, it is crucial to prove that the  penalty term is bounded\nin some suitable sense. Then we are naturally led to compute\n$$ \\left(Y_{ij}(t)-Y_{ij'}(t)+l(j,j')\\right)^-,$$\nusing It\\^o-Tanaka's formula, as done in  Hu and Tang~\\cite{HuTang}. However,\nin our present situation, the additional term $K^n$ appears in RBSDE (\\ref{penalized}),\nwhich gives rise to a serious difficulty  to derive the bound of $L^n$  in the preceding  procedure.\nIn what follows, we shall use the respresentation result for $Y^n$  in the preceding theorem\nto get around the difficulty.\n\nWe have the following lemma \\begin{lemma} For $j,j'\\in\\Pi, a(\\cdot)\\in {\\cal A}_t^i$,  we have \\begin{equation}\nn\\left(U_{j}^{a(\\cdot),n}(s)-U_{j'}^{a(\\cdot),n}(s)+l(j,j')\\right)^-\\le\n2\\|\\psi\\|_\\infty, \\quad s\\in [t,T].\\end{equation}\\end{lemma}\n\n{\\bf Proof. } We suppress the superscripts\n ${(a(\\cdot),n)}$ of $U_{j}^{a(\\cdot),n}$ and $U_{j'}^{a(\\cdot),n}$ for simplicity.\n\nThe process\n$\\bar{U}_{jj'}(s):=U_{j}(s)-U_{j'}(t)+l(j,j'),  s\\in [t,T]$ satisfies the\nfollowing BSDE: \\begin{equation}\\begin{array}{rcl} &&\\displaystyle \\bar{U}_{jj'}(s)\\\\\n&=&\\displaystyle\n\\bar{U}_{jj'}(T)+\\int_s^T \\left[\\psi(r,U_{j}(r),\nV_{j}(r),a(r),j)-\\psi(r,U_{j'}(r), V_{j'}(r), a(r), j')\\right]\\, dr\n\\\\\n&&\\displaystyle+n\\sum_{j''=1}^{m_2}\\int_s^T \\bar{U}_{jj''}(r)^-\\, dr\n-n\\sum_{j''=1}^{m_2}\\int_s^T\\bar{U}_{j'j''}(r)^-\\, dr\\\\\n&&\\displaystyle -\\int_s^T(V_{j}(r)-V_{j'}(r))\\, dW(r), \\quad s\\in [t,T]. \\end{array} \\end{equation}\n\nApplying Tanaka's formula (see, e.g. \\cite{RY}), we have\n \\begin{equation}\\begin{array}{rcl}\n&&\\displaystyle \\bar{U}_{jj'}(s)^-+n\\sum_{j''=1}^{m_2}\\int_s^T\n\\chi_{{\\cal L}_{jj'}^-}(r)\\bar{U}_{jj''}(r)^-\\,\ndr\\\\\n&&\\displaystyle-n\\sum_{j''=1}^{m_2}\\int_s^T\\chi_{{\\cal L}_{jj'}^-}(r)\\bar{U}_{j'j''}(r)^-\\,\ndr+{\\frac{1}{2}}\\int_s^T\\, d{\\widehat L}_{jj'}(r)\\\\\n&=&\\displaystyle \\int_s^T\\chi_{{\\cal L}_{jj'}^-}(r) \\left[\\psi(r,U_{j}(r),\nV_{j}(r),a(r),j)-\\psi(r,U_{j'}(r), V_{j'}(r), a(r),j')\\right]\\, dr\n\\\\ &&\\displaystyle\n-\\int_s^T\\chi_{{\\cal L}_{jj'}^-}(r)(V_{j}(r)-V_{j'}(r))\\, dW(r), \\quad s\\in [t,T]\\end{array} \\end{equation}\nwhere \\begin{equation} \\begin{array}{c}{\\cal L}_{jj'}^-:=\\{(s,\\omega)\\in [t,T]\\times \\Omega:\\bar{U}_{jj'}(s)<0\\}, \\end{array}\n\\end{equation} and ${\\widehat L}_{jj'}$ is the local time of the process $\\bar{U}_{jj'}$\nat $0$.\nWe have\n  \\begin{equation}\\label{Tanaka}\\begin{array}{rcl}\n&&\\displaystyle \\bar{U}_{jj'}(s)^-+n\\int_s^T\n\\bar{U}_{jj'}(r)^-\\,\ndr+{1\\over 2}\\int_s^T \\, d{\\widehat L}_{jj'}(r)\\\\[0.5cm]\n&=&\\displaystyle \\int_s^T {{\\cal L}}_{jj'}^-(r)\\left[\\psi(r,U_{j}(r),\nV_{j}(r),a(r),j)-\\psi(r,U_{j'}(r), V_{j'}(r), a(r), j')\\right]\\,\ndr\\\\[0.5cm]\n&&\\displaystyle-\\int_s^T{{\\cal L}}_{jj'}^-(r)\\left(V_{j}(r)-V_{j'}(r)\\right)\\,\ndW(r)\\\\\n&&\\displaystyle +n\\int_t^T({{\\cal L}}_{jj'}(r))^-(\\bar{U}_{j'j}(r))^- \\,\ndr\\\\\n&&\\displaystyle\n+n\\sum_{j''\\not= j,j''\\not=j'}\\int_s^T{{\\cal L}}_{jj'}^-(r)[\\bar{U}_{j'j''}(r)^-\n-\\bar{U}_{jj''}(r)^-]\\, dr.\n \\end{array} \\end{equation}\n\nWe claim that the integrands of the integrals in the last two\nterms of (\\ref{Tanaka}) are all less than or equal to zero. In\nfact, since\n$$\\{y\\in R^m: y_j-y_{j'}+l(j,j')<0\\}\\cap\\{y\\in R^m:y_{j'}-y_{j}+l(j',j)<0\\}=\\emptyset,$$\ndue to the fact that $$l(j,j')+l(j',j)>l(j,j)=0,$$ we have\n \\begin{equation}\n{{\\cal L}}_{jj'}^-\\bar{U}_{j'j}^- =0, \\quad\nj,j'\\in\\Pi.\\end{equation}\n\nSecondly, for $j,j',j''\\in\\Pi$,  taking into consideration both\nHypothesis \\ref{k} (ii), i.e.,\n$$l(j,j')+l(j',j'')> l(j,j''),$$\nand the elementary inequality that $x_1^--x_2^-\\le (x_1-x_2)^-$,\nfor any two real numbers $x_1$ and $x_2$, we have \\begin{equation} \\begin{array}{rcl}\n&&{{\\cal L}}_{jj'}^-\\left(\\bar{U}_{j'j''}^-\n-\\bar{U}_{jj''}^-\\right)\\\\\n &\\le&{{\\cal L}}_{jj'}^-\n\\left(\\bar{U}_{j'j''}-\\bar{U}_{jj''}\\right)^-= {{\\cal L}}_{jj'}^- \\left(U_{j'}-U_{j}+l(j',j'')-l(j,j'')\\right)^-.\n \\end{array}\\end{equation}\nThe last equality holds in the last relations, since $$ \\{y\\in\nR^m:y_j-y_{j'}+l(j,j')<0\\}\\cap\\{y\\in\nR^m:y_{j'}-y_{j}+l(j',j'')-l(j,j'')<0\\}=\\emptyset.$$\n\n\nConcluding the above, we have\n\\begin{equation}\\begin{array}{rcl}\n&&\\displaystyle \\bar{U}_{jj'}(s)^-+n\\int_s^T\n\\bar{U}_{jj'}(r)^-\\,\ndr+{1\\over2}\\int_s^T \\, d{\\widehat L}_{jj'}(r)\\\\[0.5cm]\n&=&\\displaystyle \\int_s^T {{\\cal L}}_{jj'}^-(r)\\left[\\psi(r,U_{j}(r),\nV_{j}(r),a(r),j)-\\psi(r,U_{j'}(r), V_{j'}(r), a(r), j')\\right]\\,\ndr\\\\[0.5cm]\n&&\\displaystyle-\\int_s^T{{\\cal L}}_{jj'}^-(r)\\left(V_{j}(r)-V_{j'}(r)\\right)\\,\ndW(r).\\end{array}\\end{equation}\nUsing It\\^o's formula, we have\n\\begin{equation}\\begin{array}{rcl}\n&&\\displaystyle \\mathop{\\rm exp}{(-ns)}\\bar{U}_{jj'}(s)^-\n+{1\\over2}\\int_s^T \\mathop{\\rm exp}{(-nr)}\\, d{\\widehat L}_{jj'}(r)\\\\[0.5cm]\n&=&\\displaystyle \\int_s^T \\mathop{\\rm exp}{(-nr)}{{\\cal L}}_{jj'}^-(r)[\\psi(r,U_{j}(r),\nV_{j}(r),a(r),j)-\\psi(r,U_{j'}(r), V_{j'}(r), a(r), j')]\\,\ndr\\\\[0.5cm]\n&&\\displaystyle-\\int_s^T{{\\cal L}}_{jj'}^-(r)\\mathop{\\rm exp}{(-nr)}\\left(V_{j}(r)-V_{j'}(r)\\right)\\,\ndW(r).\\end{array}\\end{equation}\nTaking the conditional expectation with respect to ${\\cal F}_s$ on the both sides of the last equality,\nin view of Hypothese~\\ref{UB}, we have\n\\begin{equation} \\begin{array}{rcl}&&\\displaystyle\n\\mathop{\\rm exp}{(-ns)}\\left(U_{j}(s)-U_{j'}(s)+l(j,j')\\right)^+\\\\\n&\\le\n&\\displaystyle\n2\\int_s^T\\mathop{\\rm exp}{(-nr)}\\|\\psi\\|_\\infty\\, dr \\\\\n&\\le &\\displaystyle  2n^{-1}\\|\\psi\\|_\\infty\n[\\mathop{\\rm exp}{(-ns)}-\\mathop{\\rm exp}{(-nT)}] \\\\\n&\\le &\\displaystyle 2n^{-1}\\|\\psi\\|_\\infty\\mathop{\\rm exp}{(-ns)}. \\end{array}\\end{equation}\nThis ends the proof. \\signed {$\\sqr69$}\n\n\nWe have the following estimates for the $L^\\infty$ bound of $Y_{ij}^n$.\n\n\\begin{lemma}\\label{2} The sequence $\\{Y_{ij}^n(t)\\}_{n=1}^\\infty$ is decreasing.\nMoreover, \\begin{equation}  -E\\left[|\\xi| |{\\cal F}_t\\right]-|\\psi|_\\infty T\n\\le Y_{ij}^n(t)\n\\le E\\left[|\\xi| |{\\cal F}_t\\right]+3|\\psi|_\\infty T\n; \\quad\nE\\left[\\mathop{\\rm sup}_t|Y_{ij}^n(t)|^2\\right]\\le c. \\end{equation} \\end{lemma}\n\n{\\bf Proof. }  In view of the comparison result for multi-dimensional RBSDEs of Tang and Zhong~\\cite{TangZhong}\n(which is a natural generalization to RBSDEs of the comparison theorem for multi-dimensional BSDEs),\nwe see that the sequence $\\{Y_{ij}^n(t)\\}_{n=1}^\\infty$ is decreasing.\n\nWe have the following two facts:\n\n(1) $U_{j}^{a(\\cdot),n}(s)\\ge -E\\left[|\\xi|\n|{\\cal F}_t\\right]-|\\psi|_\\infty T. $\n\n(2) Take $\\bar a(\\cdot)\\equiv i.$ We have \\begin{equation} U_{j}^{\\bar\na(\\cdot),n}(s)\\le  E\\left[|\\xi| |{\\cal F}_t\\right]+|\\psi|_\\infty\nT+2|\\psi|_\\infty T.\\end{equation}\nIn view of the respresentation formula in Theorem~\\ref{th2}, we conclude the proof.\n\\signed {$\\sqr69$}\n\nTherefore, from the previous representation result, we have\n\n\\begin{lemma}\\label{3} We have \\begin{equation} n\\left(Y_{ij}^n(t)-Y_{ij'}^n(t)+l(j,j')\\right)^-\\le\n2 \\|\\psi\\|_\\infty. \\end{equation} \\end{lemma}\n\n{\\bf Proof. } We have \\begin{equation} Y_{ij}^n(t)-Y_{ij'}^n(t)+l(j,j')\\le\n\\mathop{\\hbox{\\rm ess$\\,$\\rm sup$\\,$}}_{a(\\cdot)\\in\n{\\cal A}_t^{i}}\\left(U_{j}^{a(\\cdot),n}(t)-U_{j'}^{a(\\cdot),n}(t)+l(j,j')\\right).\\end{equation}\nHence, \\begin{equation}\\begin{array}{rcl} &&\\displaystyle\n\\left(Y_{ij}^n(t)-Y_{ij'}^n(t)+l(j,j')\\right)^-\\\\\n&\\le& \\displaystyle \\mathop{\\hbox{\\rm ess$\\,$\\rm sup$\\,$}}_{a(\\cdot)\\in\n{\\cal A}_t^{i}}\\left(U_{j}^{a(\\cdot),n}(t)-U_{j'}^{a(\\cdot),n}(t)+l(j,j')\\right)^-\\le\n2n^{-1}\\|\\psi\\|_\\infty.\\end{array}\\end{equation}\n\\signed {$\\sqr69$}\n\nWe have\n\n\\begin{lemma} The pair of processes $(Z_{ij}^n, K_{ij}^n)$ are uniformly bounded in $M^2\\times N^2$ for $(i,j)\\in \\Lambda\\times \\Pi$.\n\\end{lemma}\n\n{\\bf Proof. } From the RBSDE for $Y_{ij}^n$, using It\\^o's formula and Lemmas~\\ref{2} and~\\ref{3}, we have \\begin{equation}\\begin{array}{rcl}\n&&\\displaystyle E|Y_{ij}^n(0)|^2 +E\\int_0^T|Z_{ij}^n(s)|^2\\,\nds\\\\\n&\\le &\\displaystyle   E|Y_{ij}^n(T)|^2 +\nE\\int_0^T|Y_{ij}^n(s)|\nn\\left(Y_{ij}^n(s)-Y_{ij'}^n(s)+l(j,j')\\right)^-\\, ds\\\\\n&&\\displaystyle +\n E\\int_0^T|Y_{ij}^n(s)|\\cdot\n |\\psi(s,Y_{ij}^n(s), Z_{ij}^n(s),a(s),j)-\\psi(s,Y_{ij'}^n(s), Z_{ij'}^n(s),a(s),j')|\\, ds\\\\\n&&\\displaystyle  +E\\int_0^T|Y_{ij}^n(s)|\\, dK_{ij}^{n}(s)\\\\\n&\\le &\\displaystyle  E|Y_{ij}^n(T)|^2 +\n3\\|\\psi\\|_\\infty E\\int_0^T  |Y_{ij}^n(s)|\\, ds\n+E\\left[\\mathop{\\rm sup}_t|Y_{ij}^n(t)|^2K_{ij}^{n}(T)\\right]\\\\\n &\\le&\\displaystyle  C_\\epsilon +\\epsilon\nE[(K_{ij}^{n}(T))^2] \\end{array} \\end{equation} and \\begin{equation}\\begin{array}{rcl}&&\\displaystyle  E[(K_{ij}^{n}(T))^2]\\\\\n&\\le&\\displaystyle  C E|Y_{ij}^n(T)|^2+CE|Y_{ij}^n(0)|^2\n+CE\\int_0^T n\\left[\\left(Y_{ij}^n(s)-Y_{ij'}^n(s)+l(j,j')\\right)^-\\right]^2\\, ds  \\\\\n&&\\displaystyle +C E\\int_0^T\n  |\\psi(s,Y_{ij}^n(s), Z_{ij}^n(s),a(s),j)-\\psi(s,Y_{ij'}^n(s), Z_{ij'}^n(s),a(s),j')|^2\\, ds\n  \\\\\n&&\\displaystyle\n+CE\\int_0^T|Z_{ij}^n(T)|^2\\, ds\\\\\n&\\le &\\displaystyle c +CE\\int_0^T|Z_{ij}^n(T)|^2\\, ds. \\end{array}\\end{equation}\nThey together conclude the proof.\n\\signed {$\\sqr69$}\n\nDefine \\begin{equation}\n\\beta_{ij}^n(s):= n\\sum_{j'=1}^{m_2}\\left(Y_{ij}^n(s)-Y_{ij'}^n(s)+l(j,j')\\right)^-.\\end{equation}\nThen,\n\\begin{equation} L_{ij}^{n}(t)= \\int_0^t\\beta_{ij}^n(s)\\, ds.\n\\end{equation}\n\nWe have\n\n\\begin{lemma}  \\label{5} For $(i,j)\\in \\Lambda\\times \\Pi$ and integer $n$, there is a uniformly bounded process  $\\alpha_{ij}^n$  such that\n$K_{ij}^{n}$ has the following form:\n\\begin{equation}\nK_{ij}^{n}(t)= \\displaystyle  \\int_0^t\\alpha_{ij}^n(s)\\, ds, \\quad t\\in [0,T].\\end{equation}\nThe two matrix-valued processes\n$$\\{\\alpha^n\\}_{n=1}^\\infty:=\\{(\\alpha_{ij}^n; i\\in \\Lambda, j\\in\n\\Pi)\\}_{n=1}^\\infty$$\nand\n$$\\{\\beta^n\\}_{n=1}^\\infty:=\\{(\\beta_{ij}^n; i\\in \\Lambda, j\\in \\Pi)\\}_{n=1}^\\infty$$\n are  bounded in $M^2$.\\end{lemma}\n\n{\\bf Proof. } Fix the integer $n$. Consider the following penalized BSDEs:\n \\begin{equation}\\begin{array}{rcl} Y_{ij}^{n,m}(t) &=&\\displaystyle\n \\xi_{ij}+\\int_t^T\\left[\\psi(s,Y_{ij}^n(s),Z_{ij}^n(s),i,j)+\\beta_{ij}^n(s)\\right]\\,\n ds\n \\\\[3mm]\n &&\\displaystyle-m\\sum_{i'=1}^{m_1}\\int_t^T\\left(Y_{ij}^{n,m}-Y_{i'j}^{n,m}-k(i,i')\\right)^+\\,\n ds\\\\[4mm]\n &&\\displaystyle -\\int_t^TZ_{ij}^{n,m}(s)\\, dW(s),\\end{array} \\end{equation}\n with $(i,j)\\in \\Lambda\\times \\Pi$  and $t\\in [0,T]$.\n It has a unique solution, denoted by $(Y_{ij}^{n,m}, Z_{ij}^{n,m})$.\n\n Proceeding identically\n as in Lemma~\\ref{2}, we can prove that \\begin{equation}\n \\alpha^{n,m}_{ij}:=m\\sum_{i'=1}^{m_1}\\left(Y_{ij}^{n,m}-Y_{i'j}^{n,m}-k(i,i')\\right)^+\\le c. \\end{equation}\n  Therefore, $\\{\\alpha_{ij}^{n,m}\\}_{m=1}^\\infty$   has a weak limit in\n  ${\\cal L}_{\\cal F}^2(0,T)$, denoted by $\\alpha_{ij}^n$.  Then $\\alpha_{ij}^n$\n  is also uniformly bounded by the same constant $c$.\n\nDefine\n\\begin{equation}\n \\displaystyle K_{ij}^{n,m}(t):= \\int_0^t\n \\alpha_{ij}^{n,m}(s)\\, ds, \\quad t\\in [0,T].\\end{equation}\n From\n Hu and Tang~\\cite{HuTang}, we have \\begin{equation}\\begin{array}{rcl}\\displaystyle\n \\lim_{m\\to\n \\infty}Y_{ij}^{n,m}(t)&=&Y_{ij}^{n}(t), \\\\\n  \\displaystyle \\lim_{m\\to\n \\infty}Z_{ij}^{n,m}(t)&=&Z_{ij}^{n}(t), \\\\\n \\displaystyle \\lim_{m\\to\n\\infty}K_{ij}^{n,m}(t)&=&\\displaystyle \\int_0^t\\alpha_{ij}^n(s)\\, ds=K_{ij}^{n}(t). \\end{array} \\end{equation}\n\\signed {$\\sqr69$}\n\n\nWe have \\begin{lemma}    The sequence $\\{Y^n, Z^n\\}$ has a strong limit $(Y,Z)$  in $S^2\\times M^2$.\nThe two sequences $\\{\\alpha^n\\}$ and $\\{\\beta^n\\}$ have\n subsequences which converge to $\\alpha$ and $\\beta$ weakly in $M^2$, respectively.\\end{lemma}\n\n {\\bf Proof. } Note that $Y^n_{ij}$ is decreasing in $n$. In\n view of Lemma~\\ref{2}, using Lebesgue dominant convergence\n theorem, we can show the strong convergence of  $\\{Y^n\\}$ in the\n space $M^2$. Note that  $(Y^n, Z^n)$  solves the following BSDE:\n \\begin{equation}\\label{app}\\begin{array}{rcl}\\displaystyle  Y_{ij}^n(t)&=&\\displaystyle  \\xi_{ij}\n +\\int_t^T\\left(\\psi(s, Y_{ij}^n(s), Z_{ij}^n(s),i,j)+\\beta_{ij}^n(s)-\\alpha_{ij}^n(s)\\right)\\,\nds\\\\[3mm]\n &&\\displaystyle -\\int_t^TZ_{ij}^n(s)\\, dW(s),\\quad (i,j)\\in \\Lambda\\times \\Pi, \\end{array} \\end{equation}\n with $\\{\\alpha^n\\}$ and $\\{\\beta^n\\}$  being bounded in $M^2$.\n\nWe now prove the strong convergence of $Z^n$.\nUsing It\\^o's formula, we have\n\\begin{equation}\\begin{array}{rcl}&&\\displaystyle  |Y_{ij}^n(0)-Y_{ij}^m(0)|^2\n+E\\int_0^T|Z^n_{ij}(s)-Z^m_{ij}(s)|^2\\, ds\\\\\n&=&\\displaystyle\n2E\\int_0^T\\left(Y_{ij}^n(s)-Y_{ij}^m(s)\\right)\\left(\\psi(s,Y_{ij}^n(s),Z_{ij}^n(s),\ni,j)-\\alpha_{ij}^n(s)+\\beta_{ij}^n(s)\\right)\\, ds\\\\\n&&\\displaystyle-\n2E\\int_0^T\\left(Y_{ij}^n(s)-Y_{ij}^m(s)\\right)\\left(\\psi(s,Y_{ij}^m(s),Z_{ij}^m(s),\ni,j)-\\alpha_{ij}^m(s)+\\beta_{ij}^m(s)\\right)\\, ds. \\end{array} \\end{equation} Using\nCauchy-Schwartz inequality and Hypothesis~\\ref{UB}, we have \\begin{equation} \\lim_{n,m\\to\n\\infty}E\\int_0^T|Z^n-Z^m|^2\\, ds=0. \\end{equation}\n\nThen, it\nis standard to show the strong convergence of  $\\{Y^n\\}$ in the\nspace $S^2$.\n\nSince (see Lemma~\\ref{5}) we have\n \\begin{equation} \\|\\alpha^n\\|_{M^2}\\le c, \\quad\n \\|\\beta^n\\|_{M^2}\\le c, \\end{equation}\n the last assertion of the lemma is obvious. \\signed {$\\sqr69$}\n\n\n\\medskip\n\n{\\bf Proof of Theorem \\ref{existence} :}\nDefine  for $(i,j,t)\\in \\Lambda\\times \\Pi\\times [0,T]$,\n\\begin{equation} K_{ij}(t):=\\int_0^t\\alpha_{ij}(s)\\, ds, \\quad L_{ij}(t):=\\int_0^t\\beta_{ij}(s)\\, ds \\end{equation}\nand\n\\begin{equation} K:=(K_{ij}), \\quad L:=(L_{ij}).\\end{equation}\nWe shall show that $(Y,Z,K,L)$  solves RBSDE (\\ref{RBSDEi}).\n\nFirst, fixing $t\\in [0,T]$ and letting $n\\to \\infty$ in BSDE~(\\ref{app}),\nwe take the weak limit in $L^2({\\cal F}_T)$. Then,\nwe see that $(Y,Z,K,L)$ solves the following BSDE:\n\\begin{equation}\\begin{array}{rcl}\\displaystyle  Y_{ij}(t)&=&\\displaystyle  \\xi_{ij}\n+\\int_t^T\\psi(s, Y_{ij}(s), Z_{ij}(s),i,j)\\, ds-\\int_t^T\\,\ndK_{ij}(s)\\\\[3mm]\n&&\\displaystyle +\\int_t^T\\, dL_{ij}(s)-\\int_t^TZ_{ij}(s)\\, dW(s),\\quad (i,j)\\in \\Lambda\\times \\Pi. \\end{array} \\end{equation}\n\nNow we check out the boundary conditions.\nFrom Hu and Tang~\\cite{HuTang}, we have \\begin{equation}\nE\\int_0^T\\left(Y_{ij}^n(s)-\\mathop{\\rm min}_{i'\\not=i}\\{Y_{i'j}^n(s)+k(i,i')\\}\\right)^-\\alpha_{ij}^n(s)\\,\nds=0.\\end{equation} Setting $n\\to \\infty$, we have \\begin{equation}\nE\\int_0^T\\left(Y_{ij}(s)-\\mathop{\\rm min}_{i'\\not=i}\\{Y_{i'j}(s)+k(i,i')\\}\\right)^-\\alpha_{ij}(s)\\,\nds=0.\\end{equation}\nThat is,\n\\begin{equation} E\\int_0^T\\left(Y_{ij}(s)-\\mathop{\\rm min}_{i'\\not=i}\\{Y_{i'j}(s)+k(i,i')\\}\\right)^-\\,\n dK_{ij}(s)=0.\\end{equation}\n\nOn the other hand, from the construction, we have \\begin{equation}\nE\\int_0^T\\left(Y_{ij}^n(s)-\\mathop{\\rm max}_{j'\\not=j}\\{Y_{ij'}^n(s)-l(j,j')\\}\\right)^+\\beta_{ij}^n(s)\\,\nds=0.\\end{equation} Setting $n\\to \\infty$, we have \\begin{equation}\nE\\int_0^T\\left(Y_{ij}(s)-\\mathop{\\rm max}_{j'\\not=j}\\{Y_{ij'}(s)-l(j,j')\\}\\right)^+\\beta_{ij}(s)\\,\nds=0.\\end{equation}\nThat is,\n\\begin{equation} E\\int_0^T\\left(Y_{ij}(s)-\\mathop{\\rm max}_{j'\\not=j}\\{Y_{ij'}(s)-l(j,j')\\}\\right)^+\\,\n dL_{ij}(s)=0.\\end{equation}    The proof is then complete.\n\\signed {$\\sqr69$}\n\n\\section{Uniqueness}\n\nIn this section, we prove the uniqueness by a verification method.\nLet $(Y, Z, K, L)$ be a solution in the space\n$S^2\\times M^2\\times N^2\\times N^2$ to RBSDE (\\ref{RBSDEi}). We shall prove that $\\widetilde\nY$ is in fact the (vector) value for an optimal switching problem\nof RBSDEs. For this purpose, we introduce the following optimal\nswitching problem for RBSDEs.\n\nFor $a(\\cdot)\\in {\\cal A}_t^i$, we denote by $(U_j^{a(\\cdot)}, V_j^{a(\\cdot)}, K_j^{a(\\cdot)}; j\\in \\Pi)$ the uniuque\nsolution of the following RBSDEs: \\begin{equation}\\begin{array}{rcl}\nU_{j}^{a(\\cdot)}(s)&=&\\displaystyle  \\xi_{a(T)j}+\\left(A^{a(\\cdot)}(T)-A^{a(\\cdot)}(s)\\right)\\\\\n&&\\displaystyle +\\int_s^T\\psi(r, U_j^{a(\\cdot)}(r),\nV_j^{a(\\cdot)}(r),a(r),j)\\,\ndr\\\\\n&& \\displaystyle +\\int_s^T\\, dL_{j}(r)-\\int_s^TV_{j}^{a(\\cdot)}(r)\\,\ndW(r), \\quad s\\in [t, T];\\\\[3mm]\n&&\\displaystyle\n\\int_0^T\\left(U^{a(\\cdot)}_{j}-\\mathop{\\rm max}_{j'\\not=j}\\{U^{a(\\cdot)}_{j'}-l(j,j')\\}\\right)\\,\ndL_{j}(r)=0; \\quad j\\in \\Pi. \\end{array} \\end{equation}\n\nWe have the following representation for the first component of\nthe adapted solution to RBSDE (\\ref{RBSDEi}), which immediately\nimplies the uniqueness of the adapted solution to RBSDE\n(\\ref{RBSDEi}).\n\n\\begin{theorem}\\label{uniqueness} Suppose that\nHypotheses \\ref{psi} and \\ref{k} hold. Assume that  the random variable\n$$\\xi\\in L^2(\\Omega,{\\cal F}_T,P;R^m)$$ takes values in $\\bar{Q}$.\nLet $(Y, Z, K, L)$ be a solution in the space $S^2\\times M^2\\times N^2\\times N^2$ to RBSDE\n(\\ref{RBSDEi}). Then we have the representation: \\begin{equation} Y_{ij}(t)=\n\\mathop{\\mbox{\\rm essinf }}_{a(\\cdot)\\in {\\cal A}_t^{i}}U_{j}^{a(\\cdot)}(t), \\quad (i,j,t)\\in \\Lambda\\times \\Pi\\times [0,T]. \\end{equation}\n\\end{theorem}\n\n\n{\\bf Proof. } Assume $t=0$ without loss of generality.\n Otherwise, it suffices to consider the admissible\nswitching strategies starting at time $t$.\n\nFor the following $a(\\cdot)\\in {\\cal A}_0^i:$\n\\begin{equation} a(s)=i\\chi_{0}(s)+\\sum_{p=1}^N\n\\alpha_{p-1}\\chi_{(\\theta_{p-1}, \\theta_p]}(s), \\quad s\\in [0,T],\\end{equation}\nwe define for $s\\in [0,T]$ and $j\\in \\Pi$,\n\\begin{eqnarray} \\widetilde Y^{a(\\cdot)}_{j}(s)&:=&\\sum_{p=1}^N\nY_{\\alpha_{p-1},j}(s)\\chi_{[\\theta_{p-1},\n\\theta_p)}(s)+\\xi_{a(T)}\\chi_{\\{T\\}}(s),\\\\\n\\widetilde Z^{a(\\cdot)}_{j}(s)&:=&\\sum_{p=1}^N\nZ_{\\alpha_{p-1},j}(s)\\chi_{[\\theta_{p-1}, \\theta_p)}(s),\\\\\n\\widetilde K^{a(\\cdot)}_{j}(s)&:=&\\sum_{p=1}^N \\int_{\\theta_{p-1}\\wedge\ns}^{\\theta_p\\wedge s}d K_{\\alpha_{p-1},j}(r), \\\\\n\\widetilde L^{a(\\cdot)}_{j}(s)&:=&\\sum_{p=1}^N \\int_{\\theta_{p-1}\\wedge\ns}^{\\theta_p\\wedge s}d L_{\\alpha_{p-1},j}(r).\n \\end{eqnarray}\n\n\nNoting that $\\widetilde Y^{a(\\cdot)}_{j}(\\cdot)$ is a c\\`adl\\`ag process with\njump $ Y_{\\alpha_p, j}(\\theta_p)-\nY_{\\alpha_{p-1},j}(\\theta_p)$ at $\\theta_p$, $p=1,\\cdots,N-1$, we\ndeduce that\n\\begin{eqnarray*}\n& &\\widetilde Y^{a(\\cdot)}_{j}(s)-\\widetilde Y^{a(\\cdot)}_{j}(0)\\\\\n&=&\\sum_{p=1}^N \\int_{\\theta_{p-1}\\wedge s}^{\\theta_p\\wedge\ns}[-\\psi(r, Y_{\\alpha_{p-1},j}(r), Z_{\\alpha_{p-1},j}(r),\\alpha_{p-1},j)\\,\ndr+Z_{\\alpha_{p-1},j}(r)\\,\ndW(r)\\\\\n&&\\displaystyle +dK_{\\alpha_{p-1},j}(r)-d\nL_{\\alpha_{p-1},j}(r)]\\\\\n& &+\\sum_{q=1}^{N-1}[Y_{\\alpha_q, j}(\\theta_q)-\nY_{\\alpha_{q-1},j}(\\theta_q)]\\chi_{[\\theta_q,T]}(s)\\\\\n&=&\\sum_{p=1}^N \\int_{\\theta_{p-1}\\wedge s}^{\\theta_p\\wedge\ns}[-\\psi(r, \\widetilde Y_{\\alpha_{p-1},j}(r), \\widetilde Z_{\\alpha_{p-1},j}(r),\\alpha_{p-1},j)\\,\ndr+ \\widetilde Z_{\\alpha_{p-1},j}(r)\\,\ndW(r)\\\\\n&&\\displaystyle +d\\widetilde K_{\\alpha_{p-1},j}(r)-d\n\\widetilde L_{\\alpha_{p-1},j}(r)]\\\\\n & &+\\widetilde {A}^{a(\\cdot)}_{j}(s)-A^{a(\\cdot)}(s),\n\\end{eqnarray*}\nwhere \\begin{equation}\\label{UAtilde}\\widetilde\nA^{a(\\cdot)}_{j}(s)=\\sum_{p=1}^{N-1}[Y_{\\alpha_p\nj}(\\theta_p)+k(\\alpha_{p-1},\\alpha_p)- Y_{\\alpha_{p-1},\nj}(\\theta_p)]\\chi_{[\\theta_p,T]}(s), \\quad s\\in [0,T],\\end{equation} and it is an increasing\nprocess due to the fact that\n$$Y(t)\\in \\bar{Q},\\quad \\forall t\\in [0,T].$$\n Consequently, we conclude that $(\\widetilde Y^{a(\\cdot)}_{j},\\widetilde\nZ^{a(\\cdot)}_{j}, \\widetilde L_{j}^{a(\\cdot)}; j\\in \\Pi)$ is a solution of the\nfollowing RBSDE:\n\\begin{eqnarray}\\label{rbsdetilde}\n& &\\widetilde Y^{a(\\cdot)}_{j}(s)\\nonumber\\\\\n&=&\\xi_{a(T) j}-[(\\widetilde K^{a(\\cdot)}_{j}(T)+\\widetilde\nA^{a(\\cdot)}_{j}(T))-\n(\\widetilde K^{a(\\cdot)}_{j}(s)+\\widetilde A^{a(\\cdot)}_{j}(s))]\\nonumber\\\\\n&&\\displaystyle +A^{a(\\cdot)}(T)-A^{a(\\cdot)}(s)+\\widetilde L^{a(\\cdot)}_{j}(T)-\\widetilde L^{a(\\cdot)}_{j}(s)\\\\\n& &+\\int_s^T\\psi(r,\\widetilde Y^{a(\\cdot)}_{j}(r),\\widetilde\nZ^{a(\\cdot)}_{j}(r),a(r),j)\\, dr-\\int_s^T\\widetilde Z^{a(\\cdot)}_{j}(r)\\,\ndW(r), \\quad j\\in \\Pi \\nonumber\n\\end{eqnarray}\nwith the following boundary condition: \\begin{equation} \\int_0^T\\left(\\widetilde\nY^{a(\\cdot)}_{j}(r)-\\mathop{\\rm max}_{j'\\not=j}\\left\\{\\widetilde Y^{a(\\cdot)}_{j'}(r)-l(j,\nj')\\right\\}\\right)\\, d\\widetilde L_{j}^{a(\\cdot)}(r)=0, \\quad j\\in\n\\Pi. \\end{equation}\n\n\n Since both $\\widetilde K^{a(\\cdot)}$ and $\\widetilde\nA^{a(\\cdot)}$ are increasing c\\`adl\\`ag processes, from the\ncomparison theorem of Tang and Zhong~\\cite{TangZhong} for multi-dimensional RBSDEs, we conclude that\n$$\\widetilde Y^{a(\\cdot)}_j(0)\\le U^{a(\\cdot)}_{j}(0),\\quad j\\in \\Pi, a(\\cdot)\\in {\\cal A}_0^i$$\nwhich implies that\n$$Y_{ij}(0)\\le \\mathop{\\mbox{\\rm essinf }}_{a(\\cdot)\\in {\\cal A}_0^i}U^{a(\\cdot)}_{j}(0), \\quad j\\in \\Pi.$$\n\nOn the other hand, we set $\\theta_0^*=0$, $\\alpha_0^*=i$. We define the sequence\n$\\{\\theta_p^*,\\alpha_p^*\\}_{p=1}^{\\infty}$ in an inductive way as\nfollows:\n\\begin{equation}\\label{theta*}\n\\theta^*_p:=\\mathop{\\hbox{\\rm inf$\\,$}}\\{s\\ge \\theta^*_{p-1}:\n{Y}_{\\alpha^*_{p-1},j}(s)=\\mathop{\\rm min}_{p'\\not={\\alpha^*_{j-1}}}\\{{Y}_{p'j}(s)+k(\\alpha^*_{p-1},p')\\}\\wedge\nT,\n\\end{equation}\nand $\\alpha_p^*$ is an ${\\cal F}_{\\theta_p^*}$-measurable random\nvariable such that\n$$Y_{\\alpha^*_{p-1},j}(\\theta_p^*)=Y_{\\alpha_p^*j}(\\theta^*_p)+k(\\alpha_{p-1}^*,\\alpha^*_p),$$\nwith $p=1,2,\\cdots.$ We define\n\\begin{equation}\na^*(s):=i\\chi_{\\{0\\}}(s)+\\sum_{p=1}^N\n\\alpha_{p-1}^*\\chi_{(\\theta^*_{p-1},\\theta^*_p]}(s), \\quad s\\in [0,T].\n\\end{equation}\nThen, we have\n\\begin{equation} \\widetilde K_j^{a^*(\\cdot)}\\equiv 0, \\quad \\widetilde A_j^{a^*(\\cdot)}\\equiv 0. \\end{equation} Moreover, we have\n\\begin{equation} \\left(\\widetilde Y^{a^*(\\cdot)}_j,  \\widetilde Z^{a^*(\\cdot)}_j,  \\widetilde L^{a^*(\\cdot)}_j; j\\in \\Pi\\right)\\in S^2\\times M^2\\times N^2,\\end{equation}\n and it satisfies  the following RBSDE:\n\\begin{eqnarray}\\label{rbsdetilde*}\n\\widetilde Y^{a^*(\\cdot)}_{j}(s)\n&=&\\xi_{a^*(T) j}\n\\displaystyle +A^{a^*(\\cdot)}(T)-A^{a^*(\\cdot)}(s)+\\widetilde L^{a^*(\\cdot)}_{j}(T)-\\widetilde L^{a^*(\\cdot)}_{j}(s)\\nonumber \\\\\n& &+\\int_s^T\\psi(r,\\widetilde Y^{a^*(\\cdot)}_{j}(r),\\widetilde\nZ^{a^*(\\cdot)}_{j}(r),a^*(r),j)\\, dr\\\\\n&&\\displaystyle -\\int_s^T\\widetilde Z^{a^*(\\cdot)}_{j}(r)\\,\ndW(r), \\quad j\\in \\Pi.\n\\end{eqnarray}\nNow, it is standard to show from RBSDE~(\\ref{rbsdetilde*}) that $A^{a^*(\\cdot)}(T)\\in L^2({\\cal F}_T)$.\nThen, following the same arguments of Hu and Tang~\\cite{HuTang}, we see that $P$-a.s. $\\omega$,\nthere exists an integer $N(\\omega)\\in L^2({\\cal F}_T)$ such\nthat $\\theta^*_{N}=T$.\nThe  switching strategy $a^*(\\cdot)$\nis admissible, i.e., $a^*(\\cdot)\\in {\\cal A}^i_0$. Moreover,\n$$Y_i(0)=U^{a^*(\\cdot)}(0).$$\n\nThe proof is complete. \\signed {$\\sqr69$}\n\n\n\\section{Existence of the value and the saddle point. }\n\nWe denote by $(Y_{ij}, Z_{ij},K_{ij},L_{ij}; i\\in \\Lambda, j\\in \\Pi)$ the unique solution\nof RBSDEs~(\\ref{RBSDEi}). We define $(a^*(\\cdot),b^*(\\cdot))$ as follows.\n\nWe  define\n\\begin{equation}  \\theta_0^*:=0,  \\tau_0^*:=0; \\alpha_0^*:=i, \\beta_0^*:=j.     \\end{equation}\nWe  define\nstopping times $\\theta^*_p, \\tau^*_p$, $\\Lambda$-valued random variable $\\alpha^*_p$, and $\\Pi$-valued random\nvariable $\\beta^*_p$ for $p=1,2, \\ldots, $ in the following inductive manner:\n\\begin{equation} \\begin{array}{rcl}\n\\theta_p^*&:=&\\displaystyle \\mathop{\\hbox{\\rm inf$\\,$}}\\left\\{s\\ge \\theta^*_{p-1}\\wedge \\tau^*_{p-1}:\nY_{\\alpha_{p-1}^*,\\beta_{p-1}^*}(s)=\\mathop{\\rm min}_{i'\\not=i}\\{Y_{i',\\beta_{p-1}^*}(s)+k(\\alpha_{p-1}^*,i')\\}\\right\\}\\wedge T,\n\\\\[3mm]\n\\tau_p^*&:=&\\displaystyle \\mathop{\\hbox{\\rm inf$\\,$}}\\left\\{s\\ge \\theta^*_{p-1}\\wedge \\tau^*_{p-1}:\nY_{\\alpha_{p-1}^*,\\beta_{p-1}^*}(s)=\\mathop{\\rm max}_{j'\\not=j}\\{Y_{\\alpha_{p-1}^*,j'}(s)-l(\\beta_{p-1}^*,j')\\}\\right\\}\\wedge T. \\end{array}\n\\end{equation}\nThen we  choose\n$\\alpha_p^*$ such that $\\alpha_p^*:=\\alpha_{p-1}^*$ if $\\theta^*_p>\\tau^*_p$ and otherwise it is taken from $\\Lambda$ to\nsatisfy the following identity:\n\\begin{equation} Y_{\\alpha_{p-1}^*,\\beta_{p-1}^*}(\\theta_p^*)=Y_{\\alpha_p^*,\\beta_{p-1}^*}(\\theta_p^*)+k(\\alpha_{p-1}^*,\\alpha_p^*).\\end{equation}\nObviously, $\\alpha_p^*$ is ${\\cal F}_{\\theta^*_p\\wedge \\tau^*_p}$-measurable.\nWe also choose\n $\\beta_p^*$ such that $\\beta_p^*:=\\beta_{p-1}^*$ if $\\tau_p^*\\ge \\theta^*_p$ and otherwise it is taken from $\\Pi$ to\n satisfy the following identity:\n \\begin{equation} Y_{\\alpha_{p-1}^*,\\beta_{p-1}^*}(\\tau_p^*)=Y_{\\alpha_{p-1}^*,\\beta_p^*}(\\tau_p^*)-l(\\beta_{p-1}^*,\\beta_p^*).\\end{equation}\n Obviously, $\\beta_p^*$ is ${\\cal F}_{\\theta^*_p\\wedge \\tau^*_p}$-measurable.\n\n\n\\begin{definition} A sequence  $\\{i_p,j_p\\}_{p=0}^N$ where $i_p\\in \\Lambda$\nand $j_p\\in \\Pi$ for $p=0,1, \\cdots,N$ is called a loop if\n\n(i)\neither $i_{p-1}=i_p$ or $j_{p-1}=j_p$ for each $p=1, \\cdots,N-1$\n\n\\noindent and\n\n(ii) $i_N=i_0$ and $j_N=j_0$.\n\n A loop who contains no other loop (except itself) is called a\nprimary loop. \\end{definition}\n\nDefine for $i,p\\in \\Lambda, j,q\\in \\Pi,$  and either $i=p$\nor $j=q$, the following set: \\begin{equation}\\begin{array}{rcl}\nF_{ip}^{jq}&:=&\\displaystyle \\biggl\\{y=(y_{i'p'})\\in R^{m_1\\times\nm_2}: y_{ij}=y_{pj}+k(i,p) \\hbox{ \\rm if $j=q$; }\\\\\n&& \\qquad \\hbox{ \\rm or }  y_{ij}=y_{iq}-l(j,q) \\hbox{ \\rm if\n$i=p$} \\biggr\\}. \\end{array} \\end{equation}\n\nIn this section, we need the following additional assumption,\nwhich is standard in the literature of switching games\n(see, e.g., Tang and Hou~\\cite[Hypothesis 4, page 924]{TangHou}).\n\n\\begin{hypothesis}\\label{LoopCost}\nFor any loop $\\{i_p,j_p\\}_{p=0}^{N}$ with $N\\le m_1m_2$, we have\n\\begin{equation} \\sum_{p=0}^{N-1}k(i_p, i_{p+1})-\\sum_{p=0}^{N-1}l(j_p, j_{p+1})\\not=0. \\end{equation}\n\\end{hypothesis}\n\nThe assumption of no loop of zero cost implies that \\begin{equation}\n\\mathop{\\hbox{\\rm inf$\\,$}}_{y(i_pj_p)\\in F_{i_{p-1}i_p}^{j_{p-1}j_p}\\cap\\{y: |y|\\le\nY^*(T,\\omega)\\}\\cap\n\\bar{Q}}\\left(\\sum_{p=1}^N|y(i_pj_p)-y(i_{p-1}j_{p-1})|^2\\right)^{1\/2}>0\\end{equation}\nfor any loop $\\{i_p,j_p\\}_{p=0}^{N}$ and $a.s. \\omega$. This infimum\nwill be called the length of the loop, relative to  $Y(\\omega)$.\n\nSince there are only a finite number of primary loops, the least\none among all these primary loops' lengths relative to  $Y(\\omega)$\nis strictly positive $a.s.$, which will be denoted by $c(\\omega)$.\n\nOn the other hand, as $Y$ satisfies (\\ref{RBSDEi}), it is easy\nto check that \\begin{equation} E\\left[\\sum_{p=1}^\\infty |Y(\\theta_p\\wedge \\tau_p)-\nY(\\theta_{p-1}\\wedge \\tau_{p-1})|^2\\right]<\\infty.\\end{equation} As a\nconsequence, there exists $N(\\omega)$ such that $\\theta_N\\wedge \\tau_N=T$.\nOtherwise, there would be infinite number of loops, whose length\nis almost surely not less than $c(\\omega)$, and therefore it is\ncontradictory to the last inequality.\n\n\\begin{theorem} Let Hypotheses~\\ref{psi}, \\ref{k}, and~\\ref{LoopCost} be satisfied.\nLet\n$$(Y_{ij},Z_{ij},K_{ij},L_{ij}; i\\in \\Lambda, j\\in \\Pi)$$ be a solution of\nRBSDEs~(\\ref{RBSDEi}) in $S^2\\times M^2\\times N^2\\times N^2$. Then, $(Y_{ij};i\\in \\Lambda, j\\in \\Pi)$\nis the value process for our switching game, and the switching strategy $a^*(\\cdot):=(\\theta^*_p\\wedge \\tau^*_p, \\alpha^*_p)$\nfor Player $I$ and\nthe switching strategy $b^*(\\cdot):=(\\theta^*_p\\wedge \\tau^*_p, \\beta^*_p)$ for Player $II$\nis a saddle point of the switching game.\n\\end{theorem}\n\n{\\bf Proof. }\n We assume $t=0$ without loss of generality.\n Otherwise, it suffices to consider the admissible\nswitching strategies starting at time $t$.\n\nFor each pair $(a(\\cdot),b(\\cdot))\\in {\\cal A}_0^i\\times {\\cal B}_0^j$, the following BSDE\n\\begin{equation}\\begin{array}{rcl} U(s)&=&\\displaystyle\n\\xi+A^{a(\\cdot)}(T)-A^{a(\\cdot)}(s)-\\left(B^{b(\\cdot)}(T)-B^{b(\\cdot)}(s)\\right)\\\\\n&&\\displaystyle +\\int_s^T\\psi(r, U(r),V(r), a(r),b(r))\\,\ndr-\\int_s^TV(r)\\, dW(r)\\end{array}\\end{equation} has a unique solution, denoted by\n$(U^{a(\\cdot),b(\\cdot)}, V^{a(\\cdot),b(\\cdot)})\\in S^2\\times M^2$.\n\nWe shall show the following \\begin{equation} U^{a^*(\\cdot),b(\\cdot)}_0\\le\nU_0^{a^*(\\cdot),b^*(\\cdot)} \\le U_0^{a(\\cdot),b^*(\\cdot)}.\\end{equation}\n\nFor the following \\begin{equation}\\begin{array}{rcl} a(s)&=&\\displaystyle  i\\chi_{0}(s)+\\sum_{p=1}^N\n\\alpha_{p-1}\\chi_{(\\theta_{p-1}, \\theta_p]}, \\\\\nb(s)&=&\\displaystyle  j\\chi_{0}(s)+\\sum_{q=1}^N \\beta_{q-1}\\chi_{(\\tau_{q-1},\n\\tau_q]},\\end{array} \\end{equation} there is a refinement $\\{\\theta_j'\\}$ of both\nsequences of stopping times $\\{\\theta_j\\}$ and $\\{\\tau_j\\}$ such that\n\\begin{equation}\\begin{array}{rcl} a(s)&=&\\displaystyle  i\\chi_{0}(s)+\\sum_{p=1}^N\n\\alpha_{p-1}'\\chi_{(\\theta_{p-1}', \\theta_p']}, \\\\\nb(s)&=&\\displaystyle  j\\chi_{0}(s)+\\sum_{p=1}^N \\beta_{p-1}'\\chi_{(\\theta_{p-1}',\n\\theta_p']}.\\end{array} \\end{equation}\n\nThen we define\n\\begin{eqnarray} Y^{a(\\cdot), b(\\cdot)}(s)&=&\\displaystyle \\sum_{p=1}^N\nY_{\\alpha_{p-1}',\\beta_{p-1}'}(s)\\chi_{[\\theta_{p-1}',\n\\theta_p')}(s)+\\xi_{a(T)b(T)}\\chi_{\\{T\\}}(s),\\\\\nZ^{a(\\cdot), b(\\cdot)}(s)&=&\\displaystyle \\sum_{p=1}^N\nZ_{\\alpha_{p-1}',\\beta_{p-1}'}(s)\\chi_{[\\theta_{p-1}',\n\\theta_p')}(s),\\\\\n K^{a(\\cdot), b(\\cdot)}(s)&=&\\displaystyle \\sum_{p=1}^N\n\\int_{\\theta_{p-1}'\\wedge s}^{\\theta_p'\\wedge s}d\nK_{\\alpha_{p-1}',\\beta_{p-1}'}(r),  \\\\\n L^{a(\\cdot), b(\\cdot)}(s)&=&\\displaystyle \\sum_{p=1}^N\n\\int_{\\theta_{p-1}'\\wedge s}^{\\theta_p'\\wedge s}d\nL_{\\alpha_{p-1}',\\beta_{p-1}'}(r).\n \\end{eqnarray}\n\nNoting that $Y^{a(\\cdot),b(\\cdot)}(\\cdot)$ is a c\\`adl\\`ag process\nwith jump $Y_{\\alpha_p', \\beta_p'}(\\theta_p')-\nY_{\\alpha_{p-1}',\\beta_{p-1}'}(\\theta_p')$ at $\\theta_p'$,\n$p=1,\\cdots,N-1$, we deduce that\n\\begin{eqnarray*}\n& &Y^{a(\\cdot), b(\\cdot)}(s)-Y^{a(\\cdot), b(\\cdot)}(0)\\\\\n&=&-\\sum_{p=1}^N \\int_{\\theta_{p-1}'\\wedge s}^{\\theta_p'\\wedge\ns}\\psi(r, Y_{\\alpha_{p-1}',\\beta_{p-1}'}(r),\nZ_{\\alpha_{p-1}',\\beta_{p-1}'}(r),\\alpha_{p-1}',\\beta_{p-1}')\\, dr\\\\\n&&\\displaystyle +  \\sum_{p=1}^N \\int_{\\theta_{p-1}'\\wedge s}^{\\theta_p'\\wedge s}\nZ_{\\alpha_{p-1}',\\beta_{p-1}'}(r)\\,\ndW(r)\\\\\n&&\\displaystyle +\\sum_{p=1}^N \\int_{\\theta_{p-1}'\\wedge\ns}^{\\theta_p'\\wedge s}\\left[d K_{\\alpha_{p-1}',\\beta_{p-1}'}(r)-d\nL_{\\alpha_{p-1}',\\beta_{p-1}'}(r)\\right]\\\\\n& &+\\sum_{q=1}^{N-1}[Y_{\\alpha_q',\\beta_q'}(\\theta_q')-\nY_{\\alpha_{q-1}',\\beta_{q-1}'}(\\theta_q')]\\chi_{[\\theta_q',T]}(s)\\\\\n&=&-\\sum_{p=1}^N \\int_{\\theta_{p-1}'\\wedge s}^{\\theta_p'\\wedge\ns}\\psi(r,Y_{\\alpha_{p-1}',\\beta_{p-1}'}(r),\nZ_{\\alpha_{p-1}',\\beta_{p-1}'}(r),\\alpha_{p-1}',\\beta_{p-1}')\\, dr\\\\\n&&\\displaystyle +  \\sum_{p=1}^N \\int_{\\theta_{p-1}'\\wedge s}^{\\theta_p'\\wedge s}\nZ_{\\alpha_{p-1}',\\beta_{p-1}'}(r)\\,\ndW(r)\\\\\n&&\\displaystyle +\\sum_{p=1}^N \\int_{\\theta_{p-1}'\\wedge\ns}^{\\theta_p'\\wedge s}\\left[d K_{\\alpha_{p-1}',\\beta_{p-1}'}(r)-d\nL_{\\alpha_{p-1}',\\beta_{p-1}'}(r)\\right]\\\\\n&&\\displaystyle +{\\widetilde A}^{a(\\cdot),\nb(\\cdot)}(s)-A^{a(\\cdot)}(s)-\\left({\\widetilde B}^{a(\\cdot),\nb(\\cdot)}(s)-B^{ b(\\cdot)}(s)\\right),\n\\end{eqnarray*}\nwhere \\begin{equation} \\widetilde A^{a(\\cdot),\nb(\\cdot)}(s)=\\sum_{q=1}^{N-1}\\chi_{a(s)\\not=a(s+)}\\left[\nY_{\\alpha_q',\\beta_q'}(\\theta_q')+k(\\alpha_{q-1}',\\alpha_q')-\nY_{\\alpha_{q-1}',\\beta_{q-1}'}(\\theta_q')\\right]\\chi_{[\\theta_q',T]}(s),\\end{equation}\nand \\begin{equation} \\begin{array}{rcl} &&\\displaystyle \\widetilde B^{a(\\cdot),\nb(\\cdot)}(s)\\\\\n&=&\\displaystyle \\sum_{p=1}^{N-1}\\chi_{b(s)\\not=b(s+),\na(s)=a(s+)}\\left[ Y_{\\alpha_{p-1}',\\beta_{p-1}'}(\\theta_p')\n-Y_{\\alpha_p',\\beta_p'}(\\theta_p')+l(\\beta_{p-1}',\\beta_p')\\right]\\chi_{[\\theta_p',T]}(s).\\end{array}\\end{equation}\nThey are increasing processes, due to\nthe fact that\n$$Y(t)\\in \\bar{Q},\\quad \\forall t\\in [0,T].$$\nWe have \\begin{equation} \\widetilde A^{a^*(\\cdot), b(\\cdot)}=0, \\quad K^{\na^*(\\cdot), b(\\cdot)}=0, \\quad \\forall b(\\cdot)\\in {\\cal B}^{j}_0.\\end{equation}\n\nConsequently, we conclude that\n$(Y^{a^*(\\cdot), b(\\cdot)}, Z^{a^*(\\cdot), b(\\cdot)})$ solves the following BSDE:\n\\begin{eqnarray}\\label{rbsdetilde**} Y(s)\n&=&\\displaystyle \\xi_{a^*(T)b(T)}+\\left[A^{a^*(\\cdot)}(T)-A^{a^*(\\cdot)}(s)\\right]\\nonumber \\\\\n&&\\displaystyle +\\left[\\widetilde B^{a^*(\\cdot), b(\\cdot)}(T)-\\widetilde\nB^{a^*(\\cdot), b(\\cdot)}(s)\\right]-\\left[B^{b(\\cdot)}(T)-B^{b(\\cdot)}(s)\\right]\\nonumber   \\\\\n& &\\displaystyle +\\int_s^T\\psi(r, Y(r), Z(r),a^*(r),b(r))\\, dr\\nonumber \\\\\n&&\\displaystyle +\\int_s^TdL^{a^*(\\cdot), b(\\cdot)}(r) -\\int_s^T\nZ(r)\\, dW(r).\n\\end{eqnarray}\n\nFirst we  have\n\\begin{equation} A^{a^*(\\cdot)}(T)\\in L^2({\\cal F}_T), \\quad  B^{b^*(\\cdot)}(T)\\in L^2({\\cal F}_T), \\end{equation}\nwhich together with Hypothesis~\\ref{k} imply  that $a^*(\\cdot)\\in {\\cal A}_0^i$ and $b^*(\\cdot)\\in {\\cal B}_0^j$.\nWe only show the first inclusion, and the second one is shown in a symmetrical way.\n\nWe define ${\\overline b}(\\cdot)\\equiv j$. Then, we have\n\\begin{equation} \\widetilde B^{a^*(\\cdot), {\\overline b}(\\cdot)}\\equiv 0, \\quad B^{{\\overline b}(\\cdot)}\\equiv 0.\\end{equation}\n Putting the two equalities into RBSDEs~(\\ref{rbsdetilde**}), we\nsee that\n$(Y^{a^*(\\cdot), {\\overline b}(\\cdot)}, Z^{a^*(\\cdot), {\\overline b}(\\cdot)})$ satisfies the following BSDE:\n\\begin{eqnarray}\\label{BSDElast} Y^{a^*(\\cdot), {\\overline b}(\\cdot)}(s)\n&=&\\displaystyle \\xi_{a^*(T)j}+\\left[A^{a^*(\\cdot)}(T)-A^{a^*(\\cdot)}(s)\\right]\\nonumber \\\\\n& &\\displaystyle +\\int_s^T\\psi(r, Y^{a^*(\\cdot), {\\overline b}(\\cdot)}(r), Z^{a^*(\\cdot),\n{\\overline b}(\\cdot)}(r),a^*(r),j)\\, dr \\\\\n&&\\displaystyle +\\int_s^TdL^{a^*(\\cdot), {\\overline b}(\\cdot)}(r) -\\int_s^T\nZ^{a^*(\\cdot), {\\overline b}(\\cdot)}(r)\\, dW(r), \\quad s\\in [0,T]. \\nonumber\n\\end{eqnarray}\nSince\n\\begin{equation} Y^{a^*(\\cdot), {\\overline b}(\\cdot)}\\in S^2, \\quad  Z^{a^*(\\cdot), {\\overline b}(\\cdot)}\\in M^2,\n\\quad L^{a^*(\\cdot), \\overline{b}(\\cdot)}\\in N^2, \\end{equation}\n in view of Hypothesis~\\ref{psi}, it is standard to derive  from equality~(\\ref{BSDElast}) for $s=0$\nthat $A^{a^*(\\cdot)}(T)\\in L^2({\\cal F}_T)$.\n\nOn the one hand, since $ \\widetilde B^{a^*(\\cdot), b(\\cdot)}$ and $L^{\na^*(\\cdot), b(\\cdot)}$ are increasing processes, we have from the\ncomparison theorem for BSDEs that \\begin{equation} Y^{a^*(\\cdot), b(\\cdot)}(s) \\ge\nU^{a^*(\\cdot), b(\\cdot)}(s). \\end{equation} Letting $s=0$, we have \\begin{equation}\nY_{ij}(0) \\ge U^{a^*(\\cdot), b(\\cdot)}(0).\\end{equation} On the other hand,\nsince \\begin{equation} \\widetilde B^{a^*(\\cdot), b^*(\\cdot)}=0, \\quad L^{\na^*(\\cdot), b^*(\\cdot)}=0, \\end{equation} we have \\begin{equation} Y^{a^*(\\cdot),\nb^*(\\cdot)}(s) = U^{a^*(\\cdot), b^*(\\cdot)}(s). \\end{equation} Letting $s=0$,\nwe have \\begin{equation} Y_{ij}(0) = U^{a^*(\\cdot), b^*(\\cdot)}(0).\\end{equation}\n\nThe proof is complete.\n\\signed {$\\sqr69$}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nSpiral galaxies are very common in the universe. When studied with spectroscopic methods, these beautiful objects usually reveal to rotate faster than it would do if only their visible matter were taken into account. This is the missing mass problem. Then, one generally believes that there is a dark matter component surrounding their disks, making them to rotate in that way. In other words, the canonical model says that spiral galaxy rotation curves inferred from observations could be accounted for the dark matter halos that embed the galaxies \\cite[see, e.g.,][]{rubin,bahcall}.  Therefore, this additional distribution of mass causes an additional acceleration and the rotation curves can be accounted for.\n\nThis picture to the understanding of the structure of late-type morphologies finds support in the present canonical physics, that considers the general relativity the most reliable theory of gravitation, and, as a consequence, due to the correspondence principle, the Newtonian gravitation at the weak field limit.\n\nHowever, in order to avoid the dark matter problem, many authors consider modifications in the Newtonian gravity at large scales, in other words, they consider that, in galactic or even cosmological scales, the law of gravity is different from the Newtonian one. In this way, contrary to the dark matter paradigm, these authors argue that there is only a disk, but due to another kind of law of gravity  at large scales, it could well rotate in accord with the observed rotation curves \\cite[see, e.g., ][ and references therein]{milgrom1,milgrom2,moffat1,moffat2,bm2006a}.\n\nIt is worth mentioning that \\cite{bm2006b} also argue that it is possible to explain the X-ray cluster masses without dark matter using this very theory.\n\nIn particular, the present work is concerned with the study made by \\citet[][ hereafter BM]{bm2006a}, who present an alternative approach to explain the rotation curves of spiral galaxies, using an alternative theory of gravitation studied by \\citet[][ hereafter Moffatian gravity]{moffat1}. Our main aim is to probe this theory using numerical simulations.\n\nThe Moffatian gravity is a covariant theory of gravity based on the coupling of the Einsteinian gravity with a massive skew symmetric field $F_{\\alpha \\beta \\gamma}$. Here, we do not present and discuss  this alternative theory of gravity in detail, since we are only concerned with its weak field limit.\n\nBM claim that in the weak field limit the gravitational potential of a point source of mass $M$ reads\n\n\\begin{equation}\n\\Phi(r) = - \\frac{GM}{r} \\left\\{ 1 + \\sqrt{\\frac{M_0}{M}} \\left[ 1 - \\exp  \\left(- \\frac{r}{r_0} \\right) \\right] \\right\\} ,\n\\label{moffat_equation}\n\\end{equation}\n\n\\par\\noindent where $G$ is the Newtonian constant of gravity, $r$ is the distance to the source, $M_0$ is a coupling parameter and $r_0$ is a characteristic length. From the BM potential, the gravitational acceleration by a particle of mass $M$ reads\n\n\\begin{equation}\na(r) = - \\frac{GM}{r} \\left\\{ 1 + \\sqrt{\\frac{M_0}{M}} \\left[ 1 - \\exp  \\left(- \\frac{r}{r_0} \\right) \\left( 1 +  \\frac{r}{r_0} \\right)   \\right] \\right\\}.\n\\label{moffat_aceleration}\n\\end{equation}\n\nBM state that the above equations hold in any environment. To apply these equations to model their rotation curves,\nthey consider the following procedure: they integrate these equations for a spherically symmetric system. As a result, they obtain the following equations:\n\n\\begin{equation}\n\\Phi(r) = - \\frac{GM(r)}{r} \\left\\{ 1 + \\sqrt{\\frac{M_0}{M}} \\left[ 1 - \\exp  \\left(- \\frac{r}{r_0} \\right) \\right] \\right\\} ,\n\\label{moffat_equation2}\n\\end{equation}\n\n\\par\\noindent and\n\n\\begin{equation}\na(r) = - \\frac{GM(r)}{r} \\left\\{ 1 + \\sqrt{\\frac{M_0}{M}} \\left[ 1 - \\exp  \\left(- \\frac{r}{r_0} \\right) \\left( 1 +  \\frac{r}{r_0} \\right)   \\right] \\right\\}.\n\\label{moffat_aceleration2}\n\\end{equation}\n\nNow, the $M$ outside the braces becomes $M(r)$, but the $M$ inside the braces does not change. Therefore, for given values of $r_0$, $M_0$ and $M$ (the total mass of the system) the expression inside the braces depends only on $r$.\nAssuming centrifugal equilibrium and a given mass distribution, i.e. $M(r)$, BM model their rotation curves.\n\nAlso, note that Equation (\\ref{moffat_aceleration2}) obeys the superposition principle, which is an essential property if one intends to make use of $N$-body simulations.\n\nWe note from the above equations that the Moffatian gravity can be adjusted to a large range of lengths and masses, by means of the parameters $r_0$ and $M_0$, that depend on the system under investigation. Hence, they could, in principle, explain many physical effects that the canonical theory attributes to the dark matter: galaxy rotation curves, galactic cluster masses and so on, because the BM's parameters depend on the total mass of the system under study.\n\nBM apply their modified gravity to explain the rotation curves of a sample of observed spiral galaxies and yield the following set of parameters, namely, $M_0 = 96 \\times 10^{10} M_{\\odot}$ and $r_0 = 13.96 $ kpc \\citep{bm2006a}. As usual in this kind of study, they consider centrifugal equilibrium, which means in the end that their galaxy model is ``static.\"\n\nIt is well known, from the $N$-body simulation studies, that disks built with exponential-Spitzer laws (hereafter SdMH disks), when submitted to the Newtonian potentials, reveal to have secular centrifugal equilibrium, maintaining their density profiles for gigayears of simulated time \\citep{sw1999,sdmh2005}. However, under the BM's picture, we do not know how long the BM potential can maintain galactic disks in dynamical equilibrium, and in fact if it really can. This is the main aim of the present paper.\n\nIn previous works \\citep{ca2009a, ca2009b,ca2009c}, we developed an $N$-body method to study alternative theories of gravity at galactic scales. In particular, we have studied an Yukawian gravitational potential and have shown that this potential is viable only if the Yukawian parameter is such that the Yukawian potential is nearly Newtonian.\n\nIn the present work, we also use $N$-body simulations to construct and evolve spiral galaxies now submitted to the potential and acceleration given by Equations (\\ref{moffat_equation2}) and (\\ref{moffat_aceleration2}), respectively. We adopted, in these simulations, the same set of fitted parameters ($M_0 = 96 \\times 10^{10} M_{\\odot}$ and $r_0 = 13.96 $ kpc) as by BM.\n\nOur aim is to probe if the BM model is dynamically consistent, as it should be in order to be considered a realistic model. In this way, we are able to test the BM's claim and verify the reliability of this kind of law of gravity, using the concepts and techniques of the galactic dynamics studies \\citep{bt2008}.\n\nIn Section \\ref{numtech}, we present the $N$-body code used in this work and the numerical techniques used to model spiral galaxies with the BM potential; in Section \\ref{simulresul}, the results of the numerical simulation are present; and finally, in Section \\ref{conclusao}, we discuss our main findings and conclusions.\n\n\\section{Numerical Techniques}\n\\label{numtech}\n\n\\subsection{The Modified Code}\n\\label{test1}\n\nWe performed the simulations with a modified and tested version of the Gadget-2 code \\citep{springel2005}. As discussed in our previous papers \\citep{ca2009a,ca2009b,ca2009c}, this choice is based on the collisionless nature of galaxies modeled by particles, and the method used by Gadget-2 to build the recursive tree in order to compute forces and potentials.\n\nWe have modified the Gadget-2 and replaced the Newtonian potential and acceleration by the expressions given by Equations (\\ref{moffat_equation2}) and (\\ref{moffat_aceleration2}) just in the respective code's instructions. We have extensively  tested the code efficiency to calculate potentials via the tree method, using a typical initial snapshot composed by a disk embedded in a dark matter halo.\n\nIt is worth noting that we included a halo only to test the code accuracy, since in the probes of the ability of the BM gravity to generate spiral galaxies with flat rotation curves, without dark matter, the halo is not included in the simulations.\n\nThe Newtonian and the Moffatian codes were set up with the tolerance parameter $\\theta=0.8$, maximizing the tree code's performance and softening length $l_{\\rm{dh}} = 0.33$ kpc for the dark halo and $l_{\\rm{d}} = 0.15$ kpc for the baryonic disk. This set of softening lengths yields conservation of the total energy better than other sets. The procedure to choose the values for  $\\theta$, $l_{\\rm{dh}}$ and $l_{\\rm{d}}$ adopted here is analog to that used in our previous papers \\citep[][ and references therein]{ca2009a}.\n\n\n\\subsection{Initial Conditions}\n\nWe have studied three models: model I a typical exponential-Spitzer disk with a dark matter halo (Newtonian galaxy); model II a second one, derived from the first one, where the system is built and the halo is removed before running the simulation with a Moffatian potential (pseudo-Moffatian galaxy); and model III a pure BM's disk model (pure Moffatian galaxy). Below, we give the reasons why such models are chosen.\n\nModel I follows the prescriptions described by  \\citet[][ and references therein]{sdmh2005}, but with the difference that we are now including neither bulges nor gas particles, because we do not need a detailed Newtonian model, since our main aim is to investigate the dynamical properties of disk galaxies under alternative potentials.  The reason for this choice has to do with their dynamical stability and computational simplicity to model them, as explained in detail in \\citet{sdmh2005}. Besides, it is worth stressing that SdMH disks reproduce all observational features of late-type systems.\n\nIn one of our previous papers \\citep{ca2009c}, we described the principal recipes to model such galaxies. Besides\nthe parameters discussed in the previous section, the models are composed by $N_{\\rm{halo}} = 30,000$ particles for the dark matter halo and $N_{\\rm{disk}} = 30,000$ for the disk. All these figures yield a conservation of energy that turns our galaxy model reliable, as the reader will find later on in this paper. Model I is then a Newtonian model for a spiral galaxy, which is consistent with observations, that we use to compare with the Moffatian model for spiral galaxies.\n\nModel II is the same as model I but without dark halo particles, i.e., it is just the Model I from it we have extracted the halo particles. It is worth stressing that the halo potential is computed to make the rotation curve of the disk of model II as reliable as possible--like the observed ones.\n\nAfter extracting the halo, the disk should be stable under the BM's potential.  This procedure is made to test the BM's hypothesis, which claims that disks can exist under Moffat's potential without halos, explaining rotation curves without exotic non-baryonic dark matter. In this way, we are able to test this hypothesis over a secular evolution of a disk.\n\nModel III is similar to model II, but with the following differences: (1) we do not compute the halo's potential and (2) we replace the Newtonian potential originated from the disk \\cite[see, e.g.][ Equation (2.170)]{bt2008} by the BM's potential originated from  all the positions of disk particles, using the BM's parameters $M_0 = 96 \\times 10^{10} M_{\\odot}$ and $r_0 = 13.96 $ kpc, as  fitted by \\citet{bm2006a}.\n\nIn other words, we first build the positions of the particles and then compute their potentials using Equation (\\ref{moffat_equation2}) over the mesh, where the velocities will be calculated \\cite[see, e.g.,][]{ca2009c}. We use these results to integrate the Jean's equations, and then calculate the velocity dispersions. With this procedure, the initial disk is put in equilibrium with the BM's potential during the first snapshots of simulated time. We have set $N_{\\rm{disk}} = 30,000$ particles to this disk (hereafter Moffatian disk or BM's disk for short) in order to maintain the same resolution set to the Newtonian model. Note that the disk thus modeled is not necessarily consistent with observations, as the reader will see later on.\n\nThe following set of parameters is chosen to the SdMH disks (model I)): the total mass ${M_{\\rm{t}} = v^2_{200}\/(10GH_0) = 0.98 \\times 10^{12} M_{\\odot}}$, where $v_{200} = 160\\, \\rm{km\\, s^{-1}}$ is the virial velocity, $G$ is the gravitational constant, and $H_0 = 100\\, \\rm{km\\, s^{-1}\\, Mpc^{-1}}$, is the Hubble constant; the total mass of the disk $M_{\\rm{disk}} = m_{\\rm{d}} M_{\\rm{t}}$, where $m_{\\rm{d}}=0.041$ is a dimensionless fraction of the total mass; the disk scale length $h = 2.74$ kpc; the disk vertical scale height $z_0 \\sim 0.2 h$; and the spin parameter $\\lambda = 0.033$. We emphasize that this model has mass comparable to the Milk Way.\n\nFor model II we also use the same set of parameters used in model I. Recall, however, that after building the disk--halo system, we then extract the halo.\n\nIn the BM's disks, we maintain the total disk mass and the exponential-Spitzer profiles.  We built the disk particles from the density profiles, and used the BM's parameters  ($M_0 = 96 \\times 10^{10} M_{\\odot}$ and $r_0 = 13.96 $ kpc).\n\nIn the next section, we present the results of our simulations.\n\n\n\\section{Simulations and Results}\n\\label{simulresul}\n\n\\subsection{The Newtonian Model}\n\nAlthough the Newtonian model can be considered as a canonical one by the current theory of galactic astronomy \\citep{bm1998} and galactic dynamics \\citep{bt2008}, it is important to bear in mind that here the Newtonian model is not the main result of our work. We have run the Newtonian simulations to compare it with models II and III. As it is well known from simulations of canonical late-type systems \\citep{sdmh2005,ca2009c}, many observational features, as for example, the exponential radial density profile, the rotation curves, etc can be accounted for Newtonian models.\n\nIn Figures (\\ref{fig1})--(\\ref{fig3}), we display the principal results of the Newtonian model we simulated. Figure \\ref{fig1} displays a picture with four boxes: at the top left, phase space points $r \\times v$ of the initial snapshot, where $r$ is the total distance from origin (0,0,0) in kpc and $v$ is the modulus of the velocity in $\\rm{km\\, s^{-1}}$; at the top right, the relative energy conservation is $\\Delta E \/ E_0$, where $\\Delta E = E(t)-E(0)$ and $E_0 \\equiv E(0)$, where $E(t)$ denotes the total energy of the system at time $t$; at bottom left, the phase space points at final time $t = 1$ Gyr; and at the bottom right, the rotation curve is $R \\times v_r$, where R is the radial cylindrical coordinate and $v_r$ is the rotation velocity. Note the similarity between the initial and the 1 Gyr rotation curves; one concludes that early-type galaxy structure can be accounted for the Newtonian model quite well, as is well known. Later on we will see that the comparison between the density profiles at these same different times corroborates such a conclusion.\n\nFigures \\ref{fig2} and \\ref{fig3} show the particle's positions in the $z$-projection at respective times indicated in the boxes. We note from these figures that the simulation follows a typical behavior with the formation\nof spiral arms and central bar. These features are expected from simulations of late-type morphologies, as explained\nby \\citet[][ and references therein]{ca2009c}. In the first 0.33 Gyr, the disk has spiral arms and the system evolves to a central bar with two spiral arms afterward. In other words, this simulation is reliable and represents a standard typical late-type system without external influences. Energy conservation is better than $0.1 \\%$.\n\n\\subsection{Moffatian Models}\n\nWe show above that the Newtonian disks maintain their physical properties for many crossing times.  Now, when a spiral galaxy is submitted to the BM potential, an anomalous behavior occurs to the disk's structure, as we will see in this section.\n\nIn Figure \\ref{fig4}, we display the same as in Figure \\ref{fig1}.  In the top right panel of Figure \\ref{fig4}, we display the evolution of the energy violation. Note that after 1 Gyr (end of our simulation), the violation is less than 1$\\%$, showing the quality of our simulation.\n\nFrom this simulation, we also see that the particles in the phase space are more scattered as compared to the Newtonian simulation. This scattering is interpreted as a loss of information from the initial configuration. Note, at the bottom right panel, that the disk rotates slower than in its initial time, even though it still resembles a typical rotation curve. At first glance, it seems, by only examining Figure \\ref{fig4}, that the Moffatian potential can really explain BM's disks, and no dark matter is needed to maintain secularly stable exponential disks.\n\nWe display snapshots of the disk's particles in Figures \\ref{fig5} and \\ref{fig6}. Examining these snapshots, we see that at 0.08, 0.16, 0.24, and 0.32 Gyr times the system presents atypical substructures, in particular in its central region ($R \\lesssim 10.0$ kpc). As the disk evolves, an annular substructure appears, and the central region becomes almost empty of particles, a void. Subsequently, the big ring collapses and fragments into three spiral unbound arms. At this stage, there is no core.\n\nAt $t = 0.66$ Gyr, the arms merge themselves and become a bar-like structure surrounded by two weak spiral arms. This bar-like structure is small when compared to the Newtonian case. This anomalous behavior is due to the fact that the particles are not in perfect initial equilibrium with the Moffatian potential.\n\nHowever, it is important to bear in mind that we are testing the BM claim, which considers that the Moffatian potential can explain many observed rotation curves of galactic disks without dark matter halos. We recall that our simulated initial disk reproduces the observational features of real galaxies: exponential-Spitzer disk, velocity dispersions, rotation curves and baryonic mass, and has secular equilibrium in the Newtonian case. From this point of view, we would expect that, if BM hypotheses were reliable, our disk would be stable all the time in the simulation. We obtain that an equilibrium configuration is reached only after $t \\gtrsim 0.66$ Gyr. After this time, the system is similar, in some dynamical aspects, to the Newtonian case, but we will see below that this similarity is only apparent.\n\nThe results of model II's simulations bring some interesting questions. One could claim that the simulations of model II need a fine tuning of the parameters $\\rm{M_0}$ and $\\rm{r_0}$ to maintain the entire disk in equilibrium. These parameters could, therefore, be changed to avoid atypical rings and central voids during the first 0.33 Gyr. To study these sub-morphology features, we have made some tests changing the parameters $M_0$ and $r_0$, but we have not found any set of parameters that left the simulations just like the Newtonian one, which, it is worth stressing again, is consistent with observations.\n\nIt was just the results of model II  that lead us to make a pure Moffatian disk; our model III whose simulations were made with the same parameters used in the models I and II.  The results of model III can be seen in Figures \\ref{fig7}--\\ref{fig9}.\n\nIn the top right panel of Figure \\ref{fig7}, one finds the phase-space points. These points are distributed somewhat differently as compared to previous models, because the rotation curve has now a different shape. Also, after 1 Gyr of simulated time, we note that the scattering in the phase space plane is more prominent than in the model II and the initial configuration is completely lost.\n\nThis effect is neither due to simulation inaccuracies nor to the resolution of the simulation, as can be seen that the energy violation (see the top right panel of Figure \\ref{fig7}) is small, namely, $\\log{\\Delta E\/ E_0} \\lesssim -2.0$.\n\nOur results show in the end that the exponential profile is not consistent with the Moffatian potential.  As occurred with model II, the rotation curve of the model III is below the initial one. For $\\rm{R < 10}$ kpc, the rotation curve is linear and associated with the central core movement, similar to a rigid body rotation. For $\\rm{R > 10}$ kpc, the rotation curve is similar to a typical rotation curve of late-type systems, but, in analogy with model II, this similarity is only apparent, as we will see below.\n\nFigures \\ref{fig8} and \\ref{fig9} display the particle's positions at $(x,y)$ plane, where the simulated time is indicated in the respective boxes.  Note that, at initial times, the same atypical central void appears with its surrounding ring, that collapses and fragments in two small clumps, that evolves separately eventually merging at $t \\sim 0.9$ Gyr.\n\nThis simulation shows that for a typical initial equilibrium configuration, BM's disk systems evolve in such way that we cannot produce models in agreement with the observed values, since atypical configurations appear.\n\nThe mixing in the phase space, for example, shows us that our BM's models do not recover the observational characteristics and, therefore, this model cannot be considered as a good representation of disk galaxies.\n\nBut, one could argue that the snapshot at time $t=1$ Gyr resembles a typical late-type system, and this model could be considered reliable, since after $t \\gtrsim 0.9$ Gyr this configuration seems to be stable. Later on we consider this issue again by studying the density profile of the disk.\n\nWe now compare the rotation curve that comes out from our simulations with the one that\ncomes from the centrifugal equilibrium, i.e. $v^2(r)=r\\, a(r)$, where $a(r)$ is given by Equation\n(\\ref{moffat_aceleration2}). This rotation curve reads\n\n\\begin{equation}\nv(r) = \\sqrt{ \\frac{G_0 M(r)}{r}} \\left\\{ 1 + \\sqrt{\\frac{M_0}{M}} \\left[1-\\rm{exp}(-r\/r_0)(1+r\/r_0)\\right] \\right\\}^{ \\frac{1}{2}},\n\\label{rotacaolinear}\n\\end{equation}\n\n\\par\\noindent where $M(r)$ is obtained from our simulation. Note that the above equation is just the one used by BM.\n\nIn Figure \\ref{fig10}, we show the result of this comparison. Note the similarity between the rotation curves, even when considered different simulated times. The differences come from particle noise, very common in $N$-body samplings after evolving the system.  This result shows the consistency between our simulations and the calculation done by BM.\n\nAn important issue is to examine the radial density profile of the simulated disks. A reliable model of a spiral galaxy must have at the end of the simulation the same initial profile.\n\nIn Figure \\ref{fig11}, the radial density profile of the simulated disks is calculated at $t = 0$ and $t=1$ Gyr. This figure presents the logarithmic of particle counts per unit of area, i.e., $\\log(\\rho)$,  in the $(x,y)$ plane, that was divided in many concentric rings, in cylindrical coordinate system. We have also estimated the initial exponential profile analytically, which is indicated by dotted lines.\n\nFrame I indicates the Newtonian initial profile, estimated for all the particles and analytically both profiles coincide exactly. Frame II shows the final Newtonian profile; Frame III shows the final profile of the model II; and Frame IV shows the final BM profile (model III). Each frame has its initial exponential profile for comparison.  Note that the Newtonian disk maintains its initial profile, in spite of particle noise, very common in $N$-body samplings after evolving the system. This figure shows that bars and spiral arms left the exponential profile almost unchanged.\n\nFor the Frame III, however, we conclude that the simulations of the model II have minimized the central density and maximized the density at $R > 10$ kpc. In the phase space, this can be noted by the migration of particles to more distant regions, where their orbits are more stable, increasing the density there. The same happens with BM disks, as we can observe in Frame IV.\n\nIt is worth mentioning again that our BM disk has the same initial profile of the Newtonian model, and Frame I represents all initial profiles, due to the fact that all initial disks are modeled by the same exponential-Spitzer law. We note that, in spite of the final snapshot configurations of the models II and III mimic disks, their density profiles are very different from a canonical exponential-Spitzer law, from which they were built.  As a conclusion, the Moffatian potential cannot maintain exponential-Spitzer disks in dynamical equilibrium and therefore cannot be consistent with observations.\n\nIn Figure \\ref{fig12}, we present, for model III, the radial density profile for different simulated times.\nThis figure clearly shows that, even starting from a disk in equilibrium, the Moffatian gravity does not maintain the exponential-Spitzer disk with the same initial density profile. As time goes on, the profile turns gradually and is definitely different from the exponential profile. This shows that simulating models are a more robust tool than simply adjust profiles in static models, because the evolution of the system cannot be followed in this last approach.\n\nIn conclusion, an alternative gravitational law must explain not only the rotation curves of spiral galaxies,\nbut also their density profiles. In this way, simulations can be a powerful tool to deal with\nsuch an issue.\n\n\n\\subsubsection{An additional test}\n\nNote that depending on the values of $r_0$ and  $M_0$, the Newtonian gravitation would be naturally recovered.\nTherefore, as an additional test we examine such an issue, which could help one to see how the structure of\na disk galaxy would be modified by an alternative gravity law.\n\nWe start from a Moffatian disk, just the one present, for example, in the first snapshot of Figure \\ref{fig8}, whose initial rotation curve is shown in Figure \\ref{fig7}. Then, we set $r_0 = 1000 kpc$ and $M_0=1 \\times 10^{10}$ $M_{\\odot}$ and follow how the disk evolves.\n\nThe results of this simulation are shown in Figure \\ref{fig13} and \\ref{fig14}. For comparison,\nwe also performed simulations using the Newtonian Gadget-2. Both the calculations produce identical\nresults. Therefore, the Newtonian calculation is recovered.\n\nIn particular, note in Figure \\ref{fig13} the rotation curve after 1 Gyr of simulated time; for large values of $R$, the velocity is a decreasing function of $R$, as expected for a Newtonian disk.\n\nIt is also interesting to compare the Moffatian disk snapshots and Newtonian ones. Comparing Figures \\ref{fig8} and \\ref{fig14}, one clearly sees how different these configurations are.\n\n\\section{Conclusions}\n\\label{conclusao}\n\nStudies of alternative theories of gravity are usually made statically, instead of modeling ``live'' systems. For example, in the study of spiral galaxies centrifugal equilibrium is usually considered. The problem with this approach is that the secular evolution of these systems cannot be followed.\n\nIn the present paper, in particular, we follow the techniques used by \\citet{ca2009a,ca2009b,ca2009c} and probe the BM's model, verifying if the Moffatian gravity is dynamically consistent to account for spiral galaxies. A dynamical study is a robust approach, since it considers the distribution functions and Jean's equations that can maintain $N$-body systems in equilibrium.\n\nWe performed the simulations with a modified and tested version of the Gadget-2 code. We replaced the Newtonian potential and acceleration by the expressions given by Equations (\\ref{moffat_equation2}) and (\\ref{moffat_aceleration2}) just in the respective code's instructions. We have extensively  tested the code efficiency to calculate potentials via the tree method, using a typical initial snapshot composed by a disk embedded in a dark matter halo.\n\nFrom our simulations, it follows that the Moffatian potential cannot generate exponential disks in equilibrium, although the rotation curves are nearly flat throughout the whole simulated time.  The final (stable) equilibrium configuration is very different from the exponential profile, as we have explained above.  So, if we consider that exponential profiles are reliable, as it seems to be the case, since they are consistent with observations, it is hard to believe that BM model is better than the Newtonian model (baryonic disk embedded in a dark matter halo) to account for the spiral galaxy structure. Recall that \\citet{bm2006a} claim  that $\\rm{M_0}$ and $\\rm{r_0}$ fit very well many (observed) rotation curves, but they have only analyzed static models (centrifugal equilibrium configurations).\n\nFinally, it is worth mentioning that it would be very interesting to simulate the BM potential for standard elliptical systems and to probe its influence over their global properties, like we have done in this work. A forthcoming paper will consider such an issue.\n\n\n\\acknowledgments\nWe are grateful to the Brazilian agencies CAPES, CNPq and Fapesp for support. We also thank Nick Gnedin for providing us a visualization software. Last but not least, we thank the referee for useful suggestions and criticisms, which greatly improved the final version of our paper.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section*{Acknowledgments}\n\nWe thank Andy Bacher, Allena Opper, and Ed Stephenson\nfor useful discussions and encouragement, without which this work would\nnot have been completed.\nThis research was partially funded by\nFCT grant POCTI\/37280\/FNU\/2001 (ACF),\nNSF grant PHY-0457014 (AG),\nDOE grants DE-FG02-97ER41014 (GAM) and DE-FG02-04ER41338 (UvK),\nand a Sloan Research Fellowship (UvK).\n\n\n\\bibliographystyle{unsrt}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction: Four long-lived bright clumps in the D ring \\label{intro}}\nA narrow ringlet referred to as D68 lies near the inner edge of Saturn's D ring, about 67,630 km from Saturn's center.\nFrom its discovery in Voyager images \\citep{Showalter96} through much of the Cassini mission, \ninvestigation of D68 focused on its radial profile and phase angle properties \\citep{Hedman07}. \nLater studies brought attention to its longitudinal brightness variations \\citep{Hedman14}. \nIn 2014-15, four bright clumps\nformed and remained relatively evenly spaced \nwith small longitudinal variations about mean separations of 26$^{\\circ}$, 32$^{\\circ}$, and 29$^{\\circ}$ \\citep{Hedman19}. \n\\citet{Hedman19} investigated these clumps in depth and designated them T (trailing), M (middle), L (leading), and LL (leading leading). \nThe most likely explanation for the sudden increase in brightness in the four clump regions of the ringlet is that fine material was released by collisions into or among larger objects located near or within D68. \nThese hypothetical larger objects are called source bodies, whose minimum sizes can be constrained by estimating the amount of material associated with each clump from phase-corrected normal equivalent area values, and whose maximum sizes can be constrained by the fact that they have not been observed directly. \nThe range of masses that would correspond to these size constraints is $10^5-10^{10}$ kg. \nThe narrowness of the D68 ringlet and the distribution of clumps could \neither indicate that there is a collection of source bodies in a co-orbital configuration\nor imply that there is some outside force confining this material. \nIn this paper we test the first idea by modeling the D68 clumps as a co-orbital satellite system.  \n\nThe study of the dynamics of co-orbital systems is motivated by the many cases of co-orbital systems we find in our solar system. \nWe are especially interested here in systems in which the co-orbitals have comparable masses. \nThe best known of such systems are the horseshoe orbits of Janus and Epimetheus \\citep{Dermott81}.\nCo-orbital asteroids have been suggested as the source of Venus's zodiacal dust ring \\citep{Pokorny19}.\nFinally, the ring arcs in the Neptunian system have been proposed to be confined by either a corotation resonance with a moon on a separate orbit \\citep{Goldreich86, Porco91, Salo98, Namouni02} or a co-orbital resonance with an undetected moon or even multiple moons sharing the same orbit \\citep{Lissauer85, Sicardy92, Renner14}.  \n\n\nIn Section \\ref{analysis}, we analyze potential stable configurations. \nIn Section \\ref{numerical}, we describe how we use numerical simulations to investigate these scenarios.\nIn Section \\ref{disc}, we discuss some remarks for co-orbital systems as well as the possibilities for D68. \n\n\\section{Analysis of Potential Stable Configurations \\label{analysis}}\nHere we first review the theory of stable co-orbital objects, \nand we then apply the theory to the D68 clumps.\n\n\\subsection{Theory \\label{theory}}\n\n\\citet{Salo88} originally examined stationary configurations of equal-mass co-orbital satellites for small $N$ ($N \\leq 9$)\nusing a simple first-order theory, neglecting terms of the order $\\left(m\/M\\right)^{3\/2}$, \nwhere $m$ and $M$ are the masses of the satellite and the primary. \nA numerical search revealed three distinct types of stationary solutions, \nof which we are here concerned with only one, which \\citet{Salo88} label Type Ia: \nan equilibrium where all the $N$ satellites are most concentrated on the same side of the common orbit. \nThe case where $N = 2$ is the well known Trojan configuration, with an angular separation of 60$^{\\circ}$. \nType Ia configurations are stable for $2 \\leq N \\leq 8$ but are not found for $N \\geq 9$ \\citep{Salo88}.\nThis study, motivated by the D68 clumps, focuses on configurations with $N = 4$ and $N = 5$. \n\\citet{Renner04} generalized the work of \\citet{Salo88} for similar but not necessarily equal masses, \nwhich is what we expect for the D68 clumps. \n\nWhen we define $\\phi_i$ as the longitude of satellite $i$ and $\\xi_i = \\Delta r_i\/r_0$ as its relative radial excursion with respect to its average radius $r_0$,\nthe relevant equations of motion become \\citep{Renner04}\n\\begin{equation}\n\\dot{\\phi}_i = -\\frac{3}{2}\\xi_i\n\\end{equation}\nand\n\\begin{equation}\n\\dot{\\xi}_i = -2 \\displaystyle\\sum_{j \\neq i} m_j f'\\left(\\phi_i - \\phi_j\\right)\n\\end{equation}\nwhere $m_j$ is the mass of satellite $j$ and\n\\begin{equation}\nf\\left(\\phi\\right) = \\cos \\phi - \\frac{1}{2|\\sin\\frac{\\phi}{2}|}.\n\\end{equation}\nThe derivative of $f\\left(\\phi\\right)$ is \n\\begin{equation}\nf'\\left(\\phi\\right) = \\sin \\phi \\left[-1 + \\frac{1}{8|\\sin\\frac{\\phi}{2}|^3} \\right].\n\\end{equation}\nFor a co-orbital stationary configuration \\citep{Renner04}, \n\\begin{equation}\n\\xi = 0 \n\\end{equation}\nand \n\\begin{equation}\n\\displaystyle\\sum_{j \\neq i} m_j f'\\left(\\phi_i - \\phi_j\\right) = 0.\n\\label{mf}\n\\end{equation}\nEquation \\ref{mf} can be written in matrix form:\n\\begin{equation}\n\\pmatrix{ 0 & f'_{12} & \\dots & \\dots & f'_{1N} \\cr\n                -f'_{12}       & 0 & f'_{23} & \\dots & f'_{2N} \\cr\n                \\vdots & & 0 & & \\cr\n                \\vdots & & & \\ddots & \\cr\n                -f'_{1N} &  &  & & 0 }\n\\pmatrix{ m_1 \\cr\n               m_2 \\cr\n               \\vdots \\cr\n               \\vdots \\cr\n               m_N }\n= 0_{\\rm I\\!R^N}\n\\end{equation}\nBecause the $N \\times N$ matrix is antisymmetric and depends only on the longitudinal separations $\\phi_i$ between the bodies, \nfor $N \\ge 3$ one can always find a set of relative masses that satisfies these equations\nfor any given set of angular separations. \nThis solution, however, might not be physical because one or more of the masses could be negative or zero.\n\n\\subsection{Results \\label{results1}}\nWe first considered the observed configuration with four masses separated by angles of 29$^{\\circ}$, 32$^{\\circ}$, and 26$^{\\circ}$ because these are the observed separations \\citep{Hedman19}.\nThese separations are closer than the expected separations for an equal-mass situation: 41.498$^{\\circ}$, 37.356$^{\\circ}$, and 41.498$^{\\circ}$ \\citep{Salo88}.\nWe therefore solved the above equations for arbitrary masses, using Gaussian elimination, which involved re-ordering the rows, and found that the solution contains a mass $\\epsilon$ that is calculated as either zero or a small negative number on the order of $10^{-16}-10^{-19}$, \ndepending on the order in which the rows are solved (most likely a numerical issue involving the limit of double precision numbers):\n\\begin{equation}\n\\pmatrix{ m_1 \\cr\n               m_2 \\cr\n               m_3 \\cr\n               m_4 }\n= \\pmatrix{ m_T \\cr\n               m_M \\cr\n               m_L \\cr\n               m_{LL} }\n= \\pmatrix{ \\epsilon \\cr\n               0.55825 \\cr\n               0.00635 \\cr\n               0.4354 }\n\\end{equation}\nwhen we normalize the relative masses such that their sum is 1.\nThus, there does not exist a physical solution for the stationary configuration with four objects that would produce four comparably bright clumps.\n\nThere are two possible ways that the clumps could still reflect a collection of co-orbital source bodies: \nthe four source bodies could have been librating around the equilibrium location \nor there could be another massive body in the system that did not produce a visible clump.\nIt is certainly possible for there to be only four non-stationary clumps and for this to be a transient phenomenon. \nIn fact, \\citet{Hedman19} \nidentified slow changes in the clumps' azimuthal separations over time that could be evidence for libration.\nIt is unlikely, however, for the clumps to be on the edge of a libration cycle, \ndue to how azimuthally compact the whole configuration is. \nThe most compact state of a configuration of three is in the symmetric mode when the outer bodies are at their closest approach to the middle body. \nA similar symmetric mode in a system of four bodies would require the outer two bodies to converge at a faster rate than the middle two bodies.\nThe observed drift rates, however, show the opposite trend, with the middle two clumps drifting at a faster rate than the outer clumps \\citep{Hedman19}.\n\n\\begin{figure}[h]\n\\centerline{\\includegraphics[width=0.7\\columnwidth]{D68_plus_5_angular_separation}}\n\\caption{The configuration of the four D68 clumps along with the two regions where a fifth object could be. \nOne region is leading, the other is trailing. \nThe direction of orbital motion is counterclockwise.\n}\n\\label{D68p5as}\n\\end{figure}\nIf, however, the dust around four source bodies was stirred up by an object that passed nearby, \nit is certainly possible this object could have missed other source bodies in the D68 ringlet.\nWe therefore consider whether there could be an unseen fifth object, \nwhose mass could account for the angular separations we observe between the four known clumps. \nWe explore the approximately 270-degree span of longitudes ahead of Clump LL and behind Clump T.\nUsing the same equations of motion \\citep{Renner04}, we find physically realistic solutions in two regions, \none centered 33$^{\\circ}$ ahead of Clump LL and one centered 32$^{\\circ}$ behind Clump T. \nThese regions each span about 22$^{\\circ}$ in longitude and are mapped out in Figure \\ref{D68p5as}.\nThe relative masses of the clumps that correspond to these solutions are plotted in Figure \\ref{rel_mass}.\nThe horizontal axis shows the longitude of Object 5 in the same longitude reference system used by \\citet{Hedman19}.\nThe left-hand side of the split horizontal axis corresponds to a configuration in which Object 5 is trailing the other D68 clumps; \nthe right-hand side corresponds to a configuration in which Object 5 is leading the other D68 clumps. \nIn more compact configurations (when Object 5 is near longitudes -80 and 55), the middle and outer masses are greater than the second and fourth masses.  \nIn less compact configurations (when Object 5 is near longitudes -95 and 75), Object 5's mass would be more than double that of any other mass, \nand the clump farthest away from Object 5 becomes the least massive while the other three would require comparable masses. \n\n\\begin{figure}[h]\n\\centerline{\\includegraphics[width=\\columnwidth]{relative_masses}}\n\\caption{This plot shows the relative masses of the five co-orbitals for each possible configuration. \nCompact configurations are characterized by the more massive bodies in positions 1, 3, and 5. \nExtended configurations, by contrast, have the most massive object on one end, with the other masses tending to decrease with increasing distance.\n}\n\\label{rel_mass}\n\\end{figure}\n\n\\section{Numerical Investigations of the Configurations' Stability \\label{numerical}}\nTo further examine the dynamics of a co-orbital system at the semi-major axis of the D68 ringlet and to investigate stability limits, \nwe numerically simulated the motion of point masses at the longitudes of the clump peaks, \nadding in a fifth point mass at one of the locations permitted by the methods found in \\citet{Renner04}.  \nFor orbital simulations, we used the hybrid symplectic\/Bulirsch-Stoer algorithm in the Mercury6 package \\citep{Chambers99}. \nOur orbital simulations considered Saturn as the central mass and included terms up to $J_6$ in its gravitational field. \nThe constants used for these simulations were taken from \\citet{Jacobson06} and \\citet{Archinal18}, \nconverted to the units used in Mercury6, and are found in Table \\ref{tbl-1}. \nWe used a time step of 0.02 days, which for D68 corresponds to about one tenth of an orbit. \n\n\\begin{table}\n\\begin{center}\n\\caption{Parameters of Saturn used for numerical simulations, from \\citet{Archinal18} and \\citet{Jacobson06} \\label{tbl-1}}\n\\begin{tabular}{crr}\n\\tableline\n\\tableline\nParameter & Value \\\\\n\\tableline\nR$_{\\saturn}$ &60,268 km \\\\\nGM$_{\\saturn}$ &37931207.7 km$^3$ s$^{-2}$\\\\\n$J_2$ &1.629071$\\times 10^{-2}$\\\\\n$J_4$ &-9.3583$\\times 10^{-4}$ \\\\\n$J_6$ &8.614$\\times 10^{-5}$ \\\\\n\\tableline\n\\end{tabular}\n\\end{center}\n\\end{table}\n\nFor the sake of simplicity, we focused on one specific stable solution with the corresponding angular separation of Object 5 in order to do numerical simulations,\nthough other configurations were also investigated, both on the leading and trailing sides, to ensure that our conclusions are general. \nWe focus on a configuration with Object 5 ahead of Clump LL by 33$^{\\circ}$, as specified in Table \\ref{tbl-2}. \n\\begin{table}\n\\begin{center}\n\\caption{Initial parameters of co-orbitals used for numerical simulations \\label{tbl-2}}\n\\begin{tabular}{crrr}\n\\tableline\n$a$ &67,627 km \\\\\n\\tableline\n\\tableline\nBody & Mean longitude & Relative mass &\\\\\n\\tableline\nT &$122^{\\circ}$ &0.129 &\\\\\nM &$148^{\\circ}$ &0.069 &\\\\\nL &$180^{\\circ}$ &0.322 &\\\\\nLL &$209^{\\circ}$ &0.066 &\\\\\n5 &$242^{\\circ}$ &0.414 &\\\\\n\\tableline\n\\end{tabular}\n\\medskip\n\\\\ Note: Relative mass is normalized such that the sum of all five masses equals 1. \n\\end{center}\n\\end{table}\n\nWe explored perturbations to this configuration in semi-major axis and longitude, \nmodifying the initial semi-major axis or longitude for some of the bodies.\nWe also varied their absolute mass, while keeping their relative masses constant, as calculated above \\citep{Renner04}. \nAlthough the highest mass range we expect for the clump source bodies is $10^{9}-10^{10}$ kg because they have not been observed directly \\citep{Hedman19}, \nwe also consider much more massive configurations because these evolve more quickly \nand in this way clarify how these systems respond to perturbations.\nThus we consider three different situations: one with extreme masses of $10^{20}-10^{21}$ kg (i.e., similar to Enceladus, Tethys, and Dione), \none with with masses of $10^{13}-10^{14}$ kg (i.e., similar to Polydeuces, Pallene, and Daphnis), \nand one with masses of $10^{8}-10^{9}$ kg,  close to that expected for the D68 source bodies.\n\nIn each simulation, we plot the longitudinal evolution of the bodies with respect to a reference longitude, which is calculated for each timestep as\n\\begin{equation}\n\\lambda_0 = \\arctan{\\frac{\\sum_i^N{\\sin{\\lambda_i}}}{\\sum_i^N{\\cos{\\lambda_i}}}}\n\\end{equation}\nwhere $\\lambda_i$ is the mean longitude of body $i$. \nThis equation works well when the longitudinal oscillations are small. \nThis type of plot gives a quick sense of stability and of orbital evolution.\n\nWe verify that the stationary points found using the method of \\citet{Renner04} are indeed stable \nby placing objects there and finding they do not evolve in 1000-year simulations with high masses ($10^{20}-10^{21}$ kg; see Figure \\ref{large0}).\nHere we do not explore perturbations in initial longitude or semi-major axis for the high-mass case\nbecause the larger masses complicate scalings to the real system.  \n\n\\begin{figure}[h]\n\\centerline{\\includegraphics[width=\\columnwidth]{lmda_vs_t_large_0}}\n\\caption{With extremely high masses and no perturbations\n(initial angular separations of 33$^{\\circ}$, 29$^{\\circ}$, 32$^{\\circ}$, and 26$^{\\circ}$), \nthe system is stable, consistent with the analytic theory.\n}\n\\label{large0}\n\\end{figure}\n\nWe consider two types of perturbation, longitudinal and radial, in the medium-mass case,\n$10^{13}-10^{14}$ kg.\nThe objects are massive enough that it is easier to demonstrate both stable libration and more chaotic mutual encounters.\nFirst, we consider a longitudinal shift in which the system begins in a more compact configuration, \nand we find that the masses oscillate stably around the solution (see Figure \\ref{compact}).\n\n\\begin{figure}[h]\n\\centerline{\\includegraphics[width=\\columnwidth]{lmda_vs_t_med_compact}}\n\\caption{Including an initial perturbation to longitudes to start the system in a more compact configuration\n(with initial angular separations of 28$^{\\circ}$, 24$^{\\circ}$, 27$^{\\circ}$, and 21$^{\\circ}$), \nthe bodies oscillate around the stable solution, which is indicated by the dashed lines.\n}\n\\label{compact}\n\\end{figure}\n\nSecond, we consider radial perturbations in which we modify the initial semi-major axis. \nWe define a critical semi-major axis separation $\\Delta a_{\\rm{crit}}$ which separates \nsmall oscillatory motion like that shown in Figure \\ref{med50} from the sort of motion shown in Figure \\ref{med500}.\nWe explored through simulations the allowable perturbations to semi-major axis using the mode in which \nClump LL is given a positive $\\Delta a$ and Clump M is given a negative $\\Delta a$, just as in Figure \\ref{med50}. \nWe found that, for these relative masses in this specific perturbation mode, the critical semi-major axis separation's relation to absolute mass is best represented as\n\\begin{equation}\n\\frac{\\Delta a_{\\rm{crit}}}{a} \\simeq 1.06 \\left(\\frac{m_{\\rm{clumps}}}{M_{\\rm{planet}}}\\right)^{0.49}\n\\end{equation}\nFor the medium-mass case, $\\Delta a_{\\rm{crit}} = 75.8$ m, which occurs in between the cases \nshown in Figures \\ref{med50} and \\ref{med500}, namely, 50 m and 200 m. \nPerturbations of $\\Delta a = 50$ m\nare small enough that when two of the bodies approach each other, \nthey exchange energy and angular momentum in such a way as to begin receding from each other, similar to the periodic orbital swap of Janus and Epimetheus. \nPerturbations of $\\Delta a = 200$ m\nare too much for a stable configuration, which results in bodies looping around to approach the other side of the co-orbital system \nand eventual spreading into multiple orbits via gravitational interactions with the other bodies. \n\n\\begin{figure}[h]\n\\centerline{\\includegraphics[width=\\columnwidth]{lmda_vs_t_med_50}}\n\\caption{With sufficiently small initial perturbations to semi-major axes (50 m for Daphnis-scale co-orbitals), \nthe bodies oscillate around a stable solution, which is indicated by the dashed lines.\n}\n\\label{med50}\n\\end{figure}\n\n\\begin{figure}[h]\n\\centerline{\\includegraphics[width=\\columnwidth]{lmda_vs_t_med_200}}\n\\caption{With large enough initial perturbations to semi-major axes (200 m for Daphnis-scale co-orbitals), \nthe system becomes unstable when some of the bodies encounter each other.\n}\n\\label{med500}\n\\end{figure}\n\n\\begin{figure}[h]\n\\centerline{\\includegraphics[width=\\columnwidth]{lmda_vs_t_low_1}}\n\\caption{With realistic masses, \neven semi-major axis perturbations of one meter result in system instability. \nAlthough low-mass co-orbital systems are fragile, stability could be achieved with the help of an external force.\n}\n\\label{low1}\n\\end{figure}\n\nTo apply our numerical simulations to the D68 clumps, however, we must also consider the dynamics in a low-mass case,\n$10^{8}-10^{9}$ kg.\nFor the low-mass case, $\\Delta a_{\\rm{crit}} = 27$ cm, which is confirmed by Figure \\ref{low1}.\nWith only 1-m perturbations (a 2-m separation in semi-major axis), the point masses drift by each other, \nwith the three closest approaches between the centers of any two bodies as 266 m, 365 m, and 400 m.\\footnote{We re-examined \nwith a time step of $2 \\times 10^{-4}$ days any approach of two bodies within 1 km from each other, \nwhich corresponds to about 15 Hill radii for the largest mass.} \nWith a density of 0.5 g\/cm$^3$, spherical bodies of these masses would have radii ranging from 49 to 89 m. \nThus, although these closest approaches would not be collisions, they would still be close enough gravitational encounters to provide significant perturbations, in a range of 4 to 6 Hill radii for the largest mass. \nWe consider such a system to be fragile. \n\n\n\\section{Discussion and Implications \\label{disc}}\nTo emphasize how fragile the system is, we can estimate the impulse required to perturb a moonlet's semi-major axis by 1 m, similar to what has been done in \\citet{Hedman2020}. \nFor nearly circular orbits, the standard orbital perturbation equations can describe the rate of change of semi-major axis over time as \\citep{Burns76, Hedman18}\n\\begin{equation}\n\\frac{\\delta a}{\\delta t} = 2 n a \\frac{F_p}{F_G}\n\\end{equation}\nwhere the mean motion $n = \\sqrt{GM\/a^3} \\simeq 1751.7^{\\circ}\/$day, \n$F_p$ is the azimuthal component of the perturbing force, \n$F_G = GMm\/a^2 = n^2am$ is the gravitational force on the moonlet from the planet's center, \n$M$ is the planet's mass, and $m$ is the moonlet's mass. \nThe moonlet will thus undergo a semi-major axis change $\\delta a$ upon receiving an azimuthal impulse\n\\begin{equation}\nF_p \\delta t = \\frac{F_G}{2na}\\delta a = \\frac{1}{2}nm \\delta a.\n\\end{equation}\nWith the range of masses we use for our low-mass case, \nthe impulse required to perturb a moonlet's semi-major axis by 1 m ranges from \n$4.2 \\times 10^4$ kg m\/s to $2.7 \\times 10^5$ kg m\/s. \nFor any collision between a moonlet and interplanetary debris, \nthe impact speed would be comparable to the D68 orbit speed $v = na \\simeq 24$ km\/s. \nDividing the range of impulses by this orbit speed, we get a range of masses roughly from 2 kg to 10 kg for the interplanetary impactor.\nAssuming a density of 0.5 g\/cm$^3$, the piece of interplanetary debris would need to be 0.1 m to 0.2 m in radius.  \nThe estimated cumulative influx rate $\\Phi$ for debris of this size is around $10^{-17}\/\\textrm{m}^2\/\\textrm{s}$ \\citep{Tiscareno13}.\nThus the rough timescale $t \\simeq 1\/\\Phi A$ on which we can expect such a collision, \nusing the cross-sectional area $A$ for moonlets with radii of 49 m to 89 m, \ncorresponds to a range from 130,000 years to 420,000 years, but this is the impact timescale for just one of the objects. \nBecause an impact into any of the objects can break the system, we can adjust the system timescale to about 40,000 years by adding their cross-sectional areas together. \nWe therefore cannot expect a co-orbital configuration at D68 to last longer than a few tens of thousands of years. \nFor this reason we call the system fragile and find it unlikely that a co-orbital system could explain the orbital evolution of the clumps or the ringlet.\n\nConsequently, we look for other resonances that could drive the orbital evolution of the clumps or the ringlet. \nIt is unlikely that a corotation resonance with any satellite is responsible for the clumping of material into ring arcs. \nA 30$^{\\circ}$ separation between clumps would be the result of a 12-fold pattern at the D68 semi-major axis. \nA 12-fold pattern could be caused by a 13:12 corotation resonance with an external perturber or an 11:12 corotation resonance with an internal perturber.\nA 13:12 corotation resonance with an external perturber would require a perturber at a semi-major axis of 71,300 km, \nwhich is not as far out as D72, the structure closest to D68.\nAn 11:12 corotation resonance with an internal perturber would require a perturber at a semi-major axis of 63,800 km,\nwhich is a few thousand km away from Saturn's equatorial radius (60,268 km).\nThere is no evidence for any moons or ringlets in these regions.   \nMoreover, no results came from a numerical search for corotation resonances up to fourth-order between D68 and Janus, Mimas, Enceladus, Tethys, Dione, Rhea, or Titan. \n\n\\begin{table}\n\\begin{center}\n\\caption{Predicted corotation resonance locations, \nthe $r_{M14}$ column corresponding to our predictions based on the pattern periods reported in \\citet{Marley14}, \nthe $r_{M19}$ column corresponding to our predictions based on the pattern speeds reported in \\citet{Mankovich19}\n \\label{tbl-3}}\n\\begin{tabular}{ccrr}\n\\tableline\n\\tableline\n$\\ell$ & $m$ & $r_{M14}$ (km) & $r_{M19}$ (km) \\\\\n\\tableline\n8 &6  & 67,852 &67,663 \\\\\n3 &3  & 67,732 &67,932 \\\\\n2 &2  & 66,132 &67,235 \\\\\n\\tableline\n\\end{tabular}\n\\end{center}\n\\end{table}\n\nIt is possible that a resonance of some sort with Saturn itself could be responsible for the D68 clumping. \nThe outer Lindblad resonance of Saturn's $\\ell = 5$, $m=3$ oscillation mode is located in the D68 region, \nreported first at 67,625 km $\\pm$ 550 km \\citep{Marley93} and more recently at roughly 67,550 km \\citep{Marley14}.\nAlthough Lindblad resonances do not confine material, \neach such resonance can be associated with a corotation resonance, which can confine material. \nTo locate these corotation resonances, we computed the radii at which the mean motion (using the second-order equation from \\citealt{Renner06}) \nmatches the pattern speeds associated with the modes reported in \\citet{Marley14} and \\citet{Mankovich19}.\nThe modes that produce corotation resonances near D68 are listed in Table \\ref{tbl-3}.\nBecause the pattern speed is dependent on Saturn's structure, \nany of these modes could possibly be responsible for providing a corotation resonance to confine D68 material. \nMode splitting or mixing could also be involved \\citep{Fuller14}, \nallowing the locations of these resonances to fall at slightly different radii than what we can compute.\nFor a corotation resonance of one of these modes to explain the D68 clumping,\nit would require a planet-based angle that moves at a speed comparable to D68's mean motion. \nAlthough the set of angular separations among the clumps favors a 12-fold pattern,\nit is also possible for them to be confined to within the libration longitude of one or a few stable points of a lower $m$ mode, \nand then be spaced out within that external potential. \nPerhaps there is a set of co-orbital moonlets that are trapped together and librating within a larger potential, \nsimilar to \\citet{Renner14}'s model of the Neptune ring arcs. \nRadial oscillations of $\\pm$ 10 km have been observed for the D68 ringlet with an estimated period of 14-15 years \\citep{Hedman14}, \nthough the clumps are drifting more slowly than the rest of the ringlet \\citep{Hedman19}.\nThese radial oscillations could be evidence for that libration. \n\nIn conclusion, \nwe have tested and ruled out long-term stable co-orbital configurations as an explanation for the spacing of the D68 clumps. \nWe therefore predict\nthat either the clumping is a transient phenomenon,\nor that an external mechanism is trapping the clumps in this region. \n\n\\acknowledgements\nWe are grateful to many individuals for useful discussions, especially P. Ricker for guiding J. A'Hearn's initial studies on co-orbitals; \nB. Sicardy, M. El Moutamid, and M. \\'Cuk for insightful questions and clarifications at the 2019 DDA meeting; H. Salo for helpful correspondence, \nand two anonymous reviewers for their helpful comments that have improved this paper. \nWe also thank NASA for the support through the Cassini Data Analysis and Participating Scientist Program grant NNX15AQ67G.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\n\\section{Introduction} \n\n\n\n\nSome of the most intensely studied condensed matter systems are \nhighly anisotropic, see \\fref{fig:survey}: cuprates, pnictides, ruthenates, cobaltates, graphene, transition-metal dichalcogenides, and ultra-thin oxide films.\n\nIndeed, with spherical symmetry surrendered to translational invariance, the crystal-structure of periodic solids may trigger substantially anisotropic phenomena:\nThe coordination geometry of chemical constituents\ncan induce crystal-fields, the lifting of orbital degeneracies, and \nthe directional dependence of transfer integrals. \nIn layered compounds or geometrically engineered ultra-thin films, \nthe latter effects conspire to produce low-energy dispersions\nthat are---to a degree---confined to a two-dimensional plane.\nIn \\fref{fig:survey} we quantify such anisotropy for a number of prominent correlated materials\naccording to a measure $\\alpha$ of their electronic structure, which we introduce below.\n\n\n        \\begin{figure}\n            {\\includegraphics[width = 1.05\\linewidth] {material_survey.pdf}} \n            \\caption{Overview of correlated materials classified by their dimensionality $\\alpha$ (see \\protect\\eref{alpha}, \\protect\\sref{abinitio}) and the nominal valence of the transition-metal ions:  $\\alpha=1$ for the cubic case in 3D, $\\alpha=0$ corresponds to a planar 2D system. \n            Full (open) symbols indicate $\\alpha$ when the low-energy electronic structure uses the full $d$-shell (the $d_{x^2-y^2}$ orbital) of the transition metal, see also Fig.~\\protect\\ref{fig:survey2}.\n            We also include the $sp^3$-system graphite\\protect\\cite{Chung2002}.}\n            \\label{fig:survey}\n        \\end{figure}\n\n\nThe effectively reduced dimensionality in the above mentioned one-particle ingredients\nhas significant consequences for many-body effects\\cite{Aoki1992,Blundell2001,Larkin2008}:\nFirst, low dimensions are the dominion of non-local  fluctuations in space: due to the low-coordination of the\nlattice geometries, \nthe physics of the system is strongly dependent on the specific spatial configuration realized at each step of its time-evolution.\nAs a result, in comparison to the 3D isotropic case, ordering instabilities are typically suppressed in quasi-2D materials or fully obviated in purely 2D systems (cf.\\ Mermin-Wagner theorem\\cite{PhysRevLett.17.1133} or Kosterlitz-Thouless transitions\\cite{Kosterlitz1973}) by strong spatial fluctuations. The latter are reflected in significant enhancements of the corresponding susceptibilities, which  affect large regions of the parameter space in 2D, while in 3D these are typically confined to the proximity of the actual phase transitions\\cite{RevModPhys.90.025003}. \nSecond, these increased  (two-particle) fluctuations may affect one-particle spectral properties, as dictated by the Schwinger-Dyson equation of motion:\nCorresponding renormalization effects, such as static energy shifts, quasi-particle effective weights and lifetimes may acquire strong non-local variations. The most prominent example is certainly the pseudogap regime in doped cuprates\\cite{Timusk_1999,Norman2005,RevModPhys.78.17,keimer_quantum_2015},\nbut momentum-selective coherence and quasi-particle weights can in fact originate from various non-local fluctuations of spin, charge, or orbital degrees of freedom\\cite{Matthias_SVOultra}.\nAccordingly, pseudogap physics has been evidenced in a number of correlated materials, e.g.,\niron pnictides\\cite{Xu2011,PhysRevLett.109.027006,PhysRevLett.109.037002,PhysRevB.89.045101} and chalcogenides\\cite{PhysRevLett.111.217002}, iridates\\cite{Kim187}, and (layered) nickelates\\cite{PhysRevLett.106.027001}.%\n\\footnote{%\nFurther, there is indirect (=non-spectral), evidence for pseudogap physics in quasi-2D organic charge-transfer salts\n(see, e.g., Ref.~\\onlinecite{PhysRevB.80.054505} and references therein).\n}\n\n\n\nThe most successful {\\it local} approach for correlated electrons is dynamical\nmean-field theory (DMFT)\\cite{bible}, which can be combined with density functional theory, DFT+DMFT\\cite{RevModPhys.78.865}, for {\\it ab initio} material calculations. \nExact in $D=\\infty$\\cite{PhysRevLett.62.324}, DMFT is empirically found to be\nreasonable in three dimensions--- except in the vicinity of a\n(second order) phase transition (for the reasons stated above).\nHowever, recently it was shown\\cite{jmt_dga3d}\nthat even in the presence of strong non-local (anti-ferromagnetic) fluctuations,\nthe {\\it dynamical} part of many-body renormalizations (to linear order the quasi-particle weight) remains essentially local in 3D, while \na notable variation within the Brillouin zone is engendered for {\\it static} components of the self-energy.\nAs a result, the self-energy verifies---to a good approximation---a space-time separation\\cite{jmt_dga3d}:\n\\begin{equation}\n    \\Sigma(\\vek{k},\\omega)=\\Sigma_{\\text{static}}(\\vek{k})+\\Sigma_{\\text{local}}(\\omega).\n    \\label{eqn:separation}\n\\end{equation}\n\n\nIn this work we explore and quantify the limits of this approximation:\nWe monitor the momentum-dependence of many-body renormalizations \nof the doped Hubbard model as a continuous function \nof the tetragonal anisotropy in the one-particle hopping---from the cubic case in 3D to the square lattice in 2D.\nWe find that for an anisotropy \nsmaller than {roughly} one half ($\\alpha<1\/2$), a local approximation to the dynamical self-energy becomes inadequate. \nIn conjunction with the survey of\nelectronic-structure anisotropies (\\fref{fig:survey}),\nour results hence provide guidance for future first principle investigations of layered correlated systems:  We  establish a rule-of-thumb when \nthe use of techniques beyond DFT+DMFT\\cite{Tomczak2017review,RevModPhys.90.025003,Anna_ADGA,Lechermann2017} becomes a prerequisite\nfor {reliable} {\\it ab initio} calculations. \n\nThe paper is organized as follows: We detail the employed\nmethodology in \\sref{method}.\nIn \\sref{results} we present and discuss our results: \\sref{abinitioresults}\nis devoted to the {\\it ab inito} classification of materials according to the anisotropy of their electronic structure. In \\sref{modelsection}\nwe study the Hubbard model for the range of anisotropy covered by the materials in \\sref{abinitioresults}.\nWe conclude in \\sref{perspective} with a synthesis of the materials classification and our many-body findings.\n\n\n\\section{Methods}\\label{method}\n\\subsection{Ab initio calculations}\\label{abinitio}\n\nFor the {\\it ab initio} survey of materials, \\fref{fig:survey}, we (i) performed (non-spinpolarized) density functional calculations (using the PBE functional) with WIEN2k\\cite{wien2k}, (ii)\nconstructed maximally localized Wannier functions\\cite{RevModPhys.84.1419} for the $d$-orbitals of the transition metal relevant for the low-energy band-structure with wannier90\\cite{wannier90} via wien2wannier\\cite{wien2wannier}, and (iii) analyzed the hopping amplitudes of the one-particle Hamiltonian in real-space: $H=\\sum_{\\tau\\tau\\ensuremath{^\\prime}, LL\\ensuremath{^\\prime}, RR\\ensuremath{^\\prime},\\sigma}H^{\\tau\\tau\\ensuremath{^\\prime}}_{LL\\ensuremath{^\\prime}}(R-R\\ensuremath{^\\prime})\\cc_{\\tau\\ensuremath{^\\prime} L\\ensuremath{^\\prime} R\\ensuremath{^\\prime} \\sigma}\\ca_{\\tau LR\\sigma}$. Here, $\\tau$ indexes transition-metal atoms in the unit-cell, $L$\nthe $d$-orbitals (or a subset of them, see below), $R$ the unitcell, and $\\sigma$ the spin.\nThe measure $\\alpha$ is then defined as the ratio of the in-plane and out-of-plane hopping amplitudes between  nearest-neighbour transition-metal ions:\n\\begin{equation}\n    \\alpha = \\frac{\\max_{\\tau_\\perp,R_\\perp}\\sum_{LL\\ensuremath{^\\prime}}|H^{\\tau\\tau_\\perp}_{LL\\ensuremath{^\\prime}}(R_\\perp)|}{\\max_{\\tau_\\parallel,R_\\parallel}\\sum_{LL\\ensuremath{^\\prime}}|H^{\\tau\\tau_\\parallel}_{LL\\ensuremath{^\\prime}}(R_\\parallel)|}\n    \\label{alpha}\n\\end{equation}\nTo the denominator (nominator) contribute all \norbital combinations of hopping elements between transition-metal ions\nthat are nearest-neighbours in the $ab$-plane (in the $c$-direction). Taking the maximum, e.g., for the numerator, over $(\\tau_\\perp,R_\\perp)\\ne(\\tau,R)$ assures selecting transition-metal ions that are nearest-neighbours.\nFor example,\n$R_\\perp=0$ ($R_\\parallel=0$) for an intra-cell hopping between nearest neighbours $\\tau\\ne\\tau_\\perp$ ($\\tau\\ne\\tau_\\parallel$);\nwhile $R_\\perp=a_z\\, \\hat{\\vek{e}}_z$ ($R_\\parallel=a_x\\,\\hat{\\vek{e}}_x$, or, if $a_y<a_x$, $R_\\parallel=a_y\\,\\hat{\\vek{e}}_y$) \nwith $\\tau=\\tau_\\perp$ ($\\tau=\\tau_\\parallel$) indicates the hopping between the same atom but in adjacent unit-cells translated by the lattice constant $a_x$ ($a_y$) in the direction of the  respective unit-vector $\\hat{\\vek{e}}_x$ ($\\hat{\\vek{e}}_y$). \nThe anisotropy as described by \\eref{alpha} is clearly too simple to account for {\\it global} material trends.\nIn particular, $\\alpha$ largely depends on the orbital-subspace chosen to represent the pertinent low-energy dispersions (this will be detailed below for the high-$T_c$ parent compounds). However, we believe $\\alpha$ to be a good indicator for trends {\\it within} a given family of compounds. \nAs such, the anisotropy proxy of \\eref{alpha} could even serve\nas descriptor in high-throughput studies that target the optimization of some property through, e.g., chemical pressure via iso-valent substitutions\\cite{jmt_radialKI}.\n\n\n\n\n\\subsection{Many-body calculations}\\label{model}\n\n\\subsubsection{The model}\\label{modelintro}\n\n\nWe consider the one-band Hubbard model (in the usual notation)\n\\begin{eqnarray}\n    H&=&\\sum_{\\mathbf{k}\\sigma} \\epsilon_\\mathbf{k}^{\\phantom{^\\dagger}}\n     \\cc_{\\mathbf{k}\\sigma}\\ca_{\\mathbf{k}\\sigma}+U\\sum_i n_{i\\uparrow} n_{i\\downarrow}\\label{ham}\n\\end{eqnarray}\non a tetragonal lattice (see \\fref{fig:hub}):\n\\begin{eqnarray}\n    \\epsilon_{\\mathbf{k}}&=&-2t_\\alpha \n     \\left[ \\cos(k_xa_x)+\\cos(k_ya_y)+\\alpha\\cdot\\cos(k_za_z)\\right]\n     \\label{dispersion}\n\\end{eqnarray}\nwhere $k_i\\in[0,2\\pi\/a_i)$ and we set $a_i=1$ ($i=x,y,z$).\nThe parameter $\\alpha\\in[0,1]$ is the one-band equivalent of \\eref{alpha} as it scales the hopping amplitude in the $z$-direction:\nIt allows to continuously tune the system between the 3D cubic case ($\\alpha=1$) and the\n2D square lattice ($\\alpha=0$).\nSuch change in the anisotropy  directly affects, per construction, the overall kinetic energy and, hence---for a fixed interaction---the degree of electronic correlations in the system. \nAs this trivial effect would hide the most interesting trends emerging from our dimensional investigation, we chose to keep the kinetic energy essentially independent of $\\alpha$. To this end, we adjust the overall hopping amplitude $t_\\alpha$ by requiring\nthe second moment of the density of states to be constant and fix our energy units by imposing: \n\\begin{equation}\n    \\int_\\infty^{-\\infty} \\epsilon^2 N(\\epsilon)\\,d\\epsilon \\stackrel{!}{=}\\frac{1}{4},\n\\end{equation}\nwhere $N(\\epsilon)=\\sum_{\\mathbf{k}}\\delta(\\epsilon-\\epsilon_\\mathbf{k})$. \nThis requirement\nyields the analytical expression\n\\begin{equation}\n    t_{\\alpha}=\\frac{1}{2\\sqrt{4+2\\alpha^2}}.\n\\end{equation}\n\\fref{DOS} in App.~\\ref{app1} displays the corresponding density of states (DOS) varying continuously from three to two dimensions (see also Ref.~\\onlinecite{Vollhardt_Jerusalem}). \n    The DOS of a hypercubic lattice in infinite dimensions is---as a result of the central limit theorem---a Gaussian. In 3D, the DOS is still reasonably close to that Gaussian, yet with a flattened top.\n    %\n    Going to 2D, however, two significant changes appear: \n    The flat top narrows into a single peak (a logarithmic divergence: the so-called van-Hove singularity), whereas the sides develop \\enquote{knees} (which become van-Hove singularities in 1D). \n    We note that the \n    {movement} of the knees towards the band-edge and the narrowing of the central plateau is somewhat linear with the change in dimensionality, while the increase in height of the narrowing plateau starts slow, and only speeds up significantly below \n    $\\alpha\\leq 0.5$, i.e.\\ for an effective dimension D $\\le 2.5$.\n    %\n    Note that, in this work, we neglect all complications introduced by next-nearest neighbour hoppings\\cite{PhysRevLett.87.047003} or ionic potentials\\cite{PhysRevB.98.235105}.\n     \n\\begin{figure}[!t]\n            \\centering\n            {\\includegraphics[clip=true,trim=0 225 450 0, width = .75\\linewidth] {scheme_lattice.pdf}} \n            \\caption{The tetragonal lattice: In the $xy$-plane, electrons hop with an amplitude $t_\\alpha$, while the electron transfer in $z$ direction is smaller by a factor $0\\leq\\alpha\\leq1$.}\n            \\label{fig:hub}\n\\end{figure}     \n     \n\n\\subsubsection{Dynamical vertex approximation}\nIn order to be able to reliably estimate how non-local fluctuations on top of DMFT affect the space time-separation of the self-energy in Eq.~(\\ref{eqn:separation}), we use a diagrammatic extension \\cite{RevModPhys.90.025003} of DMFT, the dynamical vertex approximation (D$\\Gamma$A\\cite{toschi_dynamical_2007}) in its most used  ladder-implementation with Moriya corrections \\cite{katanin_comparing_2009,rohringer_impact_2016}. In contrast to quantum cluster methods \\cite{maier_quantum_2005}, it includes temporal and non-local correlations at all lengths scales on an equal footing. For this reason, D$\\Gamma$A has been often exploited to describe classical \\cite{rohringer_critical_2011,delRe2019} and quantum phase transitions \\cite{PhysRevLett.119.046402, schafer_pam_2019}, as well as the associated fluctuations. For further technical details on the D$\\Gamma$A calculation see App.~\\ref{app:dga}. \n\nAfter the calculation of the momentum-dependent self-energy on the Matsubara axis $\\Sigma(\\mathbf{k},i\\omega_{n})$, \nwe adapt the chemical potential $\\mu$ in order to {recover} the system's filling,\ndue to the lack of full self-consistency of ladder-D$\\Gamma$A with Moriya corrections\\cite{katanin_comparing_2009,rohringer_impact_2016}.\n\nThroughout the paper, we focus on many-body renormalizations near the Fermi level. To this end, we  perform a Taylor series expansion, specifically \n\\begin{eqnarray}\n\\Sigma(\\vek{k},\\omega)&=&\\Re\\Sigma(\\vek{k},\\omega=0)+(1-1\/Z(\\vek{k}))\\omega\\nonumber \\\\\n&&+i\\Gamma(\\vek{k})(\\omega^2+\\pi^2T^2)+\\cdots, \n\\end{eqnarray}\nwhere $\\gamma(\\vek{k})=-\\Im\\Sigma(\\vek{k},\\omega=0)=\\Gamma(\\vek{k})\\pi^2T^2+\\mathcal{O}(T^4)$ is the scattering rate and $Z(\\vek{k})$ the quasi-particle weight---provided the system realizes a Fermi-liquid.\nThese quantities are extracted from the self-energy on the Matsubara axis, $\\Sigma(\\mathbf{k},i\\omega_{n})$, as follows:\n\\begin{eqnarray}\n\\Re \\Sigma(\\mathbf{k},\\omega=0)&=&\\Re\\Sigma(\\mathbf{k},i\\omega_{n})\\big\\vert_{\\omega_{n}\\rightarrow{0}}\\label{eqS0}\\\\\n \\gamma_{\\mathbf{k}}&=&-\\Im\\Sigma(\\mathbf{k},i\\omega_{n})\\big\\vert_{\\omega_{n}\\rightarrow{0}}\\label{eqg}\\\\\n Z_{\\mathbf{k}}&=&\\left[1-\\frac{\\partial\\text{Im }\\Sigma(\\mathbf{k},i\\omega_{n})}{\\partial\\omega_{n}}\\bigg\\vert_{\\omega_{n}\\rightarrow{0}}\\right]^{-1}\\label{eqZ}\n\\end{eqnarray}\n\n\n\nTo obtain stable results for our analyzed quantities (see next subsection), we performed and averaged over polynomial fits up to order {10} before the extrapolation $i\\omega_n\\!\\rightarrow\\!0$.\n\nIn all our calculations we use $U\\!=\\!2$ in units of $t_\\alpha$,\nplacing us---at zero doping---in the intermediate-coupling regime, yielding the highest magnetic transition temperature in 3D \\cite{rohringer_critical_2011}. In this work, we will restrict ourselves to non-local correlations stemming from magnetism in the paramagnetic regime (paramagnons).\n\n\n\n\\subsubsection{Analysis tools}\n\nTo analyse our results, we introduce the following measures\n of non-locality\\cite{jmt_dga3d,jmt_pnict} for the  expansion coefficients defined above in Eqs.~(\\ref{eqS0})-(\\ref{eqZ}), $a(\\vek{k})=\\Re\\Sigma(\\vek{k},\\omega=0)$, $Z(\\vek{k})$, and $\\gamma(\\vek{k})$: \n\\begin{enumerate}[(\\roman{enumi})]\n    \\item \nthe standard deviation \n                \\begin{equation}  \\Delta_{\\kvec} a(\\kvec) = \\sqrt{1\/N_{\\kvec} \\sum_{\\kvec} {\\left| a(\\kvec) - a_{loc}  \\right|}^2   }    \\label{measure-1} \\end{equation}\nwith respect to the local (Brillouin-zone average) value\n    $a_{loc}=1\/N_\\mathbf{k}\\sum_\\mathbf{k}a(\\vek{k})$.\n\\item the standard deviation on the Fermi surface (FS), i.e.\\ $\\vek{k}=\\vek{k}_F$:\n    \\begin{equation} \\Delta_{\\kvec}^{FS} a(\\kvec) = \\sqrt{1\/N_{\\kvec_{F}} \\sum_{\\kvec_{F}} {\\left| a(\\kvec) - a_{FS}  \\right|}^2   }, \\label{measure-2}\n    \\end{equation}\nwith $a_{FS}=1\/N_{\\mathbf{k}_F}\\sum_{\\mathbf{k}_F}a(\\vek{k})$.\nIrrespective of potentially large scattering rates that may invalidate the quasi-particle picture, we determine the Fermi surface as momenta $\\vek{k}_F$ that verify the quasi-particle equation, $\\det (\\mu-\\epsilon_{\\mathbf{k}_F}-\\Re\\Sigma(\\vek{k}_F,0))=0$.\n\\item the maximum absolute difference on the Fermi surface          \n    \\begin{equation} \\delta_{\\kvec}^{FS} a(\\kvec) = \\left| {\\text{max}}_{\\mathbf{k}_F}(a(\\kvec_F)) - \\text{min}_{\\mathbf{k}_F}(a(\\kvec_F)) \\right|.    \\label{measure-3}  \\end{equation}\n                \n                \\end{enumerate}\n\n\n\\section{Results \\& Discussion}\\label{results}\n\\subsection{Dimensionality of materials: a classification}\\label{abinitioresults}\n\n\nIn \\fref{fig:survey} we classify some important correlated materials according to the dimensionality of their electronic (band) structure (following \\eref{alpha}) and their nominal valences. As low-energy model we \n{choose} the full shell of $d$-orbitals of the respective transition metal. \n\n\n        \\begin{figure}\n            \\centering\n            \\centerline{\\includegraphics[clip=true,trim=0 0 0 40,width = 1\\linewidth] {material_survey2.pdf}} \n            \\caption{Dependence of the dimensionality $\\alpha$ on the chosen low-energy orbital subspace: solid symbols correspond to the values from \\fref{fig:survey} that use the full $d$-shell of the transition metal, open symbols use a single-orbital (Cu\/Ni-$d_{x^2-y^2}$) modeling.}\n            \\label{fig:survey2}\n        \\end{figure}\n        \n        \nThe most 3D-like materials included are perovskite oxides, such as titanates, vanadates and nickelates of the form $AM$O$_3$\\cite{imada} (group 2\/3 element or rare-earth {\\it A}, transition metal {\\it M}).\nIn fact, the perfect cubic perovskite structure---realized, e.g., in SrVO$_3$---has $\\alpha=1$\nby construction.\nA dramatic dimensional reduction can, however, be achieved by deploying these materials\nin ultra-thin films. Such geometric constraints may lead to a\nreduction of out-of-plane hopping \\cite{PhysRevLett.104.147601} (i.e.\\ $\\alpha<1$) \nas well as orbital polarizations \\cite{PhysRevLett.114.246401}, to the extent\nthat, e.g., SrVO$_3$ or CaVO$_3$-thin films undergo a metal-insulator transition below a critical thickness \\cite{PhysRevLett.104.147601,Kobayashi2017,McNally2019}.\nA path to lower symmetries in bulk perovskites are \ndistortions and tiltings of the oxygen octaheadra surrounding the transition metal $M$.\nThe resulting orthorhombic structures typically exhibit an increased degree of electronic correlations.\nIndeed, for $d^1$-perovskites, the {ensuing} lifting of degeneracies quenches orbital fluctuations and triggers a Mott transition\\cite{pavarini:176403,1367-2630-7-1-188}.\nHowever, this physics is not driven by inter-orbital hybridizations\nbut dominated by changes in local crystal-fields, which do not directly affect the value of $\\alpha$. Indeed, for the Mott insulator \nYTiO$_3$ $\\alpha\\approx 1$  despite pronounced deviations from a perfect cubic structure.\n\nDimensional effects captured by Eq.~(\\ref{alpha}) are instead crucial in oxides of copper:\nWe find $\\alpha\\!\\approx\\!0.4$ for (non-superconducting) Li$_2$CuO$_2$, while the two- and one-layer high-$T_c$ parents compounds La$_2$CuO$_4$ and HgBa$_2$CuO$_4$\\cite{barisic_demonstrating_2008}\nhave consecutively smaller $\\alpha$. The quasi two-dimensional physics in cuprates seems to be\nindelibly connected with the occurrence of high-temperature superconductivity under doping (see, e.g., Ref.~\\onlinecite{Leggett2006}). \nIndeed, structural (and also chemical) variations will lead to trends in various electronic degrees of freedom, and\nempirical connections between the latter and $T_c$ have been established. Among these (interlinked) indicators are\nin-plane next-nearest neighbour hoppings \\cite{PhysRevLett.87.047003}, the charge-transfer energy \\cite{Weber_2012,PhysRevX.8.021038},\nthe $e_g$-splitting \\cite{PhysRevLett.105.057003,PhysRevB.85.064501},\nand the magnitude of in-plane magnetic exchange couplings \n\\cite{Chang_LSCO}. \nHere, we note that, for the shown cuprates, the smaller $\\alpha$, the larger $T_c$ at optimal doping.\n\n\nAlso in the case of the iron pnictides and chalcogenides (in their tetragonal structures), the dimensionality parameter $\\alpha$ follows intuition:\nThe 122-family is the most isotropic, the 111 less so, and the 11-chalcogenide is the most 2D-like, see \\fref{fig:survey}.\nThis trend neatly follows the magnitude of electronic correlations as monitored by, e.g., by the effective mass\n\\cite{Yin_pnictide} or the fluctuating\nmagnetic moment\\cite{Yin_pnictide,PhysRevB.86.064411,watzenboeck2020}\n\\footnote{A notable outlier is LaFeAsO (see \\fref{fig:survey2}), which in our measure, \\eref{alpha}, is the most 2D-like pnictide considered. \nThis finding is congruent with the fact that the Fe-Fe distance in this 1111-compound is larger by about 50\\% than in the other pnictides considered. However, effective mass renormalizations in LaFeAsO\\cite{PhysRevB.81.100502,doi:10.1143\/JPSJ.77.093714} is comparable to BaFe$_2$As$_2$, the most 3D-like pnictide here. This deviation in the trend is ascribed to the lower value of the screened interaction in the $3d$-orbital manifold of the 1111-material\\cite{Miyake2009}.}\n\n\nComparing the $\\alpha$-values of the families of cuprates and pnictides in \\fref{fig:survey}, one notices that---contrary to common belief--- the latter are overall as quasi two-dimensional as the former. As alluded to before, we stress that the measure $\\alpha$ crucially depends on the choice of the orbital subset sought to represent the low-energy electronic structure. Comparisons of materials \nare therefore most adequate when they allow for a common orbital framework, thus, in particular, when the materials belong to the same family of compounds. Indeed, in \\fref{fig:survey}, $\\alpha$ is computed from the hoppings of the full $d$-shell of the transition metal. \nContrary to the multi-orbital pnictides with nominal $d^6$-valence\\cite{Yin_pnictide},\nthe low-energy physics of $d^9$ cuprates is mostly dominated by a single orbital: the $d_{x^2-y^2}$ (see, however, e.g., Refs.~\\onlinecite{PhysRevLett.58.2794,PhysRevLett.105.057003,tabis_w_notitle_nodate}).\nThe smaller orbital space limits the possibility for other dimensional effects beyond the purely geometrical ones.\nIn \\fref{fig:survey2} we therefore report $\\alpha$-values for cuprates and the nickelate LaNiO$_2$\\cite{LaNiO2}%\n$^,$\\footnote{\nrelated to the recently discovered infinite-layer $d^9$ nickelate superconductor Nd$_{0.8}$Sr$_{0.2}$NiO$_2$\\cite{Li2019,Wilson2019}. We also include LaNiO$_2$H, a $d^8$ Mott insulating\\cite{Liang_LaNiO2} by-product in the synthesizing of LaNiO$_2$.\n}\nwithin said single-orbital framework:\nWhile trends within the cuprates remain qualitatively unchanged, the whole family substantially moves towards the 2D-limit.\nThe anisotropy of LaNiO$_2$ surpasses 90\\%, but is still small as compared to the high-$T_c$ parent compounds (making it tempting to suggest finding layered $d^9$ nickelates with smaller $\\alpha$ to potentially increase $T_c$). \n\n\nHaving established the effective dimensionality of pertinent classes of correlated materials, we now investigate how the structure-driven electronic anisotropy \naffects the non-locality of many-electron renormalizations.\nAs anticipated, we do so in the context of a simple model, allowing us to identify global trends without complicating the picture with material specific degrees of freedom.\n\n\n\n\\subsection{The Hubbard model on a tetragonal lattice}\\label{modelsection}\n\n    \\begin{figure}\n            \\centering\n            \\centerline{\\includegraphics[width = 1\\linewidth] {NeelTemp_cos.eps}} \n            \\caption{Phase diagram of the tetragonal Hubbard model (\\protect\\eref{ham}): temperature $T$ vs.\\ doping $n$ and dimensionality $\\alpha$. The blue curve corresponds to the N{\\'e}el temperature $T_{N}$ in 3D ($\\alpha=1$)\\protect\\cite{jmt_dga3d,PhysRevLett.119.046402}; the gray gradient is a guide to the eye\n            that qualitatively estimates $T_{N}$ for $\\alpha<1$ with a cosine function.\n            Coloured pins indicate the points in parameter space chosen for our investigation: $(n,\\beta)=(0.975,15)$ (red), $(0.95,17)$ (yellow), and $(0.9,20)$ (blue).\n            }\n            \\label{parameter_space}\n        \\end{figure}\n\n\\subsubsection{Results \\& Discussion}\\label{modelresults}\n\nWe solve the Hamiltonian \\eref{ham} for anisotropy parameters $\\alpha$, dopings $n$, and temperatures $T=(k_B\\beta)^{-1}$ as indicated in \\fref{parameter_space}.%\n\\footnote{As a caveat to the trends described below, let us note that with decreasing $\\alpha$, the system is located further and further away from the spin-ordered phase. Therefore, proximity effects are expected to be stronger close to 3D than for (quasi-)2D, which should be considered when discussing the overall effect of changes in the dimensionality.\n}\nEssentially, in this work we introduce with the dimensionality $\\alpha$ a new axis to the temperature vs.\\ doping phase diagram of Ref.~\\onlinecite{jmt_dga3d}.\\footnote{\nFor $\\alpha<1$ magnetic transition temperatures $T_{N}$ displayed as shadings are only guides to the eye. The $T_{N}$-curve for $\\alpha\\!=\\!1$ has been reproduced from Refs.~\\onlinecite{jmt_dga3d,PhysRevLett.119.046402}.}\n\\paragraph{Measures of non-locality.}\n\\fref{results_measures} displays the non-locality of $\\Re\\Sigma(\\vek{k},\\omega=0)$ [top], $Z(\\vek{k})$ [middle], $\\gamma(\\vek{k})$ [bottom] according to the measures  Eq.~(\\ref{measure-1}) [left], Eq.~(\\ref{measure-2}) [middle], and Eq.~(\\ref{measure-3}) [right] for the chosen regimes. For both  $Z(\\vek{k})$ and $\\gamma(\\vek{k})$ the momentum-variation is shown as a shaded region around the local value.\n\n\n\n    \\begin{figure*}\n        \\begin{subfigure}{0.8\\textwidth}\n            \\centering\n                \\begin{subfigure}{0.27\\textwidth}\n                    \\centerline{\\includegraphics[clip=true,trim=0 0 0 0,width = 1\\linewidth]{final_real_varianz.eps}}\n                    \\caption{}\n                    \\label{fig:plt:res:resigma}\n                \\end{subfigure}\n                \\hfill\n                \\begin{subfigure}{0.27\\textwidth}\n                    \\centerline{\\includegraphics[clip=true,trim=0 100 0 100,width = 0.6\\linewidth] {legend.pdf}}  \n                    \\caption{}\n                    \\label{fig:plt:res:legend}  \n                \\end{subfigure}\n                \\hfill\n                \\begin{subfigure}{0.27\\textwidth}\n                    \\centerline{\\includegraphics[clip=true,trim=0 0 0 0,width = 1\\linewidth]{final_real_minmax.eps}}\n                    \\caption{}\n                    \\label{fig:plt:res:resigma_minmax}  \n                \\end{subfigure}\n                \\vfill\n                \\end{subfigure}     \n    \n            \\begin{subfigure}{0.8\\textwidth}\n            \\centering\n                \\begin{subfigure}{0.27\\textwidth}\n                    \\centerline{\\includegraphics[width = 1\\linewidth] {final_Z_loc_varianz.eps}}\n                    \\caption{}\n                    \\label{fig:plt:res:Zalpha_BZ_varianz}\n                \\end{subfigure}\n                \\hfill\n                \\begin{subfigure}{0.27\\textwidth}\n                    \\centerline{\\includegraphics[width = 1\\linewidth] {final_Z_FS_varianz.eps}}\n                    \\caption{}\n                    \\label{fig:plt:res:Zalpha_FS_varianz}\n                \\end{subfigure}\n                \\hfill              \n                \\begin{subfigure}{0.27\\textwidth}\n                    \\centerline{\\includegraphics[width = 1\\linewidth] {final_Z_FS_minmax.eps}}\n                    \\caption{}\n                    \\label{fig:plt:res:Zalpha_FS_minmax}    \n                \\end{subfigure}         \n                \\vfill  \n                \n                \\begin{subfigure}{0.27\\textwidth}\n                    \\centerline{\\includegraphics[width = 1\\linewidth] {final_gamma_loc_varianz.eps}}\n                    \\caption{}\n                    \\label{fig:plt:res:gammaalpha_BZ_varianz}\n                \\end{subfigure}         \n                \\hfill\n                \\begin{subfigure}{0.27\\textwidth}\n                    \\centerline{\\includegraphics[width = 1\\linewidth] {final_gamma_FS_varianz.eps}}\n                    \\caption{}\n                    \\label{fig:plt:res:gammaalpha_FS_varianz}\n                \\end{subfigure}\n                \\hfill\n                \\begin{subfigure}{0.27\\textwidth}\n                    \\centerline{\\includegraphics[width = 1\\linewidth] {final_gamma_FS_minmax.eps}}\n                    \\caption{}\n                    \\label{fig:plt:res:gammaalpha_FS_minmax}    \n                \\end{subfigure} \n            \\end{subfigure}\n        %\n            \\caption{Quasi-particle parameters and their momentum-dependence:\n            $\\Re\\Sigma(\\vek{k},\\omega=0)$ (top), $Z(\\vek{k})$ (middle), $\\gamma(\\vek{k})$ (bottom) analysed in terms of the measure\n            $\\Delta^{loc}_{\\mathbf{k}}$ (Brillouin zone average, \\protect\\eref{measure-1}, left), \n            $\\Delta^{FS}_{\\mathbf{k}}$ (Fermi surface average, \\protect\\eref{measure-2}, middle)\n            $\\delta^{FS}_{\\mathbf{k}}$ (maximal absolute difference in Fermi surface,\\protect \\eref{measure-3}, right) as a function of the dimensionality $\\alpha$ for the parameters shown in panel (b).\n             \\DfsRe{} (not shown) is very similar to \\minmaxRe{}.}\n            \\label{results_measures}\n        \\end{figure*}\n\nWe first discuss the differences in these \n{\\sl local} quantities: \nThe  scattering rate $\\gamma_{loc}$ is larger the smaller the doping, i.e.\\ it is largest in direct proximity to half-filling ($n=1.0$), where an essentially insulating behavior has been evidenced by previous D$\\Gamma$A studies\\cite{rohringer_critical_2011,rohringer_impact_2016,RevModPhys.90.025003}.\nAt any doping, the scattering rate is consistently larger when averaged over the Fermi surface rather than the full Brillouin zone (compare panels (g) and (h)), as it is expected in the presence of strong non-local spin fluctuations. \nAt least at small doping the large scattering rate $\\gamma$ impedes the interpretation of $Z$ as the quasi-particle weight. For instance, for $n=0.975$ and $\\beta=15$ (data marked in red in \\fref{results_measures}), we find  $\\gamma_{loc}\\approx 65\\%$ \nof the 2D half-bandwidth, clearly invalidating the Fermi liquid picture.%\n\\footnote{Indeed, we previously noted in 3D\\cite{jmt_dga3d} that in this regime $\\gamma$ involves corrections\\cite{PhysRevB.68.155113} to Fermi liquid theory in its temperature dependence, \nas neither DMFT nor D$\\Gamma$A verify a $T^2$ behaviour.}\nAs a consequence, the fact that we find for this set of parameters the largest $Z_{loc}\\approx 0.7$ does not contradict the fact of being the most correlated regime considered.\n\nNext, we turn to the $\\alpha$-dependence of the {\\sl local} expansion coefficients:\nWhen averaged over the entire Brillouin zone, both the $Z$-factor and the scattering rate $\\gamma$ hardly change when the dimensionality is reduced.\nHowever, when limiting the momentum-average to the Fermi surface, i.e.\\ to the region where low-energy excitations are actually present, there is a significant upturn in both $Z$ and $\\gamma$ (panels (e) and (h)) when $\\alpha$ drops below $0.5$.\nThis trend is consistent with the behaviour of the underlying DOS (shown in the Appendix's \\fref{DOS}, and discussed in \\sref{modelintro}; see also the supplementary material\\cite{supps}) at low energies:\nIndeed, the density at the Fermi level drastically increases when the effective dimension drops below $2.5$ (i.e.\\ $\\alpha\\leq 0.5$), providing carriers available for electronic scattering and thus driving up the collision rate.\nIn particular at our largest dopings, this change in the trend is almost cusp-like.\n\nFinally, we address the momentum-dependence of the quasi-particle parameters.\nAs a function of $\\alpha$ the standard deviation throughout the Brillouin zone (following \\eref{measure-1}) of both $Z$ and $\\gamma$ slightly increases from 3D to 2D. Overall, however, their momentum-dependence is moderate in this measure. The momentum-variation of static shifts $\\Re\\Sigma(\\vek{k},\\omega=0)$, on the other hand,\nis already large in 3D,\nin accordance with previous results\\cite{jmt_dga3d}.\nWhen reducing $\\alpha$, this spread increases continuously for all investigated dopings and temperatures. In the most correlated case ($n=0.975$), the standard deviation $\\Delta^{loc}_{\\mathbf{k}}\\Re\\Sigma(\\vek{k},\\omega=0)$\ngrows from \n31\\% (3D) to above 50\\% (2D) of the respective half-bandwidth (cf.\\ \\fref{DOS} and \\onlinecite{supps}).\nStatic non-local renormalizations are indeed non-negligible for all studied regimes.\n\nThe variation on the Fermi surface, measured via \\eref{measure-2} \nreveals, instead, a notable enhancement towards lower dimensionality:\nWhile the standard deviation of $Z$ and $\\gamma$ (panels (e) and (h)) is virtually zero in 3D, it acquires consistently growing finite values for smaller $\\alpha$. The spread in both quantities is, however, significantly smaller than when averaged over the whole Brillouin zone (panels (d) and (g)).\n\nSince the averaging of the measures \\eref{measure-1} and \\eref{measure-2} might obfuscate a strong momentum variation carried by only a few $\\vek{k}$-points, we plot in panels (c), (f), (i) the maximal spread on the Fermi surface according to \\eref{measure-3}. The increased momentum resolution of this analysis reveals a much larger variation than in the $\\vek{k}$-averaged data for $Z$ and $\\gamma$, indicating that, as a matter of fact, the momentum dependence must be driven by {\\sl small areas} of the Fermi surface, where large deviations from the local average are found.\n\nAs to the trend with doping, we must note that the behavior of the momentum-spread of the scattering rate on the FS is inverted with respect to that on the whole Brillouin zone: the largest momentum-spread is realized for the largest doping. More quantitatively,  we find that the difference between the maximal and the minimal scattering rate surpasses 100\\% of the local value $\\gamma^{FS}$ on the Fermi surface below $\\alpha=0.5$. \n\n\n\n        \\begin{figure*}\n            \\begin{subfigure}{0.28\\textwidth}\n                \\centerline{\\includegraphics[clip=true,trim=0 40 0 50,width = 0.9\\linewidth]{Real_FS_new_20200302.eps}} \n                \\caption{}\n                \\label{fig:plt:res:FS_Real}\n            \\end{subfigure}         \n            \\hfill\n            \\begin{subfigure}{0.28\\textwidth}\n                \\centerline{\\includegraphics[clip=true,trim=0 40 0 50,width = 0.9\\linewidth]{Z_FS_20200302.eps}} \n                \\caption{}\n                \\label{fig:plt:res:FS_Z}\n            \\end{subfigure}\n            \\hfill\n            \\begin{subfigure}{0.28\\textwidth}\n                \\centerline{\\includegraphics[clip=true,trim=0 40 0 50,width = 0.9\\linewidth]{Gamma_FS_20200302.eps}} %\n                \\caption{}\n                \\label{fig:plt:res:FS_Gamma}\n            \\end{subfigure}\n            \\caption{Renormalizations on the Fermi surface: $\\Re\\left[\\Sigma(\\kvec{}_F,\\omega=0)-1\/N_{\\mathbf{k}_F}\\sum_{\\mathbf{k}_F}\\Sigma(\\vek{k}_F,\\omega=0)\\right]$, $Z(\\vek{k}_F)$,  and $\\gamma{}(\\vek{k}_F)$ for $n=0.90$, $\\beta=20$ and representative values of $\\alpha$ on the respective D$\\Gamma$A Fermi surface. \n            \\label{3DFS} }\n        \\end{figure*}\n\n\nAlso for $\\Re\\Sigma(\\vek{k},\\omega=0)$, the $\\alpha$-dependence is quite different when looking at peak values (panel (c)): instead of a continuous increase for shrinking $\\alpha$, the variation peaks at intermediate dimensionality and diminishes towards 2D.\nWe speculate that the following contributes to the reversal of the trend:\n$\\Re\\Sigma(\\vek{k},\\omega=0)$ describes the deformation of the Fermi surface with respect to the local DMFT starting point. For a metallic solution (and low enough $T$), this deformation is subject to\nconserving the Fermi surface volume (Luttinger's theorem) and C$_4$ symmetry in the ab-plane. Now, the decreasing dispersion along the c-axis introduces a strong geometric constraint that further impedes deformations (see also Appendix \\aref{sapp1}).\nFurthermore  we find, empirically, that \n\\begin{equation}\n    \\left.\\hat{\\vek{e}}_{\\mathbf{k}}\\cdot\\nabla_{\\mathbf{k}}\\Re\\Sigma(\\vek{k},\\omega=0) \/ \\hat{\\vek{e}}_{\\mathbf{k}}\\cdot \\nabla_{\\mathbf{k}}\\epsilon_{\\mathbf{k}}\\right|_{\\mathbf{k}=\\mathbf{k}_F}>0,\\label{ineq}\n    \\end{equation}\n    %\n    suggesting that the sign of the Fermi velocity perpendicular to the Fermi surface determines the direction of deformation, further limiting its freedom.\n\n    \n            \n\n\\paragraph{Renormalizations on the Fermi surface.}\nThe two measures of non-locality on the Fermi surface---\\eref{measure-2}: FS-average, \\eref{measure-3}: maximal absolute difference---yield notably different results, suggesting largely inhomogenous renormalizations.\nHere we investigate this further by explicitly plotting in \\fref{3DFS}\n$\\Re\\Sigma(\\vek{k},\\omega=0)$ (left column, with respect to the Fermi surface average, $1\/N_{\\mathbf{k}_F}\\sum_{\\mathbf{k}_F}\\Re\\Sigma(\\vek{k}_F,\\omega=0)$), $Z(\\vek{k})$ (middle), and $\\gamma(\\vek{k})$ (right) on the Fermi surface for the set of parameters given by  $n=0.9$, $\\beta=20$ and various $\\alpha$ (rows).\n\nIn 3D ($\\alpha=1$, top row in \\fref{3DFS}), the values of the $Z$-factor and the scattering rate $\\gamma$ are virtually homogeneous on the Fermi surface. \nThe momentum-differentiation grows but remains overall small down to $\\alpha=0.5$.\nBelow this mid-point between 3D and 2D, when the Fermi surface gets visibly quasi two-dimensional,\nwe notice the emergence of  a pronounced momentum-selectivity\nin $Z(\\vek{k}_F)$ and $\\gamma(\\vek{k}_F)$. Akin to hole-doped cuprates, excitations at the anti-node (($\\pi,0,k_z$)-direction and equivalent) become more short-lived than at the node ($\\pi,\\pi,k_z$).\nRecently, it has been suggested that such pseudogap physics\nis intimately linked to the proximity to a van-Hove singularity\\cite{wu2019PG}.%\n\\footnote{See Ref.~\\onlinecite{PhysRevX.8.021048} for a detailed analysis of connections between pseudogap physics and the topology of the Fermi surface in 2D.}\nIn our model, such a feature exists for $\\alpha=0$ and is located at half-filling. Regarding the trend with doping, \nthe variance throughout the Brillouin-zone is indeed largest for the smallest doping. However, as far as the momentum-selectivity of \nthe scattering rate $\\gamma$ and the \nquasi-particle lifetime $\\tau=1\/(Z\\gamma)$\non the Fermi surface is concerned, it is largest for the doping that places the chemical potential furthest away from the van-Hove singularity. \n\n\n\nNon-local renormalizations also deform the Fermi surface.\nHere (left column in \\fref{3DFS}), we show the pertinent $\\Re\\Sigma(\\vek{k}_F,\\omega=0)$ with respect to $\\overline{\\Re\\Sigma}^{FS} =1\/N_{\\mathbf{k}_F}\\sum_{\\mathbf{k}_F}\\Re\\Sigma(\\vek{k}_F,\\omega=0)$. \nThe latter Fermi-surface average can be seen as defining an effective local theory, obtained by a (partial) momentum-projection \nof the D$\\Gamma$A self-energy.\nFollowing the arguments detailed in Appendix \\aref{sapp1}, the Fermi surface expands\n(retracts) where $\\Re\\Sigma(\\vek{k}_F,\\omega=0)-\\overline{\\Re\\Sigma}^{FS}>0$ ($<0$). \nIn 3D ($\\alpha=1$), the Fermi surface is a body with tube-like openings in all axial directions.\nNon-local many-body effects shrink the tubes while expanding the inner core where the tubes intersect.\nWhen the hopping in $z$-direction decreases ($\\alpha\\leq 0.5$), the static non-local shifts of $\\Re\\Sigma$ expand the Fermi surface near the {\\sl xy}-basal \nplane, but narrow the tubal forms oriented along the z-axis, thus magnifying the effective dimensional reduction for correlated electrons.\n \n\\paragraph{Renormalizations at finite energies.}\nThe fact that the self-energy coefficients have a larger variation over the entire Brillouin zone than over the Fermi surface ($\\Delta^{loc}_{k}a(\\vek{k})>\\Delta^{FS}_{k}a(\\vek{k})$) suggests that non-local effects are larger for excitations at finite energies, $|\\epsilon_\\mathbf{k}|>0$.\nFor $\\Re\\Sigma(\\vek{k},\\omega=0)$ this can be rationalized as follows (using for simplicity a static self-energy): The momentum-derivative of $\\Re\\Sigma(\\vek{k})$ at $\\mathbf{k}_F$ reduces the effective mass (see, e.g., Eq.~(1) in Ref.~\\onlinecite{jmt_dga3d}). \nAway from the Fermi surface to first order\n$\\Re\\Sigma(\\mathbf{k})\\approx (\\vek{k}-\\vek{k}_F)\\cdot \\nabla_{\\mathbf{k}}\\Re\\Sigma(\\mathbf{k})_{\\mathbf{k}=\\mathbf{k_F}}$.\nAssuming the derivative \/ the effective mass to be roughly constant, this effect \n increases the band-width\\cite{jmt_pnict,PhysRevB.87.115110,jmt_svo_extended} via a shift  \n $\\Re\\Sigma(\\vek{k})-\\mu{> \\atop <}0$ (for $\\epsilon_\\mathbf{k}{> \\atop <}0$)\n that grows (linearly) with the distance to the Fermi surface. \n In this sense of widening dispersions, non-local correlations counteract local (dynamical) correlations.\n            \n\\section{Perspective and  Conclusions}\\label{perspective}\n\n\nWhen tackling a many-body problem---a model Hamiltonian or a strongly correlated material---knowledge about strength, nature and structure of correlation effects is a prerequisite for choosing the most efficient, yet adequate methodology:\nDoes the system allow for a perturbative\\cite{ferdi_gw,RevModPhys.74.601} or weak-coupling\\cite{RevModPhys.84.299,Hille2020} treatment? \nIs the low-energy physics known to be dominated by a single microscopic process or fluctuation channel\\cite{PhysRevLett.114.236402}? \nIs potential pseudogap physics mostly\ndriven by short-range fluctuations\\cite{ANDERSON1196,Sordi2012}?\nDoes the self-energy verify certain properties: Is it (to a sufficient degree) local (\\`a la DMFT\\cite{PhysRevLett.62.324,bible}), static (\\`a la Hartree-Fock, DFT+U\\cite{PhysRevB.44.943}), or does it approximately obey space-time separation\\cite{jmt_dga3d}, \\eref{eqn:separation}:\n$\n  \\Sigma(\\vek{k},\\omega)=\\Sigma_{\\text{static}}(\\vek{k})+\\Sigma_{\\text{local}}(\\omega)\n$?\nThe validity of \\eref{eqn:separation} in 3D for the Hubbard model\\cite{jmt_dga3d} and\nisotropic cubic materials\\cite{Anna_ADGA}, \nadvocates that the realistic methodology\nDFT+DMFT (that combines DMFT with density functional theory)\ncan be vastly improved by supplying it with an adequate static but non-local\npotential (a $\\Sigma_{\\text{static}}(\\vek{k})$ beyond DFT). The latter highlights the\nmerits of approaches such as {\\it GW}+DMFT\\cite{PhysRevLett.90.086402}$^,$\\cite{0953-8984-28-38-383001,Tomczak2017review},\nQS{\\it GW}+DMFT\\cite{jmt_sces14,Choi2016}, {SEx+DMFT}\\cite{paris_sex},\nor space-time separated {\\it GW}\\cite{jmt_dga3d},\nin all of which the non-local contribution to the self-energy (in the Wannier sense) is empirically\nfound to be essentially static\nat least at low energies.%\n\\footnote{\\eref{eqn:separation} may also lead to\nsimplifications in the dual fermion approach\\cite{PhysRevB.92.144409}.}\n\n\n\nAs a consequence, it is valuable to know the limits of validity of \\eref{eqn:separation}.\nOur results suggest that the error incurred by assuming space-time separability\nof the self-energy becomes prohibitive at an anisotropy corresponding to about half-way ($\\alpha\\approx 0.5$) between the isotropic 3D and the planar 2D case.\n\nOur findings hence support the view (the expectation) that in cuprates (d$^9$ nickelates\\cite{LaNiO2,Li2019,Wilson2019}) no local theory can account for the rich many-body physics at play. \nClearly, in 2D the self-energy does not obey space-time separation. \nAt least at half-filling other simplifications may, however, apply for the Hubbard model\\cite{PhysRevB.93.195134}.\n\nThe iron pnictides and chalcogenides reside in an $\\alpha$-regime that \nsuggests the presence of sizable non-local renormalizations, see \\fref{fig:survey}.\nConcurringly, diagrammatic fluctuation techniques recently\ndemonstrated\\cite{PhysRevLett.123.256401,bhattacharyya2020nonlocal}\nthat {\\it static} non-local renormalizations are indeed non-negligible---in line with previous experimental analyses \\cite{PhysRevLett.110.167002}.\nNon-local corrections to {\\it dynamical} renormalizations, however, were previously shown\\cite{PhysRevB.95.195115} to be small, at least away from the systems' N{\\'e}el\ntemperatures.\nThese recent results\nsubstantiate earlier claims based on perturbation theory, and strengthen the space-time separability of the self-energy in the iron pnictides and chalcogenides proposed in Refs.~\\onlinecite{jmt_pnict,jmt_sces14} \n\\footnote{Note that our study does not make any statements about non-local self-energies drawing from \nnon-local (beyond-Hubbard model) interactions. The latter can be notable\\cite{PhysRevB.95.245130} irrespective of the crystal-structure. In fact, non-local exchange self-energies are sizable in\npnictides\\cite{jmt_pnict,jmt_sces14,paris_sex}, as well as in isotropic correlated oxides\\cite{jmt_svo_extended,PhysRevB.87.115110,0295-5075-108-5-57003}. In a perturbative treatment, these non-local self-energy contributions are static, thus verifying \\eref{eqn:separation}.\nAlso, local {\\it multi-orbital} interactions\nthat can lead to non-local correlations\\cite{PhysRevB.91.235107} are (by construction) not included here.}%\n. \n\n\n\n\nAccording to the anisotropy classification of materials in \\fref{fig:survey},\nwe ascertain that perovskite oxides are firmly in the realm of \\eref{eqn:separation}: {\\it dynamical} self-energies will be essentially local.\nHowever, we found {\\it static} non-local renormalizations (beyond DMFT) to be non-negligible \nalso for cubic systems---motivating the above approaches to include a $\\Sigma_{\\text{static}}(\\vek{k})$ (beyond DFT).\nYet, most importantly, our findings strongly suggest that geometric constraints in (simulations of) oxide-based heterostructures\\cite{doi:10.1146\/annurev-matsci-070115-032057,Janson2018,Lechermann2018} or ultra-thin films\\cite{Potthoff1999,Liang_SRO,Golalikhani2018,PhysRevB.97.075107,Matthias_SVOultra,loon2020coulomb}\nmay results in strong {\\it dynamical} non-local effects---posing in particular limits on the applicability of DMFT.\nWe believe that our anisotropy measure of \\eref{alpha} can provide guidance \nto identify such systems prior to actual many-body calculations.\nMoreover, our findings call for reexamining layered systems such as sodium cobaltates and 124-ruthenates with realistic many-body approaches beyond DMFT\\cite{Anna_ADGA,Tomczak2017review,RevModPhys.90.025003}.\n\n\n\n\\section{Acknowledgements}\nThe authors gratefully acknowledge discussions with D.\\ Vollhardt.\nThe present work was supported by the Austrian Science Fund (FWF) through the Erwin-Schr\\\"odinger Fellowship J 4266 - ``{\\sl Superconductivity in the \nvicinity of Mott insulators}'' (SuMo, T.S.),\nproject ``{\\sl Simulating Transport Properties of Correlated Materials}'' (LinReTraCe P~30213-N36, J.M.T.), and\n project ``{\\sl Merging dynamical mean-field theory and functional renormalization group}'' (I 2794-N35, A.T.),\nas well as the European Research Council, with Grant No. 319286 (QMAC, T.S.) and Grant No. 725521 (TheOne, B.K.).\nCalculations were partially performed on the Vienna Scientific Cluster (VSC).\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section*{Methods}\n\n\\subsection*{Model description}\n\nThe class of models considered here are examples of\ninteracting particle systems \\cite{27} combining features of\nmultiplex networks. Given $\\ell\\in\\mathbb{N}$, a\nmultiplex network is a pair $G=(V,E)$, made of $\\ell$ layers\n$G_1,G_2,\\ldots,G_{\\ell}$ such that each layer is a simple graph\n$G_i=(V,E_i)$ with node set $V$ and edge set $E_i\\subseteq V\\times\nV$ for $i=1,\\ldots, \\ell$. Here, the node set $V$ is shared by all\nlayers and it can be either finite or infinite. The edge set of $G$\nconsists of $\\ell$ types of edges: $E=\\cup_{i=1}^{\\ell}E_i$. From\nthe perspective of graph theory, each edge between two nodes $u$ and\n$v$ in graph $G$ is a multiple edge consisting of at most $\\ell$\nparallel edges, each of which belongs to a respective layer $G_i$.\nWe assume that each layer $G_i$ has bounded degrees. Hence, each\nagent in the network $G$ has a bounded number of neighbours and at\nmost $\\ell$ types of relationship. Without loss of generality, we\nmay assume that the network $G$ is connected since one could\nconsider connected components separately in what follows.\n\nIn the Deffuant model \\cite{7,8}, two agents compromise according to the\nfollowing rules: initially (at time $t=0$), each agent $u\\in V$ is\nassigned an opinion value $X_0(u)\\in\\mathbb{R}$ identically and\nindependently distributed (i.i.d.) following some distribution\n$\\mathcal{L}(X_0)$. In the standard case, $\\mathcal{L}(X_0)$ is the\nuniform distribution over $[0,1]$. Independently of this, in the $i$th\nlayer, each edge $e\\in E_i$ is independently assigned a Poisson\nprocess with rate $\\lambda p_i$ with $p_i\\in(0,1)$ and $\\lambda>0$\nfor $i=1,\\ldots,\\ell$. We assume that $\\sum_{i=1}^{\\ell}p_i=1$\nwithout loss of generality. These Poisson processes defined on the\nedges in $E$ govern the evolution of opinions. Specifically, let\n$X_t(u)$ be the opinion value of agent $u$ at time $t\\ge0$, which\nremains unchanged as long as no Poisson event happens for any\nedge in $E$ incident to $u$. Let $d>0$, $\\alpha_1=1$ and\n$\\alpha_i\\in(0,1)$ for $i=2,\\ldots,\\ell$. When at some time $t$ the\nPoisson event occurs at an edge $e=\\{u,v\\}\\in E_i$ for some $i$,\nsuch that the pre-meeting opinions of the two agents are\n$X_{t-}(u):=\\lim_{s\\rightarrow t-}X_s(u)$ and\n$X_{t-}(v):=\\lim_{s\\rightarrow t-}X_s(v)$, we set\n\\begin{equation}\nX_t(u)=\\left\\{\\begin{array}{cc}\nX_{t-}(u)+\\mu(X_{t-}(v)-X_{t-}(u)),&\\mathrm{if}\\\n|X_{t-}(u)-X_{t-}(v)|\\le \\alpha_id;\\\\\nX_{t-}(u),&\\mathrm{otherwise},\n\\end{array}\\right.\\label{1}\n\\end{equation}\nand\n\\begin{equation}\nX_t(v)=\\left\\{\\begin{array}{cc}\nX_{t-}(v)+\\mu(X_{t-}(u)-X_{t-}(v)),&\\mathrm{if}\\\n|X_{t-}(u)-X_{t-}(v)|\\le \\alpha_id;\\\\\nX_{t-}(v),&\\mathrm{otherwise},\n\\end{array}\\right.\\label{2}\n\\end{equation}\nwhere $\\mu\\in(0,1\/2]$ is the so-called convergence parameter.\nTherefore, if the two pre-meeting opinions lie at a distance less\nthan a certain confidence bound from one another, the meeting agents\nwill come closer to each other symmetrically, by a relative amount\n$\\mu$, where $\\mu=1\/2$ implies that the two agents meet halfway through. If not,\nthen they will stay unchanged. It is worth noting that the model is well-defined since the bounded degree assumption ensures that almost\nsurely (i.e., with probability 1) none of the Poisson events will be\nsimultaneous for an infinite node set \\cite[p. 28]{27}.\n\nThe multiplexity in the above opinion model lies in two aspects.\nFirst, the interaction rates $\\lambda p_i$ in each layer can be\ndifferent. Second, the confidence bounds $\\alpha_i d$ in each layer\ncan be different too. We might as well consider distinct convergence\nparameters $\\mu=\\mu_i$ for the $i$th layer indicating different\nwillingness to change one's mind. However, it has been confirmed\nanalytically and numerically that $\\mu$ plays no role in the\nqualitative behaviour of the opinion dynamics; it rather only affects the\nconvergence time \\cite{5,7,17,18}.\n\n\\subsection*{Sharing a drink process}\n\nIn the section, we briefly review the sharing a drink (SAD) process\nproposed in \\cite{17}, which is particularly useful in later\nanalysis of the Deffuant model on $\\mathbb{Z}$; see also\n\\cite{13,18,26}.\n\nLet $k\\in\\mathbb{N}\\cup\\{0\\}$. The SAD process, denoted by\n$\\{Y_k(u)\\}_{u\\in\\mathbb{Z}}$, is a deterministic process defined\niteratively as follows: set\n\\begin{equation}\nY_0(u)=\\left\\{\\begin{array}{cl} 1,& \\mathrm{for}\\ u=0;\\\\\n0,&\\mathrm{for}\\ u\\in\\mathbb{Z}\\backslash\\{0\\}.\n\\end{array}\\right.\\label{A4}\n\\end{equation}\nFor a given sequence of nodes $u_1,u_2,\\ldots\\in\\mathbb{Z}$ and\n$\\mu\\in(0,1\/2]$, we obtain the configuration\n$\\{Y_k(u)\\}_{u\\in\\mathbb{Z}}$ for $k\\ge1$ by setting\n\\begin{equation}\nY_k(u)=\\left\\{\\begin{array}{cl}\nY_{k-1}(u)+\\mu(Y_{k-1}(u+1)-Y_{k-1}(u)),& \\mathrm{for}\\ u=u_k;\\\\\nY_{k-1}(u)+\\mu(Y_{k-1}(u-1)-Y_{k-1}(u)),& \\mathrm{for}\\ u=u_k+1;\\\\\nY_{k-1}(u),& \\mathrm{for}\\ u\\in\\mathbb{Z}\\backslash\\{u_k,u_k+1\\}.\n\\end{array}\\right.\\label{SAD}\n\\end{equation}\n\nThis procedure can be envisioned as a liquid-exchanging process on\n$\\mathbb{Z}$. A glass is put at each site $u\\in\\mathbb{Z}$. At $k=0$\nonly the glass located at the origin is full (with value 1) while\nall others are empty (with value 0). At each subsequent time step\n$k$, one picks two neighbouring glasses at $u_k$ and $u_k+1$, and\npouring liquids from the glass with higher level to that with lower\nlevel by a relative amount $\\mu$. This gives rise to the SAD\nprocess. The following lemma on unimodality can be easily proved.\n\n\\smallskip\n\\noindent\\textbf{Lemma 1.} (Unimodality) \\quad \\itshape If\n$u_j\\not=-1$ for $j=1,\\ldots,k$, then $Y_k(0)\\ge Y_k(1)\\ge\nY_k(2)\\ge\\ldots$. \\normalfont\n\\smallskip\n\nFix $t>0$ and consider the opinion model on $\\mathbb{Z}$. Note that\nthere exists a finite interval\n$[u_\\alpha,u_\\beta]\\subseteq\\mathbb{Z}$ containing 0 such that the\nPoisson events on the boundary edges $\\{u_\\alpha-1,u_\\alpha\\}\\in\nE_i$ and $\\{u_\\beta,u_\\beta+1\\}\\in E_i$ for all $i=1,\\ldots,\\ell$\nhave not happened yet up to time $t$. Let $N$ be the number of\nopinion adjustments that occur in $[u_\\alpha,u_\\beta]$ up to time $t$.\nThe times of these adjustments are arranged in the chronological\norder\n\\begin{equation}\n\\tau_{N+1}:=0<\\tau_N<\\tau_{N-1}< \\ldots<\\tau_1\\le t,\\label{a1}\n\\end{equation}\nwhere we set $\\tau_{N+1}:=0$ for convenience. For $k=1,\\ldots,N$, we\nwrite $u_k$ as the left endpoint of the edge $\\{u_k,u_k+1\\}$ for\nwhich $u_k$ and $u_k+1$ adjust opinions at time $\\tau_k$. Given the\nsequence $u_1,\\ldots,u_N$ (in this order) and $\\mu\\in(0,1\/2]$, we\nobtain a SAD process $\\{Y_k(u)\\}_{u\\in\\mathbb{Z}}$ as defined by\n(\\ref{A4}) and (\\ref{SAD}).\n\n\\smallskip\n\\noindent\\textbf{Lemma 2.} (Linear representation) \\quad \\itshape\nFor $k=0,1,\\ldots,N$,\n\\begin{equation}\nX_t(0)=\\sum_{u\\in\\mathbb{Z}}Y_k(u)X_{\\tau_{k+1}}(u).\\label{a2}\n\\end{equation}\nParticularly,\n$X_t(0)=\\sum_{u\\in\\mathbb{Z}}Y_N(u)X_0(u):=\\sum_{u\\in\\mathbb{Z}}Y_t(u)X_0(u)$.\n\\normalfont\n\\smallskip\n\nThis lemma implies that the constructed SAD process resembles the\ndynamics of the corresponding Deffuant model backwards in time so\nthat the state $X_t(0)$ in the model at any time $t>0$ can be\nexpressed as a weighted average of states at time 0 with weights\ngiven by an SAD configuration. See \\cite{17,25} for a proof.\n\n\\section*{Results}\n\n\\begin{figure}[!ht]\n    \\centering\\includegraphics[width=0.5\\textwidth]{fig1.eps}\n\\caption{\\textbf{Schematic illustration of the theoretical results in\nthe paper.}}\\label{fig_1}\n\\end{figure}\n\n\\subsection*{Opinion dynamics in 1-dimensional multiplex networks}\n\nIn this section, we will consider the multiplex opinion model on the\nintegers $\\mathbb{Z}$, focusing on the general initial opinion\ndistributions. More specifically, we take $G=(V,E)$ with\n$V=\\mathbb{Z}$ and $E_i=\\{\\{u,u+1\\}:u\\in\\mathbb{Z}\\}$ for\n$i=1,\\ldots,\\ell$. When $\\ell=1$, $G$ becomes a simplex network with\nonly one type of edges. For this interaction network, the critical\nconfidence threshold for opinion formation with i.i.d. uniform\ninitial distribution in $[0,1]$ is $d_c=1\/2$ \\cite{16,17} and later extended to the multiplex 1-dimensional\nnetworks in \\cite{25}.\n\nTo appreciate this, we first present the results for the case\n$\\ell=2$ (see Theorem 1) and then extend it to the general multiplex\ncase (see Theorem 2), as illustrated in Fig. \\ref{fig_1}. To this\nend, we take $\\ell=2$, $p=p_1$, and $\\alpha=\\alpha_2$. With these\nassumptions, the main result concerning the critical confidence\nthreshold for the 1-dimensional duplex model reads as follows.\n\n\\smallskip\n\\noindent\\textbf{Theorem 1.} (1-dimensional duplex networks)\\quad\n\\itshape Consider the above continuous opinion model ($\\ell=2$) on\n$\\mathbb{Z}$ with parameters $\\lambda,d>0$, $\\mu\\in(0,1\/2]$, and\n$\\alpha,p\\in(0,1)$ with $\\alpha>\\mu$.\n\n(a) Suppose that the initial opinion follows some bounded\ndistribution $\\mathcal{L}(X_0)$ with expected value\n$\\operatorname{E}(X_0)$, whose support is contained in the smallest\nclosed interval $[a,b]$. Let $h\\ge0$ be the length of some maximal\nopen interval $I\\subset[a,b]$ satisfying $\\operatorname{E}(X_0)\\in\nI$ and $\\operatorname{P}(X_0\\in I)=0$. Then,\n$d_c=\\max\\{(\\operatorname{E}X_0-a)(p+\\alpha(1-p))^{-1},(b-\\operatorname{E}X_0)(p+\\alpha(1-p))^{-1},h\\}$\nis the critical confidence threshold in the following sense:\n\\begin{itemize}\n\\item If $d<\\min\\{d_c,b-a\\}$, then with probability 1, there will be (infinitely\nmany) finally blocked edges, namely, $e=\\{u,u+1\\}$ satisfies\n$|X_t(u)-X_t(u+1)|>d$ for all $t$ large enough;\n\n\\item If $d>\\min\\{d_c,b-a\\}$, then with probability 1,\n$X_{\\infty}(u):=\\lim_{t\\rightarrow\\infty}X_t(u)=\\operatorname{E}(X_0)$\nfor every $u\\in\\mathbb{Z}$.\n\\end{itemize}\n\n(b) Suppose that the initial opinion distribution $\\mathcal{L}(X_0)$\nis unbounded but its expectation exists in the sense of\n$\\operatorname{E}(X_0)\\in\\mathbb{R}\\cup\\{\\pm\\infty\\}$. Then, for any\n$d>0$, with probability 1, there will be (infinitely many) finally\nblocked edges, namely, $e=\\{u,u+1\\}$ satisfying $|X_t(u)-X_t(u+1)|>d$\nfor all $t$ large enough. \\normalfont\n\\smallskip\n\nBefore proceeding with the proof, we provide a couple of remarks.\nFirstly, when the initial distribution $\\mathcal{L}(X_0)$ is bounded\nand $d<\\min\\{d_c,b-a\\}$, we will show that\n$\\{|X_t(u)-X_t(u+1)|\\}\\in\\{0\\}\\cup[d,b-a]$ for sufficiently large\n$t$ and all $u\\in\\mathbb{Z}$, and hence, the integers split into\n(infinitely many) finite clusters of neighbouring agents\nasymptotically agreeing with each other, with no global consensus achieved. Secondly, in the special case of $\\mathcal{L}(X_0)$ being\nthe standard uniform distribution in $[0,1]$, we readily reproduce\nTheorem 1 in \\cite{25}. A general $\\mathcal{L}(X_0)$ has been\nconsidered both theoretically and via simulations in \\cite{18,26} for simplex networks (i.e., $\\ell=1$). Theorem 1 can be thought of\nas an extension to multiplex networks. Finally, the assumption\n$\\alpha>\\mu$ is required here for technical reasons as in \\cite{25},\nwhich does not have a counterpart in the case of simplex networks where\n$\\mu$ only influences the convergence time of the negotiation\nprocess.\n\nThe crucial technique adopted here is the SAD process introduced in \\cite{17}. The SAD process and its basic\nproperties are briefly reviewed in the Method section. Another key\nconcept from that paper is the flat-points concept. To accommodate\nthe general distributions considered in the present paper, a slight\nextension of the definitions therein can be provided as follows.\nGiven $\\varepsilon>0$ and the initial opinion configuration\n$\\{X_0(v)\\}_{v\\in\\mathbb{Z}}$ with finite expectation, a node\n$u\\in\\mathbb{Z}$ is said to be an {\\it $\\varepsilon$-flat point to\nthe right} if for all $n\\ge0$,\n\\begin{equation}\n\\frac{1}{n+1}\\sum_{v=u}^{u+n}X_0(v)\\in\\left[\\operatorname{E}(X_0)-\\varepsilon,\\operatorname{E}(X_0)+\\varepsilon\\right].\\label{a3}\n\\end{equation}\nLikewise, $u\\in\\mathbb{Z}$ is said to be an {\\it $\\varepsilon$-flat\npoint to the left} if for all $n\\ge0$,\n\\begin{equation}\n\\frac{1}{n+1}\\sum_{v=u-n}^{u}X_0(v)\\in\\left[\\operatorname{E}(X_0)-\\varepsilon,\\operatorname{E}(X_0)+\\varepsilon\\right],\\label{a4}\n\\end{equation}\nand {\\it two-sided $\\varepsilon$-flat point} if for all $n,m\\ge0$,\n\\begin{equation}\n\\frac{1}{n+m+1}\\sum_{v=u-n}^{u+m}X_0(v)\\in\\left[\\operatorname{E}(X_0)-\\varepsilon,\\operatorname{E}(X_0)+\\varepsilon\\right].\\label{a5}\n\\end{equation}\nWe also define that $u\\in\\mathbb{Z}$ is an {\\it $\\varepsilon$-flat\npoint to the right at time $t$} if for all $n\\ge0$,\n$\\frac{1}{n+1}\\sum_{v=u}^{u+n}X_t(v)\\in\\left[\\operatorname{E}(X_0)-\\varepsilon,\\operatorname{E}(X_0)+\\varepsilon\\right].$\nSimilar definitions for {\\it $\\varepsilon$-flat point to the left at\ntime $t$} and {\\it two-sided $\\varepsilon$-flat point at time $t$}\ncan be given.\n\n\\noindent\\textbf{Proof of Theorem 1.} (a) Along the lines in\n\\cite[Section 2]{26}, we divide the proof of statement (a) into\nthree steps.\n\n\\textit{Step 1}. Suppose that the initial opinion distribution\n$\\mathcal{L}(X_0)$ is confined in [0,1] with expected value\n$\\operatorname{E}(X_0)=1\/2$. Moreover, for any $\\varepsilon>0$, we\nassume that\n$\\operatorname{P}(X_0\\not\\in[\\varepsilon,1-\\varepsilon])>0$ and\n$\\operatorname{P}(1\/2-\\varepsilon\\le X_0\\le1\/2+\\varepsilon)>0$ hold.\nThen we claim that $d_c=[2(p+\\alpha(1-p))]^{-1}$ is the critical\nconfidence threshold in the same sense as in Theorem 1(a) (with $a=0$\nand $b=1$).\n\nTo prove this claim, we need to show that the essential ingredients\nin the proof of Theorem 1 in \\cite{25} still hold true. We mention\nhere an obvious correction that the critical threshold separating\nthe subcritical and supercritical regimes therein should be\n$\\min\\{d_c,1\\}$ instead of $d_c$. For the subcritical regime, note\nthat the fact that the mass is around the expected value, i.e.,\n$\\operatorname{P}(1\/2-\\varepsilon\\le X_0\\le1\/2+\\varepsilon)>0$,\nimplies that $\\operatorname{P}(u\\ \\mathrm{is}\\\n\\varepsilon\\mbox{-}\\mathrm{flat}\\ \\mathrm{to}\\ \\mathrm{the}\\\n\\mathrm{right})=\\operatorname{P}(u\\ \\mathrm{is}\\\n\\varepsilon\\mbox{-}\\mathrm{flat}\\ \\mathrm{to}\\ \\mathrm{the}\\\n\\mathrm{left})$ $>0$ for all $\\varepsilon>0$ and $u\\in\\mathbb{Z}$ by\nsimilarly applying the coupling trick and the strong law of large\nnumbers. At time $t$ when a Poisson event occurs, define a Boolean\nrandom variable $A_t$ by $A_t=1$ with probability $p$ and\n$A_t=\\alpha$ with probability $1-p$ so that the opinion model\nconstitutes a marked Poisson process with rate $\\lambda$ \\cite{25}.\nWe can then mimic the proof for Propositions 1 and 2 in \\cite{25}\nverbatim, which employs the condition\n$\\operatorname{P}(X_0\\not\\in[\\varepsilon,1-\\varepsilon])>0$ for any\n$\\varepsilon>0$.\n\nFor the supercritical regime, we need to note that the property\n$\\operatorname{P}(u\\ \\mathrm{is}\\\n\\mathrm{two}\\mbox{-}\\mathrm{sided}\\\n\\varepsilon\\mbox{-}\\mathrm{flat})>0$ for any $\\varepsilon>0$ and\n$u\\in\\mathbb{Z}$ can now be established by keeping in mind that\n$\\operatorname{P}(1\/2-\\varepsilon\\le X_0\\le1\/2+\\varepsilon)>0$\nfollowing the same reasoning as in \\cite{17}; see also \\cite{26}. Now\nthe proof for the supercritical regime in \\cite{25} can be used,\nwhich concludes the proof of \\textit{Step 1}.\n\n\\textit{Step 2}. Suppose that the initial opinion distribution\n$\\mathcal{L}(X_0)$ is again confined in [0,1] with expected value\n$\\operatorname{E}(X_0)=1\/2$.  For any $\\varepsilon>0$, as in\n\\textit{Step 1} we assume that\n$\\operatorname{P}(X_0\\not\\in[\\varepsilon,1-\\varepsilon])>0$.\nMoreover, assume that there exists some maximal open interval\n$I\\subset[0,1]$ of length $h$ satisfying $1\/2\\in I$ and\n$\\operatorname{P}(X_0\\in I)=0$. Then, we claim that\n$d_c=\\max\\{[2(p+\\alpha(1-p))]^{-1},h\\}$ is the critical confidence\nthreshold in the same sense of Theorem 1(a) (with $a=0$ and $b=1$).\n\nWhen $d<h$, thanks to the assumption\n$\\operatorname{P}(X_0\\not\\in[\\varepsilon,1-\\varepsilon])>0$, we have\ninitial opinions both below and above $1\/2$ with probability 1. Therefore, any edges which are blocked due to\ninitial incident opinions lying on different sides of the gap $I$\nwill remain blocked for all $t$. By ergodicity, there will be\ninfinitely many such blocked edges, and thus consensus can not be reached in\nthis case.\n\nWhen $d>h$, we need to show that\n\\begin{equation}\n\\operatorname{P}(u\\ \\mathrm{is}\\ \\varepsilon\\mbox{-}\\mathrm{flat}\\\n\\mathrm{to}\\ \\mathrm{the}\\ \\mathrm{right}\\ \\mathrm{at}\\\n\\mathrm{time}\\ t)=\\operatorname{P}(u\\ \\mathrm{is}\\\n\\varepsilon\\mbox{-}\\mathrm{flat}\\ \\mathrm{to}\\ \\mathrm{the}\\\n\\mathrm{left}\\ \\mathrm{at}\\ \\mathrm{time}\\ t)>0\\label{3}\n\\end{equation}\nfor all $\\varepsilon>0$, $u\\in\\mathbb{Z}$ and for some sufficiently\nlarge $t$, since an arbitrary flat point at time $t=0$ no longer exists\ndue to the gap. Following the reasoning of \\cite[Section 2]{26}, one can then\nestablish Eq. (\\ref{3}). The only minor change that has to be made in\norder to accommodate the multiplexity is that the involved marked\nPoisson processes has rate $\\lambda p+\\lambda(1-p)=\\lambda$ instead\nof a unit rate, which does not affect the validity of the proof. Now, as\nin \\textit{Step 1}, we can mimic the proof of Propositions 1 and 2\nin \\cite{25} verbatim to settle the subcritical case. Accordingly,\n$d_c\\ge\\max\\{[2(p+\\alpha(1-p))]^{-1},h\\}$. Next, the two-sided\n$\\varepsilon$-flatness at time $t$ for any $\\varepsilon>0$ can be\nestablished similarly as in \\cite[Section 2]{26}. Hence, the\nargument in the supercritical case in \\textit{Step 1} leads to\n$d_c=\\max\\{[2(p+\\alpha(1-p))]^{-1},h\\}$, completing the proof of\n\\textit{Step 2}.\n\n\\textit{Step 3}. Now, everything is in place to prove Theorem 1(a) in\nits full generality.\n\nDefine $c:=\\max\\{\\operatorname{E}X_0-a,b-\\operatorname{E}X_0\\}$ and\nperform the linear transformation $x\\mapsto\n(x-\\operatorname{E}X_0)\/2c+1\/2$ to the dynamics of our multiplex\nDeffuant model. Using the result in \\textit{Step 2} and the fact\nthat the dynamics stays unchanged with respect to translations of\nthe initial distribution and that parameter $d$ can be re-scaled\nas per a scaling transformation of the initial distribution in order\nto recover the identical dynamics, we have\n\\begin{align}\nd_c=&2c\n\\max\\{[2(p+\\alpha(1-p))]^{-1},h\/2c\\}\\nonumber\\\\\n=&\\max\\{(\\operatorname{E}X_0-a)(p+\\alpha(1-p))^{-1},(b-\\operatorname{E}X_0)(p+\\alpha(1-p))^{-1},h\\}.\\label{a6}\n\\end{align}\nOne can see that the ultimate consensus value in\nthe supercritical regime is transformed from $1\/2$ to\n$\\operatorname{E}X_0$ in view of \\textit{Step 2}.\n\n(b) In the case of unbounded $\\mathcal{L}(X_0)$, we divide the proof\ninto two cases.\n\n\\textit{Case 1.} $\\operatorname{E}|X_0|<\\infty$.\n\nThe strong law of large numbers implies that\n\\begin{equation}\n\\operatorname{P}\\left(\\frac{1}{n+1}\\sum_{v=u}^{u+n}X_0(v)=\\operatorname{E}X_0\\right)=1\\label{a7}\n\\end{equation}\nfor any $u\\in\\mathbb{Z}$. A simple calculation shows that node $u$\nis $\\delta$-flat to the right with positive probability for some\n$\\delta>0$.\n\nFix $d>0$. Following the reasoning in \\cite[Proposition 1]{25} and\nnoting that $A_td\\le d$, we can show that if $u-1$ and $u+1$ are\n$\\delta$-flat to the left and right respectively and\n$X_0(u)\\not\\in[\\operatorname{E}X_0-\\delta-d,\\operatorname{E}X_0+\\delta+d]$\n(which happens with positive probability), then $X_t(u-1)$ and\n$X_t(u+1)$ will stay in the interval\n$[\\operatorname{E}X_0-\\delta,\\operatorname{E}X_0+\\delta]$ for all\n$t$ leaving the two edges $\\{u-1,u\\}$ and $\\{u,u+1\\}$ finally\nblocked. Since this event happens at each $u\\in\\mathbb{Z}$ with\npositive probability, it happens with probability 1 at infinitely\nmany nodes by ergodicity.\n\n\\textit{Case 2.} $\\operatorname{E}X_0\\in\\{\\pm\\infty\\}$.\n\nWithout loss of generality, we assume that\n$\\operatorname{E}X_0^+=\\infty$ and $\\operatorname{E}X_0^-<\\infty$,\nwhere $X_0^+$ and $X_0^-$ are the positive and negative parts of\n$X_0$, respectively. We may further assume that\n$\\operatorname{P}(X_0\\le0)>0$, otherwise a translation would\ntransform the problem to this case (c.f. \\textit{Step 3} above).\n\nFix $d>0$. The same argument in \\cite[Section 2]{26} can be used to show that the event $\\mathcal{E}:=\\{(1\/n)\\sum_{v=u+1}^{u+n}X_0(v)>d,\\\n\\mathrm{for}\\ \\mathrm{all}\\ n\\in\\mathbb{N}\\}$ for any\n$u\\in\\mathbb{Z}$ happens with positive probability. Similarly, along the lines of \\cite[Proposition 1]{25}, we obtain that if\n$\\mathcal{E}$ happens and $X_0(u)\\le0$ (which happens with positive\nprobability), then $X_0(u+1)>d$ for all $t$. Namely, there will\nnever be an opportunity for node $u+1$ to average with $u$. The same\nthing holds for $u-1$ by symmetry. Since the initial opinions are\ni.i.d., with positive probability we have $X_0(u)\\le0$ and\n$X_0(u-1),X_0(u+1)>d$, leaving the edges $\\{u-1,u\\}$ and $\\{u,u+1\\}$\nfinally blocked. Since this happens at every $u\\in\\mathbb{Z}$ with\npositive probability,  by ergodicity, it happens with probability 1 at infinitely\nmany nodes. $\\Box$\n\nFor a multiplex network $\\mathbb{Z}$ with $\\ell$ layers, Theorem 2\nis within easy reach by essentially using the same arguments as above.\n\n\\smallskip\n\\noindent\\textbf{Theorem 2.} (1-dimensional multiplex\nnetworks)\\quad \\itshape Consider the above continuous opinion model\non $\\mathbb{Z}$ with parameters $\\lambda,d>0$, $\\mu\\in(0,1\/2]$, and\n$p_i\\in(0,1)$ for $i=1,\\ldots,\\ell$, $\\alpha_i\\in(0,1)$ for\n$i=2,\\ldots,\\ell$ and $\\alpha_1=1$ with $\\alpha_i>\\mu$ for all\n$i$.\n\n(a) Suppose that the initial opinion follows some bounded\ndistribution $\\mathcal{L}(X_0)$ with expected value\n$\\operatorname{E}(X_0)$, whose support is contained in the smallest\nclosed interval $[a,b]$. Let $h\\ge0$ be the length of some maximal\nopen interval $I\\subset[a,b]$ satisfying $\\operatorname{E}(X_0)\\in\nI$ and $\\operatorname{P}(X_0\\in I)=0$. Then,\n$d_c=\\max\\{(\\operatorname{E}X_0-a)(\\sum_{i=1}^{\\ell}p_i\\alpha_i)^{-1},(b-\\operatorname{E}X_0)(\\sum_{i=1}^{\\ell}p_i\\alpha_i)^{-1},h\\}$\nis the critical confidence threshold in the following sense:\n\\begin{itemize}\n\\item If $d<\\min\\{d_c,b-a\\}$, then with probability 1, there will be (infinitely\nmany) finally blocked edges, namely, $e=\\{u,u+1\\}$ satisfies\n$|X_t(u)-X_t(u+1)|>d$ for all $t$ large enough;\n\n\\item If $d>\\min\\{d_c,b-a\\}$, then with probability 1,\n$X_{\\infty}(u):=\\lim_{t\\rightarrow\\infty}X_t(u)=\\operatorname{E}(X_0)$\nfor every $u\\in\\mathbb{Z}$.\n\\end{itemize}\n\n(b) Suppose that the initial opinion distribution $\\mathcal{L}(X_0)$\nis unbounded but its expectation exists in the sense of\n$\\operatorname{E}(X_0)\\in\\mathbb{R}\\cup\\{\\pm\\infty\\}$. Then, for any\n$d>0$, with probability 1, there will be (infinitely many) finally\nblocked edges, namely, $e=\\{u,u+1\\}$ satisfies $|X_t(u)-X_t(u+1)|>d$\nfor all $t$ large enough. \\normalfont\n\\smallskip\n\nSeveral observations can be drawn from Theorem 2. Firstly, when the\ninitial opinion distribution $\\mathcal{L}(X_0)$ follows the standard\nuniform distribution in $[0,1]$, we recover the previous result\n\\cite[Theorem 2]{25}. Secondly, when $\\mathcal{L}(X_0)$ is bounded,\nsince $\\sum_{i=1}^{\\ell}p_i=1$, we always have\n$d_c\\ge\\max\\{(\\operatorname{E}X_0-a),(b-\\operatorname{E}X_0),h\\}$,\nwhere the equality holds if and only if $\\ell=1$ or\n$h\\ge\\max\\{(\\operatorname{E}X_0-a)(\\sum_{i=1}^{\\ell}p_i\\alpha_i)^{-1},(b-\\operatorname{E}X_0)(\\sum_{i=1}^{\\ell}p_i\\alpha_i)^{-1}\\}$.\nThis indicates it is more difficult to reach agreement over\nmultiplex networks than simplex networks in general. When there is a large\n$h$, the critical confidence threshold $d_c$ is dominated by\n$h$ and is independent from the multiplexity; on the other hand, for\nrelatively small $h$, the threshold is determined in turn by both\nthe multiplexity and the initial distribution. When the initial\ndistribution $\\mathcal{L}(X_0)$ is unbounded, consensus cannot be\nreached regardless of the multiplexity. Thirdly, if there exists\nsome $k$ satisfying $p_k\\gg p_j$ for all $j\\not=k$, then\n$d_c\\approx\\max\\{(\\operatorname{E}X_0-a)\\alpha_k^{-1},(b-\\operatorname{E}X_0)\\alpha_k^{-1},h\\}$\nin the case of bounded $\\mathcal{L}(X_0)$. This suggests that the\ncritical confidence is governed by a frequently interacted layer in\nthe underlying network as one would expect.\n\n\n\\subsection*{Opinion dynamics in general multiplex networks}\n\nIn this section, we deal with more general multiplex networks and adopt a similar strategy by first looking into a duplex model on higher-dimensional lattices, generalising it to multiplex models and discussig on further extensions.\n\nParticularly, we take $G=(V,E)$ with $V=\\mathbb{Z}^D$ for $D\\ge2$\nand $E_i$ consists of all edges in the $D$-dimensional lattice for\n$i=1,\\ldots,\\ell$. When $\\ell=1$, $G$ becomes a simplex network with\nonly one type of edges; see \\cite[Section 3]{26}. For $\\ell=2$, we\ndenote $p=p_1$ and $\\alpha=\\alpha_2$ as in the above section. The\nmain result in this duplex case reads as follows.\n\n\\smallskip\n\\noindent\\textbf{Theorem 3.} (higher-dimensional duplex\nnetworks)\\quad \\itshape Consider the above continuous opinion model\n($\\ell=2$) on $\\mathbb{Z}^D$ with $D\\ge2$, $\\lambda>0$,\n$\\mu\\in(0,1\/2]$, and $\\alpha,p\\in(0,1)$ with $\\alpha>\\mu$.\n\nIf the initial opinion is distributed on $[a,b]$ with expected value\n$\\operatorname{E}(X_0)$ and\n$d>\\frac{1}{2}(\\operatorname{E}|2X_0-a-b|+b-a)(p+\\alpha(1-p))^{-1}$,\nthen with probability 1,\n$\\lim_{t\\rightarrow\\infty}|X_t(u)-X_t(v)|=0$ for all edges\n$\\{u,v\\}\\in E$. \\normalfont\n\\smallskip\n\nUnlike the 1-dimensional case, here we are only able to establish\nan upper bound for the critical confidence level $d_c$. In fact, as\ncommented in Remark 3.5 in \\cite{26}, the case of $D\\ge2$ is much\nmore complicated then the 1-dimensional counterpart and it is even not clear\nif there exists a critical $d_c$ separating the supercritical and\nsubcritical regimes since the ultimate consensus does not need be\nmonotonic with respect to $d$. Furthermore, we note that the\nconsensus result in Theorem 3 is weaker that in Theorems 1 and 2\n(for the supercritical regime) in the sense that only the difference\nbetween the opinions of two neighbouring individuals is required to\nconverge towards zero. It is to verify that this\nis equivalent to the convergence of each individual's opinion in a\nfinite network. For infinite networks considered in this paper,\nhowever, the picture is quite different as one may imagine a\nsituation where the opinion shows wave-like patterns on broader and\nbroader spatial scales with non-vanishing amplitude as time increases.\n\nTo prove Theorem 3, we first define the energy of node $u$ at time $t$\nas $\\mathcal{E}_t(u)=f(X_t(u))$, where\n$f:[a,b]\\rightarrow[0,\\infty)$ is some convex function. Given an\nedge $e=\\{u,v\\}\\in E$, let $T$ be the sequence of arrival times of\nthe Poisson events at $e$. The accumulated energy loss along $e$ is\ndefined as\n\\begin{equation}\n\\mathcal{E}_t^{\\dagger}(e):=\\sum_{s\\in\nT\\cap[0,t]}(\\mathcal{E}_{s-}(u)+\\mathcal{E}_{s-}(v)-\\mathcal{E}_{s}(u)-\\mathcal{E}_{s}(v)),\\label{a8}\n\\end{equation}\nwhich is nonnegative due to Jensen's inequality \\cite{26}. At time\n$t$, the total energy of node $u$ is defined as\n$\\mathcal{E}_t(u)+\\frac12\\sum_{e:e\\sim\nu}\\mathcal{E}_t^{\\dagger}(e)$, where $e\\sim u$ means $u$ is an\nend-point of $e$. Following the same argument of \\cite[Lemma\n3.2]{26} and noting that the number of Poisson rings on a single\nedge in any time period of length $\\varepsilon$ is a Poisson random\nvariable with parameter $\\lambda\\varepsilon$, we have the following\nlemma.\n\n\\smallskip\n\\noindent\\textbf{Lemma 3.} For any $u\\in\\mathbb{Z}^D$ and time\n$t\\ge0$,\n$\\operatorname{E}\\left(\\mathcal{E}_t(u)+\\frac12\\sum_{e:e\\sim\nu}\\mathcal{E}_t^{\\dagger}(e)\\right)=\\operatorname{E}\\mathcal{E}_0(0)$.\n\\normalfont\n\\smallskip\n\nThis means that the total energy at any node is conserved during the\nopinion exchange process.\n\n\\smallskip\n\\noindent\\textbf{Lemma 4.} For the above duplex opinion model on\n$\\mathbb{Z}^D$ with $D\\ge2$, $\\lambda>0$, $\\mu\\in(0,1\/2]$, and\n$\\alpha,p\\in(0,1)$. Suppose $\\alpha>\\mu$. If $d\\in(0,b-a]$, then\nwith probability 1 for every two neighbours $u,v\\in\\mathbb{Z}^D$,\neither $|X_t(u)-X_t(v)|>A_td$ for all sufficiently large $t$ (i.e.,\n$\\{u,v\\}$ is finally blocked), or\n$\\lim_{t\\rightarrow\\infty}|X_t(u)-X_t(v)|=0$. \\normalfont\n\\smallskip\n\n\\noindent\\textbf{Proof.} As commented in \\cite{25}, in the following,\nwe will use $A$ instead of $A_t$. Choose the energy function\n$f(x)=x^2$ and fix an edge $e=\\{u,v\\}$. Let $\\delta>0$. When there\nis a Poisson event at $e$ at time $t$ and $u,v$ exchange opinions,\nenergy to the amount of $2\\mu(1-\\mu)(X_{t-}(u)-X_{t-}(v))^2$ is lost\nalong the edge; see \\cite{25,26}. Hence, if\n$|X_{t-}(u)-X_{t-}(v)|\\in(\\delta,Ad]$, energy\n$\\mathcal{E}_t^{\\dagger}(e)$ will increase by the amount of at least\n$2\\mu(1-\\mu)\\delta^2$. Thanks to the memoryless property, given\n$|X_{s}(u)-X_{s}(v)|\\in(\\delta,Ad]$ at some time $s$, the first\nPoisson event after time $s$ on an edge incident to either $u$ or\n$v$ occurs at $e$ with probability $(4d-1)^{-1}$.\n\nIn view of the conditional Borel-Cantelli lemma \\cite[Corollary\n3.2]{28}, this will happen infinitely often with probability 1. If\n$|X_{t}(u)-X_{t}(v)|\\in(\\delta,Ad]$ at some sufficiently large $t$,\nthen $\\lim_{t\\rightarrow\\infty}\\mathcal{E}_t^{\\dagger}(e)=\\infty$.\nHowever, this is impossible since Lemma 3 yields\n$\\operatorname{E}(\\mathcal{E}_t^{\\dagger}(e))\\le\n2\\operatorname{E}(\\mathcal{E}_0(0))\\le2\\max\\{a^2,b^2\\}$. Thereby,\nwith probability 1, for all large enough $t$,\n$|X_t(u)-X_t(v)|\\in[0,\\delta]\\cup(Ad,b-a]$.\n\nFor small enough $\\delta>0$, $|X_t(u)-X_t(v)|$ cannot jump back and\nforth between $[0,\\delta]$ and $(Ad,b-a]$ infinitely often. This is\nbecause a single Poisson event cannot increase $|X_t(u)-X_t(v)|$ by\nmore than $\\mu d$, which for sufficiently small $\\delta$, is always\nless than the span of the gap $(\\delta,Ad]$ that needs to be crossed\ndue to $\\mu<\\alpha$. Since there are only countably many edges, the\nproof of Lemma 4 is completed. $\\Box$\n\n\\noindent\\textbf{Proof of Theorem 3.} Fix some $d\\ge(a+b)\/2$. If\n$e=\\{u,v\\}$ be a finally blocked edge, then the opinion of node $u$\nmust finally be located in one of the intervals $[a,b-Ad)$ or\n$(a+Ad,b]$. It follows from Lemma 4 that this event holds almost\nsurely for any $u$ if there are finally blocked edges. Suppose that\nthere is an edge $e$ such that\n\\begin{equation}\n\\operatorname{P}(e\\ \\mathrm{is}\\ \\mathrm{finally}\\\n\\mathrm{blocked})>0.\\label{4}\n\\end{equation}\nFollowing a similar argument as in \\cite[Lemma 3.4]{26}, we obtain with\nprobability 1 that\n$\\liminf_{t\\rightarrow\\infty}|X_t(u)-(a+b)\/2|-a-Ad\\ge(a+b)\/2$ for\nall $u\\in\\mathbb{Z}^D$.\n\nWe choose the energy function $f(x)=|x-(a+b)\/2|$. By Lemma 3 and\nFatou's lemma, we obtain\n\\begin{align}\na+[p+\\alpha(1-p)]d-\\frac{a+b}{2}\\le&\\operatorname{E}\\left(\\liminf_{t\\rightarrow\\infty}\\mathcal{E}_t(u)\\right)=\\operatorname{E}\\left(\\liminf_{t\\rightarrow\\infty}\\left|X_t(u)-\\frac{a+b}{2}\\right|\\right)\\nonumber\\\\\n\\le&\\liminf_{t\\rightarrow\\infty}\\operatorname{E}\\left|X_t(u)-\\frac{a+b}{2}\\right|\\nonumber\\\\\n\\le&\\liminf_{t\\rightarrow\\infty}\\operatorname{E}\\left(\\mathcal{E}_t(u)+\\frac12\\sum_{e:e\\sim\nu}\\mathcal{E}_t^{\\dagger}(e)\\right)\\nonumber\\\\\n=&\\operatorname{E}\\left(\\mathcal{E}_0(u)\\right)=\\operatorname{E}\\left|X_0-\\frac{a+b}{2}\\right|.\\label{a9}\n\\end{align}\nRecall that the condition of Theorem 3 implies that\n$d>\\frac{1}{2}(\\operatorname{E}|2X_0-a-b|+b-a)(p+\\alpha(1-p))^{-1}$,\nwhich leads to a contradiction. Hence, the assumption (\\ref{4}) must\nnot be true. The proof then follows from applying Lemma 4. $\\Box$\n\nTheorem 3 can be directly extended to the multiplex setting for a\ngeneral $\\ell\\ge2$.\n\n\\smallskip\n\\noindent\\textbf{Theorem 4.} (higher-dimensional multiplex\nnetworks)\\quad \\itshape Consider the above continuous opinion model\non $\\mathbb{Z}^D$ with $D\\ge2$, $\\lambda>0$, $\\mu\\in(0,1\/2]$, and\n$p_i\\in(0,1)$ for $i=1,\\ldots,\\ell$, $\\alpha_i\\in(0,1)$ for\n$i=2,\\ldots,\\ell$ and $\\alpha_1=1$. Suppose $\\alpha_i>\\mu$ for all\n$i$.\n\nIf the initial opinion is distributed on $[a,b]$ with expected value\n$\\operatorname{E}(X_0)$ and\n$d>\\frac{1}{2}(\\operatorname{E}|2X_0-a-b|+b-a)(\\sum_{i=1}^{\\ell}p_i\\alpha_i)^{-1}$,\nthen with probability 1,\n$\\lim_{t\\rightarrow\\infty}|X_t(u)-X_t(v)|=0$ for all edges\n$\\{u,v\\}\\in E$. \\normalfont\n\\smallskip\n\nSome remarks follow: firstly, it is easy to check that the\nlattice $\\mathbb{Z}^D$ in Theorem 4 can be extended to any infinite,\nlocally finite, transitive and amenable connected graph\n$G_i=(V,E_i)$ for each $i=1,\\ldots,\\ell$ by using Zygmund's ergodic\ntheorem; c.f. \\cite[Remark 3.6]{26}. Recall that a graph is locally\nfinite if every node in it has a finite degree. A graph $G=(V,E)$ is\ntransitive if for any pair of nodes $u$ and $v$ in it, there is an\nautomorphism $\\varphi:V\\rightarrow V$ such that $\\varphi(v)=u$. A\ngraph $G=(V,E)$ is amenable if there exists a sequence $S_n\\subseteq\nV$ of finite sets satisfying\n$\\lim_{n\\rightarrow\\infty}|\\partial_{E}S_n|\/|S_n|=0$, where\n$\\partial_{E}S_n$ is the edge boundary of $S_n$. The following\nresult can be established.\n\n\\smallskip\n\\noindent\\textbf{Theorem 5.} (general multiplex networks)\\quad\n\\itshape Consider the above continuous opinion model, where each\nlayer $G_i=(V,E_i)$ ($i=1,\\ldots,\\ell$) is an infinite, locally\nfinite, transitive and amenable connected graph. Let $\\lambda>0$,\n$\\mu\\in(0,1\/2]$, and $p_i\\in(0,1)$ for $i=1,\\ldots,\\ell$,\n$\\alpha_i\\in(0,1)$ for $i=2,\\ldots,\\ell$ and $\\alpha_1=1$ with\n$\\alpha_i>\\mu$ for all $i$.\n\nIf the initial opinion is distributed on $[a,b]$ with expected value\n$\\operatorname{E}(X_0)$ and\n$d>\\frac{1}{2}(\\operatorname{E}|2X_0-a-b|+b-a)(\\sum_{i=1}^{\\ell}p_i\\alpha_i)^{-1}$,\nthen with probability 1,\n$\\lim_{t\\rightarrow\\infty}|X_t(u)-X_t(v)|=0$ for all edges\n$\\{u,v\\}\\in E$. \\normalfont\n\\smallskip\n\nSecondly, note that\n$\\frac{1}{2}(\\operatorname{E}|2X_0-a-b|+b-a)<b-a$ unless (i)\n$\\operatorname{P}(X_0\\in\\{a,b\\})=1$ and (ii) $X_0$ is not constant\nwith probability 1. This indicates that the condition\n$d>\\frac{1}{2}(\\operatorname{E}|2X_0-a-b|+b-a)(\\sum_{i=1}^{\\ell}p_i\\alpha_i)^{-1}$\nin Theorem 4 stand a good chance to be nontrivial even for multiplex\nnetworks in most meaningful situations. Thirdly, we have assumed\nthroughout this paper that the initial opinions following\n$\\mathcal{L}(X_0)$ are i.i.d. However, Theorems 4 and 5 still hold\nif the initial opinions are stationary and ergodic with respect to\nthe graph automorphisms because no other specific features of i.i.d.\nvariables are used in the above proof. Finally, it seems that agents\nforming a multiplex network are more difficult to reach consensus\nfor the same reason as remarked for 1-dimensional multiplex\nnetworks in the above section. Furthermore, as we have mentioned in\nthe beginning of this section, it is generally even not clear if we\ncan still speak of critical confidence level $d_c$ in\n$D$-dimensional ($D>1$) multiplex networks and more general\nmultiplex networks.\n\n\\subsection*{Numerical results}\n\nIn this section, we conduct agent-based simulations on different\nfinite multiplex networks, including regular ones such as\n$D$-dimensional lattices which can be viewed as a truncation of\n$\\mathbb{Z}^D$ in Theorem 4, and irregular ones such as small-world\nand scale-free networks, which obviously violate the regularity\nconditions in Theorem 5 and are prominent examples of non power-law\nand power-law networks, respectively. Interestingly, we see that for\nall such networks, the critical thresholds of consensus tend to\nagree with the predicted upper bounds in Theorems 4 and 5 in the\nspecial cases of uniform $X_0$ and some choices of Poisson rates\nassociated with the multiple layers.\n\nParticularly, in Fig. \\ref{fig_percent_convergence_vs_d}, we plot\nthe percentage of convergence of opinions for five network sizes $N$\nranging from 8 to 256 with $\\ell=4$ layers each, Poisson rate\n$\\lambda p_i=0.3$ and $\\mu=0.5$ to maximise the convergence rate. At\n$t=0$, we initialise each agent $u\\in V$ by assigning an opinion\nvalue $X_0(u)\\in\\mathbb{R}$ from the uniform distribution in\n$(0,1)$. To check for convergence of opinions, we require that\n$|X_t(u)-X_t(v)|<\\hbar,\\;\\forall u,v\\in G$, where $\\hbar=10^{-5}$.\nFor each curve, we have run 100 simulations to compute the\npercentage, each time for a different set of $\\alpha$ values in Eqs.\n\\eqref{1} and \\eqref{2}. Panel (a) shows the results for regular\nlattices, whereas panel (b) for Watts-Strogatz (small-world) and (c) for Barab\\'asi-Albert (scale-free) networks. In all cases,\nwe observe that the system reaches perfect consensus (i.e. 100\\%\nopinion convergence) or almost perfect consensus (i.e. $> 90\\%$ opinion convergence), independently of the network structure, and\nthat this starts occurring for different $d$ values. It is worth\nnoting however that in all cases, convergence to consensus\nis reached for $d>0.5$ denoted by the vertical dashed line in the plots.\nParticularly, in panel (a), for $N=8$, the percentage of convergence\nstarts to increase from very small $d$ values with the tendency to\nincrease as $N$ increases, for example for $N=8$, it starts at\n$d\\approx 0.25$ whereas for $N=256$ at $d\\approx0.37$. Surprisingly,\nthe jump from very small (almost 0\\%) to very big (almost 100\\%)\npercentage of opinion convergence (reach of consensus) occurs at\n$d=0.5$. Similar conclusions can be drawn for the Watts-Strogatz\n(small-world) networks in panel (b) and Barab\\'asi-Albert\n(scale-free) networks in panel (c). This is reminiscent of a first\norder phase-transition as a function of $d$ (order parameter) that might exist for infinitely big network sizes (i.e. $N\\rightarrow\\infty$) and is\nan open question.\n\n\n\\begin{figure}[!ht]\n    \\centering\\includegraphics[width=1\\textwidth]{fig2.eps}\n\\caption{\\textbf{Percentage of opinion-convergence versus\n$\\boldsymbol{d}$ for constant $\\boldsymbol{\\mu}$ and, different\nnetwork sizes $\\boldsymbol{N}$ and topologies.} Panel (a) is for\nregular networks, panel (b) for Watts-Strogatz (small-world) and\npanel (c) for Barab\\'asi-Albert (scale-free) networks. The vertical\ndashed line at $d=0.5$ corresponds to the point around which a\nsudden jump occurs for increasing network sizes $N$. Note that in\nthese plots we have set the number of layers\n$\\ell$ to 4.}\\label{fig_percent_convergence_vs_d}\n\\end{figure}\n\n\nNote that the convergence parameter $\\mu$ considered\nabove is constant and independent of the node-degrees and thus,\nassumes that all agents have the same paces towards adjusting their\nopinions. As this is rather ideal and not in agreement with real-life situations, we consider in the following two archetypal variants\ngiven by\n\\begin{equation}\n\\mu^+(u,v)=\\frac{\\deg(u)\\deg(v)}{2\\max_{u\\sim v}\n\\{\\deg(u)\\deg(v)\\}}\\label{a10}\n\\end{equation}\nand\n\\begin{equation}\n\\mu^-(u,v)=\\frac{\\min_{u\\sim v}\n\\{\\deg(u)\\deg(v)\\}}{2\\deg(u)\\deg(v)},\\label{a11}\n\\end{equation}\nfeaturing the degree-centrality dependent scenarios,\nwhere the max and min of $\\deg(u)\\deg(v)$ is computed among all\nedges $u$, $v$ ($u\\sim v$). Namely, we replace the convergence\nparameter $\\mu$ in Eqs. (\\ref{1}) and (\\ref{2}) by $\\mu^+(u,v)$\nindicating that higher-degree nodes are more willing to adjust their\nopinions (and $\\mu^-(u,v)$ indicating the opposite way). Clearly,\n$\\mu^+(u,v)$ and $\\mu^-(u,v)$ are within the interval $(0,0.5]$ for\nconnected networks. $\\mu^+$ is close to 0.5 for a well-connected\npair of nodes, whereas $\\mu^-$ is close to 0.5 for a\npoorly-connected pair of nodes. Our choice of degree-related\nconvergence parameter here naturally reflects the idea of the number of\nneighbors\/contacts in social networks, and models the possible\nmechanisms of heterogeneous psychological, habitual and cultural\nbackgrounds in opinion spreading. Although degree-centrality\napparently depends on the structure of the layers of the multiplex\nnetwork, our numerical results indicate that the convergence\nparameters do not affect the ultimate opinion configuration. It is\nworth mentioning that there are other measures of centrality as well, such\nas eigenvector-like centralities \\cite{b4} and measures based on\nrandom walks \\cite{b5}, that have been studied in the context of multiplex\nnetworks. However, as commented above, it is reasonable to expect similar\nresults for other convergence parameters mediated by more\ncomplicated measures.\n\n\nThe simulation results for the degree-centrality dependent parameters $\\mu^+$ and $\\mu^-$ are presented in Fig.\n\\ref{fig_percent_convergence__mm_mp_vs_d}, where we have used the\nsame network sizes, number of layers, Poisson rate and number of\nsimulations as in Fig. \\ref{fig_percent_convergence_vs_d} to compute\nthe percentage of convergence of the opinions. In particular,\npanels (a) and (b) are for Watts-Strogatz (small-world)\nnetworks with $\\mu^+(u,v)$ and $\\mu^-(u,v)$, respectively and panels\n(c) and (d) for Barab\\'asi-Albert (scale-free) networks with\n$\\mu^+(u,v)$ and $\\mu^-(u,v)$, respectively. It is found that these\ndegree-dependent convergence parameters (i.e. $\\mu^+(u,v)$ and\n$\\mu^-(u,v)$) do not alter the ultimate opinion configurations and that\nthe confidence threshold still remains. This is in line with previous\nfindings for Deffuant models in the case of single-layer networks\n\\cite{7,18} and our results extend this finding to multiplex networks.\n\n\\begin{figure}[!ht]\n    \\centering\\includegraphics[width=1\\textwidth]{fig3.eps}\n\\caption{\\textbf{Percentage of opinion-convergence versus\n$\\boldsymbol{d}$ for different network sizes $\\boldsymbol{N}$ and\ntopologies using the degree-dependent convergence parameters\n$\\boldsymbol{\\mu^+(u,v)}$ and $\\boldsymbol{\\mu^-(u,v)}$ in Eqs.\n(\\ref{1}) and (\\ref{2}).} Panels (a) and (b) are for Watts-Strogatz\n(small-world) networks with $\\mu^+(u,v)$ and $\\mu^-(u,v)$,\nrespectively and panels (c) and (d) for Barab\\'asi-Albert\n(scale-free) networks with $\\mu^+(u,v)$ and $\\mu^-(u,v)$,\nrespectively. The vertical dashed line at $d=0.5$ corresponds to the\npoint around which a sudden jump occurs for increasing network sizes\n$N$. Note that in these plots we have set the number of layers\n$\\ell$ to 4.}\\label{fig_percent_convergence__mm_mp_vs_d}\n\\end{figure}\n\n\n\\section*{Discussion}\n\nIn this paper, we studied analytically and numerically opinion dynamics over\nmultiplex networks with an arbitrary number of layers, where the\nagents interact with each other with bounded confidence. In the\nliterature, agents are generally assumed to have a homogeneous\nconfidence bound and here we sought to study analytically and\nnumerically opinion evolution over multiplex networks with\nrespective confidence thresholds and general initial opinion\ndistributions. We explicitly identified the critical thresholds at\nwhich a phase transition in the long-term consensus behaviour occurs.\nWe then discussed the interaction topology of the agents by\nusing multiplex $D$-dimensional lattices and extended them to general\nmultiplex networks under some regularity conditions. Our results\nreveal the quantitative relation between the critical threshold and initial distribution. We also performed numerical\nsimulations and illustrated the consensus behaviour of the agents in\nregular lattices and, small-world and scale-free networks. We found that the numerical results agree with our theoretical ones and\nin particular, the critical thresholds of consensus tend to agree\nwith the predicted upper bounds in Theorems 4 and 5 for all\nnetwork topologies considered in the special cases of uniform $X_0$\nand some choices of Poisson rates associated with the multiple\nlayers.\n\nMoreover, we used the Deffuant opinion model represented as a\nstochastic process for the evolution of opinions that includes\nheterogenous confidence bounds and features general initial\ndistributions and, determined the critical threshold by employing\nprobability methods. The main results of our work are Theorems 2 (for $D=1$) and Theorem 4 (for $D>1$) which extend previous results in \\cite{25,26} by considering\nboth multiplex structures with $\\ell>1$ and general initial opinion\ndistribution $\\mathcal{L}(X_0)$. We show that both the initial\ndistribution and multiplex structure play an important role in\nthe phase transition of opinion evolution in an infinite\n$D$-dimensional regular lattice in the sense that the critical\nconfidence bound in the case of Theorem 2 (or an upper bound of it\nin the case of Theorem 4) is influenced by both factors. Our results\nindicate that multiplexity hinders consensus formation when the initial opinion configuration is within a\nbounded range. This is numerically found to be true in more general\nnetworks including small-world and scale-free networks,\nwhich are ubiquitous in the real world. Our results provide insight\ninto information diffusion and social dynamics in multiplex\nreal-life systems modeled by networks. However, the theoretical proof of this\nis beyond the scope of this paper as it would require the development of\nsubstantially new techniques that we leave for a future publication.\n\nIt is worth mentioning that the networks considered\nhere are static, and thus the connectivity remains fixed throughout\nopinion spreading. As a result, structural properties such as\ncentrality, correlations, homophily, and assortativity, remain the\nsame throughout opinion spreading. On the other hand, in networks of\nhuman social interactions, the interaction can be assorted according\nto, e.g., the channels used for communication such as face-to-face,\nmobile phone, and social network services \\cite{b1}. Certain social\nmechanisms such as assortativity and homophily (namely, the tendency\nof individuals to align to behaviours of their friends) are popular\nin real social networks and may play a key role in opinion formation\nand its dynamics. For instance, it is shown in \\cite{b2} that the\nhigher the homophily between individuals in a multiplex network, the\nquicker is the convergence towards cooperation in the social\ndilemma. Multiplex social ecological network analysis unravels that\nnode heterogeneity has a critical effect on community robustness\n\\cite{b3}. However, as far as convergence of the opinion spreading\nis concerned, our numerical results, for three different\ncharacteristic types of multiplex networks (regular, small-world and\nscale-free), indicate the same qualitative and almost similar\nqualitative tendency to reach consensus as a function of $d$ for different network architectures. This is in agreement with our theoretical results. In\nfact, assortativity and homophily are not included in our\ntheoretical analysis because they usually subvert the transitivity\nand amenability conditions (see Theorem 5) that form the foundation\nof our mathematical technique. In a future work, we will focus on how\nto incorporate multiplex characterisations by means of structural\nmeasures, such as homophily and assortativity of the multiplex\nnetwork, into analytically tractable opinion-formation models.\nFinally, temporal or co-evolving networks with random environments\nalso seem appealing in this respect as they might lead to\ndifferences with respect to convergence to consensus.\n\n\n\n\\section*{Acknowledgments}\nBoth authors would like to thank the support provided by the\nUniversity of Essex International Visiting Fellowships to the\nproject ``Opinion dynamics in multiplex networks''. Y. S. was partially\nsupported by the National Natural Science Foundation of\nChina (11505127) and the Shanghai Pujiang Program (15PJ1408300).\n\n\\section*{Competing financial interests}\n\nThe authors have no competing interest.\n\n\\section*{Data availability}\n\nAll data are generated by numerical simulations and they have all\nbeen reported in the paper.\n\n\n\\section*{Author Contributions statement}\n\nY. S. devised the study. Y. S. and C. G. A. analysed the results and\nwrote the main manuscript text. C. G. A. performed the numerical\nanalysis and generated the figures. Both authors reviewed the\nmanuscript.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction} \n\nLuminous Blue Variables (LBVs) represent a short-lived but important stage in the life of the most massive stars. The massive ejection of material at this stage has a great impact both on the star's fate, as the stellar mass drastically changes, and on the local interstellar environment, as large amounts of kinetic energy as well as chemical elements are injected in the surroundings. However, our knowledge of LBVs remains incomplete. The cause of these eruptions, their recurrence timescales, or their effect on the subsequent evolution of the star are still not fully understood. The most recent such eruption in our Galaxy was the famous ``Great Eruption'' in the mid-19th century of $\\eta$\\,Carinae, in which the kinetic energy of the ejected material rivaled the energy output of a weak supernova. $\\eta$\\,Carinae has a companion whose strong stellar wind collides with that of the eruptor, and such collisions provide interesting constraints on wind properties. However, for $\\eta$\\,Carinae, it has been challenging to deduce from historical records how the interaction has evolved over time. A unique opportunity to overcome this challenge occurred in 1994 with the eruption of the primary star in HD\\,5980, the most luminous multiple star system in the Small Magellanic Cloud (SMC). \n\nThe HD\\,5980\\ system lies on the outskirts of NGC\\,346. It consists of several stars. Star A, initially thought to have been an O-type supergiant \\citep{bre82}, is the component that underwent a major LBV eruption in 1994. Its optical brightness has been gradually declining since then \\citep{koe10}. The post-eruption spectrum of Star A transitioned from WN11 to WN6, which brings into question the original classification of this star as an O-type supergiant, rather favoring a WR classification \\citep{nie88,koe14}. Star A possesses a WR companion called Star B, with which it forms an eclipsing pair \\citep[$P_{A+B}$=19.3\\,d, $i$=86$^{\\circ}$, ][]{ste97,per09}. The presence of a third star, adequately named Star C, was detected through its contribution (about 30\\%) to the optical lightcurve \\citep{ste97}. Its presence was confirmed in optical\/UV spectroscopy as its photospheric absorption lines appear superposed on the WR spectra (which may have confused the original classification of Star A). These lines display velocity variations with $P_C=96.5d$ \\citep[and references therein]{sch00,foe08,koe14}. This period is too short for Star C to be in a stable orbit around the A+B pair \\citep[e.g.][]{tok04}, hence Star C is itself part of a binary, which either serendipitously lies along the line-of-sight of the A+B pair or orbits around it with a very long period.  \n\nThe emission lines in HD\\,5980\\ undergo line profile variations with $P_{A+B}$, which were attributed to colliding winds by \\citet{mof98}. It has however been argued that eclipse effects and an asymmetric wind of Star A provide alternative explanations for this variability \\citep{koe06}. Nevertheless, the presence of two powerful winds in such a massive binary system should lead to their interaction. \\citet{foe08} speculated on the geometry of the interacting region, based on the variability that was observed in the blue-shifted absorptions in the optical He\\,{\\sc i} lines. They suggested that, at distances from the A+B binary larger than the orbital separation, the interaction could be described as two spiral-like density enhancements which, when projected against the stellar continua, produced the blue-shifted absorptions. Direct evidence for colliding winds, closer to the stars, was obtained in the X-ray range: HD\\,5980\\ appears as a very bright and hard X-ray source \\citep{naz02}, two features typical of colliding winds binaries, and phase-locked variations were also detected thanks to a dedicated monitoring \\citep{naz07}.\n\nSince those observations, the wind velocity of Star A has further increased, accompanied by a further decrease in its mass-loss rate. Because the X-ray emission is sensitive to mass loss, HD\\,5980\\ provides the opportunity to determine how changes in the mass loss from Star A affect the shocked plasma in a system with a known orbital configuration, a unique astrophysical experiment. We have therefore undertaken a new monitoring in 2016 with the {\\sc{XMM}}\\emph{-Newton}\\ observatory, to document the variations in the hot shocked gas with the changing strength of the wind from Star A. Section 2 presents the data used in this paper, Section 3 presents and discusses the high-energy results, and Section 4 concludes by summarizing our results.\n\n\\section{Data} \n\n\\subsection{\\emph{Chandra}}\nHD\\,5980\\ was observed twice with \\emph{Chandra}\\ (Table~\\ref{journal}). The first dataset (ObsID=1881, PI Corcoran) was obtained in May 2001 with ACIS-I and we have analyzed it in detail in \\citet{naz02,naz03}. We reprocessed these data using CIAO 4.8 (CALDB 4.7.0). The spectrum of HD\\,5980\\ was then extracted (using {\\it specextract}) in a circle of radius 5px (2.5'') centered on the Simbad coordinates of the system, using the surrounding annulus (with radii of 5 and 15px) as background region. The second exposure (ObsID=13773, PI Ballet), taken in February 2013, was aimed at observing the nearby supernova remnant (SNR) IKT\\,16, but HD\\,5980\\ serendipitously lies within the ACIS-S field-of-view. Because of the off-axis angle, however, its PSF is wider and somewhat distorted. After reprocessing, the spectrum of HD\\,5980\\ was thus extracted in a circular region of larger radius (15px), using a larger background region (annulus with radii of 15 and 25px). We calculated specific detector response matrices (RMF and ARF) to obtain energy- and flux-calibrated spectra. Finally, these spectra were grouped to a minimum of 15 cts per bin prior to analysis.\n\n\\begin{table*}\n\\centering\n\\footnotesize\n\\caption{Journal of the X-ray observations. HJD correspond to dates at mid-exposure, and the corresponding phases $\\phi$ were calculated using the ephemeris of \\citet[$T_0$=2\\,443\\,158.705, $P_{A+B}$=19.2654\\,d]{ste97}. Phases $\\phi_C$ correspond to Star C orbit \\citep[$T_0$=2\\,451\\,183.4, $P_C$=96.56\\,d, see Table 7 of][]{koe14}. Exposure times correspond to on-axis values (for MOS1 and excluding flares, except for Rev. 0970, if {\\sc{XMM}}\\emph{-Newton}). The {\\sc{XMM}}\\emph{-Newton}\\ count rates in the 1.5--10.\\,keV band are provided in the last two columns (first one for the sum of MOS1 and MOS2 count rates, second one for pn count rates - note that the pn camera was switched off during Rev. 0970).}\n\\label{journal}\n\\begin{tabular}{lccccccc}\n\\hline\\hline\nID\/Rev & Date & $\\Delta(t)$ & HJD & $\\phi$ & $\\phi_C$ &MOS1+2 ct. rate & pn ct. rate\\\\\n   &      & (ks)        &     &        &          &\\multicolumn{2}{c}{($10^{-3}$\\,cts\\,s$^{-1}$)} \\\\\n\\hline\n\\multicolumn{6}{l}{\\emph{Chandra}} \\\\\n1881  & 2001-05-15 & 98.7 & 2452045.171 & 0.266 & 0.92 \\\\\n13773 & 2013-02-09 & 38.6 & 2456332.837 & 0.823 & 0.33 \\\\\n\\multicolumn{6}{l}{{\\sc{XMM}}\\emph{-Newton}} \\\\\n0157 & 2000-10-17 & 17.4 & 2451835.246 & 0.369 & 0.75 & 13.0$\\pm$1.1  & 14.2$\\pm$1.5   \\\\\n0357 & 2001-11-21 & 26.7 & 2452234.643 & 0.100 & 0.89 & 3.95$\\pm$0.54 & 5.29$\\pm$0.75  \\\\\n0970 & 2005-03-27 & 23.5 & 2453457.416 & 0.570 & 0.55 & 19.7$\\pm$1.6  &                \\\\\n1093 & 2005-11-27 & 17.3 & 2453701.862 & 0.259 & 0.08 & 8.09$\\pm$0.85 & 16.0$\\pm$1.3   \\\\\n1094 & 2005-11-29 & 16.4 & 2453703.806 & 0.360 & 0.10 & 13.4$\\pm$1.1  & 15.8$\\pm$1.4   \\\\\n1100 & 2005-12-11 & 13.3 & 2453716.124 & 0.999 & 0.23 & 4.23$\\pm$0.75 & 5.23$\\pm$1.03  \\\\\n3073 & 2016-09-19 & 21.4 & 2457651.187 & 0.254 & 0.98 & 24.5$\\pm$1.4  & 32.3$\\pm$1.7   \\\\\n3074 & 2016-09-21 & 26.2 & 2457653.263 & 0.362 & 0.00 & 31.9$\\pm$1.4  & 42.9$\\pm$1.8   \\\\\n3086 & 2016-10-15 & 26.8 & 2457677.143 & 0.602 & 0.25 & 36.3$\\pm$1.4  & 54.5$\\pm$1.9   \\\\\n3088 & 2016-10-19 & 21.5 & 2457681.003 & 0.802 & 0.29 & 28.0$\\pm$1.7  & 37.3$\\pm$2.4   \\\\\n3090 & 2016-10-23 & 22.7 & 2457684.806 & 0.999 & 0.33 & 13.1$\\pm$1.0  & 14.3$\\pm$1.1   \\\\\n3091 & 2016-10-25 & 13.9 & 2457686.771 & 0.101 & 0.35 & 12.5$\\pm$1.2  & 12.8$\\pm$1.3   \\\\\n3110 & 2016-12-02 & 30.9 & 2457725.322 & 0.102 & 0.75 & 13.3$\\pm$0.9  & 18.5$\\pm$1.1   \\\\\n\\hline\n\\end{tabular}\n\\end{table*}\n\n\\subsection{{\\sc{XMM}}\\emph{-Newton}}\nHD\\,5980\\ was observed 13 times by {\\sc{XMM}}\\emph{-Newton}\\ in 2000, 2001, 2005, and 2016 (PIs Bleeker, Corcoran, Parmar, Naz\\'e, respectively). The detailed journal of observations is provided in Table~\\ref{journal}. The first analyses of the 2000--2005 observations were reported in \\citet{naz04,naz07}. All datasets were reprocessed using SAS v16.0.0 using calibration files available in Winter 2016--2017 and following the recommendations of the {\\sc{XMM}}\\emph{-Newton}\\ team\\footnote{see \\\\ http:\/\/xmm.esac.esa.int\/sas\/current\/documentation\/threads\/ }. \n\n\\begin{figure}\n\\includegraphics[width=8.5cm]{hd5980col.ps}\n\\caption{EPIC image of HD\\,5980\\ (the bright source at center) and its surroundings obtained by combining all {\\sc{XMM}}\\emph{-Newton}\\ observations. Red, green, and blue correspond to 0.5--1.5\\,keV, 1.5--2.5\\,keV and 2.5--10.\\,keV energy bands, respectively. Note in particular the soft (red) emission surrounding HD\\,5980, which arises from the SNR IKT\\,18. }\n\\label{color}\n\\end{figure}\n\n\\begin{figure}\n\\includegraphics[height=8.5cm, angle=270]{newchandra_originalspec.ps}\n\\caption{The spectra recorded by \\emph{Chandra}\\ (ObsID 1881) for HD\\,5980\\ only (in green), for HD\\,5980\\ and part of the surrounding SNR (in black - 25'' circle centered on HD\\,5980, i.e., the region used for extracting {\\sc{XMM}}\\emph{-Newton}\\ spectra), and for the SNR contribution only (in red, annulus centered on HD\\,5980\\ with radii 2.5'' and 25''). Note how the SNR contribution dominates over that of HD\\,5980\\ in the {\\sc{XMM}}\\emph{-Newton}\\ region, especially below 2\\,keV. }\n\\label{chspec}\n\\end{figure}\n\n\\begin{figure*}\n\\includegraphics[height=6cm, angle=270]{newcompaxmm_minmax_2005.ps}\n\\includegraphics[height=6cm, angle=270]{newcompaxmm_minmax_2016.ps}\n\\includegraphics[height=6cm, angle=270]{newcompaxmm_minmax_2016b.ps}\n\\caption{{\\sc{XMM}}\\emph{-Newton}\\ spectra extracted over a circle of 25'' radius around HD\\,5980\\ and considering an external background (i.e. the spectra combine a contribution from HD\\,5980\\ and a contamination by the surrounding SNR emission). {\\it Left and Middle:} EPIC-pn spectra corresponding to the minimum (red crosses) and maximum (black crosses) emissions of HD\\,5980\\ in 2005 (left) and 2016 (middle), with the best-fit model (histogram-like solid lines) for the SNR contribution derived from \\emph{Chandra}\\ data in the same area. Note how the softest X-rays are not reproduced, despite being certainly due to the SNR. {\\it Right:} EPIC-pn spectra corresponding to the minimum (red crosses) and maximum (black crosses) emissions in 2016, with the best-fit model (solid line) for the SNR contribution derived from {\\sc{XMM}}\\emph{-Newton}\\ data in a nearby area. The softest X-rays are now reproduced, showing that some cross-calibration problems remain between \\emph{Chandra}-ACIS and {\\sc{XMM}}\\emph{-Newton}-EPIC. }\n\\label{compaxmmchfit}\n\\end{figure*}\n\nThe EPIC observations, taken in full-frame mode and with the medium filter (to reject optical\/UV light), were filtered to accept only the best-quality data ({\\sc{pattern}} of 0--12 for MOS and 0--4 for pn). Background flares were detected for observations taken in Revs. 0157, 0970, 3074, 3086, 3088, 3090, 3091, and 3110. These data were thus filtered to eliminate times where the count rate above 10.\\,keV was higher than 0.3\\,cts\\,s$^{-1}$ for MOS and 0.5\\,cts\\,s$^{-1}$ for pn. Unfortunately, the observation taken in Rev. 0970 was fully affected by flares: applying our flare filtering criteria results in no usable time. We have thus used these data without flare filtering but for source detection only. They were not included in the spectral analysis. Furthermore, the data taken in Rev. 1100 suffer from a slew failure at the very beginning of the observation, so we ignored the data during this interval. Finally, during Rev. 3091, there was a ground station outage during which no data could be recorded, but the data taken before and after this event were combined for further analyses.  \n\nSince HD\\,5980\\ is surrounded by the soft emission of the SNR IKT\\,18 (Fig. \\ref{color}), we performed the source detection, using the task {\\it edetect\\_chain} (for a log-likelihood of 10), only in the hard, 1.5--10\\,keV, energy band. This was done in several steps, first using a sliding box and then point spread function (PSF) fitting: the final count rates correspond to equivalent on-axis, full PSF count rates. Table~\\ref{journal} provides these EPIC count rates. \n\nWe then extracted EPIC spectra of HD\\,5980\\ using the task {\\it{especget}}, with a binning ensuring a signal-to-noise ratio of at least 3 per bin; dedicated response matrices were calculated at the same time. The source region is a circle centered on the best-fit position determined by the source detection algorithm and with a radius of 25''. Since the PSF of {\\sc{XMM}}\\emph{-Newton}\\ is much broader than that of \\emph{Chandra}, the emission of HD\\,5980\\ here appears much more blended with the surrounding SNR (compare Fig. 1 of \\citealt{naz04} with Fig. \\ref{color} above). In addition, the X-ray emission in this region is generally dominated by the SNR, not HD\\,5980\\ (Fig. \\ref{chspec}). We have thus attempted several strategies to obtain a clean estimate of the spectral parameters of HD\\,5980. First, we have used as background the SNR itself, avoiding the pn gaps: the region used is a box with 92.8'' and 107.2'' dimensions whose center is offset from HD\\,5980\\ position by 16.3'' east and 21.5'' south (i.e. -326.5\\,px in X and -430\\,px in Y), with the source region excised. This will correct for the SNR emission within the source region only if its emission is uniform, which is not exactly the case. Therefore, we also used a background region located outside the SNR, which is a circle of 25'' radius whose center is offset by 102'' east and 82'' south from HD\\,5980\\ (i.e. -2040\\,px in X and -1640\\,px in Y, as in \\citealt{naz07}). The derived background-corrected spectra were then fitted considering a (fixed) SNR contribution whose spectral parameters can be derived with \\emph{Chandra}\\ in the exact same region but after excising HD\\,5980. This procedure corrects for the SNR contribution in the exact surroundings of HD\\,5980, but it assumes a perfect cross-calibration between observatories, which is not exactly the case either (see Fig. \\ref{compaxmmchfit}). Neither solution thus is perfect, but results from both methods are in fact similar, as further discussed in the next section.\n\n\\section{The wind-wind collision in HD\\,5980} \n\n\\subsection{The X-ray lightcurve}\n\nThe first X-ray observations of HD\\,5980\\ were performed with \\emph{Chandra}\\ and {\\sc{XMM}}\\emph{-Newton}\\ in 2000 and 2001 \\citep{naz02,naz04}. These first exposures showed the source to be variable. Three additional observations obtained with {\\sc{XMM}}\\emph{-Newton}\\ in Fall 2005 confirmed the variability and further demonstrated its phased-locked nature since data taken on different dates but at the same phase appeared similar \\citep [compare also black and blue symbols in Fig. \\ref{lc}]{naz07}. In particular, the X-ray flux appeared larger around $\\phi=0.36$, when Star B is in front of Star A, than at $\\phi\\sim0$. Such orbital variations indicate that the X-ray emission of HD\\,5980\\ arises in a wind-wind collision. The absence of significant change in behaviour between data taken 5 years apart was particularly remarkable \\citep{naz07}. \n\n\\begin{figure*}\n\\includegraphics[width=6cm]{newlc1b.ps}\n\\includegraphics[width=6.5cm]{newlc2.ps}\n\\includegraphics[width=6cm]{newlc3.ps}\n\\caption{{\\it Left:} {\\sc{XMM}}\\emph{-Newton}\\ lightcurve of HD\\,5980\\ in the 1.5--10.0\\,keV energy band, for both MOS instruments combined. Data from 2000-2001 are shown as black circles, from 2005 as blue triangles, and from 2016 as red squares. Phases are computed using the ephemeris of \\citet{ste97}. This panel shows the marked brightening of the X-ray emission in 2016. {\\it Middle, from top to bottom:} {\\sc{XMM}}\\emph{-Newton}\\ lightcurve for both MOS combined using an arbitrary scaling factor to put all data points at similar levels (which demonstrates the remarkable similarity of the lightcurve shapes at different epochs); orbital separation (in units of the semimajor axis $a$); position angle (in $^{\\circ}$) defined as 0$\\degr$ when Star B is in front of Star A and 180$\\degr$ in the opposite situation. The orbital solution is that of \\citet{koe14}, i.e. $e=0.27$ and $\\omega_{peri}=134^{\\circ}$. The vertical lines in the top panel indicate the phases of the quadratures, periastron, apastron, and optical eclipses start\/end. The horizontal dashed lines in the bottom panel indicate the quadratures. {\\it Right:} Combined MOS count rates (without scaling) compared to the orbital separation (in units of the semimajor axis $a$), with their best-fit linear relations: a clear relationship is detected. }\n\\label{lc}\n\\end{figure*}\n\nThe 2016 observations extend the X-ray coverage over the entire orbit (Table \\ref{journal} and Fig. \\ref{lc}). They confirm an increasing count rate from $\\phi\\sim0$ to $\\phi=0.36$, but further show the rise in brightness to continue after the second conjunction up to the time of maximum stellar separation ($\\phi\\sim0.6$), with a decline thereafter. The tight correlation between the X-ray brightness of HD\\,5980\\ and the orbital separation between Stars A and B is further illustrated in the right panel of Fig. \\ref{lc}. The brightness variations appear to be linear in 2016, with an overall change by a factor of $\\sim$3 between periastron and apastron. In previous years, the linear fit is slightly poorer and the relative amplitude of the variations appears larger (factor of $\\sim$5), though the data point from Rev. 0970 is somewhat uncertain (see in Section 2.2). Note also that the shape of these variations does not seem to change much, as shown by the superposition of scaled lightcurves in the middle panel of Fig. \\ref{lc}. \n\nThis seems to indicate that orbital separation is the key parameter to explain variations, not orientation. Indeed, while the X-ray emission appears weak when Star A occults its companion ($\\phi$=0), the interval of low flux lasts much longer than the optical eclipses and the expected occultation during the opposite eclipse (at $\\phi$=0.36) is not observed (Fig. \\ref{lc}, middle). On the other hand, one could rather expect an increase of X-ray flux only when Star B and its more tenuous wind is on the near side of the orbit ($\\phi \\sim 0.14 - 0.77$, with a peak at the conjunction $\\phi=0.36$) but there is no such increase at those phases. This shows that neither stellar occultation nor large absorption changes in wind absorption produce the observed X-ray variations. \n\nHowever, the new observations also bring a surprise: the 2016 count rates are, on average, $2.57\\pm0.12$ times larger than in previous epochs (Table \\ref{journal} and left panel of Fig. \\ref{lc}). We recall here that all {\\sc{XMM}}\\emph{-Newton}\\ observations were analyzed in the same way, to ensure homogeneity. Besides, the reported count rates correspond to the 1.5--10.\\,keV energy band, to minimize the contamination by the SNR (see Sect. 2.2). Some remaining contamination may still exist, possibly slightly shifting upwards the count rates, but its Poissonian contribution is taken into account for the error calculation, hence the reported values should be correct, as best as possible considering the data properties and current knowledge of the instrument. The variations of HD\\,5980\\ are thus no artifact.\n\n\\subsection{Spectral properties in X-rays}\n\nThe X-ray spectra were fitted within Xspec v12.9.0i, using as reference the solar abundances of \\citet{asp09}. For {\\sc{XMM}}\\emph{-Newton}, the three EPIC spectra (MOS1, MOS2, pn) were fitted simultaneously. In these fits, the emission from HD\\,5980\\ was represented, as is adequate for massive stars, by an absorbed thermal emission model (``$apec$'') including up to two separate emission components. As mentioned in Sect. 2.2 above, two different backgrounds were considered, the immediate surroundings of HD\\,5980\\ (i.e. the SNR IKT\\,18) and an external background (in which case the SNR contamination was fixed to its best-fit in \\emph{Chandra}\\ data). For the latter case, a single thermal component was sufficient to achieve a good fit, while two components were generally required for the former case. However, to facilitate comparison and evaluate the impact of the background choice on the results, we also performed a fitting with a single component for the SNR background choice. In addition, the metallicity of the thermal emission components was fixed to the best fit values found by Hillier et al. (in prep) for Star A: mass fractions X=0.2011, Y=0.7962, Z(C)=3.05$\\times10^{-5}$, Z(N)=2.20$\\times10^{-3}$, Z(O)=1.50$\\times10^{-6}$, and Z(others)=0.20$\\times$solar, corresponding to abundances in number, with respect to Hydrogen and with respect to solar, of 11.6 for He and N, 0.05 for C, 0.001 for O, and 0.734 for other elements. Two possibilities were then envisaged for the abundances of the absorption component: solar abundances (i.e. $phabs$, all absorption occurring in the Milky Way) and abundances fixed to the same values as for the emission component(s) (i.e. $vphabs$, all absorption being circumstellar). \n\nThe results of the fits for the different methods (different absorptions, 1 or 2 thermal components, external or SNR background) are presented in Table \\ref{fit} and Fig. \\ref{fitparam}. We note that the choice of the background or the choice of absorption abundances does not change the derived phase-locked trends - it only shifts the parameters' averages. It is also immediately obvious that the flux variations derived from the count rates are confirmed. However, results for {\\sc{XMM}}\\emph{-Newton}\\ spectra taken in Revs. 0357 and 1100 differ from other ones. In fact, at these phases, the flux from HD\\,5980\\ is minimal, hence its emission is more difficult to disentangle from the SNR contamination. Clearly, some residual contamination from the SNR remains, whatever the method used for background subtraction, leading to (unrealistic) lower temperatures at these phases. Another evidence of residual contamination lies in the fluxes: those derived from \\emph{Chandra}\\ data are systematically lower than those calculated from {\\sc{XMM}}\\emph{-Newton}\\ spectra (for a similar phase and observing year). While this affects the absolute values of the spectral parameters, it does not change their variations, as SNR emission remains constant over such a short time span.\n\n\\begin{figure*}\n\\includegraphics[width=6cm]{newfitnonsol_1Textphabs.ps}\n\\includegraphics[width=6cm]{fitnonsol_1Tphabs.ps}\n\\includegraphics[width=6cm]{fitnonsol_2Tphabs.ps}\n\\caption{Evolution with phase of the spectral parameters derived for {\\sc{XMM}}\\emph{-Newton}\\ spectra considering solar absorption (see Table \\ref{fit}): left, 1T results when choosing an external background and a SNR contamination fixed to \\emph{Chandra}\\ best fit; middle and right, 1T and 2T results, respectively, for the choice of the surrounding SNR as background.  }  \n\\label{fitparam}\n\\end{figure*}\n\n\\begin{figure*}\n\\includegraphics[height=6cm, angle=270]{newcompachandra.ps}\n\\includegraphics[height=6cm, angle=270]{newcompaspecxmmphi.ps}\n\\includegraphics[height=6cm, angle=270]{newcompaspecxmm2016b.ps}\n\\caption{Spectra from HD\\,5980, corrected for the (SNR) background, as recorded by \\emph{Chandra}\\ (left) and EPIC-pn onboard {\\sc{XMM}}\\emph{-Newton}\\ (middle panel for spectra taken at a similar phase but in different years; right panel for spectra taken at minimum and maximum brightness levels in 2016). The hardening of the spectra in recent years is obvious in the middle panel.}  \n\\label{compaspec}\n\\end{figure*}\n\n\\begin{table*}\n\\centering\n\\caption{Results of the spectral fits using models of the type $(v)phabs\\times \\sum vapec_i$ . Fluxes are provided in the 0.5--10.0\\,keV energy band, the usual range used for massive stars. Limits of 1$\\sigma$ (or 68\\%) confidence intervals are provided within squared brackets but errors on fluxes should be taken with caution, as uncertainties linked to cross-calibration and remaining SNR contamination subsist for this parameter. Note that, formally, spectra from Revs. 0357 and 1100 do not require two thermal components to be well fitted (those fits are here only provided for homogeneity and completeness). }\n\\label{fit}\n\\begin{tabular}{lcccccccc}\n\\hline\\hline\nID\/Rev  & $\\phi$ & $N_{\\rm H}$ & $kT_1$ & $norm_1$ & $kT_2$ & $norm_2$ & $\\chi^2_{\\nu}$ (dof) & $F^{\\rm obs}_{\\rm X}$ \\\\\n    &        & (10$^{22}$\\,cm$^{-2}$) & (keV) & ($10^{-5}$\\,cm$^{-5}$) & (keV) & ($10^{-5}$\\,cm$^{-5}$) & & ($10^{-14}$\\,erg\\,cm$^{-2}$\\,s$^{-1}$)\\\\\n\\hline\n\\multicolumn{9}{l}{1T, solar absorption, using an external background and a fixed model for the SNR contamination}\\\\\n0157 & 0.369 &0.023 [0.015-0.033] &8.02 [6.11-11.0] &1.89 [1.75-2.02] & & & 1.29(130) &10.9 [10.1-11.2]\\\\\n0357 & 0.100 &0.016 [0.004-0.030] &1.06 [0.94-1.22] &0.69 [0.58-0.79] & & & 1.94(122) &1.80 [1.76-1.84]\\\\\n1093 & 0.259 &0.013 [0.003-0.026] &9.91 [6.85-16.2] &1.35 [1.25-1.46] & & & 1.23(119) &8.15 [7.35-8.40]\\\\\n1094 & 0.360 &0.023 [0.013-0.037] &5.47 [4.42-7.10] &1.90 [1.76-2.05] & & & 1.59(130) &9.80 [9.15-10.2]\\\\\n1100 & 0.999 &0.006 [0.000-0.018] &1.53 [1.08-3.16] &0.77 [0.63-0.91] & & & 1.52(88)  &2.20 [2.12-2.28]\\\\\n3073 & 0.254 &0.129 [0.109-0.156] &4.36 [3.90-4.86] &4.61 [4.40-4.86] & & & 1.20(189) &19.9 [19.3-20.4]\\\\\n3074 & 0.362 &0.185 [0.165-0.206] &3.05 [2.83-3.28] &7.28 [6.97-7.60] & & & 1.40(231) &25.1 [24.5-25.6]\\\\\n3086 & 0.602 &0.191 [0.178-0.205] &2.38 [2.27-2.50] &10.3 [9.97-10.6] & & & 1.37(255) &30.1 [29.5-30.7]\\\\\n3088 & 0.802 &0.074 [0.060-0.091] &5.48 [4.77-6.56] &4.73 [4.51-4.97] & & & 1.13(175) &23.5 [22.1-24.5]\\\\\n3090 & 0.999 &0.006 [0.000-0.015] &9.66 [7.14-14.5] &1.60 [1.50-1.70] & & & 1.32(147) &9.67 [9.03-9.99]\\\\\n3091 & 0.101 &0.025 [0.009-0.057] &10.9 [7.25-18.2] &1.51 [1.44-1.64] & & & 1.28(113) &9.15 [7.63-9.49]\\\\\n3110 & 0.102 &0.029 [0.021-0.039] &7.41 [6.30-9.29] &2.06 [1.96-2.16] & & & 1.61(180) &11.6 [11.1-12.0]\\\\\n\\hline\n\\multicolumn{9}{l}{1T, solar absorption, using the surrounding SNR as background}\\\\\n1881  &0.266 &0.19 [0.11-0.30] &7.62 [5.44-12.1] &0.70 [0.63-0.79] &&&1.39(17)  &3.65 [2.89-3.89] \\\\\n13773 &0.823 &0.22 [0.12-0.33] &4.51 [3.48-6.70] &5.39 [4.72-6.06] &&&1.01(29)  &22.5 [18.2-24.4] \\\\\n0157  &0.369 &0.07 [0.05-0.09] &2.58 [2.27-3.03] &3.35 [3.10-3.58] &&&1.67(54)  &11.5 [10.9-12.2] \\\\\n0357  &0.100 &0.14 [0.10-0.18] &0.83 [0.78-0.90] &2.76 [2.48-3.09] &&&1.52(39)  &5.54 [5.23-5.76] \\\\\n1093  &0.259 &0.08 [0.06-0.10] &2.23 [1.90-2.56] &2.77 [2.54-3.00] &&&2.17(44)  &8.65 [7.98-9.15] \\\\\n1094  &0.360 &0.09 [0.08-0.11] &1.97 [1.80-2.15] &3.79 [3.56-4.01] &&&1.50(60)  &10.8 [10.2-11.4] \\\\\n1100  &0.999 &0.08 [0.06-0.11] &0.99 [0.93-1.06] &2.52 [2.28-2.77] &&&0.61(19)  &5.87 [5.45-6.25] \\\\\n3073  &0.254 &0.16 [0.15-0.18] &2.56 [2.39-2.78] &6.49 [6.19-6.79] &&&1.36(124) &20.4 [19.6-21.1] \\\\\n3074  &0.362 &0.19 [0.18-0.20] &2.30 [2.18-2.43] &9.23 [8.91-9.55] &&&1.41(175) &26.4 [25.6-27.2] \\\\\n3086  &0.602 &0.22 [0.21-0.23] &1.83 [1.77-1.89] &12.7 [12.3-13.0] &&&1.63(202) &30.4 [29.8-31.2] \\\\\n3088  &0.802 &0.12 [0.11-0.14] &2.97 [2.66-3.27] &6.44 [6.13-6.75] &&&1.40(112) &22.9 [21.8-24.0] \\\\\n3090  &0.999 &0.08 [0.06-0.10] &2.42 [2.12-2.72] &2.97 [2.77-3.17] &&&2.22(65)  &9.77 [9.18-10.4] \\\\\n3091  &0.101 &0.13 [0.10-0.16] &1.94 [1.72-2.16] &3.40 [3.13-3.67] &&&2.50(41)  &9.26 [8.59-9.82] \\\\\n3110  &0.102 &0.09 [0.07-0.11] &2.56 [2.33-2.79] &3.61 [3.43-3.79] &&&2.03(103) &12.1 [11.6-12.6] \\\\\n\\multicolumn{9}{l}{2T, solar absorption, using the surrounding SNR as background}\\\\\n0157 &0.369 &0.10 [0.08-0.12] &0.86 [0.80-0.93] &2.04 [1.68-2.44] &6.57 [4.54-11.3] &1.72 [1.44-2.01] &1.07(52)  &13.5 [11.6-13.9] \\\\\n0357 &0.100 &0.19 [0.13-0.34] &0.43 [0.30-0.59] &1.74 [0.68-7.40] &0.94 [0.86-1.06] &2.08 [1.61-2.58] &1.53(37)  &5.67 [2.67-5.44] \\\\\n1093 &0.259 &0.14 [0.11-0.18] &0.83 [0.76-0.89] &2.40 [2.01-2.87] &10.8 [5.95-48.7] &1.06 [0.91-1.27] &1.14(42)  &10.9 [4.26-11.3] \\\\\n1094 &0.360 &0.14 [0.11-0.17] &0.83 [0.79-0.89] &2.39 [2.00-2.88] &3.52 [2.97-4.86] &1.90 [1.55-2.20] &0.87(58)  &12.2 [10.8-12.8] \\\\\n1100 &0.999 &0.14 [0.07-0.32] &0.36 [0.24-0.72] &1.82 [0.32-28.0] &1.08 [0.97-1.21] &2.12 [1.74-2.69] &0.58(17)  &6.05 [3.70-5.90] \\\\\n3073 &0.254 &0.22 [0.19-0.25] &1.00 [0.94-1.07] &3.49 [2.92-4.15] &4.39 [3.76-5.50] &3.79 [3.27-4.26] &1.03(122] &22.0 [20.5-22.6] \\\\\n3074 &0.362 &0.25 [0.23-0.28] &0.92 [0.84-0.98] &4.51 [3.84-5.22] &3.53 [3.15-4.07] &5.69 [5.12-6.32] &1.02(173) &28.3 [26.8-29.0] \\\\\n3086 &0.602 &0.28 [0.26-0.31] &0.92 [0.88-0.96] &7.69 [6.84-8.58] &3.16 [2.81-3.54] &6.30 [5.56-7.09] &0.99(200) &33.2 [32.2-33.8] \\\\\n3088 &0.802 &0.17 [0.14-0.20] &0.90 [0.77-0.98] &3.19 [2.56-3.86] &5.90 [4.50-8.51] &4.03 [3.51-4.66] &1.06(110) &25.7 [23.2-26.5] \\\\\n3090 &0.999 &0.13 [0.10-0.17] &0.90 [0.84-0.97] &2.36 [1.99-2.75] &11.6 [6.48-35.1] &1.26 [1.09-1.48] &1.44(63)  &12.2 [4.59-12.7] \\\\\n3091 &0.101 &0.25 [0.19-0.32] &0.80 [0.72-0.88] &3.57 [2.89-4.51] &16.0 [7.13-64.0] &1.17 [0.99-1.37] &1.36(39)  &12.3 [5.00-12.8] \\\\\n3110 &0.102 &0.13 [0.11-0.16] &0.84 [0.79-0.91] &2.32 [2.03-2.62] &6.82 [5.21-8.92] &1.87 [1.69-2.09] &1.32(101) &14.4 [13.3-14.8] \\\\\n\\multicolumn{9}{l}{2T, non-solar absorption, using the surrounding SNR as background}\\\\\n0157 &0.369 &0.014 [0.011-0.018] &0.86 [0.81-0.96] &1.69 [1.40-2.09] &6.02 [4.44-10.9] &1.79 [1.46-2.06] &1.06(52)  &13.5 [11.5-14.1] \\\\\n0357 &0.100 &0.020 [0.015-0.027] &0.83 [0.78-0.89] &2.24 [2.04-2.46] &68.4 [2.91-64.0] &0.25 [0.17-0.38] &1.52(37)  &6.49 [4.67-15.8] \\\\\n1093 &0.259 &0.021 [0.016-0.029] &0.85 [0.78-0.91] &1.96 [1.68-2.27] &9.63 [5.58-27.9] &1.11 [0.95-1.31] &1.09(42)  &11.0 [7.66-11.4] \\\\\n1094 &0.360 &0.020 [0.016-0.025] &0.86 [0.78-0.90] &1.85 [1.57-2.20] &3.52 [3.00-4.70] &1.93 [1.62-2.20] &0.87(58)  &12.3 [10.9-12.7] \\\\\n1100 &0.999 &0.019 [0.010-0.072] &0.36 [0.19-0.70] &1.16 [0.20-35.9] &1.11 [1.01-1.25] &2.01 [1.41-2.42] &0.56(17)  &6.09 [4.51-6.21] \\\\\n3073 &0.254 &0.036 [0.030-0.042] &1.06 [0.99-1.12] &2.44 [2.00-2.92] &4.44 [3.85-5.45] &3.82 [3.37-4.23] &1.03(122) &22.2 [20.9-22.9] \\\\\n3074 &0.362 &0.043 [0.038-0.048] &0.97 [0.90-1.03] &2.95 [2.45-3.47] &3.60 [3.24-4.09] &5.71 [5.21-6.22] &1.03(173) &28.6 [27.2-29.4] \\\\\n3086 &0.602 &0.049 [0.045-0.053] &0.97 [0.92-1.01] &4.96 [4.35-5.59] &3.10 [2.80-3.44] &6.61 [5.95-7.27] &0.95(200) &33.5 [32.5-34.2] \\\\\n3088 &0.802 &0.026 [0.022-0.031] &0.94 [0.82-1.02] &2.39 [1.87-2.93] &5.90 [4.68-8.25] &4.04 [3.57-4.54] &1.02(110) &25.9 [23.6-26.8] \\\\\n3090 &0.999 &0.021 [0.016-0.027] &0.93 [0.85-1.00] &2.00 [1.69-2.31] &10.8 [6.45-29.6] &1.27 [1.11-1.49] &1.43(63)  &12.3 [4.64-12.9] \\\\\n3091 &0.101 &0.045 [0.032-0.062] &0.82 [0.74-0.89] &2.71 [2.24-3.28] &12.2 [6.36-64.0] &1.23 [1.05-1.44] &1.31(39)  &12.4 [4.65-12.7] \\\\\n3110 &0.102 &0.020 [0.017-0.024] &0.87 [0.81-0.94] &1.86 [1.62-2.13] &6.56 [5.09-8.79] &1.91 [1.70-2.13] &1.27(101) &14.3 [13.4-14.9] \\\\\n\\hline\n\\end{tabular}\n\\end{table*}\n\nRegarding spectral parameters derived in 2016 (the time of our best coverage of the orbit), we find that the absorption column increases towards apastron while the temperature decreases (only that of the hottest thermal component if two were used, see Table \\ref{fit} and Fig. \\ref{fitparam}). These results are however reminiscent of the usual absorption-temperature degeneracy: spectra can be reproduced by a low-absorption, high-temperature model as well as by a high-absorption, low-temperature model \\citep[e.g.][]{naz12b}. To remove this degeneracy, we fixed the temperatures and performed the fits again: the absorption still appears larger at apastron. We also did a trial in which absorption was fixed to its mean value: the temperature still appears smaller at apastron. This shows that our result goes beyond the trade-off problem: there are truly an increase of absorption and a decrease of temperature at apastron.\n\nSince observations span over 15 years, we may also assess whether there have been any changes of the spectral parameters with time, by comparing spectra taken at similar orbital phases in different orbital cycles. For example, the first \\emph{Chandra}\\ dataset and two {\\sc{XMM}}\\emph{-Newton}\\ exposures (Revs. 1093 and 3073) were taken at $\\phi\\sim0.25$, data in Revs. 0157, 1094, and 3074 were taken at an eclipse of Star A by Star B ($\\phi\\sim0.36$) while the second \\emph{Chandra}\\ observation was taken at the same time as {\\sc{XMM}}\\emph{-Newton}\\ data of Rev. 3088. We need however to keep in mind that (1) there may be remaining cross-calibration problems and some SNR contamination in {\\sc{XMM}}\\emph{-Newton}\\ spectra, which leaves some uncertainty for any {\\sc{XMM}}\\emph{-Newton}\\ vs \\emph{Chandra}\\ comparison and (2) the lower luminosities of HD\\,5980\\ observed in 2000--2005 renders the parameter determination more difficult at those times, hence they are less precise. \n\nFor {\\sc{XMM}}\\emph{-Newton}\\ spectra, there is a clear hardening of the emission of HD\\,5980\\ in 2016 compared to that recorded in previous years (see also middle panel of Fig. \\ref{compaspec}). This hardening does not seem to be linked to changing temperatures (since they appear quite similar whatever the year considered) but rather to an increase in absorbing columns with time (Table \\ref{fit}, Fig. \\ref{fitparam}). However, the larger noise in older data complicates the comparison so our interpretation of the origin of the observed hardening should be viewed with some caution. In contrast, the fits of both \\emph{Chandra}\\ spectra, which have minimal contamination by the SNR, show only a slight increase in absorption and a slight decrease in temperature, formally not significant as they remain within 1$\\sigma$. For them, there is thus mostly a flux variation (see also left panel of Fig. \\ref{compaspec}). It is however important to note that these exposures were taken at different phases ($\\phi$=0.27 and 0.82) and the spectra are unfortunately noisier than the {\\sc{XMM}}\\emph{-Newton}\\ ones, rendering detailed comparisons difficult.\n\n\\subsection{Discussion}\n\n\\begin{table}\n\\centering\n\\footnotesize\n\\caption{Stellar parameters derived for Star A from the analysis with CMFGEN of HST data taken in 2000, 2002, 2009 \\citep{geo11} and 2014 (Hillier et al., in prep). Mass-loss rate values reported here consider a filling factor $f$ of 0.1, typical of WR stars. Simple interpolations (in italics) were calculated for 2001 and 2005. The typical errors reported in the second column correspond to formal uncertainties on the match between the CMFGEN model and the observations; they do not take into account the systematic uncertainties in the underlying assumptions inherent to all atmosphere models currently available, the non totality of the eclipses, or emission arising from the wind-wind collision.}\n\\label{starA}\n\\begin{tabular}{lccccccc}\n\\hline\\hline\nParameter & $\\sigma$ & 2000 & {\\it 2001} & 2002 & {\\it 2005} & 2009 & 2014 \\\\\n\\hline\n$\\log(L_{\\rm BOL}\/L_{\\odot})$                & 0.1 & 6.30 & {\\it 6.35} & 6.39 & {\\it 6.39} & 6.39 & 6.23 \\\\\n$T_{\\rm eff}$ (kK)                          & 3.0 & 37.3 & {\\it 38.7} & 40.0 & {\\it 41.5} & 43.0 & 43.0\\\\\n$R_*$ (R$_{\\odot}$)                         & 10\\% & 34   & {\\it 33}   & 32   & {\\it 30}   & 28   & 21.6\\\\\n$\\dot M$ ($10^{-5}$\\,M$_{\\odot}$\\,yr$^{-1}$) & 10\\% & 11.1 & {\\it 9.5}  & 7.9  & {\\it 7.6}  & 7.3  & 5.0 \\\\\n$v_{\\infty}$ (km\\,s$^{-1}$)                  & 200 & 2000 & {\\it 2100} & 2200 & {\\it 2320} & 2440 & 3000\\\\\n\\hline\n\\end{tabular}\n\\end{table}\n\nThe previous sections have helped us derive the observational picture; we now summarize the main observational results and propose our interpretation. \n\nFirst and foremost, we discover an overall flux increase over the last decade, by a factor of $\\sim$2.5, while data taken between 2000 and 2005 showed no significant differences in this respect. This brightening is accompanied by a slight hardening of the X-ray emission, possibly linked to an increased absorption. Such a brightening is totally unprecedented in colliding wind binaries. Repeatability rather is the rule \\citep[see e.g. the case of WR21a][]{gos16} - only $\\eta$\\,Car presents changes from cycle-to-cycle, but only over a very small phase interval \\citep[around periastron, see e.g.][]{ham14}. Since it is unprecedented, this secular change may cast doubt that the recorded X-rays truly are associated with the wind-wind collision in the A+B pair, but alternative explanations can all be ruled out. Indeed, the observed X-ray increase cannot be produced by the collision of the now fast wind from Star A catching up to and colliding with the slower material ejected in the eruption, since this would lead to a brightening by a constant amount at all phases (i.e. a simple shift of the lightcurve towards larger luminosities) at odds with observations showing a brightening by a similar factor at all phases (Fig. \\ref{lc}). The variation cannot be associated with Star C either: when the observed variation are phased with the Star C ephemeris \\citep[see also column $\\phi_C$ in Table \\ref{journal}]{koe14}, there is no coherent behaviour with phase (Fig. \\ref{starc}), and incompatibilities are even present at $\\phi_C\\sim0.2-0.4$. Similarly, a combination of two collisions, one in the A+B pair and one in the C+? system, can also be ruled out. Indeed, Table \\ref{journal} shows that some observations were taken with similar phases in both orbits: if the variations were due to a combination of two periodic variability schemes, data taken at similar phase pairs would be similar, but this is not the case (e.g. data taken in Revs. 1093 and 3073 widely differ in flux!). Thus the changing wind-wind collision in the A+B pair must be responsible for the observed recent changes in the X-ray emission. In this context, it is important to note that X-ray spectra from 2000 and 2005, obtained years apart but at the same orbital phase, appeared similar in brightness and spectral shape to each other, demonstrating a clear phase-locked repeatability, and indicating that the large-scale increase in X-ray brightness we observed in 2016 occurred rather suddenly after 2005.\n\n\\begin{figure}\n\\includegraphics[width=8cm]{newlc4.ps}\n\\caption{Same as Fig. \\ref{lc} but using the ephemeris relative to Star C \\citep[see Table 7 in][ and 6th column of Table \\ref{journal} above]{koe14}. There is no smooth, coherent behavior with phase.}  \n\\label{starc}\n\\end{figure}\n\nSecond, we confirm the presence of phase-locked flux variations detected earlier and refine their properties. In particular, we find a tight correlation between the X-ray brightness and the orbital separation. This result clearly indicates that the collision is not adiabatic, since a maximum at periastron is expected in such cases (see e.g. the cases of WR25 \\citealt{gosth,pan14}, or Cyg\\,OB2\\,\\#9 \\citealt{naz12c}). A similar behaviour was however recorded in several {\\it radiative} wind-wind collisions (e.g. Cyg\\,OB2\\,\\#8A - see Fig. 3 of \\citealt{caz14}, HD\\,152218 and HD\\,152248 - see Fig. 3 in \\citealt{rau16}, HD\\,166734 - see Fig. 5 in \\citealt{naz17}) though the luminosity-separation trend is here much more linear and the hysteresis around it of a much smaller amplitude (see right panel of Fig. \\ref{lc}). \n\nAnother result to point out is the absence of variations related to the orientation of the system. Not only is there no evidence of eclipses \\citep[unlike the case of V444\\,Cygni, see][but similar to the case of WR\\,20a, see \\citealt{naz08}, and CQ\\,Cep, see \\citealt{ski15}]{lom15}, but there is also no evidence of the usual absorption effects. Indeed, as far as we can tell, there is no significant increase of absorption at or close to conjunctions (i.e. at eclipse phases) nor at periastron, contrary to what is observed e.g. for the radiative collisions in V444\\,Cygni \\citep{lom15} or WR21a \\citep{gos16}. Furthermore, we fail to detect a significantly lower absorption, or larger flux, when the secondary star (Star B) and its more tenuous wind are in front of the system, though (1) this should not occur in the new monitoring as both stars now have similar wind parameters (see below) and (2) it is true that such effects mostly affect soft X-rays (see e.g. $\\gamma^2$\\,Vel, \\citealt{wil95}) which are somewhat uncertain in our case as they are most affected by any remaining SNR contamination. These facts, combined to the generally low absorbing columns found in fits, suggest that the X-ray emission zone is large compared to the stellar bodies and the innermost, opaque, parts of their stellar winds. It is however puzzling to find a larger absorption at apastron: considering that stars are then more distant, and far from conjunctions, the wind density along the line-of-sight should be smaller, not larger, underlining the peculiarity of this result.\n\n\\begin{figure*}\n\\includegraphics[width=9cm]{lc1.ps}\n\\includegraphics[width=9cm]{lc2.ps}\n\\includegraphics[width=9cm]{lc3.ps}\n\\includegraphics[width=9cm]{lc4.ps}\n\\caption{{\\it Top panels:} Results from simple equilibrium models considering the standard wind law and different properties of Star A (each year being identified by colors - see Table \\ref{starA} for details). The values of the position of the stagnation point, the pre-shock velocities, the cooling parameters $\\chi$,  the temperatures, and X-ray luminosities are shown as a function of phase. They are provided by solid or dashed lines if the primary (Star A) and the secondary (Star B), respectively, are separately considered. The total (i.e., summing contribution of both primary and secondary stars) X-ray luminosities shown on the right panels are normalized to the apastron values and compared to the MOS1+2 count rates, also normalized to apastron values (stars for 2000--2005 data, squares for 2016 data). {\\it Bottom panels:} Comparison of results with or without inclusion of radiative inhibition, considering the properties of Star A in 2005 (left) and 2014 (right). The X-ray luminosities are again normalized to the apastron values, and compared to the associated MOS1+2 count rates, also normalized to apastron values (green symbols). In all panels, the x-axes report relative separation or phase, using $\\phi-0.07$ to have 0 for periastron and 0.5 for apastron. }  \n\\label{models}\n\\end{figure*} \n\n\\begin{figure}\n\\includegraphics[width=8cm]{coneshape.ps}\n\\caption{Shape of the shock cone for 2005 and 2014, at apastron and periastron, with the part at $kT>0.2$\\,keV shown as solid lines. The axes are distances divided by the separation at the phase considered; by convention, the two stars lie at (0,0) and (1,0). }  \n\\label{cshape}\n\\end{figure}\n\n\\vspace*{7mm}\n\n\nTo further interpret these phase-locked observations in the framework of a wind-wind collision in the A+B system, it is important to examine what is predicted by models. The X-ray flux produced by the wind-wind collision is some fraction of the wind kinetic luminosity $0.5 \\dot M v^2$ \\citep{ste92}, hence the precise knowledge of the wind properties is crucial. Indeed, the change in the wind properties of Star A from a slow eruptive phase to a fast, lower-density phase could produce an observable effect on the X-ray emission. The stellar parameters of Star A were derived by comparing optical\/UV spectra with synthetic spectra computed with model atmosphere codes. To this aim, $HST$ data taken in 2000, 2002, 2009, and 2014 during the eclipse of Star B by Star A ($\\phi\\sim0$) were used. The fitting results using CMFGEN are reported by \\citet{geo11} and Hillier et al. (in prep); they are summarized in Table \\ref{starA}. Note that the larger wind terminal velocity in 2014 does not come from the CMFGEN fit itself but from the observed extension of the P Cygni profiles in the UV spectrum. The parameters fitted by \\citet{she16}, using a subset of these observations and a different model atmosphere code, agree well with those reported by \\citet{geo11} for the same data. Our X-ray observations were taken in 2000, 2001, 2005, 2013, and 2016: to get the properties of Star A in 2013 and 2016, we assume that it has not changed much since 2013, and we use the parameters from 2014; to get those for 2001 and 2005, we perform a simple interpolation (see italics in Table \\ref{starA}). \n\nContrary to Star A, Star B is assumed to have remained relatively stable over long timescales. In their Table 2, \\citet{she16} list the properties derived for Star B, which agree well with those found for Star A in 2014 (Table \\ref{starA}, Hillier et al., in prep), confirming the similarity of the two stars presented by \\citet{koe14}. Furthermore, the CMFGEN analysis results of a $HST$ spectrum taken in 2016 when Star B eclipsed Star A ($\\phi\\sim0.36$) are also consistent with Star B having similar parameters as Star A in 2014 (Hillier et al., in prep). Therefore, we adopted the 2014 parameters of Star A for Star B. We finally note that Hillier et al. (in prep) do not find any evidence for a difference in the wind clumping factor $f$ between epochs for Star A or between Star A and Star B - a value of 0.1 is here adopted in all cases.\n\nWith these parameters, if we assume in addition that the winds collide at their terminal speeds, we find that the wind kinetic luminosity of Star A has not much changed since 2000, while the X-ray flux increased by a factor $\\sim$2.5. At the same time, the wind momentum ratio $\\dot M_Av_{\\infty,A}\/\\dot M_Bv_{\\infty,B}$ declined from 1.5 to 1.0, so the opening angle of the wind-wind collision cone should have increased (to near 180$^{\\circ}$), while observations rather indicate similar lightcurve shapes over time. Finally, the wind density of Star A at a given position ($\\rho\\propto \\dot M\/v_{\\infty}$) decreased by a factor of 3.3 between 2000 and 2014, while observations suggest a potential increase of absorption in recent years. From these numbers, it appears that the changing high-energy emission of HD\\,5980\\ cannot be explained by these first, rough estimates.\n\nIn such close systems, it is however common that winds have not yet reached their terminal velocity before colliding, and the radiation from the companion may reduce the wind speed. Therefore, we performed next a more detailed calculation, based on the orbital parameters of HD\\,5980\\ \\citep{ste97,koe14}. For each orbital phase, we calculate the position of the stagnation point, where the wind momenta $\\dot M v$ equilibrate along the line joining the two stars, considering two cases for the wind acceleration: the standard wind acceleration law given by $v=v_{\\infty}[1-R_*\/r]$ and a full radiative-driving calculation \\citep{cas75,ste94,par13} taking the flux of both stars into account. Using the derived wind velocities, we then estimate the physical properties of the post-shock plasma of each wind, notably its temperature ($kT=3 mv^2\/16$) and its intrinsic X-ray luminosity \\citep[from the shocked wind formulation in Sect. 3.2 of][]{zab11} - this is an intrinsic value, i.e., without considering any wind absorption, hence it is representative of the hard emission. While a full hydrodynamic simulation (which is beyond the scope of this paper) is certainly needed, this simple modeling may clarify the general trends for the wind-wind collision properties, to be checked against data. \n\nResults of this modeling are shown in Fig. \\ref{models}. The panels in the top two rows compare the results of the simple equilibrium (using the standard wind law) for different Star A parameters, while the panels of the bottom two rows compare the modeling results with and without radiative inhibition for the cases of 2005 (left) and 2014 (right). The equilibrium position always remains close to the mid-point (or half separation), as shown in the upper left panels of each part of Fig. \\ref{models}. The collapse of the collision zone onto one of the star is thus avoided, unlike what is observed in some other eccentric binaries (e.g. $\\eta$\\,Car, \\citealt{ham14} , WR21a \\citealt{gos16}, HD\\,166734 \\citealt{naz17}). In addition, the collision always remains radiative in nature ($\\chi<1$, see notably the middle top panel of Fig. \\ref{models}), as derived from observations. \n\nWe can also reproduce, at least qualitatively, the observed correlation between hard flux and separation (e.g. rightmost panel in second row of Fig. \\ref{models}). A slightly better agreement is even obtained if radiative inhibition is included (last row of Fig. \\ref{models}). In such radiative systems, the maximum occurs at apastron rather than at periastron, despite the decrease in wind density, because the impact of higher pre-shock wind speeds. Indeed, the wind-wind collision occurs as the winds still accelerate and they reach larger speeds when they are further apart, i.e. at apastron. Moreover, the wind braking effect due to the companion radiation is less strong when stars are more separated, reinforcing the increase in wind speed at apastron. However, such larger wind speeds at apastron also imply a larger temperature for the X-ray emitting plasma, while the observations rather suggest a decrease of temperature at that phase.\n\nFinally, predictions for 2000--2005 are quite similar (top panels of Fig. \\ref{models}), again as observed, but the modeling also predicts little change in the total X-ray brightness between 2005 and 2016. In addition, the predicted temperature (e.g. middle panel of the second row in Fig. \\ref{models}) is twice as large for the primary wind in recent years because of its increased wind speed. This could explain the recent hardening, though such a large change in temperature is not obvious in spectra (fits rather favor an absorption increase). \n\nSince observations suggest the wind-wind collision zone to be large, we also calculated wind momenta equilibrium outside of the line-of-centers (without radiative inhibition). We derived the local temperature at these positions from the usual formula $kT=3mv^2\/16$, where $v$ is the velocity component perpendicular to the shock \\citep{ste92}. Considering X-rays to be emitted if $kT>0.2$\\,keV, we found that the size of the emission regions are similar in 2005 and 2014 (Fig. \\ref{cshape}), suggesting again similar flux levels at different epochs. \n\n\\vspace*{7mm}\n\nIn summary, the change in Star A's mass-loss properties {\\it alone} cannot explain the increased X-ray emission of HD\\,5980\\ in 2016. However, the X-ray properties also depend on how the shocked stellar wind material cools as it flows away from the stars. Cooling can occur either by adiabatic expansion or via radiation. A characteristic measure of the relative importance of both processes is expressed by the cooling parameter $\\chi =t_{cool}\/t_{escape} \\approx v^4 d\/\\dot M$ \\citep[with the wind speed $v$ expressed in $10^8$\\,cm\\,s$^{-1}$, the separation $d$ in $10^{12}$\\,cm and the mass-loss-rate in $10^{-7}$\\,M$_{\\odot}$\\,yr$^{-1}$, see][]{ste92}. As stated above, the wind-wind collision in HD\\,5980\\ appears to remain radiative (i.e. $\\chi<1$) throughout the orbital cycle but the value of $\\chi$ for Star A in 2000--2005 is much smaller than in 2016, by about an order of magnitude: in models using the standard wind law, $\\chi$ at the stagnation point goes from 0.004 at periastron to 0.06 at apastron in 2005 and from 0.08 to 0.39 in 2016 (Fig. \\ref{models}). These values depend of course on the chosen stellar parameters, notably wind clumping which affects the derived mass-loss rate. Using a lower wind clumping factor $f\\sim 0.025$, as in \\citet{geo11}, would lead to mass-loss rates smaller by a factor of two, hence to larger cooling parameters since $\\chi\\propto 1\/\\dot M$. A lower $f$ value would thus lead to an overall increase in $\\chi$ but would not modify its trend. Thus radiative cooling a decade ago would have been even more dominant than in 2016. \n\nA recent theoretical study by \\citet{kee14} has shown that the X-ray luminosity from highly radiative wind-wind collisions should be weaker than predictions from analytical models such as those used above. This is due to the onset of strong non-linear thin-shell instabilities in the radiatively cooling gas. In this case, the shocks become more oblique and since the post-shock temperature depends on the square of the velocity component perpendicular to the shock surface, the post-shock plasma is expected to be cooler. This theoretical study thus provides a possible explanation for the secular behavior of HD\\,5980\\ since in 2000--2005, when the collision was more strongly radiatively cooled, the X-ray emission appeared both fainter and softer than in 2016. This tentative conclusion needs however to be confirmed by detailed 3D hydrodynamical simulations specifically dedicated to HD\\,5980.\n\n\\section{Conclusion} \nTwo decades after the eruption in HD\\,5980, we have obtained new X-ray observations of the system. Surprisingly, the new data reveal a large increase of the X-ray brightness (factor of $\\sim$2.5) at all phases, coupled to a hardening of the emission. This is an unprecedented feature for colliding wind binaries. However, the lightcurve {\\it shape} has not changed significantly, with a tight linear correlation between the X-ray flux and the separation. The new data also further helped us refining our view of the phase-locked properties, enlightening the absence of eclipses and an increase of absorption at apastron coupled to a decrease in temperature at that phase. \n\nSimple analytical models of the collision, based on the known stellar and orbital parameters, explain the flux variation with separation, at least qualitatively, although they fail to reproduce some observed features - in particular, a larger temperature is rather expected at apastron. In these models, the change in stellar parameters of Star A appears consistent with the recent hardening (through an increase of plasma temperature), but the mean X-ray flux is predicted to remain similar whatever the epoch under consideration. The long-term brightness change could however be related to a recent theoretical suggestion of increased strength of thin-shell instabilities in highly radiative shocks. If confirmed by dedicated modeling, this would once more underline the exceptional astrophysical interest of the HD\\,5980\\ system.\n\n\\acknowledgments\nYN and GR acknowledge support from the Fonds National de la Recherche Scientifique (Belgium), the Communaut\\'e Fran\\c caise de Belgique, the PRODEX {\\sc{XMM}}\\emph{-Newton}\\ and Integral contracts, and an ARC grant for concerted research actions financed by the French community of Belgium (Wallonia-Brussels Federation). GK acknowledges support from CONACYT grant 252499. D.J.H. acknowledges support from Grants HST-GO-13373.001-A and HST-GO-14476.002-A. ADS and CDS were used for preparing this document. \n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\n\n\n\\subsection{Object Detection}\n\\import{Sections\/RelatedWork\/}{ObjectDetection.tex}\n\n\\subsection{General Object Detection Datasets}\n\\import{Sections\/RelatedWork\/}{GeneralObjDetectDatasets.tex}\n\n\\subsection{Maritime Vessel Detection Datasets}\n\\import{Sections\/RelatedWork\/}{ShipDetectDatasets.tex}\n\n\n\\subsection{Dataset Diversity}\nMaritime environments are inherently intricate, hence a range of factors have to be accounted for when desinging a dataset. Dataset design must ensure that the dataset characterizes well vessels in the environment. Of course,  data augmentation methods can be considered for reproducing certain environmental conditions, however authentic conditions may be difficult to anticipate.  \n\n\\textit{Background variation.}\nParticular object detection tasks are more prone to to be affected by changes in the background of the picture. For instance, facial recognition is less susceptible to background variations, because given the similar shape of most faces, it is easier to fit them into bounding boxes in a congruous manner. However, the shapes of maritime vessels are highly heterogeneous, making them more difficult to separate from the background due to a potentially vast background information in the bounding box. The accuracy of ship detection would be significantly affected if background information were classified as ship features. \nFigure~\\ref{fig:label_backgroundvariation} illustrates the background variation of images in our datasets, including urban landscapes and an opensea environment. \n\n\\begin{figure}[!htb]\n  \\centering\n \n    \\centering\n    \\includegraphics[width=1\\linewidth]{Sections\/DataAcq\/BackgroundVariationCrop.png}\n    \\caption{Example image of background variations in the ABOships dataset: (a) View of maritime vessels on Aura river including the urban landscape; (b) View of maritime vessel in the Finnish Archipelago.}\n    \\label{fig:label_backgroundvariation}\n \n\\end{figure}\n\n\\textit{Atmospheric conditions.} \nAtmospheric conditions were specific to Finnish summers, with very sunny periods, alternating with rainy intervals and cloudy skies. The dataset includes a variety of images of different atmospheric conditions throughout a day. \n\n\n\\textit{Illumination.}\nLighting variations can significantly impact image capture. Illumination throughout the day, in various geographical areas and with specific daylight hours in a given region can dramatically influence image detection. \n\n\\textit{Visible proportion.}\nA great number of the images in our dataset consists of moving ships, with objects being only partially captured in the camera field of view. However, they still represent objects that were annotated since one has to detect them as well. The annotation should comprise different visible proportions of the maritime vessels.    \n\n\\textit{Occlusion.}\nDue to the fact that our dataset has been captured in an open sea environment, in the harbor area and also comprises urban landscapes, there are many occasions when maritime vessels occlude each other or occlude other objects in the environment in the harbor area or in the urban landscape. In a subset of pictures especially in the harbor area, there is significant occlusion due to a high number of maritime vessels in the proximity of each other. Two examples of occlusion are shown in Figure~\\ref{fig:label_occlusion}.\n\n\\begin{figure}[!htb]\n  \\centering\n \n    \\centering\n    \\includegraphics[width=1\\linewidth]{Sections\/DataAcq\/OcclusionCrop.png}\n    \\caption{Example image of occlusion: (a) Boat in front of a militaryship; (b) Several sailboats occluding each other while docked, on the right half of the image.}\n    \\label{fig:label_occlusion}\n \n\\end{figure}\n\n\n\n\n\\textit{Scale variation.} \nDetection of small objects can prove to be quite difficult, especially in a complex environment like the sea, ships that occupy a small pixel area in the picture can be confused with other objects in the background. Maintaining a high level of detection for ships requires including several scales for ships sizes in the dataset. For more information regarding the annotation and the size of the bounding boxes, please refer to Section \\ref{subsec:annotation}.\n\n\nFigure~\\ref{fig:label_perspective} illustrates a saiboat from two different perspectives: a lateral and a frontal view, which shows a variation in both occupied pixel area, but also the visible proportion. \n\n\\begin{figure}[!htb]\n  \\centering\n \n    \\centering\n    \\includegraphics[width=1\\linewidth]{Sections\/DataAcq\/PerspectiveCrop.png}\n    \\caption{Example image of a sailboat, view from two perspectives: (a) Lateral; (b) Frontal.}\n    \\label{fig:label_perspective}\n \n\\end{figure}\n\n\n\n\n\\subsection{Camera System}\nThe dataset was acquired from a set of 135 videos, collected from a sightseeing watercraft by a camera  with a field of view of $65^{\\circ}$ and stored in FullHD ($1920$x$720$) resolution at $15$ FPS in MPEG format. The route of the watercraft extended from the city of Turku to Ruissalo in South-West Finland, the videos comprising the urban area along the Aura river, the port and into the Finnish Archipelago, for a duration of 13 days (26.6.2018-8.7.2018). The watercraft ran each day in the timeframe between 10.15 and 16.45. \nThe videos were captured into 30-minute long periods from the route that the watercraft took. While the route remained largely the same, the data contains a variety of typical maritime scenarios in a range of weather conditions.\n\nIn addition to camera video data, the platform had a lidar attached to it (Sick LDMRS, FoV $110$ degrees, $2$x$4$ planes, up to $300m$ detection, at $5Hz$). The data from the lidar was captured alongside the video at a rate of $5$ entries of up to $800$ points per second. Our lidar had a detection range of up to $300$ meters, which is very useful for detecting other objects in the harbor environment. Due to having only $2$ times $4$ lasers in height diretion, the data is not reliable enough for discerning what object is detected. It does allow though for detecting distances to the objects seen in the video. For the purpose of creating the dataset presented in this paper, we used the lidar data to filter out video segments that were captured in the harbor area (usually the ones that had too many points for a prolonged period of time).\n \n\nTo evaluate the models, we acquired 9880 image photos from the videos. First, we annotated all images with 11 categories: seamarks, 9 types of maritime vessels, and miscellaneous floaters. \nIn a second round, we relabelled all the inconsistencies we found, using an algorithm based on the CSRT tracker~\\cite{lukezic2017discriminative}.\n\n\\subsection{Dataset Design}\\label{subsec:design}\nThe raw data acquired from the camera on the sightseeing watercraft is captured at a resolution of $720p$, as $15$ FPS MPEG videos. The videos include some footage exhibiting content that is irrelevant for the scope of vessel detection (especially footage captured when the vessel was docked, either at the start of its route on the Aura river or at the Port of Turku) or sensitive content, such as faces of people. To address the latter issue, we performed face detection on all videos and blurred all detected faces. Addressing the former issue on the other hand, required additional data from the lidar.\n\nIn a maritime environment, lidar data is relatively sparse, authors of this study observed a high number of points detected for a prolonged duration correlates with the vessel being docked in the harbor. By setting a point threshold to detect these docked-harbor cases, we were able to filter them out in their majority and extract only the images from maritime environment. The images were extracted at an interval of 15 seconds (one image every 225 frames) and still contained some images captured during docking, but most of them were facing outwards from harbor, so the images captured that way still contain useful maritime data. As a result we acquired 9880 images of the maritime environment.\n\nThe acquired images were subsequently separated into work packages in such a way that chronologically adjacent pictures were separated into different work packages, that were then manually labelled by different people. After the initial labelling was completed, we used the CSRT tracker~\\cite{lukezic2017discriminative} to combine labels of the same object into traces - a collection of chronologically adjacent images containing a bounding-box for that object. Due to inaccuracies in the tracking process and discrepancies in labelling, the produced traces were not always accurate. After viewing the labels in these traces, we identified  the main causes for discrepancies in labelling, which were predominantly caused by different interpretations of label annotations. We refined those annotations to eliminate the discrepancies and separated the data into a second collection of work packages that were provided to annotators, who then relabelled the data, according to refined annotations. \nAfter the relabelling was completed, images were again separated into batches of 20 chronologically subsequent images. \n\n\n\n\n\n\n\\subsection{Relabelling Algorithm}\nThe labelling was performed by multiple people with different backgrounds, hence some label types were interpreted differently among them. To increase the consistency of labelling, we used the continuous nature of the raw data by tracking the labels between frames using the CSRT tracker \\cite{lukezic2017discriminative}. \n\nFor every labelled frame, we created a tracker instance. The aim was to track the object until the next labelled frame. At that point, the existing traces would be mapped onto the labels on the new frame, based on the IoU metric. During this mapping, we assumed that no labeller would confuse seamarks with vessels, so we did not map ship labels onto seamarks or vice-versa. More importantly, we did not take into account previous labels, so even if annotators labelled the same object differently in different frames, these labels would still belong to the same trace as long as the tracker could identify them. For cases where the mapping could not be found, we assigned the trace a new label - \"Unlabeled\" to denote that even though nothing was labeled in that specific case, the tracker indicated that the tracked object should belong to the trace. A visual depiction of this process is illustrated in Figure~\\ref{fig:labeling_process}.\n\n\\begin{figure}[!htb]\n  \\centering\n \n    \\centering\n    \\includegraphics[width=1\\linewidth]{Sections\/DatasetDesign\/LabelingWorkflowNew.png}\n    \\caption{Videos were separated into 48 workpackages of images (1), which were labelled (2). Using the OpenCV Tracker, the objects were tracked across frames to produce traces (3) and then relabelled to fix inconsistensies and fill in the labels that might have been skipped (4). The resulting labels were then compiled into the dataset (5).}\n    \\label{fig:labeling_process}\n \n\\end{figure}\n\nAfter a certain number of frames, either the tracker would lose the object (the most common reasons for this being object occlusion, or due to the object being either too far or going out of the frame altogether) or the tracker would have no real label mapped to it enough times (which would mean it most likely drifted onto another object or the object got too far). In both of those cases, the tracker was stopped and the resulting trace  was saved to a file for further processing.\n\nTo reduce the number of errors caused by occlusion and the tracker drifting towards other objects, we performed a second tracking in the backwards direction. By comparing labels identified in the traces we acquired from tracking videos in different directions, we could detect situations where traces could not be mapped onto each other. Those cases signify that the tracker was either occluded or drifted to another object, so we had to split traces into smaller sequences until no more conflicts could be detected.\n\nThe resulting traces (after the backwards tracking) were provided as batches for relabelling. Traces containing a single entry were batched together with other singular traces from the same category. This setup was done with the purpose of preventing and removing accidental labels (mislabeling), while at the same time providing more information about the objects being labelled. This allowed us to accurately label even the objects at a longer distance with the help of tracking history. Traces obtained in this way were then provided for relabelling as a collection of labels belonging to the same trace and participants were asked to refine the labels so that labels would correspond. Singular entries that did not have a trace were batched together with other objects of the same category. The process described above is illustrated in Figure~\\ref{fig:traces}.\n\n\\begin{figure}[!htb]\n  \\centering\n \n    \\centering\n    \\includegraphics[width=1\\linewidth]{Sections\/DatasetDesign\/RelabellingTracesVertical.png}\n    \\caption{During the relabelling participants were given a program that would show them traces of tracked images between annotation frames (1). They were then asked to either relabel everything by selecting the correct label from the right panel, or go inside an annotation (by selecting a label that stood out (2) ) and change the label of each image individually and possibly fix the bounding box to fit the object more tightly (3). Sometimes tracking would drift to another object, in which case that particular entry of the trace might have a different label from the rest (4). When the participant completed all the labels (5), the changes would be saved into a new file and the participant was provided with the next trace.}\n    \\label{fig:traces}\n \n\\end{figure}\n\n\\subsection{Dataset Statistics}\nTable~\\ref{tab:dataset_stat} shows the number of images of each category in our dataset and the number of annotations. The column Images represents the number of images that contain that particular object class and then the percentage of images that comprise that class follows. Then, the column Objects represents the number of annotations for that particular class in the dataset, along with the percentage of objects annotated for that specific class out of all the annotated objects in the dataset.   \n\nMoreover, Figure~\\ref{stat_boxplots} illustrates the distribution of annotated objects in our dataset based on occupied pixel area. \n\n\\begin{figure}\n\\centering\n    \\subfloat[]{\\label{small_obj}\\includegraphics[width=100mm,scale=1]{Sections\/DatasetDesign\/Occupied_Area_Small_944x640.png}} \\\\\n   \\subfloat[]{\\label{med_obj}\\includegraphics[width=100mm,scale=1]{Sections\/DatasetDesign\/Occupied_Area_Med_944x640.png}}\\\\\n    \\subfloat[]{\\label{big_obj}\\includegraphics[width=100mm,scale=1]{Sections\/DatasetDesign\/Occupied_Area_Big_944x640.png}}\n    \\caption{Occupied pixel area of annotated objects, divided in three categories: small: Fig.~\\ref{small_obj}, medium: Fig.~\\ref{med_obj} and big: Fig.~\\ref{big_obj}.}\n    \\label{stat_boxplots}\n\\end{figure}\n\n\n\\begin{table}[htb]\n    \\centering\n    \\begin{tabular}{ |p{2cm}|p{2cm}|p{2cm}|p{2cm}|p{2cm}| }\n\n\\hline\n\\multicolumn{5}{|c|}{Number of images and annotations for every object category} \\\\\n\\hline\nClass & Images & Percentage & Objects & Percentage\\\\\n\\hline\nSeamark       & 3744 & 37.89\\%  & 7670 & 18.27\\%  \\\\\nBoat          & 2034 & 20.58\\%  & 2913 &  6.94\\%  \\\\\nSailboat      & 3842 & 38.88\\%  & 8147 & 19.41\\%  \\\\\nMotorboat     & 4062 & 41.11\\%  & 7092 & 16.89\\%  \\\\\nPassengership & 2639 & 26.71\\%  & 4464 & 10.63\\%  \\\\\nCargoship     &  157 & 1.58\\%   &  161 &  0.38\\%  \\\\\nFerry         &  945 & 9.56\\%   & 1046 & 2.49\\%  \\\\\nMiscboat      & 2797 & 28.30\\%  & 4642 & 11.06\\%  \\\\\nMiscellaneous &  129 & 1.30\\%   & 200  &  0.47\\%  \\\\\nMilitaryship  & 2559 & 25.90\\%  & 4128 &  9.83\\% \\\\\nCruiseship    & 1347 & 13.63\\%  & 1504 &  3.58\\% \\\\\n\\hline\n\\end{tabular}\\bigskip\n    \\caption{The table shows the number of images and annotations in the ABOships dataset for every object category, along with their overall percentages.}\n    \\label{tab:dataset_stat}\n\\end{table}\n\n\n\\subsection{Annotation}\\label{subsec:annotation}\n\nTo perform the annotation task, we first investigated the captured videos and identified the vessel types that appeared most often. Due to the fact that the videos were captured at locations with a significant number of passenger ships, there is a certain level of bias for labellers towards those types of ships. This is different from the Seaships database, for instance, which comprises a higher variety of cargo ships. For the purposes of future use in machine vision, rather than using maritime terminology as such, (depicting ship scale and purpose), we selected labels that had some clearly distinct visual characteristics.\n\nThe label categories are discussed below, with more specific details for every category:\n\\begin{itemize}\n    \\item seamark (red\/blue\/black\/yellow cone-shaped metal\/plastic floater or pipe emerging from the sea);\n    \\item motorboat - primarily a speedboat, visual distinction - sleek, aerodynamic features;\n    \\item sailboat - sails-propelled boat or a boat which exhibits sails, visual distinction - sails;\n    \\item passengership, medium size ship, used to transport people on short distances, ex. restaurant boat,  visual distinction - usually it has multiple noticeable lateral windows;\n    \\item cargoship, large-scale ships used for cargo transportation, visual distinction - long ship with cargo containers or designed with container carrying capacity;\n    \\item military ship, an official ship that is either military or coast guard and includes a special hull with antennas. The coast guard usually reads \"Coast guard\" and the military ones are dark gray\/metallic\/black\/brown in colour;\n    \\item ferry, medium-sized ship, used to transport people and cars, aka waterbus\/watertaxi, another appropriate term would be cableferry, visual distinction - it includes entrances on two opposite sides and a cabin in the middle;\n    \\item miscboat - miscellaneous maritime vessel, visual distinction - generic boat that does not include any visual distinction mentioned above in the other ship categories above;\n    \\item miscellaneous - identified floaters(birds, other objects floating in the water) unidentified\/unidentifiable floaters.\n\\end{itemize}\n\n\\begin{figure}[!htb]\n  \\centering\n \n    \\centering\n    \\includegraphics[width=0.7\\linewidth]{Sections\/DatasetDesign\/LabelsCrop.png}\n    \\caption{Example images of annotated objects in the ABOships dataset: (a) boat, (b) cargoship, (c) cruiseship, (d) ferry, (e) militaryship, (f) miscboat, (g) miscellaneous (floater), (h) motorboat, (i) passengership, (j) sailboat and (k) seamark. }\n    \\label{fig:label_def}\n \n\\end{figure}\n\n\n\n\\section{Introduction}\nMaritime vessel detection from waterborne images is a crucial aspect in various fields involving maritime traffic supervision and management, marine surveillance and navigation safety. Prevailing ship detection techniques exploit either remote sensing images or radar images, which can hinder the performance of real-time applications \\cite{shao2019saliency}. Satellites can provide \\textit{near-real} time information, but satellite image acquisition   however can be unpredictable, since it is challenging to determine which satellite sensors can provide the relevant imagery in a narrow collection window \\cite{liu2014remote}. Hence, seaborne visual imagery can tremendously help in essential tasks both in civilian and military applications, since it can be collected in real-time from surveillance videos, for instance. \n\nShip detection in a traditional setting depends extensively on human monitoring, which is highly expensive and unproductive. Moreover, the complexity of the maritime environment makes it difficult for humans to focus on video footage for prolonged periods of time \\cite{shao2018seaships}.\nMachine vision, however, can take the strain from human resources and provide solutions for ship detection. Traditional methods based on feature extraction and image classification, involving background subtraction and foreground detection, as well as directional gradient histograms, are highly affected by datasets exhibiting challenging environmental factors (glare, fog, clouds, high waves, rain etc.), background noise or lighting conditions. \n\nConvolutional neural networks (CNNs) contributed massively to the image classification and object detection tasks in the past years~\\cite{deng2009imagenet,druzhkov2016survey,zhang2019recent,wu2020recent,liu2020deep}. They incorporate feature extractors and classifiers in multilayer architectures, whose number of layers regulate the selectiveness and feature invariance.  CNNs exploit convolutional and pooling layers extracting local features, and gradually advancing object representation from simple features to complex structures, across multiple layers. CNN-based detectors can subtract compelling distinguishable features automatically unlike more traditional methods which use predefined features, manually selected. However, integrating ship features into detection proves to be challenging even in this context, given the complexity of environmental factors, object occlusion, ship size variation, occupied pixel area etc. This often leads to unsatisfactory performance of detectors on ship datasets. \n\nTo address ship detection in a range of operating scenarios, including various atmospheric conditions, background variations and illumination, we introduce a new dataset consisting of 9880 images, and annotations comprising $41967$ carefully annotated objects. \n\nThe paper is organized as follows. Section~\\ref{sec:related} describes related work, including notable results in  vessel detection and maritime datasets comprising waterborne images.  Section~\\ref{sec:dataacq} describes the data acquisition and dataset diversity. In Section~\\ref{sec:design},  we discuss dataset design and our relabelling algorithm along with basic dataset statistics based on the final annotation data. In Section~\\ref{sec:basline}, we discuss evaluation criteria and present experimental results; we investigate 4 CNN-based detectors and discuss the feature extractors and object size effect on the performance of the detectors. Conclusions are presented in Section~\\ref{sec:conclusion}.\n\n\\section{Related Work}\\label{sec:related}\n\\import{Sections\/RelatedWork\/}{RelatedWork.tex}\n\n\\section{Data Acquisition}\\label{sec:dataacq}\n\\import{Sections\/DataAcq\/}{DataAcquisition.tex}\n\n\\section{Dataset Design}\\label{sec:design}\n\\import{Sections\/DatasetDesign\/}{DatasetDesign.tex}\n\n\n\\section{Experimental Results}\\label{sec:basline}\n\\subsection{Evaluation Criteria}\nTo evaluate the performance of different object detection algorithms on specific datasets, one can emply various quantitative indicators. One of the most popular measures in object detection is the \\textit{IoU} (\\textit{Intersection of Union}), which  defines the extent of overlap of two bounding boxes as the intersection between the area of the predicted bounding-box $B_p$ and the area ground truth bounding-box $B_{gt}$, over their union~\\cite{padilla2020survey}:\n\\begin{equation}\nIoU = \\frac{|B_p \\cap B_{gt}|}{|B_p \\cup B_{gt}|}\n\\end{equation}\n\nGiven an overlap threshold $t$, one can estimate whether a predicted bounding-box belongs to the background ($IoU<threshold$) or to the given classification system ($IoU>threshold$).\nWith this measure, one can proceed to assess the \\textit{average precision} ($AP$) by calculating the precision and recall. The precision reflects the capability of a given detector to identify relevant objects and it is calculated as the proportion of detected bounding-boxes, correctly identified, over the total number of detected boxes. The recall reflects the capability of a detector to identify relevant cases and it is calculated as the proportion of correct positive predictions to all ground-truth bounding-boxes. Based on these two metrics one can draw a precision-recall curve, which encloses an area representing the average precision. However, in a majority of cases, this curve is highly irregular (zigzag pattern) making it challenging to estimate the area under it, i.e. the $AP$. To address this, one can approach it as an interpolation problem, either as an 11-point interpolation or an all-point interpolation~\\cite{padilla2020survey}. \n\nThe 11-point interpolation averages the maximum values of precision over 11 recall levels that are uniformly distributed~\\cite{padilla2020survey}, as depicted below:  \n\\begin{equation}\nAP_{11} = \\sum_{R \\in \\{0,0.1,...,0.9,1\\}} P_i(R),\n\\end{equation}\nwith \n\n\\begin{equation}\nP_i(R) = \\max_{R^{*}| R^{*} \\geq R} P_i(R^{*}).\n\\end{equation} \n\n$AP_{11}$ is calculated using the maximum precision $P_i(R)$, with a recall greater than $R$. \n\n\\subsection{Baseline Detection}\nTo explore the performance of CNN-based object detectors on our dataset, we focused on prevalent detectors: one-stage(SSD~\\cite{SSD} and EfficientDet~\\cite{tan2020efficientdet}) and two-stage detectors (Faster R-CNN~\\cite{ren2015faster} and R-FCN~\\cite{R-FCN}). The detectors were previously\ntrained on the Microsoft COCO object detection dataset, which comprises a number of 91 object categories. The training dataset contains a number of $3,146$ images of marine vessels.\nWe investigated the performance of different feature extractors in the aforementioned detectors. We collect maritime vessel detection results based on SSD over different feature extractors (ResNet101, MobileNet v1, MobileNet v2). Moreover, we evaluate the performance of a new state-of-the-art detector, EfficientDet,  on our dataset, which used SSD-EfficientNet as feature extractor. We also evaluated two-stage detectors: Faster R-CNN and RFCN with different feature extractors. Combining all proposed detectors with the feature extractors, a total of 8 algorithm were investigated. All information regarding the specific configuration of these detectors can be found at~\\cite{tensorflowAPI}. \n\nWe estimated the performance of these algorithms in detecting maritime vessels, so we excluded seamark labels from our experiments and focused on detecting vessels. Moreover, we chose images with an occupied pixel area larger than $16^2$ pixels. We attained Table~\\ref{table:results}.\n\nOur experiments indicated that the object size impacts the detection accuracy. To corroborate this observation we divided all vessel labels (with an occupied pixel area larger than $16^2$ pixels) in our datasets into three categories, based on Microsoft COCO challenge's variants: small ($16^2$<area<$32^2$), medium ($32^2$<area<$96^2$) and large (area>$96^2$). Out of the annotated vessels with an occupied pixel area larger than $16^2$ pixels in our dataset, $30.25\\%$ of the annotated vessels are small, $49.37\\%$ are medium and $20.37\\%$ are large.  \n\nAnalyzing the results from our experiments, we observe that detection accuracy decreases with object size. The $AP$ for best-performing detector on the dataset (Faster R-CNN with Inception ResNet v2 as feature extractor) with a registered $AP$ of $35.18\\%$ more than doubles in size from small ($AP_S = 23.16\\%$) to large objects ($AP_L = 46.84\\%$). The second best detector on the whole dataset (EfficientDet with EfficientNet as feature extractor) however had the best performance on in the large-objects category, with an $AP_L = 55.48\\%$. In general, detecting small objects turns out to be more difficult than larger object given that there is less information associated with a smaller occupied pixel area. For medium-sized objects, the best performance is attained by SSD with ResNet101 as feature extractor ($AP_M = 31.18\\%$). For small objects, the best-performing detector, Faster R-CNN with Inception ResNet v2, outperforms the other detectors with a registered $AP_S = 23.16\\%$. Among the SSD configurations, best performing, in general, was the one having ResNet101 as feature extractor.  \n\n\n\\begin{table}[htb]\n\\centering\n\\begin{tabular}{ |p{2cm}|p{3cm}|p{1.5cm}|p{1.5cm}|p{1.5cm}| p{1.5cm}|}\n\\hline\n\\multicolumn{6}{|c|}{Detection performance of different  detectors on the ABOships dataset } \\\\\n\\hline\n\\textbf{Method} & \\textbf{Feature extractor} & $AP_S$ & $AP_M$ & $AP_L$  & $AP$\\\\\n\\hline\n\\multirow{3}{*}{Faster RCNN} &  Inception ResNet V2 & 23.16 & 30.86 & 46.84 & \\textbf{35.18} \\\\\n\\cline{2-6}\n& ResNet50 V1 & 9.76 & 20.94 & 41.65 & 26.49 \\\\\n\\cline{2-6}\n& ResNet101 & 18.42 & 25.07 & 38.17 & 30.26 \\\\\n\\hline\n\\multirow{3}{*}{SSD} &  ResNet101 V1 FPN & 21.39 & 31.18 & 42.07 & 30.03 \\\\\n\\cline{2-6}\n& MobileNet V1 FPN & 12.34 & 27.61 & 37.83  & 28.59\\\\\n\\cline{2-6}\n& MobileNet V2 & 3.01 & 17.05 & 27.37 & 17.48 \\\\\n\\hline\nEfficientDet & Efficient Net D1 & 10.94 & 29.68 & \\textbf{55.48} & 33.83\\\\\n\\hline\nRFCN & ResNet101 & 18.05 & 26.20 & 41.61 & 32.46 \\\\\n\\hline\n\\end{tabular}\\bigskip\n\\caption{Average Precision (AP) (in \\%) of the proposed CNN-based detectors on ABOships dataset, with different feature\nextractors and object sizes, for all objects with an occupied pixel area > $16^2$ pixels.}\n\\end{table}\\label{table:results}\n\n\\paragraph{\\textbf{Qualitative results}}\nFig.~\\ref{fig:qualitative} illustrates an example of detection results for the proposed methods, selecting for each the combination of feature extractor that scored the highest AP in each category. We can notice in Figure~\\ref{fig:qualitative} that Faster R-CNN with a Inception-ResNet-v2 feature extractor (a) and R-FCN with a ResNet101 feature extractor (c) provide detected regions registering high scores ranging from $0.93$ to $0.99$. The other two detectors in Figure~\\ref{fig:qualitative}, EfficientDet with EfficientNet as feature extractor (b) and SSD with ResNet101 as feature extractor (d), register satisfying results registering with scores ranging from $0.55$ to $0.67$. \n\n\\begin{figure}[!htb]\n  \\centering\n \n    \\centering\n    \\includegraphics[width=1\\linewidth]{Sections\/DatasetDesign\/ExperimentalResults.png}\n    \\caption{Qualitative detection results for the ABOships dataset on (a) Faster R-CNN and Inception-ResNet-v2 as feature extractor, (b) EfficientDet with EfficientNet as feature extractor, (c) R-FCN with ResNet101 as feature extractor, and (d) SSD with ResNet101 as feature extractor. The ground truth bounding boxes are shown as red rectangles. Predicted boxes by these methods are depicted as green bounding boxes. Each output box is associated with a class label and a score with a value in the interval [0, 1].}\n    \\label{fig:qualitative}\n \n\\end{figure}\n\\newpage\n\\section{Conclusion}\\label{sec:conclusion}\nThis paper provides a solution for addressing the annotation inconsistencies appeared as a consequence of manual labeling of images, using the CSRT tracker~\\cite{lukezic2017discriminative}. We build traces of the images in the videos they originated from and use the CSRT tracker to traverse these videos in both directions and identify the possible inconsistencies. After this step, we employed a second round of labeling and obtained a set of $41967$ carefully annotated objects, of which $9$ types of maritime vessels (boat, miscboat, cargoship, passengership, militaryship, motorboat, ferry, cruiseship, sailboat) miscellaneous floaters and seamarks. \n\nWe ensured the dataset consists of images taking into account the following factors: background variation, atmospheric conditions, illumination, visible proportion, occlusion and scale variation. We performed a comparison of the out-of-the-box performances of four state-of-the-art CNN-based detectors (Faster R-CNN~\\cite{ren2015faster}, R-FCN~\\cite{R-FCN}, SSD~\\cite{SSD} and EfficientDet~\\cite{tan2020efficientdet}. These detectors were previously trained on the Microsoft COCO dataset. We assess the performance of these detectors based on feature extractor and object size. Our experiments show that Faster R-CNN with Inception-Resnet v2 outperforms the other algorithms for objects with an occupied pixel area $>16^2$ pixels, except in the large object category where EfficientDet register the best performance with $AP = 55.48\\%$. \n\n\nFor future research, we plan to investigate different types of errors in the manual labelling, for cases where the labels still have inconsistencies, such as: fine-grained recognition (which renders it more difficult for human even to detect  objects even when they are in plain view~\\cite{russakovsky2015imagenet}, class unawareness (some annotators become unaware of certain classes as ground truth options) and insufficient training data (not enough training data for the annotators). \n\nMoreover, we plan to investigate in more detail the detection of small and very small objects, including those with an occupied pixel area $<16^2$ pixels. Also, distinguishing between different vessel types in our datasets will be an essential focus as the next steps in our experiments. In order to do this, we plan to exploit transfer learning both in the form of heterogeneous transfer learning, but also homogeneous domain adaptation. \n\nTo further our research, we will employ maritime vessel tracking detectors on the original\nvideos captured in the Finnish Archipelago and examine the impact on autonomous navigation and navigational safety.\n\n\\section{Acknowledgments}\nThe annotation of the ABOships dataset was completed with the help of the following persons: Sabina B\\\"ack, Imran Shahid, Joel Sj\\\"oberg and Alina Torbunova.  \n\n\\bibliographystyle{unsrt}  \n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{The group of autoequivalences}\nGabriel originally also characterized the autoequivalences of the category of coherent sheaves.\nIndeed, $\\Aut(\\Coh(X)) = \\Aut(X) \\ltimes \\Pic(X)$, where to form the semi-direct product we use the standard (left) action of automorphisms on line bundles: $(f,L) \\mapsto f_* L = (f^{-1})^*L$.\nWe want to now describe what happens when passing to the quotients.\n\nFor this section, assume $\\mathbf{k}$ is a finite type algebra over a field.\nLet $\\Aut_{\\geq k}(X)$ be the group of birational self-maps $X \\dashrightarrow X$ which are defined on an open subset containing $X_{\\geq k}$, modulo the obvious equivalence relation: $f \\sim g$ if the two are equal on an open subset containing all points of dimension at least $k$.\nRecall the definition of $\\Pic_{\\geq k}(X)$ from \\ref{picardo}.\n\n\\begin{thrm}\\label{autoteorema}\nThere is an isomorphism of groups \n$$ \\Aut(\\mathcal C_k(X)) \\simeq \\Aut_{\\geq k}(X)\\ltimes \\Pic_{\\geq k}(X). $$\nWhere to form the semi-direct product the action is given by $(f,L) \\mapsto (f^{-1})^*L$.\n\\end{thrm}\nLet $G = \\Aut_{\\geq k}(X)\\ltimes \\Pic_{\\geq k}(X)$.\nThere is an obvious group homomorphism $G \\to \\Aut(\\mathcats C_k)$, sending the pair $(f,L)$ to the auto-equivalence $F \\mapsto (f^{-1})^* F \\otimes L$.\nConversely, an auto-equivalence $\\Phi$ induces an automorphism of the locally ringed space $X_{\\geq k}$, and hence a birational self-map $f$, which is uniquely defined up to a subset of dimension at most $k-1$ thanks to lemma \\ref{uniqiso}. \nOne checks that the assignment $\\Phi \\mapsto (f,\\Phi(\\mathcal O_X))$ is also a group homomorphism.\nClearly, the composition $G \\to \\Aut(\\mathcats C_k(X)) \\to G$ is the identity.\nWe thus have a (split) short exact sequence\n\\begin{align*}\n\t1 \\to Q \\to \\Aut(\\mathcats C_k(X)) \\to \\Aut_{\\geq k}(X)\\ltimes \\Pic_{\\geq k}(X) \\to 1.\n\\end{align*}\nProving the theorem amounts to showing that the kernel $Q$ is trivial.\nTo this end, suppose we have an auto-equivalence $\\Phi$ such that the induced map on $X_{\\geq k}$ is the identity, and $\\Phi(\\mathcal O_X) = \\mathcal O_X$.\nWe will construct a natural isomorphism $\\tau\\colon \\Phi \\to \\Id_{\\mathcats C_k(X)}$.\n\n\\begin{lem}\\label{eximodul}\nlet $M,M'$ be coherent sheaves on $X$ and assume that $M\\otimes \\mathcal O_{X,P}=M_P = M'_P$ for a point $P \\in X$. Then there exists an open neighborhood $U$ of $P$ and an isomorphism of coherent sheaves $M_U \\simeq M'_U$.\n\\end{lem}\n\\begin{proof}\nThe same argument we used for Lemma \\ref{lociso} works here.\n\\end{proof}\n\n\\begin{lem}\\label{uniqmodul}\nlet $M,M'$ be coherent sheaves on $X$ and let $f,g\\colon M \\rightarrow M'$ be two morphisms.\nIf the localizations $f_x, g_x\\colon M_x \\rightarrow M'_x$ are equal for all points $x \\in X_{\\geq k}$, then there is an open subset $X_{\\geq k} \\subset U$ such that the restrictions $f|_U, g|_U$ are equal.\n\\end{lem}\n\\begin{proof}\nThe same argument as in Lemma \\ref{uniqiso} shows that given each such $x$ there is an open neighborhood containing $x$ where the maps are equal.\nWe can then conclude by the fact that being $0$ is an open property.\n\\end{proof}\n\n\\begin{proof}[Proof of Theorem \\ref{autoteorema}]\nWe wish to construct a natural isomorphism $\\tau\\colon \\Phi \\to \\Id_{\\mathcats C_k(X)}$, as explained earlier.\nTo start, assume that $X$ is affine.\nLet $M$ be a coherent sheaf on $X$.\nThen $$\\Hom_{\\mathcal C_k(X)}(\\mathcal O_X, M)=\\Hom_{\\mathcal C_k(X)}(\\Phi(\\mathcal O_X),\\Phi(M))=\\Hom_{\\mathcal C_k(X)}(\\mathcal O_X,\\Phi(M)).$$\n\nFix a representative $M' \\in \\Coh(X)$ of $\\Phi(M)$, and let $m'_1,\\ldots,m'_r$ generators for $M'$. To each of these corresponds a map $s'_i\\colon \\mathcal O_X \\rightarrow M'$. Let $s_1, \\ldots, s_r$ be the inverse images of these maps in $\\Hom_{\\mathcal C_k(X)}(\\mathcal O_X, M)$. By Lemma \\ref{hom} we can pick morphisms $S_1, \\ldots, S_r \\in \\Hom_U(\\mathcal O_U,M_U)$ where $U$ is an open subset of $X$ whose complement has dimension at most $k-1$, which in turn give us global sections $m_1, \\ldots, m_r$ of $M_U$. \n\nNote that the assignment $m'_i \\mapsto m_i$ induces an isomorphism between the local modules $M'_P$ and $M_P$ for all points $P$ of dimension $k$.\nBy Lemma \\ref{eximodul}, up to restricting $U$ by removing a subset of dimension at most $k-1$, we can assume that the sheaf of relations between $m'_1, \\ldots, m'_r$ is isomorphic to the sheaf of relations between $m_1, \\ldots, m_r$, and thus the assignment $m'_i \\mapsto m_i$ induces a morphism $M'_U \\rightarrow M_U$ (note that this works because every open subset is sent to itself by the induced morphism on $X_{\\geq k}$).\n\nNow fix an affine covering $U_1, \\ldots, U_s$ of $U$.\nThe assignment $m'_1, \\ldots, m'_r \\mapsto m_1, \\ldots, m_r$ induces a morphism $M'_{U_i} \\rightarrow M_{U_i}$.\nThanks to Lemma \\ref{uniqmodul}, up to restricting $U$ further, we can assume that these maps glue to a morphism $M'_U \\rightarrow M_U$.\nNote now that $m_1, \\ldots, m_r$ are generators of $M$ on an appropriate open subset due to the exact sequence $$\\mathcal O_X^r \\xrightarrow{\\oplus s'_i} M' \\rightarrow 0$$ being sent to $$\\mathcal O_X^r \\xrightarrow{\\oplus s_i} M \\rightarrow 0$$ by $\\psi^{-1}$. So doing the same process we can construct a map $M_V \\rightarrow M'_V$ for some open subset $V$ of $X$ whose complement has dimension at most $k-1$.\nOn $U \\cap V$ these two maps are inverse to each other, so we have constructed an isomorphism $\\tau_M\\colon M \\simeq M'$ in $\\mathcal C_k(X)$.\n\nNote now that on local rings this isomorphism is uniquely determined by the isomorphism $\\Hom_{\\mathcal C_P}(\\mathcal O_X,M) \\simeq \\Hom_{\\mathcal C_P}(\\mathcal O_X,M')$, for each minimal object $P$.\nUsing Lemmas \\ref{uniqmodul}, \\ref{hom} we conclude that the isomorphism $\\tau_M$ is unique and functorial.\nWe have thus proven the claim for affine schemes.\nNow note that by taking an affine covering $X_i$ of a general scheme $X$ of finite type over $\\mathbf{k}$, the maps $\\tau_{M,i}$ we construct as above will glue (up to removing closed subsets of dimension at most $k-1$), giving rise to the sought after natural isomorphism $\\tau$.\n\\end{proof}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Birational skeleton}\n\nAs we are working with irreducible schemes, we can reindex our quotient categories by codimension, by $\\mathcal C_{i}=\\mathcal C^{n-i-1}$.\n\n\\begin{defn}\nWe define the topological space $^cX$ as follows. Consider the set \n\\begin{align*}\n\t\\bigcup_{d \\leq c}\\lbrace P \\in \\mathcal C^{d}(X) \\mid P \\mbox{ is minimal}\\rbrace\n\\end{align*}\nThe closure $\\bar{P}$ of a point $P$ coming from $\\mathcal C^{d}(X)$ is defined by $\\bar{P}=\\cup_{d < j \\leq c}\\lbrace Q \\in \\mathcal C^{ j}(X) \\mid Q \\in S(P) \\rbrace$.\nThese subsets form a basis of closed subsets for $^cX$.\n\\end{defn}\n\n\\todo{da riscrivere con le referenze giuste} Propositions \\ref{min1}, \\ref{min2} give us an injection $^cX \\xrightarrow{i} |X|$.\nIt is immediate to verify that $i$ induces a bijection between $^cX$ and the subspace of $|X|$ given by points of codimension at most $c$.\n\nThe following is immediate.\n\\begin{prop}\nAn equivalence $\\mathcal C^{ c}(Y) \\simeq \\mathcal C^{ c}(X)$ induces a homeomorphism $^cX \\xrightarrow{\\phi} {^cY}$.\n\\end{prop}\n\n\\todo{opzione 1(fancy)}\n\nWe give $^cX$ a structure of ringed space in the following way. Let $S_U$ be the serre subcategory of $C^{c}$ generated by the elements $P \\mid P \\notin U$. Then we define $\\mathcal O_{^cX}(U)=\\hom_{\\mathcal C^c\/S_U}(\\mathcal O_X,\\mathcal O_X)$. Clearly the isomorphism $^cX \\xrightarrow{\\phi} ^cY$ in the previous proposition extends to this structure.\n\n\\begin{rmk}\nA map $F \\rightarrow G$ in $\\mathcal C^c\/S_U$ can be represented as a roof $F \\xleftarrow{s} E \\xrightarrow{s'} G$ where $\\ker s, \\coker s, \\coker s'$ are supported on the complement of $U$.\n\\end{rmk}\n\n\\begin{lem}\nWe have \n$$\\Hom_{\\mathcal C^c\/S_U}(F,G)= \\varinjlim_{U \\subseteq W} \\Hom_{\\Coh(W)}(F_W,G_W)$$\nwhere $W$ runs over Zariski open subsets of $X$.\n\\end{lem}\n\\begin{proof}\nWe will define mutually inverse maps $\\hom_{\\mathcal C^c\/S_U}(F,G)\\leftrightarrow \\lim_{U \\subseteq W}\\hom_{\\Coh(W)}(F_W,G_W)$, following \\cite[3.6]{MP}.\n\nLet $W$ be an open subset containing $U$, and $f:F \\rightarrow G$ a morphism. Consider a coherent sheaf $E$ such that $E_W = \\Gamma_f \\subset F \\oplus G$. Then the roof $F \\xleftarrow{\\mathrm{pr}_1} E \\xrightarrow{\\mathrm{pr}_2} G$ represents an element $f \\in \\hom_{\\mathcal C^c\/S_U}(F,G)$. One can check that this element is unique exactly as in \\emph{loc.cit.}\n\nConsider now a roof $F \\xleftarrow{s} E \\xrightarrow{s'} G$. Let $W$ be the complement of $\\mathrm{supp}(\\ker(s)) \\cup \\mathrm{supp}(\\coker(s))$. We associate to it the map $s^{-1}\\cdot s':F \\rightarrow G$. Again we can check that this element is well defined and that the two maps are inverse to each other as in \\emph{loc. cit.}\n\\end{proof}\n\\todo{bisogna scrivere una dimostrazione completa}\n\n\n\\begin{cor}\nConsider the locally ringed space $(^cX,\\mathcal O_X)$ obtained by restricting the canonical sheaf and Zariski topology to the subset $^cX$. Then we have $$(^cX,\\mathcal O_X) \\simeq (^cX, \\mathcal O_{^cX}).$$\n\\end{cor}\n\\begin{proof}\nWe have \n\t$$ \\mathcal O_{^cX}(U)=\\hom_{\\mathcal C^c\/S_U}(\\mathcal O_X,\\mathcal O_X)= \\varinjlim_{U \\subseteq W} \\hom_{\\Coh(W)}(\\mathcal O_W, \\mathcal O_W) $$\n\tand additionally $\\hom_{\\Coh(W)}(\\mathcal O_W, \\mathcal O_W)=\\mathcal O_W(W)$ which allows us to conclude immediately.\n\n\\end{proof}\n\n\\begin{lem}\\label{lociso}\nLet $R, R'$ be finitely generated and integral $A$-algebras and let $p\\subset R,p' \\subset R'$ be prime ideals. Suppose there is an isomorphism of $A$-algebras $R_p \\simeq R'_{p'}$. Then there are open subschemes $U \\subset \\Spec(R), U' \\subset \\Spec(R')$ containing $p$ and $p'$ respectively such that $U \\simeq U'$ as $\\Spec(A)$-schemes.\n\\end{lem}\n\\begin{proof}\nConsider the composition $R' \\rightarrow R'_{p'} \\rightarrow R_p$. Consider a finite set of generators $x_1, \\ldots, x_n$ for $R$ as an $A$-algebra. Their images will be in the form $r_1\/s_1, \\ldots , r_n\/s_n$ where $r_1,\\ldots,r_n$ belong to $R$ and $s_1,\\ldots,s_n$ belong to $R \\setminus p$. This shows that the image of $R'$ is contained in $S^{-1}R$ where $S=\\lbrace s_1^{i_1} \\ldots s_n^{i_n}\\rbrace$. The inclusion $\\Spec(S^{-1}R)\\subset \\Spec(R)$ is open and it contains the point $p$. This shows that we have a map $W \\rightarrow \\Spec(R')$ where $W$ is a neighbourhood of $p$, and the map send $p$ to $p'$. By restricting to open subsets $U$ of $\\Spec(R)$ and $U'$ of $\\Spec(R')$ containing $p$ and $p'$ we may further assume that the map is unramified, quasi-finite and of degree one at all points, i.e. an isomorphism.\n\\end{proof}\n\n\\begin{lem}\nSuppose we have an isomorphism of locally ringed spaces $^cX \\xrightarrow{\\phi} {^cY}$.\nThen $X \\sim^c Y$ and the morphism inducing the equivalence agrees with $\\phi$ on $^cX$.\n\\end{lem}\n\\begin{proof}\nGiven a point $p \\in ^cX$ isomorphism of local rings $\\mathcal O_{Y,\\phi(p)}\\simeq \\mathcal O_{X,p}$ induces an isomorphism $U \\simeq V$ of Zariski neighborhoods of $p, \\phi(p)$, sending $p$ to $\\phi(p)$. Moreover $X$ and $Y$ are reduced and separated so any two of these isomorphisms must agree on the intersection of the respective open neighborhoods. A simple gluing argument completes the proof.\n\\end{proof}\n\n\\todo{controllare, scrivere meglio}\n\n\\begin{cor}\nSuppose we have an isomorphism $\\psi: \\mathcal C^{ c}(Y) \\rightarrow \\mathcal C^{ c}(X)$.\nThen $X$ and $Y$ are isomorphic in codimension $c$ and the isomorphism is compatible with the isomorphism $\\psi: ^cX \\rightarrow ^cY$ of locally ringed spaces.\n\\end{cor}\n\\begin{proof}\n\n\\end{proof}\n\n\\todo{dovremmo specificare che l'isomorfismo in codimensione c non induce necessariamente la mappa sulle categorie abeliane. Sarebbe ganzo avere un esempio in cui queste categorie si possono calcolare esplicitamente.}\n\n\n\n\n\\section{Gabriel's theorem}\\label{gabrielsection}\nAs usual, let $\\mathbf{k}$ be our base ring, and let $X$, $Y$ be schemes over it.\nRecall that we write $\\mathcal C_k(X)$ for the quotient $\\Coh(X)\/\\Coh_{\\leq k-1}(X)$, with analogous notation for $Y$.\nTo prove our main theorem we will deal with the two directions separately.\n\\subsection{The ``hard'' direction}\nSuppose we have an equivalence of categories $\\mathcal C_k(X) \\simeq \\mathcal C_k(Y)$, which by default is assumed linear over $\\mathbf{k}$.\nProposition \\ref{equivlrs} implies there is an isomorphism of locally ringed spaces $X_{\\geq k} \\simeq Y_{\\geq k}$ over $\\Spec \\mathbf{k}$.\n\\begin{prop}\\label{puppa}\n\tLet $\\phi\\colon X_{\\geq k} \\to Y_{\\geq k}$ be an isomorphism of locally ringed spaces over $\\Spec \\mathbf{k}$.\n\tThere exist open subsets $U \\subset X, V \\subset Y$ containing all points of dimension $\\geq k$, and an isomorphism $f \\colon U \\rightarrow V$ of $\\mathbf{k}$-schemes, which restricts to $\\phi$.\n\\end{prop}\nLet us start with a lemma.\n\\begin{lem}\\label{lociso}\n\tLet $X, Y$ be schemes of finite type over $\\mathbf{k}$.\n\tAssume there is an isomorphism of $\\mathbf{k}$-algebras between local rings $\\mathcal O_{X,p} \\simeq \\mathcal O_{Y,q}$ where $p \\in X$, $q \\in Y$ are not necessarily closed.\n\tThen there are open subschemes $p \\in U \\subset X$, $q \\in V \\subset Y$ such that $U \\simeq V$ as $\\mathbf{k}$-schemes.\n\\end{lem}\n\\begin{proof}\nFix open neighborhoods $p \\in \\Spec(A) \\subset X, q \\in \\Spec(B)\\subset Y$, and consider the composition $B \\rightarrow B_{q} \\rightarrow A_p$.\nGiven a finite set of generators $y_1, \\ldots, y_n$ for $B$ as a $\\mathbf{k}$-algebra, their images will be in the form $x_1\/s_1, \\ldots , x_n\/s_n$ where $x_1,\\ldots,x_n$ belong to $A$ and $s_1,\\ldots,s_n$ belong to $A \\setminus I(p)$.\nHere $I(p) \\subset A$ denotes the prime ideal corresponding to the point $p \\in \\Spec A$.\n\nThe image of $B$ is thus contained in $S^{-1}A$ where $S=\\lbrace s_1^{i_1} \\ldots s_n^{i_n}\\rbrace$.\nThe inclusion $\\Spec(S^{-1}A)\\subset \\Spec(A)$ is open and contains the point $p$.\nHence we just constructed a map $W \\rightarrow \\Spec(B)$ where $W$ is a neighborhood of $p$, sending $p$ to $q$.\nBy restricting to open subsets $p \\in U \\subset \\Spec(A)$ and $q \\in V \\subset \\Spec(B)$, we may further assume this map to be unramified, quasi-finite, and of degree one at all points, i.e. an isomorphism.\n\\end{proof}\n\n\\begin{lem}\\label{uniqiso}\nLet $U=\\Spec(A), V=\\Spec(B)$ be affine schemes of finite type over $\\mathbf{k}$, and let $f, g \\colon U \\rightarrow V$ be two morphisms.\nSuppose further that there is a $p \\in U$ such that $f(p)=g(p)=q$, and $f^*_{\\mid\\mathcal O_{V,q}}=g^*_{\\mid \\mathcal O_{V,q}}$.\nThen there is an open subscheme $p \\in U' \\subset U$ such that $f$ and $g$ are equal when restricted to $U'$. \n\\end{lem}\n\\begin{proof}\nLet $y_1, \\ldots, y_n$ be a set of generators for $B$ as a $\\mathbf{k}$-algebra.\nWe know $f(x_i)=g(x_i)$ in the localization $A_{I(p)}$ for all $i$.\nAfter inverting a finite number of elements in $A \\setminus I(p)$ we will have $f(x_i)=g(x_i)$ in $S^{-1}A$, for some (finite) set $S \\subset (A \\setminus I(p))$.\nTo conclude, set $U' = \\Spec(S^{-1}A) \\subset \\Spec(A)$.\n\\end{proof}\n\n\\begin{proof}[Proof of Proposition \\ref{puppa}]\nLet $V \\subset Y$ be an affine open subset.\nAs $Y$ is of finite type over $\\mathbf{k}$, the algebra $\\mathcal O_Y(V)$ is generated by a finite set $y_1, \\ldots , y_n$ of generators.\nConsider $y_1, \\ldots, y_n$ as elements of $\\mathcal O_{Y_{\\geq k}}(V \\cap Y_{\\geq k})$.\nThere exists an open subset $U \\subset X$, containing the inverse image of $V \\cap Y_{\\geq k}$, such that the elements $\\phi^*(y_1), \\ldots, \\phi^*(y_n)$ are represented by $x_1, \\ldots, x_n \\in \\mathcal O_X(U)$.\nThis induces a morphism $\\phi_{U}\\colon U \\rightarrow V$. \n\nThe morphism $\\phi_{U}$ is an isomorphism at all the local rings of points $p \\in V \\cap Y_{\\geq k}$.\nBy Lemmas \\ref{lociso}, \\ref{uniqiso} given any point $p \\in U \\cap Y_{\\geq k}$ there is a neighborhood $U_p \\subset U$ such that $(\\phi_{U})_{\\mid U_{p}}$ is an isomorphism.\nUp to restricting $U$ and $V$ to smaller open subsets containing all points of $X_{\\geq k} \\cap U$ and $Y_{\\geq k} \\cap V$, we may then assume that $\\phi_{U}$ is an isomorphism.\n\nNote that, following the construction above, the morphism $\\phi_U$ coincides with the original $\\phi$ on all local rings in $U \\cap X_{\\geq k}$.\nExplicitly, for each $p \\in X_{\\geq k}$ we have $\\mathcal O_{X_{\\geq k}, p} = \\mathcal O_{X,p}$, and $\\phi_{U,p} = \\phi_p$.\n\nSuppose now we apply the same construction to two different open subsets, obtaining isomorphisms $\\phi_{U_i}\\colon U_i \\rightarrow V_i,\\phi_{U_j}\\colon U_j \\rightarrow V_j$.\nThe two maps must agree on local rings for all $p \\in U_i \\cap U_j \\cap Y_{\\geq k}$. \nLemma \\ref{uniqiso} informs us that, up to removing a closed subset of dimension at most $k-1$ from $U_i$ and $ U_j$ and their images, the two maps must agree.\nPick now a finite open affine cover $U_1,\\ldots,U_n$ of $X$.\nWe may iterate the procedure above to all $m$-fold intersections, with $m \\leq n$, refining our open cover and obtaining the claim.\n\\end{proof}\n\n\\subsection{The ``easy'' direction}\nThe converse direction, i.e. that a birational map induces an equivalence of categories, is actually false in the generality we have maintained so far.\n\\begin{rmk}\n\tLet $X$ be the spectrum of a DVR, with closed point $x$ and open point $\\eta$.\n\tLet $K = \\mathcal O_{X,\\eta}$ be the local ring at $\\eta$.\n\tLet $Y$ be $\\Spec K$, and write $y$ for the unique point of $Y$.\n\tThen $X$ and $Y$ are birational, in the sense that $\\mathcal O_{Y,y} \\simeq \\mathcal O_{X,\\eta}$, but $X_{\\geq 1} \\nsimeq Y_{\\geq 1}$.\n\tIndeed, the former consists of just $\\eta$, while the latter is empty.\n\tAt the categorical level, $\\mathcal C_1(X) \\simeq \\mod(K)$ while $\\mathcal C_1(Y) = 0$.\n\tThe culprit is the fact that the singleton $\\{ \\eta \\}$ is \\emph{open} in $X$, but has smaller dimension.\n\tThis phenomenon does not occur for varieties.\n\\end{rmk}\nHowever, over a field everything works just fine.\n\\begin{prop}\n\tSuppose our base ring $\\mathbf{k}$ is a finite type algebra over a field.\n\tLet $j\\colon U \\hookrightarrow X$ be an open immersion.\n\tSuppose $\\dim X \\setminus U < k$.\n\tThen $j^*$ induces an equivalence $\\mathcats C_k(X) \\simeq \\mathcats C_k(U)$.\n\\end{prop}\n\\begin{proof}\n\tThis is a consequence of \\cite[\\href{http:\/\/stacks.math.columbia.edu\/tag\/01PI}{Tag 01PI}]{sp} and Lemma \\ref{hom}.\n\\end{proof\n\n\\begin{rmk}\\label{Rydh}\nThe hypotheses in the previous proposition con be weakened. The statement holds true as long as for an open subset $U$ we always have that $U_{\\geq k} = X_{\\geq k} \\cap U$, which by \\cite[10.6.2]{EGAIV} is implied by $X$ being Jacobson, universally catenary and every irreducible component of $X$ being equicodimensional.\n\\end{rmk}\n\n\n\\subsection{The main theorem}\nAssembling our previous results together, we arrive at our main result.\nWe say two schemes $X,Y$ are \\emph{isomorphic outside of dimension $k-1$} of there exist isomorphic open subsets $U \\subset X, V \\subset Y$, such that $X_{\\geq k} \\subset U$, and $Y_{\\geq k} \\subset V$.\n\\begin{thrm}\\label{suca}\n\tAssume our base ring $\\mathbf{k}$ is a finite type algebra over a field.\n\tLet $X,Y$ be schemes of finite type over $\\mathbf{k}$.\n\tThen $X$ and $Y$ are isomorphic (over $\\mathbf{k}$) outside of dimension $k-1$ if and only if $\\mathcal C_k(X) \\simeq \\mathcal C_k(Y)$.\n\\end{thrm}\nAssume $\\mathbf{k}$ is a field, and $X$ and $Y$ are irreducible.\nIn this context it is preferable to rephrase things in terms of codimension.\nLet $\\mathcal C^c(X) = \\mathcal C_{\\dim X - c}(X)$, and similarly for $Y$.\nRecall that $X$ and $Y$ are \\emph{isomorphic in codimension $c$} if there exist isomorphic open subsets $U \\subset X$, $V \\subset Y$, such that $\\codim X \\setminus U > c$, $\\codim Y \\setminus V > c$.\n\\begin{cor}\n\tLet $\\mathbf{k}$ be a field, and let $X$ and $Y$ be irreducible (but not necessarily reduced).\n\tThen $X$ and $Y$ are isomorphic in codimension $c$ if and only if $\\mathcal C^c(X)$ and $\\mathcal C^c(Y)$ are equivalent.\n\\end{cor}\n\n\\subsection*{Previous work}\nGabriel originally proved his reconstruction theorem in \\cite{gabriel}.\nThis was later generalized considerably by various people \\cite{rosenberg, brand, jm}.\nIn \\cite{mp}, Meinhardt and Partsch were interested in constructing stability conditions on the (derived categories of the) quotient categories $\\mathcats C^c(X)$.\nAlong the way they showed the $c=0$ and $c=1$ cases of our main result, when $X,Y$ are smooth and projective over an algebraically closed field.\n\n\\subsection*{Acknowledgements}\nWe would like to thank David Rydh for reading a first draft, and providing excellent comments and corrections.\nIn particular, we thank him for pointing out a mistake in the proof of lemma \\ref{nosheaf} and for noticing that the hypotheses of our main theorem could be relaxed (see Remark \\ref{Rydh}).\nWe thank Pieter Belmans, Zinovy Reichstein, and Angelo Vistoli for useful comments.\nFinally, JC wishes to give extra-warm thanks to Richard ``TDR'' Shadrach for many many conversations throughout the years.\n\n\\subsection*{Conventions}\nFor the entirety of this paper we fix a base (commutative) noetherian ring $\\mathbf{k}$.\nAll algebras, schemes, categories, morphisms, and functors are implicitly assumed to be over $\\mathbf{k}$.\nStarting from Section \\ref{gabrielsection}, all schemes will be assumed to be of finite type over $\\mathbf{k}$.\nIf $X$ is a scheme, $U \\subset X$ is open, and $F$ is a sheaf on $X$, we will write $F_U$ for the restriction (i.e.\\ pullback) of $F$ to $U$.\nIf $R$ is a ring, we will write $\\mod(R)$ for the category of finitely generated $R$-modules.\n\n\n\n\n\n\n\\section{A locally ringed space}\nFor this section, let $X$ be a finite-dimensional noetherian scheme, which (as per our blanket conventions) is moreover defined over our base ring $\\mathbf{k}$.\nOur present goal is to define an auxiliary locally ringed space $S = S(X,k,L)$, depending on $X$, an integer $k$ and a sheaf $L$.\nThis space will control the isomorphism type of $X$ up to subsets of dimension $k-1$.\nA posteriori it will be obvious that $S$ does not depend on the sheaf $L$.\nThe reason we initially insist on the dependency on $L$ will become clear in the next section: an equivalence $\\mathcats C_k(X) \\simeq \\mathcats C_k(Y)$ need not send $\\mathcal O_X$ to $\\mathcal O_Y$.\n\nWe say a point $x \\in X$ has \\emph{dimension $k$} if $\\dim \\overline{\\{x\\}} = k$.\nPut differently, $x$ is the generic point of a subvariety of dimension $k$, which (when $X$ is equidimensional and catenary) in turn is equivalent to $\\dim \\mathcal O_{X,x} = \\dim X - k$.\nWe write $X_{\\geq k} \\subset X$ for the subset of points of dimension at least $k$.\nIn particular, $X_{\\geq k}$ always contains the generic points of the irreducible components (of dimension at least $k$) of $X$.\nWe endow $X_{\\geq k}$ with the subspace topology.\nIf $i\\colon X_{\\geq k} \\to X$ denotes the inclusion, we may view the former as a locally ringed space with structure sheaf $i^{-1}\\mathcal O_X$.\nThis space will turn out to be isomorphic to the space $S$ we are about to define.\n\nBefore we proceed, recall that $i^{-1}\\mathcal O_X$ is (by definition) obtained by sheafifying the presheaf $i^+\\mathcal O_X$, given by\n\\begin{align*}\n\tV \\mapsto \\varinjlim_{U \\supset V} \\mathcal O_X(U)\n\\end{align*}\nwhere the limit ranges over all $U \\subseteq X$ open and containing $V \\subseteq X_{\\geq k}$.\n\\begin{lem}\\label{nosheaf}\n\tThe presheaf $i^+\\mathcal O_X$ is already a sheaf, hence\n\t\\begin{align*}\n\t\ti^{-1}\\mathcal O_X(V) = \\varinjlim_{U \\supset V} \\mathcal O_X(U).\n\t\\end{align*}\n\\end{lem}\n\\begin{proof}\n\tLet $V \\subseteq X_{\\geq k}$ be open.\n\tA section in $i^+\\mathcal O_X(V)$ is represented by an $\\alpha \\in \\mathcal O_X(U)$, for $U \\subseteq X$ open and $V \\subset U$.\n\tLet $\\{V_i\\}_i$ be an open cover of $V$, such that the restriction $\\alpha|V_i = 0$ vanishes, for all $i$.\n\tWe may then find open subsets $U_i \\subset U$, such that $U_i \\cap X_{\\geq k} = V_i$ and $\\alpha|U_i = 0$.\n\tLet $U'$ be the union of the $U_i$.\n\tThen, $V \\subset U' \\subset U$ and $\\alpha|U' = 0$.\n\tHence the section $[\\alpha] \\in i^+\\mathcal O_X(V)$ must be zero.\n\t\n\tSuppose once again $\\{V_i\\}_i$ is an open cover of a given open $V \\subset X_{\\geq k}$.\n\tWe may assume the cover to be finite.\n\t\n\tLet $[\\alpha_i] \\in i^+\\mathcal O_X(V_i)$ be a collection of sections such that $[\\alpha_i]|V_i \\cap V_j = [\\alpha_j]|V_i \\cap V_j$.\n\tFor each $i$, suppose $\\alpha_i \\in \\mathcal O_X(U_i)$ for $V_i \\subset U_i \\subset X$.\n\tNow, for each pair $ij$, we may choose an open subset $V_i \\cap V_j \\subset U_{ij} \\subset U_i \\cap U_j$, such that $\\alpha_i|U_{ij} = \\alpha_j|U_{ij}$. Note that $W_{ij} = U_i \\cap U_j \\setminus U_{ij}$ has dimension at most $k-1$. \n\tWe replace $X$ with $U = \\cup_{i} U_{i} \\setminus \\cup_{i,j} \\overline{W_{ij}} $, which is open and contains $X_{\\geq k}$. Finally, we replace the open subsets $U_i$ with $U'_i = U_i \\cap U$.\n\t\n\tSince $\\mathcal O_X$ is a sheaf, and $U'_i \\cap U'_j \\subset U_{ij}$, there exists $\\alpha \\in U$ restricting to $\\alpha_i$ on $U_i'$, which gives the desired section as the complement of $U$ in $X$ has dimension at most $k-1$.\n\t\n\\end{proof}\n\n\\subsection{The topological space $S$}\nWe now come to the definition of $S$.\nLet $k \\geq 0$ be an integer.\nAs a set, $S$ consists of isomorphism classes of (non-zero) minimal objects of $\\mathcats C_d(X)$, where $d$ ranges between $k$ and $\\dim(X)$:\n\\begin{align*}\n\tS \\coloneq \\bigcup_{d \\geq k} \\left\\lbrace 0 \\neq P \\in \\mathcal C_d(X) \\mid P \\mbox{ minimal} \\right\\rbrace_{\/_\\text{iso}}\n\\end{align*}\nWe define a topology on $S$, by declaring what the closure of a point $s \\in S$ ought to be. \nLet $P \\in \\Coh(X)$ represent a nonzero minimal object in $\\mathcats C_d(X)$, with $d \\geq k$.\nDefine\n\\begin{align*}\n\tZ_P \\coloneq \n\t\t\\bigcup_{d \\geq j \\geq k} \n\t\t\\bigcap_{\n\t\t\t\\substack{ P' \\in \\mathcats C_j(X) \\\\ P' \\text{ represents } P}\n\t\t\t}\n\t\t\\left\\{ 0 \\neq Q \\in \\mathcats C_j(X) \\mid Q \\text{ is minimal, } Q \\text{ is a quotient of } P' \\right\\}_{\/_\\text{iso}}\n\\end{align*}\nWe endow $S$ with the coarsest topology containing the sets $S \\setminus Z_P$, for all $P \\in \\Coh(X)$.\n\\begin{lem}\\label{homeo}\n\tDefine a map $X_{\\geq k} \\to S$ by sending $x \\mapsto \\mathcal O_{\\overline{\\{x\\}}}$, where $\\overline{\\{x\\}}$ is equipped with the reduced scheme structure.\n\tThis map is a homeomorphism.\n\\end{lem}\n\\begin{proof}\n\tSince $X$ is noetherian, the subsets $X \\setminus \\overline{\\{x\\}}$, as $x$ varies in $X$, form a basis of the topology.\n\tProposition \\ref{geo} and Lemma \\ref{min} allow us to conclude.\n\\end{proof}\n\n\\subsection{A Picard group}\\label{picardo}\nWe introduce now the analogue of $\\Pic(X)$ for the category $\\mathcats C_k(X)$.\nWe define $\\Pic_{\\geq k}(X)$ to be the group (under tensor product) of isomorphism classes of objects $L_1 \\in \\mathcats C_k(X)$ for which there exists a representative $L \\in \\Coh(X)$ satisfying the following property: there exists an open subset $U \\subset X$, such that $X_{\\geq k}\\subset U$, and the restriction $L_U$ is an invertible sheaf.\nPut concisely: $\\Pic_{\\geq k}(X)$ consists of (the $\\mathcats C_k(X)$-isomorphism classes of) those $L$ which are line bundles away from a closed subset of dimension at most $k-1$. We will see in proposition \\ref{equivlrs} that this definition is intrinsic to the category $\\mathcats C_k(X)$.\n\n\\subsection{The locally ringed space $S$}\nFor $F \\in \\Coh(X)$, we define its \\emph{topological $k$-support} to be $\\supp_k F \\coloneq X_{\\geq k} \\cap \\supp F$.\nUsing the homeomorphism of Lemma \\ref{homeo}, we may view $\\supp_k F$ as a closed subset of $S$.\nSimilarly, if $G \\in \\mathcats C_d(X)$ with $d \\leq k$, we define $\\supp_k G = \\supp_k \\tilde G$, where $\\tilde G \\in \\Coh(X)$ is any representative of $G$. \n\nFix now $L \\in \\Pic_{\\geq k}(X)$.\nWe will now define a sheaf of rings $\\mathcal O_S$ on $S$, depending on $L$.\nFor $V \\subset S$ an arbitrary open subset, we define\n\\begin{align*}\n\t\\mathcal O_S(V) \\coloneq \\varinjlim_{U \\supset V} \\Hom_{U} (L_U,L_U)\n\\end{align*}\nwhere $U \\supset V$ runs over all open subsets of $X$ containing $V$.\n\\begin{prop}\n\tThe assignment $V \\mapsto \\mathcal O_S(V)$ defines a sheaf of rings on $S$, making $(S,\\mathcal O_S)$ into a locally ringed space.\n\tMoreover, the homeomorphism $X_{\\geq k} \\to S$ from Lemma \\ref{homeo} may be upgraded to an isomorphism of locally ringed spaces over $\\Spec \\mathbf{k}$.\n\\end{prop}\n\\begin{proof}\n\tSuppose first $L = \\mathcal O_X$.\n\tThen Lemma \\ref{homeo} and Lemma \\ref{nosheaf} imply the claim.\n\tLet $L$ be general.\n\tThere exists then an open subset $U$, with $X_{\\geq k} \\subset U$, such that $L$ becomes invertible once restricted to $U$.\n\tBut then the natural map $\\mathcal O_X \\to \\lHom_X(L,L)$ also becomes an isomorphism once restricted to $U$.\n\tThe claim then follows from the case where $L = \\mathcal O_X$.\n\\end{proof}\n\n\\subsection{Intermezzo}\nTo proceed with the final proof of this section, we must first introduce one last category.\nSuppose $0 \\neq P \\in \\mathcats C_k(X)$ is minimal, and denote by $x \\in X_{\\geq k}$ the corresponding point.\nIn particular, if $Z = \\overline{\\{x\\}}$, the sheaf $\\mathcal O_Z$ represents $P$.\nThe goal is to define a category $\\mathcats C_P$, recovering the local ring $\\mathcal O_{X,x}$ at $x$.\n\n\\begin{lem}\nGiven an element $F \\in \\Coh(X)$, the subset $\\supp_k(F) \\subset X_{\\geq k}$ only depends on the class of $F$ in $\\mathcal C_k$.\n\\end{lem}\n\\begin{proof}\nLet $P$ be a minimal object in $\\mathcal C_k$. Then by lemma (\\ref{hom}) the corresponding point $P \\in X_{\\geq k}$ belongs to $\\supp_k(F)$ if and only if $\\Hom_{\\mathcal C_k}(F,P) \\neq 0$.\n\\end{proof}\n\nConsider the collection of $E \\in S$ such that $P \\nin \\supp_k E$, and let $\\mathcal S_P \\subset \\mathcats C_k(X)$ be the smallest Serre subcategory containing them.\nWe define $\\mathcats C_P$ to be the quotient $\\mathcats C_k(X)\/\\mathcal S_P$.\nFurthermore, write $E_P$ for the class of $E$ in $\\mathcats C_P$.\n\\begin{lem}\\label{cplocale}\n\tLet $L \\in \\Pic_{\\geq k}(X)$.\n\tWe may identify the ring $\\End_{\\mathcats C_P}(L)$ with the local ring $\\mathcal O_{X,x}$.\n\tMoreover, the functor $\\mathcats C_P \\to \\mod(\\mathcal O_{X,x})$ sending $E$ to $\\Hom_{\\mathcats C_P}(L,E)$ is an equivalence.\n\\end{lem}\nIn particular we may identify the stalk $E_x \\in \\mod(\\mathcal O_{X,x})$ with the object $E_P \\in \\mathcats C_P$.\n\\begin{proof}\n\tLet $\\Sigma_x$ be the collection of closed subsets of $X$ not containing $x$.\n\tFrom the definition of $\\mathcal S_P$, we see that $\\Coh(X)\/\\mathcal S_{\\Sigma_x} = \\mathcats C_P$.\n\tUsing Lemma \\ref{hom}, we see\n\t\\begin{align*}\n\t\t\\Hom_{\\mathcats C_P}(E,F) = \\varinjlim_{x \\in U \\subset X} \\Hom_U(E_U,F_U) = \\Hom_{\\mathcal O_{X,x}}(E_x,F_x).\n\t\\end{align*}\n\tSince $L_x = \\mathcal O_{X,x}$, and $\\mathcal O_{X,x} = \\End_{\\mathcal O_{X,x}}(\\mathcal O_{X,x}) = \\End_{\\mathcats C_P}(L)$.\n\tHence the functor is fully faithful.\n\tTo prove it is essentially surjective, it suffices to observe that, given a finitely generated $\\mathcal O_{X,x}$-module $M$, there exists a coherent sheaf $E \\in \\Coh(X)$ such that $E_x = M$.\n\\end{proof}\n\\begin{rmk}\\label{funziona}\n\tWhen $X$ in the above lemma is integral, and $x$ is its generic point, then $\\mathcal O_{X,x} = K(X)$ is its function field.\n\tIt then follows that $\\mathcats C_{\\dim X}(X) = \\mathcats C_{\\mathcal O_X}$, and the functor $\\mathcats C_{\\mathcal O_X} \\to \\mod(K(X))$ is an equivalence.\n\tHence $\\mathcats C_{\\dim X}(X)$ captures the birational type of $X$.\n\\end{rmk}\n\n\\subsection{A non-commutative remark}\nFollowing up Remark \\ref{funziona}, we briefly discuss a non-commutative avenue.\nOne perspective on non-commutative geometry is that a non-commutative space should be given by an abelian category $\\mathcal A$, satisfying some niceness properties.\nMuch work has been devoted to this point of view, see for example \\cite{artinzhang,artintatevdb,rosenbergncag,vdbblow}.\nA basic question is then whether there exists a non-commutative analogue of birational geometry.\nFor example in \\cite{smith} one finds a candidate for a function field of a general $\\mathcal A$, and in \\cite{presotto} there are interesting examples of birational non-commutative surfaces.\n\nHowever, a complete and satisfactory theory of non-commutative birational geometry does not seem to exist presently.\nFor instance, to capture higher dimensional phenomena (such as flops and flips), one needs to know not just when two spaces are merely birational, but also when they are isomorphic in a certain codimension.\n\nOn the other hand, as we discussed in the previous section, the categories $\\Coh_{\\leq k}(X)$ are \\emph{intrinsic} to the category $\\Coh(X)$.\nIndeed, given an abelian category $\\mathcal A$, we may form our sequence of quotients $\\mathcal A_k$ as in \\eqref{quotients_of_cats}.\nIn light of Remark \\ref{funziona}, if there exists an $n$ such that $\\mathcal A_n \\neq 0$ but $\\mathcal A_{n+1} = 0$, we would view $\\mathcal A_n$ as the \\emph{non-commutative function field} of $\\mathcal A$.\nIt would be interesting to see the relation between this and Smith's function field \\cite{smith}.\n\nIn general, given the main theorem of this paper, the quotient categories $\\mathcal A_k$ (for $k \\geq n$) could be seen as capturing the non-commutative space $\\mathcal A$, up to a certain codimension.\n\n\n\\subsection{Equivalences}\nWe conclude this section by showing that equivalences of the quotient categories induce isomorphism of the locally ringed spaces.\n\\begin{prop}\\label{equivlrs}\n\tLet $X$, $Y$ be schemes of finite type over $\\mathbf{k}$.\n\tA $\\mathbf{k}$-linear equivalence $\\Phi\\colon \\mathcats C_k(X) \\to \\mathcats C_k(Y)$ induces an isomorphism of locally ringed spaces $\\phi\\colon X_{\\geq k} \\to Y_{\\geq k}$.\n\\end{prop}\n\\begin{proof}\n\tLet $L \\in \\Coh(Y)$ be a representative for $\\Phi(\\mathcal O_X)$.\n\tWe want to show that $L$ is a line bundle away from a small enough closed subset.\n\tNotice first that $\\mathcal O_X$ satisfies the following property: if $P$ is any non-zero minimal object of $\\mathcats C_k(X)$, then $\\Hom_{\\mathcats C_k(X)}(\\mathcal O_X,P)$ is a one-dimensional $\\End_{\\mathcats C_k(X)}(P)$ vector space.\n\tMoreover, $\\mathcal O_X$ is \\emph{locally maximal} with respect to this property: given $F$ such that $\\Hom_{\\mathcats C_k(X)}(F,P)$ is one-dimensional, any surjection $F \\to \\mathcal O_X$ in $\\mathcats C_P$ must be an isomorphism.\n\tSince these two properties are categorical (i.e. intrinsic to $\\mathcats C_k(X)$ and the minimal object $P$) they will satisfied by $L$ as well.\n\t\n\tWe claim that, for any sheaf $L$ satisfying the two properties above, there exists an open subset $U \\subset Y$, with $Y_{\\geq k} \\subset U$, such that the restriction $L_U$ is a line bundle.\n\tIndeed,\tconsider the function $\\beta\\colon Y \\to \\mathbf N$, taking $y \\in Y$ to the rank of the fibre $\\dim_{\\kappa(y)} L \\otimes \\kappa(y)$.\n\tThis function is upper semicontinuous.\n\tThe locus $\\{\\beta = 0 \\}$ is therefore open.\n\tLet $Z$ be the union of the irreducible components of dimension strictly less than $k$.\n\tWe see that $\\{\\beta = 0\\} \\cap (Y \\setminus Z)$ is empty.\n\tTherefore, the locus $U = \\{ \\beta = 1 \\} \\cap (Y \\setminus Z)$ is open, and (by assumption) contains all points of dimension $k$.\n\tBy Nakayama, the $\\mathcal O_{Y,y}$-module $L_{y}$ has rank $1$ at all points $y$ in $U$.\n\tUsing Lemma \\ref{cplocale}, and the local maximality property of $L$, we deduce that $L_y=\\mathcal O_{Y,y}$ at all points of dimension $k$, as there is by definition a surjection $\\mathcal O_{Y,y}\\rightarrow L_y$, which must be an isomorphism by maximality.\n\tFinally, the set of points $y$ such that $L_y$ is free is open.\n\tSince it contains all points of dimension $k$ it follows that $L_y$ is free of rank $1$ at all points in a subset $U'$ containing all points of dimension $k$.\n\t\n\tFrom the discussion above it follows that the equivalence $\\Phi$ induces an isomorphism of locally ringed spaces $S(X,k,\\mathcal O_X) \\simeq S(Y,k,\\Phi(\\mathcal O_X))$.\n\tThe proposition then follows: the former is isomorphic to $X_{\\geq k}$, and the latter to $Y_{\\geq k}$, as locally ringed spaces.\n\\end{proof}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Noncommutative}\n\n\\subsection*{Non-commutative birational invariants?}\n\\begin{prop}\n\tLet $R$ be a graded $\\mathbf{k}$-algebra satisfying some assumptions (like the ones for the graded Goldie theorem).\n\tThen at some point the function field shows up.\n\\end{prop}\n\\begin{proof}\n\tL'idea mi sembra abbastanza evidente, ma non riesco a renderla precisa.\n\tChiama $\\mathcal B$ il quoziente e chiama $F$ il campo delle funzioni.\n\tA ogni elemento $s^{-1}a \\in F$, con $\\deg s = d = \\deg a$, associo la mappa nel quoziente\n\t\\begin{align*}\n\t\tR \\stackrel {\\cdot s} \\leftarrow R(-d) \\stackrel {\\cdot a} \\to R\n\t\\end{align*}\n\tModulo non aver confuso destra e sinistra, questo d\\`a un morfismo $F \\to \\End_{\\mathcal B}(R)$.\n\tMa non so ancora come costruire l'inverso.\n\\end{proof}\n\n\\subsection{Example?}\n\nLet $R$ be a graded ring.\nAssume $R$ is both non-negatively graded and noetherian.\nI don't think it matters if it's generated in degree one or not.\nLet $\\grmod(R)$ be the category of finitely generated graded $R$-modules.\nLet $\\mathcal T$ be the subcategory of torsion modules.\nWhen $R$ is finite type over a field $\\mathbf{k}$, this coincides with the subcategory of modules which are finite dimensional as $\\mathbf{k}$-vector spaces.\nNotice that $\\mathcal T$ coincides with the smallest Serre subcategory containing all minimal objects.\n\nThe quotient $\\mathcal A = \\grmod(R)\/\\mathcal T$, sometimes called the \\emph{tails category}, is viewed as the category of coherent sheaves on a non-commutative proj scheme, which we denote by $X = \\Proj R$.\nThis interpretation is due to what known as Serre's theorem, which says that when $R$ is commutative there is indeed an equivalence $\\grmod(R)\/\\mathcal T = \\Coh(\\Proj(R))$.\n\n\\begin{prop}\n\tThe dimension of $\\mathcal A$ as defined earlier coincides with the Krull dimension of $R$ as a graded ring.\n\\end{prop}\n\\todo[inline,color=pink]{boh, secondo me \\`e vero}\n\nSuppose now $R$ is a ring where all non-zero homogeneous elements satisfy the Ore conditions. \\todo{or whatever the fuck it needs to satisfy}\nOne may then form its field of fractions $F$, consisting of quotients $a\/b$ where $a,b$ are both homogeneous.\nInside $F$, there is $F_0 \\subset F$, consisting of elements $a\/b$ where $\\deg a = \\deg b$.\nThis field $F_0$ is interpreted as being the function field $\\mathbf{K}(X)$ of our non-commutative scheme $X$.\n\n\\begin{prop}\\label{nonso}\n\tLet $\\mathcal B$ be the penultimate quotient.\n\tThen $\\mathcal B = \\mod(F_0)$.\n\\end{prop}\nIn particular, two non-commutative projective schemes which are isomorphic in some codimension (in our categorical sense) will be birational in the sense that they have Morita equivalent function fields.\nFor example, \\cite{presotto} provides many interesting examples of non-isomorphic but birational non-commutative surfaces.\n\\begin{proof}\n\tIt's probably easy to see that any module becomes isomorphic to $R^{\\oplus n}$ for some $n$, since everything has a presentation.\n\t\n\tWhat's a morphism $R \\rightharpoonup R$ in the quotient?\n\tWell, it's a roof $R \\leftarrow Q \\to R$ where $Q \\to R$ has kernel and cokernel being something.\n\tThis something is basically anything which is not a free module.\n\tHow do we get the function field?\n\tWell, let $a,b$ be two homogeneous elements of degree $d$.\n\tConsider $R(-d) \\to R$ given by multiplication by $b$, and $R(-d) \\to R$ given by multiplication by $a$.\n\tThen $R \\leftarrow R(-d) \\to R$ corresponds to the element $a\/b$ in the function field.\n\tIn other words, $\\End_{\\text{quotient}}(R,R) = F_0$.\n\\end{proof}\n\n\\subsubsection{Noncommutative flop}\nIt would be amazing if we could conclude this paper with an example of two algebras which weren't isomorphic as non-commutative spaces, but were isomorphic in some codimension (and not just birational).\n\nUnfortunately, the example of Atiyah does not look like it could be made non-commutative easily.\nHe takes $\\mathbf{k}[x,y,z,w]$ and looks at the singular quadric threefold $X$, given by equation $xy - zw = 0$.\nBlowing up the singular point, you get a smooth guy $W$.\nThe exceptional locus of $W \\to X$ is $\\mathbf{P}^1\\times \\mathbf{P}^1$.\nContracting the two projective lines you get two threefolds $Y^+$, $Y^-$ which are resolutions of $X$.\nThis shows in dimension three there aren't always ``minimal resolutions''.\nI think in this case $Y^+$ and $Y^-$ are actually isomorphic, but the induced birational map $Y^+ \\to X \\leftarrow Y^-$ is the famous flop (it does not extend to a regular map).\n\nOne would like to make this non-commutative, but I wouldn't know how.\nHere's one thing we may notice.\nLet $A$ be the coordinate ring of $X$.\nSince $Y^\\pm$ is projective over $X$, it may be described by a graded $A$-algebra.\nObviously this algebra is not generated in degree one.\nOne could try to make this non-commutative by declaring that some elements in degree one commute up to a root of unity, like $st = qts$ for some $q$.\nThis would roughly mean making the $\\mathbf{P}^1$ non-commutative.\nBut apparently there aren't any non-commutative $\\mathbf{P}^1$'s.\nSo maybe the thing to do is have a higher dimensional flop, where we deform the contracted $\\mathbf{P}^2$.\n\nOr maybe even the following.\nBlow up a point in affine 3-space, the exceptional locus is a $\\mathbf{P}^2$.\nDeform the $\\mathbf{P}^2$ to two different birational $\\mathbf{P}^2$s in the sense of Presotto.\nDoes this give what we want?\n\n\\section{Quotient categories}\nWe begin by briefly reviewing some standard notions and later introduce some notation.\nWe refer to \\cite{gabriel,faith} for a thorough treatment of quotients of abelian categories.\nLet $\\mathcal A$ be an abelian category.\nA subcategory $\\mathcal S \\subset \\mathcal A$ is \\emph{Serre} if, given an exact sequence $A \\to B \\to C$, then $B \\in \\mathcal S$ if and only if $A,C \\in \\mathcal S$.\nGiven a Serre subcategory $\\mathcal S \\subset \\mathcal A$, we may form the quotient $\\Phi\\colon \\mathcal A \\to \\mathcal A \/ \\mathcal S$.\nThis category is universal among all abelian categories $\\mathcal B$ equipped with an exact functor $\\Psi\\colon \\mathcal A \\to \\mathcal B$ such that $\\Psi(S) = 0$ for all $S \\in \\mathcal S$.\nRecall that the \\emph{kernel} of a functor $\\Phi$ is the full subcategory whose objects satisfy $\\Phi(M) = 0$.\nThus $\\mathcal S$ is precisely the kernel of $\\mathcal A \\to \\mathcal A\/ \\mathcal S$.\n\n\\begin{lem}\\label{iso}\n\tLet $\\mathcal S_1 \\subset \\mathcal S_2 \\subset \\mathcal A$ be two Serre subcategories of the abelian category $\\mathcal A$.\n\tThen $\\mathcal S_2 \/ \\mathcal S_1$ is a Serre subcategory of $\\mathcal A\/ \\mathcal S_1$ and the quotient $\\mathcal A \/ \\mathcal S_2$ is naturally equivalent to the iterated quotient $\\left( \\mathcal A \/ \\mathcal S_1 \\right) \/ \\left( \\mathcal S_2 \/ \\mathcal S_1 \\right)$.\n\\end{lem} \n\\begin{proof}\n\tThe first assertion follows from the definitions, while the second is a consequence of the universal property.\n\\end{proof}\n\tLet $\\mathcal A \/ \\mathcal S$ be a quotient category.\n\tLet $P, Q \\in \\mathcal A$.\n\tTo distinguish arrows in $\\mathcal A$ from arrows in $\\mathcal A \/ \\mathcal S$, we will write $P \\to Q$ for the former and $P \\rightharpoonup Q$ for the latter.\n\tIf two objects $P,Q \\in \\mathcal A$ become isomorphic in $\\mathcal A\/ \\mathcal S$, we will write $P \\equiv Q$.\n\tFinally, a morphism $f\\colon P \\to Q$ such that $\\ker f, \\coker f \\in \\mathcal S$ will be called a \\emph{weak equivalence}.\n\t\nThe quotient category $\\mathcal A \/ \\mathcal S$ may be concretely built as follows \\cite[Exercise 8.12]{ks}.\nThe objects of $\\mathcal A \/ \\mathcal S$ are the same as the objects of $\\mathcal A$.\nA morphism $P \\rightharpoonup Q$ is an equivalence class of ``roofs'', i.e. diagrams\n\\begin{center}\n\\begin{tikzcd}\n\t& U \\ar{dl}\\ar{dr} & \\\\\n\tP && Q \n\\end{tikzcd}\n\\end{center}\nwith $U \\to P$ a weak equivalence.\n\n\\begin{lem}\\label{exact}\n\tLet $\\mathcal A \/ \\mathcal S$ be a quotient category.\n\t\\begin{itemize}\n\t\t\\item $P \\equiv 0$ if and only if $P \\in \\mathcal S$.\n\t\n\t\t\\item If $0 \\rightharpoonup A \\rightharpoonup B \\rightharpoonup C \\rightharpoonup 0$ is short exact in $\\mathcal A \/ \\mathcal S$, then there exists objects $A',B',C'$ and isomorphisms (in $\\mathcal A\/ \\mathcal S$) $A \\rightharpoonup A'$, $B \\rightharpoonup B'$, $C \\rightharpoonup C'$ and a short exact sequence $0 \\to A' \\to B' \\to C' \\to 0$ in $\\mathcal A$, such that the diagram\n\t\t\\begin{center}\n\t\t\t\\begin{tikzcd}\n\t\t\t\t0 \\ar{r} & A' \\ar{r}\\ar[rightharpoonup]{d} & B' \\ar{r}\\ar[rightharpoonup]{d} & C' \\ar{r}\\ar[rightharpoonup]{d} & 0 \\\\\n\t\t\t\t0 \\ar{r} & A \\ar[rightharpoonup]{r} & B \\ar[rightharpoonup]{r} & C \\ar{r} & 0 \\\\\n\t\t\t\\end{tikzcd}\n\t\t\\end{center}\n\t\tcommutes in $\\mathcal A \/ \\mathcal S$.\n\t\\end{itemize} \n\\end{lem}\n\\begin{proof}\n\tThe first claim follows from \\cite[Prop 7.1.20 (ii)]{ks}, the second is \\cite[Cor 15.8]{faith}.\n\\end{proof}\n\n\\subsection{Dimension}\\label{subdimension}\nLet $\\mathcal A$ be an abelian category.\nWe say an object $P$ is \\emph{minimal} if it has no non-trivial sub-objects.\\footnote{What we call minimal objects are often called \\emph{simple} objects. However, by a \\emph{simple sheaf} one typically means one that has only scalar endomorphisms.}\nNote that for us the zero object is also minimal (this is slightly non-standard).\nWe let $\\mathcal S_0$ be the smallest Serre subcategory containing all minimal objects of $\\mathcal A$.\nLet $\\mathcal A_1 = \\mathcal A \/ \\mathcal S_0$.\nSince $\\mathcal A_1$ is also an abelian category, we may repeat the process.\nLet $\\mathcal S'_1$ be the smallest Serre subcategory containing all minimal objects of $\\mathcal A_1$ and let $\\mathcal A_2 = \\mathcal A_1 \/ \\mathcal S'_1$.\nWe define $\\mathcal S_1$ to be the kernel of $\\mathcal A \\to \\mathcal A_1 \\to \\mathcal A_2$.\nBy iterating we obtain a sequence of quotients\n\\begin{align}\\label{quotients_of_cats}\n\t\\mathcal A = \\mathcal A_0 \\twoheadrightarrow \\mathcal A_1 \\twoheadrightarrow \\mathcal A_2 \\twoheadrightarrow \\mathcal A_3 \\twoheadrightarrow \\cdots \\twoheadrightarrow \\{0\\}\n\\end{align}\nand, by taking kernels, we find a nested family of Serre subcategories\n\\begin{align}\\label{kers}\n\t\\{0\\} \\subset \\mathcal S_0 \\subset \\mathcal S_1 \\subset \\mathcal S_2 \\subset \\mathcal S_3  \\subset \\cdots \\subset \\mathcal A\n\\end{align}\nOf course, it need not be the case that there is a $k$ such that $\\mathcal A_k = \\{0\\}$ or that $\\mathcal S_k = \\mathcal A$.\n\\begin{defn}\n\tThe \\emph{Krull dimension} of $\\mathcal A$ is\n\t\\begin{align*}\n\t\t\\dim \\mathcal A \\coloneq \\inf \\{ k \\mid \\mathcal A_{k+1} = 0 \\} = \\inf \\{ k \\mid \\mathcal S_k = \\mathcal A \\}.\n\t\\end{align*}\n\tWe also define\n\t\\begin{align*}\n\t\t\\dim M \\coloneq \\inf \\{ k \\mid M = 0 \\text{ in } \\mathcal A_{k+1} \\} = \\inf \\{ k \\mid M \\in \\mathcal S_k \\}\n\t\\end{align*}\n\tfor any object $M \\in \\mathcal A$.\n\\end{defn}\nThese definitions are of course justified by the algebro-geometric context\n\n\n\n\\subsection{The geometric case}\nIn this subsection, $X$ denotes an arbitrary noetherian scheme.\nLet $\\mathcal C = \\Coh(X)$ be its category of coherent sheaves.\nAs before, we write $\\mathcal S_k = \\mathcal S_k(X)$ for the sequence of kernels as in \\eqref{kers}.\nDenote by $\\Coh_{\\leq k}(X)$ the category of sheaves supported in dimension at most $k$.\n\\begin{prop}\\label{geo}\n\tWe have $\\mathcal S_k(X) = \\Coh_{\\leq k}(X)$.\n\\end{prop}\nBefore we prove this result, we need an alternative way to compute morphisms in our quotients.\nWe define $\\mathcal C_0 = \\mathcal C$ and $\\mathcal C_k = \\mathcal C \/ \\mathcal S_{k-1}$ for all other $k$.\n\nLet $\\Sigma$ be a \\emph{family of supports}, i.e.\\ a collection of closed subsets of $X$ such that\n\\begin{itemize}\n\t\\item $\\emptyset \\in \\Sigma$,\n\t\\item if $Z \\in \\Sigma$, $V \\subset Z$ is closed, then $V \\in \\Sigma$,\n\t\\item if $Z_1, Z_2 \\in \\Sigma$ then $Z_1 \\cup Z_2 \\in \\Sigma$.\n\\end{itemize}\nFor example, $\\Sigma$ might be the collection of all closed subsets of $X$ of dimension at most $k$.\nWrite $\\Coh_\\Sigma(X) \\subset \\Coh(X)$ for the subcategory of sheaves $F$ such that $\\supp F \\in \\Sigma$.\nBecause of our assumptions, $\\Coh_\\Sigma(X)$ is a Serre subcategory.\nWe write $\\mathcal Q = \\Coh(X)\/\\Coh_\\Sigma(X)$ for the quotient.\n\\begin{lem}\\label{hom}\n\tLet $F,G \\in \\Coh(X)$.\n\tThen\n\t\\begin{align*}\n\t\t\\Hom_{\\mathcal Q}(F,G) = \\varinjlim_{\n\t\t\t\\substack{\\emptyset \\subset U \\subset X \\\\\n\t\t\t\t X \\setminus U \\in \\Sigma}}\n\t\t\t\\Hom_U(F_U,G_U)\n\t\\end{align*}\n\twhere $F_U, G_U$ denote the restrictions to $U$.\n\\end{lem}\n\\begin{proof}\nWe will define mutually inverse maps, following \\cite[Lemma 3.6]{mp}.\n\n\n\nWe start by constructing a map from right to left.\nAny element of the right hand side may be represented as a pair $(U,f)$, where $X \\setminus U \\in \\Sigma$ and $f\\colon F_U \\to G_U$.\nWrite $\\Gamma_f \\subset F_U \\oplus G_U$ for the graph of $f$.\nLet $E \\subset F \\oplus G$ be any coherent sheaf such that $E_U = \\Gamma_f$.\nThen $ E \\xrightarrow{\\mathrm{pr}_1} F$ is a weak equivalence in $\\mathcal Q$, so the roof $F \\xleftarrow{\\mathrm{pr}_1} E \\xrightarrow{\\mathrm{pr}_2} G$ represents an element $f \\in \\Hom_{\\mathcal Q}(F,G)$.\n\nSuppose now $f'\\colon F_{U'} \\to G_{U'}$ represents the same element as $(U,f)$.\nAs above, we construct a roof $F \\xleftarrow{\\mathrm{pr}'_1} E' \\xrightarrow{\\mathrm{pr}'_2} G$.\nSince $(U,f)$ an $(U',f')$ must eventually agree, there exists an open subset $U''$ with $f_{U''}=f'_{U''}$ and such that $E''=E \\cap E'$ is equivalent to both $E$ and $E'$ through the inclusion map.\nThe commutative diagram\n\\[ \\xymatrix @C=0.5cm @R=0.5cm { & & E'' \\ar[dl]_i \\ar[dr]^{j} & & \\\\ & E \\ar[dl]_{\\mathrm{pr}_1} \\ar[drrr]_{\\mathrm{pr}_2} & & E' \\ar[dlll]^{\\mathrm{pr}'_1} \\ar[dr]^{\\mathrm{pr}'_2} & \\\\ F & & & & G } \\]\nshows that the two roofs are equivalent precisely as in \\cite[Lemma 3.6]{mp}.\n\nConsider now a morphism $f \\in \\Hom_{\\mathcal Q}(F,G)$, represented by the roof $F \\xleftarrow{s} E \\xrightarrow{t} G$.\nSince $s$ is a weak equivalence (by definition of roof), we have $\\mathrm{supp}(\\ker(s)) \\cup \\mathrm{supp}(\\coker(s)) \\in \\Sigma$.\nLet $U$ be its complement.\nOnce we restrict to $U$, $s$ becomes an isomorphism, hence the map $t \\circ s^{-1}\\colon F_U \\rightarrow G_U$ makes sense.\nTo make sure the function $f\\mapsto t \\circ s^{-1}$ is well defined, we use that two different roofs representing the same morphism must be equivalent.\nIndeed, up to restricting to a smaller subset, we will obtain the same map.\nFinally, since we are allowed to choose ad hoc representatives, one checks the two maps just defined compose to the respective identities.\n\\end{proof}\n\n\\begin{lem}\\label{min}\n\tAn object $P \\in \\Coh(X)\/\\Coh_{\\leq k-1}(X)$ is minimal if and only if either $P \\equiv 0$ or $P \\equiv \\mathcal O_Z$ for $Z \\hookrightarrow X$ an integral closed subscheme of dimension $k$.\n\\end{lem}\n\\begin{proof}\n\tLet us write $\\mathcal Q = \\Coh(X)\/\\Coh_{\\leq k-1}(X)$.\n\tWe begin by showing that $\\mathcal O_Z$ is minimal, for $Z \\hookrightarrow X$ an integral closed subscheme.\n\t\tSuppose $\\mathcal O_Z$ sits in the middle of a short exact sequence in $\\mathcal Q$.\n\t\tBy Lemma \\ref{exact}, this means that, up to passing to an equivalent object $B \\equiv \\mathcal O_Z$, there is a short exact sequence in $\\Coh(X)$\n\t\t\\begin{align*}\n\t\t\t0 \\to A \\to B \\to C \\to 0.\n\t\t\\end{align*}\n\tThe goal is to show that either $C \\equiv 0$ or that $B \\to C$ is a weak equivalence.\n\n\t\tLet $\\eta$ be the generic point of $Z$ and let $R = \\mathcal O_{X,\\eta}$ be the corresponding local ring, with maximal ideal $\\mathfrak{m}$.\n\t\tBy taking germs, we get a short exact sequence \n\t\t$$ 0 \\to A_\\eta \\to B_\\eta \\to C_\\eta \\to 0 $$\n\t\tof $R$-modules.\n\t\tSince $B$ and $\\mathcal O_Z$ are isomorphic up to $\\Coh_{\\leq k-1}(X)$, we have $B_\\eta \\simeq \\mathcal O_{Z,\\eta} = R\/\\mathfrak{m}$.\n\t\tIt follows that either $C_\\eta = 0$ or $C_\\eta = B_\\eta$.\n\n\t\tIf $C_\\eta = 0$ then $\\dim \\supp C  < k$, which implies $C \\equiv 0$.\n\t\tIf $C_\\eta = B_\\eta$, then $\\dim \\supp A < k$, which implies $C \\equiv B \\equiv \\mathcal O_Z$.\n\n\tTo prove the other direction, let $F \\in \\Coh(X)$ represent a minimal element in $\\mathcal Q$.\n\tLet $Z$ be the scheme-theoretic support of $F$, i.e.\\ $Z = V(\\Ann(F))$.\n\tIf $\\dim Z < k$, then $F \\equiv 0$.\n\tIf not, without loss of generality we may assume $Z$ is irreducible, reduced and of dimension $k$.\n\tThis previous claim follows by minimality: we always have a surjection $F \\twoheadrightarrow F\/IF$ for $I$ the ideal sheaf of a closed subscheme of $Z$.\n\t\n\tBy the argument above, we already know $\\mathcal O_Z$ is minimal in $\\mathcal Q$.\n\tOur goal is to construct a non-zero map $\\mathcal O_Z \\rightharpoonup F$, proving $\\mathcal O_Z \\equiv F$.\n\tFor this, we use Lemma \\ref{hom}.\n\t\n\tWrite $i\\colon Z \\hookrightarrow X$ for the inclusion.\n\tWe know $F = i_* F'$, where $F' = i^*F \\in \\Coh(Z)$.\n\tLet $U \\subset Z$ be an affine open.\n\tWe must have a non-torsion element $a \\in F'_U$, otherwise the whole $F'$ would be torsion and $\\dim Z < k$.\n\tView now $a$ as a non-zero map $\\mathcal O_U \\to F'$.\n\n\tLet $W = Z \\setminus U$.\n\tSince $Z$ is irreducible, we must have $\\dim W < k$.\n\tLet $\\tilde U = X \\setminus W$.\n\tWe have\n\t\\begin{align*}\n\t\t\\Hom_{\\tilde U}\\left(\\mathcal O_{\\tilde U}, F_{\\tilde U} \\right)\n\t\t= \\Hom_{\\tilde U} \\left( \\mathcal O_{\\tilde U}, i_{U*} \\left( F'_U \\right) \\right)\n\t\t= \\Hom_U \\left( \\mathcal O_U, F'_U \\right).\n\t\\end{align*}\n\tSince the map $\\mathcal O_U \\to F'_U$ is compatible with further restricting $\\tilde U$, by Lemma \\ref{hom} we have cooked up a non-zero map $\\mathcal O_Z \\rightharpoonup F$. \n\n\tThe claim follows. \n\\end{proof}\n\n\\begin{lem}\\label{small}\n\tThe subcategory $\\Coh_{\\leq k}(X) \\subset \\Coh(X)$ is the smallest Serre subcategory containing $\\Coh_{\\leq k-1}(X)$ and all the sheaves $\\mathcal O_Z$, with $Z \\hookrightarrow X$ integral closed subscheme and $\\dim Z = k$.\n\\end{lem}\n\\begin{proof}\n\tIt is easier to work in the quotient $\\mathcal Q = \\Coh(X)\/\\Coh_{\\leq k-1}(X)$.\n\tThe claim is equivalent to showing the following.\n\tSuppose $\\mathcal S \\subset \\mathcal Q$ is a Serre subcategory containing all $\\mathcal O_Z$, for $Z \\hookrightarrow X$ an integral closed subscheme with $\\dim Z = k$, and suppose $F \\in \\Coh_{\\leq k}(X)$.\n\tThen $F \\in \\mathcal S$.\n\t\n\tIf $F = \\mathcal O_Z$, with $Z$ integral, then $F \\in \\mathcal S$ by assumption.\n\tIf $F = \\mathcal O_Y$, with $Y$ reduced but possibly reducible, let $Y_1,\\ldots,Y_r$ be its irreducible components, equipped with the reduced scheme structure.\n\tThe sheaves $\\mathcal O_{Y_i}$ all belong to $\\mathcal S$ by assumption.\n\tThere is a map $\\mathcal O_Y \\to \\mathcal O_{Y_1} \\oplus \\cdots \\oplus \\mathcal O_{Y_r}$.\n\tSince the intersection of all minimal primes in a reduced ring is zero, this map is injective.\n\tAs $\\mathcal S$ is a Serre category, it follows that $\\mathcal O_Y \\in \\mathcal S$.\n\t\n\tSuppose now $F \\in \\Coh_{\\leq k}(X)$ satisfies the following: there exists a reduced $Y$, with ideal sheaf $I$, $\\dim Y \\leq k$, and $IF = 0$.\n\tLet $i\\colon Y \\hookrightarrow X$ be the inclusion.\n\tBy assumption there is an $F' \\in \\Coh(Y)$ such that $i_*F' = F$.\n\tLet now $W \\subset Y$ be a closed subset, with $\\dim W < k$, and such that $U = Y \\setminus W$ is affine.\n\tLet $\\tilde U = X \\setminus W$.\n\tThe restriction $F'_U$ is globally generated, hence there is a surjective map $\\mathcal O_U^{\\oplus r} \\twoheadrightarrow F'_U$.\n\tBy pushing forward, there is a surjection $\\mathcal O_{\\tilde U}^{\\oplus r} \\twoheadrightarrow F_{\\tilde U}$.\n\tUsing Lemma \\ref{hom}, this induces a surjection $\\mathcal O_Y^{\\oplus r} \\twoheadrightarrow F$ in the category $\\mathcal Q$.\n\tSince $\\mathcal O_Y \\in \\mathcal S$, we see that $F \\in \\mathcal S$.\n\t\n\tSuppose now $F \\in \\Coh_{\\leq k}(X)$ is arbitrary.\n\tLet $Y$ be its scheme-theoretic support.\n\tLet $I \\subset \\mathcal O_X$ be the ideal sheaf defining $Y_\\text{red}$.\n\tBy noetherianity, there exists an $m$ such that $I^{m+1}F = 0$.\n\tWe have a string of short exact sequences\n\t\\begin{align*}\n\t\t\t\t0 \\to IF  \\to F \\to F \/ I F \\to 0 \\\\\n\t\t\t\t\t\t\t0 \\to I^2 F \\to I F \\to I F \/ I^2 F \\to 0 \\\\\n\t\t\t\t\t\t\t\t\t\t\t\t\\cdots \\\\\n\t\t0 \\to I^m F \\to I^{m-1} F \\to I^{m-1} F \/ I^m F \\to 0 \n\t\\end{align*}\n\tNow, $I(I^a F \/ I^{a+1} F) =0$ for any $a$, hence they belong to $\\mathcal S$.\n\tUsing induction starting from $I(I^m F)=0$, and using as always that $\\mathcal S$ is Serre, we see that $F \\in \\mathcal S$.\n\\end{proof}\n\n\\begin{proof}[Proof of Proposition \\ref{geo}]\n\tWe proceed by induction on $k$.\n\tBy definition, $\\mathcal S_0$ is the smallest Serre subcategory containing all the minimal objects of $\\Coh(X)$.\n\tBy Lemma \\ref{min}, these are precisely the skyscraper sheaves.\n\tBy Lemma \\ref{small}, $\\mathcal S_0 = \\Coh_{\\leq 0}(X)$.\n\t\n\tSuppose the theorem is true for $k-1$, let $\\mathcal Q = \\Coh(X)\/\\Coh_{\\leq k-1}(X) = \\Coh(X)\/\\mathcal S_{k-1}$.\n\tLet $\\mathcal S$ be the smallest Serre subcategory of $\\mathcal Q$, containing all minimal objects of $\\mathcal Q$.\n\tBy definition, $\\mathcal S_k$ is the kernel of $\\Coh(X) \\to \\mathcal Q\\to \\mathcal Q\/\\mathcal S$.\n\tBy Lemma \\ref{min}, the minimal objects of $\\mathcal Q$ are precisely the $\\mathcal O_Z$ with $Z$ integral of dimension $k$.\n\tCombined with Lemma \\ref{small}, we see that $\\mathcal S$ is the image of $\\Coh_{\\leq k}(X)$.\n\tHence, $\\mathcal S_k = \\Coh_{\\leq k}(X)$.\n\\end{proof}\nThe following follows from Lemma \\ref{hom}, and will be useful later.\n\\begin{lem}\\label{residue}\nLet $P$ be a non-zero minimal object in $\\mathcal C_k$, corresponding to a (non-necessarily closed) point $x \\in X$.\nWe have $\\Hom_{\\mathcal C_k}(P,P)=\\kappa(x)$, where $\\kappa(x)$ is the residue field of the point $x$.\n\\end{lem}\nFrom now on we will write $\\mathcats C_k = \\mathcats C_k(X) = \\Coh(X)\/\\Coh_{\\leq k-1}(X)$.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\n\n\\section{Results for AdamW (Freeze-embed)}\nIn our paper, we hypothesized that AdamW and SGD (freeze-embed) improve on SGD for the same reason---they change the embedding layer less. Based on this hypothesis, we would expect that the gains of AdamW and freeze-embed are \\emph{not complementary}. Indeed, we find that the AdamW (freeze-embed) variation performs similarly to AdamW and SGD (freeze-embed). %\n\nTables~\\ref{tab:ood_bigtable_adamw_freeze}~\\&~\\ref{tab:id_bigtable_adamw_freeze} expand on the OOD and ID accuracy results in Tables~\\ref{tab:ood_bigtable_adamw_freeze}~\\&~\\ref{tab:id_bigtable_adamw_freeze}, respectively, to include AdamW (freeze-embed).\nOverall, AdamW (freeze-embed) and SGD (freeze-embed) perform comparably for all the recent vision models, both fairly close to AdamW---although there are differences in some models and datasets.\nAveraged across all the datasets and models, AdamW, AdamW (freeze-embed), and SGD (freeze-embed) get 76\\%, 76.5\\%, and 76.7\\% accuracy, respectively, compared to 72.0\\% for SGD.\nAveraged across all the \\emph{in-distribution} datasets and models, AdamW, AdamW (freeze-embed), and SGD (freeze-embed) get 91.5\\%, 91.5\\%, and 91.3\\% accuracy, respectively, compared to 90.3\\% for SGD. Overall the performances of the three models are similar enough to suggest that these methods work well for the same reason that they tune the embedding layers less, and that freeze-embed is not an independent axis of improvement.  \n\n\\begin{table}[h!]\n \\small \n \\caption{\n\\textbf{Out-of-distribution (OOD)} accuracies with AdamW (freeze-embed). This is an expansion of  Table~\\ref{tab:ood_bigtable_final} to include AdamW (freeze-embed) OOD results for fine-tuning 7 popular models across 5 benchmark datasets. On OOD performance averaged across all models and datasets, AdamW (freeze-embed) gets slightly better accuracy than AdamW but slightly worse than SGD (freeze-embed).  \n}\n\\begin{center}\n\\resizebox{\\textwidth}{!}{\n\\begin{tabular}{cccccccccccc|cc}\n\\toprule\n &  & \\multicolumn{2}{c}{Living-17}  & \\multicolumn{2}{c}{Waterbirds}  & \\multicolumn{2}{c}{DomainNet}  & \\multicolumn{2}{c}{FMoW}  & \\multicolumn{2}{c}{Camelyon}  & \\multicolumn{2}{c}{Avg.} \\\\\n\\midrule\nCLIP ViT-B\/16 & SGD & 80.0 &  & 62.5 &  & 72.8 &  & 37.3 &  & 86.8 &  & 67.9 & \\\\\nCLIP ViT-B\/16 & AdamW & 82.8 & \\hspace{-1em}{\\hgreen{(+2.8)}} & 71.9 & \\hspace{-1em}{\\hgreen{(+9.4)}} & \\textbf{89.2} & \\hspace{-1em}{\\hgreen{(+16.4)}} & \\textbf{40.7} & \\hspace{-1em}{\\hgreen{(+3.4)}} & 95.7 & \\hspace{-1em}{\\hgreen{(+8.9)}} & \\textbf{76.0} & \\hspace{-1em}{\\hgreen{(+8.1)}}\\\\\nCLIP ViT-B\/16 & SGD (freeze-embed) & \\textbf{83.2} & \\hspace{-1em}{\\hgreen{(+3.2)}} & \\textbf{73.7} & \\hspace{-1em}{\\hgreen{(+11.2)}} & 88.2 & \\hspace{-1em}{\\hgreen{(+15.4)}} & 40.2 & \\hspace{-1em}{\\hgreen{(+2.9)}} & 94.3 & \\hspace{-1em}{\\hgreen{(+7.5)}} & \\textbf{75.9} & \\hspace{-1em}{\\hgreen{(+8.0)}}\\\\\nCLIP ViT-B\/16 & AdamW (freeze-embed) & 82.4 & \\hspace{-1em}{\\hgreen{(+2.4)}} & 69.5 & \\hspace{-1em}{\\hgreen{(+7.0)}} & 88.2 & \\hspace{-1em}{\\hgreen{(+15.4)}} & \\textbf{40.7} & \\hspace{-1em}{\\hgreen{(+3.4)}} & \\textbf{96.7} & \\hspace{-1em}{\\hgreen{(+9.9)}} & 75.5 & \\hspace{-1em}{\\hgreen{(+7.6)}}\\\\\n\\midrule\nCLIP ViT-L\/14 & SGD & 84.2 &  & 65.0 &  & 60.8 &  & 41.0 &  & 83.2 &  & 66.8 & \\\\\nCLIP ViT-L\/14 & AdamW & 88.0 & \\hspace{-1em}{\\hgreen{(+3.8)}} & \\textbf{85.2} & \\hspace{-1em}{\\hgreen{(+20.2)}} & \\textbf{93.8} & \\hspace{-1em}{\\hgreen{(+33.0)}} & 48.3 & \\hspace{-1em}{\\hgreen{(+7.3)}} & 95.9 & \\hspace{-1em}{\\hgreen{(+12.7)}} & 82.2 & \\hspace{-1em}{\\hgreen{(+15.4)}}\\\\\nCLIP ViT-L\/14 & SGD (freeze-embed) & \\textbf{90.5} & \\hspace{-1em}{\\hgreen{(+6.3)}} & 84.7 & \\hspace{-1em}{\\hgreen{(+19.7)}} & 93.1 & \\hspace{-1em}{\\hgreen{(+32.3)}} & \\textbf{49.9} & \\hspace{-1em}{\\hgreen{(+8.9)}} & \\textbf{96.5} & \\hspace{-1em}{\\hgreen{(+13.3)}} & \\textbf{83.0} & \\hspace{-1em}{\\hgreen{(+16.2)}}\\\\\nCLIP ViT-L\/14 & AdamW (freeze-embed) & 88.0 & \\hspace{-1em}{\\hgreen{(+3.8)}} & 84.6 & \\hspace{-1em}{\\hgreen{(+19.6)}} & \\textbf{93.8} & \\hspace{-1em}{\\hgreen{(+33.0)}} & 44.6 & \\hspace{-1em}{\\hgreen{(+3.6)}} & \\textbf{96.4} & \\hspace{-1em}{\\hgreen{(+13.2)}} & 81.5 & \\hspace{-1em}{\\hgreen{(+14.7)}}\\\\\n\\midrule\nSup ViT-B\/16 & SGD & \\textbf{89.5} &  & 77.4 &  & \\textbf{86.3} &  & 33.5 &  & 92.6 &  & 75.8 & \\\\\nSup ViT-B\/16 & AdamW & 88.3 & \\hspace{-1em}{\\hred{(-1.2)}} & 81.6 & \\hspace{-1em}{\\hgreen{(+4.2)}} & 84.4 & \\hspace{-1em}{\\hred{(-1.9)}} & \\textbf{35.9} & \\hspace{-1em}{\\hgreen{(+2.4)}} & 87.9 & \\hspace{-1em}{\\hred{(-4.7)}} & 75.6 & \\hspace{-1em}{\\hred{(-0.2)}}\\\\\nSup ViT-B\/16 & SGD (freeze-embed) & 88.0 & \\hspace{-1em}{\\hred{(-1.5)}} & \\textbf{82.4} & \\hspace{-1em}{\\hgreen{(+5.0)}} & \\textbf{86.3} & \\hspace{-1em}{\\lgreen{(+0.0)}} & 34.4 & \\hspace{-1em}{\\hgreen{(+0.9)}} & \\textbf{93.7} & \\hspace{-1em}{\\hgreen{(+1.1)}} & \\textbf{77.0} & \\hspace{-1em}{\\hgreen{(+1.2)}}\\\\\nSup ViT-B\/16 & AdamW (freeze-embed) & 88.1 & \\hspace{-1em}{\\hred{(-1.4)}} & \\textbf{82.4} & \\hspace{-1em}{\\hgreen{(+5.0)}} & 82.3 & \\hspace{-1em}{\\hred{(-4.0)}} & 35.7 & \\hspace{-1em}{\\hgreen{(+2.2)}} & 93.0 & \\hspace{-1em}{\\hgreen{(+0.4)}} & 76.3 & \\hspace{-1em}{\\hgreen{(+0.5)}}\\\\\n\\midrule\nDINO ViT-B\/16 & SGD & \\textbf{88.2} &  & 56.1 &  & 76.0 &  & 33.6 &  & 86.9 &  & 68.2 & \\\\\nDINO ViT-B\/16 & AdamW & 87.4 & \\hspace{-1em}{\\hred{(-0.8)}} & 61.2 & \\hspace{-1em}{\\hgreen{(+5.1)}} & 77.4 & \\hspace{-1em}{\\hgreen{(+1.4)}} & \\textbf{35.8} & \\hspace{-1em}{\\hgreen{(+2.2)}} & 91.9 & \\hspace{-1em}{\\hgreen{(+5.0)}} & 70.7 & \\hspace{-1em}{\\hgreen{(+2.5)}}\\\\\nDINO ViT-B\/16 & SGD (freeze-embed) & 86.7 & \\hspace{-1em}{\\hred{(-1.5)}} & \\textbf{67.9} & \\hspace{-1em}{\\hgreen{(+11.8)}} & \\textbf{78.4} & \\hspace{-1em}{\\hgreen{(+2.4)}} & \\textbf{35.9} & \\hspace{-1em}{\\hgreen{(+2.3)}} & 90.6 & \\hspace{-1em}{\\hgreen{(+3.7)}} & \\textbf{71.9} & \\hspace{-1em}{\\hgreen{(+3.7)}}\\\\\nDINO ViT-B\/16 & AdamW (freeze-embed) & 86.8 & \\hspace{-1em}{\\hred{(-1.4)}} & 64.5 & \\hspace{-1em}{\\hgreen{(+8.4)}} & 76.6 & \\hspace{-1em}{\\hgreen{(+0.6)}} & 35.4 & \\hspace{-1em}{\\hgreen{(+1.8)}} & \\textbf{93.5} & \\hspace{-1em}{\\hgreen{(+6.6)}} & 71.4 & \\hspace{-1em}{\\hgreen{(+3.2)}}\\\\\n\\midrule\nConvNext-Base & SGD & \\textbf{94.0} &  & 80.2 &  & 89.8 &  & 39.7 &  & 83.0 &  & 77.3 & \\\\\nConvNext-Base & AdamW & 90.3 & \\hspace{-1em}{\\hred{(-3.7)}} & 89.8 & \\hspace{-1em}{\\hgreen{(+9.6)}} & 89.5 & \\hspace{-1em}{\\hred{(-0.3)}} & 38.4 & \\hspace{-1em}{\\hred{(-1.3)}} & \\textbf{89.5} & \\hspace{-1em}{\\hgreen{(+6.5)}} & 79.5 & \\hspace{-1em}{\\hgreen{(+2.2)}}\\\\\nConvNext-Base & SGD (freeze-embed) & 92.6 & \\hspace{-1em}{\\hred{(-1.4)}} & 86.9 & \\hspace{-1em}{\\hgreen{(+6.7)}} & \\textbf{91.2} & \\hspace{-1em}{\\hgreen{(+1.4)}} & 38.2 & \\hspace{-1em}{\\hred{(-1.5)}} & 88.1 & \\hspace{-1em}{\\hgreen{(+5.1)}} & 79.4 & \\hspace{-1em}{\\hgreen{(+2.1)}}\\\\\nConvNext-Base & AdamW (freeze-embed) & 88.1 & \\hspace{-1em}{\\hred{(-5.9)}} & \\textbf{91.6} & \\hspace{-1em}{\\hgreen{(+11.4)}} & 89.8 & \\hspace{-1em}{\\lgreen{(+0.0)}} & \\textbf{41.6} & \\hspace{-1em}{\\hgreen{(+1.9)}} & 87.7 & \\hspace{-1em}{\\hgreen{(+4.7)}} & \\textbf{79.8} & \\hspace{-1em}{\\hgreen{(+2.5)}}\\\\\n\\midrule\nBiT ResNet-50 & SGD & \\textbf{84.3} &  & 76.5 &  & 80.0 &  & 34.1 &  & 90.4 &  & 73.1 & \\\\\nBiT ResNet-50 & AdamW & 83.1 & \\hspace{-1em}{\\hred{(-1.2)}} & 74.8 & \\hspace{-1em}{\\hred{(-1.7)}} & \\textbf{84.0} & \\hspace{-1em}{\\hgreen{(+4.0)}} & 33.8 & \\hspace{-1em}{\\hred{(-0.3)}} & 92.4 & \\hspace{-1em}{\\hgreen{(+2.0)}} & 73.6 & \\hspace{-1em}{\\hgreen{(+0.5)}}\\\\\nBiT ResNet-50 & SGD (freeze-embed) & 84.1 & \\hspace{-1em}{\\hred{(-0.2)}} & 75.5 & \\hspace{-1em}{\\hred{(-1.0)}} & 82.3 & \\hspace{-1em}{\\hgreen{(+2.3)}} & 35.0 & \\hspace{-1em}{\\hgreen{(+0.9)}} & \\textbf{95.2} & \\hspace{-1em}{\\hgreen{(+4.8)}} & 74.4 & \\hspace{-1em}{\\hgreen{(+1.3)}}\\\\\nBiT ResNet-50 & AdamW (freeze-embed) & 82.9 & \\hspace{-1em}{\\hred{(-1.4)}} & \\textbf{77.3} & \\hspace{-1em}{\\hgreen{(+0.8)}} & 83.3 & \\hspace{-1em}{\\hgreen{(+3.3)}} & \\textbf{36.3} & \\hspace{-1em}{\\hgreen{(+2.2)}} & 93.7 & \\hspace{-1em}{\\hgreen{(+3.3)}} & \\textbf{74.7} & \\hspace{-1em}{\\hgreen{(+1.6)}}\\\\\n\\midrule\nBiT ResNet-101 & SGD & 82.8 &  & 76.9 &  & \\textbf{86.2} &  & \\textbf{38.0} &  & 89.3 &  & 74.6 & \\\\\nBiT ResNet-101 & AdamW & 82.9 & \\hspace{-1em}{\\lgreen{(+0.1)}} & \\textbf{79.4} & \\hspace{-1em}{\\hgreen{(+2.5)}} & 83.5 & \\hspace{-1em}{\\hred{(-2.7)}} & 37.0 & \\hspace{-1em}{\\hred{(-1.0)}} & 89.7 & \\hspace{-1em}{\\hgreen{(+0.4)}} & 74.5 & \\hspace{-1em}{\\lred{(-0.1)}}\\\\\nBiT ResNet-101 & SGD (freeze-embed) & 83.1 & \\hspace{-1em}{\\hgreen{(+0.3)}} & 77.3 & \\hspace{-1em}{\\hgreen{(+0.4)}} & 86.0 & \\hspace{-1em}{\\hred{(-0.2)}} & 36.0 & \\hspace{-1em}{\\hred{(-2.0)}} & \\textbf{95.5} & \\hspace{-1em}{\\hgreen{(+6.2)}} & 75.6 & \\hspace{-1em}{\\hgreen{(+1.0)}}\\\\\nBiT ResNet-101 & AdamW (freeze-embed) & \\textbf{84.8} & \\hspace{-1em}{\\hgreen{(+2.0)}} & 78.3 & \\hspace{-1em}{\\hgreen{(+1.4)}} & 84.3 & \\hspace{-1em}{\\hred{(-1.9)}} & \\textbf{38.2} & \\hspace{-1em}{\\lgreen{(+0.2)}} & 95.0 & \\hspace{-1em}{\\hgreen{(+5.7)}} & \\textbf{76.1} & \\hspace{-1em}{\\hgreen{(+1.5)}}\\\\\n\\bottomrule\n\\end{tabular}\n}\n\\end{center}\n\\label{tab:ood_bigtable_adamw_freeze}\n\\end{table}\n\n\n\n\n\n\\begin{table}[h!]\n \\small \n\\caption{\n\\textbf{In-distribution (ID)} accuracies with AdamW (freeze-embed). This is an expansion of Table~\\ref{tab:id_bigtable_final}\nto include AdamW (freeze-embed) ID results for our 7 models and 5 datasets. AdamW, AdamW (freeze-embed), and SGD (freeze-embed) all perform comparably on ID accuracies. \n}\n\\label{tab:id_bigtable_adamw_freeze}\n\\begin{center}\n\\resizebox{\\textwidth}{!}{\n\\begin{tabular}{cccccccccccc|cc}\n\\toprule\n &  & \\multicolumn{2}{c}{Living-17}  & \\multicolumn{2}{c}{Waterbirds}  & \\multicolumn{2}{c}{DomainNet}  & \\multicolumn{2}{c}{FMoW}  & \\multicolumn{2}{c}{Camelyon}  & \\multicolumn{2}{c}{Avg.} \\\\\n\\midrule\nCLIP ViT-B\/16 & SGD & 97.8 &  & 97.2 &  & 88.8 &  & 67.0 &  & \\textbf{99.4} &  & 90.0 & \\\\\nCLIP ViT-B\/16 & AdamW & \\textbf{98.1} & \\hspace{-1em}{\\hgreen{(+0.3)}} & \\textbf{97.7} & \\hspace{-1em}{\\hgreen{(+0.5)}} & 95.0 & \\hspace{-1em}{\\hgreen{(+6.2)}} & 70.1 & \\hspace{-1em}{\\hgreen{(+3.1)}} & \\textbf{99.5} & \\hspace{-1em}{\\lgreen{(+0.1)}} & \\textbf{92.1} & \\hspace{-1em}{\\hgreen{(+2.1)}}\\\\\nCLIP ViT-B\/16 & SGD (freeze-embed) & \\textbf{98.2} & \\hspace{-1em}{\\hgreen{(+0.4)}} & \\textbf{97.8} & \\hspace{-1em}{\\hgreen{(+0.6)}} & 94.9 & \\hspace{-1em}{\\hgreen{(+6.1)}} & 70.0 & \\hspace{-1em}{\\hgreen{(+3.0)}} & \\textbf{99.5} & \\hspace{-1em}{\\lgreen{(+0.1)}} & 92.1 & \\hspace{-1em}{\\hgreen{(+2.1)}}\\\\\nCLIP ViT-B\/16 & AdamW (freeze-embed) & \\textbf{98.3} & \\hspace{-1em}{\\hgreen{(+0.5)}} & \\textbf{97.8} & \\hspace{-1em}{\\hgreen{(+0.6)}} & \\textbf{95.3} & \\hspace{-1em}{\\hgreen{(+6.5)}} & \\textbf{70.5} & \\hspace{-1em}{\\hgreen{(+3.5)}} & \\textbf{99.6} & \\hspace{-1em}{\\lgreen{(+0.2)}} & \\textbf{92.3} & \\hspace{-1em}{\\hgreen{(+2.3)}}\\\\\n\\midrule\nCLIP ViT-L\/14 & SGD & 98.4 &  & 97.3 &  & 84.3 &  & 69.0 &  & \\textbf{99.4} &  & 89.7 & \\\\\nCLIP ViT-L\/14 & AdamW & \\textbf{98.9} & \\hspace{-1em}{\\hgreen{(+0.5)}} & \\textbf{98.8} & \\hspace{-1em}{\\hgreen{(+1.5)}} & 96.9 & \\hspace{-1em}{\\hgreen{(+12.6)}} & 74.5 & \\hspace{-1em}{\\hgreen{(+5.5)}} & \\textbf{99.6} & \\hspace{-1em}{\\lgreen{(+0.2)}} & \\textbf{93.7} & \\hspace{-1em}{\\hgreen{(+4.0)}}\\\\\nCLIP ViT-L\/14 & SGD (freeze-embed) & \\textbf{98.7} & \\hspace{-1em}{\\hgreen{(+0.3)}} & \\textbf{98.9} & \\hspace{-1em}{\\hgreen{(+1.6)}} & \\textbf{97.1} & \\hspace{-1em}{\\hgreen{(+12.8)}} & 74.5 & \\hspace{-1em}{\\hgreen{(+5.5)}} & \\textbf{99.6} & \\hspace{-1em}{\\lgreen{(+0.2)}} & \\textbf{93.7} & \\hspace{-1em}{\\hgreen{(+4.0)}}\\\\\nCLIP ViT-L\/14 & AdamW (freeze-embed) & \\textbf{98.7} & \\hspace{-1em}{\\hgreen{(+0.3)}} & \\textbf{98.8} & \\hspace{-1em}{\\hgreen{(+1.5)}} & 96.7 & \\hspace{-1em}{\\hgreen{(+12.4)}} & \\textbf{75.1} & \\hspace{-1em}{\\hgreen{(+6.1)}} & \\textbf{99.6} & \\hspace{-1em}{\\lgreen{(+0.2)}} & \\textbf{93.8} & \\hspace{-1em}{\\hgreen{(+4.1)}}\\\\\n\\midrule\nSup ViT-B\/16 & SGD & \\textbf{98.6} &  & \\textbf{99.1} &  & \\textbf{91.7} &  & 64.1 &  & \\textbf{99.4} &  & 90.6 & \\\\\nSup ViT-B\/16 & AdamW & \\textbf{98.7} & \\hspace{-1em}{\\lgreen{(+0.1)}} & \\textbf{99.0} & \\hspace{-1em}{\\lred{(-0.1)}} & \\textbf{91.7} & \\hspace{-1em}{\\lred{(-0.0)}} & 66.4 & \\hspace{-1em}{\\hgreen{(+2.3)}} & \\textbf{99.5} & \\hspace{-1em}{\\lgreen{(+0.1)}} & \\textbf{91.1} & \\hspace{-1em}{\\hgreen{(+0.5)}}\\\\\nSup ViT-B\/16 & SGD (freeze-embed) & \\textbf{98.7} & \\hspace{-1em}{\\lgreen{(+0.1)}} & \\textbf{99.2} & \\hspace{-1em}{\\lgreen{(+0.1)}} & 91.5 & \\hspace{-1em}{\\hred{(-0.2)}} & 65.0 & \\hspace{-1em}{\\hgreen{(+0.9)}} & \\textbf{99.6} & \\hspace{-1em}{\\lgreen{(+0.2)}} & 90.8 & \\hspace{-1em}{\\hgreen{(+0.2)}}\\\\\nSup ViT-B\/16 & AdamW (freeze-embed) & \\textbf{98.6} & \\hspace{-1em}{\\lgreen{(+0.0)}} & \\textbf{99.0} & \\hspace{-1em}{\\lred{(-0.1)}} & 90.9 & \\hspace{-1em}{\\hred{(-0.8)}} & \\textbf{66.8} & \\hspace{-1em}{\\hgreen{(+2.7)}} & \\textbf{99.5} & \\hspace{-1em}{\\lgreen{(+0.1)}} & \\textbf{91.0} & \\hspace{-1em}{\\hgreen{(+0.4)}}\\\\\n\\midrule\nDINO ViT-B\/16 & SGD & \\textbf{98.4} &  & 97.0 &  & 88.2 &  & 62.4 &  & 99.4 &  & 89.1 & \\\\\nDINO ViT-B\/16 & AdamW & \\textbf{98.5} & \\hspace{-1em}{\\lgreen{(+0.1)}} & \\textbf{97.9} & \\hspace{-1em}{\\hgreen{(+0.9)}} & 89.4 & \\hspace{-1em}{\\hgreen{(+1.2)}} & \\textbf{66.0} & \\hspace{-1em}{\\hgreen{(+3.6)}} & \\textbf{99.6} & \\hspace{-1em}{\\hgreen{(+0.2)}} & \\textbf{90.3} & \\hspace{-1em}{\\hgreen{(+1.2)}}\\\\\nDINO ViT-B\/16 & SGD (freeze-embed) & \\textbf{98.4} & \\hspace{-1em}{\\lgreen{(+0.0)}} & 97.5 & \\hspace{-1em}{\\hgreen{(+0.5)}} & 89.0 & \\hspace{-1em}{\\hgreen{(+0.8)}} & 63.5 & \\hspace{-1em}{\\hgreen{(+1.1)}} & \\textbf{99.5} & \\hspace{-1em}{\\lgreen{(+0.1)}} & 89.6 & \\hspace{-1em}{\\hgreen{(+0.5)}}\\\\\nDINO ViT-B\/16 & AdamW (freeze-embed) & \\textbf{98.4} & \\hspace{-1em}{\\lgreen{(+0.0)}} & \\textbf{97.8} & \\hspace{-1em}{\\hgreen{(+0.8)}} & \\textbf{89.7} & \\hspace{-1em}{\\hgreen{(+1.5)}} & 65.7 & \\hspace{-1em}{\\hgreen{(+3.3)}} & \\textbf{99.6} & \\hspace{-1em}{\\hgreen{(+0.2)}} & \\textbf{90.3} & \\hspace{-1em}{\\hgreen{(+1.2)}}\\\\\n\\midrule\nConvNext-Base & SGD & \\textbf{98.7} &  & 99.0 &  & 94.8 &  & 66.3 &  & 99.4 &  & 91.6 & \\\\\nConvNext-Base & AdamW & \\textbf{98.6} & \\hspace{-1em}{\\lred{(-0.1)}} & \\textbf{99.5} & \\hspace{-1em}{\\hgreen{(+0.5)}} & 94.5 & \\hspace{-1em}{\\hred{(-0.3)}} & 68.8 & \\hspace{-1em}{\\hgreen{(+2.5)}} & \\textbf{99.7} & \\hspace{-1em}{\\hgreen{(+0.3)}} & \\textbf{92.2} & \\hspace{-1em}{\\hgreen{(+0.6)}}\\\\\nConvNext-Base & SGD (freeze-embed) & \\textbf{98.6} & \\hspace{-1em}{\\lred{(-0.1)}} & \\textbf{99.4} & \\hspace{-1em}{\\hgreen{(+0.4)}} & \\textbf{95.1} & \\hspace{-1em}{\\hgreen{(+0.3)}} & 67.4 & \\hspace{-1em}{\\hgreen{(+1.1)}} & \\textbf{99.5} & \\hspace{-1em}{\\lgreen{(+0.1)}} & 92.0 & \\hspace{-1em}{\\hgreen{(+0.4)}}\\\\\nConvNext-Base & AdamW (freeze-embed) & \\textbf{98.7} & \\hspace{-1em}{\\lgreen{(+0.0)}} & \\textbf{99.5} & \\hspace{-1em}{\\hgreen{(+0.5)}} & 94.7 & \\hspace{-1em}{\\lred{(-0.1)}} & \\textbf{69.2} & \\hspace{-1em}{\\hgreen{(+2.9)}} & \\textbf{99.6} & \\hspace{-1em}{\\hgreen{(+0.2)}} & \\textbf{92.4} & \\hspace{-1em}{\\hgreen{(+0.8)}}\\\\\n\\midrule\nBiT ResNet-50 & SGD & 97.4 &  & \\textbf{98.4} &  & \\textbf{89.3} &  & 64.6 &  & \\textbf{99.5} &  & \\textbf{89.8} & \\\\\nBiT ResNet-50 & AdamW & 97.2 & \\hspace{-1em}{\\hred{(-0.2)}} & \\textbf{98.5} & \\hspace{-1em}{\\lgreen{(+0.1)}} & \\textbf{89.2} & \\hspace{-1em}{\\lred{(-0.1)}} & \\textbf{65.1} & \\hspace{-1em}{\\hgreen{(+0.5)}} & \\textbf{99.5} & \\hspace{-1em}{\\lgreen{(+0.0)}} & \\textbf{89.9} & \\hspace{-1em}{\\lgreen{(+0.1)}}\\\\\nBiT ResNet-50 & SGD (freeze-embed) & \\textbf{97.6} & \\hspace{-1em}{\\hgreen{(+0.2)}} & \\textbf{98.5} & \\hspace{-1em}{\\lgreen{(+0.1)}} & \\textbf{89.2} & \\hspace{-1em}{\\lred{(-0.1)}} & 64.8 & \\hspace{-1em}{\\lgreen{(+0.2)}} & \\textbf{99.5} & \\hspace{-1em}{\\lgreen{(+0.0)}} & \\textbf{89.9} & \\hspace{-1em}{\\lgreen{(+0.1)}}\\\\\nBiT ResNet-50 & AdamW (freeze-embed) & \\textbf{97.4} & \\hspace{-1em}{\\lgreen{(+0.0)}} & \\textbf{98.4} & \\hspace{-1em}{\\lred{(-0.0)}} & \\textbf{89.1} & \\hspace{-1em}{\\hred{(-0.2)}} & 64.9 & \\hspace{-1em}{\\hgreen{(+0.3)}} & \\textbf{99.6} & \\hspace{-1em}{\\lgreen{(+0.1)}} & \\textbf{89.9} & \\hspace{-1em}{\\lgreen{(+0.1)}}\\\\\n\\midrule\nBiT ResNet-101 & SGD & \\textbf{98.3} &  & \\textbf{98.9} &  & \\textbf{92.0} &  & 66.0 &  & \\textbf{99.4} &  & \\textbf{90.9} & \\\\\nBiT ResNet-101 & AdamW & \\textbf{98.4} & \\hspace{-1em}{\\lgreen{(+0.1)}} & 98.6 & \\hspace{-1em}{\\hred{(-0.3)}} & 91.1 & \\hspace{-1em}{\\hred{(-0.9)}} & \\textbf{67.0} & \\hspace{-1em}{\\hgreen{(+1.0)}} & \\textbf{99.6} & \\hspace{-1em}{\\lgreen{(+0.2)}} & \\textbf{90.9} & \\hspace{-1em}{\\lred{(-0.0)}}\\\\\nBiT ResNet-101 & SGD (freeze-embed) & \\textbf{98.4} & \\hspace{-1em}{\\lgreen{(+0.1)}} & \\textbf{98.8} & \\hspace{-1em}{\\lred{(-0.1)}} & 91.5 & \\hspace{-1em}{\\hred{(-0.5)}} & 65.9 & \\hspace{-1em}{\\lred{(-0.1)}} & \\textbf{99.5} & \\hspace{-1em}{\\lgreen{(+0.1)}} & \\textbf{90.8} & \\hspace{-1em}{\\lred{(-0.1)}}\\\\\nBiT ResNet-101 & AdamW (freeze-embed) & \\textbf{98.2} & \\hspace{-1em}{\\lred{(-0.1)}} & 98.7 & \\hspace{-1em}{\\hred{(-0.2)}} & 91.1 & \\hspace{-1em}{\\hred{(-0.9)}} & 66.6 & \\hspace{-1em}{\\hgreen{(+0.6)}} & \\textbf{99.6} & \\hspace{-1em}{\\lgreen{(+0.2)}} & \\textbf{90.9} & \\hspace{-1em}{\\lred{(-0.0)}}\\\\\n\\bottomrule\n\\end{tabular}\n}\n\\end{center}\n\\end{table}\n\n\n\n\n\n\n\n\n\\section{Ablations on CLIP}\nIn  Tables~\\ref{tab:ood_clip_all_optimizers_ablation}-\\ref{tab:id_clip_all_optimizers_ablation}, we show additional ablations for the CLIP ViT-B\/16 model on all datasets.\\footnote{Running each ablation for all models on all datasets is too computationally expensive.} \nWe tried:\n\\begin{enumerate}[1.]\n    %\n    \\item SGD (freeze-embed, not layer-norm): For the CLIP model our freeze-embed variation freezes the bottom embedding layer along with the layernorm right after that. We ran an ablation where we only freeze the bottom linear embedding layer, but not the layer norm. This performs comparably with SGD (freeze-embed), which suggests that freezing the input layer is what's important, and the layer norm does not matter much.\n    \\item SGD (no momentum): Since SGD (freeze-embed, no momentum) performed very well in our experiments, we also tried fine-tuning with full SGD (no freezing), but without momentum. We found that SGD (no momentum) and SGD perform comparable.\n    \\item SGD (weight decay): Vanilla SGD is done without weight decay, but AdamW incorporates weight decay. We ran this ablation to confirm that the gains of AdamW are not because of weight decay. We used the torch SGD optimizer, and set the weight\\_decay argument to 0.01. Indeed, we found that SGD and SGD (weight decay) perform comparably, which suggests that weight\\_decay is not the reason for the improved performance of AdamW.\n    \\item Linear probing: We freeze the pretrained model, and only train a linear probe on the features of the CLIP ViT-B\/16 model. We train a logistic regression classifier using the sklearn library, sweeping over 50 regularization values in np.logspace$(-7, 2, 5)$\n\\end{enumerate}\n\n\\begin{table}[ht]\n \\small \n\\caption{\n\\textbf{Out-of-distribution (OOD)} accuracy of more optimizers on CLIP ViT-B\/16. We find that weight decay, momentum, and unfreezing the layer norm at the bottom of the model do not make much of a difference.\n}\n\\label{tab:ood_clip_all_optimizers_ablation}\n\\begin{center}\n\\resizebox{0.89\\textwidth}{!}{\n\\begin{tabular}{lccccc|c}\n\\toprule\n & Living-17 & Waterbirds & DomainNet & FMoW & Camelyon & Avg.\\\\\n\\midrule\nSGD & 80.0 & 62.5 & 72.8 & 37.3 & 86.8 & 67.9\\\\\nAdamW & 82.8 & 71.9 & 89.2 & \\textbf{40.7} & 95.7 & 76.0\\\\\n\\midrule \n\\multicolumn{6}{l|}{\\textit{Our methods}}&\\vspace{0.05in}\\\\\nSGD (freeze-embed)  & 83.2 & 73.7 & 88.2 & 40.2 & 94.3 & 75.9\\\\\nSGD (freeze-embed, no momentum) & 83.1 & \\textbf{80.4} & 89.0 & 38.8 & 93.3 & \\textbf{76.9}\\\\\nGradual-unfreezing & 81.9 & 69.1 & \\textbf{93.2} & \\textbf{40.5} & \\textbf{96.5} & 76.2\\\\\n\\midrule\n\\multicolumn{6}{l|}{\\textit{Other layerwise normalization methods}}&\\vspace{0.05in}\\\\\nLAMB & 79.5 & 64.0 & 90.4 & 38.8 & 93.4 & 73.2\\\\\nLARS & 83.9 & 48.6 & 83.8 & 38.6 & 93.3 & 69.6\\\\\n\\midrule\n\\multicolumn{6}{l|}{\\textit{Variations of SGD (freeze-embed)}}&\\vspace{0.05in}\\\\\nSGD (freeze-embed, not layer-norm) & 83.6 & 74.3 & 89.1 & 39.3 & 92.9 & 75.9\\\\\nOther freezing: Linear probing & \\textbf{86.2} & 60.4 & 89.1 & 29.0 & 92.6 & 71.5 \\\\\n\\midrule\n\\multicolumn{6}{l|}{\\textit{Variations of SGD without any freezing}}&\\vspace{0.05in}\\\\\nSGD (no momentum) & 81.4 & 59.2 & 76.7 & 37.9 & 84.3 & 67.9\\\\\nSGD (weight decay) & 83.9 & 65.1 & 67.5 & 37.1 & 85.6 & 67.8\\\\\n\\bottomrule\n\\end{tabular}\n}\n\\end{center}\n\\end{table}\n\n\n\\begin{table}[ht]\n \\small \n\\caption{\n\\textbf{In-distribution (ID)} accuracy of more optimizers on CLIP ViT-B\/16. We find that weight decay, momentum, and unfreezing the layer norm at the bottom of the model do not make much of a difference.\n}\n\\label{tab:id_clip_all_optimizers_ablation}\n\\begin{center}\n\\resizebox{0.89\\textwidth}{!}{\n\\begin{tabular}{lccccc|c}\n\\toprule\n & Living-17 & Waterbirds & DomainNet & FMoW & Camelyon & Avg.\\\\\n\\midrule\nSGD & 97.8 & 97.2 & 88.8 & 67.0 & \\textbf{99.4} & 90.0\\\\\nAdamW & 98.1 & 97.7 & 95.0 & \\textbf{70.1} & \\textbf{99.5} & \\textbf{92.1}\\\\\n\\midrule \n\\multicolumn{6}{l|}{\\textit{Our methods}}&\\vspace{0.05in}\\\\\nSGD (freeze-embed) & \\textbf{98.2} & 97.8 & 94.9 & \\textbf{70.0} & \\textbf{99.5} & \\textbf{92.1}\\\\\nSGD (freeze-embed, no momentum) & \\textbf{98.2} & 97.9 & 95.2 & \\textbf{70.1} & \\textbf{99.5} & \\textbf{92.2}\\\\\nGradual-unfreezing & \\textbf{98.3} & \\textbf{98.3} & \\textbf{96.3} & 69.2 & 99.3 & \\textbf{92.3}\\\\\n\\midrule\n\\multicolumn{6}{l|}{\\textit{Other layerwise normalization methods}}&\\vspace{0.05in}\\\\\nLAMB & \\textbf{98.2} & 97.8 & 95.1 & 67.9 & \\textbf{99.5} & 91.7\\\\\nLARS & 97.7 & 97.1 & 93.2 & 67.0 & 99.3 & 90.9\\\\\n\\midrule\n\\multicolumn{6}{l|}{\\textit{Variations of SGD (freeze-embed)}}&\\vspace{0.05in}\\\\\nSGD (freeze-embed, not layer-norm) & 98.0 & 98.0 & 95.4 & \\textbf{70.2} & \\textbf{99.5} & \\textbf{92.2}\\\\\nOther freezing: Linear probing & 97.8 & 96.6 & 94.5 & 47.2 & 96.1 & 86.4 \\\\\n\\midrule \n\\multicolumn{6}{l|}{\\textit{Variations of SGD without any freezing}}&\\vspace{0.05in}\\\\\nSGD (no momentum) & 98.0 & 97.1 & 89.5 & 66.4 & 99.3 & 90.1\\\\\nSGD (weight decay) & 97.6 & 97.2 & 87.9 & 66.4 & 99.3 & 89.7\\\\\n\\bottomrule\n\\end{tabular}\n}\n\\end{center}\n\\end{table}\n\n\\\n\n\n\\section{Additional plots on gradient norms at the pretrained initialization}\\label{app:gradnorm}\n\nIn Figure~\\ref{fig:gradient_norm_others} we show plots for the gradients norms at different layers for Living-17 and Waterbirds. These are analogous to Figure~\\ref{fig:gradient_norm} for DomainNet  in the main paper. We again see that, among the pretrained layers, the embedding layer has much higher gradients in the modern architectures. \n\nWe also consider two ways of normalizing the gradient magnitude. In the first method, we divide the norm of the gradient by the norm of the parameters in order to capture the relative ``movement'' in the first SGD step as opposed to the absolute ``movement'' which is captured by gradient norm itself. In the second method, we divide the norm of the gradient by the square root of the number of parameters. This is to check that a layer does not simply have a larger gradient because it has more parameters. The reason we use square root is as follows: suppose each parameter has gradient $\\approx c$, then the layerwise gradient norm scales with the square root of the number of parameters. Also, the first step of AdamW update is essentially a signed gradient descent step, wherein if we ignore weight decay, the per-layer ``movement'' is the square root of the number of parameters. So this normalization can be thought of as relative size of SGD update compared to AdamW in each layer at initialization. For visualization purposes, we exclude the `head' layer gradient in these plots as they often much larger than the others so the plots become hard to see if we include the `head' layer. Note that we expect the head layer to have higher gradients because it is randomly initialized~\\citep{kumar2022finetuning}. For ViT models, we omit gradients of the cls token, position embedding, and layer norm after the embedding layer.\n\n\n\n\n\\begin{figure}[htb]\n\\centering\n    \\begin{subfigure}{\\textwidth}\n    \\centering\n    \\includegraphics[width=0.76\\textwidth]{figures\/living17_grad_plot.png}\n    \\caption{Layer-wise gradient norms on Living-17 at pretrained initialization} \n    \\end{subfigure}\\\\\n    \n    \\begin{subfigure}{\\textwidth}\n    \\centering\n    \\includegraphics[width=0.76\\textwidth]{figures\/waterbirds_grad_plot.png}\n    \\caption{Layer-wise gradient norms on Waterbirds at pretrained initialization} \n    \\end{subfigure}\n    \\caption{\n    We visualize the layer-wise gradient norms our models on (a) Living-17 and (b) Waterbirds, at the pretrained initialization. The format is the same as Figure~\\ref{fig:gradient_norm}: gradient norms of ``embedding'', ``head'', and ``middle'' layers are shown as \\hred{\\textbf{red-squares}}, \\hgreen{\\textbf{green-triangles}} and \\textbf{black-circles}, respectively.\n    }\n    \\label{fig:gradient_norm_others}\n\\end{figure}\n\n\n\\begin{figure}[htb]\n\\centering\n    \\begin{subfigure}{\\textwidth}\n    \\centering\n    \\includegraphics[width=0.76\\textwidth]{figures\/domainnet_grad_plot_normalized.png}\n    \\caption{Layer-wise gradient norms divided by parameter norm, on DomainNet at pretrained initialization} \n    \\end{subfigure}\\\\\n\n    \\begin{subfigure}{\\textwidth}\n    \\centering\n    \\includegraphics[width=0.76\\textwidth]{figures\/living17_grad_plot_normalized.png}\n    \\caption{Layer-wise gradient norms divided by parameter norm, on Living-17 at pretrained initialization} \n    \\end{subfigure}\\\\\n    \n    \\begin{subfigure}{\\textwidth}\n    \\centering\n    \\includegraphics[width=0.76\\textwidth]{figures\/waterbirds_grad_plot_normalized.png}\n    \\caption{Layer-wise gradient norms divided by parameter norm, on Waterbirds at pretrained initialization} \n    \\end{subfigure}\n    \\caption{\n    We visualize the layer-wise gradient norm, \\textbf{divided by the norm of the parameters} on (a) DomainNet, (b) Living-17, and (c) Waterbirds, at the pretrained initialization. For better visualization, we omit the head from the plot, which predictably has much larger gradients than the others (since it is randomly initialized). The format is the same as Figure~\\ref{fig:gradient_norm}: gradient norms of ``embedding'' and ``middle'' layers are shown as \\hred{\\textbf{red-squares}} and \\textbf{black-circles}, respectively.  Under this normalization scheme, the embedding layer has higher gradients than the other layers in all models. However, the gradient is only slightly larger for ResNet models, and substantially larger for the Vision Transformer models---which also provides support for why freezing the embedding layer in Vision Transformers might make a larger difference.\n    }\n    \\label{fig:gradient_norm_normalized}\n\\end{figure}\n\n\n\\begin{figure}[htb]\n\\centering\n    \\begin{subfigure}{\\textwidth}\n    \\centering\n    \\includegraphics[width=0.76\\textwidth]{figures\/domainnet_grad_plot_param_normalized.png}\n    \\caption{Layer-wise gradient norms divided by $\\sqrt{\\text{\\#parameters}}$, on DomainNet at pretrained initialization} \n    \\end{subfigure}\\\\\n\n    \\begin{subfigure}{\\textwidth}\n    \\centering\n    \\includegraphics[width=0.76\\textwidth]{figures\/living17_grad_plot_param_normalized.png}\n    \\caption{Layer-wise gradient norms divided by  $\\sqrt{\\text{\\#parameters}}$, on Living-17 at pretrained initialization} \n    \\end{subfigure}\\\\\n    \n    \\begin{subfigure}{\\textwidth}\n    \\centering\n    \\includegraphics[width=0.76\\textwidth]{figures\/waterbirds_grad_plot_param_normalized.png}\n    \\caption{Layer-wise gradient norms divided by  $\\sqrt{\\text{\\#parameters}}$, on Waterbirds at pretrained initialization} \n    \\end{subfigure}\n    \\caption{\n    We visualize the layer-wise gradient norm, \\textbf{divided by the square root of the number of parameters} on (a) DomainNet, (b) Living-17, and (c) Waterbirds, at the pretrained initialization. For better visualization, we omit the head from the plot which has predictably much larger than the others (since it is randomly initialized). The format is the same as Figure~\\ref{fig:gradient_norm}: gradient norms of ``embedding'' and ``middle'' layers are shown as \\hred{\\textbf{red-squares}} and \\textbf{black-circles}, respectively. Under this normalization, we see that the gradients of the embedding layer are much larger than the other layers in all models, including ResNets. \n    }\n    \\label{fig:gradient_norm_param_normalized}\n\\end{figure}\n\n\\remove{\n\\section{Check}\n\n\\subsection{Best Results with SGD (Freeze-embed)}\n\nIn Table~\\ref{tab:sgd_freeze_best} we show the best results in our paper when using SGD (Freeze-embed).\n\n\\begin{table}[]\n\\caption{\nOur best result using SGD (Freeze-embed) outperforms previous state-of-the-art in 4\/5 datasets---all datasets except Waterbirds.\nThe difference with Table~\\ref{tab:sota_results} is that Table~\\ref{tab:sota_results} shows our best results across models and optimizers, and here we show the best results for SGD (Freeze-embed).\n}\n\\label{tab:sgd_freeze_best}\n\\vskip 0.1in\n\\begin{center}\n\\begin{tabular}{cccccc}\n\\toprule\n & Liv-17 & Waterbirds & DomainNet & FMoW & Camelyon\\\\\n\\midrule\n\\begin{tabular}{@{}c@{}}Best prior \\\\ result \\end{tabular} & 87.6 & \\textbf{89.3} & 87.2 & 47.6 & 93.3 \\\\\n\\midrule\n\\vspace{0.03in}\n\\begin{tabular}{@{}c@{}}Our best \\\\ result \\end{tabular} & \\textbf{90.5} & 86.9 & \\textbf{93.1} & \\textbf{49.9} & \\textbf{96.5} \\\\\n\\vspace{0.08in}\nOptimizer & \\begin{tabular}{@{}c@{}}SGD \\\\ Freeze-Embed \\end{tabular} & \\begin{tabular}{@{}c@{}}SGD \\\\ Freeze-Embed \\end{tabular} & \\begin{tabular}{@{}c@{}}SGD \\\\ Freeze-Embed \\end{tabular} & \\begin{tabular}{@{}c@{}}SGD \\\\ Freeze-Embed \\end{tabular} & \\begin{tabular}{@{}c@{}}SGD \\\\ Freeze-Embed \\end{tabular} \\\\\n\\vspace{0.015in}\nModel & CLIP-ViT-L\/14 & ConvNeXt-B & CLIP-ViT-L\/14 & CLIP-ViT-L\/14 & CLIP-ViT-L\/14 \\\\\n\\bottomrule\n\\end{tabular}\n\\end{center}\n\\end{table}\n}\n\\section{Profiling GPU Memory}\n\\label{app:profiling_gpu_memory}\n\nWe profiled the GPU memory consumption of 4 fine-tuning methods, on 3 models on a Titan-X GPU with 12.2 GB of GPU memory.\nThe profiling was done using Weights and Biases and measures the total GPU memory utilization (so includes weights, activations, etc.).\nThe four methods we profiled are: SGD, AdamW, SGD (freeze-embed), SGD (freeze-embed, no momentum).\nWe profiled these on three models: CLIP-ViT B\/16, CLIP ViT-L\/14, and OpenCLIP ViT-H\/14---the original CLIP model only scales up to ViT-L\/14, so we used the OpenCLIP~\\citep{ilharco_gabriel_2021_5143773} for ViT-H\/14.\nWe used a batch size of 32 and do gradient accumulation with micro-batch size of 1---i.e., we calculate the gradients one sample at a time and accumulate them before calling the optimizer update. \n\nWe show results below in Table~\\ref{tab:clip_memory_profiling_results}.  On a ViT-B\/16, AdamW uses $16\\%$ and $36\\%$ more memory than SGD (freeze-embed) and SGD (freeze-embed, no momentum), respectively. The gains are better for larger models: on a ViT-L\/14, AdamW uses $18\\%$ and $48\\%$ more memory respectively. On a ViT-H\/14, AdamW runs out of memory, while SGD (freeze-embed) and SGD (freeze-embed, no momentum) are able to run, showing that the gains are at least $20\\%$ and $60\\%$ respectively.\n\n\\begin{table}\n\\vspace{-10pt}\n\\caption{We profile the GPU memory consumption of four optimization methods on three CLIP models of varying sizes, on a Titan-X GPU. SGD (freeze-embed) gives practical gains over AdamW, especially if we drop the momentum parameter. The largest CLIP ViT-H\/14 does not fit in GPU memory when using AdamW, but fits in memory with other optimizers. Note that the freeze-embed methods perform competitively or better than AdamW on accuracy, as shown in Table~\\ref{tab:clip_results}.}\n\\label{tab:clip_memory_profiling_results}\n\\centering\n\\vspace{-7pt}\n\\resizebox{0.9\\textwidth}{!}{\n\\begin{tabular}{cccc}\n\\toprule\n     & {CLIP ViT-B\/16} & { CLIP ViT-L\/14} & { CLIP ViT-H\/14} \\\\\n\\midrule\n{ AdamW} & 2.7 GB & 7.1 GB & Out-of-Memory \\\\\n{ SGD} & 2.3 GB & 6.1 GB & 10.1 GB  \\\\\n\\midrule\nSGD (freeze-embed)  & 2.3 GB & 6.1 GB & 10.1 GB \\\\\nSGD (freeze-embed, no momentum) & \\textbf{2.0 GB} & \\textbf{4.8 GB} & \\textbf{7.6 GB} \\\\\n\\bottomrule\n\\end{tabular}\n}\n\\end{table}\n\nFor large models, it is common to use additional tricks like gradient checkpointing to reduce the activation memory. That is, for a speed penalty (at most 2x), we only need to store activations in $\\sqrt{L}$ of $L$ layers when doing backpropagation. Gradient checkpointing would further increase our gains over AdamW since they do not change the memory consumed by the weights but can substantially decrease the memory consumed by the model's activations.\n\n\\section{Additional training details} \\label{app:setup}\n\\paragraph{SGD and AdamW updates.} We start with initialization $\\theta^{(0)}=\\theta^{\\mathsf{pretrain}}$. Given hyperparameters, minibatch size $batch\\_size$ and number of epochs $num\\_epochs$, our algorithms run for $T=\\frac{num\\_epochs\\cdot{|D_{\\mathsf{train}}|}}{batch\\_size}$ steps. At steps $t=0,1,\\ldots, T$, we select a randomly shuffled minibatch $B_t$ from $D_{\\mathsf{train}}$ (reshuffled at end of each epoch) and compute the minibatch stochastic gradient $g_t =  \\frac{1}{|B_t|}\\sum_{(x,y)\\in B_t}\\nabla_\\theta l(f_\\theta(x),y)$. The parameters $\\theta^{(t)}$ are updated as follows.\n\n For SGD, in addition to gradients $g_t$ and weights $\\theta^{(t)}$, we maintain first order momentum estimate $m_t$ as optimizer state, initialized as $m_{-1}=0$. The SGD($\\eta_t,\\mu,\\lambda$) update with $\\eta_t$ learning rate, $\\mu$ momentum, and $\\lambda$ weight decay, is given by\n    \\begin{equation*}\n        \\begin{split}\n            g_t &= g_t +\\lambda \\theta^{(t)}\\\\\n            m_t &= \\mu m_{t-1}+g_t\\quad(\\text{first moment})\\\\\n            \\theta^{(t+1)} &= \\theta^{(t)}-\\eta_t m_t\n        \\end{split}\n    \\end{equation*}\n\nFor AdamW, we maintain two additional optimizer states: the first moment estimate $m_t$ and second moment estimate $v_t$, initialized as $m_{-1}= v_{-1}=0$. The AdamW($\\eta_t,\\beta_1,\\beta_2,\\lambda$) update with $\\eta_t$ learning rate, $(\\beta_1,\\beta_2)$ betas, and  $\\lambda$ weight decay is given by\n    \\begin{equation*}\n        \\begin{split}\n            %\n            m_t &= \\beta_1 m_{t-1}+ (1-\\beta_1) g_t \\; \\quad(\\text{first moment})\\\\\n            v_t &= \\beta_2 v_{t-1}+ (1-\\beta_2) g_t^{\\odot 2} \\quad(\\text{second moment})\\\\\n            \\hat{m}_t &= \\frac{m_t}{(1-\\beta_1^t)};\\;\\quad \\hat{v}_t = \\frac{v_t}{(1-\\beta_2^t)}\\\\\n            \\theta^{(t+1)} &= (1-\\eta_t\\lambda)\\theta^{(t)}-\\eta_t \\frac{\\hat{m}_t}{\\sqrt{\\hat{v}_t}+\\epsilon}\n        \\end{split}\n    \\end{equation*}\n\n\n\\paragraph{Hyperparameter details}\n\\begin{itemize}\n    \\item \\textbf{Learning rate. }We sweep over 6 learning rates for SGD ([3e-5, 1e-4, 3e-4, 1e-3, 3e-3, 1e-2]) and for AdamW ([3e-7, 1e-6, 3e-6, 1e-5, 3e-5, 1e-4]). As standard, we use smaller learning rates for AdamW because they work better. We check that the best learning rate is in the middle of this grid for each model and dataset. The best learning rate is chosen based on ID validation accuracy. \n    \\item \\textbf{Other optimizer parameters. } We use all the default options for SGD and AdamW in pytorch except set the SGD momentum parameter to 0.9. The other default options are: for SGD, the weight decay is 0 and we use momentum without dampening, and for AdamW, the weight decay is 0.01, betas are (0.9,0.999) and epsilon is 1e-08.\n    \\item \\textbf{Number of epochs. }We train for 20 epochs on Living-17, 20 epochs on Waterbirds, 50 epochs on DomainNet, 5 epochs on WILDS-FMoW, and 3 epochs on Camelyon. We use the same number of training epochs for all the optimizer methods we report results on. \n    \\item \\textbf{Learning rate schedule. } With the exception of \\textit{Gradual-unfreezing} in Table~\\ref{tab:clip_results}, we use a cosine learning rate schedule and decay the starting learning rate to $0$ over the course of $T$ training steps. Learning rates schedule for \\textit{Gradual-unfreezing} is described below. \n    \\item \\textbf{Learning rate schedule for gradual unfreezing.} For gradual unfreezing, we do not use a cosine learning rate scheduler. Instead, at epoch $t$ ($0$-indexed) out of $T$, we multiply the base learning rate by $\\exp(3.73 * (1.0 - t\/(T-1)))$. This means we multiply the learning rate by $\\exp(3.73) \\approx 41.7$ in epoch $0$, and by $1$ in the last epoch. The intuition is that when we are tuning a smaller number of layers we need a higher learning rate than for full fine-tuning---for example, the optimal learning rates for head tuning is higher than the optimal learning rate for full fine-tuning. The exact constant (3.73) was selected by comparing the optimal learning rate for head tuning with full fine-tuning on Waterbirds (the optimal learning rate for head-tuning was approximately $\\exp(3.73)$ larger for head-tuning). Without this decay schedule, for example using a vanilla cosine learning rate scheduler, gradual unfreezing worked similarly or worse than vanilla full fine-tuning.\n    \\item \\textbf{Stopping criteria. }The results presented in the paper are from models early stopped based on ID validation accuracy. We sanity checked that the conclusions are similar if we use the last epoch checkpoint. \n\\end{itemize}\n\n\n\\paragraph{Data augmentation and input preprocessing. } Additionally, we use the following preprocessing and augmentations on our input images. We use very basic augmentations (e.g., only horizontal flips for WILDS, and standard augmentations from past work), including for our state-of-the-art results:\n\\begin{enumerate}\n    \\item For WILDS-FMoW, the images are $224\\times224$, so we do not resize, and only perform a random horizontal flip augmentation. We do not perform any augmentations at test time.\n    \\item For WILDS-Camelyon, we resize the images to $224\\times 224$ with a bilinear interpolation (standard in pyorch), and only perform a random horizontal flip augmentation. We do not perform any augmentations at test time, just the resize.\n    \\item For Living-17, we follow~\\citep{kumar2022finetuning} and perform a RandomResizedCrop to $224\\times 224$ sized images (using the default options in pyorch), and then a random horizontal flip augmentation while training. At test-time we resize the image to $256\\times256$ and then take a centercrop of size $224\\times 224$.\n    \\item For DomainNet, we follow~\\citep{kumar2022finetuning} and first resize the image to $256\\times 256$ with bicubic interpolation, then take a RandomCrop of size $224\\times224$ (using the default options in pyorch), and then a random horizontal flip augmentation while training. At test-time we simply resize the image to $224\\times 224$ with bicubic interpolation.\n    \\item For Waterbirds, we resize the image to $224\\times 224$ with bicubic interpolation and then take a centercrop of size $224\\times224$. We apply the same transformation at test time.\n\\end{enumerate}\n\n\n\\paragraph{Embedding layer.} For SGD (freeze-embed), the exact layers we freeze are as follows: \n\\begin{enumerate}\n    \\item CLIP ViTs: We freeze the patch-to-token embedding layer and layernorm.\n    \\item Supervised and DINO ViTs: We freeze the patch-to-token embedding layer (there is no layernorm after this)\n    \\item BiT-ResNets: We freeze the `stem' and the first convolution block of the model. We tried freezing less and more of the model in our initial experiments, but it did not seem to help.\n    \\item ConvNeXt-B: We freeze the `stem' and the first stage of the model. \\ak{Need to see what happens if we only freeze the stem}\n\\end{enumerate}\n\n\n\\section{What could cause large ``embedding'' layer gradients?}\nGenerally speaking, we find that AdamW fine-tuning leads to better performing and more robust models than SGD fine-tuning, specially in modern architectures. \nOur algorithms to close this gap were inspired by the observation that on modern vision architectures, the embedding layers have substantially larger gradients at pretrained initialization compared to other layers. This could lead to over-training of the embedding layer when using SGD, which we hypothesized would be bad for robust fine-tuning. The success of SGD (freeze-embed) in our experiments adds further evidence to our hypothesis. But, why are the gradients of embedding layers at pretrained initialization  high in the first place? More generally, why does AdamW do better than SGD during fine-tuning? We do not have definitive answers to these questions as we are far from understanding the precise dynamics optimization algorithms on large neural networks. Below we discuss some plausible reasons. \n\n\\paragraph{Algorithmic aspects: pretraining algorithm?} Among our 7 models, the more recent models like vision transformers and ConvNeXt, which were pretrained with AdamW are also the ones with largest gaps in performance between AdamW and SGD fine-tuning. BiT ResNets that were pretrained with SGD had much smaller differences between AdamW and SGD. This strongly suggests that the {discrepancy between the algorithms in pretraining and fine-tuning} might be cause for the performance gaps. For example, it is possible that pretraining with AdamW implicitly biases towards configurations that most benefit from AdamW updates. This would also explain why other adaptive algorithms like LARS and LAMB are not competitive even though they perform some form of layer-wise normalization. On the other hand it does not explain why such effects would distinctively impact the ``embedding'' layer and not the other layers. In a related manner, it is also unclear why SGD (freeze-embed) would be able to overcome such implicit biases from AdamW pretraining. \n\n\\paragraph{Architectural aspects: ``patchifying'' first layer?} There are substantial architectural differences between the newer transformer and ConvNeXt models and the older ResNets which could also contribute to the newer models working better with AdamW. Most notably, vision transformer is fundamentally different architecture from convolutional networks. At the same time, the biggest differences between the architectures---like self-attention,  softmax non-linearity, or fully connected layers---are \\textit{not} the likely contributors to the differences between AdamW and SGD in fine-tuning. This is because, we also see gaps between these methods with the ConvNeXt models which lack the major transformer components. Rather, it is the possible that some of the designs that were adapted from transformers to ConvNeXt contributes to the differences between AdamW and SGD fine-tuning. Among these, many primarily affect the higher layers such as \n\\begin{inparaenum}[] \n\\item heavy third stage, \n\\item depthwise convolution,\n\\item inverted bottleneck, and \n\\item larger kernels,\n\\end{inparaenum} \nand  are unlikely to cause the lower layer gradients to be high. %\nThe key design change we believe is important here is the use of a ``patchify stem'' that could cause distinctive changes in the gradients for lower blocks. \n\nThe ``stem'' layer primarily controls how the input is processed for the rest of the network. ViTs and ConvNext down-samples the input images from non-overlapping patch tokens, compared to ResNets that use denser-overlapped convolution followed by max-pool. The coarser non-overlap might lead the embedding layer to attune more closely to the patch distribution in pretraining. This might not be an issue by itself as pixel\/patch level information is noisy anyways and the higher layers can extract more robust features from  across the patches. A possible dynamics in fine-tuning is as follows: On new datasets where the patch distributions are very different from pretraining, the embedding layers might have large gradients. In standard SGD, this might cause the embedding layer to moving too quickly to fit the new patches before the higher layers can adapt to the changes. Instead, freezing the embedding layer would pass along the raw patches with minimal processing and lets the model adapt to the new distribution based on higher level features, which are likely more robust. \n\n\\remove{\n\\paragraph{Dataset aspects: closeness to pretraining data? } In our experiments, we saw that Living-17 which is a dataset closest to pretraining data for non-CLIP models was also the only dataset where SGD perform better than AdamW, especially on non-CLIP models. Intuitively, it is reasonable that when the input distribution is close to the pretraining dataset, the distribution of patches seen by the embedding layer is similar and this could cause less movement of the embedding layer with SGD. This is not an entirely satisfactory explanation as even for Living-17, in Figure~\\ref{fig:gradient_norm_others} we see that the gradients of the embedding layer at least at initialization are high even on Living-17.\\sgnote{this argument is not very convincing. specially i am bothered that the gradient magnitudes of living-17 do not concur with our story overall}\n}\n\n\n\\paragraph{Summary. }In conclusion, we hypothesize that the differences between AdamW and SGD fine-tuning arise from a combination of the above reasons---pretraining with AdamW and large ``patchify'' embedding layer. Additionally, the use of GeLU activation and layer normalization might also change the optimizer dynamics although these are minor changes from ReLU activation and group normalization used in BiT ResNets.  It is of interest to systematically explore these reasons further in future work. \n\n\\section{Detailed analysis of CLIP.}\\label{sec:clip_details}\nCLIP models have strong transfer learning performance and robustness--among our 7 models, CLIP models had the best ID and OOD accuracies averaged across datasets. So we did a more detailed analysis of the CLIP ViT-B\/16 with other optimizers\\footnote{Generating Table~\\ref{tab:ood_bigtable_final} involved over 600 fine-tuning runs, so we weren't able to repeat this for every model.} which are summarized in Table~\\ref{tab:clip_results}. %\n\\begin{table}[h!]\n    \\caption{CLIP ViT-B\/16  performance with new optimizers and confidence intervals. For addition to SGD, AdamW, and SGD (freeze-embed), we provide 90\\% confidence intervals based on $3$ runs of each hyperparameter configuration. In addition,\n    we show accuracies for two of our variant methods \\textit{SGD (freeze-embed, no momentum) }and \\textit{Gradual-unfreezing}; as well as comparison to two other optimizers, LARS~\\citep{You2017LargeBT} and LAMB~\\citep{you2020large}, which use layer-wise normalization in their updates. }\n    \\label{tab:clip_results}\n    \\centering\n    \\resizebox{\\linewidth}{!}{\n    \\begin{tabular}{c|cc|cc|cc|cc|cc|cc}\n        \\toprule\n        & \\multicolumn{2}{c|}{Living-17} & \\multicolumn{2}{c|}{Waterbirds} & \\multicolumn{2}{c|}{DomainNet} & \\multicolumn{2}{c|}{FMoW} &     \\multicolumn{2}{c|}{Camelyon} & \\multicolumn{2}{c}{Avg.}\\\\\n        & ID & OOD  & ID & OOD  & ID & OOD  & ID & OOD  & ID & OOD  & ID & OOD\\\\     \n        \\midrule\n{SGD} & 97.8 (0.2) & 80.0 (1.3) & 97.2 (0.1) & 62.5 (5.0) & 88.8 (7.1) & 72.8 (18.0) & 67.0 (0.8) & 37.3 (1.1) & 99.4 (0.0) & 86.8 (1.1) & 90.0  & 67.9\\\\[5pt]\n\\midrule\n{\\begin{tabular}[c]{@{}c@{}} AdamW\\\\ \\end{tabular}} & 98.1 (0.1) & \\textbf{82.8 (1.2)} & 97.7 (0.0) & \\textbf{71.9 (2.4)} & 95.0 (0.1) & 89.2 (1.1) & \\textbf{70.1 (0.2)} & \\textbf{40.7 (0.3)} & \\textbf{99.5 (0.0)} & 95.7 (0.4) & \\textbf{92.1} & \\textbf{76.0}\\\\[5pt]\n{\\begin{tabular}[c]{@{}c@{}} SGD\\\\  (freeze-embed)\\end{tabular}} & \\textbf{98.2 (0.3)} & \\textbf{83.2 (0.8)} & 97.8 (0.1) & \\textbf{73.7 (1.1)} & 94.9 (0.3) & 88.2 (0.7) & \\textbf{70.0 (0.2)} & \\textbf{40.2 (0.7)} & \\textbf{99.5} (0.0) & 94.3 (0.3) & \\textbf{92.1} & 75.9\\\\\n\\midrule \n{\\begin{tabular}[c]{@{}c@{}} SGD\\\\  (freeze-em. no mom.)\\end{tabular}} & \\textbf{98.2} & 83.1 & 97.9 & \\textbf{80.4} & 95.2 & 89.0 & \\textbf{70.1} & 38.8 & \\textbf{99.5} & 93.3 & \\textbf{92.2} & \\textbf{76.9}\\\\[10pt]\n{Gradual-unfreezing} & \\textbf{98.3\\remove{ (0.1)}} & 81.9\\remove{ (0.1)} & \\textbf{98.3\\remove{ (0.0)}} & 69.1\\remove{ (3.4)} & \\textbf{96.3\\remove{ (0.1)}} & \\textbf{93.2\\remove{ (0.3)}} & 69.2\\remove{ (0.5)} & \\textbf{40.5\\remove{ (1.2)}} & 99.3\\remove{ (0.1)} & \\textbf{96.5\\remove{ (0.4)}} & \\textbf{92.3}& \\textbf{76.2}\\\\\n\\midrule\n{\\begin{tabular}[c]{@{}c@{}} LAMB\\\\ \\end{tabular}} & \\textbf{98.2} & 79.5 & 97.8 & 64.0 & 95.1 & 90.4 & 67.9 & 38.8 & \\textbf{99.5} & 93.4 & 91.7 & 73.2\\\\\n{\\begin{tabular}[c]{@{}c@{}} LARS\\\\ \\end{tabular}} & 97.7 & \\textbf{83.9} & 97.1 & 48.6 & 93.2 & 83.8 & 67.0 & 38.6 & 99.3 & 93.3 & 90.9 & 69.6\\\\\n\\bottomrule\n    \\end{tabular}\n    }\n\\end{table}\n\n\\remove{\n\\begin{table}[h!]\n\\parbox{.48\\linewidth}{\n\\caption{\nIn-distribution (ID) accuracies for the CLIP base model, with 90\\% confidence intervals.\nSGD (freze-embed) is better than SGD, and competitive with AdamW, but LARS and LAMB perform worse than AdamW. \\sgnote{make into one table}\n}\n\\begin{center}\n\\resizebox{\\linewidth}{!}{\n\\begin{tabular}{cccccc|c}\n\\toprule\n & Liv-17 & Waterbirds & DomainNet & FMoW & Camelyon & Avg.\\\\\n\\midrule\nSGD & 97.8 (0.2) & 97.2 (0.1) & 88.8 (7.1) & 67.0 (0.8) & 99.4 (0.0) & 90.0\\\\\nAdamW & 98.1 (0.1) & 97.7 (0.0) & 95.0 (0.1) & \\textbf{70.1 (0.2)} & \\textbf{99.5 (0.0)} & \\textbf{92.1}\\\\\n\\midrule\nSGD (Freeze-embed) & \\textbf{98.2 (0.3)} & 97.8 (0.1) & 94.9 (0.3) & \\textbf{70.0 (0.2)} & 99.5 (0.0) & \\textbf{92.1}\\\\\nGradual-unfreezing & \\textbf{98.3 (0.1)} & \\textbf{98.3 (0.0)} & \\textbf{96.3 (0.1)} & 69.2 (0.5) & 99.3 (0.1) & \\textbf{92.3}\\\\\n\\midrule\nLAMB & \\textbf{98.2} & 97.8 & 95.1 & 67.9 & \\textbf{99.5} & 91.7\\\\\nLARS & 97.7 & 97.1 & 93.2 & 67.0 & 99.3 & 90.9\\\\\n\\bottomrule\n\\end{tabular}\n}\n\\end{center}\n}\n\\hfill\n\\parbox{.48\\linewidth}{\n\\caption{\nOut-of-distribution (OOD) accuracies for the CLIP base model, with 90\\% confidence intervals.\nSGD (freze-embed) is much better than SGD, and competitive with AdamW, but LARS and LAMB performs much worse than AdamW.\n}\n\\begin{center}\n\\resizebox{\\linewidth}{!}{\n\\begin{tabular}{cccccc|c}\n\\toprule\n & Liv-17 & Waterbirds & DomainNet & FMoW & Camelyon & Avg.\\\\\n\\midrule\nSGD & 80.0 (1.3) & 62.5 (5.0) & 72.8 (18.0) & 37.3 (1.1) & 86.8 (1.1) & 67.9\\\\\nAdamW & \\textbf{82.8 (1.2)} & \\textbf{71.9 (2.4)} & 89.2 (1.1) & \\textbf{40.7 (0.3)} & 95.7 (0.4) & \\textbf{76.0}\\\\\n\\midrule\nSGD (Freeze-embed) & \\textbf{83.2 (0.8)} & \\textbf{73.7 (1.1)} & 88.2 (0.7) & \\textbf{40.2 (0.7)} & 94.3 (0.3) & 75.9\\\\\nGradual-unfreezing & 81.9 (0.1) & 69.1 (3.4) & \\textbf{93.2 (0.3)} & \\textbf{40.5 (1.2)} & \\textbf{96.5 (0.4)} & \\textbf{76.2}\\\\\n\\midrule\nLAMB & 79.5 & 64.0 & 90.4 & 38.8 & 93.4 & 73.2\\\\\nLARS & \\textbf{83.9} & 48.6 & 83.8 & 38.6 & 93.3 & 69.6\\\\\n\\bottomrule\n\\end{tabular}\n}\n\\end{center}\n}\n\\end{table}\n}\n\n\\paragraph{SGD (freeze-embed, no momentum).} Across our datasets and models, SGD (freeze-embed) performed competitively with AdamW, while using one less optimizer state\/parameter and hence less memory. We can potentially gain additional memory savings by using a more basic SGD without momentum. Note that momentum has been shown to be important when training from scratch, but its benefit while fine-tuning has not been explored much. So for our SGD (freeze-embed, no momentum) method, on all our datasets, we fine-tune a CLIP ViT-B\/16 model with SGD, freeze the embedding layer, and do not use momentum. In Table~\\ref{tab:clip_results}, we see that SGD (freeze-embed, no momentum) is competitive with SGD (freeze-embed) and AdamW, even having an edge on some datasets. This is promising, but since we only evaluated on CLIP ViT-B\/16, more experiments are needed to derive stronger conclusions about the accuracy of SGD (freeze-embed, no momentum).\n\n\\paragraph{GPU memory profiling.} We profiled all methods on a Titan-X GPU. On a ViT-B\/16, AdamW uses $16\\%$ and $36\\%$ more memory than SGD (freeze-embed) and SGD (freeze-embed, no momentum) respectively. The gains are better for larger models: on a ViT-L\/14, AdamW uses $18\\%$ and $48\\%$ more memory respectively. On a ViT-H\/14, AdamW runs out of memory, while SGD (freeze-embed) and SGD (freeze-embed, no momentum) used 10.1 GB and 7.6 GB respectively, out of 12.2 GB. Note that the gains from SGD (freeze-embed) would be even higher with gradient checkpointing, which reduces activation memory. See more details in Appendix~\\ref{app:profiling_gpu_memory}.\n\n\\paragraph{Gradual-unfreezing.} The success of freeze-embed for SGD is inline with our hypothesis that it is beneficial to change the lower layers less during fine-tuning. We could further refine this idea---instead of just freezing the first ``embedding'' layer, we could start by freezing the entire pretrained model (except for the head) and gradually unfreeze the rest of the model over training. \nStrategies like this have been explored by ULMFiT~\\citep{howard2018universal} and others~\\citep{mukherjee2019distilling, romero2020targeted}.\nAn advantage of gradual-unfreezing is that in earlier stages of training we tune a smaller fraction of the model, which speeds up training---for example, for WILDS-FMoW on an A100 GPU, the average runtime decreases by over 30\\%. \n\nFor gradual-unfreezing, we found that naively using a single learning rate throughout this process or even a cosine learning rate schedule is not sufficient to achieve good performance. Instead, we fit an exponential learning rate decay law for our experiments (see details in Appendix~\\ref{app:setup}). %\nThe performance of gradual-unfreezing on CLIP ViT-B\/16 is shown in Table~\\ref{tab:clip_results}. On some datasets, gradual-unfreezing performed particularly well. For example, it got an OOD accuracy of 93.2\\% on DomainNet and 96.5\\% on Camelyon---which are both competitive with SGD (freeze-embed) and AdamW on a much larger CLIP ViT-L\/14 model. On average, gradual-unfreezing gets slightly better results both ID  and OOD, while being $30\\%$ faster. \nOverall though, we find that the most consistent improvements come from simply freezing the embedding layer, which suggests that the embedding layer plays a particularly prominent role in modern vision architectures.\n\n\n\n\\paragraph{SGD (freeze-embed) and AdamW outperform LAMB and LARS.}\nWe also ran two alternative adaptive gradient methods, LARS~\\citep{You2017LargeBT} and LAMB~\\citep{you2020large}---also sweeping over 6 learning rates and early stopping.\nThese are alternate methods with layerwise normalization that can avoid over-training of large gradient layers. Moreover, like SGD and SGD (freeze-embed), LARS also has a lower memory footprint than AdamW. In Table~\\ref{tab:clip_results}, we see that while LARS and LAMB get higher accuracies than SGD, they do worse than SGD (freeze-embed) and AdamW both ID and OOD. In this case, our modification with freeze-embed appears to be more effective. \n\n\n\n\n\\begin{wraptable}{R}{0.34\\textwidth}\n\\vspace{-10pt}\n\\caption{Test accuracy on CIFAR-10. }\n\\label{tab:clip_cifar_results}\n\\centering\n\\vspace{-7pt}\n\\resizebox{\\linewidth}{!}{\n\\begin{tabular}{ccc}\n\\toprule\n     & {\\small CLIP ViT-B\/16} & {\\small CLIP ViT-L\/14} \\\\\n\\midrule\n{ SGD} & 98.0 & 99.0 \\\\\n{ AdamW} & 98.3 & \\textbf{99.3} \\\\\n\\midrule\n{ \\begin{tabular}{cc}SGD \\\\ (freeze-embed) \\end{tabular}} & \\textbf{98.4} & \\textbf{99.3} \\\\\n\\bottomrule\n\\end{tabular}\n}\n\\end{wraptable}\n\n\\paragraph{CIFAR-10 results.} As a proof of concept for standard (non-OOD) transfer learning, we fine-tune CLIP ViT models on CIFAR-10. In  Table~\\ref{tab:clip_cifar_results}, we see that even at high accuracies, AdamW and SGD (freeze-embed) manage to improve performance. \nSGD (freeze-embed) gets 20\\% and 30\\% lower error than SGD on CLIP ViT-B\/16 and CLIP ViT-L\/14, respectively. %\n\n\\section{Detailed experiments on freeze-embedding}\\label{sec:mainexp}\n\n\nWe consider a simple variation of SGD fine-tuning where we freeze the embedding layer and only perform SGD updates on the rest of the network---we call this method ``SGD (freeze-embed)''.  In this section we discuss the results of fine-tuning our 7 models on 5 distribution shift benchmarks mentioned in Section~\\ref{sec:problem_setting}. We compare three methods: SGD, AdamW, and our proposed SGD (freeze-embed). \nWe use the implementations of SGD and AdamW in PyTorch. For each method, we train for the same number of epochs using a cosine learning rate schedule, and sweep over 6 starting learning rates (ensuring that the optimal learning rate is in the middle of the sweep). For all datasets we follow prior work~\\citep{kumar2022finetuning} and pick the best learning rate and early stop based on the ID validation accuracy. See Appendix~\\ref{app:setup} for additional details. \n\nTable~\\ref{tab:ood_bigtable_final} and Table~\\ref{tab:id_bigtable_final} show the OOD and ID accuracies, respectively, of the three methods across our models and datasets. For each model and dataset, we also highlight the relative gain\/loss of AdamW and SGD (freeze-embed) from regular SGD in green\/red. We discuss the main observations below. \n\\begin{enumerate}[1.]\n    \\item \\textbf{AdamW outperforms SGD.} We see that AdamW largely outperforms SGD, often by substantial margins on ViT and ConvNeXt models.  Particularly in OOD evaluation, the gaps are remarkable. In the few instances where SGD performs better the gaps are much smaller. The differences between SGD and AdamW are generally more modest for older ResNet models. \n    \\item \\textbf{SGD (freeze-embed) is competitive with or better than AdamW.} \n    For each individual model, averaged across the datasets  SGD (freeze-embed) is consistently the best or tied-best method on OOD accuracy, and only minimally worse than AdamW on ID accuracy (right columns of Table~\\ref{tab:ood_bigtable_final}-\\ref{tab:id_bigtable_final}).  Averaged across all the models and datasets, SGD (freeze-embed) performs the best OOD, getting an average OOD accuracy of 76.7\\% (vs. 71.9\\% for SGD and 76.0\\% for AdamW). On ID data, SGD (Freeze-embed) closes 85\\% of the gap between SGD and AdamW, getting an average accuracy of 91.3\\% (vs. 90.2\\% for SGD and 91.5\\% for AdamW). \n    \\item \\textbf{Larger CLIP model gains more from SGD (freeze-embed) and AdamW.} On a CLIP-ViT-B\/16 we see that  SGD (freeze-embed) and AdamW get about $8\\%$ higher OOD accuracy than SGD. Upon scaling to a larger CLIP-ViT-L\/14, which is also our best model for OOD performance, we see even higher gains over SGD. On CLIP-ViT-L\/14, SGD (freeze-embed) gets a 14.3\\% higher OOD accuracy than SGD, and a 0.7\\% higher OOD accuracy than AdamW. This suggests that our findings might be even more relevant for larger models.\n    \\item \\textbf{SGD (freeze-embed) is \\textit{red} when AdamW is \\textit{red}.} Across all our models and datasets, we see that in instances where SGD (freeze-embed) is worse than SGD (i.e., highlighted as red), AdamW is also worse than SGD. This is not always the case the other way around. For example, on ViT B-\/16 fine-tuned on DomainNet and Camelyon, OOD performance of AdamW is worse than SGD, but SGD (freeze-embed) is competitive to the best of the two.  At the same time, on Living-17 OOD evaluation, multiple models have both AdamW and SGD (freeze-embed) perform significantly worse than SGD.  \n    \\item \\textbf{SGD performs well when fine-tuning data is closer to pretraining. } Breaking down the results by datasets, the main trend we see is that with models that were pretrained on ImageNet-21k (all models here except CLIP) and fine-tuned on Living-17, the performance of SGD is typically higher than AdamW and SGD (freeze-embed). Images in Living-17 are derived from ImageNet-1K and hence arguably, in this case the fine-tuning distribution is closest to pretraining. This suggests that SGD works better in instances where fine-tuning and pretraining data are similar.\n    \\item \\textbf{AdamW (freeze-embed) does not improve significantly over AdamW.} SGD (freeze-embed) was motivated by our hypothesis that AdamW would inherently avoid over-tuning of the embedding layer. If our hypothesis is true, unlike for SGD, freezing the embedding for AdamW would not be beneficial. In the Appendix, we expand the Tables~\\ref{tab:ood_bigtable_final}-\\ref{tab:id_bigtable_final} to include AdamW (freeze-embed)  and indeed we see that freeze-embed does not provide complementary gains on top of AdamW. \n\\end{enumerate}\n\n\n\n\n\n\n\n\n\\begin{table}[h!]\n \\small \n \\caption{\n\\textbf{Out-of-distribution (OOD)} accuracies for fine-tuning 7 pretrained models on 5 datasets. We compare fine-tuning with  SGD, AdamW, and SGD (freeze-embed). For each model and dataset, we also highlight the relative gain or loss of AdamW and SGD (freeze-embed) from SGD, in green (for positive) and red (for negative). Lighter shades of green\/red correspond to $<0.2$ gain\/loss. \nKey observations are: AdamW gets higher accuracies than SGD for modern ViT and ConvNeXt models. Averaged over the datasets (right column), SGD (freeze-embed) is consistently the best or tied-best method for all 7 models.\n\\label{tab:ood_bigtable_final}}\n\\begin{center}\n\\resizebox{\\textwidth}{!}{\n\\begin{tabular}{cccccccccccc|cc}\n\\toprule\n &  & \\multicolumn{2}{c}{Living-17}  & \\multicolumn{2}{c}{Waterbirds}  & \\multicolumn{2}{c}{DomainNet}  & \\multicolumn{2}{c}{FMoW}  & \\multicolumn{2}{c}{Camelyon}  & \\multicolumn{2}{c}{Avg.} \\\\\n\\midrule\nCLIP ViT-B\/16 & SGD & 80.0 &  & 62.5 &  & 72.8$^{\\dagger}$ &  & 37.3 &  & 86.8 &  & 67.9 & \\\\\nCLIP ViT-B\/16 & AdamW & 82.8 & \\hspace{-1em}{\\hgreen{(+2.8)}} & 71.9 & \\hspace{-1em}{\\hgreen{(+9.4)}} & \\textbf{89.2} & \\hspace{-1em}{\\hgreen{(+16.4)}} & \\textbf{40.7} & \\hspace{-1em}{\\hgreen{(+3.4)}} & \\textbf{95.7} & \\hspace{-1em}{\\hgreen{(+8.9)}} & \\textbf{76.0} & \\hspace{-1em}{\\hgreen{(+8.1)}}\\\\\nCLIP ViT-B\/16 & SGD (freeze-embed) & \\textbf{83.2} & \\hspace{-1em}{\\hgreen{(+3.2)}} & \\textbf{73.7} & \\hspace{-1em}{\\hgreen{(+11.2)}} & 88.2 & \\hspace{-1em}{\\hgreen{(+15.4)}} & 40.2 & \\hspace{-1em}{\\hgreen{(+2.9)}} & 94.3 & \\hspace{-1em}{\\hgreen{(+7.5)}} & \\textbf{75.9} & \\hspace{-1em}{\\hgreen{(+8.0)}}\\\\\n\\midrule\nCLIP ViT-L\/14 & SGD & 84.2 &  & 65.0 &  & 60.8$^{\\dagger}$ &  & 41.0 &  & 83.2 &  & 66.8 & \\\\\nCLIP ViT-L\/14 & AdamW & 88.0 & \\hspace{-1em}{\\hgreen{(+3.8)}} & \\textbf{85.2} & \\hspace{-1em}{\\hgreen{(+20.2)}} & \\textbf{93.8} & \\hspace{-1em}{\\hgreen{(+33.0)}} & 48.3 & \\hspace{-1em}{\\hgreen{(+7.3)}} & 95.9 & \\hspace{-1em}{\\hgreen{(+12.7)}} & 82.2 & \\hspace{-1em}{\\hgreen{(+15.4)}}\\\\\nCLIP ViT-L\/14 & SGD (freeze-embed) & \\textbf{90.5} & \\hspace{-1em}{\\hgreen{(+6.3)}} & 84.7 & \\hspace{-1em}{\\hgreen{(+19.7)}} & 93.1 & \\hspace{-1em}{\\hgreen{(+32.3)}} & \\textbf{49.9} & \\hspace{-1em}{\\hgreen{(+8.9)}} & \\textbf{96.5} & \\hspace{-1em}{\\hgreen{(+13.3)}} & \\textbf{83.0} & \\hspace{-1em}{\\hgreen{(+16.2)}}\\\\\n\\midrule\nSup ViT-B\/16 & SGD & \\textbf{89.5} &  & 77.4 &  & \\textbf{86.3} &  & 33.5 &  & 92.6 &  & 75.8 & \\\\\nSup ViT-B\/16 & AdamW & 88.3 & \\hspace{-1em}{\\hred{(-1.2)}} & 81.6 & \\hspace{-1em}{\\hgreen{(+4.2)}} & 84.4 & \\hspace{-1em}{\\hred{(-1.9)}} & \\textbf{35.9} & \\hspace{-1em}{\\hgreen{(+2.4)}} & 87.9 & \\hspace{-1em}{\\hred{(-4.7)}} & 75.6 & \\hspace{-1em}{\\hred{(-0.2)}}\\\\\nSup ViT-B\/16 & SGD (freeze-embed) & 88.0 & \\hspace{-1em}{\\hred{(-1.5)}} & \\textbf{82.4} & \\hspace{-1em}{\\hgreen{(+5.0)}} & \\textbf{86.3} & \\hspace{-1em}{\\lgreen{(+0.0)}} & 34.4 & \\hspace{-1em}{\\hgreen{(+0.9)}} & \\textbf{93.7} & \\hspace{-1em}{\\hgreen{(+1.1)}} & \\textbf{77.0} & \\hspace{-1em}{\\hgreen{(+1.2)}}\\\\\n\\midrule\nDINO ViT-B\/16 & SGD & \\textbf{88.2} &  & 56.1 &  & 76.0 &  & 33.6 &  & 86.9 &  & 68.2 & \\\\\nDINO ViT-B\/16 & AdamW & 87.4 & \\hspace{-1em}{\\hred{(-0.8)}} & 61.2 & \\hspace{-1em}{\\hgreen{(+5.1)}} & 77.4 & \\hspace{-1em}{\\hgreen{(+1.4)}} & \\textbf{35.8} & \\hspace{-1em}{\\hgreen{(+2.2)}} & \\textbf{91.9} & \\hspace{-1em}{\\hgreen{(+5.0)}} & 70.7 & \\hspace{-1em}{\\hgreen{(+2.5)}}\\\\\nDINO ViT-B\/16 & SGD (freeze-embed) & 86.7 & \\hspace{-1em}{\\hred{(-1.5)}} & \\textbf{67.9} & \\hspace{-1em}{\\hgreen{(+11.8)}} & \\textbf{78.4} & \\hspace{-1em}{\\hgreen{(+2.4)}} & \\textbf{35.9} & \\hspace{-1em}{\\hgreen{(+2.3)}} & 90.6 & \\hspace{-1em}{\\hgreen{(+3.7)}} & \\textbf{71.9} & \\hspace{-1em}{\\hgreen{(+3.7)}}\\\\\n\\midrule\nConvNext-Base & SGD & \\textbf{94.0} &  & 80.2 &  & 89.8 &  & \\textbf{39.7} &  & 83.0 &  & 77.3 & \\\\\nConvNext-Base & AdamW & 90.3 & \\hspace{-1em}{\\hred{(-3.7)}} & \\textbf{89.8} & \\hspace{-1em}{\\hgreen{(+9.6)}} & 89.5 & \\hspace{-1em}{\\hred{(-0.3)}} & 38.4 & \\hspace{-1em}{\\hred{(-1.3)}} & \\textbf{89.5} & \\hspace{-1em}{\\hgreen{(+6.5)}} & \\textbf{79.5} & \\hspace{-1em}{\\hgreen{(+2.2)}}\\\\\nConvNext-Base & SGD (freeze-embed) & 92.6 & \\hspace{-1em}{\\hred{(-1.4)}} & 86.9 & \\hspace{-1em}{\\hgreen{(+6.7)}} & \\textbf{91.2} & \\hspace{-1em}{\\hgreen{(+1.4)}} & 38.2 & \\hspace{-1em}{\\hred{(-1.5)}} & 88.1 & \\hspace{-1em}{\\hgreen{(+5.1)}} & \\textbf{79.4} & \\hspace{-1em}{\\hgreen{(+2.1)}}\\\\\n\\midrule\nBiT ResNet-50 & SGD & \\textbf{84.3} &  & \\textbf{76.5} &  & 80.0 &  & 34.1 &  & 90.4 &  & 73.1 & \\\\\nBiT ResNet-50 & AdamW & 83.1 & \\hspace{-1em}{\\hred{(-1.2)}} & 74.8 & \\hspace{-1em}{\\hred{(-1.7)}} & \\textbf{84.0} & \\hspace{-1em}{\\hgreen{(+4.0)}} & 33.8 & \\hspace{-1em}{\\hred{(-0.3)}} & 92.4 & \\hspace{-1em}{\\hgreen{(+2.0)}} & 73.6 & \\hspace{-1em}{\\hgreen{(+0.5)}}\\\\\nBiT ResNet-50 & SGD (freeze-embed) & 84.1 & \\hspace{-1em}{\\hred{(-0.2)}} & 75.5 & \\hspace{-1em}{\\hred{(-1.0)}} & 82.3 & \\hspace{-1em}{\\hgreen{(+2.3)}} & \\textbf{35.0} & \\hspace{-1em}{\\hgreen{(+0.9)}} & \\textbf{95.2} & \\hspace{-1em}{\\hgreen{(+4.8)}} & \\textbf{74.4} & \\hspace{-1em}{\\hgreen{(+1.3)}}\\\\\n\\midrule\nBiT ResNet-101 & SGD & 82.8 &  & 76.9 &  & \\textbf{86.2} &  & \\textbf{38.0} &  & 89.3 &  & 74.6 & \\\\\nBiT ResNet-101 & AdamW & \\textbf{82.9} & \\hspace{-1em}{\\lgreen{(+0.1)}} & \\textbf{79.4} & \\hspace{-1em}{\\hgreen{(+2.5)}} & 83.5 & \\hspace{-1em}{\\hred{(-2.7)}} & 37.0 & \\hspace{-1em}{\\hred{(-1.0)}} & 89.7 & \\hspace{-1em}{\\hgreen{(+0.4)}} & 74.5 & \\hspace{-1em}{\\lred{(-0.1)}}\\\\\nBiT ResNet-101 & SGD (freeze-embed) & \\textbf{83.1} & \\hspace{-1em}{\\hgreen{(+0.3)}} & 77.3 & \\hspace{-1em}{\\hgreen{(+0.4)}} & 86.0 & \\hspace{-1em}{\\hred{(-0.2)}} & 36.0 & \\hspace{-1em}{\\hred{(-2.0)}} & \\textbf{95.5} & \\hspace{-1em}{\\hgreen{(+6.2)}} & \\textbf{75.6} & \\hspace{-1em}{\\hgreen{(+1.0)}}\\\\\n\\bottomrule\n\\multicolumn{14}{l}{$^{\\dagger}$\\footnotesize{SGD fine-tuning on DomainNet with CLIP is unstable leading to high variance in the results. The variance for other datasets were small.}}\n\\end{tabular}\n}\n\\end{center}\n\\end{table}\n\n\n\n\\begin{table}[h!]\n \\small \n\\caption{\n\\textbf{In-distribution (ID)} accuracies presented in the same format as the OOD accuracies in Table~\\ref{tab:ood_bigtable_final}. We see similar high level trends: AdamW gets higher accuracies than SGD on modern vision models, but the gaps are smaller at higher accuracies; and SGD (freeze-embed) is consistently competitively with AdamW.\n\\label{tab:id_bigtable_final}}\n\\begin{center}\n\\resizebox{\\textwidth}{!}{\n\\begin{tabular}{cccccccccccc|cc}\n\\toprule\n &  & \\multicolumn{2}{c}{Living-17}  & \\multicolumn{2}{c}{Waterbirds}  & \\multicolumn{2}{c}{DomainNet}  & \\multicolumn{2}{c}{FMoW}  & \\multicolumn{2}{c}{Camelyon}  & \\multicolumn{2}{c}{Avg.} \\\\\n\\midrule\nCLIP ViT-B\/16 & SGD & 97.8 &  & 97.2 &  & 88.8$^{\\dagger}$ &  & 67.0 &  & \\textbf{99.4} &  & 90.0 & \\\\\nCLIP ViT-B\/16 & AdamW & \\textbf{98.1} & \\hspace{-1em}{\\hgreen{(+0.3)}} & \\textbf{97.7} & \\hspace{-1em}{\\hgreen{(+0.5)}} & \\textbf{95.0} & \\hspace{-1em}{\\hgreen{(+6.2)}} & \\textbf{70.1} & \\hspace{-1em}{\\hgreen{(+3.1)}} & \\textbf{99.5} & \\hspace{-1em}{\\lgreen{(+0.1)}} & \\textbf{92.1} & \\hspace{-1em}{\\hgreen{(+2.1)}}\\\\\nCLIP ViT-B\/16 & SGD (freeze-embed) & \\textbf{98.2} & \\hspace{-1em}{\\hgreen{(+0.4)}} & \\textbf{97.8} & \\hspace{-1em}{\\hgreen{(+0.6)}} & \\textbf{94.9} & \\hspace{-1em}{\\hgreen{(+6.1)}} & \\textbf{70.0} & \\hspace{-1em}{\\hgreen{(+3.0)}} & \\textbf{99.5} & \\hspace{-1em}{\\lgreen{(+0.1)}} & \\textbf{92.1} & \\hspace{-1em}{\\hgreen{(+2.1)}}\\\\\n\\midrule\nCLIP ViT-L\/14 & SGD & 98.4 &  & 97.3 &  & 84.3$^{\\dagger}$ &  & 69.0 &  & \\textbf{99.4} &  & 89.7 & \\\\\nCLIP ViT-L\/14 & AdamW & \\textbf{98.9} & \\hspace{-1em}{\\hgreen{(+0.5)}} & \\textbf{98.8} & \\hspace{-1em}{\\hgreen{(+1.5)}} & 96.9 & \\hspace{-1em}{\\hgreen{(+12.6)}} & \\textbf{74.5} & \\hspace{-1em}{\\hgreen{(+5.5)}} & \\textbf{99.6} & \\hspace{-1em}{\\lgreen{(+0.2)}} & \\textbf{93.7} & \\hspace{-1em}{\\hgreen{(+4.0)}}\\\\\nCLIP ViT-L\/14 & SGD (freeze-embed) & \\textbf{98.7} & \\hspace{-1em}{\\hgreen{(+0.3)}} & \\textbf{98.9} & \\hspace{-1em}{\\hgreen{(+1.6)}} & \\textbf{97.1} & \\hspace{-1em}{\\hgreen{(+12.8)}} & \\textbf{74.5} & \\hspace{-1em}{\\hgreen{(+5.5)}} & \\textbf{99.6} & \\hspace{-1em}{\\lgreen{(+0.2)}} & \\textbf{93.7} & \\hspace{-1em}{\\hgreen{(+4.0)}}\\\\\n\\midrule\nSup ViT-B\/16 & SGD & \\textbf{98.6} &  & \\textbf{99.1} &  & \\textbf{91.7} &  & 64.1 &  & \\textbf{99.4} &  & 90.6 & \\\\\nSup ViT-B\/16 & AdamW & \\textbf{98.7} & \\hspace{-1em}{\\lgreen{(+0.1)}} & \\textbf{99.0} & \\hspace{-1em}{\\lred{(-0.1)}} & \\textbf{91.7} & \\hspace{-1em}{\\lred{(-0.0)}} & \\textbf{66.4} & \\hspace{-1em}{\\hgreen{(+2.3)}} & \\textbf{99.5} & \\hspace{-1em}{\\lgreen{(+0.1)}} & \\textbf{91.1} & \\hspace{-1em}{\\hgreen{(+0.5)}}\\\\\nSup ViT-B\/16 & SGD (freeze-embed) & \\textbf{98.7} & \\hspace{-1em}{\\lgreen{(+0.1)}} & \\textbf{99.2} & \\hspace{-1em}{\\lgreen{(+0.1)}} & 91.5 & \\hspace{-1em}{\\hred{(-0.2)}} & 65.0 & \\hspace{-1em}{\\hgreen{(+0.9)}} & \\textbf{99.6} & \\hspace{-1em}{\\lgreen{(+0.2)}} & 90.8 & \\hspace{-1em}{\\hgreen{(+0.2)}}\\\\\n\\midrule\nDINO ViT-B\/16 & SGD & \\textbf{98.4} &  & 97.0 &  & 88.2 &  & 62.4 &  & 99.4 &  & 89.1 & \\\\\nDINO ViT-B\/16 & AdamW & \\textbf{98.5} & \\hspace{-1em}{\\lgreen{(+0.1)}} & \\textbf{97.9} & \\hspace{-1em}{\\hgreen{(+0.9)}} & \\textbf{89.4} & \\hspace{-1em}{\\hgreen{(+1.2)}} & \\textbf{66.0} & \\hspace{-1em}{\\hgreen{(+3.6)}} & \\textbf{99.6} & \\hspace{-1em}{\\hgreen{(+0.2)}} & \\textbf{90.3} & \\hspace{-1em}{\\hgreen{(+1.2)}}\\\\\nDINO ViT-B\/16 & SGD (freeze-embed) & \\textbf{98.4} & \\hspace{-1em}{\\lgreen{(+0.0)}} & 97.5 & \\hspace{-1em}{\\hgreen{(+0.5)}} & 89.0 & \\hspace{-1em}{\\hgreen{(+0.8)}} & 63.5 & \\hspace{-1em}{\\hgreen{(+1.1)}} & \\textbf{99.5} & \\hspace{-1em}{\\lgreen{(+0.1)}} & 89.6 & \\hspace{-1em}{\\hgreen{(+0.5)}}\\\\\n\\midrule\nConvNext-Base & SGD & \\textbf{98.7} &  & 99.0 &  & 94.8 &  & 66.3 &  & 99.4 &  & 91.6 & \\\\\nConvNext-Base & AdamW & \\textbf{98.6} & \\hspace{-1em}{\\lred{(-0.1)}} & \\textbf{99.5} & \\hspace{-1em}{\\hgreen{(+0.5)}} & 94.5 & \\hspace{-1em}{\\hred{(-0.3)}} & \\textbf{68.8} & \\hspace{-1em}{\\hgreen{(+2.5)}} & \\textbf{99.7} & \\hspace{-1em}{\\hgreen{(+0.3)}} & \\textbf{92.2} & \\hspace{-1em}{\\hgreen{(+0.6)}}\\\\\nConvNext-Base & SGD (freeze-embed) & \\textbf{98.6} & \\hspace{-1em}{\\lred{(-0.1)}} & \\textbf{99.4} & \\hspace{-1em}{\\hgreen{(+0.4)}} & \\textbf{95.1} & \\hspace{-1em}{\\hgreen{(+0.3)}} & 67.4 & \\hspace{-1em}{\\hgreen{(+1.1)}} & \\textbf{99.5} & \\hspace{-1em}{\\lgreen{(+0.1)}} & \\textbf{92.0} & \\hspace{-1em}{\\hgreen{(+0.4)}}\\\\\n\\midrule\nBiT ResNet-50 & SGD & 97.4 &  & \\textbf{98.4} &  & \\textbf{89.3} &  & 64.6 &  & \\textbf{99.5} &  & \\textbf{89.8} & \\\\\nBiT ResNet-50 & AdamW & 97.2 & \\hspace{-1em}{\\hred{(-0.2)}} & \\textbf{98.5} & \\hspace{-1em}{\\lgreen{(+0.1)}} & \\textbf{89.2} & \\hspace{-1em}{\\lred{(-0.1)}} & \\textbf{65.1} & \\hspace{-1em}{\\hgreen{(+0.5)}} & \\textbf{99.5} & \\hspace{-1em}{\\lgreen{(+0.0)}} & \\textbf{89.9} & \\hspace{-1em}{\\lgreen{(+0.1)}}\\\\\nBiT ResNet-50 & SGD (freeze-embed) & \\textbf{97.6} & \\hspace{-1em}{\\hgreen{(+0.2)}} & \\textbf{98.5} & \\hspace{-1em}{\\lgreen{(+0.1)}} & \\textbf{89.2} & \\hspace{-1em}{\\lred{(-0.1)}} & 64.8 & \\hspace{-1em}{\\lgreen{(+0.2)}} & \\textbf{99.5} & \\hspace{-1em}{\\lgreen{(+0.0)}} & \\textbf{89.9} & \\hspace{-1em}{\\lgreen{(+0.1)}}\\\\\n\\midrule\nBiT ResNet-101 & SGD & \\textbf{98.3} &  & \\textbf{98.9} &  & \\textbf{92.0} &  & 66.0 &  & \\textbf{99.4} &  & \\textbf{90.9} & \\\\\nBiT ResNet-101 & AdamW & \\textbf{98.4} & \\hspace{-1em}{\\lgreen{(+0.1)}} & 98.6 & \\hspace{-1em}{\\hred{(-0.3)}} & 91.1 & \\hspace{-1em}{\\hred{(-0.9)}} & \\textbf{67.0} & \\hspace{-1em}{\\hgreen{(+1.0)}} & \\textbf{99.6} & \\hspace{-1em}{\\lgreen{(+0.2)}} & \\textbf{90.9} & \\hspace{-1em}{\\lred{(-0.0)}}\\\\\nBiT ResNet-101 & SGD (freeze-embed) & \\textbf{98.4} & \\hspace{-1em}{\\lgreen{(+0.1)}} & \\textbf{98.8} & \\hspace{-1em}{\\lred{(-0.1)}} & 91.5 & \\hspace{-1em}{\\hred{(-0.5)}} & 65.9 & \\hspace{-1em}{\\lred{(-0.1)}} & \\textbf{99.5} & \\hspace{-1em}{\\lgreen{(+0.1)}} & \\textbf{90.8} & \\hspace{-1em}{\\lred{(-0.1)}}\\\\\n\\bottomrule\n\\multicolumn{14}{l}{$^{\\dagger}$\\footnotesize{SGD fine-tuning on DomainNet with CLIP is unstable leading to high variance in the results. The variance for other datasets were small.}}\n\\end{tabular}\n}\n\\end{center}\n\\end{table}\n\n\n\n\n\n\n\n\\section{Discussion.}\nWe note that the methods we consider are not complex.\nWe showed that a minor tweak of freezing the embedding layer overwhelmingly improves the performance across the board when fine-tuning with SGD. %\nWe clarify that we do not claim that SGD (freeze-embed) is a substantially better method than AdamW in terms of accuracy. Rather,  it is remarkable that with its simplicity, we can already achieve \\emph{comparable} or even slightly better accuracy than AdamW across a wide range of models and benchmarks, all while retaining the memory and computational edge of SGD over AdamW. \nThe broader point of our work is that pretrained models have a lot of useful information, but naively fine-tuning these models can lead to sub-optimal performance.\nExploring simple properties of these models such as the gradients of different layers, and designing simple modifications such as freezing just a small part of the model, can lead to state-of-the-art accuracies. %\n\\section{Introduction}\n\nFine-tuning large pretrained models on downstream tasks has become a dominant approach in deep learning~\\citep{kornblith2019better,chen2020simclr,zhai2020largescale}. The two most commonly used optimizers in current practice  are SGD and AdamW~\\citep{kingma2015adam, Loshchilov2019DecoupledWD}\\footnote{We use SGD to  refer to its usage in deep learning as minibatch stochastic gradient descent with momentum.}. While most modern vision architectures (ViTs, ConvNeXts, and variants) increasingly use AdamW for pretraining, it is still common to use SGD for fine-tuning. Part of the appeal is that SGD is more memory and compute efficient: AdamW maintains $4$ states\/parameter, while SGD only maintains $3$  states\/parameter~\\citep{Ginsburg2019TrainingDN,dettmers2022optimizers}. In training ultra-large models, the additional  memory from even $1$ extra state\/parameter can be costly. At the same time, in terms of fine-tuning accuracies, prior work \\citep{dosovitskiy2021vit, Steiner2021HowTT, kumar2022finetuning} report similar performance between AdamW and SGD on ImageNet like domains that are closer to pretraining data.  In contrast, we reach different conclusions when fine-tuning on datasets that are far from pretraining data or have substantial distribution shifts.  \n\nWe examine 7 popular models, including vision transformers~\\citep{dosovitskiy2021vit,caron2021emerging,radford2021clip}, ConvNeXts~\\citep{liu2022convnet},  and ResNets~\\citep{Kolesnikov2020BigT,he2016resnet}, of different sizes and pretraining modalities. When pretrained on a large corpus and then fine-tuned, these models achieve near state-of-the-art performance on downstream benchmarks. \nIn addition to good transfer learning, we also want our fine-tuned models to handle practical distribution shifts gracefully. So we focus on 5 distribution shift datasets that have both in-distribution (ID) and out-of-distribution (OOD) evaluations: WILDS-FMoW, WILDS-Camelyon, Waterbirds, BREEDS-Living-17, DomainNet. These  were selected to capture different types of data shifts (subpopulation shifts, spurious correlations, style shifts), including two real world shifts in medical imaging and satellite remote sensing from the WILDS benchmark~\\citep{koh2021wilds}. \n\n\\begin{figure}[thb]\n     \\centering\n     \\begin{subfigure}[b]{0.40\\textwidth}\n         %\n         \\centering\n         \\includegraphics[width=\\textwidth]{figures\/illustration-vit.png}\n         \\caption{Simplified schematic of ViT illustrating how we do freeze-embed.\n         \\label{fig:vit_simplified}}\n     \\end{subfigure}\n     \\hfill\n     \\begin{subfigure}[b]{0.55\\textwidth}\n         \\centering\n         \\label{fig:clip_results_summary}\n        %\n        %\n        %\n        %\n        %\n        %\n        %\n        %\n        %\n        %\n        \\resizebox{\\linewidth}{!}{\n         \\begin{tabular}{ccc}\n            \\toprule\n             & ID accuracy & OOD accuracy \\\\\n            \\midrule\n            SGD & {90.0\\%} & {67.9\\%}\\\\\n            AdamW & \\hgreen{(+2.1\\%)} & \\hgreen{(+8.1\\%)} \\\\\n            SGD (freeze-embed) & \\hgreen{(+2.1\\%)} & \\hgreen{(+8.0\\%)} \\\\\n            SGD (freeze-embed, no mom.) & \\hgreen{(+2.2\\%)} & \\hgreen{\\textbf{(+9.0\\%)}} \\\\            \n            Gradual-Unfreeze  & \\hgreen{\\textbf{(+2.3\\%)}} & \\hgreen{(+8.3\\%)} \\\\\n            \\bottomrule\n        \\end{tabular}\n        }\n          \\caption{Performance of different fine-tuning methods on a CLIP ViT-B\/16 averaged over $5$ distribution shift datasets.\\label{fig:three graphs}}\n     \\end{subfigure}\n     \\setlength{\\belowcaptionskip}{-10pt}\n        \\caption{We fine-tune 7 models including ViTs, DINO, CLIP, ConvNeXt, ResNet, on 5 distribution shift datasets (Living-17, Waterbirds, DomainNet, WILDS-Camelyon, WILDS-FMoW). Fine-tuning with SGD gets lower accuracies than AdamW on modern architectures (transformers and ConvNeXt), especially OOD. Interestingly, a minor tweak to SGD where we freeze the first ``embedding'' layer ($<1\\%$ of parameters---see  Figure~\\ref{fig:vit_simplified}) is competitive with AdamW while using lower GPU memory. Further, dropping the momentum state in SGD gives additional gains in accuracy at even lower memory cost. Gradually unfreezing a model's layers using a carefully annealed learning rate can further improves accuracies, while reducing computation.   %\n        }\n\\end{figure}\n\nWe find that on newer models like ViTs and ConvNeXt,  AdamW can have  significantly higher accuracies, especially OOD. For example, averaged across the datasets,  fine-tuning a CLIP ViT-B\/16 model with AdamW gets 2.1\\% higher accuracy ID and 8.1\\% higher accuracy OOD compared to SGD  (Figure~\\ref{fig:three graphs}). These gains are consistent across models too---averaged across all models and datasets, AdamW gets 1.2\\% higher accuracy ID and 4.0\\% higher accuracy OOD (Tables~\\ref{tab:ood_bigtable_final}-\\ref{tab:id_bigtable_final}).\n\nA key difference between AdamW and SGD, is that AdamW normalizes the gradient update of each parameter using an estimate of their second moments. Thus, parameters with consistently high gradients will change less when using AdamW than with SGD.\nTowards understanding these dynamics better, we examine the gradients at each layer of our pretrained models.\nWe find that for the models where AdamW significantly outperforms SGD, the gradients at pretrained initialization of the first ``embedding'' layer are much larger than the gradients of the other layers.\n\nTo test if over-training of the embedding layer is in fact why SGD performs worse than AdamW, we consider a minor modification where we freeze the embedding layer and tune the rest of the model with SGD (Figure~\\ref{fig:vit_simplified})---we call this SGD (freeze-embed).\nIn vision transformer models, the embedding layers are only a small fraction (around 0.7\\% for ViT-B\/16) of the total parameters of the model, so a priori we might not expect a substantial difference in accuracies.\nHowever, surprisingly this simple freezing of the embedding layer consistently improves SGD performance across most models and datasets and achieved ID and OOD accuracies that are competitive  with or better than AdamW (Figure \\ref{fig:three graphs}). \nAveraged across all datasets and models, {SGD (freeze-embed) gets 76.7\\% accuracy OOD (vs. 72.0\\% for SGD and 76.0\\% for AdamW)}. The analogous AdamW (freeze-embed) gets 76.5\\%, which does not improve over SGD (freeze-embed), supporting that freeze-embed may be the reason AdamW outperforms SGD (it is not an independent axis of improvement).\n\nInspired by our results from SGD (freeze-embed), we explored further variations of our method on CLIP ViTs. First, we tried a more memory efficient variation, SGD (freeze-embed, no momentum), which drops the momentum state in SGD. At least on CLIP models, we see that this variation gets better accuracy than SGD (freeze-embed) while giving additional memory gains. Second, we revisit the layerwise unfreezing technique proposed in prior work like ULMFiT~\\citep{howard2018universal} and related methods~\\citep{mukherjee2019distilling, romero2020targeted}. %\nWe find that with a certain exponential decay in learning rate, a gradual unfreezing procedure in some cases can lead to further gains in ID and OOD accuracies (Figure \\ref{fig:three graphs}).\nHowever, a large fraction of the gain over SGD appears to come from simply freezing the embedding layer, which suggests that the embedding layer plays a key role in modern vision architectures and merits further investigation. \n\nIn terms of resource comparison, our profiling on a Titan-X GPU shows that on a ViT-B\/16, AdamW uses $16\\%$ and $36\\%$ more memory than SGD (freeze-embed) and  SGD (freeze-embed, no momentum), respectively. The memory overhead of AdamW increases with the size of the models. On a ViT-L\/14 the memory overheads of AdamW are $18\\%$, and $49\\%$, respectively. Gradual-unfreezing provides a gain in computational time of about $30\\%$ over AdamW.\n\nThese methods and insights, while simple, lead to state-of-the-art accuracies on all five datasets: WILDS-Camelyon, WILDS-FMoW, DomainNet, Waterbirds, and BREEDS Living-17, while being more compute and memory efficient than AdamW.\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Acknowledgements}\n\nWe thank Percy Liang, Tengyu Ma, Yuanzhi Li, and Zhiyuan Li for helpful comments.\n\n\\clearpage\n\n\\section{Additional related works}\n\nMany works in transfer learning propose freezing parameters while fine-tuning to preserve pretrained information. For example, linear probing, which freezes the entire model except the head~\\citep{kumar2022finetuning, wortsman2021robust} and  zero-shot models~\\citep{radford2021clip} have been shown to improve OOD performance over standard fine-tuning. In NLP, methods such as prefix-tuning~\\citep{li2021prefix} and prompt-tuning~\\citep{lester2021power} have been shown to improve OOD accuracy. Other works propose regularizing parameters towards initialization, freezing \nthe first several layers, using\ndifferent learning rates for different layers, or tuning different layers for different examples~\\citep{long2013transfer, ge2017borrowing,howard2018universal, guo2019spottune,zhang2020sidetuning,zhu2020freelb,jiang2021smart,aghajanyan2021finetuning}.\nTypically, a large fraction of the model is frozen to preserve the pretrained information---a key difference in this work is that we find that freezing a very small fraction of the model (<1\\% of the parameters) can lead to substantial and consistent improvements in accuracy.\n\nOther optimizers have been proposed to reduce AdamW's memory footprint, including LARS~\\citep{You2017LargeBT} and AdaFactor~\\citep{shazeer2018adafactor}.  Our method is simpler and achieves better accuracies with same or better memory gains. A complementary line of work \\cite{dettmers2022optimizers} study quantization mechanisms for optimizer states. These tools, although developed for AdamW, can also be used with SGD (freeze-embed) to get additional gains in memory. \n\n\n\\section{Scope and setup}\\label{sec:problem_setting}\n\n\nWe use the following notation: a  network map $f_\\theta:\\mathcal{X}\\to\\mathcal{Y}$  is represented as a composition of layers as $f_\\theta = f^{(\\mathsf{head})}_{\\theta_{\\mathsf{head}}} \\circ f^{(\\mathsf{L})}_{\\theta_L} \\circ \\ldots \\circ f^{(\\mathsf{1})}_{\\theta_1} \\circ f^{(\\mathsf{embed})}_{\\theta_{\\mathsf{embed}}}$, where $\\theta=(\\theta_{\\mathsf{head}}, \\theta_L, \\ldots, \\theta_1, \\theta_{\\mathsf{embed}})$ denote all the parameters of the model. We use $f^{(\\mathsf{embed})}_{\\theta_{\\mathsf{embed}}}$ and $f^{(\\mathsf{head})}_\\theta_{\\mathsf{head}}$ to denote blocks that can conceptually be considered the ``embedding'' layer and the ``head'', respectively. %\n\n\n\\paragraph{Fine-tuning.} Consider networks that have been \\emph{pretrained} to get an initialization $\\theta^{\\mathsf{pretrain}}$. We focus on \\emph{fine-tuning} on a labeled dataset $D_{\\mathsf{train}}{\\color{olive} \\sim P_{\\mathsf{finetune}}}$ from a new task.\nConcretely, given a loss function $\\ell : \\mathcal{Y} \\times \\mathcal{Y} \\to \\mathbb{R}_{\\geq 0}$, we minimize the training loss $L(\\theta) =  \\frac{1}{\\lvert D_{\\mathsf{train}} \\rvert} \\sum_{(x, y) \\in D_{\\mathsf{train}}} \\ell(f_{\\theta}(x), y)$ using iterative optimization algorithms starting from the initialization $\\theta^{\\mathsf{pretrain}}$. %\n\nWe evaluate the accuracy of fine-tuned models on held-out in-distribution (ID) and out-of-distribution (OOD) test datasets. For ID evaluation, we use samples $D_{\\mathsf{test}}^{\\mathsf{id}} \\sim {\\color{olive} P_{\\mathsf{finetune}}}$ from the same distribution as the fine-tuning training dataset.\nTo examine whether we have learned a robust model, we consider benchmarks that also provide OOD test examples $D_{\\mathsf{test}}^{\\mathsf{ood}} \\sim {\\color{purple} P_{\\mathsf{ood}}}$, which differs from the fine-tuning distribution $ {\\color{olive}P_{\\mathsf{finetune}}}$ in practically meaningful ways. \n\n\n\\subsection{Optimizers: SGD (with momentum) and AdamW}\n %\nThe two most common optimizers for minimizing the fine-tuning loss $L(\\theta)$ from pretrained initialization are SGD (with momentum) and AdamW. We will introduce other optimizers as needed.   \nCompared to vanilla SGD (no momentum), which only stores the parameters and gradients as optimizer states, SGD (with momentum) stores 1 extra state per parameter (to track the first moment), while AdamW stores 2 extra states per parameter (to track the first and second moments)---see Appendix~\\ref{app:setup} for exact updates. This corresponds to a difference between AdamW and SGD of 4GB GPU memory per 1B parameter model during training\\footnote{The bottleneck for memory in older ResNe(X)ts is typically the number of activations, which is often much higher than the number of parameters. However, in modern large transformer training, memory requirements are of the same scale as the number of parameters. Further, techniques such as gradient accumulation and gradient checkpointing can make the activation memory small leaving the optimizer states as the main bottleneck. }. With the current scale of the models 100s of billions parameters and increasing,  such memory overheads are very costly \\citep{dettmers2022optimizers}. Thus,  understanding when and how we can use the cheaper SGD compared to AdamW can significantly improve training of ultra-large scale models.   \n\n\n\n\n\\subsection{{Datasets and model architectures}}\n\n\\paragraph{Datasets.} We choose five fine-tuning benchmarks that capture different types of data shifts (subpopulation shifts, spurious correlations, style shifts), including two real world shifts in medical imaging and satellite remote sensing from the WILDS benchmark. %\nThe datasets we consider are:\n\\begin{compactenum}\n\\item \\textbf{Living-17}~\\citep{santurkar2020breeds} is a sub-population shift dataset from the BREEDS benchmark. The goal is to classify an image as one of 17 animal categories with ID and OOD data from different sub-categories. For example,   in the  ``bear'' category, the ID dataset contains black bears and sloth bears and the OOD dataset has brown bears and polar bears.\n\\item \\textbf{Waterbirds}~\\citep{sagawa2020group} is a spurious correlation dataset where the goal is to classify an image as a ``waterbird'' or ``landbird''. In the ID dataset, ``water'' backgrounds are typically correlated with ``waterbird'' labels, but are uncorrelated in the OOD dataset.\n\\item \\textbf{DomainNet}~\\citep{peng2019moment} is a domain adaptation benchmark. The ID dataset contains \\textit{sketch} images (e.g., drawings of apples, elephants, etc), and the OOD dataset contains \\textit{real} images of the same categories. We use the version of the dataset from~\\citet{tan2020coal}.\n\\item \\textbf{WILDS-FMoW}~\\citep{christie2018fmow, koh2021wilds} consists of remote sensing satellite images. The goal is to classify a satellite image into one of 62  categories such as ``impoverished settlement'' or ``hospital''. The ID dataset contains satellite images from across the world between 2002 and 2012, and the OOD dataset contains images from Africa in 2017.\n\\item \\textbf{WILDS-Camelyon}~\\citep{bandi2018detection, koh2021wilds} is a medical images dataset. The task here is to classify whether a tissue slide has a tumor or not. The ID and OOD datasets contain slides from different hospitals.\n\\end{compactenum}\n\n\n\\paragraph{Model Architectures.} We consider seven popular pretrained models listed below that span different architectures (vision transformers and  convolutional networks), network sizes, and pretraining objectives (multi-modal, supervised, self-supervised). %\n\\begin{compactitem}\n\\item[(1-2)] CLIP ViT-B\/16 and CLIP ViT-L\/14~\\citep{radford2021clip}: CLIP vision transformers of two sizes pretrained on a custom multi-modal WebImageText (WIT) dataset.  \n\\item[(3)] ViT-B\/16~\\citep{dosovitskiy2021vit}: vision transformer pretrained on Imagenet-21k.\n\\item[(4)] DINO ViT-B\/16~\\citep{caron2021emerging}: self-supervised ViT pretrained on ImageNet-1K. \n\\item[(5)] ConvNeXt-B~\\citep{liu2022convnet}: modernized convolutional network pretrained on ImageNet-21k using advanced data augmentations and MixUp as in \\cite{Touvron2021TrainingDI}.\n\\item[(6-7)] BiT ResNet-50 and BiT ResNet-101~\\citep{Kolesnikov2020BigT}: ResNetV2 models of two sizes pretrained on ImageNet-21k.\n\\end{compactitem}\n\n\n\\textit{$f^{(\\mathsf{embed})}_{\\theta_{\\mathsf{embed}}}$ and $f^{(\\mathsf{head})}_{\\theta_{\\mathsf{head}}}$}: For vision transformer models, we consider the patch-to-token embedding layer along with its layer norm if present as the \\textit{embedding layer}. For convolutional networks, the \\textit{embedding layer} refers to the ``stem'' block along with the first stage: the ``stem'' in ResNetV2 is a $7\\times 7$ convolution with stride $2$ followed by a $2\\times 2$ MaxPool; while in ConvNeXt it is a non-overlapping $4\\times 4$ convolution with stride $4$. For each downstream task, we replace the final layer  of all the pretrained models with a randomly initialized classifier \\textit{head}  $f^{(\\mathsf{head})}_{\\theta_{\\mathsf{head}}}$.\n\n\n\n\n \n\\section{Initial observations: SGD, AdamW, and layer gradients}\n\n\nModern deep learning models increasingly use AdamW for pretraining, where it has been repeatedly shown to produce better features for downstream tasks than SGD \\citep{dosovitskiy2021vit,liu2021Swin,liu2022convnet}. %\nFor fine-tuning, on the other hand, there are no systematic studies or a definitive  answer as to whether AdamW or SGD should be used \\citep{dosovitskiy2021vit,Touvron2021TrainingDI}. %\nThe ablation study in \\citet{Touvron2021TrainingDI} even found that there is no difference in performance between AdamW and SGD when fine-tuning on ImageNet-1K. ConvNext \\citep{liu2022convnet} and Swin transformers \\citep{liu2021Swin} papers report using AdamW for fine-tuning, but they do not mention a comparison with SGD. In this work we focus on better understanding the dynamics of AdamW and SGD during the fine-tuning  phase. \nDetailed results are discussed in Section~\\ref{sec:mainexp}. \nWe first highlight some initial observations below. \n\n\\subsection{AdamW vs SGD}\n\n\\paragraph{AdamW outperforms SGD.} We find that, generally speaking, fine-tuning with AdamW produces better results than with SGD, especially for more recent models like ViT variants and ConvNeXt. The gaps are more dramatic on out-of-distribution (OOD) test accuracies compared to in-distribution (ID). See  Table~\\ref{tab:ood_bigtable_final} and Table~\\ref{tab:id_bigtable_final} for the full OOD and ID results, respectively.\nAveraged across the 7 models and 5 datasets, AdamW gets an {OOD accuracy of 76.0\\% (vs. 72.0\\% for SGD), and an ID accuracy of 91.5\\% (vs. 90.3\\% for SGD)}. We emphasize that this happens even though we sweep over 6 learning rates and early stop. Note also that these averages slightly underestimate the gaps in newer architectures as they also include the two older ResNet models, where we did not find significant differences. %\n\n\\paragraph{AdamW $\\approx$ SGD on BiT ResNets.}\nBreaking down the results by model type, we find that AdamW and SGD perform comparably for older convolutional networks, namely BiT ResNet-50 and BiT ResNet-101. For example, on a BiT ResNet-101, AdamW gets an average OOD accuracy of 74.5\\% (vs. 74.6\\% for SGD).\nHowever, for modern architectures including the modernized convolutional network (ConvNeXt), AdamW gets significantly higher accuracies.\n\n\\sgnote{I am removing accuracy gap for larger models is bigger to clip detailed evlauation section -- fr one we are not completely sure of the statement and it also does not fit in well with the initial section narrraitve --gradient outlier is smaller for clip vit-l}\n\n\\subsection{Examining layer gradients}\n\nA key operational difference between AdamW and SGD is that AdamW divides the gradient update for each parameter by a weighted running average of its second moments. So parameters with consistently high gradients will change less when using AdamW than with SGD. This suggests examining the gradients across different components of the neural network---if they vary a lot, then AdamW and SGD will behave very differently.%\n\nWe measure the average gradient norm at each layer, across minibatches of the training set.\nMore formally, recall our notation for network layers as $f_\\theta = f^{(\\mathsf{head})}_{\\theta_{\\mathsf{head}}} \\circ f^{(\\mathsf{L})}_{\\theta_L} \\circ \\ldots \\circ f^{(\\mathsf{1})}_{\\theta_1} \\circ f^{(\\mathsf{embed})}_{\\theta_{\\mathsf{embed}}}$. Given a minibatch $B=\\{ (x_1, y_1), \\ldots, (x_b, y_b) \\}$ of random samples from the training data, the stochastic gradient of layer $\\ell\\in \\{\\mathsf{embed},1,2,\\ldots,L,\\mathsf{head}\\}$ is given by: \n$g_\\ell(B) =  \\frac{1}{|B|} \\sum_{i=1}^{|B|} \\nabla_{\\theta_\\ell} l(f_{\\theta^{\\mathsf{pretrain}}}(x_i), y_i)$. \nThe norm of the stochastic gradient, $\\|g_\\ell(B)\\|_2$, roughly measures how much layer $\\ell$ will change after one step of SGD from pretrained initialization. We use the average gradient norm across all minibatches $B_1, \\ldots, B_m$ in the training set $D_{\\mathsf{train}}$ as a measure of movement in the first SGD step. \n\\begin{equation}\n    G^{\\mathsf{init}}_\\ell = \\frac{1}{m} \\sum_{t=1}^{m} \\|g_\\ell(B_t)\\|_2,\\text{  where }\\ell\\in \\{\\mathsf{embed},1,2,\\ldots,L,\\mathsf{head}\\}.\n    \\label{eq:gradnormavg}\n\\end{equation}\nNote that we measure all the gradients at the pretrained initialization $\\theta^{\\mathsf{pretrain}}$---before performing any gradient updates. In Figure~\\ref{fig:gradient_norm}, we plot the layer-wise gradient norms $G^{\\mathsf{init}}_\\ell$ on DomainNet. Similar plots for other datasets and with alternative normalization are provided in Appendix~\\ref{app:gradnorm}.\n\n \n\n\\begin{figure}[t]\n    \\centering\n    \\includegraphics[width=0.69\\textwidth]{figures\/domain_net_grad_plot.png}\n    \\caption{\n      We visualize the layer-wise gradient norms our models on DomainNet at the pretrained initialization. For each layer, we plot the average minibatch gradient norm $G^{\\mathsf{init}}_\\ell$ as computed in \\eqref{eq:gradnormavg}. We highlight two special layers: gradient norms of the ``embedding'' layer parameters $G^{\\mathsf{init}}_{\\mathsf{embed}}$  are shown as {\\color{red}\\textbf{red-squares}}, and those of the classifier ``head''s $G^{\\mathsf{init}}_{\\mathsf{head}}$ are show as {\\color{green!60!black} \\textbf{green-triangles}}. The middle layer gradient norms are shown as \\textbf{black-circles}. For transformer models, we have separate (black) points for the MLP and the attention layers. %\n      %\n    }\n    \\label{fig:gradient_norm}\n\\end{figure}\n\n\\paragraph{First-layer gradient is an outlier.}\nApriori, we expect the gradients of the classifier ``head''  (green-triangles) to be large as they are randomly initialized while the rest of the model is pretrained~\\citep{kumar2022finetuning}. The interesting observation we see in Figure~\\ref{fig:gradient_norm} (see also  Appendix~\\ref{app:gradnorm}) is that for the recent architectures (Vision Transformers and ConvNeXt), gradient norms of the ``embedding'' layers ({red-squares}) stand out as outliers with much larger gradients than the other layers. These are also the models where see big gaps between AdamW and SGD in Tables~\\ref{tab:ood_bigtable_final}-\\ref{tab:id_bigtable_final}. \nThis   suggests that the embedding layer plays a distinctive role in the transfer learning from modern architectures. \n\nWe emphasize again that Figure~\\ref{fig:gradient_norm} shows gradient norms at pretrained initialization \\emph{before} any updates have been performed. Thus, the large gradients of the embedding layer is \\textit{not the effect} of SGD vs AdamW fine-tuning. Rather, these trends hint at a potential \\textit{explanation} for the differences between them. \nSince SGD uses the same learning rate for all layers,  we will end up make substantially larger updates to the embedding layer compared to other layers---leading to either over-tuning the embedding layer or under-tuning the remaining layers (which also includes the randomly initialized head). \nOver-training of the embedding layer is undesirable  given the common wisdom that lower layers ought to be tuned less as they learn more transferable features~\\citep{kumar2022finetuning}. %\nAdamW on the other hand adaptively normalizes the movement of each parameter. %\n \nHow can we avoid the above issues with SGD? One possibility is to use different learning rates for different layers, but this might end up requiring extensive hyperparameter tuning. Another option is to use optimizers with layerwise normalization techniques like LARS \\citep{you2017crop} and LAMB \\citep{you2020large}. In our initial experiments (see Table~\\ref{tab:clip_results}), while these methods improved over SGD, they did not close the gap with AdamW. Instead, we found a much simpler modification to SGD that consistently leads to accuracies competitive with or better than AdamW while using lower memory. \n\n\n\n \n\n\\subsection{Freeze-Embedding}\n\nWhile the presence of large embedding layer gradients in Figure~\\ref{fig:gradient_norm} correlates with the observed performance gaps between AdamW and SGD, it is not clear that this observation or its potential issues discussed above are definitive \\emph{causes} for SGD performing worse than AdamW. To further test the intuition, in the next section, we consider a ``freeze-embed'' variation of fine-tuning, where we simply freeze the embedding layer to its pretrained initialization, and then fine-tune the rest of the model as usual. The embedding layer only consists of a small fraction of the model parameters (e.g., 0.7\\% in a base vision transformer), so apriori freezing the embedding layer is a very tiny tweak---and we would not expect a large change in accuracy. However, if the hypothesis above holds merit, then we would expect the modification to aid SGD but not AdamW. \n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\subsection{SoTA experiments and results}\nOur experiments get new state-of-the-art results for OOD accuracy on all 5 datasets. On 3\/5 datasets (Living-17,  WILDS-FMoW, and WILDS-Camelyon), our proposed SGD (freeze-embed) does the best, while in other 2, AdamW has a small edge. Here, state-of-the-art means that the numbers we get are better than, to our knowledge, any reported number and all numbers on the official leaderboard, and are better than standard full fine-tuning with SGD. We show the best results from our paper in Table~\\ref{tab:sota_results} with a comparison to the previous state-of-the-art. \\sgnote{what about gradual unfreezing?} %\n\n\n\n\\begin{table}[h!]\n\\caption{\nOur OOD accuracy results compared with the best reported numbers in prior work on these datasets. We restrict to methods that \\emph{do not} use OOD data for hyperparameter selection or early stopping. To  our knowledge, the previous state-of-the-art results are from~\\citet{wortsman2022modelsoups} for FMoW,~\\citet{Robey2021ModelBasedDG} for Camelyon,~\\citet{kumar2022finetuning} for Living-17 and the version of DomainNet introduced by~\\citet{tan2020coal}, and~\\citet{ghosal2022vision} for Waterbirds. The WILDS numbers and references are taken from the official WILDS leaderboard (as of 28 Sep 2022), and for Waterbirds we consider all methods that do not use group labels. {For Living-17, we omit models pretrained with ImageNet-1K as Living-17 is a subset of ImageNet-1K.} %\n}\n\\label{tab:sota_results}\n\\begin{center}\n\\resizebox{\\linewidth}{!}{\n\\begin{tabular}{cccccc}\n\\toprule\n     & Living-17 & Waterbirds & DomainNet & FMoW & Camelyon\\\\\n    \\midrule\n    Best prior result & 87.6 & 89.3 & 87.2 & 47.6 & 93.3 \\\\\n    \\midrule\n    \\vspace{0.02in}\n    Best result from our paper & \\textbf{90.5} & \\textbf{89.8} & \\textbf{93.8} & \\textbf{49.9} & \\textbf{96.5} \\\\\n    \\vspace{0.02in}\n    Optimizer for best result & \\begin{tabular}{@{}c@{}}SGD \\\\ (freeze-embed) \\end{tabular} & AdamW & AdamW & \\begin{tabular}{@{}c@{}}SGD \\\\ (freeze-embed) \\end{tabular} & \\begin{tabular}{@{}c@{}}SGD \\\\ (freeze-embed) \\end{tabular} \\\\\n    \\vspace{0.02in}\n    Model for best result & CLIPViT-L\/14 & ConvNeXt-B & CLIP ViT-L\/14 & CLIP ViT-L\/14 & CLIP ViT-L\/14 \\\\\n    \\midrule\n    SGD  (freeze-embed) result  & \\textbf{90.5} & 86.9 & {93.1} &  \\textbf{49.9} & \\textbf{96.5} \\\\\n    \\bottomrule\n\\end{tabular}\n}\n\\end{center}\n\\end{table}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nCompact star clusters around supermassive black holes (SMBHs) in galactic nuclei are the densest known stellar systems in the Universe.  Rapid exchange of angular momentum between the stars and stellar remnants in these environments causes stars to be occasionally perturbed onto low angular momentum (``loss cone'') orbits, which bring them near or within the tidal radius of the SMBH.\n\nThe vast majority of ill-fated stars descend to the SMBH on nearly parabolic orbits, from initial distances comparable to the parsec-scale SMBH sphere of influence (e.g. \\citealt{Magorrian&Tremaine99,Wang&Merritt04,Stone&Metzger16}).  Depending on the depth of the pericenter distance relative to the tidal radius, these ``plunge\" events result in a partial or complete dynamical disruption of the star accompanied by strong tidal compression, a phenomenon commonly known as a ``tidal disruption event\" (TDE; \\citealt{Hills75,Carter&Luminet83,Evans&Kochanek89,Lodato+09,Guillochon&RamirezRuiz13,Stone+13}).  \nPrompt accretion by the SMBH of the gaseous debris of the disrupted star following a TDE was predicted to power a luminous flare (\\citealt{Rees88,Phinney89,Ulmer99}), characterized by a rise time of weeks to months and followed by a $\\propto t^{-5\/3}$ power-law decay in the bolometric light curve set by the declining rate of mass fallback at late times.  \n\nDespite this initial picture, recent numerical and analytic works have demonstrated that it is non-trivial to circularize the highly eccentric stellar debris streams created by the TDE into the compact accretion disk needed to power a flare (e.g., \\citealt{Hayasaki+13,Shiokawa+15,Dai+15,Hayasaki+16,Guillochon&RamirezRuiz15,Bonnerot+16,Coughlin+16,Sadowski+16,Bonnerot+17,Tejada+17}).  This has lead to speculation that only a small fraction of TDE are accompanied by luminous flares, with most instead being ``dark\" (\\citealt{Guillochon&RamirezRuiz15,Hayasaki+16}) or producing emission through less efficient mechanisms than standard thin-disk accretion (\\citealt{Shiokawa+15,Piran+15,Miller15,Metzger&Stone16}).  This possibility is supported by the discrepency between the theoretically-predicted TDE rate (\\citealt{Stone&Metzger16,Kochanek16}) and the lower flare rate observed from optical surveys (e.g., \\citealt{vanVelzen+11,vanVelzen&Farrar14, Holoien+16}).  \n\nIn addition to those stars on parabolic plunges fated to produce TDEs, a smaller fraction approach the SMBH on more tightly bound orbits with lower eccentricities.  These ``extreme mass ratio inspirals\" (EMRIs) have received attention as gravitational wave sources for the Laser Interferometer Space Antenna (LISA; \\citealt{LISA17}).  This is especially true when the EMRI bodies are compact remnants (white dwarfs, neutron stars, and stellar mass black holes) for which tidal forces play little to no role in their inspiral evolution prior to their final plunge into the event horizon (although see \\citealt{Zalamea+10}).\n\nEMRIs of {\\it main sequence} stars have received comparatively less attention than their compact remnants, in part because matter interactions alter the gravitational wave inspiral signal  (\\citealt{Linial&Sari17}) and thus reduce their potential as pristine probes of general relativity.  If a main sequence star approaches the tidal sphere on a nearly circular orbit and begins overflowing its Roche lobe, mass transfer onto the SMBH is stable, resulting in the star being slowly accreted over millions of years  \\citep{King&Done93, Dai&Blandford13,Linial&Sari17}.  When the response of the star to mass loss is adiabatic, or once its equation of state becomes dominated by electron degeneracy after it loses sufficient mass, the radius of the star expands and its orbit evolves to {\\it larger} semi-major axes as a the result of further mass loss.  The system may be described during this phase as an extreme mass ratio ``outspiral\" (e.g., \\citealt{Dai&Blandford13,Linial&Sari17}), similar to the orbital evolution of cataclysmic variables following the period minimum.\n\nThe rate of EMRIs due to bodies entering the loss cone via two-body gravitational interactions is estimated to be $\\sim 1\\times 10^{-6}$ yr$^{-1}$ per galaxy (\\citealt{BarOr&Alexander16, Aharon&Perets16}; see also \\citealt{Hils&Bender95,Sigurdsson&Rees97,Freitag01,Ivanov02,Alexander&Hopman03,Hopman&Alexander06b}), roughly $2$ orders of magnitude lower than the TDE rate.  Due to the effects of mass segregation in nuclear clusters, this rate is also dominated by stellar mass black holes instead of lower mass main sequence stars or white dwarfs (e.g.~\\citealt{Hopman&Alexander06b,Aharon&Perets16}).  Although the eccentricities of these ``two-body\" EMRIs are much lower than those of the plunge events, they are usually still significant ($e \\gtrsim 0.5-0.9$ on scales of the tidal radius; \\citealt{Hopman&Alexander05}), such that the fate of most main sequence EMRIs delivered by two-body interactions will also be tidal disruption, though their light curves will differ from canonical $1-e\\ll 1$ TDEs \\citep{Hayasaki+13}.\n\nMore nearly circular EMRIs are created by the tidal separation of stellar binaries by the SMBH (\\citealt{Miller+05,AmaroSeoane+12}).  This is the same process hypothesized to produce the cluster of S-stars orbiting Sgr A$^{*}$ (e.g., \\citealt{Perets+09}) and hypervelocity stars from our galactic centre  (e.g.~\\citealt{Hills88,Sari+10}).  \\citet{AmaroSeoane+12} estimate the rate of circular EMRIs to be $\\sim 10^{-7}$ yr$^{-1}$ per galaxy for an assumed binary fraction of $5\\%$ (lower than in the field due to the dissociation of the soft binaries in the dense stellar environment of the nuclear cluster).  This rate could be up to two orders of magnitude higher in galactic nuclei with large numbers of massive perturbers \\citep{Perets+07} or nuclear spiral arms \\citep{Hamers&Perets17}.\n\n\nThis work develops an observable consequence of the existence of stable mass-transferring EMRIs, which is closely related to the fact that the lifetime of these systems is comparable to the average time interval between consecutive EMRIs.  As we will show, this naturally predicts a collision, or a series of quasi-periodic collisions, between every two inspiraling (or, more typically, one inspiraling and one outspiraling) stars as their semi-major axes cross on a radial scale of $\\lesssim$ few AU.  Such collisions, which happen deep within the potential well of the black hole at relative velocities up to several tenths of the speed of light, can seriously damage or completely obliterate both stars, leading to punctuated episodes of sizable gas production.  \n\nRapid accretion of this gas by the SMBH powers a single luminous flare, or a series of periodic flares lasting hundreds to thousands of years or longer, caused by multiple grazing encounters between the stars separated by intervals of decades to centuries.  Although this emission mechanism is qualitatively similar to the common picture of TDE flares, EMRI collision flares may be observed more frequently than would be naively guessed from the low EMRI rate.  This is because standard TDEs may suffer from low radiative efficiencies due either to super-Eddington accretion rates \\citep{Ayal+00, Metzger&Stone16} or, conversely,  difficulty circularizing highly eccentric debris streams \\citep{Hayasaki+13, Shiokawa+15, Guillochon&RamirezRuiz15, Hayasaki+16, Bonnerot+16}.  Indeed, small accreted masses in putative TDE flares are consistent with the low bolometric energy fluences inferred by their optical\/UV SEDs relative to the expection of thin disk accretion, as shown in Fig.~\\ref{fig:Erad} (though the minority of flares seen to exhibit luminous IR dust echoes may have higher bolometric luminosities; \\citealt{vanVelzen+16}).   Collisions among circularized EMRIs can make more judicious use of the available stellar mass budget, as accretion rates are generally sub-Eddington and collisionally liberated gas is produced on initially circular orbits.\n\n\nThis scenario raises the possibility that some nuclear transients currently identified as TDE flares may in fact be EMRI collision products, i.e. ``TDE Imposters\".  Indeed, we show that otherwise puzzling behavior seen in some of these flares, such as light curves which decay exponentially in time rather than as power-laws, may find a natural explanation in the EMRI collision model.  Viscous spreading of the accretion disks (out to radial scales of $\\gtrsim 10-100$ AU) made in regular gas production events over the prolonged interval of the collisional flaring activity also provides an alternative source for the extended reprocessing layer needed to explain the unexpectedly high optical luminosities of many TDE flares.\n\nThis paper is organized as follows.  In $\\S\\ref{sec:emri}$ we review the evolution of stable mass transfer from a main sequence star onto a SMBH.  In $\\S\\ref{sec:collision}$ we discuss the conditions required for a collision between two consecutive EMRI stars.  In $\\S\\ref{sec:transient}$ we discuss the observable signatures of such a collision, including periodic flares from SMBH accretion, compare them to the observed population of TDE flares, and discuss the interactions of main sequence EMRIs with bona fide TDEs.  In $\\S\\ref{sec:conclusions}$ we briefly summarize our conclusions.   \n \n\n\\begin{figure*}[!t]\n\\includegraphics[width=1.0\\textwidth]{Eradplot2.pdf}\n\\hspace{0.0cm}\n\\caption{\\footnotesize\n{Total radiated energy $E_{\\rm rad}$ of optically-selected TDE flare candidates as a function of SMBH mass, calculated using bolometric light curve data compiled by \\citet{Hung+17} (estimated by fitting the optical\/NUV SEDs to a single temperature blackbody), and using redshifts from the Open TDE Catalog\\footnote{\\url{http:\/\/TDE.space}}.  Purple circles show flares where the light curve is temporally resolved, while brown triangles show lower limits on $E_{\\rm rad}$ in cases when the light curve peak is not resolved.  Overplot is the expected value of $E_{\\rm rad} = 0.1 (f_{\\rm in}M_{\\star})c^{2}$ following the accretion of a $M_{\\star} = 1M_{\\odot}$ star for different accretion efficiencies $f_{\\rm in}$, but limiting the radiated luminosity at all times to the Eddington luminosity for the given SMBH mass assuming a peak flare duration equal to the expected mass fall-back time.  This is an updated version of a similar plot (Fig. 2) from \\citet{Stone&Metzger16}.  The 12 data points represent the TDE candidates iPTF16axa, ASASSN-14ae, PS1-11af, PS1-10jh, ASASSN-14li, iPTF16fnl, ASASSN-15oi, TDE1, TDE2, D1-9, D3-13, and PTF09ge.  Estimates for $M_\\bullet$ and its error range are taken from \\citet{Hung+17, Holoien+14, Chornock+14, Gezari+12, vanVelzen+16, Blagorodnova+17, Holoien+16b, vanVelzen+11, Gezari+08, Arcavi+14}.  In two cases (ASASSN-14ae and ASASSN-15oi) the discovery papers do not quote errors on SMBH mass estimates, so for these we have plotted approximate error intervals corresponding to 0.3 dex of scatter typical to galaxy scaling relations.}\n}\n\\label{fig:Erad}\n\\end{figure*}\n\n\n\n\\section{EMRI Mass Transfer Evolution}\n\\label{sec:emri}\n\n\nWe consider stars inspiraling into the SMBH on nearly circular orbits when they reach the point of Roche lobe overflow (RLOF) on radial scales of a few AU (see below).  As discussed above, this will not be satisfied for most of the main sequence EMRIs produced by two-body scattering or resonant relaxation, which instead will possess high eccentricies $e \\gtrsim 0.5$ at this separation and will likely undergo tidal disruption.  \n\nHowever, EMRIs created by tidally detached binaries will generally possess lower eccentricities $e \\sim 0.01-0.05$ when their pericenters reach these distances (\\citealt{Miller+05,AmaroSeoane+12}).  Depending on the competition between circularization of the orbit due to gravitational wave emission versus that due to tidal friction, these stars will either undergo tidal disruption or they will end up in circular, stably mass-transferring orbits (see discussion in \\citealt{AmaroSeoane+12}).  We focus on the latter case, which accounts for a large portion of the allowed parameter space for a reasonable range in the theoretically uncertain value of the $Q$ parameter controlling the rate of tidal circularization.\n\nOnce in a nearly circular orbit of semi-major axis $a$, a star of mass $M_{\\star}$ orbiting the SMBH of mass $M_{\\bullet} \\gg M_{\\star}$ loses energy to gravitational wave (GW) emission on the timescale\n\\begin{equation}\n\\tau_{\\rm GW} \\equiv \\frac{a}{|\\dot{a}|} \\simeq \\frac{5}{64}\\frac{ c^{5} a^{4}}{G^{3}M_{\\star}M_\\bullet^{2}} \\approx 1.3\\times 10^{4}\\,{\\rm yr}\\,\\,\\frac{a_{\\rm AU}^{4}}{M_{\\star,\\odot}M_{\\bullet,7}^{2}},\n\\label{eq:tauGW}\n\\end{equation}\nto lowest post-Newtonian order.  Here $M_{\\star,\\odot} \\equiv M_{\\star}\/M_{\\odot}$, $M_{\\bullet,7} \\equiv M_{\\bullet}\/10^{7}M_{\\odot}$, $a_{\\rm AU} \\equiv a\/$AU, and we have written the gravitational constant and the speed of light as $G$ and $c$, respectively.\n\nThe size of the star's Roche lobe in the extreme mass ratio limit is given by (\\citealt{Eggleton83})\n\\begin{equation}\nR_{\\rm L} \\simeq 0.462 a \\left(\\frac{M_{\\star}}{M_{\\bullet}}\\right)^{1\/3}.\n\\end{equation}\nThis becomes equal to the radius of the star $R_{\\star}$ below a critical semi-major axis\n\\begin{eqnarray}\na_{0} &\\simeq& 2.17 R_{\\star} \\left(\\frac{M_{\\bullet}}{M_{\\star}}\\right)^{1\/3} \\approx 2.16\\,{\\rm AU}\\,R_{\\star,\\odot}M_{\\bullet,7}^{1\/3}M_{\\star,\\odot}^{-1\/3} \\nonumber \\\\\n&\\approx& 2.16\\,{\\rm AU}\\,M_{\\bullet,7}^{1\/3}M_{\\star,\\odot}^{7\/15},\n\\label{eq:a0}\n\\end{eqnarray}\nwhere in the second line we have used a mass-radius relationship $R_{\\star} \\simeq R_{\\odot} M_{\\star,\\odot}^{4\/5}$ appropriate for low mass main sequence stars \\citep{Kippenhahn&Weigert90}. Below a critical black hole mass $M_{\\bullet} \\lesssim 7\\times 10^{7}M_{\\star,\\odot}^{7\/10}M_{\\odot}$, the star overflows its Roche lobe outside the innermost stable circular orbit (ISCO) $R_{\\rm isco} = 6GM_{\\bullet}\/c^{2}$ for a Schwarzschild black hole.  \n\nThe GW inspiral time at Roche contact is\n\\begin{equation}\n\\tau_{\\rm GW,0} \\equiv \\tau_{\\rm GW}(a_{0}) \\approx 2.8\\times 10^{5}\\,{\\rm yr}\\,M_{\\star,\\odot}^{13\/15}M_{\\bullet,7}^{-2\/3}.\n\\label{eq:tauGW0}\n\\end{equation}\nFollowing Roche overflow, the star loses mass, primarily through the inner L1 Lagrange point, at a characteristic rate,\n\\begin{equation}\n\\dot{M_{\\star}} \\simeq -\\frac{M_{\\star}}{\\tau_{\\rm GW}}\n\\end{equation}\nAs the star loses mass, its radius changes according to $R_{\\star} \\propto M_{\\star}^{p}$, where the value of $p$ depends on the properties of the star and its response to mass loss (see below).  \n\nCombining the above results, one finds\n\\begin{equation}\n\\frac{\\dot{M}_{\\star}}{M_{\\star}} = \\frac{-1}{\\tau_{\\rm GW,0}}\\left(\\frac{M_{\\star}}{M_{\\star,0}}\\right)^{\\frac{(7-12p)}{3}} = \\frac{-1}{\\tau_{\\rm GW,0}}\\left(\\frac{a}{a_{0}}\\right)^{\\frac{(7-12p)}{(3p-1)}},\n\\end{equation}\nwhere $M_{\\star,0}$ is the initial mass of the star.  For a fixed value of $p$, this results in the following evolution as a function of time $t$ after the onset of Roche-lobe overflow,\n\\begin{equation}\n\\frac{M_{\\star}}{M_{\\star,0}} = \\left(1 - \\frac{12p-7}{3}\\frac{t}{\\tau_{\\rm GW,0}}\\right)^{\\frac{3}{12p-7}}\n\\label{eq:Mevo},\n\\end{equation}\n\\begin{equation}\n\\frac{a}{a_0} = \\left(1 - \\frac{12p-7}{3}\\frac{t}{\\tau_{\\rm GW,0}}\\right)^{\\frac{3p-1}{12p-7}}.\n\\label{eq:aevo}\n\\end{equation}\n\nIn reality, the value of $p$ evolves in time as the star loses mass.  Following \\citet{Linial&Sari17}, for stars of initial mass $M_{\\star,0} \\lesssim 7.0M_{\\bullet,7}^{0.25} M_{\\odot}$, we have the following evolution\n\\begin{equation}\np = \\left\\{\n\\begin{array}{lr} 0.8, &\nM_{\\star} > M_{\\rm ad} \\\\\n4\/15, &\n1.2M_{\\odot}M_{\\bullet,7}^{0.1} < M_{\\star} < M_{\\rm ad}\\\\\n13\/21, &\n0.08M_{\\odot} (R_{\\star}\/0.1R_{\\odot})^{3} < M_{\\star} < 1.2M_{\\odot}M_{\\bullet,7}^{0.1}, \\\\\n\\approx 0, &\nM_{\\star} < 0.08M_{\\odot}(R_{\\star}\/0.1R_{\\odot})^{3} \\\\\n\\end{array}\n\\label{eq:p}\n\\right. ,\n\\end{equation}\nwhere $M_{\\rm ad} = 0.18M_{\\odot}(M_{\\star,0}\/M_{\\odot})^{17\/9}M_{\\bullet,7}^{-2\/9}$ is the critical mass at which the GW loss timescale $\\tau_{\\rm GW}$ equals the Kelvin-Helmholtz cooling timescale $\\tau_{\\rm KH}$.  Stars with masses initially above this critical mass evolve close to the main sequence ($p = 0.8$), before evolving with $\\tau_{\\rm GW} = \\tau_{\\rm KH}$ at lower masses, with $p = 4\/15$ or $p = 13\/21$, depending on whether the stellar envelope is radiative or convective.  In the final line of equation (\\ref{eq:p}), we have taken $p \\approx 0$ for stars below the hydrogen fusion limit to account for the radius being approximately independent of mass from brown dwarfs to Jupiter scale planets (e.g.~\\citealt{Chabrier+09}).\n\nThe above analysis assumes that mass transfer is stable, as occurs when upon mass loss the radius of the star decreases faster than its Roche lobe radius.  Mass loss from the star feeds an accretion disk around the SMBH.  A common assumption is that the gas disk transfers most of its angular momentum back to that of the orbit $J_{\\rm orb} \\simeq M_{\\star}(GM_{\\bullet}a)^{1\/2}$, which is therefore conserved\\footnote{In this stability analysis we are focusing on timescales much shorter than $\\tau_{\\rm GW}$, and therefore neglect angular momentum loss to gravitational radiation.}.  Since \n\\begin{equation} \\frac{R_{\\rm L}}{R_{\\star}} \\propto aM_{\\star}^{1\/3-p} \\underset{\\rm J_{\\rm orb}=const}\\approx  M_{\\star}^{-5\/3-p},\n\\end{equation}\nwe see that stable mass transfer requires $p > -5\/3$ in the conservative case, as is satisfied for all stages of evolution given in equation (\\ref{eq:p}).  Even if no angular momentum is placed back into the stellar orbit, $R_{\\rm L}\/R_{\\star}$ still increases upon mass loss at fixed $a$ for $p > 1\/3$, as will be satisfied until the period minimum is reached and $p \\approx 0$.\n\n\\begin{figure*}[!t]\n\\includegraphics[width=0.5\\textwidth]{emri6}\n\\includegraphics[width=0.5\\textwidth]{emrirad6}\n\\includegraphics[width=0.5\\textwidth]{emri7}\n\\includegraphics[width=0.5\\textwidth]{emrirad7}\n\\hspace{0.0cm}\n\\caption{\\footnotesize\n{Evolution of semi-major axis (left panels) and stellar radius (right panels) as a function of time since the onset of Roche-lobe overflow onto the SMBH of mass $M_{\\bullet} = 10^{6}M_{\\odot}$ (top panel) and $10^{7}M_{\\odot}$ (bottom panel), shown for stars of several initial masses as marked.  Squares and triangles denote, respectively, the point at which the stellar mass has decreased to one half and one tenth of its initial value.}\n}\n\\label{fig:orbit}\n\\end{figure*}\n\nFigure \\ref{fig:orbit} shows the evolution of the semi-major axis and stellar radius after RLOF contact as a function of time for SMBHs of mass $M_{\\bullet} = 10^{6}M_{\\odot}$ (top panel) and $10^{7}M_{\\odot}$ (bottom panel).  Squares and triangles denote, respectively, the point at which the stellar mass has decreased to one half and one tenth of its initial value.  \n\nLow mass stars $0.2-0.5M_{\\odot}$ lose half their mass in $0.1-0.5$ Myr, while higher mass $2-3M_{\\odot}$ stars require $1-3$ Myr to do the same.  These timescales decrease with increasing SMBH mass, approximately as $\\propto \\tau_{\\rm GW,0} \\propto M_{\\bullet}^{-2\/3}$ (eq.~\\ref{eq:tauGW0}).  \nThroughout most of their inspiral evolution, stars evolve from large to small semi-major axis in the convective regime $p = 13\/21$, and their semi-major axis evolves on a characteristic timescale (eqs.~\\ref{eq:Mevo}, \\ref{eq:aevo})\n\\begin{equation}\n\\tau_{\\rm GW}^{\\rm RLOF} = \\left(\\frac{a}{|\\dot{a}|}\\right)_{\\rm RLOF} = \\chi \\tau_{\\rm GW} \\underset{p = 13\/21}= \\frac{7}{2}\\tau_{\\rm GW}.\n\\label{eq:tauGWRLOF}\n\\end{equation} \nThis timescale is larger by a factor of $\\chi = 3\/(3p-1)$ than the inspiral time $\\tau_{\\rm GW}$ which occurs prior to the onset of mass transfer (eq.~\\ref{eq:tauGW0}).\n\n\nAt late times following the period minimum ($p = 0$), the orbit evolves to larger radii as $a \\propto t^{1\/7}$ while the mass decreases as $M_{\\star} \\propto t^{-3\/7}$.  This relatively slow rate of outspiral implies that the star (now technically a brown dwarf) will retain substantial mass of $\\sim 0.01-0.1M_{\\odot}$ for a period of time $\\sim 1-10$ Myr.  As we discuss below, even such a low mass (but high density) object is sufficient to damage or disrupt a more massive star in a collision given the enormous orbital velocities on these radial scales.    \n\n\\section{Collision Between Successive EMRIs}\n\\label{sec:collision}\n\nMain sequence EMRIs which undergo stable mass transfer are estimated to occur in galactic nuclei at the rate $\\mathcal{R}_{\\rm emri} \\gtrsim 10^{-7}$ yr$^{-1}$ (\\citealt{AmaroSeoane+12}).  The timescale between consecutive EMRIs of $\\sim 1\/\\mathcal{R}_{\\rm emri} \\lesssim 10$ Myr is therefore comparable to the slow outspiral timescale of a star undergoing RLOF (Fig.~\\ref{fig:orbit}).  This implies the probable existence at any time in a galactic nucleus of a $\\sim 0.1M_{\\odot}$ star or brown dwarf undergoing mass transfer evolution (e.g.~\\citealt{Linial&Sari17}).  It also raises the possibility for a strong interaction or collision between successive inspiraling\/outspiraling EMRIs. \n\nBased on the semi-major axis evolution (Fig.~\\ref{fig:orbit}) and the initial mass function of stars, one common way a collision could occur is between a star of initial mass $\\sim M_{\\odot}$ which has already already transferred most of its mass and is now migrating outwards as a brown dwarf of mass $M_{1} \\lesssim 0.1M_{\\odot}$ and radius $R_{1} \\approx 0.1R_{\\odot}$, and another star of similar initial mass $M_{2} \\sim M_{\\odot} > M_1$ and radius $R_{2} \\sim R_{\\odot} > R_1$ which has also begun RLOF but is still moving inwards and thus would cross the orbit of $M_1$ at a distance of $a \\sim 1$ AU with most of its initial mass still intact.  We consider this example as a fiducial case.  Note that, at the point of a collision, each star fills its Roche lobe and shares approximately the same semi-major axis; therefore the mean densities of the stars when they are interacting are equal, i.e. $M_{1}\/R_{1}^{3} \\simeq M_{2}\/R_{2}^{3}.$\n\nOur analysis in $\\S\\ref{sec:emri}$ assumes conservative mass transfer, which does not account for self-interaction between two EMRIs undergoing Roche-lobe overflow.  For instance, if the accretion disk of the more massive inspiraling star interferes with the ability of the less massive star's disk to feed its angular momentum back into the orbit, this could destabilize the orbit of the outspiraling brown dwarf (because $p < 1\/3$ during the outspiral phase).  It is well beyond the scope of this paper to address the complex interplay between mass transfer in three body systems, and so we leave this issue to future work.  However, we note that if the EMRI rate is higher than we have assumed (e.g.~$\\gtrsim 10^{-5}-10^{-6}$ yr$^{-1}$ due to the influence of massive perturbers; \\citealt{Perets+07}) then even a collision between two consecutive stars which are still inspiraling ($p > 1\/3$) may occur as the more massive star overtakes a less massive one.  Although the radial migration rate of the stars would differ in this case than the precise evolution predicted in $\\S\\ref{sec:emri}$, the qualitative collisional interaction we describe hereafter would not be altered.\n\nGiven the high rate of EMRIs of stellar mass black holes compared to those of main sequence ones, direct interactions between between inspiraling black holes and stars could be more common than star-star collisions.  However, we show in Appendix \\ref{sec:A} that the collision velocity is so high that the tidal or accretion interaction between the star and the black hole passing through it is probably too small to influence the evolution of the star appreciably.  \n\n\\subsection{Conditions for a Collision}\n\nNeither the orientation of the orbital plane or the orbital phases of the two stars will in general be aligned as they approach each other.  However, a physical collision\\footnote{The orbital velocities of the stars greatly exceed their surface escape speeds, so that the effect of gravitational focusing on their cross section is negligible.} is still possible once the semi-major axes of the stars cross near a value $\\sim 1$  AU (eq.~\\ref{eq:a0}), as illustrated in Fig.~\\ref{fig:cartoon}.  \n\nWe may neglect the comparatively slow outward radial motion of $M_1$ compared to the faster inspiral of $M_2$: the less massive $M_{1}$ can viewed as radially stationary for purposes of their interaction.  The more massive star $M_{2}$ migrates inward a distance $\\delta r$ on a timescale given by $(\\delta r\/a)\\tau_{\\rm GW}^{\\rm RLOF}$, where $\\tau_{\\rm GW}^{\\rm RLOF} = \\chi\\tau_{\\rm GW}$ is the characteristic inspiral time assuming that $M_{2}$ is undergoing mass transfer (eq.~\\ref{eq:tauGWRLOF}).  The number of orbits of period $\\tau_{\\rm orb} = 2\\pi (a^{3}\/GM_{\\bullet})^{1\/2}$ required for $M_2$ to migrate radially by $\\delta r$ is therefore\n\\begin{eqnarray}\n&&N_{\\rm GW} = \\frac{\\tau_{\\rm GW}^{\\rm RLOF}}{\\tau_{\\rm orb}}\\frac{|\\delta r|}{a} \\nonumber \\\\\n &\\approx& 1.3\\times 10^{6}\\chi_{3.5}\\frac{a_{\\rm AU}^{3\/2}}{M_{\\bullet,7}^{3\/2}}\\frac{R_{2,\\odot}}{M_{2,\\odot}}\\left(\\frac{\\delta r}{2R_{2}}\\right),\n\\end{eqnarray}\nwhere $\\chi_{\\rm 3.5} \\equiv \\chi\/3.5$.\n\nStars on orbits with precisely the same semi-major axis would possess identical orbital periods, modulo tiny reduced mass differences $\\sim \\mathcal{O}(M_{\\star}\/M_{\\bullet}) \\lesssim 10^{-6}$, and thus cannot collide.  However, stars on orbits with a small but finite semi-major axis difference $|\\delta a| \\lesssim R_{2}$ are still able to produce a physical collision, as their periods differ by an amount $|\\delta \\tau_{\\rm orb}|\/\\tau_{\\rm orb} \\simeq (3\/2)|\\delta a|\/a$ and thus will share the same orbital phase, on average, after $N_{\\phi} \\approx \\tau_{\\rm orb}\/(2\\delta \\tau_{\\rm orb}) \\approx a\/(3|\\delta a|)$ orbits.  \n\nA physical collision requires not only that both stars cross the same orbital phase, but also that their orbital planes cross at this phase, i.e. that they reside within a distance $l_{\\perp} \\sim 2^{3\/2}(R_{2}b)^{1\/2}$ of the line of ascending nodes of $M_{2}$ relative to the orbital plane of $M_{1}$, where $l_{\\perp}$ is the approximate length parallel to the surface of $M_{1}$ at times when the radial depth of $M_{1}$ into the atmosphere of $M_{2}$ is $b \\ll R_{2}$ ($b$ is also the impact parameter of the collision; see Fig.~\\ref{fig:cartoon}).  This coincidence will occur only a fraction $2l_{\\perp}\/(2\\pi a)$ of the times the phases cross, where the factor of 2 accounts for the two locations where the orbital planes cross.\n\nCombining results, the number of orbits required for a physical collision - once their separation $|\\delta a| \\lesssim R_2$ is sufficiently small to enable one - is given by\n\\begin{eqnarray}\nN_{\\rm coll} &\\approx& \\frac{\\pi}{2^{3\/2}}\\frac{a}{(R_{2}b)^{1\/2}}N_{\\phi}  \\nonumber \\\\\n&\\approx& 1.7\\times 10^{4} \\,\\frac{a_{\\rm AU}^{2}}{R_{2,\\odot}^{2}}\\left(\\frac{|\\delta a|}{R_{2}}\\right)^{-1}\\left(\\frac{b}{R_{2}}\\right)^{-1\/2}.\n\\label{eq:Ncoll}\n\\end{eqnarray}\nThe requisite condition for a single collision to occur before $M_2$ radially migrates past $M_1$ is that $N_{\\rm GW}\/N_{\\rm coll} \\gtrsim 1$ for $\\delta r \\lesssim 2 R_{2}$, $|\\delta a| \\sim R_{2}$, and $b \\sim R_{2}$, where \n\\begin{eqnarray}\n\\frac{N_{\\rm GW}}{N_{\\rm coll}} \\approx 77 \\frac{\\chi_{3.5}}{a_{\\rm AU}^{1\/2}}\\frac{R_{2,\\odot}^{3}}{M_{\\bullet,7}^{3\/2}M_{2,\\odot}}\\left(\\frac{|\\delta a|}{R_{2}}\\right)\\left(\\frac{\\delta r}{2R_{2}}\\right)\\left(\\frac{b}{R_{2}}\\right)^{1\/2}.\n\\label{eq:Nratio}\n\\end{eqnarray}\nThe fact that this ratio exceeds unity is a key result.  It shows that at least a single collision between the two stars is likely for all SMBH masses of interest $M_{\\bullet} \\lesssim 3\\times 10^{7}M_{\\odot}$.  \n\n\\subsection{Outcome of Stellar Collision}\n\n\n\\begin{figure}[!t]\n\\includegraphics[width=0.5\\textwidth]{cartoon.pdf}\n\\hspace{0.0cm}\n\\caption{\\footnotesize\n{Schematic diagram (not to scale) showing circular EMRI orbits around the SMBH of the effectively stationary low mass brown dwarf $M_1$ (red orbit) and the inward-migrating star $M_2 > M_1$ (blue orbit).  The diagram is looking down on the orbital plane of $M_1$, near the time when the semi-major axes of the stars differ by an amount $\\delta a \\lesssim R_{2}$ and their orbital phases match along the line of ascending nodes of $M_{2}$ (defined by the plane of $M_{1}$).  When a collision occurs, $M_1$ will penetrate inside $M_2$ with an impact parameter $b$ along a chord measuring approximately $2^{3\/2}(R_{2}b)^{1\/2}$.  For clarity, both stars are drawn here occupying approximately the same orbital plane (though the orbit of $M_2$ is slightly inclined and flattened in projection), but in general the orbital inclinations will be misaligned and hence will only overlap along the line of ascending nodes of $M_{2}$ relative to the osculating reference plane defined by the orbit of $M_{1}$.  The orbital planes will also be evolving significantly due to Lense-Thirring precession if the SMBH is spinning.}\n}\n\\label{fig:cartoon}\n\\end{figure}\n\nWe define the collision impact parameter $b$ as the distance measured from the center of $M_{1}$ to the outer edge of $M_{2}$ (Fig.~\\ref{fig:cartoon}).  The impact parameter of the first collision $b = b_{\\rm 1st}$ will on average be equal to half the inward radial distance $\\delta r$ traveled between collisions.  This value $b_{\\rm 1st} = \\delta r\/2$ is found by equating $N_{\\rm GW} = N_{\\rm coll}$ using equation (\\ref{eq:Nratio}) and assuming a grazing encounter $\\delta a \\approx R_{2}$, $b_{\\rm 1st} \\ll R_{2}$.  This gives a value\n\\begin{equation} \n\\frac{b_{\\rm 1st}}{R_{2}} \\approx 0.06 \\chi_{3.5}^{-2\/3}a_{\\rm AU}^{1\/3}M_{\\bullet,7}\\frac{M_{2,\\odot}^{2\/3}}{R_{2,\\odot}^{2}},\n\\label{eq:impact}\n\\end{equation}\nwhich is typically a few percent of the radius of $R_{2}$.  In most cases of interest $b_{\\rm 1st} \\lesssim R_{1} \\sim 0.1R_{\\odot}$ and hence $M_1$ will just graze the surface layers of $M_{2}$ instead of punching through its envelope at greater depth.\n\nThe orbital velocity at the time of collision is a mildly relativistic\\footnote{As noted before, in this work we treat the orbital dynamics of EMRI collisions in a primarily Newtonian way, which is accurate to leading order aside from the issue of nodal precession - which we account for in the post-Newtonian approximation.  However, circular orbit speeds $v_{\\rm k}\\sim 0.1c$ may motivate future, fully general relativistic treatments of this scenario.}\n\\begin{eqnarray}\nv_{\\rm k} &\\simeq& \\left(\\frac{GM_{\\bullet}}{a}\\right)^{1\/2} \\approx 9.4\\times 10^{9}\\,{\\rm cm\\,s^{-1}}M_{\\bullet,7}^{1\/2}a_{\\rm AU}^{-1\/2}. \n\\label{eq:vc}\n\\end{eqnarray}\nDepending on the mutual inclination angle $i$ of the orbital planes of $M_2$ with respect to that of $M_{1}$, the collision will occur at a relative speed \n\\begin{equation}\nv_{\\rm c} = \\sqrt{2}(1-\\cos i)^{1\/2}v_{\\rm k},\n\\end{equation}\nThis value ranges from $v_{\\rm c} = 2 v_{\\rm k}$ for a head-on collision to $v_{\\rm c} \\ll v_{\\rm k}$ for a tail-on collision.  Assuming an isotropic distribution of inclination angles, we find an average value of $\\langle v_{\\rm c}\\rangle = (4\/3)v_{\\rm k}$.\n\n\n\\begin{figure}[!t]\n\\includegraphics[width=0.5\\textwidth]{density}\n\\hspace{0.0cm}\n\\caption{\\footnotesize\n{Density $\\rho_{\\star}$ as a function of the distance below the stellar surface $b = r-R_{\\star}$ for main sequence stars of mass 0.2$M_{\\odot}$ (solid red line) and $0.5M_{\\odot}$ (solid blue line), compared to our adopted approximation $\\rho_{\\star} \\simeq 0.8(M_{\\star}\/R_{\\star}^{3})(b\/R)^{3\/2}$ (dashed lines).}\n}\n\\label{fig:density}\n\\end{figure}\n\n\n\\begin{figure}[!t]\n\\includegraphics[width=0.5\\textwidth]{collision}\n\\hspace{0.0cm}\n\\caption{\\footnotesize\n{Analytic estimate of average interval between stellar collisions $\\tau_{\\rm coll}$ in years (dashed red line; eq.~\\ref{eq:taucoll}), average mass released per collision $dM_{\\rm c} = \\delta M_{\\rm c,max}$ in solar masses (dotted blue line; eq.~\\ref{eq:dMc}) as a function of SMBH mass $M_{\\bullet}$.  We have adopted characteristic parameters for the semi-major axis of the stars $a = 1$ AU, the stellar radius $R_{2} = R_{\\odot}$ and mass $M_{2} = M_{\\odot}$, and average collision velocity $v_{\\rm c} = \\langle v_{\\rm k}\\rangle$.  For $M_{\\bullet} \\gtrsim M_{\\rm disr}$ the stars are almost completely disrupted in a single collision.  If the collisional mass loss is smaller than we have assumed ($\\delta M_{\\rm c} < \\delta M_{\\rm c,max}$), then the value of $M_{\\rm disr}$ will increase and $\\tau_{\\rm coll}$ will decrease.  Also shown are the estimated peak accretion luminosity $L_{\\rm pk}$ (brown dot-dashed line; eq.~\\ref{eq:Lpk}) and accretion timescale $t_{\\rm acc}$ (orange triple dot-dashed line; eq.~\\ref{eq:tacc}) for fiducial values of the disk viscosity $\\alpha = 0.01$, radius $r_{\\rm g} = 1$ AU, and SMBH accretion efficiency $\\eta = 0.1$. }\n}\n\\label{fig:collision}\n\\end{figure}\n\nTidal forces between the passing stars impart modest accelerations because the collision velocity is typically two to three orders of magnitude higher than the surface escape speeds of the stars (see Appendix \\ref{sec:A} for further discussion).  However, the direct, albeit grazing, physical collision between the surfaces of the stars will result in powerful shock heating, leading to gaseous mass loss.  Quantifying the collision properties, such as the total ejecta mass and its dependence on the impact parameter $b\/R_{2}$, would require 3D hydrodynamical simulations well beyond the scope of this work.  Numerical simulations of stellar collisions exist in the literature (e.g.~\\citealt{Freitag&Benz05}), however these are generally for much lower velocities than considered here and do not involve stars which already fill their Roche surfaces.  In what follows, we instead provide a rough analytic estimate of the mass loss and its impact on the subsequent evolution of the stars.  This crude treatment is justified in part because our qualitative conclusions are insensitive to the precise amount of mass loss per collision.\n\nWe approximate $M_2$ as a polytrope of index $n = 3\/2$ appropriate for a lower main sequence convective star, in which case its density at a depth $b$ from its outer edge is very approximately given by\n\\begin{equation} \\rho_{\\star} \\approx 0.8\\left(\\frac{M_{2}}{R_{2}^{3}}\\right)\\left(\\frac{b}{R_{2}}\\right)^{3\/2}, \\end{equation}  \nas is shown in Fig. \\ref{fig:density}.  A larger value of $n = 3$ would be appropriate for the outer layers of stars with radiative envelopes (such as main sequence stars more massive than the Sun), but the subsequent treatment is easily generalized.  We neglect non-spherical distortion of the star caused by its Roche-lobe filling shape.\n\nThe drag force on $M_1$ as it grazes the atmosphere of $M_2$ is approximately $F_{\\rm d} \\approx A(\\rho_{\\star}v_{\\rm c}^{2}\/2)$, where $A \\approx 2R_{1} b $ is the effective cross section of the encounter, assuming $ b \\lesssim R_{1}$ (Fig.~\\ref{fig:cartoon}).  The energy dissipated by shock heating as $M_1$ passes across the distance $l_{\\perp} \\approx 2^{3\/2}(bR_{2})^{1\/2}$ through $M_2$ is therefore \n\\begin{eqnarray}\nE_{\\rm c} &\\approx& F_{\\rm d}l_{\\perp} \\approx 2.3M_{2}v_{\\rm c}^{2}\\frac{R_{1}}{R_{2}}\\left(\\frac{b}{R_{2}}\\right)^{3}  \\\\\n&\\approx& 7.1\\times 10^{46}\\,{\\rm erg}\\,\\,\\,M_{\\bullet,7}\\frac{M_{2,\\odot}}{a_{\\rm AU}}\\left(\\frac{v_{\\rm c}}{\\langle v_{\\rm c}\\rangle}\\right)^{2}\\left(\\frac{b}{0.01R_{2}}\\right)^{3},\\nonumber\n \\label{eq:Ec}\n\\end{eqnarray}\nwhere in the second line and hereafter we take $R_{1} = 0.1R_{2}$.  This energy is usefully compared to the gravitational binding energy of the stars\n\\begin{equation}\nE_{\\rm b} \\approx \\frac{GM_{\\star}^{2}}{R_{\\star}} \\approx 4\\times 10^{48}\\,{\\rm erg}\\, \\left(\\frac{M_{\\star}}{M_{\\odot}}\\right)^{2}\\left(\\frac{R_{\\star}}{R_{\\odot}}\\right)^{-1}.\n\\label{eq:Eb}\n\\end{equation}\n The ratio of the collision heating energy $E_{\\rm c}$ to the binding energy $E_{\\rm b,2}$ of $M_2$ is given by\n\\begin{eqnarray}\n \\frac{E_{\\rm c}}{E_{\\rm b,2}} &=& 0.02\n\\frac{M_{\\bullet,7}}{a_{\\rm AU}}\\frac{R_{2,\\odot}}{M_{2,\\odot}}\\left(\\frac{v_{\\rm c}}{\\langle v_{\\rm c}\\rangle}\\right)^{2}\\left(\\frac{b}{0.01R_{2}}\\right)^{3} \\nonumber \\\\\n&\\underset{b = b_{\\rm 1st}}\\approx& 3.8 \\chi_{3.5}^{-2}\nM_{\\bullet,7}^{4}R_{2,\\odot}^{-5}M_{2,\\odot}\\left(\\frac{v_{\\rm c}}{\\langle v_{\\rm c}\\rangle}\\right)^{2},\n\\label{eq:Mmax}\n\\end{eqnarray}\nwhere in the second line we have used the characteristic impact parameter $b_{\\rm 1st}$ (eq.~\\ref{eq:impact}) of the first collision.\n\nWe expect complete disruption of the star if $E_{\\rm c} \\gg E_{\\rm b,2}$.  This occurs for collisions with impact parameter well above a critical value $b_{\\rm disr}$ given by\n\\begin{equation}\n\\frac{b_{\\rm disr}}{R_{2}} \\approx 0.039\\left(\\frac{a_{\\rm AU}}{M_{\\bullet,7}}\\frac{M_{2,\\odot}}{R_{2,\\odot}}\\right)^{1\/3}\\left(\\frac{v_{\\rm c}}{\\langle v_{\\rm c}\\rangle}\\right)^{-2\/3},\n\\label{eq:bdisrupt}\n\\end{equation}\nwhere we have taken $\\chi = 3.5$.  This condition is achieved in the first collision ($b_{\\rm disr} = b_{\\rm 1st}$; eq.~\\ref{eq:impact}) if the SMBH mass greatly exceeds a critical value\n\\begin{equation}\nM_{\\bullet,\\rm disr} = 7.1\\times 10^{6}M_{\\odot} R_{\\rm 2,\\odot}^{5\/4}M_{2,\\odot}^{-1\/4}\\left(\\frac{v_{\\rm c}}{\\langle v_{\\rm c}\\rangle}\\right)^{-1\/2}\n\\label{eq:Mdisr}\n\\end{equation}\nFor $M_{\\bullet} \\lesssim M_{\\bullet,\\rm disr}$ we instead have $E_{\\rm c} \\ll E_{\\rm b,2}$ and both stars will instead survive the first collision at least partially intact.  Although the above calculation is quite approximate and should be refined by future hydrodynamical simulations, the steep scaling $E_{\\rm c}\/E_{\\rm b, 2} \\propto M_\\bullet^4$ indicates two clear regimes for EMRI collisions: around larger SMBHs, a first encounter is likely destructive, while many encounters can occur around smaller SMBHs.\n\nAny collision will result in some mass loss from both stars.  We assume that most of the total mass loss originates from the more massive star $M_2$, motivated as follows.  As $M_{1}$ passes through the outer layers of $M_2$, the ram pressure of the interaction will drive dual shocks, through the outer layers of both stars.  However, the density $\\rho_{2}$ of $M_{2}$ at the collision depth is typically 2 orders of magnitude or more lower than the density $\\rho_{1}$ of $M_{1}$ (both stars have equal {\\it mean} densities, but the collision occurs comparatively closer to the surface of $M_{1}$ than $M_{2}$).  Since $\\rho_{1} \\gg \\rho_{2}$ the shock through $M_{1}$ will move with a velocity larger by a factor of ($\\rho_{2}\/\\rho_{1})^{1\/2}$ than the shock through $M_{2}$ and thus will carry an energy flux larger by the same factor of $\\gtrsim 10$ (which generally is comparable to or exceeds the ratio of the gravitational binding energy of the two stars).\n\nIn what follows, we make the crude approximation that the fractional mass lost from $M_{2}$ by the collision, $\\delta M_{\\rm c}$, is equal to the ratio of shock-deposited energy to the gravitational binding energy.  In other words, we take\n\\begin{equation} \\frac{\\delta M_{\\rm c}}{M_2} = \\frac{\\delta M_{\\rm c,max}}{M_2} \\approx  \\frac{E_{\\rm c}}{E_{\\rm b,2}},\n\\label{eq:dMc}\n\\end{equation}  \n as given in equation (\\ref{eq:Mmax}).  Equivalently,\n\\begin{equation} \\delta M_{\\rm c,max} \\approx 3.93M_{\\odot} \\chi_{3.5}^{-2}\nM_{\\bullet,7}^{4}R_{2,\\odot}^{-5}M_{2,\\odot}^{2}\\left(\\frac{v_{\\rm c}}{\\langle v_{\\rm c}\\rangle}\\right)^{2}. \n\\label{eq:Mc2} \\end{equation}\nEq.~\\ref{eq:dMc} provides only a rough upper limit on $\\delta M_{\\rm c}$; the true amount of mass loss could be significantly lower if low-density outer layers of $M_2$ are ejected at speeds $\\gg (GM_2\/R_2)^{1\/2}$.  As we discuss in $\\S\\ref{sec:conclusions}$, our results for the long-term evolution of the stars are not qualitatively altered if the mass loss per collision is substantially lower than this maximum (though the range of SMBH masses responsible for the most interesting behavior will shift to higher values).\n\n\\subsection{Multiple Collision Evolution}\n\nIf the stars survive their first collision ($E_{\\rm c} \\ll E_{\\rm b,2}$, $\\delta M_{\\rm c} \\ll M_{2}$), then it is natural to ask whether and when subsequent collisions will occur.  Will the stars collide again immediately, on the next orbit, or only after a significant delay required for the orbital phases to realign?  \n\nThe semi-major axes of $M_1$ and $M_2$ differ by $\\delta a \\approx R_{2}$ at the time of the first collision, and hence their orbital periods differ by a factor $\\delta \\tau_{\\rm orb}\/\\tau_{\\rm orb} = (3\/2)(\\delta a\/a) \\approx (3\/2)(R_{2}\/a)$.  The stars therefore accumulate a per-orbit phase difference of $\\delta \\phi\/2\\pi = \\delta \\tau_{\\rm orb}\/\\tau_{\\rm orb} = (3\/2)(R_{2}\/a)$, which is coincidentally comparable to the maximum phase difference between the stars that allows for a collision at the common line of ascending nodes, $\\delta \\phi_{\\rm max} =2 R_{2}\/(2\\pi a)$.  Therefore, we conclude that the second collision will not generally occur on the next immediate orbit, but will instead be delayed by at least another $\\sim N_{\\rm coll} \\gtrsim 10^{4}$ (eq.~\\ref{eq:Ncoll}) orbits, corresponding to a minimum time delay of \n\\begin{equation}\n\\tau_{\\rm coll}^{\\rm min} = N_{\\rm coll}\\tau_{\\rm orb} \\approx 57 \\,\\,{\\rm yr}\\,\\,\\,\\frac{a_{\\rm AU}^{7\/2}}{M_{\\bullet,7}^{1\/2}R_{2,\\odot}^{2}}\\left(\\frac{b}{0.01R_{2}}\\right)^{-1\/2}.\n\\label{eq:taucoll1}\n\\end{equation}\n\nEven were this differential phase accumulation insufficient to prevent an immediate second collision, Lense-Thirring precession would cause the angular momenta of the stellar orbits to precess rapidly about the spin axis of the SMBH.  This will advance the line of nodes of each orbit by a large fractional angle.  At leading post-Newtonian order, and assuming circular orbits, this angular shift is (e.g., \\citealt{Merritt+10})\n\\begin{equation}\n\\frac{\\Delta\\Omega}{2\\pi} = 2 \\chi_{\\bullet}\\left(\\frac{a}{R_{\\rm g}}\\right)^{-3\/2} \\approx 0.06 \\chi_{\\bullet} a_{\\rm AU}^{-3\/2}M_{\\bullet,7}^{3\/2},\n\\label{eq:LT}\n\\end{equation}\nwhere $R_{\\rm g} \\equiv GM_{\\bullet}\/c^{2}$ and $-1<\\chi_{\\bullet}<1$ is the dimensionless spin parameter of the SMBH.\n\n\nOver the minimum time interval before the second collision $\\tau_{\\rm coll}^{\\rm min}$, GW radiation will move $M_2$ closer to $M_1$ on average by a distance $\\delta a \\approx  2b_{\\rm 1st}$ (eq.~\\ref{eq:impact}).  If this were the whole story, then according to equation \\ref{eq:Ec}, one would expect the impact parameter of the second collision to be $\\sim 3$ times higher than the first and (according to equation \\ref{eq:Ec}) the energy released to be $\\sim 3^{3} \\sim 30$ times stronger.  Likewise, the third collision would be $\\sim 30$ times stronger than the second, and so on; this sequence would rapidly terminate in a single, final, disruption once $b \\gtrsim b_{\\rm disr}$ (eq.~\\ref{eq:bdisrupt}).  \n\nHowever, this simple runaway argument neglects the impact of mass loss on the radial separation between the stars.\\footnote{Change in the orbital energy due to collisions will cause the orbits of both $M_{1}$ and $M_{2}$ to acquire mild eccentricities.  However, the magnitude of the collision energy required to induce even a small eccentricity $e \\sim 0.01$ is comparable to the binding energy of the stars.}  \nUnder adiabatic mass loss, the radius of each star will expand slightly according to $R_{\\star} \\propto M_{\\star}^{-1\/3}$ (for assumed adiabatic index $\\gamma = 5\/3$).  More importantly, the semi-major axis of each star will increase by a much larger amount $\\delta a_{\\rm c}\/a \\approx 2 (\\delta M_{\\rm c}\/M_{\\star}).$  This expression assumes conservative mass transfer: the gaseous disk created by the collision transfers its angular momentum back into the orbit of the mass-losing star with high efficiency.\n\nGiven our expectation that the fractional mass loss from $M_{2}$ will exceed that from $M_{1}$, the main effect on the system is to increase the semi-major axis of $M_{2}$ - and thus the radial separation of the orbits of $M_{1}$ and $M_{2}$ - by an amount\n\\begin{equation}\n \\frac{ \\delta a_{\\rm c}}{R_{\\rm 2}} \\approx 2\\frac{a}{R_{2}}\\frac{\\delta M_{\\rm c}}{M_2}.\n\\label{eq:bout}\n\\end{equation}\nAgain, our assumption is that most of the angular momentum lost by $M_{2}$ is placed back into its orbit, as opposed to the orbit of $M_{1}$.  This is justified by our expectation that most of the mass loss from $M_{1}$, despite being struck obliquely by $M_{2}$, will nevertheless occur quasi-isotropically, forming a gaseous disk which lies in roughly the same orbital plane of $M_{1}$.\n\nSince this change in separation between the stars is typically much larger than the characteristic impact parameter of the first collision $b_{\\rm 1st}$ (eq.~\\ref{eq:impact}), this will introduce an additional delay until the next collision beyond the minimum value (eq.~\\ref{eq:taucoll1}) set by the double alignment of orbital phase and nodal line.  Specifically, the time required to traverse this distance $\\delta a_{\\rm c}$ through gravitational wave radiation is\n\\begin{eqnarray}\n\\tau_{\\rm coll} &\\approx& \\tau_{\\rm GW}\\left(\\frac{\\delta a_{\\rm c}}{a}\\right) \\approx 2\\tau_{\\rm GW}\\left.\\frac{\\delta M_{\\rm c}}{M_{\\rm 2}}\\right|_{b = b_{\\rm 1st}}, \\nonumber \\\\\n&\\approx & 9.9\\times 10^{4}\\,{\\rm yr}\\,\\, \\chi_{3.5}^{-2}\\frac{a_{\\rm AU}^{4}M_{\\bullet,7}^{2}}{R_{2,\\odot}^{5}}\\left(\\frac{v_{\\rm c}}{\\langle v_{\\rm c}\\rangle}\\right)^{2}\\left(\\frac{\\delta M_{\\rm c}}{\\delta M_{\\rm c,max}}\\right). \\nonumber \\\\\n\\label{eq:taucoll}\n\\end{eqnarray}\nHere we have used equations (\\ref{eq:Mmax},\\ref{eq:dMc}) and have calculated the GW inspiral time $\\tau_{\\rm GW}$ neglecting mass transfer effects (eq.~\\ref{eq:tauGW}) because, for most of its inward return, $M_{2}$ is no longer overflowing its Roche Lobe (though we have retained the $\\chi$ dependence on the impact parameter $b_{\\rm 1st}$).  This delay could be substantially shorter than estimated here if the mass loss per collision is much less than $\\delta M_{\\rm c,max}$ (\\ref{eq:dMc}).\n\nAfter this inspiral, the characteristic impact parameter and mass loss of each such collision will be similar to the first one, $b \\approx \\delta b_{\\rm 1st}$.  Each collision will also release a comparable amount of mass as the first (eq.~\\ref{eq:Mc2}).  Neglecting slow changes to the stellar properties caused by the collisions, the star will be completely destroyed over a total number of collisions very approximately given by\n\\begin{eqnarray}\nN_{\\rm c} &\\sim& \\left.\\frac{M_2}{\\delta M_{\\rm c}}\\right|_{b = b_{\\rm 1st}} \\nonumber \\\\\n&\\approx& 0.26\\chi_{3.5}^{2}\nM_{\\bullet,7}^{-4}R_{2,\\odot}^{5}M_{2,\\odot}^{-1}\\left(\\frac{v_{\\rm c}}{\\langle v_{\\rm c}\\rangle}\\right)^{-2}\\left(\\frac{\\delta M_{\\rm c}}{\\delta M_{\\rm c,max}}\\right)^{-1}.\n\\label{eq:Ncoll2}\n\\end{eqnarray}\nThis number is $\\gg 1$ for low mass SMBHs and approaches unity for $M_{\\bullet} = M_{\\bullet, \\rm disr}$ (eq.~\\ref{eq:Mdisr}).  For $M_{\\bullet} \\ll M_{\\bullet,\\rm disr}$, the total duration of the interaction is just given by the gravitational wave inspiral time\n\\begin{equation}\n\\tau_{\\rm tot} = N_{\\rm c}\\tau_{\\rm coll} \\approx 2\\tau_{\\rm GW}, \n\\label{eq:tautot}\n\\end{equation} \nwhich ranges from $\\sim 10^{4}-10^{6}$ yr, depending on the SMBH mass.  These different regimes of collisions are summarized in Figure \\ref{fig:collision}.  \n\n\\section{Transient SMBH Accretion Events}\n\\label{sec:transient}\n\nWe have shown that collisional evolution between consecutive stellar EMRIs around SMBHs of mass $\\lesssim 10^{7}M_{\\odot}$ can result in repeated, prompt episodes of gas formation; the colliding stars liberate a mass $M_{\\rm g} \\le \\delta M_{\\rm c} \\sim 10^{-4}-1M_{\\odot}$, on a radial scale of $r_{\\rm g} \\sim 0.5-2$ AU, at periodic intervals of decades to centuries or longer, and lasting for a total duration of up to a million years.  We now describe the observable signatures of these punctuated episodes of gas production.\n\nBy equating the thermal energy released per unit volume with the specific heat of radiation-dominated matter, $a T_{\\rm c}^{4} \\approx \\rho_{\\star}v_{\\rm c}^{2}\/2$, the immediate temperature of the shocked stellar material is $T_{\\rm c} \\sim 10^{8}$ K even for a head-on collision, too low to result in nuclear burning up to $^{56}$Ni.  Without such a radioactive heating source, the expanding ejecta will cool rapidly via adiabatic expansion before becoming transparent and radiating the remaining thermal energy.  This will produce a dim rapidly-evolving transient which would itself be very challenging to detect (see also \\citealt{Balberg+13}).  \n\nA more important source of luminosity is the gravitational energy liberated by the accretion of the gaseous ejecta onto the SMBH.  The ejected gas will possess an angular momentum similar to that of the stars, resulting in the formation of circular disk on a similar radial scale of $\\lesssim r_{\\rm g}$.  Internal stresses (likely magneto-turbulent in origin) within the accretion flow will transport this angular momentum outwards, allowing the gas mass $M_{\\rm g}$ to accrete inwards on the characteristic accretion timescale\n\\begin{equation}\nt_{\\rm acc} \\approx \\frac{1}{\\alpha}\\frac{1}{\\Omega_k}\\left(\\frac{h}{r_{\\rm g}}\\right)^{-2} \\approx 18.4\\,{\\rm d}\\,\\,\\frac{r_{\\rm g,AU}^{3\/2}}{\\alpha_{-1}M_{\\bullet,7}^{1\/2}}\\left(\\frac{h}{0.1r_{\\rm g}}\\right)^{-2},\n\\label{eq:tpk}\n\\end{equation}\nwhere $\\Omega_{k} = (GM_{\\bullet}\/r_{\\rm g}^{3})^{1\/2}$ is the orbital angular velocity of the disk, $\\alpha\/=0.1\\alpha_{-1}$ is the Shakura-Sunyaev viscosity parameter, and $h$ is the vertical scale height of the disk.\n\nFollowing deposition of the gaseous material, the accretion rate will quickly rise to a peak value of $\\dot{M}_{\\rm pk} \\sim M_{\\rm g}\/t_{\\rm acc}$ (\\citealt{Pringle81}).  For a radiation-pressure dominated disk with electron scattering opacity $\\kappa_{\\rm es}$, the vertical scale-height of the disk obeys $h = 3\\kappa \\dot{M}\/(8\\pi c) = \\frac{3}{2\\eta}(\\dot{M}\/\\dot{M}_{\\rm Edd})(GM_{\\bullet}\/c^{2}) \\simeq 2.2\\times 10^{13}(\\dot{M}\/\\dot{M}_{\\rm Edd})M_{\\bullet, 7}$ cm, where $\\dot{M}_{\\rm Edd} \\approx 1.6\\times 10^{25}M_{\\bullet,7}$ g s$^{-1}$ is the Eddington accretion rate for radiative efficiency $\\eta = 0.1$.  Combining results, we find that\n\\begin{equation}\n\\frac{\\dot{M}_{\\rm pk}}{\\dot{M}_{\\rm Edd}} \\approx 5.9\\times 10^{-2}\\left(\\frac{M_{\\rm g}}{10^{-3}M_{\\odot}}\\right)^{-1}M_{\\bullet,7}^{-3\/2}r_{\\rm g,AU}^{7\/2}\\alpha_{-1}^{-1}\n\\end{equation}\nwith\n\\begin{equation}\n\\frac{h}{r_{\\rm g}} \\approx 0.087\\left(\\frac{M_{\\rm g}}{10^{-3}M_{\\odot}}\\right)^{-1}M_{\\bullet,7}^{-1\/2}r_{\\rm g,AU}^{5\/2}\\alpha_{-1}^{-1}\n\\end{equation}\nand\n\\begin{equation}\nt_{\\rm acc} \\approx 24\\,{\\rm d}\\,\\,\\ \\frac{\\alpha_{-1}M_{\\bullet,7}^{1\/2}}{r_{\\rm g,AU}^{7\/2}}\\left(\\frac{M_{\\rm g}}{10^{-3}M_{\\odot}}\\right)^{2},\n\\label{eq:tacc}\n\\end{equation}\nThe resulting peak accretion luminosity is given by\n\\begin{eqnarray}\n&&L_{\\rm pk} = \\eta \\dot{M}_{\\rm pk}c^{2} \\nonumber \\\\\n&\\approx& 8.5\\times 10^{43}\\,{\\rm erg\\,s^{-1}}\\,\\frac{r_{\\rm g,AU}^{7\/2}}{M_{\\bullet,7}^{1\/2}\\alpha_{-1}}\\left(\\frac{M_{\\rm g}}{10^{-3}M_{\\odot}}\\right)^{-1},\n\\label{eq:Lpk}\n\\end{eqnarray}\nwhere we have assumed a radiative efficiency $\\eta = 0.1$.\n\nFor a characteristic value of $M_{\\rm g} \\sim 10^{-3}M_{\\odot}$, as is expected in our fiducial models for $M_{\\bullet} \\approx 2\\times 10^{6}M_{\\odot}$ (eq.~\\ref{eq:Mc2}), we predict accretion-powered transients of peak luminosity $\\approx 10^{44}$ erg s$^{-1}$ and characteristic timescales $t_{\\rm acc} \\approx 40~{\\rm days}$.  However, these properties scale sensitively with SMBH mass.\n\nThese precise scalings, in particular the unintuitive inverse relationship between peak accretion rate and gas disk mass, must be taken with caution.  It is well known that the thermal and viscous stability of radiation pressure dominated disks remains an open issue (e.g.~\\citealt{Hirose+09,Jiang+16,Sadowski&Narayan16}), and the true behavior of these disks could differ significantly from the expectations of the $\\alpha$-models.  \n\nThe evolution of the accretion rate at times after the peak ($t \\gg t_{\\rm acc}$) depends on the interaction between the outer edge of the gaseous disk and the stars.  If most of the angular momentum of the disk is transferred back into the stellar orbit, then - in contrast to circularized TDE disks \\citep{Cannizzo+90} - the disk will not freely spread outwards beyond the orbit of the star.  In this case the accretion rate will decay exponentially on a timescale set by the initial viscous timescale $t_{\\rm acc}$, i.e. we expect a bolometric light curve of the form\n\\begin{equation}\nL(t) \\sim L_{\\rm pk}e^{-t\/t_{\\rm acc}}, \\,\\,\\,\\, t \\gtrsim t_{\\rm acc}\n\\label{eq:exp}\n\\end{equation}  \n\nOn the other hand, a portion of the disk may be free to viscously spread outwards beyond the orbit of the star.  This could occur if the angular momentum of the gas produced by the collision places it into an orbital plane significantly different either star.  It could also occur if the Bardeen-Petterson effect \\citep{Bardeen&Petterson75} aligns the disk into a different plane from both stars on a timescale short compared to the disk evolution timescale; such misalignment could also be accomplished by differential nodal precession between stellar orbits and the disk.  In this case, the mass accretion rate would instead decay at late times as a power-law, $L \\propto t^{-\\alpha}$, where $\\alpha \\sim 1.1-1.3$ (e.g.~\\citealt{Cannizzo+90,Shen&Matzner14}), although we caution that this evolution could be complicated by possible thermodynamic state changes from radiation-pressure to gas-pressure dominated regimes (\\citealt{Shen&Matzner14}).  In both cases, the predicted evolution differs from the canonical $L(t) \\propto t^{-5\/3}$ prediction for the late-time decay of the mass fall-back rate in TDEs.\n\n\\subsection{TDE Imposters}\n\nThe flare timescales and bolometric luminosities we predict from colliding main sequence EMRIs overlap with those of observed tidal disruption flare candidates (\\citealt{Komossa15}).  However, unlike TDEs, which are singular events, a colliding EMRI pair may produce {\\it hundreds or thousands of mass production events} (eq.~\\ref{eq:Ncoll2}).  This large number may be sufficient to at least partially overcome the $\\sim 2-3$ order of magnitude deficit between the predicted TDE and circular EMRI rate, allowing flares of the collisional EMRIs described here to be detected.  This would especially be true if a large fraction of TDEs are ``dark\" due to inefficiency associated with accretion at super-Eddington rates (\\citealt{Strubbe&Quataert09,Metzger&Stone16}) or the process of debris circularization of the highly elliptical streams produced in most TDEs (\\citealt{Guillochon&RamirezRuiz15,Dai+15,Hayasaki+16}).    \n\nThe total radiated energy inferred from optical TDE candidate flares ranges from $E_{\\rm rad} \\approx 10^{49}$ erg for iPTF16fnl (\\citealt{Blagorodnova+17}) and $3\\times 10^{50}$ erg for PS1-11af (\\citealt{Chornock+14}) to values up to $\\sim 10$ times higher, as compiled in Figure \\ref{fig:Erad}.\nThese radiated energies correspond to accreted masses in the range $\\sim 10^{-4}-10^{-2} M_{\\odot}$ for an assumed radiative efficiency of $\\eta = 0.1$.  These often low radiated energies have been described as a ``missing energy problem'' \\citep{Piran+15}, though this problem has many possible resolutions in the TDE paradigm, including low accretion or radiative efficiencies in TDEs (e.g.~\\citealt{Piran+15, Metzger&Stone16}), large bolometric corrections \\citep{vanVelzen+16}, or the lower accreted mass of a partial tidal disruption event \\citep{Chornock+14}.  \n\nAlternatively, these low radiated energies may indicate that some of the observed TDE candidates are in fact just one in a sequence of quasi-periodic EMRI collision flares, each producing a low-mass accretion transient.  One way to distinguish these scenarios is to search for additional flares from the same galactic nucleus after the initial burst.  The required wait time will usually be too long to serve as a definitive test (up to $\\sim 10^5~{\\rm yr}$ for fiducial parameters; see eqs. \\ref{eq:taucoll1}, \\ref{eq:taucoll}), though because $\\tau_{\\rm coll} \\propto R_{2, \\odot}^{-5}$ and $\\tau_{\\rm coll}^{\\rm min} \\propto R_{2, \\odot}^{-3\/2}$, repeated flares can happen on timescales shorter than a decade if the victim star is a few times the mass of the Sun, or is a sub-solar mass star that has bloated due to shock heating.  A shorter flare duration would occur if the mass loss per collision is much less than our conservative upper limit on its value (eq.~\\ref{eq:dMc}).\n\nThe accretion timescale of the gaseous disks produced by stellar collisions, $t_{\\rm acc}$ (eq.~\\ref{eq:tacc}), is  short compared to the interval between consecutive collisions.  This implies that the light curve would appear as periodic spikes in luminosity, separated by quasi-periodic intervals ranging from decades to hundreds of millenia.  Such quasi-periodic flaring may have been observed in IC3599, a Seyfert galaxy which produced a large-amplitude nuclear X-ray outburst in the 1990s (\\citealt{Brandt+95,Grupe+95}), initially interpreted as a TDE.  However, IC3599 was then observed to repeat its flaring behavior \\citep{Grupe+15,Campana+15}.  Based on modeling the light curves and disk temperature evolution, \\citet{Campana+15} claim the existence of 3 outbursts with a separation of $\\approx$ 10 years, each reaching  a luminosity of $10^{43}$ erg s$^{-1}$.  \n\nIndividual flares produced from EMRI collisions may also be distinguished from standard TDE flares by differences in the predicted late-time light curve decay.  In cases when the orbits of the surviving stars absorb angular momentum from the gaseous disk, we could expect the light curve to decay exponentially at late times (eq.~\\ref{eq:exp}), a feature which may in fact be observed some TDE flares (e.g.,~NGC 247, \\citealt{Feng+15}; ASASSN-14li, \\citealt{Holoien+16}; iPTF16fnl, \\citealt{Blagorodnova+17}).  On the other hand, if a sizable fraction of the gas from the collision is able to viscously spread beyond the orbits of the stars, we would expect a power-lower light curve decay shallower than the canonical $\\propto t^{-5\/3}$ fall-back rate.  Such shallow decays are inferred from the X-ray sample of TDE flares compiled by \\citet{Auchettl+17}.\n\n\nOne of the most puzzling mysteries of optical TDE candidates concerns their observed color temperatures, which are roughly an order of magnitude lower than those predicted from compact disk emission models (e.g., \\citealt{Gezari+12,Cenko+12,Holoien+16}), which should instead peak in the extreme UV\/soft X-ray band (corresponding to the disk temperature at $\\sim 10R_{\\rm g}$).  One explanation for this behavior is the existence of a dense layer of gas on radial scales $\\gtrsim 10-100$ AU, which reprocesses much of the accretion power to lower frequencies (\\citealt{Loeb&Ulmer97, Roth+16}).  In TDE scenarios, this reprocessing material could be the result of a super-Eddington wind (\\citealt{Metzger&Stone16}) or highly eccentric, inefficiently circularized debris streams produced during the disruption process (e.g., \\citealt{Guillochon+14,Miller15}).  \n\nA radially extended gaseous disk is also expected to arise naturally in the EMRI collision scenario, and could provide an alternative reprocessing screen.  As discussed above, the gaseous disk deposited by a given collision at $r_{\\rm g} \\sim $ AU will in some cases carry its angular momentum outwards by viscously expanding \\citep{Cannizzo+90} beyond the orbits of the stars.  Since total angular momentum is roughly conserved in this process, the disk mass which remains when the outer edge of the disk reaches a radius $r$ is $M(r) \\sim M_{\\rm g}(r\/r_{\\rm g})^{1\/2}$.  The timescale for the gas from a collision event to reach $\\sim 10-100$ AU by viscous spreading is $\\sim 10^{3}-10^{4}$ yr \\citep{Cannizzo+90,Shen&Matzner14}, many orders of magnitude longer\\footnote{The disk evolution slows considerably once the midplane transitions from being radiation pressure dominated to being gas pressure dominated (e.g.~\\citealt{Shen&Matzner14}).} than the initial viscous timescale (flare time) and potentially comparable to the entire duration of the collisional interaction between the stars, $\\tau_{\\rm tot}$ (eq.~\\ref{eq:tautot}). \n\nThe accumulation of spreading disks produced by each of the hundreds to thousands of collisions which occur prior to a given typical collision could therefore place up to $\\sim 0.1-0.3M_{\\odot}$ over radial scales of $\\gtrsim 10-100$ AU, potentially sufficient to explain the observed reprocessing \\citep{Roth+16}. \nDue to Lense-Thirring precession of the stellar orbits, the collision plane will rotate between subsequent encounters around the SMBH spin axis (eq.~\\ref{eq:LT}), such that each gaseous disk would be created with its angular momentum pointed in a significantly different direction.  However, differential Lense-Thirring precession will generally align each transient disk with the SMBH equatorial plane \\citep{Bardeen&Petterson75}, leading to a larger scale accretion flow that is coplanar.  \n\nOnce UV\/X-ray irradiation from the flare intercepts the geometrically thin disk, the resulting heating will cause it to puff up vertically, blocking a larger fraction of the light than it would have given its initial vertical thickness.  If the timescale for the collision evolution is sufficiently long, with $\\tau_{\\rm tot} \\gg 10^{4}$ yr (eq.~\\ref{eq:tautot}), then gas accretion from a TDE in the same nucleus might destroy the fossil disk ($\\S\\ref{sec:TDEstripping}$).  However, in this case the viscous spreading-evolution of one (or several consecutive) TDE disks could itself deposit sufficient mass on large scales to replenish the repocessing disk.  Thus, our new proposed scenario for  the reprocessing layer is largely independent of the EMRI collision scenario, and could also apply to galaxies with a high TDE rate.\n\n\nThis fossil reprocessing disk could also serve as source of magnetic flux, to be swept towards the black hole by the debris from future TDEs \\citep{Tchekhovskoy+14,Kelley+14}, explaining why some TDEs are able to produce \npowerful magnetically-dominated jets (\\citealt{Giannios&Metzger11,Tchekhovskoy+14,Kelley+14}).\n\n\\subsection{Gas Ablation in True Tidal Disruption Events}\n\\label{sec:TDEstripping}\n\nThe rate of TDEs around small SMBHs is typically $\\sim 10^{-4}~{\\rm yr}^{-1}$, with a scatter of about an order of magnitude depending on the detailed properties of the specific galaxy \\citep{Stone&Metzger16}.  Given that this exceeds our fiducial rate of circular EMRI inspirals by a factor of $\\sim 10^{2}-10^{4}$, the first mass-transferring star must survive this many gas production events in a galactic nucleus in order to undergo the collisional evolution described in this paper.  This section considers whether such survival is feasible in the face of mass loss due to shock-heating from the interaction between the star and the gaseous TDE accretion disk.\n\nThe TDE results in a transient accretion event of duration $\\tau_{\\rm acc} \\sim$ months and average mass accretion rate $\\dot{M}_{\\rm acc} \\approx M_{\\rm acc}\/\\tau_{\\rm acc}$, where $M_{\\rm acc}$ is the total accreted mass.  The star's semi-major axis distance $a$ is comparable to the tidal radius of a disrupted star, and the midplane density of the gaseous TDE disk at this location is approximately given by\n\\begin{equation}\n\\rho_{\\rm d} = \\frac{\\Sigma}{2h} = \\frac{\\dot{M}_{\\rm acc}}{6\\pi \\nu h} =\\frac{M_{\\rm acc}}{6\\pi \\alpha a^{3}\\Omega(h\/a)^{3}\\tau_{\\rm acc}},\n\\end{equation}\nwhere we have used the standard relationship for a steady-state disk $\\dot{M}_{\\rm acc} = 3\\pi \\nu \\Sigma$ with viscosity $\\nu = \\alpha (h\/a)^{2}a^{2}\\Omega$.\n\nIn general the orbital plane of the gaseous disk will not be aligned with that of the star, and so the star will be impacted by a ``headwind\" of gas at a relative velocity $v_{\\rm c}$ up to twice the orbital velocity $v_{\\rm k} = a\\Omega$.  The shock created by the interaction of the star with the gaseous disk will ablate mass stellar mass.  \n\nThe shock will penetrate to a depth inside the star approximately given by equality between ram pressure and the interior pressure of the star,\n\\begin{equation}\n\\rho_{\\rm d}v_{\\rm c}^{2}\/2 = P_{\\star} = \\bar{P}_{\\star}\\left(\\frac{\\rho_{\\star}}{\\bar{\\rho}_{\\star}}\\right)^{\\gamma},\n\\end{equation}\nwhere $\\bar{\\rho}_{\\star} \\equiv M_{\\star}\/(4\\pi R_{\\star}^{3}\/3)$ is the mean stellar density and $\\bar{P}_{\\star} \\simeq 0.04 GM_{\\star}^{2}\/R_{\\star}^{4}$ for a $\\gamma = 5\/3$ polytrope.  The shocks thus initially penetrate to a depth where the stellar density equals\n\\begin{eqnarray}\n&&\\frac{\\rho_{\\star,\\rm sh}}{\\bar{\\rho}_{\\star}} \\approx 1.9\\left(\\frac{v_{\\rm c}^{2}R_{\\star}}{GM_{\\star}}\\frac{\\rho_{\\rm d}}{\\bar{\\rho}_{\\star}}\\right)^{3\/5} \\nonumber \\\\\n &&\\approx 1.9\\left[\\frac{2}{9\\alpha}\\left(\\frac{h}{a}\\right)^{-3}\\frac{1}{\\Omega \\tau_{\\rm acc}}\\left(\\frac{v_{\\rm c}}{v_{\\rm k}}\\right)^{2}\\left(\\frac{M_{\\bullet}}{M_{\\star}}\\right)\\left(\\frac{M_{\\rm acc}}{M_{\\star}}\\right)\\left(\\frac{R_{\\star}}{a}\\right)^{4}\\right]^{3\/5} \\nonumber \\\\\n&&\\approx 2.1\\times 10^{-4}\\frac{M_{\\bullet,7}^{3\/10}}{\\alpha_{-1}^{3\/5}a_{\\rm AU}^{3\/2}}\\left(\\frac{h\/a}{0.3}\\right)^{-9\/5}\\left(\\frac{v_{\\rm c}}{v_{\\rm k}}\\right)^{6\/5} \\nonumber \\\\\n&& \\times \\left(\\frac{M_{\\star}}{0.1M_{\\odot}}\\right)^{-3\/5}\\left(\\frac{M_{\\rm acc}}{M_{\\star}}\\right)^{3\/5}\\left(\\frac{R_{\\star}}{0.1R_{\\odot}}\\right)^{12\/5}\\left(\\frac{\\tau_{\\rm acc}}{{\\rm month}}\\right)^{-3\/5}.\n\\label{eq:rhopenetrate}\n\\end{eqnarray}\n\nWhile the star is within the gaseous disk, the shocks will cause mass ablation from its surface.  The maximum rate of mass ablation is approximately equal to the rate at which gas passes through the shock,\n\\begin{equation}\n\\dot{M}_{\\star} = -\\pi R_{\\star}^{2}\\rho_{\\star, \\rm sh} v_{\\rm sh},\n\\end{equation}\nwhere $\\pi R_{\\star}^{2}$ is the cross section of the star and $v_{\\rm sh} \\approx (\\rho_{\\rm d}\/\\rho_{\\star, \\rm sh})^{1\/2}v_{\\rm c}$ is the velocity of the shock bring driven into the star (\\citealt{McKee&Cowie75}).    \n\nCombining results, we find that the minimum timescale for the star to be destroyed by ablation is given by\n\\begin{eqnarray}\n&&\\frac{\\tau_{\\rm ablate}}{\\tau_{\\rm acc}} \\approx \\frac{1}{f_{\\rm d}\\tau_{\\rm acc}}\\frac{M_{\\star}}{|\\dot{M}_{\\star}|} = \\frac{4}{3f_{\\rm d}}\\frac{R_{\\star}}{\\tau_{\\rm acc} v_{\\rm c}}\\left(\\frac{\\rho_{\\star, \\rm sh}}{\\bar{\\rho}_{\\star}}\\right)^{-1\/2}\\left(\\frac{\\bar{\\rho}_{\\star}}{\\rho_{\\rm d}}\\right)^{1\/2} \\nonumber \\\\\n&=& \\frac{1}{f_{\\rm d}}\\left(\\frac{\\bar{\\rho}_{\\star}}{\\rho_{\\star, \\rm sh}}\\right)^{\\frac{1}{2}}\\left(8 \\alpha \\left(\\frac{M_{\\star}}{M_{\\rm acc}}\\right)\\left(\\frac{a}{R_{\\star}}\\right)\\left(\\frac{v_{\\rm k}}{v_{\\rm c}}\\right)^{2}\\left(\\frac{h}{a}\\right)^{3}\\frac{1}{\\tau_{\\rm acc}\\Omega}\\right)^{\\frac{1}{2}} \\nonumber \\\\\n&\\approx& \\frac{17}{f_{\\rm d}}\\left(\\frac{\\rho_{\\star, \\rm sh}}{10^{-4}\\bar{\\rho}_{\\star}}\\right)^{-1\/2}\\frac{\\alpha_{-1}^{1\/2}a_{\\rm AU}^{5\/4}}{M_{\\bullet,7}^{1\/4}}\\left(\\frac{M_{\\rm acc}}{M_{\\star}}\\right)^{-1\/2} \\nonumber \\\\\n&&\\times \\left(\\frac{h\/a}{0.3}\\right)^{3\/2}\\left(\\frac{v_{\\rm c}}{v_{\\rm k}}\\right)^{-1}\\left(\\frac{R_{\\star}}{0.1R_{\\odot}}\\right)^{-1\/2}\\left(\\frac{\\tau_{\\rm acc}}{{\\rm month}}\\right)^{-1\/2} \\nonumber \\\\\n&\\approx& 36\\frac{\\alpha_{-1}^{4\/5}a_{\\rm AU}^{2}}{M_{\\bullet,7}^{2\/5}}\\left(\\frac{M_{\\rm acc}}{M_{\\star}}\\right)^{-4\/5}\\left(\\frac{h\/a}{0.3}\\right)^{12\/5} \\left(\\frac{f_{\\rm d}}{h\/a} \\right)^{-1}\\label{eq:tauablate}  \\\\\n&&\\left(\\frac{v_{\\rm c}}{v_{\\rm k}}\\right)^{-8\/5}\\left(\\frac{R_{\\star}}{0.1R_{\\odot}}\\right)^{-17\/10}\\left(\\frac{\\tau_{\\rm acc}}{{\\rm month}}\\right)^{-1\/5}\\left(\\frac{M_{\\star}}{0.1M_{\\odot}}\\right)^{3\/10},\\nonumber\n\\end{eqnarray}\nwhere $f_{\\rm d}$ is the fraction of the star's orbit spent in the midplane of the disk, and in the final line we have substitute in equation (\\ref{eq:rhopenetrate}).   A stellar orbit misinclined relative to the plane of the disrupted star will have $f_{\\rm in} \\sim h\/a$, while one close to the same orbital plane would have $f_{\\rm in} \\sim 1$.\\footnote{However, $f_{\\rm in}$ may change even during the TDE itself due to Lense-Thirring precession of the stellar orbit (eq.~\\ref{eq:LT}).}   \n\nWe thus see that $\\tau_{\\rm ablate} \\gg \\tau_{\\rm acc}$ for fiducial parameters, and thus the star should survive at least one TDE intact.  However, the outcome is sensitive to the precise semi-major axis of the star and the total gas mass accreted in the TDE, $M_{\\rm acc}$, which could be as low as $\\sim 10^{-3}-10^{-2}M_{\\odot} \\sim 0.01-0.1M_{\\star}$ (Fig.~\\ref{fig:Erad}; \\citealt{Metzger&Stone16}), in which case $\\tau_{\\rm ablate}\/\\tau_{\\rm flare}$ would be larger by significant factor.  Also note that equation (\\ref{eq:tauablate}) is likely a lower limit on the ablation timescale, because not all matter which passes through the shock will necessarily become unbound from its surface.  This is because the specific thermal energy imparted to the shocked gas $\\sim v_{\\rm sh}^{2} \\sim (\\rho_{\\star, \\rm sh}\/\\bar{\\rho}_{\\star})v_{\\rm c}^{2} \\sim 10^{-4}(GM_{\\bullet}\/a)$ may be comparable to the gravitational binding of the star $\\sim v_{\\rm esc}^{2} \\sim GM_{\\star}\/R_{\\star}$; gas stripped from the stellar surface may therefore accumulate in a wake behind the star to become re-accreted once the flare ceases. \n\nWhat is less clear from the above numbers is whether the outspiraling brown dwarf can survive the large number of expected collisions during its lifetime.  Once a second EMRI arrives on a quasi-circular orbit, the period of the collisional interaction is $\\sim 10^{4-6}~{\\rm yr}$ (Eq. \\ref{eq:tautot}).  During this period, the expected number of TDEs is $\\sim 0.1-1000$ \\citep{Stone&Metzger16}.  Large portions of this range are likely surviveable for the brown dwarf.  The greater challenge for the brown dwarf is surviving long enough to greet an arriving second EMRI.  The fiducial circular EMRI rate of $\\sim 10^{-7}~{\\rm yr}^{-1}$ would require the brown dwarf to survive $\\sim 10^{2-4}$ tidal ablation events, which will be much more challenging (Eq. \\ref{eq:tauablate}).  However, we note that this fiducial circular EMRI rate could be enhanced by up to two orders of magnitude if binary orbits are perturbed outside the SMBH influence radius by massive objects (e.g. giant molecular clouds; \\citealt{Perets+07}) or non-axisymmetric components of the galactic potential \\citep{Hamers&Perets17}.  The large estimated galaxy-to-galaxy scatter (at fixed SMBH mass) in the classical TDE rate \\citep{Stone&Metzger16} is not necessarily correlated with the comparably large scatter in the circular EMRI rate (as most of the TDEs are sourced from $\\sim {\\rm pc}$ scales, while tidally detached binaries can come from much larger distances), making it likely that at least a subset of galactic nuclei will produce EMRI collisions which are not seriously inhibited by ablation from interloping TDEs.\n\n\n\n\n\\section{Conclusions}\n\\label{sec:conclusions}\n\nThe inevitability of stellar collisions in galactic nuclei is well-documented (e.g.~\\citealt{Ginsburg&Loeb07,Antonini+10,Balberg+13, Leigh+16a}).  However, past work has focused on singular collisions, or mergers of stars on eccentric orbits located far outside the tidal radius of the SMBH.  Given the relatively large radius of gas produced in such collisions, any resulting accretion-powered flare produced by these collisions would probably be slowly evolving and very dim.  \n\nHere we have explored a very different scenario: mildly relativistic physical colllisions between two (initially) main sequence stars on circular EMRI orbits undergoing stable Roche-lobe overflow.  Focusing on the probably most common case of an inspiraling, low mass main sequence star interacting with an outspiraling brown dwarf, we have shown that at least a single collision between the stars is inevitable, when they occupy the same orbital phase at the line of nodes where their orbital planes cross (Fig.~\\ref{fig:cartoon}).  \n\nThe initial collision is generally grazing.  Although only a tiny fraction of the stars' surfaces geometrically intersect, the enormous relative velocities ($\\sim 0.1c$) cause massive shock heating of the stellar atmospheres.  When the mass of the SMBH $M_{\\bullet} \\lesssim 7 \\times 10^{6}M_{\\odot}$, this heating is insufficient to destroy the stars in a single encounter, and only a small fraction of the stellar mass is liberated, primarily from the more massive star.  This mass loss produces a gaseous accretion disk, which feeds the SMBH and causes a transient electromagnetic flare, potentially similar in appearance to observed candidate TDE flares.  \nMass loss from the collision also causes the orbital semi-major axis of the outer star to expand, separating it from the orbit of the inner star and delaying the next interaction for at least a decade and perhaps many millenia; the stars must wait for gravitational wave inspiral to realign their orbits before they can once again collide.  The net result of the ensusing string of grazing collisions is a ``death by a thousand cuts,'' producing a series of quasi-periodic accretion-powered flares, over a total duration of thousands of years or longer.  Conversely, if $M_\\bullet \\gtrsim 7 \\times 10^6 M_\\odot$, the first collision is likely powerful enough to completely destroy one or both stars; the ensuing flare will be more analogous to a classical TDE.\n\nAlthough our estimates for the amount of shock heating and the resulting mass lost in grazing stellar collisions are crude, we expect that the qualitative features of the evolution described above should be qualitatively robust.  If the mass loss is smaller (larger) than we have assumed, this will simply increase (decrease) the total number of collisions before the stellar mass is eroded and decrease (increase) the interval between the collisions.  Future hydrodynamical simulations will better quantify the outcome of mildly relativistic stellar collisions and allow for a more accurate calibration of our model.  Future simulation work is also needed to quantify the fraction of the angular momentum of collisionally liberated gas which is fed back into the stellar orbit, accounting for the possible role of Lense-Thirring precession both on the disk and stellar orbits. \n\nBecause the lifetime of the mass transfer evolution is comparable to the expected interval between EMRIs, colliding EMRI chains should occur at a rate comparable to the circular EMRI inspiral rate of $\\gtrsim 10^{-7}$ yr$^{-1}$ per galaxy \\citep{AmaroSeoane+12}.  Although this is still 2-3 orders of magnitude smaller than the predicted or observed TDE rate, we nevertheless conclude that collisional EMRIs can still contribute an appreciable fraction of the observed TDE rate, serving as ``TDE imposters.\"  This is because a given collisional interaction may produce a number $N_{\\rm c} \\sim 1-10^{4}$ of gas production events (eq.~\\ref{eq:Ncoll2}) each of mass $\\sim M_{\\odot}\/N_{\\rm c} \\sim 10^{-4}-1M_{\\odot}$.  If accreted with high radiative efficiency, the luminosities produced after each collision could well explain those of many observed TDE flare candidates.  \n\nUnlike in TDEs, the stellar debris from colliding EMRIs is tightly bound to the SMBH, allowing it to avoid the theoretically uncertain and perhaps lossy circularization process required to accrete highly eccentric TDE debris streams.  Our model provides a natural explanation for some flare light curves which appear to decay exponentially (e.g.~iPTF-16fnl;  \\citealt{Blagorodnova+17}), or as power-laws shallower than $t^{-5\/3}$,  depending on how efficiently the gas angular momentum liberated in the collision is fed back into the stellar orbit.  Viscous spreading of the gaseous disks produced by previous collisions in the chain can also provide a natural supply of radially extended, dense gas around the site of future collisions or TDEs, providing a possible medium for reprocessing the UV\/X-ray accretion luminosity down to optical frequencies.\n\nFuture work is required to explore this new transient scenario in greater detail.  The rates of quasi-circular stellar EMRIs are quite uncertain, and hydrodynamic simulations are required to better understand mass loss in mildly relativistic stellar collisions.  They are also required to confirm whether Roche-overflowing stars can survive the substantial gas ablation expected during the many TDEs experienced between collisional interaction events ($\\S\\ref{sec:TDEstripping}$).  A population study of colliding EMRIs with a realistic distribution of initial stars and orbits would provide more accurate statistics on the expected range of outcomes.  Also deserving of future study is the role of stellar-mass black hole EMRIs, which should periodically pass through the main sequence EMRIs over the course of their mass-transfer evolution (Appendix \\ref{sec:A}). \n\n\n\\acknowledgements\n\nWe thank Itai Linial, Cole Miller, and Re'em Sari for helpful comments on an early version of this manuscript.  We also thank Tiara Hung and Suvi Gezari for providing data on optical TDE light curves.  BDM gratefully acknowledges support from the National Science Foundation (AST-1410950, AST-1615084), NASA through the Astrophysics Theory Program (NNX16AB30G) and the Fermi Guest Investigator Program (NNX15AU77G, NNX16AR73G), the Research Corporation for Science Advancement Scialog Program (RCSA 23810), and the Alfred P.~Sloan Foundation.  Financial support was provided to NCS by NASA through Einstein Postdoctoral Fellowship Award Number PF5-160145.\n\n\n\n\\clearpage\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nIn recent years, energy harvesting cognitive radio networks (EH-CRNs) has emerged as a solution to the problem of spectrum scarcity and at the same time ensures perpetual operation of the communication devices \\cite{EH_CRN}. In the literature, the EH-CRNs operating in interweave, overlay and underlay mode have been studied in  \\cite{interweave_1,interweave_2,interweave_3,interweave_4,interweave_5}, \\cite{overlay_1,overlay_2,\noverlay_3}, and \\cite{overlay_4,underlay_1,underlay_2,underlay_3}, respectively.\n\nIn \\cite{interweave_1}, authors considered an EH-CR network operating in interweave mode. The secondary user (SU) uses ambient radio signals and wireless power transfer to harvest RF energy. The authors used time homogeneous discrete Markov process to model the primary traffic. In each slot, the SU decides either to remain idle or to perform spectrum sensing and re-configures the detection threshold. The authors proposed a transmission policy such that expected total throughput of SU is maximized. They formulated the optimization problem as constrained partially observable Markov decision process (POMDP) and obtained a policy maximizing SU's throughput by designing the spectrum sensing policy and detection threshold jointly. In \\cite{interweave_2}, authors considered the system model of \\cite{interweave_1}, and used time homogeneous discrete Markov process to model the temporal correlation of the primary traffic. The authors upper bounded the achievable throughput of the SU, which is based on energy arrival rates, temporal correlation of primary traffic and detection threshold. The authors obtained an optimal detection threshold maximizing the upper bound on achievable throughput. In \\cite{interweave_3}, authors considered a CR network with energy harvesting secondary user in a single user multi-channel scenario. Based on energy availability of SU, channel conditions and belief state of PU's network, authors obtained a channel selection criterion. This channel selection criterion chooses the best subset among all the available channels for sensing. Then, using the proposed criterion, the authors constructed a channel-aware optimal and myopic sensing policy. In \\cite{interweave_4}, authors considered an EH-CR network where both the PU and SU have energy harvesting capability but they don't have any battery to store the harvested energy. So, if harvested energy is not used, it is discarded. Based on Markovian behavior of PU, to specify the interaction between the PUs and SUs in the system, authors adapted a hidden input Markov model. The authors proposed a two-dimensional spectrum and power sensing policy that improves the PU detection performance and estimates the primary transmit power level. In \\cite{interweave_5}, authors obtained an optimal detection threshold specifying a sensing policy that maximizes the expected total throughput of energy harvesting SU. Authors showed that when arrival rate of energy is less than it's expected consumption, decreased probability of accessing occupied channel may not affect the probability of accessing the vacant spectrum.\n\n\\indent In \\cite{overlay_1}, authors considered an EH-CR network operating in overlay scenario where the secondary user helps primary deliver it's data. The SU has energy harvesting capability and the authors used a discrete time queue to model the energy queue with Markov arrival and service processes. In the system model considered, when PU transmits, SU remains silent and receives some fraction of PU's data and stores it in a queue. Then, as PU becomes silent, SU relays primary data using decode and forward protocol. For the proposed system, the authors obtained inner and outer bounds on the stability region. In \\cite{overlay_2}, authors considered a CR network with joint information and energy cooperation. In the system model considered, the PU not only gives information to SU for relaying but feeds it with energy also. For such cooperation, authors proposed three schemes. In first, they considered an ideal backhaul for information and energy transfer between the two systems. Then, authors proposed power and time splitting schemes enabling the joint information-energy cooperation. Finally, authors obtained zero forcing solutions for all three schemes. In \\cite{overlay_3}, authors considered joint information and energy cooperation among the primary and secondary users. The SU relays primary data and gets spectrum access as a reward. Moreover, the PU feeds the SU with energy. The authors aim to maximize the primary rate subject to rate constraints of both the users. Then, authors analyzed the effects of SU rate constraint and finite battery on probability of cooperation and primary rate. In \\cite{overlay_4}, authors considered a scenario where PUs harvest energy from multiple access points (APs) while SUs have fixed power supply. In the \\emph{energy harvesting zone} centered at PU, nearest ST is selected to transfer energy to PU wirelessly. Also, there exist a \\emph{cooperative region} between PU and AP, where an ST is selected to relay primary data based on the channel quality between ST and AP, and in return, the ST is rewarded with some fraction of bandwidth of primary channel. Under performance constraints of primary system, authors aimed to maximize the throughput of secondary systems in a given area.\n\\begin{figure*}[!t]\n\\includegraphics[width=0.8\\linewidth]{system_model}\n\\centering\n\\caption{Primary and RF energy harvesting secondary users coexisting in an underlay scenario}\n\\label{fig:system_model}\n\\vspace{-3mm}\n\\end{figure*}\n\n\\indent In \\cite{underlay_1}, authors considered an underlay CR system where multiple PUs and energy harvesting SUs coexist. Each primary transmitter (PT) has two concentric zones, namely, \\emph{guard zone} and \\emph{harvesting zone}. If a secondary transmitter (ST) is located within the \\emph{harvesting zone}, it harvests energy from primary transmission, if it is located outside the \\emph{guard zone}, it transmits with fixed power, and otherwise remains idle. The authors considered independent homogeneous Poisson point processes (HPPP) to model PTs and STs, and maximized the spatial throughput of secondary network. In \\cite{underlay_2}, authors solved the secondary system throughput maximization problem using geometric waterfilling with peak power constraints (GWFPP). It is showed that the proposed algorithm outperforms the primal-dual interior point method (PD-IPM). In \\cite{underlay_3}, authors considered an underlay CR network where the SU harvests energy from PU's transmission. In each slot, some fraction of time is reserved for energy harvesting and remaining is reserved for information transfer. However, the authors assumed that the SU uses all of the harvested energy in that slot itself, which makes the optimization problem independent among the slots. The authors obtained an online myopic solution for  time sharing between the energy harvesting phase and information transfer phase which maximizes the achievable rate of CR system.\n\n\\indent In this paper, we consider an underlay CR network with one PU and one SU transmitter-receiver pair operating in slotted fashion. The PT transmits with a fixed power whereas in each slot, the ST spends some fraction of the slot for EH and remaining for IT. However in each slot, SU is not bound to use all of the energy harvested in that slot, rather it can save the energy for future use. The main contributions of this paper are as follows:\n\\begin{itemize}\n\\item First, we obtained optimal offline time sharing between EH and IT phase, and secondary transmit power policy maximizing the sum achievable throughput of ST under energy causality constraint of ST and interference constraint at primary receiver (PR). We formulate the optimization problem as convex optimization problem and obtained the solution in closed form.\n\\item Second, we compare the optimal offline policy with an online myopic policy where unlike the offline policy, only instantaneous channel state information (CSI) is available at ST \\cite{underlay_3}. We formulate the optimization problem as convex optimization problem and decouple it into multiple parallel subproblems and obtained the optimal time sharing parameter in closed form.\n\\end{itemize}\nThe rest of the paper is organized as follows. We present the system model and problem formulation in section II. The optimal offline and the online myopic time-sharing and transmit power policies are obtained in section III, results are given in section IV and finally, we conclude in section V.\n\n\\emph{Notation}: A bold-faced symbol with a ``bar'' (e.g. $\\vec{v}$) represents a vector of length $M$. $[x]^+$ represents $\\max(x,0)$, and $\\vec{v}\\succeq0$ means that each component of vector $\\vec{v}$, $v_i$ is greater than or equal to 0. The calligraphic symbols (e.g. $\\mathcal{S}=\\{\\vec{s}_1,\\ldots\\vec{s}_N\\}$) represents a set of $N$ vectors. And, $\\vec{0}$ and $\\vec{1}$ represent the vectors of all zeros and ones respectively.\n\n\\section{System Model}\nWe consider a scenario where a primary user (PU) and a RF energy harvesting secondary user (SU) coexist in an underlay mode as shown in Fig. \\ref{fig:system_model}. Both the primary and secondary transmitters operate in slotted fashion. The PU has a reliable power supply whereas the SU harvests RF energy from PU's transmission and stores it in an infinite sized battery. We assume that the PU transmits with a constant power of $P_p$ in each slot. However in each slot, the secondary transmitter (ST) uses some fraction of the slot for harvesting energy from primary transmission and transmits in the remaining amount of time such that the interference at primary receiver remains below an acceptable threshold $P_{int}$. The objective is to maximize the sum achievable throughput of ST by the end of $M$ slots under energy causality constraints of SU and interference constraint of PU.\\\\\n\\begin{figure*}[!t]\n\\begin{align}\nE_s^{i*}=\\alpha_i^*\\left[\\frac{1}{\\ln2(\\sum_{j=i}^M\\lambda_j^*+\\gamma_i^*h_{sp}^i)}-\\frac{1}{\\theta_i}\\right]^+\\text{and }\n\\alpha_i^*=\\theta_iE_s^{i*}\\left[\\frac{1}{\\ln2\\left(\\log_2(1+\\theta_i\\beta_i^*)-\\sum_{j=i}^M\\eta P_p\\lambda_j^*+\\gamma_i^*P_{int}-\\mu_i^*\\right)}-1\\right]^+\\label{eq:opt_sol}\n\\end{align}\nwhere $\\beta_i^*=\\left[\\frac{1}{\\ln2(\\sum_{j=i}^M\\lambda_j^*+\\gamma_i^*h_{sp}^i)}-\\frac{1}{\\theta_i}\\right]^+$ and $\\theta_i=\\frac{h_{ss}^i}{\\sigma_s^2+h_{ps}^iP_p}$.\\\\\n\\noindent\\rule{\\linewidth}{0.4pt}\n\\vspace{-.8cm}\n\\end{figure*}\n\\indent In the system model, all the channel links are assumed to be i.i.d. Rayleigh faded with variances $\\sigma_{pp}^2,\\sigma_{ps}^2,\\sigma_{sp}^2$ and $\\sigma_{ss}^2$ for PT-PR, PT-SR, ST-PR and ST-SR link respectively, so the channel power gains $h_{pp},h_{ps},h_{sp}$ and $h_{ss}$ are i.i.d. exponentially distributed. Since we have considered an offline policy, we assume that complete channel state information (CSI) is known at ST non-causally as in \\cite{non_causal_csi}. The noise at both the receivers is assumed to be zero mean additive white Gaussian with variances $\\sigma_p^2$ and $\\sigma_s^2$ at PR and SR respectively. Let $\\alpha_i$ be the time sharing parameter between EH and IT phase such that in slot $i$, the ST harvests for $(1-\\alpha_i)$ fraction of time and transmits for $\\alpha_i$ fraction of time. The slot length is assumed to be 1 second without loss of generality. Let $P_s^i$ be the transmit power of ST in $i$th slot, the achievable rate of SU in bits\/seconds\/Hz is given by Shannon capacity formula:\n\\begin{align*}\nR_i(\\alpha_i,P_s^i)=\\alpha_i\\log_2\\left(1+\\frac{h_{ss}^iP_s^i}{\\sigma_s^2+h_{ps}^iP_p}\\right),\\; i=1,\\ldots,M.\n\\end{align*}\nThe optimization problem is given as:\n\\begin{subequations}\n\\begin{align}\n\\max_{\\vec{\\pmb{\\alpha}},\\vec{P}_s} \\quad & \\sum_{i=1}^M \\alpha_i\\log_2\\left(1+\\frac{h_{ss}^iP_s^i}{\\sigma_s^2+h_{ps}^iP_p}\\right)\\label{eq:opt_1}\\\\\n\\text{s.t.}\\quad & \\sum_{j=1}^i\\alpha_jP_s^j\\leq \\sum_{j=1}^i(1-\\alpha_j)\\eta P_p,\\quad i=1,\\ldots,M \\label{eq:opt_2}\\\\\n& \\qquad\\qquad \\text{(Energy causality constraint of SU)} \\nonumber\\\\\n& h_{sp}^iP_s^i\\leq P_{int}, \\quad i=1,\\ldots,M \\label{eq:opt_3}\\\\\n&\\qquad\\qquad\\qquad\\text{(Interference constraint of PU)} \\nonumber\\\\\n& \\vec{P}_s\\succeq \\vec{0},\\,\\vec{0}\\preceq \\vec{\\pmb{\\alpha}}\\preceq \\vec{1} \\label{eq:opt_4}\n\\end{align}\n\\end{subequations}\nwhere $0\\leq\\eta\\leq1$ and $P_{int}$ are the energy harvesting efficiency of SU and interference constraint of PR respectively. The constraint (\\ref{eq:opt_2}) is the energy causality constraint of ST, which states that the total energy consumed by the end of slot $i$ must be less than or equal to total energy harvested upto slot $i$.\n\\section{Optimal Time Sharing and Transmit Policy}\n\\indent The problem (\\ref{eq:opt_1})-(\\ref{eq:opt_4}) is not a convex optimization problem as the optimization variables $\\vec{P}$ and $\\vec{\\pmb{\\alpha}}$ appear in product form. Let $E_s^i$ denotes the energy consumed by ST in $i$th slot so that $E_s^i=\\alpha_iP_s^i$. Using change of variable $P_s^i=\\frac{E_s^i}{\\alpha_i}$, the optimization problem can be rewritten as:\n\\begin{subequations}\n\\begin{align}\n\\max_{\\vec{\\pmb{\\alpha}},\\vec{E}_s} \\quad & \\sum_{i=1}^M \\alpha_i\\log_2\\left(1+\\frac{h_{ss}^iE_s^i}{\\alpha_i(\\sigma_s^2+h_{ps}^iP_p)}\\right)\\label{eq:convex_opt_1}\\\\\n\\text{s.t.}\\quad & \\sum_{j=1}^iE_s^j\\leq \\sum_{j=1}^i(1-\\alpha_j)\\eta P_p,\\quad i=1,\\ldots,M \\label{eq:convex_opt_2}\\\\\n& \\qquad\\qquad \\text{(Energy causality constraint of SU)} \\nonumber\\\\\n& h_{sp}^iE_s^i\\leq \\alpha_i P_{int}, \\quad i=1,\\ldots,M \\label{eq:convex_opt_3}\\\\\n&\\qquad\\qquad\\qquad\\text{(Interference constraint of PU)} \\nonumber\\\\\n& \\vec{E}_s\\succeq \\vec{0},\\,\\vec{0}\\preceq \\vec{\\pmb{\\alpha}}\\preceq \\vec{1} \\label{eq:convex_opt_4}\n\\end{align}\n\\end{subequations}\nwhich is a convex optimization problem as the objective function is negative of sum of relative entropies $D(p_i||q_i)$ where $p_i=\\alpha_i$ and $q_i=\\alpha_i+\\frac{h_{ss}^iE_s^i}{\\sigma_s^2+h_{ps}^iP_p}$, and the constraints are affine inequalities. Hence, it can be solved efficiently using CVX. Firstly, we propose the necessary conditions our optimal policy must satisfy and then, we will obtain the optimal offline solution.\n\\subsection{Optimality Conditions}\n\\begin{proposition*}\nThe optimal time sharing and transmit power policy must satisfy $\\sum_{i=1}^ME_s^{i*}= \\sum_{i=1}^M(1-\\alpha^*_i)\\eta P_p$, i.e., the optimal policy must use all the harvested energy by end of the transmission.\n\\end{proposition*}\n\\begin{proof}\nWe prove this using contradiction. Let $\\{\\vec{E}'_s,\\vec{\\pmb{\\alpha}}'\\}$ be an optimal policy in which we have some residual energy remaining in the battery by the end of transmission, i.e., $\\sum_{j=1}^ME_s^{'j}<\\sum_{j=1}^M(1-\\alpha'_j)\\eta P_p$. Since the objective is a concave and monotonically increasing function of consumed energy, we could have consumed more energy in previous slots without violating the energy causality constraint. This way, we can also increase the transmission power. This would result in higher achievable rate, hence contradicts with our consideration of optimality.\n\\end{proof}\n\n\\begin{figure*}[!th]\n\\centering\n\\begin{minipage}{0.3\\linewidth}\n\\centering\n  \\includegraphics[width=2.3in]{optimal_plot}\n  \\caption{Average achievable sum rate of ST ($\\vec{R}$) versus number of secondary slots ($N$).}\n  \\label{fig:opt_plot}\n\\end{minipage}\n\\hspace{3mm}\n\\begin{minipage}{0.3\\linewidth}\n\\centering\n  \\includegraphics[width=2.3in]{comparison_myopic_2}\n  \\caption{Average achievable sum rate of optimal and online myopic policies.}\n  \\label{fig:comparison}\n\\end{minipage}\n\\hspace{3mm}\n\\begin{minipage}{0.3\\linewidth}\n\\centering\n\\centering\n  \\includegraphics[width=2.3in]{diff_chan}\n  \\caption{Average achievable rate of ST ($\\vec{R}$) versus number of secondary slots ($N$) under different channel conditions ($\\eta = 0.3$ and $P_{int}=0.1$).}\n  \\label{fig:diff_chan}\n\\end{minipage}\n\\vspace{-0.3cm}\n\\end{figure*}\n\\subsection{Optimal Solution}\nThe Lagrangian of the problem (\\ref{eq:convex_opt_1})-(\\ref{eq:convex_opt_4}) is given as:\n\\begin{align}\n\\mathcal{L}(\\vec{x},\\vec{y})= &-\\sum_{i=1}^M\\alpha_i\\log_2\\left( 1+ \\frac{h_{ss}^iE_s^i}{\\alpha_i(\\sigma_s^2+h_{ps}^iP_p)}\\right)\\nonumber \\\\\n&+\\sum_{i=1}^M \\lambda_i \\left[\\sum_{j=1}^iE_s^j-\\sum_{j=1}^i(1-\\alpha_j)\\eta P_p \\right]\\nonumber \\\\\n&+\\sum_{i=1}^M\\gamma_i[h_{sp}^iE_s^i-P_{int}\\alpha_i]+\\sum_{i=1}^M\\mu_i(\\alpha_i-1) \\label{eq:lagrangian}\n\\end{align}\nwhere $\\vec{\\pmb{\\lambda}},\\vec{\\pmb{\\gamma}}$ and $\\vec{\\pmb{\\mu}}$ are the dual variables for the constraints (\\ref{eq:convex_opt_2}), (\\ref{eq:convex_opt_3}) and (\\ref{eq:convex_opt_4}) respectively and $\\vec{x}\\in\\mathcal{X}=\\{\\vec{E}_s,\\vec{\\pmb{\\alpha}}\\}$, and $\\vec{y}\\in\\mathcal{Y}=\\{\\vec{\\pmb{\\lambda}},\\vec{\\pmb{\\gamma}},\\vec{\\pmb{\\mu}}\\}$. The\nKarush-Kuhn-Tucker (KKT) stationarity conditions are:\n\\begin{align}\n\\frac{\\frac{\\theta_iE_s^{i*}}{\\alpha_i^*}}{\\ln2\\cdot\\left(1+\\frac{\\theta_iE_s^{i*}}{\\alpha_i^*}\\right)}-\\log_2\\left(1+\\frac{\\theta_iE_s^{i*}}{\\alpha_i^*}\\right)+\\sum_{j=i}^M\\eta P_p\\lambda_j^*\\nonumber\\\\\n-P_{int}\\gamma_i^*+\\mu_i^*&=0 \\label{eq:kkt_1}\\\\\n-\\frac{\\theta_i}{\\ln2\\left(1+\\frac{\\theta_iE_s^{i*}}{\\alpha_i^*}\\right)}+\\sum_{j=i}^M\\lambda_j^*+\\gamma_i^*h_{sp}^i&=0\\label{eq:kkt_2}\n\\end{align}\nwhere $\\theta_i=\\frac{h_{ss}^i}{\\sigma_s^2+h_{ps}^iP_p}$. And the complementary slackness conditions are\n\\begin{align}\n\\lambda_i^*\\left[\\sum_{j=1}^iE_s^{j*}-\\sum_{j=1}^i(1-\\alpha_j^*)\\eta P_p\\right]&=0\\\\\n\\gamma_i^*[h_{sp}^iE_s^{i*}-\\alpha_i^*P_{int}]&=0\\\\\n\\mu_i^*[\\alpha_i^*-1]&=0\n\\end{align}\nfor all $i=1,\\ldots,M$. To make computation more tractable, we neglect the dual variables associated with non-negativity constraints of $\\vec{E}_s$ and $\\vec{\\pmb{\\alpha}}$. We can include these constraints later by projecting the obtained results onto positive orthant. From (\\ref{eq:kkt_1}) and (\\ref{eq:kkt_2}) we obtain the optimal solution given in eq. (\\ref{eq:opt_sol}).\n\nSince the objective function is strictly concave, the optimal $\\vec{E}_s$ and $\\vec{\\pmb{\\alpha}}$ are unique. Since $P_s^{i*}=\\frac{E_s^{i*}}{\\alpha_i^*}$, the optimal transmit power of ST in $i$th slot is given as:\n\\begin{align}\nP_s^{i*}=\\left\\{\n\\begin{array}{cl}\n\\left[\\frac{1}{\\ln2(\\sum_{j=i}^M\\lambda_j^*+\\gamma_i^*h_{sp}^i)}-\\frac{1}{\\theta_i}\\right]^+, & 0<\\alpha_i\\leq1\\\\\n0, & \\alpha_i=0\n\\end{array}\n\\right.\\label{eq:opt_pwr}\n\\end{align}\n\\subsection{Online Myopic Policy}\n\\indent To compare our proposed scheme, we consider an online myopic policy as in \\cite{underlay_3}, in which only instantaneous channel gains are known at ST. However, unlike \\cite{underlay_3}, we consider rather more strict constraint for interference at PR. In our system model, we consider that in each slot, interference at PR must be less than the threshold $P_{int}$, whereas in \\cite{underlay_3}, authors considered that the probability of interference being greater than threshold should be arbitrarily small. As in \\cite{underlay_3}, in each slot the ST uses all the harvested energy in the same slot. Thus, in $i$th slot, the consumed energy will be $E_s^i=(1-\\alpha_i)\\eta P_p$. The optimal time sharing parameter $\\vec{\\pmb{\\alpha}}$, which maximizes the short term secondary sum rate, can be obtained by solving the following optimization problem:\n\\vspace{-0.1cm}\n\\begin{subequations}\n\\begin{align}\n\\max_{\\vec{\\pmb{\\alpha}}}\\quad & \\sum_{i=1}^M \\alpha_i \\log_2\\left(1+\\frac{1-\\alpha_i}{\\alpha_i}\\frac{h_{ss}^i\\eta P_p}{\\sigma_s^2+h_{ps}^iP_p}\\right) \\label{eq:subopt_1}\\\\\n\\text{s.t.}\\quad & h_{sp}^i(1-\\alpha_i)\\eta P_p\\leq\\alpha_i P_{int},\\quad i=1,\\ldots,M.\\label{eq:subopt_2}\\\\\n& \\vec{0}\\prec\\vec{\\pmb{\\alpha}}\\prec\\vec{1} \\label{eq:subopt_3}\n\\end{align}\n\\end{subequations}\n\\vspace{-0.1cm}\nThis is a convex optimization problem as optimization function is negative of sum of relative entropies, $D(p_i||q_i)$ where $p_i=\\alpha_i$ and $q_i=\\alpha_i+(1-\\alpha_i)\\frac{h_{ss}^i\\eta P_p}{\\sigma_s^2+h_{ps}^iP_p}$, and the constraints are linear inequalities. Since ST uses all the harvested energy in the same slot, the optimization problem becomes independent among slots and we can decouple the problem into $M$ parallel subproblems. By rearranging the terms, constraint (\\ref{eq:subopt_2}) can be written as $\\alpha_i\\leq\\frac{h_{sp}^i\\eta P_p}{P_{int}+h_{sp}^i\\eta P_p}$, which is $\\leq1$ unless $P_{int}=0$. Therefore, the constraints (\\ref{eq:subopt_2}) and (\\ref{eq:subopt_3}) can be merged together into a single constraint $0<\\alpha_i\\leq\\frac{h_{sp}^i\\eta P_p}{P_{int}+h_{sp}^i\\eta P_p}$. In $i$th slot, we solve the following convex optimization problem:\n\\begin{subequations}\n\\begin{align}\n\\max_{\\alpha_i} \\quad & \\alpha_i \\log_2\\left(1+\\frac{1-\\alpha_i}{\\alpha_i}\\frac{h_{ss}^i\\eta P_p}{\\sigma_s^2+h_{ps}^iP_p}\\right) \\label{eq:myopic_1}\\\\\n\\text{s.t.} \\quad & 0<\\alpha_i\\leq\\frac{h_{sp}^i\\eta P_p}{P_{int}+h_{sp}^i\\eta P_p}\n\\end{align}\n\\end{subequations}\nThe Lagrangian of the problem stated above is given as:\n\\begin{align*}\n\\mathcal{L}(\\alpha_i,\\lambda)=&-\\alpha_i \\log_2\\left(1+\\frac{1-\\alpha_i}{\\alpha_i}\\zeta_i\\right)+\\lambda\\left(\\alpha_i-\\Psi_i\\right)\n\\end{align*}\nwhere $\\zeta_i=\\frac{h_{ss}^i\\eta P_p}{\\sigma_s^2+h_{ps}^iP_p}$ and $\\Psi_i = \\frac{h_{sp}^i\\eta P_p}{P_{int}+h_{sp}^i\\eta P_p}$. Using the KKT optimality conditions, the closed form of optimal time sharing parameter $\\alpha_i$ is given as \\cite{myopic_solution}:\n\\begin{align}\n\\alpha_i^*=\\max\\left\\{\\frac{\\zeta_i}{\\zeta_i+z_i^*-1},\\frac{h_{sp}^i\\eta P_p}{P_{int}+h_{sp}^i\\eta P_p}\\right\\}\\label{eq:optimal_myopic}\n\\end{align}\nwhere $z_i^*>1$ is the unique solution of following equation:\n\\begin{align*}\nz_i\\ln z_i-z_i-\\zeta_i +1=0\n\\end{align*}\n\\section{Results and Discussions}\nFor simulation purpose, we initially assume that all channel links are i.i.d. Rayleigh distributed with variance $\\sigma_{pp}^2=\\sigma_{sp}^2=\\sigma_{ps}^2=\\sigma_{ss}^2=1$. The primary user transmits with power $P_p=2$ Watt in all the slots. The energy harvesting link is assumed to be static throughout the transmission and has an attenuation factor $\\eta$. Therefore, the energy harvested by ST in each slot is $E_h^i=(1-\\alpha_i)\\eta P_p$ Joule. The noise at both the receivers is assumed to be zero mean additive white Gaussian with equal variances $\\sigma_p^2=\\sigma_s^2=0.1$. In addition, we assume that there is no initial energy available in the battery. The results for Figs. \\ref{fig:opt_plot} and \\ref{fig:comparison} are obtained for the system settings as stated before, whereas for Fig. \\ref{fig:diff_chan}, we change the channel variances to analyze the effects of different channel conditions.\n\\subsection{Effects of harvesting efficiency and interference constraint}\nFig. \\ref{fig:opt_plot} shows the average achievable sum rate of ST ($\\vec{R}_{sum}$) averaged over different channel realizations versus number of secondary slots ($N$) for different values of interference threshold $P_{int}$ and energy harvesting efficiency $\\eta$ under the optimal offline policy. The average achievable sum rate is monotonically increasing with the number of secondary slots as expected. In addition, higher $\\eta$ results in higher achievable rate as with higher $\\eta$, secondary harvests more energy which allows it to transmit with higher power. Also, it can be observed that as the interference constraint at PR becomes stricter (i.e. $P_{int}$ decreases), the average achievable rate of ST reduces as the interference constraint puts an upper bound on the transmit power $\\vec{P}_s$. In addition, with stricter interference constraint, the rate at which $\\vec{R}_{sum}$ increases w.r.t. $N$, also reduces. \n\\subsection{Comparison of optimal offline and myopic policies}\nFig. \\ref{fig:comparison} shows the comparison of average achievable sum rate obtained using optimal offline and online myopic policies. The plot is obtained for a fixed energy harvesting efficiency $\\eta=0.3$. The figure clearly shows that the myopic policy with causal CSI performs worse than the optimal policy with non-causal CSI. This is expected as myopic policy does not have any knowledge of future channel gains and the ST needs to consume all the harvested energy in the same slot. Hence, the ST harvests only that much energy in a slot which it can consume in the same and tries to maximize the achievable rate of that slot only. It is observed that this difference in performances reduces as interference constraint becomes stricter. This is because in offline policy, with stricter constraint, the ST harvests very less amount of energy and transmits with very low power and saves very little amount of energy for future use.\n\\subsection{Effects of different channel conditions}\nFig. \\ref{fig:diff_chan} shows the average achievable sum rate of SU under different channel conditions. The results for weak ST-PR and weak PT-SR links are obtained by choosing $\\sigma_{sp}^2=0.1$ and $\\sigma_{ps}^2=0.1$ for ST-PR and PT-SR links respectively, whereas the variance of other channel links are assumed to be 1. From the figure it is observed that as ST-PR link becomes weak, ST's achievable rate increases significantly. This is because weak ST-PR link is as good as loose interference constraint on PR, which allows ST to transmit with higher power yielding higher sum rate. Also, weak ST-SR link decreases the sum rate as due to interference constraint, the ST transmit power is upper bounded by $\\frac{P_{int}}{h_{sp}^i}$ and small channel coefficients of ST-SR link cause the sum-rate to decrease. The effects of strong direct links ($\\sigma_{pp}^2=\\sigma_{ss}^2=1$ and $\\sigma_{ps}^2=\\sigma_{sp}^2=0.1$) and strong interference links ($\\sigma_{pp}^2=\\sigma_{ss}^2=0.1$ and $\\sigma_{ps}^2=\\sigma_{sp}^2=1$) are also shown. As expected, the system performance is much better in the case of strong direct links as compared to the case of strong interference links.\n\\section{Conclusion}\n\\indent We considered an energy harvesting underlay cognitive radio system operating in slotted fashion. We considered a case where PU is equipped with a reliable power source whereas the SU harvests energy from PU's transmission. Each secondary slot is divided into two phases: EH and IT phase. Assuming complete CSI to be available non-causally at ST, we obtained an optimal offline time sharing and transmit power policy which maximizes the sum achievable rate of SU. Also, we obtained closed form expressions for time sharing parameter $\\vec{\\pmb{\\alpha}}$ and consumed energy $\\vec{E}_s$ so that optimal values of these parameters can be found iteratively. In addition, considering only causal CSI to be available at ST, we obtained an online myopic policy maximizing the achievable sum rate of SU. We compared the performance of both the policies and it is observed that the offline policy outperforms the myopic policy. But since the myopic policy needs only causal CSI at ST, it is more practical than the offline policy.\n\n\\footnotesize\n\\bibliographystyle{IEEEtr}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Acknowledgements}\nThis work was supported in Poland by the National Science Centre under the Project No.~UMO-2017\/27\/B\/ST3\/02881. We thank Ingrid Mertig for a critical reading of the paper and insightful comments.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nOur understanding of dark matter structures has increased significantly over the recent years. This progress has mainly been driven by numerical simulations which have\nidentified a range of universalities of the dark matter structures. One of the first general\nproperties to be suggested was the radial density profile \\citep{1996ApJ...462..563N,1998ApJ...499L...5M,2004MNRAS.353..624D,2006AJ....132.2685M,2006AJ....132.2701G}, whose radial behaviour was shown to change from a fairly shallow decline in the central region to a much steeper decline in the outer regions. This behaviour has been confirmed observationally for galaxy clusters, both through X-ray observations \\citep{2004ApJ...604..116B,2005A&A...435....1P,2005A&A...441..893A,2006ApJ...640..691V,2006A&A...446..429P}, and also more recently through strong and weak lensing observations \\citep{2004ApJ...604...88S,2005ApJ...619L.143B,2006ApJ...642...39C,2008arXiv0802.4292L}.\n\nA slightly less intuitive quantity to be considered is the dark matter velocity anisotropy defined by\n\\begin{equation}\\label{eq:be}\n\\beta \\equiv 1 - \\frac{\\sigma^2_t}{\\sigma^2_r},\n\\end{equation}\nwhere $\\sigma^2_t$ and $\\sigma^2_r$ are the 1-dimensional tangential and radial velocity dispersions in a spherical system~\\citep{1987gady.book.....B}. This anisotropy was shown in pure dark matter simulations to increase radially from zero in the central region to roughly 0.5 in the outer region~\\citep{1997ApJ...485L..13C,1996MNRAS.281..716C,2006NewA...11..333H}. For collisional systems, in contrast, the velocity anisotropy is explicitly zero in the equilibrated regions. Therefore, eventually inferring $\\beta$ from observational data is an important test of whether dark matter is in fact collisionless, as assumed in the standard model of structure formation. On this note, it has been shown that the Galactic velocity anisotropy can affect the detection rates of direct dark matter searches \\citep{2008PhRvD..77b3509V}, and it is in principle measurable in a direction-sensitive detector \\citep{2007JCAP...06..016H}.\n\nThe most massive bound structures in the Universe are clusters of galaxies, which consist of an extended dark matter halo, an X-ray emitting plasma making up the intracluster medium (ICM), and the individual galaxies. While the contribution of galaxies to the total mass is small, approximately 10 $\\%$ of the cluster mass resides in the ICM. The present generation of X-ray satellites, \\emph{XMM-Newton} and \\emph{Chandra}, allows very accurate measurements of azimuthally-averaged radial profiles of density and temperature of the ICM. These are used, under the assumption of hydrostatic equilibrium and spherical symmetry of both gas and total mass distributions, to estimate total, gas, and dark matter mass profiles \\citep{1980ApJ...241..552F}.\n\nBelow we infer the radial velocity anisotropy profile of dark matter in 16 galaxy clusters using a generally applicable framework without any parametrized modeling of the clusters. In short, we assume a universal relation between the effective temperature of dark matter and the ICM temperature, which allows us to solve the dynamics of the dark matter halo using the radial gas temperature and density profiles determined from X-ray data. We investigate the shape and validity of this temperature relation in two cosmological simulations of galaxy clusters, based on independent numerical codes. We apply our method to 16 galaxy clusters from two different samples and find a velocity anisotropy significantly different from zero in the outer parts, in qualitative agreement with simulations. \n\nOur approach here is a generalization of the non-parametric analysis in \\citet{2007A&A...476L..37H} where $\\beta$ was inferred neglecting the radial dependence. We also note the parametrized analyses in \\citet{2004ApJ...611..175I} and \\citet{2007MNRAS.380.1521M}, where the total dark matter velocity dispersion was inferred assuming either $\\beta=0$, or the analytical $\\beta$-profiles of \\citet{2000ApJ...539..561C} or \\citet{1996MNRAS.281..716C} (see also \\citet{2008MNRAS.tmp..719W}). In particular, \\citet{2007MNRAS.380.1521M} found that the dark matter temperature and the ICM temperature were essentially the same in strong cooling-core clusters.\n\nThe structure of the paper is the following: In the next section, we discuss how we relate the temperature of dark matter to the observable gas temperature. In section 3 we show how we can then solve the dynamics of the dark matter. In section 4 we test the assumed temperature relation and our method on numerical simulations, and in section 5 we apply the method to observational data. Section 6 is the summary and discussion.\n\n\\section{The temperature of dark matter}\nThe equality of inertial and gravitational mass implies that the orbit of a test particle in a gravitational system is independent of mass. For example, the velocity of a circular orbit in a spherical mass distribution $v_c^2=GM(r)\/r$ depends only on the distance to the center of the system and the mass contained within that radius. Therefore it is natural to assume that, at a given radius, all species in a relaxed, spherical gravitational system have the same average specific kinetic energy. Obviously, they also have the same specific potential energy. In a gas system, equilibrium implies energy equipartition between all species. It is clear that the corresponding principle for a relaxed gravitational system is a common velocity dispersion, precisely because gravitational dynamics are independent of mass. Since the average velocity is associated with the thermal energy content, this relationship is expressed by\n\\begin{equation}\\label{eq:trl}\nT_{\\mathrm{DM}}=\\kappa T_\\mathrm{gas}.\n\\end{equation}\nThe parameter $\\kappa$ is constant as long as the impact of radiative or entropy-changing processes affecting the gas is negligible and the system is relaxed. Therefore, we allow for a radial dependence, $\\kappa=\\kappa (r\/r_{2500})$, where $r_{2500}$ is the scale radius within which the mean total density is $2500$ times the critical density at the redshift of the cluster. \n\nThe dark matter temperature in (\\ref{eq:trl}) is naturally not well-defined as there is no thermodynamic equilibrium for a collisionless gas. Instead, we simply define an effective dark matter temperature which is proportional to the three-dimensional velocity dispersion,\n\\begin{eqnarray}\nk_BT_{\\mathrm{DM}}&=&\\frac{1}{3}\\mu m_H\\sigma^2_\\mathrm{DM}\\\\\n&=&\\frac{1}{3}\\mu m_H\\left(\\sigma_r^2+2\\sigma_t^2\\right).\\label{eq:tdm}\n\\end{eqnarray}\nThe velocity dispersion has been decomposed into the contributions from the one-dimensional radial and tangential dispersions. We choose the constant of proportionality to be the mean molecular mass of the intracluster gas simply to allow $\\kappa$ to be of order unity. Equations (\\ref{eq:trl})--(\\ref{eq:tdm}) are equivalent to assuming that the specific energies of gas and dark matter particles are the same up to a factor of $\\kappa$, on average. The same conjecture was made in \\citet{2007A&A...476L..37H} but with $\\kappa=1$ explicitly. \n\nIt should be mentioned that the temperature relation (\\ref{eq:trl}) was recently analyzed in simulations by \\citet{2008ApJ...672..122E}. Whereas we allow a possible radial variation in the temperature relation, those authors considered averages within $r_{200}$ and found that\n\\begin{equation}\n\\tilde{\\kappa}_{<r_{200}}\\equiv\\frac{k_B\\overline{T_\\mathrm{gas}}\/\\mu m_H}{\\overline{\\sigma^2_\\mathrm{DM}}}=1.04\\pm0.06,\n\\end{equation}\nThis was based on their determination of $\\bar{\\kappa}^{-1}_{<r_{200}}=(0.87\\pm0.04)\\langle T_\\mathrm{spec}\/T_\\mathrm{mw}\\rangle$, where the ratio of the spectroscopic temperature to the mass-weighted temperature was $\\langle T_\\mathrm{spec}\/T_\\mathrm{mw}\\rangle=1.1\\pm0.05$ \\citep{2006ApJ...650..538N}. Then, in \\citet{2008ApJ...679L...1R} it was noted that by applying virial scaling to the WMAP5+SN+BAO results \\citep{2008arXiv0803.0547K}, an average value of $\\tilde{\\kappa}^{-1}|_{<r_{500}}=1.1$ was found. The authors concluded that the observational results indicated that the average specific energy of the ICM was close to both that of the dark matter and that of the galaxies. In section \\ref{sec:sim}, we will arrive at the same conclusion for simulated galaxy clusters. \n\n\\section{Solving the dark matter dynamics}\\label{se:th}\nEquation (\\ref{eq:trl}) allows us to estimate the total velocity dispersion profile of the dark matter structure from measurements of the radial temperature profile of the gas. In this section we discuss how we can proceed to determine the dark matter velocity anisotropy.\n\nThe collisionless Jeans equations relate the dynamical properties of the dark matter to the gravitational potential of the cluster. Assuming that the system is spherically symmetric and in a steady state, the second Jeans equation can be put in the form \\citep{1987gady.book.....B}\n\\begin{equation}\n\\frac{d (\\nu\\overline{v_r^2})}{dr}+\\frac{\\nu}{r}\\left[2\\overline{v_r^2}-\\left(\\overline{v_\\theta^2}+\\overline{v_\\phi^2}\\right)\\right]=-\\nu\\frac{GM}{r^2},\n\\end{equation}\nwhere $\\nu$ is the dark matter number density, $\\overline{v_i^2}$ is the second moment of the $i$th velocity component, and $M$ is the mass contained within radius $r$. If it is further assumed that there are no bulk flows, $\\overline{v_i}=0$, and that the tangential velocity dispersions are equal, $\\sigma_\\theta^2=\\sigma_\\phi^2\\equiv\\sigma_t^2$, the Jeans equation becomes\n\\begin{equation}\\label{eq:je}\n\\sigma_r^2\\left(\\frac{d\\ln \\rho_{\\mathrm{DM}}}{d\\ln r}+\\frac{d\\ln \\sigma_r^2}{d\\ln r}+2\\beta\\right)=-\\frac{GM(r)}{r},\n\\end{equation}\nwhere $\\rho_\\mathrm{DM}$ is the mass density, $\\sigma_r^2$ is the radial velocity dispersion, and $\\beta$ is the velocity anisotropy introduced in (\\ref{eq:be}).\n\nSimilar to the Jeans equation, the radial part of the Euler equations of the ICM expresses the condition that the thermal pressure of the gas balances the gravitational potential. This equation of hydrostatic equilibrium reads\n\\begin{equation}\\label{eq:he}\n\\frac{k_BT_\\mathrm{gas}}{\\mu m_H}\\left(\\frac{d\\ln n_e}{d\\ln r}+\\frac{d\\ln T_\\mathrm{gas}}{d\\ln r}\\right)=-\\frac{GM(r)}{r},\n\\end{equation}\nwhere $T_\\mathrm{gas}$ is the gas electron temperature and $n_e$ is the number density of electrons. This important equation has been widely used to estimate the total mass of a galaxy cluster from X-ray data. In case there is turbulence or larger scale bulk motion in the gas  additional terms of the form $(\\vec{v}\\cdot\\nabla)\\vec{v}-(v_\\theta^2+v_\\phi^2)\/r$ appear \\citep{lanlif}. Neglecting such terms may lead to an underestimate of the mass, however this is usually not a major effect for systems that appear relaxed \\citep{2008arXiv0808.1111P}.\n\nBy equating (\\ref{eq:je}) and (\\ref{eq:he}) and using (\\ref{eq:be}) and (\\ref{eq:trl}) to eliminate $\\beta$, we obtain the following differential equation for the radial velocity dispersion,\n\\begin{equation}\\label{eq:rd}\n\\sigma_r^2\\left(\\frac{d\\ln \\rho_{\\mathrm{DM}}}{d\\ln r}+\\frac{d\\ln \\sigma_r^2}{d\\ln r}+3\\right)=\\psi(r),\n\\end{equation}\nwhere the function $\\psi$ is defined by\n\\begin{equation}\n\\psi(r)=3\\kappa\\frac{k_BT_\\mathrm{gas}}{\\mu m_H}-\\frac{GM}{r}.\n\\end{equation}\nClearly, $\\psi$ is determined directly from the X-ray observables and the $\\kappa$-profile, which we discuss in section \\ref{sec:simtrl}.\n\nThe differential equation (\\ref{eq:rd}) is solved by finding an integrating factor which yields\n\\begin{equation}\\label{eq:srsq}\n\\sigma_r^2(r)=\\frac{1}{\\rho_{\\mathrm{DM}}(r)\\,r^3}\\int_0^r dr'\\, \\psi(r')\\rho_{\\mathrm{DM}}(r')r'^2.\n\\end{equation}\nThe dark matter density is determined as usual through $\\rho_{\\mathrm{DM}}=\\rho_{\\mathrm{tot}}-\\mu m_H n_e$. With the radial velocity dispersion profile determined, the velocity anisotropy is easily recovered from either the temperature relation (\\ref{eq:trl}) or the Jeans equation (\\ref{eq:je}),\n\\begin{eqnarray}\n2\\beta_{\\mathrm{tr}}&=&3\\left(1-\\kappa\\frac{k_BT_\\mathrm{gas}}{\\mu m_H \\sigma_r^2}\\right),\\label{eq:btr}\\\\\n2\\beta_{\\mathrm{Je}}&=&-\\frac{d\\ln(\\rho_{\\mathrm{DM}}\\sigma_r^2)}{d\\ln r}-\\frac{GM}{r\\sigma_r^2}.\\label{eq:bje}\n\\end{eqnarray}\nObviously these two expressions should be equal. This can be used as a consistency check on whether numerical issues related to the differentiations and integration involved are kept under control. \n\nTo summarize, the assumed relation (\\ref{eq:trl}) between the effective dark matter temperature and the gas temperature, along with the mass estimate from (\\ref{eq:he}), allows us to solve the dark matter dynamics directly from X-ray data, and determine both the radial velocity dispersion and the velocity anisotropy as functions of radius. \n\n\\section{Cluster simulations}\\label{sec:sim}\nWe use numerical simulations of the formation of galaxy clusters in the $\\Lambda$CDM cosmology to investigate the validity and shape of the temperature relation (\\ref{eq:trl}), and to test the method for determining the velocity anisotropy. In order to check systematic effects we take samples from two different simulations based on two completely independent numerical codes.\n\n\\subsection{CLEF}\nWe first consider a sample of 67 clusters taken from the CLEF simulation \\citep{Kay:2006iz}, details of which are briefly summarized here. The CLEF simulation was run with the GADGET2 $N$-body\/SPH code \\citep{2005MNRAS.364.1105S} and followed the evolution of large-scale structure within a box of comoving length, $200 h^{-1}{\\rm Mpc}$. The following cosmological parameters were assumed: $\\Omega_{\\rm m}=0.3; \\Omega_{\\Lambda}=0.7; \\Omega_{\\rm b}=0.0486; h=0.7; n_{\\rm s}=1; \\sigma_8=0.9$. Here the value of the Hubble constant is written as $100 \\, h\\,\\mathrm{km}\\,\\mathrm{s}^{-1}\\,\\mathrm{Mpc}^{-1}$ and $\\sigma_8$ is the rms mass fluctuation at the present epoch in a sphere of radius $8\\,h^{-1} \\mathrm{Mpc}$. The number of particles was set to $428^3$ for each of the gas and dark matter species, thus determining the particle masses to be $m_{\\rm DM}=7.1\\times 10^{9}h^{-1}{\\rm M}_{\\odot}$ and $m_{\\rm gas}=1.4\\times 10^{9}h^{-1}{\\rm M}_{\\odot}$ respectively. The equivalent Plummer softening length was set to $20\\,h^{-1}\\,$kpc and held fixed at all times in comoving co-ordinates. Pressure forces were calculated using the standard GADGET2 entropy-conserving version of SPH with an artificial viscosity to convert kinetic energy into thermal energy where the flow was convergent. The gas could cool radiatively assuming a fixed metallicity, $Z=0.3Z_{\\odot}$. Cold ($T<10^{5}$K) gas with $n_{\\rm H}>10^{-3}{\\rm cm}^{-3}$ either formed stars or was heated by an entropy, $\\Delta S=1000\\, {\\rm keV \\, cm}^2$. This choice was determined stochastically by selecting a random number, $r$, from the unit interval and heating the particle if $r<0.1$, i.e. a 10 per cent probability of being heated. This high level of feedback was necessary to reproduce the observed excess entropy in clusters (see \\citet{Kay:2006iz} for further details).\n\nTo select the cluster sample, we first consider all clusters at $z=0$ with X-ray temperatures, $kT>2\\, {\\rm keV}$; this produces 95 objects, with virial masses, \n$M_{\\rm vir}>1.3\\times 10^{14}h^{-1}{\\rm M}_{\\odot}$ (correspondingly, $> 15,000$ dark matter particles). We then select those clusters with 3D substructure statistic, $s<0.05$. The substructure statistic \\citep{1998MNRAS.296.1061T} measures the displacement of the centre of mass from the potential minimum of the cluster (taken to be its centre), relative to $r_{500}$, which is the scale radius within which the mean total density is $500$ times the critical density. By making this cut, we therefore exclude all clusters that show significant signs of dynamical activity, i.e. major mergers. \n\n\\subsection{V06}\nThe second sample is a subsample of the one presented in \\citet{2006NewA...12...71V} which we refer to as V06. These simulations assumed a concordance flat $\\Lambda$CDM with the same cosmological parameters as for the CLEF simulation.\n\nThe simulation ensemble of galaxy clusters was constructed according to\na procedure described in \\citep{2008arXiv0808.1111P}. Here we briefly summarize the most important aspects. The hydrodynamic simulations were run using an entropy-conserving multistep TREESPH code for a sample of 153 clusters  spanning a range from $\\simeq 1.5 \\times 10^{15} h^{-1} M_{\\odot}$ down to $M_{vir}\\simeq 1.5 \\times 10^{14} h^{-1} M_{\\odot}$. The initial conditions ($z_{in}=49$) were extracted from a set of purely N-body cosmological simulations in which clusters of galaxies were identified from the particle distribution at $z=0$ using a friends--of--friends algorithm. In order to investigate the effect of the implemented gas processes on the energy equipartition between gas end dark matter particles, we performed both adiabatic and radiative simulations. The radiative simulations are of course more realistic than the adiabatic ones, because they additionally take into account radiative cooling, star formation, energy and metal feedback \\citep{2003MNRAS.339.1117V}. More details concerning the simulation technique and the implementation of physical processes of the gas are given in \\citet{2006NewA...12...71V}.\n\nIn order to avoid contamination from dynamically perturbed clusters, we select the 20 most relaxed objects at $z=0$. The selection is based on the power ratio method, which measures the amount of substructure in X-ray surface brightness maps. The map sources a pseudo-potential which is expanded in plane harmonics, and the ratio of the third coefficient to the zeroth is a measure of substructure. More details are given in \\citet{2008arXiv0808.1111P}.  \n\n\\subsection{The temperature relation}\\label{sec:simtrl}\n\n\\begin{figure}[tbp]\n\\begin{center}\n\\epsscale{1}\n\\plotone{f1}\n\\caption{Radial profile of $\\kappa=T_{\\mathrm{DM}}\/T_{\\mathrm{gas}}$ for the samples of clusters obtained from the CLEF and V06 simulations comprising 67 and 20 clusters, respectively. We plot the median and $1\\sigma$ percentiles taken over each sample. The vertical line indicates the largest radius of the observational data sample, while the vertical lines indicate the mean and standard deviation of the $\\kappa$-profile that we use in the fiducial analysis. Note that, for the CLEF sample, only eight clusters contribute to the innermost bin.}\n\\label{fig:kap}\n\\end{center}\n\\end{figure}\n\n\nWe examine the temperature relation (\\ref{eq:trl}) in the two simulated samples by comparing the gas mass-weighted temperature to the rescaled dark matter velocity dispersion. The resulting $\\kappa$-profiles are shown in fig.~\\ref{fig:kap} and clearly $\\kappa\\approx1$ for both samples. Since we apply somewhat different criteria to select the two simulation samples, it is not surprising to find slightly different profiles. This indicates a systematic uncertainty of $\\pm0.1$ in the simulated $\\kappa$ profiles. The kinetic energy associated with bulk motions of both gas and dark matter particles is at most a few percent of the thermal energy within $2\\,r_\\mathrm{2500}$, outside which bulk motion is not negligible. This is in agreement with what was found in \\citet{2003astro.ph..5250A}.\nDue to the standard problem of limited force resolution, the simulations do not probe the innermost region reliably. Therefore we exclude data inside a cutoff radius ($56\\,h^{-1}\\,$kpc for CLEF, $0.1\\,r_{2500}$ for V06), which means we cannot estimate $\\kappa$ in the central region where gas physics can make a significant impact. \n\nThe adiabatic version of the V06 sample exhibits a larger median $\\kappa$-profile which is constant about $1.2$ within $r_{2500}$ and increases steadily to 1.4 at $r_{200}$. This is comparable with the earlier work of \\citet{2004MNRAS.351..237R}, where the specific energy of dark matter was seen to be larger than that of the gas by 20--30\\% in adiabatic simulations.  \n\n\\subsection{Reconstructing the velocity anisotropy}\n\n\\begin{figure}[htbp]\n\\epsscale{1}\n\\begin{center}\n\\plotone{f2}\n\\caption{Comparison of estimated and true values of the physical quantities involved in determining the velocity anisotropy $\\beta$ in our simulations. Top, the ratio of the reconstructed total density to the true; bottom, the ratio of the reconstructed $\\sigma_r^2$ to the true one. Error bars show the $1\\sigma$ percentiles taken over the sample members.}\n\\label{fig:simsteps}\n\\end{center}\n\\end{figure}\n\n\nIn order to test the method outlined above for determining $\\beta$, we reconstruct the anisotropy profiles observed in the simulated samples. Here, we assume $\\kappa=1$ for all radii even though we expect deviations at small radii. First we derive the integrated mass profile $M(r)$ for each cluster assuming hydrostatic equilibrium (\\ref{eq:he}), and from that the total density profile. The numerical derivatives involved are calculated using three-point quadratic interpolation. The estimated density profile, shown in fig.~\\ref{fig:simsteps} (top), displays a satisfactory agreement with the actual density profile for both the CLEF and V06 samples. The only exception is at the outermost radii above $r_{2500}$ where the density is underestimated. Next, we calculate the radial velocity dispersion (\\ref{eq:srsq}) by interpolating the integrand from $r=0$ using a four-point natural spline interpolation. We compare the resulting radial velocity dispersion with the actual in fig.~\\ref{fig:simsteps} (bottom) which shows that there is good agreement except for the deviation at large radii already seen in the density profiles.\n\n\n\\begin{figure}[tbp]\n\\begin{center}\n\\epsscale{1}\n\\plotone{f3}\n\\caption{Reconstructed velocity anisotropies for the simulated samples. The hatched bands show the actual $\\beta$-profiles of the samples. Error bars show the $1\\sigma$ percentiles taken over the sample members.}\n\\label{fig:simbeta}\n\\end{center}\n\\end{figure}\n\n\nFinally we determine the velocity anisotropy parameter $\\beta$. We find similar results whether we calculate $\\beta_{\\mathrm{Je}}$ or $\\beta_{\\mathrm{tr}}$, however the temperature relation yields less noisy results. The median velocity anisotropy profiles are shown in fig.~\\ref{fig:simbeta} together with the median actual profile. The reconstructed profile tracks the actual anisotropy well in the inner parts but overestimates $\\beta$ in the outer parts. There is also considerable noise in the results. \n\nIn order to understand the origin of the deviations at large radii and the significant scatter in our results, we investigate the systematics of the analysis, as applied to the CLEF sample (similar conditions hold for the V06 sample). First, we substitute the dark matter density estimated from hydrostatic equilibrium with the true density. The $\\beta$-profiles calculated on this basis are shown in the top panels of fig.~\\ref{fig:sys}. The agreement between the estimated and actual $\\beta$ is considerably improved, and the error bars are significantly reduced. This clearly indicates that, in the fiducial analysis, the numerical derivatives necessary to estimate $\\rho_\\mathrm{DM}$ are responsible for the large error bars. Since we do not want to do any parametrized modeling of the gas properties, the numerical derivatives are liable to amplify noise and induce systematic deviations in the outermost bin, where the quantities are only constrained to one side. Additionally, this explains why $\\beta_\\mathrm{Je}$ appears more noisy in the fiducial analysis since an additional derivative must be calculated. The test also shows that there is a deviation from hydrostatic equilibrium at large radii which is part of the reason why $\\beta$ is overestimated. As a second test, we additionally use the true three-dimensional velocity dispersion instead of using the temperature relation. This yields further improvement as to how well the reconstructed $\\beta$ tracks the true one, as shown in the bottom panels of fig.~\\ref{fig:sys}. This implies that it is possible to get the correct scale of the radial velocity dispersion, calculated as an integral from the center, despite the lack of resolved data in the inner radii. We note that, with respect to observational data, the tests we apply here can possibly be utilized in the future, e.g.~with accurate density profiles inferred from gravitational lensing, and with more detailed knowledge of $\\kappa$ from improved simulations. We conclude that the numerical simulations provide proof that our method is robust and that it is indeed possible to infer the $\\beta$-profile despite lacking knowledge of $\\kappa$ in the center.\n\n\n\\begin{figure}[tbp]\n\\begin{center}\n\\epsscale{1}\n\\plotone{f4}\n\\caption{Systematics of the reconstruction of the $\\beta$ profiles for the CLEF simulation. Again, $\\beta$ is recovered both from (\\ref{eq:btr}) (left) and (\\ref{eq:bje}) (right). The true dark matter density is substituted for the estimated, and in the bottom panels we additionally use the true total velocity dispersion instead of estimating it from $T_{\\mathrm{DM}}=\\kappa T_\\mathrm{gas}$.}\\label{fig:sys}\n\\end{center}\n\\end{figure}\n\n\n\\section{Observations}\nNext we apply our analysis to observational data from which the radial gas density and temperature profiles are recovered. This is done strictly using non-parametric methods, i.e.~no modeling of the gas properties is involved. Our data consists of the deprojected density and temperature profiles of two samples of clusters at low and intermediate redshift, respectively. The deprojected profiles were obtained from X-ray data analysis published in earlier work (details below). We consider clusters which appear relaxed and close to spherical, and for which sufficient spectroscopic data are available to analyze several annuli, so that the radial variations of the gas density and temperature are resolved with good statistics. \n\nThe first set of eleven clusters at low redshift is based on X-ray data from {\\it XMM-Newton} of the clusters: A262, A496, A1795, A1837, A2052, A4059, S\\'ersic 159$-$3, MKW3s, MKW9, NGC533, and 2A0335+096. These objects are highly relaxed cool-core (CC) clusters selected as to match the requirements described above. The objects were part of the sample analyzed in \\citet{2004A&A...413..415K} (see this paper for an extensive presentation of the data analysis), in which deprojected radial temperature and density profiles were derived from spatially resolved spectroscopy. We adopt the radial bin selection of \\citet{2005A&A...433..101P} in order to ensure a robust determination of gas temperature and density for the full radial range. Note that data for A2052 and S\\'ersic 159$-$3 were also used in the analysis by \\citet{2007A&A...476L..37H} where a constant velocity anisotropy was assumed.\n\nThe other set of five intermediate redshift X-ray galaxy clusters (RXJ1347.5, A1689, A2218, A1914, A611) is from the {\\it Chandra} sample analyzed in \\citet{Morandi:2007aw}. The radial deprojected temperature and density profiles were retrieved through resolved spectral analysis in a set of annuli, selected to collect at least 2000 net counts, by assuming spherical geometry and by using the definition of `effective volume' (see \\citet{Morandi:2007aw} for further details).\n\n\n\\section{Results}\\label{sec:res}\n\n\\begin{figure*}[tbp]\n\\begin{center}\n\\epsscale{1}\n\\plotone{f5}\n\\caption{Three steps in the calculation of the velocity anisotropy for S\\'ersic~159$-$3. Left, the inferred total density; center, the radial velocity dispersion; right, the $\\beta$-profiles. The gas density and temperature profiles are also shown. The scale radius for this cluster is estimated to be $r_{2500}=337\\pm13\\,$kpc. Error bars indicate the propagated statistical uncertainties on the ICM temperature and density profile, taken as the $1\\sigma$ percentiles of 1000 Monte Carlo samples. This is unlike in the previous figures where the error bars indicate the spread over the numerically simulated samples. In the right panel, the radial positions of $\\beta_\\mathrm{tr}$ and $\\beta_\\mathrm{Je}$ have been offset slightly for clarity.}\n\\label{fig:beAS}\n\\end{center}\n\\end{figure*}\n\n\\begin{figure*}[htbp]\n\\begin{center}\n\\epsscale{1}\n\\plotone{f6}\n\\caption{Median velocity anisotropy profiles for the remaining 15 clusters of our sample. The estimated scale radii are also shown, and the symbols are the same as in fig.~\\ref{fig:beAS}.}\n\\label{fi:bsmpl}\n\\end{center}\n\\end{figure*}\n\n\nWe determine the dark matter velocity anisotropy profile $\\beta(r)$ of each cluster according to the recipe in section \\ref{se:th} using a Monte Carlo method. For each radial bin the deprojected gas temperature and density are sampled assuming Gaussian uncertainties, i.e.~a random number is chosen from a Gaussian distribution with mean equal to the estimated temperature or density and a standard deviation equal to the uncertainty of the estimate. The bins are sampled independently. The parameter $\\kappa$ is also sampled for each bin, assuming a Gaussian distribution with a mean of 1 and a standard deviation of 0.1, which is a reasonable value according to the simulations. The sampled profiles are used to reconstruct the total mass through (\\ref{eq:he}), and then the integrand, the radial velocity dispersion, and the velocity anisotropy are calculated in each bin. The sampled set of profiles is accepted only if the temperature and density as well as the reconstructed dark matter density and radial velocity dispersion are all non-negative in all bins. For each sample, we also estimate the scale radius $r_{2500}$ and the mass $M_{2500}$ contained within that radius. Table \\ref{tb:prop} summarizes the properties of the clusters in our sample.\n\nThe numerical methods for calculating derivatives and integrals are the same as for the simulated samples, i.e.~three-point quadratic interpolation is used for derivatives and four-point spline interpolation is used for the integral in (\\ref{eq:srsq}). The integration results are stable to using two-point linear, three-point quadratic, or four-point least squares quadratic interpolation instead. \n\nIndividual steps of the reconstruction are shown in fig.~\\ref{fig:beAS} for the cluster S\\'ersic~159$-$3, and the deprojected input data are also displayed. We always plot the median and $1\\sigma$ percentiles since spurious outliers in individual Monte Carlo samples can bias the mean and standard deviation significantly. The size of the error bars is mostly determined by the uncertainties of the temperatures, to a lesser degree by the uncertainties of the ICM densities, and it is virtually insensitive to the 10\\% variation assumed for the $\\kappa$-profile.\n\nAs can be seen in the right panel of fig.~\\ref{fig:beAS}, the agreement between $\\beta_\\mathrm{tr}$ and $\\beta_\\mathrm{Je}$ indicates that numerical effects associated with the integration and differentiations are small. On the other hand, $\\beta$ becomes unphysically large in the outermost bins since the reconstructed radial velocity dispersion for some samples becomes greater than the total velocity dispersion. This result is similar to that found in the blind analysis of the simulation samples. As discussed above, this behaviour is mainly due to a deviation from hydrostatic equilibrium of the gas, and to a lesser degree to edge effects making the numerical differentiations less well determined in the outermost bin. It is possible that systematic uncertainties in the input data or radial variations in $\\kappa$ for individual clusters also play a role. In principle, we could impose $\\sigma_r^2<\\sigma^2$, thereby forcing $\\beta<1$, as another physical condition on each Monte Carlo sample, but we prefer not to do so in order to have a consistency check.\n\nWe repeat the data analysis for the remaining 15 clusters of our sample and the resulting velocity anisotropy profiles are shown in fig.~\\ref{fi:bsmpl}. In almost all cases the anisotropy is small in the inner radial bins and increases to between $0.5$ and $1.0$ in the outer parts. There is good agreement between the two derivations of $\\beta$ for all clusters, indicating that numerical issues are under control.\n\nSince the qualitative behaviour of the velocity anisotropy profiles are similar, we combine all our data into a single `stacked' profile, shown in fig.~\\ref{fi:stack}. In the region where direct comparison is possible, the measured stacked  profile is very similar to the reconstructed $\\beta$ profiles for the simulation samples (the green line), and within $r_{2500}$ there is also agreement with the actual velocity anisotropy of the simulation samples (hatched band). The velocity anisotropy is likely overestimated outside $r_{2500}$ for the same reason as for the simulated samples, i.e.~deviation from hydrostatic equilibrium, but the effect appears to be even stronger for the observational data. Interior to the cut-off radius of the numerical simulations, the observations tend to $\\beta\\sim0.3$. This is somewhat surprising since numerical simulations at all mass scales generally have very little anisotropy towards the center of structures. While we cannot exclude the possibility that cluster halos are anisotropic even at low radii, our result can also be explained by the neglected stellar contribution $\\rho_\\star$ to the total mass density. To first order, this contribution enters our analysis in the Jeans equation through the estimated dark matter density $\\widetilde{\\rho}_\\mathrm{DM}=\\rho_\\mathrm{DM}+\\rho_\\star$. In terms of $\\delta_\\star=\\rho_\\star\/\\rho_\\mathrm{DM}$, the Jeans equation becomes\n\\begin{eqnarray}\n\\sigma_r^2\\left(\\frac{d\\ln \\widetilde{\\rho}_{\\mathrm{DM}}}{d\\ln r}+\\frac{d\\ln \\sigma_r^2}{d\\ln r}+2\\beta-\\frac{d\\ln (1+\\delta_\\star)}{d\\ln r}\\right)&&\\nonumber\\\\=-\\frac{GM(r)}{r}&&,\n\\end{eqnarray}\nwhere the slope of $(1+\\delta_\\star)$ is negative since the stellar density must fall off faster than the dark matter density. This means that we overestimate the velocity anisotropy in the central region by not accounting for the stellar mass. Indeed, if we assume that 50\\% of the total mass in the innermost bin is made up of stars, the velocity anisotropy in the two innermost bins becomes consistent with zero. There is also a second order correction through the appearance of $\\widetilde{\\rho}_\\mathrm{DM}$ in (\\ref{eq:srsq}) instead of $\\rho_\\mathrm{DM}$, but this correction must be small since the density contributes to both the integrand and the normalization factor.\n\n\n\\begin{figure}[htbp]\n\\begin{center}\n\\epsscale{1}\n\\plotone{f7}\n\\caption{Median velocity anisotropy profile of all 16 clusters in our dataset. In this case the error bars denote the $1\\sigma$ percentiles of the combined probability density of all clusters within the bin. The actual and reconstructed $\\beta$-profiles from the simulations are also shown. The left vertical line is the innermost radius probed in the CLEF simulations and the right vertical line shows, roughly, the onset of significant deviations from hydrostatic equilibrium in the simulations, see fig.~\\ref{fig:simsteps}.}\n\\label{fi:stack}\n\\end{center}\n\\end{figure}\n\n\n\\begin{deluxetable}{rccc}\n\\tablewidth{0pt}\n\\tablecaption{Properties of our cluster sample\\label{tb:prop}}\n\\tablehead{\\colhead{Cluster}&{$z$}&{$r_{2500}\/$kpc}&{$M_{2500}\/M_\\odot$}}\n\\startdata\nA262 & 0.015 & $256\\pm28$ & $(2.7\\pm0.8)\\times10^{13}$\\\\\nA496 & 0.032 & $398\\pm10$ & $(1.0\\pm0.2)\\times10^{14}$\\\\\nA1795 & 0.064 & $504\\pm22$ & $(1.9\\pm0.2)\\times10^{14}$\\\\\nA1837 & 0.071 & $374\\pm26$ & $(8.0\\pm1.7)\\times10^{13}$\\\\\nA2052 & 0.036 & $362\\pm11$ & $(6.7\\pm0.6)\\times10^{13}$\\\\\nA4059 & 0.047 & $445\\pm21$ & $(1.3\\pm0.2)\\times10^{14}$\\\\\nS\\'ersic~159$-$3 & 0.057 & $337\\pm17$ & $(5.7\\pm0.8)\\times10^{13}$\\\\\nMKW3s & 0.046 & $404\\pm14$ & $(9.5\\pm0.9)\\times10^{13}$\\\\\nMKW9 & 0.040 & $279\\pm44$ & $(3.2\\pm1.5)\\times10^{13}$\\\\\nNGC533 & 0.018 & $191\\pm15$ & $(9.7\\pm2.2)\\times10^{12}$\\\\\n2A0335+096 & 0.034 & $350\\pm40$ & $(6.9\\pm2.5)\\times10^{13}$\\\\ \\hline\nA611 & 0.29 & $519\\pm52$ & $(2.5\\pm0.6)\\times10^{14}$\\\\\nA1689 & 0.18 & $609\\pm4$ & $(3.5\\pm0.7)\\times10^{14}$\\\\\nA1914 & 0.17 & $590\\pm44$ & $(3.3\\pm0.8)\\times10^{14}$\\\\\nA2218 & 0.18 & $535\\pm51$ & $(2.5\\pm0.7)\\times10^{14}$\\\\\nRXJ1347.5-1145 & 0.45 & $710\\pm60$ & $(7.3\\pm1.4)\\times10^{14}$\n\\enddata\n\\end{deluxetable}\n\n\n\\begin{figure*}[tbp]\n\\begin{center}\n\\epsscale{1}\n\\plotone{f8}\n\\caption{The effect of assuming different $\\kappa$-profiles on the stacked velocity anisotropy profile. Top left, the five $\\kappa$-profiles. Others, the resulting sample averaged $\\beta$-profiles calculated assuming the numbered $\\kappa$-profile. In all cases, $\\beta$ is greater than zero in the outer parts. }\n\\label{fi:ksys}\n\\end{center}\n\\end{figure*} \n\n\nFinally, we investigate how the assumed shape of the $\\kappa$-profile affects our results. We try five different profiles as functions of $x=r\/r_{2500}$ with noise added as before, and calculate the velocity anisotropy profiles for each. The $\\kappa$-profiles are chosen so as to mimick either the effects of gas radiative cooling or AGN heating in the central regions, or to check the results if the dark matter is generally hotter or cooler than the gas. The radially varying profiles we try are extreme cases of the simulation profiles, fig.~\\ref{fig:kap}. Typically, the result is that the $\\beta$-profile is shifted in the central regions while the outer regions are largely unaffected, as shown in fig.~\\ref{fi:ksys}. This analysis confirms that there is a significant velocity anisotropy at large radii, independent of the specific assumptions about the  temperature relation.\n\n\n\\section{Summary and discussion}\nIn this paper, we have presented a non-parametric method to infer the velocity anisotropy of dark matter in clusters of galxies from the observable temperature and density of the intracluster medium. We assume that the intracluster medium has the same specific energy as the dark matter, and we investigate the validity of this assumption in two different cosmological simulations of the formation of galaxy clusters. Both confirm the simplest possible form of the relation, namely $T_\\mathrm{DM}\\approx T_\\mathrm{gas}$ in the radial range which is resolved. \n\nWe have tested how well our method can reconstruct the actual velocity anisotropy in the simulated clusters, and we have found good agreement between the two, although the reconstruction is sensitive to systematic biases connected with deviations from hydrostatic equilibrium. \n\nWe have applied our method to the radial ICM density and temperature profiles of 16 galaxy clusters based on {\\it Chandra} and {\\it XMM-Newton} X-ray data. The shape of the velocity anisotropy profiles is always consistent with that seen in simulations, which tends to zero at the innermost radius where the temperature relation is calibrated. It then increases to about 0.5 at $r_{2500}$ and even larger in the outer regions. The same is true of the fiducial analysis applied to simulated data and is likely caused by a deviation from hydrostatic equilibrium outside $r_{2500}$. We also find a significant anisotropy even if we assume radially varying $\\kappa$-profiles, such as can be expected given the strong gas cooling and AGN heating in the core of many clusters, or if we assume $\\kappa\\ne 1$. The agreement between the observed velocity anisotropy and that predicted in numerical simulations shows that we are beginning to understand also the dynamical aspects of dark matter in halos.\n\nIn the innermost radial bins we measure a rather large anisotropy, but this is most likely an overestimation due to the neglect of the stellar mass in the center. This can be used as a means to estimate the stellar mass profile of galaxy clusters if one assumes that the velocity dispersion to be isotropic in the central regions. Similarly, our method may be used as a general test of whether a cluster is relaxed. A reconstructed velocity anisotropy which deviates significantly from the simulated profiles would be a strong hint that the data do not support the assumption of hydrostatic equilibrium.\n\nThe inferred velocity anisotropy profiles are significantly different from zero which means that the collective behaviour of dark matter is unlike that of baryonic particles in gases. This shows that dark matter is effectively collisionless on the timescale of $\\tau\\sim10^9$, the dynamical timescale of galaxy clusters. By taking typical values at $\\sim0.3\\,r_{2500}$ and allowing only a few scatterings within the time $\\tau$, this corresponds to an order--of--magnitude upper limit to the scattering cross-section of roughly $\\sigma\/m=(\\rho_{\\textrm{DM}}\\tau v)^{-1}\\lesssim1\\,$cm$^2$g$^{-1}$. This limit is similar to what has been found for merging clusters \\citep{2004ApJ...606..819M,2008arXiv0806.2320B}, and within an order of magnitude of the scattering cross-section for self-interacting dark matter proposed in \\citet{Spergel:1999mh}.\n\nWe emphasize that improvements to the numerical simulations in the near future will improve our understanding of the $\\kappa$ profile and hopefully track the impact of radiative effects in the center. We also hope that improved understanding of deviations from hydrostatic equilibrium will allow us to estimate how large the suspected bias at large radii is. On the observational side, the main problem at present is the uncertainty in the temperature profile. Improvements can be expected both with regards to the deprojection analysis and the amount of data available. Obviously, there is also the possibility of including a kinematical analysis of the galaxy clusters in our method.\n\n\\acknowledgments\nWe thank Jens Hjorth, Gary A.~Mamon, and Kristian Pedersen for comments. The Dark Cosmology Centre is funded by the Danish National Research Foundation. SE acknowledges the financial contribution from contract ASI-INAF I\/023\/05\/0 and I\/088\/06\/0.\n\n{\\it Facilities:} \\facility{XMM}, \\facility{CXO} \n\n\n\n\\bibliographystyle{apj}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\n\\label{intro}\n\n    %\n   \n    %\n \n   \n    %\n   \n    %\n   \n   \n   \n   \n   \n    %\n   \n    %\n   \n   \n   \n   \n   \nDue to the excellent performance, deep neural networks have been widely used in computer vision and nature language processing tasks. A stack of many layers is the obvious feature for deep neural networks, and how to transform and fuse information across layers efficiently is a key challenge. Neural Machine Translation(NMT), which is a multi-layer end-to-end structure, has achieved state-of-the-art performances in large-scale translation tasks \\cite{bahdanau2014neural,luong2015effective}. More recently, the system based on self-attention \\cite{vaswani2017attention} is rapidly becoming the standard component in NMT, it demonstrates both superior performance and training speed compared to previous architectures using recurrent neural network \\cite{wu2016google}.\n\nFor vanilla Transformer, it only leverages the top layers of encoder and decoder in the subsequent process, which makes it impossible to take advantage of the useful information in other layers. At the same time, in the work of \\cite{shi2016does} show that both local and global syntactic information about source sentences is captured by the encoder, and different types of syntax is stored in different layers. Fusing the outputs of different layers is helpful to improve the model performance, some promising attempts have proven to be of profound value in computer vision tasks \\cite{yu2018deep} and nature language process tasks \\cite{shen2018dense,dou2018exploiting}.\nIn this work, we continue the line of research and go towards a more efficient approach to fuse information across layers. We try to fuse information across layers through residual tree aggregation of layers, it consists of a post-order binary tree and residual connection. To achieve better model performance, we validate many kinds of aggregation formulas as well. Our contributions are threefold:\n\n\\begin{itemize}\n\t\\item Inspired by the approaches of dealing with the output of each layer in numerical analysis \\cite{peters2018deep,he2016deep,huang2017densely}, we propose an approach based on residual tree aggregation of layers(RTAL) to fuse information across layers. This overcomes the problem with the standard NMT network where only uses the top layer of encoder and decoder, and the residual connection in RTAL contributes to enrich each child node of the binary tree.\n\t\\item The proposed approach is efficient and easy to implement, and it also can be used in arbitrary multi-layer deep neural networks.\n\t\\item Our work is among the few studies \\cite{dou2019dynamic,shen2018dense,wang2018multi} which prove that the idea of fusing information across layers can have promising applications on natural language processing tasks.\n\\end{itemize}\nWe evaluate our approach on WMT14 English-to-German(En-De) and WMT17 English-to-France(En-Fr) translation tasks, and we employ the Transformer \\cite{vaswani2017attention}as the strong baseline model. Experimental results show that our approach contributes to make full use of the output of each layer in Transformer across language pairs, it outperforms the Transformer-Big model by 1.19 BLEU points on En-De translation task and 0.35 BLEU points on En-Fr translation task. In addition, our proposed approach is robust, it always outperforms the baseline model when we use different aggregation formulas.\n\n\\section{Related Work}\nTraining neural networks with multiple stacked layers is challenging, making full use of the outputs of different layers can drastically improve the model performance from computer vision tasks to nature language processing tasks. \\cite{he2016deep} propose a residual learning framework, which combine layers and encourage gradient flow by simple shortcut connections. \\cite{yu2018deep} augment standard architectures with deeper aggregation to better fuse information, and designs architectures iteratively and hierarchically merge the feature hierarchy to make networks with better accuracy and fewer parameters.\n\nConcerning nature language processing tasks, \\cite{shen2018dense} propose a densely connected NMT architecture, it not only is able to use features from all previous layers to create new features for both encoder and decoder, but also uses the dense attention structure to improve attention quality. \\cite{dou2019dynamic} propose to use routing-by-agreement strategies to aggregate layers dynamically, the algorithm learns the probability of individual layer representations assigned to aggregated representations in an iterative way. \\cite{he2018layer} propose the concept of layer-wise coordination for NMT, which explicitly coordinates the learning of hidden representations of the encoder and decoder together layer by layer, gradually from low level to high level.\n\nAlthough existing works have achieved remarkable progress in making full use of the output of each layer, there are still some shortcomings. In the work of \\cite{shen2018dense,dou2019dynamic}, they modify the encoder and decoder of the original Transformer by adding extra layers, which increases the size of model and slows down the speed of inference. In the work of \\cite{he2018layer}, it keeps the similar size of the model, but it breaks the origin structure of each layer in Transformer. In the case of ensuring that the original structure of each layer is unchanged, this approach can not be applied to arbitrary multi-layer deep neural networks. In our work, we propose a residual tree aggregation of layers for NMT, which is not suffered from the above issues.\n\n\\section{Background}\n\\subsection{Neural Machine Translation}\nThe encoder-decoder framework has noticeable effect on neural machine translation tasks \\cite{sutskever2014sequence,wu2016google}. Generally, multi-layer encoder and decoder are employed to perform the translation task through a series of nonlinear transformations. The encoder reads the source sentence denoted by $X=(x_{1}, x_{2}, ..., x_{M})$ and maps it to a continuous representation $Z=(z_{1}, z_{2}, ..., z_{M})$. Given $Z$, the decoder generates the target translation $Y=(y_{1}, y_{2}, ..., y_{N})$ conditioned on the sequence of tokens previously generated. The encoder-decoder framework model learns its parameter $\\theta$ by maximizing the log-likelihood $p(Y|X;\\theta)$, which is usually decomposed into the product of the conditional probability of each target word.\n\\begin{equation}\np(Y|X;\\theta) = \\prod_{t=1}^{N}{p(y_{t}|y_{1},...,y_{t-1},X;\\theta)}\n\\end{equation}\nwhere $N$ is the length of target sentence. Both the encoder and decoder can be implemented by different structure of neural models, such as RNN \\cite{cho2014learning}, CNN \\cite{gehring2017convolutional} and self-attention \\cite{vaswani2017attention}.\n\\subsection{Self-attention based network}\nSelf-attention networks \\cite{shaw2018self,so2019evolved} have attracted increasing attention due to their flexibility in parallel computation and dependency modeling. Self-attention networks calculate attention weights between each pair of tokens in a single sequence, and then a position-wise feed-forward network to increase the non-linearity. The self-attention is formulated as:\n\\begin{equation}\nAttention(Q,K,V) = softmax(\\frac{QK^{T}}{\\sqrt{d_{model}}})V\n\\end{equation}\nwhere $Q,K,V$ denotes the query, key and value vectors,respectively, $d_{model}$ is the dimension of hidden representations. The position-wise feed-forward network consists of a two-layer linear transformation with ReLU activation in between:\n\\begin{equation}\nFFN(x) = max(0, xW_{1} + b_{1})W_{2} + b_{2}\n\\end{equation}\nInstead of performing a single attention function, it is beneficial to capture different context features with multiple individual attention functions, namely multi-head attention. Specifically, multi-head attention model first transforms $Q,K$ and $V$ into $H$ subspaces with different linear projections:\n\\begin{equation}\nQ_{h}, K_{h}, V_{h} = QW_{h}^{Q}, KW_{h}^{K}, VW_{h}^{V}\n\\end{equation}\nwhere $Q_{h}, K_{h}, V_{h}$ are the query, key and value representation of the $h$ head,respectively. $W_{h}^{Q},W_{h}^{K},W_{h}^{V}$ denote parameter matrices associated with the $h$ head. $H$ attention functions are applied in parallel to produce the output states $(o_{1}, o_{2}, ..., o_{H})$, then the $H$ output states are concatenated and linearly transformed to produce the final state. The process is formulated as:\n\\begin{equation}\no_{h} = Attention(Q_{h}, K_{h}, V_{h})\n\\end{equation}\n\\begin{equation}\nO = [o_{1}, o_{2}, ..., o_{H}]W^{o}\n\\end{equation}\nwhere $O$ is the final output states, and $W^{o}$ is a trainable matrix.\n\n\\section{Approach}\nIn this section, we first introduce the idea of RTAL, and then we also describe the aggregation formulas used in our experiments.\n\\subsection{Residual Tree Aggregation of Layers}\n\\begin{figure*}[t]\n\t\\centering\n\t\\includegraphics[width=0.7\\textwidth]{transformer}\n\t\\caption{The Transformer-model architecture. The rectangle denotes a layer in the encoder or decoder, the circle denotes the outputs of the last layer in the encoder and decoder. The output of the last layer in the encoder is used as the input of each layer in the decoder, the output of the last layer in the decoder is used to generate the output sequence.}\n\t\\label{fig1}\n\\end{figure*}\n\\begin{figure*}[t]\n\t\\centering\n\t\\includegraphics[width=0.8\\textwidth]{RTAL}\n\t\\caption{The RTAL based on Transformer, where the encoder and decoder are both composed of stack of 4 layers. The RTAL consists of a post-order binary tree and residual connection. The blue line denotes the residual connection.}\n\t\\label{fig2}\n\\end{figure*}\nAs the figure \\ref{fig1} shows, the Transformer only uses the top layer of the encoder and decoder in the subsequent process, which makes it impossible to take advantage of the useful information in other layers. To address this issue, we propose a residual tree aggregation of layers for Transformer(RTAL), it contributes to fuse information across layers. In principle, the RTAL can be applied to any multi-layer neural models, such as Convs2s \\cite{gehring2017convolutional} and Transformer \\cite{vaswani2017attention}. In this work, we directly focus on Transformer considering that it achieves excellent performance on many translation tasks.\n\nRTAL keeps the original structure of Transformer unchanged, it only add extra layers in the encoder and decoder. As the figure \\ref{fig3} shows, RTAL are used to make full use of the output of each layer in encoder and decoder, it consists of a post-order binary tree and residual connection. Specifically, we try to fuse information by constructing a post-order binary tree. In the process of generating child nodes, the residual connection is also used except for the last node. The post-order binary tree implements hierarchical aggregation, and the residual connection provides extra information for generating child nodes. In other computer vision and nature language processing tasks, hierarchical aggregation and residual connection have also proven to be of profound value \\cite{yu2018deep,dou2018exploiting,he2016deep}.\n\nIn Transformer, the encoder and decoder are both composed of stack of 6 layers. RTAL requires the number of layer is $2^{n}$, although RTAL can be applied to the last 4 layers in Transformer and improves the model performance significantly, it also increases the size of model, these will be described in our experiments. Under the constrain of model size, we try to reduce the number of layer in the encoder and decoder and set them to be 4. This approach reduces the size of model effectively, and the loss of model performance is small.\n\n\\subsection{Aggregation formulas}\nIn order to get better model performance, we try many kinds of aggregation formulas in our RTAL. The aggregation formula is used to generate child nodes, the details are shown in figure \\ref{fig3}.\n\\begin{figure}[t]\n\t\\centering\n\t\\includegraphics[width=0.15\\textwidth]{fusion}\n\t\\caption{Generating the child nodes by aggregating the layer outputs, the circles denote layer outputs and child nodes, $AGG()$ denotes the aggregation formula.}\n\t\\label{fig3}\n\\end{figure}\n\n\\subsubsection{Aggregation by averaging}\nA simple way of aggregating all hidden representations is to average them. Given the layer outputs$(h_{i}, h_{i+1})$, the output of average-based aggregation is formulated as:\n\\begin{equation}\nAGG(h_{i}, h_{i+1}) = 0.5 * \\sum(h_{i},h_{i+1})\n\\end{equation}\nThis method does not add extra parameters and simply assigns an equal weight to the layer outputs. It is easy to implement and has little effect on model training speed.\n\\subsubsection{Aggregation by concatenation}\nA more common method is to use concatenation operation and feed-forward neural network(FFN) for fusion. Specifically, we first concatenate the layer outputs$(h_{i}, h_{i+1})$ to form a bigger vector. Then, we feed the bigger vector into the FFN layer, which only has a single hidden layer. Let $d$ be the size of the layer output vector, the formulated representation of this method is defined as:\n\\begin{equation}\nAGG(h_{i}, h_{i+1}) = FFN(concat(h_{i},h_{i+1}))\n\\end{equation}\n\nWhere $h_{i}$ and $AGG(h_{i}, h_{i+1})$ are both $d$ dimensional vectors, the concatenation of $h_{i}$ and $h_{i+1}$ is a $2*d$ dimensional vector. The FFN transforms the $2*d$ dimensional vector to a $d$ dimensional vector, and the non-linear activation functions used in FFN contributes to obtain a better result \\cite{wang2018multi}. The concatenation between layer outputs generates bigger vectors, and it takes more expensive computation in the subsequent process.\n\\subsubsection{Aggregation by element-wise production}\nIn addition to the averaging and concatenation-based methods, we design a simple and efficient aggregation functions as well. Since the concatenation operation will generate a bigger dimensional vector, we propose to use element-wise production to deal with the layer outputs and keep the dimension of the vector unchanged. At the same time, the FFN layer is also used to transform the vector generated by point-wise production. We formulate the process like this:\n\\begin{equation}\nsumb = (h_{i} + h_{i+1}) * \\beta\\label{ewp:a}\n\\end{equation}\n\\begin{equation}\nAGG(h_{i}, h_{i+1}) = FFN(LN(sumb)) + sumb\\label{ewp:b}\n\\end{equation}\n\nWhere $LN()$ denotes the layer normalization \\cite{lei2016layer}, $\\beta$ is a hyper-parameter and is trainable. Dropout \\cite{srivastava2014dropout} and residual connection \\cite{he2016deep} are also employed.\n\n\\section{Experiment}\n\n\\subsection{Setup}\n\\subsubsection{Datasets and Evaluation}\nTo make the results more convincing, we evaluate our approach on WMT14 English-to-German(En-De) and WMT17 English-to-France(En-Fr) translation tasks.\n\nOn the En-De translation task, we use the same data set with \\cite{vaswani2017attention}'s work. The dataset contains about 4.5 million sentence pairs, with 116 million English words and 110 million German works. On the En-Fr translation task, we use the significantly larger dataset consisting of 40 million sentence pairs. All the data has been tokenized and jointly byte pair encoded\\cite{sennrich2015neural} with 32k merge operations using a shared vocabulary. We use newstest2013 for validation set and newstest2014 for test set.\n\nFor evaluation, we average the last 5 checkpoints for the base model and average the last 20 checkpoints for the big model. We use beam search with a beam size of 4 and length penalty $\\alpha=0.6$. We measure case-sensitive tokenized BLEU for En-De and En-Fr translation tasks.\n\\subsubsection{Model and Hyperparameters}\nAll models were trained on the tensor2tensor \\cite{tensor2tensor} with 8 NVIDIA P100 GPUs, where each was allocated with a batch size of 4096 tokens. For our base model, the hidden dimension $d_{x}=512$, the number of attention heads are 8, and the dimension of feed-forward inner-layer is 2048. For our big model, the size of hidden dimension, the number of attention heads and the dimension of feed-forward inner-layer are double. The base model was trained for 100k steps and the big model was trained for 300k steps. All models were optimized by Adam \\cite{kingma2014adam} with $\\beta_{1}=0.9, \\beta_{2}=0.98$ and $\\epsilon=10^{-9}$, the label smoothing $\\epsilon_{ls}=0.1$ was used as regularization. In generally, we followed the configure as suggested in tensor2tensor, we employ the Transformer as the baseline model.\n\\subsection{Results}\n\\subsubsection{Results on the En-De task}\nIn table \\ref{table1}, we first show the results of the En-De translation task, which we compare to the existing systems based on self-attention. Obviously, most of the model variants outperform the Transformer significantly, although they have a larger amount of parameters. At the same time, the big-model usually has better model performance than the base-model, and it also requires more computational cost.\n\nAs for our approach, we first verify the idea of RTAL on the Transformer-Base and Transformer-Big models, where both the encoder and decoder are composed of stack of 6 layers. The results show that they outperform the Transformer baseline significantly and have 1.16 and 1.19 BLEU points higher than baseline models. Compared to the work of \\cite{dou2018exploiting}, the RTAL based on the Transformer-Big model has fewer parameters and higher BLEU points, the RTAL based on the Transformer-Base model has similar BLEU ascent and fewer parameters. Compared to the work of \\cite{he2018layer}, the RTAL based on the Transformer-Base and Transformer-Big models have higher BLEU points.\n\\begin{table*}[t]\n\\begin{center}\n\\begin{tabular}{ccccc}\nSystem & Architecture & Param & BLEU & $\\Delta$ \\\\\n\\hline\n\\cite{wu2016google}& GNMT & - & 26.30 & -\\\\\n\\cite{gehring2017convolutional} & ConvS2S & - & 26.36 & - \\\\\n\\multirow{2}*{\\cite{vaswani2017attention}} & Transformer-Base & 65M & 27.3 & - \\\\\n& Transformer-Big & 213M & 28.4 & - \\\\\n\\multirow{2}*{\\cite{shaw2018self}} & Transformer-Base + Relative Position & - & 26.8 & -0.5 \\\\\n& Transformer-Big + Relative Position & $\\dagger$210M & 29.2 & +0.8 \\\\\n\\multirow{2}*{\\cite{dou2018exploiting}} & Transformer-Base + Deep Representations & 111M & 28.78 & +1.14$\\ddagger$ \\\\\n& Transformer-Big + Deep Representations & 356M & 29.2 & +0.63$\\ddagger$ \\\\\n\\multirow{2}*{\\cite{he2018layer}} & Layer-wise Coordination-Base & - & 28.3 & +1.0 \\\\\n& Layer-wise Coordination-Big & $\\dagger$210M & 29.0 & +0.6 \\\\\n\\hline\n\\multirow{4}*{Our work} & Transformer-Base + RTAL (4L) & 59M & 27.3 & +0.0 \\\\\n& Transformer-Big + RTAL (4L)& 202M & 29.28 & +0.88 \\\\\n& Transformer-Base + RTAL (6L)& 73M & 28.46 & +1.16 \\\\\n& Transformer-Big + RTAL (6L)& 261M & 29.58 & +1.19 \t\n\\end{tabular}\n\\end{center}\n\\caption{\\label{table1} Comparing with existing NMT systems on WMT14 English-to-German translation task. \"(4L)\" denotes the encoder and decoder are both composed of stack of 4 layers. $\\dagger$ denotes an estimate value. $\\ddagger$ denotes that this model has another baseline model. \"Param\" denotes the trainable parameter size of each model(M=million).}\\smallskip\n\\end{table*}\n\nAlthough our approach contributes to get better model performance, it increases the size of model compared to the Transformer baseline model. To reduce the size of model, we also train the RTAL based on the Transformer variants, where both the encoder and decoder are composed of stack of 4 layers. As we can see, the RTAL based on the Transformer-Big variant have similar size of model compared to the Transformer-Big model, it still outperforms the Transformer-Big baseline model by 0.98 BLEU points. At the same time, it also outperforms the Big-model in the work of \\cite{dou2018exploiting,he2018layer} and has fewer parameters.\n\\subsubsection{Results on the En-Fr task}\nSeen from the En-De task, the RTAL based on the Transformer is more effective, where both the encoder and decoder have stack of 6 layers. Therefore, we evaluate our approach with the same structure on the En-Fr task. As seen in table \\ref{table2}, we retrain the Transformer-Base and Transformer-Big models, and our approach outperforms the baseline models by 0.38 and 0.35 BLEU points. It confirms that our approach is a promising strategy for fusing information across layers in Transformer.\n\\begin{table}[t]\n\\begin{center}\n\\begin{tabular}{cccc}\nSystem & Architecture & BLEU & $\\Delta$ \\\\\n\\hline\n\\multirow{4}*{Our work} & T-Base & 41.26 & - \\\\\n& T-Big & 42.58 & - \\\\\n& T-Base + RTAL(6L) & 41.64 & +0.38 \\\\\n& T-Big + RTAL(6L) & 42.93 & +0.35\n\\end{tabular}\n\\caption{\\label{table2} Experimental results on WMT17 English-to-France translation task. \"T-Base\" denotes the Transformer-Base model. \"T-Big\" denotes the Transformer-Big model. \"(6L)\" denotes the encoder and decoder are both composed of stack of 6 layers.}\\smallskip\n\\end{center}\n\\end{table}\n\\subsection{Analysis}\nWe conducted extensive analysis from different perspectives to better understand our model. All results are reported on the En-De translation task with Base-Model parameters, where the encoder and decoder are both composed of stack of 6 layers.\n\\subsubsection{Effect on Encoder and Decoder}\nBoth the encoder and decoder are multiple stacked layers, which may benefit from our proposed approach. In this experiment, we investigate the effects of RTAL on the encoder and decoder.\n\\begin{table}[h]\n\t\\begin{center}\n\t\t\\begin{tabular}{ccc}\n\t\t\tSystem & Aggregation position & BLEU \\\\\n\t\t\t\\hline\n\t\t\t\\multirow{3}*{T-Base + RTAL(6L)} & +encoder & 27.77 \\\\\n\t\t\t& +decoder & 27.82 \\\\\n\t\t\t& +both & 28.46\n\t\t\\end{tabular}\n\t\t\\caption{\\label{table3} Experimental results of applying RTAL to different components on En-De test set. \"+encoder\" denotes that  RTAL is applied to the encoder. \"+both\" denotes that RTAL is applied to the encoder and decoder.}\\smallskip\n\t\\end{center}\n\\end{table}\n\nAs shown in table \\ref{table3}, RTAL used in the encoder or decoder individually consistently outperforms the Transformer baseline model, and it further improves the model performance when using RTAL simultaneously in the encoder and decoder. These results claim that fusing information across layers is useful for both encoding the input sequence and generating output sequence.\n\n\\subsubsection{Effect of aggregation formula}\n\\begin{table}[h]\n\\begin{center}\n\\begin{tabular}{ccc}\n\tSystem & Aggregation formula & BLEU \\\\\n\t\\hline\n\t\\multirow{3}*{T-Base + RTAL(6L)} & Mean  & 27.96 \\\\\n\t& Concat + FFN & 28.23 \\\\\n\t& EWP + FFN & 28.46\t\n\\end{tabular}\n\\end{center}\n\\caption{\\label{table4}Impact of aggregation formulas for RTAL. \"Mean\" denotes aggregation by averaging. \"Concat + FFN\" denotes aggregation by concatenation. \"EWP + FFN\" denotes aggregation by element-wise production.}\\smallskip\n\\end{table}\n\nTo get better model performance, we try many kinds of aggregation formulas. In this experiment, we evaluate the impact of different choices for aggregation formulas. As shown in table \\ref{table4}, the result of averaging layer outputs to fuse information across layers is worse than two other methods, the aggregation by concatenation and aggregation by element-wise production have similar BLEU points. As seen, all the three aggregation formulas efficiently improve the model performance, which shows that our proposed approach is robust.\n\n\\subsubsection{Effect of aggregation structure}\nTo verify the superiority of our proposed approach, we also implement the approaches proposed in the previous works \\cite{dou2018exploiting,yu2018deep}. In this experiment, we retrain those different approaches with the same random seed, we also use the same aggregation formula defined in formula \\ref{ewp:a} and \\ref{ewp:b} for fusing information.\n\nAs shown in table \\ref{table5}, the linear combination of layers in Transformer outperforms the Transformer only a little, and the iterative combination of layers and CNN-Like-Tree structure in Transformer outperform the baseline model by about 0.3 and 0.4 BLEU points. Although these approaches do not outperform the baseline model significantly, they claim that fusing information across layers is a promising way to improve the model performance. At last, our proposed approach outperform the baseline model significantly, which indicates the superiority of our RTAL in fusing information across layers.\n\n\\begin{table}[h]\n\t\\begin{center}\n\t\t\\begin{tabular}{cccc}\n\t\t\tSystem & Architecture & BLEU & $\\Delta$ \\\\\n\t\t\t\\hline\n\t\t\t\\cite{vaswani2017attention} & Transformer-Base & 27.3 & - \\\\\n\t\t\t\\multirow{4}*{Our work} & +Linear Combination & 27.42 & +0.12 \\\\\n\t\t\t& +Iterative Combination & 27.62 & +0.32 \\\\\n\t\t\t& +CNN-Like-Tree & 27.76 & +0.46 \\\\\n\t\t\t& +RTAL (6L) & 28.46 & +1.16\n\t\t\\end{tabular}\n\t\\end{center}\n\t\\caption{\\label{table5} Evaluation of translation performance on WMT14 English-to-German translation task. \"+Linear Combination\" denotes the linear combination of layers. \"+Iterative Combination\" denotes the iterative combination of layers. \"+CNN-Like-Tree\" is a novel structure proposed in the work of \\cite{dou2018exploiting}. \"+RTAL\" denotes residual tree aggregation of layers.}\\smallskip\n\\end{table}\n\n\\section{Conclusion}\nIn this paper, we propose a novel approach to fuse information across layers, which consists of a post-order binary tree and residual connection. To get better model performance, we also try many kinds of aggregation formulas. Our proposed approach is efficient and easy to implement. The experimental results show that our proposed approach outperforms the baseline model significantly.\n\n\\bibliographystyle{coling}\n\n\\section{Introduction}\n\\label{intro}\n\n    %\n   \n    %\n \n   \n    %\n   \n    %\n   \n   \n   \n   \n   \n    %\n   \n    %\n   \n   \n   \n   \n   \nDue to the excellent performance, deep neural networks have been widely used in computer vision and nature language processing tasks. A stack of many layers is the obvious feature for deep neural networks, and how to transform and fuse information across layers efficiently is a key challenge. Neural Machine Translation(NMT), which is a multi-layer end-to-end structure, has achieved state-of-the-art performances in large-scale translation tasks \\cite{bahdanau2014neural,luong2015effective}. More recently, the system based on self-attention \\cite{vaswani2017attention} is rapidly becoming the standard component in NMT, it demonstrates both superior performance and training speed compared to previous architectures using recurrent neural network \\cite{wu2016google}.\n\nFor vanilla Transformer, it only leverages the top layers of encoder and decoder in the subsequent process, which makes it impossible to take advantage of the useful information in other layers. At the same time, in the work of \\cite{shi2016does} show that both local and global syntactic information about source sentences is captured by the encoder, and different types of syntax is stored in different layers. Fusing the outputs of different layers is helpful to improve the model performance, some promising attempts have proven to be of profound value in computer vision tasks \\cite{yu2018deep} and nature language process tasks \\cite{shen2018dense,dou2018exploiting}.\nIn this work, we continue the line of research and go towards a more efficient approach to fuse information across layers. We try to fuse information across layers through residual tree aggregation of layers, it consists of a post-order binary tree and residual connection. To achieve better model performance, we validate many kinds of aggregation formulas as well. Our contributions are threefold:\n\n\\begin{itemize}\n\t\\item Inspired by the approaches of dealing with the output of each layer in numerical analysis \\cite{peters2018deep,he2016deep,huang2017densely}, we propose an approach based on residual tree aggregation of layers(RTAL) to fuse information across layers. This overcomes the problem with the standard NMT network where only uses the top layer of encoder and decoder, and the residual connection in RTAL contributes to enrich each child node of the binary tree.\n\t\\item The proposed approach is efficient and easy to implement, and it also can be used in arbitrary multi-layer deep neural networks.\n\t\\item Our work is among the few studies \\cite{dou2019dynamic,shen2018dense,wang2018multi} which prove that the idea of fusing information across layers can have promising applications on natural language processing tasks.\n\\end{itemize}\nWe evaluate our approach on WMT14 English-to-German(En-De) and WMT17 English-to-France(En-Fr) translation tasks, and we employ the Transformer \\cite{vaswani2017attention}as the strong baseline model. Experimental results show that our approach contributes to make full use of the output of each layer in Transformer across language pairs, it outperforms the Transformer-Big model by 1.19 BLEU points on En-De translation task and 0.35 BLEU points on En-Fr translation task. In addition, our proposed approach is robust, it always outperforms the baseline model when we use different aggregation formulas.\n\n\\section{Related Work}\nTraining neural networks with multiple stacked layers is challenging, making full use of the outputs of different layers can drastically improve the model performance from computer vision tasks to nature language processing tasks. \\cite{he2016deep} propose a residual learning framework, which combine layers and encourage gradient flow by simple shortcut connections. \\cite{yu2018deep} augment standard architectures with deeper aggregation to better fuse information, and designs architectures iteratively and hierarchically merge the feature hierarchy to make networks with better accuracy and fewer parameters.\n\nConcerning nature language processing tasks, \\cite{shen2018dense} propose a densely connected NMT architecture, it not only is able to use features from all previous layers to create new features for both encoder and decoder, but also uses the dense attention structure to improve attention quality. \\cite{dou2019dynamic} propose to use routing-by-agreement strategies to aggregate layers dynamically, the algorithm learns the probability of individual layer representations assigned to aggregated representations in an iterative way. \\cite{he2018layer} propose the concept of layer-wise coordination for NMT, which explicitly coordinates the learning of hidden representations of the encoder and decoder together layer by layer, gradually from low level to high level.\n\nAlthough existing works have achieved remarkable progress in making full use of the output of each layer, there are still some shortcomings. In the work of \\cite{shen2018dense,dou2019dynamic}, they modify the encoder and decoder of the original Transformer by adding extra layers, which increases the size of model and slows down the speed of inference. In the work of \\cite{he2018layer}, it keeps the similar size of the model, but it breaks the origin structure of each layer in Transformer. In the case of ensuring that the original structure of each layer is unchanged, this approach can not be applied to arbitrary multi-layer deep neural networks. In our work, we propose a residual tree aggregation of layers for NMT, which is not suffered from the above issues.\n\n\\section{Background}\n\\subsection{Neural Machine Translation}\nThe encoder-decoder framework has noticeable effect on neural machine translation tasks \\cite{sutskever2014sequence,wu2016google}. Generally, multi-layer encoder and decoder are employed to perform the translation task through a series of nonlinear transformations. The encoder reads the source sentence denoted by $X=(x_{1}, x_{2}, ..., x_{M})$ and maps it to a continuous representation $Z=(z_{1}, z_{2}, ..., z_{M})$. Given $Z$, the decoder generates the target translation $Y=(y_{1}, y_{2}, ..., y_{N})$ conditioned on the sequence of tokens previously generated. The encoder-decoder framework model learns its parameter $\\theta$ by maximizing the log-likelihood $p(Y|X;\\theta)$, which is usually decomposed into the product of the conditional probability of each target word.\n\\begin{equation}\np(Y|X;\\theta) = \\prod_{t=1}^{N}{p(y_{t}|y_{1},...,y_{t-1},X;\\theta)}\n\\end{equation}\nwhere $N$ is the length of target sentence. Both the encoder and decoder can be implemented by different structure of neural models, such as RNN \\cite{cho2014learning}, CNN \\cite{gehring2017convolutional} and self-attention \\cite{vaswani2017attention}.\n\\subsection{Self-attention based network}\nSelf-attention networks \\cite{shaw2018self,so2019evolved} have attracted increasing attention due to their flexibility in parallel computation and dependency modeling. Self-attention networks calculate attention weights between each pair of tokens in a single sequence, and then a position-wise feed-forward network to increase the non-linearity. The self-attention is formulated as:\n\\begin{equation}\nAttention(Q,K,V) = softmax(\\frac{QK^{T}}{\\sqrt{d_{model}}})V\n\\end{equation}\nwhere $Q,K,V$ denotes the query, key and value vectors,respectively, $d_{model}$ is the dimension of hidden representations. The position-wise feed-forward network consists of a two-layer linear transformation with ReLU activation in between:\n\\begin{equation}\nFFN(x) = max(0, xW_{1} + b_{1})W_{2} + b_{2}\n\\end{equation}\nInstead of performing a single attention function, it is beneficial to capture different context features with multiple individual attention functions, namely multi-head attention. Specifically, multi-head attention model first transforms $Q,K$ and $V$ into $H$ subspaces with different linear projections:\n\\begin{equation}\nQ_{h}, K_{h}, V_{h} = QW_{h}^{Q}, KW_{h}^{K}, VW_{h}^{V}\n\\end{equation}\nwhere $Q_{h}, K_{h}, V_{h}$ are the query, key and value representation of the $h$ head,respectively. $W_{h}^{Q},W_{h}^{K},W_{h}^{V}$ denote parameter matrices associated with the $h$ head. $H$ attention functions are applied in parallel to produce the output states $(o_{1}, o_{2}, ..., o_{H})$, then the $H$ output states are concatenated and linearly transformed to produce the final state. The process is formulated as:\n\\begin{equation}\no_{h} = Attention(Q_{h}, K_{h}, V_{h})\n\\end{equation}\n\\begin{equation}\nO = [o_{1}, o_{2}, ..., o_{H}]W^{o}\n\\end{equation}\nwhere $O$ is the final output states, and $W^{o}$ is a trainable matrix.\n\n\\section{Approach}\nIn this section, we first introduce the idea of RTAL, and then we also describe the aggregation formulas used in our experiments.\n\\subsection{Residual Tree Aggregation of Layers}\n\\begin{figure*}[t]\n\t\\centering\n\t\\includegraphics[width=0.7\\textwidth]{transformer}\n\t\\caption{The Transformer-model architecture. The rectangle denotes a layer in the encoder or decoder, the circle denotes the outputs of the last layer in the encoder and decoder. The output of the last layer in the encoder is used as the input of each layer in the decoder, the output of the last layer in the decoder is used to generate the output sequence.}\n\t\\label{fig1}\n\\end{figure*}\n\\begin{figure*}[t]\n\t\\centering\n\t\\includegraphics[width=0.8\\textwidth]{RTAL}\n\t\\caption{The RTAL based on Transformer, where the encoder and decoder are both composed of stack of 4 layers. The RTAL consists of a post-order binary tree and residual connection. The blue line denotes the residual connection.}\n\t\\label{fig2}\n\\end{figure*}\nAs the figure \\ref{fig1} shows, the Transformer only uses the top layer of the encoder and decoder in the subsequent process, which makes it impossible to take advantage of the useful information in other layers. To address this issue, we propose a residual tree aggregation of layers for Transformer(RTAL), it contributes to fuse information across layers. In principle, the RTAL can be applied to any multi-layer neural models, such as Convs2s \\cite{gehring2017convolutional} and Transformer \\cite{vaswani2017attention}. In this work, we directly focus on Transformer considering that it achieves excellent performance on many translation tasks.\n\nRTAL keeps the original structure of Transformer unchanged, it only add extra layers in the encoder and decoder. As the figure \\ref{fig3} shows, RTAL are used to make full use of the output of each layer in encoder and decoder, it consists of a post-order binary tree and residual connection. Specifically, we try to fuse information by constructing a post-order binary tree. In the process of generating child nodes, the residual connection is also used except for the last node. The post-order binary tree implements hierarchical aggregation, and the residual connection provides extra information for generating child nodes. In other computer vision and nature language processing tasks, hierarchical aggregation and residual connection have also proven to be of profound value \\cite{yu2018deep,dou2018exploiting,he2016deep}.\n\nIn Transformer, the encoder and decoder are both composed of stack of 6 layers. RTAL requires the number of layer is $2^{n}$, although RTAL can be applied to the last 4 layers in Transformer and improves the model performance significantly, it also increases the size of model, these will be described in our experiments. Under the constrain of model size, we try to reduce the number of layer in the encoder and decoder and set them to be 4. This approach reduces the size of model effectively, and the loss of model performance is small.\n\n\\subsection{Aggregation formulas}\nIn order to get better model performance, we try many kinds of aggregation formulas in our RTAL. The aggregation formula is used to generate child nodes, the details are shown in figure \\ref{fig3}.\n\\begin{figure}[t]\n\t\\centering\n\t\\includegraphics[width=0.15\\textwidth]{fusion}\n\t\\caption{Generating the child nodes by aggregating the layer outputs, the circles denote layer outputs and child nodes, $AGG()$ denotes the aggregation formula.}\n\t\\label{fig3}\n\\end{figure}\n\n\\subsubsection{Aggregation by averaging}\nA simple way of aggregating all hidden representations is to average them. Given the layer outputs$(h_{i}, h_{i+1})$, the output of average-based aggregation is formulated as:\n\\begin{equation}\nAGG(h_{i}, h_{i+1}) = 0.5 * \\sum(h_{i},h_{i+1})\n\\end{equation}\nThis method does not add extra parameters and simply assigns an equal weight to the layer outputs. It is easy to implement and has little effect on model training speed.\n\\subsubsection{Aggregation by concatenation}\nA more common method is to use concatenation operation and feed-forward neural network(FFN) for fusion. Specifically, we first concatenate the layer outputs$(h_{i}, h_{i+1})$ to form a bigger vector. Then, we feed the bigger vector into the FFN layer, which only has a single hidden layer. Let $d$ be the size of the layer output vector, the formulated representation of this method is defined as:\n\\begin{equation}\nAGG(h_{i}, h_{i+1}) = FFN(concat(h_{i},h_{i+1}))\n\\end{equation}\n\nWhere $h_{i}$ and $AGG(h_{i}, h_{i+1})$ are both $d$ dimensional vectors, the concatenation of $h_{i}$ and $h_{i+1}$ is a $2*d$ dimensional vector. The FFN transforms the $2*d$ dimensional vector to a $d$ dimensional vector, and the non-linear activation functions used in FFN contributes to obtain a better result \\cite{wang2018multi}. The concatenation between layer outputs generates bigger vectors, and it takes more expensive computation in the subsequent process.\n\\subsubsection{Aggregation by element-wise production}\nIn addition to the averaging and concatenation-based methods, we design a simple and efficient aggregation functions as well. Since the concatenation operation will generate a bigger dimensional vector, we propose to use element-wise production to deal with the layer outputs and keep the dimension of the vector unchanged. At the same time, the FFN layer is also used to transform the vector generated by point-wise production. We formulate the process like this:\n\\begin{equation}\nsumb = (h_{i} + h_{i+1}) * \\beta\\label{ewp:a}\n\\end{equation}\n\\begin{equation}\nAGG(h_{i}, h_{i+1}) = FFN(LN(sumb)) + sumb\\label{ewp:b}\n\\end{equation}\n\nWhere $LN()$ denotes the layer normalization \\cite{lei2016layer}, $\\beta$ is a hyper-parameter and is trainable. Dropout \\cite{srivastava2014dropout} and residual connection \\cite{he2016deep} are also employed.\n\n\\section{Experiment}\n\n\\subsection{Setup}\n\\subsubsection{Datasets and Evaluation}\nTo make the results more convincing, we evaluate our approach on WMT14 English-to-German(En-De) and WMT17 English-to-France(En-Fr) translation tasks.\n\nOn the En-De translation task, we use the same data set with \\cite{vaswani2017attention}'s work. The dataset contains about 4.5 million sentence pairs, with 116 million English words and 110 million German works. On the En-Fr translation task, we use the significantly larger dataset consisting of 40 million sentence pairs. All the data has been tokenized and jointly byte pair encoded\\cite{sennrich2015neural} with 32k merge operations using a shared vocabulary. We use newstest2013 for validation set and newstest2014 for test set.\n\nFor evaluation, we average the last 5 checkpoints for the base model and average the last 20 checkpoints for the big model. We use beam search with a beam size of 4 and length penalty $\\alpha=0.6$. We measure case-sensitive tokenized BLEU for En-De and En-Fr translation tasks.\n\\subsubsection{Model and Hyperparameters}\nAll models were trained on the tensor2tensor \\cite{tensor2tensor} with 8 NVIDIA P100 GPUs, where each was allocated with a batch size of 4096 tokens. For our base model, the hidden dimension $d_{x}=512$, the number of attention heads are 8, and the dimension of feed-forward inner-layer is 2048. For our big model, the size of hidden dimension, the number of attention heads and the dimension of feed-forward inner-layer are double. The base model was trained for 100k steps and the big model was trained for 300k steps. All models were optimized by Adam \\cite{kingma2014adam} with $\\beta_{1}=0.9, \\beta_{2}=0.98$ and $\\epsilon=10^{-9}$, the label smoothing $\\epsilon_{ls}=0.1$ was used as regularization. In generally, we followed the configure as suggested in tensor2tensor, we employ the Transformer as the baseline model.\n\\subsection{Results}\n\\subsubsection{Results on the En-De task}\nIn table \\ref{table1}, we first show the results of the En-De translation task, which we compare to the existing systems based on self-attention. Obviously, most of the model variants outperform the Transformer significantly, although they have a larger amount of parameters. At the same time, the big-model usually has better model performance than the base-model, and it also requires more computational cost.\n\nAs for our approach, we first verify the idea of RTAL on the Transformer-Base and Transformer-Big models, where both the encoder and decoder are composed of stack of 6 layers. The results show that they outperform the Transformer baseline significantly and have 1.16 and 1.19 BLEU points higher than baseline models. Compared to the work of \\cite{dou2018exploiting}, the RTAL based on the Transformer-Big model has fewer parameters and higher BLEU points, the RTAL based on the Transformer-Base model has similar BLEU ascent and fewer parameters. Compared to the work of \\cite{he2018layer}, the RTAL based on the Transformer-Base and Transformer-Big models have higher BLEU points.\n\\begin{table*}[t]\n\\begin{center}\n\\begin{tabular}{ccccc}\nSystem & Architecture & Param & BLEU & $\\Delta$ \\\\\n\\hline\n\\cite{wu2016google}& GNMT & - & 26.30 & -\\\\\n\\cite{gehring2017convolutional} & ConvS2S & - & 26.36 & - \\\\\n\\multirow{2}*{\\cite{vaswani2017attention}} & Transformer-Base & 65M & 27.3 & - \\\\\n& Transformer-Big & 213M & 28.4 & - \\\\\n\\multirow{2}*{\\cite{shaw2018self}} & Transformer-Base + Relative Position & - & 26.8 & -0.5 \\\\\n& Transformer-Big + Relative Position & $\\dagger$210M & 29.2 & +0.8 \\\\\n\\multirow{2}*{\\cite{dou2018exploiting}} & Transformer-Base + Deep Representations & 111M & 28.78 & +1.14$\\ddagger$ \\\\\n& Transformer-Big + Deep Representations & 356M & 29.2 & +0.63$\\ddagger$ \\\\\n\\multirow{2}*{\\cite{he2018layer}} & Layer-wise Coordination-Base & - & 28.3 & +1.0 \\\\\n& Layer-wise Coordination-Big & $\\dagger$210M & 29.0 & +0.6 \\\\\n\\hline\n\\multirow{4}*{Our work} & Transformer-Base + RTAL (4L) & 59M & 27.3 & +0.0 \\\\\n& Transformer-Big + RTAL (4L)& 202M & 29.28 & +0.88 \\\\\n& Transformer-Base + RTAL (6L)& 73M & 28.46 & +1.16 \\\\\n& Transformer-Big + RTAL (6L)& 261M & 29.58 & +1.19 \t\n\\end{tabular}\n\\end{center}\n\\caption{\\label{table1} Comparing with existing NMT systems on WMT14 English-to-German translation task. \"(4L)\" denotes the encoder and decoder are both composed of stack of 4 layers. $\\dagger$ denotes an estimate value. $\\ddagger$ denotes that this model has another baseline model. \"Param\" denotes the trainable parameter size of each model(M=million).}\\smallskip\n\\end{table*}\n\nAlthough our approach contributes to get better model performance, it increases the size of model compared to the Transformer baseline model. To reduce the size of model, we also train the RTAL based on the Transformer variants, where both the encoder and decoder are composed of stack of 4 layers. As we can see, the RTAL based on the Transformer-Big variant have similar size of model compared to the Transformer-Big model, it still outperforms the Transformer-Big baseline model by 0.98 BLEU points. At the same time, it also outperforms the Big-model in the work of \\cite{dou2018exploiting,he2018layer} and has fewer parameters.\n\\subsubsection{Results on the En-Fr task}\nSeen from the En-De task, the RTAL based on the Transformer is more effective, where both the encoder and decoder have stack of 6 layers. Therefore, we evaluate our approach with the same structure on the En-Fr task. As seen in table \\ref{table2}, we retrain the Transformer-Base and Transformer-Big models, and our approach outperforms the baseline models by 0.38 and 0.35 BLEU points. It confirms that our approach is a promising strategy for fusing information across layers in Transformer.\n\\begin{table}[t]\n\\begin{center}\n\\begin{tabular}{cccc}\nSystem & Architecture & BLEU & $\\Delta$ \\\\\n\\hline\n\\multirow{4}*{Our work} & T-Base & 41.26 & - \\\\\n& T-Big & 42.58 & - \\\\\n& T-Base + RTAL(6L) & 41.64 & +0.38 \\\\\n& T-Big + RTAL(6L) & 42.93 & +0.35\n\\end{tabular}\n\\caption{\\label{table2} Experimental results on WMT17 English-to-France translation task. \"T-Base\" denotes the Transformer-Base model. \"T-Big\" denotes the Transformer-Big model. \"(6L)\" denotes the encoder and decoder are both composed of stack of 6 layers.}\\smallskip\n\\end{center}\n\\end{table}\n\\subsection{Analysis}\nWe conducted extensive analysis from different perspectives to better understand our model. All results are reported on the En-De translation task with Base-Model parameters, where the encoder and decoder are both composed of stack of 6 layers.\n\\subsubsection{Effect on Encoder and Decoder}\nBoth the encoder and decoder are multiple stacked layers, which may benefit from our proposed approach. In this experiment, we investigate the effects of RTAL on the encoder and decoder.\n\\begin{table}[h]\n\t\\begin{center}\n\t\t\\begin{tabular}{ccc}\n\t\t\tSystem & Aggregation position & BLEU \\\\\n\t\t\t\\hline\n\t\t\t\\multirow{3}*{T-Base + RTAL(6L)} & +encoder & 27.77 \\\\\n\t\t\t& +decoder & 27.82 \\\\\n\t\t\t& +both & 28.46\n\t\t\\end{tabular}\n\t\t\\caption{\\label{table3} Experimental results of applying RTAL to different components on En-De test set. \"+encoder\" denotes that  RTAL is applied to the encoder. \"+both\" denotes that RTAL is applied to the encoder and decoder.}\\smallskip\n\t\\end{center}\n\\end{table}\n\nAs shown in table \\ref{table3}, RTAL used in the encoder or decoder individually consistently outperforms the Transformer baseline model, and it further improves the model performance when using RTAL simultaneously in the encoder and decoder. These results claim that fusing information across layers is useful for both encoding the input sequence and generating output sequence.\n\n\\subsubsection{Effect of aggregation formula}\n\\begin{table}[h]\n\\begin{center}\n\\begin{tabular}{ccc}\n\tSystem & Aggregation formula & BLEU \\\\\n\t\\hline\n\t\\multirow{3}*{T-Base + RTAL(6L)} & Mean  & 27.96 \\\\\n\t& Concat + FFN & 28.23 \\\\\n\t& EWP + FFN & 28.46\t\n\\end{tabular}\n\\end{center}\n\\caption{\\label{table4}Impact of aggregation formulas for RTAL. \"Mean\" denotes aggregation by averaging. \"Concat + FFN\" denotes aggregation by concatenation. \"EWP + FFN\" denotes aggregation by element-wise production.}\\smallskip\n\\end{table}\n\nTo get better model performance, we try many kinds of aggregation formulas. In this experiment, we evaluate the impact of different choices for aggregation formulas. As shown in table \\ref{table4}, the result of averaging layer outputs to fuse information across layers is worse than two other methods, the aggregation by concatenation and aggregation by element-wise production have similar BLEU points. As seen, all the three aggregation formulas efficiently improve the model performance, which shows that our proposed approach is robust.\n\n\\subsubsection{Effect of aggregation structure}\nTo verify the superiority of our proposed approach, we also implement the approaches proposed in the previous works \\cite{dou2018exploiting,yu2018deep}. In this experiment, we retrain those different approaches with the same random seed, we also use the same aggregation formula defined in formula \\ref{ewp:a} and \\ref{ewp:b} for fusing information.\n\nAs shown in table \\ref{table5}, the linear combination of layers in Transformer outperforms the Transformer only a little, and the iterative combination of layers and CNN-Like-Tree structure in Transformer outperform the baseline model by about 0.3 and 0.4 BLEU points. Although these approaches do not outperform the baseline model significantly, they claim that fusing information across layers is a promising way to improve the model performance. At last, our proposed approach outperform the baseline model significantly, which indicates the superiority of our RTAL in fusing information across layers.\n\n\\begin{table}[h]\n\t\\begin{center}\n\t\t\\begin{tabular}{cccc}\n\t\t\tSystem & Architecture & BLEU & $\\Delta$ \\\\\n\t\t\t\\hline\n\t\t\t\\cite{vaswani2017attention} & Transformer-Base & 27.3 & - \\\\\n\t\t\t\\multirow{4}*{Our work} & +Linear Combination & 27.42 & +0.12 \\\\\n\t\t\t& +Iterative Combination & 27.62 & +0.32 \\\\\n\t\t\t& +CNN-Like-Tree & 27.76 & +0.46 \\\\\n\t\t\t& +RTAL (6L) & 28.46 & +1.16\n\t\t\\end{tabular}\n\t\\end{center}\n\t\\caption{\\label{table5} Evaluation of translation performance on WMT14 English-to-German translation task. \"+Linear Combination\" denotes the linear combination of layers. \"+Iterative Combination\" denotes the iterative combination of layers. \"+CNN-Like-Tree\" is a novel structure proposed in the work of \\cite{dou2018exploiting}. \"+RTAL\" denotes residual tree aggregation of layers.}\\smallskip\n\\end{table}\n\n\\section{Conclusion}\nIn this paper, we propose a novel approach to fuse information across layers, which consists of a post-order binary tree and residual connection. To get better model performance, we also try many kinds of aggregation formulas. Our proposed approach is efficient and easy to implement. The experimental results show that our proposed approach outperforms the baseline model significantly.\n\n\\bibliographystyle{coling}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n Let $n\\ge 3$ and $(U, g)$ be an $n$-dimensional Riemannian manifold and let $u$ be a scalar-valued function on $U$, not identically zero. We say that $(g, u)$ is a \\emph{static vacuum pair}   in $U$ if it solves the static vacuum system:\n\\begin{align}\\label{sv}\n\t\\left\\{ \\begin{array}{l} u\\Ric_g -\\nabla^2_g u=0\\\\\n\t\\Delta_g u=0\\end{array}\\right. \\quad \\mbox{ in } U.\n\\end{align}\nSuch  $g$ is also referred to as a  \\emph{static vacuum metric} with $u$ as a \\emph{static potential}. The study of a static vacuum pair is of fundamental importance in general relativity. It defines a spacetime  $- u^2 dt^2 + g$ on $\\mathbb{R}\\times \\{ x\\in U: u(x)>0\\}$ that solves the vacuum Einstein equation. A spacetime that takes this form is called \\emph{static}, meaning that $\\partial_t$ is a global timelike Killing vector field everywhere normal to the $t$-slices.  In a similar way, a static vacuum pair is  related to the study of Ricci curvature as $(g, u)$  defines a Ricci flat Riemannian metric $u^2 dt^2 + g$ that carries a global Killing vector field.  \n\n\nThe static vacuum metric also arises in the minimization problems of the ADM mass.  By Riemannian positive mass theorem of R.~Schoen and S.-T.~Yau~\\cite{Schoen-Yau:1979-pmt1} and Riemannian Penrose inequality of G.~Huisken and T. Ilmanen~\\cite{Huisken-Ilmanen:2001} and of H.~Bray \\cite{Bray:2001},  the ADM mass minimizing metric among an appropriately defined class of asymptotically flat metrics is  characterized by a class of static metrics, the Schwarzschild metrics. In a fundamentally related way, a static vacuum metric  appears in the study of scalar curvature deformation and gluing, see, e.g. the work of J.~Corvino~\\cite{Corvino:2000}.\n\n\n\n\n\nIn some situations, the asymptotically flat, static vacuum metrics are quite rigid. The \\emph{uniqueness of static black holes} \\cite{Israel:1968, Robinson:1977, Bunting-Masood-ul-Alam:1987} says that the Schwarzschild family of positive mass is the only asymptotically flat, static vacuum metrics that admit a static potential $u>0$ in the interior of manifold with $u=0$ on the boundary. On the other hand, the following conjecture of Bartnik asserts the existence and uniqueness of static vacuum metrics realizing arbitrarily prescribed geometric boundary data~\\cite[Conjecture 7]{Bartnik:2002}.\n\n\\begin{conjecture}[Bartnik's static extension conjecture]\\label{conjecture}\nLet $(\\Omega_0, g_0)$ be an $n$-dimensional, connected, compact Riemannian manifold with scalar curvature $R_{g_0}\\ge 0$ and with boundary $\\Sigma$. Denote by $g_0^\\intercal $ and  $H_{g_0}$ the induced metric and mean curvature on $\\Sigma$ respectively. Suppose $H_{g_0}$ is positive somewhere. Then there exists a unique  asymptotically flat, static vacuum manifold $(M, g)$ with boundary $\\partial M$ diffeomorphic to $\\Sigma$ such that  $(g^\\intercal, H_g) =  (g_0^\\intercal, H_{g_0})$, where  $g^\\intercal $ and  $H_{g}$ are respectively the induced metric and mean curvature on $\\partial M \\subset (M, g)$. \n\\end{conjecture}\n\n In the above, the mean curvatures $H_{g_0} = \\Div_\\Sigma \\nu_{g_0}$ and $H_g = \\Div_\\Sigma \\nu_g$, where  $\\nu_{g_0}$ is the outward unit normal to $\\Omega_0$ and  $\\nu_g$ is the inward unit normal  to $M$. Throughout the paper we shall refer to the geometric boundary data $(g^\\intercal, H_g)$ as the \\emph{Bartnik boundary data}, and $(M, g)$ as a static vacuum extension for the Bartnik boundary data.\n \n We remark on the assumption that  $H_{g_0}$ is  positive somewhere. Without this assumption, an asymptotically flat, static vacuum extension, if exists, would contain a closed minimal hypersurface, which violates the ``no-horizon'' condition. See Definition 4 and p. 236 in \\cite{Bartnik:2002}.  In fact, if $n=3$ and $H_{g_0}=0$, then a static vacuum extension must be a Schwarzschild metric, ~see Bray~\\cite{Bray:2001}, P.~Miao~\\cite{Miao:2005}, A.~Carlotto, O.~Chodosh, M.~Eichmair~\\cite{Carlotto-Chodosh-Eichmair:2016},  also \\cite{Mantoulidis-Schoen:2015, Huang-Martin-Miao:2018}. Let us also remark that it is not known whether the scalar curvature assumption~$R_{g_0}\\ge 0$ is needed. \n\n \n We explain uniqueness in Conjecture~\\ref{conjecture}.  If $(M, g)$ is an asymptotically flat, static vacuum  manifold, then for any diffeomorphism $\\psi:M \\to M$ that fixes the boundary and is asymptotic to a Euclidean isometry at infinity, the pull-back metric $\\psi^*g$ is also an asymptotically flat, static vacuum metric in $M$ having the same Bartnik boundary data. So the uniqueness assertion in the above conjecture should be interpreted as ``unique up to diffeomorphisms''.  \n \nConjecture~\\ref{conjecture}  originated from other conjectures of Bartnik about the quasi-local mass in general relativity, see, for example, \\cite{Bartnik:1989, Bartnik:1997, Bartnik:2002}. Nevertheless, the conjecture itself is  of independent interest.  In contrast to the existing classification results, an affirmative answer toward the conjecture will add more examples of asymptotically flat, static vacuum pairs to the current (quite limited) list (see, e.g. \\cite[Chapters 18.6 and 20.2]{Stephani:2003}).\n\n\n\n\nThere are partial progresses toward Conjecture~\\ref{conjecture}.  P. Miao~\\cite{Miao:2003} confirmed the existence for $(g_0^\\intercal, H_{g_0})$  that is sufficiently close to $(g_{S^2}, 2)$ and invariant under reflectional symmetry, where $(g_{S^2}, 2)$ is the Bartnik boundary data on a standard unit sphere, and obtained  static vacuum extensions with reflectional symmetry.  M.~Anderson and M.~Khuri formulated the problem into an elliptic boundary value problem of Fredholm index zero~\\cite{Anderson-Khuri:2013}, and  Anderson~ \\cite{Anderson:2015} confirmed the existence and local uniqueness for $(g_0^\\intercal, H_{g_0})$ sufficiently close to $(g_{S^2}, 2)$. We also note some numerical results for axial-symmetrical cases by C. Cederbaum, O. Rinne, and M. Strehlau~\\cite{Cederbaum-Rinne-Strehlau:2019}.\n\n\nIn this paper, we confirm the conjecture in that, for  larger classes of Bartnik boundary data, static vacuum extensions exist and are locally unique.  Before we state our main results, we give some definitions. Let  $\\Omega$ be a bounded open subset in $\\mathbb{R}^n$ whose boundary $\\Sigma = \\partial \\overline{\\Omega}$ is an embedded smooth hypersurface in $\\mathbb{R}^n$. Throughout the paper, we assume $\\Sigma$ is connected, unless otherwise indicated. We denote by $\\bar{g}$ a Euclidean metric in $\\mathbb{R}^n$. The definition of weighted H\\\"older spaces is given in Section~\\ref{section:basic}, and we always assume  $\\alpha\\in(0,1), q\\in (\\frac{n-2}{2} , n-2)$, unless otherwise indicated.\n\n\\begin{Definition}\\label{definition:static-regular}\nThe boundary $\\Sigma$ is said to be \\emph{static regular in  $\\mathbb R^n\\setminus\\Omega$} if  for any  pair of a symmetric $(0,2)$-tensor $h$ and a scalar-valued function $v$ satisfying $(h, v)\\in  \\C^{2,\\alpha}_{-q}(\\mathbb{R}^n\\setminus \\Omega)$ and\n\\begin{align*\nD\\Ric|_{\\bar{g}} (h)- \\nabla^2_{\\bar{g}} v&=0,  \\quad  \\Delta_{\\bar{g}} v=0 \\quad \\mbox{ in } \\mathbb{R}^n\\setminus \\Omega,\\\\\nh^\\intercal&=0, \\quad DH|_{\\bar{g}}(h)=0\\quad \\mbox{ on } \\Sigma,\n\\end{align*}\nwe must have $DA|_{\\bar{g}}(h)=0$ on~$\\Sigma$. \n\\end{Definition}\nIn the above definition, $D\\Ric|_{\\bar{g}}$ denotes the linearization of the Ricci curvature at $\\bar{g}$, and $DH|_{\\bar{g}}, DA|_{\\bar{g}}$ denote the linearizations of the mean curvature and second fundamental form on $\\Sigma$, respectively. \n\n\nOur first main result gives a general criterion for the existence and local uniqueness of static vacuum pairs in an arbitrary exterior region $\\mathbb{R}^n\\setminus \\Omega$ with Bartnik boundary data on $\\Sigma$ sufficiently close to the induced data $(\\bar{g}^\\intercal, H_{\\bar{g}})$.\n\\begin{Theorem}\\label{theorem:criterion-main}\nSuppose the boundary $\\Sigma$ is static regular in $\\mathbb{R}^n\\setminus \\Omega$. Then there exist positive constants $\\epsilon_0, C$ such that for each $\\epsilon < \\epsilon_0$, if $(\\tau, \\phi)$ satisfies~$\\| (\\tau,\\phi)- (\\bar{g}^\\intercal, H_{\\bar{g}}) \\|_{\\C^{2,\\alpha}(\\Sigma)\\times \\C^{1,\\alpha}(\\Sigma)}<\\epsilon$, then there exists an asymptotically flat  pair $(g, u)$ with $\\|(g, u) - (\\bar{g}, 1) \\|_{\\C^{2,\\alpha}_{-q}(\\mathbb{R}^n \\setminus \\Omega)} < C\\epsilon$ such that $(g, u)$ is a static vacuum pair in $\\mathbb{R}^n\\setminus \\Omega$ having the Bartnik boundary data $(g^\\intercal, H_g) = (\\tau, \\phi)$ on~$\\Sigma$.\n\nFurthermore, the solution $(g, u)$ is geometrically unique in a neighborhood $\\mathcal U$ of $(\\bar g, 1)$ in the $\\C^{2,\\alpha}_{-q}(\\mathbb{R}^n\\setminus \\Omega)$-norm; namely, there exists a unique solution satisfying both the static-harmonic gauge and the orthogonal gauge in $\\mathcal U$.\n\\end{Theorem}\n\nNote that the constants $\\epsilon_0, C$ depend on the global geometry of $\\Sigma $ and $\\mathbb{R}^n\\setminus \\Omega$. See Remark~\\ref{re:constant}.\n\nIn the above theorem, we say that an asymptotically flat pair $(g, u)$ satisfies  the \\emph{static-harmonic gauge} in $\\mathbb{R}^n\\setminus \\Omega$ if \n\\begin{align}\\label{static-harmonic}\n\\bi g + du = 0\\quad \\mbox{ in } \\mathbb{R}^n\\setminus \\Omega,\n\\end{align}\nwhere the Bianchi operator $\\bi$ (at $\\bar{g}$) is defined by $\\bi g :=- \\Div_{\\bar{g}} g + \\tfrac{1}{2} d (\\tr_{\\bar{g}} g)$. We say that $(g, u)$ satisfies the \\emph{orthogonal gauge} if \n\\begin{align}\\label{orthogonal}\n\t\\int_{\\mathbb{R}^n\\setminus \\Omega}  \\big((g- \\bar{g})\\cdot L_X\\bar{g} \\big)\\rho\\, d\\mathrm{vol}=0,\\qquad \\mbox{ for all $X\\in \\mathcal{X}_0$}\n\\end{align}\nwhere the weight function  $\\rho(x) = (1+|x|^2)^{-1}$, the dot ``$\\cdot$'' denotes the inner product of $\\bar{g}$, and $\\mathcal{X}_0$ is the subspace of $\\C^{3,\\alpha}_{\\mathrm{loc}}(\\mathbb{R}^n\\setminus \\Omega)$  vector fields, consisting of  $X$ satisfying $\\Delta_{\\bar{g}} X=0$ in $\\mathbb{R}^n\\setminus \\Omega$  with $X=0$ on $\\Sigma$ and $X- K\\in \\C^{3,\\alpha}_{2-n}(\\mathbb{R}^n\\setminus \\Omega)$ for some Killing vector field $K$ of $(\\mathbb{R}^n, \\bar{g})$. More details about $\\mathcal{X}_0$ and the gauge conditions can be found in Section~\\ref{section:gauge} and  at the beginning of Section~\\ref{section:existence}.\n\nWe remark how the imposed gauge conditions relate to uniqueness in Conjecture~\\ref{conjecture}. We show that for any $(g, u)$ sufficiently close to $(\\bar{g}, 1)$, there is a unique diffeomorphism $\\psi$ of $M$ that is close to the identity map and  fixes the boundary such that $(\\psi^* g, \\psi^* u)$ satisfies both gauges (see Lemma~\\ref{lemma:both-gauge}). Therefore, Theorem~\\ref{theorem:criterion-main} implies that the static vacuum extension with the prescribed Bartnik boundary data is unique (up to diffeomorphisms) within an open neighborhood of $(\\bar{g}, 1)$. \n\n\n\n\nPart of our proof adapts the work of Anderson and Khuri \\cite{Anderson-Khuri:2013} to formulate the  problem into finding solutions to an elliptic boundary value system of Fredholm index $0$. However, because such elliptic system always admits a nontrivial kernel and cokernel, our main argument  uses a new modified elliptic system. \n  \nMore importantly, we can show that several classes of hypersurfaces in $\\mathbb{R}^n$ are static regular. Together with Theorem~\\ref{theorem:criterion-main}, we confirm the existence and local uniqueness of Conjecture~\\ref{conjecture} for $(g_0^\\intercal, H_{g_0})$ sufficiently close to the induced Bartnik boundary data on those static regular boundaries.\n\nWe first discuss about convex surfaces in $\\mathbb{R}^3$, i.e. surfaces of positive Gauss curvature. Suppose $\\Sigma  \\subset (\\mathbb{R}^3, \\bar{g})$ is convex.   For an asymptotically flat metric $g$ in $\\mathbb{R}^3\\setminus \\Omega$ with $g^\\intercal$ sufficiently close to $\\bar{g}^\\intercal$ on $\\Sigma$, there exists an isometric embedding $f: (\\Sigma, g^\\intercal)\\to (\\mathbb{R}^3, \\bar{g})$, unique up to a rigid motion of~$\\mathbb{R}^3$. Let  $\\mathring{H}_{g^\\intercal}$  be the mean curvature of the isometric image $f(\\Sigma) \\subset (\\mathbb{R}^3,\\bar{g})$ (with respect to the unit normal $\\nu$ pointing to infinity). We can define the following functional\n\\begin{align*}\n\t\t\\mathscr{G}(g) &= -16\\pi m_{\\mathrm{ADM}}(g) + \\int_{\\mathbb{R}^3\\setminus \\Omega} R_g \\, d\\mathrm{vol}_g + \\int_\\Sigma 2(\\mathring{H}_{g^\\intercal}-H_g) \\, d\\sigma_{g}.\n\\end{align*}\nThis functional takes a similar form as the functional considered by S. Hawking and G. Horowitz~\\cite{Hawking-Horowitz:1996}, but we use it in different ways. One reason to consider  $\\mathscr{G}$ is that $\\bar{g}$ is a critical point of $\\mathscr{G}$ among any variations, partly motivated by the work of S.~Brendle, F. Marques, and A. Neves~\\cite{Brendle-Marques-Neves:2011}. It is also partly motivated by the following result of Y.~Shi and L.-F.~Tam:\n\\begin{Theorem}[Shi-Tam~\\cite{Shi-Tam:2002}]\\label{theorem:Shi-Tam}\nLet $(\\overline{\\Omega}, g_{\\I})$ be a 3-dimensional compact manifold with boundary $\\Sigma$. Suppose the scalar curvature $R_{g_{\\I}}\\ge 0$ and the boundary $(\\Sigma, (g_{\\I})^\\intercal)$ has positive Gauss curvature and positive mean curvature (with respect to outward unit normal). Then\n\\[\n\t \\int_\\Sigma  \\Big( \\mathring{H}_{(g_{\\I})^\\intercal} -H_{g_{\\I}} \\Big) \\, d\\sigma_{g_{\\I}}\\ge 0\n\\]\nwith equality if and only if $(\\overline{\\Omega}, g_{\\I})$ is isometric to a Euclidean region. \n\\end{Theorem}\nIn the above theorem, we use the subscript $\\I$ to emphasize that the metric $g_{\\I}$ and the  corresponding quantities are of the compact region $(\\overline{\\Omega}, g_{\\I})$. In particular, the integral in Theorem~\\ref{theorem:Shi-Tam} need not relate to the boundary integral of $\\mathscr{G}$, unless $g$ and $g_{\\I}$ induce the same metric and mean curvature on~$\\Sigma$. By computing the second variation of $\\mathscr{G}$ along a specific path of metrics that relates the boundary integral of $\\mathscr{G}$ with Theorem~\\ref{theorem:Shi-Tam}, we derive a \\emph{linearized} mean curvature comparison inequality.\n\n\\begin{Proposition}\\label{proposition:harmonic-function}\nLet  $\\Omega$ be a bounded open subset in $\\mathbb{R}^3$ whose boundary $\\Sigma = \\partial \\overline{\\Omega}$ has  positive Gauss curvature. If $v\\in\\C^{2,\\alpha}_{-q}(\\mathbb R^3\\setminus\\Omega)$ satisfies $\\Delta_{\\bar{g}} v=0$ in $\\mathbb{R}^n\\setminus \\Omega$, then  \n\\[\n\\int_\\Sigma 2v\\Big(D \\mathring{H}|_{\\bar{g}^\\intercal}\\big(2v\\bar{g}^\\intercal\\big) - DH|_{\\bar{g}}(2v\\bar{g}) \\Big) \\, d\\sigma_{\\bar{g}}\\geq 0,\n\\]\nwhere $D \\mathring{H}|_{\\bar{g}^\\intercal}$ denotes the linearization of $\\mathring{H}_{g^\\intercal}$ at $\\bar{g}^\\intercal$.\n\\end{Proposition}\n\n\nNote that $DH|_{\\bar{g}}(2v\\bar{g})= -vH + 2\\nu(v)$ by conformal transformation. While there is no explicit formula for the other term $D \\mathring{H}|_{\\bar{g}^\\intercal}\\big(2v\\bar{g}^\\intercal\\big)$ (since its definition involves  isometric embeddings), we know that $D \\mathring{H}|_{\\bar{g}^\\intercal}\\big(2v\\bar{g}^\\intercal\\big)$ depends only on the restriction of $v$ on $\\Sigma$. Thus, Proposition~\\ref{proposition:harmonic-function} reveals  new positivity relating the Dirichlet and Neumann boundary data of a harmonic function and may be of independent interest. Proposition~\\ref{proposition:harmonic-function} is  essential to prove the following main theorem.\n\n\n\\begin{Theorem}\\label{theorem:static-convex}\nLet  $\\Omega$ be a bounded open subset in $\\mathbb{R}^3$ whose boundary $\\Sigma=\\partial \\overline{\\Omega}$ has positive Gauss curvature. Then $\\Sigma$ is static regular in~$\\mathbb{R}^3\\setminus \\Omega$.\n\\end{Theorem}\n\n\nOur next result implies that any embedded   hypersurface can be perturbed to become static regular. We say that  a family of embedded hypersurfaces $\\{ \\Sigma_t\\}\\subset \\mathbb{R}^n$ forms a smooth \\emph{generalized foliation} if the deformation vector $X$ of $\\{ \\Sigma_t\\}$ is smooth and on each $\\Sigma_t$, $\\bar g( X, \\nu) =\\zeta$ where $\\zeta>0$, except a set of $(n-1)$-dimensional Hausdorff measure zero on $\\Sigma_t$, and $\\nu$ is the unit normal of $\\Sigma_t$. In other words, $\\{ \\Sigma_t\\}$ is slightly more general than a foliation in that they are allowed to touch each other from one-side on a set of measure zero.\n \n\\begin{Theorem}\\label{theorem:static-generic}\nLet $\\delta>0$ and let each $\\Omega_t\\subset \\mathbb{R}^n$ for $t\\in [-\\delta, \\delta]$ be a bounded open subset with connected hypersurface boundary $\\Sigma_t = \\partial\\overline{ \\Omega_t}$  embedded  in $\\mathbb{R}^n$.  Suppose  the boundaries $\\{ \\Sigma_t\\}$ form a smooth generalized foliation. Then  there is an open dense subset $J\\subset (-\\delta,\\delta)$ such that  $\\Sigma_t$  is static regular in $\\mathbb{R}^n\\setminus \\Omega_t$ for all $t\\in J$.\n\\end{Theorem}\n\n By a dilation property of the Euclidean static vacuum pair, Theorem~\\ref{theorem:static-generic} has a strong consequence.\n\\begin{Corollary}\\label{cor:dilation}\nLet  $\\Omega$ be a bounded open subset in $\\mathbb{R}^n$ whose boundary $\\Sigma=\\partial \\overline{\\Omega}$ is a star-shaped hypersurface. Then $\\Sigma$ is static regular in $\\mathbb{R}^n\\setminus \\Omega$. \n\\end{Corollary}\n\n\nSince a convex hypersurface is star-shaped, Corollary~\\ref{cor:dilation} gives a second proof to Theorem~\\ref{theorem:static-convex}.  To compare those two proofs, the dilation property used to prove Corollary~\\ref{cor:dilation} is very specific to the Euclidean static vacuum pair, while the approach to Theorem~\\ref{theorem:static-convex} can possibly be extended for a general static vacuum metric. While most of the proofs presented in this paper heavily use that the background metric is Euclidean, we are able to extend several of current results, including Theorem~\\ref{theorem:criterion-main} and Theorem~\\ref{theorem:static-generic}, to the setting that the background metric is an arbitrary asymptotically flat, static vacuum metric by incorporating new arguments in~\\cite{An-Huang:2022}. \n\nWe make some remarks about the geometric assumptions on $\\Sigma$. For all of our results, we assume the boundary $\\Sigma$ is an embedded hypersurface in $\\mathbb{R}^n$. As is pointed out in \\cite{Anderson-Khuri:2013, Anderson-Jauregui:2016}, if the boundary $\\Sigma=\\partial \\overline{\\Omega}$  is only \\emph{inner} embedded, i.e., $\\Sigma$ touches itself from the exterior region $\\mathbb{R}^n\\setminus \\Omega$, the induced data $(\\bar{g}^\\intercal, H_{\\bar{g}})$ is valid Bartnik boundary data, but $(\\bar{g}, 1)$ in $\\mathbb{R}^n\\setminus \\Omega$ is \\emph{not} a valid static vacuum extension as $\\mathbb{R}^n\\setminus \\Omega$ is not a manifold with boundary. It is clear that our arguments  don't work without embeddedness. We note that those inner embedded hypersurfaces are conjectured to be counter-examples to Conjecture~\\ref{conjecture} by \\cite[Conjecture 5.2]{Anderson-Jauregui:2016} (see also \\cite{Anderson:2019}). On the other hand, Theorem~\\ref{theorem:static-generic} implies that those possible counter-examples can be perturbed inward (with arbitrarily small perturbation) to be static regular hypersurfaces. \n\nNote the assumption that $\\Sigma$ is connected in Theorem~\\ref{theorem:criterion-main}. While Proposition~\\ref{proposition:harmonic-function}, Theorems \\ref{theorem:static-convex} and~\\ref{theorem:static-generic} can be extended for $\\Sigma$ of multiple components, we still need the connectedness assumption to derive the existence of static vacuum extensions by Theorem~\\ref{theorem:criterion-main}. This restriction seems  related to  Static $\\mathtt{n}$-body Conjecture. It asserts that an asymptotically flat static metric with $\\mathtt{n}$ boundary components for $\\mathtt{n}> 1$ must be trivial, under some separation condition of the boundary components. See, e.g.~\\cite{Beig-Schoen:2009}. Therefore, it is expected to be considerably harder, if possible at all, to find a static vacuum extension with arbitrarily prescribed Bartnik boundary data when $\\Sigma$ is not connected.  \n\nLastly, we remark that in general the ADM mass of the metric constructed Theorem~\\ref{theorem:criterion-main} may not be positive. The phenomenon seems to be closely related to another outstanding problem whether there exists a ``fill-in'' metric of nonnegative scalar curvature with prescribed Bartnik boundary data. Nevertheless, in the special situation that $\\tau= \\bar g^\\intercal$ and $\\phi \\lneq H_{\\bar g}$, the constructed static metric~$g$ in $\\mathbb{R}^n\\setminus \\Omega$ clearly has a fill-in metric  $\\bar g$ in $\\overline{\\Omega}$ (the Euclidean metric) whose  Bartnik boundary data  satisfy $g^\\intercal = \\bar g^\\intercal$ and $H_g \\lneq H_{\\bar g}$. The ADM mass of such $g$ must be positive by Miao's positive mass theorem with corners \\cite{Miao:2002}. \n\n\nThe paper is organized as follows. In Section~\\ref{section:prep}, we give basic definitions and formulas of linearized curvatures, and we identify Ricci flat deformations on a Euclidean exterior region. In Section~\\ref{section:Green}, we consider variations of the Regge-Teitelboim Hamiltonian to obtain a \\emph{Green-type identity} and other identities essential for the main results. Section~\\ref{section:existence} deals with Theorem~\\ref{theorem:criterion-main}. In Section~\\ref{section:convex}, we prove Proposition~\\ref{proposition:harmonic-function} and apply it to prove Theorem~\\ref{theorem:static-convex}. In Section~\\ref{section:generic}, we prove Theorem~\\ref{theorem:static-generic} and Corollary~\\ref{cor:dilation}.\n\n\\subsection*{Acknowledgement}\nThe first author would like to thank Michael Anderson for valuable discussions.   In a previous version of the paper, Corollary~\\ref{cor:dilation} was stated only for round spheres, and we are very grateful to Pengzi Miao for notifying us that the dilation argument extends to all star-shaped hypersurfaces.\n \n\\section{Preliminaries}\\label{section:prep}\n\\subsection{Basic definitions and notations}\\label{section:basic}\n\nLet $\\mathbb{R}^n$ be the $n$-dimensional space with the Cartesian coordinates $ x=(x_1,\\dots, x_n)$, and $n\\ge 3$. Throughout the paper we shall fix the coordinate chart $\\{ x \\}$ and reserve the notation  $\\bar{g}$ for the Euclidean metric given by $\\bar{g} =( \\delta_{ij})$ in this coordinate chart.  Let $\\Omega\\subset \\mathbb{R}^n$ be a bounded open subset whose boundary~$\\Sigma = \\partial \\Omega$ is a compact, embedded, smooth\\footnote{Most of our arguments directly extend for $\\C^{3,\\alpha}$ hypersurfaces while some others may require extra care to pinpoint the optimal regularity, e.g. Theorem~\\ref{theorem:Shi-Tam}, which we do not pursue here.} hypersurface. We assume that  $\\mathbb{R}^n\\setminus \\Omega$ is connected. Most of results in this section are valid for $\\Sigma$ having multiple components, except Corollary~\\ref{corollary:trivial}. \n\nLet $g$ be a $\\C^2_{\\mathrm{loc}}$ Riemannian metric on $\\mathbb{R}^n\\setminus \\Omega$.  For a tensor $h$ on $\\mathbb{R}^n\\setminus \\Omega$, we denote by $h^\\intercal$  the restriction of $h$ on the tangent bundle of~$\\Sigma$. We define the second fundamental form $A_g=\\tfrac{1}{2} (L_{\\nu_g} g)^\\intercal $ and the mean curvature  $H_g = \\Div_\\Sigma \\nu_g$ on $\\Sigma$, where $\\nu_g$ is the unit normal vector pointing \\emph{away} from~$\\Omega$.  \n\n\nWe will frequently work with variations of curvatures along a family of Riemannian metrics $g(s)$ on $\\mathbb{R}^n\\setminus \\Omega$ with $g(0) = g$.  We shall denote the linearization of the second fundamental form $\\left.\\ds\\right|_{s=0} A_{g(s)} $ by $DA|_g(h)$ where $h = g'(0)$. All other linearized curvature operators are denoted in the same fashion. We give a list of formulas of the linearized operators in the following lemma.  Consider a local frame $\\{ e_0, e_1, \\dots, e_{n-1}\\}$ in a collar neighborhood of $\\Sigma$ in $\\mathbb{R}^n\\setminus \\Omega$ such that $e_0$ is the parallel extension of $\\nu_g$ along itself and $e_1, \\dots, e_{n-1}$ are tangent to $\\Sigma$. We use the subscript $_{;i}$ to denote the covariant derivative by $e_i$ with respect to $g$, and all the indices are raised or lowered by $g$. \n\\begin{lemma} \\label{lemma:formula}\nThe linearizations of the Ricci tensor and scalar curvature are given by\n\\begin{align}\n\\begin{split}\\label{equation:Ricci}\n\t(D\\Ric|_g(h))_{ij} &=-\\tfrac{1}{2} g^{k\\ell} h_{ij;k\\ell} + \\tfrac{1}{2} g^{k\\ell} (h_{i k; \\ell j} + h_{jk; \\ell i} ) \\\\\n\t&\\quad - \\tfrac{1}{2} (\\tr h)_{ij}  + \\tfrac{1}{2} (R_{i\\ell} h^\\ell_j + R_{j\\ell} h^\\ell_i )\\\\\n\t&\\quad - R_{ik\\ell j} h^{k\\ell} \\qquad \\qquad  \\mbox{ for all } i, j =0,1,\\dots, n-1\n\\end{split}\\\\\n\\begin{split}\\notag\n\tD R|_g (h\n\t&=-\\Delta \\tr h + \\Div \\Div h - h \\cdot \\Ric_g.\n\\end{split}\n\\end{align}\nHere, ``$\\cdot$'' denotes the inner product with respect to $g$. \n\nWe write $\\nu = \\nu_g$ for short. We let $\\omega(e_a) = h(\\nu, e_a)$ be the one-form on the tangent bundle of $\\Sigma$ and $\\{\\Sigma_t\\}$ be the foliation by $g$-equidistant hypersurfaces to $\\Sigma$.\nFor tangential directions $a, b, c \\in \\{ 1, \\dots, n-1\\}$ on $\\Sigma$, we have \n\\begin{align}\n\tD\\nu|_g(h) &= - \\tfrac{1}{2} h(\\nu, \\nu) \\nu - g^{ab} \\omega(e_a) e_b\\label{equation:normal}\\\\\n\\begin{split}\n\tDA|_g(h) & = \\tfrac{1}{2} (L_{\\nu} h)^\\intercal - \\tfrac{1}{2} L_\\omega g^\\intercal - \\tfrac{1}{2} h(\\nu, \\nu) A_g\\\\\n\t&= \\tfrac{1}{2}( \\nabla_{\\nu} h)^\\intercal+ A_g\\circ h - \\tfrac{1}{2} L_\\omega g^\\intercal- \\tfrac{1}{2} h(\\nu, \\nu) A_g\\label{equation:linearized-sff}\n\t\\end{split}\\\\\n\tDH|_g(h) &=\\tfrac{1}{2} \\nu(\\tr h^\\intercal) - \\Div_\\Sigma \\omega - \\tfrac{1}{2} h(\\nu, \\nu) H_g\\notag.\n\\end{align}\nIn the third line $(A_g\\circ h)_{ab} = \\tfrac{1}{2} (A_{ac} h^c_{b}+A_{bc} h^c_{a})$,  and in the last equation, $\\tr h^\\intercal$ denotes the tangential trace of $h$ on $\\Sigma_t$, defined in a collar neighborhood of~$\\Sigma$. \n\nFor any family  $g(s)$ with $g(0) = g$, we let $H_{g(s)}$, defined in the collar neighborhood, be the mean curvature of $\\Sigma_t$ with respect to $g(s)$. Then on $\\Sigma$, \n\\begin{align*}\n\t\\nu (DH|_g(h))& =  - \\tfrac{1}{2} DR|_g(h) + \\tfrac{1}{2} DR^\\Sigma|_{g^\\intercal} (h^\\intercal) + A_g \\cdot (A_g \\circ h) \\\\\n\t&\\quad  - A_g \\cdot DA|_g(h  -H_g DH|_g(h) \\\\\n\t&\\quad + \\tfrac{1}{4}\\big( - R_g +R^\\Sigma_g -  |A_g|^2 - H_{g}^2\\big)h(\\nu, \\nu)\\\\\n\t&\\quad \t- \\tfrac{1}{2}\\Delta_\\Sigma h(\\nu, \\nu)  + g^{ab} \\omega (e_a) e_b(H_g)\n\\end{align*}\nwhere $R^\\Sigma_g, \\Delta_\\Sigma$ are respectively the scalar curvature and the Laplace operator of  $(\\Sigma, g^\\intercal)$. Consequently, if we assume $h(\\nu, \\nu)=0$ and $\\omega=0$, then \n\\begin{align}\\label{equation:Riccati}\n\\begin{split}\n\\nu (DH|_g(h))&=-\\tfrac{1}{2} DR|_g(h)+\\tfrac{1}{2} DR^\\Sigma|_{g^\\intercal} (h^\\intercal)  + A_g\\cdot(A_g\\circ h )  \\\\\n&\\quad - A_g\\cdot DA|_g(h) - H_gDH|_g(h).\n\\end{split}\n\\end{align}\n\\end{lemma}\n\\begin{proof}\nThe formulas for $D\\Ric|_g(h), D R|_g (h)$ can be found in, for example, \\cite[Lemma 2]{Fischer-Marsden:1975}.\n\nThe formula for the linearized unit normal vector can be derived by linearizing $g(\\nu, \\nu)=1$ and $g(\\nu, e_a)=0$. For the linearized second fundamental form, we linearize $A_g =\\tfrac{1}{2} (L_\\nu g)^\\intercal $ to get \n\\begin{align*}\n\t(DA|_g(h))_{ab} &= \\tfrac{1}{2} \\left[ (L_\\nu h)_{ab}+ (L_{D\\nu|_g(h)} g )_{ab}\\right]\\\\\n&=   \\tfrac{1}{2}(L_\\nu h)_{ab} - \\tfrac{1}{2}\\left[ e_a(\\omega( e_b) )+ e_b (\\omega( e_a))\\right] - \\tfrac{1}{2} h(\\nu, \\nu) A_{ab}\n\\end{align*}\nwhere we have used the formula for $D\\nu|_g(h)$ to get\n\\[\n \t(L_{D\\nu|_g(h)} g )_{ab}=  - e_a( \\omega( e_b))-e_b( \\omega( e_a))- h(\\nu, \\nu) A_{ab}.\n\\]\t\nThe formula for $DH|_g$ follows by linearizing $H_g = g^{ab} A_{ab}$ and applying the formula for $DA|_g(h)$. \n\nWe now compute $\\nu(DH|_g(h))$. We express $\\nu = \\eta(s) \\nu_{g(s)} + \\nu^\\intercal (s)$ on $\\Sigma$ into the normal and tangential components with respect to $g(s)$. So\n\\begin{align*}\n\t\\eta(0)=1,\\quad \\nu^\\intercal (0)= 0, \\quad \\eta'(0) =  \\tfrac{1}{2} h(\\nu, \\nu),\n\t \\quad (\\nu^\\intercal)'(0)= g^{ab} \\omega( e_a) e_b.\n\\end{align*}\nBy the variation formula of mean curvature, we have\n\\begin{align*}\n\t\\nu(H_{g(s)}) &= - \\big(|A_{g(s)}|_{g(s)}^2 + \\mathrm{Ric}_{g(s)}(\\nu_{g(s)}, \\nu_{g(s)}) \\big)\\eta(s)\\\\\n\t&\\quad - \\Delta_{(\\Sigma, g(s))} \\eta(s) + \\nu^\\intercal (s)(H_{g(s)})\\\\\n\t\t&= \\left(  - \\tfrac{1}{2} R_{g(s)} + \\tfrac{1}{2} R^\\Sigma_{g(s)} - \\tfrac{1}{2} |A_{g(s)}|_{g(s)}^2 - \\tfrac{1}{2} H_{g(s)}^2\\right) \\eta(s)\\\\\n\t\t&\\quad  - \\Delta_{(\\Sigma, g(s))} \\eta(s) + \\nu^\\intercal (s) (H_{g(s)})\n\\end{align*}\nwhere in the second line we replace the Ricci term using the Gauss equation. Differentiating the previous identity in $s$ at $s=0$ yields the desired formula. \n\\end{proof}\n\nWe also include the following general fact on the linearized mean curvature of the pull-back metric, which can be directly verified.  \n\\begin{lemma}\\label{lemma:pullback}\nLet $U_1, U_2$ be complete smooth manifolds with boundary $\\Sigma_1, \\Sigma_2$ respectively, and let $g$ be a Riemannian metric on $U_2$.  Let $\\psi: U_1\\to U_2$ be a diffeomorphism that sends $\\Sigma_1$ to $\\Sigma_2$.  Let $H_g$ be the mean curvature of $\\Sigma_2 \\subset (U_2, g)$ of $\\nu_g$ and $H_{\\psi^*g} $ be the mean curvature of $\\Sigma_1\\subset (U_1, \\psi^*g)$ of $\\psi^* \\nu_g$. Then, $H_{\\psi^*g} = \\psi^*H_g$, and for any deformation $h$ of $g$ in $U_2$, \n\\begin{align*}\n\tDH|_{\\psi^* g}\t(\\psi^* h) = \\psi^* \\left( DH|_{g} (h)\\right).\n\\end{align*} \n\\end{lemma}\n\n\nFor a Riemannian metric $g$  and a function $u$ in $\\mathbb{R}^n\\setminus \\Omega$,  we define \n\\[\n\tS(g, u) = -u\\Ric_g + \\nabla^2_g u - (\\Delta_g u )g.\n\\]\nThe right hand side is equal to $DR|_g^*(u)$, the formal $\\mathcal{L}^2$-adjoint of $DR|_g$, but  our notation $S(g, u)$ emphasizes that we will treat both $(g, u)$ as unknowns. \nWe say that the pair $(g, u)$ is \\emph{static vacuum} in $U$ if $(g, u)$ solves the following static vacuum system:\n\\begin{align*}\n\tS(g, u) = 0 \\quad \\mbox{ and } \\quad\tR_g=0\\quad \\mbox{ in }\\mathbb R^n\\setminus\\Omega.\n\\end{align*}\nThis system is equivalent to \\eqref{sv}.\n\n\nFor a symmetric $(0,2)$-tensor $h$ and a function $v$, we say that $(h,v)$ is a {\\it static vacuum deformation} at the static vacuum pair $(g,u)$ if it satisfies the linearized static vacuum system:\n\\begin{align}\\label{lsv}\n\tDS|_{(g, u)}(h, v)=0  \\quad \\mbox{ and } \\quad DR|_g(h)=0\\quad \\mbox{ in }\\mathbb R^n\\setminus\\Omega.\n\\end{align}\nIf the Ricci curvature of $g$ is zero,  we say that $h$ is a \\emph{Ricci flat deformation} (at $g$) if $D\\Ric|_g(h)=0$. In general, the Ricci flat deformation can be thought as a subclass of static vacuum deformations at a Ricci flat metric $g$ by letting $u=1$ and $v= 0$. \nOn the other hand,\nin the special case  that $(g, u)=(\\bar{g}, 1)$, any static vacuum deformation $(h, v)$ gives rise to a Ricci flat deformation $h+\\tfrac{2}{n-2} v\\bar{g}$. \nThis is because the linearized system \\eqref{lsv} at $(\\bar g, 1)$ is equivalent~to \n\\begin{align*}\n\t-D\\Ric(h) + \\nabla^2 v =0 \\quad \\mbox{ and  } \\quad \\Delta v=0 \\quad\\mbox{ in }\\mathbb R^n\\setminus\\Omega,\n\\end{align*}\nwhere we omit the subscript $\\bar{g}$,  and it \nis also equivalent to the system\n\\begin{align*}\nD\\Ric\\big(h+\\tfrac{2}{n-2} v\\bar{g}\\big)=0\\quad \\mbox{ and } \\quad \\Delta v=0 \\quad \\mbox{ in }\\mathbb R^n\\setminus\\Omega.\n\\end{align*}\n\n\n\nWe will consider the following two types of boundary conditions of a deformation $h$ on $\\Sigma$:\n\\begin{itemize}\n\\item \\emph{Bartnik boundary condition}:\n\\[\n\th^\\intercal=0 \\quad \\mbox{ and } \\quad DH|_g (h)=0 \\quad \\mbox{ on } \\Sigma. \n\\]\n\\item \\emph{Cauchy boundary condition}: \n\\[\n\th^\\intercal =0\\quad \\mbox{ and } \\quad DA|_g (h)=0 \\quad \\mbox{ on } \\Sigma. \n\\]\n\\end{itemize}\nClearly, the Cauchy boundary condition implies the Bartnik boundary condition. For a general  deformation $h$, we may refer $(h^\\intercal, DH|_g(h))$ as its Bartnik boundary data and   $(h^\\intercal, DA|_g(h))$ as its Cauchy boundary data. (In this paper we slightly abuse the terminology and use ``Bartnik boundary data'' for $(g^\\intercal, H_g)$ and also the linearization $(h^\\intercal, DH|_g(h))$, but it should be clear from context which one is referred to.)\n\nLet $k\\in \\{ 0, 1, 2, \\dots\\}$,  $\\alpha\\in (0, 1)$, and $q\\in \\mathbb{R}$. For a $\\C^{k,\\alpha}_{\\mathrm{loc}}$-function $f$ in $\\mathbb{R}^n\\setminus \\Omega$, we define the $\\C^{k,\\alpha}_{-q} (\\mathbb{R}^n\\setminus \\Omega)$-norm  by\n\\begin{align*}\n\t\\|f\\|_{\\C^{k,\\alpha}_{-q} (\\mathbb{R}^n\\setminus \\Omega)} =& \\textstyle\\sum_{|I|=0,\\dots, k} \\sup_{x\\in \\mathbb{R}^n \\setminus \\Omega} (1+|x|)^{|I|+q} |\\partial^I f(x)|  \\\\\n\t&\\quad +\\textstyle\\sum_{|I|=k} \\sup_{\\substack{x, y\\in \\mathbb{R}^n\\setminus \\Omega\\\\0<|x-y| \\le\\frac{ |x|}{2}}} (1+|x|)^{\\alpha+|I|+q} \\frac{|\\partial^I f(x) - \\partial^I f(y)|}{|x-y|^\\alpha}.\n\\end{align*} \nThe \\emph{weighted H\\\"older space} $\\C^{k,\\alpha}_{-q} (\\mathbb{R}^n\\setminus \\Omega)$ consists of those functions whose $\\C^{k,\\alpha}_{-q} (\\mathbb{R}^n\\setminus \\Omega)$-norms are finite. We slightly abuse the notation and say that a tensor is in $\\C^{k,\\alpha}_{-q}(\\mathbb{R}^n\\setminus \\Omega)$ if all of its coefficient functions in the Cartesian coordinates are in $\\C^{k,\\alpha}_{-q}(\\mathbb{R}^n\\setminus \\Omega)$. We may sometimes write $f= O^{k,\\alpha}(|x|^{-q})$ for $f\\in \\C^{k,\\alpha}_{-q}(\\mathbb{R}^n\\setminus \\Omega)$ to emphasize the asymptotic rate. \n\n\n\nFor the rest of the paper, we shall always assume\n\\[\n\tq\\in \\left(\\tfrac{n-2}{2}, n-2\\right).\n\\]\nWe say that a Riemannian metric $g$ on $\\mathbb{R}^n\\setminus \\Omega$ is \\emph{asymptotically flat} if  $g-\\bar g \\in \\C^{2,\\alpha}_{-q}(\\mathbb{R}^n\\setminus \\Omega)$. The \\emph{ADM mass of $g$} is defined as\n\\begin{align*}\n\tm_{\\mathrm{ADM}}(g)&= \\frac{1}{2(n-1)\\omega_{n-1}} \\lim_{r\\to \\infty} \\int_{|x|=r}\\sum_{i,j=1}^n \\left(\\frac{\\partial g_{ij}}{\\partial x_i} - \\frac{\\partial g_{ii}}{\\partial x_j}\\right)\\frac{x_j}{|x|} \\, d\\sigma,\n\\end{align*}\t\nwhere $d\\sigma$ is the $(n-1)$-dimensional Euclidean Hausdorff measure and $\\omega_{n-1}$ is the volume of the standard $(n-1)$-dimensional unit  sphere. \n\nIf $g$ is asymptotically flat and $u$ is a scalar-valued function such that $u-1\\in \\C^{2,\\alpha}_{-q}(\\mathbb{R}^n\\setminus \\Omega)$, we will refer to $(g, u)$ as an  \\emph{asymptotically flat pair}. We denote by $\\mathcal{M}$ an open neighborhood of $(\\bar{g}, 1)$ in the $\\C^{2,\\alpha}_{-q}(\\mathbb{R}^n\\setminus \\Omega)$-norm, consisting of asymptotically flat pairs.  \n\n\nWe will frequently use the following basic PDE lemma. See~\\cite{Lockhart-McOwen:1985, Bartnik:1986}.\n\\begin{lemma}\\label{lemma:PDE}\nLet $k\\ge 2$, $\\delta\\in \\mathbb{R}$, and let $\\Omega\\subset \\mathbb{R}^n$ be a bounded open subset whose boundary $\\Sigma$ is an embedded hypersurface. Suppose $\\mathbb{R}^n\\setminus \\Omega$ is connected (but $\\Sigma$ may be disconnected). Let $g$ be an asymptotically flat Riemannian metric on $\\mathbb{R}^n\\setminus \\Omega$ with $g- \\bar g\\in \\C^{k,\\alpha}_{-q}$. Consider the map on the space of vector fields\n\\[\n\\Delta_g:  \\left\\{ X\\in \\C^{k,\\alpha}_{\\delta} (\\mathbb{R}^n\\setminus \\Omega) : \\, X=0 \\mbox{ on } \\Sigma\\right \\} \\to \\C^{k-2,\\alpha}_{\\delta-2}(\\mathbb{R}^n\\setminus \\Omega)\n\\]\nthat sends $X$ to $\\Delta_g X$. Then \n\\begin{enumerate}\n\\item For $0< \\delta< 1$, the map is surjective and the kernel space is $n$-dimensional, spanned by $\\{ X^{(1)}, \\dots, X^{(n)}\\}$ where $X^{(i)} = \\frac{\\partial}{\\partial x_i} + O^{k,\\alpha}(|x|^{-q})$.\n\\item For $2-n<\\delta <0$, the map is an isomorphism. \n\\item For $\\delta<2-n$, the map is injective. \n\\end{enumerate}\n\\end{lemma}\n\n\n\\subsection{Ricci flat deformations}\\label{section:gauge}\n\n\nOur main result in this section is Theorem~\\ref{theorem:trivial} below where we identify all Ricci flat deformations $h \\in \\C^{2,\\alpha}_{-q}(\\mathbb{R}^n\\setminus \\Omega)$ that satisfy the Cauchy boundary condition. We remark that  Anderson and Herzlich  have proved that,  under the $H$-harmonic gauge, a Ricci flat metric $g$ is uniquely determined by its data $g^\\intercal$ and $A_g$ on the boundary~\\cite[Theorem 1.1]{Anderson-Herzlich:2008, Anderson-Herzlich:erratum}.  Implementing the similar idea, the first author proved that a Ricci flat deformation $h$, under the $H$-harmonic gauge, satisfying the Cauchy boundary condition must be zero~\\cite[Proposition 4.4]{An:2020}. We will give an alternative and elementary proof that fits to our specific setting.\n\n\nWe begin with examples of  Ricci flat deformations satisfying the Cauchy boundary condition.\n\n\\begin{example}\\label{example:vector}\nLet  $g$ be a Riemannian metric on a smooth manifold $U$ satisfying  $\\Ric_g=0$. Given a vector field $X$, let $\\psi_s$ be a family of diffeomorphisms of $U$ satisfying $\\left.\\ds\\right|_{s=0}\\psi_s=X$. Define the family  $g(s)=\\psi_s^*(g)$ and note $g'(0)= L_X g$. We call the Lie derivative $L_X g$ an \\emph{(infinitesimal) deformation generated from diffeomorphisms}. Since $\\Ric_{g(s)} = \\psi_s^* \\Ric_g=0$, differentiating in $s$ gives $D{\\Ric}|_g(L_Xg)=0$. That is, at a Ricci flat metric $g$, any deformation generated from diffeomorphisms is a Ricci flat deformation. \n\n\nNext, we find the condition of $X$ on the boundary $\\Sigma$ such that $L_X g$ satisfies the Cauchy boundary condition. We say that $K$ is a \\emph{Killing vector field} of $(U, g)$ if $L_K g=0$ in $U$.  Let $X$ satisfy $X = K$ on $\\Sigma$ for some Killing vector $K$, including the zero vector field. \nWe shall see that such $L_X g$ satisfies the Cauchy boundary condition. Let $\\{ e_0, e_1,\\dots, e_{n-1}\\}$ be a local frame in $U$ where on $\\Sigma$, $e_0=\\nu$ is a unit normal and $\\{e_1,\\dots, e_{n-1}\\}$ tangent to $\\Sigma$. We decompose $X$ as $X = \\eta \\nu +  X^\\intercal$ on $\\Sigma$. The Cauchy boundary data can be expressed as, for $a, b, c= 1, \\dots, n-1$, \n\\begin{align*\n\\begin{split}\n\t(L_X g)^\\intercal &= 2\\eta A + L_{X^\\intercal} g^\\intercal\\\\\n\t DA|_g (L_X g) (e_a, e_b)&=\\mathrm{Rm}(X, e_a, \\nu, e_b)- \\eta_{|a b} +\\tfrac{1}{2} \\eta (A_a^c A_{c b} + A_b^c A_{c a}) \\\\\n\t&\\quad +  (X^\\intercal)^c A_{c b|a} + (X^\\intercal)^c_{|b} A_{ac} + ( X^\\intercal)^c_{|a}A_{b c}\n\\end{split}\n\\end{align*}\nwhere the subscript $_{|a}$ denotes the covariant derivative of the induced metric $g^\\intercal$ on $\\Sigma$. It follows immediately from those formulas that $(L_X g)^\\intercal$  and $ DA|_g (L_X g) $  are entirely determined by $\\eta$ and $X^\\intercal$ on $\\Sigma$. Thus if $X=K$ on $\\Sigma$, then\n$\n(L_X g)^\\intercal = ( L_K g)^\\intercal =0,~\n DA|_g (L_X g)= DA|_g (L_K g) =0.\n$\n\n\\qed\n\\end{example}\n\nThe example above shows that, at a Ricci flat metric $g$, for any vector field $X$ equal to some Killing vector of $g$ on the boundary, $L_X g$ is a Ricci flat deformation satisfying the Cauchy boundary condition.  Before we  proceed to study Ricci flat deformations in greater generality, we will consider two types of gauge conditions on $h$: the \\emph{geodesic gauge} on $\\Sigma$ and the \\emph{harmonic gauge} in $U$. Both gauge conditions are well-studied in literature,  but for completeness we include the next two lemmas.  (Note we only use those lemmas for $g= \\bar g$ the Euclidean metric in this paper.)\n\nWe say that a deformation $h$  satisfies the \\emph{geodesic gauge condition on $\\Sigma$} (at $g$) if \n\\begin{align}\\label{geodesic gauge}\nh(\\nu_g, \\cdot)= 0,\\quad (\\nabla_{\\nu_g} h)(\\nu_{g}, \\cdot)=0\\quad\\mbox{ on }\\Sigma,\n\\end{align}\nwhere  $\\nu_g$ is parallelly extended along itself.  The ``dot'' means  $h(\\nu_g, e_i )= 0$  and $(\\nabla_{\\nu_g} h)(\\nu_{g}, e_i)=0$ on $\\Sigma$ for any vector $e_i$, either tangential or normal.\n\\begin{lemma}[Geodesic gauge]\\label{lemma:geodesic}\nLet $h\\in \\C^{2,\\alpha}(U)$ be a symmetric $(0,2)$-tensor. Then there exists a vector field $V\\in \\C^{3,\\alpha}(U)$ with $V=0$ on $\\Sigma$ and $V$ vanishing outside a collar neighborhood of $\\Sigma$ such that $k:= h+L_Vg$ satisfies the geodesic gauge condition and that $h, k$ have the same Cauchy boundary data on $\\Sigma$.\n\\end{lemma}\n\\begin{remark}\\label{remark:geodesic}\nIt explains the reason that $h^\\intercal=0,  DA|_g (h)=0$ on $\\Sigma$ is referred to as the Cauchy boundary condition. For such $h$,  we let $k = h+L_V g$ as in the lemma. By \\eqref{equation:linearized-sff},  $k$ must satisfy the Cauchy boundary condition on $\\Sigma$ in the usual sense: for all $i, j=0,1, \\dots, n-1$,\n\\[\nk_{ij}=0\\quad \\mbox{ and } \\quad ( \\nabla_{\\nu_g} k)_{ij}=0\\quad \\mbox{ on } \\Sigma. \n\\]\n\\end{remark}\n\\begin{proof}\nCover a collar neighborhood of $\\Sigma\\subset U$ by countably many open subsets $U_\\ell$ such that for each $\\ell$ there is a local orthonormal smooth frame $\\{ e_0, e_1, \\dots, e_{n-1}\\}$ in $U_\\ell$ where $e_0  =\\nu_g$ is parallel along itself, and  $e_1, \\dots, e_{n-1}$ are tangent to $\\Sigma$. We will simply write $\\nu=\\nu_g$. Let $\\{\\xi_\\ell\\}$ be a partition of unity subordinate to $\\{ U_\\ell\\}$.\n\n Suppose that $W =\\textstyle\\sum_{i=0}^{n-1} \\eta_i e_i$ is a vector field in $U_\\ell$ where the  coefficient function $\\eta_i=0$ on $\\Sigma$ for all $i$. The following identities hold on $U_\\ell \\cap \\Sigma$, where $a= 1,\\dots, n-1$ denotes the tangential directions:\n\\begin{align}\\label{equation:geodesic-boundary}\n\\begin{split}\n\tL_W g (\\nu, \\nu) &= 2\\nu(\\eta_0)\\\\\n\tL_W  g(\\nu, e_a) &= \\nu(\\eta_a) \\\\\n\t(\\nabla_\\nu (L_W g))(\\nu, \\nu)&=2\\nu (\\nu(\\eta_0))\\\\\n\t(\\nabla_\\nu (L_W g))(\\nu, e_a)&=\\nu(\\nu(\\eta_a)) + 2\\nu(\\eta_b) g(\\nabla_\\nu e_b, e_a)\\\\\n\t&\\quad  +e_a( \\nu(\\eta_0)) - \\nu(\\eta_b) A_{ab},\n\\end{split}\n\\end{align}\nwhere $A_{ab} = g(\\nabla_{e_a} \\nu, e_b)$. \n\nWe proceed to define the vector field $V$. We first define a local vector field  $W^{(\\ell)}$ on each $U_\\ell$. Let  $W^{(\\ell)}=\\sum_{i=0}^{n-1}\\eta_i e_i$ (we omit the superscript $\\ell$ in those $\\eta_i$). Choose $\\eta_i\\in \\C^{3,\\alpha}(U_\\ell)$ such that it vanishes outside a collar neighborhood of $\\Sigma$, and \n\\begin{align*}\n\\eta_i=0,\\;\\; 2\\nu(\\eta_0)= -h(\\nu,\\nu)\\;\\; \\mbox{ and } \\;\\; \\nu(\\eta_a) = -h(\\nu, e_a),\\quad a=1,\\dots, n-1\n\\end{align*}\non $U_\\ell\\cap\\Sigma$. We let $W= \\sum_\\ell \\xi_\\ell W^{(\\ell)}$ on $U$.\n\nNext, we define  $\\widetilde{W}^{(\\ell)}=\\sum_{i=0}^{n-1}\\widetilde\\eta_ie_i$ on $U_\\ell$ (we omit the superscript $\\ell$ in those $\\widetilde\\eta_i$). Choose $\\widetilde\\eta_i\\in \\C^{3,\\alpha}(U_\\ell)$ which vanishes outside a collar neighborhood of $\\Sigma$, and\n\\begin{align*}\n        \\widetilde\\eta_i&=0\\\\\n        \\nu(\\widetilde\\eta_i)&=0 \\\\\n\t2\\nu (\\nu(\\widetilde\\eta_0))&= -\\nabla_\\nu (h+L_Wg)(\\nu, \\nu)\\\\\n\t\\nu(\\nu(\\widetilde\\eta_a)) &=- \\nabla_\\nu (h+L_Wg)(\\nu, e_a), \\qquad a=1,\\dots, n-1\n\\end{align*}\non $U_\\ell\\cap\\Sigma$.  We define $\\widetilde{W}=\\sum_\\ell \\xi_\\ell \\widetilde W^{(\\ell)}$ and let $V= W +\\widetilde W$ on $U$. By those identities in  \\eqref{equation:geodesic-boundary}, it is direct to verify that $V$ satisfies the desired properties.\n\\end{proof}\n\n\t\n\t\n\n\n\nWe define some differential operators which will be used throughout this paper: For a  symmetric $(0,2)$-tensor $h$ and a vector field $V$, we define the \\emph{Bianchi operator} $\\bi_g$, the \\emph{Killing operator} $\\D_g$, and the $\\mathcal{L}^2$-formal adjoint  $\\bi_g^*$ by \n\\begin{align}\\label{equation:operators}\n\\begin{split}\n\t\\bi_g h &= - \\Div_g h+ \\tfrac{1}{2} d(\\tr_g h)\\\\\n\t\\D_g V &= \\tfrac{1}{2} L_V g\\\\\n\t\\bi^*_g V&= \\tfrac{1}{2} \\big( L_V g - (\\Div_{g} V) g \\big).\n\\end{split}\n\\end{align}\nWe say that \\emph{$h$ satisfies the harmonic gauge} (at $g$) if $\\bi_g h=0$.  By Ricci identity, for any vector field $V$,  $\\bi_g (L_V g)$ becomes an elliptic operator on $V$:\n\\begin{align} \\label{equation:Bianchi-vector}\n\t(\\bi_g L_V g)_i = -(\\Delta_g V)_i - \\Ric_{ij} V^j.\n\\end{align}\n\\begin{lemma}[Harmonic gauge]\\label{lemma:harmonic-gauge}\nThere is $\\epsilon>0$ such that if $g$ is asymptotically flat with $g-\\bar g \\in \\C^{3,\\alpha}_{-q}(\\mathbb{R}^n\\setminus \\Omega)$ satisfying $\\|g-\\bar{g}\\|_{\\C^{1,\\alpha}_0(\\mathbb{R}^n\\setminus \\Omega)}<\\epsilon $  and \n$\\|\\Ric_g\\|_{\\C^{0,\\alpha}_{-2}(\\mathbb{R}^n\\setminus \\Omega)}< \\epsilon$, then for any $h\\in \\C^{2,\\alpha}_{-q}(\\mathbb{R}^n\\setminus \\Omega)$, there is a vector field $V\\in \\C^{3,\\alpha}_{1-q}(\\mathbb{R}^n\\setminus \\Omega)$ with $V=0$ on $\\Sigma$  such that if we define $k= h+L_Vg$, then $\\bi_g k=0$ in $\\mathbb{R}^n\\setminus \\Omega$, and $h$ and $k$ have the same Cauchy boundary data.\n\\end{lemma}\n\n\\begin{proof}\nWe show that there is a vector field  $V$ solving\n\\begin{alignat*}{2}\n\t\\bi_g L_V g &= -\\bi_g h \\quad && \\mbox{ in } \\mathbb{R}^n\\setminus \\Omega\\\\\n\tV&=0\\quad && \\mbox{ on } \\Sigma. \n\\end{alignat*}\nThe first equation can be expressed as $-\\Delta_g V - \\Ric_g(V, \\cdot) = -\\bi_g h$  by \\eqref{equation:Bianchi-vector}. Since the operator $\\Delta$, at the Euclidean metric, is surjective by Lemma~\\ref{lemma:PDE}, for $\\epsilon$ sufficiently small,  the operator $-\\Delta_g - \\Ric_g(\\cdot, \\cdot)$ is close to $-\\Delta$ in the operator norm and thus is also surjective. Then $V \\in \\C^{3,\\alpha}_{1-q}(\\mathbb{R}^n\\setminus \\Omega)$ by elliptic regularity. \n\\end{proof}\n\n\n\n\nNow we analyze the Ricci flat deformation on a Euclidean exterior region $(\\mathbb{R}^n\\setminus \\Omega, \\bar{g})$ and prove the main result of this section. Below we shall omit the subscript $\\bar{g}$.  For example, we write $\\bi =\\bi_{\\bar{g}}, \\D = \\D_{\\bar{g}}, \\bi^* =\\bi^*_{\\bar{g}}$, and \n\\begin{align*}\n\tD\\Ric(h) = D\\Ric|_{\\bar{g}}(h)\\quad\tDA(h) = DA|_{\\bar{g}}(h).\n\\end{align*}\nUsing the above  notations and \\eqref{equation:Ricci}, we can write \n\\begin{align}\\label{equation:Ricci-gauge}\n\tD\\Ric(h) = - \\tfrac{1}{2} \\Delta h - \\D \\bi h.\n\\end{align}\nDenote the formal $\\mathcal{L}^2$-adjoint operator of  $D\\Ric$ by $(D\\Ric)^*$. Then for a symmetric $(0,2)$-tensor $\\gamma$, \n\\begin{align}\\label{equation:adjoint-Ricci}\n\t(D\\Ric)^* (\\gamma) =-\\tfrac{1}{2} \\Delta \\gamma + \\bi^* (\\Div \\gamma).\n\\end{align}\nWe say that $h$ \\emph{weakly solves $D\\Ric(h)=0$ in $\\mathbb{R}^n$} if for any symmetric $(0,2)$-tensor $\\gamma\\in \\C^\\infty_c(\\mathbb{R}^n)$, \n\\begin{align*}\n\t\\int_{\\mathbb{R}^n} h\\cdot (D\\Ric)^* (\\gamma) \\, d\\mathrm{vol}=0.\n\\end{align*}\nWe  define that $h$ weakly solves $\\bi h=0$ similarly. \n\n\\begin{theorem}\\label{theorem:trivial}\nLet $\\Omega\\subset \\mathbb{R}^n$ be a bounded open subset whose boundary $\\Sigma = \\partial \\overline{\\Omega}$ is an embedded hypersurface such that $\\mathbb{R}^n\\setminus \\Omega$ is connected (but $\\Sigma$ may be disconnected). If $h\\in \\C^{2,\\alpha}_{-q}(\\mathbb{R}^n \\setminus \\Omega)$ solves\n\\[ D\\Ric(h)= 0\\mbox{ in }\\mathbb{R}^n \\setminus \\Omega,\\quad \\mbox{ and } \\quad h^\\intercal = 0,~ DA(h) = 0\\mbox{  on }\\Sigma,\\]\nthen\n\\[ \nh=L_X \\bar{g} \n\\] \nwhere $X \\in \\C^{3,\\alpha}_{1-q}(\\mathbb{R}^n\\setminus \\Omega)$  is equal to a Killing vector field of $\\bar{g}$ (possibly the zero vector) on each connected component of $\\Sigma$.\n\\end{theorem}\n\n\\begin{proof}\nBy Lemma~\\ref{lemma:geodesic} and Remark~\\ref{remark:geodesic}, there is a vector field $V\\in \\C^{3,\\alpha}(\\mathbb{R}^n\\setminus \\Omega)$ such that $V=0$ on $\\Sigma$, $V$ vanishes outside a collar neighborhood of $\\Sigma$, and $h+L_V\\bar{g}$ satisfies the Cauchy boundary condition on $\\Sigma$ in the classical sense: for $i, j=0,1,\\dots, n-1$,  \n\\begin{align}\\label{equation:Cauchy}\n\\begin{split}\n\t\\big(h+L_V\\bar{g}\\big)(e_i, e_j) &= 0\\\\\n\t\\big(\\nabla_\\nu (h+L_V\\bar{g})\\big) (e_i, e_j) &= 0.\n\\end{split}\n\\end{align}\nWe extend $h+L_V\\bar{g}$ by defining $k\\in \\C^{1}_{-q}(\\mathbb{R}^n)$ as\n\\begin{align*}\n\tk &= h+L_V\\bar{g}\\quad \\mbox{ in } \\mathbb{R}^n\\setminus \\Omega\\\\\n\tk &=0 \\quad \\mbox{ in } \\Omega.\n\\end{align*}\n\nLet $Z\\in \\C^{1,\\alpha}_{1-q}(\\mathbb{R}^n)$ be a vector field that weakly solves $\\Delta Z = \\bi k$ in $\\mathbb{R}^n$. That is, by \\eqref{equation:Bianchi-vector}, $k+L_Z\\bar{g}$ is a weak solution to\n\\begin{align} \\label{equation:gauge}\n\t\\bi ( k+L_Z \\bar{g} )=0 \\quad \\mbox{ in } \\mathbb{R}^n.\n\\end{align}\nWe will show that  $k+L_Z\\bar{g}$ is a weak solution to $\\Delta (k+L_Z\\bar{g})=0$ in $\\mathbb{R}^n$. Observe that both $k, L_Z\\bar{g}$ weakly solve the linearized Ricci equation in $\\mathbb{R}^n$. For any $\\gamma\\in \\C^\\infty_c (\\mathbb{R}^n)$,\n\\begin{align*}\n\t\\int_{\\mathbb{R}^n}  k \\cdot (D\\Ric)^* (\\gamma) \\, d\\mathrm{vol}&= \\int_{\\mathbb{R}^n\\setminus \\Omega} (h+L_V \\bar{g}) \\cdot (D\\Ric)^* (\\gamma) \\, d\\mathrm{vol}\\\\\n\t&=  \\int_{\\mathbb{R}^n\\setminus \\Omega}  D\\Ric (h+L_V \\bar{g}) \\cdot \\gamma \\, d\\mathrm{vol}=0.\n\\end{align*}\nIn the second identity we use  integration by parts and note  the boundary integral on  $\\Sigma$ vanishes due to the boundary condition \\eqref{equation:Cauchy}. For $L_Z\\bar{g}$, we integrate by parts and directly compute $\\Div(D\\Ric)^* (\\gamma)=0$ to get\n \\[\n \t\\int_{\\mathbb{R}^n} L_{Z} \\bar{g}\\cdot (D\\Ric)^* (\\gamma)\\, d\\mathrm{vol}= \\int_{\\mathbb{R}^n} Z \\cdot \\Div(D\\Ric)^* (\\gamma)\\, d\\mathrm{vol} =0.\n \\]\nSince $k+L_Z\\bar{g}$ weakly solves the linearized Ricci equation, by the formula \\eqref{equation:adjoint-Ricci}  for $(D\\Ric)^*$,\n\\begin{align*}\n\t0 &= \\int_{\\mathbb{R}^n} (k+L_Z\\bar{g})\\cdot (D\\Ric)^* (\\gamma)\\, d\\mathrm{vol} \\\\\n\t&=\\int_{\\mathbb{R}^n} (k+L_Z\\bar{g})\\cdot \\left(-\\tfrac{1}{2} \\Delta \\gamma + \\bi^*(\\Div \\gamma) \\right)\\, d\\mathrm{vol}\\\\\n\t&=\\int_{\\mathbb{R}^n} (k+L_Z\\bar{g})\\cdot \\left(-\\tfrac{1}{2} \\Delta \\gamma \\right)\\, d\\mathrm{vol},\n\\end{align*}\nwhere we use \\eqref{equation:gauge} in the last equality. It follows that $\\Delta (k+L_Z\\bar{g})=0$ weakly in $\\mathbb{R}^n$.\n\nSince $k+L_{Z}\\bar g$ decays at the rate $O(r^{-q})$, by elliptic regularity and Lemma~\\ref{lemma:PDE},  $k+ L_{Z} \\bar{g}$ is identically zero, and thus\n\\begin{alignat*}{2}\nh&= -L_{Z+V} \\bar{g} \\quad &&\\mbox{in } \\mathbb{R}^n\\setminus \\Omega\\\\\nL_Z\\bar{g} &= 0 \\quad &&\\mbox{in } \\Omega.\n\\end{alignat*}\nLet $X = -(Z+V)$ and recall $V=0$ on $\\Sigma$. We conclude that $h=L_X \\bar{g}$, $X \\in \\C^{3,\\alpha}_{1-q}(\\mathbb{R}^n\\setminus \\Omega)$, and  $X$ is equal to a Killing vector on each connected component of $\\Sigma$. \n\n\n\\end{proof}\n\nIn view of Theorem~\\ref{theorem:trivial}, we will define the following linear spaces of vector fields with \\emph{zero} boundary value on $\\Sigma$ that are asymptotic to Killing vector fields. \n\n\\begin{definition}\\label{definition:harmonic}\nWe define $\\mathcal{X}$ to be the linear subspace of $\\C^{3,\\alpha}_{\\mathrm{loc}} (\\mathbb{R}^n\\setminus \\Omega)$ vector fields as \n\\begin{align*}\n\t\\mathcal{X} &= \\Big\\{\\mbox{ $X=0$ on $\\Sigma$, $X - K\\in  \\C^{3,\\alpha}_{1-q} (\\mathbb{R}^n\\setminus \\Omega)$ for some Killing vector $K$ of $\\bar g$}\\Big\\}.\n\\end{align*}\nLet  $\\mathcal{X}_0$ be the subspace of $\\mathcal{X}$ defined as\n\\[\n\t\\mathcal{X}_0 =  \\{ X\\in \\mathcal{X}: \\Delta X=0 \\mbox{ in } \\mathbb{R}^n\\setminus \\Omega\\}.\n\\]\n\\end{definition}\n\nDenote the number \n\\[\n\tN=\\frac{n(n+1)}{2}.\n\\] \nIt is the dimension of the space of Killing vectors on a connected open subset of $(\\mathbb{R}^n,\\bar{g})$, which is spanned by   $n$ translation vectors  $\\{ \\frac{\\partial}{\\partial x_1},\\dots, \\frac{\\partial}{\\partial x_n}\\}$ and  $\\tfrac{n(n-1)}{2}$ rotation vectors $\\{ x_i \\frac{\\partial}{ \\partial x_j} -x_j \\frac{\\partial}{ \\partial x_i} \\}_{i<j}$.\n\\begin{lemma} \\label{lemma:X_0}\nThe linear space $\\mathcal X_0$ is equal to\n\\[\n\t\\mathcal{X}_0 =\\left\\{ X: \\begin{array}{l}  \\Delta X=0 \\mbox{ in } \\mathbb{R}^n\\setminus \\Omega, \\, X = 0 \\mbox{ on } \\Sigma, \\\\\n\tX-K\\in \\C^{3,\\alpha}_{2-n}(\\mathbb{R}^n\\setminus \\Omega) \\mbox{ for some Killing vector $K$ of $\\bar{g}$}\n\t \\end{array}\\right\\}\n\\]\nand  is isomorphic to the space of Killing vector fields on $(\\mathbb R^n\\setminus\\Omega, \\bar{g})$. Hence, \n\\[\t\n\t\\Dim \\mathcal{X}_0=N.\n\\]\n\\end{lemma}\n\\begin{proof}\nLet $X\\in\\mathcal X_0$, and we write $X = K+Y$ where $K$ is a Killing vector and $Y= O^{3,\\alpha}(|x|^{1-q})$. Since a Killing vector is harmonic, the assumption $\\Delta X = 0$ implies  $\\Delta Y=0$. By harmonic expansion,  there are constants $a_1,\\dots, a_n$ such that $Y= a_i \\tfrac{\\partial}{\\partial x_i}+ O^{3,\\alpha}(|x|^{2-n})$. Replacing $K$ by $K+  a_i \\tfrac{\\partial}{\\partial x_i}$, we have $X = K+Z$ where  $Z= O^{3,\\alpha}(|x|^{2-n})$. Such decomposition of $X$ is unique. Define the homomorphism from $ \\mathcal{X}_0$ to the space of Killing vectors  by sending $X$ to $K$. The homomorphism is clearly surjective, and it is injective because if $X=O^{3,\\alpha}(|x|^{2-n})$ satisfies $\\Delta X=0$ in $\\mathbb{R}^n \\setminus \\Omega$ and $X=0$ on $\\Sigma$, then $X$ is zero by Lemma~\\ref{lemma:PDE}. \n\\end{proof}\n\n\n\nOur discussions in this section so far are valid for  boundary $\\Sigma$ having multiple components. The following corollary is  a fundamental reason that  connectedness of $\\Sigma$ is assumed in our main results. We have identified those Ricci flat deformations $h$ in Theorem~\\ref{theorem:trivial} by $h=L_X \\bar{g}$ for some vector field $X \\in \\C^{3,\\alpha}_{1-q}(\\mathbb{R}^n\\setminus \\Omega)$  equal to a Killing vector $K$ on each component of $\\Sigma$. If $\\Sigma$ is connected, then there is a \\emph{global} Killing vector field in $\\mathbb{R}^n\\setminus \\Omega$ equal to $K$ on $\\Sigma$, which we still denote by $K$. By subtracting the global Killing vector, we obtain $X-K\\in \\mathcal{X}$ and $h=L_{X-K} \\bar{g}$. It leads to the following corollary.\n  \\begin{corollary}\\label{corollary:trivial}\nLet $\\Omega\\subset \\mathbb{R}^n$ be a bounded open subset whose boundary $\\Sigma = \\partial \\overline{\\Omega}$ is an embedded hypersurface. Suppose $\\Sigma$ is connected. Let $h\\in \\C^{2,\\alpha}_{-q}(\\mathbb{R}^n \\setminus \\Omega)$. Then $h$ is a Ricci flat deformation at $\\bar g$ satisfying the Cauchy boundary condition if and only if\n$h = L_X \\bar{g}$ in $\\mathbb{R}^n\\setminus \\Omega$ for some $X\\in\\mathcal X$. In addition, such deformation satisfies the harmonic gauge  $\\bi h=0$ in $\\mathbb{R}^n\\setminus \\Omega$ if and only if $h = L_X \\bar{g}$ in $\\mathbb{R}^n\\setminus \\Omega$ for some $X\\in \\mathcal{X}_0$.\n\\end{corollary}\n\n\n\n\n\\section{Green-type identity for static vacuum deformations}\\label{section:Green}\n\n\nWe will derive several fundamental identities for static vacuum deformations in this section. Define the functional $\\mathscr{F}$ on an asymptotically flat pair  $(g, u)$ by\n\\begin{align*}\n\t\\mathscr{F}(g, u) &= -2(n-1)\\omega_{n-1} m_{\\mathrm{ADM}}(g) +\\int_{\\mathbb{R}^n\\setminus \\Omega} u R_g \\, d\\mathrm{vol}_g.\n\\end{align*}\nThis functional is a special case of the Regge-Teitelboim Hamiltonian \\cite{Regge-Teitelboim:1974}.\nThe first variation of $\\mathscr{F}$ has been computed in \\cite[Proposition 3.7]{Anderson-Khuri:2013} (Cf. \\cite[Lemma 3.1]{Miao:2007}).  The second variation formula follows from differentiating the first variation. We state both formulas below, where $A_g, H_g$ are all computed with respect to the unit normal $\\nu_g$ pointing away from $\\Omega$ and recall $S(g, u) = -u\\Ric_g + \\nabla^2_g u - (\\Delta_g u )g$.\n\n\\begin{lemma}\\label{lemma:variations}\nLet $(g(s), u(s))$ be a one-parameter family of asymptotically flat pairs. In the following formulas, we use $'$ to denote the $s$-derivative and write\n\\[\n\t(h(s), v(s)) = (g'(s), u'(s)).\n\\]\nWe may omit the argument $s$ when the context is clear. Then,\n\\begin{align*}\n\t\\ds \\mathscr{F}(g(s), u(s)) &= \\int_{\\mathbb{R}^n\\setminus \\Omega} \\Big\\langle \\big(S(g, u) + \\tfrac{1}{2} u R_g g, R_g\\big), \\big(h, v\\big)\\Big\\rangle_g \\, \\dvol_g\\\\\n\t&\\quad +\\int_\\Sigma \\Big\\langle \\big(uA_g - \\nu_g(u) g^\\intercal, 2u\\big), \\big(h^\\intercal, DH|_g(h)\\big) \\Big\\rangle_g \\, \\da_g\n\\end{align*}\nand\n\\begin{align*}\n\t&\\2ds \\mathscr{F}(g(s), u(s)) \\\\\n\t&=\\int_{\\mathbb{R}^n\\setminus \\Omega}\\bigg[ \\Big\\langle P(h, v),(h,v) \\Big\\rangle_g + \\Big\\langle \\big(S(g, u) + \\tfrac{1}{2} u R_g g, R_g\\big), (g'', u'')\\Big\\rangle_g\\bigg]\\, \\dvol_g\\\\\n\t&\\quad +\\int_\\Sigma  \\bigg[\\Big\\langle  Q(h,v), \\big(h^\\intercal, DH|_g(h)\\big)\\Big\\rangle_g+ \\Big\\langle \\big(uA_g - \\nu_g(u) g^\\intercal, 2u\\big), \\big(g''^\\intercal, H_g''\\big) \\Big\\rangle_g \\bigg] \\, \\da_g\n\\end{align*}\nwhere  the right hand sides are evaluated at $(g(s), u(s))$, the angle bracket $\\langle ,  \\rangle_g$  denotes the sum of the $g(s)$-inner product  of the corresponding components (i.e., $\\big\\langle (S_1, S_2), (h,v)  \\big\\rangle_g = S_1\\cdot h + S_2 v$),  and $P, Q$ are the linear differential operators defined by\n\\begin{align*}\n\tP(h, v) &= \\Big(\\big(S(g, u) + \\tfrac{1}{2} u R_g g\\big)', \\, R_g'\\Big) - \\Big(2\\big(S(g, u) + \\tfrac{1}{2} u R_g g\\big)\\circ h, 0\\Big) \\\\\n\t&\\quad + \\tfrac{1}{2} (\\tr_g h)  \\Big(S(g, u) + \\tfrac{1}{2} u R_g g,\\, R_g \\Big) \\quad \\quad  \\mbox{ in } \\mathbb{R}^n\\setminus \\Omega \n\t\\\\\n\tQ(h, v)&= \\Big( \\big(uA_g-\\nu_g(u) g^\\intercal\\big)', \\, 2v\\Big)  - \\Big(2\\big(uA_g-\\nu_g(u) g^\\intercal\\big)\\circ h^\\intercal, 0\\Big)\\\\\n\t&\\quad +\\tfrac{1}{2} (\\tr_g h^\\intercal )\\Big(uA_g - \\nu_g(u) g^\\intercal , \\,  2u \\Big) \\quad \\quad \\mbox{ on } \\Sigma.\n\\end{align*}\n\\end{lemma}\n\nWe can use the above variation formulas to identify the elements that are $\\mathcal{L}^2$-orthogonal to the range of the  map $(g, u)\\mapsto (-u\\Ric_g + \\nabla^2_g u, \\Delta_g u)$ and of its linearization. Recall the linear subspace of vector fields, $\\mathcal{X}$, defined in Definition~\\ref{definition:harmonic}. At an asymptotically flat pair $(g, u)$ with $u>0$, for any $X\\in \\mathcal{X}$ we define\n\\begin{align*}\n\t\\zeta_0 (X)&= \\Big( L_X g - \\big(\\Div_g X+ u^{-1} X(u) \\big)g, \\, - \\Div_g X + u^{-1} X(u)\\Big) \\\\\n\t\\kappa_0(X)&= \\Big(L_X \\bar{g} - (\\Div X)\\bar{g},  \\, -\\Div X\\Big),\n\\end{align*}\nwhere  $\\kappa_0(X)$ is obtained by evaluating $\\zeta_0(X)$ at $(g,u)=(\\bar g, 1)$. Using the operator $\\bi_g^*$ defined in \\eqref{equation:operators} and writing $\\bi^* = \\bi_{\\bar{g}}^*$, we may also re-express them as  \n\\begin{align}\\label{equation:cokernel-elements}\n\\begin{split}\n\t\\zeta_0 (X)&= \\Big( 2\\bi_g^* X - u^{-1} X(u) g, \\,- \\Div_g X + u^{-1} X(u)\\Big) \\\\\n\t\\kappa_0(X)&= \\Big(2\\bi^* X,  \\, -\\Div X\\Big).\n\\end{split}\n\\end{align}\n\n\\begin{proposition}\\label{proposition:cokernel}\nLet $(g, u)$ be an asymptotically flat pair in $\\mathbb{R}^n\\setminus \\Omega$ with $u>0$. For any deformation $(h, v) \\in \\C^{2}_{-q}(\\mathbb{R}^n\\setminus \\Omega)$ and any  $X\\in \\mathcal{X}$, the following holds:\n\\begin{align}\\label{equation:cokernel1}\n\t\\int_{\\mathbb{R}^n\\setminus \\Omega}  \\bigg\\langle \\Big(-u \\Ric_g + \\nabla^2_g u, \\Delta_g u\\Big),\\, \\zeta_0(X) \\bigg \\rangle_g\\, d\\mathrm{vol}_g =0\\\\\n\\label{equation:cokernel2}\n\t\\int_{\\mathbb{R}^n\\setminus \\Omega} \\bigg\\langle \\Big(-D\\Ric(h) + \\nabla^2 v, \\, \\Delta v\\Big),\\, \\kappa_0 (X)\\bigg\\rangle_{\\bar g} \\, d\\mathrm{vol}_{\\bar g} =0.\n\\end{align}\n\\end{proposition}\n\\begin{proof}\nFor $s$ sufficiently small, let $\\psi_s: \\mathbb{R}^n\\setminus\\Omega \\to \\mathbb{R}^n\\setminus \\Omega$ be the flow of $X$ such that $\\psi_0$ is the identity map. Let $(g_s, u_s) = (\\psi_s^* g, \\psi_s^* u)$ be the pull-back of $(g, u)$.  Because the ADM mass is invariant under such diffeomorphisms (see \\cite[Theorem 4.2]{Bartnik:1986}) and so is the volume integral in $\\mathscr{F}$, we have \n\\[\n\t\\mathscr{F}(g_s, u_s) = \\mathscr{F}(g, u).\n\\]\nUsing that $\\left. \\ps\\right|_{s=0} (g_s, u_s) = (L_X g, X(u))$, we apply the first variation formula in Lemma~\\ref{lemma:variations} and get\n\\begin{align*}\n\t0&=\\left. \\ds \\right|_{s=0}\\mathscr{F}(g_s, u_s) \\\\\n\t&=\\int_{\\mathbb{R}^n\\setminus \\Omega} \\Big\\langle \\big(-u\\Ric_g + \\nabla_g^2 u -(\\Delta_g u)g + \\tfrac{1}{2} u R_g g, R_g\\big), \\big(L_X g, X(u)\\big)\\Big\\rangle_g \\, \\dvol_g\n\\end{align*}\nwhere the boundary integral vanishes because $X=0$ on $\\Sigma$. Rearranging the integrands gives \\eqref{equation:cokernel1}. Equation \\eqref{equation:cokernel2} follows immediately by taking linearization of equation \\eqref{equation:cokernel1} at $(\\bar g,1)$.\n\\end{proof}\n\nThe above proof explores the property that the functional $\\mathscr{F}$ is invariant among diffeomorphisms  that fix both the boundary $\\Sigma$ and the ``structure of infinity'' of $\\mathbb{R}^n\\setminus \\Omega$, i.e. those diffeomorphisms generated by $X\\in \\mathcal{X}$.  The first variation formula of $\\mathscr{F}$ implies that  $(L_X g, X(u))$ is $\\mathcal{L}^2$-orthogonal to $(S(g, u) + \\tfrac{1}{2} u R_g g, R_g)$. Since we will instead work on the operator $(g, u)\\mapsto (-u\\Ric_g + \\nabla^2_g u, \\Delta_g u)$ in Section~\\ref{section:existence}, after rearranging from $(S(g, u) + \\tfrac{1}{2} u R_g g, R_g)$ to $(-u\\Ric_g + \\nabla^2_g u, \\Delta_g u)$, the ``geometric'' deformation  $(L_X g, X(u))$  leads to $\\zeta_0(X)$ which takes a less geometric form. \n\n\n\n\n\n\nAs is shown in \\cite[Proposition 3.5]{An:2020}, the next proposition says that the differential operator $(h, v)\\to \\Big(P(h, v),\\big (h^\\intercal, DH|_g(h)\\big)\\Big)$ is ``self-adjoint'' in the following sense. \n\\begin{proposition}[Green-type identity]\\label{proposition:Green}\nLet $(g, u)$ be an asymptotically flat pair in $\\mathbb{R}^n\\setminus \\Omega$. For any $(h, v), (k,w)\\in \\C^{2}_{-q}(\\mathbb{R}^n\\setminus \\Omega)$, we have \n\\begin{align}\\label{equation:Green}\n\\begin{split}\n\t&\\int_{\\mathbb{R}^n\\setminus \\Omega}\\Big\\langle P(h, v),(k,w)\\Big\\rangle_g \\, \\dvol_g-\\int_{\\mathbb{R}^n\\setminus \\Omega}\\Big\\langle  P(k, w), (h, v)\\Big\\rangle_g \\, \\dvol_g\\\\\n\t&=- \\int_\\Sigma  \\Big\\langle  Q(h,v) , \\big(k^\\intercal, DH|_g(k)\\big)\\Big\\rangle_g \\, \\da_g+\\int_\\Sigma  \\Big\\langle Q(k, w), \\big(h^\\intercal, DH|_g(h)\\big)\\Big \\rangle_g \\, \\da_g.\n\\end{split}\n\\end{align}\n\\end{proposition}\n\\begin{proof}\nLet $\\big(g(s,t), u(s,t)\\big)$ be a two-parameter family of asymptotically flat pairs defined by\n\\begin{align*}\n\tg(s,t) = g+sh+tk\\quad \\mbox{ and } \\quad u(s,t)=u+sv+tw. \n\\end{align*}\nThen we have\n\\begin{align*}\n\t(h, v )= \\ps (g(s,t), u(s,t))\\quad(k,w)= \\pt (g(s,t), u(s,t) ),\\quad\\mbox{ at }t=s=0.\n\\end{align*}\nComputing the two-derivatives of the functional $\\mathscr{F}$ at $(g(t,s), u(t,s))$ and equating\n\\[\n\t\\left. \\pst \\right|_{t=s=0}\\mathscr{F}(g(t, s), u(t, s))  = \\left.\\pts\\right|_{t=s=0}\\mathscr{F}(g(t, s), u(t, s))\n\\]\ngives \\eqref{equation:Green}. (Note that the integrands involving the two derivatives such as $\\pst (g, u)$, $\\pst (g^\\intercal, H_g)$  all cancel out by symmetry $\\pst = \\pts$.) \n\\end{proof}\n\n We shall call the identity \\eqref{equation:Green} as a \\emph{Green-type identity} (for the operator $P$). To explain this, we briefly recall the classical Green identity for the Laplace operator. Let $\\nu$ be the unit normal to $\\Sigma$ pointing away from $\\Omega$.  For any functions $v, w\\in \\C^{2}_{-q}(\\mathbb{R}^n \\setminus \\Omega)$, the Green identity says\n\\begin{align*}\n\t\\int_{\\mathbb{R}^n \\setminus \\Omega}(w\\Delta v - v\\Delta w) \\, \\dvol = \\int_\\Sigma \\left( -w \\nu(v) + v \\nu(w) \\right)\\, \\da.\n\\end{align*}\n Our Green-type identity~\\eqref{equation:Green} resembles  the classical one by replacing the operator $\\Delta$ with the operator $P$, the Dirichlet boundary data with the Bartnik boundary data, and the Neumann boundary data with the operator~$Q$. \n \nWhen the variations are evaluated at a static vacuum pair $(g, u)$, the operator $P(h,v)$ becomes\n\\begin{align*}\n\tP(h, v)&= \\bigg(\\Big(S(g, u) + \\tfrac{1}{2} u R_g g\\Big)' , R_g' \\bigg) \\\\\n\t&=  \\bigg(DS|_{(g, u)}(h,v)  , DR|_g(h)\\bigg) + \\bigg( \\tfrac{1}{2} u DR|_g(h) g, 0\\bigg).\n\\end{align*}\n\nNote that the first parenthesis in the right hand side corresponds to the linearized static vacuum system~\\eqref{lsv}. Consequently,  $(h, v)$ is a static vacuum deformation at $(g, u)$  if and only if $P(h, v)=0$. \n \n\nWe will further focus on the special case when the linearization is taken at $(\\bar{g},1)$. We often omit the subscript $g$ when the geometric quantities (such as covariant derivatives, trace, linearization, inner products, volume\/area forms, second fundamental form, etc)  are computed with respect to $g=\\bar{g}$. With respect to $(\\bar{g}, 1)$, the operators $P, Q$ take the simpler forms:\n\\vspace{-8pt}\n\\begin{align}\\label{equation:flat}\n\\begin{split}\n\tP(h, v) &=  \\Big(-D\\Ric(h) + \\nabla^2 v + \\big(- \\Delta v + \\tfrac{1}{2} DR(h) \\big)\\bar{g}, \\, DR(h) \\bigg)\\\\% \\quad  \\mbox{ in } \\mathbb{R}^n\\setminus \\Omega\\\\\n\tQ(h, v)&= \\Big( vA + DA(h) - \\nu(v) \\bar{g}^\\intercal -2A\\circ h^\\intercal+ \\tfrac{1}{2} \\tr h^\\intercal A, \\, 2v + \\tr h^\\intercal \\Big)\n\\end{split}\n\\end{align}\n\n\nWe  re-state Lemma~\\ref{lemma:variations} in the following special case that will be used in Proposition~\\ref{proposition:variation2} later.\n\\begin{corollary}\\label{corollary:variations}\nLet $(g(s), u(s))$ be a one-parameter family of asymptotically flat pairs. Assume that $g(0) = \\bar{g}$ and $u(s)\\equiv 1$. Denote by $h= g'(0)$. Then  \n\\begin{align*}\n\t\\left. \\ds\\right|_{s=0} \\mathscr{F}(g(s), 1) &=\\int_\\Sigma  \\Big(A\\cdot h^\\intercal +  2DH(h) \\Big)\\, \\da\\\\\n\t\\left. \\2ds\\right|_{s=0} \\mathscr{F}(g(s), 1) &=\\int_{\\mathbb{R}^n\\setminus \\Omega} h\\cdot \\Big(-D\\Ric(h)+\\tfrac{1}{2} DR(h) \\bar{g}\\Big) \\, \\dvol\\\\\n\t+ \\int_\\Sigma \\Big( \\big(DA(h)  - &2A\\circ h^\\intercal+ \\tfrac{1}{2} \\tr h^\\intercal A\\big)\\cdot h^\\intercal+ \\tr h^\\intercal DH(h)+  A\\cdot\\big(g''\\big)^\\intercal+2 H_g'' \\Big) \\, \\da.\n\\end{align*}\n\\end{corollary}\n\nThe next corollary is a direct consequence of Proposition~\\ref{proposition:Green}.  In particular, Equations \\eqref{equation:Green-boundary} and \\eqref{equation:Ricci-flat} below give effective formulas to compute $DA(h)$, which is essential to our main arguments in Theorems \\ref{theorem:static-convex} and \\ref{theorem:static-generic}. \n\n\\begin{corollary}\nFor any $(h, v), (k,w)\\in \\C^2_{-q}(\\mathbb{R}^n\\setminus \\Omega)$, we have\n\\begin{align*}\n\t&\\int_{\\mathbb{R}^n\\setminus \\Omega}\\Big\\langle P(h,v),(k,w)\\Big\\rangle \\, \\dvol-\\int_{\\mathbb{R}^n\\setminus \\Omega}\\Big\\langle  P(k,w), (h, v)\\Big\\rangle \\, \\dvol\\\\\n\t&=- \\int_\\Sigma  \\Big\\langle Q(h,v), \\big(k^\\intercal, DH(k)\\big) \\Big\\rangle \\, \\da+\\int_\\Sigma  \\Big\\langle Q(k, w), \\big(h^\\intercal, DH(h)\\big)\\Big \\rangle \\, \\da\n\\end{align*}\nwhere $P, Q$ take the forms \\eqref{equation:flat}. We also have the following special cases:\n\\begin{enumerate}\n\\item If both $(h, v)$ and $(k, w)$ are static vacuum deformations at $(\\bar g,1)$ and $h$ has zero Bartnik data $h^\\intercal=0, DH(h)=0$ on $\\Sigma$, then \n\\begin{align} \\label{equation:Green-boundary}\n\t\\int_\\Sigma  \\Big\\langle \\big(vA + DA(h) -\\nu(v) \\bar{g}^\\intercal, 2v \\big), \\big(k^\\intercal, DH(k)\\big) \\Big\\rangle \\, \\da=0.\n\\end{align}\n\\item If $h, k$ are both Ricci flat deformations at $ \\bar g$ and $h^\\intercal=0$ on $\\Sigma$, then  \n\\begin{align}\\label{equation:Ricci-flat}\n\t\\int_\\Sigma\\Big( DA(h)\\cdot k^\\intercal -(\\tr k^\\intercal) DH(h) \\Big)\\, d\\sigma =0.\n\\end{align} \n\\end{enumerate}\n\\end{corollary}\n\n\n\\section{Sufficient condition for existence and local uniqueness}\\label{section:existence}\n\nLet $\\Omega$ be a bounded open subset in $(\\mathbb{R}^n, \\bar{g})$ whose boundary $\\Sigma= \\partial \\overline{\\Omega}$ is an embedded hypersurface. We assume that $\\Sigma$ is connected.   Recall the differential operators $\\bi_g, \\D_g, \\bi^*_g$ defined in \\eqref{equation:operators}, and note that the subscript $g$ is omitted when $g$ is the Euclidean metric $\\bar{g}$. We also recall  the spaces of vector fields $\\mathcal{X}, \\mathcal{X}_0$ in Definition~\\ref{definition:harmonic}, and the integer $N = \\frac{n(n+1)}{2}$ denotes the dimension of $\\mathcal{X}_0$.\n\n\n\nWe prove Theorem~\\ref{theorem:criterion-main} in this section. Use  the terminologies introduced in Section~\\ref{section:basic}, we re-state the definition of a static regular boundary:\n\\begin{manualdefinition}{\\ref{definition:static-regular}}\nThe boundary $\\Sigma$ is said to be \\emph{static regular in  $\\mathbb R^n\\setminus\\Omega$}  if for any static vacuum deformation   $(h, v)\\in \\C^{2,\\alpha}_{-q}(\\mathbb{R}^n\\setminus \\Omega)$ (at $(\\bar{g}, 1)$) such that $h$ satisfies the Bartnik boundary condition on $\\Sigma$, $h$ must satisfy the Cauchy boundary condition on $\\Sigma$. \n\\end{manualdefinition}\n\n\nThe following theorem is a more precise version of Theorem~\\ref{theorem:criterion-main}.\n\n\\begin{theorem}\\label{theorem:criterion}\nSuppose the boundary $\\Sigma$ is static regular in $\\mathbb R^n\\setminus\\Omega$. Then there exist positive constants $\\epsilon_0, C>0$ such that for each $0<\\epsilon<\\epsilon_0$, if $(\\tau, \\phi)$ satisfies $\\| (\\tau,\\phi)- (\\bar{g}^\\intercal, H) \\|_{\\C^{2,\\alpha}(\\Sigma)\\times \\C^{1,\\alpha}(\\Sigma)}<\\epsilon$, there exists a unique  $(g, u)\\in \\mathcal{M}$ with $\\|(g, u) - (\\bar{g}, 1) \\|_{\\C^{2,\\alpha}_{-q}(\\mathbb{R}^n \\setminus \\Omega)} < C\\epsilon$ such that \n\\begin{align}\\label{equation:vacuum-static-gauge}\n\\begin{split}\n\t\\begin{array}{ll}\n\t\\left\\{ \\begin{array}{l}-u\\Ric_g+ \\nabla^2_g u=0\\\\\n\t\\Delta_g u=0\\\\\n\t \\bi g+du=0\t\\end{array} \\right.  &\\mbox{ in } \\mathbb{R}^n\\setminus \\Omega \\\\\n \t\\left\\{ \\begin{array}{l} \n\tg^\\intercal = \\tau\\\\\n\tH_g = \\phi\n\t\\end{array} \\right. &\\mbox{ on } \\Sigma\n\\end{array}\n\\end{split}\n\\end{align}\nand \n\\begin{align}\\label{equation:orthogonal-gauge}\n\t\\int_{\\mathbb{R}^n\\setminus \\Omega}  \\big((g-\\bar{g})\\cdot L_X \\bar{g} \\big) \\rho\\, d\\mathrm{vol} =0 \\quad \\mbox{ for all } X\\in \\mathcal{X}_0\n\\end{align}\nwhere $\\rho(x)=(1+|x|^2)^{-1}$. \n\\end{theorem}\n\n\nWe will formulate  Theorem~\\ref{theorem:criterion} into an equivalent statement, Theorem~\\ref{theorem:T}, in Section~\\ref{section:kernel}   and give its proof  at the end of Section~\\ref{section:proof}. \nBefore we proceed with the proofs to the main result, we make general remarks on the gauge conditions that appear in Theorem~\\ref{theorem:criterion}.  \n\\begin{definition}\nLet $(g, u)\\in \\mathcal{M}$.\n\\begin{enumerate}\n\\item $(g, u)$ is said to satisfy the \\emph{static-harmonic gauge}  (at $(\\bar{g},1)$) in $\\mathbb{R}^n\\setminus \\Omega$ if $\\bi g + du=0$ in $\\mathbb{R}^n\\setminus \\Omega$. \n\\item $(g, u)$ is said to satisfy the \\emph{orthogonal gauge} (at $(\\bar{g},1)$) if \\eqref{equation:orthogonal-gauge} holds. \n\\end{enumerate}\n\\end{definition}\n\n\\begin{remark}\n We remark the connection between the static-harmonic gauge and harmonic gauge. \nObviously if $u$ is constant, then those two are the same, but there is a deeper connection. If $u>0$, the warped product metric $\\mathbf{g}=\\pm u^2dt^2+g$ (either plus or minus sign in  the first term) satisfies the classical harmonic gauge at the flat metric if and only if $\\bi g+ udu=0$. Although this equation is not exactly the static-harmonic gauge equation in the above definition, it yields the same linearized gauge equation for deformations at $(\\bar{g}, 1)$.  All of the following analysis would work equally well if we would have defined the static-harmonic gauge as $\\bi g + udu=0$ instead. \n\n The choice of the positive weight function $\\rho$ in \\eqref{equation:orthogonal-gauge} is quite arbitrary. In fact, for $n>3$, we can simply let $\\rho(x)\\equiv 1$ instead. When $n=3$, we need a positive weight function that decays at least  faster than the order of  $|x|^{-\\frac{1}{2}}$ to ensure the integral is finite.\n\n\\end{remark}\n\n\n\nIt is well-known that the harmonic gauge is effective for studying the Ricci curvature equation on a closed manifold or a bounded manifold with fixed boundary conditions. But for asymptotically flat manifolds, we also need to take account of diffeomorphisms that may not be the identity map at infinity but an isometry (up to negligible terms).  More precisely, let $\\mathscr{D} $ denote the space of all $\\C^{3,\\alpha}_{\\mathrm{loc}}$ diffeomorphisms $\\psi: \\mathbb{R}^n\\setminus\\Omega \\to  \\mathbb{R}^n\\setminus\\Omega$ satisfying $\\psi|_\\Sigma = \\mathrm{Id}_{\\Sigma}$, $\\psi(x) - Ox\\in \\C^{3,\\alpha}_{1-q}(\\mathbb{R}^n\\setminus \\Omega)$ for some special orthogonal matrix $O$. Consider the canonical action $\\mathscr{D}$ on the space of asymptotically flat pairs $(g, u)\\in \\mathcal{M}$ by $\\psi\\cdot (g, u) =( \\psi^* g, \\psi^*u)$. We will see in the next lemma that the static-harmonic gauge is not sufficient to find a unique representative in the orbit $\\mathscr{D}\\cdot (g, u)$,  but there is a unique one in a neighborhood of $(\\bar{g}, 1)$ if we additionally impose the orthogonal gauge. \n\n\n\\begin{lemma}\\label{lemma:both-gauge}\n\n\nThere is an open neighborhood $\\mathcal{U}$ of $(\\bar{g}, 1)$ in $\\mathcal{M}$  and an open neighborhood $\\mathscr{D}_0$ of $\\mathrm{Id}_M$ in $\\mathscr{D}$ such that for any $(g, u)\\in \\mathcal{U}$, there exists a unique diffeomorphism $\\psi\\in \\mathscr{D}_0$ such that  $(\\psi^*g, \\psi^*u)$ satisfies both static-harmonic and orthogonal gauges.\n\\end{lemma}\n\\begin{proof}\nWe denote by $\\langle h, k\\rangle_{\\mathcal{L}^2_\\rho} = \\int_{\\mathbb{R}^n\\setminus \\Omega} (h\\cdot k )\\rho \\,d\\mathrm{vol}$, where the dot product is with respect to $\\bar{g}$ and  we recall $\\rho(x) = (1+|x|^2)^{-1}$. Since $\\mathcal{X}_0$ is $N$-dimensional (see Lemma~\\ref{lemma:X_0}),  we let $X^{(1)}, \\dots, X^{(N)}$ be a basis of $\\mathcal{X}_0$ that is orthonormal in the following sense: \n\\[\n\t\\big\\langle L_{X^{(i)}}\\bar{g} , L_{X^{(j)}}\\bar{g} \\big\\rangle_{\\mathcal{L}^2_\\rho} =\\delta_{ij}.\n\\] \nConsider the map $G: \\mathscr{D}\\times \\mathcal{M}\\to \\C^{1,\\alpha}_{-q-1}\\times \\mathbb{R}^N$ defined by\n\\[\n\tG(\\psi, (g, u))= \\big(\\bi (\\psi^* g ) + d(\\psi^* u), (b_1, \\dots, b_N) \\big)\n\\]\nwhere the numbers $b_i$ are given by\n\\[\nb_i = \\big\\langle\\psi^* g -\\bar{g},  L_{X^{(i)}}\\bar{g}\\big\\rangle_{\\mathcal{L}^2_\\rho}. \n\\]\n Linearizing $\\mathscr{D}$ in the first argument at $( \\mathrm{Id}_{\\mathbb R^n\\setminus\\Omega}, (\\bar{g}, 1))$ gives $D_1 G: \\mathcal{X} \\to \\C^{1,\\alpha}_{-q-1}(\\mathbb{R}^n\\setminus \\Omega) \\times \\mathbb{R}^N$:\n\\[\n \tD_1 G(X) =  \\big(-\\Delta X, (c_1, \\dots, c_N) \\big)\n\\]\t\nwhere  $c_i = \\langle L_X\\bar{g},  L_{X^{(i)}}\\bar{g}\\rangle_{\\mathcal{L}^2_\\rho} $. \n\nWe show that $D_1 G$ is an isomorphism: If $D_1G(X)=0$, then $\\Delta X=0$ and $c_i=0$ for all $i$. The first equation implies  $X\\in\\mathcal X_0$, and hence, the equations $c_i=0$ imply  $X=0$. Thus $D_1G$ is injective. To see that it is surjective, for any $Z\\in\\C^{1,\\alpha}_{-q-1}$ and $(c_1,\\dots, c_N)\\in  \\mathbb R^N$, there is $Y\\in \\mathcal{X}$ such that $-\\Delta Y=Z$  by Lemma~\\ref{lemma:PDE}.  Let $d_i=c_i -\\langle L_Y\\bar{g},  L_{X^{(i)}}\\bar{g}\\rangle_{\\mathcal{L}^2_\\rho} $, then $X=Y+ d_1X^{(1)}+ \\dots+ d_N X^{(N)}$ satisfies $D_1G(X)=(Z, (c_1,\\dots, c_N))$. Then the lemma follows from Implicit Function Theorem.\n\\end{proof}\n\n\\subsection{Boundary value problem under static-harmonic gauge}\\label{section:kernel}\n\nDefine the map $T:\\mathcal M\\to \\C^{0,\\alpha}_{-q-2}(\\mathbb R^n\\setminus\\Omega)\\times\\C^{1,\\alpha}(\\Sigma)\\times \\big((\\bar{g}^\\intercal, H_{\\bar{g}}) +\\mathcal{B}(\\Sigma)\\big)$ as \n\\begin{align}\\label{equation:nonlinear}\nT (g, u)=\n\\begin{array}{l}\n\t\\left\\{ \\begin{array}{l}-u\\Ric_g+ \\nabla^2_g u - \\D_g (\\bi g+du)\\\\\n\t\\Delta_g u\t\\end{array} \\right. \\quad \\mbox{ in } \\mathbb{R}^n\\setminus \\Omega \\\\\n \t\\left\\{ \\begin{array}{l} \n\t\\bi g+du\\\\\n\tg^\\intercal\\\\\n\tH_g\n\t\\end{array} \\right. \\quad \\mbox{ on } \\Sigma.\n\\end{array}\n\\end{align}\nWe note that the operator $T$ is essentially equivalent to the harmonic-gauged Ricci operator of the warped product space, introduced by Anderson and Khuri \\cite[Section 3]{Anderson-Khuri:2013}.  Let us explain the codomain of $T$. We denote by $\\C^{0,\\alpha}_{-q-2}(\\mathbb{R}^n \\setminus \\Omega)$ the codomain of the first two equations and by $\\C^{1,\\alpha}(\\Sigma)\\times  \\big((\\bar{g}^\\intercal, H_{\\bar{g}}) +\\mathcal{B}(\\Sigma)\\big)$ the codomain of the boundary equations. More specifically, $\\C^{1,\\alpha}(\\Sigma)$ consists of $\\C^{1,\\alpha}$ covectors of $\\mathbb{R}^n$ defined along $\\Sigma$ and $\\mathcal{B}(\\Sigma)$ consists of pairs $(\\tau, \\phi)$ where $\\tau\\in \\C^{2,\\alpha}(\\Sigma)$ is a symmetric $(0,2)$-tensor on the tangent bundle of $\\Sigma$  and $\\phi \\in \\C^{1,\\alpha}(\\Sigma)$ is a scalar-valued function on $\\Sigma$. \n\nThe reader should compare $T$ with~\\eqref{equation:vacuum-static-gauge}. While Theorem~\\ref{theorem:criterion} concerns with solving both \\eqref{equation:vacuum-static-gauge} and \\eqref{equation:orthogonal-gauge}, in the rest of Section~\\ref{section:existence}, we will focus on solving $T$ and an accompanying operator $\\overline{T}$ modified from $T$. It is because that, as shown in the next lemma, if we  put aside the orthogonal gauge condition  \\eqref{equation:orthogonal-gauge},   finding $(g, u)$ solving \\eqref{equation:vacuum-static-gauge} is ``equivalent'' to finding some $(g, u)$ solving $T(g,u)=(0, 0, 0, \\tau, \\phi)$ (namely, the first three equations in \\eqref{equation:nonlinear} are zero), provided that $(g, u)$ is sufficiently close to $(\\bar{g}, 1)$. \n\n\\begin{lemma}[Cf. {\\cite[Proposition 2.1]{Anderson-Khuri:2013}}]\\label{lemma:gauge}\nThere is an open neighborhood $\\mathcal{U}$ of $(\\bar{g}, 1)$ in $\\mathcal{M}$ such that for $(g, u)\\in \\mathcal{U}$,  if $(g, u)$ satisfies $T(g, u) = (0, 0, 0, \\tau, \\phi)$, then $(g, u)$ solves \\eqref{equation:vacuum-static-gauge}.\n\nConversely, if $(g, u)$ solves  \\eqref{equation:vacuum-static-gauge}, then there exists a diffeomorphism $\\psi\\in \\mathscr{D}_0$  such that $(\\psi^* g, \\psi^*u)$ satisfies $T(\\psi^* g, \\psi^*u) = (0, 0, 0, \\tau, \\phi)$. \n\\end{lemma}\n\n\\begin{proof}\nLet $(g, u)\\in \\mathcal{U}$ solve $T(g, u)=(0,0,0,\\tau,\\phi)$. We will  prove $\\bi g+du=0$ in $\\mathbb R^n\\setminus\\Omega$, provided $\\mathcal{U}$ sufficiently small. \nTaking the trace of the first tensor equation in $T(g, u)=(0,0,0,\\tau,\\phi)$ and using the harmonic equation of $u$, we get \n\\[ \n\tR_g= -u^{-1} \\tr_g \\D_g (\\bi g+du) = -u^{-1} \\Div_g (\\bi g+ du).\n\\]\nWe show that $\\bi g+du$ weakly solves the following equation:\n\\begin{align}\n\t\\bi_g \\D_g (\\bi g+du) &= -\\tfrac{1}{2} R_g du\\label{equation:1}\\\\\n\t&=\\tfrac{1}{2} u^{-1} du \\Div_g (\\bi g+ du).\\notag\n\\end{align}\nThe second identity follows by substituting $R_g$ by the earlier computation. Since by assumption $\\mathcal D_g(\\beta g+du)=-u{\\rm Ric}_g+\\nabla^2_gu$, to prove \\eqref{equation:1} it suffices to show for all vector fields $X\\in \\C^\\infty_c(\\mathbb{R}^n\\setminus \\Omega)$\n\\begin{align*}\n\t\\int_{\\mathbb{R}^n\\setminus \\Omega} (-u\\Ric_g + \\nabla_g^2 u) \\cdot \\bi_g^*X \\, d\\mathrm{vol}_g = \\int_{\\mathbb{R}^n\\setminus \\Omega}- \\tfrac{1}{2} R_g  X(u) \\, d\\mathrm{vol}_g.\n\\end{align*} \nThis identity holds due to \\eqref{equation:cokernel1} and that $u$ is harmonic. (Note that if $(g, u)$ were $\\C^3_{\\mathrm{loc}}$, we can simply apply $\\bi_g$ on the first equation of $T(g, u)=(0,0,0,\\tau,\\phi)$ to obtain \\eqref{equation:1}.)\n\n\n\nRecall that $\\bi_g \\D_gV = -\\tfrac{1}{2} \\Delta_g V -\\tfrac{1}{2} \\Ric_g(V, \\cdot)$ for any vector field $V$. If we denote by $V= \\bi g+du \\in \\C^{1,\\alpha}_{-q-1}(\\mathbb{R}^n\\setminus \\Omega)$, then the previous equation with the boundary condition $ \\bi g+du=0$ on $\\Sigma$ implies that $V$ weakly solves the boundary value problem\n\\begin{align*}\n\t\\Delta_g V + u^{-1} du \\Div_g V + \\Ric_g(V, \\cdot)&=0\\quad \\mbox{ in } \\mathbb{R}^n\\setminus \\Omega \\\\\n\tV&=0\\quad \\mbox{ on } \\Sigma.\n\\end{align*}\nBy elliptic regularity,  $V\\in \\C^{2,\\alpha}_{-q-1}(\\mathbb{R}^n\\setminus \\Omega)$. If  $\\mathcal{U}$ is sufficiently small, for $(g, u)\\in \\mathcal{U}$ the operator on $V$ in the previous first equation is sufficiently close to $\\Delta$ in the operator norm, and thus by Lemma~\\ref{lemma:PDE} the above boundary value problem has only the trivial solution $V=0$. \n\n\nFor the converse statement, we can simply take the diffeomorphism $\\psi$ obtained from Lemma~\\ref{lemma:both-gauge}.\n\\end{proof}\n\n\nAccording to the above discussions, we can reformulate Theorem~\\ref{theorem:criterion} into the following theorem for the operator $T$.\n\n\\begin{manualtheorem}{\\ref{theorem:criterion}$^\\prime$}\\label{theorem:T}\nSuppose the boundary $\\Sigma$ is static regular in $\\mathbb R^n\\setminus\\Omega$. Then there exist positive constants $\\epsilon_0, C>0$ such that for each $0<\\epsilon<\\epsilon_0$, if $(\\tau, \\phi)$ satisfies $\\| (\\tau,\\phi)- (\\bar{g}^\\intercal, H) \\|_{\\C^{2,\\alpha}(\\Sigma)\\times \\C^{1,\\alpha}(\\Sigma)}<\\epsilon$, there exists a unique $(g, u)\\in \\mathcal{M}$ with $\\|(g, u) - (\\bar{g}, 1) \\|_{\\C^{2,\\alpha}_{-q}(\\mathbb{R}^n \\setminus \\Omega)} < C\\epsilon$ such that $T(g, u) = (0, 0, 0, \\tau, \\phi)$ and $(g, u)$ satisfies the orthogonal gauge \\eqref{equation:orthogonal-gauge}.\n\\end{manualtheorem}\n\nOne may wish for a stronger statement that the linearization of $T$ at $(\\bar{g},1)$ is surjective, and thus $T(g, u)$ is solvable for \\emph{all} values sufficiently close to $T(\\bar{g},1)$  by Local Surjectivity Theorem, not only the values that take the particular form $(0, 0, 0, \\tau, \\phi)$. Unfortunately, as we will see in the next lemmas, the linearization is never surjective. \n\n\nDenote the  linearization  of $T$ at $(\\bar{g}, 1)$ by $L:\\C^{2,\\alpha}_{-q}(\\mathbb{R}^n\\setminus \\Omega) \\to\\C^{0,\\alpha}_{-q-2}(\\mathbb{R}^n\\setminus \\Omega)\\times\\C^{1,\\alpha}(\\Sigma)\\times\\mathcal{B}(\\Sigma)$. For $(h, v) \\in \\C^{2,\\alpha}_{-q}(\\mathbb{R}^n \\setminus \\Omega)$,\n\\begin{align}\\label{equation:linear}\nL(h, v)=\n\\begin{array}{l}\n\t\\left\\{ \\begin{array}{l}-D\\Ric(h)+ \\nabla^2 v - \\D(\\bi h+ dv) \\\\\n\t\\Delta v\t\\end{array} \\right. \\quad \\mbox{ in } \\mathbb{R}^n\\setminus \\Omega \\\\\n \t\\left\\{ \\begin{array}{l} \\bi h + dv\\\\\n\th^\\intercal\\\\\n\tDH(h)\n\t\\end{array} \\right. \\quad \\mbox{ on } \\Sigma.\n\\end{array}\n\\end{align}\n\nBy \\eqref{equation:Ricci-gauge}, the first equation in \\eqref{equation:linear} is reduced to the Laplace equation:\n\\begin{align*}\n-D\\Ric(h)+ \\nabla^2 v- \\D(\\bi h+ dv) = \\tfrac{1}{2} \\Delta h.\n\\end{align*}\nBased on the observation of Anderson and Khuri for the corresponding harmonic-gauged Ricci operator in the warped product space~\\cite[Proposition 3.1]{Anderson-Khuri:2013}, the map $L$ is Fredholm of index zero. \n\n\\begin{lemma}[Cf.  {\\cite[Proposition 3.1]{Anderson-Khuri:2013}}]\\label{lemma:Fredholm}\nThe linearized operator $L: \\C^{2,\\alpha}_{-q}(\\mathbb{R}^n\\setminus \\Omega) \\to \\C^{0,\\alpha}_{-q-2}(\\mathbb{R}^n \\setminus \\Omega)\\times \\C^{1,\\alpha}(\\Sigma)\\times \\mathcal{B}(\\Sigma)$ is Fredholm of index zero.\n\\end{lemma}\n\\begin{proof}\nThe proof is essentially the same as that of~\\cite[Proposition 3.1]{Anderson-Khuri:2013} for the corresponding operator.  So we only give an outline how it adapts in our setting.  By examining the principle symbol, the boundary condition of $L$ is elliptic (i.e., it satisfies the  Lopatinski-Shapiro condition), so the operator $L$ is Fredholm, see  \\cite[Theorem 20.1.2]{Hormander:2007} and  \\cite[Theorem 1.3]{Lockhart-McOwen:1985}.  To show that the index of $L$ is zero, we argue as in \\cite{Anderson-Khuri:2013} that $L$ is homotopic to a Fredholm operator of index zero through a path of Fredholm operators. Since a path is not explicitly written down in \\cite{Anderson-Khuri:2013}, we include it for completeness. Let $s\\in [0,1]$ and define the family of operators $L^{(s)}: \\C^{2,\\alpha}_{-q}(\\mathbb{R}^n\\setminus \\Omega) \\to \\C^{0,\\alpha}_{-q-2}(\\mathbb{R}^n \\setminus \\Omega)\\times \\C^{1,\\alpha}(\\Sigma)\\times \\mathcal{B}(\\Sigma)$ by varying only the boundary equations:\n\\begin{align*}\n\\begin{array}{l}\n\tL^{(s)}(h,v)=\n\t\\begin{array}{l}\\left\\{ \\begin{array}{l}\\tfrac{1}{2} \\Delta h \\\\\n\t\\Delta v\t\\end{array} \\right. \\quad \\mbox{ in } \\mathbb{R}^n\\setminus \\Omega \\\\\n \t\\left\\{ \\begin{array}{l} (1-s)(\\bi h+dv)(\\nu)+s\\nu(v)\\\\\n\t(1-s)(\\bi h+dv)^\\intercal-s\\nabla_\\nu\\omega\\\\\n\th^\\intercal\\\\\n\t(1-s)DH(h)+s(\\nabla_\\nu h)(\\nu,\\nu)\n\t\\end{array} \\right. \\quad \\mbox{ on } \\Sigma\n\\end{array}\n\\end{array}\n\\end{align*}\nwhere $\\omega(e_a) := h(\\nu, e_a)$ is a one-form defined on the tangent bundle of $\\Sigma$. By checking the principal symbol, the boundary condition in $L^{(s)}$ is elliptic and thus $L^{(s)}$ is Fredholm for each $s\\in[0,1]$. Note that $L^{(0)}=L$ and $L^{(1)}$ is the operator with a standard Dirichlet\/Neumann boundary condition. It is straightforward to verify that $L^{(1)}$ is an isomorphism, and thus of index $0$. \n\\end{proof}\n\nThe kernel space of the linearized operator $L$, however,   is never trivial and hence $L$ is not surjective.  We end this section with the fact that $\\Ker L$ always contains an $N$-dimensional subspace, generated by $\\mathcal{X}_0$. The lemma is a direct consequence of Corollary~\\ref{corollary:trivial}. A fundamental reason that $\\Ker L$ is not trivial is that, as is shown in Lemma~\\ref{lemma:both-gauge}, the static-harmonic gauge by itself cannot guarantee uniqueness of solutions to the operator $T$.\n\n\\begin{lemma}The kernel space of $L$ contains an $N$-dimensional subspace:\n\\[\n\t\\Ker L \\supseteq \\big\\{ (h, 0): h=L_X \\bar{g}, \\mbox{ for some } X \\in \\mathcal{X}_0\\big\\}.\n\\] \n\\end{lemma}\n\n\n\n\\subsection{Kernel and range for a static regular boundary}\nFrom now on, we assume the boundary $\\Sigma$ is static regular in $\\mathbb R^n\\setminus\\Omega$. The next lemma says that  the kernel elements of $L$ are exactly those arising from $X\\in \\mathcal{X}_0$. \n\n\\begin{lemma}\\label{lemma:kernel}\nIf the boundary $\\Sigma$ is static regular in  $\\mathbb R^n\\setminus\\Omega$, then \n\\[\n\t\\Ker L=\\big\\{ (h, 0): h=L_X \\bar{g}, \\mbox{ for some } X \\in \\mathcal{X}_0\\big\\}.\n\\] \nAs a direct consequence, $\\Ker L$ is isomorphic to $\\mathcal{X}_0$ and hence \n\\[\n\\Dim \\Ker L=N.\n\\] \n\\end{lemma}\n\\begin{proof}\nLet $(h, v) \\in \\Ker L$. By elliptic regularity, $(h, v)\\in \\C^\\infty_{-q}(\\mathbb{R}^n\\setminus \\Omega)$. By applying $\\bi$ on the first equation of \\eqref{equation:linear} and using $\\bi (-D\\Ric (h) + \\nabla^2 v)=0$, we have \n\\[\n\t\\Delta (\\bi h+ dv) = 0 \\quad \\mbox{ in } \\mathbb{R}^n\\setminus \\Omega.\n\\]\n Then by the  boundary condition $\\bi h+ dv=0$ on $\\Sigma$ and the fall-off rate of $\\bi h+ dv$, we see that $\\bi h+ dv$ is identically zero in $\\mathbb{R}^n\\setminus \\Omega$ and thus $(h,v)$ is a static vacuum deformation. By the assumption that $\\Sigma$ is static regular, we have $DA(h)=0$ on $\\Sigma$.  \n \n\n\n The lemma would follow by showing that $v$ vanishes identically in $\\mathbb{R}^n\\setminus \\Omega$. Once we show that $v$ is zero, we have $(h,0)\\in{\\rm Ker}L$, i.e. $h$ is a Ricci flat deformation and satisfies $\\bi h=0$, and thus $h=L_X\\bar{g}$ for some $X\\in \\mathcal{X}_0$  by Corollary~\\ref{corollary:trivial}.\n \n Recall that $v$ is a harmonic function in $\\mathbb{R}^n\\setminus \\Omega$. Although $v$ is not  assumed explicitly to satisfy a boundary condition, we observe that  other equations of $(h, v)$ imply that $v$ satisfies the following (hidden) boundary conditions on $\\Sigma$:\n\\begin{align}\n\t\\Delta_\\Sigma v + \\nu(v) H&=0  \\label{equation:boundary}\\\\\n\tA ( \\nabla^\\Sigma v, \\cdot) - d(\\nu(v))&=0 \\label{equation:boundary-extra}\n\\end{align}\nwhere $\\nabla^\\Sigma, d$ are the covariant derivative and exterior derivative on $\\Sigma$, respectively. Let $X$ be a compactly supported vector field in $\\mathbb{R}^n$. \nApplying \\eqref{equation:Green-boundary} to the static vacuum deformation pairs $(h,v)$ and $(k, w)=(L_X \\bar{g}, 0)$ and noting $DA(h)=0$, we get  \n\\[\n\t\\int_\\Sigma \\Big\\langle \\big(vA-\\nu(v) \\bar{g}^\\intercal , 2v \\big), \\big((L_X \\bar{g})^\\intercal, DH(L_X \\bar{g}) \\big) \\Big \\rangle \\, d\\sigma=0.\n\\]\nIf we let $X=\\eta \\nu$ on $\\Sigma$ for a smooth function $\\eta$, then $L_X \\bar{g} = 2\\eta A$ and $DH(L_X \\bar{g}) = -\\Delta_\\Sigma \\eta - \\eta |A|^2$ where $\\Delta_\\Sigma$ is the Laplace operator of $(\\Sigma, \\bar{g}^\\intercal)$. Therefore, the previous identity implies that \n\\[\n\t\\int_\\Sigma \\eta \\left(\\Delta_\\Sigma v + \\nu(v) H\\right)\\, d\\sigma=0.\n\\]\n  Since $\\eta$ is arbitrary, it implies \\eqref{equation:boundary}.  (See \\cite[Section 2]{Anderson:2015} for a different proof of  \\eqref{equation:boundary}.) If we let $X$ be tangential to $\\Sigma$ and apply integration by parts,  we derive\n\\begin{align*}\n\t0&= \\int_\\Sigma \\Big\\langle \\big(vA-\\nu(v) \\bar{g}^\\intercal , 2v \\big), \\big(L_{X^\\intercal} \\bar{g}^\\intercal, X^\\intercal (H) \\big) \\Big \\rangle \\, d\\sigma\\\\\n\t&=\\int_\\Sigma 2 \\Big(- \\mathrm{div}_\\Sigma \\big(vA-\\nu(v) \\bar{g}^\\intercal\\big)  + v d H\\Big)\\cdot X^\\intercal\\, d\\sigma.\n\\end{align*}\nSince $X^\\intercal$ is arbitrary and by the Codazzi equation $\\mathrm{div}_\\Sigma A = dH$ on~$\\Sigma$, we derive \\eqref{equation:boundary-extra}. Furthermore,  \\eqref{equation:boundary} and \\eqref{equation:boundary-extra} for a harmonic function are equivalent to the following boundary condition:\n\\[\n\\nabla^2 v(\\nu, \\cdot)=0 \\quad \\mbox{ on } \\Sigma.\n\\] \nTo see that, since $v$ is harmonic, we have $0=\\Delta v = \\Delta_\\Sigma v+ \\nabla^2 v(\\nu, \\nu) + H\\nu(v)$. Together with \\eqref{equation:boundary}, it implies $\\nabla^2 v(\\nu, \\nu)=0$ on $\\Sigma$.  Letting $\\{ e_1, \\dots, e_{n-1},\\nu\\}$ be a local orthonormal frame, we write \\eqref{equation:boundary-extra} as \n\\begin{align*}\n\t0 &=\\sum_b A_{ab} \\nabla_{e_b} v - \\nabla_{e_a} (\\nu (v))\\\\\n\t&=\\sum_b A_{ab} \\nabla_{e_b} v - \\big( \\nabla^2 v(e_a, \\nu) + \\nabla_{\\nabla_{e_a} \\nu} v\\big)\\\\\n\t&= - \\nabla^2 v(e_a, \\nu). \n\\end{align*}\n\nLastly, we show that a harmonic function $v\\in \\C_{-q}^{2,\\alpha}(\\mathbb{R}^n\\setminus \\Omega)$ satisfying $\\nabla^2 v(\\nu, \\cdot)=0$ on $\\Sigma$ must vanish identically in $\\mathbb{R}^n\\setminus \\Omega$. Fix $i=1,\\dots, n$ and define $f = \\frac{\\partial v}{\\partial x_i}$ where $( x_1,\\dots, x_n)$ are the Cartesian coordinates. Since $\\Delta v=0$, we also have $\\Delta f=0$ in $\\mathbb{R}^n\\setminus \\Omega$. Compute the Neumann boundary data $\\nu(f)=  \\nabla^2 v(\\nu,  \\frac{\\partial}{\\partial x_i})=0$ on $\\Sigma$. Since $f$ decays at infinity, we have $f\\equiv 0$ in $\\mathbb{R}^n\\setminus \\Omega$ and thus $\\nabla v=0$ by varying $i$. Since $v\\to 0$ at infinity, we then conclude that $v$ is identically zero. \n\n \n\n\n\n\\end{proof}\n\nIn the following we define the Banach spaces $\\mathcal{E}$ and $\\mathcal{Q}$ associated with the operator $L$. We recall that the \\emph{dual space} of a normed linear space $\\mathcal{A}$, denoted by $\\mathcal{A}^*$, is the space of all bounded linear functionals on~$\\mathcal{A}$. \n\\begin{definition}\\label{definition:spaces}\n\\begin{enumerate}\n\\item The Banach space $\\mathcal{E}$ is a complementing space of $\\Ker L$ in $\\C^{2,\\alpha}_{-q}(\\mathbb{R}^n\\setminus \\Omega)$,\nconsisting of pairs of a symmetric $(0,2)$-tensor $h$ and a scalar-valued function $v$ in $\\mathbb{R}^n\\setminus \\Omega$, defined by\n\\[\n\t\\mathcal{E}=\\left\\{ (h, v)\\in \\C^{2,\\alpha}_{-q}(\\mathbb{R}^n\\setminus \\Omega): \\int_{\\mathbb{R}^n\\setminus \\Omega} \\big( h\\cdot L_X \\bar{g}  \\big) \\rho \\, d\\mathrm{vol} =0  \\mbox{ for all } X\\in \\mathcal{X}_0\\right\\}.\n\\]\nRecall the weight function $\\rho(x) =(1+|x|^2)^{-1}$. \n\\item\n The Banach space $\\mathcal{Q}$, as a subspace of \n\n the dual space of the codomain of $L$, is defined by \n\\begin{align*\n\t\\mathcal{Q}=\\Big\\{ \\kappa\n\t\\in &\\big(\\C^{0,\\alpha}_{-q-2}(\\mathbb{R}^n \\setminus \\Omega)\\times \\C^{1,\\alpha}(\\Sigma)\\times \\mathcal{B}(\\Sigma)\\big)^*\n\t: \\\\\n\t\\kappa&=\\big(2\\bi^* X,\\, -\\Div X, \\, -2\\bi^* X(\\nu, \\cdot), \\, 0, \\, 0\\big) \\mbox{ for some } X\\in \\mathcal{X}_0\\Big\\}.\n\\end{align*}\nWe may write $\\kappa(X)$ for elements in $\\mathcal{Q}$ to emphasize its dependence on~$X$. Here, $(2\\bi^* X, -\\Div X)\\in \\big(\\C^{0,\\alpha}_{-q-2}(\\mathbb{R}^n \\setminus \\Omega)\\big)^*$ is defined on $\\mathbb{R}^n\\setminus \\Omega$, $2\\bi^* X(\\nu, \\cdot)\\in \\big(\\C^{1,\\alpha}(\\Sigma)\\big)^*$ denotes the one-form that acts on  vector fields along $\\Sigma$ (normal or tangential), and the last two components $(0,0)$ of $\\kappa(X)$ belong to $(\\mathcal{B}(\\Sigma))^*$. The evaluation of $\\kappa\\in \\mathcal{Q}$ on $\\C^{0,\\alpha}_{-q-2}(\\mathbb{R}^n \\setminus \\Omega)\\times \\C^{1,\\alpha}(\\Sigma)\\times \\mathcal{B}(\\Sigma)$ is defined by the $\\mathcal{L}^2$-pairing. \n\\end{enumerate}\n\\end{definition}\n\n\n We remark that $\\mathcal{E}$ is defined so that  $(g, u)\\in  \\big((\\bar{g}, 1)+\\mathcal{E}\\big) \\cap \\mathcal{M}$ if and only if  $(g, u)$ satisfies the orthogonal gauge.  We also remark that $\\mathcal{Q}$ is isomorphic to $\\mathcal{X}_0$, and thus $\\Dim \\mathcal{Q}=N$.\n\n\n\n\n\nLet $(\\Range L)^\\perp\\subset  \\big(\\C^{0,\\alpha}_{-q-2}(\\mathbb{R}^n \\setminus \\Omega)\\times \\C^{1,\\alpha}(\\Sigma)\\times \\mathcal{B}(\\Sigma)\\big)^*$ denote the \\emph{annihilator space  of $\\Range  L$}. Namely, $ (\\Range L)^\\perp$ consists of bounded linear functionals  whose evaluation on $L(h, v)$ is zero for all $(h, v)\\in \\C^{2,\\alpha}_{-q}(\\mathbb{R}^n\\setminus \\Omega)$. \n\n\n\\begin{lemma}\\label{lemma:perp}\nIf the boundary $\\Sigma$ is static regular in $\\mathbb{R}^n\\setminus \\Omega$, then\n\\[\n\t(\\Range  L)^\\perp=  \\mathcal{Q}.\n\\]\n\\end{lemma}\n\\begin{proof}\n\nSince $L$ is Fredholm of index $0$ by Lemma~\\ref{lemma:Fredholm}, we have $\\Dim (\\Range L)^\\perp=\\Dim\\Ker L =N$. Since $\\Dim \\mathcal{Q}=N$, we just need to show that $\\mathcal{Q}\\subset (\\Range  L)^\\perp$. Pairing $\\kappa(X)\\in \\mathcal{Q}$ with an arbitrary $(h, v)\\in\\C^{2,\\alpha}_{-q}$ gives\n\\begin{align*}\n\t \\langle \\kappa(X), L(h, v)\\rangle_{\\mathcal{L}^2}&= \\int_{\\mathbb{R}^n\\setminus \\Omega} \\Big(-D\\Ric(h)+\\nabla^2 v - \\D(\\bi h+dv)\\Big) \\cdot 2\\bi^* X \\, d\\mathrm{vol} \\\\\n\t&\\quad + \\int_{\\mathbb{R}^n\\setminus \\Omega}( \\Delta v)(-\\Div X)\\, d\\mathrm{vol} - \\int_\\Sigma (\\bi h+dv) \\cdot 2\\bi^* X(\\nu, \\cdot)\\, d\\sigma.\n\\end{align*}\nWe rearrange the integrands and compute\n\\begin{align*}\n\t\\langle \\kappa(X), L(h, v)\\rangle_{\\mathcal{L}^2} &= \\int_{\\mathbb{R}^n\\setminus \\Omega} \\Big \\langle\\big( -D\\Ric(h)+\\nabla^2 v ,  \\Delta v \\big),\\big( 2\\bi^* X, -\\Div X\\big) \\Big\\rangle \\, d\\mathrm{vol}\\\\\n\t&\\;  - \\int_{\\mathbb{R}^n\\setminus \\Omega}  \\D(\\bi h+dv) \\cdot 2\\bi^* X \\, d\\mathrm{vol}-\\int_\\Sigma (\\bi h+dv) \\cdot 2\\bi^* X(\\nu, \\cdot)\\, d\\sigma\\\\\n\t&= \\int_{\\mathbb{R}^n\\setminus \\Omega} (\\bi h + dv)\\cdot (2 \\Div \\bi^* X)\\, d\\mathrm{vol}\\\\\n\t&\\; - \\lim_{r\\to \\infty} \\int_{|x|=r}(\\bi h+dv) \\cdot 2\\bi^* X\\left(\\tfrac{x}{|x|}, \\cdot\\right)\\, d\\sigma\\\\\n\t&=0\n\\end{align*}\nwhere the integral in the first line vanishes by \\eqref{equation:cokernel2}, in the second equality we apply the integration by parts (noting that $\\nu$ on $\\Sigma$ points away from $\\Omega$), and in the last equality, we use $ 2\\Div \\bi^* X = \\Delta X=0$ and the boundary integral vanishes at infinity since the decay rate of the integrand is $O(r^{-2q-1})$. \n\\end{proof}\n\n\\begin{remark}\\label{remark:cokernelL}\nThe computation above explains the motivation for the definition of $\\kappa(X)\\in \\mathcal{Q}$. We have shown in Proposition~\\ref{proposition:cokernel} that the annihilator space of the operator $(h, v)\\mapsto (-D\\Ric(h) + \\nabla^2 v, \\Delta v)$ contains elements $\\kappa_0(X) = (2\\bi^* X, -\\Div X)$ for $X\\in \\mathcal{X}$. We let the first two components of $\\kappa(X)$ be exactly  $\\kappa_0(X)$ and use the third component $-2\\bi^* X(\\nu, \\cdot)$ of $\\kappa(X)$ to take care of the  ``gauge terms'' in the definition of $L$. \n\\end{remark}\n\n\n\n\n\nIn the next lemma, we identify a complementing space of $\\Range L$. Intuitively, one may expect that the annihilator space $\\mathcal{Q}$ complements $\\Range L$, but the elements in $\\mathcal{Q}$ may not even decay faster enough to be in the codomain of $L$. Nevertheless, we can  remedy the problem  by  multiplying a suitable weight function. \n\nLet $\\eta$ be a positive, smooth function in $\\mathbb{R}^n \\setminus \\Omega$ satisfying $\\eta = 1$ near $\\Sigma$ and {$\\eta (x) = |x|^{-1}$} outside a large ball. Then the following space $\\eta\\mathcal{Q}$ is a well-defined subspace of  $\\C^{0,\\alpha}_{-q-2}(\\mathbb{R}^n \\setminus \\Omega)\\times \\C^{1,\\alpha}(\\Sigma)\\times \\mathcal{B}(\\Sigma)$:\n\\[\n\t\\eta \\mathcal{Q}= \\{ \\eta \\kappa: \\kappa \\in \\mathcal{Q}\\}.\n\\]\nNote that $\\Dim \\eta \\mathcal{Q}=\\Dim \\mathcal{Q}= N$.  \n\n\\begin{lemma}\\label{lemma:cokernel}\nIf the boundary $\\Sigma$ is static regular in $\\mathbb{R}^n\\setminus \\Omega$,  then $\\Range L$ is complemented by $\\eta \\mathcal{Q}$. Namely,\n\\[\n\t\\C^{0,\\alpha}_{-q-2}(\\mathbb{R}^n \\setminus \\Omega)\\times \\C^{1,\\alpha}(\\Sigma)\\times \\mathcal{B}(\\Sigma)= \\Range L \\oplus \\eta \\mathcal{Q}.\n\\]\n\\end{lemma}\n\\begin{proof}\nUsing the weight function $\\eta$, we define an inner product on $\\mathcal{Q}$ as follows. Let $\\kappa(X), \\gamma(Z)\\in \\mathcal{Q}$  for $X, Z\\in \\mathcal{X}_0$. That is,\n\\begin{align*}\n\\kappa (X)&=\\big(2\\bi^* X,\\, -\\Div X, \\, -2\\bi^* X(\\nu, \\cdot), \\,0,\\, 0\\big)\\\\\n\\gamma (Z)& =\\big(2\\bi^* Z,\\, -\\Div Z, \\,-2\\bi^* Z(\\nu, \\cdot),\\, 0, \\,0\\big).\n\\end{align*}\nDefine the inner product between $\\kappa(X)$ and $\\gamma(Z)$ by \n\\begin{align*}\n& \\int_{\\mathbb{R}^n\\setminus \\Omega} \\eta (2\\bi^* X) \\cdot (2\\bi^* Z)\\, d\\mathrm{vol}  +\\int_{\\mathbb{R}^n\\setminus \\Omega} \\eta (\\Div X) (\\Div Z) \\, d\\mathrm{vol}   \\\\\n&+ \\int_\\Sigma  \\big(2\\bi^* X(\\nu, \\cdot)\\big)\\cdot \\big(2\\bi^* Z(\\nu, \\cdot)\\big)\\, d\\sigma.\n\\end{align*}\nWith respect to this inner product, we  choose an orthonormal basis of $\\mathcal{Q}$, denoted by $ \\{ \\kappa_1,\\dots, \\kappa_N\\}$.\n \n For $f\\in\\C^{0,\\alpha}_{-q-2}(\\mathbb{R}^n \\setminus \\Omega)\\times \\C^{1,\\alpha}(\\Sigma)\\times \\mathcal{B}(\\Sigma)$ and for each $\\ell = 1, \\dots, N$, define the constant $a_\\ell =\\langle  \\kappa_\\ell, f\\rangle_{\\mathcal{L}^2}$.   By construction, we have $\\langle \\kappa_m, f - \\eta \\sum_\\ell a_\\ell \\kappa_\\ell\\rangle_{\\mathcal{L}^2}=0$ for all $m$.  By Lemma~\\ref{lemma:perp} and that $\\Range L$ is closed, we have\n  \\[\n \tf- \\eta \\sum_\\ell a_\\ell \\kappa_\\ell \\in \\Range L. \n \\]\nA similar argument also shows that $(\\Range L) \\cap (\\eta \\mathcal{Q})$ contains only the zero element. This completes the proof. \n\\end{proof}\n\nLet us take a moment to summarize what we have obtained so far. We can re-express the linear map $L: \\C^{2,\\alpha}_{-q}(\\mathbb{R}^n\\setminus \\Omega)\\to \\C^{0,\\alpha}_{-q-2}(\\mathbb{R}^n \\setminus \\Omega)\\times \\C^{1,\\alpha}(\\Sigma)\\times \\mathcal{B}(\\Sigma)$ in the form:\n\\begin{align*\n\tL : \\mathcal{E} \\oplus \\Ker L \\to \\Range L \\oplus \\eta \\mathcal{Q}\n\\end{align*}\nwhere $\\mathcal{E}, \\mathcal{Q}$ are defined in Definition~\\ref{definition:spaces} and $L$ is injective on $\\mathcal{E}$. Since the elements in $\\eta \\mathcal{Q}$ all have their last two components equal to zero, $\\Range L$ must contain the subspace $\\{ 0\\}\\times \\{ 0 \\} \\times \\mathcal{B}(\\Sigma)$. In other words, for any $(\\tau,\\phi) \\in \\mathcal{B}(\\Sigma)$, there exists a unique deformation $(h, v)$ satisfying the orthogonal gauge such that $L(h, v) = (0, 0, 0, \\tau, \\phi)$. While that looks a lot like a version of Theorem~\\ref{theorem:T} for the linearized operator $L$, we cannot directly conclude that $\\Range T$ must contain $(0,0,0,\\bar{g}^\\intercal +\\tau, H+\\phi)$ for $(\\tau, \\phi)$ small. It is because that the splitting for $\\Range L$ need not imply a splitting of the nonlinear operator~$T$ (needless to say that $\\Range T$ is not even a linear subspace). Therefore, we will consider the modified operators as in the next section. \n\n\\subsection{Orthogonality for the modified nonlinear map}\\label{section:proof}\n\nWe will introduce a new operator $\\overline{T}$, modified from $T$, and prove Theorem~\\ref{theorem:T}. To motivative our approach below, we discuss the basic idea. This part of discussion holds for a general map $T$ and its linearization $L$ (say, linearized at  $0$), so the reader can ignore for now how the map $T$ was defined above. Suppose the linear map is given by\n\\[\nL : \\mathcal{E} \\oplus \\Ker L \\to \\Range L \\oplus \\mathcal{Z},\n\\]\n where $\\mathcal{E}, \\mathcal{Z}$ are Banach spaces. Without changing the codomain space, we can ``substitute'' $\\Ker L$ with $\\mathcal{Z}$ and define an isomorphism by\n\\begin{align*}\n\t\\overline{L}: \\mathcal{E}\\times \\mathcal{Z} &\\to  \\Range L \\oplus \\mathcal{Z}\\\\\n\t(e, z) &\\mapsto \\overline{L}(e, z) = L(e)+z.\n\\end{align*}\n If we define a nonlinear map by $\\overline{T}(e, z)= T(e) +z$, then $\\overline{T}$ is a local diffeomorphism at $(e, z)=0$ by Inverse Function Theorem. Thus, for $v$ sufficiently close to $T(0)$, there exists $(e, z)$ solving $T(e)+z = v$. If  $z$ is ``orthogonal'' to both $T(e)$ and $v$, then we would get $z=0$ and solve $T(e) = v$ as desired. However, this may be too rigid to get, so we would like to take the advantage of the flexibility in defining  $\\overline{T}$. We just need its linearization equal to $\\overline{L}$; in particular,  $\\overline{T}$ can be nonlinear in~$z$. Our definition of $\\overline{T}$ below  reflects this basic idea,\nwith a slight complication that we need to enlarge the space $\\mathcal{Z}$ for the domain of $\\overline{T}$ in order to impose  nonlinear conditions on $z$.\n\n Let $\\widehat{\\mathcal{X}}$ be the linear space of vectors defined similarly as $\\mathcal X$, except with less regularity:\n\\begin{align*}\n\t\\widehat{\\mathcal{X}} = \\left\\{\\mbox{ $X=0$ on $\\Sigma$, $X - K\\in  \\C^{2,\\alpha}_{1-q} (\\mathbb{R}^n\\setminus \\Omega)$ for some Killing vector $K$ of $\\bar{g}$}\\right\\}.\n\\end{align*}\nDefine the map $\\overline{T}:\\Big( \\big((\\bar{g}, 1)+\\mathcal{E}\\big)\\cap \\mathcal{M}\\Big) \\times  \\widehat{\\mathcal{X}} \\to \\C^{0,\\alpha}_{-q-2}(\\mathbb{R}^n \\setminus \\Omega)\\times \\C^{0,\\alpha}_{-q-1}(\\mathbb{R}^n \\setminus \\Omega) \\times \\C^{1,\\alpha}(\\Sigma)\\times \\Big( ( \\bar{g}^\\intercal, H) +\\mathcal{B}(\\Sigma)\\Big)$ by, for $(g, u) \\in  \\big((\\bar{g}, 1)+\\mathcal{E}\\big)\\cap \\mathcal{M}$ and $W \\in  \\widehat{\\mathcal{X}}$,\n\\begin{align*\n\t&\\overline{T} (g, u, W)\\\\\n\t&= \\begin{array}{l}\n\t\\left\\{ \\begin{array}{l}-u \\Ric_g + \\nabla_g^2 u  -\\D_g (\\bi g+du) + \\eta \\big(2\\bi_g^* W- u^{-1} W(u) g \\big)\\\\\n\t\\Delta_g u + \\eta \\big(-\\Div_g W + u^{-1} W(u)\\big)\\\\\n\t\\Div_g (2\\bi_g^* W- u^{-1} W(u) g)\n\t\\end{array}\\right. \\quad \\mbox{ in } \\mathbb{R}^n\\setminus \\Omega \\\\\n\t\\left\\{ \\begin{array}{l} \\bi g+ du - (2\\bi_g^* W - u^{-1} W(u) g)(\\nu, \\cdot)\\\\\n\tg^\\intercal\\\\\n\tH_g\n\t\\end{array}\\right.\\quad \\mbox{ on } \\Sigma\n\t\\end{array}\n\\end{align*}\nwhere $\\nu$ is the unit normal vector on $\\Sigma$ pointing away from $\\Omega$. We should compare $\\overline{T}$ to the original map $T$ in \\eqref{equation:nonlinear}. For a vector $W$, we define  \n\\begin{align*}\n\t\\zeta(W)  &=\\Big( 2\\bi_g^* W - u^{-1} W(u) g, \\,-\\Div_g W + u^{-1} W(u),\\,\\\\\n\t&\\qquad \\qquad \\qquad \\qquad  -(2\\bi_g^* W -u^{-1} W(u) g )(\\nu, \\cdot),\\, 0, \\,0\\Big).\n\\end{align*}\nThen  $\\overline{T}(g, u, W)$, after dropping the third equation for $\\Div_g (2\\bi_g^* W- u^{-1} W(u) g)$, is exactly  $T(g, u) +\\eta \\zeta(W)$. We also note that the first two components of $\\zeta(W)$ is the same as $\\zeta_0(W)$ from \\eqref{equation:cokernel-elements}. The relation between $\\zeta(W)$ and $\\zeta_0(W)$ is analogous to that of $\\kappa(X),\\kappa_0(X)$  in Remark~\\ref{remark:cokernelL}.\n\n\\begin{proposition}\\label{proposition:isom}\nSuppose the boundary $\\Sigma$ is static regular in $\\mathbb{R}^n\\setminus \\Omega$. Then the following holds:\n\\begin{enumerate}\n\\item \\label{item:1} $\\overline{T}$ is a local diffeomorphism at $\\big(\\bar{g}, 1, 0\\big)$. \n\\item \\label{item:2} If $(g,u, W)$ solves $\\overline{T}(g, u, W) = (0, 0, 0, 0, \\tau,\\phi)$; that is, $(g^\\intercal, H_g) = (\\tau, \\phi)$ on $\\Sigma$ and all other equations in $\\overline{T}$ equal zero,  then  $T(g, u) = (0, 0, 0, \\tau, \\phi)$.\n\\end{enumerate}\n\\end{proposition}\n\\begin{proof}\nDenote the linearization of $\\overline{T}$ at $(\\bar{g}, 1, 0)$ by $\\overline{L}: \\mathcal{E}\\times \\widehat {\\mathcal{X}}\\to \\C^{0,\\alpha}_{-q-2}(\\mathbb{R}^n \\setminus \\Omega)\\times\\C^{0,\\alpha}_{-q-1}(\\mathbb{R}^n \\setminus \\Omega)\\times \\C^{1,\\alpha}(\\Sigma)\\times \\mathcal{B}(\\Sigma)$. For $(h, v)\\in \\mathcal{E}$ and $X\\in  \\widehat {\\mathcal{X}}$, \n\\begin{align*}\n\t\\overline{L} (h, v, X)= \\begin{array}{l}\n\t\\left\\{ \\begin{array}{l}-D \\Ric(h) + \\nabla^2 v  -\\D (\\bi h+dv) + 2\\eta \\bi^* X\\\\\n\t\\Delta v - \\eta \\Div X \\\\\n\t2\\Div \\bi^* X\n\t\\end{array}\\right. \\quad \\mbox{ in } \\mathbb{R}^n\\setminus \\Omega \\\\\n\t\\left\\{ \\begin{array}{l} \\bi h+ dv- 2\\bi^* X (\\nu, \\cdot)\\\\\n\th^\\intercal\\\\\n\tDH(h)\n\t\\end{array}\\right.\\quad \\mbox{ on } \\Sigma. \n\t\\end{array}\n\\end{align*}\nRecall $\\kappa(X) = (2\\bi^* X, \\, -\\Div X,\\, - 2\\bi^* X (\\nu, \\cdot), 0, 0)$. \nJust as for $\\overline{T}$ and $T$,  the linear map $\\overline{L}(h, v, X)$, with the third equation for $2\\Div \\bi^* X$ dropped, is exactly $L(h, v)+ \\eta \\kappa(X)$, where $L$ is defined in~\\eqref{equation:linear}.\n\n\nWe claim that $\\overline{L}$ is an isomorphism. Then by  Inverse Function Theorem, it proves that $\\overline{T}$ is a local diffeomorphism at $\\big(\\bar{g}, 1, 0\\big)$. \n\nWe show $\\overline{L}$ is surjective: Since the third equation $2\\Div \\bi^* X = \\Delta X$ is surjective by Lemma~\\ref{lemma:PDE}, we just need to show that $\\overline{L}$ is surjective onto other components for those $X$ solving $2\\Div \\bi^* X=0$, i.e. $X\\in \\mathcal{X}_0$. It is equivalent to that $L(h, v) + \\eta \\kappa(X)$ is surjective for  $(h,v)\\in \\mathcal{E}, X\\in \\mathcal{X}_0$, which follows from Lemma~\\ref{lemma:cokernel}. We show  $\\overline{L}$ is injective: For $(h, v, X)$ solving $\\overline{L}(h , v, X)=0$,  we have $X\\in \\mathcal{X}_0$, and $(h, v, X)$ satisfies  $L(h, v) + \\eta \\kappa(X)=0$.  Again by  the decomposition from Lemma~\\ref{lemma:cokernel}, we have $L(h, v)=0$ for $(h, v)\\in \\mathcal{E}$ and $\\kappa(X)=0$, which imply that $(h, v, X)$ is zero. \n\n\n\n\nBefore we prove the second statement, we note that if $W$ satisfies $\\Div_g (2\\bi_g^* W- u^{-1} W(u) g)=0$, then $\\zeta(W)$ is $\\mathcal{L}^2$-orthogonal to $T(g, u)$:\n\\begin{align*}\n\t&\\langle T(g, u), \\zeta(W) \\rangle_{\\mathcal{L}^2}\\\\\n\t&=\\int_{\\mathbb{R}^n\\setminus \\Omega} \\Big( - u \\Ric_g + \\nabla^2_g u - \\D_g (\\bi g + du) \\Big) \\cdot \\Big(2\\bi_g^* W - u^{-1} W(u) g\\Big)\\, d\\mathrm{vol}_g \\\\\n\t&\\quad + \\int_{\\mathbb{R}^n\\setminus \\Omega} \\Delta_g u \\Big(-\\Div_g W - u^{-1} W(u)\\Big)\\, d\\mathrm{vol}_g\\\\\n\t&\\quad - \\int_\\Sigma (\\bi g+ du)\\cdot (2\\bi_g^* W - u^{-1} W(u) g ) (\\nu ,\\cdot) \\, d\\sigma_g\\\\\n\t&=\\int_{\\mathbb{R}^n\\setminus \\Omega} \\Big(- \\D_g (\\bi g + du) \\Big) \\cdot \\Big(2\\bi_g^* W - u^{-1} W(u) g\\Big)\\, d\\mathrm{vol}_g \\\\\n\t&\\quad - \\int_\\Sigma (\\bi g+ du)\\cdot (2\\bi_g^* W - u^{-1} W(u) g ) (\\nu ,\\cdot) \\, d\\sigma_g\\\\\n\t&=0\n\\end{align*} \nwhere we use \\eqref{equation:cokernel1} in the second equality and integration by parts in the last line. Note that the boundary integrals on $\\Sigma$ cancel, and the boundary integral at infinity vanishes because of the fall-off rates.\n\nNow suppose $(g, u, W)$ satisfies $\\overline{T}(g, u, W) = (0, 0, 0, 0, \\tau,\\phi)$. This is equivalent to\n\\begin{align*}\n\t\tT(g, u) + \\eta \\zeta(W) &= (0, 0, 0, \\tau, \\phi)\\\\\n\t\t\\Div_g (2\\bi_g^* W- u^{-1} W(u) g)&=0.\n\\end{align*}\n  Take the $\\mathcal{L}^2$-inner product of the first identity with $ \\zeta(W)$, and note  that $\\big\\langle T(g, u), \\zeta(W)\\big\\rangle_{\\mathcal{L}^2}=0$ as  above and  $\\big\\langle (0,0,0,\\tau, \\phi), \\zeta(W) \\big\\rangle_{\\mathcal{L}^2} =0$ trivially. So we have $\\big\\langle \\eta  \\zeta(W),  \\zeta(W) \\big\\rangle_{\\mathcal{L}^2}=0$ and thus $\\zeta(W)=0$. This implies the desired conclusion $T(g, u) = (0, 0, 0, \\tau, \\phi)$. In fact, this implies that $W$ is a Killing vector of $g$ vanishing on the boundary and thus $W\\equiv 0$ in  $\\mathbb R^n\\setminus\\Omega$.\n \\end{proof}\n\n\nCombining the above results, we give the proof of Theorem~\\ref{theorem:T}.\n\\begin{proof}[Proof of Theorem~\\ref{theorem:T}]\nFrom Item~\\eqref{item:1} in Proposition~\\ref{proposition:isom}, $\\overline{T}$ is a local diffeomorphism at $(\\bar{g}, 1)$. More specifically, there are positive constants $\\epsilon_0, C$  such that for each $\\epsilon \\in (0, \\epsilon_0)$, there is an open neighborhood $\\mathcal{U}\\times \\mathcal{V}$ of $\\big(\\bar{g}, 1, 0\\big)$ in $\\big((\\bar{g}, 1)+\\mathcal{E}\\big)\\times \\widehat {\\mathcal{X}}$ with the diameter less than $C\\epsilon$, such that $\\overline{T}$ is a diffeomorphism from $\\mathcal{U}\\times \\mathcal{V}$ onto an open ball of radius $\\epsilon$ in $\\C^{0,\\alpha}_{-q-2}(\\mathbb{R}^n \\setminus \\Omega)\\times \\C^{0,\\alpha}_{-q-1}(\\mathbb{R}^n \\setminus \\Omega)\\times\\C^{1,\\alpha}(\\Sigma)\\times \\Big( (\\bar{g}^\\intercal, H) +\\mathcal{B}(\\Sigma)\\Big)$, centered at $(0, 0, 0, 0, \\bar{g}^\\intercal, H)$. Therefore, given $(\\tau, \\phi)$ with $\\|(\\tau, \\phi)-(\\bar{g}^\\intercal, H)\\|_{\\C^{2,\\alpha}(\\Sigma)\\times \\C^{1,\\alpha}(\\Sigma)}<\\epsilon$, there exists a unique $(g, u, W)$ with $\\|(g, u) - (\\bar{g}, 1) \\|_{\\C^{2,\\alpha}_{-q} (\\mathbb{R}^n\\setminus \\Omega)} <C\\epsilon$ and  $\\| W \\|_{ \\C^{2,\\alpha}_{1-q}(\\mathbb{R}^n\\setminus \\Omega)}<C\\epsilon$  satisfying \n\\[\n\t\\overline{T}(g, u, W) = (0, 0, 0, 0, \\tau, \\phi). \n\\]\nBy Item~\\eqref{item:2} in Proposition~\\ref{proposition:isom},  $(g, u)$ satisfies $T(g, u) = (0, 0, 0, \\tau, \\phi)$. Since $(g, u) \\in (\\bar{g},1)+\\mathcal{E}$, $(g, u)$ satisfies the orthogonal gauge. \n\\end{proof}\n\\begin{remark}\\label{re:constant}\nNote that the constants $\\epsilon_0, C$ depend on the global geometry $\\Sigma$ in $\\mathbb{R}^n\\setminus \\Omega$. More precisely by the (quantitive) inverse function theorem, the constants depend on the operator norms of $\\overline{L}$, $\\overline{L}^{-1}$, as well as the second Frech\\'et derivative $D^2\\overline{T}|_{(g, u)}$ for $(g, u)$ is a neighborhood of $(\\bar g, 1)$,  between the  function spaces on $\\mathbb{R}^n\\setminus \\Omega$ and $\\Sigma$ as specified above.\n\\end{remark}\n\n\nWe explain the reason for the terminology \n\\emph{static regular}. The following discussion will not be used elsewhere in the paper. For a general $\\Omega$ whose boundary $\\Sigma = \\partial \\overline{\\Omega}$ is not necessarily static regular in $\\mathbb{R}^n\\setminus \\Omega$, define the set of static vacuum pairs (of any boundary data) near $(\\bar{g}, 1)$ by\n\\[\n\t\\mathcal{S}(\\mathbb{R}^n\\setminus \\Omega) = \\left\\{ (g, u)\\in \\mathcal{M}: -u\\Ric_g + \\nabla_g^2 u=0 \\mbox{ and } \\Delta_g u=0  \\mbox{ in } \\mathbb{R}^n\\setminus \\Omega \\right\\}.\n\\]\nAs in \\cite{Anderson-Khuri:2013}, the  \\emph{Bartnik boundary map} $\\Pi: \\mathcal{S}(\\mathbb{R}^n\\setminus \\Omega) \\to (\\bar{g}^\\intercal, H) +\\mathcal{B}(\\Sigma)$ is defined by $\\Pi(g, u)= (g^\\intercal, H_g)$. Conjecture~\\ref{conjecture} becomes finding conditions of $\\mathbb{R}^n\\setminus \\Omega$ so that the boundary map~$\\Pi$ is locally surjective. In that formulation, Theorem~\\ref{theorem:criterion} says if the boundary $\\Sigma$ is static regular in  $\\mathbb{R}^n\\setminus \\Omega$, then $(\\bar{g}, 1)$ is a \\emph{regular point} of the map~$\\Pi$. Of course, the set $\\mathcal{S}(\\mathbb{R}^n\\setminus \\Omega)$ may not be a Banach manifold for us to talk about regular points. But we can restrict the boundary map~$\\Pi$ on the subset $\\mathcal{S}_0(\\mathbb{R}^n\\setminus \\Omega)\\subset \\mathcal{S}(\\mathbb{R}^n\\setminus \\Omega)$ that  consists of  $(g, u)\\in \\mathcal{S}(\\mathbb{R}^n\\setminus \\Omega)$ satisfying both the static-harmonic gauge and orthogonal gauge. It is shown that $\\mathcal{S}_0(\\mathbb{R}^n\\setminus \\Omega)$ is a Banach manifold \\cite{An:preprint} (Cf. \\cite{Anderson-Khuri:2013}), so the above discussion can make sense. \n\n\n\n\\section{Convex surfaces in $\\mathbb{R}^3$}\\label{section:convex}\n\n\n\\subsection{Mean curvature comparison}\n\n\nLet $\\Omega$ be a bounded open subset in the Euclidean space $(\\mathbb{R}^3, \\bar{g})$ whose boundary $\\Sigma =\\partial \\overline{\\Omega}$ is a connected, embedded surface. Before we restrict to the case that $\\Sigma$ is convex, we first prove the following general lemma. Given a harmonic function $v$ in $\\mathbb{R}^3\\setminus \\Omega$, we will construct a family of scalar-flat metrics $\\gamma(s)$ in $\\mathbb{R}^3\\setminus \\Omega$ and a corresponding family of scalar-flat metrics $\\gamma_{\\I}(s)$ in the compact region $\\overline{\\Omega}$ that induce the same metrics on $\\Sigma$. We shall use the notational convention that the quantities in the compact region $\\overline{\\Omega}$ are denoted with the subscript ${\\I}$. \n\n\\begin{lemma}\\label{lemma:harmonic-generate}\nLet $v\\in \\C^{2,\\alpha}_{-q}(\\mathbb{R}^3\\setminus  \\Omega)$ solve $\\Delta v=0$ in $\\mathbb{R}^3\\setminus \\Omega$. Let $v_{\\I}\\in \\C^{2,\\alpha}\\big(\\overline{\\Omega}\\big)$ satisfy $\\Delta v_{\\I}=0$ in $\\Omega$ and $v_{\\I} = v$ on $\\Sigma$. \n\nFor $|s|$ small, there exists  a smooth family of asymptotically flat metrics $\\gamma(s)$ in $\\mathbb{R}^3\\setminus \\Omega$ and a family of metrics $\\gamma_{\\I}(s)$ in $\\overline{\\Omega}$ such that the following holds:\n\\begin{itemize}\n\\item $\\gamma(0)= \\bar{g}$, $R_{\\gamma(s)}=0$, $\\gamma'(0) = 2v\\bar{g}$ in $\\mathbb{R}^3\\setminus \\Omega$.\n\\item $\\gamma_{\\I}(0)= \\bar{g}$, $R_{\\gamma_{\\I}(s)}=0$, ${\\gamma_{\\I}}'(0) = 2v_{\\I}\\bar{g}$ in $\\overline{\\Omega}$.\n\n\\item On $\\Sigma$, $\\gamma(s)^\\intercal =\\gamma_{\\I}(s)^\\intercal= (1+sv)^2 \\bar{g}^\\intercal$ for all $s$. \n\\item Let $H_{\\gamma_{\\I}(s)}, H_{\\gamma(s)}$ be the mean curvatures of $\\Sigma$ in $\\big(\\overline{\\Omega}, \\gamma_{\\I}(s)\\big), (\\mathbb{R}^3\\setminus \\Omega, \\gamma(s))$, respectively (both with respect to the unit normals that points away from $\\Omega$). Then \n\\begin{align*}\n\t\\left. \\2ds \\right|_{s=0} \\bigg( -16\\pi m_{\\mathrm{ADM}}(\\gamma(s)) + &\\int_{\\mathbb{R}^3\\setminus \\Omega} R_{\\gamma(s)}\\, d\\mathrm{vol}_{\\gamma(s)}\\\\\n\t&+  \\int_\\Sigma 2\\Big(H_{\\gamma_{\\I}(s)} - H_{\\gamma(s)}\\Big)\\, d\\sigma_{\\gamma(s)} \\bigg)&\\ge 0.\n\\end{align*}\t \n\\end{itemize}\n\\end{lemma}\n\\begin{proof}\nConsider the asymptotically flat metric $\\hat{g}(s)=(1+sv)^2 \\bar{g}$. By the conformal  formula and that $v$ is harmonic, the scalar curvature of $\\hat{g}(s)$ is\n\\[\n\tR_{\\hat{g}(s)} = \\frac{2s^2 |\\nabla v|^2}{(1+sv)^4}.\n\\]\nFor $|s|$ sufficiently small, there exists a unique function $u(s)$ with  $\\| u(s)-1\\|_{\\C^{2,\\alpha}_{-q}(\\mathbb{R}^3\\setminus \\Omega)}\\ll 1$ that solves\n\\begin{align*}\n\t\\Delta_{\\hat{g}(s)} u(s) - \\tfrac{1}{8} R_{\\hat{g}(s)} u(s) &= 0 \\quad \\mbox{ in } \\mathbb{R}^3\\setminus \\Omega \\\\\n\tu(s) &= 1 \\quad \\mbox{ on } \\Sigma.\n\\end{align*}\nBy differentiating the above equations in $s$ and using that $\\left. \\ds\\right|_{s=0}R_{\\hat{g}(s)}=0$ and $u(0)=1$, we see that $u'(0)=O^{2,\\alpha}(|x|^{-q})$ is harmonic with $u'(0)=0$ on $\\Sigma$, and thus $u'(0)$ is identically zero.\nDefine $\\gamma(s) = u(s)^4 \\hat{g}(s)$ in $\\mathbb{R}^3\\setminus \\Omega$.  Then $\\gamma(s)$ has zero scalar curvature by the conformal formula, and we have $\\gamma'(0)  = 2v\\bar{g}$. This proves  the first item.\n\nSimilarly, we define $\\hat{g}_{\\I}(s)=(1+sv_{\\I})^2 \\bar{g}$ in $\\overline{\\Omega}$ and have\n\\[\n\tR_{\\hat{g}_{\\I}(s)} = \\frac{2s^2 |\\nabla v_{\\I}|^2}{(1+sv_{\\I})^4}.\n\\]\nThen, for $|s|$ small, we let $u_{\\I}(s)$ with  $\\| u_{\\I}(s)-1\\|_{\\C^{2,\\alpha}\\big(\\overline{\\Omega}\\big)}\\ll 1$ be the unique solution to \n\\begin{align*}\n\t\\Delta_{\\hat{g}_{\\I}(s)} u_{\\I}(s) - \\tfrac{1}{8} R_{\\hat{g}_{\\I}(s)} u_{\\I}(s) &= 0 \\quad \\mbox{ in } \\Omega \\\\\n\tu_{\\I}(s) &= 1 \\quad \\mbox{ on } \\Sigma.\n\\end{align*}\nThe same argument  as above shows that $u_{\\I}'(0)=0$, and we can define $\\gamma_{\\I}(s) = u_{\\I}(s)^4 \\hat{g}_{\\I}(s)$ corresponding. It proves the second item. \n\nBy the boundary conditions $v=v_{\\I}$ and $u(s) = 1 = u_{\\I}(s)$ on $\\Sigma$, we have \n\\[\n\t\\gamma(s)^\\intercal =\\gamma_{\\I}(s)^\\intercal= \\hat{g}(s)^\\intercal\n\\]\nand the third item follows. \n\n\nWe prove the last item. We begin by computing the ADM mass of $\\gamma(s)$. \n\\begin{align*}\n\t16\\pi m_{\\mathrm{ADM}}(\\gamma(s)) &= \\lim_{r\\to \\infty} \\int_{|x|=r}\\sum_{i,j} \\left(\\frac{\\partial \\gamma(s)_{ij}}{\\partial x_i} - \\frac{\\partial \\gamma(s)_{ii}}{\\partial x_j} \\right)\\frac{x_j}{|x|} \\, d\\sigma\\\\\n\t&= -2  \\lim_{r\\to \\infty} \\int_{|x|=r}\\left(  2s \\frac{\\partial v}{\\partial r} + 4 \\frac{\\partial u(s)}{\\partial r} \\right)\\, d\\sigma.\n\\end{align*}\n\nTo compute the mean curvatures, by the conformal formula and that $u(s)=1=u_{\\I}(s), v=v_{\\I}$ on~$\\Sigma$, we have \n\\begin{align*}\nH_{\\gamma_{\\I}(s)} &=(1+sv)^{-1} H + 2(1+sv)^{-2} s \\nu (v_{\\I}) + 4 \\nu_{\\hat{g}_{\\I}(s)}\\big(u_{\\I}(s)\\big)\\\\\n\tH_{\\gamma(s)} &= (1+sv)^{-1} H + 2(1+sv)^{-2} s \\nu (v) + 4\\nu_{\\hat{g}(s)}\\big(u(s)\\big)\n\\end{align*}\nwhere $H$ is the mean curvature $\\Sigma$ in $(\\mathbb{R}^3,\\bar{g})$ and note $\\nu_{\\hat{g}} = \\nu_{\\hat{g}_{\\I}} = (1+sv)^{-1} \\nu$. Thus, \n\\[\n\tH_{\\gamma_{\\I}(s)} - H_{\\gamma(s)}= 2(1+sv)^{-2} s \\Big( \\nu(v_{\\I})-\\nu(v) \\Big) + 4\\Big( \\nu_{\\hat{g}_{\\I}(s)}\\big(u_{\\I}(s)\\big)- \\nu_{\\hat{g}(s)}\\big(u(s)\\big)\\Big).\n\\]\nIntegrating the previous identity over $(\\Sigma, \\gamma(s)^\\intercal)$ and using that \n\\[\n\td\\sigma_{\\gamma(s)}=  (1+sv)^2 d\\sigma\n\\]\nwhere $d\\sigma$ is the area form of $(\\Sigma, \\bar{g})$, we get \n\\begin{align*}\n\t&\\int_\\Sigma \\Big( H_{\\gamma_{\\I}(s)} -H_{\\gamma(s)}\\Big)\\, d\\sigma_{\\gamma(s)} \\\\\n\t&= \\int_\\Sigma 2s \\Big( \\nu(v_{\\I})-\\nu(v) \\Big)\\, d\\sigma + \\int_\\Sigma 4\\Big( \\nu_{\\hat{g}_{\\I}(s)}\\big(u_{\\I}(s)\\big)- \\nu_{\\hat{g}(s)}\\big(u(s)\\big)\\Big)\\,d\\sigma_{\\gamma(s)}.\n\\end{align*}\nApply the divergence theorem to the right hand side  and note that the unit normals point away from $\\Omega$ (and $\\nu_{\\hat{g}_{\\I}(s)}=\\nu_{\\hat{g}(s)}=\\nu_{\\gamma(s)} = \\nu_{\\gamma_{\\I}(s)}$ on $\\Sigma$). Since $v, v_{\\I}$ are harmonic functions of $\\bar{g}$, we derive\n\\vspace{8pt}\n\\begin{align*} \n\t\\int_\\Sigma \\Big( H_{\\gamma_{\\I}(s)} -&H_{\\gamma(s)}\\Big)\\, d\\sigma_{\\gamma(s)}= -\\lim_{r\\to \\infty} \\int_{S_r} \\left( 2s\\frac{\\partial v}{\\partial r}  + 4\\frac{\\partial u(s)}{\\partial r}\\right)\\, d\\sigma \\\\\n\t&\\quad \\;\\;+ 4\\int_{\\overline{\\Omega}} \\Delta_{\\gamma_{\\I}(s)} u_{\\I}(s)\\, d\\mathrm{vol}_{\\gamma_{\\I}(s)}+ 4\\int_{\\mathbb{R}^3\\setminus \\Omega} \\Delta_{\\gamma(s)} u(s) \\, d\\mathrm{vol}_{\\gamma(s)} .\n\\end{align*}\nSubstituting the boundary integral with the ADM mass identity computed above and the Laplace terms using the differential equations that define $u(s)$ and $u_{\\I}(s)$, we obtain\n\\begin{align*}\n\t&-16\\pi m_{\\mathrm{ADM}}(\\gamma(s)) + \\int_\\Sigma 2\\Big( H_{\\gamma_{\\I}(s)}-H_{\\gamma(s)} \\Big)\\, d\\sigma_{\\gamma(s)}\\\\\n\t&\\qquad \\qquad =   \\int_{\\overline{\\Omega}} R_{\\hat{g}_{\\I}(s)} u_{\\I}(s)\\, d\\mathrm{vol}_{\\gamma_{\\I}(s)}+ \\int_{\\mathbb{R}^3\\setminus \\Omega} R_{\\hat{g}(s)} u(s) \\, d\\mathrm{vol}_{\\gamma(s)}.\n\\end{align*}\nTaking two-derivative of the previous identity in $s$ and using  $u(0)=u_{\\I}(0)=1, u'(0)=u_{\\I}'(0)=0$, $\\left.\\2ds\\right|_{s=0} R_{\\hat{g}(s)}\\ge 0$, $\\left. \\2ds\\right|_{s=0} R_{\\hat{g}_{\\I}(s)}\\ge 0$, and that $R_{\\gamma(s)}=0$ for all $s$,   we finish the proof.  \n\\end{proof}\n\n\nFor the rest of the section, we assume that $\\Sigma$ is convex, i.e., the Gauss curvature of $(\\Sigma, \\bar{g}^\\intercal)$ is positive everywhere. If $g$ is an asymptotically flat metric such that $(\\Sigma, g^\\intercal)$ has positive Gauss curvature,  it can be isometrically embedded $(\\Sigma, g^\\intercal)$ into $(\\mathbb{R}^3, \\bar{g})$.  We define $ \\mathring{A}_{g^\\intercal}, \\mathring{H}_{g^\\intercal}$ to be, respectively, the second fundamental form and mean curvature of the isometric image in $(\\mathbb{R}^3, \\bar{g})$ with respect to the unit normal pointing to infinity. They are well-defined by uniqueness (up to a rigid motion) of isometric embeddings. Define the following functional\n\\begin{align*}\n\t\t\\mathscr{G}(g) &= -16\\pi m_{\\mathrm{ADM}}(g) + \\int_{\\mathbb{R}^3\\setminus \\Omega} R_g \\, d\\mathrm{vol}_g + \\int_\\Sigma 2\\big(\\mathring{H}_{g^\\intercal} - H_g\\big) \\, d\\sigma_{g}.\n\\end{align*}\n\n\nWe will compute variations of $\\mathscr{G}$ along a family of metrics $\\gamma(s)$ whose induced metrics $\\gamma(s)^\\intercal$ vary smoothly in $s$. As a preparation, we note the following two lemmas on the smooth dependence of isometric embeddings and of diffeomorphism extensions. \n\n\n\\begin{lemma}[Smooth dependence of isometric embeddings]\\label{lemma:embedding}\nLet $s\\in (-\\epsilon, \\epsilon)$ and let $\\tau(s)\\in \\C^{2,\\alpha}(S^2)$ be a family of metrics on $S^2$ with positive Gauss curvature such that $\\tau(s)$ is $\\C^k$ in $s$. Suppose the Gauss curvatures are bounded below by a positive constant uniformly in $s$. Then there exists a  family of isometric embeddings $f(s) : (S^2, \\tau(s))\\to \\mathbb{R}^3$ such that each $f(s)\\in \\C^{2,\\alpha}(S^2)$ and $f(s)$ is $\\C^k$ in $s$. \n\\end{lemma}\n\\begin{proof}\nThe existence of an isometric embedding  $f(s)$ for each $s$ is by Nirenberg~\\cite{Nirenberg:1953}  (for  metrics of $\\C^{2,\\alpha}$ regularity, see, for example, \\cite[Theorem 2]{Heinz:1962}  and \\cite[Theorem 10.3.2]{Schulz:1990}).  Here we explain that if the family of metrics is $\\C^k$ in the parameter $s$, then the embedding $f(s)$ obtained in ~\\cite{Nirenberg:1953} is also $\\C^k$ in~$s$. Let $\\mathcal{B}_R $ consist of  maps from $S^2$ to $\\mathbb{R}^3$ defined by $\\mathcal{B}_R = \\{Z\\in \\C^{2,\\alpha}( S^2; \\mathbb{R}^3): \\| Z \\|_{\\C^{2,\\alpha}(S^2)} \\le R\\}$. Let $s_0\\in (-\\epsilon, \\epsilon)$. By Sections 5-9 in  \\cite{Nirenberg:1953}, \n there exists a constant $R>0$ such that for each $s$ sufficiently close to $s_0$ one can construct a contraction map  $F_s: \\mathcal{B}_R\\to \\mathcal{B}_R$ that sends $Z\\in \\mathcal{B}_R$ to $F_s(Z)$ where $F_s(Z)$ solves\n\\[\n\t2 df(s_0)\\cdot dF_s(Z) = \\tau(s) - \\tau(s_0) - dZ\\cdot dZ \\quad \\mbox{ on } S^2.\n\\]\nMoreover, going through the construction in \\cite{Nirenberg:1953}, one can verify that the contraction constants of $F_s$ can be chosen uniformly in~$s$, and  the family of  contraction maps $\\{F_s\\}$ is $\\C^k$ in $s$.  Let $Z(s)$ be the unique fixed point of $F_s$. A general fact about contraction maps then says that the fixed point $Z(s)$ is $\\C^k$ in $s$. Lastly, define $f(s) = f(s_0)+Z(s)$. Then $f(s)$ is $\\C^k$ in $s$ and  $df(s)\\cdot df(s) = \\tau(s)$, i.e., $f(s)$  is an isometric embedding of $(S^2, \\tau(s))$.  \n\n\n\n\\end{proof}\n\n\\begin{lemma}[Smooth dependence of diffeomorphism extensions]\\label{lemma:extension}\n Let  $f_s:\\Sigma \\to \\mathbb{R}^3$ be a smooth family of $\\C^{2,\\alpha}$ embeddings with $f_0={\\rm Id}_{\\Sigma}$. Denote by $\\Sigma_s=f_s(\\Sigma)$ and by $\\Omega_s$ the region bounded by $\\Sigma_s$.  Suppose that each $\\Sigma_s$ is a convex surface enclosing  the origin and that $\\{\\Sigma_s\\}$ is contained in a compact subset. Then, there is a smooth family of maps $\\psi_s: \\mathbb{R}^3\\setminus \\Omega \\to \\mathbb{R}^3\\setminus \\Omega_s$ such that $\\psi_0$ is the identity map, each $\\psi_s$ is a diffeomorphism for $|s|$ small, $\\psi_s|_{\\Sigma} = f_s$, and $\\psi_s$ is the identity map outside a large compact set. \n \\end{lemma}\n\\begin{proof}\nBy convexity, we can express the points of $\\Sigma\\subset \\mathbb{R}^3$ as $(r(\\theta), \\theta)$ in spherical coordinates. We can also write $f_s:\\Sigma \\to \\mathbb{R}^3$ as $f_s(r(\\theta), \\theta) = \\big(\\hat{\\rho}(\\theta, s), \\hat{\\xi}(\\theta, s)\\big)$ in spherical coordinates where $\\hat{\\rho}, \\hat{\\xi}$  are smooth in $s$ and $\\C^{2,\\alpha}$ in~$\\theta$. Since $f_0 = {\\rm Id}_{\\Sigma}$, we know $\\hat{\\xi}(\\theta, 0)=\\theta$. \n \n Since $\\{\\Sigma_s\\}$ is contained in a compact subset,  there is a positive number $a$ so that $\\hat{\\rho}(\\theta, s)< a$ for all $(\\theta, s)$. We define a family of maps  $\\zeta_s: \\mathbb{R}^3\\setminus  B_a \\to \\mathbb{R}^3\\setminus \\Omega_s$ by\n\\[\n\t\\zeta_s( r, \\theta) = \\big(\\rho(r, \\theta, s), \\xi(r,\\theta, s)\\big)\n\\] \nsuch that $\\rho(r, \\theta, s)$ and $ \\xi(r,\\theta, s)$ are smooth in $s$ and $\\C^{2,\\alpha}$ in~$(r,\\theta)$ satisfying the following properties:\n\\begin{itemize}\n\\item $\\zeta_0$ is a diffeomorphism.\n\\item $\\big(\\rho(a, \\theta, s), \\xi(a, \\theta, s) \\big)= \\big(\\hat{\\rho}(\\theta, s), \\hat{\\xi}(\\theta, s)\\big)$.\n\\item For $r>2a$, $\\big(\\rho(r, \\theta, s), \\xi(r, \\theta, s) \\big) = (r,\\theta)$ for all $s$. \n\\end{itemize}\nFor example, we can define\n\\begin{align*}\n\\rho(r, \\theta, s) &= \\Big(\\lambda(r) a^{-1} \\hat{\\rho}(\\theta, s) + \\big(1-\\lambda(r)\\big)\\Big)r\\\\\n\\xi(r,\\theta, s)&=\\hat{\\xi}(\\theta, \\lambda(r) s)\n\\end{align*}\nwhere $\\lambda(r)$ is a smooth, weakly decreasing function such that $\\lambda(a)=1$ and $\\lambda(r)\\equiv 0$ for $r>2a$. \n\nIt is direct to verify that  $\\{ \\zeta_s\\}$ is smooth in $s$, each $\\zeta_s$ is a diffeomorphism for $|s|$ sufficiently small,  $\\left.\\zeta_s\\right|_{\\partial B_a} = f_s$,  and~$\\zeta_s$ equals to the identity map outside a compact set. Then we define $\\psi_s = \\zeta_s\\circ \\zeta_0^{-1}:\\mathbb{R}^3\\setminus \\Omega\\to \\mathbb{R}^3\\setminus \\Omega_s$. \n\n\\end{proof}\n\nWe now proceed to compute the first and second variations of $\\mathscr{G}$. We'll omit the subscript~$\\bar{g}$ when the geometric quantities and linearizations are taken with respect to $\\bar{g}$. \n\n\\begin{proposition}\\label{proposition:variation2}\nLet $g(s)$ be a smooth family of asymptotically flat metrics in $\\mathbb{R}^3\\setminus \\Omega$ with $g(0)= \\bar{g}$. Denote by $h = g'(0)$. Then we have the first and second variational formulas:\n\\begin{align*}\n\t\\left.\\ds\\right|_{s=0} \\mathscr{G}&(g(s))=0\\\\\n\t\\left.\\2ds\\right|_{s=0} \\mathscr{G}&(g(s))=\\int_{\\mathbb{R}^3\\setminus \\Omega} h\\cdot\\Big(-D\\Ric(h)+\\tfrac{1}{2} DR(h) \\bar{g}\\Big)\\, d\\mathrm{vol}\\\\\n\t& +\\int_\\Sigma \\left[\\Big( D\\mathring{H}(h^{\\intercal}) - DH(h)\\Big)\\tr h^\\intercal + \\Big(DA (h)- D \\mathring{A}(h^\\intercal)\\Big)\\cdot h^\\intercal \\right] \\, d\\sigma.\n\\end{align*}\n\n\\end{proposition}\n\\begin{proof}\nWe first compute the variations for one of the boundary integrals in the functional $\\mathscr{G}$:\n\\begin{align*}\n\t&\\left.\\ds \\right|_{s=0} \\int_\\Sigma 2H_{g(s)} \\, d\\sigma_{g(s)} =\\int_\\Sigma \\Big(2DH(h) + H\\tr h^\\intercal\\Big) \\, d\\sigma\\\\\n\t&\\left.\\2ds \\right|_{s=0} \\int_\\Sigma 2H_{g(s)} \\, d\\sigma_{g(s)} \\\\\n\t&\\quad =\\int_\\Sigma \\left(\\big(2H_{g(s)}\\big)'' + \\Big(\\tr_{g(s)}g'(s)^\\intercal\\Big)'H + 2DH(h)\\tr h^\\intercal +  \\tfrac{1}{2}  \\big(\\tr h^\\intercal \\big)^2 H \\right)\\, d\\sigma.\n\\end{align*}\nHere and below,  $'$ denotes the $s$-derivative, evaluated at $s=0$. Combining those with the variations of $\\mathscr{F}$ in Corollary~\\ref{corollary:variations}, we have\n\\begin{align}\\label{equation:variation-G}\n\\begin{split}\n\t&\\left.\\ds \\right|_{s=0} \\left[\\mathscr{F}(g(s), 1) - \\int_\\Sigma 2H_{g(s)} \\, d\\sigma_{g(s)} \\right]= \\int_\\Sigma \\big(A\\cdot h^\\intercal - H \\tr h^\\intercal\\big) \\, d\\sigma\\\\\n\t\t&\\left.\\2ds\\right|_{s=0} \\left[\\mathscr{F}(g(s), 1) - \\int_\\Sigma 2H_{g(s)} \\, d\\sigma_{g(s)} \\right]\\\\\n\t\t& \\quad = \\int_{\\mathbb{R}^3\\setminus \\Omega} h\\cdot \\Big(-D\\Ric(h)+\\tfrac{1}{2} DR(h) \\bar{g}\\Big)\\, d\\mathrm{vol}\\\\\n\t\t& \\quad  +\\int_{\\Sigma}\\Big( \\big(DA(h)  - 2A \\circ h^\\intercal + \\tfrac{1}{2} \\tr h^\\intercal A \\big)\\cdot h^\\intercal +  \\tr h^\\intercal DH(h) \\Big)\\, d\\sigma \\\\\n\t\t&\\quad + \\int_\\Sigma \\left(A\\cdot \\big(g^\\intercal \\big)'' - \\Big(\\tr_{g(s)}g'(s)^\\intercal\\Big)' H - 2DH(h) \\tr h^\\intercal- \\tfrac{1}{2} \\big(\\tr h^\\intercal\\big)^2 H\\right) \\, d\\sigma.\n\t\\end{split}\n\\end{align}\n\n\nFor $|s|$ sufficiently small,  $(\\Sigma, g(s)^\\intercal)$ is convex. From Lemma~\\ref{lemma:embedding},  there is a smooth family of isometric embeddings  $f_s:(\\Sigma, g(s)^\\intercal)\\to (\\mathbb{R}^3, \\bar{g})$.  By Lemma~\\ref{lemma:extension}, there exists  $\\psi_s: \\mathbb{R}^3\\setminus  \\Omega\\to \\mathbb{R}^3$, with $\\psi_0$ being the identity map,  that is diffeomorphic onto its image such that for each $s$,  $\\psi_s|_\\Sigma = f_s$ and $\\psi_s$ is the identity map outside a compact subset. Define $\\xi(s) = (\\psi_s)^*(\\bar{g})$ on $\\mathbb{R}^3\\setminus \\Omega$, as the pull-back of the Euclidean metric, with $\\xi(0)=\\bar{g}$. By construction,  $\\xi(s)$ is isometric to the Euclidean metric and induces the same metric as $g(s)$ on $\\Sigma$:\n \\begin{align*}\n \t\\xi(s)^\\intercal&= g(s)^\\intercal.  \n\\end{align*}\nFurthermore, the second fundamental form $A_{\\xi(s)}$ of $\\Sigma$ in $(\\mathbb{R}^3, \\xi(s))$ is just $\\mathring{A}_{\\xi(s)^\\intercal}$ since $\\xi(s)$ is itself a Euclidean metric. Using that $\\xi(s)^\\intercal= g(s)^\\intercal$, we have \n\\begin{align*}\n\tA_{\\xi(s)}= \\mathring{A}_{g(s)^\\intercal},\\quad \\mbox{and}\\quad  \\ H_{\\xi(s)} = \\mathring{H}_{g(s)^\\intercal}.\n\\end{align*} \nBy differentiating the above identities in $s$, we get\n\\begin{align}\\label{equation:g-functional}\n\\begin{split}\n\t\\big(\\xi(s)^\\intercal\\big)' &= h^\\intercal\\\\\n\t\\big(\\xi(s)^\\intercal\\big)''&= \\big(g(s)^\\intercal\\big)''\\\\\n\t\\big(A_{\\xi(s)}\\big)' &= D\\mathring{A} (h^\\intercal)\\\\\n\t\\big(H_{\\xi(s)}\\big)' &=D\\mathring{H}(h^\\intercal). \n\\end{split}\n\\end{align}\n\n Apply the computations of \\eqref{equation:variation-G} to the family of asymptotically flat pairs $(\\xi(s), 1)$, and substitute the terms by \\eqref{equation:g-functional}. Because $\\mathscr{F}(\\xi(s), 1)\\equiv 0$, $D\\Ric(\\xi'(0))=0$, $DR(\\xi'(0))=0$,  we derive\n\\begin{align*}\n\t&\\left.\\ds \\right|_{s=0}  \\int_\\Sigma 2\\mathring{H}_{\\xi(s)^\\intercal} \\, d\\sigma_{g(s)} = \\int_\\Sigma (-A\\cdot h^\\intercal+ H \\tr h^\\intercal) \\, d\\sigma\\\\\n\t\t&\\left.\\2ds\\right|_{s=0}  \\int_\\Sigma 2\\mathring{H}_{\\xi(s)^\\intercal}\\, d\\sigma_{g(s)}\\\\\n\t\t&\\quad=-  \\int_{\\Sigma}\\Big( \\big(D\\mathring{A}(h^\\intercal)  -2A \\circ h^\\intercal+ \\tfrac{1}{2} \\tr h^\\intercal A \\big)\\cdot h^\\intercal +\\tr h^\\intercal D\\mathring{H}\\big(h^\\intercal \\big)  \\Big)\\, d\\sigma  \\\\\n\t\t&\\quad  + \\int_\\Sigma \\left(- A\\cdot \\big(g^\\intercal \\big)'' + \\Big(\\tr_{g(s)}g'(s)^\\intercal\\Big)' H + 2D\\mathring{H}(h^\\intercal) \\tr h^\\intercal+\\tfrac{1}{2} \\big(\\tr h^\\intercal\\big)^2 H\\right) \\, d\\sigma.\n\\end{align*}\nThe desired formulas follow from adding the above identities to \\eqref{equation:variation-G} and then simplifying. \n\\end{proof}\n\nWe now proceed to prove Proposition~\\ref{proposition:harmonic-function}. We recall its statement. \n\n\n\\begin{manualproposition}{\\ref{proposition:harmonic-function}}\nLet  $\\Omega$ be a bounded open subset in $\\mathbb{R}^3$ whose boundary $\\Sigma = \\partial \\overline{\\Omega}$ has  positive Gauss curvature. Let $v\\in\\C^{2,\\alpha}_{-q}(\\mathbb R^3\\setminus\\Omega)$ satisfy $\\Delta v=0$. Then\n\\[\n\\int_\\Sigma 2v\\Big(D \\mathring{H}\\big(2v\\bar{g}^\\intercal\\big) - DH(2v\\bar{g}) \\Big) \\, d\\sigma\\geq 0.\n\\]\n\\end{manualproposition}\n\\begin{proof}\nLet $\\gamma(s)$ be the family of metrics in $\\mathbb{R}^3\\setminus \\Omega$ constructed from   the harmonic function $v$ as in Lemma~\\ref{lemma:harmonic-generate}, and let $\\gamma_{\\I}$ be the corresponding  metrics in the compact region $\\overline{\\Omega}$.  Recall that $\\gamma'(0) = 2v\\bar{g}$. We will show that the functional~$\\mathscr{G}$ is ``concave'' at $\\bar{g}$ along $\\gamma(s)$ in the following sense:\n\\begin{align}\\label{equation:concave}\n\t\\left.\\2ds\\right|_{s=0} \\mathscr{G}(\\gamma(s)) \\ge 0.\n\\end{align}\nThe functional $\\mathscr{G}$ along $\\gamma(s)$ can be re-written as \n\\begin{align*}\n\t\\mathscr{G}(\\gamma(s))   &=-16\\pi m_{\\mathrm{ADM}}(\\gamma(s))+ \\int_{\\mathbb{R}^3\\setminus \\Omega} R_{\\gamma(s)}\\, d\\mathrm{vol}_{\\gamma(s)}+  \\int_\\Sigma 2\\Big(H_{\\gamma_{\\I}(s)} - H_{\\gamma(s)}\\Big)\\, d\\sigma_{\\gamma(s)}\\\\\n\t& \\quad + \\int_\\Sigma 2\\Big(\\mathring{H}_{\\gamma(s)^\\intercal} - H_{\\gamma_{\\I}(s)}\\Big)\\, d\\sigma_{\\gamma(s)}.\n\\end{align*}\nSince  $\\left.\\2ds\\right|_{s=0} $ of first line in the right hand side is nonnegative by Lemma~\\ref{lemma:harmonic-generate}, we just need to compute $\\left.\\2ds\\right|_{s=0}  \\int_\\Sigma 2\\Big(\\mathring{H}_{\\gamma(s)^\\intercal} - H_{\\gamma_{\\I}(s)}\\Big)\\, d\\sigma_{\\gamma(s)}$.  Using that $(\\overline{\\Omega}, \\gamma_{\\I}(s))$ has zero scalar curvature and $\\gamma_{\\I}(s)^\\intercal = \\gamma(s)^\\intercal$ on $\\Sigma$, we can apply Shi-Tam's theorem, Theorem~\\ref{theorem:Shi-Tam}, to see that \n\\[\n\\left.\\2ds\\right|_{s=0}\\int_\\Sigma 2\\Big(\\mathring{H}_{\\gamma(s)^\\intercal} - H_{\\gamma_{\\I}(s)}\\Big)\\, d\\sigma_{\\gamma(s)}\\ge 0\n\\]\nas the integral achieves the global minimum at $s=0$. Thus we prove~\\eqref{equation:concave}.\n\n\n\nFrom Proposition~\\ref{proposition:variation2},  we have the second variation formula \\begin{align*}\n\t\\left.\\2ds\\right|_{s=0} \\mathscr{G}(\\gamma(s))&=\\int_{\\mathbb{R}^3\\setminus \\Omega} h\\cdot\\Big(-D\\Ric(h)+\\tfrac{1}{2} DR(h) \\bar{g}\\Big)\\, d\\mathrm{vol}\\\\\n\t&\\quad +\\int_\\Sigma \\left[\\Big( D\\mathring{H}(h^{\\intercal}) - DH(h)\\Big)\\tr h^\\intercal + \\Big(DA (h)- D \\mathring{A}(h^\\intercal)\\Big)\\cdot h^\\intercal \\right] \\, d\\sigma\n\\end{align*}\nwhere $h=\\gamma'(0) = 2v\\bar{g}$. Since $v$ is harmonic, we have $\\bar{g}\\cdot D\\Ric (2vg) = 0$ and $DR(2vg)=0$ by the linearized curvature formulas in Lemma~\\ref{lemma:formula}. So the interior integral above is zero. For the boundary integral, obviously \n$\\Big( D\\mathring{H}(h^{\\intercal}) - DH(h)\\Big)\\tr h^\\intercal =4v\\Big( D\\mathring{ H} (2v\\bar{g}^\\intercal)- DH\\big(2v\\bar{g}\\big)\\Big)$  and \n\\begin{align*}\n\t& \\Big(D A(2v\\bar{g}) - D\\mathring{A}\\big(2v\\bar{g}^\\intercal \\big) \\Big) \\cdot h^\\intercal \\\\\n\t &= 2v \\left. \\ds\\right|_{s=0} \\Big(\\big(A_{\\gamma(s)} -\\mathring{A}_{\\gamma(s)^\\intercal}\\big) \\cdot \\gamma(s)^\\intercal\\Big)+ 2v \\big(A_{\\bar{g}} -\\mathring{A}_{\\bar{g}^\\intercal}\\big) \\cdot \\left. \\ds\\right|_{s=0} \\gamma(s)^\\intercal \\\\ \n\t &= 2v \\left. \\ds\\right|_{s=0} \\big(H_{\\gamma(s)} -\\mathring{H}_{\\gamma(s)^\\intercal}\\big) \\\\\n\t &= 2v \\Big( D H (2v\\bar{g})- D\\mathring{H}\\big(2v\\bar{g}^\\intercal\\big)\\Big).\n\\end{align*}\nTherefore,\n\\[\t\n\t\\left.\\p2s\\right|_{s=0} \\mathscr{G}(\\gamma(s))=\\int_\\Sigma 2v\\Big(D \\mathring{H}\\big(2v\\bar{g}^\\intercal\\big) - DH(2v\\bar{g}) \\Big) \\, d\\sigma.\n\\]\nCombining this with \\eqref{equation:concave} gives the proposition. \n\\end{proof}\n\n\n\n\\subsection{Rigidity of Ricci flat deformations}\n\nIn order to apply Proposition~\\ref{proposition:harmonic-function} to prove Theorem~\\ref{theorem:static-convex}, we will first derive  effective formulas to compute  $D\\mathring H (h^\\intercal)$.\n\n\n\n\nObserve that for deformations that take the form $L_X\\bar g$ (those ``generated from diffeomorphisms''), \n\\begin{align}\\label{equation:embedding}\n\tD\\mathring{H}\\big((L_X\\bar{g})^\\intercal\\big) = DH(L_X\\bar{g}). \n\\end{align}\nTo see this, we let $\\psi_t: \\mathbb{R}^3\\setminus \\Omega\\to \\mathbb{R}^3$ be be the flow of $X$. Since $\\psi_t^*\\bar{g}$ is a Euclidean metric on $\\mathbb{R}^3\\setminus \\Omega$, by definition of $\\mathring{H}$, we have $\\mathring{H}_{(\\psi_t^*\\bar{g})^\\intercal} =H_{\\psi_t^*\\bar{g}}$. Differentiating in $t$ gives the above formula. \n\n\nWe will see in the next lemma that because of the convexity of $\\Sigma$, any Ricci flat deformation $h$ must be generated from diffeomorphisms, i.e. $h=L_X\\bar g$, and any tensor $\\tau$ on $\\Sigma$ is induced from a ``unique'' Ricci flat deformation $h$ such that $h^\\intercal = \\tau$. To state the lemma more precisely, we define the Banach spaces $\\mathcal{S}_0$ and $\\mathcal{S}_1$  that consist of symmetric (0,2)-tensors in $\\mathbb{R}^n\\setminus \\Omega$:\n \\begin{align*}\n \\mathcal{S}_0&=\\{ L_Z \\bar{g}: \\mbox{$Z\\in \\C^{3,\\alpha}_{-q}(\\mathbb{R}^3\\setminus \\Omega)$  satisfies $\\Delta Z=0$ in $\\mathbb{R}^3\\setminus \\Omega$}\\}\\\\\n  \\mathcal{S}_1&=\\{ h\\in \\C^{2,\\alpha}_{-q}(\\mathbb{R}^3\\setminus \\Omega): D\\Ric(h)=0, \\bi h=0 \\mbox{ in } \\mathbb{R}^3\\setminus \\Omega\\}\/\\{L_X \\bar{g}: X\\in \\mathcal{X}_0 \\},\n \\end{align*}\n where  the equivalence relation in $\\mathcal{S}_1$ says that $h_1\\sim h_2$ if and only if $h_1-h_2 = L_X \\bar{g}$ for some  $X \\in \\mathcal X_0$. (Recall the definition of $\\mathcal{X}_0$ in Definition~\\ref{definition:harmonic}.) Geometrically, $\\mathcal{S}_0$ consists of deformations generated from diffeomorphisms  satisfying the harmonic gauge, and  $\\mathcal{S}_1$ consists of Ricci flat deformations satisfying the harmonic gauge (up to the equivalent relation).  We also define the Banach space of tensors on the boundary $\\Sigma$:\n  \\begin{align*}\n \\S_2=\\{ \\tau\\in \\C^{2,\\alpha}(\\Sigma): \\tau \\mbox{ is a symmetric $(0,2)$-tensor on } \\Sigma\\}.\n \\end{align*} \n\n\n \\begin{lemma}\\label{lemma:convex}\nLet  $\\Sigma$ be convex. Then $\\mathcal{S}_0$ is surjective onto both $\\mathcal{S}_1$ and $\\mathcal{S}_2(\\Sigma)$, and $\\mathcal{S}_1,\\mathcal{S}_2(\\Sigma)$ are isomorphic, via the maps given by\n \\begin{align*}\t\n                         &b:L_Z\\bar{g} \\in \\mathcal{S}_0\\longrightarrow  [L_Z \\bar{g}]\\in \\mathcal{S}_1\\\\\n \t \t\t&b_0:L_Z \\bar{g}\\in  \\mathcal{S}_0 \\longrightarrow  (L_Z\\bar{g})^\\intercal\\in \\S_2\\\\\n\t\t\t&b_1:[h] \\in \\mathcal{S}_1 \\longrightarrow  h^\\intercal \\in \\S_2\n \t \\end{align*}\n\n \\end{lemma}\n \\begin{remark}\n We note that the statement about surjectivity of $ b:\\mathcal{S}_0\\to \\mathcal{S}_1$ can be  compared to Corollary~\\ref{corollary:trivial}. It says that the Cauchy boundary condition in Corollary~\\ref{corollary:trivial} can be dropped if $\\Sigma$ is convex.\n \\end{remark}\n \\begin{proof}\n We will show  that $b_0$\n\n is surjective and $b_1$\n\n is injective.\n\nSince $b_0=b_1\\circ b$, it gives the desired properties. \n \nThe map $b_0$\nis surjective: Given $\\tau \\in \\S_2$, we consider the family of metrics $ \\bar{g}^\\intercal + t\\tau$ on $\\Sigma$. Since $(\\Sigma, \\bar{g}^\\intercal)$ has positive Gauss curvature, $ \\bar{g}^\\intercal + t\\tau$ also has positive Gauss curvature for $|t|$  small. So there is a smooth family of isometric embeddings $f_t: (\\Sigma, \\bar{g}^\\intercal + t\\tau) \\to (\\mathbb{R}^3, \\bar{g})$ by Lemma~\\ref{lemma:embedding}. We identify $\\Sigma$ with $f_0(\\Sigma)$. Let $V: =\\left. \\frac{\\partial }{\\partial t}\\right|_{t=0} f_t$ be the velocity vector field defined along $\\Sigma$, and we have $V\\in \\C^{2,\\alpha}(\\Sigma)$ initially. Since  $(L_V \\bar{g})^\\intercal=\\tau \\in \\C^{2,\\alpha}(\\Sigma)$ on $\\Sigma$, we can argue that in fact $V\\in \\C^{3,\\alpha}(\\Sigma)$.  Solving $Z\\in \\C^{3,\\alpha}_{-q}(\\mathbb{R}^3 \\setminus \\Omega)$ to the Dirichlet boundary value problem \n \\begin{align*}\n \t\\Delta Z&=0 \\mbox{ in } \\mathbb{R}^3 \\setminus \\Omega\\\\\n\tZ &= V \\mbox{ on } \\Sigma,\n \\end{align*}\nwe find $[L_Z\\bar{g}]\\in \\mathcal{S}_0$ so that $(L_Z\\bar{g})^\\intercal = \\tau$ on $\\Sigma$.\n\nThe map $b_1$\nis injective: If $h$ is a representative of $[h] \\in \\mathcal{S}_1$ and $h^\\intercal=0$, then we will show that $h=L_X \\bar{g}$ for some $X\\in \\mathcal{X}_0$, and thus $[h]=0$. For any $\\tau \\in \\S_2$, we have shown that there is $L_Z \\bar{g} \\in \\mathcal{S}_0$ such that $(L_Z \\bar{g})^\\intercal = \\tau$. Since both $h$ and  $L_Z\\bar{g}$ are Ricci flat deformations, we can apply \\eqref{equation:Ricci-flat} (with $k=L_Z\\bar{g}$ and hence $k^\\intercal = \\tau$)  to get\n \\[\n \t0=\\int_\\Sigma \\Big(DA(h)\\cdot \\tau-  (\\tr \\tau) DH(h)\\Big)\\, d\\sigma=\\int_\\Sigma \\big(DA(h)-DH(h){\\bar g}^\\intercal\\big)\\cdot\\tau \\, d\\sigma.\n \\]\nSince $\\tau$ is arbitrary, $DA (h)- DH(h) \\bar{g}^{\\intercal}$ must vanish identically on $\\Sigma$. Using the assumption that $h^\\intercal=0$, we see that $DH(h)$ and hence $DA(h)=0$ both vanish on $\\Sigma$. Then Corollary~\\ref{corollary:trivial} implies that $h=L_X \\bar{g}$ for some $X\\in \\mathcal{X}_0$. \n \\end{proof}\n\\begin{remark}\nWhile we will not use it elsewhere in the paper, we note the following fact.  The map~$b:\\mathcal{S}_0\\to \\mathcal{S}_1$ has an $N$-dimensional kernel space  \n\\[\n\\{ L_Z\\bar{g}\\in \\mathcal{S}_0: Z= K \\mbox{ on $\\Sigma$ for some Killing vector $K$} \\}.\n\\]\n Thus, the  maps $b$ and $b_0$  induce isomorphisms on the quotient space, $\\mathcal{S}_0$ modulo the kernel space, to  $\\mathcal{S}_1$ and $\\mathcal{S}_2(\\Sigma)$, respectively.\n\\end{remark}\n\n\nTogether with an earlier observation \\eqref{equation:embedding}, Lemma~\\ref{lemma:convex} gives the following effective formulas for $D\\mathring{H}$.  \n\n\n\\begin{corollary}\\label{corollary:convex}\nLet $\\Sigma$ be convex and  $h\\in \\C^{2,\\alpha}_{-q}(\\mathbb{R}^3\\setminus \\Omega)$ be a symmetric $(0,2)$-tensor. Then the following holds:\n\\begin{enumerate}\n\\item  $h^\\intercal = (L_X \\bar{g})^\\intercal$ on $\\Sigma$ for some $X\\in \\C^{3,\\alpha}_{-q}(\\mathbb{R}^3 \\setminus \\Omega)$ and\n\\[\n\tD\\mathring{H}(h^\\intercal) = DH(L_X\\bar{g})\\quad  \\mbox{ on }\\Sigma.\n\\]\n\\item If $h$ solves $D\\Ric(h)=0$ in $\\mathbb{R}^3\\setminus \\Omega$, then $h = L_X \\bar{g}$ in $\\mathbb{R}^3\\setminus \\Omega$ for some $X\\in \\C^{3,\\alpha}_{1-q}(\\mathbb{R}^3 \\setminus \\Omega)$ and\n\\[\n\tD\\mathring{H}(h^\\intercal) = DH(h) \\quad \\mbox{ on } \\Sigma.\n\\]\n\\end{enumerate}\n \\end{corollary}\n \\begin{proof}\n The first statement is an immediate consequence from Lemma~\\ref{lemma:convex} and \\eqref{equation:embedding}. \n\nFor the second statement,  by Lemma~\\ref{lemma:harmonic-gauge}, there is a vector field $V\\in \\C^{3,\\alpha}_{1-q}(\\mathbb{R}^3\\setminus \\Omega)$ with $V=0$ on $\\Sigma$ such that $\\bi (h+L_V \\bar{g})=0$ in $\\mathbb{R}^3\\setminus \\Omega$, and thus $[h+L_V\\bar{g}] \\in \\mathcal{S}_1$.  Since $\\mathcal{S}_0$ is surjective onto $\\mathcal{S}_1$ as shown in Lemma~\\ref{lemma:convex}, there is  $Z\\in \\C^{3,\\alpha}_{-q}(\\mathbb{R}^3\\setminus \\Omega)$ such that \n\\[\n\t[h+L_{V}\\bar{g}]= [L_Z\\bar{g}]. \n\\]\nIt is straightforward to verify that $h=L_X \\bar{g}$ for some $X= O^{3,\\alpha}(|x|^{1-q})$. The identity for the linearized mean curvature follows \\eqref{equation:embedding}.\n\\end{proof}\n\n\n\\begin{proof}[Proof of Theorem~\\ref{theorem:static-convex}]\nSuppose $(h, v)$ is a static vacuum deformation satisfying $h^\\intercal=0, DH(h)=0$ on $\\Sigma$.  We will show that $DA(h)=0$ on $\\Sigma$. \n\nIf $v$ is identically zero, then $h$ is a Ricci flat deformation.  We may assume that $h$ satisfies the harmonic gauge $\\bi h=0$ by Lemma~\\ref{lemma:harmonic-gauge}. Applying Lemma~\\ref{lemma:convex} that $\\mathcal{S}_1\\to \\mathcal{S}_2(\\Sigma)$ is isomorphic, we have $[h]=0$ and thus $h = L_X\\bar{g}$ for some $X\\in \\mathcal{X}_0$. In particular, $DA(h)=0$ on $\\Sigma$.  \n\nFrom now on, we assume that $v$ is not identically zero. \nBy Lemma~\\ref{lemma:geodesic}, we may assume that $h$ satisfies the geodesic gauge on $\\Sigma$, and therefore, by \\eqref{equation:normal}, \\eqref{equation:linearized-sff},  and \\eqref{equation:Riccati}, we have\n\\begin{align}\\label{equation:linearization-geodesic}\n\\begin{split}\n\t\\nu\\big(DH(h) \\big)&= -A \\cdot DA(h)\\\\\n\t(L_\\nu h)^\\intercal &= 2DA(h).\n\\end{split}\n\\end{align}\n\n\nLet $\\nu$ be the outward unit normal to $\\Sigma$ parallelly extended in a tubular neighborhood of $\\Sigma$.  Let $Y$ be a smooth, compactly supported vector field on $\\mathbb{R}^3$ such that $Y = \\nu$ in a tubular neighborhood of $\\Sigma$. Let $\\psi_t$ be the flow of $Y$ for $ t\\in (-\\epsilon, \\epsilon)$ and we denote by $\\Sigma_t = \\psi_t(\\Sigma)$  and $\\Omega_t = \\psi_t(\\Omega)$. Let $v_t\\in \\C^{2,\\alpha}_{-q}(\\mathbb{R}^3\\setminus \\Omega_t)$ be a $\\C^1$-family of harmonic functions such that the Dirichlet boundary value $v_t|_{\\Sigma_t} = v|_{\\Sigma_t}$ for $t\\ge 0$. In particular, $v_t = v|_{\\mathbb{R}^3\\setminus \\Omega_t}$ for $t\\ge 0$. We define the  scalar-valued  $\\C^1$-function:\n\\begin{align}\\label{equation:H1}\n\t\\mathcal{H}(t)=\\int_{\\Sigma_t} \\left[2v_t\\Big(D\\mathring{H}\\big(2v_t\\bar{g}^\\intercal\\big) - D H(2v_t\\bar{g})\\Big)\\right] \\, d\\sigma.\n\\end{align}\nApplying Proposition~\\ref{proposition:harmonic-function} on each $\\Sigma_t$,  we have  $\\mathcal{H}(t)\\ge 0$ for $t\\in (-\\epsilon, \\epsilon)$. Since $(h, v)$ is a static vacuum deformation and thus $h+2v\\bar{g}$ is a Ricci flat deformation in $\\mathbb{R}^3\\setminus\\Omega$, we can apply Corollary~\\ref{corollary:convex} to get, for each $t\\ge 0$, \n\\[\n\t D\\mathring{H}\\big( (h+2v\\bar{g})^\\intercal \\big)=DH(h+2v\\bar{g})\\quad \\mbox{ on } \\Sigma_t,\n\\]\nand thus\n\\begin{align*}\n\t  D\\mathring{H} \\big(2v\\bar{g}^\\intercal\\big)- D H(2v\\bar{g}) =   DH(h) -  D\\mathring{H} (h^\\intercal)\\quad \\mbox{ on } \\Sigma_t.\n\\end{align*}\nThis implies that the function $\\mathcal{H}(t)$ has an alternative expression for $t\\ge 0$:\n\\begin{align}\\label{equation:H2}\n\\begin{split}\n\t\\mathcal{H}(t) &= \\int_{\\Sigma_t} 2v \\Big( DH(h) -  D\\mathring{H} (h^\\intercal)\\Big) \\, d\\sigma\\\\\n\t&=\\int_\\Sigma 2\\psi_t^* \\bigg(v   \\Big( DH(h) -  D\\mathring{H} (h^\\intercal)\\Big)\\bigg) \\, d\\sigma_{\\psi_t^*(\\bar{g})}.\n\\end{split}\n\\end{align}\nNote that $\\psi_t^* \\Big(v   \\big( DH(h) -  D\\mathring{H} (h^\\intercal)\\big)\\Big) =(\\psi_t^* v )\\Big( \\psi_t^* DH(h) - \\psi_t^* D\\mathring{H}(h^\\intercal)\\Big)$ on $\\Sigma$, where $\\psi_t^*DH(h)$ on $\\Sigma$ is the pull-back of the scalar-valued function $DH|_{\\bar{g}}(h)$ on $\\Sigma_t\\subset \\mathbb{R}^n\\setminus \\Omega_t$, and similarly for $\\psi^*_tD\\mathring{H}(h^\\intercal)$. (See also Lemma~\\ref{lemma:pullback}.)  From the alternative expression~\\eqref{equation:H2}, we see that $\\mathcal{H}(0)=0$ because of the assumption $h^\\intercal=0, DH(h)=0$ on $\\Sigma$. Thus, $t=0$ is a critical point $\\mathcal{H}'(0)=0$. \n\nWe \\emph{claim} \n\\begin{align*}\n\t\\mathcal{H}'(0) &= 2\\int_\\Sigma |DA(h)|^2\\, d\\sigma.\n\\end{align*}\nOnce  the claim is verified, we can combine with the fact  $\\mathcal{H}'(0)=0$ to conclude that $DA(h)$ vanishes on $\\Sigma$, which completes the proof. \n\n\n\nWe compute $\\mathcal{H}'(0)$. Differentiating \\eqref{equation:H2} in $t$  for $t\\ge 0$ and using that $DH (h) - D\\mathring{H} (h^\\intercal)=0$ on $\\Sigma$, we get \n\\begin{align*}\n\t\\mathcal{H}'(0)&=\\int_\\Sigma \t2v \\left.\\pt\\right|_{t=0}\\psi_t^* \\left( DH (h) - D\\mathring{H} (h^\\intercal)\\right) \\, d\\sigma.\n\\end{align*}\nRecall that $\\left.\\pt\\right|_{t=0} \\psi_t = Y = \\nu$ on $\\Sigma$.  By  \\eqref{equation:linearization-geodesic}, \n\\begin{align*}\n\t\\left.\\pt\\right|_{t=0}\\psi_t^* ( DH (h) )=\\nu(DH(h))=  -A \\cdot DA(h).\n\\end{align*}\nTo compute the term involving $D\\mathring{H}$, we apply Lemma~\\ref{lemma:convex}. For each $t$, there exists a vector field $X_t$ with $h^\\intercal = (L_{X_t} \\bar{g})^\\intercal$ on $\\Sigma_t$ such that \n\\[\t\n\t D\\mathring{H}\\big(h^\\intercal\\big) = DH( L_{X_t} \\bar{g}) \\quad \\mbox{ on } \\Sigma_t.\n\\]\nThen Lemma~\\ref{lemma:pullback} implies that\n\\begin{align*}\n \t\\psi_t^* \\left(D\\mathring{H}\\big(h^\\intercal\\big) \\right)&=\\psi_t^* \\left( DH( L_{X_t} \\bar{g})\\right)\\\\\n\t&=DH|_{\\psi_t^*\\bar{g}} \\big(\\psi_t^* L_{X_t} \\bar{g}\\big)\\\\\n\t&=D\\mathring{H}|_{\\psi_t^*\\bar{g}^\\intercal} \\big(\\psi_t^* (L_{X_t} \\bar{g} )^\\intercal \\big)\\\\\n\t&= D\\mathring{H}|_{\\psi_t^*\\bar{g}^\\intercal } \\big(\\psi_t^* h^\\intercal \\big) \\quad \\mbox{ on } \\Sigma,\n\\end{align*}\nwhere in the third equation we use that $\\psi_t^* L_{X_t} \\bar{g}  = L_{\\psi_t^* X_t} (\\psi_t^* \\bar{g})$ is Ricci flat deformation  at $\\psi_t^* \\bar{g}$ and Corollary~\\ref{corollary:convex}. Differentiating the previous identity in $t$ gives\n\\begin{align*}\n\t\\left. \\pt \\right|_{t=0} \\psi_{t}^* \\left( D\\mathring{H}\\big(h^\\intercal\\big)\\right) &=\\left.\\pt\\right|_{t=0} D\\mathring{H}|_{\\psi_{t}^* \\bar{g}^\\intercal } \\big(\\psi_{t}^* h^\\intercal \\big) \\\\\n\t&=D^2 \\mathring{H}\\left((L_Y \\bar{g})^\\intercal, h^\\intercal \\right) + D\\mathring{H} \\big((L_\\nu h)^\\intercal \\big)\\\\\n\t&= D\\mathring{H} \\big(2DA(h)\\big)\n\\end{align*}\nwhere we use that $h^\\intercal=0$ on $\\Sigma$   and \\eqref{equation:linearization-geodesic}. By the first statement in Corollary~\\ref{corollary:convex}, there is a vector $Z$ such that $ 2DA(h)=(L_Z\\bar{g})^\\intercal $ and  $D\\mathring{H} \\big(2DA(h)\\big) = DH(L_Z\\bar{g})$ on $\\Sigma$. Apply \\eqref{equation:Green-boundary} for static vacuum deformations $(h, v)$ and $(L_Z\\bar{g}, 0)$. Since $h^\\intercal=0, DH(h)=0$ on $\\Sigma$ by assumption, we get\n\\begin{align*}\n\t0&=\t\\int_\\Sigma  \\Big\\langle \\big(vA + DA(h) -\\nu(v) \\bar{g}^\\intercal, 2v \\big), \\big((L_Z\\bar{g})^\\intercal, DH(L_Z\\bar{g})\\big) \\Big\\rangle \\, \\da\\\\\n\t&=\\int_\\Sigma  \\Big\\langle \\big(vA + DA(h) -\\nu(v) \\bar{g}^\\intercal, 2v \\big), \\big(2DA(h), D\\mathring{H} (2DA(h))\\big)\\Big\\rangle \\, \\da.\n\\end{align*}\t\nRearranging the terms yields\n\\[\n\t-\\int_\\Sigma 2v D\\mathring{H}\\big(2DA(h)\\big) = \\int_\\Sigma \\Big(2DA(h) \\cdot \\big(vA+ DA (h) - \\nu(v) \\bar{g}^\\intercal \\big)\\Big)\\, d\\sigma.\n\\]\n\nCombining the previous computations gives\n\\begin{align*}\n\t\t\\mathcal{H}'(0)&=\\int_\\Sigma \t2v \\left.\\pt\\right|_{t=0}\\psi_t^* \\left( DH (h) - D\\mathring{H} (h^\\intercal)\\right) \\, d\\sigma\\\\\n\t\t&=\\int_\\Sigma \\Big(-2v A\\cdot DA(h) +2   DA(h)\\cdot \\big(vA+ DA (h) - \\nu(v) \\bar{g}^\\intercal \\big) \\Big)\\, d\\sigma\\\\\n\t\t&=\\int_\\Sigma 2|DA(h)|^2\\, d\\sigma\n\\end{align*}\nwhere we have used that $ DA(h)\\cdot \\bar{g}^\\intercal  =DH(h) - A\\cdot  h^\\intercal=0$ on $\\Sigma$. It completes the proof to the claim.  \n\n\\end{proof}\n\n\n\\section{Perturbations of hypersurfaces in $\\mathbb{R}^n$}\\label{section:generic}\n\nWe prove Theorem~\\ref{theorem:static-generic} and Corollary~\\ref{cor:dilation} in this section. Let us recall the statement in which we also spell out Definition~\\ref{definition:static-regular}. \n\n\\begin{manualtheorem}{\\ref{theorem:static-generic}}\nLet $t\\in [-\\delta, \\delta]$ and each $\\Omega_t\\subset \\mathbb{R}^n$ be a bounded open subset with embedded hypersurface boundary $\\Sigma_t = \\partial\\overline{ \\Omega_t}$.  Suppose  the boundaries $\\{ \\Sigma_t\\}$ form a smooth generalized foliation. Then  there is an open dense subset $J\\subset (-\\delta,\\delta)$ such that for any $t\\in J$,  $\\Sigma_t$ is static regular in $\\mathbb{R}^n\\setminus \\Omega_t$. Namely, if $(h, v)\\in \\C^{2,\\alpha}_{-q}(\\mathbb{R}^n\\setminus \\Omega_t)$ is a static vacuum deformation at $\\bar{g}$ such that $h$ satisfies the Bartnik boundary condition on $\\Sigma_t$, then $h$ satisfies the Cauchy boundary condition on $\\Sigma_t$.\n\\end{manualtheorem}\n\nLet  $X$ be the deformation vector of $\\{ \\Sigma_t\\}$. On each $\\Sigma_t$, we can write $X= \\zeta \\nu + X^\\intercal$, where  the scalar-valued function $\\zeta>0$, except possibly zero on a set of $(n-1)$-dimensional Hausdorff measure zero, $\\nu$ is the unit normal to $\\Sigma_t$ pointing way from $\\Omega_t$, and $X^\\intercal$ is tangential.  We smoothly extend $X$ as a smooth, compactly supported vector field in $\\mathbb{R}^n$ (still denoted by $X$). Let $\\psi_t:\\mathbb{R}^n\\to \\mathbb{R}^n$ be the flow of $X$. If we denote by $\\Omega:=\\Omega_0$ and $\\Sigma:= \\Sigma_0$, then $\\Omega_t = \\psi_t(\\Omega)$ and $\\Sigma_t = \\psi_t (\\Sigma)$.  Denote by $g_t = \\psi_t^* (\\bar{g}|_{\\mathbb{R}^n\\setminus \\Omega_t})$,   the pull-back metric defined on $\\mathbb{R}^n\\setminus \\Omega$, and note $g_0 = \\bar{g}$. \nDefine the family of maps $S_t$ as, for $(h, v)\\in \\C^{2,\\alpha}_{-q}(\\mathbb{R}^n\\setminus \\Omega)$,\n\\begin{align}\\label{equation:generic}\n S_t(h,v)=\n\\begin{array}{l}\n\t\\left\\{ \\begin{array}{l}-D\\Ric|_{g_t}(h)+ \\nabla^2_{g_t} v  \\\\\n\t\\Delta_{g_t} v\t\\end{array} \\right. \\quad \\mbox{ in } \\mathbb{R}^n\\setminus \\Omega \\\\\n\t\\left\\{ \\begin{array}{l} \n         h^\\intercal\\\\\n\tDH|_{g_t}(h)\n\t\\end{array} \\right. \\quad \\mbox{ on } \\Sigma.\n\\end{array}\n\\end{align}\nTo prove that $\\Sigma_t$ is static regular, we will need  to show that for any $(h, v)\\in \\C^{2,\\alpha}_{-q}(\\mathbb{R}^n\\setminus \\Omega)$ solving $S_t(h,v)=0$,\n\n we must have $DA|_{g_t}(h)=0$ on $\\Sigma$\n \n\n\n\\subsection{Differentiation of kernel elements along perturbations}\n\nWe first prove the following theorem, which implies Theorem~\\ref{theorem:static-generic} under an extra assumption~$(\\star)$. Its proof provides an outline for the proof of the general theorem. In order  to remove that assumption, we will establish basic facts about the kernel spaces of elliptic PDE systems in Section~\\ref{section:technical} and choose the open dense subset $J\\subset (-\\delta, \\delta)$. In Proposition~\\ref{proposition:kernel},  we will show that  a ``discrete version\" of the assumption $(\\star)$ holds for $t\\in J$ and then give a complete proof  of Theorem~\\ref{theorem:static-generic}.\n\n \n\n\n\\begin{manualtheorem}{\\ref{theorem:static-generic}$^\\prime$}\\label{theorem:special}\nLet $t\\in [-\\delta, \\delta]$, $\\Omega_t$, and $\\Sigma_t$ be as in Theorem~\\ref{theorem:static-generic}. Let $(h, v)\\in \\C^{2,\\alpha}_{-q}(\\mathbb{R}^n\\setminus \\Omega)$ solve $S_a(h,v)=0$ for some $a\\in (-\\delta,\\delta)$.   Suppose the following holds:\n\\begin{align} \\tag{$\\star$}\n\\begin{split}\n&\\mbox{There exists  $(h(t), v(t))\\in \\C^{2,\\alpha}_{-q}(\\mathbb{R}^n\\setminus \\Omega)$ with $S_t(h(t), v(t))=0$ for all $t$}\\\\\n&\\mbox{such that $(h(t), v(t)) \\to (h, v)$ in $\\C^{2,\\alpha}_{-q}(\\mathbb{R}^n\\setminus \\Omega)$  as $t\\searrow  a$,}\\\\\n&\\mbox{and $(h(t), v(t))$ is differentiable at $t=a$ }\\\\\n&\\mbox{with $(h'(a), v'(a)) = (p, z)\\in \\C^{2,\\alpha}_{-q}(\\mathbb{R}^n\\setminus \\Omega)$}. \n\\end{split}\n\\end{align}\nThen $DA|_{g_a}(h)=0$ on $\\Sigma$. \n\\end{manualtheorem}\n\n\n\n\\begin{proof}\nBy re-parametrizing $t$, we may assume that $a=0$. Recall that $g_0= \\bar{g}$ in our notation. Also recall that we omit the subscript $\\bar{g}$ in linearizations and geometric operators. By Lemma~\\ref{lemma:geodesic}, we may also assume that $h$ satisfies the geodesic gauge on $\\Sigma$, and thus by \\eqref{equation:normal}, \\eqref{equation:linearized-sff},   \\eqref{equation:Riccati}, and $h^\\intercal =0$ on $\\Sigma$, we have\n\\begin{align}\\label{equation:linearization-geodesic2}\n\\begin{split}\n\t\\begin{array}{rl} \n\t(L_{\\zeta\\nu} h)^\\intercal &= \\zeta(L_{\\nu} h)^\\intercal =2\\zeta DA(h)\\\\\n\t\\nu\\big(DH(h) \\big)&= -A \\cdot DA(h)\n\t\\end{array} \\quad \\mbox{ on } \\Sigma.\n\\end{split}\n\\end{align}\n\nWe \\emph{claim} that $(p-L_X h, z-L_X v)$ is a static vacuum deformation in $\\mathbb{R}^n\\setminus \\Omega$ whose Bartnik boundary data on $\\Sigma$ satisfy\n\\begin{subequations}\n\\begin{align}\n\t(p-L_X h\\big)^\\intercal &= -2\\zeta DA(h)\\label{equation:boundary1}\\\\\n\tDH(p-L_X h)&=\\zeta A\\cdot DA(h). \\label{equation:boundary2}\n\\end{align}\n\\end{subequations}\n\n\nLet  $(h(t), v(t))$, $t>0$, be the family from the assumption~$(\\star)$. Then\n\\begin{align*}\n\t\\left\\{ \\begin{array}{rl}-\\left.D\\Ric\\right|_{g_t}(h(t)) + \\nabla^2_{g_t} v(t)&=0\\\\\n\t\\Delta_{g_t} v(t)&=0\\end{array}\\right. \\quad \\mbox{ in } \\mathbb{R}^n\\setminus \\Omega.\n\\end{align*}\nNote the assumption $t>0$ implies that $\\mathbb R^n\\setminus\\Omega_t\\subset\\mathbb R^n\\setminus\\Omega$, so the pull-back pair $(\\psi_t^*h, \\psi_t^*v)$ in $\\mathbb{R}^n\\setminus \\Omega$ from  the restriction $(h, v)|_{\\mathbb{R}^n\\setminus \\Omega_t}$ is trivially a static vacuum deformation at $g_t$:\n\\begin{align*}\n\t\\left\\{\\begin{array}{rl}-\\left.D\\Ric\\right|_{g_t}(\\psi_t^*h) + \\nabla^2_{g_t} (\\psi_t^* v)&=\\psi^*_{t} \\left(-\\left.D\\Ric\\right(h) + \\nabla^2  v\\right)=0\\\\\n\t\\Delta_{g_t} (\\psi_t^* v)&=\\psi^*_{t} (\\Delta v)=0\\end{array}\\right. \\; \\mbox{ in } \\mathbb{R}^n\\setminus \\Omega.\n\\end{align*}\nSubtracting the previous two systems yields\n\\begin{align*}\n\t\\left\\{\\begin{array}{rl}-\\left.D\\Ric\\right|_{g_t}\\big(h(t)-\\psi_t^*h\\big) + \\nabla^2_{g_t}\\big (v(t)-\\psi_t^* v\\big)&=0\\\\\n\t\\Delta_{g_t} \\big(v(t)-\\psi_t^* v\\big)&=0\\end{array}\\right. \\quad \\mbox{ in } \\mathbb{R}^n\\setminus \\Omega.\n\\end{align*}\nDifferentiating it in $t$ at $t=0$ and noting that $h(0)-\\psi_0^* h=0, v(0)-\\psi_0^*v=0$, we prove that $(p-L_X h, z-L_X v)$  is a static vacuum deformation at $\\bar{g}$ in $\\mathbb{R}^n\\setminus \\Omega$. Next, we compute the boundary data. By using $h(t)^\\intercal =0$ on $\\Sigma$ by \\eqref{equation:generic} and differentiating in $t$ at $t=0$, we see $p^\\intercal =0$ on  $\\Sigma$. Since $X= \\zeta \\nu$ on $\\Sigma$, we apply \\eqref{equation:linearization-geodesic2}  to get \\eqref{equation:boundary1}:\n\\[\n\t(p-L_X h)^\\intercal = -(L_Xh)^\\intercal = -2\\zeta DA(h).\n\\]\nTo prove \\eqref{equation:boundary2}, by the  boundary condition of \\eqref{equation:generic} and Lemma~\\ref{lemma:pullback}, we have, on $\\Sigma$,\n\\begin{align*}\n\tDH|_{g_t}(h(t))=0,\\quad\n\tDH|_{g_t}(\\psi_t^* h) = \\psi_t^* \\big(DH(h)\\big).\n\\end{align*}\nSubtracting the previous identities gives \n\\[\nDH|_{g_t}\\big(h(t)-\\psi_t^* h\\big) = -\\psi_t^* \\big(DH(h)\\big) \\mbox{ on }  \\Sigma.\n\\]\nDifferentiating it in $t$ at $t=0$ gives\n\\begin{align*}\n\tDH(p-L_Xh) =- \\zeta \\nu \\big(DH(h)\\big) = \\zeta A\\cdot DA(h) \\quad \\mbox{ on } \\Sigma\n\\end{align*}\nwhere we use \\eqref{equation:linearization-geodesic2} in the last identity. It completes the proof of the claim. \n\n\nLastly, we apply \\eqref{equation:Green-boundary} (by substituting $(k, w)=(p-L_X h, z-L_X v)$) and use the boundary condition \\eqref{equation:boundary1}, \\eqref{equation:boundary2} to obtain\n\\begin{align*}\n\t0&= \\int_\\Sigma \\Big\\langle \\big(vA + DA(h) -\\nu (v) \\bar{g}^\\intercal, 2v\\big), \\big (k^\\intercal, DH(k)\\big) \\Big\\rangle \\, d\\sigma\\\\\n\t&=\\int_\\Sigma \\Big\\langle \\big(vA + DA(h) -\\nu (v) \\bar{g}^\\intercal, 2v\\big), \\big (-2\\zeta DA(h), \\zeta A\\cdot DA(h)\\big) \\Big\\rangle \\, d\\sigma\\\\\n\t&=-\\int_\\Sigma  2 \\zeta |DA(h)|^2 \\, d\\sigma\n\\end{align*}\nwhere we have used that $\\bar{g}^\\intercal \\cdot DA(h) = DH(h) + h^\\intercal \\cdot A= 0$ on $\\Sigma$ by the Bartnik boundary condition.  Since $\\zeta>0$, except an $(n-1)$-dimensional measure zero set, we conclude that $DA(h)=0$ on $\\Sigma$.\n\\end{proof}\n\n\n \n \n\n\n\n\\subsection{Weak differentiation in an open dense parameter set}\\label{section:technical}\n\nTo remove the assumption $(\\star)$, we will need to understand how solutions of $ S_t(h(t), v(t))=0$ behave  as $t\\to a$. Instead of analyzing general solutions, it suffices to analyze only those solutions satisfying the static-harmonic gauge,  as in Section~\\ref{section:existence}. \n\nTo begin, we define $L_t:\\C^{2,\\alpha}_{-q} (\\mathbb{R}^n\\setminus \\Omega) \\to \\C^{0,\\alpha}_{-q-2} (\\mathbb{R}^n\\setminus \\Omega) \\times \\C^{1,\\alpha}(\\Sigma)\\times \\mathcal{B}(\\Sigma)$ by\n\\begin{align}\n\\label{equation:linear-t}\nL_t (h, v)=\\begin{array}{l}\n\t\\left\\{ \\begin{array}{l}-D\\Ric|_{g_t}(h)+ \\nabla^2_{g_t} v - \\D_{g_t}(\\bi_{g_t} h+ dv) \\\\\n\t\\Delta_{g_t} v\t\\end{array} \\right. \\quad \\mbox{ in } \\mathbb{R}^n\\setminus \\Omega \\\\\n \t\\left\\{ \\begin{array}{l} \\bi_{g_t} h + dv\\\\\n\th^\\intercal\\\\\n\tDH|_{g_t}(h)\n\t\\end{array} \\right. \\quad \\mbox{ on } \\Sigma\n\\end{array}\n\\end{align}\nwhere the covariant derivatives and the linearizations are taken with respect to~$g_t$ and we recall the definitions of the operators $ \\D_{g_t}, \\bi_{g_t}$ in \\eqref{equation:operators}. For ease of notation, we shall denote the codomain of $L_t$ by \n\\[\n\t\\mathcal{Z}=\\C^{0,\\alpha}_{-q-2} (\\mathbb{R}^n\\setminus \\Omega) \\times \\C^{1,\\alpha}(\\Sigma)\\times \\mathcal{B}(\\Sigma).\n\\]\nThe operator $L_t$, defined for $\\mathbb{R}^n\\setminus \\Omega$, can be viewed as the pull-back of the operator $L$ as defined in \\eqref{equation:linear} for  $\\mathbb{R}^n\\setminus \\Omega_t$ via the diffeomorphism $\\psi_t$.  In particular, since $L$ is elliptic and Fredholm by Lemma~\\ref{lemma:Fredholm}, so is each $L_t$. \n\n The following lemma, essentially a linearized version of Lemma~\\ref{lemma:gauge}, says that the space of solutions to $S_t(h,v)=0$ is equivalent to that of $L_t(h, v)=0$, up to deformations generated from diffeomorphisms.\n  \\begin{lemma}\\label{lemma:static-harmonic}\nIf $(h, v)\\in \\C^{2,\\alpha}_{-q} (\\mathbb{R}^n\\setminus \\Omega) $ solves $L_t(h, v)=0$, then $(h,v)$ satisfies $S_t(h,v)=0$. In fact, we have $S_t(h+L_V g_t, v)=0$ \nand $DA|_{g_t}(h+L_V g_t)=DA|_{g_t}(h)$ on $\\Sigma$, \nfor any $V\\in \\C^{3,\\alpha}_{1-q}(\\mathbb{R}^n\\setminus \\Omega)$ satisfying $V=0$ on $\\Sigma$.\n\nConversely, if $(h, v)$ solves $S_t(h,v)=0$,\nthen there exists a vector field $V\\in \\C^{3,\\alpha}_{1-q}(\\mathbb{R}^n\\setminus \\Omega)$ with $V=0$ on~$\\Sigma$ such that $L_t(h+L_V g_t, v)=0$  and  $DA|_{g_t}(h+L_V g_t)=DA|_{g_t}(h)$ on $\\Sigma$. \n\\end{lemma}\n\\begin{proof}\n Suppose $L_t(h,v)=0$. By the same argument as in the first paragraph in the proof of Lemma~\\ref{lemma:kernel},  $\\bi_{g_t} h + dv$ is identically zero in $\\mathbb{R}^n\\setminus \\Omega$. Hence $(h, v)$ solves $S_t(h,v)=0$. For any vector field $V$ vanishing on $\\Sigma$, we have $S_t(L_V g_t,0)=0$ and $DA|_{g_t}(L_V g_t)=0$ on $\\Sigma$ (see Example~\\ref{example:vector}).\nWe prove the first statement. \n\n\nFor the second statement, let $V$ solve $\\Delta_{g_t} V=\\bi_{g_t} h + dv$ in $\\mathbb{R}^n\\setminus\\Omega$ and $V=0$ on $\\Sigma$. Then we can verify that $\\bi_{g_t} (h+L_V g_t)+dv =0$ everywhere. The rest of the statement follows. \n\\end{proof}\n\n\nIn the next four lemmas, we establish general facts about the kernel spaces $\\Ker L_t$. For an arbitrary $a\\in (-\\delta, \\delta)$, $\\Ker L_a$ is not necessarily the limit of $\\Ker L_t$ as $t\\to a$ because the nullity of $L_a$ may ``jump up'' in the limit. Nevertheless, we shall show that  such jumping  is rare, in a precise sense that the nullity  is upper semi-continuous in $t$.  While some facts are standard for elliptic operators on bounded regions with the standard H\\\"older norms, we include some proofs in our setting for completeness. Let $\\overline{B}_R\\subset \\mathbb{R}^n$ denote the closed round ball of radius $R$.\n\n\\begin{lemma}\\label{lemma:scale-broken}\nThere are positive constants $R, C>0$, uniformly in $t\\in (-\\delta, \\delta)$, such that \n\\[\n\t\\| (h,v) \\|_{\\C^{2,\\alpha}_{-q}(\\mathbb{R}^n\\setminus \\Omega)} \\le C\\Big( \\| (h,v) \\|_{\\C^0 (\\overline{B}_R \\setminus \\Omega)} + \\| L_t (h, v) \\|_{\\mathcal{Z}} \\Big)\n\\]\n\\end{lemma}\n\\begin{proof}\nBy the boundary Schauder estimate \\cite{Agmon-Douglis-Nirenberg:1964} in the standard H\\\"older norms and a scaling argument, we have the boundary Schauder estimate in the weighted H\\\"older norms:\n\\[\n\t\\| (h,v) \\|_{\\C^{2,\\alpha}_{-q}(\\mathbb{R}^n\\setminus \\Omega)} \\le C\\Big( \\| (h,v) \\|_{\\C^0 _{-q}(\\mathbb{R}^n \\setminus \\Omega)} + \\| L_t (h, v) \\|_{\\mathcal{Z}} \\Big),\n\\]\nwhere the constant $C$ can be chosen uniformly in $t$ because the coefficients of $L_t$ are uniformly bounded in suitable weighted norms.  Following closely the arguments involving  the cut-off trick and an interpolation inequality as in \\cite[Theorem 1.10]{Bartnik:1986},  the $\\C^{0}_{-q}(\\mathbb{R}^n \\setminus \\Omega) $-norm in the right hand side can be replaced by the ``scale-broken'' $\\C^{0}(\\overline{B}_R\\setminus \\Omega)$-norm, provided that $R$ is sufficiently large.\n\\end{proof}\n\n\\begin{lemma}\\label{lemma:subsequence}\nLet $\\{t_j\\}\\subset (-\\delta,\\delta)$ be a sequence converging to $a\\in (-\\delta,\\delta)$, and let $C$ be a positive real number. Suppose $(h_j, v_j)\\in \\C^{2,\\alpha}_{-q}(\\mathbb{R}^n\\setminus \\Omega)$ and $f_j, f\\in \\mathcal{Z}$ satisfy\n\\begin{align*}\n\t\\| (h_j, v_j) \\|_{\\C^{2,\\alpha}_{-q}(\\mathbb{R}^n\\setminus \\Omega)} &< C\\\\\n\tL_{t_j} (h_j, v_j)&=f_j\\\\\n\t\\| f_j - f\\|_{\\mathcal{Z}} &\\to 0\\mbox{ as } j\\to \\infty.\n\\end{align*}\nThen there is a subsequence $(h_{j_k}, v_{j_k})$ and $(h, v)\\in \\C^{2,\\alpha}_{-q}(\\mathbb{R}^n\\setminus \\Omega)$ such that \n\\begin{align*}\n\t\\| (h_{j_k}, v_{j_k}) - (h,v)\\|_{\\C^{2,\\alpha}_{-q}(\\mathbb{R}^n\\setminus \\Omega)} &\\to 0\\mbox{ as } k\\to \\infty\\\\\n\tL_{a} (h, v)&=f.\n\\end{align*}\nFurthermore, if $\\| (h_j, v_j) \\|_{\\C^{2,\\alpha}_{-q}(\\mathbb{R}^n\\setminus \\Omega)} =1$ for all $j$, then $\\| (h, v) \\|_{\\C^{2,\\alpha}_{-q}(\\mathbb{R}^n\\setminus \\Omega)} =1$. \n\\end{lemma}\n\\begin{proof}\nIn this proof, all the weighted H\\\"older norms are taken in the region $\\mathbb{R}^n\\setminus \\Omega$.\nFix $\\alpha'<\\alpha$ and $q'<q$. By compact embedding of $\\C^{2,\\alpha}_{-q}$ in $\\C^{2,\\alpha'}_{-q'}$, there is a subsequence $(h_{j_k}, v_{j_k})$ and $(h, v)\\in \\C^{2,\\alpha}_{-q}$ such that $\\| (h_{j_k}, v_{j_k}) - (h,v)\\|_{\\C^{2,\\alpha'}_{-q'}}\\to 0$. In particular, $\\| (h_{j_k}, v_{j_k}) - (h,v)\\|_{\\C^0(\\overline{B}_R \\setminus \\Omega)}\\to 0$. Together with Lemma~\\ref{lemma:scale-broken} and the fact that the operator norm $\\|L_{t_{j_k}} -L_a \\|_{\\mathrm{op}}\\to 0$, we obtain $\\| (h_{j_k}, v_{j_k}) - (h,v)\\|_{\\C^{2,\\alpha}_{-q}}\\to 0$  as $k\\to \\infty$. The other statements follow \ndirectly. \n\\end{proof}\n\nThe following coercivity-type estimate for $(h, v)$ in a complementing space to $\\Ker L_t$ is standard if~$t$ is fixed. By slightly extending the standard argument, we show that  a \\emph{uniform} coercivity estimate holds.\n\\begin{lemma}\\label{lemma:no-kernel}\nLet $\\{t_j\\}$ be a sequence in $(-\\delta,\\delta)$ converging to $a\\in (-\\delta,\\delta)$.\nSuppose in $\\C^{2,\\alpha}_{-q}(\\mathbb{R}^n\\setminus \\Omega)$ there is a closed subspace $\\mathcal{Y}$ complementing to $\\Ker L_a$ and to $\\Ker L_{t_j}$ for each $j$, i.e.\n\\begin{align*}\n\t\\C^{2,\\alpha}_{-q}(\\mathbb{R}^n\\setminus \\Omega) =\\mathcal{Y}\\oplus \\Ker L_{a}= \\mathcal{Y}\\oplus \\Ker L_{t_j}.\n\\end{align*}\n Then there is a positive constant $C$, uniformly in $j$, such that for all $(h, v)\\in \\mathcal{Y}$, \n\\[\n\t\\| (h, v) \\|_{\\C^{2,\\alpha}_{-q}(\\mathbb{R}^n\\setminus \\Omega)} \\le C \\| L_{t_j} (h, v) \\|_{\\mathcal{Z}}.\n\\]\n\\end{lemma}\n\n\\begin{proof}\nSuppose, to get a contradiction, that there is a subsequence of $t_j$, still labelled by $j$, and  $(h_j, v_j)\\in \\mathcal{Y}$ with $\\|(h_j, v_j)\\|_{\\C^{2,\\alpha}_{-q}(\\mathbb{R}^n\\setminus \\Omega)}=1$ such that  \n\\[\n \\| L_{t_j}(h_j, v_j)\\| \\le \\tfrac{1}{j} \\|(h_j, v_j)\\|_{\\C^{2,\\alpha}_{-q}(\\mathbb{R}^n\\setminus \\Omega)}= \\tfrac{1}{j} \\to 0 \\quad \\mbox{ as } j\\to \\infty.\n \\]\nBy Lemma~\\ref{lemma:subsequence}, a subsequence of $(h_j ,v_j)$ converges to $(h, v)\\neq 0$ and $(h, v) \\in \\mathcal{Y}\\cap \\Ker L_a$. A contradiction. \n\\end{proof}\n\n\nFor the next lemma, we define the nullity function $N(t)$ for $t\\in (-\\delta,\\delta)$ by\n\\[\n\tN(t) = \\Dim \\Ker L_t.\n\\]\nWe give some definitions in order to describe the kernel spaces. Let $\\rho(x) =(1+ |x|^2)^{-1}$. We define the $\\mathcal{L}^2_\\rho$-inner product for $(h, v), (k, w)\\in \\C^{2,\\alpha}_{-q}(\\mathbb{R}^n\\setminus \\Omega)$ by\n\\[\n\t\\big\\langle (h, v), (k, w)\\big \\rangle_{\\mathcal{L}^2_\\rho} = \\int_{\\mathbb{R}^n\\setminus \\Omega} (h\\cdot k + vw)\\rho \\, d\\mathrm{vol},\n\\]\nwhere the dot product is with respect to $\\bar{g}$. The elements $(h, v)$ and $(k,w)$ are said to be \\emph{orthogonal} if $\\big\\langle (h, v), (k, w)\\big \\rangle_{\\mathcal{L}^2_\\rho} =0$. We say that the set $\\{ (h_1, v_1),\\dots, (h_\\ell, v_\\ell)\\}$  is \\emph{orthonormal}  if $\\|(h_i, v_i)\\|_{\\C^{2,\\alpha}_{-q}(\\mathbb{R}^n\\setminus \\Omega)} =1$ for all $i$, and $\\big\\langle (h_i, v_i), (h_k, v_k)\\big \\rangle_{\\mathcal{L}^2_\\rho} =0$ for all $i\\neq k$. \n\nLet  $ S_j\\subset \\C^{2,\\alpha}_{-q}(\\mathbb{R}^n\\setminus \\Omega)$ be an $\\ell$-dimensional linear space, spanned by an orthonormal basis $\\big\\{ (h_1^{(j)}, v_1^{(j)}),\\dots, (h_\\ell^{(j)}, v_\\ell^{(j)})\\big\\}$. We say that a sequence  \\emph{$\\{S_j\\}$ converges to a subspace of the linear space $S\\subset \\C^{2,\\alpha}_{-q}(\\mathbb{R}^n\\setminus \\Omega)$} if there are $ (h_1, v_1),\\dots, (h_\\ell, v_\\ell)\\in S$ such that $(h_i^{(j)}, v_i^{(j)})\\to (h_i, v_i)$ in $\\C^{2,\\alpha}_{-q}(\\mathbb{R}^n\\setminus \\Omega)$ as $j\\to \\infty$ for each $i= 1,\\dots, \\ell$. As a consequence,  $\\{ (h_1, v_1),\\dots, (h_{\\ell}, v_\\ell)\\}$ is also orthonormal and hence $\\Dim S\\ge \\ell$. If $\\Dim S=\\ell$, we simply say that \\emph{$\\{S_j\\}$ converges to $S$}, and note that in this case, any element $(h,v)\\in S$ is the limit of a sequence $(h^{(j)}, v^{(j)})\\in S_j$ in the $\\C^{2,\\alpha}_{-q}(\\mathbb{R}^n\\setminus \\Omega)$-norm.\n\n\n\n\\begin{lemma}\\label{lemma:nullity}\nThe nullity function $N(t)$ satisfies the following properties:\n\\begin{enumerate}\n\\item For any $a\\in (-\\delta,\\delta)$, $\\limsup_{t\\to a} N(t) <+\\infty$. \\label{item:finite}\n\\item Suppose $\\limsup_{t\\to a} N(t)=\\ell$ for some  $\\ell$. Then there is a sequence $\\{t_j\\}$ in $(-\\delta,\\delta)$ with $t_j\\to a$ such that $N(t_j) =\\ell$ and that $\\Ker L_{t_j}$ converges to an $\\ell$-dimensional subspace of $\\Ker L_a$.  \\label{item:convergence}\n \\item $N(t)$ is upper semi-continuous. \\label{item:upper}\n \\item Define the subset $J\\subset (-\\delta,\\delta)$ by\n \\begin{align}\\label{equation:J}\n \\begin{split}\n \tJ=\\big\\{ t\\in (-\\delta,\\delta): &  \\mbox{ the nullity function is }\\\\ \n\t&\\mbox{constant in an open neighborhood of $t$}\\big\\}.\n\\end{split}\n \\end{align}\n Then $J$ is open and dense. \\label{item:J}\n \\end{enumerate}\n\\end{lemma}\n\\begin{proof}\nWe prove Item~\\eqref{item:finite}: Suppose on the contrary that $\\limsup_{t\\to a}N(t) = +\\infty$ for some $a$. Then there is a sequence $t_j\\to a$ such that $N(t_j)$ is increasing in $j$ and $N(t_j)\\ge j$ for all positive integers~$j$.  For each~$j$, denote an orthonormal basis for $\\Ker L_{t_j}$ by\n\\[\n\t\\left\\{ \\big(h_1(t_j), v_1(t_j)\\big),  \\dots, \\big(h_{N(t_j)}(t_j), v_{N(t_j)}(t_j)\\big)\\right\\}.\n\\]\nFix a positive integer $\\ell$. For $j\\ge \\ell$, consider the $\\ell$-dimensional subspace $S_j$ of $\\Ker L_{t_j}$, spanned by $\\left\\{ \\big(h_1(t_j), v_1(t_j)\\big), \\dots, \\big(h_\\ell(t_j), v_\\ell(t_j)\\big)\\right\\}$. Letting $j\\to \\infty$ and applying Lemma~\\ref{lemma:subsequence}, after passing to a subsequence, $S_j$ converges to an $\\ell$-dimensional subspace of  $\\Ker L_a$. It implies $\\Dim \\Ker L_a\\ge \\ell$. Since $\\ell$ is arbitrary, we get  $\\Ker L_a = +\\infty$, which contradicts to that $L_a$ is Fredholm.\n\nWe prove Items~\\eqref{item:convergence} and \\eqref{item:upper}: Since $N(t)$ takes values on integers, if $\\limsup_{t\\to a} N(t) = \\ell$,  there is a sequence $t_j\\to a$ such that $N(t_j) =\\ell$ for all $j$. A similar argument as above proves Item~\\eqref{item:convergence}  (after passing to a subsequence). It also shows that \n\\[\n\t\\limsup_{t\\to a} N(t) = \\ell \\le N(a),\n\\]\nwhich implies Item~\\eqref{item:upper}. \n\n\nWe prove Item~\\eqref{item:J}: The subset $J$ is open by definition. We show that $J$ is dense. Suppose, on the contrary, that $J$ is not dense. Then there is an interval $[a,b] \\subset (-\\delta,\\delta)\\setminus J$ where $a<b$. Since $N(t)$ takes values on nonnegative integers, it attains its minimum at some $s\\in [a,b]$, say $N(s) = \\min_{[a,b]} N(t) = m$. Define the sublevel set $U = \\{ t\\in [a,b]: N(t) < m+\\frac{1}{2} \\}\\neq \\emptyset$. By upper semi-continuity of $N(t)$, $U$ is a nonempty open subset of $[a,b]$, and thus $\\Int U = U\\cap (a,b)$ is a nonempty open subset of $(-\\delta,\\delta)$.  Since $N(t)$ is constant, equal to the minimum $m$, on $\\Int U$, we see that $\\Int U\\subset J$. It contradicts to the assumption that $[a,b]$ lies in the complement of $J$.\n\\end{proof}\n\nWe have shown that for $t$ in the open dense subset $J$, the spaces $\\Ker L_t$ behaves well. In particular, for any $a\\in J$, $\\Ker L_t$ converges to $\\Ker L_a$ as $t\\to a$. This convergence fulfills the first part of the assumption $(\\star)$. The other half of the assumption $(\\star)$ requires the $t$-derivative of the kernel elements also converge, which we can replace with the convergence of the difference quotients. Thus, the following proposition can be viewed as a ``discrete version'' of the assumption $(\\star)$.\n\n\\begin{proposition}~\\label{proposition:kernel}\nLet $J\\subset (-\\delta,\\delta)$ be the open dense subset defined by \\eqref{equation:J}. Then for every $a\\in J$, $(h,v) \\in \\Ker L_a$,  there is a sequence $\\{t_j\\}$ in $J$ with $t_j \\searrow a$,  $(h(t_j), v(t_j))\\in \\Ker L_{t_j}$, and $(p, z)\\in \\C^{2,\\alpha}_{-q}(\\mathbb{R}^n\\setminus \\Omega)$ such that, as $t_j \\searrow a$, \n\\begin{align*}\n\t(h(t_j), v(t_j)) &\\to (h,v) \\\\\n\t \\frac{(h(t_j), v(t_j) )- (h, v)}{t_j - a} &\\to (p, z),\n\\end{align*}\nwhere both convergences are taken in the $\\C^{2,\\alpha}_{-q}(\\mathbb{R}^n\\setminus \\Omega)$-norm. \n\\end{proposition}\n\n\\begin{proof}\n Since $a\\in J$, there is an integer $\\ell$ such that $ N(t) = \\ell$ for $t$ in an open neighborhood of $a$ in $(-\\delta,\\delta)$. Then there is a sequence $t_j\\searrow a$ such that $N(t_j)=\\ell=N(a)$, and thus by Lemma~\\ref{lemma:nullity}, $\\Ker L_{t_j}$ converges to $\\Ker L_a$. Then for any $(h, v) \\in \\Ker L_a$, there is a sequence $(\\hat{h}(t_j), \\hat{v}(t_j))$ in $\\Ker L_{t_j}$ such that $(\\hat{h}(t_j), \\hat{v}(t_j))\\to (h, v)$ in $\\C^{2,\\alpha}_{-q}(\\mathbb{R}^n\\setminus \\Omega)$. \n\nSince the difference quotient $\\frac{ (\\hat{h}(t_j), \\hat{v}(t_j))-(h,v)} {t_j -a}$ \nneed not converge in general,  we will  modify the sequence $  (\\hat{h}(t_j), \\hat{v}(t_j))$ by adding a sequence of ``correction'' terms from $\\Ker L_{t_j}$.  To proceed, observe that since $\\Ker L_{t_j}$ converges to $\\Ker L_a$, there is a closed subspace $\\mathcal{Y}$ such that, for all $j$ sufficiently large,  $\\mathcal{Y}$ is complementing to $\\Ker L_{t_j}$ and to $\\Ker L_{a}$. Therefore, there is $\\big(k(t_j), w(t_j)\\big)\\in \\Ker L_{t_j}$ such that \n\\begin{align} \\label{equation:E}\n\t\\frac{ (\\hat{h}(t_j), \\hat{v}(t_j))-(h,v)} {t_j -a} - \\big(k(t_j), w(t_j)\\big) \\in \\mathcal{Y}.\n\\end{align}\nWe apply $L_{t_j}$ to the above element: \n\\begin{align*}\n\t&L_{t_j} \\left(\\frac{ (\\hat{h}(t_j), \\hat{v}(t_j))-(h,v)} {t_j -a} - \\big(k(t_j), w(t_j)\\big)  \\right)\\\\\n\t&=-\\frac{1}{t_j-a} L_{t_j}(h,v) = -\\frac{1}{t_j-a} (L_{t_j} - L_a)(h,v).\n\\end{align*}\nNote that $f_j:= -\\frac{1}{t_j-a} (L_{t_j} - L_a)(h,v)$ converges to some $f$ in $\\mathcal{Z}$ as $j\\to \\infty$ because $L_{t}$ is a differentiable family of operators.  Thus, $\\| f_j \\|_{\\mathcal{Z}}$ is bounded above by a constant uniformly in $j$. By Lemma~\\ref{lemma:no-kernel} and \\eqref{equation:E}, there is a positive constant $C$, uniformly in $j$, such that \n\\begin{align*}\n\t\\left\\|\\frac{ (\\hat{h}(t_j), \\hat{v}(t_j))-(h,v)} {t_j -a} - \\big(k(t_j), w(t_j)\\big)   \\right\\|_{\\C^{2,\\alpha}_{-q}(\\mathbb{R}^n\\setminus \\Omega)}&\\le C \\left\\|f_j \\right\\|_{\\mathcal{Z}}.\n\\end{align*}\nSince the right hand side is bounded from above, uniformly in $j$, we see that multiplying the inequality above with $(t_j-a)$ gives\n\\[\n\t\\|(t_j -a) \\big(k(t_j), w(t_j)\\big)\\|_{\\C^{2,\\alpha}_{-q}(\\mathbb{R}^n\\setminus \\Omega)}\\to 0 \\quad \\mbox{ as } j\\to \\infty.\n\\]\t\nAlso, by uniform boundedness of $\\| f_j \\|_{\\mathcal{Z}}$ and  Lemma~\\ref{lemma:subsequence}, there is a subsequence of $j$, which we still label by $j$, such that \n\\[\n\t\\frac{ (\\hat{h}(t_j), \\hat{v}(t_j))-(h,v)} {t_j -a} - \\big(k(t_j), w(t_j)\\big) \\to (p, z)\n\\]\n in $\\C^{2,\\alpha}_{-q}(\\mathbb{R}^n\\setminus \\Omega)$. Finally, let\n\\[\n\t(h(t_j), v(t_j)) =  (\\hat{h}(t_j), \\hat{v}(t_j))-(t_j-a) \\big(k(t_j), w(t_j)\\big).\n\\]\nWe can verify that $(h(t_j), v(t_j))$ satisfies the desired properties. \n\n\\end{proof}\n\n\nWe show how to apply Proposition~\\ref{proposition:kernel}  to remove the assumption $(\\star)$ in Theorem~\\ref{theorem:special} and thus establish Theorem~\\ref{theorem:static-generic}.\n\n\\begin{proof}[Proof of Theorem~\\ref{theorem:static-generic}]\n\n\nLet $J$ be the open dense subset of $(-\\delta, \\delta)$ from Proposition~\\ref{proposition:kernel}, and let  $a\\in J$.  By re-parametrizing~$t$, we may without loss of generality assume  $a=0$ and hence $g_a = \\bar{g}, \\Sigma_a = \\Sigma, L_a= L$.  We will show that if $(\\hat{h}, v)$ solves $S_0(\\hat{h}, v)=0$,  then $DA(\\hat{h})=0$ on $\\Sigma$. \n\nBy the second statement in Lemma~\\ref{lemma:static-harmonic}, after changing to the static-harmonic gauge, we may assume that $(\\hat{h}, v)\\in \\Ker L$. By Proposition~\\ref{proposition:kernel}, there exist  a sequence $(\\hat{h}(t_j), v(t_j))\\in \\Ker L_{t_j}$ and $(\\hat{p}, z)\\in \\C^{2,\\alpha}_{-q}(\\mathbb{R}^n\\setminus \\Omega)$ such that, as $t_j\\searrow 0$, \n\\begin{align*}\n\t(\\hat{h}(t_j), v(t_j)) &\\to (\\hat{h}, v)\\\\\n\t\\left( \\frac{\\hat{h}(t_j)-\\hat{h}}{t_j},  \\frac{v(t_j) - v}{t_j}\\right)&\\to (\\hat{p}, z)\n\\end{align*}\n in the $\\C^{2,\\alpha}_{-q}(\\mathbb{R}^n\\setminus \\Omega)$-norm. \n \n To follow closely the proof of Theorem~\\ref{theorem:special}, we will modify $(\\hat{h}, v)$ to satisfy the geodesic gauge: by Lemma~\\ref{lemma:geodesic}, there is  a  vector field $V\\in \\C^{3,\\alpha}_c(\\mathbb{R}^n\\setminus \\Omega)$ with $V=0$ on $\\Sigma$ such that $h:=\\hat{h}+L_V \\bar{g}$ satisfies the geodesic gauge on $\\Sigma$. We correspondingly define\n\\[\n\th(t_j) = \\hat{h}(t_j) + L_V g_{t_j}.\n\\]\nNote that $h(t_j)$ need not satisfy the geodesic gauge on $\\Sigma$ for $t_j\\neq 0$. By the first statement of Lemma~\\ref{lemma:static-harmonic}, $S_0(h, v)=0$\nand $S_{t_j}(h(t_j),v(t_j))=0$.\nFurthermore, we have, as $t_j\\searrow 0$, \n\\begin{align*}\n\t(h(t_j), v(t_j)) &\\to (h, v)\\\\\n\t\\left( \\frac{h(t_j)-h}{t_j},  \\frac{v(t_j) - v}{t_j}\\right)&\\to (p, z)\n\\end{align*}\n in the $\\C^{2,\\alpha}_{-q}(\\mathbb{R}^n\\setminus \\Omega)$-norm, where $p:= \\hat{p} + L_V L_X \\bar{g}$.\n\n\nFollow exactly the same argument as  in the proof of Theorem~\\ref{theorem:special}, except that wherever we need to ``differentiate in $t$ at $t=0$\", instead we ``take the limit of the difference quotients as $t_j \\to 0$''.  We see that $(p-L_X h, z-L_X v)$ is a static vacuum deformation at $\\bar{g}$ in $\\mathbb{R}^n\\setminus \\Omega$ whose  Bartnik boundary data on $\\Sigma$ satisfy\n\\begin{align*}\n\t(p-L_X h\\big)^\\intercal &= -2\\zeta DA(h)\\\\\n\tDH(p-L_X h)&=\\zeta A\\cdot DA(h).\n\\end{align*}\nThen we apply \\eqref{equation:Green-boundary} exactly as in the proof of Theorem~\\ref{theorem:special}  for $(h, v)$ and $(k, w) = (p-L_X h, z-L_X v)$ to conclude that $DA(h)=0$, and thus $DA(\\hat{h})=0$ on $\\Sigma$.\n\n\n\\end{proof}\n\n\n\\begin{proof}[Proof of Corollary~\\ref{cor:dilation}]\nLet $\\Omega$ be a bounded open subset in $\\mathbb{R}^n$ such that the boundary $\\Sigma$ is a star-shaped hypersurface (with respect to the origin).  Let $\\psi_t: \\mathbb{R}^n\\to \\mathbb{R}^n$ be the dilation map defined by $\\psi_t(x_1,\\dots, x_n) = t(x_1,\\dots, x_n)$, and define $\\Sigma_t = \\psi_t(\\Sigma)$ and $\\Omega_t=\\psi_t(\\Omega)$. \nSince $\\Sigma$ is star-shaped, for $0<\\delta<1$ small and for $ t\\in [1-\\delta, 1+\\delta]$, $\\Sigma_t$ is a smooth foliation in a tubular neighborhood of $\\Sigma$. \n\n\n\nBy Theorem~\\ref{theorem:static-generic}, there exists an open dense subset $J\\subset [1-\\delta, 1+\\delta]$ such that $\\Sigma_t$ is static regular in $(\\mathbb{R}^n\\setminus \\Omega_t, \\bar g)$ for $t\\in J$. We will use scaling from $\\Sigma_t$ to show that  $\\Sigma$ is static regular in $(\\mathbb{R}^n\\setminus \\Omega, \\bar g)$. Namely, for any $(h, v)\\in \\C^{2,\\alpha}_{-q}(\\mathbb{R}^n\\setminus \\Omega)$ satisfies \n\\begin{align*}\n\t&\\left\\{ \\begin{array}{l}  D\\Ric(h) - \\nabla^2 v =0\\\\\n\t\\Delta v=0 \\end{array}\\right. \\quad \\mbox{ in } \\mathbb{R}^n\\setminus \\Omega \\\\\n\t&\\left\\{ \\begin{array}{l} h^\\intercal =0\\\\\n\tDH(h)=0 \\end{array}\\right. \\quad \\mbox{ on } \\Sigma,\n\\end{align*}\nwe will show $DA(h)=0$ on $\\Sigma$, where the linearizations are all  taken at $\\bar g$.\n\nFor a fixed $t\\in J$, we push forward the above system by $\\psi_t$. Define $(h_t, v_t) = (\\psi_t)_* (h, v)$ and  \n\\[\n\tg_t = (\\psi_t)_* \\bar g = t^{-2} \\bar g. \n\\]\nThen the above system becomes the system for $(h_t, v_t)$ at $g_t$:\n\\begin{align*}\n\t&\\left\\{ \\begin{array}{l} - D\\Ric|_{g_t}(h_t) + \\nabla^2_{g_t} v_t =0\\\\\n\t\\Delta_{g_t} v_t=0 \\end{array}\\right. \\quad \\mbox{ in } \\mathbb{R}^n\\setminus \\Omega_t \\\\\n\t&\\left\\{ \\begin{array}{l} h_t^\\intercal =0\\\\\n\tDH|_{g_t}(h_t)=0 \\end{array}\\right. \\quad \\mbox{ on } \\Sigma_t,\n\\end{align*}\nUsing $g_t = t^{-2} \\bar g$, the linearization formulas for $D\\Ric$ and $DH$ in Lemma~\\ref{lemma:formula}, and the conformal transformation formulas for $\\nabla_{g_t}^2, \\Delta_{g_t}$, we have that $(h_t, t^{-2} v_t)$ satisfies the following linearized system at $\\bar g$:\n\\begin{align*}\n\t&\\left\\{ \\begin{array}{l}- D\\Ric|_{\\bar g}(h_t) + \\nabla^2_{\\bar g} (t^{-2} v_t )=0\\\\\n\t \\Delta_{\\bar g} v_t=0 \\end{array}\\right. \\quad \\mbox{ in } \\mathbb{R}^n\\setminus \\Omega_t \\\\\n\t&\\left\\{ \\begin{array}{l} h_t^\\intercal =0\\\\\n\t DH|_{\\bar g}(h_t)=0 \\end{array}\\right. \\quad \\mbox{ on } \\Sigma_t.\n\\end{align*}\nSince $\\Sigma_t$ is static regular in $(\\mathbb{R}^n\\setminus \\Omega_t, \\bar g)$, we have $DA|_{\\bar g} (h_t)$. To see that it implies $DA|_{\\bar g}(h)=0$ on~$\\Sigma$, we push forward it by $\\psi_t$ and get  \n\\[\n\t(\\psi_t)_* (DA|_{\\bar g} (h)) = DA|_{g_t} (h_t) = t DA|_{\\bar g} (h_t)=0 \\quad \\mbox{ on } \\Sigma_t.\n\\]\t\n\n\\end{proof}\n\n\\bibliographystyle{amsplain}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\n\\section{Introduction}\n\\label{sec:intro}\n\nThe study of human motion perception has suggested that humans perceive motion as goal-directed behaviors rather than plain pattern movements~\\cite{baldwin2001infants,gergely2002rational,woodward1998infants}. Developmental psychologists~\\cite{csibra2007obsessed} categorized such an ability into two distinct mechanisms: (1) action-effect associations that the desired effects activate the corresponding action; and (2) simulative procedures, which argues that goal attribution comes from planning under the rational action principle in others' shoes. Both mechanisms require detailed knowledge of \\textbf{action dependencies and effects}, agent's \\textbf{intents and goals} and \\textbf{beliefs about other agents}. With such knowledge playing crucial roles in human cognitive development, learning them from visual observation is pivotal for building more intelligent agents.\n\n\nTaking a closer look at how humans learn from interacting with the world, we locate objects, change their positions and manipulate them in various ways, all presumably under visual control from an egocentric perspective~\\cite{land1999roles}. This unique first-person experience provides essential visual cues for human attention and goal-oriented task understanding. Moreover, egocentric perception naturally reflects how humans reason and perform in a partially observable environment, making it the most available learning source for learning actions, tasks~\\cite{rajeswaran2022r3m}, and belief modeling~\\cite{fan2021learning}.\nThe past few years have witnessed significant progress in egocentric video understanding, especially action recognition and future anticipation~\\cite{pirsiavash2012detecting,sigurdsson2018charades,li2018eye,jia2020lemma,damen2022rescaling,grauman2022ego4d}. However, these two tasks merely cover the tip of the iceberg, considering how humans learn from visual observations to obtain knowledge for more profound tasks such as learning world models, planning for desired goals, and building beliefs about others. With their essential roles in human cognitive development, we urge the need for a benchmark that addresses these missing dimensions in egocentric activity understanding.\n\\begin{figure}[t!]\n    \\centering\n    \\includegraphics[width=0.99\\linewidth]{figure\/overview.pdf}\n    \\caption{An overview of EgoTaskQA. We show an illustrative scenario where two subjects collaborate to make and drink cereal. Based on egocentric observations, we generate questions on these seen or unseen video intervals with different {\\textcolor{type}{question types}}, {\\textcolor{semantic}{targeting semantics}} and {\\textcolor{scope}{question scopes}}. Note that we use both direct (\\eg, when the person open something...) and indirect (\\eg, the action before getting something...) references on actions and objects, where the same color indicates the same referred actions (best viewed in color).}\n    \\label{fig:overview}\n\\end{figure}\n\nHence, we present EgoTaskQA, a challenging egocentric, goal-oriented video question-answering benchmark based on the LEMMA dataset~\\cite{jia2020lemma}. The LEMMA dataset collects egocentric videos in goal-oriented and multi-agent collaborative activities with fine-grained action and task annotation. By extending the LEMMA dataset with annotations consisting of object status, human-object and multi-agent relationships, and causal dependency structures between actions, we design questions that target three specific scopes: (1) actions with world state transitions and their dependencies, (2) agents' intents and goals in task execution, and (3) agents' belief about others in collaboration to provide an in-depth evaluation metric for task understanding. These questions are procedurally generated within four types: \\textbf{descriptive}, \\textbf{predictive}, \\textbf{explanatory}, and \\textbf{counterfactual}, to systematically test models' capabilities over \\textbf{spatial}, \\textbf{temporal}, and \\textbf{causal} domains of goal-oriented task understanding. To avoid spurious correlations in questions, we include both direct and indirect references to actions and objects. We further balance the answer distribution by the reasoning type of questions and carefully design benchmarking train\/test splits to provide a systematic test on goal-oriented reasoning and indirect reference understanding; see~\\cref{fig:overview} for an example and more details in~\\cref{sec:benchmark}.\n\nAs shown in~\\cref{tab:dataset_comparison}, EgoTaskQA complements existing video reasoning benchmarks on various dimensions. With models exhibiting large performance gaps compared with humans, we devise diagnostic experiments to reveal both the easy and challenging spots in our benchmark. We hope such designs and analyses will foster new insights into goal-oriented activity understanding.\n\n\\paragraph{Contributions} In summary, our main contributions are three-fold:\n\\begin{itemize}[leftmargin=*,noitemsep,nolistsep]\n\\item We extend the LEMMA dataset with annotations of object status, human-object and multi-agent relationships to facilitate egocentric activity understanding. We further generate causal dependency structures between actions to provide ground truth for procedural task understanding.\n\\item We construct a balanced video question-answering benchmark, EgoTaskQA, to measure models' capability in understanding action dependencies and effects, intents and goals, as well as beliefs in multi-agent scenarios. We procedurally generate four challenging types of questions (descriptive, predictive, explanatory, and counterfactual) with both direct and indirect references for our benchmark and potential research on video-grounded compositional reasoning.\n\\item We devise challenging benchmarking splits over EgoTaskQA to provide a systematic evaluation of goal-oriented reasoning and indirect reference understanding. We experiment with various state-of-the-art video reasoning models, show their performance gap compared with humans, and Acknowledgementanalyze their strengths and weaknesses to promote future research on goal-oriented task understanding.\n\\end{itemize} \n\n\\section{Related Work}\\label{sec:related_work}\n\n\\paragraph{Action as Inverse Planning} Action understanding has been seen as an inverse planning problem on agents' mental states~\\cite{baker2009action, shum2019theory}. Early studies formulate it as reasoning on the first-order logic formulae that describes actions' preconditions and post-effects~\\cite{mccarthy1963situations,reiter1991frame}. This symbolic formalism is later paired with domain-specific language and algorithms to become mainstays in robotics planning~\\cite{fikes1971strips,mcdermott1998pddl}. In computer vision, similar attempts have been made to link visual observations with world states and actions~\\cite{duan2012discovering, isola2015discovering, nagarajan2018attributes}. Various methods treated actions as transformations on images to solve action-state recognition~\\cite{fathi2013modeling, fire2015learning, wang2016actions, alayrac2017joint, liu2017jointly} and video prediction~\\cite{oh2015action, vondrick2016generating, ha2018recurrent}. With the emerging interest in language-grounded understanding,~\\citet{zellers2021piglet} proposed PIGLeT to study the binding between images, world states, and action descriptions.~\\citet{padmakumar2022teach} further studies the problem of language understanding and task execution by designing an intelligent embodied agent that can chat during task execution. However, these works are mostly limited to atomic actions, missing the important action dependency in task execution. To tackle this  problem, instructional videos~\\cite{kuehne2014language,alayrac2016unsupervised,tang2019coin,miech2019howto100m} are studied with its goal-oriented multi-step activities. In these videos, external knowledge~\\cite{paulius2016functional, koupaee2018wikihow} can be used as guidance for advanced tasks like temporal dynamics learning~\\cite{epstein2021learning} and visually grounded planning~\\cite{chang2020procedure, sun2022plate}. Unfortunately, these videos highlight the instructions and include no task-level noise, which is much simpler than the partially observable, highly paralleled, multi-agent environment that humans learn from and as presented in our benchmark. These complexities make the goal-oriented action understanding a challenging task remaining to be solved.\n\n\\begin{table}[t!]\n    \\centering\n    \\caption{A comparison between EgoTaskQA and existing video question-answering benchmarks. We use ``world'' for world model-related information, including action preconditions, post-effects, and dependencies. We use FPV as short for egocentric and TPV for third-person-view videos. We use MC as short for multiple-choice question-answering, and OP for open-answer question-answering.}\n    \\resizebox{\\linewidth}{!}{%\n        \\begin{tabular}{cccccccccccc}\n        \\toprule\n        \\multirow{2}[2]{*}{Dataset} & \\multicolumn{2}{c}{Video} & \\multicolumn{3}{c}{Question Scope} & \\multicolumn{4}{c}{Question type} & \\multirow{2}[2]{*}{\\shortstack[c]{Answer \\\\ Type}} & \\multirow{2}[2]{*}{\\# questions} \\\\ \n        \\cmidrule(lr){2-3}\\cmidrule(lr){4-6}\\cmidrule(lr){7-10}\n         & View & Real-world & World & Intents \\& Goals & Multi-agent & Descriptive & Predictive & Explanatory & Counterfactual &  & \\\\\n        \\midrule\n        MarioQA~\\cite{mun2017mario} & TPV & \\ding{55} & \\ding{51} & \\ding{55} & \\ding{55} & \\ding{51} & \\ding{55} & \\ding{51} & \\ding{55}  & OP & 188K \\\\\n        Pororo-QA~\\cite{kim2017deepstory} & TPV & \\ding{55} & \\ding{51} & \\ding{55} & \\ding{55} & \\ding{51} & \\ding{55} & \\ding{51} & \\ding{55} & MC & 9K \\\\\n        CLEVRER~\\cite{yi2020clevrer} & TPV & \\ding{55} & \\ding{55} & \\ding{55} & \\ding{55} & \\ding{51} & \\ding{51} & \\ding{51} & \\ding{51} & OP+MC & 282K \\\\\n        Env-QA~\\cite{gao2021envqa} & FPV & \\ding{55} & \\ding{51} & \\ding{55} & \\ding{55} & \\ding{51} & \\ding{55} & \\ding{55} & \\ding{55} & OP & 85K \\\\\n        MovieQA~\\cite{tapaswi2016movieqa} & TPV & \\ding{51} & \\ding{55} & \\ding{55} & \\ding{55} & \\ding{51} & \\ding{55} & \\ding{51} & \\ding{55} & MC & 14K \\\\\n        Social-IQ~\\cite{zadeh2019socialiq}  & TPV & \\ding{51} & \\ding{55} & \\ding{51} & \\ding{51} & \\ding{51} & \\ding{55} & \\ding{51} & \\ding{55} & MC & 7.5K \\\\\n        TVQA~\\cite{lei2018tvqa} & TPV & \\ding{51} & \\ding{55} & \\ding{55} & \\ding{55} & \\ding{51} & \\ding{55} & \\ding{51} & \\ding{55} & MC & 152.5K \\\\\n        TVQA+~\\cite{lei2020tvqa+} & TPV & \\ding{51} & \\ding{55} & \\ding{55} & \\ding{55} & \\ding{51} & \\ding{55} & \\ding{51} & \\ding{55} & MC & 29.4K \\\\\n        MSVD-QA~\\cite{xu2017video} & TPV & \\ding{51}  & \\ding{55} & \\ding{55} & \\ding{55} & \\ding{51} & \\ding{55} & \\ding{55} & \\ding{55} & OP & 50.5K \\\\\n        MSRVTT-QA~\\cite{xu2017video} & TPV & \\ding{51} & \\ding{55} & \\ding{55} & \\ding{55} & \\ding{51} & \\ding{55} & \\ding{55} & \\ding{55} & OP & 243K \\\\\n        Video-QA~\\cite{zeng2017leveraging} & TPV & \\ding{51} & \\ding{55} & \\ding{55} & \\ding{55} & \\ding{51} & \\ding{55} & \\ding{55} & \\ding{55} & OP & 175K \\\\\n        ActivityNet-QA~\\cite{yu2019activitynet-qa} & TPV & \\ding{51} & \\ding{55} & \\ding{55} & \\ding{55} & \\ding{51} & \\ding{55} & \\ding{55} & \\ding{55} & OP & 58K\\\\\n        TGIF-QA~\\cite{jang2017tgif-qa} & TPV & \\ding{51} & \\ding{55} & \\ding{55} & \\ding{55} & \\ding{51} & \\ding{55} & \\ding{55} & \\ding{55} & MC & 165.2K \\\\\n        How2QA~\\cite{li2020hero} & TPV & \\ding{51} & \\ding{55} & \\ding{55} & \\ding{55} & \\ding{51} & \\ding{55} & \\ding{55} & \\ding{55} & MC & 44K \\\\\n        HowToVQA69M~\\cite{yang2021just} & TPV & \\ding{51} & \\ding{55} & \\ding{55} & \\ding{55} & \\ding{51} & \\ding{55} & \\ding{55} & \\ding{55} & OP & 69M\\\\\n        AGQA~\\cite{grunde-mcLaughlin2021agqa} & TPV & \\ding{51} & \\ding{55} & \\ding{55} & \\ding{55} & \\ding{51} & \\ding{55} & \\ding{55} & \\ding{55} & OP & 3.6M\\\\\n        NExT-QA~\\cite{xiao2021next} & TPV & \\ding{51} & \\ding{55} & \\ding{51} & \\ding{55} & \\ding{51} & \\ding{51} & \\ding{51} & \\ding{55} &  OP+MC & 52K\\\\\n        STAR~\\cite{wu2021star} & TPV & \\ding{51} & \\ding{51} & \\ding{55} & \\ding{55} & \\ding{51} & \\ding{51} & \\ding{55} & \\ding{55} &  MC & 60K\\\\\n        EgoVQA~\\cite{fan2019egovqa} & FPV & \\ding{51} & \\ding{55} & \\ding{55} & \\ding{51} & \\ding{51} & \\ding{55} & \\ding{55} & \\ding{55} & OP+MC & 520\\\\\n       \n        \\midrule\n        EgoTaskQA (Ours) & FPV & \\ding{51} & \\ding{51} & \\ding{51} & \\ding{51}  & \\ding{51} & \\ding{51} & \\ding{51} & \\ding{51} & OP & 40K\\\\\n        \\bottomrule\n        \\end{tabular}\n    }\n    \\label{tab:dataset_comparison}\n\\end{table}\n\n\\paragraph{Egocentric Vision} Egocentric vision offers a unique perspective for actively engaging with the world. Aside from traditional video understanding tasks like video summarization~\\cite{lee2012discovering, lu2013story}, activity recognition~\\cite{feichtenhofer2019slowfast,wu2022memvit, girdhar2022omnivore} and future anticipation~\\cite{nagarajan2020ego,furnari2020rolling,qi2018generalized,qi2020generalized,girdhar2021anticipative}, egocentric videos provide fine-grained information for tasks like human-object interaction understanding~\\cite{nagarajan2019grounded,damen2016you,cai2016understanding,bambach2015lending,qi2018learning,garcia2018first,ma2016going} and gaze\/attention prediction~\\cite{wei2018and,li2018eye}. With its natural reflectance of partial observability, egocentric videos are also used for social understanding tasks such as joint attention modeling~\\cite{fathi2012social, soo2015social}, perspective taking~\\cite{yagi2018future, ng2020you2me} and communicative modeling~\\cite{northcutt2020egocom, fan2021learning}. However, with various egocentric datasets curated over the last decade~\\cite{pirsiavash2012detecting,lee2012discovering,sigurdsson2018charades}, data and detailed annotations for human tasks are still largely missing. Large-scale daily lifelog datasets like EPIC-KITCHENS~\\cite{damen2022rescaling} and Ego4D~\\cite{grauman2022ego4d} cover certain aspects of action-dependencies, effects, and social scenarios in their recordings, but are unsuitable for detailed annotation due to their size. The other stream of datasets collects activities by providing coarse task instructions to both single actor~\\cite{rai2021home} and multiple agent collaborations~\\cite{jia2020lemma}. They annotate tasks and compositional actions to reveal agents' execution and collaboration process for multi-step goal-directed tasks. Despite all the preferred characteristics of these goal-oriented activity videos, none of them successfully addressed action-dependencies and effects, nor multi-agent belief modeling.\n\n\\paragraph{Video Question-Answering Benchmarks} Visual question-answering can be designed to evaluate a wide spectrum of model capabilities, spanning from visual concept recognition and spatial relationship reasoning~\\cite{tu2014joint,antol2015vqa, johnson2017clevr,hudson2019gqa}, abstract reasoning~\\cite{barrett2018measuring,zhang2019raven,nie2020bongard,zhang2021abstract,zhang2021acre,zhang2022learning}, to common sense reasoning~\\cite{park2020visualcomet, zellers2019recognition}. In the temporal domain, synthetic environments are used for questions that involve simple action-effect reasoning~\\cite{mun2017mario, kim2017deepstory}. Crowdsourced videos~\\cite{jang2017tgif-qa,yu2019activitynet-qa,lei2018tvqa,yang2021just} are used for collecting questions on basic spatial-temporal reasoning capabilities like event counting~\\cite{jang2017tgif-qa}, grounding~\\cite{lei2020tvqa+}, and episodic memory~\\cite{grauman2022ego4d}. Recent advances in video question-answering aim for more profound reasoning capabilities. \\citet{gao2021envqa} leverages an indoor synthetic environment to generate questions on spatial relationships and simple action-effect reasoning from an egocentric perspective. \\citet{xiao2021next} designs NExT-QA containing questions about knowledge of the past, present, and future on both temporal and causal domains. \\citet{grunde-mcLaughlin2021agqa} programmatically generates questions for compositional spatial-temporal reasoning and generalization. \\citet{wu2021star} focus on short atomic action clips for situated reasoning. \\citet{yi2020clevrer} generates synthetic videos for studying counterfactual predictions on collisions. \\citet{zadeh2019socialiq} collects questions for social intelligence evaluation. Nevertheless, none of these benchmarks addressed the aforementioned critical dimensions of goal-oriented activity understanding from a real-world egocentric perspective.\n\n\\section{The EgoTaskQA Benchmark}\\label{sec:benchmark}\nThe EgoTaskQA benchmark contains 40K balanced question-answer pairs selected from 368K programmatically generated questions generated over 2K egocentric videos. We target the crucial dimensions for understanding goal-oriented human tasks, including action effects and dependencies, intent and goals, and multi-agent belief modeling. We further evaluate models' capabilities to describe, explain, anticipate, and make counterfactual predictions about goal-oriented events.\nA detailed comparison between EgoTaskQA and existing benchmarks is shown in~\\cref{tab:dataset_comparison}. \n\n\\subsection{Data Collection}\\label{sec:data:annotation}\nWe select egocentric videos from the LEMMA dataset~\\cite{jia2020lemma} as base video sources. Compared to similar egocentric datasets, human activities in LEMMA are highly goal-oriented and multi-tasked. These activities contain rich human-object interactions and action dependencies in both single-agent and two-agent collaboration scenarios. We take advantage of these desired characteristics and augment LEMMA with ground truths of object states, relationships, and agents' beliefs about others. More specifically, we augment LEMMA on the following aspects:\n\n\\begin{figure}[t!]\n    \\centering\n    \\includegraphics[width=\\linewidth]{figure\/annotation_final}\n    \\caption{We use two actions A1:``get cup from microwave'' and A2:``put cup to the other person'' from Person 1's video in~\\cref{fig:overview} as an example to visualize annotations in EgoTaskQA. We annotate states and relationships for objects changed by actions as well as human-object and multi-agent relationships. After obtaining the ``before'' and ``after'' annotations, we examine which attributes of objects were changed by the action and what are the preconditions and post-effects. We determine the causal dependency between actions by checking if there exists an object that the post-effect of one action over this object fulfills the preconditions of another. In this case, the state change of ``cup'' determines that A1 and A2 are causally dependent (best viewed in color).}\n    \\label{fig:data_collection_stats}\n\\end{figure}\n\n\\paragraph{World States} We focus on world states consisting of object states, object-object relationships, and human-object relationships. First, we build the vocabulary of relationships and state attributes from activity knowledge defined in previous works~\\cite{paulius2016functional, ji2020action}. We manually filter irrelevant relationships and attributes by removing dataset-specific (\\eg, under the car) and detailed numerical (\\eg, cut in three) relationships. Next, we gather similar relationships to obtain 48 relationships and 14 object attributes. This vocabulary covers spatial relationships (\\eg, on top of), object affordances (\\eg, openable), and time-varying attributes (\\eg, shape). We build on top of action annotations from LEMMA and use~\\ac{amt} to annotate this information before and after the changing action for all time-varying objects. With these annotations, we reconstruct the transition chain for each interacted object and obtain their temporal status. We provide the complete list of relationships and object attributes in the \\supp.\n\n\\paragraph{Multi-agent Relationships} To capture how two agents (actor and helper) collaborate over the same task, we annotate basic information about objects' visibility and the actor's awareness of the helper. For each object that the actor operates on, we annotate its visibility to the helper by providing synchronized videos from both agents' views to~\\ac{amt} workers. For the actor's awareness of others, we instruct \\ac{amt} workers to first go through the egocentric view video of both agents to get familiar with actions performed by the actor and the helper. Next, we ask \\ac{amt} workers to replay the video of the actor and annotate, during each action segment, whether the actor can see the helper or whether the actor is aware of the helper's action if the helper is not in sight. As this annotation is usually subjective, we take the majority vote of three workers as ground truth.\n\n\\paragraph{Causal Trace} Based on the annotated transition chain of objects, we generate causal traces for each action with rules. By checking whether the post-effect of one action fulfills the preconditions of another, we define the causal relationship between two actions into unrelated, related, and causally dependent; see~\\cref{fig:data_collection_stats} for an illustration and refer to \\supp for detailed explanations. Given a video, we run this dependency check for each pair of actions. Next, we generate a video-level dependency tree by recursively checking sequential depending relationships and use it as the ground truth dependency structure for subsequent explanatory and counterfactual question generation.\n\nIn total, we augment LEMMA with 30K annotated before states, after states, and person annotation blocks as shown in~\\cref{fig:data_collection_stats}. We then segment the videos in LEMMA into clips with lengths of around 25 seconds for question generation. This design helps generate interesting clips with partially observed environmental constraints (\\eg, the cup is already washed when the person pours juice), and visual hints for future actions (\\eg, cutting watermelon into dice instead of pieces for making juice rather than eating it directly). Meanwhile, we keep our videos reasonably long, with an average of 5 actions per clip to cover sufficient information for action dependency inference and future prediction. We provide more details about data collection and annotation statistics in \\supp.\n\n\\subsection{Question-Answer Generation}\\label{sec:qa:generation}\nWe use machine-generated questions to evaluate models' task understanding capabilities. We focus on the transition chain of each interacted object, especially what actions caused changes on objects and how these changes contribute together to a multi-step task; see examples in~\\cref{fig:overview}. \n\n\\paragraph{Question Design} We design questions that pinpoint scopes, including (1) action preconditions, post-effects, and their dependencies, (2) agents' intents and goals, and (3) agents' beliefs about others. Similar to~\\cite{yi2020clevrer}, we categorize our questions over these three scopes into four types to systematically test models' capabilities over spatial, temporal, and causal domains of task understanding:\n\\begin{itemize}[leftmargin=*,noitemsep,nolistsep]\n\\item{\\textbf{Descriptive}} questions evaluate the understanding of detailed spatial-temporal information. We provide spatial-temporal references in the questions to identify a unique interval for answering queries on objects and actions. These properties include object states and changes, relationships, human actions, and multi-agent-related information. We generate this type of question by randomly sampling an interval in the video clip and then gathering all related annotations for question generation. Answers in this category are generated based on the interval annotation and contain both open-ended queries and statement verifications.\n\\item{\\textbf{Predictive}} questions aim at understanding intents and task planning. Given a video clip, we ask about possible future object states and actions for both the actor and the helper. These predictions include both direct predictions on actions and objects, as well as more challenging task-dependent predictions such as the executability of actions and the desired states of objects. Questions and answers for predictive questions are generated by gathering the future action\/object annotations in a fixed window size after the truncated interval (\\ie unseen future video) in the long original video. Answers in this category are open-ended action, object, and state queries.\n\\item{\\textbf{Counterfactual}} questions aim at understanding action preconditions and post-effects. Based on the causal trace of actions, we generate counterfactual questions with hypothetical conditions that certain actions in the sequence were not executed. Under this condition, we query both the affected and unaffected actions about their executability and whether the corresponding changes of object states associated with these actions will occur. We generate counterfactual questions by adding or removing actions in the causal trace and adjusting the depending actions' executability recursively. Answers in this category contain action executability verifications and object state queries.\n\\item{\\textbf{Explanatory}} questions evaluate the understanding of task-related object changes as well as action preconditions and post-effects. Given the object state annotations and the causal trace, we query the cause of state changes, the leading factor that satisfies the preconditions of specific actions, as well as why would the post-effect of certain actions affects other actions in the video clip.\nWe generate explanatory questions by querying both the annotations as well as the causal trace. Answers for explanatory questions contain both open-ended and verification queries.\n\\end{itemize}\n\n\\begin{figure}[t!]\n    \\includegraphics[width=\\linewidth]{figure\/stats}\n    \\caption{Generation and statistics of the question-answer pairs. (a) Question distribution according to question scope (top), question type (middle), and targeting semantics (bottom). (b) Examples of natural language questions and their corresponding executable programs. Operators and parameters of the program are represented by black and blue contour rectangles, respectively. (c) Answer distribution for the top 20 reasoning types and top 15 answers for each type before (left) and after (right) balancing, the reasoning types are abbreviated with the concatenation of their initial letters (\\eg, DWAQ for descriptive, world, action, and query).}\n    \\label{fig:qa_generation_stats}\n\\end{figure}\n\n\\paragraph{Answer Generation} In EgoTaskQA, we consider both open-answer queries and binary statement verifications. To generate question-answer pairs for these questions, we design both text templates and the corresponding functional program templates as shown in~\\cref{fig:qa_generation_stats} (b). Each program consists of a sequence of modules for querying the answer from the annotations and the causal trace. We exhaustively execute all possible program instantiations on videos to obtain answers by substituting arguments with instances in the available sample space. As all questions take action grounding as a prerequisite, we add indirect references (\\eg, the first..., the action before...) to actions and objects when making substitutions to reflect this challenge; see details in \\supp. After this initial process, we obtain 368K question-answer pairs over 2K videos as the full question set. \n\n\n\\paragraph{Answer Distribution Balancing} We balance our answer distribution to avoid shortcuts from exploiting imbalances. Following the scheme introduced in~\\cite{grunde-mcLaughlin2021agqa}, we tag each question template with its scope, type, and the targeting semantic category (\\eg, actions, objects, states) and use the composition of all tags as the unique reasoning type for each question. We balance binary verification questions to have an equal proportion of each answer within each reasoning type. For open-answer questions, we use rejection sampling to ensure that the top 20\\% frequent answers for each reasoning type do not appear as answers for more than 33.3\\% of questions in the same type. After balancing by reasoning types, we proportionally sample questions to obtain a 40K diverse and balanced question set with a 1:2 ratio of binary and open-answer questions. We visualize the statistics of questions and answers and the effect of answer balancing in~\\cref{fig:qa_generation_stats}. More details are provided in \\supp.\n\n\\paragraph{Benchmark Splits}\\label{para:split} We provide two benchmarking splits \\textit{normal} and \\textit{indirect} for video question-answering on EgoTaskQA. For the \\textit{normal} split, we randomly sample questions according to their answer distribution and reasoning types to have a 3:1:1 split over training, validation, and test sets. The \\textit{indirect} split is motivated by the fact that during task execution, actions, objects, and their changes are often strongly correlated. It leaves the chance for the model to perform well by simply over-fitting these strong correlations without thorough task understanding; see~\\cref{sec:exp:text_modality} for a more in-depth discussion. We leverage the indirect references in our question to inspect the models' capability to use the learned knowledge for multi-step reasoning and generalize them to indirect references without over-fitting. More specifically, we filter questions without indirect references and simple indirect reference questions without multiple reasoning steps (\\eg, what is the first action this person did? what did the person do before action ``putting something''?) from all question-answer pairs to form the training set, and split all indirect reference questions with multiple reasoning steps as validation and test sets. Under this setting, the \\textit{indirect} split has a portion of 2:1:1 for training, validation, and test sets, respectively. We leave the remaining discussion of the \\textit{indirect} split to~\\cref{sec:exp:indirect}.\n\n\\section{Experiments}\\label{sec:exp}\nIn this section, we evaluate and analyze the performance of video question-answering models on EgoTaskQA. We report how well models perform on different question scopes, types as well as targeting semantics on both \\textit{normal} and \\textit{indirect} splits. We also provide diagnostic experiments on the language modality to show the necessity of the \\textit{indirect} split.\n\n\\paragraph{Baselines} In our experiments, we evaluate six state-of-the-art video question-answering models: VisualBERT~\\cite{li2019visualbert}, PSAC~\\cite{li2019beyond}, HME~\\cite{fan2019heterogeneous}, HGA~\\cite{jiang2020reasoning},\nHCRN~\\cite{le2020hierarchical}, and ClipBERT~\\cite{lei2021less}. VisualBERT is a VL-BERT model designed for vision-language tasks. PSAC uses positional self-attention and co-attention network blocks to fuse visual and language features. HME uses external memory blocks for both visual inputs and questions on top of an LSTM-based encoder-decoder structure. HGA formulates video question-answering by constructing graphs for both videos and questions and aligning them. HCRN adopts a hierarchical framework by stacking relational modules over motion, question, and visual features. ClipBERT leverages sparsely sampled video clips and grid features~\\cite{jiang2020defense} in a transformer architecture and achieves state-of-the-art results on video question-answering. We formulate question-answering in EgoTaskQA as a classification problem over all answer vocabulary and use models' accuracy as the evaluation metric under different settings. We provide details on model implementation, hyperparameter selection, and the training process in \\supp.\n\n\\begin{table}[t!]\n\\caption{Model performance on the EgoTaskQA \\textit{normal} split.}\n\\label{tab:direct_split}\n\\centering\n\\resizebox{\\linewidth}{!}{%\n\\begin{tabular}{cccccccccc}\n\\toprule\n & Category & Most Likely & VisualBERT~\\cite{li2019visualbert} & PSAC~\\cite{li2019beyond} & HME~\\cite{fan2019heterogeneous} & HGA~\\cite{jiang2020reasoning} & HCRN~\\cite{le2020hierarchical} & ClipBERT~\\cite{lei2021less} & Human \\\\\n\\midrule\n\\multirow{3}{*}{\\rotatebox[origin=c]{90}{Scope}} & world & 18.62 & 39.73 & 40.76 & 41.91 & 38.82 & \\textbf{44.27} & 42.15 & 74\\\\\n& intent & 2.54 & 44.51 & 46.19 & 48.92 & 42.12 & \\textbf{49.77} & 40.94 & 82\\\\\n& multi-agent & 10.92 & 26.29 & 30.59 & 27.98 & 23.43 & \\textbf{31.36} & 27.63 & 76\\\\\n\\midrule\n\\multirow{4}{*}{\\rotatebox[origin=c]{90}{Type}} & descriptive & 18.64 & 41.99 & 40.63 & 41.45 & 38.04 & \\textbf{43.48} & 38.45 & 88\\\\\n& predictive & 1.57 & 30.37 & 31.98 & 35.88 & 25.57 & \\textbf{36.56} & 31.50 & 88\\\\\n& counterfactual & 23.62 & 41.99 & 41.89 & 44.13 & 41.94 & \\textbf{48.00} & 46.75 & 80\\\\\n& explanatory & 7.97 & 37.42 & 37.99 & 38.85 & 35.97 & 40.60 & \\textbf{42.39} & 74\\\\\n\\midrule\n\\multirow{4}{*}{\\rotatebox[origin=c]{90}{Semantic}} & action & 10.05 & 15.02 & 14.75 & 14.99 & 15.08 & 14.92 & \\textbf{22.91} & 70\\\\\n& object & 2.07 & 23.26 & 36.53 & 36.05 & 19.09 & \\textbf{45.31} & 21.80 & 82\\\\\n& state & 6.05 & 59.20 & 61.89 & 63.44 & 55.65 & \\textbf{68.28} & 54.36 & 80\\\\\n& change & 41.97 & 68.27 & 65.05 & \\textbf{68.87} & 68.38 & 67.38 & 66.58 & 82\\\\\n\\midrule\n\\multirow{3}{*}{\\rotatebox[origin=c]{90}{Overall}} & open & 0.70 & 24.62 & 26.97 & 27.66 & 22.75 & \\textbf{30.23} & 27.70 & 82\\\\\n& binary & 50.46 & 68.08 & 65.95 & 68.6 & 68.53 & \\textbf{69.42} & 67.52 & 76\\\\\n& all & 15.4 & 37.93 & 38.90 & 40.16 & 36.77 & \\textbf{42.20} & 39.87 & 80\\\\\n\\bottomrule\n\\end{tabular}\n}\n\\end{table}\n\n\\subsection{Comparative Analysis}\nWe provide experimental results of baseline models on the EgoTaskQA \\textit{normal} split in~\\cref{tab:direct_split}. Model performances are evaluated on question scopes, types, targeting semantics, and overall answer categories. To quantify the naturalness and correctness of questions and answers in the EgoTaskQA benchmark, we provide human evaluation following the consistency check introduced in~\\cite{hudson2019gqa, grunde-mcLaughlin2021agqa}. More specifically, we randomly sample 50 questions for each category and instruct \\ac{amt} workers to evaluate the quality of the generated answer. Additionally, we compare all baseline models with a simple frequency-based baseline, namely \"Most Likely\" in~\\cref{tab:direct_split}, where we select the most likely answer for each category to answer all questions in that category.  \n\nAs shown in~\\cref{tab:direct_split}, the low performance of the most likely answer proves that our answer distribution is correctly balanced. For certain categories (\\eg, change), the most likely answer has relatively high accuracy (41.97\\%) as it covers both open-answer and binary questions. Next,\nwe observe relative low human performance in certain categories (\\eg, action and explanatory). This indicates that identifying causal dependency between actions and conducting multi-step reasoning is not a trivial task for humans as also discovered in~\\cite{grunde-mcLaughlin2021agqa}. However, we still observe a large gap between state-of-the-art models and human performance. Among all models, we find HME, HCRN, and ClipBERT to perform the best. This result is reasonable since they leverage different ways to provide better visual representations and interactions between video and language. Among all question scopes, we recognize a relatively low accuracy on multi-agent-related questions among all question scopes. It implies that understanding other agents' actions during task execution is still difficult without explicit modeling. It is significant in egocentric vision as a person's view changes dramatically, and only glances can be taken to acquire others' information. Meanwhile, we notice that these models perform relatively well for questions on states and changing attributes. We conjecture that this is attributed to the task knowledge embedded in textual descriptions of questions since actions, objects, and state changes are strongly correlated, as mentioned in~\\cref{para:split}.\n\n\\subsection{The Effectiveness of Language}\\label{sec:exp:text_modality}\n\n\\paragraph{Object information} We found the object information in the texts to be highly beneficial for question-answering on task-related knowledge during initial experiments. Compared to the original LEMMA action annotation (\\eg, drinking [cereal] with [cup]), we use verbs to refer to actions in EgoTaskQA and obfuscate object information at different levels (\\eg, drink something with cup, drink something with something) as similarly done in~\\cite{grunde-mcLaughlin2021agqa,wu2021star}. While both types of action references localize to the same action interval, it contains different levels of knowledge in the language modality. Intuitively, the combination of action verbs (\\eg, cut) and targeting objects (\\eg, watermelon) provide object state information (\\eg, diced) under certain scenarios. Therefore, we compare models' performance at different levels of object information obfuscation. As shown in~\\cref{fig:obfuscation}, we recognize a significant performance gain for all models by gradually removing object information obfuscation in text, \\ie, substituting ``something'' with the original object. This result supports the hypothesis that with fine-grained action annotations, we can learn task-related knowledge reasonably well by simply exploiting texts. It shares the same conclusion with recent works on leveraging text-based knowledge for helping instructional video understanding~\\cite{lin2022learning}. To further investigate the effectiveness of the language modality, we conduct ablative experiments on the EgoTaskQA \\textit{normal} split.\n\n\\paragraph{Language-Only} Language has been shown to provide knowledge that helps visual question-answering~\\cite{goyal2017making}. To study the role of language in EgoTaskQA, we design a text-only setting for VisualBERT and HCRN, testing BERT~\\cite{devlin2018bert} and HCRN without vision against their vision-language counterparts. As shown in~\\cref{tab:without_vision}, the performance for most question categories dropped significantly. For the task of video question-answering, we should expect that dropping the vision branch will significantly affect the models' performance. As shown in~\\cref{tab:without_vision}, we observe the general performance for the two models decreased as we expected. Among all categories, the models' performance for the objects decreased the most, which is consistent with the fact that the object queries highly depend on the situation provided in the videos (\\eg, which object changed its status in the video?). However, we observe a slight performance gain on object state change questions. This further suggests that the knowledge of world state change, \\ie which object attribute could change under actions, is embedded within question texts. Models could exploit question texts to learn simple associations between attribute types and action verbs (\\eg, cleanliness and wash, emptiness and pour, shape and cut, \\etc).\n\n\n\\begin{table}[t!]\n\\centering\n\\begin{minipage}{0.52\\linewidth}\n    \\centering\n    \\includegraphics[width=\\linewidth]{figure\/acc.pdf}\n    \\captionof{figure}{Ablative study on model performance with different levels of object information obfuscation on the EgoTaskQA \\textit{normal} split.}\n    \\label{fig:obfuscation}\n\\end{minipage}\n\\hfill\n\\begin{minipage}{0.46\\linewidth}\n\\captionof{table}{Language-only question-answering results on the EgoTaskQA \\textit{normal} split.}\n\\label{tab:without_vision}\n\\resizebox{\\linewidth}{!}{\n\\begin{tabular}{ccccc}\n\\toprule\n \\multirow{2}{*}{Category} & \\multicolumn{2}{c}{BERT~\\cite{devlin2018bert}} & \\multicolumn{2}{c}{HCRN (w\/o vision)}\\\\\n \\cmidrule(lr){2-3}\\cmidrule(lr){4-5}\n & Acc. & Change & Acc.& Change\\\\\n \\midrule\nworld &  36.28 & \\dec{-8.7\\%} & 35.22 & \\dec{-20.4\\%}\\\\\nintent & 35.02 & \\dec{-21.3\\%} & 34.93 & \\dec{-29.8\\%}\\\\\nmulti-agent & 20.58 & \\dec{-21.7\\%} & 19.17 & \\dec{-38.9\\%}\\\\\n\\midrule\ndescriptive & 34.55 & \\dec{-17.7\\%} & 33.58 & \\dec{-22.8\\%}\\\\\npredictive & 24.75 & \\dec{-18.5\\%} & 24.3 & \\dec{-33.5\\%}\\\\\ncounterfactual & 41.3 & \\dec{-1.6\\%} & 40.4 & \\dec{-15.8\\%}\\\\\nexplanatory & 31.78 & \\dec{-15.1\\%} & 30.57 & \\dec{-24.7\\%}\\\\\n\\midrule\naction & 15.72 & \\inc{+4.6\\%}  & 15.64 & \\dec{-1.7\\%}\\\\\nobject & 7.43 & \\dec{-68\\%} & 6.33 & \\dec{-86.0\\%}\\\\\nstate & 45.03 & \\dec{-23.9\\%} & 42.51 & \\dec{-37.7\\%}\\\\\nchange & 69.87 & \\inc{+2.3\\%} & 68.77 & \\inc{+2.1\\%}\\\\\n\\midrule\nall & 33.92 & \\dec{-10.6\\%} & 32.51 & \\dec{-23.0\\%}\\\\\n\\bottomrule\n\\end{tabular}\n}\n\\end{minipage}\n\\end{table}\n\n\\begin{table}[t!]\n\\caption{Model performance on the EgoTaskQA \\textit{indirect} split.}\n\\label{tab:indirect_split}\n\\centering\n\\resizebox{\\linewidth}{!}{%\n\\begin{tabular}{cccccccccc}\n\\toprule\n & Category & BERT & HCRN (w\/o vision) & VisualBERT & PSAC & HME & HGA & HCRN & ClipBERT \\\\\n\\midrule\n\\multirow{3}{*}{\\rotatebox[origin=c]{90}{Scope}} & world & 34.96 & 33.61 & 40.00 & \\textbf{44.74} & 35.91 & 31.29 & 44.04 & 26.51\\\\\n& intent & 23.56 & 23.98 & 36.02 & \\textbf{48.38} & 31.73 & 20.42 & 47.02 & 14.66\\\\\n& multi-agent & 19.70 & 19.25 & 26.02 & \\textbf{35.37} & 25.07 & 17.74 & 30.11 & 20.09\\\\\n\\midrule\n\\multirow{4}{*}{\\rotatebox[origin=c]{90}{Type}} & descriptive & 33.09 & 30.73 & 38.9 & \\textbf{43.36} & 34.48 & 29.01 & 42.02 & 24.35\\\\\n& predictive & 15.58 & 13.68 & 31.37 & 29.11 & 27.79 & 15.16 & \\textbf{46.32} & 10.32\\\\\n& counterfactual & 34.59 & 34.75 & 37.63 & 39.94 & 35.07 & 33.01 & \\textbf{43.64} & 26.29\\\\\n& explanatory & 27.38 & 28.11 & 32.75 & \\textbf{42.53} & 29.16 & 24.00 & 39.69 & 22.46\\\\\n\\midrule\n\\multirow{4}{*}{\\rotatebox[origin=c]{90}{Semantic}} & action & 26.91 & 28.18 & 27.49 & \\textbf{30.06} & 25.12 & 26.15 & 29.61 & 25.25\\\\\n& object & 2.808 & 4.13 & 22.63 & 30.97 & 19.08 & 7.02 & \\textbf{32.20} & 10.49\\\\\n& state & 21.96 & 21.24 & 32.02 & \\textbf{43.29} & 31.60 & 17.67 & 41.81 & 15.29\\\\\n& change & 55.28 & 50.71 & 55.59 & \\textbf{57.20} & 47.65 & 47.22 & 56.27 & 35.26\\\\\n\\midrule\n\\multirow{3}{*}{\\rotatebox[origin=c]{90}{Overall}} & open & 11.22 & 11.38 & 21.05 & \\textbf{28.23} & 18.27 & 8.66 & 27.82 & 11.17\\\\\n& binary & 58.24 & 55.52 & 57.61 & \\textbf{60.30} & 52.55 & 53.72 & 59.29 & 40.71\\\\\n& all & 31.78 & 30.76 & 37.01 & \\textbf{42.25} & 33.06 & 28.36 & 41.56 & 24.08 \\\\\n\\midrule\n\\multicolumn{2}{c}{Performance Change} & \\dec{-6.4\\%} & \\dec{-5.4\\%} & \\dec{-2.4\\%} & \\inc{+4.9\\%} & \\dec{-17.7\\%} & \\dec{-22.9\\%} & \\dec{-1.5\\%} & \\dec{-39.6\\%}\\\\\n\\bottomrule\n\\end{tabular}\n}\n\\end{table}\n\n\\subsection{Generalizing to indirect references}\\label{sec:exp:indirect}\nOn the EgoTaskQA \\textit{indirect} split, we evaluate models' capability to leverage learned task knowledge for solving more complicated indirect reference tasks. With the \\textit{normal} split allowing for shortcuts on action-state associations, the \\textit{indirect} split forbids such exploitation by differentiating references during training and testing. As shown in~\\cref{tab:indirect_split}, we observe more significant performance drops in language-only models compared to their vision-language counterparts. More specifically, the performance of BERT and language-only HCRN dropped 20.8\\% and 26.3\\% on the ``change'' category, where we observed potential exploitation on question texts in~\\cref{sec:exp:text_modality}. This serves as a shred of evidence that the \\textit{indirect} split helps reduce the possibility of exploiting simple associations in texts. As for baseline models, we recognize a common performance decrease shared by most models on the \\textit{indirect} split. Among them, we notice a significant performance drop for ClipBERT, which conflicts with the dominating role of large-scale pretrained vision-language models on various reasoning tasks.\nWe suspect that this degeneration might originate from two lines of problems: (1) the model design on sampling fewer videos and aligning visual\/text graphs directly, which conflicts with the intuition that detailed spatial-temporal information and reasoning is indispensable for grounding indirect references; and (2) adopting large-scale pre-trained models directly to a specific domain is non-trivial, especially with challenges in grounding knowledge to visual signals. Overall, our experiments on the EgoTaskQA \\textit{indirect} split further reveals the demand for better spatial-temporal reasoning modules that solve the problem of compositional goal-oriented reasoning with indirect references.\n\n\n\n\\section{Conclusions}\n\\label{sec:conclusion}\nWe introduce the EgoTaskQA benchmark to systematically evaluate models' understanding of goal-oriented activities from an egocentric perspective. We annotate object states, relationships, and agents' beliefs on the LEMMA dataset. We generate diverse questions covering different reasoning capabilities and target the crucial dimensions of task understanding: action dependencies and effects, agents' intents and goals, and belief modeling. We evaluate state-of-the-art video question-answering models and show their gaps compared with the human on two challenging splits, \\textit{normal} and \\textit{indirect}, to promote future study on indirect reference understanding and goal-oriented reasoning.\n\n\\paragraph{Ethics} The EgoTaskQA benchmark is built upon LEMMA and contains different subjects. As noted in LEMMA, the authors obtain consent from subjects by signing an agreement form with potential impacts adequately informed before recording. We mainly annotated objective world status and multi-agent information using multiple-choice selection and asked annotators to annotate the awareness of other subjects' actions with binary options for subjective annotations. For the annotation process, the workers' agreements are obtained by the publicly available annotation service platform~\\ac{amt} we adopted. In summary, we forbid subjective comments with no personally identifiable information revealed, and all participants' agreements are well addressed.\n\n\\paragraph{Limitations} Our work is primarily limited to two aspects. (i) Constrained by the data and annotation complexity, the scope of our activities is limited to indoor goal-oriented tasks. We believe that adding more diverse activities to EgoTaskQA will further increase its value as a general benchmark.\n(ii) Although EgoTaskQA supports many more additional supervisions, we currently limit our discussions to the six state-of-the-art models to have a fair comparison. With the current result showing insights into problems and challenges in EgoTaskQA, we leave the exploration of model design to future work and briefly discuss the potential solutions as follows. Meanwhile, we will continue this data curation for a broader range of human activities. \n\n\\paragraph{Future work} We plan to investigate the following two branches in the future, (i) explicit spatial-temporal grounding for modularized video QA models and (ii) prompting large-scale pre-trained models (both visual and language) for the domain-specific video QA challenges. Firstly, egocentric data can provide finer information and ease the challenge of grounding in modularized neuro-symbolic models. This could complement existing video reasoning methods and test the potential of neuro-symbolic models on complex reasoning tasks from a real-world, multi-agent, and causal perspective. Next, with increasing efforts in adapting large-scale pre-trained models for reasoning, our experiments suggest that adopting such models directly to a specific domain is non-trivial. Compared to their capabilities in commonsense reasoning, how to enable pre-trained models with the ability to fastly adapt to complex reasoning tasks still remains an interesting problem to be solved.\n\n\\paragraph{Broader impact} With most existing intelligent robots depending on the understanding of world states to act and plan, we hope that the augmented LEMMA dataset can bridge the study of world-model learning in simulated environments and real-world complex event understanding. Additionally, we believe the EgoTaskQA benchmark proposes challenges on goal-oriented reasoning and hope such efforts can foster research in broader video understanding directions, including video-language understanding, spatial-temporal grounding, task learning, future anticipation. We also see its potential in imitation learning and knowledge acquisition, which will further drive the study of intelligent assistive robots that can perform tasks coordinately with humans.\n\n\\paragraph{Public access} We host our videos, annotations, metadata, and question-answering pairs on our website. We provide videos in .mp4 format, metadata, annotations in JSON and pandas DataFrames, and question-answer pairs with their corresponding metadata in JSON format. We make our data publicly available under the CC BY-NC-SA license, which allows reusers to distribute, remix, adapt and build upon the material for noncommercial purposes only and only so long as attribution is given to the creator. We bear all responsibility in case of violation of rights.\n\n\\paragraph{Acknowledgement} We thank all colleagues from VCLA and BIGAI for fruitful discussions. We would also like to thank the anonymous reviewers for their constructive feedback. This work reported herein was supported by National Key R\\&D Program of China (2021ZD0150200).\n\n\\bibliographystyle{unsrtnat}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nThe two-photon transition form factors of the pseudoscalar mesons\n$\\pi^0$, $\\eta$ and $\\eta^\\prime$\nhave received a great attention lately -- both from the experimental \nand the theoretical side.\nThe recent BaBar experiments~\\cite{Aubert:2009mc,:2011hk} have\nprovided us with important information in the high-$Q^2$ region\nof the photon virtuality and have triggered new insight \ninto the structure of mesons~\\cite{Dorokhov:2009jd,Kroll:2010bf,Brodsky:2011xx,Brodsky:2011yv,Klopot:2011qq,Balakireva:2011wp,Bakulev:2011rp,Dorokhov:2011dp}.\nHopefully, results from Belle experiment will soon be available,\nand will provide a very important cross-check of the BaBar  \ndata and boost a progress in the form factor phenomenology.\nA new experiment KLOE-2 at Frascati~\\cite{AmelinoCamelia:2010me,Babusci:2011bg} will soon\nbe able to provide us with the information on the pion \ntwo-photon form factor at low $Q^2$ -- in a region where no measurements\nwere available~\\cite{Babusci:2011bg}.\nAlso the transition form factors of $\\mathcal{P} = \\pi^0,\\eta,\\eta^\\prime$\n(and other) mesons will be measured in BES-III~\\cite{Asner:2008nq}\nexperiment at Beijing with high statistics.\n \nThe Monte Carlo generators based on reliable models\nare needed for data analysis and feasibility studies.\nOne of the tools in this field is the Monte Carlo generator \nEKHARA~\\cite{Czyz:2010sp,Czyz:2006dm},\nwhich is already in use by KLOE-2 Collaboration~\\cite{Babusci:2011bg}.\nA realiable simulation has to account for\nboth photon virtualities in the form factor\neven for a ``single-tag'' experiment.\nTherefore, the formulae for the form factors as functions \nof two photon virtualities are needed.\nThis criterion considerably reduces the choice\nfor the form factor,\nbecause the majority of the published formulae \nwithin different theoretical approaches \nhold only for the case with one photon being real and the other\n--- space-like and virtual.\n\nIt is worthwhile to stress that the knowledge of the transition\nform factors is important in itself, but it is also required\nfor the calculation of the hadronic light-by-light (hLbyL) scattering\npart of the anomalous magnetic moment of \nthe muon ($a_\\mu^{\\mathrm{LbyL}}$), see, \ne.g.,~\\cite{Melnikov:2003xd,Prades:2009tw,Nyffeler:2009tw,Jegerlehner:2009ry}.\n\nIn order to take full advantage from  the newly planned $g-2$\nexperiments at Fermilab~\\cite{Carey:2009zzb} and JPARC~\\cite{JParc:amm:2010}, \n it is mandatory to improve the accuracy of the hLbyL contribution. \nThis subject has been recently discussed in detail during the dedicated\nworkshop in Seattle (\\href{http:\/\/www.int.washington.edu\/PROGRAMS\/11-47w\/}{{\\tt http:\/\/www.int.washington.edu\/PROGRAMS\/11-47w\/}}).\nMany important issues related to\nthe $\\gamma^\\ast\\gamma^\\ast\\mathcal{P}$ interaction\nhave recently been discussed in~\\cite{Bernstein:2011bx}.\n\nIt has not been feasible so far to develop a rigorous \nQED\/QCD based theoretical description of the two-photon interaction of mesons,\nwhich would be applicable at an arbitrary energy scale.\nVarious methods have been used, depending on the aim of a research:\nthe Brodsky-Lepage (BL) \nhigh-$Q^2$ limit and interpolation formula~\\cite{Lepage:1980fj};\nthe Operator Product Expansion (OPE) approach \nto Vector-Vector-Pseudoscalar (VVP) and Vector-Vector-Axial (VVA)\nthree-point functions of QCD~\\cite{Knecht:2001xc};\nthe Vector Meson Dominance (VMD) models~\\cite{Silagadze:2006rt,Lichard:2010ap};\nthe holographic QCD approaches~\\cite{Grigoryan:2008up,Stoffers:2011xe,Cappiello:2010uy,Brodsky:2011xx};\nthe QCD sum rules~\\cite{Khodjamirian:1997tk,Agaev:2010aq,Klopot:2011qq,Balakireva:2011wp,Bakulev:2011rp};\nthe modified perturbative approach~\\cite{Kroll:2010bf};\nthe Regge models~\\cite{Arriola:2010aq};\nthe Dyson-Schwinger equation~\\cite{Roberts:2010rn};\nthe Nambu-Jona-Lasinio model~\\cite{Bartos:2001pg,Noguera:2011fv},\nthe constituent quark models~\\cite{Dorokhov:2009jd,Dorokhov:2011zf};\nthe Resonance Chiral Theory approach~\\cite{Mateu:2007tr,Kampf:2011ty};\nand others.\nThe research in this field is mainly dedicated \nto the high-$Q^2$ region of the form factor with one real\nand one virtual photon.\nWhen one needs to cover a wide range of the photon virtuality,\nboth high-energy and low-energy methods have to be merged in \nsome appropriate way.\n\nThe purpose of this paper is twofold:\nto develop a reliable model able to \ndescribe the two-photon form factors \nof $\\pi^0$, $\\eta$ and $\\eta^\\prime$ mesons \nwith a very small number of parameters\nand, then, to implement these form factors \nin the generator EKHARA.\n\nOur approach is described in Section~\\ref{sec:ffs}.\nWe start from the formalism of chiral effective theory \nwith resonances~\\cite{Ecker:1989yg,Ecker:1988te,Prades:1993ys} \nas a phenomenological model.\nThe masses of the particles are taken from PDG~\\cite{Nakamura:2010zzi}, \nand the $\\eta$--$\\eta^\\prime$ mixing is accounted for\naccording to Ref.~\\cite{Feldmann:1998vh,Feldmann:1999uf}.\nWe require that the form factors vanish at high $|t|$.\n  The formulae for the form factors are given in Section~\\ref{sec:results}.\n \n  In Sections~\\ref{sec:1octets} and~\\ref{sec:2octets:fit}\n  we compare the calculated form factors\n  with the experimental results of \n  CELLO~\\cite{Behrend:1990sr}, CLEO~\\cite{Gronberg:1997fj} \n  and BaBar~\\cite{Aubert:2009mc,:2011hk} experiments.\n  Furthermore, in Section~\\ref{sec:mc},\n  we discuss the implementation of the calculated form factors in \n  the Monte Carlo generator EKHARA~\\cite{Czyz:2010sp,Czyz:2006dm}\n  and simulate the single-tag visible \n  cross section, under conditions similar to those of \n  the CLEO and BaBar experiments.\n  The derived formulae allow us to study \n  the high-$Q^2$ behavior of the form factor\n  (Section~\\ref{sec:limit:high})\n  and the slope of the form factor \n  at the origin (Section~\\ref{sec:slope}).\n  Finally, in Section~\\ref{sec:summary} the main conclusions of the paper\n  are drawn. \n\n\\section{Our approach and the results}\n\\label{sec:ffs}\n\\subsection{Formulae for $F_{\\gamma^\\ast\\gamma^\\ast\\mathcal{P}}$}\n\\label{sec:results}\nThe two-photon form factor $F_{\\gamma^\\ast\\gamma^\\ast \\mathcal{P}}(t_1, t_2)$\nfor the meson of type $\\mathcal{P} = \\pi^0,\\eta,\\eta^\\prime$,\nencodes the dependence of the amplitude \n$\\mathcal{M}\\left( \\gamma^\\ast\\gamma^\\ast \\to \\mathcal{P} \\right)$\non the virtuality of the photons\n($q_1^2 = t_1$, $q_2^2 = t_2$):\n\\begin{eqnarray}\n&&\\mathcal{M}[\\gamma^\\ast(q_1, \\nu)\\, \\gamma^\\ast(q_2, \\beta) \\to \\mathcal{P}]\n \\\\ \\nonumber\n&& \\quad \\quad = \ne^2 \\epsilon_{\\mu\\nu\\alpha\\beta} q_{1}^{\\mu} q_{2}^{\\alpha} \nF_{\\gamma^\\ast\\gamma^\\ast \\mathcal{P}}(t_1, t_2)\n,\n\\end{eqnarray}\nwhere $\\epsilon_{\\mu \\nu \\alpha \\beta}$ is the totally antisymmetric\nLevi-Civita tensor.\nNote that $F_{\\gamma^\\ast\\gamma^\\ast \\mathcal{P}}(t_1, t_2) = F_{\\gamma^\\ast\\gamma^\\ast \\mathcal{P}}(t_2, t_1)$ due to Bose symmetry of the photons.\nWe obtain the formulae for the form factors \n$F_{\\gamma^\\ast\\gamma^\\ast\\mathcal{P}}(t_1,t_2)$ \non the basis of the effective chiral \nLagrangian~\\cite{Ecker:1989yg,Ecker:1988te,Prades:1993ys}\nextended to multi-octet resonance contributions,\nwith the $\\eta-\\eta'$ mixing accounted for\nas in~\\cite{Feldmann:1998vh,Feldmann:1999uf}.\nA brief summary of the model is given in Appendix~\\ref{sec:formalism}.\n We would like to remark that\n a similar approach was applied in the context of other\n processes in~\\cite{Ivashyn:2007yy,Ivashyn:2009te,Eidelman:2010ta}.  \n\n\\begin{figure}\n\\begin{center}\n\\resizebox{0.5\\textwidth}{!}{%\n     \\includegraphics{pictures\/ff-ggp.eps}\n     }\n\\end{center}\n\\caption{Diagrams for the $\\gamma^*\\gamma^*\\mathcal{P}$ transition.\nDashed lines correspond to the pseudoscalar meson $\\mathcal{P}$,\nsolid lines --- to the vector mesons \nand wavy lines to (virtual) photons.\n$V_a \\ne V_b$ for $\\pi^0$\nand $V_a = V_b$ for $\\eta$ and $\\eta^\\prime$\nform factors.}\n\\label{fig:gamgampi}\n\\end{figure}\n\nFor simplicity we neglect the mixing between the\noctets, which can be added if required by the data.\nThe diagrams describing $\\gamma^\\ast\\gamma^\\ast P$ transition \nare presented in Fig~\\ref{fig:gamgampi}. \nThe form factors read\n\\begin{widetext}\n\\begin{eqnarray}\\label{eq:ffpi_3octet}\n&& F_{\\gamma^\\ast\\gamma^\\ast\\pi^0}(t_1,t_2)=-\\frac{N_c}{12\\pi^2 f_\\pi}\n+\\sum_{i=1}^{n}\\frac{4\\sqrt{2}h_{V_i} f_{V_i}}{3f_\\pi} {t_1} \\biggl({D_{\\rho_i} (t_1)}\n+{D_{\\omega_i} (t_1) }\\biggr) \\\\\n&&+\\sum_{i=1}^{n}\\frac{4\\sqrt{2}h_{V_i} f_{V_i}}{3f_\\pi} {t_2} \\biggl({D_{\\rho_i} (t_2) }\n+{D_{\\omega_i} (t_2)}\\biggr)-\\sum_{i=1}^{n}\\frac{4\\sigma_{V_i} f_{V_i}^2}{3f_\\pi} {t_2}  {t_1}\n\\biggl({D_{\\rho_i} (t_2) }{D_{\\omega_i} (t_1) } +{D_{\\rho_i} (t_1) } {D_{\\omega_i} (t_2)\n} \\biggr) ,\\nonumber\n\\end{eqnarray}\n\\begin{eqnarray}\\label{eq:ffeta_3octet}\n&& F_{\\gamma^\\ast\\gamma^\\ast\\eta}(t_1,t_2)= \n-\\frac{N_c}{12\\pi^2f_\\pi}\n\\left( \\frac{5}{3} C_q  -  \\frac{\\sqrt{2}}{3} C_s\\right)\n\\\\\n&&+\\sum_{i=1}^{n}\\frac{4\\sqrt{2}h_{V_i} f_{V_i}}{3f_\\pi} t_1\n \\biggl(3 C_q{D_{\\rho_i} (t_1)}\n+\\frac{1}{3}C_q{D_{\\omega_i} (t_1)}\n-\\frac{2\\sqrt{2}}{3}C_s{D_{\\phi_i} (t_1)}\\biggl) \n\\nonumber\n\\\\\n&&+\\sum_{i=1}^{n}\\frac{4\\sqrt{2}h_{V_i} f_{V_i}}{3f_\\pi} t_2 \\biggl(3 C_q{D_{\\rho_i}\n(t_2)}\n+\\frac{1}{3}C_q {D_{\\omega_i}(t_2)} \n- \\frac{2\\sqrt{2}}{3} C_s {D_{\\phi_i} (t_2)}\n\\biggl)\n \\nonumber \\\\\n&& \n-\\sum_{i=1}^{n}\\frac{8\\sigma_{V_i} f_{V_i}^2}{f_\\pi} t_2 t_1 \\biggl(\n\\frac{1}{2} C_q{D_{\\rho_i} (t_2)} {D_{\\rho_i}(t_1)}\n + \\frac{1}{18} C_q {D_{\\omega_i}(t_2)} {D_{\\omega_i} (t_1)} \n - \\frac{\\sqrt{2}}{9} C_s {D_{\\phi_i} (t_2)} {D_{\\phi_i} (t_1)}\n\\biggl)  \\, .\n \\nonumber \\\\\n\\nonumber \\\\\n&&F_{\\gamma^\\ast\\gamma^\\ast\\eta^\\prime}(t_1,t_2) = \nF_{\\gamma^\\ast\\gamma^\\ast\\eta}(t_1,t_2)\n{}_{\\left|\n\\begin{array}{cr}\n C_q \\to &C_q^\\prime\\\\\n C_s \\to &-C_s^\\prime\\\\\n\\end{array}\n\\right.} , \n\\label{eq:ffetapr_3octet}\n\\end{eqnarray}\n\\end{widetext}\nwhere $n$ is a number of the vector meson resonance octets.\nThe definitions of all couplings can be found in the Appendix~\\ref{sec:formalism}.\nThe vector meson propagators $D_V$ are\n\\begin{eqnarray}\n\\label{vector-propagator-simple} D_V(Q^2) &= &[Q^2 - M_V^2 + i\n\\sqrt{Q^2} \\Gamma_{tot, V} (Q^2)]^{-1} .\n\\end{eqnarray}\nIn this paper we consider only the data in the space-like \nregion of photon virtuality, thus the modeling of \nthe vector resonance energy dependent widths\n$\\Gamma_{tot, V} (Q^2)$ is not relevant as the widths are equal to zero. \nWe take the values of the masses of all particles\naccording to PDG~\\cite{Nakamura:2010zzi}.\n\nWe require that the form factors $F_{\\gamma^\\ast\\gamma^\\ast\\mathcal{P}}(t_1,t_2)$ \ngiven in~(\\ref{eq:ffpi_3octet}), (\\ref{eq:ffeta_3octet}) \nand (\\ref{eq:ffetapr_3octet}) \nvanish when the photon virtuality $t_1$ goes to infinity for \nany value of $t_2$:\n\\begin{eqnarray}\n\\Bigl.\n\\lim_{t_1\\rightarrow -\\infty}\nF_{\\gamma^\\ast\\gamma^\\ast\\mathcal{P}}(t_1,t_2) \\Bigr|_{t_2 = const} &=& 0 .\n\\label{eq:ff_at_inf:1}\n\\end{eqnarray}\nNotice, that in this case the conditions\n\\begin{eqnarray}\n\\lim_{t\\rightarrow -\\infty}\nF_{\\gamma^\\ast\\gamma^\\ast\\mathcal{P}}(t,t)&=& 0 ,\n\\label{eq:ff_at_inf:1:pure} \\\\\n\\lim_{t\\rightarrow -\\infty}\n    F_{\\gamma^\\ast\\gamma^\\ast\\mathcal{P}}(t,0)&=& 0 \n\\label{eq:ff_at_inf:2:pure} \n\\end{eqnarray}\nare automatically satisfied,\nwhich is considered as a correct short-distance behavior of \nthe form factors (see, for example, discussion in~\\cite{Knecht:2001xc}). \nThe constraint~(\\ref{eq:ff_at_inf:1}) \nleads to the following relations for the couplings:\n\\begin{eqnarray}\n&&\n\\sqrt{2} h_{V_i} f_{V_i} -  \\sigma_{V_i}\nf_{V_i}^2 = 0 ,\\quad i=1,\\ldots,n \\, ,\n\\label{eq:qcdsum:1b} \\\\\n&& -\\frac{N_c}{ 4 \\pi^2} + 8\\sqrt{2} \\sum_{i=1}^{n} h_{V_i} f_{V_i} = 0\n\\label{eq:qcdsum:2b} \\, .\n\\end{eqnarray}\nTherefore, \nfor an ansatz with $n$ vector resonance octets\nthe two-photon form factors $F_{\\gamma^\\ast\\gamma^\\ast \\mathcal{P}}(t_1, t_2)$ \nare determined by $2n$ parameters (i.e., the products of the couplings:\n$f_{V_i}h_{V_i}$ and $\\sigma_{V_i}f_{V_i}^2$, $i=1,\\ldots,n$),\nfrom which $n-1$ are to be determined by experiment \nand the rest $n+1$ are fixed by~(\\ref{eq:qcdsum:1b}) and~(\\ref{eq:qcdsum:2b}).\nFor the one octet ansatz there are no free parameters\nand in case of the two octets ansatz there is one free parameter.\n\n\n\nOne of the main objectives of this paper was to develop a reliable model for\n the $\\gamma^*\\gamma^*\\mathcal{P}$ ($\\mathcal{P}=\\pi^0,\\eta,\\eta'$) transition form\n factors in the space-like region reflecting the experimental data and\n theoretical constrains and\n in the same time being as simple as possible.\nEven if we know that the $SU(3)$ flavor symmetry is broken we start our \n investigations using an $SU(3)$-symmetric model (apart from the\n masses of the mesons, which are fixed at their\n PDG~\\cite{Nakamura:2010zzi} values) and try to see how\n many resonance octets we have to include in order to describe the data well.\nThe existing data for the transition form factors in space-like \nregion~\\cite{Behrend:1990sr,Gronberg:1997fj,Aubert:2009mc,:2011hk}\n come from single-tag experiments, where one of the invariants \n is very close to zero (the one associated with the ``untagged'' lepton), \n thus we have information only about\n $F_{\\gamma^\\ast\\gamma^\\ast\\mathcal{P}}(t,0)$.\n It is common to define \n the $\\gamma^*\\gamma\\mathcal{P}$ form factor \n $F_{\\mathcal{P}}(Q^2,0)\\equiv F_{\\gamma^\\ast\\gamma^\\ast\\mathcal{P}}(t,0)$\n with $Q^2\\equiv -t$ (associated with the ``tagged'' lepton). \nFrom Eqs.~(\\ref{eq:ffpi_3octet}),~(\\ref{eq:ffeta_3octet}) and~(\\ref{eq:ffetapr_3octet})\n  we see that $F_{\\mathcal{P}}(Q^2,0)$ is driven by \n  $n$ parameters (i.e., the products of the couplings:\n  $f_{V_i}h_{V_i}$, $i=1,\\ldots,n$) \nand there is always only one constraint~(\\ref{eq:qcdsum:2b}) for any $n$.\nTherefore, the number of parameters in $F_{\\mathcal{P}}(Q^2,0)$\nto be determined by experiment (``free parameters'') equals to $n-1$ \n(similarly to the case of the $F_{\\gamma^\\ast\\gamma^\\ast\\mathcal{P}}(t_1,t_2)$).\nIn case of the one octet ansatz there are no free parameters\nand in the two octets case there is one free parameter.\n\n\\subsection{The one octet ansatz for the form factors}\n\\label{sec:1octets}\n\nLet us consider first the one octet ansatz. \nIn this case \n\\begin{eqnarray}\nf_{V_1} h_{V_1}= \\frac{3}{32\\pi^2 \\sqrt{2}} ,\n\\end{eqnarray}\nand the model gives a prediction for the form factors \n$F_{\\mathcal{P}}(Q^2,0)$\nwithout any possibility for adjustment.\n\n The predictions of this model are compared with experimental\n data \n in Figs.~\\ref{fig:ff-pi0}-\\ref{fig:ff-etaP} (dotted line). \n To quantify the quality of the agreement of the model predictions \n we have calculated the $\\chi^2$ values for each data set.\n  For the pion transition form factor the model agrees with\n  CELLO~\\cite{Behrend:1990sr} and CLEO~\\cite{Gronberg:1997fj} \n  and disagrees with the BaBar data~\\cite{Aubert:2009mc},\n  as can be seen from Table~\\ref{table:chi2}, which shows \n  the $\\chi^2$ values per experiment.\n  For the $\\eta$ and $\\eta^\\prime$ transition form factor the model \n  is in a perfect agreement with CELLO,\n  however for CLEO and BaBar the $\\chi^2$ is not good.\n  In total, for the one octet ansatz\n  we obtain $\\chi^2 \\approx 358$ for 116 experimental points.\n\n\n\\begin{figure}\n\\begin{center}\n\\resizebox{0.45\\textwidth}{!}{%\n     \\includegraphics{pictures\/pi-log.eps}\n     }\n\\end{center}\n\\caption{Transition form factor $\\gamma^\\ast \\gamma\\pi^0$\n compared to the data. \n The Brodsky-Lepage~\\cite{Lepage:1980fj} high-$Q^2$ limit (BL) \n is shown as a bold solid straight line at $2\\times f_\\pi=2\\times0.0924$~GeV.\n The high-$Q^2$ limit in our 1 octet ansatz and and 2 octets ansatz\n are marked as (1) and (2), correspondingly.\n }\n \\label{fig:ff-pi0}\n\\end{figure}\n\n\\begin{figure}\n\\begin{center}\n\\resizebox{0.45\\textwidth}{!}{%\n     \\includegraphics{pictures\/eta-log.eps}\n     }\n\\end{center}\n\\caption{Transition form factor $\\gamma^\\ast \\gamma\\eta$\n compared to the data.  \n The high-$Q^2$ limit is shown as a bold solid straight \n line at $2\\times f_\\eta=2\\times0.0975$~GeV,\n according to~\\cite{Gronberg:1997fj} and~\\cite{Dorokhov:2009jd} (BL).\n The limit according to the two-angle $\\eta-\\eta^\\prime$ \n mixing scheme~\\cite{Feldmann:1998vh,Feldmann:1999uf,Kroll:2010bf} (FK)\n is shown as a shaded box (green online) at $0.1705\\ldots0.1931$~GeV,\n accounting for the parameter ambiguities~(\\ref{eq:mix_angles}).\n The high-$Q^2$ limit in our 1 octet ansatz and and 2 octets ansatz\n are marked as (1) and (2), correspondingly.\n }\n \\label{fig:ff-eta}\n\\end{figure}\n\n\\begin{figure}\n\\begin{center}\n\\resizebox{0.45\\textwidth}{!}{%\n     \\includegraphics{pictures\/etaP-log.eps}\n     }\n\\end{center}\n\\caption{Transition form factor $\\gamma^\\ast \\gamma\\eta^\\prime$\n compared to the data.  \n The high-$Q^2$ limit is shown as a bold solid straight \n line at $2\\times f_{\\eta^\\prime}=2\\times0.0744$~GeV,\n according to~\\cite{Gronberg:1997fj} and~\\cite{Dorokhov:2009jd} (BL).\n The limit according to the two-angle $\\eta-\\eta^\\prime$ \n mixing scheme~\\cite{Feldmann:1998vh,Feldmann:1999uf,Kroll:2010bf} (FK)\n is shown as a shaded box (green online) at $0.29\\ldots0.31$~GeV,\n accounting for the parameter ambiguities~(\\ref{eq:mix_angles}).\n The high-$Q^2$ limit in our 1 octet ansatz and and 2 octets ansatz\n are marked as (1) and (2), correspondingly.\n }\n \\label{fig:ff-etaP}\n\\end{figure}\n\n  Even though the overall agreement of this simple model with the data is not\n  bad, there is a way to improve it, as will be discussed below.\n\n \n\\subsection{The two octet ansatz for the form factors}\n\\label{sec:2octets:fit}\n\n  In order to make the model more flexible,\n  we include the second vector meson multiplet contributions.\n  %\n  We would like to note that there are many known cases\n  when in order to improve the model predictions \n  one needs to account for the excited vector resonances,\n  the charged form factor of the pion is among the most famous examples.\n  \n  In the two octet ansatz we chose $h_{V_1}$ as a free parameter\n  and determine the value of $f_{V_1}=0.20173(86)$ using the PDG~\\cite{Nakamura:2010zzi} \n  value for the width\n\\begin{equation}\n\\label{eq:fv:pdg}\n\\Gamma(\\rho\\to ee) = \\frac{e^4 M_\\rho f_{V_1}^2}{12\\pi}\\, .\n\\end{equation}\n  \n  The fit to the data gives $\\chi^2 \\approx 140$ for 116 experimental points.\n  %\n  The obtained value of $h_{V_1}$ is\n  %\n  \\begin{eqnarray} \n  \\label{eq:hv:fit}\n  h_{V_1} = 0.03121(14) \\, ,\n  \\end{eqnarray}\n  %\n  where the error is the parabolic error given by {\\tt MINOS} package from\n  {\\tt MINUIT CERNLIB} program. \n  %\n  The remaining coupling is given by\n  %\n  \\begin{eqnarray}\n  h_{V_2}f_{V_2}=  \\frac{3}{32\\pi^2\\sqrt{2}}-h_{V_1}f_{V_1}\n  = 0.42(5) \\times10^{-3} \\, .\n  \\end{eqnarray}\n   The comparison of the two octet ansatz with the data is also shown \n   in Figs.~\\ref{fig:ff-pi0}-\\ref{fig:ff-etaP} (solid line)\n   and the $\\chi^2$ values per experiment are given in Table~\\ref{table:chi2}.\n The only data sample which is not in consistency with the model\n is the BaBar data for $\\pi^0$~\\cite{Aubert:2009mc}\n (however, for the $\\eta$ and $\\eta^\\prime$ transition form factors\n there is a perfect agreement with BaBar data~\\cite{:2011hk}).\n From the plots in Figs.~\\ref{fig:ff-pi0}-\\ref{fig:ff-etaP}\n and given the numbers in Table~\\ref{table:chi2} we conclude\n that the two octet calculation is consistent with the bulk \n of available data.\n \n\n\\begin{table}\n\\caption{The $\\chi^2$ per experiment and the total $\\chi^2$.\nNumber of data points (n.d.p.) is also given for each experiment.\nIn all given experiments the pseudoscalar meson is produced\nin a two-photon process $e^+e^- \\to e^+e^-\\mathcal{P}$, but \nthe decay channels for $\\mathcal{P}$ identification vary.\nThe ``2 octets'' column is calculated with \nthe parameter values given by the global fit. }\n\\label{table:chi2}\n\\begin{ruledtabular}\n\\begin{tabular}{ldd}\nexperiment & \\multicolumn{1}{c}{1 octet} & \\multicolumn{1}{c}{2 octets} \\\\\n           & \\multicolumn{1}{c}{$\\chi^2\/n.d.p.$}  & \\multicolumn{1}{c}{$\\chi^2\/n.d.p.$} \\\\\n\\hline\\noalign{\\smallskip}\nCELLO ($\\pi^0\\to\\gamma\\gamma$) \n& 0.29\/5 & 0.47\/5\\\\\nCLEO  ($\\pi^0\\to\\gamma\\gamma$) \n& 6.27\/15 & 20.96\/15\\\\\nBaBar  ($\\pi^0\\to\\gamma\\gamma$) \n& 124.83\/17 & 55.85\/17\\\\\n\\noalign{\\smallskip}\\hline\nCELLO ($\\eta\\to\\gamma\\gamma$) \n& 0.24\/4 & 0.13\/4\\\\\nCLEO  ($\\eta\\to\\pi^+\\pi^-\\pi^0$) \n& 19.28\/6 & 11.13\/6\\\\\nCLEO ($\\eta\\to\\gamma\\gamma$) \n& 8.55\/8 & 2.10\/8\\\\\nCLEO  ($\\eta\\to\\pi^0\\pi^0\\pi^0$) \n& 10.91\/5 & 5.63\/5\\\\\nBaBar  ($\\eta\\to\\gamma\\gamma$) \n& 89.02\/11 & 9.34\/11\\\\\n\\noalign{\\smallskip}\\hline\nCELLO ($\\eta^\\prime\\to\\gamma\\gamma$) \n& 0.11\/5 & 0.29\/5\\\\\nCLEO  ($\\eta^\\prime\\to\\gamma\\gamma\\pi^+\\pi^-$) \n& 19.90\/6 & 7.48\/6\\\\\nCLEO  ($\\eta^\\prime\\to\\gamma\\gamma\\pi^+\\pi^-\\pi^+\\pi^-$) \n& 2.61\/5 & 1.44\/5\\\\\nCLEO  ($\\eta^\\prime\\to\\gamma\\pi^+\\pi^-$) \n& 14.01\/6 & 4.64\/6\\\\\nCLEO  ($\\eta^\\prime\\to6\\gamma$) \n& 21.54\/5 & 12.62\/5\\\\\nCLEO  ($\\eta^\\prime\\to10\\gamma$) \n& 0.49\/2 & 0.23\/2\\\\\nCLEO  ($\\eta^\\prime\\to\\pi^+\\pi^-6\\gamma$) \n& 5.93\/5 & 4.80\/5\\\\\nBaBar  ($\\eta^\\prime\\to\\gamma\\gamma$) \n& 33.87\/11 & 3.10\/11\\\\\n\\noalign{\\smallskip}\\hline\ntotal & 357.87\/116 & 140.22\/116\\\\\n\\end{tabular}\n\\end{ruledtabular}\n\\end{table}\n\nIn principle the parameter $h_{V_1}$ can be estimated \n  by experiment via the value of the width\n\\begin{equation}\n\\label{eq:hv:pdg}\n\\Gamma(\\rho^0\\to \\pi^0\\gamma) = \n\\frac{4\\alpha M_\\rho^3 h_{V_1}^2}{27 f_\\pi^2}\\left(1-\\frac{m_\\pi^2}{M_\\rho^2}\\right)^3 .\n\\end{equation}\nUsing the PDG~\\cite{Nakamura:2010zzi} values for the width~(\\ref{eq:hv:pdg}),\none obtains \n\\begin{equation}\n\\label{eq:hv:pdg:value}\nh_{V_1} = 0.041(3) \\ ,\n\\end{equation}\nwhich is in tension with the value given by fit~(\\ref{eq:hv:fit}).\nThis might be a result of neglecting the higher octets \nor omission of the $SU(3)$ flavor breaking effects.\nIn order to check this we have added the third octet to \nthe model and fitted two free parameters: \n$h_{V_1}$ and $h_{V_2}f_{V_2}$. \nThe fit to the experimental data gives \n$\\chi^2 \\approx 136$ for~116 experimental points,\nso there is no essential improvement \nin the description of data by the model.\n   The fit gives $h_{V_1} = 0.03279(75)$ and $h_{V_2}f_{V_2} = -0.73(54)\\times10^{-3}$.\n   Finally, for the couplings of the third octet one gets \n   $h_{V_3}f_{V_3} = \\frac{3}{32\\pi^2\\sqrt{2}}-h_{V_1}f_{V_1}-h_{V_2}f_{V_2}\n    \\approx 7.45 \\times 10^{-3}$.\n   %\n   In this fit we observe a very high correlation\n   between the parameters $h_{V_1}$ and $h_{V_2}f_{V_2}$ \n   with the off-diagonal correlation coefficient equal to $-0.99$.\n   Notice that $h_{V_1}$ is almost unchanged as compared to~(\\ref{eq:hv:fit})\n   and we conclude that in order to accommodate the\n   value~(\\ref{eq:hv:pdg:value}) in this model we would need to allow for couplings\n   which break the $SU(3)$ flavor symmetry. \n   %\n   This is however beyond the scope of the present paper. \n   %\n   We leave the possible refinements of the model for further investigations.\n\nIn context of the discrepancy between~(\\ref{eq:hv:fit}) \nand~(\\ref{eq:hv:pdg:value}) we would like to illustrate\nthe actual experimental uncertainty in the $\\rho$ meson decay width.\nIn Table~\\ref{tab:rho_pi_gamma} we show the values of $h_{V_1}$\nwhich are deduced from different experimental values of the width:\nthe PDG constrained fit~\\cite{Nakamura:2010zzi} \nfor~$\\Gamma(\\rho^0\\to \\pi^0\\gamma)$;\nthe SND measurement~\\cite{Achasov:2003ed} \nfor~$\\Gamma(\\rho^0\\to \\pi^0\\gamma)$;\nthe PDG constrained fit and average~\\cite{Nakamura:2010zzi} \nfor~$\\Gamma(\\rho^+\\to \\pi^-\\gamma)$.\n\n\\begin{table}\n\\caption{ The $\\rho\\to \\pi\\gamma$ decay width uncertainty\n          and the corresponding values of $h_{V_1}$.\n        }\n\\label{tab:rho_pi_gamma}\n\\begin{ruledtabular}\n\\begin{tabular}{llll}\n\\multicolumn{1}{c}{decay} & \\multicolumn{1}{c}{width} &\n\\multicolumn{1}{c}{reference} & \\multicolumn{1}{c}{$h_{V_1}$} \n\\\\\n\\hline\\noalign{\\smallskip}\n$\\rho^0\\to \\pi^0\\gamma$ & $89(12)$~keV & PDG~\\cite{Nakamura:2010zzi} &  $0.041(3)$ \\\\\n$\\rho^0\\to \\pi^0\\gamma$ & $77(20)$~keV & SND~\\cite{Achasov:2003ed} &  $0.038(5)$ \\\\\n$\\rho^+\\to \\pi^+\\gamma$ & $68(7)$~keV  & PDG~\\cite{Nakamura:2010zzi} &  $0.036(2)$ \\\\\n\\end{tabular}\n\\end{ruledtabular}\n\\end{table}\n\n\\subsection{The Monte Carlo simulation}\n\\label{sec:mc}\n\nWe have implemented  the transition form factors obtained within\nthe two octet model described above \ninto the Monte Carlo generator EKHARA\n(\\href{http:\/\/prac.us.edu.pl\/\\%7Eekhara}{{\\tt http:\/\/prac.us.edu.pl\/\\~\\;\\!ekhara}}).\nFrom the technical point of view of the event generation, \nit is a straightforward generalization because \nthe mappings used in~\\cite{Czyz:2010sp} for $\\pi^0$\nwork similarly well also for $\\eta$ and $\\eta^\\prime$.\n\nWe simulate the cross sections ${d\\sigma}\/{d Q^2}$\nfor the process $e^+e^- \\to e^+e^-\\mathcal{P}$\nand compare it with existing ``single-tag'' data from the \n  CELLO~\\cite{Behrend:1990sr}, CLEO~\\cite{Gronberg:1997fj} \n  and BaBar~\\cite{Aubert:2009mc,:2011hk} experiments.\nIn a single-tag experiment,\nthe ``tagged'' lepton fixes the value of $Q^2 = -t_1$\nand the 4-momentum squared of the ``untagged'' lepton $t_2 = -q_2^2$ \nis kinematically restricted nearby zero.\nFor example, in the BaBar experiment, the actual thresholds \nfor $q_2^2$ are $0.18$~GeV$^2$ for pions~\\cite{Aubert:2009mc}\nand $0.38$~GeV$^2$ for $\\eta$ and $\\eta^\\prime$~\\cite{:2011hk}\ndue to the imposed event selection.\nThe experimental ${d\\sigma}\/{d Q^2}$ is given within these cuts,\nand, therefore, the simulated ${d\\sigma}\/{d Q^2}$ is computed within\nthe similar event selection.\nAs expected, a good agreement between the generator predictions\nand the data is observed, see Figs.~\\ref{fig:cs-pi0},~\\ref{fig:cs-eta}.\n\n\n\n\\begin{figure*}\n\\begin{center}\n\\resizebox{0.48\\textwidth}{!}{%\n     \\includegraphics{pictures\/pi-cs-BABAR09.eps}\n     }\n\\resizebox{0.48\\textwidth}{!}{%\n     \\includegraphics{pictures\/pi-cs-CLEO98.eps}\n     }\n\\end{center}\n\\caption{The cross section ${d\\sigma}\/{d Q^2}$\nfor the process $e^+e^- \\to e^+e^-\\pi^0$\n compared to BaBar~\\cite{Aubert:2009mc} (left)\n and\n CLEO~\\cite{Gronberg:1997fj} (right). \n }\n \\label{fig:cs-pi0}\n\\end{figure*}\n\n\n\\begin{figure*}\n\\begin{center}\n\\resizebox{0.48\\textwidth}{!}{%\n     \\includegraphics{pictures\/eta-cs.eps}\n     }\n\\resizebox{0.48\\textwidth}{!}{%\n     \\includegraphics{pictures\/etaP-cs.eps}\n     }\n\\end{center}\n\\caption{The cross section ${d\\sigma}\/{d Q^2}$\nfor the process $e^+e^- \\to e^+e^-\\eta$ (left) and \n$e^+e^- \\to e^+e^-\\eta^\\prime$ (right)\n compared to CLEO~\\cite{Gronberg:1997fj} and BaBar~\\cite{:2011hk} data.}\n \\label{fig:cs-eta}\n\\end{figure*}\n\n\nAn important note here is in order.\nThe values of the ${d\\sigma}\/{d Q^2}$ are the primary results of the experiment.\nThe form factor $F_{\\mathcal{P}}(Q^2,0)$ is calculated then on the basis of \nthe measured ${d\\sigma}\/{d Q^2}$ and the simulation.\nIt is known that the model dependence in the simulation leads to the uncertainty \nin the form factor, which is ``measured'' in this way.\nIn the BaBar analyses, the corresponding uncertainties in cross section\nare estimated to be at the level of $3.5~\\%$ for pions~\\cite{Aubert:2009mc}\nand $4.6~\\%$ for $\\eta$ and $\\eta^\\prime$~\\cite{:2011hk}\n(based on the simulation with the $q_2^2$--dependent\nand $q_2^2$--independent form factors).\nAs stressed in~\\cite{Aubert:2009mc}, this uncertainty \nis very sensitive to the actual $q_2^2$ cut.\n\nRecently, the effects of the $q_2^2$ cut were emphasized on \nthe level of the form factor considerations~\\cite{Lichard:2010ap}.\nThe Monte Carlo generator in hand allows us to perform a more\nconclusive study, namely to investigate the magnitude of the\ncross section uncertainties, discussed above.\nSimilarly to the method used in the BaBar~\\cite{Aubert:2009mc,:2011hk} analyses we\nperform two simulations: the first (${d\\sigma[full]}\/{dQ^2}$) \nwith the exact form factor\n$F_{\\gamma^\\ast\\gamma^\\ast \\mathcal{P}}(t_1, t_2)$\nand the second (${d\\sigma[approx]}\/{dQ^2}$)\nwith the approximated form factor \n$F_{\\gamma^\\ast\\gamma^\\ast \\mathcal{P}}(t_1, t_2)\n\\approx F_{\\gamma^\\ast\\gamma \\mathcal{P}}(t_1, 0)$,\ni.e., neglecting the momentum transfer to the untagged lepton in the form factor.\nThe relative difference of the corresponding cross sections \nis then plotted in Fig.~\\ref{fig:cs-relative}.\nOur estimations for the uncertainty\nare in a rough agreement with that of BaBar~\\cite{Aubert:2009mc,:2011hk}.\nHowever, in contrast to the estimate of BaBar,\na dependence of this uncertainty on $Q^2$ is observed\nin our simulation.\nIf this effect is not accounted for in the data it might\nresult in inducing a fake $Q^2$ dependence of the form factor.\n\nIn order to investigate the impact of the event selection \non the error estimate, we perform the simulation for $\\eta$ and $\\eta^\\prime$ mesons\nwith the direct cut on the second (untagged) invariant ($q_2^2$) \nand separately with the cut on the angle between the initial \nand final untagged lepton ($|\\cos\\theta^\\ast_{ e \\mathcal{P} }| > 0.99$),\nwhich effectively induces the cut on $q_2^2$~\\cite{Aubert:2009mc,:2011hk}.\nFrom Fig.~\\ref{fig:cs-relative} we see that the error estimate \nand its $Q^2$ dependence is very sensitive to the event selection.\n\n\n\n\\begin{figure}\n\\begin{center}\n\\resizebox{0.42\\textwidth}{!}{%\n     \\includegraphics[angle=270]{pictures\/relative_cs_all.eps}\n     }\n\\end{center}\n\\caption{The relative difference of the cross sections\n$({d\\sigma[full]} - {d\\sigma[approx]})\/{d\\sigma[full]}$\nfor the process $e^+e^- \\to e^+e^-\\pi^0$ (A) and $e^+e^- \\to e^+e^-\\eta^{(\\prime)}$ (B).\nThe approximate simulation ($d\\sigma[approx]$)\nignores the form factor dependence on the momentum transfer ($q_2^2$) \nto the untagged lepton.\nThe full simulation ($d\\sigma[full]$) accounts for the virtuality of both photons \nin the form factor.\nThe following cuts are used in the simulation: \\\\\n(1) $|q_2^2| < 0.18$~GeV$^2$~\\cite{Aubert:2009mc}; \\\\\n(2) $|\\cos \\theta^\\ast_{ e \\mathcal{P} }|>0.99$ and $|q_2^2| < 0.6$~GeV$^2$~\\cite{:2011hk}; \\\\\n(3) $|q_2^2| < 0.38$~GeV$^2$~\\cite{:2011hk}. \\\\\nThe lines denoted by~(BaBar) show estimates for the relative difference,\nas given in the BaBar papers:\nfor $\\pi^0$ --- in~\\cite{Aubert:2009mc}, \nfor $\\eta$ and $\\eta^\\prime$ --- in~\\cite{:2011hk}.\n}\n \\label{fig:cs-relative}\n\\end{figure}\n\n \n\\section{The limits of the form factors}\n\\label{sec:limits}\n\\subsection{The high-$Q^2$ limit of the form factors}\n\\label{sec:limit:high}\n\nThe issue of the asymptotic behavior of the form factors\nusually deserves an attention.\nIn our approach, the  high-$Q^2$ limits (${t\\rightarrow -\\infty}$) are the following.\nIn case of the one octet ansatz, we obtain\nfrom Eqs.~(\\ref{eq:ffpi_3octet}),~(\\ref{eq:ffeta_3octet}) and~(\\ref{eq:ffetapr_3octet}):\n\\begin{widetext}\n\\begin{eqnarray}\n    F_{\\gamma^\\ast \\gamma \\pi^0}(t,0) &=& \\frac{1}{8 \\pi^2 f_\\pi}\\frac{1}{t} (M_\\rho^2 + M_\\omega^2)\n+ \\mathcal{O} \\Bigl(\\frac{1}{t^2}  \\Bigr)\n \\label{eq:ff_limit:pi_1}   \n\n \\ ,\n\\end{eqnarray}\n\\begin{eqnarray}\n    F_{\\gamma^\\ast \\gamma^\\ast \\pi^0}(t,t) &=& \\frac{1}{4 \\pi^2 f_\\pi}\\frac{1}{t^2} M_\\rho^2 M_\\omega^2\n + \\mathcal{O} \\Bigl(\\frac{1}{t^3} \\Bigr)   \n \\label{eq:ff_limit:pi_2}   \n\n \\ ,\n\\end{eqnarray}\n\\begin{eqnarray}\n    &&F_{\\gamma^\\ast \\gamma \\eta}(t,0) = \\frac{1}{8 \\pi^2 f_\\pi} \\frac{1}{t}\n    \\bigl(3 C_q M_\\rho^2 + \\frac{1}{3} C_q M_\\omega^2 -\\frac{2\\sqrt{2}}{3} C_s M_\\phi^2 \\bigr) \n+ \\mathcal{O} \\Bigl(\\frac{1}{t^2} \\Bigr)\n\n \\label{eq:ff_limit:eta_1}   \n \\ ,\n\\end{eqnarray}\n\\begin{eqnarray}\n    &&F_{\\gamma^\\ast \\gamma^\\ast \\eta}(t,t) = \\frac{1}{8 \\pi^2 f_\\pi}\\frac{1}{t^2}\n    \\bigl(3 C_q M_\\rho^4 + \\frac{1}{3} C_q M_\\omega^4 -\\frac{2\\sqrt{2}}{3} C_s M_\\phi^4 \\bigr) \n+ \\mathcal{O} \\Bigl(\\frac{1}{t^3} \\Bigr)\n\n \\label{eq:ff_limit:eta_2}   \n \\ .\n\\end{eqnarray}\nIn case of the two octets ansatz, we obtain\n\\begin{eqnarray}\n    F_{\\gamma^\\ast \\gamma \\pi^0}(t,0) &=& \\frac{4\\sqrt{2}}{3 f_\\pi}\\frac{1}{t} \n    \\left[h_{V_1}f_{V_1} (M_\\rho^2 + M_\\omega^2)\\right.\n        + \\left.h_{V_2}f_{V_2} (M_{\\rho^\\prime}^2 + M_{\\omega^\\prime}^2)\\right]\n+ \\mathcal{O} \\Bigl(\\frac{1}{t^2} \\Bigr)\n\n \\label{eq:ff_limit:pi_3}   \n \\ ,\n\\end{eqnarray}\n\\begin{eqnarray}\n    F_{\\gamma^\\ast \\gamma^\\ast \\pi^0}(t,t) &=& \\frac{8\\sqrt{2}}{3 f_\\pi}\\frac{1}{t^2} \n    \\left[h_{V_1}f_{V_1} M_\\rho^2 M_\\omega^2\\right.\n        + \\left.h_{V_2}f_{V_2} M_{\\rho^\\prime}^2 M_{\\omega^\\prime}^2\\right]\n+ \\mathcal{O} \\Bigl(\\frac{1}{t^3} \\Bigr)\n\n \\ ,\n \\label{eq:ff_limit:pi_4}   \n\\end{eqnarray}\n\\begin{eqnarray}\n\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\n    &&\\!F_{\\gamma^\\ast \\gamma \\eta}(t,0) = \\frac{4\\sqrt{2}}{3 f_\\pi} \\frac{1}{t}\n    \\!\\!\\!\\left[\\!h_{V_1}f_{V_1} \\bigl(3 C_q M_\\rho^2 + \\frac{1}{3} C_q M_\\omega^2 -\\frac{2\\sqrt{2}}{3} C_s M_\\phi^2 \\bigr)\\right. \n       \\! + \\! \\left.h_{V_2}f_{V_2} \\bigl(3 C_q M_{\\rho^\\prime}^2 + \\frac{1}{3} C_q M_{\\omega^\\prime}^2 -\\frac{2\\sqrt{2}}{3} C_s M_{\\phi^\\prime}^2 \\bigr)\\!\\right]\n\\!\\!\\!+ \\!\\mathcal{O} \\Bigl(\\frac{1}{t^2} \\Bigr)\n \\label{eq:ff_limit:eta_3}   \n   \\;,\\ \\  \\  \\  \\  \\  \\  \\ \n\\end{eqnarray}\n\\begin{eqnarray}\n\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\n    &&\\!F_{\\gamma^\\ast \\gamma^\\ast \\eta}(t,t) = \\frac{8\\sqrt{2}}{3 f_\\pi}\\frac{1}{t^2}\n    \\!\\!\\!\\left[\\!h_{V_1}f_{V_1} \\bigl(3 C_q M_\\rho^4 + \\frac{1}{3} C_q M_\\omega^4 -\\frac{2\\sqrt{2}}{3} C_s M_\\phi^4 \\bigr) \\right.\n       \\! + \\! \\left.h_{V_2}f_{V_2} \\bigl(3 C_q M_{\\rho^\\prime}^4 + \\frac{1}{3} C_q M_{\\omega^\\prime}^4 -\\frac{2\\sqrt{2}}{3} C_s M_{\\phi^\\prime}^4 \\bigr)\\!\\right]     \n\\!\\!\\!+ \\!\\mathcal{O} \\Bigl(\\frac{1}{t^3} \\Bigr)\n \\label{eq:ff_limit:eta_4}   \n\\;. \\  \\  \\  \\  \\  \\  \\  \\ \n\\end{eqnarray}\n\\end{widetext}\nThe limits for the $\\eta^\\prime$ form factor \ncan be obtained from the above formulae \naccording to~(\\ref{eq:ffetapr_3octet}).\n\nThe expressions~(\\ref{eq:ff_limit:pi_3}) and~(\\ref{eq:ff_limit:eta_3})\nguide the high-$Q^2$ behavior of the form factors\nmeasured in single-tag experiments\n(shown in Figs.~\\ref{fig:ff-pi0},~\\ref{fig:ff-eta},~\\ref{fig:ff-etaP}).\nWe see that in our approach the asymptotic \nvalue of $Q^2|F_{\\mathcal{P}}(Q^2,0)|$\ndepends not only on the mixing parameters and decay constants\nbut also on the masses of the vector resonances.\nNumerically, for $\\pi^0$ transition form factor,\nthe value of $Q^2|F_{\\pi^0}(Q^2,0)|$ in our approach\nwith two octets is very close to that \nof the Brodsky-Lepage~\\cite{Lepage:1980fj} high-$Q^2$ limit \n$Q^2|F_{\\pi^0}(Q^2,0)| \\to 2 f_\\pi$\nshown as a bold solid line (BL) in Fig.~\\ref{fig:ff-pi0}.\n\nThe perturbative QCD prediction for the asymptotic \nof the $\\eta$ and $\\eta^\\prime$ form factors\nis often given in a simple approach in terms of the\nparameters $f_\\eta=0.0975$~GeV and\n$f_{\\eta^\\prime}=0.0744$~GeV~\\cite{Gronberg:1997fj,Dorokhov:2009jd}:\n$Q^2|F_{\\eta^{(\\prime)}}(Q^2,0)| \\to 2 f_{\\eta^{(\\prime)}}$.\nThese values are shown as bold solid line (BL)\nin Figs.~\\ref{fig:ff-eta},~\\ref{fig:ff-etaP}\nand in this case one can notice no coincidence \nwith the values given by~(\\ref{eq:ff_limit:eta_3}).\nSometimes it is also called the Brodsky-Lepage limit,\nwith a reference to~\\cite{Brodsky:1981rp}.\nHowever, we would like to remark that \nin~\\cite{Brodsky:1981rp} the $SU(3)$ flavor\nbreaking effects are not considered and the assumed $\\eta-\\eta^\\prime$\nmixing may be not consistent with modern data.\nAn attempt to interpret the results of~\\cite{Brodsky:1981rp}\nby means of\n$f_{\\eta^\\prime}=0.0744$~GeV\nis in tension with the data for $\\eta^\\prime$ form factor, as\none can see in Fig.~\\ref{fig:ff-etaP} \nand as noticed in~\\cite{Dorokhov:2009jd}.\n\nIn principle, there are other ways to apply the formulae\nof~\\cite{Brodsky:1981rp} to the form factors of physical states\n$\\eta$ and $\\eta^\\prime$~\\cite{Ametller:1991jv}.\nFor example, the limit for the $\\eta$ and $\\eta^\\prime$ \ntransition form factors can be calculated \naccording to the two-angle $\\eta-\\eta^\\prime$ \nmixing scheme~\\cite{Feldmann:1999uf,Kroll:2010bf}.\nThe latter values are shown as a shaded box (FK)\nin Figs.~\\ref{fig:ff-eta},~\\ref{fig:ff-etaP} (green online).\nNotice, that numerically this limit for $\\eta$  meson \nis very close to that of BL approach and \nalso to the value given by our model,\nhowever for $\\eta^\\prime$ all the three values are different. \n\n \n\\subsection{The slope of the form factor at the origin}\n\\label{sec:slope}\n\nSometimes it is convenient to define the so-called \nslope of the transition form factor at the origin\n$a_{\\mathcal{P}}$ (``slope parameter''):\n\\begin{eqnarray}\na_{\\mathcal{P}} &\\equiv& \\left.\\frac{1}{F_{\\gamma^\\ast\\gamma^\\ast\\mathcal{P}}(0,0)}\n\\frac{d\\ F_{\\gamma^\\ast\\gamma^\\ast\\mathcal{P}}(t,0)}{d\\ x}\\right|_{t=0},\n\\end{eqnarray}\nwhere $x\\equiv t\/m^2_{\\mathcal{P}}$.\nNotice that being defined this way, $a_{\\mathcal{P}}$\nis a dimensionless quantity,\nwhich is related to the effective region\nof the $\\gamma\\gamma^\\ast\\mathcal{P}$ interaction\n$\\langle{r^2_{\\mathcal{P}}}\\rangle = 6 a_{\\mathcal{P}}\/m^2_{\\mathcal{P}}$. \nThe average experimental value for $a_{\\pi}$ \nlisted in PDG~\\cite{Nakamura:2010zzi}\n(linear coefficients of the $\\pi^0$ electromagnetic \nform factor) \nis mainly driven not by a direct measurement,\nbut by an extrapolation done in Ref.~\\cite{Behrend:1990sr}.\nThe direct measurements of $a_{\\pi}$\nare less precise~\\cite{Farzanpay:1992pz,MeijerDrees:1992qb}.\nThe experimental knowledge of $a_{\\eta}$ is much better\nand recently the new experiments contributed: \nMAMI-C~\\cite{Berghauser:2011zz} and NA60~\\cite{:2009wb,Usai:2011zz}. \nFor $a_{\\eta^\\prime}$ we were not able to find a result of\na direct measurement.\n\nFrom Eqs.~(\\ref{eq:ffpi_3octet}),~(\\ref{eq:ffeta_3octet}) and~(\\ref{eq:ffetapr_3octet})\none obtains the following model prediction for the slope parameters:\n\\begin{eqnarray}\n\\label{eq:slope:pi}\na_{\\pi} &=&\n\\frac{16\\sqrt{2}\\pi^2m^2_{\\pi}}{N_C}\n\\sum_{i=1}^{n}{\n     h_{V_i} f_{V_i} \\left( \\frac{1}{M^2_{\\rho_i}} + \\frac{1}{M^2_{\\omega_i}}\\right)\n    }\n    \\;,\n\\end{eqnarray}\n\\begin{eqnarray}\na_{\\eta} &=&\n\\frac{16\\sqrt{2}\\pi^2 m^2_{\\eta}}{N_C}\n\\left( \\frac{5}{3}C_q - \\frac{\\sqrt{2}}{3}C_s \\right)^{-1}\n\\nonumber\\\\\n&&\\!\\!\\!\\!\\!\\!\\times\n \\sum_{i=1}^{n}{\n     h_{V_i} f_{V_i} \\left( \\frac{3C_q}{M^2_{\\rho_i}} + \\frac{C_q}{3M^2_{\\omega_i}} - \\frac{2\\sqrt{2}C_s}{3 M^2_{\\phi_i}}\\right)\n    }\n    \\;,\n\\label{eq:slope:eta}\n\\end{eqnarray}\n\\begin{eqnarray}\na_{\\eta^\\prime} &=&\n\\frac{16\\sqrt{2}\\pi^2 m^2_{\\eta^\\prime}}{N_C}\n\\left( \\frac{5}{3}C_q^\\prime + \\frac{\\sqrt{2}}{3}C_s^\\prime \\right)^{-1}\n\\nonumber\\\\\n&&\\!\\!\\!\\!\\!\\!\\times\n \\sum_{i=1}^{n}{\n     h_{V_i} f_{V_i} \\left( \\frac{3C_q^\\prime}{M^2_{\\rho_i}} + \\frac{C_q^\\prime}{3M^2_{\\omega_i}} + \\frac{2\\sqrt{2}C_s^\\prime}{3 M^2_{\\phi_i}}\\right)\n    }\n    \\;.\n\\label{eq:slope:etaP}\n\\end{eqnarray}\n\nThe numerical values for $a_{\\mathcal{P}}$ \nare listed in Table~\\ref{table:slope}.\nOn its basis we conclude that there is a reasonable agreement \nbetween model predictions and experiments.\n\nWe would like to remark that in the limit of the \nequal masses for vector resonances within the octet, \nEqs.~(\\ref{eq:slope:pi}),~(\\ref{eq:slope:eta}),~(\\ref{eq:slope:etaP})\nlead to the following relation between the slope parameters:\n$a_{\\pi}\/m^2_{\\pi} = a_{\\eta}\/m^2_{\\eta} = a_{\\eta^\\prime}\/m^2_{\\eta^\\prime}$.\n\n\\begin{table*}\n\\caption{\nModel prediction for the slope parameters\n$a_{\\mathcal{P}}$ and two most recent experimental\nvalues.\nThe ``2 octets'' column is calculated with \nthe parameter values given by our global fit. \nThe first error in experimental value is due to\nstatistics and the second one is systematics.\n}\n\\label{table:slope}\n\\begin{ruledtabular}\n\\begin{tabular}{lllll}\n& \\multicolumn{1}{c}{1 octet} & \\multicolumn{1}{c}{2 octets} & \\multicolumn{2}{c}{experiments}\\\\\n\\hline\\noalign{\\smallskip}\n$a_{\\pi}$         & 0.03003(1) & 0.02870(9) & 0.026(24)(48)~\\text{\\cite{Farzanpay:1992pz}}& 0.025(14)(26)~\\text{\\cite{MeijerDrees:1992qb}}\\\\\n$a_{\\eta}$        & 0.546(9) & 0.521(2) & 0.576(105)(39)~\\text{\\cite{Berghauser:2011zz}}& 0.585(18)(13)~\\text{\\cite{Usai:2011zz}}\\\\\n$a_{\\eta^\\prime}$ & 1.384(3) & 1.323(4) & \\text{---}& \\text{---}\\\\\n\\end{tabular}\n\\end{ruledtabular}\n\\end{table*}\n\n\n\\section{Summary}\n\\label{sec:summary}\n\nUsing the scheme of the $\\eta$--$\\eta^\\prime$ mixing\nwith two decay parameters ($f_0$ and $f_8$) \nand two mixing angles ($\\theta_0$, $\\theta_8$)~\\cite{Feldmann:1998vh,Feldmann:1999uf}\nand following the approach of chiral effective theory with resonances~\\cite{Ecker:1989yg,Ecker:1988te,Prades:1993ys},\nwe derive the expressions for the two-photon transition form \nfactors of the $\\mathcal{P}=\\pi^0$, $\\eta$, $\\eta^\\prime$ mesons.\nThe tree-level contributions within this effective field theory\napproach are considered.\n  For the case of the one octet ansatz there are no free parameters\n  and we obtain the model prediction for the form factors.\n\n  For the case of the two octet ansatz the model parameter\n  is fitted to the data.\nWe find that the two octet calculation is consistent with the bulk \n of available data.\nThe high-$Q^2$ limits of the form factors\nin our approach are compared to those of\nthe Brodsky and Lepage~\\cite{Lepage:1980fj,Gronberg:1997fj,Dorokhov:2009jd}\nand to those of Feldmann and Kroll~\\cite{Feldmann:1999uf,Kroll:2010bf}.\nThe slope of the transition form factor at the origin,\n$a_{\\mathcal{P}}$, is calculated and compared to available data.\nA reasonable agreement between model predictions and experiments is found.\n\nThe obtained form factors are implemented in the EKHARA Monte Carlo\ngenerator.\nAs a test of the generator, the cross-section ${d\\sigma}\/{d Q^2}$ is simulated \nfor the process $e^+e^- \\to e^+e^-\\mathcal{P}$ and compared to data\nusing the event selections,\nwhich mimic the ``single-tag'' experimental conditions. \n\nUsing the Monte Carlo simulation we investigate the\nimpact of neglecting the momentum transfer to the untagged lepton \n($t_2$) on the cross section and form factor measurements.\nThe uncertainty in the visible cross section due to the\nsimplification of the form factor \n$F_{\\gamma^\\ast\\gamma^\\ast\\mathcal{P}}(t_1,t_2) \\approx F_{\\gamma^\\ast\\gamma^\\ast\\mathcal{P}}(t_1,0)$\nis estimated for the phase space cuts \nsimilar to the experimental ones.\n\nDue to very small number of free parameters \nand good agreement with data, the approach presented\nin this work is a good starting point\nfor further model adjustments, e.g., for including the\n$SU(3)$ flavor symmetry breaking in the couplings.\nUsing the developed generator one will be able to study, \ne.g., a possible manifestation of such effects \nin the cross section ${d\\sigma}\/{d Q^2}$ within \na realistic phase space cuts.\n\n\\section*{Acknowledgments}\nWe would like to kindly acknowledge our colleagues from\nexperimental collaborations, whose feedback about the EKHARA generator\nand the physics case of $e^+e^- \\to e^+e^-\\mathcal{P}$ gave a motivation\nof the current research. \nWe greatly profited from discussions\nwith Danilo Babusci, Achim Denig, Dario Moricciani, Matteo Mascolo, \nElisabeta Prencipe and Graziano Venanzoni.\nWe also thank Fred Jegerlehner and Andreas Nyffeler for discussions.\n\n\nThis research was partly supported by Marie Curie Intra\nEuropean Fellowship within the 7th European Community Framework Programme\n(FP7-PEOPLE-2009-IEF),\nby Polish Ministry of Science and High Education\nfrom budget for science for years 2010-2013: grant number N~N202~102638,\nby National Academy of Science\nof Ukraine under contract $50\/53$---$2011$,\nby Sonderforschungsbereich SFB1044 of the Deutsche Forschungsgemeinschaft,\nby Spanish Consolider Ingenio 2010 Programme\nCPAN (CSD2007-00042) as well by MEC (Spain) under Grants {FPA2007-60323},\n{FPA2011-23778}.\nThis work is a part of the activity of the ``Working Group on Radiative\nCorrections and Monte Carlo Generators for Low \nEnergies''(\\href{http:\/\/www.lnf.infn.it\/wg\/sighad\/}{{\\tt http:\/\/www.lnf.infn.it\/wg\/sighad\/}})~\\cite{Actis:2010gg}.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nThe Yau-Tian-Donaldson (YTD) conjecture predicts that the existence of cscK metrics is equivalent to a purely algebro-geometric condition called K-polystability. In Fano case, the conjecture is solved by Chen-Donaldson-Sun \\cite{CDS} and Tian \\cite{T2} independently and algebro-geometers made remarkable progress in K-stability of log Fano varieties in ten years. For example, Fujita \\cite{Fjt} and Li \\cite{Li3} established the valuative criterion for K-stability of $\\mathbb{Q}$-Fano varieties independently. Fujita and Odaka \\cite{FO} introduced the $\\delta$-invariant and Blum and Jonsson \\cite{BlJ} proved that $\\delta(X)\\ge1$ if and only if a $\\mathbb{Q}$-Fano variety $(X,-K_X)$ is K-semistable. In the definition of K-stability, $(X,L)$ is K-semistable if and only if the Donaldson-Futaki invariant of any ample test configuration $(\\mathcal{X},\\mathcal{L})$ is nonnegative. Even if $X=\\mathbb{P}^2$, it had been difficult to prove K-stability of $(X,L)$ in a purely algebro-geometric way at first. However, thanks to the progress, we can characterize K-stable Fano varieties without using test configurations. On the other hand, Odaka \\cite{O2} proved K-stability of Calabi-Yau varieties and canonically polarized varieties in a purely algebro-geometric way.\n\nIn K\\\"{a}hler geometry, the existence problem of cscK metrics on fibrations is studied by \\cite{J2}, \\cite{Fine}, \\cite{JSS}, \\cite\n{DS2}. In contrast, K-stability of algebraic fiber spaces is not known so much. In this paper, fibrations mean algebraic fiber spaces $f:X\\to B$, where $f$ are morphisms of varieties with connected general fiber (cf., Definition \\ref{algfib}). We call an algebraic fiber space $f:X\\to B$ a {\\it Calabi-Yau fibration} if $K_X$ is relatively trivial over $B$. Fine \\cite{J2}, \\cite{Fine} and Dervan-Sektnan studied fibrations when all fibers have cscK metrics. On the other hand, Jian-Shi-Song \\cite{JSS} showed existence of a cscK metric of a Calabi-Yau fibration $f:X\\to B$ when $K_X\\sim_{\\mathbb{Q}}f^*L_0$, $L_0$ is ample and $f$ might have a singular fiber. They also proved the existence problem for canonically polarized fibrations over curves. By the result of Berman-Darvas-Lu \\cite{BDL}, we can conclude that Jian-Shi-Song \\cite{JSS} proved adiabatic K-(poly)stability (cf., Definition \\ref{algfibk}) of such fibrations. In this paper, we generalize the results of Jian-Shi-Song in a purely algebro-geometric way. \n\nOn the other hand, it is known for a polarized algebraic space that adiabatic K-semistability of the total space implies twisted K-semistability of the base space by the work of Dervan-Ross \\cite[Corollary 4.4]{DR2}. We also obtain a modification of the result of Dervan-Ross for lc-trivial fibrations (cf., Definition \\ref{ambro model}) and that some stability (log-twisted K-stability in Definition \\ref{deflogtwisted}) of the base gives a sufficient condition for adiabatic K-stability of the total space.\n \n Moreover, we also study K-stability of rational elliptic surfaces. Rational elliptic surfaces are important objects both in algebraic geometry and K\\\"{a}hler geometry respectively. In algebraic geometry, Miranda \\cite{Mi2}, \\cite{Mi} constructed two moduli spaces of rational elliptic surfaces by applying GIT. In the two results, Miranda realized that GIT-stability of a rational elliptic surface $X$ depends on what fiber $X$ has, and furthermore, he conjectured in \\cite{Mi3} that asymptotic Chow stability of rational Weierstrass fibrations would be decided by their fibers similarly to the results of \\cite{Mi2}, \\cite{Mi}. In this paper, we prove that if rational elliptic surfaces have only reduced fibers, then they are adiabatically K-stable and have cscK metrics. In contrast, we also prove that if a rational elliptic surface has at least one\"bad\" fiber, then it is adiabatically K-unstable. \n\n\\vspace{0.3cm}\n\nThe main purpose of the paper is to classify adiabatic K-stable Calabi-Yau and canonically polarized fibrations over curves. In \\cite[Theorem C]{Hat2}, we showed that Calabi-Yau fibrations are $\\mathfrak{f}$-stable. Taking \\cite[Conjecture]{Hat2} into account, it is natural to conjecture that adiabatic K-stability of Calabi-Yau fibrations is decided by K-stability of the bases in a certain sense. We prove the following main theorems \\ref{ff} and \\ref{dd}.\n\\begin{teiri}[Theorem \\ref{can}]\\label{ff}\nLet $f:(X,\\Delta)\\to C$ be a lc-trivial fibration. Let also $(C,L)$ be a polarized normal variety and $(X,\\Delta,H)$ be a polarized pair. Let\n\\[\nK_{X}+\\Delta=f^*(K_C+M_C+B_C)\n\\]\nbe the canonical bundle formula (cf., Definition \\ref{ambro model}). If the moduli divisor $M_C$ is $\\mathbb{Q}$-Cartier and $(X,\\Delta,H)$ is adiabatically K-semistable, then $(C,B_C,M_C,L)$ is log-twisted K-semistable (see Appendix and Definition \\ref{deflogtwisted}).\n\\end{teiri} \nThe proof of Theorem \\ref{ff} is inspired by \\cite[Proposition 4.3]{DR2}. As in the proof of \\cite[Proposition 4.3]{DR2}, we take test configurations $\\mathcal{X}$ for $X$ and $\\mathscr{C}$ for $C$ respectively. Applying the canonical bundle formulae of Fujino-Mori \\cite{FM2}, Ambro \\cite{A} and Fujino-Gongyo \\cite{FG} and results in MMP (cf., \\cite{HX}, \\cite{Has}) to the morphism $\\mathcal{X}\\to\\mathscr{C}$, we analyze the relative canonical bundle $K_{\\mathcal{X}\/\\mathscr{C}}$ more explicitly. \n\nIf the base is a curve, we obtain the following partial converse of Theorem \\ref{ff}.\n\n\n\\begin{teiri}[Theorem \\ref{ctn3}, Corollary \\ref{chikichiki}]\\label{dd}\nLet $f:(X,\\Delta,H)\\to (C,L)$ be a polarized algebraic fiber space pair such that $C$ is a smooth curve, $(X,\\Delta)$ is klt and $K_X+\\Delta\\sim_{C,\\mathbb{Q}}0$. Let $M$ be the moduli divisor and $B$ be the discriminant divisor on $C$ as Definition \\ref{ambro model}. \n\nThen, $(X,\\Delta,H)$ is uniformly adiabatically K-stable over $(C,L)$ if and only if $(C,B,M,L)$ is log-twisted K-stable. Here, uniform adiabatic K-stability is a stronger condition than adiabatic K-stability (Definition \\ref{algfibk}). Moreover, if $\\Delta=0$, $X$ is smooth and $(C,B,M,L)$ is log-twisted K-stable, then $(X,\\epsilon H+L)$ has a cscK metric for sufficiently small $\\epsilon>0$.\n\\end{teiri}\n\n\n\n\\begin{rem}\nWe introduce the log-twisted K-stability of log-twisted pairs as a generalization of log and twisted K-stability of \\cite{OS} and \\cite{De2} respectively.\n \\end{rem}\n \n This theorem states that adiabatic K-stability of any Calabi-Yau fibration over a curve depends almost completely on the log-twisted K-stability of the base. We explain the proof briefly here. Let $f:(X,\\Delta,H)\\to (B,L)$ be a polarized algebraic fiber space pair. Here, we do not assume that $B$ is a curve. Then, we introduce the twisted canonical class $\\kappa_B$ of $B$ (Definition \\ref{kappainv}) after Fine \\cite[Theorem 1.1]{Fine}. We can check that bases of fibrations treated in \\cite{JSS} have  the ample twisted canonical classes and generalize the result of Jian-Shi-Song for surfaces by applying results of the author \\cite[Theorem A]{Hat} on J-stability as follows,\n\n\n\\begin{teiri}[Corollary \\ref{j2e}]\\label{oo}\nLet $f:(X,\\Delta)\\to B$ be a lc algebraic fiber space pair over a smooth curve $B$. Suppose that $\\mathrm{dim}\\,X=2$ and $K_X+\\Delta$ is $f$-ample. If $\\mathrm{deg}\\,\\kappa_B\\ge 0$ for $(X,K_X+\\Delta)\\to B$, then $(X,\\Delta,K_X+\\Delta)$ is adiabatically K-semistable over $(B,L)$ (if furthermore $\\mathrm{deg}\\,\\kappa_B> 0$ or $(X,\\Delta)$ is klt, then it is adiabatically uniformly K-stable), where $L$ is a line bundle of degree $1$ on $B$. Moreover, if $\\mathrm{deg}\\,\\kappa_B\\ge0$, $\\Delta=0$ and $X$ is smooth, $(X,\\epsilon K_X+f^*L)$ has cscK metrics for sufficiently small $\\epsilon>0$.\n \\end{teiri}\n \n \\begin{rem}\n Theorem \\ref{oo} is different from \\cite[Theorem 1.2]{JSS} in that we do not assume that $K_X+\\Delta$ is ample here. We also treat the case when $\\mathrm{deg}\\,\\kappa_B<0$ in Corollary \\ref{d2e}.\n \\end{rem}\n\n\n Note that $\\mathrm{deg}\\,\\kappa_C\\ge0$ if and only if the log-twisted pair $(C,B,M,L)$ is general type or Calabi-Yau in Theorem \\ref{dd}. In this case, the theorem is solved in \\cite[Theorem G]{Hat}. \n \n Next, we discuss adiabatic K-stability of klt-trivial fibrations $f:(X,\\Delta)\\to C$ such that $\\kappa_C$ is not ample (e.g., rational elliptic surfaces). Here, klt (resp., lc)-trivial fibration is an algebraic fiber space pair such that $K_X+\\Delta\\sim_{C,\\mathbb{Q}} 0$ and $(X,\\Delta)$ is klt (resp., lc). By the argument before Theorem \\ref{oo}, it remains to prove Theorem \\ref{dd} in this case. We study klt-trivial fibrations over curves such that $-\\kappa_C$ is ample by applying the calculation of $\\delta$-invariant (cf., \\cite{BlJ}) and Theorem \\ref{ff}. \n\nThe proof of Theorem \\ref{dd} when $\\mathrm{deg}\\,\\kappa_C<0$ consists of two parts. First, we prove the following theorem.\n\n\n\\begin{teiri}[Theorems \\ref{ctn}, \\ref{ctnd}]\\label{ee}\nLet $f:(X,\\Delta,H)\\to (C,L)$ be a polarized algebraic fiber space pair. Suppose that $(X,\\Delta)$ is a klt pair, $C$ is a smooth curve and $\\mathrm{deg}\\, L=1$. Then \n\\begin{align*}\n\\lim_{\\epsilon\\to 0}\\alpha_{(X,\\Delta)}(\\epsilon H+f^*L)&=\\inf_{p\\in C} \\mathrm{lct}_{(X,\\Delta)}(f^{-1}(p)),\\\\\n\\lim_{\\epsilon\\to 0}\\delta_{(X,\\Delta)}(\\epsilon H+f^*L)&=2\\inf_{p\\in C} \\mathrm{lct}_{(X,\\Delta)}(f^{-1}(p)).\n\\end{align*} \n\\end{teiri} \nHere, we explain $\\alpha$-invariant and $\\delta$-invariant. $\\alpha$-invariant is introduced by Tian \\cite{T3} for Fano manifolds to detect the existence of K\\\"{a}hler-Einstein metrics. $\\alpha$ and $\\delta$-invariants are important for K-stability in the following sense. It is known that a polarized klt (resp., Fano) pair with a large $\\alpha$ (resp., $\\delta$)-invariant is automatically uniformly K-stable. See the details in \\cite{OS2}, \\cite{BlJ} and Definitions \\ref{defalpha}, \\ref{defdelta}. It has been known that $\\alpha$-invariant and $\\delta$-invariant are continuous on the ample cone by Dervan \\cite{Der} and Zhang \\cite{Z2} respectively. In other words, this theorem states that we can extend them to the fiber class continuously. On the other hand, we prove the following fact that was already known in the case when $X$ is smooth and $\\Delta=0$ by \\cite[Corollary 2.11]{BJ2}.\n\\begin{fact}[Theorem \\ref{bhjz}]\\label{pp}\nLet $(X,\\Delta,L)$ be a polarized klt pair. Then \n\\[\n\\delta_{(X,\\Delta)}(L)\\le\\sup\\{\\lambda>0|H_{\\Delta}^\\mathrm{NA}(\\mathcal{X},\\mathcal{L})\\ge\\lambda(I^\\mathrm{NA}(\\mathcal{X},\\mathcal{L})-J^\\mathrm{NA}(\\mathcal{X},\\mathcal{L}))\\}\n\\]\nwhere $(\\mathcal{X},\\mathcal{L})$ runs over the normal semiample test configurations for $(X,L)$.\n\\end{fact}\n\n\nCombining Theorem \\ref{ee} with Fact \\ref{pp}, we prove that $(X,\\Delta,H)$ is uniformly adiabatically K-stable over $(C,L)$ if $(C,B,M,L)$ is log-twisted K-stable. Next, we want to prove the converse of Theorem \\ref{dd}. Indeed, we can prove the following stronger result than Theorem \\ref{ff} when the base is log-twisted Fano. \n\n\n\\begin{teiri}[Theorem \\ref{stp1}]\\label{gg}\nNotations as in Theorem \\ref{ff}. Suppose that $(X,\\Delta,H)$ is uniformly adiabatically K-stable over $(C,L)$, $(X,\\Delta)$ is klt and $L=-(K_C+M+B)$ is ample. Then $(C,B,M,L)$ is log-twisted K-stable.\n\\end{teiri} \n\nWe prove Theorem \\ref{gg} by applying Theorem \\ref{ff} and the following fact.\n\\begin{fact}[Theorem \\ref{appendix}]\\label{aa}\nLet $(X,B,T,L=-K_X-B-T)$ be a strictly K-semistable log-twisted Fano pair such that $T$ is semiample (i.e., $(X,B,T,L)$ is K-semistable but not K-stable). Then there exists a $\\mathbb{Q}$-Cartier divisor $D\\in |L|_{\\mathbb{Q}}$ such that $(X,B+\\epsilon D,T,(1-\\epsilon) L)$ is a K-unstable log-twisted Fano pair for sufficiently small $\\epsilon>0$.\n\\end{fact}\n\nAs an application of Theorem \\ref{dd} to rational elliptic surfaces, we obtain the following,\n\n\n\\begin{kei}[Corollary \\ref{II}, \\ref{234}]\\label{hh}\nLet $X$ be a smooth rational elliptic surface and $S$ be a relatively ample divisor over $\\mathbb{P}^1 $. Let also the multiplicity of a unique multiple fiber be $m\\ge 1$. Then the following hold.\n\\begin{enumerate}\n\\item If $m=1$ and $X$ has at most reduced fibers, then $(X,S)$ is uniformly adiabatically K-stable and has cscK metrics.\n\\item If $m=2$ and $X$ has no $II^*$ or $III^*$-type fiber, then $(X,S)$ is uniformly adiabatically K-stable and has cscK metrics.\n\\item If $m=3$ and $X$ has no $II^*$-type fiber, then $(X,S)$ is uniformly adiabatically K-stable and has cscK metrics.\n\\item If $m\\ge 4$, then $(X,S)$ is uniformly adiabatically K-stable and has cscK metrics.\n\\end{enumerate}\n On the other hand,\n\\begin{enumerate}[(i)]\n\\item If $m=1$ and $X$ admits one of $II^*$, $III^*$ or $IV^*$-type fibers, then $(X,S)$ is adiabatically K-unstable over $\\mathbb{P}^1$.\n\\item If $m=2$ and $X$ admits a $II^*$-type fiber, then $(X,S)$ is adiabatically K-unstable over $\\mathbb{P}^1$.\n\\end{enumerate}\n\\end{kei}\n\n\n On the other hand, recall Miranda's conjecture on Chow-stability \\cite[Conjecture 2.3]{Mi3}.\n \\begin{conjj}[Miranda]\n Let $W$ be a rational Weierstrass fibration, $X$ be the minimal resolution and $H$ be an arbitrary ample line bundle. Then, $(W,H)$ is \n\\begin{enumerate}[(a)]\n\\item asymptotically Chow-stable if $X$ has only reduced fibers,\n\\item asymptotically Chow-semistable if $X$ has only reduced and $I_N^*$-type fibers for $N\\ge0$,\n\\item asymptotically Chow-unstable if $X$ has at least one $II^*$, $III^*$ or $IV^*$-type fiber.\n\\end{enumerate}\n\\end{conjj}\n \n We also partially settle this conjecture.\n\\begin{teiri}[Theorems \\ref{jj1}, \\ref{Mirconj}]\\label{jj}\nLet $X$ be a smooth rational elliptic surface and $S$ be a section over $\\mathbb{P}^1 $. Let $W$ be the log canonical model of $(X,S)$ over $\\mathbb{P}^1$. Let $f:W\\to\\mathbb{P}^1$ be the fibration and $L=\\mathcal{O}_{\\mathbb{P}^1}(1)$. Then the following hold;\n\\begin{enumerate}\n\\item If $W$ has at least one $II^*$, $III^*$, $IV^*$-type fiber, then $(W,aS+f^*L)$ is K-unstable for $0<a<1$.\n\\item If $W$ has at least one $I_N^*$-type fiber for some $N>0$, then $(W,aS+f^*L)$ is asymptotically Chow-unstable for sufficiently small $a>0$.\n\\end{enumerate}\n\\end{teiri} \n\nThus, Conjecture (c) holds for $W$ such that the Picard number $\\rho(W)=2$ and (b) fails in general.\n\n\n\\subsection*{Outline of this paper}\nWe consider a relationship between adiabatic K-stability of the total space and log-twisted K-stability of the base variety (Definition \\ref{algfibk}). In \\S 3, we introduce the twisted canonical class after Fine \\cite[Theorem 1.1]{Fine} and we prove that the positivity of the twisted canonical classes of the base spaces implies adiabatic K-stability of some algebraic fiber spaces. In \\S4, we consider Calabi-Yau fibrations with the negative twisted canonical classes. In \\S\\S4.1, we prove Theorem \\ref{ee} and obtain that log-twisted K-stability of the base Fano curve implies adiabatic K-stability of the total space of a klt-trivial fibration as in Theorem \\ref{dd}. In \\S\\S4.2, we prove Theorem \\ref{ff} and Theorem \\ref{dd} completely by applying MMP. Finally, we apply the theorems we obtain to rational elliptic surfaces in \\S5. In \\S\\S5.1, we classify log-twisted K-stable Fano curves associated with rational elliptic surfaces and have Corollary \\ref{hh}. In \\S\\S 5.2, \\S\\S5.3, we compare adiabatic K-stability of rational elliptic surfaces with results on GIT of Miranda \\cite{Mi} and \\cite{Mi2} respectively by the CM line bundle and a method of \\cite{OSS}. In \\S\\S 5.4, we apply the methods used in \\S\\S4.2, \\S\\S 5.2 and results in differential geometry to show Theorem \\ref{jj}.\n\nIn Appendix, we introduce log-twisted K-stability and state the results on this. In \\S\\S A.1, we define log-twisted K-stability as Dervan defined twisted K-stability in \\cite{De2}. In \\S\\S A.2, we consider log-twisted Ding stability when the twist $T$ is not nef and deduce Fact \\ref{pp}. In \\S\\S A.3, we prove Fact \\ref{aa}. To do this, we generalize the results of Li-Xu \\cite{LX}, Fujita \\cite{Fjt} and Blum-Liu-Zhou \\cite{BLZ} to the case when $T$ is semiample.\n\n\\section*{Acknowledgements}\nThe author is grateful to his research advisor Professor Yuji Odaka for a lot of suggestive advices and productive discussions. The author would like to thank him and Doctor Keita Goto for careful reading my draft. The author is also grateful to Professor Yoshinori Hashimoto for giving him a lecture on the paper \\cite{Z}. The author would like to thank Doctor Eiji Inoue for fruitful discussions and letting him know Fact \\ref{pp} in the smooth case in \\cite{BJ2}. Discussions with them led the author to improve Theorem \\ref{dd}. The author is grateful to Professor Kenta Hashizume for productive discussions and letting the author know his great paper \\cite{Has}, which led the author to improve the proof of Proposition \\ref{model}. \n\n\\section{Notation}\n\\label{Notat}\nIn this paper, we work over the complex number field $\\mathbb{C}$. $(X,\\Delta,L)$ is a polarized log pair if $X$ is a proper normal variety over $\\mathbb{C}$, $\\Delta$ is a $\\mathbb{Q}$-divisor on $X$ such that $K_{(X,\\Delta)}\\mathrel{\\mathop:}= K_X+\\Delta$ is $\\mathbb{Q}$-Cartier and $L$ is a $\\mathbb{Q}$-ample line bundle. For any divisorial valuation $v=c\\, \\mathrm{ord}_F$ where $F$ is a prime divisor over $X$, the log discrepancy of $(X,\\Delta)$ with respect to $v$ is defined as follows. We choose a log resolution $\\sigma:Y\\to X$ of $(X,\\Delta)$ such that $F$ is a smooth prime divisor on $Y$ and then define\n\\[\nA_{(X,\\Delta)}(v)=v(K_Y+F-\\sigma^*(K_X+\\Delta)).\n\\] See the details in \\cite[\\S 1.5]{BHJ}. In this section, we define K-stability of polarized log pairs as \\cite{BHJ}. For deminormal schemes, we follow the definitions of test configuration, J-stability and K-stability from \\cite{Hat2}.\n\n\n Fix a polarized log pair $(X,\\Delta,L)$.\n\n\\begin{de}[Test configuration]\nLet $\\pi:\\mathcal{X}\\to \\mathbb{A}^1$ be a proper and flat morphism such that $\\pi$ is $\\mathbb{G}_m$-equivariant. Here, $\\mathbb{A}^1$ admits the canonical $\\mathbb{G}_m$-action. If $\\pi^{-1}(1)\\cong X$, $\\pi$ or $\\mathcal{X}$ is called a {\\it test configuration}. Let $\\mathcal{X}$ be a test configuration. Assume that $\\mathcal{X}$ admits an $\\mathbb{A}^1$-(semi)ample $\\mathbb{G}_m$-linearized $\\mathbb{Q}$-line bundle $\\mathcal{L}$ such that $\\mathcal{L}|_{\\pi^{-1}(1)}=L$. Then, $(\\mathcal{X},\\mathcal{L})$ is called a {\\it (semi)ample test configuration} for $(X,L)$.\n \n If $(\\mathcal{X},\\mathcal{L})$ is $\\mathbb{G}_m$-equivariantly isomorphic to $(X\\times\\mathbb{A}^1,L\\otimes\\mathcal{O}_{\\mathbb{A}^1})$, then we call this the trivial test configuration and denote it by $(X_{\\mathbb{A}^1},L_{\\mathbb{A}^1})$. \n\\end{de}\n\n\\begin{ntt}\nFor simplicity, we will denote line bundles and divisors interchangeably and write self intersection products as $m$-th power of divisors. For instance, $$L^m\\cdot (kH+D)=L^{\\cdot m}\\cdot (H^{\\otimes k}\\otimes \\mathcal{O}_X(D)).$$ \n\n\\end{ntt}\n\nWe define the following functionals and the Donaldson-Futaki invariant as in \\cite{BHJ}.\n\n\\begin{de}[{\\cite[\\S 6, \\S 7]{BHJ}}]\\label{nadef}\nLet $\\mathrm{dim}\\,X=n$ and take an arbitrary normal semiample test configuration $(\\mathcal{X},\\mathcal{L})$. To consider intersection products, we need the {\\it $\\mathbb{G}_m$-equivariant compactification over $\\mathbb{P}^1$}, $(\\overline{\\mathcal{X}},\\overline{\\mathcal{L}})$ (See \\cite[Definition 2.7]{BHJ}). Then, let $\\rho:\\overline{\\mathcal{X}}\\dashrightarrow X_{\\mathbb{P}^1}$ be the canonical $\\mathbb{G}_m$-equivariant birational map such that $\\rho$ is identical on $X\\times\\mathbb{P}^1\\setminus0$, where $(X_{\\mathbb{P}^1},L_{\\mathbb{P}^1})=(\\overline{X_{\\mathbb{A}^1}},\\overline{L_{\\mathbb{A}^1}})$, the canonical compactification. If $\\rho$ satisfies the above condition, $\\rho$ is called the {\\it canonical} map. Assume that $\\rho$ is a morphism by resolution of indeterminacy. Furthermore, let $\\mathcal{X}_{0,\\mathrm{red}}$ be the reduced central fiber, $\\Delta_{\\overline{\\mathcal{X}}}$ be the closure of $\\Delta\\times \\mathbb{A}^1\\setminus0$ in $\\overline{\\mathcal{X}}$ and $K^{\\mathrm{log}}_{(\\overline{\\mathcal{X}},\\Delta_{\\overline{\\mathcal{X}}})\/\\mathbb{P}^1}=K_{(\\overline{\\mathcal{X}},\\Delta_{\\overline{\\mathcal{X}}})\/\\mathbb{P}^1}+(\\mathcal{X}_{0,\\mathrm{red}}-\\mathcal{X}_0)$.\nThen we define\n\\begin{itemize}\n\\item \n$I^{\\mathrm{NA}}(\\mathcal{X},\\mathcal{L})=(L^n)^{-1}(\\overline{\\mathcal{L}}\\cdot \\rho^*L_{\\mathbb{P}^1}^n)-(L^n)^{-1}(\\overline{\\mathcal{L}}- \\rho^*L_{\\mathbb{P}^1})\\cdot\\overline{\\mathcal{L}}^n,$\n\\item\n$E^{\\mathrm{NA}}(\\mathcal{X},\\mathcal{L})=\\frac{\\overline{\\mathcal{L}}^{n+1}}{(n+1)(L^n)},$\n\n\\item \n$J^{\\mathrm{NA}}(\\mathcal{X},\\mathcal{L})=(L^n)^{-1}(\\overline{\\mathcal{L}}\\cdot \\rho^*L_{\\mathbb{P}^1}^n)- E^{\\mathrm{NA}}(\\mathcal{X},\\mathcal{L}),$\n\\item \n$(\\mathcal{J}^H)^{\\mathrm{NA}}(\\mathcal{X},\\mathcal{L})=(L^n)^{-1}(\\rho^*H_{\\mathbb{P}^1}\\cdot \\overline{\\mathcal{L}}^{n})- (L^n)^{-1}(nH\\cdot L^{n-1})E^{\\mathrm{NA}}(\\mathcal{X},\\mathcal{L})$\nwhere $H$ is a $\\mathbb{Q}$-line bundle on $X$,\n\\item\n$H_\\Delta^{\\mathrm{NA}}(\\mathcal{X},\\mathcal{L})=(L^n)^{-1}\\sum_EA_{(X,B)}(v_E)(E\\cdot\\overline{\\mathcal{L}}^n)=(L^n)^{-1}(K^{\\mathrm{log}}_{(\\overline{\\mathcal{X}},\\Delta_{\\overline{\\mathcal{X}}})\/\\mathbb{P}^1}-\\rho^*K_{(X_{\\mathbb{P}^1},\\Delta_{\\mathbb{P}^1})\/\\mathbb{P}^1})\\cdot\\overline{\\mathcal{L}}^n$\nwhere $E$ runs over the irreducible components of $\\mathcal{X}_0$, $v_E=$ the restriction of $b_E^{-1}\\mathrm{ord}_E$ to the quotient field of $X$ and $b_E=\\mathrm{ord}_E(\\mathcal{X}_0)$ ($v_E$ is known to be a divisorial valuation or trivial. See the details in \\cite[\\S4]{BHJ}),\n\\item\n$R_\\Delta^{\\mathrm{NA}}(\\mathcal{X},\\mathcal{L})=(\\mathcal{J}^{K_{(X,\\Delta)}})^{\\mathrm{NA}}(\\mathcal{X},\\mathcal{L})+\\frac{n(K_{(X,\\Delta)}\\cdot L^{n-1})}{(L^n)}E^{\\mathrm{NA}}(\\mathcal{X},\\mathcal{L}),$\n\\item\n$\\mathrm{DF}_{\\Delta}(\\mathcal{X},\\mathcal{L})=(L^n)^{-1}(K_{(\\overline{\\mathcal{X}},\\Delta_{\\overline{\\mathcal{X}}})\/\\mathbb{P}^1}-\\rho^*K_{(X_{\\mathbb{P}^1},\\Delta_{\\mathbb{P}^1})\/\\mathbb{P}^1})\\cdot\\overline{\\mathcal{L}}^n+(\\mathcal{J}^{K_{(X,\\Delta)}})^{\\mathrm{NA}}(\\mathcal{X},\\mathcal{L}),$\n\\item \n$M_\\Delta^{\\mathrm{NA}}(\\mathcal{X},\\mathcal{L})=H_\\Delta^{\\mathrm{NA}}(\\mathcal{X},\\mathcal{L})+(\\mathcal{J}^{K_{(X,\\Delta)}})^{\\mathrm{NA}}(\\mathcal{X},\\mathcal{L}).$\n\\end{itemize}\nFor simplicity, we will denote $(\\overline{\\mathcal{X}},\\overline{\\mathcal{L}})$ as $(\\mathcal{X},\\mathcal{L})$.\nA semiample test configuration $(\\tilde{\\mathcal{X}},\\tilde{\\mathcal{L}})$ {\\it dominates} $(\\mathcal{X},\\mathcal{L})$ via $\\mu$ if there exists a $\\mathbb{G}_m$-equivariant morphism $\\mu:\\tilde{\\mathcal{X}}\\to\\mathcal{X}$ such that $\\mu$ is identical on $\\mathcal{X}\\setminus\\mathcal{X}_0$ and $\\tilde{\\mathcal{L}}=\\mu^*\\mathcal{L}$.\nIt is proved in \\cite[\\S\\S 7.5]{BHJ} that those functionals are {\\it pullback invariant} as defined in \\cite{BHJ}. For example, if $(\\tilde{\\mathcal{X}},\\tilde{\\mathcal{L}})$ dominates $(\\mathcal{X},\\mathcal{L})$ then $J^{\\mathrm{NA}}(\\tilde{\\mathcal{X}},\\tilde{\\mathcal{L}})=J^{\\mathrm{NA}}(\\mathcal{X},\\mathcal{L})$. We define two semiample test configurations $(\\mathcal{X}_i,\\mathcal{L}_i)$ $(i=1,2)$ are equivalent if there exists a test configuration $(\\mathcal{X}_3,\\mathcal{L}_3)$ that dominates both of $(\\mathcal{X}_i,\\mathcal{L}_i)$. Then let $\\mathcal{H}^{\\mathrm{NA}}(L)=\\{$semiample test configurations for $(X,L)\\}\/\\sim$ be the set of all {\\it non-Archimedean positive metrics} for $(X,L)$ where $\\sim$ means the equivalence relation. It is well-known that the functionals above are independent of the choice of representatives of $\\phi\\in\\mathcal{H}^{\\mathrm{NA}}(L)$. See the details in \\cite[\\S6,7]{BHJ}.\n\\end{de}\n\n\\begin{rem}\nWe can define the (log) Donaldson-Futaki invariant for nonnormal test configurations by using the $\\mathbb{G}_m$-actions on $\\mathcal{X}$ and $\\Delta_{\\mathcal{X}}$. See the details in \\cite{OS}, \\cite{De2} and \\cite{BHJ}.\n\\end{rem}\n\n\n\\begin{de}[K-stability and J-stability]\nLet $(X,\\Delta,L)$ be a polarized log pair. $(X,\\Delta,L)$ is {\\it K-stable} if \n\\[\n\\mathrm{DF}_{\\Delta}(\\mathcal{X},\\mathcal{L})>0\n\\]\nfor any non-trivial normal semiample test configuration. On the other hand, $(X,\\Delta,L)$ is {\\it uniformly K-stable} (resp, {\\it K-semistable}) if there exists a constant $c>0$ (resp., $c\\ge 0$) such that\n\\[\n\\mathrm{DF}_{\\Delta}(\\mathcal{X},\\mathcal{L})\\ge c I^{\\mathrm{NA}}(\\mathcal{X},\\mathcal{L})\n\\]\nfor any semiample test configuration. The above condition is known by \\cite[\\S8]{BHJ} to be equivalent to\n\\[\nM^{\\mathrm{NA}}_{\\Delta}(\\mathcal{X},\\mathcal{L})\\ge c I^{\\mathrm{NA}}(\\mathcal{X},\\mathcal{L}).\n\\]\nThe definition of J$^H$-stability is of similar form (cf., \\cite{Hat}). On the other hand, $(X,\\Delta,L)$ is {\\it K-polystable} if for any normal semiample test configuration $(\\mathcal{X},\\mathcal{L})$,\n\\[\n\\mathrm{DF}_{\\Delta}(\\mathcal{X},\\mathcal{L})=0\n\\]\nif and only if $(\\mathcal{X},\\Delta_{\\mathcal{X}})$ is isomorphic to $(X\\times\\mathbb{A}^1,\\Delta\\times\\mathbb{A}^1)$ as an abstract log pair.\n\nNote that $I^{\\mathrm{NA}}$, $J^{\\mathrm{NA}}$ and $I^{\\mathrm{NA}}-J^{\\mathrm{NA}}$ are nonnegative and equivalent norms in the sense of \\cite[Proposition 7.8]{BHJ}.\n\\end{de}\n\nLet us recall the scaling action to a non-Archimedean metric \\cite[\\S6.2]{BHJ}.\n\n\\begin{de}\\label{rramamra}\nLet $(\\mathcal{X},\\mathcal{L})$ be a normal semiample test configuration and $d\\in\\mathbb{Z}_{>0}$. Then let $(\\mathcal{X}^{(d)},\\mathcal{L}^{(d)})$ be the normalized base change of $(\\mathcal{X},\\mathcal{L})$ via the $d$-th power map $\\mathbb{A}^1\\ni t\\mapsto t^d\\in\\mathbb{A}^1$. If $\\phi\\in\\mathcal{H}^{\\mathrm{NA}}(L)$ is represented by $(\\mathcal{X},\\mathcal{L})$, we denote by $\\phi^{(d)}$ the metric represented by $(\\mathcal{X}^{(d)},\\mathcal{L}^{(d)})$.\n\nIt is well-known that $d\\,M^{\\mathrm{NA}}_{\\Delta}(\\phi)=M^{\\mathrm{NA}}_{\\Delta}(\\phi_d)$ and $d\\,\\mathrm{DF}_{\\Delta}(\\phi)\\ge\\mathrm{DF}_{\\Delta}(\\phi_d)$ for any $d\\in\\mathbb{Z}_{>0}$. See the details in \\cite[Proposition 7.16]{BHJ}.\n\\end{de}\n \n\nIn this paper, we study K-stability of algebraic fiber spaces.\n\n\\begin{de}\\label{algfib}\\label{algfibk}\nA {\\it polarized algebraic fiber space pair} $f:(X,\\Delta,H)\\to (B,L)$ is a surjective morphism of polarized normal pairs with connected geometric fibers such that $H$ is an $f$-ample line bundle.\nLet $f:(X,\\Delta,H)\\to (C,L)$ be a polarized algebraic fiber space pair. Then $(X,\\Delta,H)$ is {\\it adiabatically K-stable over} $(C,L)$ if $(X,\\epsilon H+f^*L)$ is K-stable for sufficiently small $\\epsilon>0$. Adiabatic K-semistability and K-polystability is defined in a similar way. Furthermore, $(X,\\Delta,H)$ is\n\n\\begin{itemize}\n\\item {\\it uniformly adiabatically K-stable over} $(C,L)$ if there exist positive constants $\\delta>0$ and $\\epsilon_0>0$ such that if $0<\\epsilon<\\epsilon_0$ and $\\epsilon H+f^*L$ is ample on $X$, $\\mathrm{DF}_{\\Delta}(\\mathcal{X},\\mathcal{M})\\ge \\delta J^{\\mathrm{NA}}(\\mathcal{X},\\mathcal{M})$ for any semiample test configuration $(\\mathcal{X},\\mathcal{M})$ for $(X,\\epsilon H+f^*L)$,\n\\item {\\it adiabatically K-unstable over} $(C,L)$ if for sufficiently small $\\epsilon>0$, there exists a semiample test configuration $(\\mathcal{X},\\mathcal{M})$ for $(X,\\epsilon H+f^*L)$ such that $\\mathrm{DF}_{\\Delta}(\\mathcal{X},\\mathcal{M})<0$.\n\\end{itemize}\n\nWe call $(X,H)$ {\\it has cscK metrics adiabatically} if $X$ is smooth, $\\Delta=0$ and $(X,\\delta H+f^*L)$ has a cscK metric for sufficiently small $\\delta>0$. Here, we will denote $H+f^*L$ by $H+L$ for simplicity. See the detail in Notation after \\cite[Remark 2.17]{Hat2}.\n\\end{de}\n\nOn the other hand, we need to recall the geometric invariant theory (GIT).\n\\begin{de}\\label{degit}\nLet $(X,L)$ be a polarized variety and $G$ be a reductive algebraic group. Here, we do not assume that $X$ is proper. We define $G$-linearization of a $\\mathbb{Q}$-line bundle where $G$ is an algebraic group (cf., \\cite[Definition 1.6]{GIT}) as follows. Let $X$ be a proper scheme and $G$ act on $X$. A $\\mathbb{Q}$-line bundle $L$ on $X$ is {\\it $G$-linearized} if $mL$ has a $G$-linearization for sufficiently divisible $m\\in\\mathbb{Z}_{>0}$. If $(X,L)$ is a polarized scheme, an algebraic group $G$ act on $X$ and $L$ is $G$-linearized, then we call that $G$ acts on $(X,L)$.\n\nSuppose that $G$ acts on $(X,L)$. For any $s\\in H^0(X,mL)$, we define $X_s=\\{x\\in X|s(x)\\ne0\\}$. If $r\\in\\mathbb{Z}$ satisfies that $rL$ is $\\mathbb{Z}$-Cartier, then $R(X,rL)\\mathrel{\\mathop:}= \\bigoplus_{m\\ge0}H^0(X,mrL)$ has the induced $G$-action. $x\\in X$ is {\\it GIT-semistable} (resp., {\\it GIT-polystable; GIT-stable}) if there exist $m>0$ and a non-constant $G$-invariant function $f\\in H^0(X,mrL)$ such that $X_f$ is affine and $x\\in X_f$ (resp., $G\\cdot x$ is closed in $X_f$; $x$ is GIT-polystable and its stabilizer group is finite). See the details in \\cite[Definition 1.6]{GIT}.\n\nFurther, suppose that $X$ is proper. A {\\it one-parameter subgroup} (1-PS) $\\lambda$ of $G$ is an algebraic group homomorphism $\\lambda:\\mathbb{G}_m\\to G$. The {\\it Hilbert-Mumford weight} ({\\it GIT-weight}) $\\mu^L(x,\\lambda)$ of $\\lambda$ and $x\\in X$ defined as follows. Let $y=\\lim_{q\\to0}\\lambda(q)\\cdot x$ and $\\mu^L(x,\\lambda)$ is the weight of $L|_{y}$. Then, $x\\in X$ is GIT-(semi)stable if and only if $\\mu^L(x,\\lambda)>0$ (resp., $\\ge0$) for any 1-PS $\\lambda$ (cf., \\cite[Theorem 2.1]{GIT}). See the detail in \\cite{GIT}.\n\nLet $(V,M)$ be a polarized variety. First, suppose that $M$ is very ample and $V\\hookrightarrow \\mathbb{P}^N$ is the corresponding embedding. Then, $(V,M)$ is {\\it Chow-(semi)stable} if its Chow form is GIT-(semi)stable in the Chow variety with the natural polarization with respect to the natural $SL(N+1)$-action. Secondly, suppose that $M$ is a $\\mathbb{Q}$-ample line bundle. Then, $(V,M)$ is {\\it asymptotically Chow-(semi)stable} if $(V,M^n)$ is Chow-(semi)stable for $n\\gg0$. See the detail and the definition of asymptotic Hilbert-stability in \\cite{M} or \\cite{RT07}.\n\\end{de}\n\nNext, let us recall the definition of the log CM line bundle (cf., \\cite{FR}) as follows. \n\\begin{de}\\label{logcmd}\nLet $f:X\\to B$ be a flat and proper morphism with a generic geometric integral fiber of relative dimension $n\\ge 1$ and $L$ be an $f$-ample line bundle on $X$ such that $R^if_*L=0$ for $i>0$. Let also\n\\[\n\\mathrm{det}(f_*L^{\\otimes m})=\\bigotimes_{j=0}^{n+1}\\mathcal{M}_j^{\\otimes\\binom{m}{j}}\n\\]\nbe the Knudsen-Mumford expansion \\cite[Theorem 4]{KM}. The {\\it Hilbert bundle} of $f$ is the line bundle \n\\[\n\\mathrm{Hilb}(X,L^{\\otimes r},k)=\\mathrm{det}(f_*L^{\\otimes kr})^{\\otimes p(r)}\\otimes \\mathrm{det}(f_*L^{\\otimes r})^{\\otimes -kp(kr)}\n\\]\nfor $k, r>0$ where $p(r)=$ rank $f_*L^{\\otimes r}$. The {\\it Chow bundle} of $f$ is the $\\mathbb{Q}$-line bundle\n\\[\n\\mathrm{Chow}(X,L)=\\mathcal{M}_{n+1}\\otimes\\mathrm{det}(f_*L)^{\\otimes-\\frac{a_0(n+1)!}{p(1)}},\n\\]\nwhere $p(m)=a_0m^n+a_1m^{n-1}+O(m^{n-2})$. In \\cite{FR}, $\\mathrm{Chow}(X,L)^{\\otimes p(1)}$ is called the Cornalba-Harris line bundle. It is easy to check that the Hibert bundle and the Chow bundle do not change when we replace $L$ by $L\\otimes f^*\\sigma$ for any line bundle $\\sigma$ on $B$ (cf., \\cite{FR}). Let $\\mu=\\frac{2a_1}{a_0}$. Then, the {\\it CM line bundle} of $f:(X,L)\\to B$ is\n\\[\n\\lambda_{\\mathrm{CM}}(X,L)=\\mathcal{M}_{n+1}^{\\otimes \\mu+n(n+1)}\\otimes \\mathcal{M}_n^{\\otimes -2(n+1)}.\n\\]\n\nLet $D$ be a relative Cartier divisor of $X$ over $B$ such that $R^i(f|_{D})_*L|_{D}=0$ for $i>0$. We define the following line bundle\n\\[\n\\mathrm{RelHilb}_{D}(X,L^{\\otimes r},k)=\\mathrm{det}((f|_{D})_*L|_{D}^{\\otimes kr})^{\\otimes p(r)}\\otimes \\mathrm{det}(f_*L^{\\otimes r})^{\\otimes -kq(kr)}\n\\]\nfor $k, r>0$ where $q(r)=$ rank $(f|_{D})_*L|_{D}^{\\otimes r}$. Let also\n\\[\n\\mathrm{det}((f|_{D})_*L|_{D}^{\\otimes m})=\\bigotimes_{j=0}^{n}\\mathcal{N}_j^{\\otimes \\binom{m}{j}}\n\\]\nbe the Knudsen-Mumford expansion. Let $\\hat{a_0}$ be the rational number such that $q(m)=\\hat{a_0}m^{n-1}+O(m^{n-2})$ and $\\hat{\\mu}=\\frac{\\hat{a_0}}{a_0}$. Then, the following line bundle \n\\[\n\\mathcal{N}_{n}^{\\otimes (n+1)}\\otimes \\mathcal{M}_{n+1}^{\\otimes -\\hat{\\mu}}\n\\]\nis the leading order term of $\\mathrm{RelHilb}_{D}(X,L^{\\otimes r},k)$ for $k\\gg r\\gg 0$. More precisely, we can prove that there exist line bundles $\\epsilon_{i,j}$ on $B$ such that\n\\[\n\\mathrm{RelHilb}_{D}(X,L^{\\otimes r},k)=(\\mathcal{N}_{n}^{\\otimes (n+1)}\\otimes \\mathcal{M}_{n+1}^{\\otimes -\\hat{\\mu}})^{\\otimes \\frac{a_0}{(n+1)!}k^{2n}r^n}\\otimes \\bigotimes_{0\\le i\\le 2n-1,0\\le j\\le n-1}\\epsilon_{i,j}^{\\otimes k^ir^j}\n\\]\nas \\cite[Proposition 3.1]{FR}. Then, we define the {\\it log CM line bundle}  of $f:(X,D,L)\\to B$ as\n\\[\n\\lambda_{\\mathrm{CM}}(X,D,L)=\\lambda_{\\mathrm{CM}}(X,L)\\otimes (\\mathcal{N}_{n}^{\\otimes (n+1)}\\otimes \\mathcal{M}_{n+1}^{\\otimes -\\hat{\\mu}}).\n\\]\n\\end{de}\n\n\\begin{lem}[cf., {\\cite[Lemma 2.5, Proposition 3.1]{FR}}]\\label{logcml}\nNotations as in Definition \\ref{logcmd}. Suppose also that a reductive algebraic group $G$ acts on $f:(X,D,L)\\to B$. Then, $\\mathcal{M}_j$ and $\\mathcal{N}_i$ are $G$-linealized. Further, suppose that $G=\\mathbb{G}_m$ and $b\\in B$ is a fixed point. Write the weight of $H^0(D_b,L|_{D_b}^{\\otimes k})$ as a polynomial in k with coefficients (cf., \\cite[Theorem 3.1]{BHJ})\n\\[\n\\hat{w}(k)=\\hat{b_0}k^n+O(k^{n-1}).\n\\]\nThen, the weight of $\\mathcal{N}_{n}^{\\otimes (n+1)}\\otimes \\mathcal{M}_{n+1}^{\\otimes -\\hat{\\mu}}|_b$ is\n\\[\n\\frac{(n+1)!}{a_0}(\\hat{b_0}a_0-\\hat{a_0}b_0).\n\\]\n\\end{lem}\nThe proof is straightforward (cf., \\cite{FR}). Thus, we can conclude that the weight of $\\lambda_{\\mathrm{CM}}(X,D,L)|b$ is a positive multiple of the log Donaldson-Futaki invariant (cf., \\cite[Definition 3.25]{De2}). More precisely,\n\\begin{prop}\\label{cml}\nNotations as in Definition \\ref{logcml}. Write the weight of $H^0(X_b,L_b^{\\otimes k})$ as a polynomial in k with coefficients \n\\[\nw(k)=b_0k^{n+1}+b_1k^n+O(k^{n-2}).\n\\]\nThen, the weight of $\\lambda_{\\mathrm{CM}}(X,D,L)|b$ is\n\\[\n2\\frac{(n+1)!}{a_0}(b_1a_0-a_1b_0)+\\frac{(n+1)!}{a_0}(\\hat{b_0}a_0-\\hat{a_0}b_0).\n\\]\n\\end{prop}\n\nWe will apply this proposition in Example \\ref{lcbaseext}.\n\nOn the other hand, recall the definition of the log canonical threshold and the $\\alpha$-invariant. We also recall the $\\delta$-invariant in \\cite{FO}.\n\\begin{de}\nLet $(X,B)$ be a normal pair and $D$ be an effective $\\mathbb{Q}$-Cartier divisor. Suppose that there exists $l\\in\\mathbb{Q}$ such that $(X,B+lD)$ is sublc. Then the {\\it log canonical threshold} of $(X,B)$ with respect to $D$ is\n\\[\n\\mathrm{lct}_{(X,B)}(D)=\\inf\\{r\\in\\mathbb{Q}|(X,B+rD) \\,\\mathrm{is}\\,\\mathrm{sublc}\\}=\\min_v\\frac{A_{(X,B)}(v)}{v(D)}\n\\]\nwhere $v$ runs over the divisorial valuations such that $v(D)\\ne0$.\n\\end{de}\n\n\\begin{de}\\label{defalpha}\nLet $(X,B,L)$ be a polarized klt pair. Then the {\\it alpha invariant} of $(X,B)$ with respect to $L$ is\n\\[\n\\alpha_{(X,B)}(L)=\\inf_D \\mathrm{lct}_{(X,B)}(D)\n\\] where $D$ runs over the elements of $|L|_{\\mathbb{Q}}$. Here, $|L|_{\\mathbb{Q}}$ is the set of effective $\\mathbb{Q}$-Cartier $\\mathbb{Q}$-divisors $D$ such that $D\\sim_{\\mathbb{Q}}L$.\n\\end{de}\n\n\\begin{de}\\label{defdelta}\nLet $(X,B,L)$ be a polarized klt pair. Suppose first that $L$ is a line bundle. $D\\in |L|_{\\mathbb{Q}}$ is a divisor of {\\it $m$-basis type} if there exists a basis $\\{f_j\\}_{j=1}^{h^0(X,mL)}\\subset H^0(X,mL)$ such that \n\\[\nD=(mh^0(X,mL))^{-1}\\sum_{j=1}^{h^0(X,mL)}\\mathrm{div}(f_j).\n\\]\nThen, we set\n\\[\n\\delta^m_{(X,B)}(L)=\\inf_D\\mathrm{lct}_{(X,B)}(D)\n\\]\nwhere $D\\in |L|_{\\mathbb{Q}}$ runs over the divisors of $m$-basis type. $|L|_{m-\\mathrm{basis}}$ denotes the set of all $m$-basis type divisor $D$ with respect to $L$. By the result of \\cite{BlJ}, $\\{\\delta^m_{(X,B)}(L)\\}_{m=1}^{\\infty}$ converges and let the {\\it delta invariant} of $(X,B)$ with respect to $L$ be\n\\[\n\\delta_{(X,B)}(L)=\\lim_{m\\to\\infty}\\delta^m_{(X,B)}(L).\n\\]\nMoreover, thanks to \\cite{BlJ}, we can extend the definition for ample $\\mathbb{Q}$-line bundles $L$ as\n\\[\n\\delta_{(X,B)}(L)=\\inf_v\\frac{A_{(X,B)}(v)}{S_{L}(v)}\n\\]\nwhere $v$ runs over the divisorial valuations on $X$ and $S_{L}(v)=\\mathrm{vol}(L)^{-1}\\int_0^{\\infty}\\mathrm{vol}(L-xv)dx$. See the details in \\cite{BlJ}.\n\\end{de}\n\n$\\alpha$ and $\\delta$-invariants are important in K-stability in the following sense. \n\n\\begin{thm}[{\\cite[Proposition 9.16]{BHJ}}, \\cite{BlJ}]\nLet $(X,B,L)$ be a klt polarized pair. Then the following hold.\n\\begin{enumerate}\n\\item $\\alpha_{(X,B)}(L)>0$ and \\[\nH^{\\mathrm{NA}}_{B}(\\mathcal{X},\\mathcal{L})\\ge\\alpha_{(X,B)}(L)I^{\\mathrm{NA}}(\\mathcal{X},\\mathcal{L})\n\\]\nfor any semiample normal test configuration for $(X,L)$,\n\\item Suppose that $K_X+B=-L$. Then, $(X,B,L)$ is uniformly K-stable (resp., K-semistable) if and only if $\\delta_{(X,B)}(L)>1$ (resp., $\\ge1$).\n\\end{enumerate}\n\\end{thm}\n  \n \n\\section{K-stability of a fibration via the twisted canonical class of the base}\n \nIn this section, we consider some sufficiency conditions of adiabatic K-stability of algebraic fiber spaces. Dervan and Sektnan \\cite{DS4} introduced their fibration stability and conjectured that this algebraic condition implies the existence of optimal symplectic connections as K-stability and YTD conjecture. Taking \\cite[Theorems 1.1, 1.2]{DS2} into account, it is also natural to conjecture that fibration stability (or $\\mathfrak{f}$-stability) and twisted K-stability of base spaces imply adiabatic K-stability of fibrations over curves also when the fibration has a singular fiber. However, this conjecture does not hold even when the general fiber is Calabi-Yau as we will mention in Corollary \\ref{bdbd} below. On the other hand, Fine \\cite{Fine} introduced the twisted canonical class of the base as we will define below and he gave a sufficiency condition of existence of cscK metrics on smooth fibrations adiabatically. We study the relationship of his twisted canonical class with adiabatic K-stability in Calabi-Yau and canonically polarized fibrations as we stated in \\S1. Finally, we obtain the main result \\ref{j2e} in this section. This is a generalization of a result of Jian-Shi-Song \\cite[Theorem 1.2]{JSS}.\n  \nFirst, we introduce the following invariant of polarized algebraic fiber spaces,\n \n\\begin{de}\\label{kappainv}\nLet $\\pi:(X,\\Delta,H)\\to (B,L)$ be a polarized algebraic fiber space pair. Suppose that $n=\\mathrm{dim}\\, B$ and $m=\\mathrm{rel.dim}\\, \\pi$. Then {\\it the twisted canonical class} of $B$ with respect to $\\pi$ is the divisor class on $B$ as follows:\n\\[\n\\kappa_{(X,\\Delta,H)\/(B,L)}=\\frac{\\pi_*\\left[\\left( (m+1)K_{(X,\\Delta)}-m\\frac{K_{(X_b,\\Delta_b)}\\cdot H_b^{m-1}}{H_b^m}H\\right)\\cdot H^m\\right]}{(m+1)H_b^m},\n\\]\nwhere $b\\in B$ is a general point. We will denote the twisted canonical class of $B$ by $\\kappa_B=\\kappa_{(X,\\Delta,H)\/(B,L)}$ simply.\n\\end{de}\n\nSuppose that $\\kappa_B$ as above is nef (resp. ample) then $$\\left((m+1)K_{(X,\\Delta)}-m\\frac{K_{(X_b,\\Delta_b)}\\cdot H_b^{m-1}}{H_b^m}H\\right)\\cdot H^m\\cdot L^{n-1}\\ge 0 \\quad (\\mathrm{resp.}\\, >0).$$ On the other hand, we can see that\n\\begin{align*}\nS(X_b,\\Delta_b,H_b)-S(X,\\Delta,H+jL)&=\\frac{j^{-1}n}{(m+1)H^m\\cdot L^n} \\left((m+1)K_{(X,\\Delta)}-m\\frac{K_{(X_b,\\Delta_b)}\\cdot H_b^{m-1}}{H_b^m}H\\right)\\\\\n&\\cdot H^m\\cdot L^{n-1}+O(j^{-2}).\n\\end{align*}\n We can conclude that nefness of $\\kappa_B$ tells us that $S(X_b,\\Delta_B,H_b)-S(X,\\Delta,H+jL)\\ge 0$ for $j\\gg 0$. We summarize the argument as follows,\n \\begin{prop}\n Notations as above. If $\\kappa_B$ is nef (resp., ample), then\n \\[\n S(X_b,\\Delta_b,H_b)-S(X,\\Delta,H+jL)\\ge0, \\,(\\mathrm{resp.,} \\, >0)\n \\]\n for $j\\gg0$. If $B$ is a curve and $H$ is ample, then the converse holds.\n \\end{prop}\n \nLet $\\pi: (X,H)\\to (B,L)$ be a smooth fibration of polarized smooth varieties, $\\mathrm{dim}\\,B=n$ and $\\mathrm{rel.dim}\\,\\pi=m$. Fine used the Grothendieck-Riemann-Roch theorem \\cite[Theorem 15.2]{F} and introduced the rationally equivalent class of divisor $\\kappa_B$ in \\cite{Fine} as follows, (cf., \\cite[(2.4)]{FR})\n\\begin{align*}\n\\kappa_B&=\\frac{\\mathrm{c}_1(L_\\mathrm{CM})}{(m+1)\\pi_*(\\mathrm{c}_1(H)^m)}-\\mathrm{c}_1(B) \\\\\n&=\\frac{\\pi_*\\left[\\left( (m+1)K_{X\/B}-m\\frac{K_{X_b}\\cdot H_b^{m-1}}{H_b^m}H\\right)\\cdot H^m\\right]}{(m+1)H_b^m}+K_B \\\\\n&\\equiv\\frac{\\pi_*\\left[\\left( (m+1)K_{X}-m\\frac{K_{X_b}\\cdot H_b^{m-1}}{H_b^m}H\\right)\\cdot H^m\\right]}{(m+1)H_b^m},\n\\end{align*}\nwhere $b\\in B$ is a general point and $L_\\mathrm{CM}=\\lambda_{\\mathrm{CM}}(X,H)$ is the CM line bundle on $B$. \n\nNext, we see that if the twisted canonical class is positive, then an algebraic fiber space is adiabatically K-stable under some conditions. Recall the following fact,\n \n\\begin{thm}[cf., {\\cite[Theorem 1.1]{JSS}}, {\\cite[Theorem 1.1]{Sj}}, {\\cite[Corollary 1.3]{S}}, {\\cite[Theorem 8.15]{Hat}}]\\label{minimal}\nLet $(X,\\Delta)$ be a klt minimal pair i.e., $K_X+\\Delta$ is nef. Suppose that $f:X\\to B$ is an algebraic fiber space such that there exists a $\\mathbb{Q}$-ample line bundle $L$ on $B$ that satisfies $K_X+\\Delta\\sim_{\\mathbb{Q}}f^*L $. Then $f:(X,\\Delta,H)\\to (B,L)$ is uniformly adiabatically K-stable with respect to any relative polarization $H$ on $X$ over $B$. Moreover, if $\\Delta=0$, $X$ is smooth, then $(X,H)$ has cscK metrics adiabatically.\n\\end{thm}\n\nSuppose that $f:(X,\\Delta,H)\\to (B,L)$ is a polarized algebraic fiber space. If $K_X+\\Delta=f^*L_0$ where $L_0$ is a line bundle on $B$, then\n\\[\n\\kappa_B\\equiv L_0.\n\\]\nIt is well-known. In fact, for any integral curve $C$ of $B$,\n\\[\n\\kappa_B\\cdot C=\\frac{1}{(H_b)^{m}}C\\cdot L_0\\cdot \\pi_*(H^m)=L_0\\cdot C\n\\]\nwhere $b\\in B$ is a general point. Therefore, we can interpret Theorem \\ref{minimal} as follows:\n\\begin{cor}\\label{min2}\nNotations as above. If $\\kappa_B$ is ample, then $(X,\\Delta,H)$ is uniformly adiabatically K-stable over $(B,\\kappa_B)$.\n\\end{cor}\n \nTherefore, we conclude that positivity of $\\kappa_B$ implies adiabatically K-stability of Iitaka fibrations of klt minimal models.\n\n\\begin{rem}\\label{itso}\nNote that $\\kappa_B\\equiv K_B+D+M$ where $D$ is the discriminant and $M$ is the moduli divisor (cf., \\S\\ref{unstadiab}, Definition \\ref{ambro model}). Thus, $\\kappa_B$ is ample if and only if the log-twisted pair $(B,D,M)$ is canonically polarized. Then $(B,D,M,\\kappa_B)$ is log-twisted K-stable (see \\S\\S \\ref{ltp} for the definition and Proposition \\ref{bhj9394}).\n\\end{rem}\n\n\\begin{rem}\nWe remark that Theorem \\ref{minimal} does not hold for lc pairs. Let $X=\\mathbb{P}_{\\mathbb{P}^1}(\\mathcal{O}\\oplus\\mathcal{O}(e))$ for $e>0$, $C_0$ be a section such that $C_0^2=-e$ and $f$ be the fiber class. Recall that $L=mC_0+nf$ is ample iff $n>me$ (cf., \\cite[V Theorem 2.17]{Ha}). Suppose that $n>me$. Note that\n\\[\n2\\frac{L\\cdot f}{L^2}L-f=C(2C_0+ef),\n\\]\nwhere $C>0$ is a positive constant. Since $(2C_0+ef)\\cdot C_0=-e$, $(X,L)$ is not slope $\\mathrm{J}^f$-semistable by \\cite[Corollary B]{Hat} and $C_0$ is a destabilizing subvariety. By \\cite[V Corollary 2.11]{Ha}, $K_X\\equiv -2C_0-(e+2)f$. If an ample divisor $H\\equiv C_0+(e+3)f$ is general, $(X,C_0+H)$ is plt (cf., \\cite[Definition 2.34]{KoMo}) and $K_X+C_0+H\\equiv f$. If $(\\mathcal{X},\\mathcal{L})$ is the deformation to the normal cone of $C_0$ for $(X,L)$, then $H_{(C_0+H)}^{\\mathrm{NA}}(\\mathcal{X},\\mathcal{L})=0$ and $M_{(C_0+H)}^{\\mathrm{NA}}(\\mathcal{X},\\mathcal{L})=(\\mathcal{J}^f)^{\\mathrm{NA}}(\\mathcal{X},\\mathcal{L})$. Therefore, $(X,C_0+H,L)$ is K-unstable for $n\\gg 0$. This means that $(X,C_0+H,C_0)$ is adiabatically K-unstable over $\\mathbb{P}^1$. Note also that the base curve is log-twisted general type in the sense of Remark \\ref{itso} and hence log-twisted K-stable. Thus we obtain the following fact.\n\\begin{prop}\\label{counter}\nThere exists a polarized lc minimal pair $(X,\\Delta,L)\\to \\mathbb{P}^1$ such that $(X,\\Delta,\\epsilon L+K_X+\\Delta)$ is slope K-unstable for sufficiently small $\\epsilon>0$ and $\\kappa_{\\mathbb{P}^1}$ is ample. \n\\end{prop}\n\nFor any normal surface $X$, let $f$ be a nef $\\mathbb{Q}$-Cartier divisor such that $f^2=0$ and $L$ be an ample divisor. Then, for any curve $C$,\n\\[\n2(f\\cdot (L+mf))((L+mf)\\cdot C)-(L+mf)^2(f\\cdot C)=2(f\\cdot L)(L\\cdot C)-(L^2)(f\\cdot C).\n\\]\nTherefore, $\\mathrm{J}^f$-stability of $(X,L)$ coincides with that of $(X,L+mf)$ for $m>0$. We can conclude that:\n\\begin{prop}\nIf a polarized lc minimal surface pair $(X,\\Delta,H)$ that is $\\mathrm{J}^{K_X+\\Delta}$-stable (resp., $\\mathrm{J}^{K_X+\\Delta}$-semistable) satisfies that $(K_X+\\Delta)^2=0$, then $(X,\\Delta,\\epsilon H+K_X+\\Delta)$ is K-stable (resp., K-semistable) for $\\epsilon>0$.\n\\end{prop}\n\n\\end{rem}\n\n \nOn the other hand, we can relate the twisted canonical class to the result of Jian-Shi-Song \\cite[Theorem 1.2]{JSS} and generalize their theorem to singular varieties (see also Corollary \\ref{j2e}).\n \n\\begin{thm}[cf., \\cite{JSS}]\\label{JS}\nLet $(X,\\Delta ,H)$ be an $n$-dimensional polarized lc variety and $f:(X,\\Delta ,H)\\to (B,L)$ be a polarized algebraic fiber space pair over a smooth curve $B$. Suppose that $K_{(X,\\Delta)}$ is nef (resp., $K_{(X,\\Delta)}$ is ample or $K_{(X,\\Delta)}$ is nef and $(X,\\Delta)$ is klt) and $H\\equiv K_{(X,\\Delta)}+f^*L_0$ where $L_0$ is a line bundle on $B$. \n\nThen $(X,\\Delta, H)$ is adiabatically K-semistable (resp. uniformly adiabatically K-stable) for sufficiently small $\\delta>0$. Moreover, if $\\Delta=0$, $X$ is smooth and $K_{(X,\\Delta)}$ is nef, then $(X,H)$ has cscK metrics adiabatically.\n\\end{thm}\nWe need the following lemma.\n\n\\begin{lem}\\label{uuuse}\nLet $f:(X,\\Delta ,H)\\to (B,L)$ be an $n$-dimensional polarized algebraic fiber space pair over a smooth curve $B$. Suppose that $K_{(X,\\Delta)}$ is $f$-ample and $H\\equiv K_{(X,\\Delta)}+f^*L_0$ where $L_0$ is a line bundle on $B$. Then the following are equivalent.\n\\begin{enumerate}\n\\item $\\kappa_B=\\kappa_{(X,\\Delta ,H)\/(B,L)}$\nis nef (resp., ample),\n\\item\n\\[\nn\\frac{K_{(X,\\Delta)}\\cdot (L+\\delta H)^{n-1}}{(L+\\delta H)^n}(L+\\delta H)-(n-1)K_{(X,\\Delta)}\n\\]\nis nef (resp., ample),\n\\item $(K_X+\\Delta)^n\\ge0$ (resp., $>0$).\n\\end{enumerate}\n Furthermore, if $\\kappa_B$ is ample, then there exists a positive constant $a>0$ such that\n\\[\nn\\frac{K_{(X,\\Delta)}\\cdot (L+\\delta H)^{n-1}}{(L+\\delta H)^n}(L+\\delta H)-(n-1)K_{(X,\\Delta)}-a(L+\\delta H)\n\\] is ample for sufficiently small $\\delta>0$.\n\\end{lem}\n\n\\begin{proof}\nSince \\[\n\\kappa_B=\\frac{\\pi_*(K_{(X,\\Delta)}^{m+1})}{(m+1)(K_{(X_b,\\Delta_b)})^m}\n\\] holds, $(1)\\Leftrightarrow (3)$ follows. A direct computation shows that\n\\begin{align*}\n-S(X,\\Delta,L+\\delta H)=&n\\frac{\\delta^{n-1}K_{(X,\\Delta)}\\cdot H^{n-1}+(n-1)\\delta^{n-2}K_{(X,\\Delta)}\\cdot H^{n-2}\\cdot f^*L}{\\delta^nH^n+n\\delta^{n-1}(H^{n-1}\\cdot L)} \\\\\n=&\\frac{n-1}{\\delta}+(\\delta H^n+n H^{n-1}\\cdot L)^{-1}H^n\\left(\\frac{K_{(X,\\Delta)}\\cdot H^{n-1}}{H^n}-\\frac{n-1}{n}\\right) \\\\\n=&\\frac{n-1}{\\delta}+n^{-1}(\\delta H^n+n H^{n-1}\\cdot L)^{-1}H^n(S(X_b,\\Delta_b,H_b)-S(X,\\Delta,H)),\n\\end{align*}\nwhere $S(X_b,\\Delta_b,H_b)=-n+1$. Let $c_\\delta=-S(X,\\Delta,L+\\delta H)$. Then, we have \n\\begin{align}\nc_\\delta& (f^*L+\\delta H)-(n-1)K_{(X,\\Delta)}=c_\\delta f^*L+\\delta c_\\delta (H-K_{(X,\\Delta)}) \\nonumber \\\\\n&+n^{-1}(\\delta H^n+n H^{n-1}\\cdot L)^{-1} H^n(S(X_b,\\Delta_b,H_b)-S(X,\\Delta,H))K_{(X,\\Delta)} \\nonumber \\\\\n=&c_\\delta f^*L +\\delta c_\\delta f^*L_0+n^{-1}(\\delta H^n+n H^{n-1}\\cdot L)^{-1}\\cdot H^n(S(X_b,\\Delta_b,H_b)-S(X,\\Delta,H))(H-f^*L_0)\\nonumber \\\\\n=&f^*(c_\\delta L+O(\\delta^0)L_0)+n^{-1}(\\delta H^n+n H^{n-1}\\cdot L)^{-1}(H^n)(S(X_b,\\Delta_b,H_b)-S(X,\\Delta,H))H.\\nonumber\n\\end{align}\nTherefore, $c_\\delta (f^*L+\\delta H)-(n-1)K_{(X,\\Delta)}$ is nef (resp., ample) for sufficiently small $\\delta>0$ if and only if $S(X_b,\\Delta_b,H_b)-S(X,\\Delta, H)\\ge 0$ (resp., $>0$). Recall that $\\kappa_B$ is nef (resp. ample) if and only if $S(X_b,\\Delta_b,H_b)-S(X,\\Delta, H+jL)\\ge 0$ (resp., $>0$) for sufficiently large $j>0$. On the other hand, we can see that\n\\[\nS(X_b,\\Delta_b,H_b)-S(X,\\Delta,H+jL)=\\frac{j^{-1}}{n(H^{n-1}\\cdot L)}  \\left(nK_{(X,\\Delta)}-(n-1)\\frac{K_{(X_b,\\Delta_b)}\\cdot H_b^{n-2}}{H_b^{n-1}}H\\right)\\cdot H^{n-1}.\n\\]\n Therefore, $S(X_b,\\Delta_b,H_b)-S(X,\\Delta, H+jL)$ is a monotone function in $j>0$. Hence, $\\kappa_B$ is nef (resp. ample) if and only if $S(X_b,\\Delta_b,H_b)-S(X,\\Delta, H)\\ge 0$ (resp., $>0$). Thus, the lemma holds.\n\\end{proof}\n \n \\begin{proof}[Proof of Theorem \\ref{JS}]\n If $K_{(X,\\Delta)}$ is big and nef, then $\\kappa_B$ is ample. On the other hand, suppose that $K_{(X,\\Delta)}^n=0$. Then, we have $\\kappa_B\\equiv0$.  \n Thus, the theorem immediately follows from the criterion for J-stability of Song-Weinkove \\cite[Theorem 1.1]{SW} (cf., \\cite[Theorem 8.2, Theorem 8.12]{Hat}). The last assertion follows from \\cite[Theorem 6.11]{Li}.\n\\end{proof}\n \nWe remark that \\cite[Theorem 1.2]{JSS} treats the case when $K_X$ is ample in Theorem \\ref{JS}.\nNext, we will prove that Theorem \\ref{JS} also holds for lc surface pair such that $\\kappa_B$ is ample by \\cite[Corollary 6.4]{Hat} even when $K_X+\\Delta$ is not ample. More precisely, we prove the generalization of Theorem \\ref{JS} assuming the following conjecture,\n\n\\begin{conj}\\label{swconj}\nLet $(X,L)$ be an $n$-dimensional polarized normal variety and $H$ be a pseudoeffective line bundle. If \n\\[\nn\\frac{H\\cdot L^{n-1}}{L^n}L-(n-1)H\n\\]\nis nef, then $(X,L)$ is $\\mathrm{J}^H$-semistable.\n\\end{conj}\n\nWe remark that this conjecture is solved when $n=2$ by the author \\cite[Theorem A]{Hat}.\n \n\n  \\begin{thm}\\label{jne}\nLet $f:(X,\\Delta)\\to B$ be an $n$-dimensional polarized algebraic fiber space pair over a smooth curve $B$. Suppose that $(X,\\Delta)$ is lc and assume Conjecture \\ref{swconj}. Suppose also that $K_X+\\Delta$ is $f$-ample. If $\\kappa_B=\\kappa_{(X,\\Delta,K_X+\\Delta)\/(B,L)}$ is nef (resp., $\\kappa_B$ is ample, or $\\kappa_B$ is nef and $(X,\\Delta)$ is klt), then $(X,\\Delta,K_X+\\Delta)$ is adiabatically K-semistable over $(B,L)$ (resp., $(X,\\Delta,K_X+\\Delta)$ is uniformly adiabatically K-stable), where $L$ is a line bundle of degree $1$ on $B$. Moreover, if $\\Delta=0$, $\\kappa_B$ is nef and $X$ is smooth, $(X,\\epsilon K_X+L)$ has cscK metrics for sufficiently small $\\epsilon>0$.\n\\end{thm}\n\n\\begin{proof}\nFirst, we claim that if $(K_X+\\Delta)^n>0$ (resp., $\\ge0$), then $K_X+\\Delta$ is big (resp., pseudoeffective). Indeed, $K_{X\/B}+\\Delta$ is nef due to \\cite[Theorem 1.11]{Fjn}. Hence, it suffices to show the assertions when $B=\\mathbb{P}^1$. Suppose first that $(K_X+\\Delta)^n>0$. Since and $-f^*K_{\\mathbb{P}^1}$ is nef, we have\n\\[\n((K_{X\/\\mathbb{P}^1}+\\Delta)-n(-f^*K_{\\mathbb{P}^1}))\\cdot (K_{X\/\\mathbb{P}^1}+\\Delta)^{n-1}=(K_X+\\Delta)^n>0.\n\\]\nBy \\cite[Theorem 2.2.15]{Laz}, $K_X+\\Delta$ is big. If $(K_X+\\Delta)^n\\ge0$, note that $K_{X}+\\Delta-\\epsilon f^*K_{\\mathbb{P}^1}$ is big for sufficiently small $\\epsilon>0$ by what we have shown.\nThus, the theorem follows form Lemma \\ref{uuuse} and Conjecture \\ref{swconj} immediately as the proof of Theorem \\ref{JS}. The last assertion follows from \\cite[Theorem 6.11]{Li}.\n\\end{proof}\n\nSince Conjecture \\ref{swconj} holds when $n=2$, we obtain the main result in this section,\n \n \\begin{cor}\\label{j2e}\nLet $f:(X,\\Delta)\\to B$ be a lc algebraic fiber space over a smooth curve $B$. Suppose that $\\mathrm{dim}\\,X=2$ and $K_X+\\Delta$ is $f$-ample. If $\\mathrm{deg}\\,\\kappa_B\\ge 0$ for $(X,K_X+\\Delta)\\to B$, then $(X,\\Delta,K_X+\\Delta)$ is adiabatically K-semistable over $(B,L)$ (if $\\mathrm{deg}\\,\\kappa_B> 0$ or $\\mathrm{deg}\\,\\kappa_B\\ge 0$ and $(X,\\Delta)$ is klt, then it is adiabatically uniformly K-stable), where $L$ is a line bundle of degree $1$ on $B$. Moreover, if $\\mathrm{deg}\\,\\kappa_B\\ge0$, $\\Delta=0$ and $X$ is smooth, $(X,\\epsilon K_X+L)$ has cscK metrics for sufficiently small $\\epsilon>0$.\n\\end{cor}\n\nWe will show that if $\\kappa_B$ is not nef in Corollary \\ref{j2e}, adiabatic K-stability does not hold in general (see Example \\ref{lcbaseext}).\n\n\\section{K-stability of fibrations via log-twisted K-stability}\nAs we saw in the previous section, Calabi-Yau fibrations over curves are adiabatic K-stable if the twisted canonical classes of the bases are nonnegative. It is also natural to consider the case when the twisted canonical class is negative. For example, it is well-known that $\\mathrm{deg}\\,\\kappa_{\\mathbb{P}^1}<0$ for rational elliptic surfaces. Let $f:(X,\\Delta,H)\\to (C,L)$ be a polarized lc-trivial fibration such that $C$ is a curve, i.e., $K_X+\\Delta\\sim_{C,\\mathbb{Q}}0$ and $(X,\\Delta)$ is lc (cf., Definition \\ref{ambro model}). Note that if $\\mathrm{deg}\\,\\kappa_C<0$, then $C=\\mathbb{P}^1$ since the sum of the discriminant divisor and the moduli divisor $(M+B)$ is nef (cf., \\S 7.2).  In this section, we discuss such fibrations and prove Theorem \\ref{dd}.\n\\subsection{Convergence of $\\alpha$ and $\\delta$ invariant}\\label{Weier}\nIn this section, we calculate $\\lim_{\\epsilon\\to 0}\\delta_{X}(\\epsilon H+L)$ when $f:(X,H)\\to (C,L)$ is a polarized algebraic fiber space and $C$ is a curve in Theorem \\ref{ctnd}. Next, we apply this theorem and obtain the sufficiency condition of adiabatic K-stability of such fibrations in Theorem \\ref{ctn3}.\n\n \\begin{de}\\label{dalpha}\nLet $\\pi:(X,\\Delta)\\to C$ be an algebraic fiber space pair over a smooth curve $C$. Suppose that $(X,\\Delta)$ is klt. Let $L$ be a $\\mathbb{Q}$-ample line bundle on $C$ with $\\mathrm{deg}\\, L=s>0$. Then \n\\[\n\\alpha_{(X,\\Delta)}(L)\\mathrel{\\mathop:}= s^{-1}\\inf_{p\\in C}\\mathrm{lct}_{(X,\\Delta)}(\\pi^{-1}(p)).\n\\]\n\\end{de}\n \n\\begin{thm}\\label{ctn}\nLet $\\pi:(X,\\Delta,H)\\to (C,L)$ be an $n$-dimensional polarized algebraic fiber space pair over a smooth curve $C$. Let also $\\alpha_{(X,\\Delta)}(L)=\\alpha$ and suppose that $(X,\\Delta)$ is klt and $\\mathrm{deg}\\, L=1$. For sufficiently small $\\delta>0$, there exists $\\epsilon_0>0$ such that if $0<\\epsilon\\le \\epsilon_0$, then $\\alpha_{(X,\\Delta)}(L+\\epsilon H)\\ge \\alpha-\\delta$. In other words, \n\\[\n\\lim_{\\epsilon\\to 0}\\alpha_{(X,\\Delta)}(L+\\epsilon H)=\\alpha_{(X,\\Delta)}(L).\n\\]\n\\end{thm}\n \n\\begin{proof}\nIt is easy to see that $\\alpha_{(X,\\Delta)}(L+\\epsilon H)\\le\\alpha_{(X,\\Delta)}(L)$ for $\\epsilon>0$. Put $\\lambda=\\alpha-\\delta$ and note that $\\alpha_{(X,\\Delta)}(L+\\epsilon H)\\le \\alpha$ for $\\epsilon>0$. Assume that the theorem fails for some $\\delta$. Then, for sufficiently small $\\epsilon>0$, there exists $D_\\epsilon\\in |L+\\epsilon H|_{\\mathbb{Q}}$ such that $(X,\\Delta+\\lambda D_\\epsilon)$ is not a log canonical pair. Take a log resolution $g:X'\\to X$ of $(X,\\Delta)$ and let $\\pi'=\\pi\\circ g$. Further, we may assume that the reduced structure of any irreducible component of fiber of $\\pi'$ is smooth. Let $\\Delta'$ be a $\\mathbb{Q}$-divisor on $X'$ such that \n\\[\nK_{X'}+\\Delta'=g^*(K_X+\\Delta).\n\\]\nFix a very ample divisor $A$ on $X'$. Let also $D_\\epsilon'=g^*D_\\epsilon$. Then, we can decompose $\\lambda D'_\\epsilon=D_{\\mathrm{vert}}+D_{\\mathrm{hor}}$. Here, $D_{\\mathrm{vert}}$ has a zero dimensional image of $\\pi'$ and $D_{\\mathrm{hor}}$ is flat over $C$. \n\nLet $D_{\\mathrm{vert}}=\\sum_{i=1}^r a_iF_i+B$, where each $F_i$ is a fiber of $\\pi'$, $B$ is effective and $a_i\\ge 0$. We may assume that $B$ contains no fiber. Then, we will show that $B$'s coefficients are small for sufficiently small $\\epsilon>0$. In fact, there exist finitely many reducible fibers of $X$. Let $\\pi'^{-1}(c)=\\sum_{j=0}^k n_jC_j$ be the one of them and $\\sum b_jC_j$ be the restriction of $B$ to the neighborhood of $\\pi'^{-1}(c)$ where $c\\in C$. Here, one of $b_j$ is zero. We may assume that $b_0=0$. Cutting by $(n-2)$-hyperplane sections in $|A|$, we may also assume that $X'$ is a surface. Thanks to Zariski lemma \\cite[Lemma 1]{LX}, $\\mathrm{Span}_{\\mathbb{R}}\\langle C_j \\rangle_{j=1}^k$ with $\\sqrt{-(\\cdot ,\\cdot)}$ is a finite dimensional normed space and hence the norm $\\sqrt{-(\\cdot ,\\cdot)}$ is equivalent to the supremum norm, where $(\\cdot ,\\cdot)$ is the intersection product. Therefore, there exists a constant $\\beta>0$ such that $-(\\sum b_jC_j,\\sum b_jC_j)\\ge \\beta (\\max\\{ b_j\\})^2$. Since $D'_\\epsilon$ is ample,\n\\[\n\\sum_{j=1}^k b_jC_j\\cdot \\left(\\sum_{j=1}^k b_jC_j +D_{\\mathrm{hor}}\\right)=\\sum_{j=1}^k b_jC_j\\cdot (D_{\\mathrm{vert}}+D_{\\mathrm{hor}})>0.\n\\]\nNote that if $m=\\min_jn_j>0$ then $\\frac{\\max\\{ b_j\\}}{m}(\\pi')^{-1}(c)-\\sum_{j=1}^k b_jC_j$ is effective. Thus, we obtain\n\\begin{align*}\n\\frac{\\max\\{ b_j\\}}{m}\\lambda\\epsilon H\\cdot L&=\\lambda\\frac{\\max\\{ b_j\\}}{m}\\pi^{-1}(c)\\cdot D_\\epsilon\\\\\n&=\\frac{\\max\\{ b_j\\}}{m}(\\pi')^{-1}(c)\\cdot D_{\\mathrm{hor}}\\\\\n&\\ge \\sum_{j=1}^k b_jC_j\\cdot D_{\\mathrm{hor}}\\\\\n&>-\\left(\\sum b_jC_j,\\sum b_jC_j\\right)\\\\\n&\\ge\\beta (\\max\\{ b_j\\})^2.\n\\end{align*}\nTherefore, $\\max\\{ b_j\\}\\le \\frac{\\lambda\\epsilon}{m\\beta}H\\cdot L$. The constants $\\beta$ and $m$ depend only on finitely many choices of irreducible components of $(\\pi')^{-1}(c)$. Hence, for any $B$, there exists a constant $M>0$ such that $M\\epsilon\\sum_{j=1}^l(\\pi')^{-1}(c_j)-B$ is effective for some $l$ and $c_j\\in C$. Here, $l$ is the number of singular fibers.\n \nOn the other hand, we will prove that the multiplicity of $D_{\\mathrm{hor}}$ is sufficiently small at every point $P\\in X'$. If $\\gamma$ is a curve that passes through $P$ and is not contained in $D_{\\mathrm{hor}}$, then\n\\[\n\\gamma\\cdot D_{\\mathrm{hor}}\\ge \\mathrm{mult}_P(D_{\\mathrm{hor}}).\n\\] The theorem follows from the next claim.\n\\begin{claim}\nThere exists a positive constant $M>0$ such that for any $D_\\epsilon$, for any closed point $P$ of $X'$ and for any irreducible component $G$ of fiber of $\\pi'$ that contains $P$, there exists a curve $\\gamma$ satisfying the following:\n\\begin{enumerate}\n \\item  $P\\in\\gamma \\subset(\\pi')^{-1}(\\pi'(P))$,\n \\item $\\gamma \\not\\subset D_{\\mathrm{hor}}$\n \\item $\\gamma\\cdot D_{\\mathrm{hor}}\\le M\\epsilon$.\n\\end{enumerate}\n\\end{claim}\nLet us prove the claim. Note that $G$ is smooth as we assumed in the first paragraph of the proof. Since $D_{\\mathrm{hor}}$ is flat over $C$, $G\\not\\subset D_{\\mathrm{hor}}$. Let $\\mu:\\tilde{X}\\to X'$ be the blow up at $P$ and $E$ be the exceptional divisor. By Seshadri's theorem \\cite[Theorem 1.4.14]{Laz}, we can take a positive constant $m_1$ independent from $P$ and $G$ such that $\\mu^*(mA-K_{X'}-G)-(n-1)E$ and $\\mu^*(mA)-E$ are ample for $m\\ge m_1$. Indeed, since there are only finitely many numerical classes of such $G$, we can apply \\cite[Corollary 1.2.24]{Laz}. If $\\tilde{G}=\\mu^{-1}_*G$, then $m\\mu^*A_{|\\tilde{G}}-K_{\\tilde{G}}-E_{|\\tilde{G}}$ is ample. Therefore, by the effective base point freeness theorem \\cite{Ko3}, there exists a constant $m_2\\ge m_1$ independent from $P$ and $G'$ such that $m(m\\mu^*A_{|\\tilde{G}}-E_{|\\tilde{G}})$ is ample and generated by global sections for $m\\ge m_2$. We now fix such $m$. Then there exist $D_1,\\cdots,D_{n-2}\\in |m(m\\mu^*A_{|\\tilde{G}}-E_{|\\tilde{G}})|$ such that $\\gamma'=\\bigcap_{i=1}^{n-2}D_i\\subset \\tilde{G}$ is a smooth curve intersecting with $E$ properly by Bertini's theorem and is not contained in $\\mu^*D_{\\mathrm{hor}}$. Note that\n\\begin{align*}\n\\gamma'\\cdot \\mu^*D_{\\mathrm{hor}}&=m^{2(n-2)}A^{n-2}\\cdot G\\cdot D_{\\mathrm{hor}}\\\\\n&\\le m^{2(n-2)}A^{n-2}\\cdot L\\cdot D_{\\mathrm{hor}}\\\\\n&=m^{2(n-2)}A^{n-2}\\cdot L\\cdot (\\lambda D'_{\\epsilon}-D_{\\mathrm{vert}})\\\\\n&= \\lambda\\epsilon m^{2(n-2)}A^{n-2}\\cdot L\\cdot H.\n\\end{align*}\nThus, there exists a positive constant $m_3$ such that $\\gamma'\\cdot \\mu^*D_{\\mathrm{hor}}\\le m_3\\epsilon$. Therefore, let $\\gamma=\\mu_*\\gamma'$ and we obtain the claim.\n\nNow we can prove the theorem. By the claim, there exists a positive constant $M$ independent from $D'_\\epsilon$ and $\\lambda D'_\\epsilon=\\sum a_iF_i+ D''$, where $D''$ is the effective divisor such that $\\mathrm{mult}_P(D'')\\le M\\epsilon$ for any point $P$ of $X'$. We can take $\\epsilon>0$ so small that $M\\epsilon<\\frac{\\delta}{2}$. Note that if $\\sum a_iF_i\\equiv a\\,(\\pi')^*L$ for $a>0$,\n\\[\naA^{n-1}\\cdot L\\le \\lambda A^{n-1}\\cdot (\\epsilon H+L).\n\\]\nThen we can take $\\epsilon>0$ so small that $a<\\alpha-\\frac{\\delta}{2}$. By the assumption, $(X',\\Delta'+\\sum a_iF_i)$ is subklt. If $\\epsilon$ is so small, $(X',\\Delta'+D''+\\sum a_iF_i)$ is subklt by the same argument of the proof of \\cite[Lemma 8.10]{Ko1}. This is a contradiction and we complete the proof.\n\\end{proof}\n\\begin{rem}\nThere is a fact that the alpha invariant is continuous on the K\\\"{a}hler cone \\cite[Theorem 3.5]{Der}. Due to this theorem, we can extend the alpha invariant continuously to the fiber class.\n\\end{rem}\nAs Theorem \\ref{ctn}, we obtain the following theorem for $\\delta$-invariant.\n\\begin{thm}\\label{ctnd}\nLet $f:(X,\\Delta,H)\\to (B,L)$ be a polarized algebraic fiber space pair. Suppose that $(X,\\Delta)$ is a klt pair, $B$ is a smooth curve and $\\mathrm{deg}\\, L=1$. Then\n\\[\n\\lim_{\\epsilon\\to 0}\\delta_{(X,\\Delta)}(\\epsilon H+L)=2\\alpha_{(X,\\Delta)}(L).\n\\]\n\\end{thm}\n\\begin{proof}\nFirst, let $G=\\epsilon^{-1}$ and we may assume that $L$ is a $\\mathbb{Z}$-divisor since $\\delta$-invariant is independent from numerical equivalence. Thanks to \\cite{BlJ},\n\\[\n\\delta_{(X,\\Delta)}(\\epsilon H+L)=G\\delta_{(X,\\Delta)}(H+GL)=\\lim_{m\\to\\infty}\\inf_{D_m\\in|(H+GL)|_{m-\\mathrm{basis}}}\\mathrm{lct}_{(X,\\Delta)}(G^{-1}D_m).\n\\]\nNext we will estimate the multiplicity of $D_m$ along vertical divisors. Let $\\pi:X'\\to X$ be a log resolution of singularities of $(X,\\Delta)$, $F=f^{-1}(p)$ and $F_0$ be an irreducible component of $\\pi^*F$. Fix $A$ a very ample divisor on $X'$. By the same argument of the proof of \\cite[Lemma 2.2]{FO} and \\cite{BlJ}, if $\\pi^*F=\\sum_{i=0}^r l_i F_i$ and $D'_m=G^{-1}\\pi^*D_m$,\n\\[\n\\mathrm{mult}_{F_0}(D'_m)\\le\\frac{\\sum_{k=1}^\\infty h^0(X',\\pi^*(mH+mGL)-kF_0)}{l_0mGh^0(X,mH+mGL)}.\n\\]\nThe denominator of the right hand side is\n\\[\nl_0mG\\left(\\frac{1}{n!}(mH+mGL)^n+O(m^{n-1})\\right)\\ge \\frac{l_0m^{n+1}G^2}{(n-1)!}(H^{n-1}\\cdot L)+O(m^n).\n\\]\nOn the other hand, there exists an effective divisor $N$ on $X'$ such that\n\\[\nH^0(X',N+P)\\ne 0\n\\]\nfor any numerically trivial divisor $P$ due to \\cite[Lemma 2.2.42]{Laz}. Since $\\pi^*H$ is big, there exists $s\\in\\mathbb{Z}_{\\ge0}$ such that $s\\pi^*H-N$ is big. Therefore,\n\\begin{align*}\nh^0(X',m\\pi^*H+mG\\pi^*L-kF_0)&\\le h^0(X',m\\pi^*H+mG\\pi^*F-kF_0+N)\\\\\n&\\le h^0(X',(m+s)\\pi^*H+mG\\pi^*F-kF_0)\n\\end{align*}\nfor any $G,k$ and for sufficiently divisible $m$ that $mG$ is an integer. Then, it is easy to see that there exists an effective $\\mathbb{Q}$-divisor $E$ such that\n\\[\nE=\\sum_{i=0}^r e_i F_i\n\\]\nand\n\\[\nF_i\\cdot A^{n-2}\\cdot (\\pi^*H+E)=0\n\\]\nfor $i>0$. Here, recall that $\\pi^*F=\\sum_{i=0}^r l_i F_i$ is the irreducible decomposition. We may assume that $s$ is sufficiently divisible that $sE$ is a $\\mathbb{Z}$-divisor. Thus we obtain\n\\[\nh^0(X',(m+s)\\pi^*H+mG\\pi^*F-kF_0)\\le h^0(X',(m+s)(\\pi^*H+E)+mG\\pi^*F-kF_0).\n\\]\nOn the other hand,\n\\[\n\\sum_{i=0}^{l_0-1}h^0(X',(m+s)(\\pi^*H+E)+mG\\pi^*F-(kl_0+i)F_0)\\le l_0h^0(X',(m+s)(\\pi^*H+E)+mG\\pi^*F-kl_0F_0)\n\\]\nand\n\\[\n(m+s)(\\pi^*H+E)+mG\\pi^*F-kl_0F_0=(m+s)(\\pi^*H+E)+(mG-k)\\pi^*F+k\\sum_{i=1}^r l_i F_i.\n\\]\nHere, we claim that\n\\[\nh^0(X',(m+s)(\\pi^*H+E)+mG\\pi^*F-kl_0F_0)=h^0(X',(m+s)(\\pi^*H+E)+(mG-k)\\pi^*F).\n\\]\nIndeed, if $F'=\\sum_{i=1}^r a_i F_i\\ne0$ is effective, then\n\\[\nF_i\\cdot ((m+s)(\\pi^*H+E)+(mG-k)\\pi^*F+F')\\cdot A^{n-2}=F_i\\cdot F'\\cdot A^{n-2}\n\\]\nfor $i>0$ and there exists $i>0$ such that\n\\[\nF_i\\cdot F'\\cdot A^{n-2}<0\n\\]\nby the Zariski lemma \\cite[Lemma 1]{LX}. This means that $F_i$ is contained in the fixed part of $|(m+s)(\\pi^*H+E)+(mG-k)\\pi^*F+F'|$. Thus the claim follows from the induction on the multiplicity of $F'$.\n \nWe have shown that\n\\begin{align*}\n\\sum_{k=1}^\\infty h^0(X',m\\pi^*H+mG\\pi^*L-kF_0)&\\le l_0\\sum_{k=1}^\\infty h^0(X',(m+s)(\\pi^*H+E)+(mG-k)\\pi^*F)\\\\\n&=l_0\\sum_{k=1}^{mG} h^0(X',(m+s)(\\pi^*H+E)+(mG-k)\\pi^*F)\\\\\n&+l_0\\sum_{k=1}^\\infty h^0(X',(m+s)(\\pi^*H+E)-k\\pi^*F).\n\\end{align*}\nIt is easy to see that there exists a positive constant $C_1>0$ such that\n\\[\nl_0\\sum_{k=1}^\\infty h^0(X',(m+s)(\\pi^*H+E)-k\\pi^*F)< C_1m^{n+1}\n\\]\nfor $m>0$. We may assume that the very ample divisor $A$ on $X'$ satisfies that $H^i(X',A+M)=0$ for any $i>0$ and for any nef divisor $M$ by the Fujita vanishing theorem \\cite[Theorem 1.4.35]{Laz}. We may also assume that there exists a positive integer $s'\\in\\mathbb{Z}_{>0}$ such that $s'\\pi^*F-E$ is effective. Therefore,\n\\[\nh^0(X',(m+s)(\\pi^*H+E)+(mG-k)\\pi^*F)\\le h^0(X',(m+s)(\\pi^*H+s'\\pi^*F)+(mG-k)\\pi^*F+A)\n\\]\nand we obtain\n\\[\nh^0(X',(m+s)(\\pi^*H+s'\\pi^*F)+(mG-k)\\pi^*F+A)=\\chi(X',(m+s)(\\pi^*H+s'\\pi^*F)+(mG-k)\\pi^*F+A)\n\\]\nfor $1\\le k\\le mG$. Thus we have\n\\begin{align*}\n&\\sum_{k=1}^{mG} h^0(X',(m+s)(\\pi^*H+E)+(mG-k)\\pi^*F)\\\\\n&\\le \\sum_{k=1}^{mG}\\chi(X',(m+s)(\\pi^*H+s'\\pi^*F)+(mG-k)\\pi^*F+A)\\\\\n&=\\frac{1}{n!}\\int_{0}^{mG}(m(\\pi^*H+s'\\pi^*F)+(mG-t)\\pi^*F)^ndt+O(m^n)\\\\\n&<\\frac{m^{n+1}G^2}{2(n-1)!}(H^{n-1}\\cdot L)+C_2m^{n+1}G+C_3m^{n+1}+O(m^n)\n\\end{align*}\nwhere $C_2>0$ and $C_3>0$ are positive constants. Therefore, we have\n\\[\n\\mathrm{mult}_{F_0}(D'_m)\\le\\frac{1}{2}+\\left(\\frac{l_0}{(n-1)!}(H^{n-1}\\cdot L)\\right)^{-1}(C_2\\epsilon+(C_1+C_3)\\epsilon^2)+O(m^{-1}).\n\\]\nWe claim that we can take a positive constant $C'$ such that\n\\[\n\\mathrm{mult}_{F_0}(D'_m)\\le\\frac{1}{2}+C'\\epsilon+O(m^{-1})\n\\]\nholds uniformly for any irreducible component $F_0$ of $\\pi^*F$ and for any point of $p\\in B$. Indeed, since $f:X\\to B$ is an algebraic fiber space, the number of reducible fiber is finite. On the other hand, for irreducible fiber $\\pi^*F=F_0$,\n\\begin{align*}\n\\sum_{k=1}^\\infty h^0(X',\\pi^*(mH+mGL)-kF_0)&\\le\\sum_{k=1}^\\infty h^0(X',\\pi^*((m+s)H+(mG-k)L))\\\\\n&=\\sum_{k=1}^\\infty h^0(X,(m+s)H+(mG-k)L)\\\\\n&=\\frac{1}{n!}\\int_{0}^{\\infty}\\mathrm{vol}((m+s)\\pi^*H+(mG-t)\\pi^*L)dt+O(m^n).\n\\end{align*}\nTherefore, we prove the claim and there exists a positive constant $C'$ such that\n\\[\n\\mathrm{mult}_{F_0}(D'_m)\\le\\frac{1}{2}+C'\\epsilon+O(m^{-1})\n\\]\nfor any irreducible component $F_0$ of $\\pi^*F=(f\\circ\\pi)^{-1}(p)$ and for any point of $p\\in B$. There exists a positive constant $C''>0$ such that for any point $q\\in X$, $\\mathrm{mult}_qD'_{m,\\mathrm{hor}}\\le C''\\epsilon $ uniformly by the proof of Claim in the proof of Theorem \\ref{ctn}. If $K_{X'}+\\Delta'=\\pi^*(K_X+\\Delta)$, there exists a positive constant $C'''>0$ such that\n\\begin{align*}\n\\mathrm{lct}_{(X,\\Delta)}((mG)^{-1}D_m)&=\\mathrm{lct}_{(X',\\Delta')}(D'_m)\\\\\n&\\ge \\left(\\frac{1}{2}+C'''\\epsilon+O(m^{-1})\\right)^{-1}\\alpha_{(X,\\Delta)}(L).\n\\end{align*}\nThus we obtain\n\\[\n\\delta_{(X,\\Delta)}(\\epsilon H+L)\\ge\\left(\\frac{1}{2}+C'''\\epsilon\\right)^{-1}\\alpha_{(X,\\Delta)}(L)\n\\]\nand hence\n\\[\n\\liminf_{\\epsilon\\to0}\\delta_{(X,\\Delta)}(\\epsilon H+L)\\ge2\\alpha_{(X,\\Delta)}(L).\n\\]\nOn the other hand,\n\\[\n\\limsup_{\\epsilon\\to0}\\delta_{(X,\\Delta)}(\\epsilon H+L)\\le2\\alpha_{(X,\\Delta)}(L)\n\\]\nis easier to prove. Indeed, for any fiber $F=f^{-1}(p)$, there exists a divisor of $m$-basis type $D_m\\in|(H+GL)|_{\\mathbb{Q}}$ such that\n\\[\nl^{-1}\\mathrm{mult}_{F_0}(D_m)\\ge\\frac{\\sum_{k=1}^\\infty h^0(X,mH+mGL-kF)}{m\\,h^0(X,mH+mGL)}\n\\]\nfor any irreducible component $F_0$ of $F$ with multiplicity $l=\\mathrm{mult}_{F_0}(F)$ and\n\\begin{align*}\n\\liminf_{\\epsilon\\to0}\\lim_{m\\to\\infty}\\frac{\\sum_{k=1}^\\infty h^0(X,mH+mGL-kF)}{mGh^0(X,mH+mGL)}=\\liminf_{\\epsilon\\to0}\\frac{\\int^{\\infty}_0\\mathrm{vol}(\\epsilon H+L-xL)dx}{(\\epsilon H+L)^n}\\ge\\frac{1}{2}.\n\\end{align*}\nThus we complete the proof.\n\\end{proof}\n\nLet $f:X\\to \\mathbb{P}^1$ be a Calabi-Yau fibration such that $X$ is smooth. Suppose that $B$ is the discriminant divisor and $M$ is the moduli divisor on $\\mathbb{P}^1$ (cf., \\S \\ref{unstadiab}, Definition \\ref{ambro model}). Now, we will prove that if $(\\mathbb{P}^1,B,M,L=-(K_{\\mathbb{P}^1}+B+M))$ is uniformly log-twisted K-stable, $X$ has cscK metrics and hence adiabatically K-stable by calculating the $\\delta$-invariant and applying the results of K. Zhang \\cite{Z2}, \\cite{Z}. First, we prove the following,\n  \n\\begin{lem}\\label{Jcomp}\nLet $f:(X,H)\\to (C,L)$ be a polarized algebraic fiber space where $\\mathrm{deg}\\,L=1$. Suppose that $H$ is ample. Then for sufficiently small $\\epsilon>0$ and for any semiample test configuration $(\\mathcal{X},\\mathcal{M})$ for $(X,\\epsilon H+L)$,\n\\[\n(\\mathcal{J}^{-f^*L})^{\\mathrm{NA}}(\\mathcal{X},\\mathcal{M})> -(I^\\mathrm{NA}(\\mathcal{X},\\mathcal{M})-J^\\mathrm{NA}(\\mathcal{X},\\mathcal{M})).\n\\]\n\\end{lem}\n \n\\begin{proof}\nIt suffices to show that $(X,\\epsilon H+L)$ is J$^{H}$-stable since $-L+(\\epsilon H+L)= \\epsilon H$. By the ampleness of\n\\[\nn\\frac{H\\cdot (\\epsilon H+L)^{n-1}}{(\\epsilon H+L)^n}(\\epsilon H+L)-(n-1)H=\\frac{\\epsilon^2(H^n)H+n((n-1)H^{n-1}\\cdot L+\\epsilon H^n)L}{n\\epsilon H^{n-1}\\cdot L+\\epsilon^2 H^n}\n\\]\nand by the Song-Weinkove criterion for J-stability \\cite[Theorem 8.12]{Hat}, $(X,\\epsilon H+L)$ is uniformly J$^{H}$-stable.\n\\end{proof}\n \n \n\\begin{thm}\\label{ctn3}\nLet $f:(X,\\Delta,H)\\to(\\mathbb{P}^1,L)$ be a polarized algebraic fiber space pair such that $K_X+\\Delta\\sim_{\\mathbb{P}^1,\\mathbb{Q}}0$. Suppose that $(X,\\Delta)$ is klt, $M$ is the moduli divisor, $B$ is the discriminant divisor (cf., Definition \\ref{ambro model}), $\\mathrm{deg}\\, L=1$ and $-(K_{\\mathbb{P}^1}+M+B)$ is ample. Then $(X,\\Delta,H)$ is uniformly adiabatically K-stable if $(\\mathbb{P}^1,B,M,L)$ is uniformly K-stable. Moreover, if $\\Delta=0$, $X$ is smooth and $(\\mathbb{P}^1,B,M,L)$ is uniformly adiabatically K-stable, then $(X,H)$ has cscK metrics adiabatically.\n\\end{thm}\n\\begin{proof}\nLet $\\mathrm{deg}\\, (B+M)=s\\ge0$ and then\n\\[\n\\max_{p\\in\\mathbb{P}^1}\\mathrm{mult}_p(B)<\\frac{s}{2}.\n\\]\nby Corollary \\ref{appc}. Since $\\mathrm{mult}_p(B)=1-\\mathrm{lct}_{(X,\\Delta)}(f^{-1}(p))$, we have\n\\[\n\\alpha_{(X,\\Delta)}(L)>1-\\frac{s}{2}.\n\\]\nTherefore, by Theorem \\ref{ctnd}\n\\[\n\\lim_{\\epsilon\\to 0}\\delta_{(X,\\Delta)}(\\epsilon H+L)>2-s\n\\]\nholds. On the other hand, \n\\[\nK_X+\\Delta\\sim_{\\mathbb{Q}}f^*(K_{\\mathbb{P}^1}+M+B)\\equiv -(2-s)f^*L.\n\\]\n Therefore,\n\\[\n(\\mathcal{J}^{K_X+\\Delta})^\\mathrm{NA}(\\mathcal{X},\\mathcal{M})\\ge -(2-s)(I^\\mathrm{NA}(\\mathcal{X},\\mathcal{M})-J^\\mathrm{NA}(\\mathcal{X},\\mathcal{M}))\n\\]\nfor any semiample test configuration $(\\mathcal{X},\\mathcal{M})$ for $(X,\\epsilon H+L)$ by Lemma \\ref{Jcomp}. Thus the first assertion follows from Theorem \\ref{bhjz}.\n \nOn the other hand, we can prove that the K-energy is coercive by \\cite[Proposition 3.5]{Z} and \\cite[Theorem 1.1]{C} as the proof of the first assetion. Then the second assertion follows from \\cite[Theorem 4.1]{Ch}.\n\\end{proof}\n\n\\begin{rem}\nThe converse to Theorem \\ref{ctn3} holds as we will see in Corollary \\ref{chikichiki}.\n\\end{rem}\n\n\\begin{cor}\\label{chiki1}\nLet $f:(X,\\Delta,H)\\to (C,L)$ be a polarized algebraic fiber space pair such that $K_X+\\Delta\\sim_{C,\\mathbb{Q}}0$ over a smooth curve $C$. Suppose that $M$ is the moduli divisor and $B$ is the discriminant divisor. Then $(X,\\Delta,H)$ is uniformly adiabatically K-stable if $(C,B,M,L)$ is uniformly K-stable. \n\\end{cor}\n\n\\begin{proof}\nBy Corollary \\ref{min2} and Theorem \\ref{ctn3}, we have only to consider the case when $K_C+B+M\\equiv0$. Then, $K_X+\\Delta\\equiv 0$ and hence $(X,\\Delta,H)$ is uniformly adiabatically K-stable.\n\\end{proof}\n\n \\begin{cor}\\label{ctn1}\nLet $f:X\\to \\mathbb{P}^1$ be an algebraic fiber space such that $K_X\\sim_{\\mathbb{P}^1,\\mathbb{Q}}0$ and $X$ is smooth. Suppose that $\\mathrm{deg}\\, f_*\\omega_{X\/\\mathbb{P}^1}^{\\otimes m}\\ne0$ for sufficiently large $m>0$ and $f$ has only reduced and slc fiber. Then $(X,H)$ is uniformly adiabatically K-stable for any polarization $H$ on $X$ and has cscK metrics adiabatically.\n\\end{cor}\n \n\\begin{proof}\nLet $H$ be an ample divisor on $X$. By the assumptions and Kawakita's inversion of adjunction of log canonicity \\cite{K} (see also \\cite[Corollary 1.1]{OX}), we have $\\alpha_{X}(L)=1$, where $L$ is a degree 1 line bundle on $\\mathbb{P}^1$. In other words, the discriminant divisor $B=0$. On the other hand, the moduli divisor $M\\ne0$ is ample by the assumption. Therefore, the assertion follows from Corollary \\ref{ctn3} and Corollary \\ref{appc}.\n\\end{proof}\n\nWe also obtain the following result on canonically polarized fibrations over curves,\n\\begin{thm}\\label{dne}\nLet $f:(X,\\Delta)\\to C$ be an $n$-dimensional polarized algebraic fiber space pair over a smooth curve $C$. Suppose that $(X,\\Delta)$ is klt and assume Conjecture \\ref{swconj}. Suppose also that $K_X+\\Delta$ is $f$-ample. Let $B$ be the discriminant divisor i.e., $B$ is defined as\n\\[\nB=\\sum_{P\\in B}(1-\\mathrm{lct}_{(X_P,\\Delta)}(f^*P))P\n\\] \nas Definition \\ref{ambro model}. Let also $M=\\kappa_C-K_C-B$ and call this the moduli divisor. If $K_C+B+M$ is antiample and $(C,B,M,L)$ is log-twisted uniformly Ding-stable, then $(X,\\Delta,K_X+\\Delta)$ is uniformly adiabatically K-stable. \n\\end{thm}\n\n\\begin{proof}\nLet $\\mathrm{deg}\\,L=1$. It follows from \\cite[Theorem 1.11]{Fjn} that $\\mathrm{deg}\\, (B+M)=s\\ge0$ and $C=\\mathbb{P}^1$. Therefore, if $(C,B,M,L)$ is log-twisted Fano and uniformly Ding-stable, then $\\delta_{(C,B)}(L)>2-s$ by Corollary \\ref{lastcoro}. By Theorem \\ref{ctnd}, we also have $d_{\\epsilon}\\mathrel{\\mathop:}=\\delta_{(X,\\Delta)}(\\epsilon (K_X+\\Delta)+L)> 2-s$ for sufficiently small $\\epsilon>0$ by Corollary \\ref{appc}. By Lemma \\ref{Jcomp}, $I^{\\mathrm{NA}}(\\mathcal{X},\\mathcal{M})-J^{\\mathrm{NA}}(\\mathcal{X},\\mathcal{M})\\ge (\\mathcal{J}^L)^{\\mathrm{NA}}(\\mathcal{X},\\mathcal{M})$ holds for any semiample test configuration $(\\mathcal{X},\\mathcal{M})$ for $(X,\\epsilon (K_X+\\Delta)+L)$. On the other hand, we have $H^{\\mathrm{NA}}_\\Delta(\\mathcal{X},\\mathcal{M})\\ge d_\\epsilon(I^{\\mathrm{NA}}(\\mathcal{X},\\mathcal{M})-J^{\\mathrm{NA}}(\\mathcal{X},\\mathcal{M}))$ by Theorem \\ref{bhjz}. Therefore, we have to show that $(X,\\epsilon (K_X+\\Delta)+L)$ is uniformly J$^{K_X+\\Delta+d_\\epsilon f^*L}$-stable. As Theorem \\ref{jne}, it suffices to show that $(K_X+\\Delta+d_\\epsilon f^*L)^n\\ge0$. Then, we can calculate \n\\[\n(K_X+\\Delta+d_\\epsilon f^*L)^n=(K_X+\\Delta)^n+nd_\\epsilon (L\\cdot (K_X+\\Delta)^{n-1})\n\\]\nand \n\\[\ns-2=\\mathrm{deg}\\,(\\kappa_C)=\\frac{(K_X+\\Delta)^n}{n(L\\cdot (K_X+\\Delta)^{n-1})}.\n\\]\nThus, \n\\[\n(K_X+\\Delta+d_\\epsilon f^*L)^n=n(d_\\epsilon-(2-s)) (L\\cdot (K_X+\\Delta)^{n-1})>0.\n\\]\nTherefore, $K_X+\\Delta+d_\\epsilon f^*L$ is big and the assetion follows from Conjecture \\ref{swconj}.\n\\end{proof}\n  We have the following as Corollary \\ref{j2e}.\n   \\begin{cor}\\label{d2e}\nLet $f:(X,\\Delta)\\to C$ be a klt algebraic fiber space pair over a smooth curve $C$. Suppose that $\\mathrm{dim}\\,X=2$ and $K_X+\\Delta$ is $f$-ample. Define $B$ and $M$ as in Theorem \\ref{dne}. If $(C,B,M,L)$ is log-twisted Fano and uniformly Ding-stable, then $(X,\\Delta,K_X+\\Delta)$ is uniformly adiabatically K-stable. \n\\end{cor}\n\n\n  \n\\subsection{Klt-trivial fibrations and log-twisted stability}\\label{unstadiab}\nIn this subsection, we see that log-twisted K-stability of the base of a klt (lc)-trivial fibrations (cf., Definition \\ref{ambro model}) is a necessary condition for adiabatic K-stability under some conditions (cf., Theorems \\ref{can} and \\ref{stp1}). Moreover, we also prove a partial converse to Theorem \\ref{ctn3}.\n\n First, we prove the following modified version of \\cite[Corollary 4.4]{DR2} for lc-trivial fibrations even when the fibrations have singular fibers.\n\\begin{thm}\\label{can}\nLet $f:(X,\\Delta)\\to C$ be an algebraic fiber space pair such that $K_{(X,\\Delta)}\\sim_{C,\\mathbb{Q}}0$. Let also $(C,L)$ be a $b$-dimensional polarized variety and $(X,\\Delta,H)$ be an $n$-dimensional polarized lc pair. Let\n\\[\nK_{(X,\\Delta)}=f^*(K_C+M_C+B_C)\n\\]\nbe the canonical bundle formula (cf., Definition \\ref{ambro model}). If the moduli divisor $M_C$ is $\\mathbb{Q}$-Cartier and $(X,\\Delta,H)$ is adiabatically K-semistable, then $(C,B_C,M_C,L)$ is log-twisted K-semistable.\n\\end{thm}\nFor the definition and properties of log-twisted pair, see Appendix \\ref{appendices}.\n\n\\begin{rem}\nBy Proposition \\ref{counter}, the converse does not hold in lc cases. \n\\end{rem}\n\nTo prove Theorem \\ref{can}, we shall recall the canonical bundle formula (cf, \\cite{FM2}, \\cite{A} and \\cite{FG}).\n \n\\begin{de}[Klt-trivial fibrations, canonical bundle formula, Ambro model]\\label{ambro model}\nLet $f:(X,\\Delta)\\to C$ be an algebraic fiber space. $f$ is a {\\it sublc-trivial fibration} (resp. {\\it subklt-trivial fibration}) if $(X,\\Delta)$ is a sublc (resp. subklt) pair satisfying the following properties:\n\\begin{itemize}\n\\item $ K_{(X,\\Delta)}\\sim_{C,\\mathbb{Q}}0$,\n\\item rank $f_*\\mathcal{O}_X(\\lceil \\mathbf{A}^*(X,\\Delta)\\rceil)=1$ (resp., rank $f_*\\mathcal{O}_X(\\lceil \\mathbf{A}(X,\\Delta)\\rceil)=1$),\n\\item $(X,\\Delta)$ is lc (resp. klt) over the generic point of $C$,\n\\end{itemize} \nwhere $\\mathbf{A}(X,\\Delta)$ is the discrepancy b-divisor (cf., \\cite[\\S 1.3]{A}) and\n \\[\n\\mathbf{A}^*(X,\\Delta)=\\mathbf{A}(X,\\Delta)+\\sum_{A_{(X,\\Delta)}(E)=0}E.\n\\]\nWe call $f$ is a {\\it lc (resp., klt)-trivial fibration} if $(X,\\Delta)$ is a lc (resp., klt) pair and $ K_{(X,\\Delta)}\\sim_{C,\\mathbb{Q}}0$. It is easy to see that if $f$ is a lc (resp., klt)-trivial fibration, then $f$ is a sublc (resp., subklt)-trivial fibration. Note that sublc-trivial fibrations are called lc-trivial fibrations in \\cite{FG}. \n\n If $f$ is a sublc-trivial fibration, then the following {\\it canonical bundle formula} (of Ambro) holds (cf., \\cite{A}, \\cite{FG}). There exist a positive integer $b\\in\\mathbb{Z}_{>0}$ and a rational function $\\varphi\\in K(X)$ such that\n\\[\nK_{(X,\\Delta)}+\\frac{1}{b}(\\varphi)= f^*(K_C+M_{(X,\\Delta)\/C}+B_{(X,\\Delta)\/C}),\n\\]\nwhere $M_{(X,\\Delta)\/C}$ and $B_{(X,\\Delta)\/C}$ are well-defined $\\mathbb{Q}$-divisors (see \\cite{A}, \\cite{FG}). We will denote the formula as $K_{(X,\\Delta)}= f^*(K_C+M_{(X,\\Delta)\/C}+B_{(X,\\Delta)\/C})$ for simplicity. If there is no afraid of confusion, we will denote $M_{(X,\\Delta)\/C}$ and $B_{(X,\\Delta)\/C}$ by $M_C$ and $B_C$ for short. $B_C$ is called the {\\it discriminant divisor} of $f$ defined by the equation\n\\[\nB_C=\\sum_D(1-\\mathrm{lct}_{(X_D,\\Delta)}(f^*D))D,\n\\]\nwhere $X_D$ is a neighborhood of the fiber of generic point of $D$ and $D$ runs over the prime divisors of $C$. $M_C$ is called the {\\it moduli divisor}. For any birational morphism $h:C'\\to C$ of projective normal varieties, we can take a sublc-trivial fibration $f':(X',\\Delta')\\to C'$ such that the following commutative diagram holds\n\\[\\xymatrix{\nX' \\ar[d]^{f'} \\ar[r]^{\\tilde{h}} \\ar@{}[dr]|\\circlearrowleft &X \\ar[d]^{f} \\\\\nC' \\ar[r]^{h} & C \\\\\n}\\]\nand $K_{X'}+\\Delta'=h^*(K_X+\\Delta)$. Then $M_C=h_*M_{C'}$. In other words, the moduli divisors $\\{M_C\\}_C$ forms a $\\mathbb{Q}$-{\\it b} divisor (see the details in \\cite[Lemma 2.6]{A} and \\cite{FG}). For any birational map $h':C''\\dashrightarrow C$, we say that a $\\mathbb{Q}$-divisor $\\mathcal{M}$ on $C''$ is {\\it induced by} $M_C$ if the following holds. Let $\\Gamma$ be the normalization of the graph of $h''$ and let $M_{\\Gamma}$ be the moduli divisor of a sublc-trivial fibration induced by $f$. Then, $\\mathcal{M}=p_*M_{\\Gamma}$ where $p:\\Gamma\\to C$ is the projection.\n\n If $M_C$ is {\\it stable under birational base change} i.e., $M_C$ is $\\mathbb{Q}$-Cartier and\n\\[\nM_{C'}=h^*M_C\n\\]\nfor any birational morphism $h:C'\\to C$, $f:(X,\\Delta)\\to C$ or $C$ is an {\\it Ambro model} (cf., \\cite[\\S2]{FL}). If moreover, $M_C$ is semiample, we call $f:(X,\\Delta)\\to C$ a {\\it good Ambro model} in this paper. Thanks to \\cite[Theorem 1.1]{FG}, there exists an Ambro model $C'$ dominating $C$ for any sublc-trivial fibration and then the moduli divisor is abundant (cf., \\cite[Definition 1.2]{FG}). For subklt-trivial fibrations, this fact is originally proved in \\cite{Am}.\n\\end{de}\n\n\\begin{rem}\nThe definition of an Ambro model of Floris-Lazi\\'{c} \\cite{FL} as follows. Let $f:(X,\\Delta)\\to C$ be a sublc-trivial fibration such that the moduli divisor $M$ is stable under birational base change and the discriminant divisor $B$ is simple normal crossing. In this paper, we do not assume that the discriminant $B$ is snc or $C$ is smooth.\n\\end{rem}\n\\begin{rem}\nIt immediately follows from the same argument of \\cite[\\S 5]{A} that if $f:(X,\\Delta)\\to C$ is a sublc-trivial fibration over a curve $C$, then $C$ is an Ambro model.\n\\end{rem}\n\n\nThe following conjecture is still open in general setting.\n \n\\begin{conj}[B-semiampleness conjecture, cf., {\\cite[\\S3]{FL}}]\nFor any sublc-trivial fibration, there exists a good Ambro model.\n\\end{conj}\n \n Nevertheless, we are particularly interested in Ambro models such that the bases are log-twisted Fano and we can prove the following,\n \n\\begin{lem}\\label{abund}\nLet $f:(X,\\Delta)\\to C$ be a subklt-trivial fibration and $M$ and $B$ are the moduli divisor and the discriminant divisor on $C$ respectively. If $C$ is an Ambro model and $-(K_C+B)$ or $-(K_C+B+M)$ is big and nef, then $C$ is also a good Ambro model.\n\\end{lem}\n \n\\begin{proof}\nThanks to \\cite{Am}, $M$ is abundant and in particular nef. Therefore, if $-(K_C+B+M)$ is big and nef, then so is $-(K_C+B)$. Hence, we may assume that $-(K_C+B)$ is big and nef. On the other hand, $(C,B)$ is klt due to \\cite[Theorem 3.1]{A}. Thus, $M$ is semiample by \\cite[Theorem 3.3]{KoMo}.\n\\end{proof}\n\nBefore proving Theorem \\ref{can}, we explain the proof briefly. Let $\\mathcal{X}$ and $\\mathscr{C}$ be test configurations for $X$ and $C$ and $\\Pi:\\mathcal{X}\\to\\mathscr{C}$ be a morphism of test configurations. Roughly speaking, in the proof of \\cite[Proposition 4.3]{DR2}, $K_{\\mathcal{X}\/\\mathscr{C}}$ yields the twisted part of the CM line bundle in the Donaldson-Futaki invariant of the base when we consider klt-trivial fibrations. Here, we want to analyze $K_{\\mathcal{X}\/\\mathscr{C}}$ by the canonical bundle formula more explicitly. If $\\mathcal{X}$ and $\\mathscr{C}$ have a reduced and snc fiber over $0\\in\\mathbb{A}^1$, then we have by assumption and the argument of \\cite{A},\n\\[\nK_{\\mathcal{X}}=\\Pi^*(K_{\\mathscr{C}}+\\rho^*M_C+\\rho^{-1}_* B_C)+D\n\\]\nwhere $\\rho:\\mathscr{C}\\to C_{\\mathbb{A}^1}$ is a birational morphism of test configurations and $D$ is a $\\mathbb{Q}$-divisor as \\cite[Theorem 4.5]{FM2}. Here, we want to ignore $D$ when we compare the log-twisted Donldson-Futaki invariant of $\\mathscr{C}$ and the Donaldson-Futaki invariant of $\\mathcal{X}$ as Dervan-Ross did. Then, we apply the result of Hacon-Xu \\cite[Theorem 1.1]{HX} to contract $D$.\n\n \nNow, we prove Theorem \\ref{can}. The following is the key step to prove Theorem \\ref{can}:\n \n\\begin{prop}\\label{model}\nLet $f:(X,\\Delta)\\to C$ be a lc-trivial fibration and $\\pi:Y\\to X$ be a log resolution of $(X,\\Delta)$ and $\\Theta=\\pi^{-1}_*\\Delta+\\sum F$ be an effective $\\mathbb{Q}$-divisor on $Y$ where $F$ runs over the irreducible and reduced $\\pi$-exceptional divisors. Let also $H$ be an ample divisor on $X$ and $L$ be an ample divisor on $C$. Let $\\mathcal{Y}$ be a smooth test configuration for $Y$ dominating $Y_{\\mathbb{A}^1}$ such that $(\\rho_{\\mathcal{Y}})^{-1}_*\\Theta_{\\mathbb{A}^1}+\\mathcal{Y}_{0,\\mathrm{red}}$ is an snc divisor, where $\\rho_{\\mathcal{Y}}:\\mathcal{Y}\\to Y_{\\mathbb{A}^1}$ is the canonical morphism. Let also $\\mathscr{C}$ be a normal test configuration such that $\\rho_{\\mathscr{C}}:\\mathscr{C}\\to C_{\\mathbb{A}^1}$ be the dominant morphism. Suppose that the canonical rational map $\\Pi:\\mathcal{Y}\\to\\mathscr{C}$ is a morphism. \n\nThen, $(\\mathcal{Y},(\\rho_{\\mathcal{Y}})^{-1}_*\\Theta_{\\mathbb{A}^1}+\\epsilon(\\pi\\times\\mathrm{id}_{\\mathbb{A}^1}\\circ\\rho_{\\mathcal{Y}})^*H_{\\mathbb{A}^1})$ has a $\\mathbb{G}_m$-equivariant common relative log canonical model $\\eta:\\mathcal{Y}\\dashrightarrow\\mathcal{X}$ over $\\mathscr{C}$ for sufficiently small $\\epsilon>0$. Moreover, there is a $\\mathbb{Q}$-factorial good minimal model $(\\mathcal{W},\\Delta_\\mathcal{W})$ of $(\\mathcal{Y},(\\rho_{\\mathcal{Y}})^{-1}_*\\Theta_{\\mathbb{A}^1})$ over $\\mathscr{C}$ and a birational contraction $\\xi: \\mathcal{W}\\dashrightarrow\\mathcal{X}$.\n\\end{prop}\n \n\\begin{proof}\nFirst, note that $(Y,\\Theta)$ has a relative good minimal model over $C$. Indeed, by the proof of the existence of a $\\mathbb{Q}$-factorial dlt modification $(Z,\\Delta_Z)$ over $X$ (cf., \\cite[4.1]{Fuj}), the $(K_Y+\\Theta)$-MMP with scaling of an ample divisor will yield a minimal model $(Z,\\Delta_Z)$ of $(Y,\\Theta)$ over $X$. Let $g:Z\\to X$ be the structure morphism. Since $K_Z+\\Delta_Z=g^*(K_X+\\Delta)$ and $K_X+\\Delta\\sim_{\\mathbb{Q},C}0$, $(Z,\\Delta_Z)$ is also a good minimal model of $(Y,\\Theta)$ over $C$. On the other hand, $(Z,\\Delta_Z+\\epsilon g^*H)$ is also a good minimal model of $(Y,\\Theta+\\epsilon\\pi^*H)$ for suficiently small $\\epsilon>0$. Indeed, if $\\psi:Y\\dashrightarrow Z$ is the birational contraction, for any $\\psi$-exceptional divisor $F$, $A_{(Y,\\Theta)}(F)<A_{(Z,\\Delta_Z)}(F)$. Furthermore, $K_Z+\\Delta_Z+\\epsilon g^*H$ is semiample over $C$. Hence, $(Z,\\Delta_Z+\\epsilon g^*H)$ is a good minimal model. \n \nWe have just show that $(\\mathcal{Y},(\\rho_{\\mathcal{Y}})^{-1}_*\\Theta)$ has a relative good minimal model if we restrict the base to $\\mathscr{C}\\setminus \\mathscr{C}_0$. Since $(\\mathcal{Y},(\\rho_{\\mathcal{Y}})^{-1}_*\\Theta+\\mathcal{Y}_{0,\\mathrm{red}})$ is log smooth and dlt, every lc center of the log pair $(\\mathcal{Y},(\\rho_{\\mathcal{Y}})^{-1}_*\\Theta)$ meets $\\mathcal{Y}\\setminus\\mathcal{Y}_0$. Therefore, we can apply \\cite[Theorem 1.1]{HX} and there exists a good minimal model $(\\mathcal{W},\\Delta_{\\mathcal{W}})$ of $(\\mathcal{Y},(\\rho_{\\mathcal{Y}})^{-1}_*\\Theta_{\\mathbb{A}^1})$ over $\\mathscr{C}$. Note that the model $(\\mathcal{W},\\Delta_{\\mathcal{W}})$ is $\\mathbb{G}_m$-equivariant by the argument of \\cite{A2} and \\cite{LX}. \n\n \nLet $h:\\mathcal{Y}\\dashrightarrow \\mathcal{W}$ be the birational contraction. We may assume that $h$ is the output of a minimal model program with scaling by \\cite[Corollary 2.9]{HX}. Therefore, $\\mathcal{W}$ is $\\mathbb{Q}$-factorial and $(\\mathcal{W},\\Delta_{\\mathcal{W}}+\\epsilon_1 h_*(\\pi\\times\\mathrm{id}_{\\mathbb{A}^1}\\circ\\rho_{\\mathcal{Y}})^*H_{\\mathbb{A}^1})$ is also dlt and has no lc center in $\\mathcal{W}_0$ for sufficiently small $\\epsilon_1>0$. By \\cite[Theorem 1.1]{HX}, $(\\mathcal{W},\\Delta_{\\mathcal{W}}+\\epsilon_2 h_*(\\pi\\times\\mathrm{id}_{\\mathbb{A}^1}\\circ\\rho_{\\mathcal{Y}})^*H_{\\mathbb{A}^1})$ also has a good minimal model over $\\mathscr{C}$ for $0\\le \\epsilon_2<\\epsilon_1$. Thanks to \\cite[Lemma 2.5]{Has}, there exists a birational contraction $\\xi:\\mathcal{W}\\dashrightarrow\\mathcal{X}$ such that for sufficiently small $0<\\epsilon\\ll\\epsilon_1$, the pair $(\\mathcal{X},\\xi_*(\\Delta_{\\mathcal{W}})+\\epsilon\\,\\xi_*(h_*(\\pi\\times\\mathrm{id}_{\\mathbb{A}^1}\\circ\\rho_{\\mathcal{Y}})^*H_{\\mathbb{A}^1}))$ is the lc model of $(\\mathcal{W},\\Delta_{\\mathcal{W}}+\\epsilon h_*(\\pi\\times\\mathrm{id}_{\\mathbb{A}^1}\\circ\\rho_{\\mathcal{Y}})^*H_{\\mathbb{A}^1})$ over $\\mathscr{C}$.\nFinally, we can take $\\eta=\\xi\\circ h$. Note that $(\\mathcal{W},\\Delta_{\\mathcal{W}}+\\epsilon h_*(\\pi\\times\\mathrm{id}_{\\mathbb{A}^1}\\circ\\rho_{\\mathcal{Y}})^*H_{\\mathbb{A}^1})$ has the canonical $\\mathbb{G}_m$-action and hence $\\mathcal{X}$ is a test configuration for $X$.\n\\end{proof}\n \nNote that the log canonical model $(\\mathcal{X},\\Delta_{\\mathcal{X}})$ of $(\\mathcal{Y},(\\rho_{\\mathcal{Y}})^{-1}_*\\Theta+\\epsilon(\\pi\\circ\\rho_{\\mathcal{Y}})^*H_{\\mathbb{A}^1})$ is a test configuration of $X$ for sufficiently small $\\epsilon>0$ by the previous proof. Since $\\mathscr{C}$ is projective, $K_{\\mathcal{X}}+\\Delta_{\\mathcal{X}}$ is ample over $\\mathscr{C}$ and it is $\\mathbb{Q}$-linearly equivalent over $\\mathscr{C}$ to $\\epsilon H_{\\mathbb{A}^1}$ on $\\mathcal{X}\\setminus\\mathcal{X}_0$, we conclude that $\\mathcal{X}$ admits a $\\mathbb{G}_m$-linearized ample $\\mathbb{Q}$-line bundle $\\mathcal{H}$ that is an extention of $(H+rL)_{\\mathbb{A}^1}|_{X\\times(\\mathbb{A}^1\\setminus0)}$ for sufficiently large $r>0$.\n\n \n\\begin{prop}\\label{cp}\nLet $f:(X,\\Delta)\\to C$ be a lc-trivial fibration and $H$ and $L$ be ample line bundles on $X$ and $C$ respectively. Let $\\phi\\in\\mathcal{H}^\\mathrm{NA}(L)$ be a positive non-Archimedean metric (cf., Definition \\ref{nadef}). Then, for sufficiently divisible $d$, there exists a representative $(\\mathscr{C},\\mathcal{L})$ of $\\phi_d$ that satisfies the following (cf., Definition \\ref{rramamra}).\n\\begin{enumerate}\n\\item $(\\mathscr{C},\\mathcal{L})$ is a semiample, normal and projective test configuration $(\\mathscr{C},\\mathcal{L})$ over $(C,L)$ such that $\\mathscr{C}$ dominates $C_{\\mathbb{A}^1}$. Let the dominant morphism be $\\rho_{\\mathscr{C}}$.\n\\item $\\mathscr{C}_0$ is reduced and all irreducible components of $\\mathscr{C}_0$ are $\\mathbb{Q}$-Cartier.\n\\item  There exist a sufficiently large integer $r>0$ and a lc test configuration $(\\mathcal{X},\\mathcal{H})$ for $(X,H+rL)$ that has a $\\mathbb{G}_m$-equivariant morphism $\\Pi:(\\mathcal{X},\\mathcal{H})\\to(\\mathscr{C},\\mathcal{L})$ and satisfies the following equation,\n\\[\nK_{(\\mathcal{X},(\\rho_{\\mathcal{X}})_*^{-1}\\Delta_{\\mathbb{A}^1})}=\\Pi^*(K_{\\mathscr{C}}+\\mathcal{M}+ (\\rho_{\\mathscr{C}})_*^{-1}B_{(X_{\\mathbb{A}^1},\\Delta_{\\mathbb{A}^1})\/ C_{\\mathbb{A}^1}})+\\mathcal{D}\n\\]\nwhere $\\rho_{\\mathcal{X}}:\\mathcal{X}\\dashrightarrow X_{\\mathbb{A}^1}$ is the canonical birational map, $\\mathcal{M}$ is induced by $M_{(X_{\\mathbb{A}^1},\\Delta_{\\mathbb{A}^1})\/C_{\\mathbb{A}^1}}$ and $\\mathcal{D}$ is a $\\mathbb{Q}$-divisor such that $\\mathrm{codim}_{\\mathscr{C}}\\Pi(\\mathcal{D})\\ge2$. Here, $\\Pi$ coincides with $f$ over $1\\in\\mathbb{A}^1$.\n\\end{enumerate}\n\\end{prop}\n\n\\begin{proof}\nLet $\\pi:Y\\to X$ be the log resolution and $\\Theta$ be the divisor on $Y$ as in Proposition \\ref{model}. Then $(Y,\\Theta)$ is a subdlt and log smooth pair. First, take a normal and projective representative $(\\mathscr{C},\\mathcal{L})$ of $\\phi$ such that $\\mathscr{C}$ dominates $C_{\\mathbb{A}^1}$. Let the dominant morphism be $\\rho_{\\mathscr{C}}$. We can prove that $(C,B)$ is an lc pair as \\cite[Theorem 3.1]{A}. Applying \\cite[Theorem 1.1]{OX} to the log pair $(\\mathscr{C},(\\rho_{\\mathscr{C}})^{-1}_*B_{\\mathbb{A}^1}+\\mathscr{C}_{0,\\mathrm{red}})$, we may assume that $(\\mathscr{C},(\\rho_{\\mathscr{C}})^{-1}_*B_{\\mathbb{A}^1}+\\mathscr{C}_{0,\\mathrm{red}})$ is lc. \n\n Let $\\mathcal{Y}$ be a $\\mathbb{G}_m$-equivariant blow up of $Y_{\\mathbb{A}^1}$ that is a smooth and projective test configuration over $Y$ dominates $\\mathscr{C}$ (cf., \\cite{Ko2}, \\cite{LX}). Let $g:\\mathcal{Y} \\to X_{\\mathbb{A}^1}$ be the canonical birational morphism and let $\\rho_{\\mathcal{Y}}:\\mathcal{Y}\\to Y_{\\mathbb{A}^1}$ be the morphism. Furthermore, by taking the normalized base change $(\\mathscr{C}^{(d)},\\mathcal{L}^{(d)})$ via $g_d:\\mathbb{A}^1\\ni x\\mapsto x^d\\in\\mathbb{A}^1$ for sufficiently divisible $d$, we may assume that $\\mathscr{C}$ has a reduced central fiber and $\\mathrm{Ex}(g)+\\mathrm{red}(g^{-1}_*(\\Delta_{\\mathbb{A}^1}))+\\mathcal{Y}_0$ is a reduced snc divisor on $\\mathcal{Y}$ (cf., \\cite{KKMS}, \\cite[Theorem 7.17]{KoMo}) by replacing $\\mathscr{C}$ by $\\mathscr{C}^{(d)}$. In fact, we can do this $\\mathbb{G}_m$-equivariantly by the argument of \\cite[Lemma 5]{LX}. By the generic smoothness \\cite[III Corollary 10.7]{Ha}, $\\mathcal{Y}$ has reduced fibers and $\\mathcal{Y}_0+\\rho_{\\mathcal{Y}}^*\\Theta_{\\mathbb{A}^1}$ is simple normal crossing over the generic points of the irreducible components of the central fiber $\\mathscr{C}_0$ over $0\\in\\mathbb{A}^1$. By \\cite[Proposition 5.7]{Hat2}, there exists a $\\mathbb{G}_m$-equivariant birational morphism $\\varphi:\\mathscr{C}'\\to\\mathscr{C}$ of test configurations such that $\\varphi$ is an isomorphism in codimension 1 and irreducible components of central fiber $\\mathscr{C}'_0$ are $\\mathbb{Q}$-Cartier since $(\\mathscr{C},(\\rho_{\\mathscr{C}})^{-1}_*B_{\\mathbb{A}^1}+\\mathscr{C}_0)$ is lc. Let $\\beta:\\mathcal{Y}'\\to \\mathcal{Y}$ be a resolution of indeterminacy of $\\mathcal{Y}\\dashrightarrow \\mathscr{C}'$ such that $\\mathcal{Y}'$ is smooth and $\\mathrm{Ex}(\\beta\\circ g)+\\mathrm{red}((\\beta\\circ g)^{-1}_*(\\Delta_{\\mathbb{A}^1}))+\\mathcal{Y}'_{0,\\mathrm{red}}$ is an snc divisor. Note that we can take $\\mathcal{Y}' $ isomorphic to $\\mathcal{Y}$ over codimension 1 points of $\\mathscr{C}'$. Therefore, $\\mathcal{Y}'$ also has semistable fibers over the generic points of the irreducible components of the central fiber $\\mathscr{C}'_0$. We may replace $\\mathscr{C}$ by $\\mathscr{C}'$ and $\\mathcal{Y}$ by $\\mathcal{Y}'$. Thus, $(\\mathscr{C},\\mathcal{L})$ satisfies the conditions (1) and (2) in the assumption now. Moreover, $\\mathcal{Y}$ satisfies the following;\n \\begin{itemize}\n \\item $\\mathcal{Y}$ is a smooth test configuration for $Y$ dominating $Y_{\\mathbb{A}^1}$. Let $\\rho_{\\mathcal{Y}}:\\mathcal{Y}\\to Y_{\\mathbb{A}^1}$ be the canonical morphism. Moreover, $\\mathcal{Y}_0$ is reduced over generic points of irreducible components of $\\mathscr{C}_0$ and $\\mathrm{Ex}(g)+\\mathrm{red}(g^{-1}_*(\\Delta_{\\mathbb{A}^1}))+\\mathcal{Y}_0$ is snc where $g=\\pi\\times\\mathrm{id}_{\\mathbb{A}^1}\\circ \\rho_{\\mathcal{Y}}:\\mathcal{Y}\\to X_{\\mathbb{A}^1}$.\n \\end{itemize}\n \nNote that $\\mathcal{M}=M_{(\\mathcal{Y},g^{-1}_*\\Delta_{\\mathbb{A}^1}+\\sum(1-A_{(X_{\\mathbb{A}^1},\\Delta_{\\mathbb{A}^1})}(E))E)\/\\mathscr{C}}$ by the assumption. Then the canonical bundle formula of Ambro \\cite{A} is\n\\begin{align*}\nK_{\\mathcal{Y}}+g^{-1}_*\\Delta_{\\mathbb{A}^1}+\\sum(1-A_{(X_{\\mathbb{A}^1},\\Delta_{\\mathbb{A}^1})}(E))E&=\\Pi_{\\mathcal{Y}}^*\\rho_{\\mathscr{C}}^*(K_{C_{\\mathbb{A}^1}}+M_{(X_{\\mathbb{A}^1},\\Delta_{\\mathbb{A}^1})\/ C_{\\mathbb{A}^1}}+ B_{(X_{\\mathbb{A}^1},\\Delta_{\\mathbb{A}^1})\/ C_{\\mathbb{A}^1}})\\\\\n&=\\Pi_{\\mathcal{Y}}^*(K_{\\mathscr{C}}+\\mathcal{B}+\\mathcal{M}),\n\\end{align*}       \nwhere $E$ runs over the $g$-exceptional divisors and $\\mathcal{B}$ is the discriminant divisor of $(\\mathcal{Y},g^{-1}_*\\Delta_{\\mathbb{A}^1}+\\sum(1-A_{(X_{\\mathbb{A}^1},\\Delta_{\\mathbb{A}^1})}(E))E)\\to \\mathscr{C}$. \n\nTherefore, we obtain\n\\[\nK_{\\mathcal{Y}}+(\\rho_{\\mathcal{Y}})^{-1}_*\\Theta_{\\mathbb{A}^1}=\\Pi_{\\mathcal{Y}}^*(K_{\\mathscr{C}}+\\mathcal{M}+ \\mathcal{B})-\\sum_E(1-A_{(X_{\\mathbb{A}^1},\\Delta_{\\mathbb{A}^1})}(E))E+\\sum_FA_{(X_{\\mathbb{A}^1},\\Delta_{\\mathbb{A}^1})}(F)F\n\\]\nwhere $E$ and $F$ run over all $g$-exceptional divisors such that $E\\subset\\mathcal{Y}_0$ and $F\\not\\subset \\mathcal{Y}_0$ respectively. Since $\\mathcal{Y}_0$ is reduced over the generic points of irreducible components $P$ of $\\mathscr{C}_0$ and $\\mathcal{Y}_0+(\\rho_{\\mathcal{Y}})^{-1}_*\\Theta_{\\mathbb{A}^1}$ is snc, we have\n\\[\n\\mathrm{lct}_{(\\mathcal{Y}_P,(\\rho_{\\mathcal{Y}})^{-1}_*\\Theta_{\\mathbb{A}^1})}(\\Pi_{\\mathcal{Y}}^*P)=0.\n\\]\nOn the other hand, \n\\[\n\\mathrm{mult}_P(\\mathcal{B})=1-\\mathrm{lct}_{\\left(\\mathcal{Y}_P,g^{-1}_*\\Delta_{\\mathbb{A}^1}+\\sum_E(1-A_{(X_{\\mathbb{A}^1},\\Delta_{\\mathbb{A}^1})}(E))E\\right)}(\\Pi_{\\mathcal{Y}}^*P)=\\max_F(1-A_{(X_{\\mathbb{A}^1},\\Delta_{\\mathbb{A}^1})}(F))\n\\]\nwhere $E$ runs over the $g$-exceptional divisors and $F$ runs over the divisors dominating $P$. Thus, there exists a $\\mathbb{Q}$-divisor $D$ such that\n\\[\nK_{\\mathcal{Y}}+(\\rho_{\\mathcal{Y}})^{-1}_*\\Theta_{\\mathbb{A}^1}=\\Pi_{\\mathcal{Y}}^*(K_{\\mathscr{C}}+\\mathcal{M}+ (\\rho_{\\mathscr{C}})_*^{-1}B_{(X_{\\mathbb{A}^1},\\Delta_{\\mathbb{A}^1})\/ C_{\\mathbb{A}^1}})+D\n\\]\nwhere $D_+$ is of insufficient fiber type in the sense of \\cite[2.9]{Lai} and $\\mathrm{codim}_{\\mathscr{C}}\\Pi_{\\mathcal{Y}}(D_-)\\ge2$.\nHere, $D_+$ and $D_-$ are effective divisors such that they have no common component and $D=D_+-D_-$. Note that $K_{\\mathscr{C}}+\\mathcal{M}+ (\\rho_{\\mathscr{C}})_*^{-1}B_{(X_{\\mathbb{A}^1},\\Delta_{\\mathbb{A}^1})\/ C_{\\mathbb{A}^1}}$ is $\\mathbb{Q}$-Cartier by the condition (2). If $Q$ is a prime divisor on $\\mathcal{Y}$ dominating $P$, then \n\\[\n\\mathrm{mult}_Q(D)=A_{(X_{\\mathbb{A}^1},\\Delta_{\\mathbb{A}^1})}(Q)-\\min_FA_{(X_{\\mathbb{A}^1},\\Delta_{\\mathbb{A}^1})}(F)\\ge0\n\\]\n where $F$ runs over the prime divisors dominating $P$.\n\nNote that $(\\mathcal{Y},(\\rho_{\\mathcal{Y}})^{-1}_*\\Theta_{\\mathbb{A}^1}+\\mathcal{Y}_{0,\\mathrm{red}})$ is dlt and log smooth. By Proposition \\ref{model}, we can take a $\\mathbb{G}_m$-equivariant common relative log canonical model $(\\mathcal{X},\\Delta_{\\mathcal{X}})$ of $(\\mathcal{Y},(\\rho_{\\mathcal{Y}})^{-1}_*\\Theta_{\\mathbb{A}^1}+\\epsilon(\\pi_{\\mathbb{A}^1}\\circ\\rho_{\\mathcal{Y}})^*H_{\\mathbb{A}^1})$ for $0<\\epsilon \\ll 1$ over $\\mathscr{C}$. Moreover, there is also a good minimal model $(\\mathcal{W},\\Delta_\\mathcal{W})$ of $(\\mathcal{Y},(\\rho_{\\mathcal{Y}})^{-1}_*\\Theta_{\\mathbb{A}^1})$ and a birational contraction $\\xi: \\mathcal{W}\\dashrightarrow\\mathcal{X}$ by Proposition \\ref{model}. Then, \n\\[\nK_{\\mathcal{W}}+\\Delta_{\\mathcal{W}}=\\Pi_{\\mathcal{W}}^*(K_{\\mathscr{C}}+\\mathcal{M}+ (\\rho_{\\mathscr{C}})^{-1}_*B_{(X_{\\mathbb{A}^1},\\Delta_{\\mathbb{A}^1})\/ C_{\\mathbb{A}^1}})+D'\n\\]\nwhere $\\Pi_{\\mathcal{W}}:\\mathcal{W}\\to \\mathscr{C}$ is the structure morphism and $D'$ is the strict transformation of $D$. Now, we can prove that $\\mathrm{codim}_{\\mathscr{C}}\\Pi_{\\mathcal{W}}(D')\\ge 2$. Indeed, we can check this as follows. Take a prime divisor $Q$ on $\\mathcal{Y}$ dominating $P$ such that $Q\\not\\subset \\mathrm{Supp}\\, D_+$. It is easy to see that $Q\\not\\in \\mathbf{B}(K_{\\mathcal{W}}+\\Delta_{\\mathcal{W}}\/U_P)$ over a neighborhood $U_P$ of the generic point of $P$. Thus, $Q$ is not contracted by $\\mathcal{Y}\\dashrightarrow\\mathcal{W}$ by \\cite[2.4]{HX} and $D'_+$ is also of insufficient fiber type \\cite[2.9]{Lai} over $U_P$. Therefore, $D'=0$ over $\\mathscr{C}\\setminus\\Pi_{\\mathcal{Y}}(D_-)$ by \\cite[2.10]{Lai} and \\cite[2.4 (1)]{HX}. Take a polarization $\\mathcal{H}$ on $\\mathcal{X}$ as the argument after Proposition \\ref{model}. Let $\\mathcal{D}$ be the strict transformation of $D'$ on $\\mathcal{X}$. Then, it is easy to check that $\\mathcal{X}$ and $\\mathcal{D}$ satisfy the condition (3).\n\\end{proof}\n \n\\begin{proof}[Proof of Theorem \\ref{can}]\nLet $\\phi\\in\\mathcal{H}^\\mathrm{NA}(L)$ be a positive non-Archimedean metric such that \n\\[\nM^\\mathrm{NA}_{(B,M)}(\\phi)<0.\n\\]\n Then, take $(\\mathcal{X},\\mathcal{H})$ and $(\\mathscr{C},\\mathcal{L})$ as in Proposition \\ref{cp}. We may replace $H$ by $H+rL$ where $r$ is as in Proposition \\ref{cp}. For sufficiently large $m\\in\\mathbb{Q}$,\n\\[\n\\frac{-K_X\\cdot (H+mL)^{n-1}}{ (H+mL)^n}=-\\frac{b(K_C+M_{(X,\\Delta)\/C}+B_{(X,\\Delta)\/C})\\cdot L^{b-1}}{nL^{b}}m^{-1}+O(m^{-2}).\n\\]\nThen\n\\begin{align*}\n(H+mL)^n&\\mathrm{DF}_{\\Delta}(\\mathcal{X},\\mathcal{H}+m\\Pi^*\\mathcal{L})\\\\\n&=K_{(\\mathcal{X},(\\rho_\\mathcal{X})^{-1}_*\\Delta_{\\mathbb{P}^1})\/\\mathbb{P}^1}\\cdot (\\mathcal{H}+m\\Pi^*\\mathcal{L})^n+\\frac{n}{n+1}\\frac{-K_X\\cdot (H+mL)^{n-1}}{ (H+mL)^n}(\\mathcal{H}+m\\Pi^*\\mathcal{L})^{n+1}\\\\\n&=K_{(\\mathcal{X},(\\rho_\\mathcal{X})^{-1}_*\\Delta_{\\mathbb{P}^1})\/\\mathbb{P}^1}\\cdot (\\mathcal{H}+m\\Pi^*\\mathcal{L})^n\\\\\n&-\\frac{n}{n+1}\\left(\\frac{b(K_C+M_{(X,\\Delta)\/C}+B_{(X,\\Delta)\/C})\\cdot L^{b-1}}{nL^{b}}m^{-1}+O(m^{-2})\\right)(\\mathcal{H}+m\\Pi^*\\mathcal{L})^{n+1}.\n\\end{align*}\nSince \\[\nK_{(\\mathcal{X},(\\rho_{\\mathcal{X}})_*^{-1}\\Delta_{\\mathbb{P}^1})}=\\Pi^*(K_{\\mathscr{C}}+\\mathcal{M}+ (\\rho_{\\mathscr{C}})_*^{-1}B_{(X_{\\mathbb{P}^1},\\Delta_{\\mathbb{P}^1})\/ C_{\\mathbb{P}^1}})+\\mathcal{D}\n\\]\nholds over $\\mathscr{C}$ where $\\mathrm{codim}_{\\mathscr{C}}\\Pi(\\mathcal{D})\\ge 2$ and $\\rho_{\\mathscr{C}}^*M_{(X_{\\mathbb{P}^1},\\Delta_{\\mathbb{P}^1})\/ C_{\\mathbb{P}^1}}-\\mathcal{M}$ is effective by the negativity lemma \\cite[Lemma 3.39]{KoMo} and \\cite[Theorem 1.1]{FG}, the $m^{b+1}$-term of $(H+mL)^n\\mathrm{DF}_{\\Delta}(\\mathcal{X},\\mathcal{H}+m\\Pi^*\\mathcal{L})$ vanishes and the $m^b$-term is smaller than or equals to\n\\[\n\\binom{n}{b}(H^{n-b}\\cdot L^b)\\mathrm{DF}_{(B,M)}(\\mathscr{C},\\mathcal{L})<0,\n\\]\nHere, we use $\\mathrm{DF}_{(B,M)}(\\mathscr{C},\\mathcal{L})=M^\\mathrm{NA}_{(B,M)}(\\mathscr{C},\\mathcal{L})$ since $\\mathscr{C}_0$ is reduced. Thus, $\\mathrm{DF}_{\\Delta}(\\mathcal{X},\\mathcal{H}+m\\Pi^*\\mathcal{L})<0$ for sufficiently large $m\\gg0$.\n\\end{proof}\n \nIf $C$ is a curve, $C$ is necesarily an Ambro model for any sublc-trivial fibration $f:(X,\\Delta)\\to C$. Thus, we obtain the following by Theorem \\ref{can}:\n\\begin{cor}\nLet $f:(X,\\Delta)\\to C$ be a lc-trivial fibration. Suppose that $C$ is a smooth curve with an ample line bundle $L$ and $H$ is a $f$-ample divisor. Let\n\\[\nK_{(X,\\Delta)}=f^*(K_C+M+B)\n\\]\nbe the canonical bundle formula where $M$ (resp., $B$) is the moduli (resp., discriminant) divisor. If $(X,\\Delta,H)$ is adiabatically K-semistable over $(C,L)$, then $(C,B,M,L)$ is log-twisted K-semistable.\n\\end{cor}\n \nNext, we state a partial result that uniform adiabatic K-stability of the total space implies uniform K-stability of the log-twisted Fano base of a klt-trivial fibration.\n \n\\begin{lem}\\label{unad}\nLet $f:X\\to C$ be a lc-trivial fibration and suppose that $(X,\\Delta,H)$ is uniformly adiabatically K-stable over $(C,L)$. Then there exist three positive constants $\\delta_1$, $\\delta_2$ and $\\epsilon_0$ such that if $0<\\epsilon<\\epsilon_0$ and $\\epsilon H+f^*L$ is ample on $X$, \n\\[\nM_{\\Delta}^{\\mathrm{NA}}(\\mathcal{X},\\mathcal{M})\\ge \\delta_1H_{\\Delta}^{\\mathrm{NA}}(\\mathcal{X},\\mathcal{M}) +\\delta_2 J^{\\mathrm{NA}}(\\mathcal{X},\\mathcal{M})\n\\]\n for any semiample test configuration $(\\mathcal{X},\\mathcal{M})$\n over $(X,\\epsilon H+f^*L)$.\n\\end{lem}\n \n\\begin{proof}\nDecompose $M_{\\Delta}^{\\mathrm{NA}}(\\mathcal{X},\\mathcal{M})=H_{\\Delta}^{\\mathrm{NA}}(\\mathcal{X},\\mathcal{M})+(\\mathcal{J}^{K_{(X,\\Delta)}})^{\\mathrm{NA}}(\\mathcal{X},\\mathcal{M})$. By the assumption there exist positive constant $\\epsilon_0$ and $\\delta$ such that $H_{\\Delta}^{\\mathrm{NA}}(\\mathcal{X},\\mathcal{M})\\ge-(\\mathcal{J}^{K_{(X,\\Delta)}})^{\\mathrm{NA}}(\\mathcal{X},\\mathcal{M})+\\delta J^{\\mathrm{NA}}(\\mathcal{X},\\mathcal{M})$ for $0<\\epsilon\\le \\epsilon_0$ such that $\\epsilon H+f^*L$ is ample and for any semiample test configuration $(\\mathcal{X},\\mathcal{M})$ over $(X,\\epsilon H+f^*L)$. For $0<b\\ll1$,\n\\[\n(1-b)H_{\\Delta}^{\\mathrm{NA}}(\\mathcal{X},\\mathcal{M})\\ge-(\\mathcal{J}^{K_{(X,\\Delta)}})^{\\mathrm{NA}}(\\mathcal{X},\\mathcal{M})+\\delta(1-b) J^{\\mathrm{NA}}(\\mathcal{X},\\mathcal{M})+b(\\mathcal{J}^{K_{(X,\\Delta)}})^{\\mathrm{NA}}(\\mathcal{X},\\mathcal{M}).\n\\]\nOn the other hand, let $T$ be a $\\mathbb{Q}$-line bundle such that $K_{(X,\\Delta)}\\sim_{\\mathbb{Q}}f^*T$. There exists a positive constant $G>0$ independent from $(\\mathcal{X},\\mathcal{M})$ such that for any sufficiently small $\\epsilon>0$,\n\\begin{align*}\n&\\left(n\\frac{f^*T\\cdot (f^*L+\\epsilon H)^{n-1}}{(f^*L+\\epsilon H)^n}+\\frac{G}{n}\\right)(f^*L+\\epsilon H)-\\frac{n^2-1}{n}f^*T\\\\\n&=\\left(\\frac{G}{n}+c\\frac{f^*T\\cdot f^*L^{c-1}\\cdot H^{n-c}}{f^*L^{c}\\cdot H^{n-c}}+O(\\epsilon) \\right)f^*L-\\frac{n^2-1}{n}f^*T\\\\\n&+\\epsilon\\left(\\frac{G}{n}+c\\frac{f^*T\\cdot f^*L^{c-1}\\cdot H^{n-c}}{f^*L^{c}\\cdot H^{n-c}}+O(\\epsilon) \\right)H\n\\end{align*}\nis ample, where $X$ (resp., $C$) is $n$ (resp., $c$)-dimensional. By the proof of \\cite[Corollary 8.5]{Hat},\n\\[\nG(I^{\\mathrm{NA}}(\\mathcal{X},\\mathcal{M})-J^{\\mathrm{NA}}(\\mathcal{X},\\mathcal{M}))+(\\mathcal{J}^{K_{(X,\\Delta)}})^{\\mathrm{NA}}(\\mathcal{X},\\mathcal{M})\\ge0.\n\\]\nTherefore, there exists a positive constant $\\delta_2>0$ such that for sufficiently small $\\epsilon>0$ and $b>0$,\n\\[\n\\delta(1-b) J^{\\mathrm{NA}}(\\mathcal{X},\\mathcal{M})+b(\\mathcal{J}^{K_{(X,\\Delta)}})^{\\mathrm{NA}}(\\mathcal{X},\\mathcal{M})\\ge \\delta_2J^{\\mathrm{NA}}(\\mathcal{X},\\mathcal{M}).\n\\]\nHence,\n\\[\nH_{\\Delta}^{\\mathrm{NA}}(\\mathcal{X},\\mathcal{M})\\ge-(\\mathcal{J}^{K_{(X,\\Delta)}})^{\\mathrm{NA}}(\\mathcal{X},\\mathcal{M})+bH_{\\Delta}^{\\mathrm{NA}}(\\mathcal{X},\\mathcal{M})+\\delta_2J^{\\mathrm{NA}}(\\mathcal{X},\\mathcal{M}).\n\\]\nHere, let $\\delta_1=b$.\n\\end{proof}\n \n\\begin{thm}\\label{stp1}\nLet $f:(X,\\Delta)\\to C$ be a klt-trivial fibration, $M$ be the moduli divisor and $B$ be the discriminant divisor on $C$. Suppose that $C$ is an Ambro model, $(X,\\Delta,H)$ is uniformly adiabatically K-stable over $(C,L)$ and $L=-(K_C+M+B)$ is ample. Then $(C,B,M,L)$ is log-twisted K-stable.\n\\end{thm}\n \n\\begin{proof}\nNote that $C$ is also a good Ambro model by Lemma \\ref{abund}. By Theorem \\ref{can}, it suffices to show that there exists no non-almost trivial semiample test configuration for $(C,B,M,L)$ whose log-twisted Donaldson-Futaki invariant is 0. Assume that the assertion would fail and hence there would exist a non almost trivial semiample test configuration $(\\mathscr{C},\\mathcal{L})$ over $(C,B,M,L)$ such that $\\mathrm{DF}_{(B,M)}(\\mathscr{C},\\mathcal{L})=0$. By Theorem \\ref{appendix}, we may assume that there exists a $\\mathbb{Q}$-Cartier $\\mathbb{Q}$-divisor $D\\sim_{\\mathbb{Q}}L$ on $X$ such that $\\mathrm{DF}_{(B+aD,M)}(\\mathscr{C},\\mathcal{L})<0$ for sufficiently small $a>0$. Take a test configuration $(\\mathcal{X},\\mathcal{H})$ for $(X,H)$ and we may assume that there exists a morphism $\\Pi:\\mathcal{X}\\to\\mathscr{C}$ as in Proposition \\ref{cp}. Then $\\mathrm{DF}_{\\Delta+af^*D}(\\mathcal{X},\\mathcal{H}+m\\mathcal{L})<0$ for sufficiently small $a>0$ and for sufficiently large $m\\gg0$. However, since $(X,\\Delta,H)$ is uniformly adiabatically K-stable over $(C,L)$, there exist three positive constants $\\delta_1$, $\\delta_2$ and $\\epsilon_0$ such that if $0<m^{-1}<\\epsilon_0$ and $H+mf^*L$ is ample on $X$,\n\\[\nM_{\\Delta}^{\\mathrm{NA}}(\\mathcal{X},m^{-1}\\mathcal{H}+\\mathcal{L})\\ge \\delta_1H_{\\Delta}^{\\mathrm{NA}}(\\mathcal{X},m^{-1}\\mathcal{H}+\\mathcal{L}) +\\delta_2 J^{\\mathrm{NA}}(\\mathcal{X},m^{-1}\\mathcal{H}+\\mathcal{L})\n\\]\nby Lemma \\ref{unad}. Then\n\\begin{align*}\nM_{\\Delta+af^*D}^{\\mathrm{NA}}(\\mathcal{X},m^{-1}\\mathcal{H}+\\mathcal{L})&\\ge \\delta_1H_{(\\Delta+\\frac{a}{\\delta_1}f^*D)}^{\\mathrm{NA}}(\\mathcal{X},m^{-1}\\mathcal{H}+\\mathcal{L})\\\\\n&+\\delta_2(J^{\\mathrm{NA}}(\\mathcal{X},m^{-1}\\mathcal{H}+\\mathcal{L})+(\\mathcal{J}^{(\\frac{a}{\\delta_2}f^*D)})^{\\mathrm{NA}}(\\mathcal{X},m^{-1}\\mathcal{H}+\\mathcal{L})).\n\\end{align*}\nFor sufficiently small $a>0$ that $(X,\\Delta+\\frac{a}{\\delta_1}f^*D)$ is klt, $H_{(\\Delta+\\frac{a}{\\delta_1}f^*D)}^{\\mathrm{NA}}(\\mathcal{X},m^{-1}\\mathcal{H}+\\mathcal{L})>0$ by \\cite[9.16]{BHJ}. Similarly to the proof of Lemma \\ref{unad}, by \\cite[Corollary 8.5]{Hat},\n\\[\nJ^{\\mathrm{NA}}(\\mathcal{X},m^{-1}\\mathcal{H}+\\mathcal{L})+(\\mathcal{J}^{(\\frac{a}{\\delta_2}f^*D)})^{\\mathrm{NA}}(\\mathcal{X},m^{-1}\\mathcal{H}+\\mathcal{L})>0\n\\]\nfor sufficiently small $a>0$. Therefore, $$\\mathrm{DF}_{\\Delta+af^*D}(\\mathcal{X},\\mathcal{H}+m\\mathcal{L})\\ge M_{\\Delta+af^*D}^{\\mathrm{NA}}(\\mathcal{X},m^{-1}\\mathcal{H}+\\mathcal{L})>0$$ for sufficiently small $a>0$ and for sufficiently large $m\\gg0$. This is a contradiction. \n\\end{proof}\n \nIt immediately follows from the last theorem and Corollary \\ref{ctn3}:\n \n\\begin{cor}\\label{chikichiki}\nLet $f:(X,\\Delta)\\to \\mathbb{P}^1$ be a klt-trivial fibration, $M$ be the moduli divisor and $B$ be the discriminant divisor on $\\mathbb{P}^1$. Suppose that $L=-(K_{\\mathbb{P}^1}+M+B)$ is ample. Then, $(X,\\Delta,H)$ is uniformly adiabatically K-stable over $(\\mathbb{P}^1,L)$ if and only if $(\\mathbb{P}^1,B,M,L)$ is uniformly K-stable. Moreover, if $\\Delta=0$, $X$ is smooth and $(\\mathbb{P}^1,B,M,L)$ is uniformly K-stable, then $(X,H)$ has cscK metrics adiabatically.\n\\end{cor}\n\n Here, we remark that Theorem \\ref{dd} follows immediately from Corollaries \\ref{chiki1} and \\ref{chikichiki}.\n  \n\\section{Applications to Rational Elliptic Surfaces}\\label{RES}\n\nWe apply the general theory established in \\S8 to rational elliptic surfaces.\n\\subsection{Log-twisted K-stability of the base curve}\nIn this subsection, we classify K-stable log-twisted Fano curve $(\\mathbb{P}^1,B,M,L)$ associated with a rational elliptic surface.\n\nLet $f:X\\to C$ be a smooth (relatively minimal) elliptic surface. Thanks to Kodaira's canonical bundle formula (cf., \\cite{FM2}), the relative canonical divisor $K_{X\/C}$ is expressed as\n\\[\nK_{X\/C}=\\frac{1}{12}f^*J^*\\mathcal{O}_{\\mathbb{P}^1}(1)+\\sum_Q (1-\\mathrm{lct}_{X}(f^*(Q)))f^*(Q),\n\\]\nwhere $J:C\\to \\mathbb{P}^1$ is the {\\it functional invariant} (cf., \\cite[V.8]{BHPV}). We call $T=\\frac{1}{12}f^*J^*\\mathcal{O}_{\\mathbb{P}^1}(1)$ and $B=\\sum_Q (1-\\mathrm{lct}_{X}(f^*(Q)))f^*(Q)$ to be the moduli divisor and the discriminant divisor, respectively (cf., \\cite{A}). Recall that Kodaira classified the singular fibers of elliptic surfaces. See the details in \\cite[V. \\S7]{BHPV}. Let $F$ be a fiber. Then the following holds,\n\\begin{itemize}\n\\item if $F$ is a smooth or $I_N$-type fiber, then $\\mathrm{lct}_X(F)=1$.\n\\item if $F$ is a $II$-type fiber, then $\\mathrm{lct}_X(F)=\\frac{5}{6}$.\n\\item if $F$ is a $III$-type fiber, then $\\mathrm{lct}_X(F)=\\frac{3}{4}$.\n\\item if $F$ is a $IV$-type fiber, then $\\mathrm{lct}_X(F)=\\frac{2}{3}$.\n\\item if $F$ is an $I_N^*$-type fiber for some $N\\ge 0$, then $\\mathrm{lct}_X(F)=\\frac{1}{2}$.\n\\item if $F$ is an $II^*$-type fiber, then $\\mathrm{lct}_X(F)=\\frac{1}{6}$.\n\\item if $F$ is an $III^*$-type fiber, then $\\mathrm{lct}_X(F)=\\frac{1}{4}$.\n\\item if $F$ is an $IV^*$-type fiber, then $\\mathrm{lct}_X(F)=\\frac{1}{3}$.\n\\item if $F$ is a multiple fiber with multiplicity $m$, then $\\mathrm{lct}_X(F)=\\frac{1}{m}$.\n\\end{itemize}\n\n\n\nNext, we consider rational elliptic surfaces $f:X\\to C$. By \\cite[Proposition 3.23]{FM}, we see that $C=\\mathbb{P}^1$ and $\\mathrm{deg}\\,f_*\\omega_{X\/\\mathbb{P}^1}=1$. Let $M$ and $B$ be the moduli and discriminant divisors on $\\mathbb{P}^1$ respectively. Then, $(\\mathbb{P}^1,B,M,-(K_{\\mathbb{P}^1}+M+B))$ is a log-twisted Fano curve such that $(\\mathbb{P}^1,B)$ is klt. Moreover, it is easy to see that a rational elliptic surface $X$ is isomorphic to a blow up of $\\mathbb{P}^2$ at nine points (cf., \\cite[Theorem 5.6.1]{CD}) and $X$ has at most one multiple fiber. \n\\begin{de}\nIf $X$ has one multiple fiber, the {\\it multiplicity} $m(X)$ of $X$ is the multiplicity $m$ of the multiple fiber. If $X$ has no multiple fiber, we define $m(X)=1$.\n\\end{de}\nThen we have the following.\n\\begin{thm}\\label{P^1}\nLet $f:X\\to\\mathbb{P}^1$ be a rational elliptic surface, $B=\\sum_{Q\\in\\mathbb{P}^1}s_QQ$ be the discriminant and $M$ be the moduli divisor. Then $(\\mathbb{P}^1,B,M,L=\\mathcal{O}(1))$ is uniformly K-stable (resp, K-semistable, K-unstable) iff $\\max_Qs_Q< \\frac{2m-1}{2m}$ (resp., $\\max_Qs_Q= \\frac{2m-1}{2m}$, $\\max_Qs_Q> \\frac{2m-1}{2m}$).\n\\end{thm}\n\\begin{proof}\nImmediately from Corollary \\ref{appc}.\n\\end{proof}\nAccording to this, the GIT-stability of a rational elliptic surface admitting a section in the sense of Miranda \\cite{Mi2}, \\cite{Mi} coincides with the log-twisted K-stability of the base. More precisely, we obtain the following by the Kodaira canonical bundle formula.\n\\begin{cor}\\label{II}\nLet $f:X\\to \\mathbb{P}^1$ be a smooth rational elliptic surface with multiplicity $m(X)=m$. Then the log twisted pair $(\\mathbb{P}^1,B,M,L)$ induced by $f$ is K-unstable iff one of the following holds.\n\\begin{itemize}\n \\item $m=1$ and $X$ admits one of $II^*$, $III^*$ or $IV^*$-type fibers,\n \\item $m=2$ and $X$ admits at least one $II^*$-type fiber.\n \\end{itemize}\n\\end{cor}\n\\begin{proof}\nThis follows immediately from Theorem \\ref{P^1}. We remark that if $f^{-1}(P)$ is the multiple fiber of multiplicity $m>1$, $P\\in\\mathbb{P}^1$ does not destabilize the log-twisted pair since $\\frac{m-1}{m}< \\frac{2m-1}{2m}$.\n\\end{proof}\nTherefore, we obtain the following by Theorem \\ref{can}, Corollary \\ref{II} and the canonical bundle formula:\n\\begin{cor}\\label{234}\nLet $X$ be a smooth rational elliptic surface with multiplicity $m(X)=m$. Then the following hold,\n\\begin{enumerate}\n\\item If $m=1$ and $X$ admits one of $II^*$, $III^*$ or $IV^*$-type fibers, then $X$ is adiabatically K-unstable for any relative polarization over $\\mathbb{P}^1$.\n\\item If $m=2$ and $X$ admits at least one $II^*$-type fiber, then $X$ is adiabatically K-unstable for any relative polarization over $\\mathbb{P}^1$.\n\\end{enumerate}\n\\end{cor}\n\nBy \\cite[Theorem C]{Hat2} and Corollary \\ref{234}, we conclude as follows.\n\\begin{ex}\\label{DRF}\nLet $X$ be a rational elliptic smooth surface over $\\mathbb{P}^1$ admitting a section and a $IV^*$-type fiber. There is a polarization $H=\\epsilon H+F$ on $X$, where $H$ is relatively ample and $F$ is the fiber class. Let the line bundle $T=M+B=\\frac{\\mathrm{c}_1(\\lambda_\\mathrm{CM}(X,H))}{8\\pi_*(\\mathrm{c}_1(H))}$ on $\\mathbb{P}^1$. Here, $T$ has degree 1 and $M$ (resp., $B$) is the moduli (resp., discriminant) divisor on $\\mathbb{P}^1$. Then, it is easy to see that the twisted pair $(\\mathbb{P}^1,T,L)$ is uniformly twisted K-stable for $L=T$. Here, the twisted pair $(\\mathbb{P}^1,T,L)$ denotes the log-twisted pair $(\\mathbb{P}^1,0,T,L)$. Then $(\\mathbb{P}^1,T,L)$ is a uniformly K-stable twisted Fano pair. By twisted Yau-Tian-Donaldson conjecture (cf., \\cite{Z}), we conclude that there exists a unique twisted K\\\"{a}hler-Einstein metric in $\\mathrm{c}_1(L)$. By \\cite[Theorem C]{Hat2} and Corollary \\ref{234}, we obtain the following.\n\\begin{cor}\\label{bdbd}\nThere exists a polarized algebraic fiber space $f:(X,H)\\to (B,L)$ such that $\\mathrm{dim}\\,X=2$, $\\mathrm{dim}\\,B=1$, $X$ is smooth and $K_X\\sim_{B,\\mathbb{Q}}0$, and $f$ satisfies the following. If $T=\\frac{\\mathrm{c}_1(\\lambda_\\mathrm{CM}(X,H))}{8\\pi_*(\\mathrm{c}_1(H))}$, then the twisted pair $(B,T,L)$ has a unique twisted cscK metric and $f$ is $\\mathfrak{f}$-stable but adiabatically K-unstable. \n\\end{cor}\nThus, we see that Conjecture in \\S1 does not hold for fibrations with at least one singular fiber.\n Hence, it is necessary to consider the relation of K-stability of klt-trivial fibrations and log-twisted K-stability of the base scheme (cf., Theorem \\ref{can}, \\cite[Corollary 4.3]{DR2}).\n\\end{ex}\n\n\nConversely, we obtain the following by Theorems \\ref{ctn3} and \\ref{P^1} and the canonical bundle formula,\n\\begin{cor}\\label{K-fib}\nLet $X$ be a smooth rational elliptic surface and $S$ be a relatively ample divisor over $\\mathbb{P}^1 $. Let also the multiplicity of a unique multiple fiber be $m\\ge 1$.\n\\begin{enumerate}\n\\item If $m=1$ and $X$ has at most reduced fibers, then $(X,S)$ is uniformly adiabatically K-stable and has cscK metrics.\n\\item If $m=2$ and $X$ has no $II^*$ or $III^*$-type fiber, then $(X,S)$ is uniformly adiabatically K-stable and has cscK metrics.\n\\item If $m=3$ and $X$ has no $II^*$-type fiber, then $(X,S)$ is uniformly adiabatically K-stable and has cscK metrics.\n\\item If $m\\ge 4$, then $(X,S)$ is uniformly adiabatically K-stable and has cscK metrics.\n\\end{enumerate}\n\\end{cor}\n\n \nFrom the next section, we will compare the log-twisted K-stability of the base curves with adiabatic K-stability of rational elliptic surfaces.\n\n\n\n\\subsection{GIT-stability in the sense of Miranda I: Weierstrass fibrations}\\label{wss1}\nMiranda \\cite{Mi} constructs the family $ M_1$ of rational Weierstrass fibrations as below and applies GIT to obtain the moduli of rational Weierstrass fibrations. Comparing Theorem \\ref{P^1} with Miranda's results (e.g., \\cite[Theorem 6.2]{Mi}), we realize that the log-twisted K-stability of the base curves coincides with GIT-stability in the sense of Miranda. In this subsection, by comparing the Hilbert-Mumford weight with the weight of CM line bundle, we will prove Theorem \\ref{can} when $X$ is a rational Weierstrass fibration in another way. \n\nFirst, recall rational Weierstrass fibrations.\n\n\\begin{de}[cf., {\\cite{Mi}}]\nLet $f:X\\to \\mathbb{P}^1$ be a rational elliptic surface. If $m(X)=1$, it is known that $X$ has at least one section $S$ for $f$ \\cite[Example 5.12]{FM}. $\\pi:W\\to \\mathbb{P}^1$ is the {\\it rational Weierstrass form} of $(X,S)$ if $W$ is the relative lc model of $(X,S)$ over $\\mathbb{P}^1$. \n\nOn the other hand, let $\\pi:W\\to\\mathbb{P}^1$ be an algebraic fiber space such that $W$ is a normal and irreducible surface that has only canonical singularities (see \\cite[\\S4.2]{KoMo}). We call $\\pi:W\\to\\mathbb{P}^1$ is a {\\it Weierstrass fibration} if there exists a rational elliptic surface $f:X\\to \\mathbb{P}^1$ admitting a section $S$ such that $W$ is the Weierstrass form of $(X,S)$. See the details in \\cite{Mi}. For simplicity, we will also call $W$ a Weierstrass fibration.\n\nWe will say that $W$ has a $*$-type fiber over $p\\in\\mathbb{P}^1$ if the minimal resolution $X$ has a $*$-type fiber over the same point. (Here, $*=I_N,I_N^*,\\cdots,$ etc.)\n\\end{de}\n\nNext, let us recall the setting of \\cite{Mi}. Let $V_N=H^0(\\mathbb{P}^1,\\mathcal{O}_{\\mathbb{P}^1}(N))$ parametrizes $A\\in H^0(\\mathbb{P}^1,\\mathcal{O}(4))$ and $B\\in H^0(\\mathbb{P}^1,\\mathcal{O}(6))$. Let $\\mathcal{U}'\\subset(V_4\\times V_6\\setminus 0) \\times \\mathbb{P}_{\\mathbb{P}^1}(\\mathcal{O}_{\\mathbb{P}^1}(-2)\\oplus\\mathcal{O}_{\\mathbb{P}^1}(-3)\\oplus \\mathcal{O})$ be the closed subscheme defined by $y^2z=x^3+\\mathcal{A}xz^2+\\mathcal{B}z^3$, where $\\mathcal{A}\\in H^0((V_4\\times V_6)\\times\\mathbb{P}^1,\\mathcal{O}_{\\mathbb{P}^1}(4))$ and $\\mathcal{B}\\in H^0((V_4\\times V_6)\\times\\mathbb{P}^1,\\mathcal{O}_{\\mathbb{P}^1}(6))$ are the universal sections and $\\mathcal{U}'$ is flat over $(V_4\\times V_6\\setminus\\{0\\})\\times\\mathbb{P}^1$. Note that \\[\n\\mathbb{P}_{\\mathbb{P}^1}(\\mathcal{O}_{\\mathbb{P}^1}(-2)\\oplus\\mathcal{O}_{\\mathbb{P}^1}(-3)\\oplus \\mathcal{O})=\\mathcal{P}roj_{\\mathbb{P}^1}\\left(\\bigoplus_n\\mathcal{S}ym^n(\\mathcal{O}_{\\mathbb{P}^1}(-2)\\oplus\\mathcal{O}_{\\mathbb{P}^1}(-3)\\oplus \\mathcal{O})\\right),\n\\]\nand $x\\in H^0(\\mathcal{O}_{\\mathbb{P}_{\\mathbb{P}^1}(\\mathcal{O}_{\\mathbb{P}^1}(-2)\\oplus\\mathcal{O}_{\\mathbb{P}^1}(-3)\\oplus \\mathcal{O})}(1)\\otimes\\mathcal{O}_{\\mathbb{P}^1}(2))$, $y\\in H^0(\\mathcal{O}_{\\mathbb{P}_{\\mathbb{P}^1}(\\mathcal{O}_{\\mathbb{P}^1}(-2)\\oplus\\mathcal{O}_{\\mathbb{P}^1}(-3)\\oplus \\mathcal{O})}(1)\\otimes\\mathcal{O}_{\\mathbb{P}^1}(3))$ and $z\\in H^0(\\mathcal{O}_{\\mathbb{P}_{\\mathbb{P}^1}(\\mathcal{O}_{\\mathbb{P}^1}(-2)\\oplus\\mathcal{O}_{\\mathbb{P}^1}(-3)\\oplus \\mathcal{O})}(1))$ are the canonical coordinate functions. Here, we remark that $(V_4\\times V_6) \\times \\mathbb{P}_{\\mathbb{P}^1}(\\mathcal{O}_{\\mathbb{P}^1}(-2)\\oplus\\mathcal{O}_{\\mathbb{P}^1}(-3)\\oplus \\mathcal{O})$ and $\\mathcal{U}'$ admit canonical $GL(2)$-actions equivariantly over $V_4\\times V_6\\times\\mathbb{P}^1$. There exists a section $\\mathcal{S}'$ of $\\Pi':\\mathcal{U}'\\to (V_4\\times V_6\\setminus\\{0\\})\\times\\mathbb{P}^1$ defined by $[0:1:0]$. Note that $\\mathcal{S}'$ is a relative Cartier divisor since $\\mathcal{U}'$ is smooth around $\\mathcal{S}'$. Furthermore, $\\Pi'^*((V_4\\times V_6\\setminus\\{0\\})\\times [0:1])$ and $\\mathcal{O}_{\\mathbb{P}_{\\mathbb{P}^1}}(1)|_{\\mathcal{U}'}$ admit the canonical $GL(2)$-linearizations. Here, we can take a $GL(2)$-linearized relative polarization $\\mathcal{L}'_a=\\Pi'^*((V_4\\times V_6\\setminus\\{0\\})\\times [0:1])+\\frac{a}{3}\\mathcal{O}_{\\mathbb{P}_{\\mathbb{P}^1}}(1)|_{\\mathcal{U}'}$ for $0<a<1$ and $\\mathcal{L}'_a$ has 0-weight of the center $\\mathbb{G}_m\\subset GL(2)$. Then we can take the quotient by the center of $GL(2)$. Let $$M_1=(V_4\\times V_6\\setminus 0)\/\\mathbb{G}_m\\cong \\mathbb{P}(2:2:2:2:2:3:3:3:3:3:3:3)$$ be the universal geometric quotient (cf., \\cite[Definition 0.7]{GIT}) and we obtain\n\\[\n\\Pi:(\\mathcal{U},\\mathcal{L}_a)= (\\mathcal{U}',\\mathcal{L}'_a)\/\\mathbb{G}_m\\to M_1\n\\]\n an $SL(2)$-linearized polarized family equivariant over $M_1$ by the following.\n\\begin{lem}\\label{gittig}\n$(\\mathcal{U},\\mathcal{L}_a)$ exists and is the universal geometric quotient of $(\\mathcal{U}',\\mathcal{L}'_a)$ by $\\mathbb{G}_m$. Here, $\\mathcal{L}_a$ is the quotient of $\\mathcal{L}'_a$ in the sense that if $q:\\mathcal{U}'\\to\\mathcal{U}$ is the quotient map, $q^*\\mathcal{L}_a=\\mathcal{L}'_a$.\n\\end{lem}\n\\begin{proof}\nRecall the construction of $p:V_4\\times V_6\\setminus 0\\to\\mathbb{P}(2:2:2:2:2:3:3:3:3:3:3:3)$. For any nonconstant $\\mathbb{G}_m$-eigenfunction $f\\in H^0(V_4\\times V_6\\setminus0,\\mathcal{O})$, any point of $ (V_4\\times V_6)_f$ is GIT-stable and $p| (V_4\\times V_6)_f$ is a GIT-quotient (see Definition \\ref{degit}). For the first assertion, we have to show that any point of $(V_4\\times V_6)_f \\times \\mathbb{P}_{\\mathbb{P}^1}(\\mathcal{O}_{\\mathbb{P}^1}(-2)\\oplus\\mathcal{O}_{\\mathbb{P}^1}(-3)\\oplus \\mathcal{O})$ is GIT-stable with respect to $\\mathcal{O}_{\\mathbb{P}}(1)$ by \\cite[Theorem 1.10]{GIT}. Since the action of $\\mathbb{G}_m$ has a finite group stabilizer on any point, it suffices to show that any point of $\\mathcal{U}'$ is GIT-semistable by \\cite[Amplification 1.11]{GIT}. This immediately follows from that $x,y$ and $z$ are $\\mathbb{G}_m$-invariant sections and $\\{x=y=z=0\\}=\\emptyset$. \n\nOn the other hand, we can construct $\\mathcal{L}_a$ similarly to the proof of \\cite[Theorem 1.10]{GIT}. \n\\end{proof}\nRecall that $GL(2)$ acts on $\\mathcal{S}'$ equivariantly and let $\\mathcal{S}=\\mathcal{S}'\/\\mathbb{G}_m$. Note also that $3\\mathcal{S}=\\mathcal{O}_{\\mathbb{P}_{\\mathbb{P}^1}}(1)|_{\\mathcal{U'}}\/\\mathbb{G}_m$ as a relative Cartier divisor on $\\mathcal{U}$ over $M_1\\times\\mathbb{P}^1$. On the other hand, by Lemma \\ref{gittig}, the fiber of $\\Pi'$ over $p\\in (V_4\\times V_6\\setminus0)$ coincides with the fiber of $\\Pi$ over $p\\in M_1$ where $p$ is the image of $p'$. Therefore, the fiber $(\\mathcal{U}_s,\\mathcal{L}_{a}|_{\\mathcal{U}_s})$ of $\\Pi$ over $s\\in M_1$ can be denoted as $(W,F+a S)$ where $W$ is the corresponding rational Weierstrass fibration (or the corresponding singular surface), $S$ is the section induced by $\\mathcal{S}$ and $F$ is the fiber class since $\\mathcal{L}_a$ is the sum of $a\\mathcal{S}$ and the fiber over $\\mathbb{P}^1$ of the canonical morphism $\\mathcal{U}\\to\\mathbb{P}^1$. Moreover, $\\Pi$ is flat by \\cite[III Theorem 9.9]{Ha}. By Theorem \\ref{P^1}, GIT-stability of rational Weierstrass fibrations $W$ in the sense of Miranda \\cite[Theorem 6.2]{Mi} coincides with log-twisted K-stability of the base curves. Furthermore, if $W$ is GIT-stable in the sense of Miranda, $(W,S)$ is uniformly adiabatically K-stable as Corollary \\ref{K-fib}. Then, we obtain the following,\n\\begin{thm}\\label{W-unst}\nLet $W$ be a rational Weierstrass fibration. If the minimal resolution of $W$ has one of $II^*,III^*$ or $IV^*$-type fibers, then $(W,F+\\epsilon S)$ is K-unstable for sufficiently small $\\epsilon>0$. If $W$ has a singular fiber type $I^*_N$, then $(W,F+\\epsilon S)$ is not K-stable. Here, $F$ is the fiber class and $S$ is a section.\n\\end{thm}\n\\begin{proof}\nIt is well-known that the Picard number $\\rho(M_1)=1$. Moreover, there exists $s\\in M_1$ such that $(\\mathcal{U}_s,\\mathcal{L}_{\\epsilon}|_{\\mathcal{U}_s})$ is uniformly K-stable for sufficiently small $\\epsilon>0$ by Corollary \\ref{K-fib}. Since $SL(2)$ is a semisimple algebraic group, we can apply \\cite[Theorem 3.4]{OSS} and \\cite[Theorem 6.2]{Mi} to conclude that there exist $r\\in\\mathbb{Q}_{>0}$ and a $\\mathbb{G}_m$-equivariant isomorphism $\\Lambda_{CM}\\cong r\\mathcal{O}_{M_1}(6)$ where $\\Lambda_{CM}=\\lambda_{CM}(\\mathcal{U},\\mathcal{L}_\\epsilon)$ is the CM line bundle with respect to the family $(\\mathcal{U},\\mathcal{L}_\\epsilon)\\to M_1$. Therefore, GIT-stability of $(M_1,\\mathcal{O}_{M_1}(6))$ coincides with that of $(M_1,\\Lambda_{CM})$.\n\\end{proof}\n\\begin{rem}\nThe first assertion also follows from Corollary \\ref{234}. On the other hand, the second one does not follow directly from the general theory we constructed in \\S \\ref{Weier} and \\S \\ref{unstadiab}.\n\\end{rem}\n\n\\begin{rem}\\label{2,3,4}\n\nAs in Theorem \\ref{W-unst}, we can prove the following weaker assertion in a more concrete way when $X$ is a smooth rational elliptic surface than Corollary \\ref{234}.\n\n\\begin{claim}\nLet $X$ be a smooth rational elliptic surface with $m(X)=1$ and $F$ be the fiber class. Suppose that $X$ has one of $II^*,III^*$ or $IV^*$-type fibers. Then for sufficiently small $\\epsilon>0$, there exists an ample line bundle $H$ on $X$ such that $(X,\\epsilon H+F)$ is K-unstable.\n\\end{claim}\n \n\\begin{proof}\nWithout loss of generality, we may assume that $\\pi:X\\to \\mathbb{P}^1$ has a $II^*,III^*$ or $IV^*$-type fiber at $[0:1]\\in \\mathbb{P}^1$. Fix a section $\\tilde{S}$ of $\\pi$ and let $W$ be the Weierstrass fibration of $(X,\\tilde{S})$. Let $S$ be the strict transformation of $\\tilde{S}$ on $W$. By Theorem \\ref{W-unst} and the proof of \\cite[Proposition 5.1]{Mi}, a 1-PS\n\\[\n\\left(\n\\begin{array}{cc}\nq & 0 \\\\\n0 & q^{-1}\n\\end{array}\n\\right)\\in SL(2)\n\\]\ninduces an ample test configuration $(\\mathcal{W},\\mathcal{M})$ of $(W,F+\\epsilon S)$ such that $\\mathrm{DF}(\\mathcal{W},\\mathcal{M})<0$. It is easy to check that $\\mathcal{W}_0$ is a rational Weierstrass fibration with the same type fiber over $[0:1]$ of $X$ and its functional invariant $J$ is constant. Furthermore, $\\mathcal{W}_0$ has a $II,III$ or $IV$-type fiber over $[1:0]$ if $X$ has a $II^*,III^*$ or $IV^*$-type fiber over $[0:1]$ respectively (cf., \\cite[\\S5, \\S9]{Mi}). Then, we can take a test configuration $\\mathcal{X}$ for $X$ by blowing up $\\mathcal{W}$ along horizontal singularities over $\\mathbb{A}^1$ as follows. Let $p\\in W$ be one of the canonical singularities. Let $T$ be the closure of $\\{p\\}\\times\\mathbb{A}^1\\setminus0$ and $\\mathscr{I}$ be the ideal sheaf corresponding to $T$. By \\cite[III Proposition 9.8]{Ha}, $T$ is a flat subvariety over $\\mathbb{P}^1$. Therefore, $T\\cong\\mathbb{A}^1$ and hence the restriction $\\mathscr{I}_0$ to the fiber of $0\\in\\mathbb{A}^1$ coincides with the maximal ideal corresponding to $T_0$. \n \n\\begin{sclaim}\nThe exceptional divisor of the blow up along $T$ is the $\\mathbb{P}^1$-bundle over $\\mathbb{P}^1$.\n\\end{sclaim}\nIt suffices to show that $\\mathscr{I}^n\/\\mathscr{I}^{n+1}$ is flat over $\\mathbb{A}^1$ for $n$. For $n\\gg0$ and $t\\in \\mathbb{A}^1$, $\\mathrm{dim}((\\mathcal{O}_{\\mathcal{W}}\/\\mathscr{I}^n)_t)$ is a constant because all rational double points are hypersurface singularities of $\\mathbb{A}^3$. Therefore, $(\\mathcal{O}_{\\mathcal{W}}\/\\mathscr{I}^n)$ is flat and so is $\\mathscr{I}^n\/\\mathscr{I}^{n+1}$ for $n$. Therefore the exceptional divisor $\\mathcal{P}roj(\\bigoplus\\mathscr{I}^n\/\\mathscr{I}^{n+1})$ is flat over $\\mathbb{A}^1$. We complete the proof of Subclaim. \n \nThus, we can take a test configuration $\\mathcal{W}_1$ by blowing up along $T$. By the inductive argument, we also obtain the test configuration $\\mathcal{X}$ for $X$. $\\mathcal{X}$ is the blow up along the horizontal canonical singularities of $\\mathcal{W}$. All fiber of $\\mathcal{W}$ over $\\mathbb{A}^1$ is a Gorenstein scheme and hence $K_{\\mathcal{W}}$ is Cartier. Hence, it is easy to see that $\\mathcal{X}$ is a normal test configuration of $X$ and $\\rho:\\mathcal{X}\\to \\mathcal{W}$ is crepant. \n  \nWe will prove that there exists a polarization $H$ of $X$ such that $\\mathrm{DF}(\\mathcal{X},\\mathcal{M})<0$ for $(X,\\epsilon H+F)$. Let $\\mathcal{L}$ be the pullback under $\\rho$ of the polarization of $\\mathcal{W}$ and $-\\mathcal{E}$ be a $\\rho$-ample exceptional divisor. Note that $\\mathcal{L}-\\delta\\mathcal{E}$ is a $\\mathbb{G}_m$-linearized polarization of $\\mathcal{X}$ and\n\\[\n\\mathrm{DF}(\\mathcal{X},\\mathcal{L}-\\delta\\mathcal{E})<0\n\\]\nfor sufficiently small $\\epsilon>0$. Thus, $(X,F+\\epsilon S-\\delta E)$ is K-unstable for $0\\ll\\delta\\ll\\epsilon\\ll 1$, where $E=\\mathcal{E}_1$ is the fiber over $1\\in\\mathbb{A}^1$.\n\\end{proof}\n\\end{rem}\n\nOn the other hand, we can prove as Theorem \\ref{W-unst} that there exists the following counter example to Theorem \\ref{j2e} when $\\kappa_B$ is not nef.\n\n\\begin{ex}\\label{lcbaseext}\nThere exists an adiabatically K-unstable klt algebraic fiber space pair $(W,\\Delta_W)$ over $\\mathbb{P}^1$ such that $\\mathrm{dim}\\,W=2$ and $K_W+\\Delta_W$ is relative ample over $\\mathbb{P}^1$. Let $W$ be a rational Weierstrass fibration admitting a section $S$ and a $IV^*$-type fiber. Let also $\\Delta=\\delta S$ for $0<\\delta\\ll1$ and $L$ be a degree 1 line bundle on $\\mathbb{P}^1$. Then, we claim that $(W,\\Delta_W,\\epsilon(K_W+\\Delta)+L)$ is K-unstable for sufficiently small $\\epsilon>0$. Note also that $\\epsilon(K_W+\\Delta)+L\\sim_{\\mathbb{Q}}(1-\\epsilon)L+\\epsilon\\delta S$ and $K_W+\\Delta$ is relatively ample over $\\mathbb{P}^1$.\n\nConsider the family $(\\mathcal{U},\\mathcal{L}_{\\frac{\\epsilon\\delta}{1-\\epsilon}})\\to M_1$ and the log CM line bundle $\\lambda_{\\mathrm{CM}}(\\mathcal{U},\\delta\\mathcal{S},\\mathcal{L}_{\\frac{\\epsilon\\delta}{1-\\epsilon}})$ on $M_1$. By the argument of Theorem \\ref{W-unst}, it suffices to prove that $(X,\\Delta,\\epsilon(K_X+\\Delta)+L)$ is K-stable for smooth rational elliptic surface $X$ that has at most $I_1$-type fiber and sufficiently small $\\epsilon>0$. Here, $S$, $L$ and $\\Delta$ on $X$ are as defined in $W$. Note that $\\alpha_{(X,\\Delta)}(L)=1$ (cf., Definition \\ref{dalpha} and Theorem \\ref{ctn}) and take $\\delta$ sufficiently small that $(K_X+\\Delta +\\frac{5}{4}L)^2>0$. Then, $(X,\\Delta,\\epsilon(K_X+\\Delta)+L)$ is K-stable for sufficiently small $\\epsilon>0$. Indeed, we can show by Lemma \\ref{uuuse} that for any ample test configuration $(\\mathcal{X},\\mathcal{M})$ for $(X,\\epsilon(K_X+\\Delta)+L)$,\n\\[\n(\\mathcal{J}^{K_X+\\Delta+\\frac{5}{4}L})^{\\mathrm{NA}}(\\mathcal{X},\\mathcal{M})\\ge 0.\n\\]\nNote that $\\epsilon(K_X+\\Delta)+L=(1-\\frac{5}{4}\\epsilon)L+\\epsilon(K_X+\\Delta+\\frac{5}{4}L)$ and hence\n\\begin{align*}\n(\\mathcal{J}^{K_X+\\Delta})^{\\mathrm{NA}}(\\mathcal{X},\\mathcal{M})&\\ge -\\frac{5}{4}(\\mathcal{J}^L)^{\\mathrm{NA}}(\\mathcal{X},\\mathcal{M})\\\\\n&\\ge-\\frac{5}{4-5\\epsilon}(I^{\\mathrm{NA}}(\\mathcal{X},\\mathcal{M})-J^{\\mathrm{NA}}(\\mathcal{X},\\mathcal{M})).\n\\end{align*}\nIf $\\epsilon$ is so small that $\\alpha_{(X,\\Delta)}(\\epsilon(K_X+\\Delta)+L)>\\frac{5}{6}$, then\n \\[\n\\frac{3}{2}\\alpha_{(X,\\Delta)}(\\epsilon(K_X+\\Delta)+L)>\\frac{5}{4-5\\epsilon}\n\\]\nand hence $(X,\\Delta,\\epsilon(K_X+\\Delta)+L)$ is K-stable by Theorem \\ref{ctn}, \\cite[Proposition 9.16]{BHJ} and $\\frac{2}{3}I^\\mathrm{NA}\\ge I^\\mathrm{NA}-J^\\mathrm{NA}$. \n\\end{ex}\n\n\nFinally, we prove the next lemma.\n\\begin{lem}\\label{aut}\nLet $\\pi :W\\to \\mathbb{P}^1$ be the rational Weierstrass fibration admitting a section $S$, $\\mathrm{Aut}_0(W)$ be the identity component of the group of automorphisms of $W$. If $W$ is not GIT-polystable but GIT-semistable, $\\mathrm{Aut}_0(W)$ is trivial.\n\\end{lem}\n\\begin{proof}\nLet $J$ be the functional invariant of $W$. First, we consider the case when $J$ is not constant. By rigidity lemma (cf., \\cite[Lemma 1.6]{KoMo}), we can conclude that any smooth fiber $F$ satisfies that $\\pi\\circ\\varphi(F)$ is a point for $\\varphi\\in \\mathrm{Aut}_0(W)$. Furthermore, since $J$ is not constant, we obtain $\\pi\\circ\\varphi(F)=\\pi(F)$. Note that $\\varphi(S)=S$ since $S$ is a $(-1)$-curve. Therefore, $\\varphi$ is the identity morphism on a nonempty open subset of $W$.\n\nSecondly, we consider the case when $J$ is a constant. By \\cite[Proposition 8.4]{Mi}, $J=0$ or $1$. Moreover, by the argument of \\cite[\\S 9]{Mi}, if $J=1$, $W$ has one $I_0^*$-type fiber and two $III$-type fibers. On the other hand, if $J=0$, $W$ has $I_0^*$, $II$ and $IV$-type fibers or has an $I^*_0$ and three $II$-type fibers. Thus, $W$ has at least three singular fibers. By rigidity lemma, if $\\mathrm{Aut}_0(W)$ is nontrivial, $\\varphi\\in\\mathrm{Aut}_0(W)$ induces an automorphism of $\\mathbb{P}^1$ leaving three points fixed. Therefore, $\\varphi$ is trivial again.\n\\end{proof}\n\nThus, by \\cite[\\S8, \\S9]{Mi} and Lemma \\ref{aut}, we can conclude that if a rational Weierstrass fibration $(W,S)$ with a section $S$ is adibatically K-polystable but not K-stable, then $W$ has two $I_0^*$-type fiber.\n\n\\subsection{GIT-stability in the sense of Miranda II: Cubic pencils}\n\nIn this subsection, we obtain some results of the existence of cscK metrics on a smooth rational elliptic surface when the base curve is log-twisted K-semistable by comparing the weight of CM line bundle with the Hilbert-Mumford weight in \\cite{Mi2}.\n\nIn \\cite{Mi2}, all pencils of cubic curves in $\\mathbb{P}^2$ are parametrized by $G$ the $16$-dimensional Grassmannian variety. Recall the fact that the Picard number of $G$ is 1 (cf., \\cite[Lemma 11.1]{Do}). It is easy to see that a locus $F$ of $G$ consisting of pencils that have fixed parts is of codimension $\\ge 2$. In fact, $G$ is the geometric quotient of the GIT-stable locus of $\\mathrm{Mat}(2,10)$ with respect to $GL(2)$ (cf., \\cite[7.1, 7.2]{Mu}). Let $F'$ be a locally closed subset that consists of GIT-stable pencils with fixed parts. Then, $F'$ is the union of $12$ and $15$-dimensional subvarieties and $F$ is the image of $F'$. Note that $F'$ is $GL(2)$-stable and $\\mathrm{codim}\\,F\\ge2$. Let $U=G\\setminus F$. Let $\\mathscr{W}$ be the universal 2-dimensional sub vector space of $\\Gamma(\\mathbb{P}^2\\times G, \\mathcal{O}(3))$ and $\\mathscr{I}$ be the image of\n\\[\n\\mathscr{W}\\otimes \\mathcal{O}(-3)\\to \\mathcal{O}.\n\\]\nIn fact, $(\\mathscr{I}\/\\mathscr{I}^{n+1})|_{\\mathbb{P}^2\\times U}$ is flat over $U$ for $n>0$ since $\\mathcal{O}\/\\mathscr{I}$ has length $9$ over $U$ and by \\cite[Theorem 17.7]{Ma} and the fact that $\\mathscr{I}$ is locally generated by two sections, $\\mathscr{I}^n\/\\mathscr{I}^{n+1}$ has the same length over $U$. Then we blow up $\\mathbb{P}^2\\times U$ along $\\mathscr{I}|_{U}$ and obtain a flat family $\\mathscr{V}$ over $U$. Note that $\\mathscr{V}$ is the family of smooth rational elliptic surfaces admitting a section generically over $U$. We also remark that $\\Pi^*\\mathcal{O}(3)-(1-\\epsilon)E$ is ample for sufficiently small $\\epsilon>0$, where $\\Pi: \\mathscr{V}\\to\\mathbb{P}^2\\times U$ is the morphism and $E$ is the exceptional divisor. It is easy to see that $(\\mathscr{V},\\Pi^*\\mathcal{O}(3)-(1-\\epsilon)E)$ admits an $SL(3)$-action equivariant over $ U$. Then we study the CM line bundle $\\Lambda_{\\mathrm{CM}}=\\lambda_{\\mathrm{CM}}(\\mathscr{V},\\Pi^*\\mathcal{O}(3)-(1-\\epsilon)E)$ on $U$. By the same argument in the proof of \\cite[Converse 1.13]{GIT}, it follows that there exists an $SL(3)$-linearized extension of $\\Lambda_{\\mathrm{CM}}$ to $G$ and let us call this $\\overline{\\Lambda_{\\mathrm{CM}}}$. We claim the following:\n\\begin{prop}\\label{cmext} \n$\\overline{\\Lambda_{\\mathrm{CM}}}$ is ample on $G$.\n\\end{prop}\n\n\nSince $\\rho(G)=1$, to show $\\overline{\\Lambda_{\\mathrm{CM}}}$ is ample, we have to show that there exists $s\\in G$ such that $\\mathscr{V}_s$ is adiabatically K-stable and a nontrivial 1-PS $\\lambda$ such that $\\lim_{t\\to 0}\\lambda(t)\\cdot s\\in U$ as Theorem \\ref{W-unst}. Indeed, if we can show this, the Hilbert-Mumford weight of $\\Lambda_{\\mathrm{CM}}$ with respect to $\\lambda$ and $s$ coincides with a positive multiple of that of an ample line bundle on $G$ by \\cite[Theorem 7.1]{Mi2} and Corollary \\ref{K-fib}.\nHere, we can take a cubic pencil $P$ corresponding to a rational elliptic surface whose all fibers are integral as follows. \n\\begin{itemize}\n\\item Two smooth cubic curves $A$ and $B$ generate $P$ such that $A=X^3+Y^3+Z^3$. Furthermore, $B(X,Y,0)$ and $A(X,Y,0)$ have no common component where $[X:Y:Z]$ is the canonical projective coordinate. \n\\end{itemize}\nThen if 1-PS\n\\[\n\\lambda(q)=\\left(\n\\begin{array}{ccc}\nq^{-1}& 0& 0 \\\\\n0& q^{-1} & 0 \\\\\n0& 0 & q^2\n\\end{array}\n\\right)\\in SL(3)\n\\]\nfor $q\\in \\mathbb{C}^*$, then $P_\\infty=\\lim_{q\\to 0}\\lambda(q)\\cdot P$ has no fixed part and hence $P_\\infty\\in U$. Note that $P$ corresponds to a smooth rational elliptic surface $X$ that has at most integral fibers is GIT-stable due to \\cite[Theorem 7.1]{Mi2}. Recall that $X$ is adaibatically K-stable by Corollary \\ref{K-fib}. Therefore, if such $P$ exists, then Proposition \\ref{cmext} follows. The existence of such $P$ follows from the next lemma.\n\n\n\\begin{lem}\nLet $A$ be a smooth cubic curve such that \n\\[\nA:X^3+Y^3+Z^3=0.\n\\]\nThen there exists a smooth cubic curve $B\\in H^0(\\mathbb{P}^2,\\mathcal{O}(3))$ such that the pencil $P$ generated by $A$ and $B$ corresponds to a rational elliptic surface whose all fibers are integral. \n\\end{lem}\n\\begin{proof}\nConsider $B$ as a point on $Y=\\mathbb{A}^{h^0(\\mathbb{P}^2,\\mathcal{O}(3))}$. Note that there exists the universal section $\\mathcal{B}\\in H^0(Y\\times\\mathbb{P}^2,\\mathcal{O}(3))$ on $Y$. Take an affine open subset $T\\subset Y$ such that for any $B\\in T$, the locus of $B=0$ is smooth, $B(X,Y,0)$ and $A(X,Y,0)$ have no common component and the pencil $P=\\langle A,B\\rangle$ has different 9 base points. There exists a morphism $h:T\\to U$ that maps $B$ to $P=\\langle A,B\\rangle$. By replacing $T$ by a smaller open affine subscheme, we may assume that there is an elliptic fibration\n\\[\n\\pi:\\mathscr{V}\\times_UT\\to T\\times \\mathbb{P}^1\n\\] defined by $A$ and $\\mathcal{B}$. Here, $P_t=\\langle A,B_t\\rangle$ induces the elliptic surface $(\\mathscr{V}\\times_UT)_{t}$ for $B_t\\in T$. Since $\\pi$ is flat and proper, we may assume that all the fiber of $\\pi$ is integral by replacing $T$ by a smaller open neighborhood of $0$ (cf., \\cite[12.2.1]{EGA}). \n\\end{proof}\n\n Next, \\cite[Proposition 5.2]{Mi2} and the calculations show that the following lemma.\n\\begin{lem}\\label{abovelem}\nLet $P$ be a smooth pencil of cubic curves in $\\mathbb{P}^2$ i.e., the general member of $P$ is smooth. Suppose that the elliptic surface corresponding to $P$ has no $I_N^*$-type fiber for any $N\\ge 1$. If $P$ is GIT-semistable but not properly stable, there exists a nontrivial 1-PS $\\lambda$ such that $\\lim_{q\\to 0}\\lambda(q)\\cdot P\\in U$ and the Hilbert-Mumford weight of $\\lambda$ is $0$.\n\\end{lem}\n\\begin{proof}\nNotations as in \\cite[Proposition 5.2]{Mi2}. Let us check the following five cases one by one.\n\\begin{itemize}\n\\item[(6)] If $P$ has a triple line, $P$ is GIT-unstable by \\cite[Proposition 5.1]{Mi2}. Thus, we may assume that $A=x^2z$ and $B$ generate $P$. A 1-PS \\[\n\\lambda(q)=\\left(\n\\begin{array}{ccc}\nq^2& 0& 0 \\\\\n0& q^{-1} & 0 \\\\\n0& 0 & q^{-1}\n\\end{array}\n\\right)\\in SL(3)\n\\]\nhas the Hilbert-Mumford weight 0 (with respect to $P$) by \\cite[Proposition 4.4]{Mi2}. Therefore, $\\lim_{q\\to 0}\\lambda(q)\\cdot B\\in \\langle z^3,yz^2,y^2z,y^3\\rangle$ i.e., $\\lim_{q\\to 0}\\lambda(q)\\cdot B$ is contained in the subspace of $H^0(\\mathbb{P}^2,\\mathcal{O}(3))$ generated by $z^3,yz^2,y^2z,y^3$. If $B$ has the nonzero coefficient of $y^3$, then $\\lim_{q\\to 0}\\lambda(q)\\cdot P\\in U$ and has a closed orbit due to \\cite[Proposition 8.3]{Mi2}. Assume that $B$ has the zero coefficient of $y^3$. If $\\lim_{q\\to 0}\\lambda(q)\\cdot B=az^3+byz^2+cy^2z$, then $c\\ne 0$. In fact, if $c=0$, then the semistable pencil $\\lim_{q\\to 0}\\lambda(q)\\cdot P$ is contained in Case (2) in \\cite[Proposition 5.1]{Mi2}. This is a contradiction. Hence we may assume that $a=0$ and $c=1$ by projective coordinate change. Hence $B\\in \\langle y^2z,yz^2,xy^2,xyz,xz^2,x^2y,x^2z,x^3 \\rangle$. Then a 1-PS\n\\[\n\\lambda'(q)=\\left(\n\\begin{array}{ccc}\nq& 0& 0 \\\\\n0& q^{-2} & 0 \\\\\n0& 0 & q\n\\end{array}\n\\right)\\in SL(3)\n\\]\nhas the Hilbert-Mumford weight 0 and $\\lim_{q\\to 0}\\lambda'(q)\\cdot B\\in \\langle y^2z, xy^2\\rangle$. If $\\lim_{q\\to 0}\\lambda'(q)\\cdot P$ has a fixed part, then $B$'s coefficient of $xy^2$ equals to 0. Therefore, $$B\\in \\langle y^2z,yz^2,xyz,xz^2,x^2y,x^2z,x^3 \\rangle.$$ Then a 1-PS\n\\[\n\\lambda''(q)=\\left(\n\\begin{array}{ccc}\n1& 0& 0 \\\\\n0& q^{-1} & 0 \\\\\n0& 0 & q\n\\end{array}\n\\right)\\in SL(3)\n\\]\nhas the Hilbert-Mumford weight 0 and $\\lim_{q\\to 0}\\lambda''(q)\\cdot B\\in \\langle y^2z, x^2y\\rangle$. If $\\lim_{q\\to 0}\\lambda''(q)\\cdot P$ has a fixed part, then $B$'s coefficient of $x^2y$ equals to 0. Therefore, $$B\\in \\langle y^2z,yz^2,xyz,xz^2,x^2z,x^3 \\rangle.$$ Then a 1-PS\n\\[\n\\lambda'''(q)=\\left(\n\\begin{array}{ccc}\nq^{-1}& 0& 0 \\\\\n0& q^{-4} & 0 \\\\\n0& 0 & q^5\n\\end{array}\n\\right)\\in SL(3)\n\\]\nhas the Hilbert-Mumford weight 0 and $\\lim_{q\\to 0}\\lambda'''(q)\\cdot B\\in \\langle y^2z, x^3\\rangle$. If $\\lim_{q\\to 0}\\lambda''(q)\\cdot P$ has a fixed part, then $B$'s coefficient of $x^3$ equals to 0. Therefore, $B\\in \\langle y^2z,yz^2,xyz,xz^2,x^2z \\rangle$. Therefore, $P$ has a fixed part and it is a contradiction.\n\\item[(7)] In this case, $P$ contains no smooth curve. Thus, it is unnecessary to consider this case.\n\\item[(8)] By projective coordinate change, we may assume that $P$ is generated by $A=xy^2+x^2z$ and $B\\in \\langle y^2z,y^3,xz^2,xyz,xy^2,x^2z,x^2y, x^3\\rangle$. Then a 1-PS\n\\[\n\\lambda(q)=\\left(\n\\begin{array}{ccc}\nq& 0& 0 \\\\\n0& 1 & 0 \\\\\n0& 0 & q^{-1}\n\\end{array}\n\\right)\\in SL(3)\n\\]\nhas the Hilbert-Mumford weight 0 and $\\lim_{q\\to 0}\\lambda(q)\\cdot B\\in \\langle y^2z, xz^2\\rangle$. If $\\lim_{q\\to 0}\\lambda(q)\\cdot B\\ne y^2z+ xz^2$, there is nothing to prove. Suppose that $\\lim_{q\\to 0}\\lambda(q)\\cdot B= y^2z+ xz^2$. Then $B$ intersects with the conic $y^2+xz$ at $[0:0:1]$ with multiplicity more than 2. Therefore, $A$ and $B$ intersect with multiplicity $m>3$ at $[0:0:1]$. Thus, the elliptic surface corresponding to $P$ has a $I_N^*$-type fiber for some $N\\ge 1$.\n\\item[(9)] By projective coordinate change, we may assume that $P$ is generated by $A=y^3-x^2y$ and $B\\in \\langle yz^2,y^2z,y^3,xz^2,xyz,xy^2,x^2z,x^2y,x^3\\rangle$. Then a 1-PS\n\\[\n\\lambda(q)=\\left(\n\\begin{array}{ccc}\nq& 0& 0 \\\\\n0& q & 0 \\\\\n0& 0 & q^{-2}\n\\end{array}\n\\right)\\in SL(3)\n\\]\nhas the Hilbert-Mumford weight 0 and $\\lim_{q\\to 0}\\lambda(q)\\cdot B\\in \\langle yz^2, xz^2\\rangle$. If $\\lim_{q\\to 0}\\lambda(q)\\cdot B\\ne yz^2\\pm xz^2$, or $yz^2$, there is nothing to prove. By projective coordinate change, we may assume that $\\lim_{q\\to 0}\\lambda(q)\\cdot B= yz^2$. Then $B$ is tangent to the line $y$ at the point of concurrency. Therefore, $A$ and $B$ intersect with multiplicity $m\\ge 3$ at $[0:0:1]$. Thus, the elliptic surface corresponding to $P$ has a $I_N^*$-type fiber for some $N\\ge 1$.\n\\item[(10)] In this case, $P$ contains no smooth curve. There is nothing to prove.\n\\end{itemize}\nWe have completed the proof.\n\\end{proof}\nAccording to this lemma, we obtain the following theorem.\n\\begin{thm}\nLet $\\pi:X\\to \\mathbb{P}^1$ be a smooth rational elliptic surface admitting a section without $I_N^*$-type fiber for $N\\ge 1$. Suppose that $f:X\\to \\mathbb{P}^2$ is a blow up at $9$ points. Let $H=f^*\\mathcal{O}_{\\mathbb{P}^2}(3)$ be the $\\pi$-ample line bundle. If $X$ has one and only one $I_0^*$-type fiber, then $(X,\\epsilon H+ \\pi^*\\mathcal{O}_{\\mathbb{P}^1}(1))$ is not K-polystable for sufficiently small $\\epsilon>0$ and has no cscK metric.\n\\end{thm}\n\n\\begin{proof}\n Suppose that $X$ has an $I_0^*$-type fiber but no $I_N^*$-type fiber for $N\\ge 1$. Then as Lemma \\ref{aut}, we can prove that $X$ has two $I^*_0$-type fiber if and only if $\\mathrm{Aut}_0(X)$ is non-trivial. They are also equivalent to that the corresponding pencil $P$ is GIT-polystable by \\cite[Theorem 8.5]{Mi2}. By \\cite[Theorem 8.1]{Mi2}, Lemma \\ref{abovelem} and Proposition \\ref{cmext}, if $X$ has one and only one $I_0^*$-type fiber, there exists an ample test configuration $(\\mathcal{X},\\mathcal{M})$ for $(X,\\epsilon H+ \\pi^*\\mathcal{O}_{\\mathbb{P}^1}(1))$ such that $\\mathrm{DF}(\\mathcal{X},\\mathcal{M})=0$ for sufficiently small $\\epsilon>0$. Therefore, the first assertion follows. The last assertion follows from \\cite[Theorem 1.6]{BDL}.\n\\end{proof}\n\n\\subsection{Applications to Miranda's conjecture on Chow stability}\\label{miranda}\n\nIn \\cite{Mi3}, Miranda conjectured as follows:\n\\begin{conj}[{\\cite[Conjecture 2.3]{Mi3}}]\\label{mi3conj}\nLet $W$ be a rational Weierstrass fibration and $H$ be an arbitrary ample line bundle on $W$. Then, $(W,H)$ is \n\\begin{enumerate}[(a)]\n\\item asymptotically Chow-stable if $W$ has only reduced fibers,\n\\item asymptotically Chow-semistable if $W$ has only reduced and $I_N^*$-type fibers for $N\\ge0$,\n\\item asymptotically Chow-unstable if $W$ has at least one $II^*$, $III^*$ or $IV^*$-type fiber.\n\\end{enumerate}\n\\end{conj}\n\nHe also proved that if $W$ has at least one $II^*$, $III^*$ or $IV^*$-type fiber, there exists an ample divisor $H$ such that $(W,H)$ is asymptotically Chow-unstable in \\cite{Mi3}. In this subsection, we will prove that Conjecture \\ref{mi3conj} (c) holds when $\\rho(W)=2$ and Conjecture \\ref{mi3conj} (b) fails.\n\nFirst, we show the following.\n \n\\begin{lem}\\label{up}\nLet $f:(X,\\Delta,H)\\to \\mathbb{P}^1$ be a polarized $n$-dimensional klt-trivial fibration over $\\mathbb{P}^1$ such that $B$ is the discriminant divisor and $M$ is the moduli divisor of $f$. Suppose that $H$ is an ample divisor on $X$ and $(\\mathbb{P}^1,B,M,L=-(K_{\\mathbb{P}^1}+M+B))$ is a log-twisted Fano curve. Let $(\\mathscr{C},\\mathcal{L})$ be a special log-twisted test configuration for $(\\mathbb{P}^1,B,M,L)$ (cf., Definition \\ref{sfp}) and $(\\mathcal{X},\\mathcal{H})$ be a normal semiample test configuration for $(X,H)$ such that there exists a $\\mathbb{G}_m$-equivariant morphism $\\Pi:\\mathcal{X}\\to \\mathscr{C}$. Suppose also that\n\\[\nK_{(\\mathcal{X},\\mathcal{D})}=\\Pi^*(K_{\\mathscr{C}}+\\rho_{\\mathscr{C}}^*M_{\\mathbb{A}^1}+(\\rho_{\\mathscr{C}})^{-1}_*B_{\\mathbb{A}^1})\n\\]\nwhere $\\mathcal{D}$ is the strict transformation of $\\Delta_{\\mathbb{A}^1}$. Here, $\\rho_{\\mathscr{C}}^*M_{\\mathbb{A}^1}$ is defined as the strict transformation of $M_{1,\\mathbb{A}^1}$ of general $M_1\\in|M|_{\\mathbb{Q}}$ (see Definition \\ref{detw}). Then, there exist positive constants $C_0>0$ and $C_1>0$ such that\n\\[\n((H+mL)^n)^2\\mathrm{DF}_{\\Delta}(\\mathcal{X},\\mathcal{H}+m\\mathcal{L})=\\mathrm{DF}_{(B,M)}(\\mathscr{C},\\mathcal{L})(C_0m^2+C_1m)+(H^n)^2\\mathrm{DF}_{\\Delta}(\\mathcal{X},\\mathcal{H}).\n\\]\n\\end{lem}\n \n\\begin{proof}\nAs in the last part of the proof of Theorem \\ref{can}, we can prove that there exists a positive constant $C_0>0$ such that\n\\[\n((H+mL)^n)^2\\mathrm{DF}_{\\Delta}(\\mathcal{X},\\mathcal{H}+m\\mathcal{L})=C_0m^2\\mathrm{DF}_{(B,M)}(\\mathscr{C},\\mathcal{L})+O(m).\n\\]\n By an easy computation, the $m^1$-term of this polynomial is\n\\begin{align*}\nn&(L\\cdot H^{n-1})(K_{(\\mathcal{X},\\mathcal{D})\/\\mathbb{P}^1}\\cdot \\mathcal{H}^n)+nH^n(K_{(\\mathcal{X},\\mathcal{D})\/\\mathbb{P}^1}\\cdot \\mathcal{L}\\cdot\\mathcal{H}^{n-1})-n(K_{(X,\\Delta)}\\cdot H^{n-1})\\mathcal{L}\\cdot\\mathcal{H}^{n}\\\\\n=&n((L\\cdot H^{n-1})((K_{(\\mathcal{X},\\mathcal{D})\/\\mathbb{P}^1}+\\mathcal{L})\\cdot \\mathcal{H}^n)+H^n(K_{(\\mathcal{X},\\mathcal{D})\/\\mathbb{P}^1}\\cdot \\mathcal{L}\\cdot\\mathcal{H}^{n-1})).\n\\end{align*}\nSince $(\\mathscr{C},\\mathcal{L})$ is a special log-twisted test configuration for $(\\mathbb{P}^1,B,M,L)$, we may assume that $K_{\\mathscr{C}}+\\rho_{\\mathscr{C}}^*M_{\\mathbb{P}^1}+(\\rho_{\\mathscr{C}})^{-1}_*B_{\\mathbb{P}^1}=-\\mathcal{L}$. Hence,\n\\begin{align*}\n\\mathrm{deg}\\, L\\,\\mathrm{DF}_{(B,M)}(\\mathscr{C},\\mathcal{L})=\\left(K_{\\mathscr{C}\/\\mathbb{P}^1}+\\rho_{\\mathscr{C}}^*M_{\\mathbb{P}^1}+(\\rho_{\\mathscr{C}})^{-1}_*B_{\\mathbb{P}^1}+\\frac{1}{2}\\mathcal{L}\\right)\\cdot\\mathcal{L}=\\frac{1}{2}(4\\mathscr{C}_0-\\mathcal{L})\\cdot\\mathcal{L}.\n\\end{align*}\nTherefore, the $m^1$-term is\n\\begin{align*}\nn(H^n(K_{(\\mathcal{X},\\mathcal{D})\/\\mathbb{P}^1}\\cdot \\mathcal{L}\\cdot\\mathcal{H}^{n-1})+2(L\\cdot H^{n-1})(H^n))&=nH^n(4\\mathcal{X}_0-\\mathcal{L})\\cdot\\mathcal{L}\\cdot\\mathcal{H}^{n-1}\\\\\n&=2n(H^n)(H^{n-1}\\cdot L)\\mathrm{DF}_{(B,M)}(\\mathscr{C},\\mathcal{L}).\n\\end{align*}\n\\end{proof}\n \n \nNext, we explain the following important application of Lemma \\ref{up}:\n \n\\begin{thm}[cf., {\\cite[Conjecture 2.3 (c)]{Mi3}}]\\label{jj1}\nLet $W$ be a rational Weierstrass fibration that has one of $II^*$, $III^*$ or $IV^*$-type fibers and $S$ be a section. Then $(W,aS-K_W)$ is K-unstable for $0<a<1$. In particular, $(W,-K_W+aS)$ is also asymptotically Chow(Hilbert)-unstable. Moreover, if the Picard number $\\rho(W)=2$, $W$ is K-unstable for any polarization.\n\\end{thm}\n \n\\begin{proof}\nWithout loss of generality, we may assume that the fiber $F$ over $[0:1]$ is a $II^*$, $III^*$ or $IV^*$-type fiber. Consider the family $(\\mathcal{U},\\mathcal{L}_a)\\to M_1$ constructed in \\S\\ref{wss1} for $0<a<1$. There exists a 1-PS $\\lambda$ of $SL(2)$ such that $W$ degenerates to a $J$-constant elliptic surface via $\\lambda$ (cf., Remark \\ref{2,3,4} and the proof of Claim) and $\\lambda$ has the negative Hilbert-Mumford weight with respect to $P$. Then, we obtain the test configuration $\\mathcal{W}\\subset\\mathcal{U}$ for $W$ and the product test configuration $\\mathscr{C}$ for $\\mathbb{P}^1$ corresponding to $\\lambda$. Let $\\Pi:\\mathcal{W}\\to\\mathscr{C}$ be the $\\mathbb{G}_m$-equivariant structure morphism. Note that $\\Pi$ is flat and all fibers are integral. Then $\\mathcal{W}$ is a klt minimal model over $\\mathscr{C}$ and $(\\mathcal{W},\\mathcal{W}_0)$ is plt by the inversion of adjunction since the central fiber $\\mathcal{W}_0$ of $\\mathcal{W}$ has only canonical singularities over $0\\in\\mathbb{A}^1$. Note that $\\mathcal{W}$ has only canonical singularities since $\\mathcal{W}$ is Gorenstein and klt and $\\mathscr{C}$ is smooth. By the canonical bundle formula of \\cite{FM2}, we obtain\n\\[\nK_{\\mathcal{W}}=\\Pi^*(K_{\\mathscr{C}}+\\mathcal{M}+(\\rho_{\\mathscr{C}})^{-1}_*B_{\\mathbb{A}^1})\n\\]\nwhere $\\mathcal{M}$ is the moduli divisor. We claim that $\\mathcal{M}=\\rho_{\\mathscr{C}}^*M_{\\mathbb{A}^1}$. It suffices to show that the coefficient of $\\mathcal{M}$ is 0 on $\\mathscr{C}_0$ and this can be checked directly as follows. Let $s=\\mathrm{mult}_{\\mathscr{C}_0}(\\mathcal{M})$. Let also $\\overline{\\mathscr{C}}$ be the canonical compactification of $\\mathscr{C}$ over $\\mathbb{P}^1$ and $\\mathscr{H}$ be a very ample general hypersurface of $\\overline{\\mathscr{C}}$ such that $\\Pi_*\\mathcal{O}(cK_{\\mathcal{W}\/\\mathscr{C}})|_{\\mathscr{H}}=(\\Pi|_{\\mathscr{H}})_*\\mathcal{O}(cK_{\\mathcal{W}\\cap\\mathscr{H}\/\\mathscr{H}})$ for sufficiently divisible $c\\in\\mathbb{Z}_{>0}$ as \\cite[(2.6.i)]{Mo}. Then, by \\cite[Proposition 2.2]{FM2}, $\\mathcal{M}|_{\\mathscr{H}}$ is the moduli divisor of $\\Pi_{\\mathscr{H}}$. Since $\\rho_{\\mathscr{C}}$ induces the finite surjection $\\rho:\\mathscr{H}\\to \\mathbb{P}^1$ of smooth curves, we have $\\rho^*M=\\mathcal{M}|_{\\mathscr{H}}$ by \\cite[Corollary 2.5]{FM2}. Thus, $s=0$ and $\\mathcal{M}=\\rho_{\\mathscr{C}}^*M_{\\mathbb{A}^1}$.\n\nWe will show that $(\\mathscr{C},\\mathcal{L}=-(K_{\\mathscr{C}}+\\rho_{\\mathscr{C}}^*M_{\\mathbb{A}^1}+(\\rho_{\\mathscr{C}})^{-1}_*B_{\\mathbb{A}^1}))$ is the log-twisted special test configuration corresponding to $\\lambda$. Furthermore, $\\mathrm{DF}_{(B,M)}(\\mathscr{C},\\mathcal{L})<0$. Indeed, let $h:\\mathbb{P}^1_{\\mathbb{A}^1}\\dashrightarrow\\mathscr{C}$ be the $\\mathbb{G}_m$-equivariant birational map. Recall that $\\lambda$ acts on $\\mathbb{P}^1$ as \n\\[\n[T_0:T_1]\\to [qT_0:q^{-1}T_1], \\, \\mathrm{for} \\, q\\in\\mathbb{G}_m.\n\\]\nThen, $\\tilde{\\mathscr{C}}=\\mathrm{Bl}_{([0:1],0)}(\\mathbb{P}^1_{\\mathbb{A}^1})$ is the resolution of the indeteminacy of $h$ and let $\\psi:\\tilde{\\mathscr{C}}\\to\\mathscr{C}$. If $E$ is the exceptional divisor of $\\rho_{\\tilde{\\mathscr{C}}}:\\tilde{\\mathscr{C}}\\to\\mathbb{P}^1_{\\mathbb{A}^1}$, then\n \\begin{align*}\nK_{\\mathscr{C}\/\\mathbb{A}^1}+\\rho_{\\mathscr{C}}^*M+(\\rho_{\\mathscr{C}})^{-1}_*B&=\\psi_*(K_{\\tilde{\\mathscr{C}}\/\\mathbb{A}^1}+\\rho_{\\tilde{\\mathscr{C}}}^*M+(\\rho_{\\tilde{\\mathscr{C}}})^{-1}_*B)\\\\\n&=\\psi_*(K_{\\tilde{\\mathscr{C}}\/\\mathbb{A}^1}+\\rho_{\\tilde{\\mathscr{C}}}^*M+(\\rho_{\\tilde{\\mathscr{C}}})^{-1}_*B-s\\,(\\rho_{\\tilde{\\mathscr{C}}})^{-1}_*(\\mathbb{P}^1\\times\\{0\\}))\\\\\n&\\sim_{\\mathbb{A}^1,\\mathbb{Q}}\\psi_*(\\rho_{\\tilde{\\mathscr{C}}}^*\\mathcal{O}_{\\mathbb{P}^1_{\\mathbb{A}^1}}(-1)\\otimes\\mathcal{O}_{\\tilde{\\mathscr{C}}}(E)).\n\\end{align*}\nHere, $\\frac{1}{2}<s=\\mathrm{mult}_{[0:1]}B<1$ and $\\rho_{\\tilde{\\mathscr{C}}}^*\\mathcal{O}_{\\mathbb{P}^1_{\\mathbb{A}^1}}(1)\\otimes\\mathcal{O}_{\\tilde{\\mathscr{C}}}(-E)$ is the pullback of an $\\mathbb{A}^1$-ample line bundle on $\\mathscr{C}$. Thus, $-(K_{\\mathscr{C}}+\\rho_{\\mathscr{C}}^*M_{\\mathbb{A}^1}+(\\rho_{\\mathscr{C}})^{-1}_*B_{\\mathbb{A}^1})$ is $\\mathbb{A}^1$-ample. It is easy to see that $(\\mathscr{C},\\rho_{\\mathscr{C}}^*M_{\\mathbb{A}^1}+(\\rho_{\\mathscr{C}})^{-1}_*B_{\\mathbb{A}^1}+\\psi_*E)$ is plt and \n\\[\n\\mathrm{DF}_{(B,M)}(\\mathscr{C},-(K_{\\mathscr{C}\/\\mathbb{A}^1}+\\rho_{\\mathscr{C}}^*M+(\\rho_{\\mathscr{C}})^{-1}_*B))=\\mathrm{DF}_{(B,M)}(\\tilde{\\mathscr{C}},-\\rho_{\\tilde{\\mathscr{C}}}^*\\mathcal{O}_{\\mathbb{P}^1_{\\mathbb{A}^1}}(-1)\\otimes\\mathcal{O}_{\\tilde{\\mathscr{C}}}(E))<0\n\\]\nby the above equation. Therefore, $\\mathcal{W}$ satisfies the assumption of Lemma \\ref{up}. Similarly to the argument in Theorem \\ref{W-unst}, we only have to show that the Donaldson-Futaki invariant of the corresponding test configuration $(\\mathcal{W},\\mathcal{L}_{1-\\epsilon}|_{\\mathcal{W}})$ of $(W, (1-\\epsilon)S-K_W)$ is negative for sufficiently small $\\epsilon>0$. Indeed, since the test configuration $\\mathcal{W}$ has only canonical singularities and has the relative trivial canonical divisor over the special log-twisted test configuration $\\mathscr{C}$ for $\\mathbb{P}^1$ whose Donaldson-Futaki invariant is negative, $\\mathrm{DF}(\\mathcal{W},\\mathcal{L}_{1-\\epsilon}|_{\\mathcal{W}})>\\mathrm{DF}(\\mathcal{W},\\mathcal{L}_{a}|_{\\mathcal{W}})$ for $a<1-\\epsilon$ by Lemma \\ref{up}.\n \nNext, we prove $\\mathrm{DF}(\\mathcal{W},\\mathcal{L}_{1-\\epsilon}|_{\\mathcal{W}})<0$ as follows. By the same argument of Theorem \\ref{W-unst}, we only have to prove that $(W',(1-\\epsilon)S'-K_{W'})$ for elliptic surfaces $W'$ corresponding to general points in $M_1$. Here, $S'$ is the restriction of $\\mathcal{S}$ on $\\mathcal{U}$ to $W'$. Recall that such rational Weierstrass fibration $W'$ has at most irreducible fiber and is smooth, and it is easy to see that $-K_{W'}+S'$ is semiample and the ample model of $-K_{W'}+S'$ is a smooth del Pezzo surface $Y$ of degree 1. Thanks to \\cite{T}, smooth del Pezzo surfaces of degree 1 have the K\\\"{a}hler-Einstein metrics and hence $(W',(1-\\epsilon)S'-K_{W'})$ has a cscK metric due to the main theorem of \\cite{AP}. As Lemma \\ref{aut}, we can prove that $\\mathrm{Aut}_0(W')$ is trivial since the functional invariant $J$ of $W'$ is not constant. Thanks to \\cite{St}, $(W',(1-\\epsilon)S'-K_{W'})$ is also K-stable. We complete the proof of the first assertion. The rest follows immediately from the first assertion. \n\\end{proof}\n\n\\begin{rem}\nWe emphasize that this theorem is different from Theorem \\ref{W-unst} in that this theorem states not only about adiabatic K-unstability. We also remark that for example, if a rational Weierstrass fibration $W$ has a $II^*$-type fiber, then $\\rho(W)=2$. Thus, we conclude that \\cite[Conjecture 2.3 (c)]{Mi3} holds for such rational Weierstrass fibrations.\n\\end{rem}\n \nNext, we recall the fact in \\cite[\\S 8]{Mi} that there exist the test configurations $\\mathcal{W}$ for rational Weierstrass fibrations $W$ admitting $I_N^*$-type fibers for some $N\\ge 1$ such that the central fiber is a GIT-polystable singular surface $W_0$. Here, $W_0$ is defined by the equation $y^2z=(x-T_0T_1z)^2(x+2T_0T_1z)$ in $\\mathbb{P}_{\\mathbb{P}^1}(\\mathcal{O}(-2)\\oplus \\mathcal{O}(-3)\\oplus \\mathcal{O})$. $W_0$ has general nodal rational curve fiber and two cuspidal rational curve fibers. Therefore, $W_0$ is Gorenstein. The normalization of $W_0$ equals to $X_0=\\mathbb{P}_{\\mathbb{P}^1}(\\mathcal{O}\\oplus \\mathcal{O}(1))$ since $W_0$ has a $(-1)$-section $S$ and since the singular locus is isomorphic to the base curve. Thus $W_0$ is an slc surface. Note that the conductor scheme $D\\subset X_0$ is smooth, irreducible and linearly equivalent to $2S+2f$, where $f$ is the fiber class, since $K_{W_0}\\sim -f$.\n\\begin{prop}\\label{WW-unst}\n$(W_0,S)$ is adiabatically K-unstable over $\\mathbb{P}^1$.\n\\end{prop}\n \n\\begin{proof}\nIt suffices to show that $D$ is a destabilizing subvariety of $(X_0,D,L=S+mf)$ for $m\\gg 0$ i.e., $Z$ induces a semiample deformation to the normal cone $(\\mathcal{X},\\mathcal{L})$ such that $\\mathrm{DF}_B(\\mathcal{X},\\mathcal{L})<0$ (cf., \\cite[Definition 4.17]{RT07}). Let $(\\mathcal{X},\\mathcal{L}=L_{\\mathbb{A}^1}-\\epsilon E)$ be the non almost trivial semiample deformation to the normal cone of $D$ for $(X,L)$ where $E$ is the exceptional divisor and $\\epsilon>0$. Since $D$ is a log canonical center, $H_D^{\\mathrm{NA}}(\\mathcal{X},\\mathcal{L})=0$. Note also that $$\\left(2\\frac{f\\cdot (S+mf)}{(S+mf)^2}(S+mf)-f\\right)\\cdot D>0.$$ Then, it follows that $(\\mathcal{J}^f)^{\\mathrm{NA}}(\\mathcal{X},\\mathcal{L})>0$ from the similar argument of the proof of \\cite[Theorem 6.3]{Hat}. Since $K_{X_0}+D\\sim-f$, we can conclude that $M_D^{\\mathrm{NA}}(\\mathcal{X},\\mathcal{L})=(\\mathcal{J}^{-f})^{\\mathrm{NA}}(\\mathcal{X},\\mathcal{L})<0$ for sufficiently small $\\epsilon$.\n\\end{proof}\n \nWe deduce the following theorem from Proposition \\ref{WW-unst}.\n\\begin{thm}[cf. Conjecture 2.3 (b) \\cite{Mi3}]\\label{Mirconj}\nRational Weierstrass fibrations admitting an $I_N^*$-type fiber for some $N\\ge1$ are asymptotically Chow unstable.\n\\end{thm}\nTo prove this, recall the following lemma, \n\n\\begin{lem}\\label{lastlem}\nLet $f:(X,L)\\to B$ be a projective morphism of projective varieties and $L$ be a $f$-ample line bundle such that $R^if_*L=0$ for $i>0$. Suppose that a semisimple algebraic group $G$ acts on $(X,L)$ and $B$ equivariantly. Let $\\Lambda$ be the Chow bundle $\\mathrm{Chow}(X,L)$. Suppose that the Picard number $\\rho(B)=1$ and there exists a non trivial $1$-PS $\\lambda$ of $G$ and a point $s\\in B$ such that a Chow-stable variety $(X_s,L_s)$ degenerates non trivially via $\\lambda$.\n\n Then, $\\Lambda$ is ample and a point $b\\in B$ is GIT (poly, semi)stable with respect to $\\Lambda$ if $(X_b,L_b)$ is Chow-(poly, semi)stable.\n\\end{lem}\nReplacing the CM line bundle by the Chow bundle, the lemma follows similarly to \\cite[Theorem 3.4]{OSS}.\n\nWe also need the following lemma in GIT:\n\\begin{lem}\\label{github1}\nLet $G$ be a reductive algebraic group and act on a polarized variety $(X,L)$. Suppose that there exists $1$-PS $\\lambda$ such that $\\lim_{q\\to0}\\lambda(q)\\cdot x=y$. If the Hilbert-Mumford weight of $\\lambda$ is $0$ and $x$ is GIT-semistable, then $y$ is also GIT-semistable. \n\\end{lem}\n\n\\begin{proof}\nFor the readers' convenience, we prove this here. As in the first part of the proof of \\cite[Theorem 2.1]{GIT}, we may assume that $(X,L)=(\\mathbb{P}^{n-1},\\mathcal{O}(1))$. Let $\\pi:\\mathbb{A}^n\\setminus\\{0\\}\\to\\mathbb{P}^{n-1}$ be the canonical projection and $x^*\\in\\mathbb{A}^n\\setminus\\{0\\}$ such that $\\pi(x^*)=x$. By the assumption and \\cite[Proposition 2.3]{GIT}, there exists $y^*=\\lim_{q\\to 0}\\lambda(q)\\cdot x^*\\ne0$. On the other hand, $z\\in\\mathbb{P}^{n-1}$ is GIT-semistable if and only if there exists a nonconstant $G$-invariant homogeneous polynomial $F$ such that $F(z^*)\\ne0$ for $z^*\\in\\mathbb{A}^n\\setminus\\{0\\}$ such that $\\pi(z^*)=z$ (see the proof of \\cite[Proposition 2.2]{GIT}). Therefore, we can find a nonconstant $G$-invariant homogeneous polynomial $F$ such that $F(x^*)\\ne0$. By the definition of $y^*$, we have that $F(y^*)\\ne0$. Therefore, $y$ is also GIT-semistable.\n\\end{proof}\n\n\\begin{proof}[Proof of Theorem \\ref{Mirconj}]\nNote that the general points of $M_1$ corresponds to polarized smooth rational elliptic surfaces admitting cscK metrics adiabatically by Corollary \\ref{K-fib}. Due to \\cite[Corollary 4]{Dn}, those surfaces are asymptotically Hilbert stable. By \\cite{Fo} (cf., \\cite[p.215]{GIT}) and \\cite{Mb} or \\cite[Theorem 2.5]{Od}, asymptotic Hilbert stability coincides with asymptotic Chow stability. Therefore, we can apply Lemma \\ref{lastlem} to $M_1$ and $(\\mathcal{U},\\mathcal{L}_\\epsilon)$ for sufficiently small $\\epsilon>0$ to conclude that $\\mathrm{Chow}(\\mathcal{U},\\mathcal{L}_\\epsilon^{\\otimes k})$ is ample for sufficiently large $k>0$ as in Theorem \\ref{W-unst}. Let $W$ be a rational Weierstrass fibration admitting an $I_N^*$-type fiber for some $N\\ge 1$. Take a 1-PS $\\lambda$ whose corresponding test configuration $\\mathcal{W}$ satisfies the following property.\n\\begin{itemize}\n\\item The general fiber of $\\mathcal{W}$ is $W$ over $\\mathbb{A}^1$.\n\\item The central fiber of $\\mathcal{W}$ is $W_0$ as in Proposition \\ref{WW-unst}.\n\\item The weight of $\\mathcal{O}(6)$ with respect to $\\lambda$ is 0 for sufficiently small $\\epsilon>0$ and for sufficiently large $k>0$.\n\\end{itemize} \n Then, we conclude that the weight of $\\mathrm{Chow}(\\mathcal{U},\\mathcal{L}_\\epsilon^{\\otimes k})$ for sufficiently small $\\epsilon>0$ and for sufficiently large $k>0$ with respect to $\\lambda$ is 0. Assume that the theorem would fail for $W$. Since the Hilbert-Mumford weight of $\\mathrm{Chow}(\\mathcal{U},\\mathcal{L}_\\epsilon^{\\otimes k})$ with respect to $\\lambda$ and $W$ is 0, $W_0$ is also Chow semistable with respect to the embedding $W_0\\hookrightarrow \\mathbb{P}(H^0(\\mathcal{L}_{\\epsilon|W_0}^{\\otimes k}))$ for sufficiently small $\\epsilon>0$ and for sufficiently large $k>0$ by Lemma \\ref{github1}. However, $(W_0,S)$ is adiabatically K-unstable by Proposition \\ref{WW-unst} and hence $(W_0,\\epsilon S+L)$ is asymptotically Chow unstable for sufficiently small $\\epsilon>0$. It is a contradiction. Hence, we complete the proof.\n\\end{proof}\nThus, we have proved that \\cite[Conjecture 2.3 (b)]{Mi3} fails in general.\n  \n \n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction and Main Results}\n\nThe system of Rotating Shallow Water (RSW) equations is a widely\nadopted 2D approximation of the 3D incompressible Euler equations\nand the Boussinesque equations in the regime of large scale\ngeophysical fluid motion (\\cite{Pedlosky}). It is also regarded as\nan important extension of the compressible Euler equations with\nadditional rotational forcing.\n\nStart with the following formulation,\n\\begin{align}\n\\label{RSWh}\\partial_t h+\\nabla\\!\\!\\cdot\\!(h{\\bf u})&=0,\\\\\n\\label{RSWu} \\partial_t{\\bf u}+{\\bf u}\\!\\cdot\\!\\!\\nabla{\\bf u}+\\nabla h+{\\bf u}^\\perp&=0,\n\\end{align}\nwhere $h=h(t,x_1,x_2)$  and $ {\\bf u}=(u_1(t,x_1,x_2),u_2(t,x_1,x_2))^T$ denote the total height and\nvelocity of the fluids, respectively, and ${\\bf u}^\\perp:=(-u_2, u_1)^T$\ncorresponds to the rotational force. For mathematical convenience,\nall physical parameters are scaled to the unit (cf. \\cite{Majda} for\ndetailed discussion on scaling).\n\nSince $(h, {\\bf u})=(1,0)$ is a steady-state solution of  (\\ref{RSWh}), (\\ref{RSWu}), we introduce the perturbations\n$(\\rho,{\\bf u}):=(h-1,{\\bf u})$ and arrive at\n\\begin{align}\n\\label{RSWrho}\\partial_t\\rho+\\nabla\\!\\!\\cdot\\!(\\rho{\\bf u})+\\nabla\\!\\!\\cdot\\!{\\bf u}&=0,\\\\\n\\label{RSWurho} \\partial_t{\\bf u}+{\\bf u}\\!\\cdot\\!\\!\\nabla{\\bf u}+\\nabla\\rho+ {\\bf u}^\\perp&=0,\n\\end{align}\nsubject to initial data\n\\begin{equation}\\label{RSWinit}\\quad\\rho(0,\\cdot)=\\rho_0,\\;{\\bf u}(0,\\cdot)={\\bf u}_0.\\end{equation}\n\nAn important feature of the RSW system is that the relative\nvorticity \\(\\theta:=\\nabla\\times{\\bf u}-\\rho=(\\partial_1 u_2-\\partial_2 u_1)-\\rho\\) is\nconvected by ${\\bf u}$, \\begin{equation}\\label{theta}\\partial_t\\theta+\\nabla\\!\\!\\cdot\\!(\\theta{\\bf u})=0.\\end{equation}\nIndeed, $\\nabla\\times$(\\ref{RSWurho})$-$(\\ref{RSWrho}) readily leads\nto (\\ref{theta}). The linearity of (\\ref{theta}) then suggests that\n$\\theta\\equiv0$ be an invariant with respect to time (as long as\n${\\bf u}\\in C^1$), i.e. \\begin{equation}\\label{inv}\n\\theta_0\\equiv0\\iff\\theta(t,\\cdot)\\equiv0\\iff\\nabla\\times{\\bf u}\\equiv\\rho.\n\\end{equation}\n\nBefore stating the main theorems, we fix some notations. For $1\\leq\np\\leq \\infty$, let $L^p$ denote the standard $L^p$ space on $\\mbox{R}^2$.\nFor $l \\geq 0$ and $s\\geq 0$, define the weighted Sobolev norm\nassociated with the space $H^{l,s}$ as\n\\begin{align}\\label{def:weighted}\n\\|v\\|_{H^{l,s}}:=\\|(1+|x|^2)^{s\/2}(1-\\Delta)^{l\/2} v\\|_{L^2}.\n\\end{align}\nAlso, denote the standard Sobolev space $H^l:=H^{l,0}$.\n\n\\begin{theorem}\\label{mainTh}\nConsider the RSW system (\\ref{RSWrho}), (\\ref{RSWurho}), (\\ref{RSWinit}) with initial data ${\\bf u}_0=(u_{1,0}, u_{2,0})^T\\in H^{k+2,k}$ for $k\\ge52$ and zero relative vorticity,\n\\[\n\\rho_0=\\partial_1 u_{2,0}-\\partial_2 u_{1,0}.\n\\]\nThen, there exists a universal constant $\\delta_0>0$ such that the\nRSW system admits a unique\nclassical solution $(\\rho, {\\bf u})$ for all time, provided that the\ninitial data satisfy\n\\begin{equation*}\\label{Isizeu}\n\\|{\\bf u}_0\\|_{H^{k+2,k}}=\\delta<\\delta_0.\n\\end{equation*}\nMoreover, there exists a free solution ${\\bf u}^+(t,\\cdot)$ such that \n\\[\\|{\\bf u}(t,\\cdot)-{\\bf u}^+(t,\\cdot)\\|_{H^{k-15}}+\\|\\partial_t{\\bf u}(t,\\cdot)-\\partial_t{\\bf u}^+(t,\\cdot)\\|_{H^{k-16}}\\leq C(1+t)^{-1},\\]\nwhere\n${\\bf u}^+(t,\\cdot):=(\\cos(1-\\Delta)^{1\/2}\nt){\\bf u}_0^++\\left((1-\\Delta)^{-1\/2}\\sin((1-\\Delta)^{1\/2}\nt)\\right){\\bf u}_1^+$ for some ${\\bf u}_{0}^+\\in H^{k-15}$ and ${\\bf u}_1^+\\in H^{k-16}$.\n\\end{theorem}\nThe proof is a straightforward combination of Lemma \\ref{symm}, Theorem \\ref{KGsystem} and Theorem \\ref{asymTh} below.\n\nThis result shows fundamentally different lifespan of the classical\nsolutions for the RSW system in comparison with the compressible\nEuler systems. Note that the relative vorticity $\\theta=\\nabla\\times{\\bf u}-\\rho$ in the RSW\nequations plays a very similar role as vorticity $\\nabla\\times{\\bf u}$ in the compressible\nEuler equations. Correspondingly, RSW solutions with zero relative\nvorticity $\\theta$ is an analogue of irrotational solutions for the\ncompressible Euler equations. However, the life span for 2D\ncompressible Euler equations with zero vorticity was proved to be\nbounded from below (Sideris \\cite{Sideris:2D}) and above (Rammaha\n\\cite{Rammaha}) by $O({1}\/{\\delta^2})$. Here, $\\delta$ indicates the\nsize of the initial data. Sideris also showed that the life span in\nthe 3D case is bounded from below by $O(e^{1\/\\delta})$ in\n\\cite{Sideris:3D} and from above by $O(e^{1\/{\\delta^2}})$ in\n\\cite{Sideris:3D:singularity}. Our result for 2D RSW system, on the other hand, is global in time due to the additional rotating\nforce. Consult \\cite{Babin, LiTa:rotation, ChTa:SIAM} for related results on global and long-time existence of classical RSW solutions in various regimes.\n\nA key ingredient of the proof is to treat the RSW system as a system\nof quasilinear Klein-Gordon equations -- cf. Lemma \\ref{symm} for a formal\ndiscussion. Such reformulation allows us to utilize the fruitful\nresults on nonlinear Klein-Gordon equations appearing in recent decades. To\nmention a few, for spatial dimensions $N\\ge5$, Klainerman and\nPonce \\cite{KlainermanP} and Shatah \\cite{Shatahev} showed that the\nKlein-Gordon equation admits a unique, global solution for small\ninitial data and that the solution approaches the free solution of\nthe linear Klein-Gordon equation as $t\\rightarrow \\infty$. The\nproofs in \\cite{KlainermanP, Shatahev} are based on $L^p-L^q$ decay\nof the linear Klein-Gordon equations. The global existence for  quasilinear Klein-Gordon equations in dimensions $N=4,3$ was proved\nindependently by Klainerman in \\cite{Klainerman} using the vector\nfields approach and Shatah in \\cite{Shatah} using the normal forms.\nIn the $N=2$ case, global existence of classical solutions become increasingly subtle due\nto the $(1+t)^{-1}$ decay rate of solutions to linear Klein-Gordon equations. Nevertheless, it has been proved by Ozawa et al in\n\\cite{OzawaSL} for semilinear, scalar equations. The authors combined\nthe vector fields approach and the normal form method after partial\nresults in \\cite{GeorgievP, Georgiev, Simon}. The result of global existence on\nquasilinear, scalar Klein-Gordon equations was announced in\n\\cite{OzawaQL}. Recently, Delort et al obtained global existence for\na two dimensional system of two Klein-Gordon equations in\n\\cite{Fang}, where the authors transformed the problem using\nhyperbolic coordinates and then studied it with the vector fields\napproach, which was restricted to compactly supported initial data.\nFor applications of the Klein-Gordon equations in fluid equations,\nwe refer to \\cite{Guo} by Y. Guo on global existence of three\ndimensional Euler-Poisson system. Note that the irrotationality\ncondition used there plays a counterpart of the\nzero-relative-vorticity constraint in our result.\n\nFor general initial data, we have the following theorem on the\nlifespan of classical solutions. Its proof is given in Section 5.\n\\begin{theorem}\\label{perTh}\nConsider the RSW system (\\ref{RSWrho}), (\\ref{RSWurho}), (\\ref{RSWinit}) with initial data $(\\rho_0,{\\bf u}_0)\\in H^{k+1,k}$ for $k\\ge52$. Let $\\delta$ denote the size of the initial data \\begin{equation*}\n\\delta=\\|(\\rho_0,{\\bf u}_0)\\|_{H^{k+1,k}},\n\\end{equation*}\n and $\\varepsilon$ the size of the initial relative vorticity,\n \\[\\varepsilon=\\|(\\partial_1 u_{2,0}-\\partial_2 u_{1,0})-\\rho_0\\|_{H^2}.\\]\nThen, there exists a universal constant $\\delta_0>0$ such that, for any $\\delta\\le\\delta_0$, the\nRSW system admits a unique\nclassical solution $(\\rho, {\\bf u})$ for \\begin{equation}\\label{span:general}t\\in[0,\nC_1\\varepsilon^{-\\frac{1}{1+C_2\\delta}}].\\end{equation}\nHere, $C_1$ and $C_2$ are constants independent of $\\delta$ and $\\varepsilon$. \n\\end{theorem}\n\nThis theorem confirms the key role that relative vorticity plays in the studies of Geophysical Fluid Dynamics (\\cite{Pedlosky}). In fact, having two uncorrelated scales $\\varepsilon$ and $ \\delta$ in (\\ref{span:general})\nallows us to solely let the size of the initial relative vorticity\n$\\varepsilon\\to0$ and achieve a very long lifespan of classical solutions, regardless of the total size of initial data. The proof, given in Section \\ref{sec:general}, treats the full RSW system as perturbation to the zero-relative-vorticity one and utilizes the standard energy methods for symmetric hyperbolic PDE systems. The sharp estimates of Theorem \\ref{mainTh} play a crucial role in controlling the total energy growth. We note that a similar problem of the compressible Euler equations is studied by Sideris in \\cite{Sideris:2D}.\n\nWe also note by passing that the study of hyperbolic PDE systems\nwith small initial data is closely related, if not entirely\nequivalent, to the singular limit problems with large initial data.\nSee \\cite{Majda}, \\cite{Babin} and references therein for results on\nthe particular case of inviscid RSW equations. A recent survey paper\n\\cite{Bresch} contains a collection of open problems and recent\nprogress on viscous Shallow Water Equations and related models.\n\nWe finally comment that all results in this paper should be true for two dimensional compressible Euler equations with rotating force and general pressure law. The proof remains largely the same except for the symmetrization part and associated energy estimates.\n\nThe structure of the rest of the paper is outlined as following. In\nSection 2, we reformulate the RSW system into a symmetric hyperbolic\nsystem of first order PDEs. Under the zero-relative-vorticity\nconstraint, it is further transformed into a system of quasilinear\nKlein-Gordon equations with symmetric quasilinear part. Section 3 is\ndevoted to the local wellposedness of the RSW equation with general\ninitial data and zero-relative-vorticity initial data. Section 4\ncontains the proof of Theorem \\ref{mainTh} in a series of lemmas,\nadapting results from \\cite{Georgiev}, \\cite{Shatah}, \\cite{OzawaSL}.\nThe discussion and proof of Theorem \\ref{perTh} on general initial\ndata can be found in Section 5. The Appendix contains the proof of a\ntechnical proposition used in Section 4.\n\n\\section{Reformulation of the Problem}\n\nIn order to obtain local wellposedness for (\\ref{RSWrho}) and\n(\\ref{RSWurho}), we first symmetrize the system into a symmetric\nhyperbolic system. This will also be used for proving global\nexistence, where we need to reduce (\\ref{RSWrho}) and(\\ref{RSWurho})\nto a symmetric quasilinear Klein-Gordon system.\n\nIntroduce a symmetrizer $m:=2\\left(\\sqrt{1+\\rho}-1\\right)$ such that\n$\\rho=m+{1\\over4}m^2$, then (\\ref{RSWrho}), (\\ref{RSWurho}) are\ntransformed into a symmetric hyperbolic PDE system,\n\\begin{align}\n\\label{RSWm}\\partial_t m+{\\bf u}\\!\\cdot\\!\\!\\nabla m+\\dfrac{1}{2} m\\nabla\\!\\!\\cdot\\!{\\bf u}+\\nabla\\!\\!\\cdot\\!{\\bf u}&=0,\\\\\n\\label{RSWum} \\partial_t{\\bf u}+{\\bf u}\\!\\cdot\\!\\!\\nabla{\\bf u}+\\dfrac{1}{2} m\\nabla m+\\nabla m+{\\bf u}^\\perp&=0.\n\\end{align}\n\nThe following lemma asserts that, under the invariant (\\ref{inv}),\nthe above system amounts to a system of Klein-Gordon equations with\nsymmetric quasilinear terms.\n\n\\begin{lemma}\\label{symm}\nUnder the invariant (\\ref{inv}), $\\nabla\\times{\\bf u}=\\rho$, and\ntransformation $m=2\\left(\\sqrt{\\rho+1}-1\\right)$, the solution to\nthe RSW system (\\ref{RSWrho}), (\\ref{RSWurho}) satisfies the\nfollowing symmetric system of quasilinear Klein-Gordon equations for\n${\\bf U}:=\\left(\\begin{matrix} m\\\\{\\bf u}\\end{matrix}\\right)$, \\begin{equation}\\label{ptt}\n\\partial_{tt}{\\bf U}-\\Delta{\\bf U}+{\\bf U}=\\sum_{i,j=1}^2A_{ij}({\\bf U})\\partial_{ij}{\\bf U} +\n\\sum_{j=1}^2A_{0j}({\\bf U})\\partial_{0j}{\\bf U}+R(\\tilde{\\vU}\\otimes\\tilde{\\vU}), \\end{equation} where\nlinear functions $A_{ij}$ and $A_{0j}$ map $\\mbox{R}^3$ vectors to\n\\emph{symmetric} $3\\times 3$ matrices and satisfy $A_{ij}=A_{ji}$.\nThe remainder term $R$ depends linearly on the tensor product\n$\\tilde{\\vU}\\otimes\\tilde{\\vU}$ with $\\tilde{\\vU}:=({\\bf U}^T,\\partial_t{\\bf U}^T,\\partial_1{\\bf U}^T,\\partial_2{\\bf U}^T)$.\n\\end{lemma}\n\nHere and below, for notational convenience, we use both $\\partial_t$ and\n$\\partial_0$ to denote the time derivatives.\n\\begin{proof}\nRewrite (\\ref{RSWm}), (\\ref{RSWum}) into a matrix-vector form,\n\\begin{equation}\n\\label{RSWmatrix}\\partial_t{\\bf U}+\\sum_{a=1,2}(u_aI+\\dfrac{1}{2} mJ_a)\\partial_a{\\bf U}={\\mathcal L}({\\bf U}),\n\\end{equation}\nwhere\n\\begin{equation}\\label{defJL}\nJ_1:=\\bma0&1&0\\\\1&0&0\\\\0&0&0\\end{matrix}\\right),\\,\\,\\,\\, J_2:=\\bma0&0&1\\\\0&0&0\\\\1&0&0\\end{matrix}\\right),\n\\,\\,\\, \\, \\text{and}\\,\\, {\\mathcal L}({\\bf U}):=-\\left(\\begin{matrix} \\nabla\\!\\!\\cdot\\!{\\bf u}\\\\\n\\nabla m+{\\bf u}^\\perp\\end{matrix}\\right). \\end{equation}\n\nBy taking time derivative on the above system, we have\n\\[\\partial_{tt}{\\bf U}+N({\\bf U})={\\mathcal L}^2({\\bf U}),\\]\nwhere the nonlinear term\n\\begin{equation}\\label{N1N2}\\begin{split}\nN({\\bf U})&=\\partial_t\\sum_{a=1,2}(u_aI+\\dfrac{1}{2}\nmJ_a)\\partial_a{\\bf U}+{\\mathcal L}\\left(\\sum_{a=1,2} (u_aI+\\dfrac{1}{2}\nmJ_a)\\partial_a{\\bf U}\\right)\\\\&:=N_1+N_2.\n\\end{split}\\end{equation}\nThe ${\\mathcal L}^2$ term, using the  calculus identity\n$\\nabla(\\nabla\\cdot{\\bf u}) -\\nabla^\\perp(\\nabla\\times{\\bf u})=\\Delta{\\bf u}$,\nis\n\\begin{equation}\\label{N3}\\begin{split}\n{\\mathcal L}^2({\\bf U})&=\\left(\\begin{matrix} \\nabla\\!\\!\\cdot\\!(\\nabla m+{\\bf u}^\\perp)\\\\ \\nabla(\\nabla\\!\\!\\cdot\\!{\\bf u})+\n(\\nabla m+{\\bf u}^\\perp)^\\perp\\end{matrix}\\right)\\\\\n&=\\left(\\begin{matrix}(\\Delta-1)m -(\\nabla\\times{\\bf u}-m)\\\\\n(\\Delta-1){\\bf u}+\\nabla^\\perp(\\nabla\\times{\\bf u}-m)\\end{matrix}\\right)\\\\\n&=(\\Delta-1){\\bf U}+\\left(\\begin{matrix}-(\\nabla\\times{\\bf u}-\\rho+{1\\over4} m^2)\\\\\n\\nabla^\\perp(\\nabla\\times{\\bf u}-\\rho+{1\\over 4} m^2)\\end{matrix}\\right) \\qquad\n\\mbox{since  } \\rho=m+{1\\over4}m^2 \\\\\n&=(\\Delta-1){\\bf U}+{1\\over4}\\left(\\begin{matrix}-m^2\\\\\n\\nabla^\\perp(m^2)\\end{matrix}\\right)\\qquad\\mbox{ under (\\ref{inv})}\\\\\n&:=(\\Delta-1){\\bf U}+N_3.\\end{split}\\end{equation}\n\nNow that we've revealed the Klein-Gordon structure of (\\ref{ptt}),\nit suffices to show that the other terms $N_1,N_2,N_3$ can all be\nsplit into a symmetric second order part and a remainder lower order\npart as given in (\\ref{ptt}).\n\\begin{itemize}\n\\item The $N_1$ term in (\\ref{N1N2}). It is easy to see that the lower\norder terms (with less than second order derivatives) in $N_1$ are\nquadratic in $\\tilde{\\vU}$, i.e. linear in $\\tilde{\\vU}\\otimes\\tilde{\\vU}$. The terms\nwith second order derivatives are\n\\[\\sum_{a=1,2}(u_aI+\\dfrac{1}{2} mJ_a)\\partial_t\\partial_a{\\bf U}\\]\nwhere matrices $I$, $J_a$ are all symmetric. \n\\item The $N_2$ term in (\\ref{N1N2}).\nObserve that the linear operator ${\\mathcal L}$ partially comes from a\nlinearization of the nonlinear terms in (\\ref{RSWmatrix}) and it\nindeed can be represented as\n\\[{\\mathcal L}({\\bf U})=\\sum_{a=1,2}J_a\\partial_a{\\bf U}+K{\\bf U}\\]\nwith constant matrix $K$. Thus, manipulate the $N_2$ term,\n\\[\\begin{split}\nN_2&=\\sum_{a=1,2}J_a\\partial_a\\left(\\sum_{b=1,2}(u_bI+\\dfrac{1}{2} mJ_b)\\partial_b{\\bf U}\\right)\n+K\\left(\\sum_{b=1,2}(u_bI+\\dfrac{1}{2} mJ_b)\\partial_b{\\bf U}\\right)\\\\\n&=\\sum_{a=1,2}\\sum_{b=1,2}(u_bJ_{a}+\\dfrac{1}{2} mJ_{a}J_b)\\partial_{a}\\partial_b{\\bf U}+\n\\mbox{ quadratic terms of }\\tilde{\\vU}.\n\\end{split}\\]\nThe quasilinear terms above have the desired symmetric structure\nsince for each index pair $(a,b)$, the coefficient of\n$\\partial_{a}\\partial_b{\\bf U}=\\partial_b\\partial_{a}{\\bf U}$ is\n\\[\n\\frac{1}{2}(u_bJ_{a}+\\dfrac{1}{2}\nmJ_{a}J_b)+\\frac{1}{2}(u_{a}J_{b}+\\dfrac{1}{2} mJ_{b}J_{a})\n\\]\nwhich, by the definition of $J_a$, is symmetric.\n\\item The $N_3$ term in (\\ref{N3}). By\ndefinition, this term has no second order derivatives and is\nquadratic in $\\tilde{\\vU}$.\n\\end{itemize}\n\\end{proof}\n\n\n\\section{Local Wellposedness}\n\nAs in the previous section, let\n$\\partial_0=\\frac{\\partial}{\\partial t}$, $\\partial_1=\\frac{\\partial}{\\partial x_1}$, $\\partial_2=\n\\frac{\\partial}{\\partial x_2}$. Define the vector fields\n\\begin{align}\\label{def:vfields}\n\\Gamma:=\\{\\Gamma_j\\}_{j=1}^6=\\{\\partial_0,\\partial_1,\\partial_2, L_1, L_2,\n\\Omega_{12}\\},\n\\end{align}\nwhere\n\\begin{align*}\nL_j:=x_j\\partial_t+t\\partial_j,\\,\\,j=1,2; \\quad\n\\Omega_{12}:=x_1\\partial_2-x_2\\partial_1.\n\\end{align*}\nWe abbreviate\n\\begin{align*}\n\\partial^{\\alpha}=\\partial_t^{\\alpha_1}\\partial_1^{\\alpha_2}\\partial_2^{\\alpha_3}\n\\quad\\mbox{ for } \\alpha=(\\alpha_1,\\alpha_2,\\alpha_3),\n\\end{align*}\nand\n\\begin{align*}\n\\Gamma^{\\beta}=\\Gamma_1^{\\beta_1}\\cdots\\Gamma_6^{\\beta_6}\n\\quad\\mbox{ for } \\beta=(\\beta_1,\\cdots, \\beta_6).\n\\end{align*}\n\nThe local wellposedness of (\\ref{RSWm}) and (\\ref{RSWum})\nand their regularity are contained in the following theorem.\n\\begin{theorem}\\label{LocalTh}\n\\begin{itemize}\n\\item[(i)] Let $(m_0,{\\bf u}_0)\\in H^n$ with $n\\ge3$. Then, there exists a $T>0$\ndepending only on $\\|(m_0,{\\bf u}_0)\\|_{H^3}$ such that (\\ref{RSWm}), (\\ref{RSWum})\nadmit a unique solution ${\\bf U}=\\left(\\begin{matrix} m \\\\ {\\bf u}\\end{matrix}\\right) $  on $[0, T]$ satisfying\n\\begin{align}\n{\\bf U}\\in  \\bigcap_{j=0}^{n} C^j([0,T], H^{n-j}).\n\\end{align}\n\\item[(ii)] Under the assumptions in (i), also assume\n$\\rho_0=\\partial_1 u_{2,0}-\\partial_2 u_{1,0}$. Then,\n\\begin{align}\\label{Zerovor2}\n\\rho(t,\\cdot)=\\partial_1 u_2(t,\\cdot)-\\partial_2 u_1(t,\\cdot)\\,\\, \\text{for}\\,\\, t\\in[0,T],\n\\end{align}\nand ${\\bf U}=\\left(\\begin{matrix} m\\\\{\\bf u}\\end{matrix}\\right)=\\bma2\\left(\\sqrt{\\rho+1}-1\\right)\\\\{\\bf u}\\end{matrix}\\right)$\nsatisfies the Klein-Gordon system (\\ref{ptt}). If the initial data\nbelong to the weighted Sobolev space (cf. definition\n(\\ref{def:weighted})) such that ${\\bf U}_0\\in H^{k+1,k}$ with $k\\geq\n3$, then the above solution ${\\bf U}$ satisfies\n\\begin{align}\\label{locvec}\n\\Gamma^{\\alpha}{\\bf U},\\,\\Gamma^{\\alpha}\\partial{\\bf U}\\in C([0,T], L^2),\\qquad(1+|x|)\\Gamma^{\\beta}{\\bf U}\\in C([0,T],L^\\infty),\n\\end{align}\nfor any multi-indices $|\\alpha|\\leq k$ and $|\\beta|\\leq k-3$.\n\\end{itemize}\n\\end{theorem}\n\n\\begin{proof}\nThe proof of (i) follows from the standard local wellposedness and\nregularity theory for symmetric hyperbolic system, cf. \\cite{Kato,\nMajda84}.\n\nFor part (ii), (\\ref{Zerovor2}) comes from the derivation of\n(\\ref{inv}). Then, it follows from Lemma \\ref{symm} that ${\\bf U}$\nsolves (\\ref{ptt}). Finally, the proof of (\\ref{locvec}) is based on\nthe arguments in \\cite{KlainermanP, Shatahev}. Note that, using\n(\\ref{RSWm}),  (\\ref{RSWum}), one has\n\\begin{align*}\n  {\\bf U}(0,\\cdot)\\in H^{k+1,k} \\text{ and } \\partial_t^l {\\bf U}(0,\\cdot)\\in H^{k+1-l,k}\n   \\end{align*}\nas long as ${\\bf u}_0\\in H^{k+2,k}$ and $\\rho_0=\\nabla\\times{\\bf u}_0$.\n\\end{proof}\n\n\\section{Global Existence and Asymptotic Behavior with Zero Relative Vorticity}\n\nThroughout this section, we focus on the solutions with zero\nrelative vorticity -- cf. (\\ref{inv}). Theorem \\ref{LocalTh}, part\n(ii), suggests that the RSW system be treated as a system of\nquasilinear Klein-Gorden equations. To this end, it is convenient to\nintroduce the following generalized Sobolev norms associated with\nvector fields $\\Gamma$ defined in (\\ref{def:vfields}).\n\\begin{align*}\n|{\\bf U}\\RGN{l,d}(t):=&\n\\sum_{|\\alpha|\\leq l }|(1+t+|x|)^{-d}\\Gamma^{\\alpha} {\\bf U}(t,x)|_{L^{\\infty}_x},\\\\\n\\|{\\bf U}\\RGNN{l}(t):=&\\sum_{|\\alpha|\\leq l}\n\\|\\Gamma^{\\alpha} {\\bf U}(t,x) \\|_{L^2_x}.\n\\end{align*}\n\nTo extend the local solution of (\\ref{RSWm}) and (\\ref{RSWum})\nglobally in time, we need to derive the decay and energy estimates\nof solutions to (\\ref{ptt}). We start with\ndefining a functional (see e.g. \\cite{OzawaSL}) measuring the size of the solution at time\n$t\\ge0$,\\begin{equation}\\label{def:X}\\begin{split}\nX(t):=&\\sup_{s\\in[0,t]}\\left\\{|{\\bf U}\\RGN{k-25,-1}(s)+\\|{\\bf U}\\RGNN{k-9}(s)+\\|{\\partial} {\\bf U}\\RGNN{k-9}(s)\\right.\\\\\n&\\left.+(1+s)^{-\\sigma}\\|{\\bf U}\\RGNN{k}(s)+(1+s)^{-\\sigma}\\|{\\partial}\n{\\bf U}\\RGNN{k}(s)\\right\\},\n\\end{split}\\end{equation}\nhere, pick any fixed $\\sigma\\in(0,1\/2)$ and $k\\ge52$.\n\nWe then state and prove the following global existence result regarding any symmetric quasilinear system of Klein-Gordon equations in 2D. Two key lemmas used in the proof will be discussed immediately after this.\n\\begin{theorem}\\label{KGsystem}Consider a two dimensional $n$ by $n$ system (\\ref{ptt}) satisfying the conclusion of Lemma \\ref{symm}. Then, for any $k\\ge52$, there exists a universal constant $\\delta_0$ such that the system admits a unique classical solution for all times if\n\\begin{equation*}\n\\|{\\bf U}_0\\|_{H^{k+1,k}}=\\delta<\\delta_0.\n\\end{equation*}\nIn particular, $X(t)\\le C\\delta$ uniformly for all positive times.\n\\end{theorem}\n\\begin{proof}By the definition of $X(t)$ and local existence (\\ref{locvec}) of Theorem \\ref{LocalTh}, there exists $T$ such that\n\\begin{equation}\\label{XT}\n  X(T)\\leq 4C_1\\delta.\n \\end{equation}\n Here, we choose constant $C_1$ to be greater than all constants appearing in Lemma \\ref{lemL} and \\ref{lemenergy} below. Then, choose $\\delta$ to be sufficiently small so that the assumptions of Lemma \\ref{lemL} and \\ref{lemenergy} are satisfied, which in turn implies\n\\[\nX(T)\\leq 2C_1\\delta+32C_1^3\\delta^2.\n\\]\nImpose one more smallness condition on $\\delta$ so that $X(T)\\le 3C_1\\delta$ in the above estimate. Finally, by the continuity argument, we can extend $T$ in (\\ref{XT}) to infinity, i.e., have $X(t)\\le 4C_1\\delta$ uniformly for all positive times. \n\\end{proof}\n\nThe following lemmas provide estimates on the lower order norms and highest order norms of $X(t)$ respectively. The quadratic term $X^2(t)$, rather than linear term, on the RHS of these estimates guarantees that we can extend such estimates to global times as long as $X(t)$ stays sufficiently small.\n\\begin{lemma}\\label{lemL}\nAssume\n$\\|{\\bf U}_0\\|_{H^{k+1,k}}\\le1$, $X(t)\\le1$. Then, the solution ${\\bf U}$\nof (\\ref{ptt}) satisfies \\begin{equation}\\label{Lestimate}\n|{\\bf U}\\RGN{k-25,-1}(t)+\\|{\\bf U}\\RGNN{k-9}(t)+\\|\\partial{\\bf U}\\RGNN{k-9}(t)\\leq\nC(\\|{\\bf U}_0\\|_{H^{k+1,k}}+ X^2(t)). \\end{equation} as long as the solution\nexists. Here, constant $C$ is independent of $\\delta$ and $t$.\n\\end{lemma}\n\\begin{proof}\nWe start the proof with defining the bilinear form associated with\nkernel $Q(y,z)$,\n\\\n\\begin{split}\n[G,Q,H](x):=&\\int_{\\mbox{R}^2\\times\\mbox{R}^2}G^T(y)Q(x-y,x-z)H(z)\\,dydz\\\\\n=&{1\\over(2\\pi)^4}\\int_{\\mbox{R}^2\\times\\mbox{R}^2}e^{i(\\xi+\\eta)\\cdot\nx}\\hat{G}^T(\\xi)\\hat{Q}(\\xi,\\eta)\\hat{H}(\\eta)\\,d\\xi\nd\\eta.\n\\end{split}\n\\]\nHere, $G(\\cdot)$, $H(\\cdot)$ are any $(2\\times1)$-vector-valued\nfunctions defined on $\\mbox{R}^2$ and $Q(\\cdot,\\cdot)$ is $(2\\times\n2)$-matrix-valued distribution defined on $\\mbox{R}^2\\times\\mbox{R}^2$.\nFourier transform is denoted with $\\hat{\\;\\;}$ for both $\\mbox{R}^2$ and\n$\\mbox{R}^2\\times\\mbox{R}^2$.\n\nThe same notation will be used for scalars\n\\[\n[g,q,h](x):=\\int_{\\mbox{R}^2\\times\\mbox{R}^2}g(y)q(x-y,x-z)h(z)\\,dydz.\n\\]\n\nThen, we follow the construction of \\cite{Shatah} to transform\n(\\ref{ptt}) in terms of the new variable\n\\begin{equation}\\label{def:V}{\\bf V}=(V_1,V_2,V_3)^T={\\bf U}+{\\bf W}={\\bf U}+(W_1,W_2,W_3)^T\\end{equation}\nwhere\n\\begin{equation}\\label{def:vi}\nW_k:=\\sum_{i,j=1}^3\\left[\\left(\\begin{matrix} U_i\\\\\n\\partial_t U_i\\end{matrix}\\right),Q^{ij}_k,\\left(\\begin{matrix} U_j\\\\\n\\partial_t U_j\\end{matrix}\\right)\\right],\\quad k=1,2,3.\n\\end{equation}\nThe kernels $Q^{ij}_k$ are to be determined later so that the new\nvariable ${\\bf V}$ satisfies a Klein-Gordon system with \\emph{cubic}\nnonlinearity for which estimate (\\ref{Lestimate}) will be proved\nusing techniques from \\cite{OzawaSL}, \\cite{Georgiev}.\n\nWithout loss of generality, we will demonstrate the proof using\n$V_1,U_1,W_1,Q^{ij}_1$ associated with the mass equation. From now\non, the subscript ``1'' is neglected for simplicity.\n\n\\emph{Step 1.} We claim that there exists kernels $Q^{ij}$ in\n(\\ref{def:vi}) such that $V=U+W$ satisfies the following\nKlein-Gordon equation with cubic and quadruple nonlinearity,\n\\begin{equation}\\label{cubicKG}(\\partial_{tt}-\\Delta +1)V=S\\end{equation} where the\nRHS\n\\begin{equation}\\label{cubicRHS}\n\\begin{split}S:=&\\sum_{|\\alpha|+|\\beta|+|\\gamma|\\le4\\atop\n\\max\\{|\\alpha|,|\\beta|,|\\gamma|\\}\\le3}\\sum_{a,b,c=1}^3[\\partial^\\alpha\nU_a\\partial^\\beta U_b,q_{\\alpha\\beta\\gamma}^{abc},\\partial^\\gamma\nU_c]\\\\&+\\sum_{|\\alpha|+|\\beta|+|\\gamma|+|\\zeta|\\le4\\atop\n\\max\\{|\\alpha|,|\\beta|,|\\gamma|,|\\zeta|\\}\\le3}\\sum_{a,b,c,d=1}^3[\\partial^\\alpha\nU_a\\partial^\\beta U_b,q_{\\alpha\\beta\\gamma\\zeta}^{abcd},\\partial^\\gamma\nU_c\\partial^\\zeta U_d] \\end{split}\\end{equation} with $q_{\\alpha\\beta\\gamma}^{abc}$,\n$q_{\\alpha\\beta\\gamma\\zeta}^{abcd}$ being linear combinations of the\nentries of all $Q_k^{ij}$'s. Moreover, all the $Q_k^{ij}$'s  satisfy\nthe growth\ncondition\\begin{equation}\\label{growth:Q}\\left|D^N\\hat{Q}(\\xi,\\eta)\\right|\\le\nC_N(1+|\\xi|^4+|\\eta|^4)\\end{equation} for any nonnegative integer N.\n\nIndeed, substitute $V$ on the LHS of (\\ref{cubicKG}) with $V=U+W$\nwhere $W$ is defined in (\\ref{def:vi}) for $k=1$ and $U$ satisfies\nthe first equation of (\\ref{ptt}),\n\\begin{equation}\\label{form:0}(\\partial_{tt}-\\Delta+1)V=(\\partial_{tt}-\\Delta+1)U+(\\partial_{tt}-\\Delta+1)W.\\end{equation}\n\n\nThen, apply the normal form transform on each term of the RHS of the\nabove equation. By (\\ref{ptt}), the\n$U$ term amounts to\n\\begin{equation}\\label{form:1}(\\partial_{tt}-\\Delta+1)U=\\sum_{i,j=1}^3\\left[\\left(\\begin{matrix} U_i\\\\\\partial_t\nU_i\\end{matrix}\\right),P^{ij},\\left(\\begin{matrix} U_j\\\\\\partial_t U_j\\end{matrix}\\right)\\right]\\end{equation} with\n$\\hat{P}^{ij}(\\xi,\\eta)$ being $2\\times2$ matrices of polynomials\nwith degree less than or equal to 2. By (\\ref{def:vi}), the $W$ term amounts to\n\\begin{equation}\\label{form:2}(\\partial_{tt}-\\Delta+1)W=\\sum_{i,j=1}^3\\left[\\left(\\begin{matrix} U_i\\\\\\partial_t\nU_i\\end{matrix}\\right),{\\mathcal A} Q^{ij},\\left(\\begin{matrix} U_j\\\\\\partial_t U_j\\end{matrix}\\right)\\right]+S\\end{equation} with $S$\nsatisfying (\\ref{cubicRHS}) and linear transform ${\\mathcal A}$ defined as\n\\begin{equation}\\label{form:3}\\widehat{{\\mathcal A}\nQ}(\\xi,\\eta):=\\bma0&|\\xi|^2+1\\\\-1&0\\end{matrix}\\right)\n\\hat{Q}\\bma0&-1\\\\|\\eta|^2+1&0\\end{matrix}\\right)+(2\\xi\\cdot\\eta-1)\\hat{Q}\\end{equation}\n\nCombining (\\ref{form:0}) --- (\\ref{form:3}), we find that, for\nproving (\\ref{cubicKG}) --- (\\ref{growth:Q}), it suffices to show\nthat there exist solutions $\\hat{Q}^{ij}(\\xi,\\eta)$ to\n\\begin{equation}\\label{eq:Q}\\hat{P}^{ij}(\\xi,\\eta)+\\widehat{{\\mathcal A}\n{Q}}^{ij}(\\xi,\\eta)\\equiv0\\end{equation} that satisfy the growth condition\n(\\ref{growth:Q}). This part of the calculation only involves basic\nLinear Algebra and Calculus so we neglect the details.\n\n\\emph{Step 2.} We apply the decay estimate of Georgiev in\n\\cite{Georgiev} to obtain the $L^\\infty$ estimate for $V$ and\ntherefore $U$ with $(1+t)^{-1}$ decay in time.\n\\begin{theorem}[{\\cite[Theorem 1]{Georgiev}}]\\label{thm:Georgiev}Suppose $u(t,x)$\nis a solution of\\[(\\partial_{tt} -\\Delta +1)u=F(t,x).\\]Then, for $t\\ge0$, we\nhave\n\\[\n\\begin{split}|(1+t+|x|)u(t,x)|\\le& C\\sum_{n=0}^\\infty\\sum_{|\\alpha|\\le4}\n\\sup_{s\\in(0,t)}\\phi_n(s)\\|(1+s+|y|)\\Gamma^{\\alpha} f(s,y)\\|_{L^2_y}\\\\\n&+C\\sum_{n=0}^\\infty\\sum_{|\\alpha|\\le5}\\|(1+|y|)\\phi_n(y)\\Gamma^{\\alpha}\nu(0,y)\\|_{L^2_y}.\n\\end{split}\n\\]\nHere, $\\{\\phi_n\\}_{n=0}^\\infty$ is a Littlewood-Paley partition of\nunity,\n\\[\\sum_{n=0}^\\infty\\phi_n(s)=1,\\quad s\\ge0;\\quad\\phi_n\\in C^\\infty_0(\\mbox{R}),\n\\quad\\phi_n\\ge0\\quad\\mbox{for all }n\\ge0\\]\n\\[\\mbox{supp}\\,\\phi_n=[2^{n-1},2^{n+1}]\\quad\\mbox{for }n\\ge1,\n\\quad\\mbox{supp}\\,\\phi_0\\cap\\mbox{R}_+=(0,2].\\]\n\\end{theorem}\n\nApply $\\Gamma^{\\alpha}$ on the Klein-Gordon equation (\\ref{cubicKG}) and use the\ncommutation properties of the vector fields to obtain \\((\\partial_{tt}\n-\\Delta +1)\\Gamma^{\\alpha} V=\\Gamma^{\\alpha} S\\) so that by the above theorem,\n\\[\n\\begin{split}|(1+t+|x|)\\Gamma^{\\alpha} V(t,x)|\\le& C\\sum_{n=0}^\\infty\n\\sum_{|\\beta|\\le|\\alpha|+4}\n\\sup_{s\\in(0,t)}\\phi_n(s)\\|(1+s+|y|)\\Gamma^{\\beta} S(s,y)\\|_{L^2_y}\\\\\n&+C\\sum_{n=0}^\\infty\\sum_{|\\beta|\\le|\\alpha|+5}\n\\|(1+|y|)\\phi_n(y)\\Gamma^{\\beta} V(0,y)\\|_{L^2_y}.\n\\end{split}\n\\]\nBy definition $V=U+W$, we immediately have estimates for\n$(1+t+|x|)\\Gamma^{\\alpha} U$. After taking summation over all $\\alpha$'s with\n$|\\alpha|\\le k-25$, we arrive at \\begin{equation}\\label{estimate:infty}\n\\begin{split}| U\\RGN{k-25,-1}(t)\\le& | W\\RGN{k-25,-1}+\nC\\left(\\sum_{n=0}^\\infty\\sum_{|\\beta|\\le k-21}\n\\sup_{s\\in(0,t)}\\phi_n(s)\\|(1+s+|y|)\\Gamma^{\\beta} S(s,y)\\|_{L^2_y}\\right.\\\\\n&\\left.+\\|U(0,y)\\|_{H^{k+1,k}}+\\sum_{|\\beta|\\le k-20}\\|(1+|y|)\\Gamma^{\\beta}\nW(0,y)\\|_{L^2_y}\\right).\\end{split} \\end{equation}\n\nTo obtain estimate on each term, we use the following proposition,\nthe proof of which is given in the Appendix.\n\\begin{proposition}\\label{prop:est}\nLet the scalar function $q:\\mbox{R}^2\\times\\mbox{R}^2\\mapsto\\mbox{R}$ satisfy the\ngrowth condition (\\ref{growth:Q}) in terms of its Fourier transform.\nLet $f(t,x)$, $g(t,x)$, $h(t,x)$ be functions with sufficient\nregularity. Consider $p=\\infty$ (respectively $p=2$). Then, at each\n$t\\ge0$, for $a=|\\beta|+8$ (respectively $a=|\\beta|+6$) and\n$b=\\left\\lceil {|\\beta|\\over2}\\right\\rceil+7$,\n\\begin{align}\\label{Estimate1prop}\n\\left\\|(1+t+|x|)\\Gamma^{\\beta}[f,q,g]\\right\\|_{L^p_x}\\le& C\\left(\\|\nf\\RGNN{a}| g\\RGN{b,-1}+| f\\RGN{b,-1}\\| g\\RGNN{a}\\right),\n\\end{align}\n\\begin{align}\\label{Estimate2prop}\n\\begin{split}\n\\left\\|(1+t+|x|)\\Gamma^{\\beta}[f,q,gh]\\right\\|_{L^p_x}\\le &C(1+t)^{-1}\n\\left(\\| f\\RGNN{a}| g\\RGN{b,-1}| h\\RGN{b,-1}\\right.\\\\\n&+\\left.| f\\RGN{b,-1}\\left(\\| g\\RGNN{a}|\nh\\RGN{\\lceil{a\\over2}\\rceil,-1} +|\ng\\RGN{\\lceil{a\\over2}\\rceil,-1}\\| h\\RGNN{a}\\right)\\right).\n\\end{split}\n\\end{align}\n\\end{proposition}\n\nApply this proposition (with $p=\\infty$) on the first term, and\n(with $p=2$) on the second and fourth terms of the RHS of\n(\\ref{estimate:infty}) and use the definition of $W$ in\n(\\ref{def:vi}) and $S$ in (\\ref{cubicKG}),\n\\[\\begin{split}| U\\RGN{k-25,-1}(t)\\le& CX^2(t)+C\\sum_{n=0}^\\infty\n\\sup_{s\\in(0,t)}\\phi_n(s)(1+s)^{-1}(X^3(t)+X^4(t))\\\\\n&+ C\\left(\\|U_0\\|_{H^{k+1,k}}+\\|{\\bf U}_0\\|^2_{H^{k+1,k}}\\right).\n\\end{split}\\]\n\nWe finish the $L^\\infty$ estimate part of this Lemma\nusing the fact that \\[\\sum_{n=0}^\\infty\\sup_{s\\in\n(0,t)}\\phi_n(s)(1+s)^{-1}<{5\\over2}\\]and the assumptions\n$\\|{\\bf U}_0\\|_{H^{k+1,k}}\\le1$, $X(t)\\le1$.\n\n\\emph{Step 3.} We obtain the $L^2$ estimate part regarding the terms\n\\(\\|{\\bf U}\\RGNN{k-9}(t)+\\|{\\partial} {\\bf U}\\RGNN{k-9}(t)\\) using a very\nsimilar approach as in Step 2. In fact, apply $\\Gamma^{\\alpha}$ on the\nKlein-Gordon equation (\\ref{cubicKG}) and use the commutation\nproperties of vector fields to obtain \\((\\partial_{tt} -\\Delta +1)\\Gamma^{\\alpha} V=\\Gamma^{\\alpha}\nS\\). Then, we take the inner product of this equation with $\\partial_t\\Gamma^{\\alpha}\nV$, sum over all $\\alpha$ with $|\\alpha|\\le k-9$ to obtain\n\\[\n\\begin{split}\\| V(t,x)\\RGNN{k-9}+&\\| \\partial V(t,x)\\RGNN{k-9}\\le\nC\\int_0^t\\| S(s,x)\\RGNN{k-9}ds\\\\&+C\\left(\\| V(0,x)\\RGNN{k-9}\n+\\| \\partial V(0,x)\\RGNN{k-9}\\right).\\end{split}\n\\]\nSince $V=U+W$, we have\n\\[\n\\begin{split}\\| U(t,x)\\RGNN{k-9}+&\\| \\partial U(t,x)\\RGNN{k-9}\n\\le\\left(\\| W(t,x)\\RGNN{k-9}+\\| \\partial W(t,x)\\RGNN{k-9}\\right)\\\\&\n+C\\int_0^t\\| S(s,x)\\RGNN{k-9}ds\\\\&+ C\\left(\\|\nU(0,x)\\RGNN{k-9}+\\| \\partial U(0,x)\\RGNN{k-9}+\n\\| W(0,x)\\RGNN{k-9}+\\| \\partial W(0,x)\\RGNN{k-9}\\right)\\\\\n&=:I+II+III.\\end{split}\n\\]\n\nThe estimate of the $I,II,III$ terms above follows closely to that of\n(\\ref{estimate:infty}), which evokes Proposition \\ref{prop:est} repeatedly,\n\\[\\begin{split}I\\le&(1+t)^{-1}\\sum_{|\\beta|\\le k-9}\\left\\|(1+t+|x|)\\Gamma^{\\beta} W(t,x)\n\\right\\|_{L^2_x}\\\\\n\\le&C(1+t)^{-1}\\| {\\bf U}(t,x)\\RGNN{k}| {\\bf U}(t,x)\\RGN{k-25,-1}\n\\quad\\mbox{ by (\\ref{def:vi}) and Prop. \\ref{prop:est}}\\\\\n\\le&C(1+t)^{-1+\\sigma}X^2(t),\\\\\nII\\le&\\int_0^t(1+s)^{-1}\\sum_{|\\beta|\\le k-9}\\left\\|(1+s+|x|)\\Gamma^{\\beta}\nS(s,x)\n\\right\\|_{L^2_x}ds\\\\\n\\le&C\\int_0^t(1+s)^{-2}\\| {\\bf U}(s,x)\\RGNN{k} \\left(|\n{\\bf U}(s,x)\\RGN{k-25,-1}^2\n+| {\\bf U}(s,x)\\RGN{k-25,-1}^3\\right)\\\\\n&\\phantom{C\\int_0^t(1+s)^{-2+\\sigma}\\left(X^3(s)+X^4(s)\\right)\\,ds}\n\\quad\\mbox{ by (\\ref{cubicRHS}) and Prop. \\ref{prop:est}}\\\\\n\\le&C\\int_0^t(1+s)^{-2+\\sigma}\\left(X^3(s)+X^4(s)\\right)\\,ds,\\\\\nIII\\le&C\\left(\\|U(0,y)\\|_{H^{k+1,k}}+\\|{\\bf U}(0,y)\\|^2_{H^{k+1,k}}\\right),\n\\end{split}\n\\]\nwhere we use $k\\geq 52$ and that $S$ contains at most third order\nderivatives of ${\\bf U}$. These estimates shall finish the proof of\nLemma \\ref{lemL} given assumptions \n$\\|{\\bf U}_0\\|_{H^{k+1,k}}\\le1$ and $X(t)\\le1$.\n\\end{proof}\n\nNote that in the above estimates for $I$ and $II$, we use the $k-th$\norder norms to bound all lower order norms. In order to get the\nglobal a priori estimate, we have to close the estimates for the\nhighest order norms. For the RSW system, this is achieved by the\nenergy estimates on the highest order Sobolev norms\n$\\|\\cdot\\RGNN{k}$,\n where its symmetric structure shown in Lemma\n\\ref{symm} plays a crucial role.\n\\begin{lemma}\\label{lemenergy}\nAssume $\\|A_{ij}({\\bf U})\\|_{L^\\infty}\\leq 1\/4$.\nThen, the solution ${\\bf U}$ of (\\ref{ptt}) satisfies\n\\begin{align*}\\label{Enestimate}\n(1+t)^{-\\sigma}(\\|{\\bf U}\\RGNN{k}(t)+\\|{\\partial} {\\bf U}\\RGNN{k}(t))\\leq\nC(\\|{\\bf U}_0\\|_{H^{k+1,k}}+X^2(t))\n\\end{align*}\nas long as the solution exists. Here, constant $C$ is independent of $\\delta$ and $t$.\n\\end{lemma}\n\n\\begin{proof}\nThe proof of this lemma combines the ideas in \\cite{Hormander,\nOzawaSL, Fang} for energy estimates for the Klein-Gordon equations\ntogether.\n\nDefine an energy functional\n\\begin{align*} F(t):=&\\frac{1}{2}\\sum_{|\\alpha|\\leq k} (\\|\\partial_t\\Gamma^{\\alpha}\n{\\bf U}\\|_{L^2}^2+\\|\\nabla \\Gamma^{\\alpha} {\\bf U}\\|_{L^2}^2+\\|\\Gamma^{\\alpha}{\\bf U}\\|_{L^2}^2)(t)\\\\\n&+\\frac{1}{2}\\sum_{|\\alpha|\\leq k}\n\\sum_{i,j=1}^2\\left\\langle A_{ij}({\\bf U})\\partial_i\\Gamma^{\\alpha} {\\bf U}, \\partial_j\\Gamma^{\\alpha}{\\bf u}\\right\\rangle(t),\n\\end{align*} where $\\langle, \\rangle$ is the $L^2$ inner product defined by\n$\\left\\langle f, g\\right\\rangle=\\int_{\\mbox{R}^2}f^Tgdx$.\nClearly, by the commutation property of $\\Gamma$, $\\partial$ and the assumption $\\|A_{ij}({\\bf U})\\|_{L^\\infty}\\leq\n\\frac{1}{4}$, we have \\begin{equation}\\label{MEE} C_1\\sqrt{F(t)}\\leq \\|{\\bf U}\\RGNN{k}(t)+\\|{\\partial} {\\bf U}\\RGNN{k}(t)\\leq C_2\\sqrt{F(t)}, \\end{equation} which ensures the equivalence\nof $\\sqrt{F(t)}$ and $\\|{\\bf U}\\RGNN{k}(t)+\\|{\\partial} {\\bf U}\\RGNN{k}(t)$.\n\nApplying $\\Gamma^{\\alpha}$ to (\\ref{ptt})  and taking the $L^2$\ninner product on  the resulting system with $\\partial_t\\Gamma^{\\alpha}{\\bf U}$\ni.e. $\\partial_0\\Gamma^{\\alpha}{\\bf U}$ , it follows from Leibniz's rule that\n\\begin{align*}\n\\left\\langle(\\partial_{tt}-\\Delta+1)\\Gamma^{\\alpha}{\\bf U}, \\partial_t\\Gamma^{\\alpha}{\\bf U}\\right\\rangle=\n&\\sum_{i,j=1}^2\\left\\langle\nA_{ij}({\\bf U})\\partial_{ij}\\Gamma^{\\alpha}{\\bf U},\n\\partial_0\\Gamma^{\\alpha}{\\bf U}\\right\\rangle \\\\+\\sum_{j=1}^2\\left\\langle A_{0j}\n\\partial_{0j}\\Gamma^{\\alpha}{\\bf U},\n\\partial_0\\Gamma^{\\alpha}{\\bf U}\\right\\rangle&+\\sum_{|\\beta|+|\\gamma|\\le |\\alpha|}\\left\\langle\nR^{\\beta\\gamma}(\\Gamma^{\\beta}\\tilde{\\vU}\\otimes\\Gamma^{\\gamma}\\tilde{\\vU}),\n\\partial_0\\Gamma^{\\alpha}{\\bf U} \\right\\rangle,\n\\end{align*}\nwhere all $R^{\\beta\\gamma}$'s are linear functions. Here\n$\\tilde{\\vU}=({\\bf U}^T,\\partial_t{\\bf U}^T,\\partial_1{\\bf U}^T,\\partial_2{\\bf U}^T)$.\n\nUpon integrating by parts, one has\n\\begin{equation}\\label{Highenergyest}\n\\begin{split}\n&\\frac{1}{2}\\partial_t(\\|\\partial_t\\Gamma^{\\alpha} {\\bf U}\\|_{L^2}^2+\\|\\nabla \\Gamma^{\\alpha} {\\bf U}\\|_{L^2}^2\n+\\|\\Gamma^{\\alpha}{\\bf U}\\|_{L^2}^2)\\\\\n=&-\\sum_{i,j=1}^2 \\left\\langle\nA_{ij}\\partial_i\\Gamma^{\\alpha}{\\bf U},\\partial_0\\partial_j\\Gamma^{\\alpha}{\\bf U}\\right\\rangle -\\left\\langle\n\\partial_jA_{ij}({\\bf U})\\partial_i\\Gamma^{\\alpha}{\\bf U},\\partial_0\\Gamma^{\\alpha}{\\bf U}\\right\\rangle\\\\\n& -\\frac{1}{2}\\sum_{j=1}^2\\left\\langle\\partial_j\nA_{0j}\\partial_0\\Gamma^{\\alpha}{\\bf U},\\partial_0\\Gamma^{\\alpha}{\\bf U}\\right\\rangle +\\sum_{|\\beta|+|\\gamma|\\le\n|\\alpha|}\\left\\langle R^{\\beta\\gamma}(\\Gamma^{\\beta}\\tilde{\\vU}\\otimes\\Gamma^{\\gamma}\\tilde{\\vU}),\n\\partial_0\\Gamma^{\\alpha}{\\bf U}\\right\\rangle\n\\end{split}\n\\end{equation}\nand the first terms on the RHS above\\begin{align*} \\left\\langle\nA_{ij}\\partial_i\\Gamma^{\\alpha}{\\bf U},\\partial_0\\partial_j\\Gamma^{\\alpha}{\\bf U}\\right\\rangle=&-\\frac{1}{2}\\partial_t\\left\\langle\nA_{ij}\\partial_i\\Gamma^{\\alpha}{\\bf U},\\partial_j\\Gamma^{\\alpha}{\\bf U}\\right\\rangle+\\frac{1}{2}\\left\\langle \\partial_0A_{ij}\n\\partial_i\\Gamma^{\\alpha}{\\bf U},\\partial_j\\Gamma^{\\alpha}{\\bf U}\\right\\rangle.\n\\end{align*}\nThe equations above hold because $A_{ij}$ and $A_{0j}$ are symmetric\nmatrices.  Summing for all indices $\\alpha$ with $|\\alpha|\\leq k$ in\nthe above equations and using the fact that $\\min\\{|\\beta|,|\\gamma|\\}\\le k\/2<k-25$ in the last terms on the RHS of\n(\\ref{Highenergyest}), one obtain\n\\[\\partial_t F(t)\\leq C|{\\bf U}\\RGN{k-25,0}F(t).\\]\nThus,\n\\[\n\\partial_t \\sqrt{F(t)}\\leq C(1+t)^{-1}|{\\bf U}\\RGN{k-25,-1}(1+t)^\\sigma\n(1+t)^{-\\sigma}\\sqrt{F(t)}.\n\\]\n\nBy the virtue of (\\ref{MEE}) and the definition of $X(t)$ in (\\ref{def:X}), we have\n\\begin{align*}\n\\|{\\bf U}\\RGNN{k}+\\|{\\partial} {\\bf U}\\RGNN{k}\\leq &C\\left(\\sqrt{F(t)}-\\sqrt{F(0)}\\right)+C\\sqrt{F(0)}\\\\\n\\leq&\nC\\int_0^t(1+s)^{-1}{X(s)}(1+s)^{\\sigma}\nX(s)ds +C\\|{\\bf U}_0\\|_{H^{k+1,k}}\\\\\n\\leq& C(1+t)^{\\sigma}X^2(t)+C\\|{\\bf U}_0\\|_{H^{k+1,k}},\n\\end{align*}\nwhich finishes the proof of Lemma \\ref{lemenergy}.\n\\end{proof}\n\nThe proofs of these lemmas also help reveal the asymptotic behavior of $U$ in the following theorem\n\\begin{theorem}\\label{asymTh}\nUnder the same assumptions as in Theorem \\ref{KGsystem}, there exists a free solution ${\\bf U}^+$\nsuch that \\begin{equation}\\label{KGasym} \n\\|{\\bf U}(t,\\cdot)-{\\bf U}^+(t,\\cdot)\\|_{H^{k-15}}+\\|\\partial_t{\\bf U}(t,\\cdot)-\\partial_t{\\bf U}^+(t,\\cdot)\\|_{H^{k-16}}\\leq C(1+t)^{-1}, \\end{equation} where\n\\begin{equation}\\label{free:U}{\\bf U}^+(t,\\cdot):=\\cos((1-\\Delta)^{1\/2}\nt){\\bf U}_0^++(1-\\Delta)^{-1\/2}\\sin((1-\\Delta)^{1\/2}\nt){\\bf U}_1^+\\end{equation} for some initial data ${\\bf U}_{0}^+\\in H^{k-15}$ and ${\\bf U}_1^+\\in H^{k-16}$.\n\\end{theorem}\n\\begin{proof}\nRecall the definitions of ${\\bf V}$ in (\\ref{def:V}), ${\\bf W}$ in (\\ref{def:vi}) and $S$ in (\\ref{cubicKG}), (\\ref{cubicRHS}). Without loss of generality, switch the notations in (\\ref{cubicKG}) to boldface ${\\bf V}$ and ${\\bf S}$ while cubic nonlinearity of ${\\bf S}$ remains valid. Then, Theorem \\ref{KGsystem} and Proposition \\ref{prop:est} imply that\n\\begin{equation}\\label{est:S}\n   \\|{\\bf S}(s,\\cdot)\\|_{H^{k-15}}\\leq C(1+s)^{-2}.\n   \\end{equation}\n   Therefore, we can define\n\\begin{equation}\\label{free:U0}\n {\\bf U}^+_{0}:={\\bf V}(0,\\cdot)-\\int_0^{\\infty}(1-\\Delta)^{-1\/2}\\sin\n ((1-\\Delta)^{1\/2}s){\\bf S}(s,\\cdot) ds\\in H^{k-15}\n \\end{equation}\nand\n\\begin{equation}\\label{free:U1}\n {\\bf U}^+_{1}:=\\partial_t{\\bf V}(0,\\cdot)+\\int_0^{\\infty}\\cos\n ((1-\\Delta)^{1\/2}s){\\bf S}(s,\\cdot) ds\\in H^{k-16}.\n \\end{equation}\n \nThen, apply the Duhamel's principle on (\\ref{cubicKG}),\n\\begin{align*} {\\bf V}(t,\\cdot) =&\\cos((1-\\Delta)^{1\/2}\nt){\\bf V}(0,\\cdot)+(1-\\Delta)^{-1\/2}\\sin((1-\\Delta)^{1\/2}\nt)\\partial_t{\\bf V}(0,\\cdot)\\\\\n&+\\int_0^t(1-\\Delta)^{-1\/2}\\sin((1-\\Delta)^{1\/2}(t-s)){\\bf S}(s,\\cdot)ds,\n\\end{align*}\nand combine it with (\\ref{free:U}), (\\ref{free:U0}), (\\ref{free:U1}) so that\nwe have\n\\begin{align*}\n{\\bf V}(t,\\cdot)-{\\bf U}^+(t,\\cdot)=&\\int_t^{\\infty}(1-\\Delta)^{-1\/2}\\sin\n ((1-\\Delta)^{1\/2}(t-s)){\\bf S}(t,\\cdot)ds.\n\\end{align*}\nTherefore, by (\\ref{est:S}), \\[\\|{\\bf V}(t,\\cdot)-{\\bf U}^+(t,\\cdot)\\|_{H^{k-15}}+\\|\\partial_t{\\bf V}(t,\\cdot)-\\partial_t{\\bf U}^+(t,\\cdot)\\|_{H^{k-16}}\\le C(1+t)^{-1}\\]\n\nFinally, apply Theorem \\ref{KGsystem} and Proposition \\ref{prop:est} to ${\\bf W}$ defined in (\\ref{def:vi}) and arrive at\n\\[\n   \\|{\\bf W}(t,\\cdot)\\|_{H^{k-15}}+\\|\\partial_t{\\bf W}(t,\\cdot)\\|_{H^{k-16}}\\leq C(1+t)^{-1}.\n   \\]\nWe then conclude that ${\\bf U}-{\\bf U}^+={\\bf V}-{\\bf U}^+-{\\bf W}$ also decays like $C(1+t)^{-1}$ as in (\\ref{KGasym}).\n\\end{proof}\n\n\n\n\\section{Lifespan of Classical Solutions with General Initial Data}\\label{sec:general}\nThe proof of Theorem \\ref{perTh} combines standard energy methods with the sharp estimates from previous section, in particular the $(1+t)^{-1}$ decay rate of $L^\\infty$ norms.\n\nWe start with extracting the zero-relative-vorticity part $(\\rho^K_0,{\\bf u}^K_0)$ from the general initial data $(\\rho_0,{\\bf u}_0)$ --- supscript ``$K$'' here stands for Klein-Gordon. This can be achieved by $L^2$ projection of $(\\rho_0,{\\bf u}_0)$ onto the function space with zero relative vorticity but we find it easier to use the complementary projection,\n\\begin{align*}\\rho_0-\\rho^K_0:=&(\\Delta-1)^{-1}(\\partial_1\nu_{2,0}-\\partial_2u_{1,0}-\\rho_0),\\\\\n{\\bf u}_0-{\\bf u}^K_0:=&\\left(\\begin{matrix}-\\partial_2\\\\ \\partial_1\\end{matrix}\\right)(\\rho_0-\\rho^K_0).\\end{align*}\n\nUsing the notations from Theorem \\ref{perTh}, we have\n\\begin{align}\\label{size:E0}\\|(\\rho_0,{\\bf u}_0)-(\\rho^K_0,{\\bf u}^K_0)\\|_{H^3}\\le& C\\|\\partial_1\nu_{2,0}-\\partial_2u_{1,0}-\\rho_0\\|_{H^2}=C\\varepsilon,\\\\\n\\label{size:K0}\\|(\\rho^K_0,{\\bf u}^K_0)\\|_{H^{k+1,k}}\\le& C\\|(\\rho_0,{\\bf u}_0)\\|_{H^{k+1,k}}=C\\delta.\n\\end{align}\n\nLet $m^{K}:=2(\\sqrt{1+\\rho^K}-1)$, then ${\\bf U}^{K}:=(m^K, u_1^K, u_2^K)^T$\nsolves the symmetrized RSW system (\\ref{RSWm}), (\\ref{RSWum}) as well as the Klein-Gordon system (\\ref{ptt}). By choosing $\\delta$ in (\\ref{size:K0}) to be sufficiently small, we have ${\\bf U}^K_0$ satisfy the assumptions in\nTheorem \\ref{KGsystem}, so that there exists a unique global solution\n${\\bf U}^K$ to (\\ref{RSWm}) and (\\ref{RSWm})\nassociated with the initial data ${\\bf U}_0^K$. In addition, the following estimate holds true\\begin{equation}\\label{Irest}\n |{\\bf U}^{K}|_{W^{k-25,\\infty}}\\leq \\frac{C\\delta}{1+t}.\n\\end{equation}\n\nNow it remains to estimate the difference ${\\bf E}:={\\bf U}-{\\bf U}^K$. To this end, we write the symmetrized RSW system (\\ref{RSWm}), (\\ref{RSWum}) into the following compact form \\begin{equation}\\label{cptfeq}\n\\partial_t{\\bf U}+\\sum_{j=1}^2B_j({\\bf U})\\partial_j{\\bf U}={\\mathcal L}({\\bf U}), \\end{equation} with symmetric matrices\n\\[ B_j({\\bf U}):=u_jI+\\frac{1}{2}m J_j\\]\nand $J_j$, ${\\mathcal L}$ defined in (\\ref{defJL}).\n\nSince both ${\\bf U}={\\bf U}^K+{\\bf E}$ and ${\\bf U}^{K}$ satisfy the same system (\\ref{cptfeq}), straightforward calculation shows that ${\\bf E}$ satisfies\n\\[\\partial_t{\\bf E}+\\sum_{j=1}^2B_j({\\bf E})\\partial_j{\\bf E}+\\sum_{j=1}^2B_j({\\bf U}^K)\\partial_j{\\bf E}+\\sum_{j=1}^2B_j({\\bf E})\\partial_j{\\bf U}^K={\\mathcal L}({\\bf E}),\\]\nsubject to initial data ${\\bf E}_0=(2\\sqrt{1+\\rho_0}-2\\sqrt{1+\\rho^K_0}, {\\bf u}_0-{\\bf u}^K_0)$.\n\nEmploying the standard energy method, we have $e(t):=\\|{\\bf E}(t,\\cdot)\\|_{H^3}$ satisfy an energy inequality,\n\\[e'(t)\\le Ce(t)(|\\nabla{\\bf E}|_{L^\\infty}+|{\\bf U}^K|_{W^{4,\\infty}}).\\]\nWe then use Sobolev inequalities and estimate (\\ref{Irest}) to further the above inequality as\n\\[e'(t)\\le Ce(t)\\left(e(t)+{\\delta\\over1+t}\\right).\\]\nDivide it with $e^2(t)$,\n\\[\\left(-e^{-1}(t)\\right)'\\le C\\left(1+{\\delta\\over1+t}e^{-1}(t)\\right)\\]which is linear in terms of $e^{-1}(t)$. We finally arrive at\n\\[e(t)\\le(1+t)^{C\\delta}\\left(e^{-1}(0)-{C\\over 1+C\\delta}\\left[(1+t)^{1+C\\delta}-1\\right]\\right)^{-1}.\\]\n\nBy (\\ref{size:E0}), the initial value is bounded by $e(0)\\le C\\varepsilon$. Thus, $e(t)$ remains bounded as long as\\[t<\\left({1+C\\delta\\over C\\varepsilon}+1\\right)^{1\\over1+C\\delta}-1\\]\nwhich is of the same order as (\\ref{span:general}) in Theorem \\ref{perTh} under the smallness assumption on $\\delta$ and the fact that $\\varepsilon\\le\\delta$.\n\n\\section{Appendix: Proof of Proposition \\ref{prop:est}}\nWe first claim the following estimate on kernel $q(y,z)$.\n\\begin{proposition}\\label{prop:kernel}Let $q(y,z)$ satisfy the growth\ncondition (\\ref{growth:Q}), i.e.\n\\begin{equation}\\label{growth:q}|D^N\\hat{q}(\\xi,\\eta)|\\le\nC_N(1+|\\xi|^4+|\\eta|^4)\\end{equation} for any $N\\geq 0$. Then, for\n\\begin{equation}\\label{def:q1}q_1:=(1-\\Delta_y)^{-3}(1-\\Delta_z)^{-3}q(y,z)\\end{equation}\nthe following estimate holds true\n\\[\n(1+|y|+|z|)^lq_1(y,z)\\in L^1(\\mbox{R}^2_y\\times\\mbox{R}^2_z),\n\\]\nfor any $l\\geq 0$.\n \\end{proposition}\n\\begin{proof}By (\\ref{growth:q}), we have a growth condition for $q_1$,\n\\[|\\hat{q}_1(\\xi,\\eta)|={|\\hat{q}(\\xi,\\eta)|\\over(1+|\\xi|^2)^3(1+|\\eta|^2)^3}\n\\le C(1+|\\xi|^2)^{-1}(1+|\\eta|^2)^{-1}\\] and inductively,\n\\[|D^{N}\\hat{q}_1(\\xi,\\eta)|\\le C(1+|\\xi|^2)^{-1}(1+|\\eta|^2)^{-1}\\in\nL^2(\\mbox{R}^2_\\xi\\times\\mbox{R}^2_\\eta)\\] for any integer $N\\ge0$. Therefore, by the Plancherel Theorem\n$(1+|y|^N+|z|^N)q_1(y,z)\\in L^2_{yz}$, which readily implies\n\\[\n(1+|y|+|z|)^l q_1(y,z)=(1+|y|+|z|)^{l-N}\\cdot\n(1+|y|+|z|)^Nq_1(y,z)\\in L^1(\\mbox{R}^2_y\\times\\mbox{R}^2_z).\n\\]\n\\end{proof}\n\nNote that (\\ref{Estimate2prop}) is a direct consequence of\n(\\ref{Estimate1prop}), it suffices to prove (\\ref{Estimate1prop}).\nWe then show the following estimate that serves as a slightly\nstronger version of Proposition \\ref{prop:est}: for any kernel\n$q(y,z)$ satisfying the growth condition (\\ref{growth:q}) and\n$f(t,x)$, $g(t,x)$ with sufficient regularity, there exists a\nconstant $C$ independent of $f,g$ such that\n\\begin{equation}\\label{prop:proof}\\left\\|(1+t+|x|)\\Gamma^{\\beta}[f,q,g]\\right\\|_{L^p_x}\\le\nC\\sum_{i+j=|\\beta|}\\min\\left\\{\\| f\\RGNN{i+\\gamma}|\ng\\RGN{j+6,-1},\\,| f\\RGN{i+6,-1}\\| g\\RGNN{j+\\gamma}\\right\\}\\end{equation}\nwhere $\\gamma=6$ if $p=2$ or $\\gamma=8$ if $p=\\infty$.\n\nWe prove it by induction.\n\n\\emph{Step 1.} Set $|\\beta|=0$. By integrating by parts, the LHS of\n(\\ref{prop:proof})\n\\begin{equation}\\label{step0:1}\\left\\|(1+t+|x|)[f,q,g]\\right\\|_{L^p_x}\n=\\left\\|(1+t+|x|)[f_1,q_1,g_1]\\right\\|_{L^p_x}\\end{equation} where $q_1$ is defined in (\\ref{def:q1}) and\n\\[\\begin{split} f_1(t,y):=&(1-\\Delta_y)^3f(t,y),\\qquad\ng_1(t,z):=(1-\\Delta_z)^3g(t,z)\\end{split}.\\]\nContinue from (\\ref{step0:1}),\n\\[\\begin{split}\n&\\left\\|(1+t+|x|)[f,q,g]\\right\\|_{L^p_x}\\\\\n=\n&\\left\\|(1+t+|x|)\\int_{\\mbox{R}^2\\times\\mbox{R}^2}f_1(t,x-y)\nq_1(y,z)g_1(t,x-z)\\,dydz\\right\\|_{L^p_x}\\\\\n\\le&\\left\\|\\int_{\\mbox{R}^2\\times\\mbox{R}^2}\\left|(1+t+|x-y|)\nf_1(t,x-y)q_1(y,z)g_1(t,x-z)\\right|\\,dydz\\right\\|_{L^p_x}\\\\\n&+\\left\\|\\int_{\\mbox{R}^2\\times\\mbox{R}^2}\\left|f_1(t,x-y)yq_1(y,z)\ng_1(t,x-z)\\right|\\,dydz\\right\\|_{L^p_x}\\\\\n\\le&\\left|(1+t+|y|)f_1(t,y)\\right|_{L^\\infty_y}\\|q_1(y,z)\\|_{L^1_{yz}}\n\\|g_1(t,z)\\|_{L^p_z}\\\\&+\\left|f_1(t,y)\\right|_{L^\\infty_y}\n\\|yq_1(y,z)\\|_{L^1_{yz}}\\|g_1(t,z)\\|_{L^p_z}\n\\end{split}\\]\nby Young's inequality. Combined with Proposition \\ref{prop:kernel}\n(and Sobolev inequality if $p=\\infty$), this\nimplies\\[\\left\\|(1+t+|x|)[f,q,g]\\right\\|_{L^p_x}\\le C|\nf_1\\RGN{0,-1}\\|g_1\\|_{L^p}\\le C| f\\RGN{6,-1}\\|\ng\\RGNN{\\gamma}.\\]The same estimate holds if we switch $f$ and $g$.\nThus, we proved (\\ref{prop:proof}) for $|\\beta|=0$.\n\n\\emph{Step $2$.} Suppose (\\ref{prop:proof}) is true for all $(n-1)$-th order\nvector fields. Now pick any $n$-th order vector field\n$\\Gamma^{\\beta}:=\\Gamma^{\\beta'}\\Gamma^1$ where $|\\beta'|=n-1$ and\n$\\Gamma^1\\in\\{\\partial_t,\\,\\partial_1,\\,\\partial_2,\\,t\\partial_1+x_1\\partial_t,\\,t\\partial_2+x_2\\partial_t,\n\\,x_1\\partial_2-x_2\\partial_1\\}$. By product rule and the definition of normal\nforms (\\ref{def:vi}), for any $\\partial\\in\\{\\partial_t,\\partial_1,\\partial_2\\}$,\nwe have\n\\[\n\\begin{split}\\partial[f,q,g]=&[\\partial f,q,g]+[f,q,\\partial g],\\\\\nt\\partial[f,q,g]=&[t\\partial f,q,g]+[f,q,t\\partial g],\\\\\nx_i\\partial[f,q,g]=&\\left([x_i\\partial f(t,x),q(y,z),g(t,x)]\n+[\\partial f(t,x),y_iq(y,z),g(t,x)]\\right)\\\\\n&+\\left([f(t,x),q(y,z),x_i\\partial g(t,x)]+[f(t,x),z_iq(y,z),\\partial g(t,x)]\\right),\n\\end{split}\n\\]\nwhich immediately implies that\n\\begin{equation}\\label{productrule}\n\\begin{split}\\Gamma^{\\beta}[f,q,g]=&\\Gamma^{\\beta'}\\left([\\Gamma^1 f,q,g]\n+[f,q,\\Gamma^1g]\\right)\\\\ &+\\Gamma^{\\beta'}\\sum_{i=0}^2\n\\sum_{j=1}^2C_{ij}\\left([\\partial_if,y_jq,g]+[f,z_jq,\\partial_ig]\\right).\n\\end{split}\n\\end{equation} Here, the kernels are $q(y,z)$, $y_jq(y,z)$, $z_jq(y,z)$, all\nsatisfying the growth condition (\\ref{growth:q}). Therefore, the\ninductive hypothesis is true and we apply (\\ref{prop:proof}) with\n$|\\beta'|=n-1$ on (\\ref{productrule}) to conclude that\n(\\ref{prop:proof}) also holds for $|\\beta|=n$. This finishes the proof.\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{Introduction}\nUnmanned Aerial Vehicles (UAVs) have been recently used in many civilian, commercial and military applications \\cite{Razi_Asilomar17,Peng2018,Mousavi_INFOCOM18,Afghah_ACC18,Khaledi_SECON18,Afghah_NWRCS}. With recent advances in design and production of UAVs, the global market revenue of UAVs is expected to reach \\$11.2 billion by 2020 \\cite{Gartner_UAV}. \n\nSpectrum management is one of the key challenges in UAV networks, since spectrum shortage\ncan impede the operation of\nthese networks. In particular, in applications involving a low-latency video streaming, the UAVs may require additional spectrum to complete their mission. The conventional spectrum sharing mechanism such as spectrum sensing may not be very practical in UAV systems noting the considerable required energy for spectrum sensing or the fact that they cannot guarantee a continuous spectrum access. The property-right spectrum sharing techniques operate based on an agreement between the licensed and unlicensed users where the spectrum owners lease their spectrum to the unlicensed ones in exchange for certain services such as cooperative relaying or energy harvesting.\n\nIn this paper, we studied the problem of limited spectrum in UAV networks and considered a relay-based cooperative spectrum leasing scenario in which a group of UAVs in the network cooperatively forward data packet for a \nground primary user (PU) in exchange for spectrum access. The rest of the UAVs in the network utilize the obtained spectrum for transmission and completion of the remote sensing operation. Thus, the main problem is to partition the UAV network into two task groups in a distributed way.\n\nIt is worth noting that cooperative spectrum sharing has been studied previously in the context of cognitive radio networks \\cite{Stanojev08,Afghah_CDC2013,Korenda_CISS,Namvar15}. The existing models are mostly centralized and the set of relay nodes\nis typically chosen by the PU. Such solutions, however, are not applicable to UAV networks, due to their distributed\ninfrastructure and autonomous functionality.\n\nTo tackle this problem, we utilize multi-agent reinforcement learning \\cite{Guestrin2002,Lauer2000,mousavi2016deep,Chalkiadakis2003, mousavi2017traffic, mousavi2016learning, mousavi2017applying}, which is an effective tool for designing algorithms in distributed systems, where the environment is unknown and a reliable communication among agents is not guaranteed. The main problems in distributed multi-agent reinforcement learning\ninclude dealing with state space complexity and\nthe lack of complete information about other agents. There have been proposals in the literature to address these issues through message passing or simplifying assumptions. For instance, \\cite{Guestrin2002} assumes that the decision of an agent depends only on a limited group of other agents, which decomposes the state space and simplifies the problem. In another work \\cite{Chalkiadakis2003}, a Bayesian setting is proposed where each agent has some distributional knowledge about other agents' decisions. Such simplifications, however, are not applicable to the distributed UAV network environment.\n\nIn this paper, we propose a distributed multi-agent reinforcement learning algorithm for task allocation among UAVs. Each UAV either joins a relaying group to provide relaying service for the PU or performs data transmission to the UAV fusion center. In this approach, each UAV maintains a local table about the\nrespective rewards for\nits actions in different states. The tables are updated locally based on a feedback from PU receiver and the UAV fusion node. We define utilities for both the PU and the UAV network, and the objective is to maximize the total utility of the system (sum utility of the PU and the UAV network). We discuss the convergence of our learning algorithm and we present simulation results to verify the convergence to the optimal solution.\n\n\nThe remainder of this paper is organized as follows. In Section \\ref{sec:SystemModel}, the system model and the assumptions of the proposed model are described. In Section \\ref{sec:ProposedMethod}, we propose a distributed multi-agent learning algorithm to solve the spectrum sharing problem.  In Section \\ref{sec:Simulations}, we present simulation results and discuss the performance of our distributed learning algorithm. Finally, we make concluding remarks in Section \\ref{sec:Conclusion}.\n\n\n\\section{System Model}\\label{sec:SystemModel} \nWe consider a licensed primary user (PU) who is willing to share a part of its spectrum with a network of UAVs, in exchange for receiving a cooperative relaying service. The UAV network consists of $N$ UAVs which can be partitioned into two sets depending on the task of the UAV. In fact, UAVs either relay for the PU or utilize the spectrum to transmit their own packets to the fusion center. Let $K$ be the number of nodes who perform the relaying task and $N - K$ denote the number of  UAVs that transmit packets to their fusion center. In this paper, we assume that both the PU's transmitter and receiver are terrestrial, while UAVs are operating in high elevation. Also, we assume no reliable direct link exists between the PU's transmitter and receiver. Moreover, there is zero chance for direct transmission between the UAVs' source and fusion, due to their distances from the fusion center. Fig. \\ref{fig:System Model} illustrates a sample scenario with 6 total nodes, where\nthe nodes are partitioned into a set of 4 \nrelay nodes for the fusion center, and 2 other nodes relay information for the PU receiver on the ground.\n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=0.7\\linewidth]{systemModel.jpg}\n\\caption{System Model: A sample Scenario with 6 UAVs, where four UAVs handle  packets relaying between the Source and Fusion Center and two UAVs relay packets for the Primary User. \n\t\\vspace{-10pt}\n\\label{fig:System Model}\n\\end{figure}\n\nThe PU's transmitter intends to send its packet to a designated\nreceiver, which is far away from its location. Hence, a single or a number of UAVs are required to deliver its information to the receiver. In addition, we assume that the UAVs' spectrum is  congested or unreliable, therefore the UAVs  are required to lease additional spectrum from the PU to communicate with their fusion. \nBy delivering the PU's packet, the UAVs  gain  spectrum access to send their own packets. All the UAVs transmitters and receivers are assumed to be equipped with a single antenna. Also, we assume that the channels between UAVs, source, fusion, and PU transmitter and receiver are slow Rayleigh fading with a constant coefficient over one time slot. \nThe channel coefficients are defined as follows: i) $h_{PT,U_i}$ refers to the channel parameters between the PU's transmitter and $i^{th}$ UAV; ii) $h_{U_i,PR}$ denotes the parameters between the $i^{th}$ UAV and the PU's receiver; iii) $h_{S,U_i}$ and $h_{U_i,F}$, respectively denote the channel coefficients between the Source and the $i^{th}$ UAV, and between the $i^{th}$ UAV and the fusion center.\nFor the sake of simplicity, the instant Channel State Information (CSI) are assumed to be available for all UAVs\nfollowing similar works in \\cite{stanojev2013improving,wu2011information,al2016enhancingg, afghah2018reputation,shamsoshoara2015enhanced}\n\nThe source of the noise at the receivers\nis considered as\na symmetric \\textit{normally} distributed random variable, denoted by \n$z \\sim CN(0, \\sigma^2)$. Many works such as \\cite{roberge2013comparison, mozaffari2016optimal,shamsoshoara2015enhanced} optimized the power consumption and nodes' lifetime in this area. On the other hand, power optimization is not the purpose of this work, hence we assume constant powers during the transmissions. However, the transmission power for the Source and the PU transmitter is less than\nthose of the UAVs. Half-duplex strategy is utilized in this work. Without loss of generality, time-division notations are characterized in order to ensure the half-duplex operations. After these assumptions, the channel and system model for a single relay is shown in Fig. \\ref{fig:Sinle Relay model}. In this model, all UAVs and terminals utilize a single antenna for transmission.\n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=0.7\\linewidth]{SingleRelay.jpg}\n\\caption{Communication channels for a single relay}\n\t\\vspace{-10pt}\n\\label{fig:Sinle Relay model}\n\\end{figure}\n\nIn the first half of a transmission \ncycle, \nthe source transmits its packet and\nthe relay UAVs receive the information.\nThe channel model for the first half is presented as follows:\n\\begin{align}\\label{eq:1}\ny_{U}[n] = h_{S, U}x_{S}[n] + z_{r}[n],\t\n\\end{align}\nwhere $x_{S}$ is the source's transmitted signal and $y_{U}$ is the UAV's received signal. Then, in the second half of the transmission, the UAV sends the received packet\nin the previous time slot. \nWe can write the second half as another model for the received signal as follow\n\\begin{align}\\label{eq:2}\ny_{F}[n] = h_{U, F}x_{U}[n] + z_{F}[n],\t\n\\end{align}\nwhere $x_{U}$ is the UAV's transmitted signal and $y_{F}$ is the destination's received signal. \n\nIn equations (\\ref{eq:1}) and (\\ref{eq:2}), the CSI parameters $h_{ij}$ represent the effects of the path loss and likewise $z_{j}$ represents the effect of noise and interference terms at the receiver, where $i \\in \\{\\textnormal{Source}, \\textnormal{PU-Transmitter}, \\textnormal{UAV}\\}$ and $j \\in \\{\\textnormal{Fusion}, \\textnormal{PU-R}, \\textnormal{UAV} \\}$. In our scenario, $h_{ij}$ is calculated by the proper receiver. \n\\\\\nBased on equations (\\ref{eq:1}) and (\\ref{eq:2}), the throughput capacity of the non-degraded discrete memoryless broadcast channel is expressed in (\\ref{eq:3}) \\cite{AnalyzingAFZhang}: \n\\begin{align}\\label{eq:3}\nC_{Throughput} = \\max_{w \\rightarrow x \\rightarrow y_{d}} \\{I(w;y_{d})\\},\n\\end{align}\nwhere $d \\in \\{\\textnormal{Fusion}, \\textnormal{PU-Receiver}\\}$, $w$ is the message word and $x$ is the codeword which has been assigned to each message by the encoder. Preferably, equation (\\ref{eq:3}) should be solved for the optimal joint distribution of both $w$ and $x$. \nHowever, as discussed in \\cite{shafiee2007achievable}, we can achieve the suboptimal throughput rate in (\\ref{eq:4}), with the aid of assumption $x = w$. Also, $p(x)$ denotes the probability mass function (pmf) for the codeword.\n\\begin{align}\\label{eq:4}\nR_{Throughput} = \\max_{p(x)} \\{I(x;y_d)\\}\n\\end{align}\nIn scenarios, where \nusers can exploit the existence of UAVs, different cooperation protocols such as Decode and Forward (DF) and Amplify and Forward (AF) can be used \\cite{laneman2001efficient}.\nThe idea behind the concept of \\textit{cooperative relaying} is that a set of relay nodes decode, amplify and collectively ``beam-form\" the signal received from the source node (potentially with help of source node itself) towards a designated destination in order to exploit transmission diversity and increase the overall throughput of the system \\cite{levin2012amplify}. \n\\\\\nConsidering an AF cooperation, \neach  UAV first amplifies the signals from the source and then cooperates with source to send its information to the fusion center or to the PU-Receiver. According to \\cite{laneman2004cooperative}, the mutual information for i) the first set of source, UAV, fusion and ii) the second set of PU-T, UAV, PU-R can be written as equations (\\ref{eq:5}) and (\\ref{eq:6}), respectively. In these equations, $P_S$ denotes the transmitter power from the source of the UAV network and $i$ specifies the index for the UAV.\n\\begin{align}\\label{eq:5}\n&I_{SF_{AF}} = \\log_2(1 + P_{S}|h_{SF}|^2  \\\\\n\\nonumber\n&+ \\frac{P_{S}|h_{S,U_i}|^2\\: P_{U_i}|h_{U_i, F}|^2}{1 +P_{S}|h_{S,U_i}|^2 + P_{U_i}|h_{U_i, F}|^2})\n\\end{align}\n\\begin{align}\\label{eq:6}\n&I_{PU(TR)_{AF}} = \\log_2(1 + P_{PT}|h_{PT,PR}|^2  \\\\\n\\nonumber\n&+ \\frac{P_{PT}|h_{PT,U_i}|^2 \\: P_{U_i}|h_{U_i, PR}|^2}{1 +P_{PT}|h_{PT,U_i}|^2 + P_{U_i}|h_{U_i, PR}|^2})\n\\end{align}\nWe denote the throughput rate for both primary users and source-fusion users as (\\ref{eq:7}) and (\\ref{eq:8}), respectively.\n\\begin{align}\\label{eq:7}\nR_{PU} = I_{PU(TR)_{AF}} \n\\end{align}\n\\begin{align}\\label{eq:8}\nR_{SF} = I_{SF_{AF}} \n\\end{align}\nIt is noteworthy\nthat these equations\nare valid only \nfor cooperation with a single Relay or UAV. However, the\nobjective \nof this paper is dealing with Multi-UAV or Multi-Agent relays. Fig. \\ref{fig:Groups} demonstrates the distribution of $N$ UAVs into two groups\nincluding $K$ UAVs facilitating the air source-to-fusion communication and $N - K$ UAVs providing relaying service for a ground-based primary transmitter-receiver pair.\nHence, the equations for multi-UAV should be changed to (\\ref{eq:9}) and (\\ref{eq:10}). In (\\ref{eq:9}), $i$ defines the lower bound for the first UAV in the source-fusion pair and $i + N - K$ denotes the upper bound.\n\\begin{figure}[t]\n\\centering\n\\includegraphics[height=5in, width=0.5\\linewidth]{Groups.jpg}\n\\caption{System Model: Dividing UAVs into $K$ and $N-K$ groups, for cooperating in two sets of Source-Fusion and Primary Transmitter-Receiver.}\n\t\\vspace{-10pt}\n\\label{fig:Groups}\n\\end{figure}\n\\begin{align}\\label{eq:9}\n&R_{SF}\\textnormal{(Multi-UAV)} = \\log_2(1 + P_{S}|h_{SF}|^2  \\\\\n\\nonumber\n&+ \\sum_{j = i}^{i+N-K} \\frac{P_{S}|h_{S,U_j}|^2 \\: P_{U_j}|h_{U_j, F}|^2}{1 +P_{S}|h_{S,U_j}|^2 + P_{U_j}|h_{U_j, F}|^2})\n\\end{align}\nHere, $R_{SF}\\textnormal{(Multi-UAV)}$ is the achievable rate\nfor the fusion center.\nThis rate is achieved with the help of ($N-K$) UAVs. \n$P_S$ and $P_{U_i}$ are transmission powers for the source and the $i^{th}$ UAV, respectively. Also, $h_{SF}$ denotes the channel coefficient for the pair of source-fusion center, $h_{S,U_j}$ stands for the channel between the source and $j^{th}$ UAV, and finally $h_{U_j, F}$ denotes CSI for the $j^{th}$ UAV and the fusion. In (\\ref{eq:10}), $m$ and $m + K$ define the lower and upper bound for the first  and last UAV in the source-fusion pair respectively.\n\\begin{align}\\label{eq:10}\n&R_{PU}\\textnormal{(Multi-UAV)} = \\log_2(1 + P_{PT}|h_{PT,PR}|^2  \\\\\n\\nonumber\n&+ \\sum_{l = m}^{m + K} \\frac{P_{PT}|h_{PT,U_l}|^2 \\: P_{U_l}|h_{U_l, PR}|^2}{1 +P_{PT}|h_{PT,U_l}|^2 + P_{U_l}|h_{U_l, PR}|^2})\n\\end{align}\nIn (\\ref{eq:10}), $R_{PU}\\textnormal{(Multi-UAV)}$ is the achievable rate for the primary transmitter-receiver pair with the aid of $K$ UAVs. \n$P_{PT}$ and $P_{U_l}$ are transmission power for the primary user and the $i^{th}$ UAV, respectively. Moreover, $h_{PT,PR}$, denotes the channel coefficients for primary transmitter and receiver. $h_{PT,U_l}$ stands for the primary transmitter and $l^{th}$ UAV. Finally $h_{U_l, PR}$ is CSI parameters for the $l^{th}$ UAV and the primary receiver.\nBased on the assumption of long distance between the source and the fusion center and also the long distance between the primary transmitter and receiver, we can assume that $h_{SF}$ and $h_{PT,PR}$ are negligible. \n\n\n\n\n\n\n\\iffalse\nI don't want this to print\n\nConsidering a Amplify-and-Forward cooperation from the relay UAVs, the utility of the PU is defined as\n\\begin{equation}\\label{eq:UtilityPU}\nU_P(k)= \\min ( R_{PU}(k), R_{UP}(k) )\n\\end{equation}\n\\noindent where $R_{PU}(k)$ is the rate achievable between the primary transmitter and the relay UAVs. This rate is dominated by the worst channel from the primary transmitter to the relay UAVs. Let UAV $k$ be the one with the worst channel and $h_{PU,k}$ be the channel gain between the primary transmitter and the UAV $k$. Also, let $P_p$ be the transmission power of the primary transmitter, we have\n\\begin{equation}\nR_{PU}(k)=\\log_{2}\\Big( 1+ \\frac{|h_{PU,k}|^2 P_p}{N_0}\\Big)\n\\end{equation}\n\\noindent The rate between the $k$ relay UAVs and the primary receiver, $R_{UP}(k)$, is defined as\n\\begin{equation}\nR_{UP}(k)=\\log_{2}\\Big( 1+\\sum_{i=1}^{k} \\frac{|h_{UP,i}|^2 P_i}{N_0}\\Big)\n\\end{equation}\n\\noindent where $P_i$ is the transmission power of the UAV $i$ and $h_{UP,i}$ is the channel gain between the UAV $i$ and the primary receiver. \n\nFor the UAV network we have a source that broadcasts to the $N-k$ UAV nodes who transmit to the fusion center. Therefore, the rate of the network can be defined as the minimum of the broadcast rate from the source to the $N-k$ UAV nodes ($R_{B}(N-k)$) and the multicast rate from these $N-k$ UAVs to the fusion center (($R_{M}(N-k)$)). We should note that the UAV network also incurs a power cost for relaying the PU's message which should be taken into account in the utility function of the network.\n\\begin{equation}\\label{eq:UtilityUAV}\nU_U(N-k)= \\min ( R_{B}(N-k), R_{M}(N-k) ) - c \\sum_{i=1}^{k} P_i\n\\end{equation}\n\\noindent where $c$ is the unit power cost and we have\n\\begin{equation}\nR_{B}(N-k)=\\sum_{i=k+1}^{N} \\log_{2}\\Big( 1+ \\frac{|h_{SU,i}|^2 P_i}{\\sum_{j=k+1, j\\neq i}^{N} P_j+N_0}\\Big)\n\\end{equation}\n\\noindent where $h_{SU,i}$ is the channel gain between the source and the UAV $i$. Let $h_{UF,i}$ be the channel gain between the UAV $i$ and the fusion center, the multicast rate $R_{M}(N-k)$ is defined as\n\\begin{equation}\nR_{M}(N-k)=\\log_{2}\\Big( 1+ \\sum_{i=k+1}^{N}\\frac{|h_{UF,i}|^2 P_i}{N_0}\\Big)\n\\end{equation}\n\\fi\n\nTime is slotted and at the end of each time slot, the fusion center and the primary receiver send feedback to the UAVs informing them about the achieved accumulated rates. This information is used by each UAV to decide on joining a task group. The goal is to find the optimal task allocation for UAVs in a fully distributed way such that the total utility of the system (i.e. sum utility of UAV network (\\ref{eq:9}) and the PU (\\ref{eq:10})) is maximized. We assume that the UAVs decide locally with no information exchange among themselves. \n\nIt is noteworthy that in some cases,\nthe maximum throughput is achieved \nwhen all UAVs join the same set and deliver packets only for one set, which is not consistent with the proposed model. \nIf all UAVs are distributed in the set of source-fusion, then the total throughput rate is zero because there is no available spectrum for UAVs to utilize for their transmission. Also, if all UAVs are partitioned in the primary set, then the sum throughput rate is equal to the rate of the primary user. In this case the proposed method handles this issue by considering the Jain fairness index \\cite{lan2010axiomatic}. \nBased on the fact that we only have two sets and based on the Jain index definition, (\\ref{eq:11}) describes the fairness for the proposed method in our system model. \n\\begin{align}\\label{eq:11}\nJ(x) = \\frac{1}{n} \\times \\frac{(\\sum_{i} x_{i})^2}{\\sum_{i} x_{i}^{2}},\n\\end{align}\n\n\\noindent Here, $n$ is equal to 2 and $i \\in \\{0,1\\}$ which indicates the set of source-fusion or Primary Users. We assume that $x_0$ and $x_1$ are equal to the number of UAVs in the Fusion-Source set and the Primary Users set, respectively. Therefore, we can define the fairness as (\\ref{eq:12}).\n\\begin{align}\\label{eq:12}\nFariness = \\frac{1}{2} \\times \\frac{(\\#\\textnormal{U}_{F} + \\#\\textnormal{U}_{P})^2}{(\\#\\textnormal{U}_{F})^2 + (\\#\\textnormal{U}_{P})^2}\n\\end{align}\nNow, if all UAVs are distributed in one set, then the fairness will be minimum ($0.5$), and if the UAVs are partitioned equally among two sets, then the fairness will be maximum ($1$). \n\nBased on these definitions, we define (\\ref{eq:13}), as the gain value for each time slot which indicates the efficiency and performance for the distributed UAVs in two sets. \n\\begin{align}\\label{eq:13}\n& Gain = \\gamma_{1} \\times \\Delta(Rate_{Fusion})\\\\\n\\nonumber\n&+\\gamma_{2} \\times \\Delta(Rate_{Primary}) + \\gamma_{3} \\times (Fairness)\n\\end{align}\n\nIn (\\ref{eq:13}), $\\Delta(\\textnormal{Rate}_{\\textnormal{Fusion}})$ is the difference between the rate at time $t$ and the average of previous rates for the fusion center and $\\Delta\\textnormal{(Rate}_{\\textnormal{Primary}})$ is the difference between the rate at time $t$ and the average of previous rates for the primary user. Also, $\\gamma_{1}$, $\\gamma_{2}$, and $\\gamma_{3}$ are defined to control the gain value. Then, we use this gain in our proposed method as described in section \\ref{sec:ProposedMethod}.\n\n\\color{black}\n\\section{The Distributed Learning Algorithm for Task Allocation}\\label{sec:ProposedMethod}  \n\n\n\nThe proposed method is a general form of the Q-learning algorithm \\cite{watkins1992q} for a distributed multi-agent environment.\n\nLet $a^{i}_{t}$ denote the action chosen by UAV $i$ at time $t$, and let $A^i$ denote the set of all possible actions for UAV $i$. We consider two possible actions for a UAV that correspond to either joining the relaying task group or the fusion task partition. Therefore, the set of possible actions are identical across UAVs. We denote the action vector of UAVs at time $t$ by $u_t=(a^{1}_{t}, a^{2}_{t},\\cdots, a^{N}_{t})$, and we refer to the set of all possible action vectors by $\\mathcal{U}$. There is a finite set of states $\\mathcal{S}$, where state $s\\in \\mathcal{S}$ corresponds to the current task partition. A deterministic transition rule $\\delta$ governs the transition between states, i.e. $\\delta: \\mathcal{S}\\times \\mathcal{U} \\rightarrow \\mathcal{S}$. The reward function $r$ maps the current state and action vector to a real value, that is $r: \\mathcal{S}\\times \\mathcal{U} \\rightarrow \\mathbb{R}$.\nAt the beginning of each time step, the UAVs observe the current state (this information is obtained by the feedback from the previous step). Then, each UAV independently decides on its action (i.e. which task group to join) without knowing any information about actions of the other agents. The rewards associated with the UAVs' actions are computed\nby the PU receiver and the UAV fusion. The reward is basically the gain obtained from the task partitioning, taking into account the utilities of the PU and the UAV network. After the reward is calculated, a feedback message from the PU receiver and the UAV fusion is broadcasted to the UAVs. This feedback message contains the reward and the current task partitions. \n\nThe feedback information is used to update and maintain local Q-tables at each UAV. A Q-table basically represents the quality of different actions for a given state. For instance, $q^{i}_{t}(s,a)$ denotes the quality of action $a$ at state $s$ for UAV $i$ at time $t$. Individual Q-tables are updated as follows. At first, the tables are initialized with $q^{i}_{0} (s,a) = 0 $. Then, the following equation is used to update the Q-tables:\n\n\\begin{align} \\label{eq:qUpdates}\nq^{i}_{t+1} (s,a) = \n\\begin{cases}\nq^{i}_{t} (s,a),  \\qquad\\qquad\\qquad \\textnormal{if} \\;s\\neq s_t\\; \\textnormal{or} \\;\\;a\\neq a^{i}_{t},  \\\\\n\\\\\n(1-\\alpha) \\:q^{i}_{t} (s,a) +\\\\\n\\qquad \\alpha \\cdot \\big(r_{t} +\\beta \\cdot \\max_{a'\\in A^i} q^{i}_{t} (\\delta(s_{t},u_t), a') \\big), \\\\\n\\qquad\\qquad\\qquad\\qquad\\qquad \\textnormal{otherwise},\n\\end{cases}\n\\end{align}\n\\noindent where $0\\leq \\alpha <1$ is the learning rate, $r_{t}$ is the reward or the gain obtained at time $t$, as defined in the system model, and $0\\leq \\beta <1$ is the discount factor to control the weight of future rewards in the current decisions.\n\n\nThe main idea is that in our distributed environment, the UAVs are unable to keep a global Q-table,\ncorresponding to the \ncurrent action vectors, i.e. $Q: \\mathcal{S}\\times \\mathcal{U} \\rightarrow \\mathbb{R}$. Instead, each UAV $i$ keeps a local (and considerably smaller) Q-table which cares about its own current action, i.e. $q^i: \\mathcal{S}\\times A^i \\rightarrow \\mathbb{R}$. This approach significantly reduces the complexity of the algorithm and eliminates the need for coordination (or sharing information) with other UAVs at the time of decision making. However, we need a projection method that compresses the information of the global Q-table into the local small tables. \n\nThe results in \\cite{Lauer2000} prove that in a deterministic multi-agent Markov decision process and for the same sequence of states and actions, if every independent learner chooses locally optimal actions, the result would be the same as choosing the optimal action from a global table. We utilize this result and consider an optimistic projection method that assumes each UAV chooses the maximum quality action from its local table. This reasonable assumption is a necessary condition for the optimality of the learning algorithm. It is worth noting that the existence of a unique optimal solution is the sufficient condition for the optimality of this algorithm. It means that there should be a unique task partition, which results in the maximum total utility. If multiple task partitions yield the maximum utility, it is possible that the UAVs act optimally and choose the optimal actions in their local Q-tables, but the combination of their actions may not be optimal. In this case, message passing among UAVs is needed as they need to coordinate decisions at every step.\n\nIt should also be noted that in learning algorithms we need a balance between exploring new actions and exploiting the previously learned quality of actions. Therefore, a greedy strategy that always exploits the Q-table and chooses the optimal action from the Q-table may not provide enough exploration for the UAV to guarantee an optimal performance.\nA very common approach is to add some randomness to the policy \\cite{Singh2000}. We use $\\epsilon$-greedy with a decaying exploration, in which a UAV chooses a random exploratory action at state $s$ with probability $\\epsilon(s)=c\/n(s)$, where $0<c<1$ and $n(s)$ is the number of times the state $s$ has been observed so far. The UAV exploits greedily from its Q-table with probability of $1-\\epsilon(s)$. In this approach, the probability of exploration decays over time as the UAVs learn more.\n\nSimilar to the original Q-learning for a single agent environments, the proposed learning algorithm converges if the state-action pairs are observed infinitely many times.\nAlso, the time complexity \nof\nthe algorithm is in \nthe order of \n$O(|\\mathcal{S}|\\times |A^i|)$, where $|\\mathcal{S}|$ is the size of the state space,\nand $|A^i|$ is\nthe size of action space for UAV $i$. Since there are only two possible actions in our application, the complexity can be expressed as $O(|\\mathcal{S}|)$. In terms of space complexity, each UAV $i$ needs to keep a table of size $|\\mathcal{S}|\\times |A^i|$.\n\n\n\n\\section{Simulation Results} \\label{sec:Simulations}\nIn this section, we present the simulation results to evaluate the performance of the proposed method. We simulate our system model for\na ground-based primary transmitter-receiver pair along with the pair of source and fusion for the UAV network. \nThe location of primary users, source and fusion are fixed during the simulation. However, the UAVs are distributed randomly in the environment. The channels between nodes $i$ and $j$ are obtained from $h_{i,j} \\sim CN(0,d^{-2}_{i,j})$, where $d_{i,j}$ is the distance between nodes $i$ and $j$. The duration of one time slot, $T$, is assumed to be equal to 1. The values of $\\gamma_{1}$ , $\\gamma_{2}$ and $\\gamma_{3}$ are set to $2$, $2$ and $0.4$, respectively. \n\\vspace{-3mm}\n\\subsection*{Scenario I: 2 UAVs}\nIn the first scenario, we consider two UAVs to be partitioned into two task groups. \n\\begin{figure}[t]\n\t\\centering\n\t\\includegraphics[width=.92\\columnwidth]{2nodesSetting.jpg}\n\t\\caption{\\footnotesize Topology for 2 UAVs in a 100 x 100 mission area.}\n\n\t\t\\vspace{-10pt}\n    \\label{fig:2nodesSetting}\n\\end{figure}\nThe network topology for this scenario is demonstrated in Fig. \\ref{fig:2nodesSetting}. Since in this scenario we only have 2 nodes, the possible states for task allocation is equal to $2^{2} = 4$. Hence, the Q-tables will be learned after a few iterations. Fig. \\ref{fig:2nodesRate} illustrates the summation of the obtained throughput.\nThe convergence to the optimal task allocation \noccurs after\nthe 35$^{th}$ iteration, since the number of states is relatively small\n\\begin{figure}[t]\n\t\\centering\n\t\\includegraphics[width=.92\\columnwidth]{2nodesRate.jpg}\n\n\t\\caption{\\footnotesize Sum Rate for 2 UAVs for 100 iterations}\n\n\t\t\\vspace{-10pt}\n    \\label{fig:2nodesRate}\n\\end{figure}\nThe matrix below shows the final task allocation values for these UAVs.\n \\[\n\\begin{bmatrix} \\label{matrix:2nodes}\n   0 & 1\n\\end{bmatrix}\n\\]\nIn this notation, 0 corresponds to the set of source-fusion and 1 means the set of the primary users. UAV1 who has a lower relative distance to the source-fusion, is allocated to the fusion set, while UAV2 is allocated to the another set to relay for the primary network. \n\\vspace{-3mm}\n\\subsection*{Scenario II: 6 UAVs}\nIn this scenario, we consider 6 UAVs to show that the convergence of the proposed method is achieved\nafter more iterations compared to the case of 2 UAVs in the first scenario, since the number of states with 6 nodes is equal to $2^6 = 64$. This means, at least 64 iterations are required for the algorithm to just test all the states. \n\nFig. \\ref{fig:6nodesSetting} demonstrates the network topology with these 6 UAVs\n\\begin{figure}[t]\n\t\\centering\n\t\\includegraphics[width=.92\\columnwidth]{6nodestopology.jpg}\n\t\\caption{\\footnotesize Topology for 6 UAVs in 100 x 100 simulation field}\n\n\t\t\\vspace{-10pt}\n    \\label{fig:6nodesSetting}\n\\end{figure}\nfor the primary user and the fusion. As we can see in Fig. \\ref{fig:6nodesRate}, the convergence to \nthe \nbest task allocation occurred \nafter \n240 iterations. This implies that the more UAVs \nare \nadded to the model, the more iterations will be taken to the convergence\nepoch. \nMoreover, Fig. \\ref{fig:6nodesswitch} shows the number of UAVs switching their actions (i.e. task partitions) in this scenario. After the 240$^{th}$ iteration, when the convergence happens, we see that no UAV changes its task partition, and the number of switches stays at zero.  \n\\begin{figure}[t]\n\t\\centering\n\t\\includegraphics[width=.92\\columnwidth]{6nodesrate.jpg}\n\t\\caption{\\footnotesize Sum Rate for 6 UAVs for 1000 Iterations}\n\t\\vspace{-10pt}\n    \\label{fig:6nodesRate}\n\\end{figure}\n\n\\begin{figure}[t]\n\t\\centering\n\t\\includegraphics[width=.92\\columnwidth]{6nodeswitch.jpg}\n\t\\caption{\\footnotesize Number of Switching UAVs for 1000 iterations}\n\n\t\t\\vspace{-10pt}\n    \\label{fig:6nodesswitch}\n\\end{figure}\n\nAlso, task matrix shown below denotes the final task allocation for the 6 UAVs.\n \\[\n\\begin{bmatrix} \\label{matrix:6nodes}\n   \t 1 & 0 & 1 & 0 & 0 & 1\n\\end{bmatrix}\n\\]\nBased on this matrix, $\\textnormal{UAV}_i$; $i \\in \\{2,4,5\\}$ are considered for the set of source-fusion and \nthe rest of UAVs\nare assigned to the relay task group for the primary network. This allocation makes sense considering the location of UAVs and their relative distances.\n\n\n\\color{black}\n\n\n\n\n\n\n\n\n\n\n\n\\section{Conclusion}\\label{sec:Conclusion}\nIn this paper, we studied the task allocation problem for spectrum management in UAV networks. We considered a cooperative relay system in which a group of UAVs \nprovide relaying service\nfor a \nground-based \nprimary user in exchange for spectrum access. The borrowed spectrum is not necessarily used by the relay UAV, rather\nis used by other UAVs \nto transmit their own information\nto a fusion center. \nThis makes a win-win situation for both networks. \nWe defined utilities for both the UAV network and the \nground-based primary network\nbased on the achieved rates. \nNext, we proposed a distributed learning algorithm by which the UAVs take proper decisions by joining the relaying or fusion task groups without the need for information exchange or knowledge about other UAV's decisions. \nThe algorithm converges to the optimal task partitioning that maximizes the total utility of the system. Simulation results were presented in different scenarios to verify the convergence of the proposed algorithm. \n\n\n\n\n\n\n\\bibliographystyle{elsarticle-harv}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction\\protect\\bigskip}\n\nThe notion that our universe may have begun from a hot big-bang forms the\npillar of the standard cosmological model, a model that plots the time\nevolution of the universe. This model holds that soon after the big-bang $%\n\\left( \\thicksim 10^{-36}s\\right) $ the temperature of the universe drops\nbelow the critical grand-unified theory temperature $\\left( T_{GUT}\\thicksim\n10^{14}GeV\\right) $. In this over-cool sub-stable state, the universe is\ndominated by vacuum energy-density with an effective cosmological constant \n\\[\n\\Lambda =\\frac{2}{15}\\pi ^{3}\\left( \\frac{T_{GUT}^{2}}{m_{p}}\\right) ^{2}\n\\]\nwhere $m_{p}$ is the Planck mass $\\left( \\thicksim 10^{19}\\,GeV\\right) $.\nThis energy then drives the same universe into a de Sitter-like exponential\ninflation $\\left[ 1\\right] $.\\ The standard cosmological model has been very\nsuccessful in explaining the origins of most of the properties of our\nuniverse: from the very small to the very big, from nucleosynthesis to large\nscale structure formation and the expansion of the universe.\n\nThis standard model of cosmology has not, however, been able to address the\nphysics of the events prior to the inflationary era. Understanding this very\nshort but very important era of the universe requires a theory that\nincorporates General Relativity and Quantum Theory. Such an effort\ntranslates into unifying gravity with the other three forces of nature.\nJudging by what has been learned from such candidates as Supergravity $\\left[\n2\\right] $ and now Superstrings $\\left[ 3\\right] $, it appears that such a\nunifying theory will likely require the universe to be multi-dimensional. In\nSuperstrings, for example, the view is that the universe started out in a\nhigher-dimensional phase. Four of the dimensions then expand leaving behind\nthe rest in the regime of the planck length. In the field theoretic-limit of\nSuperstring theories, gravity is described reasonably accurately by\nmulti-dimensional Einstein field equations $\\left[ 3\\right] $. In this\nunification scheme, internal symmetries can be traced to the spacetime\nassociated with the extra dimensions $\\left[ 4\\right] $. Gauge invariance\nthen assumes the same status as spacetime invariance and internal quantum\nnumbers such as electric charge, believed to result from symmetry motions in\nthe extra dimensions, are now brought to the same footing as energy and\nmomentum.\n\nIn searching for possible effects of the extra dimensions it makes sense,\ntherefore, to evolve the Friedman-Robertson-Walker model of our universe\nback in time to the early de Sitter phase where one reaches energies at\nwhich such extra dimensions may be resolvable (see [4] and Refs. therein).\nCoincidentally, this too turns out to be the era when primordial black holes\nmay have been produced $\\left[ 5-9\\right] $. For this \\ and other reasons\n(like the distinctly high curvature nature of the spacetime around them)\nblack holes will continue to be important probes in any quantum theory of\ngravity, and in the quest to understand the role of multi-dimensionality in\nthe early (and possibly present? (see $\\left[ 10\\right] $)) evolution of our\nuniverse. The recent findings of links between black hole entropy and\ntopological structures of the extra dimensions (see for example $\\left[ 11%\n\\right] $ and $\\left[ 12\\right] $) not only puts high-dimensionality to a\nstronger footing but also emphasizes the role black holes will play as\nprobes in such future research. Naturally this calls for an understanding of\nthe dynamics of black holes in higher dimensions, particularly under\nconditions that mimic those of \\ the early universe. In this paper we begin\na study of\\ the dynamics of such a black hole in such a setting.\n\nSeveral solutions to the Einstein equations of localized sources in higher\ndimensions have been obtained in the recent years. This includes the higher\ndimensional generalizations of the Schwarzschild and the Reisner-Nordstrom\nsolutions $\\left[ 13,14\\right] $, the Kerr solution $\\left[ 15,16\\right] $\nand the Vaidya solution $\\left[ 17\\right] $. Recently the metric of a\nradiating black hole in a de Sitter background, that is a generalization of\nthe Mallett $\\left[ 18\\right] $ metric, has been written down $\\left[ 19,20%\n\\right] $. In the present work our aim is to demonstrate that the dynamics\nof a radiating black hole in a higher dimensional cosmological background\ncan be sensibly discussed. First, we seek to identify and locate the various\nhorizons. After this we then go on to study the structures and discuss the\ndynamics of such horizons. It is shown, at each stage, that all the results\nwe obtain reduce to the well known Mallett $\\left[ 21\\right] $ results as we\ngo down to four dimensions, and to make this transparent our analysis is\nclosely modelled to that of Mallett.\n\nIn Section II we introduce the working metric and the theoretical\nbackground. In Section III we derive equations for the horizons. We solve\nthese equations and use the solutions to identify and locate the various\nhorizons in the problem. In Section IV we take up the issue of the structure\nof such horizons and study their dynamics. In Section V we conclude the\ndiscussion.\n\n\\section{The metric and the theory}\n\n\\subsection{The Metric}\n\nIn this treatment we wish to consider a radiating black hole introduced in\nan N dimensional de Sitter space-time. We suppose, for simplicity, that such\na black hole is reasonably modelled by an imploding shell of\nnegative-energy-density null fluid in the de Sitter universe. In advanced\ntime, comoving, coordinates the metric [19, 20] is given by the line element\n,\n\n\\begin{equation}\nds^{2}=-\\left[ 1-\\frac{2G_{N}m\\left( v\\right) }{nr^{n}}-\\frac{2\\Lambda }{%\n\\left( n+1\\right) \\left( n+2\\right) }r^{2}\\right] dv^{2}+2dvdr+d\\Omega\n_{n+1}^{2}  \\eqnum{2.1}\n\\end{equation}\nwhere $n=N-3$, $m\\left( v\\right) $, the mass, is a monotonically decreasing\nfunction of the advanced time coordinate $v$, $G_{N}$ is the N-dimensional\ngravitational constant, $\\Lambda $ is the cosmological constant and \n\\begin{equation}\nd\\Omega _{n+1}^{2}=r^{2}\\left( d\\theta _{1}^{2}+\\sin ^{2}\\theta _{1}d\\theta\n_{2}^{2}+.....+\\sin ^{2}\\theta _{1}\\sin ^{2}\\theta _{2}....\\sin ^{2}\\theta\n_{n}d\\theta _{n+1}^{2}\\right)  \\eqnum{2.2}\n\\end{equation}\n\\ is the line element on the $\\left( n+1\\right) $-sphere. For N = 4 the\ngeometry reduces to that of the Vaidya-Mallett space-time [18].\n\nIn the absence of the cosmological background, $\\Lambda $, the usual\nluminosity, $L_{0}$, of the black hole is defined [19] from the only\nnon-vanishing component of the energy-momentum tensor, \n\\begin{equation}\nT_{v}^{r}=T_{vv}=\\frac{\\left( n+1\\right) }{8\\pi n}\\frac{\\dot{m}\\left(\nv\\right) }{r^{n+1}}  \\eqnum{2.3}\n\\end{equation}\nas \n\\begin{equation}\nL_{0}=-\\dot{m}\\   \\eqnum{2.4}\n\\end{equation}\nHere and henceforth $\\dot{m}\\left( v\\right) $ and $m^{\\prime }\\left(\nv\\right) $ denote, as usual, derivatives with respect to the time and space\ncoordinates, respectively. The Luminosity which is bounded from above, $%\nL_{0}<1$, is measured in regions where $\\frac{d}{dv}$ is time-like.\n\nOne can introduce a basis of vectors at every point in this spacetime. \\ Two\nsuch vectors $\\beta _{a}$ and $l_{a}$ span the radial-temporal subspace and\nare given by \n\\begin{equation}\n\\beta _{a}=\\delta _{a}^{v}  \\eqnum{2.5}\n\\end{equation}\nand \n\\begin{equation}\nl_{a}=-\\frac{1}{2}\\left[ 1-\\frac{2G_{N}m\\left( v\\right) }{nr^{n}}-\\frac{%\n2\\Lambda }{\\left( n+1\\right) \\left( n+2\\right) }r^{2}\\right] \\delta\n_{a}^{v}+\\delta _{a}^{r}  \\eqnum{2.6}\n\\end{equation}\nwhile the rest of the $\\left( N-2\\right) $ vectors are defined on the $n+1$%\n-sphere and induce on the latter a tensor field $\\gamma _{ab}$ of the form \n\\begin{equation}\n\\gamma _{ab}=r^{2}\\left( \\delta _{a}^{\\theta _{1}}\\delta _{b}^{\\theta\n_{1}}+\\sin ^{2}\\theta _{1}\\delta _{a}^{\\theta _{2}}\\delta _{b}^{\\theta\n_{2}}+.....+\\sin ^{2}\\theta _{1}\\sin ^{2}\\theta _{2}....\\sin ^{2}\\theta\n_{n}\\delta _{a}^{\\theta _{n+1}}\\delta _{b}^{\\theta _{n+1}}\\right) . \n\\eqnum{2.7}\n\\end{equation}\n\\ \\ \\ The vectors satisfy the conditions\n\n\\begin{equation}\n\\beta _{a}\\beta ^{a}=l_{a}l^{a}=0,\\;\\gamma _{ab}\\beta ^{b}=\\gamma\n_{ab}l^{b}=0,\\;\\beta _{a}l^{a}=-1.  \\eqnum{2.8}\n\\end{equation}\n\nOne can do a null-vector decomposition of the above metric in this basis so\nthat\n\n\\begin{equation}\ng_{ab}=-\\beta _{a}l_{b}-l_{a}\\beta _{b}+\\gamma _{ab}.  \\eqnum{2.9}\n\\end{equation}\n\n\\subsection{Deformation of relativistic Membranes}\n\nThe structure and dynamics of horizons of such non-static metrics can be\napproached from the non-perturbative description of deformation of\nrelativistic membranes. In general, one considers the evolution of such\ndeformations of an arbitrary D-dimensional membrane in an arbitrary\nN-dimension space-time. A significant amount of literature has been written\non the subject of such deformations [22, 23, 24]. The quantities that\ncharacterize how a variation in the symmetry of a membrane evolves are the\nexpansion rate, $\\theta $, the shear rate, $\\sigma $, and the vorticity\n(twist), $\\omega $. For the general D-dimensional membrane case, the\nequations for the deformations have usually yielded no simple clear\ninterpretation of these quantities [23]. Recently, Zafiris [24] has made\nsome progress in addressing the problem. However, for the special $D=1$ case\nin an $N$ dimensional background one can still generalize the Carter [22]\nform of the Rachaudhuri [25] equation so that\n\n\\begin{equation}\n\\frac{d\\theta }{dv}=\\kappa \\theta -\\left( \\gamma _{c}^{c}\\right) ^{-1}\\theta\n^{2}-\\sigma _{ab}\\sigma ^{ab}+\\omega _{ab}\\omega ^{ab}-R_{ab}l^{a}l^{b} \n\\eqnum{2.10}\n\\end{equation}\nwith $\\theta $, $\\sigma $, and $\\omega $ taking their usual physical\nmeaning. Here $R_{ab}$ is the N-dimensional Ricci tensor, $\\gamma _{c}^{c}$\nis the trace of the projection tensor for null geodesics and $\\kappa $ is to\nbe identified as the surface gravity. The latter is given by \n\\begin{equation}\n\\kappa =-\\beta ^{a}l^{b}\\nabla _{b}l_{a},  \\eqnum{2.11}\n\\end{equation}\nwhere $\\nabla $ is the covariant derivative operator. For infinitesimally\nneighboring members of the congruence separated by a relative separation\nvector $d{\\bf x}$ the rate of the change of separation is given [22]\\ by \n\\begin{equation}\n\\frac{1}{2}(ds^{2}\\dot{)}=\\theta _{ab}dx^{a}dx^{b}.  \\eqnum{2.12}\n\\end{equation}\nThe expansion rate $\\theta $, of a null geodesic congruence is then given by\nthe trace of the expansion tensor as \n\\begin{equation}\n\\theta =\\theta _{a}^{a}=\\gamma ^{ab}\\nabla _{a}l_{b}.  \\eqnum{2.13}\n\\end{equation}\nIt follows then that \n\\begin{equation}\n\\nabla _{a}l^{a}=\\frac{\\partial l_{a}}{\\partial x^{a}}+\\Gamma\n_{ac}^{a}l^{c}=\\kappa +\\theta .  \\eqnum{2.14}\n\\end{equation}\nAnd clearly in flat space-time $\\theta $ vanishes since the connection\ncoefficients $\\Gamma _{ac}^{a}$ will.\n\n\\section{Location of the Horizons}\n\nSpherically symmetric irrotational space-times, such as under consideration,\nare vorticity \\ and the shear free. The structure and dynamics of the\nhorizons are then only dependent on the expansion, $\\theta $. Following York\n[26]\\ we note that to $O(L_{0})$ the evolution of an apparent horizons ($AH$%\n) is to satisfy the requirement that $\\theta \\simeq 0$, while that of an\nevent horizons ($EH$) is to satisfy the requirement that $\\frac{d\\theta }{dv}%\n\\simeq 0$.\n\n\\subsection{The Apparent Horizons}\n\nWe have written the general expression for the expansion $\\theta $\\ as \n\\begin{equation}\n\\theta =\\gamma ^{ab}\\nabla _{a}l_{b}  \\eqnum{3.1}\n\\end{equation}\nUsing equations $\\left( 2.1\\right) $, $\\left( 2.6\\right) $ and $\\left(\n2.7\\right) $ in $\\left( 3.1\\right) $ yields \n\\begin{equation}\n\\theta \\left( r\\right) =\\frac{n+1}{2r}\\left[ 1-\\frac{2G_{N}m\\left( v\\right) \n}{nr^{n}}-\\frac{2\\Lambda }{\\left( n+1\\right) \\left( n+2\\right) }r^{2}\\right]\n.  \\eqnum{3.2}\n\\end{equation}\n\nConsider now the function $f\\left( r\\right) =-\\frac{2}{n+1}r\\theta $. Since\nthe York conditions require that at the ($AHs$) $\\theta $ and hence $f$\nvanish, it follows from equation (3.2) that these surfaces will satisfy \n\\begin{equation}\nr^{n+2}-\\frac{\\left( n+1\\right) \\left( n+2\\right) }{2\\Lambda }r^{n}+\\frac{%\n\\left( n+1\\right) \\left( n+2\\right) }{n}\\frac{G_{N}m\\left( v\\right) }{%\n\\Lambda }=0.  \\eqnum{3.3}\n\\end{equation}\nAs it stands equation (3.3) will obviously not admit simple closed form\nsolutions. It is possible, however, to put this equation in a useful form\nthat yields solutions which for practical purposes can, reasonably and\njustifiably, be taken as the working solutions to the problem at hand. To\nthis end it is helpful to first gain some insight in the nature of the\nfunction $f\\left( r\\right) $ whose roots satisfy (3.3). One notices that the\nturning points for the function $f\\left( r\\right) $ are located at points\nwhere\n\n\\begin{equation}\n\\left[ r^{2}-\\frac{n\\left( n+1\\right) }{2\\Lambda }\\right] r^{n-1}=0. \n\\eqnum{3.4}\n\\end{equation}\nThe derivative $\\frac{df}{dr}$ then vanishes at points $\\ r=0$ and $r=\\pm %\n\\left[ \\frac{n\\left( n+1\\right) }{2\\Lambda }\\right] ^{\\frac{1}{2}}$. From\nthis one then notices that between $r=+\\infty $ and $r=+\\left[ \\frac{n\\left(\nn+1\\right) }{2\\Lambda }\\right] ^{\\frac{1}{2}}$ the function changes sign\nfrom positive to negative which implies that the function has a root\nimbedded in between these two values. \\ One notices further, that between \\ $%\nr=+\\left[ \\frac{n\\left( n+1\\right) }{2\\Lambda }\\right] ^{\\frac{1}{2}}$ and $%\nr=0$, $f\\left( r\\right) $ does again change signs this time from negative to\npositive, and this betrays the existence of yet a second root. We then\nestablish that the function $f\\left( r\\right) $\\ has at least two real\nroots. More importantly we establish that for $r>0$ this function has two,\nand only two, real roots. This is an encouraging result \\ consistent with\nour expectations [21] that there should be two apparent horizons $r_{AH}^{-}$\nand $r_{AH}^{+}$ associated with the black hole and the cosmological\nconstant, respectively.\n\nThe case of the $r^{n-1}$ turning points at $r=0$ suggests the existence of\nother roots for the function $f\\left( r\\right) $. Such roots are actually\ncomplex and would suggest the existence of {\\it ghost horizons} (at least\nfor an observer in our four dimensional spacetime). I will defer, for now,\nfurther speculation of their significance and other issues about the turning\npoints for a future discussion. It is, however, interesting to note that\nsuch {\\it ghost horizons }only{\\it \\ }start to appear at five dimensions $%\n\\left( n=2\\right) $ and persist for higher dimensions.\n\nTo put equation (3.3) in a more manageable form, it is useful to institute a\nchange of variables. Thus setting\\ \n\\begin{equation}\nr=k\\xi ,  \\eqnum{3.5}\n\\end{equation}\nwhere $k=\\sqrt{\\frac{\\left( n+1\\right) \\left( n+2\\right) }{2\\Lambda }}$\nequation (3.3) can be cast in a form \n\\begin{equation}\n\\xi ^{n+2}-\\xi ^{n}+\\beta \\left( n\\right) =0,  \\eqnum{3.6}\n\\end{equation}\nwhere \n\\begin{equation}\n\\beta \\left( n\\right) =\\frac{2G_{N}m\\left( v\\right) }{n}\\left[ \\frac{%\n2\\Lambda }{\\left( n+1\\right) \\left( n+2\\right) }\\right] ^{\\frac{n}{2}}. \n\\eqnum{3.7}\n\\end{equation}\nOne notes that since in our model $\\Lambda >0$, and so $\\beta \\left(\nn\\right) >0$, then (3.6), along with the necessary positive reality of $r$,\nimply that $0<\\xi <1$ and $0<\\beta <1$.\n\nNow set $\\xi =1-x$, where $0<x<1$ to cast equation (3.6) in the form \n\\begin{equation}\nx\\left( 2-x\\right) \\left( 1-x\\right) ^{n}-\\beta \\left( n\\right) =0. \n\\eqnum{3.8}\n\\end{equation}\n\nThe above limits imposed on $\\xi $ and hence $x$ suggest one can seek\napproximate solutions through expansion of the quantity $\\left( 1-x\\right)\n^{n}$\\ in equation (3.8). Thus keeping in the equation terms up to $x^{4}$\ngives \n\\begin{equation}\nx^{4}+ax^{3}+bx^{2}+cx+d=0,  \\eqnum{3.9}\n\\end{equation}\nwhere \n\\[\na=-\\frac{6n}{\\left( n-1\\right) \\left( 2n-1\\right) },\\;b=\\frac{6\\left(\n2n+1\\right) }{n\\left( n-1\\right) \\left( 2n-1\\right) },\\;c=-\\frac{12}{n\\left(\nn-1\\right) \\left( 2n-1\\right) },\\; \n\\]\n\nand$\\;$%\n\\begin{equation}\nd=\\frac{6}{n\\left( n-1\\right) \\left( 2n-1\\right) }\\beta \\left( n\\right) . \n\\eqnum{3.10}\n\\end{equation}\n\nWe should mention that this expansion is strictly valid for $n>2$. The\nsolutions for the 5-dimensional $\\left( n=2\\right) $ case are actually\nexact. Further, the solutions have the right limits when $n=1$ and for this\ncase one does recover the exact Mallett's cubic equation [21] for the\n4-dimensional case from either of the equations (3.3) or 3.6). In general\nthis approximation is good in the limit $x\\rightarrow 0$, $\\xi \\rightarrow 1$%\n. It, however, breaks down when $x\\rightarrow 1$, $\\xi \\rightarrow 0$. In\nthis latter limit the solution to (3.6) simply goes to $\\xi =\\left[ \\beta\n\\left( n\\right) \\right] ^{\\frac{1}{n}}\\Longrightarrow r=\\left( \\frac{2Gm}{n}%\n\\right) ^{\\frac{1}{n}}$and the geometry loses knowledge of the cosmological\nconstant.\n\nEquation (3.9) has a resolvent cubic equation that can\\ be that can be\nwritten in the form \n\\begin{equation}\ny^{3}+py+q=0  \\eqnum{3.11}\n\\end{equation}\nwhere $p=\\frac{1}{3}\\left[ \\left( 3ac-b^{2}\\right) -12d\\right] $ and$\\;q=%\n\\frac{1}{27}\\left[ 9abc-2b^{3}-27c^{2}-9\\left( 3a^{2}+b\\right) d\\right] $.\nEquation (3.11) admits three solutions. One such solution that is real for\nthe parameters $p$, $q$ as defined above can be written as\n\n\\begin{equation}\ny_{1}=2\\sqrt{\\frac{-p}{3}}\\cos \\frac{1}{3}\\varphi ,  \\eqnum{3.12}\n\\end{equation}\nwhere $\\frac{\\pi }{2}<\\varphi <\\pi $\\ is given by $\\varphi =\\arccos \\left( \n\\frac{3q}{2\\sqrt{\\frac{-p^{3}}{3}}}\\right) $.\n\nThere are four solutions to the quartic equation (3.9) in $x$. The only\nphysically interesting solutions to equation (3.9) consistent with $r=k\\xi\n=k\\left( 1-x\\right) $ should be real and satisfy $x<1$. There are two such\nsolutions obtained by letting \n\\[\nR=\\sqrt{\\frac{a^{2}}{4}-b+\\left( 2\\sqrt{\\frac{-p}{3}}\\right) \\cos \\frac{1}{3}%\n\\varphi } \n\\]\nand \n\\begin{equation}\nD=\\sqrt{\\frac{3a^{2}}{4}-R^{2}\\left( p,\\varphi \\right) -2b+\\frac{4ab-8c-a^{3}%\n}{4R\\left( p,\\varphi \\right) }}  \\eqnum{3.13}\n\\end{equation}\nThese are: \n\\begin{equation}\nx_{\\pm }=-\\frac{1}{4}a+\\frac{1}{2}R\\left( p,\\varphi \\right) \\pm \\frac{1}{2}%\nD\\left( p,\\varphi \\right) .  \\eqnum{3.14}\n\\end{equation}\nNote that all the time dependency of $R$ and so of $D$ is expressed in $p$\nand $\\varphi $ via $d\\left( \\beta \\left( m\\left( v\\right) \\right) \\right) $.\n\nUsing the solutions in equation (3.14) and recalling that $\\xi =1-x$ \\ we\nobtain two values $\\xi _{1}$ and $\\xi _{2}$.\\ On applying these results to\nequation (3.5) i.e. $r=k\\xi =k\\left( 1-x\\right) $ we find two solutions $%\nr_{1}=r_{AH}^{-}\\left( v\\right) $ and $r_{2}=r_{AH}^{+}\\left( v\\right) $\nsuch that\n\n\\begin{equation}\nr_{AH}^{-}\\left( v\\right) =r_{1}\\simeq k\\left\\{ 1-\\frac{1}{4}\\left[ 2D\\left(\np,\\varphi \\right) +2R\\left( p,\\varphi \\right) -a\\right] \\right\\} , \n\\eqnum{3.15}\n\\end{equation}\nand \n\\begin{equation}\nr_{AH}^{+}\\left( v\\right) =r_{2}\\simeq k\\left\\{ 1+\\frac{1}{4}\\left[ 2D\\left(\np,\\varphi \\right) -2R\\left( p,\\varphi \\right) +a\\right] \\right\\} \n\\eqnum{3.16}\n\\end{equation}\n\\ \n\nIn the limit $n\\rightarrow 1$ one recovers the well known solutions [21].\nThus \n\\begin{equation}\n\\mathrel{\\mathop{\\lim }\\limits_{n=1}}%\nr_{AH}^{-}\\left( v\\right) =-\\left( \\frac{2}{\\sqrt{\\Lambda }}\\right) \\cos\n\\left( \\frac{1}{3}\\Psi +\\frac{1}{3}\\pi \\right)  \\eqnum{3.17}\n\\end{equation}\nand \n\\begin{equation}\n\\mathrel{\\mathop{\\lim }\\limits_{n=1}}%\nr_{AH}^{+}\\left( v\\right) =\\left( \\frac{2}{\\sqrt{\\Lambda }}\\right) \\cos\n\\left( \\frac{1}{3}\\Psi \\right)  \\eqnum{3.18}\n\\end{equation}\nwhere now $\\left( \\frac{\\pi }{2}<\\Psi \\left( v\\right) <\\pi \\right) =\\arccos %\n\\left[ -3m\\left( v\\right) \\sqrt{\\Lambda }\\right] $. (Note, however, that $%\n\\varphi $ is not simply related to $\\Psi $ by taking the limit $n\\rightarrow\n1$). Further, in the limit $\\Lambda \\longrightarrow 0$ and $n\\rightarrow 1$\nwe recover the black hole apparent horizon $r_{AH}^{-}\\left( v\\right)\n\\rightarrow 2m\\left( v\\right) $ and in the limit $m\\rightarrow 0$ and $%\nn\\rightarrow 1$ we find, as one expects that $r_{AH}^{+}\\left( v\\right)\n\\rightarrow \\sqrt{\\frac{\\Lambda }{3}}$. Hence our solutions reduce to all\nthe well known solutions [21] in a four dimensional spacetime. Consequently,\nwe identify the locus of $r_{AH}^{-}\\left( v\\right) $\\ in equation (3.15) as\nthe black hole apparent horizon $\\left( AH^{-}\\right) $ while we identify\nthe locus of $r_{AH}^{+}\\left( v\\right) $ in equation\\ (3.16) as the de\nSitter apparent horizon $\\left( AH^{+}\\right) $, for a black hole in an\nN-dimensional background with a cosmological constant.\n\n\\subsection{The Event Horizons}\n\nThe event horizons are null surfaces. \\ To $O(L_{0})$\\ the evolution of\nthese surfaces can be determined from the second of the York conditions that \n$\\frac{d\\theta }{dv}\\simeq 0$. We first show that for a black hole imbedded\nin an N-dimensional de Sitter background this condition is satisfied. We\nshall then solve the resulting equations to locate the event horizons and\nstudy their structure.\n\nThe surface gravity $\\kappa $ in this spacetime is given from (2.11) by \n\\begin{equation}\n\\kappa =\\frac{G_{N}m\\left( v\\right) }{r^{n+1}}-\\frac{2\\Lambda }{\\left(\nn+1\\right) \\left( n+2\\right) }r.  \\eqnum{3.17}\n\\end{equation}\n\\bigskip Consider now the acceleration of the geodesics $\\frac{d^{2}r}{dv^{2}%\n}$ \\ in our space-time, parametrized by $v$. Since from the line element\n(2.1) we have that \n\\begin{equation}\n\\frac{dr}{dv}=\\frac{1}{2}\\left[ 1-\\frac{2G_{N}m\\left( v\\right) }{nr^{n}}-%\n\\frac{2\\Lambda }{\\left( n+1\\right) \\left( n+2\\right) }r^{2}\\right] \n\\eqnum{3.18}\n\\end{equation}\nthen \n\\begin{equation}\n\\frac{d^{2}r}{dv^{2}}=\\frac{G_{N}L_{0}}{nr^{n}}+\\kappa \\frac{dr}{dv} \n\\eqnum{3.19}\n\\end{equation}\nEquations (3.2) and (3.18) in (3.19) give \n\\begin{equation}\n\\frac{d^{2}r}{dv^{2}}=\\frac{G_{N}L_{0}}{nr^{n}}+\\kappa \\theta \\frac{r}{n+1}.\n\\eqnum{3.20}\n\\end{equation}\nBut the event horizon is a null surface and satisfies the general\nrequirement that null-geodesic congruencies have a vanishing acceleration. \\\nThus at the event horizon (3.20) takes the form \n\\begin{equation}\n\\kappa \\theta _{EH}+\\frac{n+1}{n}\\frac{G_{N}L_{0}}{\\left( r_{EH}\\right)\n^{n+1}}=0.  \\eqnum{3.21}\n\\end{equation}\n\nNow the Einstein field equations for the $N(N=n+3)$-dimensional space-time\nare [4]\n\n\\begin{equation}\nR_{ab}=8\\pi G_{N}\\left[ T_{ab}-\\frac{1}{n+1}g_{ab}T_{c}^{c}\\right] +\\frac{%\n2\\Lambda }{n+1}g_{ab}.  \\eqnum{3.22}\n\\end{equation}\nUsing equations $\\left( 2.3\\right) $, $\\left( 2.6\\right) $, and $\\left(\n2.8\\right) $ with $\\left( 3.22\\right) $ one finds that \n\\begin{equation}\nR_{ab}l^{a}l^{b}=\\frac{\\left( n+1\\right) }{n}\\frac{G_{N}}{r^{n+1}}\\dot{m}%\n\\left( v\\right)  \\eqnum{3.23}\n\\end{equation}\n\nFor a spherically symmetric irrotational space-time, such as under\nconsideration, the vorticity $\\omega $\\ and the shear $\\sigma $ vanish and\nthe Raychauduri equation (2.10) reduces to \n\\begin{equation}\n\\frac{d\\theta }{dv}=\\kappa \\theta -R_{ab}l^{a}l^{b}-\\left( \\gamma\n_{c}^{c}\\right) ^{-1}\\theta ^{2}.  \\eqnum{3.24}\n\\end{equation}\nEquations (3.21) and (3.23) when substituted in (3.24) show (on neglecting\nthe term in $\\theta ^{2}$) that, indeed, the York condition for the event\nhorizon is satisfied and we have \n\\begin{equation}\n\\left( \\frac{d\\theta }{dv}\\right) _{EH}\\simeq 0.  \\eqnum{3.25}\n\\end{equation}\nThe event horizons $(EHs)$ in our problem are therefore located by equation\n(3.25). Using equation (3.2) equation (3.25) can be written in the form \n\\begin{equation}\nr^{n+2}-\\frac{\\left( n+1\\right) \\left( n+2\\right) }{2\\Lambda }r^{n}+\\frac{%\n\\left( n+1\\right) \\left( n+2\\right) }{n}\\frac{G_{N}m^{\\ast }\\left( v\\right) \n}{\\Lambda }=0.  \\eqnum{3.26}\n\\end{equation}\nwhere $m^{\\ast }$ is some effective mass given by \n\\begin{equation}\nm^{\\ast }\\left( v\\right) =m\\left( v\\right) -\\frac{L_{0}}{\\kappa } \n\\eqnum{3.27}\n\\end{equation}\n\nEquation (3.26)\\ for the location of event horizons is exactly of the same\nform as its counterpart equation (3.3) for location of the apparent horizons\nwith the mass $m$ replaced by the effective mass $m^{\\ast }$ as defined in\nequation (3.27). Hence borrowing from our previous techniques in solving\nequation (3.3) we can immediately write down the solutions to 3.26 as \n\\begin{equation}\nr_{EH}^{-}\\left( v\\right) \\simeq k\\left\\{ 1-\\frac{1}{4}\\left[ 2D^{\\ast\n}\\left( p,\\varphi \\right) +2R^{\\ast }\\left( p,\\varphi \\right) -a^{\\ast }%\n\\right] \\right\\} ,  \\eqnum{3.28}\n\\end{equation}\nand \n\\begin{equation}\nr_{EH}^{+}\\left( v\\right) \\simeq k\\left\\{ 1+\\frac{1}{4}\\left[ 2D^{\\ast\n}\\left( p,\\varphi \\right) -2R^{\\ast }\\left( p,\\varphi \\right) +a^{\\ast }%\n\\right] \\right\\} ,  \\eqnum{3.29}\n\\end{equation}\nwhere $\\ast $\\ means $m\\left( v\\right) \\rightarrow m^{\\ast }\\left( v\\right)\n=m\\left( v\\right) -\\frac{L_{0}}{\\kappa }$ and $\\frac{1}{2}\\pi <\\varphi\n^{\\ast }<\\pi $. In the limit $n\\rightarrow 1$ equation (3.28) reduces to, \n\\begin{equation}\n\\mathrel{\\mathop{\\lim }\\limits_{n=1}}%\nr_{EH}^{-}\\left( v\\right) =-\\left( \\frac{2}{\\sqrt{\\Lambda }}\\right) \\cos\n\\left( \\frac{1}{3}\\Psi ^{\\ast }+\\frac{1}{3}\\pi \\right)  \\eqnum{3.30}\n\\end{equation}\nwhile \n\\begin{equation}\n\\mathrel{\\mathop{\\lim }\\limits_{n=1}}%\nr_{EH}^{+}\\left( v\\right) =\\left( \\frac{2}{\\sqrt{\\Lambda }}\\right) \\cos\n\\left( \\frac{1}{3}\\Psi ^{\\ast }\\right)  \\eqnum{3.31}\n\\end{equation}\nwhere $\\left( \\frac{\\pi }{2}<\\Psi ^{\\ast }\\left( v\\right) <\\pi \\right)\n=\\arccos \\left[ -3m^{\\ast }\\left( v\\right) \\sqrt{\\Lambda }\\right] $. These\nlimiting cases then reproduce exactly the known equations [21] for locations\nof the event horizons in such a four dimensional spacetime. Further, as one\nswitches $\\Lambda $ off one finds from equations (3.17) and (3.3) that the\nsurface gravity of the black hole measured at the apparent horizon becomes\n\n\\begin{equation}\n\\kappa =\\frac{n}{2}\\left[ \\frac{2G_{N}m\\left( v\\right) }{n}\\right] ^{-\\frac{1%\n}{n}}  \\eqnum{3.32}\n\\end{equation}\nEquation (3.26) along with equation (3.32) imply that in the limit $\\Lambda\n\\rightarrow 0$, then \n\\begin{equation}\nr_{EH}^{-}\\left( v\\right) \\rightarrow \\left( \\frac{2G_{N}}{n}\\right) ^{\\frac{%\n1}{n}}\\left[ m\\left( v\\right) -\\frac{2}{n}\\left( \\frac{2G_{N}m\\left(\nv\\right) }{n}\\right) ^{\\frac{1}{n}}L_{0}\\right]  \\eqnum{3.33}\n\\end{equation}\nEquation (3.33) locates the event horizon of a radiating black hole imbedded\nin a higher dimensional \\ Schwarzschild spacetime. And as the dimensionality\nis reduced to four, it is clear that \n\\begin{equation}\nr_{EH}^{-}\\left( v\\right) \\rightarrow 2Gm\\left( v\\right) \\left(\n1-4GL_{0}\\right) .  \\eqnum{3.34}\n\\end{equation}\nThis is the exact result originally obtained by York [26] and later verified\nby Mallett [18]. It follows then that in equation (3.28) $r_{EH}^{-}\\left(\nv\\right) $\\ does indeed represent the locus of the black hole event horizon\nin a higher dimensional spacetime with a cosmological constant. Further, the \n$r_{EH}^{+}\\left( v\\right) $ in equation (3.29) is seen to give the locus of\nthe cosmological event horizon \\ $r_{EH}^{+}\\left( v\\right) $.\n\nFor an observer positioned at $r_{AH}^{-}\\left( v\\right) <r<r_{AH}^{+}\\left(\nv\\right) $, the region $EH^{-}\\cap AH^{-}$ represents the quantum ergosphere\n[26]. The ordering of the horizons can now be made. One finds from our\nresults that $EH^{-}<AH^{-}<AH^{+}<EH^{+}$. The finite location of the\ncosmological event horizon $EH^{+}$ is, as can be inferred from equation\n(3.29), due to the presence of the cosmological constant. One finds, indeed,\nthat in the event $\\Lambda \\rightarrow 0$ then $EH^{+}\\rightarrow \\infty $.\nAgain all these results are consistent with the known results for the $N=4$\ncase.\\ \n\n\\section{Structure and Dynamics of the Horizons}\n\nWe now turn to the problem of the structure and motion of the various\nhorizons obtained in our results above.\n\n\\subsection{Structure of the Apparent Horizons}\n\nIn the foregoing discussion we have found the locations of the apparent\nhorizons. We now deduce the structure of these surfaces. Since at the\napparent horizons the expansion $\\theta $ vanishes then from equations (3.3)\nand (2.1) the metric on such surfaces will take the form $%\nds^{2}=2dvdr+d\\Omega _{n+1}^{2}$. One finds that equations (2.1), and \\\n(3.15) as defined by equations (3.13) and (3.14) will induce on the surface $%\nr_{AH}^{-}$ a metric of the form \n\\begin{equation}\nds^{2}\\mid _{r=r_{AH}^{-}}=\\frac{k}{2}\\alpha _{-}\\left( p,\\varphi \\right) %\n\\left[ \\rho \\frac{\\sin \\frac{1}{3}\\varphi }{\\sin \\varphi }+\\sigma \\cos \\frac{%\n1}{3}\\varphi \\right] \\frac{dm}{dv}dv^{2}+d\\Omega _{n+1}^{2},  \\eqnum{4.1}\n\\end{equation}\nwhere \n\\begin{eqnarray}\n\\alpha _{-}\\left( p,\\varphi \\right) &=&\\left( DR\\right) ^{-1}\\left[ 2R+\\frac{%\n\\left( 4ac-8c-a^{3}\\right) }{4R}-1\\right] ,\\;\\rho =\\frac{1}{3}\\sqrt{\\frac{-p%\n}{3}}\\frac{d}{dm}\\left( \\frac{3q}{2\\sqrt{\\frac{-p^{3}}{3}}}\\right) , \n\\nonumber \\\\\n\\;\\sigma &=&\\frac{2d}{m\\left( v\\right) \\sqrt{-3p}}.  \\eqnum{4.2}\n\\end{eqnarray}\nNoting that $\\alpha ,\\rho ,\\sigma $ are all positive quantities then for $%\n\\frac{\\pi }{2}<\\varphi <\\pi $, $\\frac{dm}{dv}<0$ contributes the only\nnegative quantity in equation (4.2). We conclude, on this basis, that\naccording to equation (4.1) the apparent horizon surface $r_{AH}^{-}\\left(\nv\\right) $ of an evaporating black hole in a higher dimensional de Sitter\nspacetime is timelike.\n\nSimilarly, for $\\frac{\\pi }{2}<\\varphi <\\pi $ \\ equations (2.1), and \\\n(3.16) as defined by equations (3.13) and (3.14) will induce on the surface $%\nr_{AH}^{+}$ a metric of the form \n\\begin{equation}\nds^{2}\\mid _{r=r_{AH}^{+}}=-\\frac{k}{2}\\alpha _{+}\\left( p,\\varphi \\right) %\n\\left[ \\rho \\frac{\\sin \\frac{1}{3}\\varphi }{\\sin \\varphi }+\\sigma \\cos \\frac{%\n1}{3}\\varphi \\right] \\frac{dm}{dv}dv^{2}+d\\Omega _{n+1}^{2},  \\eqnum{4.3}\n\\end{equation}\nwhere $\\rho $ and $\\sigma $ are as given in equation (4.2) and \n\\begin{equation}\n\\alpha _{+}\\left( p,\\varphi \\right) =\\left( DR\\right) ^{-1}\\left[ 2R+\\frac{%\n\\left( 4ac-8c-a^{3}\\right) }{4R}+1\\right]  \\eqnum{4.4}\n\\end{equation}\nIt follows, then from use of equations \\ (4.2) and (4.4) that the\ncosmological apparent horizon $r_{AH}^{+}$\\ in a higher dimensional\nspacetime (as located by equation (4.3)) is spacelike.\n\nIn the limit $n\\rightarrow 1$, one finds that \n\\begin{equation}\nds^{2}\\mid _{r=r_{AH}^{-}}=-\\frac{4\\sin \\left( \\frac{1}{3}\\Psi +\\frac{1}{3}%\n\\pi \\right) }{\\sin \\Psi }\\frac{dm}{dv}dv^{2}+d\\Omega _{n+1}^{2},  \\eqnum{4.5}\n\\end{equation}\nand \n\\begin{equation}\nds^{2}\\mid _{r=r_{AH}^{+}}=\\frac{4\\sin \\left( \\frac{1}{3}\\Psi \\right) }{\\sin\n\\Psi }\\frac{dm}{dv}dv^{2}+d\\Omega _{n+1}^{2},  \\eqnum{4.6}\n\\end{equation}\nwith $\\frac{\\pi }{2}<\\Psi \\left( v\\right) <\\pi =\\arccos \\left[ -3m\\left(\nv\\right) \\sqrt{\\Lambda }\\right] $.\\ \\ Equations (4.5) and (4.6) are exactly\nthe known results [21] for the four dimensional $\\left( n=1\\right) $ case.\n\n\\subsection{The dynamics of the Horizons}\n\nThe manner in which the various horizons move can now be inferred. One can\nrewrite equations (4.1) and (4.3) in the form \n\\begin{equation}\nds^{2}\\mid _{r=r_{AH}^{\\pm }}\\simeq \\pm 2L_{0}\\Gamma _{\\left( AH\\right)\nn}^{\\pm }dv^{2}+d\\Omega _{n+1}^{2}  \\eqnum{4.7}\n\\end{equation}\nwhere \n\\begin{equation}\n\\Gamma _{\\left( AH\\right) n}^{\\pm }=\\frac{k}{2}\\alpha _{\\left( \\pm \\right)\n}\\left( p,\\varphi \\right) \\left[ \\rho \\frac{\\sin \\frac{1}{3}\\varphi }{\\sin\n\\varphi }+\\sigma \\cos \\frac{1}{3}\\varphi \\right] .  \\eqnum{4.8}\n\\end{equation}\nWe see then that to $O\\left( L\\right) $\\ the apparent horizons $r_{AH}^{\\pm\n} $\\ move with velocities given by \n\\begin{equation}\n\\frac{dr_{AH}^{\\pm }}{dv}\\simeq \\pm 2L_{0}\\Gamma _{\\left( AH\\right) n}^{\\pm }\n\\eqnum{4.9}\n\\end{equation}\n\nSimilarly the motion of the event horizons $r_{EH}^{\\pm }$ can be deduced\nfrom equations (3.28) and (3.29). One finds that the velocities of these\nsurfaces are given by \n\\begin{equation}\n\\frac{dr_{EH}^{\\pm }}{dv}\\simeq \\pm 2L_{0}\\Gamma _{\\left( EH\\right) n}^{\\pm }\n\\eqnum{4.10}\n\\end{equation}\nwhere the $\\Gamma _{\\left( EH\\right) n}^{\\pm }$ are obtained by applying to\nequation (4.8) the transformation $m\\left( v\\right) \\rightarrow m^{\\ast\n}\\left( v\\right) =m\\left( v\\right) -\\frac{L_{0}}{\\kappa }$ that turns\napparent horizon quantities to event horizon quantities. For the range $%\n\\left( \\frac{1}{2}\\pi <\\varphi ,\\varphi ^{\\ast }<\\pi \\right) $ considered\nthe quantities $\\Gamma _{\\left( AH\\right) n}^{\\pm }$ and $\\Gamma _{\\left(\nEH\\right) n}^{\\pm }$ are positive. It follows then from (4.9) and (4.10)\nthat for the observer in the region $r_{AH}^{-}<r<r_{AH}^{+}$ both the black\nhole horizons $EH^{-}$ and $AH^{-}$ move with respective velocities $%\n-2L_{0}\\Gamma _{\\left( EH\\right) n}^{-}$ and $-2L_{0}\\Gamma _{\\left(\nAH\\right) n}^{-}$. For such an observer these velocities are negative \\\nConsequently such motion represents in each case a contraction of the\nrespective black hole horizon. Conversely, the same equations show that for\nthe same observer, the cosmological horizons $EH^{+}$ and $AH^{+}$ are\nexpanding at velocities given by $2L_{0}\\Gamma _{\\left( EH\\right) n}^{+}$\nand $2L_{0}\\Gamma _{\\left( AH\\right) n}^{+}$, respectively.\n\nIn the event $\\left( n\\rightarrow 1\\right) $ one recovers the known results\n[21] for the four dimensional case. And in this limit the results are also\nconsistent with the usual results that as $\\Lambda \\rightarrow 0$, we\nrecover from equations (4.7) and (4.8) the relations \n\\begin{equation}\n\\mathrel{\\mathop{%\n\\mathrel{\\mathop{\\lim }\\limits_{n\\rightarrow 1}}}\\limits_{\\Lambda \\rightarrow 0}}%\n\\frac{dr_{AH}^{-}}{dv}=-2L_{0}  \\eqnum{4.11}\n\\end{equation}\nand \n\\begin{equation}\n\\mathrel{\\mathop{%\n\\mathrel{\\mathop{\\lim }\\limits_{n\\rightarrow 1}}}\\limits_{\\Lambda \\rightarrow 0}}%\n\\frac{dr_{EH}^{-}}{dv}=-2L_{0}  \\eqnum{4.12}\n\\end{equation}\n\\ \n\n\\section{Conclusion}\n\nIn this discussion we have examined black holes of the Vaidya type in an\nspatially flat higher dimensional spacetime with a cosmological constant.\nThe analysis revealed the existence of four horizons associated with such a\nspacetime and identified as the event horizon, $EH^{-}$ and the apparent\nhorizon $AH^{-}$ for the black hole, and their cosmological counterparts, $%\nEH^{+}$ and $AH^{+}$, respectively. We have pointed out, to good order of\naccuracy, the location of these horizons. Further, from our results, we\ndeduced the structure and discussed the dynamics of these horizons. All our\nresults reduce to already known results under various limits. In particular,\nit was shown at each stage that for the $n=1$, our results reduced exactly\nto those previously obtained [21] for the four dimensional case. It is seen\nthen that the problem of the dynamics of a radiating blackhole in a higher\ndimensional cosmological background can be sensibly discussed.\n\nAn application of our results to the Hawking radiation problem will be the\ntopic of a future discussion.\n\n\\ \n\n\\ \\ \n\n{\\bf Acknowledgments:}\n\nI would like to thank Fred Adams for some insightful discussions and Gordy\nKane, Marty Einhorn and Jean Krisch for their constructive comments.\n\nThis work was supported by funds from The University of Michigan.\n\n\\ \\newpage\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction} \\label{sec:intro}\n\nThe architectures of planetary systems give insight into their formation and dynamical histories.  For example, interactions with the protoplanetary disk tend to drive adjacent planets into first-order, mean-motion resonances (MMRs, such as the 2:1), while simultaneously damping their eccentricities to values that are difficult to measure \\citep{LeePeale02,Tinney06}.  On the other hand, planet-planet scattering \\citep{Chatterjee08,FordRasio08} or Kozai-Lidov oscillations \\citep{Kozai62,Lidov62,Fabrycky07} can produce single planets with eccentric orbits.  While not all planetary systems must pass through these phases of disk migration or eccentricity growth, the system architectures that they produce rarely occur from in-situ formation.  Thus, reliable estimates of their frequencies will reveal the relative importance of these processes in planet formation and evolution in general.\n\nFor radial velocity (RV) observations in particular, the challenge in identifying the true system architecture is a degeneracy between two models---one with a single planet with eccentric orbits (single eccentrics) and one with two planets with circular orbits at the 2:1 (circular doubles) \\citep{Angelada10,Wittenmyer13}.  Historically, the single-planet model has been favored on the grounds of Occam's razor \\citep{Kurster15}, since a system with a single planet is simpler than a system with two.  However, the circular double model has the same number of model parameters as single eccentrics (it is just as simple) and it is a consequence of dynamical processes known to occur.  These facts motivate careful scrutiny of existing discoveries in order to properly characterize the systems.  If circular doubles are more common than currently suggested, then disk-migration may be more important than previously thought \\citep{Tinney06}.\n\nThe source of the degeneracy between these models is in a first-order expansion of the RV signal of a single eccentric planet\n\\begin{equation}\nRV_{\\mathrm{single}} \\approx K \\cos (M + \\omega) + K e \\cos (2M + \\omega) + \\mathcal{O}(e^2),\n\\label{eq:SingleKeplerian}\n\\end{equation}\nwhere $RV_{\\mathrm{single}}$ is the observed radial velocity, $K$ is the velocity semi-amplitude, $e$ is the eccentricity, $\\omega$ is the longitude of periastron, and $M$ is the mean anomaly\\Edit{, which is a function of time}.  By comparison, the signal of a circular double is\n\\begin{equation}\nRV_{\\mathrm{double}} = K_{\\mathrm{out}} \\cos (M_{\\mathrm{out}}) + K_{\\mathrm{in}} \\cos(M_{\\mathrm{in}}),\n\\label{eq:CircularDouble}\n\\end{equation}  \nwhere $RV_{\\mathrm{double}}$ is the observed radial velocity, $K_{\\mathrm{out}}$ and $K_{\\mathrm{in}}$ are the velocity semi-amplitudes, and $M_{\\mathrm{out}}$ and $M_{\\mathrm{in}}$ are the mean anomalies.  At the 2:1 MMR, $M_{\\mathrm{in}}=2 M_{\\mathrm{out}}$ and the inner planet signal ($K_{\\mathrm{in}}$) masquerades as the eccentricity signal ($K e$) of the single planet.\n\nThis degeneracy is widely known though rarely addressed.  Nevertheless, there is precedent for reconsidering certain systems.  For example, \\citet{Kurster15} reanalyzed RV data for HD~27894 and found that a circular double model was a better fit than the reported single eccentric model.  Also, \\citet{Angelada10, Wittenmyer13}, found similar results for several RV systems.  At the same time, new measurements from the \\textit{Kepler}\\ mission show that planet pairs near 2:1 are quite common.  For example, using the method of \\citet{Steffen15} on the \\textit{Kepler} DR25 catalog \\citep{Thompson17}, we estimate that 20$\\%$ of \\textit{Kepler}'s transiting adjacent planet pairs with period ratios between 1 and 6 are within ten percent of 2---including the most prominent peak of the period ratio distribution at 2.17 \\citep{Steffen15}.  Motivated by these new facts and the results of previous studies, we reanalyze a sample of 60\\, single eccentric planetary systems using a new Bayesian analysis pipeline in an effort to discover their true architectures.\n\n\\section{\\Edit{Methods}}\\label{sec:setup}\n \nOur sample contains 60\\ systems and comprises every non-transiting RV system from the NASA Exoplanet Archive (NASA Archive), as of November 2016 \\citep{Akeson13}, that is listed as having only a single planet, orbiting a main sequence star, and whose system properties were derived from a single data set.  \\Edit{Our pipeline did not have the capability to analyze multiple datasets when the analysis began.  We did not limit our sample by eccentricity.}  Our main results will focus on the main sequence stars, but we will also report on an extended sample which ignores stellar type.  The extended sample contains 95\\ systems, which is nearly a quarter of all RV-discovered single-planet systems.   Figure~\\ref{fig:astellarcomparison-entire} shows how we determine stellar types based on their reported surface gravity and how we select the main sequence sample.\n\n\\subsection{\\Edit{The Pipeline}}\\label{sec:pipeline}\nOur pipeline estimates the Bayes factor---the ratio of the probabilities of the RV data given the circular double model and the single eccentric model---to quantitatively compare the two models.  For each system we test four planetary system models: a single eccentric; two circular doubles (one with a period ratio fixed at 2 and the other fixed at 2.17---where there are two large peaks in the period ratio distribution from \\textit{Kepler}\\ \\citep{Steffen15}); and a ``floating'' circular double with no period ratio constraint.  This last model has an additional model parameter, but the Bayes factor calculation can account for different numbers of model parameters.  Our primary results work with the two fixed models given the compelling theoretical and observational reasons to consider them, the fact that the number of model parameters are identical (and thus more directly comparable), and because a narrow-band signal at a fixed period ratio is less susceptible to a false positive detection from stellar RV jitter or statistical noise.  \n\n\\begin{figure}\n\\includegraphics[width=\\columnwidth]{Figure1.pdf}\n\\caption{Stellar effective temperature vs.$\\,\\,$stellar surface gravity for the RV multi-planet systems from the NASA Archive as grey crosses and the \\textit{Kepler}\\ multi-planet systems as blue circles.  Our sample of 95 stars, ignoring stellar type, are orange diamonds.  For our sample of main sequence stars we selecte those with $\\mathrm{log}\\,g\\geq3.825$, there are 60 main sequence stars in the main sample and 95 stars in the entire sample.}\n\\label{fig:astellarcomparison-entire}\n\\end{figure}\n\nThe parameters for the single eccentric model are: period ($P$), velocity semi-amplitude ($K$), eccentricity ($e$), longitude of periastron ($\\omega$), and mean anomaly at the time of the earliest RV measurement ($M_\\mathrm{o}\\equiv M(t_\\mathrm{o})$).  The parameters for the circular double models are: outer planet period ($P_{\\mathrm{out}}$), the velocity semi-amplitude for the outer\/inner planets ($K_{\\mathrm{out}}$\/$K_{\\mathrm{in}}$), and mean anomaly at the time of the earliest RV measurement for the outer\/inner planets ($M_{\\mathrm{o,out}}$\/$M_{\\mathrm{o,in}}$). The floating circular double model includes the inner planet orbital period ($P_{\\mathrm{in}}$).  Each model also has a linear trend ($A$) and a velocity offset ($C$).   \n\nOur model fitting is a three step process.  First, we determine the starting values for our Markov Chain-Monte Carlo (MCMC) by maximizing the likelihood function:\n\\begin{equation}\n\\mathrm{ln} \\; P(t,RV,\\sigma_{RV}|\\theta) = -\\frac{1}{2} \\sum_{n} \\left [ \\frac{(RV_n - \\mathcal{M}(\\theta,t_n))^2}{\\sigma_{RV,n}^2} + \\mathrm{ln} \\; 2\\pi\\, \\sigma_{RV,n}^2 \\right ]\n\\label{eq:likelihood}\n\\end{equation}\nwhere $t$, $RV$, and $\\sigma_{RV}$ are the observed times, RV measurements, and RV errors.  $\\mathcal{M}$ is the RV model and $\\theta$ are its associated parameters.  We draw our set of initial conditions for the maximum likelihood estimation (MLE) from the NASA Archive.  \\Edit{The time of periastron passage is used to determine the initial $M_{\\mathrm{o}}$.}  We first fit for $C$, fixing the other parameters at their nominal values and setting $A=0$.  We next fit for $A$ and $C$ simultaneously.  Some systems did not have $K$, $\\omega$, and\/or \\Edit{the time of periastron passage} reported on the NASA Archive.  In those cases, an MLE was done with the missing quantities as the only free parameters.  We initialized the fixed circular double models to their first-order, single eccentric equivalent values using equations (\\ref{eq:SingleKeplerian}) and (\\ref{eq:CircularDouble}).  For the floating circular double, the inner planet orbital period is initialized to either the 2:1 or the 2.17:1, depending on which fixed model produced a larger Bayes factor.\n\nThe second step in our pipeline estimates the posterior distributions of the model parameters using an ensemble sampler MCMC \\citep{DFM13}.  Each run has thirty Markov chains, thins the chains every hundred steps, and ignores the first 20\\% of the chain as burn-in. We allow the chains to evolve until they yield a set of at least ten thousand independent samples per model per RV dataset.  \\Edit{We measure the autocorrelation length after each run to determine the number of independent samples.  If the number of independent samples falls short of ten thousand, then the autocorrelation length is used to determine how many additional steps are needed to yield ten thousand independent samples and the \\EditTwo{MCMC} is rerun with the new number of steps.}  The different chains were initialized using the parameter values from the MLE, with each parameter scattered by a sufficiently small amount to allow the ensemble sampler to fill the posterior mode.  \n\n\\Move{We impose a modified Jeffery's prior for the orbital period and velocity semi-amplitude: $p(X)=[(1+X)\\times\\mathrm{ln}(1+X_{\\mathrm{max}}\/X_0)]^{-1}$ with bounds between 0--10,000~days and 0--2,000~m~s$^{-1}$ respectively and $X_0$ equal to 1 day and 1 m s$^{-1}$ respectively.  We use this prior because it is normalizable, objective, and intended for scalable parameters that could have zero as a value.  We use uniform priors for the remaining parameters, ($e$, $\\omega$, $M_0$), because they are also normalizable and objective.  We sample the parameters for the single eccentric model in \\{$P$, $K$, $\\sqrt{e} \\sin (\\omega)$, $\\sqrt{e} \\cos (\\omega)$, $\\omega+M_0$\\}-space in order to maintain uniform priors \\citep{DFMBen}. \n\nThe prior bounds for $K$, $K_{\\mathrm{out}}$, and $K_{\\mathrm{in}}$ are between 0 and 2,000 m s$^{-1}$.  The prior bounds for $P$, $P_{\\mathrm{out}}$, and $P_{\\mathrm{in}}$ are between 0 and 10,000 days. The prior bounds for $\\sqrt{e} \\, \\sin (\\omega)$ and $\\sqrt{e} \\cos (\\omega)$ are such that $0 < (\\sqrt{e} \\sin (\\omega))^2 + (\\sqrt{e} \\cos (\\omega))^2<1$, i.e.~$0<e<1$.  The prior bounds for $(M_0+\\omega)$, $M_{\\mathrm{0,out}}$, and $M_{\\mathrm{0,in}}$ are between -2$\\pi$ and 4$\\pi$.  These limits allow the Markov chains to cross the 0 and 2$\\pi$ coordinate singularities while remaining well-behaved.\nFurthermore, these values are modded by 2$\\pi$ before doing any calculations.\nThe prior bounds for $C$ are between -100,000 m s$^{-1}$ and 100,000 m s$^{-1}$ to accommodate the wide range in offset values in the real sample.}\n\nFinally, we estimate the Bayes factors between the single eccentric model and the circular double models by taking the ratio of the fully marginalized likelihoods (FML, i.e. Bayesian evidence) for the two models.  We approximate the FML using an importance sampling algorithm where the sampling distribution is informed by a set of posterior samples taken from the aforementioned MCMC \\citep{Nelson16}.  For each system we take the larger of the Bayes factor for the two fixed circular double models.\n\n\\Move{In this context, importance sampling is essentially a general form of Monte Carlo integration to estimate the fully marginalized likelihood, $\\mathcal{Z}$.\nThe value of $\\mathcal{Z}$ is the integral over the prior probability distribution $p(\\theta)$ times the likelihood function $\\mathcal{L}(\\theta) \\equiv p(t, RV, \\sigma_{RV}|\\theta)$, i.e., \n\\begin{equation}\n\\mathcal{Z} = \\int p(\\theta)\\mathcal{L}(\\theta)d\\theta\n\\end{equation}\nWe multiply the numerator and denominator of the integrand by $g(\\theta)$, a distribution over the model parameters with a known normalization.\n\\begin{equation}\n\\mathcal{Z}=\\int \\frac{\\mathcal{L}(\\theta)p(\\theta)}{g(\\theta)} g(\\theta) d\\theta.\n\\label{eq-mcmcis-1}\n\\end{equation}\nEquation \\ref{eq-mcmcis-1} is in a form such that $\\mathcal{Z}$ can be estimated numerically by drawing $N$ samples from $g(\\vec{\\theta})$, \n\\begin{equation}\n\\widehat{\\mathcal{Z}} \\approx \\frac{1}{N}\\sum\\limits_{\\theta_i \\sim g(\\theta)}\\frac{\\mathcal{L}(\\theta_i)p(\\theta_i)}{g(\\theta_i)}.\n\\label{eq-mcmcis-2}\n\\end{equation} \n\nThe key to an accurate and efficient estimate of $\\widehat{\\mathcal{Z}}$ lies in choosing an appropriate $g(\\theta)$.\nAssuming our parameter space contains one dominant posterior mode, we choose a multivariate normal $\\mathcal{N}(\\vec{\\mu}_g, \\vec{\\Sigma}_g)$, where $\\vec{\\mu}_g$ and $\\vec{\\Sigma}_g$ describe the mean vector and covariance matrix of the model parameters respectively.\nAfter we perform an MCMC on a particular model\/dataset, we can estimate $\\vec{\\mu}_g$ and $\\vec{\\Sigma}_g$ using a set of posterior samples.  That information is fed into our importance sampling algorithm to estimate $\\widehat{\\mathcal{Z}}$ for that model.\n\\citet{Nelson16}, \\citet{Guo12}, and \\citet{Weinberg13} provide more detailed prescriptions and investigations of this method.}\n\n\\subsection{\\Edit{Pipeline Characterization}}\\label{sec:pipelinecharacter}\nWe characterized the pipeline efficiency with an ensemble of 1,000 synthetic RV time series whose system and data properties match the real systems.  \\Move{We use the Bayes factors of these synthetic systems to characterize our model comparison pipeline.  This Monte Carlo simulation was initialized as follows.\n\nThe start time ($t_0$) is \\Edit{a uniform} random \\Edit{draw} between 1 and 1,000 days.  The number of observations are drawn from the real systems with a normally-distributed adjustment with a standard deviation 10\\% of the nominal value rounded to the nearest whole number.  \nThe observation time series is produced by selecting a set of observation differences ($t_{i}-t_{i-1})$ from the real distribution of observation differences with a similar, normally-distributed 10\\% variation added to each difference.  The number of orbits is the number of orbits of a randomly chosen real system with a normally distributed 10\\% variation.\n\nWe determine the orbital period ($P$) using the selected number of orbits and the observation time series.\nThe velocity semi-amplitude ($K$) and the eccentricity ($e$) are separate random draws from the real systems.  The mean anomaly of the start time ($M_0$) and argument of periastron ($\\omega$) are randomly drawn between 0 and $2\\pi$. \nThe linear trend ($A$) is a 10\\% variation to a random draw from the real systems. \n\nWe assume that the RV errors are normally distributed with a standard deviation that is the quadrature sum of stellar jitter and instrumental and photon noise.  The instrumental and photon noise ($\\sigma_{\\text{RV}}$) are drawn randomly from the RV errors of the real systems and our error bars are assigned to this value.  Steller jitter is selected from a log uniform distribution between 0.5 and 5 m s$^{-1}$.   \nThe observation errors are added to the synthetic RV measurement---not to the error in the RV measurement.  Figure~\\ref{fig:parametercomparison} shows the parameter distributions for the 1,000 synthetic time series and the real systems as reported in the NASA Archive.}\n\n\\begin{figure}\n\\includegraphics[width=\\columnwidth]{Figure2.pdf}\n\\caption{Property distributions for our sample of 95 real systems from the NASA Exoplanet Archive in orange and the 1,000 synthetic time series in blue.}\n\\label{fig:parametercomparison}\n\\end{figure}\n\nThe resulting Bayes factors from this characterization are shown as the blue distribution in Figure~\\ref{fig:bayesrealvssynthMS}.  The vertical lines denote the 95th and 90th percentiles of the distribution. The shape of the distribution is not symmetric, and the vast majority of our synthetic datasets favor the single-planet case---as expected since the synthetic systems were constructed to be single eccentrics.  \nReal systems with Bayes factors larger than those thresholds may be circular double systems mischaracterized as single eccentrics. \n\nThe approach outlined above is different from earlier studies.  For example, \\citet{Wittenmyer13} used the reduced $\\chi^2$ to determine the preferred model and refined their results with stability tests using the N-body integrator \\textit{Mercury} \\citep{Chambers97}.  \\citet{Angelada10} randomized individual sets of data to calculate the false positive rate per system.  Their model selection was also based on the reduced $\\chi^2$ of least-squares fitting.  In this work, we use a fully marginalized likelihood to calculate the Bayes factor for the model comparison and we estimate our false positive rate by analyzing a large simulated dataset with our pipeline.\n \n\\begin{figure}\n\\includegraphics[width=\\columnwidth]{Figure3.pdf}\n\\caption{The log Bayes factor distribution for the 1,000 synthetic single eccentric time series in blue and 60 real systems hosted by main sequence stars in orange.  Here, we compare only the single eccentric model to the fixed, circular double model with the largest Bayes factor.  The 95th and 90th percentiles are indicated with the dotted lines near a Bayes factor of 24 and 8, respectively.\n}\n\\label{fig:bayesrealvssynthMS}\n\\end{figure}\n \n\\section{Results}\\label{sec:results}\n\nAfter analyzing our synthetic systems, we ran our sample of 60\\ real systems (95\\ systems for the extended sample) through the same pipeline.\nFigure~\\ref{fig:bayesrealvssynthMS} shows that the Bayes factor distributions for the synthetic and real systems (in orange) are not similar.  We find that 15\\ (25\\%)  of the systems have Bayes factors larger than the 95th percentile of the synthetic systems.  (For the extended sample of 95\\ systems the numbers are 30\\ and 31\\%\\ of the entire sample respectively.)  9\\ of these systems prefer the 2.17:1 model (22\\ of the extended sample) while the remaining 6\\ (8\\ from the extended sample) prefer the 2:1 model.  Assuming a false positive rate of 5\\% from our 95\\% confidence level, our estimate of the number of false positives is 0.75 $\\pm$ 0.87 (1.5 $\\pm$ 1.2 for the extended sample).  \\Edit{The systems from the extended sample that prefer the fixed circular double model, the model parameters, and Bayes factors are shown in Table~\\ref{tab:Results}.}  A CSV file containing the model parameters with errors for all four models, Bayes factors between the circular double models and the single eccentric model, and percentile of the best fixed model for each system in the extended sample are available online as \\Edit{Table~2}.\n\nWe examine the consequences of these potential discoveries on several distributions of planet properties.\nFigure~\\ref{fig:massvsperiodMS} shows the planet mass vs.$\\,\\,$orbital period for known RV planets along with the new planets favored by our analysis orbiting main sequence stars.  These potential new systems lie well within the range of values measured in known systems.\nWe point out, however, that some systems may yet be false positives.  For instance, there are a few \\Edit{candidate circular double} systems that \\EditTwo{would be} hot Jupiters (planet with $P \\lesssim 10$ days) with interior companions.  Presently, there is only a single known system (WASP-47 \\citep{Becker15}) where a hot Jupiter has a known interior companion.  And the period ratio in this case is over 5:1---far from the degeneracy we consider here.  However, \\Edit{the hot Jupiter has an} outer companion \\Edit{with} a period ratio near 2.17.  Figure~\\ref{fig:PRHistogramPlusMS} shows how the predicted mass ratios for the main sequence systems that favor the two-planet model compare with the mass ratios for RV systems on the NASA Archive. \n\n\\begin{figure}\n\\includegraphics[width=\\columnwidth]{Figure4.pdf}\n\\caption{Orbital period vs.$\\,\\,$planetary mass for all RV planets.  Systems from the NASA Exoplanet Archive are in teal, with multi- and single-planet systems as open circles and crosses, respectively.  Each system orbiting a main sequence star with a measured Bayes factor larger than 95th percentile of the synthetic systems are plotted in orange.  Each putative system is represented by a line on the plot, with the diamonds as the inner planet and the circles are the outer companion.  Systems that remain in the 95th percentile after including a white noise stellar jitter term are in black.  We note that these results lie well within normal parameter values of known systems.}\n\\label{fig:massvsperiodMS}\n\\end{figure}\n\n\\begin{figure}\n\\includegraphics[width=\\columnwidth]{Figure5.pdf}\n\\caption{Mass ratio distribution for all RV adjacent planet pairs in grey.  The stacked, orange distribution are our systems with Bayes factors larger than 24 (95th percentile) around main sequence stars.  The nature of the signal favors more massive outer planets.  The black outline shows the systems that have stellar jitter included and still had Bayes factors larger than the 95th percentile.}\n\\label{fig:PRHistogramPlusMS}\n\\end{figure}\n\n\\onecolumn\n\\begin{table}\n\\begin{center}\n\\fontsize{7.5}{7.2}\\selectfont\n\n\\rotatebox{90}{\n\\begin{tabular}{ |c|c|c|c|c|c|c|c|c|c|c|c| }\n\\hline\nStar Name\t&\tStar Type\t&\tModel\t&\t\t$P_{\\mathrm{out}}$ [days]\t\t&\t\t$K_{\\mathrm{out}}$ [m s$^{-1}$]\t\t&\t\t$M_{\\mathrm{out}}$ [rad]\t\t&\t\t$P_{\\mathrm{in}}$ [days]\t\t&\t\t$K_{\\mathrm{in}}$ [m s$^{-1}$]\t\t&\t\t$M_{\\mathrm{in}}$ [rad]\t\t&\t\tA [\\EditTwo{k}m s$^{-1}$\\EditTwo{day$^{-1}$}]\t\t&\t\tC [m s$^{-1}$]\t\t&\tBayes Factor\t\\\\\n\\hline\n\\hline\nHD 240237\t&\tnMS\t&\t2.17:1\t&\t770.45\t$\\pm$\t3.09\t&\t69.75\t$\\pm$\t1.59\t&\t5.45\t$\\pm$\t0.04\t&\t355.05\t$\\pm$\t3.09\t&\t57.52\t$\\pm$\t2.57\t&\t1.43\t$\\pm$\t0.08\t&\t9.96\t$\\pm$\t2.49\t&\t82.31\t$\\pm$\t1.54\t&\t1.09 $\\times$ 10$^{85}$\t\\\\\nHD 2952\t&\tnMS\t&\t2.17:1\t&\t318.83\t$\\pm$\t0.27\t&\t23.16\t$\\pm$\t0.94\t&\t4.13\t$\\pm$\t0.05\t&\t146.92\t$\\pm$\t0.27\t&\t15.90\t$\\pm$\t0.71\t&\t4.41\t$\\pm$\t0.10\t&\t6.37\t$\\pm$\t0.62\t&\t3.15\t$\\pm$\t0.74\t&\t6.88 $\\times$ 10$^{57}$\t\\\\\n$\\alpha$ Arietis\t&\tnMS\t&\t2.17:1\t&\t372.60\t$\\pm$\t0.52\t&\t29.67\t$\\pm$\t0.52\t&\t4.31\t$\\pm$\t0.04\t&\t171.70\t$\\pm$\t0.52\t&\t10.25\t$\\pm$\t0.46\t&\t6.17\t$\\pm$\t0.10\t&\t-9.5\t$\\pm$\t0.64\t&\t0.91\t$\\pm$\t0.45\t&\t9.80 $\\times$ 10$^{49}$\t\\\\\nHD 96127\t&\tnMS\t&\t2.17:1\t&\t632.39\t$\\pm$\t2.19\t&\t96.66\t$\\pm$\t1.50\t&\t5.95\t$\\pm$\t0.04\t&\t291.42\t$\\pm$\t2.19\t&\t41.02\t$\\pm$\t1.26\t&\t1.41\t$\\pm$\t0.07\t&\t22.83\t$\\pm$\t3.49\t&\t-923.95\t$\\pm$\t0.93\t&\t9.63 $\\times$ 10$^{43}$\t\\\\\nHD 95089\t&\tnMS\t&\t2.17:1\t&\t496.57\t$\\pm$\t2.88\t&\t20.65\t$\\pm$\t0.55\t&\t2.40\t$\\pm$\t0.05\t&\t228.83\t$\\pm$\t2.88\t&\t6.73\t$\\pm$\t0.57\t&\t4.64\t$\\pm$\t0.11\t&\t-4.47\t$\\pm$\t1.39\t&\t0.23\t$\\pm$\t0.35\t&\t3.75 $\\times$ 10$^{16}$\t\\\\\n11 Ursae Minoris\t&\tnMS\t&\t2.17:1\t&\t513.22\t$\\pm$\t0.75\t&\t185.38\t$\\pm$\t1.15\t&\t4.30\t$\\pm$\t0.02\t&\t236.51\t$\\pm$\t0.75\t&\t19.43\t$\\pm$\t1.15\t&\t4.80\t$\\pm$\t0.08\t&\t-11.14\t$\\pm$\t2.18\t&\t-10.39\t$\\pm$\t0.84\t&\t1.06 $\\times$ 10$^{15}$\t\\\\\nHD 136418\t&\tnMS\t&\t2.17:1\t&\t474.99\t$\\pm$\t1.26\t&\t41.62\t$\\pm$\t0.39\t&\t5.63\t$\\pm$\t0.03\t&\t218.89\t$\\pm$\t1.26\t&\t10.41\t$\\pm$\t0.47\t&\t2.86\t$\\pm$\t0.08\t&\t-6.93\t$\\pm$\t1.1\t&\t-4.99\t$\\pm$\t0.42\t&\t8.03  $\\times$ 10$^{12}$\t\\\\\nHD 81688\t&\tnMS\t&\t2.17:1\t&\t184.08\t$\\pm$\t0.17\t&\t60.05\t$\\pm$\t0.92\t&\t5.47\t$\\pm$\t0.04\t&\t84.83\t$\\pm$\t0.17\t&\t6.96\t$\\pm$\t0.97\t&\t5.44\t$\\pm$\t0.15\t&\t4.91\t$\\pm$\t1.53\t&\t-1.21\t$\\pm$\t0.67\t&\t1.41  $\\times$ 10$^{11}$\t\\\\\nHIP 57050\t&\tMS\t&\t2.17:1\t&\t41.31\t$\\pm$\t0.01\t&\t29.43\t$\\pm$\t0.62\t&\t2.60\t$\\pm$\t0.11\t&\t19.04\t$\\pm$\t0.01\t&\t6.98\t$\\pm$\t0.64\t&\t3.86\t$\\pm$\t0.22\t&\t9.01\t$\\pm$\t0.53\t&\t-15.77\t$\\pm$\t0.83\t&\t4.27 $\\times$ 10$^{9}$\t\\\\\nHD 206610\t&\tnMS\t&\t2.17:1\t&\t628.84\t$\\pm$\t6.03\t&\t34.50\t$\\pm$\t0.55\t&\t1.67\t$\\pm$\t0.05\t&\t289.79\t$\\pm$\t6.03\t&\t6.92\t$\\pm$\t0.79\t&\t3.06\t$\\pm$\t0.09\t&\t-17.3\t$\\pm$\t1.7\t&\t19.30\t$\\pm$\t0.47\t&\t1.59  $\\times$ 10$^{7}$\t\\\\\nHD 216770\t&\tMS\t&\t2:1\t&\t121.77\t$\\pm$\t0.38\t&\t14.05\t$\\pm$\t2.97\t&\t4.30\t$\\pm$\t0.33\t&\t60.88\t$\\pm$\t0.38\t&\t33.86\t$\\pm$\t3.44\t&\t0.79\t$\\pm$\t0.18\t&\t36.91\t$\\pm$\t6.87\t&\t31146.80\t$\\pm$\t1.92\t&\t1.61  $\\times$ 10$^{5}$\t\\\\\nHD 114386\t&\tMS\t&\t2.17:1\t&\t955.46\t$\\pm$\t15.81\t&\t33.79\t$\\pm$\t1.55\t&\t3.92\t$\\pm$\t0.11\t&\t440.31\t$\\pm$\t15.81\t&\t14.08\t$\\pm$\t1.96\t&\t2.23\t$\\pm$\t0.22\t&\t-5.24\t$\\pm$\t3.82\t&\t33367.71\t$\\pm$\t1.28\t&\t8.80  $\\times$ 10$^{4}$\t\\\\\no Coronae Borealis\t&\tnMS\t&\t2:1\t&\t187.61\t$\\pm$\t0.13\t&\t33.64\t$\\pm$\t0.67\t&\t6.00\t$\\pm$\t0.05\t&\t93.81\t$\\pm$\t0.13\t&\t8.64\t$\\pm$\t0.67\t&\t4.66\t$\\pm$\t0.10\t&\t1.92\t$\\pm$\t0.57\t&\t1.82\t$\\pm$\t0.49\t&\t8.66 $\\times$ 10$^{4}$\t\\\\\nHD 101930\t&\tMS\t&\t2.17:1\t&\t71.30\t$\\pm$\t0.17\t&\t17.93\t$\\pm$\t0.41\t&\t1.38\t$\\pm$\t0.04\t&\t32.86\t$\\pm$\t0.17\t&\t2.37\t$\\pm$\t0.50\t&\t1.49\t$\\pm$\t0.20\t&\t-10.72\t$\\pm$\t3.43\t&\t18363.19\t$\\pm$\t0.35\t&\t3.00  $\\times$ 10$^{4}$\t\\\\\n14 Andomedae\t&\tnMS\t&\t2.17:1\t&\t185.89\t$\\pm$\t0.22\t&\t99.80\t$\\pm$\t1.41\t&\t4.90\t$\\pm$\t0.04\t&\t85.66\t$\\pm$\t0.22\t&\t6.56\t$\\pm$\t1.61\t&\t4.60\t$\\pm$\t0.22\t&\t-7.64\t$\\pm$\t2.96\t&\t1.68\t$\\pm$\t1.24\t&\t2.59 $\\times$ 10$^{4}$\t\\\\\nHD 180902\t&\tnMS\t&\t2.17:1\t&\t482.22\t$\\pm$\t2.92\t&\t28.68\t$\\pm$\t1.16\t&\t0.30\t$\\pm$\t0.04\t&\t222.22\t$\\pm$\t2.92\t&\t4.20\t$\\pm$\t0.88\t&\t1.05\t$\\pm$\t0.23\t&\t-3.74\t$\\pm$\t2.69\t&\t9.27\t$\\pm$\t0.54\t&\t1.83 $\\times$ 10$^{4}$\t\\\\\nHD 218566\t&\tMS\t&\t2:1\t&\t225.54\t$\\pm$\t0.14\t&\t7.60\t$\\pm$\t0.24\t&\t0.90\t$\\pm$\t0.07\t&\t112.77\t$\\pm$\t0.14\t&\t2.47\t$\\pm$\t0.26\t&\t0.17\t$\\pm$\t0.17\t&\t0.61\t$\\pm$\t0.11\t&\t0.82\t$\\pm$\t0.19\t&\t1.70 $\\times$ 10$^{4}$\t\\\\\nGJ 649\t&\tMS\t&\t2:1\t&\t601.38\t$\\pm$\t2.17\t&\t11.55\t$\\pm$\t0.31\t&\t3.41\t$\\pm$\t0.11\t&\t300.69\t$\\pm$\t2.17\t&\t2.88\t$\\pm$\t0.39\t&\t1.83\t$\\pm$\t0.32\t&\t0.86\t$\\pm$\t0.31\t&\t6.18\t$\\pm$\t0.46\t&\t1.10 $\\times$ 10$^{4}$\t\\\\\n75 Ceti\t&\tnMS\t&\t2.17:1\t&\t694.41\t$\\pm$\t1.40\t&\t37.12\t$\\pm$\t0.74\t&\t3.84\t$\\pm$\t0.04\t&\t320.01\t$\\pm$\t1.40\t&\t3.31\t$\\pm$\t0.70\t&\t4.32\t$\\pm$\t0.21\t&\t4.59\t$\\pm$\t0.48\t&\t0.40\t$\\pm$\t0.47\t&\t1.04 $\\times$ 10$^{4}$\t\\\\\nHD 27894\t&\tMS\t&\t2.17:1\t&\t17.97\t$\\pm$\t0.01\t&\t58.40\t$\\pm$\t0.49\t&\t5.04\t$\\pm$\t0.07\t&\t8.28\t$\\pm$\t0.01\t&\t3.91\t$\\pm$\t0.73\t&\t0.50\t$\\pm$\t0.17\t&\t-29.59\t$\\pm$\t10.58\t&\t82907.65\t$\\pm$\t1.94\t&\t6118.83\t\\\\\nHD 32518\t&\tnMS\t&\t2.17:1\t&\t157.45\t$\\pm$\t0.19\t&\t117.90\t$\\pm$\t2.19\t&\t6.00\t$\\pm$\t0.03\t&\t72.56\t$\\pm$\t0.19\t&\t9.90\t$\\pm$\t2.38\t&\t4.92\t$\\pm$\t0.17\t&\t13.84\t$\\pm$\t4.14\t&\t-11.90\t$\\pm$\t1.26\t&\t3259.36\t\\\\\nHD 231701\t&\tMS\t&\t2.17:1\t&\t141.30\t$\\pm$\t0.35\t&\t41.56\t$\\pm$\t1.48\t&\t1.32\t$\\pm$\t0.10\t&\t65.11\t$\\pm$\t0.35\t&\t12.81\t$\\pm$\t3.37\t&\t2.98\t$\\pm$\t0.23\t&\t-8.44\t$\\pm$\t6.3\t&\t-0.02\t$\\pm$\t1.81\t&\t993.37\t\\\\\n$\\gamma_1$ Leonis\t&\tnMS\t&\t2:1\t&\t428.87\t$\\pm$\t0.17\t&\t206.77\t$\\pm$\t0.62\t&\t1.09\t$\\pm$\t0.01\t&\t214.43\t$\\pm$\t0.17\t&\t31.92\t$\\pm$\t0.70\t&\t4.99\t$\\pm$\t0.03\t&\t9.44\t$\\pm$\t0.86\t&\t178.66\t$\\pm$\t0.53\t&\t598.25\t\\\\\nHD 2638\t&\tMS\t&\t2.17:1\t&\t3.45\t$\\pm$\t0.00\t&\t66.75\t$\\pm$\t0.45\t&\t5.26\t$\\pm$\t0.09\t&\t1.59\t$\\pm$\t0.00\t&\t1.36\t$\\pm$\t0.46\t&\t4.44\t$\\pm$\t0.35\t&\t49.46\t$\\pm$\t13.72\t&\t9619.32\t$\\pm$\t2.32\t&\t407.32\t\\\\\nHD 31253\t&\tMS\t&\t2:1\t&\t464.44\t$\\pm$\t0.64\t&\t10.75\t$\\pm$\t0.34\t&\t0.27\t$\\pm$\t0.06\t&\t232.22\t$\\pm$\t0.64\t&\t3.58\t$\\pm$\t0.33\t&\t2.87\t$\\pm$\t0.13\t&\t0.59\t$\\pm$\t0.17\t&\t1.96\t$\\pm$\t0.25\t&\t295.22\t\\\\\nHD 221287\t&\tMS\t&\t2.17:1\t&\t452.51\t$\\pm$\t1.00\t&\t73.50\t$\\pm$\t1.37\t&\t2.10\t$\\pm$\t0.04\t&\t208.53\t$\\pm$\t1.00\t&\t12.95\t$\\pm$\t2.46\t&\t2.34\t$\\pm$\t0.05\t&\t10.72\t$\\pm$\t2.36\t&\t-21861.09\t$\\pm$\t1.29\t&\t124.62\t\\\\\nHD 190647\t&\tMS\t&\t2:1\t&\t931.10\t$\\pm$\t76.66\t&\t30.25\t$\\pm$\t2.44\t&\t3.81\t$\\pm$\t0.44\t&\t465.55\t$\\pm$\t76.66\t&\t7.90\t$\\pm$\t1.04\t&\t2.94\t$\\pm$\t0.93\t&\t-20.43\t$\\pm$\t5.67\t&\t-40266.66\t$\\pm$\t1.65\t&\t82.74\t\\\\\nHD 220773\t&\tMS\t&\t2:1\t&\t2877.77\t$\\pm$\t87.74\t&\t14.60\t$\\pm$\t1.51\t&\t1.63\t$\\pm$\t0.12\t&\t1438.88\t$\\pm$\t87.74\t&\t10.15\t$\\pm$\t1.42\t&\t4.80\t$\\pm$\t0.25\t&\t2.89\t$\\pm$\t0.72\t&\t-4.96\t$\\pm$\t1.05\t&\t54.06\t\\\\\nHD 330075\t&\tMS\t&\t2.17:1\t&\t3.39\t$\\pm$\t0.00\t&\t106.81\t$\\pm$\t0.73\t&\t5.87\t$\\pm$\t0.01\t&\t1.56\t$\\pm$\t0.00\t&\t0.49\t$\\pm$\t0.49\t&\t3.22\t$\\pm$\t1.21\t&\t-13.2\t$\\pm$\t4.57\t&\t61278.58\t$\\pm$\t0.40\t&\t37.21\t\\\\\nHIP 79431\t&\tMS\t&\t2.17:1\t&\t113.99\t$\\pm$\t0.40\t&\t155.87\t$\\pm$\t2.20\t&\t2.19\t$\\pm$\t0.02\t&\t52.53\t$\\pm$\t0.40\t&\t30.65\t$\\pm$\t1.82\t&\t4.00\t$\\pm$\t0.09\t&\t-385.58\t$\\pm$\t24.21\t&\t10.53\t$\\pm$\t1.90\t&\t26.99\t\\\\\n\\hline\t\t\t\t\t\t\t\n\\end{tabular}\n}\n\\caption{The preferred fixed circular double models of the extended sample that has Bayes factor larger than the 95th percentile of the synthetic systems.  Under star type, MS and nMS refer to Main Sequence and non Main Sequence stars, respectively.  The order of the table is by Bayes factor.}\n\\label{tab:Results}\n\\end{center}\n\\end{table}\n\\twocolumn\n\nOur primary analysis does not include stellar jitter (even though our synthetic data has jitter added to the simulated data).  We made this choice for a number of reasons.  One is that since we are considering a fixed period ratio, only noise that occurs at that specific frequency could produce a spurious signal.  Most sources of stellar noise occur on much different time-scales.  The stellar rotation periods (typically ranging from 4--40 days, \\citep{McQuillan14}) are shorter than the inner planet periods for most of these systems.  Stellar p-mode oscillations have typical time-scales of 5-15 minutes \\citep{Haywood15}.  And, surface granulation variations last minutes to hours, with the largest granules remaining on the surface of stars for about a day \\citep{DelMoro04,Haywood15}.  The time-scales of long term stellar activity arising from the cyclical appearance of starspots are on the order of years to decades \\citep{Strassmeier09}.\n\nThese facts support the interpretation that stellar noise is not the cause of the inner companion signal for the majority of our systems.  Nevertheless, we did a separate analysis that included a white noise jitter term to all models and found that 5\\ of the 15\\ systems still remain in the 95th percentile of likely two-planet systems, 4\\ of which prefer the 2.17:1 architecture.  Thus, even if we adopt the much more conservative approach---which assumes stellar jitter does indeed affect our data at precisely the relevant time scales---we still see a number of systems that favor the two-planet models.\n\nWhile our results are primarily from the fixed circular double models, we examined the results of a floating circular double model in order to estimate the likely distribution of orbital periods for the inner companion.  We analyzed the real and synthetic systems with the floating circular double model and find an even larger portion of the systems that have Bayes factors above the 95th percentile---19\\ systems, 32\\%\\ of the main sequence sample, (41\\ systems, 43\\%, of the extended sample) with an estimated false positive rate of 0.95 $\\pm$ 0.97 (2.1 $\\pm$ 1.4 for the extended sample).  Of these 19\\ systems, 9\\ prefer the floating circular double model, 6\\ prefer the fixed 2.17:1 model, and the remaining 4\\ prefer the 2:1 double circular model.  4\\ systems remain in the 95th percentile when including stellar jitter in the model as a white noise term.\n\n\n\nWe show the period ratio posteriors that result from this analysis for these 19\\ systems and the synthetic systems in Figure~\\ref{fig:PRHistogramMS}.  These histograms show the combined, period ratio posterior distribution from fitting the circular double model without a constraint on orbital periods to the 19\\ systems and to the synthetic systems.  The distribution for the synthetic systems clearly shows the degeneracy at the location of the 2:1 MMR.  If the real systems (in orange) were single-planet systems, then the expected distribution should be the same as for the synthetics.  However, the two distributions differ significantly.  In fact, the distribution for the real systems mirrors the period ratio distribution from the \\textit{Kepler}\\ data \\citep{Steffen15}.  Most of the combined posteriors favor period ratios just wide of the 2:1 or between 2.15 and 2.2.  Only a few systems preferred the circular double model near the 2:1 because the degeneracy is located at the 2:1 and the power to distinguish between the models diminishes.  Thus, in that regime, more data with appropriate phase-sampling is essential to distinguish between the models.\n\n\\begin{figure}\n\\includegraphics[width=\\columnwidth]{Figure6.pdf}\n\\caption{The posterior distributions for the period ratio when considering the inner planet period as a free parameter.  The blue distribution is the 1,000 synthetic single eccentric systems.  This distribution peaks at 2:1---the location of the degeneracy.  The orange distribution is the 19 systems with Bayes factors larger than the 95th percentile thresholds that are hosted by main sequence stars.}\n\\label{fig:PRHistogramMS}\n\\end{figure}\n\n\\section{Discussion \\& Conclusions} \\label{sec:d&c}\n\nThere are currently 395 confirmed \\EditTwo{solitary} RV planets\\EditTwo{,} \nand we reanalyzed about fifteen percent of them.  Our extended sample contains nearly a quarter of the confirmed RV planets.  \\Edit{The distribution of eccentricities, periods, velocity semi-amplitudes, etc. of our extended sample is shown in Figure~\\ref{fig:parametercomparison}.}  If the 15\\ systems in the main sequence sample (30\\ systems in the extended sample) that we identify are indeed circular doubles, then they would increase the number of RV multi-planet systems by $\\sim$12.5\\% ($\\sim$25\\%) since there are 120 confirmed systems reported with at least \\Edit{two planets} discovered by RV.  They would also significantly alter the estimated mixture of these two architectures---shifting the relative importance of their implied dynamical histories.\n\n\\Edit{If the fraction of misidentified single eccentrics in the entire NASA Archive is similar to the misidentification fraction seen in our sample}, then there could be as many as $\\sim$100 planets missing, or $\\sim$15\\% \nof the overall confirmed RV planets ($\\sim$120 in the extended sample, or $\\sim$18\\% of the overall confirmed RV planets).    \nMoreover, the apparent propensity for some systems to cluster around period ratios near 2.17 is a further indication that there is something fundamental, but still unknown, that attracts planet pairs into this period ratio.  We encourage observers to consider planning follow-up observations of these systems and make additional measurements at phases where the degeneracy is at its weakest.  New observations near these phases could confirm or refute the existence of these putative interior companions.  The success of such a campaign opens the door to identifying the architectures of the systems where the preferred model is still ambiguous.\n\n\n\n\\bibliographystyle{mnras}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nAccurate knowledge of the Point Spread Function (PSF) is a necessary requirement for performing high-quality photometric and astrometric\nanalysis on astronomical imaging.\nStandard methods used to model the PSF directly from the image typically exploit the wealth of information available in the observations of\ndense stellar systems like globular clusters. In this kind of imaging, each star is the representation of the same PSF, but at a different pixel-phase.\nBy selecting isolated stars with a high signal-to-noise ratio (S\/N) and combining their light profiles together, it is thus possible to recover a\ngood model for the actual shape of the PSF, which is then fit to all of the other sources in the image in order to measure their positions and magnitudes\n(see e.g.~\\citealt{stetson87, diolaiti00, anderson00}).\n\nHowever, these standard methods are becoming less effective as we approach the era of Extremely Large Telescopes (ELTs), when observations will be routinely assisted by \nAdaptive Optics (AO) techniques. In fact, these techniques will enable diffraction-limited observations from the ground, at the cost of making the PSF variable across\nthe field of view (FoV). Because of the variability, fewer stars restricted to smaller portions of the FoV can be used as independent representations of the same PSF.\nThis is why, in sparse fields, very often it is the case that no stars at all other than the natural guide star will be available to model the PSF. \nThis is when techniques for modelling the PSF which do not use the information coming from the images become most significant, such as PSF-reconstruction (PSF-R, \\citealt{veran97}).\n\nPSF-R is a technique that relies on AO control-loop data to determine the shape of the PSF potentially at any spatial location in the FoV. \nDespite its being theoretically well established (e.g. \\citealt{joli18, ragland18}), so far, PSF-R has never surpassed the performance obtained \nby standard methods when applied to real astronomical imaging, not even in the case of AO-assisted data (e.g.~\\citealt{turri15, massari16a, massari16b, monty18}).\n\nIn this Letter, we exploit a new hybrid technique known as PRIME (\\citealt{beltramo19}), which combines PSF-R with image fitting to perform\na photometric and astrometric analysis of a small stellar field in the Galactic globular cluster NGC~6121 observed with SPHERE\/ZIMPOL (\\citealt{schmid18}).\nWhile the theoretical background has been provided in Beltramo-Martin et al. (2020; hereafter BMS), here we focus on the results in terms of the\nphotometric and astrometric precision achieved on these real data. The comparison with results obtained using standard PSF-modelling methods shows, for the first\ntime, a significant improvement and demonstrates the potential of PRIME towards the advent of future ELT observations.\n\nThe Letter is organised as follows. In Section~\\ref{data}, the observations of NGC~6121 that form the basis of this work are presented. In Section~\\ref{onaxis}, we describe the methods\nemployed to perform the analysis and we show the results obtained on the on-axis guide star. In Section~\\ref{off}, the photometric and astrometric precision is quantified for \nthe off-axis sources. Finally, a discussion and the conclusions are provided in Section~\\ref{concl}.\n\n\\section{Data analysis}\\label{data}\n\nIn this work, we focus on imaging data of the Galactic globular cluster NGC~6121 taken with SPHERE\/ZIMPOL as part of a technical proposal following the ESO Calibration Workshop \n2017\\footnote{{\\tt www.eso.org\/sci\/meetings\/2017\/calibration2017.html}}, under the program ID 60.A-9801(S).\nThe dataset consists of 12 exposures in the V filter, each with a duration of $200$ s (NDIT$=2$, DIT$=100$ s). Since ZIMPOL is capable of performing simultaneous observations through two cameras, we effectively end up with\n24 independent measurements of the positions and magnitude of the stars in the FoV. Observations were taken on the night of 26 of June 2018, at an average airmass of $\\sim1.2$.\n\nThe FoV is a small squared window with a size of $3.5\\times3.5$ arcsec$^2$, sampled with a pixel-scale of $7.2$ mas\/pixel after data processing. Overall, five stars have been detected within the field. The brightest one\n(V$\\simeq10.9$ mag, \\citealt{anderson08}) has been used as the natural guide star for the Adaptive Optics system, which has provided a fairly good correction throughout the observations, with an average Strehl ratio\nin H-band of SR$=65$\\% (this corresponding to SR$\\sim2$\\% in the adopted V-band) and an on-axis full-width at half maximum (FWHM) of FWHM$=33$ mas.\nThe other four stars are significantly fainter, having magnitudes in the range of V$=16.6-18.5$ and  with a position located at a distance of about 2 arcsec from the guide star.\nFor the sake of visualisation, in Fig.~\\ref{fov} we show the averaged image of all of the available exposures, with the location of the five stars are marked in red. \n\n\\begin{figure}\n    \\includegraphics[width=\\columnwidth]{fov_cam1_cut.pdf}\n    \\caption{Median of the 24 exposures sampling the field of NGC~6121. The five detected stars are marked with red symbols. The orientation of the field and the physical scale \n    are also highlighted.}\\label{fov}\n\\end{figure}\n\nThe data reduction, including bias subtraction and flat-fielding, was performed using the SPHERE Data and Reduction Handling\npipeline (\\citealt{pavlov08}). Bad-pixel correction was applied using dedicated Python routines, as described in BMS.\n\nThe photometric and astrometric analysis is performed in two ways. The first uses standard PSF modelling techniques that only exploit the information\ncoming directly from the images and relies on the DAOPHOT-II suite of software (\\citealt{stetson87}). The second uses the hybrid method encoded in PRIME instead.\nIn all of the exposures, DAOPHOT-II can only use the on-axis guide star to model the PSF as all of the other sources are too faint to actually provide further constraints\nto the model. The way DAOPHOT-II works is described in detail in \\cite{stetson87}. Briefly, the PSF is modelled by means of a Moffat function plus a table of residuals,\nboth of which are determined by fitting the stellar light profile enclosed within an aperture of an eight-pixel radius. Then the PSF that has been\nrecovered in this way is applied via ALLSTAR to all of the five stars within an aperture of a 30-pixel radius and, in our case, without any possibility for \nincluding PSF spatial variation in the measurements of magnitude and position. \nWe note that this is a rather extreme case for resolved stellar population science that usually can rely on the presence of hundreds or thousands of stars in the field\nto model the PSF. However, this is not uncommon at all in the investigations of individual extragalactic objects, such as the lensed Active Galactic Nuclei \n(e.g. \\citealt{auger10, spingola19}), or in planetary science (e.g. \\citealt{bonnefoy11}). This is why, despite the conditions are particularly poor to\nperform the analysis with DAOPHOT-II, it is still important to compare its results with those coming from a better-suited method like PRIME.\n\nOn the other hand, PRIME exploits atmospheric and AO control-loop data to build a first-guess PSF model,\nwhose parameters are then adjusted based on the fitting of the natural guide star. Also in this case, the same model is then fit to all of the five stars.\nAtmospheric parameters were simultaneously measured by the MASS-DIMM at Paranal (\\citealt{butterley18, tokovinin07}), \nby the stereo-SCIDAR (\\citealt{osborn18}), and the AO real time computer known as SPARTA (\\citealt{suarezvalles12}), which all confirmed\nthat the observing conditions had been stable over the night, with the seeing at zenith varying between 0.7-0.9 arcsec.\nTwo 30 sec-long full sets of AO telemetry data were also obtained at the beginning of ZIMPOL observations (see BMS for details).\n\nFinally, for the sake of comparison we also analysed the data on-axis using a simple aperture method. The magnitude measured in this way is given by the sum \nof the counts within a circular aperture of a 30-pixel radius in order to be consistent, thus, with the prescriptions used in the other two methods. \nPositions were derived within the same aperture radius as intensity-weighted centroids. All the individual exposures were treated independently during the data reduction and the analysis that followed. \n\n\\section{Results: on-axis performance}\\label{onaxis}\n\n\\begin{figure*}\n    \\includegraphics[width=\\textwidth]{davide_onaxis.pdf}\n    \\caption{Photometric (top panel) and astrometric (bottom panel) precision achieved on the guide star when using the aperture method (left column), \n    DAOPHOT-II (middle column) and PRIME (right column).}\\label{on}\n\\end{figure*}\n\nThe first target of the analysis is the natural guide star that has been used to model the PSF by DAOPHOT-II and to adjust the reconstructed PSF model by PRIME.\nThe results of the analysis are shown in Fig. \\ref{on}, which compares the photometric (top panels) and astrometric (bottom panels) precision achieved\nwith the three methods employed. The precision ($\\sigma$) is defined as the root mean square error around the mean value of the distributions of the $24$ independent magnitude and position\nmeasurements of the guide star. We remark that a small degree of intrinsic photometric variation could be due to the temporal changes in the atmospheric conditions,\nfor example, because the airmass of our exposures ranges from $1.1$ at the beginning of the observations to $1.3$ at the end. However, this intrinsic variation\nshould be very small (sky transparency during the night was stable within a 2\\% level and the airmass changed by very little) and, in any case, it affects the measurements \nobtained with the three methods in the same way. Moreover, the magnitudes measured in the two ZIMPOL cameras are slightly offset due to the different throughput of the two\nchannels. We corrected for such an offset by imposing that the average magnitudes of the guide star over the 12 exposures observed with each cameras are the same.\n\nIn terms of photometry, the aperture method and DAOPHOT-II perform similarly well, achieving a precision of $\\sigma=0.172$ mag and $\\sigma=0.160$ mag, respectively.\nThis may seem surprising as PSF-fitting usually produces much more precise results than aperture photometry. However, it should be considered that\nthe guide star is very bright and isolated, which are also the circumstances under which aperture photometry performs well.\nOn the other hand, it is interesting to note that PRIME achieves a precision one order of magnitude better than the previous two methods ($\\sigma=0.015$ mag).\nThis level of precision matches the typical performance obtained on more stable, seeing-limited observations of dense stellar fields, where \nthe modelling of the PSF is strongly facilitated by the large number of sources describing the same PSF (e.g. \\citealt{massarim3, savino18}). \n\nIn terms of astrometry, the precision achieved by the aperture method ($\\sigma=0.038$ pixels) and DAOPHOT-II ($\\sigma=0.046$ pixels) are again comparable for the same\nreasons as those described in the photometry case. Moreover, also in this case, PRIME yields the best performance, achieving a precision of $\\sigma=0.009$ pixels. The improvement,\ntherefore, amounts to about a factor of four compared to the other methods. When translated to an angular size, such a precision corresponds to $\\sim70 ~\\mu$as. \nThis is a remarkable result as this level of performance is higher than that achieved so far by other Adaptive Optics facilities, which have otherwise achieved precision of the order of $150-200 ~\\mu$as, at best, on resolved stellar population science cases \n (see e.g. \\citealt{ghez08, neichel14, massari16b}). Our findings match the results typically obtained by the best space-based \nastrometric facilities, such as {\\it Gaia} (\\citealt{brown18}) and the {\\it Hubble Space Telescope} (e.g. \\citealt{bellini14}).\n\n\\section{Results: off-axis performance}\\label{off}\n\nFour other stars, other than the guide star, are located in the FoV. They represent a very good opportunity to test PRIME in different conditions as they\nare off-axis by about 2 arcsec and because they are much fainter than the guide star. Star-3 is in fact $\\sim8$ mag fainter (see Fig.~\\ref{fov}) and this is why \nin some of the exposures, its peak falls below the DAOPHOT-II detection limit. In these cases (17 out of 24 exposures), neither photometry nor astrometry for Star-3 were measured by DAOPHOT-II.\nMoreover, since the aperture method is poorly suited to perform the analysis in the case of very faint stars, we restrict the following discussion \nto DAOPHOT-II and PRIME results only. In both cases, the positions and magnitude of the off-axis stars were measured after the subtraction of the on-axis star model.\n\n\\begin{figure}\n    \\includegraphics[width=\\columnwidth]{davide_offaxis_photometry_on.pdf}\n    \\caption{Comparison between the photometric precision achieved by DAOPHOT-II (black empty histograms) and PRIME (red histograms) on the four\n    individual off-axis stars (each shown in a different panel). The values of the precision are also quoted for sake of comparison.}\\label{offphoto}\n\\end{figure}\n\nFigure~\\ref{offphoto} shows the comparison between the photometric precision achieved by the two methods (black empty histograms for DAOPHOT-II measurements and \nred histograms for PRIME estimates) for each of the four off-axis stars.\nAlso in this case, PRIME seems to give better results compared to DAOPHOT-II, though not as strikingly as on the natural guide star. \nThe photometric precision ranges from $\\sigma\\sim0.05$ mag for the two brightest sources (Star-1 and Star-2) up to $\\sigma\\sim0.17$ mag for the two faintest ones (Star-3 and Star-4).\nThe gain with respect to standard methods thus amounts to a factor of two to three in the best cases, while the performance is comparable for the faintest stars.\n \nExisting Adaptive Optics photometry has been usually performed on K-band images. To compare it with our results, we first consider theoretical isochrones (\\citealt{basti06})\nfor an old, metal-intermediate (\\citealt{marino08}) globular cluster to find that the V-band magnitudes of our four off-axis stars lie in the range K$=15-16.5$ mag.\nAt these levels of K-band brightness and with similar exposure times ($\\sim160$ s), the Multi-Conjugate Adaptive Optics (MCAO) camera GeMS\/GSAOI (\\citealt{rigaut12}) has\nachieved a photometric precision of $\\sim0.06$ mag (see \\citealt{massari16a, saracino16}), which is remarkably similar to the PRIME performance described in this work. Nonetheless, we stress \nthat our results were obtained in much more challenging conditions as $i$) we could only rely on one star to model the PSF (compared to the hundreds available in the quoted papers); $ii$) the SR of our V-band images is $\\sim2$\\% compared to SR$=20-30$\\% for the GeMS IR data (e.g. \\citealt{neichel14, massari16a, dalessandro16}); and $iii$) the PSF stability granted by \nGeMS MCAO system ($\\sim15$\\% FWHM variability over a $85\\times85$ arcsec$^2$ FoV, see \\citealt{bernard16}) is better than the one achieved by ZIMPOL ($\\sim15$\\% FWHM variability but over a much smaller\n$3.5\\times3.5$ arcsec$^2$ FoV).\n\n\\begin{figure}\n    \\includegraphics[width=\\columnwidth]{davide_offaxis_astrometry_on.pdf}\n    \\caption{Astrometric precision achieved on off-axis stars with DAOPHOT-II and PRIME (the colour-coding is the same as in Fig.\\ref{offphoto}).}\\label{offastro}\n\\end{figure}\n\nOn the other hand, Fig.~\\ref{offastro} compares the results obtained in terms of astrometric precision. As in the case of the guide star,\nPRIME seems to maintain some advantage over DAOPHOT-II in recovering precise stellar positions. The improvement decreases, however, when moving from a factor\nof four on-axis to a factor of $\\sim2$ for the two brightest off-axis stars and to then disappear for the two faintest sources.\nIn the case of Star-3, the performance achieved by DAOPHOT-II seems to be a factor of two better. Although it is based on only a few measurements and, thus, could be affected by\nsmall number statistics issues, it is, rather, the PRIME precision that is worse than expected. The large dispersion of PRIME measurements seems to be driven by few outliers, with the \nbulk of the distribution overlapping nicely with that from DAOPHOT-II. We therefore assess the two distributions as comparable.\n\nFinally, also in this case we can compare our astrometric precision with that achieved by GeMS\/GSAOI MCAO observations (having similar exposure times) on \nanother Galactic globular cluster (NGC~6681, see \\citealt{massari16b}). \nFor stars in the same magnitude range, GeMS obtained a positional precision of the order of $0.5-1$ mas. Despite the lower S\/N of our images caused by the less efficient \nAO correction in the optical bands, these results match ours well as the precision we achieved on Star-1 and Star-2 amounts to $\\sim0.7$ mas.\n\n\\section{Summary and Conclusions}\\label{concl}\n\nIn this Letter, we carried out an experimental comparison of the photometric and astrometric performance achieved using both standard PSF-modelling techniques and the PSF-R-based hybrid technique known as PRIME \non real observations of a small field in the Galactic globular cluster NGC~6121.\nThe conditions of the experiment are rather extreme, with only one star in the field bright enough to be suitable for PSF-modelling. Despite\nbeing a rather uncommon condition in the case of resolved stellar populations studies, such an occurrence is more typical for investigations of individual extragalactic objects\nand it is, therefore, valuable for a wealth of science cases.\n\nOn-axis, the hybrid approach followed by PRIME that starts from a PSF model reconstructed using AO control-loop data and then adjusts its parameters\nvia image fitting (see the detailed description of the method in \\citealt{beltramo19}) achieves remarkable results. The photometric precision\nis $\\sigma=0.015$ mag, about one order of magnitude better than what is obtained using more standard methods like aperture photometry or DAOPHOT-II.\nAlso in terms of astrometric precision, the performance ensured by PRIME is excellent ($\\sigma\\simeq70~\\mu$as), leading to a gain of about a factor of four\ncompared to the other methods.\n\nOn the other hand, when assessing the performance on the four, faint off-axis sources, PRIME achieves results that are comparable with those obtained\nby DAOPHOT-II on both photometry and astrometry. When the comparison refers to existing analysis of GeMS MCAO observations of stars in a similar magnitude range, \nthe results in terms of precision are again similar, despite the unfavourable conditions to perform the analysis. \n\nIn conclusion, these results demonstrate the potential of PRIME as a powerful tool for analysing future Adaptive Optics-assisted observations with  ELTs. This is especially true \nin cases where there is a poor availability of point-like sources in the field for modelling the PSF. Future experiments will test PRIME performance on observational cases that\nare well suited for an analysis with standard methods as well. \n\n\\begin{acknowledgements}\n\nBased on observations collected at the European Organisation for Astronomical Research in the Southern Hemisphere.\nGF has been supported by the Futuro in Ricerca 2013 (grant RBFR13J716).\n\n\\end{acknowledgements}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\n\n\\subsection{The subject}\n\nThis small review is based on papers \\cite{BGMS-75},\n\\cite{GKST-Springer}, \\cite{TESLA-04}. It deals with high-energy\n$\\gamma\\gamma$ collisions, this is a new and promising area in\nhigh-energy physics related to fundamental problems of strong and\nelectro-weak interactions.\n\nOur knowledge about elementary particles and their interactions is\nmainly obtained from particle collisions. Accelerators with\ncolliding beams are called now {\\it colliders}. Most of fundamental\nresults in particle physics have been obtained from experiments at\nthe $pp,\\; p\\bar p, \\;e^+e^-$ and $ep$ colliders.\n\nPrincipal characteristics of colliders are:\n\\begin{enumerate}\n\\item{the {\\it energy} in the center-of-mass system (c.m.s)\n$E_{\\rm cm}$;}\n\\item{{\\it luminosity} of a collider $L$ which determines\ncollision rate $\\dot N$ of events with the cross section $\\sigma$ by\nrelation $\\dot N = L\\, \\sigma$;}\n\\item{{\\it types} of colliding particles.}\n\\end{enumerate}\n\nThe progress on high-energy colliders can be seen from Table\n\\ref{T:1}. Up to now and in the nearest future, the $pp$ and $ p\\bar\np$ colliders are the machines with the highest energy. That is why\nsuch epochal discoveries as $W$ and $Z$ bosons (responsible for weak\ninteraction) and $t$ quark had been performed at the S$p\\bar p$S and\nthe TEVATRON, respectively. It is not excluded that at future $pp$\ncollider LHC the Higgs boson $ H$ (thought to be responsible for the\norigin of the particle masses) will be discovered.\n\n\\begin{table}[th]\n\\centering\n \\caption{High energy colliders}\n\\begin{tabular}{lccl}\\hline\nCollider & Type & $ E_{\\rm cm} $, TeV & Start date \\\\  \\hline\n&&& \\\\[-2mm]\nS$p\\bar p$S & $p\\bar p$ & 0.6 & 1981 \\\\\nTEVATRON & $p\\bar p$ & 2 & 1987 \\\\\nLHC & $ p p$ & 14 & 2007 \\\\[2mm]\nHERA & $e p$ & 0.31 & 1992 \\\\[2mm]\nSLC & $e^+e^-$ & 0.1 & 1989 \\\\\nLEP-I & $e^+e^-$ & 0.1 & 1989 \\\\\nLEP-II &$ e^+e^- $ & 0.2 & 1999 \\\\[2mm]\nLinear collider & $e^+e^-$ & 0.5 & 2010? \\\\\nPhoton collider & $\\gamma \\gamma,\\; \\gamma e$ & 0.4 & 2010+?\n\\\\[2mm]\nMuon collider & $\\mu^+\\mu^-$ & 0.1$\\div$ 3 & ?? \\\\\n\\end{tabular}\n \\label{T:1}\n\\end{table}\n\nFor detail study of new phenomena, it is important not only the\nenergy but also types of colliding particles. The $e^+e^-$\ncolliders, being less energetic then $pp$ colliders, have some\nadvantages over proton colliders due to much lower background and\nsimpler initial state. Well known example --- the study of $Z$\nboson. It was discovered at $ p\\bar p$ collider S$p\\bar p$S where\nabout 100 events with $Z$ bosons were found among about $10^{11}$\nbackground events, while the detailed study of $Z$ boson had been\nperformed at the $e^+e^-$ colliders LEP and SLC which provided us\nwith more than $10^7$ $Z$-events with a very low background.\n\nAbout thirty years ago a new field of particle physics ---\nphoton-photon interactions --- has\nappeared~\\cite{Bal71}--\\cite{BKT}. Up to now $\\gamma \\gamma$\ninteractions were studied in collisions of virtual (or equivalent)\nphotons at $e^+e^-$ storage rings. It was obtained a lot of\ninteresting results. However, the number of the equivalent photons\nis by 2 order of magnitude less than the number of electrons.\n\nA new possibility in this field is connected with high-energy\n$e^{\\pm}e^-$ linear colliders which are now under development. The\nelectron bunches in these colliders are used only ones. This makes\npossible to ``convert'' electrons to real high-energy photons, using\nthe Compton back-scattering of laser light, and thus to obtain\n$\\gamma \\gamma$ and $\\gamma e$ colliders with real\nphotons~\\cite{GKST81a}--\\cite{GKPST84}. The luminosity and energy of\nsuch colliders will be comparable to those of the basic $e^{\\pm}e^-$\ncolliders.\n\nPhysical problems, which are now investigated in the $\\gamma \\gamma$\ncollisions, are mainly connected with strong interaction at large\n$\\sim \\hbar \/(m_\\pi c)$ and moderate small distances $\\sim \\hbar\n\/p_\\bot$, where $p_\\bot \\; \\stackrel{<}{\\sim} 10$ GeV\/c. In future\nit will be a continuation of the present day experiments plus\nphysics of gauge $(W^\\pm,\\; Z)$ bosons and Higgs $H$ bosons, i.e. it\nwill be problems of the electro-weak interactions, Standard Model\nand beyond, search of new particles and new interactions. In other\nwords, physics at high-energy $\\gamma \\gamma$ and $\\gamma e$\ncollisions will be very rich and no less interesting than at $pp$ or\n$e^+e^-$ collisions. Moreover, some phenomena can best be studied at\nphoton collisions.\n\nAt the end of this subsection it will be appropriate to cite some\nwords from the article ``Gamma-Ray Colliders and Muon Colliders'' of\nthe former President of the American Physical Society A.~Sessler in\n{\\it Physics Today}~\\cite{Sessler}:\n\n\\begin{quotation}\n\nIn high-energy physics, almost all of the present accelerators are\ncolliding-beam machines.\n\nIn recent decades these colliders have produced epochal discoveries:\nStanford SPEAR electron-positron collider unveiled the charmed-quark\nmeson and $\\tau$ lepton in 1970s. In the realm of high-energy\nproton-antiproton colliders, the Super Proton Synchrotron at CERN\ngave us the W$^{\\pm}$ and Z$^0$ vector bosons of electroweak\nunification in 1990s, and in 1999s the Tevatron at Fermilab finally\nunearthed the top quark, which is almost 200 times heavier than the\nproton.\n\n...What about other particles? Beam physicists are now actively\nstudying schemes for colliding photons with one another and schemes\nfor colliding a beam of short-lived $\\mu^+$ leptons with a beam of\ntheir $\\mu^-$ antiparticles.\n\nIf such schemes can be realized, they will provide extraordinary new\nopportunities for the investigation of high-energy phenomena.\n\nThese exotic collider ideas first put forward in Russia more that 20\nyears ago...\n\n\\end{quotation}\n\n\\begin{figure}[!h]\n\\centering\n\\includegraphics[width=8cm,angle=0]{f1_1.eps}\n\\caption\nFeynman diagrams for the elastic $\\gamma \\gamma$  scattering in QED}\n \\label{F1.1}\n\\end{figure}\n\n\\subsection{Interaction of photons in the Maxwell theory and QED }\n\nThe Maxwell's equations are linear in the strengths of the electric\nand magnetic fields. As a result, in the classical Maxwell theory of\nelectromagnetism, rays of light do not interact with each other. In\nquantum electrodynamics (QED) photons can interact via virtual\n$e^+e^-$ pairs. For example, an elastic $\\gamma \\gamma$ scattering\nis described by Feynman diagrams of Fig.~\\ref{F1.1}. The maximal\nvalue of the cross section is achieved at the c.m.s. photon energy\n$\\omega \\sim m_ec^2$ and is large enough:\n\\begin{equation}\n\\max \\;\\sigma_{\\gamma \\gamma \\to \\gamma \\gamma} \\sim \\alpha^4\n\\left({\\hbar\\over m_ec}\\right)^2= \\alpha^2 r^2_e\\sim 4\\cdot\n10^{-30}\\mbox{ cm}^2. \\label{1.3}\n\\end{equation}\nHowever, at low energies, $\\omega \\ll m_ec^2$, this cross section is\nvery small\n\\begin{equation}\n\\sigma_{\\gamma \\gamma \\to \\gamma \\gamma}=0.031 \\alpha^2 r^2_e\n\\left({\\omega\\over m_ec^2}\\right)^6\\,. \\label{1.1}\n\\end{equation}\nFor example, for visible light, $\\omega \\sim 1$ eV,\n\\begin{equation}\n\\sigma_{\\gamma \\gamma \\to \\gamma \\gamma} \\sim 10^{-65} \\; \\mbox{cm}\n^2\\,. \\label{1.2}\n\\end{equation}\nIt is too small to be measured even with the most powerful modern\nlasers, though there were such attempts. In recent\npaper~\\cite{Bernard} it was obtained an upper limit of the cross\nsection of $\\sigma(\\gamma \\gamma \\to \\gamma \\gamma)_{\\rm\nLim}=1.5\\times 10^{-48}$ cm$^2$  for the photon c.m.s. energy $0.8$\neV (see Fig.~\\ref{cro} from ~\\cite{Bernard}).\n\\begin{figure}[!htb]\n\\centering\n\\includegraphics[width=9cm,angle=0]{cro.eps}\n\\caption{Elastic photon cross section as a function of photon c.m.s.\nenergy}\n \\label{cro}\n\\end{figure}\n\nAt energies $\\omega > m c^2$, two photons can produce a pair of\ncharged particles. The cross section of the characteristic process\n$\\gamma \\gamma \\to \\mu^+ \\mu^-$ (Fig.\\ref{F1.2}a) is equal to\n\\begin{equation}\n\\sigma_{\\gamma \\gamma \\to \\mu^+ \\mu^-} = 4\\pi r_e^2\n\\frac{m_{e}^2c^4}{s} \\, \\ln{s\\over m^2_\\mu c^4} \\;\\;\\;\\; {\\rm at}\n\\;\\;\\;\\; s=(2\\omega)^2 \\gg 4m^2_\\mu c^4\\,.\n \\label{1.5}\n\\end{equation}\n\\begin{figure}[!htb]\n\\centering\n\\includegraphics[width=8cm,angle=0]{f1_2.eps}\n\\caption\nFeynman diagrams for a) $\\gamma \\gamma \\to \\mu^+\\mu^-$ and b)\n$e^+e^- \\to \\mu^+\\mu^-$} \\label{F1.2}\n\\end{figure}\nIt is larger than the ``standard'' cross section for the production\nof the same pair in the $e^+ e^-$ collisions (Fig.\\ref{F1.2}b via a\nvirtual photon only)\n\\begin{equation}\n\\sigma_{e^+e^- \\to \\mu^+ \\mu^-} = \\frac{4}{3}\\,\\pi r_e^2\\,\n\\frac{m_{e}^2c^4}{s}\\,.\n \\label{1.5a}\n\\end{equation}\n\n\\subsection{Collisions of equivalent photons at $e^+ e^-$\nstorage rings}\n\nUnfortunately, there are no sources of intense high-energy photon\nbeams (like lasers at low energies). However, there is indirect way\nto get such beams --- to use equivalent photons which accompanied\nfast charged particles. Namely this methods was used during last\nthree decades for investigation of two-photon physics at $e^+e^-$\nstorage rings. The essence of the equivalent photon approach can be\nexplained in the following way~\\cite{Fermi,WW} (see also~\\cite{BLP}\n\\S 99). The electromagnetic field of an ultra-relativistic electron\nis similar to the field of a light wave. Therefore, this field can\nbe described as a flux of the {\\it equivalent} photons with energy\ndistribution $dn_\\gamma \/ d\\omega$. The number of these photons per\none electron with the energy $E$ is\n\\begin{equation}\ndn_\\gamma\\sim {2\\alpha\\over \\pi}\\;\\ln{E\\over \\omega}\\,\n{d\\omega\\over\\omega} \\label{1.6}\n\\end{equation}\nor approximately\n\\begin{equation}\ndn_\\gamma\\sim 0.03\\;{d\\omega\\over\\omega}\\,.\n \\label{1.7}\n\\end{equation}\nAt the $e^+e^-$ colliders the equivalent photons also collide and\ncan produce some system of\n\\begin{figure}[!htb]\n\\centering\n\\includegraphics[width=11cm,angle=0]{f1_3.eps}\n\\caption\nProduction of system $X$ a) by two equivalent photons $\\gamma^*$\nwith 4-momenta (energies) $q_1\\;(\\omega_1)$ and $q_2\\; (\\omega_2)$\nemitted by an electron and a positron and b) in the annihilation\nprocess $e^+e^- \\to X$ } \\label{F1.3}\n\\end{figure}\nparticles $X$ (see Fig.~\\ref{F1.3}a, $\\gamma^*$ denotes the\nequivalent photon)\n\\begin{equation}\ne^+ e^-\\to e^+ e^-\\gamma^\\ast\\gamma^\\ast\\to e^+ e^- X\\,. \\label{1.8}\n\\end{equation}\nThus, this process is directly connected with the subprocess\n$\\gamma^* \\gamma^* \\to X$. Strictly speaking, the equivalent photons\nare not real photons, they are virtual ones. The 4-momentum squared\nof such a photon $q_i^2$ (which is equal to $m^2 c^2$ for usual\nparticle) is not equal zero, $q_i^2 \\neq 0$. But for large part of\nthe cross section $|q^2_i|$ is very small, therefore, the most of\nequivalent photons are almost real.\n\nThe cross section for two-photon production of  $e^+ e^-$ in\ncollisions of two fast particles with charges $Z_1e$ and $Z_2e$,\ni.e. for the $Z_1 Z_2 \\to Z_1 Z_2 e^+ e^-$ process, was calculated\nby Landau and Lifshitz \\cite{LL} in 1934 (see also \\cite{BLP} \\S\n100). In fact, it was the PhD of E.M.~Lifshitz.\n\nAt first sight, the cross sections of the two-photon processes at\n$e^+ e^-$ colliders (Fig.~\\ref{F1.3}a) are very small since they are\nthe 4-order processes: $ \\sigma_{\\rm two-phot} \\propto \\alpha^4\\,, $\nwhile for the annihilation processes of Fig.~\\ref{F1.3}b the cross\nsections $ \\sigma_{\\rm annih} \\propto \\alpha ^2. $ However, the\nannihilation cross sections decrease with increase of the energy\n(compare with (\\ref{1.5a}))\n\\begin{equation}\n\\sigma_{\\rm annih}\\sim{\\alpha^2}\\,{\\hbar^2 c^2\\over s},\\quad\ns=(2E)^2,\n \\label{1.9}\n\\end{equation}\nwhile the two-photon cross sections increase\n\\begin{equation}\n\\sigma_{\\rm two-phot}\\sim{\\alpha^4}\\,{\\hbar^2\\over m^2_{\\rm\nchar}c^2}\\, \\ln^n{s}\\,.\n \\label{1.10}\n\\end{equation}\nHere $n=3\\div 4$ depending on the process, and the characteristic\nmass $ m_{\\rm char}$ is constant (for example, $ m_{\\rm char}\\sim\nm_\\mu$ for $X= \\mu^+ \\mu^-$ and $ m_{\\rm char}\\sim m_\\pi$ for $X=\nhadrons$). As a result, already at $\\sqrt{s}> 2$ GeV\n\\begin{equation}\n\\sigma_{e^+ e^-\\to e^+ e^-\\mu^+\\mu^-} > \\sigma_{e^+\ne^-\\to\\mu^+\\mu^-}. \\label{1.11}\n\\end{equation}\nAnother  example, at the LEP-II electron-positron\n\\begin{figure}[htb]\n\\centering\n\\hspace*{5mm}\\includegraphics[width=11cm,angle=0]{f1_4.eps}\n\\caption{\nCross sections for some annihilation and two-photon processes in\n$e^+e^-$ collisions}\n \\label{F1.4}\n\\end{figure}\ncollider with the energy $\\sqrt{s}= 200$ GeV, the number of events\nfor two-photon production of hadrons with the c.m.s. energy\n$W_{\\gamma \\gamma} > 2$ GeV is by a three order of magnitude larger\nthan that in the annihilation channel (Fig.\\ref{F1.4}).\n\nAt $e^+e^-$ storage rings the first two-photon processes $e^+e^- \\to\ne^+e^- e^+e^-$ had been observed in 1970 (Novosibirsk~\\cite{Bal71}).\nThe importance of two-photon processes for the lepton and hadron\nproduction at $e^+e^-$ storage rings had been emphasized in the\npapers Arteage-Romero, Jaccarini, Kessler and Parisi~\\cite{Kes},\nBalakin, Budnev and Ginzburg~\\cite{BBG} and Brodsky, Kinoshita and\nTerazawa~\\cite{BKT}. In the papers~\\cite{BBG} it  was shown that\n$e^+e^-$ colliding beam experiments can give information about a new\nfundamental process $\\gamma^* \\gamma^* \\to hadrons$ and the\nnecessary formulae and estimations were obtained.\n\nAt that time there were a lot of theoretical investigations of\nvarious aspects of two-photon physics , but only a few experimental\nresults~\\cite{Bal71,Fras} have been obtained related mainly to the\nprocesses $\\gamma\\gamma \\to e^+ e^-$, $\\gamma\\gamma \\to\\mu^+ \\mu^-$.\nThis period of two-photon physics was summarized in review by\nBudnev, Ginzburg, Meledin and Serbo~\\cite{BGMS-75}.\n\nA few years later (approximately from 1977) it was shown in a number\nof theoretical papers that the two-photon processes are very\nconvenient for the test and detailed study of the Quantum\nChromodynamics (QCD) including investigation of\n\\begin{itemize}\n\\item a photon structure function (Witten \\cite{Witten}),\n\\item  a jet production in the $\\gamma \\gamma$ collisions\n(Llewelyn Smith \\cite{LS}; Brodsky, De Grand, Gunion and Weis\n\\cite{BGGW}; Baier, Kuraev and Fadin \\cite{BKF}),\n\\item  the $\\gamma \\gamma \\to c {\\bar c} c {\\bar c}$ process and\nthe problem of the perturbative Pomeron (Balitsky and Lipatov\n\\cite{BL}).\n\\end{itemize}\n\nA new wave of experimental activity in this field was initiated by\nthe experiment at SLAC~\\cite{ATel79} which demonstrated that\ntwo-photon processes can be successfully studied without detection\nof the scattered electrons and positrons. After that there was a\nflow of experimental data from almost all detectors at the $e^+e^-$\nstorage rings. It should be noted a special detector MD-1 in\nNovosibirsk with a transverse magnetic field in the interaction\nregion and system of registration of scattered at small angles\nelectrons which was developed for two-photon experiments (see review\n\\cite{MD1}). This period was reviewed by Kolanoski \\cite{Kol84}.\n\\begin{figure}[!h]\n\\centering\n\\includegraphics[width=12cm,angle=0]{savinov.eps}\n\\caption{Results for the process $\\gamma \\gamma \\to \\eta_c$}\n\\label{savinov}\n\\end{figure}\n\n\\section{Results obtained in virtual $\\gamma^* \\gamma^* $\ncollisions}\n\nIn experiments at $e^+ e^-$ storage rings a lot of interesting\nresults about $\\gamma^* \\gamma^* $ collisions have been obtained\n(see reviews \\cite{Kol84,Pope,MPW94} and Proceedings of Workshops on\nPhoton-Photon Collisions), among them:\n\\begin{itemize}\n\\item production of $C$-even resonances in $\\gamma ^* \\gamma ^*\n$ collisions, such as $\\pi ^0,\\; \\eta,$ $ \\eta ^{\\prime}, \\; f_2,\\;\na_2$, $ \\eta_c$, $\\chi_c$, ...  and measurement of their $\\gamma\n\\gamma$ width;\n\\item measurement of the total $\\gamma \\gamma \\to hadrons$\ncross section up to c.m.s. energy $W_{\\gamma \\gamma} $ about 150\nGeV;\n\\item measurement of the total $\\gamma ^* \\gamma ^* \\to hadrons$\ncross section with large values of $W^2_{\\gamma \\gamma} $ and photon\nvirtualities $-q^2_1 \\sim -q^2_2 \\sim 10$ (GeV\/c)$^2$;\n\\item a number of exclusive reactions: $\\gamma ^* \\gamma ^*\n\\to \\pi \\pi, \\; K \\bar K,\\; p \\bar p,\\; \\rho \\rho,\\; \\rho \\omega,$\netc.;\n\\item  investigation of the photon structure function in the\ncollision of almost real photon and highly virtual photon with\n$-q^2$ up to about 1000 (GeV\/c)$^2$;\n\\item jet production in $\\gamma \\gamma$ collisions.\n\\end{itemize}\nAs an example, let us present the beautiful result (Fig.\n\\ref{savinov}) for the study of two-photon process  $\\gamma \\gamma\n\\to \\eta_c$ at CLEO (Cornell).\n\nUnfortunately, the number of equivalent photons per one electron is\nrather small, and correspondingly the $\\gamma ^* \\gamma ^*$\nluminosity is about $3\\div 4$ orders of magnitude smaller than that\nin $e^+e^-$ collisions. Therefore, it is not surprising that the\nmost important results at $e^+e^-$ storage rings were obtained in\nthe $e^+e^-$ annihilation.\n\n\\section{Linear $e^{\\pm} e^-$ collider }\n\nNew opportunities for two-photon physics are connected with future\nlinear $e^{\\pm}e^-$ colliders. Projects of such accelerators are now\nunder development in several laboratories. A linear collider\nconsists of several main systems (see Fig. \\ref{schema-ses-ee}\nfrom~\\cite{Sessler}): electron injectors, pre-accelerators, a\npositron source, two damping rings, bunch compressors, main linacs,\ninteraction regions, a beam dump.\n\\begin{figure}[!h]\n\\begin{center}\n\\includegraphics[width=14cm,angle=0]{schema_ses_2.eps}\n\\end{center}\n \\caption{Schema of a linear $e^+e^-$ collider}\n  \\label{schema-ses-ee}\n \\end{figure}\n\nSince 1988 this field is developed in a very tight international\ncollaboration of physicists from many countries. In 1996-97 three\nprojects NLC (North America), JLC (Asia) and TESLA (Europe) have\npublished their Conceptual Design Reports~\\cite{Projects} of the\nlinear colliders in the energy range from a few hundred GeV to about\none TeV; in 2001 the TESLA Technical Design Report~\\cite{TESLA},\n\\cite{TESLA-04} has been published. One team at CERN is working now\non the conception of multi-TeV Compact Linear Collider (CLIC).\nCurrent parameters of these projects are presented in\nTable~\\ref{t2.1}. Parameters of the projects NLC and JLC are\npresented in one column since their teams developed a common set of\nthe collider parameters.\n\\begin{table}[!h]\n \\caption{Some parameters of the linear colliders  NLC\/JLC and TESLA}\n \\vspace{.2cm}\n\\renewcommand{\\arraystretch}{1} \\setlength{\\tabcolsep}{1.1mm}\n\\begin{center}\n\\begin{tabular}{lccc} \\hline &&NLC\/JLC&TESLA\n\\\\ \\hline \\hline\nC.m.s. energy $2E_0$& [TeV]& 0.5 & 0.5 \\\\\nLuminosity $L$& [$10^{34}$\/(cm$^{2}$s)]&  2.2 & 3 \\\\\nRepetition rate $f_r$ &[Hz]&  120 & 5 \\\\\nNo. bunch\/train $n_b$&  & 190& 2820\\\\\nNo. particles\/bunch $N_e$& [$10^{10}$] &  0.75 & 2 \\\\\nCollision rate $\\nu$& kHz &  22.8 & 14.1 \\\\\nBunch spacing $\\Delta t_b$& [ns] &  1.4 & 337  \\\\\nAccel. gradient $G$ & [MeV\/m]&  50 & $\\sim 25$\\\\\n{Linac length} $L_l$ & {[km]}& { 10} &{ 20} \\\\\nBeams power $2P_b$& [MW]& 14 & 22.5 \\\\\nIP beta-function $\\beta_x\/\\beta_y$& [mm] &   8\/0.1& 15\/0.4 \\\\\n{R.m.s. beam size at IP} $\\sigma_x\/\\sigma_y$ \\hspace{-5mm}\n& {[nm]}& {245\/2.7}& {555\/5} \\\\\nR.m.s. beam length $\\sigma_z$ & [$\\mu$]&  110&300 \\\\\n \\hline\n\\end{tabular}\n\\end{center}\n\\label{t2.1}\n\\end{table}\n\nNow the project of International Linear Collider (ILC) is under\ndevelopment.\n\nA few special words have to be said about luminosity of the linear\ncolliders, which is determined as\n  \\begin{equation}\nL=\\nu\\;{N_{e^+}\\, N_{e^-}\\over S_{\\rm eff}}\\,,\n \\end{equation}\nwhere the effective transverse bunch aria $S_{\\rm eff}\\sim \\sigma_x\n\\sigma_y$. In the next table we present a comparison of the storage\nring LEP-II and linear colliders:\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|c|c|c|}\n \\hline\nCollider & $ 2E_{e}$, &  $L$, $10^{33}$ & $\\nu$, & $N_{e^{\\pm}},$ &\n$\\sigma_x$, &$\\sigma_y$, \\\\ &  GeV &1\/(cm$^{2}$s) &\n kHz &  $10^{10}$ &\n$\\mu$ & nm\\\\  \\hline LEP-II & 200 & 0.05 & 45 & 30 & 200 & 8000\\\\\n\\hline NLC\/JLC & 500 & 22 & 22.8 & 0.75 & 0.245  & 2.7 \\\\\n\\hline TESLA   & 500 & 30  & 14.1   & 2    & 0,555 & 5 \\\\\n\\hline\n\\end{tabular}\n\\end{center}\nNote  transverse bunch sizes\n  $$\n{\\rm LEP-II}: \\;\\;\\;\\;\\;\\;\\sigma_x \\sigma_y \\;\\;\\sim \\;\\; 10^{-5}\n{\\rm cm}^2\\,,\n  $$\n $$\n{\\rm TESLA}: \\;\\; \\sigma_x \\sigma_y \\sim  3\\cdot 10^{-11} {\\rm\ncm}^2\\,.\n $$\n\n So, it is likely that a first linear collider  will have energy\nabout 500 GeV with some possible extension up to 1.5 TeV. Compared\nto the LEP, these colliders of the so called next generation are\ndesigned on 2.5--7 times higher energy and four orders of magnitude\nhigher luminosity!\n\n\n\\section{Photon collider on a base of a linear\n$e^{\\pm} e^-$ collider}\n\n\\subsection{Idea of high-energy $\\gamma\\gamma$ and\n$\\gamma e$ colliders with real photons}\n\n\nUnlike the situation in storage rings, {\\it in linear colliders each\n$e^{\\pm}$ bunch is used only once}. It makes possible to ``convert''\nelectrons to high-energy photons and to obtain the $\\gamma \\gamma$\nor $\\gamma e$ colliding beams with approximately the same energy and\nluminosity\\footnote{ This can not be done at usual $e^+e^-$ storage\nrings where a high luminosity is provided by large number of\ncollisions ($\\sim 10^9-10^{11} $) of the same $e^+$ and $e^-$\nbunches. Conversion of the electron and positron bunches into the\n$\\gamma $ bunches at the storage ring gives only a single collision\nof gamma bunches. The resulting luminosity will be very low because\nobtaining of new $e^+$ and $e^-$ bunches at storage rings takes long\ntime.} as in the basic $e^{\\pm}e^-$ collisions. Moreover, $\\gamma\n\\gamma$ luminosity may be even larger due to absence of some\ncollisions effects.\n\nThis idea was put forward by Novosibirsk group in 1981--1983\n(Ginzburg, Kotkin, Serbo and Telnov \\cite{GKST81a}--\\cite{GKST83})\nand was further developed in detail.\n\nAmong various methods of $e\\to \\gamma$ conversion (bremsstrahlung,\nondulator radiation, beamstrahlung and so on), the best one is the\nCompton scattering of laser light on high-energy electrons. In this\nmethod a laser photon is scattered backward taking from the\nhigh-energy electron a large fraction of its energy. The scattered\nphoton travels along the direction of the initial electron with an\nadditional angular spread $\\theta \\sim 1\/\\gamma_e$ where $\\gamma_e\n=E\/(m_e c^2)$ is the Lorentz factor of the electron. This method was\nknown long time ago~\\cite{ATM63} and has been realized in a number\nof experiments, e.g.~\\cite{FIAN64,SLAC69}. However, the conversion\ncoefficient of electrons to high-energy photons $k=N_\\gamma \/ N_e$\nwas very small in all these experiments. For example, in the SLAC\nexperiment \\cite{SLAC69} it was $\\sim 10^{-7}$.\n\nIn our papers \\cite{GKST81a,GKST83} it was shown that at future\nlinear $e^{\\pm}e^-$ colliders {\\it it will be possible to get}\n$k\\sim 1$ at a quite reasonable laser flash energy of a few Joules.\n\nTherefore, two principal facts, which make possible a photon\ncollider, are:\n\n\\begin{itemize}\n\n\\item linear colliders are single-pass accelerators, the\nelectron beams are used here only once;\n\n\\item obtaining of conversion coefficient $k\\sim 1$ is\ntechnically feasible.\n\n\\end{itemize}\n\nIt should be noted that {\\it positrons are not necessary for photon\ncolliders}, it is sufficient and much easier to use the\nelectron-electron colliding beams.\n\n\nThe problems of the $\\gamma \\gamma$ and $\\gamma e$ colliders were\ndiscussed on many conferences: Photon-Photon Collisions, Linear\nColliders, and dedicated $\\gamma \\gamma$ Workshops~\\cite{gg94,gg00}.\nVery rich physics, potentially higher than in $e^+e^-$ collisions\nluminosity, simplification of the collider (positrons are not\nrequired) are all attractive to physicists. Progress in development\nof linear $e^+e^-$ colliders and high power lasers (both\nconventional and free-electron lasers) makes it possible to consider\nphoton colliders as very perspective machines for investigation of\nelementary particles.\n\nThis option has been included in Conceptual~\\cite{Projects} and\nTechnical~\\cite{TESLA} Designs of linear colliders. All these\nprojects foresee the second interaction regions for the $\\gamma\n\\gamma$ and $\\gamma e$ collisions.\n\n\n\\subsection{Scheme of a photon collider}\n\n\nTo create a $\\gamma \\gamma$ or $\\gamma e$ collider with parameters\ncomparable to those in $e^+e^-$ colliders, the following\nrequirements should be fulfilled:\n\n\\begin{itemize}\n\\item the photon energy $\\omega \\approx E_0$ from $100$ GeV\nto several TeV;\n\\item the number of high-energy photons\n$N_\\gamma \\sim N_e \\sim 10^{10}$;\n\\item photon beams should be focused on the spot with transverse sizes\nclose to those which electron bunches would have at the interaction\npoint $\\sigma_x \\times \\sigma_y \\sim 10^{-5}$ cm $\\times 10^{-7}$\ncm.\n\\end{itemize}\n\n\\begin{figure}[!h]\n\\centering\n\\includegraphics[width=11cm,angle=0]{f3_2.eps}\n\\caption{A principal scheme of a photon collider. High-energy\nelectrons scatter on laser photons (in the conversion region C) and\nproduce high-energy $\\gamma$ beam which collides with similar\n$\\gamma$ or $e$ beam at the interaction point IP.} \\label{F3.2}\n\\end{figure}\n\nThe best solution for this task is to use a linear $e^{\\pm}e^-$\ncollider as a basis and convert the $e^{\\pm}$ beams into $\\gamma$\nbeams by means of the backward Compton\nscattering~\\cite{GKST81a}---\\cite{GKPST84}.\n\nThe principal scheme is shown in Fig.~\\ref{F3.2}. An electron beam\nafter the final focus system is traveling towards the interaction\npoint IP. At the distance $b\\, {\\sim} 0.1 \\div 1$ cm from the\ninteraction point, the electrons collide with the focused laser beam\nin the conversion region C. The scattered high-energy photons follow\nalong the initial electron trajectories (with small additional\nangular spread $\\sim 1\/\\gamma_e$), hence,  {\\bf the high-energy\nphotons are also focused at the interaction point IP}. This very\nfeature is one of the most attractive point in the discussed scheme.\nThe produced $\\gamma $ beam collides downstream with the oncoming\nelectron or a similar $\\gamma $ beam.~\\footnote{To reduce\nbackground, the ``used\" electrons can be deflected from the\ninteraction point by an external magnetic field. In the scheme\nwithout magnetic deflection background is somewhat higher, but such\nscheme is simpler and allows to get higher luminosity.}\n\nMore details about the conversion region are shown in Fig.\n\\ref{conversion_ses_2} from~\\cite{Sessler}.\n\\begin{figure}[!htb]\n\\centering\n\\includegraphics[width=14cm,angle=0]{conversion_ses_2.eps}\n\\caption{Conversion region}\n \\label{conversion_ses_2}\n\\end{figure}\n\nIt is very important that modern laser technology allows to convert\nmost of electrons to high-energy photons. This means that the\n$\\gamma \\gamma$ luminosity will be close to the luminosity of the\nbasic $e^{\\pm}e^-$ beams.\n\n\n\\subsection{Compton scattering as a basic process for $e\\to\n\\gamma$ conversion}\n\nProperties of the linear and non-linear Compton scattering are\nconsidered in details in~\\cite{GKPST84}, \\cite{KPS98}, \\cite{IKS04}.\nIn the conversion region a laser photon with energy $\\omega_{0} \\sim\n1$ eV scatters on an electron with energy $E_0 \\sim 100$ GeV at a\nsmall collision angle $\\alpha _0$ (Fig.~\\ref{F6.1}) and produces a\nfinal photon with the energy $\\omega$ and the emission angle\n$\\theta$. Kinematics of the backward Compton scattering\n \\begin{equation}\n  e(p_0) +\\gamma_0 (k_0) \\to e(p) +\\gamma (k)\n \\label{6.1}\n  \\end{equation}\nis characterized by two dimensionless  variables $x$ and $y$:\n\\begin{equation}\nx\\;={2p_0 k_0\\over m^2c^2}\\approx {4E_0\\omega_{0}\\over m^{2}\nc^4}\\cos ^{2} {{\\alpha_{0}\\over 2}}\\,, \\;\\;\\; y\\; = {kk_0\\over\np_0k_0}\\approx {\\omega \\over E_0}\\,.\n \\label{6.2}\n\\end{equation}\n\\begin{figure}[htb]\n\\begin{center}\n\\vspace{-0mm}\n\\hspace*{-1cm}\\includegraphics[width=7cm,angle=0]{f6_1.eps}\n\\vspace{-0mm} \\caption{Kinematics of the backward Compton\nscattering} \\label{F6.1} \\vspace{-2mm}\n\\end{center}\n\\end{figure}\nThe maximum energy of the scattered photon $\\omega_{m}$ and the\nmaximum value of the parameter $y$ are:\n\\begin{equation}\n\\omega \\leq \\omega_{m}\\;= \\; {x\\over x+1}\\, E_0\\,, \\;\\;\\; y\\leq \\;\ny_m\\; =\\; {x\\over x+1} = {\\omega_{m}\\over E_0}\\;\\,. \\label{6.7}\n\\end{equation}\nThe energy of the scattered electron is $E= (1-y)E_0$ and its\nminimum value is $E_{\\min} = E_0\/(x+1)$.\n\nTypical example: in collision of the photon with $\\omega_0 = 1.17$\neV ($\\lambda =1.06\\;\\mu$m --- the region of the most powerful solid\nstate lasers) and the electron with $E_0= 250$ GeV, the parameter\n$x=4.5$ and the maximum photon energy\n  \\begin{equation}\n\\omega_m =0.82 E_0 = 205\\;\\;{\\rm GeV}\n \\end{equation}\nis close enough to the initial electron energy $E_0$.\n\nA photon emission angle is very small, $\\theta\\sim 1\/\\gamma_e =\n2\\cdot 10^{-6}$.\n\nThe total Compton cross section is\n $$\n\\sigma_{\\mathrm c}\\;=\\;\\sigma^{\\mathrm{up}}_{\\mathrm c}\\;+\n\\;2\\lambda_{e} P_{\\mathrm c} \\,\\tau_{\\mathrm c}\\, ,\n $$\n where $\\lambda_{e} $ is the mean helicity of the initial\nelectron, $P_c $ is that of the laser photon and\n \\begin{eqnarray}\n\\sigma^{\\mathrm{up}}_{{\\mathrm c}}&=&{2\\sigma_{0}\\over x}\n\\left[\\left(1-{4\\over x} - {8\\over x^{2}}\\right)\\ln (x+1) +{1\\over\n2} +{8\\over x} - {1\\over 2(x+1)^{2}}\\right] \\, ,\n \\label{6.8}\n \\\\\n \\tau_{\\mathrm c}&=& {2\\sigma_{0}\\over x} \\left[\\left(1+{2\\over\nx}\\right) \\ln (x+1)-{5\\over 2}+{1\\over x+1}-{1\\over\n2(x+1)^{2}}\\right]\\,,\n  \\nonumber\\\\\n  \\sigma_{0}&=& \\pi r_e^2=2.5\\cdot 10^{-25}\\;\\mbox {cm} ^{2}\\,.\n  \\nonumber\n \\end{eqnarray}\nNote, that polarizations of initial beams influence the total cross\nsection (as well as the spectrum) only if both their helicities are\nnonzero, i.e. at $\\lambda_{e} P_{\\mathrm c} \\neq 0$. The\nfunctions $\\sigma^{\\mathrm{up}}_{{\\mathrm c}}$, corresponding to the\ncross section of unpolarized beams, and $\\tau_{{\\mathrm c}}$,\ndetermining the spin asymmetry, are shown in Fig.~\\ref{F6.2}.\n\\begin{figure}[htb]\n\\begin{center}\n\\vspace{-9mm}\n\\includegraphics[width=10cm,angle=0]{f6_2.eps}\n\\vspace{-8mm} \\caption{Cross section $\\sigma^{\\mathrm{up}}_{{\\mathrm\nc}}$ for unpolarized photons and $\\tau_{{\\mathrm c}}$, related to\nspin asymmetry (see (\\ref{6.8}))}\n \\label{F6.2}\n\\end{center}\n\\end{figure}\nIn the region of interest $x= 1 \\div 5$ the  total  cross section is\nlarge enough\n \\begin{equation}\n\\sigma_{\\mathrm c}\\sim \\sigma_0=2.5\\cdot 10^{-25}\\;\\;{\\rm cm}^2\n \\label{6.12}\n \\end{equation}\nand only slightly depends  on  the  polarization of the initial\nparticles,  $|\\tau_{\\mathrm c}|\/\\sigma^{\\mathrm{up}}_{\\mathrm c}<\n0.1$.\n\nOn the contrary, the energy spectrum does essentially depend on the\nvalue of $\\lambda_{e} P_{\\mathrm c}$. The energy spectrum of\nscattered photons is defined by the differential Compton cross\nsection:\n \\begin{equation}\n{d\\sigma _{{\\mathrm c}}\\over dy} ={2\\sigma_{0}\\over x} \\left[{1\\over\n1-y} + 1-y - 4r(1-r) - 2\\lambda_{e} P_c\\;{y(2-y)\\over\n1-y}\\,(2r-1)\\right]\\,.\n \\label{6.13}\n \\end{equation}\nIt is useful to note that $r=y\/[x(1-y)] \\leq 1$ and $r\\to 1$ at\n$y\\to y_m$.\n\\begin{figure}[!h]\n\\begin{center}\n\\vspace{-8mm}\n\\includegraphics[width=10cm,angle=0]{f6_3.eps}\n\\vspace{-8mm} \\caption{Energy spectrum of scattering photons at\n$x=4.8$} \\label{F6.3}\n\\end{center}\n\\end{figure}\n\n\\begin{figure}[!h]\n\\begin{center}\n\\vspace{-5mm}\n\\hspace*{-0.5cm}\\includegraphics[width=7cm,angle=0]{f6_4a.eps}\\hspace*{-0.7cm}\n\\includegraphics[width=7cm,angle=0]{f6_4b.eps}\n\\vspace{-5mm}\n \\caption{Energy spectrum of scattering photons at\n$x=1$ (left figure) and $x=20$ (right figure)}\n \\label{F6.4}\n \\vspace{-2mm}\n\\end{center}\n\\end{figure}\nThe ``quality'' of the photon beam, i.e. the relative number of hard\nphotons, is better for the negative value of $\\lambda_{e}\nP_{\\mathrm c}$. For $2\\lambda_{e} P_c=-1$ the peak value of the\nspectrum at $\\omega =\\omega_{m}$ nearly doubles improving\nsignificantly the monochromaticity of the $\\gamma$ beam (cf. curves\n$a$ and $b$ in Fig.~\\ref{F6.3}).\n\nIn order to increase the maximum photon energy, one should use the\nlaser with larger frequency. This also increases a fraction of hard\nphotons (cf. Figs.~\\ref{F6.4}  and \\ref{F6.3}). Unfortunately, at\nlarge $x$ the high energy photons disappear from the beam producing\n$e^{+} e^{-}$ pairs in collisions with laser photons. The threshold\nof this reaction, $\\gamma\\gamma_L\\to e^+e^-$, corresponds to $x\n\\approx 4.8$. Therefore, it seems that the value $x\\approx 5$ is the\nmost preferable.\n\n\\subsubsection{Angular distribution}\n\nThe energy of a scattered photon depends on its emission angle\n$\\theta $ as follows:\n \\begin {equation}\n\\omega ={\\omega _m \\over 1+(\\theta \/\\theta_0)^2}; \\quad \\theta_{0}\n\\; = \\; {m c^2 \\over E_0}\\; \\sqrt{x+1}\\,.\n \\label{6.16}\n \\end{equation}\nNote, that photons with the maximum energy $\\omega_m$ scatter at\nzero angle. The angular distribution of scattered protons has a very\nsharp peak in the  direction of the incident electron momentum.\n\nAfter the Compton scattering, both electrons and photons travel\nessentially along the original electron beam direction. The photon\nand electron scattering angles ($\\theta$ and $\\theta_e$) are unique\nfunction of the photon energy:\n \\begin {equation}\n\\theta (y)=\\theta_0\\; \\sqrt{{y_m\\over y}-1}, \\;\\;\\; \\;\n\\theta_e={\\theta_0\\; \\sqrt{y(y_m-y)} \\over 1-y}\\; .\n \\label{6.21}\n \\end {equation}\n\\begin{figure}[!h]\n\\begin{center}\n\\vspace{-4mm}\n\\includegraphics[width=8cm,angle=0]{f6_5.eps}\n\\vspace{-6mm} \\caption{ Photon and electron scattering angles vs the\nphoton energy $\\omega$  at $x=4.8$} \\label{F6.5} \\vspace{-6mm}\n\\end{center}\n\\end{figure}\nFor $x=4.8$ these  functions are plotted in Fig.~\\ref{F6.5}. It is\nremarkable that electrons are only slightly deflected from their\noriginal direction and scatter into a narrow cone:\n\\begin {equation}\n\\theta_e \\leq {x \\over 2\\gamma} = {2\\omega _0 \\over m c^2} .\n\\label{6.22}\n\\end {equation}\n\n\\subsubsection{Polarization of final photons}\n\nUsing the polarized initial electrons and laser photons, one can\nobtain the high-energy photons with various polarization. Let us\npresent two typical examples.\n\n\\begin{figure}[!h]\n\\begin{center}\n\\vspace{-8mm}\n\\includegraphics[width=9cm,angle=0]{f6_6.eps}\n\\vspace{-5mm} \\caption{Mean helicity of the scattered photons\n$\\lambda_\\gamma$ vs. $\\omega \/E_0$ for various laser photon\nhelicities $P_c$ and electron helicities $\\lambda _e$ at $x= 4.8$ }\n\\label{F6.6} \\vspace{-2mm}\n\\end{center}\n\\end{figure}\nThe mean helicity of the final photon $\\lambda_\\gamma$ in dependence\non the final photon energy $\\omega$ is shown in Figs.~\\ref{F6.6}.\nCurves $a,\\; b$ and $c$ in Fig.~\\ref{F6.6} correspond to spectra\n$a,\\; b$ and $c$ in Fig.~\\ref{F6.3}. In the case $2\\lambda_e\nP_c = -1$ (the case of good monochromaticity -- see curves $a$ in\nFig.~\\ref{F6.3}) almost all high-energy photons have a high degree\nof circular polarization. In the case $b$ the electrons are\nunpolarized, and the region, where high-energy photons have a high\ndegree of polarization, is much narrow.\n\n\\begin{figure}[!h]\n\\begin{center}\n\\vspace{-8mm}\n\\includegraphics[width=9cm,angle=0]{f6_8.eps}\n\\vspace{-5mm}\n \\caption{Average linear polarization of the scattered\nphotons $l_\\gamma$  vs. $\\omega\/E_0$ at $x=1,\\; 2,\\; 3$ and $4.8$\n(the degree of linear polarization of the laser photon $P_l = 1$)}\n \\label{F6.8}\n\\end{center}\n\\end{figure}\nIf the  laser light has a linear polarization, then high-energy\nphotons are polarized in the same direction. The average degree of\nthe linear polarization of the final photon $l_\\gamma$ in dependence\non the final photon energy $\\omega$ is shown in Figs.~\\ref{F6.8}.\nThe linear polarization of the colliding photons is very important\nfor study of the nature of Higgs boson.\n\n\\section{Physics of $\\gamma\\gamma$ Interactions }\n\n\n\\begin{figure}[!ht]\n\\centering\n\\includegraphics[width=14cm,angle=0]{all_cs.eps}\n \\caption{Cross sections for some interesting processes in\n $e^+e^-$, $\\gamma e$ and $\\gamma \\gamma$ collisions}\n \\label{all-cross}\n \\end{figure}\n\nPhysical potential of such $\\gamma \\gamma$ and $\\gamma e$ colliders\nwill be on the same level with future $e^+e^-$ and $pp$ colliders.\nMoreover, there is a number of problems in which photon colliders\nare beyond competition. The comparison of cross sections in the\n$e^+e^-$, $\\gamma \\gamma$ and $\\gamma e$ colliders are given in\nFig.~\\ref{all-cross}.\n\nPhoton collider (PC) makes it possible to investigate both problems\nof new physics and of ``classical\" hadron physics and QCD.\n\nSince photon couple directly to all fundamental charged particles\n--- leptons, quarks, $W$ bosons, super-symmetric particles, etc.\n--- a PC can provide a possibility to test every aspect of the\nStandard Model (SM) and beyond.\n\nBesides, photons can couple to neutral particles (gluons, $Z$\nbosons,  Higgs bosons, etc.) through charged particles box diagrams\n(Fig. \\ref{gg-to-higgs}).\n \\begin{figure}[!ht]\n \\centering\n \\includegraphics[width=7cm,angle=0]{gg-to-higgs.eps}\n  \\caption{Higgs boson production in $\\gamma \\gamma$ collision}\n   \\label{gg-to-higgs}\n  \\end{figure}\n\nOn the other hand, in a number of aspects photons are similar to\nhadrons, but with simpler initial state. Therefor, PC will be\nperfect in studying of QCD and other problems of hadron physics.\n\nLet us list the problems in which the photon colliders have a high\npotential or some advantages:\n\n{\\it Higgs hunting and investigation}. PC provides the opportunity\nto observe the Higgs boson at the smallest energy as a resonance in\nthe $\\gamma \\gamma$ system, moreover, PC is out of competition in\nthe testing of Higgs nature.\n\n\\begin{figure}[!t]\n\\centering\n\\includegraphics[width=15cm,angle=0]{ggtoSS.eps}\n \\caption{Cross sections for charged pair production in\n $e^+e^-$ and $\\gamma \\gamma$ collisions: here $S$ -- scalars, $F$ -- fermions,\n$\\sigma= (\\pi \\alpha^2\/M^2)\\, f(x)$}\n \\label{ggtoSS}\n\\end{figure}\n\n{\\it Beyond SM}. PC provides the excellent opportunities for finding\nof various particles beyond the SM: SUSY partners, charged Higgses,\nexcited leptons and quarks, leptoquarks,... In particular, $\\gamma\ne$ collider will be the best machines for discovery of selectron or\nexcited electron. Cross sections for the charged pair production in\n$\\gamma \\gamma$ collisions are larger than in $e^+e^-$ collisions\n--- see Fig. \\ref{ggtoSS}.\n\n{\\it Electroweak gauge boson physics}. The electroweak theory is the\nsubstantial part of the SM which pretend for precise description\nlike QED. PC will be $W$ factories with a rate about $10^7\\; W$\nbosons per year. In addition, the $\\gamma e \\to W \\nu$ process will\nproduce  single $W$s which is very attractive for $W$ decay's study.\nThus, PCs provide one of the best opportunity to test the precise\npredictions of the electroweak theory.\n\n{\\it QCD and hadron physics}. The photon colliders provide the\nunique possibility to investigate the problems of hadron physics and\nQCD in the new type of collisions and with the simplest structure of\ninitial state. The principal topics here are the following:\n\\begin{itemize}\n\\item{the $t \\bar t $ production in different partial waves;}\n\\item{the photon structure functions;}\n\\item{the semihard processes;}\n\\item{the jet production;}\n\\item{the total $\\gamma \\gamma \\to hadrons$ cross section.}\n\\end{itemize}\n\nTo clarify many of these points it will be very useful  to compare\nresults from $\\gamma \\gamma,\\; \\gamma e,\\; ep$ and $pp$ colliders.\n\nBesides the high-energy $\\gamma \\gamma $ and $\\gamma e$ collisions,\nPCs provide {\\it some additional options}:\n\n{\\it (i)} The region of conversion $e\\to \\gamma$ can be treated as\n$e\\gamma_L$ collider (here $\\gamma _L$ is the laser photon) with\nc.m.s. energy $\\sim 1 $ MeV but with enormous luminosity $\\sim\n10^{38}\\div10^{39}$ cm$^{-2}$s$^{-1}$.  It can be used, for example,\nfor search of weakly interacting light particles, like invisible\naxion.\n\n{\\it (ii)} In the conversion region one can test non-linear QED\nprocesses, like the $e^+e^-$ pair production in collision of\nhigh-energy photon with a few laser photons.\n\n{\\it (iii)} The used high-energy photon beams can be utilized for\nfixed-target experiments.\n\n\n\\section{Concluding remarks}\n\n\\subsection{Summary from TESLA TDR}\n\nThe TESLA Technical Design Report~\\cite{TESLA} contains the part\ndevoted to the high-energy photon collider~\\cite{TESLA-04}. It will\nbe useful to cite the  Physics Summary from this part:\n\n \\begin{quotation}\n\nTo summarize, the Photon Collider will allow us to study the physics\nof the electroweak symmetry breaking in both the weak-coupling and\nstrong-coupling scenario.\n\n Measurements of the two-photon Higgs width of the $h,\\,H$ and $A$\nHiggs states provide a strong physics motivation for developing the\ntechnology of the $\\gamma \\gamma$ collider option.\n\nPolarized photon beams, large cross sections and sufficiently large\nluminosities allow to significantly enhance the discovery limits of\nmany new particles in SUSY and other extensions of the Standard\nModel.\n\nMoreover, they will substantially improve the accuracy of the\nprecision measurements of anomalous W boson and top quark couplings,\nthereby complementing and improving the measurements at the $e^+e^-$\nmode of TESLA.\n\nPhoton colliders offer a unique possibility for probing the photon\nstructure and the QCD Pomeron.\n\n \\end{quotation}\n\n\\subsection{Prediction of Andrew Sessler~\\cite{Sessler} in 1998 }\n\n\\begin{quotation}\n\nAt present, Europe has the lead in electron colliders (LEP), hadron\ncolliders (LHC) and hadron-electron colliders(HERA). Stanford and\nJapan's High Energy Research Organization (KEK) are jointly working\non a TeV $e^+e^-$ collider disign, as in DESY. Japan and\/or Germany\nseem to be the most likely location for the next-generation $e^+e^-$\nmachine. Looking broadly, and also contemplating what US will do in\nhigh-energy physics, one may imagine a $\\mu^+\\mu^-$ collider in the\nUS, early in the next century.\n\n\\end{quotation}\n\n\\subsection{Conclusion of Karl von Weizs\\\"acker for young physicists}\n\nI would like to tell you a little real story. In 1991 the First\nInternational Conference on Physics devoted to Andrej Sakharov held\nin Moscow. A number of great man have participated in this\nConference including a dozen of Nobel prize winners. Among others\nwas Karl von Weizs\\\"acker.\n\nI personally was very interesting in application of the\nWeizs\\\"acker-Williams method to the two-photon processes at\ncolliding beams and I even have published a few articles on this\nsubject. So, I would like to see and to speak with the author of\nthis method. At the beginning of our conversation, Weizs\\\"acker told\nme a history of this invention.\n\nIn 1934 Weizs\\\"acker was in Copenhagen as an assistant of Prof.\nNiels Bohr. And just at that moment there was some international\nConference on Physics in the N.~Bohr Institute. Williams was the\nfirst who suggested the idea of the equivalent photon approximation\nfor QED processes. But the final result appeared only after a lot of\nheat discussions in which Weizs\\\"acker, Williams, Niels Bohr, Lev\nLandau, Edward Teller and some others participated.\n\nAfter the Conference all people, but Weizs\\\"acker, went back to\ntheir home institutes. So, it was quite natural that some day\nN.~Bohr invited  Weizs\\\"acker and said: ``And you, young man, you\nshould write a paper on the discussed subject''.\n\nAnd  Weizs\\\"acker did and brought the paper to Bohr.  A few days\nafter that Bohr invited him to discuss the work. ``At the beginning\nof this meeting, --- Weizs\\\"acker told me, --- I was young and\nexited, but Professor seemed to me was old and tired''.\n\nBohr said:``It is an excelent work, a very clean and perfect paper,\nbut... I have a small remark about the second page''. So they have\ndiscussed this small remark. After that they have discussed another\nremark, and another remark, and another...\n\n``After four or five hours of discussion. ---  Weizs\\\"acker\ncontinued, --- I was young and tired, but Professor was old and\nexcited''. At the end Bohr said:``Now I see that you wrote quite\ncontrary to what you think, you should rewrite your paper''.\n\n Weizs\\\"acker's conclusion was: ``I think that it is the best way for a\n young scientist to study physics: you should have a good problem\n and a possibility to discuss this problem during hours with a great man''.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{sec:1}\n In a typical double-slit experiment interference fringes are formed\non a screen placed behind the slits which are then traversed by\nparticles of suitable wavelength.  By blocking one of the slits, the\nspatial interference pattern disappears.\n In a recent wave-vector space double slit experiment\n\\cite{pursehousedynamic} a photo-electron wave packet receding from\na single atom   is forced to pass through two energy levels within the atom, so that the levels play the role of the double-slit. As illustrated  in\nFig.\\,\\ref{fig0}, this is achieved by two low-intensity\ncontinuous-wave lasers which  excite the $5p$ and $6p$ states of a\nRubidium atom.  The infrared light field can concurrently ionize the\n$6p$ state and the blue laser the $5p$ state.   We image the interference in\nwave-vector space by scanning the photoelectron angular distribution.\nThe interference pattern disappears if only one state is excited,\ndemonstrating  that the phase relationship between the interfering\nwaves is imprinted by the atom. The two-lasers are not phase locked.\n\nThe $5p$ and $6p$ states thus represent  \"slits\", which can be closed by\ndetuning the respective laser fields, leading to non-resonant\nexcitation and damped occupation  of the specific intermediate state.\nThis \"damping\" is however different from thermal damping,\nas it does not swiftly destroy the coherence. The interference in\nthis case is affected because the two interfering amplitudes\nhave largely different strengths.\n\nIn the experiment the interference term and the associated\n\\begin{figure}[t]\n\\includegraphics[width=1.0\\columnwidth]{Fig0}\n\\caption{(a) Schematic representation of the atomic \"double slit\nexperiment\" in Rubidium. The slits are represented by the bound and\ninitially unoccupied $5p$ and $6p$ intermediate states which are\nresonantly excited by the infrared and blue laser pulses. The two\nlasers subsequently ionize the intermediate levels, giving rise to two\nionization pathways $t_1$ and $t_2$. The photoelectron is transferred\ninto the continuum with a finite kinetic energy $E_f$ without knowing\nwhich pathway was taken. (b) Closing one slit by detuning one of the\ntwo lasers which then inhibits the occupation of the respective\nintermediate state (in this example the $6p$ state). We hence\nobtain a conventional two-color photoionization process via a single\nionization pathway. The energy level of the final state is now shifted down due\nto the detuning decreasing effectively $\\varepsilon_f=\\hbar\\omega_{\\rm BL}+\\hbar\\omega_{\\rm IR}+\\varepsilon_{5s}$.}\n\\label{fig0}\n\\end{figure}\ndifference in the phases of the amplitudes  were  recorded by\nessentially three measurements. First, both laser fields were set resonant\nto the $5p$ and $6p$ dipolar excitations leading to an ionization\namplitude $t_1+t_2$ [cf.\\,\\ref{fig0}(a)]. In the second and third\nmeasurements, one of the two laser fields was  detuned to\nclose one of the ionization pathways [cf.\\,\\ref{fig0}(b)] to extract\nthe individual amplitudes $t_1 (t_2)$. The ground state energy of the\n$5s$ Rb state is -4.177\\,eV while $E_{6p}=-1.589\\,$eV and\n$E_{5p}=-2.950\\,$eV. Thus, no other states in the bound spectrum of\nRb are accessible by single-photon processes.  An interesting point is\nthat in the experiment the two \\emph{cw} (continuous wave) lasers (cf. Fig.\\ref{fig0}) were\nnot phase locked and were weak. The calculations show that a\nrandom phase between these two lasers does not affect the\ninterference.  Thus the observed interference has to stem from the\natom, while the photoelectron wave is propagating out to the\ndetectors at infinity.\n\nThe question we are considering here is how the interference\nbehaves when the number of photons (the laser intensity) increases,\nand hence one would expect an increase in the probability that a\nblue and red photon are absorbed at the same time. The same process applies when\nwe consider  much shorter pulses. In this case one would expect the\nphase relation between the laser pulses to be important, and it then becomes possible\nto access the time scale on which the interference pattern  builds up and evolves.\n Unfortunately, due to experimental limitations we are currently not in a\nposition to investigate these ideas in the laboratory, and therefore the\ncurrent work is mostly theoretical.\n\nIn the final sections of this paper we discuss mechanisms for\ncontrolling and manipulating the interference phenomena between both\nionization pathways. Atomic units are used throughout the paper.\n\n\n\\section{Propagation on a space-time grid}\n\\label{sec:2}\n\nWithin the single-particle picture the Rubidium atom is well\ndescribed by the angular-dependent model potential introduced by\nMarinescu \\emph{et al.} \\cite{marinescu1994dispersion}:\n\\begin{equation}\nV_\\ell(r)=-\\frac{Z_\\ell(r)}{r} -\n\\frac{\\alpha_c}{2r^4}\\left[1-e^{-(r\/r_c)^6}\\right],\n\\end{equation}\nwhere $\\alpha_c$ is the static dipole polarizability of the\npositive-ion core and the effective radial charge $Z_\\ell(r)$ is\ngiven by\n\\begin{equation}\nZ_\\ell(r)=1+(z-1)e^{-a_1r}-r(a_3+a_4r)e^{-a_2r},\n\\end{equation}\nwith the nuclear charge $z$ and the cut-off parameter $r_c$ as well\nas the parameters $a_1-a_4$ fitted to experimental values. The\noptimized parameters are tabulated in\nRef.\\,\\cite{marinescu1994dispersion}.\n\nOn a very fine space grid ($\\Delta r=0.005$\\,a.u.) the radial wave\nfunctions of the atomic eigenstates\n${\\langle\\pmb{r}|\\phi_i\\rangle=R_{n_i,\\ell_i}(r)Y_{\\ell_i,m_i}(\\Omega_{\\pmb{r}})}$\nare found by (numerical) diagonalization of the matrix corresponding\nto the time-independent Hamiltonian\n$\\hat{H}_0=-(1\/2)\\partial_r^2+{\\ell(\\ell+1)\/(2r^2)} + V_\\ell(r)$. In\nthe presence of solenoidal and moderately intense electromagnetic\nfields the light-matter interaction Hamiltonian is given by $\\hat{H}_{\\rm\nint}(t)=-\\pmb{A}(\\pmb{r},t)\\cdot\\hat{\\pmb{p}}$ where\n$\\pmb{A}(\\pmb{r},t)$ is the vector potential and $\\hat{\\pmb{p}}$ is\nthe momentum operator.\n\nTo account for all multi-photon and multipole effects, a numerical\npropagation of the ground state wave function in the external vector\npotential is necessary, e.g. by the matrix iteration scheme\n\\cite{nurhuda1999numerical}. Exploiting the spherical symmetry of the\natomic system, the time-dependent wave function is decomposed in\nspherical harmonics, i.e. $\\Psi(\\pmb{r},t)=\\sum_{\\ell,m}^{\\ell_{\\rm\nmax}}b_{\\ell,m}(r,t)Y_{\\ell,m}(\\Omega_{\\pmb{r}})$. We therefore have to\npropagate $(\\ell_{\\rm max} + 1)^2$ channel functions\n$b_{\\ell,m}(r,t)$ which are coupled through the corresponding\n(dipole) matrix elements\n$\\langle\\ell'm'|\\pmb{A}\\cdot\\hat{\\pmb{p}}|m\\ell\\rangle$. Initially,\nthe ground state channel is fully occupied by the 5s Rubidium\norbital, i.e. ${\\Psi(\\pmb{r},t\\rightarrow-\\infty)=\n\\langle\\pmb{r}|\\phi_{5s}\\rangle}$ meaning\n${b_{0,0}(r,t-\\infty)=R_{n_i=5,\\ell_i=0}(r)}$. Introducing a time\n$T_{\\rm obs}$ where the external electromagnetic field perturbation\nis off, the wave function $\\Psi(\\pmb{r},t)$ is propagated to a\ntime $t>T_{\\rm obs}$ where the photoelectron wave packet is fully\nformed. The radial grid is extended to $10^4$\\,a.u. to avoid\nnonphysical reflections at the boundaries. Additionally, we\nimplemented absorbing boundary conditions by using an imaginary\npotential. The resulting simulation shows that the\nelectron density at the final grid point $r_N$ is then smaller than the\nnumerical error at the considered propagation times. \\\\\nTo obtain the scattering properties of the liberated electron, we\nproject  $\\Psi(\\pmb{r},T_{\\rm obs})$ onto a set of continuum wave\nfunctions which are given by the partial wave decomposition:\n\\begin{equation}\n\\langle\\pmb{r}|\\varphi_{\\pmb{k}}^{(-)}\\rangle=\\sum_{\\ell,m}i^\\ell\nR_{k\\ell}(r)e^{-i\\delta_\\ell(k)}\nY^*_{\\ell,m}(\\Omega_{\\pmb{k}})Y_{\\ell,m}(\\Omega_{\\pmb{r}}).\n\\label{eq:continuum}\n\\end{equation}\nHere, the kinetic energy is defined by $E_k = k^2\/2$,\n$\\delta_\\ell(k)$ are the scattering phases and $R_{k\\ell}(r)$ are\nradial wave functions satisfying the stationary radial Schr\u00f6dinger\nequation for positive energies in the same pseudopotential\n$V_\\ell(r)$ which is used for obtaining the bound spectrum. The\nscattering phases $\\delta_\\ell(k)={\\rm\narg}[\\Gamma(1+\\ell-i\/k)]+\\eta_\\ell(k)$ consist of the well-known\nCoulomb phases (first term) and phase $\\eta_\\ell(k)$ characterizing\nthe atomic-specific short-ranged deviation from the Coulomb\npotential. The radial wave functions are normalized to $\\langle\nR_{k\\ell}|R_{k'\\ell}\\rangle= \\delta(E_k-E_{k'})$. Finally, the\nprojection coefficients are given by\n\\begin{equation}\n\\begin{split}\na_{\\ell,m}(k) =&\\,e^{i(E_kT_{\\rm obs} + \\delta_\\ell(k) - \\ell\\pi\/2)}\n\\\\\n&\\times\\int_{r>r_a}{\\rm d}r\\,b_{\\ell,m}(r,T_{\\rm obs})R_{k\\ell}(r).\n\\end{split}\n\\end{equation}\nHere, we introduce the core radius $r_a$ and ensure that the\nintegration region is outside the residual ion. The photoionization probability (differential cross\nsection, abbreviated as DCS in the following) is defined as\n\\begin{equation}\n\\begin{split}\n{\\rm DCS} = \\frac{{\\rm d}\\sigma}{{\\rm d}\\Omega_{\\pmb{k}}}(E_k,\\Omega_{\\pmb{k}})\n\\propto &\\sum_{\\ell,\\ell'}\\sum_{m,m'} a^*_{\\ell',m'}(k)a_{\\ell,m}(k)\n\\\\ &\\times\nY^*_{\\ell',m'}(\\Omega_{\\pmb{k}})Y_{\\ell,m}(\\Omega_{\\pmb{k}}).\n\\end{split}\n\\end{equation}\nwhile the total cross section is\n$\\sigma(E_k)\\propto\\sum_{\\ell,m}\\sigma_{\\ell,m}(E_k)$ with\n$\\sigma_{\\ell,m}(E_k)=|a_{\\ell,m}(k)|^2$. Note that this treatment\ngives explicit insight into the population of the individual angular\nchannels and their contributions to the photoelectron wave packet.\n\nIn the following two-color ionization of Rubidium, the electric\nfields of both pulses are modeled according to\n\\begin{equation}\nE_p(t) =\n\\epsilon_p\\mathcal{E}_p\\Omega(t+\\Delta_p)\\cos(\\omega_p(t+\\Delta_p)+\\phi_p)\n\\end{equation}\nwith $p = {\\rm IR,BL}$ standing for the infrared and blue laser\npulses, respectively. The polarization vectors are $\\epsilon_p$, the\ntemporal envelope is given by $\\Omega(t)=\\cos^2(\\pi t\/T_d^p)$ for\n$t\\in[-T_d^p\/2,T_d^p\/2]$ with the pulse duration $T_d^p=2\\pi\nn_p\/\\omega_p$\ndetermined by the number of optical cycles $n_p$. Further we\nintroduce a temporal difference $\\Delta_p$ and a phase difference\n$\\phi_p$ to account for both laser fields originating from\ndifferent sources, and which are hence not phase-locked. Without loss of\ngenerality we set $\\phi_{\\rm IR}=0$ and $\\Delta_{\\rm IR}=0$. Both\nlaser fields are assumed to be linearly polarized in the $z$-direction\nso that the\nazimuthal angular quantum number $m$ is conserved. The number\nof angular channels then reduces to $\\ell_{\\rm max}+1$ and in the\nfollowing treatment we omit the subscript $m$ for brevity.\n\\begin{figure}[t]\n\\includegraphics[width=1.0\\columnwidth]{Fig1}\n\\caption{Occupation numbers of the $5p$ and $6p$ intermediate states\nafter laser excitation, which depend on the time delay $\\Delta$ and\nthe phase difference  between both pulses. (a) shows the results of\nthe numerical propagation for six optical cycles. (b) corresponds to\nten optical cycles. Dots indicate the results for $\\phi=0$ radians and\ncrosses belong to $\\phi=2\/3\\pi$.}\n\\label{fig1}\n\\end{figure}\nTo balance the differences in the oscillator strengths between the\n$5s\\rightarrow5p$ and $5s\\rightarrow6p$ channels we used the field strengths\n$\\mathcal{E}_{\\rm BL}=0.05$ a.u. and $\\mathcal{E}_{\\rm IR}=0.007$\na.u. in the following simulations.\n\nThe aim here is to determine the influence of the phase\ndifferences on the resulting photoionization and occupation\nprobabilities. In Fig.\\,\\ref{fig1} we present the occupation numbers\nof the intermediate $5p$ and $6p$ states for two different laser\nconfigurations at a time $T_{\\rm obs}$ after laser excitation, meaning both\nlight pulses are completely extinguished and the photoelectron wave packet\npropagates freely in the Coulomb field of the residual Rubidium ion.\nPanel (a) corresponds to the case where both laser pulses have a\nlength of six optical cycles. We see that both occupation numbers\nshow a strong dependence on the temporal difference $\\Delta$ as well\nas on a random phase difference $\\phi$. The dots belong to $\\phi=0$\nwhile crosses indicate the results for a non-zero phase difference.\nHere, we show the occupation numbers for $\\phi=2\/3\\pi$, which show a\nlarge difference to the case of $\\phi=0$. In addition, we repeated\nthe simulation for other numbers of the random phase difference and\nobtained similarly pronounced discrepancies. The situation changes\ncompletely when increasing the number of optical cycles as shown in\npanel (b). For ten optical cycles the influence of both\nthe temporal and phase differences on the resulting occupation\nnumbers is drastically reduced, pointing to the transition into the\n\\emph{cw} limit.\n\nAlready from the \\emph{bounded} properties we suspect the rather fast\nconvergence of the photoionization process into a description\nwithin the frequency domain, which is characterized by infinitely long\nlaser pulses characterised by delta distribution-like bandwidths. We can\nunderline this observation by looking at the characteristics of the\nejected photoelectron wave packet. In Fig.\\,\\ref{fig2}(a) we present\nthe ionization probability of the angular channel $\\ell=2$\ncharacterized by the partial cross section\n$\\sigma_2=\\left|a_{\\ell=2}(k)\\right|^2$. As expected, the probability\ncurve sharpens under an increase in the pulse lengths of both laser\nfields. While for $n_p=6$ a manifold of energy states in the\ncontinuum is excited, from $n_p=10$ upwards we see clearly the\nunfolding of a Gaussian-like peak around the final energy\n$E_f=E_{5s}+\\hbar\\omega_{\\rm IR} + \\hbar\\omega_{\\rm BL}$. In the case\nof 35 optical cycles the FHWM of the probability peak is around\n0.07\\,eV.\n\nIt is also interesting to study the evolution of the quantum phase associated with\nthe photoelectron wave packet, which can be expressed as\n$\\varphi(k,\\vartheta_k)={\\rm\narg}\\left[\\sum_{\\ell}a_\\ell(k)Y_{\\ell,0}(\\Omega_{\\pmb{k}})\\right]$.\nFor 6 optical cycles, the angular channels with $\\ell=0$ and\n$\\ell=2$ already represent the dominant contributions to the photoelectron\nat the final energy around 0.4 eV, as expected for a two-color\nphotoionization process of an initial $s$-state. The quantum phase is\na result of interference between both partial waves and for the long\npulse limit it may be mathematically expressed by\n\\cite{watzel2014angular}\n\\begin{equation}\n\\varphi(k,\\vartheta_k)=\\arctan\\left[\\frac{\\sum_{\\ell=0,2}S_\\ell(k,\\vartheta_k)\\sin(\\phi_\\ell(k))}{\\sum_{\\ell=0,2}\nS_\\ell(k,\\vartheta_k)\\cos(\\phi_\\ell(k))}\\right]\n\\label{eq:phase}\n\\end{equation}\nwith\n$S_\\ell(k,\\vartheta_k)=\\left|a_\\ell(k)\\right|Y_{\\ell,0}(\\Omega_{\\pmb{k}})$\nand $\\phi_\\ell(k)={\\rm arg}[a_\\ell(k)]\\simeq\\delta_\\ell(k)-\\ell\\pi\/2$.\n\\begin{figure}[t]\n\\includegraphics[width=1.0\\columnwidth]{Fig2}\n\\caption{Properties of the ejected photoelectron. (a) Ionization\nprobability of the angular channel $\\ell=2$ for different pulse\nlengths of the incident IR and blue laser fields. (b) Angular\nvariation of the averaged quantum phase\n$\\overline{\\varphi}(\\vartheta_k)$ in dependence on the number of\noptical cycles.}\n\\label{fig2}\n\\end{figure}\nThe quantum phase hence depends crucially on the scattering\nphases $\\delta_\\ell(k)$ and on the ratio between the transitions\nstrengths into $\\ell=2$ and $\\ell=0$ angular channels respectively.\nWe note that in principle $\\varphi(k,\\vartheta_k)$ is an experimentally accessible\nquantity, since it can be recovered by integrating the Wigner time\ndelay in photoionization defined as $\\tau_{\\rm\nW}(E_k,\\vartheta_k)=({\\rm d}\/{\\rm d}E_k)\\varphi(k,\\vartheta_k)$.\nThis can be extracted from delay measurements that are possible due\nto recent experimental advances within the attosecond timeframe\n\\cite{schultze2010delay, isinger2017photoionization}. The measured atomic time delay $\\tau_{\\rm a}$ consists of\nan \"intrinsic\" contribution (Wigner time delay $\\tau_{\\rm\nW}$) upon the absorption of an XUV photon which can be interpreted\nas the group delay of the outgoing photoelectron wave packet due to the collision process. As mentioned above, it contains information about the internal quantum phase. The second term $\\tau_{\\rm cc}$ arises from continuum-continuum transitions due to the interaction of the laser probe field\nwith the Coulomb potential and depends crucially on the experimental parameters. Hence, the difference $\\tau_{\\rm W}=\\tau_{\\rm a} - \\tau_{\\rm cc}$ provides access to the phase information $\\varphi(k,\\vartheta_k)$.\n\nIn Fig.\\ref{fig2}(b) it is shown how the\nphase develops by increasing the pulse lengths (number of optical\ncycles). Here, we show the phase averaged over the ionization probability (total cross section\n$\\sigma(E_k)$): $\\overline{\\varphi}(\\vartheta_k)=\\int{\\rm\nd}E_k\\,\\sigma(E_k)\\varphi(k,\\vartheta_k)\/\\int{\\rm\nd}E_k\\,\\sigma(E_k)$. Interestingly for very short pulses\n($n_p=6$) where the cross section is far from being centered\naround a final energy $E_f=E_{5s}+\\hbar\\omega_{\\rm IR} +\n\\hbar\\omega_{\\rm BL}$, the shape of the phase matches that extracted\nfrom simulations with longer pulses. As anticipated from earlier\nresults, from ten cycles upwards the results converge quickly into the\n\\emph{cw} limit. As an example, this is seen since the discrepancy between the quantum\nphase for 12 and 35 optical cycles is smaller than 5\\%.\n\n\\section{From short pulses to the \\emph{cw}-limit}\n\\label{sec:3}\n\n\\begin{figure*}[t!]\n\\includegraphics[width=0.95\\textwidth]{Fig3}\n\\caption{(a) Comparison between photoionization pathways via $5p$ and\n$6p$ intermediate states in the case of detuning $\\delta\\omega_{\\rm\nIR}=0.15$\\,eV of the infrared field. The curves are extracted from\nEq.\\,\\eqref{eq:d2_2}. (b) Time dependence of the first-order product\n$\\Re\\{[F_{\\omega_0}^{(1)}(t,-\\omega_{\\rm\nIR},0)]^*F_{\\omega_0}^{(1)}(t,-\\omega_{\\rm BL},\\Delta_{\\rm BL})\\}$\nfor different final energies. (c) Time dependent occupation numbers\n$c_{5p}(t)$ and $c_{6p}(t)$ extracted from perturbation theory and\na full numerical treatment. The grey curve indicates the temporal\nvariation of the IR laser field.}\n\\label{fig3}\n\\end{figure*}\n\n\nBy considering the two-color ionization process using perturbation\ntheory, we express the time-dependent wave function as\n$\\Psi(\\pmb{r},t)=\\sum_{\\nu\\ell} d_{\\nu\\ell}(t)e^{-{\\rm\ni}E_{\\nu\\ell}t}\\phi_{\\nu\\ell}(\\pmb{r})$. Note that the quantum number\n$\\nu$ includes both the bound and the continuum states. The first\norder amplitude is given by:\n\\begin{equation}\n\\begin{split}\nd^{(1)}_{0\\rightarrow\nf}(t)=&-\\frac{1}{i}\\sum_{\\lambda=\\pm1}\\left[\\langle f|D_{\\rm\nIR}|n\\rangle \\mathcal{E}_{\\rm\nIR}F_{\\omega_{0f}}^{(1)}(t,\\lambda\\omega_{\\rm IR},\\Delta=0) \\right. \\\\\n&\\left.+ e^{i\\lambda\\Delta\\omega_{\\rm BL}} \\langle f|D_{\\rm\nBL}|n\\rangle \\mathcal{E}_{\\rm\nBL}F_{\\omega_{0f}}^{(1)}(t,\\lambda\\omega_{\\rm\nBL},\\lambda\\Delta)\\right],\n\\end{split}\n\\label{eq:d1}\n\\end{equation}\nwhere $D_i=\\hat{\\epsilon}_i\\cdot\\hat{d}$ ($i={\\rm IR,BL}$) is the\ndipole operator, $\\omega_{0f}=E_f-E_0$ and\n\\begin{equation}\nF_{\\omega_{0f}}^{(1)}(t,\\omega,\\Delta)= \\int_{-\\infty}^{t}\n\\Omega(t'+\\Delta)e^{{\\rm i}(\\omega_{0f}+\\omega)t'} {\\rm d}t'.\n\\end{equation}\nWithout loss of generality we assume both laser fields are\ndescribed by the same temporal function $\\Omega(t)=\\cos^2[\\pi t\/T_p]$\nso that both pulses have the same pulse length\n$T_p=2n_p\\pi\/\\omega_{\\rm IR}$. In the following discussion the number of\noptical cycles $n_p$ hence refers to the infrared laser field. For the\nsquared-cosine shaped envelope $\\Omega(t)$ of the pulses, the\nfunction $F_{\\omega_{0}}^{(1)}(t,\\omega,\\Delta)$ can be obtained\nanalytically and converges against\n$F_{\\omega_{0}}^{(1)}(t,\\omega,\\Delta)\\rightarrow\\delta(\\omega_{0}-\\omega)$\nfor $T_p\\rightarrow\\infty$. The second order amplitude yields the\nfollowing expression:\n\\begin{equation}\n\\begin{split}\nd^{(2)}_{0\\rightarrow\nf}(t)=&-\\sum_{i,j}\\sum_{\\lambda,\\lambda'=\\pm1}\\sum_n\ne^{{\\rm i}(\\lambda\\Delta_j\\omega_j+\\lambda'\\Delta_i\\omega_i)}\\\\\n&\\times\\mathcal{E}_j\\mathcal{E}_i \\frac{\\langle f|D_j|n\\rangle\n\\langle n|D_i|0\\rangle}{\\omega_{0n}-\\omega_i} \\\\\n&\\times\nF^{(2)}_{\\omega_{nf},\\omega_{0n}}(t,\\lambda\\omega_j,\\lambda'\\omega_i,\\lambda\\Delta_j,\\lambda'\\Delta_i),\n\\end{split}\n\\label{eq:d2_2}\n\\end{equation}\nwhere again $i,j={\\rm IR,BL}$. The second-order temporal function is\ndefined as\n\\begin{equation}\n\\begin{split}\nF&^{(2)}_{\\omega_{nf},\\omega_{0n}}(t,\\omega_j,\\omega_i,\\Delta_j,\\Delta_i)=(\\omega_{0n}-\\omega_i)\\\\\n&\\times\\int_{-\\infty}^{t}{\\rm d}t'\\,\\Omega(t'+\\Delta_j)e^{{\\rm\ni}(\\omega_{nf}+\\omega_j)t'}F^{(1)}_{\\omega_{0n}}(t',\\omega_i,\\Delta_i).\n\\end{split}\n\\end{equation}\nA closed expression for the second-order temporal function $F^{(2)}$\ncannot be obtained analytically. However, a solution for $t>T_p\/2$\n(time of switch off) can be found and investigated for\n$T_p\\rightarrow\\infty$ (the continuous wave limit). It follows that in\nthe many optical cycle limit we find energy conservation, i.e.\n\\begin{equation}\n\\lim_{n_p\\rightarrow\\infty}\nF^{(2)}_{\\omega_{nf},\\omega_{0n}}(t,-\\omega_j,-\\omega_i,\\Delta_j,\\Delta_i)\n= \\frac{3\\pi}{4}\\delta(\\omega_{0f} - \\omega_{\\rm j} - \\omega_{\\rm i})\n\\label{eq:F2_asymptotic}\n\\end{equation}\nFurther, in this limit the individual temporal shifts $\\Delta_j$ and\n$\\Delta_i$ have no influence on the (second-order) transition\namplitude. Thus, in the long (\\emph{cw}) pulse limit and for energies around\n$E_f=E_0+\\hbar\\omega_{0f}$, this behavior allows us to rewrite the\nresulting second-order transition amplitude:\n\\begin{equation}\n\\begin{split}\nd^{(2)}_{0\\rightarrow\nf}\\underset{n_p\\rightarrow\\infty}{\\rightarrow}&\\frac{3\\pi}{4}\\mathcal{E}_{\\rm\nIR}\\mathcal{E}_{\\rm BL}e^{{\\rm i}\\Delta\\omega_{\\rm\nBL}}\\sum_n\\left[\\frac{\\langle f|D_{\\rm IR}|n\\rangle\\langle n|D_{\\rm\nBL}|0\\rangle}{\\omega_{n0}-\\omega_{\\rm BL}}\\right. \\\\\n&\\left.+ \\frac{\\langle f|D_{\\rm BL}|n\\rangle\\langle n|D_{\\rm\nIR}|0\\rangle}{\\omega_{n0}-\\omega_{\\rm IR}} \\right],\n\\end{split}\n\\label{eq:2PhotonMatrix}\n\\end{equation}\nwhich is similar to the traditional form of the two-photon matrix\nelement \\cite{toma2002calculation}. From here we see directly that a\nrandom phase $\\phi=\\Delta\\omega_{\\rm BL}$ does not play any role\n(especially when analyzing the cross sections\n$\\sim|d^{(2)}_{0\\rightarrow f}|^2$.\n\nLet us now come back to the two-color ionization process in the Rubidium\natom. It is crucial for the first and second-order amplitudes to\nprecisely evaluate the dipole matrix elements \\cite{amusia2013atomic}\n\\begin{equation}\n\\langle\nf|D_i|n\\rangle=(-1)^{m_f+\\ell_>}\\begin{pmatrix}\\ell_f&1&\\ell_n \\\\\n-m_f&0&m_n \\end{pmatrix}\\sqrt{\\ell_>}\\langle f||\\hat{d}||n\\rangle.\n\\end{equation}\nwhere $\\ell_>={\\rm max}(\\ell_f,\\ell_n)$ and the reduced radial matrix\nelement $d_{\\ell_f\\ell_n}=\\langle\nf||\\hat{d}||n\\rangle=\\int_0^\\infty{\\rm d}r\\,R_f(r)D_iR_n(r)$. \nOne way to account, at least partially, for the the interaction between the valence  and the core electrons \\cite{hameed1968core}\n is to modify the operator  $\\hat{Q}_L$ as \n\\begin{equation}\n\\hat{Q}_L\\rightarrow\\hat{Q}_L\\left[1-\\frac{a_c^{(L)}}{r^{2L+1}}\\left(1-e^{-(r\/r_c')^{2L+1}}\\right)\\right],\n\\label{eq:core_Q}\n\\end{equation}\nwhere $a_c^{(L)}$ is the  $2^L$ tensor core polarizability and $r_c'$ is an empirical cut-off radius (for Rb $r_c'=4.339773$ a.u.\n\\cite{marinescu1994dispersion}). This physical picture behind the corrections is roughly that  the valence electron with a dipole moment $\\pmb{d}$ induces  (by virtue of its field)  a (core) dipole moment $-\\alpha_c\\pmb{d}\/r^3$ (and higher multipoles).  Then, the complete dipole moment becomes $\\pmb{d}(1-\\alpha_c\/r^3)$. Note that in our case the Dipole operator $\\hat{d}=\\hat{Q}_1$. Using the modified dipole operator delivers very accurate matrix elements near the ionization threshold in comparison with experiments and more sophisticated theoretical models\n\\cite{petrov2000near}.\n\nWe learn from Eq.\\,\\eqref{eq:2PhotonMatrix} that changing the\nelectric field amplitudes $\\mathcal{E}_{\\rm IR}$ and\n$\\mathcal{E}_{\\rm BL}$ will not balance any differences between the\nmatrix element products $\\langle f|D_{\\rm IR}|n\\rangle\\langle\nn|D_{\\rm BL}|0\\rangle$ and $\\langle f|D_{\\rm BL}|n\\rangle\\langle\nn|D_{\\rm IR}|0\\rangle$. Thus, to reach equipollent ionization\npathways $E_{5s}\\rightarrow E_{5p}\\rightarrow E_f$ and\n$E_{5s}\\rightarrow E_{6p}\\rightarrow E_f$ we have to detune the laser\nfrequency corresponding to the stronger bound-bound transition. Given\nthe reduced matrix elements $\\langle 5p||\\hat{d}||5s\\rangle=-5.158$\nand $\\langle 6p||\\hat{d}||5s\\rangle=0.468$, we have to detune the\ninfrared field as shown in Fig\\,\\ref{fig3}(a). The curves are\nobtained from a numerical integration of Eq.\\,\\eqref{eq:d2_2}. For\n$n=35$ optical cycles, a detuning of\n$\\delta\\omega_{\\rm IR}=+0.15$\\,eV is required to allow the second-order\ntransition amplitudes within each pathway to have the same magnitude.\nNote that the value of $\\Delta\\omega_{\\rm IR}$ decreases by\nincreasing the pulse lengths. In this vein we performed the\nsimulation for 75 optical cycles and found that a detuning of only\n0.07 eV is needed to reach equipollent ionization pathways (not shown\nfor brevity).\n\nLet us consider the time-dependent first order probability\n$P_{0\\rightarrow f}^{(1)}(t)=\\left|d^{(1)}_{0f}(t)\\right|^2$ which\nreads explicitly\n\\begin{equation}\n\\begin{split}\nP_{0\\rightarrow f}^{(0)}(t)=&\\left|\\langle f|D_{\\rm IR}|n\\rangle\n\\mathcal{E}_{\\rm IR} F_{\\omega_{0f}}^{(1)}(t,-\\omega_{\\rm\nIR},0)\\right|^2 \\\\\n&+ \\left|\\langle f|D_{\\rm BL}|n\\rangle \\mathcal{E}_{\\rm\nBL}F^{(1)}_{\\omega_{0f}}(t,-\\omega_{\\rm BL},\\Delta_{\\rm BL})\\right|^2\n\\\\\n&+2\\mathcal{E}_{\\rm IR}\\mathcal{E}_{\\rm BL} \\langle f|D_{\\rm\nIR}|n\\rangle\\langle f|D_{\\rm BL}|n\\rangle \\\\\n&\\times\\Re\\left\\{F_{\\omega_{0f}}^{(1)}(t,-\\omega_{\\rm\nIR})\\left[F_{\\omega_{0f}}^{(1)}(t,-\\omega_{\\rm BL},\\Delta_{\\rm\nBL})\\right]^*\\right\\}.\n\\end{split}\n\\end{equation}\nHere, we have to emphasize that all terms containing $+\\omega_i$ are\nnegligibly small which we refer to as the rotating wave approximation.\nThe last term in the third line might look like a two-photon process\nbut a closer inspection of the function\n$F_{\\omega_0}^{(1)}(t,-\\omega,\\Delta)$ reveals that it sharply peaks\naround $\\omega_0-\\omega=0$ even for times close to $T_p\/2$. Since\n$\\omega_{f0}$ is fixed, the product between both $F^{(1)}$ functions\nis zero, so that\n\\begin{equation}\n\\begin{split}\n\\lim_{T_p\\rightarrow\\infty} F_{\\omega_{f0}}^{(1)}&(T_p\/2,-\\omega_{\\rm\nIR},0)\\left[F_{\\omega_{f0}}^{(1)}(T_p\/2,-\\omega_{\\rm BL},\\Delta_{\\rm\nBL})\\right]^* \\\\\n&= \\delta(\\omega_{f0}-\\omega_{\\rm BL})\\delta(\\omega_{f0}-\\omega_{\\rm\nBL}).\n\\end{split}\n\\end{equation}\nAs confirmation, in Fig.\\,\\ref{fig3}(b) we show the time-dependent\nproduct of the functions $F_{\\omega_0}^{(1)}(t,-\\omega_{\\rm IR},0)$\nand $F_{\\omega_0}^{(1)}(t,-\\omega_{\\rm BL},\\Delta_{\\rm BL})$ for\ndifferent $\\omega_{f0}$. As stated above, all situations have in\ncommon that the product is zero at the time when the pulse is\nswitched off, representing energy conservation. Finally, the\nfirst-order transitions for $t=T_p\/2$ are given by\n\\begin{equation}\n\\begin{split}\nd^{(1)}_{0\\rightarrow f}(t>T_p\/2)=&-\\frac{1}{2i}\\left[\\mathcal{E}_{\\rm\nIR}\\langle f|D_{\\rm IR}|0\\rangle\\delta(w_{f0}-\\omega_{\\rm IR})\\right.\n\\\\\n&\\left.+ e^{{\\rm i}\\Delta\\omega_{\\rm BL}}\\mathcal{E}_{\\rm BL}\\langle\nf|D_{\\rm BL}|0\\rangle\\delta(w_{f0}-\\omega_{\\rm BL})\\right].\n\\end{split}\n\\end{equation}\nThe energies of the blue and red photons are not sufficiently high to\nreach the continuum and only the two-photon matrix element developed\nin Eq.\\,\\eqref{eq:d2_2} gives insight into the properties of the\nphotoelectron.\n\nAs a consequence, the amplitude $d^{(1)}$ describes the\nphotoexcitation process of the intermediate $f=5p$ and $f=6p$ states.\nHowever, due to the sharp laser pulses the same final state $f$\ncannot be excited by both photons. Hence, in the resulting\nphotoexcitation probabilities $c_f=|d^{(1)}_{0\\rightarrow\nf}(t>T_p\/2)|^2$ the random phase $\\phi=\\Delta\\omega_{\\rm BL}$ does\nnot play a role, which confirms the full-numerical results shown in\nFig.\\,\\ref{fig1}(b).\n\\begin{figure*}[t!]\n\\includegraphics[width=0.95\\textwidth]{Fig4}\n\\caption{(a) Differential cross section (DCS) when the blue laser field\nis off-resonance, thereby effectively closing the second ionization pathway.\nComparison between two theoretical models (PT:red and\npropagation:gray) and experimental results. For the propagation, the\nblue field was detuned by $\\delta\\omega_{\\rm BL}=0.13$\\,eV. Further,\nthe infrared field was slightly detuned by $\\delta\\omega_{\\rm\nIR}=0.04$\\,eV to manipulate the ratio between both two-photon\namplitudes (see discussions in Sec.\\,\\ref{sec:3}). (b) Same as in (a)\nwith the infrared field completely off-resonance, effectively closing\nthe first ionization pathway. The detuning amount is $\\delta\\omega_{\\rm\nIR}=0.14$\\,eV. (c) Both laser fields resonant with associated\npathways. To achieve pathway amplitudes that are comparable with experiment\nthe infrared field was slightly detuned by $\\delta\\omega_{\\rm\nIR}=0.04$\\,eV. The inset shows a polar plot of the DCS. All\ntheoretical curves are scaled by the same factor and are shifted by a\nconstant offset of 0.1 to aid  comparison with the experimental data.\nThe data points are reproduced from Ref.\\cite{pursehousedynamic}.}\n\\label{fig4}\n\\end{figure*}\nIn panel \\ref{fig3}(c) we present the occupation numbers\n$c_{5p}(t)=|\\langle\\phi_{5p}|\\Psi(t)\\rangle|^2$ and\n$c_{6p}(t)=|\\langle\\phi_{6p}|\\Psi(t)\\rangle|^2$ extracted from the\nperturbative treatment (PT) in Eq.\\eqref{eq:d1} and the numerical\nsimulation scheme developed in Sec.\\,\\ref{sec:2}. We find a remarkable\nagreement between the PT results and the full numerical treatment,\ndemonstrating the transition into the \\emph{cw} limit and the\nvalidity of the perturbative treatment of the two-color ionization\nproblem. Due to the detuning of the IR field the $c_{5s}(t)$\nterm decreases at the end of the pulse while the $c_{6p}(t)$ term belongs to\nresonant excitation ($5s\\rightarrow6p$) in the blue laser field. This\nis the reason for the nearly monotonous increase that is seen. We note that the corresponding\nRabi frequencies $\\Omega_{5s-5p}$ and $\\Omega_{5s-6p}$ are very small which means the occupation numbers in\nFig.\\,\\ref{fig3}(c) represent only the first segment of the first Rabi Cycle which can be identified by\nthe characteristic quadratic dependence on the time.\n\n\\section{Interference between both ionization pathways}\n\\label{sec:4}\n\nAs revealed by the two-photon matrix element in\nEq.\\,\\eqref{eq:2PhotonMatrix}, both laser fields act simultaneously\nto produce the same final photoelectron state, and so we have to deal\nwith two transition amplitudes $t_1$ and $t_2$ which are presented by\nthe two terms. As already demonstrated in Sec.\\,\\ref{sec:2} the final\nstate $|f\\rangle$ is mainly described by a superposition of two\nangular channels $\\ell=0$ and $\\ell=2$. Thus, we may write\n\\begin{equation}\n\\begin{split}\nd^{(2)}_{0\\rightarrow\nf}(\\vartheta_{\\pmb{k}})&= t_1(\\vartheta_{\\pmb{k}})+t_2(\\vartheta_{\\pmb{k}}) \\\\\n&=\\underbrace{S_{\\ell=0}^{(t_1)}(\\vartheta_{\\pmb{k}})e^{i\\phi_{\\ell=0}(k_f)}\n+\nS_{\\ell=2}^{(t_1)}(\\vartheta_{\\pmb{k}})e^{i\\phi_{\\ell=2}(k_f)}}_{t_1=|t_1|e^{i\\varphi_1}}\n\\\\\n&\\quad+\\underbrace{S_{\\ell=0}^{(t_2)}(\\vartheta_{\\pmb{k}})e^{i\\phi_{\\ell=0}(k_f)}\n+\nS_{\\ell=2}^{(t_2)}(\\vartheta_{\\pmb{k}})e^{i\\phi_{\\ell=2}(k_f)}}_{t_2=|t_2|e^{i\\varphi_2}}.\n\\end{split}\n\\end{equation}\nThe final state is in the continuum and is defined by\nEq.\\,\\eqref{eq:continuum}. Consequently, we can define\n\\begin{equation}\n\\begin{split}\nS^{(t_i)}_{\\ell_f}(\\vartheta_{\\pmb{k}})=&(-1)^{2+\\ell_f\/2}\\sqrt{1+\\ell_f\/2}\\begin{pmatrix}\\ell_f&1&1\n\\\\ 0&0&0 \\end{pmatrix}\\begin{pmatrix}1&1&0\\\\0&0&0 \\end{pmatrix}\\\\\n&\\times\\sum_n \\frac{\\langle E_f\\ell_f||\\hat{d}||n\\rangle\\langle\nn||\\hat{d}||5s\\rangle}{\\omega_{0n}-\\omega_1} \\\\\n&\\times \\mathcal{E}_{\\rm IR}\\mathcal{E}_{\\rm BL}\nF^{(2)}_{\\omega_{nf},\\omega_{0n}}(t\\rightarrow\\infty,-\\omega_2,-\\omega_1,0,0)\\\\\n&\\times Y_{\\ell_f,0}(\\vartheta_{\\pmb{k}}).\n\\end{split}\n\\end{equation}\nHere without loss of generality we set the time delays $\\Delta_{\\rm IR\/BL}$ to\nzero since we are in the \\emph{cw} limit. For pathway $t_1$, $\\omega_1=\\omega_{\\rm IR}$ and $\\omega_2=\\omega_{\\rm\nBL}$, while for pathway $t_2$ the opposite is required. Further,\n$\\phi_\\ell(k)=\\delta_\\ell(k)-\\ell\\pi\/2$ while the corresponding pathway phases $\\varphi_{\\rm i}$ are already defined\nin Eq.\\,\\eqref{eq:phase}. Note, that $|t_1(\\vartheta)|^2$ and\n$|t_2(\\vartheta)|^2$ define the DCS for the individual pathways while\nthe interference term between both pathways is given by\n\\begin{equation}\n\\begin{split}\n{\\rm DCS}_{\\rm interf.} &=\nt_1(\\vartheta_{\\pmb{k}})t_2^*(\\vartheta_{\\pmb{k}}) +\nt_2(\\vartheta_{\\pmb{k}})t_1^*(\\vartheta_{\\pmb{k}}) \\\\\n&=\\left|t_1(\\vartheta_{\\pmb{k}})+t_2(\\vartheta_{\\pmb{k}})\\right|^2 -\n(\\left|t_1(\\vartheta_{\\pmb{k}})\\right|^2 +\n\\left|t_2(\\vartheta_{\\pmb{k}})\\right|^2)\n\\end{split}\n\\end{equation}\nThe \\emph{phase difference} related to ${\\rm DCS}_{\\rm interf.}$ is given by\n\\begin{equation}\n\\begin{split}\n\\Delta\\varphi_{12}(\\vartheta_{\\pmb{k}})=\\cos^{-1}\\left[\\frac{t_1(\\vartheta_{\\pmb{k}})t_2^*(\\vartheta_{\\pmb{k}})\n+\nt_2(\\vartheta_{\\pmb{k}})t_1^*(\\vartheta_{\\pmb{k}})}{2\\left|t_1(\\vartheta_{\\pmb{k}})\\right|\n\\left|t_2(\\vartheta_{\\pmb{k}})\\right|}\\right].\n\\end{split}\n\\end{equation}\nThe interference term is the result of differences in the ratios\n$S^{(t_1)}_{\\ell=2}\/S^{(t_1)}_{\\ell=0}$ and\n$S^{(t_2)}_{\\ell=2}\/S^{(t_2)}_{\\ell=0}$ between the $s$- and $d$-\npartial waves associated with the individual pathways $t_1$ and\n$t_2$. These ratios depend crucially on the reduced bound-continuum\ndipole matrix elements $\\langle E_f,\\ell||\\hat{d}||5p\\rangle$ and\n$\\langle E_f,\\ell||\\hat{d}||6p\\rangle$ as well as the bound-bound\ndipole matrix elements $\\langle 5p||\\hat{d}||5s\\rangle$ and $\\langle\n6p||\\hat{d}||5s\\rangle$.\n\nThe perturbative treatment of the two-pathway ionization process shares  some parallels with\nthe theoretical description of the recently developed attosecond measurement techniques \\cite{dahlstrom2013theory}. For instance, the occurrence and spectral characteristics of the $2q$th sideband in the RABBITT scheme \\cite{muller2002reconstruction} stem from the interference between two ionization pathways: the absorption of harmonic $H_{2q-1}$ or $H_{2q+1}$ plus absorption or emission of a laser photon with $\\hbar\\omega$. Hence, similarly to the effects studied in this work, the measured intensity of the side band  depends on the phase difference between the quantum paths. In contrast to typical attosecond experiments, we create a bound wave packet upon absorption of the first photon. This case was studied in photoionization of Potassium where the spectral properties of the initially created bound wave packet was used to eliminate the influence of the dipole phase in the angle-integrated photoelectron spectrum making it possible to fully characterize the attosecond pulses \\cite{pabst2016eliminating}. Similar to the investigated experiment by Pursehouse and Murray, the underlying physical principle is the quantum interference of pathways corresponding to ionization from different energy levels. Moreover, a realistic many-body treatment revealed that correlation effects have only a minor influence in Alkali atoms which supports our theoretical treatment in this work.\n\nIn Fig.\\,\\ref{fig4} we present the individual $t_1$ DCS (a), $t_2$\nDCS (b) and the DCS corresponding to the coherent summation $t_1+t_2$\n(c). In all panels we compare the ionization probabilities extracted\nfrom the full numerical and perturbative treatment with experimental\nresults from measurements performed by Pursehouse and Murray\n\\cite{pursehousedynamic}. In the experiment the individual pathway\ncross sections are obtained by the appropriate  detuning of the\nrespective laser fields: To extract $|t_1|^2$\n($5s\\rightarrow5p\\rightarrow E_f$) the blue laser beam is detuned to block\noccupation of the $6p$ state. To obtain the $|t_2|^2$ amplitude\n($5s\\rightarrow6p\\rightarrow E_f$), the infrared laser field is tuned\nto be off-resonant to the $5p$ transition. To obtain the total\namplitude $|t_1+t_2|^2$ both laser pulses are on resonance to the\nrespective $5s\\rightarrow np$ transitions. As explained in\nSec.\\,\\ref{sec:3} the infrared laser field is always slightly detuned by\na fixed $\\delta\\omega_{\\rm IR}$ so that both two-photon\namplitudes $t_1$ and $t_2$ are of the same magnitude.\n\nIn panel \\ref{fig4}(a) we show the DCS of the individual pathway 1.\nFor the full numerical treatment with a number of 70 optical cycles\nwe used\na blue detuning of $\\delta\\omega_{\\rm BL}=0.13$\\,eV while the red\nfield detuning amounts to $\\delta\\omega_{\\rm IR}=0.04$\\,eV. In the\nperturbative treatment ($n_p\\rightarrow\\infty$) we need a much\nsmaller detuning (in the range of meV). Both theoretical models agree\nextraordinary well with the experiment in general, while minor discrepancies can be found\naround the maxima. A possible explanation is the shift of the final\nenergy by nearly 0.17\\,eV in the full-numerical propagation (finite\nnumber of optical cycles) due to the detuning of both fields which\nchanges slightly the ratios $S^{(t_1)}_{\\ell=2}\/S^{(t_1)}_{\\ell=0}$.\nIn comparison with the experiment, the theoretical models predict the\ncorrect shape of the DCS. They underestimate the data around\n$\\vartheta_{\\pmb{k}}=90^\\circ$ and $\\vartheta_{\\pmb{k}}=270^\\circ$,\nhowever agree well at $\\vartheta_{\\pmb{k}}=0^\\circ$ and\n$\\vartheta_{\\pmb{k}}=180^\\circ$ (along the polarization vectors). The\nsame observations apply for the DCS of the second ionization pathway\n$|t_2|^2$ in panel \\ref{fig4}(b). Figure \\ref{fig4}(c) shows the\nphotoionization probability $|t_1+t_2|^2$ when both laser fields are\nset to resonance with the respective $5s\\rightarrow np$ transitions.\nHere, the infrared field is again detuned by a small\n$\\delta\\omega_{\\rm IR}$ so that both amplitudes $t_1$ and $t_2$\nhave the same magnitude. In addition, we show here the result for\nrather short laser pulses with 12 optical cycles.\n\\begin{figure}[t]\n\\includegraphics[width=1.0\\columnwidth]{Fig5}\n\\caption{(a) Interference term ${\\rm DCS}_{\\rm interf.}$ for  two\ntheoretical models and experimental data. Parameters are the same as\nin Fig.\\,\\ref{fig4}. (b) Phase difference $\\varphi_{12}$ between both\nionization pathway amplitudes $t_1(\\vartheta_{\\pmb{k}})$ and\n$t_2(\\vartheta_{\\pmb{k}})$. The yellow line presents a fit of the\nexperimental data to the symmetry-adapted function\n$f(\\vartheta_{\\pmb{k}})=\\sum_{n=0}^2\na_{2n}\\cos^{2n}(\\vartheta_{\\pmb{k}})$. The data points are reproduced from Ref.\\cite{pursehousedynamic}.}\n\\label{fig5}\n\\end{figure}\nSurprisingly, the DCS extracted from the short pulse calculations\nagree very well with the smaller maxima around\n$\\vartheta_{\\pmb{k}}=90^\\circ$ and $\\vartheta_{\\pmb{k}}=270^\\circ$.\nHowever, this rather accidental agreement must be considered within the\nexperimental uncertainties.\n\nIn Fig.\\,\\ref{fig5}(a) we present the interference term ${\\rm\nDCS}_{\\rm interf.}$ for the two developed theoretical models and the\nexperimental data. We see that the amplitude of the interference term\nis clearly non-zero and varies from $13\\%$ to $55\\%$ of the\nnormalized signal shown in Fig.\\,\\ref{fig4}(c). In panel\nFig.\\,\\ref{fig5}(b) we present the corresponding phase difference\n$\\varphi_{12}(\\vartheta_{\\pmb{k}})$ between the two-photon transition\npathways $t_1$ and $t_2$. In comparison to all amplitudes, the\nagreement is less satisfactory which can be explained by the relatively large\nuncertainties due to error propagation through the arccos function\n\\cite{pursehousedynamic}. Interestingly the average value of the phase\nshift is accurately reproduced by both calculations. Under these conditions the predicted angular variation is not very\npronounced and ranges from $110^\\circ$ to $122^\\circ$. Surprisingly,\n the models do not agree as well as for the DCS in\nFig.\\,\\ref{fig4}, which points to the extreme sensitivity of the\nquantum phase to small changes in the transition matrix elements. The yellow curve represents a fit of the experimentally\nobtained phase difference to the symmetry-adapted function $\\sum_{n=0}^2 a_{2n}\\cos^{2n}(\\vartheta_{\\pmb{k}})$.\nIt highlights the agreement with the theoretical prediction with respect to the general shape.\n\nIn the next section we will present mechanisms to manipulate, decrease and\nincrease this modulation.\n\n\\section{Manipulation of the quantum interference}\n\\subsection{Role of the energy gap}\n\\label{sec:5}\n\\begin{figure}[t]\n\\includegraphics[width=1.0\\columnwidth]{Fig6}\n\\caption{Interference cross section (a) and phase difference (b)\n$\\varphi_{12}(\\vartheta_{\\pmb{k}})$ for different intermediate state\npairs extracted from the two-photon matrix element in\nEq.\\,\\eqref{eq:2PhotonMatrix}. The IR field is detuned to reach\nequipollent pathway strengths.\nThe numbers in panel (a) represent the ratio\n$\\nu=(d^{(1)}_2\/d^{(1)}_0):(d^{(2)}_2\/d^{(2)}_0)$.}\n\\label{fig6}\n\\end{figure}\n\\begin{figure*}[t]\n\\includegraphics[width=0.95\\textwidth]{Fig7}\n\\caption{(a) Addition of a third laser pulse (orange) with amplitude\n$\\mathcal{E}_3$ creates an additional third (direct) ionization pathway $t_{\\rm\ncontr}$ to the final energy $E_{f}$. The blue and IR pulses are the same as\nin Fig.\\,\\ref{fig0}. (b) Interference term ${\\rm DCS}_{\\rm interf}$\nobtained from Eq.\\,\\eqref{eq:interference_mod} for different\nstrengths of the amplitude $t_{\\rm contr}$ relative to\n$t_1+t_2$. (c) Angular dependence of the associated interference\nphase $\\varphi_{12}$ between $t_1$ and $t_2$. All results are\nobtained by numerically calculation with pulse lengths of 75 optical\ncycles while the infrared and blue laser fields have the same field\namplitudes and detunings as in the previous sections.}\n\\label{fig7}\n\\end{figure*}\nIn Fig.\\,\\ref{fig6} we present interference studies on different\nintermediate state pairs defining the two-color ionization amplitudes\n$t_1$ and $t_2$. The blue and the red curves present the interference\nbetween $5p\/7p$ ($\\Delta E= 1.88$\\,eV) and $5p\/8p$ ($\\Delta E=\n2.13$\\,eV) states which have an increasing energy gap $\\Delta E$ between them. The quantity $\\nu=(d^{(t_1)}_{\\ell=2}\/d^{(t_1)}_{\\ell=0})\/\n(d^{(t_2)}_{\\ell=2}\/d^{(t_2)}_{\\ell=0})$ represent the ratios of the\nbound-continuum reduced radial matrix elements between both pathways.\nAs expected for higher Rydberg states this ratio converges quickly to\nthe same number, i.e. the coupling of the $7p$ and $8p$ state to the\ncontinuum is similar. The interference for both state pairs is\nhence comparable. Note that the final energy\n$E_f=E_{5s}+\\hbar\\omega_1 + \\hbar\\omega_2$ is larger in comparison to\nthe $5p\/6p$ case (0.84\\,eV and 1.09\\,eV). Interestingly, ${\\rm\nDCS}_{\\rm interf.}$ has a different shape and changes its sign in\nboth cases, which has an impact on the angular modulation of the phase\ndifference $\\varphi_{12}$. Surprisingly while the amplitude is\nsmaller in comparison with the $5p\/6p$ case shown in\nFig.\\,\\ref{fig5}, the angular variation of the phase is significantly more\npronounced, ranging from $80^\\circ$ to $100^\\circ$. This highlights\nthe change of sign of the interference term (negative sign of the\nargument of the arccos function means a phase larger than\n$90^\\circ$).\n\nThe green and orange curves represent cases when we decrease the\nenergy gap. We chose the intermediate state pairs $6p\/7p$ ($\\Delta\nE=0.51$\\,eV, $E_f=2.22$\\,eV) and $7p\/8p$ ($\\Delta E=0.25$\\,eV,\n$E_f=2.97$\\,eV). Intriguingly, the interference cross section remain\nunaffected when changing to higher lying state pairs. The amplitudes\nof ${\\rm DCS}_{\\rm interf.}$ ranges in both cases from 5\\% to\n50\\% and is negative. However, the angular modulation of the phase\ndifference $\\varphi_{12}$ decreases drastically. We address this\ndevelopment with the dipole matrix element ratio $\\nu$ which\nconverges rapidly to 1, meaning the ratio between the $\\ell=0$\nand $\\ell=2$ angular channels is  nearly the same for both\nionization pathways $t_1$ and $t_2$. According to\nEq.\\,\\eqref{eq:phase} the individual phase shapes are then\ncomparable.\n\nFrom these results we learn that the angular modulation of the phase\ndifference depends critically on the quantity $\\nu$, while the overall\namplitude of the interference term (${\\rm DCS}_{\\rm interf.}$) is\nmore robust and reveals a dependence on the energy gap between the\nintermediate state pair defining $t_1$ and $t_2$.\n\n\\subsection{laser-driven perturbation of ionization pathways}\n\nAnother method for dynamic control of the interference phenomena is\nthe addition of a third control pulse. As represented in the scheme\nin Fig.\\,\\ref{fig7}(a) the corresponding parameters are chosen in a way\nthat  initiates a weak one-photon process directly into the\ncontinuum, so that $\\hbar\\omega_{\\rm contr.}=(E_f-E_{5s})\/\\hbar$.\nThe field amplitude $\\mathcal{E}_{\\rm contr}$ is hence chosen in a\nway that the corresponding transition amplitude $t_{\\rm contr}$ is of\nthe same magnitude as that of the two-color pathways $t_1$\nand $t_2$. As indicated in the modified scheme in\nFig.\\,\\ref{fig7}(a), in the presence of all fields the total\namplitude is given by the coherent sum $t_{\\rm all}=t_1+t_2+t_{\\rm\ncontr.}$. To access the desired interference term ${\\rm\nDCS}_{\\rm interf}=t_1t_2^* + t_1^*t_2$ one has to then perform four\ndifferent measurements: (i) with all fields on resonance to the\nintermediate and final states respectively, (ii) with the blue\nfield to the $6p$ state off-resonance, (iii) with the red field\nto the $5p$ state off-resonance, and (iv) with both red and blue\nfields off-resonant (thereby blocking both\npathways $t_1$ and $t_2$). The interference term is then given by\n\\begin{equation}\n{\\rm DCS}_{\\rm interf}=|t_{\\rm all}|^2 - (|t_{\\rm (ii)}|^2 + |t_{\\rm\n(iii)}|^2 - |t_{\\rm (iv)}|^2)\n\\label{eq:interference_mod}\n\\end{equation}\nwith $t_{\\rm (ii)}=t_1 + t_{\\rm contr}$, $t_{\\rm (iii)}=t_2 + t_{\\rm\ncontr}$ and $t_{\\rm (iv)}=t_{\\rm contr}$.\n\nIn Fig.\\,\\ref{fig7}(b) we present the interference term for\ndifferent strengths of the perturbation by the third (control)\nlaser field. As expected, the additional one-photon ionization route\nhas a large impact on the quantum interference between $t_1$ and $t_2$.\nHere, the field amplitude $\\mathcal{E}_{\\rm contr}$ has to be very\nlow so that the associated $t_{\\rm contr}$ is of the same magnitude\nas the two-photon pathways $t_1$ and $t_2$. In comparison to the\nunperturbed case shown in Fig.\\ref{fig5}(a), the magnitude of the\ninterference is increased in the presence of the control field. In\nstrong contrast to the previous findings, even the sign of the ${\\rm\nDCS}_{\\rm interf}$ can be changed by the effect of the additional\none-photon process when the field amplitude is sufficiently large.\nIt is therefore not surprising that the large impact seen here is directly\ntransferred to the phase $\\varphi_{12}$ associated with the quantum\ninterference. As shown in Fig.\\,\\ref{fig7}(c), the angular variation is\ndrastically increased due to the action of the controlling field. In the\noriginal experiment and theoretical treatment the modulation in the\npolar angle was smaller than 20$^\\circ$. Now we obtain strongly\npronounced phase peaks and a rather complex angular structure of\n$\\varphi_{12}$ with a variation covering more than 140$^\\circ$.\n\nThe addition of the third laser field helps to emphasize that the\ninterference effect is not robust to statistical fluctuations, but is\nunique for every set of laser parameters.\nFor this purpose we slightly detuned the blue laser field so that\nthe $6p$ state was not excited [cf.\\,Fig\\,\\ref{fig8}(a)]. The\nresulting interference phenomenon in the ionization channel then stems\nfrom the superposition of the two-photon pathway 1 (via $5p$\nphotoexcitation) and the one-photon direct photoionization amplitude\nmediated by $E_{\\rm contr.}$. By tuning the parameters of the third field so\nthat the transition strength of the amplitude $t_{\\rm contr.}$ is\nequal to $t_1$ we obtain a characteristic interference as shown by\nthe dark blue curve in Fig.\\,\\ref{fig8}(b). We then varied the\namplitude of the control field in a way that $t_{\\rm\ncontr.}\\in[-1.0t_1,1.0t_1]$ by use of a random number generator. The\nadditional curves in the figure show a statistical average based on the total number of\nrandom amplitudes input to the model. One can clearly see a trend that increasing the\nnumber of random events decreases the interference effect. This shows\nthat for an infinite number of measurements the resulting\ninterference would disappear.\n\\begin{figure}[t]\n\\includegraphics[width=0.95\\columnwidth]{Fig8}\n\\caption{Stochastic nature of the interference effect. In this example one of\nthe ionization pathways is closed by slight detuning of the laser\nfield. We hence\nfind interference between the two-photon amplitude (pathway 1) and\nthe one-photon transition initiated by the third control field\n$E_3(t)$ (scheme panel a). (b) The stochastic nature of the interference\neffect: the amplitude of the one-photon process is randomly varied between\n$-1.0t_1$ and $+1.0t_1$. The curves show the dependence on the number\nof random amplitudes with the reference to $t_{\\rm contr.}=1.0t_1$\n(blue line). Numbers in the round brackets denote the number of\namplitudes used for the statistical average (see text for details).}\n\\label{fig8}\n\\end{figure}\n\n\\section{Conclusions}\n\nIn this paper we have presented a theoretical investigation of\nexperimental interference studies in a single Rubidium atom.\nWe have systematically demonstrated the transition from the short-pulse into\nthe continuous wave regime, and the evolution of the occupation numbers,\nphotoionization probability and quantum phase under an increase of\nthe pulse lengths. We find that for pulse lengths of more than ten\noptical cycles the theoretical description of the ionization scheme\nvia the two-photon matrix element in the frequency regime is\nsufficient, and that this provides all the physical information\nrequired for interference studies.\n\nOur theoretical model provides generally good agreement with the\nexperimental data and predicts a pronounced interference amplitude\n${\\rm DCS}_{\\rm interf.}$ while the angular variation of the\nassociated phase difference $\\varphi_{12}$ is relatively weak. In this\ntreatment we have developed various strategies to manipulate the quantum\ninterference between both photoionization pathways $t_1$ and $t_2$.\nAs an example, we can change the populations of the intermediate\nstates by laser detuning which introduces an imbalance between both\npathways so as to change the interference phenomena. Further,\nby choosing different state pairs $n_1p$ and $n_2p$ we change the\nenergy difference and coupling to the continuum, which again markedly\nchanges the quantum interference. A new method which does not\nchange the intermediate state pairs and the parameters of the blue\nand infrared laser fields is the addition of a third control laser\nfield which perturbs the original transition pathways $t_1$ and $t_2$. An\nappropriate tuning of the corresponding one-photon transition\namplitude into the continuum can even invert the sign of the\ninterference amplitude, as well as produce a much more pronounced\nangular variation of the interference phase. There is no analogy to\nthis third control in the conventional double slit experiments.\n\nAs well as the addition of a third pulse, there are several other\npossibilities to explore the behaviour of the interference phenomenon.\nAs an example, one can show that in Alkali atoms quadrupole\ntransitions into the continuum reveal Cooper minima at kinetic\nenergies below 1eV. Thus, one can choose intermediate state pairs\nwhich lead to final energies in the region of such Cooper minima\nwhile such quadrupole transitions at low intensity are generally\naccessible by structured light fields \\cite{schmiegelow2016transfer}.\nFurther work will hence be dedicated to studying these two-pathway interference\neffects with inhomogeneous light-induced quadrupole transitions.\n\n\\section*{Acknowledgements}\n\nThis work was partially supported by the DFG through SPP 1840 and SFB TRR 227. The EPSRC\nU.K. is acknowledged for current funding through Grant No. R120272.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nThe Dark Energy Survey (DES) is a next generation\ndeep, wide, multi-band imaging survey that will map 5000 square degrees \nof sky in 525 nights over five years beginning in late 2012.  DES relies on the new \n570-Megapixel, 2.2 degree field-of-view \nDark Energy Camera, known as DECam \\citep{decam}, now installed at the prime focus\nof the 4 m Blanco telescope at the Cerro Tololo Inter-American Observatory (CTIO).  \n\nModern imaging surveys require highly accurate and precise photometry to meet stringent requirements placed on the data \nby the science goals.\nThe requirement for accurate and precision photometry in turn requires accurate and precise calibration of the data products.  \nFor example, DES has a goal of achieving photometry accurate to 1\\% (0.01 mag)\nover the entire survey region.  \nIn order to make measurements with this level of accuracy, care must be taken to \ncalibrate the data appropriately and to remove any spurious effects due to \nweather, the earth's atmosphere, noise and other features of \nthe detector system, and changes in instrumental throughput or response.\nThese calibrations must be made in addition to standard astronomical photometric corrections \nsuch as photometric zeropoint corrections and airmass and color terms.\n\nDECal is a complete calibration system for DECam that will \nallow for the correction of two of these sources of error.  DECal will\nprovide daily flat fields for DECam to correct the pixel-to-pixel variations across the \nCCD detectors; it will also measure the relative throughput\nof the complete telescope+instrument system as a function of wavelength. \nThis spectrophotometric calibration system will allow for the monitoring of the \ninstrumental throughput (e.g., to determine whether optical coatings evolve with time); \nit will also provide accurate knowledge of the filter transmission functions\nthat can be used to calculate precise Supernova k-corrections and improved photometric redshift measurements. \nWe plan to use DECal to monitor the throughput of the telescope at regular\nintervals (i.e., $\\sim$once per month) during the five-year survey to monitor the\ninstrumental performance.\nDECal will be installed permanently at the Blanco telescope and will be available for general use. \n\nSimilar spectrophotometric calibration systems have been tested and\ninstalled on the CTIO 4 m Blanco telescope \\citep{stubbs2007} \nand the PanSTARRS telescope \\citep{panstarrs}.\nBoth of these systems use a tunable laser as a light source. The DECal system uses a \nmonochromator-based light source in place of a tunable laser, which \ngenerally requires less maintenance and other personnel attention than a tunable \nlaser, and is well suited for routine use over an extended period of time. \nThe DECal system also has the advantage of its extended and continuous \nwavelength coverage.  A system such as DECal can operate from the far UV into the infrared (250 $<\\lambda<$ 2400 nm) \nwith relatively minor modifications.  The disadvantage \nof a monochromator-based light source is that it cannot produce as much light\nas a tunable laser, so integration times must be longer and the dome must be kept very dark.  \n\nWe have successfully deployed a prototype of the DECal spectrophotometric calibration system at the Swope \n1 m and du Pont 2.5 m telescopes at Las Campanas Observatory in Chile. \nWe measured the throughput of the $u$, $g$, $r$, $i$, $B$, $V$, $Y$, $J$, $H$, and $K_s$ filters used in the WIRC and RetroCam instruments\nduring the Carnegie Supernova Project with an accuracy of \n1\\% \\citep{stritz}.\nA forthcoming paper on the completed DECal system will present the results of the calibration \nof the DECam instrument using the final DECal system once both DECam and DECal are installed at the CTIO 4m telescope.  \n\nThe complete DECal system consists of a new \nflat field screen, a daily dome flat field illumination system, and \nthe spectrophotometric calibration system.  These systems are described in detail below.\nA schematic of the entire calibration system is shown in Figure \\ref{fig:overview}.\nMore details on the final system may be found in a recent paper by \\citet{decal2012}.\n\n\n\\begin{figure}\n\\plotone{marshall_j_1.eps}\n\\caption{Schematic drawing of the DECal system.\n\\label{fig:overview}\n}\n\\end{figure}\n\n\n\\section{Flat Field Screen}\n\nThe flat field screen is an important part of any dome-based calibration\nsystem and should be carefully considered when designing the calibration scheme.  \nAn ideal flatfield screen would have 100\\% reflectivity at all wavelengths \nof interest and would reflect light only within the acceptance angle of the \ntelescope optics. Since the latter condition is almost impossible to satisfy, \nwe have chosen to provide uniform illumination of the top end of the telescope\nby producing an illumination pattern with a Lambertian profile.  \nA Lambertian surface reflects incident light such that  \nthe surface luminance of the screen as seen by the telescope is isotropic.   A flat field screen with Lambertian scattering\nproperties ensures that all points on the screen illuminate the focal plane\nwith the same angular profile regardless of where the illumination source is \nplaced. If the telescope is well baffled, illumination of this type should be \nequivalent to illuminating the telescope with incident light at only the acceptance angle of the telescope.\n\nThe DECal flat field screen is made of a 2x4 grid of 4 foot by 8 foot \nlightweight aluminum honeycomb panels coated with \na white, highly reflective, almost perfectly Lambertian coating.   \nThese panels are mounted on an extruded aluminum structure using screws that have \nbeen coated with the same coating as the screen.  \nThe white circular portion of the screen is slightly oversized (4.64 m) and is surrounded by a black ring made of \nsheet metal painted with flat black spray paint by Maaco (note that we do not recommend this \nvendor for black coatings).  Our studies show that it is important to \ninclude this outer ring on the flat field screen to minimize stray light reflected into the beam by the \nshiny internal surface of the dome.  We have also determined that any stray light \nreflected by the seams between the two panels or the heads of the screws is minimal \nand has almost no effect ($<<$1\\%) on the precision of photometric measurements that \nhave been flattened with this system; these results will be presented in a forthcoming paper describing the \ndeployed DECal system.\n\nSeveral candidate flat field screen coatings were tested for absolute reflectivity as well as reflectivity \nas a function of incidence angle (Lambertian-ness).\nMore details of this study are provided by \\citet{decal2010}.  \nOut of several candidate screen coatings we finally selected the nearly ideal \n``Duraflect'' coating provided by Labsphere, which provides high reflectivity \n(roughly 95\\% from 350 to 1200nm and greater than 85\\%  for 300 $<\\lambda<$ 2200nm).\nThis coating also boasts a nearly perfectly Lambertian scattering surface.  It is relatively durable and can be \n(gently) cleaned, an important quality for the dome environment.\n\n\n\\section{Dome Flat Field Illumination}\n\nThe DECal daily flat field calibration system provides for daily flat fields to be taken at \nthe telescope.  It uses high-power light-emitting diodes (LEDs) as the illumination source.  \nWe have selected one LED to illuminate each filter bandpass used in the DES survey,\nusually centered on the filter bandpass.   We have also provided LEDs at a wavelength that can \nilluminate a planned u-band filter, which will be added to DECam by CTIO in the near future. Figure \\ref{fig:led}\nshows the spectra of the selected LEDs along with the DES filter bandpasses.\n\n\\begin{figure}\n\\plotone{marshall_j_2.eps}\n\\caption{Spectra of selected DECal LEDs overplotted with the DES $grizy$ (and $u$) filter response curves. The normalization is arbitrary.\n\\label{fig:led}\n}\n\\end{figure}\n\nThe seven LEDs selected to illuminate the DECam filter bandpasses are manufactured by Roithner Lasertechnik and Luxeon Star.  \nThe model numbers of the LEDs are given in Table \\ref{table:leds}.\nThey are positioned at four locations around the top of the telescope ring and provide adequately flat illumination \nof the flat field screen.  \nThe flat field screen does not need to be perfectly uniformly illuminated to produce \na reasonably uniform illumination of the focal plane; however, large scale gradients \nshould be avoided to minimize the need to remove such trends from the data.  \nThe calibration system \ncan be tuned so that the relative power in each bandpass is about the same, i.e., DECam \nexposure times will be the same for each filter.\n\n\\begin{table}[!ht]\n\\label{table:leds}\n\\caption{DECal LEDs}\n\\smallskip\n\\begin{center}\n{\\small\n\\begin{tabular}{lll}\n\\tableline\n\\noalign{\\smallskip}\nManufacturer & Model Number & Central Wavelength\\\\\n\\noalign{\\smallskip}\n\\tableline\n\\noalign{\\smallskip}\nRoithner & H2A1-H365-S & 365 nm\\\\\nRoithner & H2A1-H650 & 650 nm\\\\\nRoithner & H2A1-H780 & 780 nm\\\\\nRoithner & H2A1-H905 & 905 nm\\\\\nRoithner & H2A1-H970 & 970 nm\\\\\nRoithner & H2A1-H1030 & 1030 nm\\\\\nLuxeon Star & Warm white (3100k) Rebel LED & broad\\\\\n\\noalign{\\smallskip}\n\\tableline\n\\end{tabular}\n}\n\\end{center}\n\\end{table}\n\nTesting of similar LED-based dome flat field systems \\citep{ff2005} shows that it \nis not necessary to illuminate the entire filter bandpass to provide adequate \nflat fields.  The exception to this, of course, is in the case that the detector\nquantum efficiency changes rapidly with wavelength.  This is usually not \nthe case for modern CCD detectors.  \n\n\n\\section{Spectrophotometric Calibration System}\n\nIn addition to broadband flat fields, DECal also projects nearly monochromatic (1--10nm bandwidth)\nflat field light onto the flat field screen to be used to measure the relative throughput of the instrument \nas a function of wavelength.  This spectrophotometric calibration is accomplished by \nimaging the monochromatic light incident on the flat field screen with DECam while at the same time\nmonitoring the amount of light on the screen with calibrated photodiodes placed around the top ring of the telescope.  \nThe signal received by the photodiodes is proportional to the light received by DECam; note however that we do not attempt to \nestimate the total amount of light gathered by the telescope, i.e., \nthis is a relative calibration, not an absolute measure of instrumental throughput.\nThe DECam images are compared with the photodiode output to determine the \nrelative sensitivity of the entire telescope+instrument optical system as \na function of wavelength.  \nThis part of the DECal system will be used to spectrophotometrically calibrate \nthe DES survey data, and to monitor the DECam instrument's sensitivity as a function of time.\n\nWe calculate that this system will have a peak output power of 2 mW resulting in approximately \n800 photons\/second\/pixel when imaged by DECam.  This flux level will require exposure times with DECam of \napproximately one minute to obtain adequate signal-to-noise ratio (S\/N) measurements\nof the DECam focal plane; given the relatively bright environment of the Blanco dome we anticipate that \nthe DECal measurements will best be made during a cloudy night: to minimize scattered light from the Sun during the daytime and \nto avoid interfering with regular nighttime observations.\n\n\\subsection{Monochromator}\nThe monochromatic light is produced by a fully automated Czerny-Turner monochromator \n(manufactured by Horiba, model iHR-320). We adjust the \nbandwidth of the light by varying the input slit width. The output slit width is fixed to \n900 microns (the width of the linear fiber bundle that feeds light to the top of the telescope). \nThe monochromator holds two light sources, multiple gratings on a turret, and all of the necessary \norder sorting filters to properly produce the monochromatic light at the appropriate wavelength. \nIt is completely remotely controlled by a Labview interface developed in our lab and is highly flexible \nin producing light of the appropriate wavelength, bandwidth, and interval between requested wavelengths. \n\n\\subsection{Fiber bundle}\nWe have designed and manufactured a custom 75 m long ``line-to-spot'' fiber bundle (assembled by Fibertech Optica) that \nuses a broad-spectrum fiber (Polymicro FBP-300660710). This fiber has excellent \ntransmission both in the UV and the IR, in contrast with standard optical fibers that have either good \ntransmission  in the infrared (due to low OH$^-$ content) or in the UV \n(high OH$^-$ content) but not at both ends of the spectrum simultaneously. \n\nThe bundle consists of 87 fibers with 300 micron cores. At the input end, the fibers are \narranged in 3 parallel lines of 29 fibers each.  The bundle brings the light from \nthe calibration room located beneath the telescope main floor through the telescope cable wrap and\nto the telescope top ring. At a \nlength of 65 m, the fiber bundle splits into four branches, each 10 m long and containing \n21 fibers. These 21 fibers are placed in a compact circular arrangement and coupled to the\nprojection system. Each of the four branches sample the monochromator slit evenly to ensure \nthat they all have the same intensity and spectral content.\n\nA fifth branch bifurcates from the main bundle 1 m from the input slit. It contains the \nthree central fibers and is fed to the monitor spectrometer (see below). To ensure that \nany absorption present in the fiber does not change the measured wavelength profile, \nthis fifth branch is also 75 m long even though the spectrometer is located only 1 m from the monochromator. \n\n\\subsection{Projection system}\n\nTo ensure a uniform \nillumination of the focal plane area, the output from the four fiber ends is collimated \nand then an engineered diffuser (manufactured by RPC Photonics)\nis used to diffuse the light projected onto the flat field screen by the fibers. \nThis type of diffuser can be designed \nto distribute incident collimated light in almost any pattern; in this case \nwe chose a 20 degree angle cone diffuser that projects the light in a top hat shape with a \ndiverging half-angle of 20 degrees, with more than 80\\% of the \nlight exiting the fiber bundle falling within this 40 degree cone of light. Engineered diffusers such as the one used in DECal are a far \nsuperior way to diffuse incident light as compared to a standard ground glass diffuser that would also produce an adequately\ndiffuse light pattern but would scatter much of the light away from the screen. \n\n\\subsection{Spectrometer}\n\nTo monitor in real time the spectral content of the illumination source, \na branch of three fibers bifurcates from the main fiber bundle and is fed to an echelle spectrometer \n(manufactured by Optomechanics Research, model SE-100). \nThe spectrometer measures both the central wavelength \nand the full width at half maximum (FWHM) of the light. For each DECal exposure, a spectrum of the DECal light is obtained and automatically analyzed \nto measure the central wavelength and FWHM of the illumination light with a precision of 0.1 nm. \nThe spectrometer calibration is verified with a Mercury calibration lamp.\n\n\\subsection{Reference photodiodes}\n\nThe goal of the spectrophotometric calibration system is to measure the relative throughput of the \ntelescope+instrument optical system at each wavelength.  This is accomplished by imaging the \nmonochromatic light incident on the flat field screen with the DECam imager while at the same time \nmonitoring the light on the screen to remove any fluctuations in brightness.  \nThe monitoring of the screen is done with the reference photodiodes.  \n\nThe reference photodiodes are mounted \nin the same locations as the fiber projection units at four points around the top ring of the telescope, facing the\nscreen (see insert on Figure \\ref{fig:overview}).   We use 10 mm diameter \nsilicon photodiodes (manufactured by Hamamatsu, model S2281) to measure the light in the range 300 $<\\lambda<$ 1100nm.\nWe acquire the signal from the photodiodes with a data acquisition unit and record the data with the Labview interface.\n\nThe photodiodes we use exhibit a slight change in sensitivity with temperature \nat the far red end of their sensitivity range. This effect is negligible below 1000nm but can reach a \nsensitivity change of up to 1\\%\/\\deg C at 1100nm. We monitor the photodiode temperature in real time \nand correct for it in the data reduction process. \n\n\n\\section{Results}\n\nWe have used a prototype of the DECal spectrophotometric calibration system to measure the \nsystem throughput as a function of wavelength for the WIRC infrared imager on the 2.5 m du Pont telescope and Retrocam optical and infrared imager on the 1 m \nSwope telescope at Las Campanas Observatory, to improve the calibration of the \nCarnegie Supernova Project (CSP) data obtained on these telescopes.\nA description of the DECal prototype used for these measurements is presented by \\citet{decal2010} and the \nresults of the successful calibration are applied to the optical CSP data and presented by \\citet{stritz}.\nIn brief, the DECal prototype used at the du Pont and Swope telescopes included a new flat field screen for each telescope \nwith the same properties but smaller in size than the flat field screen described above, a monochromator smaller but functionally the same \nas the final DECal monochromator, a custom line-to-spot fiber bundle using the same high throughput fibers used in DECal, \nand a monitor photodiode mounted behind the secondary mirror of each telescope.\nHere we present the results of the optical and infrared measurements made using the DECal prototype.\n\nFigure \\ref{fig:results_optical} shows the scans of \nthe six optical filters used to make CSP photometric measurements on \nthe Swope telescope, as well as the scan we made of the telescope+instrument system with no filter in place.   \nThe figures presented in this section give the relative throughput of the telescope, including losses by the primary and secondary mirrors, \ncorrector plate, filter, dewar window and CCD quantum efficiency.  \n\nThe scan of the CSP Retrocam $u$-band filter was particularly \nuseful in analyzing the CSP photometric data.  The manufacturer of the $u$-band filter did not provide \na complete throughput curve with the filter.  An assumption (which turned out to be not entirely correct)\nwas made about the spectral response of the blue edge of the $u$-band filter in the initial data reduction of CSP data; \nwe were able to measure the true spectral response of the $u$-band filter to thereby improve the CSP photometry with our measurements.\n\nFigures \\ref{fig:results_swopeir} and  \\ref{fig:results_irdupont} show scans of the infrared CSP \nbandpasses used on the Swope and du Pont telescopes, respectively.  \nEach infrared filter scan has been normalized independently, i.e., the relative throughput from filter to filter is not \nreflected in these figures.\nThese data will also be used to calibrate the CSP infrared photometric measurements, to be published in a forthcoming \nCSP data release.\nThe infrared system scan measurements had slightly lower signal-to-noise (S\/N) than the optical measurements; \nthis is the source of the noise seen at the red end of the $K_s$-band data in Figure \\ref{fig:results_irdupont}.\nHowever, we note that with multiple measurements or increased signal from a brighter light source or even a higher throughput \nmonochromator, higher S\/N measurements of the infrared filters could be made. \nThat is, a system such as DECal could be used to determine the spectral response of an imaging system well into the infrared\nwith only minimal modification.\n\n\\begin{figure}\n\\plotone{marshall_j_3.eps}\n\\caption{DECal prototype scan of the optical filters used by the CSP survey in the Retrocam instrument on the Swope telescope.\n\\label{fig:results_optical}\n}\n\\end{figure}\n\n\n\\begin{figure}\n\\plotone{marshall_j_4.eps}\n\\caption{DECal prototype scan of the infrared filters used by the CSP survey in the Retrocam instrument on the Swope telescope.\n\\label{fig:results_swopeir}\n}\n\\end{figure}\n\n\\begin{figure}\n\\plotone{marshall_j_5.eps}\n\\caption{DECal prototype scan of the infrared filters used by the CSP survey in the WIRC instrument on the du Pont telescope.\n\\label{fig:results_irdupont}\n}\n\\end{figure}\n\n\\section{Summary}\n\nDECal is a new calibration system that provides broadband and monochromatic flat fields for the DECam instrument \non the CTIO Blanco  4 m telescope.  DECal will\nbe used to flatten optical DECam images and to monitor the transmission function of the instrument+telescope system \nand provide spectrophotometric calibration of the system.\n\nWe have fully tested the components of DECal and have deployed a prototype of the system on the Las Campanas Observatory \nSwope and du Pont telescopes.  This prototype has informed the design of the final \nDECal system and was used to successfully calibrate the Carnegie Supernova Project optical and infrared photometry.\n\nThe DECal and DECam systems are currently in the final stages of installation on the CTIO Blanco 4 m telescope and will\nbe commissioned in Fall 2012.  We will use the DECal broadband flat field system to produce daily flat fields \nfor the DECam instrument, and the DECal spectrophotometric calibration system to produce monthly scans of the \nspectral response of the complete telescope+instrument system, through each DES filter and at all wavelengths.  \nThese data will be used to measure the \nspectral response of the instrument, which will enable the highly accurate and precise photometric measurements required by DES.  \nThe DECal system will also be used to monitor any changes in the spectral response of the instrument, \nincluding evolution of filter functions, degradation of optical coatings, and off-band optical leaks in the filters.  \n\n\n\\acknowledgements Texas A\\&M University thanks Charles R. '62 and Judith G. Munnerlyn, \nGeorge P. '40 and Cynthia Woods Mitchell, and their families for support of astronomical \ninstrumentation activities in the Department of Physics and Astronomy.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\\label{sec1}\n\\setcounter{equation}{0}\\setcounter{Theorem}{0}\\setcounter{Definition}{0}\n\nThe assumption of global conformal invariance -- which says that we are\ndealing with a single valued  representation of $SU(2,2)$ rather than\nwith a representation of its covering -- in  (4-dimensional) Minkowski\nspace has surprisingly strong consequences \\cite{NT01}. Combined with\nthe Wightman axioms, it implies \\emph{Huygens locality}, which yields\nthe vertex-algebra-type condition\n\\begin{equation}\\label{eq1.1}\n\\bigl(({\\mathrm{x}}-{\\mathrm{y}})^2\\bigr)^n \\, \\bigl[\\phi({\\mathrm{x}}), \\psi({\\mathrm{y}})\\bigr] = 0\n\\quad \\text{for} \\quad n \\gg 0 \\end{equation} for any pair $\\phi, \\psi$ of local\nBose fields ($n\\gg 0$ meaning ``$n$ sufficiently large''). Huygens\nlocality and energy positivity imply, in turn, rationality of\ncorrelation functions. A GCI quantum field theory (QFT) that admits a\nstress-energy tensor (something, we here assume) necessarily involves\ninfinitely many conserved symmetric tensor currents in the operator\nproduct expansion (OPE) of any Wightman field with its conjugate.\nThe twist two contributions give rise to a harmonic {\\it bifield}\n$V({\\mathrm{x}},{\\mathrm{y}})$, which is an important tool in the study of GCI QFT models\n\\cite{NST02,NST03,NRT05,BNRT07,NRT07}.  The spectacular development of\n2-dimensional (2D) \\emph{conformal field theory} in the 1980's\nis based on the preceding study of infinite dimensional (Kac--Moody\nand Virasoro) Lie algebras and their representations. A\nstraightforward generalization of this tool did not seem to apply\nin higher dimensions. After the first attempts\nto construct (4D) Poincar{\\'e} invariant Lie fields led to examples\nviolating energy positivity \\cite{L67}, it was proven \\cite{B76},\nthat scalar Lie fields do not exist in three or more dimensions.\nIt is therefore important to realize that the argument does not pass to\n{\\em bi\\\/}fields, and that the above mentioned harmonic bifields do\ngive rise to infinite dimensional Lie algebras.\n\nConsider bilocal fields of the form\n\\begin{equation} \\label{eq1.1a}\nV_M({\\mathrm{x}},{\\mathrm{y}}) \\, = \\, \\sum_{ij} \\, M_{ij} \\ {:}\\varphi_i({\\mathrm{x}})\\varphi_j({\\mathrm{y}}){:}\n\\,, \\end{equation}\nwhere $M$ is a real matrix and $\\varphi_j$ are a system of independent\nreal massless free fields. According to Wick's theorem, the\ncommutator of $V_{M_1}({\\mathrm{x}}_1,{\\mathrm{x}}_2)$ and $V_{M_2}({\\mathrm{x}}_3,{\\mathrm{x}}_4)$ is:\n\\begin{eqnarray}\n\\label{eq2.1n}\n&& \\hspace{-20pt}\n\\bigl[V_{M_1}({\\mathrm{x}}_1,{\\mathrm{x}}_2), V_{M_2}({\\mathrm{x}}_3,{\\mathrm{x}}_4)\\bigr]\n\\, = \\,\n\\Delta_{2,3} \\, V_{M_1M_2}({\\mathrm{x}}_1,{\\mathrm{x}}_4)\n\\, + \\,\n\\Delta_{2,4} \\, V_{M_1\\trp{M_2}}({\\mathrm{x}}_1,{\\mathrm{x}}_3)\n\\nonumber \\\\ && \\hspace{-20pt}\n\\qquad \\hspace{10pt}\n\\, + \\\n\\Delta_{1,3} \\, V_{\\trp{M_1}M_2}({\\mathrm{x}}_2,{\\mathrm{x}}_4)\n\\, + \\,\n\\Delta_{1,4} \\, V_{\\trp{M_1}\\trp{M_2}}({\\mathrm{x}}_2,{\\mathrm{x}}_3)\n\\nonumber \\\\ && \\hspace{-20pt}\n\\qquad \\hspace{10pt}\n\\, + \\\n\\mathrm{tr} \\, (M_1 M_2) \\Delta_{12,34} + \\mathrm{tr} \\, (\\trp{M_1}\nM_2) \\Delta_{12,43}, \n\\end{eqnarray}\nwhere $\\trp{M}$ is the transposed matrix, $\\Delta_{j,k}$ is the free field\ncommutator, $\\Delta_{j,k}$ $=$ $\\Delta^+_{j,k} - \\Delta^+_{k,j}$,\\ and\n$\\Delta_{jk,lm}$ $=$ $\\Delta^+_{j,m}\\Delta^+_{k,l} - \\Delta^+_{m,j}\\Delta^+_{l,k}$\\\nfor\\ $\\Delta^+_{j,k}$ $:=$ $\\Delta_+({\\mathrm{x}}_j$ $-$ ${\\mathrm{x}}_k)$, the two point\nmassless scalar correlation function.\n\nIt is one of the main results of \\cite{NRT07} that the same abstract\nstructure can be derived from first principles in GCI quantum field theory.\nMore precisely, the twist two bilocal fields appearing in the OPE of\nany two scalar fields of dimension 2 can be linearly labeled by\nmatrices $M$ such that the commutation relations \\eqref{eq2.1n} hold. {}\nFrom this, the representation \\eqref{eq1.1a} can be deduced.\nIn the present paper we shall consider only finite size matrices; in\ngeneral, the system of independent massless free fields can be infinite\nand then the $M$'s should be assumed to be Hilbert--Schmidt operators.\n\nThe question arises, whether there are nontrivial linear subspaces\n$\\mathcal{M}$ of real matrix algebras upon which the commutation\nrelations of the corresponding bifields $V_M$ ($M \\in \\mathcal{M}$) close.\nWe shall call such systems of bifields \\textit{Lie systems},\nor, \\textit{Lie bifields}.\nIt follows from (\\ref{eq2.1n}) that if $\\mathcal{M}$ is a {\\em $t$-subalgebra}\n(i.e., a subalgebra closed under transposition)\nof the real matrix algebra, then $\\{V_M\\}_{M \\, \\in \\, \\mathcal{M}}$\nis a Lie system. Conversely, any Lie system corresponds to a subalgebra\n$\\mathcal{M}$ such that $\\trp{M_1}M_2$, $M_1\\trp{M_2}$,\n$\\trp{M_1}\\trp{M_2}$ $\\in\\mathcal{M}$ whenever $M_1,M_2 \\in\\mathcal{M}$.\nIn particular, if $\\mathcal{M}$ contains the identity matrix,\nthen it is a $t$-subalgebra.\n\n\\section{$t$-subalgebras of real matrix algebras}\\label{sec2a}\n\\setcounter{equation}{0}\\setcounter{Theorem}{0}\\setcounter{Definition}{0}\n\nLet us consider $t$-subalgebras $\\mathcal{M}$ of the matrix algebra $Mat(L,{\\mathbb R})$,\nwhere $L$\nis a positive integer (equal to the number of fields $\\varphi_j$).\nThe classification of all such $\\mathcal{M}$ is a classical mathematical problem,\nwhich goes back to F.G.~Frobenius, I.~Schur, and J.H.M.~Wedderburn\n(see, e.g., \\cite[Chapter XVII]{L02} and \\cite[Chapter 9, Appendix II]{B82}).\n\nWe first observe that $\\mathcal{M}$ is equipped with the Frobenius inner product\n\\begin{equation}\\label{eq2.1xx}\n\\langle M_1, M_2 \\rangle = \\mathrm{tr} \\, (\\trp{M_1}M_2) = \\sum_{ij} \\,\n(M_1)_{ij} \\, (M_2)_{ij} \\,,\n\\end{equation}%\nwhich is symmetric, positive definite, and has the property\n$\\langle M_1 M_2,\\,M_3 \\rangle=\\langle M_1,$ $M_3\\;\\trp{M_2} \\rangle$.\nThis implies that for every right ideal $\\mathcal{I} \\subset\\mathcal{M}$,\nthe orthogonal complement $\\mathcal{I}^\\perp$ is again a right ideal.\nNote also that $\\mathcal{I}$ is a right ideal if and only if\n$\\trp{\\mathcal{I}}$ is a left ideal.\nTherefore, $\\mathcal{M}$ is a {\\em semisimple} algebra (i.e., a direct sum of\nleft ideals), and every module over $\\mathcal{M}$ is a direct sum of\nirreducible ones.\n\nNow assume, without loss of generality, that the algebra\n$\\mathcal{M} \\subset End_{\\,{\\mathbb R}}\\ \\mathcal{L} \\cong Mat(L,{\\mathbb R})$\nacts irreducibly on the vector space $\\mathcal{L} \\cong {\\mathbb R}^L$.\nLet $\\mathcal{M}'\\subset End_{\\,{\\mathbb R}}\\ \\mathcal{L}$ be the {\\em commutant} of $\\mathcal{M}$,\ni.e., the set of all matrices $M$ commuting with all elements of $\\mathcal{M}$.\nThen by Schur's lemma\n(whose real version \\cite{L02} is much less popular than the complex one),\n$\\mathcal{M}'$ is a real division algebra.\nBy the Frobenius theorem, $\\mathcal{M}'$ is isomorphic to ${\\mathbb R}$, ${\\mathbb C}$, or ${\\mathbb H}$\nas a real algebra (where ${\\mathbb H}$ denotes the algebra of {\\em quaternions}).\nFinally, the classical Wedderburn theorem gives that\n$\\mathcal{M}$ is isomorphic to the matrix algebra $End_{\\mathcal{M}'}\\mathcal{L}$.\nIn addition, since $\\mathcal{M}$ is closed under transposition, then $\\mathcal{M}'$\nis also a $t$-algebra, and the transposition in $\\mathcal{M}'$\ncoincides with the conjugation in ${\\mathbb R}$, ${\\mathbb C}$, or ${\\mathbb H}$, respectively.\n\nObserve that, since $\\mathcal{M} \\cong End_{\\mathbb F}\\,\\mathcal{L}$\n(where ${\\mathbb F}={\\mathbb R}$, ${\\mathbb C}$, or ${\\mathbb H}$),\nwe can view $\\mathcal{L}$ as a left ${\\mathbb F}$-module on which $\\mathcal{M}$ acts\n${\\mathbb F}$-linearly. Alternatively, $\\mathcal{L}$ can be made an $(\\mathcal{M},{\\mathbb F})$-bimodule\nby setting $M \\cdot f \\cdot M' := M \\, (\\trp{M'}) \\, f$ for\n$f \\in \\mathcal{L}$, $M \\in \\mathcal{M}$ and $M' \\in \\mathcal{M}' \\cong {\\mathbb F}$.\nThen the embedding ${\\mathbb F}\\subset End_{\\mathbb F}\\,\\mathcal{L}\\cong\\mathcal{M}$ endows $\\mathcal{L}$\nwith the structure of an ${\\mathbb F}$-bimodule. In other words,\nwe have two commuting copies,\nleft and right, of ${\\mathbb F}$ in $End_{\\,{\\mathbb R}}\\ \\mathcal{L}$,\nwhich are subalgebras of $\\mathcal{M}$ and $\\mathcal{M}'$, respectively.\nMoreover, denoting $N=dim_{{\\mathbb F}} \\, \\mathcal{L}$, we have:\n$L = dim_{{\\mathbb R}} \\, \\mathcal{L} = N \\, dim_{{\\mathbb R}} \\, {\\mathbb F} =N,2N$ or $4N$\nwhen ${\\mathbb F}={\\mathbb R},{\\mathbb C}$ or ${\\mathbb H}$, respectively.\n\nIf $\\mathcal{M}$ is not an irreducible $t$-subalgebra of $Mat(L,{\\mathbb R})$,\ni.e., $\\mathcal{L} \\cong {\\mathbb R}^L$ is not an irreducible $\\mathcal{M}$-module,\nthen $\\mathcal{L}$ splits into irreducible submodules, each of them of the above\nthree types:\n\\begin{equation}\\label{e1.3}\n\\mathcal{L} \\, = \\, \\mathcal{L}_{{\\mathbb R}} \\, \\oplus \\, \\mathcal{L}_{{\\mathbb C}} \\, \\oplus \\, \\mathcal{L}_{{\\mathbb H}} \\,,\n\\end{equation} where each $\\mathcal{L}_{{\\mathbb F}}$ (${\\mathbb F}={\\mathbb R},{\\mathbb C},{\\mathbb H}$)\nis an ${\\mathbb F}$-module such that $\\mathcal{M}$ acts on it ${\\mathbb F}$-linearly.\n In our QFT application, the space $\\mathcal{L}$ is the real linear span of the\nreal massless scalar fields $\\varphi_j$, and then a Lie system of bifields\n$V_{M}$ splits into three subsystems: of types ${\\mathbb R}$, ${\\mathbb C}$ and ${\\mathbb H}$.\nThe first two cases were considered in a previous paper \\cite{BNRT07}\nand led to gauge groups of type $U(N,{\\mathbb R}) = O(N)$ and $U(N,{\\mathbb C}) = U(N)$, respectively,\nwhere $N=dim_{{\\mathbb F}} \\, \\mathcal{L}$.\nHere we are going to consider the third case in which, as we shall see,\nthe gauge groups that arise are of type $U(N,{\\mathbb H}) = Sp(2N)$,\nthe compact real form of the {\\em symplectic group}.\n\nIn each of the three cases, the associated infinite-dimensional\nLie algebra (\\ref{eq2.1n}) has a central charge proportional to the\norder $N$ of the gauge group $U(N,{\\mathbb F})$.\n\n\n\\section{Irreducible Lie bifields and associated dual pairs}\\label{sec2}\n\\setcounter{equation}{0}\\setcounter{Theorem}{0}\\setcounter{Definition}{0}\n\nIn this section we consider Lie bifields $\\{V_{M}\\}_{M\\in\\mathcal{M}}$\ncorresponding to irreducible $t$-subalgebras $\\mathcal{M}$ of $Mat(L,{\\mathbb R})$.\nAs discussed in the previous section, we have\n$\\mathcal{M}\\cong End_{\\mathcal{M}'}\\mathcal{L}$, where $\\mathcal{L} \\cong {\\mathbb R}^L$ and\nthe commutant $\\mathcal{M}'\\cong$ ${\\mathbb R}$, ${\\mathbb C}$, or~${\\mathbb H}$.\n\nIn the case when $\\mathcal{M}'\\cong{\\mathbb R}$ and $dim_{\\mathbb R}\\, \\mathcal{L}=1$, we have one bifield\n\\begin{equation}\n\\label{eq2.1a}\nV({\\mathrm{x}},{\\mathrm{y}}) \\, = \\, {:}\\varphi({\\mathrm{x}}) \\varphi({\\mathrm{y}}){:} \\,. \\end{equation}\nMore generally, $V$ can be taken a sum of $N$ independent copies\nof Lie bifields of type (\\ref{eq2.1a}),\n\\begin{equation}\n\\label{eq2.2a}\nV({\\mathrm{x}},{\\mathrm{y}}) \\equiv V_{(N)}({\\mathrm{x}},{\\mathrm{y}}) \\, = {:}\\mbf{\\varphi}({\\mathrm{x}})\n\\mbf{\\varphi}({\\mathrm{y}}){:} \\, =  \\mathop{\\sum}\\limits_{j \\, = \\, 1}^N\n{:}\\varphi_j({\\mathrm{x}}) \\varphi_j({\\mathrm{y}}){:} \\,,\n\\end{equation}\nwhich is invariant under the \\textit{gauge group} $O(N)$ (including\nreflections). Here $L=N$ and $O(N)$ is realized as\nthe group of linear automorphisms of $\\mathcal{L} =\nSpan_{{\\mathbb R}} \\{\\varphi_j\\}$ preserving the quadratic form\n(\\ref{eq2.2a}) in $\\varphi_j$. In this case the\n\\textit{field Lie algebra} (i.e., the Lie algebra of field modes\ncorresponding to the eigenvalues of the one-particle energy, see the Appendix)\nis isomorphic to a central extension of $sp(\\infty,{\\mathbb R})$ of central charge $N$; see \\cite{BNRT07}.\n\nThe case when $\\mathcal{M}' \\cong {\\mathbb C}$ and $dim_{\\mathbb C}\\, \\mathcal{L}=1$ is given by two\nreal bifields, $V_{\\Mbf{1}}$ and $V_{\\varepsilon}$ that correspond to the $2\n\\times 2$ matrices\n\\begin{equation}\\label{eq2.3a}\n\\Mbf{1} \\, = \\, \\left(\\hspace{-4pt} \\begin{array}{cc} 1 & 0 \\\\ 0 & 1 \\end{array} \\hspace{-2pt}\\right)\n\\, , \\quad\n\\varepsilon \\, = \\, \\left(\\hspace{-4pt} \\begin{array}{rc} 0 & 1 \\\\ -1 & 0 \\end{array} \\hspace{-2pt}\\right) .\n\\end{equation}\nThey are thus generated by two independent real massless fields $\\varphi_1 ({\\mathrm{x}})$ and $\\varphi_2({\\mathrm{x}})$:\n\\begin{eqnarray}\\label{eq2.4a}\nV_{\\Mbf{1}} ({\\mathrm{x}},{\\mathrm{y}}) \\, = && \\hspace{-15pt} {:}\\varphi_1({\\mathrm{x}}) \\varphi_1({\\mathrm{y}}){:} + {:}\\varphi_2({\\mathrm{x}}) \\varphi_2({\\mathrm{y}}){:} \\,,\n\\nonumber \\\\\nV_{\\varepsilon} ({\\mathrm{x}},{\\mathrm{y}}) \\, = && \\hspace{-15pt} {:}\\varphi_1({\\mathrm{x}}) \\varphi_2({\\mathrm{y}}){:} - {:}\\varphi_2({\\mathrm{x}}) \\varphi_1({\\mathrm{y}}){:}\n\\,.\n\\end{eqnarray}\nCombining $\\varphi_1$ and $\\varphi_2$ into one complex field\n$\\mbf{\\varphi} ({\\mathrm{x}}) = \\varphi_1 ({\\mathrm{x}}) + i \\varphi_2({\\mathrm{x}})$ we get that\n$V_{\\Mbf{1}}$ and $V_\\varepsilon$ are the real and the imaginary parts of the\ncomplex bifield\n\\begin{equation}\\label{eq2.5a}\nW ({\\mathrm{x}},{\\mathrm{y}}) \\, = \\, {:}\\mbf{\\varphi}^* ({\\mathrm{x}}) \\mbf{\\varphi} ({\\mathrm{y}}){:}\n\\, = \\, V_{\\Mbf{1}} ({\\mathrm{x}},{\\mathrm{y}}) + i \\, V_\\varepsilon ({\\mathrm{x}},{\\mathrm{y}}) .\n\\end{equation}\nTaking again $N$ independent copies of such Lie bifields,\n\\begin{equation}\\label{eq2.6a}\nW_{(N)} ({\\mathrm{x}},{\\mathrm{y}}) \\, = \\, \\mathop{\\sum}\\limits_{j \\, = \\, 1}^N  {:}\\mbf{\\varphi}^*_j ({\\mathrm{x}}) \\mbf{\\varphi}_j ({\\mathrm{y}}){:}\n\\, , \\quad\n\\mbf{\\varphi}_j ({\\mathrm{x}}) \\, = \\, \\varphi_{1,j} ({\\mathrm{x}}) + i \\, \\varphi_{2,j} ({\\mathrm{x}}),\n\\end{equation}\nwe get a {\\em gauge group} $U (N)$, where $L=2N$.\nThe {\\em field Lie algebra} in this second case is\nisomorphic to a central extension of $u(\\infty,\\infty)$ again of central charge $N$ (\\cite{BNRT07}).\n\nFinally, for $\\mathcal{M}' = {\\mathbb H}$ the minimal size of the matrices in\n$\\mathcal{M}$ is four. We can formally derive the basic bifields $V_{M}$ in\nthis case as in the above complex case~(\\ref{eq2.5a}). Let us combine\nthe four independent scalar fields $\\varphi_j ({\\mathrm{x}})$ ($j=0,1,2,3$)\nin a single ``quaternionic-valued'' field and its conjugate:\n\\begin{eqnarray}\n\\label{eq2.6}\n\\mbf{\\varphi} ({\\mathrm{x}}) \\, = && \\hspace{-15pt} \\varphi_0 ({\\mathrm{x}}) + \\varphi_1 ({\\mathrm{x}}) \\, I + \\varphi_2 ({\\mathrm{x}}) \\, J + \\varphi_3 ({\\mathrm{x}}) \\, K\n, \\quad\n\\nonumber \\\\\n\\mbf{\\varphi}^+ ({\\mathrm{x}}) \\, = && \\hspace{-15pt} \\varphi_0 ({\\mathrm{x}}) - \\varphi_1 ({\\mathrm{x}}) \\, I - \\varphi_2 ({\\mathrm{x}}) \\, J - \\varphi_3 ({\\mathrm{x}}) \\, K,\n\\end{eqnarray}\nwhere $I, J, K$ are the (imaginary) quaternionic units\nsatisfying $IJ = K = -JI,\nI^2 = J^2 = K^2 = -1$. This allows us to write a quaternionic bifield $Y$ as\n\\begin{equation}\n\\label{eq2.7}\nY({\\mathrm{x}},{\\mathrm{y}}) \\, = \\, {:}\\mbf{\\varphi}^+({\\mathrm{x}}) \\, \\mbf{\\varphi}({\\mathrm{y}}){:}\n\\, = \\,  V_0({\\mathrm{x}},{\\mathrm{y}}) + V_1({\\mathrm{x}},{\\mathrm{y}}) \\, I + V_2({\\mathrm{x}},{\\mathrm{y}}) \\, J + V_3({\\mathrm{x}},{\\mathrm{y}}) \\, K,\n\\end{equation}\nwhere the components $V_\\alpha$ ($\\alpha = 0, 1, 2, 3$) of $Y$ can be\nfurther expressed in terms of the 4-vectors $\\mbf{\\varphi}$ and a\n$4\\times 4$ matrix realization of the quaternionic units in a manner\nsimilar to~(\\ref{eq2.4a}):\n\\begin{eqnarray}\n\\label{eq2.8}\n&\nV_\\alpha({\\mathrm{x}},{\\mathrm{y}}) \\, \\equiv \\, V_{\\ell_{\\alpha}} \\bigl({\\mathrm{x}},{\\mathrm{y}}\\bigr) \\,\n\\, = \\, {:}\\mbf{\\varphi}(x) \\,\\ell_\\alpha \\, \\Mbf{\\varphi}(y){:} \\, , \\quad\n&\n\\nonumber \\\\\n&\n\\nonumber \\\\\n&\n\\ell_0 = \\Mbf{1} ,\\qquad\n\\ell_1 \\, = \\, \\left(\\hspace{0pt} \\begin{array}{rrrr}                                                                                                                                       0 & 1 & 0 & 0 \\\\\n{\\hspace{-8pt}}-{\\hspace{-2pt}}1 & 0 & 0 & 0 \\\\\n0 & 0 & 0 & {\\hspace{-8pt}}-{\\hspace{-2pt}}1 \\\\\n0 & 0 & 1 & 0\n\\end{array}\n\\right)\n,\\qquad\n&\n\\nonumber \\\\\n&\n\\ell_2 \\, = \\,\n\\left(\\hspace{0pt}\n\\begin{array}{rrrr}\n0 & 0 & 1 & 0 \\\\\n0 & 0 & 0 & 1 \\\\\n{\\hspace{-8pt}}-{\\hspace{-2pt}}1 & 0 & 0 & 0 \\\\\n0 &{\\hspace{-8pt}}-{\\hspace{-2pt}}1 & 0 & 0\n\\end{array} \\right)\n,\\qquad\n\\ell_3 \\, = \\,\n\\left(\\hspace{0pt}\n\\begin{array}{rrrr}\n0 & 0 & 0 & 1 \\\\\n0 & 0 &{\\hspace{-8pt}}-{\\hspace{-2pt}}1 & 0 \\\\\n0 & 1 & 0 & 0 \\\\\n{\\hspace{-8pt}}-{\\hspace{-2pt}}1 & 0 & 0 & 0\n\\end{array} \\right) .\n\\end{eqnarray}\nIt is straightforward to check that the $4\\times 4$ matrices $\\ell_{\\alpha}$\ngenerate the quaternionic algebra ${\\mathbb H} \\cong \\mathcal{M}$.\nThe commutant $\\mathcal{M}'$ in $Mat(4,{\\mathbb R})$ is spanned by the unit matrix and\nanother realization of the imaginary quaternionic units as a set of real\nantisymmetric $4\\times 4$ matrices $r_k$ $(k=1,2,3)$.\nThe two sets $\\{r_k\\}_{k\\, = \\, 1}^3$ and $\\{\\ell_k\\}_{k\\, = \\, 1}^3$\ncorrespond to the splitting of the Lie algebra $so(4)$\ninto a direct sum of two $so(3)$ algebras:\n\\begin{eqnarray}\n\\label{eq2.9}\n&\n\\ell_1 \\, = \\, \\sigma_3 \\otimes \\varepsilon\n\\, , \\quad\n\\ell_2 \\, = \\, \\varepsilon \\otimes \\Mbf{1}\n\\, , \\quad\n\\ell_3 \\, = \\, \\ell_1 \\, \\ell_2 \\, = \\, \\sigma_1 \\otimes \\varepsilon \\,,\n\\quad\n&\n\\nonumber \\\\\n&\nr_1 \\, = \\, \\varepsilon \\otimes \\sigma_3\n\\, , \\quad\nr_2 \\, = \\, \\Mbf{1} \\otimes \\varepsilon\n\\, , \\quad\nr_3 = r_1 \\, r_2 \\, = -r_2 \\,r_1 \\, = \\, \\varepsilon \\otimes \\sigma_1  \\,,\n\\quad\n&\n\\end{eqnarray}\nwhere $\\sigma_k$\nare the Pauli matrices and $\\varepsilon = i\\sigma_2$\nas in (\\ref{eq2.3a}).\n\nWe shall demonstrate that the quaternionic field $Y$ (\\ref{eq2.7}) generates\na central extension of\nthe Lie algebra\\footnote{%\nFor a description of the Lie algebra $so^* (2n)$ of the noncompact\ngroup $SO^*(2n)$ \nand of its highest weight representations, see \\cite{EHW83}. For an oscillator realization of the\nLie superalgebra $osp(2m^*\\vert 2n)$ (with even subalgebra $so^*(2m)\\times\nsp(2n)$), see \\cite{GS91}.\nIf we view $so^* (4\\infty)$ as an inductive limit of $so^*(4n)$\nthen the central extension is trivial.}\n$so^* (4\\infty)$.\nTo this end, we represent $Y$ by a pair of complex bifields\n\\begin{equation}\\label{eq3.11it}\n\\begin{array}{l}\nW ({\\mathrm{x}},{\\mathrm{y}}) = \\, \\frac{1}{2} \\, \\Bigl(V_0 ({\\mathrm{x}},{\\mathrm{y}}) + i V_3\n({\\mathrm{x}},{\\mathrm{y}})\\Bigr) \\,\n\\\\\n\\qquad \\qquad \\qquad\n= \\, :{\\hspace{-2pt}} \\psi^*_1 ({\\mathrm{x}}) \\, \\psi_1 ({\\mathrm{y}}) {\\hspace{-2pt}}:\n+ :{\\hspace{-2pt}} \\psi^*_2 ({\\mathrm{x}}) \\, \\psi_2 ({\\mathrm{y}}) {\\hspace{-2pt}}:\n\\, = \\, W ({\\mathrm{y}},{\\mathrm{x}})^*\n\\, , \\quad\n\\\\\n\\hfill\n\\\\\nA ({\\mathrm{x}},{\\mathrm{y}}) = \\, \\frac{1}{2} \\, \\Bigl(V_1 ({\\mathrm{x}},{\\mathrm{y}}) - i V_2({\\mathrm{x}},{\\mathrm{y}})\\Bigr)\n\\\\\n\\qquad \\qquad \\qquad\n= \\, \\psi_1 ({\\mathrm{x}}) \\, \\psi_2 ({\\mathrm{y}}) - \\psi_2 ({\\mathrm{x}}) \\,\n\\psi_1 ({\\mathrm{y}}) \\, = \\, - A ({\\mathrm{y}},{\\mathrm{x}})\n\\, , \\qquad\n\\end{array}\n\\end{equation}\nand their conjugates, where $\\psi_{\\alpha}$ are complex linear\ncombinations of $\\varphi_{\\nu}$:\n\\begin{equation}\\label{eq3.12it}\n\\psi_1 \\, = \\, \\frac1{\\sqrt{2}} \\Bigl( \\varphi_0 + i \\varphi_3 \\Bigr)\n\\, , \\quad\n\\psi_2 \\, = \\, \\frac1{\\sqrt{2}} \\Bigl( \\varphi_1 - i \\varphi_2 \\Bigr)\\,.\n\\end{equation}\n\nSubstituting as above each $\\varphi_{\\nu}$ (respectively $\\psi_{\\alpha}$) by\nan $N$-vector of commuting free fields we can write the nontrivial local\ncommutation relations (CR) of $W(1,2)$ $\\equiv W ({\\mathrm{x}}_1,{\\mathrm{x}}_2)$ and $A(1,2)$\nin the form\n\\begin{eqnarray}\\label{eq3.13it}\n\\bigl[ W^* (1,2), W(3,4) \\bigr]\n\\, = && \\hspace{-15pt}\n\\Delta_{1,3} \\, W(2,4) + \\Delta_{2,4} \\, W^* (1,3) + 2N \\Delta_{12,43} \\, ; \\quad\n\\\\ \\label{eq3.14it}\n\\bigl[ W(1,2), A(3,4) \\bigr]\n\\, = && \\hspace{-15pt}\n\\Delta_{1,3} \\, A(2,4) - \\Delta_{1,4} \\, A (2,3) \\, , \\quad\n\\nonumber \\\\\n\\bigl[ W(1,2), A^*(3,4) \\bigr]\n\\, = && \\hspace{-15pt}\n\\Delta_{2,3} \\, A^* (1,4) - \\Delta_{2,4} \\, A^* (1,3) \\, , \\quad\n\\nonumber \\\\\n\\bigl[ A^*(1,2), A(3,4) \\bigr]\n\\, = && \\hspace{-15pt}\n\\Delta_{1,3} \\, W(2,4) - \\Delta_{1,4} \\, W (2,3)\n+\\Delta_{2,4} \\, W(1,3)\n\\nonumber \\\\ && \\hspace{-15pt}\n- \\, \\Delta_{2,3} \\, W (1,4) + 2N \\bigl(\\Delta_{12,43}-\\Delta_{12,34}\\bigr)\n\\,. \\quad\n\\end{eqnarray}\nIn particular, $W$\ncoincides with $W_{(2N)}$ in (\\ref{eq2.6a}) and\ngenerates the $u (\\infty,\\infty)$ algebra (of even central charge),\nwhich contains the compact Cartan subalgebra of $so^*\n(4\\infty)$; see Appendix~A. On the other hand, it is\nstraightforward to display the gauge group in the original\npicture as the invariance group of the quaternionic valued bifield\n$Y$~(\\ref{eq2.7}) viewed as a quaternionic form in the\n$N$-dimensional space of real quaternions. We obtain the group\nof $N\\times N$ unitary matrices with quaternionic entries\n\\begin{equation}\\label{eq3.15it}\nU (N,{\\mathbb H}) \\, = \\, Sp(2N) \\,\\equiv\\, USp(2N) \\,,\n\\end{equation}\ni.e., the compact group of unitary complex\nsymplectic $2N \\times 2N$ matrices.\n\n\\section{Unitary positive energy representations and\nsuperselection  structure}\\label{sec4}\n\n\\setcounter{equation}{0}\\setcounter{Theorem}{0}\\setcounter{Definition}{0}\n\nTwo important developments, one in QFT, the other in representation theory,\noriginated half a century ago from the talks of Rudolf Haag and Irving\nSegal at the first Lille conference \\cite{Lille57} on mathematical\nproblems in QFT. Later they gradually drifted apart and lost sight of\neach other. The work of the Hamburg--Rome--G\\\"ottingen school on the\noperator algebra approach to local quantum physics \\cite{H92}\nculminated in the theory of (global) gauge groups and superselection\nsectors \\cite{DR90,BDLR92}. The parallel development of the theory of\nhighest weight modules of semisimple Lie groups (and of the related\ndual pairs) can be traced back from \\cite{EHW83, H89, S90}. Here we aim at\ncompleting the task, undertaken in \\cite{BNRT07} of (restoring and)\ndisplaying the relationship between the two developments.\n\nBefore formulating the main result of this section we shall rewrite\nthe CR (\\ref{eq3.13it}), (\\ref{eq3.14it}) in terms of the discrete\nmodes of $W, A$ and $A^*$ and introduce along the way the conformal\nHamiltonian. We first list the $u(\\infty, \\infty)$ modes of $W$\n\\cite{BNRT07} and write down their CR. Here belong the generators\n$E_{ij}^\\epsilon$ $(\\epsilon=+, -)$ of the maximal compact subalgebra\n$u(\\infty) \\oplus u(\\infty)$ of $u(\\infty, \\infty)$ and of the\nnoncompact raising and lowering operators $X_{ij}$ and $X_{ij}^*$, respectively\n($i,j=1,2,\\dots$)\nsatisfying\n\\begin{eqnarray}\n[E^+_{ij},E^+_{kl}] = \\delta_{jk} E^+_{il} - \\delta_{il} E^+_{kj}, \\hspace{10pt}\n[E^-_{ij},E^-_{kl}] = \\delta_{jk} E^-_{il} - \\delta_{il} E^-_{kj}, \\hspace{10pt}\n[E^+_{ij},E^-_{kl}] = 0, \\hspace{-16pt} \\nonumber\n\\end{eqnarray} \\vskip-9mm\n\\begin{eqnarray}\n[E^+_{ij},X^*_{kl}] = \\delta_{jl} X^*_{ki},\\qquad\n[E^+_{ij},X_{kl}] = - \\delta_{il} X_{kj},\\nonumber \\\\[1mm]\n[E^-_{ij},X^*_{kl}] = \\delta_{jk} X^*_{il}, \\qquad\n[E^-_{ij},X_{kl}] = - \\delta_{ik} X_{jl}, \\nonumber\n\\end{eqnarray} \\vskip-9mm\n\\begin{eqnarray} \\label{eq4.1it}\n[X_{ij},X_{kl}^*] = \\delta_{ik} E^+_{lj} + \\delta_{jl} E^-_{ki} \\;.\n\\end{eqnarray}\nThe commuting diagonal elements $E_{ii}^\\epsilon$ span a compact\n{\\it Cartan subalgebra}. The {\\it antisymmetric bifield} $A$ gives\nrise to an {\\it abelian algebra} spanned by the {\\it raising\n  operators} $Y_{ij}^+ = -Y_{ji}^+$, the {\\it lowering operators}\n$(Y_{ij}^-)^* = -(Y_{ji}^-)^*$ and the operators $F_{ij}$;\nthe modes of $A^*$ are hermitian conjugate to those of $A$.\nThe above $E$'s together with the $F_{ij}$ and their conjugates,\n$F_{ij}^*$, give rise to the maximal compact subalgebra $u(2\\infty)$\nof $so^*(4\\infty)$.\nThe additional nontrivial CR can be restored (applying\nwhen necessary hermitian conjugation) from the following ones:\n\\begin{eqnarray}\\label{eq4.2it}\n[E_{ij}^- , F_{kl}] = && \\hspace{-15pt} \\delta_{jk} F_{il},\\ \\ [F_{ij}, E_{kl}^+] = \\delta_{jk} F_{il},\\ \\\n[F_{ij}, F_{kl}^* ] = \\delta_{jl} E_{ik}^- - \\delta_{ik} E_{lj}^+ ;\n\\nonumber \\\\ {}\n[X_{ij}, F_{kl}] = && \\hspace{-15pt} \\delta_{ik} Y_{jl}^+,\\ \\ [X_{ij}, F_{kl}^* ] = -\\delta_{jl} Y_{ik}^- ,\n\\nonumber \\\\ {}\n[Y_{ij}^\\epsilon , E_{kl}^\\epsilon ] = && \\hspace{-15pt} \\delta_{jk} Y_{il}^\\epsilon - \\delta_{ik}Y_{jl}^\\epsilon,\\ \\\n\\nonumber \\\\ {}\n[Y_{ij}^+ , X_{kl}^* ] = && \\hspace{-15pt} \\delta_{il} F_{kj} - \\delta_{jl} F_{ki},\\ \\\n\\nonumber \\\\ {}\n[Y_{ij}^- , X_{kl}^*] = && \\hspace{-15pt} \\delta_{jk} F_{il}^* - \\delta_{ik} F_{jl}^*;\n\\nonumber \\\\ {}\n[Y_{ij}^\\epsilon , (Y_{kl}^\\epsilon)^* ] = && \\hspace{-15pt} \\delta_{ik} E_{lj}^\\epsilon - \\delta_{jk} E_{li}^\\epsilon\n+ \\delta_{jl} E_{ki}^\\epsilon - \\delta_{il} E_{kj}^\\epsilon;\n\\nonumber \\\\ {}\n[Y_{ij}^+ , F_{kl}^* ] = && \\hspace{-15pt} \\delta_{il} X_{kj} - \\delta_{jl} X_{ki},\\ \\\n\\nonumber \\\\ {}\n[Y_{ij}^- , F_{kl} ] = && \\hspace{-15pt} \\delta_{jk} X_{il} - \\delta_{ik} X_{jl}.\n\\end{eqnarray}\nWe note that the CR (\\ref{eq4.1it}) and (\\ref{eq4.2it}) do not depend\non the ``central charge'' $2N$ of the inhomogeneous terms in\nEqs. (\\ref{eq3.13it}) and (\\ref{eq3.14it}) that is absorbed in the\ndefinition of $E_{ii}^\\epsilon$ (cf.\\ Eq.\\ (\\ref{eqA.3}) of Appendix\nA). The parameter $N$ reappears, however, in the expression for the\n{\\it conformal Hamiltonian} $H_c$ which involves an infinite sum of\nCartan modes -- and hence only belongs to an appropriate completion of\n$u(\\infty, \\infty)\\subset so^*(4\\infty)$:\n\\begin{equation}\n\\label{eq4.3}\nH_c = \\mathop{\\sum}\\limits_{i \\, = \\, 1}^\\infty\\varepsilon_i (E_{ii}^+ + E_{ii}^- - 2N).\n\\end{equation}\nHere the energy eigenvalues $\\varepsilon_i$ form an increasing\nsequence of positive integers (in $D=4$: $\\varepsilon_1 = 1, \\varepsilon_2 =\n\\cdots = \\varepsilon_5 = 2, \\, \\varepsilon_6 = \\cdots =  \\varepsilon_{14} =\n3$, etc.).\nThe {\\it charge} $Q$ and the {\\it number operator} $C_1^u$\nwhich span the centre of $u(\\infty, \\infty)$ and of $u(2\\infty)$,\nrespectively, also involve infinite sums of Cartan modes:\n\\begin{equation}\n\\label{eq4.4}\nQ = \\mathop{\\sum}\\limits_{i \\, = \\, 1}^\\infty (E_{ii}^+ - E_{ii}^-), \\quad\nC_1^u = \\mathop{\\sum}\\limits_{i \\, = \\, 1}^\\infty (E_{ii}^+ + E_{ii}^- - 2N).\n\\end{equation}\nA priori $N$ is a (positive) real number.\nIt has been proven in \\cite{NST02, NRT07},\nhowever, that in a unitary positive energy realization of any algebra of\nbifields generated by local scalar fields of scaling dimension two,\n$N$ must be a natural number.\n\nLet us define the {\\it vacuum representation} of the bifields $W$ and\n$A^{(*)}$ obeying the CR (\\ref{eq3.13it}) and (\\ref{eq3.14it}) as the\n{\\it unitary irreducible positive energy representation}\n(UIPER) of $so^*(4\\infty)$ in which $H_c$ is well\ndefined and has eigenvalue zero on the ground state\n$|vac\\rangle$ (the {\\it vacuum state}). We are now ready to\nstate our main result.\n\n\\medskip\n\n\\noindent {\\bf Theorem 4.1} {\\it In any UIPER {\\rm(}of\\\/ fixed $N${\\rm)} of\\\/\n$so^*(4\\infty)$ we have{\\rm:}\n\n{\\rm(i)} $N$ is a nonnegative integer and all UIPERs of\\\/ $so^*(4\\infty)$ are\nrealized {\\rm(}with multiplicities{\\rm)}\nin the Fock space $\\mathcal{F}_{2N}$ of\\\/\n$2N$ free complex massless scalar fields {\\rm(}see Appendix A{\\rm)}.\n\n{\\rm(ii)} The ground states of equivalent UIPERs of\\\/ $so^*(4\\infty)$ in\n$\\mathcal{F}_{2N}$ form irreducible representations of the gauge group\n$Sp(2N)$. This establishes a one-to-one correspondence between UIPERs\nof\\\/ $so^*(4\\infty)$ occurring in the Fock space and the irreducible\nrepresentations of\\\/ $Sp(2N)$.}\n\n\\medskip\n\nThe {\\it proof} parallels that of Theorem 1 in \\cite[Sect.~2]{BNRT07}\nusing the results of Appendix A. We shall only note that each UIPER of\n$so^*(4\\infty)$ is expressed in terms of the fundamental weights\n$\\Lambda_\\nu$ of $so^*(4n)$ (for large enough $n$, exceeding $N$):\n\\begin{equation}\n\\label{eq4.5}\n\\Lambda = \\mathop{\\sum}\\limits_{\\nu \\, = \\, 0}^{2n-1}k_\\nu\n\\Lambda_\\nu, \\quad k_\\nu\\leq 0.\n\\end{equation}\nIn particular, the vacuum representation has weight $-2N\\Lambda_0$\n(see (\\ref{eqA.16})). Thus, each UIPER remains irreducible when\nrestricted to some $so^*(4n)$, so that we are effectively dealing with\nrepresentations of finite dimensional Lie algebras. We also note that\nthe bifield $W$ has a vanishing vacuum expectation value in view of\n(\\ref{eqA.15}), in accord with its definition as a sum of twist two\nlocal fields.\n\nThe outcome of Theorem 4.1\nand of Theorems 1 and 3 of \\cite{BNRT07} was expected in\nview of the abstract results of the\nDoplicher--Haag--Roberts theory of superselection\nsectors \\cite{H92, DR90, BDLR92}. However,\nconsiderable technical difficulties are\nencountered in relating the extension theory of bifields with the\nrepresentations of the corresponding nets. Our study provides an independent\nderivation of DHR-type results in the field theoretic framework, advancing at\nthe same time the program of classifying globally conformal invariant quantum\nfield theories in four dimensions.\n\n\\section*{Acknowledgments}\n\nThe results of the present paper were reported at three conferences\nduring the summer of 2007: ``LT7 -- Lie Theory and Its Applications in\nPhysics'' (Varna, June 18--23); ``Infinite Dimensional Algebras and\nQuantum Integrable Systems'' (Faro, July 23--27); ``SQS'07 --\nSupersymmetries and Quantum Symmetries'' (Dubna, July 30 -- August\n4). I.T. thanks the organizers of all three events for hospitality and\nsupport. N.M.N.\\ and K.-H.R. also acknowledge the invitation to the\nVarna meeting. B.B.\\ was partially supported by NSF grant\nDMS-0701011. N.M.N.\\ and I.T.\\ were supported in part by the Research\nTraining Network of the European Commission under contract\nMRTN-CT-2004-00514 and by the Bulgarian National Council for Scientific\nResearch under contract PH-1406. K.-H.R.\\ thanks the\nAlexander-von-Humboldt foundation for financial support.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\\label{se:intro}\n\nThe most widely used extension to rectangular matrices\nof the notion of inverse of square matrices is\nthe so called Moore-Penrose inverse. For a full rank\nmatrix $A\\in\\mathbb{R}^{m\\times n}$ this is defined as\n$A^\\dagger:=(A^{\\rm T} A)^{-1}A^{\\rm T}$ if $m\\geq n$,\nand as $A^\\dagger:=A^{\\rm T} (AA^{\\rm T})^{-1}$, otherwise.\nImmediate applications of $A^\\dagger$ include the solution\nof least square problems\n\\begin{equation}\\label{eq:ls}\n    \\min_{x\\in\\mathbb{R}^n}\\|Ax-b\\|^2,\n\\end{equation}\nwith $b\\in\\mathbb{R}^m$ and $m>n$, or of smallest solutions of\nunderdetermined systems\n\\begin{equation}\\label{eq:us}\n    \\min_{x\\mid Ax=b}\\|x\\|^2\n\\end{equation}\nwhen $n>m$.\nIn both cases, the solution is given by $x=A^\\dagger b$.\nWell known results in error analysis show that the accuracy in the\ncomputation of $A^\\dagger$, or in the computation of the solution\n$x$ for the problems above, crucially depends on the {\\em condition\nnumber} $\\kappa(A):=\\|A\\|\\,\\|A^\\dagger\\|$ of $A$, where $\\|A\\|$\ndenotes the spectral norm (see~\\cite[Ch.~19]{higham:96}). Accuracy\nanalysis is not the only source of interest in $\\kappa(A)$.\nAlgorithms such as the conjugate gradient method produce approximate\nsolutions of linear systems $Px=c$ ---here $P\\in\\mathbb{R}^{m\\times m}$ is a\npositive definite matrix and $c\\in\\mathbb{R}^m$--- with a number of\niterations proportional to $\\sqrt{\\kappa(P)}$ and, in many cases,\nthe matrix $P$ has been obtained as $P=AA^{\\rm T}$ for some matrix\n$A\\in\\mathbb{R}^{m\\times n}$. In those cases, $\\sqrt{\\kappa(P)}=\\kappa(A)$\nand one is again interested in the latter, this time by complexity\nconsiderations.\n\nThe condition number $\\kappa(A)$ is not directly readable\nfrom $A$, and its computation seems to require that\nof $A^\\dagger$. This is a common situation in numerical\nanalysis. A way out of it, proposed as early as 1951\nby von Neumann and Goldstine~\\cite{vNGo51} and\nmore recently pioneered by\nDemmel~\\cite{Demmel88} and Smale~\\cite{Smale97},\nconsists of randomizing the matrix~$A$ ---say, by endowing\n$\\mathbb{R}^{m\\times n}$ with a multivariate standard Gaussian\ndistribution $N(0,\\mathrm{I})$--- and considering\nits condition number as a derived random variable.\n\nIn Chen and Dongarra~\\cite{ChDo:05} the\nfollowing tail estimates on $\\kappa(A)$ were shown\nfor $A\\in\\mathbb{R}^{m\\times n}$ with $n\\ge m$:\nfor $x\\ge n-m+1$ we have\n\\begin{equation}\\label{eq:chen-dong}\n \\frac1{\\sqrt{2\\pi}}\\, \\Big(\\frac{1}{5x}\\Big)^{n-m+1}\\ \\leq\\\n \\mathop{\\rm Prob}_{A\\sim N(0,\\mathrm{I})}\n \\Big\\{\\kappa(A)\\geq\\frac{x}{1-\\lambda}\\Big\\}\\\n  \\leq\\ \\frac1{\\sqrt{2\\pi}}\\, \\Big(\\frac{7}{x}\\Big)^{n-m+1} .\n\\end{equation}\nMoreover, the expectation $\\mathbb{E}(\\kappa(A))$ can be bounded as a function\nof the {\\em elongation} $\\frac{m-1}{n}$ only, independently of~$n$.\n(We remark that this is not true for Demmel's scaled\ncondition number $\\|A\\|_F\\,\\|A^\\dagger\\|$, compare~\\cite{edelm:92}.)\nMore precisely,\nfor a sequence $(m_n)$ of integers such that\n$\\lim_{n\\to\\infty}m_n\/n=\\lambda\\in(0,1)$ and a\nsequence of standard Gaussian random matrices $A_n \\in\\mathbb{R}^{m_n\\times n}$,\nwe have in almost sure convergence\n\\begin{equation}\\label{eq:Edelman_kappa}\n  \\kappa(A_n)\\stackrel{\\rm a.s.}{\\longrightarrow}\n  \\frac{1+\\sqrt{\\lambda}}{1-\\sqrt{\\lambda}} .\n\\end{equation}\nThis follows from Geman~\\cite{geman:80} and Silverstein~\\cite{silver:85}\n(see Edelman~\\cite{edelm:88} for more precise results).\n\nThe above results provide theoretical reasons of why least squares\nproblems such\nas~\\eqref{eq:ls} or underdetermined systems such as~\\eqref{eq:us}\nare solved to great accuracy or why the conjugate gradient method is\nso efficient in practice. In fact, it follows from~\\eqref{eq:Edelman_kappa}\nthat the expected number of iterations of\nthe conjugate gradient method on the random input $P=AA^{\\rm T} $ remains\nbounded in terms of the {\\em elongation} $m\/n$ as $n\\to\\infty$ and\n$A\\in\\mathbb{R}^{m\\times n}$ is standard Gaussian. Our main result stated\nbelow implies that this phenomenon is still true for any matrix\nthat is only slightly perturbed.\n\nThe choice of $N(0,\\mathrm{I})$ as underlying data distribution\nis pervasive in the {\\em average-case analysis} of condition\nnumbers\n(and other quantities occurring in numerical analysis). It has the\nvirtue of simplicity as a first approach to understanding\nwhich condition numbers one may expect. But it has been\ncriticized due to the loose relationship of the Gaussian\n$N(0,\\mathrm{I})$ to the measures that may be governing\ndata drawing in practice.\nIn particular, it has been observed\nthat the use of Gaussians may be `optimistic' in the sense\nthat they may put more probability mass on the instances where\nthe values of the function $\\psi$ under consideration are small.\nSuch an optimism would produce yield an expectation\n$\\mathbb{E}(\\psi)$ smaller than the true one.\n\nAn alternate, more conservative,\nform of analysis has been proposed by Spielman and Teng\nunder the name of {\\em smoothed analysis}. It replaces\nthe Gaussian measure $N(0,\\mathrm{I})$ by the measures\n$N(\\overline{A},\\sigma^2\\mathrm{I})$ where $\\overline{A}$ is arbitrary. The idea is\nthen to replace the unlikely `average data' by a (usually\nsmall) perturbation of any possible occurring data. The\nrationale for this form of analysis is offered in a number\nof papers~\\cite{ST-simplex:04,sst:06,ST:06,ST:09} and\nwe won't repeat it here in full.\nWe note, nonetheless,\nthat the local nature of randomization in smoothed analysis,\ncoupled with its worst-case dependence on the input data,\nremoves from smoothed analysis the possible optimism\nwe mentioned above for average-case analysis.\nIn recent years,\ndifferent aspects of algorithm behavior for a variety of\nproblems have been analyzed this way. These include\ncondition numbers of square matrices with\nreal~\\cite{Wsch:04} or $\\{-1,1\\}$\ncoefficients~\\cite{TaoVu:07}, complexity of\ninterior-point methods~\\cite{DuSpTe:09}, and machine\nlearning~\\cite{ArVa:09}. The typical satisfying result\nis {\\em polynomial smoothed complexity}\n(see~\\cite[Def.~2]{ST:09}), consisting of a bound\nof the form\n\\begin{equation}\\label{eq:sc}\n  \\sup_{\\overline{A}} \\mathbb{E}_{A\\sim N(\\overline{A},\\sigma^2\\mathrm{I})} \\psi(A)\n  \\leq c \\sigma^{-k_1} \\mathsf{size}(A)^{k_2}\n\\end{equation}\nwhere $\\psi$ is the function whose behavior we are\nanalyzing and $c,k_1,k_2$ are positive constants.\n\nIn this paper we provide a smoothed analysis for\nMoore-Penrose inversion, extending~\\eqref{eq:chen-dong}\nfrom the average-case analysis to smoothed analysis.\nTo state the results we need to introduce some notations.\nWe assume $1\\le m\\le n$ throughout the paper.\nFor a standard Gaussian $X\\in\\mathbb{R}^{m\\times n}$ we put\n\\begin{equation}\\label{eq:defQ}\n Q(m,n) := \\frac1{\\sqrt{n}}\\, \\mathbb{E}(\\|X\\|) .\n\\end{equation}\n(Lemma~\\ref{pro:Espnorm} shows that $Q(m,n) \\le 6$ .)\nWe define for $\\lambda \\in (0,1)$ the quantity\n\\begin{equation}\\label{eq:defc}\n c(\\lambda) \\ :=\\   \\sqrt{\\frac{1+\\lambda}{2(1-\\lambda)}} .\n\\end{equation}\nNote that $c(\\lambda)$ is monotonically increasing,\n$\\lim_{\\lambda\\to 0}c(\\lambda) = \\frac1{\\sqrt{2}}$ and\n$\\lim_{\\lambda\\to 1}c(\\lambda) = \\infty$.\nFurther, for $1\\le m\\le n$ and $0<\\sigma\\le 1$, we\ndefine the {\\em elongation}\n$\\lambda := \\frac{m-1}{n}$ and introduce the quantity\n\\begin{equation}\\label{eq:defB}\n \\zeta_\\sigma(m,n) \\ :=\\ \\Big(Q(m,n) + \\frac1{\\sigma\\sqrt{n}}\\Big)\\\n  c(\\lambda)^{\\frac1{n-m+1}}  .\n\\end{equation}\n\nOur main result is the following tail bound on the condition\nnumber of rectangular matrices under local Gaussian perturbations.\n\n\\begin{theorem}\\label{thm:tailbound}\nSuppose that $\\overline{A}\\in\\mathbb{R}^{m\\times n}$ satisfies\n$\\|\\overline{A}\\| \\le 1$ and let $0<\\sigma\\le 1$.\nPut $\\lambda := \\frac{m-1}{n}$.\nThen, for $z \\ge \\zeta_\\sigma(m,n)$, we have \n$$\n  \\mathop{\\rm Prob}_{A\\sim N(\\overline{A},\\sigma^2\\mathrm{I})}\n   \\Big\\{\\kappa(A)\\geq\\frac{ez}{1-\\lambda}\\Big\\}\\\n  \\leq\\\n   2c(\\lambda)\\bigg[\n   \\Big(Q(m,n) + \\sqrt{2\\ln(2z)} + \\frac1{\\sigma\\sqrt{n}}\\Big)\\,\n    \\frac1z\\bigg]^{n-m+1} .\n$$\n\\end{theorem}\n\n\\begin{remark}\n{\\bf 1.} The decay in~$z$ in this tail bound is the same as\nin~\\eqref{eq:chen-dong} up to the logarithmic factor $\\sqrt{\\ln z}$.\nWe believe that the latter is an artefact of our proof that could be omitted.\nIn fact, the exponent $n-m+1$ is just the codimension of the set\n$\\Sigma := \\{A\\in\\mathbb{R}^{m\\times n} \\mid \\mathrm{rk} A < m\\}$\nof rank deficient matrices, cf.~\\cite{harr:95}.\nMoreover, it is known~\\cite{higham:96} that\n$\\|A^\\dagger\\| = 1\/\\mathrm{dist}(A,\\Sigma)$\nwhere the distance is measured in the Euclidean norm.\nFrom the interpretation of $\\mathop{\\rm Prob}\\{\\kappa(A)\\geq t\\}$\nas the volume of a tube around $\\Sigma$, as discussed in~\\cite{BCL:06a},\none would therefore expect a decay of order $1\/z^{n-m+1}$.\n\n{\\bf 2.} When $\\sigma=1$ and $\\overline{A}=0$, Theorem~\\ref{thm:tailbound}\nyields tail bounds for the usual average case. One may therefore\ncompare these bounds with~\\eqref{eq:chen-dong}.\nIn doing so, we see that\nthe bound in Theorem~\\ref{thm:tailbound}\nhas the additional factor~$c(\\lambda)$ (going to $\\infty$\nfor $\\lambda\\to 1$). However, we note that the bound~\\eqref{eq:chen-dong}\nholds only for\n$x=ez\\ge n-m + 1$,\nwhile our bound holds for any $z \\ge \\zeta_\\sigma(m,n)$.\nFurthermore, if we fix\n$\\lambda\\in(0,1)$ and let $(m_n)$ be a sequence of positive integers\nsuch that $\\lim m_n\/n=\\lambda$,\nit follows from~\\cite{geman:80} that\n$$\n\\lim_{n\\to\\infty} Q(m_n,n)= 1 + \\sqrt{\\lambda}.\n$$\nThis implies that\n$\\lim_{n\\to\\infty} \\zeta_\\sigma(m_n,n) = 1 + \\sqrt{\\lambda}$\nfor fixed $\\sigma \\in (0,1]$ and, in particular, that\n$\\zeta_\\sigma(m_n,n) \\le 2$ for sufficiently large~$n$ .\nThat is, for large $n$, the tail bound in Theorem~\\ref{thm:tailbound}\nis valid for any $z\\ge 2$.\n\\end{remark}\n\nTheorem~\\ref{thm:tailbound} easily implies the following\nbound on expectations.\n\n\\begin{corollary}\\label{cor:main}\nFor all $\\lambda_0\\in(0,1)$\nthere exists $n_0$ such that for all\n$1\\le m\\le n$ such that\n$\\lambda = \\frac{m-1}{n} \\le \\lambda_0$ and\n$n\\ge n_0$ we have\nfor all $\\sigma$ with $\\frac1{\\sqrt{m}}\\le \\sigma\\le 1$,\nand all\n$\\overline{A}\\in\\mathbb{R}^{m\\times n}$ with $\\|\\overline{A}\\| \\le 1$, that\n$$\n \\mathbb{E}_{A\\sim N(\\overline{A},\\sigma^2\\mathrm{I})}(\\kappa(A))\n  \\ \\leq\\ \\frac{20.1}{1-\\lambda} .\n$$\n\\end{corollary}\n\nAs for the average-case analysis, this bound\nis independent of~$n$ and depends only on the bound $\\lambda_0$\non the elongation.\nThus we have a bound of type~\\eqref{eq:sc} with $k_2=0$.\nSurprisingly, the smoothed complexity bound in Corollary~\\ref{cor:main}\nis also independent of~$\\sigma$. We thus add reasons\n---and we will become more specific in\nSection~\\ref{sec:applications}---\nto the current understanding of the accuracy\nin least squares or underdetermined system solving\nor the complexity of the conjugate gradient method.\n\nA first approach to the smoothed analysis of Moore-Penrose\ninversion appears in~\\cite{CDW:05}. The bounds obtained in\nthat paper are worse by an order of magnitude\nthan those we obtain here.\nIn Section~\\ref{sec:numerical} we compare these bounds\n{\\sf with ours} \nas well as with actual averages obtained,\nfor specific values of $n,m$ and $\\sigma$, in numerical\nsimulations.\n\nOur proof techniques are an extension of methods\nemployed by Sankar et al.~\\cite{sst:06}.\n\n\\medskip\n\n\\noindent{\\bf Acknowledgements.}\nThis work was carried out during the special semester on Foundations\nof Computational Mathematics in the fall of 2009.\nWe thank the Fields Institute in Toronto for hospitality and\nfinancial support.\n\n\\section{Preliminaries}\n\n\\subsection{Some definitions and notation}\n\nThe {\\em spectral norm} of a matrix $A\\in\\mathbb{R}^{m\\times n}$ is defined as\n$\\|A\\| := \\sup_{\\|x\\|=1} \\|Ax\\|$, where $\\|x\\|$ denotes the Euclidean norm.\nThe {\\em Frobenius norm} of $A$ is defined as the Euclidean norm of $A$\nwhen interpreted as a vector.\n\nSuppose that $A\\in\\mathbb{R}^{m\\times n}$ is of maximal rank and $m\\le n$.\nThe {\\em Moore-Penrose inverse} of $A$\nis defined as\n$A^\\dagger := A^{\\rm T}  (AA^{\\rm T})^{-1}\\in\\mathbb{R}^{n\\times m}$.\nIt can also be characterized as follows.\nFor any $v\\in\\mathbb{R}^m$ the vector $w=A^\\dagger v$ is orthogonal to\nthe kernel of~$A$ and satisfies $Aw=v$.\nThe {\\em condition number} $\\kappa(A)$ is defined as\n$\\kappa(A) :=\\|A\\| \\cdot \\|A^\\dagger\\|$.\n\nLet $\\overline{A}\\in \\mathbb{R}^{m\\times n}$ and $\\sigma>0$.\nThe {\\em isotropic normal distribution} $N(\\overline{A},\\sigma \\mathrm{I})$ with center~$\\overline{A}$\nand covariance matrix $\\sigma^2\\mathrm{I}$ is the probability distribution on\n$\\mathbb{R}^{m\\times n}$ with the density\n$$\n \\rho_{\\overline{A},\\sigma}(A) := \\frac1{(2\\pi)^{\\frac{mn}{2}}}\\,\n  e^{-\\frac{\\|A-\\overline{A}\\|_F^2}{2\\sigma^2}} .\n$$\n\n\\begin{lemma}\\label{le:gbound}\nFor $\\lambda\\in (0,1)$ we have\n$\\lambda^{-\\frac{\\lambda}{1-\\lambda}} \\le e$.\n\\end{lemma}\n\n{\\noindent\\sc Proof. \\quad}\nWriting $u=1\/\\lambda$ the assertion is equivalent to\n$u^{\\frac1{u-1}} \\le e$ or $u\\le e^{u-1}$,\nwhich is certainly true for $u \\ge 1$.\n{\\mbox{}\\hfill\\qed}\\medskip\n\n\\subsection{Concentration on spheres}\n\nLet $\\mathbb{S}^{m-1}:=\\{x\\in\\mathbb{R}^m \\mid \\|x\\|=1\\}$ denote the unit sphere in $\\mathbb{R}^m$.\nWe denote by ${\\cal O}_{m-1}$ its volume, which is given by\n${\\cal O}_{m-1} = 2\\pi^{m\/2}\/\\Gamma(\\frac{m}{2})$.\n\nThe following estimate tells us how likely a random point on $\\mathbb{S}^{m-1}$\nwill lie in a fixed spherical cap.\n\n\\begin{lemma}\\label{lem:s}\nLet $u\\in\\mathbb{S}^{m-1}$ be fixed, $m\\geq 2$. Then, for all $\\xi\\in[0,1]$,\n$$\n   \\mathop{\\rm Prob}_{v\\sim U(\\mathbb{S}^{m-1})}\n   \\big\\{\\big|u^{\\rm T} v\\big|\\geq \\xi\\big\\} \\geq\n   \\sqrt{\\frac{2}{\\pi m}}\\; (1-\\xi^2)^{\\frac{m-1}{2}}.\n$$\n\\end{lemma}\n\n{\\noindent\\sc Proof. \\quad}\nWe put $\\theta=\\arccos\\xi$ and let $\\mathsf{cap}(u,\\theta)$ denote\nthe spherical cap in $\\mathbb{S}^{m-1}$ with center~$u$ and angular radius~$\\theta$.\nUsing the bounds in Lemmas~2.1 and 2.2 of~\\cite{bcl:08a} we get\n$$\n   \\mathop{\\rm Prob}_{v\\sim U(\\mathbb{S}^{m-1})}\n   \\big\\{\\big|u^{\\rm T} v\\big|\\geq \\xi\\big\\}\n  = \\frac{2\\,\\mathsf{vol\\,}\\mathsf{cap}(u,\\theta)}{\\mathsf{vol\\,}\\mathbb{S}^{m-1}}\n  \\geq \\frac{2{\\cal O}_{m-2}}{{\\cal O}_{m-1}}\\;\n   \\frac{(1-\\xi^2)^{\\frac{m-1}{2}}}{(m-1)}.\n$$\nUsing the formula for ${\\cal O}_{m-1}$ and the recursion\n$\\Gamma(x+1)=x\\Gamma(x)$ we have\n$$\n    \\frac{{\\cal O}_{m-2}}{{\\cal O}_{m-1}}\n  = \\frac{1}{\\sqrt{\\pi}}\n   \\frac{\\Gamma\\big(\\frac{m}{2}\\big)}\n   {\\Gamma\\big(\\frac{m-1}{2}\\big)}\n  = \\frac{1}{\\sqrt{\\pi}}\n   \\frac{\\Gamma\\big(\\frac{m+1}{2}\\big)}\n   {\\Gamma\\big(\\frac{m-1}{2}\\big)}\n   \\frac{\\Gamma\\big(\\frac{m}{2}\\big)}\n   {\\Gamma\\big(\\frac{m+1}{2}\\big)}\n   = \\frac{m-1}{2\\sqrt{\\pi}}\n   \\frac{\\Gamma\\big(\\frac{m}{2}\\big)}{\\Gamma\\big(\\frac{m+1}{2}\\big)}.\n$$\nThe assertion follows now from the estimate\n\\begin{equation}\\label{eq:Gine}\n \\frac{\\Gamma\\big(\\frac{m}{2}\\big)}{\\Gamma\\big(\\frac{m+1}{2}\\big)}\n \\ \\ge\\ \\sqrt{\\frac{2}{m}}.\n\\end{equation}\nThis estimate can be quickly seen as follows.\nSuppose that $Z\\in\\mathbb{R}^{m}$ is standard normal distributed.\nUsing polar coordinates and the variable\ntransformation $u=\\rho^2\/2$ we get\n\\begin{eqnarray}\\notag\n \\mathbb{E}(\\|Z\\|) &= &\\frac{{\\cal O}_{m-1}}{(2\\pi)^{\\frac{m}{2}}}\n \\int_0^\\infty \\rho^m e^{-\\frac{\\rho^2}{2}} d\\rho\n \\:=\\: \\frac{{\\cal O}_{m-1}}{(2\\pi)^{\\frac{m}{2}}}\\, 2^{\\frac{m-1}{2}}\n    \\int_0^\\infty u^{\\frac{m-1}{2}} e^{-u} du \\\\ \\label{eq:remember}\n &=& \\frac{{\\cal O}_{m-1}}{(2\\pi)^{\\frac{m}{2}}}\\, 2^{\\frac{m-1}{2}}\n    \\Gamma(\\frac{m+1}{2})\n \\:=\\: \\sqrt{2}\\,\\frac{\\Gamma(\\frac{m+1}{2})}\n   {\\Gamma(\\frac{m}{2})},\n\\end{eqnarray}\nwhere we used the definition of the Gamma function\nfor the second last equality.\nTo complete the proof of \\eqref{eq:Gine} we note that\n$\\mathbb{E}(\\|Z\\|) \\le \\sqrt{\\mathbb{E}(\\|Z\\|^2)} =\\sqrt{m}$.\n{\\mbox{}\\hfill\\qed}\\medskip\n\nFor later use we note that~\\eqref{eq:remember} implies\n$$\n \\frac{\\Gamma\\big(\\frac{m+1}{2}\\big)}{\\Gamma\\big(\\frac{m}{2}\\big)}\n = \\frac{\\Gamma\\big(\\frac{m+2}{2}\\big)}{\\Gamma\\big(\\frac{m}{2}\\big)}\n      \\frac{\\Gamma\\big(\\frac{m+1}{2}\\big)}{\\Gamma\\big(\\frac{m+2}{2}\\big)}\n = \\frac{m}{2}\\, \\frac{\\Gamma\\big(\\frac{m+1}{2}\\big)}{\\Gamma\\big(\\frac{m+2}{2}\\big)}\n \\ \\ge\\  \\frac{m}{2}\\, \\sqrt{\\frac{2}{m+1}},\n$$\nusing~\\eqref{eq:Gine} for the right-hand inequality. Therefore\n\\begin{equation}\\label{eq:chi-lb}\n \\mathbb{E}(\\|Z\\|)\\ \\ge\\ \\frac{m}{\\sqrt{m+1}} .\n\\end{equation}\n\n\\subsection{Large deviations}\n\nWe will use a powerful large deviation result.\nLet $F:\\mathbb{R}^N\\to\\mathbb{R}$ be a Lipschitz continous function with Lipschitz\nconstant~$L$, so that $|F(x)-F(y)| \\le L \\|x-y\\|$ for all $x,y\\in\\mathbb{R}^N$,\nwhere $\\|\\ \\|$ denotes the Euclidean norm.\nNow suppose that $x\\in\\mathbb{R}^N$ is a standard Gaussian random vector such that\n$\\mathbb{E}(F(x))$ exists. Then it is known~\\cite[(1.4)]{letala:91} that for all $t>0$\n\\begin{equation}\\label{eq:concentration}\n  \\mathop{\\rm Prob}\\{F(x)\\geq \\mathbb{E}(F) +t\\}\\leq e^{-\\frac{t^2}{2L^2}}.\n\\end{equation}\n(We note that in ~\\cite[(1.4)]{letala:91} this is only stated for the median, but\nthe inequality holds as well for the expectation. See also~\\cite{ledoux:01}.)\n\n\\subsection{A bound on the expected spectral norm}\n\nThe function $\\mathbb{R}^{m\\times n}\\to\\mathbb{R}$ mapping a matrix $X$ to its\nspectral norm~$\\|X\\|$ is Lipschitz continuous with Lipschitz constant~$1$,\nas $\\|X-Y\\| \\le \\|X-Y\\|_F$.\nThe concentration bound~\\eqref{eq:concentration}, together\nwith~\\eqref{eq:defQ}, implies that for $t>0$,\n\\begin{equation}\\label{eq:spnb}\n\\mathop{\\rm Prob}\\Big\\{\\|X\\| \\ge Q(m,n)\\sqrt{n} + t\\}\\ \\le\\ e^{-\\frac{t^2}{2}} .\n\\end{equation}\nThis tail bound easily implies the following large deviation result.\n\n\\begin{proposition}\\label{le:enorm}\nLet $\\overline{A}\\in\\mathbb{R}^{m\\times n}$ with $m\\le n$, $\\|\\overline{A}\\| \\le 1$, and $\\sigma\\in(0,1]$.\nIf $A\\in\\mathbb{R}^{m\\times n}$ follows the law $N(\\overline{A},\\sigma^2\\mathrm{I})$, then, for $t>0$,\n$$\n \\mathop{\\rm Prob}_{A\\sim N(\\overline{A},\\sigma^2\\mathrm{I})}\\Big\\{\\|A\\|\\geq Q(m,n)\\sigma\\sqrt{n}+t+1\\Big\\}\n \\ \\leq\\  e^{-\\frac{t^2}{2\\sigma^2}}.\n$$\n\\end{proposition}\n\n{\\noindent\\sc Proof. \\quad}\nWe note that\n$\\|A\\|\\geq Q(m,n)\\sigma\\sqrt{n}+t+1$ implies that\n$\\|A -\\overline{A}\\| \\geq \\|A\\| -\\|\\overline{A}\\| \\ge Q(m,n)\\sqrt{n} + t$.\nMoreover, if $A\\in\\mathbb{R}^{m\\times n}$ follows the law $N(\\overline{A},\\sigma^2\\mathrm{I})$, then\n$X:=\\frac{A-\\overline{A}}{\\sigma}$ is standard Gaussian in $\\mathbb{R}^{m\\times n}$.\nThe assertion follows from~\\eqref{eq:spnb}.\n{\\mbox{}\\hfill\\qed}\\medskip\n\nWe derive now an upper bound on $Q(m,n)$.\nSuch result should be well-known but we could not locate in the literature.\n\n\\begin{lemma}\\label{pro:Espnorm}\nFor $n>1$ we have\n$\\sqrt{\\frac{n}{n+1}}\n\\le Q(m,n)\\le\n 2\\Big( 1 + \\sqrt{\\frac{2\\ln (2m-1)}{n}} + \\frac1{\\sqrt{n}}\\Big) \\le 6$.\n\\end{lemma}\n\nThe proof relies on the following lemma.\n\n\\begin{lemma}\\label{le:maxchi}\nLet $r_1,\\ldots,r_n$ be independent random variables with nonnegative values\nsuch that $r_i^2$ is $\\chi^2$-distributed with $f_i$ degrees of freedom.\nThen,\n$$\n \\mathbb{E}\\Big(\\max_{1\\le i \\le n} r_i\\Big) \\le \\max_{1\\le i\\le n}\\sqrt{f_i}\n   + \\sqrt{2\\ln n} +  1 .\n$$\n\\end{lemma}\n\n{\\noindent\\sc Proof. \\quad}\nWe start by a large deviation estimate for $\\chi^2$-distributed random variables.\nNote that $\\mathbb{R}^f\\to\\mathbb{R}$, $x\\mapsto \\|x\\|$, is Lipschitz continuous with\nLip\\-schitz constant~$1$.\nFrom \\eqref{eq:concentration} we know that for standard Gaussian $x\\in\\mathbb{R}^n$\nand all $t>0$,\n$$\n  \\mathop{\\rm Prob}\\{\\|x\\| \\geq \\mathbb{E}(\\|x\\|) +t\\}\\ \\leq\\ e^{-\\frac{t^2}{2}}.\n$$\nSince $\\mathbb{E}(\\|x\\|)\\le \\sqrt{\\mathbb{E}(\\|x\\|^2)} = \\sqrt{f}$, this implies for all $t>0$,\n\\begin{equation}\\label{eq:chi-conc}\n  \\mathop{\\rm Prob}\\{\\|x\\| \\geq \\sqrt{f} +t \\}\\ \\leq\\ e^{-\\frac{t^2}{2}}.\n\\end{equation}\n\nWe suppose now that $r_1,\\ldots,r_n$ are\nindependent random variables with nonnegative values\nsuch that $r_i^2$ is $\\chi^2$-distributed with $f_i$ degrees of freedom.\nPut $f:=\\max_i f_i$.\nEquation~\\eqref{eq:chi-conc} tells us that for all $i$ and all $t>0$,\n$$\n \\mathop{\\rm Prob}\\{ r_i \\ge \\sqrt{f} + t \\}\\ \\le\\  e^{-\\frac{t^2}{2}}\n$$\nand hence, by the union bound,\n$$\n \\mathop{\\rm Prob}\\Big\\{\\max_{1\\le i\\le n} r_i \\ge \\sqrt{f} + t \\Big\\}\\\n  \\le\\ n e^{-\\frac{t^2}{2}} .\n$$\nFor a fixed parameter~$b\\ge 1$ (to be determined later), this implies\n\\begin{eqnarray*}\n \\mathbb{E}(\\max_{1\\le i\\le n} r_i) &\\le& \\sqrt{f} + b + \\int_{\\sqrt{f} + b}^\\infty\n                       \\mathop{\\rm Prob}\\{\\max_{1\\le i\\le n} r_i \\ge T \\}\\, dT \\\\\n  &=& \\sqrt{f} + b + \\int_{b}^\\infty \\mathop{\\rm Prob}\\{\\max_{1\\le i\\le n}  r_i \\ge \\sqrt{f} + t \\}\\, dt \\\\\n  &\\le& \\sqrt{f} + b + n \\int_{b}^\\infty  e^{-\\frac{t^2}{2}}\\,dt .\n\\end{eqnarray*}\nUsing the well-known estimate\n$$\n \\frac1{\\sqrt{2\\pi}}\\int_b^\\infty  e^{-\\frac{t^2}{2}}\\,dt\n \\le \\frac1{b\\sqrt{2\\pi}}\\,e^{-\\frac{b^2}{2}}\n  \\le \\frac1{\\sqrt{2\\pi}}\\,e^{-\\frac{b^2}{2}}\n$$\nwe obtain\n$$\n \\mathbb{E}(\\max_{1\\le i\\le n} r_i) \\le \\sqrt{f} + b + n e^{-\\frac{b^2}{2}} .\n$$\nFinally, choosing $b:=\\sqrt{2\\ln n}$ we get\n$$\n \\mathbb{E}(\\max_{1\\le i\\le n} r_i) \\le \\sqrt{f} + \\sqrt{2\\ln n} + 1,\n$$\nas claimed.\n{\\mbox{}\\hfill\\qed}\\medskip\n\n\\proofof{Lemma~\\ref{pro:Espnorm}}\nA general matrix $X\\in\\mathbb{R}^{m\\times n}$ can be transformed into a\nbidiagonal matrix of the form\n$$\nY:=\\begin{bmatrix}\nv_n    &       &         &      & 0     &\\cdots &0     \\\\\nw_{m-1}&v_{n-1}&         &      & \\vdots&       &\\vdots\\\\\n       &\\ddots &\\ddots   &      & \\vdots&       &\\vdots\\\\\n       &       & w_1     &v_{n-m+1}&0   &\\cdots &0\n\\end{bmatrix}\n$$\nwith $v_i,w_j\\ge 0$ by performing Householder transformations\nfrom the left and right hand side of~$X$, cf.~\\cite[\\S5.4.3]{golloan:83}.\nIn particular, $\\|X\\| = \\|Y\\|$.\nAn analysis of this transformation shows that if we start\nwith a standard Gaussian matrix $X$, then\nthe $v_n,\\ldots,v_{n-m+1},w_{m-1},\\ldots,w_1$\nare independent random variables such that $v_i^2$ and $w_i^2$ are\n$\\chi^2$-distributed with $i$~degrees of freedom, cf.~\\cite{silver:85}.\n\nThe spectral norm of $Y$ is bounded by\n$\\max_{i}v_i + \\max_j w_j \\le 2 r$, where\n$r$~denotes the maximum of the values $v_i$ and $w_j$.\nLemma~\\ref{le:maxchi} implies that, for $n>1$,\n$$\n \\mathbb{E}(r) \\le \\sqrt{n} + \\sqrt{2\\ln (2m-1)} + 1\n  \\le 3\\sqrt{n} .\n$$\nThis shows the claimed upper bound on $Q(m,n)$.\nFor the lower bound we note that\n$\\|Y\\| \\ge |v_n|$ which gives\n$\\mathbb{E}(\\|Y\\|) \\ge \\mathbb{E}(|v_n|)$.\nThe claimed lower bound now follows from~\\eqref{eq:chi-lb},\nwhich states that\n$\\mathbb{E}(|v_n|) \\ge \\sqrt{\\frac{n}{n+1}}$.\n{\\mbox{}\\hfill\\qed}\\medskip\n\n\\section{Proof of the main results}\n\nThe main work consists of deriving tail bounds on $\\|A^\\dagger\\|$,\nwhich is done in the next subsection.\n\n\\subsection{Tail bounds for $\\|A^\\dagger\\|$}\\label{se:tb-Adag}\n\n\n\\begin{proposition}\\label{thm:main_tail}\nLet $\\overline{A}\\in\\mathbb{R}^{m\\times n}$, $\\sigma>0$, and put\n$\\lambda :=\\frac{m-1}{n}$.\nFor random $A\\sim N(\\overline{A},\\sigma^2\\mathrm{I})$\nwe have, for any $t>0$,\n$$\n \\mathop{\\rm Prob}_{A\\sim N(\\overline{A},\\sigma^2\\mathrm{I})}\\Big\\{\\|A^\\dagger\\|\\geq \\frac{t}{1-\\lambda}\\Big\\}\n  \\ \\leq\\ c(\\lambda)\\,\n       \\bigg(\\frac{e}{\\sigma\\sqrt{n}\\, t}\\bigg)^{(1-\\lambda)n}.\n$$\n\\end{proposition}\n\nWe first show the following result.\n\n\\begin{proposition}\\label{prop:bound1}\nFor all $v\\in \\mathbb{S}^{m-1}$,\n$\\overline{A}\\in\\mathbb{R}^{m\\times n}$, $\\sigma>0$, and $\\xi>0$\nwe have\n$$\n  \\mathop{\\rm Prob}_{A\\sim N(\\overline{A},\\sigma^2\\mathrm{I})}\n  \\big\\{\\|A^\\dagger v\\|\\geq \\xi \\big\\}\\ \\leq\\\n       \\frac{1}{(\\sqrt{2\\pi})^{n-m+1}}\\,\n               \\frac{{\\cal O}_{n-m}}{n-m+1}\\,\n               \\Big(\\frac{1}{\\sigma \\xi}\\Big)^{n-m+1}.\n$$\n\\end{proposition}\n\n\n{\\noindent\\sc Proof. \\quad}\nWe first claim that, because of unitary invariance,\nwe may assume that $v=e_m:=(0,\\ldots,0,1)$.\nTo see this, take $\\Phi\\in U(m)$ such that $v=\\Phi e_m$.\nConsider the isometric map\n$A\\mapsto B=\\Phi^{-1}A$ which transforms the density\n$\\rho_{\\overline{A},\\sigma}(A)$ into a density of the same form, namely\n$\\rho_{\\Phi^{-1}\\overline{A},\\sigma}(B)$.\nThus the assertion for $e_m$ and random $B$\nimplies the assertion for $v$ and $A$, noting that\n$A^\\dagger v=B^\\dagger e_m$. This proves the claim.\n\nWe are going to characterize the norm of $w:=A^\\dagger e_m$\nin a geometric way.\nLet $a_i$ denote the $i$th row of $A$. Almost surely,\nthe rows $a_1,\\ldots,a_m$ are linearly independent;\nhence, we assume so in what follows.\nLet\n$$\n  R:=\\mathsf{span}\\{a_1,\\ldots,a_{m}\\},\\ S:=\\mathsf{span}\\{a_1,\\ldots,a_{m-1}\\} .\n$$\nLet $S^\\perp$ denote the orthogonal\ncomplement of $S$ in $\\mathbb{R}^n$. We decompose\n$a_m=a_m^\\perp + a_m^S$,\nwhere $a_m^\\perp$ denotes the orthogonal projection\nof $a_m$ onto $S^\\perp$ and $a_m^S\\in S$. Then $a_m^\\perp\\in R$\nsince both $a_m$ and $a_m^S$ are in $R$. It follows that\n$a_m^\\perp\\in R\\cap S^\\perp$.\n\nWe claim that $w\\in R\\cap S^\\perp$ as well. Indeed,\nnote that $R$ equals the orthogonal complement of the\nkernel of $A$ in $\\mathbb{R}^n$. Therefore, by definition of the\nMoore-Penrose inverse, $w=A^\\dagger e_m$ lies in $R$.\nMoreover, since $A A^\\dagger=\\mathrm{I}$, we have\n$\\langle w,a_i\\rangle =0$ for $i=1,\\ldots,m-1$\nand hence $w\\in S^\\perp$ as well.\n\nIt is immediate to see that $\\dim R\\cap S^\\perp=1$.\nIt then follows that $R\\cap S^\\perp=\\mathbb{R} w = \\mathbb{R} a_m^\\perp$.\nSince $\\langle w,a_m\\rangle =1$, we get\n$1 = \\langle w,a_m\\rangle = \\langle w,a_m^\\perp \\rangle\n      = \\|w\\|\\, \\|a_m^\\perp\\|$\nand therefore\n\\begin{equation}\\label{eq:star}\n \\|A^{\\dagger}e_m\\| = \\frac1{\\|a_m^\\perp\\|}.\n\\end{equation}\n\nLet $A_m\\in\\mathbb{R}^{(m-1)\\times n}$ denote the matrix\nobtained from~$A$ by omitting $a_m$. The density\n$\\rho_{\\overline{A},\\sigma}$ factors as\n$\\rho_{\\overline{A},\\sigma}(A) =\\rho_1(A_n)\\rho_2(a_n)$ where\n$\\rho_1$ and $\\rho_2$ denote the density functions of\n$N(\\overline{A}_m,\\sigma^2\\mathrm{I})$ and\n$N(\\bar{a}_m,\\sigma^2\\mathrm{I})$, respectively\n(the meaning of $\\overline{A}_m$ and $\\bar{a}_m$\nbeing clear). Fubini's Theorem combined\nwith \\eqref{eq:star} yield, for $\\xi>0$,\n\\begin{eqnarray}\\label{eq:WX}\n  \\mathop{\\rm Prob}_{N(\\overline{A},\\sigma^2\\mathrm{I})}\\big\\{\\|A^{\\dagger}e_m\\|\\geq \\xi\\big\\}\n   &=&\n     \\int_{\\|A^{\\dagger}e_m\\| \\ge \\xi}\n     \\rho_{\\overline{A},\\sigma^2\\mathrm{I}}(A)\\, dA\\\\\n  &=&\n     \\int_{A_{m}\\in\\mathbb{R}^{(m-1)\\times n}}\n     \\rho_1(A_{m})\n     \\cdot  \\left(\\int_{\\|a_m^\\perp\\|\\leq 1\/\\xi}\n     \\rho_2(a_m)\\, da_m\\right) dA_{m}.\\nonumber\n\\end{eqnarray}\nTo complete the proof it is sufficient to show the bound\n\\begin{equation}\\label{eq:C4}\n    \\int_{\\|a_m^\\perp\\|\\leq\\frac1\\xi}\n    \\rho_2(a_m)\\, da_m\n    \\leq   \\frac{1}{(\\sqrt{2\\pi})^{n-m+1}} \\,\n               \\frac{{\\cal O}_{n-m}}{n-m+1}\\,\n               \\Big(\\frac{1}{\\sigma \\xi}\\Big)^{n-m+1}\n\\end{equation}\nfor fixed, linearly independent $a_1,\\ldots,a_{m-1}$ and $\\xi>0$.\n\nTo show~\\eqref{eq:C4} note that\n$a_m^\\perp\\sim N(\\bar{a}_m^\\perp,\\sigma^2\\mathrm{I})$\nin $S^\\perp\\simeq\\mathbb{R}^{n-m+1}$\nwhere $\\bar{a}_m^\\perp$\nis the orthogonal projection of $\\bar{a}_m$ onto\n$S^\\perp$.\nLet $B_r$ denote the ball of radius~$r$ in $\\mathbb{R}^p$ centered\nat the origin. It is easy to see that $\\mathsf{vol\\,} B_r = {\\cal O}_{p-1}r^p\/p$.\nFor any\n$\\bar{x}\\in\\mathbb{R}^p$ and any $\\sigma>0$ we have\n\\begin{eqnarray*}\n  \\mathop{\\rm Prob}_{x\\sim N(\\bar{x},\\sigma^2\\mathrm{I})}\\big\\{\\|x\\|\\leq \\varepsilon\\big\\}\n  &\\leq & \\mathop{\\rm Prob}_{x\\sim N(0,\\sigma^2\\mathrm{I})}\\big\\{\\|x\\|\\leq \\varepsilon\\big\\}\n  \\;=\\; \\frac{1}{(\\sigma\\sqrt{2\\pi})^p}\n          \\int_{\\|x\\|\\leq \\varepsilon} e^{-\\frac{\\|x\\|^2}{2\\sigma^2}} dx\\\\\n  &\\stackrel{\\scriptstyle x=\\sigma z}{=}&\n    \\frac{1}{(\\sqrt{2\\pi})^p}\n          \\int_{\\|z\\|\\leq \\frac{\\varepsilon}{\\sigma}}\n         e^{-\\frac{\\|z\\|^2}{2}} dz\\\\\n  &\\leq & \\frac{1}{(\\sqrt{2\\pi})^p}\\,\n               \\mathsf{vol\\,} B_{\\frac{\\varepsilon}{\\sigma}}\n  \\;=\\; \\frac{1}{(\\sqrt{2\\pi})^p}\n               \\Big(\\frac{\\varepsilon}{\\sigma}\\Big)^p\\,\\mathsf{vol\\,} B_1\\\\\n   &=& \\frac{1}{(\\sqrt{2\\pi})^p}\n               \\Big(\\frac{\\varepsilon}{\\sigma}\\Big)^p\\,\n               \\frac{{\\cal O}_{p-1}}{p}.\n\\end{eqnarray*}\nTaking $\\bar{x}=\\bar{a}_m^\\perp$,\n$\\varepsilon=\\frac{1}{\\xi}$, and $p=n-m+1$\nthe claim~\\eqref{eq:C4} follows.\n{\\mbox{}\\hfill\\qed}\\medskip\n\n\\proofof{Proposition~\\ref{thm:main_tail}}\nThe proof is based on an idea in~\\cite{sst:06}.\nFor $A\\in\\mathbb{R}^{m\\times n}$ there exists $u_A\\in\\mathbb{S}^{m-1}$ such that\n$\\|A^{\\dagger}\\|=\\|A^{\\dagger}u_A\\|$.\nMoreover, for almost all $A$, the vector $u_A$ is uniquely determined\nup to sign.\nUsing the singular value decomposition it is easy to show that,\nfor all $v\\in\\mathbb{S}^{m-1}$,\n\\begin{equation}\\label{eq:sankar}\n  \\|A^{\\dagger}v\\| \\geq \\|A^{\\dagger}\\| \\cdot |u_A^{\\rm T} v|.\n\\end{equation}\n\nNow take $A~\\sim N(\\overline{A},\\sigma^2\\mathrm{I})$ and\n$v\\sim U(\\mathbb{S}^{m-1})$ independently.\nThen, for any $s\\in(0,1)$ and $t>0$ we have\n\\begin{align*}\n  \\mathop{\\rm Prob}_{A,v}\\big\\{\\|A^{\\dagger}v\\|\\geq\\, t & \\sqrt{1-s^2}\\big\\}\n  \\;\\geq\\;\n  \\mathop{\\rm Prob}_{A,v}\\Big\\{\\|A^{\\dagger}\\|\\geq t\\ \\&\\\n    |u_A^{\\rm T} v|\\geq \\sqrt{1-s^2}\\Big\\}\\\\\n  =\\; & \\mathop{\\rm Prob}_{A}\\big\\{\\|A^{\\dagger}\\|\\geq t\\big\\} \\cdot\n     \\mathop{\\rm Prob}_{A,v}\\Big\\{|u_A^{\\rm T} v|\\geq \\sqrt{1-s^2}\n    \\;\\Big|\\; \\|A^{\\dagger}\\| \\geq t\\Big\\}\\\\\n  \\geq\\; & \\mathop{\\rm Prob}_{A}\\big\\{\\|A^{\\dagger}\\|\\geq t\\big\\} \\cdot\n   \\sqrt{\\frac{2}{\\pi m}}\\,s^{m-1},\n\\end{align*}\nthe last line by Lemma~\\ref{lem:s} with $\\xi=\\sqrt{1-s^2}$.\nNow we use Proposition~\\ref{prop:bound1} with\n$\\xi=t\\sqrt{1-s^2}$ to deduce that\n\\begin{eqnarray}\\label{eq:bound2}\n  \\mathop{\\rm Prob}_{A}\\big\\{\\|A^{\\dagger}\\|\\geq t \\big\\}\n  &\\leq& \\sqrt{\\frac{\\pi m}{2}}\\;\n   \\frac{1}{s^{m-1}}\n  \\mathop{\\rm Prob}_{A,v}\\{\\|A^{\\dagger}v\\|\\geq t\\sqrt{1-s^2}\\}\\\\\n  &\\leq&  \\frac{\\sqrt{m}}{2s^{m-1}}\\;\n       \\frac{1}{(\\sqrt{2\\pi})^{n-m}}\\,\n               \\frac{{\\cal O}_{n-m}}{n-m+1}\\,\n               \\Big(\\frac{1}{\\sigma t\\sqrt{1-s^2}}\\Big)^{n-m+1}.\n               \\nonumber\n\\end{eqnarray}\n\nWe next choose $s\\in(0,1)$ to minimize the bound above.\nTo do so amounts to maximize\n$(1-x)^{\\frac{n-m+1}{2}} x^{\\frac{m-1}{2}}$ where\n$x=s^2\\in(0,1)$, or yet, to maximize\n$$\n   g(x)=\\Big((1-x)^{\\frac{n-m+1}{2}}\n    x^{\\frac{m-1}{2}}\\Big)^{\\frac2n}\n     =(1-x)^{\\frac{n-m+1}{n}} x^{\\frac{m-1}{n}}\n     =(1-x)^{1-\\lambda} x^\\lambda .\n$$\nWe have\n$\n   \\frac{d}{dx}\\ln g(x)= \\frac{\\lambda}{x}-\\frac{1-\\lambda}{1-x}\n$\nwith the only zero attained at $x^*=\\lambda$.\n\nReplacing $s^2$ by $\\lambda$ in~\\eqref{eq:bound2} we obtain the bound\n$$\n  \\mathop{\\rm Prob}_{A}\\big\\{\\|A^{\\dagger}\\|\\geq t \\big\\} \\ \\leq\\\n  \\frac{\\sqrt{\\lambda n+1}}\n  {2\\lambda^{\\frac{\\lambda n}{2}}}\\;\n       \\frac{1}{(\\sqrt{2\\pi})^{n-m}}\\,\n               \\frac{{\\cal O}_{n-m}}{(1-\\lambda)n}\\,\n               \\bigg(\\frac{1}{\\sigma t\\sqrt{1-\\lambda}}\\bigg)^{(1-\\lambda)n} .\n$$\nLemma~\\ref{le:gbound} implies\n$$\n\\lambda^{-\\frac{\\lambda n}{2}}\n = \\Big( \\lambda^{-\\frac{\\lambda}{2(1-\\lambda)}}\\Big)^{(1-\\lambda)n}\n \\le e^{\\frac{(1-\\lambda)n}{2}} .\n$$\nSo we get\n\\begin{align*}\n\\mathop{\\rm Prob}_{A}\\big\\{& \\|A^{\\dagger}\\|\\geq t\\big\\} \\ \\leq\\\n\\frac{\\sqrt{\\lambda n+1}}{2}\\;\n       \\frac{1}{(\\sqrt{2\\pi})^{n-m}}\\,\n               \\frac{{\\cal O}_{n-m}}{(1-\\lambda)n}\\,\n               \\bigg(\\frac{\\sqrt{e}}{\\sigma t\\sqrt{1-\\lambda}}\\bigg)^{(1-\\lambda)n} \\\\\n  \\;=\\;&\n  \\; \\frac{\\sqrt{\\lambda n +1}}{2}\\;\n    \\bigg(\\frac{e}{1-\\lambda}\\bigg)^{\\frac{(1-\\lambda)n}{2}}\\;\n       \\frac{1}{(\\sqrt{2\\pi})^{n-m}}\\,\n               \\frac{{\\cal O}_{n-m}}{(1-\\lambda)n}\\,\n               \\bigg(\\frac{1}{\\sigma t}\\bigg)^{(1-\\lambda)n}\\\\\n =\\;& \\frac1{2(1-\\lambda)}\\;\n \\sqrt{\\lambda +\\frac1{n}}\\;\\frac1{\\sqrt{n}}\n\\bigg(\\frac{e}{1-\\lambda}\\bigg)^{\\frac{(1-\\lambda)n}{2}}\\;\n       \\frac{{\\cal O}_{n-m}}{(\\sqrt{2\\pi})^{n-m}}\\,\n               \\bigg(\\frac{1}{\\sigma t}\\bigg)^{(1-\\lambda)n}\\\\\n\\le\\;& \\frac{\\sqrt{\\lambda + 1}}{2(1-\\lambda)}\\;\\frac1{\\sqrt{n}}\n  \\bigg(\\frac{e}{1-\\lambda}\\bigg)^{\\frac{(1-\\lambda)n}{2}}\\;\n       \\frac{2\\pi^{\\frac{n-m+1}{2}}}\n       {\\Gamma\\big(\\frac{n-m+1}{2}\\big)(\\sqrt{2\\pi})^{n-m}}\\,\n               \\bigg(\\frac{1}{\\sigma t}\\bigg)^{(1-\\lambda)n}\\\\\n = \\;& \\frac{\\sqrt{1+\\lambda}}{1-\\lambda}\\; \\frac1{\\sqrt{n}}\\;\n  \\bigg(\\frac{e}{1-\\lambda}\\bigg)^{\\frac{(1-\\lambda)n}{2}}\\;\n       \\frac{\\sqrt{2\\pi}}\n     {\\Gamma\\big(\\frac{n(1-\\lambda)}{2}\\big)2^{\\frac{(1-\\lambda)n}{2}}}\\,\n               \\bigg(\\frac{1}{\\sigma t}\\bigg)^{(1-\\lambda)n}.\n\\end{align*}\nWe next estimate $\\Gamma\\big(\\frac{(1-\\lambda)n}{2}\\big)$. To do so,\nrecall Stirling's bound\n$$\n  \\sqrt{2\\pi}x^{x+\\frac12}e^{-x}<\\Gamma(x+1)\n  <\\sqrt{2\\pi}x^{x+\\frac12}e^{-x+\\frac{1}{12x}}\n \\qquad\\mbox{for all $x>0$}\n$$\nwhich yields, using $\\Gamma(x+1)=x\\Gamma(x)$,\nthe bound $\\Gamma(x) > \\sqrt{2\\pi\/x}\\, (x\/e)^{x}$.\nWe use this with $x=\\frac{(1-\\lambda)n}{2}$ to obtain\n$$\n  \\Gamma\\Big(\\frac{(1-\\lambda)n}{2}\\Big)\n  \\geq \\sqrt{\\frac{4\\pi}{(1-\\lambda)n}}\\\n  \\Big(\\frac{(1-\\lambda)n}{2e}\\Big)^\\frac{(1-\\lambda)n}{2}.\n$$\nPlugging this into the above we obtain\n(observe the crucial cancellation of $\\sqrt{n}$)\n\\begin{align*}\n  \\mathop{\\rm Prob}_{A}\\big\\{&\\|A^{\\dagger}\\|\\geq t\\big\\} \\\\\n  \\leq\\ & \\sqrt{\\frac{1+\\lambda}{(1-\\lambda)^2}}\\; \\frac1{\\sqrt{n}}\\;\n  \\bigg(\\frac{e}{1-\\lambda}\\bigg)^{\\frac{(1-\\lambda)n}{2}}\\;\n     \\, \\sqrt{2\\pi}\\, \\sqrt{\\frac{(1-\\lambda)n}{4\\pi}}\\,\n   \\Big(\\frac{e}{(1-\\lambda)n}\\Big)^\\frac{(1-\\lambda)n}{2}\n               \\bigg(\\frac{1}{\\sigma t}\\bigg)^{(1-\\lambda)n}\\\\\n  =\\ &\n  c(\\lambda)\\, \\bigg(\\frac{e}{1-\\lambda}\\bigg)^{(1-\\lambda)n}\\;\n   \\Big(\\frac{1}{n}\\Big)^{\\frac{(1-\\lambda)n}{2}}\n               \\bigg(\\frac{1}{\\sigma t}\\bigg)^{(1-\\lambda)n}\n  = c(\\lambda)\\,\\bigg(\\frac{e}{\\sigma\\sqrt{n}(1-\\lambda)t}\\bigg)^{(1-\\lambda)n}\\;              ,\n\\end{align*}\nwhich completes the proof of the proposition.\n{\\mbox{}\\hfill\\qed}\\medskip\n\n\\subsection{Proof of Theorem~\\ref{thm:tailbound}}\n\nTo simplify notation we write\n$c:=c(\\lambda)$ and $Q:=Q(m,n)$.\nProposition~\\ref{thm:main_tail} implies that for any $\\varepsilon>0$ we have\n\\begin{equation}\\label{eq:e_Adagger}\n\\mathop{\\rm Prob}_{A\\sim N(\\overline{A},\\sigma^2\\mathrm{I})}\\Big\\{\n  \\|A^\\dagger\\|\\geq \\frac{e}{1-\\lambda}\\,\n    \\frac{1}{\\sigma\\sqrt{n}}\\,\n   \\Big(\\frac{c}{\\varepsilon}\\Big)^{\\frac{1}{(1-\\lambda)n}} \\Big\\}\n   \\ \\leq\\ \\varepsilon.\n\\end{equation}\nSimilarly, letting $\\varepsilon=e^{-\\frac{t^2}{2\\sigma^2}}$ in\nProposition~\\ref{le:enorm}\nand solving for $t$ we deduce that, for any $\\varepsilon\\in(0,1]$,\n\\begin{equation}\\label{eq:e_A}\n\\mathop{\\rm Prob}\\Big\\{\\|A\\|\\geq  Q\\sigma\\sqrt{n}+\\sigma\\sqrt{2\\ln\\frac{1}{\\varepsilon}} + 1\n     \\Big\\}\\ \\leq\\ \\varepsilon.\n\\end{equation}\nWe conclude that\n\\begin{equation}\\label{eq:tail_kappa}\n   \\mathop{\\rm Prob}_{A\\sim N(\\overline{A},\\sigma^2\\mathrm{I})}\\Big\\{\\kappa(A)\\geq\n   \\frac{ez(\\varepsilon)}{1-\\lambda}\\Big\\}\n   \\ \\leq\\ 2\\varepsilon,\n\\end{equation}\nwhere we have have set, for $\\varepsilon\\in(0,1]$,\n\\begin{equation}\\label{def:z(e)}\n  z(\\varepsilon):=\n     \\Bigg(Q + \\sqrt{\\frac{2}{n} \\ln\\frac1{\\varepsilon}} + \\frac1{\\sigma\\sqrt{n}}\\Bigg)\\,\n        \\Big(\\frac{c}{\\varepsilon}\\Big)^{\\frac{1}{(1-\\lambda)n}}.\n\\end{equation}\nWe note that $z(1)=\\zeta:=\\zeta_\\sigma(m,n)$, cf.~Equation~\\eqref{eq:defB}.\nMoreover, $\\lim_{\\varepsilon\\to 0}z(\\varepsilon) =\\infty$ and $z$ is decreasing in the\ninterval $(0,1]$.\nHence, for $z\\ge \\zeta$, there exists $\\varepsilon=\\varepsilon(z)\\in (0,1]$ such that $z=z(\\varepsilon)$.\n\nWe need to upper bound $\\varepsilon(z)$ as a function of $z$.\nTo do so, we start with a weak lower bound on $\\varepsilon(z)$ and\nclaim that\n\\begin{equation}\\label{eq:claim}\n\\frac1{n}\\,\\ln\\frac1{\\varepsilon}\\ \\le\\ \\ln (2z(\\varepsilon)) .\n\\end{equation}\nTo show this, recall that\n$Q\\ge \\sqrt{\\frac{n}{n+1}} \\ge \\frac1{\\sqrt{2}}$\ndue to Lemma~\\ref{pro:Espnorm}.\nHence\n$\\zeta \\ge Q \\ge 1\/\\sqrt{2}$ and it follows that\n$\\sqrt{2} z \\le 1$ for  $z \\ge \\zeta$.\nThus, Equation~\\eqref{def:z(e)} implies that\n$$\n z(\\varepsilon)\\ \\ge\\ \\frac{1}{\\sqrt{2}}\\,\n  \\Big(\\frac{c}{\\varepsilon}\\Big)^{\\frac{1}{(1-\\lambda)n}}.\n$$\nUsing $c\\ge \\frac1{\\sqrt{2}}$ we get\n$$\n (\\sqrt{2} z)^n \\ \\ge\\ (\\sqrt{2} z)^{(1-\\lambda)n}\n \\ge\\ \\frac{c}{\\varepsilon}\\ \\ge\\ \\frac1{\\sqrt{2}\\,\\varepsilon}.\n$$\nHence $(2z)^n \\ge 1\/\\varepsilon$, which shows\nthe claimed inequality~\\eqref{eq:claim}.\n\nUsing the bound~\\eqref{eq:claim} in Equation~\\eqref{def:z(e)} we get,\nagain writing $z=z(\\varepsilon)$, that\n$$\n z\\ \\le\\\n \\Big(Q + \\sqrt{2\\ln (2z)} + \\frac1{\\sigma\\sqrt{n}}\\Big)\\,\n        \\Big(\\frac{c}{\\varepsilon}\\Big)^{\\frac{1}{(1-\\lambda)n}},\n$$\nwhich means\n$$\n \\varepsilon\\ \\le\\\n  c \\bigg[\n   \\Big(Q + \\sqrt{2\\ln(2z)} + \\frac1{\\sigma\\sqrt{n}}\\Big)\\,\n    \\frac1z\\bigg]^{(1-\\lambda)n}.\n$$\nBy~\\eqref{eq:tail_kappa} this completes the proof.\n{\\mbox{}\\hfill\\qed}\\medskip\n\n\\subsection{Proof of Corollary~\\ref{cor:main}}\n\nFix $\\lambda_0\\in (0,1)$ and put $c:=c(\\lambda_0)$.\nSuppose that $m\\le n$ satisfy $\\lambda =(m-1)\/n \\le \\lambda_0$.\nThen $n-m +1 = (1-\\lambda)n \\ge (1-\\lambda_0) n$ and\nin order to have $n-m$ sufficiently large\nit suffices to require that $n$ is sufficiently large.\nThus, $c^{\\frac1{n-m+1}} \\le 1.1$\nif $n$ is sufficiently large. Similarly, because of\nLemma~\\ref{pro:Espnorm},\n$Q(m,n)\\leq 2.1$\nfor large enough~$n$. This implies that,\nfor $\\frac1{\\sqrt{m}} \\le \\sigma\\le 1$,\nwe have\n$$\n Q(m,n)  + \\frac1{\\sigma\\sqrt{n}}\\ \\le\\\n 2.1 + \\frac1{\\sigma\\sqrt{n}} \\ \\le\\\n 2.1 + \\sqrt{\\frac{m}{n}} \\ \\le\\\n 2.1 + \\sqrt{\\lambda_0 +\\frac1{n}}\\ \\le\\ 3.1 ,\n$$\nprovided $n$ is large enough. Then\n$\\zeta_\\sigma(m,n) \\le 3.1\\cdot 1.1 = 3.41$.\n\nBy Theorem~\\ref{thm:tailbound},\nthe random variable\n$Z :=  (1-\\lambda)\\kappa(A)\/e$\nsatisfies, for any $\\overline{A}$ with $\\|\\overline{A}\\|\\le 1$ and\nany $z\\ge 3.41$,\n\\begin{eqnarray*}\n \\mathop{\\rm Prob}_{A\\sim N(\\overline{A},\\sigma^2\\mathrm{I})}\\big\\{ Z \\ge z \\big\\}\n  &\\leq& 2c\\,\\bigg[\\Big(Q(m,n)+ \\sqrt{2\\ln (2z)} + \\frac1{\\sigma\\sqrt{n}}\\Big)\\,\n    \\frac1z\\bigg]^{n-m+1} \\\\\n  &\\leq& 2c\\,\\bigg[\\Big(3.1+ \\sqrt{2\\ln (2z)}\\Big)\\,\n    \\frac1z\\bigg]^{n-m+1}.\n\\end{eqnarray*}\nSince\n$3.1 + \\sqrt{2\\ln (2z)}\\le e\\sqrt{z}$\nfor $z\\ge 4$\nwe deduce that, for all such $z$,\n$$\n \\mathop{\\rm Prob}_{A\\sim N(\\overline{A},\\sigma^2\\mathrm{I})}\\big\\{ Z \\ge z \\big\\}\n  \\ \\leq\\ 2c\\,\\Big(\\frac{e}{\\sqrt{z}}\\Big)^{n-m+1} .\n$$\nUsing this tail bound to compute $\\mathbb{E}(Z)$ we get\n\\begin{eqnarray*}\n \\mathbb{E}(Z) &=& \\int_0^\\infty \\mathop{\\rm Prob}\\{Z\\ge z\\}\\, dz\\\n  \\;\\le\\; e^2 + 2c\n  \\int_{e^2}^\\infty\n  \\left(\\frac{e^2}{z}\\right)^{\\frac{n-m+1}{2}}dz \\\\\n  &\\stackrel{\\scriptstyle z=e^2y}{=}&\n    e^2 + 2c\\int_1^\\infty\n  \\left(\\frac{1}{y}\\right)^{\\frac{n-m+1}{2}}e^2dy\n  \\;=\\; e^2+ \\frac{4ce^2}{n-m-1}.\n\\end{eqnarray*}\nWe can now conclude since\n$$\n \\mathbb{E}((1-\\lambda)\\kappa(A))\n  = \\mathbb{E}(eZ) =e\\mathbb{E}(Z)\\leq e^3+ \\frac{4ce^3}{n-m-1}\n  \\leq 20.1\n$$\nthe inequality, again, by taking $n$ large enough.\n{\\mbox{}\\hfill\\qed}\\medskip\n\n\\section{Applications}\n\\label{sec:applications}\n\nWe next briefly discuss the two applications of our main result\nmentioned in the introduction.\n\n\\subsection{Accuracy of Linear Least Squares}\n\nRecall the problem~\\eqref{eq:ls} described in the introduction, namely,\nto compute the minimum of $\\|Ax-b\\|^2$ over $x\\in\\mathbb{R}^n$ \nfor given $A\\in\\mathbb{R}^{m\\times n}$ and $b\\in\\mathbb{R}^m$ (with $m>n$).\nIt is well known\nthat the loss of precision $\\mathsf{LoP}(A^\\dagger b)$\n---that is, the number of correct digits in\nthe entries of the data $(A,b)$ minus the same number for the computed\nsolution $A^\\dagger b$--- satisfies\n(cf.~\\cite{Wedin73} and \\cite[Ch.~19]{higham:96}) \n$$\n  \\mathsf{LoP}(A^\\dagger b)\\leq \\log m n^{3\/2} + 2\\log\\kappa(A)+{\\cal O}(1).\n$$\n Corollary~\\ref{cor:main}, combined with Jensen's inequality,\nimplies that $\\mathbb{E}(\\log\\kappa(A)) \\le \\log(20.1\/(1-\\lambda)) = {\\cal O}(1)$\nunder the assumptions stated in the corollary.\nHence for sufficiently elongated, large matrices $\\overline{A}$, the\nexpected loss of precision in the computation of the solution\n$A^\\dagger b$ over all small perturbations $A$ of $\\overline{A}$ is\ndominated by the term $\\log m n^{3\/2}$.\n\n\\subsection{Complexity of the Conjugate Gradient Method}\n\nIf $P\\in\\mathbb{R}^{m\\times m}$ is a symmetric positive definite\nmatrix and $c\\in\\mathbb{R}^m$, the system $Px=c$ can be solved by\nthe Conjugate Gradient Method (CGM), cf.~\\cite{hest-stiefel:52}.\nThis is an iterative algorithm which performs at most $m$ iterations but\nmay require less. Indeed, it is known\n(see, e.g., \\cite[Lecture~38]{ThrefethenBau})\nthat an $\\varepsilon$-approximation\nof the solution~$x$ can be computed in at most\n$\\frac12\\sqrt{\\kappa(P)}|\\ln\\varepsilon| $\niterations\n($\\varepsilon$ measures the\nrelative error of the approximation with respect to the\nEuclidean norm).\n\nIn many cases the matrix $P$ arises as $P=AA^{\\rm T}$ for some\nmatrix $A\\in\\mathbb{R}^{m\\times n}$ with $n>m$. \nIf $A$ is standard Gaussian distributed, \nthen the resulting distribution of $P$, \ncalled {\\em Wishart distribution}, \nhas been extensively studied in multivariate statistics. \nHowever, in our case of interest, $A$ is noncentered \nand much less is known about the resulting \ndistribution of $P$ \n(called noncentral Wishart). \nFortunately, using the fact that $\\sqrt{\\kappa(P)}=\\kappa(A)$, \nwe can directly apply our tail bounds for $\\kappa(A)$ \nfor a noncentral, isotropic Gaussian \ndistribution of $A$,  \nto derive bounds for the expected number\nof iterations of CGM.\n\nTo do so we use again Corollary~\\ref{cor:main}.\nIt shows that for all $\\lambda_0\\in(0, 1)$ and all\n$0 <\\sigma \\le 1$ there exists $n_0$ such that for\nall $1\\leq m<n$ we have\n$$\n \\sup_{\\|\\overline{A}\\| \\le 1} \\mathbb{E}_{A\\sim N(\\overline{A},\\sigma^2\\mathrm{I})}(\\kappa(A))\n  \\ \\leq\\ \\frac{20.1}{1-\\lambda},\n$$\nprovided $\\lambda = \\frac{m-1}{n}\\leq \\lambda_0$ and $n\\geq n_0$.\nIt follows that if $P$ is obtained as $AA^{\\rm T}$ for a large, elongated,\nrectangular matrix $A$ then, we should expect to\ncompute a solution with the desired accuracy with\nabout $\\frac{1}{2}\\frac{20.1}{1-\\lambda}|\\ln\\varepsilon|$ iterations.\nIt is known that\neach iteration of CGM takes $6n^2 +{\\cal O}(n)$ arithmetic operations.\nTherefore, the expected cost of running CGM on $P$ is\n$$\n  3n^2 \\frac{20.1}{1-\\lambda}|\\ln\\varepsilon|\n +{\\cal O}(n)=\\frac{60.3n^2}{1-\\lambda}|\\ln\\varepsilon| +{\\cal O}(n).\n$$\nThe leading term in this expression is smaller than the\n$\\frac23 n^3$ operations performed\nby Gaussian elimination as long as\n$$\n  \\varepsilon\\geq e^{-\\frac{n(1-\\lambda)}{91}}.\n$$\nFor large $n$ (and $\\lambda$ not too close to 1) this bound produces\nvery small values of $\\varepsilon$\nand therefore, CGM yields, on the average (both for a Wishart distribution\nof data $P$ and for Wishart perturbations of arbitrary data), \nremarkably good approximations of the solution $x=P^{-1}c$.\n\n\\section{Some Numerical Simulations}\n\\label{sec:numerical}\n\nSection~6 in~\\cite{CDW:05} describes the result of numerical computations\nproducing experimental values for $\\mathbb{E}(\\ln\\kappa(A))$ (for certain\nchoices of $\\overline{A}$ and $\\sigma$), which are\ndenoted by $\\mathbf{Avr}(\\ln\\kappa(A))$ and compared with\nthe upper bound for $\\mathbb{E}(\\ln\\kappa(A))$\n$$\n  {\\mu(m,n,\\sigma)}:=\n  \\ln\\left(m\n  +\\sigma m\\sqrt{5n}\\right)\n  +\\ln \\frac{2.35}{\\sigma} + \\frac{1}{r}\n  +\\sqrt{\\frac{e\\pi}{5}}\n$$\nobtained there. The data in\nTables~\\ref{tab:2} to~\\ref{tab:5}\nis taken from~\\cite{CDW:05}.\nEach row in these tables corresponds to a pair $(m,n)$.\nFor each row, 500 random matrices $A\\in\\mathbb{R}^{m\\times n}$\nwere computed following the distribution $N(\\sqrt{m}\\,\\overline{A},\\mathrm{I})$,\nwhere $\\overline{A}$ was chosen as\n$$\n  \\overline{A} := \\frac{{\\sf ones(m,n)}}{\\|{\\sf ones(m,n)}\\|},\n$$\nand {\\sf ones(m,n)} denotes the $m\\times n$ matrix all of whose\nentries are 1.\nThe column with header ${{\\mathbf{Avr}\\left(\\ln\\kappa(A)\\right)}}$ shows the empirical average of\n$\\ln\\kappa(A)$ for the 500 chosen random matrices~$A$.\nSince $\\kappa(A)$ is scale invariant we note that this corresponds\nto random matrices chosen from $N(\\overline{A},\\sigma\\mathrm{I})$,\nwhere $\\sigma=1\/\\sqrt{m}$.\n\n\\begin{table}[h]\n\\begin{center}\n\\begin{tabular}{||c|c|c|c|c||}\n \\hline\n $m$ & $n$ &${{\\mathbf{Avr}\\left(\\ln\\kappa(A)\\right)}}$ & ${\\mu(m,n,\\sigma)}$ & $\\ln (20.1\/(1-\\lambda))$\\\\\n\\hline \\hline\n10&15& 1.88278226808667 &7.73190477060415 & \\\\\n\\cline{1-4} 20&30&  2.04718612539162 & 8.74083698937094 &\n\\\\\n\\cline{1-4}\n40&60 &2.13539482051851&9.75820027818245& 4.0993321\n\\\\\n\\cline{1-4}\n80 & 120 & 2.19377719811291 & 10.78180469776403 &\n\\\\\n \\cline{1-4}\n160& 240&2.23119383890675&11.80997066079053&\n\\\\\n\\hline\n\\end{tabular}\\caption{$n=1.5m.$}\\label{tab:2}\n\\end{center}\n\\end{table}\n\n\\begin{table}[h]\n\\begin{center}\n\\begin{tabular}{||c|c|c|c|c||}\n \\hline\n $m$ & $n$ &${{\\mathbf{Avr}\\left(\\ln\\kappa(A)\\right)}}$ & ${\\mu(m,n,\\sigma)}$ & $\\ln (20.1\/(1-\\lambda))$\\\\\n\\hline \\hline\n 5& 10 & 1.28204418194521&6.35902343647518&\\\\\n\\cline{1-4}\n10&20& 1.48669849397793 &7.36178009761038&  \\\\\n\\cline{1-4}\n20&40 &  1.59394635398509&8.37451330180407 & \\\\\n \\cline{1-4}\n 40& 80&1.64896402420115& 9.39470162365532& 3.693866\\\\\n \\cline{1-4}\n80&160&1.69565973841311&10.42037692088400&\\\\\n \\cline{1-4}\n 160 &320 &1.72154032592663 &11.45004561375610&\\\\\n \\hline\n\\end{tabular} \\caption{$n=2m.$}\\label{tab:3}\n\\end{center}\n\\end{table}\n\n\\begin{table}[h]\n\\begin{center}\n\\begin{tabular}{||c|c|c|c|c||}\n \\hline\n $m$ & $n$ &${{\\mathbf{Avr}\\left(\\ln\\kappa(A)\\right)}}$ & ${\\mu(m,n,\\sigma)}$ & $\\ln (20.1\/(1-\\lambda))$\\\\\n\\hline \\hline\n10&25& 1.24167342192086 &7.46370799208199&\\\\\n\\cline{1-4}\n20&50 &1.34213347902230&8.47908853717777 &  \\\\\n \\cline{1-4}\n 40& 100&1.40120155287858&9.50123344342563& 3.511545 \\\\\n \\cline{1-4}\n80&200&1.44120596017225&10.52833707967242&\\\\\n \\cline{1-4}\n 160 &400 &1.45928497502137&11.55903912197539&\\\\\n \\hline\n\\end{tabular} \\caption{$n=2.5 m.$}\\label{tab:4}\n\\end{center}\n\\end{table}\n\n\\begin{table}[h]\n\\begin{center}\n\\begin{tabular}{||c|c|c|c|c||}\n \\hline\n $m$ & $n$ &${{\\mathbf{Avr}\\left(\\ln\\kappa(A)\\right)}}$ & ${\\mu(m,n,\\sigma)}$ & $\\ln (20.1\/(1-\\lambda))$\\\\  \\hline \\hline\n 5& 15 &  0.98741849882614& 6.37209092337754 &  \\\\\n\\cline{1-4}\n10&30&1.10550395287499 &7.38102314214432&  \\\\\n\\cline{1-4}\n20&60 &1.18790345922560&8.39838643095583 & \\\\\n \\cline{1-4}\n 40& 120& 1.23914387557043&9.42199085053742& 3.406185 \\\\\n \\cline{1-4}\n80&240& 1.27096561714092&10.45015681356392&\\\\\n \\cline{1-4}\n 160 &480  &1.28600775609989&12.14829242876138&\\\\\n \\hline\n\\end{tabular}\\caption{$n=3m.$}\\label{tab:5}\n\\end{center}\n\\end{table}\n\nIn~\\cite{CDW:05} it is\nobserved that ``one sees that when one fixes $m$ and lets $n$ increase\nthe quantity $\\mathbf{Avr}(\\ln\\kappa(A))$ decreases. This is in contrast\nwith the behaviour of $\\mu(m,n,\\sigma)$. It appears that our methods\nare not sharp enough to capture the behaviour of\n$\\mathbb{E}(\\ln\\kappa(A))$.'' Compare now with the\nresults of the present paper. It follows from\nCorollary~\\ref{cor:main}, by Jensen's inequality, that,\nfor sufficiently large $n$,\n$\\mathbb{E}(\\ln\\kappa(A))\\leq \\ln \\frac{20.1}{1-\\lambda}$. But if $m$ is\nheld fixed then, when $n$ increases,\n$\\lambda$ decreases and so does $\\ln \\frac{20.1}{1-\\lambda}$.\n\nOne still observes a difference between the bound $\\ln \\frac{20.1}{1-\\lambda}$\nand the values of ${{\\mathbf{Avr}\\left(\\ln\\kappa(A)\\right)}}$. Part of this difference comes from the\nasymptotic character of this bound and the fact that our data\nis limited to $m\\leq 160$. One sees on the tables that larger values\nof $m$ would approach ${{\\mathbf{Avr}\\left(\\ln\\kappa(A)\\right)}}$ to $\\ln \\frac{20.1}{1-\\lambda}$.\nWe conjecture that, in addition to the possible loss of sharpness\ncoming from the use of Jensen's inequality, the difference above is due to\nthe roughness of the constant 20.1.\n\n{\\small\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\n\\section{The relation between 2D and 3D physical parameters}\nIn cold atomic gases where atoms move freely in two dimensions but are strongly confined along the axial direction by a one-dimensional harmonic trap of frequency $\\omega_z$, the confined dimension still has a residual effect because of the populations of the bound molecule states in the excited levels even for an extremely strong trap. To capture this residual effect, we follow Ref.~\\cite{Duan1} to convert the actual 3D Hamiltonian to an effective 2D Hamitonian of Eq.~(1) in the main text with a renormalized interaction between the atoms in the axial ground state and the dressed molecules. The 2D bare parameters $\\delta_b$, $\\alpha_b$ and $V_b$ in the model Hamiltonian are connected to the 2D physical parameters $\\delta_p$, $\\alpha_p$ and $V_p$ via the renormalization relation ($\\hbar=1$)\n\\begin{eqnarray}\n\\frac{1}{V_b + \\frac{\\alpha_b^2}{2\\mu-\\delta_b}} = \\frac{1}{V_{\\rm eff}^p} - \\sum_{\\bf k}\\frac{1}{2\\epsilon_{\\bf k}+\\omega_z},\n\\end{eqnarray}\nwhere $V_{\\rm eff}^p=V_p+\\frac{\\alpha_p^2}{2\\mu-\\delta_p}$.\n\nThe 2D physical parameters $\\delta_p$, $\\alpha_p$ and $V_p$ are determined by matching the two-body bound state from the original 3D Hamiltonian, with the 3D dimensionless physical parameters $U_p=4\\pi a_{\\rm bg}\/a_z$, $g_p^2=\\mu_{\\rm co}WU_p\/\\omega_z$, and $\\nu_p=\\mu_{\\rm co}(B-B_0)\/\\omega_z$. Here, $a_{\\rm bg}$ is background scattering length, $a_z=\\sqrt{1\/m\\omega_z}$ is the characteristic length for axial motion, $\\mu_{\\rm co}$ is the difference in magnetic moments between the open and close channels, $W$ is the resonance width, and $B_0$ is the resonance position. To compare with the experiments of 2D $^{6}{\\rm Li}$ atomic gases~\\cite{Moritz1}, we will use the parameters $a_{bg}=-1405a_{0}$, $W=300$G, and $\\mu_{\\rm co}=2\\mu_{B}$, where $a_0$ and $\\mu_B$ are respectively the Bohr radius and Bohr magneton. Two functions $S_p(E)$ and $\\sigma_p(E)$ turn out to be important for matching the two-body bound state and the parameter renormalization~\\cite{Duan1}:\n\\begin{eqnarray}\nS_p(E) &=& -\\frac{1}{2^{5\/2}\\pi}\\int_0^\\infty dx\\Big[\\frac{\\Gamma(x-E\/2)}{\\Gamma(x+1\/2-E\/2)} - \\frac{1}{\\sqrt{x}}\\Big],\\notag\\\\\n\\sigma_p(E) &=& \\frac{{\\rm ln}|E|}{2^{5\/2}\\pi^{3\/2}}.\n\\end{eqnarray}\nFor example, the two-body binding energy $E_B$ is determined by the solution of the eigen-equation\n\\begin{eqnarray}\n1\/U_{\\rm eff}^p(E_B)=S_p(E_B),\n\\end{eqnarray}\nwhere $U_{\\rm eff}^p(E) = U_p+\\frac{g_p^2}{E-\\nu_p}$.\n\nIn brief, the first matching of the bound state energy in the off-resonance BCS limit leads to~\\cite{Duan1}\n\\begin{eqnarray}\n\tV_{p}^{-1}=\\sqrt{2\\pi}(U_{p}^{-1}-C_{p}),\n\\end{eqnarray}\nwhere $C_p = {\\rm lim}_{\\nu_p\\to\\infty}[S_p(E_B)-\\sigma_p(E_B)]$. Then, matching the binding energy at an arbitrary detuning yields\n\\begin{eqnarray}\n\\delta_{p} & = & E_B - \\frac{\\sigma_p(E_B)}{\\partial_E\\Big[1\/U_{\\rm eff}^p(E)-\\big(S_p(E)-\\sigma_p(E)\\big)\\Big]\\Big|_{E=E_B}} \\bigg(1-\\frac{\\sigma_p(E_B)}{U_p^{-1}-C_p}\\bigg),\\notag\\\\\n\\alpha_p^2& = & \\frac{1\/\\sqrt{2\\pi}}{\\partial_E\\Big[1\/U_{\\rm eff}^p(E)-\\big(S_p(E)-\\sigma_p(E)\\big)\\Big]\\Big|_{E=E_B}}\n\\bigg(1-\\frac{\\sigma_p(E_B)}{U_p^{-1}-C_p}\\bigg)^2.\n\\label{eq:3dpara}\n\\end{eqnarray}\n\n\\section{Gaussian pair fluctuation theory}\nAdopting the functional path-integral formalism over imaginary time $\\tau$ and applying the Hubbard-Stratonovich transformation in real space~\\cite{Hu-20}, we decouple the interaction term by introducing an auxiliary bosonic field $\\Delta(x)=\\alpha_{b}\\phi(x)+\\widetilde{\\Delta}(x)$ with $\\phi(x)=\\langle d(x)\\rangle$, $\\widetilde{\\Delta}(x)=V_{b}\\langle a_{\\downarrow}(x)a_{\\uparrow}(x)\\rangle$, and $x = ({\\bf x}, \\tau)$ the short-hand notation. The effective action is then given in term of the fermionic Nambu spinor $A(x)=[a_{\\uparrow}(x), a_{\\downarrow}^{\\dagger}(x)]^{T}$ as\n\\begin{eqnarray}\nS & = & \\beta\\sum_{\\mathbf{k}}\\xi_{\\mathbf{k}}+\\int dxdx' A(x)^{\\dagger}[-\\mathcal{G}_{a}^{-1}(x,x')] A(x')-\\frac{|\\Delta(x)|^{2}}{V_{b}}\\nonumber\\\\\n& &+\\overline{d}(x)\\left(\\partial_{\\tau}+\\widetilde{\\omega}_{\\bf q}-\\frac{\\alpha_{b}^{2}}{V_{b}}\\right)d(x)+\\frac{\\alpha_{b}}{V_{b}}[\\overline{\\phi}(x)\\Delta(x) + {\\rm H.C.}].\n\\end{eqnarray}\nHere, $\\widetilde{\\omega}_{\\mathbf{q}}=\\mathbf{q}^{2}\/4m-2\\mu+\\delta_{b}$, and $\\mathcal{G}_{a}(x,x')$ is the Green's function (GF) of atoms. Under the mean-field approximation $\\Delta(x)=\\Delta$ and $\\phi(x)=\\phi_{0}$, the inverse GF is given by\n\\begin{eqnarray}\n\\mathcal{G}_{a}^{-1}(x,x')=\\left(\n                        \\begin{array}{cc}\n                          -\\partial_{\\tau}-H_{0}^{a} & -\\Delta \\\\\n                          -\\Delta & -\\partial_{\\tau}+H_{0}^{a} \\\\\n                        \\end{array}\n                      \\right)\\delta_{xx'}\\delta_{\\tau\\tau_{'}},\n\\end{eqnarray}\nwhere $H_{0}^{a}={-\\nabla^{2}}\/{2m}-\\mu$ is the kinetic energy of atoms.\nBy minimizing the mean-field effective action with respect to $\\phi_{0}$, we obtain the zero-temperature thermodynamic potential ($\\Omega_{\\rm MF}=k_BT S_{\\rm MF}$)\n\\begin{eqnarray}\n\\Omega_{\\rm MF}=\\sum_{\\mathbf{k}}\\left(\\xi_{\\mathbf{k}}-E_{\\mathbf{k}}+\\frac{\\Delta^{2}}{2\\epsilon_{\\mathbf{k}}+\\omega_{z}}\\right)-\\frac{\\Delta^{2}}{V_{\\rm eff}^p},\n\\end{eqnarray}\nwhere $E_{\\mathbf{k}}=\\sqrt{\\xi_{\\mathbf{k}}^{2}+\\Delta^{2}}$ is the quasiparticle dispersion, and $V_{\\rm eff}^p=V_{p}+\\frac{\\alpha_{p}^{2}}{2\\mu-\\delta_{p}}$ is the renormalized effective interaction. Note that the bare parameters are replaced by the physical ones by the renormalization relation $1\/{V_{\\rm eff}^b}=1\/{V_{\\rm eff}^p}-\\sum_{\\bf k}\\frac{1}{2\\epsilon_{\\bf k}+\\omega_z}$, with $V_{\\rm eff}^b=V_{b}+\\frac{\\alpha_{b}^{2}}{2\\mu-\\delta_{b}}$ the bare effective interaction. The saddle point conditions of $\\Omega_{\\rm MF}$ lead to the gap and number equations in the mean-field level\n\\begin{eqnarray}\n\\frac{1}{V_{\\rm eff}^p} & = & -\\sum_{\\mathbf{k}}\\left(\\frac{1}{2E_{\\mathbf{k}}}-\\frac{1}{2\\epsilon_{\\mathbf{k}}+\\omega_{z}}\\right),\n\\label{eqn:gap} \\\\\nn_{\\rm MF} & = & -\\frac{\\partial\\Omega_{\\rm MF}}{\\partial\\mu} = 2\\phi_{0}^{2}+\\sum_{\\mathbf{k}}\\left(1-\\frac{\\xi_{\\mathbf{k}}}{E_{\\mathbf{k}}}\\right),\n\\end{eqnarray}\nwhere $\\phi_{0}^{2}={\\Delta^{2}}\/{[\\alpha_{p}+{V_{p}}(2\\mu-\\delta_{p})\/{\\alpha_{p}}]^{2}}$ is the population of dressed molecules.\n\nWe then introduce the fluctuation of the pairing order parameters $\\Delta(x)=\\Delta+\\delta\\psi(x)$\nand $\\phi(x)=\\phi+\\delta\\phi(x)$, and calculate the contribution to the effective action up to the second order. This leads to the following Gaussian term at zero temperature\n\\begin{eqnarray}\nS_{\\rm GPF} & = & -\\frac{1}{2}\\sum_{q}\\overline{\\Phi}_{m}(q)D_{0}^{-1}(q)\\Phi_{m}(q)-\\frac{1}{2}\\overline{\\Psi}(q)\\Gamma_0^{-1}(q)\\Psi(q)\\nonumber\\\\\n& &+\\frac{\\alpha_{b}}{2V_{b}}\\left[\\overline{\\Phi}_{m}(q)\\Psi(q) + {\\rm H.C.}\\right],\n\\end{eqnarray}\nwhere the bosonic Nambu spinors are defined as $\\Phi_{m}(q)=(\\delta\\phi(q), \\overline{\\delta\\phi}(-q))^{T}$ and $\\Psi (q)=(\\delta\\psi(q), \\overline{\\delta\\psi}(-q))^{T}$, with $q = ({\\bf q}, \\nu_n)$ denoting both the 2D pairing momentum ${\\bf q}$ and the bosonic Matsubara frequency $\\nu_n$. The inverse of the bosonic GF of dressed molecule reads\n\\begin{eqnarray}\nD_{0}^{-1}(q)=\\left(\n                \\begin{array}{cc}\n                  i\\nu_{n}-\\omega_{\\mathbf{q}}+\\frac{\\alpha_{p}^{2}}{V_{p}} & 0 \\\\\n                  0 & -i\\nu_{n}-\\omega_{\\mathbf{q}}+\\frac{\\alpha_{p}^{2}}{V_{p}} \\\\\n                \\end{array}\n              \\right),\n\\end{eqnarray}\nwith $\\omega_{\\mathbf{q}}={\\bf q}^{2}\/4m-2\\mu+\\delta_{p}$. The elements of the $2\\times 2$ inverse vertex function $\\Gamma_0^{-1}$ are\n\\begin{eqnarray}\n\\Gamma_{0,11}^{-1}(q) & = & \\Gamma_{0,22}^{-1}(-q) =  + \\sum_{k}\\mathcal{G}_{11}(k)\\mathcal{G}_{22}(k-q)+\\frac{1}{V_b},\n\\nonumber \\\\\n\\Gamma_{0,12}^{-1}(q) & = & \\Gamma_{0,21}^{-1}(q) = - \\sum_{k}\\mathcal{G}_{12}(k)\\mathcal{G}_{12}(k-q),\n\\end{eqnarray}\nwhere $\\mathcal{G}$ is the saddle-point GF with the notation $k=({\\bf k},\\omega_m)$,\n\\begin{eqnarray}\n\\mathcal{G}(k)=\\frac{1}{(i\\omega_{m})^{2}-E_{\\bf k}^{2}}\\left(\n  \\begin{array}{cc}\n    i\\omega_{m}+\\xi_{\\mathbf{k}} & -\\Delta \\\\\n    -\\Delta & i\\omega_{m}-\\xi_{\\mathbf{k}} \\\\\n  \\end{array}\n\\right).\n\\end{eqnarray}\nBy integrating over the two bosonic Nambu spinors~\\cite{Hu-20}, we find that the thermodynamic potential $\\Omega_{\\rm GPF} = k_BT S_{\\rm GPF}$ contributed by quantum pair fluctuations is given by\n\\begin{equation}\n\\Omega_{\\textrm{GPF}}=\\frac{1}{2}\\sum_{q}\\ln\\left[M_{11}\\left(q\\right)M_{22}\\left(q\\right)-M_{12}\\left(q\\right)M_{21}\\left(q\\right)\\right]e^{i\\nu_{n}0^{+}},\\label{eq:OmegaGPF}\n\\end{equation}\nwhere an extra factor $e^{i\\nu_{n}0^{+}}$ is added to ensure the convergence of the summation over $q$. The expressions of $M_{ij}(q)$($i,j=1,2$) can be worked out explicitly and at zero temperature lead to\n\\begin{eqnarray}\nM_{11}(q)&=&\\sum_{\\bf k} \\left(\\frac{u_{{\\bf k}+}^{2}u_{{\\bf k}-}^{2}}{i\\nu_{n}-E_{{\\bf k}+}-E_{{\\bf k}-}}-\\frac{v_{{\\bf k}+}^{2}v_{{\\bf k}-}^{2}}{i\\nu_{n}+E_{{\\bf k}+}+E_{{\\bf k}-}}+\\frac{1}{2\\epsilon_{\\bf k}+\\omega_{z}}\\right)-\\frac{1}{V_{\\rm eff}^p(q)}, \\nonumber \\\\\nM_{12}(q)&=&\\sum_{\\bf k}\\left(\\frac{u_{{\\bf k}+}v_{{\\bf k}+}u_{{\\bf k}-}v_{{\\bf k}-}}{i\\nu_{n}+E_{{\\bf k}+}+E_{{\\bf k}-}}-\\frac{u_{{\\bf k}+}v_{{\\bf k}+}u_{{\\bf k}-}v_{{\\bf k}-}}{i\\nu_{n}-E_{{\\bf k}+}-E_{{\\bf k}-}}\\right),\n\\end{eqnarray}\nwhere  $V_{\\rm eff}^p(q)=V_{p}+\\frac{\\alpha_{p}^{2}}{i\\nu_{n}-(\\mathbf{q}^{2}\/4m-2\\mu+\\delta_{p})}$.\n\nIn the expressions above, we define $\\mathbf{k}_{\\pm}=\\mathbf{q}\/2 \\pm \\mathbf{k}$, and BCS superfluid parameters $u_{\\mathbf{k}\\pm}^{2}=(1+{\\xi_{{\\bf k}\\pm}}\/{E_{{\\bf k}\\pm}})\/2$ and $v_{\\mathbf{k}\\pm}^{2}=(1-{\\xi_{{\\bf k}\\pm}}\/{E_{{\\bf k}\\pm}})\/2$ to simplify notation, where $\\xi_{{\\bf k}\\pm}={\\bf k}_{\\pm}^2\/2m-\\mu$ and $E_{{\\bf k}\\pm}=\\sqrt{\\xi_{{\\bf k}\\pm}^{2}+\\Delta^{2}}$. With that, the number equation including the Gaussian fluctuation effect reads\n\\begin{eqnarray}\n\\label{eqn:num}\nn=-\\frac{\\partial\\Omega_{\\rm MF}}{\\partial\\mu}-\\frac{\\partial\\Omega_{\\rm GPF}}{\\partial\\mu},\n\\end{eqnarray}\nwhich is solved self-consistently with the mean-field gap equation numerically to determine the chemical potential $\\mu$ and order parameter $\\Delta$.\n\n\\begin{figure}[htbp]\n\\centering\n\\includegraphics[width=0.8\\linewidth]{figureS1.eps}\\hfill{}\\caption{\\label{figS1}(Color online) (a) Superfluid gap and (b) chemical potential versus the 2D interaction strength ${\\rm ln}(k_Fa_{\\rm 2D})$. Results obtained from Q2D GPF theory (red solid line) are compared with the experimental data from $^{6}{\\rm Li}$ atomic gases (red triangles)~\\cite{Moritz1} and the 2D QMC results (red dots)~\\cite{Zhang} with the correction of Q2D binding energy $E_B$~\\cite{Levinsen}.}\n\\end{figure}\n\n\n\\section{Superfluid gap and chemical potential}\nBased on the coupled Eqs.~(3) and (6) in the main text, we determine the superfluid gap $\\Delta$ and the chemical potential $\\mu$ self-consistently in a wide range of crossover, where the interaction strength of pure 2D and Q2D systems can be  parameterized by the 2D dimensionless interaction parameter ${\\rm ln}(k_Fa_{\\rm 2D})$. We compare the numerical result of superfluid gap with the experimental data of a Q2D $^{6}{\\rm Li}$ Fermi gas~\\cite{Moritz1} with the same axial trapping frequency $\\omega_z=1\\omega_0$ in Fig.~\\ref{figS1}(a). The numerical results of chemical potential are compared with 2D auxiliary field QMC calculations~\\cite{Zhang} with modified Q2D binding energy $E_B$~\\cite{Levinsen} in Fig.~\\ref{figS1}(b).\nNote that in the experiment the order parameter is measured in two ways, one is directly extracted from the integrated dynamic structure factor $S(\\omega)=\\int S(q,\\omega)qdq$ in the BCS regime, the other is from the dynamic structure factor $S(q,\\omega)$ at a small momentum transfer in the BEC regime. The chemical potential in the experiment is determined by the auxiliary field QMC calculations with the modification binding energy $E_B$ for a Q2D geometry, which is given by\n\\begin{eqnarray}\n\t\\frac{a_z}{a_s} = \\int_0^{+\\infty}\\frac{du}{\\sqrt{4\\pi u^3}}\\left[\n\t1 - \\frac{e^{-\\frac{E_B}{\\omega_z}u}}{\\sqrt{(1-e^{-2u})\/(2u)}}\n\t\\right].\n\t\\label{eq:S1}\n\\end{eqnarray}\n\nWe find that the experimental data of superfluid gap agree well with our theoretical prediction of a Q2D system in both the BCS and BEC regimes, while the deviation between them is more notable in the unitary region owing to significant fluctuations.\nA self-consistent calculation of the chemical potential for a Q2D system is in excellent agreement with the 2D QMC results when the binding energy $E_B$ is corrected by the Q2D result~\\cite{Levinsen}. This indicates that the correction plays an essential role in the connection between Q2D and purely 2D systems.\n\n\\begin{figure}[htbp]\n\\centering\n\\includegraphics[width=0.8\\linewidth]{figureS2.eps}\\hfill{}\\caption{\\label{figS2}(Color online) Dimensionless pair size $k_F\\xi$ for (a) Q2D (red solid line) and 2D (gray dotted line) systems as a function of 2D interaction strength ${\\rm ln}(k_Fa_{\\rm 2D})$ and (b) 3D (blue dashed-dotted line) system as a function of 3D interaction strength $1\/k_Fa_s$. Results obtained from GPF theory are compared with the experiment data obtained from the onset momentum $k_o$ of the pair breaking continuum for Q2D (gray dots)~\\cite{Moritz1} and 3D~\\cite{Moritz2} (gray diamonds) gases of $^{6}{\\rm Li}$ atoms. The red diamonds and blue dots represent the Q2D and 3D pair sizes calculated via Eq.~(\\ref{eq:xi2d}) and Eq.~(\\ref{eq:xi3d}) using the superfluid gap and the chemical potential obtained from Ref.~\\cite{Moritz1}. The black squares show the results from a pair size measurement implemented in 3D $^{6}{\\rm Li}$ atomic gases using RF spectra~\\cite{Ketterle}.}\n\\end{figure}\n\n\n\n\\section{pair size}\nTo estimate the pair size of Fermi gases with different dimensionalities, we follow the definition of Eq.~(7) as stated in the main text. We plot the numerical results of $k_F\\xi$ for Q2D (red solid line) and 2D (gray dotted line) systems by tuning the conventional 2D interaction parameter ${\\rm ln}(k_Fa_{\\rm 2D})$ in Fig.~\\ref{figS2}(a), and for 3D (blue dashed-dotted line) system by varying the 3D scattering length $1\/k_Fa_s$ in Fig.~\\ref{figS2} (b). Analytically, the pair size can be obtained as\n\\begin{eqnarray}\n\\xi^{2}=\\frac{1}{4m\\Delta}\\bigg[\\frac{\\mu}{\\Delta} + \\frac{\\mu^2+2\\Delta^2}{\\mu^2+\\Delta^2}\\Big(\\frac{\\pi}{2}+{\\rm arctan}\\frac{\\mu}{\\Delta}\\Big)^{-1}\\bigg],\n\\label{eq:xi2d}\n\\end{eqnarray}\nin two dimensions~\\cite{Mohit} and\n\\begin{eqnarray}\n\\xi^{2}=\\frac{1}{16m\\Delta^2}\\bigg[2\\mu + \\frac{5\\Delta^2+2\\mu^2}{\\sqrt{\\mu^2+\\Delta^2}}\\bigg],\n\\label{eq:xi3d}\n\\end{eqnarray}\nin three dimensions~\\cite{Marini}.\n\nWe note that the pair size of Q2D $^{6}{\\rm Li}$ atomic gases is not directly measured in experiments~\\cite{Moritz1}. Instead, one determines the onset momentum $k_o$ of the pair breaking continuum from the intersection point of a bilinear fit of the dynamic structure factor at a transferred energy of $\\omega=2\\Delta$, which is related to pair size via $k_o=\\alpha\/\\xi$. The prefactor $\\alpha$ is evaluated by comparing the experimental results of $1\/k_o$ to theoretical predictions of the pair size of 2D or 3D Fermi gases. The raw data of pair size in Q2D~\\cite{Moritz1} and 3D~\\cite{Moritz2} $^{6}{\\rm Li}$ Fermi gases are shown in Fig.~\\ref{figS2} by gray dots and gray diamonds, respectively. We find that the raw experimental data fit better with the theoretical predictions of a 2D Fermi gas than a Q2D system. However, by substituting the superfluid gap and chemical potential measured in experiment~\\cite{Moritz1} to Eq.~(\\ref{eq:xi2d}) and Eq.~(\\ref{eq:xi3d}), we can obtain a modified pair size. These modified results for Q2D (red diamonds) and 3D (blue dots) $^6$Li gases agree well with our Q2D and 3D predictions from GPF theory. In addition, the black squares represent the pair size of 3D cold atomic gases of $^{6}{\\rm Li}$ performed in Ref.~\\cite{Ketterle} obtained from radio-frequency spectra, also showing a good match to our prediction.\n\n\n\\begin{figure}[htbp]\n\\centering\n\\includegraphics[width=0.8\\linewidth]{figureS3.eps}\\hfill{}\\caption{\\label{figS3}(Color online) Collective mode spectrum $\\omega_q$ versus $q$ for Q2D systems with different trapping frequencies and interactions of (a) $1\/k_Fa_s=-1$ and (b) $1\/k_Fa_s=1$. Three Q2D configurations with $\\omega_z = \\omega_0$ (red solid line), $3\\omega_0$ (yellow dashed line), and $5 \\omega_0$ (blue dashed-dotted line) are shown for comparison. The shades represent the continuum of pair breaking excitations.}\n\\end{figure}\n\n\\section{Collective mode}\n\nThe low-energy collective mode spectrum $\\omega_q$ is well-known as the Goldstone mode arising from the broken of $U(1)$ symmetry, which is determined by the pole of the inverse Gaussian fluctuation boson propagator $M$, i.e. the solution of ${\\rm det}[M(q,\\omega_q)]=0$. The two-particle continuum begins at $E_{c}(q)={\\rm min}_{\\bf k}(E_{+}+E_{-})$, corresponding to the single particle excitation from the pair breaking threshold.\n\nIn Fig.~\\ref{figS3}, we present the Goldstone mode by lines and pair breaking continuum by shades with different trapping frequencies on the BCS side with the interaction strength $1\/k_Fa_s=-1$ [Fig.~\\ref{figS3}(a)], and on the BEC side with the interaction strength $1\/k_Fa_s=1$ [Fig.~\\ref{figS3}(b)], where the trapping frequencies are $1\\omega_0$ by red solid lines and red shades, $3\\omega_0$ by yellow dashed lines and yellow shades, $5\\omega_0$ by blue dashed-dotted lines and blue shades. In the long-wavelength limit, the dispersions exhibit linear shape for all configurations, the slope of which determines the sound velocity. We find that $\\omega_q$ is always below the two-particle continuum, indicating the existence of a two-body bound state in a Q2D system over a wide range of crossover.\n\nAs shown in Fig.~\\ref{figS3}(a), in the BCS regime, we observe a wide pair breaking continuum and the pairs are weakly bound. When the transferred energy is strong enough to break a Cooper pair, the atoms will be excited to the continuum state with a sharp energy threshold of $2\\Delta$. In particular, for large transferred momentum, the Goldstone mode touches the pair breaking continuum and bends down instead of keeping a linear slope. However, in the BEC regime where the atom pairs condense as tightly-bound molecules, the continuum is notably lifted until moves out of the spectra, as shown in Fig.~\\ref{figS3}(b).\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{The Constructions in Detail}\n\\label{sec-from-all}\n\n\\subsection{The $(n,n)$-reconstruction protocol \\label{sec-from-all-details}}\nWe start our formalization of the intuition presented above with the\ncase where all $n$ out of the $n$ agents participate in the state\nreconstruction.  The protocol for this case, which we call $\\nnPi$, is presented\nbelow.\n\n\\vspace{-.1in}\n\\paragraph{Protocol $\\nnPi$.} The protocol consists of three phases:\n\n\\vspace{.1in}\n\n\\noindent {\\em Initialization.} The dealer secret-shares among the agents \na secret value for each state, such that the value for the initial\nstate is $1$ and for all the other states is $0$. This is done as follows. \nAgent $A_i$ ($1\\leq i\\leq n$) is given a a random binary string\n$x^i_1x^i_2...x^i_m$, with the constraints that \n$$x^1_{\\init} + x^2_{\\init} + ... + x^n_{\\init}\\equiv 1 \\bmod 2,$$ \nwhere $\\init$ is the index of the initial state of the\ncomputation, and for every $1\\leq j \\neq init \\leq m$, \n\n$$x^1_j + x^2_j + ... + x^n_j \\equiv 0 \\bmod 2.$$ \nEach agent then proceeds to assign its state labels as\n$\\ell_j^i \\leftarrow x_j^i$.\n\n\n\\vspace{.1in}\n\\noindent {\\em Event Processing.}  Each agent runs Algorithm~\\ref{algo:calc},\nupdating its labels and computing the new seeds for the PRG.  \nLet ${\\cal T}$ be the set of all possible agents' pairs. For line 8\nof Algorithm \\ref{algo:calc}, each agent $A_i$ now computes\n$$\nR_j= \\sum_{\\substack{T\\in {\\cal T}, A_i\\in T}} (b_j^T)_r.\n$$\n\n\n\\begin{figure*} [ht]\n\\begin{center}\n\\includegraphics[width=7cm]{comp_scheme1}\n\\caption{\\sl The internal state of agent $A_i$ before a transition.}\n\\end{center}\n\\label{fig:comp_scheme1}\n\\end{figure*}\n\n\\vspace{.1in}\n\\noindent {\\em Reconstruction.}  All agents submit their internal states to the\ndealer, who reconstructs the secrets  corresponding to each state, by \nadding (mod 2) the shares of each state, and determines and outputs \nthe currently active state (the one whose reconstructed secret is $1$).\n\n\n\n\\begin{figure*} [ht]\n\\begin{center}\n\\includegraphics[width=11cm]{comp_scheme2}\n\\caption{\\sl The internal state of agent $A_i$ after an $\\alpha$ transition.}\n\\end{center}\n\\label{fig:comp_scheme2}\n\\end{figure*}\n\n\\vspace{.1in}\nBefore proving the correctness and privacy achieved by the protocol, \nwe illustrate the operation of the online (Event Processing) phase\nwith the following example;\nrefer to Figures 1\nand 2.\nThe two figures describe the execution of the\nprotocol\non an automaton with four states and two possible\ninputs. \nFigure 1 presents the \ninternal state of agent $A_i$ after the $(r-1)$-th clock cycle. \nThe agent holds the original automaton and has a label for each of \nthe four states, \n$(\\ell^i_1)_{r-1}$, $(\\ell^i_2)_{r-1}$, $(\\ell^i_3)_{r-1}$ and\n$(\\ell^i_4)_{r-1}$.\n\n\n\nFigure 2\nshows the changes in the agent's internal state compared to Figure 1\nafter the $r$-th clock cycle. We also assume\nthat in this clock cycle the agents receive an input symbol\n$\\alpha$. The new labels for each state are the sum of old labels and\npseudo-random values. The labels in the sum are the old labels of\nall the states that transition to the current state given the\ninput. Thus, the new $(\\ell_2^i)_r$ includes a sum of the old\n$(\\ell^i_2)_{r-1}$ and the old $(\\ell^i_4)_{r-1}$, while the new\n$(\\ell_3^i)_r$ doesn't include any labels in its sum because there is\nno state that transitions to $s_3$ after an $\\alpha$ input. The \npseudo-random addition to each state $j=1,\\ldots,4$ is the sum $\\sum_{T, i\n\\in T} (b_j^T)_{r}$.\n\n\nWe start by proving  the correctness of the construction.\n\n\n\\begin{proposition}\n\\label{prop-all}\nAt every Event Processing step of protocol$\\nnPi$'s,\nthe secret corresponding\nto the current state in the computation is $1$ and for all other states \nthe secret is $0$.\n\\end{proposition}\n\\begin {proof}\nThe proof is\nby induction on the number of steps $r$ that the automaton performs, i.e., \nthe number of clock cycles.\n\nFor the base case, if we consider the state of the protocol\nafter the initialization step and before the first clock cycle, i.e.,\nat $r=0$, then the statement is true by our definition of the label\nassignments. Let us first consider the case where at the $r$-th step\nan input symbol $\\gamma _r$ from $\\Gamma$ is received.  \nFollowing the protocol,\nagent $A_i$'s new label for state $j$ becomes\n$$ \\ell ^i_j \\longleftarrow\n\\sum_{\\substack{k~ :\\\\ \\mu(s_k, \\gamma_r)=s_j}} \\ell^i_k + \\sum _{A_i \\in\nT} (b_j^T)_r.\n$$\n\nConsider now the next state of the computation in the automaton; we\nwish to show that the secret corresponding to that state will be 1.\nLet $\\curr$ be the index of the current state of the automaton, and\n$\\next$ be the index corresponding to the next state; by definition,\n$\\mu(s_{\\curr},\\gamma_r)=s_{\\next}$. Then, \n$$\\ell^i_{\\next} \\longleftarrow\n\\sum_{\\substack{k~:\\\\ \\mu(s_k, \\gamma_r)=s_{\\next}}} \\ell^i_k = \\\\\n$$ \n$$\n\\ell^i_{\\curr}+ \\sum_{\\substack{k\\neq \\curr~:\\\\ \\mu(s_k,\n\\gamma_r)=s_{\\next}}} \\ell^i_k + \\sum _{i \\in T} (b_j^T)_r.$$ \n\nBy the induction hypothesis, we know that \n$$\\sum_{i=0}^n \\ell^i_{\\curr} \\equiv 1\n\\mbox{ (mod 2)}$$ \nand for $k\\neq \\curr$, \n$$\\sum_{i=0}^n \\ell^i_k \\equiv 0 \\mbox{ (mod 2)}.$$ \n\n\\noindent Thus, if we will sum over all the agents:\n\n\\begin{eqnarray*}\n\\lefteqn{ \\sum_{i=0}^n \\left( \\ell^i_{\\curr}+  \\sum_{\\substack{k\\neq \\curr~:\\\\ \\mu(s_k, \\gamma_r)=s_{\\next}}} \\ell^i_k + \\sum _{i \\in T} (b_j^T)_r \\right)} \\\\\n&=& \\sum_{i=0}^n  \\ell^i_{\\curr} + \\sum_{\\substack{k \\neq \\curr~:\\\\ \\mu(s_k, \\gamma_r)=s_{\\next}}} \\sum_{i=0}^n \\ell^i_k  \\\\\n&+& \\sum_{i=0}^n \\sum _{i \\in T} (b_j^T)_r  \\equiv 1 + 0  \\equiv 1 \\mbox{ (mod 2)}.\n\\end{eqnarray*}\n\n\\noindent This is because in $ \\sum_{i=1}^n \\sum _{i \\in T} (b_j^T)_r$, \nevery $(b_j^T)_r$ appears exactly twice in this sum, once for\nevery \nelement in $T$. \nUsing similar arguments one can see that all the other\nstates will resolve to 0.  \n\nIn the case that in the $r$-th step no input symbol is received,\ndue to the fact that we just add the\nrandom strings in the same way as in the case above,\nwe again get\nthat the secret corresponding to the current\nstate of the computation is 1, and for all others is 0.\n\\end {proof}\n\n\n\\begin{proposition}\n\\label{prop-pcm-all}\nProtocol \n$\\nnPi$ is $(n-1)$-private in the PCM model according to \nDefinition~\\ref{defProgressive}.\n\\end{proposition}\n\\begin {proofsk}\nRecall that the underlying observation is that when a corruption takes place\n(which cannot happen during the label-update procedure), the agent's state\nincludes the current labels and PRG seeds which have already been evolved, \nand hence cannot be correlated with the label shares previously\ngenerated.\n\nWithout loss of generality, consider the case where \\Ad\\ corrupts\nall but one agent according to an arbitrary corruption timeline,\nand assume, say, agent $A_1$ is not corrupted.  We argue that the \nview of the adversary is indistinguishable from a \nview corresponding to\n(randomly) initialized agents $A_2,...,A_n$ on the given automaton and\nany initial state.  In other words, the view of the adversary is\nindistinguishable from the view he would obtain if he corrupted the\nagents simultaneously and before any input was processed.  Once we\nprove that, the proposition follows.\n\nThe view of each corrupted agent includes $n-1$ seeds that he shares\nwith other agents and the FSA labels which are secret shares of $0$ or\na $1$.  We argue that, from the point of view of the adversary,\nthese labels are {\\em random} shares \nof either $0$ or $1$.  This follows\nfrom the PRG property that an evolved seed cannot be correlated with\na prior output of the PRG, and from the fact that $A_1$ remains uncorrupted.\nIndeed, the newly generated ``empty'' states' labels look random since\nthe adversary cannot link them to the PRG seeds in his view.  The other\nstates' labels look random to the adversary since they are XORed\nwith $A_1$'s label.\n\nThus, the total view of the adversary consists of random shares of\n$0$ and $1$, and is hence indistinguishable from the one corresponding\nto the initial state.\n\\end{proofsk}\n\n\n\n\nWe now calculate time and storage complexity of $\\nnPi$.\nAt every step of the\ncomputation, each agent pseudo-randomly generates and XORs $n-1$ strings.\nFurther, each agent holds a small constant-length label for each\nautomaton state, and $n-1$ PRG seeds, yielding an\n$O(m+n)$\nmemory requirement.\n\n\n\n\\subsection{The $(t+1,n)$-reconstruction protocol}\n\\label{sec-from-t}\nRecall that in this case, up to $t$ of the agents might not take part\nin the reconstruction, and thus $n>2t$.\n\n\nA straightforward (albeit costly) solution to this scenario would be to execute $\\nnPi$ independently for every subset of agents of size $t+1$.  This would involve each agent $A_i$ holding ${n-1 \\choose t}$ copies of the automaton ${\\cal A}$, one copy for each such subset which includes $A_i$, and updating them all, as in $\\nnPi$, according to the same input symbol.  \nNow, during the reconstruction, the dealer can recover the\noutput from any subset of $t+1$ agents. The cost of this approach would be as follows.  Every agent holds ${n-1 \\choose t}$ automata (one for every $t+1$ tuple that includes this agent), and executes $\\nnPi$, which requires $O(m+t)$ memory, resulting in a total cost of $O \\big( {n-1\\choose t}\\cdot (m+t) \\big)$,\nwith the cost of computation per input symbol being proportional to storage's.\n \n\nWe now present $\\tnPi$, an improved $(t+1,n)$ reconstruction scheme,\nwhose intuition was already presented in Section~\\ref{outline}. The protocol\nuses Shamir's secret-sharing scheme~\\cite{Sha79}, which we now \nbriefly review. Let $\\mathbb{F}$ be a field of\nsize greater than $n$,\nand $s\\in \\mathbb{F}$ be the secret.\nThe dealer randomly generates coefficients $c_1,c_2,...,c_t$ from\n$\\mathbb{F}$ and construct the following polynomial of degree $t$,\n$f(x)=s+c_1x+c_2x^2+...+c_{t}x^{t}$. \nThe dealer gives each participant $A_i$, $1 \\leq i \\leq n$, the value $f(i)$.  It can be easily seen that one can reconstruct the\nsecret from any subset of at least $t+1$ points, and no information\nabout the secret is revealed by \n$t$ points (or less).\n\n\\vspace{-.15in}\n\\paragraph{Protocol $\\tnPi$.} As before, the protocol consists of\nthree phases:\n\n\\vspace{.05in}\n\\noindent {\\em Initialization.}  Using Shamir's secret sharing as\ndescribed above, the dealer shares a secret $1$ for the \ninitial state and $0$ for all other states.\nIn addition, the dealer generates a random seed for every\nset of $n - (t-1) = n-t+1$ agents, and gives each agent the seeds for the sets it\nbelongs to. \nLet ${\\cal T}$ be the set of all possible subsets of\n$n-t+1$ agents.\n\n\n\\vspace{.06in}\n\\noindent {\\em Event Processing.} Each agent runs Algorithm~\\ref{algo:calc} updating its labels, as follows.  \n\nLet $T\\in {\\cal T}$ and $j$, $1 \\le j\\le m$, be a \nstate of the automaton.  Upon obtaining value $b^T_j$ (refer to Algorithm~\\ref{algo:calc}),\nthe agents in $T$ (individually) construct a\ndegree-$t$ polynomial, $P^T_j$, by defining its value on the following\n$t+1$ field points: $0$, all the points $i$ such that $A_i \\not \\in T$,\nand $k$ such that $k$ is the minimal agent's index in $T$ (the choice of \nwhich point in $T$ is arbitrary). Now define $P^T_j(0)=0$, \n$P^T_j(i)=0 ~ \\forall A_i \\not \\in T$, and $P^T_j(k)=b^T_j$.\n\nObserve that by this definition, {\\em every} agent $A_i\\in T$ can use\npolynomial interpolation to compute $P^T_j(i)$, since the only required\ninformation is $b^T_j$ (and the knowledge of set membership).\n\nLet polynomial $P_j$ be defined as\n$P_j=\\sum_{T \\in {\\cal T}} P^T_j$. Each agent $A_i$ now computes \n$P_j(i)$ (note that this is possible since the values corresponding to\nsets the agent does not belong to is set to $0$), and updates the \n$j$-th label, $1 \\leq j \\leq m$, in Algorithm \\ref{algo:calc} by \nsetting $R_j=P_j(i)$ in line $8$.\n\n\\vspace{.06in}\n\\noindent {\\em Reconstruction.} At least $t+1$ agents submit their \ninternal state to the dealer, who, for every $j=1,\\ldots,m$, views the\n$j$-th labels of $t+1$ agents as shares in a Shamir secret-sharing\nscheme. The dealer reconstructs all the $m$ secrets using \nthe scheme's reconstruction procedure, and determines and outputs the\ncurrently active state (whose recovered secret is equal to $1$).\n\n\n\n\\begin{proposition}\n\\label{prop-threshold}\nAt every Event Processing step of protocol $\\tnPi$, the shared secret\nfor the current state in the computation is $1$ and for all the other\n(inactive) states, the shared secret is $0$. Furthermore, $t+1$ agents\ncan jointly reconstruct all secrets.\n\\end{proposition}\n\n\\begin {proof}\nWe prove the proposition by induction on the number of clock cycles $r$. We show that at each clock cycle $r$, for every state $s_j$, the $n$ labels $\\ell^1_j,\\ldots,\\ell^n_j$ are points on a degree $t$ polynomial $Q_j$ whose free coefficient is $1$ if $j$ is the current state and $0$ otherwise.\n\nAt initialization, the \nclaim is true by our definition of the label assignments. \n\nAssume that the induction hypothesis is correct after $r-1$. We prove\nthe hypothesis for the $r$-th step. Assume first that in this step the\nagents receive an input letter $\\gamma_r$, and denote the current\nstate by $s_{\\curr}$. By our definition, the new label of the state $j$\nof agent $i$ is\n\n$$ \\ell ^i_j \\longleftarrow \\sum_{\\substack{k~:\\\\ \\mu(s_k, \\gamma_r)=s_j}} \\ell^i_k + P_j(i),$$\n\\noindent or, equivalently,\n$$ \\ell ^i_j \\longleftarrow \\sum_{\\substack{k~:\\\\ \\mu(s_k, \\gamma_r)=s_j}} Q_k(i) + P_j(i).$$\n\nFor every $j, 1 \\leq j \\leq m$, define polynomial $Q'_j$ as\n$$Q'_j=\\sum_{\\substack{k~:\\\\ \\mu(s_k, \\gamma_r)=s_j}} Q_k + P_j.$$\nTherefore, $Q'_j(i)=\\ell ^i_j$ for every $j$ and every $i$. In\naddition, since every $Q_k$ is of degree $t$ and so is $P_j$, we deduce\nthat $Q'_j$ is also of degree $t$. We finish proving the induction step by\nshowing that $Q'_j(0)=1$ only for the correct state.\n\nLet $\\mu(s_{\\curr},\\gamma_r)=s_{\\next}$. By induction, $Q_{curr}(0)=1$\nand $Q_j(0)=0$ for any $j \\neq \\curr$. Furthermore, by construction\n$P_j(0)=0$, and therefore $Q'_{curr}(0)=1$. Since $Q_j(0)=0$ for any\n$j \\neq \\curr$, we have that $Q'_j(0)=0$ for any $j \\neq \\next$.\n\nIf the agents do not receive any input symbol in the $r$-th clock cycle, \nthen \nthe claim follows by similar arguments as above.\n\\end {proof}\n\n\n\n\\begin{proposition}\n\\label{prop-pcm-tn}\n$\\tnPi$ is $t$-private in the PCM model according to Definition~\\ref{defProgressive}.\n\\end{proposition}\n\\noindent At a high level, the proof follows the steps of the proof of \nProposition~\\ref{prop-pcm-all}. The full details of the privacy\nanalysis are presented in Section \\ref{sec-priv-ana}.\n\n\n\nWe now calculate the costs incurred by the protocol.  The space\ncomplexity of each agent is as follows. An agent holds a label for\nevery state, i.e. $m\\cdot (\\lceil log |\\mathbb{F}| \\rceil +1)$\nbits. Additionally every agent holds $ {n-1 \\choose n-t}={n-1 \\choose \\ t-1}$ seeds, where every seed is of size $\\len$. Thus, in total we have ${n-1 \\choose \\ t-1}\\cdot \\len + m\\cdot (\\lceil log |\\mathbb{F}| \\rceil +1)$\nbits. Each step of the Event Processing phase requires $O(m {n-1 \\choose \\ t-1})$ time for seed manipulation and field\noperations. Reconstruction (by the dealer)\nis just interpolation of $m$ polynomials of degree $t$.\n\n\\newpage\n\\section*{Supplementary Material:}\n\\section{Privacy Analysis in Detail \\label{sec-priv-ana}}\n\n\\ignore{\n\\todo{I started reading the proofs but got confused a little.  I modified and moved the definitions part to prelims.  I wrote my own proofs in previous sections.  I am not 100\\% convinced by my proofs, so please adjust and\/or check if they make sense. I think the rest of this section can go.\n\n Also, you guys pay a lot of attention to the PRG syntax (and other minute details).  Usually in crypto papers it is taken for granted, and people don't write it all out.}\n\n}\n\nWe show that each of our schemes in Sections\n\\ref{sec-from-all-details} and \\ref{sec-from-t} is computationally\nprivate in the PCM in two stages. In the first stage we construct for\neach scheme $\\Pi$ and every possible corruption timeline $\\rho$ an\nintermediate scheme, $I(\\Pi, \\rho)$. We prove that if the corruption\ntimeline is $\\rho$ then the view of an adversary in $I(\\Pi, \\rho)$ is\nindependent of the state of the automaton. In other words, the\nadversary's view is distributed identically for any initial state and\nany sequence of input symbols.\n\nIn the second stage we prove that the view of an adversary in $\\Pi$\nwith {\\em any} efficiently constructible corruption timeline $\\rho$\nand any efficiently constructible input stream is computationally\nindistinguishable from the adversary's view in $I(\\Pi, \\rho)$. We\ndeduce that $\\Pi$ is computationally private in the PCM.\n\n\\subsection{Constructing $I(\\Pi, \\rho)$}\n\n\\begin{notation}\nLet $\\nnPi$ denote the scheme of Section \\ref{sec-from-all} that\nrequires all the agents for reconstruction, and let $\\tnPiNaive$ \nand $\\tnPi$\ndenote the threshold schemes of Section \\ref{sec-from-t}. We say that\nan adversary is {\\em appropriate} for the scheme $\\nnPi$ if it\ncorrupts at most $n-1$ agents. We say that an adversary is {\\em\nappropriate} for the $\\tnPiNaive$ and $\\tnPi$ if schemes it corrupts\nat most $t$ agents.\n\\end{notation}\n\n\nLet $\\Pi$ be one of the schemes $\\nnPi, \\tnPiNaive$ or $\\tnPi$. $\\Pi$ defines initial data that an agent $A$ stores: a description of the automaton, a label for each node in the automaton and random seeds that are shared with other agents. For each scheme the domain of seeds is $\\{0,1\\}^{len}$ while the domain of labels is a field $\\mathbb{F}$. For example, in $\\nnPi$ the field is $\\mathbb{F}=\\mbox{GF}(2)$. The subsets of agents that share a single seed are specific to each scheme. The description of $I(\\Pi, \\rho)$ follows.\n\n\\smallskip\n\\noindent\n{\\bf Initialization:} An agent $A_i$ is initialized with a description of the automaton as in $\\Pi$. For every subset of agents $T$ such that $A_i \\in T$, if the agents in $T$ share a seed in $\\Pi$ that other agents do not have then $A_i$ is initialized with $m+1$ elements, $seed^T_0$, $R^T_1,\\ldots,R^T_m$. $seed^T_0$ is chosen uniformly at random from $\\{0,1\\}^{len}$, while $R^T_1,\\ldots,R^T_m$ are chosen uniformly at random and independently from $\\mathbb{F}$. $A_i$ computes the initial label of the $j$-th state (indexed from $1$ to $m$) over $\\mathbb{F}$ as \n$$\\ell^i_j = \\sum_{T, A \\in T} \\alpha^T R^T_j,$$\nfor fixed coefficients $\\alpha_T \\in \\mathbb{F}-\\{0\\}$.\n\nThe agent stores $seed^T_0$ but $R^T_1$,$\\ldots$,$R^T_0$ are deleted.\n\n\\smallskip\n\\noindent\n{\\bf Processing:} Each of the schemes $\\Pi$ defines data processing for every clock cycle. This processing includes computing new values for each seed and new values for each node label. \n\nComputing new labels and seeds in $I(\\Pi, \\rho)$ depends on the corruption timeline $\\rho$, which is defined by a sequence $\\rho=((A_1, \\tau_1),\\ldots,(A_t, \\tau_t))$ such that the adversary corrupts agent $A_i$ at time $\\tau_i$ for $i=1,\\ldots,t$ and $\\tau_1\\leq\\tau_2 \\leq \\ldots \\leq \\tau_t$. \n\nAn agent $A$ in $I(\\Pi, \\rho)$ begins updating $seed^T_0$ only after the corruption of the first agent $A_i$ that holds $seed^T_0$. Therefore, at any time $\\tau$, $\\tau \\leq \\tau_i$, we have that $seed^T_{\\tau} = seed^T_0$. If $\\tau \\geq \\tau_i+1$ the agent modifies $seed^T_{\\tau}$ as $\\Pi$ specifies for updating a seed.\n \nAn agent $A_i$ in $I(\\Pi, \\rho)$ begins updating a state label $\\ell^i_j$ only after $A_i$ is corrupted at time $\\tau_i$. Therefore, when the adversary corrupts $A_i$ it obtains the original label $\\ell^i_j$. For every clock cycle after $\\tau_i$ the agent modifies $\\ell^i_j$ as $\\Pi$ specifies for updating a label.\n\n\\subsection{Privacy of $I(\\Pi, \\rho)$}\nWe show that if the corruption timeline is fixed to $\\rho$ then $I(\\Pi, \\rho)$ is private in the information-theoretic sense. In order to do so, we introduce the following definition and lemma.\n\\begin{definition}\nLet ${\\cal A}$ be a set of agents, let $H=({\\cal A}, {\\cal E})$ be a hypergraph and let $\\mathbb{F}$ be a finite field. We call $H$ a {\\em distribution hypergraph}, if for every $T \\in {\\cal E}$, there is an element $R^T$ chosen uniformly from $\\mathbb{F}$, such that every $A\\in T$ holds $R^T$ and every $A\\notin T$ has no information on $R^T$.\n\\end{definition}\n\n\\begin{lemma} \n\\label{prop-privacy}\nLet ${\\cal A}$ be a set of agents and let $H=({\\cal A}, {\\cal E})$ be a distribution hypergraph over a finite field $\\mathbb{F}$. Assume that each agent $A_i$ uses a fixed set of public elements $\\{\\alpha^T\\}_{A_i \\in T}$, such that $\\forall T, \\alpha^T \\in \\mathbb{F}-\\{0\\}$,  to compute a label \n$$\\ell^i=\\sum_{T, A_i \\in T} \\alpha^T R^T.$$\nAssume that an adversary that corrupts an agent $A_j$ obtains both the agent's label $\\ell^j$ and its random strings $R^T$ for all $T$ such that $A_j \\in T$ and its . Then, the label of any uncorrupted agent $A_i$ is distributed independently of the adversary's view if and only if for any subset of agents $K$ that the adversary corrupts and for any agent $A_i, A_i \\not \\in K$, there exists a hyper-edge $T\\in {\\cal E}$, such that $A_i \\in T$ and $T\\cap K=\\emptyset$. \n\\end{lemma}\n\n\\begin {proof}\nLet $A_i$ be an agent such that $A_i \\not \\in K$ and there exists a hyper-edge $T\\in {\\cal E}$, such that $A_i \\in T$ and $T\\cap K=\\emptyset$. Since $R^T$ is chosen uniformly at random from $\\mathbb{F}$, which is a field, and $\\alpha^T \\neq 0$ we have that $\\ell^i=\\sum_{T, A_i \\in T} \\alpha^T R^T$ is distributed uniformly at random in $\\mathbb{F}$ and furthemore $\\ell^i$ is independent of the adversary's view.  \n\nConversely, assume that there exists an agent $A_i \\not \\in K$ such that for every $T, A_i \\in T$, there exists an agent $A$, $A \\neq A_i$,  such that $A \\in K \\cap T$. Then, the adversary obtains $R^T$ for every $T$ such that $A_i \\in T$ and can therefore compute $\\ell^i$ without corrupting $A_i$.\n\\end {proof} \n\n\n\n\\begin{proposition}\n\\label{prop:priv_intermediate}\nFor each of the three possible schemes $\\nnPi, \\tnPiNaive, \\tnPi$ and every corruption timeline $\\rho$, if the adversary is appropriate for the scheme $\\Pi$ $(\\Pi \\in \\{\\nnPi, \\tnPiNaive, \\tnPi\\})$ then $I(\\Pi, \\rho)$ is private in the following sense. For  every two states $s_1, s_2 \\in ST$ and for any two input streams $X_1, X_2\\in \\Gamma^\\ast$,\n$\\mbox{VIEW}_{\\rho}^{I(\\Pi,\\rho)}(X_1, s_1)=\\mbox{VIEW}_{\\rho}^{I(\\Pi,\\rho)}(X_2, s_2)$. Furthermore, all the state labels are random and independent elements in a finite field $\\mathbb{F}$.\n\\end{proposition}\n\\begin{proof}\nThe view of an adversary $I(\\Pi, \\rho)$ is made up of the description of the automaton, the seeds and the labels. All of these are obtained from an agent at the moment of corruption. The description of the automaton is static. The distribution of the seeds depends only on $\\rho$ and as a consequence is independent of the initial state $s$ and the input stream $X$.\n\nTherefore, the only data elements in the adversary's view that could depend on the initial state and the input stream are the state labels. \n\nHowever, just prior to the adversary corrupting an agent $A$, since the adversary is {\\em appropriate} there is a subset of uncorrupted agents $T$ such that $A \\in T$ and all agents in $T$ share a seed that is not known to any agent outside $T$. In $I(\\nnPi, \\rho)$, an appropriate adversary corrupts a total of at most $n-1$ agents and just prior to corrupting an agent there are at least two uncorrupted agents. By the definition of $\\nnPi$ this pair of agents, which we denote by $T$, shares a seed. Therefore, by the definition of $I(\\nnPi, \\rho)$ the two agents share the elements $seed^T_0, R^T_1,\\ldots,R^T_m$, which are all random and independent of the adversary's view. Such subsets $T$ of uncorrupted agents also exist in $I(\\tnPiNaive, \\rho)$ and $I(\\tnPi, \\rho)$. \n\nBy the construction of $I(\\nnPi, \\rho)$, before the corruption of $A$, the state labels of $A_i$ have their initial value $\\ell^i_j = \\sum_{T, A \\in T} \\alpha^T R^T_j$ and therefore, by Lemma \\ref{prop-privacy} these labels are random and independent of the adversary's view.\n\nTherefore, the adversary's view in $I(\\Pi_i, \\rho)$ with corruption timeline $\\rho$ is distributed identically for any initial state $s$ and any input stream $X$.\n\\end{proof}\n\n\\subsection{Computational Privacy of $\\nnPi, \\tnPiNaive$ and $\\tnPi$}\n\nWe complete the analysis by proving that $\\nnPi, \\tnPiNaive$ and $\\tnPi$ are computationally private in the PCM. \n\n\\begin{notation}\nLet $\\kappa$ be a security parameter, let $\\mathbb{F}$ be a field and let $q(\\kappa)$ be a polynomial. Let $m, t, n$ and $len$ be parameters such that $t^2{n-1 \\choose t-1}(m\\abs{\\mathbb{F}}+len) < q(\\kappa)$ and let $G:\\{0,1\\}^{len} \\longrightarrow \\{0,1\\}^{m\\abs{\\mathbb{F}}+len}$ be a pseudo-random generator. Denote the uniform distribution on $\\{0,1\\}^{m\\abs{\\mathbb{F}}+len}$ by $U$. \n\n\\end{notation}\n\nWe regard $\\mbox{VIEW}_{\\rho}^{\\Pi}(X, s)$ as a random variable that represents the whole view of the adversary in $\\Pi$ and regard $\\mbox{VIEW}_{\\rho[\\tau]}^{\\Pi}(X, s)$ as a reduction of that view to the first $\\tau$ clock cycles. Similarly, $\\mbox{VIEW}_{\\rho[\\tau]}^{I(\\Pi,\\rho)}(X, s)$ represents the adversary's view of the first $\\tau$ clock cycles in $I(\\Pi,\\rho)$.\n\n\\begin{notation}\nLet $\\Pi$ be one of the schemes $\\nnPi,\\tnPiNaive$ or $\\tnPi$. For every $\\tau=0,\\ldots,q(\\kappa)$ define $H_{\\tau}$ to be a hybrid scheme which is identical to $I(\\Pi,\\rho)$ for any clock cycle $\\tau'$ such that $\\tau' \\leq \\tau$ and is identical to $\\Pi$ for any clock cycle $\\tau''$ such that $\\tau'' > \\tau$. Define a sequence of random variables $Y_0, Y_1, \\ldots,Y_{q(\\kappa)}$ as follows. Select an arbitrary initial state $s$, select an input stream $X$ from $\\Gamma^{q(\\kappa)}$ and select a corruption timeline $\\rho=((A_1,\\tau_1),\\ldots,(A_t,\\tau_t))$ from $\\rho^{q(\\kappa)}$. $Y_{\\tau}$ is the view of an adversary for the scheme $H_{\\tau}$ given the choices of $s$, $X$ and $\\rho$.\n\\end{notation}\n\nIt follows from the definition of the schemes $H_0,\\ldots,H_{q(\\kappa)}$ that $H_0$ is $\\Pi$ and $H_{q(\\kappa)}$ is $I(\\Pi,\\rho)$. Therefore, $Y_0=\\mbox{VIEW}_{\\rho}^{\\Pi}(X, s)$ and $Y_{q(\\kappa)}=\\mbox{VIEW}_{\\rho}^{I(\\Pi,\\rho)}(X, s)$. \n\nNote that $H_{\\tau}$ is well defined for any $\\tau$ since the memory contents and the inputs of $\\Pi$ and $I(\\Pi, \\rho)$ are all in the same domain (although the distribution of the memory contents is not identical). The only difference between $\\Pi$ and $I(\\Pi, \\rho)$ is the processing at each clock cycle. \n\n\n\\begin{proposition}\n\\label{prop:indis_pi_int}\nLet $\\Pi$ be one of the schemes $\\nnPi, \\tnPiNaive$ or $\\tnPi$. If the adversary is appropriate for $\\Pi$ then $\\mbox{VIEW}_{\\rho}^{\\Pi}(X, s) \\stackrel{c}{\\equiv} \\mbox{VIEW}_{\\rho}^{I(\\Pi,\\rho)}(X, s)$ for  any initial state $s$, any efficiently constructible corruption timeline $\\rho \\in \\rho^{q(\\kappa)}$ and any efficiently constructible input stream $X \\in \\Gamma^{q(\\kappa)}$. \n\\end{proposition}\n\\begin{proof}\nWe assume towards a contradiction that the views of an adversary in $\\Pi$ and in $I(\\Pi,\\rho)$ are not computationally indistinguishable.\nTherefore, there exist a probabilistic, polynomial time algorithm $D$ and a polynomial $p(\\cdot)$ such that \n$$ \\abs{D_{\\Pi} - D_{I(\\Pi,\\rho)}} - \\frac{1}{p(\\kappa)},$$\nfor an infinite number of values $\\kappa$. $D_{\\Pi}$ denotes $\\mbox{Pr}[D(\\mbox{VIEW}_{\\rho}^{\\Pi}(X, s))=1]$ and $D_{I(\\Pi,\\rho)}$ denotes $\\mbox{Pr}[D(\\mbox{VIEW}_{\\rho}^{I(\\Pi,\\rho)}(X, s))=1]$.\n\n\nWe construct an algorithm $\\bar{D}$ that distinguishes between $t^2 {n-1 \\choose t-1}$ independent samples of $U$ and $t^2 {n-1 \\choose t-1}$ independent samples of $G(seed)$ for a random $seed \\in \\{0,1\\}^{len}$. Since $t^2 {n-1 \\choose t-1}$ is at most a polynomial in $\\kappa$, the algorithm $\\bar{D}$ contradicts the assumption that $G$ is a pseudo-random generator, thus proving the proposition. Denote the distribution on $t^2 {n-1 \\choose t-1}$ independent samples of $U$ by $U_{long}$ and denote the distribution on $t^2 {n-1 \\choose t-1}$ independent samples of $G(seed)$ by $G_{long}$.\n\n\\medskip\n\\noindent\n{\\bf Description of $\\bar{D}$:} the algorithm receives as input a description of the automaton, $\\kappa$, $n$ and $t$. In addition, the algorithm receives as input a binary string $z$ of length $t^2 {n-1 \\choose t-1} (m\\abs{\\mathbb{F}}+len)$ and decides whether it is chosen from $U_{long}$ or $G_{long}$ by performing the following steps.\n\\begin{enumerate}\n\\item Choose a random initial state $s$, select an input stream $X$ from $\\Gamma^{q(\\kappa)}$ and select a corruption timeline $\\rho=((A_1,\\tau_1),\\ldots,(A_t,\\tau_t))$ from $\\rho^{q(\\kappa)}$.\n\\item Choose a random $\\tau$ in the range $1,2,\\ldots,q(\\kappa)$.\n\\item Simulate the operation of the agents $A_1,\\ldots,A_t$ in the scheme $I(\\Pi, \\rho)$ for the first $\\tau-1$ clock cycles.\n\\item In the $\\tau$-th clock cycle all the agents that have already been corrupted, i.e. in cycles $1$ to $\\tau-1$, execute $\\Pi$ (which is identical to $I(\\Pi, \\rho)$ for a corrupted agent). For any uncorrupted player, including those that are corrupted in the $\\tau$-th cycle do the following:\n\\begin{enumerate}\n\\item Update any seed that is shared with a corrupted player as specified by $\\Pi$ (which is identical to the update process of $I(\\Pi, \\rho)$ for such seeds).\n\\item For any seed that is shared by set of uncorrupted agents $T$, select a fresh string of length $m\\abs{\\mathbb{F}}+len$ from $z$ and parse it as $B^T||S^T$ for $S^T \\in \\{0,1\\}^{len}$ and $B^T=b_1^T,\\ldots,b_m^T$. Replace the previous seed with $S^T$. \n\\item Recall that in every $\\Pi$ the label of the $j$-th state, $j=1,\\ldots,m$, is updated by a linear combination of previous state labels and of elements $b_j^T$ derived from expanded seeds. $\\bar{D}$ updates the label in a similar way, except that for every $T$ such that $S^T$ is shared by uncorrupted agents, $b_j^T$ is selected from $z$ as described in the previous step instead of being selected from an expanded seed.  \n\\end{enumerate}\n\\item Simulate the operation of the agents $A_1,\\ldots,A_t$ in the scheme $\\Pi$ for the last $q(\\kappa)-\\tau$ clock cycles.\n\\item Throughout the simulation of the agents simulate the actions of an adversary with corruption timeline $\\rho$.\n\\item Run $D$ on the adversary's view and return the result of $D$.\n\\end{enumerate}\n\nWe argue that if $z$ is chosen from $U_{long}$ then the view of the adversary that $\\bar{D}$ simulates is $Y_{\\tau}$, while if $z$ is chosen from $G_{long}$ then the view of that adversary is $Y_{\\tau-1}$. Obviously, the view that the adversary obtains in the first $\\tau-1$ clock cycles is identical to the view in $I(\\Pi, \\rho)$ and the view in the last $q(\\kappa)-\\tau$ clock cycles is identical to the view in $\\Pi$. Therefore, we need to prove that the view in the $\\tau$-th clock cycle is identical to $I(\\Pi, \\rho)$ if $z$ is uniformly random and identical to $\\Pi$ if $z$ is selected from $G_{long}$.\n\n$I(\\Pi, \\rho)$ specifies identical processing to $\\Pi$ for corrupted agents and seeds shared by corrupted agents. Therefore, the  differences are in seeds that are shared only by uncorrupted agents and in state labels of uncorrupted agents. \n\nIn the $\\tau$-th clock cycle, $\\bar{D}$ replaces seeds that are shared by uncorrupted agents with strings selected from $z$. If $z$ is uniformly random then these seeds are uniformly random. Therefore, in this case the distribution of the seeds is identical to the distribution if $I(\\Pi, \\rho)$ is executed in the previous clock cycle, $\\tau-1$. If $z$ is a sequence of elements of the form $G(seed)$, where $seed$ is random, then the new seed, $S^T$ is exactly as specified by $\\Pi$ after a single clock tick. That is the expected distribution if the agents run $\\Pi$ in the previous clock tick, $\\tau-1$.\n\nThe state labels are updated by a linear combination in which the coefficients of each $b_j^T$ are non-zero. If $z$ is uniformly random then each $b_j^T$ is a random field element in $\\mathbb{F}$ and therefore each state label is a a random field element in $\\mathbb{F}$. By Proposition \\ref{prop:priv_intermediate} that is identical to the distribution of state lables in $I(\\Pi, \\rho)$. If $z$ is a sequence of elements of the form $G(s)$, where $s$ is random, then the new label is exactly as specified by $\\Pi$ after a single clock tick.\n\nThe argument above shows that once $\\tau$ is given, $\\bar{D}$ distinguishes between a sequence of uniform elements and a sequence of pseudo-random elements with the same probability that $D$ distinguishes between $Y_{\\tau}$ and $Y_{\\tau-1}$. Since $\\tau$ is chosen randomly in the range $1,2,\\ldots,q(\\kappa)$ and since $Y_0=\\mbox{VIEW}_{\\rho}^{\\Pi}(X, s)$ and $Y_{q(\\kappa)}=\\mbox{VIEW}_{\\rho}^{I(\\Pi,\\rho)}(X,s)$ we have that \n$$\\mbox{Pr}[\\bar{D}(G_{long}=1)]=\\frac{1}{q(\\kappa)} \\sum_{\\tau=1}^{q(\\kappa)} \\mbox{Pr}[D(Y_{\\tau})=1],$$\nand\n$$\\mbox{Pr}[\\bar{D}(U_{long})=1]=\\frac{1}{q(\\kappa)} \\sum_{\\tau=1}^{q(\\kappa)} \\mbox{Pr}[D(Y_{\\tau-1})=1].$$\n\nTherefore,\n\\begin{flalign*}\n& \\abs{\\mbox{Pr}[\\bar{D}(G_{long}=1)] - \\mbox{Pr}[\\bar{D}(U_{long})=1]} & = \\\\\n& \\frac{1}{q(\\kappa)} \\abs{\\mbox{Pr}[D(Y_{q(\\kappa)})=1] - \\mbox{Pr}[D(U)=1]} & > \\\\\n& \\frac{1}{q(\\kappa)\\cdot p(\\kappa)}\n\\end{flalign*}\n\n\t \nfor an infinite number of values $\\kappa$. Since $\\bar{D}$ distinguishes between $U_{long}$ and $G_{long}$ we deduce that $G$ is not a pseudo-random generator and have thus reached a contradiction.\n\\end{proof}\n\n\\begin{theorem}\nIf the adversary is appropriate then the schemes $\\nnPi, \\tnPiNaive$ and $\\tnPi$ are all computationally private in the PCM. \n\\end{theorem}\n\\begin{proof}\nBy proposition \\ref{prop:indis_pi_int} if the adversary is appropriate then for every efficiently constructible corruption timeline $\\rho$, $\\rho \\in \\rho^{q(\\kappa)}$, every two initial states $s_1, s_2$ and every two efficiently constructible input streams $X_1, X_2$, such that $X_1, X_2 \\in \\Gamma^{q(\\kappa)}$ we have $\\mbox{VIEW}_{\\rho}^{\\Pi}(X_1, s_1) \\stackrel{c}{\\equiv} \\mbox{VIEW}_{\\rho}^{I(\\Pi,\\rho)}(X_1, s_1)$ and $\\mbox{VIEW}_{\\rho}^{\\Pi}(X_2, s_2) \\stackrel{c}{\\equiv} \\mbox{VIEW}_{\\rho}^{I(\\Pi,\\rho)}(X_2, s_2)$.\n\nBy Proposition \\ref{prop:priv_intermediate} we know that\n$\\mbox{VIEW}_{\\rho}^{I(\\Pi,\\rho)}(X_1,\ns_1)=\\mbox{VIEW}_{\\rho}^{I(\\Pi,\\rho)}(X_2, s_2)$. Therefore,\n$$\\mbox{VIEW}_{\\rho}^{\\Pi}(X_1, s_1) \\stackrel{c}{\\equiv}\n\\mbox{VIEW}_{\\rho}^{\\Pi}(X_2, s_2).$$\n\\end{proof}\n\n\\ignore{\n\\subsection{Minimizing the number of the seeds}\n\\todo{VLAD: this sect. needs to be reworked}\nIn the first scheme which was described in \\ref{sec-from-all}, as the seed distribution we used a clique on $n$ vertices $K_n$. In total we needed ${n \\choose 2}=\\frac {n(n-1)}{2}$ seeds. \nAn interesting question is, are we able to minimize our number of seeds. By proposition \\ref{prop-privacy}, in the first scheme, if we know that our adversary can catch at most $t$ agents, then a $t$-regular graph would be enough and will be necessary. Such a graph will have $O(tn)$ edges.\nWhat can we say about the second scheme?\n}\n\n\n{\\small\n\n\n\\section{Introduction}\n\nDistributed computing has become an integral part of a variety of\nsystems, including cloud computing and ``swarm'' computing, where $n$\nagents perform a computation\non common inputs.  In these emerging computing paradigms, security\n(i.e., privacy and correctness) of the computation is of a primary\nconcern.  Indeed, in\nswarm computing, often considered in military contexts (e.g., unmanned\naerial vehicle (UAV) operation), security \nof the data and program state is of paramount importance;\nsimilarly, one persistent challenge in the field of cloud\ncomputing is ensuring the privacy of users' data, demanded by\ngovernment, commercial, and even individual cloud users.\n\nIn this work, we revisit the notion of {\\em never-ending}\nprivate distributed computation, first considered by Dolev, Garay, Gilboa and\nKolesnikov~\\cite{DGGK11}. In such a computation, an unbounded sequence\nof commands (or inputs) are interpreted by several machines (agents) in \na way that no information about the inputs as well as the state of\nthe computation is revealed to an adversary who is able to ``corrupt''\nthe agents and examine their internal state, as long as up to a predetermined\nthreshold of the machines are corrupted.\n\nDolev {\\em et al.}\nwere able to provide very strong (unconditional, or information-theoretic)\nsecurity for computations performed by a finite-state machine (FSA),\nat the price however of the computation being efficient only for a small\nset of functions, as in general the complexity of the computation\nis exponential in the size (number of states) of the FSA computing\nthe function.\n\n\nIn this work, we {\\em minimally}\\footnote{Indeed, the existence\nof one-way functions is considered a minimal assumption in contemporary\ncryptography. In particular, we do not allow the use of public-key \ncryptography.}\nweaken the original model by\nadditionally assuming the existence of one-way functions (and hence\nconsider polynomial-time adversaries---in the security parameter; more\ndetails below), and in return achieve very high efficiency as a\nfunction of the size of the FSA.  We stress that we still consider\ncomputation on {\\em a priori} unbounded number of inputs, and\nwhere the online (input-processing) phase incurs {\\em no communication}.\nWe now describe the model in more detail.\n\n\n\\vspace{-.1in}\n\\paragraph{The setting.}\nAs in~\\cite{DGGK11}, we consider a distributed computation setting in\nwhich a party, whom we refer to as {\\it the dealer}, has a finite\nstate automaton (FSA) ${\\cal A}$ which accepts an ({\\em a priori}\nunbounded) stream of inputs $x_1, x_2, \\ldots$ received from an\nexternal source. The dealer delegates the computation to agents\n$A_1,\\ldots, A_n$, by furnishing them with an implementation of \\A.\nThe agents receive, in a synchronized manner, all\nthe inputs for ${\\cal A}$ during the online input-processing phase,\nwhere no communication whatsoever is allowed.  Finally, given a signal\nfrom the dealer, the agents terminate the execution, submit their \n internal state to the dealer, who computes the  state of ${\\cal A}$  and returns it as output.\n\nWe consider an attack model where an entitiy,\ncalled the adversary, \\Ad, is able to adaptively ``corrupt'' agents\n(i.e., inspect their internal state) during the online execution\nphase, up to a threshold\\footnote{We note that more general access \nstructures may be naturally employed with our constructions.} \n$t < n$.  We do not aim at maintaining the privacy of the\nautomaton \\A; however, we wish to protect the secrecy of the state of\n\\A\\ and the inputs' history.  We note that \\Ad\\ may have external\ninformation about the computation, such as partial inputs or length of\nthe input sequence, state information, etc.  This auxiliary\ninformation, together with the knowledge of \\A, may exclude the\nprotection of certain configurations, or even fully determine \\A's\nstate.  We stress that this cannot be avoided in any implementation,\nand we do not consider this an insecurity.  Thus, our goal is to\nprevent the leakage or derivation by\n\\Ad\\ of any knowledge from seeing the execution traces which \\Ad\\ did\nnot already possess.\n\nAs mentioned above, our constructions relying on one-way functions\ndictates that the computational power of entities (adversary, agents),\nbe polynomially bounded (in\n$\\kappa$, the security parameter).\nSimilarly, our protocols run on input streams of polynomial length.\nAt the same time, we do not impose an {\\it a priori} bound on its length; \nmoreover, the size of the agents' state is independent of it.\nThis allows  to use agents of the same (small) complexity (storage and \ncomputational power) in all situations.\n\n\n\\vspace{-.1in}\n\n\\paragraph{Our contributions.}\nOur work is the first significant extension of the work\nof~\\cite{DGGK11}.  Towards our goal of making never-ending and private\ndistributed computation practical, we introduce an additional (minimal)\nassumption of existence of one-way functions (and hence pseudo-random\nnumber generators [PRGs]), and propose the following constructions:\n\\begin{tiret}\n\\item  A scheme with $(n,n)$ reconstruction (where all $n$ agents participate in reconstruction), where\nthe storage and processing time per input symbol is $O(mn)$ for each\nagent. The reconstruction complexity is $O(mn$).\n\\item A scheme with $(t+1,n)$ reconstruction (where $t$ corrupted agents do not \ntake part in the reconstruction), where the above costs are \n$O (m{n-1 \\choose t-1})$.\\footnote{For some values of $t$, e.g. $t=\\frac{n}{2}$, this quantity would be exponential in $n$. This does not contradict our\nassumption on the computational power of the participants; it simply\nmeans that, given $\\kappa$, for some values of $n$ and $t$ this\nprotocol cannot be executed in the allowed time.}\n\\end{tiret} \nRegarding tools and techniques, the carefully orchestrated use of PRGs\nand secret-sharing techniques~\\cite{Sha79} allows our protocols to hide the state of the computation against an adaptive adversary by using share re-randomization. \nTypically, in the context of secret sharing, this is simply done by the addition of a suitable (i.e., passing through the origin) random polynomial. However, due to the no-communication requirement, share re-randomization is a lot more challenging in our setting. This is particularly so in the more general case of the $(t+1,n)$-reconstruction protocol. We achieve share re-randomization by sharing PRG seeds among the players in a manner which allows players to achieve sufficient synchronization of their randomness, which is resilient to $t$ corruptions.\n\n\n\\vspace{-.1in}\n\n\\paragraph{Related work.} \nReflecting a well-known phenomenon in distributed computing, where\na single point of failure needs to be avoided,\na team of agents (e.g., UAVs)  that collaborate in a mission\nis more robust than a single agent trying\nto complete a mission by itself (e.g., \\cite{Bamberger,BDDS10}). \nSeveral techniques have been  suggested for this purpose;\nanother related line of work is that of automaton splitting and\nreplication, yielding\ndesigns that can tolerate faults and as well as provide\nsome form of privacy of the computation\n(see, e.g.,~\\cite{DKS07,DLY07,DGKPS10,DGGK09,DGGK11}).\nAs mentioned above, \nonly~\\cite{DGGK11}\naddresses the unbounded-input-stream scenario.\n\\ignore{\nTechniques for information-theoretic communication among the participants of \nsuch systems have been investigated as well~\\cite{DO00,BD03,DGGN08,DGKPS10},\nsome of which might be amenable for enhancement and application to our\nscenario as well.\\todo{VLAD: which techniques can be amenable? We have unbounded-input non-interactive online phase}\n}\n\n\nRecall that in {\\em secure multi-party\ncomputation}~\\cite{DBLP:conf\/stoc\/GoldreichMW87,BGW88,CCD88}, $n$\nparties, some of which might be corrupted, are to compute an $n$-ary\n(public) function on their inputs, in such a way that no information\nis revealed about them beyond what is revealed by the function's\noutput.  At a high level, we similarly aim in our context to ensure\nthe correctness and privacy of the distributed computation. However,\nas explained in~\\cite{DGGK11}, our setting is significantly different\nfrom that of MPC, and MPC definitions and solutions cannot be directly\napplied here.  The reason is two-fold: MPC protects players {\\em\nindividual} inputs, whereas in our setting the inputs are common to\nall player.  Secondly, and more importantly, MPC operates on inputs of\nfixed length, which would require an {\\it a priori} estimate on the \nmaximum input size $s_{max}$ and agents' storage linear in $s_{max}$.  \nWhile unbounded inputs could be processed, \nby for example processing them ``in blocks,'' this would require\ncommunication during the online phase, which is not allowed\nin our setting. Refer to~\\cite{DGGK11} for a more detailed discussion\non the unbounded inputs setting {\\em vis-\\`{a}-vis} MPC's.\n\n\nFinally, we note that using recently proposed fully-homomorphic encryption\n(FHE)~\\cite{DBLP:conf\/stoc\/Gentry09} (and follow-ups) trivially solves\nthe problem we pose, as under FHE the agents can simply compute arbitrary\nfunctions. In fact, plain additively homomorphic encryption \n(e.g.,~\\cite{EC:Paillier98}) can\nbe used to encrypt the current state of the FSA and non-interactively\nupdate it as computation progresses, in a manner similar to what is\ndescribed in our constructions (see the high-level intuition in\nSection~\\ref{outline}).  We note that, firstly, public-key encryption\nand, dramatically so, FHE, suffer from orders-of-magnitude\ncomputational overhead, as compared to the symmetric-key operations that\nwe rely on.  More importantly, in this work we aim at minimizing\nthe assumptions needed for efficient unbounded private distributed\ncomputation.\n\n\\ignore{\n\\todo{Move the MPC discussion below to appendix}\n\n\\subsection{Inapplicability of MPC to our setting}\n\\label{app-mpc}\nFirstly, MPC aims to solve a different problem, that of protecting the\nplayers' individual inputs from \\Ad, who can corrupt some of them,\nlearn their input and observe the communication they receive.  In\ncontrast, in our problem the inputs are common to all the players (but\nnot {\\it a priori} known to \\Ad, or revealed in case of corruption),\nand the goal is to protect the state of, as well as the inputs to, the\ncomputation.  (Therefore, we cannot in particular treat the common\ninput as public information, and the shares received from the dealer\nas MPC input.)\n\nOf course, an adequate representation (circuit-based, for example) of\nthe MPC computation would be able to evaluate \\A, with respect to a\nsubset of corrupted players, and at least for the basic MPC setting,\nwhere there is a single (tuple of secret) input(s) out of which an\noutput (tuple) is produced.  But then comes our main feature, of\nmultiple, possibly unbounded number of input symbols.  This is\nreminiscent of secure {\\it reactive} systems (e.g.,~\\cite{PW00}),\nwhere the computation is not limited to ``one shot'' as above, but\ninstead processes inputs ``in blocks'' throughout several rounds of\ninteraction. However, because all MPC solutions (and definitions) are\nexplicitly tied to the length of the input, being able to handle\nunbounded number of inputs without communication does not seem\nimmediate.  This is what our Krohn-Rhodes-based approach achieves, at\nthe expense of solving a narrower problem.\n\n\n\\todo{VLAD: end of appendix block}\n\n\n\\paragraph{Private perennial computation without communication from Krohn-Rhodes decomposition.}\nDolev et al.~\\cite{DGGK11}\n show the feasibility of achieving private  computation  on unbounded inputs\nwith non-interactive input-processing phase, at the cost of limiting the class of  \nfunctions that can be evaluated---specifically, those carried out by \nfinite-state automata (FSA).  Their cost is also in general exponential in the FSA size.\n\nIn this work, we aim to achieve much more practical efficiency for distributed FSA evaluation.  We {\\em minimally} weaken our model by additionally assuming the existence of one-way functions (and hence consider polytime adversaries), and in exchange achieve efficiency linear in the FSA size.  We stress that we still consider computation on {\\em a priori} unbounded inputs, where the online (input-processing) phase incurs not communication.\n\n}\n\n\n\n\n\n\n\\vspace{-.1in}\n\n\\paragraph{Organization of the paper.} The remainder of the paper is\norganized as follows. In Section~\\ref{sec-Prelim_and_outline} we present\nin more detail the model, definitions and building blocks that we use \nthroughout the paper. We dedicate Section~\\ref{outline} to a high-level\ndescription of our constructions, while in Section \\ref{sec-from-all} we \npresent them in detail. The full privacy analysis is presented in Section \\ref{sec-priv-ana}. \n\\ignore{The first scheme $\\Pi_1$, in which we need all of the agents to reconstruct the current state of the computation is described in section~\\ref{sec-from-all-details}. In section \\ref{sec-from-t} we present two schemes $\\Pi_2$ and $\\Pi_3$, in which we need only a subset of the agents to reconstruct the current state of the computation. Sections \\ref{sec-from-all-details} and \\ref{sec-from-t} contain proofs of correctness and security.\n}\n\n\n\\section{Model and Definitions}\n\\label{sec-Prelim_and_outline}\n\nA {\\it finite-state automaton} (FSA) \\A\\ has a finite set of states\n$ST$, a finite alphabet $\\Sigma$, and a transition\nfunction $\\mu: ST \\times \\Sigma \\longrightarrow ST$. In this work we\ndo not assume an initial state or a terminal state for the automaton,\ni.e., it may begin its execution from any state and does not\nnecessarily stop.\n\nWe already described in the previous section the distributed\ncomputation setting---dealer, agents, adversary, and unbounded input\nstream---under which the FSA is to be executed. In more detail, we\nassume a {\\em global clock} to which all agents are synchronized.  We\nwill assume that no more than one input symbol arrives during any\nclock tick.  By {\\em input stream}, \nwe mean a sequence of input symbols arriving at a certain schedule of\nclock ticks.  Abusing notation, we will sometimes refer to the input\nwithout explicit reference to the schedule.  (We note that the global\nclock requirement can in principle be removed if we allow the input\nschedule to be leaked to \\Ad.)\n\\ignore{\nAs in~\\cite{DGGK11}, we consider a distributed computation setting in\nwhich a party, which we refer to as {\\it the dealer}, has an FSA \\A\\\nwhich accepts an ({\\it a priori} unbounded) stream of inputs $x_1,\nx_2,\n\\ldots$ received from an external source. The dealer delegates the \ncomputation to a set of agents $A_1,\n\\ldots, A_n$, by \nfurnishing them with an implementation of \\A.\nThe agents, each receiving\nall the inputs for \\A, execute their \nimplementation of \\A\\ without communicating with each other, and,\nat a given signal from the dealer, terminate the\nexecution, compute the current state of \\A\\ and return it as output. \nWe will typically use $\\rmPi$ to refer to distributed computation \nschemes thus described.\n\nWe consider \nan adversarial model where an entity, \ncalled the adversary \\Ad, is allowed to corrupt agents as the\nexecution of the protocol proceeds. We consider the so-called\n{\\em passive} or {\\em semi-honest} adversary model, where corrupted\nagents can combine their views in order to learn protected information,\nbut are not allowed to deviate from the protocol. Each agent can be\ncorrupted only once during an execution. \\Ad\\ can view the entire\ncontents of a corrupted agent's memory, but does not obtain any of the\nglobal inputs. Incidentally, we consider event processing by an agent\nas an {\\em atomic operation}.  That is, agents cannot be corrupted\nduring an execution of  \nstate update.  This\nis a natural and easily achievable assumption, which allows us to not\nworry about some tedious details.  The computation is then considered\nto be secure, if any two executions (possibly on different inputs and\ninitial states---defined more formally\nbelow) \nare ``similarly'' distributed.  }\n\nWe also\nmentioned that \\Ad\\ is\nallowed to corrupt agents as the execution of the protocol\nproceeds. We consider the so-called {\\em passive} or {\\em semi-honest}\nadversary model, where corrupted agents can combine their views in\norder to learn protected information, but are not allowed to deviate\nfrom the protocol. Furthermore, each agent can be corrupted only once\nduring an execution. When it does, \\Ad\\ can view the entire contents\nof a corrupted agent's memory, but does not obtain any of the global\ninputs. \n\nIncidentally, we consider event processing by an agent as an\n{\\em atomic operation}.  That is, agents cannot be corrupted during an\nexecution of\nstate update.  This\nis a natural and easily achievable assumption, which allows us to not\nworry about some tedious details.  The computation is then considered\nto be secure, if any two executions (possibly on different inputs and\ninitial states---defined more formally\nbelow) \nare ``similarly''\ndistributed.  \n\nThis model of security for distributed computation on unbounded input\nstreams was introduced by Dolev {\\em et al.}~\\cite{DGGK11} as the {\\it\nprogressive corruption} model (PCM), allowing \\Ad\\ to be\ncomputationally unbounded, and in particular requiring that the\ndistributions of the two executions (again, more formally defined\nbelow) be identical.\n\nIn this work we use a variant of PCM, applying the following two\nweakenings to the PCM definition:\n\\begin{newenum}\n\\item Rather than requiring that the distributions of executions be identical,\nwe require them to be {\\em computationally} indistinguishable.\nThis means that we guarantee security only against polynomial-time-bounded\nadversaries.\n\n\\item We \nrequire indistinguishability of executions for the {\\em same} corruption\ntimeline (and, of course, different input streams).  This means that,\nfor example, agent IDs are now allowed to be included in the agents'\nviews. (We use agent IDs in one of our contructions.)\nWe stress that this is not a significant security weakening,\nas essentially we only allow the adversary to differentiate among the\nagents' identities;\nthe inputs and current state of the computation remain computationally\nhidden.\n\\end{newenum}\n\n\nWe now present our amended PCM definition.  We first formalize the\nnotion of \n{\\em corruption timeline} and  the view of the adversary.\n\n\n\n\\begin{definition}\nA {\\em corruption timeline} $\\rho$ is a sequence $\\rho=((A_1,\n\\tau_1),\\ldots,(A_k, \\tau_k))$, where $A_1,\\ldots,A_k$ are the\ncorrupted agents and $\\tau_1,\\ldots,\\tau_k$ ($\\tau_1 \\leq \\ldots \\leq \\tau_k$)\n denote the time when the corresponding corruption took place.\nThe {\\em length} of\na corruption timeline is $|\\rho|=k$.\n\\end{definition} \n\n\n\nWe denote by $\\view_{\\rho}^{\\rmPi}(X, s)$ the probability\ndistribution of the aggregated internal states of corrupted agents at\nthe time of corruption, when executed on input $X$ and\ninitial state $s$.\n\n\n\n\n\\begin{definition} [\\bf Computational Privacy in the Progressive Corruption Model]\n\\label{defProgressive}\nWe say that a distributed computation scheme $\\rmPi$ is {\\rm\n$t$-private in the Progressive Corruption Model (PCM)} if for every\ntwo states $s_1, s_2 \\in ST$, polynomial-length input streams $X_1,\nX_2$, and any corruption timeline $\\rho$, $|\\rho| \\leq\nt$,\n\\[\n\\view_{\\rho}^{\\rmPi}(X_1, s_1) \\compind \\view_{\\rho}^{\\rmPi}(X_2, s_2).\n\\]\n\\end{definition}\n\\noindent Here, `$\\compind$' denotes the computational indistinguishability\nof two distributions.\n\n\n\n\n\\subsection{Tools and Building Blocks}\n\\label{sec:tools-app}\nA pseudo-random generator (PRG) $G : X \\rightarrow Y$, where $X$ and $Y$ are typically of the\nform $\\{0,1\\}^k$ and $\\{0,1\\}^{k+l}$, respectively, for some positive\nintegers $k,l$. Recall that PRGs are known to exist based on the\nexistence of one-way functions, and that the security property of a\nPRG guarantees that it is computationally infeasible to distinguish\nits output on a value chosen uniformly at random from $X$ from a value\nchosen uniformly at random from $Y$ (see,\ne.g.,~\\cite{Goldreich2000}). In our setting, we will further assume that\nthe old values\nof the PRG seeds are securely erased by the agents upon use and hence are\nnot included in the view of the adversary.\n\nThe other basic tool that our protocols make use of \nis\n{\\em secret sharing}~\\cite{Sha79}, where\nessentially, a secret piece of information is ``split'' into shares\nand handed out to a set of players by a distinguished player called\n{\\em the dealer}, in such a way that up to a threshold $t < n$ of the\nplayers pulling together their shares are not able to learn anything\nabout it, while $t + 1$ are able to reconstruct the secret. We\npresent the specific instantiations of secret sharing as needed\nin the \ncorresponding sections.\n\n\\subsection{Overview of Our Approach}\n\\section{Overview of Our Approach}\n\\label{outline}\nLet $\\cal A$ be a publicly known automaton with $m$ states. We assume\nthat we have some ordering of the states of \\A, which are denoted\nby corresponding labels.  Every agent\nstores the description of the automaton. In addition, during the\ncomputation, for every state $s_j$ of \\A, every agent $A_i$ computes\nand stores its current label $\\ell _j ^ i$. \nAs mentioned above, all agents receive a global input stream\n$\\Gamma=\\gamma_1,\\gamma_2,...\\gamma_i,...$ and perform computation in\nsynchronized time steps.\n\nAt a high level, the main idea behind our constructions is that the\nstate labels will be shares ({\\em \\`{a} la} secret\nsharing~\\cite{Sha79}) of a secret which identifies the currently\nactive state of $\\cal A$.  More specifically, for each of the $m$\nautomaton states, the $n$ state labels (held by the $n$ agents) will\nbe shares of a $1$ if the state is currently active, and shares of a\n$0$ otherwise.  We will show how the players' local computation on\ntheir shares will ensure that this property is maintained throughout\nthe computation on the entire input stream $\\Gamma$.  When the input\nstream $\\Gamma$ is fully processed (or a stop signal is issued), the\nagents recover the current state by reconstructing the secrets\ncorresponding to each automaton state.  At the same time, shares of\nthe secrets (when not taken all together) reveal no information on the\ncurrent state of $\\cal A$.\n\n\nWe now present additional high-level details on two variants of\nthe approach above. Recall that we consider the semi-honest adversary model, \nwhere corrupted players are not allowed to deviate from the protocol, but \ncombine their views in order to learn protected information.\n\n\\vspace{-.1in}\n\\paragraph{$(n,n)$-reconstruction.}  In this scenario, we require that all \n$n$ agents participate in the reconstruction of the secret (corrupted\nplayers are considered semi-honest and hence honestly provide their computed\nshares).\n\nAt the onset of computation, the shares are initialized using an $(n,n)$\nadditive secret-sharing scheme, such that the initial state labels are the\nsharing of $1$, and labels of each of the other states are shares of $0$.\nWhen processing a global input symbol $\\gamma$, each agent computes a\nnew label for a state $s$ by summing the previous labels of all states\n$s'$ such that $\\mu(s',\\gamma)=s$.  It is easy to see that, due to the\nfact that we use additive secret sharing, the newly computed shares\nwill maintain the desired\nsecret-sharing property.  Indeed, say that on input symbol\n$\\gamma$, $u$ states transition into state $s$.  If all of them\nwere inactive and their labels were shares of $0$'s, then the newly computed \nshares \nwill encode a $0$ (as the sum of $u$ zeros).  Similarly, if one of the $u$\npredecessor states was active and and its label shared a $1$, then  \nthe new active state $s$ will also correspond to a share a $1$.\n\nA technical problem arises in the case of ``empty'' states, i.e.,\nthose that do not have incoming transitions for symbol $\\gamma$, and\nhence their labels are undefined. Indeed, to hide the state of the\nautomaton from the adversary who corrupts agent(s), we need to ensure that\neach label is a random share of the appropriate secret.  Hence, we\nneed to generate a random $0$-share for each empty state without\ncommunication among the agents.\n\n\n\n\n\nIn the $(n,n)$ sharing and reconstruction scenario, we will non-interactively\ngenerate these labels pseudo-randomly as follows.  Each pair of\nagents $(A_i,A_j)$ will be assigned a random PRG seed $\\seed_{ij}$ \nThen, at each event (e.g., processing input symbol\n$\\gamma$), each agent $A_i$ will pseudo-randomly generate a string\n$r_j$ using each of the seeds $\\seed_{ij}$, and set the label of the\nempty state to be the sum of all strings $r_j$. \nThis is done for each empty state independently.  The PRG\nseeds are then (deterministically) ``evolved'' thereby erasing from the\nagent's view the knowledge of the labels' provenance, and making them all\nindistinguishable from random.  As all agents are synchronized with\nrespect to the input and the shared seeds, it is easy to see that the\nshares generated this way reconstruct a $0$, since each string $r_j$ will be\nincluded twice in the total sum, and hence will cancel out (we will use an\nappropriate [e.g., XOR-based] secret-sharing scheme such that \nthis is ensured.).\n\nFinally, and intuitively, we observe that  PCM security\nwill hold since\nthe view of each corrupted agent only includes pseudo-randomly\ngenerated labels for each state and the current PRG seed value.  As noted\nabove, even when combined with the views of other corrupted players, the labels\nare still indistinguishable from random.\n\n\\vspace{-.1in}\n\\paragraph{$(t+1, n)$-reconstruction.} In this scenario, up to $t$ corrupted \nagents do not take part in the reconstruction (this is motivated by the\npossibility of agents (UAVs) being captured or destroyed by the\nadversary).  Agents who submit their inputs are doing so correctly.\nThus, here we require $n>2t$.\n\nWe will take our $(n, n)$-reconstruction solution as the basis, and\nadapt and expand it as follows.  First, in order to enable\nreconstruction with $t+1$ agents, we will use $(t+1,n)$ additive\nsecret-sharing (such as Shamir's~\\cite{Sha79}).  Second, as before,\nwe will use a PRG to generate labels, but now we will have a\nseparate seed for each subset of agents of size $n-t+1$.  Then, at each\nevent (e.g., processing of an input symbol),\n\\ignore{\n $\\gamma$\n\\textcolor{blue}{\\todo{Lena: clock cycle with or without input letter}\nVLAD: We need to agree on the detail how we schedule inputs. I think\nit should be at most one input per clock cycle, right.  We should\nwrite and justify it in the model section above.},\n}\neach agent $A_i$,\nfor each of the groups he belongs to, will update its shares by\ngenerating a random $(t+1,n)$-secret sharing of a $0$ using the randomness generated by applying $G$\nto the group's seed.  Then, agent $A_i$ will use the share thus generated for\nthe $i$-th agent as its own,\nand set the label of the empty state to be the sum of all such shares.\n\nHere we note that, since agents are excluded from some of the groups,\nand that in this scenario up to $t$ agents might not return their\nstate during reconstruction, special care must be taken in the\ngeneration of the re-randomizing polynomials so that all agents have\ninvariantly consistent shares, even for groups they do not belong to,\nand that any set of agents of size $t+1$ enable the reconstruction of\nthe secrets.  (See Section~\\ref{sec-from-t} for details.)\nThe above is done for each empty state independently.\nAs before, the PRG seeds are then (deterministically) evolved,\nmaking them all indistinguishable from random.\n\n\\ignore{\n\n{\\bf $(t+1, n)$-reconstruction using additively homomorphic encryption.}  In our third scenario, we will make use of additively homomorphic public-key encryption, such as Goldwasser-Micali~\\cite{C:GolMic88} (for $(n,n)$ reconstruction) or Paillier~\\cite{EC:Paillier98}.\n\nOur main idea here is to encrypt the state labels with the public key known to the dealer (or reconstructible by a threshold of players).  The output is then obtained by running a secure multiparty computation on the player's current states.   Now, during FSA computation, since the labels are encrypted, they hide the state update history, and empty states can be simply labeled with encryptions of $0$ (for $(n,n)$ reconstruction) and with Shamir shares of $0$ derived from a single seed.\n\n}\n\n\\begin{algorithm*} [htb]\n\\caption{Template algorithm for agent $A_i$, $1\\leq i \\leq n$, for label \nand state update.\\label{alg:updating}}\n\\label{algo:calc}\n\\begin{algorithmic}[1]\n\\REQUIRE An input symbol $\\gamma$.\n\\ENSURE New labels for every state.\n\\IF{$\\gamma$ is initialized} \n\\STATE $ \\ell ^i_j := \\sum_{k, \\mu(s_k, \\gamma)=s_j} \\ell^i_k$ (the sum is calculated over some field $\\mathbb{F}$, depending on the scheme).\n\\ENDIF\n\\FOR {every $T\\in {\\cal T}$ s.t. $A_i\\in T$}\n\\STATE Compute  $B^T S^T \\leftarrow G(\\seed^T_r)$, where $B^T=b^T_1b^T_2...b^T_m$, and $b^T_j \\in \\mathbb {F}$, $1\\le j \\le m$.\n\\STATE $\\seed^T_{r+1} := S^T$.\n\\FOR {$j=1$ to $m$}\n\\STATE $\\ell_j^i := \\ell_j^i+ R_j$, where $R_j$ is a scheme-specific pseudo-random quantity.\n\\ENDFOR\n\\ENDFOR\n\\end{algorithmic}\n\\end{algorithm*}\n\n\\begin{remark}\n{\\em This approach reveals the length and schedule of the input\n$\\Gamma$ processed by the players.  Indeed, the stored seeds (or more\nprecisely, their evolution which is traceable by the adversary simply\nby corrupting at different times players who share a seed) do reveal\nto the adversary the number of times the update function has been invoked.\nWe hide this information by\nby requiring the agents to run updates at each\nclock tick.\n} \n\\end{remark}\n\n\nAlgorithm~\\ref{alg:updating} \nsummarizes the update operations performed by agent $A_i$ ($1\\le i \\le\nn$) during the $r$-th clock cycle. The key point is the generation of\n$R_j$, the label re-randomizing quantity. Notice also that in every\nclock cycle, there may or may not be an input symbol received by the\nagent; if the agent did not receive any input, we assume that the\ninput symbol is not initialized.\n\n\n\\ignore{\n\n\\todo{VLAD: I haven't gone through the algorithm. I suspect it needs some care.}\n\n\\todo{need to add text explaining that add-randomness does, depending on the instance of the protocol... what does it do?}\n\n\\todo{add-randomness is not a good name}\n\n\\todo{Juan: I tried to simplify the notation, and made add-randomness explicit. I'll probably change it further once I go over the next section.}\n\n}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction} \n\nAstronomical observations have established that collimated supersonic\noutflow, or jets, are ubiquitous. Young Stellar Objects (YSO),\npost-AGB stars, X-ray binaries and active radio galaxies show jets.\nThe ``central engines'' of these flow cannot be directly observed\ndue to insufficient telescope resolution.  The launch and collimation\nof jets has been modelled in terms of a combination of accretion,\nrotation and magnetic mechanisms (Pudritz et al.~\\citeyear{pudritz07}).\nThe study of magnetized supersonic jets has recently reached the\nlaboratory, where experiments have provided scale models\nof the launch and propagation of magnetized jets with dimensionless\nparameters relevant for astrophysical systems (Lebedev\net~al.~\\citeyear{lab1}; Ciardi et~al.~\\citeyear{ciardi9}; Suzuki-Vidal\net~al.~\\citeyear{suzuki}).  In combination with numerical simulations\n(Ciardi et~al.~\\citeyear{ciardi7}; Huarte-Espinosa et~al.~\\citeyear{we}),\nthese experiments help us to resolve unanswered questions on jet\nphysics. In particular the distinction between the physics of jet\nlaunch and that of jet propagation far from the engine. Since we\ncannot observe the former, it is important to identify distinct\nfeatures of jets in the asymptotic propagation regime that can\ndistinguish different engine paradigms. While both simulations and\nexperiments now consistently reveal the promise, if not essentiality,\nof dynamically significant magnetic fields for jet launch, the\ncorrelation between the initial jet magnetic configuration and the\nstability of the flow far from the launching region is unclear.\n\nThe importance of the magnetic field flux relative to the\nflows' kinetic energy divides jets into (i) Poynting flux dominated\n(PFD; \\citealp{shibata86}; \\citealp{bell96}; \\citealp{ustyugova00};\n\\citealp{lovelace02}; Nakamura \\& Meier, 2004), in which magnetic\nfields dominate the jet structure, (ii) magneto-centrifugally launched\n(MCL; \\citealp{blandford82}; Ouyed \\& Pudritz, 1997; \\citealp{blackman01};\n\\citealp{mohamed07}), in which magnetic fields only dominate out\nto the Alfv\\'en radius.  The observable differences between PFD and\nMCL jets are unclear, as are the effects that cooling\nand rotation have on PFD jets.\n\n\\section{Laboratory experiments}\n\\label{exps}\n\\vskip-.3cm\n\nLebedev et~al.~(\\citeyear{lab1}), Ciardi et al. (2007,2009) and\nSuzuki-Vidal et~al. (2010) have carried out laboratory experiments\nof magnetized jets in the pulsed power facilities of Imperial College\nLondon. Lebedev et~al.~(\\citeyear{lab1}) adjusted the dimensionless\nnumbers (Reynolds, magnetic Reynolds and Peclet) of the plasma\nin these experiments in appropriate regimes for astrophysics. \nA critical ingredient of these laboratory\nexperiments is the significant\nthermal energy loss of both the jets and the ambient medium plasmas;\nit plays a critical role in many astrophysical jet environments, e.g. YSOs.\nThe\nexperimental setup of Lebedev et~al.~(\\citeyear{lab1}) consisted of a pair of\nconcentric electrodes connected by a conical array of tungsten\nwires, of 13\\,$\\mu$m in diameter, inside a vacuum chamber. A TW\nelectrical pulse (1\\,MA, 250\\,ns) was applied to this circuit. Such\ncurrent causes ablation of the wires which results in the formation\nof a background ambient plasma (Figure~1a).  This material is pushed\nabove the wires by Lorentz forces then, while resistive diffusion\nkeeps the current close to the wires. The current induces a toroidal\nmagnetic field which at this stage is confined around the central\nelectrode, below the wires. \nAfter the complete ablation of the\nsection of the wires near the central electrode, the current switches\nto the plasma and creates a magnetic cavity (Figure~1b) containing a\ncentral jet.  The jet's core is confined and accelerated upward by\ntoroidal magnetic field pressure. The return current flows along\nthe walls of the magnetic cavity which is in turn confined by both\nthe thermal pressure and the inertia of the ambient medium plasma.\n\n\\begin{figure}[hb]\n\\vspace{-00pt}\n\\begin{center}\n\\includegraphics[width=\\textwidth]{exp}\n~~~~~~~~\n(a)\n~~~~~~~~~~~~~~\n(b)\n~~~~~~~~~~~~~~\n(c)\n~~~~~~~~~~~~~~\n(d)\n~~~~~~~~~~~~~~\n(e)\n~~~~~~~~~~~~~~\n(f)\n~~~~~~ \\\\\n\\vspace{-0pt}\n\\caption{Soft X-rays images showing a time sequence of the formation\nof the background plasma (a), the expansion of the magnetic cavity (b), the\nlaunch of the jet (c), the development of instabilities in the central\njet column (d) and the fragmentation of the jet (e,f). \nThe wire array shows in the bottom part of the figures.\nImage taken from Suzuki-Vidal et~al.~(\\citeyear{suzuki}).}\n\\end{center}\n\\vspace{-08pt}\n\\end{figure}\n\nThen, the magnetic cavity opens up, the jet becomes detached and\nit propagates away from the source, in a collimated fashion, at\nvelocities of order 200\\,km\\,s$^{-1}$ (Figure~1c). \n\nLebedev et~al.~(\\citeyear{lab1}) found that the body of the jets in\ntheir experiments was significantly affected by the development of\ninstabilities (Figure~1e). The outflows where fragmented into a well\ncollimated set of clumps, or ``bullets'', with characteristic axial\nnon-uniformities (Figure~1e,f; Lebedev et~al.~\\citeyear{lab1}).  Ciardi\net~al.~(\\citeyear{ciardi7}) reproduced these experiments using 3D\nnon-ideal MHD numerical simulations carefully designed to model the\nlaboratory components (electrodes and wires) and all the plasma\nevolution phases. The simulations of Ciardi et~al.~(\\citeyear{ciardi7})\nreproduced the experimental results of Lebedev et~al.~(\\citeyear{lab1})\nvery well. The simulations clearly showed that during the final\nunstable phase of jet propagation the magnetic fields in the central\njet adopted a twisted helical structure. Thus Ciardi\net~al.~(\\citeyear{ciardi7}) concluded the nano-metric jets of Lebedev\net~al.~(\\citeyear{lab1}) are affected by normal $m=\\,$1 mode perturbations.\n\n\n\\section{Simulations}\n\\label{model}\n\nWe use the Adaptive Mesh Refinement code AstroBEAR2.0 (Cunningham\net~al.~\\citeyear{bear}; Carroll-Nellenback et~al.~\\citeyear{bear2}) to solve\nthe equations of MHD in 3D with cooling source terms.  The grid represents\n160$\\times$160$\\times$400\\,AU divided into 64$\\times$64$\\times$80\ncells plus 2 adaptive refinement levels.  We use extrapolation boundary\nconditions at the four vertical faces of the domain and at the top\none, as well as a combination of MHD reflective and\ninflow conditions at the bottom face.\n\nWe will now briefly describe the setup of our simulations,\nsee Huarte-Espinosa et~al.~(2012b)\nfor a full description of the implementation.\nInitially, the gas is static and has an ideal gas equation of state\n($\\gamma=\\,$5$\/$3), a number density of 100\\,cm\\,s$^{-1}$ and a\ntemperature of 10000\\,K.  The magnetic field is helical, centrally\nlocalized (${\\bf B}\\ne 0$ within $r,z<r_e$) and the magnetic pressure exceeds the thermal pressure\ninside the magnetized region. \nWe use source terms to continually inject magnetic or kinetic energy\nat cells where $r,z<r_e$. We have carried out six simulations: three PFD\nmagnetic tower jets and three hydrodynamic jets. For each case we\ncalculated an adiabatic, a cooling and a rotating case. The\nhydrodynamic runs were implemented is such as way as to have same\ntime average propagation speed and energy flux as the adiabatic PFD\njet run. We use the cooling tables of Dalgarno \\& McCray~(\\citeyear{dm})\nfor the cooling case. For the rotating runs we applied a Keplerian\nrotation profile to the gas and magnetic fields located within\n$r,z<r_e$. \n\n\\section{Results}\n\n\\subsection{Jet structure and stability}\n\nOur magnetic tower simulations consistently show that magnetic\npressure gradients, alone, push field lines and plasma upward,\nforming magnetic cavities with low densities (Figure~2, all but\nright panel). PFD jets decelerate relative to the hydro ones;\nalthough both the PFD and the hydro jets have the same injected\nenergy flux, the PFD case produces both axial and radial expansion\ndue to magnetic pressure.  \\\\\n\n\\begin{wrapfigure}[22]{r}{.73\\textwidth}\n\\vspace{-30pt}\n\\begin{center}\n\\includegraphics[width=.70\\textwidth]{f2} \\\\\n\\vspace{-10pt}\n\\caption{Logarithmic false color density maps of the adiabatic \n(1st column), rotating\n(2nd column) and cooling (3rd column) PFD jets. Hydrodynamic jet \n(4th column). From top to bottom the time is 42, 84 and 118\\,yr.} \n\\end{center} \n\\vspace{+00pt}\n\\end{wrapfigure}\n\nThe cores of the PDF jets are confined\nby magnetic hoop stress, while their surrounding cavities are\ncollimated by external thermal pressure.  In contrast, the\npre-collimated hydro jets can only expand via a much lower thermal\npressure. Thus for our setup all of the energy flux in the\nhydro-case is more efficiently directed to axial mechanical power.\nIt is worth mentioning that our hydro cases can emulate to some\ndegree the asymptotic propagation regime of a jet that was\nmagneto-centrifugally launched (e.g. Blackman 2007) which is distinct\nfrom a PFD jet. \\\\\n\n\n\nRegarding the structure of the jets, we find that the PFD jet cores are\nthin and unstable, whereas the hydro jet beams are thicker, smoother\nand stable. These are observationally distinguishable features.\nThe base rotation and the cooling conditions that we explore (see end\nof Section~\\ref{model}) have mild effects on the propagation of the\nhydro jets, but strong effects on the stability of PFD jets. \\\\\n\nInstabilities in our PDF jets are current driven; the jets carry\nhigh axial currents which return along their outer contact\ndiscontinuity.  Lebedev et~al.~(\\citeyear{lab1}) saw this current density \ndistribution in their magnetic tower experiments.  \nWe find that the thermal to magnetic pressure ratio\ndistribution, $\\beta({\\bf X},t)$, consists of a central high beta\nplasma column --the jets' core-- surrounded by the low beta plasma of the\ncavities (Figure~3). \nThe growth rate of the current driven\ninstabilities is accelerated by cooling, firstly, and by base\nrotation, secondly.  \n\n\\begin{figure}[hb]\n\\begin{center}\n$\\log (\\beta)$ \\\\\n\\includegraphics[width=.9\\columnwidth]{f3.ps} \\\\\n\\caption{Thermal to magnetic pressure ratio of out PDF magnetic\ntowers at time$=$84\\,yr.  These are logarithmic false gray-scale\nmaps of the adiabatic (left), the rotating (middle) and the cooling\n(right) jets.}\n\\label{beta}\n\\end{center}\n\\end{figure}\n\nAt the core of these jets the axial\nmagnetic field dominates over the toroidal one. $|B_{\\phi} \/ B_z |\n\\ll 1$.  Thus the columns' instability condition is given by\n\\begin{equation}\n   \\left| \\frac{B_{\\phi}}{B_z} \\right| > | (\\beta_z - 1)k r_{jet}  |,\n   \\label{insta}\n\\end{equation}\n\\noindent where $\\beta_z=2 \\mu_0 P \/ B_z^2$, $\\mu_0$ is the magnetic\nvacuum permeability, $P$ is the plasma thermal pressure and $k^{-1}$\nis the characteristic wavelength of the current-driven perturbations\n(Huarte-Espinosa et~al., 2012b).\n\nThe cooling jet core consistently shows $\\beta_z \\sim\\,$1\nwhich means that because of thermal energy losses it does not have sufficient\nthermal energy to damp the toroidal magnetic pressure. Hence kink perturbations\ngrow exponentially.\nA different path to instability operates in the case in which we\napply a Keplerian rotation profile at the base of the PDF jet. Rotation\ncauses a slow amplification of the toroidal magnetic field, hence\nthe left hand side of equation~(\\ref{insta}) increases slowly and\nso do the kink mode perturbations. The rotating jet is not completely\ndestroyed by these perturbations, and their amplitude is about twice\nthe radius of the central jet (see Huarte-Espinosa et~al., 2012b\nfor details), in agreement with the Kruskal-Shafranov criterion\n(Kruskal et~al.~\\citeyear{kruskal}; Shafranov~\\citeyear{shafranov}).\n\n\\subsection{Energy flux distribution}\n\nIn Figure~\\ref{pfd} we show logarithmic color maps of the distribution\nof the jet Poynting to kinetic flux ratio $Q({\\bf x},t)= f_P\/f_k$\nas a function of time, where \n\\begin{equation}\n   \\begin{array}{c l l}\n   f_P = & \\int\\limits_s \\, [{\\bf B \\times (V \\times B ) }]_z \\, dS, \\\\\n   f_k = & \\int\\limits_s  \\, \\frac{1}{2} \\, \\rho \\, |{\\bf V}|^2 \\, V_z \\, dS.\n   \\end{array}\n\\label{fluxratio} \n\\end{equation}\n\\noindent These integrals are taken over the area of the jets' beam.\n\nWe consistently see the core of the jets is dominated by kinetic\nenergy flux ($Q < \\, $1, blue region), while the bulk of the beams\nis PFD ($Q > \\, $1, red region).  Such distribution is consistent\nwith the one in the laboratory jets of \\citet[][section~\\ref{exps}]{lab1}.\nThe time average mean $Q$ of our magnetic tower beams is $\\sim\\,$6,\nin agreement with the magnetic towers of \\citep[][see their Fig.~3b,\nbottom]{kato}. \\\\\n\n\\newpage\n\\begin{wrapfigure}[38]{r}{.5\\textwidth}\n\\vspace{-15pt}\n\\begin{center}\n~~~~~~~~~~~$\\log \\left| Q({\\bf x},t) \\right|$ \\\\\n\\includegraphics[width=.4\\columnwidth]{f4.ps} \\\\\n\\caption{Logarithmic color maps of the distribution and evolution\nof the magnetic towers Poynting to kinetic flux ratio.}\n\\label{pfd}\n\\end{center}\n\\end{wrapfigure}\n\nA very interesting result of our simulations is the long term\nevolution of PFD jets which yields a series of collimated clumps,\nthe magnetization properties of which vary over time and distance\nfrom the engine. PFD flows may thus eventually evolve into HD jets\nat large distances from the central engine.  A future avenue for\nthis investigation is a comparison of the $Q({\\bf X},t)$ distribution\nbetween our magnetic towers --where $Q>$\\,1 for the parameters\nexplored-- and models of jets created by MCL\nprocesses; while MCL jets begin with $Q>\\,$1 on scales less than\nthe Alfv\\'en radius, in the asymptotic limit the kinetic energy\nflux comes to dominate the flux of electromagnetic energy leading.\nSimulations of MCL launching in which the flow is cold and gas\npressure can be ignored show typical values of $Q \\sim \\, $0.7 at\nobservationally-resolved distances from the engine (Krasnopolsky\net al. 1999, 2003). \n\nChanges in the distribution of $Q({\\bf X},t)$ are quite relevant\nfor astrophysical jets. E.g. Hartigan et al. 2007 have shown that in YSO,\nand perhaps in planetary nebulae jets as well, some mechanism may be\nneeded to reduce the magnetization of plasma close to the jet source.\nIf these flows began as magnetic towers then the disruption of the\ncentral jets via kink modes may provide a means to produce collimated\nhigh beta clumps of material as is observed in HH flows. Our\ncalculations are thus the fists steps towards distinguishing between\ndifferent launch mechanisms by providing descriptions of asymptotic\nflow characteristics where observations might be possible, specially\nwith the next generation of telescopes, e.g. ALMA.\n\n\\section{Conclusions}\n\nOur magnetic tower jet simulations are in good agreement with the\nlaboratory experiments of \\citet{lab1}. In both investigations jets:\n(1) carry axial\ncurrents which return along the contact discontinuities, (2) the\ncores have a high $\\beta$ plasma, (3) the beams and cavities are PFD, \n(4) are eventually corrugated by\ncurrent driven instabilities becoming a collimated chain of \nmagnetized ``clumps'' or ``bullets''.  We stress the\nsimilarity between our models and the laboratory experiments because \nour implementation was not tuned to\nrepresent the laboratory results at all.  \nThe laboratory\nexperiments and the simulations support each other then, as well as the\nconclusion that both are revealing generic properties of PFD outflows.\n\nOur simulations show that PFD jet beams are lighter, slower and\nless stable than pre-collimated asymptotically hydrodynamic jets.\nThe latter might represent the asymptotic propagation regimes of\nmagneto-centrifugally launched jets. We find current-driven\nperturbations in the PFD jets, which for the regimes studied are\namplified by cooling, firstly, and by base rotation, secondly. This\nhappens because shocks and thermal pressure support are weakened\nby cooling, making the jets more susceptible to kinking. Base\nrotation, on the other hand, amplifies the toroidal magnetic field\nwhich in turn exacerbates the kink instability.  Eventually the\ninstabilities cause the corrugation of the jets' beam. PFD jets\nbecome a collimated chain of magnetized ``clumps'' or ``bullets''\nthen.  \\\\\n\n\\textit{Financial support for this project was provided\nby: the Space Telescope Science Institute grants HST-AR-11251.01-A\nand HST-AR-12128.01-A; the National Science Foundation under\naward AST-0807363 and grant PHY0903797; the Department of Energy \nunder award DE-SC0001063; Cornell University grant 41843-7012;\nthe Laboratory for Laser Energetics of Rochester NY grant \nDE-FC52-08NA28302.}\n\n\n\n\n\n\n\\bibliographystyle{elsarticle-harv}\n\n\n\n ","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\n\n\\section{Introduction}\n\nThe look-and-say (\\textsf{LS}) sequence \\cite{conway2012book}, also known as the \\emph{Morris} or the \\emph{Conway sequence} \\cite{conway1987weird,hilgemeier1996one,ekhad1997proof} is a recreational integer sequence having very intriguing properties.\\smallskip\n\nA \\textsf{LS} sequence is obtained iteratively by reading off the digits of the current value, and counting the number of digits in groups of the identical digit.\\smallskip\n\nStarting with \\texttt{1}, the sequence reads (\\textsf{OEIS A005150}): \\texttt{1}, \\texttt{11}, \\texttt{21}, \\texttt{1211}, \\texttt{111221}, \\texttt{312211}, etc. Starting with any digit $d \\neq {\\dig{1}}$ gives Conway's sequence (\\textsf{OEIS A006715}): $d$, ${\\dig{1}}d$, ${\\dig{111}}d$, ${\\dig{311}}d$, ${\\dig{13211}}d$, etc. Conway popularised these sequences and studied some of their properties. For example, an \\textsf{LS} sequence contains only the digits $\\dig{1}, \\dig{2}, \\dig{3}$, and satisfies a so-called \\emph{cosmological decay} \\cite{ekhad1997proof}, if $L_n$ denotes the number of digits of the $n^{\\mbox{\\scriptsize{th}}}$ term of the sequence, then \n\\begin{equation*}\n\\lim _{n\\to \\infty }{\\frac {L_{n+1}}{L_{n}}}=\\lambda\\simeq 1.303577\n\\end{equation*}\nwhere $\\lambda$ is the only real root of a degree-$71$ polynomial \\cite[\u00a76.12]{finch2003mathematical}. \nConway showed that $\\exists N\\in\\mathbb{N}^*$ such that every term of the \\textsf{LS} term decays in at most $N$ rounds to a compound of \\enquote{common} and \\enquote{transuranic} terms.\n\nFollowing Conway's work, numerous variants of \\textsf{LS} sequences were proposed and studied.\nFor instance, Pea Pattern sequences \\cite{pea}, Sloane's sequences\\cite{slo} or Kolakoski sequences \\cite{kola,kola2}.\nIn this paper we consider a new \\textsf{LS} sequence and study some of its properties.\nThe concerned variant, called \\textit{\\enquote{look-and-say again}} sequence, consists in repeating each \\textsf{LS} digit twice. We prove that the such sequences contain only the digits ${\\dig{1}},{\\dig{2}},{\\dig{4}},{\\dig{6}},d$, where $d$ is the starting digit.\n\n\\section{Notations and definitions}\nIn this paper we assume that numbers are written in base 10. Any integer $T$ can thus be written $T = t_1t_2 \\cdots t_k$ with $t_1, \\dotsc, t_k \\in \\{{\\dig{0}}, {\\dig{1}}, \\dotsc, {\\dig{9}}\\}$. To avoid any ambiguity, $ab$ will denote the \\emph{concatenation} of the numbers $a$ and $b$; accordingly $a^b$ indicates that a digit $a$ is repeated $b$ times. If we want to emphasise concatenation we use $a\\|b$ instead of $ab$.\n\n\\begin{definition}[Run-length representation]\nLet $T\\in \\mathbb{N}^*$, we can write \n\\begin{equation*}\n    T = \\underbrace{a_1\\ldots a_1}_{n_1}\\underbrace{a_2\\ldots a_2}_{n_2}\\ldots\\underbrace{a_k\\ldots a_k}_{n_k}\n\\end{equation*}\nwith $a_1\\neq a_2, a_2\\neq a_3, \\dotsc, a_{k-1}\\neq a_{k}$. The \\emph{run-length representation} of $T$ is the sequence $\\mathsf{RunL}(T)= a_1^{n_1}a_2^{n_2} \\cdots a_k^{n_k}$. Conversely, any finite sequence of couples $(a_i, b_i)_i$ where $a \\in \\mathbb N^*$ and ${\\dig{0}}\\leq b_i\\leq {\\dig{9}}$ is such that $b_{i-1}\\neq b_i\\neq b_{i+1}$, corresponds to an integer with run-length representation $(a_i^{b_i})_i$. \n\\end{definition}\nNote that the run-length representation of an integer is unique. \n\n\\begin{definition}[Pieces]\nIf $N = (a_i^{b_i})$ is a run-length encoded integer, we call each $a_i^{b_i}$ a \\emph{piece} of $N$.\n\\end{definition}\n\n\\begin{definition}[Look-and-say-again sequence]\nLet $T_0$ be a decimal digit, and for each $T_n$ define\n\\begin{equation*}\n    T_{n+1} = n_1n_1a_1a_1n_2n_2a_2a_2\\cdots n_kn_ka_ka_k.\n\\end{equation*}\nwhere $(a_i^{n_i})_i = \\mathsf{RunL}(T_n)$. We call the sequence $(T_k)_{k \\in \\mathbb N}$ the \\emph{look-and-say-again sequence} of seed $T_0$, and denote it by $\\mathsf{LSA}(T_0)$.\n\\end{definition}\n\n\\begin{example}\n\n\\noindent\n$\n\\mathsf{LSA}(1)=\n\\small{\n    {\\dig{1}}\\to\n    {\\dig{1111}}\\to\n    {\\dig{4411}}\\to\n    {\\dig{22442211}}\\to\n    {\\dig{2222224422222211}}\\to\n    {\\dig{6622224466222211}}\\to \\dotsc\n}$\n\n\\noindent\n$\\mathsf{LSA}(2)= \n\\small{\n{\\dig{2}}\\to\n{\\dig{1122}}\\to\n{\\dig{22112222}}\\to\n{\\dig{222222114422}}\\to\n{\\dig{6622221122442222}}\\to\n{\\dig{226644222211222222444422}}\\to\n\\dotsc}$\n\\end{example}\n\n\n\\section{The look-and-say-again sequence}\n\n\\begin{theorem}[Digits of $\\mathsf{LSA}$]\nOnly the digits $\\dig{1}, \\dig{2}, \\dig{3}, \\dig{4}, \\dig{6}, d$ appear in $\\mathsf{LSA}(d)$.\n\\end{theorem}\n\n\\begin{proof}\nLet $n \\in \\mathbb N$ and $(a_i^{n_i})_i = \\mathsf{RunL}(n)$, we write\n\\begin{equation*}\nP(n):=\\{\\forall i, \\,a_i \\in \\{\\dig{1}, \\dig{2}, \\dig{4}, \\dig{6}\\} \\text{ and } n_i \\in \\{2, 4, 6\\}\\\n\\end{equation*}\n(the matter of $d$ will be settled further down). Assume that $P(T_n)$ is true, and let $a_i^{n_i} \\in \\mathsf{RunL}(T_n)$.\nWe have four situations:\n\\begin{enumerate}[\\text{Case} 1:]\n    \\item  $\\mathsf{RunL}(T_n) = a_i^{n_i}$, in other terms $T_n = a_ia_i \\cdots a_i$ and there is no other digit. Then the next term in the sequence is $T_{n+1}=n_in_ia_ia_i$, which clearly satisfies $P(T_{n+1})$ since $a_i\\in\\{\\dig{1},\\dig{2},\\dig{4}, \\dig{6}\\}$ and $n_i\\in\\{2, 4, 6\\}$.\n\n\\item $i=1$, i.e., $T_n$ starts with the repeated digits $a_i$. In this case\n\\begin{equation*}\nT_{n+1}=n_1n_1a_1a_1n_{2}n_{2}a_{2}a_{2} \\cdots\n\\end{equation*}\nwith $a_k\\neq a_{k+1}$ for all $k$. It is clear that no $a_{k}$ can be contained in a piece that also contains $a_{k+1}$, therefore the possible pieces that $n_1$ and $a_1$ can take part to are either $n_1n_1a_1a_1n_{2}n_{2}$ (in the case $n_1=a_1=n_{2}$), or $n_1n_1a_1a_1$ (in the case $n_1=a_1\\neq n_{2}$), or $n_1n_1$ and $a_1a_1$ \/ $a_1a_1n_{2}n_{2}$ (respectively in the cases $n_1\\neq a_1\\neq n_{2}$ and $n_1\\neq a_1=n_{2}$). \n\nIn conclusion $a_1$ and $n_1$ generate pieces made of digits in $\\{\\dig{1},\\dig{2},\\dig{4}, \\dig{6}\\}$ with multiplicity either 2, 4, or 6. \n\\item $T_n$ ends with $a_i$; this is analogous to case 2 above.\n\n\\item The piece $a_i^{n_i}$ is neither at the end nor at the beginning of $T_n$. The next term in the sequence is:\n\\begin{equation*}\nT_{n+1}=\\cdots a_{i-2}n_{i-1}n_{i-1} a_{i-1}a_{i-1}n_in_ia_ia_in_{i+1}n_{i+1}a_{i+1}a_{i+1}\\cdots\n\\end{equation*}\nBy definition of the run-length representation, $a_{i-2},a_{i-1},a_i,a_{i+1}$ contains no consecutive values, hence the possible pieces resulting for $n_i$ and $a_i$ are:\n\\begin{enumerate}\n    \\item $n_{i-1}n_{i-1}a_{i-1}a_{i-1}n_{i}n_i$, and either $a_ia_i$ or $a_ia_in_{i+1}n_{i+1}$\n    \\item $a_{i-1}a_{i-1}n_{i}n_i$, and either $a_ia_i$ or $a_ia_in_{i+1}n_{i+1}$\n    \\item $n_{i}n_i$, and either $a_ia_i$ or $a_ia_in_{i+1}n_{i+1}$\n    \\item $n_in_ia_ia_i$ or $n_in_ia_ia_in_{i+1}n_{i+1}$\n    \\item $n_in_i$ and $a_ia_i$\n\\end{enumerate}\n In each case, since $\\forall k, \\,a_k\\in\\{\\dig{1},\\dig{2},\\dig{4},\\dig{6}\\}$ and $n_k\\in\\{2, 4, 6\\}$,\n\n we see that $n_i$ and $a_i$ can appear only in pieces that are of multiplicity 2, 4, or 6, and which contain numbers $\\in\\{\\dig{1},\\dig{2},\\dig{4},\\dig{6}\\}$.\n\\end{enumerate}\nIf $T_0 = d \\in \\{\\dig{1},\\dig{2},\\dig{4},\\dig{6}\\}$, then $P(T_0)$ holds and by the above case exhaustion argument $P(T_n)$ hold for all $n$.\n\nIt remains to discuss the case $T_0 = d \\notin \\{\\dig{1},\\dig{2},\\dig{4},\\dig{6}\\}$. Writing the first few terms of the resulting sequence shows that this is easily dealt with:\n \\begin{align*}\n      T_0 & =  d \n      & T_1 & = \\dig{11}dd \n      & T_2 & = \\dig{221122}dd\n      & T_3 & = \\dig{22222211222222}dd & \\cdots\n     \n     \n \\end{align*}\n Indeed, save for the first term, the digit $d$ only appears as $dd$ at the end of $T_n$. The rest of $T_n$ satisfies $P(T_n)$ discussed previously.\n \n To prove this, assume that $T_n = k \\| dd$, which means that $T_n$ starts with an integer $k$ and ends with the two digits $dd$, and further assume that $P(k)$ is true and $k$'s last piece is $\\dig{2}^k$, with $k\\in\\{2,4,6\\}$.\n Then the next term in the sequence is $T_{n+1} = k' \\| \\dig{22}dd$, where $k'$ is an integer that ends with the digit \\dig{2} and such that $P(k')$ is true (thanks to what we have proved in the first part of the theorem). Let $S = k'\\|\\dig{22}$. Consider three cases, as a function of the last piece of $k$, denoted $\\dig{2}^{\\omega}$:\n \\begin{center}\n\\begin{tabular}{ccccccr}\n$\\omega={2}$&$\\Rightarrow k'$ ends with &$\\dig{2222}$ \n&$\\Rightarrow S$ ends with & $\\dig{222222}$&$\\Rightarrow P(S) \\mbox{~holds}$\\\\\n$\\omega={4}$&$\\Rightarrow k'$ ends with &$\\dig{4422}$ \n&$\\Rightarrow S$ ends with & $\\dig{2222}$&$\\Rightarrow P(S) \\mbox{~holds}$\\\\\n$\\omega={6}$&$\\Rightarrow k'$ ends with &$\\dig{6622}$ \n&$\\Rightarrow S$ ends with & $\\dig{2222}$&$\\Rightarrow P(S) \\mbox{~holds}$\\\\\n \\end{tabular}\n\\end{center}\n Since $P(k')$ holds, the only problem in $S$ was at the interface between the ending \\dig{2}'s of $k'$ and the couple \\dig{22} at the end of $S$. With the exhaustion argument above we have shown that in each possible case $P(S)$ holds. Finally the number $S\\|dd$ is such that $d$ only appears as a couple $dd$ at the end, whereas $S$ is made of digits belonging to $\\{\\dig{1},\\dig{2},\\dig{4},\\dig{6}\\}$ with multiplicities in $\\{2, 4, 6\\}$.\n \n$T_0$ and $T_1$ contain only digits $\\in\\{\\dig{1}, \\dig{2}, \\dig{4}, \\dig{6},d\\}$. Since $T_2$ is of the form $k\\|dd$ with $P(k)$ true and $k$'s last digit being $\\dig{2}$, the above argument shows that all subsequent terms in the sequence can be written in this way, and the proof is completed.\\qed \n\\end{proof}\n\n\\begin{corollary}\nIf $T_0 = d \\neq \\dig{1}, \\dig{2}$ then $\\mathsf{LSA}(T_0)$ gives the same sequence, save for the two last digits of each term which are $dd$.\n\\end{corollary}\n\n\n\\begin{remark}\nThe length sequences for $T_0 = \\dig{1}, \\dig{2}, d$ are respectively:\n\\begin{align*}\n    & 1, 4, 4, 8, 16, 16, 24, 40, 48, 56, 88, 104, 120, 176, 224, 280, 392, 520, 648, 864, 1168 \\\\\n    & 1, 4, 8, 12, 16, 24, 32, 40, 48, 72, 92, 112, 156, 204, 264, 352, 464, 592, 784, 1036, 1320 \\\\\n    & 1, 4, 8, 16, 16, 24, 40, 48, 56, 88, 104, 120, 176, 224, 280, 392, 520, 648, 864, 1168, 1432\n\\end{align*}\n\\end{remark}\n\n\\begin{remark}\nFor all seeds $s$, $\\mathsf{LSA}(d)$ grows to infinity, namely the ratio of lengths $\\lambda_n = \\ell_{n+1}\/\\ell_n$ for two consecutive terms of sequence (which is between $1$ and $2$) tends towards Conway's constant $\\lambda \\simeq 1.303577$, regardless of the seed $T_0$. The following numerical simulation backs up this intuition:\n\\begin{center}\n\\input{convergence.tex}\n\\end{center}\n\\end{remark}\nThis is in fact a consequence of the following result:\n\\begin{theorem}\nConsider the following operations on pieces:\n$$\nC: a^b \\mapsto ba ~~~~~~~~~~ L: a^b \\mapsto b^2 a^2 ~~~~~~~~~~ \\eta: a^b \\mapsto \\kappa(a)^{2b}\n$$\nwith $\\kappa: a \\mapsto \\dig{2}a$ for $a\\in\\{\\dig{1}, \\dig{2}, \\dig{3}\\}$ and $\\kappa(a) = \\dig{1}$ otherwise. Then $L \\circ \\eta = \\eta \\circ C$.\n\\end{theorem}\n\n\\begin{proof}\nLet $a^b$ be a piece,\n\\begin{align*}\n    (\\eta \\circ C)(a^b) & = \\eta(b^1 a^1) = \\eta(b^1) \\eta(a^1) = \\kappa(b)^2 \\kappa(a)^2 \\\\\n    (L \\circ \\eta)(a^b) & = L(\\kappa(a)^{\\kappa(b)}) = \\kappa(b)^2 \\kappa(a)^2.\n\\end{align*}\n\\qed \n\\end{proof}\nA result of this theorem is that $\\mathsf{LSA}$ is equivalent to Conway's sequence, where $\\kappa$ and $\\eta$ allow us to translate from one to the other. Then $\\mathsf{LSA}$ inherits many of the properties that are known of Conway's sequence: decomposition into \\enquote{elements}, convergence to $\\lambda$, and so forth.\n\\begin{remark}\nThis would also work with $L(a^b) = b^3 a^3$ and $\\kappa(a) = \\dig{3}a$ for $a \\in \\{\\dig{1}, \\dig{2}, \\dig{3}\\}$ (i.e. an $\\mathsf{LSA}$ variant where elements are repeated three times instead of two). However the argument breaks down for a \\enquote{look-and-say four times} sequence. We leave this sequence and the study of its properties open for further research.\n\\end{remark}\n\n\n\n\\bibliographystyle{alpha}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{INTRODUCTION}\n\nThe idea of spontaneous birth of the universe by quantum tunneling from a de\nSitter instanton has been coupled by Vilenkin \\cite{vilenkin} to the\nbeginning of the inflationary expansion in a de Sitter spacetime with space\nsections of positive curvature. We propose an extension of this process,\nthrough topology change and inflation, to arrive at a Friedmann model with\nclosed hyperbolic spaces (CHS) as spatial sections. We are motivated by\nprevious work by Gott\\cite{gott1} and Bucher et al. \\cite{bucher} in the\nmatter of inflation leading to an $\\Omega $$_{0}<1$ universe, but assume a\nmechanism different from theirs, relying on De Lorenci et al.'s \\cite\n{lorenci} formalism to quantify the probability of changes in cosmic global\ntopology. There have been suspicions of incompatibility of CHS universes\nwith maps of the cosmic microwave background (CMB)\\cite{pogosyan}; we refer\nto a recent paper by Cornish et al. \\cite{cornish} to answer to this\ncriticism.\n\n\\section{CREATION AND EVOLUTION}\n\nWe use Planck units $c=G=\\hbar =1.$\n\nDe Sitter instanton has the metric (cf. \\cite{vilenkin})\n\n\\begin{equation}\nds^{2}=dt^{\\prime 2}+r_{0}^{2}\\cos {}^{2}(t^{\\prime }\/r_{0})(d\\chi ^{2}+\\sin\n{}^{2}\\chi \\ d\\Omega ^{2}),  \\label{metricInstanton}\n\\end{equation}\nwhere $t^{\\prime }=-it$, with $t$ imaginary time, $d\\Omega ^{2}=d\\theta\n^{2}+\\sin {}^{2}\\theta \\ d\\varphi ^{2}$, $(\\chi ,\\theta ,\\varphi )$ are\nspherical coordinates, and $r_{0}$ is a constant of the order of Planck's\nlength or time. But instead of being defined on a four-sphere $S^{4}$, our\ngravitational instanton has a more general topology: note that (i) the $%\nS^{4} $ instanton is obtained by analytic continuation into imaginary time\nof de Sitter spacetime with topology $R\\times S^{3}$, where the real line $R$\nis the time axis and the three-sphere $S^{3}$ is the spatial section \\cite\n{GandH}; (ii) de Sitter's metric of positive spatial curvature can be\nassigned to any topology of the form $R\\times M$, where $M=S^{3}\/\\Gamma $ is\nthe quotient space of $S^{3}$ by a discrete group of isometries $\\Gamma ,$\nwhich acts on $S^{3}$ without fixed points - see, for example, Ellis \\cite\n{ellis}, or Lachi\\`{e}ze-Rey and Luminet \\cite{LaLu}; and (iii) the same\nprocess of analytic continuation into imaginary time can be done on these\nnew spacetime topologies. So we assume as a generalization of de Sitter's\ninstanton the metric of Eq. \\ref{metricInstanton} defined on the quotient\nspaces $S^{4}\/\\Gamma $ (which may not be manifolds but rather orbifolds -\nsee \\cite{scott}, \\S 2). The volume of $M,$ with its curvature normalized to\nunity, is $2\\pi ^{2}\/($order of $\\Gamma );$ its fundamental group $\\pi\n_{1}(M)$ is isomorphic to $\\Gamma ,$ hence, except for the case $M=S^{3}$\n(trivial $\\Gamma ),M$ is multiply connected.\n\nWhen this instanton becomes a real de Sitter spacetime, we get the\ninflationary metric of positive spatial curvature (cf. \\cite{vilenkin}, \\cite\n{gott1}), starting at $t=0,$ \n\\begin{equation}  \\label{deSitterSph}\nds^2=-dt^2+r_0\\cosh {}^2(t\/r_0)(d\\chi ^2+\\sin {}^2\\chi \\ d\\Omega ^2),\n\\end{equation}\nand the topology of the direct product $R\\times M$, as implied above. The\nparameter $r_0\\,$ is related in \\cite{gott1} to a constant energy density,\nwhich is now interpreted as the initial value of the inflaton potential $%\nV(\\phi )$. As in \\cite{bucher}, during this epoch the inflaton field $\\phi $\nis stuck in a false vacuum (region $A$ in Fig. 1), causing a growth that\nerases inhomogeneities in $M$. But in our case this stage is very short. In\nthe numerical example of next Section, $t_f=r_0$ is the endtime of this\nepoch, which therefore may hardly be called inflationary; see below.\n\nAt $t=t_{f}$ we assume that a topology and metric transition takes place;\nthe latter becomes\n\n\\begin{equation}\nds^{2}=-d\\tau ^{2}+r_{0}^{2}\\sinh {}^{2}(\\tau \/r_{0})(d\\chi ^{\\prime\n2}+\\sinh {}^{2}\\chi ^{\\prime }d\\Omega ^{2}),  \\label{deSitterHyp}\n\\end{equation}\nwith initial $\\tau =\\tau _{i}$ to be determined below. This is similar to\nthe metric change in \\cite{gott1}, but here we shall assume it to happen as\na quantum transition in minisuperspace. The change in metric in a closed\nspace implies a change in topology, since the spherical space $S^{3}\/\\Gamma $\ncannot support the hyperbolic metric in the spatial part of Eq. \\ref\n{deSitterHyp} -- cf. \\cite{scott}, Theorem 5.2. This metric and topology\nchange is signaled by the small bump in the potential $V(\\phi )$ -- region $B\n$ in Fig. 1; we hope a mechanism for this process can be adapted from the\none recently developed by De Lorenci et al. \\cite{lorenci}, who estimated\nthe probabilities of a few similar transitions. Their results, interpreted\nwith some liberty, indicate that a spherical cosmology is unstable against\nquantum transitions, and has a good chance of becoming hyperbolic, as in our\ncase.\n\nHere the spatial section becomes the CHS $M^{\\prime }=H^3\/\\Gamma ^{\\prime },$\nwhere $H^3$ is hyperbolic three-space, and $\\Gamma ^{\\prime }\\cong \\pi\n_1(M^{\\prime })$ is a discrete group of isometries acting on $H^3$ without\nfixed points -- cf. \\cite{LaLu}. Spacetime topology becomes $R\\times\nM^{\\prime }.$ Since this is a quantum process, we need not demand continuity\nin the five-dimensional pseudoeuclidean space where de Sitter spacetime is\nimbedded, as done in \\cite{gott1}. But, as a working hypothesis, we\npostulate that energy, hence physical volume, is conserved in the\ntransition, and we expect the change in the expansion factor to be small;\ntherefore the normalized volumes of $M$ and $M^{\\prime }$ should be of the\nsame order of magnitude. This can be arranged, and $\\tau _i$ calculated, as\nwill be seen in the next Section.\n\nWhen space becomes $M^{\\prime }$ it may again have density irregularites.\n(To see this, imagine a uniform distribution of a thin film of matter over a\ntwo-sphere, which suddenly gets a handle and becomes a torus; the film takes\na time to spread itself evenly over the new surface.) In the light of this,\nit is not so essential that the previous stage gets homogenized, as it is\nthat its duration be short so that $M$ will not grow too much and $M^{\\prime\n}$ may become smooth; see next Section. At this point one might wonder why\nis stage $M$ necessary at all. Perhaps because there is no gravitational\ninstanton \\cite{hawking} that would actualize directly as a closed\nhyperbolic universe.\n\nThus $M^{\\prime }$ is neither equivalent to the nucleated bubble in \\cite\n{gott1} or \\cite{bucher}, nor to the smooth patch in Kolb and Turner's \\cite\n{KandT} basic picture of inflation, for either of these is already\nhomogeneous as it appears on the scene. Besides, $M^{\\prime }$ need not grow\nto encompass the observable universe; when it does not, as in the example\nbelow, the model predicts observable effects of the nontrivial topology. The\nmost obvious of these, but still difficult to verify, is the production of\nmultiple images of each source, so as to mimic the uniform distribution of\nan open Friedmann model. See \\cite{hvf1983};\\cite{LaLu} and references\ntherein.\n\nIn \\cite{bucher} this is the time of slow roll inflation, with $V(\\phi )$\nsloping as $-\\mu ^3\\phi $ towards the true vacuum. For simplicity here we\nprefer the potential in \\cite{gott1}, which has a plateau (region $C$ in\nFig. 1) of about the same height as in the false vacuum, so that inflation\nproceeds at the same rate as before. This epoch is $\\tau _i\\leq \\tau \\leq\n\\tau _1$, with $\\tau _1$ being determined by a continuity condition with the\nnext phase.\n\nFinally \\cite{gott1} a phase transition is arranged, the equation of state\ngoing from $p=-\\rho $ to $p=\\rho \/3,$ which ends the inflationary period.\nThere is reheating around the true vacuum ($D$ in Fig. 1), and spacetime\ngets the usual Friedmann metric of negative spatial curvature\n\n\\begin{equation}  \\label{FriedmannHyp}\nds^2=-d\\tau ^2+a^2(\\tau )(d\\chi ^{\\prime 2}+\\sinh {}^2\\chi ^{\\prime }d\\Omega\n^2).\n\\end{equation}\nThus we reach the epoch of standard cosmology, except for the effects of the\ncompactness and multiple connectedness of the spatial sections -- see \\cite\n{LaLu}.\n\n\\section{NUMBERS}\n\nA convenient family of candidates for $M=S^3\/\\Gamma $ are the lens spaces $%\nL(p,q),$ where $p,q$ are coprime integers with $1\\leq q\\leq p\/2$ \\cite{SandT}%\n. Their fundamental group, hence also $\\Gamma ,$ is of order $p$, so their\nvolume is $2\\pi ^2\/p.$ Hyperbolic manifolds $M^{\\prime }$ are known to exist\nwith normalized volumes from $0.94$ up \\cite{weeks}.\n\nAs an example, let us take $M=L(50,1),$ with normalized volume $%\nv_{sph}=0.394784$, and as $M^{\\prime }$ the smallest known CHS,\nWeeks-Matveev-Fomenko manifold (cf. \\cite{hvf1996}), with $v_{hyp}=0.942707.$\nThen the universe's largest half-diameter, $R_{\\max }(t)=(\\pi \/4)r_0\\cosh\n(t\/r_0),$ is smaller than the radius of the event horizon, $R_H(t)=2r_0\\cosh\n(t\/r_0)\\{\\tan {}^{-1}[\\exp (t\/r_0)]-\\pi \/4\\}$ for $t>0.8814r_0.$ Let us take \n$t_f=r_0$ as the endtime of this epoch, which is enough to homogenize $M$.\nBy the conservation of physical volume, $v_{sph}\\cosh {}^3(1)=v_{hyp}\\sinh\n{}^3(\\tau _i\/r_0),$ whence $\\tau _i=0.9865r_0.$\n\nTo find $\\tau _{1}$, first we have to make sure that there is enough time\nfor inflation to smoothen an initial inhomogeneity in $M^{\\prime }$. The\ncircumscribing radius of the Dirichlet domain given in \\cite{weeks} is $%\nr_{\\max }=0.752470$ in comoving (or normalized) units; the maximal distance\nbetween points in $M%\n{\\acute{}}%\n$ is of this order of magnitude. The horizon's radius is $r_{H}(\\tau )=\\ln\n[\\tanh (\\tau \/2r_{0})\/\\tanh (\\tau _{i}\/2r_{0})].$ We can have $r_{H}(\\tau\n)\\gtrsim r_{\\max }$ only if $\\tanh (\\tau _{i}\/2r_{0})<\\exp (-r_{\\max }),$ or \n$\\tau _{i}<1.0232r_{0}.$ With $\\tau _{i}$ as obtained above, we must have $%\n\\tau _{1}>4.17r_{0}.$ This value is compatible with the one we now obtain\nfrom the continuity of the expansion factor, $r_{0}\\sinh (\\tau\n_{1}\/r_{0})=a(\\tau _{1}).$ In the beginning of the Friedmann radiation era\nthat follows reheating, $a(\\tau )=(2b_{\\ast }\\tau )^{1\/2}$ (cf. \\cite\n{ohanian}), where $b_{\\ast }=(8\\pi G\\rho _{rad,0}\/3c^{2})^{1\/2}a_{0}^{2},$\nwith $\\rho $$_{rad,0}=$ present density of radiation energy $=4.6477\\times\n10^{-34}$ g\\ cm$^{-3}$ \\cite{KandT}, and $a_{0}=$ present value of $a(\\tau )$%\n. Taking $r_{0}=1.6160\\times 10^{-33}$ cm (Planck's length), $\\Omega\n_{0}=0.3,$ and Hubble's constant $H_{0}=65$ km s$^{-1}$Mpc$^{-1}$, the\ncontinuity condition gives $\\tau _{1}=71.1r_{0},$ close to the value $%\n69r_{0} $ in \\cite{gott1}.\n\nToday's physical volume of $M^{\\prime }$ would be $4.64\\times 10^{84}$ cm$^3$%\n, while the observable universe -- interpreted as the region of repeated\ncosmic images -- is about 200 times larger.\n\n\\section{REMARKS}\n\nIn the above reasoning the dynamics of spontaneous birth and inflation\nprocesses in the modified scenario was presumed to be adaptable from\nprevious results. We plan to elaborate on this point, as also on the matter\nof topology changes, and to present a more detailed picture in the future.\nThermodynamics considerations, prevalent in \\cite{gott1} and not touched\nupon here, may also be addressed in the wider study.\n\nThe question of the influence of space closure on the generation and growth\nof primordial fluctuations has been left out, to be examined elsewhere. Here\nwe only comment on doubts that have appeared on whether a closed hyperbolic\nuniverse would be compatible with the long wavelength modes of the CMB --\nsee \\cite{pogosyan}. As pointed out in \\cite{cornish}, in hyperbolic\nuniverses there is no low cutoff for these modes. The confusion seems to\narise from mistaking the separations between equivalent points in the space\nof images $H^3$ with maximal wavelengths of perturbations; but the latter\ncan be spread over closed geodesics, which may form knotted patterns of\nincreasing length inside the CHS. See also \\cite{roukema}, \\cite{FandG}.\nTherefore these models are probably compatible with the spotted maps of the\nCMB obtained by NASA's COBE satellite, but of course more research is needed\nin this direction.\n\nOne of us (S. S. C.) thanks Funda\\c c\\~ao de Amparo \\`a Pesquisa do Estado\nde S\\~ao Paulo (FAPESP) for a doctoral scholarship. H. V. F. is grateful to\nRuben Aldrovandi for conversations on geometry, and to Conselho Nacional de\nDesenvolvimento Cient\\'{\\i}fico e Tecnol\\'ogico (CNPq - Brazil) for partial\nfinancial support.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\nThere is an ever-growing interest in single and few photon scattering from a one-dimensional (1D) continuum because of their possible applications in quantum information processing. This subject has been studied both theoretically \\cite{c1,c2,c3,c4,c5,c6,c7,c8,c9,c10,c11,c12,c13,c14,c15,c16,c17,c18,c19,c20,c21,c22,c23,c24,c25,c26,c27,c28,c29,c30,c31,c32,c33,c34,c35,c36,c37} and experimentally \\cite{c38,c39,c40,c41,c42,c43,c44,c45,c46,c47,c48} from a variety of perspectives and reviewed quite elaborately in Reference \\cite{c49}. Some commonly used 1D waveguides are conducting nanowires \\cite{c38,c39}, photonic crystal waveguides \\cite{c47}, and superconducting microwave transmission lines \\cite{c43,c44}. Collective effects emerging from a periodic array of two-level atoms coupled to a waveguide can lead to Fano minima in the reflection spectrum \\cite{c28,c29,c37}, superradiant decays \\cite{c47}, rearrangement of the optical band structure \\cite{c50} and realization of Bragg mirrors \\cite{c51,c52}. The role of spatial separation between the atoms in the context of photon scattering from a 1D continuum has been manifested in recent works \\cite{c5,c31,c32,c51,c52,c53,c54,c55,c56,c57}. Very recently, multiple Fano interference channels due to the waveguide-mediated phase coupling between the atoms leading to the appearance of transparency points were demonstrated in \\cite{c37}.\n\nTheoretical analysis of photon scattering from a collection of disparately detuned atoms coupled to a waveguide is a hard problem in real space, because of the waveguide mediated phase coupling between the atoms. Here, we address the collective behavior of such a system by constraining the spatial periodicity to be an integral or half-integral multiple of the resonance wavelength. Working at this phase substantially simplifies the problem and provides insight into the various kinds of collective properties of the output radiation. Since our interest is primarily in the realizability of induced transparency, we demonstrate the emergence of new Fano minima in the reflection spectrum for a system of even number of emitters via appropriate choice of detunings. For an odd chain size, the system manifests an effective re-emergence of single-atom behavior. These are new effects that were not observed in \\cite{c37}, as the atoms were all supposed to have identical transition frequencies. Our results are strong enough to withstand dissipative effects as long as the dissipative channel is weak compared to the waveguide channel and the atomic detunings are large compared to the decay rate. In other words, much like the phenomenon of EIT \\cite{harris1991,eitreview}, while perfect transparency emerges in an ideal dissipation-free scenario, a fairly high degree of transparency can still be observed in the presence of weak dissipation. However, in the usual scenario for EIT, the system comprises of non-interacting atoms, whereas in a waveguide, an effective phase coupling between the emitters persists even at long spatial separations (of the order of a wavelength). Another important feature that surfaces at this choice of phase is the reciprocity in optical transport, i.e. with regard to both the reflected and transmitted amplitudes. More generally, the transport is insensitive to the order in which the atoms are placed in the chain. This is starkly different from the usual scenario, where the phase coupling makes the system strongly non-reciprocal. Reciprocity relations for a large class of one-dimensional systems were derived in \\cite{rec1}. A general discussion on the subject of Lorentz reciprocity for lossless systems can be found in \\cite{recrev}.\n\nThe paper is organized as follows: in Sec. \\ref{s2}, we review the transport model for a single photon through a waveguide coupled to a chain of non-identical emitters. In Sec. \\ref{s3}, we demonstrate how one can observe transparent behavior for an atomic chain consisting of two disparately detuned atoms, when $kL$ is chosen to be an integral multiple of $\\pi$. In Sec. \\ref{s4}, we write down the analytical forms for the reflection and transmission coefficients for an arbitrary chain size and generalize some of the observations to the case of an even chain size. Concurrently, we indicate, for an odd chain size, how one can effect a complete suppression of collective effects so as to recover single-atom behavior. In Sec. \\ref{s5}, we argue that the commutativity between the transfer matrices leads to reciprocal transport properties Finally, in Sec. \\ref{s6}, we probe the effect of including radiative decay into nonwaveguide modes on the spectral characteristics, with particular emphasis on the observability of transparent behavior. Section \\ref{s7} summarizes the key ideas explored.\n\n\\section{Single-photon transport in a waveguide coupled to $N$ non-identical emitters} \\label{s2}\n \nFor a periodic 1-D array of $N$ two-level emitters evanescently coupled to a 1-D continuum (Fig. \\ref{f1}), when the atomic transition frequency far exceeds the waveguide cutoff frequency, one can write down the Hamiltonian of the system in real space as:\n\\begin{align}\n\\mathcal{H}=i\\hbar v_g\\int_{-\\infty}^{\\infty}\\dd{x}\\bigg(a_L^{\\dagger}(x)\\dfrac{\\partial a_L(x)}{\\partial x}-a_R^{\\dagger}(x)\\dfrac{\\partial a_R(x)}{\\partial x}\\bigg) \\notag \\\\+\\hbar\\sum_{j=1}^{N}(\\omega_j-i\\Gamma_0)\\ket{e}_j\\bra{e}+ \\notag \\\\ \\hbar\\sum_{j=1}^N\\bigg[\\{\\mathcal{J}a_L(x_j)+\\mathcal{J}a_R(x_j)+\\text{h.c.}\\}\\ket{e}_j\\bra{g}\\bigg],\n\\end{align}\nwhere $a_L(x)$ and $a_R(x)$ describe the real-space Bosonic operators corresponding to the left and the right propagating fields, $\\omega_j$  corresponds to the transition frequency of the $j^{\\text{th}}$ atom and $x_j=(j-1)L$ to its location along the waveguide, $v_g$ is the group velocity of the propagating photon and $\\mathcal{J}$ denotes the coupling strength between the propagating field and any of the emitters. $\\Gamma_0$ denotes the rate of spontaneous emission into all modes outside of the waveguide continuum, assumed equal for all the atoms. We disregard the dipole-dipole interaction (DDI) between the atoms by assuming the interatomic separation to be comparable or larger than the resonance wavelength.\n\n\\begin{figure}[!t]\n \\captionsetup{justification=raggedright,singlelinecheck=false}\n\\centering\n\\includegraphics[scale=0.80]{f1.pdf} \n\\caption{\\small  {Atomic array coupled to a waveguide.}} \\label{f1} \n\\end{figure}\n\nThe scattering eigenstate has the form\n\\begin{align}\n\\ket{\\mathcal{E}_k}=\\int_{-\\infty}^{\\infty}\\dd{x}\\bigg[\\phi_{kL}(x)a_L^{\\dagger}(x)+\\phi_{kR}(x)a_R^{\\dagger}(x)\\bigg]\\ket{\\Psi} \\notag\\\\\n+\\sum_{j=1}^Nc_k^{(j)}\\ket{0;e_j},\n\\end{align}\nwhere $\\ket{\\Psi}$ refers to the state with the field in vacuum and all atoms in the ground state, and $\\ket{0;e_j}$ to the one where the field is still in vacuum but only the $j^{\\text{th}}$ atom has been excited. $\\phi_{kL}$ and $\\phi_{kR}$ denote the wavefunctions corresponding to left- and right-propagating photonic excitations respectively. The Schr$\\ddot{\\text{o}}$dinger equation $\\mathcal{H}\\ket{\\mathcal{E}_k}=\\hbar v_g k\\ket{\\mathcal{E}_k} $ leads to a system of ODEs for the various probability amplitudes:\n\\begin{align}\n\\bigg(-iv_g\\dfrac{\\dd}{\\dd x}-v_gk\\bigg)\\phi_{kR}(x)+\\mathcal{J}\\sum_{j=1}^{N}c_k^{(j)}\\delta(x-x_j)&=0, \\notag\\\\\n\\bigg(iv_g\\dfrac{\\dd}{\\dd x}-v_gk\\bigg)\\phi_{kL}(x)+\\mathcal{J}\\sum_{j=1}^{N}c_k^{(j)}\\delta(x-x_j)&=0, \\notag\\\\\n(\\Omega_j-v_gk)c_k^{(j)}+\\mathcal{J}\\phi_{kL}(x_j)+\\mathcal{J}\\phi_{kR}(x_j)&=0,\n\\end{align}\n\nwhere $\\Omega_j=\\omega_j-i\\Gamma_0$. For a wave incident from the left, one can solve these ODEs subject to the boundary condition that $\\phi_{kL\/kR}(x_j)=\\frac{1}{2}[\\phi_{kL\/kR}(x_j^+)+\\phi_{kL\/kR}(x_j^-)]$ and also the discontinuity imposed on the wavefunctions due to the the delta-function source, i.e. $-iv_g[\\phi_{kR}(x_j^+)-\\phi_{kR}(x_j^-)]+\\mathcal{J}c_k^{(j)}=iv_g[\\phi_{kL}(x_j^+)-\\phi_{kL}(x_j^-)]+\\mathcal{J}c_k^{(j)}=0$. The solutions take the form\n\n\\begin{align}\n\\phi_{kL}(x)&=\\begin{cases} r_1e^{-ikx}, & x<0 \\\\\nr_{j+1}e^{-ik(x-jL)}, & (j-1)L\n<x<jL\\\\\n0, & x>(N-1)L\n\\end{cases}\\hspace{1mm},\n\\end{align}\n\n\\begin{align}\n\\phi_{kR}(x)&=\\begin{cases} e^{ikx}, & x<0 \\\\\nt_{j}e^{ik(x-jL)}, & (j-1)L<x<jL\\\\\nt_Ne^{ik(x-NL)}, & x>(N-1)L\n\\end{cases}\\hspace{1mm},\n\\end{align}\n\nwhich subsequently lead to a system of simultaneous equations involving the transmission and reflection coefficients and the atomic excitation amplitudes. Eliminating the excitation amplitudes from the system engenders in a recursive matrix relation\n\n\\begin{align}\n\\begin{bmatrix}\nr_j\\\\\nt_{j-1}\\\\\n\\end{bmatrix}=\\mathcal{L}_j\\begin{bmatrix}\nr_{j+1}\\\\\nt_j\\\\ \\end{bmatrix},\n\\end{align}\nwhere\n\\begin{align}\n\\mathcal{L}_j=\\begin{bmatrix}\ne^{ikL}(1-i{\\delta^{-1}_{k (j)}}) & -ie^{-ikL}\\delta^{-1}_{k (j)}\\\\\nie^{ikL}\\delta^{-1}_{k (j)} & e^{-ikL}(1+i\\delta^{-1}_{k (j)}) \\\\\n\\end{bmatrix}, \\label{E1}\n\\end{align}\n\nwith ${\\delta_{k (j)}}=\\dfrac{\\omega_{k}-\\Omega_j}{\\Gamma}$ and $\\Gamma=\\dfrac{\\mathcal{J}^2}{v_g}$ . Iterative use of this relation, conjoined with appropriate boundary conditions, yields the reflection and transmission coefficients:\n\\begin{align}\nr=\\dfrac{\\bigg(\\prod_{j=1}^N\\mathcal{L}_j\\bigg)_{12}}{\\bigg(\\prod_{j=1}^N\\mathcal{L}_j\\bigg)_{22}}, \\hspace{5mm}t=\\dfrac{e^{-ikNL}}{\\bigg(\\prod_{j=1}^N\\mathcal{L}_j\\bigg)_{22}}. \\label{E2}\n\\end{align}\n\nBecause of the presence of the phase factors $e^{\\pm ikL}$ and the differential detunings assigned to the emitters, it is analytically hard to find out compact expressions for the above. The matrix product, however, becomes simple to evaluate for $kL=n\\pi$, $n$ being a natural number. \n\n\\section{Transparency and collective behavior for a two-atom system} \\label{s3}\n\nLet us first consider the simpler scenario of two differentially detuned atoms in a waveguide and define the mean laser detuning $\\overline{\\Delta}=\\omega_k-\\frac{1}{2}(\\omega_1+\\omega_2)$ and relative atomic detuning $s=\\omega_1-\\omega_2$.  It turns out that while $[\\mathcal{L}_1\\mathcal{L}_2]_{22}$ is symmetric in $s$, $[\\mathcal{L}_1\\mathcal{L}_2]_{12}$ is not, owing to the phase coupling between the emitters mediated by the waveguide. In other words, in view of Eq. \\ref{E2}, even though transmission is perfectly reciprocal, reflection is not. For this system, the reflection and transmission coefficients reduce to\n\\begin{align}\nr&=\\dfrac{-i\\Gamma[(e^{i\\alpha}+1)(\\overline{\\Delta}+i\\Gamma_0)-(e^{i\\alpha}-1){s}\/{2}]-\\Gamma^2(e^{i\\alpha}-1)}{[\\overline{\\Delta}+i(\\Gamma+\\Gamma_0)]^2+\\Gamma^2e^{i\\alpha}-(s\/2)^2}, \\notag\\\\\nt&=\\dfrac{(\\overline{\\Delta}+i\\Gamma_0+s\/2)(\\overline{\\Delta}+i\\Gamma_0-s\/2)}{[\\overline{\\Delta}+i(\\Gamma+\\Gamma_0)]^2+\\Gamma^2e^{i\\alpha}-(s\/2)^2}, \\label{E4}\n\\end{align}\n\nwhere we have taken $\\alpha=2kL$. We note, however, in the special case $kL=n\\pi$, that the above expressions turn symmetric in $s$, thereby leading to reciprocity in the transport properties. In subsequent considerations, we analyse the results pertaining to this special choice of phase.\n\nWe also assume an idealized scenario where $\\Gamma_0$ can be ignored. The effect of $\\Gamma_0$ on the transmission is studied later in Sec. \\ref{s6}. With $\\Gamma_0$ set to $0$, one observes an emergence of transparent behavior when the two atoms are equally detuned, albeit in opposite directions, with respect to the laser frequency. In other words, $t$ becomes unity when $\\overline{\\Delta}$ equals zero, or $\\omega_k-\\omega_1=\\omega_2-\\omega_k$, i.e. for a pair of antisymmetrically detuned emitters. It follows, from Eq. \\ref{E4} and the plots in Fig. \\ref{figure2}, that there is a transmission peak at $\\overline{\\Delta}=0$, while there are two roots of the profile at $\\overline{\\Delta}=\\frac{s}{2}$ (or $\\omega_k=\\omega_1$) and $\\overline{\\Delta}=-\\frac{s}{2}$ (or $\\omega_k=\\omega_2$) corresponding to perfect reflection.  The peak has unit height in the absence of decay, signifiying transparent behavior. The height of this peak is strictly less than unity for any other choice of phase, as can be verified from \\ref{E4} (for instance, in the specific scenario, when $kL=\\frac{n\\pi}{2}$ with odd $n$, the height of this peak is $\\bigg[\\dfrac{s^2}{s^2+8\\Gamma^2}\\bigg]^2$ - see Fig. \\ref{f3}). For a sufficiently small yet non-zero value of $\\abs{\\omega_1-\\omega_2}$, one finds a very narrow window of size $s$ over which the system is capable of demonstrating both opacity as well as transparency. As $s\\rightarrow 0$, the two roots come progressively closer. Fig. \\ref{figure2} illustrates this scenario for various choices of $\\abs{\\omega_1-\\omega_2}$.\n\n\\begin{figure}[!t]\n \\captionsetup{justification=raggedright,singlelinecheck=false}\n\\centering\n\\includegraphics[scale=0.65]{f2-eps-converted-to.pdf} \n\\caption{\\small  {Transmission for a two-atom system without decay for a couple of values of $s=\\omega_1-\\omega_2$ and with $kL=n\\pi$. Perfect transparency is observed at $\\omega_k=\\frac{1}{2}(\\omega_1+\\omega_2)$ (zero mean detuning), unless $\\omega_1=\\omega_2$, in which case, the system is perfectly reflecting at zero detuning. The two roots of the transmission come closer as the atomic frequencies approach each other.}} \\label{figure2} \n\\end{figure}\n\nWe also note, in passing, that the poles of the transmission and reflection demonstrate features remindful of level attraction. These poles occur at \n\\begin{align}\n\\overline{\\Delta}_{\\pm}^{(\\text{p})}=-i\\Gamma\\pm\\sqrt{\\bigg(\\frac{s}{2}\\bigg)^2-\\Gamma^2}\\hspace{0.5mm}.\n\\end{align}\nLevel attraction is typically observed between the normal modes of two coupled oscillators when one of the bare modes has negative energy and the modes have comparable decay rates. When the coupling strength equals or exceeds a critical value, the level separation vanishes. As a direct analogy, we see, in our case, that the real parts of the transmission poles coincide and become $0$ in the region $\\frac{s}{2}\\leq\\Gamma$, while the imaginary parts expand and shrink respectively. The point of transition where the coupling equals this critical value is referred to as an exceptional point where the complex eigenfrequencies coincide \\cite{E1}. Realizing level attraction has been quite a challenge from an experimental perspective and consequently, there is burgeoning interest in level attraction and phenomena occuring in the vicinity of exceptional points. Recently, level attraction has been observed in a variety of systems and topological behavior around an exceptional point has also been explored \\cite{L1,L2,L3,L4,L5,L6,L7}.\n\n\\begin{figure}[!t]\n \\captionsetup{justification=raggedright,singlelinecheck=false}\n\\centering\n\\includegraphics[scale=0.65]{f3-eps-converted-to.pdf} \n\\caption{\\small  {Comparison of transmission spectra corresponding to $kL=\\pi$ and $kL=\\frac{\\pi}{2}$, with $s=1.5\\Gamma$. The transmission peak attains unit height for $kL=\\pi$ whereas it is much shorter than unity for $kL=\\frac{\\pi}{2}$.}} \\label{f3} \n\\end{figure}\n\nIn the waveguide case with two atoms, we see that the transmission has zeros at\n\\begin{align}\n\\overline{\\Delta}_{\\pm}^{(\\text{r})}=\\pm\\frac{s}{2},\n\\end{align} \nwhich simultaneously determine the peaks of the reflection spectrum. These zeros are close to the the real parts of the poles when $\\frac{s}{2}\\gg\\Gamma$. In the complementary regime, when $\\frac{s}{2}$ is comparable to $\\Gamma$, the real parts of the poles become small compared to the respective imaginary parts, as a consequence of which, the resolution between the two levels (or the two poles) becomes difficult. This problem of resolution arises fundamentally because $\\Gamma$ not only appears in the discriminant of the poles, but also acts as a natural broadening term.\\\\\n\n\\begin{figure}[!t]\n \\captionsetup{justification=raggedright,singlelinecheck=false}\n\\centering\n\\includegraphics[scale=0.65]{f4-eps-converted-to.pdf} \n\\caption{\\small  {Transmission at $kL=n\\pi$ for $\\frac{s}{2}=\\Gamma-\\eta$ with $\\eta=0.1\\Gamma$ and $\\eta=0.25\\Gamma$.}} \\label{f4} \n\\end{figure}\n\nThe shrinking of the transmission width as $\\frac{s}{2}$ goes below $\\Gamma$ is clearly reflected in the transmission plots shown in Fig. \\ref{f4}. Letting $\\frac{s}{2}=\\Gamma-\\eta$, where $\\frac{\\eta}{\\Gamma}\\sim 10^{-1}$, we find that the pole $\\overline{\\Delta}_{+}^{(\\text{p})}$ shrinks in width, with the relevant width given by $\\bigg(1-\\sqrt{\\frac{2\\eta}{\\Gamma}}\\bigg)\\Gamma$. Thus, the transmission window becomes narrower as $\\eta$ becomes larger.\n\n\\section{Transparency in a multiatom chain} \\label{s4}\n\nWe now bring out some interesting features of the spectral behavior collectively induced by a chain of multiple emitters with non-identical detunings, corresponding to a spatial periodicity so chosen that $kL=n\\pi$. In Ref. \\cite{c37}, the analytical expressions for the spectral amplitudes were derived for a system of identical emitters for which the transfer matrices were also identical. Through a diagonalization procedure, the matrix product was calculated. However, $kL=n\\pi$ is an exceptional point of the transfer matrices, since the eigenvalues coincide and become $(-1)^n$. Hence, diagonalization fails. However, in the special case when $kL$ is an integral multiple of $\\pi$, one can derive exact analytical expressions for the relevant matrix product in \\ref{E2}:\n\\begin{align}\n\\prod_{j=1}^N\\mathcal{L}_j=(-1)^{nN}\\begin{bmatrix}\n1-i\\Gamma\\sum_{j=1}^{N}\\delta^{-1}_{k (j)} & -i\\Gamma\\sum_{j=1}^{N}\\delta^{-1}_{k (j)}\\\\\ni\\Gamma\\sum_{j=1}^{N}\\delta^{-1}_{k (j)} & 1+i\\Gamma\\sum_{j=1}^{N}\\delta^{-1}_{k (j)}\n\\end{bmatrix}.\n\\end{align}\n\nIt is easy to check this result for $N=2$, and the general result for arbitrary $N$ can be be verified using the procedure of mathematical induction. We then have the following transmission and reflection coefficients:\n\n\\begin{align}\nt=\\frac{1}{1+i\\Gamma\\sum_{j=1}^{N}(\\omega_k-\\omega_j+i\\Gamma_0)^{-1}}\\hspace{1mm}, \\notag\\\\\nr=-\\frac{i\\Gamma\\sum_{j=1}^{N}(\\omega_k-\\omega_j+i\\Gamma_0)^{-1}}{1+i\\Gamma\\sum_{j=1}^{N}(\\omega_k-\\omega_j+i\\Gamma_0)^{-1}}\\hspace{1mm}.\\label{E5}\n\\end{align}\n\n\n\\begin{figure}[!t]\n \\captionsetup{justification=raggedright,singlelinecheck=false}\n\\centering\n\\includegraphics[scale=0.70]{f5.pdf} \n\\caption{\\small  {Even number of emitters with equal and opposite detunings assigned in pairs generates transparency. The order of the atoms is not important, so the arrangement shown here is just one of the possible permutations.}} \\label{f5} \n\\end{figure}\n\nThe collective effects due to emission from multiple periodically spaced emitters is clearly embodied in the aforementioned expressions.  The spectral dependence on the detunings has a close resemblance with that in the single-emitter scenario. The key factor that modifies the spectrum is $\\sum_{j=1}^{N}(\\omega_k-\\Omega_j)^{-1}$, an additive effect of the inverse detunings pertaining to the individual emitters. The expression is, of course, not as simple for other choices of phase. As a further simplification, let us focus on the case where $\\Gamma_0$ is small enough to be dropped from consideration. This, in principle, entails in the possibility of generating perfect transmission through suitable arrangements of the individual detunings. The condition for transparency ($r=0$ and $t=1$)is given by\n\n\\begin{align}\n\\sum_{j=1}^{N}\\frac{1}{\\Delta_{k(j)}}=0.\n\\end{align}\n\nwhere $\\Delta_{k(j)}=\\omega_k-\\omega_j$ is the laser detuning relative to the transition frequency of the $j^{\\text{th}}$ atom. For a single emitter, this relation is clearly impossible to satisfy and therefore, a single atom in a waveguide does not lead to transparent behavior. One must have multiple atoms to enable the achievement of transparency. In a double-emitter scenario, where $N=2$, the condition translates to $2\\omega_k=\\omega_1+\\omega_2$, which implies an exactly antisymmetric assignment of detunings to the two emitters. This is in line with what was highlighted in the previous section dedicated to the study of a two-atom chain (see Fig. \\ref{f2}). \n\nFor $N=3$, the corresponding constraint appears as a quadratic equation\n\n\\begin{align}\n3\\omega_k^2-2(\\omega_1+\\omega_2+\\omega_3)\\omega_k+\\omega_1\\omega_2+\\omega_2\\omega_3+\\omega_3\\omega_1=0,\n\\end{align}\nwith roots given by \n$$\\frac{1}{3}\\bigg[\\omega_1+\\omega_2+\\omega_3 \\pm\\sqrt{\\omega_1^2+\\omega_2^2+\\omega_3^2-\\omega_1\\omega_2-\\omega_2\\omega_3-\\omega_3\\omega_1}\\bigg].$$ The discriminant can be re-expressed as $\\frac{1}{2}[(\\omega_1-\\omega_2)^2+(\\omega_2-\\omega_3)^2+(\\omega_3-\\omega_1)^2]$, which, being a sum of squares, is strictly non-negative, and hence, the roots are real.\n\n In general, it is easy to see that for even number of emitters in the chain, it is always possible to make the system transparent if the atomic transition frequencies can be so adjusted that for any atom detuned by a certain amount, there exists another atom in the chain detuned by the same amount but in the opposite sense (Fig. \\ref{f5}). In other words, for a chain of $2l$ atoms, an assignment of the frequency detunings $+\\Delta_1$, $-\\Delta_1$, $+\\Delta_2$, $-\\Delta_2$,... $+\\Delta_l$, $-\\Delta_l$, in no particular order, would give rise to transparency in the system. Such an asymmetric pairwise assignment of detunings lead to Fano minima in the reflection spectrum, which signifies a destructive interference between the reflected waves emanating from the emitters. Concomitantly, the transmitted waves constructively interfere, leading to perfect transmission. This extreme resonant inhibition of the reflection amplitude relative to the single-atom emission is a new phenomenon that is not observed at $kL=n\\pi$ for a system of identically detuned emitters. \n\nOn the other hand, if one has odd number of emitters in the chain, one can recover the single-atom emission spectra by assigning pairwise asymmetric detunings to any randomly chosen $(N-1)\/2$ emitter pairs, leaving out a single atom. It then follows from Eqs. \\ref{E5}, that the remaining atom completely determines the spectral characteristics. That is, if this particular atom has a transition frequency $\\omega_0$, the transmitted spectrum due to the entire atomic chain reduces simply to\n\\begin{align}\nt=\\frac{1}{1+i\\Gamma(\\omega_k-\\omega_0)^{-1}}\\hspace{1mm},\n\\end{align}\n\nwhich is identical to the transmission coefficient with just that single atom coupled to the waveguide (Fig. \\ref{f6}). Stated differently, when an even number of atoms with a pairwise asymmetric assignment of frequency detunings are added, in a periodic fashion, to a single atom coupled to a waveguide, no discernible collective effects emerge. The order of this arrangement and therefore, the location of the odd atom are not important. This makes sense from the perspective of Fano interference, since the reflected waves from the appended atoms destructively interfere, while that from the residual atom effectively goes through unperturbed. As a consequence, if the odd atom is in resonance with the laser frequency, the system resembles a perfectly reflecting mirror, regardless of the frequency detunings of the other atoms.\n\n\\begin{figure}\n \\captionsetup{justification=raggedright,singlelinecheck=false}\n\\centering\n\\includegraphics[scale=0.80]{f6.pdf} \n\\caption{\\small  {A system of three atoms, out of which two carry equal and opposite detunings $+\\Delta$ and $-\\Delta$. The odd one out (the middle one, in this figure) with a detuning of $\\bm{\\Delta_0}$ determines the spectral behavior, and no collective effects exist. This behavior transcends to the case of any odd number of emitters in the chain with a commensurate assignment of frequency detunings. When $\\bm{\\Delta_0}$$=0$ , the system behaves as a perfectly reflecting mirror.}} \\label{f6} \n\\end{figure}\n\n\nFinally, in the event that all the atoms have identical frequencies, one can observe Dicke-type superradiant behavior due to enhancement of the reflection amplitude. If the atomic frequencies are all set equal to $\\omega_0$, the corresponding transmitted spectrum is obtained to be\n\n\\begin{align}\nt=\\frac{1}{1+iN\\Gamma(\\omega_k-\\omega_0)^{-1}}\\hspace{1mm},\n\\end{align}\n\n\nwhich pertains to a Lorentzian with a half-width of $N\\Gamma$. This superradiant behavior observed for a collection of identical emitters, at $kL=n\\pi$, was also derived in \\cite{c37}. However, the possibility of controlling collective effects by tuning the individual atomic frequencies was not explored in that work. Having this added flexibility of adjusting the atomic frequencies brings out different types of interesting radiant behavior that one can observe, in principle. We do not merely encounter the possibility of superradiant reflection, but also come across new points of transparency. In particular, if we have an even number of emitters coupled to the waveguide, a pairwise asymetric allotment of detunings generates new Fano minima. For an odd chain size, we see how a similar assignment of frequencies to any $N-1$ of the atoms can lead to a complete disappearance of collective behavior and give rise to a spectrum governed entirely by the transition property of the leftover atom.\n\n\\section{Reciprocal behavior for $kL=n\\pi$} \\label{s5}\n\nAs has been discussed previously, in the context of a two-atom system, reciprocity entails for the choice of phase $kL=n\\pi$. It follows, quite generally, from the expressions in \\ref{E5}, that $kL=n\\pi$ ensures perfect optical reciprocity for any arbitrary chain size. In fact, this choice of phase is both necessary and sufficient for reciprocity in both the reflection and the transmission. The fundamental property that brings about this reciprocal character is the commutativity between any two transfer matrices, i.e.\n\\begin{align}\n[\\mathcal{L}_j,\\mathcal{L}_k]=0\\hspace{2mm} \\forall \\hspace{1mm}j,k \\label{E3}.\n\\end{align}\n\nThe commutation relation follows from  the form in \\ref{E1}. As an essential implication of this, one finds that the matrix product $\\prod_{j=1}^N\\mathcal{L}_j$ is insensitive to the order in which the individual matrices are multiplied. Consequently, no matter what the order of the atoms is, one has the same transmission and reflection coefficients. Of course, this result is valid under the assumption that each emitter couples identically to the left as well as the right-propagating fields, as far as the two coupling strengths are concerned. It is easy to verify that Eq. \\ref{E3} holds true for any arbitrary assisgnment of detunings if and only if $kL=n\\pi$.\n\n\\section{Effect of dissipation into nonwaveguide modes on transparency} \\label{s6}\n\nThe simplistic results laid out in the preceding discussions in Secs. \\ref{s3} and \\ref{s4} hold only when $\\Gamma_0$ is small enough to be ignored. However, we can look at more realistic scenarios with dissipation included ($\\Gamma_0\\neq 0$) and examine the effect of the same on those observations. For a two-atom chain, we discover that the behavior changes drastically depending on how the relative detuning between the atomic frequencies compares to this decay rate. Fig. \\ref{f7} shows the plots for $\\abs{t}^2$ vs. $\\abs{\\frac{\\overline{\\Delta}}{\\Gamma}}$ for a good-quality waveguide with a weak dissipative channel ($\\Gamma_0=0.1\\Gamma$), for varying values of $\\abs{\\omega_1-\\omega_2}$. It is observed that for sufficiently small values of the latter, the transmission peak almost disappears, whereas for large values, the height of the peak approaches unity. In other words, by adjusting the relative frequency detuning between the emitters, one can achieve either high opacity or high transparency around $\\omega_k=\\frac{1}{2}(\\omega_1+\\omega_2)$. For perfectly matched up atomic frequencies, i.e. $\\omega_1=\\omega_2$, one observes a diametrically opposite behavior as the two roots coincide - the central peak is replaced by a trough. This is a Dicke-type superradiant effect - for negligible decay, the transmission profile is a vertically inverted Lorentzian with a half-width of $2\\Gamma$.\n\n\\begin{figure}[!t]\n \\captionsetup{justification=raggedright,singlelinecheck=false}\n\\centering\n\\includegraphics[scale=0.70]{f7-eps-converted-to.pdf} \n\\caption{\\small  {Effect of dissipation on the transmission of a two-atom system. If the dissipative channel is weak compared to the waveguide channel, the profile closely resembles the dissipation-free spectrum, except when the frequency mismatch between the atoms is smaller than or comparable to the rate of dissipation.}} \\label{f7} \n\\end{figure}\n\nOne can analytically understand this behavior by considering two specific regimes, (i) $s\\ll 2\\Gamma_0$ and (ii) $s\\gg 2\\Gamma_0$. At $\\overline{\\Delta}=0$, one obtains\n\n\\begin{align}\nt=\\frac{(\\frac{s}{2})^2+\\Gamma_0^2}{(\\frac{s}{2})^2+\\Gamma_0(\\Gamma_0+2\\Gamma)}\\hspace{1mm}.\n\\end{align}\n\nFor small relative detuning between the atoms, i.e. $s\\ll 2\\Gamma_0$, the approximate form is given as $\\abs{t}^2\\approx \\frac{\\Gamma_0^2}{4\\Gamma^2}$, which is vanishingly dimunitive, as long as the decay rate is much less than $\\Gamma$. In the opposite scenario when $s\\gg 2\\Gamma_0$, we get $\\abs{t}^2\\approx 1-\\mathcal{O}(\\frac{8\\Gamma\\Gamma_0}{\\delta^2})$. Thus, a fairly high degree of transparency can be achieved by specifically working with a large relative detuning $\\abs{\\omega_1-\\omega_2}$.\n\nFor a system of even number of emitters, in which half of them have detuning $+\\Delta$ whereas the other half have detuning $-\\Delta$, the transmission goes as\n\n\\begin{align}\nt=\\frac{\\Delta^2+\\Gamma_0^2}{\\Delta^2+\\Gamma_0(\\Gamma_0+N\\Gamma)}\\hspace{1mm}.\n\\end{align}\n\nFor $\\Delta\\ll\\Gamma_0$, $\\abs{t}^2\\approx (\\frac{\\Gamma_0}{N\\Gamma})^2$, which testifies to highly reflecting behavior. However, when $\\Delta\\gg \\Gamma_0$, we have $\\abs{t}^2\\approx 1-\\mathcal{O}(\\frac{4N\\Gamma\\Gamma_0}{\\Delta^2})$, implying near-transparency. The situation here is reminiscent of EIT where perfect transparency emerges in the absence of dissipative transitions \\cite{harris1991,eitreview}.\n\nSimilarly, when there are odd number of emitters, with $(N-1)\/2$ emitters each having a detuning of $+\\Delta$ and $(N-1)\/2$ other emitters each detuned by $-\\Delta$, one has, for the transmission coefficient\n\n\\begin{align}\nt=\\frac{1}{1+\\frac{(N-1)\\Gamma\\Gamma_0}{\\Delta^2+\\Gamma_0^2}+\\frac{i\\Gamma}{\\omega_k-\\omega_0+i\\Gamma_0}}\\hspace{1mm},\n\\end{align}\n\nwhere $\\omega_0$ is the frequency of the remaining atom. When $\\Delta$ is large compared to $\\Gamma_0$, one can discern a re-emergence of single-atom behavior.\n\n\\section{Conclusions} \\label{s7}\n\nTo summarize, we have thrown light on new possibilities that emerge in relation to the collective effects of a chain of atoms side-coupled to a waveguide when the interemitter separation is fixed to satisfy $kL=n\\pi$, where $n$ is an integer. For a chain of $N$ atoms, we have demonstrated the emergence of new Fano minima (transparency points) in the reflection spectrum for negligible dissipation. When $N$ is even, we have seen how transparency can be generated by assigning equal and opposite detunings to the atoms in pairs, while for odd $N$, we have highlighted the possibility of reproducing single-atom behavior through a similar assignment, so that the odd one out completely determines the emission spectrum. A system of identically detuned emitters, on the other hand, demonstrates superradiant behavior. We have also shown that the optical system demonstrates reciprocal behavior with respect to both transmission and reflection. In general, the system turns out to be insensitive to the order in which the atoms are arranged. Finally, it has been demonstrated, both analytically and graphically, that when dissipation into nonwaveguide modes cannot be neglected, one can still produce highly transparent behavior by implementing a considerable disparity in the atomic transition frequencies. For a small mismatch in the frequencies, one, however, observes predominantly opaque behavior in its place.\\\\\n\n\n\\begin{acknowledgments}\n\nD. M. is supported by the Herman F. Heep and Minnie Belle Heep Texas A\\&M University Endowed Fund held\/administered by the Texas A\\&M Foundation.\n\n\n\n\\end{acknowledgments}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section*{I.Introduction}\n\\baselineskip 0.32in\n\\indent Gravity in 1+2 dimensional spacetime has been a\npopular subject of\ndiscussion from a decade ago\\cite{s1}-\\cite{s6}. Though some\nof the models\nare toy  models, the studies may shed light on the understanding\nof not\nonly quasi-(1+2) dimensional physics (e.g. QHE and high $T_{c}$),\nbut also\nthe realistic 1+3 dimensional gravity. In any gauge theory in\nodd dimensions,\nthere exists  a special term, i.e. the Chern-Simons term,\nwhich can be\nincorporated into the model Lagrangian. The concept of\ngravitational anyons\nis a simple non-Abelian generalization of the U(1)\nChern-Simons theory to\nnon-compact gauge group\\cite{s1}\\cite{s2}. Using his sloution\nto the\nlinearized field equations, Deser studied the mass and spin of the\ngravitational anyons\\cite{s4}. The conclusion states that the\ngravitational\nand inertial quantities are not equal to each other in general and\nthus\nthe equivalence principle is violated. Hence, there exist much\ndifference\nbetween gravitational anyons and 1+3 Einstein gravity.\\\\\n\\indent It seems to us that in order to understand the difference,\nwe should\nhave a well-defined definition of gravitational conservative\nquantities, or\nin other words, we should have generally covariant conservation\nlaws. In\nour previous work\\cite{s7}, we have obtained a generally covariant\nconservation law of angular-momentum for gravitational anyons.\nAs suggested\nin\\cite{s7}, the present paper is to study the generally covariant\nconservation law of energy-momentum in the approach proposed\nin\\cite{s8}.\nThe paper is arranged as follows. In section II, we give a general\ndescription of the scheme for establishing generally covariant\nconservation\nlaws in general relativity. In section III, we use the general\ndisplacement\ntransform and the scheme to obtain a generally covariant\nconservation law\nof energy-momentum. In this section, we will use a first order\nLagrangian\ninstead of the original one which is in second order. In section\nIV, we\ncalculate the total energy-momentum for Deser's,\nCle$\\acute{e}$ment's\nsolutions, and a solution in the vanishing Chern-Simons coupling\nlimit.\nThe last section is devoted to some remarks and discussions.\\\\\n\\section*{II. General Scheme for Conservation Laws in General\nRelativity}\n\\indent As in 1+3 Einstein gravity, conservation laws are also\nthe consequence\nof the invariance of the action corresponding to some transforms.\nIn order\nto study the covariant energy-momentum of more complicated systems,\nit is\nbenifecial to discuss conservation laws by Noether theorem in general.\nSuppose that the spacetime is of dimension $D=1+d$ and the Lagrangian\nis in the first order formalism, i.e.\n\\begin{equation}\nI=\\int_{G}{\\cal L}(\\phi^{A}, \\partial_{\\mu}\\phi^{A})d^{D}x\n\\end{equation}\nwhere $\\phi^{A}$ denotes the generic fields. If the action is invariant\nunder the infinitesimal transforms\n\\begin{equation}\nx^{\\prime\\mu}=x^{\\mu}+\\delta x^{\\mu}\\,\\,\\,\\,\\,\\,\\,\\,  \\phi^{\\prime A}\n(x^{\\prime})=\\phi^{A}(x)+\\delta\\phi^{A}(x)\n\\end{equation}\n (it is not required that $\\delta\\phi^{A}_{\\mid\\partial G}=0$), then following relation\nholds\\cite{s8}-\n\\cite{s10}.\n\\begin{equation}\n\\partial_{\\mu}({\\cal L}\\delta x^{\\mu}+\\frac{\\partial{\\cal L}}{\\partial\\p_{\\mu}\\phi^{A}}\n\\delta_{0}\\phi^{A}\n)+[{\\cal L}]_{\\phi^{A}}\\delta_{0}\\phi^{A}=0\n\\end{equation}\nwhere\n\\begin{equation}\n[{\\cal L}]_{\\phi^{A}}=\\frac{\\partial{\\cal L}}{\\partial\\phi^{A}}-\\partial_{\\mu}\n\\frac{\\partial{\\cal L}}{\\partial\\p_{\\mu}\\phi^{A}}\n\\end{equation}\nand $\\delta_{0}\\phi^{A}$ is the Lie variation of $\\phi^{A}$\n\\begin{equation}\n\\delta_{0}\\phi^{A}=\\phi^{\\prime A}(x)-\\phi^{A}(x)=\n\\delta\\phi^{A}(x)-\\partial_{\\mu}\n\\phi^{A}\\delta x^{\\mu}\n\\end{equation}\n\\indent If ${\\cal L}$ is the total Lagrangian of the system,\nthe field equations of\n$\\phi^{A}$ is just $[{\\cal L}]_{\\phi^{A}}=0$. Hence from eq.(3), we\ncan obtain the\nconservation equation corresponding to transform eq.(2)\n\\begin{equation}\n\\partial_{\\mu}({\\cal L}\\delta x^{\\mu}+\\frac{\\partial{\\cal L}}{\\partial\\p_{\\mu}\\phi^{A}}\n\\delta_{0}\n\\phi^{A})=0\n\\end{equation}\nIt is important to recognize that if ${\\cal L}$ is not the total\nLagrangian\n, e.g. the gravitational part ${\\cal L}_{g}$, then so long as the\naction of\n${\\cal L}_{g}$ remains invariant under transform eq.(2), eq.(3) is\nstill valid\nyet eq.(6) is no longer admissible because of\n$[{\\cal L}_{g}]_{\\phi^{A}}\\not=0$.\\\\\n\\indent Suppose that $\\phi^{A}$ denotes the Riemann tensors\n$\\phi^{A}_{\\mu}$\nand Riemann scalars $\\psi^{A}$ (for gravitational anyons,\nthey are dreibein\n$e^{a}_{\\mu}$, SO(1,2) connection $\\omega^{a}_{\\mu}$, the Lagrangian\nmultiplier $\\lambda^{a}_{\\mu}$ and the matter field $\\psi^{A}$).\nEq.(3) reads\n(suppose that ${\\cal L}_{g}$ does not contain $\\psi^{A}$)\n\\begin{equation}\n\\partial_{\\mu}({\\cal L}_{g}\\delta x^{\\mu}+\\frac{\\partial{\\cal L}_{g}}{\\partial\\p_{\\mu}\\phi^{A}_{\\nu}}\n\\delta_{0}\n\\phi^{A}_{\\nu})+[{\\cal L}_{g}]_{\\phi^{A}_{\\mu}}\\delta_{0}\\phi^{A}_{\\mu}=0\n\\end{equation}\nUnder transforms eq.(2), the Lie variations are\n\\begin{equation}\n\\delta_{0}\\phi^{A}_{\\nu}=-\\delta x^{\\alpha}_{,\\nu}\\phi^{A}_{\\alpha}\n-\\phi^{A}_{\\nu,\\alpha}\\delta x^{\\alpha}\n\\end{equation}\nwhere the dot \",\" denotes partial derivative. So eq.(7) reads\n\\begin{equation}\n\\partial_{\\mu}[{\\cal L}_{g}\\delta x^{\\mu}-\\frac{\\partial{\\cal L}_{g}}{\\partial\\p_{\\mu}\n\\phi^{A}_{\\lambda}}\n(\\delta x^{\\nu}_{,\\lambda}\\phi^{A}_{\\nu}+\\phi^{A}_{\\lambda ,\\nu}\n\\delta x^{\\nu})]\n-[{\\cal L}_{g}]_{\\phi^{A}_{\\lambda}}(\\delta x^{\\nu}_{,\\lambda}\\phi^{A}_{\\nu}+\n\\phi^{A}_{\\lambda ,\\nu}\\delta x^{\\nu})=0\n\\end{equation}\nComparing the coefficients of $\\delta x^{\\nu},\n\\delta x^{\\nu}_{,\\lambda}\n$ and $\\delta x^{\\nu}_{,\\mu\\lambda}$, we may obtain an identity\n\\begin{equation}\n\\partial_{\\lambda}([{\\cal L}_{g}]_{\\phi^{A}_{\\lambda}}\\phi^{A}_{\\nu})=\n[{\\cal L}_{g}]_{\\phi^{A}\n_{\\lambda}}\\phi^{A}_{\\lambda,\\nu}\n\\end{equation}\nThen eq.(9) can be written as\n\\begin{equation}\n\\partial_{\\mu}[{\\cal L}_{g}\\delta x^{\\mu}-\\frac{\\partial{\\cal L}_{g}}{\\partial\\p_{\\mu}\n\\phi^{A}_{\\lambda}}\n(\\delta x^{\\nu}_{,\\lambda}\\phi^{A}_{\\nu}+\\phi^{A}_{\\lambda ,\\nu}\n\\delta\nx^{\\nu})-[{\\cal L}_{g}]_{\\phi^{A}_{\\mu}}\\phi^{A}_{\\nu}\\delta x^{\\nu}]=0\n\\end{equation}\nor\n\\begin{equation}\n\\partial_{\\mu}[({\\cal L}_{g}\\delta^{\\mu}_{\\nu}-\\frac{\\partial{\\cal L}_{g}}{\\partial\\p_{\\mu}\n\\phi^{A}_{\\lambda}}\n\\phi^{A}_{\\lambda,\\nu}-[{\\cal L}_{g}]_{\\phi^{A}_{\\mu}}\\phi^{A}_{\\nu})\n\\delta x^{\\nu}\n-\\frac{\\partial{\\cal L}_{g}}{\\partial\\phi^{A}_{\\lambda,\\mu}}\\phi^{A}_{\\nu}\\delta\nx^{\\nu}_{,\\lambda}]=0\n\\end{equation}\nBy definition, we introduce\n\\begin{equation}\n\\tilde{I}^{\\mu}_{\\nu}=-({\\cal L}_{g}\\delta^{\\mu}_{\\nu}-\n\\frac{\\partial{\\cal L}_{g}}{\\partial\\p_{\\mu}\n\\phi^{A}_{\\lambda}}\\phi^{A}_{\\lambda,\\nu}-\n[{\\cal L}_{g}]_{\\phi^{A}_{\\mu}}\\phi^{A}_{\\nu})\n\\end{equation}\n\\begin{equation}\n\\tilde{Z}^{\\lambda\\mu}_{\\nu}=\\frac{\\partial{\\cal L}_{g}}{\\partial\\phi^{A}_{\\lambda,\\mu}}\n\\phi^{A}_{\\nu}\n\\end{equation}\nThen eq.(12) gives\n\\begin{equation}\n\\partial_{\\mu}(\\tilde{I}^{\\mu}_{\\nu}\\delta x^{\\nu}+\n\\tilde{Z}^{\\lambda\\mu}_{\\nu}\n\\delta x^{\\nu}_{,\\lambda})=0\n\\end{equation}\nSo by comparing the coefficients of $\\delta x^{\\nu},\n\\delta x^{\\nu}_{,\\mu}$\nand $\\delta x^{\\nu}_{,\\mu\\lambda}$, we have the following from eq.(15)\n\\begin{equation}\n\\partial_{\\mu}\\tilde{I}^{\\mu}_{\\nu}=0\n\\end{equation}\n\\begin{equation}\n\\tilde{I}^{\\lambda}_{\\nu}=-\\partial_{\\mu}\\tilde{Z}^{\\lambda\\mu}_{\\nu}\n\\,\\,\\,\\,\\,\\,\\,\\,\n\\tilde{Z}^{\\mu\\lambda}_{\\nu}=-\\tilde{Z}^{\\lambda\\mu}_{\\nu}\n\\end{equation}\nEq.(16)-(17) are fundamental to the establishing of conservation law of\nenergy-momentum.\\\\\n\\section*{III. Conservation Law of Energy-momentum for Gravitational\nAnyons}\n{\\it 3.1. General Displacement Transform}\\\\\n\\indent In 1+3 Einstein gravity, a generally covariant conservation law\nof energy-momentum was obtained by means of whant we usually call\nthe {\\it general displacement transform}\\cite{s8}\n\\begin{equation}\n\\delta x^{\\mu}=e^{\\mu}_{a}\\epsilon^{a}\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\n(\\epsilon^{a}\n=const.)\n\\end{equation}\nIt really represents an infinitesiaml displacement while\n\\begin{equation}\n\\delta x^{\\mu}=b^{\\mu}=const.\n\\end{equation}\ndoes not because $x^{\\mu}$ can be any coordinates. For instance,\nit can be\nspherical coordinates, in which case, the resulting conservation\nlaw, if\nexists, should be that of angular-momentum other than energy-momentum.\nUsing\nthe invariance of the action with respect to eq.(18) and Einstein\nequations\n, the following general covariant conservation law of energy-momentum\nis\nobtained\n\\begin{equation}\n\\nabla_{\\mu}(T^{\\mu}_{a}+t^{\\mu}_{a})=0\n\\end{equation}\nand it was shown that there exist superpotentials $V^{\\mu\\nu}_{a}$\n\\begin{equation}\nT^{\\mu}_{a}+t^{\\mu}_{a}=\\nabla_{\\nu}V^{\\mu\\nu}_{a}\n\\end{equation}\n\\begin{equation}\nV^{\\mu\\nu}_{a}=\\frac{c^{4}}{8\\pi G}[e^{\\mu}_{b}e^{\\nu}_{c}\n\\omega_{a}\\,^{bc}\n+(e^{\\mu}_{a}e^{\\nu}_{b}-e^{\\mu}_{b}e^{\\nu}_{a})\\omega^{b}]\n\\end{equation}\nThis definition of energy-momentum has the following main\npropertities:\\\\\n1). It is a covariant definition with respect to general coordinate\ntransforms. But the energy-momentum tensor is not covariant under\nlocal\nLorentz transforms, this is reasonable because of the equivalence\nprinciple.\\\\\n2). For closed system, the total energy-momentum does not depend\non the\nchoice of Riemann coordinates and transforms in the covariant way\n\\begin{equation}\nP_{a}^{\\prime}=L_{a}\\,^{b}P_{b}\n\\end{equation}\nunder local Lorentz transform $\\Lambda^{a}\\,_{b}$ which is constant\n$L^{a}\\,\n_{b}$ at spatial infinity.\\\\\n3). For a closed system with static mass center, the total\nenergy-momentum is\n$P_{a}=(Mc,0,0,0)$.\\\\\n4). For a rather concentrated matter system, the gravitational\nenergy\nradiation is\\cite{s11}\n\\begin{equation}\n-\\frac{\\partial E}{\\partial t}=\\frac{G}{45c^{5}}(\\stackrel{\\cdots}{D})^{2}\n\\end{equation}\n5). For Bondi's plane wave, the energy current is\\cite{s11}\n\\begin{equation}\nt^{\\mu}_{0}=(\\frac{1}{4\\pi}\\beta^{\\prime 2},\n\\frac{1}{4\\pi}\\beta^{\\prime 2}, 0,0)\n\\end{equation}\n6). For the solution of gravitational solitons, we can obtain\nfinite energy\nwhile the Landau-Lifshitz definition leads to infinite energy\n\\cite{s12}.\\\\\n7). In Ashtekar's complex formalism of general relativity,\nthe energy-momentum and angular-momentum constitute a 3-Poincare\nalgebra\nand the energy coincides with the ADM energy\\cite{s10}.\\\\\nWith these foundations, we next use 1+2 dimensional transform\neq.(18) to\nobtain conservative energy-momentum for gravitational anyons.\\\\\n{\\it 3.2 The Energy-momentum for Gravitational Anyons}\\\\\n\\indent We take the Lagrangian for gravitational anyons to be\nin the\nfirst order ($\\omega^{a}_{\\mu}=\\frac{1}{2}\\epsilon^{abc}\n\\omega_{\\mu bc},\n\\omega_{\\mu bc}$ is the SO(1,2) connection.)\n\\begin{equation}\n{\\cal L}={\\cal L}_{g}+{\\cal L}_{m}\n\\end{equation}\nwhere\n\\begin{equation}\n{\\cal L}_{g}=-\\frac{1}{\\kappa}\\epsilon^{\\mu\\nu\\alpha}\\omega^{a}_{\\nu}\\partial_{\\mu}\ne_{\\alpha a}+\\frac{1}{2\\kappa}\\epsilon^{\\mu\\nu\\alpha}e_{\\alpha a}\n\\epsilon^{abc}\\omega_{\\mu b}\\omega_{\\nu c}\n+{\\cal L}_{c-s}+\\epsilon^{\\mu\\nu\\alpha}\\lambda^{a}_{\\mu}(\\partial_{\\nu}\ne_{\\alpha a}\n+\\epsilon_{abc}\\omega^{b}_{\\nu}e^{c}_{\\alpha})\n\\end{equation}\n\\begin{equation}\n{\\cal L}_{c-s}=\\frac{1}{2\\kappa\\mu}\\epsilon^{\\mu\\nu\\alpha}(\\omega_{\\mu a}\n\\partial_{\\nu}\\omega^{a}_{\\alpha}+\\frac{1}{3}\\epsilon_{abc}\\omega^{a}_{\\mu}\n\\omega^{b}_{\\nu}\\omega^{c}_{\\alpha})\n\\end{equation}\nand ${\\cal L}_{m}$ denotes the matter part. The field equations for\n$e^{a}_{\\mu},\n\\omega^{a}_{\\mu}$ and $\\lambda^{a}_{\\mu}$ are $[{\\cal L}]_{e^{a}_{\\mu}}=0,\n[{\\cal L}]_{\\omega^{a}_{\\mu}}=0 $ and $[{\\cal L}]_{\\lambda^{a}_{\\mu}}=0$, i.e.\n\\begin{equation}\n\\frac{1}{2\\kappa}\\epsilon^{\\mu\\nu\\alpha}\\epsilon^{abc}\\omega_{\\mu b}\n\\omega_{\\nu c}+\\epsilon^{\\mu\\nu\\alpha}\\lambda^{b}_{\\mu}\\epsilon_{abc}\n\\omega^{c}_{\\nu}+\\frac{1}{\\kappa}\\epsilon^{\\mu\\nu\\alpha}\\partial_{\\mu}\n\\omega^{a}_{\\nu}+\\epsilon^{\\mu\\nu\\alpha}\\partial_{\\mu}\\lambda_{\\nu}^{a}=\n-[{\\cal L}_{m}]_{e^{a}_{\\alpha}}\n\\end{equation}\n\\begin{equation}\n\\epsilon^{\\mu\\nu\\alpha}(\\partial_{\\nu}e_{\\alpha a}+\\epsilon_{abc}\n\\omega^{b}_{\\nu}\ne^{c}_{\\alpha})=0\n\\end{equation}\n$$\n\\frac{1}{\\kappa}\\epsilon^{\\nu\\mu\\alpha}(\\partial_{\\mu}e^{a}_{\\alpha}+\n\\epsilon^{a}\\,\n_{bc}\\omega^{b}_{\\mu}e^{c}_{\\alpha})+\\frac{1}{\\kappa\\mu}\n\\epsilon^{\\nu\\mu\\alpha}\n(\\partial_{\\mu}\\omega^{a}_{\\alpha}$$\n\\begin{equation}\n+\\frac{1}{2}\\epsilon^{abc}\\omega_{\\mu b}\\omega_{\\alpha c})+\n\\epsilon^{\\nu\\mu\n\\alpha}\\lambda_{b \\mu}\\epsilon^{abc}e_{\\alpha c}\n=-[{\\cal L}_{m}]_{\\omega^{a}_{\\nu}}\n\\end{equation}\nUsing eq.(30), eq.(31) can be rewritten as\n\\begin{equation}\n\\frac{1}{\\kappa\\mu}\\epsilon^{\\nu\\mu\\alpha}\n(\\partial_{\\mu}\\omega^{a}_{\\alpha}\n+\\frac{1}{2}\\epsilon^{abc}\\omega_{\\mu b}\\omega_{\\alpha c})+\n\\epsilon^{\\nu\\mu\n\\alpha}\\lambda_{b \\mu}\\epsilon^{abc}e_{\\alpha c}\n=-[{\\cal L}_{m}]_{\\omega^{a}_{\\nu}}\n\\end{equation}\nThese equations are the same as those given in\\cite{s13}.\\\\\n\\indent From eq.(14)\n$$\\tilde{Z}^{\\lambda\\mu}_{\\nu}=\\frac{\\partial{\\cal L}_{g}}{\\partial e^{a}\n_{\\lambda,\\mu}}e^{a}_{\\nu}\n+\\frac{\\partial{\\cal L}_{g}}{\\partial\\omega^{a}_{\\lambda,\\mu}}\\omega^{a}_{\\nu}\n+\\frac{\\partial{\\cal L}_{g}}{\\partial\\lambda^{a}_{\\lambda,\\mu}}\\lambda^{a}_{\\nu}$$\n\\begin{equation}\n=-\\frac{1}{\\kappa}\\epsilon^{\\mu\\alpha\\lambda}\\omega_{\\alpha a}\ne^{a}_{\\nu}\n+\\frac{1}{2\\kappa\\mu}\\epsilon^{\\alpha\\mu\\lambda}\\omega_{\\alpha a}\n\\omega^{a}_{\\nu}+\\epsilon^{\\alpha\\mu\\lambda}\\lambda_{\\alpha a}\ne^{a}_{\\nu}\n\\end{equation}\nFor transform eq.(18), eq.(15) implies\n\\begin{equation}\n\\partial_{\\mu}(\\tilde{I}^{\\mu}_{\\nu}e^{\\nu}_{a}+\\tilde{Z}^{\\lambda\\mu}_{\\nu}\ne^{\\nu}_{a,\\lambda})=0\n\\end{equation}\nDefine\n\\begin{equation}\n\\tilde{I}^{\\mu}_{a}=-\\partial_{\\lambda}\\tilde{Z}^{\\lambda\\mu}_{a},\\,\\,\\,\\,\n\\,\\,\\,\\,\\,\n\\tilde{Z}^{\\lambda\\mu}_{a}=\\tilde{Z}^{\\lambda\\mu}_{\\nu}e^{\\nu}_{a}\n\\end{equation}\nwe then have\n\\begin{equation}\n\\partial_{\\mu}\\tilde{I}^{\\mu}_{a}=0\n\\end{equation}\nSince $[{\\cal L}_{g}]_{e^{a}_{\\mu}}=-[{\\cal L}_{m}]_{e^{a}_{\\mu}}$, and\n$T^{\\mu}_{a}\n=-\\frac{1}{e}[{\\cal L}_{m}]_{e^{a}_{\\mu}}$, we have from eq.(13)\n\\begin{equation}\n\\tilde{I}^{\\mu}_{\\nu}=-({\\cal L}_{g}\\delta^{\\mu}_{\\nu}-\n\\frac{\\partial{\\cal L}_{g}}{\\partial e^{a}_{\\lambda,\n\\mu}}e^{a}_{\\lambda,\\nu}-\\frac{\\partial{\\cal L}_{g}}{\\partial \\omega^{a}_{\\lambda,\n\\mu}}\\omega^{a}_{\\lambda,\\nu}\n-[{\\cal L}_{g}]_{\\omega^{a}_{\\mu}}\\omega^{a}_{\\nu}-\n[{\\cal L}_{g}]_{\\lambda^{a}_{\\mu}}\n\\lambda^{a}_{\\nu})+eT^{\\mu}_{a}e^{a}_{\\nu}\n\\end{equation}\nDefine $t^{\\mu}_{a}$ by\n\\begin{equation}\n\\tilde{I}^{\\mu}_{\\nu}e^{\\nu}_{a}+\\tilde{Z}^{\\lambda\\mu}_{\\nu}\ne^{\\nu}_{a,\\lambda}=e(T^{\\mu}_{a}+t^{\\mu}_{a})\n\\end{equation}\nThen we have\n\\begin{equation}\ne(T^{\\mu}_{a}+t^{\\mu}_{a})=e\\nabla_{\\lambda}Z^{\\lambda\\mu}_{a}\n\\end{equation}\nwhere $\\tilde{Z}^{\\lambda\\mu}_{a}=eZ^{\\lambda\\mu}_{a}$. Eq.(39) is the\ndesired general covariant conservation law of energy-momentum for\ngravitational anyons. The total energy-momentum is\n\\begin{equation}\nP_{a}=\\int e(T^{0}_{a}+t^{0}_{a}) d^{2}x=\\int \\partial_{i}\\tilde{Z}^{i0}_{a}\nd^{2}x\n\\end{equation}\n{\\it 3.3 The iso(1,2) Algebra}\\\\\n\\indent The pure Einstein case is restored by setting\n$\\mu\\rightarrow\\infty$\nand ${\\cal L}_{m}=0$. In this limit, we have $\\lambda^{a}_{\\mu}=0$\nand the\nsuperpotential is simply\n\\begin{equation}\n\\tilde{Z}^{\\mu\\nu}_{a}=-\\frac{1}{\\kappa}\\epsilon^{\\mu\\nu\\alpha}\n\\omega_{\\alpha\na}\n\\end{equation} and the total energy-momentum is\n\\begin{equation}\nP_{a}=\\frac{1}{\\kappa}\\oint_{\\partial\\Sigma}\\omega_{a}\n\\end{equation}\nwhere $\\partial\\Sigma$ is the spatial infinity which is 1 dimensional.\nFrom the\nangular-momentum\\cite{s7}\n\\begin{equation}\nJ_{a}=-\\frac{1}{\\kappa}\\oint_{\\partial\\Sigma}e_{a}=\\frac{1}{\\kappa}\\int\n\\epsilon^{ij}\\epsilon_{abc}\\omega^{b}_{i}e^{c}_{j} d^{2}x\n\\end{equation} and the Poisson brackets given in \\cite{s5}\n$$\\{\\omega^{a}_{i}(x),e^{b}_{j}(y)\\}=\\kappa\\epsilon_{ij}\\eta^{ab}\n\\delta^{2}(x-y)\n$$\n\\begin{equation}\n\\{\\omega^{a}_{i}(x),\\omega^{b}_{j}(y)\\}=\\{e^{a}_{i}(x),e^{b}_{j}\n(y)\\}=0\n\\end{equation}\nwe have\n\\begin{equation}\n\\{J_{a},J_{b}\\}=-\\frac{1}{\\kappa^2}\\{\\oint e_{a}(x), \\int\n\\epsilon^{ij}\n\\epsilon_{bcd}\\omega^{c}_{i}e^{d}_{j} d^{2}y\\}=\\epsilon_{abc}J^{c}\n\\end{equation}\n\\begin{equation}\n\\{P_{a}, P_{b}\\}=0\n\\end{equation}\n\\begin{equation}\n\\{J_{a}, P_{b}\\}=\\{\\frac{1}{\\kappa}\\int \\epsilon^{ij}\\epsilon_{acd}\n\\omega^{c}_{i}e^{d}_{j}d^{2}x, \\frac{1}{\\kappa}\\oint\\omega_{b}(y)\\}\n=\\epsilon_{abc}P^{c}\n\\end{equation}\nThus the ${\\it iso}(1,2)$ algebra can be restored.\n\\section*{IV. Examples}\n\\indent We now consider the special case that $[{\\cal L}_{m}]_{\\omega^{a}\n_{\\mu}}=0$. Using the identities in 3-dim Riemann geometry\n$$\nR_{\\alpha\\beta\\gamma\\delta}=g_{\\alpha\\gamma}\\tilde{R}_{\\beta\\delta}+\ng_{\\beta\\delta}\\tilde{R}_{\\alpha\\gamma}-g_{\\alpha\\delta}\\tilde{R}\n_{\\beta\\gamma}-g_{\\beta\\gamma}\\tilde{R}_{\\alpha\\delta}$$\n$$\\tilde{R}_{\\mu\\nu}=R_{\\mu\\nu}-\\frac{1}{4}g_{\\mu\\nu}R\n\\,\\,\\,\\,\\,\\,\\,\\, R^{\\alpha\\beta}\\,_{\\gamma\\delta}=\n-\\epsilon^{\\alpha\\beta\\mu}\n\\epsilon_{\\gamma\\delta\\nu}G^{\\nu}_{\\mu}\\,\\,\\,\\,\\,\\,\\, G^{\\mu\\nu}=\nR^{\\mu\\nu}\n-\\frac{1}{2}g^{\\mu\\nu}R$$\nwe have\n\\begin{equation}\n-\\frac{1}{\\kappa\\mu}\\epsilon^{\\mu\\alpha\\beta}(\\partial_{\\alpha}\n\\omega_{\\beta a}\n+\\frac{1}{2}\\epsilon_{abc}\\omega^{b}_{\\alpha}\\omega^{c}_{\\beta})=\n\\frac{1}{\\kappa\\mu}eG^{\\mu}_{a}\n\\end{equation}\nSo\n\\begin{equation}\n\\lambda^{a}_{\\mu}=-\\frac{1}{\\kappa\\mu}\\tilde{R}^{a}_{\\mu}\n\\end{equation}\nSubstitute into eq.(29), we have\n\\begin{equation}\nG^{\\mu}_{a}+\\frac{1}{\\mu}C^{\\mu}_{a}=-\\kappa T^{\\mu}_{a}\n\\end{equation}\nwhere $C^{\\mu}_{a}$ is the Cotton tensor. For Deser's solution\n\\cite{s4}.\n$$e^{0}_{0}\\simeq 1,\\enspace\ne^{1}_{1}=e^{2}_{2}\\simeq \\sqrt{\\frac{m\\kappa^{2}}{\\pi}}\nln^{\\frac{1}{2}}r$$\n $$e^{0}_{2}\\simeq\n-\\frac{\\kappa^{2}}{\\mu}\\frac{(m+\\mu\\sigma)}{2\\pi}\\frac{x}\n {r^{2}},\\enspace e^{0}_{1}\\simeq\n\\frac{\\kappa^{2}(m+\\mu\\sigma)}{2\\pi\\mu}\n\\frac{y}{r^{2}}$$\n\\begin{equation}\ne^{1}_{2}\\simeq\n-\\frac{\\kappa^{4}(m+\\mu\\sigma)^{2}}{4\\pi^{2}\\mu^{2}\\sqrt{\\frac\n{m\\kappa^{2}}{\\pi}}}\\frac{xy}{r^{4}ln^{1\/2}r}\n\\end{equation}\n we can obtain the asymptotical behaviour of the\nspin connection \\begin{equation}\n\\omega_{\\mu ab}\\simeq \\frac{1}{rf(r)}\n\\end{equation}\n where$f(r)$ represents some monototically\nincreasing functions of $r$. Thus we  have\n\\begin{equation}\n\\oint_{\\partial\\Sigma}\\omega_{\\mu ab}dx^{\\mu}=0\n\\end{equation}\nThus the total energy-momentum vanishes.\nFor Cl$\\acute{e}$ment's \\cite{s14}\n self-dual exact solution\n\\begin{equation}\n ds^{2}=A^{-1}[dt-(\\omega_{0}+A)d\\theta]^{2}-dr^{2\n}-Ad\\theta^{2} \\end{equation}\n$$A=a+ce^{-\\mu r}$$\n where $a,c$ are constants and $\\mu$ should be\npositive since $r\\in (0,+\\infty)\n$. In terms of rectangular coordinates, it takes\nthe following form\n $$ds^{2}=A^{-1}dt^{2}-2(\\omega_{0}A^{-1}+1)(\\frac\n{x}{r^{2}}dtdy-\\frac{y} {r^{2}}dtdx)$$\n $$+\\{[A^{-1}(\\omega_{0}+A)^{2}-A]\\frac{x^{2}}{r^{\n4}}-1\\}dy^{2}+\\{[A^{-1}\n(\\omega_{0}+A)^{2}-A]\\frac{y^{2}}{r^{4}}-1\\}dx^{2}$$\n\\begin{equation}\n-[A^{-1}(\\omega_{0}+A)^{2}-A]\\frac{2xy}{r^{4}}dydx\n\\end{equation}\n We  obtain the following\n asymptotical dreibein \\cite{s7}\n $$e^{0}_{0}=\\frac{1}{\\sqrt{a}}, \\enspace\ne^{0}_{1}=\\sqrt{a}(1+\\frac{\\omega\n_{0}}{a})\\frac{y}{r^{2}}$$\n\\begin{equation}\n e^{0}_{2}=-\\sqrt{a}(1+\\frac{\\omega_{0}}{a})\\frac{\nx}{r^{2}}, \\enspace e^{1}_{1}\n=1+\\frac{ay^{2}}{2r^{4}}\n\\end{equation}\n $$ e^{1}_{2}=\\frac{axy}{r^{4}}, \\enspace\ne^{2}_{2}=1+\\frac{ax^{2}}{2r^{4}}$$\nHence it can be shown that\n$$\\lim_{r\\rightarrow \\infty}r\\omega_{\\mu ab}=0$$\nThus\n$$\\int_{\\partial\\Sigma}\\omega_{\\mu ab}dx^{\\mu}=0$$\nSo the total energy vanishes also. In the limit,\n$\\mu\\rightarrow\\infty$, eq.(50) has a solution with\ndreibein and spin connection\\cite{s15}\n$$e^{0}=dt+\\frac{\\kappa J}{2\\pi r^{2}}{\\bf r}\\times d{\\bf r}\n\\,\\,\\,\\,\\,\\,\\,\n{\\bf e}=(1-\\frac{\\kappa m}{2\\pi})d{\\bf r}+\\frac{\\kappa m}\n{2\\pi r^{2}}{\\bf r}\n({\\bf r}\\cdot d{\\bf r})$$\n\\begin{equation}\n\\omega^{0}=\\frac{\\kappa m}{2\\pi r^{2}}{\\bf r}\\times d{\\bf r}\n\\,\\,\\,\\,\\,\\,\\,\n\\omega^{i}=0\n\\end{equation}\nwe have\n\\begin{equation}\nP_{a}=(m,0,0)\n\\end{equation}\nwhich is the same as in\\cite{s16}.\n\\section*{V. Discussions}\n\\indent As an end, we make some discussions. First, general\ncovariance is a\nfundamental demand for conservation laws in general relativity.\nour definition\neq.(39) (40) of energy-momentum is coordinate independent. As\nthe definition\nof angular-momentum\\cite{s7}, under local SO(1,2) transform\n$e^{a}\\rightarrow\n\\Lambda^{a}\\,_{b}(x)e^{b}$, where $\\Lambda^{a}\\,_{b\\mid\\partial\\Sigma}\n=L^{a}\\,_{b}\n=const.$, we have $P^{a}\\rightarrow L^{a}\\,_{b}P^{b}$. Second,\nit is worth\nwhile noting that for Deser's solution, the source stress-energy\ntensor\nof which is given {\\it a priori}(the energy-momentum vanishes\nwhile Deser's\ngravitational mass vanishes only when $m+\\sigma\\mu=0$). This is\nquite\ndifferent from the solution eq.(57) in the limit\n$\\mu\\rightarrow\\infty$.\nThe reason is that, though the form of eq.(51) agrees with eq.(57),\nthe\nfall-off is substantially different. Remember that in Deser's\nsolution,\nthe metric is linearized $g_{\\mu\\nu}=\\eta_{\\mu\\nu}+h_{\\mu\\nu}$,\nwhich\nis a good approximation on condition that $\\mid h_{\\mu\\nu}\\mid\\ll 1$.\nYet in Deser's\nsolution, $h_{ij}=\\phi\\delta_{ij}$ and $\\phi\\sim \\ln r$, so\n$h_{\\mu\\nu}$ does\nnot satisfy the confition. We expect a solution with the same\n$T^{\\mu\\nu}$ as Deser's while without the difficulty.\n\n\\vskip 0.3in\n\\underline{\\bf Acknowledgement} S.S. Feng is indebted to Prof. S.\nRandjbar-Daemi for his invitation for working at ICTP for three\nmonths.This work is supported by the National Science\nFoundation of China under Grant No. 19805004 and the Funds for Young Teachers\nof Shanghai Education Council.\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n\\subsection{Overview and Related Work}\n\nThe problem of reaching consensus on the ordering of acts by participants in a distributed system has been investigated for four decades~\\cite{shostak1982byzantine}, with  efforts in the last decade falling into two categories:  \\emph{Permissioned}, where the set of participants is determined by some authority, and \\emph{permissionless}, where anyone may join and participate provided that they pass some `sybil-proof' test, notably proof-of-work~\\cite{bitcoin} or proof-of-stake~\\cite{kiayias2017ouroboros}.  Two leaders-of-the-pack in the permissioned category are the State-Machine-Replication protocol (SMR, consensus on an ordering of proposals) for the eventual-synchrony model -- Hotstuff~\\cite{yin2019hotstuff} and its extensions and variations~\\cite{cohen2021aware}, and the Byzantine Atomic Broadcast protocol (BAB, consensus on an ordering of all proposals made by correct participants) for the asynchronous model -- DAG-Rider~\\cite{keidar2021need} and its extensions and variations~\\cite{giridharan2022bullshark}.  Since the emergence of Bitcoin~\\cite{bitcoin}, followed by Ethereum with its support for smart contracts~\\cite{buterin2014next}, permissionless consensus protocols have received the spotlight.\n\nRecent conceptual and computational advances, notably stake-based sampling, have allowed permissioned consensus protocols to join the cryptocurrency fray  (e.g. Cardano~\\cite{kiayias2017ouroboros} and Algorand~\\cite{gilad2017algorand}), offering much greater efficiency and throughput compared to proof-of-work protocols.  According to this approach, in every epoch (which could be measured in minutes or weeks) a new set of \\emph{miners} \nis chosen in a random auction, where the probability of being an auction winner is correlated with the stake bid by the miner.\nMechanism design ensures that miners benefit from performing the protocol well, benefit less if they perform the protocol less well, and lose their stake if they subvert the protocol.  \n\n\n\n\\begin{table}[t] \\smaller\n\\makebox[\\linewidth]{\n\\begin{tabular}{|c|cccc|c|} \n\\hline\n\\multirow{3}{*}{\\textbf{Protocol}}                                                                             & \\multicolumn{4}{c|}{\\textbf{Latency} (number of communication rounds)}                                                                                                                                              & \\multirow{3}{*}{\\textbf{\\begin{tabular}[c]{@{}c@{}}Message \\\\ complexity\\end{tabular}}} \\\\ \\cline{2-5}\n                                                                                                               & \\multicolumn{2}{c|}{Eventual synchrony}                                                                                           & \\multicolumn{2}{c|}{Asynchrony}                &                                                                                         \\\\ \\cline{2-5}\n                                                                                                               & \\multicolumn{1}{c|}{Good case} & \\multicolumn{1}{c|}{Expected case}                                                               & \\multicolumn{1}{c|}{Good-case} & Expected case &                                                                                         \\\\ \\hline\n\\multicolumn{1}{|l|}{Cordial Miners}                                                                           & \\multicolumn{1}{c|}{3}         & \\multicolumn{1}{c|}{4.5}                                                                         & \\multicolumn{1}{c|}{6}         & $9$            & amortized $O(n)$                                                                                  \\\\ \\hline\n\\multicolumn{1}{|l|}{\\begin{tabular}[c]{@{}l@{}}Bullshark~\\cite{giridharan2022bullshark} (same   \\\\  protocol as DAG-Rider~\\cite{keidar2021need} \\\\ in asynchrony)\\end{tabular}} & \\multicolumn{1}{c|}{4}         & \\multicolumn{1}{c|}{\\begin{tabular}[c]{@{}c@{}}Fallback \\\\ to asynchrony\\end{tabular}} & \\multicolumn{1}{c|}{8}         & 12            & amortized $O(n)$                                                                                  \\\\ \\hline\n\\end{tabular}\n}\n\t\\caption{\\textbf{Performance summary.} Latency is measured in communication rounds.\n\tThe Cordial Miners protocols for both models have better latency than Bullshark in both the good-case and the expected case. All protocols have the same concrete and asymptotic message complexity. Note that each round of Bullshark and DAG-Rider uses reliable broadcast, meaning two rounds of communication.\n\t}\n\t\\vspace{-2em}\n\t\\label{table:performance}\n\\end{table}\n\nWith this in mind, the expectation is that miners will do their best, not their worst, to execute the protocol, and hence the focus of analyses of permissioned consensus protocols has shifted from worst-case complexity to  \\emph{good-case complexity}~\\cite{abraham2021good,giridharan2022bullshark}, where miners are generally expected to behave as well as they can, given compute and network limitations, as opposed to as bad as they can.  Still, standard protections against a malicious adversary are needed, for example to prevent a double-spending, a hostile takeover, or a meltdown of the cryptocurrency supported by the consensus protocol.\n\nThe use of a DAG-like structure to solve consensus has been introduced in previous works, especially in asynchronous networks~\\cite{moser1999byzantine}.\nHashgraph~\\cite{RN284} builds an unstructured DAG, with each block containing two references to previous blocks, and on top of the DAG the miners run an inefficient binary agreement protocol.\nThis leads to expected exponential time complexity.\nAleph~~\\cite{gkagol2018aleph} builds a structured round-based DAG, where miners proceed to the next round once they receive $2f+1$ DAG nodes from other miners in the same round.\nOn top of the DAG protocols run a binary agreement protocol to decide on the order of vertices to commit. Nodes in the DAG are reliably broadcast.\nBlockmania~\\cite{danezis2018blockmania} uses a variant of PBFT~\\cite{RN581} in the eventual synchrony model.\n\n\n\nDAG-Rider~\\cite{keidar2021need} is a BAB protocol for the asynchronous model.  It assumes an adaptive adversary that eventually delivers messages between any two correct miners.\nIn DAG-Rider the miners jointly build a DAG of blocks, with blocks as vertices and pointers to previously-created blocks as edges, divided into strong and weak edges.\nStrong edges are used for the commit rule, and weak edges are used to ensure fairness.\nThe protocol employs an underlying reliable broadcast protocol of choice, which ensures that eventually the local DAGs of all correct miners converge and equivocation is excluded.\nEach miner independently converts its local DAG to an ordered sequence of blocks, with the use of threshold signatures to implement a global coin that retrospectively chooses one of the miners as the leader for each round.\nThe decision rule for delivering a block is if the vertex created by the leader is observed by at least $2f+1$ miners three rounds after it is created.\nThe DAG is divided to waves, each consisting of the nodes of four rounds. When a wave ends, miners locally check whether a decision rule is met, similar to our protocol.\nDAG-Rider has an expected amortized linear message complexity, and expected constant latency.\nTusk~\\cite{danezis2021narwhal} is an implementation based on DAG-Rider.\nBullshark~\\cite{giridharan2022bullshark} is the current state-of-the-art dual consensus protocol based on DAG-Rider that offers a fast-track to commit nodes every two rounds in case the network is synchronous.\nOther DAG-based consensus protocols include~\\cite{chockler1998adaptive,dolev1993early,sompolinsky2015secure,RN349}.\n\n\n\nHotStuff~\\cite{yin2019hotstuff} is an SMR protocol designed for the eventual synchrony model. \nThe protocol employs all-to-leader, leader-to-all communication: In each round, a deterministically-chosen designated leader proposes a block to all and collects from all signatures on the block.  Once the leader has $2f+1$ signatures, it can combine them into a threshold signature~\\cite{boneh2001short} which it sends back to all.\nThe decision rule for delivering a block is three consecutive correct leaders.\nThis leads to a linear message complexity and constant latency in the good case.   The protocol delivers a block if there are three correct leaders in a row, which is guaranteed to happen after GST.\nHotStuff is based on Tendermint~\\cite{buchman2016tendermint} and is also the core of several other consensus protocols~\\cite{kamvar2019celo,cypherium,flow,thunder}.\nIn this model, there are number of leader-based protocols such as DLS~\\cite{dwork1988consensus}, PBFT~\\cite{RN581}, Zyzzyva~\\cite{kotla2007zyzzyva}, and SBFT~\\cite{gueta2019sbft}.\n\n\nIt is within this context that we introduce \\textbf{Cordial Miners} -- a family of simple, efficient, self-contained Byzantine Atomic Broadcast~\\cite{cachin2001secure} protocols, and present two of its instances: Retrospective random leader selection for the asynchronous model and deterministic leader selection for the eventual synchrony model (See Table \\ref{table:performance} for their performance).\n\nWe believe that the simplicity-cum-efficiency of the Cordial Miners protocols stems from the use of the blocklace data structure and its analysis for all key algorithmic tasks (the following refers to correct miners):\n\\begin{enumerate}\n    \\item \\textbf{The Blocklace}~\\cite{shapiro2021multiagent} is a partially-ordered generalization of the totally-ordered blockchain (Def. \\ref{definition:block}), that consists of cryptographically-signed blocks, each containing a payload and a finite number of cryptographic hash pointers to previous blocks.  The blocklace induces a DAG, as cryptographic hash pointers cannot form cycles by a compute-bound adversary.  The DAG induces a partial order $\\succ$ (Def. \\ref{definition:acknowledge}) on the blocks that includes Lamport's `happened-before' causality relation~\\cite{lamport1978time} among correct miners.\n   \n    The globally-shared blocklace is constructed incrementally and cooperatively by all miners, who cordially disseminate it to each other.\n    \n    \\item \\textbf{Ordering}: The ordering algorithm (Algorithm \\ref{alg:CMO}) is used locally by each miner to topologically-sort its locally-known part of the blocklace into a totally-ordered output sequence of blocks, excluding equivocation along the way.   This conversion is monotonic (Prop. \\ref{proposition:tau-finality}) -- the output sequence is extended as the miner learns of or produces ever-larger portions of the global blocklace, and in this sense every output block of each miner is final.  We say that two sequences are \\emph{consistent} if one is a prefix of the other (Def. \\ref{definition:prefix-consistent}), a notion stronger than the common prefix property of Ouroboros~\\cite{kiayias2017ouroboros}.  We\n    assume that less than one-third of the miners are faulty,\n    and prove that the following holds for the remaining correct miners of the Cordial Miners protocols:\n    \\begin{itemize}\n        \\item \\textbf{Safety}: Outputs of different miners are consistent (Prop. \\ref{proposition:safety}). \n        \\item \\textbf{Liveness:} A block created by one is eventually output by everyone (Proposition \\ref{proposition:CMAbd-liveness}).\n       \n       \n    \\end{itemize}\n\n    \\item  \\textbf{Dissemination:} Any new block created by a miner $p$ acknowledges blocks known to $p$ by including pointers to the tips (DAG sources) of $p$'s local blocklace.  Correspondingly, a miner $p$ will buffer, rather than include in its blocklace, any received block with dangling pointers -- pointers to blocks not known to $p$.\n    Hence, a block $b$ by $p$ informs any recipient $q$ of blocks not known to $p$ at the time of $b$'s creation.  Thus $q$, being cordial, when sending to $p$ a new $q$-block, will include with it blocks $q$ knows but, to the best of $q$'s knowledge, are not yet known to $p$ and have not already been sent by $q$ to $p$, thus ensuring block dissemination (Prop. \\ref{proposition:CMAbd-dissemination}). \n    \n    \\item  \\textbf{Equivocation exclusion}:  An \\emph{equivocation} (Def. \\ref{definition:da-faulty-consistent}) is a pair of blocks by the same miner that are not causally-related -- have no path of pointers from one to the other;  a miner that creates an equivocation is considered faulty and is referred to as an \\emph{equivocator}.  The shared blocklace will eventually include any equivocating block known to a correct miner, and hence eventually known to all correct miners. The question is:  What should miners do with this knowledge?   \n    \n    A block $b$ \\emph{acknowledges} block $b'$ if there is a (possibly empty) path from $b$ to $b'$, namely $b \\succeq b'$ (Def. \\ref{definition:acknowledge}).   Let $[b]$ denote the set of blocks acknowledged by $b$, also referred to as the \\emph{closure} of $b$ (Def. \\ref{definition:closure}). A block $b$ \\emph{approves} block $b'$ if it acknowledges $b'$ and does not acknowledge any block $b''$ equivocating with $b'$ (Def. \\ref{definition:approval}, see Fig. \\ref{figure:approve-sr}.A).  A key observation is that a miner cannot approve both blocks of an equivocation without being itself an equivocator (Ob. \\ref{observation:approve-da}).  Hence, if less than one-third of the miners are equivocators, then no equivocation will ever receive an approval from blocks created by a \\emph{supermajority}  (at least two-thirds) of the miners.  This is the basis of equivocation-exclusion by the blocklace: A miner finalizes a block $b$ once its local blocklace includes blocks that approve $b$ by a supermajority (Algorithm \\ref{alg:CMO}).\n\\begin{figure}[ht]\n\\centering\n\\includegraphics[width=14cm]{Figs\/approve-sr.jpeg}\n\\caption{\nAcknowledgement, Approval, Equivocation, Ratification, Super-Ratification:  (A) Observing an Equivocation: The `drop shape' depicts a blocklace with a block at the tip of the drop, where all the blocks that the tip block acknowledges being the entire volume of the drop. Initial blocks of the blocklace are at the bottom, and drop inclusion implies $\\succ$. Assume $b_1, b_2$ are an equivocation (Def. \\ref{definition:da-faulty-consistent}) by the red miner. According to the figure, $b''$ approves $b_2$ (Def. \\ref{definition:approval}) since it acknowledges $b_2$ (Def. \\ref{definition:acknowledge}, namely $b'' \\succ b_2$) and does not acknowledge any conflicting red block, in particular it does not acknowledge $b_1$.  However, since $b'$ acknowledges $b''$  it acknowledges both $b_1$ and $b_2$ and hence does not approve the equivocating $b_1$ (nor $b_2$). (B) \\emph{Ratified}:  The block  (black dot) at round $r$ is ratified by another block (gray dot), if the black block is approved by a supermajority (green horizontal line) that is acknowledged by the gray block.  (C) \\emph{Super-Ratified}:  A block (blue dot) at round $r$ is  super-ratified if there is a supermajority (light green horizontal line) at round $r+\\alpha$, each member of which ratifies the blue block at round $r$ by acknowledging a supermajority (dark green horizontal line) that approves the blue block. In the case of eventual synchrony, we also require the acknowledging supermajority of round $r+\\beta$ to include the leader of that round (black dot).  In asynchrony, the leader is only elected in the next round (grey dot). The parameters are $\\alpha=1, \\beta =2$ for deterministic leader selection in the eventual synchrony model and $\\alpha=2, \\beta=5$ for retrospective random leader selection via a shared coin in the asynchronous model.  Hence the \\emph{wave length} (gap between leaders) is 3 for eventual synchrony and 5 for asynchrony.}\n\n\n\\label{figure:approve-sr}\n\\end{figure}\n   \n    \\item \\textbf{Cordial Miners}: The \\emph{depth}, or \\emph{round}, of a block $b$ is the maximal length of any path emanating from $b$ (Def. \\ref{definition:block}). A \\emph{round} is a set of blocks of the same depth. Miners are cordial in two respects.  First, as explained above, in informing other miners of blocks they believe the other miner lacks.  Second, awaiting a supermajority of round $d$ before producing a block of round $d+1$ (Def. \\ref{definition:cordial}).\n    \n    \\item \\textbf{Leader Selection}:  The Cordial Miners protocol for the eventual synchrony model  employs deterministic leader selection (e.g. via a shared pseudorandom function). In the asynchronous model, the adversary has complete control over the order of message delivery, indefinitely.  The panacea to such an adversary, employed for example by DAG-Rider~\\cite{keidar2021need}, is to use a shared random coin~\\cite{cachin2005random} and elect the leader retroactively.    \n   \n    \n    \n    \\item \\textbf{Ratified and Super-Ratified Leaders}: A block $b$ of round $r$ is \\emph{ratified} by block $b'$ if  $[b']$ includes a supermajority of round $r+\\alpha$ that approves $b$. A leader block $b$ of round $r$ is \\emph{super-ratified} if there is a supermajority of round $r+\\beta$ that include the leader, each member of which acknowledges a supermajority of round $r+\\alpha$ that approves $b$. In the case of eventual synchrony, we also require the acknowledging supermajority of round $r+\\beta$ to include the leader of that round.  In asynchrony, the leader is only elected in the next round. The parameters are $\\alpha=1, \\beta=1$ for deterministic leader selection in the eventual synchrony model and $\\alpha=2, \\beta=5$ for retrospective random leader selection via a shared coin in the asynchronous model.  Hence the \\emph{wave length} (gap between leaders) is 3 for eventual synchrony and 6 for asynchrony. (Def. \\ref{definition:ratified-leaders}, see Fig. \\ref{figure:approve-sr}).\n    \n    \\item \\textbf{Common Core}: Since miners are cordial, the notion of \\emph{common core}~\\cite{canetti1996studies} can be applied to prove a lower bound on the probability of a leader being super-ratified despite such a powerful adversary.  This  approach is applied in the Cordial Miners protocol for asynchrony, at the cost of delaying leader selection and hence expected finality to 4 rounds.  As the approach uses threshold signatures, a secure method for key distribution is needed~\\cite{damgaard2001practical}.\n    \n    The common core property implies that there is a set $V$ of at least $2f+1$ blocks in round $r$ such that every correct miner that issues block $b$ in round $r+2$ has $ b \\succeq V$.  Hence, if a miner issues a block at round $r+3$ then it has a set $U$ of at least $2f+1$ blocks in round $r+3$ such that for $u \\in U$ and $v \\in V$ $u\\succ v$. \n\n    \n    \\item  \\textbf{Ordering by Leaders}:  We assume a topological sort procedure that takes a blocklace as an input and produces a sequence of its blocks while respecting their causal partial order $\\succ$.  With it, a recursive \\emph{ordering function} $\\tau$, applied to a leader block $b$, is defined as follows (Def. \\ref{definition:tau}): Let $b'$ be the highest-depth leader block in $[b]$ ratified by $b$.\n    If there is no such $b'$, output the topological sort of $[b]$ and terminate.  Else \n    recursively call the ordering function with $b'$ and output its result following by the topological sort  of $[b]\\setminus [b']$.\n    \n    \\item  \\textbf{Finality by Super-Ratified Leaders}:   The key insight to finality is this (Proposition \\ref{proposition:finality-of-dr}): A super-ratified leader will be ratified by any subsequent leader (see Figure \\ref{figure:b-hat}).  Hence, the ordering algorithm (Algorithm \\ref{alg:CMO}) used by any miner is as follows:  When identifying a new super-ratified leader $b$ in its blocklace, apply the ordering function $\\tau$ to $b$ and output the newly added suffix since the previous super-ratified leader.\n    As the ordering function is guaranteed to be called with the previous super-ratified leader in one of its recursive calls, it can be optimized to return if it does, rather than recompute the prefix it has already delivered.\n\\begin{figure*}[ht]\n\\centering\n\\includegraphics[width=10cm]{Figs\/finality.jpeg}\n\\caption{Finality of a Super-Ratified Leader (Definition \\ref{definition:leaders}):  Assume\nthat the blue leader block of round $r$ (blue dot) is super-ratified.  We show that it is ratified by any subsequent cordial leader.\n(A) If the cordial leader is at round $r+\\beta$ (black dot, eventual synchrony instance) then it is part of the ratifying supermajority (light green) by definition. Else if the cordial red leader is at round $r+\\beta+1$  then it acknowledges a supermajority (red horizontal line) of  \nthe previous round $r+\\beta$, which must have a correct agent (purple) in common with the ratifying supermajority at round $r+\\beta$ (light green horizontal line), which acknowledges the approving supermajority at round $r+\\alpha$, and  via which the red leader at $r+\\beta+1$ ratifies the black leader at $r$.\n(B) Otherwise,  there are two purple blocks $b_1$, $b_2$ by the same correct purple agent in the red $\\hat{B}$ and light green $B'$ supermajorities, which are connected via a path  since the purple agent is correct, connecting the red leader $\\hat{b}$ to the blue leader $b$ via the approving supermajority $B$ (dark green horizontal line) at round $r+\\alpha$. Namely, the red leader $\\hat{b}$ ratifies the blue leader $b$.}\n\n\\label{figure:b-hat}\n\\end{figure*}\n\n\n    \\item  \\textbf{Identification and exclusion of faulty miners}:  Any faulty $p$-block known to some correct miner will eventually be known to all, resulting in correct miners suspending any further communication with $p$.  For example, any miner can easily verify whether another miner $p$ is cordial in the second sense by examining the blocklace.  In addition, an equivocation by $p$, with each block of the pair known to a different correct miner, will eventually be known to all and result in the exclusion of $p$.\n   \n    \\item   \\textbf{Exclusion of nonresponsive miners}:    A miner $p$ need not be cordial to a miner $q$ as long as $q$ has not acknowledged a previous block $b$ sent to $q$ by $p$. If $q$ fail-stopped, then $p$ should definitely not waste resources on $q$; if $q$ is only suspended or delayed, then eventually it will send to $p$ a block acknowledging $b$, following which $p$---being cordial---will send to $q$ all the backlog $p$ has previously refrained from sending it, and is not acknowledged by the new block received from $q$.\n    \n    \n    \n\\end{enumerate}\nMiners accomplish all the above by simple and efficient analyses of their local blocklace.\n\nThe protocols are presented as algorithmic components written in pseudo-code:\nAlgorithm \\ref{alg:blocklace} contains a description of a block in a blocklace and blocklace utilities.  Algorithm \\ref{alg:CMO} describes the local conversion each miner executes with the function $\\tau$ converting its local copy of the partially-ordered blocklace into a totally-ordered sequence of blocks.  Algorithm \\ref{alg:CMAbd} incorporates Algorithms \\ref{alg:blocklace} and \\ref{alg:CMO}, \ndescribes how a miner creates a block and sends it to all correct miners and\nincludes any backlog needed for dissemination.  It has two instances that specify the conditions to create a block, one for asynchrony and one for eventual synchrony.\n\n\n\n\\subsection{Models, Problem,  Safety and Liveness Proof Outline}\nWe assume $n\\ge 3$ miners $\\Pi$, of which $f < n\/3$ may be faulty (act arbitrarily, be \\emph{Byzantine}), and the rest are \\emph{correct}, and that a message sent from one correct miner to another will eventually arrive. We assume that every miner has a single and unique key-pair (PKI) and can cryptographically sign messages.\nEach miner $p \\in \\Pi$ has an input call $\\textit{payload}()$ that returns a payload (e.g., a proposal from a user or a mempool), and an output call $\\textit{deliver}(b)$ where $b$ is a block.\n\n\\begin{definition}\\label{definition:models}\nWe consider two models of communication with an adversary:\n\\begin{itemize}\n    \\item \\emph{Asynchrony}, where an adversary controls the finite message delay of each message. \n      \\item \\emph{Eventual Synchrony}~\\cite{dwork1988consensus}, where an adversary can delay arbitrarily an unknown but finite number of messages, beyond which messages arrive possibly under adversarial control but within a bounded delay.\n\\end{itemize}\n\\end{definition}\n\n\nIn each model the protocol addresses Byzantine Atomic Broadcast~\\cite{cachin2001secure}.  \n\n\n\\begin{definition}[Prefix, $\\preceq$, Consistent Sequences]\\label{definition:prefix-consistent}\nA sequence $x$ is a \\emph{prefix} of a sequence $x'$, $x \\preceq x'$ if $x'$ can be obtained from $x$ by appending to it zero or more elements. Two sequences $x, x'$ are \\emph{consistent} if  $x \\preceq x'$ or $x'\\preceq x$.\n\\end{definition}\n\n\\begin{definition}[Safety and Liveness]\\label{definition:safety-liveness}\nThese are the requirements of a blocklace-based Byzantine Atomic Broadcast protocol: \n\\begin{itemize}\n    \\item \\emph{Safety}: Outputs of correct miners are consistent.\n    \\item \\emph{Liveness:} A block created by a correct miner is eventually output by every correct miner with probability 1.\n\\end{itemize}\n\\end{definition}\nWe note that these safety and liveness requirements, combined with the uniqueness of a block in a blocklace, imply the standard Byzantine Atomic Broadcast guarantees: Agreement, Integrity, Validity, and Total Order~\\cite{bracha1987asynchronous, keidar2021need}.\n\nThe proofs of safety and liveness  proceed as follows, assuming  less than one-third of the miners are faulty, and referring to the remaining correct miners.\\\\\n\\textbf{Safety}: \n        \\begin{enumerate}\n            \\item Prove that the function $\\tau$ that converts a blocklace to a sequence of blocks is monotonic with respect to the superset relation (Prop. \\ref{proposition:tau-finality}).\n            \\item Observe that if two sequences are each a prefix of a third sequence, then they are consistent (Ob.\\ref{observation:consistent-triplet}).  Given any two local blocklaces $B, B'$ of miners $p, p'$, then due to the monotonicity of $\\tau$, both $\\tau(B)$ and $\\tau(B')$ are prefixes of $\\tau(B\\cup B')$. Therefore they are consistent (Corollary \\ref{corollary:consistency}).\n            \\item Argue that Algorithm \\ref{alg:CMO} correctly implements $\\tau$ and hence is safe (Prop. \\ref{proposition:safety}).\n        \\end{enumerate}\n\\textbf{Liveness:} \n        \\begin{enumerate}\n            \\item Observe that the conversion function $\\tau$, applied to a super-ratified leader block $b$, includes in the output sequence any block known to that leader, namely any block in $[b]$ (Ob. \\ref{observation:fairness}).\n             \\item Given a block $b$ known to a miner at some point $t$ in the computation, argue that $b$ will eventually be known to every miner at some later point $t'$ in the computation (Prop. \\ref{proposition:CMAbd-dissemination}).\n             \\item Argue that eventually some leader block $b'$ of a miner will be ratified at a point later than $t'$ with probability 1.\n             \\item By construction, $b \\in [b']$, hence $b$ is included in the output of $\\tau$ applied to $b'$ (Prop.  \\ref{proposition:CMAbd-liveness}).\n        \\end{enumerate}\n\nFor asynchrony, we employ a shared random global coin.\n\\begin{definition} [Shared random coin] \\label{def:sharedCoin}\nWe use a \\emph{global perfect coin}, which is unpredictable by the\nadversary. \nIn round $r$, $r \\in \\mathbb{N}$, of the coin is invoked by miner $p_i \\in \\Pi$ by calling $\\textit{leader}_i(r)$.\nThis call returns a miner $p_j \\in \\Pi$, which is the chosen leader for round $r$.\nLet $X_r$ be the random variable that represents the probability that the coin returns miner $p_j$ as the return value of the call $\\textit{leader}_i(r)$.\nThe global perfect coin has the following guarantees:\n\\begin{description}\n    \\item[Agreement] If two correct miners call $\\textit{leader}_i(r)$ and $\\textit{leader}_j(r)$ with respective return values $p_1$ and $p_2$, then $p_1=p_2$.\n    \\item [Termination] If at least $f+1$ miners call $\\textit{leader}(r)$, then every $\\textit{leader}(r)$ call eventually returns.\n    \\item[Unpredictability] As long as less than $f+1$ miners call $\\textit{leader}(r)$, the return value is indistinguishable from a random value except with negligible probability $\\epsilon$.\n    Namely, the probability $pr$ that the adversary can guess the returned\n    miner $p_j$ of the call $\\textit{leader}(r)$ is $pr \\leq\n    \\Pr [X_r = p_j] + \\epsilon$. \n    \\item[Fairness] The coin is fair, i.e., $\\forall r \\in \\mathbb{N}, \\forall p_j \\in \\Pi \\colon \\Pr[X_r = p_j] = 1\/n$.\n\\end{description}\n\\end{definition}\nExamples of such a coin implementation using a PKI and threshold signatures~\\cite{boneh2001short,libert2016born,shoup2000practical} are in~\\cite{cachin2005random,keidar2021need}.\nSee DAG-Rider~\\cite{keidar2021need} on details on how to implement such a coin as part of a distributed blocklace-like structure.\n\n\\section{The Blocklace: A Partially-Ordered Generalization of the Totally-Ordered Blockchain}\n\nThe blocklace was introduced in reference~\\cite{shapiro2021multiagent}. For completeness we include here needed definitions and results.\n\n\\begin{definition}[Block]\\label{definition:block}\nGiven a set of \\emph{miners} $\\Pi$ and a set of \\emph{payloads} $\\mathcal{A}$, a \\emph{block} is a triple $b=(p,a,H)$, referred to as a \\emph{$p$-block}, $p \\in \\Pi$, with $a \\in \\mathcal{A}$ being the \\emph{payload} of $b$, and $H$ is a (possibly empty) finite set of hash pointers to blocks, namely for each $h \\in H$, $h=\\textit{hash}(b')$ for some block $b'$.  Such a hash pointer $h$ is a \\emph{$q$-pointer} if $b'$ is a $q$-block, and \\emph{same-miner} if $h$ is a $p$-pointer, in which case $b'$ is the \\emph{immediate predecessor} of $b$.\nThe set $H$ may have at most one $q$-pointer for any miner $q \\in \\Pi$, and if $H$ has no same-miner pointer  then $b$ is called \\emph{initial}.  The \\emph{depth} of $b$, $\\textit{depth}(b)$, is the maximal length of any path emanating from $b$.\n\\end{definition}\n\nNote that $\\textit{hash}$ being cryptographic implies that a cycle cannot be effectively computed.\n\n\\begin{definition}[Dangling Pointer, Closed]\nA hash pointer $h=\\textit{hash}(b)$ for some block $b$ is \\emph{dangling} in $B$ if $b \\notin B$.  \nA set of blocks $B$ is \\emph{closed} if no block $b \\in B$ has a pointer dangling in B.  \n\\end{definition}\n\nThe non-dangling pointers of a set of blocks $B$ induce finite-degree directed graph $(B, E)$, $E \\subset B \\times B$, with blocks $B$ as vertices and directed edges $(b,b') \\in E$ if $b, b' \\in B$ and $b$ includes a hash pointer to $b'$. We overload $B$ to also mean its induced graph $(B,E)$. \n\nIn the following we assume a given set of payloads  $\\mathcal{A}$.\n\n\\begin{definition}[Blocklace]\nLet $\\mathcal{B}$ be the maximal set of blocks over $\\mathcal{A}$ and $\\textit{hash}$ for which the induced directed graph $(\\mathcal{B},\\mathcal{E})$ is acyclic. A \\emph{blocklace} over $\\mathcal{A}$ is a set of blocks $B \\subset \\mathcal{B}$.\n\\end{definition}\n\n\n\n\n\nThe two key blocklace notions used in our protocols are \\emph{acknowledgement} and \\emph{approval}.\n\\begin{definition}[$\\succ$, Acknowledge]\\label{definition:acknowledge}\nGiven a blocklace $B \\subseteq \\mathcal{B}$, the strict partial order $\\succ_B$  is defined by $b'\\succ_B b$ if $B$ has a non-empty path of directed edges from $b'$ to $b$ ($B$ is omitted if $B = \\mathcal{B}$).  Given a blocklace $B$, $b'$ \\emph{acknowledges $b$ in} $B$ if $b' \\succ_B b$. Miner $p$ \\emph{acknowledges $b$ via} $B$ if there is a $p$-block $b' \\in B$ that acknowledges $b$, and a group of miners $Q \\subseteq \\Pi$ \\emph{acknowledge $b$ via} $B$ if for every miner $p \\in Q$ there is a $p$-block $b' \\in B$ that acknowledges $b$.\n\\end{definition}\n\n\\begin{definition}[Closure, Tip]\\label{definition:closure}\nThe \\emph{closure of $b \\in \\mathcal{B}$ wrt $\\succ$} is the set $[b] := \\{b'\\in \\mathcal{B} : b \\succeq b' \\}$. The \\emph{closure of $B\\subset \\mathcal{B}$  wrt $\\succ$} is the set $[B] := \\bigcup_{b \\in B} [b]$.\nA block $b \\in \\mathcal{B}$ is a \\emph{tip} of $B$ if $[b] =[B] \\cup \\{b\\}$.\n\\end{definition}\nNote that a set of blocks is closed iff it includes its closure (and thus is identical to it):\n\\begin{observation}\n$B\\subset \\mathcal{B}$ is closed iff $[B] \\subseteq B$. \n\\end{observation}\n\n\n\n\nWith this, we can define the basic notion of \\emph{equivocation} (aka \\emph{double-spending} when payloads are conflicting financial transactions).\n\\begin{definition}[Equivocation, Equivocator]\\label{definition:da-faulty-consistent}\nA pair of $p$-blocks $b\\ne b'\\in \\mathcal{B}$, $p \\in \\Pi$, form an \\emph{equivocation} of $p$ if they are not consistent wrt $\\succ$, namely $b' \\not\\succ b$ and $b \\not\\succ b'$. A miner $p$ is an \\emph{equivocator} in $B$ if $[b]$ has an equivocation of $p$. \n\\end{definition}\nNamely,  a pair of $p$-blocks form an equivocation of $p$ if they do not acknowledge each other in $\\mathcal{B}$. In particular, two initial $p$-blocks constitute an equivocation by $p$. As $p$-blocks are cryptographically signed by $p$, an equivocation of $p$ is a volitional fault of $p$.  \n\n\n\\begin{definition}[Approval]\\label{definition:approval}\nGiven a blocklace $B$, a block $b$ \\emph{approves $b'$ in $B$} if $b$ acknowledges $b'$ in $B$ and does not acknowledge any block $b''$ in $B$ that together with $b'$ forms an equivocation. An miner $p$ \\emph{approves $b'$ in $B$} if there is a $p$-block $b$ that approves $b'$ in $B$, in which case we also say that \\emph{$p$ approves $b'$ in $B$ via $b$}. A set of miners $Q \\subseteq \\Pi$ \\emph{approve $b'$ via $B'$ in $B$} if every miner $p \\in Q$ approves $b'$ in $B$ via some $p$-block $b \\in B'$, $B'\\subseteq B$.\n\\end{definition}\n\n\\begin{observation}\nApproval is monotonic wrt $\\supset$.\n\\end{observation}\nNamely, if $b$ or $p$ approve $b'$ in $B$ they also approve $b'$ in $B' \\supset B$.\n\nA key observation is that a miner cannot approve an equivocation of another miner without being an equivocator itself (Fig. \\ref{figure:approve-sr}.A):\n\n\\begin{observation}\\label{observation:approve-da}[Approving an Equivocation]\nIf miner $p \\in \\Pi$ approves an equivocation $b_1, b_2$ in a blocklace $B \\subseteq \\mathcal{B}$, then $p$ is an equivocator in $B$.\n\\end{observation}\n\n\n\nAs equivocation is a fault, at most $f$ miners may equivocate.  \n\\begin{definition}[Supermajority]\nA set of miners $P \\subset \\Pi$ is a \\emph{supermajority} if $|P| \\ge 2f+1$.  A set of\nblocks $B$ is a \\emph{supermajority} if the set $\\{p\\in \\Pi : b \\in B \\text{ is a $p$-block}\\}$ is a supermajority.\n\\end{definition}\n\n\n\\begin{lemma}[No Supermajority Approval for Equivocation]\\label{lemma:no-double-majority}\nIf there are at most $f$ equivocators in a blocklace $B\\subset \\mathcal{B}$ with an equivocation $b, b' \\in B$, then not both $b, b'$ have  supermajority approval in $B$.\n\\end{lemma}\n\nBlocklace utilities that realize these definition are presented in Algorithm \\ref{alg:blocklace}.\n\n\\begin{algorithm*}[t!] \n    \\caption{\\textbf{Cordial Miners\\xspace: Blocklace Utilities} \n    \\\\ pseudocode for miner $p \\in \\Pi$}\n   \n    \\label{alg:blocklace}\n    \\small\n    \\begin{algorithmic}[1] \\smaller\n       \n        \\Statex \\textbf{Local variables:}\n        \\StateX struct $\\textit{block } b$: \\Comment{The struct of the most recent block $b$ created by miner $p$} \n       \n        \\StateXX $b.\\textit{creator}$ -- the miner that created $b$ \n        \\StateXX $b.\\textit{payload}$ -- a set of transactions\n \n        \\StateXX $b.\\textit{pointers}$ -- a possibly-empty set of hash pointers to other blocks\n        \\StateX \\textit{blocks} $\\gets \\{\\}$  \n      \n        \n\t\t\n\t\t\\vspace{0.5em}\n\t\t\\Procedure{$\\textit{create\\_block}$}{\\textit{blocks'}}  \\Comment create into $b$ a new block pointing to the sources of \\textit{blocks}$'$\n\t\t\\State $b.\\textit{payload} \\gets \\textit{payload}()$ \\Comment e.g. dequeue a payload from a queue of proposals (aka mempool)\n\t\t\\State $b.\\textit{creator} \\gets p$\n\t\t\\State $b.\\textit{pointers} \\gets \\{hash(b') ~:~ b' \\in \\textit{blocks}', \\text{ $b'$ has no incoming pointers in \\textit{blocks}}' \\}$ \n\t      \\State $\\textit{blocks} \\gets \\textit{blocks}~ \\cup \\{ b\\}$ \t\n\t      \\State $\\textit{deliver\\_blocks}()$  \\label{alg:CMA:addedOutgoingBlock}\n\t      \\State \\Return $b$\n\t\t\\EndProcedure\n\t\t\n\t\t\n        \\vspace{0.5em}\n\t\t\\Procedure{\\textit{hash}}{$b$}\n\t    \\Return collision-free cryptographic hash pointer(s) to block(s) $b$ \\Comment{Returns a set if applied to a set}\n        \\EndProcedure\n        \n\t\t\n        \\vspace{0.5em}\n        \\Procedure{\\textit{path}}{$b,b'$} \n        \\Return $\\exists b_1,b_2,\\ldots,b_k \\in \\textit{blocks}$, $k\\ge 1$,  s.t.\\\n        $b_1 = b $, $b_k = b'$ and $\\forall i \\in [k-1] \\colon b_{i+1} \\in b_i.\\textit{pointers}$\n        \\EndProcedure\n        \n        \\vspace{0.5em}\n        \\Procedure{\\textit{depth}}{$b$} \n        \\Return max~ $\\{k : \\exists b' \\in$ \\textit{blocks} and \\textit{path$(b,b')$} of length $k$\\}. \\Comment{Depth of block $b$}\n       \n       \n        \\EndProcedure\n        \n        \\vspace{0.5em}\n        \\Procedure{\\textit{depth}}{\\textit{blocks}} \n        \\Return $\\max~ \\{\\textit{depth}(b) : b \\in \\textit{blocks}\\}$ \\Comment{Depth of set of blocks}\n        \\EndProcedure\n        \n\t    \n        \\vspace{0.5em}\n        \\Procedure{\\textit{blocks\\_prefix}}{$d$}\n        \\Return  $\\{b \\in \\textit{blocks} : \\textit{depth}(b) \\le d\\}$\n        \\EndProcedure\n     \n     \n        \n        \\vspace{0.5em}\n        \\Procedure{\\textit{closure}}{$b$} \\label{alg:SMR:closure}\n        \\Return \\{$b' \\in \\textit{blocks} : \\textit{path}(b,b')$\\} \\Comment{also referred to as $[b]$. $b$ could also be a set of blocks}\n       \n        \\EndProcedure\n       \n      \n        \\vspace{0.5em}\n        \\Procedure{\\textit{leader}}{$d$} \n        \\Return \\Comment{Returns a leader at depth $d$ if there is one: Predetermined for every even $d$ for eventual synchrony; random\/unpredictable using a shared coin for every $d$ divisible by 5 for asynchrony (Def.~\\ref{def:sharedCoin}).} \n        \n       \n        \\EndProcedure\n        \n        \n    \n        \\vspace{0.5em}\n        \\Procedure{\\textit{leaders}}{\\textit{blocks}} \n        \\Return $\\{b \\in \\textit{blocks} : b.\\textit{creator} = \\textit{leader}(\\textit{depth}(b))\\}$\n        \\EndProcedure\n        \n        \n        \n\t\t\\vspace{0.5em}  \n\t    \\Procedure{\\textit{equivocation}}{$b_1,b_2$} \\label{alg:SMR:da}\n            \\Return \n            $b_1.\\textit{creator} = b_2.\\textit{creator} \\wedge\n            b_1 \\notin [b_2] \\wedge\n            b_2 \\notin [b_1]\n           $ \\Comment{See Figure \\ref{figure:approve-sr}.A} \n          \n          \n        \\EndProcedure\n\t\t\n\t\t\n\t\t\\vspace{0.5em}  \n\t    \\Procedure{\\textit{equivocator}}{$q$} \\label{alg:SMR:dactor}\n            \\Return \n            $\\exists b_1,b_2 \\in \\textit{blocks} \\wedge\n            b_1.\\textit{creator} = b_2.\\textit{creator} = q \\wedge\n          \\textit{equivocation}(b_1,b_2)\n           $ \\Comment{See Figure \\ref{figure:approve-sr}.A}\n        \\EndProcedure\n        \n        \n        \t \\vspace{0.5em}\t\n\t\t\\Procedure{\\textit{approved}}{$b_1,b$} \\label{alg:SMR:approved}\n            \\Return $b_1 \\in [b] \\wedge \\forall b_2 \\in [b] : \\lnot$\\textit{equivocation}$(b_1,b_2)$  \\Comment{See Figure \\ref{figure:approve-sr}.A}\n        \\EndProcedure\n          \n      \n\t\t \\vspace{0.5em}\t\n\t\t\\Procedure{\\textit{ratified}}{$b_1,b_2$} \\label{alg:SMR:sm_approved} \n         \\Return\n            $|\n            \\{b.\\textit{creator} : b \\in \\textit{blocks} \\wedge \n             b \\in [b_2] \\wedge\n              \\textit{approved}(b_1,b) \\}\n            | \\ge 2f+1$ \\Comment{See Figure \\ref{figure:approve-sr}.B}\n            \\EndProcedure\n       \n       \n\t \n        \\vspace{0.5em}\n        \\Procedure{\\textit{cordial\\_block}}{$b$}\n        \\Return  \n        $|\\{b'\\in [b] : \\textit{depth}(b') = \\textit{depth}(b)-1\\}| \\geq 2f+1$\n        \\EndProcedure\n        \n\t \n\t\t\\vspace{0.5em}  \n\t    \\Procedure{\\textit{faulty}}{$q$} \\label{alg:faulty}\n            \\Return \n            $\\textit{equivocator}(q) \\vee\n            (\\exists b \\in \\textit{blocks} \\wedge\n            b.\\textit{creator} = q \\wedge\n            \\lnot\\textit{cordial\\_block}(b))$ \\Comment{Can be expanded}\n        \\EndProcedure\n        \n\t\n        \\vspace{0.5em}\n\t\t\\Procedure{$\\textit{cordial\\_round}()$}{} \\label{alg:CMO:cr} \\Comment{Returns recent new cordial round, singleton or the empty set, Def. \\ref{definition:cordial}}\n              \\State \\Return $\\textit{arg}_{r \\in R} \\max~ \\textit{depth}(r)$  where  $R =$\n              \\label{alg:CMA:computeCordial}\n              \\State  $\\{ r \\in \\textit{depth}(\\textit{blocks}) : \n              |\\{b.\\textit{creator} : b \\in \\textit{blocks} \\wedge b.\\textit{depth} = r \\wedge \\lnot\\textit{equivocator}(b.\\textit{creator})\\}| \\ge 2f+1  \\wedge\n              \\not\\exists b \\in \\textit{blocks} : (b.\\textit{creator} = p \\wedge \\textit{depth}(b) \\ge r)\\}$ \n           \\EndProcedure    \n          \n        \\alglinenoNew{counter}\n        \\alglinenoPush{counter}\n\n    \\end{algorithmic}\n\\end{algorithm*}\n\n\n\n\\section{Converting a Blocklace into a Sequence of Blocks}\n\nHere we present a deterministic function $\\tau$ that incrementally converts a blocklace into a sequence of some of its blocks, respecting $\\succ$, and show that it is monotonic wrt to the subset relation, provided no more than $f$ miners equivocate. Each miner in the Cordial Miners protocol employs $\\tau$ to locally compute the final output sequence of blocks from its local copy of the blocklace as input, as realized by Algorithm \\ref{alg:CMO}. The monotonicity of $\\tau$ ensures finality, as it implies that the output sequence will only extend while the input local blocklace increases over time. It also ensures the safety of a protocol that uses $\\tau$ (Proposition \\ref{proposition:tau-finality} below).  \nTo ensure liveness, one has to argue that every block in the input blocklace will eventually be in the output of $\\tau$; this argument is made in Proposition \\ref{proposition:CMAbd-liveness}.\n\n\\begin{definition}[Cordial]\\label{definition:cordial}\nA block $b\\in \\mathcal{B}$ is \\emph{cordial} if the set $|\\{b'\\in [b] : \\textit{depth}(b') = \\textit{depth}(b)-1\\}|$ is a supermajority.\nMiner $p$ is \\emph{cordial} in blocklace $B \\subseteq \\mathcal{B}$ if every $p$-block $b \\in B$ is cordial.\n\\end{definition}\n\n\\begin{definition}[Round, Leader, Leader Block]\\label{definition:leaders}\nGiven a blocklace $B \\subseteq \\mathcal{B}$, then a \\emph{round} $r \\ge 1$ in $B$ is the set of blocks \n$\\{b\\in B : \\textit{depth}(b)=r\\}$. We assume a leader selection function $\\textit{leader}: \\mathbb{N} \\mapsto \\Pi \\cup \\{\\bot\\}$ that is defined only for certain depths.  If $\\textit{leader}(r)=p$ then $p$ is the \\emph{leader} of round $r$, and if, in addition, $b\\in B$ is a $p$-block of depth $r$, then $b$ is a \\emph{leader block} of round $r$ in $B$.\n\\end{definition}\n\n\nSee Figure \\ref{figure:approve-sr}.B for the following definition. We employ parameters \n$\\alpha$ and $\\beta$, where $\\alpha=1, \\beta=1$ for deterministic leader selection in the eventual synchrony model and $\\alpha=2, \\beta=5$ for retrospective random leader selection via a shared coin in the asynchronous model.\n\\begin{definition}[Ratified and Super-Ratified Blocks]\\label{definition:ratified-leaders}\nA block $b \\in \\mathcal{B}$  is \\emph{ratified} if there is a supermajority of blocks $B$ of depth $d(b)+\\alpha$ (light green in Figure \\ref{figure:approve-sr}.B) that approves $b$;  $b$ is \\emph{super-ratified} if, in addition, there is a supermajority of blocks $B'$ of depth $d(b)+\\beta$ (dark green in Figure \\ref{figure:approve-sr}.B) such that each member of $B'$ acknowledges $B$. \nIn the case of eventual synchrony, we also require the acknowledging supermajority of round $r+\\beta$ to include the leader of that round.  \n\\end{definition}\nIn asynchrony, the leader is only elected in the next round.  Hence the \\emph{wave length} (gap between leaders) is 3 for eventual synchrony and 6 for asynchrony.\n\nThe following proposition ensures that a super-ratified block is ratified by any subsequent cordial block (Fig. \\ref{figure:b-hat}).\n\n\\begin{proposition}[Finality of Super-Ratification]\\label{proposition:finality-of-dr}\nIf a leader block $b$ is super-ratified in $B$, then it is ratified by any subsequent cordial leader block $b'$. \n\\end{proposition}\n\\begin{proof}\nAssume a leader block $b$ super-ratified via supermajorities $B'$ and  $B$ (Fig. \\ref{figure:approve-sr}.B), and assume that $\\hat{b}$ is a subsequent cordial leader block.\nBeing cordial, $\\hat{b}$ acknowledges a supermajority of the preceding round $\\hat{B}$.  \nWe consider three cases (Figure \\ref{figure:b-hat}): \n\\begin{enumerate}\n    \\item In the eventual synchrony instance,  $d(b') = d(b) + \\beta$,  (Figure \\ref{figure:b-hat}.A),  $\\hat{b}$ (black dot) is part of the supermajority that acknowledges $B$.\n    \\item  Consider the asynchrony instance. If $d(b') = d(b) + \\beta+1$  (Figure \\ref{figure:b-hat}.A), then $\\hat{B}$ and $B'$ are supermajorities of the same depth and hence must have a block (purple dot Figure \\ref{figure:b-hat}.A) in  in their intersection, via which $b'$ acknowledges $B$. \n    \\item Else, $d(b') > d(b) + \\beta + 1$  (Figure \\ref{figure:b-hat}.B), then, by counting,  at least one  block $b_1 \\in B'$ and one block  $b_2 \\in \\hat{B}$ are both $p$-blocks by the same correct (purple) miner $p \\in \\Pi$. Since $p$ is not an equivocator $b_2 \\succ b_1$.  Hence $\\hat{b} \\succ b_2 \\succ b_1$ and $b_1$ acknowledges $B$, thus $\\hat{b}$ acknowledges $B$ and hence ratifies $b$.\n\\end{enumerate}\n\n\\end{proof}\n\nProposition \\ref{proposition:finality-of-dr} ensures that given a blocklace $B$,  a super-ratified leader $b$  in $B$ will be ratified by any subsequent cordial leader, hence included in the sequence of final leaders. Hence, the final sequence up to a super-ratified leader $b$ is fully-determined by $b$ itself independently of the (continuously changing) identity of the last super-ratified leader.  Hence the final sequence up to a super-ratified leader $b$  can be `cached' and will not change as the blocklace increases.\n\n\nSuper-ratified leaders are the anchors of finality in a growing chain,  each `writes history' backwards till the preceding ratified leader.  We use the term `Okazaki fragments'~\\cite{okazaki2017days} for the sequences computed backwards from each super-ratified leaders to its predecessor, acknowledging the analogy with the way one of the DNA strands of a replicated DNA molecule is elongated via the stitching of backwards-synthesized Okazaki-fragments.\n\n\nThe following recursive ordering function $\\tau$ maps a blocklace into a sequence of blocks. Formally, the entire sequence is computed backwards from the last super-ratified leader, afresh by each application of $\\tau$.  Practically, a sequence up to a super-ratified leader is final (Prop. \\ref{proposition:tau-finality}) and hence can be cached, allowing $\\tau$ to be computed one `Okazaki-fragment' at a time, from the new super-ratified leader backwards to the previous super-ratified leader.  \n\n\n\n\n\n\\begin{definition}[$\\tau$]\\label{definition:tau}\nWe assume an fixed topological sort function (e.g. lexicographic) $s$ that maps a blocklace to a sequence of blocks consistent with $\\succ$. The function $\\tau: 2^{\\mathcal{B}} \\xrightarrow{} \\mathcal{B}^*$ is defined for a blocklace $B\\subset \\mathcal{B}$ backwards, from the last output element to the first, as follows:\nIf $B$ has no super-ratified leaders then $\\tau(B) :=\\Lambda$.\nElse let $b$ be the last super-ratified leader in $B$. Then $\\tau(B) := \\tau'(b)$, where $\\tau'$ is defined recursively:\n$$\n\\tau'(b) := \n\\begin{cases}\ns([b]) \\text{\\ \\   if $[b]$ has no leader ratified by $b$, else }\\\\\n\\tau'(b') \\cdot s([b]\\setminus [b']) \\text{\\ \\   if $b'$ is the last leader ratified by $b$ in $[b]$}\n\\end{cases}\n$$\n\\end{definition}\nNote that $\\tau'$ uses the notion of a leader ratified by another leader, not a super-ratified leader.\n\n\nProposition \\ref{proposition:finality-of-dr} shows that if there are less than $f$ equivocators, a super-ratified leader is final, in that it will be ratified by any subsequent leader, super-ratified or not.  Hence the following:\n\n\\begin{proposition}[Monotonicity and Finality of $\\tau$]\\label{proposition:tau-finality}\nIf there are at most $f$ equivocators, the function $\\tau$ is monotonic wrt $\\subseteq$, namely $B \\subseteq B'$ for two blocklaces with at most $f$ equivocators implies that $\\tau(B) \\preceq \\tau(B')$, and in this sense membership in $\\tau$ is final. \n\\end{proposition}\n\\begin{proof\nAssume given blocklaces $B \\subset B' \\subseteq \\mathcal{B}_{21}$.\nIf $\\tau(B)=\\Lambda$ the claim holds vacuously.\nSo let $b$ be the last super-ratified leader in $B$.  If $B'$ has no super-ratified leader not in $B$, then $\\tau(B) = \\tau(B')$ and the claim holds vacuously.  So let $b_1,\\ldots, b_k$, $k\\ge 1$, be \nthe sequence of final leaders in $B' \\setminus B$.\nThen by the definition of $\\tau$,  $\\tau(B') = \\tau(B) \\cdot s([b_1]\\setminus [b]) \\cdot \\ldots \\cdot s([b_k]\\setminus  [b_{k-1}])$, namely $\\tau(B) \\preceq \\tau(B')$.\n\\end{proof}\n\\begin{observation}[Consistent triplet]\\label{observation:consistent-triplet}\nGiven three sequences $x, x', x''$, if $x' \\preceq x$ and $x'' \\preceq x$ then $x'$ and $x''$ are consistent.\n\\end{observation}\n\nThe following Corollary ensures that the output sequences computed  by miners based on their local blocklaces would be consistent as long as there are at most $f$ equivocators.\n\n\\begin{corollary}[Consistency of Output Sequences]\\label{corollary:consistency}\nIf $B, B'\\subset \\mathcal{B}$  are closed with at most $f$ equivocators in $B := B' \\cup B''$ then $\\tau(B')$ and $\\tau(B'')$ are consistent.\n\\end{corollary}\n\\begin{proof\nBy monotonicity of $\\tau$ with at most $f$ equivocators (Prop.  \\ref{proposition:tau-finality}), $\\tau(B') \\preceq \\tau(B)$ and  $\\tau(B'') \\preceq \\tau(B)$. By Observation \\ref{observation:consistent-triplet}, $\\tau(B')$ and $\\tau(B'')$ are consistent.\n\\end{proof}\n\nWhile $\\tau$ does not output all the blocks in its input, as blocks not acknowledged by the\nlast super-ratified leader in its input are not delivered, the following observation reminds us of the half-full glass:\n\\begin{observation}[Fairness of $\\tau$]\\label{observation:fairness}\nIf a block $b \\in B$ is acknowledged by the last super-ratified leader in $B$, then it is included in $\\tau(B)$.\n\\end{observation}\nIf every block known to a correct miner will  be known to all correct miners, \nand if  every super-ratified leader has a successor, then\nevery block will be eventually acknowledged by some super-ratified leader\nand therefore be delivered by $\\tau$.\n\n\n\n\\begin{algorithm*}[t] \n\t\\caption{\\textbf{Cordial Miners\\xspace: Conversion of Blocklace to Sequence of Blocks with $\\tau$} \\\\ pseudocode for miner $p \\in \\Pi$}\n\t\\label{alg:CMO}\n\t\\small\n\t\\begin{algorithmic}[1] \n\t\\alglinenoPop{counter} \n\n\t\t\\vspace{0.5em}\n\t\t\\Statex \\textbf{Local Variable:}\n\t\t\\StateX $\\textit{deliveredBlocks} \\gets \\{\\}$\n\t\t\\StateX $\\textit{currentLeader} \\gets \\{\\}$\n\t\t\\StateX $\\alpha$ \\Comment{A constant; $\\alpha\\gets 1$ for eventual synchrony and $\\alpha\\gets 2$ for asynchrony}\n\t\t\\StateX $\\beta$ \\Comment{A constant; $\\beta\\gets 2$ for eventual synchrony and $\\beta\\gets 5$ for asynchrony}\n\t\t\n\t\t\\vspace{0.5em}\n\t\t\n\t\t\\Procedure{\\textit{deliver\\_blocks}()}{} \n\t\t\\label{alg:CMO:waveCompletion}\n\t\t\\If{$\\textit{super\\_ratified\\_leader}() \\ne \\textit{currentLeader}$} \n\t\t  $\\textit{tau}(\\textit{super\\_ratified\\_leader}())$\n\t\t \\EndIf\n\t    \\EndProcedure\n\t\t\\vspace{0.5em}\n\t\t\n\t\t\n\t\n\t\t\\label{alg:CMO:commitrule}\n\t\t\n\t\t \\Procedure{\\textit{tau}}{$b_1$} \\label{alg:CMO:da}\n\t\t \t\\If{$b_1 \\in \\textit{deliveredBlocks} \\vee b_1 = \\emptyset$}\n\t\t  \\Return\n\t\t \\EndIf\n\t\t \\State  $b_2 \\gets \\textit{arg}_{r \\in R} \\max~  \\textit{depth}(r)$\n          \\textit{ where } $R = \\{ b \\in \\textit{leaders}(\\textit{blocks}) : \\textit{ratified}(b,b_1)\\}$ \n             \\Comment{Previous ratified leader or $\\emptyset$,  Fig. \\ref{figure:b-hat}}\n             \\label{alg:CMO:previous_ratified}\n         \\State  \\textit{tau}$(b_2)$ \\Comment{Recursive call to $tau$}\n\t\t \\For{$\\textbf{every} ~b \\in [b_1] \\setminus [b_2]$ s.t. $\\textit{approved}(b,b_1)$, ordered by topological sort}                    \\Comment{Deliver a new equivocation-free `Okazaki-fragment'}\n\t  \t\t  \\State \\textbf{deliver} $(b)$  \n\t  \t\t  \\label{alg:CMO:Okazaki}\n\t\t      \\State $\\textit{deliveredBlocks} \\gets \\textit{deliveredBlocks} \\cup \\{b\\}$\n\t\t  \\EndFor\n\t\n\t\t  \\EndProcedure\n\t\t\n        \\vspace{0.5em}\n\t\t\\Procedure{$\\textit{super\\_ratified\\_leader}()$}{} \\label{alg:CMO:dr} \\Comment{Returns last super-ratified leader, if any, Figure \\ref{figure:approve-sr}.B}\n              \\State \\Return $\\textit{arg}_{u \\in U} \\max~ \\textit{depth}(u)$  where  $U =$\n              \\State  $\\{ b \\in \\textit{leaders}(\\textit{blocks}) : \n              B = \\{b' : \\textit{depth}(b) +\\alpha = \\textit{depth}(b') \\wedge\n                          \\textit{approved}(b,b')\\}\n              \\wedge |B| \\ge 2f +1 \\wedge$ \n             $|\\{b' : \\textit{depth}(b) +\\beta = \\textit{depth}(b') \\wedge\n                          B \\subseteq [b']\\}|  \\ge 2f +1  \n                          \\wedge (\\alpha = 1 \\xrightarrow{} B \\subseteq  [\\textit{leader\\_block}(r+\\beta)])\n                          \\}$ \n      \n                 \n        \\EndProcedure\n       \n\t\n\t\t\n\t\\alglinenoPush{counter}\t\n\t\\end{algorithmic}\n\\end{algorithm*}\n\n\n\n\n\n\nAlgorithm \\ref{alg:CMO} is a fairly literal implementation of the mathematics described above:\nIt maintained \\emph{deliveredBlocks} that includes the prefix of the output of $\\tau$ that it has already computed.  Upon adding a new block to its blocklace (\\cref{alg:CMO:waveCompletion}), it computes the most-recent super-ratified leader $b_1$ according to Definition \\ref{definition:ratified-leaders}, and applies \\emph{tau} to it, which is intended to be a literal realization of the mathematical definition of $\\tau$ (Def. \\ref{definition:tau}), with the optimization, discussed above, that a recursive call with a delivered block is returned.  Hence the following proposition:\n\n\\begin{proposition}[Tau implements $\\tau$]\\label{proposition:tau-tau}\nThe procedure \\emph{tau} in Algorithm \\ref{alg:CMO} correctly implements $\\tau$ in Definition \\ref{definition:tau}.\n\\end{proposition}\n\nAnd based on it, we conclude the following, which ensures the safety of a protocol that employs Algorithm \\ref{alg:CMO} for ordering a blocklace.\n\\begin{proposition}\\label{proposition:safety}\nAlgorithm \\ref{alg:CMO} satisfied the safety requirement of Definition \\ref{definition:safety-liveness}.\n\\end{proposition}\n\\begin{proof\nProposition \\ref{proposition:tau-tau} establishes that\nAlgorithm \\ref{alg:CMO} implements $\\tau$ correctly. Together with Corollary \\ref{corollary:consistency}, we conclude that the outputs of all miners using Algorithm \\ref{alg:CMO} are consistent, satisfying the safety requirement. \n\\end{proof}\n\n\n\n\n\n\n\n\n\n\\section{Cordial Miners Protocols}\n\n\n    \n\nThe Cordial Miners protocols employ the blocklace for dissemination. To do so, each miner maintains a \\emph{history} array that records its communication history with the other miners, and updates it upon receiving a block (\\cref{alg:CMAbd:receive}) and upon sending one (\\cref{alg:CMS:historySend}).\nA miner $p$ sends the block $b$ it has created, which includes pointers to the blocks $p$ knows, to each non-faulty miner $q$, and if $q$ is responsive, it includes in the package all the blocks that $p$ knows but, to the best of $p$'s knowledge, $q$ does not, namely $[b] \\setminus [\\textit{history}[q]]$ .  This package is constructed in \\cref{alg:CMS:CreatePackage}. Excluded from the closure of $b$ it is the closure of all $q$-blocks known to $p$ and all blocks $p$ has sent to $q$, both recorded in $history[q]$ of miner $p$.\nA miner $q$ is considered \\emph{responsive} (\\cref{alg:CMS:responsive}) by $p$ if $q$ has responded to the last block $p$ has sent it.\n\n\n\n\\begin{algorithm*}[t]\n\t\\caption{\\textbf{Cordial Miners\\xspace: The Generic Protocol and Instances for Asynchrony and Eventual Synchrony}\\\\ pseudocode for miner $p \\in \\Pi$, including Algorithms \\ref{alg:blocklace} \\& \\ref{alg:CMO}}\n\t\\label{alg:CMAbd}\n\t\\small\n\t\\begin{algorithmic}[1]\n\t\\alglinenoPop{counter} \n\t\t\n\n\t\\Statex \\textbf{Local variables:}\n\t\\StateX array $\\textit{history}[n]$, initially $\\forall k \\in [n] \\colon \\textit{history}[k] \\gets \\{ \\}$  \\Comment{Communication history of $p$ with others} \n\t\\StateX \\textit{timer} \\Comment{A timer that can be \\textbf{reset} and tested}\n    \\StateX $\\Delta$ \\Comment{A constant; set $\\Delta$ to  $0$ for asynchrony and to the expected network round time after GST for eventual synchrony}\n    \n\t\\vspace{0.5em}\n\t\\State \\textbf{reset} \\textit{timer}\n\t\\Upon{\\textbf{receipt} of $b$} \\Comment{\\textbf{send} and \\textbf{receipt} are simple messaging on a reliable link}\n\t\\State $\\textit{buffer} \\gets \\textit{buffer} \\cup \\{ b \\}$  \\label{alg:CMA:buffer}\n\t\\EndUpon\n\t\n\t\\While{True}\n\t\\For{$b \\in \\textit{buffer} \\colon \\textit{b.pointers} \\subseteq \\textit{hash}(\\textit{blocks})\n\t\\wedge \\textit{cordial\\_block}(b)$\n\t    } \\Comment{Accept cordial block with no dangling pointers}\n\t    \\State $\\textit{buffer} \\gets \\textit{buffer}~ \\setminus \\{ b \\}$\n    \t\\State $\\textit{blocks} \\gets \\textit{blocks}~ \\cup \\{ b\\}$ \t\\label{alg:CMAbd:receive}\n        \\State $\\textit{history}[b.\\textit{creator}] \\gets \\textit{history}[b.\\textit{creator}] \\cup \\{ b \\}$ \\label{alg:CMAbd:historyReceive}\n\n    \t\\State $\\textit{deliver\\_blocks}()$\n\t\\EndFor\n\t    \n\t    \\If{ $\\textit{can\\_proceed}()$}   \\label{alg:CMA:isCordial} \\Comment{Defined at \\cref{alg:CMO:cr}}\n\t    \\State $b \\gets \\textit{create\\_block}(\\textit{blocks\\_prefix}(\\textit{cordial\\_round}()))$   \\Comment{Create a cordial block} \n\t\t\t\\For{$q \\in \\Pi \\wedge q \\ne p \\wedge \\lnot\\textit{faulty}(q)$} \n\t\t\t \\If{$\\textit{responsive}(q)$} \n\t        \\State $\\textit{package} \\gets [b] \\setminus [\\textit{history}[q]]$\n\t\t\\label{alg:CMS:CreatePackage}\n\t\n\t\t\\Else{  $\\textit{package} \\gets b$} \t\\label{alg:CMS:CreateBlockPackage}\n\t\t \\EndIf\n\t\t \\State $\\textit{history}[q] \\gets \\textit{history}[q] \\cup \\textit{package}$ \\label{alg:CMS:historySend}\n\t\t \\State \\textbf{send} \\textit{package} to $q$\n\t\t \\State \\textbf{reset} \\textit{timer}\n\t\t   \\EndFor\n\t\t   \\EndIf\n\t \\EndWhile\n\t \n\t \t\\vspace{0.5em}\n        \n        \\Procedure{\\textit{responsive}}{$q$} \\label{alg:CMS:responsive}\n        \\Return  \n        $\\forall b \\exists b'\\in \\textit{history}[q] : b.\\textit{creator} =  p \\wedge b'.\\textit{creator} = q \\wedge b \\in [b'] $\n        \\EndProcedure\n\n      \n        \\vspace{0.5em}\n        \\Procedure{$\\textit{can\\_proceed}()$}{}\n        \\Return  $\\textit{cordial\\_round}()$ \\Comment{Instance for asynchrony}\n\t\t\\EndProcedure\\\n\t\t\n        \\vspace{0.5em}\n        \\Procedure{$\\textit{can\\_proceed}()$}{}\n         \\Return  $ r \\in \\textit{cordial\\_round}() \n        \\wedge (timer \\ge \\Delta \\vee \\textit{leader}(r))$\n        \\Comment{Instance for eventual synchrony}\n        \n\t\t\\EndProcedure\n\t\t\n      \n\t\t\n\t\t\\alglinenoPush{counter}\n\t\\end{algorithmic}\n\t\n\\end{algorithm*}\n\n\n\nBased on this, we argue the following:\n\\begin{proposition}[Algorithm \\ref{alg:CMAbd} Dissemination]\\label{proposition:CMAbd-dissemination}\nIn any run of Algorithm \\ref{alg:CMAbd}, if a correct miner knows a block $b$, then eventually every correct miner will know $b$.\n\\end{proposition}\n\\begin{proof\nConsider a correct miner $p$ with block $b \\in \\emph{blocks}$, and miner $q$.  If $q$ is correct then eventually it will send a block to $p$, and consider a subsequent round $r$. If at round $r$ the communication history of $p$ with $q$ shows that $q$ knows $b$, we are done. Else, $p$ will include $b$ in its package to $q$, together with all blocks in $[b]$ for which $p$ has no evidence that $q$ knows, based on their communication history.  Then $q$ will eventually receive the package.  The package has no blocks with dangling pointers since $p$ includes in the package everything $q$ might miss, hence $q$ can receive it and include it together with $b$ in its \\emph{blocks}.\n\\end{proof}\n\n\\begin{proposition}[Liveness of the Cordial Miners Protocol Instances for Asynchrony and Eventual Synchrony]\\label{proposition:CMAbd-liveness}\nThe instances of Algorithm \\ref{alg:CMAbd} for asynchrony and eventual synchrony satisfy the liveness requirement of Definition \\ref{definition:safety-liveness}.\n\\end{proposition}\n\\begin{proof\nConsider a suffix of an infinite computation of Algorithm \\ref{alg:CMAbd}.  According to Proposition \\ref{proposition:CMAbd-dissemination}, every block $b$ known to a correct miner will be known eventually to every correct miner. The probability of each leader block in this suffix being super-ratified is greater than some fixed $\\epsilon$ (specifically,  $\\epsilon = \\frac{2}{3}$ for asynchrony and for eventual asynchrony after GST). Hence the probability measure of an infinite computation in which no correct leader block being super-ratified is zero. \nHence, for any block $b$ and point $t$ in the computation, some leader block $b'$ that acknowledges $b$ will be super-ratified at a point later than $t'$ with probability 1.  Thus for any block $b$ known to a correct miner, there will be a subsequent super-ratified leader block $b'$ that acknowledges $b$ and hence delivers $b$, satisfying the liveness requirement.\n\\end{proof}\n\n\\section{Performance analysis}\n\\textbf{Latency} (See Table \\ref{table:performance}).\nIn the asynchronous instance of the protocol, each wave (rounds till leader finality) consists of six rounds.\nTo calculate the probability that the decision rule is met in a wave, we use the common-core~\\cite{canetti1996studies} abstraction that is also used (and proved) in DAG-Rider and Bullshark.\n\\begin{claim} [Blocklace common-core]\nEventually, for every miner that completes round $r+5$ there exists a set $V$ of blocks in round $r+5$ and a set $U$ of blocks in round $r+2$ such that $|V| \\geq 2f+1, |U| \\geq 2f+1$ and for any $u \\in U, v \\in V$ there exists a path from $u$ to $v$.\n\\end{claim}\n\nTherefore, for every wave $w$, any miner that completes round $r+5$ has a supermajority (the blocks in $U$) that acknowledge a supermajority in round $r+2$ (the blocks in $V$).\nThus, the probability that the decision rule is met in wave $w$ is the probability that the blocks in $V$ acknowledge the retrospective elected leader block in round $r$.\nSince each block in $b \\in V$ acknowledges at least $2f+1$ blocks in round $r+1$, which in turn acknowledge $2f+1$ blocks in round $r$, then the probability of such a block $b$ to acknowledge a supermajority that approves a correct leader of round $r$ is $(1 - (\\frac{f}{3f+1})^{2f+1})^{2f+1}$, which is $1 - \\epsilon$ where $\\epsilon$ is a function of $f$ that converges exponentially to $0$ as $f$ increases. Hence, the probability that each of the blocks in $V$ acknowledges the elected leader block is close to 1 if the elected leader is a correct process, and $\\frac{2}{3}-\\varepsilon$ overall.\n\nTherefore, in the good-case, where the decision rule is met in wave $w$, the latency is $6$ rounds of communication (the length of a single wave in asynchrony).\nIn the expected case, the decision rule is met on average every $1.5$ waves, and therefore the expected latency is $9$ rounds of communication.\n\nNote that the adversary can equivocate up to $f$ times, but after each Byzantine process $p$ equivocates, all correct processes eventually detect the equivocation and do not consider $p$'s blocks as part of their cordial rounds when building the blocklace.\nThus, in an infinite run, equivocations do not affect the overall expected latency.\n\nIn the eventual synchrony case, the probability that the decision rule is met in each wave is the probability that the elected leader is a correct miner, therefore, it is at least $2\/3$.\nThus, in the good case, the latency is $3$ rounds of communication (the length of a single wave in eventual synchrony), and in the expected case it is $1.5$ waves, i.e., $4.5$ rounds of communication.\n\n\n\\textbf{Bit complexity.}\nEach block in the blocklace is linear in size, since it has a linear number of hash pointers to previous blocks.\nIn the worst-case, each block is sent to all miners by all the other miners, i.e., in the worst-case the bit complexity is $O(n^3)$ per block.\nBut, if we batch per block a linear number of transactions, when the decision rule is met, a quadratic number of transactions is committed each time.\nThus, the amortized bit complexity per decision is $O(n)$.\nThis is on par with the amortized bit complexity of DAG-Rider and Bullshark.\n\nThe concrete message complexity of the protocols depends on the security parameter of the hash function and the implementation of the threshold signatures in the shared random coin for the asynchronous protocol.\nIn any case, using the same security parameters as DAG-Rider and Bullshark achieves the same concrete amortized bit complexity as those protocols as well, and not just the same asymptotic bit complexity.\n\n\n\n\n\\section{Optimizations and Future Instances of the Cordial Miners Protocol Family}\n\nSeveral optimizations are possible to the protocol instances presented.\nFirst, as faulty miners are uncovered, they are excommunicated and therefore need not be counted as parties to the agreement, which means that the number of remaining faulty miners, initially bounded by $f$, decreases.  As a result, the supermajority needed for finality is not  $\\frac{n+f}{2n}$ (namely $2f+1$ votes in case $n = 3f+1$), but $\\frac{n+f-2f'}{2(n-f')}$, where $f'$ is the number of uncovered faulty miners, which converges to simple majority ($\\frac{1}{2}$) among the correct miners as more faulty miners are uncovered and $f'$ tends to $f$.\n\nSecondly, once faulty miners are uncovered and excommunicated, their slots as leaders could be taken by correct miners, improving  good-case and expected complexity.\n\nThirdly, a hybrid protocol in the spirit of Bullshark~\\cite{giridharan2022bullshark} can be explored. Such a protocol would employ two leaders per round -- deterministic and random,  try to achieve quick finality with the deterministic leader, and fall back to the randomly-selected leader if this attempt fails.\n\nWe also intend to explore the use of reliable broadcast~\\cite{bracha1987asynchronous} in the first round of each wave as it may improve the good case and expected case latency of our protocol in the asynchronous model.\n\n\\section{Conclusions}\n\nThe Cordial Miners protocols are simple and efficient.  We believe simplicity has many ramifications when practical applications are considered:  Simpler algorithms are easier to debug, to optimize, to  make robust, and to extend.\n\nAn important next step towards making Cordial Miners a useful foundation for cryptocurrencies is to design a mechanism that will encourage miners to cooperate---as needed by these protocols---as opposed to compete, which is the current standard in mainstream cryptocurrencies.\n\n\\newpage\n\n\\section{Introduction}\n\\subsection{Overview and Related Work}\n\nhttps:\/\/arxiv.org\/pdf\/1809.01620.pdf\n\nThe problem of reaching consensus on the ordering of acts by participants in a distributed system has been investigated for four decades~\\cite{shostak1982byzantine}, with  efforts in the last decade falling into two categories:  \\emph{Permissioned}, where the set of participants is predetermined by an outside authority, and \\emph{permissionless}, where anyone may join and participate provided that they pass some `sybil-proof' test, notably proof-of-work~\\cite{bitcoin} or proof-of-stake~\\cite{kiayias2017ouroboros}.  Two leaders-of-the-pack in the permissioned category are the State-Machine-Replication protocol (SMR, consensus on an ordering of proposals) for the eventual-synchrony model -- Hotstuff~\\cite{yin2019hotstuff} and its extensions and variations~\\cite{cohen2021aware}, and the Byzantine Atomic Broadcast protocol (BAB, consensus on an ordering of all proposals made by correct participants) for the asynchronous model -- DAG-Rider~\\cite{keidar2021need} and its extensions and variations~\\cite{giridharan2022bullshark}.  Since the emergence of Bitcoin~\\cite{bitcoin}, followed by Ethereum with its support for smart contracts~\\cite{buterin2014next}, permissionless consensus protocols have received the spotlight.\n\nRecent conceptual and computational advances, notably stake-based sampling, have allowed permissioned consensus protocols to join the cryptocurrency fray  (e.g. Cardano~\\cite{kiayias2017ouroboros} and Algorand~\\cite{gilad2017algorand}), offering much greater efficiency and throughput compared to proof-of-work protocols.  According to this approach, in every epoch (which could be measured in minutes or weeks) a new set of \\emph{miners} \nis chosen in a random auction, where the probability of being a winner is correlated with the stake bid by the miner.\nMechanism design ensures that miners benefit from performing the protocol well, benefit less if they perform the protocol less well, and lose their stake if they subvert the protocol.  \n\nWith this in mind, the expectation is that miners will do their best, not their worst, to execute the protocol, and hence the focus of analyses of permissioned consensus protocols has shifted from worst-case complexity to  \\emph{good-case complexity}~\\cite{abraham2021good,giridharan2022bullshark}, where miners are generally expected to behave as well as they can, given compute and network limitations, as opposed to as bad as they can.  Still, standard protections against a malicious adversary are needed, for example to prevent a double-spending, a hostile takeover, or a meltdown of the cryptocurrency supported by the consensus protocol.\n\nIt is within this context that we present \\textbf{Cordial Miners} -- a family of Byzantine Atomic Broadcast~\\cite{cachin2001secure} protocols that  includes simpler, yet as efficient, counterparts to both DAG-Rider and Hotstuff.\n\nWe believe that the simplicity-cum-efficiency of the Cordial Miners protocols stems from the data structure and associated ordering algorithm that they share:  \n\\begin{enumerate}\n    \\item \\textbf{The Blocklace}~\\cite{shapiro2021multiagent}: The shared data structure is a partially-ordered generalization of the totally-ordered blockchain, termed \\emph{blocklace} (Def. \\ref{definition:block}), that consists of cryptographically-signed blocks, each containing a payload and a finite number of cryptographic hash pointers to previous blocks.  The blocklace induces a DAG, as cryptographic pointers cannot form cycles even by a compute-bound adversary.  The DAG induces a partial order $\\succ$ (Def. \\ref{definition:acknowledge}) on the blocks that includes Lamport's `happened-before' causality relation~\\cite{lamport1978time} among correct miners.\n   \n    The globally-shared blocklace is constructed incrementally and cooperatively by all miners, who also disseminate it to each other.\n    \n    \\item \\textbf{Ordering}: The ordering algorithm (Algorithm \\ref{alg:CMO}) is used locally by each miner to topologically-sort its locally-known part of the blocklace into a totally-ordered output sequence of blocks, excluding equivocation along the way.   This conversion is monotonic (Prop. \\ref{proposition:tau-finality}) -- the output sequence is extended as the miner learns of or produces ever-larger portions of the global blocklace, and in this sense every output block of each miner is final.  We say that two sequences are \\emph{consistent} if one is a prefix of the other (Def. \\ref{definition:prefix-consistent}), a notion stronger than the common prefix property of Ouroboros~\\cite{kiayias2017ouroboros}.  We\n    assume that less than one third of the miners are faulty,\n    and prove that the following holds for the remaining correct miners of the Cordial Miners protocols:\n    \\begin{itemize}\n        \\item \\textbf{Safety}: Outputs of different miners are consistent (Prop. \\ref{proposition:safety}). \n        \\item \\textbf{Liveness:} A block created by one is eventually output by everyone (Props. \\ref{proposition:aCM-liveness}, \\ref{proposition:sCM-liveness}, \\ref{proposition:CMAbd-liveness}).\n       \n       \n    \\end{itemize}\n\\end{enumerate}\n\nThe simplicity of the protocols in the Cordial Miners family stems from their use of the blocklace and its analysis for all key algorithmic tasks (the following refers to correct miners):\n\\begin{enumerate}\n    \\item  \\textbf{Dissemination:} Any new block created by a miner $p$ acknowledges blocks known to $p$ by including pointers to the tips (DAG sources) of $p$'s local blocklace, as well as a pointer to $p$'s previous block.  Correspondingly, a miner $p$ will buffer, rather than include in its blocklace, any received block with dangling pointers -- pointers to blocks not known to $p$.\n    Hence, a block $b$ by $p$ informs any recipient $q$ of blocks not known to $p$ at the time of $b$'s creation.  Thus $q$, being cordial, when sending to $p$ a new $q$-block, will include with it blocks $q$ knows but, to the best of $q$'s knowledge, are not yet known to $p$ and have not already been sent to $p$, thus ensuring block dissemination (Prop. \\ref{proposition:sCM-dissemination}). \n    \n    \\item  \\textbf{Equivocation exclusion}:  An \\emph{equivocation} (Def. \\ref{definition:da-faulty-consistent}) is a pair of blocks by the same miner that are not causally-related -- have no path of pointers from one to the other; such blocks are \\emph{conflicting} and a miner that creates them is considered faulty and is referred to as an \\emph{equivocator}.  The shared blocklace will eventually include any conflicting block known to a correct miner, and hence eventually known to all correct miners. The question is:  What should miners do with this knowledge?   \n    \n    A block $b$ \\emph{acknowledges} block $b'$ if there is a (possibly empty) path from $b$ to $b'$, namely $b \\succeq b'$ (Def. \\ref{definition:acknowledge}).   Let $[b]$ denote the set of blocks acknowledged by $b$, also referred to as the \\emph{closure} of $b$ (Def. \\ref{definition:closure}). A block $b$ \\emph{approves} block $b'$ if it acknowledges $b'$ and does not acknowledge any block $b''$ conflicting with $b'$ (Def. \\ref{definition:approval}, see Fig. \\ref{figure:approve-sr}.A).  A key observation is that a miner cannot approve both blocks of an equivocation without being itself an equivocator (Ob. \\ref{observation:approve-da}).  Hence, if less than one-third of the miners are equivocators, then no equivocation will ever receive an approval from blocks created by a \\emph{supermajority}  (at least two-thirds) of the miners.  This is the basis of equivocation-exclusion by the blocklace: A miner finalizes a block $b$ once its local blocklace includes blocks that approve $b$ by a supermajority (Algorithm \\ref{alg:CMO}).\n\\begin{figure}[ht]\n\\centering\n\\includegraphics[width=15cm]{Figs\/approve-sr.jpeg}\n\\caption{\\oded{A few reviewers said that image A isn't clear. Maybe remove it?} Acknowledgement, Approval, Equivocation, Ratification, Super-Ratification:  (A) Observing an Equivocation: Initial blocks are at the bottom, inclusion implies $\\succ$. Assume $b_1, b_2$ are an equivocation (Def. \\ref{definition:da-faulty-consistent}) by the red miner. According to the figure, $b''$ approves $b_2$ (Def. \\ref{definition:approval}) since it acknowledges $b_2$ (Def. \\ref{definition:acknowledge}, namely $b'' \\succ b_2$) and does not acknowledge any conflicting red block, in particular it does not acknowledge $b_1$.  However, since $b'$ acknowledges $b''$  it acknowledges both $b_1$ and $b_2$ and hence does not approve the equivocating $b_1$ (nor $b_2$). (B) \\emph{Ratified}:  The block  (black dot) at round $r$ is ratified by the block (gray dot) at round $\\ge r+2$, as the black block is approved by a supermajority at round $r+1$ (green horizontal line) that is acknowledged by the gray block.  (C) \\emph{Finalized}:  A leader block (blue dot) at round $r$ is  finalized if there is a supermajority (light green horizontal line) that includes a leader block  (red dot) at round $r+2$, each member of which ratifies the blue leader at round $r$, namely each acknowledges a supermajority (dark green horizontal line) at round $r+1$ that approves the blue leader.}\n\n\n\\label{figure:approve-sr}\n\\end{figure}\n   \n    \\item \\textbf{Cordial Miners}: The \\emph{depth}, or \\emph{round}, of a block $b$ is the maximal length of any path emanating from $b$ (Def. \\ref{definition:block}). A \\emph{round} is a set of blocks of the same depth. Miners are cordial in two respects.  First, as explained above, in informing other miners of blocks they believe the other miner lacks.  Second, in waiting for a supermajority of round $d$ before producing a block of round $d+1$ (Def. \\ref{definition:cordial}).\n    \n    \n    \\item \\textbf{Leader Selection}:  Our protocols assume a fair \\emph{leader selection} function, that chooses a miner $p$ as \\emph{leader} for every round $d$, in which case a $p$-block of depth $d$ is a \\emph{leader block}.  The leader selection function can be round-robin, pseudo-random, or random, selecting the leader retroactively using a shared coin (as in DAG-Rider~\\cite{keidar2021need}). The communication model (asynchrony, eventual synchrony) and the leader selection function determine the strength of the adversary (bounded, unbounded) the protocol can withstand.  In particular, to withstand an unbounded adversary in the asynchronous model, the leader needs to be selected randomly and retroactively via a shared coin.\n    \n    \\item \\textbf{Ratified and Finalized Leaders}: A block $b$ of round $r$ is \\emph{ratified} by block $b'$ if  $[b']$ includes a supermajority of round $r+1$ that approves $b$. A leader block $b$ of round $r$ is \\emph{final} if there is a supermajority of round $r+2$ that include a leader, each member of which acknowledges a supermajority of round $r+1$ that approves $b$. (Def. \\ref{definition:ratified-leaders}, see Fig. \\ref{figure:approve-sr}).\n\n    \n    \\item  \\textbf{Ordering by Leaders}:  We assume a given topological sort procedure that takes a blocklace as an input and produces a sequence of its blocks while respecting their causal partial order $\\succ$.  With it, a recursive \\emph{ordering function} $\\tau$, applied to a leader block $b$, is defined as follows (Def. \\ref{definition:tau}): Let $b'$ be the highest-depth leader block in $[b]$ ratified by $b$.\n    If there is no such $b'$, output the topological sort of $[b]$ and terminate.  Else \n    recursively call the ordering function with $b'$ and output its result following by the topological sort  of $[b]\\setminus [b']$.\n    \n    \\item  \\textbf{Finality by Finalized Leaders}:   The key insight to finality is this (Proposition \\ref{proposition:finality-of-dr}): A finalized leader will be ratified by any subsequent leader (see Figure \\ref{figure:b-hat}).  Hence, the ordering algorithm (Algorithm \\ref{alg:CMO}) used by any miner is as follows:  When identifying a new finalized leader $b$ in its blocklace, apply the ordering function $\\tau$ to $b$ and output the newly added suffix since the previous finalized leader.\n    As the ordering function is guaranteed to select the previous finalized leader in one of its recursive calls, it can be optimized to return if it does so, rather than recompute the prefix it has already delivered.\n\\begin{figure*}[ht]\n\\centering\n\\includegraphics[width=14cm]{Figs\/finality.jpeg}\n\\caption{Finality of a Finalized Leader (\\ref{definition:leaders}):  If a blue leader block of round $r$ (blue dot) is finalized then it is ratified by any subsequent leader leader (red dot).\n(A) If the red leader is at round $r+2$ then by definition of super-ratification it acknowledges a supermajority that approves the blue leader. (B+C) If the red leader is at a subsequent round $r+k$, $k>2$, then, being cordial, it acknowledges a supermajority (red horizontal line) of the previous round $r+k-1$, which must have a correct agent (purple) in common with the super-ratification supermajority at round $r+2$ (light green horizontal line).\n(B) If these supermajorities are of the same round  $k=2$,  then there is a single purple block that connects the red leader to the blue leader via the approving supermajority (dark green horizontal line) at round $r+1$.  (C) Otherwise $k>2$,  there are two purple blocks $b_1$, $b_2$ by the purple agent in the red $\\hat{B}$ and light green $B'$ supermajorities, which are connected via a path  since the purple agent is correct, connecting the red leader $\\hat{b}$ to the blue leader $b$ via the approving supermajority $B$ (dark green horizontal line) at round $r+1$. In all cases, the red leader $\\hat{b}$ ratified the blue leader $b$.}\n\n\\label{figure:b-hat}\n\\end{figure*}\n\n\n    \\item  \\textbf{Identification and exclusion of faulty miners}:  Any faulty $p$-block known to some correct miner will eventually be known to all, resulting in correct miners suspending any further communication with $p$.  For example, any miner can easily verify whether another miner $p$ is cordial in the second sense by examining the blocklace.  In addition, an equivocation by $p$, with each block of the pair known to a different correct miner, will eventually be known to all and result in the exclusion of $p$.\n   \n    \\item   \\textbf{Exclusion of nonresponsive miners}: Being cordial endows legitimacy to ignore nonresponsive miners.   A miner $p$ need not send new blocks to a miner $q$ as long as $q$ has not acknowledged a previous block $b$ sent to $q$ by $p$. If $q$ fail-stopped, then $p$ should definitely not waste resources on $q$; if $q$ is only suspended or delayed, then eventually it will send to $p$ a block acknowledging $b$, following which $p$---being cordial---will send to $q$ all the backlog $p$ has previously refrained from sending it, and is not acknowledged by the new block received from $q$.\n    \n    \n    \\item \\textbf{Retroactive Leader Election via Shared Coin}\n    In the model of asynchrony the adversary has complete control over the order of message delivery, indefinitely.  The panacea to such an adversary, employed for example by DAG-Rider~\\cite{keidar2021need}, is to use a shared random coin~\\cite{cachin2005random} and elect the leader retroactively, and to allow more rounds for a finalized leader to emerge.  Since miners are cordial, the notion of \\emph{common core}~\\cite{canetti1996studies} can be applied to prove a lower bound on the probability of a leader being finalized despite such a powerful adversary.  The same approach can be applied here for the Cordial Miners protocols, at the cost of delaying expected finality from 2 rounds to 4 rounds, as shown in Figure \\ref{figure:coin-core}.  As the approach uses threshold signatures, a secure method for key distribution is needed~\\cite{damgaard2001practical}.\n\n\\begin{figure*}[ht]\n\\centering\n\\includegraphics[width=12cm]{Figs\/coin-core.png}\n\\caption{Finality with a Shared Coin: }\n\n\\label{figure:coin-core}\n\\end{figure*}\n\n    \n\\end{enumerate}\nMiners accomplish all the above by simple and efficient analyses of their local blocklace.\n\nWe present three simple instances of the Cordial Miners protocol family (Figure \\ref{figure:family}):  A counterpart to DAG-Rider~\\cite{keidar2021need}, a BAB protocol in the asynchronous model; a counterpart to HotStuff~\\cite{yin2019hotstuff}, an SMR protocol in the eventual synchrony model; and a hybrid protocol that takes the best of both.\n\\begin{figure*}[ht]\n\\centering\n\\includegraphics[width=15cm]{Figs\/cm-family.png}\n\\caption{\\oded{Add the new algorithm 2 variant for the asynchronous setting} The Cordial Miners Protocols and their Algorithmic Components. Algorithms 1 \\& 2 are shared.  Algorithm 1 includes basic blocklace utilities, Algorithm 2 realizes blocklace ordering.  Algorithm 3 incorporates Algorithms 1 \\& 2 with a modular asynchronous data dissemination protocol to realize Byzantine Atomic Broadcast in the model of eventual asynchrony\/asynchrony, depending on the leader selection function employed.  Algorithm 4 incorporates Algorithms 1 \\& 2 with cordial all-to-leader-to-all blocklace dissemination to realize Byzantine Atomic Broadcast in the eventual synchrony model. Algorithm 5 integrates ideas from the first two:  It incorporates Algorithms 1 \\& 2 with cordial all-to-all block communication and cordial leader-to-all blocklace dissemination to realize Byzantine Atomic Broadcast in the model of eventual asynchrony\/asynchrony, depending on the leader selection function employed.  Length in lines of pseudocode is noted for each. The asynchronous data-dissemination protocol employed by the Algorithm 3 is not shown.}\n\\label{figure:family}\n\\end{figure*}\nThe protocols are presented as algorithmic components written in pseudo-code, with some components shared between them:\n\\begin{enumerate}\n    \\item The \\emph{Asynchronous Cordial Miners protocol with modular Asynchronous Data Dissemination}  consists of Algorithms \\ref{alg:blocklace}, \\ref{alg:CMO}, and \\ref{alg:CMA}.\n    \\item The \\emph{Synchronous Cordial Miners protocol with Blocklace Dissemination}  consists of Algorithms \\ref{alg:blocklace}, \\ref{alg:CMO}, and \\ref{alg:CMS}.\n    \\item The \\emph{Asynchronous Cordial Miners protocol with Blocklace Dissemination}  consists of Algorithms \\ref{alg:blocklace}, \\ref{alg:CMO}, and \\ref{alg:CMAbd}.\n    \\item Optimizations for all, which exclude equivocating and nonresponsive miners, are described in Algorithm \\ref{alg:CMOpt}.\n\\end{enumerate}\nFigure \\ref{figure:family} presents the shared (gray) and distinct (green\/blue\/red) components of three protocols in the family. Table \\ref{table:intro} compares their various properties with DAG-Rider and HotStuff.\n\\begin{table*}[]\n\\centering\n\n \\begin{tabular}{ | m{9em} | m{6em}| m{6em}|   m{6em} | m{6em} |m{6em} |} \n\n    \\hline\n          & \\textbf{\\textsc{DAG-Rider}}   &  \\textbf{\\textsc{Asynch. Cordial Miners + ADD}}& \\textbf{\\textsc{HotStuff}} &  \\textbf{\\textsc{Synch. Cordial Miners + BD}}&  \\textbf{\\textsc{Asynch. Cordial Miners + BD}} \\\\\n    \\hline\\hline\n        \\textbf{Problem}  & BAB & BAB &  SMR & BAB & BAB \\\\\n    \\hline\n        \\textbf{Model}  & Asynchrony &  Asynchrony\/ Eventual  Asynchrony & Eventual Synchrony & Eventual Synchrony  &  Asynchrony\/ Eventual  Asynchrony\\\\ \n       \n        \n         \\hline\n        \\textbf{Cryptography} & Threshold signatures  & Threshold signatures\/PKI & Threshold signatures & PKI & Threshold signatures\/PKI \\\\\n          \\hline\n        \\textbf{Data Types} & DAG of blocks, weak and strong edges   & Blocklace  & Blocks    & Blocklace & Blocklace \n        \\\\\n   \n    \\hline\n         \\textbf{Message Types} & Block, Threshold signature & Signed Block, Threshold signature\n          & Block, Signature, Threshold signature  & Blocklace & Blocklace, w\/Threshold signatures\n        \\\\\n         \\hline\n        \\textbf{Communication}  & All-to-All  & All-to-All  & Leader-based    & Leader-based & All-to-All + Leader-based backlog\n        \\\\\n    \\hline\n         \\textbf{Underlying Protocol} &  Reliable Broadcast & Asynchronous Data Dissemination &  Reliable Links  &  Reliable Links  &  Reliable Links \n         \\\\\n          \\hline\n         \\textbf{Leader Election} & Random & Random\/ Deterministic  &  Deterministic   &  Deterministic  &  Random\/ Deterministic  \n        \\\\\n    \\hline\n     \\textbf{Decision Rule}  & Supermajority of $4^{th}$ round observing $1^{st}$ round leader & Super-ratified leader & Three consecutive correct leaders  &  Super-ratified leader &  Super-ratified leader\n        \\\\\n       \n      \\hline\n        \\textbf{Expected Latency} &   $\\frac{1}{(2\/3)*8}$ O(1) & $\\frac{1}{(2\/3)^3}$ O(1) & $\\frac{1}{(2\/3)^3}$ O(1) & $\\frac{1}{(2\/3)^3}$ O(1)& $\\frac{1}{(2\/3)^3}$ O(1)\\textbf{}\\\\\n        \\hline\n        \\textbf{Bit complexity}  & Amortized expected linear & Amortized expected linear & Good case linear & Amortized linear & Amortized linear\\\\\n          \\hline\n          \n  \n   \n\n\\hline\n         \n        \n    \n    \n\n\n\n\\end{tabular}\n\\caption{A comparison between DAG-Rider, Asynchronous Cordial Miners with Modular Dissemination, Hotstuff, Synchronous Cordial Miners, and Asynchronous Cordial Miners with blocklace Dissemination \\oded{Like we talked, need to think if removing the table is better than leaving it.}}\n\\label{table:intro}\n\\end{table*}\n\n\nAlgorithm \\ref{alg:blocklace} contains a description of a block in a blocklace and blocklace utilities.  Algorithm \\ref{alg:CMO} describes the local conversion each miner executes with the function $\\tau$ converting its local copy of the partially-ordered blocklace into a totally-ordered sequence of blocks.  Then there are two algorithms: Algorithm \\ref{alg:CMA}, an all-to-all communication protocol that is analogue to DAG-Rider~\\cite{keidar2021need}, and uses asynchronous data dissemination (e.g. ADD~\\cite{Das2021ADD}) as the underlying communication protocol; and Algorithm \\ref{alg:CMS}, a leader-based communication protocol that is analogue to HotStuff~\\cite{yin2019hotstuff}.  Algorithm \\ref{alg:CMAbd} combines the best of both Algorithm \\ref{alg:CMA} and Algorithm \\ref{alg:CMS}:  It uses all-to-all communication as in  Algorithm \\ref{alg:CMA}, but eschews an external dissemination protocol by the leader of each round sending any backlog needed for dissemination as in Algorithm \\ref{alg:CMS}.\nAlgorithm \\ref{alg:CMOpt} presents several optimizations that exclude faulty and nonresponsive leaders in all protocols.  \n\n\n\n\\subsection{Models, Problem,  Safety and Liveness Proof Outline}\nWe assume $n\\ge 3$ miners $\\Pi$, of which $f < n\/3$ may be faulty (act arbitrarily, be \\emph{Byzantine}), and the rest are \\emph{correct}, and that a message sent from one correct miner to another will eventually arrive. We assume that every miner has a single and unique key-pair (PKI) and can cryptographically sign messages.\nEach miner $p \\in \\Pi$ has an input call $\\textit{payload}()$ that returns a payload (e.g., a proposal from a user or a mempool), and an output call $\\textit{deliver}(b)$ where $b$ is a block.\n\n\\begin{definition}\\label{definition:models}\nWe consider three models of communication with an adversary:\n\\begin{itemize}\n    \\item \\emph{Asynchrony}, where an adversary controls the finite message delay of each message. \n      \\item \\emph{Eventual Synchrony}~\\cite{dwork1988consensus}, where an adversary controls message delay for an unknown but finite number of messages, beyond which messages arrive within bounded delay.\n      \\item \\emph{Eventual Random Asynchrony}, where an adversary controls message delay for an unknown but finite number of messages, beyond which messages arrive with random delay of finite expectation.\n    \\item \\change{\\emph{Asynchrony}, where the adversary controls the delivery times of the messages.}\n\\end{itemize}\n\\end{definition}\n\nEventual asynchrony is a natural counterpart to eventual synchrony, yet novel to the best of our knowledge.\nThe Synchronous Cordial Miners protocol is devised for eventual synchrony and the Asynchronous Cordial Miners protocols are can be applied for eventual asynchrony with pseudorandom leader selection and to asynchrony with random retroactive shared-coin leader selection \\oded{Maybe cite some works about a common coin}.\nAll protocols address Byzantine Atomic Broadcast~\\cite{cachin2001secure}.  \n\n\n\\begin{definition}[Prefix, $\\preceq$, Consistent Sequences]\\label{definition:prefix-consistent}\nA sequence $x$ is a \\emph{prefix} of a sequence $x'$, $x \\preceq x'$ if $x'$ can be obtained from $x$ by appending to it zero or more elements. Two sequences $x, x'$ are \\emph{consistent} if  $x \\preceq x'$ or $x'\\preceq x$.\n\\end{definition}\n\n\\begin{definition}[Safety and Liveness]\\label{definition:safety-liveness}\nThese are the requirements of correct miners of a blocklace-based Byzantine Atomic Broadcast protocol: \n\\begin{itemize}\n    \\item \\emph{Safety}: Outputs of miners are consistent.\n    \\item \\emph{Liveness:} A block created \n   \n    by a correct miner is eventually output by every miner with probability 1.\n\\end{itemize}\n\\end{definition}\nWe note that these safety and liveness requirements, combined with the uniqueness of a block in a blocklace, imply the standard Byzantine Atomic Broadcast guarantees: Agreement, Integrity, Validity, and Total Order~\\cite{bracha1987asynchronous, keidar2021need}.\n\nProofs of safety and liveness of the two protocols proceed as follows, assuming  less than one third of the miners are faulty, and referring to the remaining correct miners.\\\\\n\\textbf{Safety}: \n        \\begin{enumerate}\n            \\item Prove that the function $\\tau$ that converts a blocklace to a sequence of blocks is monotonic with respect to the superset relation (Prop. \\ref{proposition:tau-finality}).\n            \\item Observe that if two sequences are each a prefix of a third sequence, then they are consistent (Ob.\\ref{observation:consistent-triplet}).  Given any two local blocklaces $B, B'$ of miners $p, p'$, then due to the monotonicity of $\\tau$, both $\\tau(B)$ and $\\tau(B')$ are prefixes of $\\tau(B\\cup B')$. Therefore they are consistent (Corollary \\ref{corollary:consistency}).\n            \\item Argue that Algorithm \\ref{alg:CMO} correctly implements $\\tau$ and hence is safe (Prop. \\ref{proposition:safety}).\n        \\end{enumerate}\n\\textbf{Liveness:} \n        The proofs of the protocols are different but have a common structure.\n        \\begin{enumerate}\n            \\item Observe that the conversion function $\\tau$, applied to a finalized leader block $b$, includes in the output sequence any block known to that leader, namely any block in $[b]$ (Ob. \\ref{observation:fairness}).\n             \\item Given a block $b$ known to a miner at some point $t$ in the computation, argue that $b$ will eventually be known to every miner at some later point $t'$ in the computation  (follows from the correctness of the underlying asynchronous data dissemination protocol used by Algorithm \\ref{alg:CMA}; claimed by Props. \\ref{proposition:sCM-dissemination},\\ref{proposition:CMAbd-dissemination}  for Algs. \\ref{alg:CMS}, \\ref{alg:CMAbd}).\n             \\item Argue that eventually some leader block $b'$ of a miner will be ratified at a point later than $t'$ with probability 1.\n             \\item By construction, $b \\in [b']$, and hence $b$ is included in the output of $\\tau$ applied to $b'$ (Props. \\ref{proposition:aCM-liveness}, \\ref{proposition:sCM-liveness}, \\ref{proposition:CMAbd-liveness}).\n        \\end{enumerate}\n\n\n\\subsection{Additional Related Work}\nThe use of a DAG-like structure to solve consensus has been introduced in previous works, especially in asynchronous networks~\\cite{moser1999byzantine}.\nHashgraph~\\cite{RN284} builds an unstructured DAG, with each block containing two references to previous blocks, and on top of the DAG the miners run an inefficient binary agreement protocol.\nThis leads to expected exponential time complexity.\nAleph~~\\cite{gkagol2018aleph} builds a structured round-based DAG, where miners proceed to the next round once they receive $2f+1$ DAG nodes from other miners in the same round.\nOn top of the DAG protocols run a binary agreement protocol to decide on the order of vertices to commit. Nodes in the DAG are reliably broadcast.\n\nDAG-Rider~\\cite{keidar2021need} is also a structured round-based DAG protocol that proceeds in rounds.\nNodes are reliably broadcast.\nThe DAG is divided to waves, each consisting of the nodes of four rounds. When a wave ends, miners locally check whether a decision rule is met, similar to our protocol.\nTusk~\\cite{danezis2021narwhal} is an implementation based on DAG-Rider.\nBullshark is a dual consensus protocol based on DAG-Rider that offers a fast-track to commit nodes every two rounds in case the network is synchronous.\nOther DAG-based consensus protocols include~\\cite{chockler1998adaptive,dolev1993early,sompolinsky2015secure,RN349}.\n\nHotStuff~\\cite{yin2019hotstuff} has a similar commit rule to our protocol of a finalized leader.\nThe commit rule is met when there are three consecutive correct leaders in a row.\nHotStuff is based on Tendermint~\\cite{buchman2016tendermint} and is also the core of several other consensus protocols~\\cite{kamvar2019celo,cypherium,flow,thunder}.\nUnlike HotStuff, Cordial miners guarantees fairness, i.e., each block proposed by a miner is eventually guaranteed to be included in the blockchain.\nHotStuff works in eventually synchronous networks and is a leader-based consensus protocol.\nIn this model, there are number of leader-based protocols such as DLS~\\cite{dwork1988consensus}, PBFT~\\cite{RN581}, Zyzzyva~\\cite{kotla2007zyzzyva}, SBFT~\\cite{gueta2019sbft}, and more.\n\n\\section{The Blocklace: A Partially-Ordered Generalization of the Totally-Ordered Blockchain}\n\nThe blocklace was introduced in reference~\\cite{shapiro2021multiagent}. For completeness we include the needed definitions and results, and for space considerations relegate them to  Appendix \\ref{appendix:section:blocklace}. Blocklace utilities that realize these definition are presented in Algorithm \\ref{alg:blocklace}.\n\n\n\\begin{algorithm*}[t]\n    \\caption{\\textbf{Cordial Miners\\xspace: Blocklace Utilities} \n    \\\\ pseudocode for miner $p \\in \\Pi$}\n   \n    \\label{alg:blocklace}\n    \\small\n    \\begin{algorithmic}[1]\n        \\Statex \\textbf{Local variables:}\n        \\StateX struct $\\textit{block } b$: \\Comment{The struct of the most recent block $b$ created by miner $p$} \n       \n        \\StateXX $b.\\textit{creator}$ -- the miner that created $b$ \n        \\StateXX $b.\\textit{payload}$ -- a set of transactions\n        \\StateXX $b.\\textit{share}$ -- a share of the random coin, if needed \n        \\StateXX $b.\\textit{pointers}$ -- a possibly-empty set of hash pointers to other blocks\n        \\StateX \\textit{blocks} $\\gets \\{\\}$  \n      \n        \n\t\t\n\t\t\\vspace{0.5em}\n\t\t\\Procedure{$\\textit{create\\_block}$}{\\textit{blocks'}}  \\Comment create into $b$ a new block pointing to the sources of \\textit{blocks}$'$\n\t\t\\State $b.\\textit{payload} \\gets \\textit{payload}()$ \\Comment e.g. dequeue a payload from a queue of proposals (aka mempool)\n\t\t\\State $b.\\textit{creator} \\gets p$\n\t\t\\State $b.\\textit{pointers} \\gets \\{hash(b') ~:~ b' \\in \\textit{blocks}', \\text{ $b'$ has no incoming pointers in \\textit{blocks}}' \\}$ \n\t\t\\State $b.\\textit{share} \\gets \\textit{choose\\_leader}_p(\\textit{depth}(b)+3)$ \\Comment If a shared coin is employed\n\t      \\State $\\textit{blocks} \\gets \\textit{blocks}~ \\cup \\{ b\\}$ \t\\label{alg:CMA:addedOutgoingBlock}\n\t      \\State $\\textit{deliver\\_blocks}()$\n\t      \\State \\Return $b$\n\t\t\\EndProcedure\n\t\t\n\t\t\n        \\vspace{0.5em}\n\t\t\\Procedure{\\textit{hash}}{$b$}\n\t    \\Return collision-free cryptographic hash pointer(s) to block(s) $b$ \\Comment{Returns a set if applied to a set}\n        \\EndProcedure\n        \n\t\t\n        \\vspace{0.5em}\n        \\Procedure{\\textit{path}}{$b,b'$} \n        \\Return $\\exists b_1,b_2,\\ldots,b_k \\in \\textit{blocks}$, $k\\ge 1$,  s.t.\\\n        $b_1 = b $, $b_k = b'$ and $\\forall i \\in [k-1] \\colon b_{i+1} \\in b_i.\\textit{pointers}$\n        \\EndProcedure\n        \n        \\vspace{0.5em}\n        \\Procedure{\\textit{depth}}{$b$} \n        \\Return the max~ $\\{k : \\exists b' \\in$ \\textit{blocks} and \\textit{path$(b,b')$} of length $k$\\}. \\Comment{Depth of block $b$}\n       \n       \n        \\EndProcedure\n        \n        \\vspace{0.5em}\n        \\Procedure{\\textit{depth}}{\\textit{blocks}} \n        \\Return $\\max~ \\{\\textit{depth}(b) : b \\in \\textit{blocks}\\}$ \\Comment{Depth of set of blocks}\n        \\EndProcedure\n        \n\t    \n        \\vspace{0.5em}\n        \\Procedure{\\textit{blocks\\_prefix}}{$d$}\n        \\Return  $\\{b \\in \\textit{blocks} : \\textit{depth}(b) \\le d\\}$\n        \\EndProcedure\n     \n     \n        \n        \\vspace{0.5em}\n        \\Procedure{\\textit{closure}}{$b$} \\label{alg:SMR:closure}\n        \\Return \\{$b' \\in \\textit{blocks} : \\textit{path}(b,b')$\\} \\Comment{also referred to as $[b]$. $b$ could also be a set of blocks}\n       \n        \\EndProcedure\n       \n      \n        \\vspace{0.5em}\n        \\Procedure{\\textit{leader}}{$d$} \n        \\Return an index in $[n]$ or $\\bot$.  \n        \\oded{Otherewise return $\\bot$? Why not just use round robin? odd is not defined, better to use $d \\bmod 2 = 1$ or ``if d is odd''}\n        \\udi{Let's discuss how to define it parametric for all scenarios including shared coin.}\n        \n       \n        \\EndProcedure\n        \n        \n    \n        \\vspace{0.5em}\n        \\Procedure{\\textit{leaders}}{\\textit{blocks}} \n        \\Return $\\{b \\in \\textit{blocks} : b.\\textit{creator} = \\textit{leader}(\\textit{depth}(b))\\}$\n        \\EndProcedure\n        \n        \n        \n\t\t\\vspace{0.5em}  \n\t    \\Procedure{\\textit{equivocation}}{$b_1,b_2$} \\label{alg:SMR:da}\n            \\Return \n            $b_1.\\textit{creator} = b_2.\\textit{creator} \\wedge\n            b_1 \\notin [b_2] \\wedge\n            b_2 \\notin [b_1]\n           $ \\Comment{See Figure \\ref{figure:approve-sr}.A} \n          \n          \n        \\EndProcedure\n\t\t\n\t\t\n        \t \\vspace{0.5em}\t\n\t\t\\Procedure{\\textit{approved}}{$b_1,b$} \\label{alg:SMR:approved}\n            \\Return $b_1 \\in [b] \\wedge \\forall b_2 \\in [b] : \\lnot$\\textit{equivocation}$(b_1,b_2)$  \\Comment{See Figure \\ref{figure:approve-sr}.A}\n        \\EndProcedure\n          \n      \n\t\t \\vspace{0.5em}\t\n\t\t\\Procedure{\\textit{ratified}}{$b_1,b_2$} \\label{alg:SMR:sm_approved} \n         \\Return\n            $|\n            \\{b.\\textit{creator} : b \\in \\textit{blocks} \\wedge \n             b \\in [b_2] \\wedge\n              \\textit{approved}(b_1,b) \\}\n            | \\ge 2f+1$ \\Comment{See Figure \\ref{figure:approve-sr}.B}\n            \\EndProcedure\n       \n       \n\t \n        \\vspace{0.5em}\n        \\Procedure{\\textit{cordial\\_block}}{$b$}\n        \\Return  \n        $|\\{b'\\in [b] : \\textit{depth}(b') = \\textit{depth}(b)-1\\}| \\geq 2f+1$\n        \\EndProcedure\n        \n\t \n        \\vspace{0.5em}\n\t\t\\Procedure{$\\textit{cordial\\_round}()$}{} \\label{alg:CMO:cr} \\Comment{Returns recent new cordial round, singleton or the empty set, Def. \\ref{definition:cordial}}\n              \\State \\Return $\\textit{arg}_{r \\in R} \\max~ \\textit{depth}(r)$  where  $R =$\n              \\label{alg:CMA:computeCordial}\n              \\State  $\\{ r \\in \\textit{depth}(\\textit{blocks}) : \n              |\\{b.\\textit{create} : b \\in \\textit{blocks} \\wedge b.\\textit{depth} = r\\}| \\ge 2f+1  \\wedge\n              \\not\\exists b \\in \\textit{blocks} : (b.\\textit{creator} = p \\wedge \\textit{depth}(b) \\ge \\textit{depth}(r))\\}$ \n           \\EndProcedure    \n          \n       \n        \\vspace{0.5em}\n\t\t\\change{\\Procedure{$\\textit{choose\\_leader}$}{$r$} \\Comment{A random shared coin}\n\t\t    \\State \\Return a random leader process drawn by the coin \n\t\t   \\State \\oded{Need to add somewhere in the text the properties of the coin: doesn't return until at least f+1 processes call it, the adversary cannot guess the outcome of the coin in advance, the probability of each process to be chosen is equal, the coin returns the same results for each $r$.}}\n\t\t   \n        \n        \\alglinenoNew{counter}\n        \\alglinenoPush{counter}\n\n    \\end{algorithmic}\n\\end{algorithm*}\n\n\n\n\\section{Converting a Blocklace into a Sequence of Blocks}\n\nHere we present a deterministic function $\\tau$ that incrementally converts a blocklace into a sequence of some of its blocks, and show that it is monotonic wrt to the subset relation, provided no more than $f$ miners equivocate. The intention is that each miner in a blockchain consensus protocol employs $\\tau$ to locally compute the final output sequence of blocks from its local copy of the blocklace as input, as realized by Algorithm \\ref{alg:CMO}. Monotonicity ensures finality, as it implies that the output sequence will only extend while the input local blocklace increases over time. Monotonicity is a sufficient condition for the safety of each protocol that uses $\\tau$ (Proposition \\ref{proposition:tau-finality} below).  To ensure liveness, one has to argue that every block in the input blocklace will eventually be in the output of $\\tau$; this argument is protocol-specific and argued separately for the asynchronous protocol (Prop. \\ref{proposition:aCM-liveness}) and the synchronous protocol (Prop. \\ref{proposition:sCM-liveness}).\n\n\\begin{definition}[Cordial]\\label{definition:cordial}\nLet $b', b\\in \\mathcal{B}$ be two consecutive $p$-blocks.\nThen block $b$ is \\emph{cordial} if $[b] \\setminus ( [b'] \\cup \\{b\\})$ is a supermajority.\nMiner $p$ is \\emph{cordial} in blocklace $B \\subseteq \\mathcal{B}$ if every $p$-block $b \\in B$ is cordial.\n\\end{definition}\n\n\\begin{definition}[Round, Leader, Leader Block]\\label{definition:leaders}\nGiven a blocklace $B \\subseteq \\mathcal{B}$, then a \\emph{round} $r \\ge 1$ in $B$ is the set of blocks \n$\\{b\\in B : \\textit{depth}(b)=r\\}$. We assume a leader selection function $\\textit{leader}: \\mathbb{N} \\mapsto \\Pi \\cup \\{\\bot\\}$ that is defined only for depths.  If $\\textit{leader}(r)=p$ then $p$ is the \\emph{leader} of round $r$, and if, in addition, $b\\in B$ is a $p$-block of depth $r$, then $b$ is a \\emph{leader block} of round $r$ in $B$.\n\\end{definition}\n\n   \n    \nSee Figure \\ref{figure:approve-sr}.B for the following definition.\n\\begin{definition}[Ratified and Finalized Leader Blocks]\\label{definition:ratified-leaders}\nA block $b \\in \\mathcal{B}$  is \\emph{ratified} if there is a supermajority of depth $d(b)+1$ that approves $b$.  A leader block $b$ of depth $r$ is \\emph{final} if there is a supermajority $B$ of depth $d(b)+2$ that includes a leader block such that each member of $B$ ratifies $b$.\n\\end{definition}\n\nThe following proposition ensures that a finalized block is ratified by any subsequent cordial block (Fig. \\ref{figure:b-hat}).\n\n\\begin{proposition}[Finality of Super-Ratification]\\label{proposition:finality-of-dr}\nIf $b$ is finalized leader in $B$, then it is ratified by any subsequent leader \n$b' \\succ b$ in $B$. \n\\end{proposition}\n\\begin{proof}\nAssume a leader block $b$ finalized via supermajorities $B'$ and  $B$ (Fig. \\ref{figure:approve-sr}.B), and assume that $\\hat{b}$ is some subsequent leader block that, being cordial, acknowledges a supermajority of the preceding round $\\hat{B}$.  By counting,  at least one  block $b_1 \\in B'$ and one block  $b_2 \\in \\hat{B}$ are both $p$-blocks by the same correct miner $p \\in \\Pi$ (Fig. \\ref{figure:b-hat}). Since $p$ is not an equivocator $b_2 \\succ b_1$.  Hence $\\hat{b} \\succ b_2 \\succ b_1$ and $b_1$ ratifies $b$, thus $\\hat{b}$ ratifies $b$.\n\\end{proof}\n\nProposition \\ref{proposition:finality-of-dr} ensures that given a blocklace $B$,  a finalized leader $b$  in $B$ will be ratified by any subsequent finalized leader, hence included in the sequence of final leaders. Hence, the final sequence up to a finalized leader $b$ is fully-determined by $b$ itself independently of the (continuously changing) identity of the last finalized leader.  Hence the final sequence up to a finalized leader $b$  can be `cached' and will not change as the blocklace increases.\n\n\nSuper-ratified leaders are the anchors of finality in a growing chain,  each `writes history' backwards till the preceding ratified leader.  We use the term `Okazaki fragments'~\\cite{okazaki2017days} for the sequences computed backwards from each finalized leaders to its predecessor, acknowledging the analogy with the way one of the DNA strands of a replicated DNA molecule is elongated via the stitching of backwards-synthesized Okazaki-fragments.\n\n\nThe following recursive ordering function $\\tau$ maps a blocklace into a sequence of blocks. Formally, the entire sequence is computed backwards from the last finalized leader, afresh by each application of $\\tau$.  Practically, a sequence up to a finalized leader is final (Prop. \\ref{proposition:tau-finality}) and hence can be cached, allowing $\\tau$ to be computed in  `Okazaki-fragment' increments from the new finalized leader backwards to the previous finalized leader.  \n\n\n\n\n\n\\begin{definition}[$\\tau$]\\label{definition:tau}\nWe assume an fixed ordering function (e.g. lexicographic) $s$ that maps a blocklace to a sequence of blocks consistent with $\\succ$. The function $\\tau: 2^{\\mathcal{B}} \\xrightarrow{} \\mathcal{B}^*$ is defined for a blocklace $B\\subset \\mathcal{B}$ backwards, from the last output element to the first, as follows:\nIf $B$ has no finalized leaders then $\\tau(B) :=\\Lambda$.\nElse let $b$ be the last finalized leader in $B$. Then $\\tau(B) := \\tau'(b)$, where $\\tau'$ is defined recursively:\n$$\n\\tau'(b) := \n\\begin{cases}\ns([b]) \\text{\\ \\   if $[b]$ has no leader ratified by $b$, else }\\\\\n\\tau'(b') \\cdot s([b]\\setminus [b']) \\text{\\ \\   if $b'$ is the last leader ratified by $b$ in $[b]$}\n\\end{cases}\n$$\n\\end{definition}\nNote that $\\tau'$ uses the notion of a leader ratified by another leader, not a finalized leader.\n\n\nProposition \\ref{proposition:finality-of-dr} shows that if there are less than $f$ equivocators, a finalized leader is final, in that it will be ratified by any subsequent leader, finalized or not.  Hence the following:\n\n\\begin{proposition}[Monotonicity and Finality of $\\tau$]\\label{proposition:tau-finality}\nIf there are at most $f$ equivocators, the function $\\tau$ is monotonic wrt $\\subseteq$, namely $B \\subseteq B'$ for two blocklaces with at most $f$ equivocators implies that $\\tau(B) \\preceq \\tau(B')$, and in this sense membership in $\\tau$ is final. \n\\end{proposition}\n\\begin{proof}[Proof of Proposition \\ref{proposition:tau-finality} ]\nAssume given blocklaces $B \\subset B' \\subseteq \\mathcal{B}_{21}$.\nIf $\\tau(B)=\\Lambda$ the claim holds vacuously.\nSo let $b$ be the last finalized leader in $B$.  If $B'$ has no finalized leader not in $B$, then $\\tau(B) = \\tau(B')$ and the claim holds vacuously.  So let $b_1,\\ldots, b_k$, $k\\ge 1$, be \nthe sequence of final leaders in $B' \\setminus B$.\nThen by the definition of $\\tau$,  $\\tau(B') = \\tau(B) \\cdot s([b_1]\\setminus [b]) \\cdot \\ldots \\cdot s([b_k]\\setminus  [b_{k-1}])$, namely $\\tau(B) \\preceq \\tau(B')$.\n\\end{proof}\n\n\n\\begin{observation}[Consistent triplet]\\label{observation:consistent-triplet}\nGiven three sequences $x, x', x''$, if $x' \\preceq x$ and $x'' \\preceq x$ then $x'$ and $x''$ are consistent.\n\\end{observation}\n\nThe following Corollary ensures that the output sequences computed  by miners based on their local blocklaces would be consistent as long as there are there are at most $f$ equivocators.\n\n\\begin{corollary}[Consistency of Output Sequences]\\label{corollary:consistency}\nIf $B, B'\\subset \\mathcal{B}$  are grounded with at most $f$ equivocators in $B := B' \\cup B''$ then $\\tau(B')$ and $\\tau(B'')$ are consistent.\n\\end{corollary}\n\\begin{proof}\nBy monotonicity of $\\tau$ with at most $f$ equivocators (Prop.  \\ref{proposition:tau-finality}), $\\tau(B') \\preceq \\tau(B)$ and  $\\tau(B'') \\preceq \\tau(B)$. By Observation \\ref{observation:consistent-triplet}, $\\tau(B')$ and $\\tau(B'')$ are consistent.\n\\end{proof}\n\nWhile $\\tau$ does not output all the blocks in its input, as blocks not acknowledged by the\nlast finalized leader in its input are not delivered, the following observation reminds us of the half-full glass:\n\\begin{observation}[Fairness of $\\tau$]\\label{observation:fairness}\nIf a block $b \\in B$ is acknowledged by the last finalized leader in $B$, then it is included in $\\tau(B)$.\n\\end{observation}\nIf every block known to a correct miner will  be known to all correct miners, \nand if  every finalized leader has a successor, then\nevery block will be eventually acknowledged by some finalized leader\nand therefore be delivered by $\\tau$.\n\n\n\n\\begin{algorithm*}[t]\n\t\\caption{\\textbf{Cordial Miners\\xspace: Conversion of Blocklace to Sequence of Blocks with $\\tau$} \\\\ pseudocode for miner $p \\in \\Pi$}\n\t\\label{alg:CMO}\n\t\\small\n\t\\begin{algorithmic}[1]\n\t\\alglinenoPop{counter} \n\n\t\t\\vspace{0.5em}\n\t\t\\Statex \\textbf{Local Variable:}\n\t\t\\StateX $\\textit{deliveredBlocks} \\gets \\{\\}$\n\t\t\\vspace{0.5em}\n\t\t\n\t\t\\Upon{ $\\textit{deliver\\_blocks}()$} \\Comment{Signal that a new block was added to \\textit{blocks}  (\\cref{alg:CMA:addedOutgoingBlock}, \\cref{alg:CMA:receive}}, \\cref{alg:CML:addedIncomingBlock})\n\t\t\\label{alg:CMO:waveCompletion} \n\t\t\n\t\t  \n\t\t\\EndUpon\n\t\t\\vspace{0.5em}\n\t\t\n\t\t\n\t\n\t\t\\label{alg:CMO:commitrule}\n\t\t\n\t\t \\Procedure{\\textit{tau}}{$b_1$} \\label{alg:CMO:da}\n\t\t \t\\If{$b_1 \\in \\textit{deliveredBlocks} \\vee b_1 = \\emptyset$}\n\t\t  \\Return\n\t\t \\EndIf\n\t\t \\State  $b_2 \\gets \\textit{arg}_{r \\in R} \\max~  \\textit{depth}(r)$\n          \\textit{ where } $R = \\{ b \\in \\textit{leaders}(\\textit{blocks}) : \\textit{ratified}(b,b_1)\\}$ \n             \\Comment{Previous ratified leader or $\\emptyset$,  Fig. \\ref{figure:b-hat}}\n             \\label{alg:CMO:previous_ratified}\n         \\State  \\textit{tau}$(b_2)$ \\Comment{Recursive call to $tau$}\n\t\t \\For{$\\textbf{every} ~b \\in [b_1] \\setminus [b_2]$, ordered by topological sort}                    \\Comment{Deliver a new `Okazaki-fragment'}\n\t  \t\t  \\State \\textbf{deliver} $(b)$  \n\t  \t\t  \\label{alg:CMO:Okazaki}\n\t\t      \\State $\\textit{deliveredVertices} \\gets \\textit{deliveredVertices} \\cup \\{b\\}$\n\t\t  \\EndFor\n\t\n\t\t  \\EndProcedure\n\t\t\n        \\vspace{0.5em}\n\t\t\\Procedure{$\\textit{super\\_ratified\\_leader}()$}{} \\label{alg:CMO:dr} \\Comment{Returns last finalized leader, singleton or the empty set, Figure \\ref{figure:approve-sr}.B}\n              \\State \\Return $\\textit{arg}_{u \\in U} \\max~ \\textit{depth}(u)$  where  $U =$\n              \\State  $\\{ b \\in \\textit{leaders}(\\textit{blocks}) : \\exists b', b'' \\in \\textit{leaders}(\\textit{blocks}) \\wedge$ \n              $\\textit{depth}(b) +4 = \\textit{depth}(b')+2  = \\textit{depth}(b'')  \\wedge$ \n              $\\textit{ratified}(b,b') \\wedge \\textit{ratified}(b',b'')\\}$ \n          \n                 \n        \\EndProcedure\n      \n      \n\t\t \n\t\\alglinenoPush{counter}\t\n\t\\end{algorithmic}\n\\end{algorithm*}\n\n\n\n\n\n\nAlgorithm \\ref{alg:CMO} is a fairly literal implementation of the mathematics described above:\nIt maintained \\emph{deliveredBlocks} that includes the prefix of the output of $\\tau$ that it has already computed.  Upon adding a new block to its blocklace (\\cref{alg:CMO:waveCompletion}), it computes the most-recent finalized leader $b_1$ according to Definition \\ref{definition:ratified-leaders}, and applies \\emph{tau} to it, which is intended to be a literal realization of the mathematical definition of $\\tau$ (Def. \\ref{definition:tau}), with the optimization, discussed above, that a recursive call with a delivered block is returned.  Hence the following proposition:\n\n\\begin{proposition}[Tau implements $\\tau$]\\label{proposition:tau-tau}\nThe procedure \\emph{tau} in Algorithm \\ref{alg:CMO} correctly implements $\\tau$ in Definition \\ref{definition:tau}.\n\\end{proposition}\n\nAnd based on it, we conclude the following, which ensures the safety of every protocol that employs Algorithm \\ref{alg:CMO} for ordering a blocklace; in particular, the safety of the asynchronous and synchronous Cordial Miners protocols.\n\\begin{proposition}\\label{proposition:safety}\nAlgorithm \\ref{alg:CMO} satisfied the safety requirement of Definition \\ref{definition:safety-liveness}.\n\\end{proposition}\n\\begin{proof}\nProposition \\ref{proposition:tau-tau} establishes that\nAlgorithm \\ref{alg:CMO} implements $\\tau$ correctly. Together with Corollary \\ref{corollary:consistency}, we conclude that the outputs of all miners using Algorithm \\ref{alg:CMO} are consistent, satisfying the safety requirement. \n\\end{proof}\n\n\n\n\n\n\n\n\n\n\\section{Asynchronous Cordial Miners Protocol}\n\nDAG-Rider~\\cite{keidar2021need} is a BAB protocol designed for the asynchronous model.  It assumes an adaptive adversary that eventually delivers messages between any two correct miners.\nIn DAG-Rider the miners jointly build a DAG of blocks, with blocks as vertices and pointers to previously-created blocks as edges, divided into strong and weak edges.\nStrong edges are used for the commit rule, and weak edges are used to ensure fairness.\nThe protocol employs an underlying reliable broadcast protocol of choice, which ensures that eventually the local DAGs of all correct miners converge and equivocation is excluded.\nEach miner independently converts its local DAG to an ordered sequence of blocks, with the use of threshold signatures to implement a global coin that retrospectively chooses one of the miners as the leader for each round.\nThe decision rule for delivering a block is if the vertex created by the leader is observed by at least $2f+1$ miners three rounds after it is created.\nDAG-Rider has an expected amortized linear message complexity, and expected constant latency.\n\nThe Asynchronous Cordial Miners protocol is our DAG-Rider counterpart. It assumes an underlying data dissemination protocol (e.g. \\cite{Das2021ADD}), consists of Algorithms  \\ref{alg:blocklace}, \\ref{alg:CMO} \\& \\ref{alg:CMA}.\nAlgorithm \\ref{alg:CMA}  assume an asynchronous data dissemination  protocol as a sub-protocol.  For example, it could use ADD~\\cite{Das2021ADD}. To do so, when a miner executing the Cordial Miners protocol calls \\textbf{disseminate}$(b)$, it sends the block $b$ to all other miners.  This ensures that $2f+1 > f+1$ correct miners receive $b$, satisfying condition for ADD to operate correctly. A miner that receives a proper block $b$ (signed, cordial, with no dangling pointers) for the first time initiates the ADD protocol with $b$.  Every shard of $b$ created by the ADD protocol is tagged with the hash value of $b$, so that multiple instances of the ADD protocol on multiple blocks can proceed concurrently, namely asynchronously and in parallel, without confusing between the blocks of the shards.  When the ADD protocol reconstructs a block $b'$, the \\textbf{receipt} condition of the Cordial Miners protocol is fulfilled, with block $b'$.  The correctness of ADD ensures that every block disseminated by a correct miner will eventually be reconstructed and received by every correct miner.\n\n\n\nThe Asynchronous Cordial Miners protocol is designed for the model of eventual asynchrony (Def. \\ref{definition:models}).\nThe dissemination component is called with $\\textit{disseminate}(m)$ for some block $b$ and can output $\\textit{receipt}(b)$.  We assume that the component guarantees that if a correct miner calls $\\textit{disseminate}(b)$ or $\\textit{receipt}(b)$ then eventually all correct miners perform $\\textit{receipt}(b)$.\n\n\\begin{algorithm*}[t]\n\t\\caption{\\textbf{Cordial Miners\\xspace: Asynchronous Protocol with Modular Block Dissemination}\\\\ pseudocode for miner $p \\in \\Pi$, including Algorithms \\ref{alg:blocklace} \\& \\ref{alg:CMO}}\n\t\\label{alg:CMA}\n\t\\small\n\t\\begin{algorithmic}[1]\n\t\\alglinenoPop{counter} \n\t\t\n\n\n\t\n\t\\vspace{0.5em}\n\t\n\t\\Upon{$\\textbf{receipt}(b)$} \\Comment{\\textbf{disseminate} and \\textbf{receipt} are calls to the Asynchronous Data Dissemination (ADD) protocol~\\cite{Das2021ADD}} \\oded{The ADD guarantees in [13] are different from the ones needed here} \\udi{ Please explain\/let's discuss}\n\t\\State $\\textit{buffer} \\gets \\textit{buffer} \\cup \\{ b \\}$  \\label{alg:CMA:buffer}\n\t\\EndUpon\n\t\n\t\\While{True}\n\t\\For{$b \\in \\textit{buffer} \\colon \\textit{b.pointers} \\subseteq \\textit{hash}(\\textit{blocks})\n\t\\wedge \\textit{cordial\\_block}(b)$\n\t    } \\Comment{Accept cordial block with no dangling pointers}\n    \t\\State $\\textit{blocks} \\gets \\textit{blocks}~ \\cup \\{ b\\}$ \t\\label{alg:CMA:receive}\n    \t\\State $\\textit{buffer} \\gets \\textit{buffer} \\setminus \\{ b \\}$\n    \t\\State $\\textit{deliver\\_blocks}()$\n\t\\EndFor\n\t    \n\t    \\If{ $\\textit{cordial\\_round}()$}   \\label{alg:CMA:isCordial}\n\t    \\State $b \\gets \\textit{create\\_block}(\\textit{blocks\\_prefix}(\\textit{cordial\\_round}()))$ \n\t    \\State $\\textbf{disseminate}(b)$ \n\t    \\EndIf\n\t \\EndWhile\n\t \n                 \n\t\t\\alglinenoPush{counter}\n\t\\end{algorithmic}\n\t\n\\end{algorithm*}\n\n\n   \nAlgorithm \\ref{alg:CMA} is short and sweet.  All it does is buffer incoming blocks (\\cref{alg:CMA:buffer}), receive them when they have no dangling pointers  (\\cref{alg:CMA:receive}), and if it observes a new cordial round  (\\cref{alg:CMA:isCordial}), it creates a block for the most recent cordial round in which it has not participated (\\cref{alg:CMA:computeCordial}) and disseminates it.  That's it.\n\nThe dissemination protocol ensures that all correct miners eventually the same blocks.  The only concern is that the adversary, being familiar with the `celebrated FLP theorem'~\\cite{fischer1985impossibility}, will order message delivery so that in every round the supermajority first observed by all correct miners will not include the leader of that round.  This way no ratified leader, let alone finalized one, will emerge.  Fortunately, in the eventual asynchrony model our adversary will wane after controlling delivery times for some finite number of messages, and random network delays will rule thereafter.  Hence, from any point in the computation,  a finalized leader will eventually emerge and all blocks acknowledged by it and not yet delivered will be delivered.  Hence the following:\n\n\n\\begin{proposition}[Liveness of Asynchronous Cordial Miners protocol with Modular Asynchronous Data Dissemination]\\label{proposition:aCM-liveness}\nAlgorithm \\ref{alg:CMA} satisfies the liveness requirement of Definition \\ref{definition:safety-liveness}.\n\\end{proposition}\n\\begin{proof}\nConsider a suffix of an infinite computation of Algorithm \\ref{alg:CMA} beyond the control of the adversary.  The underlying asynchronous data dissemination protocol ensures that every block $b$ known to a miner will be known eventually to every miner. The probability of each leader block in this suffix being ratified by its successor is greater than some fixed $\\epsilon$,  hence the probability of a leader block being finalized is greater than some smaller but fixed $\\epsilon'$. Hence the probability measure of an infinite computation in which no correct leader block being finalized is zero. \nHence, for any block $b$ and point $t$ in the computation, some leader block $b'$ that acknowledges $b$ will be finalized at a point later than $t'$ with probability 1.  Thus for any block $b$ known to a correct miner, there will be a subsequent finalized leader block $b'$ that acknowledges $b$ and hence delivers $b$, satisfying the liveness requirement.\n\\end{proof}\n\n\n  \n\\section{Synchronous Cordial Miners Protocol}\n\nHotStuff~\\cite{yin2019hotstuff} is an SMR protocol designed for the eventual synchrony model. \nThe protocol employs all-to-leader, leader-to-all communication: In each round, a deterministically-chosen designated leader proposes a block to all and collects from all signatures on the block.  Once the leader has $2f+1$ signatures, it can combine them into a threshold signature~\\cite{boneh2001short} which it sends back to all.\nThe decision rule for delivering a block is three consecutive correct leaders.\nThis leads to a linear message complexity and constant latency in the good case.   The protocol delivers a block if there are three correct leaders in a row, which is guaranteed to happen after GST.\n\n\nThe Synchronous Cordial Miners protocol is our Hotstuff counterpart, s designed for the model of eventual synchrony (Def. \\ref{definition:models}).  It consists of Algorithms \n\\ref{alg:blocklace}, \\ref{alg:CMO} \\& \\ref{alg:CMS}).\n\n\n\\begin{algorithm*}[t]\n\t\\caption{\\textbf{Cordial Miners\\xspace: Synchronous Protocol with Blocklace Dissemination} \\\\ pseudocode for miner $p \\in \\Pi$, including Algorithms \\ref{alg:blocklace} \\& \\ref{alg:CMO}}\n\t\\label{alg:CMS}\n\t\\small\n\t\\begin{algorithmic}[1]\n\t\\alglinenoPop{counter} \n\t\t\n\n\t\\Statex \\textbf{Local variables:}\n\t\\StateX array $\\textit{history}[1..n]$, initially $\\forall k \\in [n] \\colon \\textit{history}[k] \\gets \\{ \\}$\n\t\\Comment{Communication history of $p$ with other miners} \n\t\\StateX $\\textit{buffer} \\gets \\{ \\}$\n    \n\n\n\t\n\t\\vspace{0.5em}\n\t\n\t\\Upon{\\textbf{receipt} of $b$}      \\Comment{\\textbf{send} and \\textbf{receipt} are simple messaging on a reliable link} \n\t\\State $\\textit{buffer} \\gets \\textit{buffer} \\cup \\{b \\}$ \\Comment{New blocks are buffered}\n\t\\EndUpon\n\t\n\t\\While{True}\n\t    \\For{$b \\in \\textit{buffer} \\colon   \\textit{b.pointers} \\subseteq     \\textit{hash}(\\textit{blocks})\n\t    \\wedge \\textit{cordial\\_block}(b)$\n\t    } \\Comment{Accept cordial block with no dangling pointers}\n\t        \\State $\\textit{history}[b.\\textit{creator}] \\gets \\textit{history}[b.\\textit{creator}] \\cup \\{ b \\}$ \\label{alg:CMS:historyReceive}\n\t        \\State $\\textit{blocks} \\gets \\textit{blocks}~ \\cup \\{ b\\}$ \n\t        \\label{alg:CML:addedIncomingBlock}\n\t        \\State $\\textit{buffer} \\gets \\textit{buffer}~ \\setminus \\{ b \\}$\n\t        \\State $\\textit{deliver\\_blocks}()$\n\t    \\EndFor\n\t   \n\t        \\If{$p= \\textit{leader}(\\textit{cordial\\_round}())$} \\Comment{Cordial leader sends packaged new block to all responsive miners}  \\label{alg:CMS:codrialLeader} \n\t       \n\t       \n\t            \\State $b \\gets \\textit{create\\_block}(\\textit{blocks\\_prefix}(\\textit{cordial\\_round}()))$  \n\t\t\t    \\For{$q \\in \\Pi \\wedge q \\ne p \\wedge  \\textit{responsive}(q) $}\n\t\t\t      \\label{alg:CMAbd:isCordial}\n\t            \\State $\\textit{package} \\gets [b] \\setminus [\\textit{history}[q]]$ \n\t           \n\t           \n\t\t        \\label{alg:CMS:CreatePackage}\n\t\t        \\State $\\textit{history}[q] \\gets \\textit{history}[q] \\cup \\textit{package}$ \\label{alg:CMS:historySendPackage}\n\t\t        \\State \\textbf{send} \\textit{package} to $q$\n\t\t        \\EndFor\n\t\t    \\EndIf\n            \\If{$p \\neq \\textit{leader}(\\textit{depth}(\\textit{blocks})+1)~ \\vee$ \\textit{timeout}()}   \\Comment{Non-leader sends new block to leader} \n            \\udi{It seems that this can be taken erroneously due to a Byzantine block}\n           \n                \\State $b \\gets \\textit{create\\_block}(\\textit{blocks}))$  \n\t            \\State $\\textit{history}[q] \\gets \\textit{history}[q] \\cup \\{b\\}$ \\label{alg:CMS:historySendBlock}\n\t\t        \\State \\textbf{send} $b$ to \\textit{leader}(\\textit{depth}$(b)$)\n            \\EndIf     \n           \n\t\\EndWhile\t   \n    \n    \n\n \n  \n          \n        \n \n\n \n \n \n \n      \n        \n        \n        \\Procedure{\\textit{responsive}}{$q$} \\label{alg:CMS:responsive}\n        \\Return  \n        $\\forall b \\exists b'\\in \\textit{history}[q] : b.\\textit{creator} =  p \\wedge b'.\\textit{creator} = q \\wedge b \\in [b'] $\n        \\EndProcedure\n      \n      \n   \n    \n     \n      \n   \n        \n      \n\t\t\\alglinenoPush{counter}\n\t\\end{algorithmic}\n\t\n\\end{algorithm*}\n\n\nUnlike the Asynchronous Cordial Miners protocol (Algorithm \\ref{alg:CMA}), which relies on another protocol to perform dissemination, the Synchronous Cordial Miners protocol in Algorithm \\ref{alg:CMS} employs the the blocklace structure to perform dissemination. To achieve that, each miner maintains a \\emph{history} array that records its communication history with the other miners, and updates it upon receiving a block (\\cref{alg:CMS:historyReceive}) and upon sending one (\\cref{alg:CMS:historySend}).\nAn ordinary miner $p$ simply sends the block $b$ it has created, which includes pointers to the blocks $p$ knows, to the leader (\\cref{alg:CMS:CreateBlockPackage}). A leader $p$, on the other hand, sends to each responsive miner $q$ not only the block $b$, but also all the blocks that $p$ knows but, to the best of $p$'s knowledge, $q$ does not, namely $[b] \\setminus [\\textit{history}[q]$ .  This package is constructed in \\cref{alg:CMS:CreatePackage}. Excluded from the closure of $b$ it is the closure of all $q$-blocks known to $p$ and all blocks $p$ has sent to $q$, both recorded in $history[q]$ of miner $p$.\nA miner $q$ is considered \\emph{responsive} (\\cref{alg:CMS:responsive}) by $p$ if $q$ has responded to the last block $p$ has sent it.\n\n\nWhen sending a block to miner $q$, the miner $p$ uses their communication history to create a package of all blocks $p$ knows, which $q$ may not know (\\cref{alg:CMS:SendPackage}) and sends the whole package.  Based on this, we argue the following:\n\n\n\n\n\n\\begin{proposition}[Algorithm \\ref{alg:CMS} Dissemination]\\label{proposition:sCM-dissemination}\nIn any run of Algorithm \\ref{alg:CMS}, if a correct miner knows a block $b$, then eventually every correct miner will know $b$.\n\\end{proposition}\n\\begin{proof}\nConsider a correct miner $p$ with block $b \\in \\emph{blocks}$, and miner $q$.  If $q$ is correct then eventually $p$ will send a block to $q$, with one of them being a leader.  If at that time the communication history of $p$ with $q$ shows that $q$ knows $p$, we are done. Else, $p$ will include $b$ in its package to $q$, together with all blocks in $[b]$ for which $p$ has no evidence that $q$ knows, based on their communication history.  Then $q$ will eventually receive the package.  The package has no blocks with dangling pointers since $p$ includes in the package everything $q$ might miss, hence $q$ can receive it and include it together with $b$ in its \\emph{blocks}.\n\\end{proof}\n\nUnlike Algorithm \\ref{alg:CMA}, in which block communication is all-to-all, here in Algorithm \\ref{alg:CMS} block communication is from all-to-leader-to-all.  This requires less synchronization, as only leaders need to be cordial, not all miners as in Algorithm \\ref{alg:CMA}, and in the good case reduces message complexity (but not bit complexity), since the leader of each round serves as a relay for all miners for that round.  As a result, the argument for the liveness of the protocol is a bit different.\n\n\\begin{proposition}[Liveness of Synchronous Cordial Miners protocol]\\label{proposition:sCM-liveness}\nAlgorithm \\ref{alg:CMS} satisfies the liveness requirement of Definition \\ref{definition:safety-liveness}.\n\\end{proposition}\n\\begin{proof}\nProposition \\ref{proposition:sCM-dissemination} ensures that every block known to a correct miner will be known to every correct miner.\nConsider a suffix of an infinite computation beyond the control of the adversary. Every miner will get there despite the adversary thanks to the timeout (\\cref{alg:CMS:timeout}).\nIn this synchronous suffix, the probability of a leader block being ratified by its successor is $(\\frac{2}{3})^2$, and  the probability of a leader block being finalized is $(\\frac{2}{3})^3$. \nHence the probability measure of an infinite computation in which no correct leader block being finalized is zero. \n\nHence, for any block $b$ and any point $t$ in the computation, some leader block $b'$ that acknowledges $b$ will be finalized at a point later than $t'$ with probability 1.  Thus for any block $b$ known to a correct miner, there will be a subsequent finalized leader block $b'$ that acknowledges $b$ and hence delivers $b$, satisfying the liveness requirement.\n\\end{proof}\n\n\n\n\\section{Asynchronous Cordial Miners with Blocklace Dissemination}\n\nAlgorithm \\ref{alg:CMAbd} is a hybrid of Algorithm \\ref{alg:CMA} and Algorithm \\ref{alg:CMS}.\nIt employs all-to-all block communication as Algorithm \\ref{alg:CMA}.  But rather than employing an underlying asynchronous data dissemination protocol, it\nperforms dissemination \\emph{en passant}, similarly to Algorithm \\ref{alg:CMS}\nAn ordinary miner $p$ simply sends the block $b$ it has created, which includes pointers to the blocks $p$ knows, to all miners (\\cref{alg:CMS:CreateBlockPackage}). A leader $p$, on the other hand, operates similarly to a leader in Algorithm \\ref{alg:CMA}: It sends to each responsive miner $q$ not only the block $b$, but also all the blocks that $p$ knows but, to the best of $p$'s knowledge, $q$ does not, namely $[b] \\setminus [\\textit{history}[q]$ .  This package is constructed in \\cref{alg:CMS:CreatePackage}. Excluded from the closure of $b$ it is the closure of all $q$-blocks known to $p$ and all bloks $p$ has sent to $q$, both recorded in $history[q]$ of miner $p$.\nA miner $q$ is considered \\emph{responsive} (\\cref{alg:CMS:responsive}) by $p$ if $q$ has responded to the last block $p$ has sent it.\n\n\\begin{algorithm*}[t]\n\t\\caption{\\textbf{Cordial Miners\\xspace: Asynchronous Protocol with Blocklace Dissemination}\\\\ pseudocode for miner $p \\in \\Pi$, including Algorithms \\ref{alg:blocklace} \\& \\ref{alg:CMO}}\n\t\\label{alg:CMAbd}\n\t\\small\n\t\\begin{algorithmic}[1]\n\t\\alglinenoPop{counter} \n\t\t\n\n\t\\Statex \\textbf{Local variables:}\n\t\\StateX array $\\textit{history}[n]$, initially $\\forall k \\in [n] \\colon \\textit{history}[k] \\gets \\{ \\}$  \\Comment{Communication history of $p$ with others} \n\n\t\\vspace{0.5em}\n\t\\Upon{\\textbf{receipt} of $b$} \\Comment{\\textbf{send} and \\textbf{receipt} are simple messaging on a reliable link}\n\t\\State $\\textit{buffer} \\gets \\textit{buffer} \\cup \\{ b \\}$  \\label{alg:CMA:buffer}\n\t\\EndUpon\n\t\n\t\\While{True}\n\t\\For{$b \\in \\textit{buffer} \\colon \\textit{b.pointers} \\subseteq \\textit{hash}(\\textit{blocks})\n\t\\wedge \\textit{cordial\\_block}(b)$\n\t    } \\Comment{Accept cordial block with no dangling pointers}\n\t    \\State $\\textit{buffer} \\gets \\textit{buffer}~ \\setminus \\{ b \\}$\n    \t\\State $\\textit{blocks} \\gets \\textit{blocks}~ \\cup \\{ b\\}$ \t\\label{alg:CMAbd:receive}\n        \\State $\\textit{history}[b.\\textit{creator}] \\gets \\textit{history}[b.\\textit{creator}] \\cup \\{ b \\}$ \\label{alg:CMAbd:historyReceive}\n\n    \t\\State $\\textit{deliver\\_blocks}()$\n\t\\EndFor\n\t    \n\t    \\If{ $\\textit{cordial\\_round}()$}   \\label{alg:CMA:isCordial} \\Comment{Defined at \\cref{alg:CMO:cr}}\n\t    \\State $b \\gets \\textit{create\\_block}(\\textit{blocks\\_prefix}(\\textit{cordial\\_round}()))$   \\Comment{Create a cordial block} \n\t\t\t\\For{$q \\in \\Pi \\wedge q \\ne p $} \n\t\t\t \\If{$p = \\textit{leader}(\\textit{depth}(b)) \\wedge \\textit{responsive}(q)$}  \\Comment{Defined at \\cref{alg:CMS:responsive}} \\label{alg:CMAbd:isCordial}\n\t        \\State $\\textit{package} \\gets [b] \\setminus [\\textit{history}[q]]$\n\t\t\\label{alg:CMS:CreatePackage}\n\t\n\t\t\\Else{  $\\textit{package} \\gets b$} \t\\label{alg:CMS:CreateBlockPackage}\n\t\t \\EndIf\n\t\t \\State $\\textit{history}[q] \\gets \\textit{history}[q] \\cup \\textit{package}$ \\label{alg:CMS:historySend}\n\t\t \\State \\textbf{send} \\textit{package} to $q$\n\t\t   \\EndFor\n\t\t   \\EndIf\n\t \\EndWhile\n\t\t\\alglinenoPush{counter}\n\t\\end{algorithmic}\n\t\n\\end{algorithm*}\n\n\n\n\\begin{proposition}[Algorithm \\ref{alg:CMAbd} Dissemination]\\label{proposition:CMAbd-dissemination}\nIn any run of Algorithm \\ref{alg:CMAbd}, if a correct miner knows a block $b$, then eventually every correct miner will know $b$.\n\\end{proposition}\n\\begin{proof}\nConsider a correct miner $p$ with block $b \\in \\emph{blocks}$, and miner $q$.  If $q$ is correct then eventually it will send a block to $p$, and consider a subsequent round $r$ in which $p$ is a cordial leader. If at round $r$ communication history of $p$ with $q$ shows that $q$ knows $p$, we are done. Else, $p$ will include $b$ in its package to $q$, together with all blocks in $[b]$ for which $p$ has no evidence that $q$ knows, based on their communication history.  Then $q$ will eventually receive the package.  The package has no blocks with dangling pointers since $p$ includes in the package everything $q$ might miss, hence $q$ can receive it and include it together with $b$ in its \\emph{blocks}.\n\\end{proof}\n\n\\begin{proposition}[Liveness of Asynchronous Cordial Miners protocol with Blocklace Dissemination]\\label{proposition:CMAbd-liveness}\nAlgorithm \\ref{alg:CMAbd} satisfies the liveness requirement of Definition \\ref{definition:safety-liveness}.\n\\end{proposition}\n\\begin{proof}\nConsider a suffix of an infinite computation of Algorithm \\ref{alg:CMAbd} beyond the control of the adversary.  According to Proposition \\ref{proposition:CMAbd-dissemination}, every block $b$ known to a miner will be known eventually to every miner. \nThe rest of the proof is identical to the proof of Proposition \\ref{proposition:aCM-liveness}.\n\\end{proof}\n\n\n\n\\section{Optimizations}\n\nThe optimizations in Algorithm \\ref{alg:CMOpt} apply to all Cordial Miners  protocols.  They  exclude  miners that are faulty, either by equivocation or by  not being cordial.  These can be easily taken further in various directions.  A simple example is to improve leader utilization by bypassing immediately (without timeout) rounds with a faulty leader.\n\n\\begin{algorithm*}[t]\n\t\\caption{\\textbf{Cordial Miners\\xspace:  Optimizations} \\\\ pseudocode for miner $p \\in \\Pi$}\n\t\\label{alg:CMOpt}\n\t\\small\n\t\\begin{algorithmic}[1]\n\t\\alglinenoPop{counter} \n\t\n\t\t \n        \\vspace{0.5em}\n        \\Procedure{\\textit{proper}}{$b$}\n        \\Return  \n        $\\textit{b.pointers} \\subseteq \\textit{hash}(\\textit{blocks}) \\wedge \n        (\\textit{leader}(b) \\rightarrow \\textit{cordial\\_block}(b)) \\wedge\n        \\lnot \\textit{equivocator}(b.\\textit{creator})$\n        \\EndProcedure\n        \n\t\t\\vspace{0.5em}\n\t\t\\Procedure{\\textit{send\\_blocks}}{$q$} \n\t\t\\If{$\\textit{equivocator}(q) \\vee \\textit{nonresponsive}(q) $} \\Return \\Comment{Don't bother sending to Byzantine or nonresponsive miners}\n\t\t\\EndIf \n\t    \\State $\\ldots$\n\t\t\\EndProcedure\n\t\t\n\t\n        \\vspace{0.5em}\n        \\Procedure{\\textit{proper}}{$b$}\n        \\Return  \n        $\\textit{b.pointers} \\subseteq \\textit{hash}(\\textit{blocks}) \\wedge \n        (\\textit{leader}(b) \\rightarrow \\textit{cordial\\_block}(b)) \\wedge\n        \\lnot \\textit{equivocator}(b.\\textit{creator})$\n        \\EndProcedure\n     \n     \n     \n\t\n\t\t\\vspace{0.5em}  \n\t    \\Procedure{\\textit{equivocator}}{$q$} \\label{alg:SMR:dactor}\n            \\Return \n            $\\exists b_1,b_2 \\in \\textit{blocks} \\wedge\n            b_1.\\textit{creator} = b_2.\\textit{creator} = q \\wedge\n          \\textit{equivocation}(b_1,b_2)\n           $ \\Comment{See Figure \\ref{figure:approve-sr}.A}\n        \\EndProcedure\n     \n        \\vspace{0.5em}\n\n      \n\t\t\\alglinenoPush{counter}\n\t\\end{algorithmic}\n\t\n\\end{algorithm*}\n\n\n\n\n\n\\section{Performance analysis}\nFor the Synchronous Cordial Miners protocol, every block in the blocklace is sent from the miner that creates it to the leader.\nIn the worst-case, there is linear a number of Byzantine leaders in a row, meaning at most that each block is sent $O(n^2)$ times.\nThe block size is linear, due to its hash pointers, meaning a bit complexity of $O(n^3)$ per block.\nThe leader sends to all miners the blocks it receives, i.e., it sends a linear number of blocks, each linear in size, resulting in a bit complexity of $O(n^3)$ as well.\nBut, if we batch per block a linear number of transactions, when the commit rule is met a quadratic number of transactions is committed each time.\nThus, the amortized bit complexity per decision is $O(n)$.\nHotStuff also achieves this this complexity in the good-case, i.e., when all miners are synchronized to the same round.\n\nFor the Asynchronous Cordial Miners protocol, each block is disseminated using an $O(n^2)$ message complexity dissemination protocol.\nSince each block is linear-sized, the bit complexity of the dissemination each node is $O(n^3)$.\nIn a similar way to the synchronous protocol, each block can also be batched with a linear number of transactions.\nThus, when the commit rule is met, a quadratic number of transactions is committed, leading to an amortized $O(n)$ bit complexity per decision as well.\nThis is the same amortized bit complexity as DAG-Rider.\n\n\\section{Conclusions}\n\nThe Cordial Miners protocol family is simple and diverse.  We believe simplicity has many ramifications when practical applications are considered:  Simpler algorithms are easier to debug, to optimize, to  make robust, and to extend.\n\nAn interesting next step towards making Cordial Miners a useful foundation for cryptocurrencies is to design a mechanism that will encourage miners to cooperate---as needed by these protocols---as opposed to compete, which is the current standard in mainstream cryptocurrencies.\n\n\n\n\n\n\\newpage\n\\bibliographystyle{ACM-Reference-Format}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nOngoing and upcoming large galaxy surveys~\\cite{Aghamousa:2016zmz,Laureijs:2011gra,Hill:2008mv,Adams:2010un,Dawson:2015wdb,Ellis:2012rn,Spergel:2013tha} provide us a wealth of information about the initial condition of density perturbations. However, accurately modeling the low redshift matter distribution is a challenge in modern cosmology, as the history of structure formation suffers from the nonlinear evolution of density fields. Although various improvement has been made for perturbation theories~\\cite{10.1093\/mnras\/203.2.345,Fry:1983cj,Suto:1990wf,Makino:1991rp,Jain:1993jh,Scoccimarro:1996se,Bernardeau:2001qr,Crocce:2005xy,McDonald:2006hf,McDonald:2006hf,Crocce:2007dt,Taruya:2007xy,Matarrese:2007aj,Matarrese:2007wc,Takahashi:2008yk,Bernardeau:2008fa,Carlson:2009it,Shoji:2009gg,Matsubara:2007wj,Pietroni:2011iz,Tassev:2011ac,Taruya:2012ut,Sugiyama:2013mpa}, small scale matter power spectrum is still difficult to model without additional fitting parameters~\\cite{Baumann:2010tm,Carrasco:2012cv,Pajer:2013jj,Manzotti:2014loa,Carroll:2013oxa,Porto:2013qua}.\n$N$-body simulations that numerically solve the evolution of all representative matter particles in the Universe can be a solution.\nHowever, this is computationally expensive, making it challenging to browse all possible parameter spaces to find the best fit cosmological parameters to the observed data.\nAlternatively, a method to reduce the nonlinear effect in the observed galaxy distribution can potentially ease modeling the late time field and extracting cosmological information from large galaxy surveys. \n\n\nBaryon acoustic oscillation~(BAO) reconstruction has been developed in this context~\\cite{Eisenstein:2006nk}, and a series of both numerical and analytic investigations have been conducted to understand and utilize the method~\\cite{Padmanabhan:2008dd,Noh:2009bb,Tassev:2012hu,White:2015eaa,Schmittfull:2015mja,Seo:2015eyw,Hikage:2017tmm,Schmittfull:2017uhh,Chen:2019lpf,Hada:2018fde,Hada:2018ziy}.\nThe standard BAO reconstruction pioneered by Ref.~\\cite{Eisenstein:2006nk} showed that shifting the observed particles along the gradient of the smoothed density efficiently recovers the linear BAO signals.\nThe standard reconstruction works because we reduce the degradation effect due to the free streaming by minimizing displacement that the particles traveled by moving them back.\nThe reconstructed field shows an augmented correlation with the initial field, but it is not quite the same as the initial linear density field because it cannot fully reverse all the nonlinearities~\\cite{Padmanabhan:2008dd,Schmittfull:2015mja}. \nIn other words, the BAO reconstruction partially brings the cosmological information of higher-order statistics back to the two-point statistics, making the reconstructed field closer to be Gaussian~\\cite{Schmittfull:2015mja}.\nThe resulting discrepancy returns the broadband spectrum and redshift space distortion~(RSD) that deviates from the linear prediction and the prediction of the nonlinear density field. \nThis procedure introduces a challenge from the modeling perspective,\nconsequently making it difficult to combine the BAO analysis with other large-scale structure analyses. Recently, there have been a few promising studies that address the challenge and construct perturbation theory models of postreconstruction full clustering~\\cite{White:2015eaa,Hikage:2017tmm,Chen:2019lpf}. In parallel, there have been many studies on initial density field reconstruction by forward modeling~\\cite{Lavaux:2019fjr,Schmidt:2020viy,Nguyen:2020yuc,Seljak:2017rmr,Feng:2018for,Modi:2021acq}. However, we should also note that, in principle, we cannot recover the initial field without solving the equation of motion~(EoM) backward, which is not reasonably possible due to the shell crossing. \nThen a natural question arises: can we find a more consistently defined late time field that is more correlated with the initial condition and less degraded than the density perturbations and try to model the field rather than reconstruct such a patchy initial density field?\n\n\nIn this work, we argue that the Lagrangian displacement field can be an alternative to reconstructing matter density perturbations. As our $N$-body simulations show in the top panel of Fig.~\\ref{fig00}, the late time displacement divergence and initial linear density field are indeed 99\\% correlated for $k{\\rm Mpc}\/h\\lesssim 0.2$ and 95\\% for $k{\\rm Mpc}\/h \\lesssim 0.5$ at $z=0.6$~\\cite{Baldauf:2015tla}~(see Sec.~\\ref{simsec} and Sec.~\\ref{sec5} for the details of our simulations.). \nMoreover, the top and middle panels imply that the power spectrum is amplified rather than damped. \nThus, both the nonlinear instability and degradation effect are reduced for the displacement field compared to the density field, and we can potentially extract a wealth of information from the displacement field if they can be precisely measured and accurately modeled.\nIn practice, measuring the displacement fields for actual surveys is nontrivial because we do not know the initial galaxy positions a priori.\nHence, we are interested in reconstructing the displacement field from the observed galaxy distributions, which was the motivation of Ref.~\\cite{Schmittfull:2017uhh}, while their primary focus was the BAO feature.  \nIf we could adequately reconstruct the displacement field, the broadband power spectrum would also be consistently reconstructed. In this paper, we construct a model for the broadband power spectrum for the reconstructed displacement field.\n\nWhile there have been a few comparable approaches developed to iteratively reconstruct the nonlinear displacement field~\\cite{Zhu:2016xyy,Zhu:2016sjc,Hada:2018fde}, the ``iterative reconstruction'' introduced by Ref.~\\cite{Schmittfull:2017uhh} iteratively moves the particles along the density gradient by progressively reducing the smoothing radius step by step until we get the almost zero density field.\nThe final position is an estimated Lagrangian position, and the displacement from the original, i.e., the observed Eulerian position, is the reconstructed displacement field.\nRef.~\\cite{Schmittfull:2017uhh} simulated the postreconstruction displacement field and showed that it is 95$\\%$ correlated with the initial linear density up to $k{\\rm Mpc}\/h=$0.35 at $z$=0.\n\nAlthough the top panel in Fig.~\\ref{fig00} shows the linearity of both the displacement field and post-iterative reconstruction field, it does not necessarily mean that we can model those fields within the linear theory.\nThe displacement field power spectrum has about an 8\\% discrepancy from the linear matter field at $k{\\rm Mpc}\/h=0.2$ due to the shift term, i.e., the term correlated directly with the linear density field as shown in the middle panel of Fig.~\\ref{fig00}.\nIn addition, the post-iterative reconstruction estimator does not perfectly recover the true displacement field even for the BAO scale.\nOn the other hand, the bottom panel shows that the power spectrum of the errors from the linear matter field is similar to the nonlinear\/reconstructed displacement field despite the discrepancy in the power spectrum in the middle panel.\nThus, the relation among those fields is not simple.\n\nThe authors of Ref.~\\cite{Schmittfull:2017uhh} called the displacement field reconstruction ``$\\mathcal O(1)$'' reconstruction, and they further reconstruct the initial density field from the reconstructed displacement field perturbatively, calling it ``$\\mathcal O(2)$''. \nThis paper takes a rigorous approach to directly model the ``$\\mathcal O(1)$'' estimator using perturbation theory.\nWe present a theoretical modeling of the reconstruction estimator in Lagrangian perturbation theory~(LPT) up to third order for dark matter without redshift space distortions just for simplicity.\nRef.~\\cite{Schmittfull:2017uhh} presented a motivation for perturbation theory modeling for their iterative reconstruction.\nHowever, they did not consider modeling the iterative steps but provided a way to find the Zel'dovich displacement from the measured density.\nThis paper will model the iterative steps explicitly and find $n$-th step displacement field up to 1-loop order perturbations.\n\n\n\nThis paper is organized as follows.\nSec.~\\ref{simsec} provides the details of our two simulations and their measurements.\nIn Sec.~\\ref{sec2}, we explain the difficulty of perturbation theory for the displacement field towards an application to the iterative reconstruction.\nSec.~\\ref{moderec} provides the reconstruction algorithm in the LPT perspective.\nIn, Sec.~\\ref{sec4}, we present our LPT modeling of the iterative reconstruction and give a comparison with the simulations in Sec.~\\ref{sec5}.\nThen we give a summary and conclusions in the last section.\n\n\n\\begin{figure}\n  \\includegraphics[width=\\linewidth]{nl-disp-lin_00}\n  \\caption{\\textit{Top}: The cross correlation coefficient $r_X\\equiv P_{X\\delta_{\\rm L}}\/\\sqrt{P_X P_{{\\rm L}}}$ for the nonlinear density field $\\delta_{\\rm NL}$, nonlinear displacement field potential $\\phi$, standard reconstruction estimator $\\delta_{\\rm rec}$ and 9th step iterative reconstruction estimator $\\phi_{\\rm rec}^{(9)}$.\n   The nonlinear displacement field is 95\\% correlated with the linear field for $k{\\rm Mpc}\/h\\lesssim 0.5$, which is better than that for the standard density field reconstruction for $R=20$Mpc\/$h$.\n   \\textit{Middle}: power spectrum of the nonlinear fields normalized by the linear matter power spectrum $P_{\\rm L}$. The shift term is not reconstructed well in the iterative reconstruction.\n   \\textit{Bottom:} $E(X,\\delta_{\\rm L})$ is the power spectrum of the error field, which is the difference between the nonlinear\/reconstructed displacement field and $\\delta_{\\rm L}$ normalized by $P_{\\rm L}$~(see Sec.~\\ref{simsec} and \\ref{sec5} for the details of simulations.).}\n  \\label{fig00}\n\\end{figure}\n\n\n\n\\section{Measurement of the simulated displacements}\\label{simsec}\n\n\n\\begin{table*}\n\\caption{\\label{table1}Simulation and sampling parameters. \nBoth simulations assume a flat $\\Lambda$CDM cosmology based on Ref.~\\cite{Ade:2015xua} with $\\Omega_{\\rm m} = 0.3075$, $\\Omega_{\\rm b}h^2=0.0223$, $h=0.6774$, and $\\sigma_8=0.8159$.}\n\\begin{ruledtabular}\n\\begin{tabular}{lccccc}\nName & subsampling \\% & \\# of meshes  & Box size [Mpc$\/h]^3$ & \\# of particles &\\# of simulations \\\\\n\\hline\nL500 &4 & $512^3$  & $500^3$ & $1536^3$ & 5\\\\\nsubL500 &0.15 & $512^3$  & $500^3$ & $1536^3$ & 5\\\\\nL1500 &4 \t & $1024^3$ & $1500^3$ & $1536^3$ &1  \\\\\nfullL1500 &100  & $1536^3$  & $1500^3$ & $1536^3$ &1 \\\\\n\\end{tabular}\n\\end{ruledtabular}\n\\end{table*}\n\nTo compare our theoretical model with realistic nonlinear\/reconstructed displacement fields, we need to minimize nonphysical numerical artifacts in measuring the simulated displacement fields. \nIn this section, we explain the setup of our numerical simulations.\nReaders who are only interested in the theoretical implementation can skip this section.\n\n\n\n\\medskip\nMeasuring the displacement field or velocity field is challenging due to these vector fields' discrete and nonuniform sampling. \nMeasuring the displacement using mass tracers gives the mass-weighted displacement field, giving no measurement in the locations without mass, while perturbation theories derive a volume-weighted quantity. \nThe Delaunay tessellation method (e.g., Ref.~\\cite{Pueblas:2008uv}) works for a volume-weighted measurement, which is more robust to the discreteness effect. \nIn this paper, however, we adopt the mass-weighted measurement, following the convention of Ref.~\\cite{Schmittfull:2017uhh}, for both calculating the Lagrangian displacement field and the reconstructed displacement field.\nAs a caveat, one could instead directly model the mass-weighted quantity in perturbation theory~\\cite{Vlah:2012ni}.\nWe test the sanity of the mass-weighted measurement using a convergence test as a function of various levels of discreteness effect against a reference simulation without the discreteness effect. \nThe mass-weighted displacement can be written as\n\\def \\mathbf x{\\mathbf x}\n\\begin{align}\n\t\\mathbf \\Psi^{\\rm obs.}_p \n\t&= \\frac{\\sum_i W_{\\rm CIC}({\\mathbf x_p,\\mathbf x_i}) \\mathbf \\Psi^{\\rm obs.}(\\mathbf x_i)  }{\\sum_i W_{\\rm CIC}({\\mathbf x_p,\\mathbf x_i}) }\n\t,\\label{mweight}\n\\end{align}\nwhere $\\mathbf \\Psi^{\\rm obs.}(\\mathbf x_i)$ is the observed displacement field of $i$-th particle, $W_{\\rm CIC}$ is the pixel window function indicating that we are using the Cloud-in-Cell assignment, and $\\mathbf \\Psi^{\\rm obs.}_p$ is the mass-weighted displacement field for a pixel centered at $\\mathbf x_p$. For pixels with no particles found, we incorrectly set $\\mathbf \\Psi^{\\rm obs.}_p=0$. \nThis operation biases the result; the overall amplitude is reduced by the fraction of the zeroed pixels, which we correct with a simple multiplicative rescaling. However, this prescription also introduces a spurious small-scale power correlated with the pixel window function and the nonuniform particle spacing, which we cannot straightforwardly correct. Note that $\\mathbf \\Psi$ is evaluated at the initial Lagrangian position, which is fixed at $\\mathbf x_{p}$, and that this is fundamentally different from estimating the velocity divergence field, which we evaluate in the Eulerian position.\n\nWe show the subsampling parameters~(i.e., particle spacing) and pixel window function of our two numerical simulations in Tab.~\\ref{table1}. Simulations assume a flat $\\Lambda$\\rm{CDM}~cosmology based on Ref.~\\cite{Ade:2015xua} with $\\Omega_{\\rm m} = 0.3075$, $\\Omega_{\\rm b}h^2=0.0223$, $h=0.6774$, and $\\sigma_8=0.8159$. \nThis is the same cosmology used in Ref.~\\cite{Schmittfull:2017uhh}.  \nFull $N$-body simulations were produced using the MP-Gadget~\\cite{Springel:2000yr,Springel:2005mi,2018zndo...1451799F} with the box size of $500{\\rm Mpc}\/h$ and $1500{\\rm Mpc}\/h$. For $500{\\rm Mpc}\/h$, we use the average of five simulations. Both simulations use a particle resolution of $1536^3$. The simulation evolves $1536^3$ particles from $z=99$ by computing forces in a grid of $1536^3$, and we subsample 4\\% of the output particles at $z=0.6$ to make L1500 (for a box of $1500{\\rm Mpc}\/h$) and L500 (for a box of $500{\\rm Mpc}\/h$). We use a grid of $512^3$ to Fourier-transform, reconstruct this nonlinear field for L500 and subL500, and use a grid of $1024^3$ for fullL1500 and L1500.\n\n\nHere, fullL1500 is the simulation of $1500{\\rm Mpc}\/h$ without subsampling, therefore uniformly being distributed in the Lagrangian position. The mass-weighted displacement for this simulation should be equivalent to the volume-weighted measurement, and we expect no discreteness and no pixel window function effect for this set. We do not use this complete catalog for reconstruction, as, first, it is computationally expensive for iterative steps, and second, the reconstructed density field will suffer the discreteness and the pixel window function effect even if we used this complete set, as we could not perfectly recover the Lagrangian, uniform position even after reconstruction.\nWe use this complete set as our reference to estimate the convergence of the simulation L500, which is our main set to test the broadband shape after reconstruction.\nWe find that the displacement field of the two simulations is convergent up to  $k{\\rm Mpc}\/h\\lesssim 0.2$ for those mesh resolutions within 1.1\\%. We find that further subsampling and\/or increasing the mesh size breaks this convergence.\nGiven the mesh resolution and the particle resolution of these two simulations in  Tab.~\\ref{table1},  we consider 1 ${\\rm Mpc}\/h$ as the limiting resolution and account for it in our LPT model.\nThe convergence of the displacement field does not straightforwardly imply the convergence of the post-iterative reconstruction displacement field.  \nHowever, in this paper, we shelve the plans for 100\\% sampling for the iterative reconstruction mainly because of our numerical resources.\nTo see the numerical stability of the iterative reconstruction, we also introduced subL500, which is a 0.15\\% sampling of the $512^3$ simulation and the rest of the parameters are the same as those for L500.\n\n\nFor the BAO feature comparison, L500 is not optimal due to its small box size, i.e., $500{\\rm Mpc}\/h$.\nFor this reason, we apply the iterative reconstruction on a pair of L1500, the one generated with an initial condition with BAO and the one generated without BAO with the same white-noise fields~\\cite{Wojtak:2016sxr,Schmittfull:2017uhh,Ding:2017gad}, which will be shown in Fig.~\\ref{fig2}. By pairing and dividing, we cancel out the cosmic variance and the spurious effect due to the discreteness aforementioned.\n\nAs a caveat, note that the observed field from galaxy surveys will be subject to many orders of magnitude more severe discreteness effects than our default simulation L500. Our companion paper~\\cite{Seo:2021} discusses a surrogate reconstruction method to avoid the discreteness in reconstruction. \nIn this paper, however, we focus on modeling the reconstructed displacement in the shot-noiseless limit without redshift space distortion to test the model with minimal complications.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{The UV mistake of 3LPT displacement}\\label{sec2}\n\n\nThis paper aims at modeling the iterative reconstruction up to 1-loop order, which can also be referred to as ``3LPT'' as we expand the equation of motion up to third order in the linear density field.\nWe also call the Zel'dovich approximation, i.e., linear perturbations, ``1LPT'' in this paper.\n\nOne might wonder if the perturbation theory for the displacement field is more convergent and more manageable than that for the density field, since we mentioned that the simulated nonlinear displacement field is more correlated with the linear field than the density field.\nHowever, it does not work as we expected because of the ``UV-mistake'', which we explain in this section.\nThe UV mistake, first pointed out by Ref.~\\cite{Baldauf:2015tla}, is an issue about the cut-off dependence of a loop calculation for the displacement field.\n\n\\medskip\nLet $\\mathbf x$ and $\\mathbf q$ be the Eulerian and Lagrangian coordinates.\nThe relation between these two coordinates is written as \n\\begin{align}\n\t\\mathbf x = \\mathbf q + \\mathbf \\Psi(\\mathbf q),\\label{psidef}\n\\end{align}\nand $\\mathbf \\Psi$ is called the displacement field.\nThe Eulerian density perturbation is exponentiated by the Lagrangian displacement field as~\\cite{Matsubara:2007wj}\n\t\\begin{align}\n   \\delta_{\\rm NL}(\\mathbf x)\n    &=  \\int \\frac{d^3kd^3 q}{(2\\pi)^3}  e^{i\\mathbf k\\cdot (\\mathbf x-\\mathbf q)} \\left [ e^{ -i\\mathbf k\\cdot \\mathbf \\Psi(\\mathbf q)} \n    -\n    1\\right].\\label{14}\n\\end{align}\n\n\n\n\n\\medskip\nWe introduce the displacement field potential $\\phi$ that satisfies $i\\mathbf k\\phi =  \\mathbf \\Psi$ in Fourier space.\nThe 1LPT spectrum of $\\phi$ is straightforwardly written by the linear matter power spectrum $P_{\\rm L}$:\n\\begin{align}\n    P^{\\rm 1LPT}_{\\phi}=& \\frac{P_{\\rm L}}{k^4},\n\\end{align}\nThen, the 3LPT power spectrum is given as\n\\begin{align}\n    P^{\\rm 3LPT}_{\\phi}=& P^{\\rm 1LPT}_{\\phi}+ P_{\\phi13}+ P_{\\phi22} \\label{LPT:disp},\n\\end{align}\nwhere we defined~\\cite{Baldauf:2015tla}\n\\begin{align}\nP_{\\phi22}=& \\frac{9}{98}\\int  dxd\\mu \\frac{P_{\\rm L}(kx)P_{\\rm L}(ky)}{4\\pi^2k}\n\t\t \\frac{x^2(1-\\mu^2)^2}{y^4},\\label{P22}\\\\\nP_{\\phi13}=&\n\\frac{10}{21}P_{\\rm L}(k)\\int  dxd\\mu  \\frac{P_{\\rm L}(k x)}{4\\pi^2k}   \n\t\t \\frac{x^2(1-\\mu^2)^2}{y^2 },\n\t\t\\label{P13}\n\\end{align}\nwith $y\\equiv(1 - 2x \\mu+x^2)^{1\/2}$.\nThe integral variables $x$ and $\\mu$ run from $0$ to $\\infty$ and $-1$ to 1 respectively.\nWe show $P^{\\rm 3LPT}_{\\phi}$, $P_{\\phi13}$ and $P_{\\phi22}$ and their standard perturbation theory~(SPT) counterpart for the density field in Fig.~\\ref{fig0}~(see e.g., Ref.~\\cite{Bernardeau:2001qr} for the SPT 1-loop term).\nThe 3LPT correction is dominated by $P_{\\phi13}$, which is about 10\\% correction to the linear power spectrum even at the BAO scale.\nWe explain why $P_{\\phi13}\\gg P_{\\phi22}$ as follows.\nIn the large loop momentum $x\\to \\infty$ limit, we get~\\cite{Baldauf:2015tla}\n\\begin{align}\n\tk^4P_{\\phi22}\\to & \\frac{24}{245}\\frac{k^3}{4\\pi^2}\\int  \\frac{dx}{x^2} P_{\\rm L}(kx)^2\\, ,\\\\\nk^4P_{\\phi13} \\to &\n\t\t \t\t \\frac{32}{63}P_{\\rm L}(k)\\frac{k^3}{4\\pi^2}\\int dx P_{\\rm L}(k x) \\label{UVmistake}\\, ,\n\\end{align}\nand the integral in Eq.~\\eqref{UVmistake} is sensitive to the large $x$ contributions.\nAs we see the gray curve in Fig.~\\ref{fig0}, $P_{\\phi}^{\\rm 1LPT}$ overestimates the displacement field at high $k$, and thus the loop integral contains the incorrect UV modes.\nAs a result, $P_{\\phi13}$, i.e., the shift term is overestimated.\nThis is why the authors in Ref.~\\cite{Baldauf:2015tla} called this problem ``UV-mistake''.\nThe UV mistake around the BAO scale is specific to the displacement field.\nWhile we have the integral like Eq.~\\eqref{UVmistake} in the 3SPT corrections, they are canceled, and the net correction is suppressed.\nOn the other hand, the fact that $|P_{\\phi 13}|\\gg |P_{\\delta13}|$, while $P_{\\phi 13}$ is positive and $P_{\\delta13}$ is negative, and $P_{\\phi 22}\\ll P_{\\delta22}$ is the reason that the displacement field is correlated with the linear field very well and that the phase shift in the BAO is negligible for the displacement field.\n\n\n\n\n\\begin{figure}\n  \\includegraphics[width=\\linewidth]{lptspt.pdf}\n  \\caption{\\textit{Top}: Comparison of the simulation displacement~(gray) and density field~(black), the 1-loop displacement power sepctrum~(blue) and 1-loop density power spectrum~(orange). We evaluated at $z$=0.6. The second order LPT is shown in pink.\n  The EFT~(green) is defined in Eq.~\\eqref{EFT1}.\n  All power spectra are normalized by the linear matter power spectrum to better illustrate the difference.\n    \\textit{Bottom}: 1-loop corrections for the density~(SPT) and displacement fields~(LPT) are shown. While $|P_{\\delta13}| \\sim P_{\\delta22} \\gg k^4P_{\\phi13}\\gg k^4P_{\\phi22}$, we have $P_{\\rm L}\\sim P^{\\rm 3SPT}_{\\delta}<k^4P^{\\rm 3LPT}_{\\phi}$ for $k{\\rm Mpc}\/h<0.1$, and thus the low $k$ displacement field is sensitive to the high $k$ loop contributions.}\n  \\label{fig0}\n\\end{figure}\n\n\n\\medskip\nWhy do we have the UV mistake?\nIn the LPT, we solve the following equation of motion (EoM)~\\cite{Bernardeau:2001qr}:\n\\begin{align}\n\t\\frac{d^2 \\mathbf \\Psi}{d\\eta^2} - \\mathcal H \\frac{d\\mathbf \\Psi}{d\\eta} &= -\\nabla_{ x} \\phi_{\\rm N}(\\mathbf q+\\mathbf \\Psi),\\label{eom1}\n\\end{align}\nwhere $\\mathcal H$, $\\phi_{\\rm N}$, and $\\eta$ are the conformal Hubble parameter, the gravitational potential, and the conformal time, respectively.\nThis equation is based on Newton's equation of a point particle in the gravitational potential.\nTo close the EoM, we impose the Poisson equation\n\\begin{align}\n\t\\nabla_x^2 \\phi_{\\rm N}  &= \\frac{3}{2}\\mathcal H^2 \\delta_{\\rm NL}.\\label{HC}\n\\end{align}\nThe RHS of Eq.~\\eqref{HC} assumes the pressure-less perfect fluid approximation, which ignores the gravitational interactions between the individual fluid elements.\nThe UV mistake is a consequence of this simplification, i.e., gravitational scattering of the individual fluid elements happens at a small scale, which affects the large-scale configuration of the elements. Those effects are canceled out to some extent for the density field but are not for the displacement.\n$N$-body simulations (or more advanced hydrodynamical simulations) can directly account for these effects, but their analytic implementation is generally complicated and impossible.\nOne solution to this problem is to use an effective field theory, where we introduce all possible higher-order gradient corrections to the energy-momentum tensor and modify Eq.~\\eqref{HC} to include the many-body effects.\nThe 1-loop level effective displacement field is written as~\\cite{Porto:2013qua}\n\\begin{align}\n\t\\mathbf \\Psi_{\\rm eff.} = (1+\\alpha k^2  )\\mathbf  \\Psi_{(1)} + \\mathbf  \\Psi_{(2)} + \\mathbf \\Psi_{(3)} +\\cdots,\\label{tobias_eft}\n\\end{align}\nand $\\alpha \\sim -1$ at $z=0.6$ is read from Fig.~11 in Ref.~\\cite{Baldauf:2015tla}, where a subscript means the order of the perturbations.\nNote that we take the upper bound of the loop integral $k_{\\rm max}$ as $k_{\\rm max}\/k = \\infty$.\nIn this paper, we fit $\\langle \\delta_{\\rm L}\\phi_{\\rm sim.} \\rangle\/\\langle \\delta_{\\rm L}^2 \\rangle $ with\n\\begin{align}\n\tf(k;\\alpha)\n\t \\equiv 1+ \\alpha k^2 +  \\frac{P_{\\phi13}}{2P^{\\rm 1LPT}_\\phi} ,\\label{EFT1}\n\\end{align}\nusing the data in $0.02\\lesssim k{\\rm Mpc}\/h\\lesssim 0.2$, and get $\\alpha\\sim-1.27$, which is slightly different from $\\alpha$ in Ref.~\\cite{Baldauf:2015tla}.\nThe sample variance can be canceled by dividing the nonlinear spectrum by the initial realizations in Eq.~\\eqref{EFT1}.\nThe advantage of fitting the cross power spectrum instead of the auto power spectrum is to avoid overfitting noise and $P_{\\phi22}$.\nOur EFT is shown in Fig.~\\ref{fig0} as the green dotted curve.\nNote that the authors in the reference got $\\alpha$ by fitting the data at the field level while we got them at the power spectrum level for simplicity.\nThe comparison of these two approaches and the overfitting issue are discussed in their paper.\n\n\n\n\n\\medskip\nTo summarize, the 3LPT, mainly the shift term, suffers the ``UV mistake,'' which requires correcting the mistake beyond the perturbation theory framework.\nDo we also have the same problem with the LPT reconstruction? \nIndeed, we will show that the iterative procedure partly cancels the UV mistake, and the postreconstruction displacement field agrees with the true displacement field up to much higher $k$.\n\n\n\n\\section{Displacement field reconstruction}\\label{moderec}\n\nIn this section, we review the standard and iterative reconstruction algorithm from the viewpoint of the Lagrangian perturbation theory, following Ref.~\\cite{Schmittfull:2017uhh}\n\n\\subsection{Standard reconstruction algorithm}\\label{stdbao}\n\nThe standard density field reconstruction algorithm is summarized as follows~\\cite{Eisenstein:2006nk,Padmanabhan:2008dd}: \n\\begin{enumerate}\n\t\\item Firstly, measure the density field $\\delta_{\\rm NL}$ in the Eulerian frame $\\mathbf x$. The Lagrangian expression of the nonlinear density field is given in Eq.~\\eqref{14}.\nThe Fourier transform of the density field is then\n\t  \\begin{align}\n     \\delta_{\\rm NL}(\\mathbf k)\n    &=   \\int d^3 qe^{-i\\mathbf k\\cdot \\mathbf q} \\left [ e^{ -i\\mathbf k\\cdot \\mathbf \\Psi(\\mathbf q)} \n    -\n    1\\right].\\label{17}\n\\end{align}\n\n\t\\item Smooth the density field as $  \\delta_{\\rm NL}(\\mathbf k) \\to S(k)  \\delta_{\\rm NL}(\\mathbf k)$. $S(k)$ is a smoothing window in Fourier space, and we usually choose a Gaussian window. We smooth the density field to remove the nonlinear part of the density field to use the linear continuity equation to estimate the displacement.  \n\t\\item Introduce the shift vector as the gradient of the smoothed density field. In Fourier space this operation is equivalent to \n\t\\begin{align}\n\t\t  {\\mathbf s}(\\mathbf k) \\equiv  \\frac{-i\\mathbf k}{k^{2}}S(k)  \\delta_{\\rm NL}(\\mathbf k).\\label{shiftdef}\n\t\\end{align} \n\t\\item Introduce the displaced density field \n\t\\begin{align}\n\t   \\delta_{\\rm d}(\\mathbf k) \\equiv    \\int d^3 qe^{-i\\mathbf k\\cdot \\mathbf q} \\left [ e^{ -i\\mathbf k\\cdot (\\mathbf \\Psi(\\mathbf q)+\\mathbf s(\\mathbf x))} \n    -\n    1\n    \\right].\\label{23}\n\\end{align}\n\t$\\mathbf s(\\mathbf x)$ is the smoothed linear negative displacement and the linear part of the original displacement field is partly subtracted.\n\tNote that the above Fourier transform is with respect to $\\mathbf q$, while the shift vector in the exponential is defined at $\\mathbf x$~\\cite{Schmittfull:2015mja}.\n\\end{enumerate}\nIn the standard reconstruction, we also introduce the shifted uniform\/reference field \n\\begin{align}\n\t   \\delta_{\\rm s}(\\mathbf k) \\equiv    \\int d^3 qe^{-i\\mathbf k\\cdot \\mathbf q} \\left [ e^{ -i\\mathbf k\\cdot \\mathbf s(\\mathbf q)} \n    -\n    1\n    \\right],\n\\end{align}\nand the reconstruction estimator is defined as \n\\begin{align}\n\t\\delta_{\\rm rec} \\equiv \\delta_{\\rm d} - \\delta_{\\rm s}.\n\\end{align}\nExpanding the estimator with respect to $\\mathbf \\Psi$, we get~\\cite{Padmanabhan:2008dd}\n\\begin{align}\n\t\\delta_{\\rm rec} = -i\\mathbf k\\cdot \\mathbf \\Psi(\\mathbf k) + \\mathcal O(\\Psi^2).\n\\end{align}\nRef.~\\cite{Padmanabhan:2008dd} showed that we could not remove the pure second-order contribution in $\\mathcal O(\\Psi^2)$. Therefore, $\\delta_{\\rm rec}$ does not coincide with the linear field at nonlinear order.\n\n\n\n\n\\subsection{Iterative reconstruction algorithm}\\label{itrbao}\n\nIn the above first step reconstruction, we reduce the original displacement $\\mathbf \\Psi(\\mathbf q)$ to $\\mathbf \\Psi(\\mathbf q)+\\mathbf s(\\mathbf x)$.\nIn Ref.~\\cite{Schmittfull:2017uhh}, they repeat the above steps 1 to 4 by updating the nonlinear field by the displaced density field until we get almost zero displacement in the end.\nHereafter, we put indices in the above equations as\n\\begin{align}\n    \\mathbf \\Psi \\to \\mathbf \\Psi^{(0)},~S\\to S^{(0)},~\\mathbf s \\to \\mathbf s^{(0)}, ~\\delta_{\\rm NL}\\to \\delta^{(0)}_{\\rm NL},\n\\end{align}\nand generalize 0 to $n$.\nWe also write $ \\delta^{(n)}_{\\rm NL}=  \\delta^{(n-1)}_{\\rm d}$.\nWe may rephrase step 4 with respect to the displacement field as follows:\n\\begin{enumerate}\n\\setcounter{enumi}{4}\n\t\\item[4'.] We define the new displacement field\n\t\\begin{align}\n\t\t\\mathbf \\Psi^{(1)}(\\mathbf q) \\equiv  \\mathbf \\Psi^{(0)}(\\mathbf q)+\\mathbf s^{(0)}(\\mathbf q+\\mathbf \\Psi^{(0)}(\\mathbf q)) \\label{dispupd}\n\t\\end{align}  and update the nonlinear field as \n\t\\begin{align}\n   \\delta^{(1)}_{\\rm NL}(\\mathbf x)\n    &\\equiv    \\int \\frac{d^3kd^3 q}{(2\\pi)^3}  e^{i\\mathbf k\\cdot (\\mathbf x-\\mathbf q)} \\left [ e^{ -i\\mathbf k\\cdot \\mathbf \\Psi^{(1)}(\\mathbf q)} \n    -\n    1\\right].\n\\end{align}\n\\end{enumerate}\n\nWe then repeat 1 to 4' until we get $\\delta^{(n)}_{\\rm NL}(\\mathbf x) \\approx 0$. The final positions are the estimated Lagrangian positions where the Eulerian density field is zero. Generalizing Eq.~\\eqref{dispupd}, the $n$-th step displacement field is recursively defined as \n\t\\begin{align}\n\t\t\\mathbf \\Psi^{(n)}(\\mathbf q) \\equiv  \\mathbf \\Psi^{(n-1)}(\\mathbf q)+\\mathbf s^{(n-1)}(\\mathbf q+\\mathbf \\Psi^{(n-1)}(\\mathbf q)).\\label{psin}\n\t\\end{align}\n\tWe reduce the smoothing radius $R$ step by step, and $n$-th step smoothing function is defined as\n\\begin{align}\n\tS^{(n)} \\equiv & \\exp\\left(-\\frac{R_n^2k^2}{4}\\right),\\\\\nR_n\\equiv &\\epsilon^{n} R.\\label{Rdef}\n\\end{align}\n\tNote that the field level smoothing scale is given as $R\/\\sqrt{2}$ in the present convention. \nIn this paper, we use $\\epsilon = 1\/\\sqrt{2}$ unless otherwise stated.\nThe remaining process is summarized as follows.\n\n\\begin{enumerate}\n\\setcounter{enumi}{4}\n\n\t\n\\item Evaluate the total negative displacement:\n\\begin{align}\n\t\\mathbf s^{(n)}_{\\rm tot.}(\\mathbf x) \\equiv &\\mathbf s^{(0)}(\\mathbf q+\\mathbf \\Psi^{(0)}(\\mathbf q)) + \\mathbf s^{(1)}(\\mathbf q+\\mathbf \\Psi^{(1)}(\\mathbf q))\\notag \\\\\n\t&+\\cdots + \\mathbf s^{(n-1)}(\\mathbf q+\\mathbf \\Psi^{(n-1)}(\\mathbf q)).\\label{totn}\n\\end{align}\nNote that $\\mathbf s^{(n)}_{\\rm tot.}$ is evaluated at the initial (i.e., the observed) Eulerian position $\\mathbf x$, so, if the reconstruction is nearly perfect, we reconstruct the negative displacement that satisfies\n\\begin{align}\n\t\\mathbf x +\\mathbf s^{(n)}_{\\rm tot.}(\\mathbf x) = \\mathbf q.\n\\end{align}\n\t\\item Evaluate the negative displacement as a function of the estimated Lagrangian position, which is given by\n\t\\begin{align}\n\t\t{\\mathbf \\Psi}^{(n)}_{\\rm rec}(\\mathbf q) = -\\mathbf s^{(n)}_{\\rm tot.}(\\mathbf q+\\mathbf \\Psi(\\mathbf q)).\n\t\\end{align}  \n\\end{enumerate}\nThis ${\\mathbf \\Psi}^{(n)}_{\\rm rec}$ is the output of the iterative reconstruction.\nWe usually compute the divergence of ${\\mathbf \\Psi}^{(n)}_{\\rm rec}$ to compare it with the density fields.\nThis paper also refers to the displacement field's divergence as ``displacement field'' or $k^2\\phi$ unless otherwise stated.\n\n\n\n\n\\section{Modeling the iterative reconstruction with LPT}\\label{sec4}\n\n\nOur modeling strategy is as follows. Firstly, we put forward a perturbative expansion ansatz of the $n$-th step displacement field in the most general way. Then, following steps presented in Sec.~\\ref{stdbao} and \\ref{itrbao}, we compute the displacement field at $n+1$th step.\nComparing it with the $n$th step ansatz, we derive the recurrence relation for the expansion kernels.\nReaders who are only interested in the main result may skip the detailed derivation and refer to Eqs.~\\eqref{1-loop:power} and \\eqref{1-loop:prop}.\n\n\n\n\\subsection{Recurrence relation}\n\\subsubsection{Displacement field}\nLet us define the potential field of the observed particles after the $n$ iterations as \n\\begin{align}\n\t\\mathbf \\Psi^{(n)}(\\mathbf k) & =i\\mathbf k  \\phi^{(n)}(\\mathbf k).  \\end{align}\n The curl component is at least at third order, which does not contribute to the divergence in the end.\nThen the iterative reconstruction displacement estimator ${\\mathbf \\Psi}_{\\rm rec}^{(n)}$ is defined as the difference between and the true nonlinear displacement field $\\mathbf \\Psi^{(0)}(\\mathbf q)$ and the  $n$-th step displacement field \n\\begin{align}\n\t{\\mathbf \\Psi}_{\\rm rec}^{(n)}(\\mathbf q) = - \\mathbf s^{(n)}_{\\rm tot} (\\mathbf x)=  \\mathbf \\Psi^{(0)}(\\mathbf q) - \\mathbf \\Psi^{(n)}(\\mathbf q),\n\\end{align}\nwhere we used Eq.~\\eqref{psin} and Eq.~\\eqref{totn}.\nThe reconstructed displacement potential in Fourier space is then defined as\n\\begin{align}\n\t\\phi^{(n)}_{\\rm rec} \\equiv \\phi^{(0)} - \\phi^{(n)}.\\label{def:rec}\n\\end{align}\nWe put forward the $n$-th step ansatz as a series expansion with respect to the zeroth step potential field $\\phi = \\phi^{(0)}$ as\n\\begin{align} \n\t&\\phi^{(n)}(\\mathbf k) \n\t=A^{(n)}(\\mathbf k) \\phi(\\mathbf k) \\notag \\\\\n\t+&\n\t\\frac{1}{2!}\\int\\frac{d^3k_1d^3k_2}{(2\\pi)^6}(2\\pi)^3 \\delta^{(3)}_{\\rm D}(\\mathbf k-\\mathbf k_1-\\mathbf k_2) \n\t\\notag \\\\\n\t&\\times A^{(n)}(\\mathbf k_2,\\mathbf k_3)\\phi(\\mathbf k_2)\\phi(\\mathbf k_3) \\notag \\\\\n\t +&\n\t\\frac{1}{3!}\\int\\frac{d^3k_1d^3k_2d^3k_3}{(2\\pi)^9}(2\\pi)^3 \\delta^{(3)}_{\\rm D}(\\mathbf k-\\mathbf k_1-\\mathbf k_2-\\mathbf k_3) \n\t\\notag \\\\\n\t&\\times A^{(n)}(\\mathbf k_1,\\mathbf k_2,\\mathbf k_3)\\phi(\\mathbf k_1)\\phi(\\mathbf k_2)\\phi(\\mathbf k_3),\\label{eq27}\n\\end{align}\nwhere, by definition, the zeroth step kernels are given as\n\\begin{align}\n    A^{(0)}(\\mathbf k) &= 1,\\label{A1_0}\t\\\\\n\tA^{(0)}(\\mathbf k_2,\\mathbf k_3) &=A^{(0)}(\\mathbf k_1,\\mathbf k_2,\\mathbf k_3)= 0\\label{A23_0}.\n\\end{align}\nNote that Eq.~\\eqref{eq27} is the most general form of the expansion at the 1-loop order.\nThen, we will get $\\phi^{(n+1)}$ to find the recurrence relations for $A$.\nTo simplify our notation, we introduce the following notation:\n\\begin{align} \n\t\\phi^{(n)}_a =A^{(n)}_a{}^b \\phi_b \n\t+\n\t\\frac{1}{2}A^{(n)}_a{}^{bc}\\phi_b\\phi_c  +\n\t\\frac{1}{3!}A^{(n)}_a{}^{bcd}\\phi_b\\phi_c\\phi_d,\\label{abs28}\n\\end{align}\nwhere we defined\n\\begin{align}\n    A^{(n)}_a{}^b \\equiv & A^{(n)}(\\mathbf k_b) (2\\pi)^3 \\delta^{(3)}_{\\rm D}(\\mathbf k_a-\\mathbf k_b) ,\t\\\\\n\tA^{(n)}_a{}^{bc} \\equiv & A^{(n)}(\\mathbf k_b,\\mathbf k_c)(2\\pi)^3 \\delta^{(3)}_{\\rm D}(\\mathbf k_a-\\mathbf k_b-\\mathbf k_c) . \\\\\n\tA^{(n)}_a{}^{bcd} \\equiv & A^{(n)}(\\mathbf k_b,\\mathbf k_c,\\mathbf k_d)\\notag \\\\\n\t&\\times (2\\pi)^3 \\delta^{(3)}_{\\rm D}(\\mathbf k_a-\\mathbf k_b-\\mathbf k_c-\\mathbf k_d),\n\\end{align}\nand we integrate the momentum for a pair of repeated upper and lower indices.\n\n\n\\subsubsection{Nonlinear density}\n\nGiven a displacement field~\\eqref{abs28}, we calculate the nonlinear density field from Eq.~\\eqref{14}.\nEq.~\\eqref{14} can be expanded into\n\\if0\n\\begin{align}\n\t&\\delta_{\\rm NL}^{(n)}(\\mathbf k) = -\n\tik_j\\Psi_j^{(n)}(\\mathbf k)\n\t\\notag \\\\\n\t&-\n\t\\frac{k_{j}k_{k}}{2}\\int\\frac{d^3k_1d^3k_2}{(2\\pi)^6}\\notag \\\\\n\t&\\times (2\\pi)^3 \\delta^{(3)}_{\\rm D}(\\mathbf k-\\mathbf k_1-\\mathbf k_2) \\Psi_{j}^{(n)}(\\mathbf k_1) \\Psi_{k}^{(n)}(\\mathbf k_2)\\notag \\\\\n\t&\n\t+\n\t\\frac{ik_{i}k_{j}k_{k}}{3!}\\int\\frac{d^3k_1d^3k_2d^3k_3}{(2\\pi)^6}\\notag \\\\\n\t&\\times (2\\pi)^3 \\delta^{(3)}_{\\rm D}(\\mathbf k-\\mathbf k_1-\\mathbf k_2-\\mathbf k_3) \\notag \\\\\n\t&\\times  \\Psi_{i}^{(n)}(\\mathbf k_1) \\Psi_{j}^{(n)}(\\mathbf k_2)\\Psi_{k}^{(n)}(\\mathbf k_3)\n\t+\\mathcal O(\\Psi^4).\n\\end{align}\nThis equation can be simplified to\n0\\fi\n\\begin{align}\n\t\\delta^{(n)}_{{\\rm NL},a} &= X_a{}^b \\phi^{(n)}_b \n\t+\n\t\\frac{1}{2}X_a{}^{bc}\\phi^{(n)}_b\\phi^{(n)}_c  \\notag \\\\\n\t&+\n\t\\frac{1}{3!}X_a{}^{bcd}\\phi^{(n)}_b\\phi^{(n)}_c\\phi^{(n)}_d,\\label{abs34}\n\t\\end{align}\nwhere we defined\n\\begin{align}\n\tX_a{}^b &=(\\mathbf k_a\\cdot \\mathbf k_b)(2\\pi)^3\\delta^{(3)}_{\\rm D}(\\mathbf k_a-\\mathbf k_b),\\\\\n\tX_a{}^{bc} &=(\\mathbf k_a \\cdot \\mathbf k_b)(\\mathbf k_a \\cdot \\mathbf k_c)\n\t(2\\pi)^3\\delta^{(3)}_{\\rm D}(\\mathbf k_a-\\mathbf k_b-\\mathbf k_c),\\\\\n\tX_a{}^{bcd} &= (\\mathbf k_a \\cdot \\mathbf k_b)(\\mathbf k_a \\cdot \\mathbf k_c)(\\mathbf k_a \\cdot \\mathbf k_d)\n\t\\notag \\\\\n\t&\\times \n\t(2\\pi)^3\\delta^{(3)}_{\\rm D}(\\mathbf k_a-\\mathbf k_b-\\mathbf k_c-\\mathbf k_d).\n\\end{align}\nNote that we get the 3SPT density power spectrum when we evaluate the 1-loop density power spectrum for the expansion~\\eqref{abs34}, and there is no resummation for this truncation.\nSubstituting Eqs.~\\eqref{abs28} into \\eqref{abs34}, one finds\n\\begin{align}\n\t&\\delta^{(n)}_{{\\rm NL},a} \n\t = X_a{}^b A^{(n)}_b{}^p \\phi_p \n\t\\notag \\\\\n\t&+\n\t\\frac{1}{2}\\left( X_a{}^bA^{(n)}_b{}^{pq}\n\t+\n\tX_a{}^{be}A^{(n)}_b{}^p  \n\tA^{(n)}_e{}^q \n\t\\right)\\phi_p\\phi_q  \n\t\\notag \\\\\n\t&\n\t+\n\t\\frac{1}{3!}\\left( X_a{}^bA^{(n)}_b{}^{pqr}\n\t+\n\t3X_a{}^{be}A^{(n)}_b{}^p  A^{(n)}_e{}^{qr} \\right.\\notag \\\\\n\t& \\left.+\n\tX_a{}^{bdf}A^{(n)}_b{}^p \n\tA^{(n)}_d{}^q\n\tA^{(n)}_f{}^r\\right)\\phi_p \\phi_q \\phi_r. \\label{deltaNLexp}\n\t\\end{align}\n\tWe obtained the connection between the expansion of the potential field and the nonlinear density field. \n    Our next step is to introduce the smoothing factor \n     and then the shift vector.\n    \n\t\n\t\\subsubsection{Shift vector}\nLet us \nintroduce the shift vector and find its perturbative expansion with respect to $\\phi$.\nWe generalize Eq.~\\eqref{shiftdef} for $\\delta^{(n)}_{\\rm NL}$ obtained in Eq.~\\eqref{deltaNLexp} and find\n\\begin{align}\n    \\mathbf s^{(n)}(\\mathbf k) = -\\frac{i\\mathbf k}{k^2}S^{(n)}(k)\\delta^{(n)}_{\\rm NL}(\\mathbf k).\n\\end{align}\nAs a caveat, we cannot simply update the Fourier space displacement field as $\\mathbf \\Psi^{(n)}(\\mathbf k) + \\mathbf s^{(n)}(\\mathbf k)$ because we are mixing the Lagrangian field $\\mathbf \\Psi^{(n)}(\\mathbf q)$ and the Eulerian field $\\mathbf s^{(n)}(\\mathbf x)$ in Eq.~\\eqref{psin}. \nIndeed, the Fourier transform of  Eq.~\\eqref{psin} with respect to $\\mathbf q$ is\n\\begin{align}\n    \\mathbf \\Psi^{(n+1)}(\\mathbf k) =\\mathbf \\Psi^{(n)}(\\mathbf k) +\\mathbf t^{(n)}(\\mathbf k)\\label{psinexetk},\n\\end{align}\nwhere we introduced \n\\begin{align}\n    \\mathbf t^{(n)}(\\mathbf k) \\equiv \\int d^3q e^{-i\\mathbf k\\cdot \\mathbf q}\\mathbf s^{(n)}(\\mathbf q+\\mathbf \\Psi^{(n)}(\\mathbf q)).\\label{deftk}\n\\end{align}\nThen,  we expand the above equation in terms of $\\mathbf \\Psi$, and identify $\\int d^3q e^{-i\\mathbf k\\cdot \\mathbf q}\\mathbf s^{(n)}(\\mathbf q)$ with $\\mathbf s^{(n)}(\\mathbf k)$. Note that $\\mathbf q$ in this integral is nothing but a dummy variable. \nThen, Eq.~\\eqref{deftk} is expanded into\n\\begin{align}\n    \\mathbf Y_a{}^s\\delta^{(n)}_{{\\rm NL}s}\n    +\n    \\mathbf Y_a{}^{bs} \\phi^{(n)}_b  \\delta^{(n)}_{{\\rm NL}s}\n    +\n    \\frac{1}{2}\\mathbf Y_a{}^{bcs}  \\phi^{(n)}_b \\phi^{(n)}_c \\delta^{(n)}_{{\\rm NL}s},\n\\end{align}\nwhere the expansion kernels are given as\n\\begin{align}\n\t\\mathbf Y_a{}^{b}&=\n     \\frac{-i\\mathbf k_b}{k_b^2} S^{(n)}(k_b)(2\\pi)^3\\delta^{(3)}_{\\rm D}(\\mathbf k_a-\\mathbf k_b),\\\\\n\t\\mathbf Y_a{}^{bc}&=\n     -\\mathbf k_{b}\\cdot \\mathbf k_{c} \\frac{-i\\mathbf k_c}{k_c^2} S^{(n)}(k_c)\\notag \\\\\n    & \\times (2\\pi)^3\\delta^{(3)}_{\\rm D}(\\mathbf k_a-\\mathbf k_b-\\mathbf k_c),\\\\\n    \\mathbf Y_a{}^{bcd}& = (\\mathbf k_{b}\\cdot \\mathbf k_{d})(  \\mathbf k_{c} \\cdot  \\mathbf   k_{d} ) \\frac{-i\\mathbf k_d}{k_d^2} S^{(n)}(k_d)\\notag \\\\\n    &\\times (2\\pi)^3\\delta^{(3)}_{\\rm D}(\\mathbf k_a-\\mathbf k_b-\\mathbf k_c-\\mathbf k_d).\n\\end{align}\nThen, expanding with respect to $\\phi$ using Eq.~\\eqref{deltaNLexp}, we get\n\\begin{align}\n\t\\mathbf t_a &= \\mathbf Y_a{}^s\nX_{s}{}^p A_p{}^b \\phi_b \n\t+\n\t\\frac{1}{2}\\mathbf Y_a{}^s X_{s}{}^p A_p{}^{bc}\\phi_b\\phi_c  \n\t\\notag \\\\\n\t&\t\n\t+ \\frac{1}{2}\\mathbf Y_a{}^s X_s{}^{pq}\n\tA_p{}^b  A_q{}^c \\phi_b\\phi_c\n\t\t\\notag \\\\\n\t&+\n    \\frac12\\mathbf Y_a{}^{ts}X_{s}{}^p \n    A_t{}^{(b}  A_p{}^{c)} \\phi_b \\phi_c \n    \\notag \\\\\n\t&\n\t+\n\t\\frac{1}{3!}\\mathbf Y_a{}^s X_{s}{}^pA_p{}^{bcd} \\phi_b  \\phi_c  \\phi_d\n\t\t\\notag \\\\\n\t&\t\n\t+\\frac{1}{3!}\\mathbf Y_a{}^s  X_s{}^{pq}A_p{}^{(bc} A_q{}^{d)}  \\phi_b  \\phi_c  \\phi_d\n\t\t\t\\notag \\\\\n\t&\t\n    +\\frac{1}{3!}\n    \\mathbf Y_a{}^{ts}  \n\t X_{s}{}^p A_p{}^{(cd} A_t{}^{b)} \\phi_b \\phi_c\\phi_d\n    \\notag \\\\\n    &\n   + \\frac{1}{3!} \\mathbf Y_a{}^{ts} X_{s}{}^p  A_t{}^{(bc}  A_p{}^{d)} \\phi_b\\phi_c \\phi_d\n   \t\t\\notag \\\\\n\t&\t\n+\\frac{1}{3!}\n    \\mathbf Y_a{}^{ts}  X_s{}^{pq}\n\tA_p{}^{(c}  A_q{}^d A_t{}^{b)} \\phi_b \\phi_c\\phi_d\n    \t\\notag \\\\\n\t&\t\n  +\n\t\\frac{1}{3!}\\mathbf Y_a{}^sX_s{}^{pqr} A_p{}^bA_q{}^cA_r{}^d  \\phi_b  \\phi_c  \\phi_d\n\t\t\\notag \\\\\n\t&\t\n+\n    \\frac{1}{3!}\\mathbf Y_a{}^{tus} X_{s}{}^p A_t{}^{(b}A_u{}^cA_p{}^{d)} \\phi_b \\phi_c   \\phi_d,\\label{shiftvecn}\n\\end{align}\nwhere the parenthesis implies that we take all possible permutations without duplication.\nThus, we obtained the perturbative expansion of the shift vector in Fourier space.\nWe are ready to shift the density field using Eq.~\\eqref{shiftvecn}.\n\n\\subsubsection{Next step displacement}\n\nFinally, we update the displacement field by following Eq.~\\eqref{psin}.\nIgnoring the curl component, Eq.~\\eqref{psinexetk} yields\n\\begin{align}\n\t\\phi^{(n+1)} = \\phi^{(n)} + \\frac{-i\\mathbf k}{k^2}\\cdot \\mathbf t^{(n)}.\n\\end{align}\nFor all $\\mathbf Y$ we introduce\n\\begin{align}\n\tY_{a}{}^{b\\cdots} \\equiv  \\frac{-i\\mathbf k_a}{k_a^2}\\cdot \\mathbf Y_{a}{}^{b\\cdots}.\n\\end{align}\nThen we find the following recurrence relations:\n\\begin{align}\n\tA^{(n+1)}_a{}^b &= A^{(n)}_a{}^b\n\t+ Y_a{}^s X_{s}{}^p A^{(n)}_p{}^b ,\\label{rec1_}\n\t\\\\\n\tA^{(n+1)}_a{}^{bc} &=A^{(n)}_a{}^{bc} \n\t+\n\tY_a{}^s X_{s}{}^p A^{(n)}_p{}^{bc}\n\t\t\\notag \\\\\n\t&\n\t+\n\t Y_a{}^s X_s{}^{pq}\n\tA^{(n)}_p{}^b  A^{(n)}_q{}^c \n\t\\notag \\\\\n\t&\n\t+Y_a{}^{ts}X_{s}{}^p \n    A^{(n)}_t{}^{(b}  A^{(n)}_p{}^{c)}  ,\\label{rec2_2}\n\t \\\\\n    A^{(n+1)}_a{}^{bcd} &=A^{(n)}_a{}^{bcd}\n    +Y_a{}^s X_{s}{}^pA^{(n)}_p{}^{bcd}\n\t\\notag \\\\\n\t&\n\t+Y_a{}^s  X_s{}^{pq}A^{(n)}_p{}^{(bc} A^{(n)}_q{}^{d)} \n\t\t\\notag \\\\\n\t&\n    +\n    Y_a{}^{ts}  \n\t X_{s}{}^p A^{(n)}_p{}^{(cd} A^{(n)}_t{}^{b)} \n    \\notag \\\\\n    &\n   + Y_a{}^{ts} X_{s}{}^p  A^{(n)}_t{}^{(bc}  A^{(n)}_p{}^{d)} \n       \\notag \\\\\n    &\n   \t+\n    Y_a{}^{ts}  X_s{}^{pq}\n\tA^{(n)}_p{}^{(c}  A^{(n)}_q{}^d A^{(n)}_t{}^{b)} \n    \\notag \\\\\n    &\n      +Y_a{}^sX_s{}^{pqr} A^{(n)}_p{}^bA^{(n)}_q{}^cA^{(n)}_r{}^d  \n          \\notag \\\\\n    &\n+Y_a{}^{tus} X_{s}{}^p A^{(n)}_t{}^{(b}A^{(n)}_u{}^cA^{(n)}_p{}^{d)},\\label{rec3_3}\n\\end{align}\nwhere the parentheses mean that we take the possible permutations to symmetrize the indices.\nThus, we obtain the recurrence relation for the expansion kernels defined in Eq.~\\eqref{eq27}.\nEven though the recurrence relations misleadingly appear complicated, they only depend on the smoothing scales and are independent of cosmology. We can therefore precompute them.\nUsing the above recurrence relation and the initial conditions~\\eqref{A1_0} and \\eqref{A23_0}, we can compute the $n$-th step displacement field systematically.\nExplicit forms of Eqs. \\eqref{rec1_} to \\eqref{rec3_3} without index are presented in Appendix~\\ref{woind}.\nEq.~\\eqref{rec1_} straightforwardly reduces to Eq.~\\eqref{Eq.A1}, and we get \n\\begin{align}\n    A^{(n)} = (1 - S^{(n-1)})(1 - S^{(n-2)})\\cdots (1 - S^{(0)}).\n\\end{align}\nThus, $A^{(n)}\\to 0$ for $n\\to \\infty$ and the displacement fields in the later steps converge to zero as approaching the estimated Lagrangian position.\nThe second and third-order kernels are more complicated, but we will see that we only use a few specific momentum configurations in the following calculations.\n\n\n\n\\subsection{Post iterative reconstruction power spectrum}\n\nFrom Eqs~\\eqref{def:rec} and \\eqref{abs28}, we get the reconstructed displacement potential\n\\begin{align}\n\t\\phi^{(n)}_{{\\rm rec},a} =& \\bar A^{(n)}_a{}^b \\phi_b \n\t-\n\t\\frac{1}{2}A^{(n)}_a{}^{bc}\\phi_b\\phi_c  -\n\t\\frac{1}{3!}A^{(n)}_a{}^{bcd}\\phi_b\\phi_c\\phi_d,\n\\end{align}\nwhere $\\bar A^{(n)}_a{}^b = \\delta_a{}^b - A^{(n)}_a{}^b $, with $\\delta_a{}^b\\equiv (2\\pi)^3\\delta^{(3)}_{\\rm D}(\\mathbf k_a-\\mathbf k_b)$.\nThe 1-loop power spectrum of the estimator is \n\\begin{align}\n\t\\langle \\phi^{(n)}_{{\\rm rec,}a}  \\phi^{(n)}_{{\\rm rec,}e}\\rangle \n\t&= \\bar A^{(n)}_a{}^b\\bar A^{(n)}_e{}^f \\langle \\phi_b \n\t \\phi_f \\rangle\n\t\\notag \\\\\n\t&-\n\t\\bar A^{(n)}_a{}^b  A^{(n)}_e{}^{fg}\\langle \\phi_b\\phi_f\\phi_g \\rangle  \\notag \\\\\n\t&-\\frac{1}{3}\n\t\\bar A^{(n)}_a{}^b A^{(n)}_e{}^{fgh}\\langle \\phi_b\\phi_f\\phi_g\\phi_h \\rangle\n\t\\notag \\\\\n\t&+\n\t\\frac{1}{4}A^{(n)}_a{}^{bc}\n\tA^{(n)}_e{}^{fg}\\langle \\phi_b\\phi_c\\phi_f\\phi_g \\rangle.\\label{eq64}\n\\end{align}\nWe compute Eq.~\\eqref{eq64} by expanding $\\phi$ with respect to $\\delta_{\\rm L}$:\n\\begin{align}\n\t\\phi_a =& L_a{}^b\\delta_{{\\rm L},b} +\\frac{1}{2}L_a{}^{bc}\\delta_{{\\rm L},b}\n\t\\delta_{{\\rm L},c}\n\t+\\frac{1}{3!}L_a{}^{bcd}\\delta_{{\\rm L},b}\n\t\\delta_{{\\rm L},c}\n\t\\delta_{{\\rm L},d}\\,,\\label{phiex}\n\\end{align}\nwhere the Lagrangian kernels are written as~\\cite{Matsubara:2007wj,Baldauf:2015tla}\n\\begin{align}\n\t L_a{}^b&= \\frac{1}{k_a^2}(2\\pi)^3 \\delta^{(3)}_{\\rm D}(\\mathbf k_a-\\mathbf k_b),\\label{L1}\\\\\n\t L_a{}^{bc}&=\\frac{3}{7}\\frac{1}{k_a^2}\\frac{(\\mathbf k_b\\times \\mathbf k_c)^2}{|\\mathbf k_b|^2|\\mathbf k_c|^2} (2\\pi)^3  \\delta^{(3)}_{\\rm D}(\\mathbf k_a-\\mathbf k_b-\\mathbf k_c),\\label{L2}\\\\\n\t L_a{}^{bcd}&=\\left[\t-\\frac{1}{3}\\frac{1}{k_a^2}\\frac{[\\mathbf k_d\\cdot (\\mathbf k_b \\times \\mathbf k_c)]^2}{|\\mathbf k_d|^2|\\mathbf k_b|^2|\\mathbf k_c|^2} \\right. \\notag \\\\\n\t &\\left. +\\frac{5}{21}\\frac{1}{k_a^2}\\frac{ [\\mathbf k_b\\times (\\mathbf k_c+\\mathbf k_d)]^2}{|\\mathbf k_b|^2(\\mathbf k_c+\\mathbf k_d)^2}\\frac{(\\mathbf k_c\\times \\mathbf k_d)^2}{|\\mathbf k_c|^2|\\mathbf k_d|^2}+2~{\\rm perms}\\right]\n\t \\notag \\\\\n\t &\\times (2\\pi)^3\\delta^{(3)}_{\\rm D}(\\mathbf k_a-\\mathbf k_b-\\mathbf k_c-\\mathbf k_d)\n.\\label{L3}\n\\end{align}\n\n\nThen the 1LPT auto-power spectrum of $\\phi^{(n)}_{\\rm rec}$ and its cross-power spectrum with $\\delta_{\\rm L}$ are straightforwardly obtained as\n\\begin{align}\n\t P^{{\\rm 1LPT}}_{\\phi^{(n)}_{\\rm rec}} = A^{(n)2}P^{\\rm 1LPT}_{\\phi},~ P^{{\\rm 1LPT}}_{\\delta_{\\rm L} \\phi^{(n)}_{\\rm rec}} = A^{(n)}P^{\\rm 1LPT}_{\\phi}.\\label{rec:zel:pow}\n\\end{align}\nThen the 3LPT postreconstruction power spectrum is given as follows: \n\\begin{align}\n\t&P^{{\\rm 3LPT}}_{\\phi^{(n)}_{\\rm rec}}=\t \n\tP^{{\\rm 3LPT}}_{\\phi^{(n)}_{\\rm rec}11}\n\t+\n\tP^{{\\rm 3LPT}}_{\\phi^{(n)}_{\\rm rec}13}\n\t+\n\tP^{{\\rm 3LPT}}_{\\phi^{(n)}_{\\rm rec}22} ,\\label{1-loop:power}\n\\end{align}\t\nwhere we defined\n\\begin{align}\n    P^{{\\rm 3LPT}}_{\\phi^{(n)}_{\\rm rec}11} = & \\bar A^{(n)}(k)^2 P^{\\rm 3LPT}_\\phi(k), \\\\\n    P^{{\\rm 3LPT}}_{\\phi^{(n)}_{\\rm rec}22} =\t& - \\int dx  d\\mu\\frac{P_{\\rm L}\\left(ky\\right)P_{\\rm L}(kx)}{4\\pi^2k}\\notag \\\\\n\t \\times &\\left(\\frac{3}{7y^4} (1-\\mu^2)\\bar A^{(n)}(k) \\alpha^{(n)} - \\frac{\\alpha^{(n)2} }{2x^2y^4} \\right) ,\n\t \\\\\nP^{{\\rm 3LPT}}_{\\phi^{(n)}_{\\rm rec}13}=\t& -\n\t\\bar A^{(n)}(k)P_{\\rm L}(k)\\int dx  d\\mu \\frac{P_{\\rm L}(kx)}{4\\pi^2k} \\notag \\\\\n\t\\times &\n\t\\left( \n\t\\frac{\\beta^{(n)}}{x^2} +    \\frac{6}{7y^2}(1-\\mu^2) \\alpha^{(n)}  \\right) ,\n\\end{align}\nwhere $k'=kx$ is the loop momentum, and $\\mu$ is the cosine defined for $\\mathbf k$ and the loop momentum, $y\\equiv \\sqrt{1-2x\\mu+x^2}$, and $\\alpha$, $\\beta$ and $\\gamma$ are given as \n\\begin{align}\n\t\\alpha^{(n)}(k,k',\\mu) &\\equiv A^{(n)}(\\mathbf k-\\mathbf k',\\mathbf k')\/k^2,\\\\\n\t\\beta^{(n)}(k,k',\\mu) &\\equiv A^{(n)}(\\mathbf k,\\mathbf k',-\\mathbf k')\/k^4, \\\\\n\t\\gamma^{(n)}(k,k',\\mu) &\\equiv  A^{(n)}(\\mathbf k,-\\mathbf k')\/k^2\\, .\n\\end{align}\n\\eqref{A2} and \\eqref{rec3_3_s} can be reduced to \n\\begin{align}\n\t\\alpha^{(n+1)} &=\\alpha^{(n)}\n\t-S^{(n)}(k)\\alpha^{(n)} \\notag \\\\\n\t&-\n\tx\\mu( 1-x\\mu) \n\tS^{(n)}(k)A^{(n)}(kx) A^{(n)}(ky) \n\t\\notag \\\\\n\t&- x( x- \\mu  ) (1-x\\mu)   S^{(n)}(ky)\n\tA^{(n)}(kx)  A^{(n)}(ky) \\notag \\\\\n\t&\n\t- x^2\\mu( x- \\mu  )  S^{(n)}(kx) A^{(n)}(kx)  A^{(n)}(ky),\\label{recalpha}\n\\\\\n\t\\beta^{(n+1)} &= \\beta^{(n)}\n    -S^{(n)}(k)\\beta^{(n)}\n     \\notag \\\\\n    &\n      +2 x^2\\mu^2  S^{(n)}(k)  A^{(n)}(k)A^{(n)}(kx)^2\n\t\\notag \\\\\n\t&\n\t+  2x^4\\mu^2  S^{(n)}(kx) A^{(n)}(k)A^{(n)}(kx)^2\n\t\\notag \\\\\n    &\n\t-2x\\mu\n\t(1-x\\mu) S^{(n)}(k) A^{(n)}(kx) \\gamma^{(n)}  \n\t\\notag \\\\\n    &\n   - 2x^2\\mu(x-\\mu) S^{(n)}(kx) A^{(n)}(kx)\\gamma^{(n)}\n   \\notag \\\\\n\t&   -\n\t2x(x-\\mu )(1-x\\mu) S^{(n)}(ky)  A^{(n)}(kx)\\gamma^{(n)} \n   \t\\notag \\\\\n\t&-\n    2\\frac{ x^2(x-\\mu)^2 (1-x\\mu)^2  }{y^2} S^{(n)}(ky)\n\tA^{(n)}(k)A^{(n)}(kx)^2 \n  ,\\label{recbeta}\n\t\n\\\\\n\t\\gamma^{(n+1)} &=\\gamma^{(n)} \n\t-S^{(n)}(ky)\\gamma^{(n)} \\notag \\\\\n\t&- \n\t\\frac{x(x-\\mu) (1-x\\mu)  }{y^2} \n\tS^{(n)}(ky) A^{(n)}(k) A^{(n)}(kx) \n\t\\notag \\\\\n\t&-\\frac{x^2\\mu}{y^2}  (x-\\mu) S^{(n)}(kx) A^{(n)}(k)  A^{(n)}(kx) \\notag \\\\\n\t&-\\frac{x\\mu}{y^2} (1-x\\mu) S^{(n)}(k) \n    A^{(n)}(k)  A^{(n)}(kx)\n\t\t.\\label{recgamma}\n\\end{align}\nNote that Eq.~\\eqref{recbeta} is valid under the premise that we integrate $\\beta^{(n)}$ with $\\mu^2$ from $\\mu=-1$ to 1, and its original expression was the symmetric version of Eq.~\\eqref{recbeta} with respect to $\\mu \\to -\\mu$.  \nSimilarly, we get the 3LPT cross-power spectrum\n\\begin{align}\n\t&P^{{\\rm 3LPT}}_{\\delta_{\\rm L}\\phi^{(n)}_{\\rm rec}} \n\t=\\bar A^{(n)}(k) \\left[P^{\\rm 1LPT}_{\\phi}+\\frac{P_{\\phi13}}{2} \\right] +\\frac{1}{2}P^{{\\rm 3LPT}}_{\\phi^{(n)}_{\\rm rec}13}\n .\\label{1-loop:prop}\n\\end{align}\n\nTo summarize, although we have derived a strenuous set of equations so far, we have straightforwardly followed the steps in Sec.~\\ref{moderec}, expanding the equations up to third order in the perturbations.\nReaders who are only interested in the main result can refer to Eqs.~\\eqref{1-loop:power} and \\eqref{1-loop:prop} and the associated recurrence relations.\n\n\n\\subsection{Post iterative reconstruction propagator and cross-correlation coefficient}\n\nThe propagator is defined as the cross-correlation function between a late time field $X$ and the linear matter perturbation $\\delta_{\\rm L}$ normalized by the linear spectrum:\n\\begin{align}\n\tC_{X}\\equiv \\frac{P_{\\delta_{\\rm L} X}}{P_{\\rm L}}.\\label{cc}\n\\end{align}\nWhile not directly observable in real data, this quantity is a useful metric for density field reconstruction techniques, since it quantifies how much of the initial linear perturbation has been recovered from the (measured) late time density as a function of the scale $k$.\nOne also uses the cross-correlation coefficient \n\\begin{align}\n\tr_{XY}\\equiv \\frac{P_{XY}}{\\sqrt{P_{XX}P_{YY}}},\\label{rr}\n\\end{align}\nto quantify the reconstruction efficiency.\nThe difference between Eqs~\\eqref{cc} and \\eqref{rr} is the normalization, and the two represent different information as explained below.\nConsider $X=\\delta_{\\rm L}$ and $Y=k^2\\phi$.\nThe 1-loop cross-correlation coefficient is\n\\begin{align}\n\tr^{\\rm 3LPT}_{\\delta_{\\rm L} k^2\\phi} = \\frac{P_{\\phi11}+\\frac{1}{2}P_{\\phi13}}{\\sqrt{P_{\\phi11}(P_{\\phi11}+P_{\\phi13}+P_{\\phi22})}}. \\label{eqcrossco}\n\\end{align}\nThen, $P_{\\phi 11}\\gg|P_{\\phi13}|, P_{\\phi22}$ yields\n\\begin{align}\n\tr^{\\rm 3LPT}_{\\delta_{\\rm L} k^2\\phi} \\sim 1 -\\frac{1}{2}\\frac{P_{\\phi22}}{P_{\\phi11}}+\\mathcal O\\left(\\frac{P^2_{\\phi22}}{P^2_{\\phi11}},\\frac{P^2_{\\phi13}}{P^2_{\\phi11}}\\right).\n\\end{align}\nThus, $P_{\\phi13}$ vanishes at the leading order even if $|P_{\\phi13}|\\gtrapprox P_{\\phi22}$.\nHence, the cross-correlation coefficient is a measure of $P_{\\phi22}$, i.e., the mode coupling.\nIn simulations or observations, noises also appear in the same way.\nOn the other hand, the propagator is given as\n\\begin{align}\n\tC^{\\rm 3LPT}_{k^2\\phi} = 1 + \\frac{1}{2}\\frac{P_{\\phi13}}{P_{\\phi11}},\n\\end{align}\nbeing a measure of $P_{\\phi13}$, i.e., the shift.\nIn the following section, we will present both the propagator and the cross-correlation coefficient to quantify how well the LPT modeling works.\n\n\n\n\n\n\\subsection{Theory perspective of the iterative reconstruction as a regularization method}\n\n\n\n\n\nAs we already mentioned, the prereconstruction 3LPT is known to suffer the UV mistake when modeling the nonlinear displacement field.\nIn field theory, we have a similar situation.\nWe often encounter superficial divergences while actual physical quantities are finite.\nSuch divergences appear only at intermediate stages of calculation, which can be canceled after regularization.\nThe prereconstruction 3LPT loop integral UV sensitivity is a similar issue, so it is a natural question to ask how we regularize the loop integral to compute the observable properly.\n\n\n\nAssuming that the iterative reconstruction correctly returns the true displacement field in the end, we may think of the procedure as another way of calculating the displacement field from the density field.\nThe reconstructed displacement field is not necessarily the same as the naively defined displacement field in Eq.~\\eqref{psidef}, and there is no prior reason to believe that the latter is the correct one as the 3LPT prereconstruction includes the UV mistake.\nIndeed, the iterative reconstruction can be a more clever alternative to estimate the Lagrangian displacement since we avoid directly computing the nonlinearity.\nThat is, we smooth the nonlinear part of $\\delta_{\\rm NL}$ step by step, and at every step, the linear approximation is quite good for the continuity equation.\nTherefore, we may think of the iterative reconstruction to correct the UV mistake, i.e., it can work as a regularization of the UV-sensitive loop integral.\n\nOn the analogy of field theory, we may regard the displacement field in Eq.~\\eqref{psidef} as a ``bare'' quantity, which can be divergent at the intermediate stage of the calculation.\nThen the reconstructed displacement field $\\phi^{(n)}_{\\rm rec}$ is ``physical'' because the reconstruction procedures are based on the observed energy-momentum tensor, i.e., the mass density.\nThe iterative procedure naturally introduces the counter term as  Eq.~\\eqref{1-loop:power} can be recast into\n\\begin{align}\n    P^{\\rm 3LPT}_{\\phi^{(n)}_{\\rm rec}} &= P^{\\rm 3LPT}_{\\phi} - P^{{\\rm ct}(n)},\n\\end{align}\nwhere we introduced\n\\begin{align}\n        P^{\\rm ct(n)} \\equiv & (2A^{(n)}-A^{(n)2})P^{\\rm 3LPT}_{\\phi} - P^{{\\rm 3LPT}}_{\\phi^{(n)}_{\\rm rec}13}\n\t-\n\tP^{{\\rm 3LPT}}_{\\phi^{(n)}_{\\rm rec}22}.\n\\end{align}\nBelow, we will see that $P^{\\rm 3LPT}_{\\phi^{(n)}_{\\rm rec}}$ agrees with the true displacement field up to $k{\\rm Mpc}\/h<0.2$; therefore, this is a physically well-motivated way to introduce a counter term.\nAs a caveat, a simple sharp cut-off in the integral \\eqref{P13} violates the momentum space transnational symmetry and cannot reproduce the true displacement field. \nThe regularization would work up to some $n$ corresponding to the UV cut-off scale of the theory described by Eq.~\\eqref{eom1}.\nIn this scheme, we progressively include the modes from IR to UV and find the scale where the UV modes are no more reliable.\nIndeed, the true displacement field has a peak at $k{\\rm Mpc}\/h\\sim 0.2$ and is quickly damped for higher $k$ in Fig.~\\ref{fig00}.\nTherefore, the linear displacement is overestimated for $k{\\rm Mpc}\/h> 0.2$ in the perturbation theory, and so is the loop integral.\n\n\n\\section{Comparison with simulations}\n\\label{sec5}\n\nIn this section, we compare our LPT modeling with simulations.\nLet us first present the simulation results and see if the iterative procedures are successfully modeled.\n\n\\subsection{Simulation of post reconstruction field}\\label{simulation:sec}\n\nWe choose the initial ($n=0$) smoothing scale of $20 {\\rm Mpc}\/h$ to reconstruct L500 and subL500, and we reduce the smoothing scale incrementally by $\\epsilon=1\/\\sqrt{2}$ until we reach $n=9$.\nIn Fig.~\\ref{fig2}, we reproduced Fig.~5 in Ref.~\\cite{Schmittfull:2017uhh}, adding displacement fields.\nWe plot the nonlinear fields and postreconstruction fields normalized by the corresponding no-wiggle spectra. The no-wiggle spectra are defined for each simulation, calculated from the paired no-wiggle simulation in Sec.\\ref{simsec}, and thus the nonlinear and numerical effect on the broadband shape is removed. Hence, we can single out the degradation effect in the BAO.\nAs we see in the plot, the initial density, nonlinear displacement, and the iterative reconstruction estimators agree well below $k{\\rm Mpc}\/h<0.3$. We overplot the standard reconstruction $\\delta_{\\rm rec}$ with the smoothing scale of $20 {\\rm Mpc}\/h$ (blue lines); its reconstructed BAO feature is less distinct than $n>7$ mainly because the smoothing scale is smaller than the effective smoothing scale of the iterative reconstructions~\\cite{Seo:2021}.\n\n\n\\begin{figure}\n  \\includegraphics[width=\\linewidth]{wg_nwg}\n  \\caption{power spectrum divided by its matched no-wiggle one. The data set is L1500. The no-wiggle spectra are defined for each simulation and thus any nonlinear and numerical effect on the broadband shape are removed. $\\delta_{\\rm rec}$ is the standard reconstruction with $R=20$Mpc\/$h$. }\n  \\label{fig2}\n\\end{figure}\n\n\nTo see the rest of the nonlinear effects, we plotted the nonlinear power spectrum normalized by the linear one in Fig.~\\ref{fig22}.\nOur $n=9$ visually agrees with Fig.14 in Ref.~\\cite{Schmittfull:2017uhh}. The higher noise in the $\\delta_{\\rm NL}$ power spectrum in the figure, compared to the power spectra of displacement field tracers, reflects its poor cross-correlation with the initial field. The iterative reconstruction appears to converge to the displacement field rather than to the linear power spectrum as $n$ approaches seven and then approaches closer to the linear shape as $n$ increases.\nTo better quantify how close the reconstructed field is to the nonlinear displacement field or to the linear field, \nwe introduce the error power spectrum of $X$ and $Y$ as\n\\begin{align}\n    E(X,Y) = \\frac{P_{X-Y}}{P_Y},\\label{erdef}\n\\end{align}\nwhere $P_{X-Y}$ and $P_Y$ are the auto-power spectrum of $X-Y$ and $Y$, respectively. \nThen we show $E(k^2\\phi^{(n)}_{\\rm rec},k^2\\phi)$ and $E(k^2\\phi^{(n)}_{\\rm rec},\\delta_{\\rm L})$ in Fig.~\\ref{fig222}.\nWe can see $E(k^2\\phi^{(n)}_{\\rm rec},k^2\\phi)<E(k^2\\phi^{(n)}_{\\rm rec},\\delta_{\\rm L})$, and thus the postreconstruction field is relatively ``closer'' to the displacement field than the linear field as iteration increases in terms of the error power spectrum.\nFig.~\\ref{fig222} shows that the iterative reconstruction works as expected, i.e., gradually reduces the error power spectrum progressively to a smaller scale; most of the convergence is reached around $n=7$, and there is a slight improvement for $n>7$.\n\n\n\n Figs \\ref{1-loopres1}, \\ref{1-loopres1.1} and \\ref{1-loopres3.1} show the power spectrum, the propagator, and the cross-correlation coefficient in each step of iteration. At $n=6$ and 7, we indeed see a slight positive shift term, similar to that of the displacement (grey), but such feature is very weak and disappears after $n>7$. In terms of the cross-correlation coefficient, again, the reconstructed field converges to the displacement field up to $k {\\rm Mpc}\/h \\sim 0.2$  for $n>5$. \n \n In summary, the iterative reconstruction converges to the displacement field in mode coupling terms, but it deviates in terms of the shift term. For the shift term, the reconstructed field appears to converge to the linear field ($k {\\rm Mpc}\/h \\sim 0.2$). \n\n\n\\begin{figure}\n  \\includegraphics[width=\\linewidth]{nl-disp-lin}\n  \\caption{Power spectra normalized by the linear matter power spectrum for L500. One can see the effect of reconstruction on the full shape.}\n  \\label{fig22}\n\\end{figure}\n\n\n\n\\begin{figure}\n  \\includegraphics[width=\\linewidth]{nl-disp-itrec}\n  \\caption{The error power spectrum $E(k^2\\phi^{(n)}_{\\rm rec},Y)$ for $Y=k^2\\phi$~(solid) and $Y=\\delta_{\\rm L}$ (dotted). A lower error power spectrum implies a better agreement. The postreconstruction field is ``closer'' to the displacement field than the linear field. Iterative reconstruction is convergent around $n=6$ and the error is at most 3\\% for $k$Mpc\/$h<$0.3.}\n  \\label{fig222}\n\\end{figure}\n\n\\subsection{LPT modeling and simulations}\n\nIn this section, we compare our LPT model derived in Sec.~\\ref{sec5} with the simulation result and check if the discrepancy between the iterative reconstruction and the displacement field can be explained. \nUsing Eqs.~\\eqref{1-loop:power} and \\eqref{1-loop:prop} we obtained the solid orange curves in Figs.~\\ref{1-loopres1},~\\ref{1-loopres1.1}, and \\ref{1-loopres3.1}.\nFor $n< 4$, except for the first step, the 1LPT (dashed orange curves) and 3LPT (solid orange curves) solutions are very similar, and they reproduce the simulation result (solid blue lines) in Figs.~\\ref{1-loopres1},~\\ref{1-loopres1.1}, and~\\ref{1-loopres3.1}. That is, the model explains the discrepancy between the postreconstruction field and the displacement field (gray lines) properly.\nFor higher iterations, LPT predictions begin to deviate, and only the 3LPT can predict the excess above the linear power spectrum, which is reasonable as we are trying to reconstruct smaller scale information that requires a higher-order description.\nNevertheless, we overall observe that the LPT models do not describe the numerical results very well. Looking at the propagator (Fig.~\\ref{1-loopres1.1})  and the cross-correlation coefficient (Fig.~\\ref{1-loopres3.1}), we find that the main deviation happens in the propagator, while the prediction of the cross-correlation coefficient is in excellent agreement for $n < 6$ and within less than 1\\% at $k{\\rm Mpc}\/h \\sim 0.2$ even at $n=9$. \n\nOn the other hand, we find that the 3LPT up to $n = 6$ is converging to the true displacement obtained in the simulation. The 3LPT model for $n>6$ then deviates from the true displacement field to the 3LPT model of the displacement field. This shows that the perturbation theory predicts that the iterative process we adopted should recover the true displacement field up to $k{\\rm Mpc}\/h\\lesssim 0.2$.\n The relatively weak convergence of the simulated post-iterative reconstruction field to the true nonlinear displacement field after $n=4$ implies the iterative reconstruction simulation includes some unknown numerical or technical effect that is not described in Sec.~\\ref{stdbao} and \\ref{itrbao}, particularly in the propagator.\nGiven that the offset between the model and the simulations is similarly parametrized as Eq.~\\eqref{tobias_eft}, we conduct a quick test of an EFT model by simply adding a counter term\n\\begin{align}\n\tP^{{\\rm 3EFT}}_{\\phi^{(n)}_{\\rm rec}} = P^{{\\rm 3LPT}}_{\\phi^{(n)}_{\\rm rec}} + 2\\alpha k^2 P^{{\\rm 1LPT}}_{\\phi^{(n)}_{\\rm rec}},\n\t\\label{3EFT}\n\\end{align}\nand show the green curves in Figs.~\\ref{1-loopres1},~\\ref{1-loopres1.1}, and \\ref{1-loopres3.1}. $\\alpha$ is the same as for the displacement field. This test indicates that the offset between the model and the reconstructed field in the propagator is less likely mitigated by a simple counter term, and again the perturbation theory model expects that the iterative reconstruction converges to the displacement field in the end.\n\nAnother likely source of the discrepancy is the aforementioned numerical noise\/artifact. Earlier, we justified subsampling particles (i.e., L500) by inspecting the convergence of the displacement field measurement on the L1500  (Sec.~\\ref{simsec}).\nHowever, the convergence in the measurements does not necessarily mean convergence in the process of reconstruction. \nThe UV sensitivity of the loop integral implies that the displacement field in a simulation is sensitive to the particle resolution\/density, and the effect may have been more complicated\/more significant in the backward evolution of reconstructing the field based on the subsampled tracers in the iterative reconstruction. Our numerical resource is limited conducting the iterative reconstruction on fullL1500, but we present the case of 0.15\\% subsampling (subL500) in Figs.~\\ref{1-loopres1},~\\ref{1-loopres1.1}, and \\ref{1-loopres3.1}, as dotted lines to show the greater sensitivity to the particle subsampling in the process of reconstruction compared to the displacement field. Including a shot noise effect in the theoretical model and implementing a pixel window function de-convolution in the simulated field will be crucial for further comparison, which we plan to investigate in a future paper.\n\nTo account for the expected breakdown due to various missing small-scale physics of our subsampled catalog, we manually impose a minimum smoothing scale in the reconstruction in an attempt to control such an effect before reaching this breakdown. \nFig.~\\ref{fig222} showed that the reconstruction changes the convergence behavior around $n=6$, which is suggestive of the effective minimum smoothing scale in the simulation and the theory to be the smoothing scale around this step, which is $3.5.{\\rm Mpc}\/h$.\nTherefore, instead of Eq.~\\eqref{Rdef}, we set\n\\begin{align}\nR_n =  &{\\rm max}(\\epsilon^{n} R,R_{\\rm min}),\n\\end{align}\nwith $R_{\\rm min}=3.5.{\\rm Mpc}\/h$ and show the results in Figs.~\\ref{1-loopres2}, \\ref{1-loopres2.1} and \\ref{1-loopres3.2}.\nIntroducing the minimum smoothing scale derives a convergence at $n=7$, as we cannot extract the information from smaller scales than the smoothing scale. However, the discrepancy between the theory and the simulation remains.  \n\n\nBased on the lack of UV modes that we cannot explain, we introduce a phenomenological ad hoc model that we call 3LPT*.\nIn this model, we subtract all UV sensitive integrals from the 3LPT power spectrum and define \n\\begin{align}\n\t&P^{{\\rm 3LPT*}}_{\\phi^{(n)}_{\\rm rec}}\\equiv \t \n\t\\bar A^{(n)2}\\left [ \n\tP^{\\rm 1LPT}_{\\phi}\n\t+\n\tP_{\\phi22}\n\t\\right]\n\t+\n\tP^{{\\rm 3LPT}}_{\\phi^{(n)}_{\\rm rec}22},\\label{1-loop:power22}\n\\end{align}\t\nand we show the red curves in Figs.~\\ref{1-loopres1},~\\ref{1-loopres1.1}, and \\ref{1-loopres3.1} and Figs.~\\ref{1-loopres2}, \\ref{1-loopres2.1} and \\ref{1-loopres3.2}. This model takes the propagator of 1LPT, which better describes the observed propagator of the reconstructed field than 3LPT.  \nAs a caveat, there is no reasonable explanation about this ad hoc truncation, and this is an inconsistent treatment as either loop expansion or perturbative expansion.\nHowever, we found the 3LPT* fits the simulated iterative reconstruction at 1\\% accuracy for $n>6$ for both cases with and without the minimum smoothing scale.\n\n\n\n\n\n\n\n\n\n\\begin{figure*}\n  \\includegraphics[width=\\linewidth]{1_pow_lpt_rec_kmax_inf}\n  \\caption{The comparison between the simulated postreconstruction power spectra~(blue), simulated true displacement~(gray) and the 3LPT~\\eqref{1-loop:power}, 1LPT~\\eqref{rec:zel:pow}, 3EFT~\\eqref{3EFT} and 3LPT*~\\eqref{1-loop:power22} models.\n  Solid and doted curves for the simulation are 4\\% and 0.15\\% subsampling, respectively.\n  Our 3LPT prediction agrees with the true displacement field rather than the simulated postreconstruction displacement up to $n=6$.\n  3LPT* is comparable to the simulated post reconstruction displacement field within 1\\% accuracy.\n  The EFT slowly converges to the true displacement field. \n  }\n  \\label{1-loopres1}\n   \\includegraphics[width=\\linewidth]{1_pow_lpt_rec_kmax_inf35}\n  \\caption{The power spectrum in Fig.~\\ref{1-loopres1} for the minimum smoothing scale $R_{\\rm min}=3.5{\\rm Mpc}\/h$ Plots are the same with Fig.~\\ref{1-loopres1} up to $n=6$ and are almost fixed at $n=7$. \n  }\n  \\label{1-loopres2}\n \\end{figure*}\n\n\n\\begin{figure*}\n   \\includegraphics[width=\\linewidth]{1_proc_lpt_rec_kmax_inf}\n  \\caption{The propagator counterparts of Fig.~\\ref{1-loopres1}. The simulated displacement field, 3LPT, 3EFT show the shift term, but the effect is very small for the simulated postreconstruction.}\n  \\label{1-loopres1.1}\n\n  \\includegraphics[width=\\linewidth]{1_proc_lpt_rec_kmax_inf35}\n  \\caption{The propagator in Fig.~\\ref{1-loopres1.1} with the minimum smoothing scale $R_{\\rm min}\\sim 3.5$Mpc\/$h$.\n  Plots are the same with Fig.~\\ref{1-loopres1.1} up to $n=6$.\n  }\n  \\label{1-loopres2.1}\n\\end{figure*}\n\n\n\n\n\\begin{figure*}\n   \\includegraphics[width=\\linewidth]{1_coef_lpt_rec_kmax_inf}\n  \\caption{The cross-correlation coefficient counterparts of Fig.~\\ref{1-loopres1}.\n  3LPT, 3EFT, the true displacement and the simulation post reconstruction displacement agree well, while $r=1$ for 1LPT, i.e., the linear perturbations.\n  While the cross-correlation coefficient should be always bounded by 1, the EFT is divergent for high $k$ since we introduced $-k^2$ term.  \n  }\n  \\label{1-loopres3.1}\n\n  \\includegraphics[width=\\linewidth]{1_coef_lpt_rec_kmax_inf35}\n  \\caption{The cross correlation coefficient in Fig.~\\ref{1-loopres1.1} with the minimum smoothing scale $R_{\\rm min}\\sim 3.5$Mpc\/$h$. Plots are the same with Fig.~\\ref{1-loopres3.1} up to $n=6$.}\n  \\label{1-loopres3.2}\n\\end{figure*}\n\n\n\n\n\n\n\n\n\n\n\\section{Conclusions}\\label{secdisc}\nAs a recent study~\\cite{Baldauf:2015tla} has pointed out, the nonlinear displacement at low redshift maintains a high cross-correlation with the initial field, contrary to the corresponding nonlinear density field. \nThus, if one could measure the nonlinear displacement field directly in the late time Universe, the measurement would contain the BAO feature almost without the nonlinear degradation, and it could also allow us to consistently use the information beyond the BAO, i.e., the full clustering shape, based on nonlinear perturbation theory.  \n\nRef. \\cite{Schmittfull:2017uhh} then developed a  non-standard extension to the density reconstruction technique, which was designed to reproduce the displacement field from the observed nonlinear density fluctuations by iteratively moving mass tracers along the gradient of the progressively smoothed density field to the Lagrangian positions. \nThis paper investigated how closely the method returned the nonlinear displacement field, focusing on the BAO and the full shape. Then, we derived an LPT-based model for the reconstructed displacement field and tested the model against the simulated reconstructed displacement field.  We summarize our main results below.\n\nFirst, we confirmed that the displacement field is highly correlated with the linear field; the degradation effect and the phase shift on the BAO are negligible, and the nonlinear instability is milder than the nonlinear density field. However, the full shape of the nonlinear displacement field is different from the linear matter power spectrum on the BAO scale by $\\sim 8\\%$, and modeling with a higher-order perturbation theory with a correction term for the UV sensitive loops, such as the EFT model, is required for precision cosmology.\n\n\nWe found that the postreconstructed displacement field approaches closer to the nonlinear displacement field with increasing iterations but does not perfectly recover the nonlinear displacement field in the end. There is an 8\\% discrepancy between the true displacement field and the simulated postreconstruction displacement field at $k{\\rm Mpc}\/h\\sim 0.2$ due to unknown effects specific to the iterative process in the simulation. The discrepancy is mainly in the propagator (i.e., the shift term), while the cross-correlation coefficient (i.e., mode-coupling term) agrees well.\nAt the final iteration, the postreconstruction field tends to converge closer to the linear power spectrum. \n\nWe built perturbation theory models for the reconstructed field to understand the difference.\nIn this process, we realized that the UV mistake should largely cancel for the reconstructed displacement for moderate iterations in the process of estimating the displacement field, allowing us to avoid introducing the EFT counter term up to $k{\\rm Mpc}\/h\\lesssim 0.2$ at $z=0.6$.  We, therefore, adopt the 1-loop LPT approach.\n\nWe modeled the iterative reconstruction as follows: firstly, we put forward the $n$th reconstruction step displacement field ansatz in the most general form, expanding the equations up to third order in the 0th step, i.e., the nonlinear displacement field corresponding to the observed nonlinear density. Then we derived the $(n+1)$th step displacement field and obtained the recurrence relation for the expansion kernels.\nThis allows us to systematically compute an arbitrary step's postreconstruction 1-loop spectrum using the recurrence relation.\n\nOur model predicts that the postreconstruction displacement field should converge to the true displacement field, contrary to what we found in our simulation. The model provides a good description of the cross-correlation coefficient, while the difference in the propagator is mainly responsible for the discrepancy between the model and the simulation.  \n\n\n\nWhile we could not identify the reason for the discrepancy between the model and the reconstructed field in the present paper, we suspect this is potentially related to the subsampling of the tracers. Including such an effect in the theoretical model and implementing a pixel window function correction will help identify the discrepancies. We leave such an extension for future papers. \n\nOur tentative solution to the discrepancy is to drop all UV-sensitive integrals from the 3LPT spectrum, and the ad hoc model fits the postreconstruction displacement field with 1\\% accuracy for $n>6$.\n\n\n\n\n\nThis paper presents the proof of concept on how we can model the iterative reconstruction steps using the real-space mass distribution with little shot noise effect.\nImproving the agreement between the model and the simulation in future papers, we expect it would be straightforward to generalize our modeling to the redshift space. \nOn the other hand, an extension to biased tracers would be nontrivial, based on our investigation in the companion paper~\\cite{Seo:2021}.  While the model we built appears quite strenuous, we note that all iteration kernels are cosmology-independent and could be precomputed and interpolated. A short come is that we find that the speed of 1-loop calculation is still slow. We confirmed that the prereconstruction 3LPT calculation could be optimized with FFTlog, using the technique presented in Ref.~\\cite{Schmittfull:2016jsw}. However, we found that the same algorithm does not apply to the postreconstruction spectrum because of the complex structure of the iteration kernels.\nThus, the iterative reconstruction requires 2D standard quadrature integration. Speeding up those calculations would be critical for actual data analysis. Additionally, it would be essential to extract a more compressed and practical form of the model that can be more efficiently utilized for data analysis, which we plan to do to implement the redshift space distortions and galaxy bias. \n\n\n\nFinally, we note that our work is still in an early stage compared to the modeling of the standard reconstruction density field using the 1-loop SPT in Ref.~\\cite{Hikage:2017tmm} and the Zeldovich approximation in Ref.~\\cite{Chen:2019lpf}, etc. Moreover, Ref.~\\cite{Seo:2021} implies that the iterative displacement reconstruction needs to be better optimized in the presence of high shot noise to take full advantage of it. We nevertheless believe that reconstructing the nonlinear displacement field could be pretty helpful in a deficient shot noise regime that can be available in future galaxy surveys due to its high correlation with the initial condition and, therefore, worth further investigation. \n\n\\begin{acknowledgments}\n\nThe authors would like to thank Marcel Schmittfull for providing simulations and valuable discussions.\nThe authors also would like to thank Stephen Chen for their helpful discussions.\nAO would like to thank Masaru Hongo for valuable discussions. AO and H.-J.S. are supported by the U.S.~Department of Energy, Office of Science, Office of High Energy Physics under DE-SC0019091. \nSS was supported in part by World Premier International Research Center Initiative (WPI Initiative), MEXT, Japan. \nThis project has received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation program (grant agreement 853291). FB is a University Research Fellow.\n\n\\end{acknowledgments}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\par\nThe study of rare kaon decays \\cite{kdecay-rare} has played an important role \nin the establishment of the standard model (SM) in particle physics. \nIt has also been crucial for understanding the phenomenon of CP violation \\cite{kdecay-CP}. \nThe rare decay $K_L^0\\rightarrow\\pi^0\\nu\\bar{\\nu}\\;$ \\cite{litt89,isidori,buras} is a direct CP violation process caused by\na flavor-changing neutral current (FCNC) with transition from a strange to down quark.\n\\par\nThe unique characteristic of the $K_L^0\\rightarrow\\pi^0\\nu\\bar{\\nu}\\;$ decay is that the branching ratio can be \ncalculated with very small theoretical uncertainties.  In the SM\nbased on the Cabibbo-Kobayashi-Maskawa (CKM) matrix \\cite{ckm} for quark flavor mixing, \nthe $K_L^0\\rightarrow\\pi^0\\nu\\bar{\\nu}\\;$ branching ratio is predicted to be $(2.49\\pm 0.39) \\times 10^{-11}$ \\cite{mescia}. \nThe uncertainty of the prediction is dominated by the\nallowed range of the imaginary part of a CKM matrix element $V_{td}$, \nwhich is determined by other measurements,\nand the intrinsic theoretical uncertainty is only 1-2\\% \\cite{buras}.\n\\par\nThe determination of CKM matrix elements has been greatly improved\nin the past ten years\nby measuring various B-decay properties \\cite{bdecay}; \nall measurements were consistent with each other within the standard model parameterization. \nBy measuring the $K_L^0\\rightarrow\\pi^0\\nu\\bar{\\nu}\\;$ decay precisely, we can check the consistency against the \ncurrently predicted value in the SM.\nA deviation indicates new physics beyond the SM \nbecause of the very small uncertainty in deriving the imaginary part of $V_{td}$\nfrom the branching ratio of the $K_L^0\\rightarrow\\pi^0\\nu\\bar{\\nu}\\;$ decay.\n\\par\nThe decay $K_L^0\\rightarrow\\pi^0\\nu\\bar{\\nu}\\;$ is one of the processes expected to have a significant impact on \nnew physics searches because it is an FCNC process that can proceed through loop diagrams, \nincluding the interactions at short distance and large mass scales.\nGrossman and Nir \\cite{grossman} pointed out the importance of studying new physics by using \nboth the $K_L^0\\rightarrow\\pi^0\\nu\\bar{\\nu}\\;$ and $K^+\\rightarrow\\pi^+\\nu\\bar{\\nu}\\;$  decays, where the decay $K^+\\rightarrow\\pi^+\\nu\\bar{\\nu}\\;$ is another rare kaon decay with \nsmall theoretical uncertainties \\cite{buras,mescia,brod}. \nVarious new physics models have been developed and used to predict \nthe $K_L^0\\rightarrow\\pi^0\\nu\\bar{\\nu}\\;$ and $K^+\\rightarrow\\pi^+\\nu\\bar{\\nu}\\;$ branching ratios \\cite{buras, isidori,mesciaweb}.\n\\par \nThe best upper limit obtained in previous experiments was BR($K_L^0\\rightarrow\\pi^0\\nu\\bar{\\nu}$) $<$\n5.9 $\\times$ 10$^{-7}$ (90\\% CL) \\cite{ktev}. \nThe ultimate goal of our experimental study is to determine the $K_L^0\\rightarrow\\pi^0\\nu\\bar{\\nu}\\;$ branching ratio \nwith an accuracy less than 10\\% of the value predicted in the SM.\nThe goal can be achieved by performing a series of experiments with improved and refined \ndetection methods. \n\\par\nThe E391a experiment, which was carried out at the KEK 12-GeV proton synchrotron \n(KEK-PS), is the first step of this approach. The main objectives of E391a were not only \nto investigate the decay with a dedicated detector, but also to test and confirm our basic \nexperimental methods for $K_L^0\\rightarrow\\pi^0\\nu\\bar{\\nu}\\;$ at the highest possible sensitivity.  \nData collection of the E391a experiment started in February 2004 and continued \nuntil November 2005, which was one month before the shutdown of the KEK-PS. \nThe total running time was about twelve months and it was divided into three periods \n(Run-1, Run-2, and Run-3). \nEarly results from the first and second periods were reported in \nRef.~\\cite{run1-oneweek} and Ref.~\\cite{run2}, respectively. In this article, we report \nthe final results on the $K_L^0\\rightarrow\\pi^0\\nu\\bar{\\nu}\\;$ decay as obtained from E391a.\n\n\n\\section{Experimental method}\n\\subsection{Basic method of $K_L^0\\rightarrow\\pi^0\\nu\\bar{\\nu}\\;$ detection}\nThe key signature of the $K_L^0\\rightarrow\\pi^0\\nu\\bar{\\nu}\\;$ decay is detection of exactly two photons and \nnothing else.  \nThe \\pizs decays dominantly into two photons ($\\gamma\\gamma$), \nand the neutrinos are undetectable.  \nBecause detection of the incident $K_L^0\\;$ is difficult, measurements can only be made of \nthe energy and position of the two photons in a calorimeter located downstream of the $K_L^0\\;$ decay region, \nwithout having direct information of the incident particle.\nKinematic constraints are weak for definitive identification of the decay. \nInstead, the decay must be isolated by eliminating all possible backgrounds. \n\\par\nBecause the signal mode is identified as the final state of two photons and nothing else,\nprocesses that make two or more photons can cause background events.\nThe $K_L^0\\;$ decays such as $K_L^0\\rightarrow\\pi^0\\pi^0\\;$ and $K_L^0\\rightarrow\\pi^+\\pi^-\\pi^0\\;$ become backgrounds when\nextra photons or charged particles escape detection.\nThe $K_L^0\\rightarrow\\gamma\\gamma\\;$ decay can be a background source because it has only two photons in the final state,\nalthough it is well suppressed by kinematical constraints.\nAnother background process is hadronic interactions of beam neutrons \nwith the residual gas in the beamline or in detectors near the beam.\nAny \\pizs or $\\eta$ produced in these interactions decay into two photons and\nproduce backgrounds. \nHyperons produced at the target can cause backgrounds through processes \nsuch as $\\Lambda \\to \\pi^0 n$;\nthese hyperon events are strongly suppressed because most of them decay in a 10 m long neutral beamline.\n\\par\nThe most important tool for reducing the background is a hermetic detector\nsystem to detect and veto extra particles. \nBecause all of the other $K_L^0\\;$ decays, except for $K_L^0\\rightarrow\\gamma\\gamma\\;$, are accompanied with \nat least two additional photons or charged particles, \nthe detector should be highly sensitive to photons and charged particles. \n\\par\nThe signature of $K_L^0\\rightarrow\\pi^0\\nu\\bar{\\nu}\\;$ can be provided by using a small-diameter $K_L^0\\;$ beam \n(called a ``pencil beam\") and measuring precisely the energy and position of the two photons.\nAlthough $K_L^0\\;$ flux is reduced with a pencil beam, it has several advantages.  \nFirst, the beam hole at the center of the calorimeter, which compromises hermeticity, can be minimized. \nSecondly, the $K_L^0\\;$ decay vertex position ($Z_{\\textrm{VTX}}$) can be assumed to be on the beam axis.\nThe vertex position, which is same as the \\pizs decay position due to the short lifetime of $\\pi^0$,\nis obtained from the kinematics of the two photons from its decay (See Sec.\\ III.B).\nThe transverse momentum of the \\pizs ($P_T$) with respect to the $K_L^0\\;$ \nbeam axis is also obtained. \n\\par\nThe signal region for $K_L^0\\rightarrow\\pi^0\\nu\\bar{\\nu}\\;$ decay can be defined with $Z_{\\textrm{VTX}}\\;$ and $P_T$.\nRequiring a sufficiently large missing transverse momentum \neliminates contamination from the $K_L^0\\rightarrow\\gamma\\gamma\\;$ decay and also reduces the contamination \nof other $K_L^0\\;$ decays. Most of the $K_L^0$'s decay into multiple particles that have \nlow momenta in the $K_L^0\\;$ rest frame, and hence low $P_T\\;$ in the laboratory frame. \nThe $Z_{\\textrm{VTX}}\\;$ should be in the region away from beam counters \nand $P_T\\;$ should be in the range from 120 to 240 MeV\/$c$, \nwhere the maximum momentum of $\\pi^0$'s in the $K_L^0$-rest frame is 231 MeV\/$c$ for $K_L^0\\rightarrow\\pi^0\\nu\\bar{\\nu}$. \n\\par\nProduction of $\\pi^0$'s through beam-gas interactions in the decay fiducial region are \nreduced by having a high vacuum.  Hit-rates of the beam halo (mostly neutrons) \nwith the surrounding detectors are minimized by a sharp collimation of the beam. \nIn addition, as discussed in Sec.\\ III.A and in the subsequent analysis of the\ndata, mis-combinations of two photons from background processes and\/or \nmis-measurements of energies and positions were found to be major sources \nof backgrounds in the E391a experiment. \nThe photons were often produced from interactions of the beam halo and can lead to incorrect determinations of $Z_{\\textrm{VTX}}$.  \nSome backgrounds from beam interactions can be reduced by detecting low-energy deposits from recoil \nparticles emitted after the interactions.  More detailed descriptions of the \ndetection method have been reported in the original proposal of E391a \\cite{design}. \n\\par\nInefficiencies of photon and charged particle detection provide backgrounds. \nThe decay $K_L^0\\rightarrow\\pi^0\\pi^0\\;$ is the most serious background source arising from an inefficiency \nbecause it has only two extra photons in the final state. \nThe inefficiency of photon detection was investigated in a series of experiments \n\\cite{phineff}, which showed that the inefficiency monotonically decreased with the \nincident photon energy. \nIt was also found that photon detection with a very low energy threshold is necessary \nto achieve a small overall inefficiency, even for high-energy photons. If an extremely \nlow energy threshold around 1 MeV was set, the backgrounds from other $K_L^0\\;$ decays were \nreduced to a negligible level within the sensitivity of the experiment. \n\\par\nThe following sections describe the application of the basic methods in the\nE391a experiment.\n\n\n\\subsection{Beamline}\nThe production target of the $K_L^0\\;$ beam was a platinum rod with a length of 60 mm and \na diameter of 8 mm. The profile of the primary beam was $\\sigma$ = 3.3 mm and \n$\\sigma$ = 1.1 mm along the $X$ (horizontal) and $Y$ (vertical) axes.\nThe neutral beam was extracted at an angle of 4$^{\\circ}$ with respect to the primary \nproton beam. The target rod and beamline elements were aligned along a straight line \nin the direction of the neutral beam. The total length of the beamline was 10 m. \nThe neutral beamline consisted of two dipole magnets (D1, D2) to sweep charged particles \nout of the beam, with field strengths of 4 T$\\cdot$m and 3 T$\\cdot$m, respectively, and six collimators \n(C1 - C6) to collimate the beam, as shown in Fig.~\\ref{fig_beamline}.\n\n\\begin{figure}[htbp]\n\\begin{center}\n\\includegraphics[width=.48\\textwidth]{.\/fig1.eps}\n\\end{center}\n\\caption{Schematic view of the neutral beamline: (top) the arrangements of the components and \n(bottom) the collimation scheme. C1-C3, C5, and C6 were tungsten collimators, assembled as a \nstack of cylindrical blocks 5-cm thick with circular holes of different diameters.\nEach collimator approximated the cone-shaped aperture as indicated by the A-E lines \nin the bottom figure.  The $Z = 0$ coordinate is at the center of the production target.}\n\\label{fig_beamline}\n\\end{figure}\n\n\\par\nFive collimators (C1-C3, C5, and C6) were made of tungsten. \nThe C2 and C3 collimators in the upstream end were used to define the beam with the designed \napertures, which were arranged to form a half cone angle of 2 mrad from the target center.  \nThe last two collimators, C5 and C6, were used to trim the beam halo.\nThe most upstream collimator C1 reduced the size of the beam immediately after the target \nwithout producing a large penumbra.  The total thickness of these five collimators \nwas approximately 6 m. A thermal-neutron absorber, which was made of polyethylene terephthalate \n(PET) sheets containing Gd$_2$O$_3$ 40\\% in weight, was used for C4. \nThe aperture of C4 was set to be larger than  that of the other collimators.\n\\par\nMovable absorbers, made of lead (Pb) and beryllium (Be), were placed between C1 and C2 \nto reduce the number of photons and neutrons relative to the $K_L^0$'s. The absorbers were \n10-mm-diameter rods with the lengths of 5 cm and 30 cm for Pb and Be, respectively. \nThe downstream region, starting with a stainless steel window 100 $\\mu$m thick\nat the upstream end of C4, was evacuated to approximately 1 Pa.\n\\par\nThe primary beam on the target was monitored by a secondary-emission chamber (SEC) placed \nupstream of the target, and a target monitor (TM) that was a counter telescope\nwhich viewed the target center at 90 degrees.  \nThe primary beam position at the target was adjusted with steering magnets \nby monitoring the $K_L^0\\rightarrow\\pi^0\\pi^0\\pi^0\\;$ decay events.\nIn the $3\\pi^0$ events, six photons were detected by a CsI calorimeter\nwhich will be described in Sec.\\ II.C.\nThe center of energy was defined to be $\\bm{r}_c = \\sum E_i \\, \\bm{r}_i$, where $E_i$ are the photon \nenergies and $\\bm{r}_i$ are the photon hit positions at the CsI calorimeter. \nBecause the six-photon events were mostly $K_L^0\\rightarrow\\pi^0\\pi^0\\pi^0\\;$ decays, the peak position \nof the center of energy should be on the axis of the beamline.\nThe position of the neutral beam was maintained to be within 0.2 mm \nfrom the center throughout the entire running period.\nThe beam size at the CsI calorimeter, was also monitored by the distribution of the center of energy.\nIts diameter was $\\sigma = 40$ mm, which was consistent with the beam divergence.\n\\par\nThe peak momentum of the $K_L^0\\;$ as determined from beamline simulations was 2 GeV\/$c$ at the exit of C6, \nas shown in Fig.~\\ref{fig_spectrum}.\n\n\\begin{figure}[htbp]\n\\begin{center}\n\\includegraphics[width=.43\\textwidth]{.\/fig2.eps}\n\\end{center}\n\\caption{\nMomentum spectrum of $K_L^0\\;$ at the exit of C6, obtained from beamline simulations.}\n\\label{fig_spectrum}\n\\end{figure}\n\nThe initial $P_T\\;$ spread due to the beam divergence of 2 mrad was approximately 4 MeV\/$c$. \nThe neutron-to-$K_L^0\\;$ ratio was 60 and the halo-to-core ratio was approximately 10$^{-5}$ \nfor both neutrons and photons with energies above 1 MeV, as shown in Fig.~\\ref{fig_profile}. \nBy inserting a Be absorber, the neutron-to-$K_L^0\\;$ ratio was reduced to 40, \nwith roughly a 45\\% loss in $K_L^0\\;$ flux.\nProfiles of the $K_L^0\\;$ beam are shown in Fig.~\\ref{fig_klprofile}.\n\n\\begin{figure}[htbp]\n\\begin{center}\n\\includegraphics[width=.43\\textwidth]{.\/fig3.eps}\n\\end{center}\n\\caption{\nBeam profiles of neutrons and photons above 1 MeV at the exit of C6 collimator, \nobtained from beamline simulations.}\n\\label{fig_profile}\n\\end{figure}\n\n\\begin{figure}[htbp]\n\\begin{center}\n\\includegraphics[width=.43\\textwidth]{.\/fig4.eps}\n\\end{center}\n\\caption{\nBeam profile of $K_L^0\\;$ at the exit of C6 collimator, obtained from beamline simulations.\nFilled and open circles show the case with and without the Be absorber, respectively.\n}\n\\label{fig_klprofile}\n\\end{figure}\n\n\nPunch-through muons were emitted in the direction parallel to the beam axis. \nTheir position distribution was almost flat and the flux density was larger than the \ncosmic ray flux by roughly one order of magnitude.\nDetails of the beamline has been reported elsewhere \\cite{beamline}.\n\n\\subsection{Detectors}\nThe E391a detection system was located at the end of the beamline.\nThe detector subsystems were cylindrically arranged around the beam axis, and most of \nthem were placed inside a large vacuum vessel, as shown in Fig.~\\ref{fig_apparatus}.\nFrom here on, the origin of the coordinate system is defined to be at the upstream end of the \nE391a detector, as shown in Fig.~\\ref{fig_apparatus}.\nThis position was approximately 12 m from the production target.\n\n\\begin{figure*}[htbp]\n\\begin{center}\n\\rotatebox{-90}{\\includegraphics[width=.3\\textwidth]{.\/fig5.eps}}\n\\end{center}\n\\caption{Detection system.}\n\\label{fig_apparatus}\n\\end{figure*}\n\n\n\\subsubsection{CsI calorimeter}\nThe energy and hit position of photons, such as the two photons\nthat would come from the \\pizs decay in the $K_L^0\\rightarrow\\pi^0\\nu\\bar{\\nu}\\;$ signal, were measured by \nusing a calorimeter placed at the downstream end of the decay region. \nAs shown in Fig.~\\ref{fig_csi}, the calorimeter was made of 496 undoped CsI crystals\neach with a dimension of 7 $\\times$ 7 $\\times$ 30 cm$^3$ (Main CsI)~\\cite{e162-csi} and \nassembled in a cylindrical shape with an outer diameter of 1.9 m.\nThere was a 12 cm $\\times$ 12 cm beam hole at the center of the calorimeter.\nThe beam hole was surrounded by a tungsten-scintillator collar counter (CC03) and\n24 CsI modules with dimensions of 5 $\\times$ 5 $\\times$ 50 cm$^3$ (KTeV CsI)~\\cite{ktev-csi}.\nThe gaps at the periphery of the cylinder were filled with \n56 CsI modules with trapezoidal shape and 24 modules \nof lead-scintillator sandwich counters (Lead\/Scintillator Sandwich).\nThese modules were tightly stacked such that the gaps between them were less than 0.1 mm. \nAlthough the CC03 and the sandwich counters were used only in the veto system, \nthe other parts, which consisted of CsI crystals, were used for photon measurements. \nThe average light yield of the main CsI modules was 16 pe\/MeV, where pe\/MeV is the unit \nof the number of photo-electrons emitted from the photomultiplier tube (PMT) cathode \nfor an energy deposit of 1 MeV in the detector.  Details of the CsI calorimeter have \nbeen reported in Ref.~\\cite{csi}.\n\n\n\n\\begin{figure}[htbp]\n\\begin{center}\n\\includegraphics[width=.47\\textwidth]{.\/fig6.eps}\n\\end{center}\n\\caption{Assembly of detector subsystems in the downstream end-cap.}\n\\label{fig_csi}\n\\end{figure}\n\n\n\\subsubsection{Charged veto counter}\nA set of plastic scintillation counters (named CV) were placed in front of the CsI \ncalorimeter to identify events that included the emission of charged particles,\nsuch as $K_L^0 \\rightarrow \\pi^-e^+\\nu\\;$ decay~\\cite{chineff}.\nA total of 32 sector-shaped modules (outer CV) were placed at a distance of 50 cm\nfrom the front face of the CsI calorimeter. \nThe outer CV modules were arranged to have overlaps between adjacent modules.\nThe beam region from the outer CV to the \nCsI was covered by the inner CV, which was a square pipe formed with four plastic scintillator plates. \nThe inner and outer CVs were closely connected with aluminum fixtures.\nIn order to eliminate gaps between the outer and inner modules, \nthe inner modules were extended to cover the edge of the outer modules.\nThe edges of the outer CV and the inner CV were located \nclose to the beam, and neutrons in the beam halo frequently interacted with them. \n\n\n\\subsubsection{Barrel counters}\nThe decay region was surrounded with two large lead-scintillator sandwich counters: \nthe main barrel (MB) and the front barrel (FB), composed of 32 and 16 modules, \nrespectively. The modules were tightly assembled with small gaps \\cite{kumitate}, \nas shown in Fig.~\\ref{fig_barrels}. \n\n\\begin{figure}[htbp]\n\\begin{center}\n\\includegraphics[width=.35\\textwidth]{.\/fig7.eps}\n\\end{center}\n\\caption{Cross-sectional end view of the MB (outer ring) and FB (inner ring).}\n\\label{fig_barrels}\n\\end{figure}\n\nThe lamination was parallel to the beam axis, and the lengths of the MB and FB modules \nwere 5.5 m and 2.75 m, respectively. We developed a new type of plastic scintillator\nmade of a resin mechanically strengthened with a mixture of styrene and methacrylate \\cite{ci-scinti}. \nThe scintillator counters were fabricated by extrusion.\nThe thickness of the MB and FB modules were 13.5$X_0$ and 17.2$X_0$, respectively. \nThe emitted light was transmitted through wavelength-shifting (WLS) fibers, \nwhich were glued to each scintillator plate with a pitch of 10 mm.  \nThe fibers had a double-cladded structure with Y11 as a dopant. \nThe green light emitted from Y11 was read by a newly developed photomultiplier tube \nwith a high quantum efficiency for green light (EGP-PMT)~\\cite{pmt}. \nFor readout, each module was divided into an outer and an inner parts. \nThe MB module was viewed from both ends, and the FB module was viewed from the upstream end.  \nFigure~\\ref{fig_mblight} shows the light yields for 4 readout channels of one MB module, \nobtained from a cosmic-ray test. \nThe light yield and attenuation of the FB module were similar to those shown in \nFig.~\\ref{fig_mblight} for the MB module. Details of the barrel counters has been \nreported in Ref.~\\cite{barrel}.\n\n\n\\begin{figure}[htbp]\n\\begin{center}\n\\includegraphics[width=.43\\textwidth]{.\/fig8.eps}\n\\end{center}\n\\caption{Light yield and attenuation for a MB module.}\n\\label{fig_mblight}\n\\end{figure}\n\n\nA layer of plastic scintillator was placed inside of the MB modules to identify charged particles. \nIt consisted of 32 modules and was called the barrel charged veto (BCV).\nA module was made of a 10-mm-thick plastic scintillator, \nand the light signal was read from both ends by an EGP-PMT through WLS fibers glued \nto the scintillator with a 5-mm pitch. \n\n\n\\subsubsection{Counters close to the beam}\nMultiple collar-shaped counters, CC02 -- CC07, were placed along the beam axis.  \nCounter CC02 at the entrance of the decay region was a lead-scintillator \nsandwich of the Shashlik type, with WLS fibers piercing the lead and \nscintillator layers.  Counter CC03 at the end of the decay region was a \ntungsten-scintillator sandwich. Counters CC04 -- CC07 covered the solid-angle in \nthe downstream direction. Counters CC04 and CC05 were lead-scintillator\nsandwiches with WLS fibers glued on each scintillator plate at a pitch of 10 mm, \nwhich was the same as for the MB and FB.  \nCounters CC06 and CC07 were made of SF5 lead-glass.  \nThe direction of lamination was perpendicular to the beam for \nCC02, CC04, and CC05, and was parallel to the beam for CC03.  The signals \nfrom CC02 -- CC05 were read by EGP-PMTs. \n\\par\nBeginning with Run-2, another collar counter called \nCC00 was installed in front of the FB and outside the vacuum vessel to reduce\nthe effects of halo neutrons. It consisted of 11 layers of a 5-mm-thick plastic \nscintillator interleaved with 10 layers of 20-mm-thick tungsten. \nHowever, CC00 did not significantly reduce the neutron halo because it was \nnot placed as close to the beam as the other collar counters. \n\\par\nA beam-plug counter, back-anti (BA), was placed at the end of detector system along \nthe beam axis. In the first two data-taking periods, Run-1 and Run-2, the BA consisted \nof six superlayers, where each superlayer had six plastic-scintillator layers\ninterleaved with lead sheets, and a single layer of quartz. In Run-3, the lead \nand plastic layers were replaced with PWO crystals as shown in Fig~\\ref{fig_ba},\nwith the intention to separate electromagnetic showers from the neutron hits.\n\\par\nA thin layer of plastic scintillator, the beam hole charged veto (BHCV), \nwas placed in front of the BA. It was effective in removing the \n$K_L^0\\rightarrow\\pi^0\\pi^+\\pi^-\\;$ decay. \nThe thickness of the BHCV was 1~mm for Run-1, and 3~mm for Run-2 and Run-3.\n\n\\begin{figure}[htbp]\n\\begin{center}\n\\includegraphics[width=.32\\textwidth]{.\/fig9a.eps} \\\\\n\\includegraphics[width=.32\\textwidth]{.\/fig9b.eps}\n\\end{center}\n\\caption{BA used in Run-2 (top) and Run-3 (bottom).}\n\\label{fig_ba}\n\\end{figure}\n\n\\par\nThe upstream counters, CC02 -- CC04, were placed inside the vacuum vessel, \nand the downstream CC05 -- BA were placed outside the vessel. A vacuum duct, \nwhich was directly connected to the vacuum vessel, penetrated CC05 -- CC07 \nand the vacuum region was extended up to the front face of BHCV.\n\\par   \nThe size and basic parameters of CC02 -- CC07, BHCV, and BA are summarized in Table~\\ref{tab:table1}.\n\n\\begin{table*}\n\\caption{%\n\\label{tab:table1}\nSize and basic parameters of detectors along the beam.\nThe origin of the $z$ position corresponds to the start of the E391a detector.\nIn the sizes of cross section and hole, dia. and sq. represents diameter and square, respectively.\nVisible fraction ($R_{vis}$) is defined as the ratio of the energy deposit in \nthe active material to that in the whole volume. \nThe thickness is expressed in units of radiation length.\nThe recorded number of photo-electrons per 1-MeV visible energy deposit \nwas 10 or more for sandwich counters (CC02-05), and \nat least 0.5 for lead glass detectors (CC06-07), respectively.}\n\\begin{ruledtabular}\n\\begin{tabular}{lccclccc}\ndetector & z position (cm) & outer dimension (cm) & inner dimension (cm) \n& configuration & $R_{vis}$ & thickness ($X_0$) \\\\\n\\hline\nCC02 & 239.1 & 62.0~dia. & 15.8~dia. \n& lead\/scint. & 0.32 & 15.7 \\\\\nCC03 & 609.8 & 25.0~sq. & 12.0~sq. \n& tungsten\/scint & 0.23 & 7.6\\footnotemark[1] \\\\\nCC04 & 710.3 & 50.0~sq. & 12.6~sq. \n& lead\/scint. & 0.28 & 11.8 \\\\\nCC05 & 874.1 & 50.0~sq. & 12.6~sq. \n& lead\/scint. & 0.28 & 11.8 \\\\\nCC06 & 925.6 & 30.0~sq. & 15.0~sq. \n& lead glass & 1.0 & 6.3  \\\\\nCC07 & 1000.6 & 30.0~sq. & 15.0~sq. \n& lead glass & 1.0 & 6.3 \\\\\nBHCV & 1029.3 & 23.0~sq. & no\n& plastic scint. & 1.0 & 0.007 \\\\\nBA(Run-2) & 1059.3 & 24.5~sq. & no \n      & quartz & 1.0 & 1.5 \\\\\n& & & & lead\/scint. & 0.31 & 13.3 \\\\\nBA(Run-3) & 1059.3 & 24.5~sq. & no\n      & quartz & 1.0 & 1.2 \\\\\n& & & & PWO & 1.0 & 16.8 \\\\\n\\end{tabular}\n\\end{ruledtabular}\n\\footnotetext[1]{The lamination of CC03 was parallel to the beam.}\n\\end{table*}\n\n\\subsubsection{Vacuum}\nThe vacuum pressure in the decay region had to be maintained below 10$^{-5}$ Pa in order \nto reduce the \\pizs backgrounds produced by beam-gas interactions to a negligible level \nat the sensitivity corresponding to the SM prediction of $K_L^0\\rightarrow\\pi^0\\nu\\bar{\\nu}$.\nThe amount of dead material along the path of particles from the decay vertex to the detector had to be \nminimized in order to achieve highly efficient detection even for low-energy particles. \nA differential pumping method was adopted for this purpose. \n\\par\nThe entire detection system was placed in a large vacuum vessel, as shown in\nFig.~\\ref{fig_apparatus}.  The pressure in the outer part, where the detectors were placed, \nwas around 1 Pa, and the pressure in the inner part, through which the beam passed, was \n$1 \\times 10^{-5}$ Pa.  \nThe two regions were separated by a laminated membrane sheet \nwith a thickness of 20 mg\/cm$^2$. \nAs shown in Fig.~\\ref{fig_membrane},\nthe sheet was a lamination of four films. \nThe EVAL film had low transmission for oxygen gas (mostly air) and the nylon film strengthened the sheet. \nThe polyethylene layers on both sides were used to make a tight connection by using a heat iron press.\nThe bag-shaped membrane covered the inner surface of the CsI detectors by using a \nskeleton structure of thin aluminum pipe, similar to a camping tent. \n\\par\nTwo sets of rotary and root pump systems were connected through a manifold\nand eight ports to the outer-vacuum part. The pumping speed of each system was \n1200 m$^3$\/hour. Four turbo molecular pumps, each having the pumping speed of 800 l\/sec, \nwere connected between the inner vacuum part and the manifold.  They produced \nthe necessary pressure difference of an additional five orders of magnitude.\n\\par  \nFor the PMTs installed in vacuum, the pressure had to be less than 10 Pa to prevent \nhigh voltage discharges.  The PMT operation in vacuum additionally caused a cooling \nproblem due to the absence of convection. We modified the configuration of the\nresistor chains of the PMTs, and cooled them with a water circulation system \nas described in Ref.~\\cite{csi}. The temperature was stable within $\\pm 0.1^{\\circ}$ \nfor the CsI calorimeter.\n\n\n\\begin{figure}[htbp]\n\\begin{center}\n\\includegraphics[width=.3\\textwidth]{.\/fig10.eps}\n\\end{center}\n\\caption{\nMembrane used for the vacuum separation.\n}\n\\label{fig_membrane}\n\\end{figure}\n\n\n\\subsubsection{Electronics}\nAbout 1000 PMTs were used as readout devices for all detectors. \nSignals from the detectors in the vacuum vessel were extracted through individual \nfeed-through connectors. A connector was made of a coaxial cable simply molded to a metal \nflange with a resin. \n\\par\nAll detector signals were fed to Amplifier-Discriminator (AD) modules, \nwhich were newly developed for the experiment. An AD module accepted 16 PMT signals \nand generated 16 analog signals, 16 discriminated signals, and two 8-channel linear sums.  \nBecause early discrimination prevents time deterioration due to distortion of the signal \nshape in the cable, the AD modules were placed near the vacuum vessel to shorten the cables.\n\\par\nThe analog output signal was derived from each input signal with a throughput of 95$\\%$. \nIt was sent to a charge-sensitive analog-to-digital converter (ADC) in the counting hut \nthrough a 90-m coaxial cable, while the cable length of the other outputs was 30 m. \nThe discriminator output was generated with a very low threshold of 1 mV, which corresponded \nto an energy deposit of 1 MeV for the CsI calorimeter and below 1 MeV for the other detectors.\nThe discriminator output was sent to a time-to-digital converter (TDC) in the counting hut \nthrough 30 m of twisted-pair cable after passing through a fixed delay of 300 ns in the AD \nmodule. The signal summed over 8 inputs was sent to the counting hut through 30 m of \ncoaxial cable and used to form trigger signals for data acquisition. \n\\par\nThe ADC module, LeCroy FASTBUS 1885F, had 96 input channels, \neach channel having the equivalent dynamic range of a 15-bit ADC in its 12-bit data\nby using a bi-linear technique.\nThe typical resolution at a low energy range was\n0.13 MeV\/bit for the CsI calorimeter, and 0.035 MeV\/bit for photon veto detectors.\nThe gate width for the CsI calorimeter was 200 ns. \nThe TDC module in TKO (Tristan KEK Online-system), which was an electronics platform developed in KEK~\\cite{tko},\nwas operated at a \nfull range of 200 ns and a resolution of 50 ps. The analog and discriminated signals were \nalready delayed by 300 ns (60-m cable with a propagation velocity of 20 cm\/ns) \ncompared to the timing of linear-sum signal at the entrance of the counting hut. \nAll cables to the counting hut were placed inside trays covered with copper-plated iron sheets \nto minimize ground noise caused by alternating magnetic fields. \nThe pedestal widths of almost all the ADC channels were less than 1 bit. \n\\par\nAny failures in the PMTs lead to serious problems in the experiment. Almost all \nPMTs were operated at voltages below 60\\% of the rated voltage. Such a large margin was \nmade possible by using very sensitive ADCs and very low thresholds for the TDCs. \nAll PMTs operated properly during the entire running time.\n\\par\nMulti-hit TDCs (MTDC) were used for the BHCV and BA to cope with high counting rates. \nWe did not benefit from using MTDCs in Run-1 because the input pulse width was set \nto 100 ns.  For Run-2 and Run-3, the pulse width was shortened to 25 ns.\n\n\n\\subsection{Trigger and data acquisition}\nThe signals from eight adjacent CsI blocks (a segment) were collected into a summed\nsignal; there were 72 such segments as shown in Fig.~\\ref{fig_hwclustering}. \n\n\\begin{figure}[htbp]\n\\begin{center}\n\\includegraphics[width=.43\\textwidth]{.\/fig11.eps}\n\\end{center}\n\\caption{Segments of CsI blocks.}\n\\label{fig_hwclustering}\n\\end{figure}\n\nThe segments were used to form a trigger signal.\nA threshold, which corresponded to an energy deposit of 80 MeV, was applied to\neach summed signal, and the number of segments whose summed signal was above the threshold, \nN$_{\\rm HC}$, was counted. \nWe required N$_{\\rm HC} \\ge 2$ for the $K_L^0\\rightarrow\\pi^0\\nu\\bar{\\nu}\\;$ trigger.\nWe also required an anti-coincidence of several veto counters, CV, MB, FB, and CC02 -- CC05 \nwith a rather high threshold. \nLater, in the offline analysis, these conditions were set tighter: \na higher threshold for photon detection and lower thresholds for the vetos. \n\n\\par\nIn addition to the trigger for $K_L^0\\rightarrow\\pi^0\\nu\\bar{\\nu}$, we prepared triggers for light pulsars: \nxenon lamps for CsI and LEDs for the other detectors. They were used to monitor\nshort-term drifts in the PMT gains. Moreover, they were useful for studying the\neffects of beam loading by flashing them within and outside the beam spills. \nTriggers for cosmic ray muons and punch-through muons were also used for \ndetector calibration. \n\\par\nWe prepared two types of triggers to record accidental hits in the detector system \nby using an electronic pulse generator and the TM, which was a counter telescope near the target. \nThe accidental activities observed in the two types of triggers were consistent with each other\nfor all detectors except the BA and BHCV. \nWhile the pulsar trigger had no correlation with the event time and was randomized \nwith respect to the asynchronous timing of events, \nthe TM trigger reflected intensity variations of the primary beam. \nThe TM trigger data were normally used because we observed \nthe micro time-structure of the beam extraction.\n\\par\nAccidental losses were estimated by applying the analysis cuts used for \n$K_L^0\\rightarrow\\pi^0\\nu\\bar{\\nu}\\;$ to the TM trigger data.  \nThe estimated value was compared with the value estimated from Monte Carlo simulations \nwith imposition of the TM trigger data, and consistency was confirmed. \n\\par\nData were collected through multiple parallel systems of the VME crates\noperated with a CPU to control them. \nEnvironmental data such as temperatures at 100 locations, \nvacuum pressure at several places, and single counting rates of \nsampled channels were accumulated by using PCs.\nThe CPU and PCs were distributed on the network.\nThe basic software used in the E391a experiment was MIDAS \\cite{midas}. \n\\par\nThe dead time of the DAQ system was around 600 $\\mu$s\/event. \nIn a typical run, in which the proton intensity was $2.5\\times 10^{12}$ POT (protons on target) \nin a 2-s spill every 4 s, the trigger rate was 300 per spill (150 Hz), \nand the live time was 91\\%.  The data size was 3 Mbytes\/spill and the typical \ndata size collected per day was 60 GB.\n \n\n\\subsection{Calibration}\nGood linearity between the energy deposit and the ADC output was observed in all \ndetector subsystems. \nTheir gains were calibrated through a constant in units of MeV\/bit.\nThe gain constants of all detectors were basically calibrated in situ, \nafter assembling and in vacuum, by using cosmic-ray muons and\/or punch-through muons \ncoming from the upstream region of the primary beamline. While the cosmic-ray muons \nprimarily traveled in the downward direction, the punch-through muons were parallel \nto the beam. In the case of sandwich detectors, we selected the muons so that their \nprimary direction was perpendicular to the lamination.\n\\par\nThe CsI modules were initially calibrated by using cosmic-ray tracks \nsuch as the example shown in Fig.~\\ref{fig_cosmic}. \n\n\\begin{figure}[htbp]\n\\begin{center}\n\\includegraphics[width=.43\\textwidth]{.\/fig12.eps}\n\\end{center}\n\\caption{Cosmic ray track used for calibration. The outer ring shows the MB.}\n\\label{fig_cosmic}\n\\end{figure}\n\nThe punch-through muons were used to cross-check the cosmic-ray calibration because their \ndirections were perpendicular to each other and their penetration lengths were different \n(7 cm for cosmic-ray muons and 30 cm for punch-through muons). The gain constants were \nrefined by using the two photons from a \\pizs produced in a special run in which an aluminum \nplate was installed on the beam axis. \nFinally, the gain constants were refined by an iteration process based on the\nkinematic constraints of $K_L^0\\rightarrow\\pi^0\\pi^0\\pi^0\\;$ decay.\nThe short-term variation of the gain was corrected by using xenon-lamp light pulses. \nThe reconstructed $K_L^0\\;$ mass and the width were well stabilized, as shown in Fig.~\\ref{fig_gainstab}. \n\n\n\\begin{figure}[htbp]\n\\begin{center}\n\\includegraphics[width=.45\\textwidth]{.\/fig13a.eps} \\\\\n\\includegraphics[width=.45\\textwidth]{.\/fig13b.eps}\n\\end{center}\n\\caption{\nStability of the kinematic variables over the entire period of Run-3.\nThe upper and lower graphs show the peak and width of the effective mass distribution \nof 3\\pizs from the $K_L^0\\rightarrow\\pi^0\\pi^0\\pi^0\\;$ decay, respectively.}\n\\label{fig_gainstab}\n\\end{figure}\n\n\\par\nIn the Monte Carlo simulations, we smeared the energy deposit of each\nphoton by the function $a\/\\sqrt{E(\\rm{GeV})}+b$.  The first term was given as \n$0.008\/\\sqrt{E(\\rm{GeV})}$, which was consistent with the statistical fluctuation \nof photoelectron yields of 16 pe\/MeV. The parameter $b$ was determined to be \n0.004 by tuning it to reproduce the $K_L^0\\rightarrow\\pi^0\\pi^0\\pi^0\\;$ invariant mass distribution.\nBecause this smearing was applied to the deposited energy instead of the incident \nphoton energy,\nthe small value indicates good understanding of the calibration.\n \\par\nThe beam loading effect was examined by flashing the light pulse during and between \nthe beam spills. \nIt was negligibly small in the current experiment for all detectors, \nexcept for the BHCV and BA in which the PMT gain was shifted by 10\\%.\n\\par\nThe timings were determined relative to one of the photon clusters in the CsI calorimeter. \nFirst, TDC constants (ns\/count) were measured for all TDC channels by using a fixed \ndelay, and the time-zero value was calibrated from the data. Cosmic-ray and\/or punch-through \nmuons were also used for the time-zero calibration. They were determined step-by-step by \nutilizing the overlapping parts of different detectors with respect to the muon tracks.\n\\par\nThe time-zero values among CsI modules were determined by using cosmic ray tracks.\nFinally the calibration was refined by using six \nphotons from $K_L^0\\rightarrow\\pi^0\\pi^0\\pi^0\\;$ decays. The time difference between two photons was determined \nwith a standard deviation of 0.3 ns, as shown in Fig.~\\ref{fig_csitiming}, where the \ntiming of a photon cluster was estimated from the timing of the central block that had \nthe local maximum energy deposit.\n\n\\begin{figure}[htbp]\n\\begin{center}\n\\includegraphics[width=.47\\textwidth]{.\/fig14.eps}\n\\end{center}\n\\caption{Time difference between two photons from the $K_L^0\\rightarrow\\pi^0\\pi^0\\pi^0\\;$ decay,\nobtained by subtracting the timing of lower energy photon from that of higher energy photon.\n}\n\\label{fig_csitiming}\n\\end{figure}\n\n\n\n\\section{Data analysis}\n\\subsection{Summary of three runs and outline of present analysis}\nDuring the course of the experiment, there were $(1 - 2.5 ) \\times 10^{12}$ \nPOT per spill, with a total yield of $4.6 \\times 10^{18}$ POT for physics runs.\nA summary of three physics runs is listed in Table~\\ref{table_runs}.\nThe experimental setup was modified during intervals between the runs, \nand the running parameters were refined on the basis of the results of a previous run.\n\n\\begin{table}[htbp]\n\\caption{Summary of three physics runs.}\n\\label{table_runs}\n\\begin{tabular}{ccccccc}\n\\hline \\hline\nRun & \\hspace{0.15cm}& Run period & \\hspace{0.15cm} & POT & \\hspace{0.15cm} & Remarks \\\\\n\\hline\nRun-1 & & Feb.-Jun. 2004 & & $2.1 \\times 10^{18}$ & & Membrane problem \\\\\nRun-2 & & Feb.-Apr. 2005 & & $1.4 \\times 10^{18}$ & & Be absorber \\\\\nRun-3 & & Oct.-Dec. 2005 & & $1.1 \\times 10^{18}$ & & New BA \\\\\n\\hline \\hline\n\\end{tabular}\n\\end{table}\n\n\\par\nThe data quality of Run-1 was severely affected because the \nmembrane for vacuum separation drooped into the beam core, causing many beam interactions. \nThe results obtained from a 10\\% data sample of Run-1 have been published \nelsewhere \\cite{run1-oneweek}. The single-event sensitivity (S.E.S.) was \n$(9.11\\pm 0.20_{\\textrm{stat}}\\pm 0.64_{\\textrm{syst.}})\\times10^{-8}$ \nwith $1.9\\pm 1.0$ background events expected in the signal box ($N_{\\textrm{bg}}$). \nThe complete data sample obtained in Run-1 was analyzed with a  blind analysis. \nIn the blind analysis, the signal candidate events were not examined \nuntil after the final selection criteria were set, and the criteria were not changed later.  \nThe results of the complete data sample of Run-1 were found to be essentially \nconsistent with those obtained from the 10\\% data sample. \nThe S.E.S. was $(5.14\\pm0.25)\\times10^{-8}$ with \nN$_{\\textrm{bg}}=2.10$ \\cite{run1-full}. \nThe number of observed events in the signal box (N$_{\\rm{ob}}$) \nwas 0 for the 10\\% sample and 1 for the complete data sample. \nThe corresponding upper limits for the branching \nratio of $K_L^0\\rightarrow\\pi^0\\nu\\bar{\\nu}\\;$ mode at the 90\\% confidence level (C.L.) were found to be \n$2.1\\times10^{-7}$ and $1.26\\times10^{-7}$, respectively. \nBecause the sensitivity of the Run-1 data sample was not as high as those of \nRun-2 and Run-3, and the running condition of Run-1 was considerably \ndifferent from the later runs, the results of Run-1 are not included \nin the results in this article.\n\\par\nThe membrane problem was rectified before Run-2. \nAs described in Sec.\\ II.C.4,\nthe plastic scintillator of the BHCV was replaced with a thicker one, \nand the discriminated pulse width for the BHCV and BA was reduced. \nAn additional collar counter, CC00, was installed in front of the FB outside \nthe vacuum vessel. In Run-2 and Run-3, a neutron absorber made of beryllium \nwas inserted into the beam line.  The results for the complete Run-2 data \nsample, S.E.S. = $(2.91\\pm 0.31)\\times10^{-8}$ with N$_{\\textrm{bg}}=0.42\\pm 0.14$ \nand N$_{\\rm{ob}}=0$, and the upper limit of BR($K_L^0\\rightarrow\\pi^0\\nu\\bar{\\nu}$) $= 6.7\\times10^{-8}$ \nat the 90\\% CL, have been published in a letter \\cite{run2}.\n\\par\nFrom the previous Run-2 analysis, the dominant source of the background was found to be \nthe interaction of the beam halo (mostly neutrons) with the detectors near the beam \n(``halo neutron background\"). The halo neutron background was from three sources:\n$\\pi^0$'s produced by the interaction with CC02, and $\\pi^0$'s and $\\eta$'s \nproduced by the interaction with the CV.  \nIn the case of CC02-\\pizs background, the computed $Z_{\\textrm{VTX}}\\;$ could be shifted downstream due to shower \nleakage or photo-nuclear interactions in the CsI calorimeter. \nIn the case of CV-\\pizs background, $Z_{\\textrm{VTX}}\\;$ could be shifted upstream due to the fusion \nof multiple photons or the overlap of other hits in the calorimeter. \nIn the case of CV-$\\eta$ background, \n$Z_{\\textrm{VTX}}\\;$ could be shifted upstream by applying the \\pizs assumption in the reconstruction of the two photons.\n\\par\nIn the previous Run-2 analysis, the background level N$_{\\textrm{bg}}$ was estimated \nwith a different method for each background source. A special run with an Al plate was used \nto estimate the background level for CC02-$\\pi^0$. \nA bifurcation method with a pair of cut sets was used to estimate the CV-\\pizs background.\nAn MC calculation was used to estimate the CV-$\\eta$ background.\nThe difference in estimating these backgrounds made it difficult \nto carry out optimizations with the objective of obtaining the best signal-to-noise ratio, S\/N. \n\\par\nTo improve the analysis of Run2-3 data as described in this paper, \nwe developed a method to generate a large number of beam halo interactions \nby using the hadron-interaction code FLUKA~\\cite{fluka}, whose reliability was confirmed \nby the data from the beam survey \\cite{beamline}. \nThe simulation generated beam-halo interactions with approximately 8 times \nthe total number of real data obtained in Run-3. \nFor the CV-$\\eta$ background case, we recycled the event seeds \nto boost the $\\eta$ production and obtained approximately 80 times larger statistics \nas compared to the Run-3 data.\nAlthough the simulation was carried out for Run-3, the events were\nused for Run-2 with minor modifications. The running conditions were almost \nthe same in Run-2 and Run-3, except that the super-layers of the lead scintillator \nsandwich of the BA were replaced with PWO hodoscopes in Run-3.\n\\par\nIn the analysis presented in this article, we first carefully checked the MC events \nagainst the data from the aluminum plate run to confirm the hadronic interaction model. \nNext, we optimized the selection criteria by monitoring the S\/N ratio of MC data. \nHere, S is the number of events inside the signal region \n(340 $<$ $Z_{\\textrm{VTX}}\\;$ $<$ 500 cm and 120 $<$ $P_T\\;$ $<$ 240 MeV\/$c$) \nfor the $K_L^0\\rightarrow\\pi^0\\nu\\bar{\\nu}\\;$ MC events, and N is the MC background events caused \nby beam-halo interactions~\\cite{jiasen}. \nThe optimization based on the MC is also effective to avoid potential bias \nin reanalyzing the Run-2 data.\nFor the data processing, we first masked the events in the signal region \nfor both Run-2 and Run-3. \nOnce we fixed all the selection criteria by the optimization described above, \nwe compared the data and MC for the events outside the signal region. \nAfter confirming the consistency, we finally examined the events inside the signal region.\n\n\n\n\\subsection{Event reconstruction}\n\\subsubsection{Photon clustering}\nThe energy of each photon was measured by forming a cluster of CsI blocks with finite energy deposits.  \nFirst, a search was made for a block with the locally maximum energy, \n{\\em i.e.} the maximum energy among the five blocks geometrically sharing a side. \nAn envelope surrounding the local-maximum block was developed \nby gathering blocks that shared one side and had lower energies. \nAn envelope with a fixed boundary was required\nto reject extra particles that hit the outsides of the two photon clusters. \nIt was also necessary to determine that there was only one local maximum within \nthe envelope, to discriminate clusters that arose from the fusion of \nmore than one photon.\nFigure~\\ref{fig_ncluster} shows the distribution of the number of crystals contained in one photon cluster\nby comparing the six photon events of data and $K_L^0\\rightarrow\\pi^0\\pi^0\\pi^0\\;$ MC.\nThe number of crystals was obtained by counting the crystals having the energy deposit greater than 5 MeV.\nThe size of the cluster was well reproduced by the simulation.\n\n\\begin{figure}[htbp]\n\\begin{center}\n\\includegraphics[width=.44\\textwidth]{.\/fig15.eps}\n\\end{center}\n\\caption{\nDistribution of the number of crystals having the energy deposit greater than 5 MeV in a photon cluster, \ncomparing the data and $K_L^0\\rightarrow\\pi^0\\pi^0\\pi^0\\;$ MC.\nThe top portion shows a comparison between the number of events in the data and MC, \nand the bottom portion shows the ratio between them.\nAll analysis cuts for the $K_L^0\\rightarrow\\pi^0\\pi^0\\pi^0\\;$ mode, as will be described in Sec.\\ V.B., are imposed.\n}\n\\label{fig_ncluster}\n\\end{figure}\n\n\n\\subsubsection{Energy and position correction}\nThe hit position of a photon at the front of the CsI calorimeter was first estimated \nby determining the center of energy of hit blocks in each photon cluster.\nThe $Z_{\\textrm{VTX}}\\;$ coordinate was calculated by using two photon clusters and assuming the \\pizs mass. \nThe energy, position, and injection angle were obtained for each photon. \n\\par\nIn this experiment, the energy and position of each photon \nhad to be corrected for the injection angle to obtain better resolution.\nThe thickness of the CsI crystal (16.2X$_0$) was not sufficient to prevent a \nsmall leakage of energy from its downstream end, and the energy leakage \ndepended on the injection angle. \nWe accumulated a large sample of MC data for a single photon with various energies,\npositions, and angles, \nand compiled look-up tables to correct energy and position of each photon.\nThe correction was iteratively carried out by using data from the tables. \nFigure~\\ref{fig_hslee_corr} shows a result of the correction.  \n\n\n\\begin{figure}[htbp]\n\\begin{center}\n\\includegraphics[width=.44\\textwidth]{.\/fig16.eps}\n\\end{center}\n\\caption{\nDifference of the reconstructed $x$-position of the photons on the calorimeter\nfrom the true value ($\\Delta x$), obtained by $K_L^0\\rightarrow\\pi^0\\nu\\bar{\\nu}\\;$ MC simulation. \nThe $x$ coordinate denotes the horizontal position at the face of the CsI.\nThe resolution was improved after the energy and position corrections.\n}\n\\label{fig_hslee_corr}\n\\end{figure}\n\n\n\\subsubsection{Sorting of events}\nThe events were sorted into several samples according to the number of \nreconstructed photon clusters.\nThe two-photon sample was used to search for the $K_L^0\\rightarrow\\pi^0\\nu\\bar{\\nu}\\;$ decay and for monitoring \nthe $K_L^0\\rightarrow\\gamma\\gamma\\;$ decay. The four-photon and six-photon samples were used for monitoring \nthe $K_L^0\\rightarrow\\pi^0\\pi^0\\;$ and $K_L^0\\rightarrow\\pi^0\\pi^0\\pi^0\\;$ decays, respectively. \n\n\\subsubsection{\\pizs reconstruction}\nBy using the position and energy of two photon clusters, \nthe decay vertex ($Z_{\\textrm{VTX}}$) and the momentum of the \\pizs were reconstructed\nwith assumptions that the invariant mass of two photons was equal to the \\pizs mass \nand that the decay vertex was on the beam axis.\nThe opening angle of two photons ($\\theta$) was \ncalculated from the equation\n\\begin{equation}\nM_{\\pi^0}^2 = 2E_1 E_2 (1 - \\cos{\\theta}) \\:,\n\\end{equation}\nwhere $E_1$ and $E_2$ are the energies of the two photons. \nThe momentum and transverse momentum ($P_T$) of the \\pizs were calculated \nfrom the decay vertex.\n\n\n\\subsection{Monte Carlo simulations}\nWe generated Monte Carlo events that had the same structure as the recorded events, \nand analyzed them with the same code as for the real events. \nBecause a full shower simulation would require extensive computation time, \nthe calculations were separated into several stages and streams, as shown \nin Fig.~\\ref{fig_mcscheme}. The first stage was the beamline simulation in \nwhich the target, collimators, magnetic field, and absorbers were introduced. \nThe GEANT3 package~\\cite{geant3} with the GFLUKA plug-in code for hadronic \ninteractions was used for this simulation. The momentum, position in the \ntransverse directions, and angle distributions at the exit of the last \ncollimator C6 were obtained for various beam particles.\n\n\\begin{figure}[htbp]\n\\begin{center}\n\\includegraphics[width=.47\\textwidth]{.\/fig17.eps}\n\\end{center}\n\\caption{\nSchematic diagram of simulation.\n}\n\\label{fig_mcscheme}\n\\end{figure}\n\n\\par\nFor $K_L^0$, functions for the momentum and position \ndistributions at the C6 exit were obtained by fitting the Monte Carlo data.\nThe $K_L^0\\;$ beam beyond C6 was subsequently \ngenerated with respect to the momentum spectrum and targeting angle at C6.\nThe generated $K_L^0$'s were used for \nthe study of various $K_L^0\\;$ decays through detector simulations with the GEANT3 \npackage. \nIn the generation of the $K_L^0\\rightarrow\\pi^0\\nu\\bar{\\nu}\\;$ events, we assumed V-A interactions.\nIn the detector simulations, particles were traced until their \nenergy decreased below the cutoff value (e.g. 0.05 MeV for photons and electrons).\nTypical results of $K_L^0\\;$ decay simulations: reconstructed mass, vertex position,\nmomentum, and transverse momentum for the six-photon data samples are shown in Fig.~\\ref{fig_kpithree}.\nSmall corrections to the momentum and radial position of the $K_L^0\\;$ were included.\nEach distribution is reproduced by the simulation of the $K_L^0\\rightarrow\\pi^0\\pi^0\\pi^0\\;$ decays.\nThe method used to reconstruct the six-photon events and \nthe event selections are described in a later section.\n\n\\begin{figure*}[htbp]\n\\begin{center}\n\\includegraphics[width=.44\\textwidth]{.\/fig18a.eps}\n\\includegraphics[width=.44\\textwidth]{.\/fig18b.eps} \\\\\n\\includegraphics[width=.44\\textwidth]{.\/fig18c.eps}\n\\includegraphics[width=.44\\textwidth]{.\/fig18d.eps}\n\\end{center}\n\\caption{\nDistributions of reconstructed mass (top left), vertex position (top right), momentum (bottom left), \nand transverse momentum (bottom right) obtained from the \nsix-photon data samples, compared to the data and the $K_L^0\\rightarrow\\pi^0\\pi^0\\pi^0\\;$ MC. \nIn each plot, the top portion shows a comparison between the number of \nevents in the data and MC, and the bottom portion shows the ratio between them.\nAll the veto cuts and kinematic selections, except those for the \nrespective abscissa variable for the $K_L^0\\rightarrow\\pi^0\\pi^0\\pi^0\\;$ mode, are applied in these plots. \nDetails of the cuts are described in Sec.\\ V.B.}\n\\label{fig_kpithree}\n\\end{figure*}\n\n\n\\par\nAnother analysis stream from the MC particles at C6 was \nthe generation of halo neutrons. \nFirst, the core neutrons were removed from the neutron data sample. \nThe remaining halo neutrons were used multiple times as seeds. \nDuring the generation, we introduced Gaussian fluctuations\nto the momentum, position, and angle of \neach new event to prevent duplications of the same event. \nWe used the hadron code FLUKA to simulate the interaction between halo neutrons\nand the detector.\nSecondary particles generated by FLUKA were fed into the GEANT3 detector simulation.\n\nThe detector response was simulated by using the GEANT3 package.\nThe energy deposit and the timing in each detector subsystem was stored. \nTrigger conditions were simulated according to the energy deposits of the detector elements,\nincluding the segments of the CsI shown in Fig.~\\ref{fig_hwclustering}.\n\n\n\n\\subsection{Reproducibility of the MC simulations}\nIn addition to the distributions of kinematic variables for the $K_L^0\\rightarrow\\pi^0\\pi^0\\pi^0\\;$ decay mode, \nthe events with four photons reconstructed in the calorimeter were analyzed\nin order to verify the detection inefficiency of photon counters in the simulation.\nFigure~\\ref{fig_4gmass} shows the reconstructed mass distribution for four-cluster events.\nIn addition to the $K_L^0\\rightarrow\\pi^0\\pi^0\\;$ events at the $K_L^0\\;$ mass, there was a tail in the lower \nmass region due to contamination from $K_L^0\\rightarrow\\pi^0\\pi^0\\pi^0$, two out of six photons of which \nescaped detection. The number of events in the tail was reproduced by our\nsimulations.\n\n\\begin{figure}[htbp]\n\\begin{center}\n\\includegraphics[width=.47\\textwidth]{.\/fig19.eps}\n\\end{center}\n\\caption{Reconstructed invariant-mass distribution of events with four photons in \nthe calorimeter. The points show the data and the histograms indicate the \ncontribution of $K_L^0\\rightarrow\\pi^0\\pi^0\\;$ and $K_L^0\\rightarrow\\pi^0\\pi^0\\pi^0\\;$ decays (and their sum), as expected from \nthe simulation, normalized by the number of events in the $K_L^0\\rightarrow\\pi^0\\pi^0\\;$ peak.\nAll veto and kinematic cuts \nexcept for the cut on the 4$\\gamma\\;$ invariant mass \nfor the $K_L^0\\rightarrow\\pi^0\\pi^0\\;$ mode are applied.\nIt includes the shower-leakage correction, but does not include the correction for photo-nuclear interactions.}\n\\label{fig_4gmass}\n\\end{figure}\n\nA simulation with the FLUKA hadronic-interaction package to a dedicated run \n(``Al plate run\")  was carried out in order to confirm the reproducibility of hadronic interactions. \nIn the Al plate run, a 0.5-cm-thick aluminum plate was inserted \ninto the beam at 6.5 cm downstream of the rear end of CC02, as shown in \nFig.~\\ref{fig_pi0run_setup}. Because the position of the Al plate is known, \nthe invariant mass of two photons can be reconstructed from the energy and \nposition of two photons. Figure~\\ref{fig_alrun} demonstrates the simulation \nin which the invariant mass distribution (from \\pizs mass to $\\eta$ mass) \nof the events from the Al plate run was reproduced with two photons in the calorimeter.\n\nNext, by assuming the \\pizs mass and reconstructing the vertex position in the Al plate run,\nthe effect of shower leakage and photo-nuclear interactions in the CsI was examined.\nFigure~\\ref{fig_alrun_vtx} shows the distribution of $Z_{\\textrm{VTX}}\\;$ obtained from the Al plate run\nby comparing the data and MC.\nIf the photon energy is incorrectly measured as being low due to these processes, \nthe \\pizs vertex position is reconstructed downstream of the true position.\nBecause the true $Z_{\\textrm{VTX}}\\;$ is known in the Al plate run, we can estimate the effect \nof these processes by using the tail in the distribution of the reconstructed $Z_{\\textrm{VTX}}$.\n\nPhoto-nuclear interactions are not implemented in GEANT3, and therefore the\neffects were estimated from GEANT4 simulations~\\cite{geant4}. \nWe estimated the probability of photo-nuclear interactions by taking the difference \nbetween MC samples with and without photo-nuclear interactions.\nThe energies of two photons were smeared according to the probability of the interaction.\nAs shown in Fig.~\\ref{fig_alrun_vtx},\na tail on the downstream side of the peak is reproduced\nby the simulations with photo-nuclear interactions.\n\n\n\\begin{figure}[htbp]\n\\begin{center}\n\\includegraphics[width=.38\\textwidth]{.\/fig20.eps}\n\\end{center}\n\\caption{\nSchematic layout of the Al plate run.\n}\n\\label{fig_pi0run_setup}\n\\end{figure}\n\n\\begin{figure}[htbp]\n\\begin{center}\n\\includegraphics[width=.47\\textwidth]{.\/fig21.eps}\n\\end{center}\n\\caption{Reconstructed invariant mass distribution of the two-photon event \nin the Al plate run with all the veto cuts imposed. \nThe points represent data and the solid line is from FLUKA simulations. \nThe peaks at 0.14 GeV\/c$^{2}$ 0.55 GeV\/c${^2}$ correspond \nto $\\pi^0$ and $\\eta$ particles, respectively.}\n\\label{fig_alrun}\n\\end{figure}\n\n\\begin{figure}[htbp]\n\\begin{center}\n\\includegraphics[width=.47\\textwidth]{.\/fig22.eps}\n\\end{center}\n\\caption{Reconstructed $Z_{\\textrm{VTX}}\\;$ distribution of two photon events in the Al plate run.\nThe points represent the real data and the solid (dashed) line is obtained from \nthe FLUKA MC simulaiton with (without) the effects of photo-nuclear interaction.\nIn both cases, the shower-leakage correction was applied.\n}\n\\label{fig_alrun_vtx}\n\\end{figure}\n\n\n\\section{Event Selection and background estimation}\n\n\\subsection{Event selection}\nEvent selection consisted of veto cuts and kinematic selections.\nVetoing extra activities was the primary method used to isolate the $K_L^0\\rightarrow\\pi^0\\nu\\bar{\\nu}\\;$ signal events\nfrom the background events. Kinematic selections were applied to obtain further \nrejection of background events.\n\n\\subsubsection{Veto cuts}\nParticle veto with a hermetic detector system is the primary method to reject \npossible background sources. For example, events made by the $K_L^0\\rightarrow\\pi^0\\pi^0\\;$ or \n$K_L^0\\rightarrow\\pi^+\\pi^-\\pi^0\\;$ mode can be rejected by veto cuts because these modes have \nat least two extra activities. \nThe method also works for halo neutron backgrounds because the hadronic \ninteractions of halo neutrons are often accompanied by extra particles \nsuch as protons or pions.\n\nEnergy thresholds for veto cuts were set at two levels. \nFor loose veto thresholds (typically, 10 MeV), no timing cut was made \nto reduce inefficiencies caused by accidental hits before the event time. \nFor tighter veto thresholds (typically, 1-2 MeV), events with a TDC value \nwithin roughly $\\pm5\\sigma$ from the on-time peak were rejected.\nCombining these conditions, we balanced the efficiencies of detecting extra \nparticles and the suppression of the acceptance loss caused by accidental hits.\n\n\n\\textbf{CsI veto cut:}\nIn addition to its main role of measuring the energy and position of photons, \nthe CsI also served as a veto detector for extra photons.\nExtra activities that were not reconstructed as photon clusters, {\\it i.e.} \n``single crystal hits,\" were rejected with this cut.\nHowever, a photon occasionally creates a single crystal hit near its genuine \ncluster due to fluctuations in the electromagnetic processes.\nThus, applying a tight energy threshold for a single crystal hit near the \nphoton cluster can cause acceptance loss for signal candidates. \nTo recover this loss, the energy threshold for a single crystal hit was \ndetermined as a function of the distance $d$ to the closest cluster \nas shown in Fig.~\\ref{fig_csiveto}:\n\\begin{itemize}\n  \\item $E_{\\textrm{thres.}} = 10$ MeV \\;\\;\\; for $d < 17 $ cm, \n  \\item $E_{\\textrm{thres.}} = 5 - (3\/8)(d-17)$ MeV for  $17 < d < 25$ cm,\n  \\item $E_{\\textrm{thres.}} = 2$ MeV \\;\\;\\; for $d>25$ cm,\n\\end{itemize}\nwhere $E_{\\textrm{thres.}}$ is the energy threshold for a single crystal hit.\n\n\n\\begin{figure}[htbp]\n\\begin{center}\n\\includegraphics[width=.4\\textwidth]{.\/fig23a.eps}\\\\\n\\includegraphics[width=.4\\textwidth]{.\/fig23b.eps}\n\\end{center}\n\\caption{\nEnergy deposition in a single crystal hit versus the distance\nfrom the nearest photon cluster \nof the two-photon events from the $K_L^0\\rightarrow\\pi^0\\nu\\bar{\\nu}\\;$ (top) and $K_L^0\\rightarrow\\pi^0\\pi^0\\;$ (bottom) simulations.\nEvents above the solid line were rejected.\n}\n\\label{fig_csiveto}\n\\end{figure}\n\n\n\\textbf{Other photon veto cuts:}\nPhoton veto detectors consisted of barrel counters (MB and FB), beam collar counters \n(CC00, CC02 - CC07), and a beam hole counter (BA).\n\nBecause the MB was read out at both the upstream and downstream ends, \nthe energy deposit was determined as the geometrical mean of the visible energies in both sides\nin order to cancel the position dependence of the light yield. \nThe timing requirement for the MB was loosened when a tighter energy threshold \nwas used because the timing was broadly distributed in the halo neutron backgrounds.\nWhen a photon hit the CsI calorimeter, low-energy photons and electrons that were created \nin an electromagnetic shower occasionally went backward and hit the MB. \nThis process caused a larger acceptance loss in the MB as compared to the loss\nin other veto detectors.\n\nFor the collar counters in the downstream region (CC06 and CC07), \nboth tighter and looser energy thresholds were\nset higher than for other detectors,\nin order to reduce accidental losses due to beam-induced activities.\nThe BA located inside the beamline rejected photons escaping into the beam hole. \nThe energy thresholds for these counters are summarized in Table~\\ref{table_veto}.\n\n\\textbf{Charged particle veto cuts:}\nBackground events produced by charged $K_L^0\\;$ decay modes were mostly rejected \nby cuts on the charged particle veto detectors, mainly CV, BCV, and BHCV.\nAs in the case of MB, an energy deposit in the BCV was determined as \nthe geometrical mean of the hits in the upstream and downstream ends, and the timing \nrequirement was loosened.\nThe energy thresholds for these \ncounters are summarized in Table~\\ref{table_veto}.\n\n\\begin{table}[htbp]\n   \\caption{Summary of tighter veto energy threshold for each detector.}\n   \\label{table_veto}\n      \\begin{tabular}{lc}\n\t\\hline \\hline\t\n\tDetector & Threshold \\\\\n\t\\hline \\hline\n\tCC00 & 2 MeV\\\\\n\t\\hline\n\tFB      & 1 MeV \\footnotemark[1] \\\\\n\t\\hline\n\tCC02 & 1 MeV \\\\\n\t\\hline\n\tBCV   & 0.75 MeV \\footnotemark[2] \\\\\n\t\\hline\n\tMB Inner   & 1 MeV \\footnotemark[2] \\\\\n\t\\hline\n\tMB Outer   & 1 MeV \\footnotemark[2] \\\\\n\t\\hline\n\tCV Outer   & 0.3 MeV  \\\\\n\t\\hline\n\tCV Inner   & 0.7 MeV \\\\\n\t\\hline\t\n\tCC03     & 2 MeV \\\\\n\t\\hline\n\tCsI         & depends on $d$ \\footnotemark[3] \\\\\n\t\\hline\n\tSandwich & 2 MeV \\\\\n\t\\hline\n\tCC04 Scintillator & 0.7 MeV \\\\\n\t\\hline\n\tCC04 Calorimeter & 2 MeV \\\\\n\t\\hline\n\tCC05 Scintillator & 0.7 MeV \\\\\n\t\\hline\n\tCC05 Calorimeter & 3 MeV \\\\\n\t\\hline\n\tCC06 & 10 MeV \\\\\n\t\\hline\n\tCC07 & 10 MeV \\\\\n\t\\hline\n\tBHCV & 0.1 MeV \\\\\n\t\\hline\n\tBA Scintillator (Run-2) & 20 MeV \\footnotemark[4] \\\\\n\tBA PWO (Run-3) & 50 MeV \\footnotemark[4] \\\\\n\tBA Quartz & 0.5 MIPs \\footnotemark[5] \\\\\n\t\\hline \\hline\n   \\end{tabular}\n   \\footnotetext[1]{Sum of inner and outer layers.}\n   \\footnotetext[2]{Geometrical mean of upstream and downstream ends.}\n   \\footnotetext[3]{Detailed in the text.}\n   \\footnotetext[4]{Summed over layers.}\n   \\footnotetext[5]{Determined by AND logic of scintillator\/PWO and quartz.}\n\\end{table}\n\n\n\\subsubsection{Kinematic selections}\nKinematic selections were applied to discriminate the signal from background events \nby using the information of two photons in the CsI calorimeter. \nThese selections can be categorized into three types: photon-cluster quality \nselections, \nselection on photons, and \\pizs selections.\n\n\\textbf{Photon-cluster quality selections:}\nPhoton-cluster quality selections were developed from the information of each photon cluster.\nThe energy was required to be \n$E_H > 250$ MeV and $E_L > 150$ MeV, where $E_H$ and $E_L$ are\nthe energies of the higher- and lower-energy photons, respectively.\nWe also made selections on the incident position, size, profile of the energy distribution,\nand timing dispersion of each photon cluster.\nThese cuts were applied to ensure that two photon clusters were from the $K_L^0\\rightarrow\\pi^0\\nu\\bar{\\nu}\\;$ decay\nand were distinguished from fake clusters created by hadronic showers or photons produced by\nhalo neutron interactions.\n\nIn addition, two neural network selections were developed pertaining to the shape \nof a photon cluster. The first neural network was used to reduce events with \nclusters that overlapped those from other photons or associated particles and reconstructed \nas a single cluster (``fusion cluster\"). In the case of $K_L^0\\rightarrow\\pi^0\\pi^0\\;$ background, a fusion cluster \nresults in an inefficiency to extra photons; in the case of CV-\\pizs background, \nit results in a larger photon energy such that the event vertex \ncomes into the signal region. This neural network was trained by single \nand fusion clusters obtained from the $K_L^0\\rightarrow\\pi^0\\pi^0\\;$ MC samples. \nThe second neural network selection was used to reject the CV-$\\eta$ background.\nBecause $\\eta$ particles were produced near the front end of CsI and had a larger invariant \nmass than a $\\pi^0$, the two photons produced by $\\eta$ decay tended to \nhave large incident angles to the CsI calorimeter.\nThis neural network was trained by the MC samples of $K_L^0\\rightarrow\\pi^0\\nu\\bar{\\nu}\\;$ and CV-$\\eta$ background. \nThe rejection power of these two neural network selections is\nshown in Fig.~\\ref{fig_nncuts}, with the distribution of the output values from \nthe neural network for the signal and background modes.\n\n\\begin{figure}[htbp]\n\\begin{center}\n\\includegraphics[width=.47\\textwidth]{.\/fig24a.eps}\\\\\n\\includegraphics[width=.47\\textwidth]{.\/fig24b.eps}\n\\end{center}\n\\caption{\nDistributions of the output values obtained by two neural network selections \nfor the signal mode (hollow) and the background events (hatched).\nThe top figure shows the fusion neural network selection, in which the background \nis CV-$\\pi^0$. The bottom figure shows the CV-$\\eta$ neural network selection, \nin which the background is CV-$\\eta$. In both plots, all analysis cuts except \nfor the respective neural network cut are imposed for the signal mode. \nSeveral cuts are loosened for the background case in order to enhance the events.\n}\n\\label{fig_nncuts}\n\\end{figure}\n\n\\textbf{Selection on photons:}\nWe required the distance between two photons to be greater \nthan 15 cm, and the timing difference between them to be \n$-9.6 < T(E_H) - T(E_L) < 18.4$ ns, where $T(E_H)$ and $T(E_L)$ are the \ntiming of the higher and lower energy photons, respectively.\nIn addition, the energy balance of two photons defined as \n$(E_{H}-E_{L})\/(E_{H}+E_{L})$ , was required to be less than 0.75.\nThese selections were needed mainly to suppress accidental activities.\n\n\\textbf{\\pizs selections:}\nBy using the information for reconstructed $\\pi^0$'s, several selection criteria were imposed. \nWe required the kinetic energy of the reconstructed $\\pi^0$'s to be less \nthan 2 GeV to suppress neutron backgrounds with high energy.\nIn addition, the reconstructed $\\pi^0$'s had to be kinematically consistent \nwith a $K_L^0\\rightarrow\\pi^0\\nu\\bar{\\nu}\\;$ decay within the proper $K_L^0\\;$ momentum range.\nA neural-network selection was applied to the discrepancy between two angles.\nOne angle was calculated by connecting the incident position of photons and the position of reconstructed $\\pi^0$'s,\nand the other angle was estimated by the neural network based on the cluster shape.\nThis selection suppressed CV-\\pizs and CV-$\\eta$ backgrounds \nbecause the discrepancy of the angle increased in these processes.\nTo reduce the $K_L^0\\rightarrow\\gamma\\gamma\\;$ background, we calculated the opening angle between \nthe two photon directions projected onto the CsI calorimeter plane, \nand required it to be less than 135 degrees (``acoplanarity angle cut\").\n\n\n\\subsubsection{Signal region}\nWe set the signal region in the $Z_{\\textrm{VTX}}$-$P_T\\;$ plane to be \n$340 \\leq Z_{VTX} \\leq 500 $ cm and $0.12 \\leq P_T \\leq 0.24$ GeV\/$c$.\nThe requirement on $Z_{\\textrm{VTX}}\\;$ was determined to avoid background $\\pi^0$'s \ncoming from the CC02 (upstream) and CV (downstream).\nThe lower limit for $P_T\\;$ was set to reduce the $K_L^0\\rightarrow\\pi^0\\pi^0\\;$ and CV-$\\eta$ \nbackgrounds; these events cluster in the low $P_T\\;$ region due to kinematics \nand a large halo neutron flux near the beam center, respectively. \nThe upper bound for $P_T\\;$ was determined by the kinematic limit of the \n$K_L^0\\rightarrow\\pi^0\\nu\\bar{\\nu}\\;$ decay, whose maximum value is 0.231 GeV\/$c$.\n\n\n\\subsection{Background estimation}\n\n\\subsubsection{Halo neutron background}\nThe halo neutron background was estimated from combined MC simulations \nwith FLUKA and GEANT3.\n\nFigure~\\ref{fig_bgestim} shows the distribution of background events in the \n$Z_{\\textrm{VTX}}$-$P_T\\;$ plane as estimated from the MC samples.\nThe outside of the signal box was divided into four regions, \nas shown in Fig.~\\ref{fig_bgestim}. \nThe number of events in each region was dominated by the halo neutron backgrounds.\nThe numbers of halo neutron backgrounds were normalized \nto the number of events in the upstream region (shown as Region-(1)). \nThe numbers for events outside the signal box \nwere compared between the data and the MC, as listed in Table~\\ref{table_outside}.\nThe numbers of events in each region are consistent within the statistical uncertainties\nbetween the data and the MC.\nPossible impacts of discrepancies outside the signal box on the estimation inside the signal box\nare included in the systematic uncertainties below, and will be discussed in detail \nin a later section (Sec.\\ VI.B.).\n\nFor estimating the CC02-\\pizs background, the effects of shower leakage and photo-nuclear\ninteractions described in Sec.\\ III.D were taken into account.\nAfter applying all event selections, we estimated $0.66 \\pm 0.33_{\\textrm{stat}} \n\\pm 0.20_{\\textrm{syst}}$ background events in the signal box for the combined \ndata sample of Run-2 and Run-3. \n\nFor CV-\\pizs events, a fusion cluster or a cluster made by a neutron produces \nbackground. After applying all event selections, no events inside \nthe signal box were obtained with 3 times larger statistics as compared to \nthe combined data sample of Run-2 and Run-3. An upper limit of 0.36 events \nat the $1 \\sigma$ level was set for this process.\n\nCV-$\\eta$ events were strongly suppressed by the dedicated neural-network cut \non the cluster shape. A value of $0.19 \\pm 0.08_{\\textrm{stat}} \\pm \n0.10_{\\textrm{syst}}$ events was estimated for this process.\n\n\n\\begin{figure}[htbp]\n\\begin{center}\n\\includegraphics[width=.47\\textwidth]{.\/fig25.eps}\n\\end{center}\n\\caption{\nScatter plot of $P_{T}$ vs.\\ the reconstructed $Z$ position after applying \nall cuts to the MC samples for background simulation, including halo neutron and $K_L^0\\rightarrow\\pi^0\\pi^0\\;$ \nbackgrounds. The region bounded by the solid line shows the signal box.\nEvents around $Z_{VTX} = 275$ cm are halo neutron events reconstructed at \nthe position of CC02, and those around $Z_{VTX} = 560$ cm are events \nreconstructed at the CV. The numbers of events in each region are listed \nin Table~\\ref{table_outside}.\n}\n\\label{fig_bgestim}\n\\end{figure}\n\n\\begin{table}[htbp]\n\\caption{\nEstimated numbers of events outside the signal region, compared with the\ncombined data samples of Run-2 and Run-3.\nNote that the numbers of events estimated by the MC simulation were normalized in Region-(1).\nErrors in the MC estimates indicate statistical uncertainties.\n}\n\\label{table_outside}\n\\begin{tabular}{rlcc}\n\\hline \\hline\n\\multicolumn{2}{c}{Region} & Data & MC estimation \\\\\n\\hline\nRegion-(1) & (CC02)       & 360 & $360.0 \\pm 15.6$ \\\\\nRegion-(2) & (CV)            & 101 & $77.2 \\pm 5.6$ \\\\\nRegion-(3) & (upstream)  & 8     &  $5.9 \\pm 1.1$  \\\\\nRegion-(4) &  (low-$P_T$)     & 8      & $2.9 \\pm 0.9$   \\\\\n\\multicolumn{2}{c}{Signal box} & (masked) & $0.87 \\pm 0.34$ \\\\\n\\hline\\hline\n\\end{tabular}\n\\end{table}\n\n\n\\subsubsection{$K_L^0\\;$ background}\nThe $K_L^0\\;$ background was estimated by using GEANT3-based MC simulations \nthat were performed separately for Run-2 and Run-3 conditions.\nAmong the backgrounds from $K_L^0\\;$ decay, the $K_L^0\\rightarrow\\pi^0\\pi^0\\;$ mode made the \nlargest contribution. \nThe amount of Monte Carlo statistics for the $K_L^0\\rightarrow\\pi^0\\pi^0\\;$ mode was roughly 70 (60) times \nthat for the real data of the Run-2 (Run-3) sample.\nIn Run-2, two events remained after applying all event selections in \nthe simulations, and in Run-3, no events remained. \nThe two remaining events both corresponded to the case that \ntwo photons in the CsI came from the same \\pizs.\nOne extra photon with high energy ($\\sim$ 1 GeV) went through the CsI (``punch-through\"), \nand the other extra photon with low energy ($\\sim$ 10 MeV) hit the MB and failed \nto be detected by the fluctuation in an electromagnetic shower.\nThe estimated number of background events from $K_L^0\\rightarrow\\pi^0\\pi^0\\;$ decay was \nestimated as $(2.4 \\pm 1.8_{\\textrm{stat}} \\pm 0.2_{\\textrm{syst}}) \n\\times 10^{-2}$ for the combined data sample of Run-2 and Run-3.\n\nFor the MC $K_L^0\\rightarrow\\gamma\\gamma\\;$ samples whose statistics corresponded \nto 3.8 and 8.7 times the Run-2 and Run-3 data, respectively,\nall of the event selection cuts except for the acoplanarity angle cut were \nimposed and no events remained in the signal region.\nFurthermore, the acoplanarity angle cut strongly suppressed the $K_L^0\\rightarrow\\gamma\\gamma\\;$ events, \nwith a typical rejection of $1\\times 10^5$.\nWe concluded that the background due to $K_L^0\\rightarrow\\gamma\\gamma\\;$ was negligible in this analysis.\n\nFor charged modes, the branching ratios of the $K_L^0\\rightarrow\\pi^+\\pi^-\\pi^0\\;$ and \n$K_L^0\\rightarrow\\pi l\\nu\\;$ $(l=e, \\mu)$ modes are too large to generate sufficient statistics \nin the simulations.\nThe background from these modes \nwere estimated by assuming that the inefficiency of charged-particle vetos was \n$1.0\\times10^{-4}$ for the CV, and $1.0\\times10^{-3}$ for BCV and BHCV \\cite{chineff}. \nBy applying the event weight calculated from the inefficiency, the numbers of \nbackground events from these modes were estimated to be \n$4.2 \\times 10^{-4}$ for $K_L^0\\rightarrow\\pi^- e^+\\nu\\;$ and less than $1.0 \\times 10^{-4}$  for $K_L^0\\rightarrow\\pi^+\\pi^-\\pi^0$.\nHere, the $K_L^0\\rightarrow\\pi^- e^+\\nu\\;$ mode had the largest contribution to the background events among \n$K_L^0\\rightarrow\\pi l\\nu\\;$ modes because a charge-exchange interaction of a $\\pi^-\\;$ ($\\pi^- + p \n\\rightarrow \\pi^0 + n$) and annihilation of $e^+$ prevent the detection of \ncharged particles.\n\n\n\\subsubsection{Other background sources}\nIn addition to the backgrounds from halo neutron and $K_L^0\\;$ decay, two other \npossible background sources were considered. The first one is the \n{\\it backward-going \\pizs background}. \nWhen the halo neutron interacts in the end cap of the vacuum vessel located \n2 m downstream of the CsI, $\\pi^0$'s are occasionally produced in the upstream direction. \nBecause we were unable to discriminate whether photons came from the front or the back, \nthese events were reconstructed as coming from the front and ended up inside the \nsignal box.\nWe estimated this background by using FLUKA simulations with 20 times the amount \nof statistics as compared to the combined data sample of Run-2 and Run-3.\nNo events remained after applying all cuts, and the number of background events \nwas determined to be less than 0.05 events.\n\nThe second additional background source is the {\\it residual gas background} \nthat occurs from interactions of beam neutrons with the residual gas.\nThis process is well suppressed by a high-vacuum system that provided \na very low pressure of $10^{-5}$ Pa.\nTo estimate the background, we carried out a dedicated run at atmospheric \npressure, accumulating statistics that were roughly equivalent to 0.6\\% of the \ncombined data sample of Run-2 and Run-3. We obtained 6867 candidate events \nwith loose event selections. By considering a reduction factor of $10^{-10}$ \ndue to the air pressure, this background was concluded to be negligible.\n\nWe investigated possible contributions from unknown background sources by\nloosening the selection criteria for the data and MC. \nThe numbers of events in the data sample with loosened cuts were consistent \nwith the prediction with the known background sources.\n\nIn total, the estimated number of background events was $0.87 \\pm 0.41$, \nwhere the CV-\\pizs and the backward-going \\pizs backgrounds were not included.\nTable~\\ref{table_bgestim} summarizes estimates for the numbers of background \nevents.\n\n\n\\begin{table}\n\\caption{\nEstimated number of background events.\n}\n\\label{table_bgestim}\n\\begin{tabular}{ccc}\n\\hline \\hline\n\\multicolumn{2}{c}{Background source}          & Estimated number of BG\\\\\n\\hline\nhalo neutron BG  & CC02-$\\pi^0$       & $0.66 \\pm 0.39$ \\\\\n & CV-$\\pi^0$       & $<0.36$ \\\\\n & CV-$\\eta$       & $0.19 \\pm 0.13$ \\\\\n\\hline\n$K_L^0\\;$ decay BG  & $K_L^0\\rightarrow\\pi^0\\pi^0$       & $(2.4 \\pm 1.8)\\times 10^{-2}$ \\\\\n                           & $K_L^0\\rightarrow\\gamma\\gamma\\;$       & negligible \\\\\n                           & charged modes       & negligible (${\\cal O}(10^{-4})$) \\\\\n\\hline\nother BG           & backward $\\pi^0$       & $< 0.05$ \\\\\n                          & residual gas       & negligible (${\\cal O}(10^{-4})$)\\\\  \n\\hline\n\\multicolumn{2}{c}{total}          & $0.87 \\pm 0.41$ \\\\\n\\hline \\hline\n\\end{tabular}\n\\end{table}\n\n\n\\section{Sensitivity and results}\n\n\n\\subsection{Principle of normalization}\n\nThe single event sensitivity (S.E.S.) is represented as \n\\begin{equation}\n\\textrm{S.E.S.} (K_L^0\\rightarrow \\pi^0\\nu\\bar{\\nu}) = \n\\frac{1}{N(K_L^0\\:\\textrm{decays}) \\times A_{\\textrm{signal}}} \\:,\n\\label{eq_SES}\n\\end{equation}\nwhere $N(K_{L}^{0}\\:\\textrm{decays})$ denotes the number of $K_L^0$'s that decayed \nin the fiducial region and $A_{\\textrm{signal}}$ denotes the acceptance for the signal mode.\nThe value of $N(K_{L}^{0}\\:\\textrm{decays})$ is determined from the normalization mode as\n\\begin{equation}\n N(K_L^0\\:\\textrm{decays}) = \\frac{N_{\\textrm{norm.}}^{\\textrm{data}}}{A_{\\textrm{norm.}}\n\\times BR_{\\textrm{norm.}}}   \\;,\n\\label{eq_nkl}\n\\end{equation}\nwhere $N_{\\textrm{norm.}}^{\\textrm{data}}$ is the number of observed events from \nthe normalization mode and $A_{\\text{norm.}}$ and $BR_{\\textrm{norm.}}$ represent \nthe acceptance and branching ratio of the mode, respectively.\nSubstituting Eq.~\\ref{eq_nkl} into Eq.~\\ref{eq_SES}, the S.E.S. is obtained \nfrom the number of remaining events in the normalization mode and the ratio of\nacceptances between the signal and the normalization modes.\nBy taking this ratio, uncertainties arising from variations in beam condition, \n{\\em etc.}, can be canceled.\n\nWe examined three decay modes, namely, $K_L^0\\rightarrow\\pi^0\\pi^0\\pi^0$, $K_L^0\\rightarrow\\pi^0\\pi^0$, and $K_L^0\\rightarrow\\gamma\\gamma\\;$, to obtain \nthe number of $K_L^0\\;$ decays, because they were fully reconstructed and clearly identified.\nBecause these three decay modes have different numbers of photons in the final states, \nwe could cross-check the reliability of the clustering method. \nThe mean value of the accepted $K_L^0\\;$ momentum was also different, which provided a \ncheck in the full range of the accepted $K_L^0\\;$ momentum for the $K_L^0\\rightarrow\\pi^0\\nu\\bar{\\nu}\\;$ mode.\n\n\n\\subsection{Analysis of normalization modes}\n\\subsubsection{$K_L^0\\;$ reconstruction}\nFor the $K_L^0\\rightarrow\\pi^0\\pi^0\\pi^0\\;$ and $K_L^0\\rightarrow\\pi^0\\pi^0\\;$ modes, $K_L^0$'s were reconstructed from \nthe six and four photons in the CsI calorimeter, respectively.\nIn the reconstruction, the number of possibile combinations of pairs of \nthe two photons was $(_6C_4 \\times _4C_2 \\times _2C_2)\/3! = 15$ for \n$K_L^0\\rightarrow\\pi^0\\pi^0\\pi^0\\;$ and $(_4C_2 \\times _2C_2)\/2! = 3$ for $K_L^0\\rightarrow\\pi^0\\pi^0$. \nThe decay vertex location can be calculated from the energy and position \nof the two photons of each pair by assuming the \\pizs mass.\nTo find the best combination of the photons, the variance in the \nreconstructed vertex points was calculated, named ``pairing $\\chi_Z^2$,\" \nfor all the possible combinations.\nThe pairing $\\chi_Z^2$ was defined as\n\\begin{equation}\n\\chi_Z^2 = \\sum_{i=1}^n \\frac{  ( Z_i - \\bar{Z} )^2}{ \\sigma_i^2} \\:,\n\\label{eq_pairing_chi}\n\\end{equation}\nwhere $i$ runs over the two-photon pairs reconstructing $\\pi^0$, \n({\\it e.g.}, $n=3$ for $K_L^0\\rightarrow\\pi^0\\pi^0\\pi^0\\;$ and $n=2$ for $K_L^0\\rightarrow\\pi^0\\pi^0$.), \n$Z_i$ is the vertex point of $i$-th two-photon pair, $\\sigma_i$ is the \nresolution in reconstructing $Z_i$ calculated from the energy and position \nresolutions of the two photons, and \n\\begin{equation}\n\\bar{Z} =  \\frac{ \\sum_{i=1}^n  Z_i\/\\sigma_i^2}{ \\sum_{i=1}^n 1\/\\sigma_i^2} \\:.\n\\label{eq_barZ}\n\\end{equation}\nThe decay vertex of the $K_L^0\\;$ was determined as $\\bar{Z}$ for the combination \nwith minimum $\\chi_Z^2$.\nWe required the minimum pairing $\\chi_Z^2$ to be less than 3.0 and the difference \nbetween the next-to-minimum one to be greater than 4.0, in order to reduce \nincorrect paring.\n\nFor the $K_L^0\\rightarrow\\gamma\\gamma\\;$ mode, $K_L^0$'s were reconstructed from the two photons in the \nCsI calorimeter by assuming the $K_L^0\\;$ mass.\n\nSeveral analysis cuts were imposed on the events in each decay mode to remove \ncontaminations of other decay modes. \nPhoton veto cuts were important to detect the modes with extra photons,\nsuch as $K_L^0\\rightarrow\\pi^0\\pi^0\\pi^0\\;$ to the $K_L^0\\rightarrow\\pi^0\\pi^0\\;$ events, and $K_L^0\\rightarrow\\pi^0\\pi^0\\;$ to the $K_L^0\\rightarrow\\gamma\\gamma\\;$ events.\nThe energy threshold for the veto in each subsystem was the same as that \nused in the $K_L^0\\rightarrow\\pi^0\\nu\\bar{\\nu}\\;$ analysis (see Table~\\ref{table_veto}).\nThe reconstructed mass of six photons in the $K_L^0\\rightarrow\\pi^0\\pi^0\\pi^0\\;$ events and four photons \nin the $K_L^0\\rightarrow\\pi^0\\pi^0\\;$ events had to be consistent with the $K_L^0\\;$ mass (from 0.481 to \n0.513 GeV\/$c^2$). \nThe decay vertex point of $K_L^0\\;$ also had to be located in the fiducial region \n(from 340 to 500 cm), as in the case of $K_L^0\\rightarrow\\pi^0\\nu\\bar{\\nu}$.\nSome additional cuts, such as the neural network fusion cluster selection, were also\nimposed on these decay modes.\n\n\n\\subsubsection{Number of $K_L^0\\;$ decays}\nThe acceptance of each mode was estimated from Monte-Carlo simulations.\nBecause, in the simulations, $K_L^0$'s were generated at the exit of the \nlast collimator (C6), the probability of $K_L^0\\;$ decay in the fiducial region, \n$(2.14 \\pm 0.02)$\\%, was calculated separately\nand was taken into account in calculating the number of $K_L^0\\;$ decays.\nLosses due to accidental activities in the detector \nwere also included in the acceptance.\n\nTable~\\ref{table_flux} shows a summary of the estimated numbers of $K_L^0\\;$ decays \nfrom the three decay modes $K_L^0\\rightarrow\\pi^0\\pi^0\\pi^0$, $K_L^0\\rightarrow\\pi^0\\pi^0$, and $K_L^0\\rightarrow\\gamma\\gamma$.\nThe difference among the three modes was within the systematic uncertainties,\nand considered to come from the CsI veto, because it has the largest systematic uncertainties\nas well as a dependence on the number of photons.\nWe adopted the number obtained from $K_L^0\\rightarrow\\pi^0\\pi^0$,\n$(8.70 \\pm 0.17_{\\textrm{stat}} \\pm 0.59_{\\textrm{syst}}) \\times 10^9$ for the combined \ndata sample of Run-2 and Run-3, as the normalization of this analysis \nbecause the energy distribution of photons in the CsI calorimeter from $K_L^0\\rightarrow\\pi^0\\pi^0\\;$\nwas similar to that expected to $K_L^0\\rightarrow\\pi^0\\nu\\bar{\\nu}$.\nEstimates of the systematic uncertainties are described later.\n\n\n\\begin{table*}[htbp]\n\\caption{\nEstimated numbers of $K_L^0\\;$ decays calculated from the three decay modes in the \ncombined sample of Run-2 and Run-3 data.\nUncertainties in the acceptances are statistical ones due to the amount of the MC samples.\n$N_{\\textrm{norm}}^{\\textrm{data}}$ is the number of events obtained from the \nreal data after imposing all the analysis cuts. \nIn the $K_L^0\\rightarrow\\pi^0\\pi^0\\;$ mode, $N_{norm}^{data}$ is obtained by subtracting the \ncontamination from the $K_L^0\\rightarrow\\pi^0\\pi^0\\pi^0\\;$ mode.\nStatistical uncertainties in the number of $K_L^0\\;$ decays include an ambiguity \nin $N_{\\textrm{norm}}^{\\textrm{data}}$, and systematic ones include an \nambiguity from the reproducibility of the MC (described later) and\nstatistical uncertainties of the MC in the acceptance estimate.\n}\n\\label{table_flux}\n\\begin{tabular}{cccc}\n\\hline\nmode & acceptance & $N_{norm}^{data}$ & $N(K_L^0 \\mathrm{decays})$ \\\\\n\\hline\n$K_L^0\\rightarrow\\pi^0\\pi^0\\pi^0\\;$ $\\;$& $\\;$ $ (7.21 \\pm 0.06) \\times 10^{-5} $ $\\;$ & $\\;$ 118334 $\\;$ \n  & $\\;$ $ (8.41 \\pm 0.03_{\\textrm{stat}} \\pm 0.53_{\\textrm{syst}}) \\times 10^{9} $ \\\\\n$K_L^0\\rightarrow\\pi^0\\pi^0\\;$  $\\;$  & $\\;$ $ (3.42 \\pm 0.03) \\times 10^{-4} $ $\\;$ & $\\;$ 2573.9 $\\;$ \n  & $\\;$ $ (8.70 \\pm 0.17_{\\textrm{stat}} \\pm 0.59_{\\textrm{syst}}) \\times 10^{9} $ \\\\\n$K_L^0\\rightarrow\\gamma\\gamma\\;$        $\\;$ & $\\;$ $ (7.18 \\pm 0.03) \\times 10^{-3} $ $\\;$ & $\\;$ 35367 $\\;$ \n  & $\\;$ $ (9.02 \\pm 0.05_{\\textrm{stat}} \\pm 0.51_{\\textrm{syst}}) \\times 10^{9} $ \\\\\n\\hline\n\\end{tabular}\n\\end{table*}\n\n\n\\subsection{Acceptance and single event sensitivity}\n\\subsubsection{Signal acceptance}\nThe acceptance for the $K_L^0\\rightarrow\\pi^0\\nu\\bar{\\nu}\\;$ decay was estimated from Monte-Carlo simulations.\nThe raw acceptance was calculated by dividing the number of remaining events\nafter imposing all the analysis cuts by the number of $K_L^0\\rightarrow\\pi^0\\nu\\bar{\\nu}\\;$ decays generated in \nthe simulation.\nIt was 1.40\\% for the Run-2 and 1.39\\% for the Run-3 data, respectively.\n\nIn the raw acceptance, the losses caused by geometrical acceptance, veto cuts, \nkinematic selections, and selection on fiducial region were included. \nThe geometrical acceptance to detect two photons in the CsI was approximately 20\\%.\nThe loss by veto cuts, or self-vetoing, was about 50\\%, which was dominated by the CsI and MB vetoes.\nThe loss in the CsI veto was caused by hits accompanied by the genuine photon cluster, as shown in \nFig.~\\ref{fig_csiveto}, and the loss in the MB was caused by photons escaping from the\nfront face of the CsI and hitting the MB.\nThe acceptance of the kinematic and signal region selections was approximately 15\\%.\nThese values were estimated through MC simulations of the $K_L^0\\rightarrow\\pi^0\\nu\\bar{\\nu}\\;$ mode.\nThe evaluation of the acceptance loss was supported by the fact that\nthe acceptance losses in the normalization modes ($K_L^0\\rightarrow\\pi^0\\pi^0\\pi^0$, $K_L^0\\rightarrow\\pi^0\\pi^0$, and $K_L^0\\rightarrow\\gamma\\gamma$)\nwere reproduced by the simulation.\n\nThe acceptance loss due to accidental activities in the detector \nwas estimated from real data taken with the TM trigger. \nThe accidental loss was estimated to be 20.6\\% for the Run-3 data,\nin which the losses in MB (7.4\\%) and BA (6.4\\%) were major contributions.\nFor Run-2, the accidental loss was estimated to be 17.4\\%; the difference \nbetween Run-2 and Run-3 was due to the difference in the BA counters used \nin the data taking. \nThe acceptance loss caused by \nthe selections on the timing dispersion of each photon cluster and \non the timing difference between two photons\nwas estimated separately by using real data and was obtained to be 8.9\\%. \nThus, the total acceptance for the $K_L^0\\rightarrow\\pi^0\\nu\\bar{\\nu}\\;$ was $(1.06\\pm0.08)$\\% for Run-2 and\n$(1.00\\pm0.06)$\\% for Run-3 case, where the errors are dominated by the \nsystematic uncertainties that are discussed later. Figure~\\ref{fig_ptz_signal} \nshows the distribution of the MC $K_L^0\\rightarrow\\pi^0\\nu\\bar{\\nu}\\;$ events in the scatter plot of \n$P_T$-$Z_{\\textrm{VTX}}\\;$ after imposing all of the other cuts.\n\n\n\\begin{figure}[htbp]\n\\begin{center}\n\\includegraphics[width=.47\\textwidth]{.\/fig26.eps}\n\\end{center}\n\\caption{\nDensity plot of $P_T\\;$ vs. the reconstructed $Z$ position for the $K_L^0\\rightarrow\\pi^0\\nu\\bar{\\nu}\\;$ Monte Carlo events\nafter imposing all of the analysis cuts.\nThe box indicates the signal region for $K_L^0\\rightarrow\\pi^0\\nu\\bar{\\nu}$.}\n\\label{fig_ptz_signal}\n\\end{figure}\n\n\n\\subsubsection{Single event sensitivity}\nBy using the number of $K_L^0\\;$ decays and the total acceptance, \nthe single event sensitivity for $K_L^0\\rightarrow\\pi^0\\nu\\bar{\\nu}\\;$ was\n$(1.84 \\pm 0.05_{\\textrm{stat}} \\pm 0.19_{\\textrm{syst}} )\\times 10^{-8}$ for Run-2,\n$(2.80 \\pm 0.09_{\\textrm{stat}} \\pm 0.23_{\\textrm{syst}} )\\times 10^{-8}$ for Run-3,\nand\n$(1.11 \\pm 0.02_{\\textrm{stat}} \\pm 0.10_{\\textrm{syst}} )\\times 10^{-8}$ in total.\n\n\n\\subsection{Results}\nAfter finalizing all of the event selection cuts, the candidate events inside the \nsignal region were examined. No events were observed in the signal region, as shown \nin Fig.~\\ref{fig_final}. \nAn upper limit for the $K_L^0\\rightarrow\\pi^0\\nu\\bar{\\nu}\\;$ branching ratio was set to be $2.6\\times10^{-8}$ at \nthe 90\\% confidence level, based on Poisson statistics. \nThe result improves the limit previously published~\\cite{run2}\nby a factor of 2.6.\n\n\\begin{figure}[htbp]\n\\begin{center}\n\\includegraphics[width=.47\\textwidth]{.\/fig27.eps}\n\\end{center}\n\\caption{Scatter plot of $P_T\\;$ vs.\\ the reconstructed $Z$ position for the events \nwith all of the selection cuts imposed. The box indicates the signal region for $K_L^0\\rightarrow\\pi^0\\nu\\bar{\\nu}$.}\n\\label{fig_final}\n\\end{figure}\n\n\n\n\\section{Systematic uncertainties}\nAlthough the systematic uncertainties were not taken into account \nin setting the current upper limit on the branching ratio,\nwe will describe our treatment for them \nto provide a thorough understanding of the experiment. \nIn particular, systematic uncertainties of the single event sensitivity and \nbackground estimates due to halo neutrons are discussed in order.\n\n\\subsection{Uncertainty of the single event sensitivity}\nThe systematic uncertainty of the single event sensitivity was evaluated \nby summing the uncertainties of the number of $K_L^0\\;$ decays \nand the acceptance of the $K_L^0\\rightarrow\\pi^0\\nu\\bar{\\nu}\\;$ decay. \nBecause the calculation of the former also includes \nthe acceptance of the normalization modes, \nthe acceptance of both the normalization mode and the $K_L^0\\rightarrow\\pi^0\\nu\\bar{\\nu}\\;$ mode are relevant to the acceptance evaluation\nby the Monte Carlo simulations.\nTo estimate the uncertainties in the acceptance calculation,\nwe utilized the fractional difference between data and the simulation in each selection criterion,\ndefined by the equation\n\\begin{equation}\nF^i = \\frac{A_{\\textrm{data}}^i - A_{\\textrm{MC}}^i}{A_{\\textrm{data}}^i} \n\\end{equation}\nwhere $A_{\\textrm{data}}^i$ and $A_{\\textrm{MC}^i}$ denote the acceptance values of the $i$-th cut, \ncalculated as the ratio of numbers of events with and without the cut,\nfor the data and MC simulations, respectively. \nIn $F^i$, the acceptance was calculated with all the other cuts imposed.\nThe systematic uncertainty of the acceptance was evaluated\nby summing all the fractional differences in quadrature, \nweighted by the effectiveness of each cut, as \n\\begin{equation}\n\\sigma_{\\textrm{syst.}}^2 = \\frac{\\sum_{i=\\textrm{cuts}} \n(F^i\/A_{\\textrm{data}}^i)^2}{\\sum_{i=\\textrm{cuts}}(1\/A_{\\textrm{data}}^i)^2} \\:.\n\\end{equation}\n\nFor the three decay modes used in the normalization, \n$K_L^0\\rightarrow\\pi^0\\pi^0\\pi^0$, $K_L^0\\rightarrow\\pi^0\\pi^0$, and $K_L^0\\rightarrow\\gamma\\gamma$, \nthe calculated uncertainties were 5.2\\%, 5.7\\%, and 3.6\\% (in Run-3), respectively.\nThe acceptances of the CsI veto cut had the largest uncertainties in all decay modes.\nThe number of $K_L^0\\;$ decays was obtained by using the $K_L^0\\rightarrow\\pi^0\\pi^0\\;$ mode, \nand thus its uncertainty is quoted as the same value for the $K_L^0\\rightarrow\\pi^0\\pi^0\\;$ mode.\n\nFor the acceptance of the $K_L^0\\rightarrow\\pi^0\\nu\\bar{\\nu}\\;$ mode, the same systematic uncertainties as the \n$K_L^0\\rightarrow\\pi^0\\pi^0\\;$ mode were adopted because there were no signal candidates in the data \nto be compared with the MC simulations.\n\nThe systematic error of the single event sensitivity was evaluated to be a\nquadratic sum of the uncertainties of the number of $K_L^0\\;$ decays and the \nacceptance of the $K_L^0\\rightarrow\\pi^0\\nu\\bar{\\nu}\\;$ decay.  It was 10.3\\% in Run-2 and 8.2\\% in Run-3, \nrespectively.\n\n\n\\subsection{Uncertainty of the halo neutron backgrounds}\nThe systematic errors of halo neutron backgrounds were also estimated \nby utilizing fractional differences between data and the simulations.\nThere were two kinds of cuts in our analysis, \nveto cuts and kinematic selections.\nFor the former, the method was unable to be used directly because \nveto cuts had been applied in the early stages of the MC simulations  \nto save the computing time. \nThus, for the systematic uncertainty due to veto cuts, \nthe same value obtained in the $K_L^0\\rightarrow\\pi^0\\pi^0\\;$ analysis was assigned.\nFor the kinematic selections, the same method as described in \nthe previous section was used.\nThe acceptance of each cut was calculated \nwith all veto cuts imposed except kinematic selections.\nIn total, the systematic uncertainties due to fractional differences \nwere calculated to be 31\\%, 32\\%, and 44\\% \nfor CC02-$\\pi^0$, CV-$\\pi^0$, and CV-$\\eta$ backgrounds, respectively.\n\nIn addition, the uncertainties in the normalizations of the MC \nsimulations were taken into account. \nBecause the normalization was determined by \nusing the number of events in the CC02 region (Region-(1)),\nthere can be ambiguity in estimating the CV-related backgrounds. \nAs shown in Table~\\ref{table_outside}, \nthere was a 24\\% difference between data and MC simulation \nin the downstream region (Region-(2)).\nEven though they were statistically consistent, we assigned the \ndifference as an additional systematic uncertainty of CV-$\\pi^0$.\nFor the CV-$\\eta$ case, further ambiguity due to the reproducibility \nof $\\eta$ production should be considered.\nIt was estimated to be 24\\% from the difference between data and MC\nsimulations in the numbers of $\\eta$ events in the Al plate run, \nas shown in Fig.~\\ref{fig_alrun}.\n\nThe total systematic uncertainties were calculated by summing up\nthe contributions quadratically, \nto be 31\\%, 40\\%, and 55\\% for CC02-$\\pi^0$, CV-$\\pi^0$, and CV-$\\eta$\nbackgrounds, respectively.\n\n\n\\section{Conclusion and discussion}\nThe E391a experiment at the KEK-PS was the first dedicated experiment \nfor the $K_L^0\\rightarrow\\pi^0\\nu\\bar{\\nu}\\;$ decay. Combining the periods of Run-2 and Run-3, \nthe single event sensitivity reached $1.11\\times10^{-8}$.\nNo events were observed inside the signal region and \nthe new upper limit on the branching ratio of the $K_L^0\\rightarrow\\pi^0\\nu\\bar{\\nu}\\;$ decay was set \nto be BR $< 2.6 \\times 10^{-8}$ at the 90\\% confidence level.\nThe result improves the previous published limit given by the Run-2 \nanalysis by a factor of 2.6, \nand the E391a experiment as a whole has improved the limit \nfrom previous experiments by a factor of 20. \n\nThe E391a experiment was also the first step of our step-by-step approach\ntoward the accurate measurement of the $K_L^0\\rightarrow\\pi^0\\nu\\bar{\\nu}\\;$ decay. The experiment was planned \nto confirm our experimental approach, and this purpose was well achieved. \nSeveral points need to be noted.\n\nFirst, we found solutions to several technical issues, \nsuch as the pencil beamline, differential pumping for ultra-high vacuum, \nlow-threshold particle detection with a hermetic configuration, in situ calibration, {\\it etc.},\nwhich can be successively used in the next step. \nWe encountered some technical problems that exist in the current apparatus, \nsuch as insufficient thickness and segmentation of the CsI calorimeter, \nthe structure of the CV, the limitation of the BA in an environment with higher counting rate, {\\it etc.}.\nThey will be improved in the next experiment.\n  \nSecondly, we were able to control the systematic uncertainties to be small in the estimate \nof the single event sensitivity, as described in the previous section. \nA small systematic error is essential for an accurate determination of the \nbranching ratio. We are confident that the branching ratio of the decay $K_L^0\\rightarrow\\pi^0\\nu\\bar{\\nu}\\;$ \ncan be measured accurately with this method. \n\nA third point concerns the understanding and estimation of backgrounds. \nIn the experiment for the decay $K_L^0\\rightarrow\\pi^0\\nu\\bar{\\nu}$, elimination of all possible backgrounds \nis the only effective way to identify the decay, \nand it can be achieved by a profound understanding of backgrounds.\n\nThe dominant source of backgrounds was the result of beam interactions.  \nAlthough the background from this source was considered to be as \nserious as those from $K_L^0\\;$ decays in the experiment, \nits understanding and estimation were very difficult to assess before the work \nreported here.  The background mechanisms were clearly understood and divided \ninto three different sources, CC02, CV-$\\pi^0$, and CV-$\\eta$. \nMethods were developed to estimate them with rather small systematic errors. \nBased on this experience, we clearly know the direction of upgrades to minimize \nthe beam backgrounds in the next step. \n\nThe backgrounds from other $K_L^0\\;$ decays were reduced to be negligibly small in \nthe current experiment, mainly due to the  success of applying tight vetoes with \na hermetic configuration.  One of the important results is the invariant mass \ndistribution of the events with four photons, shown in Fig.~\\ref{fig_4gmass}. \nThe distribution was well reproduced by the simulations. In particular, the\nlow-mass region was well described by $K_L^0\\rightarrow\\pi^0\\pi^0\\pi^0\\;$ decays with a small contamination \nby mis-combination events of four photons from $K_L^0\\rightarrow\\pi^0\\pi^0\\;$ decays. The relation between \n$K_L^0\\rightarrow\\pi^0\\pi^0\\pi^0\\;$ and $K_L^0\\rightarrow\\pi^0\\pi^0\\;$ is similar to the relation between $K_L^0\\rightarrow\\pi^0\\pi^0\\;$ and $K_L^0\\rightarrow\\pi^0\\nu\\bar{\\nu}$. \nBy taking their branching ratios into account, it was found that the low-mass \nregion could to be reproduced after reducing the $K_L^0\\rightarrow\\pi^0\\pi^0\\pi^0\\;$ yield by several orders \nof magnitude.  Reproduction could not be achieved without good simulations of \nsmall signals from the detector, to which a tight veto was applied.  \nThis achievement indicates that a similar direction will be promising in the next step.   \n\nThe E391a experiment was also able to study other decay modes\nincluding $\\pi^0$'s in the final state, such as \n$K_L^0 \\rightarrow \\pi^0 \\pi^0 \\nu \\bar{\\nu}$ and \n$K_L^0 \\rightarrow \\pi^0 \\pi^0 X$ $(X\\rightarrow \\gamma \\gamma)$.\nThe \\pizs reconstruction method and the hermetic veto system are also valid \nin the analysis of these modes.\nResults based on portions of the data have already been published\n\\cite{ppnn,ppgg}, and further studies are in progress with the entire data set.\n\nThe next step, the KOTO experiment \\cite{koto} at the new J-PARC accelerator \\cite{jparc}, \nis now in preparation.  \nMost of the improvements pointed out above are being implemented. \nThe new beamline has been constructed; runs to evaluate properties \nof the beamline started in October, 2009.\n\n\n\n\n\n\n\\begin{acknowledgments}\nWe are grateful to the crew of the KEK 12-GeV proton synchrotron\nfor successful beam operation during the experiment. We express our sincere thanks to KEK staff, \nin particular \nthe directors: Professors H. Sugawara, Y. Totsuka, A. Suzuki, S. Yamada, M. Kobayashi, \nF. Takasaki, K. Nishikawa, K. Nakai, and K. Nakamura\nfor their continuous encouragement and support. \nWe also express our sincere thanks \nto the many colleagues in universities, institutes, and the kaon-physics\ncommunity for their continuous encouragement and support.\n\nThis work was partly supported by \nGrant-in-Aids from MEXT and JSPS in Japan, \ngrants from NSF and DOE of US, \ngrants from NSC in Taiwan,\ngrants from KRF in Korea,\nand the ISTC project from CIS countries.\n\\\\\n\\end{acknowledgments}\n\n\n\n\\noindent  \n\n\n\\bibliographystyle{plain}\n\\noindent\n$^a$Present address:  Laboratory of Nuclear Problem, Joint Institute for Nuclear Research, \nDubna, Moscow Region, 141980 Russia \\\\\n$^b$Present address: KEK, Tsukuba, Ibaraki, 305-0801 Japan. \\\\\n$^c$Present address: CERN, CH-1211 Geneva 23, Switzerland. \\\\\n$^d$Deceased \\\\\n$^e$Present address:  SPring-8, Japan. \\\\\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\n\nThe recent interest in modelling low dimensional diffusion-limited reactions\nhas been stimulated in part by the experimental observation of anomalous \nkinetics\nin low-dimensional systems \\cite{Privman97}. The traditional approach to \nchemical \nreactions\nis based on mean-field theory, i.e., rate equations for the densities of the\nvarious reactants. The latter describe well the reaction kinetics in three\ndimensions because diffusive transport of reactants allows to eliminate the\nspatial\nfluctuations of the concentrations. However, in lower dimensions, due to the\nlack of phase space, \nthe reactants spatial fluctuations can grow and develop \ninhomogeneities in the concentrations. \nFuthermore, even in the spatially homogeneous case, the rate equations are not\napplicable in less than three dimensions; for example, in \nthe two-species diffusion-limited annihilation the \nconcentration of\nthe particles decays (for identical initial concentrations) slowlier than the\nmean-field theory predicts. Thus\nthe fluctuation-dominated dynamics is beyond the classical theories, yet can be\naccounted for by simple one-dimensional models of hard-core particles. \nThe latter are solved\neither numerically or analytically by applying techniques from  \n(classical or quantum) statistical mechanics \\cite{Privman97}. \nIn particular, exact solutions\nhave been obtained by inter-particle distribution methods, by relating the\nsystems\nto dual and solvable 1D models \\cite{Glauber,Ra,Family,Droz} (Kinetic Ising and Potts) and \nby mapping the\ndiffusion-reaction processes to an imaginary-time dynamics \nof quantum spin chains with non-hermitian Hamiltonians \\cite{Droz}. An alternative\nand fruitful approach has been developed by Cardy and collaborators \n\\cite{Cardy1,Cardy2,Cardy3,Lee}: \nthe idea is to\nreformulate the original problem in terms of a field theory of interacting \nbosons \nand subsequently use renormalization group techniques. This is \na powerful method as it\napplies to arbitrary dimension and low densities of particles, a regime where \nuniversal behavior (scaling) is usually observed.    \nDespite the progress achieved in this field, the multi-species case\nis still poorly understood. Furthermore, when the density of particles is high, \nthe hard-core constraint on the dynamics of the diffusing particles becomes \nimportant.\nExperimentally the single-species fusion model used to describe\nthe photoluminescence of excitons diffusing along one dimensional chains\nis seen to apply only as long as the initial exciton densities are small \\cite{Privman97}.\nTo our knowledge, there has been no systematic effort yet to investigate \ntheoretically the regime\nof high densities of reactants.\\\\\nThis is the first in a series of technical papers where \nwe will explore a new \napproach\nin an attempt to remedy the difficulties encountered so far. \nWe propose to start from a quantum spin chain formulation of the master \nequation,\nfermionize and subsequently\napply renormalization group techniques to deal with\nthe interaction terms arising in the multi-species case. Note that\nin the single species case, the method\napplies at arbitrary densities as the hard-core of the classical particles is\nautomatically accounted for by the Fermi statistics.\nWhen various species react and diffuse, it is appropriate to distinguish two\ncases: i) the various species have infinite on-site\nrepulsion with themselves only; this can be treated readily following the \nmethods outlined in the following sections; ii) the particles of different \nspecies all have a hard core constraint; this is far more difficult and will be\ninvestigated elsewhere. \nAs with the other methods devised so far, \nthere is a price to pay: the calculations involved are sometimes extremely\ntedious.\\\\ \nThe purpose of this first paper is modest as we will focus on the\nsingle-species case for which fermion-fermion interactions do not arise as a consequence of mapping\nclassical particles to fermions.\nWe want to show explicitly how by elementary means we can\nreproduce known results \n(the long-time behaviour of the density), and\nderive some new results, i.e., the explicit and exact analytical form of the \ntwo-point correlation function of one and two-time.  \nThe paper is intended as an introduction to the fermionic functional\nintegration approach which we will combine in forthcoming papers\nwith the renormalization techniques to treat the multi-species case.\\\\\nIn the first section, we define the model, introduce notations and\nreview an extension of a method developed by Lushnikov. The second section is\ndevoted to the application of an elegant technique introduced \nby Grynberg et {\\it al.} in a different context to evaluate the two-point (one-time) correlation function.\nThe third section\ndeals with a new and powerful formulation of the problem\nin terms of path integrals of pseudo-fermionic\nvariables. We illustrate the technique by computing the two-time correlation function. \nA brief discussion concludes the paper.\n  \n  \n\n\\section{ Lushnikov's approach}\nThis section introduces notations, summarizes and extends\nLushnikov's genuine approach.\nWe consider a lattice of $N$ (even) sites (length $L=Na,\\, a=1$ and \nassume $\\frac{N}{2}$ even), with periodic boundary conditions, on which \nclassical \n(spinless) particles with  hard-core can diffuse (annihilate)\nto\nadjacent empty (occupied) sites with rate $D$. Whenever the arrival site is\noccupied, an annihilation-reaction ($1+1\\rightarrow \\O$)\ntakes place. A source of intensity $J$ injects pairs of particles on adjacent\nsites ($\\O\\rightarrow 1+1$).\nLushnikov \\cite{Lushnikov} has managed to rewrite the master equation that \ndescribes the\nannihilation and diffusion processes described above in terms of \nan imaginary-time\nSchr\\\"odinger equation:\n\\begin{eqnarray}\n\\label{1.4}\n\\displaystyle\n\\frac{d}{dt}|\\psi (t)\\rangle = {\\mathcal L}|\\psi (t)\\rangle\\;\\; ,\n\\end{eqnarray}\nwhere ${\\mathcal L}$ denotes the so called ``Liouvillian\",\nwhich by abuse of language we will call\n``non-hermitian Hamiltonian''. $|\\psi (t)\\rangle$\nrepresents the state of the system at time t,\n\\begin{eqnarray}\n\\label{1.5}\n\\displaystyle\n\\ |\\psi (t)\\rangle =\\sum_{\\{n\\}} \\Bigr( P(\\{n\\},t)\\prod_{m'(\\{n\\})} \n\\sigma^+_{m'(\\{n\\})}\n\\Bigr)|0\\rangle\\;\\;,\n\\end{eqnarray}\nwhere $m'(\\{n\\})$ represents the sites of configuration $\\{n\\}$\nwhich are occupied. The Liouvillian is given by \\cite{Lushnikov} \n\\begin{eqnarray}\n\\displaystyle\n\\label{1.7}\n{\\mathcal{L}}=(J+D)\\sum_{m}\\Big(\\sigma_m^+\\sigma_{m+1}^+ \n+\\sigma_{m}^+\\sigma_{m+1}^- + \\sigma_{m}^-\\sigma_{m+1}^+ +\n\\sigma_{m}^-\\sigma_{m-1}^-\\Big) +D\\sum_{m} \\Big(\\sigma_{m+1}^-\\sigma_{m}^- -\n\\sigma_{m+1}^+\\sigma_{m}^+   -2\\sigma_{m}^+\\sigma_{m}^- \\Big)-JN\n\\end{eqnarray}\nWhen $J=0$, i.e., no source, in addition to diffusive processes with rate $D$, \nonly irreversible reactions ($1+1\\rightarrow \\O$), with\nrate $2D$ take place.\\\\\nFor finite ($J>0$) source, the diffusive processes ($1+\\O\\rightarrow \\O +1$ and \n$\\O + 1\\rightarrow 1 +\\O $) take place with rate $J+D$ and we also have  reversible reactions:\n particles are\nannihilated ($1+1\\rightarrow \\O$) with rate $J+2D$ and created ($\\O \\rightarrow 1+1$)  with rate\n$J$. It is worth emphasizing that these rates are not independent, and are chosen such that \nthe Liouvillian is quadratic in \nthe spin variables for a single\nspecies (the higher order terms cancel due to the properties of Pauli matrices).\nWe also point out that in this model the ``annihilation rate\" ($J+2D$) is always bigger \nto the ``creation rate\" ($J$).\nIn the n-species case, this property does no longer hold if we assume\nhard core repulsion between all species. Indeed, one obtains a spin\nHamiltonian ($S=n\/2$) that is a polynomial of higher order in the \nspin operators and in general cannot be solved exactly. \nIf however, we assume infinite on-site repulsion only bewteen\nparticles of the same species, we can rewrite the Hamiltonian as a quadratic\nform of coupled spins $1\/2$. The latter can be solved by the techniques\npresented below.\\\\\nTo solve the Schr\\\"odinger equation in imaginary time, Lushnikov\nperforms a \nJordan-Wigner transformation and introduces the fermionic operators\n $a_m =\\prod_{j<m}(1-2 n_j)\\sigma_{m}^- $.\\\\\nBecause of the form of the resulting non-hermitian Hamiltonian, it is\nappropriate to work with Fourier modes $a_q=\\frac{e^{i\\frac{\\pi}{4}}}{\\sqrt{N}}\\sum_{m} a_m e^{-iqm}$.\nThe antiperiodic boundary conditions \\cite{Mattis} lead to:\n$q=\\pm (2l-1)\\pi\/N\\;, \\;l=1,2,\\ldots,N\/2$.\nOn Fourier transformation, the evolution operator reads\n${\\mathcal L}=\\sum_{q>0} {\\mathcal L}_q$, where ($n_q\\equiv a_q^+a_q$)\n\\begin{eqnarray}\n\\label{1.14}\n\\displaystyle\n{\\mathcal L}_q=2(J+D) \\Bigr \\{\\cos{q(n_{q}+n_{-q})}+\\sin{q(a_{q}a_{-q}- \na_{q}^+a_{-q}^+)}\\Bigr \\} \n +2D \\Bigr \\{\\sin{q(a_{q}a_{-q}+ a_{q}^+a_{-q}^+)}  -(n_{q}+n_{-q}) \\Bigr \n\\}-JN\\;\\;,\n\\end{eqnarray}\nNow, by a BCS-like Ansatz, \n\\begin{eqnarray}\n\\label{1.15}\n\\displaystyle\n\\ |\\psi(t)\\rangle=\\prod_{q>0}|\\psi_q (t)\\rangle  =\\prod_{q>0}\\Bigr( A_q (t) \na_q^+a_{-q}^+ + B_q\n(t)\\Bigr)|0\\rangle,\n\\end{eqnarray}\nLushnikov \\cite{Lushnikov} is able to decouple the dynamical equation as\n\\begin{eqnarray}\n\\label{1.16}\n\\displaystyle\n\\  \\frac{d}{dt}|\\psi_q (t)\\rangle = {\\mathcal L}_q|\\psi_q (t)\\rangle\n\\;,\\;\\forall q>0\n\\end{eqnarray}\nFor a lattice which is initially completely occupied, i.e., $A_q(0)=1, \nB_q(0)=0$, \none solves the above equations by\n\\begin{eqnarray}\n\\displaystyle\n\\label{1.17}\n\\ A_{q}(t)=\\frac{1}{4(J+2D)\\sin^2{(\\frac{q}{2})}}(p_{2}{\\rm e}^{p_{2}t} - \np_{1}{\\rm e}^{p_{1}t}) \\;\\;,\\;\\; \\ B_{q}(t)=\\frac{-(J+2D)\\sin{q}}{2(J+2D)\\sin^2{(\\frac{q}{2})}}({\\rm e}^{p_{2}t} \n- {\\rm e}^{p_{1}t})\n\\end{eqnarray}\nwhere\n\\begin{eqnarray}\n\\displaystyle\n\\label{1.19}\n\\ p_{1}=-2(J+2D)(1-\\cos{q}) \\;\\;,\\;\\; \\ p_{2}=2J (1+\\cos{q})\n\\end{eqnarray}\nIn the absence of source ($J=0$), the solution simplifies considerably to\n\\begin{eqnarray}\n\\displaystyle\n\\label{1.20}\n\\ A_{q}(t)=\\exp(-4Dt(1-\\cos{q})) \\;\\;,\\;\\; B_{q}(t)=\\cot{(\\frac{q}{2})}\\Bigr(\\exp(-4Dt(1-\\cos{q})-1)\\Bigr)\n\\end{eqnarray}\nAt this point it is worth noting that the ket\n$|\\psi(t)\\rangle$ \ncharacterizes the state of the system at any time without however being an \neigenvector \nof ${\\mathcal L}$ (The Liouvillian is not normal, however see below).\\\\ \nIn ref.\\cite{Lushnikov}, Lushnikov calculates the density of particles \nby the method of the\ngenerating function, which we extend in order to evaluate the two-point\ncorrelation function.\nThe density reads\n\\begin{eqnarray}\n\\displaystyle\n\\label{2.1}\n\\  \\rho(t) = \\sum_{\\{n\\}} \\widetilde n_i(\\{n\\})\\,\\, P(\\{n\\},t), \\forall i,\n\\end{eqnarray}\nSimilarly, the two-point correlation function is written as \n\\begin{eqnarray}\n\\displaystyle\n\\label{2.2}\n\\ {\\mathcal G}_r(t)\\equiv\\langle\\widetilde n_i \\widetilde n_{i+r} \\rangle (t)= \n\\sum_{\\{n\\}} \\widetilde n_i(\\{n\\})\\widetilde n_{i+r}(\\{n\\})\\,\\, P(\\{n\\},t) \\,,\n\\end{eqnarray} \nwhere the translational symmetry of the system has been used.\nWe observe that in  \nthis formalism  \n\\begin{eqnarray}\n\\displaystyle\n\\label{2.4}\n\\ \\rho (t)= \\langle 0|\\exp(\\sum_n \\sigma_n^-)\\,\\, n_i |{\\psi}(t)\\rangle \\;\\;,\\;\\;\n{\\mathcal G}_r(t) =\\langle 0| \\exp(\\sum_n \\sigma_n^-)\\,\\, n_i n_{i+r}  \n|{\\psi}(t)\\rangle\n\\end{eqnarray}\nas one can check using the explicit form of $|\\psi(t)\\rangle$ (develop the \nexponential, order\neach term and perform a Jordan-Wigner transformation \n\\cite{Lushnikov}).\nIt is appropriate to consider the following generating function:\n\\begin{eqnarray}\n\\displaystyle\n\\label{2.6}\n\\ G(x,y,z,t)= \\langle 0| exp \\Bigr(x \\sigma_i^- + y\\sigma_{i+r}^- + \nz \\sum_{n\\neq i,i+r}\\sigma_n^-  \\Bigr)|\\psi(t)\\rangle= \\sum_{\\{n\\}}x^{\\widetilde \nn_i} y^{\\widetilde\nn_{i+r}}z^{\\sum_{j}\\widetilde n_j - \\widetilde n_i - \\widetilde n_{i+r}} P(\\{n\\},t)\n\\end{eqnarray}\nSo we have,\n\\begin{eqnarray}\n\\displaystyle\n\\label{2.7}\n \\rho(t)=\\frac{\\partial}{\\partial x}G(x,z,z,t)\\Big|_{x,y,z=1}\\;\\;,\\;\\;\n {\\mathcal G}_r(t)=\\frac{\\partial^2}{\\partial x \\partial y}G(x,y,z,t)\n \\Big|_{x,y,z=1}\n\\end{eqnarray}\nTo compute the generating function, we rewrite the state as\n\\begin{eqnarray}\n\\displaystyle\n\\label{2.9}\n|\\psi(t)\\rangle=\\prod_{q>0}\\Bigr( A_q (t) a_q^+a_{-q}^+ + B_q(t)\\Bigr)|0\\rangle=\n\\prod_{q>0}\\Bigr( B_q(t)-\\frac{2A_q(t)}{N}\\sum_{n>m} a_m^+a_n^+ \n\\sin{q(n-m)}\\Bigr)|0\\rangle\n\\end{eqnarray}\nNote the normalisation condition due to the conservation of probability:\n\\begin{eqnarray}\n\\displaystyle\n\\label{2.10}\n\\sum_{\\{n\\}}P(\\{n\\},t)=\\prod_{q>0}\\Big(B_q(t)-A_q(t) \\cot{\\frac{q}{2}}\\Big)=1\n\\end{eqnarray}\nNext expand the argument of the exponential as \n\\begin{eqnarray}\n\\displaystyle\n\\label{2.11}\nG(x,y,z,t)=\\langle 0| \\Bigg(\\openone+(x a_i+y a_{i+r} + z\\sum_{n\\neq i,i+r}a_n) \n+\n\\Big\\{z^2\\sum_{i+r\\neq n>n'\\neq i}a_n a_{n'} + \nxz(a_i\\sum_{n<i}a_n+\\sum_{n>i,n\\neq i+r}a_n a_i)+\\nonumber\\\\\n+yz(\\sum_{n>i+r}a_n a_{i+r}+ a_{i+r}\\sum_{\nn<i+r, n\\neq i} a_n ) +xy a_{i+r}a_i\n     \\Big\\}+\\ldots \\Bigg) \n     \\prod_{q>0} \\Biggr( 2\\frac{A_q(t)}{N} \\sum_{n>m} a_m^+a_n^+ \\sin{q(n-m)}  -\n     B_q(t)\\Biggr)  |0\\rangle\n\\end{eqnarray}\n\nIn this expression, only the terms proportional to ``$xy$\" contribute to  \n${\\mathcal G}_r(t)$. Let us call the first of these terms  $G_1$,\n\\begin{eqnarray}\n\\displaystyle\n\\label{2.12}\nG_1=\\langle 0| xy a_{i+r}a_i\n     \\sum_{q>0} \\Biggr\\{ 2\\frac{A_q(t)}{N} \\sum_{n>m} a_m^+a_n^+\n     \\frac{\\sin{q(n-m)}}{A_q(t)\\cot{\\frac{q}{2}}-B_q(t)} \\prod_{q\\neq\n     q'>0} \\Biggr(\\frac{B_{q'}(t)}{B_{q'}(t)-A_{q'}(t)\\cot{\\frac{q'}{2}}}\\Biggr) \n       \\Biggr\\}  |0\\rangle = \\nonumber\\\\ =xy \n\\frac{2}{N}\\Biggr(\\sum_{q>0}\\frac{\\sin{qr}}{\\cot{\\frac{q}{2}}-\n       \\frac{B_q(t)}{A_q(t)}}\\Biggr)\\Biggr(\\prod_{q\\neq \nq'>0}\\frac{1}{1-\\frac{A_{q'}(t)}{B_{q'}(t)}\\cot{\\frac{q}{2}}  } \\Biggr)\n\\end{eqnarray}\nIn the absence of source, we have $A_q(t)\\rightarrow 0$ and\n$B_q(t)\\rightarrow -\\cot{\\frac{q}{2}} $ exponentially fast (see \\ref{1.20}), so \nthat in the thermodynamic limit $(N\\rightarrow\n\\infty)$, the asymptotic behaviour of ${\\mathcal G}_r(t)$ follows as\n\\begin{eqnarray}\n\\displaystyle\n\\label{2.13}\n{\\mathcal G}_r(t)\\sim \\frac{\\partial ^2}{\\partial x \\partial y}G_1 \\rightarrow \n\\frac{1}{\\pi} \\int_{0}^{\\pi} \\frac{dq \n\\sin{qr}}{\\cot{\\frac{q}{2}}-\\frac{B_q(t)}{A_q(t)}}\n\\end{eqnarray}\nWe can do the same for the density and in the thermodynamic limit, \none obtains\n\\begin{eqnarray}\n\\displaystyle\n\\label{2.15}\n\\rho(t)=\\frac{\\partial}{\\partial x}  \\prod_{q>0} \\langle 0| \n\\Biggr(\\openone+\\Big(x a_i + z\\sum_{n\\neq i}a_n \\Big) +\n\\Big\\{z^2\\sum_{i\\neq n,n', n> n'}a_n a_{n'} + \nxz(a_i\\sum_{n<i}a_n+\\sum_{n>i} a_n a_i) + \\ldots \\Big \\}  \\Biggr) \\times \n\\nonumber\\\\ \n     \\times \\Biggr( 2\\frac{A_q(t)}{N} \\sum_{n>m} a_m^+a_n^+ \\sin{q(n-m)}  -\n     B_q(t)\\Biggr)  |0\\rangle\\Big|_{z=1,x=0}\n     \\rightarrow \\frac{1}{\\pi}\\int_{0}^{\\pi}\\frac{dq}{1-\\frac{B_q(t)}{A_q(t) \n\\cot{\\frac{q}{2}}}}\n\\end{eqnarray}\nIn the above, we used the following identities:\n\\begin{eqnarray}\n\\label{2.16}\n\\displaystyle\n &\\sum_{m>i}&\\sin{q(m-i)}=\\frac{1}{2}\\Bigr(\\cot{\\frac{q}{2}}+\n\\frac{\\cos{q(i-\\frac{1}{2})}}{\\sin{\\frac{q}{2}}}\\Bigr)\\;\\;\\;,\\;\\;\\;\n\\sum_{m<i}\\sin{q(m-i)}=\\frac{1}{2}\\Bigr(\\cot{\\frac{q}{2}}-\\frac{\\cos{q(i-\\frac{1\n}{2})}}{\\sin{\\frac{q}{2}}}\\Bigr)\\nonumber \\\\ &\\Rightarrow & \\sum_{n>m}\\sin{q(n-m)}=\\frac{N}{2} \\cot{\\frac{q}{2}}\n\\end{eqnarray}\nThe evaluation of the two-point correlation function  \nrequires the calculation of all the terms proportional to\n$xy$, which in general is a very hard task. \nIn the following sections we will be able \nto solve this\ndifficulty by reformulating the problem in a different language.\\\\\nUsing the explicit form of $A_q(t)$ and $B_q(t)$ (\\ref{1.17}-\\ref{1.20}) and the \nresults of appendix A, we find the\nasymptotic behaviour of the density in the\n({\\it irreversible}) {\\it critical case} as \\cite{Lushnikov}\n\\begin{eqnarray}\n\\label{2.19}\n\\displaystyle\n\\rho(t)=\\frac{e^{-4Dt}}{\\pi}\\int_{0}^{\\pi} dq \\, e^{4Dt\n\\cos{q}}=e^{-4Dt}I_0(4Dt)\\sim\\frac{1}{\\sqrt{8\\pi Dt}}\n\\end{eqnarray}\nSimilarly the asymptotic\nbehaviour of the two-point correlation function is \n\\begin{eqnarray}\n\\label{2.20}\n\\displaystyle\n\\ {\\mathcal G}_r (t)&\\sim & \\frac{e^{-4Dt}}{\\pi} \\int_{0}^{\\pi} dq \n\\frac{\\sin{qr}}{\\sin{q}}\n(1-\\cos{q})e^{4Dt\\cos{q}} \\nonumber\\\\ \n&=& e^{-4Dt}\\sum_{0\\leq n <r}\\Bigr\\{ I_{2n-r+1}(4Dt) - I_{2n-r} (4Dt) \\Bigr\\}  \\sim \\frac{\\pi r}{(8 \\pi Dt)^{\\frac{3}{2}}}\n\\end{eqnarray}\nwhich implies\n\\begin{eqnarray}\n\\label{2.21}\n\\displaystyle\nC_r(t)\\sim  -\\frac{1}{8\\pi Dt}\n\\end{eqnarray}\nUnfortunately, in the massive case (when the source intensity is finite), \nthis method applies only to the computation of the\ndensity. Assuming \n$Dt,Jt \\gg 1$, we find\n\\begin{eqnarray}\n\\label{2.22}\n\\displaystyle\n\\rho(t)=\\rho_{eq}+2(J+2D)\\int_{t}^{\\infty} dt'\n e^{-4(J+D)t'}\\Big(I_0(4Dt') -I_1(4Dt')\\Big)  \\\\ \n\\sim\\frac{\\sqrt{J}}{\\sqrt{J}+\\sqrt{J+2D}}+\\Big(1+\\frac{J}{2D}\\Big)\n\\frac{e^{-4Jt}\n}{8Jt\\sqrt{8\\pi Dt}}\n\\end{eqnarray}\nwhere $\\rho_{eq}= \\frac{\\sqrt{J}}{\\sqrt{J}+\\sqrt{J+2D}}$ represents \nthe equilibrium\nvalue of the density, in agreement with Lushnikov's result \\cite{Lushnikov}.\nIn the next section we provide the full two-point correlation function\nwhen $J>0$.\n\\section{The pseudo-fermionic approach}\nIn this section we evaluate the full two-point correlation function \nin the general case by means of a powerful\nformalism. The central idea is to perform on the fermionic non-hermitian (and\nnon-normal)\nHamiltonian\na  generalized Bogoliubov transformation which allows us to work with a diagonal evolution operator\n(see Ref.\\cite{Stinch1,Stinch2}).\nFollowing  previous works \\cite{Stinch1,Stinch2,Schutz,Droz}, we denote each of the $2^N$ possible configurations  \n  by a ket $|n\\rangle $:\n\\begin{eqnarray}\n\\label{3.1}\n\\displaystyle\n\\langle n|n'\\rangle = \\delta_{n,n'}\\;\\;\\;,\\;\\;\\;\n\\sum_{n}|n\\rangle \\langle n|= \\openone\n\\end{eqnarray}\nIn this Fock space, we can efficiently record\nthe probabilities for the various configurations in the ket\n\n\\begin{eqnarray}\n\\label{3.2}\n\\displaystyle\n|P(t)\\rangle\\equiv \\sum_{n} P(n,t)|n\\rangle\n\\end{eqnarray}\nThe master equation governing the\ndynamics of the annihilation and diffusion processes \ndescribed in the previous section\ncan be rewritten as\n\\begin{eqnarray}\n\\label{3.3}\n\\displaystyle\n\\frac{\\partial}{\\partial t}|P(t)\\rangle={\\cal U}|P(t)\\rangle =\n\\sum_{n}\\partial_t P(n,t)|n\\rangle =\\sum_{n,n'}\\Big(A(n'\\rightarrow n) P(n',t)\n- A(n\\rightarrow n') P(n,t)\\Big)|n\\rangle\n\\end{eqnarray}\nwhere ${\\cal U}$ denotes the evolution operator, $A(n'\\rightarrow n)$ \nand $A(n\\rightarrow n')$\nrepresent the transition rates $J$ and $D$ in Lushnikov's formulation. \nThe matrix\nelements for the operator ${\\cal U}$ are \n\\begin{eqnarray}\n\\label{3.4}\n\\displaystyle\n\\langle n'|{\\cal U}|n\\rangle = A(n \\rightarrow n'), \\forall n'\\neq n \\\\\n\\langle n|{\\cal U}|n\\rangle =-\\sum_{n'\\neq n} A(n \\rightarrow n')\n\\end{eqnarray}\nThis Fock-space formulation was used in Ref. \\cite{Stinch1,Stinch2}\nto study a stochastic  adsorption-desorption problem.\nIn what follows, we will specifically focus on the  reaction-diffusion \nproblem in one dimension.\\\\\nLet us now introduce the left and right steady-states, respectively \n\\begin{eqnarray}\n\\label{3.5}\n\\displaystyle\n\\langle\\widetilde\\chi| \\equiv \\sum_n \\langle n|\\;\\;\\;,\\;\\;\\;|\\chi\\rangle \\equiv \\sum_n P(n,eq)|n\\rangle\n\\end{eqnarray}\nwhere $P(n,eq)$ denotes the probability for a configuration $|n\\rangle$ at\nequilibrium. It is easy to check that\n$e^{{\\cal L}t}$ has no effect on $ |\\chi \\rangle$ and $\\langle \\widetilde \\chi|$, \nand the\nconservation of probability leads to $\\langle \\widetilde \\chi|\\chi \\rangle \n=1$.\\\\\nThe transition probability from a configuration $|n\\rangle$ to $|n'\\rangle$ \nis simply: $W_{n,n'}(t)\\equiv \\langle n'| e^{{\\cal L}t}|n\\rangle$.\\\\\nWe intend to calculate the density and two-point one-time correlation functions \nof  a system initially in\nthe state $|\\phi_0\\rangle \\equiv \\sum_{n}P(n,t=0)|n\\rangle$.\nThe occupation number operator $n_r$ being diagonal in the basis \n$\\{|n\\rangle\\}$, we have \n\\begin{eqnarray}\n\\label{3.7}\n\\displaystyle\n\\rho(t)=\\sum_{n,n'}\\langle n'|n_r|n'\\rangle W_{n,n'}(t) P(n,t=0)=\n\\sum_{n,n'}\\langle n'|n_r e^{{\\cal L}t}|n\\rangle  P(n,t=0)=\n\\langle \\widetilde \\chi| n_r e^{{\\cal L}t}|\\phi_o\\rangle\n\\end{eqnarray}\nand similarly,\n\\begin{eqnarray}\n\\label{3.8}\n\\displaystyle\n{\\cal G}_r(t)=\\sum_{n,n'}\\langle n'|n_{l} n_{m}|n'\\rangle W_{n,n'}(t) P(n,t=0)=\n\\sum_{n,n'}\\langle n'|n_{l} n_{m} e^{{\\cal L}t}|n\\rangle  P(n,t=0)=\n\\langle \\widetilde \\chi|n_{l} n_{m} e^{{\\cal L}t}|\\phi_o\\rangle, \\, \n\\end{eqnarray}\nwhere $r=|m-l|$.\nAt this point, we perform a  generalized Bogoliubov \ntransformation (rotation supplemented by a rescaling),\n\\begin{eqnarray}\n\\label{3.9}\n\\displaystyle\n \\left(\n \\begin{array}{c}\n \\xi_q^+  \\\\\n \\xi_{-q} \n\\end{array}\\right)\\\n=\n\\left(\n \\begin{array}{c c}\n \\alpha \\cos{\\theta_q} & \\alpha^{-1} \\sin{\\theta_q}  \\\\\n  -\\alpha \\sin{\\theta_q} & \\alpha^{-1} \\cos{\\theta_q}\n\\end{array}\\right)\\\n\\left(\n \\begin{array}{c}\n a_q^+  \\\\\n a_{-q} \n\\end{array}\\right)\\\n\\end{eqnarray}\nin order to obtain a diagonal representation for the evolution operator.\\\\\nThis transformation is orthogonal, i.e., invertible \nand  canonical, in the sense \nthat it preserves the anticommutation relations of the\n$a_q$'s, namely:\n\\begin{eqnarray}\n\\label{3.10}\n\\displaystyle\n\\{\\xi_q^+,\\xi_{q'}\\}=\\delta_{q,q'}\\,, \\{\\xi_q^+,\\xi_{q'}^+\\}=\\{\\xi_q,\\xi_{q'}\\} \n=0\n\\end{eqnarray}\nDespite the fact that the $\\xi_q$ \nand $\\xi_q ^+$ {\\it are\nnot adjoint} of each other, this representation will be very useful in \nthe following.\\\\\nWe set $\\alpha=\\Big(\\frac{J}{J+2D}\\Big)^{\\frac{1}{4}}$, \nso that the mode $q$ evolution operator becomes\n\\begin{eqnarray}\n\\label{3.12}\n\\displaystyle\n-{\\cal L}_q=2\\Big(D(1-\\cos{q})-J\\Big)\\Big((\\xi_q^+\\xi_q + \\xi_{-q}^+\\xi_{-q})\n+\\sin{\\theta_q}^2(\\xi_{-q}\\xi_{-q}^+ + \\xi_q \\xi_q^+) +2(\\xi_{-q}^+\\xi_{q}^+ + \n\\xi_q \\xi_{-q})\\Big)\n\\nonumber\\\\+\\sqrt{J(J+2D)}\\sin{q}\\Big(\\cos{2\\theta_q} (\\xi_{-q}^+\\xi_{q}^+ + \n\\xi_q \\xi_{-q})\n+ \\sin{2\\theta_q}(\\xi_{q}\\xi_{q}^+ + \\xi_{-q}^+ \\xi_{-q})  \\Big) +2J\n\\end{eqnarray}\nTo get rid of the terms that do not conserve the number of  \npseudo-particles, we choose $\\theta_q$ as\n\\begin{eqnarray}\n\\label{3.13}\n\\displaystyle\n\\tan{2\\theta_q}=\\frac{\\sqrt{J(J+2D)}\\sin{q}}{(J+D)\\cos{q}-D}\n\\end{eqnarray}\nso that the Hamilton operator becomes\n\\begin{eqnarray}\n\\label{3.16}\n\\displaystyle\n{\\cal L}= -\\sum_{q>0}\\lambda_q \\Big(\\xi_q^+ \\xi_q + \\xi_{-q}^+ \n\\xi_{-q}\\Big)=-\\sum_{q} \\lambda_q  \\xi_q^+ \\xi_q\n\\end{eqnarray}\nwhere\n\\begin{eqnarray}\n\\label{3.15}\n\\displaystyle\n\\lambda_q= 2\\Big(D(1-\\cos{q})+J\\Big)\n\\end{eqnarray}\non account of the periodic boundary conditions ($\\sum_{q>0} \n\\cos{q}=0$).\nNow it is readily seen that $\\langle \n\\widetilde \\chi|$ and \n$|\\chi\\rangle$ act, respectively, as left and right vacua, i.e.,\n$\\xi_q|\\chi\\rangle=0$ and \n$\\langle \\widetilde \\chi|\\xi_q^+ =0$.\nTo simplify the calculations, we\nexpress the initial ket-state $|\\phi_0\\rangle$ in terms of the \nsteady state \n$|\\chi\\rangle$.\nWe consider here two kinds of initial conditions:\\\\\ni) The  whole lattice is initially filled.\nWe write $|\\phi_0\\rangle=|all\\rangle$ and immediately\nconclude $a_q^+|all\\rangle =0$.\nUsing the inverse of (\\ref{3.9}), one can check that\n\\begin{eqnarray}\n\\label{3.18}\n\\displaystyle\n|all\\rangle = \\prod_{q>0}\\Big(1-\\cot{\\theta_q}\\xi_q^+ \\xi_{-q}^+ \n\\Big)|\\chi\\rangle =\n\\exp{\\Big(-\\sum_q\\frac{\\cot{\\theta_q}}{2} \\xi_q^+ \\xi_{-q}^+}\\Big)|\\chi\\rangle\n\\end{eqnarray}\nii) The lattice is initially empty; one can check in the same way\n\\cite{Stinch2} that :\n\\begin{eqnarray}\n\\label{3.19}\n\\displaystyle\n|\\phi_0\\rangle=|0\\rangle = \\prod_{q>0}\\Big(1+\\tan{\\theta_q}\\xi_q^+ \n\\xi_{-q}^+\\Big)|\\chi\\rangle =\n\\exp{ \\Big(\\sum_q\\frac{\\tan{\\theta_q}}{2} \\xi_q^+ \\xi_{-q}^+}\\Big)|\\chi\\rangle\n\\end{eqnarray}\nThe time dependence of $\\xi_k(t)$ and $\\xi_k^+(t)$ follow as\n\\begin{eqnarray}\n\\label{3.20}\n\\displaystyle\n\\xi_k(t)=e^{-{\\cal L}t} \\xi_q e^{{\\cal L}t} = e^{-\\lambda_q t} \\xi_k \\\\\n\\xi_k^+(t)=e^{-{\\cal L}t} \\xi_q^+ e^{{\\cal L}t} = e^{\\lambda_q t} \\xi_k^+ \\\\\n\\end{eqnarray}\nFuthermore, we have \n\\begin{eqnarray}\n\\label{3.21}\n\\displaystyle\n\\langle \\xi_{k_1}\\xi_{k_2}\\rangle (t=0)\\equiv \\langle \\widetilde \n\\chi|\\xi_{k_1}\\xi_{k_2}|all\\rangle=\n\\cot{\\theta_{k_1}}\\delta_{k_1,-k_2}\n\\end{eqnarray}\nand\n\\begin{eqnarray}\n\\label{3.22}\n\\displaystyle\n\\langle \\xi_{k_1}\\xi_{k_2}\\xi_{k_3}\\xi_{k_4}\\rangle (t=0)=\n\\cot{\\theta_{k_1}}\\cot{\\theta_{k_3}}\\delta_{k_1,-k_2}\\delta_{k_3,-k_4}+\n\\cot{\\theta_{k_1}}\\cot{\\theta_{k_2}}\\Big(\\delta_{k_1,-k_4}\\delta_{k_2,-k_3} -   \n\\delta_{k_1,-k_3}\\delta_{k_2,-k_4} \\Big)\n\\end{eqnarray}\nas one can check by applying Wick's theorem (see also  section {\\rm IV}).\nUsing the properties of Fourier transform and the generalized Bogoliubov \ntransformation \n(\\ref{3.9}), the expression of the density and the two-point\ncorrelation function become respectively (for a lattice initially  \nfilled, $|\\phi_0\\rangle=|all\\rangle$):\n\\begin{eqnarray}\n\\label{3.23}\n\\displaystyle\n\\rho (t)= \n\\frac{1}{N}\\sum_{k} \\sin^2{\\theta_k} -\n \\sum_{k,k'}\\frac{e^{i(k'-k)l}}{N}\\sin{\\theta_k} \\cos{\\theta_{k'}} \n \\langle \\widetilde \\chi|\\xi_{-k}\\xi_{k'}e^{{\\cal L}t}|all\\rangle\n\\end{eqnarray}\nand one can do the same for unconnected one-time correlation function \n${\\cal G}_r (t)$ (\\ref{3.8}).\\\\\nTo derive tractable formulas, we have performed tedious but straightforward \ncalculations. Indeed, we have\nextracted the time-dependence of $\\rho(t)$ and  ${\\cal G}_r (t)$\nusing (\\ref{3.20}) and\ncommuted all the pseudo-creation operators to the\nleft of the pseudo-annihilation operators. In the expression \n$\\rho(t)$ and ${\\cal G}_r (t)$ \nonly terms like $\\langle\\widetilde \\chi|\\xi_{-k}\\xi_{k'}|all\\rangle=\\langle \n\\xi_{-k}\\xi_{k'}\\rangle (t=0)$\nand $\\langle\\widetilde \\chi|\\xi_{-k}\\xi_{k'} \\xi_{-q}\\xi_{q'} |all\\rangle=\\langle \n\\xi_{-k}\\xi_{k'}\\xi_{-q}\\xi_{q'} \\rangle (t=0)$ survive:\nthese were evaluated with the help of (\\ref{3.21}, \\ref{3.22}).\\\\\nIn the thermodynamic limit, we arrive at\n\\begin{eqnarray}\n\\label{3.25}\n \\rho(t)&=&\\frac{2}{N}\\sum_{k>0} \\Big(\\sin^2{\\theta_k}+ e^{-2\\lambda_k t} \n\\sin{\\theta_k}\\cos{\\theta_k}\\cot{\\theta_k}\\Big) \\nonumber \\\\\n & \\rightarrow & \\frac{1}{\\pi}\\int_{0}^{\\pi} dq \\Big(\\sin^2{\\theta_q}+ \ne^{-2\\lambda_q t}\\cos^2{\\theta_q}\\Big)  \n = \\frac{\\sqrt{J}}{\\sqrt{J}+\\sqrt{J+2D}}+2(J+2D)\\int_{t}^{\\infty} dt' \n\\Big(I_0(4Dt')-I_1(4Dt')\\Big)e^{-4(J+D)t'}\n \\end{eqnarray}\nwhich coincides with (\\ref{2.22}).\\\\\nIt is worth emphasizing that this result is general and works for both, the \n{\\it massive} ($J\\neq 0$) and the {\\it critical} ($J=0$) cases.\nThe point here is that the limit $J\\rightarrow 0$ is not singular, despite\nthe divergence of $\\cot\\theta\\rightarrow-\\infty$. In fact, integration over \n$k$ and $k'$ of terms\nproportional to $\\sin{\\theta_k}\\cot{\\theta_{k'}}$ yields finite results. \nTherefore, we can perform the computations (\\ref{3.9}) at $J$ finite and set\nsubsequently $J=0$ in  \n$\\rho(t)$ and ${\\cal G}_r(t)$. \\\\\nSimilarly, the two-point correlation function (of one-time) is evaluated\nas ($ r=|m-l|$):\n %\n\\begin{eqnarray}\n\\label{3.26}\n\\displaystyle\n{\\cal G}_r(eq)=\\rho_{eq}^2+\\frac{1}{\\pi^2}\\Big(\\int_{0}^{\\pi} dq \\,\n\\sin^2{\\theta_q}\\cos{qr}\\Big)\\Big(\\int_{0}^{\\pi}dq' \\,\n\\cos^2{\\theta_{q'}}\\cos{q'r}\\Big) +\\Big(\\frac{1}{2\\pi}\\int_{0}^{\\pi} \n dq \\,\\sin{2\\theta_q}\\sin{qr}\\Big)^2 \n \\end{eqnarray}\n %\n %\n\\begin{eqnarray}\n\\label{3.27.0}\n\\displaystyle\n{\\cal G}_r(t)-{\\cal G}_r(eq)&=&\\Big(\\rho(t)^2 - \\rho_{eq}^2\\Big)+ \n\\frac{1}{\\pi^2}\\Big(\\int_{0}^{\\pi}dq \\, e^{-2\\lambda_q\nt}\\cos^2{\\theta_q}\\cos{qr}\\Big)\\Big( \\int_{0}^{\\pi} dq' \\, \n\\cos^2{\\theta_{q'}}\\cos{q'r}\\Big)\n\\nonumber\\\\ &+& \\frac{1}{2\\pi^2}\\Big(\\int_{0}^{\\pi} dq \\,\n\\sin{2\\theta_q}\\sin{qr}\\Big)\n\\Big(\\int_{0}^{\\pi}dq'\\, e^{-2\\lambda_{q'}t}\\cos^2{\\theta_{q'}}\\cot{\\theta_{q'}}\\sin{q'r}\\Big)\\nonumber\\\\\n&-& \\frac{1}{\\pi^2}\\Big(\\int_{0}^{\\pi} dq \\, \\sin^2{\\theta_q}\\cos{qr}\\Big)\n\\Big(\\int_{0}^{\\pi}dq'\\, e^{-2\\lambda_{q'}t}\\cos^2{\\theta_{q'}}\\cos{q'r}\\Big)\\nonumber\\\\\n&-& \\frac{1}{4\\pi^2}\\Big(\\int_{0}^{\\pi} dq \\, \\sin{2\\theta_q}\\sin{qr}\\Big)\n\\Big(\\int_{0}^{\\pi}dq' \\, e^{-2\\lambda_{q'}t}\\sin{2\\theta_{q'}}\\sin{q'r}\\Big)\\nonumber\\\\\n&-& \\Big( \\frac{1}{\\pi}\\int_{0}^{\\pi}dq \\, e^{-2\\lambda_qt}\\cos^2{\\theta_q}\\cos{qr}\\Big)^2\n-\\Big(\\frac{1}{\\pi^2} \\int_{0}^{\\pi} dq  \\, e^{-2\\lambda_{q}t}\\cos^2{\\theta_{q}}\\cot{\\theta_{q}}\\sin{qr}  \\Big)\n \\nonumber\\\\ \n&\\times & \\Big( \\int_{0}^{\\pi}dq  \\, e^{-2\\lambda_{q}t}\\sin^2{\\theta_{q}}\\cot{\\theta_{q}}\\sin{qr}  \\Big)\\;\\; ,\n \\end{eqnarray}\nwhere we separated the static contribution to the correlation function from the\ndynamic one. Using known properties of modified Bessel functions given (see appendix)  \nand writing  $I'_m(x)\\equiv \\frac{d}{dx} I_m (x)$, we finally infer \n(notice that ${\\cal G}_r(eq)=\\rho_{eq}^2$) \n\\begin{eqnarray}\n\\label{3.27.1}\n\\displaystyle\n{\\cal G}_r(t) &=& \\rho(t)^2 + 2(J+2D) (A_0 - C_0) \n\\int_{t}^{\\infty}dt' e^{-4(J+D)t'}\\Big\\{I_r(4Dt')-I'_r(4Dt')\\Big\\} \\nonumber\\\\ \n&+& \\frac{\\sqrt{J(J+2D)}B_0}{2}\\int_{t}^{\\infty}dt' e^{-4(J+D)t'} \\frac{r}{2Dt'} I_{r}(4Dt')\n \\nonumber\\\\ \n &-&  \\frac{\\Big(J+2D\\Big)^{\\frac{3}{2}}}{\\sqrt{J}}\\sum_{0\\leq n<r}\\Big(\\int_{t}^{\\infty} dt'\\, \ne^{-4(J+D)t'}\\{2I_{2n-r+1}(4Dt')- 2I_{2n-r}(4Dt')\n+I'_{2n-r}(4Dt')\\}\\Big) \\nonumber\\\\\n&-& \\Big(2(J+2D)\\int_{t}^{\\infty} dt'\\, e^{-4(J+D)t'} \\Big\\{I_r(4Dt')-I'_r(4Dt')\\Big\\}\\Big)^2 \\nonumber\\\\\n&+& 2(J+2D)^2 \\Big(\\sum_{0\\leq n<r}(2n-r)\\Big(\\int_{t}^{\\infty} dt'\\, \ne^{-4(J+D)t'}\\frac{I_{2n-r}(4Dt')}{2Dt'} \\Big)\\Big) \\nonumber\\\\\n& \\times & \\Big(\\sum_{0\\leq n<r}\\Big(\\int_{t}^{\\infty} dt'\\, \ne^{-4(J+D)t'}\\{2I_{2n-r+1}(4Dt')- 2I_{2n-r}(4Dt')\n+I'_{2n-r}(4Dt')\\}\\Big)\\Big)\n \\end{eqnarray}\n %\nIn the above formula $A_0$, $B_0$, $C_0$ have been defined by:\n %\n\\begin{eqnarray}\n\\label{3.28}\n\\displaystyle\nA_0\\equiv \\frac{1}{\\pi}\\int_{0}^{\\pi} dq\\,\\cos^2{\\theta_q}\\cos{qr},\\;\\;\nB_0\\equiv  \\frac{1}{\\pi} \\int_{0}^{\\pi}dq\\, \\sin{2\\theta_q}\\sin{qr},\\;\\;\nC_0 \\equiv \\frac{1}{\\pi} \\int_{0}^{\\pi}dq\\, \\sin^2{\\theta_q}\\cos{qr} \n\\end{eqnarray}\nor more explicitly, in the massive case (with help of (\\ref{A.16})),\n %\n\\begin{eqnarray}\n\\label{3.28.bis}\n\\displaystyle\nA_0 &=& \\frac{(J+2D)}{\\pi}\\int_{0}^{\\pi}\\frac{dq}{\\lambda_q} \n\\,\\Big(\\cos{qr}-\\frac{\\cos{q(r+1)}+\\cos{q(r-1)}}{2}\\Big)= \n-(J+2D)\\zeta^{r-1}\\frac{(1-\\zeta)^2}{4\\sqrt{J^2+2JD}},\\\\\nB_0 &=& \\frac{\\sqrt{J(J+2D)}}{\\pi}  \\int_{0}^{\\pi}\\frac{dq}{\\lambda_q}\n\\,\\Big(\\cos{q(r+1)}-\\cos{q(r-1)}\\Big)  = \n-\\zeta^{r-1}\\frac{(1-\\zeta^2)}\n{2}, \\\\\nC_0 &=& \\frac{J}{\\pi} \\int_{0}^{\\pi}\\frac{dq}{\\lambda_q}\n\\,\\Big(\\cos{qr}+\\frac{\\cos{q(r+1)}+\\cos{q(r-1)}}{2}\\Big)  \n=J\\zeta^{r-1}\\frac{(1+\\zeta)^2}{4\\sqrt{J^2+2JD}}\n\\end{eqnarray}\nwith the notation \n\\begin{eqnarray}\n\\label{3.29}\n\\displaystyle\n\\zeta\\equiv \\frac{D}{(J+D)+\\sqrt{J^2+2JD}}\\leq 1\n\\end{eqnarray}\nNote that the above result, though obtained by elementary\nmeans, is new. The formula can be evaluated numerically. \nThe connected two-point \ncorrelation function ${\\cal C}_r (t)\\equiv {\\cal G}_r(t)-\\rho(t)^2$\nfollows immediately from the above. \nTo establish a connection with the results derived previously \\cite{Droz}\nwe explicitly evaluate ${\\cal C}_r (t)$ when $J=0$. Taking due attention to the\napparent singularities occuring in this limit, we find\n\\begin{eqnarray}\n\\label{3.30}\n\\displaystyle\n{\\cal C}_r(t)&=& \\frac{1}{2\\pi^2}\\int_{0}^{\\pi} dq \\sin{2\\theta_q}\\sin{qr}\n\\int_{0}^{\\pi}dq' e^{-2\\lambda_{q'}t}\\cot{\\theta_{q'}}\\sin{q'r} \\nonumber\\\\\n&-& \\Big( \\frac{1}{\\pi}\\int_{0}^{\\pi}dq \\, e^{-2\\lambda_qt}\\cos{qr}\\Big)^2 \n- \\Big(\\frac{1}{\\pi} \\int_{0}^{\\pi}dq\\,e^{-2\\lambda_{q}t}\\cot{\\theta_{q}}\\sin{qr}  \n\\Big) \n\\Big(\\frac{1}{\\pi} \\int_{0}^{\\pi}dq \\, e^{-2\\lambda_{q}t}\\sin^2{\\theta_{q}}\\cot{\\theta_{q}}\\sin{qr}\\Big)\n\\end{eqnarray}\nor in terms of elementary functions,\n\\begin{eqnarray}\n\\label{3.30bis}\n\\displaystyle\n{\\cal C}_r(t)\n&=& \\sum_{0\\leq n<r} e^{-4Dt}\\Big(I_{2n-r+1}(4Dt)-I_{2n-r}(4Dt) \\Big) \\nonumber\\\\\n&+& \\Big(\\sum_{0\\leq n<r}e^{-4Dt} I_{2n-r}\\Big)^2 - \\Big(\\sum_{0\\leq n<r}e^{-4Dt} I_{2n-r+1}\\Big)^2\n- \\Big(e^{-4Dt}I_r(4Dt)\\Big)^2\n\\end{eqnarray}\nThis expression is equivalent to the result derived by a well-known mapping of Lushnikov's model \nto Glauber's $1D$ Ising model \\cite{Droz}.\\\\\nFinally, we compute (using formula (\\ref{A.15})),  \nthe asymptotic behaviour of the two-point correlation\nfunction ${\\cal C}_r(t)$ when $Dt \\gg 1$,\\,$Jt\\gg 1$, and $r<\\infty$\n\\begin{eqnarray}\n\\label{3.31}\n\\displaystyle\n{\\cal C}_r (t)\\sim -\\frac{\\zeta^{r-1} e^{-4Jt}}{8Jt\\sqrt{8\\pi Dt}}\\Biggr\\{ (J+2D)(1-\\zeta)^2 +\nJ(1+\\zeta)^2 + \\frac{rJ(J+2D)}{D}(1-\\zeta^2) \n  \\Biggr\\} \\frac{1}{4\\sqrt{J(J+2D)}}\n\\end{eqnarray}\nAs expected for a finite source intensity $J>0$, the density and one-time correlation\nfunction decay exponentially with time. Morever, the one-time correlation function also decays\nexponentially with distance.\\\\\nA similar calculation can be performed for the case of an initial empty lattice\n ($|\\phi_0 \\rangle \\equiv |0\\rangle$). In this case the density is\n  $\\rho(t)=2J \\int_{t}^{\\infty} dt'\\Big(I_0(4Dt')+I_1(4Dt')\\Big)e^{-4(J+D)t'}\\sim \n  \\rho_{eq}-e^{-4Jt}\/\\sqrt{8\\pi Dt}$ {\\cite {Lushnikov}}. It has been shown \\cite{Stinch1}, via the mapping to the\n  Glauber dynamics, that the nearest neighbor connected function decays ($ Dt,Jt\\gg 1$)\n  as $C(r=1,t)\\sim \\rho_{eq}e^{-4Jt}\/ \\sqrt{2\\pi Dt}$.\\\\\nThe large-time and large-distance behavior of the one-time correlation\nfunction can also be studied. In fact, in the {\\it critical} case it has been shown \\cite{Family,Droz} that the \none-time correlation function obeys a scaling form.\\\\\nHere we investigate the behavior of ${\\cal C}_r(t)$ in the limit where $r$ and $Dt \\rightarrow \\infty$\nwith $u\\equiv \\frac{r^2}{8Dt}$  finite. In the {\\it massive} case,\nwe do not expect a scaling form :\n\\begin{eqnarray}\n\\label{3.32}\n\\displaystyle\n{\\cal C}_r (t)\\sim -\\sqrt{1+\\frac{2D}{J}}\\zeta^{r-1}(1-\\zeta^2)\\frac{g(r^2, u)}{4r^2},\n\\end{eqnarray}\nwhere\n\\begin{eqnarray}\n\\label{3.33}\n\\displaystyle\ng(r^2, u)\\equiv \\sqrt{\\frac{u^3}{\\pi}}e^{-\\frac{Jr^2}{2Du}}\n\\end{eqnarray}\nThe effect of the source  is to disrupt the spatial correlations, i.e. to make\nthem short-range. In this sense the finite source prohibits\nthe formation of arbitrarly large vacancy domains.\n\\\\\n\nIn the next section we will introduce a field-theoretical approach to deal with \ntwo-time correlation function. Our approach and the results obtained thereby complement the\nexact treatments of the critical case \\cite{Stinch3}.\nIt is worth pointing out, that Glauber calculated in his pioneering work the two-time two-points \nof spin correlations functions\\cite{Glauber,Bray}.\n\\section{Field theoretical Reformulation}\n \nThe purpose of this section is to reformulate the results of the previous \nsection in a field theoretical language, i.e., in terms\nof path integrals of fermionic variables (see for example \\cite{Negele}). We define the Grassmann numbers $\\eta_{q},\\eta_{q}^{\\ast}$ \nwhich anticommute with each \nother and with the fermionic operators introduced in the previous section, i.e., \n$\\{\\eta_q,\\eta_{q'}\\}=\\{\\eta_q^{\\ast},\\eta_{q'}^{\\ast}\\}\n=\\{\\eta_q^{\\ast},\\eta_{q'}\\}= 0$ and\n$\\{\\eta_q,\\xi_{q'}\\}= \n\\{\\eta_q^{\\ast},\\xi_{q'}^+\\}=\\{\\eta_q^{\\ast},\n\\xi_{q'}\\}=\\{\\eta_q^{\\ast},\\xi_{q'}^+\\}= 0$.\nWe follow standard practice and\nconsider the {\\it coherent states} associated to the fermionic variables.\nWe recall that pseudo-fermionic operators required the introduction \nof a {\\it left vacuum}  \n$\\langle \\widetilde \\chi|$ and a {\\it right vacuum} \n$| \\chi\\rangle$, ergo we define the right  \n$|\\eta\\rangle$ and the left coherent states \n$ \\langle \\widetilde \\eta|$, respectively, as $|\\eta\\rangle =\ne^{-\\sum_q \\eta_q \\xi_q^{+}}|\\chi\\rangle\\;\\;\\;,\\;\\;\\;\\langle \\widetilde\\eta| =e^{-\\sum_q \\xi_q\\eta_q^{\\ast}\n}$\\\\\nDespite the fact that $\\xi $ and $\\xi^{+}$ are not adjoint of each other,\nwe find the familiar results:\\\\ $\\langle \\widetilde \\eta|\\chi\\rangle=\\langle \\widetilde \\chi|\\eta\\rangle=1\\;\\;\\;,\\;\\;\\;\n\\langle \\widetilde \\eta|\\eta\\rangle=e^{\\sum_q \\eta_q^{\\ast}\\eta_q}$ and \n$\\xi_q|\\eta\\rangle=\\eta_q|\\eta\\rangle \\;\\;,\\;\\;\\; \\langle \\widetilde \n\\eta|\\xi_q=\\frac{\\partial}{\\partial \\eta_q^{\\ast}}\\langle\\widetilde\n\\eta| \\;\\;,\\;\\;\\;\n\\langle \\widetilde \\eta| \\xi_q^{+}= \\langle \\widetilde \\eta|\\eta_q^{\\ast}\\;\\;,\\;\\;\\; \n\\xi_q^{+}|\\eta\\rangle=-\\frac{\\partial}{\\partial \\eta_q}\n|\\eta\\rangle $.\\\\\nMost importantly, the closure relation holds true, i.e.,\n\\begin{eqnarray}\n\\label{4.7.0}\n\\displaystyle\n\\int \\prod_{q}d\\eta_q^{\\ast}d\\eta_q e^{-\\sum_q \\eta_q^{\\ast}\\eta_q} |\\eta\\rangle \n\\langle \\widetilde \\eta | =\\openone\n\\end{eqnarray}\nAt this point, we know from field theory \nhow to calculate $\\langle \\widetilde \\chi| n_m e^{{\\cal L} \nt}|all\\rangle$ and \n$\\langle \\widetilde \\chi| n_m n_{m+r} e^{{\\cal L} t}|all\\rangle$ using \nthe path-integral formalism. \nWe discretize time in $M$ infinitesimal intervals\nof width $\\epsilon \\equiv \\lim_{M\\rightarrow \\infty} \\frac{t}{M}$. ${\\cal L}$ is normal\n ordered, and the closure relation is inserted into the above formulas. We have for the density\n\n\\begin{eqnarray}\n\\label{4.7}\n\\displaystyle\n\\rho(t)=\\int \\prod_{q,\\alpha=1\\ldots M}d\\eta_{q,\\alpha}^{\\ast}d\\eta_{q,\\alpha} e^{-\\sum_{q,\\alpha}\n \\eta_{q,\\alpha}^{\\ast}\\eta_{q,\\alpha}}   \n\\langle \\widetilde \\chi| n_m|\\eta_M\\rangle\n\\langle\\widetilde\\eta_M|e^{{\\cal{L}}\\epsilon_M}|\\eta_{M-1}\\rangle \\ldots \n\\langle\\widetilde\\eta_2|e^{{\\cal{L}}\\epsilon_2}|\\eta_{1}\\rangle\n\\langle\\widetilde\\eta_1|e^{{\\cal{L}}\\epsilon_1}|all\\rangle \\;\\;,\n\\end{eqnarray}\nwhere\n\\begin{eqnarray}\n\\label{4.8}\n\\displaystyle\n\\lim_{M\\rightarrow \\infty} \\langle\\widetilde\\eta_1|e^{{\\cal{L}}\\epsilon_1}|all\\rangle=\ne^{-\\frac{1}{2}\\sum_{q}\\cot{\\theta_{q}} \\eta_{1,q}^{\\ast}\\eta_{1,-q}^{\\ast}  } \n\\\\\n \\langle \\widetilde \\chi| n_m|\\eta_M\\rangle=\\frac{1}{N}\\Big(\\sum_{k}\\sin^2{\\theta_k}-\\sum_{k,k'}\\sin{\\theta_k}\n\\cos{\\theta_{k'}}\n \\langle \\widetilde \\chi| \\xi_{-k}\\xi_{k'}|\\eta_m\\rangle\\Big)\\nonumber\\\\\n=\\frac{1}{N}\\Big(\\sum_{k}\\sin^2{\\theta_k}-\\sum_{k,k'}\\sin{\\theta_k}\\cos{\\theta_{\nk'}}\\,\n\\, \\eta_{M,k} \\eta_{M,k'} \\Big)\n\\end{eqnarray}\nTaking the continuum limit \\cite{Negele}, we arrive at\n\\begin{eqnarray}\n\\label{4.9}\n\\displaystyle\n\\rho(t)=\\frac{1}{N}\\int \\prod_{q}d\\eta_q^{\\ast}(t)d\\eta_q(t) \n\\Big(\\sum_{k}\\sin^2{\\theta_k}-\\sum_{k,k'}\\sin{\\theta_k}\\cos{\\theta_{k'}}\\,\\,\n\\eta_{q}(t) \\eta_{q'}(t) \n\\Big)e^{-\\sum_{q}\\frac{\\cot{\\theta_q}}{2}\\eta_q^{\\ast}(0)\\eta_{-q}^{\\ast}(0)} \ne^{-S_0(\\eta^{\\ast}(t),\\eta(t))} \\;\\;,\n\\end{eqnarray}\nwhere, the Liouvillian operator \n${\\cal L}$ acts as the {\\it Hamiltonian} of Field theory. \nBy abuse of language we call\n$S_0(\\eta^{\\ast}(t),\\eta(t))$ the {\\it Euclidian action} of our \nproblem, which is defined by\n\\begin{eqnarray}\n\\label{4.10}\n\\displaystyle\nS_0(\\eta^{\\ast}(t),\\eta(t))\\equiv \\int_{0}^{t}dt'\\,\\, \n\\sum_{q}\\Big(\\eta_q^{\\ast}(t')\\partial_{t'} \\eta_q (t') -\n{\\cal \nL}(\\eta_q^{\\ast}(t'),\\eta_q(t'))\\Big)+\\sum_{q'}\\eta_{q'}^{\\ast}(0)\\eta_{q'}(0)\n\\end{eqnarray}\nTaking averages $\\langle \\ldots \\rangle_{S_0}$ on the gaussian distribution, the density\ncan be rewritten as\n\\begin{eqnarray}\n\\label{4.11}\n\\displaystyle\n\\rho(t)=\\frac{1}{N}\\Big \\langle \n\\Big(\\sum_{k}\\sin^2{\\theta_k}-\\sum_{k,k'}\\sin{\\theta_k}\\cos{\\theta_{k'}}\\,\\,\n\\eta_{q}(t) \\eta_{q'}(t) \n\\Big)e^{-\\sum_{q}\\frac{\\cot{\\theta_q}}{2}\\eta_q^{\\ast}(0)\\eta_{-q}^{\\ast}(0)} \n\\Big \\rangle_{S_0}\n\\end{eqnarray}\nWe proceed to discretize \nthe free Euclidian action $S_0$ which is bilinear, as\n\\begin{eqnarray}\n\\label{4.12}\n\\displaystyle\nS(q)=\n\\left(\n \\begin{array}{c c c c c c}\n 1 & 0 & 0 & \\ldots & \\ldots & 0 \\\\\n -a & 1 & 0 & \\ldots & \\ldots & 0 \\\\\n 0 & -a & 1 & 0 & \\ldots&0 \\\\\n \\vdots &\\ddots &\\ddots &\\ddots &\\vdots &\\vdots\\\\\n 0 &\\ddots &\\ddots &\\ddots &\\vdots &0 \\\\\n 0 &\\ldots &\\ldots &0 &-a &1\n\\end{array}\\right)\\\n\\end{eqnarray}\nwhere, $a\\equiv 1-\\frac{t}{M} \\lambda_q$.\\\\\nFollowing Ref. \\cite{Negele}, we find  $\\det(S(q))=1$\nand \n\\begin{eqnarray}\n\\label{4.11.1}\n\\displaystyle\n\\lim_{M\\rightarrow \\infty} \nS^{-1}_{\\alpha,\\beta}(q)=e^{-\\lambda_q(t_{\\alpha}-t_{\\beta})},\\,\\, \n\\forall \\alpha = 1\\ldots M \\geq \\beta,\\,\\,\\,\\, t_{\\alpha}=\\frac{\\alpha t}{M}\n\\end{eqnarray}\nFuthermore we have, in the continuum limit \n$\\Bigg(\\int\\prod_{q} d\\eta_{q}^{\\ast}(t)d\\eta_{q}(t) \\,\\, e^{-S_0} =1\\Bigg)$\n\\begin{eqnarray}\n\\label{4.13}\n\\displaystyle\n\\langle \\eta_p(t_2) \\eta_{p'}^{\\ast}(t_1)\\rangle_{S_0}=\\int \n\\prod_{q}d\\eta_{q}^{\\ast}(t)d\\eta_{q}(t)\\,\\, \\eta_p(t_2)\\eta_{p'}^{\\ast}(t_1) e^{-S_0}=\n\\delta_{p,p'}e^{-\\lambda_p(t_2-t_1)}\n\\end{eqnarray}\nApplying Wick's theorem, leads to\n\\begin{eqnarray}\n\\label{4.14}\n\\displaystyle\n\\langle \\xi_{p} \\xi_{p'}\\rangle (t)= \\Big \\langle \\eta_{p}(t) \\eta_{p'}(t) e^{-\\frac{1}{2}\\sum_{q} \\cot{\\theta_q}\n\\eta_{q}^{\\ast}(0) \\eta_{-q}^{\\ast}(0)} \\Big \\rangle_{S_0} =\\delta_{p,-p'}\\cot{\\theta_p}e^{-2\\lambda_p t}\n\\end{eqnarray}\nWith this result, it is straightforward to calculate the density using (\\ref{4.14}) and (\\ref{4.11}).\n\nThe computation of correlation function requires evaluation of terms like\n $\\langle \\xi_{k1} \\xi_{k2} \\xi_{k3} \\xi_{k4} \\rangle (t) $ . We  generate in the standard\n fashion the multi-point correlation functions. The generating functional in discretized time reads:\n\\begin{eqnarray}\n\\label{4.15}\n\\displaystyle\nZ[g_{q,\\alpha}; g^{\\ast}_{q',\\alpha'}]=\\Biggr\\langle \ne^{\\sum_q\\Big\\{-\\frac{1}{2}\\cot{\\theta_q}\\eta_{q,1}^{\\ast} \n\\eta_{-q,1}^{\\ast} + \\sum_{\\alpha\\geq 2}(g_{q,\\alpha}^{\\ast}\\eta_{q,\\alpha} + \n\\eta_{q,\\alpha}^{\\ast} g_{q,\\alpha} ) \\Big\\}}\n\\Biggr\\rangle_{S_0}\n\\end{eqnarray}\nhere $g_{q,\\alpha},g_{q,\\alpha}^{\\ast} $ denote (Grassman-numbers) the\ncoefficient of the source \nterms. Note that this functional differs from the usual field theoretical one\n\\cite{Negele} by a term that codes the initial condition, i.e.,\n$\\prod_{q>0}e^{\\cot{\\theta_q}\\eta_{q,1}^{\\ast}\\eta_{-q,1}^{\\ast}}$.\nTaking this term into account however is no trouble, i.e.,  \n\\begin{eqnarray}\n\\label{4.16}\n\\displaystyle\nZ[g_{q,\\alpha}; \ng^{\\ast}_{q',\\alpha'}]=\\Biggr\\{\\prod_{q>0}\\Big(e^{g_{-q,1}^{\\ast} \ng_{q,1}^{\\ast}\\cot{\\theta_q}} \n+  g_{-q,1}^{\\ast} g_{q,1}^{\\ast}g_{-q,1} g_{q,1}\\Big)\n\\Biggr\\}\\prod_{q,q'}\\prod_{i,j=2}^{M}e^{g_{q,i}^{\\ast}S_{i,j}(q)^{-1}g_{q',j}\\delta_{q,q'}}\n\\end{eqnarray}\nConsidering that the source term will be set to zero in the end of the\ncalculation and noting that \nfor $\\alpha=1 $, only pseudo-creation operators contribute, \nwe find it convenient to work with\n\\begin{eqnarray}\n\\label{4.17}\n\\displaystyle\n\\widetilde Z[g_{q,\\alpha}; g^{\\ast}_{q',\\alpha'}]=\\prod_{q}e^{\\Big(\\frac{1}{2} \ng_{-q,1}^{\\ast}\ng_{q,1}^{\\ast}\\cot{\\theta_q}+\\sum_{q'}\\sum_{i,j>1}g_{q,i}^{\\ast}e^{\\lambda_q \n(t_j-t_i)}g_{q',j} \\delta_{q,q'}   \\Big)}\n\\end{eqnarray}\nFrom the definition, it follows then that\n\\begin{eqnarray}\n\\label{4.18.0}\n\\displaystyle\n\\langle\\xi_{k_1}\\xi_{k_2}\\rangle (t)=\\frac{ \\delta^6 \\widetilde \nZ[g_{q,\\alpha};g_{q',\\alpha'}^{\\ast}]}{\\delta g_{-q,1}\n\\delta g_{q,1}^{\\ast}  \\delta g_{k_1,M}^{\\ast} \\delta g_{k_2,M}^{\\ast} \\delta g_{q,2} \n\\delta g_{-q,2}}\\Biggr|_{g=g^{\\ast}=0}\n\\end{eqnarray}\nor more generally,\n\\begin{eqnarray}\n\\label{4.18.1}\n\\displaystyle\n\\langle\\xi_{k_1}\\xi_{k_1'}\\ldots \\xi_{k_n}\\xi_{k_n'} \\rangle (t)\\equiv\n\\langle \\widetilde \\chi| \\xi_{k_1}\\xi_{k_1'}\\ldots \\xi_{k_n}\\xi_{k_n'} e^{{\\cal L}t}|all \\rangle =\n \\langle \\widetilde \\chi| \\xi_{k_1}\\xi_{k_1'}\\ldots \\xi_{k_n}\\xi_{k_n'} e^{{\\cal L}t}e^{-\\frac{1}{2}\\sum_q{\\cot{\\theta_q}\n \\xi_q^{+}\\xi_{-q}^{+}}}|\\chi \\rangle \n\\nonumber\\\\=\n\\frac{ \\delta^{6n} \\widetilde Z[g_{q,\\alpha};g_{q',\\alpha'}^{\\ast}]}{\\delta \ng_{-q_n,1}^{\\ast} \n\\delta g_{q_n,1}^{\\ast} \\ldots  \\delta g_{-q_1,1}^{\\ast} \\delta g_{q_1,1}^{\\ast} \n\\delta g_{k_1,M}^{\\ast} \\delta g_{k_1',M}^{\\ast}  \\ldots \\delta g_{k_n,M}^{\\ast} \\delta g_{k_n',M}^{\\ast} \n\\delta g_{q_1,2} \\delta g_{-q_1,2} \\ldots  \\delta g_{q_n,2} \n\\delta g_{-q_n,2}^{\\ast}} \\Biggr|_{g=g^{\\ast}=0}\n\\end{eqnarray}\nAs an application of this field-theoretical formulation we will now extend  our approach\nto the computation of the two-time correlation function, for an initial state $|\\phi_0\\rangle$,\n\n\\begin{eqnarray}\n\\label{4.19}\n\\displaystyle\n{\\cal G}_r(t,t')\\equiv \\langle n_{l+r}(t+t') n_l (t) \\rangle =\\langle \\widetilde \\chi| n_l e^{{\\cal L}t'} n_{l+r} e^{{\\cal\nL}t}|\\phi_0 \\rangle  =\n\\sum_{n,n'}\\langle n'|n_{l+r} e^{{\\cal L}t'} n_l e^{{\\cal L}t} |n \\rangle  P(n,0) \\nonumber \\\\\n= \\sum_{n,n',n''}\\langle n'|n_{l+r} |n'\\rangle  \\langle n''|n_{l} |n''\\rangle W_{n'n''}(t') W_{n''n}(t) P(n,0)\n\\end{eqnarray}\nAs above, we consider an initially filled lattice ($|\\phi_0\\rangle = |all \\rangle$).\\\\\nTo perform the computation of ${\\cal G}(r,t,t')$ we need to evaluate terms like \n\\begin{eqnarray}\n\\label{4.21}\n\\displaystyle\n\\langle \\widetilde \\chi | \\xi_p \\xi_{p'} e^{{\\cal L}t'}\\xi_q \\xi_{q'} e^{{\\cal L}t}|all \\rangle =\n\\Big \\langle \\eta_p (t+t') \\eta_{p'} (t+t') \\eta_q (t) \\eta_{q'} (t) e^{-\\frac{1}{2}\\sum_k \\cot{\\theta_k}\n\\eta_k^{\\ast}(0)\\eta_{-k}^{\\ast}(0) }  \\Big \\rangle_{S_0}\n\\end{eqnarray}\nThis expression can be calculated by using the generating functional, where as before $t$\n ($t'$) is discretized into $M$ ($N$) infinitesimal\ntime steps (respectively), with $M,N \\rightarrow \\infty$:\n\\begin{eqnarray}\n\\label{4.22}\n\\displaystyle\n\\langle \\widetilde \\chi | \\xi_p \\xi_{p'} e^{{\\cal L}t'}\\xi_q \\xi_{q'} e^{{\\cal L}t}|all \\rangle &=& \n\\frac{ \\delta^{12} \\widetilde Z[g_{q,\\alpha};g_{q',\\alpha'}^{\\ast}]}{\\delta g_{-q_2,1}^{\\ast} \n\\delta g_{q_2,1}^{\\ast} \\delta g_{-q_1,1}^{\\ast} \\delta g_{q_1,1}^{\\ast} \n\\delta g_{q,M}^{\\ast} \\delta g_{q',M}^{\\ast} \\delta g_{p,M+N}^{\\ast} \\delta g_{p',M+N}^{\\ast}\n\\delta g_{q_1,2} \\delta g_{-q_1,2} \\delta g_{q_2,2} \n\\delta g_{-q_2,2}  } \\Big|_{g=g^{\\ast}=0}\\nonumber\\\\\n&=& e^{-(\\lambda_p + \\lambda_{p'})(t+t')-(\\lambda_q + \\lambda_{q'})t}\n\\Biggr (\\cot{\\theta_p}\\cot{\\theta_q}\\delta_{p,-p'}\\delta_{q,-q'} \\nonumber\\\\ &+& \\cot{\\theta_p}\\cot{\\theta_{p'}}\n\\Big(\\delta_{-p,q'}\\delta_{-q,p'} - \\delta_{-q,p}\\delta_{-p',q'} \\Big)\\Biggr)\n\\end{eqnarray}\nwhere  the continuum limit for the time has been taken.\\\\\nApplying the same technique to each term of ${\\cal G}(r,t,t')$, yields:\n\\begin{eqnarray}\n\\label{4.23}\n\\displaystyle\n{\\cal G}_r(t,t')&=&\\rho_{eq}^2 +2(J+2D)\\rho_{eq}\\int_{t}^{\\infty}dT e^{-4(J+D)T}\\Big(I_0(4DT)-\nI_1(4DT)\\Big)\\nonumber \\\\\n&+& J(J+2D)\\Biggr(\\int_{t'}^{\\infty} dT e^{-2(J+D)T}\\Big\\{I_r(2DT)+I'_r(2DT)\\Big\\}\\Biggr) \n \\nonumber \\\\ &\\times &\n\\Biggr(\\int_{t'}^{\\infty} dT e^{-2(J+D)T}\\Big\\{I_r(2DT)- I'_r(2DT)\\Big\\}\\Biggr)\\nonumber \\\\\n&+& \\frac{J(J+2D)}{4}\\Biggr(\\int_{t'}^{\\infty}dT e^{-2(J+D)T} \\frac{r}{DT}I_r (2DT)\\Biggr)^2  \\nonumber\\\\\n&-& 2(J+2D)^2 \\Biggr( \\int_{t'}^{\\infty}dT e^{-2(J+D)T}\n\\Big\\{I_r(2DT)-I'_r(2DT)\\Big\\}\\Biggr)\n \\nonumber \\\\ &\\times &\n\\Biggr(\\int_{t}^{\\infty} dT e^{-2(J+D)(2T+t')}\\Big\\{I_{r}(2D(2T+t'))-I'_{r}(2D(2T+t'))\\Big\\}\n \\Biggr)\\nonumber\\\\\n&+& (J+2D)^2\\Biggr( \\int_{t'}^{\\infty} dT \ne^{-2(J+D)T}\\frac{r}{2D(2T+t')}I_r(2DT) \\Biggr)\n \\nonumber \\\\ &\\times &\n\\Biggr(\\sum_{0\\leq n<r}\\int_{t}^{\\infty} dT e^{-2(J+D)(2T+t')}\\Big\\{2I_{2n-r+1}(2D(2T+t'))-2I_{2n-r}(2D(2T+t'))\n+ I'_{2n-r}(2D(2T+t'))\\Big\\}\\Biggr) \\nonumber \\\\\n&+& J(J+2D) \\Biggr(\\int_{t}^{\\infty} dT e^{-2(J+D)(2T+t')} \\Big(I_0(2D(2T+t'))- I_1(2D(2T+t'))\\Big)\\Biggr) \\nonumber \\\\\n& \\times & \\Biggr(\\int_{t'}^{\\infty}dT e^{-2(J+D)T} \\Big(I_0(2DT)+ I_1(2DT)\\Big)\\Biggr)\n\\nonumber \\\\ &-& \n\\frac{J(J+2D)}{2}\\Biggr(\\int_{t}^{\\infty}dT e^{-2(J+D)(2T+t')} \n\\frac{r}{D(2T+t')}I_{r}(2D(2T+t'))\\Biggr) \\Biggr(\\int_{t'}^{\\infty}dT e^{-2(J+D)T} \\frac{r}{DT}I_r(2DT)\\Biggr)\\nonumber\\\\\n&-& 2J(J+2D) \\Biggr(\\int_{t'}^{\\infty}dT e^{-2(J+D)T}\n\\Big\\{I_r (2DT)+I'_r (2DT)\\Big\\}\\Biggr) \\nonumber \\\\ &\\times&\n\\Biggr(\\int_{t}^{\\infty}dT e^{-2(J+D)(2T+t')}\\Big\\{I_{r}(2D(2T+t'))- I'_{r}(2D(2T+t'))\\Big\\}\\Biggr) \n\\nonumber \\\\ \n&+& 4(J+2D)^2 \\Biggr(\\int_{t}^{\\infty}dT e^{-2(J+D)(2T+t')} \\Big(I_0(2D(2T+t'))- I_1(2D(2T+t'))\\Big) \\Biggr)^2 \n\\nonumber \\\\\n&-& 2(J+2D)^2 \\Biggr(\\int_{t}^{\\infty}dT e^{-2(J+D)(2T+t')} \\Big\\{ I_{r}(2D(2T+t'))-I'_{r}(2D(2T+t'))\n\\Big\\}\\Biggr)^2 \\nonumber\\\\\n&+&2(J+2D)^2 \\Biggr(\\sum_{0\\leq n<r}(2n-r)\\int_{t}^{\\infty} dT\\, e^{-2(J+D)(2T+t')}\n\\frac{I_{2n-r}(2D(2T+t')) }{D(2T+t')}\\Biggr)  \\nonumber \\\\\n&\\times & \\Big(\\sum_{0\\leq n<r}\\int_{t}^{\\infty} dT\\, e^{-2(J+D)(2T+t')}\n\\{2I_{2n-r+1}(2D(2T+t')- 2I_{2n-r}(2D(2T+t'))\n+I'_{2n-r}(2D(2T+t'))\\}\\Big)\n\\end{eqnarray}\nIn the critical case some care is required when taking $J \\rightarrow 0$,\n\\begin{eqnarray}\n\\label{4.24}\n\\displaystyle\n{\\cal G}_r(t,t')&=& e^{-4D(2t+t')}\\Biggr\\{\\Big(I_0(2D(2t+t'))\\Big)^2 - \\Big(I_r(2D(2t+t'))\\Big)^2\\Biggr\\} \\nonumber\\\\ &+&\nDe^{-2D(2t+t')}\\Biggr\\{ \\int_{t'}^{\\infty} dT e^{-2DT}\\frac{r}{DT} I_{r}(2DT) \\Biggr\\} \\nonumber\\\\\n&\\times & \\Biggr\\{\\sum_{0\\leq n <r} \\Big(I_{2n-r+1}(2D(2t+t')) - I_{2n-r}(2D(2t+t'))\\Big)   \\Biggr\\} \\nonumber \\\\ &+&\ne^{-4D(2t+t')}\\Biggr\\{ \\Big(\\sum_{0\\leq n<r} I_{2n-r}(2D(2t+t'))\\Big)^2 -  \\Big(\\sum_{0\\leq n<r} I_{2n-r+1}(2D(2t+t'))\\Big)^2 \\Biggr\\}\n\\end{eqnarray}\nTo study the behavior at finite distance of the two-time correlation function ${\\cal C}_r(t,t')\\equiv{\\cal G}_r(t,t') -\\rho(t)\\rho(t')$ \n , we distinguish  the massive and the critical regime.\nWe begin with the massive case and  consider $Dt,Dt' \\gg 1$, $Jt, Jt'\\gg 1$  and $r< \\infty$. \n With help of (\\ref{A.12}-\\ref{A.15}), we have:\n\\begin{eqnarray}\n\\label{4.25.0}\n\\displaystyle\n{\\cal G}_r(t,t') &\\sim & \\rho_{eq}^2 + \\Big(1+\\frac{J}{2D}\\Big)\\rho_{eq}\\frac{e^{-4Jt}}{8Jt \\sqrt{8\\pi Dt}}\\nonumber \\\\\n{\\cal C}_r(t,t') &\\sim & - \\Big(1+\\frac{J}{2D}\\Big)\\rho_{eq} \\frac{e^{-4Jt'}}{8Jt'\\sqrt{8\\pi Dt'}}\n\\end{eqnarray}\nWhen $r < \\infty,\\,Dt$ and $Jt$ are finite and $Dt', Jt' \\gg 1$ we obtain: \n\\begin{eqnarray}\n\\label{4.25.1}\n\\displaystyle\n{\\cal G}_r(t,t') &\\sim & \\rho_{eq}^2 + \\frac{J+2D}{128J}\\frac{e^{-4Jt'}}{(Dt')^2}+ \\frac{J+2D}{4JD}\n\\frac{e^{-4J(t+t')}}{16 \\pi D \\sqrt{t'}(2t+t')^{\\frac{3}{2}}}\\nonumber \\\\\n{\\cal C}_r(t,t') &\\sim & -\\Big(1+\\frac{J}{2D}\\Big)\\rho_{eq} \\frac{e^{-4Jt'}}{8Jt' \\sqrt{8\\pi Dt'}}\n\\end{eqnarray}\nIn the critical case ($r<\\infty$), both asymptotic  behavior $Dt, Dt'\\gg 1$ and \n $Dt$ finite with $Dt'\\gg 1$ of the disconnected and connected \ncorrelation function are given by:\n\\begin{eqnarray}\n\\label{4.26}\n\\displaystyle\n{\\cal G}_r(t,t') &\\sim & \\frac{r^2 }{4\\pi D^2\n(2t+t')^2}\\Big(1+\\frac{1}{8}\\sqrt{1+\\frac{2t}{t'}}\\Big)  \\nonumber \\\\\n{\\cal C}_r(t,t') &\\sim & -\\frac{1}{8\\pi D \\sqrt{t t'}}\n\\end{eqnarray}\n\\\\\nNear the initial state (the density of particles is high), when $Dt, Dt' \\ll 1$  the decay is linear \n and independ of $r$, i.e., \n\\begin{eqnarray}\n\\label{4.28}\n\\displaystyle\n{\\cal G}_r(t,t') &\\sim & 1-4D(2t+t')  \\nonumber \\\\\n{\\cal C}_r(t,t') &\\sim & -4Dt\n\\end{eqnarray}\nWe now provide a scaling form for the two-time correlation function ${\\cal C}_r (t,t')$.\nIt is known, from the duality with Glauber's model, that the single-time correlation function obeys a \nscaling form for large-time and long-distances ($Dt$ and \n$ r \\rightarrow \\infty$ with the  ratio $r^2\/Dt$ held finite \\cite{Family,Droz}).\nThe scaling form is found as ${\\cal C}_r(t)\\sim r^{-2} f(r^{2}\/4Dt)$, where the exponent $-2$ \nis believed to be universal \\cite{Droz}.\nWe further assume the long-time and large-distance scaling form  \n${\\cal C}_r(t,t')\\sim r^{-y} h(u,v)$, where $r$, $Dt$, $Dt' \\rightarrow \\infty $ with\n$u\\equiv r^2\/4Dt$ and $v\\equiv r^2\/4Dt'$ held finite and arrive at\n\\begin{eqnarray}\n\\label{4.29}\n\\displaystyle\n{\\cal C}_r(t,t') &\\sim & \\frac{1}{\\pi r^2}\\Biggr\\{K^{\\frac{3}{2}}e^{-K}\n\\Big( \\sqrt{\\pi}\\, {\\rm erf}{\\sqrt{v}} - 2\\sqrt{K}\\Big) +K\\Big(1-e^{-2K}\\Big)\n-\\frac{\\sqrt{uv}}{2}\\Biggr\\},\n\\end{eqnarray}\nwhere  $0< K\\equiv \\frac{uv}{u+2v} < \\infty$.\n\\\\\nAt this point, it is appropriate to review  what we have achieved so far: \nwe have been able to reformulate the problem of the evaluation of the \nmulti-point\ncorrelation functions in a language that parallels the field theoretical one.\nThis allows us to compute in an efficient and systematic way physical quantities\nof interest despite some technical differences to the standard approach. \nWhile \nthis paper deals with a free ``field theory'' of pseudo-fermions, it is tempting\n to apply the same formalism to the multi-species case\nwhere two- or multi-body interactions arise. The latter however will be\ninvestigated in a future work \nby pertubative renormalization group techniques, as no exact\nsolutions are available. \n\\section{Concluding remarks}\nWe have studied three different approaches to the\nproblem of  diffusion-annihilation of\nclassical hard-core particles moving on\na one-dimensional ring. \nThough Lushnikov's contributions to the problem are genuine and undisputable, \nwe have\nshown how an extension of his generating function method to evaluate the\ntwo-point correlation function can be cumbersome in\npractice, even in the simplest case available of a single-species. \nWe have seen that it is advantageous to apply a\n generalized Bogoliubov transformation first used by Grynberg et {\\it al.}\n\\cite{Stinch1,Stinch2} in a different context. The evolution operator \ncan be diagonalized, i.e., expressed as a quadratic form of two operators \nthat are not adjoint of each other. Despite this fact\nthe formalism resembles the standard one and\nappears as a powerful tool. Indeed, we were able\nto compute for the first time the full one-time and  two-time correlation functions \n for an initially fully occupied lattice\n(other initial conditions can also be studied) in the presence of a finite\nsource. We derived a scaling form for the two-time correlation function.\nWe used the results of the first section (algebraic decay in the \ncritical case and exponential in the presence of source) to check the\nasymptotic behavior of the density and two-point correlation function.\nWe discovered that while in the absence of source, the modes at long wave \nlengths\nfully control the long-time asymptotics of the density and correlation\nfunctions, in the presence of a finite source all modes contribute. This means\nthat in the general case, the long wave-length approximations that were so\nsuccessfull in strongly\ncorrelated systems (such as bosonization or conformal field theory techniques) \ndo not work for the problem at hand.\nMoreover, the idea of exploiting the integrability of some spin Hamiltonians \non which the\nmulti-species Liouvillian maps, might turn out more elusive than expected.\nIn view of the above remark, it would seem extremely difficult if not\nimpossible to\nextract from the exact Bethe-ansatz solution of the non-hermitian\nspin Hamiltonian the relevant matrix\nelements that in turn allow the evaluation of correlation functions \\cite{Droz}.   \nWe propose to tackle the multi-species problem in terms of fermion\nfunctionals, the main difficulties arising from the two- and\/or multi-body\ninteractions occuring in the process of mapping classical particles to fermions. \nWe will illustrate the power of the formalism in a\nforthcoming paper, where we will apply the renormalization group scheme.\nBesides the obvious advantage of formulating \nthe problem in a\nfield theoretical language (perturbation theory,...), the method is \napplicable to arbitrary densities of particles, and thus\ncomplements the approach developed by Cardy and collaborators \\cite{Cardy1,Cardy2,Cardy3,Lee}.\nIn higher dimensions, however, Fermi statistics requires the introduction of a\ngauge field that is strongly coupled to the fermions. We also intend to explore\nthis line of research in the future.  \n\\section*{ACKNOWLEDGMENTS}\nWe thank M. Droz, L. Frachebourg, Ch. Gruber, S. Gyger, Ph. Nozi\\`{e}res and T.M. Rice for stimulating\ndiscussions.\nWe are grateful to the referee for drawing our attention to references\n\\cite{Stinch3,Bray}.\nThe support of the Swiss National Fonds is gratefully acknowledged.\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{introduction}\n\n\n\nThe experimental observation of charge-parity ({\\ensuremath{C\\!P}}\\xspace) violation in the neutral-kaon system in 1964 by Cronin, Fitch {\\it et al.}~\\cite{Cronin1964} caused a real shock in the physics community. Just a few years before, Landau presented a convincing theory showing {\\ensuremath{C\\!P}}\\xspace conservation to play  a key role in understanding particle-antiparticle symmetry~\\cite{Landau}.  At that time, no one had disagreed with this fundamental idea and, as a consequence, there was no theory predicting the  {\\ensuremath{C\\!P}}\\xspace symmetry breaking.  Among several arguments for {\\ensuremath{C\\!P}}\\xspace to be a good symmetry,  a compelling one was that, assuming {\\ensuremath{C\\!PT}}\\xspace symmetry to be exact (a consequence of Lorentz invariance),  the reversibility of time would naturally lead to {\\ensuremath{C\\!P}}\\xspace conservation.  The observation of {\\ensuremath{C\\!P}}\\xspace violation meant, for the first time in physics, the irreversibility of some processes at the fundamental level.\n\n\n\nIt was only in 1973 that Kobayashi and Maskawa proposed the  nowadays well-accepted description~\\cite{Kobayashi1973} for the observed {\\ensuremath{C\\!P}}\\xspace violation in the kaon system. They included a third  quark family, which extended  the $2\\times 2$ Cabibbo mixing matrix \\cite{Cabibbo:1963yz} to a $3\\times 3$ matrix,  the now called Cabibbo-Kobayashi-Maskawa (CKM) matrix, allowing the presence of a complex phase in the hadronic sector of the weak interaction. The two new quarks, beauty and top, were experimentally observed in 1977 \\cite{Herb1977} and 1995 \\cite{D01995,CDF1995}, respectively, and finally {\\ensuremath{C\\!P}}\\xspace violation was observed in {\\ensuremath{\\PB}}\\xspace decays, as predicted, in 2001~\\cite{Babar2001,Belle2001}. Kobayashi and Maskawa received the Nobel Prize in 2008 for the confirmation of their {\\ensuremath{C\\!P}}\\xspace violation mechanism within the Standard Model (SM).\n\nA crucial question remains unanswered until today: what is the dynamical origin of the quark mixing matrix and, in particular, the weak phase? \nUnlike parity, whose breaking for all fermions comes from the universal vector--axial  nature of the weak currents, {\\ensuremath{C\\!P}}\\xspace breaking in not the same for all quarks.\n\n\nFrom its observation, {\\ensuremath{C\\!P}}\\xspace violation represented a  new paradigm, opening the path for the emergence of ideas and concepts. One of them was introduced by Sakharov in 1967 \\cite{Sakharov1967}, stating {\\ensuremath{C\\!P}}\\xspace violation as one of the necessary conditions for baryogenesis. Among others, there was also the notion of ``mirror universe'' where {\\ensuremath{C\\!P}}\\xspace violation would be the door between our universe and its {\\ensuremath{C\\!P}}\\xspace mirror version \\cite{Okun}, with several implications \\cite{Okun2006}. According to some authors, this universe would be the source of the Dark Matter \\cite{Ibe2019}. \n\n\nEven a few years before the CKM ansatz, a different approach  was  proposed by Wolfenstein \\cite{Wolfenstein1964} and required a new interaction that was called {\\it superweak}. It was able to explain the observed {\\ensuremath{C\\!P}}\\xspace asymmetry associated to oscillation in the neutral-kaon system  but did not allow for {\\ensuremath{C\\!P}}\\xspace violation coming from the decay process itself  -- what is called {\\it direct} {\\ensuremath{C\\!P}}\\xspace violation. It was only in the early 2000s  that this idea was definitively ruled out by the observation of  direct {\\ensuremath{C\\!P}}\\xspace violation in kaons by the KTeV \\cite{KTeV_1999,KTeV_2003} and NA48 \\cite{NA48_2001,NA48_2002} collaborations, through the observable $\\Re(\\epsilon'\/\\epsilon)$ which, measured to be $\\sim10^{-3}$, implies  a direct {\\ensuremath{C\\!P}}\\xspace asymmetry of $\\sim10^{-5}$ \\cite{Sozzi2003}.\n\nGiven the CKM matrix hierarchy the natural sectors to observe {\\ensuremath{C\\!P}}\\xspace violation are processes involving strange and beauty hadrons. This can be better seen through the Wolfenstein parameterisation~\\cite{Wolfenstein_CKMparam_1983}, which describes the CKM elements in powers of $\\lambda \\approx 0.223$~\\cite{PDG2019}. Up to ${\\cal O}(\\lambda^3)$ it is written as\n\\begin{equation}\nV_{\\rm CKM} = \\left( \\begin{array}{c c c} V_{{\\ensuremath{\\Pu}}\\xspace{\\ensuremath{\\Pd}}\\xspace} & V_{{\\ensuremath{\\Pu}}\\xspace{\\ensuremath{\\Ps}}\\xspace} & V_{{\\ensuremath{\\Pu}}\\xspace{\\ensuremath{\\Pb}}\\xspace} \\\\\n                            V_{{\\ensuremath{\\Pc}}\\xspace{\\ensuremath{\\Pd}}\\xspace} & V_{{\\ensuremath{\\Pc}}\\xspace{\\ensuremath{\\Ps}}\\xspace} & V_{{\\ensuremath{\\Pc}}\\xspace{\\ensuremath{\\Pb}}\\xspace} \\\\\n                            V_{{\\ensuremath{\\Pt}}\\xspace{\\ensuremath{\\Pd}}\\xspace} & V_{{\\ensuremath{\\Pt}}\\xspace{\\ensuremath{\\Ps}}\\xspace} & V_{{\\ensuremath{\\Pt}}\\xspace{\\ensuremath{\\Pb}}\\xspace} \\end{array} \\right) = \\left( \\begin{array}{ccc}\n1-\\lambda^2\/2 &  \\lambda         &  A\\lambda^3(\\rho -i\\eta) \\\\\n-\\lambda      &   1-\\lambda^2\/2  & A\\lambda^2 \\\\\nA\\lambda^3(1-\\rho -i\\eta) & -A\\lambda^2 & 1\n\\end{array}\\right) ~. \n\\label{CKM}\n\\end{equation}\nThe other three parameters are $A$, $\\rho$ and $\\eta$. The latter is responsible for the complex nature of the CKM matrix and appears at order $\\lambda^3$ in the matrix elements $V^{\\phantom{*}}_{{\\ensuremath{\\Pt}}\\xspace{\\ensuremath{\\Pd}}\\xspace}$ and $V^{\\phantom{*}}_{{\\ensuremath{\\Pu}}\\xspace{\\ensuremath{\\Pb}}\\xspace}$. The CKM matrix is almost diagonal, breaking at the $\\lambda$ level, and almost real, breaking at the $\\lambda^3$ level. Being unitary, it leads to orthogonality relations of the type $\\sum_{k=u,c,t} V^{\\phantom{*}}_{ij}V^*_{ik} = \\delta_{jk}$ ($j,k = d,s,b$). The well-known {\\it Unitarity Triangle}, constructed from the relation $V^{\\phantom{*}}_{{\\ensuremath{\\Pu}}\\xspace{\\ensuremath{\\Pd}}\\xspace}V^*_{{\\ensuremath{\\Pu}}\\xspace{\\ensuremath{\\Pb}}\\xspace}+ V^{\\phantom{*}}_{{\\ensuremath{\\Pc}}\\xspace{\\ensuremath{\\Pd}}\\xspace}V^*_{{\\ensuremath{\\Pc}}\\xspace{\\ensuremath{\\Pb}}\\xspace}+V^{\\phantom{*}}_{{\\ensuremath{\\Pt}}\\xspace{\\ensuremath{\\Pd}}\\xspace}V^*_{{\\ensuremath{\\Pt}}\\xspace{\\ensuremath{\\Pb}}\\xspace}=0$, defines three angles in the complex plane: $\\alpha \\equiv \\arg\\left(-\\frac{V^{\\phantom{*}}_{{\\ensuremath{\\Pt}}\\xspace{\\ensuremath{\\Pd}}\\xspace}V^*_{{\\ensuremath{\\Pt}}\\xspace{\\ensuremath{\\Pb}}\\xspace}}{V^{\\phantom{*}}_{{\\ensuremath{\\Pu}}\\xspace{\\ensuremath{\\Pd}}\\xspace}V^*_{{\\ensuremath{\\Pu}}\\xspace{\\ensuremath{\\Pb}}\\xspace}}\\right)$, \n $\\beta \\equiv \\arg\\left(-\\frac{V^{\\phantom{*}}_{{\\ensuremath{\\Pc}}\\xspace{\\ensuremath{\\Pd}}\\xspace}V^*_{{\\ensuremath{\\Pc}}\\xspace{\\ensuremath{\\Pb}}\\xspace}}{V^{\\phantom{*}}_{{\\ensuremath{\\Pt}}\\xspace{\\ensuremath{\\Pd}}\\xspace}V^*_{{\\ensuremath{\\Pt}}\\xspace{\\ensuremath{\\Pb}}\\xspace}}\\right)$ and  $\\gamma \\equiv \\arg\\left(-\\frac{V^{\\phantom{*}}_{{\\ensuremath{\\Pu}}\\xspace{\\ensuremath{\\Pd}}\\xspace}V^*_{{\\ensuremath{\\Pu}}\\xspace{\\ensuremath{\\Pb}}\\xspace}}{V^{\\phantom{*}}_{{\\ensuremath{\\Pc}}\\xspace{\\ensuremath{\\Pd}}\\xspace}V^*_{{\\ensuremath{\\Pc}}\\xspace{\\ensuremath{\\Pb}}\\xspace}}\\right)$. Then, up to ${\\cal O}(\\lambda^3)$, $V^{\\phantom{*}}_{{\\ensuremath{\\Pt}}\\xspace{\\ensuremath{\\Pd}}\\xspace} = |V^{\\phantom{*}}_{{\\ensuremath{\\Pt}}\\xspace{\\ensuremath{\\Pd}}\\xspace}|e^{-i\\beta}$ and $V_{{\\ensuremath{\\Pu}}\\xspace{\\ensuremath{\\Pb}}\\xspace} = |V^{\\phantom{*}}_{{\\ensuremath{\\Pu}}\\xspace{\\ensuremath{\\Pb}}\\xspace}|e^{-i\\gamma}$.\n\n\nSince the top quark does not hadronise, the manifestation of $\\arg(V_{{\\ensuremath{\\Pt}}\\xspace{\\ensuremath{\\Pd}}\\xspace})$ at short distances occurs through box or loop (``penguin'' type) diagrams in ${\\ensuremath{\\Ps}}\\xspace$- and ${\\ensuremath{\\Pb}}\\xspace$-hadron processes.  Given the top mass is very large, \\footnote{The top mass was predicted to be large after the the observation of ${\\ensuremath{\\B^0}}\\xspace-{\\ensuremath{\\Bbar{}^0}}\\xspace$ oscillations \\cite{UA1_Boscill_1987,ARGUS_Boscill_1987,CLEO_Boscill_1989}. Its contribution in the one-loop process is important due to the non-decoupling nature of the SM, manifesting a violation of the Applequist--Carazzone theorem \\cite{AppelquistCarazzone}.} its contribution to these diagrams is much more important than those mediated by up and charm quarks, which also almost cancel each other due to the GIM mechanism~\\cite{GIM}. From box-type diagrams,  {\\it indirect} {\\ensuremath{C\\!P}}\\xspace violation\\footnote{The term {\\it indirect} {\\ensuremath{C\\!P}}\\xspace violation here refers to processes where {\\ensuremath{C\\!P}}\\xspace violation \\cite{PDG2019} requires mixing to occur, and it can arise between the interference of particle and antiparticle decaying to a common final state, or it can appear through the interference between mixing and decay.}   may arise in  neutral meson processes. On the other hand the effect of $\\arg(V_{{\\ensuremath{\\Pu}}\\xspace{\\ensuremath{\\Pb}}\\xspace})=-\\gamma$ can also take place at tree level for $b$-hadron decays. \nIt can also be read from Eq.~\\ref{CKM} that the possible effects of {\\ensuremath{C\\!P}}\\xspace violation in charm are expected to be tiny, potentially arising due to  $\\arg(V_{{\\ensuremath{\\Pu}}\\xspace{\\ensuremath{\\Pb}}\\xspace})$ in box or loop diagrams or, at tree level, via a $\\lambda^5$ correction in $V_{{\\ensuremath{\\Pc}}\\xspace{\\ensuremath{\\Pd}}\\xspace}$.\n\nIt is interesting to note the time gap of almost forty years between the discovery of {\\ensuremath{C\\!P}}\\xspace violation in mixing and direct {\\ensuremath{C\\!P}}\\xspace violation for kaon decays --- a consequence of the tiny effect of the latter in the kaon system.  A similar situation, but not so drastic, has occurred in {\\ensuremath{\\PB}}\\xspace decays, where {\\ensuremath{C\\!P}}\\xspace asymmetry in mixing  was observed in ${\\ensuremath{\\B^0}}\\xspace \\ensuremath{\\rightarrow}\\xspace {\\ensuremath{{\\PJ\\mskip -3mu\/\\mskip -2mu\\Ppsi\\mskip 2mu}}}\\xspace{\\ensuremath{\\kaon^0_{\\rm\\scriptscriptstyle S}}}\\xspace$  by the BaBar \\cite{Babar2001} and Belle \\cite{Belle2001} collaborations while  direct {\\ensuremath{C\\!P}}\\xspace asymmetry was observed (with more than 5 standard deviations) more than ten years later \\cite{BaBar2012, LHCb-PAPER-2013-018}.  Interestingly, in the charm sector direct {\\ensuremath{C\\!P}}\\xspace violation has very recently been observed by the LHCb collaboration~\\cite{LHCb-PAPER-2019-006} --- and yet there is no sign for indirect {\\ensuremath{C\\!P}}\\xspace violation associated to mixing in {\\ensuremath{\\D^0}}\\xspace decays \\cite{LHCb-PAPER-2017-046,LHCb-PAPER-2019-001,LHCb-PAPER-2019-032}.\n\nFrom the theory side, there is an even higher challenge regarding estimates for direct {\\ensuremath{C\\!P}}\\xspace violation asymmetries. They depend crucially on the size of the strong phases involved in the decays~\\cite{Bigi_Book}.\nIn fact, since the first theoretical proposal for the study of direct {\\ensuremath{C\\!P}}\\xspace violation in an inclusive way, by Bander, Silverman and Soni~\\cite{BSS} and known as the  BSS model, hundreds of papers were published, based on different mechanisms, aiming to  understand the importance of strong interaction phases, mainly in exclusive {\\ensuremath{\\PB}}\\xspace-meson decays. There is fair consensus that the strong phases coming from short-distance effects should be small. If these are assumed to be  the primary source of strong phases, the level of direct {\\ensuremath{C\\!P}}\\xspace violation should be small. On the other hand, if one  considers that long-distance, final-state interaction (FSI) processes in non-leptonic decays introduce significant strong phase shifts, potentially large  direct {\\ensuremath{C\\!P}}\\xspace violation effects should be expected.\n         \n                  Many good reviews exist addressing  {\\ensuremath{C\\!P}}\\xspace violation in hadrons  (see, for example,   \\cite{TimVava, EricaAlberto, Bevan2014,Artuso_CPBs_2016, Hou_Review_2017}). This work is dedicated exclusively to discussing direct {\\ensuremath{C\\!P}}\\xspace violation in ${\\ensuremath{\\Pb}}\\xspace$-  and ${\\ensuremath{\\Pc}}\\xspace$-hadron systems. New, interesting, and even unexpected results on direct {\\ensuremath{C\\!P}}\\xspace violation emerged in the recent years which, confronting with the development of new insights from the theory side,  motivate  this review. \nFor the {\\ensuremath{\\PB}}\\xspace meson sector, we concentrate the discussion only on well-established results on direct {\\ensuremath{C\\!P}}\\xspace violation and their implications. For $b$ baryons and charm hadrons  the overall {\\ensuremath{C\\!P}}\\xspace violation picture is just in its initial stages, and we address the current state-of-the-art.  BaBar, Belle and a few other experiments have contributed significantly in this area, nevertheless the LHCb collaboration, with unprecedented samples of beauty and charm hadrons, is currently the large source of new experimental results. Belle II  and the LHCb Upgrade I  will soon enter the game. \n                  \n                  This review is organised as follows. In Sect.~\\ref{chapter2_CP_CPT}, we discuss the fundamental ideas regarding direct {\\ensuremath{C\\!P}}\\xspace violation and call attention on how {\\ensuremath{C\\!PT}}\\xspace conservation enters  as a constraint  for decay modes with the same final-state flavour content. In Sect.~\\ref{chapter3}  the major experiments and main strategies  for measuring direct {\\ensuremath{C\\!P}}\\xspace violation observables are discussed. Particular emphasis is given on the importance of multi-body decays, for which {\\ensuremath{C\\!P}}\\xspace violation effects can appear, and  be enhanced, in specific regions of their phase space.  Direct {\\ensuremath{C\\!P}}\\xspace violation in charmless $b$-hadron decays  is presented in Sect.~\\ref{chapter4}, followed by Sect.~\\ref{chapter5} where a complementary discussion on double-charm contributions to the charmless final states is considered. Section~\\ref{chapter6} is dedicated to {\\ensuremath{\\PB}}\\xspace decays to open charm, which represent the ideal environment for the measurement of $\\gamma$. Section~\\ref{chapter7} addresses the charm sector, contextualising the field after the first observation of direct {\\ensuremath{C\\!P}}\\xspace violation  and discussing the  current status and prospects for many other modes. Finally, we present our conclusions in Sect.~\\ref{chapter8}.\n                  \n                        \n\n      \n \\section{{\\ensuremath{C\\!P}}\\xspace violation in decays and {\\ensuremath{C\\!PT}}\\xspace symmetry}\n\\label{chapter2_CP_CPT}\n\n{\\ensuremath{C\\!P}}\\xspace violation in decays -- or {\\it direct} {\\ensuremath{C\\!P}}\\xspace violation -- is manifested\nwhen the decay width of a particle $P$ to a given final state $f$ differs from the conjugate process:\n\\begin{equation}\n\\label{width_difference}\n\\Delta \\Gamma(P\\ensuremath{\\rightarrow}\\xspace f) \\equiv   \\Gamma(P\\ensuremath{\\rightarrow}\\xspace f) - \\Gamma(\\bar P\\ensuremath{\\rightarrow}\\xspace \\bar f)  \\ne 0~.\n\\end{equation}\n\nIn terms of the decay amplitudes, \n\\begin{equation}\n\\label{ampl_difference}\n|{\\cal A} (P\\ensuremath{\\rightarrow}\\xspace f) |^2 - |{\\cal A} (\\bar P\\ensuremath{\\rightarrow}\\xspace \\bar f) |^2 \\ne 0~.\n\\end{equation}\n\n\n\n\nThe origin of {\\ensuremath{C\\!P}}\\xspace violation relies on the presence of complex terms in the Lagrangian density, which translate to phases in transition amplitudes. If these phases are {\\ensuremath{C\\!P}}\\xspace-odd -- i.e. change sign under a {\\ensuremath{C\\!P}}\\xspace transformation -- {\\ensuremath{C\\!P}}\\xspace violation can appear. Clearly, for phases to become   observables, there should be at least two possible interfering paths producing a final state $f$. \n\nThe usual example \\cite{Bigi_Book, Sozzi_Book, Branco_Book} is to consider two decay amplitudes producing a final state $f$ such that\n\\begin{equation}\n{\\cal A} (P\\ensuremath{\\rightarrow}\\xspace f) = |{\\cal A}_1| e^{i\\phi_1} + |{\\cal A}_2| e^{i\\phi_2}~,\n\\label{eq-amplweakphase}\n\\end{equation}\nwhere  ${\\cal A}_ 1$ and ${\\cal A}_2$ have  different phases $\\phi_1$ and $\\phi_2$. Considering both $\\phi_1$ and $\\phi_2$ are CP-odd phases, the antiparticle process is given by \n${\\cal A} (\\bar P\\ensuremath{\\rightarrow}\\xspace\\bar  f)  = |{\\cal A}_1| e^{-i\\phi_1} + |{\\cal A}_2| e^{-i\\phi_2}$. It is easy to \nshow that the decay rate difference, Eq.~\\ref{ampl_difference},  gives a null result, and {\\ensuremath{C\\!P}}\\xspace violation is not observed. In contrast, if the decay amplitude is given by \n\\begin{equation}\n{\\cal A} (P\\ensuremath{\\rightarrow}\\xspace f) = |{\\cal A}_1| e^{i\\phi_1} e^{i\\delta_1}+ |{\\cal A}_2| e^{i\\phi_2}e^{i\\delta_2}~,\n\\label{eq-amplphases}\n\\end{equation}\nwhere $\\delta_1$ and $\\delta_2$ are {\\ensuremath{C\\!P}}\\xspace-even phases, then the decay rate difference is\n\\begin{equation}\n|{\\cal A} (P\\ensuremath{\\rightarrow}\\xspace f) |^2 - |{\\cal A} (\\bar P\\ensuremath{\\rightarrow}\\xspace \\bar f) |^2= 2|{\\cal A}_1| |{\\cal A}_2| \\sin(\\phi_1 -\\phi_2)\\sin(\\delta_1-\\delta_2).\n\\label{eq-diffamplsquare}\n\\end{equation}\nUsually the observable is the {\\ensuremath{C\\!P}}\\xspace asymmetry defined as \n\\begin{equation}\nA_{\\ensuremath{C\\!P}}\\xspace = \\frac {|{\\cal A} (P\\ensuremath{\\rightarrow}\\xspace f) |^2 - |{\\cal A} (\\bar P\\ensuremath{\\rightarrow}\\xspace \\bar f) |^2}{|{\\cal A} (P\\ensuremath{\\rightarrow}\\xspace f) |^2 + |{\\cal A} (\\bar P\\ensuremath{\\rightarrow}\\xspace \\bar f) |^2}\n\\label{CPasym_def}\n\\end{equation}  that for the amplitude given in Eq.~\\ref{eq-amplphases} it is given by \n\\begin{equation}\nA_{\\ensuremath{C\\!P}}\\xspace = \\frac{2|{\\cal A}_2\/{\\cal A}_1|\\sin(\\delta_1-\\delta_2)\\sin(\\phi_1-\\phi_2)}{1 + |{\\cal A}_2\/{\\cal A}_1|^2 + 2 |{\\cal A}_2\/{\\cal A}_1|\\cos(\\delta_1-\\delta_2)\\cos(\\phi_1-\\phi_2)}~.\n\\label{ACP_sindelta_sinphi}\n\\end{equation}\nTherefore both {\\ensuremath{C\\!P}}\\xspace-odd and {\\ensuremath{C\\!P}}\\xspace-even phases are necessary for {\\ensuremath{C\\!P}}\\xspace violation to be observed. Although this is a simple academic example, the conclusion in quite general.  Moreover, the size of the  {\\ensuremath{C\\!P}}\\xspace asymmetry depends  not only on the phase differences but also on the relative size of the two amplitudes  -- if, for instance, $|{\\cal A}_1| \\gg |{\\cal A}_2|$ the expected {\\ensuremath{C\\!P}}\\xspace asymmetry would be small even with large phase differences.\nWithin the context of the SM, {\\ensuremath{C\\!P}}\\xspace-odd phases can only come from weak processes, while {\\ensuremath{C\\!P}}\\xspace-even phases may appear through strong (or electromagnetic) interactions. For this reason, {\\ensuremath{C\\!P}}\\xspace-odd and {\\ensuremath{C\\!P}}\\xspace-even phases are usually called {\\it weak} and {\\it strong}  phases, respectively. \n\nFor a neutral meson decaying to {\\ensuremath{C\\!P}}\\xspace eigenstates like ${\\ensuremath{\\pion^+}}\\xspace{\\ensuremath{\\pion^-}}\\xspace$, ${\\ensuremath{\\kaon^+}}\\xspace{\\ensuremath{\\kaon^-}}\\xspace$, ${\\ensuremath{\\D^0}}\\xspace{\\ensuremath{\\Dbar{}^0}}\\xspace$ (or final states accessible for both particle and antiparticle, such as ${\\ensuremath{\\kaon^0_{\\rm\\scriptscriptstyle S}}}\\xspace{\\ensuremath{\\pion^0}}\\xspace$) it is necessary to disentangle direct {\\ensuremath{C\\!P}}\\xspace violation and  {\\ensuremath{C\\!P}}\\xspace violation from oscillation and in this case the time-dependent {\\ensuremath{C\\!P}}\\xspace asymmetry, arising from the so-called {\\it master equations} \\cite{Bigi_Book} need to be used. Assuming {\\ensuremath{C\\!PT}}\\xspace invariance, for a $P^0-\\bar P^0 $ system with a mass difference $\\Delta M$ between the mass eigenstates (but no width difference), we have\n\\begin{equation}\nA_{\\ensuremath{C\\!P}}\\xspace(t) = \\frac {|{\\cal A} (P^0(t)\\ensuremath{\\rightarrow}\\xspace f) |^2 - |{\\cal A}(\\bar P^0(t)\\ensuremath{\\rightarrow}\\xspace \\bar f) |^2}{|{\\cal A} (P^0(t)\\ensuremath{\\rightarrow}\\xspace f) |^2 + |{\\cal A} (\\bar P^0(t)\\ensuremath{\\rightarrow}\\xspace \\bar f) |^2} = {\\cal S}_f \\sin(\\Delta M t) - {\\cal C}_f \\cos(\\Delta M  t)\n\\label{CPasym_time}\n\\end{equation} \nwhere $t$ is the decay time of $P$ decaying to $f$.\nThe ${\\cal C}_f$ and ${\\cal S}_f$ observables are sensitive to direct and indirect {\\ensuremath{C\\!P}}\\xspace violation, respectively, with $0 \\le {\\cal S}_f^2 +{\\cal C}_f^2 \\le 1$ \\cite{Bevan2014}.  Here we are only interested in ${\\cal C}_f$, which is given by\n\\begin{equation}\n{\\cal C}_f = \\frac {|{\\cal A} (P^0\\ensuremath{\\rightarrow}\\xspace f) |^2 - |{\\cal A} (\\bar P^0\\ensuremath{\\rightarrow}\\xspace f) |^2}{|{\\cal A} (P^0\\ensuremath{\\rightarrow}\\xspace f) |^2 + |{\\cal A} (\\bar P^0\\ensuremath{\\rightarrow}\\xspace  f) |^2}.\n\\label{CPasym_C}\n\\end{equation} \n\nGenerally, while the origin of the weak phase can be pinned down at the quark level to the CKM matrix elements, making it predictable for which decays {\\ensuremath{C\\!P}}\\xspace violation could {\\it potentially} be manifested, the size of the direct {\\ensuremath{C\\!P}}\\xspace asymmetry is usually hard to estimate due in part to the rather limited knowledge of the strong phases. \nThere are two possible sources of strong phases: one associated with the short-distance penguin contribution at quark level and  another coming from final-state interactions, e.g. hadronic rescatterings in non-leptonic decays. An overview of these mechanisms  is given below. \n\n\n\\subsection{Hadronic strong phases and {\\ensuremath{C\\!PT}}\\xspace constraints on {\\ensuremath{C\\!P}}\\xspace violation}\n\\label{CPT}\n\n\nFor a particle decaying only weakly, the initial state is stable under strong (and electromagnetic) interactions. On the other hand, following the weak transition the final state is the potential result of strong processes associating states with the same  quantum numbers. Strong phases are the result of these final-state interactions. \nFor instance, the total amplitude of the decay of $P$ to a given final state $f$ may include contributions as $P\\ensuremath{\\rightarrow}\\xspace f_i\\ensuremath{\\rightarrow}\\xspace f$ ($i = 1,2,...,n$) where $P\\ensuremath{\\rightarrow}\\xspace f_i$ incorporates the weak transition and $f_i\\ensuremath{\\rightarrow}\\xspace f$ the strong processes (elastic or inelastic scatterings):\n\\begin{equation}\n{\\cal A} (P\\ensuremath{\\rightarrow}\\xspace f) = {\\cal A}_1(P\\ensuremath{\\rightarrow}\\xspace f_1){\\cal A}(f_1\\ensuremath{\\rightarrow}\\xspace f) + {\\cal A}_2(P\\ensuremath{\\rightarrow}\\xspace f_2){\\cal A}(f_2\\ensuremath{\\rightarrow}\\xspace f)+ .....+ {\\cal A}_n(P\\ensuremath{\\rightarrow}\\xspace f_n){\\cal A}(f_n\\ensuremath{\\rightarrow}\\xspace f)~.\n\\label{eq-f1f2}\n\\end{equation}\nAt this point, it is interesting to discuss {\\ensuremath{C\\!P}}\\xspace violation in the context of {\\ensuremath{C\\!PT}}\\xspace constraints, as follows.\n\n\nThe {\\ensuremath{C\\!PT}}\\xspace symmetry, expected to be an exact symmetry in field theories, establishes that the lifetimes -- thus the total decay widths -- of a particle and its antiparticle are the same. On the other hand, {\\ensuremath{C\\!P}}\\xspace violation allows partial decay widths to be different.  Clearly, preserving the equality of total widths for particle and antiparticle while allowing partial widths to be different requires a  ``communication'' between the different decay modes, and this can only happen within modes that have the same flavour quantum numbers. \n\n\nThus, and in accordance to what was discussed previously, final-state interactions, connecting states via strong (or electromagnetic) interactions, provide the natural strong phases for {\\ensuremath{C\\!P}}\\xspace violation to be observable, and are also the key ingredient to allow the preservation of  {\\ensuremath{C\\!PT}}\\xspace symmetry \\cite{Bigi_Book, Sozzi_Book, Branco_Book,Marshak_Book}.\n\nThe constraint from {\\ensuremath{C\\!PT}}\\xspace can become more apparent by considering the arguments from Wolfenstein in \\cite{Wolfenstein1990} (see also \\cite{Bigi_Book} from which  the notation below is partly adopted). Suppose that particle $P$ can decay to only two final states $f_\\alpha$ and $f_\\beta$ with the same conserved  quantum numbers. The states $f_\\alpha$ and $f_\\beta$ can be connected through strong interactions, meaning they can rescatter into each other. This interaction may be described by a scattering matrix ${\\cal S}_{\\rm strong}$ given by\n\\begin{equation}\n\\label{Smatrix_alphabeta}\n{\\cal S}_{\\rm strong} = \\left( \\begin{array}{cc} e^{2i\\delta_\\alpha} & i{\\cal T}e^{i(\\delta_\\alpha+ \\delta_\\beta)} \\\\  i{\\cal T}e^{i(\\delta_\\alpha+ \\delta_\\beta)} & e^{2i\\delta_\\beta} \\end{array} \\right), \n\\end{equation}\nwhere ${\\delta_\\alpha}$ and ${\\delta_\\beta}$ are the elastic scattering phases, ${\\cal T}$ is assumed to be small so as to be treated only to first order, ${\\cal S}_{\\rm strong}$ is unitary to that order and symmetrical  to ensure time reversal symmetry. The two states $f_\\alpha$ and $f_\\beta$ form a complete set of states given their flavour content, and using  ${\\cal A}_{\\alpha(\\beta)}(P\\ensuremath{\\rightarrow}\\xspace f_{\\alpha(\\beta)}) = e^{i\\delta_{\\alpha(\\beta)}} A_{\\alpha(\\beta)}$ it can be shown that \\cite{Bigi_Book} \n\\begin{eqnarray}\n{\\cal A}(P\\ensuremath{\\rightarrow}\\xspace f_\\alpha)& = & e^{i\\delta_\\alpha}\\left( A_\\alpha+ i {\\cal T}  A_\\beta \\right)  ~, \\nonumber \\\\\n{\\cal A} (\\bar P \\ensuremath{\\rightarrow}\\xspace \\bar f_\\alpha) & =&  e^{i\\delta_\\alpha}\\left(  A^*_\\alpha+ i {\\cal T} A^*_\\beta \\right) ~, \n\\end{eqnarray}\nwhere  $A_{\\alpha(\\beta)}$ is defined through ${\\cal A}_{\\alpha(\\beta)}(P\\ensuremath{\\rightarrow}\\xspace f_{\\alpha(\\beta)}) = e^{i\\delta_{\\alpha(\\beta)}} A_{\\alpha(\\beta)}$. The second equation above follows from imposing {\\ensuremath{C\\!PT}}\\xspace invariance. Analogous equations are given for ${\\cal A}(P\\ensuremath{\\rightarrow}\\xspace f_\\beta)$ and $ {\\cal A}(\\bar P\\ensuremath{\\rightarrow}\\xspace \\bar f_\\beta)$. So, it can be seen that \n\\begin{eqnarray}\n|{\\cal A}(\\bar P\\ensuremath{\\rightarrow}\\xspace \\bar f_\\alpha)|^2 - |{\\cal A}( P\\ensuremath{\\rightarrow}\\xspace f_\\alpha)|^2& = & 2{\\cal T}\\Im ({\\cal A}_\\alpha^* {\\cal A}_\\beta ) ~, \\nonumber \\\\\n|{\\cal A}(\\bar P\\ensuremath{\\rightarrow}\\xspace \\bar f_\\beta)|^2 -|{\\cal A}( P\\ensuremath{\\rightarrow}\\xspace f_\\beta)| ^2& = & 2{\\cal T}\\Im ({\\cal A}_\\beta^* {\\cal A}_\\alpha )\n\\end{eqnarray}\nwhich leads to $\\Delta\\Gamma(\\bar P\\ensuremath{\\rightarrow}\\xspace \\bar f_\\alpha) = - \\Delta\\Gamma((\\bar P\\ensuremath{\\rightarrow}\\xspace \\bar f_\\beta)$ or, alternatively,  \n\\begin{equation}\n\\sum_{i = \\alpha, \\beta} \\Delta \\Gamma (P \\ensuremath{\\rightarrow}\\xspace f_i) = 0~. \n\\end{equation}\n \n Even for decays with more channels with same  quantum numbers, the conclusion remains the same \\cite{Bigi_Book},\n \\begin{equation}\n \\sum_{i} \\Delta \\Gamma (P \\ensuremath{\\rightarrow}\\xspace f_i) = 4 \\sum_i  \\sum_{j\\ne i} {\\cal T}_{ij} \\Im ({\\cal A}_i^* {\\cal A}_j) = 0~. \n \\end{equation}\n Note that, as made explicit by this Wolfenstein mechanism \\cite{Wolfenstein1990}, it is the interference terms between  (at least) two hadronic rescattering amplitudes that generate an observable {\\ensuremath{C\\!P}}\\xspace asymmetry.\n \n  The main lesson here is that {\\ensuremath{C\\!PT}}\\xspace constrains not only the total widths of particle and antiparticle to be the same, but also the sum of partial widths to final states with the same quantum numbers.  For instance, if one finds a sizeable positive {\\ensuremath{C\\!P}}\\xspace asymmetry in a given decay mode, there should be strongly-coupled final states for which the {\\ensuremath{C\\!P}}\\xspace asymmetry has negative sign to compensate. \n  One can also immediately expect,  as a consequence, that if a given final state is stable under strong interactions, e.g. without possible rescattering process, direct {\\ensuremath{C\\!P}}\\xspace violation  would not be observed, even if the weak transition involves a weak phase. This is  the case of a semi-leptonic decay of a charged hadron such as $B^+\\ensuremath{\\rightarrow}\\xspace \\pi^0\\mu^+\\nu_\\mu$.\\footnote{Unless explicitly stated otherwise, charge conjugation is implied throughout the text.}   Another example is the decay $K^+\\ensuremath{\\rightarrow}\\xspace \\pi^+\\pi^0$ which,  due to G-parity, cannot rescatter to any other final state,  implying  $\\Gamma({\\ensuremath{\\kaon^+}}\\xspace\\ensuremath{\\rightarrow}\\xspace{\\ensuremath{\\pion^+}}\\xspace{\\ensuremath{\\pion^0}}\\xspace)=\\Gamma({\\ensuremath{\\kaon^-}}\\xspace\\ensuremath{\\rightarrow}\\xspace {\\ensuremath{\\pion^-}}\\xspace{\\ensuremath{\\pion^0}}\\xspace)$. Analogously,  the sum of the partial widths of the two 3-pion final states for $K^+$ and $K^-$ has to be the same~\\cite{Sozzi_Book, Marshak_Book}, or equivalently \n\\begin{equation}\n\\Gamma(K^+\\ensuremath{\\rightarrow}\\xspace \\pi^+\\pi^-\\pi^+)+\\Gamma(K^+\\ensuremath{\\rightarrow}\\xspace\\pi^+\\pi^0\\pi^0) = \\Gamma(K^-\\ensuremath{\\rightarrow}\\xspace \\pi^-\\pi^+\\pi^-)+\\Gamma(K^-\\ensuremath{\\rightarrow}\\xspace\\pi^-\\pi^0\\pi^0)~.\n\\end{equation}\n\nClearly the {\\ensuremath{C\\!PT}}\\xspace constraint gains flexibility when the number of possible connected final states increase. For non-leptonic charm decays, in principle the families can be divided in terms of the strangeness quantum number $S$: Cabibbo-favoured transitions, $c\\ensuremath{\\rightarrow}\\xspace su\\bar d$ , with $\\Delta S =+1$; Cabibbo-suppressed transitions, $c\\ensuremath{\\rightarrow}\\xspace du\\bar d$ or $c\\ensuremath{\\rightarrow}\\xspace su\\bar s$ , with $\\Delta S = 0$; and doubly-Cabibbo suppressed decays, $c\\ensuremath{\\rightarrow}\\xspace du\\bar s$, with $\\Delta S =-1$.   For beauty decays, there are in principle eight different families, considering the combinations of transitions with $\\Delta S =0,\\pm 1$ and $\\Delta C = 0, \\pm 1$ ($C$ being the charm quantum number). Since the b-hadron masses are higher than those of charm hadrons,  the multiplicity of decay channels for each family is larger, making it difficult to foresee where and how the compensation of an eventual {\\ensuremath{C\\!P}}\\xspace asymmetry in a particular channel would appear in the other strongly-coupled channels. \n\n\\subsection{Short-distance effects and inclusive {\\ensuremath{\\PB}}\\xspace decays}\n\nAs discussed above, the presence of at least two amplitudes, with different weak and strong phases, is a requirement for the manifestation of {\\ensuremath{C\\!P}}\\xspace violation in decays. While weak phases in the SM appear from specific CKM matrix elements, strong phases can appear from different sources. At short distances, a strong phase can be produced from loop (second-order) processes. This effect is described for inclusive charmless  {\\ensuremath{\\PB}}\\xspace decays by the BSS model \\cite{BSS}. The tree amplitude with $ {\\ensuremath{\\Pb}}\\xspace \\ensuremath{\\rightarrow}\\xspace {\\ensuremath{\\Pu}}\\xspace $ transition has the CKM weak phase $\\gamma$, and a strong, short-distance phase may arise from a penguin amplitude.  Both contributions are shown in Fig.~\\ref{treepenguin}. \n\n\\begin{figure}[htb]\n\\centering\n\\includegraphics[width=.6\\columnwidth,angle=0]{Figure1_Btreepenguin.png}\n\\caption{ (Left) Tree and (right) penguin diagram contributions to inclusive charmless ${\\ensuremath{\\Pb}}\\xspace \\ensuremath{\\rightarrow}\\xspace {\\ensuremath{\\Pd}}\\xspace, {\\ensuremath{\\Ps}}\\xspace$ transition. The index $i$ on the right plot  sums over internal quarks ${\\ensuremath{\\Pu}}\\xspace, {\\ensuremath{\\Pc}}\\xspace, {\\ensuremath{\\Pt}}\\xspace$.  }\n\\label{treepenguin}\n\\end{figure}\n\nFor the penguin operator to produce a strong phase, the intermediate particles need to be on-shell \\cite{Sozzi_Book,Wolfenstein1990}. With an internal charm line, this can take place  in a small part of the phase space, as discussed in Ref.~\\cite{gerardhou1991} and shown in Fig.~\\ref{penguinphase}. For charmless $b \\ensuremath{\\rightarrow}\\xspace s u \\bar u $ it occurs  only for $q^2 > 4m_c^2$ ($q$ being the gluon momentum and $m_c$ the charm quark mass), while, on the other hand,  the tree amplitude is real. \n\n\\begin{figure}[htb]\n\\centering\n\\includegraphics[width=.5\\columnwidth,angle=0]{Figure2.png}\n\\caption{Absorptive and dispersive penguin amplitudes, in arbitrary units, as a function of the ratio of the gluon momentum transfer  and the internal quark mass \\cite{gerardhou1991}. }\n\\label{penguinphase}\n\\end{figure}\n\n\nFor a given inclusive flavour content of a {\\ensuremath{\\PB}}\\xspace decay, for example $\\Delta S=1$ and  $\\Delta C=0$,  quark-hadron duality \\cite{Bigi1987,Uraltsev1992} involves the summation over all channels with final states with the same quantum numbers, as\n\\begin{equation}\n\\Delta\\Gamma = \\Gamma({\\ensuremath{\\overline \\bquark}}\\xspace \\ensuremath{\\rightarrow}\\xspace {\\ensuremath{\\overline \\squark}}\\xspace{\\ensuremath{\\Pu}}\\xspace{\\ensuremath{\\overline \\uquark}}\\xspace) - \\Gamma( {\\ensuremath{\\Pb}}\\xspace \\ensuremath{\\rightarrow}\\xspace {\\ensuremath{\\Ps}}\\xspace{\\ensuremath{\\overline \\uquark}}\\xspace{\\ensuremath{\\Pu}}\\xspace) = -4\\Im(\\lambda_i\\lambda^*_j)\n\\sum_{{f}} \\Im\\left[A_i(b \\ensuremath{\\rightarrow}\\xspace f)\\cdot A^*_j(b \\ensuremath{\\rightarrow}\\xspace f)\\right]~,\n\\label{inclusive}\n\\end{equation}\nwhere $f$ are the possible final states for the process, $A_i$ and $A_j$ are the decay amplitudes for each  final state $f$, and  $\\lambda_i, \\lambda^*_j$  are the combinations of the relevant CKM matrix elements associated to the transition and are factored out. The assumption of quark-hadron duality in principle allows the quoted {\\ensuremath{C\\!P}}\\xspace asymmetries to be free of  hadronic uncertainties.  \n\nHowever, as discussed previously, {\\ensuremath{C\\!PT}}\\xspace conservation demands  the sum of partial width differences for particle and antiparticle to be zero within a family of final states with the same quantum numbers. In particular for the example being treated,  both charmless {\\ensuremath{\\PB}}\\xspace decays with $\\Delta S=1$ and  $\\Delta S=0$ must each have the total $A_{{\\ensuremath{C\\!P}}\\xspace} = 0$. G\u00e9rard and Hou \\cite{gerardhou1991, gerardhou1989} noticed this inconsistency in the BSS mechanism and presented a new proposal considering the {\\ensuremath{C\\!PT}}\\xspace constraint, mostly including ${\\cal O}(\\alpha_S ^2) $ contributions.\\footnote{An interesting relationship between these ${\\cal O}(\\alpha_S ^2) $ contributions and the recent direct {\\ensuremath{C\\!P}}\\xspace symmetry distributions observed in charmless three-body {\\ensuremath{\\PB}}\\xspace decays can be found in \\cite{Hou_2019}. Such calculations can also be found in  \\cite{Lenz1999}. } \nIn any case, for a large number of decay channels within a flavour family, the concept of an  inclusive {\\ensuremath{C\\!P}}\\xspace asymmetry has no practical experimental usage other than to give an estimate of the order of magnitude expected for {\\ensuremath{C\\!P}}\\xspace violation.\n\n\\subsection{Exclusive {\\ensuremath{\\PB}}\\xspace decays and final-state interaction phases}\n\nThe situation is quite different from the above when considering  exclusive decays, where important contributions may come from hadronic rescattering, as presented in Eq.~\\ref{eq-f1f2}. In principle, these FSI  can introduce strong phase shifts which are not restricted to be small according to  some authors \\cite {Bigi1987,Suzuki2007,Soni2005,Donoghue1996,Smith2003}.\n\nBut this is far from being a consensus.  From a different standpoint, the authors of Ref. \\cite{Beneke1999,Beneke2000} propose a factorisation based on Quantum Chromodynamics (QCD) that assumes that,  in the limit of the beauty mass $m_b \\ensuremath{\\rightarrow}\\xspace  \\infty$, the number of intermediate physical states is arbitrarily large and the soft rescattering phases vanish at this limit.\\footnote{The heavy-quark limit  is also used in a different approach known as soft-collinear effective theory (SCET) \\cite{Zupan2006}.} \nThus the complications associated to the non-perturbative nature of   hadronic {\\ensuremath{\\PB}}\\xspace decays  can be treated systematically as $1\/m_b$  power corrections, which vanish at the heavy quark limit through systematic cancellations \\cite{Neubert2000}.  Moreover, the large number of intermediate states is used as an argument for not taking the {\\ensuremath{C\\!PT}}\\xspace constraint into  account. Very recently the calculation has been extended to next-to-next-leading-order  (NNLO) in QCD factorisation with a sizeable impact  \\cite{Bell2020}.  \n         \n         \n                \n This long-term discussion has been using the ${\\ensuremath{\\PB}}\\xspace \\ensuremath{\\rightarrow}\\xspace \\PK \\pi$  system as the preferred place for confrontation. These charmless {\\ensuremath{\\PB}}\\xspace decays have branching fractions at ${\\cal O}(10^{-5})$  and clean experimental signatures. Indeed the ${\\ensuremath{\\B^0}}\\xspace \\ensuremath{\\rightarrow}\\xspace {\\ensuremath{\\kaon^-}}\\xspace{\\ensuremath{\\pion^+}}\\xspace$   decay was the first charmless {\\ensuremath{\\PB}}\\xspace decay to be observed, and the first that had its direct {\\ensuremath{C\\!P}}\\xspace asymmetry well determined experimentally.  The theoretical interest comes from the dominance of the penguin contribution in all   ${\\ensuremath{\\PB}}\\xspace \\ensuremath{\\rightarrow}\\xspace \\PK \\pi$ decays,  and  the  contribution of a tree component in most of them, with the $ {\\ensuremath{\\Pb}}\\xspace \\ensuremath{\\rightarrow}\\xspace {\\ensuremath{\\Pu}}\\xspace$  process providing the weak phase $\\gamma$. \n      \n                   Besides the short-distance contribution mentioned above, there are two possible classes of hadronic rescattering  contributions for ${\\ensuremath{\\PB}}\\xspace \\ensuremath{\\rightarrow}\\xspace \\PK \\pi$ decays, one coming from a process involving charm hadronic intermediate states, the other involving only  light-meson final states. A suitable division for these contributions  is proposed in Ref.~\\cite{Buras1997}: the first group is defined as $ {\\ensuremath{\\PB}}\\xspace \\ensuremath{\\rightarrow}\\xspace  f_c  \\ensuremath{\\rightarrow}\\xspace \\PK\\pi$, with $f_c = {\\ensuremath{\\PD}}\\xspace^{(*)} {\\ensuremath{\\PD}}\\xspace_s^{(*)}$ and also called  hard hadronic rescattering  \\cite{Suzuki2007,Soni2005,Smith2003}. The second group involves   ${\\ensuremath{\\PB}}\\xspace \\ensuremath{\\rightarrow}\\xspace  f_s \\ensuremath{\\rightarrow}\\xspace \\PK\\pi$, with $f_s = {\\ensuremath{\\kaon^*}}\\xspace \\pi, {\\ensuremath{\\kaon^*}}\\xspace\\rho, {\\ensuremath{\\kaon^*}}\\xspace\\eta$, among others, and is  called soft rescattering \\cite{Soni2005,Donoghue1996,Buras1997,AtwoodSoni1997}.  Although the rescattering involving the $f_s$ family of intermediate states is more likely than that of the $f_c$ family, the two orders of the magnitude larger branching fractions of the latter family relative to the former  may allow both possibilities to be competitive.\n \n           The advantage of the short-distance QCD factorisation approach  is the possibility to predict the level of direct  {\\ensuremath{C\\!P}}\\xspace asymmetry for several {\\ensuremath{\\PB}}\\xspace exclusive final states  ~\\cite{Beneke1999,Beneke2000} to be compared to those from experimental data. On the other hand, the hadronic rescattering approaches to ${\\ensuremath{\\PB}}\\xspace \\ensuremath{\\rightarrow}\\xspace \\PK \\pi$ (or two-body decays in general) do not allow a similar direct comparison. However, as discussed in Sect.~\\ref{chapter4} and \\ref{chapter5}, multi-body {\\ensuremath{\\PB}}\\xspace decays can shed light on this debate.\n           \n\\subsection{What to expect in charm decays}\n\\label{charm_cap2}\n\n\\begin{figure}\n\\centering\n\\includegraphics[width = 0.9\\textwidth]{Figure3_D_CS.png}\n\\caption{Tree and penguin topologies for a charm-hadron CS decay, where $q$ can be an $s$ or a $d$ quark.}\n\\label{CS-diagrams}\n\\end{figure}\n\nAt a mass scale around $2\\ensuremath{\\mathrm{\\,Ge\\kern -0.1em V}}\\xspace$,\\footnote{Natural units are used.} charm hadrons lie in a region of semi-perturbative QCD, at most, for which the heavy quark limit mentioned above is not such a good tool.  For general reviews on charm physics, see for instance Refs.~\\cite{Bianco_2003,Artuso_Charm_2008}. Strong interactions at low energy play a major role which makes it a challenge, from the theoretical point of view, to describe the dynamics of their hadronic decays. \n\nCharm transitions at tree level are driven by the 2$\\times$2 Cabibbo ``sub-space'' of the CKM matrix, which is  unitary and real up to ${\\cal O}(\\lambda^4)$. Loop and box processes allow, however, the phenomena of ${\\ensuremath{\\D^0}}\\xspace-{\\ensuremath{\\Dbar{}^0}}\\xspace$ mixing and {\\ensuremath{C\\!P}}\\xspace violation to occur at very low rates. Oscillations in the neutral charm system have been observed for the first time in 2007~\\cite{BaBar_DOscill_2007,Belle_DOscill_2007,CDF_DOscill_2008} and is now very well established \\cite{LHCb-PAPER-2017-046,LHCb-PAPER-2019-001} but so far there is no evidence for {\\ensuremath{C\\!P}}\\xspace violation associated to mixing~\\cite{LHCb-PAPER-2019-032}. \n\nWithin the CKM mechanism, direct {\\ensuremath{C\\!P}}\\xspace violation in charm is only possible in singly Cabibbo-suppressed (CS) decays. As for the case of {\\ensuremath{\\PB}}\\xspace decays, both tree and penguin amplitudes can produce final states with the same flavour content, as depicted in Fig.~\\ref{CS-diagrams}. \nContrary to {\\ensuremath{\\PB}}\\xspace decays, though, the penguin amplitude is expected to be very small.  Within the Wolfenstein parameterisation~\\cite{Wolfenstein_CKMparam_1983}, the tree topology amplitude is proportional to $\\lambda$ and for the penguin topology the relevant contribution is with the ${\\ensuremath{\\Pb}}\\xspace$ quark in the loop, which carries the weak phase $\\gamma$ at $\\lambda^5$ level, so the interference term appears with a $\\lambda^4$ dependence relative to the leading term. No weak phases are expected from penguin contributions with ${\\ensuremath{\\Pd}}\\xspace$ and ${\\ensuremath{\\Ps}}\\xspace$ quarks in the loop, which moreover almost cancel due to the GIM mechanism.\nA tree-level interference  between ${\\ensuremath{\\Pc}}\\xspace\\ensuremath{\\rightarrow}\\xspace {\\ensuremath{\\Pd}}\\xspace{\\ensuremath{\\overline \\dquark}}\\xspace{\\ensuremath{\\Pu}}\\xspace$ and  ${\\ensuremath{\\Pc}}\\xspace\\ensuremath{\\rightarrow}\\xspace {\\ensuremath{\\Ps}}\\xspace{\\ensuremath{\\overline \\squark}}\\xspace{\\ensuremath{\\Pu}}\\xspace$ is also sensitive to the complex nature of the CKM matrix since $V_{cd}$ gets an imaginary term at order ${\\cal O}(\\lambda^5)$. The imaginary part of the interference term is then $\\frac{V^{\\phantom{*}}_{ud}V^*_{cd}}{V^{\\phantom{*}}_{us}V^*_{cs}}$ which, by the CKM orthogonality relations,  is also at the $\\lambda^4$ level relative to the leading term.  This tree-level effect can appear for certain final states such as ${\\ensuremath{\\D^0}}\\xspace\\ensuremath{\\rightarrow}\\xspace {\\ensuremath{\\kaon^0_{\\rm\\scriptscriptstyle S}}}\\xspace\\KS,\\eta'{\\ensuremath{\\pion^0}}\\xspace, {\\ensuremath{\\kaon^+}}\\xspace{\\ensuremath{\\kaon^-}}\\xspace{\\ensuremath{\\pion^+}}\\xspace{\\ensuremath{\\pion^-}}\\xspace$, and ${\\ensuremath{\\D^+}}\\xspace\\ensuremath{\\rightarrow}\\xspace {\\ensuremath{\\pion^-}}\\xspace{\\ensuremath{\\pion^+}}\\xspace\\pip, {\\ensuremath{\\kaon^-}}\\xspace{\\ensuremath{\\kaon^+}}\\xspace{\\ensuremath{\\pion^+}}\\xspace$ and can also play a role through rescattering for ${\\ensuremath{\\D^0}}\\xspace\\ensuremath{\\rightarrow}\\xspace{\\ensuremath{\\pion^-}}\\xspace{\\ensuremath{\\pion^+}}\\xspace, {\\ensuremath{\\kaon^-}}\\xspace{\\ensuremath{\\kaon^+}}\\xspace$ \\cite{Grossman_2019,Bianco_Bigi_2020}. \n\nA general form for the amplitude of a CS decay can be written as \\cite{Lenz_CHARM2013}\n\\begin{equation}\n{\\cal A}_{\\rm CS} = \\frac{G_F}{\\sqrt{2}}\\lambda_d \\,T \\left[1+\\frac{\\lambda_b}{\\lambda_d}\\frac{P}{T}\\right],\n\\end{equation}\nwhere $G_F$ is the Fermi weak coupling constant, $\\lambda_q = V^*_{cq}V_{uq}$ ($q=d,s,b$) is the product of the relevant CKM matrix elements (and by unitarity $\\lambda_s = -\\lambda_d - \\lambda_b$) and $T$ and $P$ are the contributions  driven by $\\lambda_d$ and $\\lambda_b$, respectively, thus the latter carrying the weak phase. By computing the {\\ensuremath{C\\!P}}\\xspace asymmetry, \nas in Eq.~\\ref{ACP_sindelta_sinphi} (and neglecting the $|P\/T|$ terms in the denominator when compared to 1) it is easy to see that\n\\begin{equation}\nA_{\\ensuremath{C\\!P}}\\xspace^{\\rm (CS)} = 2 \\frac{\\lambda_b}{\\lambda_d}\\sin\\gamma\\sin\\delta_{\\rm s}\\left|\\frac{P}{T}\\right|\n\\label{ACP_CS}\n\\end{equation}\nwhere the weak ($\\gamma$) and strong ($\\delta_s$) phases are shown explicitly. The ratio $\\left|\\frac{P}{T}\\right|$ is unknown, as well as the strong phase. While this expression is rather general, it refers to the direct {\\ensuremath{C\\!P}}\\xspace violation contribution. For  {\\ensuremath{\\D^0}}\\xspace decays, the observed time-integrated asymmetry also receives a contribution from indirect {\\ensuremath{C\\!P}}\\xspace violation (as discussed  in Sect.~\\ref{Dto2body}). For multibody decays the asymmetries can also be accessed throughout the phase space while Eq.~\\ref{ACP_CS} refers to the phase-space integrated asymmetry. With the current values for $\\lambda_b$, $\\lambda_d$ and $\\gamma$ ~\\cite{CKMfitter2015}, $A_{\\ensuremath{C\\!P}}\\xspace^{\\rm (CS)} \\approx 1.4\\times 10^{-3} \\sin\\delta_{\\rm s}\\left|\\frac{P}{T}\\right|$ is obtained.\n\nAn estimate of the ratio $\\left|\\frac{P}{T}\\right|$ is challenging in the SM, and many authors arrive to different results, depending on the assumptions made. Short-distance QCD gives $\\left|\\frac{P}{T}\\right| \\approx \\frac{\\alpha_s}{\\pi} \\approx 0.1$ which leads to  $A_{\\ensuremath{C\\!P}}\\xspace^{\\rm (CS)} \\approx 10^{-4}$ (assuming $\\sin\\delta_{\\rm s}\\approx 1$) or even less \\cite{Grossman_CPV_CS_2007,Khodjamirian_Petrov_2017}. Yet non-perturbative effects may give rise to penguin enhancements, flavour SU(3) symmetry breaking and rescattering effects which could push asymmetries to as high as a few per mille (see for instance \\cite{Golden_Grinstein_1989,Cheng_Chiang_2012_1,Brod_ConsistentPenguins_2012,Grossman_2019}). This is further discussed in Sect.~\\ref{chapter7}. \n\nDirect {\\ensuremath{C\\!P}}\\xspace violation has now been observed  through  the difference in {\\ensuremath{C\\!P}}\\xspace asymmetries between the decays ${\\ensuremath{\\D^0}}\\xspace\\ensuremath{\\rightarrow}\\xspace{\\ensuremath{\\kaon^+}}\\xspace{\\ensuremath{\\kaon^-}}\\xspace$ and ${\\ensuremath{\\D^0}}\\xspace\\ensuremath{\\rightarrow}\\xspace{\\ensuremath{\\pion^+}}\\xspace{\\ensuremath{\\pion^-}}\\xspace$~\\cite{LHCb-PAPER-2019-006} at the level of ${\\cal O}(10^{-3})$. This is an impressive achievement  and further stimulates the search for signs of {\\ensuremath{C\\!P}}\\xspace violation in other decays. The measured value is higher than those from short-distance expectations and in principle favours  models with hadronic enhancements. It also give some fuel for new physics (NP) models \\cite{Nir_Implications_CharmCPV_2019,Chala_2019,Calibbi_2019}.\n\nIndeed the search for {\\ensuremath{C\\!P}}\\xspace violation in charm is an excellent place to look for new dynamics due to the low ``background'' from the SM. There are many channels for which the experimental uncertainties are still above a few percent and can be tested for sources of {\\ensuremath{C\\!P}}\\xspace violation beyond the SM.\n\nIn any case, SM or beyond, the interplay of weak and hadronic amplitudes still dictates the level of {\\ensuremath{C\\!P}}\\xspace violation to be observed. Moreover, as discussed previously, for particle and antiparticle decays the {\\ensuremath{C\\!PT}}\\xspace invariance constrains the sum of their partial widths to final states with same flavour content to be the same. The multiplicity of final states here, when compared to {\\ensuremath{\\PB}}\\xspace decays, is not so high and thus the  {\\ensuremath{C\\!PT}}\\xspace constraint can turn out to be more enlightening and helpful than that in the $b$-quark sector. \n\n   \\section{Experimental facilities and analysis tools} \n \\label{chapter3}\n \n As discussed previously, it took a long time since the first observation of {\\ensuremath{C\\!P}}\\xspace violation in the neutral kaon system to the start of a rich programme of {\\ensuremath{C\\!P}}\\xspace violation measurements, occurring at the beginning of the $21^{\\rm st}$ century with the measurement of $\\sin{2\\beta}$  in  ${\\ensuremath{\\B^0}}\\xspace\\ensuremath{\\rightarrow}\\xspace {\\ensuremath{{\\PJ\\mskip -3mu\/\\mskip -2mu\\Ppsi\\mskip 2mu}}}\\xspace{\\ensuremath{\\kaon^0_{\\rm\\scriptscriptstyle S}}}\\xspace$ decays by the Babar~\\cite{Babar2001} and Belle collaborations~\\cite{Belle2001}. Now, two decades has passed and we have witness how a coherent picture of the {\\ensuremath{C\\!P}}\\xspace violation phenomena has emerged from a wide range of results, with a remarkable consistency with the CKM description~\\cite{CKMfitter2015,UTfit-UT}. \n Within the last 10 years or so, many experiments, such as D$\\emptyset$, CDF, CLEO, and BESIII,  have contributed to  direct {\\ensuremath{C\\!P}}\\xspace violation measurements, but largely the bulk of results comes from  BaBar, Belle and LHCb. Here we briefly describe some of their main features,  and then discuss Belle II and the Upgrade LHCb, which will lead the field for the next decade. The principal aspects and techniques for the measurement of direct {\\ensuremath{C\\!P}}\\xspace violation observables are presented afterwards.  \n\n\n\n \\subsection{Main facilities for {\\ensuremath{C\\!P}}\\xspace violation in beauty and charm hadrons}\n \n The fundamental aspects of a detector dedicated to beauty and charm physics are: excellent vertexing and tracking capabilities, in order to provide a good reconstruction of charged particles and a precise determination of the $b$- or $c$-hadron  decay vertex; a reliable particle identification system, fundamental to distinguish the final-state particles (protons, kaons, pions, muons, etc); good momentum and time resolutions. For the study of neutral {\\ensuremath{\\PB}}\\xspace and {\\ensuremath{\\PD}}\\xspace mesons it is also fundamental a trustful {\\it tagging} system, {\\it i.e.} the knowledge of the flavour of the neutral meson at production (or at a given time), to allow the study of its time evolution and its decay to self-{\\ensuremath{C\\!P}}\\xspace-conjugated final states. \n \n In the quest for the first signs of {\\ensuremath{C\\!P}}\\xspace violation in the beauty sector, the $B$ factories, BaBar~\\cite{Aubert:2001tu,TheBABAR:2013jta} and Belle~\\cite{Abashian:2000cg}, took the leading role for many years. They both relied on a copious production of ${\\ensuremath{\\Bu}}\\xspace{\\ensuremath{\\Bub}}\\xspace$ and ${\\ensuremath{\\B^0}}\\xspace{\\ensuremath{\\Bbar{}^0}}\\xspace$ pairs at the ${\\Y4S}$ bottonium resonance, corresponding to a centre-of-mass energy of $10.58\\ensuremath{\\mathrm{\\,Ge\\kern -0.1em V}}\\xspace$. The PEPII machine  provided $\\en$ and ${\\ensuremath{\\Pe^+}}\\xspace$ beams to BaBar at energies of 9.0 and 3.1\\,\\ensuremath{\\mathrm{\\,Ge\\kern -0.1em V}}\\xspace, respectively, while KEKB machine, for Belle,  had $\\en$ and ${\\ensuremath{\\Pe^+}}\\xspace$ energies at 8.0 and 3.5\\,\\ensuremath{\\mathrm{\\,Ge\\kern -0.1em V}}\\xspace, respectively. The energy-asymmetric beams were designed to provide the ${\\ensuremath{\\PB}}\\xspace\\bar{B}$ pair a boost in the laboratory, in order for the {\\ensuremath{\\PB}}\\xspace-decay vertices to be well separated from the production vertex. The ${\\ensuremath{\\B^0}}\\xspace{\\ensuremath{\\Bbar{}^0}}\\xspace$ pair were produced as an entangled quantum system, and by measuring the flavour of one {\\ensuremath{\\PB}}\\xspace meson at its decay vertex through a flavour-specific mode it was possible to determine the flavour of the other (conjugate) {\\ensuremath{\\PB}}\\xspace meson. Since charm mesons are common decay products of {\\ensuremath{\\PB}}\\xspace mesons, BaBar and Belle had also a significant impact on charm physics. With $\\en{\\ensuremath{\\Pe^+}}\\xspace$ collisions producing low hadronic background, and an almost $4\\pi$ of geometrical coverage,  the experiments were very efficient for reconstructing both charged and neutral final-state particles. \n The BaBar experiment ran from 1999 to 2008 and cumulated about $424\\ensuremath{\\mbox{\\,fb}^{-1}}\\xspace$ of data at the ${\\Y4S}$ resonance while Belle ran from 1999 to 2010, cumulating $711\\ensuremath{\\mbox{\\,fb}^{-1}}\\xspace$ at the same energy. \\footnote{Although the majority of the data taking occurred at the  ${\\Y4S}$  resonance, both BaBar and Belle took data off-resonance to study background. BaBar also had runs at ${\\Y2S}$ and${\\Y3S}$, while Belle had a dedicated ${\\Y5S}$ run which allowed the production of ${\\ensuremath{\\B^0_\\squark}}\\xspace{\\ensuremath{\\Bbar{}^0_\\squark}}\\xspace$ pairs.}  For further details, a nice review can be found in Ref.~\\cite{Bevan2014}.\n  \n The LHCb experiment~\\cite{LHCb-DP-2008-001,LHCb-DP-2014-002} has entered the game with the start of the LHC in 2010 with proton-proton collisions at $7\\ensuremath{\\mathrm{\\,Te\\kern -0.1em V}}\\xspace$, and since the last 5 years is leading the experimental field of  {\\ensuremath{C\\!P}}\\xspace violation.   The LHCb is a single-arm forward detector with an angular coverage corresponding to  the pseudorapidity in the range $2<\\eta<5$,  taking advantage that $b$ and $c$ hadrons are produced at relatively small angles with respect to the ${\\ensuremath{\\Pp}}\\xspace\\proton$ beam direction. In contrast to  $\\en{\\ensuremath{\\Pe^+}}\\xspace$ machines, the initial state  is not known for ${\\ensuremath{\\Pp}}\\xspace\\proton$ collisions at the LHC; the main mechanism for heavy-flavour production is  gluon-gluon fusion, and a typical event may produce a hundred charged particles. For illustration, a  comparison of the event profiles at  Belle and LHCb is shown in Fig.~\\ref{BellevsLHCb_eventdisplay}.  \n To distinguish the $b$- and $c$-hadron decay products from a large background, LHCb relies on an efficient and reliable trigger system, capable of full online reconstruction, exploiting  the topological characteristics of the desired decay processes. The reconstruction of neutrals, nevertheless, are not very efficient when compared to that from low-occupation $\\en{\\ensuremath{\\Pe^+}}\\xspace$ collisions. Another important difference, very relevant for {\\ensuremath{C\\!P}}\\xspace violation studies, is that the initial state is not {\\ensuremath{C\\!P}}\\xspace symmetric and this leads to particle-antiparticle production asymmetries that have to be considered (subtracted) for the study of (or search for)  {\\ensuremath{C\\!P}}\\xspace asymmetries. \nThe data was taken in two periods: run I with collisions at a centre-of-mass energy of $7\\ensuremath{\\mathrm{\\,Te\\kern -0.1em V}}\\xspace$ in 2011 and at $8\\ensuremath{\\mathrm{\\,Te\\kern -0.1em V}}\\xspace$ in 2012, with integrated luminosities of $1\\ensuremath{\\mbox{\\,fb}^{-1}}\\xspace$ and $2\\ensuremath{\\mbox{\\,fb}^{-1}}\\xspace$, respectively; and run II, from 2015 to 2018, at a centre-of-mass energy of $13\\ensuremath{\\mathrm{\\,Te\\kern -0.1em V}}\\xspace$ with an integrated luminosity of $6\\ensuremath{\\mbox{\\,fb}^{-1}}\\xspace$. \n While the great majority of LHCb analyses based on run I data are completed, many studies based on run II data are still under progress, and we shall expect new results throughout the next few years. \n \n \\begin{figure}[!htbp]\n \\centering\n \\includegraphics[align=c, width=0.4\\textwidth]{Figure4_Belle2_event_display.png}\\hspace*{0.05\\textwidth}\n  \\includegraphics[align=c,width=0.4\\textwidth]{Figure4_LHCb_eventdisplay.png}\n \\caption{(left) A typical event from $\\en{\\ensuremath{\\Pe^+}}\\xspace$ collisions at Belle II, as obtained from simulation (source: image courtesy of Belle II\/KEK).  (right) A ${\\ensuremath{\\Pp}}\\xspace\\proton$-collision event recorded by the LHCb experiment (source: taken from ~\\cite{Ciezarek2017}).}\n \\label{BellevsLHCb_eventdisplay}\n \\end{figure}\n\n \n Meanwhile, the Belle II experiment~\\cite{BelleII_TDR,BelleII_PhysicsBook_2018} has started its physics run in 2019 and will continue until 2025. All components of the detector are new or highly upgraded with respect to the former Belle detector to be able to operate at an event rate 40 times higher than before, now provided by the SuperKEKB accelerator \\cite{SuperKEKB}. As KEKB, the  $\\en{\\ensuremath{\\Pe^+}}\\xspace$ collisions occur at centre-of-mass energies in the region of the  $\\PUpsilon$ resonances, mainly the ${\\Y4S}$. The expected integrated luminosity is $50 \\rm ab^{-1}$. \n \n The  LHCb Upgrade I~\\cite{LHCb-TDR-012} is currently being commissioned for the start of run III of LHC in 2021. It is a major upgrade, with a factor of five higher luminosity as compared to run II,  with the goal to collect $50\\ensuremath{\\mbox{\\,fb}^{-1}}\\xspace$ by the end of run IV in 2028. The LHCb Upgrade II~\\cite{LHCb-PII-EoI,LHCb-PII-Physics} is planned from 2031 onwards (run V) during the High Luminosity LHC (HL-HLT) operation period.  \n \n LHCb and Belle II will allow comprehensive and complementary programmes on quark-flavour physics in the next decade. While Belle II will be able to efficiently study  final states with neutral particles and benefit from a very efficient tagging system, LHCb will have much larger yields for final states with charged particles and the capability to produce all $b$-hadron species, specially ${\\ensuremath{\\B^0_\\squark}}\\xspace$, ${\\ensuremath{\\B_\\cquark^+}}\\xspace$  and $b$-baryons.  \n \n  \n  \\subsection{Direct {\\ensuremath{C\\!P}}\\xspace violation in the decay rate}\nAs discussed in Sect.~\\ref{chapter2_CP_CPT}, the usual observable when measuring or searching for {\\ensuremath{C\\!P}}\\xspace violation in decays is the charge asymmetry, as defined in Eq.~\\ref{CPasym_def}. For two-body decays of {\\ensuremath{\\PD}}\\xspace and {\\ensuremath{\\PB}}\\xspace mesons, this is the only {\\ensuremath{C\\!P}}\\xspace violation observable. Experimentally, though, one measures the particle and antiparticle {\\it yields},  $N(P\\ensuremath{\\rightarrow}\\xspace f)$ and $N(\\bar P\\ensuremath{\\rightarrow}\\xspace \\bar f)$, respectively,  which are affected by a chain of effects from production, reconstruction, and final selection of events. The observed  {\\it raw} asymmetry is obtained,\n\\begin{equation}\nA_{\\rm raw} = \\frac{N(P\\ensuremath{\\rightarrow}\\xspace f) - N(\\bar P\\ensuremath{\\rightarrow}\\xspace \\bar f)}{N(P\\ensuremath{\\rightarrow}\\xspace f) + N(\\bar P\\ensuremath{\\rightarrow}\\xspace \\bar f)}~,\n\\label{Araw_def}\n\\end{equation}\nwhich needs then to be corrected for eventual production, $A_{\\rm P}$, and detection, $A_{\\rm D}$, asymmetries. Usually these constitute small effects, of a few percent at most, which allow the approximation \\cite{Gersabeck_Interplay_2012}\n\\begin{equation}\nA_{\\ensuremath{C\\!P}}\\xspace \\approx A_{\\rm raw} - A_{\\rm P} - A_{\\rm D}~.\n\\end{equation}\nThe precise measurement of these nuisance asymmetries is thus fundamental. Usually they are measured through control channels for which no {\\ensuremath{C\\!P}}\\xspace violation effects are expected.\n\nFor baryons and for multi-body decays other measurements can also be accessed, due to  polarisation and phase space, as  discussed in the following.\n\n \\subsection{Direct {\\ensuremath{C\\!P}}\\xspace violation in multi-body decays}\n \\label{multibody}\n \n\nFor three or more particles in the final state, besides the measurement of the (total) charge asymmetry, {\\ensuremath{C\\!P}}\\xspace violation can be studied through the phase space distribution of the decay. Usually in {\\ensuremath{\\PB}}\\xspace and {\\ensuremath{\\PD}}\\xspace mesons decays, the process is dominated by the formation of resonances as intermediate states, which then decay strongly or electromagnetically to form the detectable final state. The distribution of events throughout the phase space is the result of the superposition of the various amplitudes, and the interference pattern depends directly on the strong and weak phases involved. The rich dynamics potentially allow different sources of strong phases to appear.  It is natural to expect localised {\\ensuremath{C\\!P}}\\xspace asymmetries to be stronger than phase-space integrated ones, and can even change sign. Altogether these features make multi-body decays an excellent tool for studying  {\\ensuremath{C\\!P}}\\xspace violation in the hadronic sector. \n\nTo measure or to search for {\\ensuremath{C\\!P}}\\xspace violation over the phase space, two approaches are pursued: model-dependent and model-independent analyses.  The first is based on an amplitude analysis fit;  the decay amplitude is modelled as a coherent sum of intermediate states such that their relative contributions can differ for particle and antiparticle decays. In  model-independent strategies the phase space distributions for particle and antiparticle decays are directly compared to look for regions where there are statistically significant differences, and localised asymmetries can be measured. The two approaches are complementary: while model-independent techniques can pin down the regions of the phase space where {\\ensuremath{C\\!P}}\\xspace violation is manifest, amplitude analysis can identify its dynamical source. \n\n\\subsubsection{Amplitude analysis}\n\\label{AmAn}\n\nDecays of {\\ensuremath{\\PD}}\\xspace and {\\ensuremath{\\PB}}\\xspace mesons to three pseudo-scalar mesons in the final state have their dynamics fully described in terms of only  two invariants.  By writing a general decay as $P\\ensuremath{\\rightarrow}\\xspace h_1h_2h_3$ ($P={\\ensuremath{\\PB}}\\xspace, {\\ensuremath{\\PD}}\\xspace$ and $h_{1,2,3}$ light mesons)  this two-dimensional phase space can be constructed in terms of two-body squared invariant masses such as $s_{12} = m^2(h_1h_2)$ and $s_{23} = m^2(h_2h_3)$, the so-called Dalitz plot. The distribution of events in the Dalitz plot is directly proportional to the total amplitude squared, with no phase-space factors involved. \n\nThe simplest and most common approach for an amplitude analysis is the so-called isobar model where the total amplitude is described as a coherent sum of resonant amplitudes,  for instance $P\\ensuremath{\\rightarrow}\\xspace R_{12} h_3$, $R_{12}$ being a resonance decaying to $h_1h_2$; the ``companion'' hadron $h_3$ is assumed to not interact with the $h_1h_2$ system. \n\nThe intermediate resonance is described by a propagator, usually a relativistic Breit--Wigner, which is a good representation only for narrow and non-overlapping resonances, having the form\n\\begin{equation}\n{\\it BW_{\\! R} }=\\frac{1}{m_R^2 - s_{ij} -i m_R \\Gamma_R(s_{ij})}~,\n\\label{BW}\n\\end{equation}\nwhere $\\Gamma_R$ is the energy-dependent width, which introduces a strong phase variation of $180^\\circ$ around the nominal resonance mass $m_R$, and $i,j$ represent the label of the hadron pair it decays to. Each resonant amplitude is described as \n\\begin{equation}\n{\\cal A}_R= {\\it  {^J\\!F}_{\\! P}  {\\it ^J\\!F}_{\\! R}  \\times {^J\\!M}_{\\! R} \\times  BW_{\\! R} }~,\n\\label{AM}\n\\end{equation}\nwhere $J$ is the spin of the resonance, the factors ${\\it ^J\\!F}_{\\! P,R}$  are the Blatt--Weisskopf damping factors \\cite{Blatt:1952ije}, accounting for the finite size of the $P$ and $R$ mesons, and ${\\it ^J\\!M}_{\\! R}$ is an angular factor, ensuring angular-momentum conservation \\cite{Argus}. Explicitly, ${^0\\!M}_{\\! R} = 1$, ${^1\\!M}_{\\! R} =- 2|p_k||p_i|\\cos\\theta_{ki}^{R_{ij}}$ and \\mbox{${^2\\!M}_{\\! R} =4(|p_3||p_2|)^2(3\\cos^2\\theta_{ki}^{R_{ij}} -1)$} for resonances of spin 0, 1 and 2, respectively, where $p_{i(k)}$ is the modulus of the momentum of particle $h_{i(k)}$ and $\\theta_{ki}^{R_{ij}}$ is the angle between the momenta of particles $h_k$ and $h_i$ in the $h_ih_j$ rest frame.  \n\nThe total amplitude is then written as a superposition of individual amplitudes\n\\begin{equation}\n{\\cal A}(s_{12}, s_{23}) = a_{\\rm nr}e^{i\\delta_{\\rm nr}} {\\cal A}_{\\rm nr}(s_{12},s_{23})+ \\sum_{{n =1}}^{N} a_n e^{i\\delta_n}{\\cal A}_{R_n}(s_{12},s_{23})~,\n\\label{Coherent}\n\\end{equation}\nwhere a magnitude $a_n$ and a phase $\\delta_n$ are associated to each resonant contribution $n$; these are parameters to be determined by the amplitude fit,  the former giving the relative strength  and the latter representing an attributed phase incorporating potentially the effects of both  weak and strong phases. The first term allows for a possible non-resonant contribution, with magnitude $a_{\\rm nr}$, phase $\\delta_{\\rm nr}$ and the shape ${\\cal A}_{\\rm nr}$ usually assumed to be a  constant for {\\ensuremath{\\PD}}\\xspace decays and an exponential or a single-pole function for {\\ensuremath{\\PB}}\\xspace decays \\cite{Argus,Belle_bkpipi,Tobias_CPT2,LHCb-PAPER-2018-051}. The Dalitz plot is fitted allowing for {\\ensuremath{C\\!P}}\\xspace violation, that is, $a_n$ and $\\delta_n$, for the decay of $P$ meson,  and $\\bar a_n$ and $\\bar \\delta_n$, for the decay of $\\bar P$ meson, are independent fit parameters.\n\nThe isobar approach, while simple, has limitations. One of them is the assumption that the companion hadron (that not resulting from the decaying resonance) does not take part in the dynamics. This is the $2$+$1$ or quasi-two-body approximation and is discussed below in Sect.~\\ref{quasi2body}. The other is that it is known to be fairly good only for narrow, non-overlapping resonances with the same quantum numbers. There are examples of  three- or four-body {\\ensuremath{\\PB}}\\xspace and {\\ensuremath{\\PD}}\\xspace decays where low-mass ($\\lesssim 2 \\ensuremath{\\mathrm{\\,Ge\\kern -0.1em V}}\\xspace$) broad scalar resonances are very important or even dominant. Assuming the  $2$+$1$ approximation to be exact, the simple sum of relativistic Breit--Wigners would not preserve two-body unitarity.\\footnote{However the validity of the isobar model finds support  with arguments  in Ref.~\\cite{Svec2000}.} To overcome such a problem, some analyses have been using alternative approaches for the parameterisation of the S-wave. One is a quasi-model-independent approach, where the S-wave amplitude is binned throughout the two-body invariant mass range to fit for its amplitude and phase in each bin, as in Refs.~\\cite{E791_Brian,Focus_PWA,BaBar_ds3pi,LHCb-PAPER-2019-017}. Another one is the K-matrix approach~\\cite{Aitchison_Kmatrix}, which imposes two-body unitarity by using external data from scattering experiments, as in Refs.~\\cite{LHCb-PAPER-2019-017,FOCUS_Kmatrix_3pi}. \n   \nYet, while approximate, the simple and elegant isobar model can reveal important features related to what has been discussed in Sect.~\\ref{chapter2_CP_CPT}. To illustrate this, we show a simple example with only two resonant amplitudes appearing in the $h_1h_2$ channel, interfering in a particular region of the Dalitz plot. Suppose one is a vector resonance, for example $\\rho(770)^0$ and the other a scalar,  $f_0(980)$.\\footnote{Scalar plus vector amplitudes are the dominant configuration in the low-mass regions in three-body heavy meson decays.} For simplicity and without loss of generality, we represent each resonant amplitude by a Breit--Wigner, and consider only the relevant $\\cos\\theta$ factor for the $\\rho(770)^0$ amplitude ($\\cos\\theta^{R_{12}}_{13}$ in this case).\\footnote{Unless otherwise stated, wherever $\\cos\\theta$ appears throughout the text, $\\theta$ refers to the angle $\\theta_{ki}^{R_{ij}}$ defined earlier, that is, that of the direction of the companion hadron $h_k$ with $h_i$ in the $h_ih_j$ rest frame.} The  two charge-conjugate amplitudes are written  as follows:\n\\begin{eqnarray}\n{\\cal A} = \\ensuremath{a_\\rho } e^{i \\ensuremath{\\delta_\\rho }} \\ensuremath{BW_{\\rho}} \\cos\\theta + \\ensuremath{a_{f_0} } e^{i \\ensuremath{\\delta_{f_0} }} \\ensuremath{BW_{f_0}}  \\nonumber \\\\\n\\overline{\\cal A} = \\ensuremath{{\\bar a}_\\rho } e^{i \\ensuremath{\\bar\\delta_\\rho }} \\ensuremath{BW_{\\rho}} \\cos\\theta + \\ensuremath{{\\bar a}_{f_0} } e^{i \\ensuremath{\\bar\\delta_{f_0} }} \\ensuremath{BW_{f_0}}~.\n\\label{Arho_f0}\n\\end{eqnarray}\nBesides the inherent (and energy-dependent) Breit--Wigner phases, the phases $\\delta$ and $\\bar\\delta$ represent, as mentioned before,  the net effect of contributions from the strong phase due to on-shell penguin diagrams, final-state (soft) hadron-hadron interactions, and the underlying weak phase from the CKM matrix.  \nThe difference of the total amplitude squared for particle and antiparticle is  given by\n\\begin{eqnarray}\n\\Delta |{\\cal A}|^2= |{\\cal A}|^2 - |\\overline{\\cal A}|^2 = [\\ensuremath{a_\\rho }^2 - \\ensuremath{{\\bar a}_\\rho }^2]\\ensuremath{|BW_{\\rho}|^2}\\cos^2\\theta + [\\ensuremath{a_{f_0} }^2 - \\ensuremath{{\\bar a}_{f_0} }^2]\\ensuremath{|BW_{f_0}|^2} + \\nonumber \\\\ \n2 \\ensuremath{|BW_{\\rho}|^2} \\ensuremath{|BW_{f_0}|^2} \\cos\\theta \\cdot   \\nonumber  \\\\\n\\Big\\{ \\left[(\\ensuremath{m_{\\rho}}^2 -s_{12})(\\ensuremath{m_{f_0}}^2 -s_{12})+  (\\ensuremath{m_{\\rho}} \\ensuremath{\\Gamma_{\\rho}})(\\ensuremath{m_{f_0}} \\ensuremath{\\Gamma_{f_0}})\\right]  \\left[ \\ensuremath{a_\\rho } \\ensuremath{a_{f_0} } \\cos(\\ensuremath{\\delta_\\rho }-\\ensuremath{\\delta_{f_0} }) - \\ensuremath{{\\bar a}_\\rho } \\ensuremath{{\\bar a}_{f_0} } \\cos(\\ensuremath{\\bar\\delta_\\rho }-\\ensuremath{\\bar\\delta_{f_0} })\\right] \\nonumber \\\\\n- \\left[\\ensuremath{m_{\\rho}} \\ensuremath{\\Gamma_{\\rho}}(\\ensuremath{m_{f_0}}^2 -s_{12}) - \\ensuremath{m_{f_0}} \\ensuremath{\\Gamma_{f_0}}(\\ensuremath{m_{\\rho}}^2 -s_{12})\\right] \\left[\\ensuremath{a_\\rho } \\ensuremath{a_{f_0} } \\sin(\\ensuremath{\\delta_\\rho }-\\ensuremath{\\delta_{f_0} }) - \\ensuremath{{\\bar a}_\\rho } \\ensuremath{{\\bar a}_{f_0} } \\sin(\\ensuremath{\\bar\\delta_\\rho }-\\ensuremath{\\bar\\delta_{f_0} })\\right]  \\Big\\}~.\n\\label{AcpAmplitude}\n\\end{eqnarray}\n\nThe first two terms correspond to a {\\ensuremath{C\\!P}}\\xspace asymmetry associated to each of the resonances individually. A short-distance tree-penguin interference would manifest through them (assuming the resonance can be formed by both processes).  It would be the analogous case of a two-body asymmetry, but here as a quasi-two-body state. Due to its vector nature, the signature from the first term is the $\\cos^2 \\theta$  distribution, and this feature shall be better discussed in Sect.~\\ref{quasi2body}.  \nFrom the amplitude fit, a {\\ensuremath{C\\!P}}\\xspace asymmetry for each individual amplitude can be defined in terms of the magnitudes. For instance, for the $\\rho(770)^0$ amplitude\n\\begin{equation}\nA_{{\\ensuremath{C\\!P}}\\xspace}^\\rho = \\frac{a_\\rho^2 - \\bar a_\\rho^2}{a_\\rho^2 + \\bar a_\\rho^2}~.\n\\label{ACP_r}\n\\end{equation}\n\n The interference term between the two resonant amplitudes gives rise to the third term of Eq.~\\ref{AcpAmplitude}, and is proportional to $\\cos\\theta$. As such, this term drives a change of sign of the asymmetry throughout the phase space, the inflexion region being around $\\cos\\theta \\approx 0$.  Yet, this factor multiplies terms related to the complex nature of the Breit--Wigner. When  $s_{12}$ crosses the resonance nominal mass $m_n^2$ (either $\\rho(770)^0$ or $f_0(980)$), the {\\ensuremath{C\\!P}}\\xspace asymmetry associated with this contribution tends to change sign. This effect has been seen on data in the decays  $ {\\ensuremath{\\PB}}\\xspace^\\pm \\ensuremath{\\rightarrow}\\xspace \\pi^\\pm \\pi^+\\pi^+$ and $ {\\ensuremath{\\PB}}\\xspace^\\pm \\ensuremath{\\rightarrow}\\xspace K^\\pm \\pi^+\\pi^+$ as discussed later in Sect.~\\ref{CPfromcostheta}. \nThis S- and P-wave interference term, integrated in the whole phase space, must be equal to zero due to orthogonality, and consequently the net {\\ensuremath{C\\!P}}\\xspace asymmetry coming from the third term should be equal to zero. \nThe integrated asymmetry may still receive contributions from the first and second terms though. In Sect.~\\ref{quasi2body}, we discuss that, by  strictly  assuming the $2$+$1$ approximation,  {\\ensuremath{C\\!P}}\\xspace violation involving light vector mesons should be very small. \n\n\n\\subsubsection{The quasi-two-body approximation}\n\\label{quasi2body}\n\n Any approximation neglecting hadron interactions  in multi-hadron decays, such as  the $2$+$1$ approximation,  has to be taken with parsimony. Weak decays are point-like interactions, and strong forces mediate the final state  formation. The validity of the  approximation may depend on specific cases but in general one would expect to be better for {\\ensuremath{\\PB}}\\xspace than for {\\ensuremath{\\PD}}\\xspace decays, and with less hadrons in the final state. \n       \nIn fact, despite the large use of the $2$+$1$ approximation in three-body {\\ensuremath{\\PD}}\\xspace amplitude analyses, some problems have been observed with this approach in experimental studies. The Fermilab E791  \\cite{E791_Brian} and FOCUS~\\cite{Focus_PWA} experiments have performed a quasi-model-independent partial wave amplitude (QMI-PWA) approach to measure the ${\\ensuremath{\\kaon^-}}\\xspace{\\ensuremath{\\pion^+}}\\xspace$ S-wave in the ${\\ensuremath{\\D^+}}\\xspace\\ensuremath{\\rightarrow}\\xspace{\\ensuremath{\\kaon^-}}\\xspace{\\ensuremath{\\pion^+}}\\xspace\\pip$ decays.  Both results  deviate from the ${\\ensuremath{\\kaon^-}}\\xspace{\\ensuremath{\\pion^+}}\\xspace$  S-wave measured by the LASS scattering experiment \\cite{LASS}, as shown in Fig.~\\ref{e791PWA} for the E791 result. The  ${\\ensuremath{\\kaon^-}}\\xspace{\\ensuremath{\\pion^+}}\\xspace$  system, if isolated, should present the same phase behaviour independently of the production process, according to  the Watson theorem~\\cite{Watson}. A plausible explanation for the difference in the ${\\ensuremath{\\kaon^-}}\\xspace{\\ensuremath{\\pion^+}}\\xspace$  behaviour is the interaction with the companion {\\ensuremath{\\pion^+}}\\xspace in the ${\\ensuremath{\\D^+}}\\xspace\\ensuremath{\\rightarrow}\\xspace{\\ensuremath{\\kaon^-}}\\xspace{\\ensuremath{\\pion^+}}\\xspace\\pip$ decay. \n\nThe QMI-PWA of the $ {\\ensuremath{\\D^+_\\squark}}\\xspace\\ensuremath{\\rightarrow}\\xspace{\\ensuremath{\\pion^-}}\\xspace{\\ensuremath{\\pion^+}}\\xspace\\pip$~\\cite{BaBar_ds3pi} and  ${\\ensuremath{\\D^+_\\squark}}\\xspace\\ensuremath{\\rightarrow}\\xspace {\\ensuremath{\\kaon^-}}\\xspace{\\ensuremath{\\kaon^+}}\\xspace{\\ensuremath{\\pion^+}}\\xspace$~\\cite{BaBar_dskkpi} decays have also presented a  different behaviour for the ${\\ensuremath{\\pion^-}}\\xspace{\\ensuremath{\\pion^+}}\\xspace$ and ${\\ensuremath{\\kaon^-}}\\xspace{\\ensuremath{\\kaon^+}}\\xspace$ S-wave, respectively,  when compared to those from two-bodies elastic scattering~\\cite{CERN-Munich,Hadron2017}. Altogether these results reinforce the idea that the companion particle in three-body {\\ensuremath{\\PD}}\\xspace decays takes a role in the final state. \n\nThe importance of three-body final-state interactions (FSI) in heavy-meson decays has been investigated by different theoretical groups~\\cite{Boito, Pat_Rob2011,Pat_Rob2015,Nakamura,Guimaraes2014}. They use a simple model implementing three-body unitarity, solving the Faddeev equation. The first-order rescattering  interaction is represented through a vector-resonance intermediate state as shown in Fig.~\\ref{pat2015}. Additional contributions  can be added by including higher-order loops in a  transition matrix that sums the connected scattering series from  a ladder graph~\\cite{Guimaraes2014,Mikhasenko_private}, as shown in Fig.~\\ref{Mikhashenko1}.\nIn this figure, three-body rescattering effects in these decays have been classified as a direct term, with a small correction to the Breit--Wigner lineshape~\\cite{Pat_Ig}, and an induced term, with a possible  presence of a resonant term to  the same three-body final state \\cite{Mikhasenko}. Figure~\\ref{Mikhashenko2} presents the Dalitz plot distribution for these two terms.\n\n \n\n\\begin{figure}[!htb]\n\\centering\n\\includegraphics[width=.45\\columnwidth,angle=0]{Figure5_E791kpipiphase.png}\n\\caption{The S-wave phase as a function of the ${\\ensuremath{\\kaon^-}}\\xspace{\\ensuremath{\\pion^+}}\\xspace$  mass from the QMI-PWA\n $ {\\ensuremath{\\D^+}}\\xspace\\ensuremath{\\rightarrow}\\xspace{\\ensuremath{\\kaon^-}}\\xspace{\\ensuremath{\\pion^+}}\\xspace\\pip$ Dalitz plot fit, taken from \\cite{E791_Brian}. The band shows the result of the Isobar model fit in a previous work \\cite{E791_kappa}. The hashed vertical line shows the elastic range according\nto LASS \\cite{LASS}. }\n\\label{e791PWA}\n\\end{figure}\n\n\\begin{figure}[!htb]\n\\centering\n\\includegraphics[width=.4\\columnwidth,angle=0]{Figure6_rescattering.png}\n\\caption{Leading vector-current contribution to the $ {\\ensuremath{\\D^+}}\\xspace\\ensuremath{\\rightarrow}\\xspace{\\ensuremath{\\kaon^-}}\\xspace{\\ensuremath{\\pion^+}}\\xspace\\pip$ decay~\\cite{Pat_Rob2015}.}\n\\label{pat2015}\n\\end{figure}\n\n\n\n\\begin{figure}[htb]\n\\centering\n\\includegraphics[width=.85\n\\columnwidth,angle=0]{Figure7_Direct_Induced.png}\n\\caption{ Direct and Induced terms produced in a three-body  rescattering \\cite{Mikhasenko_private}. }\n\\label{Mikhashenko1}\n\\end{figure}\n\n\\begin{figure}[htb]\n\\centering\n\\includegraphics[width=.85\n\\columnwidth,angle=0]{Figure8_Dalitz_Direct_Induced.png}\n\\caption{ Signatures in the Dalitz plot  for the direct and Induced terms in re-scattering \\cite{Mikhasenko_private}.}\n\\label{Mikhashenko2}\n\\end{figure}\n\n\nFor three-body charmless {\\ensuremath{\\PB}}\\xspace decays, with a much larger phase space, the $ 2\\!+\\!1$ approximation would be expected to be less problematic. However, depending on the momentum distribution of the final-state particles in the Dalitz plot, the validity of the approximation is better  in some regions than in others. \nFor instance, in the {\\ensuremath{\\PB}}\\xspace-meson rest frame, the corners of the Dalitz plot (see Fig.~\\ref{Dalitz_momentum}) correspond to a configuration in which one particle is at rest. The midway between the corners corresponds to a configuration where two of the particles are colinear, thus having the same scalar momentum as the third particle. The centre of the Dalitz plot has all three particles with the same scalar momentum. Thus the validity of the $2$+$1$ approximation may be  better in some regions than in others. The participation of the third particle in the two-body interaction can change the two-body strong phase over the Dalitz plot and should be taken into account in {\\ensuremath{C\\!P}}\\xspace violation studies.\n\nThe characteristics of the three-body momentum distribution over the phase space for the specific process of $ {\\ensuremath{\\Bu}}\\xspace\\ensuremath{\\rightarrow}\\xspace{\\ensuremath{\\pion^-}}\\xspace{\\ensuremath{\\pion^+}}\\xspace\\pip$ have been addressed in Refs.~\\cite{Mannel1,Mannel2}. The authors show that the peripheral regions of the Dalitz plot, where light resonances appear, are essentially dominated by non-perturbative effects. On the other hand, the central region of the Dalitz is dominated by large  momentum transfer, allowing for a quasi-perturbative treatment of QCD. \n\n\\begin{figure}[htb]\n\\centering\n\\includegraphics[width=.25\\columnwidth,angle=0]{Figure9_Dalitz_momentum.png}\n\\caption{A sketch of the relative momenta  for particles 1, 2 and 3 in the Dalitz plot constructed by $s_{12}$ and $s_{23}$ invariants. Taken from~\\cite{Bediaga:2012up}.}\n\\label{Dalitz_momentum}\n\\end{figure}\n\n    \n\\subsubsection{The {\\it Miranda} techniques}    \n\\label{miranda}\n\nThe use of amplitude analyses allows to access amplitude differences -- both their magnitudes and phases -- between particle and antiparticle decays, but relies on an adequate modelling of the possible intermediate states and, as discussed in Sect.~\\ref{AmAn},  the description of three- and also four-body decays is a complicated task with non-perturbative QCD effects possibly playing a major role depending on the region of phase space.  \nThe study of {\\ensuremath{C\\!P}}\\xspace violation in multi-decays can also be carried out by model-independent techniques by comparing the distribution of events for particle and antiparticle over the phase space. \n\nAs a search tool for {\\ensuremath{C\\!P}}\\xspace violation, which is particularly relevant for {\\ensuremath{\\PD}}\\xspace decays, the {\\it Miranda} technique~\\cite{bigi1} consists in dividing the phase space in bins of the relevant variables (for instance, the two Dalitz plot variables in a three-body decay) and for each bin $i$ the number of observed events $N_i(P)$ and $N_i(\\bar P)$ for the decay of particle $P$ and its {\\ensuremath{C\\!P}}\\xspace conjugate process, respectively,  are obtained. The bin sizes can be uniform across the phase space, or adaptively defined according to a given criterion (for example, number of events in each bin). An asymmetry significance is calculated for each bin\n\\begin{equation}\nS_{\\ensuremath{C\\!P}}\\xspace(i) = \\frac{N^i(P) -\\alpha N^i(\\bar P)}{\\sqrt{\\alpha(N^i(P) +N^i(\\bar P))}}~,\n\\label{SCP}\n\\end{equation}\nwhere  the factor \\mbox{$\\alpha = \\sum_i N^i(P)\/\\sum_i N^i (\\bar P)$} is used to normalise the total sample size of $\\bar P$ decays to that of $P$. It serves to correct for any global asymmetries,  such as those coming from production and detection effects, but also includes the net effect of the integrated {\\ensuremath{C\\!P}}\\xspace asymmetry.  The method is highly sensitive to {\\it local} asymmetries, expected to be more pronounced that the integrated one. Under  {\\ensuremath{C\\!P}}\\xspace conservation, the distribution of $S_{\\ensuremath{C\\!P}}\\xspace(i)$ follows a Gaussian distribution, with mean and width equal to zero and one, respectively.\\footnote{The number of entries in each bin is ensured to be large enough to guarantee Gaussian uncertainties.} Any deviation is indication of {\\ensuremath{C\\!P}}\\xspace violation. In this case, the distribution pattern of $S_{\\ensuremath{C\\!P}}\\xspace(i)$ through the phase space evidences the regions where the effect appears. A p-value for the {\\ensuremath{C\\!P}}\\xspace conservation hypothesis can be obtained by constructing a $\\chi^2 \\equiv \\sum_i S^2_{\\ensuremath{C\\!P}}\\xspace(i)$ and the number of degrees of freedom, which is equal to the number of bins minus one (due to the $\\alpha$ normalisation). Figure~\\ref{mirandaD3pi_original}(a) shows a typical Miranda distribution where a simulated sample of about 1 million ${\\ensuremath{\\PD}}\\xspace^\\pm \\ensuremath{\\rightarrow}\\xspace\\pi^\\pm{\\ensuremath{\\pion^-}}\\xspace{\\ensuremath{\\pion^+}}\\xspace$  decays with a 1\\% ($3.6^\\circ$) phase difference is included in the $\\rho(770)^0{\\ensuremath{\\pion^+}}\\xspace$ component with respect to that of $\\rho(770)^0{\\ensuremath{\\pion^-}}\\xspace$~\\cite{bigi1}. \n\nIn the case of presence of a measurable {\\ensuremath{C\\!P}}\\xspace violation effect, the method above is very sensitive to it, giving p-values incompatible with the null hypothesis, and it is possible to distinguish the regions in the Dalitz plot with substantial $S_{\\ensuremath{C\\!P}}\\xspace$ values, but otherwise gives no quantification. A ``second generation'' Miranda technique~\\cite{bigi2} is dedicated to the measurement of {\\ensuremath{C\\!P}}\\xspace asymmetries across the phase space. The method strengthens local effects by obtaining asymmetries through the division of the Dalitz plot in bins of equal population for the combined samples of particle and antiparticle decays: each bin has $ N_i^{\\rm tot} = N_i(P) + N_i (\\bar P)$ events. In Fig.~\\ref{2ndMiranda} a simulation of ${\\ensuremath{\\B^0}}\\xspace\\ensuremath{\\rightarrow}\\xspace{\\ensuremath{\\kaon^0_{\\rm\\scriptscriptstyle S}}}\\xspace{\\ensuremath{\\pion^+}}\\xspace{\\ensuremath{\\pion^-}}\\xspace$ decays is shown, where {\\ensuremath{C\\!P}}\\xspace violation was introduced as a $60^\\circ$ phase difference in the $\\rho(770)^0{\\ensuremath{\\kaon^0_{\\rm\\scriptscriptstyle S}}}\\xspace$ mode. In the left plot, localised positive and negative asymmetries are clearly seen, reaching values as high as 80\\%, although the total integrated asymmetry is only about 1\\%. In the right plot, the distribution of asymmetries is shown. This is a clear example evidencing how {\\ensuremath{C\\!P}}\\xspace violation effects can be substantially stronger in regions of the phase space when compared to the total, integrated, {\\ensuremath{C\\!P}}\\xspace effect. \n\\begin{figure}[htb]\n\\centering\n\\includegraphics[width=.6\n\\columnwidth,angle=0]{Figure10_MirandaD3pi_original.png}\n\\caption{Example of $S_{\\ensuremath{C\\!P}}\\xspace$ distribution for simulated ${\\ensuremath{\\PD}}\\xspace^\\pm \\ensuremath{\\rightarrow}\\xspace\\pi^\\pm{\\ensuremath{\\pion^-}}\\xspace{\\ensuremath{\\pion^+}}\\xspace$ decays where a 1\\% phase difference was introduced between $\\rho(770)^0\\pi^+$ and $\\rho(770)^0{\\ensuremath{\\pion^-}}\\xspace$ amplitudes. From the top-left plot clockwise: $S_{\\ensuremath{C\\!P}}\\xspace$ across the Dalitz plot, distribution of $S_{\\ensuremath{C\\!P}}\\xspace$ for the whole Dalitz plot, $S_{\\ensuremath{C\\!P}}\\xspace$ for regions II, and I, as taken from Ref.~\\cite{bigi1}. }\n\\label{mirandaD3pi_original}\n\\end{figure}     \n\nWe discuss the use of these techniques in Sects.~\\ref{chapter4} and \\ref{chapter6}.\n\n\n\\begin{figure}[htb]\n\\centering\n\\includegraphics[width=.45\\columnwidth, height =.3\\columnwidth, angle=0]{Figure11_2ndMiranda_ACP}\\includegraphics[width=.45\\columnwidth,height =.3\\columnwidth,angle=0]{Figure11_2ndMiranda_ACPdistr}\n\\caption{(left) Charge asymmetries across the Dalitz plot from  simulated  ${\\ensuremath{\\B^0}}\\xspace\\ensuremath{\\rightarrow}\\xspace {\\ensuremath{\\kaon^0_{\\rm\\scriptscriptstyle S}}}\\xspace{\\ensuremath{\\pion^+}}\\xspace{\\ensuremath{\\pion^-}}\\xspace$ decays; (right) distribution of the charge asymmetries in different regions of the phase space~\\cite{bigi2}.}\n\\label{2ndMiranda}\n\\end{figure}       \n          \n\\subsubsection{Unbinned techniques}\n\\label{unbinned}\n\nThe search for {\\ensuremath{C\\!P}}\\xspace violation within the phase space in a model-independent way can also be performed through unbinned, event-by-event techniques. One such technique is the  so-called {\\it energy test} \\cite{Aslan_energytest1_2005,Aslan_energytest2_2005,Williams_energytest_2011}. The method relies on a test statistic, $T$,  for a comparison of distances of pairs of events in the phase space,\n\\begin{equation}\nT = \\sum_{i,j>i}^N \\frac{\\psi_{ij}}{N(N-1)} + \\sum_{i,j>i}^{\\bar N} \\frac{\\psi_{ij}}{\\bar N(\\bar N-1)} - \\sum_{i}^N \\sum_j^{\\bar N}\\frac{\\psi_{ij}}{N\\bar N}. \n\\label{T_energytest}\n\\end{equation}\nThe first and second terms correspond to the average weighted-distance between events in the sample of particle $P$ (with $N$ total events) and antiparticle $\\bar P$ ($\\bar N$ events), respectively. The third term measures the same quantity but now between $P$ and $\\bar P$ events. The metric function $\\psi_{ij}$ is usually chosen as a Gaussian, $\\psi_{ij} = e^{-d^2_{ij}\/2\\delta^2}$ where $d_{ij}$ is the ``distance'' (measured in \\ensuremath{{\\mathrm{\\,Ge\\kern -0.1em V^2\\!\/}c^4}}\\xspace) between the $i$ and $j$ events in the phase space. The parameter $\\delta$ is an effective radius which defines the region where the distributions for $P$ and $\\bar P$  are being compared. If localised {\\ensuremath{C\\!P}}\\xspace asymmetries are present, the last term in Eq.~\\ref{T_energytest} is smaller, since the average distances are larger, when compared to a scenario with no asymmetries, and the $T$ value will be larger. To do this comparison and access a p-value for the hypothesis of {\\ensuremath{C\\!P}}\\xspace  conservation, the nominal $T$ value is compared to a distribution of $T$ values obtained from permutation samples (associating randomly the ``flavour'' -- $P$ or $\\bar P$ -- to all events for each permutation sample). When compared to the binned technique, the energy test has been shown to have equivalent or higher sensitivity to {\\ensuremath{C\\!P}}\\xspace violation effects~\\cite{Williams_energytest_2011}.\n\nAn alternative unbinned technique is the k-nearest neighbour (kNN) method~\\cite{Henze_kNN_1988,Schilling_kNN_1986}. Instead of using a metric function to define an effective region around each event, a given number of  nearest neighbours is defined and among them  the number of particle and antiparticle events are evaluated. Similar to  the energy-test method, a test statistic $T$ is constructed for the whole phase space (or for phase-space regions) \n\\begin{equation}\nT = \\frac{1}{n_k(N+\\bar N)}\\sum\\limits_{i=1}^{N+\\bar N}\\sum\\limits_{k=1}^{n_k}I(i,k),\n\\end{equation}\nwhere  $I(i,k)=1$ if the $i^{\\rm th} $event and its $k^{\\rm th}$nearest neighbour have\nthe same charge and $I(i,k)=0$ otherwise;  $N$ ($\\bar N$) is the number of $P$ ($\\bar P$) events. The test statistic $T$ is the mean fraction of like-charged neighbour pairs\nin the combined $P$ and $\\bar P$  samples. The advantage of \nthe kNN method, in comparison with energy-test, is that the calculation of $T$ is simple and fast \nand the expected distribution of $T$ is well known: for the null hypothesis \nit follows a Gaussian distribution with mean $\\mu_T$ and \nvariance $\\sigma^2_T$  calculated from known parameters of the distributions. A p-value for the null hypothesis can then be obtained. \n\nBoth unbinned methods have been used for searches of {\\ensuremath{C\\!P}}\\xspace violation in three- and four-body charm decays, as discussed in Sect.~\\ref{chapter6}. As in the case of the binned Miranda method given by Eq.~\\ref{SCP}, these methods are search tools and  rely on the assumption that potential nuisance asymmetries, those coming from production or detection effects, do not vary across the phase space. Typically control modes, for which {\\ensuremath{C\\!P}}\\xspace violation signals are not expected, are studied prior to the modes of interest, to guarantee this premise.  \n\\subsubsection{Triple products}\n\\label{tripleproducts}\n\nFor four-body decays or for baryon three-body decays (due to the baryon polarisation), other {\\ensuremath{C\\!P}}\\xspace-violating observables can be accessed. A triple-product in terms of the final-state particle momenta can be constructed,  \n\\begin{equation}\nC_T \\equiv \\vec{p}_1\\cdot {\\vec p}_2 \\times {\\vec p}_3, \n\\label{C_T}\n\\end{equation}\nwhere subscripts $1$, $2$ and $3$ represent the final-state particles (three out of three for baryon decays, or three out of four for meson decays). The $C_T$ quantity constitutes a T-odd observable ---  which reverses sign under reversal of momenta and spin of all involved particles (it is not the same as a true time reversal odd variable). For a particle $P$ (meson or baryon) decaying to a final state $f$, the triple product T-odd asymmetry  is defined as \n\\begin{equation}\nA_T = \\frac{\\Gamma(P\\ensuremath{\\rightarrow}\\xspace f; C_T>0) - \\Gamma(P\\ensuremath{\\rightarrow}\\xspace f;C_T<0)}{\\Gamma(P\\ensuremath{\\rightarrow}\\xspace f;C_T>0) + \\Gamma(P\\ensuremath{\\rightarrow}\\xspace f;C_T<0)}\n\\label{A_T}\n\\end{equation}\nand analogously for the $\\bar P$ process, \n\\begin{equation}\n{\\bar A}_T = \\frac{\\Gamma(\\bar P\\ensuremath{\\rightarrow}\\xspace \\bar f; -C_T>0) - \\Gamma(\\bar P\\ensuremath{\\rightarrow}\\xspace \\bar f;-C_T<0)}{\\Gamma(\\bar P\\ensuremath{\\rightarrow}\\xspace \\bar f;-C_T>0) + \\Gamma(\\bar P\\ensuremath{\\rightarrow}\\xspace \\bar f;-C_T<0)}.\n\\label{ATbar}\n\\end{equation}\nIn the presence of final-state interactions, $A_T$ and ${\\bar A}_T$ are generally different from zero even in the absence of {\\ensuremath{C\\!P}}\\xspace violation, but a  {\\ensuremath{C\\!P}}\\xspace violation observable can be constructed through the  asymmetry \n\\begin{equation}\na_{\\ensuremath{C\\!P}}\\xspace^{T-\\rm odd} \\equiv \\frac{1}{2}\\left(A_T - {\\bar A}_T\\right), \n\\label{a_CP_Todd}\n\\end{equation}\nwhich, under {\\ensuremath{C\\!PT}}\\xspace conservation,  is different from zero if {\\ensuremath{C\\!P}}\\xspace is violated --- even in the absence of strong phases. Indeed, not relying on strong phases constitutes a clear advantage of this {\\ensuremath{C\\!P}}\\xspace observable, making this kind of measurement complementary to the usual asymmetries. Moreover, by construction $a_{\\ensuremath{C\\!P}}\\xspace^{T-\\rm odd}$ as well as $A_T$ and ${\\bar A}_T$ are insensitive to production and detection asymmetries.\n\nIn Sect.~\\ref{b-baryons} and \\ref{Dto4body} results of searches for {\\ensuremath{C\\!P}}\\xspace violation using triple products are presented.  \n  \\section{Direct {\\ensuremath{C\\!P}}\\xspace violation in charmless {\\ensuremath{\\PB}}\\xspace decays}     \n\\label{chapter4}\n\nFor $b$-hadron decays, $b\\ensuremath{\\rightarrow}\\xspace u$ transitions -- leading to charmless final states --   are highly suppressed due to the small value of $|V_{ub}|$, when compared to $b\\ensuremath{\\rightarrow}\\xspace c$ processes. The weak phase $\\gamma$ enters at the tree-level amplitude. Penguin transitions  leading to the same final states can be of comparable size as the tree amplitude due to the  top quark in the loop, thus interference terms for which {\\ensuremath{C\\!P}}\\xspace violation could be assessed are naively expected to be large, depending on the size of the strong phases.\n\nThere are a variety of two-, three- and multi-body charmless final states from $b$-hadron decays. The measurements of {\\ensuremath{C\\!P}}\\xspace violation effects for these channels make use of some of the techniques described in the previous section. For two-body decays, $A_{{\\ensuremath{C\\!P}}\\xspace}$ is obtained  directly from the event yields from particle and antiparticle decays, correcting for nuisance asymmetries when applicable. The same goes for phase-space integrated measurements of multi-body decays, while for resonant intermediate states or for asymmetries in localised regions of the phase space the use of amplitude analyses and model-independent techniques is required. \n\n\n\\subsection{Two-body decays}\n\n\nAlthough there are many two-body charmless {\\ensuremath{\\PB}}\\xspace decays, for only a very few of them the measured {\\ensuremath{C\\!P}}\\xspace asymmetry is statistically significant. In 2004,  BaBar \\cite{BaBar_CPKpi} and Belle \\cite{Belle_CPKpi} collaborations presented the first experimental evidence for {\\ensuremath{C\\!P}}\\xspace violation in $  {\\ensuremath{\\B^0}}\\xspace \\ensuremath{\\rightarrow}\\xspace {\\ensuremath{\\kaon^+}}\\xspace {\\ensuremath{\\pion^-}}\\xspace$  decay,  with a statistical significance around $4 \\sigma$ (where $\\sigma$ is one standard deviation). It was only in 2012 that  the BaBar collaboration \\cite{BaBar2012}  observed this decay channel with more than $5 \\sigma$, followed by direct {\\ensuremath{C\\!P}}\\xspace asymmetry observation of  {\\ensuremath{\\B^0_\\squark}}\\xspace to the same final state from the LHCb collaboration \\cite{LHCb-PAPER-2013-018}.  The ${\\ensuremath{\\kaon^+}}\\xspace{\\ensuremath{\\pion^-}}\\xspace$ and ${\\ensuremath{\\kaon^-}}\\xspace{\\ensuremath{\\pion^+}}\\xspace$ mass spectra obtained are shown in Fig.~\\ref{LHCB_CP_BsKpi} where one can clearly see that the yields for both {\\ensuremath{\\B^0}}\\xspace and {\\ensuremath{\\B^0_\\squark}}\\xspace are higher that those for {\\ensuremath{\\Bbar{}^0}}\\xspace and {\\ensuremath{\\Bbar{}^0_\\squark}}\\xspace.     \n\\begin{figure}[!htb]\n\\centering\n\\includegraphics[width=.65\\columnwidth]{Figure12_BKpi_LHCb.png}\n\\caption{Invariant-mass distributions for (a) ${\\ensuremath{\\B^0}}\\xspace\\ensuremath{\\rightarrow}\\xspace{\\ensuremath{\\kaon^+}}\\xspace {\\ensuremath{\\pion^-}}\\xspace$, (b) ${\\ensuremath{\\Bbar{}^0}}\\xspace\\ensuremath{\\rightarrow}\\xspace{\\ensuremath{\\kaon^-}}\\xspace {\\ensuremath{\\pion^+}}\\xspace$, (c) ${\\ensuremath{\\B^0_\\squark}}\\xspace\\ensuremath{\\rightarrow}\\xspace{\\ensuremath{\\kaon^-}}\\xspace{\\ensuremath{\\pion^-}}\\xspace$ and (d) ${\\ensuremath{\\Bbar{}^0_\\squark}}\\xspace\\ensuremath{\\rightarrow}\\xspace{\\ensuremath{\\kaon^-}}\\xspace{\\ensuremath{\\pion^+}}\\xspace$. The legend describes the fit components \\cite{LHCb-PAPER-2013-018}. }\n\\label{LHCB_CP_BsKpi}\n\\end{figure}\n\nThe best measurement for these decays, with {\\ensuremath{C\\!P}}\\xspace violation fully established, comes from the recent update from the LHCb experiment \\cite{LHCb-PAPER-2018-006}:\n\\begin{eqnarray}\nA_{{\\ensuremath{C\\!P}}\\xspace}(  {\\ensuremath{\\B^0}}\\xspace \\ensuremath{\\rightarrow}\\xspace {\\ensuremath{\\kaon^+}}\\xspace {\\ensuremath{\\pion^-}}\\xspace)= -0.084 \\pm 0.004 \\pm 0.003~,\\nonumber \\\\\nA_{{\\ensuremath{C\\!P}}\\xspace}(  {\\ensuremath{\\B^0_\\squark}}\\xspace \\ensuremath{\\rightarrow}\\xspace {\\ensuremath{\\kaon^+}}\\xspace {\\ensuremath{\\pion^-}}\\xspace )= 0.213 \\pm 0.015\\pm 0.007~,\n\\label{ACP_Bd}\n\\end{eqnarray} \nwhere the first uncertainty is statistical and the second systematic.\\footnote{Unless otherwise stated, whenever measurements appear with two uncertainties, the first is statistical and the second systematic.}\n\nThe theoretical prediction from  NNLO approach  of the {\\ensuremath{C\\!P}}\\xspace asymmetry for the  ${\\ensuremath{\\B^0}}\\xspace \\ensuremath{\\rightarrow}\\xspace {\\ensuremath{\\kaon^+}}\\xspace {\\ensuremath{\\pion^-}}\\xspace$  decay is $(8.08 \\pm1.52 \\pm 2.60)\\%$ (not including long-distance effects)\\cite{Beneke2015}.\\footnote{Curiously enough, this prediction   is very close to one made 30 years ago in Ref.~\\cite{Bigi1987}.} The experimental value above, with opposite sign with respect to the theory prediction, opened  speculations about the possibility of new physics (NP) effects.  An interesting and robust relationship between the $ A_{{\\ensuremath{C\\!P}}\\xspace}$ of this decay and the $ A_{{\\ensuremath{C\\!P}}\\xspace}$ of ${\\ensuremath{\\B^0_\\squark}}\\xspace \\ensuremath{\\rightarrow}\\xspace {\\ensuremath{\\kaon^+}}\\xspace {\\ensuremath{\\pion^-}}\\xspace $ decay was proposed by Lipkin~\\cite{Lipkin2005} to test  whether the result was consistent with the SM.  Using U-spin symmetry \\cite{Gronau2000}, the interference terms from these two decays are expected to be very similar, producing the approximate equality  \\mbox{$ | A({\\ensuremath{\\B^0_\\squark}}\\xspace \\ensuremath{\\rightarrow}\\xspace \\pi^+ K^-)|^2 - | A(\\bar {\\ensuremath{\\B^0_\\squark}}\\xspace \\ensuremath{\\rightarrow}\\xspace \\pi^- K^+)|^2 \\approx | A(\\bar {\\ensuremath{\\B^0}}\\xspace \\ensuremath{\\rightarrow}\\xspace \\pi^+ K^-)|^2 - | A(  {\\ensuremath{\\B^0}}\\xspace \\ensuremath{\\rightarrow}\\xspace \\pi^- K^+)|^2 $}, which can be rewritten as \n\\begin{eqnarray}\n\\Delta =  \\frac{A_{{\\ensuremath{C\\!P}}\\xspace}(  {\\ensuremath{\\B^0}}\\xspace \\ensuremath{\\rightarrow}\\xspace {\\ensuremath{\\kaon^+}}\\xspace {\\ensuremath{\\pion^-}}\\xspace)}{A_{{\\ensuremath{C\\!P}}\\xspace}(  {\\ensuremath{\\B^0_\\squark}}\\xspace \\ensuremath{\\rightarrow}\\xspace {\\ensuremath{\\kaon^+}}\\xspace {\\ensuremath{\\pion^-}}\\xspace )} +  \\frac{ {\\cal B} (  {\\ensuremath{\\B^0_\\squark}}\\xspace \\ensuremath{\\rightarrow}\\xspace {\\ensuremath{\\kaon^+}}\\xspace {\\ensuremath{\\pion^-}}\\xspace ) \\tau_d}{ {\\cal B} (  {\\ensuremath{\\B^0}}\\xspace \\ensuremath{\\rightarrow}\\xspace {\\ensuremath{\\kaon^+}}\\xspace {\\ensuremath{\\pion^-}}\\xspace) \\tau_s} \\approx 0, \n\\label{Delta}\n\\end{eqnarray} \nwhere $\\tau_d$ and $\\tau_s$ are the {\\ensuremath{\\B^0}}\\xspace and {\\ensuremath{\\B^0_\\squark}}\\xspace mean lifetimes, respectively. Thus, from the measurements in  Eq.~\\ref{ACP_Bd},  the LHCb collaboration quotes $ \\Delta = 0.11\\pm 0.04 \\pm 0.03$ \\cite{LHCb-PAPER-2018-006}, which is consistent with Eq.~\\ref{Delta} within experimental uncertainties and favouring the SM nature of {\\ensuremath{C\\!P}}\\xspace violation in these decays.   \n\nA third charmless two-body {\\ensuremath{\\PB}}\\xspace decay with well established  direct {\\ensuremath{C\\!P}}\\xspace violation is the ${\\ensuremath{\\B^0}}\\xspace\\ensuremath{\\rightarrow}\\xspace{\\ensuremath{\\pion^+}}\\xspace{\\ensuremath{\\pion^-}}\\xspace$ mode. Since this final state is  self-{\\ensuremath{C\\!P}}\\xspace conjugate, the measurement was performed with the time-dependent formalism presented in Eq.~\\ref{CPasym_time}. The parameters ${\\cal C}_{{\\ensuremath{\\pion^+}}\\xspace{\\ensuremath{\\pion^-}}\\xspace}$ and ${\\cal S}_{{\\ensuremath{\\pion^+}}\\xspace{\\ensuremath{\\pion^-}}\\xspace}$  were measured by Belle \\cite{Belle_pipi}, BaBar \\cite{BaBar2012} and more recently by LHCb \\cite{LHCb-PAPER-2018-006} collaborations. \n\nThe time-dependent distribution for $A_{\\ensuremath{C\\!P}}\\xspace$ obtained by the Belle collaboration is presented in Fig.~\\ref{Belle_time}, from where the parameters ${\\cal C}_{{\\ensuremath{\\pion^+}}\\xspace{\\ensuremath{\\pion^-}}\\xspace}$ and ${\\cal S}_{{\\ensuremath{\\pion^+}}\\xspace{\\ensuremath{\\pion^-}}\\xspace}$ were extracted.\\footnote{${\\ensuremath{\\B^0}}\\xspace$ and ${\\ensuremath{\\Bbar{}^0}}\\xspace$ are produced as entangled pairs at Belle, so $t$ in Eq.~\\ref{CPasym_time} is replaced by $\\Delta t$, the time elapsed since the other {\\ensuremath{\\PB}}\\xspace (used for tagging) decayed.}\n\\begin{figure}[htbp]\n\\centering\n\\includegraphics[width=.4\\columnwidth,angle=0]{Figure13_BelleBpipi.png}\n\\caption{ Background subtracted time-dependent fit results for $B^0 \\ensuremath{\\rightarrow}\\xspace \\pi^+\\pi^ -$  \\cite{Belle_pipi}. (a) $\\Delta t$ distribution for {\\ensuremath{\\B^0}}\\xspace (solid blue line) and {\\ensuremath{\\Bbar{}^0}}\\xspace (dashed red line) (b) Time-dependent asymmetry  distribution.}\n\\label{Belle_time}\n\\end{figure}\n\n\n\nThe combined results from HFLAV \\cite{HFLAV19} are shown in Fig.~\\ref{Bpipi_HFLAV}  with the average result for direct {\\ensuremath{C\\!P}}\\xspace violation  ${\\cal C}_{{\\ensuremath{\\pion^+}}\\xspace{\\ensuremath{\\pion^-}}\\xspace}=-0.32 \\pm 0.04$ .\n\n\\begin{figure}[htbp]\n\\centering\n\\includegraphics[width=.4\\columnwidth,angle=0]{Figure14_pi+pi-S_CP.png}\n\\includegraphics[width=.4\\columnwidth,angle=0]{Figure14_pi+pi-C_CP.png}\n\\caption{ Averages of (left)   ${\\cal S}_{{\\ensuremath{\\pion^+}}\\xspace{\\ensuremath{\\pion^-}}\\xspace}$ and (right) ${\\cal C}_{{\\ensuremath{\\pion^+}}\\xspace{\\ensuremath{\\pion^-}}\\xspace}$ for ${\\ensuremath{\\B^0}}\\xspace\\ensuremath{\\rightarrow}\\xspace{\\ensuremath{\\pion^+}}\\xspace{\\ensuremath{\\pion^-}}\\xspace$ decay mode \\cite{HFLAV19}.}\n\\label{Bpipi_HFLAV}\n\\end{figure}\n\n\nMany other two-body channels have been studied, with just one of them presenting  a {\\ensuremath{C\\!P}}\\xspace asymmetry with more than $3\\sigma$ significance: $A_{{\\ensuremath{C\\!P}}\\xspace}({\\ensuremath{\\Bu}}\\xspace\\ensuremath{\\rightarrow}\\xspace{\\ensuremath{\\kaon^+}}\\xspace\\eta)$ has been measured by Belle  \\cite{Belle_CPKeta} and BaBar \\cite{BaBar_CPKeta}, with consistent values. The current average values for $A_{\\ensuremath{C\\!P}}\\xspace$ from the PDG~\\cite{PDG2019} for different channels are reproduced in Table~\\ref{ACP_2body}.\n\n\n\\begin{table}[htpb]\n\\begin{center}\n\\caption{Average $A_{\\ensuremath{C\\!P}}\\xspace$ values for two-body charmless {\\ensuremath{\\PB}}\\xspace decays \\cite{PDG2019}.}\n\\begin{tabular}{| l c |}\\hline\nChannel & $A_{\\ensuremath{C\\!P}}\\xspace$ \\\\ \\hline \\hline\n${\\ensuremath{\\Bu}}\\xspace \\ensuremath{\\rightarrow}\\xspace {\\ensuremath{\\kaon^0_{\\rm\\scriptscriptstyle S}}}\\xspace {\\ensuremath{\\pion^+}}\\xspace$ &  $-0.017 \\pm 0.016$ \\\\ \\hline\n${\\ensuremath{\\Bu}}\\xspace \\ensuremath{\\rightarrow}\\xspace {\\ensuremath{\\kaon^+}}\\xspace {\\ensuremath{\\pion^0}}\\xspace$   & $ -0.037 \\pm 0.021$\\\\ \\hline\n${\\ensuremath{\\Bu}}\\xspace \\ensuremath{\\rightarrow}\\xspace {\\ensuremath{\\kaon^+}}\\xspace \\eta'$   &  $ 0.004 \\pm 0.011$\\\\ \\hline\n${\\ensuremath{\\Bu}}\\xspace \\ensuremath{\\rightarrow}\\xspace {\\ensuremath{\\kaon^+}}\\xspace \\eta$    & $ -0.37 \\pm 0.08$ \\\\ \\hline\n${\\ensuremath{\\Bu}}\\xspace \\ensuremath{\\rightarrow}\\xspace {\\ensuremath{\\kaon^+}}\\xspace {\\ensuremath{\\kaon^0}}\\xspace$    & $ -0.04 \\pm 0.14$\\\\ \\hline\n${\\ensuremath{\\Bu}}\\xspace \\ensuremath{\\rightarrow}\\xspace {\\ensuremath{\\kaon^+}}\\xspace {\\ensuremath{\\kaon^0_{\\rm\\scriptscriptstyle S}}}\\xspace$ &  $-0.21 \\pm 0.14$\\\\ \\hline\n${\\ensuremath{\\Bu}}\\xspace \\ensuremath{\\rightarrow}\\xspace {\\ensuremath{\\pion^+}}\\xspace {\\ensuremath{\\pion^0}}\\xspace$  &  $0.03 \\pm 0.04$\\\\ \\hline\n${\\ensuremath{\\Bu}}\\xspace \\ensuremath{\\rightarrow}\\xspace  {\\ensuremath{\\pion^+}}\\xspace\\eta$  &  $-0.14 \\pm 0.07$\\\\ \\hline\n${\\ensuremath{\\Bu}}\\xspace\\ensuremath{\\rightarrow}\\xspace {\\ensuremath{\\pion^+}}\\xspace\\eta'$  &   $ 0.06 \\pm 0.16$\\\\ \\hline \\hline\n $ {\\ensuremath{\\B^0}}\\xspace \\ensuremath{\\rightarrow}\\xspace {\\ensuremath{\\Kbar{}^0}}\\xspace{\\ensuremath{\\kaon^0}}\\xspace$ & $-0.6 \\pm 0.7$ \\\\ \\hline\n$ {\\ensuremath{\\B^0}}\\xspace \\ensuremath{\\rightarrow}\\xspace {\\ensuremath{\\kaon^+}}\\xspace {\\ensuremath{\\pion^-}}\\xspace$  & $-0.082 \\pm 0.006$ \\\\ \\hline \\hline\n$ {\\ensuremath{\\B^0_\\squark}}\\xspace \\ensuremath{\\rightarrow}\\xspace {\\ensuremath{\\kaon^+}}\\xspace {\\ensuremath{\\pion^-}}\\xspace $ & $ 0.26 \\pm 0.04$ \\\\ \\hline \\hline\n\\end{tabular}\n\\label{ACP_2body}\n\\end{center}\n\\end{table}\n\nFrom the theoretical point of view, branching fraction and direct {\\ensuremath{C\\!P}}\\xspace asymmetry estimates for charmless two-body {\\ensuremath{\\PB}}\\xspace decays are challenging.  For branching fractions, there is quite  reasonable agreement between calculations based on factorisation techniques and the experimental results, as can be found in Ref.~\\cite{ChengChiangKuo2014}. Yet, for  {\\ensuremath{C\\!P}}\\xspace asymmetries the  consistency between calculations and measured values is currently not so good. Despite the theoretical developments and the experimental improvements over the last decade, the origin of the strong phase leading to {\\ensuremath{C\\!P}}\\xspace asymmetries is not clear yet~\\cite{Beneke2015}. \n\n\nOne crucial example is the difference between the well-measured {\\ensuremath{C\\!P}}\\xspace asymmetry in the  $ {\\ensuremath{\\Bbar{}^0}}\\xspace \\ensuremath{\\rightarrow}\\xspace {\\ensuremath{\\kaon^-}}\\xspace{\\ensuremath{\\pion^+}}\\xspace$  and  $ {\\ensuremath{\\Bu}}\\xspace \\ensuremath{\\rightarrow}\\xspace {\\ensuremath{\\kaon^+}}\\xspace{\\ensuremath{\\pion^0}}\\xspace$ channels.   From the short-distance point of view, these decay channels are equivalent --- the dominant quark-level diagrams coincide with the only difference being the  spectator quark, ${\\ensuremath{\\overline \\dquark}}\\xspace$ quark for {\\ensuremath{\\Bbar{}^0}}\\xspace and ${\\ensuremath{\\overline \\uquark}}\\xspace$ quark for {\\ensuremath{\\Bu}}\\xspace and the colour-suppressed tree contribution. As such,  in principle they would have similar behaviour with respect to {\\ensuremath{C\\!P}}\\xspace violation, in particular one should expect $\\Delta A_{\\ensuremath{C\\!P}}\\xspace({\\ensuremath{\\PB}}\\xspace\\ensuremath{\\rightarrow}\\xspace \\PK\\pi) \\equiv  A_{{\\ensuremath{C\\!P}}\\xspace}( {\\ensuremath{\\B^0}}\\xspace \\ensuremath{\\rightarrow}\\xspace {\\ensuremath{\\kaon^-}}\\xspace{\\ensuremath{\\pion^+}}\\xspace) -A_{{\\ensuremath{C\\!P}}\\xspace}({\\ensuremath{\\Bu}}\\xspace \\ensuremath{\\rightarrow}\\xspace {\\ensuremath{\\kaon^+}}\\xspace{\\ensuremath{\\pion^0}}\\xspace)\\approx 0$. Instead, it is measured to be $\\Delta A_{{\\ensuremath{C\\!P}}\\xspace}({\\ensuremath{\\PB}}\\xspace\\ensuremath{\\rightarrow}\\xspace \\PK\\pi)= -0.122 \\pm 0.022$ ~\\cite{PDG2019}. This result also seems to point towards a large influence of long-distance, final-state interaction effects, with hadronic re-scattering processes taking a leading role.                     \n                       \n                        Invoking the {\\ensuremath{C\\!PT}}\\xspace constraint, as highlighted by Wolfenstein \\cite{Wolfenstein1990} and discussed in Sect.~\\ref{chapter2_CP_CPT},  the significant negative {\\ensuremath{C\\!P}}\\xspace asymmetry observed in $ {\\ensuremath{\\B^0}}\\xspace \\ensuremath{\\rightarrow}\\xspace {\\ensuremath{\\kaon^-}}\\xspace{\\ensuremath{\\pion^+}}\\xspace$ decays has to be compensated by other channels in the same flavour family.  If one also assumes that  a two-body final state   would preferably rescatter to other two-body final states \\cite{Smith2003}, the most probable candidates to present a positive {\\ensuremath{C\\!P}}\\xspace asymmetry are the channels  ${\\ensuremath{\\kaon^0}}\\xspace{\\ensuremath{\\pion^0}}\\xspace$, ${\\ensuremath{\\kaon^0}}\\xspace\\eta$ and ${\\ensuremath{\\kaon^0}}\\xspace\\eta'$. The {\\ensuremath{\\B^0}}\\xspace decay to these final states also has a penguin contribution plus a colour-suppressed tree component carrying the weak phase $\\gamma$. For the decay ${\\ensuremath{\\Bu}}\\xspace \\ensuremath{\\rightarrow}\\xspace {\\ensuremath{\\kaon^+}}\\xspace{\\ensuremath{\\pion^0}}\\xspace$, the most probable rescattered two-body final states would be ${\\ensuremath{\\kaon^0}}\\xspace{\\ensuremath{\\pion^+}}\\xspace, {\\ensuremath{\\kaon^+}}\\xspace\\eta$ and ${\\ensuremath{\\kaon^+}}\\xspace\\eta'$. For these, however, the {\\ensuremath{\\Bu}}\\xspace decay has contribution from the penguin diagram only, carrying no weak phase.  This means that, if {\\ensuremath{C\\!P}}\\xspace violation is observed in any of these final states, it can have its origin from rescattering effects, while preserving {\\ensuremath{C\\!PT}}\\xspace. \n                                       \n                                         \n It is not trivial, however, to estimate the impact of  rescattering effects in the aforementioned processes and consequently in the value of $\\Delta A_{{\\ensuremath{C\\!P}}\\xspace}({\\ensuremath{\\PB}}\\xspace \\ensuremath{\\rightarrow}\\xspace K\\pi)$.  But, in any case, it seems plausible that the presence of strong hadronic phases, appearing through final-state interactions and not taken into account in short-distance calculations, are key ingredients to understand the  discrepancies between experimental results and theory predictions --- although there are no available theory nor experimental results to support how important these FSI phases are, especially above 2 GeV.  The experimental measurement of {\\ensuremath{C\\!P}}\\xspace asymmetries for $ {\\ensuremath{\\B^0}}\\xspace \\ensuremath{\\rightarrow}\\xspace {\\ensuremath{\\kaon^0}}\\xspace{\\ensuremath{\\pion^0}}\\xspace$, $ {\\ensuremath{\\B^0}}\\xspace \\ensuremath{\\rightarrow}\\xspace {\\ensuremath{\\kaon^0}}\\xspace\\eta $, and  $ {\\ensuremath{\\B^0}}\\xspace \\ensuremath{\\rightarrow}\\xspace {\\ensuremath{\\kaon^0}}\\xspace\\eta'$ can shed  light on this puzzle.  In this respect, Belle II experiment~\\cite{BelleII_PhysicsBook_2018} will be the main actor. \n\n\n\n   \\subsection{Three-body decays}\n \\label{Bhhh}\n \n As discussed in Sect.~\\ref{multibody}, decays of three or more particles in the final state offer a richer environment for  the study of {\\ensuremath{C\\!P}}\\xspace violation, since the phase space can be scanned and observables other than the total {\\ensuremath{C\\!P}}\\xspace asymmetry exist. \n \n \n{\\ensuremath{C\\!P}}\\xspace violation was first observed in 2013 by the LHCb collaboration in local regions of the Dalitz plot for the charmless decays ${\\ensuremath{\\Bu}}\\xspace\\ensuremath{\\rightarrow}\\xspace {\\ensuremath{\\kaon^+}}\\xspace{\\ensuremath{\\pion^+}}\\xspace{\\ensuremath{\\pion^-}}\\xspace$, $ {\\ensuremath{\\Bu}}\\xspace\\ensuremath{\\rightarrow}\\xspace {\\ensuremath{\\kaon^+}}\\xspace\\Kp{\\ensuremath{\\kaon^-}}\\xspace$, ${\\ensuremath{\\Bu}}\\xspace\\ensuremath{\\rightarrow}\\xspace {\\ensuremath{\\pion^+}}\\xspace\\pip{\\ensuremath{\\pion^-}}\\xspace$ and  ${\\ensuremath{\\Bu}}\\xspace\\ensuremath{\\rightarrow}\\xspace {\\ensuremath{\\pion^+}}\\xspace{\\ensuremath{\\kaon^+}}\\xspace{\\ensuremath{\\kaon^-}}\\xspace$ ~\\cite{LHCb-PAPER-2013-027,LHCb-PAPER-2013-051}. Integrated {\\ensuremath{C\\!P}}\\xspace asymmetries were then measured by the same collaboration in 2014,  using the total available sample of $3\\ensuremath{\\mbox{\\,fb}^{-1}}\\xspace$ from run I. \nThe studies have shown sizeable integrated {\\ensuremath{C\\!P}}\\xspace asymmetries but also confirmed rich structures of {\\ensuremath{C\\!P}}\\xspace asymmetries across the phase space~\\cite{LHCb-PAPER-2014-044}. The results for the integrated {\\ensuremath{C\\!P}}\\xspace asymmetries were\n\\begin{eqnarray}\nA_{{\\ensuremath{C\\!P}}\\xspace}({\\ensuremath{\\Bu}}\\xspace\\ensuremath{\\rightarrow}\\xspace {\\ensuremath{\\kaon^+}}\\xspace{\\ensuremath{\\pion^+}}\\xspace{\\ensuremath{\\pion^-}}\\xspace) & = +0.025 \\pm 0.004 \\pm 0.004 \\pm 0.007,  \\nonumber  \\\\\nA_{{\\ensuremath{C\\!P}}\\xspace}( {\\ensuremath{\\Bu}}\\xspace\\ensuremath{\\rightarrow}\\xspace {\\ensuremath{\\kaon^+}}\\xspace\\Kp{\\ensuremath{\\kaon^-}}\\xspace) & = -0.036 \\pm 0.004 \\pm 0.002 \\pm 0.007, \\nonumber\\\\\nA_{{\\ensuremath{C\\!P}}\\xspace}( {\\ensuremath{\\Bu}}\\xspace\\ensuremath{\\rightarrow}\\xspace {\\ensuremath{\\pion^+}}\\xspace\\pip{\\ensuremath{\\pion^-}}\\xspace) & = +0.058 \\pm 0.008 \\pm 0.009 \\pm 0.007, \\nonumber\\\\\nA_{{\\ensuremath{C\\!P}}\\xspace}( {\\ensuremath{\\Bu}}\\xspace\\ensuremath{\\rightarrow}\\xspace {\\ensuremath{\\pion^+}}\\xspace{\\ensuremath{\\kaon^+}}\\xspace{\\ensuremath{\\kaon^-}}\\xspace) & = -0.123 \\pm 0.017 \\pm 0.012 \\pm 0.007, \n\\label{B3h_ACP}\n\\end{eqnarray}\nwhere the first uncertainties are statistical, the second systematic, and the third due to the limited knowledge of the {\\ensuremath{C\\!P}}\\xspace asymmetry in the ${\\ensuremath{\\Bu}}\\xspace\\ensuremath{\\rightarrow}\\xspace {\\ensuremath{\\kaon^+}}\\xspace{\\ensuremath{{\\PJ\\mskip -3mu\/\\mskip -2mu\\Ppsi\\mskip 2mu}}}\\xspace$, used as a reference mode. The statistical significances of the {\\ensuremath{C\\!P}}\\xspace asymmetries were respectively 2.8,  4.3, 4.2, and 5.6$\\sigma$.   The sample sizes were about 180 thousand decays for ${\\ensuremath{\\Bu}}\\xspace\\ensuremath{\\rightarrow}\\xspace {\\ensuremath{\\kaon^+}}\\xspace{\\ensuremath{\\pion^+}}\\xspace{\\ensuremath{\\pion^-}}\\xspace $, 110 thousand for ${\\ensuremath{\\Bu}}\\xspace\\ensuremath{\\rightarrow}\\xspace {\\ensuremath{\\kaon^+}}\\xspace\\Kp{\\ensuremath{\\kaon^-}}\\xspace  $, 25 thousand for $ {\\ensuremath{\\Bu}}\\xspace\\ensuremath{\\rightarrow}\\xspace {\\ensuremath{\\pion^+}}\\xspace\\pip{\\ensuremath{\\pion^-}}\\xspace $ and 6 thousand for $ {\\ensuremath{\\Bu}}\\xspace\\ensuremath{\\rightarrow}\\xspace {\\ensuremath{\\pion^+}}\\xspace{\\ensuremath{\\kaon^+}}\\xspace{\\ensuremath{\\kaon^-}}\\xspace $. \n\nThe {\\ensuremath{C\\!P}}\\xspace asymmetries across the phase space for the four channels were obtained through the second Miranda technique~\\cite{bigi2}, as presented in Sect.~\\ref{miranda}, and are reproduced here in Fig.~\\ref{Mirandizing4channels}. Rich {\\ensuremath{C\\!P}}\\xspace violation patterns can be seen along the Dalitz plots, with positive (red) and negative (blue) {\\ensuremath{C\\!P}}\\xspace asymmetries coexisting in the same final state. Since the source of weak phase for all these charmless {\\ensuremath{\\PB}}\\xspace decays is the CKM angle $\\gamma$, the {\\ensuremath{C\\!P}}\\xspace asymmetry variations over the Dalitz plot have to come from the variation of the strong phase. The Miranda distributions evidence the richness involved in the {\\ensuremath{C\\!P}}\\xspace violation dynamics of these decays.   A discussion on these features follows. \n\\begin{figure}[!htb]\n\\centering\n \\includegraphics[width=.4\\columnwidth,angle=0]{Figure15_KKK_LHCb.png} \n\\includegraphics[width=.4\\columnwidth,angle=0]{Figure15_Kpipi_LHCb.png}  \\\\\n \\includegraphics[width=.4\\columnwidth,angle=0]{Figure15_pipipi_LHCb.png} \n \\includegraphics[width=.4\\columnwidth,angle=0]{Figure15_KKpi_LHCb.png} \n\\caption{{\\ensuremath{C\\!P}}\\xspace asymmetry distributions with the Miranda technique, with  background-subtracted and acceptance-corrected events for: (top left) ${\\ensuremath{\\Bu}}\\xspace\\ensuremath{\\rightarrow}\\xspace{\\ensuremath{\\kaon^+}}\\xspace\\Kp{\\ensuremath{\\kaon^-}}\\xspace$, (top right) ${\\ensuremath{\\Bu}}\\xspace\\ensuremath{\\rightarrow}\\xspace{\\ensuremath{\\kaon^+}}\\xspace{\\ensuremath{\\pion^+}}\\xspace{\\ensuremath{\\pion^-}}\\xspace$, (bottom left)  ${\\ensuremath{\\Bu}}\\xspace\\ensuremath{\\rightarrow}\\xspace{\\ensuremath{\\pion^+}}\\xspace\\pip{\\ensuremath{\\pion^-}}\\xspace$ and (bottom right)   ${\\ensuremath{\\Bu}}\\xspace\\ensuremath{\\rightarrow}\\xspace{\\ensuremath{\\pion^+}}\\xspace{\\ensuremath{\\kaon^+}}\\xspace{\\ensuremath{\\kaon^-}}\\xspace$ \\cite{LHCb-PAPER-2014-044}. }\n\\label{Mirandizing4channels}\n\\end{figure}\n\n\n\n\n\n\\subsubsection{Interference between different angular-momentum waves}     \n\\label{CPfromcostheta}\n\nBy inspecting the low ${\\ensuremath{\\pion^+}}\\xspace{\\ensuremath{\\pion^-}}\\xspace$ invariant-mass region in both ${\\ensuremath{\\Bu}}\\xspace\\ensuremath{\\rightarrow}\\xspace{\\ensuremath{\\kaon^+}}\\xspace{\\ensuremath{\\pion^+}}\\xspace{\\ensuremath{\\pion^-}}\\xspace$ and  ${\\ensuremath{\\Bu}}\\xspace\\ensuremath{\\rightarrow}\\xspace{\\ensuremath{\\pion^+}}\\xspace\\pip{\\ensuremath{\\pion^-}}\\xspace$ Dalitz plots, the results from the LHCb experiment \\cite{LHCb-PAPER-2014-044} show a peculiar {\\ensuremath{C\\!P}}\\xspace asymmetry distribution, with  positive and negative $A_{{\\ensuremath{C\\!P}}\\xspace}$ around the $\\rho(770)^0$ mass. As discussed in Sect.~\\ref{AmAn} (see Eq.~\\ref{AcpAmplitude}), this is a signature from  the interference term between scalar and vector amplitudes, proportional to $\\cos \\theta $. This can be better seen in Fig.~\\ref{SP_CPdistribution}, where the  ${\\ensuremath{\\pion^+}}\\xspace{\\ensuremath{\\pion^-}}\\xspace$ invariant-mass combination\\footnote{In ${\\ensuremath{\\Bu}}\\xspace\\ensuremath{\\rightarrow}\\xspace{\\ensuremath{\\pion^+}}\\xspace\\pip{\\ensuremath{\\pion^-}}\\xspace$, with two identical pions, $m({\\ensuremath{\\pion^+}}\\xspace {\\ensuremath{\\pion^-}}\\xspace)_{\\rm low\\,(high)}$  is the lower (higher) ${\\ensuremath{\\pion^+}}\\xspace{\\ensuremath{\\pion^-}}\\xspace$ mass among the two possible combinations. The same subscript $\\rm low\\, (high)$ is used for other decays with identical particles.} for $\\cos\\theta<0$ and $\\cos\\theta>0$ are shown for both ${\\ensuremath{\\Bu}}\\xspace\\ensuremath{\\rightarrow}\\xspace{\\ensuremath{\\pion^+}}\\xspace\\pip{\\ensuremath{\\pion^-}}\\xspace$ and ${\\ensuremath{\\Bub}}\\xspace\\ensuremath{\\rightarrow}\\xspace{\\ensuremath{\\pion^-}}\\xspace\\pim{\\ensuremath{\\pion^+}}\\xspace$ decays --- the subtraction of the histograms is also shown, evidencing the charge distribution difference. \nBesides, the observed zeros in the bottom plots in Fig.\\ref{SP_CPdistribution} around the $\\rho(770)^0$ nominal mass indicate a dominance of the real part of the Breit--Wigner in this process. \n       \n\\begin{figure}[!htb]\n\\centering\n \\includegraphics[width=.4\\columnwidth,angle=0]{Figure16_a.png} \n\\includegraphics[width=.4\\columnwidth,angle=0]{Figure16_b.png}  \\\\\n \\includegraphics[width=.4\\columnwidth,angle=0]{Figure16_c.png} \n \\includegraphics[width=.4\\columnwidth,angle=0]{Figure16_d.png} \n\\caption{Projections of the $m({\\ensuremath{\\pion^+}}\\xspace {\\ensuremath{\\pion^-}}\\xspace)_{\\rm low}$  variable  for  $B^\\pm \\ensuremath{\\rightarrow}\\xspace \\pi^\\pm {\\ensuremath{\\pion^+}}\\xspace{\\ensuremath{\\pion^-}}\\xspace  $  signal events  separated by charge for (top left) $\\cos \\theta < 0 $ and (top right)  $\\cos \\theta > 0$. The respective subtracted  {\\ensuremath{\\Bu}}\\xspace and {\\ensuremath{\\Bub}}\\xspace distributions are shown in the bottom plots.\\ \\cite{LHCb-PAPER-2014-044}.}\n\\label{SP_CPdistribution}\n\\end{figure}\n\nThe LHCb collaboration very recently has extended the study by performing a full amplitude analysis of the decay ${\\ensuremath{\\Bu}}\\xspace\\ensuremath{\\rightarrow}\\xspace{\\ensuremath{\\pion^+}}\\xspace\\pip{\\ensuremath{\\pion^-}}\\xspace$ to understand the features being observed in data \\cite{LHCb-PAPER-2019-017,LHCb-PAPER-2019-018}. Three different models for the scalar amplitude were used: the isobar model with $ \\sigma(500)$ pole and a $ \\pi\\pi-\\PK\\PK$ rescattering amplitude, the K-matrix formalism, and the QMI-PWA method. Resonant amplitudes for spin-1 and spin-2 were included through the isobar model for the three approaches. The general agreement between these three approaches was good. The dominant amplitudes for this decay were found to be the $\\rho(770)^0{\\ensuremath{\\pion^+}}\\xspace$ amplitude, with about 50\\% fit fraction, followed by the S-wave amplitude with about 25\\% fit fraction.            \n           Within the isobar model, the phase difference between the charged conjugated decays ${\\ensuremath{\\Bub}}\\xspace \\ensuremath{\\rightarrow}\\xspace \\sigma(500){\\ensuremath{\\pion^-}}\\xspace $ and ${\\ensuremath{\\Bu}}\\xspace \\ensuremath{\\rightarrow}\\xspace\\sigma(500){\\ensuremath{\\pion^+}}\\xspace $, measured with respect to that of the $\\rho(770)^0 \\pi^\\pm$ components, was measured to be about  $60^\\circ$ \\cite{LHCb-PAPER-2019-017}, confirming the interference trend  presented in Fig.~\\ref{SP_CPdistribution},  with zeros at  the $\\rho(770)^0$ nominal mass. The conservation of {\\ensuremath{C\\!P}}\\xspace symmetry  in the interference between S- and P-wave was excluded with a significance of $25\\sigma$.\n\n\n        A similar result for the interference distribution was observed in  the ${\\ensuremath{\\pion^-}}\\xspace {\\ensuremath{\\pion^+}}\\xspace$ invariant mass for the   \\mbox{$ {\\ensuremath{\\Bu}}\\xspace\\ensuremath{\\rightarrow}\\xspace {\\ensuremath{\\kaon^+}}\\xspace {\\ensuremath{\\pion^+}}\\xspace{\\ensuremath{\\pion^-}}\\xspace$} decay \\cite{LHCb-PAPER-2014-044} as can be seen in Fig.~\\ref{SP_CPinterferenceKpipi}. However, there are  important differences between these two decays. In ${\\ensuremath{\\Bu}}\\xspace\\ensuremath{\\rightarrow}\\xspace{\\ensuremath{\\pion^+}}\\xspace\\pip{\\ensuremath{\\pion^-}}\\xspace$ decay, the $\\rho(770)^0$ amplitude is dominant, with a smaller  scalar amplitude \\cite{LHCb-PAPER-2019-017}. For the $ {\\ensuremath{\\Bu}}\\xspace\\ensuremath{\\rightarrow}\\xspace {\\ensuremath{\\kaon^+}}\\xspace {\\ensuremath{\\pion^+}}\\xspace{\\ensuremath{\\pion^-}}\\xspace$  decay, instead,  the scalar resonance $f_0(980)$ is dominant, with a low  $\\rho(770)^0$ amplitude. In the bottom right plot in Fig.~\\ref{SP_CPinterferenceKpipi}, the presence of two zeros is clearly seen, one around the $\\rho(770)^0$ mass and another near the  scalar resonance $f_0(980)$.  Another remarkable difference is the apparent absence of {\\ensuremath{C\\!P}}\\xspace asymmetry below the $\\rho(770)^0$ mass for $\\cos \\theta > 0$ (see bottom right plot in Fig.~\\ref{SP_CPinterferenceKpipi}), as observed for the equivalent region in  ${\\ensuremath{\\Bu}}\\xspace\\ensuremath{\\rightarrow}\\xspace{\\ensuremath{\\pion^+}}\\xspace\\pip{\\ensuremath{\\pion^-}}\\xspace$ decay.  There is yet no amplitude analysis with this data set from LHCb.  BaBar \\cite{BaBar_bkpipi} and Belle   \\cite{Belle_bkpipi} collaborations   presented  results for the $ {\\ensuremath{\\Bu}}\\xspace \\ensuremath{\\rightarrow}\\xspace {\\ensuremath{\\kaon^+}}\\xspace {\\ensuremath{\\pion^-}}\\xspace {\\ensuremath{\\pion^+}}\\xspace$  with large statistical uncertainties in the phase for both  ${\\ensuremath{\\kaon^+}}\\xspace\\rho(770)$    and ${\\ensuremath{\\kaon^+}}\\xspace f_0(980) $ amplitudes. In any case, larger data samples coming from LHCb run II may shed light on these differences. \n                \n \\begin{figure}[!htb]\n\\centering\n \\includegraphics[width=.4\\columnwidth,angle=0]{Figure17_a.png} \n\\includegraphics[width=.4\\columnwidth,angle=0]{Figure17_b.png}  \\\\\n \\includegraphics[width=.4\\columnwidth,angle=0]{Figure17_c.png} \n \\includegraphics[width=.4\\columnwidth,angle=0]{Figure17_d.png} \n\\caption{Projections of the $m({\\ensuremath{\\pion^+}}\\xspace {\\ensuremath{\\pion^-}}\\xspace)$  variable  for  $B^\\pm \\ensuremath{\\rightarrow}\\xspace K^\\pm {\\ensuremath{\\pion^+}}\\xspace{\\ensuremath{\\pion^-}}\\xspace  $ signal events  separated by charge for (top left) $\\cos \\theta < 0 $ and (top right)  $\\cos \\theta > 0$. The respective subtracted  {\\ensuremath{\\Bu}}\\xspace and {\\ensuremath{\\Bub}}\\xspace distributions are shown in the bottom plots \\cite{LHCb-PAPER-2014-044}.}               \n\\label{SP_CPinterferenceKpipi}\n\\end{figure}\n\n\n\\subsubsection{${\\ensuremath{\\pion^-}}\\xspace {\\ensuremath{\\pion^+}}\\xspace  \\ensuremath{\\rightarrow}\\xspace {\\ensuremath{\\kaon^-}}\\xspace {\\ensuremath{\\kaon^+}}\\xspace$ re-scattering}\n\\label{kkpipiresc}\n\n       A very interesting result in charmless ${\\ensuremath{\\Bu}}\\xspace$ decays obtained from LHCb   \\cite{LHCb-PAPER-2013-027,LHCb-PAPER-2013-051,LHCb-PAPER-2014-044} was the pattern of {\\ensuremath{C\\!P}}\\xspace asymmetries observed in the ${\\ensuremath{\\pion^-}}\\xspace {\\ensuremath{\\pion^+}}\\xspace $  and $ {\\ensuremath{\\kaon^-}}\\xspace {\\ensuremath{\\kaon^+}}\\xspace$ mass regions between 1.0 and 1.5 \\ensuremath{\\mathrm{\\,Ge\\kern -0.1em V}}\\xspace for the flavour coupled-channels ${\\ensuremath{\\Bu}}\\xspace\\ensuremath{\\rightarrow}\\xspace {\\ensuremath{\\pion^+}}\\xspace{\\ensuremath{\\kaon^+}}\\xspace{\\ensuremath{\\kaon^-}}\\xspace$ and ${\\ensuremath{\\Bu}}\\xspace\\ensuremath{\\rightarrow}\\xspace {\\ensuremath{\\pion^+}}\\xspace\\pip{\\ensuremath{\\pion^-}}\\xspace$,  and also  ${\\ensuremath{\\Bu}}\\xspace\\ensuremath{\\rightarrow}\\xspace {\\ensuremath{\\kaon^+}}\\xspace\\Kp{\\ensuremath{\\kaon^-}}\\xspace$ and ${\\ensuremath{\\Bu}}\\xspace\\ensuremath{\\rightarrow}\\xspace {\\ensuremath{\\kaon^+}}\\xspace{\\ensuremath{\\pion^+}}\\xspace{\\ensuremath{\\pion^-}}\\xspace$. Figure~\\ref{B-KK} shows the distributions for ${\\ensuremath{\\Bu}}\\xspace\\ensuremath{\\rightarrow}\\xspace {\\ensuremath{\\kaon^+}}\\xspace\\Kp{\\ensuremath{\\kaon^-}}\\xspace$  and ${\\ensuremath{\\Bu}}\\xspace\\ensuremath{\\rightarrow}\\xspace {\\ensuremath{\\pion^+}}\\xspace{\\ensuremath{\\kaon^+}}\\xspace{\\ensuremath{\\kaon^-}}\\xspace$, separated by charge. \n              \n                  \n       In Table~\\ref{ACP_rescattering}, the resulting {\\ensuremath{C\\!P}}\\xspace asymmetries for the ${\\ensuremath{\\pion^+}}\\xspace{\\ensuremath{\\pion^-}}\\xspace$ and ${\\ensuremath{\\kaon^+}}\\xspace{\\ensuremath{\\kaon^-}}\\xspace$ invariant-mass  region between $1.0$ and $1.5 \\ensuremath{\\mathrm{\\,Ge\\kern -0.1em V}}\\xspace$ are shown. The values are larger than those integrated in the phase space (Eq.~\\ref{B3h_ACP}) and evidence the change in sign among the pairs of final states with same flavour content. \n\\begingroup\n\\begin{table}[httb] \n\\caption{Charge asymmetries in the rescattering ${\\ensuremath{\\pion^-}}\\xspace {\\ensuremath{\\pion^+}}\\xspace  \\ensuremath{\\rightarrow}\\xspace {\\ensuremath{\\kaon^-}}\\xspace {\\ensuremath{\\kaon^+}}\\xspace$ region --- between $1.0$ and $1.5 \\ensuremath{\\mathrm{\\,Ge\\kern -0.1em V}}\\xspace$. Uncertainties are first statistical, second systematic, and the third is due to the {\\ensuremath{C\\!P}}\\xspace asymmetry of the  ${\\ensuremath{\\Bu}}\\xspace\\ensuremath{\\rightarrow}\\xspace {\\ensuremath{{\\PJ\\mskip -3mu\/\\mskip -2mu\\Ppsi\\mskip 2mu}}}\\xspace {\\ensuremath{\\kaon^+}}\\xspace$  reference mode\\cite{LHCb-PAPER-2014-044}.}\n\\begin{center} \n\\begin{tabular}{lc}\n\\hline \n Decay & $A_{{\\ensuremath{C\\!P}}\\xspace}$ \\\\ \n\\hline \n$ {\\ensuremath{\\Bu}}\\xspace\\ensuremath{\\rightarrow}\\xspace {\\ensuremath{\\kaon^+}}\\xspace{\\ensuremath{\\pion^+}}\\xspace{\\ensuremath{\\pion^-}}\\xspace$  &  $+0.123 \\pm 0.012 \\pm 0.017 \\pm 0.006$ \\\\ \n$ {\\ensuremath{\\Bu}}\\xspace\\ensuremath{\\rightarrow}\\xspace {\\ensuremath{\\kaon^+}}\\xspace\\Kp{\\ensuremath{\\kaon^-}}\\xspace$  &   $-0.209 \\pm 0.011 \\pm 0.004 \\pm 0.006$ \\\\ \n$ {\\ensuremath{\\Bu}}\\xspace\\ensuremath{\\rightarrow}\\xspace {\\ensuremath{\\pion^+}}\\xspace\\pip{\\ensuremath{\\pion^-}}\\xspace$ & $+0.173 \\pm 0.021 \\pm 0.015 \\pm 0.006$ \\\\ \n$ {\\ensuremath{\\Bu}}\\xspace\\ensuremath{\\rightarrow}\\xspace {\\ensuremath{\\pion^+}}\\xspace{\\ensuremath{\\kaon^+}}\\xspace{\\ensuremath{\\kaon^-}}\\xspace$  & $-0.326 \\pm 0.028 \\pm 0.029 \\pm 0.006$ \\\\ \n\\hline \n\\end{tabular} \\end{center} \n\\label{ACP_rescattering}\n\\end{table} \n\\endgroup\n\n \n\\begin{figure}[htb]\n\\centering\n \\includegraphics[width=.4\\columnwidth,angle=0]{Figure18_a.png} \n\\includegraphics[width=.4\\columnwidth,angle=0]{Figure18_b.png}  \\\\\n\\caption{${\\ensuremath{\\kaon^+}}\\xspace{\\ensuremath{\\kaon^-}}\\xspace$ invariant mass distributions for (left) ${\\ensuremath{\\PB}}\\xspace^\\pm\\ensuremath{\\rightarrow}\\xspace\\PK^\\pm{\\ensuremath{\\kaon^+}}\\xspace{\\ensuremath{\\kaon^-}}\\xspace$ and (right) ${\\ensuremath{\\PB}}\\xspace^\\pm\\ensuremath{\\rightarrow}\\xspace\\pi^\\pm{\\ensuremath{\\kaon^+}}\\xspace{\\ensuremath{\\kaon^-}}\\xspace$ decays, separated by charge. The inset in the left plot magnifies the $\\phi(1020)$ mass region \\cite{LHCb-PAPER-2014-044}.}\n\\label{B-KK}\n\\end{figure}     \n\nA possible explanation for this effect is $\\pi\\pi-KK$ rescattering.  A well-known  ${\\ensuremath{\\pion^-}}\\xspace {\\ensuremath{\\pion^+}}\\xspace $   elastic-scattering data  result \\cite{CERN-Munich} shows that the Argand plot for the scalar amplitude departs from the unitary circle at about $1\\ensuremath{\\mathrm{\\,Ge\\kern -0.1em V}}\\xspace$ and returns at around $1.5\\ensuremath{\\mathrm{\\,Ge\\kern -0.1em V}}\\xspace$, as shown in Fig.~\\ref{CERN-Munich}(a). Also in the region $1.0-1.5\\ensuremath{\\mathrm{\\,Ge\\kern -0.1em V}}\\xspace$ a bump is observed in the inelastic interaction  ${\\ensuremath{\\pion^-}}\\xspace {\\ensuremath{\\pion^+}}\\xspace  \\ensuremath{\\rightarrow}\\xspace {\\ensuremath{\\kaon^-}}\\xspace {\\ensuremath{\\kaon^+}}\\xspace$, as shown in  Fig.~\\ref{CERN-Munich}(b) \\cite{Cohen1980}. \nIn other words, $\\pi\\pi-KK$ rescattering may be at least partially responsible for the opposite {\\ensuremath{C\\!P}}\\xspace asymmetries found above, and would be in line with the Wolfenstein mechanism for {\\ensuremath{C\\!P}}\\xspace violation \\cite{Wolfenstein1990} through coupled channels with the same quantum number \\cite{Bediaga2013,Nogueira2015}. \n    \n \\begin{figure}[htb]\n \\centering\n\\includegraphics[width=.35\\columnwidth,angle=0]{Figure19_Argand.png}\n\\includegraphics[width=.43\\columnwidth,angle=0]{Figure19_pipiKK.png}\n\\caption{  (left) Argand circle of the scalar ${\\ensuremath{\\pion^-}}\\xspace {\\ensuremath{\\pion^+}}\\xspace$ elastic scattering  amplitude \\cite{CERN-Munich}. (right) Modulos of the scalar $\\pi \\pi  \\ensuremath{\\rightarrow}\\xspace \\PK\\PK$ amplitude  distribution \\cite{Cohen1980}.}\n\\label{CERN-Munich}\n\\end{figure}            \n\n\n\nRecently, the LHCb collaboration published the first amplitude analysis for the decay ${\\ensuremath{\\Bu}}\\xspace\\ensuremath{\\rightarrow}\\xspace {\\ensuremath{\\pion^+}}\\xspace{\\ensuremath{\\kaon^+}}\\xspace {\\ensuremath{\\kaon^-}}\\xspace$ with run I data \\cite{LHCb-PAPER-2018-051}. Apart from a dominant non-resonant component and a few resonant contributions, an explicit amplitude parameterisation for the $\\pi\\pi-KK$ rescattering has been included, following Ref.~\\cite{Pelaez2004}. The fit model shows good description of the data, as can be seen in Fig.~\\ref{KKPI_AmplitudeAnalysis}. The ${\\ensuremath{\\PB}}\\xspace^\\pm$ average fit fraction for the rescattering contribution  is found to be about 23$\\%$  while presenting the largest {\\ensuremath{C\\!P}}\\xspace asymmetry observed for a single amplitude, with a measured value of  $A_{{\\ensuremath{C\\!P}}\\xspace} =( -66 \\pm 4\\pm 2)$. \n\n \\begin{figure}[htb]\n\\centering\n\\includegraphics[width=.65\\columnwidth,angle=0]{Figure20.png}\n\\caption{${\\ensuremath{\\kaon^+}}\\xspace {\\ensuremath{\\kaon^-}}\\xspace$ squared mass distribution for (top) ${\\ensuremath{\\Bu}}\\xspace\\ensuremath{\\rightarrow}\\xspace {\\ensuremath{\\pion^+}}\\xspace {\\ensuremath{\\kaon^+}}\\xspace {\\ensuremath{\\kaon^-}}\\xspace$  and (bottom) ${\\ensuremath{\\Bub}}\\xspace\\ensuremath{\\rightarrow}\\xspace{\\ensuremath{\\pion^-}}\\xspace {\\ensuremath{\\kaon^-}}\\xspace {\\ensuremath{\\kaon^+}}\\xspace$ decays.  The total amplitude fit model, including a term for the $\\pi\\pi-KK$ rescattering, is overlaid \\cite{LHCb-PAPER-2018-051}.}\n\\label{KKPI_AmplitudeAnalysis}\n\\end{figure}   \n\n          The same amplitude is also included in the amplitude analysis for the  ${\\ensuremath{\\Bu}}\\xspace\\ensuremath{\\rightarrow}\\xspace {\\ensuremath{\\pion^+}}\\xspace\\pip{\\ensuremath{\\pion^-}}\\xspace$ decay \\cite{LHCb-PAPER-2019-017,LHCb-PAPER-2019-018}. The  total contribution for this channel is much lower than that obtained in ${\\ensuremath{\\Bu}}\\xspace\\ensuremath{\\rightarrow}\\xspace  {\\ensuremath{\\pion^+}}\\xspace {\\ensuremath{\\kaon^+}}\\xspace {\\ensuremath{\\kaon^-}}\\xspace$. However, when compared to the  ${\\ensuremath{\\kaon^+}}\\xspace {\\ensuremath{\\kaon^-}}\\xspace$  mass spectrum in the rescattering region, the ${\\ensuremath{\\pion^+}}\\xspace{\\ensuremath{\\pion^-}}\\xspace$ mass spectrum has different resonant contributions, including part of the broad and scalar $\\sigma(500)$ and the tensor $f_2(1270)$, both with significant important positive {\\ensuremath{C\\!P}}\\xspace asymmetries, as outlined in the next sub-section. It would be important to perform a coupled-channel amplitude analysis -- in particular a simultaneous fit for ${\\ensuremath{\\Bu}}\\xspace\\ensuremath{\\rightarrow}\\xspace {\\ensuremath{\\pion^+}}\\xspace\\pip{\\ensuremath{\\pion^-}}\\xspace$ and ${\\ensuremath{\\Bu}}\\xspace\\ensuremath{\\rightarrow}\\xspace  {\\ensuremath{\\pion^+}}\\xspace {\\ensuremath{\\kaon^+}}\\xspace {\\ensuremath{\\kaon^-}}\\xspace$ decays -- to clarify better a possible correlation between $\\pi\\pi$ and $\\PK\\PK$ amplitudes through rescattering as a way to understand the {\\ensuremath{C\\!P}}\\xspace violation mechanisms involved.\n          \n           \n\\subsubsection{Quasi-two-body decays}\n\\label{Bquasi2body}\n\nAs discussed in Sects.~\\ref{AmAn} and \\ref{quasi2body}, the dynamics of a three-body decay are usually dominated by the formation of resonances. For a {\\ensuremath{\\PB}}\\xspace-meson decay these are quasi-two-body processes with a scalar, vector or tensor resonance  plus a pseudo-scalar. Although the resonance formation is an intermediate {\\it amplitude} path -- and as such one cannot strictly measure a decay rate or a charge asymmetry associated to it -- it is still useful to define such quantities in terms of its squared amplitude. A {\\ensuremath{C\\!P}}\\xspace asymmetry can be defined as presented in Eq.~\\ref{ACP_r}, exemplified for a  $\\rho(770)^0$ intermediate resonance. \n\nA few values of {\\ensuremath{C\\!P}}\\xspace asymmetries for  quasi-two body modes are presented in Table~\\ref{ACP_quasi2body}, as quoted by the PDG~\\cite{PDG2019}. \n\n\\begin{table}[htbp]\n\\centering\n\\caption{{\\ensuremath{C\\!P}}\\xspace asymmetries for a few quasi two-body $B$ decays~\\cite{PDG2019}.}\n\\begin{tabular}{lc} \\hline\nMode & $A_{\\ensuremath{C\\!P}}\\xspace$  \\\\ \\hline\n${\\ensuremath{\\Bu}}\\xspace\\ensuremath{\\rightarrow}\\xspace {\\ensuremath{\\kaon^+}}\\xspace f_2(1270)$ & $ -0.68 \\pm 0.19$ \\\\\n${\\ensuremath{\\Bu}}\\xspace\\ensuremath{\\rightarrow}\\xspace {\\ensuremath{\\kaon^+}}\\xspace\\rho(770)^0$ & $\\phantom{-}0.37 \\pm 0.10$ \\\\ \n${\\ensuremath{\\Bu}}\\xspace\\ensuremath{\\rightarrow}\\xspace {\\ensuremath{\\kaon^{*+}}}\\xspace \\eta$ & $ \\phantom{-}0.19 \\pm 0.05$ \\\\ \\hline\n\\end{tabular}\n\\label{ACP_quasi2body}\n\\end{table}\n\nRecent amplitude analyses performed by the  LHCb collaboration using run I data   were able to further  determine the following {\\ensuremath{C\\!P}}\\xspace asymmetries associated to resonant intermediate decays:\n\n\\begin{enumerate}\n\\item $A_{{\\ensuremath{C\\!P}}\\xspace}(  {\\ensuremath{\\B^0}}\\xspace \\ensuremath{\\rightarrow}\\xspace {\\ensuremath{\\kaon^*}}\\xspace(892)^+ {\\ensuremath{\\pion^-}}\\xspace)= -0.30 \\pm 0.06$ in ${\\ensuremath{\\B^0}}\\xspace\\ensuremath{\\rightarrow}\\xspace {\\ensuremath{\\kaon^0_{\\rm\\scriptscriptstyle S}}}\\xspace{\\ensuremath{\\pion^+}}\\xspace{\\ensuremath{\\pion^-}}\\xspace$ decays\\cite{LHCb-PAPER-2017-033};\n\\item $A_{{\\ensuremath{C\\!P}}\\xspace}({\\ensuremath{\\Bu}}\\xspace \\ensuremath{\\rightarrow}\\xspace \\sigma(500){\\ensuremath{\\pion^+}}\\xspace)= -0.16 \\pm 0.02 \\pm 0.03$   in ${\\ensuremath{\\Bu}}\\xspace\\ensuremath{\\rightarrow}\\xspace{\\ensuremath{\\pion^+}}\\xspace\\pip{\\ensuremath{\\pion^-}}\\xspace$ decays \\cite{LHCb-PAPER-2019-017};\n\\item $A_{{\\ensuremath{C\\!P}}\\xspace}( {\\ensuremath{\\Bu}}\\xspace \\ensuremath{\\rightarrow}\\xspace f_2(1270){\\ensuremath{\\pion^+}}\\xspace)= -0.47 \\pm 0.06 \\pm 0.6$  in ${\\ensuremath{\\Bu}}\\xspace\\ensuremath{\\rightarrow}\\xspace{\\ensuremath{\\pion^+}}\\xspace\\pip{\\ensuremath{\\pion^-}}\\xspace$ decays  \\cite{LHCb-PAPER-2019-017}.\n\\item $A_{{\\ensuremath{C\\!P}}\\xspace}^0(  {\\ensuremath{\\B^0}}\\xspace \\ensuremath{\\rightarrow}\\xspace \\rho(770)^0{\\ensuremath{\\kaon^*}}\\xspace(892)^0) = -0.62\\pm 0.09 \\pm 0.09$, for the longitudinal amplitude component in ${\\ensuremath{\\B^0}}\\xspace\\ensuremath{\\rightarrow}\\xspace ({\\ensuremath{\\pion^+}}\\xspace{\\ensuremath{\\pion^-}}\\xspace)({\\ensuremath{\\kaon^+}}\\xspace{\\ensuremath{\\kaon^-}}\\xspace)$ decays \\cite{LHCb-PAPER-2018-042}.\n\\end{enumerate}\n\nAlthough with the mentioned caveat on the interpretation of {\\ensuremath{C\\!P}}\\xspace asymmetries in quasi two-body decays, it is very interesting to note that the values of asymmetries here are considerably larger than those of two-body {\\ensuremath{\\PB}}\\xspace decays presented in Table~\\ref{ACP_2body} (not considering those with large uncertainties). This is probably related to the richness of the hadronic processes in a multi-body decay, where different sources of strong phases, necessary to observe {\\ensuremath{C\\!P}}\\xspace violation, are possible. The {\\ensuremath{C\\!PT}}\\xspace  constraint has to enter somehow connecting final states with the same flavour content: resonant states usually have more than one decay channel, so a natural interconnection of {\\ensuremath{C\\!P}}\\xspace asymmetries may arise. Moreover, possible rescattering involving the companion pseudoscalar, and as such breaking the $2\\!+1\\!$ approximation,   can be of some importance, at least in a part of the phase space.  \n\nInterestingly enough, the quasi-two-body decay ${\\ensuremath{\\Bu}}\\xspace\\ensuremath{\\rightarrow}\\xspace \\rho(770)^0{\\ensuremath{\\pion^+}}\\xspace$, which is the dominant resonant contribution of the decay ${\\ensuremath{\\Bu}}\\xspace\\ensuremath{\\rightarrow}\\xspace{\\ensuremath{\\pion^+}}\\xspace\\pip{\\ensuremath{\\pion^-}}\\xspace$, has a {\\ensuremath{C\\!P}}\\xspace asymmetry compatible with zero: within the isobar model in Ref.~\\cite{LHCb-PAPER-2019-017} it was measured to be $A_{{\\ensuremath{C\\!P}}\\xspace}= (0.7 \\pm 1.1 \\pm 2.2) \\%$. This value contradicts some theoretical predictions with a high value for this quantity \\cite{ChengChiangKuo2014,Zhou2017}.\\footnote{A very recent work though, using QCD factorisation, presents a result consistent with no {\\ensuremath{C\\!P}}\\xspace violation in this intermediate mode ~\\cite{Cheng:2020hyj}.} Yet, the $\\rho(770)^0$ resonance has only one possible decay,   $\\rho(770)^0 \\ensuremath{\\rightarrow}\\xspace {\\ensuremath{\\pion^-}}\\xspace {\\ensuremath{\\pion^+}}\\xspace$,  since it cannot decay to two neutral pions.  So, if one takes the $2$+$1$ approximation at its limit of validity, the {\\ensuremath{C\\!P}}\\xspace asymmetry should indeed be zero, since there is no possible re-scattering to interconnect to another channel to produce an opposite {\\ensuremath{C\\!P}}\\xspace asymmetry \\cite{Josue}. \n\nIn this sense, within the $2$+$1$ approximation, the LHCb observation of a large negative {\\ensuremath{C\\!P}}\\xspace asymmetry  \\cite{LHCb-PAPER-2017-033} for the   $  {\\ensuremath{\\B^0}}\\xspace \\ensuremath{\\rightarrow}\\xspace {\\ensuremath{\\kaon^*}}\\xspace(892)^+ {\\ensuremath{\\pion^-}}\\xspace$ decay, with ${\\ensuremath{\\kaon^*}}\\xspace(892)^+\\ensuremath{\\rightarrow}\\xspace {\\ensuremath{\\kaon^0}}\\xspace {\\ensuremath{\\pion^+}}\\xspace $,  should be compensated by a similarly large positive {\\ensuremath{C\\!P}}\\xspace asymmetry in the  channel $  {\\ensuremath{\\B^0}}\\xspace \\ensuremath{\\rightarrow}\\xspace {\\ensuremath{\\kaon^*}}\\xspace(892)^+ {\\ensuremath{\\pion^-}}\\xspace$ with  $ {\\ensuremath{\\kaon^*}}\\xspace(892)^+ \\ensuremath{\\rightarrow}\\xspace {\\ensuremath{\\kaon^+}}\\xspace {\\ensuremath{\\pion^0}}\\xspace $. There is yet no analysis of the final state ${\\ensuremath{\\B^0}}\\xspace\\ensuremath{\\rightarrow}\\xspace {\\ensuremath{\\kaon^+}}\\xspace{\\ensuremath{\\pion^-}}\\xspace{\\ensuremath{\\pion^0}}\\xspace$ where this {\\ensuremath{C\\!P}}\\xspace asymmetry could be measured. With two neutral pions in the final state, this study is challenging for LHCb but less so for Belle II.\n       \n\nTwo- and quasi two-body modes have been given quite some attention from the theory side.  In general, estimates for  {\\ensuremath{C\\!P}}\\xspace asymmetries involve final states with  two pseudoscalars (${\\ensuremath{\\PB}}\\xspace\\ensuremath{\\rightarrow}\\xspace PP$) or one pseudoscalar and one low-mass vector resonance (${\\ensuremath{\\PB}}\\xspace\\ensuremath{\\rightarrow}\\xspace PV$). Yet and again recalling the caveat on quasi-two-body decays, to measure the {\\ensuremath{C\\!P}}\\xspace asymmetry  in a ${\\ensuremath{\\PB}}\\xspace\\ensuremath{\\rightarrow}\\xspace PV$ process  it is necessary to manage  the complexity of three-body decay, with interferences of different resonant amplitudes, with different spins,  across the phase space.\n\nIn certain situations, to avoid much of this complexity, a model-independent analysis method \\cite{Josue} has been proposed, taking into account the angular momentum distribution for  ${\\ensuremath{\\PB}}\\xspace\\ensuremath{\\rightarrow}\\xspace PV$, the relatively low invariant mass (assuming $V$ to be a resonance below $\\sim 2 \\ensuremath{\\mathrm{\\,Ge\\kern -0.1em V}}\\xspace$) where this amplitude is dominant and the proximity with a possible light scalar resonance. The method takes a slice around the nominal  mass of the light vector resonance including the region of interference with the  light scalar resonance. The potential of this method was tested successfully with fast Monte Carlo~\\cite{Josue}. \n\n\n\\subsubsection{{\\ensuremath{\\PB}}\\xspace decays involving baryons} \n\nWith a mass exceeding 5 \\ensuremath{\\mathrm{\\,Ge\\kern -0.1em V}}\\xspace, {\\ensuremath{\\PB}}\\xspace mesons can  also decay to final states involving a nucleon and an antinucleon. Three-body processes such as ${\\ensuremath{\\PB}}\\xspace\\ensuremath{\\rightarrow}\\xspace h{\\ensuremath{\\Pp}}\\xspace{\\ensuremath{\\overline \\proton}}\\xspace $ (where $h$ is a light, stable meson) can proceed through intermediate resonances such as ${\\ensuremath{\\Pc}}\\xspace{\\ensuremath{\\overline \\cquark}}\\xspace$ (charmonium) states and excited ${\\ensuremath{\\Lbar}}\\xspace$ states and may also provide further insights into the {\\ensuremath{C\\!P}}\\xspace violation mechanisms. At the quark level the weak process, with both tree and penguin diagrams,  is essentially the same as those of the charmless {\\ensuremath{\\PB}}\\xspace decays discussed in the previous sections, and in particular the origin of the weak phase is the same. Diagrams involving charmonium states, however,  bring no associated weak phase. With ${\\ensuremath{\\Pp}}\\xspace{\\ensuremath{\\overline \\proton}}\\xspace$ in the final state, there are interesting new features in the decay dynamics,  associated with the spin-1\/2 particles, and potentially different hadronic interactions involving the production of the baryon pair. These features can be important to understand the role of FSI in the {\\ensuremath{C\\!P}}\\xspace asymmetry in non-leptonic B decays.  \n\nThe first evidence for {\\ensuremath{C\\!P}}\\xspace violation in this kind of decays was obtained by the LHCb collaboration in 2014 \\cite{LHCb-PAPER-2014-034} in the decay mode ${\\ensuremath{\\Bu}}\\xspace\\ensuremath{\\rightarrow}\\xspace {\\ensuremath{\\kaon^+}}\\xspace{\\ensuremath{\\Pp}}\\xspace{\\ensuremath{\\overline \\proton}}\\xspace$, in a study which also included the Cabibbo-suppressed decay ${\\ensuremath{\\Bu}}\\xspace\\ensuremath{\\rightarrow}\\xspace {\\ensuremath{\\pion^+}}\\xspace{\\ensuremath{\\Pp}}\\xspace{\\ensuremath{\\overline \\proton}}\\xspace$. The sample comprised about 19 thousand decays of ${\\ensuremath{\\Bu}}\\xspace\\ensuremath{\\rightarrow}\\xspace {\\ensuremath{\\kaon^+}}\\xspace{\\ensuremath{\\Pp}}\\xspace{\\ensuremath{\\overline \\proton}}\\xspace$ and about 2 thousand decays of ${\\ensuremath{\\Bu}}\\xspace\\ensuremath{\\rightarrow}\\xspace {\\ensuremath{\\pion^+}}\\xspace{\\ensuremath{\\Pp}}\\xspace{\\ensuremath{\\overline \\proton}}\\xspace$.\n\n\nThe Dalitz plot distributions for these two channels are shown in Fig.~\\ref{Dalitz_ppk_pppi}. Except for the charmonium resonances, which  play an important role in these decays, the event distribution accumulates at the edges of the phase space. Only one resonance was clearly identified in the ${\\ensuremath{\\kaon^+}}\\xspace{\\ensuremath{\\overline \\proton}}\\xspace$ mass spectrum, the ${\\ensuremath{\\kern 0.1em\\overline{\\kern -0.1em\\PLambda}}}\\xspace(1520)$.  No resonance was identified at low ${\\ensuremath{\\Pp}}\\xspace{\\ensuremath{\\overline \\proton}}\\xspace$ invariant mass.  There is no amplitude analysis for these decays yet.\n\n\n\n\\begin{figure}[htbp]\n\\centering\n\\includegraphics[width=.45\\columnwidth,angle=0]{Figure21_a.png}\n\\includegraphics[width=.45\\columnwidth,angle=0]{Figure21_b.png}\n\\caption{ Dalitz plot distributions for the decays (left) ${\\ensuremath{\\Bu}}\\xspace\\ensuremath{\\rightarrow}\\xspace {\\ensuremath{\\kaon^+}}\\xspace{\\ensuremath{\\Pp}}\\xspace{\\ensuremath{\\overline \\proton}}\\xspace$ and (right) ${\\ensuremath{\\Bu}}\\xspace\\ensuremath{\\rightarrow}\\xspace {\\ensuremath{\\pion^+}}\\xspace{\\ensuremath{\\Pp}}\\xspace{\\ensuremath{\\overline \\proton}}\\xspace$  from LHCb run I data \\cite{LHCb-PAPER-2014-034}. }\n\\label{Dalitz_ppk_pppi}\n\\end{figure}\n\nThe evidence for {\\ensuremath{C\\!P}}\\xspace violation in the ${\\ensuremath{\\Bu}}\\xspace\\ensuremath{\\rightarrow}\\xspace {\\ensuremath{\\kaon^+}}\\xspace{\\ensuremath{\\Pp}}\\xspace{\\ensuremath{\\overline \\proton}}\\xspace$ mode comes mainly from the low ${\\ensuremath{\\Pp}}\\xspace{\\ensuremath{\\overline \\proton}}\\xspace$ mass region. Figure~\\ref{CP_ppk} shows the asymmetry for two regions of  $ m^2_{{\\ensuremath{\\kaon^+}}\\xspace{\\ensuremath{\\overline \\proton}}\\xspace}$, above and below $10 \\ensuremath{\\mathrm{\\,Ge\\kern -0.1em V}}\\xspace\\gev$, dividing the phase space in two bands.  The {\\ensuremath{C\\!P}}\\xspace asymmetry as a function of the squared ${\\ensuremath{\\Pp}}\\xspace{\\ensuremath{\\overline \\proton}}\\xspace$ invariant mass shows a clear sign inversion like that  observed at low ${\\ensuremath{\\pion^+}}\\xspace{\\ensuremath{\\pion^-}}\\xspace$ mass for ${\\ensuremath{\\Bu}}\\xspace\\ensuremath{\\rightarrow}\\xspace{\\ensuremath{\\pion^+}}\\xspace\\pip{\\ensuremath{\\pion^-}}\\xspace$ and ${\\ensuremath{\\Bu}}\\xspace\\ensuremath{\\rightarrow}\\xspace{\\ensuremath{\\kaon^+}}\\xspace{\\ensuremath{\\pion^+}}\\xspace{\\ensuremath{\\pion^-}}\\xspace$ decays, which has its origin from the strong phase difference between the S and P-waves.\n\n\\begin{figure}[!htb]\n\\centering\n\\includegraphics[width=.4\\columnwidth,angle=0]{Figure22.png}\n\\caption{ Distribution of $N(B^-) - N(B^+)$ in the squared ${\\ensuremath{\\Pp}}\\xspace{\\ensuremath{\\overline \\proton}}\\xspace$ invariant mass for $ m^2_{Kp} < 10 \\ensuremath{\\mathrm{\\,Ge\\kern -0.1em V^2}}\\xspace$ (black filled circles) and  $ m^2_{Kp} > 10 \\ensuremath{\\mathrm{\\,Ge\\kern -0.1em V^2}}\\xspace$ (open triangles) \\cite{LHCb-PAPER-2014-034}.}\n\\label{CP_ppk}\n\\end{figure}\n\nIn the region $m_{{\\ensuremath{\\Pp}}\\xspace{\\ensuremath{\\overline \\proton}}\\xspace} < 2.85\\ensuremath{\\mathrm{\\,Ge\\kern -0.1em V}}\\xspace$ and  $ m^2_{{\\ensuremath{\\kaon^+}}\\xspace{\\ensuremath{\\overline \\proton}}\\xspace} > 10\\ensuremath{\\mathrm{\\,Ge\\kern -0.1em V}}\\xspace$, the {\\ensuremath{C\\!P}}\\xspace asymmetry  was  measured to be $0.096\\pm 0.024 \\pm 0.004$, representing an effect of about 4$\\sigma$ significance. For the decay ${\\ensuremath{\\Bu}}\\xspace\\ensuremath{\\rightarrow}\\xspace {\\ensuremath{\\pion^+}}\\xspace{\\ensuremath{\\Pp}}\\xspace{\\ensuremath{\\overline \\proton}}\\xspace$ there were not enough data to perform a similar study. With LHCb data from run II such a study should be possible and in both decays amplitude analysis would be the ideal way to study the sources of {\\ensuremath{C\\!P}}\\xspace violation. \n \n\n\\subsubsection{Charmless $b$-baryon decays} \n\\label{b-baryons}\nAlthough {\\ensuremath{C\\!P}}\\xspace violation in baryons is an essential issue for baryogenesis, the study of these decays presents challenges from both the theoretical~\\cite{Lu:2009cm,Hsiao:2014mua,Durieux:2016nqr,Zhao:2018zcb,Roy:2019cky} and experimental sides. \n\nThe first difficulty for the experimental observation of {\\ensuremath{C\\!P}}\\xspace violation in beauty baryons comes from  production,  which is almost exclusive to hadronic machines. At the LHC, though, their production is favoured at low transverse momenta \\cite{LHCb-PAPER-2014-004} which turns $b$-baryon selection efficiencies lower than those of {\\ensuremath{\\PB}}\\xspace-meson decays \\cite{TimVava}.  \nAs in charmless {\\ensuremath{\\PB}}\\xspace decays, the underlying weak phase responsible for any potential {\\ensuremath{C\\!P}}\\xspace asymmetry comes from the $b\\ensuremath{\\rightarrow}\\xspace u$ transition, and the branching ratios are of ${\\cal O}(10^{-5})$ or less. \n\nIn two-body decays, the LHCb collaboration, using Run I data, has searched for {\\ensuremath{C\\!P}}\\xspace violation in the modes ${\\ensuremath{\\Lz^0_\\bquark}}\\xspace\\ensuremath{\\rightarrow}\\xspace {\\ensuremath{\\Pp}}\\xspace{\\ensuremath{\\kaon^-}}\\xspace$ and ${\\ensuremath{\\Lz^0_\\bquark}}\\xspace\\ensuremath{\\rightarrow}\\xspace {\\ensuremath{\\Pp}}\\xspace{\\ensuremath{\\pion^-}}\\xspace$, with signal yields of about respectively 9 and 6 thousand decays~\\cite{LHCb-PAPER-2018-025}. No significant {\\ensuremath{C\\!P}}\\xspace asymmetries were found, \n\\begin{eqnarray}\nA_{{\\ensuremath{C\\!P}}\\xspace}( {\\ensuremath{\\Lz^0_\\bquark}}\\xspace\\ensuremath{\\rightarrow}\\xspace {\\ensuremath{\\Pp}}\\xspace{\\ensuremath{\\kaon^-}}\\xspace)  = & \\!\\!-0.020 \\pm 0.013 \\pm 0.019, \\nonumber \\\\\nA_{{\\ensuremath{C\\!P}}\\xspace}({\\ensuremath{\\Lz^0_\\bquark}}\\xspace\\ensuremath{\\rightarrow}\\xspace {\\ensuremath{\\Pp}}\\xspace{\\ensuremath{\\pion^-}}\\xspace)  = & \\!\\! -0.035 \\pm 0.017 \\pm 0.020, \\nonumber\n\\end{eqnarray}\nyet the measurements are about 4 times more precise then those performed previously by the CDF collaboration~\\cite{CDF2014}. \n\nSimilarly to {\\ensuremath{\\PB}}\\xspace-meson decays, {\\ensuremath{C\\!P}}\\xspace violation studies become richer in three- or four-body decays. In 2016, with data from run I, the LHCb collaboration reported a first evidence for {\\ensuremath{C\\!P}}\\xspace violation in the baryon sector by analysing the decay ${\\ensuremath{\\Lz^0_\\bquark}}\\xspace\\ensuremath{\\rightarrow}\\xspace {\\ensuremath{\\Pp}}\\xspace{\\ensuremath{\\pion^-}}\\xspace{\\ensuremath{\\pion^+}}\\xspace{\\ensuremath{\\pion^-}}\\xspace$  \\cite{LHCb-PAPER-2016-030}, with a sample of about 6 thousand events. The analysis made use of the triple-product observables discussed in Sect.~\\ref{tripleproducts} such that a {\\ensuremath{C\\!P}}\\xspace asymmetry $a_{\\ensuremath{C\\!P}}\\xspace^{T-\\rm odd}$ as in Eq.~\\ref{a_CP_Todd} was measured as a function of phase space bins and the angle $\\Phi$, which is the angle between the decay planes formed by the pairs ${\\ensuremath{\\Pp}}\\xspace{\\ensuremath{\\pion^-}}\\xspace_{\\rm fast}$ and ${\\ensuremath{\\pion^+}}\\xspace{\\ensuremath{\\pion^-}}\\xspace_{\\rm slow}$ (slow and fast referring to the ${\\ensuremath{\\pion^+}}\\xspace$ momenta). The distribution of $a_{\\ensuremath{C\\!P}}\\xspace^{T-\\rm odd}$ in terms of $\\Phi$ presented a p-value of $10^{-3}$ of compatibility with  {\\ensuremath{C\\!P}}\\xspace symmetry, translating to a 3.3$\\sigma$ evidence for {\\ensuremath{C\\!P}}\\xspace violation in this decay. No evidence for {\\ensuremath{C\\!P}}\\xspace violation was found in the measurement of $a_{\\ensuremath{C\\!P}}\\xspace^{T-\\rm odd}$ integrated through the phase space. \n\nAn update of this measurement has just been released by the LHCb collaboration with Run I and part of Run II data, with a sample size comprising about 27 thousand ${\\ensuremath{\\Lz^0_\\bquark}}\\xspace\\ensuremath{\\rightarrow}\\xspace {\\ensuremath{\\Pp}}\\xspace{\\ensuremath{\\pion^-}}\\xspace{\\ensuremath{\\pion^+}}\\xspace{\\ensuremath{\\pion^-}}\\xspace$ decays  \\cite{LHCb-PAPER-2019-028}. The study was carried out by means of measuring the  $a_{\\ensuremath{C\\!P}}\\xspace^{T-\\rm odd}$ observable -- following the same strategy as above -- as well as using the energy test method, sensitive to local {\\ensuremath{C\\!P}}\\xspace effects in the phase space distribution,  as outlined in Sect.~\\ref{unbinned}. The evidence of {\\ensuremath{C\\!P}}\\xspace violation previously reported  \\cite{LHCb-PAPER-2016-030} has not been confirmed, although altogether the results are marginally compatible with {\\ensuremath{C\\!P}}\\xspace conservation.\n\nThe search for {\\ensuremath{C\\!P}}\\xspace violation by means of the $a_{\\ensuremath{C\\!P}}\\xspace^{T-\\rm odd}$ observable both integrated and in regions of the phase space has been extended to the ${\\ensuremath{\\Lz^0_\\bquark}}\\xspace \\ensuremath{\\rightarrow}\\xspace {\\ensuremath{\\Pp}}\\xspace {\\ensuremath{\\kaon^-}}\\xspace {\\ensuremath{\\pion^+}}\\xspace{\\ensuremath{\\pion^-}}\\xspace$, ${\\ensuremath{\\Lz^0_\\bquark}}\\xspace \\ensuremath{\\rightarrow}\\xspace {\\ensuremath{\\Pp}}\\xspace  {\\ensuremath{\\kaon^-}}\\xspace {\\ensuremath{\\kaon^+}}\\xspace{\\ensuremath{\\kaon^-}}\\xspace$ and ${\\ensuremath{\\Xires_\\bquark}}\\xspace \\ensuremath{\\rightarrow}\\xspace {\\ensuremath{\\Pp}}\\xspace {\\ensuremath{\\kaon^-}}\\xspace {\\ensuremath{\\pion^+}}\\xspace{\\ensuremath{\\kaon^-}}\\xspace$ modes in Ref.~\\cite{LHCb-PAPER-2018-001} using Run I data from LHCb. The sample sizes were  about 20 thousand, 5 thousand and 700 decays, respectively. No evidence for {\\ensuremath{C\\!P}}\\xspace violation was found.\n\nMore recently, a comprehensive  study including four ${\\ensuremath{\\Lz^0_\\bquark}}\\xspace$ and two ${\\ensuremath{\\Xires_\\bquark}}\\xspace$ 4-body modes provided measurements of integrated {\\ensuremath{C\\!P}}\\xspace asymmetries \\cite{LHCb-PAPER-2018-044}. Each {\\ensuremath{C\\!P}}\\xspace asymmetry was measured with respect to that of a ${\\ensuremath{\\Lz^+_\\cquark}}\\xspace$ formed as an intermediate state and leading to the same final state, for example $\\Delta A_{\\ensuremath{C\\!P}}\\xspace({\\ensuremath{\\Lz^0_\\bquark}}\\xspace\\ensuremath{\\rightarrow}\\xspace {\\ensuremath{\\Pp}}\\xspace{\\ensuremath{\\pion^-}}\\xspace{\\ensuremath{\\pion^+}}\\xspace{\\ensuremath{\\pion^-}}\\xspace)\\equiv A_{\\ensuremath{C\\!P}}\\xspace({\\ensuremath{\\Lz^0_\\bquark}}\\xspace\\ensuremath{\\rightarrow}\\xspace {\\ensuremath{\\Pp}}\\xspace{\\ensuremath{\\pion^-}}\\xspace{\\ensuremath{\\pion^+}}\\xspace{\\ensuremath{\\pion^-}}\\xspace) - A_{\\ensuremath{C\\!P}}\\xspace({\\ensuremath{\\Lz^0_\\bquark}}\\xspace\\ensuremath{\\rightarrow}\\xspace ({\\ensuremath{\\Lz^+_\\cquark}}\\xspace\\ensuremath{\\rightarrow}\\xspace{\\ensuremath{\\Pp}}\\xspace{\\ensuremath{\\pion^-}}\\xspace{\\ensuremath{\\pion^+}}\\xspace){\\ensuremath{\\pion^-}}\\xspace)$. In this way, corrections due to instrumental and $b$-baryon production asymmetries cancel out to first order. Further corrections, such as kinematic weighting, were also used to guarantee control over nuisance asymmetries. The resulting asymmetries are presented in Table~\\ref{ACP_Lambdab}. No evidence for {\\ensuremath{C\\!P}}\\xspace violation was found.\n\n\\begin{table}[tb] \n\\caption{Phase-space integrated {\\ensuremath{C\\!P}}\\xspace asymmetries for 4-body ${\\ensuremath{\\Lz^0_\\bquark}}\\xspace$ and ${\\ensuremath{\\Xires_\\bquark}}\\xspace$ charmless decays from LHCb collaboration  \\cite{LHCb-PAPER-2018-044}.}\n\\begin{center} \n\\begin{tabular}{lc}\n\\hline \n Decay & $\\Delta A_{{\\ensuremath{C\\!P}}\\xspace}(\\%)$ \\\\ \n\\hline \n${\\ensuremath{\\Lz^0_\\bquark}}\\xspace \\ensuremath{\\rightarrow}\\xspace {\\ensuremath{\\Pp}}\\xspace {\\ensuremath{\\kaon^-}}\\xspace {\\ensuremath{\\pion^+}}\\xspace{\\ensuremath{\\pion^-}}\\xspace$  &  $+ 3.2 \\pm 1.1 \\pm 0.6$ \\\\ \n${\\ensuremath{\\Lz^0_\\bquark}}\\xspace \\ensuremath{\\rightarrow}\\xspace {\\ensuremath{\\Pp}}\\xspace  {\\ensuremath{\\pion^-}}\\xspace {\\ensuremath{\\pion^+}}\\xspace{\\ensuremath{\\pion^-}}\\xspace$ &   $+ 1.1 \\pm 2.5 \\pm 0.6$ \\\\ \n${\\ensuremath{\\Lz^0_\\bquark}}\\xspace \\ensuremath{\\rightarrow}\\xspace {\\ensuremath{\\Pp}}\\xspace  {\\ensuremath{\\kaon^-}}\\xspace {\\ensuremath{\\kaon^+}}\\xspace{\\ensuremath{\\pion^-}}\\xspace$  & $-6.9  \\pm 4.9  \\pm 0.8$ \\\\ \n${\\ensuremath{\\Lz^0_\\bquark}}\\xspace \\ensuremath{\\rightarrow}\\xspace {\\ensuremath{\\Pp}}\\xspace  {\\ensuremath{\\kaon^-}}\\xspace {\\ensuremath{\\kaon^+}}\\xspace{\\ensuremath{\\kaon^-}}\\xspace$  & $+ 0.2 \\pm 1.8 \\pm 0.6$ \\\\ \n${\\ensuremath{\\Xires_\\bquark}}\\xspace \\ensuremath{\\rightarrow}\\xspace {\\ensuremath{\\Pp}}\\xspace {\\ensuremath{\\kaon^-}}\\xspace {\\ensuremath{\\pion^+}}\\xspace{\\ensuremath{\\kaon^-}}\\xspace$  &  $-6.8 \\pm 8.0 \\pm 0.8$ \\\\ \n${\\ensuremath{\\Xires_\\bquark}}\\xspace \\ensuremath{\\rightarrow}\\xspace {\\ensuremath{\\Pp}}\\xspace {\\ensuremath{\\kaon^-}}\\xspace {\\ensuremath{\\pion^+}}\\xspace{\\ensuremath{\\pion^-}}\\xspace$  &  $17 \\pm 11 \\pm 1$ \\\\ \n\\hline \n\\end{tabular} \\end{center} \n\\label{ACP_Lambdab}\n\\end{table} \n\n\nOverall, the current status is that while {\\ensuremath{C\\!P}}\\xspace violation is well established in the {\\ensuremath{\\PB}}\\xspace-meson sector, no compelling evidence exists for {\\ensuremath{C\\!P}}\\xspace violation in $b$-baryon decays to date. Yet most of the studies rely on LHCb data from run I. It is reasonable to expect that  signs of {\\ensuremath{C\\!P}}\\xspace violation from $b$-baryons may still arise with a comprehensive study using the full Run I and Run II data.\n\n \n \n\\section{Hadronic double-charm interaction in charmless {\\ensuremath{\\PB}}\\xspace decays}\n\\label{chapter5}\n\nCharmless {\\ensuremath{\\PB}}\\xspace decays, as discussed in the previous section, can potentially produce {\\ensuremath{C\\!P}}\\xspace violation due to the tree level transition $ {\\ensuremath{\\Pb}}\\xspace \\ensuremath{\\rightarrow}\\xspace {\\ensuremath{\\Pu}}\\xspace $, involving the weak phase $\\gamma$, interfering with different weak transitions such as the penguin-type diagram depicted in Fig.~\\ref{treepenguin}. Nevertheless, a charmless final state can also be reached through the formation of a double-charm.  By simply considering the CKM hierarchy, the inclusive process  ${\\ensuremath{\\Pb}}\\xspace \\ensuremath{\\rightarrow}\\xspace {\\ensuremath{\\Pc}}\\xspace {\\ensuremath{\\overline \\cquark}}\\xspace  {\\ensuremath{\\Pq}}\\xspace$ is about two orders of magnitude more likely than  those with ${\\ensuremath{\\Pb}}\\xspace\\ensuremath{\\rightarrow}\\xspace {\\ensuremath{\\Pu}}\\xspace  {\\ensuremath{\\Pq}}\\xspace{\\ensuremath{\\overline \\quark}}\\xspace'$ ($q, q'={\\ensuremath{\\Pu}}\\xspace,{\\ensuremath{\\Pd}}\\xspace,{\\ensuremath{\\Ps}}\\xspace$) transitions.\n\nThe effective contribution of a double-charm amplitude to charmless {\\ensuremath{\\PB}}\\xspace decays can be twofold. The first one is by direct formation of a charmonium resonance decaying to a pair of pions or kaons, in a three- or four-body  decays. An example is  $ {\\ensuremath{\\Bu}}\\xspace\\ensuremath{\\rightarrow}\\xspace \\chi_{c0}{\\ensuremath{\\pion^+}}\\xspace$, with $\\chi_{c0}\\ensuremath{\\rightarrow}\\xspace{\\ensuremath{\\pion^+}}\\xspace{\\ensuremath{\\pion^-}}\\xspace$. The other is by production of a ${\\ensuremath{\\PD}}\\xspace{\\kern 0.2em\\overline{\\kern -0.2em \\PD}{}}\\xspace'$ pair rescattering to $h \\bar h'$ ($h, h'$ being  light mesons), a long-distance process expected to be suppressed (otherwise ${\\ensuremath{\\PB}}\\xspace \\ensuremath{\\rightarrow}\\xspace {\\ensuremath{\\PD}}\\xspace{\\kern 0.2em\\overline{\\kern -0.2em \\PD}{}}\\xspace$ and $ {\\ensuremath{\\PB}}\\xspace \\ensuremath{\\rightarrow}\\xspace h \\bar h'$ branching ratios would be similar) but nevertheless suitable to  have an impact giving the aforementioned higher inclusive process.\n\nThe possibility of this long-distance rescattering with the same quantum numbers, naturally providing a different strong phase, refers back to the discussion in Sect.~\\ref{CPT}. Many authors have contributed to estimate the effect of double-charm component to charmless {\\ensuremath{\\PB}}\\xspace decays \\cite{Suzuki2007,Soni2005,Smith2003,Buras1997}. \nMost of these models treat only the rescattering of the double-charm to  two- or quasi-two-body {\\ensuremath{\\PB}}\\xspace decays, while comparing to other models based on short-distance approximation or light meson  rescattering. But since the only observables are total rates (and asymmetries) there is no way to disentangle  the relative contribution of each process from the experimental results. \n\n\n\nHere again three- and four-body  decays, due to their rich dynamics across the phase space, may provide signatures of possible contributions associated with double-charm intermediate states, as discussed in the following. \n\n\\subsection{The charmonium ${\\ensuremath{\\Bu}}\\xspace\\ensuremath{\\rightarrow}\\xspace h^+ \\chi_{c0}$ decay}\n\nThe main charmonium resonance contributing to a charmless {\\ensuremath{\\Bu}}\\xspace three-meson final state is the scalar $\\chi_{c0}$, with mass $(3414.7\\pm 0.3)\\ensuremath{\\mathrm{\\,Me\\kern -0.1em V}}\\xspace$ and width $(10.5\\pm 0.8)\\ensuremath{\\mathrm{\\,Me\\kern -0.1em V}}\\xspace$~\\cite{PDG2019}. \nThe  ${\\ensuremath{\\Bu}}\\xspace\\ensuremath{\\rightarrow}\\xspace h^+ \\chi_{c0}$ ($h = {\\ensuremath{\\PK}}\\xspace, {\\ensuremath{\\Ppi}}\\xspace$) decays weakly at tree level through a ${\\ensuremath{\\Pb}}\\xspace \\ensuremath{\\rightarrow}\\xspace {\\ensuremath{\\Pc}}\\xspace {\\ensuremath{\\overline \\cquark}}\\xspace {\\ensuremath{\\Pq}}\\xspace$ (${\\ensuremath{\\Pq}}\\xspace = {\\ensuremath{\\Ps}}\\xspace, {\\ensuremath{\\Pd}}\\xspace$) transition,  with the $\\chi_{c0}$ meson then decaying strongly to a pair of mesons. \nThe branching fraction of the decay ${\\ensuremath{\\Bu}}\\xspace \\ensuremath{\\rightarrow}\\xspace {\\ensuremath{\\kaon^+}}\\xspace \\chi_{c0}$,  for example,  is ${\\cal B}({\\ensuremath{\\Bu}}\\xspace \\ensuremath{\\rightarrow}\\xspace {\\ensuremath{\\kaon^+}}\\xspace \\chi_{c0})= (1.49 \\pm 0.15)\\times 10^{-4} $  with ${\\cal B}(\\chi_{c0}\\ensuremath{\\rightarrow}\\xspace{\\ensuremath{\\pion^+}}\\xspace{\\ensuremath{\\pion^-}}\\xspace) \\sim 6 \\times 10^{-3}$ \\cite{PDG2019},.  \n\n\nIn what concerns theoretical calculations, the critical factor here is that this decay does not include factorisable contributions; the $\\chi_{c0}$, being scalar, cannot be produced by a  matrix element \\mbox{$\\langle \\chi_{c0}|{\\ensuremath{\\overline \\cquark}}\\xspace\\bar \\gamma_\\mu(1-\\gamma_5){\\ensuremath{\\Pc}}\\xspace |0\\rangle  = 0$} due to conservation of the axial-vector current \\cite{Melic2004}. On the other hand, the non-factorisable contributions from the soft hadronic matrix elements, based on the light-cone approach, are too small to accommodate the experimental data \\cite{Wang2003}. It has been shown, however,  that rescattering effects, involving charm meson intermediate states, can reproduce the observed branching ratio  through a mechanism  shown in Fig.~\\ref{B-chiK} \\cite{Colangelo2003}. \n \\begin{figure}[!htb]\n\\centering\n\\includegraphics[width=0.4\\columnwidth,angle=0]{Figure23_BchiK.png}\n\\caption{Possible long-distance contribution to the decay $B^\\pm \\ensuremath{\\rightarrow}\\xspace K^\\pm  \\chi_{c0}$ \\cite{Colangelo2003} .}\n\\label{B-chiK}\n\\end{figure}\n\n\nThe contribution of the intermediate state ${\\ensuremath{\\kaon^+}}\\xspace\\chi_{c0}$ has been observed at a  fit fraction level of 1\\% in amplitude analyses performed by the Babar collaboration in both final states ${\\ensuremath{\\Bu}}\\xspace\\ensuremath{\\rightarrow}\\xspace {\\ensuremath{\\kaon^+}}\\xspace{\\ensuremath{\\pion^+}}\\xspace{\\ensuremath{\\pion^-}}\\xspace$~\\cite{BaBar_bkpipi} and ${\\ensuremath{\\Bu}}\\xspace\\ensuremath{\\rightarrow}\\xspace {\\ensuremath{\\kaon^+}}\\xspace {\\ensuremath{\\kaon^+}}\\xspace{\\ensuremath{\\kaon^-}}\\xspace$~\\cite{BaBar_B3K}. It has been seen to contribute to the same final states from LHCb run I data  \\cite{LHCb-PAPER-2014-044} as can be seen in  Fig.~\\ref{Chi_c_LHCb} as a clear narrow band at  around $11.6 \\ensuremath{\\mathrm{\\,Ge\\kern -0.1em V^2}}\\xspace$ in $m^2_{{\\ensuremath{\\kaon^+}}\\xspace{\\ensuremath{\\kaon^-}}\\xspace}$ and $m^2_{{\\ensuremath{\\pion^+}}\\xspace{\\ensuremath{\\pion^-}}\\xspace}$ in the Dalitz plot of the decays ${\\ensuremath{\\Bu}}\\xspace\\ensuremath{\\rightarrow}\\xspace {\\ensuremath{\\kaon^+}}\\xspace {\\ensuremath{\\kaon^+}}\\xspace{\\ensuremath{\\kaon^-}}\\xspace$ and ${\\ensuremath{\\Bu}}\\xspace\\ensuremath{\\rightarrow}\\xspace {\\ensuremath{\\kaon^+}}\\xspace{\\ensuremath{\\pion^+}}\\xspace{\\ensuremath{\\pion^-}}\\xspace$, respectively.\nIt is also possible to see a band with lack of events just above the $\\chi_{c0}$ mass indicating a pattern of interference with the non-resonant contribution underneath. \n\n\\begin{figure}[!htb]\n\\centering\n\\includegraphics[width=0.45\\columnwidth,angle=0]{Figure24_a.png}\n\\includegraphics[width=0.45\\columnwidth,angle=0]{Figure24_b.png}\n\\caption{Scalar $\\chi_{c0}$ signature in the Dalitz plots of (left) ${\\ensuremath{\\Bu}}\\xspace\\ensuremath{\\rightarrow}\\xspace{\\ensuremath{\\kaon^+}}\\xspace\\Kp{\\ensuremath{\\kaon^-}}\\xspace$  and  (right) ${\\ensuremath{\\Bu}}\\xspace\\ensuremath{\\rightarrow}\\xspace{\\ensuremath{\\kaon^+}}\\xspace{\\ensuremath{\\pion^+}}\\xspace{\\ensuremath{\\pion^-}}\\xspace$ decays  \\cite{LHCb-PAPER-2014-044}.}\n\\label{Chi_c_LHCb}\n\\end{figure}       \n    \nThe decay amplitude ${\\ensuremath{\\pion^+}}\\xspace\\chi_{c0}$ was  not identified in the Dalitz plot analysis of the decay ${\\ensuremath{\\Bu}}\\xspace\\ensuremath{\\rightarrow}\\xspace{\\ensuremath{\\pion^+}}\\xspace\\pip{\\ensuremath{\\pion^-}}\\xspace$ made recently by the LHCb collaboration~\\cite{LHCb-PAPER-2019-017} but it is likely to be observed from run  II data, at the level of a few hundred events or so. \n\n A possible interference between the ${\\ensuremath{\\Bu}}\\xspace\\ensuremath{\\rightarrow}\\xspace {\\ensuremath{\\pion^+}}\\xspace \\chi_{c0} $ amplitude  and the dominant ${\\ensuremath{\\Bu}}\\xspace\\ensuremath{\\rightarrow}\\xspace{\\ensuremath{\\pion^+}}\\xspace\\rho(770)^0$ (which has a total {\\ensuremath{C\\!P}}\\xspace asymmetry consistent with zero, as discussed in Sect.~\\ref{quasi2body}) could be used for a measurement of $\\gamma$ \\cite{Gobel1998}, neglecting contributions from other amplitudes in their interference region, otherwise other components may as well be included \\cite{Gobel2000}. \nA similar argument  is proposed in Ref.~\\cite{Gronau1995}, this time exploiting a measurement of  the {\\ensuremath{C\\!P}}\\xspace asymmetry for the ${\\ensuremath{\\Bu}}\\xspace\\ensuremath{\\rightarrow}\\xspace {\\ensuremath{\\pion^+}}\\xspace\\chi_{c0}$ through the interference with a non-resonant amplitude.\nSince $\\gamma\\approx 70^\\circ$, this interference term possibly changes sign by  charge conjugation, appearing as constructive interference for {\\ensuremath{\\Bu}}\\xspace decay and destructive for {\\ensuremath{\\Bub}}\\xspace decay (or {\\it vice-versa}).  \nThey also have generalised this approach for other final states involving  vector light mesons and the charmonium $\\eta_c$.\n\n\\subsection{Double-charm and ${\\ensuremath{\\Bu}}\\xspace\\ensuremath{\\rightarrow}\\xspace {\\ensuremath{\\kaon^+}}\\xspace\\Kp{\\ensuremath{\\kaon^-}}\\xspace$ final state }\n\\label{doublecharmKKK}\nThe amplitude analysis of  ${\\ensuremath{\\Bu}}\\xspace\\ensuremath{\\rightarrow}\\xspace{\\ensuremath{\\kaon^+}}\\xspace\\Kp{\\ensuremath{\\kaon^-}}\\xspace$ from both the BaBar and Belle collaborations  \\cite{BaBar_B3K,Belle_3h_2004} showed a dominant  non-resonant contribution. In a recent paper \\cite{Pat2017}  the authors  propose a non-resonant contribution to this decay which includes rescattering hadronic component involving double-charm-meson intermediate states, in a similar direction as proposed for the ${\\ensuremath{\\Bu}}\\xspace\\ensuremath{\\rightarrow}\\xspace {\\ensuremath{\\kaon^+}}\\xspace\\chi_{c0}$ in Ref.~\\cite{Colangelo2003}. \n\nThe LHCb experiment collected more than a hundred thousand ${\\ensuremath{\\Bu}}\\xspace\\ensuremath{\\rightarrow}\\xspace {\\ensuremath{\\kaon^+}}\\xspace\\Kp{\\ensuremath{\\kaon^-}}\\xspace$ decays from \\mbox{run I} period  \\cite{LHCb-PAPER-2014-044} and the Dalitz plot distribution can be seen in Fig.~\\ref{Chi_c_LHCb}. \nThe narrow resonant contributions are clearly seen,  ${\\ensuremath{\\Bu}}\\xspace\\ensuremath{\\rightarrow}\\xspace {\\ensuremath{\\kaon^+}}\\xspace\\phi(1020)$ and ${\\ensuremath{\\Bu}}\\xspace\\ensuremath{\\rightarrow}\\xspace {\\ensuremath{\\kaon^+}}\\xspace\\chi_{c0}$, together with a large non-resonant contribution, populating  the whole phase space. Although no amplitude analysis was yet performed, the distribution appears to be consistent with the results from BaBar and Belle amplitude analyses \\cite{BaBar_B3K,Belle_3h_2004}, which had much smaller samples. This significant presence of the non-resonant amplitude makes this decay different from the other charmless three-body {\\ensuremath{\\PB}}\\xspace decays (with low occupation at higher masses), and potentially offers sensitivity, through interference, to study high-mass intermediate resonances, including   {\\ensuremath{C\\!P}}\\xspace signatures. \n\nAnother interesting feature observed in the ${\\ensuremath{\\Bu}}\\xspace\\ensuremath{\\rightarrow}\\xspace {\\ensuremath{\\kaon^+}}\\xspace\\Kp{\\ensuremath{\\kaon^-}}\\xspace$ Dalitz plot distribution is the localised {\\ensuremath{C\\!P}}\\xspace asymmetry at low ${\\ensuremath{\\kaon^+}}\\xspace{\\ensuremath{\\kaon^-}}\\xspace$ mass, already discussed in Sect.~\\ref{kkpipiresc}. Figure~\\ref{KKK_projection} (left) shows the events for ${\\ensuremath{\\Bu}}\\xspace$ and ${\\ensuremath{\\Bub}}\\xspace$ integrated in $m({\\ensuremath{\\kaon^+}}\\xspace{\\ensuremath{\\kaon^-}}\\xspace)_{\\rm low}$  \\cite{LHCb-PAPER-2014-044}. The two peaks correspond to the angular distribution associated with the vector resonance $\\phi(1020)$. By subtracting both distributions, as shown in Fig.~\\ref{KKK_projection} (right), the charge difference associated to {\\ensuremath{C\\!P}}\\xspace violation can be seen. It is possible to identify the negative {\\ensuremath{C\\!P}}\\xspace asymmetry located in the $\\pi\\pi-KK$ rescattering region, but also the fact that the {\\ensuremath{C\\!P}}\\xspace violation pattern changes sign,  crossing zero at about 4\\ensuremath{\\mathrm{\\,Ge\\kern -0.1em V}}\\xspace, which is near the ${\\ensuremath{\\PD}}\\xspace{\\kern 0.2em\\overline{\\kern -0.2em \\PD}{}}\\xspace$ threshold. Moreover, the LHCb data distribution for ${\\ensuremath{\\Bu}}\\xspace\\ensuremath{\\rightarrow}\\xspace {\\ensuremath{\\kaon^+}}\\xspace{\\ensuremath{\\pion^+}}\\xspace{\\ensuremath{\\pion^-}}\\xspace$ also shows a similar,  although opposite,  sign inversion around 4 \\ensuremath{\\mathrm{\\,Ge\\kern -0.1em V}}\\xspace.\\footnote{See the Supplemental material associated to \\cite{LHCb-PAPER-2014-044} at https:\/\/cds.cern.ch\/record\/1751517\/files.} The same behaviour appears in the pair of channels ${\\ensuremath{\\Bu}}\\xspace\\ensuremath{\\rightarrow}\\xspace{\\ensuremath{\\pion^+}}\\xspace\\pip{\\ensuremath{\\pion^-}}\\xspace$ and ${\\ensuremath{\\Bu}}\\xspace\\ensuremath{\\rightarrow}\\xspace{\\ensuremath{\\pion^+}}\\xspace{\\ensuremath{\\kaon^+}}\\xspace{\\ensuremath{\\kaon^-}}\\xspace$. \n\n\n\n\n\\begin{figure}[htbp]\n\\centering\n\\includegraphics[width=0.45\\columnwidth,angle=0]{Figure25_a.png}\n\\includegraphics[width=0.45\\columnwidth,angle=0]{Figure25_b.png}\n\\caption{ (left) Projection of $m(KK)_{\\rm high}$ distribution for $B^\\pm \\ensuremath{\\rightarrow}\\xspace K^\\pm K^+K^-$ decay  separated by charge. (right)  Subtracted distribution of the {\\ensuremath{\\Bub}}\\xspace and {\\ensuremath{\\Bu}}\\xspace $m(KK)_{\\rm high}$ projections from the left plot. \\cite{LHCb-PAPER-2014-044} }\n\\label{KKK_projection}\n\\end{figure}   \n\nThis  {\\ensuremath{C\\!P}}\\xspace-asymmetry sign inversion can be explained  by a hadronic charm loop process shown in Fig.~\\ref{LoopKKK}, according to the model presented in Ref.~\\cite{Pat2017}.\nThe corresponding amplitude  has a strong-phase sign change at the ${\\ensuremath{\\PD}}\\xspace{\\kern 0.2em\\overline{\\kern -0.2em \\PD}{}}\\xspace$ threshold: the phase changes from about $-\\pi\/2$ to $\\pi\/2$ close to the region where the {\\ensuremath{C\\!P}}\\xspace asymmetry changes sign in Fig.~\\ref{KKK_projection}. Although this signature favours the interpretation of  double-charm rescattering, the model has more complexity and only a full amplitude analysis could test for the validity of this mechanism.\n\n\\begin{figure}[htbp]\n\\centering\n\\includegraphics[width=.4\\columnwidth,angle=0]{Figure26_hadronloop.png}\n\\caption{Possible long-distance contribution to the ${\\ensuremath{\\Bu}}\\xspace\\ensuremath{\\rightarrow}\\xspace {\\ensuremath{\\kaon^+}}\\xspace\\Kp{\\ensuremath{\\kaon^-}}\\xspace$ decay \\cite{Pat2017}. }\n\\label{LoopKKK}\n\\end{figure}    \n \n\n\\subsection{Possible consequence for the ${\\ensuremath{\\Bu}}\\xspace\\ensuremath{\\rightarrow}\\xspace{\\ensuremath{\\pion^+}}\\xspace\\pip{\\ensuremath{\\pion^-}}\\xspace$ decay} \n\n \nAs discussed previously, the short-distance contribution for the ${\\ensuremath{\\Bu}}\\xspace \\ensuremath{\\rightarrow}\\xspace {\\ensuremath{\\pion^+}}\\xspace\\pip{\\ensuremath{\\pion^-}}\\xspace$ decay is dominated by the tree contribution with a weak phase $\\gamma$. On the other hand, a  long-distance contribution can also be possible by a similar process to that in Fig.~\\ref{LoopKKK} in the decay ${\\ensuremath{\\Bu}}\\xspace\\ensuremath{\\rightarrow}\\xspace{\\ensuremath{\\kaon^+}}\\xspace\\Kp{\\ensuremath{\\kaon^-}}\\xspace$. The interference between these short-distance tree component  and  double-charm rescattering, carrying no weak phase, may lead to a signature of {\\ensuremath{C\\!P}}\\xspace violation spread throughout the phase space. By inspecting the bottom-left plot in Fig.~\\ref{Mirandizing4channels}, indeed there is a significant {\\ensuremath{C\\!P}}\\xspace asymmetry in the high ${\\ensuremath{\\pion^+}}\\xspace{\\ensuremath{\\pion^-}}\\xspace$ mass regions.\n\n\n\n\n\nRecently a phenomenological study for the ${\\ensuremath{\\Bu}}\\xspace\\ensuremath{\\rightarrow}\\xspace{\\ensuremath{\\pion^+}}\\xspace\\pip{\\ensuremath{\\pion^-}}\\xspace$ mode has explored the interference between the tree component, carrying the weak phase, and a modelled amplitude for the ${\\ensuremath{\\PD}}\\xspace\\bar{D}$ and $\\pi\\pi$ coupled channels, with   $\\chi_{c0}$ as  complex pole in the ${\\ensuremath{\\PD}}\\xspace\\bar{D}$ channel \\cite{Bediaga:2020ztp}. As discussed above in Sect.~\\ref{doublecharmKKK}, the  latter amplitude has a strong-phase sign change when it passes from off-shell to on-shell, and this effect could be   responsible for the {\\ensuremath{C\\!P}}\\xspace asymmetry pattern  observed  in the bottom-left plot in Fig.~\\ref{Mirandizing4channels} near the ${\\ensuremath{\\PD}}\\xspace\\bar{D}$ thresholds. A simulation for this model, using Laura++ \\cite{Laura++} generator, is shown in Fig.~\\ref{B3pi_DD}. \nAnother recent study ~\\cite{Mannel:2020abt}  also discusses the potential role  of open-charm threshold amplitudes to the observed {\\ensuremath{C\\!P}}\\xspace violation in this decay channel. In a similar way,  the model exploits  the interference between light resonances with both open-charm  ${\\ensuremath{\\PD}}\\xspace\\bar{D}$  amplitudes and high-mass charmonium  resonances.  The {\\ensuremath{C\\!P}}\\xspace asymmetry pattern obtained through the  interference between $\\rho(770)^0$ and the propagator for $\\chi_{c0}(3860)$ scalar resonance is illustrated in Fig.~\\ref{MannelB3pi}.\n\n\n\n\n\\begin{figure}[htbp]\n\\centering\n\\includegraphics[width=.5\\columnwidth,angle=0]{Figure27.png}\n\\caption{{\\ensuremath{C\\!P}}\\xspace asymmetry pattern for the decay ${\\ensuremath{\\Bu}}\\xspace\\ensuremath{\\rightarrow}\\xspace{\\ensuremath{\\pion^+}}\\xspace\\pip{\\ensuremath{\\pion^-}}\\xspace$, obtained through the Miranda method via simulation according to the model presented in Ref.~\\cite{Bediaga:2020ztp}. }\n\\label{B3pi_DD}\n\\end{figure}  \n\n\\begin{figure}[htbp]\n\\centering\n\\includegraphics[width=.5\\columnwidth,angle=0]{Figure28.png}\n\\caption{{\\ensuremath{C\\!P}}\\xspace asymmetry pattern for the decay ${\\ensuremath{\\Bu}}\\xspace\\ensuremath{\\rightarrow}\\xspace{\\ensuremath{\\pion^+}}\\xspace\\pip{\\ensuremath{\\pion^-}}\\xspace$ considering the interference between the $\\rho(770)^0$ and $\\chi_{c0}(3860)$ \\cite{Mannel:2020abt}. The variables $s_{12}$ and $s_{23}$ here represent $m^2({\\ensuremath{\\pion^+}}\\xspace{\\ensuremath{\\pion^-}}\\xspace)_{\\rm low}$ and $m^2({\\ensuremath{\\pion^+}}\\xspace{\\ensuremath{\\pion^-}}\\xspace)_{\\rm high}$, respectively.}\n\\label{MannelB3pi}\n\\end{figure}  \n\n\n\n\n\nA possible observation of the $B^\\pm \\ensuremath{\\rightarrow}\\xspace \\pi^\\pm \\chi_{c0}$ mode with the LHCb run II data may show a {\\ensuremath{C\\!P}}\\xspace violation pattern in this charmonium state as predicted by Ref.~\\cite{Gronau1995}, and, if so, one may infer the importance of a double charm re-scattering in this decay. In other words, if the {\\ensuremath{C\\!P}}\\xspace violation pattern for both  the $\\chi_{c0}$ component and the non-resonant component (covering a wider mass region) are similar, this would point towards the interpretation of the non-resonant amplitude receiving an important contribution from the double-charm loop mechanism. \n\n\n\n\\subsection{Search for double-charm signature in $ {\\ensuremath{\\B_\\cquark^+}}\\xspace\\ensuremath{\\rightarrow}\\xspace{\\ensuremath{\\pion^+}}\\xspace{\\ensuremath{\\kaon^+}}\\xspace{\\ensuremath{\\kaon^-}}\\xspace$ decay} \n\nRecently LHCb, using Run I data,  presented evidence of the decay ${\\ensuremath{\\B_\\cquark^+}}\\xspace \\ensuremath{\\rightarrow}\\xspace {\\ensuremath{\\pion^+}}\\xspace\\chi_{c0}$,  with $\\chi_{c0} \\ensuremath{\\rightarrow}\\xspace {\\ensuremath{\\kaon^+}}\\xspace{\\ensuremath{\\kaon^-}}\\xspace$ decay, with a branching ratio  of $ \\left[\\sigma({\\ensuremath{\\B_\\cquark^+}}\\xspace)\/\\sigma({\\ensuremath{\\Bu}}\\xspace)\\right] \\times {\\cal B}({\\ensuremath{\\Bu}}\\xspace\\ensuremath{\\rightarrow}\\xspace{\\ensuremath{\\pion^+}}\\xspace\\chi_{c0} )= (9.8 \\pm 3.4) \\times 10^{-6}$  \\cite{LHCb-PAPER-2016-022}. The collaboration searched also for a possible non-charmonium component to ${\\ensuremath{\\B_\\cquark^+}}\\xspace\\ensuremath{\\rightarrow}\\xspace {\\ensuremath{\\pion^+}}\\xspace{\\ensuremath{\\kaon^+}}\\xspace{\\ensuremath{\\kaon^-}}\\xspace$ decay, but no significant contribution was found. From short-distance calculations  \\cite{Liu2009}, this contribution indeed  should be helicity suppressed. This opens  the possibility to   observe ${\\ensuremath{\\B_\\cquark^+}}\\xspace \\ensuremath{\\rightarrow}\\xspace {\\ensuremath{\\pion^+}}\\xspace{\\ensuremath{\\kaon^+}}\\xspace{\\ensuremath{\\kaon^-}}\\xspace$  events coming from a long-distance double charm rescattering contribution similar to those shown in Figs.~\\ref{B-chiK} and \\ref{LoopKKK}. \n\n\n   A  simulation with  amplitudes corresponding to the long-distance contributions from the double-charm rescattering processes ${\\ensuremath{\\D^0}}\\xspace{\\ensuremath{\\Dbar{}^0}}\\xspace \\ensuremath{\\rightarrow}\\xspace {\\ensuremath{\\kaon^+}}\\xspace{\\ensuremath{\\kaon^-}}\\xspace$ and ${\\ensuremath{\\D^+}}\\xspace{\\ensuremath{\\D^-_\\squark}}\\xspace\\ensuremath{\\rightarrow}\\xspace{\\ensuremath{\\pion^+}}\\xspace{\\ensuremath{\\kaon^-}}\\xspace$, as from Ref.~\\cite{Pat2018}, presents a  distinguished  signature for the Dalitz plot distribution, as shown in Fig.~\\ref{B_cDalitz}. This is quite different from the distribution observed in charmless three-body {\\ensuremath{\\Bu}}\\xspace decays where the events are concentrated at the edges of the Dalitz plot, as in Fig.~\\ref{Chi_c_LHCb}. \n   \n   \n\\begin{figure}[!htb]\n\\centering\n\\includegraphics[width=0.5\\columnwidth,angle=0]{Figure29_BcDalitz.png}\n\\caption{Simulated Dalitz plot  for the decay $ {\\ensuremath{\\B_\\cquark^+}}\\xspace\\ensuremath{\\rightarrow}\\xspace  {\\ensuremath{\\pion^+}}\\xspace{\\ensuremath{\\kaon^+}}\\xspace{\\ensuremath{\\kaon^-}}\\xspace$ considering double-charm long-distance contributions \\cite{Pat2018}, using Laura++ \\cite{Laura++}.}\n\\label{B_cDalitz} \n\\end{figure} \n\n\n\n\n\n \n\\section{Direct {\\ensuremath{C\\!P}}\\xspace violation in {\\ensuremath{\\PB}}\\xspace decays to open charm}\n\\label{chapter6}\n\nAt tree level, the ${\\ensuremath{\\Pb}}\\xspace$ quark dominantly decays to a ${\\ensuremath{\\Pc}}\\xspace$ quark, with no weak phase.  This process makes decays to open charm those with higher branching ratios. Some of these final states, after the further weak decay of the ${\\ensuremath{\\Pc}}\\xspace$ hadron,  can also be reachable through the quark coupling $ {\\ensuremath{\\Pb}}\\xspace \\ensuremath{\\rightarrow}\\xspace {\\ensuremath{\\Pu}}\\xspace{\\ensuremath{\\overline \\cquark}}\\xspace{\\ensuremath{\\Pq}}\\xspace$ (${\\ensuremath{\\Pq}}\\xspace = {\\ensuremath{\\Ps}}\\xspace, {\\ensuremath{\\Pd}}\\xspace$) involving the CKM weak phase $\\gamma$, which is also a tree-level process, with no penguin diagrams involved. The interference between them can  allow the experimental determination of  the angle $\\gamma$. From the theoretical point of view, these decays are extremely clean \\cite{BrodZupan_2013}.\n\nThe main mechanisms proposed to exploit this feature as a source of direct {\\ensuremath{C\\!P}}\\xspace violation are the Gronau--London--Wyler (GLW) \\cite{Gronau1990,Gronau1991},  Atwood--Dunietz--Sony (ADS) \\cite{Atwood1996,Atwood2000}, and Giri--Grossman--Soffer--Zuppan (GGSZ) \\cite{GGSZ} methods, as described next, with their main associated experimental results. The value of $\\gamma$ is obtained from the combination of these different methods.\n\n \\subsection{The GLW method}\n \nThe GLW method makes use of the interference between the processes ${\\ensuremath{\\Bu}}\\xspace\\ensuremath{\\rightarrow}\\xspace {\\ensuremath{\\D^0}}\\xspace h^+$ and ${\\ensuremath{\\Bu}}\\xspace\\ensuremath{\\rightarrow}\\xspace {\\ensuremath{\\Dbar{}^0}}\\xspace h^+$ ($h=\\PK,\\pi$). The first is dominant, proceeding through a ${\\ensuremath{\\Pb}}\\xspace\\ensuremath{\\rightarrow}\\xspace {\\ensuremath{\\Pc}}\\xspace$ transition, while the second is suppressed given by the process $ {\\ensuremath{\\Pb}}\\xspace \\ensuremath{\\rightarrow}\\xspace {\\ensuremath{\\Pu}}\\xspace {\\ensuremath{\\overline \\cquark}}\\xspace {\\ensuremath{\\Ps}}\\xspace$. \n  These decays can interfere  if  ${\\ensuremath{\\D^0}}\\xspace$ and ${\\ensuremath{\\Dbar{}^0}}\\xspace$ (referred to as ${\\ensuremath{\\PD}}\\xspace$ -- a general superposition of both) decay to {\\ensuremath{C\\!P}}\\xspace eigenstates, such as  ${\\ensuremath{\\PD}}\\xspace\\ensuremath{\\rightarrow}\\xspace {\\ensuremath{\\kaon^+}}\\xspace{\\ensuremath{\\kaon^-}}\\xspace$ ({\\ensuremath{C\\!P}}\\xspace-even) or ${\\ensuremath{\\PD}}\\xspace\\ensuremath{\\rightarrow}\\xspace{\\ensuremath{\\kaon^0_{\\rm\\scriptscriptstyle S}}}\\xspace{\\ensuremath{\\pion^0}}\\xspace$ ({\\ensuremath{C\\!P}}\\xspace-odd),  as shown in Fig.~\\ref{B-DK} \\cite{TimVava}. \n  The resulting {\\ensuremath{C\\!P}}\\xspace asymmetry, as in Eq.~\\ref{ACP_sindelta_sinphi}, depends on the ratio $r_{\\ensuremath{\\PB}}\\xspace$ of the sub-leading ${\\ensuremath{\\Bu}}\\xspace\\ensuremath{\\rightarrow}\\xspace {\\ensuremath{\\Dbar{}^0}}\\xspace{\\ensuremath{\\kaon^+}}\\xspace$ to the leading ${\\ensuremath{\\Bu}}\\xspace\\ensuremath{\\rightarrow}\\xspace {\\ensuremath{\\D^0}}\\xspace{\\ensuremath{\\kaon^+}}\\xspace$ amplitude as well as on their relative strong phase $\\delta_{\\ensuremath{\\PB}}\\xspace$,\n\\begin{equation}\nA_{\\ensuremath{C\\!P}}\\xspace = \\frac{2\\eta^{\\ensuremath{C\\!P}}\\xspace r_{\\ensuremath{\\PB}}\\xspace \\sin (\\delta_{\\ensuremath{\\PB}}\\xspace) \\sin (\\gamma)}{1+r_{\\ensuremath{\\PB}}\\xspace^2 +2\\eta_{\\ensuremath{C\\!P}}\\xspace r_{\\ensuremath{\\PB}}\\xspace\\cos(\\delta_{\\ensuremath{\\PB}}\\xspace)\\cos(\\gamma)}~,\n\\label{ACP_GLW}\n\\end{equation}\nwhere $\\eta^{\\ensuremath{C\\!P}}\\xspace=\\{+1,-1\\}$ depending on whether the ${\\ensuremath{\\D^0}}\\xspace$ decays to a {\\ensuremath{C\\!P}}\\xspace-even or {\\ensuremath{C\\!P}}\\xspace-odd final state, respectively. The experimental analyses can obtain $r_{\\ensuremath{\\PB}}\\xspace$, $\\delta_B$ and $\\gamma$ using measured {\\ensuremath{C\\!P}}\\xspace asymmetries from different ${\\ensuremath{\\D^0}}\\xspace$ final states. \n\n  \\begin{figure}[htbp]\n\\centering\n\\includegraphics[width=.8\\columnwidth,angle=0]{Figure30_BDK.png}\n\\caption{Leading diagrams for the decays ${\\ensuremath{\\Bu}}\\xspace\\ensuremath{\\rightarrow}\\xspace {\\ensuremath{\\D^0}}\\xspace {\\ensuremath{\\kaon^+}}\\xspace $ and ${\\ensuremath{\\Bu}}\\xspace \\ensuremath{\\rightarrow}\\xspace {\\ensuremath{\\Dbar{}^0}}\\xspace{\\ensuremath{\\kaon^+}}\\xspace $, where $\\left[f\\right]_D$ is a final state accessible by both ${\\ensuremath{\\D^0}}\\xspace$ and ${\\ensuremath{\\Dbar{}^0}}\\xspace$ \\cite{TimVava}.}.\n\\label{B-DK}\n\\end{figure}\n\nThe Babar, Belle, CDF and LHCb collaborations have reported $A_{\\ensuremath{C\\!P}}\\xspace$ and $\\gamma$ measurements based on GLW modes (see for example \\cite{Aaltonen:2009hz,delAmoSanchez:2010ji,Aihara:2012aw,LHCb-PAPER-2014-041}).  The latest and most precise measurement of $A_{\\ensuremath{C\\!P}}\\xspace$ in \\mbox{${\\ensuremath{\\Bu}}\\xspace\\ensuremath{\\rightarrow}\\xspace{\\ensuremath{\\D^0}}\\xspace{\\ensuremath{\\kaon^+}}\\xspace$} comes from LHCb using full run I and part of run II data~\\cite{LHCb-PAPER-2017-021}. Figure~\\ref{B-DK-hh} shows the relevant mass distributions for ${\\ensuremath{\\Bu}}\\xspace\\ensuremath{\\rightarrow}\\xspace{\\ensuremath{\\PD}}\\xspace{\\ensuremath{\\kaon^+}}\\xspace$ where ${\\ensuremath{\\PD}}\\xspace\\ensuremath{\\rightarrow}\\xspace{\\ensuremath{\\kaon^+}}\\xspace{\\ensuremath{\\kaon^-}}\\xspace$ and ${\\ensuremath{\\PD}}\\xspace\\ensuremath{\\rightarrow}\\xspace{\\ensuremath{\\pion^+}}\\xspace{\\ensuremath{\\pion^-}}\\xspace$. A visible difference in signal yields for {\\ensuremath{\\Bu}}\\xspace and {\\ensuremath{\\Bub}}\\xspace is seen in both final states. The combined  {\\ensuremath{C\\!P}}\\xspace asymmetry of the two final states was measured to be \n\\begin{equation}\nA_{{\\ensuremath{C\\!P}}\\xspace} = - (12.4\\pm  1.2\\pm 0.2)\\%\n\\end{equation}\nwith {\\ensuremath{C\\!P}}\\xspace violation clearly established in this mode. \n\n\\begin{figure}[htbp]\n\\centering\n\\includegraphics[align = c, width=0.7\\columnwidth,angle=0]{Figure31_a.png}\n\\includegraphics[align = c, width=.25\\columnwidth,angle=0]{Figure31_b.png}\n\\caption{Invariant mass distributions of selected  (top) ${\\ensuremath{\\PB}}\\xspace^\\pm \\ensuremath{\\rightarrow}\\xspace [{\\ensuremath{\\kaon^+}}\\xspace{\\ensuremath{\\kaon^-}}\\xspace]_{\\ensuremath{\\PD}}\\xspace\\PK^\\pm$ and (bottom) ${\\ensuremath{\\PB}}\\xspace^\\pm \\ensuremath{\\rightarrow}\\xspace [{\\ensuremath{\\pion^+}}\\xspace{\\ensuremath{\\pion^-}}\\xspace]_D\\PK^\\pm$ decays, separated by\ncharge \\cite{LHCb-PAPER-2017-021}. The different fit components are displayed in the legend on the right. }\n\\label{B-DK-hh}\n\\end{figure}\n  \n  \n  \\subsection{The ADS method}\n  \nThe ADS method \\cite{Atwood1996} is similar to GLW but instead uses the interference between Cabibbo-favoured (CF) ${\\ensuremath{\\D^0}}\\xspace$ decays and Doubly--Cabibbo-suppressed (DCS) ${\\ensuremath{\\Dbar{}^0}}\\xspace$ decays (and vice versa), for example ${\\ensuremath{\\D^0}}\\xspace\\ensuremath{\\rightarrow}\\xspace{\\ensuremath{\\kaon^-}}\\xspace{\\ensuremath{\\pion^+}}\\xspace$ and ${\\ensuremath{\\Dbar{}^0}}\\xspace\\ensuremath{\\rightarrow}\\xspace{\\ensuremath{\\kaon^-}}\\xspace{\\ensuremath{\\pion^+}}\\xspace$. In this case, two new parameters get in place: $r_{\\ensuremath{\\PD}}\\xspace$ and $\\delta_{\\ensuremath{\\PD}}\\xspace$, respectively  the ratio of the DCS to the CF amplitude and their relative phase ($\\bar D$ to ${\\ensuremath{\\PD}}\\xspace$ to the same final state). Equation~\\ref{ACP_GLW} gets modified to\n\\begin{equation}\nA_{\\ensuremath{C\\!P}}\\xspace = \\frac{ 2r_{\\ensuremath{\\PD}}\\xspace r_{\\ensuremath{\\PB}}\\xspace \\sin (\\delta_{\\ensuremath{\\PD}}\\xspace+\\delta_{\\ensuremath{\\PB}}\\xspace) \\sin (\\gamma)}{r_{\\ensuremath{\\PD}}\\xspace^2+r_{\\ensuremath{\\PB}}\\xspace^2 +2r_{\\ensuremath{\\PD}}\\xspace r_{\\ensuremath{\\PB}}\\xspace \\cos(\\delta_{\\ensuremath{\\PD}}\\xspace+\\delta_{\\ensuremath{\\PB}}\\xspace)\\cos(\\gamma)}~.\n\\label{ACP_ADS}\n\\end{equation}\n\nMany ADS modes have been studied, such as in Refs.~\\cite{PhysRevD.84.091504,PhysRevLett.106.231803, LHCb-PAPER-2016-003}. \nThe measured {\\ensuremath{C\\!P}}\\xspace asymmetry obtained for the ${\\ensuremath{\\PD}}\\xspace\\ensuremath{\\rightarrow}\\xspace \\PK^\\mp\\pi^\\pm$ channels is quite large \\cite{LHCb-PAPER-2016-003}, \n\n\\begin{equation}\nA_{{\\ensuremath{C\\!P}}\\xspace} = - 0.403 \\pm  0.056~,\n\\end{equation}\nwhich can also be seen from Fig.~\\ref{B-DK-Kpi}.\n  \n\\begin{figure}[htb]\n\\centering\n\\includegraphics[width=.8\\columnwidth,angle=0]{Figure32.png}\n\\caption{Invariant mass distributions of selected  ${\\ensuremath{\\Bu}}\\xspace\\ensuremath{\\rightarrow}\\xspace[{\\ensuremath{\\kaon^-}}\\xspace{\\ensuremath{\\pion^+}}\\xspace]_{{\\ensuremath{\\PD}}\\xspace} {\\ensuremath{\\kaon^+}}\\xspace $ separated by\ncharge. The signal contribution to the fits3 is displayed through the red curve. For details, see \\cite{LHCb-PAPER-2016-003}.}\n\\label{B-DK-Kpi}\n\\end{figure}  \n  \nThe ADS method can also be applied for three- or four-body modes \\cite{Atwood2000}, for example ${\\ensuremath{\\PD}}\\xspace\\ensuremath{\\rightarrow}\\xspace{\\ensuremath{\\kaon^-}}\\xspace{\\ensuremath{\\pion^+}}\\xspace{\\ensuremath{\\pion^0}}\\xspace$ and ${\\ensuremath{\\PD}}\\xspace\\ensuremath{\\rightarrow}\\xspace{\\ensuremath{\\kaon^-}}\\xspace{\\ensuremath{\\pion^+}}\\xspace{\\ensuremath{\\pion^-}}\\xspace{\\ensuremath{\\pion^+}}\\xspace$, with the introduction of an extra parameter, the coherence factor $\\kappa$, multiplying the $2r_{\\ensuremath{\\PD}}\\xspace r_{\\ensuremath{\\PB}}\\xspace$ terms in Eq.~\\ref{ACP_ADS} varying between 0 and 1 depending on the resonant structure of the decay. It has been also used by several analyses such as in Refs.~\\cite{PhysRevD.84.012002}  (using ${\\ensuremath{\\PD}}\\xspace\\ensuremath{\\rightarrow}\\xspace{\\ensuremath{\\kaon^-}}\\xspace{\\ensuremath{\\pion^+}}\\xspace{\\ensuremath{\\pion^0}}\\xspace$) and \\cite{LHCb-PAPER-2016-003} (using ${\\ensuremath{\\PD}}\\xspace\\ensuremath{\\rightarrow}\\xspace{\\ensuremath{\\kaon^-}}\\xspace{\\ensuremath{\\pion^+}}\\xspace{\\ensuremath{\\pion^-}}\\xspace{\\ensuremath{\\pion^+}}\\xspace$). \n\n \n\\subsection{The GGSZ method}\n\n\n\nThe GGSZ method \\cite{GGSZ} exploits the strong phase variation across the phase space of a three- or four-body self-conjugate {\\ensuremath{\\PD}}\\xspace decay in the process ${\\ensuremath{\\Bu}}\\xspace\\ensuremath{\\rightarrow}\\xspace {\\ensuremath{\\PD}}\\xspace h^+$. The benchmark mode is ${\\ensuremath{\\PD}}\\xspace\\ensuremath{\\rightarrow}\\xspace {\\ensuremath{\\kaon^0_{\\rm\\scriptscriptstyle S}}}\\xspace{\\ensuremath{\\pion^+}}\\xspace{\\ensuremath{\\pion^-}}\\xspace$ but there are  others such as, for example,  ${\\ensuremath{\\PD}}\\xspace\\ensuremath{\\rightarrow}\\xspace{\\ensuremath{\\kaon^0_{\\rm\\scriptscriptstyle S}}}\\xspace{\\ensuremath{\\kaon^+}}\\xspace{\\ensuremath{\\kaon^-}}\\xspace$ and ${\\ensuremath{\\PD}}\\xspace\\ensuremath{\\rightarrow}\\xspace{\\ensuremath{\\kaon^0_{\\rm\\scriptscriptstyle S}}}\\xspace{\\ensuremath{\\pion^+}}\\xspace{\\ensuremath{\\pion^-}}\\xspace{\\ensuremath{\\pion^0}}\\xspace$. It compares the distribution of events in the phase space for {\\ensuremath{\\Bu}}\\xspace and {\\ensuremath{\\Bub}}\\xspace  and is the most sensitive method for obtaining $\\gamma$. It depends on the prior -- external -- knowledge of the strong phase which could have been measured through an amplitude analysis (model dependent) or  direct measurement (model independent) via the quantum correlation of ${\\ensuremath{\\D^0}}\\xspace{\\ensuremath{\\Dbar{}^0}}\\xspace$ pairs from $\\psi(3770)$, as obtained by the CLEO collaboration \\cite{CLEO_DKSpipi_phase_2010} for ${\\ensuremath{\\PD}}\\xspace\\ensuremath{\\rightarrow}\\xspace {\\ensuremath{\\kaon^0_{\\rm\\scriptscriptstyle S}}}\\xspace{\\ensuremath{\\pion^+}}\\xspace{\\ensuremath{\\pion^-}}\\xspace$ for instance.\\footnote{Just recently, the BESIII collaboration released their results on the strong phase across the Dalitz plot for ${\\ensuremath{\\PD}}\\xspace\\ensuremath{\\rightarrow}\\xspace {\\ensuremath{\\kaon^0_{\\rm\\scriptscriptstyle S}}}\\xspace{\\ensuremath{\\pion^+}}\\xspace{\\ensuremath{\\pion^-}}\\xspace$ decays \\cite{BESIII_Kspipi2020_1,BESIII_Kspipi2020_2}, with the same binning scheme as CLEO, and which is expect to improve the corresponding uncertainty on $\\gamma$ by approximately a factor of 3.} \nIn Fig.~\\ref{cleo_Dkspipiphasemap} the model-independent strong phase difference for {\\ensuremath{\\D^0}}\\xspace and {\\ensuremath{\\Dbar{}^0}}\\xspace decays is shown across the Dalitz plot, where the eight (coloured) regions represent different phase intervals. Although the model-dependent approach may provide better statistical uncertainties, it relies on amplitude models (full Dalitz plot analysis) which suffer from irreducible systematics. The model-independent approach, on the other hand, is only statistically limited.  \n\nThe observables in the GGSZ method are the variables $x_\\pm$ and $y_\\pm$ which related to the physical observables as\n\\begin{eqnarray}\nx_\\pm =&\\!\\!  r_B\\cos(\\delta_{\\ensuremath{\\PB}}\\xspace\\pm\\gamma) \\nonumber \\\\\ny_\\pm =&\\!  r_B\\sin(\\delta_{\\ensuremath{\\PB}}\\xspace\\pm\\gamma) ~.\n\\end{eqnarray}\n\nThe GGSZ method has been applied for many modes, both using model-dependent (see for instance \\cite{PhysRevLett.105.121801,PhysRevD.81.112002,LHCb-PAPER-2014-017})  and model-independent approaches~\\cite{PhysRevD.85.112014,LHCb-PAPER-2014-041,10.1093\/ptep\/ptw030,LHCb-PAPER-2016-006,LHCb-PAPER-2019-044}. The most precise measurement of $\\gamma$ from a single analysis comes from the LHCb collaboration, using the full run I and part of run II data,  with a model-independent approach for the decay modes ${\\ensuremath{\\PD}}\\xspace\\ensuremath{\\rightarrow}\\xspace{\\ensuremath{\\kaon^0_{\\rm\\scriptscriptstyle S}}}\\xspace{\\ensuremath{\\pion^+}}\\xspace{\\ensuremath{\\pion^-}}\\xspace$ and ${\\ensuremath{\\PD}}\\xspace\\ensuremath{\\rightarrow}\\xspace{\\ensuremath{\\kaon^0_{\\rm\\scriptscriptstyle S}}}\\xspace{\\ensuremath{\\kaon^+}}\\xspace{\\ensuremath{\\kaon^-}}\\xspace$  \\cite{LHCb-PAPER-2018-017}, to get  $\\gamma=(87^{+11}_{-12})^\\circ$.\nBelle collaboration has very recently performed the first GGSZ analysis using the four-body mode ${\\ensuremath{\\PD}}\\xspace\\ensuremath{\\rightarrow}\\xspace{\\ensuremath{\\kaon^0_{\\rm\\scriptscriptstyle S}}}\\xspace{\\ensuremath{\\pion^+}}\\xspace{\\ensuremath{\\pion^-}}\\xspace{\\ensuremath{\\pion^0}}\\xspace$ \\cite{Resmi:2019ajp}.\n\n\\begin{figure}[htbp]\n\\centering\n\\includegraphics[width=0.4\\textwidth]{Figure33_CLEO_binning.png}\n\\caption{Map of the strong phase difference across the Dalitz plot between {\\ensuremath{\\D^0}}\\xspace and {\\ensuremath{\\Dbar{}^0}}\\xspace decays in the final state ${\\ensuremath{\\kaon^0_{\\rm\\scriptscriptstyle S}}}\\xspace{\\ensuremath{\\pion^+}}\\xspace{\\ensuremath{\\pion^-}}\\xspace$, as obtained by CLEO. The Dalitz variables are $m^2_+ = m^2({\\ensuremath{\\kaon^0_{\\rm\\scriptscriptstyle S}}}\\xspace{\\ensuremath{\\pion^+}}\\xspace)$ and $m^2_- = m^2({\\ensuremath{\\kaon^0_{\\rm\\scriptscriptstyle S}}}\\xspace{\\ensuremath{\\pion^-}}\\xspace)$~\\cite{CLEO_DKSpipi_phase_2010}.}\n\\label{cleo_Dkspipiphasemap}\n\\end{figure}\n\n\\subsection{The combination result of $\\gamma$}\n\nFrom combining the values of $\\gamma$ obtained by the different methods and different final states, the world average from HFLAV \\cite{HFLAV19} is\n\\begin{equation}\n\\gamma = (71.1^{+4.6}_{-5.3})^\\circ~, \n\\end{equation}\nwhich is also shown in Fig.~\\ref{gamma_hflav} separated for GLW, ADS and GGSZ methods. The best sensitivity is clearly seen to come from the GGSZ method. Overall the LHCb measurements largely dominate the result. \n\nWhen compared to indirect determinations coming from fits with CKM constraints, such as from CKMFitter~\\cite{CKMfitter2015} and UTFit~\\cite{UTfit-UT} collaborations, the central value of $\\gamma$ above is  higher and marginally compatible. The smaller indirect values are driven by the determination of the mass split in ${\\ensuremath{\\PB}}\\xspace-\\bar{B}$ system through mixing measurements~\\cite{BlankeBuras_2018,KingLenz_2019}. If NP effects appear at tree level, the value of $\\gamma$ from direct determination can differ substantially from that of indirect measurements~\\cite{BrodLenz_2014}. The expected precision of about $1^\\circ$ from the LHCb Upgrade and Belle II in the upcoming years will be able to tackle this apparent tension.\n\n\n\\begin{figure}[htb]\n\\centering\n\\includegraphics[width =0.5\\textwidth]{Figure34_hflav_gammacombo.pdf}\n\\caption{World average of $\\gamma$ split by method ~\\cite{HFLAV19}.}\n\\label{gamma_hflav}\n\\end{figure}\n\n\n\n\n \n\\section{Direct {\\ensuremath{C\\!P}}\\xspace violation in Charm}\n\\label{chapter7}\n\nAs discussed in Sect.~\\ref{charm_cap2},  {\\ensuremath{C\\!P}}\\xspace violation in the charm sector is highly suppressed from the CKM ansatz. Nevertheless, during the last two decades, one could witness the charm available samples to grow enormously: in the early 2000s there were just a few thousand reconstructed ${\\ensuremath{\\D^+}}\\xspace\\ensuremath{\\rightarrow}\\xspace {\\ensuremath{\\kaon^+}}\\xspace{\\ensuremath{\\kaon^-}}\\xspace$ in FOCUS \\cite{FOCUS_CPV_2000} and CLEO \\cite{CLEO_CPV_2002} experiments,  while the current available sample from LHCb in run II comprises tens of millions decays in the same channel \\cite{LHCb-PAPER-2019-006,LHCb-PAPER-2019-032}. \n\nBy the end of 2011, there appeared some evidence for {\\ensuremath{C\\!P}}\\xspace violation through the {\\ensuremath{C\\!P}}\\xspace asymmetry difference in the decays of {\\ensuremath{\\D^0}}\\xspace\\ensuremath{\\rightarrow}\\xspace{\\ensuremath{\\kaon^+}}\\xspace{\\ensuremath{\\kaon^-}}\\xspace and {\\ensuremath{\\D^0}}\\xspace\\ensuremath{\\rightarrow}\\xspace{\\ensuremath{\\pion^+}}\\xspace{\\ensuremath{\\pion^-}}\\xspace, first announced by the LHCb experiment \\cite{LHCb-PAPER-2011-023} then by CDF~\\cite{CDF_2_DCPV_D_KK_pipi_2012} and Belle~\\cite{Belle_DACP_2012}. Although the effect soon after faded  with new measurements from LHCb~\\cite{LHCb-PAPER-2013-003, LHCb-CONF-2013-003} it created a lot of excitement in the theoretical community in order to try to understand whether the effect --  almost at the percent level -- could still be explained by the SM, via penguin enhancements, $SU(3)_F$ breaking, long-distance effects \\cite{Brod_ConsistentPenguins_2012,Brod_SizePenguins_2012,Grossman_Robinson_JHEP_2013,Atwood_Soni_2013,Cheng_Chiang_2012_2,Cheng_Chiang_2012_1,Cheng_2012,Pirtskhalava:2011va,Bhattacharya:2012ah,Li_2012,Hiller_2013,Feldmann:2012js,Franco:2012ck}\n or would require the contribution of new physics dynamics ~\\cite{DaRold:2012sz,Hiller:2012wf,Altmannshofer:2012ur,Isidori:2011qw,Rozanov:2012zz,Grossman:2012eb,Wang:2011uu,Hochberg:2011ru,Chang:2012gna,Giudice:2012qq,Delaunay:2012cz,Chen:2012am,Chen:2012usa}. This huge amount of effort, the ``$\\Delta A_{\\ensuremath{C\\!P}}\\xspace$ saga'', has stimulated further the quest for {\\ensuremath{C\\!P}}\\xspace violation in charm. \n \nNow direct {\\ensuremath{C\\!P}}\\xspace violation has finally been established from the same observable, by the LHCb experiment in 2019~\\cite{LHCb-PAPER-2019-006}. This is a milestone in Particle Physics and it is natural to expect that the upcoming years, with further analyses in different decay modes, and increasing data samples from LHCb and Belle II,  will bring more surprises in the charm sector. \nIn the following we present the state-of-the-art on study and search for direct {\\ensuremath{C\\!P}}\\xspace violation.\n\n\n\n\\subsection{Two-body modes}\n\\label{Dto2body}\n\nThe decays of {\\ensuremath{\\PD}}\\xspace mesons to two particles offer the simplest topology for {\\ensuremath{C\\!P}}\\xspace violation studies in charm. The majority of the two-particle final states  are composed of pseudo-scalar (stable) mesons; a few ``quasi'' two-body modes can also be addressed. Although a few decay modes, such as ${\\ensuremath{\\D^+}}\\xspace\\ensuremath{\\rightarrow}\\xspace \\phi{\\ensuremath{\\pion^+}}\\xspace$ and ${\\ensuremath{\\D^+}}\\xspace\\ensuremath{\\rightarrow}\\xspace{\\ensuremath{\\kaon^+}}\\xspace{\\ensuremath{\\kaon^-}}\\xspace$, have reached the impressive sensitivity of a few to several $10^{-4}$, most of the measurements are still testing for asymmetries at the level of percent. There is yet plenty of room for searches of NP effects  with run II LHCb data as well as with the upcoming data from  Belle II and the upgraded LHCb.\n\n\\subsubsection*{${\\ensuremath{\\D^0}}\\xspace\\ensuremath{\\rightarrow}\\xspace h^-h^+$} \n\n\n\nAs mentioned previously, a big breakthrough has come recently with the announcement, from the LHCb collaboration, of the first observation of {\\ensuremath{C\\!P}}\\xspace violation in the charm sector \n\\cite{LHCb-PAPER-2019-006}. The measurement was made through the Cabibbo-suppressed decay modes \\decay{{\\ensuremath{\\D^0}}\\xspace}{{\\ensuremath{\\kaon^-}}\\xspace{\\ensuremath{\\kaon^+}}\\xspace} and \\decay{{\\ensuremath{\\D^0}}\\xspace}{{\\ensuremath{\\pion^-}}\\xspace{\\ensuremath{\\pion^+}}\\xspace}. Since the final states are {\\ensuremath{C\\!P}}\\xspace symmetrical -- and thus reachable from both {\\ensuremath{\\D^0}}\\xspace and {\\ensuremath{\\Dbar{}^0}}\\xspace  mesons -- these decay modes permit, in principle, direct {\\ensuremath{C\\!P}}\\xspace violation as well as through mixing and interference between mixing and decay. \n\nThe {\\ensuremath{\\D^0}}\\xspace\\ensuremath{\\rightarrow}\\xspace{\\ensuremath{\\kaon^-}}\\xspace{\\ensuremath{\\kaon^+}}\\xspace and {\\ensuremath{\\D^0}}\\xspace\\ensuremath{\\rightarrow}\\xspace{\\ensuremath{\\pion^-}}\\xspace{\\ensuremath{\\pion^+}}\\xspace decays are among the most important channels for the study of {\\ensuremath{C\\!P}}\\xspace violation due to their relatively large branching ratios and, from the experimental side, the simplicity of two charged particles in the final state. Within the past decade or so, the search for direct {\\ensuremath{C\\!P}}\\xspace violation with \\decay{{\\ensuremath{\\D^0}}\\xspace}{{\\ensuremath{\\kaon^-}}\\xspace{\\ensuremath{\\kaon^+}}\\xspace} and \\decay{{\\ensuremath{\\D^0}}\\xspace}{{\\ensuremath{\\pion^-}}\\xspace{\\ensuremath{\\pion^+}}\\xspace} decays had contributions from Babar~\\cite{BaBar_DCPV_D2h_2008}, Belle~\\cite{Belle_DCPV_D2h_2008}, CDF~\\cite{CDF_1_DCPV_D_KK_pipi_2012, CDF_2_DCPV_D_KK_pipi_2012} and LHCb~\\cite{LHCb-PAPER-2011-023, LHCb-PAPER-2013-003, LHCb-PAPER-2014-013, LHCb-PAPER-2015-055}.  \n\nAs for any neutral mode, the key experimental ingredient is the determination of the {\\ensuremath{\\PD}}\\xspace flavour ({\\ensuremath{\\D^0}}\\xspace or {\\ensuremath{\\Dbar{}^0}}\\xspace) at production -- the {\\it tagging} strategy. Usually the decay  ${\\ensuremath{\\D^{*+}}}\\xspace\\ensuremath{\\rightarrow}\\xspace {\\ensuremath{\\D^0}}\\xspace\\pi^+_s$ is used ($ \\pi^+_s$ notation used since this emerging pion is slow) and this is referred to as $\\pi$-tagging. LHCb also uses {\\ensuremath{\\PD}}\\xspace decays coming from semi-leptonic {\\ensuremath{\\PB}}\\xspace decays, $B\\ensuremath{\\rightarrow}\\xspace{\\ensuremath{\\D^0}}\\xspace\\mu^-X$ (where $X$ is whatever particles not reconstructed in the decay) where the  Z.                                                                                                                                                                                                      muon charge gives the flavour of the {\\ensuremath{\\PD}}\\xspace ~-- referred to as $\\mu$-tagging.\n\nWhen studying these modes, an interesting observable is the difference of the time-integrated {\\ensuremath{C\\!P}}\\xspace asymmetries \n\\begin{equation}\n\\Delta A_{\\ensuremath{C\\!P}}\\xspace({\\ensuremath{\\PK}}\\xspace\\kaon-{\\ensuremath{\\Ppi}}\\xspace\\pion) = A_{\\ensuremath{C\\!P}}\\xspace(\\decay{{\\ensuremath{\\D^0}}\\xspace}{{\\ensuremath{\\kaon^-}}\\xspace{\\ensuremath{\\kaon^+}}\\xspace}) - A_{\\ensuremath{C\\!P}}\\xspace(\\decay{{\\ensuremath{\\D^0}}\\xspace}{{\\ensuremath{\\pion^-}}\\xspace{\\ensuremath{\\pion^+}}\\xspace})~,\n\\label{dacp}\n\\end{equation}\nwhich is a robust measurement for two reasons: production asymmetries are expected to cancel out since they do not depend on the final state; and detection asymmetries, due only to the flavour-tagging particle (since the final states are {\\ensuremath{C\\!P}}\\xspace symmetric) are also expected to cancel out. Besides it is quite possible the difference in the {\\ensuremath{C\\!P}}\\xspace asymmetries to be enhanced comparatively to that of the individual asymmetries. For instance, under the limit of exact  U-spin symmetries the decays \\decay{{\\ensuremath{\\D^0}}\\xspace}{{\\ensuremath{\\kaon^-}}\\xspace{\\ensuremath{\\kaon^+}}\\xspace} and \\decay{{\\ensuremath{\\D^0}}\\xspace}{{\\ensuremath{\\pion^-}}\\xspace{\\ensuremath{\\pion^+}}\\xspace} would have {\\ensuremath{C\\!P}}\\xspace asymmetries equal in size and opposite in sign \\cite{Grossman_CPV_CS_2007,Hiller_2013}.  The two final states are also connected through final-state rescattering, as discussed in Sect.~\\ref{chapter2_CP_CPT}, and the level of {\\ensuremath{C\\!P}}\\xspace violation in one may be correlated to the other.\n\nThe time-integrated {\\ensuremath{C\\!P}}\\xspace asymmetry is expected to be mostly sensitive to direct {\\ensuremath{C\\!P}}\\xspace violation \\cite{Gersabeck_Interplay_2012,CDF_1_DCPV_D_KK_pipi_2012}, \n\\begin{equation}\nA_{\\ensuremath{C\\!P}}\\xspace(f) = a_{\\ensuremath{C\\!P}}\\xspace^{\\rm dir}(f) - \\frac{\\langle t(f)\\rangle}{\\tau({\\ensuremath{\\D^0}}\\xspace)}A_\\Gamma~.\n\\end{equation}\nThe direct {\\ensuremath{C\\!P}}\\xspace-violating term $a_{\\ensuremath{C\\!P}}\\xspace^{\\rm dir}(f)$ depends on the final state $f$; $\\langle t(f)\\rangle$ is the measured (reconstructed) mean decay time, and $A_\\Gamma$ is the indirect {\\ensuremath{C\\!P}}\\xspace violation parameter, currently compatible with zero at the per-mille level \\cite{LHCb-PAPER-2016-063} and not expected to be final-state dependent. Thus, one can write \n\\begin{equation}\n\\Delta A_{\\ensuremath{C\\!P}}\\xspace(KK-\\pi\\pi) = \\Delta a_{\\ensuremath{C\\!P}}\\xspace^{\\rm dir}(KK-\\pi\\pi) - \\frac{\\Delta\\langle t\\rangle}{\\tau(D^0)}A_\\Gamma~,\n\\end{equation}\nwhere $\\Delta\\langle t\\rangle$ is the difference of the reconstructed decay times of \\decay{{\\ensuremath{\\D^0}}\\xspace}{{\\ensuremath{\\kaon^+}}\\xspace{\\ensuremath{\\kaon^-}}\\xspace} and \\decay{{\\ensuremath{\\D^0}}\\xspace}{{\\ensuremath{\\pion^-}}\\xspace{\\ensuremath{\\pion^+}}\\xspace} modes. \n\nUsing run II data, the LHCb collaboration analysed large samples of 44 (9) million decays of ${\\ensuremath{\\D^0}}\\xspace\\ensuremath{\\rightarrow}\\xspace{\\ensuremath{\\kaon^+}}\\xspace{\\ensuremath{\\kaon^-}}\\xspace$ and 14 (3) million decays of ${\\ensuremath{\\D^0}}\\xspace\\ensuremath{\\rightarrow}\\xspace{\\ensuremath{\\pion^+}}\\xspace{\\ensuremath{\\pion^-}}\\xspace$ for the $\\pi$-tagged ($\\mu$-tagged) samples, as shown in Fig.~\\ref{LHCb_DKK_Dpipi}. \n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=0.3\\textwidth]{Figure35_DACP_DKK_pitag} \\includegraphics[width=0.3\\textwidth]{Figure35_DACP_Dpipi_pitag}\n\n\\includegraphics[width=0.3\\textwidth]{Figure35_DACP_DKK_mutag} \\includegraphics[width=0.3\\textwidth]{Figure35_DACP_Dpipi_mutag}\n\\end{center}\n\\caption{Invariant mass distributions of (left) ${\\ensuremath{\\D^0}}\\xspace\\ensuremath{\\rightarrow}\\xspace{\\ensuremath{\\kaon^+}}\\xspace{\\ensuremath{\\kaon^-}}\\xspace$ and (right)  ${\\ensuremath{\\D^0}}\\xspace\\ensuremath{\\rightarrow}\\xspace{\\ensuremath{\\pion^+}}\\xspace{\\ensuremath{\\pion^-}}\\xspace$ candidates from (top) $\\pi$-tagged and (bottom) $\\mu$-tagged samples. For the $\\pi$-tagged samples the invariant mass corresponds to that of the ${\\ensuremath{\\D^{*+}}}\\xspace\\ensuremath{\\rightarrow}\\xspace {\\ensuremath{\\D^0}}\\xspace{\\ensuremath{\\pion^+}}\\xspace$ candidate, while for the $\\mu$-tagged sample it corresponds to that of the ${\\ensuremath{\\D^0}}\\xspace$ candidates~\\cite{LHCb-PAPER-2019-006}.}\n\\label{LHCb_DKK_Dpipi}\n\\end{figure}\n\nThe measured values obtained are  $\\Delta A_{\\ensuremath{C\\!P}}\\xspace(KK-\\pi\\pi)^{\\rm \\pi-tagged} =(-18.2\\pm 3.2\\pm 0.9)\\times 10^{-4}$ and $\\Delta A_{\\ensuremath{C\\!P}}\\xspace(KK-\\pi\\pi)^{\\rm \\mu-tagged} =(-9\\pm 8\\pm 5)\\times 10^{-4}$, for $\\pi$-tagged and $\\mu$-tagged samples, respectively. By performing a full combination of these results with previous LHCb measurements~\\cite{LHCb-PAPER-2011-023, LHCb-PAPER-2013-003, LHCb-PAPER-2014-013, LHCb-PAPER-2015-055},  the final result is\n\\begin{equation}\n\\Delta A_{\\ensuremath{C\\!P}}\\xspace(KK-\\pi\\pi) =(-15.4\\pm 2.9)\\times 10^{-4}, \n\\end{equation}\nwhich is a $5.3\\sigma$ effect and constitutes the first observation of {\\ensuremath{C\\!P}}\\xspace violation in the charm sector. \nAfter correcting by the small effect of indirect {\\ensuremath{C\\!P}}\\xspace violation, using the experimental value of $\\Delta\\langle t\\rangle$ and the measurement of $A_\\Gamma$ from LHCb~\\cite{LHCb-PAPER-2016-063}, the value for direct {\\ensuremath{C\\!P}}\\xspace violation obtained is $\\Delta a_{\\ensuremath{C\\!P}}\\xspace^{\\rm dir}(KK-\\pi\\pi) = (-15.7\\pm 2.9)\\times 10^{-4}$. By performing a world average combination the HFLAV group~\\cite{HFLAV19} gets\n\\begin{equation}\n\\Delta a_{\\ensuremath{C\\!P}}\\xspace^{\\rm dir}(KK-\\pi\\pi) = (-1.64\\pm 0.28)\\times 10^{-3},\n\\end{equation}\nwhich can also be observed from the combination plot of direct $\\Delta a_{\\ensuremath{C\\!P}}\\xspace^{\\rm dir}(KK-\\pi\\pi)$ and indirect $a_{\\ensuremath{C\\!P}}\\xspace^{\\rm ind}(KK-\\pi\\pi)$ {\\ensuremath{C\\!P}}\\xspace asymmetry results  in Fig.~\\ref{DeltaaCP_hflav}.\n\\begin{figure}\n\\centerline{\\includegraphics[width=0.5\\textwidth]{Figure36_DeltaACP_AGamma_HFLAV.png}}\n\\caption{The combination plot for $\\Delta a_{\\ensuremath{C\\!P}}\\xspace^{\\rm dir}(KK-\\pi\\pi)$ against $a_{\\ensuremath{C\\!P}}\\xspace^{\\rm ind}(KK-\\pi\\pi)$ from the HFLAV group~\\cite{HFLAV19}.}\n\\label{DeltaaCP_hflav}\n\\end{figure}\n\nBesides the {\\ensuremath{C\\!P}}\\xspace asymmetry differences, it is clearly important to access the individual {\\ensuremath{C\\!P}}\\xspace asymmetries.\nThe LHCb collaboration measured $A_{\\ensuremath{C\\!P}}\\xspace({\\ensuremath{\\kaon^-}}\\xspace{\\ensuremath{\\kaon^+}}\\xspace) $ \\cite{LHCb-PAPER-2016-035} with $\\pi$-tagged samples using run I data, with 3.0\\ensuremath{\\mbox{\\,fb}^{-1}}\\xspace  -- but results are yet to come from run II.  \n\nThe  value for $\\Delta a_{\\ensuremath{C\\!P}}\\xspace^{\\rm dir}(KK-\\pi\\pi)$  puts {\\ensuremath{C\\!P}}\\xspace violation in the charm sector at the level of $10^{-3}$, which is higher than naive expectations at ${\\cal O}(10^{-4})$ ~\\cite{Grossman_CPV_CS_2007,Khodjamirian_Petrov_2017} and some authors consider contributions from NP dynamics \\cite{Nir_Implications_CharmCPV_2019,Chala_2019,Calibbi_2019}. Nevertheless it is\nconsistent with models that include non-perturbative effects, with QCD enhancements and $SU(3)_F$ symmetry breaking ~\\cite{Buccella_2019,Cheng_2019,Li_ImplicationsCPV_2019,Grossman_2019}.  An interesting effect discussed in ~\\cite{Soni_CPenhancement_2019} considers  {\\ensuremath{C\\!P}}\\xspace violation enhancement due to a nearby resonance, $f_0(1710)$ playing the role of the {\\it communicator} between the channels and, being close to the {\\ensuremath{\\D^0}}\\xspace mass, providing the necessary strong phase.\n\n One should consider more generally the potential effects from $KK\\leftrightarrow\\pi\\pi$ rescattering~\\cite{Bianco_Bigi_2020}. This was discussed in Sect.~\\ref{kkpipiresc} in the context of charmless three-body {\\ensuremath{\\PB}}\\xspace decays and, for charm decays, rescattering can be expected to be even more important. In this sense, three-body  Cabibbo-suppressed decays, especially those with  large samples such as  ${\\ensuremath{\\D^+}}\\xspace\\ensuremath{\\rightarrow}\\xspace {\\ensuremath{\\kaon^-}}\\xspace{\\ensuremath{\\kaon^+}}\\xspace{\\ensuremath{\\pion^+}}\\xspace$ and ${\\ensuremath{\\D^+}}\\xspace\\ensuremath{\\rightarrow}\\xspace{\\ensuremath{\\pion^-}}\\xspace{\\ensuremath{\\pion^+}}\\xspace\\pip$, are excellent candidates for observation of {\\ensuremath{C\\!P}}\\xspace violation with the already available data from LHCb, as it is discussed below in Sect.~\\ref{Dto3body}.  \n\n In any case, this outstanding result from ${\\ensuremath{\\D^0}}\\xspace\\ensuremath{\\rightarrow}\\xspace{\\ensuremath{\\kaon^+}}\\xspace{\\ensuremath{\\kaon^-}}\\xspace$ and ${\\ensuremath{\\D^0}}\\xspace\\ensuremath{\\rightarrow}\\xspace{\\ensuremath{\\pion^+}}\\xspace{\\ensuremath{\\pion^-}}\\xspace$ decays calls for a  comprehensive study of potential charm processes where direct {\\ensuremath{C\\!P}}\\xspace violation might appear at a similar level. \n\n\\subsubsection*{${\\ensuremath{\\PD}}\\xspace^+_{(s)}\\ensuremath{\\rightarrow}\\xspace h^0 h^+$}\n\nRecently, the LHCb collaboration has searched for CP violation in the Cabibbo-suppressed decays ${\\ensuremath{\\D^+_\\squark}}\\xspace\\ensuremath{\\rightarrow}\\xspace{\\ensuremath{\\kaon^0_{\\rm\\scriptscriptstyle S}}}\\xspace{\\ensuremath{\\pion^+}}\\xspace$, ${\\ensuremath{\\D^+}}\\xspace\\ensuremath{\\rightarrow}\\xspace {\\ensuremath{\\kaon^0_{\\rm\\scriptscriptstyle S}}}\\xspace{\\ensuremath{\\kaon^+}}\\xspace$ and ${\\ensuremath{\\D^+}}\\xspace\\ensuremath{\\rightarrow}\\xspace \\phi(1020){\\ensuremath{\\pion^+}}\\xspace$ with $3.8\\,\\rm fb^{-1}$ from run II $pp$ collision data \\cite{LHCb-PAPER-2019-002}. The  data samples for these decays are huge, comprising respectively 600 thousand, 5.1 million and 53.3 million candidate decays. The {\\ensuremath{\\kaon^0_{\\rm\\scriptscriptstyle S}}}\\xspace meson is reconstructed through its {\\ensuremath{\\pion^+}}\\xspace{\\ensuremath{\\pion^-}}\\xspace final state, which is {\\ensuremath{C\\!P}}\\xspace symmetric. The ${\\ensuremath{\\D^+}}\\xspace\\ensuremath{\\rightarrow}\\xspace \\phi{\\ensuremath{\\pion^+}}\\xspace$ constitutes a quasi-two-body mode; the measurement is made selecting {\\ensuremath{\\kaon^+}}\\xspace{\\ensuremath{\\kaon^-}}\\xspace pairs in the vicinity of the \\Pphi mass with no attempt to separate from other contributions of the decay ${\\ensuremath{\\D^+}}\\xspace\\ensuremath{\\rightarrow}\\xspace{\\ensuremath{\\kaon^-}}\\xspace{\\ensuremath{\\kaon^+}}\\xspace{\\ensuremath{\\pion^+}}\\xspace$. To allow the measurement of the {\\ensuremath{C\\!P}}\\xspace asymmetry, LHCb uses a set of Cabibbo-favoured control modes, $D^+\\ensuremath{\\rightarrow}\\xspace K_S^0\\pi^+$, $D_s^+\\ensuremath{\\rightarrow}\\xspace K_S^0K^+$ and $D_s^+\\ensuremath{\\rightarrow}\\xspace \\phi(1020)\\pi^+$, in order to provide a cancellation of production and detection asymmetries.  As such, the {\\ensuremath{C\\!P}}\\xspace asymmetries to a good approximation are given by\n\\begin{eqnarray}\nA_{{\\ensuremath{C\\!P}}\\xspace}({\\ensuremath{\\D^+_\\squark}}\\xspace\\ensuremath{\\rightarrow}\\xspace {\\ensuremath{\\kaon^0_{\\rm\\scriptscriptstyle S}}}\\xspace{\\ensuremath{\\pion^+}}\\xspace)&  \\approx&  A_{\\rm raw}({\\ensuremath{\\D^+_\\squark}}\\xspace\\ensuremath{\\rightarrow}\\xspace {\\ensuremath{\\kaon^0_{\\rm\\scriptscriptstyle S}}}\\xspace{\\ensuremath{\\pion^+}}\\xspace) - A_{\\rm raw}({\\ensuremath{\\D^+_\\squark}}\\xspace\\ensuremath{\\rightarrow}\\xspace \\phi{\\ensuremath{\\pion^+}}\\xspace)\\nonumber  \\\\\nA_{{\\ensuremath{C\\!P}}\\xspace}({\\ensuremath{\\D^+}}\\xspace\\ensuremath{\\rightarrow}\\xspace {\\ensuremath{\\kaon^0_{\\rm\\scriptscriptstyle S}}}\\xspace{\\ensuremath{\\kaon^+}}\\xspace) &  \\approx&   A_{\\rm raw}({\\ensuremath{\\D^+}}\\xspace\\ensuremath{\\rightarrow}\\xspace {\\ensuremath{\\kaon^0_{\\rm\\scriptscriptstyle S}}}\\xspace{\\ensuremath{\\kaon^+}}\\xspace) - A_{\\rm raw}({\\ensuremath{\\D^+}}\\xspace\\ensuremath{\\rightarrow}\\xspace {\\ensuremath{\\kaon^0_{\\rm\\scriptscriptstyle S}}}\\xspace {\\ensuremath{\\pion^+}}\\xspace)\\nonumber \\\\  \n & - & A_{\\rm raw}({\\ensuremath{\\D^+_\\squark}}\\xspace\\ensuremath{\\rightarrow}\\xspace {\\ensuremath{\\kaon^0_{\\rm\\scriptscriptstyle S}}}\\xspace{\\ensuremath{\\kaon^+}}\\xspace)  + A_{\\rm raw}({\\ensuremath{\\D^+_\\squark}}\\xspace\\ensuremath{\\rightarrow}\\xspace \\phi{\\ensuremath{\\pion^+}}\\xspace)  \\\\\nA_{{\\ensuremath{C\\!P}}\\xspace}({\\ensuremath{\\D^+}}\\xspace\\ensuremath{\\rightarrow}\\xspace \\phi\\pi^+) & \\approx&    A_{\\rm raw}({\\ensuremath{\\D^+}}\\xspace\\ensuremath{\\rightarrow}\\xspace \\phi{\\ensuremath{\\pion^+}}\\xspace) - A_{\\rm raw}({\\ensuremath{\\D^+}}\\xspace\\ensuremath{\\rightarrow}\\xspace {\\ensuremath{\\kaon^0_{\\rm\\scriptscriptstyle S}}}\\xspace{\\ensuremath{\\pion^+}}\\xspace)~, \\nonumber\n\\end{eqnarray}\nwhere $A_{\\rm raw}$ are the raw asymmetries from  ${\\ensuremath{\\PD}}\\xspace_{(s)}^+$ and ${\\ensuremath{\\PD}}\\xspace_{(s)}^-$ yields. \nA weighting procedure to match the distribution of kinematic quantities is performed to guarantee accurate cancellations. In the final states with a {\\ensuremath{\\kaon^0_{\\rm\\scriptscriptstyle S}}}\\xspace state, both its detection and {\\ensuremath{C\\!P}}\\xspace asymmetries are subtracted from the measured values \\cite{LHCb-PAPER-2014-013}. The results obtained for 2015--2017 data \\cite{LHCb-PAPER-2019-002} are combined to previous LHCb results from run I, leading to the measurements\n\\begin{eqnarray}\nA_{{\\ensuremath{C\\!P}}\\xspace}({\\ensuremath{\\D^+_\\squark}}\\xspace\\ensuremath{\\rightarrow}\\xspace {\\ensuremath{\\kaon^0_{\\rm\\scriptscriptstyle S}}}\\xspace{\\ensuremath{\\pion^+}}\\xspace) & =& (1.6\\pm 1.7\\pm 0.5)\\times 10^{-3} \\nonumber\\\\\nA_{{\\ensuremath{C\\!P}}\\xspace}({\\ensuremath{\\D^+}}\\xspace\\ensuremath{\\rightarrow}\\xspace {\\ensuremath{\\kaon^0_{\\rm\\scriptscriptstyle S}}}\\xspace{\\ensuremath{\\kaon^+}}\\xspace)  & =& (-0.04\\pm 0.61\\pm 0.45)\\times 10^{-3} \\nonumber\\\\\nA_{{\\ensuremath{C\\!P}}\\xspace}({\\ensuremath{\\D^+}}\\xspace\\ensuremath{\\rightarrow}\\xspace \\phi\\pi^+)& =& (0.03\\pm 0.40\\pm 0.29)\\times 10^{-3}~,\n\\end{eqnarray}\nwhich  are all consistent with {\\ensuremath{C\\!P}}\\xspace conservation. For the $D^+\\ensuremath{\\rightarrow}\\xspace \\phi\\pi^+$ mode, the result represents  the most precise {\\ensuremath{C\\!P}}\\xspace asymmetry measurement obtained from a single measurement, with a combined statistical and systematic uncertainty of only $5\\times 10^{-4}$. The decays of {\\ensuremath{\\D^+}}\\xspace and {\\ensuremath{\\D^+_\\squark}}\\xspace to {\\ensuremath{\\kaon^0_{\\rm\\scriptscriptstyle S}}}\\xspace{\\ensuremath{\\kaon^+}}\\xspace have  also been studied by the BESIII collaboration in 2019~\\cite{BESIII_DsKSLKp_2019} with a much smaller sample, so LHCb dominates these measurements. The novelty is the inclusion of {\\ensuremath{\\kaon^0_{\\rm\\scriptscriptstyle L}}}\\xspace{\\ensuremath{\\kaon^+}}\\xspace final states -- with results consistent with {\\ensuremath{C\\!P}}\\xspace conservation at $\\sim 3\\%$ level. \n\nOther interesting modes for charged {\\ensuremath{\\PD}}\\xspace's involve the production of \\ensuremath{\\Peta^{\\prime}}\\xspace, such as {\\ensuremath{\\D^+}}\\xspace\\ensuremath{\\rightarrow}\\xspace\\ensuremath{\\Peta^{\\prime}}\\xspace{\\ensuremath{\\pion^+}}\\xspace, which is Cabibbo-suppressed, and {\\ensuremath{\\D^+_\\squark}}\\xspace\\ensuremath{\\rightarrow}\\xspace\\ensuremath{\\Peta^{\\prime}}\\xspace{\\ensuremath{\\pion^+}}\\xspace, which is Cabibbo-favoured. A few years ago, Belle~\\cite{Belle_Detaprpi_2011} and CLEO \\cite{CLEO_Dsetaprpi_2013} have measured {\\ensuremath{C\\!P}}\\xspace asymmetries for {\\ensuremath{\\D^+}}\\xspace\\ensuremath{\\rightarrow}\\xspace\\ensuremath{\\Peta^{\\prime}}\\xspace{\\ensuremath{\\pion^+}}\\xspace and {\\ensuremath{\\D^+_\\squark}}\\xspace\\ensuremath{\\rightarrow}\\xspace\\ensuremath{\\Peta^{\\prime}}\\xspace{\\ensuremath{\\pion^+}}\\xspace above the 1\\% and 2\\% level, respectively. Despite the difficulty of reconstructing  \\ensuremath{\\Peta^{\\prime}}\\xspace in a hadron machine, the  LHCb collaboration has performed a study for these decays with 3\\ensuremath{\\mbox{\\,fb}^{-1}}\\xspace from run I data \\cite{LHCb-PAPER-2016-041}. Control modes {\\ensuremath{\\D^+}}\\xspace\\ensuremath{\\rightarrow}\\xspace{\\ensuremath{\\kaon^0_{\\rm\\scriptscriptstyle S}}}\\xspace{\\ensuremath{\\pion^+}}\\xspace and {\\ensuremath{\\D^+_\\squark}}\\xspace\\ensuremath{\\rightarrow}\\xspace\\Pphi{\\ensuremath{\\pion^+}}\\xspace are used to deal with detection and production asymmetries. The results\n\\begin{eqnarray}\nA_{\\ensuremath{C\\!P}}\\xspace({\\ensuremath{\\D^+}}\\xspace\\ensuremath{\\rightarrow}\\xspace\\ensuremath{\\Peta^{\\prime}}\\xspace{\\ensuremath{\\pion^+}}\\xspace)  = (-0.61\\pm 0.72\\pm 0.53\\pm 0.12)\\% \\nonumber \\\\\nA_{\\ensuremath{C\\!P}}\\xspace({\\ensuremath{\\D^+_\\squark}}\\xspace\\ensuremath{\\rightarrow}\\xspace\\ensuremath{\\Peta^{\\prime}}\\xspace{\\ensuremath{\\pion^+}}\\xspace)  = (-0.82\\pm 0.36\\pm 0.22\\pm 0.27)\\%\n\\end{eqnarray}\nare the most precise to date. The last uncertainty is due to external measurements of the asymmetry of the control modes.\n\n\nThe Cabibbo-suppressed mode ${\\ensuremath{\\D^+}}\\xspace\\ensuremath{\\rightarrow}\\xspace{\\ensuremath{\\pion^+}}\\xspace{\\ensuremath{\\pion^0}}\\xspace$ has the same dominant tree diagram as {\\ensuremath{\\D^0}}\\xspace\\ensuremath{\\rightarrow}\\xspace{\\ensuremath{\\pion^-}}\\xspace{\\ensuremath{\\pion^+}}\\xspace, with the only change in the spectator quark. \nBut since the final state has isospin $I=2$, it cannot proceed through a gluonic-penguin amplitude. Thus, in this case there is no weak phase contributing from the SM and the expected {\\ensuremath{C\\!P}}\\xspace asymmetry is null;  any sign of {\\ensuremath{C\\!P}}\\xspace violation would be indication of New Physics mechanisms.\n In 2018, Belle searched for  {\\ensuremath{C\\!P}}\\xspace violation in this mode with its full  data sample, with an integrated luminosity of 921\\ensuremath{\\mbox{\\,fb}^{-1}}\\xspace sample \\cite{Belle_Dpipi0_2018}. The ${\\ensuremath{\\D^+}}\\xspace\\ensuremath{\\rightarrow}\\xspace{\\ensuremath{\\pion^+}}\\xspace{\\ensuremath{\\pion^0}}\\xspace$ decay was reconstructed both from ${\\ensuremath{\\D^{*+}}}\\xspace\\ensuremath{\\rightarrow}\\xspace{\\ensuremath{\\D^+}}\\xspace{\\ensuremath{\\pion^0}}\\xspace$ (tagged) and directly from the interaction point (not tagged). The raw asymmetry $A_{\\rm raw}^{{\\ensuremath{\\pion^+}}\\xspace{\\ensuremath{\\pion^0}}\\xspace}$ was obtained from signal yields, and corrected by the {\\ensuremath{\\pion^+}}\\xspace detection asymmetry and forward--backward asymmetry.\\footnote{Forward--backward asymmetry arises due to the interference of amplitudes mediated by virtual photon, {\\ensuremath{\\PZ}}\\xspace and higher orders in \\en{\\ensuremath{\\Pe^+}}\\xspace collisions.} \nThe result,\n\\begin{equation}\nA_{\\ensuremath{C\\!P}}\\xspace({\\ensuremath{\\D^+}}\\xspace\\ensuremath{\\rightarrow}\\xspace{\\ensuremath{\\pion^+}}\\xspace{\\ensuremath{\\pion^0}}\\xspace) = (+2.3\\pm 1.2\\pm 0.2)\\%,  \\nonumber\n\\end{equation}\nis consistent with the SM and with {\\ensuremath{C\\!P}}\\xspace conservation at the 1\\% level, which still places no strong constraint. The LHCb collaboration may be able to provide results from run II with precision at the level of Belle. Belle II, with the full $50{\\rm ab}^{-1}$, will be able to reach the 0.2\\% level~\\cite{BelleII_PhysicsBook_2018}.\n\n\\subsubsection*{${\\ensuremath{\\D^0}}\\xspace\\ensuremath{\\rightarrow}\\xspace h^0h^0$}\n\nThe ${\\ensuremath{\\D^0}}\\xspace\\ensuremath{\\rightarrow}\\xspace{\\ensuremath{\\kaon^0_{\\rm\\scriptscriptstyle S}}}\\xspace\\KS$ channel has a small branching fraction, $(1.41\\pm 0.05)\\times 10^{-4}$~\\cite{PDG2019}, about 30 times smaller than the ${\\ensuremath{\\kaon^+}}\\xspace{\\ensuremath{\\kaon^-}}\\xspace$ mode. This is due to the fact that it can only occur through  $W$-exchange diagrams with ${\\ensuremath{\\Pc}}\\xspace{\\ensuremath{\\overline \\uquark}}\\xspace\\ensuremath{\\rightarrow}\\xspace {\\ensuremath{\\Ps}}\\xspace{\\ensuremath{\\overline \\squark}}\\xspace$ and  ${\\ensuremath{\\Pc}}\\xspace{\\ensuremath{\\overline \\uquark}}\\xspace\\ensuremath{\\rightarrow}\\xspace {\\ensuremath{\\Pd}}\\xspace{\\ensuremath{\\overline \\dquark}}\\xspace$ which would  cancel in the limit of exact  $SU(3)_F$  symmetry, or via a penguin-type annihilation. The contributions, with different  weak phases, are expected to be of similar size, so the interference term is comparatively larger than for other charm modes. According to Ref.~\\cite{Nierste_DKSKS_2015}, the {\\ensuremath{C\\!P}}\\xspace asymmetry could be as large as 1\\% within the SM. \n\nWithin the last 5 years, the search for {\\ensuremath{C\\!P}}\\xspace violation in this mode  was performed by LHCb~\\cite{LHCb-PAPER-2015-030, LHCb-PAPER-2018-012} and by Belle~\\cite{Belle_D0KSKS_2017}  -- the latter leading with best precision so far, $A_{\\ensuremath{C\\!P}}\\xspace({\\ensuremath{\\D^0}}\\xspace\\ensuremath{\\rightarrow}\\xspace{\\ensuremath{\\kaon^0_{\\rm\\scriptscriptstyle S}}}\\xspace\\KS) = (-0.02\\pm 1.5)\\%$. The combined uncertainty, though, is still not sufficient to test the SM prediction. Belle II expects to lower this uncertainty \\cite{BelleII_PhysicsBook_2018} to 0.66\\% and 0.23\\% with integrated luminosities of $5\\,\\rm ab^{-1}$ and  $50\\, \\rm ab^{-1}$, respectively.\n\nMore recently the authors of Ref.~\\cite{Nierste_DKSKS_2015} extended the work to study the quasi two-body decay \\mbox{${\\ensuremath{\\D^0}}\\xspace\\ensuremath{\\rightarrow}\\xspace{\\ensuremath{\\kaon^0_{\\rm\\scriptscriptstyle S}}}\\xspace{\\ensuremath{\\kaon^*}}\\xspace$} \\cite{Nierste_DKKst_2017} to estimate a {\\ensuremath{C\\!P}}\\xspace asymmetry that can be as high as  $0.3\\%$. In any case, the adequate treatment for this mode should be addressed through a full Dalitz plot investigation to the {\\ensuremath{\\D^0}}\\xspace\\ensuremath{\\rightarrow}\\xspace{\\ensuremath{\\kaon^0_{\\rm\\scriptscriptstyle S}}}\\xspace{\\ensuremath{\\kaon^-}}\\xspace{\\ensuremath{\\pion^+}}\\xspace final state. This is discussed in Sect.~\\ref{Dto3body}. \n\nThe ${\\ensuremath{\\D^0}}\\xspace\\ensuremath{\\rightarrow}\\xspace{\\ensuremath{\\pion^0}}\\xspace\\piz$ is another interesting channel and it gains interest also with its connection  to ${\\ensuremath{\\D^0}}\\xspace\\ensuremath{\\rightarrow}\\xspace{\\ensuremath{\\kaon^+}}\\xspace{\\ensuremath{\\kaon^-}}\\xspace$ and ${\\ensuremath{\\D^0}}\\xspace\\ensuremath{\\rightarrow}\\xspace{\\ensuremath{\\pion^+}}\\xspace{\\ensuremath{\\pion^-}}\\xspace$ final states (see for instance  \\cite{MullerNierste_2015}).\nBelle studied the decay {\\ensuremath{\\D^0}}\\xspace\\ensuremath{\\rightarrow}\\xspace{\\ensuremath{\\pion^0}}\\xspace\\piz with 966\\ensuremath{\\mbox{\\,fb}^{-1}}\\xspace, with a sample of 34 thousand signal events, and also studied the Cabibbo-favoured {\\ensuremath{\\D^0}}\\xspace\\ensuremath{\\rightarrow}\\xspace{\\ensuremath{\\kaon^0_{\\rm\\scriptscriptstyle S}}}\\xspace{\\ensuremath{\\pion^0}}\\xspace, with a sample of 467 thousand signal events~\\cite{Belle_Dpi0pi0_2014}. The results are $A_{\\ensuremath{C\\!P}}\\xspace({\\ensuremath{\\D^0}}\\xspace\\ensuremath{\\rightarrow}\\xspace{\\ensuremath{\\pion^0}}\\xspace\\piz) = (-0.03\\pm 0.64\\pm 0.10)\\%$ and $A_{\\ensuremath{C\\!P}}\\xspace({\\ensuremath{\\D^0}}\\xspace\\ensuremath{\\rightarrow}\\xspace{\\ensuremath{\\kaon^0_{\\rm\\scriptscriptstyle S}}}\\xspace{\\ensuremath{\\pion^0}}\\xspace) = (-0.10\\pm 0.16\\pm 0.07)\\%$, where the last result is already corrected for {\\ensuremath{\\kaon^0}}\\xspace asymmetries. The {\\ensuremath{\\pion^0}}\\xspace\\piz final state represents a challenge for LHCb but in the upcoming years Belle II is expected to bring improvement, with an expected sensitivity of $0.3\\%$ with the first $5 \\,\\rm ab^{-1}$.\n\n\n\\subsection{Three-body modes}\n\\label{Dto3body}\n\nAs discussed in Sect.~\\ref{multibody}, for multi-body decay modes {\\ensuremath{C\\!P}}\\xspace violation  effects can be studied through the phase space of the decay, and can be potentially larger than integrated ones. Even if, when compared to {\\ensuremath{\\PB}}\\xspace decays, the available phase space is relatively small, there are many resonances with masses below the {\\ensuremath{\\PD}}\\xspace mass, and usually the decay dynamics are very rich and with rather interesting interference patterns. {\\ensuremath{C\\!P}}\\xspace violation effects may be enhanced in local regions and these local asymmetries can be higher than the phase-space integrated results \\cite{bigi1,Bigi_CP-2-3-4Body_2011,Bigi_Charm_2015}.\n\nFor {\\ensuremath{\\PD}}\\xspace decaying to stable pseudo-scalar mesons in a three-body final state,  the dynamics are directly read from the distribution of events in the Dalitz plot. In particular, interference effects are visually apparent. If sizeable localised {\\ensuremath{C\\!P}}\\xspace violation effects exist, they would be clearly visible, as for the case of three-body  charmless {\\ensuremath{\\PB}}\\xspace decays  discussed in Sect.~\\ref{Bhhh}. For {\\ensuremath{\\PD}}\\xspace decays, with very small asymmetries expected within the SM, the search for {\\ensuremath{C\\!P}}\\xspace violation within the phase space requires very large samples and a very careful control of nuisance asymmetries, due to production and detection effects. \n\nAlthough a few Dalitz analyses searched for {\\ensuremath{C\\!P}}\\xspace violation more than 10 years ago with limited number of events, it was in 2008 when the Babar collaboration presented a search for {\\ensuremath{C\\!P}}\\xspace violation with about 82 thousand  ${\\ensuremath{\\D^0}}\\xspace\\ensuremath{\\rightarrow}\\xspace {\\ensuremath{\\pion^-}}\\xspace{\\ensuremath{\\pion^+}}\\xspace{\\ensuremath{\\pion^0}}\\xspace$ and 11 thousand ${\\ensuremath{\\D^0}}\\xspace\\ensuremath{\\rightarrow}\\xspace{\\ensuremath{\\kaon^-}}\\xspace{\\ensuremath{\\kaon^+}}\\xspace{\\ensuremath{\\pion^0}}\\xspace$ decay candidates, with high purity~\\cite{BaBar_CPVD3body_2008}. The study  was performed through both  amplitude analysis and model-independent search, by calculating the residuals of the difference in population in bins of the Dalitz plot for {\\ensuremath{\\D^0}}\\xspace and {\\ensuremath{\\Dbar{}^0}}\\xspace (similar to the Miranda technique presented in Sect.~\\ref{chapter3}). No sign for {\\ensuremath{C\\!P}}\\xspace violation was found. Through the model-dependent approach, for the ${\\ensuremath{\\D^0}}\\xspace\\ensuremath{\\rightarrow}\\xspace {\\ensuremath{\\pion^-}}\\xspace{\\ensuremath{\\pion^+}}\\xspace{\\ensuremath{\\pion^0}}\\xspace$ (with larger samples) the uncertainties for {\\ensuremath{C\\!P}}\\xspace violation in phases and magnitudes were about 2$^\\circ$ and higher, and about $2\\%$ and higher, respectively.\n\nA more recent model-dependent study where {\\ensuremath{C\\!P}}\\xspace violation  was carried out by the LHCb collaboration in the channels ${\\ensuremath{\\D^0}}\\xspace\\ensuremath{\\rightarrow}\\xspace {\\ensuremath{\\kaon^0_{\\rm\\scriptscriptstyle S}}}\\xspace{\\ensuremath{\\kaon^-}}\\xspace{\\ensuremath{\\pion^+}}\\xspace$ and ${\\ensuremath{\\D^0}}\\xspace\\ensuremath{\\rightarrow}\\xspace{\\ensuremath{\\kaon^0_{\\rm\\scriptscriptstyle S}}}\\xspace{\\ensuremath{\\kaon^+}}\\xspace{\\ensuremath{\\pion^-}}\\xspace$ \\cite{LHCb-PAPER-2016-026}, using  3\\ensuremath{\\mbox{\\,fb}^{-1}}\\xspace  (full run I data) with signal yields of about 113 thousand and 76 thousand, respectively.  No evidence for {\\ensuremath{C\\!P}}\\xspace violation was found, probing at or higher than 2$^\\circ$ and $2\\%$  for phase and magnitude differences, respectively.\n\nUsing the first set of data collected from the LHCb experiment in 2010, a search for {\\ensuremath{C\\!P}}\\xspace violation was performed for ${\\ensuremath{\\D^+}}\\xspace\\ensuremath{\\rightarrow}\\xspace {\\ensuremath{\\kaon^-}}\\xspace{\\ensuremath{\\kaon^+}}\\xspace{\\ensuremath{\\pion^+}}\\xspace$ decays  \\cite{LHCb-PAPER-2011-017}. A total of about 370 thousand signal candidates were analysed. The approach used the Miranda technique ~\\cite{bigi1} described in Sect.~\\ref{miranda} where the Dalitz plot was divided in bins, both uniformly or adaptively. The Cabibbo-favoured decays ${\\ensuremath{\\D^+_\\squark}}\\xspace\\ensuremath{\\rightarrow}\\xspace {\\ensuremath{\\kaon^\\pm}}\\xspace{\\ensuremath{\\kaon^+}}\\xspace{\\ensuremath{\\pion^+}}\\xspace$ and ${\\ensuremath{\\D^+}}\\xspace\\ensuremath{\\rightarrow}\\xspace{\\ensuremath{\\kaon^-}}\\xspace{\\ensuremath{\\pion^+}}\\xspace\\pip$, for which {\\ensuremath{C\\!P}}\\xspace violation is not expected,  were used as {\\it control channels} to check for nuisance asymmetries, with no observable effects. The ${\\ensuremath{\\D^+}}\\xspace\\ensuremath{\\rightarrow}\\xspace {\\ensuremath{\\kaon^-}}\\xspace{\\ensuremath{\\kaon^+}}\\xspace{\\ensuremath{\\pion^+}}\\xspace$ Dalitz plot was then studied for a few combinations of binning schemes. In Fig.~\\ref{LHCb_KKpi_2011} the significances are shown across the Dalitz plot for 106 adaptive bins; the corresponding $\\chi^2\\equiv \\sum_i [S_{\\ensuremath{C\\!P}}\\xspace(i)]^2$ and number of degrees of freedom  resulted  in a p-value of about 10\\%, compatible with the hypothesis of {\\ensuremath{C\\!P}}\\xspace conservation. Other variations in binning schemes lead to the same conclusion. \n\\begin{figure}\n\\centerline{\\includegraphics[width = 0.4\\textwidth]{Figure37_LHCb_DpKKpi_2011.pdf} \\includegraphics[width = 0.4\\textwidth]{Figure37_LHCb_DKKpi_Gauss.pdf}}\n\\caption{(Left) {\\ensuremath{C\\!P}}\\xspace asymmetry significances, $S_{\\ensuremath{C\\!P}}\\xspace$, across the Dalitz plot for candidate decays of ${\\ensuremath{\\D^+}}\\xspace\\ensuremath{\\rightarrow}\\xspace {\\ensuremath{\\kaon^-}}\\xspace{\\ensuremath{\\kaon^+}}\\xspace{\\ensuremath{\\pion^+}}\\xspace$ in 2010 data from the LHCb experiment. (Right) Distribution of the $S_{\\ensuremath{C\\!P}}\\xspace$ values. See \\cite{LHCb-PAPER-2011-017} for details.}\n\\label{LHCb_KKpi_2011}\n\\end{figure}\n\nThe same technique was applied to ${\\ensuremath{\\D^+}}\\xspace\\ensuremath{\\rightarrow}\\xspace{\\ensuremath{\\pion^-}}\\xspace{\\ensuremath{\\pion^+}}\\xspace\\pip$, with a sample of about 3 million candidates with 82\\% purity~\\cite{LHCb-PAPER-2013-057} collected by LHCb during 2011. Besides the binned Miranda approach, the kNN  unbinned technique, described in Sect.~\\ref{unbinned},  was also used. Sensitivity studies based on pseudo-experiments showed that with the given data sample size  both methods would be sensitive to asymmetries of about 2\\% and phase differences of about $1^\\circ-2^\\circ$ for specific resonant contributions (${\\ensuremath{\\D^+}}\\xspace\\ensuremath{\\rightarrow}\\xspace \\rho^0(770)$ for example). The modes ${\\ensuremath{\\D^+_\\squark}}\\xspace\\ensuremath{\\rightarrow}\\xspace {\\ensuremath{\\pion^-}}\\xspace{\\ensuremath{\\pion^+}}\\xspace\\pip$ and ${\\ensuremath{\\D^+}}\\xspace\\ensuremath{\\rightarrow}\\xspace{\\ensuremath{\\kaon^-}}\\xspace{\\ensuremath{\\pion^+}}\\xspace\\pip$ were used as control modes, with no sign of local asymmetries. When analysing the signal region, no evidence for {\\ensuremath{C\\!P}}\\xspace violation in  ${\\ensuremath{\\D^+}}\\xspace\\ensuremath{\\rightarrow}\\xspace{\\ensuremath{\\pion^-}}\\xspace{\\ensuremath{\\pion^+}}\\xspace\\pip$ was found. In Fig.~\\ref{LHCb_D3pi}  the $S_{\\ensuremath{C\\!P}}\\xspace(i)$ distribution across the Dalitz plot is shown for 106 bins, as well as  the resulting p-values for the kNN method.\n\\begin{figure}\n\\centerline{\\includegraphics[width= 0.4\\textwidth]{Figure38_LHCb_D3pi_Miranda}\\includegraphics[width= 0.4\\textwidth]{Figure38_LHCb_D3pi_kNN}}\n\\caption{Results for the {\\ensuremath{C\\!P}}\\xspace violation search in ${\\ensuremath{\\D^+}}\\xspace\\ensuremath{\\rightarrow}\\xspace{\\ensuremath{\\pion^-}}\\xspace{\\ensuremath{\\pion^+}}\\xspace\\pip$ decays by LHCb collaboration with 2011 data. (Left) $S_{\\ensuremath{C\\!P}}\\xspace$ distribution with 106 adaptive bins. (Right) Distribution of p-values of the test statistic $T$ for different regions in the Dalitz plot. See \\cite{LHCb-PAPER-2013-057} for details.}\n\\label{LHCb_D3pi}\n\\end{figure}\n\nThere are no results yet with LHCb run II data for {\\ensuremath{C\\!P}}\\xspace violation searches across the Dalitz plot for ${\\ensuremath{\\D^+}}\\xspace\\ensuremath{\\rightarrow}\\xspace{\\ensuremath{\\pion^-}}\\xspace{\\ensuremath{\\pion^+}}\\xspace\\pip$ and ${\\ensuremath{\\D^+}}\\xspace\\ensuremath{\\rightarrow}\\xspace {\\ensuremath{\\kaon^-}}\\xspace{\\ensuremath{\\kaon^+}}\\xspace{\\ensuremath{\\pion^+}}\\xspace$ decays. The expected sample sizes are ${\\cal O}(10^8)$ events~\\cite{LHCb-PII-Physics} which would allow {\\ensuremath{C\\!P}}\\xspace asymmetries to be tested at the $10^{-3}$ level. \nGiven the recent observation of {\\ensuremath{C\\!P}}\\xspace violation in ${\\ensuremath{\\D^0}}\\xspace\\ensuremath{\\rightarrow}\\xspace{\\ensuremath{\\kaon^+}}\\xspace{\\ensuremath{\\kaon^-}}\\xspace$ and ${\\ensuremath{\\D^0}}\\xspace\\ensuremath{\\rightarrow}\\xspace{\\ensuremath{\\pion^+}}\\xspace{\\ensuremath{\\pion^-}}\\xspace$ it is likely  that direct  {\\ensuremath{C\\!P}}\\xspace violation would be soon observed in these decays too. Moreover, the study across the phase space, being potentially more sensitive to local asymmetries, may provide information regarding the eventual sources of {\\ensuremath{C\\!P}}\\xspace asymmetries,  e.g. resonance enhancements, S- and P- wave interference, and $\\PK\\PK-\\pi\\pi$ rescattering \\cite{Bigi3,Bigi_Charm_2015,Bianco_Bigi_2020,Li_ImplicationsCPV_2019}. In Fig.~\\ref{DalitzD3pi_LHCb} the (diagonal-folded) Dalitz plot distribution for ${\\ensuremath{\\D^+}}\\xspace\\ensuremath{\\rightarrow}\\xspace{\\ensuremath{\\pion^-}}\\xspace{\\ensuremath{\\pion^+}}\\xspace\\pip$ is shown evidencing the richness of the dynamics of the decay --- in particular one can see the large interference between the low-mass S-wave, dominated by the $\\sigma(500){\\ensuremath{\\pion^+}}\\xspace$ amplitude, and the $\\rho(770)^0{\\ensuremath{\\pion^+}}\\xspace$ amplitude which makes the higher mass lobe of the $\\rho(770)^0$ to almost disappear. \n\n\\begin{figure}\n\\centerline{\\includegraphics[width = 0.5\\textwidth]{Figure39_DalitzD3pi_LHCb}}\n\\caption{Dalitz plot distribution of ${\\ensuremath{\\D^+}}\\xspace\\ensuremath{\\rightarrow}\\xspace{\\ensuremath{\\pion^-}}\\xspace{\\ensuremath{\\pion^+}}\\xspace\\pip$ candidate decays, from 2011 LHCb data. The Dalitz variables $s_{\\rm low}$ and $s_{\\rm high}$ correspond to $m^2({\\ensuremath{\\pion^-}}\\xspace{\\ensuremath{\\pion^+}}\\xspace)_{\\rm low}$ and $m^2({\\ensuremath{\\pion^-}}\\xspace{\\ensuremath{\\pion^+}}\\xspace)_{\\rm high}$, respectively ~\\cite{LHCb-PAPER-2013-057}.}\n\\label{DalitzD3pi_LHCb}\n\\end{figure}\n\nAnother important and promising decay is ${\\ensuremath{\\D^0}}\\xspace\\ensuremath{\\rightarrow}\\xspace {\\ensuremath{\\pion^-}}\\xspace{\\ensuremath{\\pion^+}}\\xspace{\\ensuremath{\\pion^0}}\\xspace$. With  data collected in 2012, the LHCb collaboration searched for  {\\ensuremath{C\\!P}}\\xspace violation in this decay. The sample comprised 660 thousand candidate decays and used the ${\\ensuremath{\\D^*}}\\xspace\\ensuremath{\\rightarrow}\\xspace {\\ensuremath{\\D^0}}\\xspace{\\ensuremath{\\pion^+}}\\xspace$ decay to tag the flavour of {\\ensuremath{\\D^0}}\\xspace at production.  The search was carried out in the Dalitz plot via the energy test technique described in Sect.~\\ref{unbinned}. The metric function $\\psi_{ij}$, defined in Eq.~\\ref{T_energytest} was chosen as a Gaussian, $\\psi_{ij} = e^{-d^2_{ij}\/2\\delta^2}$ where $d_{ij}$ is the ``distance'' (measured in \\ensuremath{{\\mathrm{\\,Ge\\kern -0.1em V^2\\!\/}c^4}}\\xspace) between the $i$ and $j$ events in the 3D-space spanned by the two-body  mass combinations of the three final-state particles.  From pseudo-experiments with the same size as the data, the energy test showed sensitivity of about 2--4\\% in magnitudes and 1--3$^\\circ$ in phases, depending on which intermediate amplitude {\\ensuremath{C\\!P}}\\xspace violation were introduced. When compared to the Miranda technique, the energy-test method showed similar or better precision. When applied to the ${\\ensuremath{\\D^0}}\\xspace\\ensuremath{\\rightarrow}\\xspace {\\ensuremath{\\pion^-}}\\xspace{\\ensuremath{\\pion^+}}\\xspace{\\ensuremath{\\pion^0}}\\xspace$ data sample, the $T$ value obtained is consistent with {\\ensuremath{C\\!P}}\\xspace conservation, with a p-value of $(2.6\\pm 0.5)\\%$. A limitation of the method is the computation price: with samples of ${\\cal O}(10^{7-8})$ events  (current sample sizes for some Cabibbo-suppressed channels in LHCb run II) the permutations needed to access the p-value can be very  time consuming.\n\n\n\n\\subsection{Four-body modes}\n\\label{Dto4body}\n\nThe main four-body  hadronic modes for the search for {\\ensuremath{C\\!P}}\\xspace violation are the Cabibbo-suppressed decays ${\\ensuremath{\\D^0}}\\xspace\\ensuremath{\\rightarrow}\\xspace{\\ensuremath{\\kaon^+}}\\xspace{\\ensuremath{\\kaon^-}}\\xspace{\\ensuremath{\\pion^+}}\\xspace\\pip$ and ${\\ensuremath{\\D^0}}\\xspace\\ensuremath{\\rightarrow}\\xspace{\\ensuremath{\\pion^+}}\\xspace{\\ensuremath{\\pion^-}}\\xspace{\\ensuremath{\\pion^+}}\\xspace\\pip$.\nThe dynamics of four-body decay modes involving only mesons in the initial and final state are fully described by  a five-dimensional phase space. As in the case for three-body  modes, the search for {\\ensuremath{C\\!P}}\\xspace violation can be pursued in model-dependent and model-independent ways, with the further complication of dealing with 5 independent variables. Clearly, given the limited sizes of decay samples, the sensitivity to charge asymmetries may vary substantially depending on the phase space region. Since there is no hint in which region  {\\ensuremath{C\\!P}}\\xspace violation may be potentially enhanced, the study is challenging. \nFor instance, for the Miranda technique bins should in principle be defined in 5 dimensions, preferably with adaptive binning to control the minimum number of events in each bin, with a careful study of the binning scheme in order not  to lose sensitivity. This also applies for the  energy-test technique where the parameter $\\delta$ defines an effective radius in phase space. Model-dependent searches rely on a full amplitude analysis, usually using the Isobar model to disentangle the different intermediate contributions by fitting their magnitudes and phases separately for particle and antiparticle. \n\nAs discussed in Sect.~\\ref{tripleproducts} in a four-body  decay other {\\ensuremath{C\\!P}}\\xspace-violating observables can be accessed, such as the $a_{\\ensuremath{C\\!P}}\\xspace^{T-\\rm odd}$ asymmetry defined in terms of a triple-product of momenta of the final-state particles, defined in Eqs.~\\ref{C_T} to \\ref{a_CP_Todd}. \nThe first searches for {\\ensuremath{C\\!P}}\\xspace violation using this T-odd observable were carried out in the  2000s by the FOCUS~\\cite{FOCUS_KKpipiTodd_2005} and BaBar~\\cite{Babar_D0KKpipi_Todd_2010} collaborations with the final state ${\\ensuremath{\\D^0}}\\xspace\\ensuremath{\\rightarrow}\\xspace{\\ensuremath{\\kaon^+}}\\xspace{\\ensuremath{\\kaon^-}}\\xspace{\\ensuremath{\\pion^+}}\\xspace{\\ensuremath{\\pion^-}}\\xspace$. In 2014, the LHCb collaboration   also performed a search for T-odd correlations with about 170 thousand ${\\ensuremath{\\D^0}}\\xspace\\ensuremath{\\rightarrow}\\xspace{\\ensuremath{\\kaon^+}}\\xspace{\\ensuremath{\\kaon^-}}\\xspace{\\ensuremath{\\pion^+}}\\xspace{\\ensuremath{\\pion^-}}\\xspace$ signal decays from run I data with three complementary approaches~\\cite{LHCb-PAPER-2014-046}: by measuring the phase-space integrated value of  $A_T$, $\\bar A_T$ and $a_{\\ensuremath{C\\!P}}\\xspace^{T-\\rm odd}$, as well as in different regions of the phase-space and in different decay-time bins. The quantity $C_T$ was defined with particles 1, 2 and 3 being respectively {\\ensuremath{\\kaon^+}}\\xspace, {\\ensuremath{\\pion^+}}\\xspace and {\\ensuremath{\\pion^-}}\\xspace. The value obtained for $a_{\\ensuremath{C\\!P}}\\xspace^{T-\\rm odd}$ was $(0.18\\pm 0.29\\pm0.04)\\%$, consistent with {\\ensuremath{C\\!P}}\\xspace conservation, as well as the values obtained in bins of phase space and decay time. Interestingly enough, the integrated values of $A_T$ and ${\\bar A}_T$ are $\\approx (-7\\pm0.5)\\%$, with also relatively large variations across the phase space, pointing to the significant final-state interactions due to the rich resonant contributions. Belle has performed a similar study, for the first time for the decay ${\\ensuremath{\\D^0}}\\xspace\\ensuremath{\\rightarrow}\\xspace {\\ensuremath{\\kaon^0_{\\rm\\scriptscriptstyle S}}}\\xspace{\\ensuremath{\\pion^+}}\\xspace{\\ensuremath{\\pion^-}}\\xspace{\\ensuremath{\\pion^0}}\\xspace$, with a total of about 1.7 million candidates~\\cite{Belle_D0Kspipipi0_2017}. This final state receives contributions from  both Cabibbo-favoured and doubly-Cabibbo-suppressed transitions. The result for phase-space integrated T-odd {\\ensuremath{C\\!P}}\\xspace asymmetry, as well in different regions of the phase space, has shown consistency with {\\ensuremath{C\\!P}}\\xspace conservation with a precision of 0.14\\%. \n\nIn a very recent analysis of the ${\\ensuremath{\\D^0}}\\xspace\\ensuremath{\\rightarrow}\\xspace{\\ensuremath{\\kaon^+}}\\xspace{\\ensuremath{\\kaon^-}}\\xspace{\\ensuremath{\\pion^+}}\\xspace\\pip$ mode, Belle~\\cite{Belle_D0KKpipiCPV_2019} has extended the concept of the asymmetry  $A_X$ for different kinematic variables $X$, so that \n\\begin{equation}\na_X^{\\ensuremath{C\\!P}}\\xspace \\equiv \\frac{1}{2}\\left( A_X - \\eta_X^{\\ensuremath{C\\!P}}\\xspace {\\bar A}_{\\bar X}\\right)~, \n\\end{equation}\nwith $\\bar A_{\\bar X}$ defined the same way as $A_X$ (changing sign convention with respect to that  in Eq.~\\ref{ATbar}) and $\\eta_X^{\\ensuremath{C\\!P}}\\xspace$ is the {\\ensuremath{C\\!P}}\\xspace eigenvalue specific to $X$. The kinematic variables used were $\\sin2\\Phi$, $\\cos\\theta_1\\cos\\theta_2\\sin\\Phi$, $\\sin\\Phi$ (these with $\\eta_X^{\\ensuremath{C\\!P}}\\xspace=-1$) , and $\\cos\\Phi$ and $\\cos\\theta_1\\cos\\theta_2\\cos\\Phi$ (with $\\eta_X^{\\ensuremath{C\\!P}}\\xspace=+1$), where $\\theta_1$ ($\\theta_2$) is the angle of the {\\ensuremath{\\kaon^+}}\\xspace ({\\ensuremath{\\pion^+}}\\xspace) momentum and the direction opposite of the {\\ensuremath{\\D^0}}\\xspace in the {\\ensuremath{\\kaon^+}}\\xspace{\\ensuremath{\\kaon^-}}\\xspace ({\\ensuremath{\\pion^+}}\\xspace{\\ensuremath{\\pion^-}}\\xspace) rest frame, and $\\Phi$ is the angle between the decay planes of {\\ensuremath{\\kaon^+}}\\xspace{\\ensuremath{\\kaon^-}}\\xspace and {\\ensuremath{\\pion^+}}\\xspace{\\ensuremath{\\pion^-}}\\xspace pairs in the {\\ensuremath{\\D^0}}\\xspace rest frame. This set of variables is expected to be sensitive to {\\ensuremath{C\\!P}}\\xspace violation in the interference between the S- and P-wave productions of the ${\\ensuremath{\\kaon^+}}\\xspace{\\ensuremath{\\kaon^-}}\\xspace$ and ${\\ensuremath{\\pion^+}}\\xspace{\\ensuremath{\\pion^-}}\\xspace$ pairs. With a sample of about 108 thousand signal decays, all $a_X^{\\ensuremath{C\\!P}}\\xspace$ asymmetries were found consistent with zero, at the level of 0.4\\% (including both statistical and systematic uncertainties).\n\nSearches for {\\ensuremath{C\\!P}}\\xspace violation using model-dependent and model-independent analysis over the phase space were also performed for both ${\\ensuremath{\\D^0}}\\xspace\\ensuremath{\\rightarrow}\\xspace{\\ensuremath{\\kaon^+}}\\xspace{\\ensuremath{\\kaon^-}}\\xspace{\\ensuremath{\\pion^+}}\\xspace\\pip$ and ${\\ensuremath{\\D^0}}\\xspace\\ensuremath{\\rightarrow}\\xspace{\\ensuremath{\\pion^+}}\\xspace{\\ensuremath{\\pion^-}}\\xspace{\\ensuremath{\\pion^+}}\\xspace\\pip$ modes. Full amplitude analyses for these decays were published in 2017 making use of CLEO data~\\cite{dArgent_D0hhhh_2017} with limited samples sizes of a few thousand events, resulting in integrated asymmetries probed only at the level of 1.5--2\\%, and in different sub-channels at 5--10\\% or higher. \n\nThe LHCb collaboration studied both decays with much larger samples. In a first work with 1\\ensuremath{\\mbox{\\,fb}^{-1}}\\xspace of data from run I ~\\cite{LHCb-PAPER-2013-041},  $5.7\\times 10^4$ and $3.3\\times 10^5$  signal decays for ${\\ensuremath{\\D^0}}\\xspace\\ensuremath{\\rightarrow}\\xspace{\\ensuremath{\\kaon^+}}\\xspace{\\ensuremath{\\kaon^-}}\\xspace{\\ensuremath{\\pion^+}}\\xspace\\pip$  and ${\\ensuremath{\\D^0}}\\xspace\\ensuremath{\\rightarrow}\\xspace {\\ensuremath{\\pion^-}}\\xspace{\\ensuremath{\\pion^+}}\\xspace{\\ensuremath{\\pion^-}}\\xspace{\\ensuremath{\\pion^+}}\\xspace$ modes, respectively, both tagged through ${\\ensuremath{\\D^{*+}}}\\xspace\\ensuremath{\\rightarrow}\\xspace{\\ensuremath{\\D^0}}\\xspace{\\ensuremath{\\pion^+}}\\xspace$,  were analysed and {\\ensuremath{C\\!P}}\\xspace violation was searched for  via the Miranda technique over the phase space  defined by five invariants chosen as a set of  2- and three-body  mass combinations. The p-values under the no {\\ensuremath{C\\!P}}\\xspace violation hypothesis were found to be 9.1\\% and 41\\%, indicating consistency with {\\ensuremath{C\\!P}}\\xspace symmetry. \nWith the full run I data sample, comprising about 1 million signal decays, the  ${\\ensuremath{\\D^0}}\\xspace\\ensuremath{\\rightarrow}\\xspace {\\ensuremath{\\pion^-}}\\xspace{\\ensuremath{\\pion^+}}\\xspace{\\ensuremath{\\pion^-}}\\xspace{\\ensuremath{\\pion^+}}\\xspace$ mode was then studied by LHCb via the energy test technique --- and also dividing the total sample into four sub-samples  depending on  the sign of a triple-product $C_T$  and on the  {\\ensuremath{\\D^0}}\\xspace flavour~\\cite{LHCb-PAPER-2016-044}. As such, the analysis could be sensitive to both $P$-even and $P$-odd {\\ensuremath{C\\!P}}\\xspace asymmetries. From simulation, sensitivity for {\\ensuremath{C\\!P}}\\xspace violation effects is found to be typically 4--5\\% in amplitudes or 3--4$^\\circ$ in phases for the main intermediate states. The results have  p-values of $(4.3\\pm 0.6)\\%$ and $(0.6\\pm 0.2)\\%$ for $P$-even and $P$-odd {\\ensuremath{C\\!P}}\\xspace violation tests, respectively, obtained from the distribution of permutation T-values.  Interestingly,  the $P$-odd {\\ensuremath{C\\!P}}\\xspace-violation test was found to be only marginally consistent with {\\ensuremath{C\\!P}}\\xspace conservation --- this can be seen in Fig.~\\ref{LHCb-D4pi-energytest}, where the right plot indicates the $\\pi^-\\pi^+$ invariant-mass region where the discrepancies are found. \n\n\\begin{figure}\n\\centerline{\\includegraphics[align=c, width = 0.4\\textwidth, height = 0.4\\textwidth]{Figure40_LHCb_D04pi_a.png} \\includegraphics[align=c, width = 0.4\\textwidth, height = 0.42\\textwidth]{Figure40_LHCb_D04pi_b.png}}\n\\caption{Results of the energy test for ${\\ensuremath{\\D^0}}\\xspace\\ensuremath{\\rightarrow}\\xspace {\\ensuremath{\\pion^-}}\\xspace{\\ensuremath{\\pion^+}}\\xspace{\\ensuremath{\\pion^-}}\\xspace{\\ensuremath{\\pion^+}}\\xspace$ decays by the LHCb collaboration \\cite{LHCb-PAPER-2016-044}.  (left) Permutations of T-value for the $P$-odd {\\ensuremath{C\\!P}}\\xspace violation test. (right) The $\\pi^-\\pi^+$ invariant-mass projection showing the significance of the $T$-value contribution in different regions.}\n\\label{LHCb-D4pi-energytest}\n\\end{figure}\n\nThe decay ${\\ensuremath{\\D^0}}\\xspace\\ensuremath{\\rightarrow}\\xspace{\\ensuremath{\\kaon^+}}\\xspace{\\ensuremath{\\kaon^-}}\\xspace{\\ensuremath{\\pion^+}}\\xspace\\pip$ was also studied very recently by the LHCb collaboration through a full amplitude analysis from the full run I data sample (3\\ensuremath{\\mbox{\\,fb}^{-1}}\\xspace) using semileptonic b-hadron decays into ${\\ensuremath{\\D^0}}\\xspace\\mun X$ final states \\cite{LHCb-PAPER-2018-041}. The sample has almost 200 thousand candidates with 83\\% purity. In total, 26 different intermediate states were considered for the phase-space fit and, for each, {\\ensuremath{C\\!P}}\\xspace magnitude asymmetries, relative phase differences, and {\\ensuremath{C\\!P}}\\xspace fraction asymmetries were obtained. The sensitivity for {\\ensuremath{C\\!P}}\\xspace violation varied depending on the sub-channel, from about 1\\% to 15\\%, with results compatible with {\\ensuremath{C\\!P}}\\xspace conservation.\n\nThere are yet no results using run II data from LHCb. The sensitivity expected for $a_{\\ensuremath{C\\!P}}\\xspace^{T-\\rm odd}$, for example, could be as low as $2.4\\times 10^{-4}$~\\cite{LHCb-PII-Physics}.\n\n\n\\subsection{Charm Baryons}\n\n    In general, the charm baryon sector is so far much less studied than the charm meson sector. Within the last 5 years or so, though, this scenario began to change, with many new results  on production and branching ratios \\cite{BESIII_LambdaCS_2016, BESIII_LambdaAbsolute_2016, BESIII_LambdaAbsoluteSL_2017, BESIII_Lambdac_nKSpi_2016, BESIII_eeLcLcbarProd_2018, Acharya:2017lwf, Acharya:2017kfy,LHCb-PAPER-2012-041,LHCb-PAPER-2017-026, LHCb-PAPER-2018-040, LHCb-PAPER-2019-035}, lifetimes~\\cite{LHCb-PAPER-2018-019,LHCb-PAPER-2018-028,LHCb-PAPER-2019-008}, decay asymmetries and polarisation \\cite{BESIII_LambdaDecayAsymmetries_2019}, and new states\\cite{LHCb-PAPER-2017-002, LHCb-PAPER-2017-018}. In what concerns {\\ensuremath{C\\!P}}\\xspace violation, there is a long way to go \\cite{Bigi_CharmBaryons_2012} but the direct {\\ensuremath{C\\!P}}\\xspace violation observed in the meson sector is now adding fuel and pushing towards the study of charm baryons \\cite{Wang_2019,Shi_Bigi_2019,Grossman_USpin_baryons_2019}. \n    \n    Within the CKM ansatz, the underlying mechanism for the appearance of a weak phase in Cabibbo-suppressed charmed baryon decays is the same as for charmed mesons, $c\\ensuremath{\\rightarrow}\\xspace du\\bar u$ and $c\\ensuremath{\\rightarrow}\\xspace su\\bar u$ transitions. \nThe main actors for the search of {\\ensuremath{C\\!P}}\\xspace violation are, in principle,  the two-body  ${\\ensuremath{\\Lz^+_\\cquark}}\\xspace\\ensuremath{\\rightarrow}\\xspace {\\ensuremath{\\PLambda}}\\xspace{\\ensuremath{\\kaon^+}}\\xspace$ and three-body  ${\\ensuremath{\\Lz^+_\\cquark}}\\xspace\\ensuremath{\\rightarrow}\\xspace {\\ensuremath{\\Pp}}\\xspace{\\ensuremath{\\kaon^+}}\\xspace{\\ensuremath{\\kaon^-}}\\xspace$ and  ${\\ensuremath{\\Lz^+_\\cquark}}\\xspace\\ensuremath{\\rightarrow}\\xspace {\\ensuremath{\\Pp}}\\xspace{\\ensuremath{\\pion^+}}\\xspace{\\ensuremath{\\pion^-}}\\xspace$.  From both the theoretical and experimental sides, the study involving three-body  baryon decays is challenging due to polarisation, five-dimensional phase space and short lifetime.  \n\nThe LHCb collaboration performed the first search for {\\ensuremath{C\\!P}}\\xspace violation in the decays ${\\ensuremath{\\Lz^+_\\cquark}}\\xspace\\ensuremath{\\rightarrow}\\xspace {\\ensuremath{\\Pp}}\\xspace{\\ensuremath{\\kaon^+}}\\xspace{\\ensuremath{\\kaon^-}}\\xspace$ and  ${\\ensuremath{\\Lz^+_\\cquark}}\\xspace\\ensuremath{\\rightarrow}\\xspace {\\ensuremath{\\Pp}}\\xspace{\\ensuremath{\\pion^+}}\\xspace{\\ensuremath{\\pion^-}}\\xspace$, using run I data~\\cite{LHCb-PAPER-2017-044}. The decays were selected through the decay chain ${\\ensuremath{\\Lz^0_\\bquark}}\\xspace\\ensuremath{\\rightarrow}\\xspace {\\ensuremath{\\Lz^+_\\cquark}}\\xspace\\mun X$ (with $X$ being any additional unreconstructed particles) to diminish the level of background. The sample consisted of about 25 thousand ${\\ensuremath{\\Lz^+_\\cquark}}\\xspace\\ensuremath{\\rightarrow}\\xspace {\\ensuremath{\\Pp}}\\xspace{\\ensuremath{\\kaon^+}}\\xspace{\\ensuremath{\\kaon^-}}\\xspace$ decays and 160 thousand ${\\ensuremath{\\Lz^+_\\cquark}}\\xspace\\ensuremath{\\rightarrow}\\xspace {\\ensuremath{\\Pp}}\\xspace{\\ensuremath{\\pion^+}}\\xspace{\\ensuremath{\\pion^-}}\\xspace$ decays. The invariant mass distributions and the Dalitz plots for both final states are shown in Fig.~\\ref{LcDeltaACP}.\n\\begin{figure}\n\\centering\n\\includegraphics[width = 0.35\\textwidth]{Figure41_LcDeltaACP_a.png}\n\\includegraphics[width = 0.35\\textwidth]{Figure41_LcDeltaACP_b.png} \\\\\n\\includegraphics[width = 0.35\\textwidth]{Figure41_LcDeltaACP_c.png}\n\\includegraphics[width = 0.35\\textwidth]{Figure41_LcDeltaACP_d.png} \n\\caption{Invariant mass distributions for (top left) ${\\ensuremath{\\Lz^+_\\cquark}}\\xspace\\ensuremath{\\rightarrow}\\xspace {\\ensuremath{\\Pp}}\\xspace{\\ensuremath{\\kaon^+}}\\xspace{\\ensuremath{\\kaon^-}}\\xspace$ and (top right) ${\\ensuremath{\\Lz^+_\\cquark}}\\xspace\\ensuremath{\\rightarrow}\\xspace {\\ensuremath{\\Pp}}\\xspace{\\ensuremath{\\pion^+}}\\xspace{\\ensuremath{\\pion^-}}\\xspace$ candidates. The corresponding Dalitz plots (background-subtracted and efficiency-corrected) are shown in the bottom plots~\\cite{LHCb-PAPER-2017-044}.}\n\\label{LcDeltaACP}\n\\end{figure}\nTo cancel the ${\\ensuremath{\\Lz^0_\\bquark}}\\xspace$ production asymmetry as well as muon and proton detection asymmetries, the differences of raw asymmetries are measured, once the kinematics of the ${\\ensuremath{\\Lz^+_\\cquark}}\\xspace\\ensuremath{\\rightarrow}\\xspace {\\ensuremath{\\Pp}}\\xspace{\\ensuremath{\\kaon^+}}\\xspace{\\ensuremath{\\kaon^-}}\\xspace$ decays are matched to those of ${\\ensuremath{\\Lz^+_\\cquark}}\\xspace\\ensuremath{\\rightarrow}\\xspace {\\ensuremath{\\Pp}}\\xspace{\\ensuremath{\\pion^+}}\\xspace{\\ensuremath{\\pion^-}}\\xspace$. This procedure may alter the numerical value of the true physics asymmetry, thus the measured asymmetry for ${\\ensuremath{\\Lz^+_\\cquark}}\\xspace\\ensuremath{\\rightarrow}\\xspace {\\ensuremath{\\Pp}}\\xspace{\\ensuremath{\\kaon^+}}\\xspace{\\ensuremath{\\kaon^-}}\\xspace$ is called a  {\\it weighted} asymmetry, and the measurement is defined as\n\\begin{equation}\n\\Delta A_{\\ensuremath{C\\!P}}\\xspace^{\\rm wgt} \\equiv A_{\\ensuremath{C\\!P}}\\xspace({\\ensuremath{\\Pp}}\\xspace{\\ensuremath{\\kaon^+}}\\xspace{\\ensuremath{\\kaon^-}}\\xspace) - A_{\\ensuremath{C\\!P}}\\xspace^{\\rm wgt} ({\\ensuremath{\\Pp}}\\xspace{\\ensuremath{\\pion^+}}\\xspace{\\ensuremath{\\pion^-}}\\xspace) \\approx A_{\\rm raw}({\\ensuremath{\\Pp}}\\xspace{\\ensuremath{\\kaon^+}}\\xspace{\\ensuremath{\\kaon^-}}\\xspace) - A_{\\rm raw}^{\\rm wgt}({\\ensuremath{\\Pp}}\\xspace{\\ensuremath{\\pion^+}}\\xspace{\\ensuremath{\\pion^-}}\\xspace) ~.\n\\end{equation}\nThe result is\n\\begin{equation}\n\\Delta A_{\\ensuremath{C\\!P}}\\xspace^{\\rm wgt}  = (0.30\\pm 0.91\\pm 0.61)\\%~,\n\\end{equation}\nshowing consistency with {\\ensuremath{C\\!P}}\\xspace symmetry at the level of 1\\%. It represents the first search for {\\ensuremath{C\\!P}}\\xspace violation in a three-body  baryon mode. New results are expected from LHCb using run II data not only on ${\\ensuremath{\\Lz^+_\\cquark}}\\xspace$ but also, potentially,  ${\\ensuremath{\\Xires_\\cquark}}\\xspace$ decay modes. Yet, to match the current precision in mesons the required amount of data will be possible only by the end of run V of LHCb, with 300\\ensuremath{\\mbox{\\,fb}^{-1}}\\xspace ~\\cite{LHCb-PII-Physics}.\n\n \n\\section{Conclusion}\n\\label{chapter8}\n\n       More than half a century after the discovery of {\\ensuremath{C\\!P}}\\xspace violation,  many theoretical and experimental achievements led to a better understanding of this phenomenon. The CKM mechanism has proved to be  very successful in describing the current experimental data, in particular by the consistency among many different measurements leading to a  coherent  picture of the Unitarity Triangle. The measurement of $\\gamma$ with a current precision of about $5^\\circ$ is a remarkable result and $\\gamma$ is, indeed, the only angle that have been measured exclusively from a direct {\\ensuremath{C\\!P}}\\xspace violation measurement, involving only tree-level amplitudes in {\\ensuremath{\\PB}}\\xspace decays to {\\ensuremath{\\PD}}\\xspace plus light mesons. \n              \n        However, direct {\\ensuremath{C\\!P}}\\xspace violation in non-leptonic decays is in general a complex subject due to the interplay between the weak process and the underlying strong interactions. Although, on  the one hand, the strong phases are as much important as the weak phases for the observability of {\\ensuremath{C\\!P}}\\xspace violation, from the other hand it is very difficult to know a priori the size of the strong phases and, as a consequence, the expected level of {\\ensuremath{C\\!P}}\\xspace violation  effects in exclusive modes. Indeed, there is a long-term discussion about the relative importance of the short- versus long-distance contributions as sources of the strong phases. On this subject, much progress was obtained  within the  last years, mainly due to the recent experimental results. \n\nSome results in the {\\ensuremath{\\PB}}\\xspace-meson sector are particularly interesting. {\\ensuremath{C\\!P}}\\xspace violation is well established in  the two body modes  $ {\\ensuremath{\\B^0_{(\\squark)}}}\\xspace \\ensuremath{\\rightarrow}\\xspace{\\ensuremath{\\kaon^-}}\\xspace{\\ensuremath{\\pion^+}}\\xspace$ and ${\\ensuremath{\\B^0}}\\xspace\\ensuremath{\\rightarrow}\\xspace{\\ensuremath{\\pion^+}}\\xspace{\\ensuremath{\\pion^-}}\\xspace$. In quasi two-body modes,  first observations of {\\ensuremath{C\\!P}}\\xspace asymmetry were made for ${\\ensuremath{\\Bu}}\\xspace\\ensuremath{\\rightarrow}\\xspace\\sigma(500){\\ensuremath{\\pion^+}}\\xspace$, $ {\\ensuremath{\\Bu}}\\xspace\\ensuremath{\\rightarrow}\\xspace  f_2(1270) {\\ensuremath{\\pion^+}}\\xspace$ and $ {\\ensuremath{\\B^0}}\\xspace\\ensuremath{\\rightarrow}\\xspace {\\ensuremath{\\kaon^{*+}}}\\xspace {\\ensuremath{\\pion^-}}\\xspace$ decays while no significant asymmetry was found in the $ {\\ensuremath{\\Bu}}\\xspace\\ensuremath{\\rightarrow}\\xspace \\rho(770){\\ensuremath{\\pion^+}}\\xspace$  mode.\nBig surprises came from three-body charmless {\\ensuremath{\\Bu}}\\xspace decays. Rich patterns of large {\\ensuremath{C\\!P}}\\xspace asymmetries across the Dalitz plot were observed, with regions of positive and negative $A_{\\ensuremath{C\\!P}}\\xspace$. Since the source of weak phase from the SM in these decays comes only from the angle $\\gamma$, the structures  need to arise from the variation of the strong phase. Moreover, these results point towards  the importance of long-distance hadronic phases. A particular result is the large {\\ensuremath{C\\!P}}\\xspace asymmetry of about 66\\% associated to the rescattering region $\\PK\\PK\\leftrightarrow \\pi\\pi$ in the decay ${\\ensuremath{\\Bu}}\\xspace\\ensuremath{\\rightarrow}\\xspace{\\ensuremath{\\kaon^-}}\\xspace{\\ensuremath{\\kaon^+}}\\xspace{\\ensuremath{\\pion^+}}\\xspace$. Another unforeseen result was the pattern of large {\\ensuremath{C\\!P}}\\xspace asymmetries at high $\\pi\\pi$ masses in the decay ${\\ensuremath{\\Bu}}\\xspace\\ensuremath{\\rightarrow}\\xspace{\\ensuremath{\\pion^+}}\\xspace\\pip{\\ensuremath{\\pion^-}}\\xspace$, in a region not dominated by resonances. \n\nIn the charm sector, a new era is starting with the observation of direct {\\ensuremath{C\\!P}}\\xspace violation  in the decays ${\\ensuremath{\\D^0}}\\xspace\\ensuremath{\\rightarrow}\\xspace{\\ensuremath{\\kaon^+}}\\xspace{\\ensuremath{\\kaon^-}}\\xspace$ and ${\\ensuremath{\\D^0}}\\xspace\\ensuremath{\\rightarrow}\\xspace{\\ensuremath{\\pion^+}}\\xspace{\\ensuremath{\\pion^-}}\\xspace$, with a  {\\ensuremath{C\\!P}}\\xspace-asymmetry difference at the level of $10^{-3}$. The result, higher than expected from naive short-distance expectations, can be accommodated by models predicting strong, hadronic enhancements, although approaches based on NP models are also suggested. \nThis outstanding result pushes further forward the study of other modes where effects of {\\ensuremath{C\\!P}}\\xspace violation may be at a similar level. This is the case of three- and four-body Cabibbo-suppressed modes such ${\\ensuremath{\\D^+}}\\xspace\\ensuremath{\\rightarrow}\\xspace{\\ensuremath{\\kaon^-}}\\xspace{\\ensuremath{\\kaon^+}}\\xspace{\\ensuremath{\\pion^+}}\\xspace$,  ${\\ensuremath{\\D^+}}\\xspace\\ensuremath{\\rightarrow}\\xspace{\\ensuremath{\\pion^-}}\\xspace{\\ensuremath{\\pion^+}}\\xspace\\pip$ and ${\\ensuremath{\\D^0}}\\xspace\\ensuremath{\\rightarrow}\\xspace{\\ensuremath{\\kaon^+}}\\xspace{\\ensuremath{\\kaon^-}}\\xspace{\\ensuremath{\\pion^+}}\\xspace\\pip$. From the beautiful lesson learnt from charmless {\\ensuremath{\\Bu}}\\xspace decays, localised {\\ensuremath{C\\!P}}\\xspace asymmetries in the phase space can be potentially larger than integrated ones. The role of $\\PK\\PK\\leftrightarrow \\pi\\pi$ rescattering can be potentially more important for charm-hadron decays too, and deserves attention.\n\n\nAn interesting and relevant discussion is the impact of {\\ensuremath{C\\!PT}}\\xspace symmetry -- assuming it to be exact --  on constraining {\\ensuremath{C\\!P}}\\xspace violation in exclusive modes. Final-state interactions, fundamentally hadronic,  are the carriers of this constraint: any observed {\\ensuremath{C\\!P}}\\xspace violation in one mode should be compensated by other(s) mode(s) with the same quantum numbers and, in particular, flavour content. Although the concept is clear, its practical consequences are not obvious when the multiplicity of decay channels is large, which is the case of {\\ensuremath{\\PB}}\\xspace decays. Yet, the experimental data, in general with a concentration near the edges of the Dalitz plot, may seem to indicate that {\\ensuremath{C\\!PT}}\\xspace can be more useful than anticipated \\cite{Bigi_Charm_2015}. While quantitative statements are hard to make, the idea here is to  use it to enlighten  {\\ensuremath{C\\!P}}\\xspace violation studies  by trying to correlate different measurements or to search for new paths.  \n\nA not-so-welcome surprise is the absence, so far, of a compelling evidence for {\\ensuremath{C\\!P}}\\xspace violation in baryons. The LHCb latest results in ${\\ensuremath{\\Lz^0_\\bquark}}\\xspace\\ensuremath{\\rightarrow}\\xspace{\\ensuremath{\\Pp}}\\xspace{\\ensuremath{\\pion^-}}\\xspace{\\ensuremath{\\pion^+}}\\xspace{\\ensuremath{\\pion^-}}\\xspace$ decay with a larger sample have neither confirmed nor denied the hint for {\\ensuremath{C\\!P}}\\xspace violation in an earlier study. It is curious that {\\ensuremath{C\\!P}}\\xspace violation, being a necessary condition for baryogenesis, has only shown up clearly in the meson sector. \n\nThe last years were naturally dominated by results from the LHCb collaboration, with  BaBar, Belle, and  BES III also contributing. New results, and hopefully  some surprises, are awaited from  run II LHCb  data, since  many decay modes are still being analysed. This is the case in particular of multi-body decays discussed in this review, including baryons. Belle II is now entering the game, with the ability to provide good physics impact to final states with neutrals. One, from many examples, is the possibility to soon bring new elements to the $\\Delta A_{\\ensuremath{C\\!P}}\\xspace({\\ensuremath{\\PB}}\\xspace\\ensuremath{\\rightarrow}\\xspace \\PK\\pi)$ puzzle from the analyses of decay modes $ {\\ensuremath{\\B^0}}\\xspace \\ensuremath{\\rightarrow}\\xspace {\\ensuremath{\\kaon^0}}\\xspace{\\ensuremath{\\pion^0}}\\xspace$, $ {\\ensuremath{\\B^0}}\\xspace \\ensuremath{\\rightarrow}\\xspace {\\ensuremath{\\kaon^0}}\\xspace\\eta $, and  $ {\\ensuremath{\\B^0}}\\xspace \\ensuremath{\\rightarrow}\\xspace {\\ensuremath{\\kaon^0}}\\xspace\\eta'$. Similarly, three-body {\\ensuremath{\\PB}}\\xspace decay modes with neutrals in the final states may help to better understand the sources of direct {\\ensuremath{C\\!P}}\\xspace violation, such as rescattering contributions and the S- and P-wave interference. \n\n\nIn a near future, with further accumulation of Belle II data and the start of the upgraded LHCb from run III (and beyond), flavour physics and in particular {\\ensuremath{C\\!P}}\\xspace violation will be entering into an era of high precision.  Besides testing the validity of the SM and representing a tool for NP searches, precise measurements of direct {\\ensuremath{C\\!P}}\\xspace violation, especially coming from multi-body decays, could be a very interesting instrument for the understanding of hadronic interactions. In any scenario, {\\ensuremath{C\\!P}}\\xspace violation is expected to maintain a leading role in Particle Physics. \n\n\n            \n            \n                           \n        \n\\section*{Aknowledgements}\nThe authors want to thank S. Amato, J. Miranda, I. Nasteva, E. Polycarpo and A. Reis for the useful  discussions while preparing this review. \nThis work was partially supported from the Brazilian funding agencies {\\it Coordena\u00e7\u00e3o de Aperfei\u00e7oamento de Pessoal de N\u00edvel Superior} (CAPES) [Finance Code 001], {\\it Conselho Nacional de Desenvolvimento Cient\u00edfico e Tecnol\u00f3gico} (CNPq), and {\\it Funda\u00e7\u00e3o Carlos Chagas Filho de Amparo e Pesquisa do Estado do Rio de Janeiro} (FAPERJ). \n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{I.~~Introduction}\n\nThere is increasing evidence that our universe is presently in a\nstate of cosmic accelerating expansion~\\cite{phys1}. This result has\nbeen confirmed with observations via Supernovae Ia~\\cite{phys2},\nCosmic Microwave Background anisotropies~\\cite{phys3}, Large Scale\nStructure formation~\\cite{phys4}, baryon oscillations~\\cite{phys5}\nand weak lensing~\\cite{phys6}, etc. These observations strongly\nsuggest that the universe is spatially flat and is dominated by an\nextra component with negative pressure, dubbed as dark\nenergy~\\cite{phys7,phys8}. In the recent years, various candidates\nof dark energy have been proposed. The simplest candidate is the\ncosmological constant~\\cite{phys9}. But this scenario is plagued by\nthe severe fine-tuning problem and the coincidence problem. Other\npossible forms of dark energy include quintessence~\\cite{phys10},\nk-essence~\\cite{phys11}, phantom~\\cite{phys12}, Born-Infeld\nscalars~\\cite{phys13}, quintom~\\cite{phys14}, tachyon\nfield~\\cite{phys15}, holographic dark energy~\\cite{phys16}, and so\non. Additionally, the conjecture that dark energy and dark matter\ncan be unified by using the so-called Chaplygin gas (CG) obeying an\nexotic equation of state (EoS)\n\\begin{equation}\np_{g}=-\\frac{A}{\\rho_{g}}\n\\end{equation}\nhas been investigated in several literatures~\\cite{phys17}.\nRecently, the CG model was generalized to some possible\nforms~\\cite{phys18}. As an alternative model, the EoS of the\ngeneralized Chaplygin gas (GCG)~\\cite{phys19} reads\n\\begin{equation}\np_{g}=-\\frac{A}{\\rho_{g}^{\\alpha}},\n\\end{equation}\nwhere $0\\leq\\alpha\\leq1$. It is clear that the case $\\alpha=1$\ncorresponds to the CG. Within the framework of\nFriedmann-Robertson-Walker (FRW) cosmology, this EoS leads, after\ninserted into the relativistic energy conservation equation, to an\nenergy density evolving as\n\\begin{equation}\n\\rho_{g}=(A+\\frac{B}{a^{3(1+\\alpha)}})^{\\frac{1}{1+\\alpha}},\n\\end{equation}\nwhere $B$ is an integration constant. Hence, we see that at early\ntime the energy density behaves as a dust-like matter, while at late\ntime it behaves like a cosmological constant. So the GCG model can\nalso be interpreted as an entangled mixture of dark matter and dark\nenergy. This dual role is at the heart of the surprising properties\nof the GCG model. However, these so-called unified dark matter\nmodels have been ruled out because they produce oscillations or\nexponential blowup of the matter power spectrum inconsistent with\nobservation~\\cite{phys69}. But no observation so far rule out the\npossibility of the GCG as dark energy though it is disfavored as\ndark matter. In fact, the dynamical properties of the GCG have been\nstudied in Ref.~\\cite{phys20}, which indicates that the EoS of the\nGCG may cross the so-called phantom divide $w=-1$ if there exists an\ninteraction between the GCG and dark matter. Therefore, it is\nworthwhile to further understand such unified dark matter models by\nstudying another candidate for the generalization of the CG,\nreferred to as the modified Chaplygin gas\n(MCG)~\\cite{phys21,phys22}, which is characterized by a simple EoS\n\\begin{equation}\np_{g}=A\\rho_{g}-\\frac{B}{\\rho_{g}^{\\alpha}},\n\\end{equation}\nwhere $A$, $B$ and $\\alpha$ are constants and $0\\leq\\alpha\\leq1$.\nThe attractive feature of this model is that the EoS looks like\nthat of two fluids, one obeying a perfect EoS $p=A\\rho$ and the\nother being the GCG. From Eq. (4), it is easy to see that the MCG\nreduces to the GCG if $A=0$ and to the perfect fluid if $B=0$.\nAccordingly, the evolution of the energy density is given by\n\\begin{equation}\n\\rho_{g}=(\\frac{B}{1+A}+\\frac{C}{a^{3(1+A)(1+\\alpha)}})^{\\frac{1}{1+\\alpha}},\n~~~~ (A\\neq-1)\n\\end{equation}\nwhere $C$ is the constant of integration. Thus the MCG behaves as a\nradiation (when $A=1\/3$) or a dust-like matter (when $A=0$) at early\nstage, while as a cosmological constant at later stage.\n\nAs we all know, observations at the level of the solar system\nseverely constrain non-gravitational interactions of baryons,\nnamely, non-minimal coupling between dark energy and ordinary matter\nfluids is strongly restricted by the experimental tests in the solar\nsystems~\\cite{phys23}. However, since the nature of dark sectors\nremains unknown, it is possible to have non-gravitational\ninteractions between dark energy and dark matter. In this paper, we\nstudy the dynamical evolution of the interacting MCG model by\nconsidering an interaction term between the MCG and cold dark\nmatter. In our scenario, we find that there exists a stable scaling\nsolution, which is characterized by a constant ratio of the energy\ndensities of the MCG and cold dark matter. This provides the\npossibility to alleviate the coincidence problem. Moreover, we find\nthat the final state is determined by the parameters of the MCG,\n$\\alpha$, $A$, and the coupling constants $c_{1}$ and $c_{2}$. The\nlatter two parameters represent the transfer strength between the\nMCG and dark matter. Interestingly, we find that the EoS of the MCG\n$w_{g}$ tends to a constant, which is only determined by the\ncoupling constants $c_{1}$ and $c_{2}$. However, both the EoS of the\ntotal cosmic fluid $w$ and the deceleration parameter $q$ tend to\n$-1$, which are independent of the choice of values for the\nparameters. This indicates that the cosmic doomsday is avoided and\nthe universe enters to a de Sitter phase and thus accelerates\nforever. Further, we investigate the effect of the coupling\nconstants $c_{1}$ and $c_{2}$ on the dynamical evolution of this\nmodel from the statefinder viewpoint. By our analysis, we see that\nthe coupling cantants $c_{1}$ and $c_{2}$ play a significant role\nduring the dynamics of the interacting MCG model. It is also\nworthwhile to note that, we can distinguish this interacting model\nfrom other dark energy models in the $s-r$ plane.\n\n\nIn Sec. II, we study the dynamical evolution of the interacting MCG.\nThen we apply the statefinder diagnosis to the interacting modified\nChaplygin gas model for various different parameters in Sec. III.\nThe conclusions are summarized in Sec. IV.\n\n\\section{II.~~Dynamics analysis}\n\nIn our scenario, the universe contains the MCG $\\rho_{g}$ (as dark\nenergy), cold dark matter $\\rho_{m}$ and baryonic matter $\\rho_{b}$.\nFor a spatially flat universe, the Friedmann equation is\n\\begin{equation}\nH^{2}=\\frac{\\kappa^{2}}{3}\\rho=\\frac{\\kappa^{2}}{3}(\\rho_{g}+\\rho_{m}+\\rho_{b}),\n\\end{equation}\nwhere $H$ is the Hubble parameter, $\\kappa^{2}\\equiv8\\pi G$ and\n$\\rho$ is the total energy density (nature units $c=\\hbar=1$ is used\nthroughout the paper). Then differentiating the above equation with\nrespect to cosmic time $t$ and using the total energy conservation\nequation\n\\begin{equation}\n\\dot{\\rho}+3H(\\rho+p)=0,\n\\end{equation}\nwhere $p$ is the total pressure of the background fluid, we can get\nthe Raychaudhuri equation\n\\begin{equation}\n\\dot{H}=-\\frac{\\kappa^{2}}{2}(\\rho_{g}+p_{g}+\\rho_{m}+p_{b}),\n\\end{equation}\nin which $p_{g}$ represents the pressure of the MCG.\n\nWe postulate that the two dark sectors interact through the\ninteraction term $Q$ and the baryonic matter only interacts\ngravitationally with the dark sectors. Then the continuity equation\nis written as\n\\begin{equation}\n\\dot{\\rho}_{b}+3H(\\rho_{b})=0,\n\\end{equation}\n\\begin{equation}\n\\dot{\\rho}_{m}+3H(\\rho_{m})=Q,\n\\end{equation}\n\\begin{equation}\n\\dot{\\rho}_{g}+3H(\\rho_{g}+p_{g})=-Q.\n\\end{equation}\nClearly, $Q$ is the rate of the energy density exchange in the dark\nsectors and the sign of $Q$ determines the direction of energy\ntransfer. A positive $Q$ corresponds to the transfer of energy from\ndark energy to dark matter, while a negative $Q$ represents the\nother way round. Due to the unknown nature of dark sectors, there is\nas yet no basis in fundamental theory for a special coupling between\ntwo dark sectors. So the interaction term $Q$ discussed currently\nhas to be chosen in a phenomenological way~\\cite{phys40}. One\npossible choice for the interaction term is\n$Q=3H(c_{1}\\rho_{m}+c_{2}\\rho_{g})$~\\cite{phys41}, where $c_{1}$ and\n$c_{2}$ are coupling constants. This form was first proposed in\n~\\cite{phys42} and it is a more general form than those found\nin~\\cite{phys40,phys43}, which can be obtained when $c_{1}=c_{2}=c$,\n$c_{1}=0$ or $c_{2}=0$.\n\n\\subsection{A.~~~~Stability Analysis}\n\nTo analyze the evolution of the dynamical system, we introduce the\nfollowing dimensionless variables:\n\\begin{equation}\nx\\equiv\\frac{\\kappa^{2}\\rho_{g}}{3H^{2}},~~~~~~~~y\\equiv\\frac{\\kappa^{2}p_{g}}{3H^{2}},~~~~~~~~z\\equiv\n\\frac{\\kappa^{2}\\rho_{m}}{3H^{2}}.\n\\end{equation}\nAccordingly, the density of the baryonic matter is determined by the\nFriedmann constraint (6) as\n\\begin{equation}\n\\frac{\\kappa^{2}\\rho_{b}}{3H^{2}}=1-x-z,\n\\end{equation}\nwhich implies that $0\\leq x+z\\leq1$ and $0\\leq x,z\\leq1$.\nFurthermore, using these variables, the EoSs of the MCG and the\ntotal cosmic fluid are respectively given by\n\\begin{eqnarray}\n&&w_{g}=\\frac{p_{g}}{\\rho_{g}}=\\frac{y}{x},\\\\\n&&w=\\frac{p}{\\rho}=\\frac{p_{g}}{\\rho_{g}+\\rho_{m}+\\rho_{b}}=y.\n\\end{eqnarray}\nThe sound velocity and the deceleration parameter respectively read\n\\begin{eqnarray}\n&&c^{2}_{s}=\\frac{\\partial\np_{g}}{\\partial\\rho_{g}}=-\\alpha\\frac{y}{x}+(1+\\alpha)A,\\\\\n&&q=-\\frac{\\ddot{a}a}{\\dot{a}^{2}}=-1+\\frac{3}{2}(1+y).\n\\end{eqnarray}\nNote that the condition for acceleration is $w<-\\frac{1}{3}$ and the\nphysically meaningful range of the sound velocity is $0\\leq\nc^{2}_{s}<1$. Using Eqs. (6)-(11), we can obtain the following\nautonomous system:\n\\begin{eqnarray}\nx^{'}&=&-3[(1+c_{2})x+y+c_{1}z]+3x(1+y),\\\\\ny^{'}&=&-3[(1+c_{2})x+y+c_{1}z][-\\alpha\\frac{y}{x}+(1+\\alpha)A]+3y(1+y),\\\\\nz^{'}&=&-3[-c_{2}x+(1-c_{1})z]+3z(1+y),\n\\end{eqnarray}\nwhere the prime denotes a derivative with respect to $N\\equiv lna$.\nWe set the current scale factor by $a_{0}=1$. Then the current value\nof $N$ reads $N_{0}=0$. Setting $x^{'}=y^{'}=z^{'}=0$, we can obtain\nthe critical points $(x_{*},y_{*},z_{*})$ of the autonomous system\nas follows:\n\\begin{itemize}\n  \\item \\textbf{point $(a)$}:~~~\n($\\frac{A-c_{1}+c_{2}}{2A}(1+x_{s})$,~~$\\frac{A-c_{1}+c_{2}}{2}(1+x_{s})$,~~$1-\\frac{A-c_{1}+c_{2}}{2A}(1+x_{s})$),\\\\\n  \\item \\textbf{point $(b)$}:~~~\n($\\frac{A-c_{1}+c_{2}}{2A}(1-x_{s})$,~~$\\frac{A-c_{1}+c_{2}}{2}(1-x_{s})$,~~$1-\\frac{A-c_{1}+c_{2}}{2A}(1-x_{s})$),\\\\\n  \\item \\textbf{point $(c)$}:~~~\n($\\frac{1-c_{1}}{1-c_{1}+c_{2}}$,~~$-1$,~~$\\frac{c_{2}}{1-c_{1}+c_{2}}$).\n\\end{itemize}\nHere the parameter $x_{s}$ is defined by\n\\begin{equation}\nx_{s}=\\sqrt{1+\\frac{4Ac_{1}}{(A-c_{1}+c_{2})^{2}}}.\n\\end{equation}\nThus, we can constrain the parameters in the model under the\nphysically meaningful conditions, namely, $0<x_{*}\\leq1$ and $0\\leq\nc^{2}_{s*}<1$. Concretely, we analyze the existence conditions for\nthese three points respectively:\n\n$\\bullet$ \\textbf{points $(a)$ and $(b)$}: Considering the\nphysically meaningful range of $x_{*}=\\frac{A-c_{1}+c_{2}}{2A}(1\\pm\nx_{s})$, we can obtain that $0<\\frac{A-c_{1}+c_{2}}{2A}(1\\pm\nx_{s})\\leq1, A\\neq0$, and moreover, $x_{s}$ is real. Together with\nthe constraint of the sound velocity $c^{2}_{s*}=A$, namely, $0\\leq\nA<1$, we can conclude that the existence conditions of points $(a)$\nand $(b)$ are $c_{1}<0$, but $c_{1}\\neq -1$,\n$-(1-\\sqrt{-c_{1}})^{2}<c_{2}<0$ and\n$(\\sqrt{-c_{1}}+\\sqrt{-c_{2}})^{2}\\leq A<1$.\n\n$\\bullet$ \\textbf{point $(c)$}: According to the meaningful range of\n$x_{*}=\\frac{1-c_{1}}{1-c_{1}+c_{2}}$, we know that\n$0<\\frac{1-c_{1}}{1-c_{1}+c_{2}}\\leq1$, and therefore, $c{1}<1,\nc_{2}\\geq0$ or $c{1}>1, c_{2}\\leq0$. Furthermore, the constraint of\nthe sound velocity $c^{2}_{s*}$ implies that the range of $A$ is\n$-\\frac{\\alpha}{1+\\alpha}\\frac{1-c_{1}+c_{2}}{1-c_{1}}\\leq\nA<\\frac{1}{1+\\alpha}(1-\\alpha\\frac{1-c_{1}+c_{2}}{1-c_{1}})$. Thus,\nthe existence conditions of point $(c)$ are obtained. Results from\nour above analysis are concluded in Table I.\n\n\n\\begin{table}[tbp]\n\\begin{center}\n\\begin{tabular}{|c||c|c|c|c|c|}\n  \\toprule\n \n  Point & Existence & Eigenvalues & Stability & $w_{*}$ & Acceleration\n\\\\\\hline\\hline\n  $~$ & $c_{1}<0$, but $c_{1}\\neq -1$ & $\\lambda_{1}=3(A-c_{1}+c_{2})x_{s}$, & ~ & ~ &\n  ~\n  \\\\\n  $(a)$ & $-(1-\\sqrt{-c_{1}})^{2}<c_{2}<0$ & $\\lambda_{2}=\\frac{3}{2}(A-c_{1}+c_{2})(1+x_{s})$, & Unstable &\n  $\\frac{A-c_{1}+c_{2}}{2}(1+x_{s})$ &\n  No\n  \\\\\n  $~$ & $(\\sqrt{-c_{1}}+\\sqrt{-c_{2}})^{2}\\leq A<1$ & $\\lambda_{3}=3(1+\\alpha)[1+\\frac{(A-c_{1}+c_{2})}{2}(1+x_{s})]$\n  & ~ & ~ &\n  ~\n  \\\\\\hline\n  $~$ & $c_{1}<0$, but $c_{1}\\neq -1$ & $\\lambda_{1}=-3(A-c_{1}+c_{2})x_{s}$, & ~ & ~ &\n  ~\n  \\\\\n  $(b)$ & $-(1-\\sqrt{-c_{1}})^{2}<c_{2}<0$ & $\\lambda_{2}=\\frac{3}{2}(A-c_{1}+c_{2})(1-x_{s})$, & Saddle &\n  $\\frac{A-c_{1}+c_{2}}{2}(1-x_{s})$ &\n  No\n  \\\\\n  $~$ & $(\\sqrt{-c_{1}}+\\sqrt{-c_{2}})^{2}\\leq A<1$ & $\\lambda_{3}=3(1+\\alpha)[1+\\frac{(A-c_{1}+c_{2})}{2}(1-x_{s})]$\n  & ~ & ~ &\n  ~\n  \\\\\\hline\n  $~$ & ~ & ~ & Stable if & ~ &\n  ~\n  \\\\\n  $~$ & ~ & ~ & $c_{1}<1,c_{2}\\geq 0$ and & ~ &\n  ~\n  \\\\\n  $~$ & $c_{1}<1,c_{2}\\geq 0$ or & $\\lambda_{1}=-3$, & $-\\frac{\\alpha}{1+\\alpha}\\frac{1-c_{1}+c_{2}}{1-c_{1}}\\leq A$\n  & ~ &\n  ~\n  \\\\\n  $(c)$ & $c_{1}>1,c_{2}\\leq 0$ & $\\lambda_{2,3}=-\\frac{3}{2(c_{1}-1)}\\{-\\alpha (1-c_{1}+c_{2})$ &\n  $<\\frac{1}{1+\\alpha}(1-\\alpha\\frac{1-c_{1}+c_{2}}{1-c_{1}})$ & $-1$ &\n  All\n  \\\\\n  $~$ & $-\\frac{\\alpha}{1+\\alpha}\\frac{1-c_{1}+c_{2}}{1-c_{1}}\\leq A$ & $-(1-c_{1})[2-c_{1}+c_{2}+(1+\\alpha)A]\n  \\pm \\lambda_{s}\\}$ & Saddle if & ~ &\n  $\\alpha, A, c_{1}$ and $c_{2}$\n  \\\\\n  $~$ & $<\\frac{1}{1+\\alpha}(1-\\alpha\\frac{1-c_{1}+c_{2}}{1-c_{1}})$ & ~ & $c_{1}>1,c_{2}\\leq 0$ and & ~ &\n  ~\n  \\\\\n  $~$ & ~ & ~ & $-\\frac{\\alpha}{1+\\alpha}\\frac{1-c_{1}+c_{2}}{1-c_{1}}\\leq A$ & ~ &\n  ~\n  \\\\\n  $~$ & ~ & ~ & $<\\frac{1}{1+\\alpha}(1-\\alpha\\frac{1-c_{1}+c_{2}}{1-c_{1}})$ & ~ &\n  ~\n  \\\\\n  \\botrule\n\\end{tabular}\n\\end{center}\n\\caption{The properties of the critical points for the interacting\nmodified Chaplygin gas model. Here, the parameter $\\lambda_{s}$ is\ndefined in Eq. (25).}\n\\end{table}\n\n\n\nTo study the stability of the critical points for the autonomous\nsystem, we substitute linear perturbations $x\\rightarrow\nx_{*}+\\delta x$, $y\\rightarrow y_{*}+\\delta y$ and $z\\rightarrow\nz_{*}+\\delta z$ about the critical points into the autonomous system\nEqs. (18)-(20). To first-order in the perturbations, one gets the\nfollowing evolution equations of the linear perturbations:\n\\begin{eqnarray}\n\\delta x^{'}&=&3(y_{*}-c_{2})\\delta x+3(x_{*}-1)\\delta y-3c_{1}\\delta z,\\\\\n\\delta y^{'}&=&-3[\\alpha \\frac{y_{*}}{x_{*}}(\\frac{y_{*}}{x_{*}}+c_{1}\\frac{z_{*}}{x_{*}})+(1+c_{2})(1+\\alpha)A]\n\\delta x \\nonumber\\\\\n&&+3[1-(1+\\alpha)A+(1+c_{2})\\alpha+2y_{*}+\\alpha (2\\frac{y_{*}}{x_{*}}+c_{1}\\frac{z_{*}}{x_{*}})]\\delta y-3c_{1}\n[-\\alpha \\frac{y_{*}}{x_{*}}+(1+\\alpha)A]\\delta z,\\\\\n\\delta z^{'}&=&3c_{2}\\delta x+3z_{*}\\delta y+3(y_{*}+c_{1})\\delta z.\n\\end{eqnarray}\nThe three eigenvalues of the coefficient matrix of Eqs. (22)-(24)\ndetermine the stability of the critical points. We list the three\neigenvalues for each point in Table I. Moreover, the condition for\nstability and acceleration (i.e., $w_{*}<-\\frac{1}{3}$) are also\nsummarized in the same table. For convenience, we introduce the\nparameter\n\\begin{equation}\n\\lambda_{s}=\\{[\\alpha\n(1-c_{1}+c_{2})+(1+\\alpha)(1-c_{1})A]^2+2(1-c_{1})(c_{1}+c_{2})[\\alpha\n(1-c_{1}+c_{2})+(1+\\alpha)(1-c_{1})A]+[(1-c_{1})(c_{1}-c_{2})]^2\\}^{\\frac{1}{2}}.\n\\end{equation}\nFrom Table I, we can clearly see that points $(a)$ and $(b)$ are not\nstable if they exist, and point $(c)$ is stable under conditions\n$c_{1}<1,c_{2}\\geq 0$ and\n$-\\frac{\\alpha}{1+\\alpha}\\frac{1-c_{1}+c_{2}}{1-c_{1}}\\leq\nA<\\frac{1}{1+\\alpha}(1-\\alpha\\frac{1-c_{1}+c_{2}}{1-c_{1}})$.\nFurthermore, the stable attractor is a scaling solution since the\nenergy density of the MCG remains proportional to that of cold dark\nmatter, $\\frac{\\rho_{g*}}{\\rho_{m*}}=\\frac{1-c_{1}}{c_{2}}$, when\n$c_{2}\\neq0$. Thus, point $(c)$ is an accelerated scaling solution\nthat probably alleviates the coincidence problem. Additionally, for\nthe stable attractor point $(c)$, the location of the point depends\nonly on the coupling constant $c_{1}$ and $c_{2}$. The EoS of the\ntotal cosmic fluid reads $w_{*}=-1$, which indicates that point\n$(c)$ is a accelerated attractor ($w_{*}<-\\frac{1}{3}$) and implies\nthat the universe will enter to a de sitter phase and accelerate\nforever, and therefore, there is no singularity in the finite future\nfor any parameters. However, the EoS of the MCG is expressed by\n$w_{g*}=-\\frac{1-c_{1}+c_{2}}{1-c_{1}}=-1-\\frac{c_{2}}{1-c_{1}}$,\nthat is to say, point $(c)$ has a phantom equation of state\n($w_{g*}<-1$) for $c_{1}<1,c_{2}\\geq 0$. According to Eq. (17), we\ncan obtain that $q_{*}=-1+\\frac{3}{2}(1+y_{*})=-1$ for the stable\nattractor point $(c)$, which also implies that the universe will\nenter to a de sitter phase and accelerate forever.\n\n\n\n\\begin{figure}[tbp]\n\\includegraphics[width=0.3\\textwidth]{c.EPS}\n\\caption{The stable regions in the $(\\alpha, A)$ parameter space.\nThe solid and dotted lines respectively denote the region for\n$c_{1}=c_{2}=0$ (the modified Chaplygin gas model without\ninteraction) or $c_{1}\\neq0,c_{2}=0$ (coupling between dark sectors\nonly proportional to the energy density of dark matter) and\n$c_{1}=c_{2}=0.2$. In the region I (the region between the two same\nlines), point $(c)$ is a stable accelerated attractor. The region II\nrepresents the region of solution without physical meaning. }\n\\label{Fig.1}\n\\end{figure}\n\n\n\n\n\\begin{figure}[tbp]\n\\includegraphics[width=0.3\\textwidth]{c1.EPS}%\n\\includegraphics[width=0.3\\textwidth]{c2.EPS}%\n\\includegraphics[width=0.3\\textwidth]{c3.EPS}\n\\caption{The stable regions in the parameter space for the fixed\n$c_{1}$ or $c_{2}$. The left two correspond to $c_{1}=0$ (coupling\nbetween dark sectors only proportional to the energy density of dark\nenergy) and $c_{1}=0.4$, respectively. The right is for\n$c_{2}=0.02$. The region between the two colorful planes represents\nthat point $(c)$ is a stable accelerated attractor. The other\nregions correspond to the regions of solution without physical\nmeaning. } \\label{Fig.2}\n\\end{figure}\n\n\n\\begin{figure}[tbp]\n\\includegraphics[width=0.3\\textwidth]{wgla.EPS}%\n\\includegraphics[width=0.3\\textwidth]{wga.EPS}\\\\\n\\includegraphics[width=0.3\\textwidth]{wgc1.EPS}%\n\\includegraphics[width=0.3\\textwidth]{wgc2.EPS}\n\\caption{The evolution of the EoSs of the modified Chaplygin gas\n$w_{g}$ for different values of various parameters. The top two\nplots correspond to the cases for different values of the modified\nChaplygin gas parameters $\\alpha$ or $A$, when $c_{1}= 0.2$ and\n$c_{2}=0.4$. In addition, we respectively fix the parameter $A=0.1$\nand $\\alpha=0.5$ in these two plots. The bottom two plots represent\nthe case for different values of the coupling constants $c_{1}$ or\n$c_{2}$, when $\\alpha=0.5$ and $A=0.1$. Moreover, in these two\nplots, the parameter $c_{2}$ and $c_{1}$ are fixed to be $c_{2}=0.4$\nand $c_{1}=0.2$ respectively.} \\label{Fig.3}\n\\end{figure}\n\n\n\n\n\\subsection{B.~~~~Numerical Results}\n\nIn what follows, we numerically study the dynamical results from the\nautonomous system (18)-(20) to clearly confirm the complicated\nstability condition for the stable accelerated attractor point\n$(c)$. In Figs.1 and 2, we depict the parameter space for point\n$(c)$ to be stable. We suppose that $c_{1}=c_{2}=c$ or\n$c_{1}\\neq0,c_{2}=0$ in Fig.1. From the figure, we see that the\nparameter space is independent of the coupling constant $c_{1}$ when\n$c_{2}=0$, namely, the stable region of the modified Chaplygin gas\nwithout interaction ($c_{1}=c_{2}=0$) is the same as that of the\ninteracting MCG, in which the interaction term is solely\nproportional to the energy density of dark matter\n($c_{1}\\neq0,c_{2}=0$). Fig.2 shows that the stable regions in the\nparameter space for the fixed $c_{1}$ or $c_{2}$. In the figure, the\nleft two plots is respectively for $c_{1}=0$ and $c_{1}=0.4$,\nmeanwhile, the range of $c_{2}$ is specially chosen as $0\\leq\nc_{2}<2$ to present the results more transparently though\n$c_{2}\\geq0$ is permitted. The right plot is shown by choosing\n$c_{2}=0.02$, and also the range of $c_{1}$ is selected as $-1\\leq\nc_{1}<1$ for clarity although its allowed range is $c_{1}<1$.\n\n\n\\begin{figure}[tbp]\n\\includegraphics[width=0.3\\textwidth]{wla.EPS}%\n\\includegraphics[width=0.3\\textwidth]{wa.EPS}\\\\\n\\includegraphics[width=0.3\\textwidth]{wc1.EPS}%\n\\includegraphics[width=0.3\\textwidth]{wc2.EPS}\n\\caption{The evolution of the EoSs of the total cosmic fluid $w$ for\ndifferent values of various parameters. In these four plots, we\nrespectively select the same parameters as those in Fig.3.}\n\\label{Fig.4}\n\\end{figure}\n\n\n\n\nIn Figs.3-4, we plot the evolution of the EoSs of the MCG $w_{g}$\nand the total cosmic fluid $w$ for different parameters in the\nstable region to illustrate the evolutional feature of our universe.\nMeanwhile, the initial condition in these figures is taken to be\n$x_{0}=\\Omega_{g0}=0.7$, $y_{0}=-0.7$ (we assume $w_{g0}=-1$) and\n$z_{0}=\\Omega_{mo}=0.26$ to compare with the observation data. As\nexample, we choose several sets of values for the parameters\n$\\alpha$, $A$, $c_{1}$ and $c_{2}$ for clarity. From Fig.3, we see\nthat in the final state the EoS of the MCG, $w_{g}$, tends to a\nconstant, which is only determined by the coupling constants $c_{1}$\nand $c_{2}$. Furthermore, the final EoS of the MCG, $w_{g*}$, is\nalways below $-1$ for different parameters in the stable region,\ni.e., the MCG has a phantom equation of state in the final state. In\nFig.4, we find that the final EoS of the total cosmic fluid $w_{*}$\ntends to $-1$, which is independent of any parameters in the stable\nregion. This indicates that the cosmic doomsday is avoided and the\nuniverse accelerates forever. Note that the evolution of $w$ is\nexotic when a set of parameters is chosen as $\\alpha=0.5$, $A=-0.1$,\n$c_{1}=0.2$ and $c_{2}=0.4$. In this case, the value of $w$ is\nsmaller than $-1$ at the high redshift, i.e., the interacting\nmodified Chaplygin gas model behave as phantom ($w<-1$) at the not\nvery early universe.\n\nAs the most significant parameter from the viewpoint of\nobservations, the deceleration parameter $q$ is also discussed. We\nexhibit the evolution of the deceleration parameter $q$ for various\nparameters in the stable region in Fig.5. The sets of values for all\nparameters in this model are selected as those in Fig.3-4. From the\nfigure, we find that in the final state the deceleration parameter\n$q$ tends to $-1$ for different values of various parameters. This\nfurther indicates the universe enters to a de Sitter phase in the\nfinal state. Moreover, varying any one of the MCG parameters\n$\\alpha$, $A$ and the coupling constants $c_{1}$, $c_{2}$, and\nfixing the rest, we find that the larger the variational parameter\nis, the smaller the transition redshift $z_{t}$ is, namely, the\nlater the transition from the deceleration phase to the acceleration\nphase is. In addition, it is worthwhile to note that when we fix the\nparameters $\\alpha=0.5$, $c_{1}=0.2$ and $c_{2}=0.4$, and select\n$A=-0.1$, there are two transition redshift: the first one denotes\nthe transition from the acceleration phase to the deceleration\nphase; the second one represents the transition point from the\ndeceleration phase to the acceleration phase. This provides the\npossibility that the evolution of our universe from acceleration to\ndeceleration and then from deceleration to acceleration.\n\n\n\n\\begin{figure}[tbp]\n\\includegraphics[width=0.3\\textwidth]{qla.EPS}%\n\\includegraphics[width=0.3\\textwidth]{qa.EPS}\\\\\n\\includegraphics[width=0.3\\textwidth]{qc1.EPS}%\n\\includegraphics[width=0.3\\textwidth]{qc2.EPS}\n\\caption{The evolution of the deceleration parameter $q$ for\ndifferent values of various parameters. Also, the parameters in\nthese four plots are respectively chosen as those in Fig.3 for\nclarity.} \\label{Fig.5}\n\\end{figure}\n\n\n\n\n\n\n\\begin{figure}[tbp]\n\\includegraphics[width=0.3\\textwidth]{rsc1.EPS}%\n\\includegraphics[width=0.3\\textwidth]{rsc2.EPS}%\n\\includegraphics[width=0.3\\textwidth]{rsc22.EPS}\n\\caption{The evolution of the statefinder for the interacting MCG\nmodel in the $s-r$ plane with different coupling constants $c_{1}$\nand $c_{2}$, which is illustrated in the left one and the right two\npanels, respectively. To be clarity, we extract a part of the middle\npanel to be the right one. In these three plots, we fix the\nparameters $\\alpha=0.5$ and $A=-0.1$. The solid points show the\ncurrent values of the statefinder parameters $(s_{0},r_{0})$. }\n\\label{Fig.6}\n\\end{figure}\n\n\n\n\n\n\n\\section{III.~~STATEFINDER DIAGNOSIS}\n\nIn this section, we study the dynamics of the interacting modified\nChaplygin gas model from the statefinder viewpoint. The so-called\nstatefinder parameter was first introduced by Sahni et\nal.~\\cite{phys44} in order to discriminate among more and more\ncosmological models. It is constructed from the scalar $a$ and its\nderivatives up to the third order. The statefinder pair $\\{r,s\\}$ is\ndefined as\n\\begin{equation}\nr\\equiv\\frac{\\dddot{a}}{aH^{3}},\\mbox{}\\hspace{15pt}s\\equiv\\frac{r-1}{3(q-1\/2)}.\n\\end{equation}\nSince different cosmological models exhibit qualitatively different\ntrajectories of evolution in the $s-r$ plane, the statefinder\nparameter is a good tool to distinguish cosmological models. It has\na remarkable property for the basic spatially flat $\\Lambda$CDM\nmodel and the matter dominated universe SCDM, i.e., the statefinder\npair $\\{r,s\\}$ for $\\Lambda$CDM model takes the constant value\n$(0,1)$, while SCDM model corresponds to the fixed point $(1,1)$. We\ncan clearly identify the ``distance'' from a given cosmological\nmodel to $\\Lambda$CDM model in the $s-r$ plane, such as the\nquintessence, the phantom, the Chaplygin gas, the holographic dark\nenergy models, the interacting dark energy models, and so forth,\nwhich have been shown in the literatures~\\cite{phys45}.\n\nGenerally, according to the reexpression of the deceleration\nparameter $q$\n\\begin{equation}\nq=-1-\\frac{\\dot{H}}{H},\n\\end{equation}\nwe can also rewrite the statefinder pair $\\{r,s\\}$ in terms of the\nHubble parameter $H$ and its first and second derivatives with\nrespect to the cosmic time $t$, $\\dot{H}$ and $\\ddot{H}$, as\n\\begin{eqnarray}\nr&=&1+3\\frac{\\dot{H}}{H^2}+\\frac{\\ddot{H}}{H^3},\\\\\ns&=-&\\frac{3\\frac{\\dot{H}}{H^2}+\\frac{\\ddot{H}}{H^3}}{3(\\frac{\\dot{H}}{H^2}+\\frac{3}{2})}.\n\\end{eqnarray}\nNow we apply the statefinder parameter to the interacting MCG model.\nBy using Eqs. (6)-(11), the statefiner parameter can be concretely\nexpressed as\n\\begin{eqnarray}\nr&=&1+\\frac{9}{2}[(1+c_{2})x+y+c_{1}z][-\\alpha\\frac{y}{x}+(1+\\alpha)A],\\\\\ns&=&\\frac{[(1+c_{2})x+y+c_{1}z][-\\alpha\\frac{y}{x}+(1+\\alpha)A]}{y}.\n\\end{eqnarray}\n\nThe evolution of statefinder for the interacting MCG model in the\n$s-r$ plane is plotted in Fig.6. The arrows in the figure denote the\nevolution directions of the statefinder trajectories. In this\nfigure, we also use the initial condition: $x_{0}=\\Omega_{g0}=0.7$,\n$y_{0}=-0.7$ (we assume $w_{g0}=-1$) and $z_{0}=\\Omega_{mo}=0.26$.\nSince the evolution trajectory in the $s-r$ plane will be\ninterrupted when $y=0$ (i.e.,$w=0$), which is not our favorite case,\nwe carefully choose the parameter $A=-0.1$ based on the model\nparameters selected before to avoid the interruption. This choice\ncan be seen from Fig.4, in which we can avoid the possibility of\n$w=0$ during its evolution only if $A\\geq0$ based on the model\nparameters we selected. So in Fig.6, we take $A=-0.1$ and\n$\\alpha=0.5$ as example to investigate the effect of the coupling\nconstants $c_{1}$ and $c_{2}$, and furthermore, to discriminate\nbetween the interacting MCG model and other dark energy models. From\nFig.6, we see that the evolution trajectories not only cross the\n$\\Lambda$CDM fixed point but also end at the fixed point for various\nvalues of $c_{1}$ and $c_{2}$. However, the $r(s)$ curve could not\ntraverse the SCDM fixed point for any case. In addition, in the left\nplot, the $r(s)$ curves are very different with various $c_{1}$ and\nthe fixed $c_{2}$, especially, the current values of $(s,r)$ are\ndifferent for various values of $c_{1}$. Also, we can clearly see\nthe tremendous difference between the current values $(s_{0},r_{0})$\nand the $\\Lambda$CDM fixed point, and the larger the value of\n$c_{1}$ is, the greater the ``distance'' from the current value\n$(s_{0},r_{0})$ to the fixed point is. On the other hand, fixing the\ncoupling constant $c_{1}$ and varying $c_{2}$ as illustrated in the\nright two plots of Fig.6, we find that the $r(s)$ curves are very\ndifferent from each other with various $c_{2}$, and the current\nvalues of $(s,r)$ are distinct for different values of $c_{2}$.\nAlso, we can clearly see the difference between the current values\n$(s_{0},r_{0})$ and the $\\Lambda$CDM fixed point, and the larger the\nvalue of $c_{2}$ is, the greater the ``distance'' from\n$(s_{0},r_{0})$ to the fixed point is. Therefore, we can conclude\nthat the statefinder diagnosis can not only discriminate the\ninteracting MCG model with different coupling constant but also\ndistinguish the interacting MCG model from other dark energy models.\n\n\n\n\n\n\\section{IV.~~Conclusions}\n\nIn previous sections, we have studied some physical properties of\nthe interacting MCG model. By considering an interaction term\nbetween the MCG and cold dark matter, we study the dynamical\nevolution of this model and pay our attention to the final state of\nthe universe. By our analysis, there exists a stable scaling\nsolution, which is characterized by a constant ratio of the energy\ndensities of the MCG and dark matter. This provides the possibility\nto alleviate the coincidence problem. Furthermore, we see that the\nfinal state is determined by the parameters of the MCG $\\alpha$, $A$\nand the coupling constants $c_{1}$ and $c_{2}$.\n\nInterestingly, we find that the EoS of the MCG, $w_{g}$, tends to a\nconstant, which is only determined by the coupling constants $c_{1}$\nand $c_{2}$. But both the EoS of the total cosmic fluid $w$ and the\ndeceleration parameter $q$ tend to $-1$, which are independent of\nthe choice of values for the parameters. This indicates that the\ncosmic doomsday is avoided and the universe enters to a de Sitter\nphase and thus accelerates forever. The transition from the\ndeceleration phase to the acceleration phase also depends on the MCG\nparameters $\\alpha$, $A$ and the coupling constants $c_{1}$,\n$c_{2}$. Varying one of these four parameters and fixing the rest,\nwe find that the the larger the variational parameter is, the\nsmaller the transition redshift $z_{t}$ is, namely, the later the\ntransition from the deceleration phase to the acceleration phase is.\nFurther, we study the effect of the coupling constants $c_{1}$ and\n$c_{2}$ on the dynamical evolution of this model from the\nstatefinder viewpoint. We clearly see that the coupling constants\n$c_{1}$ and $c_{2}$ play a significant role during the dynamics of\nthe interacting MCG model. The evolution trajectories of the\nstatefinder for the interacting MCG model not only cross the\n$\\Lambda$CDM fixed point but also end at the fixed point for various\nvalues of $c_{1}$ and $c_{2}$. However, the $r(s)$ curve could not\ntraverse the SCDM fixed point for any case. Moreover, fixing one of\nthe coupling constants $c_{1}$ and $c_{2}$, and varying the other\none, we find that the larger the variational parameter is, the\ngreater the ``distance'' from the current value $(s_{0},r_{0})$ to\nthe fixed point is. Thus, the statefinder parameter can discriminate\nthe interacting MCG model with different coupling constant. It is\nalso worthwhile to note that, we can distinguish this interacting\nmodel from other dark energy models in the $s-r$ plane.\n\n\n\\section{Acknowledgements}\n\nThis work is a part of project 10675019 supported by NSFC.\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction} \\label{1. Introduction}\n\nType Ia supernovae (SNe Ia) are used as distance indicators successfully to determine the cosmological parameters (e.g. Riess et al. 1998; Perlmutter et al. 1999) and play a key role in the chemical evolution of galaxies (e.g. Greggio \\& Renzini 1983; Matteucci \\& Greggio 1986). They are suggested to originate from thermonuclear runaway of carbon-oxygen white dwarfs (CO WDs) when the WDs grow in mass approaching the Chandrasekhar-mass limit (${M}_{\\rm Ch}$; e.g. Hoyle \\& Fowler 1960; Woosley, Taam \\& Weaver 1986).\n\nSo far, the single-degenerate model and the double-degenerate model are two popular progenitor models for the formation of SNe Ia, which have been discussed for many decades (e.g. Whelan \\& Iben 1973; Webbink 1984). In the single-degenerate model, a CO WD accretes H- or He-rich material from its non-degenerate companion (e.g. a main-sequence star, a red-giant star or a He star) and finally an SN Ia may be produced if the WD grows in mass close to ${M}_{\\rm Ch}$ (e.g. Hachisu, Kato \\& Nomoto 1996; Li \\& van den Heuvel 1997; Han \\& Podsiadlowski 2004; Wang 2018). In the double-degenerate model, an SN Ia may be produced if the total mass of the double WDs exceeds ${M}_{\\rm Ch}$ (e.g. Iben \\& Tutukov 1984).\n\nIn the single-degenerate model, CO WD+He star systems are thought to be one of the promising ways for producing SNe Ia with short delay times (e.g. Wang et al. 2009; Ruiter, Belczynski \\& Fryer 2009; Liu et al. 2010). The He-rich material is transferred from the companion star on to the surface of the CO WD, and burns into carbon and oxygen, resulting in the mass-increase of the WD. During the mass-transfer process, the mass-accretion rate ($\\dot{M}_{\\rm acc}$) is a key parameter for the evolution of CO WDs. The accumulated He-rich material can burn into carbon and oxygen stably if $\\dot{M}_{\\rm acc}$ is in the steady burning region. In this case, the CO WD can grow in mass steadily to ${M}_{\\rm Ch}$ and explodes as an SN Ia (e.g. Piersanti, Tornamb\\'{e}, \\& Yungelson 2014; Wang et al. 2015; Wu et al. 2016).\n\nWang, Podsiadlowski \\& Han (2017) found that there exists a critical mass-accretion rate ($\\dot{M}_{\\rm cr}$) in the steady burning region, and carbon under the He-shell ignites before the mass of the WD reaches ${M}_{\\rm Ch}$ if $\\dot{M}_{\\rm acc}>\\dot{M}_{\\rm cr}$ (see also Saio \\& Nomoto 1998; Brooks et al. 2016). Previous studies usually assumed that the inwardly propagating carbon flames transform the CO WDs into ONe cores, and subsequently the ONe cores may collapse into neutron stars (NSs) by the electron-capture of $^{\\rm 24}{\\rm Mg}$ and $^{\\rm 20}{\\rm Ne}$ if the ONe cores can increase their masses continuously (e.g. Saio \\& Nomoto 1985, 1998; Nomoto \\& Kondo 1991; Brooks et al. 2017; Liu \\& Li 2017). However, in the previous studies the inwardly propagating carbon flames did not reach the centre of the WDs, leading to some uncertainties on the final fates of the He-accreting CO WDs.\n\nIn this article, we aim to simulate the long-term evolution of off-centre carbon burning in He-accreting CO WDs and to investigate the final outcomes of the CO WDs. In Sect. 2, we provide our basic assumptions and methods for the numerical simulations. The results of our simulations are shown in Sect. 3. Finally, we present discussion and conclusions in Sect. 4.\n\n\\section{Numerical Methods}\\label{Methods}\n\nWe use the stellar evolution code \\texttt{MESA} (version 7624; see Paxton et al. 2011, 2013, 2015) to simulate the off-centre carbon burning in He-accreting CO WDs. The OPAL opacity is adopted (e.g. Iglesias \\& Rogers 1996). This is applicable to extra carbon and oxygen during the He burning. We adopted \\texttt{co\\_burn.net} as the nuclear reaction network in our simulations. This nuclear reaction network includes the isotopes needed for helium, carbon and oxygen burning (i.e., $^{\\rm 3}{\\rm He}$, $^{\\rm 4}{\\rm He}$, $^{\\rm 7}{\\rm Li}$, $^{\\rm 7}{\\rm Be}$, $^{\\rm 8}{\\rm B}$, $^{\\rm 12}{\\rm C}$, $^{\\rm 14}{\\rm N}$, $^{\\rm 15}{\\rm N}$, $^{\\rm 16}{\\rm O}$, $^{\\rm 19}{\\rm F}$, $^{\\rm 20}{\\rm Ne}$, $^{\\rm 23}{\\rm Na}$, $^{\\rm 24}{\\rm Mg}$, $^{\\rm 27}{\\rm Al}$ and $^{\\rm 28}{\\rm Si}$), which are coupled by 57 nuclear reactions.\n\nFirst, we construct a hot CO core with initial mass of ${M}_{\\rm WD}^{\\rm i}=0.9{M}_\\odot$. In our simulations, the hot CO core is built in idealized methods, in which the mass-fractions of $^{\\rm 12}{\\rm C}$ and $^{\\rm 16}{\\rm O}$ are both $50\\%$ (e.g. Saio \\& Nomoto 1998; Schwab, Quataert \\& Kasen 2016). During the formation process, all nuclear reactions are neglected, resulting in an uniform elemental abundance distribution of the core from the inside out. Then, we cool down the CO core for a given time of ${t}_{\\rm cool}=10^{6}\\,\\mbox{yr}$. At this moment, the initial WD model has been constructed, in which the effective temperature is ${\\rm {log}_{10}}({T}_{\\rm eff}\/{\\rm K})=5.367$ and the central temperature is ${\\rm {log}_{10}}({T}_{\\rm c}\/{\\rm K})=7.865$.\n\nSecondly, we simulate the long-term evolution of He-accreting CO WDs. The mass-fraction of He and metallicity of the accreted material is set to be 0.98 and 0.02, respectively. We use a constant mass-accretion rate of $\\dot{M}_{\\rm acc}=4\\times10^{-6}\\,{M}_\\odot\\,\\mbox{yr}^{-1}$ in our simulations, in which the WD undergoes stable He-shell burning. At beginning of the He-shell burning, the energy release rate is relatively high, resulting in a He-shell flash. We adopted the super-Eddington wind as the mass-loss mechanism during the first He-shell flash (see also Denissenkov et al. 2013a; Wu et al. 2017; Wu \\& Wang 2018). In the super-Eddington wind assumption, if the luminosity ($L_{\\rm eff}$) on the surface of the WD exceeds the Eddington luminosity ($L_{\\rm Edd}$), the super-Eddington wind would be triggered and blow away part of the accumulated material outside the CO core.\n\n\\section{Numerical Results}\\label{Results}\n\nFigs\\,1 to 3 show the results of off-centre carbon burning, in which ${M}_{\\rm WD}^{\\rm i}=0.9{M}_\\odot$ and $\\dot{M}_{\\rm acc}=4.0\\times10^{-6}\\,{M}_\\odot\\,\\mbox{yr}^{-1}$. The accreted He-rich material burns into carbon and oxygen, resulting in the mass-increase of the CO WD, and the surface of CO core under the He-shell is ignited when the core mass increases to about $1.277{M}_\\odot$.\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=0.7\\textwidth]{a.pdf}\n \\caption{Results of off-centre carbon burning at the moment when carbon under the He-shell ignites, in which ${M}_{\\rm WD}^{\\rm i}=0.9{M}_\\odot$ and $\\dot{M}_{\\rm acc}=4\\times10^{-6}\\,{M}_\\odot\\,\\mbox{yr}^{-1}$. Panel (a): the elemental abundance distribution profile. Panel (b): the density-temperature profile, in which the dashed line shows the separation of degenerate and non-degenerate regions (${\\varepsilon}_{\\rm F}\/{\\rm k}{T}\\approx4$, where ${\\varepsilon}_{\\rm F}$ is the Fermi energy), light blue curves indicates the convective region, the grey curve signifies the non-convective region, the blue, green and red curves show the energy production rates of nuclear reactions, respectively.}\n  \\end{center}\n\\end{figure}\n\nFig.\\,1 presents the abundance distribution and the density-temperature profile of the CO core at the moment when the carbon under the He-shell is ignited. The carbon burning produces an outwardly extending convective zone, leading to the expansion of the star. However, the convective zone does not extend to the outside convective zone produced by He-shell burning and subsequently the He-shell burning begins to quench. As the burning releases energy outwardly, the carbon burning zone shrinks gradually. At the same time, the off-centre carbon flame begins to propagate inward.\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=0.7\\textwidth]{b.pdf}\n \\caption{Similar to Fig.\\,1, but for the moment when the off-centre carbon burning flame reached the position where the mass-coordinate ${M}_{\\rm r}=1.0{M}_\\odot$.}\n  \\end{center}\n\\end{figure}\n\nOwing to the high temperature of the carbon flame, $^{\\rm 20}{\\rm Ne}$ is transformed into $^{\\rm 28}{\\rm Si}$ once it appears, resulting in an extremely thin neon flame (about $1\\,\\mbox{km}$ thick). Fig.\\,2 shows the abundance distribution and the density-temperature profile of the CO core when the carbon burning flame reaches the position where the mass-coordinate ${M}_{\\rm r}\\approx1{M}_\\odot$. At this moment, the abundance of $^{\\rm 16}{\\rm O}$ and $^{\\rm 28}{\\rm Si}$ near the flame are almost the same.\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=0.7\\textwidth]{c.pdf}\n \\caption{Similar to Fig.\\,1, but for the moment after the off-centre carbon burning flame reached the centre. Panel (c): The kippenhahn diagram during the carbon flame propagating phase, in which x-axis represents evolution time and y-axis is mass-coordinate.}\n  \\end{center}\n\\end{figure}\n\nAs the carbon burning flame keeps propagating inward, the density of the burning region becomes higher, and the nuclear reaction rate increases at the same time, resulting in more $^{\\rm 28}{\\rm Si}$ produced. In Fig.\\,3, we present the abundance distribution (panel a) and the density-temperature profile (panel b) of the CO core when the carbon flame reached the centre. From this figure, we can see that the core is transformed into an OSi core which is surrounded by an ONe-shell and a thin He-rich envelope. It takes about one hundred years for the carbon flame to propagate from the surface to the centre. The dynamical time-scale of the WD is $\\tau_{\\rm dyn}\\approx({R}^{3}\/{\\rm G}M)^{0.5}\\approx2.7\\,\\mbox{s}$ in this example, whereas the flame propagating time-scale ${\\tau}_{\\rm f}\\approx100\\,\\mbox{yr}$. This indicates that ${\\tau}_{\\rm f}\\gg{\\tau}_{\\rm dyn}$ and the flame propagating process does not require special hydrodynamic treatment. The Kippenhahn diagram during the flame propagating phase is shown in panel (c). From this figure we can see that the carbon flame moves inwardly in an unstable way, resulting in the fluctuations of the elemental abundance distributions of carbon, neon and magnesium in panel (a). After the burning front reaches the centre, the carbon flame quenches gradually and the OSi core begins to cool down at the same time. On the surface of the OSi core, the He-shell is reignited owing to the continuation of mass-accretion. As a result, the OSi core reaccumulates a CO-shell until next carbon ignition occurs. The final outcomes of the CO core can be divided into two cases as follows: (1) If the OSi core is not massive enough to trigger an off-centre silicon burning, the core would cool down to form a massive OSi WD. This implies a new class of WDs may exist in the Universe. (2) If the WD accretes material continuously, the off-centre silicon ignition may be triggered and transforms silicon into iron. Finally, a NS would be formed through the iron-core-collapse SN (Fe-CCSN), likely relating to the formation of some faint SNe. The present work may increase the birthrates of OSi WDs and Fe-CCSNe in the Universe.\n\n\\section{Discussion and conclusions}\\label{Discussion and conclusions}\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=0.7\\textwidth]{d.pdf}\n \\caption{Similar to panel (a) of Fig.\\,2, but for the increase of temporal and spatial resolution.}\n  \\end{center}\n\\end{figure}\n\nIt is difficult to resolve the flame that is propagating inside the star, and the computational results may be sensitive to the temporal and spatial resolution. We limited the time-step of each model and increased the mass of zones by decreasing the value of ``varcontrol targer'' and ``mesh delta coeff''. We made several convergence tests to confirm the fact that our results are independent of the choice of time-step and zoning. In Fig.\\,4, we replot the elemental abundance distribution profile when the carbon burning flame propagates to the position where the mass-coordinate ${M}_{\\rm r}=1.0{M}_\\odot$ (similar to panel (a) of Fig.\\,2) by both limiting the time-step and increasing the number of mass zone. In this case, the number of mass zone is increased from about $1700$ to more than $3500$. From this figure, we can see that the elemental abundances are similar to those in Fig.\\,2. Our tests indicate that the present results are indeed numerically converged.\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=0.7\\textwidth]{e.pdf}\n \\caption{Results of off-centre carbon burning, in which ${M}_{\\rm WD}^{\\rm i}=0.9{M}_\\odot$ and $\\dot{M}_{\\rm acc}=3\\times10^{-6}\\,{M}_\\odot\\,\\mbox{yr}^{-1}$. In this case, the inwardly propagating carbon flame reaches the position where ${M}_{\\rm r}=0.1{M}_\\odot$. Panel (a): the elemental abundance distribution profile. Panel (b): the density-temperature profile.}\n  \\end{center}\n\\end{figure}\n\nIn our simulations, we set the mass-accretion rate as $4\\times10^{-6}\\,{M}_\\odot\\,\\mbox{yr}^{-1}$. However, $\\dot{M}_{\\rm acc}$ may have influence on the final outcomes of the He-accreting WDs. Here, we use ${M}_{\\rm WD}^{\\rm i}=0.9{M}_\\odot$ and $\\dot{M}_{\\rm acc}=3\\times10^{-6}\\,{M}_\\odot\\,\\mbox{yr}^{-1}$ as a comparison. In Fig.\\,5, we present the abundance distribution and the density-temperature profile of the $0.9{M}_\\odot$ CO WD when the inwardly propagating carbon flame reached the position where the mass-coordinate ${M}_{\\rm r}=0.1{M}_\\odot$. The WD increases its mass to $1.323{M}_\\odot$ when the off-centre carbon is ignited, in which the mass of the core is higher than the case in Sect.\\,3. From this figure, we can see that the mass-fraction of $^{\\rm 28}{\\rm Si}$ produced by carbon and neon burning becomes higher as $\\dot{M}_{\\rm acc}$ decreases. This is because the degeneracy of the CO-shell grows as $\\dot{M}_{\\rm acc}$ decreases, resulting in the higher temperature of burning front. In this case, we stop our calculation when the burning wave reaches the position where ${M}_{\\rm r}=0.05{M}_\\odot$, because the calculation becomes extremely time-consuming when the flame propagates to the innermost part of the core. The inwardly propagating carbon flame would finally reach the centre, transforming the CO WD into a Si core. The final fate of the core would be similar to the case in Sect.\\,3. For the lower mass-accretion rate (e.g. we simulated the evolution of the accreting WD by using $\\dot{M}_{\\rm acc}=2\\times10^{-6}\\,{M}_\\odot\\,\\mbox{yr}^{-1}$), a rapidly outwardly extending convective zone caused by carbon burning would extend to the He-burning shell once the off-centre carbon is ignited, resulting in a large energy release. Subsequently, the gravitational binding energy may not bound such a high energy generation rate (the surface velocity in such a simulation becomes extremely high), likely triggering a dynamical process along with a strong mass-loss on the surface of the WD. To understand the final outcomes of the He-accreting WDs under this condition, further dynamical simulations are needed.\n\nBesides the mass-accretion rate, the initial mass of the WD may also be important for the final fates of He-accreting CO WDs. Kato, Saio \\& Hachisu (2018) investigated the evolution of He-accreting CO WDs that experience He-flashes. They found that the mass-fraction of $^{\\rm 28}{\\rm Si}$ becomes obviously higher in He-burning ashes for more massive initial WDs. For the present work, the mass of initial WD may also influence the evolution of off-centre carbon burning. For a massive WD, the temperature of accumulated He-shell is higher owing to the high surface gravitational acceleration, leading to a thinner He-shell when carbon is ignited. The inwardly propagating carbon flame takes more time to reach the centre and the mass-fraction of $^{\\rm 28}{\\rm Si}$ may become higher in carbon burning ashes owing to the high density of the core.\n\nIn our simulations, we did not consider the convective boundary mixing. Denissenkov et al. (2013a,b) investigated the inwardly propagating carbon flame in the CO cores by considering a convective boundary mixing assumption, and found that the mixing effect can prevent the carbon flames from reaching the centre, resulting in the formation of hybrid CONe WDs (see also Farmer, Fields \\& Timmes 2015). If the convective boundary mixing is considered in our simulations, the final structure of the CO WD may be a COSi core surrounded by an ONe-shell and a thin He-rich envelope.\n\nHD 49798\/RX J0648.0-4418 (e.g. Mereghetti et al. 2011) is a binary system consisting of a H depleted subdwarf star and a compact companion. Recent studies suggested that the compact star in this system is a WD rather than a NS (e.g. Mereghetti et al. 2013; Liu et al. 2015, 2018). The WD (${M}_{\\rm WD}=1.28\\pm0.05{M}_\\odot$) is accreting He-rich material from its companion (${M}_{\\rm He}=1.50\\pm0.05{M}_\\odot$) through the stellar wind at a rate of $10^{-11}<\\dot{M}_{\\rm acc}<10^{-10}\\,{M}_\\odot\\,\\mbox{yr}^{-1}$ (e.g. Bisscheroux et al. 1997; Mereghetti et al. 2009). Wang \\& Han (2010) estimated that the He star will fill its Roche lobe in about $4\\times10^{4}\\,\\mbox{yr}$, and the WD will grow to ${M}_{\\rm Ch}$. Brooks et al. (2017) simulated the binary evolution of ONe WD+He star system, in which $1.1\\leq{M}_{\\rm WD}\\leq1.3{M}_\\odot$, ${M}_{\\rm He}=1.5{M}_\\odot$ and $3\\,\\mbox{hr}\\leq{P}_{\\rm orb}\\leq8\\,\\mbox{d}$, and found that the mass-accretion rate of the WD is higher than $3\\times10^{-6}\\,{M}_\\odot\\,\\mbox{yr}^{-1}$ when the He donor fills its Roche lobe. According to our present work, RX J0648.0-4418 may experience off-centre carbon burning and then form a NS through Fe-CCSN if the compact star is a CO WD. Thus, we suggest that the final fate of HD 49798\/RX J0648.0-4418 will be a NS system whether the primary star is a WD (through Fe-CCSN if it is a CO WD or through electron capture SN if it is a ONe WD) or a NS (see also Brooks, Kupfer \\& Bildsten 2017).\n\nThe OSi WDs may also be produced in the double-degenerate model. Some recent studies suggested that the off-centre carbon burning would occur during the merging of two CO WDs, and the inwardly propagating flame may transform the CO WD into a Si core if the merge of double CO WDs satisfies some conditions. For example, the mass of the merger remnants is massive enough (see Schwab, Quataert \\& Kasen 2016) or the mass-accretion rate during the merger process is in a certain limits (see Wu, Wang \\& Liu 2019). Eventually, the Si core may cool down to form a Si WD if the off-centre Si burning does not occur. Alternatively, the Si core may form a NS through Fe-CCSN if the core mass is high enough to form an iron core.\n\n\nBy employing the stellar evolution code MESA, we investigated the long-term evolution of off-centre carbon burning of CO WDs by accreting He-rich material. We found that the off-centre carbon burning transforms CO WDs into OSi cores, in which the mass-fraction of Si is sensitive to the mass accretion rate, i.e. the lower $\\dot{M}_{\\rm acc}$ results in the higher mass-fraction of Si. The present work indicates that the CO WDs accreting He-rich material will form OSi WDs or experience Fe-CCSNe rather than ECSNe if the off-centre carbon ignition occurs. The present work increases the birthrates of OSi WDs and Fe-CCSNe in the Universe. In order to further understand the evolutions of He-accreting WDs, more CO WD+He star systems are hoped to be detected, and more numerical simulations on off-centre carbon burning are needed.\n\n\\section*{Acknowledgments}\n\nWe acknowledge the referee, Christopher Tout, for the valuable comments that helped us to improve the paper.\nThis study is supported by the National Natural Science Foundation of China (Nos 11873085, 11673059 and 11521303),\nthe Chinese Academy of Sciences (Nos KJZD-EW-M06-01 and QYZDB-SSW-SYS001),\nand the Natural Science Foundation of Yunnan Province (Nos 2018FB005 and 2017HC018).\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\hskip\\parindent\nKnown paradoxes in mathematics  \nare either considered issues of mathematical \nlogic \\cite{Bena}, or treated as deducible\nstatements which are counterintuitive but \ntrue \\cite{Czyz, Szek}, where ``true'' implies ``without contradiction''. \nTreating deducible statements as \nresults without contradiction is not grounded\non mathematical justification. Such treatment is usually based \non a popular belief, i.e., \nmathematics is free from contradiction as it is.\nThe belief is an issue of mathematical philosophy, \nwhich is not our concern.\nIn this paper, we show that a contradictory ``equation'' $1\/2 = 0$ \nis deducible in measure-theoretic probability, where\nthe two sides of the ``equation''\nrepresent probabilities. Unlike the paradoxes discussed in \\cite{Czyz, Szek},\nthe contradictory ``equation'' cannot be explained away and actually indicates that\nmeasure-theoretic probability is inconsistent.\n\nAfter a brief discussion about why measure-theoretic probability is very successful and\nyet inconsistent (Section \\ref{sec-2}), we review some key notions relevant to revealing the inconsistency, such as\ninadequacy of the $\\sigma$-algebra generated by coordinate variables of stochastic\nprocesses (Section \\ref{sec-3}), weak convergence and tightness of probability measures on the $\\sigma$-algebra\n${\\mathcal B}({\\mathbb R})$ of subsets of\nthe real line ${\\mathbb R}$ (Section \\ref{sec-4}), weak convergence of probability measures on the $\\sigma$-algebra \n${\\mathcal B}(\\overline{\\mathbb R})$ of subsets of\nthe extended real line $\\overline{\\mathbb R}$ (Section \\ref{sec-5}), and the difference between probabilities and\nnumbers in the closed unit interval $[0, 1]$ (Section \\ref{sec-6}), in particular, \nthe difference between limiting probabilities and limits of sequences \nin $[0, 1]$ (Section \\ref{sec-7}).\nBased on the above preparation, we demonstrate\nthe deducibility of the contradictory ``equation'' $1\/2 = 0$ (Section \\ref{sec-8}). \nWe conclude in Section \\ref{sec-9}.\n\n\\section{Measure-Theoretic Probability: Success and Inconsistency}\n\\label{sec-2}\n\\hskip\\parindent\nFor measure-theoretic probability, \nthe ultimate criterion of success \nis whether the probability of a real event (i.e., an event in the physical world)\npredicted by the theory\ncan be verified empirically. By this criterion, measure-theoretic\nprobability is very successful. And yet,\nmeasure-theoretic probability is \ninconsistent, as we shall demonstrate. \nHow can measure-theoretic probability \nbe both successful and inconsistent? A brief answer is as follows.\nAs a consequence of ``ideal events'',\nthe inconsistency of measure-theoretic probability \nappears only in the theory, but \nnot in the physical world. \nExpressed as an\ninfinite union or intersection of different\nevents of positive probability, an \n``ideal event'' is a result of infinitely repeated or\ninfinitely fine operations.\n\n\\begin{exam}\n\\label{ex-ideal1}\n{\\em\nAn absolutely accurate measurement of a physical\nquantity with its relative magnitude represented by an irrational number\nis an ideal event. Corresponding to picking up a point from a set with the\ncardinality of the\ncontinuum, such measurement requires\ninfinitely fine operations,\nand hence is not physically realizable. \n$\\Box$\n}\n\\end{exam}\n\n\nThe criterion of success of measure-theoretic probability \nimplies a constraint: Since ``ideal events'' do not exist physically, \nor since it is impossible to verify the\nexistence of ``ideal events'' in the physical world,\nphysical existence of theoretical properties derived by\nmanipulating\nprobabilities of ``ideal events''\nshould not be inferred from experiments.\nThis constraint serves as\na ``firewall'' of measure-theoretic probability. When the constraint is satisfied,\nit is safe to apply measure-theoretic probability to solve problems in the real world.\nIn practice, however,\nviolating the constraint can cause\nconfusion of real and ideal events.\nA typical violation is to use \na finite set of experimental data to ``prove'' or\n``verify''\nphysical existence of theoretical\nproperties obtained by calculating probabilities of ``ideal events''.\n \n\n\n\n\\section{Inadequacy of $\\sigma$-Algebra Generated by Coordinate Variables}\n\\label{sec-3}\n\\hskip\\parindent\nA real-valued random variable is usually described by the \nprobability measure (i.e., the distribution) induced by the random variable on the real line ${\\mathbb R}$.\nIn general, a stochastic process is usually described by the probability measures \n(i.e., the finite-dimensional distributions) induced by the process in $k$-dimensional\nEuclidean spaces ${\\mathbb R}^k, k = 1, 2, \\cdots$. \nThe construction of a process in measure-theoretic probability begins with a given\nsystem of finite-dimensional distributions. However, the system of\nfinite-dimensional distributions may not completely capture the\nproperties of a process \\cite{Bill}.\n\nA stochastic process is a collection of real-valued random variables labeled by an\nindex variable taking values from a set $T$. The points in the set $T$ usually represent time.\nWe consider only $T=\\{1, 2, \\cdots\\}$. Denote by\n${\\mathbb R}^T$ the set of all real sequences.\nFor each $n\\in T$, define a mapping $W_n: {\\mathbb R}^T\\to {\\mathbb R}$ by\n\\[\nW_n(\\omega) = \\omega(n) = \\omega_n.\n\\]\nFor a given $n\\in T$, the mapping $W_n$ (also referred to as a coordinate function)\nmaps a real sequence $\\omega\\in {\\mathbb R}^T$ to the $n$th coordinate $\\omega(n) = \\omega_n$ of $\\omega$. \nWe may also write $W_n(\\omega)$ as $W(n, \\omega)$. For a fixed $\\omega$, \n$W(\\cdot, \\omega) = \\omega(\\cdot)$ is just $\\omega$ itself.\nDenote by ${\\mathcal B}({\\mathbb R}^T)$ \nthe $\\sigma$-algebra (also referred to as $\\sigma$-field)\ngenerated by all the coordinate functions $W_n, n\\in T$:\n${\\mathcal B}({\\mathbb R}^T) = \\sigma\\{W_n: n\\in T\\}$, which is the intersection of all the $\\sigma$-algebras\ncontaining the sets of the form \n\\[\n\\{\\omega\\in {\\mathbb R}^T: W_n(\\omega)\\in H\\} = \\{\\omega\\in {\\mathbb R}^T: \\omega_n\\in H\\}\n\\]\nfor $n\\in T$ and $H\\in {\\mathcal B}({\\mathbb R})$. In other words,\n${\\mathcal B}({\\mathbb R}^T)$ is the ``smallest'' class of the sets which is closed under the finite and countable\nset-theoretic operations.\n\nLet ${\\mathbb R}^T$ be a sample space represented by $\\Omega$, and\nlet a probability measure be\ndefined on ${\\mathcal B}({\\mathbb R}^T)$. \nNow $\\omega\\in \\Omega = {\\mathbb R}^T$ represents a sample point, and $W_n$ can be interpreted as\nthe $n$th coordinate variable $U_n$ of a random sequence\n$(U_n)_{n\\geq 1}$. \nA realization of $(U_n)_{n\\geq 1}$ at a given sample point $\\omega$ is\na real sequence $(U_n(\\omega))_{n\\geq 1}$, which can also be expressed as\n$U(\\cdot, \\omega)$ and is just $\\omega$ itself. \n\nIf $(U_n)_{n\\geq 1}$ is a non-decreasing random sequence, \n$U_n(\\omega)$ may tend to infinity as $n\\to\\infty$ at\nsome $\\omega$. Since\n$\\{\\infty\\}\\not\\in {\\mathcal B}({\\mathbb R}^T)$, \na probability measure\non the measurable space \n$({\\mathbb R}^T, {\\mathcal B}({\\mathbb R}^T))$ \nis not useful\nto describe the limiting behavior of $(U_n)_{n\\geq 1}$.\nClearly, to define the limiting random variable of \n$(U_n)_{n\\geq 1}$ on a probability space, the\nlimiting random variable should be measurable with respect\nto the corresponding probability measure,\nwhich requires the values of the limiting random variable to\nform a measurable set on the probability space.\nFortunately, \nthe probability measures \n$\\rho_n$ induced by $U_n$ constitute a sequence $(\\rho_n)_{n\\geq 1}$\nsuch that,\ncorresponding to $(U_n)_{n\\geq 1}$ and $(\\rho_n)_{n\\geq 1}$, \na limiting random variable $U$ and a\nlimiting probability measure $\\rho$ induced by $U$ exist on the\nextended real line $\\overline{\\mathbb R}$ and the corresponding $\\sigma$-algebra\n${\\mathcal B}(\\overline{\\mathbb R})$, respectively.\nFor a \nconvergent sequence of probability\nmeasures, the corresponding  \nrandom variables can be on different probability spaces \\cite{Bill2}. Consequently,\nconvergence of probability measures does not require\nthe involved random variables to form a convergent\nsequence on the same probability space. Convergence of probability measures on ${\\mathcal B}(\\overline{\\mathbb R})$ is a generalization\nof convergence of probability measures on \n${\\mathcal B}({\\mathbb R})$. If a sequence of probability measures converges on ${\\mathcal B}({\\mathbb R})$, the sequence is necessarily\ntight on ${\\mathcal B}({\\mathbb R})$. We shall review these notions\nin the next two sections. \n\n\n\n\\section{Weak Convergence and Tightness of Probability Measures on \n${\\mathcal B}({\\mathbb R})$}\n\\label{sec-4}\n\\hskip\\parindent\nLet $(\\rho_n)_{n\\geq 1}$ be a sequence of probability measures on ${\\mathcal B}({\\mathbb R})$. The probability measures $\\rho_n$ are distributions induced by \nreal-valued random variables $U_n$, such that\n$U_n$ converge in distribution to a limiting random variable $U$\nas $n\\to\\infty$ (notation $U_n\\Rightarrow U$).\nEquivalently,\nthe probability measures $\\rho_n$\nconverge weakly to\na limiting probability measure $\\rho$\ninduced by $U$ (notation $\\rho_n\\Rightarrow\\rho$). \nIf $(U_n)_{n\\geq 1}$ is non-decreasing,\n$U$ may take $\\infty$\nas its value with positive probability, i.e., $\\rho$\nmay assign positive probability to\n$\\{\\infty\\}$. \nOne way to define convergence in distribution of\na sequence of real random variables is by convergence of the corresponding\ndistribution functions at each continuous point of the limiting distribution function,\nwhich may not necessarily be a distribution function of a real random variable \\cite{Loev}.\nThis definition has some disadvantages, for example, see \\cite{Poll}.\n \nAn equivalent definition of weak convergence of \nprobability measures (i.e., convergence in distribution of \nthe corresponding random variables)\nis as follows \\cite{Bill2,Poll}.\nSuppose ${\\mathbb R}$ is the common range for all the random variables involved.\nThe sequence $(\\rho_n)_{n\\geq 1}$ of probability measures induced by the\nrandom variables converges weakly to a\nprobability measure $\\rho$, if and only if for each $f\\in {\\mathbb C}_b({\\mathbb R})$,\n\\[\n\\int_{\\mathbb R}fd\\rho_n\\to \\int_{\\mathbb R}fd\\rho,\\; n\\to\\infty\n\\]\nwhere ${\\mathbb C}_b({\\mathbb R})$ is the class of bounded, continuous functions on ${\\mathbb R}$.\nClearly, such convergence is not only a property of\nthe probability measures $\\rho_n$ or corresponding random variables $U_n$, but also relevant to the common range space\nof the involved\nrandom variables on which weak convergence is defined. \nIf the common range space is defined to be ${\\mathbb R}$, and if the limiting random variable is not\nreal-valued, then weak convergence or convergence in distribution as defined above does not hold, even if\na limiting random variable and the corresponding probability measure exist on $\\overline{\\mathbb R}$ and \n${\\mathcal B}(\\overline{\\mathbb R})$, respectively.\n\n\\begin{exam}\n\\label{ex-Dirac}\n{\\em\nLet $(U_n)_{n\\geq 1}$ be a sequence of degenerate random variables such that\n$U_n = n$. The distribution, i.e., the probability measure $\\rho_n$ induced by $U_n$ is\nthe Dirac measure concentrated at $\\{n\\}$. As $n\\to\\infty$, \n$U_n\\Rightarrow U =\\infty\\in\\overline{\\mathbb R}$ and\n$\\rho_n\\Rightarrow\\rho$ where $\\rho$ is the Dirac measure concentrated on \n$\\{\\infty\\}\\in{\\mathcal B}(\\overline{\\mathbb R})$. But $(U_n)_{n\\geq 1}$ does not converge on ${\\mathbb R}$, and\n$(\\rho_n)_{n\\geq 1}$ does not converge weakly on ${\\mathcal B}({\\mathbb R})$. \n$\\Box$\n}\n\\end{exam}\n\n\nOn the one hand,\nas illustrated by the above example,\nfor a sequence of probability measures, mass may ``escape to infinity'' as $n\\to\\infty$,\nand meanwhile, in an extended sense,\nthe notion of\nlimiting probability measure remains meaningful. \nIn particular, for a non-decreasing random sequence such as $(U_n)_{n\\geq 1}$ in \nExample \\ref{ex-Dirac}, a unique\nlimiting random variable exists, which may not necessarily be\nreal-valued. By Helly's selection theorem (e.g., see \\cite{Bill}), the limiting random variable has \nunique distribution function and probability measure, such that the limiting distribution function may\nhave a lower bound larger than 0 or an upper bound less than 1, and the limiting probability measure $\\rho$ is on\n${\\mathcal B}(\\overline{\\mathbb R})$ and may be a subprobability measure when restricted to ${\\mathcal B}({\\mathbb R})$,\ni.e., $\\rho({\\mathbb R}) < 1$. The cause of $\\rho({\\mathbb R}) < 1$ is mass ``escaping to infinity''.\n\nOn the other hand, probability measures defined on ${\\mathcal B}({\\mathbb R})$ are particularly\nimportant for measure-theoretic probability and its applications.\nTo prevent mass from ``escaping to infinity'',\na condition must be imposed on a sequence of probability measures before we consider\nwhether the sequence converges to a limiting probability measure on ${\\mathcal B}({\\mathbb R})$.\n\nTightness of probability measures on ${\\mathcal B}({\\mathbb R})$ is the condition to prevent\nmass from ``escaping to infinity''.\nIf a sequence of probability measures\n$(\\rho_n)_{n\\geq 1}$ is tight on\n$({\\mathbb R}, {\\mathcal B}({\\mathbb R}))$, then\nfor each $\\epsilon > 0$, there is an interval $I_{\\epsilon}\\subset {\\mathbb R}$,\nsuch that $\\sup_n\\rho_n(I_{\\epsilon}^c) < \\epsilon$ where \n$I_{\\epsilon}^c = {\\mathbb R}\\setminus I_{\\epsilon}$. If such a tight sequence\nof probability measures $(\\rho_n)_{n\\geq 1}$ converges weakly to $\\rho$,\nthen the weak limit\n$\\rho$ is also a probability measure on $({\\mathbb R}, {\\mathcal B}({\\mathbb R}))$.\nGiven $f\\in {\\mathbb C}_b({\\mathbb R})$, even if $f(x)$ does not converge as $x\\to \\pm\\infty$,\nso long as $(\\rho_n)_{n\\geq 1}$ is tight on $({\\mathbb R}, {\\mathcal B}({\\mathbb R}))$\nand $\\rho_n\\Rightarrow\\rho$, we have\n$\\rho(\\{\\pm\\infty\\}) = 0$ and \n$\\int_{{\\mathbb R}} fd\\rho$ exists. \nSuppose $\\rho_n\\Rightarrow\\rho$. If not only all $\\rho_n$ but also $\\rho$ are\nrequired to be probability measures on $({\\mathbb R}, {\\mathcal B}({\\mathbb R}))$,\nthen $(\\rho_n)_{n\\geq 1}$ is necessarily tight on $({\\mathbb R}, {\\mathcal B}({\\mathbb R}))$.\n\n\\begin{exam}\n\\label{ex-tight}\n{\\em\nLet $\\lambda, \\lambda_n, n = 1, 2, \\cdots$ \nbe probability measures on \n$({\\mathbb R}, {\\mathcal B}({\\mathbb R}))$, \nsuch that $\\lambda, \\lambda_1, \\lambda_2, \\cdots$ are\ndistributions of i.i.d. random variables $X, X_1, X_2, \\cdots$, \nrespectively. The probability mass functions of $X$ and $X_n$ are the same:\n\n\\[\n\\lambda(\\{j\\}) =\\lambda_n(\\{j\\}) =  \\left\\{\n\\begin{array}{llr}\n1\/2, & j = 0\\\\\n1\/2, & j = 1\\\\\n0,   & \\mbox{otherwise.} \n\\end{array}\\right.\n\\]\nIf we define $\\lambda(\\{\\pm\\infty\\})=\\lambda_n(\\{\\pm\\infty\\})=0$, then $\\lambda, \\lambda_n, n = 1, 2, \\cdots$ \nare also probability measures on \n$(\\overline{{\\mathbb R}}, {\\mathcal B}(\\overline{\\mathbb R}))$ trivially. \nThe sequence $(\\lambda_n)_{n\\geq 1}$ is tight on $({\\mathbb R}, {\\mathcal B}({\\mathbb R}))$, and\nconverges weakly to $\\lambda$. Equivalently, $X_n$ converge in distribution to $X$.\n$\\Box$\n}\n\\end{exam}\n\n\n\\begin{exam}\n\\label{ex-notight}\n{\\em\nConsider a random sequence\n$(Z_n)_{n\\geq 1}$. \nDenote by $\\mu_n$ the probability measure (i.e., the distribution) induced by $Z_n$, and let \nthe probability mass function of $Z_n$ be\n\\begin{equation}\n\\label{eq-1}\n\\mu_n(\\{j\\}) = \\left\\{\n\\begin{array}{llr}\n2^{-(n - j + 1)}, & 1\\leq j < n\\\\\n2^{-1} + 2^{-n},  & j = n\\\\\n0,                & \\mbox{otherwise.} \n\\end{array}\\right.\n\\end{equation}\nFor each $n$, $Z_{n+1}$ takes larger values with larger probabilities.\nThe values of $Z_{n+1}$ include $n+1$ and all the values of $Z_n$.\n\nThe sequence $(\\mu_n)_{n\\geq 1}$ is not tight on $({\\mathbb R}, {\\mathcal B}({\\mathbb R}))$.\nTo see this,\nconsider $\\epsilon = 2^{-1}$. By (\\ref{eq-1}), \nfor all $n$, $\\mu_n(\\{n\\}) = 2^{-1} + 2^{-n} > 2^{-1}$.\nIn particular, for all real number $b > 0$ and all $n > N_b$, \n$\\mu_n(\\{n\\}) > 2^{-1}$, where $N_b= \\lceil b\\rceil$, and\n$\\lceil b\\rceil$ is the smallest integer larger than or equal to $b$.\nConsequently, for each interval $I_b=(-\\infty, b)$, \n$\\sup_n\\mu_n(I_b^c) > 2^{-1}$.\nThe probability mass function (\\ref{eq-1}) \nexplains not only why $(\\mu_n)_{n\\geq 1}$ fails to be tight on $({\\mathbb R}, {\\mathcal B}({\\mathbb R}))$\nbut also where all the mass goes as $n\\to\\infty$.\nFor a fixed $j$, $\\lim_n\\mu_n(\\{j\\})=0$. \nTo see what happens when $j$ increases with $n$, \nwrite $j = n - k$, where $0\\leq k < n$ is fixed.\nWe then have\n$\\mu_n(\\{n-k\\})=1\/2^{k+1}$ for $k > 0$ and \n$\\mu_n(\\{n\\})=2^{-1} + 2^{-n}$.\nSince $(n-k)\\to\\infty$ as $n\\to\\infty$,  \n$1\/2^{k+1}, k = 1, 2, \\cdots$ constitute one half\nof the mass that\n``escapes to infinity''. \nThe other half of the mass at $\\{\\infty\\}$ in the\nlimit as $n\\to\\infty$\ncomes from $\\mu_n(\\{n\\})$.\nLet a vertical line segment of\nlength $\\mu_n(\\{j\\})$ at $j$ represent\nthe mass assigned to $\\{j\\}$ by $\\mu_n$.\nAs $n\\to\\infty$, \nthe length of the segment at a fixed $j$ diminishes towards 0, the segment at $j = n - k$ with a fixed $0< k < n$\nmaintains an unchanged length $2^{-(k + 1)}$, and the length of the segment at $n$ is $2^{-1} + 2^{-n}$,\nwhich decreases gradually to 1\/2. \nFigure 1 illustrates the case for $k > 0$ with $n = 2$ and 3. \nThe total ``escaped'' mass then is\n$\\sum_{k\\geq 0}1\/2^{k+1} = 1$. \nActually, by Helly's theorem, each subsequence of $(\\mu_n)_{n\\geq 1}$ converges\nto a subprobability measure $\\mu$ on $({\\mathbb R}, {\\mathcal B}({\\mathbb R}))$ such that $\\mu({\\mathbb R}) = 0$,\nfor $\\mu({\\mathbb R}) > 0$ contradicts (\\ref{eq-1}). So $\\mu_n\\Rightarrow\\mu$ where \n$\\mu$ is the Dirac measure on \n$(\\overline{\\mathbb R}, {\\mathcal B}(\\overline{\\mathbb R}))$\nwith $\\mu(\\{\\infty\\}) = 1$. \nEquivalently, $Z_n\\Rightarrow Z$, where $Z$ is the\nextended random variable corresponding to $\\mu$.\nWhen restricted to $({\\mathbb R}, {\\mathcal B}({\\mathbb R}))$, $\\mu$ is\na zero measure. \n$\\Box$\n}\n\\end{exam}\n\nAs we shall see later, \n$(\\lambda_n)_{n\\geq 1}$ in Example \\ref{ex-tight} has an intrinsic connection to \n$(\\mu_n)_{n\\geq 1}$ in Example \\ref{ex-notight}, though the former is tight \non $({\\mathbb R}, {\\mathcal B}({\\mathbb R}))$ and the\nlatter is not. \n\n\n\n\\begin{figure*}\n\\setlength{\\unitlength}{0.4mm}\n\\begin{picture}(100,110)\n\n\\put(20,60){\\vector(1,0){100}}\n\\put(62,60){\\line(0,1){26}}\n\\put(62,93){\\makebox(0,0){$1\/4$}}\n\\put(50,52){\\makebox(0,0){$Z_2 = 1$}}\n\\put(128,60){\\makebox(0,0){$x$}}\n\n\\put(160,60){\\vector(1,0){100}}\n\\put(189,60){\\line(0,1){13}}\n\\put(230,60){\\line(0,1){26}}\n\\put(189,80){\\makebox(0,0){$1\/8$}}\n\\put(230,95){\\makebox(0,0){$1\/4$}}\n\\put(177,52){\\makebox(0,0){$Z_3=1$}}\n\\put(219,52){\\makebox(0,0){$Z_3=2$}}\n\\put(268,60){\\makebox(0,0){$x$}}\n\n\\put(295,60){\\vector(1,0){100}}\n\\put(350,52){\\makebox(0,0){$Z<\\infty$}}\n\\put(403,60){\\makebox(0,0){$x$}}\n\n\\put(210,25){\\makebox(0,0){Figure 1. \nFor any fixed $k\\geq 0$, mass at $\\{n-k\\}$ ``escapes\nto infinity'' as $n\\to\\infty$, so \n$\\mu({\\mathbb R})= 0$.}}\n\\end{picture}\n\\end{figure*}\n\n\n\n\n\\section{Weak Convergence of Probability Measures on \n${\\mathcal B}(\\overline{\\mathbb R})$}\n\\label{sec-5}\n\\hskip\\parindent\nThe definition of $\\rho_n\\Rightarrow\\rho$ can be generalized \nsuch that, for each $n$, $\\rho_n$ is still a probability measure on \n$({\\mathbb R}, {\\mathcal B}({\\mathbb R}))$, but $(\\rho_n)_{n\\geq 1}$ may not\nbe tight on $({\\mathbb R}, {\\mathcal B}({\\mathbb R}))$, and $\\rho$ is a probability measure\non $(\\overline{\\mathbb R}, {\\mathcal B}(\\overline{\\mathbb R}))$ with $\\rho(\\{\\pm\\infty\\}) > 0$ (see Example \\ref{ex-Dirac}).\nIn the above generalized definition of $\\rho_n\\Rightarrow\\rho$, ${\\mathbb C}_b({\\mathbb R})$ in the\noriginal definition is replaced by ${\\mathbb C}(\\overline{\\mathbb R})$, the class of continuous functions\non $\\overline{\\mathbb R}$. The reason for such replacement is shown\nby the example below. Let ${\\bf 1}_A$ represent the indicator function, i.e.,\n\\[\n\\funbi{{\\bf 1}_A(x) =}{1,}{$x\\in A$}\n{0,}{otherwise.}\n\\]\n\\begin{exam}\n\\label{ex-Dirac2}\n{\\em\nConsider $(\\rho_n)_{n\\geq 1}$ such that $\\rho_n$ is the Dirac measure\nconcentrated at $\\{n\\}$ as in Example \\ref{ex-Dirac}. The distribution function corresponding to $\\rho_n$ is\n${\\bf 1}_{[n, \\infty)}(\\cdot)$ with ${\\bf 1}_{[n, \\infty)}(x)\\to 0$ as $n\\to\\infty$\nfor $\\lim_n[n, \\infty)=\\cap_n[n, \\infty)=\\emptyset$. By Helly's selection theorem and\nthe generalized definition of $\\rho_n\\Rightarrow\\rho$, $\\rho$ is the Dirac\nmeasure concentrated at $\\{\\infty\\}$. However, for some $f\\in {\\mathbb C}_b({\\mathbb R})$ such as $f =\\sin$,\n$\\int_{{\\mathbb R}} fd\\rho_n = f(n) = \\sin(n)$, which does not converge \nas $n\\to\\infty$. \n$\\Box$ \n}\n\\end{exam}\n\nLet $\\overline{\\mathbb R}$ be equipped with a topology induced by a fixed metric.\nA function from one metric space to another is called a homeomorphism if the function\nis continuous, one-to-one, onto, and its inverse function is also continuous.\nIf such a function exists, then the two metric spaces are said to be homeomorphic.\nWith a homeomorphism from $\\overline{\\mathbb R}$ onto\n$[0, 1]$, ${\\mathbb C}(\\overline{\\mathbb R})$ can be identified with ${\\mathbb C}([0, 1])$,\nthe space of continuous functions on $[0, 1]$. \n\\begin{exam}\n\\label{ex-homeo}\n{\\em\nA homeomorphism $h: \\overline{\\mathbb R}\\to[0, 1]$ from $\\overline{\\mathbb R}$ onto\n$[0, 1]$ is\n\\[\nh(x) =  \\left\\{\n\\begin{array}{llr}\n0, & x = -\\infty\\\\\n1-\\pi^{-1}\\cot^{-1}(x), & x\\in{\\mathbb R}\\\\\n1,   & x=\\infty.\n\\end{array}\\right.\n\\]\nThe inverse of $h$ is $h^{-1}: [0, 1]\\to\\overline{\\mathbb R}$ given by\n\\[\nh^{-1}(y) =  \\left\\{\n\\begin{array}{llr}\n-\\infty, & y = 0\\\\\n\\cot(\\pi(1-y)), & y\\in(0, 1)\\\\\n\\infty,   & y=1.\n\\end{array}\\right.\n\\]\nA usual metric on $\\overline{\\mathbb R}$ is $d(x, y) = |\\tan^{-1}(x) - \\tan^{-1}(y)|$.\n$\\Box$\n}\n\\end{exam}\n\n\nEach $f\\in {\\mathbb C}([0, 1])$ has limits\nat the boundary points.\nWeak convergence of probability measures\non ${\\mathcal B}(\\overline{\\mathbb R})$ can then be identified with\nweak convergence of probability measures on ${\\mathcal B}([0, 1])$\n(the $\\sigma$-algebra of subsets of $[0, 1]$), and hence\na limiting probability measure \non ${\\mathcal B}([0, 1])$ which assigns mass to a boundary point of $[0, 1]$\nis identified with the corresponding limiting probability measure\non ${\\mathcal B}(\\overline{\\mathbb R})$ which assigns the same amount of mass to\n$\\{\\infty\\}$. Since the probability measures on ${\\mathcal B}([0, 1])$\nmay not necessarily be tight on ${\\mathcal B}((0, 1))$ (the $\\sigma$-algebra of subsets of\n$(0, 1)$),\nweak convergence of such probability measures on ${\\mathcal B}([0, 1])$\ndoes not necessarily imply tightness of the corresponding\nprobability measures on ${\\mathcal B}({\\mathbb R})$.\n\\begin{exam}\n\\label{ex-homeo2}\n{\\em\nConsider $(Z_n)_{n\\geq 1}$ in Example \\ref{ex-notight} again. Write\n$Z'_n = h(Z_n)$ where $h$ is given in Example \\ref{ex-homeo}. \nDenote by $\\mu'_n$ the probability measure induced by $Z'_n$.\nAs $n\\to\\infty$,\n$Z_n\\Rightarrow Z$. Accordingly $Z'_n\\Rightarrow Z'$. Let $\\mu'$ be the probability measure\ninduced by $Z'$. Now $\\mu'_n\\Rightarrow\\mu'$ on\n$([0, 1], {\\mathcal B}([0, 1]))$ corresponds to \n$\\mu_n\\Rightarrow\\mu$ on\n$(\\overline{\\mathbb R}, {\\mathcal B}(\\overline{\\mathbb R}))$.\nConsequently,\n$\\mu'(\\{1\\}) = 1$ and\n$\\mu'((0, 1)) = 0$ correspond to \n$\\mu(\\{\\infty\\}) = 1$ and $\\mu({\\mathbb R}) = 0$, respectively.\nAlthough $\\overline{\\mathbb R}$ and $[0, 1]$ are homeomorphic,\n$(\\mu_n)_{n\\geq 1}$ is still not tight on $({\\mathbb R}, {\\mathcal B}({\\mathbb R}))$. Clearly, the correspondence between \n$Z'=1$ and $Z=\\infty$ does not imply that $Z'$\nand $Z$ are equivalent in magnitude. The trivial tightness of $(\\mu'_n)_{n\\geq 1}$\non $([0, 1], {\\mathcal B}([0, 1])$ will not change the fact\nthat $(\\mu_n)_{n\\geq 1}$ is not tight on $({\\mathbb R}, {\\mathcal B}({\\mathbb R}))$. \n$\\Box$\n}\n\\end{exam}\n\n\\section{Probabilities and Real Numbers in $[0, 1]$}\n\\label{sec-6}\n\\hskip\\parindent\nThe notion of real numbers and the notion of probability can \nboth be formulated axiomatically. However, the axioms\nfor real numbers cannot formulate the\nnotion of probability. As a set function in measure-theoretic probability\nformulated by the axioms of probability,\na probability measure assigns a numerical probability \n(also referred to as probability mass) represented by a number in the closed unit interval $[0, 1]$ \nto an event characterized by a measurable set on a probability space.\nThe numerical probability \nquantifies the possibility of the event.  In other words,\nassociated with a measurable set on a\nspecified probability space,\na (numerical) probability describes quantitatively how likely\nan event will occur. \nReal numbers without the constraint imposed by the axioms of probability\ndo not have such unique property of probabilities.\nTherefore, a number in $[0, 1]$ is not necessarily\na probability of\nan event on a given\nprobability space.\nProbabilities must comply with the constraint imposed by\nthe axioms of probability.\nNumbers in $[0, 1]$ are not necessarily subject to\nsuch constraint. Consequently,\nalthough the amount or relative magnitude of a \nprobability is represented by a real\nnumber in $[0, 1]$,\nprobabilities cannot be treated as such numbers.\n\n\nReal numbers are an ordered field. In such a field,\ncomparison of real numbers is defined by the axioms of order.\n``Possibility'' is a quantifiable attribute of an event.\nProbabilities are compared as quantified possibilities.\nSince the amount of a probability is represented by a real number\nin $[0, 1]$, comparing probabilities is equivalent to\ncomparing their amounts. \nIn contrast, if a real number in $[0, 1]$ fails\nto comply with the axioms of probability, then the number\nis not a quantified possibility of an event on the\nunderlying probability space, and hence is not a probability.\nComparing such a number with a probability is meaningless. \nSuch comparison is not defined\nby the axioms for either probability or real numbers.  \nActually, relations on ${\\mathbb R}$ \nsuch as $<, >, =, \\leq$ and $\\geq$\nare all defined for and only for real numbers, where real numbers\nare merely ``points''\nin the ordered field. No other\ninterpretation is attached to such abstract points.\n\\begin{exam}\n\\label{ex-unitpro}\n{\\em\nUnit probability is a unique quantity for\nindicating certainty or almost sure occurrence of\nan event.\nIn contrast, the real number ``1''\nis merely a point \nin an absolute space ${\\mathbb R}$ formulated as a set based on the axioms of real numbers without any other interpretation. \n``An event occurs with probability 1'' is certainly\na meaningful statement. However, if ``with probability 1'' is replaced by\n``with the real number 1'', the statement\n``an event occurs with the real number 1'' is meaningless.\nUnit probability and the real number ``1'' are of the same amount, but\nnot of the same kind, and hence\nnot comparable. $\\Box$\n}\n\\end{exam}\n\n\\begin{exam}\n\\label{ex-info}\n{\\em\nFrom information theory we can also see\nthe difference between probabilities and\nnumbers in $[0, 1]$.\nAs shown in information\ntheory, the notion of probability\nis closely related to the notion of ``information''. \nWithout the constraint imposed by the axioms of \nprobability,\nnumbers in $[0, 1]$ have no such \nconnection to ``information''.\n$\\Box$\n}\n\n\\end{exam}\n \nNevertheless, since the amount of a probability is\nrepresented by a real number in $[0, 1]$,\naccording to the axioms of real numbers,\nwe can still compare the amount of a probability with a real number.\nHowever, when making such comparison,\nwe must keep in mind that a probability and its amount\n(i.e., its relative magnitude represented by a real number)\nare different.\n\n\n\n\n\nAn operation defined on a set is said to be\n``closed'' with respect to\na given property of some or all elements in the\nset, if every result of the operation is an\nelement of the set and possesses the property.\nThe values of probabilities are represented by numbers in $[0, 1]$. However,\nalthough operations defined for numbers in $[0, 1]$ may be\nclosed with respect to numbers, i.e., results of\nsuch operations are also numbers, the operations may not\nnecessarily be closed with respect to probabilities.\nAny closed operation defined for probabilities\nmust obey the axioms of probability.\nThis requirement will not change the closeness of an operation\ndefined for numbers. When a closed operation defined for\nnumbers violates the axioms of probability,\nthe result of the operation is not a probability, but is\nstill a number.\n\\begin{exam}\n\\label{ex-4}\n{\\em\nA probability space has four elementary events\nrepresented by $\\omega_i$ with probabilities\n\\[\nP(\\{\\omega_i\\}) = 1\/8, i = 1, 2, 3,\\; \\mbox{and}\\; P(\\{\\omega_4\\}) = 5\/8.\n\\]\nFor $A = \\{\\omega_1, \\omega_2\\}$ and $B = \\{\\omega_2, \\omega_3\\}$,\nthe condition for $P(A)$ and $P(B)$ to be additive is\nviolated by $A\\cap B\\not=\\emptyset$. Consequently\n$P(A) + P(B) = 1\/2$ is not a probability of any event on the\nprobability space. Nevertheless, as the sum of two numbers, $P(A) + P(B)$\nis still a number.   \n$\\Box$ \n}\n\\end{exam}\n\n\nLet $\\rho$ and\n$\\rho_n, n = 1, 2, \\cdots$ be probability measures on a common measurable space\n$(\\overline{\\mathbb R}, {\\mathcal B}(\\overline{\\mathbb R}))$, such that\n$\\rho_n\\Rightarrow\\rho$, i.e., $\\rho_n$ converges weakly to $\\rho$ as $n\\to\\infty$,\nwhere $\\rho(\\{-\\infty\\}) = 0$ and \n$\\rho_n({\\mathbb R}) = 1$ for each $n$.\nLet $\\alpha$ and $\\alpha_n, n = 1, 2, \\cdots$ be measurable sets in\n${\\mathcal B}(\\overline{\\mathbb R})$, such that $\\alpha = \\lim_n\\alpha_n, n\\to\\infty$, \n$\\alpha_n \\in {\\mathcal B}({\\mathbb R})\n\\subset {\\mathcal B}(\\overline{\\mathbb R})$, and\n$\\rho_n(\\alpha_n) < 1$ for all $n$. \nSuppose $\\lim_n\\rho_n(\\alpha_n)$ exists.\nIf $\\alpha = {\\mathbb R}$, and if $\\lim_n\\rho_n(\\alpha_n)$ is treated\nas the limit of a real sequence, i.e., if $\\lim_n\\rho_n(\\alpha_n)$ is\ninterpreted as $\\lim_n[\\rho_n(\\alpha_n)]$, \nthen the result of such treatment may not necessarily\nbe a probability. \n\\begin{exam}\n\\label{ex-limp}\n{\\em\nLet $\\rho_n$ be $\\mu_n$ as given in Example \\ref{ex-notight}.\nSuppose $\\alpha_n = I(x_n) = (-\\infty, x_n]$. Consider $x_n = n-1$.\n\\[\n\\rho_n(\\alpha_n) = \\mu_n(I(n-1))\n=\\sum_{j=1}^{n-1}\\mu_n(\\{j\\}) = \\sum_{j=1}^{n-1}\\left(\\frac{1}{2^{n-j+1}}\\right)\n= \\frac{1}{2} - \\frac{1}{2^n}.\n\\]\nSuppose $\\lim_n\\mu_n(I(n-1))$ is treated as \n\\[\n\\lim_{n\\to\\infty}[\\mu_n(I(n-1))] = \\lim_{n\\to\\infty}\\left(\\frac{1}{2} - \\frac{1}{2^n}\\right)\n= \\frac{1}{2}.\n\\]\nHowever, no set in  \n${\\mathcal B}(\\overline{\\mathbb R})$ is assigned\none half of unit mass by\nany of the involved probability measures $\\mu, \\mu_1, \\mu_2, \\cdots$.\nSo $\\lim_n[\\mu_n(I(n-1))] = 1\/2$ is not a probability, and the operation of taking limit is\nnot closed with respect to probabilities. Nevertheless, $\\lim_n[\\mu_n(I(n-1))] = 1\/2$ is\nstill the limit of a real sequence. The operation of taking limit remains to be closed with\nrespect to numbers for any convergent sequence in $[0, 1]$.\n$\\Box$\n}\n\\end{exam}\n\n\n\n\\section{Limiting Probabilities on ${\\mathcal B}({\\mathbb R})$ and\nLimits of Real Sequences in $[0, 1]$}\n\\label{sec-7}\n\\hskip\\parindent\nLimiting probabilities are not necessarily limits of probabilities. \nThe former must be probabilities. But the latter may or may not be\nprobabilities. \nA limiting probability is determined by the corresponding limiting \nmeasure and event. The event may not necessarily be a continuity set of the\nlimiting measure \\cite{Bill2}.\nSuppose $\\rho_n\\Rightarrow\\rho$ and $\\lim_n\\alpha_n = \\alpha$\nwhere for all $n$, $\\rho_n({\\mathbb R}) = 1$ and\n$\\rho_n(\\alpha_n) < 1$ with $\\alpha_n\\in {\\mathcal B}({\\mathbb R})$,\n$\\rho$ is a probability measure on \n${\\mathcal B}(\\overline{\\mathbb R})$ with $\\rho(\\{-\\infty\\}) = 0$, and\n$\\alpha\\in {\\mathcal B}({\\mathbb R})$. If $\\lim_n\\rho_n(\\alpha_n)$\nis a probability, $\\lim_n[\\rho_n(\\alpha_n)]$ may not necessarily be a\nprobability. There is a subtle difference\nbetween $\\lim_n\\rho_n(\\alpha_n)$ and $\\lim_n[\\rho_n(\\alpha_n)]$.\nAs shown in Example \\ref{ex-notight} and Example \\ref{ex-limp},\nif $\\rho({\\mathbb R}) = 0$ and if\n$\\lim_n[\\rho_n(\\alpha_n)] > 0$ exists, then $\\lim_n[\\rho_n(\\alpha_n)]$ is\nnot a probability. On the other hand, \nif $\\rho({\\mathbb R}) = 1$, then $\\lim_n[\\rho_n(\\alpha_n)]$ is\nalso a probability and coincides with\n$\\lim_n\\rho_n(\\alpha_n)$. \nA limiting probability cannot always be expressed as the limit of a real sequence.\nTaking limit for a real sequence is not necessarily a closed operation\nwith respect to probabilities. Nevertheless,\nthe definition of the limit of a real sequence remains unchanged, \neven if the limit is not a probability. \nWe shall prove the above claims in this section.\nThe case $0< \\rho({\\mathbb R}) < 1$ may be reduced to\n$\\rho'({\\mathbb R}) = 1$ where $\\rho' = \\rho\/\\rho({\\mathbb R})$ such that $\\rho'_n\\Rightarrow\\rho'$ and\n$\\rho'_n = \\rho_n\/\\rho_n({\\mathbb R})$ \\cite{Loev}. \n\nTo clarify the ambiguity about $\\lim_n\\rho_n(\\alpha_n)$,\nwe introduce a mapping.\nLet $\\psi$ be a probability measure \non ${\\mathcal B}(\\overline{\\mathbb R})$, and\n$\\eta$ a measurable set in ${\\mathcal B}(\\overline{\\mathbb R})$.\nDenote by ${\\mathscr P}$ a mapping,\nwhich maps $(\\psi, \\eta)$ to\n$\\psi(\\eta)$, i.e.,\n${\\mathscr P}(\\psi, \\eta)= \\psi(\\eta)$. \nWith ${\\mathscr P}$, $\\lim_n[\\rho_n(\\alpha_n)]$ can be expressed by\n$\\lim_n{\\mathscr P}(\\rho_n, \\alpha_n)$.\nIf $\\lim_n\\rho_n(\\alpha_n)$ is a probability,\nwe use $[\\lim_n\\rho_n(\\alpha_n)]_m$ and \n$[\\lim_n\\rho_n(\\alpha_n)]_s$ \nto represent the respective\nprobability measure and measurable set \nwhich produce $\\lim_n\\rho_n(\\alpha_n)$.\n\n\\begin{rema}\n{\\em\nIn general, the argument of $[\\cdot]_m$ and\n$[\\cdot]_s$ corresponds to\na procedure for calculating a probability. The procedure\nmay simply be a specific probability\nassignment such as $\\psi(\\eta)$, or is given by an expression for computing a\nlimiting probability, which can eventually be reduced to the\nform of $\\rho(\\alpha)$, where $\\rho$ is the measure of the limiting\nprobability and $\\alpha$ the associated $\\rho$-measurable set,\ndetermined respectively by weak convergence of the sequence of \ninvolved probability measures $\\rho_n$ and the corresponding measurable \nsets $\\alpha_n$ in the context. Procedures for calculating limiting probabilities \nmay not necessarily be the same as those for calculating limits of real sequences. \nA procedure for calculating the limit of a real sequence may not necessarily yield \na limiting probability as shown by Example \\ref{ex-limp}. $\\Box$\n}\n\\end{rema}\n\n\n\\begin{lemm}\n\\label{le-1}\nSuppose $\\rho_n\\Rightarrow\\rho$ with $\\rho(\\{-\\infty\\}) = 0$, $\\lim_n\\alpha_n = \\alpha$, and \n$\\rho_n(\\alpha_n) < 1$ with $\\alpha_n\\in {\\mathcal B}({\\mathbb R})$\nand $\\rho_n({\\mathbb R}) = 1$ for all $n$.\n\n(a) If $\\lim_n\\rho_n(\\alpha_n)$ is a probability, \n\\[\n\\lim_{n\\to\\infty}\\rho_n(\\alpha_n) = {\\mathscr P}\\left(\n\\rho, \\alpha\\right).\n\\]\n\n(b) If both $\\lim_n\\rho_n(\\alpha_n)$ and\n$\\lim_n{\\mathscr P}(\\rho_n, \\alpha_n)$ are probabilities,\n\\\n\\label{eq-lemb}\n\\lim_{n\\to\\infty}\\rho_n(\\alpha_n) = {\\mathscr P}\\left(\n\\rho, \\alpha\\right)\n= \\lim_{n\\to\\infty}{\\mathscr P}(\\rho_n, \\alpha_n).\n\\\n\\end{lemm}\n\n\\proof\n(a) If $\\lim_n\\rho_n(\\alpha_n)$ is a probability, then\nfrom $\\rho_n\\Rightarrow\\rho$ and $\\lim_n\\alpha_n=\\alpha$\nwe have\n\\[\n\\left[\\lim_{n\\to\\infty}\\rho_n(\\alpha_n)\\right]_m = \\rho\\;\\; \n\\mbox{and}\\;\\;\n\\left[\\lim_{n\\to\\infty}\\rho_n(\\alpha_n)\\right]_s = \\alpha.\n\\]\nImmediately from the definitions of $[\\cdot]_m$, $[\\cdot]_s$ and ${\\mathscr P}$,\n\\[\n\\lim_{n\\to\\infty}\\rho_n(\\alpha_n) =\n\\left[\\lim_{n\\to\\infty}\\rho_n(\\alpha_n)\\right]_m\n\\left(\\left[\\lim_{n\\to\\infty}\\rho_n(\\alpha_n)\\right]_s\\right)\n= \\rho(\\alpha)={\\mathscr P}(\\rho, \\alpha).\n\\]\n\n(b) If $\\lim_n\\rho_n(\\alpha_n)$ and\n$\\lim_n{\\mathscr P}(\\rho_n, \\alpha_n)$ are both probabilities,\nwe also have $[\\lim_n{\\mathscr P}(\\rho_n, \\alpha_n)]_m = \\rho$, \n$[\\lim_n{\\mathscr P}(\\rho_n, \\alpha_n)]_s = \\alpha$, and\n\\[\n\\lim_{n\\to\\infty}{\\mathscr P}(\\rho_n, \\alpha_n) \n= \\left[\\lim_{n\\to\\infty}{\\mathscr P}(\\rho_n, \\alpha_n)\\right]_m\n\\left(\\left[\\lim_{n\\to\\infty}{\\mathscr P}(\\rho_n, \\alpha_n)\\right]_s\\right)\n= \\rho(\\alpha)={\\mathscr P}(\\rho, \\alpha)\n\\]\nwhich together with (a) proves (b).\n$\\Box$\\vspace{10pt}\n\nThus, if $\\lim_n\\rho_n(\\alpha_n)$ is a probability,\na general expression, or so to speak, a synonym of\n$\\lim_n\\rho_n(\\alpha_n)$ is ${\\mathscr P}(\\rho, \\alpha)$.\n\\begin{rema}\n\\label{re}\n{\\em \nBy Lemma \\ref{le-1}, given $\\rho_n\\Rightarrow\\rho$ and $\\lim_n\\alpha_n = \\alpha$,\nif both $\\lim_n\\rho_n(\\alpha_n)$ and $\\lim_n[\\rho_n(\\alpha_n)]$ are probabilities,\nthen they can be expressed respectively as ${\\mathscr P}(\\rho, \\alpha)$ and\n$\\lim_n{\\mathscr P}(\\rho_n, \\alpha_n)$, and the order of applying ${\\mathscr P}$ and\ntaking limit is interchangeable.\nLabeled by a single index variable $n$, $\\lim_n\\rho_n(\\alpha_n)$ is not a\ndouble limit. A double limit involves a double sequence labeled by two independent\nindex variables (e.g., $(a_{ij})_{i\\geq 1, j\\geq 1}$). Issues about\na double limit, such as whether different orders of taking limit will\nyield the same result, are irrelevant to $\\lim_n\\rho_n(\\alpha_n)$. $\\Box$\n}\n\\end{rema}\n\nThe following is a variant of Lemma \\ref{le-1}\nin some specific settings\nwith $\\alpha_n = I(x_n) = (-\\infty, x_n]$, where\n$(x_n)_{n\\geq 1}$ is a real\nsequence, such that \n$\\rho_n(I(x_n)) < 1$\nfor all $n$,\nand as $n\\to\\infty$,\n$x_n\\uparrow\\infty$ or $x_n\\to x \\in{\\mathbb R}$.\n\n\n\\begin{lemm}\n\\label{le-2}\nAssume that $\\lim_n\\rho_n(I(x_n))$ is a probability,\n$\\rho_n\\Rightarrow\\rho$ with $\\rho(\\{-\\infty\\}) = 0$, and \n$\\rho_n(I(x_n)) < 1$ \nfor all $n$.\n\n(a) If $x_n\\to x \\in{\\mathbb R}$ as $n\\to\\infty$ where $\\rho(\\{x\\}) = 0$, \nthen $\\lim_n{\\mathscr P}(\\rho_n, I(x_n))$ is a probability, and\n\\[\n\\lim_{n\\to\\infty}\\rho_n(I(x_n)) \n= {\\mathscr P}(\\rho, I(x))\n= \\lim_{n\\to\\infty}{\\mathscr P}(\\rho_n, I(x_n))\n\\]\nwhere $I(x) = \\lim_nI(x_n) = (-\\infty, x]$.\n\n(b) Suppose\n$x_n\\uparrow\\infty$ as $n\\to\\infty$. Then\n\\[\n\\lim_{n\\to\\infty}\\rho_n(I(x_n)) \n= {\\mathscr P}(\\rho, {\\mathbb R}) \\leq 1.\n\\]\nIf $(\\rho_n)_{n\\geq 1}$ is tight on\n$({\\mathbb R}, {\\mathcal B}(\\mathbb R))$ and if\n$\\lim_n{\\mathscr P}(\\rho_n, I(x_n))$ is a probability,\n\\[\n\\lim_{n\\to\\infty}{\\mathscr P}(\\rho_n, I(x_n)) \n= {\\mathscr P}(\\rho, {\\mathbb R}) = 1.\n\\]\nIf $(\\rho_n)_{n\\geq 1}$ is not tight on\n$({\\mathbb R}, {\\mathcal B}(\\mathbb R))$, then\n$\\lim_n{\\mathscr P}(\\rho_n, I(x_n))$ \n(if the limit exists) is not a probability, and\n\\[\n{\\mathscr P}(\\rho, {\\mathbb R})\n= 1-\\rho(\\{\\infty\\})<1.\n\\]\n\\end{lemm}\n\n\\proof\n(a) Since $\\rho_n\\Rightarrow \\rho$ and since $x_n\\to x\\in{\\mathbb R}$,\ngiven $\\epsilon > 0$, there exists an $N_{\\epsilon}$\nsuch that for all $n \\geq N_{\\epsilon}$,\n\\[\n|\\rho_n(I(x_n)) - \\rho(I(x))|\\leq \n|\\rho_n(I(x_n)) - \\rho_n(I(x))| + |\\rho_n(I(x)) - \\rho(I(x))|\n< \\epsilon\/2 + \\epsilon\/2 = \\epsilon.\n\\]\nSo\n\\[\n\\lim_{n\\to\\infty}{\\mathscr P}(\\rho_n, I(x_n))\n= \\rho(I(x)) \n= {\\mathscr P}(\\rho, I(x)).\n\\]\nOn the other hand,\n\\[\n\\lim_{n\\to\\infty}\\rho_n(I(x_n))\n=\\left[\\lim_{n\\to\\infty}\\rho_n(I(x_n))\\right]_m\n\\left(\\left[\\lim_{n\\to\\infty}\\rho_n(I(x_n))\\right]_s\\right)\n= \\rho(I(x)).\n\\]\nThis completes the proof of (a).\n\n(b) Since $\\lim_nI(x_n)= \\cup_{n\\geq 1}I(x_n) = {\\mathbb R}$, \n\\[\n\\lim_{n\\to\\infty}\\rho_n(I(x_n)) \n=\\left[\\lim_{n\\to\\infty}\\rho_n(I(x_n))\\right]_m\n\\left(\\left[\\lim_{n\\to\\infty}\\rho_n(I(x_n))\\right]_s\\right)\n=\\rho({\\mathbb R}) = {\\mathscr P}(\\rho, {\\mathbb R}) \\leq 1.\n\\]\nIf $(\\rho_n)_{n\\geq 1}$ is tight on\n$({\\mathbb R}, {\\mathcal B}({\\mathbb R}))$, \nno mass ``escapes to infinity'' in probability\nassignment given by $\\rho$, and hence\n${\\mathscr P}(\\rho, {\\mathbb R}) =1$.\nMoreover,\nfor each $\\epsilon > 0$, there is an interval \n$I_{b(\\epsilon)} = (-\\infty, b(\\epsilon)]$,\nsuch that \n\\[\n\\sup_{n\\geq 1}\\rho_n\\left(I^c_{b(\\epsilon)}\\right)\n= \\sup_{n\\geq 1}{\\mathscr P}\\left(\\rho_n, I^c_{b(\\epsilon)}\\right)\n< \\epsilon\n\\]\nwhere $I^c_{b(\\epsilon)} = {\\mathbb R}\\setminus I_{b(\\epsilon)}$.\nFor all $n\\geq N_{\\epsilon}$ where $N_{\\epsilon} = \\min\\{n\\in {\\mathbb N}: x_n > b(\\epsilon)\\}$,\n$I(x_n) \\supset I_{b(\\epsilon)}$, \n$I^c(x_n)(= {\\mathbb R}\\setminus I(x_n))\\subset I^c_{b(\\epsilon)}$, and hence\n\\[\n{\\mathscr P}(\\rho_n, I^c(x_n))\\leq \n{\\mathscr P}\\left(\\rho_n, I^c_{b(\\epsilon)}\\right) \n\\leq \\sup_{n\\geq 1}{\\mathscr P}\\left(\\rho_n, I^c_{b(\\epsilon)}\\right)\n< \\epsilon.\n\\] \nIn other words, ${\\mathscr P}(\\rho_n, I^c(x_n))\\to 0$ as $n\\to\\infty$. Consequently,\n${\\mathscr P}(\\rho_n, I(x_n))\\to 1$.\n\nNow assume that $(\\rho_n)_{n\\geq 1}$ is not tight on\n$({\\mathbb R}, {\\mathcal B}({\\mathbb R}))$.\nIn probability assignment given\nby $\\rho_n$ for each $n$, \nmass must be conserved on the\nextended real line. Consequently\n\\begin{equation}\n\\label{eq-lem2c}\n\\rho_n(I(x_n)) \n= \\rho_n(\\overline{\\mathbb R})\n- \\rho_n(I^c(x_n)) - \\rho_n(\\{\\infty\\}).\n\\end{equation}\nNote $\\rho_n(I^c(x_n))+\\rho_n(\\{\\infty\\})\n=\\rho_n(\\overline{\\mathbb R}\\setminus I(x_n))$.\nIn (\\ref{eq-lem2c}), $\\rho_n(\\{\\infty\\})=0$ and\n$\\rho_n(\\overline{\\mathbb R})=1$,\nsince by assumption, for each $n$, $\\rho_n({\\mathbb R}) = 1$.\nThus, with ${\\mathscr P}$, we can write (\\ref{eq-lem2c}) as \n\\[\n{\\mathscr P}(\\rho_n, I(x_n)) \n= 1- {\\mathscr P}(\\rho_n, I^c(x_n)).\n\\]\nTaking limit on both sides of this equation yields\n(if the limits exist)\n\\begin{equation}\n\\label{eq-lem2clim}\n\\lim_{n\\to\\infty}{\\mathscr P}(\\rho_n, I(x_n))\n= 1\n- \\lim_{n\\to\\infty}{\\mathscr P}(\\rho_n, I^c(x_n)).\n\\end{equation}\nIf $\\lim_n{\\mathscr P}(\\rho_n, I(x_n))$ is a probability, so\nis $\\lim_n{\\mathscr P}(\\rho_n, I^c(x_n))$, such that\n\\[\n\\left[\\lim_{n\\to\\infty}{\\mathscr P}(\\rho_n, I(x_n))\\right]_s \n= \\lim_{n\\to\\infty}I(x_n) = {\\mathbb R}\n\\]\n\\[\n\\left[\\lim_{n\\to\\infty}{\\mathscr P}(\\rho_n, I^c(x_n))\\right]_s\n=\\lim_{n\\to\\infty}I^c(x_n) = \\bigcap_{n\\geq 1}I^c(x_n) = \\emptyset\n\\]\nand (\\ref{eq-lem2clim}) implies \n\\[\n\\left[\\lim_{n\\to\\infty}{\\mathscr P}(\\rho_n, I(x_n))\\right]_m\n=\\left[\\lim_{n\\to\\infty}{\\mathscr P}(\\rho_n, I^c(x_n))\\right]_m\n=\\rho\n\\]\nwith $\\rho({\\mathbb R}\\cup\\emptyset)= \\rho({\\mathbb R})=1$.\nHowever, given that $(\\rho_n)_{n\\geq 1}$\nis not tight on $({\\mathbb R}, {\\mathcal B}({\\mathbb R}))$, we have\n$\\rho(\\{\\infty\\}) > 0$.\nConsequently, $\\rho({\\mathbb R}) = 1$ implies\n\\[\n\\rho({\\mathbb R}) + \\rho(\\{\\infty\\})\n= \\rho(\\overline{\\mathbb R})> 1\n\\]\nand hence violates the \nconstraint imposed by the\naxioms of probability. As a probability measure on\n${\\mathcal B}(\\overline{\\mathbb R})$,\n$\\rho$ assigns unit mass to $\\overline{\\mathbb R}$.\nSo $\\lim_n{\\mathscr P}(\\rho_n, I(x_n))$ is\nnot a probability. \nSimilarly, from\nmass conservation on $\\overline{\\mathbb R}$\nin probability assignment given\nby $\\rho$,\n\\[\n\\rho({\\mathbb R}) = \\rho(\\overline{\\mathbb R})\n- \\rho(\\{\\infty\\})\n= {\\mathscr P}(\\rho, {\\mathbb R}) = 1 -\\rho(\\{\\infty\\})<1\n\\]\nfollows immediately. \n$\\Box$\\vspace{10pt}\n\n\n\n\\section{The Inconsistency}\n\\label{sec-8}\n\\hskip\\parindent\nNow we demonstrate the inconsistency of measure-theoretic probability.\nLet $\\overline{\\mathbb R}$ be equipped with a topology \ninduced by a fixed metric (e.g., see Example \\ref{ex-homeo}). \nLet $\\lambda, \\mu, \\lambda_n, \\mu_n, n = 1, 2, \\cdots$ \nbe probability measures on \n$(\\overline{{\\mathbb R}}, {\\mathcal B}(\\overline{\\mathbb R}))$, \nsuch that $\\lambda, \\lambda_1, \\lambda_2, \\cdots$ are\ndistributions of i.i.d. random variables $X, X_1, X_2, \\cdots$, \nrespectively,\nas given in Example \\ref{ex-tight}, and\n$\\mu_n$ are distributions of\n\\begin{equation}\n\\label{eq-Zn}\nZ_n = \\max\\{i: X_i = Y_n,\\; i\\in \\{1, 2, \\cdots, n\\}\\}\n\\end{equation}\nwhere\n\\begin{equation}\n\\label{eq-Yn}\nY_n = \\max\\{X_i: i\\in\\{1, 2, \\cdots, n\\}\\}.\n\\end{equation}\nBy (\\ref{eq-Zn}), $Z_n, n = 1, 2, \\cdots$ form a non-decreasing random sequence $(Z_n)_{n\\geq 1}$.\nAs we can readily see, \nfor each $n$, the probability mass function of $Z_n$ is given\nby (\\ref{eq-1}) in Example \\ref{ex-notight}.\nAs $n\\to\\infty$, $Z_n$ converges in distribution to an extended random variable\n$Z$. Equivalently,\n$\\mu_n\\Rightarrow\\mu$, i.e.,\n$\\mu_n$ converges weakly to $\\mu$ induced by $Z$. As a probability measure on\n${\\mathcal B}(\\overline{\\mathbb R})$, \n$\\mu$ is the Dirac measure concentrated at $\\{\\infty\\}$, i.e.,\n$\\mu(\\{\\infty\\}) = 1$. \nHowever, when restricted to ${\\mathcal B}({\\mathbb R})$, $\\mu$ \nis a subprobability measure \nwith $\\mu({\\mathbb R}) = 0$. As shown in Example \\ref{ex-tight} and\nExample \\ref{ex-notight},\n$(\\lambda_n)_{n\\geq 1}$ is tight on \n$({\\mathbb R}, {\\mathcal B}({\\mathbb R}))$ but\n$(\\mu_n)_{n\\geq 1}$ is not (see Sections \\ref{sec-4} and \\ref{sec-5}).\nBy (\\ref{eq-Zn}) and (\\ref{eq-Yn}),\n\\begin{equation}\n\\label{eq-Vnn}\nZ_n \\leq n\n\\end{equation}\nand\n\\begin{equation}\n\\label{eq-XVY}\nX_{Z_n} = Y_n.\n\\end{equation}\n\n\n\nFor the random sequences $(X_n)_{n\\geq 1}$ and $(Y_n)_{n\\geq 1}$,\na probability measure ${\\mathbb P}$ can be defined on the measurable space\n$({\\mathbb R}^T, {\\mathcal B}({\\mathbb R}^T))$\nwhere $T = \\{1, 2, \\cdots\\}$.\nThis probability measure ${\\mathbb P}$ can also express probabilities given by the\ndistribution $\\mu_n$ induced by $Z_n$ for each $n$. However, as shown in Section \\ref{sec-3},\nno probability measure can be defined on $({\\mathbb R}^T, {\\mathcal B}({\\mathbb R}^T))$\nto capture the impact of the infinite limit of\n$(Z_n)_{n\\geq 1}$ on limiting probabilities given by $\\mu$. \n\nIn contrast, for the notion of weak convergence, \nprobability spaces  \nwhich are the domains\nof the involved random variables \ncan all be distinct. The domain spaces\nare not essential, and actually remain\noffstage \\cite{Bill2,Loev}.\nSo we can focus on the probability measures\nand their weak limits, as long as the range space and the topology on it are the same for\nall the random variables involved \n\\cite{Bill2}.\nThe weak limit of $\\lambda_n$ is trivially $\\lambda$, i.e., \n$\\lambda_n\\Rightarrow\\lambda$\nwith $\\lambda(\\{0\\}) = 1\/2$.\nHowever, the following theorem is also provable.\n\n\\begin{theo}\n\\label{th}\nFor $\\lambda_n$ given in Example \\ref{ex-tight}, $\\lambda_n\\Rightarrow\\lambda$ with \n$\\lambda(\\{0\\}) = \\mu({\\mathbb R}) = 0$. \n\\end{theo}\n\n\n\\proof\nBy (\\ref{eq-Vnn}), \n\\begin{equation}\n\\label{eq-2p}\n\\lambda_n(\\{0\\}) = {\\mathbb P}(X_n = 0) =  \n{\\mathbb P}(X_n = 0, Z_n < n) +\n{\\mathbb P}(X_n = 0, Z_n = n).\n\\end{equation}\nConsider the limiting probabilities on both\nsides of (\\ref{eq-2p}).\nWe first show $\\lim_n{\\mathbb P}(X_n =0, Z_n = n)=0$.\n\\begin{eqnarray}\n\\label{eq-rec}\n&&\\lim_{n\\rightarrow\\infty}\n{\\mathbb P}(X_n =0, Z_n = n) \\nonumber\\\\\n&& =\n\\lim_{n\\rightarrow\\infty}\n{\\mathbb P}(X_{Z_n} =0, Z_n = n)\\nonumber \\\\\n&& =\n\\lim_{n\\rightarrow\\infty}\n{\\mathbb P}(Y_n =0, Z_n = n)\n\\;(\\mbox{by (\\ref{eq-XVY}))} \\nonumber\\\\\n&& = 0. \n\\end{eqnarray}\nThe last equality follows from \n${\\mathbb P}(\\lim_{n\\to\\infty}Y_n = 1)= 1$,\n\\[\n\\{Y_n =0, Z_n = n\\}\n\\subset \\{Y_n =0\\},\\;\\;\n{\\mathbb P}(Y_n =0, Z_n = n)\n\\leq {\\mathbb P}(Y_n = 0)\n\\]\nand \n\\[\n\\lim_{n\\to\\infty}{\\mathbb P}(Y_n= 0)\n= {\\mathbb P}\\left (\\bigcap_{n\\geq 1}\\{Y_n=0\\}\\right )\n(\\mbox{since $\\{Y_n =0\\}\\supset\\{Y_{n+1}=0\\}$\nas implied by (\\ref{eq-Yn})}).\n\\]\nSimilarly, \nfrom (\\ref{eq-2p}), (\\ref{eq-rec}), and \n\\[\n\\{X_n = 0, Z_n < n\\}\n= \\{Z_n < n\\}\n\\]\nwe have \n\\[\n{\\mathbb P}\\{X_n =0, Z_n < n\\} = {\\mathbb P}\\{Z_n < n\\}\n=\\mu_n(I(n-1))\n\\]\nand \n\\[\n\\lambda(\\{0\\}) = \\lim_{n\\to\\infty}\\lambda_n(\\{0\\})\n= \\lim_{n\\to\\infty}{\\mathbb P}(X_n=0, Z_n < n)\n= \\lim_{n\\to\\infty}\\mu_n(I(n-1)). \n\\]\nSince $\\lambda(\\{0\\})$ is a probability, so is $\\lim_n\\mu_n(I(n-1))$, and\n\\[\n\\lambda(\\{0\\}) \n=\\lim_{n\\to\\infty}\\mu_n(I(n-1)) \n= {\\mathscr P}(\\mu, {\\mathbb R}) = \\mu({\\mathbb R}) = 0\n\\]\nfollows from\nLemma \\ref{le-2}(b) with $x_n = n-1$, \n$\\rho_n = \\mu_n$ and $\\rho(\\{\\infty\\})= \\mu(\\{\\infty\\})=1$.\n$\\Box$\\vspace{10pt}\n\\begin{rema}\n\\label{re-2}\n{\\em\nBy definition, ``1\/2'' in \nthe probability mass function ``$\\lambda(\\{0\\})= 1\/2$'' (see Example \\ref{ex-tight})\nrepresents a (numerical) probability. In contrast, \nas explained in Example \\ref{ex-notight} and Example \\ref{ex-limp},\n``1\/2'' in ``$\\lim_n[\\mu(I(n-1))]= \\lim_n{\\mathscr P}(\\mu_n, I(n-1)) = 1\/2$'' is not\na probability. As we have discussed in Sections \\ref{sec-6} and \\ref{sec-7},\nwithout representing the value of a quantity in the physical world, a real number\nis only a point in a set formulated by a system of axioms. It makes no sense at all\nto treat a probability, which is a quantified attribute of an event in the\nphysical world, as a point\nof a set, which is merely a mathematical \n(or metaphysical)\nentity still considered questionable \nby some mathematicians even nowadays \\cite{Brid,Rose}. \nSince  $\\lambda(\\{0\\}) = \\lim_n\\mu(I(n-1))$ as shown in the proof of\nTheorem \\ref{th}, and since $\\lambda(\\{0\\})$ is a probability, \n$\\lim_n\\mu(I(n-1))$ must be a probability, and this probability is\n${\\mathscr P}(\\mu, {\\mathbb R}) = \\mu(\\mathbb R) = 0$ by Lemma \\ref{le-2}. Since ``1\/2'' in\n``$\\lim_n[\\mu(I(n-1))]= \\lim_n{\\mathscr P}(\\mu_n, I(n-1)) = 1\/2$''\nis a real number, i.e., a point on the abstract real line ${\\mathbb R}$ but\nnot a probability, \nit is meaningless to equate $\\lambda(\\{0\\})$ with $\\lim_n[\\mu(I(n-1))] = \\lim_n{\\mathscr P}(\\mu_n, I(n-1))$\n(see also Example \\ref{ex-unitpro} and Example \\ref{ex-info}). \n$\\Box$\n}\n\\end{rema}\n\n\\section{Concluding Remarks}\n\\label{sec-9}\n\\hskip\\parindent\nIn general, the presupposition \nfor questions of identity (expressed by ``$=$'') in a theory is that the two sides\ninquired about both belong to the same category \\cite{Bena2}.\nAn expression ``$x = y$'' where\n$x$ and $y$ are in the same category but numerically different is erroneous \nonly when the theory is proved to be contradiction-free as requested by Hilbert.\nUnfortunately, according to G{\\\" o}del's theorems, \nmeasure-theoretic probability is one of the theories for which such a proof\ndoes not exist. Nevertheless, if one side of ``$=$'' is a probability, then\nthe other side must also be a probability. This requirement is necessary\nto prevent confusion of probabilities and arbitrarily calculated numbers\nwhich are not eligible to serve as probabilities. \nWhether a number is eligible to serve as a probability is \ndetermined by whether the number is the result of \napplying a probability measure to a measurable set in the context.\nIn particular, when we calculate\nlimiting probabilities, the two sides of ``$=$'' must both be limiting probabilities.\nSince procedures for calculating limits of real sequences \nmay not necessarily yield limiting probabilities, the limit of a real sequence which\nis not a probability is irrelevant to whatever implied by calculations of\nlimiting probabilities.\n\nConsider \n$\\lim_n\\mu_n(I(n-1))$ (see the proof of Theorem \\ref{th}). \nThe probability measures $\\mu_n$ and the associated measurable sets \n$I(n-1)$ approach their respective limits simultaneously. \nThis implies two different orders of operations for calculating\n$\\lim_n\\mu_n(I(n-1))$:\n\\begin{itemize}\n\\item[(a)]\ntaking the limits of the probability measures $\\mu_n$\nand the measurable sets $I(n-1)$ \nsimultaneously first, then applying the limiting measure $\\mu$ to \nthe limiting measurable set ${\\mathbb R}$, i.e.,\n\\[\n\\lim_{n\\to\\infty}\\mu_n(I(n-1)) = {\\mathscr P}(\\mu, {\\mathbb R}) = \\mu({\\mathbb R})\n\\]\n\\item[(b)] \napplying the measures $\\mu_n$ to the corresponding measurable sets \n$I(n-1)$ to obtain a sequence $(\\mu_n(I(n-1)))_{n\\geq 1}$ of \nnumbers first, then taking the limit of the real sequence, i.e.,\n\\[\n\\lim_{n\\to\\infty}\\mu_n(I(n-1))=\\lim_{n\\to\\infty}{\\mathscr P}(\\mu_n, I(n-1)).\n\\]\n\\end{itemize}\nOrder (a) produces a probability, but order \n(b) yields a number which is not a probability (see Example \\ref{ex-limp}). So the meaning of \n$\\lim_n\\mu_n(I(n-1))$ is ambiguous. \n\nSince $\\lim_n\\mu_n(I(n-1))$ can only be written as a ``linear'' string of symbols,\nsuch a ``linear'' way of writing might result in an impression as if \n(b) were the only option for calculating $\\lim_n\\mu_n(I(n-1))$. \nHowever, when standing in the relation ``$=$'' to\na probability (i.e., $\\lambda(\\{0\\})$), $\\lim_n\\mu_n(I(n-1))$ has to be calculated according to\n(a) rather than (b) to yield a probability as well, for\nthe two sides of ``$=$'' must both be\nprobabilities, even if they are numerically different. \nAs we have\ndemonstrated, ``two numerically different\nprobabilities equal each other'' is a provable statement, which shows that\nmeasure-theoretic probability is inconsistent. \n\nThe revealed contradiction of measure-theoretic probability does not exist in the physical world.\nIt appears only in the theory. Practical applications of\nmeasure-theoretic probability will not be affected \nby the inconsistency as long as ideal events in the theory\nare not mistaken for real events in the physical world. On the other hand,\nthe inconsistency of measure-theoretic probability cannot be explained away, and must be resolved.\nConstructive mathematics (e.g., see \\cite{Brid,Rose}) can avoid such inconsistency. There is no contradiction \nreported in constructive mathematics.\n\n\n\\small\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\n  \\pagenumbering{arabic}\nIn many physical systems, even the\nqualitative features along a line of phase transitions may\ndepend on the choice of the coupling constants which\nparametrize the bare interaction energies. Quite often one\nencounters the case where a continuous higher order\ntransition changes into a discontinuous first-order one. In\nmost cases, the cross-over behaviour is\ncharacterized by a tricritical point (TCP)\\cite{TCP}. In\nfield theoretic language, the origin of a TCP is easy\nto understand on the basis of a Landau expansion of the\neffective potential\nin terms of\na order- (or disorder-) parameter. One source for a\nTCP is the sign-change in the quartic term (assuming a\nstabilizing sextic term) in some range of the parameter\nspace. A positive quartic term  leads to a second-order\ntransition, and a negative one to a first-order transition. A\nvanishing quartic term is associated with the TCP. Another\nimportant mechanism arises from the presence of a cubic\nterm. Such a cubic term may be generated,\nin some range of the parameter space, by fluctuations\nof other fields. This always drives the transition\nfirst-order. While the analysis of the Landau expansion\nis very elementary, the\nnon-trivial problem is, of course, its derivation\nfrom the underlying physical system with\nquite complicated microscopic forces (the most prominent\nexample being the Gorkov derivation \\cite{gorkov} of the effective\nGinzburg-Landau theory of superconductivity). It is\ntherefore desirable to find alternative properties of the system which\nallow at least for a semi-quantitative\nunderstanding of the appearance of\nTCP's. In this lecture we exhibit such properties for compact U(1)\nlattice models, having in mind quite different physical\nsystems (D=dimension):\n\n1. D=4 Quantumelectrodynamics (U(1)-LGT)\n\n2. D=3 Superfluid Helium (XY-model)\n\n3. D=3 Defect models of melting\n\nThe common point of these systems is that they can be studied in\n\na)\nthe weak-coupling or ``defect'' expansion. This is formulated\nin terms of lines\\footnote\n{When the dimension is reduced by one, $D \\rightarrow\nD-1$, ``lines'' have to be replaced by ``points''.}\nwith long-range Biot-Savart-like interactions. This is why all three\nsystems can be described, alternatively, by Abelian Higgs models\nin which the complex fields account for the line-like disorder of the systems\nand the gauge fields for the long-range interactions \\cite{book}.\n\nb) The\nstrong-coupling or ``stress'' expansion. This is also formulated in\nterms of geometrical objects (surfaces, lines), but with\nno long-range interactions. This simplifies their\nstatistical behaviour and will provide for the desired alternative\ncriterion for the existence of TCP's.\n\nFor the specific example\nof U(1) lattice gauge theory the different representations,\nbeing dual to each other, and their corresponding\nfield-theoretic descriptions are summarized in\nfig.1. The diagrams for the other systems are\ncompletely analogous.\n  \\begin{figure}\n     \\vspace{7.0cm}\n     \\caption[1]{Interrelation between the dual ``defect''\n              (weak-coupling) and ``stress''\n              (strong-coupling) representations of U(1)\n              lattice gauge theory and their equivalent\nfield theoretic descriptions.}\n   \\end{figure}\n  \\section{The ``Defect'' Representation}\nLet us first look at the weak-coupling expansion of the various\nU(1) lattice models. They all contain topological excitations with\nlong-range interactions caused by the Nambu-Goldstone\nbosons of the U(1) symmetry. In a three-dimensional solid, the\ntopological excitations are line-like defects, dislocations and\ndisclinations \\cite{defect} , and the Nambu-Goldstone bosons\nare elastic\ndistortions (phonons), leading to a stress-field around each\ndefect and to Biot-Savart-like interactions between defect\nelements. In superfluid Helium in three dimensions, the ``defects''\nare vortex-lines \\cite{vortex} and the long-range forces are\ncaused by the superflow. In four-dimensional U(1)-LGT\nfinally, the ``defects'' are world lines of magnetic\nmonopoles \\cite{monopole} and the forces are due to electromagnetism.\nThe upper right box in fig.1 symbolizes the\nequivalent disorder field description of these ``defect'' lines\nin terms of a complex field\n$\\Psi$ interacting with a gauge field $A_{\\mu}$. In\nsuperfluid Helium and U(1)-LGT, the disorder field theory of\nline-like defects turns out \\cite{hk3d,hk4d} to be an\nAbelian Higgs model. In a solid, it is a more complicated field theory\nof a similar type, involving two disorder fields, one for dislocation\nand one for disclination lines \\cite{book}.\n\nLet us briefly recall the\nphysical content of the dual equivalence. From Feynman's\npath-integral representation of quantum mechanics, we know\nthat the statistical mechanics of one\nfluctuating line (``orbit'') corresponds to the quantum\nmechanics of one particle. It is then easy to see that the\ngrand-canonical ensemble of fluctuating lines corresponds to\nthe quantum mechanics of a\nmany-particle system. This in turn is described most conveniently by\na second quantized\nfield theory.\nThe long-range Biot-Savart-like nature of the interactions\nbetween line-elements is what permits their description by\nan Abelian gauge-field\n$A_{\\mu}$. The minimal coupling of $A_{\\mu}$\nto the disorder field $\\Psi$ leads then immediately to an\nAbelian Higgs model (scalar electrodynamics).\nIndeed, in the lattice formulation of\ninteracting defect lines, these steps can be carried out\nrigorously \\cite{hk3d,hk4d}, leading to the Higgs action\n\\begin{equation}\n{\\cal A} = \\int d^{D}\\!x \\left[ \\frac{1}{4}F_{\\mu\\nu}^{2} + \\frac{1}{2}\n|(\\partial\n_{\\mu} - ieA_{\\mu}) \\Psi|^{2} + \\frac{1}{2} m^{2} |\\Psi|^{2} +\n\\frac{1}{4} g |\\Psi|^{4} + \\ldots \\right]\n\\label{eq:1}\n\\end{equation}\nwhere $F_{\\mu\\nu} = \\partial _{\\mu} A_{\\nu} - \\partial _{\\nu}\nA_{\\mu}$, and $e,m^{2},g>0$ can in principle be calculated from\nthe couplings of\nthe original compact U(1) lattice model, indicated by the circle in\nthe center of fig.1. According to the famous\nColeman-Weinberg argument \\cite{coleman,halperin},\nthis Abelian Higgs model should\nalways undergo a first-order transition when the mass\nparameter turns negative.\nThe argument goes\nas follows: Assuming $\\Psi = $ const., the gauge fields can\nbe integrated out, yielding a one-loop correction to (\\ref{eq:1})\n\\begin{equation}\n{\\rm tr} \\log (k^{2} + e^{2}|\\Psi|^{2})\n\\label{eq:1.1}\n\\end{equation}\nThis amounts to additional $\\Psi$ interactions of the\nform\n\\begin{equation}\n{\\rm ~~}-|\\Psi|^{3} {\\rm ~~~~~~~~~~in~D=3}\n\\label{eq:1.2}\n\\end{equation}\nand\n\\begin{equation}\n|\\Psi|^{4} \\log |\\Psi|        {\\rm  ~~~~~~~in~D=4}\n\\label{eq:1.3}\n\\end{equation}\nIn both cases, the additional term should drive the transition\nfirst-order.\n\nIn three\ndimensions, this conclusion is in clear contradiction to the\nwell established fact that the original D=3 XY model has a continuous\nphase transition \\cite{3dxy,dasgupta}. The crucial\npoint in the whole argument is, of course, the assumption of\nan almost constant $\\Psi$-field. If this assumption breaks\ndown, then also the generation of a cubic term (in D=3) is\nno longer reliable and the transition may stay continuous\neven with $A_{\\mu}$-field fluctuations. The decisive parameter is\nthe ratio of the length-scales of\n$A_{\\mu}$- and $\\Psi$- fluctuations,\n\\begin{equation}\n\\kappa = \\frac{1}{\\sqrt{2}} \\frac{{\\rm penetration~depth }(A_{\\mu})}\n{{\\rm coherence~length }\n(\\Psi)}\n\\label{eq:1.4}\n\\end{equation}\nIn three dimensions, it was possible to show that there\nexists a tricritical value of $\\kappa \\approx 1\/\\sqrt{2}$,\ni.e. near the separation line between type-I and type-II\nsuperconductivity \\cite{hk3d}.\nFor small $\\kappa$, the Coleman-Weinberg mechanism is\nvalid, leading to a first-order transition,\nwhereas for large $\\kappa$, the transition stays second\norder.\n\nThis led to the suggestion that also four-dimensional scalar\nelectrodynamics could have a TCP \\cite{hk4db}. By the above duality\narguments, this would also hold for U(1)-LGT.\nIndeed, the\nfirst-order nature predicted by the Coleman-Weinberg mechanism\nwas at odds with early Monte Carlo\ninvestigations of the dual U(1)-LGT which all claimed evidence\nfor a continuous transition and even reported estimates for\ncritical indices \\cite{earlyu1}. More recent work favors the existence\nof both transition regions, first-order {\\em and} second-order, in\nagreement with ref.\\cite{hk4db}. Only one paper claims the validity\nof the Coleman-Weinberg mechanism everywhere.\nThe present status\nwill be reviewed in more detail in the next section.\n\nThe goal of this lecture is to present semi-quantitative\nexplanations for these TCP's, working within the dual\nstrong-coupling expansions of the various U(1) models. This\nhas the advantage that, since the geometrical\nobjects appearing in this expansion have no long-range\ninteractions, subtleties of the type (\\ref{eq:1.1})--(\\ref{eq:1.3})\nare absent.\n  \\section{The ``Stress'' Representation}\nLet us start by recalling the definitions of the various\nU(1) models mentioned in the introduction. The partition\nfunctions\n\\begin{equation}\nZ = \\sum_{\\{\\gamma{\\rm - conf.}\\}} (\\prod B)\n\\label{eq:2}\n\\end{equation}\nare products of local Boltzmann factors which may be chosen\nas\n\\begin{equation}\nB = \\left\\{\n\\begin{array}{ll}\ne^{\\beta \\cos \\Theta} & {\\rm Wilson}\\\\\ne^{\\beta \\cos \\Theta + \\gamma \\cos 2\\Theta} & {\\rm Mixed}\\\\\n\\sum_{n}e^{-\\frac{\\beta_{V}}{2}(\\Theta - 2 \\pi n)^{2}} & {\\rm\nVillain}\n\\end{array}\n\\right\\} {\\rm action}\n\\label{eq:3}\n\\end{equation}\nwith $\\Theta$ standing short for\n\\begin{equation}\n\\Theta = \\left\\{\n\\begin{array}{ll}\n\\nabla_{i}\\gamma_{j} - \\nabla_{j}\\gamma_{i} \\equiv\n      (\\nabla_{i}\\gamma_{j})^{A} & {\\rm Gauge}\\\\\n\\nabla_{i}\\gamma\n& {\\rm XY}\\\\\n\\nabla_{i}\\gamma_{j} + \\nabla_{j}\\gamma_{i} \\equiv\n      (\\nabla_{i}\\gamma_{j})^{S} & {\\rm Melting}\n\\end{array}\n\\right\\} {\\rm model}\n\\label{eq:4}\n\\end{equation}\nThe product in (\\ref{eq:2}) runs over the plaquettes or\nlinks of the lattice, and\n$\\nabla_{i}$ are the usual lattice derivatives\n($\\nabla_{i}f(\\vec x) = f(\\vec x + \\vec i) - f(\\vec x))$.\nThe $\\gamma$-configurations are, in U(1)-LGT,\nthe euclidean\nelectromagnetic fields, in the case of superfluid\nHelium, the phase angles of the condensate, and in the\nmelting model, the\natomic displacements.\nThe common important property of these models\nis the U(1) invariance\n\\begin{equation}\n\\gamma \\rightarrow \\gamma + 2 \\pi\n\\label{eq:5}\n\\end{equation}\nsuggesting a Fourier (``character'') expansion of the\nBoltzmann factors\n\\begin{equation}\nB(\\Theta) = \\sum_{b=-\\infty}^{\\infty} W_{b} e^{ib\\Theta}\n\\label{eq:6}\n\\end{equation}\nwith Fourier coefficients\n\\begin{equation}\nW_{b} =\n\\int_{-\\pi}^{\\pi}\\frac{d\\Theta}{2\\pi}B(\\Theta)e^{-ib\\Theta}\n\\label{eq:7}\n\\end{equation}\nInserting (\\ref{eq:6}) into the partition function and\nsumming over the $\\gamma$-configurations, one obtains the\nstrong-coupling expansion $(\\bar{W}_{b} \\equiv\nW_{b}\/W_{0})$\n\\begin{equation}\nZ = (\\prod W_{0}) \\sum_{\\{b{\\rm-conf.}\\}} (\\prod \\bar{W}_{b})\n\\label{eq:8}\n\\end{equation}\nThe algebraic structure of $\\Theta$ in eq.(\\ref{eq:4}) leads\nto conservation laws which constraint the admissible\n$b$-configurations. It is most transparent to characterize\nthem in terms of geometrical objects:\n\\begin{equation}\n\\{b{\\rm-conf.}\\} = {\\rm closed} \\left\\{\n\\begin{array}{ll}\n{\\rm surfaces} & {\\rm Gauge}\\\\\n{\\rm lines} & {\\rm XY}\\\\\n{\\rm complicated~lines} & {\\rm Melting}\n\\end{array}\n\\right.\n\\label{eq:8.1}\n\\end{equation}\nIn the U(1)-LGT, the $b$-variables correspond to the\nelectromagnetic field strengths, in the superfluid to the\nsuperfluid currents, and in the melting model to the\nphysical stress. This is why we call the expansion (\\ref{eq:8})\ngenerically the ``stress''-representation. The geometrical\nobjects in (\\ref{eq:8.1}) are subject only to short-range\ncontact interactions.\nNotice that the geometrical characterization of these\n``stress''-graph configurations does not depend on the\ndimension D. This is in contrast to the\n``defect''-representation where the dimensionality of the\ngeometrical objects depends on the dimension in which\nthe duality transformation is performed.\n  \\section{Analytical and Numerical Results}\nLet us now\nanalyze the ``stress'' representation (\\ref{eq:8}) in some detail.\nTo each choice of action in (\\ref{eq:3}) corresponds a\nspecial ``natural'' set of weights $\\bar {W}_{b}$ in\n(\\ref{eq:8}). Their relative importance with increasing\nstrength $b=\\pm 1, \\pm 2, \\ldots$ can be studied\nconveniently by simulating each model with different actions and\ncomparing their thermodynamic quantities such as their\ninternal energies.\n\nWe start with the comparison\nbetween Wilson's and Villain's action for which the weights\nare\n$\\bar{W}_{b}^{W} = I_{b}(\\beta)\/I_{0}(\\beta)$ ($I_{b}:$ modified\nBessel function) and $\\bar{W}_{b}^{V} =\n\\exp(-b^{2}\/(2\\beta_{V}))$, respectively. The two actions\nare made as similar as possible by equating the\n$b=\\pm 1$ weights which amounts to relating the\nVillain parameter $\\beta_{V}$ to the Wilson parameter\n$\\beta$ as follows (see fig.2)\n\\begin{equation}\n\\beta_{V}(\\beta) = -\\frac{1}{2 \\log\n[I_{1}(\\beta)\/I_{0}(\\beta)]}\n\\label{eq:9}\n\\end{equation}\nThis relation was first written down by Villain\n\\cite{villain} when analyzing the Wilson type of\naction of the XY model in terms of the discrete Gaussian model\n(Villain approximation).\nIf furthermore the overall normalizations of the partition\nfunctions are adjusted, then the difference between the\ntwo actions lies all in the higher weights,\n$\\bar{W}_{b}^{W}(\\beta)=I_{b}(\\beta)\/I_{0}(\\beta) \\neq\n[I_{1}(\\beta)\/I_{0}(\\beta)]^{b}=\\exp(-b^{2}\/(2\\beta_{V}))=\n\\bar{W}_{b}^{V}$ for $b \\geq 2$.\n  \\begin{figure}\n    \\vspace{7.5cm}\n    \\caption[2]{The parameter $\\beta_{V}$ of the Villain\napproximation versus $\\beta$ of the corresponding Wilson\naction. The curve results from the requirement of equal\nweights for graphs of\nstrength-1 in the ``stress'' representation for both\nactions.}\n   \\end{figure}\nIn order to measure their importance,\nwe have performed Monte Carlo simulations in the\nrepresentation (\\ref{eq:2})\nwith both Wilson's and Villains's action \\cite{jkvill}. As a typical\nexample, we compare in fig.3 our results for the internal energy of\nthe U(1) lattice gauge model.\nThe excellent agreement for\nlow $\\beta$ up to the phase transition around $\\beta \\approx\n1$ demonstrates that, in this range, the systems are dominated\nby $b=\\pm1$ excitations.\n   \\begin{figure}\n      \\vspace{7.5cm}\n      \\caption[3]{The internal energy of D=4 U(1) lattice gauge model\nwith Wilson action in comparison with the Villain approximation.}\n   \\end{figure}\nIt is therefore not surprising that also the\ntransition temperatures of the Villain action (with\n$\\beta_{V}$ as a free parameter) are mapped by (\\ref{eq:9})\nvery precisely onto the corresponding ones of the Wilson\naction (see table 1). Furthermore, from fig.3 we read off\nthat at and\n   \\input{egertab1}\nabove the phase transition the graphs of higher strengths\nproliferate much stronger using the Wilson action.\nAmong these we expect the graphs of strength-2 to be most\nimportant. This suggests that the weight ratio\n$\\bar{W}_{2}\/\\bar{W}_{1}$ should be the relevant distinguishing\nfeature of the different actions. In the sequel, it will be\ncalled the ``2:1 ratio''. In the Wilson action, the ``2:1\nratio'' is much larger than in the Villain action as is\nshown in table 2.\n\nIt is easy to convince ourselves that the\nhigher ratios $\\bar{W}_{3}\/\\bar{W}_{1},\\ldots$ are\npractically irrelevant near the phase transition.\n   \\input{egertab2}\nFor this purpose we compare the Villain action\nwith further Monte Carlo simulations of the mixed\n$\\beta-\\gamma$ action (see eq.(\\ref{eq:3})), which is chosen\nin such a way that the ``2:1 ratio'' coincides with that of\nthe Villain action. The coincidence is achieved by requiring\n\\begin{eqnarray}\n\\frac{I_{1}(\\beta,\\gamma)}{I_{0}(\\beta,\\gamma)} & = &\n\\exp(-\\frac{1}{2\\beta_{V}}) \\label{eq:10a}\\\\\n\\frac{I_{2}(\\beta,\\gamma)}{I_{0}(\\beta,\\gamma)} & = &\n\\exp(-\\frac{4}{2\\beta_{V}})\n\\label{eq:10b}\n\\end{eqnarray}\nwhere $I_{b}(\\beta,\\gamma)$ are generalized modified Bessel\nfunctions\n\\begin{equation}\nI_{b}(\\beta,\\gamma) = \\int_{-\\pi}^{\\pi} \\frac{d\\phi}{2\\pi}\n\\cos(b\\phi) e^{\\beta \\cos(\\phi) + \\gamma \\cos(2\\phi)}\n\\label{eq:11}\n\\end{equation}\nHere, it is convenient to solve eqs.(\\ref{eq:10a},\\ref{eq:10b}) in the\nform $\\beta(\\beta_{V}),\\gamma(\\beta_{V})$ which we shall\ncall the ``Villain locus'' in the $\\beta-\\gamma$\nplane. The comparison of our Monte Carlo data for the\ninternal energy of U(1)-LGT\nin fig.4 clearly\ndemonstrates that the differences due to graphs of higher strength\n$b = \\pm 3,\\ldots$\n(which still have different weights) are completely\nnegligible - the data for the Villain action and the\nmixed $\\beta-\\gamma$ action\n(along the ``Villain locus'') fall almost\nperfectly on top of each other.\n  \\begin{figure}\n     \\vspace{8.5cm}\n     \\caption[4]{The internal energy of U(1)-LGT with Villain\naction. The lower data points are obtained from simulations\nof the $\\beta\\cos\\Theta$ model transformed according to\neq.(\\ref{eq:9}). The\nstars which fall practically on top of the Villain\ndata are from a simulation with mixed action\n$\\beta\\cos\\Theta + \\gamma\\cos 2\\Theta$ treated according to\neqs.(\\ref{eq:10a},\\ref{eq:10b}).}\n   \\end{figure}\n\nAs a conclusion, the\ndifferences between the Wilson and the Villain action\nnear the phase transition are completely\nexplained by the different weights of strength-2 ``stress''\ngraphs, whose proliferation is much more pronounced in the Wilson\ncase.\n\nThat an admixture of strength-2 graphs with large enough weight\ncan in principle drive the transition first-order, can be demonstrated\neasily in the disorder field theory of ``stress'' lines (i.e. in the\nmean-field formulation of the D=3 XY model with mixed action). This is\ndescribed by an effective action \\cite{jkxy}\n\\begin{equation}\n{\\cal A} = a|\\phi_{1}|^{2} + b|\\phi_{1}|^{4} + c|\\phi_{2}|^{2}\n-\\frac{1}{2} [ \\phi_{1}^{2} \\phi_{2}^{+} + c.c.] + {\\rm gradients}\n+ \\ldots\n\\label{eq:last}\n\\end{equation}\nwhere the complex fields $\\phi_{1},\\phi_{2}$ represent strength-1 and\nstrength-2 ``stress'' lines, respectively. The coupling $\\phi_{1}\n\\phi_{1} \\phi_{2}^{+}$ corresponds to the merging of two lines of strength-1\ninto one line of strength-2. The complex conjugate coupling describes\nthe reversed process. Integrating out the field $\\phi_{2}$, one obtains\nan additional quartic term in $\\phi_{1}$, $-\\frac{1}{4c} |\\phi_{1}|^{4}$,\nsuch that the total quartic term in $\\phi_{1}$ may change sign, signalizing\na first-order transition.\n\nThis explains at least qualitatively why in careful\nMonte Carlo studies with the Wilson action a\nfirst-order transition was reported while with the Villain\naction evidence for a continuous transition was claimed.\nMoreover, this picture is consistent with Monte Carlo\nstudies of the mixed $\\beta-\\gamma$ action by\nJers\\'{a}k et al.\\cite{jersak}\nwho located a TCP for slightly negative $\\gamma$.\nLooking at fig.5, we see that our ``Villain locus'' indeed\ncrosses the phase transition line in the range\nof second-order transitions. It would be interesting to\ninvestigate whether the TCP is connected with a universal\ntricritical ``2:1 ratio'' $\\bar{W}_{2}\/\\bar{W}_{1}$.\n   \\begin{figure}\n     \\vspace{9.5cm}\n     \\caption[5]{The parameters $\\beta,\\gamma$ of the mixed\naction $\\beta\\cos\\Theta + \\gamma \\cos2\\Theta$ which can be\nstudied by means of the improved Villain approximation\n(``Villain locus''). The\nfat and dashed lines show an interpolation to the critical\npoints found by Jers\\'{a}k et al.\\cite{jersak}. The dotted line was estimated\nby those authors to be the locus of the Villain model in a\ndifferent way from ours (which we believe to be less\naccurate).}\n   \\end{figure}\n\nThe most recent Monte Carlo renormalization group (MCRG) studies\nare inconsistent with this nice picture. They are\ninterpreted as evidence for a first-order transition, with both\nthe Wilson and the Villain action. A historic summary is\ncompiled in table 3.\n\\input{egertab3}\nThe latest study \\cite{hase} even speculates that the transition is always\nfirst-order, downto $\\gamma=-\\infty$.\nThe last result would imply that in the whole parameter range of the D=4\nAbelian Higgs model which is covered by the dual\n$\\beta-\\gamma$ action, the Coleman-Weinberg mechanism is\nindeed working. We feel, however, that even in view of this\nimpressive list of work in table 3, the\nanswer has not yet settled and much more high precision\nstudies are necessary to provide the final answer.\n\nWhile in D=4 the situation is still quite\ncontroversial, in D=3 dimensions the numerical evidence is\nclearly against the analog of the Coleman-Weinberg\nmechanism, as advanced by Halperin,Lubensky and Ma\n\\cite{halperin}. Here, it\nwas confirmed many times by different methods that both\nWilson's and Villain's action undergo continuous\ntransitions.\nThe\nquestion was then whether there exists at all a parameter range in the mixed\n$\\beta-\\gamma$ action which shows first-order transitions.\nAccording to our above analysis, a good candidate was only\nthe range $\\gamma \\geq 0$. We therefore concentrated on this\nrange and found \\cite{jkxy}, first in a mean-field (MF) treatment,\nfirst-order\ntransitions for $0.166 \\leq \\gamma \\leq 0.375$. The full\nMF phase diagram is displayed in fig.6 (use the\nlabelings on the right and top axis).\n  \\begin{figure}\n     \\vspace{10.5cm}\n     \\caption[6]{Phase diagram of the D=3 XY model with mixed action as\nobtained by mean-field methods and Monte Carlo simulations.}\n   \\end{figure}\nSince already in the pure\nXY model ($\\beta = 0$ or $\\gamma=0$ axis), the MF\ntransition temperatures are off by a factor\n$0.45\/0.33=1.36$, we have rescaled the MF curves by this\nfactor when comparing with our Monte Carlo simulations which\nwe run in the range where the rescaled MF results show the\nlargest entropy jump, $\\gamma \\approx 0.35...0.40$.\nThe numerical results confirmed the first-order nature of\nthe transition. As a typical signal, we show in fig.7 the\ndouble peak structure in the internal energy histogram,\ncorresponding to tunnelings between two metastable states.\n  \\begin{figure}\n    \\vspace{6.0cm}\n    \\caption[7]{Energy histogram near the first-order\ntransition line of\nthe D=3 XY model with mixed action.}\n  \\end{figure}\nSince the observed first-order transition is\nvery weak and the $\\gamma$-range is probably very small, we\nwere not able to locate the two tricritical points with\nresonable accuracy. We can therefore only claim evidence for\na short line of first-order transitions in the range $\\gamma\n\\approx 0.35...0.40$.\n\nIn defect models of melting, no TCP seems to exist. The reason is the\nvery large activation energy of the graphs in the stress expansion\nso that there are no pretransitional excitations up to the point\nwhere the free energy intercepts the defect expansion. This is\nresponsible for a jump in the slope of the free energy, i.e. for\nfirst-order transitions \\cite{book}.\n\\newpage\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Background}\n\nIn this section, we present a brief review of time series classification, and make a focus on shapelet methods. From now on, we use calligraphic font (\\({\\cal X}\\)) to denote a collection of elements (e.g. a set of time series), capital (\\(X\\)) for one element (e.g. a time series), and lowercase (\\(x\\)) for a value of this element. In this work we consider supervised classification: the ensemble of input time series will be denoted by \\({\\cal X}=\\{X_1, ..., X_n\\}\\) with \\(X_i=\\{x_1, ..., x_m\\}\\) a time series and \\(Y=\\{y_1, ..., y_n\\}\\) their respective classes.\n\n\\subsection{Time series classification}\n\\label{sec:TSC}\nWe present a brief overview of the algorithms identified as state-of-the-art and used in our experimental section, and we report the reader to a recent review \\cite{TSCreview} for a more detailed view of the field.\n\\begin{itemize}\n\\item \\textbf{Shapelet Transform Classifier (STC)} \\cite{STC}, is regarded as a state of the art for shapelet algorithms in terms of accuracy. This algorithm iteratively initializes new shapelets, assesses their discriminative power, and removes those that are too similar. The goal being to maximize the discriminative power of an ensemble of shapelets. A Rotation Forest is then applied as a classifier.\n\\item \\textbf{Temporal Dictionary Ensemble (TDE)} \\cite{TDE} is an ensemble of dictionary-based classifiers. It uses some variants of the BOSS classifier \\cite{BOSS} and WEASEL \\cite{WEASEL}, as base estimators and optimizes their parameters through a Gaussian process.\n\\item \\textbf{Diverse Representation Canonical Interval Forest Classifier (DrCIF)} \\cite{HC2}, is an extension of the CIF algorithm \\cite{CIF}. After selecting random intervals from different representations of the time series, it uses the Catch22 \\cite{catch22} method  to extract a feature matrix.\n\\item \\textbf{RandOm Convolutional KErnel Transform (ROCKET)} \\cite{Rocket}, randomly generates a huge set of convolutional kernels, and extracts as features the maximum and the proportion of positive values of the convolution for each time series and each kernel. It is followed by a Ridge classifier. An ensemble version of this method, called ARSENAL, was introduced by \\cite{HC2}.\n\\item \\textbf{Inception-Time} \\cite{Inception} is an ensemble of Inception networks, which introduce Inception modules as replacement for traditional fully convolutional layers, notably to mitigate the vanishing gradient problem.\n\\item \\textbf{Hierarchical Vote Collective of Transformation-based Ensembles (HC1)} \\cite{HC1}is a meta-ensemble classifier using a variety of time series classifiers, such as STC, with a novel ensemble learning scheme which estimate the weight of each base classifier in the final decision. This estimation is based on performance in a 10-fold validation scheme.\n\nVariants of this method were developed, such as TS-CHIEF \\cite{TSCHIEF} and HC2 \\cite{HC2}, that both modify the set of base classifiers to improve accuracy. HC2 also modified the meta-ensemble procedure, using a Out-Of-Bag estimate instead of a 10-fold validation to estimate the performance of each base classifier, which improved scalability compared to HC1.\n\\end{itemize}\n\n\n\\subsection{Shapelets}\n\\label{sec:shapelet}\nShapelets \\cite{Shapelets} were originally defined as time series subsequences representative of class membership. In the following, we define a shapelet $S$ as a vector $S=\\{s_1, ..., s_l\\}$ with $l$ its length. All shapelet-based algorithms have the same protocol to extract features from a shapelet $S$ and a time series $X=\\{x_1, ..., x_m\\}$, by using a distance vector $f(S,X) = \\{f_1, ..., f_{m-(l-1)}\\} $ defined as :\n\n\\begin{equation}\nf_i = \\sqrt{\\sum_{j=1}^{l} (X_{i+(j-1)} - s_j)^2}\n\\end{equation}\n\nIn this definition, a point $f_i$ is simply the Euclidean distance between $S$ and the subsequence of length $l$ starting at index $i$ in $X$. The minimum value of $f(S,X)$ is then extracted as a feature, which can be interpreted as an indicator of the presence of the pattern represented by $S$ in $X$. A popular variant of this distance function consists in using a z-normalized Euclidean distance, where $S$ and all subsequences of $X$ are z-normalized independently, allowing to add scale invariance to the translation invariance of the initial formulation.\nThen, as presented in the Shapelet Transform \\cite{ST}, by using a set of shapelets $\\cal S$, one can then transform an ensemble of time series $\\cal X$ into a feature matrix of shape $(|{\\cal X}|,|{\\cal S}|)$, and use it as input in a non-temporal classifier such as a decision tree.\\newline\n\nThe step of generating and selecting shapelet candidates is the main difference between most approaches. In order to speed up the exhaustive search, Fast Shapelet \\cite{FastShp} use input discretization, while Ultra Fast Shapelet \\cite{UFShp} use random sampling. FLAG \\cite{FLAG} build shapelets location indicators from the data to reduce the set of admissible candidates, and GENDIS \\cite{GENDIS} use an evolutionary algorithm initialized by a clustering on the set of possible candidates. Learning Time Series Shapelet \\cite{LTS} use a gradient-descent optimization that iteratively change the values of a set of shapelets. MrSEQL \\cite{MrSEQL}, while not strictly speaking a shapelet algorithm, searches for discriminative symbolic sequences in a variety of symbolic representations of the inputs.\n\nSince the publication of Localized Random Shapelet (LRS) \\cite{LRS}, which showed the benefit of extracting $argmin\\ d(S,X)$ to discriminate time series based on the location of the minimum between $S$ and $X$, it has been included in most recent approaches. Based on their results, we will also use this feature in our method.\n\\section{Conclusions and future work}\nThe Random Dilated Shapelet Transform introduces new ways of increasing the global performance of shapelet algorithms, notably through the use of dilation, allowing the use of small non-contiguous subsequences as shapelets, efficiently covering areas of interest in the data. We have shown in our experiments that this new method improves on the state-of-the-art for shapelet algorithms with a good scalability compared to most of the approaches. This work offers many perspectives for future work, notably a generalized version to process uneven length or multivariate time series, as well as modifications of the shapelet generation process to better leverage class information. A more formal explainability framework is also one of our main priorities with this work, since being able to extract clear and visual explanations for domain experts is an extremely desirable property.\n\\section{Experiments}\n\\label{sec:Experiments}\nOur focus in this section is to study the influence of the four parameters of our method on classification accuracy, as well as comparing its performance to recent state-of-the-art approaches. All the experiments were run on a DELL PowerEdge R730 on Debian 9 with 2 XEON E5-2630 Corei7 (92 cores) and 64GB of RAM. We provide a python package\\footnote{https:\/\/github.com\/baraline\/convst} using community standards to run the method and the interpretability tool on any dataset, along with all result tables, and reproducibility instructions for our experiments.\n\nIn the following, we use the 112 univariate datasets from the UCR archive \\cite{Dataset} and when comparing to state-of-the-art results, we use the same resamples scheme as the one used in their experiments. We use critical difference diagrams to display the mean ranks of objects, with cliques (formed by horizontal bars) computed using the Wilcoxon-Holm post-hoc analysis \\cite{CriticalDiagram}, with a p-value of $0.05$. A clique indicates that the accuracy difference between objects is not statistically significant. \n\n\\subsection{Sensitivity Analysis}\nWe conduct a sensitivity analysis on the four input parameters of our algorithm and their effect on classification accuracy on 40 datasets selected randomly, with raw results and selected datasets in a specific file in the online repository. \nFor each parameter analysis, all other parameters remain fixed at the following default values : $n\\_shapelets = 10000$, $p\\_norm=0.9$, $L=[7,9,11]$, $P1=5$ and $P2=15$. Figure \\ref{fig:sensi_len_nshp} and Figure \\ref{fig:sensi_bound_pnorm} give the mean accuracy ranks of each method over the 40 datasets, with the accuracy of each method and each dataset computed as the mean of the same 10 resamples.\nGiven the tested set of values, the most impactful parameter is the number of shapelets, with a noticeable increase in performance above 10000 shapelets. All other parameters only display minor gains and thus seem to be stable. Based on those results, for all further experiments we set as default parameters $n\\_shapelets = 10000$, $p\\_norm=0.8$, $L=[11]$ and $P1=5$, $ P2=10$, and report results for datasets used in sensitivity analysis and the others.\n\n\\begin{figure}[h]\n  \\includegraphics[width=1.\\textwidth]{len.png}\n  \\centering\n  \\caption{Accuracy ranks for (a) different number of shapelets, and (b) different shapelet lengths}\n  \\label{fig:sensi_len_nshp}\n\\end{figure}\n\n\\begin{figure}[h]\n  \\includegraphics[width=1.\\textwidth]{ppnorm.png}\n  \\centering\n  \\caption{Accuracy ranks for (a) different percentiles bounds, and (b) proportion of z-normalized shapelets}\n  \\label{fig:sensi_bound_pnorm}\n\\end{figure}\n\n\\subsection{Scalability}\nWe perform a comparison of the scalability of our approach against Hive-Cote 1.0 (HC1), Hive-Cote 2.0 (HC2), DrCIF, ROCKET, and the Shapelet Transform Classifier (STC). Note that when used as a component in HC1 and HC2, STC is by default subject to a time contract of two hours. Except from this default configuration in HC1 and HC2, we are not setting any time contract in other algorithms. Both STC and RDST are by default sampling 10000 shapelets, and ROCKET use 10000 kernels.\n\nWe are aware that the runtime of HC1, HC2 and STC could be reduced with time contracts. But, as our goal in this section is to contextualize the gain in classification accuracy against the time complexity of each method, we present the results with the time contracts used to generate the accuracy results of the next section.\n\nWe use the Crop Dataset and the Rock Dataset of the UCR archive for evaluating the scalability respectively on the number of time series and their length. As all competing algorithms implemented in the sktime package of \\cite{sktime} can use parallel processing, we set each algorithm to use 90 cores. Figure \\ref{fig:scal} reports the mean training time over 10 resamples, showing the very competitive scalability of RDST. Note that due to job time limitation on our machine and the computational cost of HC2, we could not consider all samples for the Crop dataset. We report the reader interested in the implementation details of our algorithm to the web page of the project.\n\n\\begin{figure}[h]\n  \\includegraphics[width=1.\\textwidth]{scalability.png}\n  \\centering\n  \\caption{Result of the scalability study of the competing algorithms for current state-of-the-art, for (a) number of time series and (b) time series length. Y-axis use log-scale.}\n  \\label{fig:scal}\n\\end{figure}\n\n\\subsection{Comparative study}\nWe present the results of our comparative study using the mean accuracy over the same 30 resamples for each of the 112 datasets as HC2 \\cite{HC2} used in their study, and compare our approach against their experimental result. Figure \\ref{fig:ranksdiv} gives the mean accuracy rank of each method over the 40 datasets used for setting the defaults parameters in sensitivity analysis, and for the 72 others. The full result tables including standard deviation per dataset and more visualizations of the results are available online as supplementary materials. \n\n\\begin{figure}[h]\n  \\includegraphics[width=1.0\\textwidth]{ranks2.png}\n  \\centering\n  \\caption{Mean accuracy ranks of each method for the 40 dataset used in sensitivity analysis and the 72 others.}\n  \\label{fig:ranksdiv}\n\\end{figure}\n\nGiven the scalability and simplicity of our method, having an accuracy comparable to the prior developments of HC2 and to deep learning approaches is a very promising result. Notably for future developments where focus would shift to accuracy rather than scalability. For reference, using RDST without any distance normalization is equivalent to STC in terms of mean accuracy rank, with the same protocol as above.\n\n\\subsection{Interpretability}\nGiven a set of $M$ shapelets, RDST generates $3M$ features. Each feature is linked to a weight for each class in the Ridge classifier, as it is trained in a one-vs-all fashion. \nGiven a class, we can then visualize either global or local information. Locally, we can inspect a shapelet to show how it discriminates the current class, and where the shapelet is positioned with either training or testing data, as shown in Figure \\ref{fig:interp}. Globally, we can display the distribution of weights for each feature type ($min$, $\\argmin$ and $SO$) or by shapelet characteristics such as length, dilation, or use of normalization as shown in Figure \\ref{fig:interp2}. \nWhile this only provides a basic interpretation of the results, we believe a more formal framework could be developed to extract explanations from this data.\n\\begin{figure}[h!]\n  \\includegraphics[width=1.0\\textwidth]{interp1.png}\n  \\centering\n  \\caption{The most important shapelet for class 0 of the Coffee dataset, according to weights of the Ridge classifier, with distribution displayed on the testing data, and two testing samples for visualization.}\n  \\label{fig:interp}\n\\end{figure}\n\\begin{figure}[h!]\n  \\includegraphics[width=1.\\textwidth]{inter2.png}\n  \\centering\n  \\caption{A global interpretation of RDST, with (a) distribution of weights for each type of feature, and (b) distribution of weights per dilation.}\n  \\label{fig:interp2}\n\\end{figure}\n\n\\section{Introduction}\nTime series occur in a multitude of domains, covering a wide range of applications, which have impacts in many parts of society. The ever-increasing quantity of data and the publications of laws regarding models interpretability are setting new constraints for applications across industries. \n\nRecent research in time series classification produced highly accurate classifiers, using either deep learning approaches \\cite{Inception}, or meta-ensemble methods \\cite{HC1,HC2}. Despite being the most accurate approaches, they are among the slowest, which make them hard to apply on use-cases with huge amount of data. \nOther methods, based on random approaches, notably the RandOm Convolutional KErnel Transform (ROCKET) \\cite{Rocket}, achieve comparable accuracy with extreme scalability. Even though recent works on post-hoc methods and specific frameworks \\cite{XAITS} improved the interpretability of those approaches, they lack a \"by design\" interpretability.\n\nOn the other hand, time series shapelets \\cite{Shapelets} have been widely used in time series classification for their ease of interpretation, which is a critical aspect to some application domains such as health and security. \nThe downside is that shapelet algorithms are often outperformed by recent approaches, both in terms of accuracy and scalability. Most Shapelet approaches tried to solve the scalability issues at the expense of some classification accuracy, notably through the use of symbolic approximation techniques \\cite{FastShp}, while others used random shapelets \\cite{UFShp}. Recently, a symbolic sequence ensemble learning \\cite{MrSEQL} method was proposed, which improved the predictive power of approximation-based methods, while other work focused on finding a new discriminative feature \\cite{LRS} to consider during the extraction process.\n\nIn this work, we present the Random Dilated Shapelet Transform, an adaptation of time series shapelets that includes the notion of dilation, one of the core mechanism of the success of convolutional kernel approaches. We also extend on the work of \\cite{LRS} and introduce a new feature to enhance the discriminative power of shapelets. Our contributions can be summarized as follows:\n\n\\begin{itemize}\n\\item an adaptation of time series shapelets allowing the use of dilation, and a feature to capture a new discriminative property of shapelets, \n\\item an interpretable, scalable and accurate shapelet algorithm, which allows shapelet based algorithm to catch-up with the state-of-the-art,\n\\item an experimental study about the sensitivity of our method parameters and a comparative study against the state-of-the-art algorithms for time series classification.\n\\end{itemize}\n\n\n\\section{Proposed method}\n\\label{sec:RDST}\nIn this section, we introduce the main components of our method: the use of dilation in the shapelet formulation and the features extracted from the distance vector between a shapelet and a time series. We put emphasis on the dilation and on the Shapelet Occurrence feature that are new contributions to shapelet algorithms. We give some simple visual examples to illustrate these notions, and report the visualization on real data to the experimental section.\n\n\n\\subsection{Dilated Shapelets}\nTo introduce the notion of dilation in shapelets, we define now a shapelet $S$ as $S=\\{ \\{v_1, ..., v_l\\}, d \\}$ with $l$ the length parameter and $d$ the dilation parameter. In practice, the dilation is used in the distance function $f$, where each value of the shapelet will be compared to a dilated subsequence of the input time series. More formally, consider a time series $X=\\{x_1, ..., x_m\\}$ and a dilated shapelet $S$, we now define $f(S,X) = \\{f_1, ..., f_{m-(l-1)\\times d}\\}$ as :\n\\begin{equation}\nf_i = \\sqrt{\\sum_{j=1}^{l} (X_{i+(j-1) \\times d} - s_j)^2}\n\\end{equation}\n\nThe interest of using dilation in shapelets is to make them non-contiguous subsequences. It allows a shapelet to either match a non-contiguous pattern, or a contiguous one, by focusing on key points of the pattern without covering it entirely, as illustrated in Figure \\ref{fig:ex_dil_shp}. Note that this formulation is equivalent to the original shapelet formulation when $d=1$.\n\n\\begin{figure}[h]\n  \\includegraphics[width=0.9\\textwidth]{ex_dil_shp.png}\n  \\centering\n  \\caption{An example of two possible shapelets (in orange), positioned on a synthetic pattern (in blue): (a) one without dilation, and (b) a much smaller one but with dilation}\n  \\label{fig:ex_dil_shp}\n\\end{figure}\n\n\\subsection{Shapelet Occurrence feature}\n\\label{sec:SO}\nIf we consider a shapelet $S$ and two time series $X_1$ and $X_2$, we can imagine multiple ways of discriminating $X_1$ and $X_2$ using $S$.\n\\begin{itemize}\n\\item $S$ can be present in $X_1$ but not in $X_2$. This is captured by $min\\ f(S,X_i)$, with smaller distances indicating better matches between the shapelet and the series.\n\\item $S$ can be present in both series, but not at the same place. This is captured by the $argmin$ feature introduced by LRS \\cite{LRS}.\n\\item $S$ can be present in both series, but not at the same scale. In this case, a normalized distance would not be able to discriminate the series.\n\\item $S$ can be present in both series, but occurs a different number of times in $X_1$ compared to $X_2$. This is captured by a new feature, called Shapelet Occurrence (SO).\n\\end{itemize}\n\nThose points are illustrated in Figure \\ref{fig:shp_disc}. Deciding whether scaling is important or not is highly dependent on the application, but without prior knowledge, one cannot know which to choose. For this reason, we introduce a parameter in Section \\ref{sec:RandDST} allowing to tune the amount of normalized shapelets.\n\n\\begin{figure}[h]\n  \\includegraphics[width=1.\\textwidth]{class_diff.png}\n  \\centering\n  \\caption{Synthetic examples of possible discriminative properties between two classes}\n  \\label{fig:shp_disc}\n\\end{figure}\n\nTo the best of our knowledge, the number of occurrences of a shapelet has never been considered as a feature. This requires another modification to the definition of $S$ as $S=\\{ \\{v_1, ..., v_l\\}, d, \\lambda \\}$, with $\\lambda$ a threshold allowing us to compute the Shapelet Occurrence ($SO$) feature as $SO = |\\{i| f(S,X)_i <\\lambda\\}|$. Although the parameter $\\lambda$ could be set randomly, we discuss in Section \\ref{sec:RandDST} a method to set the value of this threshold.\n\n\\subsection{Random Dilated Shapelet Transform (RDST)}\n\\label{sec:RandDST}\nOur objective for this algorithm is to produce an accurate but scalable approach. As our shapelet formulation adds attributes compared to the initial formulation \\cite{Shapelets}, optimizing a set of dilated shapelets with a threshold $\\lambda$ will be costly, and this explains why we choose a random approach.\n\nFor simplicity, we present our approach in the context of univariate and even length time series, with ${\\cal X} = \\{X_1, ..., X_n\\}$  a set of time series ($X_i = \\{x_1, ..., x_m \\}$) and $Y=\\{y_1, ..., y_n\\}$ their respective classes. Our method takes as input four parameters that are: $n\\_shapelets$ the number of shapelets to generate, $L$ a set of possible lengths for the shapelets, $p\\_norm$ the proportion of shapelets that will use z-normalization, and $(P_1, P_2) \\in [0,100]$ a pair used as percentile bounds for the sampling of the threshold $\\lambda$. \n\nGiven the definition of a shapelet $S=\\{ \\{v_1, ..., v_l\\}, d, \\lambda \\}$, we initialize each parameter as follows:\n\\begin{itemize}\n\\item the length $l$ is uniformly drawn from $L$,\n\\item the dilation $d$, in the same way as ROCKET \\cite{Rocket}, is set to $d = \\left\\lfloor 2^x \\right\\rfloor$ with $x$ uniformly drawn in $[0, log_2 \\frac{m}{l}]$,\n\\item we randomly choose whether the shapelet will use a z-normalized distance with probability $p\\_norm$,\n\\item for setting the values, a sample $X$ is uniformly drawn from ${\\cal X}$, and an admissible start point $i$ (given $l, d$) is randomly selected. Then, values are set to $[X_i, ..., X_{i+(l-1)\\times d}]$.\n\\item finally, given a shapelet $S$, to fix the value of $\\lambda$,  we take a sample $X$ from the same class as the one used for extracting the shapelet value, and uniformly draw a value between the two percentiles $(P_1, P_2)$ of $f(S,X)$.\n\\end{itemize}\n\nThe strategy employed to find $\\lambda$ is a classic trade-off between time and accuracy. If scalability was not a focus, we could compute the distance vector for more samples in ${\\cal X}$, and optimize the value of $\\lambda$ based on an information measure.\nAfter computing the distance vector between all pairs of time series and shapelets, the output of our method is a feature matrix of size $(|{\\cal X}|,3 \\times n\\_shapelets)$, with the three features extracted from the distance vector $f(S,X)$ being the $min, argmin, SO(S,X)$.\n\nFollowing the arguments of the authors of ROCKET \\cite{Rocket}, we use a Ridge Classifier after the transformation of ${\\cal X}$, as the L2 regularization used in Ridge is of critical importance due to the high number of features that are generated, while being scalable and interpretable.\n\\subsubsection*{Acknowledgements}\nThis work is supported by the ANRT CIFRE grant n\u00b02019\/0281 in partnership with Worldline and the University of Orl\u00e9ans.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nWe present a graphical hidden variable framework for modeling non-negative sequential data with hierarchical structure. Our framework is intended for applications where the observed data is non-negative and is well-modeled as a non-negative linear combination of underlying non-negative components. Provided that we are able to adequately model these underlying components individually, the full model will then be capable of representing any observed additive mixture of the components due to the linearity property. This leads to an economical modeling representation, since a compact parameterization can explain any number of components that combine additively. Thus, in our approach, we do not need to be concerned with explicitly modeling the maximum number of observed components nor their relative weights in the mixture signal.\n\nTo motivate the approach, consider the problem of computational auditory scene analysis (CASA), which involves identifying auditory ``objects'' such as musical instrument sounds, human voice, various environmental noises, etc, from an audio recording. Speech recognition and music transcription are specific examples of CASA problems. When analyzing audio, it is common to first transform the audio signal into a time-frequency image, such as the spectrogram (i.e., magnitude of the short-time Fourier transform (STFT)). We empirically observe that the spectrogram of a mixture of auditory sources is often well-modeled as a linear combination of the spectrograms of the individual audio sources, due to the sparseness of the time-frequency representation for typical audio sources. For example, consider a recording of musical piece performed by a band. We empirically observe that the spectrogram of the recording tends to be well-approximated as the sum of the spectrograms of the individual instrument notes played in isolation. If one could construct a model using our framework that is capable of representing any individual instrument note played in isolation, the model would then automatically be capable of representing observed data corresponding to arbitrary non-negative linear combinations of the individual notes. Likewise, if one could construct a model under our framework capable of representing a recording of a single human speaker (possibly including a language model), such a model would then be capable of representing an audio recording of multiple people speaking simultaneously. Such a model would have obvious applications to speaker source separation and simultaneous multiple-speaker speech recognition. We do not attempt to construct such complex models in this paper, however. Rather, our primary objective here will be to construct models that are simple enough to illustrate the interesting properties of our approach, yet complex enough to show that our approach is noise-robust, capable of learning from training data, and at least somewhat scalable. We hope that the results presented here will provide sufficient motivation for others to extend our ideas and begin experimenting with more sophisticated PFNs, perhaps applying them to the above-mentioned CASA problems. \n\nAn existing area of research that is related to our approach is non-negative matrix factorization (NMF) and its extensions. NMF is a data modeling and analysis tool for approximating a non-negative matrix $X$ as the product of two non-negative matrices $W$ and $H$ so that the reconstruction error between $X$ and  $W H$ is minimized under a suitable cost function. NMF was originally proposed by Paatero as \\emph{positive matrix factorization} \\cite{paatero_1994}. Lee and Seung later developed robust and simple to implement multiplicative update rules for iteratively performing the factorization \\cite{Lee_seung}. Various sparse versions of NMF have also been recently proposed \\cite{hoyer_sparse_NMF}, \\cite{cichockiNMF}, \\cite{nsNMF2006}. NMF has recently been applied to many applications where a representation of non-negative data as an additive combination of non-negative basis vectors seems reasonable. Such applications include object modeling in computer vision, magnitude spectra modeling of audio signals \\cite{wang_mag_spectrogram_NMF}, and various source separation applications \\cite{NMFParis}. The non-negative basis decomposition provided by NMF is, by itself, not capable of representing complex model structure. For this reason, extensions have been proposed to make NMF more expressive. Smaragdis extended NMF in \\cite{NMFParis} to model the temporal dependencies in successive spectrogram time slices. His NMF extension, which he termed \\emph{Convolutive NMF}, also appears to be a special case of one of our example models in Section~\\ref{sec:sparseHierarchicalModel}.  We are unaware of any existing work in the literature that allow for the general graphical representation of complex hidden variable models, particularly sequential data models, that is provided by our approach, however. \n\nA second existing area of research that is related to our approach is probabilistic graphical models \\cite{jordan_graphical_models} and in particular, dynamic Bayesian networks (DBNs) \\cite{Murphy_Thesis}, \\cite{Friedman1999}, which are probabilistic graphical models for sequential data. We note that the Hidden Markov Model (HMM) is a special case of a DBN. DBNs are widely used for speech recognition and other sequential data modeling applications. Probabilistic graphical models are appealing because they they can represent complex model structure using an intuitive and modular graphical modeling representation. A drawback is that the corresponding exact and\/or approximate inference and learning algorithms can be complex and difficult to implement, and overcoming tractability issues can be a challenge.\n\nOur objective in this paper is to present a framework for modeling non-negative data that retains the non-negative linear representation of NMF, while also supporting more structured hidden variable data models with a graphical means for representing variable interdependencies analogously to that of the probabilistic graphical models framework. We will be particularly interested in developing models for sequential data consisting of the spectrograms of audio recordings. Our framework is essentially a modular extension of NMF in which the full graphical model corresponds to several coupled NMF sub-models. The overall model then corresponds to a system of coupled vector or matrix factorization equations. Throughout this paper, we will refer to a particular system of factorizations and the corresponding graphical model as a \\emph{positive factor network (PFN)}. We will refer to the dynamical extension of a PFN as a \\emph{dynamic positive factor network (DPFN)}. Given an observed subset of the PFN model variables, we define inference as solving for the values of the hidden subset of variables and learning as solving for the model parameters in the system of factorization equations. \n\nNote that our definition of inference is distinct from the probabilistic notion of inference. In a PFN, inference corresponds to solving for actual values of the hidden variables, whereas in a probabilistic model inference corresponds to solving for probability distributions over the hidden variables given the values of the observed variables. Performing inference in a PFN is therefore more analogous to computing the MAP estimates for the hidden variables in a probabilistic model. One could obtain an analogous probabilistic model from a PFN by considering the model variables to be non-negative continuous-valued random vectors and defining suitable conditional probability distributions that are consistent with the non-negative linear variable model. Let us call this class of models \\emph{probabilistic PFNs}.  Exact inference is generally intractable in such a model since the hidden variables are continuous-valued and the model is not linear-Gaussian. However, one could consider deriving algorithms for performing approximate inference and developing a corresponding EM-based learning algorithm. We are unaware of any existing algorithms for performing tractable approximate inference in a probabilistic PFN. It is possible that our PFN inference algorithm may also have a probabilistic interpretation, but exploring the idea further is outside the scope of this paper. Rather, in this paper our objective is to to develop and motivate the inference and learning algorithms by taking a modular approach in which existing NMF algorithms are used and coupled in a way that seems to be intuitively reasonable. We will be primarily interested in empirically characterizing the performance of the proposed inference and learning algorithms on various example PFNs and test data sets in order to get a sense of the utility of this approach to interesting real-world applications.\n\nWe propose general joint inference and learning algorithms for PFNs which correspond to performing NMF update steps independently (and therefore potentially also in parallel) on the various factorization equations while simultaneously enforcing coupling constraints so that variables that appear in multiple factorization equations are constrained to have identical values. Our empirical results show that the proposed inference and learning algorithms are fairly robust to additive noise and have good convergence properties. By leveraging existing NMF multiplicative update algorithms, the PFN inference and learning algorithms have the advantage of being straightforward to implement, even for relatively large networks. Sparsity constraints can also be added to a module in a PFN model by leveraging existing sparse NMF algorithms. We note that the algorithms for performing inference and learning in PFNs should be understandable by anyone with a knowledge of elementary linear algebra and basic graph theory, and do not require a background in probability theory. Similar to existing NMF algorithms, our algorithms are highly parallel and can be optimized to take advantage of parallel hardware such as multi-core CPUs and potentially also stream processing hardware such as GPUs. More research will be needed to determine how well our approach will scale to very large or complex networks.\n\nThe remainder of this paper has the following structure. In Section~\\ref{sec:pfn_main_section}, we present the basic PFN model. In Section~\\ref{sec:main_factored_model}, we present an example of how a DPFN can be used to represent a transition model and present empirical results. In Section~\\ref{sec:main_hierarchical_state_model}, we present an example of using a PFN to model sequential data with hierarchical structure and present empirical results for a regular expression example. In Section~\\ref{sec:target_tracking}, we present a target tracking example and provide results for synthetic observation data which serve to illustrate the interesting properties of PFNs and motivate their potential usefulness in applications such as music transcription, source separation, and speech recognition. We show how a target process characterized by a hierarchical state transition model can be represented by a PFN. Our results illustrate that a PFN which is defined in terms of a single target observation can then be used to effectively track the states of multiple simultaneous targets in the observed data. In Section~\\ref{sec:hierarchicalSeqDecomp} we present results for an example in which meaningful hierarchical features are extracted from a spectrogram. Such a hierarchical representation could be useful for music transcription and source separation applications. In Section~\\ref{sec:main_language_model}, we propose a DPFN for modeling the sequence of words or characters in a text document as an additive factored transition model of word features. We also propose slightly modified versions Lee and Seung's update rules to avoid numerical stability issues. The resulting modified update rules are presented in Appendix~\\ref{appendix_nmf}. \n\n\n\\section{Positive Factor Networks}\n\\label{sec:pfn_main_section}\n\nIn this section, we specify the basic data model and present a graphical representation. We then propose inference algorithms for solving for the hidden variables and learning the model parameters. \n\n\\subsection{Model specification}\n\\label{sec:model_specification}\n\nWe now specify a data model for a set of non-negative continuous vector-valued variables $\\{x_i: i = 1, \\dots, N\\}$ where the dimension of each $x_i$, in general, can be distinct. We will refer to the set $\\{x_i\\}$ as the \\emph{model variables}. We assume that a subset $X_E$ of the model variables is observed and the rest of the variables comprise the hidden subset, $X_H$. \n\nThe model is specified by a system of $Q$ of non-negative factorization equations where the $j$'th equation is given by:\n\n\\begin{align}\nx_{f(j,0)} =& \\sum_{k=1}^{P_j} W^j_k x_{f(j,k)} \\notag \\\\\n=&  \\left[ \\begin{array}{cccc} W^j_1 & W^j_2 & \\dots & W^j_{P_j} \\end{array} \\right] \\left[ \\begin{array}{c} x_{f(j,1)}\\\\\nx_{f(j,2)} \\\\\n\\vdots \\\\\nx_{f(j,{P_j})} \\end{array} \\right] \\notag \\\\\n=& W^j x^j\n\\label{eq:factorization_equation}\n\\end{align}\n\nwhere $P_j \\geq 1$ for all $j \\in \\{1, \\dots, Q\\}$ and all $W^j_k$ are non-negative matrices with a possibly distinct number of columns for each $k$. The function $f(j,k)$ maps each $(j,k)$ above into a corresponding index $i \\in \\{1, \\dots, N\\}$ and satisfies $f(j,k) = f(m,n)$ implies $j \\ne m$. Thus, $x_{f(j,k)}$ refers to one of the model variables $x_i$, and it is possible for a given variable $x_i$ to appear in multiple equations above but only at most once in any given equation. The matrix $W^j$ is defined as the horizontal concatenation of the non-negative $W^j_k$ matrices.\n\nSince there are no constant terms in the above equations, the system corresponds to a a homogeneous system of linear equations, subject to a non-negativity constraint on the variables \\{$x_i$\\}, which can be written as:\n\n\\begin{align}\nA y = 0 \\text{ , where } y = \\left[ \\begin{array}{c} x_1\\\\\nx_2 \\\\\n\\vdots \\\\\nx_N \\end{array} \\right] \\geq 0\n\\end{align}\n\nNote that $A$ will contain both negative and positive values, however, since all of the terms are grouped together on one side. It follows that our model satisfies the linearity property subject to non-negativity constraints (non-negative superposition property): if $y_1$ and $y_2$ are solutions to the system, then $\\alpha_1 y_1 + \\alpha_2 y_2$ is also a solution, for any choice of scalars $\\alpha_1 > 0, \\alpha_2 > 0$. \n\n\n\n\\subsection{Graphical representation}\nWe now develop a graphical representation to facilitate the visualization of the local linear relationships between the model variables in the various factorization equations. Hidden variables correspond to shaded nodes in the graph, and observable variables correspond to unshaded nodes. We construct the graphical model such that the $j$'th factorization equation corresponds to a subgraph with $x_{f(j,0)}$ as the child node and the $\\{x_{f(j,k)} : k = 1, \\dots, P_j\\}$ as the parent nodes. Thus, a child node variable is a linear function its parent nodes in the subgraph. We then arrive at the complete graphical model by superimposing the subgraphs corresponding to the various factorization equations. We allow the same variable $x_i$ to correspond to the child node in multiple subgraphs (i.e., distinct factorization equations), provided that its parents do not overlap between the subgraphs. That is, we do not allow a single arc to correspond to multiple linear relationships. We annotate arcs with dash marks where necessary, to disambiguate subsets of parent nodes that correspond to the same factorization equation. Arcs annotated with the same number of dash marks connecting a child node to its parents denote a subset of parent nodes that corresponds to a single factorization equation. The pseudocode in Algorithm~\\ref{alg:create_graph} outlines a procedure for creating a graphical model representation from a system of factorization equations. \n\n\\begin{algorithm}\n\\caption{Create a directed graph from a system of factorizations.}\n\\label{alg:create_graph}\n\\begin{tabbing}\n{\\bf for} \\= {$j$ = 1 to $Q$}  \\\\\n\\> \/\/ for each factorization equation $x_{f(j,0)} = \\sum_{k=1}^{P_j} W^j_k x_{f(j, k)}$ \\\\\n\\> Create a node for the corresponding variable $x_{f(j, 0)}$, if it does not already exist. \\\\\n\\> $dashCount \\gets$ 1 + maximum dash count on any existing arc from a parent node of $x_{f(j, 0)}$ to $x_{f(j, 0)}$. \\\\\n\\> {\\bf for} \\= {$k$ = 1 to $P_j$} \\\\\n\\> \\> {\\bf if} \\= a node corresponding to $x_{f(j, k)}$ does not already exist \\\\\n\\> \\> \\> Create a node corresponding to $x_{f(j, k)}$ \\\\\n\\> \\> \\>  {\\bf if} \\= $x_{f(j, k)}$ is observed \\\\\n\\> \\> \\> \\> Shade the node corresponding to $x_{f(j, k)}$ \\\\\n\\> \\> \\> {\\bf end} \\\\\n\\> \\> {\\bf end} \\\\\n\\> \\> Create a directed arc from the $x_{f(j,k)}$ node to the $x_{f(j, 0)}$ node. \\\\\n\\> \\> Annotate the arc with the number of dash marks given by the current value of $dashCount$. \\\\\n\\> \\> Annotate the arc with $W^j_k$ \\\\\n\\> {\\bf end} \\\\\n{\\bf end} \n\\end{tabbing}\n\\end{algorithm}\n\n\nAs an example, consider the set of variables \\{$x_1, x_2, \\dots, x_{12}$\\} that are described by the following system of factorization equations:\n\n\\begin{align}\n\\label{eqn:factor1}\nx_1 =& W_1 x_5 + W_2 x_6 \\\\\n\\label{eqn:factor2}\nx_2 =& W_3 x_6 + W_4 x_7 \\\\\n\\label{eqn:factor3}\nx_2 =& W_5 x_8 + W_6 x_9 \\\\\n\\label{eqn:factor4}\nx_2 =& W_7 x_{10} \\\\\nx_3 =& W_8 x_{10} \\\\\n\\label{eqn:factor6}\nx_3 =& W_9 x_{11} \\\\\n\\label{eqn:factor7}\nx_4 =& W_{10} x_{11} \\\\\nx_6 =& W_{11} x_{12} \\\\\nx_{10} =& W_{12} x_{12} \\\\\n\\label{eqn:factor10}\nx_{11} =& W_{13} x_{12}\n\\end{align}\n\nFigure~\\ref{fig:sampleModel1} shows the graphical model associated with the above system of factorizations. Note that nodes $x_3$ and $x_4$ are connected by a dashed line. This is an optional annotation that specifies that the corresponding connected variables have \\emph{forced factored co-activations}, or simply \\emph{forced co-activations}. So far the only constraint we have placed on the parameter matrices \\{$W_i$\\} is that they be non-negative. However, in some models we might wish to place the additional constraint that the columns $\\{w_n\\}$ of $W_i$ are normalized in some way. For example, we might consider requiring that each $w_n$ have column sum = 1. Consider the subgraph corresponding to Equations (\\ref{eqn:factor6}) and (\\ref{eqn:factor7}). The dashed line connecting $x_3$ and $x_4$ specify that these variables are factorized in terms of a common parent ($x_{11}$ in this case) and that the columns of the corresponding \\{$W_i$\\} ($W_9$ and $W_{10}$ in this case) are normalized to have unit sum. This constraint ensures that these three variables will have equal column sums. Thus, if any one of these variables is observed in the model, then we can infer that all three must have the same column sum. An activation of the parent $x_{11}$ implies that its children $x_3$ and $x_4$ will then be activated with the same column sum (activated together), hence the term forced co-activations.\n\nConsider the subgraph corresponding to Equation (\\ref{eqn:factor1}). An activation in $x_1$ (i.e., $x_1$ is nonzero) could be explained by just one of the parents being nonzero. Now consider the subgraph consisting of $x_2$ and its parents, corresponding to Equations (\\ref{eqn:factor2}),  (\\ref{eqn:factor3}), and (\\ref{eqn:factor4}). In this case, an activation of $x_2$ corresponds to at least one of $x_6$ and $x_7$ being activated, at least one of $x_8$ and $x_9$ being activated, as well as $x_{10}$ being activated.\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=60ex]{.\/figures\/sampleModel1.pdf}\n\\caption{A graphical model corresponding to the example system of factorizations in Equations~\\ref{eqn:factor1} - \\ref{eqn:factor10}.}\n\\label{fig:sampleModel1}\n\\end{figure}\n\n\nWe borrow the notion of a \\emph{plate} from the probabilistic graphical models formalism \\cite{jordan_graphical_models}, in which plates are used to represent replicated random variables. A plate consists of a box that is drawn around the replicated variables, with the replication count specified in a corner. Figure~\\ref{fig:simpleNmfPlate} shows an example of using the graphical plate notation. This graphical model corresponds to the following system of factorization equations:\n\n\\begin{align}\n\\label{eqn:nmf_vector}\nx^1_1 =& W_1 x^2_1 \\notag \\\\\nx^1_2 =& W_1 x^2_2 \\notag \\\\\n\\vdots \\notag \\\\\nx^1_N =& W_1 x^2_N\n\\end{align}\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=25ex]{.\/figures\/simpleNmfPlate.pdf}\n\\caption{An example of using plate notation, corresponding to $N$ observable variables \\{$x^1_i$\\} and $N$ hidden variables \\{$x^2_i$\\}. This model corresponds to standard NMF.}\n\\label{fig:simpleNmfPlate}\n\\end{figure}\n\nLetting $X^1 = \\left[ \\begin{array}{cccc} x^1_1 & x^1_2 & \\dots & x^1_N \\end{array} \\right]$, and letting $X^2 = \\left[ \\begin{array}{cccc} x^2_1 & x^2_2 & \\dots & x^2_N \\end{array} \\right]$, we can then write the system of factorization equations more compactly as the matrix equation:\n\n\\begin{align}\n\\label{eqn:nmf_matrix}\nX^1 = W_1 X^2\n\\end{align}\n\nNote that this corresponds to standard NMF, since the observed $X^1$ matrix is factored as the product of two non-negative matrices. \n\n\n\\subsection{Algorithms for Inference and Learning}\n\\label{sec:inference_and_learning}\n\n\n\nTypically, a subset of the model variables $X_E = \\{x_{E_n}: n = 1, \\dots, N_E\\}$ is observed, and we are interested in solving for the values of the hidden variables $X_H = \\{x_{H_n}: n = 1, \\dots, N_H\\}$, which we refer to as the \\emph{inference problem}. We are also interested in solving for the values of the model parameters $\\theta = \\{W^1, W^2, \\dots, W^Q\\}$, which we refer to as the \\emph{learning problem}. The observed variables may deviate from the modeling assumptions and\/or contain noise so that an exact solution will not be possible in general. We will thus seek an approximate solution by optimizing a reasonable cost function.\n\nGiven a subset of observable variables and the model parameters, we define the inference solution as the values of the hidden variables that minimize some reasonable cost function $g(\\theta, X_H, X_E)$. That is, we consider $X_E$ and $\\theta$ to be fixed and solve for $X_H$:\n\n\\begin{align}\nX_H = \\argmin_{X_H} g(\\theta, X_H, X_E)\n\\end{align}\n\nThe learning problem corresponds to solving for $\\theta$ given observations $X_E$. The joint learning and inference problem corresponds to minimizing $g(\\theta, X_H, X_E)$ jointly for $\\theta$ and $X_H$ given $X_E$: \n\n\\begin{align}\n(\\theta, X_H) = \\argmin_{\\theta, X_H} g(\\theta, X_H, X_E)\n\\end{align}\n\nThe cost function should have the property that its value approaches zero as the approximation errors of the system of factorization equations approach zero. One possibility is the following function, specified as the sum of the squared approximation errors of the factorization equations (\\ref{eq:factorization_equation}):\n\n\\begin{align}\ng(\\theta, X_H, X_E) =& \\sum_{j = 1}^Q ||x_{f(j,0)} -W^j x^j ||^2 \\notag \\\\\n=& \\sum_{j = 1}^Q ||x_{f(j,0)} -W^j \\left[ \\begin{array}{c} x_{f(j,1)}\\\\\nx_{f(j,2)} \\\\\n\\vdots \\\\\nx_{f(j,{P_j})} \\end{array} \\right] ||^2 \n\\end{align}\n\nOther possibilities could involve using the generalized KL-divergence (\\ref{eq:kl_div}), for example. However, we will not develop inference and learning algorithms by directly optimizing any particular cost function, and so will not be too concerned with its exact form. Rather, we propose algorithms that seem intuitively reasonable and for which we have observed good empirical convergence properties on test data sets, but do not offer any proof of convergence, even to a local minimum of any particular cost function. We will only make use of a particular cost function in order to quantify the empirical performance of our algorithms.\n\n\n\nWe require that the graphical model correspond to a directed acyclic graph (DAG). Since there are no cycles, the graph can be arranged so that it flows from top to bottom. We then rename each of the model variables $\\{x_n : n=1, \\dots, N\\}$ to $x^l_i$ where $l = 1,\\dots, L$ denotes the level of the variable (vertical index), and $i$ is its position within the level (horizontal index). We then have a graph with $L$ levels, such that the top level nodes (level $L$) have no parents and the bottom level nodes (level 1) have no children. A level is defined such that no pair of nodes with a parent-child relationship are allowed to be in the same level. That is, for any pair of variables $(x^l_i, x^l_j), i \\ne j$ in the same level $l$, we disallow that $x^l_i$ is the parent of $x^l_j$ or vice versa.  If an arc exists between a higher level variable and a lower level variable, then the higher level variable must be the parent of the lower level variable. That is, we require that for any two variables $(x^m_i, x^n_j), \\text{ s.t }. m > n$, $x^m_i$ cannot be a child of $x^n_j$. For example, the graph in Figure~\\ref{fig:sampleModel1} corresponds to an $L = 3$ level graph with the following renamed variables. Variables $(x_1, \\dots, x_4)$ would be renamed to $(x^1_1, \\dots, x^1_4)$. Variables $(x_5, \\dots, x_{11})$ would be renamed to $(x^2_1, \\dots, x^2_7)$. Variable $x_{12}$ would be renamed to $x^3_1$.\n\n\nThe inference and learning algorithm will require a set of \\emph{local variables} \\{$v^j_k$\\}. We use the term local variable because a given $v^j_k$ is associated only with the $j$'th factorization equation, unlike a model variable $x_{f(j,k)}$ which is allowed to appear in multiple factorization equations. Specifically, a distinct $v^j_k$ will be associated with  each allowable combination of $j$ and $k$ that appears in the factorization equations in (\\ref{eq:factorization_equation}). Thus, several distinct $v^j_k$ may be associated with a given model variable and this will be the case when a given model variable appears in more than one factorization equation. Replacing the $x_{f(j,k)}$ with the associated $v^j_k$, we then have $Q$ factorization equations $FactorSystem = \\{eq_j : j = 1, \\dots, Q\\}$  where equation $eq_j$ is given by:\n\n\\begin{align}\nv^j_0 =& \\sum_{k=1}^{P_j} W^j_k v^j_k \\notag \\\\\n=&  \\left[ \\begin{array}{cccc} W^j_1 & W^j_2 & \\dots & W^j_{P_j} \\end{array} \\right] \\left[ \\begin{array}{c} v^j_1\\\\\nv^j_2 \\\\\n\\vdots \\\\\nv^j_{P_j} \\end{array} \\right] \\notag \\\\\n=& W^j v^j\n\\label{eq:factorization_equation333}\n\\end{align}\n\nWe say that the above equations are in a \\emph{consistent state} if each model variable $x_{f(j,k)}$ and all of its associated local variables \\{$v^j_k$\\} have the same value. Otherwise, we say that the equations are in an \\emph{inconsistent state}. Note being in a consistent state does not imply that the corresponding $x_{f(j,k)}$ actually constitute a solution to the system.\n\nThe basic idea of our approach is to learn a generative model of data by iteratively performing inference and learning in a bottom-up pass through the network and then perform data generation through activation propagation in a top-down pass. In the inference and learning pass, parent node values (activations) and parameters are updated based on the values of their child node variables. These inference and learning updates are performed locally using NMF algorithms. Once the top level nodes have been updated, we then propagate values downward to the lowest level nodes in a data generation pass. This is performed by computing the new child node values as the value of the right hand side of the corresponding factorization equations in which they appear. Throughout this process, the multiple $v^j_k$ variables that correspond to a single model variable $x_{f(j,k)}$ are repeatedly replaced by their mean value in order to put the system of factorization equations back into a consistent state. This process of a bottom-up inference and learning step followed by a top-down value propagation (data generation) step is iterated until convergence. \n\nAlgorithm~\\ref{alg:inference_learning1} and the corresponding procedures in Algorithm~\\ref{alg:upStepDownStep} and Algorithm~\\ref{alg:averagingProcedures} show the pseudocode for the basic inference and learning algorithm that was used to obtain all of the empirical results in this paper. We start by initializing the hidden variables $X_H$ to small random positive values. We then make the system consistent by copying the value of each model variable $x_{f(j,k)}$ into each of its corresponding local variables $v^j_k$. The only distinction between hidden and observed variables from the perspective of the learning and inference algorithm is that model variables in the observed set $X_E$ are never modified in the algorithm. In computing the mean values of the variables, the new mean for the model variables is computed only from the subset of the local variables that were updated in the corresponding inference or value propagation step. After performing the value propagation step, the updated values of all lower level variables in the model are a function of the top level variables.\n\nWhen a parameter matrix $W$ is tied across multiple vector factorization equations, the corresponding equations can be merged into a single matrix factorization equation, as we did in going from (\\ref{eqn:nmf_vector}) to (\\ref{eqn:nmf_matrix}). This merging will also be possible in the dynamic models that we present starting in Section~\\ref{sec:main_factored_model}. In this case, the learning update of the upStep() procedure is simply modified to perform a single NMF left update instead of multiple left up steps. Appendix~\\ref{appendix_inf_learn_2} describes a similar inference and learning algorithm in which the mean values of the variables are computed differently.\n\n\\begin{algorithm}\n\\caption{Perform inference and learning}\n\\label{alg:inference_learning1}\n\\begin{tabbing}\nInitialize hidden variables to random positive values \\\\\n\/\/ Main loop \\\\\n{\\bf repe}\\={\\bf at}  \\\\\n\\> \/\/ Bottom-to-top inference and learning \\\\\n\\> {\\bf for} \\= $l$ = 1 to $L -1$ \\\\\n\\> \\> upStep($l$) \\\\\n\\> \\> averageParents($l$) \\\\\n\\> {\\bf end} \\\\\n\\> \/\/ Top-to-bottom value propogation \\\\\n\\> {\\bf for} $l$ = $L -1$ downto 1 \\\\\n\\> \\> downStep($l$) \\\\\n\\> \\> averageChildren($l$) \\\\\n\\> {\\bf end} \\\\\n{\\bf until} convergence\n\\end{tabbing}\n\\end{algorithm}\n\n\n\n\\begin{algorithm}\n\\caption{upStep() and downStep() procedures}\n\\label{alg:upStepDownStep}\n\\begin{tabbing}\n\/\/ Using the values of the child variables at level $l$, update the values of the parent variables and \\\\\n\/\/ update the parameter matrices by performing NMF update steps. \\\\\n\/\/ Let $X_l$ denote the set of model variables \\{$x^l_1, x^l_2, \\dots, x^l_{LevelCount_l}$\\} corresponding to the level $l$ nodes.\\\\\n\/\/ Let $FactorSystem_l$ denote the subset of the factorization equations from (\\ref{eq:factorization_equation333})\\\\\n\/\/ \\{$eq_j: $ such that $v^j_0$ in $eq_j$ corresponds to an $x^l_i \\in X_l$\\}.  \\\\\n\/\/ Let $duplicationSetChild(i,l)$ = \\{$j : eq_j \\in FactorSystem_l$ and $v^j_0$ corresponds to $x^l_i$\\}. \\\\\n{\\bf upSt}\\={\\bf ep}($l$) \\\\\n\\> {\\bf for} \\={\\bf each} $j \\in FactorSystem_l$ \\\\\n\\> \\> {\\bf if} \\= learning is enabled \\\\\n\\> \\> \\> Learning update: Using $v^j_0 = W^j v^j$, perform a left NMF update on $W^j$, using, e.g. (\\ref{eqn:left_nmf_update}) \\\\\n\\> \\> {\\bf end} \\\\\n\\> \\> Inference update: Using $v^j_0 = W^j v^j$, perform a right NMF update on $v^j$, using, e.g. (\\ref{eqn:right_nmf_update}) \\\\\n\\> {\\bf end} \\\\\n{\\bf end} \\\\\n\\\\\n\/\/ Using the values of the parent variables, update the values of the level $l$ child variables by performing \\\\\n\/\/ value propagation. \\\\\n{\\bf down}\\={\\bf step}($l$) \\\\\n\\> {\\bf for} \\={\\bf each} $j \\in FactorSystem_l$ \\\\\n\\> \\> \/\/ Perform value propagation \\\\\n\\> \\> $v^j_0 \\gets W^j v^j$ \\\\\n\\> {\\bf end} \\\\\n{\\bf end} \n\\end{tabbing}\n\\end{algorithm}\n\n\\begin{algorithm}\n\\caption{averageChildren() and averageParents() procedures}\n\\label{alg:averagingProcedures}\n\\begin{tabbing}\n\/\/ Let $X_l$ denote the set of model variables \\{$x^l_1, x^l_2, \\dots, x^l_{LevelCount_l}$\\} corresponding to the level $l$ nodes.\\\\\n\/\/  Let $duplicationCountChild(i,l)$ = $|duplicationSetChild(i,l)|$.\\\\\n\/\/ Let $duplicationSet(i,l)$ = \\{$(j,k) : eq_j \\in FactorSystem$ and $v^j_k$ corresponds to $x^l_i$\\}. \\\\\n\/\/ Update the value of each $x^l_i \\in X_l$ as the mean value of the corresponding $v^j_0$ \\\\\n\/\/ that appear in $FactorSystem_l$. Then set all $v^j_k$ corresponding to $x^l_i$ to this value as well.\\\\\n{\\bf aver}\\={\\bf ageChildren}($l$) \\\\\n\\> {\\bf for} \\= $i$ = 1 to $LevelCount_l$ \\\\\n\\> \\> \/\/ For each (child) variable $x^l_i \\in X_l$ \\\\\n\\> \\> {\\bf if} \\=$x^l_i$ is hidden \\\\\n\\> \\> \\> meanValue = $\\frac{1}{duplicationCountChild(i,l)} \\sum_{j \\in duplicationSetChild(i,l)} v^j_0$ \\\\\n\\> \\> \\> {\\bf for} \\= {\\bf each} $(j,k) \\in duplicationSet(i,l)$ \\\\\n\\> \\> \\> \\> $v^j_k \\gets meanValue $ \\\\\n\\> \\> \\> {\\bf end} \\\\\n\\> \\> \\> $x^l_i \\gets meanValue $ \\\\\n\\> \\> {\\bf else if} $x^l_i$ is observed \\\\\n\\> \\> \\> {\\bf for each} $(j,k) \\in duplicationSet(i,l)$ \\\\\n\\> \\> \\> \\> $v^j_k \\gets x^l_i $ \\\\\n\\> \\> \\> {\\bf end} \\\\\n\\> \\> {\\bf end} \\\\\n\\> {\\bf end} \\\\\n{\\bf end} \\\\\n\\\\\n\/\/ Let $X_{l+1}$ denote the set of model variables \\{$x^{l+1}_1, x^{l+1}_2, \\dots, x^{l+1}_{LevelCount_{l+1}}$\\} corresponding to the level $l+1$ nodes.\\\\\n\/\/ Note that it is possible for a level $l$ variable to have some or all of its parents in a level higher than $l+1$, in\\\\\n\/\/ which case the set $X_{l+1}$ will only represent a proper subset of the parents of level $l$. \\\\\n\/\/ Let $duplicationSetParent(i,l+1)$ = \\{$(j,k) :  eq_j \\in FactorSystem_l$ and $v^j_k, k \\geq 1$ corresponds to $x^{l+1}_i$\\}. \\\\\n\/\/ Let $duplicationCountParent(i,l+1) = |duplicationSetParent(i,l+1)|$. \\\\\n\/\/ For each $x^{l+1}_i \\in X_{l+1}$, update the corresponding $v^j_k, k \\geq 1$ to their mean value.\\\\\n{\\bf aver}\\={\\bf ageParents}($l$) \\\\\n\\> {\\bf for} \\= $i$ = 1 to $LevelCount_{l+1}$ \\\\\n\\> \\> \/\/ For each (parent) variable $x^{l+1}_i \\in X_{l+1}$ \\\\\n\\> \\> {\\bf if} \\=$x^{l+1}_i$ is hidden \\\\\n\\> \\> \\> meanValue = $\\frac{1}{duplicationCountParent(i,l+1)} \\sum_{(j,k) \\in duplicationSetParent(i,l+1)} v^j_k$ \\\\\n\\> \\> \\> {\\bf for} \\= {\\bf each} $(j,k) \\in duplicationSet(i,l+1)$ \\\\\n\\> \\> \\> \\> $v^j_k \\gets meanValue $ \\\\\n\\> \\> \\> {\\bf end} \\\\\n\\> \\> \\> $x^{l+1}_i \\gets meanValue$ \\\\\n\\> \\> {\\bf else if} $x^{l+1}_i$ is observed \\\\\n\\> \\> \\> {\\bf for each} $(j,k) \\in duplicationSet(i,l+1)$ \\\\\n\\> \\> \\> \\> $v^j_k \\gets x^{l+1}_i $ \\\\\n\\> \\> \\> {\\bf end} \\\\\n\\> \\> {\\bf end} \\\\\n\\> {\\bf end} \\\\\n{\\bf end} \n\\end{tabbing}\n\\end{algorithm}\n\n\nNote that the only distinction the algorithm makes between $X_E$ and $X_H$ is that the values of the $X_E$ variables are not updated during inference. If we wish to make some previously hidden variables observed or vice versa, is is then trivial to modify the algorithm to handle this by simply enabling or disabling updates on those variables. \n\nWe also note that typically, some kind of normalization of the parameter matrices $W$ will be performed during the left NMF update steps. For example, one could normalize the column of $W$ to have unit sum. \n\nNote also that inference and learning are performed jointly. We can perform inference only by simply disabling the learning updates. If a subset of the parameters $\\theta$ are known, then the learning updates can be disabled for those parameters.\n\nThe computational cost of performing a single iteration of the inference and learning algorithm is given by the sum of the costs of performing the learning and inference NMF updates and the value propagation multiplication on each factorization equation. The number of iterations to convergence can depend on several factors, such as the longest path in the graph, NMF algorithm employed, etc.\n\n\n\n\n\n\n\\section{Factored sequential data models}\n\\label{sec:main_factored_model}\n\nWe now consider PFNs for modeling sequential data, which we will refer to as dynamic positive factor networks (DPFNs). A DPFN provides for the modeling of a data sequence as an additive combination of realizations of an underlying process model. Analogously to a dynamic Bayesian network (DBN) \\cite{Friedman1999}, a DPFN is created by horizontally replicating a PFN subgraph. Thus, a DPFN is simply a PFN that has a regular repeating graphical structure. The corresponding parameters are also typically replicated. We only require that the particular subgraph that is replicated correspond to a valid PFN. We will refer to the subgraph that is replicated as corresponding to a time slice. Although we use terminology that implies a time series, such an interpretation is not required in general. For example, the data could correspond to a biological sequence, or the characters or words in a text document, for example.\n\n\n\nOur models will make use of an non-negative linear state representation. Intuitively, one can think of the model as supporting the simultaneous representation of any additive combination of allowable realizations of an underlying state transition model, or more generally, an underlying dynamic process model. A state variable in the model is defined such that the dimensionality of the variable corresponds to the number of possible states. We allow these variables to be general non-negative vectors, so that the non-zero-valued components of the variable represent a (positive) weight or strength of the corresponding state. Any given non-zero valued state variable then corresponds to some superposition of states. A given pair of state variables in adjacent time slices is then factored in terms of a set of basis vectors that specify the allowable transitions, and a corresponding encoding vector. The factored state representation seems to allow significant representational power, while making use of a compact parameter set.\n\nIn this section we present a factored state transition model and present empirical results to illustrate the performance of the inference and learning algorithms.\n\n\\subsection{Model}\n\\label{sec:fact_state_tran_model}\n\nConsider a transition model with $M$ states and $R$ possible state transitions. Figure~\\ref{fig:fsm1} shows an example state transition diagram for a 4-state automaton. This transition model contains $M = 4$ states, labeled $S_1, \\dots, S_4$ and $R = 6$ state transitions, labeled $t_1, \\dots, t_6$. \n\n\\begin{figure}\n\\centering\n\\includegraphics[width=60ex]{.\/figures\/fsm1.pdf}\n\\caption{A state transition diagram for a 4-state automaton.}\n\\label{fig:fsm1}\n\\end{figure}\n\nFigure~\\ref{fig:1layerDynamic} shows the DPFN that we will use to model a process that evolves according to the transition model given in Figure~\\ref{fig:fsm1}. This DPFN corresponds to a system of $T-1$ factorization equations where for each $t \\in 1, \\dots, T-1$ we have:\n\n\\begin{align}\n\\left[ \\begin{array}{c} \nx_t\\\\\nx_{t+1}  \\end{array} \\right] =&\\ W h_t \\notag \\\\\n =&\\ \\left[ \\begin{array}{c}\nW_1 \\\\\nW_2 \\end{array} \\right] h_t \n\\label{eqn:single_timeslice_factorization}\n\\end{align}\n\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=60ex]{.\/figures\/1layerDynamic.pdf}\n\\caption{The DPFN corresponding to the factorized state model in Equation (\\ref{eqn:key_factorization}). The first four time slices are shown. The state at time slice $t$ is represented by $x_t$.  We place constrains on states at adjacent time slices such that the states $x_t$, $x_{t+1}$ are represented as an additive combination of the allowable state transition vectors. $h_t$ represents the encoded transitions for $x_t, x_{t+1}$.}\n\\label{fig:1layerDynamic}\n\\end{figure}\n\n\nSince the same parameter matrix $W$ appears in each equation, the above system of $T-1$ vector factorization equations can be expressed as a single matrix factorization equation:\n\n\\begin{align}\n\\label{eqn:expanded_key_factorization}\n\\left[ \\begin{array}{ccccc}\nx_1 & x_2 & x_3 & \\dots & x_{T-1} \\\\\nx_{2} & x_{3} & x_{4} & \\dots & x_{T} \\end{array} \\right] =&\\ W \\left[ \\begin{array}{ccccc} h_1 & h_2 & h_3 & \\dots & h_{T-1} \\end{array} \\right] \\notag \\\\\n=&\\ \\left[ \\begin{array}{c}\nW_1 \\\\\nW_2 \\end{array} \\right] \\left[ \\begin{array}{ccccc} h_1 & h_2 & h_3 & \\dots & h_{T-1} \\end{array} \\right]\n\\end{align}\n\n\n\nLet us define $x_{c_t}$ as the vertical concatenation of any two adjacent state variables $x_t$, $x_{t+1}$ so that:\n\n\\begin{align}\nx_{c_t} = \\left[ \\begin{array}{c} \nx_t\\\\\nx_{t+1}  \\end{array} \\right]\n\\end{align}\n\nWe define the $2 M$ x T-1 matrix $X_c$ as the following horizontal concatenation of the time-ordered $x_{c_t}$ vectors:\n\n\\begin{align}\nX_c =& \\left[ \\begin{array}{cccccc} x_{c_1} & x_{c_2} & x_{c_3} & \\dots & x_{c_{T-1}} \\end{array} \\right] \\notag \\\\\n=& \\left[ \\begin{array}{ccccc}\nx_1 & x_2 & x_3 & \\dots & x_{T-1} \\\\\nx_{2} & x_{3} & x_{4} & \\dots & x_{T} \\end{array} \\right]\n\\end{align}\n\nWe define the $R$ x T-1 matrix $H$ as the following horizontal concatenation of the time-ordered \\{$h_t$\\} vectors:\n\n\\begin{align}\nH = \\left[ \\begin{array}{cccccc} h_1 & h_2 & h_3 & \\dots & h_{T-1} \\end{array} \\right]\n\\end{align}\n\nUsing the above matrix definitions, we can then write the matrix factorization equation (\\ref{eqn:expanded_key_factorization}) more compactly as:\n\n\\begin{align}\nX_c = W H\n\\label{eqn:key_factorization}\n\\end{align}\n\nWe now illustrate how this DPFN can be used to represent the transition model in Figure~\\ref{fig:fsm1}. We let vector $x_t \\in  \\mathbb{R}^M$ represent the model state at time $t$. The sequence, $\\{x_t : t = 1, \\dots, T\\}$ is then modeled as a realization of the transition model specified in the state transition diagram. The parameter matrix $W$ specifies the transition model, and the vector $h_t$ represents an encoding of $(x_t, x_{t+1})$ in terms of the basis columns of $W$.\n\nThe parameter matrix $W$ is constructed directly from the state transition diagram as follows. For each state $S_i$, we define a corresponding \\emph{state basis vector} $s_i \\in  \\mathbb{R}^M$ such that the $i$'th component is 1 and all other components are zero. States $S_1, \\dots, S_4$ then correspond to the following state basis vectors, respectivley:\n\n\\begin{align}\ns_1=& \\left[ \\begin{array}{c} \n1\\\\\n0\\\\\n0\\\\\n0  \\end{array} \\right], \ns_2 = \\left[ \\begin{array}{c} \n0\\\\\n1\\\\\n0\\\\\n0  \\end{array} \\right], \ns_3 = \\left[ \\begin{array}{c} \n0\\\\\n0\\\\\n1\\\\\n0  \\end{array} \\right], \ns_4 = \\left[ \\begin{array}{c} \n0\\\\\n0\\\\\n0\\\\\n1  \\end{array} \\right]\n\\end{align}\n\nFor each transition $t_k$ in the state transition diagram that represents a transition from state $S_i$ to state $S_j$, we define the corresponding \\emph{transition basis vector} $w_k$ as the vertical concatenation of the corresponding state basis vectors $s_i$ on top of $s_j$. Each of the $R$ transitions $t_k$ will then have a corresponding $2 M$ x 1 transition basis vector $w_k$ given by:\n\n\\begin{align}\nw_k =  \\left[ \\begin{array}{c} \ns_i\\\\\ns_j  \\end{array} \\right]\n\\end{align}\n\nFor example, transition $t_2$ in the diagram, which represents a transition from state $S_2$ to state $S_3$ has a corresponding transition basis vector $w_2$ given by:\n\n\\begin{align}\nw_2 =  \\left[ \\begin{array}{c} \ns_2\\\\\ns_3  \\end{array} \\right] = \\left[ \\begin{array}{c} \n0\\\\\n1\\\\\n0\\\\\n0\\\\ \\hline\n0\\\\\n0\\\\\n1\\\\\n0  \\end{array} \\right]\n\\end{align}\n\nLet the $2 M$ x $R$ \\emph{transition basis matrix} $W$ be defined as the horizontal concatenation of the transition basis vectors \\{$w_i$\\}. Any ordering of the columns is possible. The columns of $W$ then specify the allowable transitions in our model:\n\n\\begin{align}\nW =& \\left[ \\begin{array}{ccccc} w_1 & w_2 & w_3 & \\dots & w_R \\end{array} \\right]\n\\end{align}\n\nWe will find it useful to partition $W$ into upper and lower sub-matrices $W_1$ and $W_2$ such that the $M$ x $R$ upper sub-matrix $W_1$ represents the time slice $t-1$ state basis vectors and the $M$ x $R$ lower sub-matrix $W_2$ represents the corresponding time slice $t$ state basis vectors. One possible $W$ corresponding to the transition diagram in Figure~\\ref{fig:fsm1} is given by:\n\n\\begin{align}\nW =& \\left[ \\begin{array}{cccccc} w_1 & w_2 & w_3 & w_4 & w_5 & w_6 \\end{array} \\right] \\notag \\\\\n=&\\ \\left[ \\begin{array}{c}\nW_1 \\\\\nW_2 \\end{array} \\right] \\notag\\\\\n=&\\ \\left[ \\begin{array}{cccccc}\ns_1 & s_2 & s_3 & s_4 & s_3 & s_2 \\\\\ns_2 & s_3 & s_4 & s_1 & s_3 & s_4 \\end{array} \\right] \\notag \\\\\n=& \\left[ \\begin{array}{cccccc}\n1 & 0 & 0 & 0 & 0 & 0 \\\\\n0 & 1 & 0 & 0 & 0 & 1 \\\\\n0 & 0 & 1 & 0 & 1 & 0 \\\\\n0 & 0 & 0 & 1 & 0 & 0 \\\\ \\hline\n0 & 0 & 0 & 1 & 0 & 0 \\\\\n1 & 0 & 0 & 0 & 0 & 0 \\\\\n0 & 1 & 0 & 0 & 1 & 0 \\\\\n0 & 0 & 1 & 0 & 0 & 1 \\end{array} \\right] \n\\label{eqn:W_non_deterministic}\n\\end{align}\n\nConsider the following state sequence of length $T=10$, which is a valid sequence under the example transition diagram:\n\n\\begin{align}\n\\label{eq:example_state_seq}\nseq_1 = (S_1, S_2, S_3, S_4, S_1, S_2, S_3, S_4, S_1, S_2)\n\\end{align}\n\nNow suppose we construct a sequence of state variables $\\{x_t: t=1,\\dots,T\\}$ corresponding to this state sequence by setting $x_t$ equal to the corresponding state basis vector $s_i$ for state $S_i$ at time $t$:\n\n\\begin{align}\nx_1 = s_1 = \\left[ \\begin{array}{c} \n1\\\\\n0\\\\\n0\\\\\n0  \\end{array} \\right], \nx_2 = s_2 = \\left[ \\begin{array}{c} \n0\\\\\n1\\\\\n0\\\\\n0  \\end{array} \\right],\nx_3 = s_3 = \\left[ \\begin{array}{c} \n0\\\\\n0\\\\\n1\\\\\n0  \\end{array} \\right],\n\\dots,\nx_{10} = s_2 = \\left[ \\begin{array}{c} \n0\\\\\n1\\\\\n0\\\\\n0  \\end{array} \\right]\n\\label{eqn:simple_state_seq}\n\\end{align}\n\n\nLet us define the $M$ x $T$ matrix $X$ as the horizontal concatenation of the state variables $x_t$ as:\n\n\\begin{align}\nX =& \\left[ \\begin{array}{ccccc} x_1 & x_2 & x_3 & \\dots & x_T \\end{array} \\right]\n\\end{align}\n\n\nWe then have the following sequence $X$ corresponding to the state sequence (\\ref{eq:example_state_seq}):\n\n\n\\begin{align}\nX =& \\left[ \\begin{array}{cccccccccc} x_1 & x_2 & x_3 & x_4 & x_5 & x_6 & x_7 & x_8 & x_9 & x_{10} \\end{array} \\right] \\notag \\\\\n=& \\left[ \\begin{array}{cccccccccc} \n1 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 1 & 0\\\\\n0 & 1 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 1\\\\\n0 & 0 & 1 & 0 & 0 & 0 & 1 & 0 & 0 & 0\\\\\n0 & 0 & 0 & 1 & 0 & 0 & 0 & 1 & 0 & 0 \\end{array} \\right]\n\\label{eqn:fsm1_determ_X1}\n\\end{align}\n\nUsing the above $X$ and (\\ref{eqn:key_factorization}), we then have the following model factorization:\n\n\\begin{align}\nX_c =& W H \\notag \\\\\n\\left[ \\begin{array}{ccccccccc} \n1 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 1\\\\\n0 & 1 & 0 & 0 & 0 & 1 & 0 & 0 & 0\\\\\n0 & 0 & 1 & 0 & 0 & 0 & 1 & 0 & 0\\\\\n0 & 0 & 0 & 1 & 0 & 0 & 0 & 1 & 0\\\\ \\hline\n0 & 0 & 0 & 1 & 0 & 0 & 0 & 1 & 0\\\\\n1 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 1\\\\\n0 & 1 & 0 & 0 & 0 & 1 & 0 & 0 & 0\\\\\n0 & 0 & 1 & 0 & 0 & 0 & 1 & 0 & 0 \\end{array} \\right] =&\n\\left[ \\begin{array}{cccccc}\n1 & 0 & 0 & 0 & 0 & 0 \\\\\n0 & 1 & 0 & 0 & 0 & 1 \\\\\n0 & 0 & 1 & 0 & 1 & 0 \\\\\n0 & 0 & 0 & 1 & 0 & 0 \\\\ \\hline\n0 & 0 & 0 & 1 & 0 & 0 \\\\\n1 & 0 & 0 & 0 & 0 & 0 \\\\\n0 & 1 & 0 & 0 & 1 & 0 \\\\\n0 & 0 & 1 & 0 & 0 & 1 \\end{array} \\right] \n\\left[ \\begin{array}{ccccccccc} \n1 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 1\\\\\n0 & 1 & 0 & 0 & 0 & 1 & 0 & 0 & 0\\\\\n0 & 0 & 1 & 0 & 0 & 0 & 1 & 0 & 0\\\\\n0 & 0 & 0 & 1 & 0 & 0 & 0 & 1 & 0\\\\\n0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\\\n0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\end{array} \\right]\n\\end{align}\n\n Note that given $X_C$ and $W$, only one solution for $H$ is possible. We see that each pair of state vectors $x_{c_t}$ in $X_c$ corresponds to exactly one transition basis vector in the resulting factorization, since the value of each $x_t$ was chosen equal to one of the $s_i$ and $X$ corresponds to a valid state sequence. The columns of $H$ then given an encoding of $X$ in terms of the basis vectors of $W$ so that exactly one component of each column $h_t$ has value 1, corresponding to the transition that explains $x_{c_t}$. We say that a sequence $X$ of state vector observations form an \\emph{elementary state sequence} if there exists a factorization $X_C = W H$ such that each column of $H$ contains at most one nonzero component. \n\n\nNow let us return to the case where the $x_t$ is no longer constrained to be equal to one the state basis vectors. We only require $x_t$ to be a non-negative vector in $\\mathbb{R}^M$. Although $W$ is specified in the example above as containing only 0 and 1 valued components, in general we only place the constraint that the matrices of the factorization equation be non-negative. Note that since the model is a PFN, the linearity property from Section~\\ref{sec:model_specification} holds, so that any additive combination of solutions is also a solution. That is, any additive mixture of state sequences that are valid under the transition model of $W$ are representable.  We can also see that this property follows directly from (\\ref{eqn:key_factorization}). For non-negative scalars $\\alpha$, $\\beta$, we have:\n\n\\begin{align}\nX_{c_a} =& W H_a \\notag \\\\\nX_{c_b} =& W H_b \\notag \\\\\n\\Leftrightarrow \\alpha X_{c_a} + \\beta X_{b_c} =& W (\\alpha H_a + \\beta H_b)\n\\end{align}\n\nFor the case where multiple components of $x_t$ are positive, the interpretation is that multiple states are simultaneously active at time $t$. For example, consider an example where $x_t = (\\alpha, 0, 0, \\beta)^T$. That is, at time $t$, the model is in state $S_1$ with magnitude $\\alpha$ and is simultaneously in state $S_4$ with magnitude $\\beta$. Suppose that $x_{t+1}$ is hidden. We can see from the factorization equation above that the solution is $x_{t+1} = (\\beta, \\alpha, 0, 0)^T$. That is, our factored transition model specifies that if the model is in state $S_1$ with magnitude $\\alpha$ at time $t$ then the model must be in state $S_2$ with magnitude $\\alpha$ at time $t+1$. Likewise, if the model is in state $S_4$ with magnitude $\\beta$ at time $t$ then the model must be in state $S_1$ with magnitude $\\beta$ at time $t+1$. Due to the factored state representation, the model can represent both realizations simultaneously. A sequence of state vectors is modeled as an non-negative linear combination of the allowable transition basis vectors.\n\nNote also that since we do not impart a probabilistic interpretation, all allowable outgoing transitions from a given state can be considered equally likely. For example, suppose $x_{t+1}$ is hidden and that $x_t = (0, \\alpha, 0, 0)$, corresponding to the model being in state $S_2$ with magnitude $\\alpha$ at time $t$. Since there are multiple outgoing transitions from $S_2$ ($t_2$ and $t_6$), multiple solutions for $x_{t+1}$ are possible. We will explore the issue of performing inference when multiple solutions are possible in the following results sections.\n\nA learning and inference algorithm for this network can be obtained by applying Algorithm~\\ref{alg:inference_learning1} to the system of factorizations (\\ref{eqn:key_factorization}). Example pseudocode is presented in Appendix~\\ref{sec:simpleDyn1_inf_learn}.\n\n\n\n    \n \n  \n\\subsection{Empirical results}\n\\label{sec:dyn_model}\n\nIn this section, we perform inference and learning on the dynamic network in Figure~\\ref{fig:1layerDynamic}, for two distinct transition models, using synthetic data sets. We present results for the cases of fully and partially observed input sequences. We also present results for sequences that have been corrupted by noise. We first consider the case where the underlying transition model is known. We then present results for the case where the transition model is learned from training data.\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=60ex]{.\/figures\/fsm1_deterministic.pdf}\n\\caption{A deterministic state transition diagram for a 4-state automaton.}\n\\label{fig:fsm1_deterministic}\n\\end{figure}\n\n\n\\subsubsection{Fully observed input sequences under a known transition model}\n\nWe start with a simple deterministic transition model, and will later revisit the earlier nondeterministic model from  Figure~\\ref{fig:fsm1}.  Figure~\\ref{fig:fsm1_deterministic} shows the state transition diagram for a deterministic model that is obtained by removing transitions $t_5$ and $t_6$ from the earlier nondeterministic model. Using the previously outlined procedure, we obtain a parameter matrix $W$ corresponding to this transition diagram as follows. First we construct a transition basis vector $w_k$ corresponding to each transition $t_k$:\n\n\\begin{align}\nw_1=& \\left( \\begin{array}{c} s_1\\\\\ns_2 \\end{array} \\right) ,\nw_2= \\left( \\begin{array}{c} s_2\\\\\ns_3 \\end{array} \\right) ,\nw_3= \\left( \\begin{array}{c} s_3\\\\\ns_4 \\end{array} \\right) ,\nw_4= \\left( \\begin{array}{c} s_4\\\\\ns_1 \\end{array} \\right) ,\n\\end{align}\n\nThe transition basis matrix $W$ is then constructed as the horizontal concatenation of the transition basis vectors (again, the column ordering is arbitrary):\n\n\\begin{align}\nW = \\left[ \\begin{array}{cccc} w_1 & w_2 & w_3 & w_4 \\end{array} \\right]\n\\end{align}\n\nWe start by considering the case where the state sequence $X$ is fully observed, and the transition sequence $H$ is hidden. We then wish to infer the values of $H$ given the observed $X$.  Suppose we wish to specify an input sequence $X$ that corresponds to the following sequence of states: $(S_1, S_2, S_3, S_4, S_1, S_2, S_3, S_4, S_1, S_2)$, which is a valid state sequence under the transition diagram in Figure~\\ref{fig:fsm1_deterministic}. This sequence can be represented by setting $X =\\alpha \\left[ \\begin{array}{cccccccccc} s_1 & s_2 & s_3 & s_4 & s_1 & s_2 & s_3 & s_4 & s_1 & s_2 \\end{array} \\right]$, where $\\alpha$ is any positive scalar. For example, the choice $\\alpha = 1$ results in:\n\n\\begin{align}\nX = \\left[ \\begin{array}{cccccccccc} \n1 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 1 & 0\\\\\n0 & 1 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 1\\\\\n0 & 0 & 1 & 0 & 0 & 0 & 1 & 0 & 0 & 0\\\\\n0 & 0 & 0 & 1 & 0 & 0 & 0 & 1 & 0 & 0 \\end{array} \\right]\n\\label{eqn:not_used}\n\\end{align}\n\nWe will be dealing with matrices that are non-negative, typically sparse, and such that many or all of the non-zero components will take on the same value. For visualization purposes, the particular value will typically not be relevant. For these reasons we find that, rather than simply printing out the numerical values of a matrix as above, an image plot can be a more visually appealing way to take in the relevant information. In the remainder of this paper, we will therefore make extensive use of image plots of the matrices that appear in the various PFN factorization equations. In displaying the image plots, we make use of the ``hot'' colormap, so that zero-valued components appear black, small values appear red, larger values yellow, and the maximum valued component(s) in a matrix appear white. Figure~\\ref{fig:fsm1_determ_X1} shows an image plot of the observed $X$ from above.\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=60ex]{.\/figures\/fsm1_determ_X1.jpg}\n\\caption{An image plot of the observed $X$ from Equation~\\ref{eqn:fsm1_determ_X1}, corresponding to the state sequence: $(S_1, S_2, S_3, S_4, S_1, S_2, S_3, S_4, S_1, S_2)$. Row $i$ of $X$ corresponds to state $S_i$. Column $i$ corresponds to the $i$'th time slice. Here and throughout this paper we use the \"hot\" color map shown on the right, so that components with a minimum component value are displayed in black, components with the maximum component value are displayed in white, and values in between are displayed in shades of red, orange, and yellow.}\n\\label{fig:fsm1_determ_X1}\n\\end{figure}\n\nRecall that the network in Figure~\\ref{fig:1layerDynamic} corresponds to the matrix factorization Equation (\\ref{eqn:key_factorization}), which we reproduce here:\n\n\\begin{align}\nX_C = W H\n\\label{eqn:repeatedDPN1}\n\\end{align}\n\nRecall also that $X_C$ is defined as the vertical stacking of adjacent columns of $X$ and so only $H$ is unknown (hidden). We solve for the values of $H$ by applying the inference algorithm from Appendix~\\ref{sec:simpleDyn1_inf_learn}, which is a special case of the general learning and inference algorithm specified in Algorithm~\\ref{alg:inference_learning1}. In solving for $H$, this corresponds to first initializing $H$ to random positive values and then iterating the inference algorithm (learning is disabled since $W$ is known) until convergence. For all empirical results in this paper, hidden variables are initialized to random positive values uniformly distributed between 0 and $10^{-6}$. Each iteration of the inference algorithm in this case corresponds to performing an NMF right update step, e.g., using (\\ref{eqn:right_nmf_update}).\n\nFigure~\\ref{fig:fsm1_determ_factor} shows a plot of the matrices in the factorization, where $H$ is shown after convergence of the inference algorithm. For this input sequence, a few hundred iterations was sufficient for the RMSE (root mean squared error between $X$ and the reconstruction $W H$) to drop below $10^-4$. The inference computation, implemented in Java and Python, took approximately 1 second to run on a desktop PC with a 3 GHz Core 2 Duo processor, which is representative of the time required to run each of the examples in this section.\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=80ex]{.\/figures\/fsm1_determ_factor.jpg}\n\\caption{A plot of the matrices of the factorization $X_C = W H$, after solving for $H$ Note that the upper half submatrix of $X_c$ is equal to $X$, which corresponds to the state sequence: $(S_1, S_2, S_3, S_4, S_1, S_2, S_3, S_4, S_1, S_2$). Here, $W$ corresponds to the deterministic state transition diagram in Figure~\\ref{fig:fsm1_deterministic}}\n\\label{fig:fsm1_determ_factor}\n\\end{figure}\n\nWe now add some noise consisting of uniformly distributed values in $[0, 0.1]$ to $X$. Figure~\\ref{fig:fsm1_determ_factor_noise} shows the resulting estimate for $H$. The noisy observations can no longer be represented exactly as a linear combination of the transition basis vectors, yielding an RMSE of 0.034. We observe empirically that the approximate factorization found by the NMF updates appears to be relatively insensitive to small amounts of additive noise, and produces an inference result that appears visually similar to applying additive noise to the previous noiseless solution for $H$. \n\\begin{figure}\n\\centering\n\\includegraphics[width=80ex]{.\/figures\/fsm1_determ_factor_noise.jpg}\n\\caption{A plot of the matrices of the factorization $X_C = W H$, where the observations $X$ are corrupted by additive noise. The inferred hidden transition sequence $H$ is shown after convergence of the inference algorithm. Here, $W$ corresponds to the deterministic state transition diagram in Figure~\\ref{fig:fsm1_deterministic}}\n\\label{fig:fsm1_determ_factor_noise}\n\\end{figure}\n\nWe now consider an example where the observation sequence corresponds to a mixture of realizations of the underlying transition model. Recall that the if observation sequence $X_a$ and the hidden sequence $H_a$ are one solution to the model, and observation sequence $X_b$ and the hidden sequence $H_b$ are another solution to the model, then the observation sequence $X = X_a + X_b$ and the hidden sequence $H = H_a + H_b$ are also a solution of the model. As an example, consider the observation sequence $X = X_a + X_b$, where $X_a$ and $X_b$ are given by:\n\n\\begin{align}\n X_a =& 0.5 \\left[ \\begin{array}{cccccccccc} s_1 & s_2 & s_3 & s_4 & s_1 & s_2 & s_3 & s_4 & s_1 & s_2 \\end{array} \\right] \\notag \\\\\n X_b =& 1.0 \\left[ \\begin{array}{cccccccccc} s_3 & s_4 & s_1 & s_2 & s_3 & s_4 & s_1 & s_2 & s_3 & s_4 \\end{array} \\right]\n \\label{eqn: superposition1}\n \\end{align}\n\nFigure~\\ref{fig:fsm1_determ_X1_superpos} shows an image plot of the observed sequence $X = X_a + X_b$. Note that both sequences $X_a$ and $X_b$ correspond to a valid realization of the underlying transition model of the network. From the superposition property, we know that the sum sequence $X$ has a corresponding hidden sequence $H$ which satisfies the factorization Equation~\\ref{eqn:repeatedDPN1} with equality. The expressiveness of this network is such that any observations and hidden variables corresponding to any non-negative superposition of valid realizations of the underlying transition model specified by $W$ are representable. This does not necessarily mean that our particular choice of inference algorithm will always be able find the corresponding solution, however. We do observe, though, that for this particular network choice, repeated runs of our inference algorithm always converged to an exact solution (specifically, inference was stopped when the RMSE dropped below a certain threshold: $10^-4$). Figure~\\ref{fig:fsm1_determ_factor_superpos} shows an image plot of the inferred $H$ along with the other matrices in Equation~\\ref{eqn:repeatedDPN1}. \n\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=60ex]{.\/figures\/fsm1_determ_X1_superpos.jpg}\n\\caption{An image plot of an observation sequence  $X = X_a + X_b$ consisting of an additive combination of two sequences $X_a$ and $X_b$, which are each a realization of the underlying network transition model. Here, $X_1$ begins in state $s_1$ in the first time slice, and corresponds to the orange components since it has half the magnitude of sequence $X_b$. Sequence $X_b$ begins in state $s_3$ and corresponds to the white components.}\n\\label{fig:fsm1_determ_X1_superpos}\n\\end{figure}\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=80ex]{.\/figures\/fsm1_determ_factor_superpos.jpg}\n\\caption{An image plot of the matrices of the factorization $X_C = W H$, where the observations are an additive combination of two sequences: $X = X_a + X_b$. $H$ is shown after convergence of the inference algorithm. Here, $W$ corresponds to the deterministic state transition diagram in Figure~\\ref{fig:fsm1_deterministic}}\n\\label{fig:fsm1_determ_factor_superpos}\n\\end{figure}\n\n\\subsubsection{Partially observed input sequences under a known deterministic transition model}\n\\label{sec:example_deterministic_fsm}\n\n\n\n\nWe now consider the case where the sequence $X$ is partially observed so that we are given the values of $x_t$ for some subset of time slices, and the values of $x_t$ for the remaining time slices are hidden. We would then like to infer the values of all hidden variables in the model, which will consist of $H$ along with the hidden subset of $\\{x_t\\}$. \n\nSuppose that $X$ represents a sequence of length 10 where the first time slice $x_1$ is observed and corresponds to state $S_1$. That is, $x_1 = \\alpha s_1$. We arbitrarily choose $\\alpha = 1$. We then wish to infer the values of the future time slices of $X$ as well as the values of $H$. That is, we wish to perform prediction. We again use the same inference algorithm and initialize all hidden variables to small positive random values.\n\nFigure~\\ref{fig:fsm1_determ_factor_X1_after_various_iterations} shows the initial $X$ and the estimates after 10, 50, and 500 iterations of the inference algorithm. We see that convergence is effectively reached by 500 iterations. Figure~\\ref{fig:fsm1_determ_factor_partial} shows the corresponding plots of the matrices of the network factorization equation after 500 iterations. We observe that the inference estimates for $X$ appear to converge outward in time from the observed time slice $x_1$. The local variable averaging step of the inference algorithm causes the transition model to propagate a positive value one time slice forward and backward for each iteration. Note also that given the observed time slice, there is only one possible solution for the hidden variables (and that our inference algorithm successfully finds it) since the transition model is deterministic.\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=80ex]{.\/figures\/fsm1_determ_factor_X1_after_various_iterations.jpg}\n\\caption{A plot of the estimate for $X$ after various iteration counts of the inference algorithm. The top image shows the initial $X$ which only has the first time slice (time index 1 in the figure) set to state $S_1$ (i.e., $x_1 = s_1$). The future time slices $(x_2 \\dots x_{10})$ are initialized to small random values, which appear black in the figure since their maximum value of $10^-6$ is small compared to the largest value of 1 in the figure. The next lower figures show the $X$ estimates after 10, 50, and 500 iterations, respectively. We see that 500 iterations is sufficient to effectively reach convergence.}\n\\label{fig:fsm1_determ_factor_X1_after_various_iterations}\n\\end{figure}\n\nAlthough the first time slice $x_1$ was chosen as the only observed variable above, we could have just as easily chosen any subset of time slices (or even any subset of components) of $X$ and $H$ as the observed subset $X_E$. For example, Figure~\\ref{fig:fsm1_determ_factor_X1_after_various_iterations2} shows the estimates of $X$ for various iteration counts for the case where the time slice $x_4$ is observed and all other time slices are hidden. As expected, we observe that the estimates propogate outward (both backward and forward) from $x_4$ with increasing iteration count.\n\n\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=80ex]{.\/figures\/fsm1_determ_factor_partial.jpg}\n\\caption{An image plot of the matrices of the factorization $X_C = W H$, after solving for $H$ and the hidden time slices $(x_2 \\dots x_{10})$ of $X$. Here, $W$ corresponds to the deterministic state transition diagram in Figure~\\ref{fig:fsm1_deterministic}}\n\\label{fig:fsm1_determ_factor_partial}\n\\end{figure}\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=80ex]{.\/figures\/fsm1_determ_factor_X1_after_various_iterations2.jpg}\n\\caption{A plot of the estimate for $X$ after various iteration counts of the inference algorithm. The top image shows the initial $X$ in which $x_4$ is observed and all other variables are hidden. Here we let $x_4$ correspond to state $S_1$ by setting $x_4 = s_1$. All other variables (i.e, $(x_1,\\dots,x_3), (x_5,\\dots,x_{10}), (h_1,\\dots,h_{10})$) are initialized to small positive random values.  We see that convergence is effectively reached by 500 iterations.}\n\\label{fig:fsm1_determ_factor_X1_after_various_iterations2}\n\\end{figure}\n\n\\subsubsection{Partially observed input sequences under a known nondeterministic transition model}\n\nIn this section we perform experiments to observe how the inference algorithm performs when multiple solutions for the hidden variables are possible. We will employ a nondeterministic state transition model and supply partial state observations so that multiple solutions will be possible for at least some of the hidden states. We will now use the nondeterministic state transition diagram in Figure~\\ref{fig:fsm1}, which corresponds to the transition basis matrix $W$ from Equation~\\ref{eqn:W_non_deterministic}.\n\nWe first perform an experiment to verify that the inferred values for the hidden transitions $H$ are correct for a short fully observed sequence of observations $X$.  Figure~\\ref{fig:fsm1_nondeterm_X1} shows an image plot for  $X = \\left[ \\begin{array}{cccccccccc} s_1 & s_2 & s_3 & s_3 & s_4 & s_1 & s_2 & s_4 & s_1 & s_2 \\end{array} \\right]$, which represents a valid sequence of state transitions under the transition model in Figure~\\ref{fig:fsm1}. We then perform inference on $H$ using the algorithm from Appendix~\\ref{sec:simpleDyn1_inf_learn}, which is a special case of the general learning and inference algorithm specified in Algorithm~\\ref{alg:inference_learning1}. The inference results are shown in Figure~\\ref{fig:fsm1_nondeterm_factor} and correspond to the correct factorization.\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=60ex]{.\/figures\/fsm1_nondeterm_X1.jpg}\n\\caption{An image plot of $X$ corresponding to the state sequence: $(S_1, S_2, S_3, S_3, S_4, S_1, S_2, S_4, S_1, S_2)$. This is a valid realization under the transition diagram in Figure~\\ref{fig:fsm1}.}\n\\label{fig:fsm1_nondeterm_X1}\n\\end{figure}\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=80ex]{.\/figures\/fsm1_nondeterm_factor.jpg}\n\\caption{An image plot of the matrices of the factorization $X_C = W H$, after solving for $H$. Note that $X$ is equal to the upper half submatrix of $X_C$ and corresponds to the state sequence $(S_1, S_2, S_3, S_3, S_4, S_1, S_2, S_4, S_1, S_2)$. Here, $W$ corresponds to the nondeterministic state transition diagram in Figure~\\ref{fig:fsm1}}\n\\label{fig:fsm1_nondeterm_factor}\n\\end{figure}\n\nWe now consider a partially observed sequence of state observations for $X$. Let\n\n\\begin{align}\nX = \\left[ \\begin{array}{cccccccccc}? & s_2 & ? & s_3 & s_4 & ? & ? & ? & ? & ?\\end{array} \\right]\n\\label{eqn:x1PartiallyObserved1}\n\\end{align}\n\nwhere the \"?\" state vectors denotes a hidden state. We initialize $H$ and all hidden variables in $X$ to small positive random values and run the inference algorithm to convergence. Figure~\\ref{fig:fsm1_nondeterm2_X1} shows $X$ before and after performing inference. An exact factorization is found and is shown in Figure~\\ref{fig:fsm1_nondeterm2_factor}. We observed that repeated runs always converged to an exact solution, although different (but valid) result was obtained each time. Observe that some of the hidden variables correspond to exactly one state, while others correspond to a superposition of states, such that the column sums are equal. We do not explicitly normalize the columns to have equal column sums in the inference procedure, but rather, the equality follows from the choice of $W$ and the observed variables. Looking at the transition diagram in Figure~\\ref{fig:fsm1}, we see that the inferred variables with a single state component (i.e., $x_t: t \\in  \\{1, 3, 6, 7\\}$) occur at the time slices for which there is exactly one possible state that satisfies the transition model, given the observed variables in $X$. For example, time slice 2 has an observed state $x_2 = s_2$, corresponding to $S_2$. From the transition diagram we see that the only state that can transition to $S_2$ is $S_1$ and therefore the hidden state at time slice 1 must correspond to state $S_1$. The variables with multiple distinct state components (i.e., $x_t, t \\in  \\{ 8,9,10\\}$) correspond to time slices for which there are multiple possible transitions that satisfy the transition model. We can think of these time slices simultaneously being in multiple distinct states. For example, a valid solution for hidden state $x_8$ can consist of any additive combination of state vectors $s_3, s_4$ such that the column sum of $x_8$ is 1.\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=80ex]{.\/figures\/fsm1_nondeterm2_X1.jpg}\n\\caption{The top image plot shows $X$ after initialization, corresponding to the partially observed state sequence: $(?, S_2, ?, S_3, S_4, ?, ?, ?, ?, ?)$, where \"?\" denotes a hidden state. The bottom image plot shows the inference results for $X$ in which the hidden variables have now been estimated. Observe that variables ${x_1, x_3, x_6, x_7}$ correspond to exactly one state and variables $x_8,x_9,x_{10}$ correspond to a superposition of states, since multiple solutions are possible for $x_8,x_9,x_{10}$.}\n\\label{fig:fsm1_nondeterm2_X1}\n\\end{figure}\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=80ex]{.\/figures\/fsm1_nondeterm2_factor.jpg}\n\\caption{An image plot of the matrices of the factorization $X_C = W H$, after solving for both $H$ and the hidden variables of $X$.}\n\\label{fig:fsm1_nondeterm2_factor}\n\\end{figure}\n\nSuppose that we now make the final time slice observed by setting $x_{10} = s_2$, corresponding to the partially observed sequence $(?, S_2, ?, S_3, S_4, ?, ?, ?, ?, S_2)$. In this case, there is only one possible configuration of hidden states that is constant with the observed states and with the underlying transition model. Figures~\\ref{fig:fsm1_nondeterm3_X1} and \\ref{fig:fsm1_nondeterm3_factor} show the corresponding inference results. We observed that repeated runs of the algorithm always converged to the correct solution.\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=80ex]{.\/figures\/fsm1_nondeterm3_X1.jpg}\n\\caption{The top image plot shows $X$ after initialization, corresponding to the partially observed state sequence: $(?, S_2, ?, S_3, S_4, ?, ?, ?, ?, S_2)$, where \"?\" denotes a hidden state. The bottom image plot shows $X$ again after running the inference algorithm to infer the hidden states. In this case, there is only one possible configuration of hidden states that is constant with the observed states and with the underlying transition model.}\n\\label{fig:fsm1_nondeterm3_X1}\n\\end{figure}\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=80ex]{.\/figures\/fsm1_nondeterm3_factor.jpg}\n\\caption{An image plot of the matrices of the factorization $X_C = W H$, after solving for $H$. This is for the case where $X$ corresponds to the paritally observed sequence: $(?, S_2, ?, S_3, S_4, ?, ?, ?, ?, S_2)$. In this case, there is only one possible configuration of hidden states that is constant with the observed states and with the underlying transition model.}\n\\label{fig:fsm1_nondeterm3_factor}\n\\end{figure}\n\nIn the case where multiple valid solutions are possible for the hidden variables, it is interesting to consider modifying the inference algorithm to attempt to find sparse solutions. One possibility that is simple to implement and seems to provide robust solutions consists of replacing the standard NMF update steps of the basic inference algorithm specified in Algorithm~\\ref{alg:inference_learning1} with corresponding sparse NMF update steps. For these experiments, we chose to use the nonsmooth non-negative matrix factorization (nsNMF) algorithm \\cite{nsNMF2006} which is described in Appendix~\\ref{appendix_sparse_nmf} for reference. Using the partially observed sequence from Equation~\\ref{eqn:x1PartiallyObserved1} and the modified inference algorithm with an nsNMF sparseness value of 0.1, we obtain the results shown in Figures~\\ref{fig:fsm1_nondeterm4_X1} and \\ref{fig:fsm1_nondeterm4_factor}. Repeated runs of the algorithm appear to produce distinct solutions such that each inferred hidden state $x_t$ has only a single nonzero component, corresponding to a single active state in any given time slice. For sparseness values much lower than 0.1, we observe that a solution consisting of a superposition of states can still occur, but most of the weight tends to be concentrated in a single state for each time slice.\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=80ex]{.\/figures\/fsm1_nondeterm4_X1.jpg}\n\\caption{The top image plot shows $X$ after initialization, corresponding to the partially observed state sequence: $(?, S_2, ?, S_3, S_4, ?, ?, ?, ?, ?)$, where \"?\" denotes a hidden state. The bottom image plot shows $X$ again after running the inference algorithm to infer the hidden states. Sparse NMF with sparseness = 0.1 was used. We observe that multiple solutions are possible (since $x_{10}$ is now hidden), but the sparsity constraint leads to solutions in which only 1 state is active in any given time slice.}\n\\label{fig:fsm1_nondeterm4_X1}\n\\end{figure}\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=80ex]{.\/figures\/fsm1_nondeterm4_factor.jpg}\n\\caption{An image plot of the matrices of the factorization $X_C = W H$, after solving for $H$. Note that $X$ is equal to the upper half submatrix of $X_C$ and corresponds to the state sequence $(?, S_2, ?, S_3, S_4, ?, ?, ?, ?, ?)$.  Sparse NMF with sparseness = 0.1 was used. We observe that multiple solutions are possible  (since $x_{10}$ is now hidden), but the sparsity constraint leads to solutions in which only 1 state is active in any given time slice.}\n\\label{fig:fsm1_nondeterm4_factor}\n\\end{figure}\n\nWe now perform an experiment to see what will happen if all model variables are made hidden (i.e., $X$ and $H$ are initialized to positive random values), and we perform inference using the sparse NMF updates. The top plot of Figure~\\ref{fig:fsm1_nondeterm5_X1} shows $X$ after being initialized to random positive values uniformly distributed between 0 and 1. The bottom plot shows the result of one run of the inference algorithm using a nsNMF sparseness value of 0.1. Figure~\\ref{fig:fsm1_nondeterm5_factor} shows the resulting factorization, which converged to an exact solution. We observe that repeated runs of the inference algorithm produce factorizations corresponding to a randomly sampling of the underlying transition model. This interesting result only appears to occur when we use the sparse inference algorithm. The non-sparse algorithm also converges to an exact factorization, but the inferred $X$ and $H$ then tend to consist of a superposition of many states in any given time slice.\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=80ex]{.\/figures\/fsm1_nondeterm5_X1.jpg}\n\\caption{The top image plot shows $X$ after being initialized to random positive values. The bottom image plot shows $X$ again after convergence of the inference algorithm, using sparse NMF with sparseness = 0.1}\n\\label{fig:fsm1_nondeterm5_X1}\n\\end{figure}\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=80ex]{.\/figures\/fsm1_nondeterm5_factor.jpg}\n\\caption{An image plot of the matrices of the factorization $X_C = W H$, after solving for $X$ and $H$, which were initialized to random positive values. Even though all model variables are hidden, the sparse NMF updates caused the inference algorithm to converge to a sparse solution corresponding to an valid state sequence under the state transition diagram in Figure~\\ref{fig:fsm1} in which only a single state is active in any given time slice.}\n\\label{fig:fsm1_nondeterm5_factor}\n\\end{figure}\n\n\\subsubsection{Learning the transition model from training data}\n\\label{sec:learn_tran_model_from_data}\n\n\nWe now attempt to learn a transition model from training data. The training data will consist of an observed sequence $X$, and $H$ will be hidden. The model parameters $W$ are therefore unknown and will be learned from the observed sequence $X$. We first consider the case where the input sequence $X$ is an elementary sequence, so that any two adjacent state vectors $x_t, x_{t+1}$ can be represented by a single transition basis vector in $W$. That is, any state vector $x_t$ contains only one nonzero component and therefore corresponds to exactly one state in the transition model. We let $X$ consist of a sequence of state basis vectors, such as:\n\n\\begin{align}\nX = \\left[ \\begin{array}{ccccc}s_1 & s_2 & s_3 & s_3 & \\dots \\end{array} \\right]\n\\end{align}\n\nwhere the states are constrained to evolve according to the transition diagram in Figure~\\ref{fig:fsm1}. The training sequence is generated as follows: We choose the initial state $x_1$ randomly with uniform probability. When multiple outgoing transitions from a state are possible, the next state vector is chosen randomly with equal probability from the set of allowable transitions. For all experiments in this section, we use a training sequence length of 1000. \n\n\n\nRecall that inference and learning is performed jointly, so that in the process of learning $W$, we also end up with an estimate for $H$ (the transition activations for the training sequence). The column count of $W$ is a free parameter that must be specified before performing learning. Since we already know the transition model that was used to generate the training data, we know that we must specify at least 6 columns for $W$ in order to have any hope of achieving an exact factorization. We will therefore specify that $W$ have 6 columns. We then run the inference and learning algorithm from Appendix~\\ref{sec:simpleDyn1_inf_learn} to convergence. Figure~\\ref{fig:fsm1_learnW_1component_6basis_factor} shows the inference and learning results on a length 10 sub-sequence of $X$. Observe that the learned $W$ contains the 6 correct transition basis vectors. However, we have observed that it can sometimes take a few runs of the algorithm in order to find an exact factorization. When the column count of $W$ was then increased slightly to 8, we found that the inference and learning algorithm always converged to an exact factorization. Figure~\\ref{fig:fsm1_learnW_1component_8basis_factor} shows the inference and learning results for a $W$ with 8 columns. Note that although the transition basis vectors are correct, there are now some duplicate columns in $W$. This is due to the normalization step in which the columns of $W$ are normalized to have unit sum as part of the NMF left (learning) update step. \n\nWe have observed that the number of duplicate columns in $W$ can be reduced if we use sparse NMF updates in the inference and learning algorithm and also remove the column sum normalization step from the NMF left update step. We instead normalize each column of $W$ so the the upper and lower sub-columns (corresponding to $W_1$ and $W_2$) are constrained have equal sum, which may be 0. This allows unneeded columns of $W$ to tend towards zero during the learning updates. Using an nsNMF sparseness value of 0.1, we obtain the results in Figure~\\ref{fig:fsm1_learnW_1component_8basis_factorSparse}. We observe that even though it was specified that $W$ have two more columns than required to represent the transition model, the learned $W$ contains only the required 6 columns and two zero-valued columns. Note also that since we have omitted the step of normalizing all columns to have sum equal to one, the nonzero columns of $W$ no longer sum to 1. If we wish for the nonzero columns to sum to the same value (e.g., 1), we could consider adding column normalization for the nonzero columns as a post-processing step. We have also observed that the use of the nsNMF algorithm leads to a small approximation error so that the resulting factorization is no longer exact.\n\n\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=80ex]{.\/figures\/fsm1_learnW_1component_6basis_factor.jpg}\n\\caption{An image plot of the matrices of the factorization $X_C = W H$, after learning $W$ and solving for $H$, which were initialized to random positive values. $W$ was constrained to have 6 transition basis vectors. The training sequence $X$ consisted of state basis vectors that evolved according to the transition diagram in Figure~\\ref{fig:fsm1}.}\n\\label{fig:fsm1_learnW_1component_6basis_factor}\n\\end{figure}\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=80ex]{.\/figures\/fsm1_learnW_1component_8basis_factor.jpg}\n\\caption{An image plot of the matrices of the factorization $X_C = W H$, after learning $W$ and solving for $H$, which were initialized to random positive values. $W$ was constrained to have 6 transition basis vectors. Since there are two more columns than necessary to represent the transition model, observe that there are two duplicate columns in the learned $W$. The training sequence $X$ consisted of state basis vectors that evolved according to the transition diagram in Figure~\\ref{fig:fsm1}.}\n\\label{fig:fsm1_learnW_1component_8basis_factor}\n\\end{figure}\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=80ex]{.\/figures\/fsm1_learnW_1component_8basis_factorSparse.jpg}\n\\caption{An image plot of the matrices of the factorization $X_C = W H$, after learning $W$ and solving for $H$. $W$ was constrained to have 6 transition basis vectors. There are two more columns than necessary to represent the transition model, but we remove the normalization step from the $W$ update and set the nsNMF sparseness value to 0.1. The learning algorithm then learns only the 6 required columns. That is, we see that two columns of the learned $W$ are zero-valued.}\n\\label{fig:fsm1_learnW_1component_8basis_factorSparse}\n\\end{figure}\n\n\nWe now consider the more challenging learning problem in which the training sequence consists of an additive mixture of realizations of the transition model in Figure~\\ref{fig:fsm1}. We construct the training sequence $X$ as the following additive mixture of three scaled elementary state transition realization sequences:\n\n\\begin{align}\nX = 0.5*Xelem_1 + 1.0*Xelem_2 + 1.5*Xelem_3\n\\end{align}\n\nwhere the $Xelem_i$ are independent elementary state sequences, each consisting of a sequence of state basis vectors that conform to the transition model in Figure~\\ref{fig:fsm1}. Figure~\\ref{fig:fsm1_learnW_mixture_component_X1} shows the first 10 time slices of the training sequence. Note that each time slice of $X$ now corresponds to a state vector that represents a superposition of three states. Given this training sequence, we perform inference and learning using 8 columns for $W$. The algorithm converges to an exact factorization, which is shown in Figure~\\ref{fig:fsm1_learnW_mixture_component_factor} and we see that the underlying transition model was learned, even though the training data consisted of an additive mixture of elementary state sequences. Thus, even though a realization of the transition model was not presented individually as training data, the model still was able to learn an underlying transition model that explained the mixture training sequence.\n\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=80ex]{.\/figures\/fsm1_learnW_mixture_component_X1.jpg}\n\\caption{An image plot showing the first 10 time slices of the length 1000 training sequence $X$ consisting of an additive mixture of 3 elementary state transition sequences.}\n\\label{fig:fsm1_learnW_mixture_component_X1}\n\\end{figure}\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=80ex]{.\/figures\/fsm1_learnW_mixture_component_factor.jpg}\n\\caption{An image plot of the matrices of the factorization  $X_C = W H$, after learning $W$ and solving for $H$ where the $X$ training sequence consists of an additive mixture of three state transition sequences. $W$ was specified to have 8 transition basis vectors. The underlying transition model used to generate the training data corresponds to the transition diagram in Figure~\\ref{fig:fsm1}.}\n\\label{fig:fsm1_learnW_mixture_component_factor}\n\\end{figure}\n\nWe now add a noise component to the training sequence so that:\n\n\\begin{align}\nX = 0.5*Xelem_1 + 1.0*Xelem_2 + 1.5*Xelem_3 + \\epsilon\n\\end{align}\n\nwhere $\\epsilon$ is a 4 x $L$ matrix of uniformly distributed noise between 0 and 0.1. Figure~\\ref{fig:fsm1_learnW_mixture_noise_component_X1} shows a portion of the noisy training sequence. The inference and learning algorithm converges to the approximate factorization shown in Figure~\\ref{fig:fsm1_learnW_mixture_noise_component_factor}. We see that the underlying transition model was still recovered in $W$, although with a small error. These experiments indicate that it is possible to learn an underlying transition model, given only noisy training data corresponding to an additive mixture of realizations of the underlying model.\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=80ex]{.\/figures\/fsm1_learnW_mixture_noise_component_X1.jpg}\n\\caption{An image plot showing the first 10 time slices of the length 1000 training sequence $X$ consisting of an additive mixture of 3 elementary state transition sequences and a noise component.}\n\\label{fig:fsm1_learnW_mixture_noise_component_X1}\n\\end{figure}\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=80ex]{.\/figures\/fsm1_learnW_mixture_noise_component_factor.jpg}\n\\caption{An image plot of the matrices of the matrices of the factorization $X_C = W H$, after learning $W$ and solving for $H$ where the training sequence $X$ consists of an additive mixture of three state transition sequences plus a noise component. $W$ was specified to have 8 transition basis vectors.  The underlying transition model used to generate the training data corresponds to the transition diagram in Figure~\\ref{fig:fsm1}.}\n\\label{fig:fsm1_learnW_mixture_noise_component_factor}\n\\end{figure}\n\n\n\n\n\\section{Hierarchical factored state model}\n\\label{sec:main_hierarchical_state_model}\n\n\n\nIn this section we present DPFNs for modeling sequential data with hierarchical structure. The general DPFN framework allows for the representation of dynamical models with complex hierarchical structure. We present multilevel hierarchical networks in which only certain combinations of upper level and lower level variables and state transitions may be simultaneously active. We show how such networks provide for the modeling of factored hierarchical state transition models.\n\nThe models that we present in this section can be thought of as two copies of the network from Figure Figure~\\ref{fig:1layerDynamic} to represent multiple levels in a hierarchy, along with an additional coupling module to enforce desired coupling constraints between pairs of variables within a time slice. As a concrete example, we present a hierarchical state model for a regular expression and present empirical results.\n\n\n\\subsection{Model}\n\\label{sec:dyn2level}\nConsider a 2 level network in which each level represents a state transition model. One can think of each of the levels as having a corresponding state transition diagram. The active state transitions in the 2 levels are then coupled so that only certain combinations of level 2 and level 1 state transitions may be simultaneously active. Figure~\\ref{fig:2layerDynamic} shows a 2-level network for modeling a factored hierarchical state model with two levels in the transition hierarchy. The model immediately extends to an arbitrary number of levels. The level 2 state variables are denoted by \\{$x^2_t: t = 1, \\dots, T$\\} and the level 1 state variables are denoted by $\\{x^1_t: t=1, \\dots, T\\}$. The level 2 transition variables $\\{h^2_t: t=1, \\dots, T-1 \\}$ are coupled to the level 1 transition variables $\\{h^1_t: t = 1, \\dots, T -1\\}$ via the coupling variables $\\{v_t: t = 1, \\dots, T -1\\}$.\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=60ex]{.\/figures\/2layerDynamic.pdf}\n\\caption{A DPFN with a coupled 2-level transition model. The level 2 state variables are denoted by $x^2_t$ and the level 1 state variables are denoted by $x^1_t$. The level 2 transition variables $h^2_t$ are coupled to the level 1 transition variables $h^1_t$. The first four time slices of the $T$ time slice network are shown. The dashed lines represent (optional) forced factored co-activations, which are enforced if all sub-columns of the parameter matrices are normalized to have equal column sums.}\n\\label{fig:2layerDynamic}\n\\end{figure}\n\nNote the symmetry between levels 1 and 2. One can think of the level 1 and level 2 transition models as executing independently, except for where the coupling constraints forbid it by constraining the combinations of level 1 and level 2 state transitions that can be simultaneously activated. Any two consecutive time slices ($t, t+1$) of Figure~\\ref{fig:2layerDynamic} correspond to the following three factorizations:\n\nFor level 1, we have:\n\\begin{align}\n\\left[ \\begin{array}{c}\nx^1_t \\\\\nx^1_{t+1} \\end{array} \\right] =&\\ W^1 h^1_t\n\\nonumber \\\\\n =&\\ \\left[ \\begin{array}{c}\nW^1_1 \\\\\nW^1_2 \\end{array} \\right] h^1_t\n\\end{align}\n\nFor level 2, we have:\n\\begin{align}\n\\left[ \\begin{array}{c}\nx^2_t \\\\\nx^2_{t+1} \\end{array} \\right] =&\\ W^2 h^2_t\n\\nonumber \\\\\n =&\\ \\left[ \\begin{array}{c}\nW^2_1 \\\\\nW^2_2 \\end{array} \\right] h^2_t\n\\end{align}\n\nFor the level 1 to level 2 transition coupling, we have:\n\\begin{align}\n\\left[ \\begin{array}{c}\nh^2_t \\\\\nh^1_t \\end{array} \\right] =&\\ U v_t\n\\nonumber \\\\\n =&\\ \\left[ \\begin{array}{c}\nU_1 \\\\\nU_2 \\end{array} \\right] v_t\n\\end{align}\n\nFor $T$ time slices, we then have the  following three matrix factorization euqations:\n\nFor level 1, we have:\n\\begin{align}\n\\left[ \\begin{array}{ccccc}\nx^1_1 & x^1_2 & x^1_3 & \\dots & x^1_{T-1} \\notag \\\\\nx^1_{2} & x^1_{3} & x^1_{4} & \\dots & x^1_{T} \\end{array} \\right] =&\\ W^1 \\left[ \\begin{array}{ccccc} h^1_1 & h^1_2 & h^1_3 & \\dots & h^1_{T-1} \\end{array} \\right] \\\\\n=&\\ \\left[ \\begin{array}{c}\nW^1_1 \\\\\nW^1_2 \\end{array} \\right] \\left[ \\begin{array}{ccccc} h^1_1 & h^1_2 & h^1_3 & \\dots & h^1_{T-1} \\end{array} \\right]\n\\end{align}\n\nwhich we can write more concisely as:\n\n\\begin{align}\n\\label{eq:dynamic1_fact1}\nX^1_c =& W^1 H^1 \\notag \\\\\n=& \\left[ \\begin{array}{c}\nW^1_1 \\\\\nW^1_2 \\end{array} \\right] H^1\n\\end{align}\n\nFor level 2, we have:\n\\begin{align}\n\\left[ \\begin{array}{ccccc}\nx^2_1 & x^2_2 & x^2_3 & \\dots & x^2_{T-1} \\notag \\\\\nx^2_{2} & x^2_{3} & x^2_{4} & \\dots & x^2_{T} \\end{array} \\right] =&\\ W^2 \\left[ \\begin{array}{ccccc} h^2_1 & h^2_2 & h^2_3 & \\dots & h^2_{T-1} \\end{array} \\right] \\\\\n=&\\ \\left[ \\begin{array}{c}\nW^2_1 \\\\\nW^2_2 \\end{array} \\right] \\left[ \\begin{array}{ccccc} h^2_1 & h^2_2 & h^2_3 & \\dots & h^2_{T-1} \\end{array} \\right]\n\\end{align}\n\nwhich we can write more concisely as:\n\n\\begin{align}\n\\label{eq:dynamic1_fact2}\nX^2_c =& W^2 H^2 \\notag \\\\\n=& \\left[ \\begin{array}{c}\nW^2_1 \\\\\nW^2_2 \\end{array} \\right] H^2\n\\end{align}\n\nFor the level 1 to level 2 transition coupling, we have:\n\\begin{align}\n\\left[ \\begin{array}{ccccc}\nh^2_1 & h^2_2 & h^2_3 & \\dots & h^2_{T-1} \\notag \\\\\nh^1_1& h^1_3 & h^1_4 & \\dots & h^1_{T-1} \\end{array} \\right] =&\\ U \\left[ \\begin{array}{ccccc} v_1 & v_2 & v_3 & \\dots & v_{T-1} \\end{array} \\right] \\\\\n=& \\left[ \\begin{array}{c}\nU_1 \\\\\nU_2 \\end{array} \\right] \\left[ \\begin{array}{ccccc} v_1 & v_2 & v_3 & \\dots & v_{T-1} \\end{array} \\right]\n\\end{align}\n\nDefining $V = \\left[ \\begin{array}{ccccc} v_1 & v_2 & v_3 & \\dots & v_{T-1} \\end{array} \\right]$, we can then write the above factorization more concisely as:\n\n\\begin{align}\n\\label{eq:dynamic1_fact3}\n\\left[ \\begin{array}{c}\nH^2 \\\\\nH^1 \\end{array} \\right] =& U V \\notag \\\\\n=& \\left[ \\begin{array}{c}\nU_1 \\\\\nU_2 \\end{array} \\right] V\n\\end{align}\n\nThe columns of $U$ define the coupling basis. The positive components of each column $u_i$ specify components of $h^2_t$ and $h^1_t$ that can be co-activated. That is, the stacked vector of $h^2_t$ and $h^1_t$ is representable as a non-negative combination of the columns of $U$, with $v_t$ giving the encoding in terms of the basis columns. A learning and inference algorithm for this network can be obtained by applying Algorithm~\\ref{alg:inference_learning1} to the system of factorizations (\\ref{eq:dynamic1_fact1}),(\\ref{eq:dynamic1_fact2}), and (\\ref{eq:dynamic1_fact3}). Example pseudocode is presented in Appendix~\\ref{sec:inf_learn_dyn2level}.\n\n\n\n\n\nWe now consider a regular expression example similar in spirit to the one used by Murphy in \\cite{Murphy_Thesis}. Murphy showed how a Hierarchical hidden Markov model (HHMM) \\cite{murphy01linear} could be used to model the hierarchical transition model corresponding to a regular expression. In this section, we show how a regular expression can be modeled in a somewhat analogous manner by using DPFN with hierarchical structure. We stress that there are many significant differences, however. For example, in a HHMM, each possible state transition is assigned a probability, which is not the case here for the DPFN. Also, in a HHMM, the hidden states are discrete-valued, whereas they are non-negative continuous-valued here. Figure~\\ref{fig:hfsm1_crop} shows a state transition diagram for the DPFN in Figure~\\ref{fig:2layerDynamic} that models the regular expression ``a+b(de)*c(de)+''. Under this transition model, an exact state factorization is only possible if the input sequence satisfies the given regular expression. This regular expression specifies that we have one or more repetitions of ``a'' followed by ``b'' followed by zero or more repetitions of ``de'' followed by ``c'' followed by one or more repetitions of ``de.'' We allow the expression to repeat after reaching the end state. Observe that states $S^2_3$ and $S^2_5$ of the top level transition model (which we refer to as the level 2 transition model) refine to the same lower level transition model (the level 1 transition model). Some states (the ones with letters under them) are \\emph{production states} that correspond to the production of the corresponding letters. The initial state is $S^2_1$, and states $S^2_6$ and $S^1_3$ denote explicit ends states. Although explicit end states are used here, they are not needed in general, as we will see in the next section. The semantics is such that when a transition to a refining state, such as $S^2_3$ or $S^2_5$ occurs, the refinement executes while the parent remains in the same state. We will choose the basis vectors for the coupling activations $U$ so that at any time step, the sum of active state values in level 2 is equal to the corresponding sum of active state values in level 1. For example, the transition $t^2_3$ correspond to a transition from $S^2_2$ to $S^2_3$ in level 2 and a simultaneous transition from the off\/end state to the starting state $S^1_1$ in level 1. The next transition must correspond to a transition from $S^1_1$ (production of ``d'' to $S^1_2$ (production of ``e''), while level 2 remains in the parent state $S^2_3$. The hierarchical transition diagram places implicit constraints on transitions that may simultaneously occur (co-occur) in levels 1 and 2. For example, the level 2 transition $t^2_5$ can co-occur with the level 1 transition $t^1_2$. An implicit self-loop transition from $S^2_3$ to $S^2_3$ can co-occur with level 1 transitions $t^1_1$ and $t^1_3$. Figure~\\ref{fig:hfsm1_detailed_crop} shows the same transition diagram, annotated to include the implicit transitions. Table~\\ref{table:trans_coupling} shows the transition coupling constraints that specify the pairs of transitions in levels 1 and 2 that can co-occur. For example, note that when the refinement is not producing a ``d'' or ``e'' the end state remains in a self-loop.\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=60ex]{.\/figures\/hfsm1_crop.pdf}\n\\caption{A state transition diagram for the regular expression ``a+b(de)*c(de)+''.}\n\\label{fig:hfsm1_crop}\n\\end{figure}\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=60ex]{.\/figures\/hfsm1_detailed_crop.pdf}\n\\caption{The state transition diagram for the regular expression ``a+b(de)*c(de)+''. The implicit transitions are shown in blue.}\n\\label{fig:hfsm1_detailed_crop}\n\\end{figure}\n\n\n\\begin{table}\n\\begin{center}\n\\begin{tabular}{| c ||c | c | c | c | c | c | c | c | c | c | c | c | c |}\n    \\hline\n    Level 2 transition & $t^2_3$ & $t^2_{10}$ & $t^2_{10}$ & $t^2_5$ & $t^2_6$ & $t^2_9$ & $t^2_9$ & $t^2_7$ & $t^2_{11}$ & $t^2_8$ & $t^2_1$ & $t^2_2$ & $t^2_4$ \\\\ \\hline\n    Level 1 transition & $t^1_4$ & $t^1_1$ & $t^1_3$ & $t^1_2$  & $t^1_4$  & $t^1_1$ & $t^1_3$ & $t^1_2$ & $t^1_5$ & $t^1_5$ & $t^1_5$ & $t^1_5$ & $t^1_5$ \\\\ \\hline\n\\end{tabular}\n \\caption{Transition coupling constraints for the example hierarchical transition model. Each column specifies a level 2 transition and a level 1 transition that can co-occur.}\n \\label{table:trans_coupling}\n \\end{center}\n \\end{table}\n\nThe level 1 transition model, level 2 transition model, and coupling constraints from the state transition diagram directly map into the corresponding parameters $W^1$, $W^2$, and $U$ as follows. The $W^1$ and $W^2$ matrices are constructed following the procedure outlined in Section~\\ref{sec:fact_state_tran_model}. Each $w^1_i$ is the vertical concatenation of the two state basis vectors associated transition $t^1_i$. $W^1$ is then constructed as the horizontal concatenation of the transition basis vector columns \\{$w^1_i$\\}. We then have:\n\n\\begin{align}\nW^1 =& \\left[ \\begin{array}{ccccc} w^1_1 & w^1_2 & w^1_3 & w^1_4 & w^1_5\\end{array} \\right] \\notag \\\\\n=&\\ \\left[ \\begin{array}{c}\nW^1_1 \\\\\nW^1_2 \\end{array} \\right] \\notag\\\\\n=&\\ \\left[ \\begin{array}{ccccc}\ns^1_1 & s^1_2 & s^1_2 & s^1_3 & s^1_3 \\\\\ns^1_2 & s^1_3 & s^1_1 & s^1_1 & s^1_3 \\end{array} \\right] \\notag \\\\\n=& \\left[ \\begin{array}{cccccc}\n1 & 0 & 0 & 0 & 0 \\\\\n0 & 1 & 1 & 0 & 0 \\\\\n0 & 0 & 0 & 1 & 1 \\\\ \\hline\n0 & 0 & 1 & 1 & 0 \\\\\n1 & 0 & 0 & 0 & 0 \\\\\n0 & 1 & 0 & 0 & 1 \\end{array} \\right] \n\\label{eqn:W_1hfsm}\n\\end{align}\n\nNote that $W^1_1$ is given by:\n\n\\begin{align}\nW^1_1 =& \\left[ \\begin{array}{ccccc} s^1_1 & s^1_2 & s^1_2 & s^1_3 & s^1_3 \\end{array} \\right] \\notag \\\\\n=& \\left[ \\begin{array}{cccccc}\n1 & 0 & 0 & 0 & 0 \\\\\n0 & 1 & 1 & 0 & 0 \\\\\n0 & 0 & 0 & 1 & 1\\end{array} \\right] \n\\end{align}\n\nand $W^1_2$ is given by:\n\n\\begin{align}\nW^1_2 =& \\left[ \\begin{array}{ccccc} s^1_2 & s^1_3 & s^1_1 & s^1_1 & s^1_3 \\end{array} \\right] \\notag \\\\\n=& \\left[ \\begin{array}{cccccc}\n0 & 0 & 1 & 1 & 0 \\\\\n1 & 0 & 0 & 0 & 0 \\\\\n0 & 1 & 0 & 0 & 1\\end{array} \\right] \n\\end{align}\n\nWe construct $W^2$ as:\n\n\\begin{align}\nW^2 =& \\left[ \\begin{array}{ccccccccccc} w^2_1 & w^2_2 & w^2_3 & w^2_4 & w^2_5 & w^2_6 & w^2_7 & w^2_8 & w^2_9 & w^2_{10} & w^2_{11} \\end{array} \\right] \\notag \\\\\n=&\\ \\left[ \\begin{array}{c}\nW^2_1 \\\\\nW^2_2 \\end{array} \\right] \\notag\\\\\n=&\\ \\left[ \\begin{array}{ccccccccccc}\ns^2_1 & s^2_1 & s^2_2 & s^2_2 & s^2_3 & s^2_4 & s^2_5 & s^2_6 & s^2_5 & s^2_3 & s^2_6 \\\\\ns^2_1 & s^2_2 & s^2_3 & s^2_4 & s^2_4 & s^2_5 & s^2_6 & s^2_1 & s^2_5 & s^2_3 & s^2_6 \\end{array} \\right] \\notag \\\\\n=& \\left[ \\begin{array}{ccccccccccc}\n1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n0 & 0 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 1 & 0 \\\\\n0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\\\\n0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 1 & 0 & 0 \\\\\n0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 1 \\\\ \\hline\n1 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\\\\n0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\\\\n0 & 0 & 0 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 1 & 0 & 0 \\\\\n0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 1 \\end{array} \\right] \n\\label{eqn:W_2hfsm}\n\\end{align}\n \nLikewise, $W^2_1$ corresponds to the upper sub-matrix of $W^2$ and $W^2_2$ corresponds to the lower sub-matrix.\n\nWe construct the transition coupling matrix $U$ as follows. We start with the $Q$ pairs of level 2 - level 1 couplings specified in Table~\\ref{table:trans_coupling}. We construct $U$ from this table such that the $i$'th column of $U$ corresponds to the transition pair in the $i$'th column of the table. For the level 2 transitions, we replace each transition $t^2_j$ in the table with the column vector $u_{upper_j} \\in \\mathbb{R}^{R2}$ that contains a 1 in the component corresponding to the row of $H^2$ that activates the corresponding transition in $W^2$. Likewise, for the level 1 transitions, we replace each transition $t^1_j$ in the table with the column vector $u_{lower_j} \\in \\mathbb{R}^{R1}$ that contains a 1 in the component corresponding to the row of $H^1$ that activates the corresponding transition in $W^1$. We then form the $j$'th column $u_j$ of $U$ as the vertical concatenation of $u_{upper_j}$ on top of $u_{lower_j}$ so that we have for the $(R2 + R1)$ x $Q$ matrix U:\n\n\\begin{align}\nU =& \\left[ \\begin{array}{ccccc} u_1 & u_2 & u_3 & \\dots & u_Q \\end{array} \\right] \\notag \\\\\n=& \\left[ \\begin{array}{c} U_1 \\\\\nU_2 \\end{array} \\right] \\notag \\\\\n=& \\left[ \\begin{array}{ccccc} u^2_{upper_1} & u^2_{upper_2} &u^2_{upper_3} & \\dots & u^2_{upper_Q} \\\\\nu^1_{lower_1} & u^1_{lower_2} & u^1_{lower_3} & \\dots & u^1_{lower_Q} \\end{array} \\right]\n\\end{align}\n\nwhere the $R2$ x $Q$ submatrix $U_1$ is given by:\n\n\\begin{align}\nU_1 = \\left[ \\begin{array}{ccccc} u_{upper_1} & u_{upper_2} &u_{upper_3} & \\dots & u_{upper_Q} \\end{array} \\right]\n\\end{align}\n\nand the $R1$ x $Q$ submatrix $U_2$ is given by:\n\n\\begin{align}\nU_2 = \\left[ \\begin{array}{ccccc} u_{lower_1} & u_{lower_2} & u_{lower_3} & \\dots & u_{lower_Q} \\end{array} \\right]\n\\end{align}\n\n\n\nFrom Table~\\ref{table:trans_coupling}, we then have the following for $U$:\n\n\\begin{align}\n\\label{eq:U_coupling}\nU = \\left[ \\begin{array}{ccccccccccccc}\n0 &  0 &  0 &  0 &  0 &  0 &  0 &  0 &  0 &  0 &  1 &  0 &  0 \\\\\n 0 &  0 & 0 &  0 &  0 &  0 &  0 &  0 &  0 &  0 &  0 &  1  &  0 \\\\\n 1 &  0 &  0 &  0 &  0 &  0 &  0 &  0 &  0 &  0 &  0 &  0 &  0 \\\\\n 0 &  0 &  0 &  0 &  0 &  0 &  0 &  0 &  0 &  0 &  0 &  0 &  1 \\\\\n 0 &  0 &  0 &  1 &  0 &  0 &  0 &  0 &  0 &  0 &  0 &  0 &  0 \\\\\n 0 &  0 &  0 &  0 &  1 &  0 &  0 &  0 &  0 &  0 &  0 &  0 &  0 \\\\\n 0 &  0 &  0 &  0 &  0 &  0 &  0 &  1 &  0 &  0 &  0 &  0 &  0 \\\\\n 0 &  0 &  0 &  0 &  0 &  0 &  0 &  0 &  0 &  1 &  0 &  0 &  0 \\\\\n 0 &  0 &  0 &  0 &  0 &  1 &  1 &  0 &  0 &  0 &  0 &  0 &  0 \\\\\n 0 &  1 &  1 &  0 &  0 &  0 &  0 &  0 &  0 &  0 &  0 &  0 &  0 \\\\\n 0 &  0 &  0 &  0 &  0 &  0 &  0 &  0 &  1 &  0 &  0 &  0 &  0 \\\\ \\hline\n 0 &  1 &  0 &  0 &  0 &  1 &  0 &  0 &  0 &  0 &  0 &  0 &  0 \\\\\n 0 &  0 &  0 &  1 &  0 &  0 &  0 &  1 &  0 &  0 &  0 &  0 &  0 \\\\\n 0 &  0 &  1 &  0 &  0 &  0 &  1 &  0 &  0 &  0 &  0 &  0 &  0 \\\\\n 1 &  0 &  0 &  0 &  1 &  0 &  0 &  0 &  0 &  0 &  0 &  0 &  0 \\\\\n 0 &  0 &  0 &  0 &  0 &  0 &  0 &  0 &  1 &  1 &  1 &  1 &  1 \\end{array} \\right] \n\\end{align}\n\nNote that $U^1$ activates $H^2$ (activations for $W^2$) and $U^2$ activates $H^1$ (activations for $W^1$)\n\n\n\\subsection{Empirical results}\n\nIn this section, we perform inference on the network in Figure~\\ref{fig:2layerDynamic} using the parameter matrices for the regular expression hierarchical transition diagram in Figure~\\ref{fig:hfsm1_detailed_crop}. We use parameter matrices  $W^1$, $W^2$, and $U$ from Equations (\\ref{eqn:W_1hfsm}),(\\ref{eqn:W_2hfsm}), and (\\ref{eq:U_coupling}) respectively.\n\nWe choose to model a length 10 sequence for ease of diplay of the results. We let the observed set $X_E$ consist of $\\{x^2_2 = s^2_2, x^2_7 = s^2_4\\}$. We let the hidden set $X_H$ consist of all other model variables. That is, we are only given that in time slice 2, the level 2 state is $S^2_2$ (production state ``b'') in Figure~\\ref{fig:hfsm1_detailed_crop} and that in time slice 7, the level 2 state is $S^2_4$ (production state ``c''). We are given no information about the level 1 states, transitions, or coupling variable states. However, even given this limited information, the constraints impose by the hierarchical transition model result in a single possible solution for the hidden variables in time slices $t=1,\\dots,9$. Multiple solutions (due to multiple outgoing transitions) are possible for time slice 10. The reader can verify this by plugging in the observed states and studying the transition diagram to trace the patch of allowable transitions.\n\nWe now perform inference using an instance of the general inference and learning algorithm given in Appendix~\\ref{sec:inf_learn_dyn2level}. All hidden variables are initialized to random positive values, and the algorithm is run for 500 iterations, which is sufficient for convergence to an exact factorization (RMSE less than $10^-4$) for all three factorization equations. We observed that repeated runs always converged, although a different solution for time slice 10 was reached in each case (since multiple solutions are possible for this time slice).\n\nFigure~\\ref{fig:hdyn1_X1_X2_results} shows both the initialized and inferred values for $X^1$ and $X^2$.  Model variables $x^2_2 = s^2_2$ and $x^2_7 = s^2_4$ were observed and all other model variables were hidden and estimated by the inference algorithm. Note that the hidden values of $X^2$ (all time slices except $t=2,7$) appear black in the figure because the initialized random values are very small (order of $10^-6$) compared to the maximum value of the observed time slices in the figure. The inference results for $X^1$ and $X^2$ are shown in the rightmost image plots. We see that the inference algorithm successfully found the single possible solution for time slices $t=1,\\dots,9$. In time slice 10 there are two possible solution states for both $x^1_{10}$ and $x^2_{10}$ and we see that the inferred values represent a mixture of the two possible state configurations. That is, in time slice 9, the inferred state is $x^2_9 = s^2_5$ and $x^1_9 = s^1_2$. According to the transition diagram in Figure~\\ref{fig:hfsm1_detailed_crop}, we see that in the next time slice (slice 10), we can either remain in level 2 state $S^2_5$ while level 1 transitions to $S^1_1$, or both level 2 and level 1 can transition to their respective end states: $S^2_6$ and $S^1_3$. We observe that the inferred state values represent a superposition of these two possibilities. We also note that when sparse NMF updates were used in inference, the solution tended to chose one alternative or the other, but not a superposition of the two.\n\nFigure~\\ref{fig:hdyn1_X_W_H_factor} shows the inference results for $X^2_C$ and $H^2$ in the factorization $X^2_C = W^2 H^2$ in the top three image plots. The bottom three image plots show the inference results for $X^1_C$ and $H^1$ in the factorization $X^1_C = W^2 H^1$.  Figure~\\ref{fig:hdyn1_H_U_V_factor} shows an image plot of the inference results for the matrices in the transition coupling factorization.\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=80ex]{.\/figures\/hdyn1_X1_X2_results.jpg}\n\\caption{The initialized $X^2$ is shown in the upper left. Model variables $x^2_2 = s^2_2$ and $x^2_7 = s^2_4$ were observed and all other model variables were hidden and estimated by the inference algorithm. The initialized $X^1$ is shown in the lower left. All hidden variables were initialized to random positive values. The inference results for $X^1$ and $X^2$ are shown in the rightmost image plots. We note that given the two observed variables, only one possible solution exists for time slices $t=1,\\dots,9$. Multiple solutions exist for the final time slice $t=10$.}\n\\label{fig:hdyn1_X1_X2_results}\n\\end{figure}\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=80ex]{.\/figures\/hdyn1_X_W_H_factor.jpg}\n\\caption{The top three matrices show the inference results for $X^2_C$ and $H^2$ in the factorization $X^2_C = W^2 H^2$. The bottom three matrices show the inference results for $X^1_C$ and $H^1$ in the factorization $X^1_C = W^2 H^1$. Model variables $x^2_2$ and $x^2_7$ were observed and all other model variables were hidden and inferred by the inference algorithm. Parameter matrices $W^1$ and $W^2$ are given in Equations (\\ref{eqn:W_1hfsm}) and (\\ref{eqn:W_2hfsm}), respectively.}\n\\label{fig:hdyn1_X_W_H_factor}\n\\end{figure}\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=80ex]{.\/figures\/hdyn1_H_U_V_factor.jpg}\n\\caption{An image plot of the inference results for the matrices in the transition coupling factorization. All variables $H^1, H^2, V$ were hidden and inferred. Parameter matrix $U$ is given in Equation (\\ref{eq:U_coupling}).}\n\\label{fig:hdyn1_H_U_V_factor}\n\\end{figure}\n\n\n\n\\section{Multiple target tracking model}\n\\label{sec:target_tracking}\n\nWe now consider a multiple target tracking model that is motivated by the problem of tracking auditory objects in computational auditory scene analysis (CASA) \\cite{ref:bregman}. This example serves to motivate the potential usefulness of our approach to problems such as CASA. Recall that a DPFN supports the representation of an additive combination of realizations of underlying dynamical process models. Such a representation seems to fit very naturally within CASA problems, since the sound pressure waves that arrive at the eardrum are well modeled as an additive mixture of scaled versions of the sound pressure waves produced by each auditory object alone. If it turns out to be possible to use a DPFN to model auditory objects such as musical instruments, environmental noises, speech, etc., then the linearity property would imply that any additive mixture of the various supported auditory objects at various sound levels would then be representable by the model as well. Of course, this does not imply that any particular inference and\/or learning algorithm will successfully be able to deal with such a mixture, but the fact that an additive mixture is representable at all seems to be quite powerful. Our approach has the desirable property that nowhere in our model do we need to explicitly specify anything about the maximum number of simultaneous targets\/objects or the allowable magnitude range (volume levels) for targets.\n\nWe will specify a dynamical model for a single target, and then observe how a scaled mixture of multiple targets is then automatically supported. This illustrates how the factored representation of a dynamical system allows for a compact parameter set, while still being expressive enough to support multiple simultaneous scaled realizations of the underlying process model. All parameters in the target model will be manually specified here for ease of illustration, although in principle one could attempt to learn them from training data as well. We therefore perform inference only in the results that follow. We will also work with synthetic target observation data, rather than actual audio recordings, again for ease of explanation and illustration. It is hoped that after reading this paper, the interested reader will find it straightforward to come up with ideas for developing more complex extensions of these models that may then be applied to practical problems involving real-world data sets. \n\n\n\nThe problem setup is as follows. We are given a sequence of non-negative observation column vectors, one for each time slice. When concatenated horizontally, they form an image such that the time axes runs from left to right. The vertical axis then corresponds to sensor position or frequency bin index, depending on the interpretation. We therefore use position and frequency bin interchangeably in the following discussion. In the context of CASA, each observation vector can be considered a time slice of a magnitude spectrogram such that the components of an observation vector represent the short-time spectra of an audio signal.  In the CASA interpretation, we can then think of the observation image as a magnitude time frequency image, such as a spectrogram.\n\nThe observations may also be potentially corrupted by additive non-negative noise. Our goal is to infer the types and positions of all targets given only the target observation sequence. We may also be interested in predicting future target trajectories and\/or inferring missing observations data. \n\n\nSpecifically, we wish to model a target that has the following properties:\n\\begin{enumerate}\n\n\\item A new target can come come into existence (become alive) at any time instant. \n\n\\item Each target has a certain localized \\emph{pattern signature}. That is, its position\/frequency evolves in time according to an underlying transition model. Target classes are distinguished by their position signatures. For example, one can think of two distinct musical instruments: one produces a certain type of vibrato regardless of the note pitch, while the other instrument produces a short chirp followed by a more steady pitch.\n\n\\item A target's pattern signature is modeled as being independent of the target's overall position\/location\/frequency bin index. Thus, we model a target's location as a position shift or frequency shift of the pattern signature. We only allow a target's overall position to change at certain constrained points in the position signature transition model. In the context of CASA, a the target's overall location might correspond to the pitch of a note played by an instrument, which is independent of the instrument's pattern signature (e.g., each note is performed with a vibrato).\n\n\\item A target may cease to exist (die) after some number of time slices which may or may not be deterministic. \n\n\\item When a target comes into existence, it can take on any positive magnitude value, which remains constant during the time that the target is alive. This corresponds to an auditory object, such as an instrument note event, that can begin sounding at an arbitrary sound level or loudness and remains at the same sound level until the note ends. It is possible to relax this constraint, but we enforce it in this example.\n\n\\item Any number of targets may be simultaneously present, each potentially having a distinct magnitude. This corresponds to multiple auditory objects simultaneously occurring, such as several instruments performing together, or several people talking at the same time, for example.\n\n\\end{enumerate}\n\n\n\n\\subsection{Model}\n\nFigure~\\ref{fig:targetTracking1} shows the DPFM for the target tracker. $x^1_t, t=1...T$ are observed and all other variables are hidden. We now describe the role of each of the 7 factorization equations that specify this model. We make use of the matrix notation developed in Section \\ref{sec:fact_state_tran_model} so that the factorizations may be expressed concisely.\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=60ex]{.\/figures\/targetTracking1.pdf}\n\\caption{The DPFN for the the multiple target tracker. The target observations variables $x^1_t, t=1...T$ are observed and all other variables are hidden. The first two time slices are shown.}\n\\label{fig:targetTracking1}\n\\end{figure}\n\n\\subsubsection{Target state transition model}\n\n\n\nThe target state transition model corresponds to the factorization:\n\\begin{align}\n\\label{eq:target_state_transition_model}\nX^4_c =& W^6 H^3 \\notag \\\\\n=& \\left[ \\begin{array}{c}\nW^6_1 \\\\\nW^6_2 \\end{array} \\right] H^3\n\\end{align}\n\n\nFigure~\\ref{fig:fsm_target_tracker1} shows the target state transition diagram. Observe that states $S^2_1$ through $S^2_5$ correspond to a ``A-type'' target and states $S^2_6$ through $S^2_9$ correspond to a ``B-type'' target. The initial state when a ``A-type'' target comes into existence is $S^2_1$, and the minimum duration (i.e., target lifetime) is 5 time slices, since the target must pass consecutively through states $S^2_1$ through $S^2_5$ before it is possible for the target to die (i.e., transition to the 0\/off state). We see that when the current state is $S^2_5$ the following state can be $S^2_2$ or the 0 state. Recall that this is not a probabilistic model. Thus, multiple outgoing transitions from a given state can be thought of as transition possibilities that are equally likely. The particular transition (or superposition of transitions) that is chosen during inference depends on the values of the observed variables as well as any other constraints imposed by other parts of the model that are coupled to the transition model. The state transition model for a ``B-type'' target is similar but only consists of 4 states. We construct $W^6$ directly by inspection of the state transition diagram using the procedure described in Section \\ref{sec:fact_state_tran_model}. Note that there are 9 states and 13 transitions. The ``off'' state is not an explicit state, and is represented by the 9-dimensional 0 vector in the transition basis matrix. Thus $W^6$ has size 18 x 13. The state basis vectors \\{$s^2_i$\\} are therefore 9-dimensional column vectors, as are the target state variables \\{$x^4_t$\\}. Each $w^6_i$ is the vertical concatenation of the two basis state vectors associated transition $t^2_i$. $W^6$ is then constructed as the horizontal concatenation of the transition basis vector columns \\{$w^1_i$\\}. We then have for $W^6$:\n\n\\begin{align}\nW^6 =& \\left[ \\begin{array}{ccccccccccccc} w^6_1 & w^6_2 & w^6_3 & w^6_4 & w^6_5 & w^6_6 & w^6_7 & w^6_8  & w^6_9 & w^6_{10} & w^6_{11} & w^6_{12} & w^6_{13} \\end{array} \\right] \\notag \\\\\n=&\\ \\left[ \\begin{array}{c}\nW^6_1 \\\\\nW^6_2 \\end{array} \\right] \\notag\\\\\n=&\\ \\left[ \\begin{array}{ccccccccccccc}\ns^2_1 & s^2_2 & s^2_3 & s^2_4 & s^2_5  & s^2_5 & 0      & s^2_6 & s^2_7 & s^2_8 & s^2_9 & s^2_9 & 0\\\\\ns^2_2 & s^2_3 & s^2_4 & s^2_5 & 0      & s^2_2 & s^2_1  & s^2_7 & s^2_8 & s^2_9 & 0     & s^2_7 & s^2_6\\end{array} \\right] \\notag \\\\\n\\label{eqn:W_6target}\n\\end{align}\n\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=80ex]{.\/figures\/fsm_target_tracker1.pdf}\n\\caption{The state transition diagram for the target state variables $x^4_t$. The corresponding state emission vectors $x^3_t$ are also shown.}\n\\label{fig:fsm_target_tracker1}\n\\end{figure}\n\n\n\n\\subsubsection{Target label to target state coupling}\n\nThe target label to target state coupling corresponds to the factorization:\n\n\\begin{align}\n\\label{eq:target_label_to_target_state_coupling}\n\\left[ \\begin{array}{c}\nX^5 \\\\\nX^4 \\end{array} \\right] =& W^7 U^3 \\notag \\\\\n=& \\left[ \\begin{array}{c}\nW^7_1 \\\\\nW^7_2 \\end{array} \\right] U^3\n\\end{align}\n\nThis module performs the task of assigning a target type label $x^5_t$ to a target state $x^4_t$. From Figure~\\ref{fig:fsm_target_tracker1} we see that there are two distinct target types. We let $x^5_t$ denote the 2-dimensional column vector such that the first (top) component denotes the strength of an ``A-type'' target and the second (bottom) component denotes the strength of a ``B-type'' target. We will choose the basis columns $U$ such that \n\nWe wish to add coupling constraints so that the ``A-type'' basis vector $s^3_1 = (1,0)^T$ can co-occur with any of the basis state vectors \\{$s^2_1, s^2_2, s^2_3, s^2_4, s^2_5$\\} and the ``B-type'' basis vector $s^3_2 = (0,1)^T$ can co-occur with any of the basis state vectors \\{$s^2_6, s^2_7, s^2_8, s^2_9$\\}. We construct each column of $W^7$ as the vertical concatenation of an $s^3_i$ on top of and $S^2_j$ that can co-occur. Noting that the column ordering is arbitrary, we have for the 7 x 9 matrix $W^7$:\n\n\\begin{align}\nW^7 =& \\left[ \\begin{array}{c} W^7_1 \\\\\nW^7_2 \\end{array} \\right] \\notag \\\\\n=& \\left[ \\begin{array}{ccccccccc} s^3_1 & s^3_1 & s^3_1 & s^3_1 & s^3_1 & s^3_2 & s^3_2 & s^3_2 & s^3_2 \\\\\ns^2_1 & s^2_2 & s^2_3 & s^2_4 & s^2_5 & s^2_6 & s^2_7 & s^2_8 & s^2_9 \\end{array} \\right]\n\\end{align}\n\n\n\\subsubsection{Target state to target emission coupling}\n\nThe target state to target emission coupling corresponds to the factorization:\n\n\\begin{align}\n\\label{eq:target_state_to_target_emission_coupling}\n\\left[ \\begin{array}{c}\nX^4 \\\\\nX^3 \\end{array} \\right] =& W^3 U^2 \\notag \\\\\n=& \\left[ \\begin{array}{c}\nW^3_1 \\\\\nW^3_2 \\end{array} \\right] U^2\n\\end{align}\n\nThis module performs the task of coupling the target state variable $x^4_t$ to the corresponding production (emission) variable $x^3_t \\in \\mathbb{R}^{patternSize}$, where $patternSize = 5$ in this example. Figure~\\ref{fig:fsm_target_tracker1} shows the emission basis vector $s^e_i \\in \\mathbb{R}^{patternSize}$ below the corresponding state $S^2_i$. If a given state $S^2_i$ is present in $x^4_t$ with magnitude $\\alpha$, then we wish for the corresponding emission variable $s^e_i$ to also be present in $x^3_t$ with magnitude $\\alpha$ and vice versa. We construct each column of $W^3$ as the vertical concatenation of a basis state vector $s^2_i$ on top of the corresponding emission basis vector $s^e_i$. The 10 x 9 matrix $W^3$ is then given as:\n\n\\begin{align}\nW^3 =& \\left[ \\begin{array}{c} W^3_1 \\\\\nW^3_2 \\end{array} \\right] \\notag \\\\\n=& \\left[ \\begin{array}{ccccccccc} s^2_1 & s^2_2 & s^2_3 & s^2_4 & s^2_5 & s^2_6 & s^2_7 & s^2_8 & s^2_9 \\\\\ns^e_1 & s^e_2 & s^e_3 & s^e_4 & s^e_5 & s^e_6 & s^e_7 & s^e_8 & s^e_9 \\end{array} \\right]\n\\end{align}\n\n\\subsubsection{Pattern translation transition model}\n\nThe pattern translation transition model corresponds to the factorization:\n\n\\begin{align}\n\\label{eq:pattern_translation_transition_model}\nX^2_c =& W^2 H^1 \\notag \\\\\n=& \\left[ \\begin{array}{c}\nW^2_1 \\\\\nW^2_2 \\end{array} \\right] H^1\n\\end{align}\n\nThis module represents the transition model for the amount by the target emission pattern is translated (i.e., target position) between time steps $t$ and $t+1$. Figure~\\ref{fig:targetTrackingPosition} shows the state transition diagram for the position shift variables $x^2_t$ with $P$ possible distinct position shift amounts. Note that $x^2_t$ is therefore a $P$ dimensional column vector, with the $i$'th component corresponding to the magnitude of $S^1_i$ at time $t$. Each state $S^1_i$ corresponds to a distinct target position. The index $i$ representing the downward translation amount that the target pattern will be translated when it appears in the observation vector $x^1_t$. The state $S^1_1$ corresponds to the minimum downward translation amount, so that the target emission pattern appears as the top $patternSize$ components of $x^1_t$. The state $S^1_P$ corresponds to the maximum downward translation amount, so that the target emission pattern appears as the bottom $patternSize$ components of $x^1_t$.\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=80ex]{.\/figures\/targetTrackingPosition.pdf}\n\\caption{The state transition diagram for the position shift variables $x^2_t$ with $P$ possible distinct position shift amounts.}\n\\label{fig:targetTrackingPosition}\n\\end{figure}\n\nWe specify in the state transition diagram that a transition $t_{on_i}$ from the off state at time $t$ to to any of the $P$ possible position shift amounts $S^1_i$ at time $t+1$ is allowed. Let $t_{on}$ denote the set \\{$t_{on_i}, i = 1, \\dots, P$\\} of all ``on'' transitions. We specify that a transition $t_{off_i}$ from any of the $P$ possible positions $S^1_i$ to the off state is allowed. Let $t_{off}$ denote the set \\{$t_{off_i}, i = 1, \\dots, P$\\} of all ``off'' transitions.  A self-loop transition $t_{self_i}$ from any position state $S^1_i$ to itself is allowed. Let $t_{self}$ denote the set \\{$t_{self_i}, i = 1, \\dots, P$\\} of all ``self-loop'' transitions.  Finally, we also allow a transition $t_{inc_i}$ from state $S^1_i$ to $S^1_{i+1}$, as well as a transition $t_{dec_i}$ from state $S^1_{i+1}$ to $S^1_{i}$. Let $t_{inc}$ denote the set \\{$t_{inc_i}, i = 1, \\dots, P-1$\\} of all ``position increment'' transitions. Let $t_{dec}$ denote the set \\{$t_{dec_i}, i = 1, \\dots, P-1$\\} of all ``position decrement'' transitions. Let $t_{change}$ denote the union of $t_{inc}$ and $t_{dec}$, which is the set of all ``position change'' transitions. The complete set of all position transitions is then $Tpos =\\{t_{self}, t_{change}, t_{on}, t_{off}\\}$.\n\nWe construct $W^2$ directly by inspection of the state transition diagram using the procedure described in Section \\ref{sec:fact_state_tran_model}. Using our convention for basis state vectors, state $S^1_i$ in the transition diagram corresponds to a 9-dimensional column vector $s^1_i$ such that the $i$'th component is 1 and all other components are 0.  The ``off'' state is not an explicit state, and is represented by the 9-dimensional 0 vector in the transition basis matrix. Since there are $P$ states, the state basis vectors \\{$s^1_i$\\} are 9-dimensional column vectors. Each column of the $2 P$ x $|Tpos|$ matrix $W^2$ then corresponds to the vertical concatenation of the two basis state vectors associated with each allowable transition in $Tpos$. Again, note that the column ordering of the transition basis vectors is arbitrary. For example, the column of $W^2$ corresponding to transition $t_{inc_2}$ would consist of the vertical concatenation of $s^1_2$ on top of $s^1_3$.\n\n\n\n\\subsubsection{Target state transition to position shift transition coupling}\n\nWe now place coupling constraints on the target state transition and pattern translation transition models. Without these constraints, it would be possible for a target to have a positive state magnitude but a zero-valued corresponding position magnitude and vice versa. However, we wish for these magnitudes to be identical for a given target. Specifically, we add coupling constraints so that a target state ``on'' transition co-occurs with a target position ``on'' transition, and likewise for the ``off'' transitions. We also constrain the target position value so that a pattern position change (pattern translation) can co-occur with the repetition transitions ($t^2_6, t^2_{13}$) for the target pattern, but is otherwise disallowed.\n\nAlthough it would be possible to couple the target state transitions $h^3_t$ directly with the target position transitions $h^1_t$, each of the transitions $t^2_i$ would then have couplings for each of the $P$ positions, resulting in a potentially large number of coupling basis vectors. We can reduce the number of required coupled transitions by introducing an intermediate high-level position transition variable $h^2_t$. The target state transition variable $h^3_t$ is coupled to $h^2_t$ via the factorization:\n\n\\begin{align}\n\\label{eq:h3Toh2CouplingTarget}\n\\left[ \\begin{array}{c}\nH^3 \\\\\nH^2 \\end{array} \\right] =& W^5 V^2 \\notag \\\\\n=& \\left[ \\begin{array}{c}\nW^5_1 \\\\\nW^5_2 \\end{array} \\right] V^2\n\\end{align}\n\nand $h^2_t$ is coupled to the target position transition variable $h^1_t$ via the factorization:\n\n\\begin{align}\n\\label{eq:h2Toh1CouplingTarget}\n\\left[ \\begin{array}{c}\nH^2 \\\\\nH^1 \\end{array} \\right] =& W^4 V^1 \\notag \\\\\n=& \\left[ \\begin{array}{c}\nW^4_1 \\\\\nW^4_2 \\end{array} \\right] V^1\n\\end{align}\n\n\nLet $h^2_t$ correspond to a 4-dimensional column vector such that each component corresponds to a distinct type of position transition. Specifically, we let $h^2_t = a_1 = (1,0,0,0)^T$ represent ``remaining at the same position.'' We let   $h^2_t = a_2 = (0,1,0,0)^T$ represent ``position change.'' We let   $h^2_t = a_3 = (0,0,1,0)^T$ represent ``position turn on.''  We let   $h^2_t = a_4 = (0,0,0,1)^T$ represent ``position turn off.''\n\nTable~\\ref{table:trans_coupling_target_tran_abstract_pos} shows the transition coupling constraints between $h^3_t$ and $h^2_t$. We construct the parameter matrix $W^5$ from this table as follows. We see from \\ref{eqn:W_6target} that setting $h^3_t$ equal to the standard unit basis vector $e_i \\in \\mathbb{R}^9$ will activate transition $t^2_i$. We then construct the $j$'th column of $W^5$ as the vertical concatenation of $e^j$ on top of $a_k$ such that $e^j$ and $a_k$ correspond to the $j$'th target state transition and high-level position transition in the table, respectively. We then arrive at the following 9 x 15 matrix for $W^5$:\n\n\\begin{align}\nW^5 =& \\left[ \\begin{array}{ccccccccccccccc} e_6 & e_6 & e_{12} & e_{12} & e_{7} & e_{13} & e_5 & e_{11} & e_1 & e_2 & e_3 & e_4 & e_8 & e_9 & e_{10}\\\\\na_2 & a_1 & a_2 & a_1 & a_3 & a_3 & a_4 & a_4 & a_1 & a_1& a_1& a_1& a_1& a_1& a_1 \\end{array} \\right]\n\\end{align}\n\n\n\n\\begin{table}\n\\begin{center}\n\\begin{tabular}{| c ||c | c | c | c | c | c | c | c |}\n    \\hline\n    Target state transition & $t^2_6$ & $t^2_6$ & $t^2_{12}$ & $t^2_{12}$ & $t^2_7$ & $t^2_{13}$ & $t^2_5$  & $t^2_{11}$ \\\\ \\hline\n    High-level position transition & $t_{change}$ & $t_{self}$ & $t_{change}$ & $t_{self}$  & $t_{on}$  & $t_{on}$ & $t_{off}$  & $t_{off}$ \\\\ \\hline\n\\end{tabular}\n\n\\begin{tabular}{| c ||c | c | c | c | c | c | c | }\n    \\hline\n    Target state transition & $t^2_1$ & $t_2$ & $t_3$ & $t_4$ & $t_8$ & $t_9$ & $t_{10}$ \\\\ \\hline\n    High-level position transition & $t_{self}$ & $t_{self}$ & $t_{self}$ & $t_{self}$ & $t_{self}$ & $t_{self}$ & $t_{self}$ \\\\ \\hline\n\\end{tabular}\n \\caption{Transition coupling constraints between the target state transitions $h^3_t$ and the high-level position transitions $h^2_t$. Each column specifies a target state transition and high-level position transition that can co-occur.}\n \\label{table:trans_coupling_target_tran_abstract_pos}\n \\end{center}\n \\end{table}\n\nWe construct $W^4$ as follows. Note that the top 4 rows of $W^4$ correspond to the submatrix $W^4_1$, which contain the basis activations for $h^2_t$. The bottom submatrix $W^4_2$ of $W^4$ contains the activations for $h^1_t$. Note that setting $h^1_t$ equal to the standard unit basis vector $e_i \\in \\mathbb{R}^{|Tpos|}$ activates the $i$'th column of $W^2$, corresponding to a position transition in $Tpos =\\{t_{self}, t_{change}, t_{on}, t_{off}\\}$. For each $i \\in 1 , \\dots, |Tpos|$, we add a corresponding column $w^4_i$ to $W^4$ as follows. If the $h^1_t = e_i$ activates a transition in $W^2$ in the set $t_{self}$, let $w^4_i$ equal the vertical concatenation of $a_1$ on top of $e_i$. Otherwise, if $h^1_t = e_i$ activates a transition in $W^2$ in the set $t_{change}$, let $w^4_i$ equal the vertical concatenation of $a_2$ on top of $e_i$. Otherwise, if $h^1_t = e_i$ activates a transition in $W^2$ in the set $t_{on}$, let $w^4_i$ equal the vertical concatenation of $a_3$ on top of $e_i$. Otherwise, if $h^1_t = e_i$ activates a transition in $W^2$ in the set $t_{off}$, let $w^4_i$ equal the vertical concatenation of $a_4$ on top of $e_i$.\n\n\\subsubsection{Target pattern translation coupling}\n\nThe target pattern translation coupling corresponds to the factorization:\n\n\\begin{align}\n\\label{eq:target_pattern_translation_coupling}\n\\left[ \\begin{array}{c}\nX^3 \\\\\nX^2 \\\\\nX^1 \\end{array} \\right] =& W^1 U^1 \\notag \\\\\n=& \\left[ \\begin{array}{c}\nW^1_1 \\\\\nW^1_2 \\\\\nW^1_3 \\end{array} \\right] U^1\n\\end{align}\n\nwhere $W^1_1$ has $patternSize$ rows, $W^1_2$ has $P$ rows, and $W^1_3$ has $P -1 + patternSize$ rows.\n\n\nWe wish for the target emission vectors to appear translated (vertically shifted) in the observation vectors. An observation vector $x^1_t$ has dimension greater than the target emission vector $x^3_t$ so that $x^3_t$ may appear as a subvector in $x^1_t$. The number of components by which $x^3_t$ is translated is specified by the pattern shift amount $x^2_t$. If component $i$ of $x^3_t$ has value $\\alpha$ and component $j$ of $x^2_t$ also has value $\\alpha$ then we wish for component $k = i + j$ of $x^1_t$ to also have value $\\alpha$. We then construct $W^1$ as follows. For each combination of $i \\in 1, \\dots, dimension(x^3_t)$ and $j \\in 1, \\dots, dimension(x^2_t)$ add a column $q_{i_j}$ to $W^1$. The subcolumn of $q_{i_j}$ corresponding to $W^1_1$ has only the $i$'th component equal to 1 and all other components 0-valued. The subcolumn of $q_{i_j}$ corresponding to $W^1_2$ has only the $j$'th component equal to 1 and all other components 0-valued. The subcolumn of $q_{i_j}$ corresponding to $W^1_3$ has only the $k$'th component equal to 1 and all other components 0-valued. \n\nFigure~\\ref{fig:W1W2W3} shows an image plot of the submatrices of $W^1$ for the case where $patternSize = 5$, $dimension(x^2_t) = 15$, and $dimension(x^1_t) = P -1 + patternSize = 19$. Note that $W^1$ has $patternSize P = 75$ columns. Note that each column in $W^1_1, W^1_2, W^1_3$ has column sum equal to 1 so that $x^3_t$, $x^2_t$, and $x^1_t$ are modeled as having equal column sums. In our model, $x^1_t$ is observed and all other variables are hidden. Note that if a given component of $x^1_t$ has some positive value, then there are typically many basis columns in $W^1$ that can explain it, meaning that many solutions for $x^2_t, x^3_t$ are generally possible. However, the other factorizations in the model will add enough additional constraints that a unique solution for the hidden variables can still be possible.\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=100ex]{.\/figures\/W1W2W3.jpg}\n\\caption{Submatrices $W^1_1$, $W^1_2$, and $W^1_3$ of $W^1$.}\n\\label{fig:W1W2W3}\n\\end{figure}\n\n\n\nA learning and inference algorithm for this network can be obtained by applying Algorithm~\\ref{alg:inference_learning1} to the system of factorizations Equations (\\ref{eq:target_state_transition_model}) - (\\ref{eq:target_pattern_translation_coupling}). Appendix~\\ref{sec:learn_inf_target_tracking} shows the psuedocode for the algorithm. Since all parameters matrices are known, we run the algorithm in inference-only mode with learning disabled in the following section. \n\n\\subsection{Empirical Results}\n\nWe now present empirical results for the target tracking model. We make use of the parameter matrices $\\{W^i\\}$ presented in the previous section so that the model parameters can be considered known. We present inference-only results for the case where the observation sequence $X^1$ is either fully or partially observed.\n\n\\subsubsection{Clean target observations}\n\\label{sec:clean_target_obs}\nWe first consider the case where $X^1$ is fully observed and consists of noiseless observations. All other model variables are hidden and will be inferred. The bottom image plot in Figure~\\ref{fig:hdyn2_clean_X5X4X3X2X1} shows the target observations. There are three target objects represented: two A-type targets and one B-type target. Observe that there are two targets present during time slices $t = 4,5,6,7$. The targets have magnitudes of 1.0, 0.8, and 0.6. These target objects were chosen such that they conform to the target model and therefore are exactly representable under the model. We used $P = 15$ position values for these experiments, yielding a pattern shift amount from -7 to 7. That is, $P = 8$ corresponds to the 0 shift amount where a pattern is in the center of the observation vector. \n\n\nWe used the inference algorithm described in Appendix~\\ref{sec:learn_inf_target_tracking}, which is a special case of the general inference and learning algorithm presented in Algorithm~\\ref{alg:inference_learning1}. The top four image plots in Figure~\\ref{fig:hdyn2_clean_X5X4X3X2X1} show the inference results for $X^2$, $X^3$, $X^4$, $X^5$. We note that only $X^1$ was observed, so that higher-level variables $U^1$, $U^2$, $U^3$, $H^1$, $H^2$, $H^3$, $V^1$, and $V^2$ were also hidden, but are not shown here since they are less relevant.\n\nWe found that 10000 iterations were sufficient for all factorization equations to converge completely (RMSE less than 10e-4), which took approximately 1 minute to run on an a PC with a 3.0 GHz Intel Core2 Duo CPU. Given the observed sequence, exactly 1 solution is possible. All results for this model were obtained by running the inference algorithm for 10000 iterations. We observed that on repeated runs, the inference algorithm always converged to the correct exact solution (within our RMSE tolerance). We see that the observed sequence $X^1$ was successfully decomposed into the target label ($X^5$), target state ($X^4$), target emission pattern ($X^3$), pattern shift amount ($X^2$), and the other higher-level hidden variables which are not shown for space reasons.\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=100ex]{.\/figures\/hdyn2_clean_X5X4X3X2X1.jpg}\n\\caption{Inference results for $X^2$, $X^3$, $X^4$, $X^5$. Note that only the bottom $X^1$ is observed and consist of clean target observations. All other variables are hidden and were correctly inferred.}\n\\label{fig:hdyn2_clean_X5X4X3X2X1}\n\\end{figure}\n\n\n\\subsubsection{Noisy target observations}\n\nWe now add some noise to the previous clean target observations to see how inference will perform. The bottom image plot in Figure~\\ref{fig:hdyn2_noisy_X5X4X3X2X1_0_sparseness} shows the observed noisy target observations. Noise consisting of uniformly distributed values in the range $[0,0.06]$ was added to the clean target observations from the previous section to form the noisy target observations $X^1$. The maximum noise amount is thus one tenth of the smallest target magnitude.\n\n\nFigure~\\ref{fig:hdyn2_noisy_X5X4X3X2X1_0_sparseness} shows the inference results  for $X^2$, $X^3$, $X^4$, $X^5$. As before, the results for the other variables in the model are not shown for space reasons. We observe that the inferred values also appear somewhat noisy, since the model is also trying to explain the noise in the target observations. We observed that the visual cleanness of the results appears to degrade gradually as more noise is added. Small amounts of noise are tolerated quite well. A noise amount less than 0.02 or so produces visually clean results.\n \n\n\\begin{figure}\n\\centering\n\\includegraphics[width=100ex]{.\/figures\/hdyn2_noisy_X5X4X3X2X1_0_sparseness.jpg}\n\\caption{Inference results for $X^2$, $X^3$, $X^4$, $X^5$. Note that only the bottom $X^1$ is observed and consist of target observations corrupted by additive noise.}\n\\label{fig:hdyn2_noisy_X5X4X3X2X1_0_sparseness}\n\\end{figure}\n\n\n\n\\subsubsection{Noisy target observations using sparse inference}\n\nGiven the previous results for the case of noisy observations, we might like for the inference algorithm to attempt to ignore as much of the noise as possible while still accounting for the non-noise components. We modify the inference algorithm so that the NMF update steps are replaced by corresponding sparse NMF update steps using the nsNMF algorithm \\cite{nsNMF2006}, which is describe in Appendix~\\ref{appendix_sparse_nmf}. We have observed empirically that the quality of the inference results seems to be improved if the sparseness parameter value is gradually increased during the inference procedure. For these results, the sparseness value was increased linearly from 0 at iteration 5000 to 0.05 at iteration 10000.\n\nThe input observations are generated with the same targets plus additive noise as in the previous section. Figure~\\ref{fig:hdyn2_noisy_X5X4X3X2X1_gradual_sparseness} shows the inference results, where the bottom subplot shows the noisy target observations for reference. We observe that the inference results are visually less cluttered with noise compared to the previous non-sparse inference case. However, we observe that the magnitudes of the inference results are off by approximately a factor of 2, so that some renormalization would be needed as a post-processing step.\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=100ex]{.\/figures\/hdyn2_noisy_X5X4X3X2X1_gradual_sparseness.jpg}\n\\caption{Inference results for $X^2$, $X^3$, $X^4$, $X^5$. Note that only the bottom $X^1$ is observed and consist of target observations corrupted by additive noise. Gradually increasing sparseness was used in the inference algorithm.}\n\\label{fig:hdyn2_noisy_X5X4X3X2X1_gradual_sparseness}\n\\end{figure}\n\n\\subsubsection{Noisy target observations using modified sparse inference}\n\nAs an experiment, we now modify the previous sparse inference procedure so that the observation sequence $X^1$ is made to be hidden (i.e., all model variables are now hidden) during the final several inference iterations. The motivation for this is that after basically reaching convergence on the noisy observations data, the model has settled on a somewhat sparse solution. However, the noise in $X^1$ cannot be ignored completely, and so the model tries to find a configuration of the hidden variables that best explain the noise to some extent. However, if we stop updating $X^1$ on each iteration with the actual observations, the model can then be free to ignore the noise and converge to a sparser solution. We need to be careful, though. If the model is allowed to run a large number of iteration with $X^1$ now hidden, it may gradually ``forget'' $X^1$ and settle on some other configuration of variables.\n\nWe again supply the same target observations corrupted by additive noise as the previous examples. Sparseness was 0 for iterations 0 through 5000, and then gradually increased linearly reaching a maximum sparseness of 0.05 at iteration 9980. For the final 20 iterations, $X^1$ was made hidden. This was accomplished by simply disabling the re-copying of the observations data into the corresponding $X^1$ local variable during inference. Thus, on each iteration, the down propagated values for $X^1$ would be reused on the successive up propagation. \n\nFigure~\\ref{fig:hdyn2_noisy_X5X4X3X2X1} shows the inference results for $X^1$, $X^2$, $X^3$, $X^4$, $X^5$. Note that the inference results appear quite clean. However, the relative magnitudes of two of the targets are not correct. Note that the lowest-magnitude target in the figure corresponds to an to an inferred target with with magnitude greater than one of the other inferred targets.\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=100ex]{.\/figures\/hdyn2_noisy_X5X4X3X2X1.jpg}\n\\caption{Inference results for $X^1$, $X^2$, $X^3$, $X^4$, $X^5$. Gradually increasing sparseness was used. Here $X^1$ was observed for the first 9980 iterations and then made to be hidden for the last 20 iterations. Note that since $X^1$ was hidden for several iteration, the inferred values for $X^1$ here differ from those of the actual observed $X^1$ in the bottom image plot.}\n\\label{fig:hdyn2_noisy_X5X4X3X2X1}\n\\end{figure}\n\n\\subsubsection{Prediction}\n\nWe now consider the case where some of the target observations are hidden. We will then infer their values are part of the inference procedure, along with the other hidden variables. We first consider the case where time slice 25 of the clean $X^1$ from Section~\\ref{sec:clean_target_obs} is hidden. The regular inference algorithm was used (i.e.,the usual non-sparse NMF updates were used).\n\nFigure~\\ref{fig:hdyn2_clean_X5X4X3X2X1_prediction1} shows the inference results for $X^1$, $X^2$, $X^3$, $X^4$, $X^5$. We observe that the inferred values for $x^1_{25}$ are the same for the case of fully observed $X^1$. This makes sense since the the transition model in Figure~\\ref{fig:targetTracking1} specifies that target state 9 ($x^4_{25}$) must follow target state 8 ($x^4_{24}$) and a pattern shift is not allowed during this transition (i.e., $X^2$ must remain in the same state). Thus, given the observed time slices of $X^1$ before and after 25, there is only one possible configuration of variable values for time slice 25 that satisfies the model, and our inference algorithm successfully solved for it.\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=100ex]{.\/figures\/hdyn2_clean_X5X4X3X2X1_prediction1.jpg}\n\\caption{Inference results for $X^1$, $X^2$, $X^3$, $X^4$, $X^5$. $x^1_{25}$ was hidden and all other time slices of $X^1$ are observed. We observe that the inference results are the same as for the case of a fully observed $X^1$ since only a single solution is possible.}\n\\label{fig:hdyn2_clean_X5X4X3X2X1_prediction1}\n\\end{figure}\n\nWe now set the last 5 time slices of $X^1$ ($x^1_{22}, x^1_{23}, x^1_{24}, x^1_{25}, x^1_{26}$) to be hidden and run the inference algorithm. Figure~\\ref{fig:hdyn2_clean_X5X4X3X2X1_prediction2} shows the corresponding inference results. We observe that the infered values for the final 5 time slices now correspond to a superposition of states since multiple solutions are possible.\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=100ex]{.\/figures\/hdyn2_clean_X5X4X3X2X1_prediction2.jpg}\n\\caption{Inference results for $X^1$, $X^2$, $X^3$, $X^4$, $X^5$. Time slices 22 through 26 of $X^1$ are hidden. We observe that the infered values for the final 5 time slices now correspond to a superposition of states since multiple solutions are possible.}\n\\label{fig:hdyn2_clean_X5X4X3X2X1_prediction2}\n\\end{figure}\n\n\nIn performing these experiments, we observed that the inference procedure appears to be quite robust, since for the noiseless case, complete convergence (to a very small RMSE) was achieved on every run that we performed. Even in the noisy case, the noise in the inferred results appeared to increase gradually in proportion to the noise in the observation sequence. A possible drawback is that a large number of iterations were needed in order for the algorithm to converge to a solution. We have not yet explored how the number of iterations to convergence varies with network size or graphical structure.\n\n\n\\section{Sparse hierarchical sequential data model}\n\\label{sec:hierarchicalSeqDecomp}\n\n\n\nIn this section we consider a multilevel hierarchical DPFN for sequential data in which the activations required in order to explain a given sequence (or non-negative superposition of sequences) become more sparse we we ascend the hierarchy. We then present learning and inference results on magnitude spectrogram data.\n\n\\subsection{Model}\n\\label{sec:sparseHierarchicalModel}\n\nIn the sequential data models presented in the previous sections, we modeled $x_t$ as being simultaneously representable as a both a linear function of parent $h_{t-1}$ and another linear function of parent $h_t$. Thus, a nonzero value of the child node $x_t$ implied nonzero values for all of its parents and vice versa. In this section, we consider an alternative network structure for sequential data in which the value of a child node $x_t$ is modeled as sum of linear functions of multiple parents. Under this representation, a nonzero value of a child node implies that at least one parent node must also have a nonzero value and vice versa. Thus, a nonzero value of a single parent node can explain a sequence of possibly several nonzero child node values. \n\n\n \nFigure~\\ref{fig:1hiddenLayerRep2} shows a DPFN with two layers. We note that Non-Negative Matrix Factor Deconvolution \\cite{NMFParis} appears to be a special case of this network.  The variables are non-negative vectors such that $x^i_t \\in \\mathbb{R}^{M_i}$.  Typically, $\\{x^1_t\\}$ would correspond to the observed data sequence and $\\{x^2_t\\}$ would be hidden. A node $x^1_t$ with parents $x^2_{t-1}$ and $x^2_t$ corresponds to the factorization:\n\n\\begin{align}\nx^1_t =& W_1 x^2_t + W_2 x^2_{t-1} \\notag \\\\\n=& \\left[ \\begin{array}{cc} W_1 & W_2 \\end{array} \\right] \\left[ \\begin{array}{c} x^2_t\\\\\nx^2_{t-1} \\end{array} \\right] \n\\end{align}\n\nFor time slices $x^1_1 \\dots x^1_T$ we then have:\n\n\\begin{align}\n\\left[ \\begin{array}{ccccc} x^1_1 & x^1_2 & x^1_3 & \\dots & x^1_T \\end{array} \\right] = \\left[ \\begin{array}{cc} W_1 & W_2 \\end{array} \\right] \\left[ \\begin{array}{ccccc}\nx^2_1 & x^2_2 & x^2_3 & \\dots & x^2_T \\\\\nx^2_0 & x^2_1 & x^2_2 & \\dots & x^2_{T-1} \\end{array} \\right] \n\\end{align}\n\nwhere we note that $x^2_0 = 0$. Denoting the right-most matrix above as $X^2_S$, we can then express the factorization more concisely as:\n\n \\begin{align}\n X^1 =&  \\left[ \\begin{array}{cc} W_1 & W_2 \\end{array} \\right] X^2_S\\\\\n=& W X^2_S\n \\end{align}\n\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=60ex]{.\/figures\/1hiddenLayerRep2.pdf}\n\\caption{A 2-level DPFN for a sparse hierarchical sequence model.}\n\\label{fig:1hiddenLayerRep2}\n\\end{figure}\n\nNote the duplicated variables in $X^2_S$. Since $x^2_j$ appears in two consecutive columns of $X^2_S$, we see that activating $x^2_j$ will correspond to the $j$'th column of $W_1$ being activated in time slice $j$ and the $j$'th column of $W_2$ being activated in time slice $j+1$. Thus, one can think of corresponding columns of $W_1$ and $W_2$ as comprising a transition basis pair. \n\nWe might consider normalizing $W$ so that all columns sum to 1. Alternatively, we could consider normalizing $W$ so that the sum of the corresponding columns of all the $W_j$ sum to 1. That is, we can view the matrix consisting of the j'th column of each consecutive $W_i$ as a basis sequence. \n\n\n\nFigure~\\ref{fig:2hiddenLayerRep2V2} shows a 3-level network. A sequence of nonzero values $x^1_t, x^1_{t+1}, x^1_{t+2}, x^1_{t+3}$ in level 1 can be represented by  two nonzero values $x^2_t, x^2_{t+2}$ in level 2, and by a single nonzero value of $x^3_t$ in level 3. That is, a single nonzero value $x^{i+1}_t$ in level $i+1$ represents a pair of consecutive values in level $i$. If viewed in a generative setting, a single nonzero level 3 variable $x^1_1$ can cause the level 2 variables $x^2_1, x^2_3$ to be nonzero, which in turn can cause the level 1 variables $x^1_1, x^1_2, x^1_3, x^1_4$ to be nonzero.\n\nSuppose that the level 1 variables $x^1_1, \\dots, x^1_T$ are observed. Provided that the parameter matrices $W^1 = \\left[ \\begin{array}{cc} W^1_1 & W^1_2 \\end{array} \\right]$ and $W^2 = \\left[ \\begin{array}{cc} W^2_1 & W^2_2 \\end{array} \\right]$ contain sufficient basis columns to represent the possible observation pairs $(x^1_t, x^1_{t+1})$, a given sequence can be represented such that only every other level 2 variable is nonzero, and only every 4th level 3 variable is nonzero. Thus, as we ascend the hierarchy, the activations become increasingly sparse.\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=60ex]{.\/figures\/2hiddenLayerRep2V2.pdf}\n\\caption{A 3-level DPFN for a sparse hierarchical sequence model. The first five time slices are shown.}\n\\label{fig:2hiddenLayerRep2V2}\n\\end{figure}\n\nWe represent each level 1 variable in Figure~\\ref{fig:2hiddenLayerRep2V2} as:\n\n\\begin{align}\nx^1_t =& W^1_1 x^2_t + W^1_2 x^2_{t-1} \\notag \\\\\n=& \\left[ \\begin{array}{cc} W^1_1 & W^1_2 \\end{array} \\right] \\left[ \\begin{array}{c} x^2_t\\\\\nx^2_{t-1} \\end{array} \\right] \n\\end{align}\n\nLikewise, we represent each level 2 variable as:\n\n\\begin{align}\nx^2_t =& W^2_1 x^3_t + W^2_2 x^3_{t-2} \\notag \\\\\n=& \\left[ \\begin{array}{cc} W^2_1 & W^2_2 \\end{array} \\right] \\left[ \\begin{array}{c} x^3_t\\\\\nx^3_{t-2} \\end{array} \\right] \n\\end{align}\n\nFor a network with $T$ time slices, we then have the following factorizations:\n\n\\begin{align}\n\\left[ \\begin{array}{ccccc} x^1_1 & x^1_2 & x^1_3 & \\dots & x^1_T \\end{array} \\right] = \\left[ \\begin{array}{cc} W^1_1 & W^1_2 \\end{array} \\right] \\left[ \\begin{array}{ccccc}\nx^2_1 & x^2_2 & x^2_3 & \\dots & x^2_T \\\\\nx^2_0 & x^2_1 & x^2_2 & \\dots & x^2_{T-1} \\end{array} \\right] \n\\end{align}\n\nNote that $x^2_0 = 0$.\n\n\\begin{align}\n\\left[ \\begin{array}{ccccc} x^2_1 & x^2_2 & x^2_3 & \\dots & x^2_T \\end{array} \\right] = \\left[ \\begin{array}{cc} W^2_1 & W^2_2 \\end{array} \\right] \\left[ \\begin{array}{ccccc}\nx^3_1 & x^3_2 & x^3_3 & \\dots & x^3_T \\\\\nx^3_{-1} & x^3_0 & x^3_1 & \\dots & x^3_{T-2} \\end{array} \\right] \n\\end{align}\n\nNote that $x^3_{-1}  = x^3_0 = 0$.\n\n\n\nThe network in Figure~\\ref{fig:2hiddenLayerRep2V2} extends immediately to networks with an arbitrary number of levels, arbitrary numbers of parent variables, and arbitrary separation (in time slices) between consecutive child nodes in any given level. Consider a network with $L$ levels and such that the variables $x^{i+1}_t$ in level $i+1$ each have $p$ child nodes with a separation of $q$ time slices between nodes. Let $x^{i+1}_t$ at each time slice $t$ be an $M_{i+1}$ dimensional column vector and let $x^i_t$ at each time slice $t$ be $M_i$ dimensional column vector. For each time slice $t$ and $i \\in \\{1, \\dots, L-1\\}$, $x^i_t$ is then expressed by the factorization:\n\n\\begin{align}\nx^i_t =& W^i_1 x^{i+1}_t + W^i_2 x^{i+1}_{t-q} + W^i_3 x^{i+1}_{t - 2 q} + \\dots + W^i_p x^{i+1}_{t- (p-1) q}\\\\\n=& \\left[ \\begin{array}{ccccc} W^i_1 & W^i_2 & W^i_3 & \\dots & W^i_p \\end{array} \\right] \\left[ \\begin{array}{c} x^{i+1}_t\\\\\nx^{i+1}_{t-q}\\\\\nx^{i+1}_{t - 2 q}\\\\\n\\vdots\\\\\nx^{i+1}_{t- (p-1) q} \\notag \\\\\n \\end{array} \\right] \n\\end{align}\n\nFor a network with $T$ time slices, we then have the following factorization equation for each $i \\in \\{1, \\dots, L-1\\}$, $x^i_t$:\n\n\\begin{align}\n\\left[ \\begin{array}{ccccc} x^i_1 & x^i_2 & x^i_3 & \\dots & x^i_T \\end{array} \\right] = \\left[ \\begin{array}{ccccc} W^i_1 & W^i_2 & W^i_3 & \\dots & W^i_p \\end{array} \\right] \\left[ \\begin{array}{ccccc}\nx^{i+1}_1 & x^{i+1}_2 & x^{i+1}_3 & \\dots & x^{i+1}_T \\\\\nx^{i+1}_{1-q} & x^{i+1}_{2-q} & x^{i+1}_{3-q} & \\dots & x^{i+1}_{T-q} \\\\\nx^{i+1}_{1-2 q} & x^{i+1}_{2-2 q} & x^{i+1}_{3-2 q} & \\dots & x^{i+1}_{T-2 q}\\\\\n\\vdots & \\vdots & \\vdots & \\vdots & \\vdots \\\\\nx^{i+1}_{1-(p-1) q} & x^{i+1}_{2-(p-1) q} & x^{i+1}_{3-(p-1) q} & \\dots & x^{i+1}_{T-(p-1) q} \\\\\n \\end{array} \\right] \n\\end{align}\n\nAll $x^{i+1}_j$ such that $j < 1$ are zero-valued.\n\nLetting \n\\begin{align}\nX^i = \\left[ \\begin{array}{ccccc} x^i_1 & x^i_2 & x^i_3 & \\dots & x^i_T \\end{array} \\right]\n\\end{align}\n\n\\begin{align}\nW^i = \\left[ \\begin{array}{ccccc} W^i_1 & W^i_2 & W^i_3 & \\dots & W^i_p \\end{array} \\right]\n\\end{align}\n\n\\begin{align}\nX^{i+1}_S = \\left[ \\begin{array}{ccccc}\nx^{i+1}_1 & x^{i+1}_2 & x^{i+1}_3 & \\dots & x^{i+1}_T \\\\\nx^{i+1}_{1-q} & x^{i+1}_{2-q} & x^{i+1}_{3-q} & \\dots & x^{i+1}_{T-q} \\\\\nx^{i+1}_{1-2 q} & x^{i+1}_{2-2 q} & x^{i+1}_{3-2 q} & \\dots & x^{i+1}_{T-2 q}\\\\\n\\vdots & \\vdots & \\vdots & \\vdots & \\vdots \\\\\nx^{i+1}_{1-(p-1) q} & x^{i+1}_{2-(p-1) q} & x^{i+1}_{3-(p-1) q} & \\dots & x^{i+1}_{T-(p-1) q}\\\\\n\\end{array} \\right]\n\\end{align}\n \nwe can express the factorization more concisely as:\n \n \\begin{align}\n\\label{eq:sparse_hierarchical_fact}\n X^i = W^i X^{i+1}_S\n \\end{align}\n\nNote that $X^i$ has size $M_i$ by $T$. Each $W^i_j$ has $M_{i+1}$ columns. The matrix $W^i$ has size $M_i$ by $ M_{i+1}$. The matrix $X^{i+1}_S$ has size $p M_{i+1}$ by $T$.\nFor example, a network with $L = 2$ levels, $p = 3$ children and $q = 2$ seperation will correspond to the factorization:\n\n\\begin{align}\n\\left[ \\begin{array}{ccccccccc} x^1_1 & x^1_2 & x^1_3 & x^1_4 & x^1_5 & x^1_6 & x^1_7 & \\dots & x^1_T \\end{array} \\right] = \\left[ \\begin{array}{ccc} W^1_1 & W^1_2 & W^1_3 \\end{array} \\right] \\left[ \\begin{array}{ccccccccc}\nx^2_1 & x^2_2 & x^2_3 & x^2_4 & x^2_5 & x^2_6 & x^2_7 & \\dots & x^2_T \\\\\n0 & 0 & x^2_1 & x^2_2 & x^2_3 & x^2_4 & x^2_5 & \\dots & x^2_{T-2} \\\\\n0 & 0 & 0 & 0 & x^2_1 & x^2_2 & x^2_3 & \\dots & x^2_{T-4} \\\\\n\\end{array} \\right] \n\\end{align}\n\nA learning and inference algorithm for a network with $L$ levels can be obtained by applying Algorithm~\\ref{alg:inference_learning1} to the system of factorizations equations (\\ref{eq:sparse_hierarchical_fact}). Appendix~\\ref{sec:sparse_hierarchical_inf_alg} shows the pseudocode for the algorithm.\n\n\\subsection{Empirical results}\n\nWe now present learning and inference results on magnitude spectrogram data. We let the lowest level variables $X^1$ correspond to a magnitude spectrogram such that $x^1_t$ represents time slice $t$ of a spectrogram. For this experiment, we use a 15 second audio clip from Beethoven's Piano Sonata No. 8 in C minor, op. 13, Adagio cantabile, performed by the author. The corresponding spectrogram, with 512 frequency bins and 622 time slices (41 time slices per second) is shown in the bottom plot of Figure~\\ref{fig:actual_vs_inferred_spectrogram_0sparseness}.\n\nWe make use of a DPFN with the following network parameters: We use a 4-level network with $X^1$ observed and $X^2$, $X^3$, and $X^4$ hidden. Let $x^2_t \\in \\mathbb{R}^{50}$ have $p^2 = 4$ child nodes and a separation of $q^2 = 1$ time slice. Let $x^3_t \\in \\mathbb{R}^{40}$ have $p^3 = 4$ child nodes and a separation of $q^3 = 4$ time slices. Let $x^4_t \\in \\mathbb{R}^{40}$ have $p^4 = 4$ child nodes and a separation of $q^4 = 16$ time slices.\n\nWe use the learning and inference algorithm described in Appendix~\\ref{sec:sparse_hierarchical_inf_alg}, which is a special case of the general algorithm described in Algorithm~\\ref{alg:inference_learning1}. We start by setting $X^1$ to the spectrogram data values and initializing all other variables and model parameters $W^i$ to random positive values. We have observed that model convergence can sometimes be improved by performing the parameter learning update steps more frequently on parameter matrices that are closer to the observed data in the graphical model. In this case, we perform $W^1$ more frequently than $W^2$, and perform $W^2$ updates more frequently then $W^3$ and so on. For each level, $W^i$ was normalized so that the sum of the corresponding columns of all the $W^i_j$ sum to 1.\n\n\nFigure~\\ref{fig:inferred_spectrogram_0sparseness} Shows the observed spectrogram $X^1$ and inferred activations for $X^2$, $X^3$, and $X^4$ after running the learning and inference algorithm for 800 iterations. Note that the activations become progressively more sparse as we ascend the hierarchy from $X^2$ to $X^4$, even though sparse NMF update steps were not used in the inference and learning algorithm. \n\n\\begin{figure}\n\\centering\n\\includegraphics[width=100ex]{.\/figures\/inferred_spectrogram_0sparseness.jpg}\n\\caption{Observed spectrogram $X^1$ and inferred activations for $X^2$, $X^3$, and $X^4$. Sparseness was 0.}\n\\label{fig:inferred_spectrogram_0sparseness}\n\\end{figure}\n\nFigure~\\ref{fig:actual_vs_inferred_spectrogram_0sparseness} shows the actual observed spectrogram as well as the approximation to the spectrogram obtained by using only the inferred top-level activations $X^4$ and propagating them downward to form an approximation to $X^1$. The resulting RMSE was 0.07. Only the first 250 frequency bins are shown since the higher frequency bins have negligible energy.\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=100ex]{.\/figures\/actual_vs_inferred_spectrogram_0sparseness.jpg}\n\\caption{Actual vs approximated spectrogram $X^1$ for the 0 sparseness case.}\n\\label{fig:actual_vs_inferred_spectrogram_0sparseness}\n\\end{figure}\n\n\nFigure~\\ref{fig:inferred_spectrogram_sparse} Shows the observed spectrogram $X^1$ and inferred activations for $X^2$, $X^3$, and $X^4$ for the case where sparse NMF updates were used in the inference and learning algorithm. Here, we ran the inference and learning algorithm for 400 iterations with 0 sparseness, and then increased the nsNMF sparseness parameter gradually from 0 to 0.2 at 800 iterations.\n\n\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=100ex]{.\/figures\/inferred_spectrogram_sparse.jpg}\n\\caption{Observed spectrogram $X^1$ and inferred activations for $X^2$, $X^3$, and $X^4$. Sparseness was gradually increased to a value of 0.2 at 800 iterations.}\n\\label{fig:inferred_spectrogram_sparse}\n\\end{figure}\n\n\nFigure~\\ref{fig:actual_vs_inferred_spectrogram_sparse} shows the actual observed spectrogram as well as the approximation to the spectrogram obtained by using only the inferred top-level activations $X^4$ and propagating them downward to form an approximation to $X^1$. The resulting RMSE was 0.2. Only the first 250 frequency bins are shown since the higher frequency bins have negligible energy. This is for gradually increasing sparse NMF. We observe that the results using sparse NMF updates appear slightly more sparse than those using the non-sparse NMF updates. However, the reconstruction error using sparse updates is also higher.\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=100ex]{.\/figures\/actual_vs_inferred_spectrogram_sparse.jpg}\n\\caption{Actual vs approximated spectrogram $X^1$ for the 0 sparseness case.}\n\\label{fig:actual_vs_inferred_spectrogram_sparse}\n\\end{figure}\n\n\n\nWe feel that it may be interesting to consider applying this or a similar DPFN to problems where a sparse hierarchical representation of sequential data could be useful, such as compression, noise reduction, and transcription.\n  \n\n\n\n\n\n\n\\section{A DPFN for language modeling}\n\\label{sec:main_language_model}\n\n\nIn this section we propose a data model for the sequence of words in a text document, but do not present any corresponding empirical results. In this model, we perform dimensionality reduction by extracting low dimensional feature vectors corresponding to each distinct word. We then make use of a transition model for the feature vectors. By performing dimensionality reduction to extract features, we can use a much more compact transition model than would be possible by operating directly on the words themselves. The idea of extracting low-dimensional word features in order to model sequential word data in text documents and make predictions of future words was used in \\cite{bengioy-journal}.\n\nFigure~\\ref{fig:wordFeatureMode} shows the graphical model. In this model, the observed \\{$y_t : t = 1, \\dots, T$\\} correspond to the sequence of words that appear in some text document. Suppose our vocabulary size is $N$ words. Then let $y_t$ denote an $N$ dimensional vector such that the $t$'th word $i \\in \\{1, \\dots, N\\}$ in the document is represented by setting the $i$'th component of $y_t$ to 1 and all other components to zero.  This model can also be used for prediction of future words, given a sequence of words up to the present $t_p$ by simply considering the word vector nodes for future time slices \\{$y_{t_p}, y_{{t_p}+1}, \\dots$\\} to be hidden.\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=60ex]{.\/figures\/wordFeatureModel.pdf}\n\\caption{A DPFN representing a transition model over word features. The observed $y_t$ denotes a word vector, $x_t$ denotes a word feature vector, and $h_t$ denotes a word feature transition. The first four time slices are shown.}\n\\label{fig:wordFeatureMode}\n\\end{figure}\n\nWe perform dimensionality reduction to extract low-dimensional word features $x_t$ by left multiplying each word vector $y_t$ by a \\emph{word feature matrix} $B$ so that we have:\n\n\\begin{align}\nx_t = B y_t\n\\end{align}\n\nThe non-negative matrix $B$ then represents the word feature matrix, where the $i,j$'th component represents the amount of feature $i$ contained by word $j$. One might consider normalizing the columns of $B$ to have unit column sum. Letting $Y = \\left[ \\begin{array}{ccccc} y_1 & y_2 & y_3 & \\dots & y_T \\end{array} \\right]$, we then have the following matrix factorization for the word features:\n\n\\begin{align}\nX = B Y\n\\label{eqn:word_feature_factorization}\n\\end{align}\n\nThe factorization equations that describe this model are exactly the same as those for the model in Figure~\\ref{fig:1layerDynamic} with the additional above factorization for the word features in terms of the word vectors. Thus, the complete system of factorizations for this model can be expressed as the two matrix factorizations (\\ref{eqn:key_factorization}) and (\\ref{eqn:word_feature_factorization}).\n\n\nSuupose, for example, that the word ``cat'' has the features ``animal'' and ``has claws.'' The feature transition model (parametrized by $W_1$, $W_2$) could possibly have distinct basis transitions containing the ``animal'' and ``has claws'' features. For a word such as ``cat'' that has both features, both basis transitions could be simultaneously activated due to the non-negative superposition property (Section~\\ref{sec:model_specification}). Thus, a given word sequence could be explained such that each word has possibly multiple features, and that multiple distinct basis transitions in the feature transition model could simultaneously be activated to explain the words in a sequence as an additive combination of basis feature transitions. One could then think of the word feature transition model as operating on the individual word features independently and simultaneously such that the observed or inferred word sequence is consistent with them. We feel that such an additive factored language model might be a powerful way of modeling word and\/or character sequences. However preliminary empirical results suggest that additional sparseness and\/or orthogonality constraints might need to be imposed in order to obtain meaningful solutions. \n\n\n\n\n\\section{Discussion}\n\nWe have presented a framework for modeling non-negative data that retains the linear representation of NMF, while providing for more structured representations using (non-probabilistic) graphical models. The graphical model provides a visual representation of the factored correlation constraints between variables. We have presented algorithms for performing the inference and learning that leverage existing non-negative matrix factorization (NMF) algorithms. Since the algorithms are inherently parallel, we expect that optimized implementations for hardware such as multi-core CPUs and GPUs will be possible. These algorithm have the advantage of being quite straightforward to implement, even for relatively complex networks structures. We expect that they can be understood and implemented by anyone with a basic understanding of computer programming, matrix algebra, and graph theory, and require no background in probability theory. \n\nWe presented empirical results for several example networks that illustrated the robustness of the inference and learning algorithms. Our goal was to provide a sufficient variety of model structures so that the interested reader will hopefully be able to extend this work and apply it to interesting and practical applications. We observed that in all models in which noiseless synthetic data sets were used, learning and\/or inference converged to an essentially exact factorization, corresponding to a correct solution for the hidden variables. In the case of additive noise, the quality of the solutions tended to degrade gradually as the noise level was increased. In particular, in Section~\\ref{sec:learn_tran_model_from_data} we observed the somewhat surprising result that it was possible to learn an underlying transition model even when the training sequence contained only an additive mixture of realizations of the underlying model. In our magnitude spectrogram modeling example, we observed that the inference and learning solution appeared to correspond to a hierarchical sparse decomposition, with the solution sparseness increasing as we ascend the model hierarchy. As with existing NMF algorithms, however, there is no guarantee that convergence to a global optimum for any particular cost function will occur. We did observe good convergence properties in our empirical results, however. We also proposed a PFN for language modeling and plan to experiment with related models as future research. \n\nIt is still unknown how well models using this framework will perform on hard real-world problems such as speech recognition, language modeling, and music transcription, and this will be an interesting area of future research. It could also be interesting to explore the use of dynamic networks that are directional in time, such as DPFNs with a similar state space model to an HMM or HHMM, for example.  Source code for implementing the models in this paper will be made available by the author.\n\n\\section{Acknowledgments}\n\nI would like to thank Michael I. Jordan for reviewing a draft of this paper and offering helpful suggestions.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nApart from giving deep insight to strongly coupled gauge theories\nand string\/M-theory compactifications in various dimensions, the\nAdS\/CFT correspondence has been recently used to explain the entropy\nof asymptotically $AdS_4$ black holes\n\\cite{Zaffaroni_BH1,Zaffaroni_BH2,Zaffaroni_BH3}. In this context,\nthe black hole entropy is computed using topologically twisted index\nof three-dimensional superconformal field theories (SCFTs)\ncompactified on a Riemann surface $\\Sigma_2$\n\\cite{twisted_index1,twisted_index2,twisted_index3,twisted_index4,twisted_index5}.\nIn the dual gravity solutions, the black holes interpolate between\nthe asymptotically $AdS_4$ and the near horizon $AdS_2\\times\n\\Sigma_2$ geometries. These can be interpreted as RG flows from\nthree dimensional SCFTs in the form of Chern-Simons-Matter (CSM)\ntheories possibly with flavor matters to superconformal quantum\nmechanics corresponding to the $AdS_2$ geometry.\n\\\\\n\\indent Along this line of research, BPS black hole solutions in\nfour-dimensional gauged supergravity, in particular near horizon\ngeometries, with known higher dimensional origins are very useful.\nMost of the solutions have been studied within $N=2$ gauged\nsupergravities\n\\cite{BH_M_theory1,BH_M_theory2,AdS4_BH1,AdS4_BH2,AdS4_BH3,AdS4_BH4,AdS4_BH5},\nfor recent results, see \\cite{Guarino_BH1,Guarino_BH2}. Many of\nthese solutions can be uplifted to string\/M-theory since these $N=2$\ngauged supergravities can be obtained either from truncations of the\nmaximal $N=8$ gauged supergravity, whose higher dimensional origin\nis known, or direct truncations of M-theory on Sasaki-Einstein\nmanifolds.\n\\\\\n\\indent In this work, we give an evidence for a new class of BPS\nblack hole solutions in the half-maximal $N=4$ gauged supergravity\nwith known higher dimensional origin by finding a number of new\n$AdS_2\\times \\Sigma_2$ solutions. This gauged supergravity has\n$SO(3)\\ltimes (\\mathbf{T}^3,\\hat{\\mathbf{T}}^3)$ gauge group and can\nbe obtained from a compactification of M-theory on a tri-sasakian\nmanifold \\cite{N010_truncation_Cassani}. Holographic RG flows and\nsupersymmetric Janus solutions, describing $(1+1)$-dimensional\ninterfaces in the dual SCFTs have recently appeared in\n\\cite{trisasakian_flow}. In the present paper, we will look for\nsupersymmetric solutions of the form $AdS_2\\times \\Sigma_2$ within\nthis tri-sasakian compactification.\n\\\\\n\\indent Apart from giving this type of solutions in gauged\nsupergravity with more supersymmetry, to the best of the author's\nknowledge, the results are the first example of $AdS_2\\times\n\\Sigma_2$ solutions from the truncation of M-theory on a\ntri-sasakian manifold. The truncation given in\n\\cite{N010_truncation_Cassani} gives a reduction ansatz for\neleven-dimensional supergravity on a generic tri-sasakian manifold\nincluding massive Kaluza-Klein modes. Among this type of manifolds,\n$N^{010}$ with isometry $SU(2)\\times SU(3)$ is of particular\ninterest. In this case, there is a non-trivial two-form giving rise\nto an extra vector multiplet, see \\cite{N3_spectrum1} and\n\\cite{N3_spectrum2} for the Kaluza-Klein spectrum of $AdS_4\\times\nN^{010}$. This background, discovered long ago in\n\\cite{Castellani_Romans}, preserves $N=3$ out of the original $N=4$\nsupersymmetry. There is another supersymmetric $AdS_4$ vacuum with\n$SO(3)$ symmetry and $N=1$ supersymmetry, the one broken by\n$AdS_4\\times N^{010}$. This vacuum corresponds to $AdS_4\\times\n\\tilde{N}^{010}$ geometry, with $\\tilde{N}^{010}$ being a squashed\nversion of $N^{010}$.\n\\\\\n\\indent Not much is known about the dual $N=1$ SCFT, but the dual\n$N=3$ SCFT has been proposed in a number of previous works\n\\cite{Ring_N3_superfield,Shadow_N3_multiplet}, see also\n\\cite{Hanany_Zaffaroni1,Hanany_Zaffaroni2}. This SCFT takes the form\nof a CSM theory with $SU(N)\\times SU(N)$ gauge group. It is a theory\nof interacting three hypermultiplets transforming in a triplet of\nthe $SU(3)$ flavor symmetry, and each hypermultiplet transforms as a\nbifundamental under the $SU(N)\\times SU(N)$ gauge group and as a\ndoublet of the $SU(2)_R\\sim SO(3)_R$ R-symmetry. There are also a\nnumber of previous works giving holographic studies of this theory\nboth in eleven-dimensional context and in the effective $N=3$ and\n$N=4$ gauged supergravities\n\\cite{trisasakian_flow,N3_flow_Ahn1,N3_flow_Ahn2,Yi_4D_flow,N3_SU2_SU3,N3_Janus}.\nSolutions given in these works are holographic RG flows, Janus\nsolutions and supersymmetric $AdS_2\\times \\Sigma_2$ solutions with magnetic\ncharges.\n\\\\\n\\indent In this work, we consider $N=4$ gauged supergravity\nconstructed in the embedding tensor formalism in\n\\cite{N4_gauged_SUGRA}. This construction is the most general\nsupersymmetric gaugins of $N=4$ supergravity in which both the\n``electric'' vector fields, appearing in the ungauged Lagrangian, and\ntheir magnetic duals can participate. Therefore, magnetic and dyonic gaugings are allowed\nin this formulation. Furthermore, this formulation\ncontains the ``purely electric'' gauged $N=4$ supergravity\nconstructed long time ago in \\cite{Eric_N4_4D} and the non-trivial\n$SL(2,\\mathbb{R})$ phases of \\cite{de_Roo_N4_4D,N4_Wagemans} as\nspecial cases. We will look for supersymmetric $AdS_2\\times\n\\Sigma_2$ solutions in the $N=4$ gauged supergravity with a dyonic\ngauging of the non-semisimple group $SO(3)\\ltimes\n(\\mathbf{T}^3,\\hat{\\mathbf{T}}^3)$. The solutions are required to\npreserve $SO(2)\\subset SO(3)_R$, so only a particular combination of\nvector fields corresponding to this $SO(2)$ residual gauge symmetry\nappears in the gauge covariant derivative. The strategy is essentially\nsimilar to the wrapped brane solutions of\n\\cite{Maldacena_Nunez_nogo}, implementing the twist by cancelling\nthe spin connections on $\\Sigma_2$ by the $SO(2)$ gauge connection.\n\\\\\n\\indent These $AdS_2\\times \\Sigma_2$ solutions should appear as near\nhorizon geometries of supersymmetric black holes in asymptotically\n$AdS_4$ space-time. Since the $N=4$ gauged supergravity admits two\nsupersymmetric $AdS_4$ vacua with unbroken $SO(3)_R$ symmetry and\n$N=1,3$ supersymmetries, the $AdS_2\\times \\Sigma_2$ solutions should\nbe RG fixed points in one dimension of the dual $N=1,3$ SCFTs.\nAlthough the structure of the dual $N=1$ SCFT is presently not\nclear, we expect that there should be RG flows between these twisted\n$N=1,3$ SCFTs on $\\Sigma_2$ to one-dimensional superconformal\nquantum mechanics dual to the $AdS_2\\times \\Sigma_2$ solutions. In\nthis sense, the entropy of these black holes would possibly be computed from\nthe topologically twisted indices of the dual $N=1,3$ SCFTs.\nFurthermore, these solutions should provide a new class of $AdS_2$\ngeometries within M-theory.\n\\\\\n\\indent The paper is organized as follow. In section \\ref{N4theory},\nwe review $N=4$ gauged supergravity coupled to vector\nmultiplets and relevant formulae for uplifting the resulting solutions to eleven dimensions. The analysis of BPS equations for $SO(2)\\subset SO(3)_R$ singlet scalars and Yang-Mills equations, for static black hole ansatze consistent with the symmetry of $\\Sigma_2$, will be carried out in section \\ref{flow_eq}. In section \\ref{AdS2_solution}, we will explicitly give $AdS_2\\times \\Sigma_2$ solutions to the BPS flow equations. We separately consider the $N=1$ and $N=3$ cases and end the section with the uplift formulae for embedding the solutions in eleven dimensions. We finally give some conclusions and\ncomments on the results in section \\ref{conclusions}. In the appendix, we collect the convention regarding `t Hooft matrices and give the explicit form of the Yang-Mills and BPS equations.\n\n\\section{$N=4$ gauged supergravity with dyonic gauging}\\label{N4theory}\nIn this section, we review $N=4$ gauged supergravity in the\nembedding tensor formalism following \\cite{N4_gauged_SUGRA}. We\nmainly focus on the bosonic Lagrangian and supersymmetry\ntransformations of fermions which provide basic ingredients for\nfinding supersymmetric solutions. Since the gauged supergravity\nunder consideration is known to arise from a tri-sasakian truncation\nof eleven-dimensional supergravity, we will also give relevant\nformulae which are useful to uplift four-dimensional solutions to\neleven dimensions. The full detail of this truncation can be found\nin \\cite{N010_truncation_Cassani}.\n\n\\subsection{$N=4$ gauged supergravity coupled to vector multiplets}\nIn the half-maximal $N=4$ supergravity in four\ndimensions, the supergravity multiplet consists of the graviton\n$e^{\\hat{\\mu}}_\\mu$, four gravitini $\\psi^i_\\mu$, six vectors\n$A_\\mu^m$, four spin-$\\frac{1}{2}$ fields $\\chi^i$ and one complex\nscalar $\\tau$. The complex scalar can be parametrized by the $SL(2,\\mathbb{R})\/SO(2)$ coset. The supergravity\nmultiplet can couple to an arbitrary number $n$ of vector multiplets containing a vector\nfield $A_\\mu$, four gaugini $\\lambda^i$ and six scalars $\\phi^m$.\nThe scalar fields can be parametrized by the $SO(6,n)\/SO(6)\\times\nSO(n)$ coset.\n\\\\\n\\indent Space-time and tangent space indices are denoted\nrespectively by $\\mu,\\nu,\\ldots =0,1,2,3$ and\n$\\hat{\\mu},\\hat{\\nu},\\ldots=0,1,2,3$. Indices $m,n=1,\\ldots, 6$ and\n$i,j=1,2,3,4$ respectively describe the vector and spinor\nrepresentations of the $SO(6)_R\\sim SU(4)_R$ R-symmetry or\nequivalently a second-rank anti-symmetric tensor and fundamental\nrepresentations of $SU(4)_R$. The $n$ vector multiplets are labeled by\nindices $a,b=1,\\ldots, n$. All the fields in the vector multiplets\nwill accordingly carry an additional index in the form of\n$(A^a_\\mu,\\lambda^{ia},\\phi^{ma})$.\n\\\\\n\\indent All fermionic fields and the supersymmetry parameters\ntransform in the fundamental representation of $SU(4)_R$ R-symmetry\nwith the chirality projections\n\\begin{equation}\n\\gamma_5\\psi^i_\\mu=\\psi^i_\\mu,\\qquad \\gamma_5\\chi^i=-\\chi^i,\\qquad \\gamma_5\\lambda^i=\\lambda^i\\, .\n\\end{equation}\nSimilarly, for the fields transforming in the anti-fundamental\nrepresentation of $SU(4)_R$, we have\n\\begin{equation}\n\\gamma_5\\psi_{\\mu i}=-\\psi_{\\mu i},\\qquad \\gamma_5\\chi_i=\\chi_i,\\qquad \\gamma_5\\lambda_i=-\\lambda_i\\, .\n\\end{equation}\n\\indent General gaugings of the matter-coupled $N=4$ supergravity\ncan be efficiently described by the embedding tensor $\\Theta$ which\nencodes the information about the embedding of any gauge group $G_0$\nin the global or duality symmetry $SL(2,\\mathbb{R})\\times SO(6,n)$.\nThere are two components of the embedding tensor $\\xi^{\\alpha M}$\nand $f_{\\alpha MNP}$ with $\\alpha=(+,-)$ and $M,N=(m,a)=1,\\ldots,\nn+6$ denoting fundamental representations of $SL(2,\\mathbb{R})$ and\n$SO(6,n)$, respectively. The electric vector fields\n$A^{M+}=(A^m_\\mu,A^a_\\mu)$, appearing in the ungauged Lagrangian,\ntogether with their magnetic dual $A^{M-}$ form a doublet under\n$SL(2,\\mathbb{R})$. These are denoted collectively by $A^{M\\alpha}$.\nIn general, a subgroup of both $SL(2,\\mathbb{R})$ and $SO(6,n)$ can\nbe gauged, and the magnetic vector fields can also participate in\nthe gauging. However, in this paper, we only consider gaugings with\nonly $f_{\\alpha MNP}$ non-vanishing. We then set $\\xi^{\\alpha M}$ to\nzero from now on. This also considerably simplifies many formulae\ngiven below.\n\\\\\n\\indent The full covariant derivative can be written as\n\\begin{equation}\nD_\\mu=\\nabla_\\mu-gA_\\mu^{M\\alpha}f_{\\alpha M}^{\\phantom{\\alpha\nM}NP}t_{NP}\n\\end{equation}\nwhere $\\nabla_\\mu$ is the space-time covariant derivative including the spin connections.\n$t_{MN}$ are $SO(6,n)$ generators which can be chosen as\n\\begin{equation}\n(t_{MN})_P^{\\phantom{P}Q}=2\\delta^Q_{[M}\\eta_{N]P},\n\\end{equation}\nwith $\\eta_{MN}$ being the $SO(6,n)$ invariant tensor. The gauge\ncoupling constant $g$ can be absorbed in the embedding tensor\n$\\Theta$. The original gauging considered in \\cite{Eric_N4_4D} only\ninvolves electric vector fields and is called electric gauging. In\nthis case, only $f_{+MNP}$ are non-vanishing. In the following\ndiscussions, we will consider dyonic gauging involving both electric\nand magnetic vector fields. In this case, both $A^{M+}$ and $A^{M-}$\nenter the Lagrangian, and $f_{\\alpha MNP}$ with $\\alpha=\\pm$ are\nnon-vanishing. Consistency requires the presence of two-form fields\nwhen magnetic vector fields are included. In the present case with\n$\\xi^{\\alpha M}=0$, these two-forms transform as an anti-symmetric\ntensor under $SO(6,n)$ and will be denoted by $B^{MN}_{\\mu\\nu}=B^{[MN]}_{\\mu\\nu}$.\nThe two-forms modify the gauge field strengths to\n\\begin{equation}\n\\mathcal{H}^{M\\pm}=dA^{M\\pm}-\\frac{1}{2}\\eta^{MQ}f_{\\alpha QNP}A^{N\\alpha}\\wedge A^{P\\pm}\\pm\\frac{1}{2}\\eta^{MQ}f_{\\mp QNP}B^{NP}\\, .\n\\end{equation}\nNote that for non-vanishing $f_{-MNP}$ the field strengths of\nelectric vectors $\\mathcal{H}^{M+}$ have a contribution from the two-form\nfields.\n\\\\\n\\indent Before moving to the Lagrangian, we explicitly give the parametrization of the scalar coset\nmanifold $SL(2,\\mathbb{R})\/SO(2)\\times SO(6,n)\/SO(6)\\times SO(n)$.\nThe first factor can be described by a coset representative\n\\begin{equation}\n\\mathcal{V}_\\alpha=\\frac{1}{\\sqrt{\\textrm{Im} \\tau}}\\left(\n                                         \\begin{array}{c}\n                                           \\tau \\\\\n                                           1 \\\\\n                                         \\end{array}\n                                       \\right)\\label{Valpha}\n\\end{equation}\nor equivalently by a symmetric matrix\n\\begin{equation}\nM_{\\alpha\\beta}=\\textrm{Re} (\\mathcal{V}_\\alpha\\mathcal{V}^*_\\beta)=\\frac{1}{\\textrm{Im}\n\\tau}\\left(\n                                    \\begin{array}{cc}\n                                      |\\tau|^2 & \\textrm{Re} \\tau \\\\\n                                      \\textrm{Re} \\tau & 1 \\\\\n                                    \\end{array}\n                                  \\right).\n\\end{equation}\nIt should also be noted that $\\textrm{Im}(\\mathcal{V}_\\alpha\\mathcal{V}^*_\\beta)=\\epsilon_{\\alpha\\beta}$.\nThe complex scalar $\\tau$ can also be written in terms of the dilaton $\\phi$ and the axion $\\chi$ as\n\\begin{equation}\n\\tau=\\chi+ie^\\phi\\, .\n\\end{equation}\n\\indent For the $SO(6,n)\/SO(6)\\times SO(n)$ factor, we can introduce the\ncoset representative $\\mathcal{V}_M^{\\phantom{M}A}$ transforming by\nleft and right multiplications under $SO(6,n)$ and $SO(6)\\times\nSO(n)$, respectively. The $SO(6)\\times SO(n)$ index will be split as\n$A=(m,a)$ according to which the coset representative can be written as\n$\\mathcal{V}_M^{\\phantom{M}A}=(\\mathcal{V}_M^{\\phantom{M}m},\\mathcal{V}_M^{\\phantom{M}a})$.\nAs an element of $SO(6,n)$, the matrix $\\mathcal{V}_M^{\\phantom{M}A}$ also\nsatisfies the relation\n\\begin{equation}\n\\eta_{MN}=-\\mathcal{V}_M^{\\phantom{M}m}\\mathcal{V}_N^{\\phantom{M}m}+\\mathcal{V}_M^{\\phantom{M}a}\\mathcal{V}_N^{\\phantom{M}a}\\,\n.\n\\end{equation}\nAs in the $SL(2,\\mathbb{R})\/SO(2)$ factor, the $SO(6,n)\/SO(6)\\times\nSO(n)$ coset can also be parametrized in term of a symmetric matrix\ndefined by\n\\begin{equation}\nM_{MN}=\\mathcal{V}_M^{\\phantom{M}m}\\mathcal{V}_N^{\\phantom{M}m}+\\mathcal{V}_M^{\\phantom{M}a}\\mathcal{V}_N^{\\phantom{M}a}\\,\n.\n\\end{equation}\n\\indent The bosonic Lagrangian of the $N=4$ gauged supergravity is given by\n\\begin{eqnarray}\ne^{-1}\\mathcal{L}&=&\\frac{1}{2}R+\\frac{1}{16}\\mathcal{D}_\\mu M_{MN}\\mathcal{D}^\\mu\nM^{MN}-\\frac{1}{4(\\textrm{Im}\\tau)^2}\\partial_\\mu \\tau \\partial^\\mu \\tau^*-V\\nonumber \\\\\n& &-\\frac{1}{4}\\textrm{Im}\\,\\tau M_{MN}\\mathcal{H}^{M+}_{\\mu\\nu}\\mathcal{H}^{N+\\mu\\nu}-\\frac{1}{8}\\textrm{Re}\\,\\tau e^{-1}\\epsilon^{\\mu\\nu\\rho\\sigma}\\eta_{MN}\\mathcal{H}^{M+}_{\\mu\\nu}\\mathcal{H}^{N+}_{\\rho\\sigma}\\nonumber \\\\\n& &-\\frac{1}{2}e^{-1}\\epsilon^{\\mu\\nu\\rho\\sigma}\\left[f_{-MNP}A^{M-}_\\mu A^{N+}_\\nu\\partial_\\rho A^{P-}_\\sigma +\\frac{1}{4}f_{\\alpha MNR}f_{\\beta PQS}\\eta^{RS}A^{M\\alpha}_\\mu A^{N+}_\\nu A^{P\\beta}_\\rho A^{Q-}_\\sigma \\right.\\nonumber \\\\\n& &-\\frac{1}{4}f_{-MNP}B^{NP}_{\\mu\\nu}\\left(2\\partial_\\rho A^{M-}_\\sigma -\\frac{1}{2}\\eta^{MS}f_{\\alpha SQR}A^{Q\\alpha}_\\rho A^{R-}_\\sigma\\right)\\nonumber \\\\\n& &\\left.-\\frac{1}{16}f_{+MNR}f_{-PQS}\\eta^{RS}B^{MN}_{\\mu\\nu}B^{PQ}_{\\rho\\sigma}\n\\right]\n\\end{eqnarray}\nwhere $e$ is the vielbein determinant. The scalar potential is given\nby\n\\begin{eqnarray}\nV&=&\\frac{g^2}{16}\\left[f_{\\alpha MNP}f_{\\beta\nQRS}M^{\\alpha\\beta}\\left[\\frac{1}{3}M^{MQ}M^{NR}M^{PS}+\\left(\\frac{2}{3}\\eta^{MQ}\n-M^{MQ}\\right)\\eta^{NR}\\eta^{PS}\\right]\\right.\\nonumber \\\\\n& &\\left.-\\frac{4}{9}f_{\\alpha MNP}f_{\\beta\nQRS}\\epsilon^{\\alpha\\beta}M^{MNPQRS}\\right]\n\\end{eqnarray}\nwhere $M^{MN}$ is the inverse of $M_{MN}$, and $M^{MNPQRS}$ is\ndefined by\n\\begin{equation}\nM_{MNPQRS}=\\epsilon_{mnpqrs}\\mathcal{V}_{M}^{\\phantom{M}m}\\mathcal{V}_{N}^{\\phantom{M}n}\n\\mathcal{V}_{P}^{\\phantom{M}p}\\mathcal{V}_{Q}^{\\phantom{M}q}\\mathcal{V}_{R}^{\\phantom{M}r}\\mathcal{V}_{S}^{\\phantom{M}s}\\label{M_6}\n\\end{equation}\nwith indices raised by $\\eta^{MN}$. The covariant derivative of $M_{MN}$ is defined by\n\\begin{equation}\n\\mathcal{D}M_{MN}=dM_{MN}+2A^{P\\alpha}\\eta^{QR}f_{\\alpha QP(M}M_{N)R}\\, .\n\\end{equation}\n\\indent It should be pointed out here that the magnetic vectors and\nthe two-forms do not have kinetic terms. They are auxiliary fields\nwhose field equations give rise to the duality relation between\ntwo-forms and scalars and the electric-magnetic duality of $A^{M+}$\nand $A^{M-}$, respectively. Together with the Yang-Mills equations\nobtained from the variation with respect to $A^{M+}$, these\nequations are given by\n\\begin{eqnarray}\n\\eta_{MN}*\\mathcal{D}\\mathcal{H}^{N-}&=&-\\frac{1}{4}{f_{+MP}}^NM_{NQ}\\mathcal{D}M^{QP},\\label{YM1}\\\\\n\\eta_{MN}*\\mathcal{D}\\mathcal{H}^{N+}&=&\\frac{1}{4}{f_{-MP}}^NM_{NQ}\\mathcal{D}M^{QP},\\label{YM2}\\\\\n\\mathcal{H}^{M-}&=&\\textrm{Im}\\, \\tau M^{MN}\\eta_{NP}*\\mathcal{H}^{P+}-\\textrm{Re}\\, \\tau\\mathcal{H}^{M+}\\label{YM3}\n\\end{eqnarray}\nwhere we have used differential form language for later\ncomputational convenience. By substituting $\\mathcal{H}^{M-}$ from\n\\eqref{YM3} in \\eqref{YM1}, we obtain the usual Yang-Mills equations\nfor $\\mathcal{H}^{M+}$ while equation \\eqref{YM2} simply gives the\nrelation between the Hodge dual of the three-form field strength and\nthe scalars due to the usual Bianchi identity of the gauge field\nstrengths\n\\begin{equation}\n\\mathcal{F}^{M\\pm}=dA^{M\\pm}-\\frac{1}{2}\\eta^{MQ}f_{\\alpha QNP}A^{N\\alpha}\\wedge A^{P\\pm}\n\\end{equation}\n\\indent In this paper, we are interested in $N=4$ gauged\nsupergravity coupled to three vector multiplets. The gauge group of\ninterest here is a non-semisimple group $SO(3)\\ltimes\n(\\mathbf{T}^3,\\hat{\\mathbf{T}}^3)\\subset SO(6,3)$ described by the\nfollowing components of the embedding tensor\n\\begin{eqnarray}\nf_{+IJ,K+6}&=&-f_{+I+3,J+6,K+6}=-2\\sqrt{2}\\epsilon_{IJK},\\qquad I,J,K=1,2,3,\\nonumber \\\\\nf_{+I+6,J+6,K+6}&=&6\\sqrt{2}k\\epsilon_{IJK},\\qquad f_{-I,J+6,K+6}=-4\\epsilon_{IJK}\\, .\\label{embedding_tensor}\n\\end{eqnarray}\nThe constant $k$ is related to the four-form flux along the\nfour-dimensional space-time, see equation \\eqref{4_form_flux} below.\n\\\\\n\\indent We should also remark that we follow the convention of\n\\cite{N010_truncation_Cassani} in all of the computations carried\nout here. In particular, the $SO(6,3)$ tensor $\\eta_{MN}$ is\noff-diagonal\n\\begin{equation}\n\\eta_{MN}=\\left(\n                     \\begin{array}{ccc}\n                       -\\mathbf{I}_3 & \\mathbf{0}_3 & \\mathbf{0}_3 \\\\\n                       \\mathbf{0}_3 & \\mathbf{0}_3 & \\mathbf{I}_3 \\\\\n                       \\mathbf{0}_3 & \\mathbf{I}_3 & \\mathbf{0}_3 \\\\\n                     \\end{array}\n                   \\right)\n\\end{equation}\nwhere $\\mathbf{0}_3$ and $\\mathbf{I}_3$ denote $3\\times 3$ zero and\nidentity matrices, respectively. As a result, the computation of\n$M_{MNPQRS}$ in \\eqref{M_6} and parts of the supersymmetry\ntransformations given below which involve $\\mathcal{V}_M^{\\phantom{M}m}$ and\n$\\mathcal{V}_M^{\\phantom{M}a}$ must be done with the projection to the negative and\npositive eigenvalues of $\\eta_{MN}$, respectively. This can be achieved by using the projection matrix\n\\begin{equation}\nP=\\left(\n            \\begin{array}{ccc}\n                       \\mathbf{0}_3 & \\sqrt{2}\\tilde{P}_3 & \\mathbf{0}_3 \\\\\n                       -\\tilde{P}_3 & \\mathbf{0}_3 & \\tilde{P}_3 \\\\\n                       \\tilde{P}_3 & \\mathbf{0}_3 & \\tilde{P}_3 \\\\\n                     \\end{array}\n                   \\right)\n\\end{equation}\nwhere the $3\\times 3$ matrix $\\tilde{P}_3$ is given by\n\\begin{equation}\n\\tilde{P}_3=\\frac{1}{\\sqrt{2}}\\left(\n            \\begin{array}{ccc}\n                       0 & 0 & 1 \\\\\n                       0 & 1 & 0 \\\\\n                       1 & 0 & 0 \\\\\n                     \\end{array}\n                   \\right).\n\\end{equation}\n\\indent We now turn to the supersymmetry transformations of\nfermionic fields. These are given by\n\\begin{eqnarray}\n\\delta\\psi^i_\\mu &=&2D_\\mu \\epsilon^i-\\frac{2}{3}gA^{ij}_1\\gamma_\\mu\n\\epsilon_j+\\frac{i}{4}(\\mathcal{V}_\\alpha)^*{\\mathcal{V}_M}^{ij}\\mathcal{H}^{M\\alpha}_{\\nu\\rho}\\gamma^{\\nu\\rho}\\gamma_\\mu\\epsilon_j,\\\\\n\\delta \\chi^i &=&i\\epsilon^{\\alpha\\beta}\\mathcal{V}_\\alpha D_\\mu\n\\mathcal{V}_\\beta\\gamma^\\mu \\epsilon^i-\\frac{4}{3}igA_2^{ij}\\epsilon_j+\\frac{i}{2}\\mathcal{V}_\\alpha{\\mathcal{V}_M}^{ij}\\mathcal{H}^{M\\alpha}_{\\mu\\nu}\\epsilon_j,\\\\\n\\delta \\lambda^i_a&=&2i\\mathcal{V}_a^{\\phantom{a}M}D_\\mu\n\\mathcal{V}_M^{\\phantom{M}ij}\\gamma^\\mu\\epsilon_j+2igA_{2aj}^{\\phantom{2aj}i}\\epsilon^j\n-\\frac{1}{4}\\mathcal{V}_\\alpha\\mathcal{V}_{Ma}\\mathcal{H}^{M\\alpha}_{\\mu\\nu}\\gamma^{\\mu\\nu}\\epsilon^i\\,\n.\n\\end{eqnarray}\nThe fermion shift matrices are defined by\n\\begin{eqnarray}\nA_1^{ij}&=&\\epsilon^{\\alpha\\beta}(\\mathcal{V}_\\alpha)^*\\mathcal{V}_{kl}^{\\phantom{kl}M}\\mathcal{V}_N^{\\phantom{N}ik}\n\\mathcal{V}_P^{\\phantom{P}jl}f_{\\beta M}^{\\phantom{\\beta M}NP},\\nonumber\n\\\\\nA_2^{ij}&=&\\epsilon^{\\alpha\\beta}\\mathcal{V}_\\alpha\\mathcal{V}_{kl}^{\\phantom{kl}M}\\mathcal{V}_N^{\\phantom{N}ik}\n\\mathcal{V}_P^{\\phantom{P}jl}f_{\\beta M}^{\\phantom{\\beta M}NP},\\nonumber\n\\\\\nA_{2ai}^{\\phantom{2ai}j}&=&\\epsilon^{\\alpha\\beta}\\mathcal{V}_\\alpha\n\\mathcal{V}^M_{\\phantom{M}a}\\mathcal{V}^N_{\\phantom{N}ik}\\mathcal{V}_P^{\\phantom{P}jk}f_{\\beta\nMN}^{\\phantom{\\beta MN}P}\n\\end{eqnarray}\nwhere $\\mathcal{V}_M^{\\phantom{M}ij}$ is defined in terms of the `t Hooft\nmatrices $G^{ij}_m$ and $\\mathcal{V}_M^{\\phantom{M}m}$ as\n\\begin{equation}\n\\mathcal{V}_M^{\\phantom{M}ij}=\\frac{1}{2}\\mathcal{V}_M^{\\phantom{M}m}G^{ij}_m\n\\end{equation}\nand similarly for its inverse\n\\begin{equation}\n\\mathcal{V}^M_{\\phantom{M}ij}=-\\frac{1}{2}\\mathcal{V}_M^{\\phantom{M}m}(G^{ij}_m)^*\\,\n.\n\\end{equation}\n$G^{ij}_m$ satisfy the relations\n\\begin{equation}\nG_{mij}=(G^{ij}_m)^*=\\frac{1}{2}\\epsilon_{ijkl}G^{kl}_m\\, .\n\\end{equation}\nThe explicit form of these matrices is given in the appendix. It should also be noted that the scalar\npotential can be written in terms of $A_1$ and $A_2$ tensors as\n\\begin{equation}\nV=-\\frac{1}{3}A^{ij}_1A_{1ij}+\\frac{1}{9}A^{ij}_2A_{2ij}+\\frac{1}{2}A_{2ai}^{\\phantom{2ai}j}\nA_{2a\\phantom{i}j}^{\\phantom{2a}i}\\, .\n\\end{equation}\n\\indent With the explicit form of $\\mathcal{V}_\\alpha$ given in\n\\eqref{Valpha} and equation \\eqref{YM3}, it is straightforward to\nderive the following identities\n\\begin{eqnarray}\ni\\mathcal{V}_\\alpha{\\mathcal{V}_M}^{ij}\\mathcal{H}^{M\\alpha}_{\\mu\\nu}\\gamma^{\\mu\\nu}&=&-(\\mathcal{V}_-)^{-1}{\\mathcal{V}_M}^{ij}\\mathcal{H}^{M+}_{\\mu\\nu}\\gamma^{\\mu\\nu}(1-\\gamma_5),\\\\\ni\\mathcal{V}_\\alpha{\\mathcal{V}_M}^a\\mathcal{H}^{M\\alpha}_{\\mu\\nu}\\gamma^{\\mu\\nu}&=&-(\\mathcal{V}_-)^{-1}{\\mathcal{V}_M}^a\\mathcal{H}^{M+}_{\\mu\\nu}\\gamma^{\\mu\\nu}(1+\\gamma_5),\\\\\ni(\\mathcal{V}_\\alpha)^*{\\mathcal{V}_M}^{ij}\\mathcal{H}^{M\\alpha}_{\\mu\\nu}\\gamma^{\\mu\\nu}\\gamma_\\rho\n&=&(\\mathcal{V}_-)^{-1}{\\mathcal{V}_M}^{ij}\\mathcal{H}^{M+}_{\\mu\\nu}\\gamma^{\\mu\\nu}\\gamma_\\rho(1-\\gamma_5).\n\\end{eqnarray}\nIn obtaining these results, we have used the following relations for\nthe $SO(6,n)$ coset representative \\cite{Eric_N4_4D}\n\\begin{eqnarray}\n\\eta_{MN}&=&-\\frac{1}{2}\\epsilon_{ijkl}{\\mathcal{V}_M}^{ij}{\\mathcal{V}_N}^{kl}+{\\mathcal{V}_M}^{a}{\\mathcal{V}_N}^{a},\\qquad {\\mathcal{V}_M}^a{\\mathcal{V}^M}_{ij}=0,\\nonumber \\\\\n{\\mathcal{V}_M}^{ij}{\\mathcal{V}^{M}}_{kl}&=&-\\frac{1}{2}(\\delta^i_k\\delta^j_l-\\delta^i_l\\delta^j_k),\\qquad {\\mathcal{V}_M}^a{\\mathcal{V}^M}_b=\\delta^a_b\\, .\n\\end{eqnarray}\nThese relations are useful in simplifying the BPS equations\nresulting from the supersymmetry transformations. Note also that\nthese relations are slightly different from those given in\n\\cite{N4_gauged_SUGRA} due to a different convention on $\\mathcal{V}_\\alpha$ in term of the scalar\n$\\tau$. In more detail, $\\mathcal{V}_\\alpha$ used in this paper and in\n\\cite{N010_truncation_Cassani} satisfies $\\mathcal{V}_+\/\\mathcal{V}_-=\\tau$ while $\\mathcal{V}_\\alpha$ used in\n\\cite{N4_gauged_SUGRA} gives $\\mathcal{V}_+\/\\mathcal{V}_-=\\tau^*$. This results in some sign changes in the\nabove equations compared to those of \\cite{N4_gauged_SUGRA}.\n\n\\subsection{Uplift formulae to eleven dimensions}\nAs mentioned above, four-dimensional $N=4$ gauged supergravity\ncoupled to three vector multiplets with $SO(3)\\ltimes\n(\\mathbf{T}^3,\\hat{\\mathbf{T}}^3)$ gauge group has been obtained\nfrom a truncation of eleven-dimensional supergravity on a\ntri-sasakian manifold in \\cite{N010_truncation_Cassani}. We will\nbriefly review the structure and relevant formulae focusing on the\nreduction ansatz which will be useful for embedding four-dimensional\nsolutions. Essentially, we simply collect some formulae without\ngiving detailed explanations for which we refer the interested readers\nto \\cite{N010_truncation_Cassani}.\n\\\\\n\\indent The eleven-dimensional metric can be written as\n\\begin{equation}\nds^2_{11}=e^{2\\varphi}ds^2_4+e^{2U}ds^2(B_\\textrm{QK})+g_{IJ}(\\eta^I+A_1^I)(\\eta^J+A_1^J)\\,\n.\n\\end{equation}\nThe three-dimensional internal metric $g_{IJ}$ can be written in\nterms of the vielbein as\n\\begin{equation}\ng=Q^TQ\\, .\n\\end{equation}\nFollowing \\cite{N010_truncation_Cassani}, we will parametrize the\nmatrix $Q$ in term of a product of a diagonal matrix $V$ and an\n$SO(3)$ matrix $O$\n\\begin{equation}\nQ=VO,\\qquad V=\\textrm{diag}(e^{V_1},e^{V_2},e^{V_3})\\, .\\label{Q_def}\n\\end{equation}\nThe scalar $\\varphi$ is chosen to be\n\\begin{equation}\n\\varphi=-\\frac{1}{2}(4U+V_1+V_2+V_3)\n\\end{equation}\nin order to obtain the Einstein frame action in four dimensions.\n$B_{\\textrm{QK}}$ denotes a four-dimensional quaternionic Kahler\nmanifold whose explicit metric is not needed in the following\ndiscussions.\n\\\\\n\\indent The ansatz for the four-form field is given by\n\\begin{eqnarray}\nG_4&=&H_4+H_{3I}\\wedge (\\eta+A_1)^I+\\frac{1}{2}\\epsilon_{IJK}\\tilde{H}_2^I\\wedge(\\eta+A_1)^J\n\\wedge (\\eta+A_1)^K+4\\textrm{Tr}c\\,\\textrm{vol}(\\textrm{QK})\\nonumber \\\\\n& &H_{1IJ}\\wedge (\\eta+A_1)^I\\wedge J^I+\\frac{1}{6}\\epsilon_{IJK}d\\chi\\wedge(\\eta+A_1)^I\\wedge (\\eta+A_1)^J\\wedge (\\eta+A_1)^K\\nonumber \\\\\n& &+H_{2I}\\wedge\nJ^I+\\epsilon_{IJL}\\left[(\\chi+\\textrm{Tr}c)\\delta_{LK}-2c_{(LK)}\\right](\\eta+A_1)^I\\wedge(\\eta+A_1)^J\\wedge\nJ^K\\, .\\nonumber \\\\\n& &\n\\end{eqnarray}\n$c_{IJ}$ is a $3\\times 3$ matrix and $\\textrm{Tr}\nc=\\delta^{IJ}c_{IJ}$. The volume form of $B_{\\textrm{QK}}$,\n$\\textrm{vol}(\\textrm{QK})$, can be written in terms of the two-forms $J^I$\nas\n\\begin{equation}\n\\textrm{vol}(\\textrm{QK})=\\frac{1}{6}J^I\\wedge J^I\\, .\n\\end{equation}\nVarious forms in the above equation are defined by\n\\begin{eqnarray}\nH_4&=&dc_3+c_{2I}\\wedge F^I_2,\\qquad H_{3I}=Dc_{2I}+\\epsilon_{IJK}F^J_2\\wedge \\tilde{c}_{1K},\\nonumber \\\\\n\\tilde{H}_{2I}&=&D\\tilde{c}_{1I}-2c_{2I}+\\chi F_{2I},\\qquad H_{2I}=Dc_{1I}+2c_{2I}+c_{JI}F^J_2,\\nonumber \\\\\nH_{1IJ}&=&Dc_{IJ}+2\\epsilon_{IJK}(c_{1K}+\\tilde{c}_{1K})\n\\end{eqnarray}\nwith the $SO(3)$ covariant derivative\n\\begin{equation}\nDc_{I_1\\ldots I_n}=dc_{I_1\\ldots I_n}+2\\sum_{l=1}^n\\epsilon_{JI_lK}A^J_1\\wedge c_{I_1\\ldots K\\ldots I_n}\\, .\n\\end{equation}\nThe $SO(3)_R$ field strengths are defined by\n\\begin{equation}\nF^I_2=dA^I_1-\\epsilon_{IJK}A^J_1\\wedge A^K_1\\, .\n\\end{equation}\nIt is useful to note here that the $SL(2,\\mathbb{R})\/SO(2)$ scalars\nare given by\n\\begin{equation}\n\\tau=\\chi+ie^{V_1+V_2+V_3}\\, .\n\\end{equation}\n\\indent Although we will not directly need the explicit form of\n$ds^2(B_{\\textrm{QK}})$ and $\\eta^I$'s in the remaining parts of\nthis paper, it is useful to give some information on the $N^{010}$\ntri-sasakian manifold. $N^{010}$ is a 7-manifold with $SU(2)\\times\nSU(3)$ isometry. The $SU(2)$ is identified with the R-symmetry of\nthe dual $N=3$ SCFT while $SU(3)$ is the flavor symmetry. A simple\ndescription of $N^{010}$ can be obtained in term of a coset manifold\n$SU(3)\/U(1)$. With the standard Gell-Mann matrices, the $SU(3)$\ngenerators can be chosen to be $-\\frac{i}{2}\\lambda_\\alpha$,\n$\\alpha=1,\\ldots, 8$. The coset and $U(1)$ generators are\naccordingly identified as\n\\begin{equation}\nK_i=-\\frac{i}{2}(\\lambda_1,\\lambda_2,\\lambda_3,\\lambda_4,\\lambda_5,\\lambda_6,\\lambda_7),\\qquad\nH=-\\frac{i\\sqrt{3}}{2}\\lambda_8\\, .\n\\end{equation}\nThe vielbein on $N^{010}$ can eventually be obtained from the decomposition of the Maurer-Cartan one-form\n\\begin{equation}\nL^{-1}dL=e^iK_i+\\omega H\n\\end{equation}\nwhere $L$ is the coset representative for $SU(3)\/U(1)$, and $\\omega$ is\nthe corresponding $U(1)$ connection.\n\\\\\n\\indent Following \\cite{N010_truncation_Cassani}, we can use the tri-sasakian structures of the form\n\\begin{eqnarray}\n\\eta^I&=&\\frac{1}{2}(e^1,e^2,e^7),\\nonumber\\\\\nJ^I&=&\\frac{1}{8}(e^4\\wedge e^5-e^3\\wedge e^6,-e^3\\wedge e^5-e^4\\wedge e^6,e^5\\wedge e^6-e^3\\wedge e^4).\n\\end{eqnarray}\nFrom these, we find the metric on the Quaternionic-Kahler base $B_{\\textrm{QK}}$ to be\n\\begin{equation}\nds^2(B_{\\textrm{QK}})=\\frac{1}{256}\\left[(e^3)^2+(e^4)^2+(e^5)^2+(e^6)^2\\right]\n\\end{equation}\nwith the volume form given by\n\\begin{equation}\n\\textrm{vol}(\\textrm{QK})=\\frac{1}{6}J^I\\wedge J^I=-\\frac{1}{64}e^3\\wedge e^4\\wedge e^5\\wedge e^6\\, .\n\\end{equation}\nAs mentioned before, all of the fields appearing in the reduction of \\cite{N010_truncation_Cassani} are $SU(3)$ singlets.\n\n\\section{BPS flow equations}\\label{flow_eq}\nIn this section, we perform the analysis of Yang-Mills equations\nand supersymmetry transformations in order to obtain BPS equations\nfor the flows between $AdS_4$ vacua and possible $AdS_2\\times\n\\Sigma_2$ geometries. We set all fermions to zero and truncate the\nbosonic fields to $SO(2)\\subset SO(3)_R$ singlets. This $SO(2)$ is\ngenerated by\n\\begin{equation}\n\\hat{X}=X_{9+}+X_{6+}+X_{3-}\n\\end{equation}\nwhere the gauge generators are defined by\n\\begin{equation}\nX_{M\\alpha}=-f_{\\alpha MNP}t^{NP}\\, .\n\\end{equation}\nWe see that a combination of the electric vectors $A^{9+}$, $A^{6+}$\nand the magnetic vector $A^{3-}$ becomes the corresponding $SO(2)$\ngauge field.\n\\\\\n\\indent We are interested in supersymmetric solutions of the form\n$AdS_2\\times \\Sigma_2$ with $\\Sigma_2=S^2,H^2$. We will then take\nthe ansatz for the four-dimensional metric to be\n\\begin{equation}\nds^2_4=-e^{2f(r)}dt^2+dr^2+e^{2g(r)}(d\\theta^2+F(\\theta)^2d\\phi^2)\n\\end{equation}\nwith\n\\begin{equation}\nF(\\theta)=\\sin\\theta\\qquad \\textrm{and}\\qquad F(\\theta)=\\sinh\\theta\n\\end{equation}\nfor the $S^2$ and $H^2$, respectively. We will also use the\nparameter $\\kappa=\\pm 1$ to denote the $S^2$ and $H^2$ cases. The\nfunctions $f(r)$, $g(r)$ and all other fields only depend on the\nradial coordinate $r$ for static solutions. With the obvious\nvielbein\n\\begin{equation}\ne^{\\hat{t}}=e^{f}dt,\\qquad e^{\\hat{r}}=dr,\\qquad e^{\\hat{\\theta}}=e^{g}d\\theta,\\qquad e^{\\hat{\\phi}}=e^gFd\\phi,\n\\end{equation}\nit is now straightforward to compute the spin connections of the above metric\n\\begin{eqnarray}\n\\omega^{\\hat{t}\\hat{r}}&=&f'e^{\\hat{t}},\\qquad \\omega^{\\hat{\\theta}\\hat{r}}=g'e^{\\hat{\\theta}},\\nonumber \\\\\n\\omega^{\\hat{\\phi}\\hat{r}}&=&g'e^{\\hat{\\phi}},\\qquad \\omega^{\\hat{\\theta}\\hat{\\phi}}=\\frac{F'(\\theta)}{F(\\theta)}e^{-g}e^{\\hat{\\phi}}\\, .\n\\end{eqnarray}\nIn the above expressions, we have used the hat to denote ``flat''\nindices while $'$ stands for the $r$-derivative with the only\nexception that $F'(\\theta)=\\frac{dF(\\theta)}{d\\theta}$. The ansatz for electric and magnetic vector fields are given by\n\\begin{eqnarray}\nA^{M+}&=&\\mathcal{A}^M_tdt- p^MF'(\\theta)d\\phi, \\\\\nA^{M-}&=&\\tilde{\\mathcal{A}}^M_tdt- e_MF'(\\theta)d\\phi\\label{AM_minus}\n\\end{eqnarray}\nwhere we have chosen the gauge such that $A^{M\\alpha}_r=0$. $p^M$\nand $e_M$ correspond to magnetic and electric charges, respectively.\nIn the present case, only $A^{M\\alpha}$ with $M=3,6,9$ are relevant.\n\\\\\n\\indent We finally give the explicit form of the scalar coset representative for $SO(6,3)\/SO(6)\\times SO(3)$. The parametrization of \\cite{N010_truncation_Cassani} which is directly related to the higher dimensional origin is given by\n\\begin{equation}\n\\mathcal{V}=\\mathcal{C}\\mathcal{Q}\n\\end{equation}\nwhere the matrices $\\mathcal{Q}$ and $\\mathcal{C}$ are defined by\n\\begin{equation}\n\\mathcal{Q}=\\left(\n                     \\begin{array}{ccc}\n                       \\mathbf{I}_3 & \\mathbf{0}_3 & \\mathbf{0}_3 \\\\\n                       \\mathbf{0}_3 & e^{-2U}Q^{-1} & \\mathbf{I}_3 \\\\\n                       \\mathbf{0}_3 & \\mathbf{0}_3 & e^{2U}Q^T \\\\\n                     \\end{array}\n                   \\right),\n                   \\qquad\n\\mathcal{C}=\\text{exp}\\,\\left(\n                     \\begin{array}{ccc}\n                       \\mathbf{0}_3 & \\sqrt{2}c^T & \\mathbf{0}_3 \\\\\n                       \\mathbf{0}_3 & \\mathbf{0}_3 & \\mathbf{0}_3 \\\\\n                       \\sqrt{2}c & a & \\mathbf{0}_3 \\\\\n                     \\end{array}\n                   \\right).\n\\end{equation}\nFor $SO(2)$ invariant scalars, the $3\\times 3$ matrices $c$ and $a$ are given by\n\\begin{equation}\nc=\\left(\n                     \\begin{array}{ccc}\n                       Z_1 & Z_3 & 0 \\\\\n                       -Z_3 & Z_1 & 0 \\\\\n                       0 & 0 & Z_2 \\\\\n                     \\end{array}\n                   \\right),\n                   \\qquad\na=\\left(\n                     \\begin{array}{ccc}\n                       0 & \\Phi & 0 \\\\\n                       -\\Phi & 0 & 0 \\\\\n                       0 & 0 & 0 \\\\\n                     \\end{array}\n                   \\right)\n\\end{equation}\nwhile $Q$ can be obtained from \\eqref{Q_def} with $V_2=V_1$ and $O$ being\n\\begin{equation}\nO=\\textrm{exp}\\left(\n                     \\begin{array}{ccc}\n                       0 & \\beta & 0 \\\\\n                       -\\beta & 0 & 0 \\\\\n                       0 & 0 & 0 \\\\\n                     \\end{array}\n                   \\right).\n\\end{equation}\nThis is a generalization of the coset representative of the\n$SO(3)_R$ singlet scalars used in \\cite{trisasakian_flow} in which\n$\\Phi=\\beta=Z_3=0$, $Z_1=Z_2$ and $V_1=V_2=V_3$. In the following,\nwe will rename the scalars $V_3\\rightarrow V_2$ such that the\ncomplex scalar $\\tau$ becomes\n\\begin{equation}\n\\tau=\\chi+ie^{2V_1+V_2}\\, .\n\\end{equation}\n\\indent  We now give the scalar potential for $SO(2)$ singlet scalars\n\\begin{eqnarray}\nV&=&e^{-3(4U+2V_1+V_2)}\\left[e^{4(U+V_2)}(e^{4U}+2e^{4V_1})+9k^2+4\\chi^2e^{4U+2V_1} \\right.\\nonumber\\\\\n& &-4e^{6U+4V_1+2V_2}(6+e^{2(U-V_1)}-e^{-2(U-V_1)})+24k\\chi Z_1+16\\chi^2Z_1^2\\nonumber \\\\\n& &+8\\chi Z_2e^{4U+2V_1}-12k\\chi Z_2+(16\\chi^2-24k)Z_1Z_2+32\\chi Z_1^2Z_2\\nonumber \\\\\n& &+4Z_2^2e^{4U+2V_1}+4\\chi^2Z_2^2+8\\chi Z_1Z_2^2+16Z_1^2Z_2^2-4\\chi Z_2^3 -8Z_1Z_2^3\\nonumber \\\\\n& &\\left.+6kZ_2^2+Z_2^4+2e^{2V_2}\\left[e^{4U}(\\chi+2Z_1-Z_2)^2+2e^{4V_1}(2Z_1+Z_2)^2\\right]\\right].\n\\end{eqnarray}\nThe scalars $\\beta$, $\\Phi$ and $Z_3$ do not appear in the\npotential. It can also be checked that setting $\\beta=\\Phi=Z_3=0$ is\na consistent truncation. In fact, $\\beta$ never appears in any\nequations, so we can set it to zero. On the other hand, the\nYang-Mills equations, to be given later, demand that $\\Phi$ and\n$Z_3$ must be constant. Since we are interested in the flow\nsolutions interpolating between $AdS_2\\times \\Sigma_2$ and $AdS_4$\nvacua, and at supersymmetric $AdS_4$ critical points, both $\\Phi$\nand $Z_3$ vanish. We then choose $Z_3=\\Phi=0$. \n\\\\\n\\indent The kinetic terms for\nthe remaining scalars read\n\\begin{eqnarray}\n\\mathcal{L}_{\\textrm{kin}}&=&-6{U'}^2-2U'(2V_1'+V_2')-2V_1^{\\prime 2}-V'_1V'_2\\nonumber\\\\\n& &-\\frac{1}{4}\\left[3V_2^{\\prime 2}+e^{-2(2V_1+V_2)}{\\chi'}^2+4e^{-2(2U+V_1)}Z_1^{\\prime 2}+2e^{-2(2U+V_2)}Z_2^{\\prime 2}\\right].\n\\end{eqnarray}\nWe now redefine the scalars such that the kinetic terms are diagonal\n\\begin{equation}\n\\tilde{V}=2V_1+V_2,\\qquad \\tilde{U}_1=2U+V_1,\\qquad \\tilde{U}_2=2U+V_2\n\\end{equation}\nin terms of which we find\n\\begin{equation}\n\\mathcal{L}_{\\textrm{kin}}=-\\frac{1}{4}\\left(4\\tilde{U}_1^{\\prime 2}+2\\tilde{U}_2^{\\prime 2}+\\tilde{V}^{\\prime 2}+e^{-2\\tilde{V}}{\\chi'}^2+4e^{-2\\tilde{U}_1}Z_1^{\\prime 2}\n+2e^{-2\\tilde{U}_2}Z_2^{\\prime 2} \\right).\n\\end{equation}\nThese new scalars will also be useful in the analysis of the BPS equations below.\n\\\\\n\\indent The above scalar potential admits two supersymmetric $AdS_4$\nvacua with $N=1$ and $N=3$ supersymmetries\n\\cite{N010_truncation_Cassani}. At these vacua the symmetry is\nenhanced from $SO(2)$ to $SO(3)$. For convenience, before carry out\nthe analysis of the Yang-Mills and BPS equations, we review the\n$N=3$ and $N=1$ $AdS_4$ critical points in terms of the new scalars\ndefined above:\n\\begin{eqnarray}\nN=3&:&\\qquad \\tilde{V}=\\tilde{U}_1=\\tilde{U}_2=\\frac{1}{2}\\ln k,\\qquad  V_0=-12|k|^{-\\frac{3}{2}},\\qquad k>0, \\\\\nN=1&:&\\qquad \\tilde{U}_1=\\tilde{U}_2=\\ln 5+\\frac{1}{2}\\ln \\left[-\\frac{k}{15}\\right],\\qquad  \\tilde{V}=\\frac{1}{2}\\ln \\left[-\\frac{k}{15}\\right],\\nonumber \\\\\n\\qquad & &\\qquad  V_0=-12|k|^{-\\frac{3}{2}}\\sqrt{\\frac{3^7}{5^5}},\\qquad k<0\\, .\n\\end{eqnarray}\n$V_0$ is the cosmological constant related to the $AdS_4$ radius by\n\\begin{equation}\nL^2=-\\frac{3}{V_0}\\, .\n\\end{equation}\n\n\\subsection{The analysis of Yang-Mills equations}\nWe now solve the equations of motion for the gauge fields given in\n\\eqref{YM1}, \\eqref{YM2} and \\eqref{YM3}. We should emphasize that, in the reduction of\n\\cite{N010_truncation_Cassani}, the magnetic vectors $A^{M-}$ with\n$M=4,5,6$ do not appear in the reduction ansatz. These might arise\nfrom the reduction of the dual internal seven-dimensional metric.\nFurthermore, in this reduction, the two-form fields corresponding to\nthese magnetic vectors do not appear. \n\\\\\n\\indent Although the present analysis involves $A^{6+}$, we will truncate out the $A^{6-}$ in order to use the reduction ansatz of \\cite{N010_truncation_Cassani} to uplift the resulting solutions to eleven dimensions. This amounts to setting $e_6$ and $\\tilde{\\mathcal{A}}_t^6$ in \\eqref{AM_minus} to zero. It turns out that this truncation is consistent provided that the two-form fields are properly truncated. Therefore, we will set $e_6=\\tilde{\\mathcal{A}}_t^6=0$ in the following analysis. Note also that the vanishing of $A^{6-}$ does not mean\nthe covariant field strength $\\mathcal{H}^{6-}$ vanishes although the\nusual gauge field strength $\\mathcal{F}^{6-}$ vanishes. This is due to\nthe fact that $\\mathcal{H}^{6-}$ gets a contribution from the two-form\nfields. \n\\\\\n\\indent In order to consistently remove\n$A^{6-}$, we truncate the two-form fields to only $B^{18}$ and $B^{78}$.\nWith the symmetry of $AdS_2\\times \\Sigma_2$ background and a\nparticular choice of tensor gauge transformations\n\\begin{equation}\nB^{MN}\\rightarrow B^{MN}+d\\Xi^{MN},\n\\end{equation}\nwe will take the ansatz for the two-forms to be\n\\begin{equation}\nB^{78}=B(r)F(\\theta)d\\theta \\wedge d\\phi,\\qquad B^{18}=\\tilde{B}(r)F(\\theta)d\\theta \\wedge d\\phi\\, .\n\\end{equation}\n\\indent With the explicit form of the embedding tensor, we can compute the\ncovariant field strengths\n\\begin{eqnarray}\n\\mathcal{H}^{3+}&=&{\\mathcal{A}_t^3}'dr\\wedge dt+(p^3+4B)F(\\theta)d\\theta\\wedge d\\phi,\\nonumber\\\\\n\\mathcal{H}^{6+}&=&{\\mathcal{A}_t^6}'dr\\wedge dt+(p^6-4\\tilde{B})F(\\theta)d\\theta\\wedge d\\phi,\\nonumber\\\\\n\\mathcal{H}^{9+}&=&{\\mathcal{A}_t^9}'dr\\wedge dt+p^9F(\\theta)d\\theta\\wedge d\\phi,\\nonumber\\\\\n\\mathcal{H}^{3-}&=&\\tilde{\\mathcal{A}}^{3\\prime }_tdr\\wedge dt+(e_3-2\\sqrt{2}\\tilde{B})F(\\theta)d\\theta\\wedge d\\phi,\\nonumber\\\\\n\\mathcal{H}^{6-}&=&-6\\sqrt{2}kBF(\\theta)d\\theta\\wedge d\\phi,\\nonumber\\\\\n\\mathcal{H}^{9-}&=&\\tilde{\\mathcal{A}}^{9\\prime}_tdr\\wedge dt+(e_9-2\\sqrt{2}B)F(\\theta)d\\theta\\wedge d\\phi\\, .\n\\end{eqnarray}\nNote the non-vanishing covariant field strength $\\mathcal{H}^{6-}$,  as mentioned above, due to the contribution from the two-form fields despite $A^{6-}=0$.\n\\\\\n\\indent Equations arising from \\eqref{YM1} and \\eqref{YM2} are explicitly\ngiven in the appendix. They can be solved by imposing the following\nconditions\n\\begin{eqnarray}\nZ_3'&=&0,\\qquad \\Phi'=2Z_1Z_3'-2Z_3Z_1',\\nonumber \\\\\nB'F(\\theta)dr\\wedge d\\theta \\wedge d\\phi&=&\\sqrt{2}e^{-4(2U+V_1)}(3k*A^{9+}+*A^{6+}-\\sqrt{2}*A^{3-}),\\nonumber \\\\\n\\tilde{B}'F(\\theta)dr\\wedge d\\theta \\wedge d\\phi&=&4Z_1e^{-4(2U+V_1)}(3k*A^{9+}+*A^{6+}-\\sqrt{2}*A^{3-}).\\label{Con_YM1}\n\\end{eqnarray}\nThe first condition implies that $Z_3$ is constant. As mentioned\nabove, this allows to set $Z_3=0$. The second condition then\nrequires that $\\Phi$ is constant. We can also set $\\Phi=0$. Together\nwith $\\beta=0$, we are left with only six scalars $(U,V_1,V_2,\\chi,Z_1,Z_2)$ or equivalently\n$(\\tilde{U}_1,\\tilde{U}_2,\\tilde{V},\\chi,Z_1,Z_2)$.\n\\\\\n\\indent We move to the last two conditions in \\eqref{Con_YM1}. First\nof all, the $dt\\wedge dr\\wedge d\\theta$ component gives\n\\begin{equation}\n3kp^9+p^6-\\sqrt{2}e_3=0 \\label{twist1}\n\\end{equation}\nwhile the $dr\\wedge d\\theta\\wedge d\\phi$ component leads to\nfirst-order differential equations for $B$ and $\\tilde{B}$\n\\begin{eqnarray}\nB'&=&\\sqrt{2}e^{-4(2U+V_1)+2g-f}(3k\\mathcal{A}^9_t+\\mathcal{A}^6_t-\\sqrt{2}\\tilde{\\mathcal{A}}^3_t),\\label{2-form-eq1}\\\\\n\\tilde{B}'&=&-4Z_1e^{-4(2U+V_1)+2g-f}(3k\\mathcal{A}^9_t+\\mathcal{A}^6_t-\\sqrt{2}\\tilde{\\mathcal{A}}^3_t).\\label{2-form-eq2}\n\\end{eqnarray}\n\\indent After solving all of the Yang-Mills equations and Bianchi\nidentities, we now consider the duality equation for electric and\nmagnetic vector fields. These equations whose explicit form is given\nin the appendix lead to the relations between\n$(\\mathcal{A}^{M\\prime}_t,\\tilde{\\mathcal{A}}^{M\\prime}_t)$ and scalars. We\ncan accordingly express the former in terms of the latter. These\nrelations are given by\n\\begin{eqnarray}\n\\mathcal{A}^{3\\prime}_t&=&e^{f-2g-2(U+V_1)-3V_2}\\left[e^{4U+2V_2}\\left[e_3+\\sqrt{2}e_9Z_2-4BZ_2+\\chi(p^3+4B+\\sqrt{2}Z_2)\\right] \\right.\\nonumber \\\\& &+Z_2^2[2(e_3+p^3\\chi)+\\sqrt{2}Z_2(e_9+p^9\\chi)]-4Z_2B(3k-2\\chi Z_2+Z_2^2)\\nonumber \\\\\n& &\\left.-2\\sqrt{2}\\tilde{B}(e^{4U+2V_2}+2Z_2\\chi+2Z_2^2)+\\sqrt{2}p^6Z_2\\chi\\right],\\label{Ap1}\\\\\n{\\mathcal{A}_t^6}'&=&e^{f-2g-2(2U+V_1)-3V_2}\\left[(2\\sqrt{2}B-e_9-p^9\\chi)e^{8U+4V_2}-p^6Z_2^2\\chi \\right.\\nonumber \\\\\n& &-e^{4U+2V_2}Z_2[\\sqrt{2}e_3-4\\tilde{B}+2e_9Z_2+\\chi(\\sqrt{2}p^3+2p^9Z_2)]\\nonumber\\\\\n& &+4\\tilde{B}Z_2^2(\\chi+Z_2)-Z_2^3[\\sqrt{2}(e_3+p^3\\chi)+Z_2(e_9+p^9\\chi)]\\nonumber \\\\\n& &\\left.+4\\sqrt{2}BZ_2e^{4U+2V_2}(Z_2-\\chi)+2\\sqrt{2}BZ_2^2(3k-2\\chi Z_2+Z_2^2)\\right],\\\\\n{\\mathcal{A}_t^9}'&=&-e^{f-2g-2(2U+V_1)-3V_2}\\left[Z_2(\\sqrt{2}e_3-4\\tilde{B}+e_9Z_2)-2\\sqrt{2}B(3k-2\\chi Z_2 +Z_2^2) \\right.\\nonumber \\\\\n& &\\left. +\\chi(p^6-4\\tilde{B}+\\sqrt{2}Z_2+p^9Z_2^2)\\right],\\\\\n\\tilde{\\mathcal{A}}^{3\\prime}_t&=&\\frac{e^{f-2g-2V_1-V_2}}{Z_2}\\left[-e^{4V_1+2V_2}[\\sqrt{2}e^{4U+2V_2}p^9+Z_2(p^3+4B+\\sqrt{2}p^9Z_2)] \\right.\\nonumber \\\\\n& &+\\chi Z_2[e_3-2\\sqrt{2}\\tilde{B}+\\sqrt{2}e_9Z_2-4BZ_2+\\chi(p^3+4B+\\sqrt{2}p^9Z_2)]\\nonumber \\\\\n& &\\left.+\\chi e^{4U+2V_2}\\left[\\sqrt{2}(e_9+p^9\\chi)-4B\\right]\\right],\\\\\n\\tilde{\\mathcal{A}}^{9\\prime}_t&=&\\frac{e^{f-2g-2V_1-V_2}}{Z_2^2} \\left[e^{4(U+V_1+V_2)}p^9-e^{4U+2V_1}\\chi(e_9-2\\sqrt{2}B+p^9\\chi) \\right.\\nonumber \\\\\n& &-\\chi Z_2[\\sqrt{2}e_3-4\\tilde{B}+4\\sqrt{2}B(\\chi-Z_2)+2e_9Z_2+\\chi(\\sqrt{2}p^3+2p^9Z_2)]\\nonumber \\\\\n& &\\left.\n+e^{4V_1+2V_2}Z_2(\\sqrt{2}p^3+4\\sqrt{2}B+2p^9Z_2)\\right].\\label{Ap6}\n\\end{eqnarray}\nIt turns out that only $\\mathcal{A}_t^{9}$, $\\mathcal{A}_t^{6}$ and\n$\\tilde{\\mathcal{A}}_t^{3}$ appear in other equations while the remaining\nones only appear through their derivatives. Therefore, these fields\ncan be integrated out.\n\n\\subsection{BPS equations for $SO(2)$ invariant scalars}\nWe now use the ansatz for all the fields given in the previous\nsection to set up the BPS equations for finding supersymmetric\nsolutions. We will use Majorana representation for the gamma\nmatrices in which all $\\gamma_\\mu$ are real, and\n\\begin{equation}\n\\gamma_5=i\\gamma_{\\hat{0}}\\gamma_{\\hat{r}}\\gamma_{\\hat{\\theta}}\\gamma_{\\hat{\\phi}}\n\\end{equation}\nis purely imaginary. We then have, for example,\n\\begin{equation}\n\\epsilon^i=\\frac{1}{2}(1+\\gamma_5)\\epsilon^i_M,\\qquad \\epsilon_i=\\frac{1}{2}(1-\\gamma_5)\\epsilon^i_M\n\\end{equation}\nwith $\\epsilon^i_M$ being four-component Majorana spinors. It follows that $\\epsilon_i=(\\epsilon^i)^*$.\n\\\\\n\\indent We first consider the gravitino transformations. As in other\nholographic solutions involving twisted compactifications of the dual\nSCFTs, the strategy is to use the gauge connection to cancel the\nspin connection on $\\Sigma_2$. Equations from\n$\\delta\\psi^i_{\\hat{\\theta}}=0$ and $\\delta\\psi^i_{\\hat{\\phi}}=0$\nthen reduce to the same equation. The gauge connection enters the\ncovariant derivative of $\\epsilon^i$ through the composite\nconnection ${Q_j}^i$. With the $SO(2)$ singlet scalars, we find that\n${Q_j}^i$ takes the form of\n\\begin{equation}\n{Q_j}^i=\\frac{1}{2}\\hat{A}\\left(\n                     \\begin{array}{cccc}\n                       0 & 1 & 0 & 0 \\\\\n                       -1 & 0 & 0 & 0 \\\\\n                       0 & 0 & 0  & 0\\\\\n                        0 & 0 & 0  & 0\\\\\n                     \\end{array}\n                   \\right)\n\\end{equation}\nwhere $\\hat{A}$ is given by\n\\begin{equation}\n\\hat{A}=\\sqrt{2}e^{-2(2U+V_1)}(3kA^{9+}+A^{6+}-\\sqrt{2}A^{3-}-4e^{4U+2V_1}A^{9+}).\n\\end{equation}\nFrom the form of ${Q_i}^j$, we can see that supersymmetry\ncorresponding to $\\epsilon^{3,4}$ is broken for spherical and\nhyperbolic $\\Sigma_2$ since we cannot cancel the spin connections\nalong $\\epsilon^{3,4}$. The $N=4$ supersymmetry is then broken to\n$N=2$.\n\\\\\n\\indent After using the condition \\eqref{twist1} in the\n${Q_{\\hat{\\phi}i}}^j$ components, the twist is achieved by imposing\nthe projection\n\\begin{equation}\n\\gamma^{\\hat{\\theta}\\hat{\\phi}}\\epsilon^{\\hat{i}}={\\epsilon^{\\hat{i}}}_{\\hat{j}}\\epsilon^{\\hat{j}}\\label{projector1}\n\\end{equation}\nprovided that we impose the following twist condition\n\\begin{equation}\n2\\sqrt{2}\\kappa p^9=1\\, .\n\\end{equation}\nIndices $\\hat{i},\\hat{j}=1,2$ denote the Killing spinors\ncorresponding to the unbroken supersymmetry. From equation\n\\eqref{projector1}, the chirality condition on $\\epsilon^{\\hat{i}}$\nimplies that\n\\begin{equation}\n\\gamma^{\\hat{0}\\hat{r}}\\epsilon^{\\hat{i}}=-i{\\epsilon^{\\hat{i}}}_{\\hat{j}}\\epsilon^{\\hat{j}}\\, .\\label{projector2}\n\\end{equation}\nWith these projections, we can write the $\\delta\n\\psi^i_{\\hat{\\theta}}=0$ equation, which is the same as $\\delta\n\\psi^i_{\\hat{\\phi}}$ equation, as\n\\begin{equation}\ng'\\gamma_{\\hat{r}}\\epsilon^{\\hat{i}}-\\frac{2}{3}A_1^{\\hat{i}\\hat{j}}\\epsilon_{\\hat{j}}+\\frac{i}{2}(\\mathcal{V}_\\alpha)^*{\\mathcal{V}_M}^{\\hat{i}\\hat{j}}(i\\mathcal{H}^{M\\alpha}_{\\hat{0}\\hat{r}}-\\mathcal{H}^{M\\alpha}_{\\hat{\\theta}\\hat{\\phi}}){\\epsilon_{\\hat{j}}}^{\\hat{k}}\\epsilon_{\\hat{k}}=0\n\\end{equation}\nwhere we have multiplied the resulting equation by $\\gamma^{\\hat{\\theta}}$. We further impose the projector\n\\begin{equation}\n\\gamma_{\\hat{r}}\\epsilon^{\\hat{i}}=e^{i\\Lambda}\\delta^{\\hat{i}\\hat{j}}\\epsilon_{\\hat{j}}\\label{projector3}\n\\end{equation}\nin which $e^{i\\Lambda}$ is an $r$-dependent phase. By equation \\eqref{projector2}, this projector implies\n\\begin{equation}\n\\gamma_{\\hat{0}}\\epsilon^{\\hat{i}}=ie^{i\\Lambda}\\epsilon^{\\hat{i}\\hat{j}}\\epsilon_{\\hat{j}}\\, .\\label{projector4}\n\\end{equation}\nIt should be noted that there are only two independent projectors\ngiven in \\eqref{projector1} and \\eqref{projector3}. Therefore, the\nentire flows preserve $\\frac{1}{4}$ supersymmetry. On the other\nhand, the $AdS_2\\times \\Sigma_2$ vacua is $\\frac{1}{2}$\nsupersymmetric since the $\\gamma_{\\hat{r}}$ projection is not needed\nfor constant scalars.\n\\\\\n\\indent As a next step, we introduce the ``superpotential'' $\\mathcal{W}$\nand ``central charge'' $\\mathcal{Z}$ defined respectively by the\neigenvalues of\n\\begin{equation}\n\\frac{2}{3}A^{\\hat{i}\\hat{j}}_1=\\mathcal{W}_{\\hat{i}}\\delta^{\\hat{i}\\hat{j}}\n\\end{equation}\nand\n\\begin{equation}\n-\\frac{i}{2}(\\mathcal{V}_\\alpha)^*{\\mathcal{V}_M}^{\\hat{i}\\hat{j}}(i\\mathcal{H}^{M\\alpha}_{\\hat{0}\\hat{r}}\n-\\mathcal{H}^{M\\alpha}_{\\hat{\\theta}\\hat{\\phi}}){\\epsilon_{\\hat{j}}}^{\\hat{k}}=\\mathcal{Z}_{\\hat{i}}\\delta^{\\hat{i}\\hat{k}}\\,\n.\n\\end{equation}\nIt should be emphasized that no summation is implied in the above two equations.\n\\\\\n\\indent With all these, we obtain the BPS equation from $\\delta\\psi^{\\hat{i}}_{\\hat{\\theta}}=0$ equation\n\\begin{equation}\ne^{i\\Lambda}g'-\\mathcal{W}_i-\\mathcal{Z}_i=0\n\\end{equation}\nwhich gives\n\\begin{equation}\ng'=|\\mathcal{W}_i+\\mathcal{Z}_i|\\qquad \\textrm{and}\\qquad e^{i\\Lambda}=\\frac{\\mathcal{W}_i+\\mathcal{Z}_i}{|\\mathcal{W}_i+\\mathcal{Z}_i|}\\, .\n\\end{equation}\n\\\\\n\\indent Using all of these results, we find that equation $\\delta\\psi^{\\hat{i}}_{\\hat{0}}=0$ gives\n\\begin{equation}\ne^{i\\Lambda}(f'+i\\hat{A}_te^{-f})-\\mathcal{W}_i+\\mathcal{Z}_i=0\\, .\n\\end{equation}\nTaking the real and imaginary parts leads to the following BPS equations\n\\begin{equation}\nf'=\\textrm{Re}[e^{-i\\Lambda}(\\mathcal{W}_i-\\mathcal{Z}_i)]\\label{f_prime}\n\\end{equation}\nand\n\\begin{equation}\n\\hat{A}_t=e^f\\textrm{Im}[e^{-i\\Lambda}(\\mathcal{W}_i-\\mathcal{Z}_i)].\\label{At_constraint}\n\\end{equation}\nWe now come to $\\delta\\psi^{\\hat{i}}_{\\hat{r}}=0$ equation which\ngives the $r$-dependence of the Killing spinors. When combined with\n$\\delta\\psi^{\\hat{i}}_{\\hat{0}}=0$ equation, this equation reads\n\\begin{equation}\n2\\epsilon^{\\hat{i}\\prime}-f'-i\\hat{A}_te^{-f}\\epsilon^{\\hat{i}}=0\n\\end{equation}\nwhich can be solved by\n\\begin{equation}\n\\epsilon^{\\hat{i}}=e^{\\frac{f}{2}+\\frac{i}{2}\\int \\hat{A}_te^{-f}dr}\\tilde{\\epsilon}^{\\hat{i}}.\n\\end{equation}\n$\\tilde{\\epsilon}^{\\hat{i}}$ are constant spinors satisfying the projections\n\\begin{equation}\n\\gamma_{\\hat{r}}\\tilde{\\epsilon}^{\\hat{i}}=\\delta^{\\hat{i}\\hat{j}}\\tilde{\\epsilon}_{\\hat{j}},\\qquad\n\\gamma_{\\hat{\\theta}\\hat{\\phi}}\\tilde{\\epsilon}^{\\hat{i}}={\\epsilon^{\\hat{i}}}_{\\hat{j}}\\tilde{\\epsilon}^{\\hat{j}}\\,\n.\n\\end{equation}\n\\indent Using the $\\gamma_{\\hat{r}}$ projector, we obtain the\nfollowing BPS equations from $\\delta\\chi^i$ and $\\delta\\lambda^i_a$\n\\begin{eqnarray}\n-e^{i\\Lambda}\\epsilon^{\\alpha\\beta}\\mathcal{V}_\\alpha \\mathcal{V}'_\\beta\\delta_{\\hat{i}\\hat{j}}-\\frac{4i}{3}A^{\\hat{j}\\hat{i}}_2\n+i\\mathcal{V}_\\alpha{\\mathcal{V}_M}^{\\hat{i}\\hat{k}}\\epsilon^{\\hat{k}\\hat{j}}(i\\mathcal{H}^{M\\alpha}_{\\hat{0}\\hat{r}}+\\mathcal{H}^{M\\alpha}_{\\hat{\\theta}\\hat{\\phi}})&=&0,\\\\\n{\\mathcal{V}_a}^M{\\mathcal{V}_M}^{ij\\prime}e^{-i\\Lambda}+\\frac{1}{4}\\mathcal{V}_\\alpha \\mathcal{V}_{Ma}(\\mathcal{H}^{M\\alpha}_{\\hat{0}\\hat{r}}+i\\mathcal{H}^{M\\alpha}_{\\hat{\\theta}\\hat{\\phi}})\\delta^i_{\\hat{i}}\\delta^j_{\\hat{j}}\\epsilon^{\\hat{i}\\hat{j}}\n+{A_{2aj}}^i&=&0\\, .\n\\end{eqnarray}\nNote that there are four equations from $\\delta\\lambda^i_a$ for each\nvalue of $a=1,2,3$, but $\\delta\\lambda^{i=3,4}_a$ do not get any\ncontribution from the gauge fields. However, the scalars appearing\nin these equations cannot be consistently set to zero since\n${A_{2aj}}^i$ is not diagonal in $ij$ indices.\n\\\\\n\\indent It should be pointed out that the $N=3$ supersymmetric\n$AdS_4$ vacuum corresponds to the Killing spinors $\\epsilon^{2,3,4}$\nwhile $\\epsilon^1$ is the Killing spinor of the $N=1$ $AdS_4$\ncritical point. In the next section, we will look for possible\n$AdS_2\\times \\Sigma_2$ solutions to the above BPS equations. As\nmentioned before, in the twist given above, the supersymmetry\ncorresponding to $\\epsilon^{3,4}$ is broken. Therefore, the\nresulting $AdS_2\\times \\Sigma_2$ solutions will preserve only two\nsupercharges or half of the $N=1$ supersymmetry corresponding to\neither $\\epsilon^1$ or $\\epsilon^2$. We will analyze these two cases\nseparately.\n\n\\section{Supersymmetric $AdS_2\\times \\Sigma_2$ solutions}\\label{AdS2_solution}\nIn this section, we look for the $AdS_2\\times \\Sigma_2$ fixed points\nof the above BPS flow equations with constant scalars. These\nsolutions should correspond to IR fixed points of the RG flows from\ntwisted compactifications of the dual $N=3$ and $N=1$ SCFTs in three\ndimensions. They also describe near horizon geometries of BPS black\nholes arising from M2-branes wrapped on $\\Sigma_2$. Before giving\nthe solutions, we first discuss the conditions for obtaining the\n$AdS_2$ fixed points.\n\\\\\n\\indent At the $AdS_2\\times \\Sigma_2$ geometries, the scalars are\nconstant, and we can choose the gauge in which $A_t^{M\\alpha}\\sim\n0$. Furthermore, the warped factor $g(r)$ is required to be\nconstant, $g'(r)=0$. Let $r_h$ be the position of the horizon, we\ncan summarized the conditions for $AdS_2\\times \\Sigma_2$ solutions\nand their properties as follow\n\\begin{eqnarray}\nf(r_h)=\\frac{r_h}{L_{AdS_2}},\\qquad e^{g(r_h)}&=&L_{\\Sigma_2},\\qquad \\textrm{Im}[e^{-i\\Lambda}(\\mathcal{W}_i-\\mathcal{Z}_i)]=0,\\nonumber \\\\\n|\\mathcal{W}_i+\\mathcal{Z}_i|&=&0,\\qquad \\frac{4}{3}A_2^{\\hat{i}\\hat{j}}=\\mathcal{V}_\\alpha {\\mathcal{V}_M}^{\\hat{i}\\hat{k}}\\epsilon^{\\hat{k}\\hat{j}}(i\\mathcal{H}^{M\\alpha}_{\\hat{0}\\hat{r}}+\\mathcal{H}^{M\\alpha}_{\\hat{\\theta}\\hat{\\phi}}),\\nonumber \\\\\n\\frac{i}{4}\\mathcal{V}_\\alpha \\mathcal{V}_{Ma}(-i\\hat{H}^{M\\alpha}_{\\hat{0}\\hat{r}}+\\mathcal{H}^{M\\alpha}_{\\hat{\\theta}\\hat{\\phi}})\\epsilon^{\\hat{i}\\hat{j}}&=&-{A_{2a\\hat{j}}}^{\\hat{i}} \\qquad {A_{2a\\tilde{j}}}^{\\hat{i}}=0,\\quad \\tilde{j}=3,4\n\\end{eqnarray}\nwhere $L_{AdS_2}$ and $L_{\\Sigma_2}$ are respectively the radii of\n$AdS_2$ and $\\Sigma_2$. These conditions can be viewed as attractor\nequations for the scalars at the black hole horizon.\n\n\\subsection{Solutions in the $N=3$ case}\nWe begin with the $N=3$ case. The $AdS_2\\times \\Sigma_2$ solutions\nwill describe the fixed points of the RG flows from $N=3$ SCFTs dual\nto the $N^{010}$ compactification of eleven-dimensional supergravity\nto supersymmetric CFT$_1$'s dual to the $AdS_2\\times \\Sigma_2$\ngeometries. These flows are examples of the twisted\ncompactifications of the $N=3$ SCFT on $\\Sigma_2$.\n\\\\\n\\indent In this case, the superpotential and central charge are\ngiven in term of the redefined scalars\n$(\\tilde{U}_1,\\tilde{U}_2,\\tilde{V})$ by\n\\begin{eqnarray}\n\\mathcal{W}_2&=&\\frac{1}{2}e^{-\\frac{1}{2}(4\\tilde{U}_1+2\\tilde{U}_2+\\tilde{V})}\\left[e^{2\\tilde{U}_2}+4e^{\\tilde{U}_1+\\tilde{U}_2}-2e^{\\tilde{U}_2+\\tilde{V}}+4e^{\\tilde{U}_1+\\tilde{V}}\n\\right.\\nonumber \\\\\n& &  -3k+2iZ_2e^{\\tilde{U}_2}+4iZ_2e^{\\tilde{U}_1}-4iZ_1(e^{\\tilde{U}_2}+e^{\\tilde{V}}+iZ_2)\\nonumber \\\\\n& &\\left.-2iZ_2e^{\\tilde{V}}-Z_2^2+2\\chi(2ie^{\\tilde{U}_1}-ie^{\\tilde{U}_2}+2Z_1+Z_2) \\right],\\\\\n\\mathcal{Z}_2&=&\\frac{1}{4}e^{-\\frac{1}{2}(4g+2\\tilde{U}_2+\\tilde{V})}\\left[2e_3e^{\\tilde{U}_2}-\\sqrt{2}ie_9e^{2\\tilde{U}_2}+2ie_3\\chi+2p^3\\chi e^{\\tilde{U}_2} \\right.\\nonumber\\\\\n& &-\\sqrt{2}ip^9\\chi(e^{2\\tilde{U}_2}+3k)-4\\sqrt{2}\\tilde{B}[e^{\\tilde{U}_2}+e^{\\tilde{V}}+i(\\chi+Z_2)]\\nonumber\\\\\n& &+2ie_3Z_2+2\\sqrt{2}e_9Z_2e^{\\tilde{U}_2}+2ip^3\\chi Z_2+2\\sqrt{2}p^9\\chi Z_2e^{\\tilde{U}_2}\\nonumber\\\\\n& &+\\sqrt{2}i(e_9+p^9\\chi)Z_2^2+4iB(e^{2\\tilde{U}_2}-2e^{\\tilde{U}_2+\\tilde{V}}-3k)\\nonumber \\\\\n& &+4B[2\\chi (e^{\\tilde{U}_2}+iZ_2)+Z_2(e^{\\tilde{V}}-2e^{\\tilde{U}_2}-iZ_2)]\\nonumber \\\\\n& &+e^{\\tilde{V}}(2e_3-3\\sqrt{2}p^9-\\sqrt{2}p^9e^{2\\tilde{U}_2}+2p^3Z_2+\\sqrt{2}p^9Z_2^2)\\nonumber \\\\\n& &\\left. -2ie^{\\tilde{U}_2+\\tilde{V}}(p^3+\\sqrt{2}p^9Z_2)\\right]\n\\end{eqnarray}\nin which the subscript $2$ on $\\mathcal{W}_2$ and $\\mathcal{Z}_2$ refers to the\nsuperpotential and central charge associated to the Killing spinor\n$\\epsilon^2$.\n\\\\\n\\indent The BPS equations are given by\n\\begin{eqnarray}\nf'&=&\\textrm{Re}[e^{-i\\Lambda}(\\mathcal{W}_2-\\mathcal{Z}_2)],\\qquad e^{i\\Lambda}=\\frac{\\mathcal{W}_2+\\mathcal{Z}_2}{|\\mathcal{W}_2+\\mathcal{Z}_2|},\\\\\ng'&=&|\\mathcal{W}_2+\\mathcal{Z}_2|,\n\\end{eqnarray}\n\\begin{eqnarray}\ne^{i\\Lambda}\\tilde{V}'-ie^{-\\tilde{V}+i\\Lambda}\\chi'&=&\\frac{1}{2}\n\\left[e^{-\\frac{\\tilde{V}}{2}-\\tilde{U}_2-2\\tilde{U}_1}\\left[2e^{\\tilde{U}_2}+8e^{2\\tilde{U}_1}\n-6k+Z_2(8Z_1-2Z_2)\n\\right]\\right.\\nonumber \\\\\n &\n&-e^{-2g-2\\tilde{U}_1+\\frac{\\tilde{V}}{2}}\\left[4e^{2g}+2e^{2\\tilde{U}_1}(p^3+4B+\\sqrt{2}p^9Z_2)\\right]\\nonumber\n\\\\\n& & \\left.\n+4\\chi(2Z_1+Z_2)e^{-\\frac{\\tilde{V}}{2}-\\tilde{U}_2-2\\tilde{U}_1}+\\sqrt{2}e_9e^{\\tilde{U}_2-2g-\\frac{\\tilde{V}}{2}}\\right]\n \\nonumber \\\\\n& &+\\frac{1}{2}e^{-2g-\\tilde{U}_2-\\frac{\\tilde{V}}{2}}\\left[\\sqrt{2}Z_2(4\\tilde{B}-e_9Z_2)-2e_3(\\chi+Z_2)+4\\sqrt{2}\\chi \\tilde{B} \\right.\\nonumber\\\\\n& &-4B(e^{2\\tilde{U}_2}-3k+2\\chi Z_2-Z_2^2)\n+\\sqrt{2}p^9\\chi(e^{\\tilde{U}_2}+3k)\\nonumber \\\\\n& &\\left.-Z_2\\chi(2p^3+\\sqrt{2}p^9Z_2) \\right]\\nonumber \\\\\n&\n&-\\frac{i}{2}e^{-\\tilde{U}_2-\\frac{\\tilde{V}}{2}}\\left[4e^{\\tilde{U}_2-2\\tilde{U}_1}(Z_2-2Z_1-\\chi)\n-4e^{\\tilde{V}-2\\tilde{U}_1}(2Z_1+Z_2) \\right.\\nonumber \\\\\n&\n&-2e^{\\tilde{U}_2-2g}\\left[Z_2(\\sqrt{2}e_9-4B-2\\sqrt{2}\\tilde{B})+\\chi\n(p^3+4B+\\sqrt{2}p^9Z_2)\n\\right]\\nonumber \\\\\n& &+\ne^{\\tilde{V}-2g}\\left[2e_3-4\\sqrt{2}\\tilde{B}-\\sqrt{2}p^9(3k+e^{2\\tilde{U}_2})\n-4\\sqrt{2}\\tilde{B}\\right.\\nonumber\n\\\\\n&\n&\\left.\\left.+Z_2(2p^3+8B+\\sqrt{2}p^9Z_2)\\right]-2e^{\\tilde{U}_2-2g}e_3\\right],\n\\end{eqnarray}\n\\begin{eqnarray}\ne^{-i\\Lambda}\\tilde{U}_2'+ie^{-\\tilde{U}_2-i\\Lambda}Z_2'&=&\\frac{1}{2}e^{-2g-\\tilde{U}_2\n-2\\tilde{U}_1-\\frac{\\tilde{V}}{2}}\\left[2e^{2(g+\\tilde{U}_2)}+\\sqrt{2}ie_9e^{2(\\tilde{U}_1+\\tilde{U}_2)}\n+6ke^{2g} \\right.\\nonumber \\\\\n& &-2ie_3\\chi e^{2\\tilde{U}_1}+\\sqrt{2}ip^9\\chi e^{2(\\tilde{U}_1+\\tilde{U}_2)}\n+3\\sqrt{2}ikp^9\\chi e^{2\\tilde{U}_1}\\nonumber \\\\\n& &+8iZ_2e^{2g+\\tilde{U}_1}-2ie_3Z_2e^{2\\tilde{U_1}}-4\\chi Z_2e^{2g}-2ip^3\\chi Z_2e^{2\\tilde{U}_1}\\nonumber \\\\\n&\n&-8Z_1Z_2e^{2g}+2Z_2^2e^{2g}-\\sqrt{2}ie_9Z_2^2e^{2\\tilde{U}_1}-8\\chi\nZ_1 e^{2g}\n\\nonumber \\\\\n&\n&-4iBe^{2\\tilde{U}_1}\\left[e^{2\\tilde{U}_2}-3k+Z_2(2\\chi-Z_2-2ie^{\\tilde{V}})\\right]+8i\\chi\ne^{2g+\\tilde{U}_1}\\nonumber \\\\\n&\n&+4\\sqrt{2}\\tilde{B}e^{2\\tilde{U}_1}(e^{\\tilde{V}}+i\\chi+iZ_2)-\\sqrt{2}ip^9\\chi\nZ_2^2e^{2\\tilde{U}_1}\n\\nonumber \\\\\n& &\n-4ie^{2g+\\tilde{V}}(2Z_1+Z_2)-Z_2e^{2\\tilde{U}_1+\\tilde{V}}(2p^3+\\sqrt{2}p^9Z_2)\n\\nonumber \\\\\n&\n&\\left.-e^{\\tilde{U}_1+\\tilde{V}}\\left[8e^{2g}+e^{\\tilde{U}_1}\\left[2e_3-\\sqrt{2}p^9(e^{2\\tilde{U}_2}+3k)\\right]\\right]\n\\right],\n\\end{eqnarray}\n\\begin{eqnarray}\ne^{-i\\Lambda}\\tilde{U}_1'-ie^{-\\tilde{U}_1-i\\Lambda}Z_1'&=&e^{-\\tilde{U}_2-2\\tilde{U}_1-\\frac{\\tilde{V}}{2}}\\left[2e^{\\tilde{U}_2+\\tilde{V}}-e^{2\\tilde{U}_2}-2e^{\\tilde{U}_1}(e^{\\tilde{U}_2}+e^{\\tilde{V}})+3k \\right.\\nonumber \\\\\n& &-4iZ_1(e^{\\tilde{U}_2}+e^{\\tilde{V}}-iZ_2)+2i\\chi (e^{\\tilde{U}_1}-e^{\\tilde{U}_2}+2iZ_1+iZ_2)\\nonumber \\\\\n& &\\left.+2iZ_2(e^{\\tilde{U}_2}+e^{\\tilde{U}_1}-e^{\\tilde{V}})+Z_2^2\n\\right]\n\\end{eqnarray}\nwhere we have used the relation \\eqref{twist1} to express $p^6$ in terms of $p^9$ and $e_3$.\n\\\\\n\\indent To obtain the complete flow solutions, we have to solve\nthese equations together with the two-form equations\n\\eqref{2-form-eq1}, \\eqref{2-form-eq2} and the equations for the\ngauge fields \\eqref{Ap1} to \\eqref{Ap6} as well as the algebraic\nconstraint given by equation \\eqref{At_constraint}. These equations\nare very complicated even with the numerical technique not to\nmention the analytic solutions. In what follow, we will present only\nthe $AdS_2\\times\\Sigma_2$ solutions and will not give the numerical\nflow solutions which may be obtained by suitable boundary\nconditions. In principle, the horizon is characterized by the values\nof the scalars as functions of the electric and magnetic charges.\nHowever, due to the complexity of the BPS equations, it is more\nconvenient to solve the horizon conditions for the charges in terms\nof the scalar fields although inverting the solutions to express the\nscalars in terms of the charges is desirable.\n\\\\\n\\indent In the present case, although it is straightforward to solve\nthe above equations for $(B,\\tilde{B},\\chi, Z_1,p^9,p^3,e_3,e_9)$ in\nterms of $(\\tilde{U}_1, \\tilde{U}_2, \\tilde{V},Z_2)$, the resulting\nexpressions turn out to be cumbersome and not very illuminating.\nAccordingly, we refrain from giving the general result here but\ninstead present some solutions with specific values of the\nparameters. These are obtained from truncating the full result and\nrepresent some examples of $AdS_2\\times \\Sigma_2$ geometries within the solution space.\n\\\\\n\\indent Examples of $AdS_2\\times \\Sigma_2$ solutions are as\nfollow:\n\\begin{itemize}\n\\item We begin with a simple solution with vanishing pseudoscalars.\nIn the M-theory point of view, only scalars coming from the eleven-dimensional metric are turned on.\nThe solution is given by\n\\begin{eqnarray}\nk&=&\\frac{1}{5},\\qquad \\chi=Z_1=Z_2=0,\\qquad e_9=0,\\qquad \\tilde{V}=\\frac{1}{2}\\ln \\left[\\frac{27}{5}\\right],\\nonumber \\\\\n\\tilde{U}_1&=&\\frac{1}{2}\\ln \\left[\\frac{27}{80}\\right],\\qquad\n\\tilde{U}_2=-\\frac{1}{2}\\ln \\left[\\frac{5}{3}\\right],\\qquad\n\\tilde{B}=\\frac{1}{20}(5\\sqrt{2}e_3-27p^9),\\nonumber \\\\\ng&=&\\frac{1}{2}\\ln\\left[-\\frac{81}{80}\\sqrt{\\frac{3}{10}}\\kappa\np^9\\right],\\qquad B=-\\frac{p^3}{4},\\qquad\nL_{AdS_2}=\\frac{3^{\\frac{9}{4}}}{32(5)^{\\frac{3}{4}}}\\, .\n\\end{eqnarray}\nIt is clearly seen that only the hyperbolic horizon ($\\kappa=-1$) is possible otherwise\n$g(r_h)$ will become complex. Therefore, we find that this is an $AdS_2\\times H^2$ solution.\n\\item We next consider a solution with scalars and pseudoscalars turned on.\nIn the eleven-dimensional context, the solution involves scalar fields from both the metric\nand the four-form field. This solution is characterized by\n\\begin{eqnarray}\nk&=&1,\\qquad Z_1=Z_2=\\tilde{U}=0,\\qquad \\tilde{U}=\\tilde{V}=\\ln \\left[\\frac{12}{7}\\right],\\nonumber \\\\\np^3&=&\\frac{41e_9+220p^9}{41\\sqrt{2}},\\qquad\nB=-\\frac{41e_9+136p^9}{164\\sqrt{2}},\\qquad\n\\tilde{B}=\\frac{e_3}{2\\sqrt{2}}-\\frac{111}{41}p^9,\\nonumber \\\\\n\\chi&=&-\\frac{1}{7},\\qquad\ng=\\frac{1}{2}\\ln\\left[-2^{\\frac{5}{2}}\\kappa p^9\\sqrt{\n\\frac{21}{41}}\\right],\\qquad L_{AdS_2}=\\frac{\\sqrt{21}}{19}\\, .\n\\end{eqnarray}\nThis solution is also $AdS_2\\times H^2$.\n\\item As a final example, we consider a solution with more scalars turned on\nand hence more general than the previous two solutions. This solution is given by\n\\begin{eqnarray}\nZ_1&=&0,\\qquad Z_2=-\\frac{2\\sqrt{k}}{7},\\qquad \\chi=-\\frac{\\sqrt{k}}{7},\\qquad \\tilde{U}_1=\\tilde{U}_2\n=\\frac{1}{2}\\ln k,  \\nonumber \\\\\np^3&=&\\frac{128,447k-104,895}{4,116\\sqrt{2k}}p^9,\\qquad e_9=\\frac{32,723k-13,923}{4,116\\sqrt{2k}}p^9,\n \\nonumber \\\\\n\\tilde{B}&=&\\frac{e_3}{2\\sqrt{2}}+\\frac{567-667k}{98}p^9,\\qquad\ng=\\frac{1}{2}\\ln \\left[\\frac{21(1-k)\\sqrt{k}\\kappa\np^9}{2\\sqrt{2}}\\right],\\nonumber \\\\\n\\tilde{V}&=&\\ln (2\\sqrt{k}),\\qquad\nB=-25p^9\\left[\\frac{3,809k-2,961}{16,464\\sqrt{2k}}\\right], \\qquad\nL_{AdS_2}=\\frac{k^{\\frac{3}{4}}}{3\\sqrt{2}}\\, .\\nonumber \\\\\n& &\n\\end{eqnarray}\nIn this case, the flux parameter $k$ is not fixed, and there are two\ntypes of solutions, $AdS_2\\times S^2$ and $AdS_2\\times H^2$,\ndepending on the value of $k$. For $k>1$, we have an $AdS_2\\times\nH^2$ solution with $\\kappa=-1$ while the solution with $k<1$ is\n$AdS_2\\times S^2$ for which $\\kappa=1$.\n\\end{itemize}\n\n\\subsection{Solutions in the $N=1$ case}\nWe now repeat a similar analysis for the $N=1$ case in which the\n$N=1$ $AdS_4$ vacuum arises from the squashed $N^{010}$ manifold.\nThis critical point exists only for $k<0$, and the $AdS_2\\times\n\\Sigma_2$ solutions would be IR fixed points of the twisted\ncompactifications of the dual $N=1$ SCFT. The superpotential and\ncentral charge are given by\n\\begin{eqnarray}\n\\mathcal{W}_1&=&\\frac{1}{2}e^{-\\tilde{U}_2-2\\tilde{U}_1-\\frac{\\tilde{V}}{2}}\\left[e^{2\\tilde{U}_2}\n-4e^{\\tilde{U}_1+\\tilde{U}_2}-2e^{\\tilde{V}}(e^{\\tilde{U}_2}+2e^{\\tilde{U}_1}) +4Z_1(Z_2-ie^{\\tilde{U}_2}-ie^{\\tilde{V}})\\right. \\nonumber \\\\\n&\n&\\left.-3k+iZ_2(2e^{\\tilde{U}_2}-4e^{\\tilde{U}_1}-2e^{\\tilde{V}}+iZ_2)+2\\chi\n(2Z_1+Z_2-ie^{\\tilde{U}_2}-2ie^{\\tilde{U}_1})\\right],\\nonumber \\\\\n& &\\\\\n\\mathcal{Z}_1&=&\\frac{1}{4}e^{-2g-\\tilde{U}_2-\\frac{\\tilde{V}}{2}}\\left[2e_3(e^{\\tilde{U}_2}+i\\chi)-\\sqrt{2}ie_9e^{2\\tilde{U}_2}+2p^3\\chi e^{\\tilde{U}_2}-3\\sqrt{2}ikp^9\\chi \\right.\\nonumber \\\\\n& &-\\sqrt{2}ip^9 \\chi e^{2\\tilde{U}_2}-4\\sqrt{2}\\tilde{B}(e^{\\tilde{U}_2}+e^{\\tilde{V}+i\\chi+i Z_2})+2ie_3Z_2\\nonumber \\\\\n& &+2\\sqrt{2}e_9Z_2e^{\\tilde{U}_2}+2ip^3\\chi Z_2+2\\sqrt{2}p^9\\chi Z_2+\\sqrt{2}ie_9Z_2^2\\nonumber \\\\\n& &+\\sqrt{2}ip^9\\chi Z_2^2+4B[2\\chi(e^{\\tilde{U}_2}+iZ_2)+i(e^{2\\tilde{U}_2}-2e^{\\tilde{U}_2+\\tilde{V}}-3k)]\\nonumber \\\\\n& &+4BZ_2(2e^{\\tilde{V}}-2e^{\\tilde{U}_2}-iZ_2)-2ie^{\\tilde{U}_2+\\tilde{V}}(p^3+\\sqrt{2}p^9Z_2)\\nonumber \\\\\n& &\\left. +e^{\\tilde{V}}(2e_3-6\\sqrt{2}p^9-\\sqrt{2}p^9e^{2\\tilde{U}_2}+2p^3Z_2+\\sqrt{2}p^9Z_2^2)\\right].\n\\end{eqnarray}\n\\indent The procedure is essentially the same, so we will just\npresent the result of $AdS_2\\times \\Sigma_2$ solutions and leave the\nexplicit form of the corresponding BPS equations to the appendix. In\nthis case, it turns out to be more difficult to find the solutions\nin particular we have not found any solutions without the\npseudoscalars turned on. With some effort, we obtain the following\nsolutions:\n\\begin{itemize}\n\\item We begin with a simple solution in which all scalars have the same value as the $N=1$\nsupersymmetric $AdS_4$ vacuum\n\\begin{eqnarray}\nk&=&-\\frac{18}{11},\\qquad Z_1=Z_2=\\chi=0,\\qquad \\tilde{U}_1=\\tilde{U}_2=\\ln5-\\frac{1}{2}\\ln \\left[\\frac{55}{6}\\right],\\nonumber \\\\\n\\tilde{V}&=&-\\frac{1}{2}\\ln \\left[\\frac{55}{6}\\right],\\qquad B=-\\frac{p^3}{4},\\qquad \\tilde{B}=\\frac{e_3}{2\\sqrt{2}},\\qquad e_9=-\\frac{14p^3}{5\\sqrt{2}},\\nonumber \\\\\ng&=&\\frac{1}{2}\\ln \\left[-\\frac{10}{11}\\sqrt{\\frac{15}{11}}\\kappa p^9\\right],\\qquad L_{AdS_2}=\\frac{5^{\\frac{5}{4}}}{2^{\\frac{5}{4}}(3^{\\frac{1}{4}})(11^{\\frac{3}{4}})}\\, .\n\\end{eqnarray}\nThe solution is of the $AdS_2\\times H^2$ form.\n\\item We now give a more complicated solution\n\\begin{eqnarray}\nk&=&-\\frac{18}{11},\\qquad Z_1=\\chi=0,\\qquad \\tilde{U}_1=\\tilde{V}=\\ln\\left[7\\sqrt{-\\frac{3k}{319}}\\right],\\nonumber \\\\\np^3&=&\\sqrt{\\frac{3}{638}}\\left(\\frac{p^9}{3,190\\sqrt{-k}}\\right)(567,365k-1,002,298),\\nonumber \\\\\nB&=&\\sqrt{\\frac{3}{638}}\\left(\\frac{p^9}{89,320\\sqrt{-k}}\\right)(13,987,355k-27,368,286),\\nonumber \\\\\n\\tilde{B}&=&\\frac{e_3}{2\\sqrt{2}}+\\frac{3p^9}{8,932}(63,162-32,267k),\\qquad Z_2=-5\\sqrt{-\\frac{3k}{319}},\n\\nonumber \\\\\ng&=&\\ln\\left[7\\left(\\frac{3}{638}\\right)^{\\frac{1}{4}}\\sqrt{(k-2)\\sqrt{-k}\\kappa\np^9}\\right],\\nonumber \\\\\n\\tilde{U}_2&=&\\frac{1}{2}\\ln \\left[-\\frac{588k}{319}\\right],\\qquad\nL_{AdS_2}=\\frac{21(3^{\\frac{1}{4}})}{11}\\sqrt{\\frac{7}{21}}\\left(\\frac{2}{29}\\right)^{\\frac{3}{4}}\\,\n.\n\\end{eqnarray}\nThis solution also gives $AdS_2\\times H^2$ geometry. To show that\nthis leads to real solutions, we explicitly give one example of the\npossible solutions\n\\begin{eqnarray}\nZ_1&=&\\chi=0,\\qquad e_9=54.35,\\qquad p^3=-11.56,\\qquad \\tilde{U}_1=\\tilde{V}=-0.14,\\nonumber \\\\\n\\tilde{U}_2&=&0.55,\\qquad Z_2=-0.62,\\qquad B=10.66,\\qquad\n\\tilde{B}=-13.77+0.35e_3,\\nonumber \\\\\n g&=&1.06\\, .\n\\end{eqnarray}\n\\end{itemize}\n\n\\subsection{Uplift formulae}\nWe end this section by giving the uplift formulae for embedding the\npreviously found $AdS_2\\times \\Sigma_2$ solutions in eleven\ndimensions. We first identify the vector and tensor fields in the\n$N=4$ gauged supergravity and those obtained from the dimensional\nreduction of eleven-dimensional supergravity on a tri-sasakian\nmanifold\n\\begin{eqnarray}\nA^3_1&=&\\sqrt{2}A^{9+},\\qquad a^3_1=-\\sqrt{2}A^{6+},\\qquad c^3_1=A^{3+},\\qquad \\tilde{a}^3_1=-A^{3-},\\nonumber \\\\\n\\tilde{c}^3_1&=&\\sqrt{2}A^{9-},\\qquad a^{12}_2=\\sqrt{2}B^{18},\\qquad c^3_2=B^{78}\\, .\n\\end{eqnarray}\nWith this identification and the ansatz for the scalars and vector fields, the eleven-dimensional metric and the four-form field are given by\n\\begin{eqnarray}\nds^2_{11}&=&e^{-\\frac{1}{3}(4\\tilde{U}_1+2\\tilde{U}_2+\\tilde{V})}\\left[-e^{2f}dt^2+dr^2+e^{2g}\n(d\\theta^2+F(\\theta)^2d\\phi^2)\\right]\\nonumber \\\\\n& &+e^{\\frac{1}{3}(2\\tilde{U}_1+\\tilde{U}_2-\\tilde{V})}ds^2(B_{\\textrm{QK}})+e^{\\frac{2}{3}(\\tilde{U}_1-\\tilde{U}_2+\\tilde{V})}\n\\left[(\\eta^1)^2+(\\eta^2)^2\\right]\\nonumber \\\\\n& &+e^{\\frac{2}{3}(\\tilde{V}-2\\tilde{U}_1-2\\tilde{U}_2)}(\\eta^3+\\sqrt{2}\\mathcal{A}^9_tdt\n-\\sqrt{2}p^9F'(\\theta)d\\phi)^2\n\\end{eqnarray}\nand\n\\begin{eqnarray}\nG_4&=&-\\left[6ke^{-(4\\tilde{U}_1+2\\tilde{U}_2+\\tilde{V})+f+2g}-\\sqrt{2}B\\mathcal{A}^{9\\prime}_t-\\sqrt{2}B'\\mathcal{A}^9_t\\right]F(\\theta)dt\\wedge dr\\wedge d\\theta \\wedge d\\phi\\nonumber \\\\\n& &+B'F(\\theta)dr\\wedge d\\theta \\wedge d\\phi\\wedge \\eta^3+dZ_1\\wedge (\\eta^1\\wedge J^1+\\eta^2\\wedge J^2)\\nonumber \\\\\n& &[\\sqrt{2}(\\tilde{\\mathcal{A}}^{9\\prime}_t+\\chi \\mathcal{A}^{9\\prime}_t)dr\\wedge dt+\\sqrt{2}(e_9+\\chi p^9-\\sqrt{2}B)F(\\theta)d\\theta \\wedge d\\phi]\\wedge \\eta^1\\wedge \\eta^2\\nonumber \\\\\n& &[(\\mathcal{A}^{3\\prime}_t+\\sqrt{2}Z_2\\mathcal{A}^{9\\prime}_t)dr\\wedge dt+(p^3+\\sqrt{2}p^9Z_2+2B)F(\\theta)d\\theta \\wedge d\\phi ]\\wedge J^3\\nonumber \\\\\n& &+2(\\chi+2Z_1)\\eta^1\\wedge \\eta^2\\wedge J^3+(dZ_2\\wedge J^3+d\\chi \\wedge \\eta^2\\wedge \\eta^2)\\wedge (\\eta^3-\\sqrt{2}p^9F(\\theta)d\\phi)\\nonumber \\\\\n& &+2[(\\mathcal{A}^3_t+\\sqrt{2}\\tilde{\\mathcal{A}}^9_t)dt-(\\sqrt{2}e_9+p^3)F(\\theta)d\\phi+4(2Z_1+Z_2)\\textrm{vol}(B_{\\textrm{QK}})\n\\nonumber \\\\\n& &\n+(\\chi+Z_2)(\\eta^3+\\sqrt{2}\\mathcal{A}^9_tdt-\\sqrt{2}p^9F(\\theta)d\\phi)]\\wedge(\\eta^1\\wedge\nJ^2-\\eta^2\\wedge J^1).\\label{4_form_flux}\n\\end{eqnarray}\n\n\\section{Conclusions}\\label{conclusions}\nIn this paper, we have found a number of $AdS_2\\times \\Sigma_2$\nsolutions in $N=4$ gauged supergravity with $SO(3)\\ltimes\n(\\mathbf{T}^3,\\hat{\\mathbf{T}}^3)$ gauge group. The solutions can be\nuplifted to M-theory since the $N=4$ gauged supergravity is a\nconsistent truncation of eleven-dimensional supergravity on a\ntri-sasakian manifold. These $AdS_2\\times \\Sigma_2$ gemetries are\nexpected to arise from the near horizon limit of certain dyonic BPS\nblack holes which can be identified as holographic RG flows from\ntwisted compactifications of the dual $N=1,3$ SCFTs in the UV to\nsuperconformal quantum mechanics corresponding to the $AdS_2$ geometry\nin the IR. We have found that most of the solutions have hyperbolic\nhorizons, but some of them have spherical horizons depending on the\nvalues of the four-form flux parameter. These solutions provide examples\nof $AdS_2$ geometries from M-theory compactified on a tri-sasakian\nmanifold such as $N^{010}$ and are hopefully useful in the\nholographic study of the $N=1,3$ Chern-Simons-Matter theories\nin three dimensions. They should also be useful in the study of\nblack hole entropy along the line of recent results in \\cite{Zaffaroni_BH_entropy,BH_entropy_benini,BH_entropy_Passias}. In\nthis aspect, the near horizon solutions given here are enough\nalthough we have not constructed the full black hole solutions,\nnumerically. It would be interesting to compute the topologically\ntwisted index in the dual $N=1,3$ SCFTs and compare with the black\nhole entropy computed from the area of the horizon $A\\sim\nL^2_{\\Sigma_2}$.\n\\\\\n\\indent The solutions found here might constitute only a small\nnumber of all possible solutions due to the complexity of the\nresulting BPS equations. It could be interesting to look for more\nsolutions or even to identify all possible black hole solutions to\nthis $N=4$ gauged supergravity similar to the analysis in $N=2$\ngauged supergravity. For the case of $N^{010}$ manifold, there\nexists an invariant two-form in addition to the universal forms on a\ngeneric tri-sasakian manifold. This leads to an additional vector\nmultiplet, called Betti multiplet, in $N=4$ gauged supergravity.\nThis vector multiplet corresponds to a baryonic symmetry in the dual\nSCFTs. Finding a reduction that includes the Betti multiplet and\n$SU(3)$ non-singlet fields would be very useful in order to find\nmore interesting black hole and other holographic solutions. We\nleave all these issues for future work. \\vspace{1cm}\n\\\\\n{\\large{\\textbf{Acknowledgement}}} \\\\\nThe author would like to thank Davide Cassani for useful\ncorrespondences and the Abdus Salam Centre for Theoretical Physics\nfor hospitality while most of this work has been done. This work is\nsupported by The Thailand Research Fund (TRF) under grant\nRSA5980037.\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nHuman society benefits greatly from the discovery and design of new drug candidates with desired properties. Promising drug candidates (also known as ligands) can be small molecules that interact with macromolecules like proteins to form complexes. Such interactions modulate the cellular functions in biological processes for the treatment of diseases. As shown in Figure~\\ref{fig:task}, in many related tasks like molecular property prediction and protein-ligand binding affinity prediction, 3D structures of small molecules or macromolecule complexes are used to predict the related properties including internal energy, binding affinity, etc. These properties can be traditionally either measured by experiments or computed based on simulation methods like density functional theory (DFT)~\\cite{becke2014perspective} and molecular dynamics (MD)~\\cite{karplus2002molecular}. However, such approaches are time-consuming. Recently, novel machine learning approaches, especially deep learning (DL) algorithms, play an increasingly important role in accelerating drug discovery~\\cite{chen2018rise}. Among various DL methods, Graph Neural Networks (GNNs) have shown superior performance by treating each 3D structure as a graph and performing message passing scheme on it~\\cite{wu2020comprehensive,sun2020graph,atz2021geometric,xia2021geometric,li2021structure}. Molecular graphs can naturally represent the structural information in molecules by treating atoms (or set of atoms like functional groups) as nodes, and chemical bonds (or any predefined pairwise interactions) between atoms as edges. In addition, the success of GNNs relies on the implementation of the underlying physics laws, which are symmetry and rotate invariance, associated with chemical molecules~\\cite{noe2020machine,klicpera_dimenet_2020}. To better capture molecular structures and increase the expressive power of models, previous GNNs have adopted auxiliary information such as chemical properties~\\cite{duvenaud2015convolutional,kearnes2016molecular,yang2019analyzing}, atomic pairwise distances in 3D space~\\cite{gilmer2017neural,schutt2018schnetpack,unke2019physnet}, and angular information~\\cite{klicpera_dimenet_2020,klicpera_dimenetpp_2020,shui2020heterogeneous,zhang2020molecular,li2021structure}. \n\nIn spite of their successes, the application of GNNs in 3D representation learning for drug discovery is still in its infancy. Existing GNN models suffer from several limitations: \\textbf{(1) All interactions within or between molecules are modeled by the same message passing scheme in each GNN.} Such design fails to consider the diversity of interactions (e.g. covalent or non-covalent, local or non-local) that are modeled in physics-aware approaches like molecular mechanics~\\cite{schlick2010molecular}. \\textbf{(2) The adoption of additional auxiliary information in GNNs requires higher computational complexity.} For example, by incorporating angular information, the resulting GNNs~\\cite{klicpera_dimenet_2020,klicpera_dimenetpp_2020,shui2020heterogeneous,li2021structure} require at least $O(Nk^2)$ messages to be computed, where $N$ is the number of nodes and $k$ is the average number of nearest neighbors per node. As a comparison, the less powerful GNNs without angular information require only $O(Nk)$ messages~\\cite{schutt2018schnetpack,unke2019physnet}. With restricted computational resources, those computational expensive GNNs will exhibit limited expressive power or even fail when applied to macromolecules like proteins and nucleic acids. \n\\textbf{(3) Most works only focus on predicting scalar properties while ignore vectorial properties.} Although scalar properties like energy and binding affinity are essential in drug discovery, there are still many important vectorial properties like dipole moment and force. The flexibility of predicting such vectorial quantities is highly demanded in various tasks~\\cite{mailoa2019fast,kouza2018role}. \n\nTo tackle the aforementioned limitations, we propose a novel GNN framework, known as the Multiple\\underline{X} \\underline{M}essage \\underline{P}assing (XMP\\xspace), that enables the flexibility of using different message passing schemes to handle different kinds of interactions in either 2D or 3D molecules. In particular, we use multiplex graphs to represent molecular structures. Each plex (or layer) in a multiplex graph contains one type of interaction. With such input, we assign a different message passing scheme to be operated on each plex accordingly to model the interactions and update node embeddings. \n\nBased on our proposed XMP\\xspace, we are inspired by the ideas from physics and build an efficient and accurate GNN, \\underline{P}hysics-\\underline{a}ware Multiple\\underline{x} Graph Neural \\underline{Net}work (PaxNet\\xspace), for the 3D representation learning of both small molecules and macromolecule complexes. PaxNet\\xspace takes advantage of XMP\\xspace and is guided by molecular mechanics~\\cite{schlick2010molecular} to model local and non-local interactions in 3D molecular structures differently. With this flexibility, PaxNet\\xspace achieves efficiency by incorporating the angular information \\textit{only} in the local interactions to avoid using expensive angle-related computations on all interactions. In addition to scalar properties, PaxNet\\xspace can predict vectorial properties by capturing the geometric vectors in molecular structures that originate from quantum mechanics~\\cite{veit2020predicting}.\n\nTo comprehensively verify the effectiveness of PaxNet\\xspace, we conduct experiments on benchmark datasets that involve small molecules and macromolecule complexes. For small molecules, we use QM9~\\cite{ramakrishnan2014quantum} whose task is to predict quantum chemical properties of small organic molecules. For macromolecule complexes, we choose PDBbind~\\cite{wang2004pdbbind} whose task is to predict the binding affinities between proteins and their small-molecule ligands. On both datasets, our model outperforms the state-of-the-art baselines in terms of not only accuracy but also efficiency regarding inference time and memory consumption. Thus, PaxNet\\xspace is suitable to large-scale machine learning for drug discovery. We summarize the main contributions as follows:\n\\begin{itemize}[leftmargin=10pt]\n\\item We propose a novel GNN framework, Multiplex Message Passing (XMP\\xspace), that builds multiplex graphs to represent molecular structures and uses diverse message passing schemes to model different kinds of molecular interactions.\n\\item We build Physics-aware Multiplex Graph Neural Network (PaxNet\\xspace) based on XMP\\xspace for the representation learning of 3D molecular structures. PaxNet\\xspace differently models the local and non-local molecular interactions with an effective and efficient design. PaxNet\\xspace is also extended to directly predict vectorial values besides scalar values.\n\\item Comprehensive experiments related to small molecules and macromolecule complexes are conducted to demonstrate the accuracy and efficiency of our proposed PaxNet\\xspace.\n\\end{itemize}\n\n\n\\section{Related Work}\n\\textbf{Graph Neural Networks for Molecular Structures.}\nGraph Neural Networks (GNNs) have been proposed~\\cite{duvenaud2015convolutional,niepert2016learning,kipf2017semi} to learn the representation of graph-structured data using neural networks. The architectures of GNNs implicitly encode translational, rotational, and permutation invariance, which are physics laws obeyed by molecules~\\cite{noe2020machine,klicpera_dimenet_2020}. These motivate researchers to use GNNs for learning robust and generalizable representations of molecular structures in many challenging tasks such as molecular property prediction~\\cite{kearnes2016molecular,gilmer2017neural,schutt2018schnetpack,klicpera_dimenet_2020}, protein-ligand binding affinity prediction~\\cite{li2021structure}. Initial related works treat chemical bonds in molecules as edges and atoms as nodes to create graphs for molecules~\\cite{duvenaud2015convolutional,kearnes2016molecular}. These GNNs also integrate many hand-picked chemical features to improve performance. However, they do not take account of the 3D structures of molecules, which are critical for molecular representations~\\cite{townshend2020atom3d}. Thus later works turn to take atomic positions into consideration to compute interatomic distances as edge features between atoms~\\cite{gilmer2017neural,schutt2017quantum,schutt2018schnetpack,unke2019physnet,liu2021transferable}. Usually, a cutoff distance is used to create molecular graphs instead of using a complete graph to reduce computational complexity and overfitting. However, GNNs may fail to distinguish certain molecules due to the choice of cutoff distance~\\cite{klicpera_dimenet_2020}. To solve this issue, angular information in 3D molecular structures is further incorporated in GNNs to achieve higher expressive power~\\cite{klicpera_dimenet_2020,klicpera_dimenetpp_2020, shui2020heterogeneous,li2021structure}. However, those angle-aware GNNs have significantly higher computational complexity than the previous works. With limited computational resources, they are hard to be scaled to macromolecules or large-batch learning.\n\nOur proposed model reduces the high complexity caused by angle-related computations: As inspired by molecular mechanics, we exclude the use of angular information in the modeling of non-local interactions in molecules. Other 3D geometric-related information is carefully incorporated.\n\n\\textbf{Multiplex Graph.}\nA multiplex graph (also known as a multi-view graph) consists of multiple types of edges among a set of nodes. Informally, it can be considered as a collection of graphs, where each type of edge with the same set of nodes forms a graph or a layer. To get the representation of each node, both intra-layer relationships and cross-layer relationships have to be addressed properly. In practice, various methods have been proposed to learn the embedding of the multiplex graph~\\cite{zhang2018scalable,schlichtkrull2018modeling,cen2019representation} and the multiplex graph can be applied in many fields~\\cite{lee2020heterogeneous,wang2020abstract}. For the representation learning on molecules, previous work~\\cite{shi2020graph} implicitly represents molecular graphs as multiplex graphs and passes messages according to the edge types. Different from the existing work, we explicitly represent molecules as multiplex graphs based on the 3D geometric information in molecules. Moreover, we propose different message passing schemes for different layers in the multiplex graph.\n\n\\textbf{Neural Networks with Geometric Vectors.}\nA geometric vector is a feature in $\\mathbb{R}^3$ that has a direction as well as a magnitude. In the real world, geometric vectors are used to define many properties such as force, dipole moment, position vector. For the tasks related to geometric vectors, the adoption or prediction of geometric vectors is crucial in the design of neural network architectures. Especially for molecules, neural networks have been proposed to deal with geometric vectors: ~\\cite{mailoa2019fast} predicts force vectors by treating them as sets of scalars and makes scalar predictions separately with expensive data augmentation to approximate rotational invariance. In other physics-informed approaches~\\cite{chmiela2018towards,schutt2018schnetpack,unke2019physnet,klicpera_dimenet_2020}, the force vectors are computed by taking gradients of a predicted scalar energy field since the force field satisfies the conservation of energy. However, many geometric vectors do not have the underlying conservative scalar field. Another approach for predicting geometric vectors is to use equivariant networks with components that are all carefully designed to implement the equivariance of geometric vectors~\\cite{thomas2018tensor,fuchs2020se,schutt2021equivariant}. However, for molecule-related tasks, usually the molecular representations only need to be invariant rather than equivariant. Thus the complicated equivariance may not be necessary. Recently, geometric vectors like position vectors are used to better encode the geometric features in 3D protein backbone structures~\\cite{jing2020learning} using GNN. The node embeddings are updated by geometric vector features to capture more information. \n\nDifferent from the previous works, when predicting geometric vectors, we deal with general 3D molecular structures and directly make an extension based on GNN that encode invariance without complicatedly designing all components to be equivariant. Particularly, we initialize vectorial node-level contributions with position vectors to be updated by node embeddings. The finally updated vectorial node-level contributions are vector summed as the vectorial prediction.\n\n\\section{Preliminaries}\n\\textbf{Notations.}\nLet $G = ( V , E )$ be a graph with $N=|V|$ nodes and $M = |E|$ edges. The nearest neighbors of node $i$ are defined as $\\mathcal { N } ( i ) = \\{ j | d ( i , j ) = 1 \\}$, where $d ( i , j )$ is the shortest distance between node $i$ and $j$. The average number of the nearest neighbors of each node is $k = 2M \/ N$. In later formulations, we will use $\\boldsymbol{h}_{i}$ as the embedding of node $i$, $\\boldsymbol{e}_{j i}$ as the edge embedding between node $i$ and $j$, $\\boldsymbol{m}_{ji}$ as the message being sent from node $j$ to node $i$ in the message passing scheme~\\cite{gilmer2017neural}, $F$ as the hidden dimension in our model, $\\mathrm{MLP}$ as multi-layer perceptron, $\\concat$ as concatenation operation, $\\odot$ as element wise production and $\\boldsymbol{W}$ as weight matrix. \n\nHere we give the definition of a multiplex molecular graph, which is the input of our model, as follows:\n\\begin{definition} \\label{def:Multiplex} \\textbf{Multiplex Molecular Graph.} \nWe denote a molecular structure as an $(L+1)$-tuple $G = (V, E^1, \\ldots, E^L)$ where $V$ is the set of nodes (atoms) and for each $l\\in\\{1, 2, \\ldots, L\\},$ $E^l$ is the set of edges (molecular interactions) in type $l$ that between pairs of nodes (atoms) in $V$. By defining the graph $G^l = (V,E^l)$ which is also called a plex or a layer, the multiplex molecular graph can be seen as the set of graphs $G = \\{G^1, G^2, ..., G^L\\}$.\n\\end{definition}\nNext we introduce the message passing scheme~\\cite{gilmer2017neural} which is the basis of our model and a framework widely used in spatial-based GNNs~\\cite{wu2020comprehensive}:\n\\begin{definition} \\label{def:MP} \\textbf{Message Passing Scheme.} \nGiven a graph $G$, the node feature of each node $i$ is $\\boldsymbol{x}_i$, and the edge feature for each node pair $j$ and $i$ is $\\boldsymbol{e}_{ji}$. The message passing scheme iteratively updates message $\\boldsymbol{m}_{ji}$ and node embedding $\\boldsymbol{h}_{i}$ for each node $i$ using the following functions:\n\\begin{align}\n\\boldsymbol{m}_{ji}^{t} &= f_{\\text {m}}(\\boldsymbol{h}_{i}^{t-1}, \\boldsymbol{h}_{j}^{t-1}, \\boldsymbol{e}_{ji}), \\\\\n\\boldsymbol{h}_{i}^{t} &= f_{\\text {u}}(\\boldsymbol{h}_{i}^{t-1}, \\sum\\nolimits_{j \\in \\mathcal{N}(i)} \\boldsymbol{m}_{ji}^{t}),\n\\end{align}\nwhere superscript $t$ denotes the $t$-step iteration, $f_{\\text {m}}$ and $f_{\\text {u}}$ are learnable functions. For each node $i$, $\\boldsymbol{x}_i$ is the input node embedding $\\boldsymbol{h}_{i}^{0}$.\n\\end{definition}\nIn recent works~\\cite{klicpera_dimenet_2020,klicpera_dimenetpp_2020,shui2020heterogeneous,li2021structure}, the message passing scheme has been modified to capture the angular information in a molecular graph $G = (V, E)$ with $N$ nodes and position vectors $\\boldsymbol{r}=\\left\\{\\boldsymbol{r}_{1}, \\ldots, \\boldsymbol{r}_{N}\\right\\}$, where $\\boldsymbol{r}_{i} \\in \\mathbb{R}^3$ is the position of node $i$. We here analyze their computational complexity by addressing the number of angles in $G$:\n\\begin{theorem} \\label{theo}\nGiven a molecular graph $G$ with position vectors $\\boldsymbol{r}$, an angle is defined by a pair of adjacent edges that share a common node in $G$. By definition, there are at least $O(Nk^2)$ angles in $G$, where $N$ is the number of nodes and $k$ is the average number of nearest neighbors for each node.\n\\end{theorem}\nProof of Theorem~\\ref{theo} can be found in Appendix~\\ref{A1}. Based on Theorem~\\ref{theo}, we have:\n\\begin{corollary} \\label{corollary}\nFor a message passing-based GNN that requires at least one message to encode an angle, its computational complexity is at least $O(Nk^2)$ for each graph in a message passing iteration.\n\\end{corollary}\nWith Corollary~\\ref{corollary}, we find that the related works~\\cite{klicpera_dimenet_2020,klicpera_dimenetpp_2020,shui2020heterogeneous,li2021structure} all have at least $O(Nk^2)$ complexity.\n\n\n\\begin{figure*}[ht]\n    \\begin{center}\n\t\\centerline{\\includegraphics[width=1.6\\columnwidth]{framework.png}}\n\t\\vskip -0.05in\n\t\\caption{\\label{fig:framework} Illustration of XMP\\xspace framework. XMP\\xspace consists of three steps including (1) the construction of multiplex molecular graph, (2) the assignment of message passing schemes, and (3) the communication and fusion of updated node embeddings.}\n\t\\end{center}\n\t\\vskip -0.25in\n\\end{figure*}\n\n\\section{Methodology}\nIn this section, we present PaxNet\\xspace for efficient and accurate representation learning of 3D molecular structures. We first describe the underlying XMP\\xspace framework of PaxNet\\xspace. Then we describe the components of PaxNet\\xspace comprehensively.\n\n\\subsection{Multiplex Message Passing Framework}\nIn this work, we propose a novel GNN framework, Multiplex Message Passing (XMP\\xspace), that enables the modeling of different molecular interactions with different message passing schemes to learn the representations of 2D\/3D molecular structures. \n\nAs shown in Figure~\\ref{fig:framework}, XMP\\xspace consists of three parts: (1) Given molecular structures, the corresponding multiplex molecular graphs are constructed based on predefined interactions in 2D\/3D molecules. In detail, for each kind of interaction, we use a plex to contain them accordingly. The resulting multiplex molecular graphs are considered as input of XMP\\xspace. (2) For each plex in a multiplex molecular graph, we assign a corresponding message passing scheme to model the interactions and update the node embeddings in it. (3) To address the connections across plexes, XMP\\xspace communicates the updated node embeddings in different plexes. All node embeddings are finally fused together to get graph-level representations to be fed into downstream tasks.\n\n\\subsection{Physics-aware Multiplex Graph Neural Network}\\label{PaxNet}\nBased on XMP\\xspace, we focus on learning the representations of 3D molecular structures and build Physics-aware Multiplex Graph Neural Network (PaxNet\\xspace) as an efficient and accurate approach. The design of PaxNet\\xspace is inspired by physics-based knowledge to be a novel instance of XMP\\xspace.\n\nIn this section, we will first introduce the input graphs of PaxNet\\xspace, which are two-plex multiplex molecular graphs based on 3D molecular structures. Next we will present the components in PaxNet\\xspace specifically designed for the two-layer multiplex molecular graphs to learn 3D molecular representations. For vectorial property prediction, we will describe our geometric vector-based approach in detail, which is used for the prediction of dipole moments based on quantum mechanics as a practical application. Finally, we give a theoretical analysis of the computational complexity of PaxNet\\xspace to demonstrate the efficiency.\n\n\\textbf{Multiplex Molecular Graphs Inspired by Molecular Mechanics. } \\label{sec:multiplex}\nIn molecular mechanics~\\cite{schlick2010molecular}, the molecular energy $E$ is modeled with a separate consideration of local and non-local interactions: $E=E_{\\text {local}}+E_{\\text {nonlocal}}$. Local interactions $E_{\\text {local}}=E_{\\text {bond}}+E_{\\text {angle}}+E_{\\text {dihedral}}$ models local, covalent interactions including $E_{\\text {bond}}$ that depends on bond lengths, $E_{\\text {angle}}$ on bond angles, and $E_{\\text {dihedral}}$ on dihedral angles. Non-local interactions $E_{\\text {nonlocal}}=E_{\\text {electro}}+E_{\\text {vdW}}$ models non-local, non-covalent interactions including electrostatic and van der Waals interactions which depend on interatomic distances. For the geometric information in molecular mechanics, the local interactions need pairwise distances and angles, while the non-local interactions only need pairwise distances. These inspire us to avoid using expensive angle-related computations when modeling non-local interactions to achieve efficiency. \n\nTo reach our goal, we first decouple the local and non-local interactions in 3D molecular structures by using different cutoff distances when defining those interactions or just treating chemical bonds as local interactions. With the interactions, we then construct a two-plex multiplex molecular graph $G = \\{G_{global}, G_{local}\\}$ as shown in Figure~\\ref{fig:method}a. The local plex $G_{local}$ contains only local, covalent interactions, while the global layer $G_{global}$ further includes non-local, non-covalent interactions besides local interactions. As for the geometric information, $G_{local}$ captures the related adjacency matrix $\\textbf{A}_{local}$, pairwise distances $d _{local}$ and angles $\\theta _{local}$, while $G_{global}$ contains only the related adjacency matrix $\\textbf{A}_{global}$ and pairwise distances $d _{global}$. As illustrated in Figure~\\ref{fig:geometric}, given a node $i$ in $G$, we use the geometric information within the two-hop neighborhoods around $i$.\n\n\\begin{figure*}[t]\n    \\centering\n\t\\includegraphics[width=1.7\\columnwidth]{method.png}\n\t\\vskip -0.05in\n\t\\caption{\\label{fig:method}Illustration of PaxNet\\xspace method. (a) Given a 3D molecular structure, a two-plex molecular graph $G$ is constructed to be the input of PaxNet\\xspace. (b) The overall architecture of PaxNet\\xspace. (c) The detailed components in PaxNet\\xspace.}\n\t\\vskip -0.1in\n\\end{figure*}\n\n\\textbf{Overall Architecture of PaxNet\\xspace. }\nAs shown in Figure~\\ref{fig:method}b, with the aforementioned multiplex molecular graph $G$ as input, PaxNet\\xspace uses \\textit{Global Message Passing} and \\textit{Local Message Passing} for $G_{global}$ and $G_{local}$ in $G$ accordingly. The node embeddings $\\boldsymbol{h}$ are updated in each plex and communicated across the two plexes in $G$. To fuse $\\boldsymbol{h}$ for a final representation or prediction of each molecular structure, we use an \\textit{Attention Module} for each hidden layer of PaxNet\\xspace and finally sum the results together.\n\n\\textbf{Global Message Passing. }\nIn this module, we update node embeddings in the global plex $G_{global}$ by capturing the pairwise distances $d_{global}$ based on the message passing in Definition~\\ref{def:MP}. Each message passing operation is:\n\\begin{align}\n\\boldsymbol{m}_{ji}^{t-1} &= \\mathrm{MLP}_m([\\boldsymbol{h}_{j}^{t-1} \\concat \\boldsymbol{h}_{i}^{t-1} \\concat \\boldsymbol{e}_{j i}]),\\label{message_embedding}\\\\\n\\boldsymbol{h}_{i}^{t} &= \\boldsymbol{h}_{i}^{t-1} + \\sum\\nolimits_{j \\in \\mathcal{N}(i)} \\boldsymbol{m}_{ji}^{t-1}\\odot \\phi_{d}(\\boldsymbol{e}_{j i}), \\label{node_update_g}\n\\end{align} \nwhere $i, j \\in G_{global}$ are connected nodes that define a message embedding, $\\phi_{d}$ is a learnable function for pairwise distance. The edge embedding $\\boldsymbol{e}_{j i}$ encodes the corresponding pairwise distance information. In total, this step needs $O(Nk)$ messages.\n\nAfter the message passing, an update function containing multiple residual modules is used to get the node embeddings for the next layer as well as an output module for this layer. The output module is a three-layer MLP to get the output $\\boldsymbol {h}_{global}$ to be fused together. Illustrations of these operations are shown in Figure~\\ref{fig:method}c.\n\n\\textbf{Local Message Passing. } \nFor the updates of node embeddings in the local plex $G_{local}$, we incorporate both pairwise distances $d_{local}$ and angles $\\theta_{local}$. When updating the embedding of node $i$, we consider the one-hop neighbors $\\{j\\}$ and the two-hop neighbors $\\{k\\}$ of $i$. Specifically for the angles related to those nodes, we show an example in Figure~\\ref{fig:geometric} and cluster them depending on the edges around $i$: (a) The \\textit {one-hop angles} are angles between the one-hop edges ($\\theta_{1}$, $\\theta_{2}$ and $\\theta_{3}$ in Figure~\\ref{fig:geometric}). (b) The \\textit {two-hop angles} are angles between the one-hop edges and two-hop edges ($\\theta_{4}$, $\\theta_{5}$ and $\\theta_{6}$ in Figure~\\ref{fig:geometric}). In previous GNNs that incorporate angular information, they either only address the two-hop angles~\\cite{klicpera_dimenet_2020,klicpera_dimenetpp_2020} or only captures the one-hop angles~\\cite{shui2020heterogeneous}. In our model, we contain all those angles to encode more related geometric information. \n\nTo perform message passing, we use the same way as Equation (\\ref{message_embedding}) to compute the message embeddings $\\boldsymbol{m}$. The message passing operation in the $t$-th iteration is:\n\\begin{align}\n\\boldsymbol{m}_{ji}^{'t-1} = \\boldsymbol{m}_{ji}^{t-1} &+ \\sum_{j' \\in \\mathcal{N}(i)\\setminus\\{j\\}} \\boldsymbol{m}_{j'i}^{t-1} \\odot \\phi_{d}(\\boldsymbol{e}_{j'i}) \\odot \\phi_{\\theta}(\\boldsymbol{\\theta}_{j'i, j i}) \\nonumber\\\\\n&+ \\sum_{k \\in \\mathcal{N}(j)\\setminus\\{i\\}} \\boldsymbol{m}_{kj}^{t-1} \\odot \\phi_{d}(\\boldsymbol{e}_{kj}) \\odot \\phi_{\\theta}(\\boldsymbol{\\theta}_{k j, j i}), \\label{message_update} \\\\\n\\boldsymbol{h}_{i}^{t} = \\boldsymbol{h}_{i}^{t-1} &+ \\sum_{j \\in \\mathcal{N}(i)} \\boldsymbol{m}_{ji}^{'t-1}\\odot \\phi_{d}(\\boldsymbol{e}_{ji}) , \\label{node_update_l}\n\\end{align}\nwhere $i, j, k \\in G_{local}$, $\\phi_{d}$ is a learnable function for pairwise distance, $\\phi_{\\alpha}$ is a learnable function for angles. $\\boldsymbol{e}_{j i}$ encodes the corresponding pairwise distance information. $\\boldsymbol{\\theta}_{k j, j i}$ encodes the angle $\\theta_{k j, j i}=\\angle k j i$ accordingly.\n\nIn Equation (\\ref{message_update}), we use two summation terms to separately encode the one-hop and two-hop angles with the associated pairwise distances to update $\\boldsymbol{m}_{j i}$. Motivated by Theorem~\\ref{theo}, there are both $O(Nk^2)$ one-hop and two-hop angles. Thus this step needs $O(2Nk^2)$ messages. In Equation (\\ref{node_update_l}), a similar operation as Equation (\\ref{node_update_g}) is used to update node embedding $\\boldsymbol{h}_{i}$, which requires $O(Nk)$ messages.\n\n\\begin{figure*}[t]\n    \\begin{center}\n\t\\centerline{\\includegraphics[width=1.6\\columnwidth]{geometry.png}}\n\t\\vskip -0.05in\n\t\\caption{\\label{fig:geometric} An example of the geometric information in $G$. By defining the one-hop neighbors $\\{j\\}$ and two-hop neighbors $\\{k\\}$ of node $i$, we can define the pairwise distances $d$ and the related angles $\\theta$ including one-hop angles and two-hop angles.}\n\t\\end{center}\n\t\\vskip -0.25in\n\\end{figure*}\n\nAs shown in Figure~\\ref{fig:method}c, we design the remaining functions in Local Message Passing similarly to those in Global Message Passing to get $\\boldsymbol {h}_{local}$ to be fused and the input for the next iteration.\n\n\\textbf{Cross-layer Communication. }\nTo address the cross-layer relations between $G_{global}$ and $G_{local}$, we let the information in those layers communicate with each other as depicted in Figure~\\ref{fig:method}b. We first perform Global Message Passing on $G_{global}$ in each iteration. Then the updated node embeddings are transferred to $G_{local}$ for Local Message Passing. Finally, the further updated node embeddings are passed back to $G_{global}$ for the next iteration. \n\n\\textbf{Attention Module. }\nAs shown in Figure~\\ref{fig:method}b, to fuse the node embeddings for a final graph-level representation or prediction as shown in the next, we design Attention Module with attention mechanism for each hidden layer $t$ of PaxNet\\xspace to get the corresponding node-level scalar prediction ${y}_{out}^t$ using $\\boldsymbol {h}_{global}^t$ and $\\boldsymbol {h}_{local}^t$ computed by the output modules in hidden layer $t$. \n\nWe first compute the attention weight $\\alpha_{m,i}$ that measures the contribution of $\\boldsymbol {h}_{m,i}$, which belongs to node $i$ on plex $m$ in $G$:\n\\begin{align}\n\\alpha _ {m,i}^ {t} = \\frac { \\exp (\\operatorname { LeakyReLU } (\\boldsymbol{W}_m^t\\boldsymbol {h}_{m,i}^t)) } { \\sum _ { m} \\exp ( \\operatorname { LeakyReLU } (\\boldsymbol{W}_m^t\\boldsymbol {h}_{m,i}^t)) },\\label{softmax}\n\\end{align}\nwhere $m$ can be either global or local, $\\boldsymbol{W}_m^t \\in \\mathbb{R}^{1\\times F}$ is a learnable weight matrix different for each hidden layer $t$ and plex $m$. With $\\alpha _ {m,i}^ {t}$, we then compute the node-level scalar prediction ${y}_{out,i}^t$ of node $i$ using weighted summation:\n\\begin{align}\n{y}_{out,i}^t = \\sum\\nolimits _ { m } \\alpha _ {m,i}^ {t} (\\boldsymbol{W}_{out_{m}}^t\\boldsymbol {h}_{m,i}^t),\\label{weight_sum}\n\\end{align}\nwhere $\\boldsymbol{W}_{out_{m}}^t \\in \\mathbb{R}^{1\\times F}$ is a learnable weight matrix different for each hidden layer $t$ and plex $m$. The node-level predictions ${y}_{out,i}^t$ are used to compute the final graph-level predicted result ${y}$:\n\\begin{align}\n{y} = \\sum\\nolimits_{i=1}^{N}\\sum\\nolimits_{t=1}^{T} {y}_{out,i}^t.\\label{sum}\n\\end{align}\n\n\\textbf{Geometric Vector-Based Approach for Predicting Vectorial Properties. }\nTo enable the prediction of vectorial properties instead of only scalar properties for 3D molecular structures, we design an extension of PaxNet\\xspace which can also be applied to any message-passing-based GNNs.\n\nWe extend the idea of learning scalar node-level contributions to be summed together to compute the final scalar property using GNNs. To predict vectorial properties, we aim to have \\textit {vectorial node-level contributions} to be \\textit {vector summed} together. We define each vectorial node-level contribution $\\vec{y_i}$ to be the multiplication of the scalar atomic contribution $y_i$ and an associated vector $\\vec{v}_i$. To learn $\\vec{v}_i$, we propose an update function to be added in message passing operations:\n\\begin{align}\n\\vec{v}_{i}^t = f_{\\vec{v}}(\\boldsymbol{h}^{t}, \\vec{r}),\\label{vector}\n\\end{align}\nwhere $\\boldsymbol{h}^{t}=\\{\\boldsymbol{h}_{1}^{t}, \\ldots, \\boldsymbol{h}_{N}^{t}\\}$, $\\vec{r}=\\{\\vec{r}_{1}, \\ldots, \\vec{r}_{N}\\}$, $\\boldsymbol{h}_{i}^{t}$ is the learned embedding of node $i$ in $t$-th iteration, and $\\vec{r}_{i} \\in \\mathbb{R}^3$ is the position vector of node $i$. In concept, $f_{\\vec{v}}$ is a function that outputs $\\vec{v}_i$ to be equivariant with respect to an arbitrary composition $R$ of rotations and reflections in $\\mathbb{R}^3$. For example, given Equation (\\ref{vector}), there should be\n\\begin{align}\nR(\\vec{v}_{i}^t)=f_{\\vec{v}}(\\boldsymbol{h}^t, R(\\vec{r})). \\label{equivariant}\n\\end{align}\nSuch constraint is necessary for vectorial property prediction and is physics-aware since the vectorial properties are also equivariant with respect to $R$ in the real world. \n\nTo have a final predicted vectorial value $\\vec{y}$, we modify Equation (\\ref{weight_sum}) and (\\ref{sum}) to be:\n\\begin{align}\n\\vec{y}_{out,i}^t &= \\sum\\nolimits _ { m } \\alpha _ {m,i}^ {t} (\\boldsymbol{W}_{out_{m}}^t\\boldsymbol {h}_{m,i}^t)\\vec{v}_{m,i}^t,\\\\\n\\vec{y} &= \\sum\\nolimits_{i=1}^{N}\\sum\\nolimits_{t=1}^{T} \\vec{y}_{out,i}^t,\n\\end{align}\nwhere $\\vec{v}_{m,i}^t$ is the vector for node $i$ on plex $m$ in the $t$-th iteration. We multiply $\\vec{v}_{m,i}^t$ with the learned scalar atomic contributions. The final prediction $\\vec{y}$ is a vector sum of all vectorial contributions $\\vec{y}_{out,i}^t$.\n\nIn practice, we apply our approach to predict dipole moments $\\vec{\\mu}$ using two different $f_{\\vec{v}}$ as motivated by different quantum mechanics approximations~\\cite{veit2020predicting}:\n\n1) With the approximation that electronic charge densities are concentrated at each atomic position, we can compute $\\vec{\\mu}=\\sum\\nolimits_{i}\\vec{r}_{c,i}q_i$, where $q_i$ is the partial charge of node $i$, and $\\vec{r}_{c,i}=\\vec{r}_i - (\\sum\\nolimits_{i}\\vec{r}_i)\/N$. We treat $q_i$ to be modeled by the scalar node-level contribution, and define $f_{\\vec{v}}(\\boldsymbol{h}, \\vec{r})=\\vec{r}_{c,i}$ to be the vector $\\vec{v}_{i}$ associated with each atom $i$.\n\n2) With the approximation of adding dipoles onto atomic positions in the distributed multipole analysis (DMA) approach~\\cite{stone1981distributed}, we can write $\\vec{\\mu}=\\sum\\nolimits_{i}(\\vec{r}_{c,i}q_i+\\vec{\\mu}_i)$, where $\\vec{\\mu}_i$ is the associated partial dipole of node $i$. We can rewrite the equation as $\\vec{\\mu}=\\sum\\nolimits_{i}f(\\vec{r}_{i})q_i$, where $q_i$ can be modeled by the scalar node-level contribution. We treat $f(\\vec{r}_{i})$ as $\\vec{v}_{i}$ and define $f_{\\vec{v}}(\\boldsymbol{h}, \\vec{r})=\\sum\\nolimits_{j \\in \\mathcal{N}(i)}|m_{i j}|(\\vec{r}_{i} - \\vec{r}_{j})$ to compute $\\vec{v}_{i}$.\n\nTo be noted that both of the two $f_{\\vec{v}}$ guarantee the equivariance of $\\vec{v}_{i}$ as given by Equation (\\ref{equivariant}). The reason is that $f_{\\vec{v}}$ only contains linear combination of $\\vec{r}_{i}$ to compute $\\vec{v}_{i}$.\n\n\\textbf{Computational Complexity. }\nWe here analyze the computational complexity of PaxNet\\xspace by addressing the number of messages as an approximation: We denote the cutoff distance when creating the edges as $d_g$ and $d_l$ in $G_{global}$ and $G_{local}$. The average number of the nearest neighbors per node is $k_g$ in $G_{global}$ and is $k_l$ in $G_{local}$. As mentioned in previous sections, the message passings in PaxNet\\xspace require the computation of $O(Nk_g+2N{k_l}^2+Nk_l)$ messages, while previous approaches~\\cite{klicpera_dimenet_2020,klicpera_dimenetpp_2020, shui2020heterogeneous,li2021structure} require $O(N{k_g}^2)$ messages. For 3D molecular structures, we have $k_g \\propto {d_g}^3$ and $k_l \\propto {d_l}^3$. With proper choices of $d_l$ and $d_g$, we have $k_l \\ll k_g$ (e.g. $d_l=2\\text{\\normalfont\\AA}$ and $d_g=5\\text{\\normalfont\\AA}$). In such cases, our PaxNet\\xspace requires much fewer messages in message passings than those related GNNs.\n\n\n\\begin{table*}[t]\n\\caption{Performance comparison on QM9. We report the averaged results together with the standard deviations for PaxNet\\xspace. We mark the best results in bold and the second-best results with underline.}\n\\label{table:QM9}\n\\centering\n\\begin{tabular}{lcccccc}\n\\toprule\n{Target} & {SchNet} & {PhysNet} & {MGCN} & {HMGNN} & {DimeNet++} & \\textbf{PaxNet\\xspace}\\\\\n\\midrule\n$\\mu$ (D)          & \\underline{0.021} & 0.0529 & 0.056 & 0.0272 & 0.0297 & \\textbf{0.0108} (0.0001)\\\\\n$\\alpha$ ($a_0^3$)  & 0.124 & 0.0615 & \\textbf{0.030} & 0.0561 & \\underline{0.0435} & 0.0447 (0.0003)\\\\\n$\\epsilon_{\\text{HOMO}}$ (meV)  & 47 & 32.9 & 42.1 & 24.78 & \\underline{24.6} & \\textbf{22.8} (0.3)\\\\\n$\\epsilon_{\\text{LUMO}}$ (meV)  & 39 & 24.7 & 57.4 & 20.61 & \\underline{19.5} & \\textbf{19.2} (0.2)\\\\\n$\\Delta\\epsilon$ (meV)          & 74 & 42.5 & 64.2 & 33.31 & \\underline{32.6} & \\textbf{31.0} (0.3)\\\\\n$\\left\\langle R^{2}\\right\\rangle$ ($a_0^2$)  & 0.158 & 0.765 & \\underline{0.11} & 0.416 & 0.331 & \\textbf{0.093} (0.020)\\\\\nZPVE (meV)   & 1.616 & 1.39 & \\textbf{1.12} & 1.18 & 1.21 & \\underline{1.17} (0.02)\\\\\n$U_0$ (meV)  & 12 & 8.15 & 12.9 & \\underline{5.92} & 6.32 & \\textbf{5.90} (0.12)\\\\\n$U$ (meV)    & 12 & 8.34 & 14.4 & 6.85 & \\underline{6.28} & \\textbf{5.92} (0.14)\\\\\n$H$ (meV)    & 12 & 8.42 & 16.2 & \\underline{6.08} & 6.53 & \\textbf{6.04} (0.14)\\\\\n$G$ (meV)    & 13 & 9.40 & 14.6 & 7.61 & \\underline{7.56} & \\textbf{7.14} (0.12)\\\\\n$c_v$ ($\\frac{\\mathrm{cal}}{\\mathrm{mol} \\mathrm{K}}$)  & 0.034 & 0.0280 & 0.038 & 0.0233 & \\textbf{0.0230} & \\underline{0.0231} (0.0002)\\\\\n\\midrule\nstd. MAE ($\\%$ ) & 1.78 & 1.37 & 1.89 & 1.00 & \\underline{0.98} & \\textbf{0.83}\\\\\n\\bottomrule\n\\end{tabular}\n\\end{table*}\n\n\\section{Experiments}\nIn this section, we present our experiments using PaxNet\\xspace on two benchmark datasets to answer the following questions:\n\\begin{itemize}[leftmargin=10pt]\n\\item{\\textbf{Q1}:} How accurate is our proposed PaxNet\\xspace compared with the state-of-the-art models?\n\\item{\\textbf{Q2}:} How efficient is PaxNet\\xspace compared with the state-of-the-art models?\n\\item{\\textbf{Q3}:} How does PaxNet\\xspace perform for predicting vectorial property?\n\\item{\\textbf{Q4}:} Do all components in PaxNet\\xspace contribute to the performance?\n\\end{itemize}\n\n\\subsection{Experimental Setup}\n\\subsubsection{Datasets} To comprehensively evaluate the performance of PaxNet\\xspace, we use two datasets to address the 3D molecular structures in different scales from small organic molecules to protein-ligand complexes.\n\n\\textbf{QM9. }\nQM9 is a widely used benchmark for the prediction of 12 molecular properties in\nequilibrium~\\cite{ramakrishnan2014quantum}. It consists of around 130k small organic molecules with up to 9 non-hydrogen atoms. Following~\\cite{klicpera_dimenet_2020}, we randomly use 110000 molecules for training, 10000 for validation and 10831 for testing. Mean absolute error (MAE) and mean standardized MAE (std. MAE)~\\cite{klicpera_dimenet_2020} are used for quantitative evaluation of the target properties.\n\n\\textbf{PDBbind. }\nPDBbind is a database of experimentally measured binding affinities for protein-ligand complexes~\\cite{wang2004pdbbind}. The goal is to predict the binding affinity of each protein-ligand complex based on its 3D structure. We use the same subsets in PDBbind v2016 dataset which contains ~4k structures and the same data splitting approach as in~\\cite{li2021structure}. We preprocess each original complex to a structure that contains around 300 nonhydrogen atoms on average with only the ligand and the protein residues within 6$\\text{\\normalfont\\AA}$ around it. To comprehensively evaluate the performance, following~\\cite{li2021structure}, we use Root Mean Square Error (RMSE), Mean Absolute Error (MAE), Pearson's correlation coefficient (R), and the standard deviation (SD) in regression.\n\n\\subsubsection{Baselines}\nOn QM9, we compare PaxNet\\xspace with 5 baseline models: \\textbf{SchNet}~\\cite{schutt2018schnetpack}, \\textbf{PhysNet}~\\cite{unke2019physnet}, \\textbf{MGCN}~\\cite{lu2019molecular}, \\textbf{HMGNN}~\\cite{shui2020heterogeneous}, and \\textbf{DimeNet++}~\\cite{klicpera_dimenetpp_2020}. On PDBbind, PaxNet\\xspace is compared with 4 baselines including \\textbf{D-MPNN}~\\cite{yang2019analyzing}, \\textbf{CMPNN}~\\cite{song2020communicative}, \\textbf{DimeNet}~\\cite{klicpera_dimenet_2020}, and \\textbf{SIGN}~\\cite{li2021structure}. In all experiments, we use the original results reported in the related works for baselines. More descriptions of the baselines can be found in Appendix~\\ref{A2}.\n\n\\subsubsection{Model Setup}\nIn our message passing operations, we define $\\phi_{d}(\\boldsymbol{e})=\\boldsymbol{W}_{\\boldsymbol{e}}\\boldsymbol{e}$ and $\\phi_{\\alpha}(\\boldsymbol{\\alpha})=\\mathrm{MLP}_{\\alpha}(\\boldsymbol{\\alpha})$, where $\\boldsymbol{W}_{\\boldsymbol{e}}$ is a weight matrix, $\\mathrm{MLP}_{\\alpha}$ is a multi-layer perceptrons (MLP). For the MLPs used in our model, they all have 2 layers to take advantage of the approximation capability of MLP. For all activation functions, we use the self-gated Swish activation function as in~\\cite{klicpera_dimenet_2020}. All results are reported with the averages and the standard deviations based on five random runs. More details of the settings are included in Appendix~\\ref{A3}.\n\n\n\\subsection{Result}\n\\textbf{Performance on small molecule dataset (Q1). }\nWe show and compare PaxNet\\xspace with baseline methods on QM9 in Table~\\ref{table:QM9}. PaxNet\\xspace achieves 9 best and 2 second best results among all 12 properties. It also achieves a new state-of-the-art result regarding std. MAE, which evaluates the overall performance. From the results, we can observe that the models that incorporate only atomic pairwise distance $d$ as geometric information like SchNet, PhysNet, and MGCN perform worse than those models that incorporate both $d$ and related angles $\\theta$ like HMGNN, DimeNet++, and our PaxNet\\xspace. This shows the importance of capturing rich geometric information when representing 3D small molecules. \n\nBesides, although PaxNet\\xspace takes advantage of the concept in molecular mechanics, which is mainly used for predicting potential energies, PaxNet\\xspace works also very well for properties that are not energy-related like electronic structure properties ($\\epsilon_{\\text{HOMO}}$, $\\epsilon_{\\text{LUMO}}$, and $\\Delta\\epsilon$), electronic spatial extent $\\left\\langle R^{2}\\right\\rangle$, heat capacity $c_v$, and dipole moment $\\mu$. The superior performance on those properties demonstrates the power of separating the modeling of different interactions in molecules based on XMP\\xspace and the effectiveness of the message passing schemes involved in PaxNet\\xspace. \n\n\\textbf{Performance on macromolecule dataset (Q1). }\nOn PDBbind, we compare the results of PaxNet\\xspace and baselines in Table~\\ref{table:PDBBind}. PaxNet\\xspace achieves the best performance regarding all 4 evaluation metrics in our experiments. When compared with the second-best model, SIGN, PaxNet\\xspace performs significantly better (p-value < 0.05 as shown in Appendix~\\ref{A4}). These results clearly demonstrate the accuracy of our model when learning representations of 3D macromolecules. The success of PaxNet\\xspace relies on the separate modeling of local and non-local interactions based on XMP\\xspace. For protein-ligand complexes, the local interactions mainly capture the interactions inside a protein and a ligand, while the non-local interactions can capture the interactions between protein and ligand. PaxNet\\xspace is able to effectively handle the diverse interactions and achieve accurate results.\n\nAmong all models, D-MPNN and CMPNN only implicitly encode the geometric information in 3D structures. While DimeNet, SIGN, and our PaxNet\\xspace all explicitly use pairwise distances $d$ and related angles $\\theta$. With explicitly encoded geometric information, DimeNet performs excellently on small molecule datasets. However, we find it loses superiority when compared to models designed for macromolecules like CMPNN and SIGN. This shows that DimeNet does not generalize well on macromolecules. As for SIGN which performs the best among all baselines on PDBbind, its algorithm is not directly suitable for small molecules. As a contrast, our proposed PaxNet\\xspace is generalizable for both small molecules and macromolecule complexes, and can both achieve state-of-the-art performance in our experiments.\n\n\\begin{table}[t]\n\\vspace{-0.5em}\n\\caption{Performance comparison on PDBbind. We report the averaged results together with the standard deviations. For the evaluation metrics, $\\downarrow$ denotes the lower the better, while $\\uparrow$ denotes the higher the better. We mark the best results in bold and the second-best results with underline.}\n\\label{table:PDBBind}\n\\centering\n\\small\n\\vskip -0.05in\n\\begin{tabular}{ccccc}\n\t\\toprule\n\tModel & RMSE $\\downarrow$ & MAE $\\downarrow$ & SD $\\downarrow$ & R $\\uparrow$ \\\\\n\t\\midrule\n\tD-MPNN & 1.493 (0.016) & 1.188 (0.009) & 1.489 (0.014) & 0.729 (0.006)\\\\\n\tCMPNN & 1.408 (0.028) & 1.117 (0.031) & 1.399 (0.025) & 0.765 (0.009)\\\\\n\tDimeNet & 1.453 (0.027) & 1.138 (0.026) & 1.434 (0.023) & 0.752 (0.010)\\\\\n\tSIGN & \\underline{1.316} (0.031) & \\underline{1.027} (0.025) & \\underline{1.312} (0.035) & \\underline{0.797} (0.012)\\\\\n\t\\midrule\n\t\\textbf{PaxNet\\xspace} & \\textbf{1.263 (0.017)} & \\textbf{0.987 (0.013)} & \\textbf{1.261 (0.015)} & \\textbf{0.815 (0.005)}\\\\\n\t\\bottomrule\n\\end{tabular}\n\\vskip -0.05in\n\\end{table}\n\n\\begin{figure}[t]\n  \\centering\n  \\includegraphics[width=\\columnwidth]{result.png}\n  \\vskip -0.05in\n  \\caption{\\textbf{Results of efficiency evaluation.} (a): Std. MAE vs. memory consumption on QM9. (b): Memory consumption vs. the largest cutoff distance $d$ on PDBbind.}\n  \\label{fig:efficiency}\n  \\vskip -0.15in\n\\end{figure}\n\n\\textbf{Efficiency Evaluation (Q2). }\nTo analyze the efficiency of PaxNet\\xspace, we first use QM9 to compare PaxNet\\xspace with SchNet, PhysNet, and DimeNet++ for small molecules. The model configurations of the baselines are the same as those in their original papers. We depict the std. MAE and the memory consumption during training of each model in Figure~\\ref{fig:efficiency}a. For SchNet and PhysNet, although they require less memory than PaxNet\\xspace, they perform significantly worse than PaxNet\\xspace with 114$\\%$ and 65$\\%$ higher std. MAE. When compared with DimeNet++, PaxNet\\xspace uses 73$\\%$ less memory and reduces the std. MAE by 15$\\%$. Thus PaxNet\\xspace achieves high efficiency with superior accuracy. As for the training time per epoch, PaxNet\\xspace is 26$\\%$ slower than DimeNet++ by having two successive message passing schemes in each iteration. However, since PaxNet\\xspace is more memory-efficient than DimeNet++, PaxNet\\xspace can be trained with a larger batch size than DimeNet++ to achieve a speedup with the same computational resources.\n\nWe further evaluate the efficiency on PDBbind to compare PaxNet\\xspace with DimeNet and SIGN that explicitly use 3D information for macromolecules. In macromolecules, the cutoff distance $d$ plays an important role to capture inter-molecular interactions~\\cite{li2021structure} and directly affects the computational complexity. Thus we compare the memory consumption when using different $d$ in related models in Figure~\\ref{fig:efficiency}b. From the results, we observe that the memory consumed by DimeNet and SIGN increases much faster than PaxNet\\xspace when $d$ increases. When fixing $d=5\\text{\\normalfont\\AA}$ as an example, PaxNet\\xspace requires 80$\\%$ and 71$\\%$ less memory than DimeNet and SIGN, respectively. Thus PaxNet\\xspace is much more memory-efficient and is able to capture longer-range interactions than these baselines with restricted resources. When comparing SIGN with PaxNet\\xspace, both of which use the configurations to achieve the results in Table~\\ref{table:PDBBind}, we find that PaxNet\\xspace outperforms SIGN while reducing the memory consumption by 33$\\%$ and the inference time by 85$\\%$. This clearly demonstrates the accuracy and efficiency of PaxNet\\xspace.\n\n\\textbf{Performance on Vectorial Property Prediction (Q3). }\nOn QM9 when predicting dipole moment $\\mu$, we can simply predict $\\mu$ as a scalar property, or predict the original vectorial value $\\vec{\\mu}$ and get its magnitude: $\\mu = |\\vec{\\mu}|$. We compare three different methods of using PaxNet\\xspace to predict $\\mu$: 1) No geometric vectors are used in PaxNet\\xspace. 2) Geometric vectors are defined as $\\vec{v}=\\vec{r}_{c,i}$. 3) Geometric vectors are defined as $\\vec{v}=\\sum\\nolimits_{j \\in \\mathcal{N}(i)}|m_{i j}|(\\vec{r}_{i} - \\vec{r}_{j})$. Method 1 directly predict scalar values, while method 2 and 3 are those we proposed in Section~\\ref{PaxNet} and can directly predict vectorial values.\n\nAmong the three methods, method 3 achieves the lowest MAE with 0.0108D, which significantly outperforms the best baseline result (0.021D) in Table~\\ref{table:QM9}. Method 2 gets an MAE of 0.0120D, which is still much better than the other baselines. On the contrary, the scalar-based method 1 get 0.0240D which is the worst one among our three methods based on PaxNet\\xspace. The success of method 3 relies on our geometric vector-based approach and better physics-aware approximation compared to the one used in method 2.\n\n\\begin{table}[t]\n\\caption{Ablation study of PaxNet\\xspace on QM9. We compare the variants with the original PaxNet\\xspace and report the differences of MAE on different targets.} \\label{table:ablation}\n\\vskip -0.05in\n\\centering\n\\small\n\\newcommand{\\tabincell}[2]{\\begin{tabular}{@{}#1@{}}#2\\end{tabular}}\n\\begin{tabular}{lrr}\n\t\\toprule\n\t\\multirow{2}{*}{PaxNet\\xspace Variant} & \\multicolumn{2}{c}{MAE on Target}\\\\\n\t & $U_0$, $U$, $H$, $G$ & $\\mu$\\\\\n\t\\midrule\n\tNo Attention Module   & +4\\% & +3\\%\\\\\n\tNo Communication between MPs   & +10\\% & +21\\%\\\\\n    Global MP + Local MP (One-hop $\\theta$) & +4\\% & +8\\%\\\\\n    Global MP + Local MP (Two-hop $\\theta$) & +6\\% & +11\\%\\\\\n\tGlobal MP   & +28\\% & +29\\%\\\\\n    Local MP (all $\\theta$)  & +49\\% & +102\\%\\\\\n    Local MP (One-hop $\\theta$)  & +57\\% & +107\\%\\\\\n    Local MP (Two-hop $\\theta$)  & +64\\% & +111\\%\\\\\n\t\\bottomrule\n\\end{tabular}\n\\vskip -0.05in\n\\end{table}\n\n\\textbf{Ablation Study for (Q4). }\nTo test whether all of the components in PaxNet\\xspace, including attention module, message passing schemes, and communications across plexes, contribute to the success of PaxNet\\xspace, we conduct ablation study on QM9 by designing PaxNet\\xspace variants: Without the attention module, we use an average of all output results. By removing the communications across plexes, we perform the message passing on two plexes in parallel without communication. We also remove either Global Message Passing or Local Message Passing. For Local Message Passing, we design variants by considering different $\\theta$. The performances of all variants are evaluated on scalar targets including $U_0$, $U$, $H$ and $G$, and vectorial target $\\mu$. The results in Table~\\ref{table:ablation} show that all variants decrease the performance of the original PaxNet\\xspace. These results validate the contributions of those components. For $\\theta$, we find that using all $\\theta$ performs the best. It demonstrates the importance of addressing more geometric information.\n\n\\section*{Conclusion}\nIn this work, we propose XMP\\xspace, which is a novel GNN framework to better address the diverse molecular interactions with different message passing schemes than state-of-the-art algorithms. Based on XMP\\xspace, we are inspired by the ideas in physics and propose to separate the modeling of local and non-local interactions in molecules. Such design makes it possible to effectively model the interactions and avoid many computational expensive operations. The resulting GNN, PaxNet\\xspace, can also predict vectorial properties by learning an associated vector for each node. The experiments conducted that involve both small molecules and macromolecule complexes clearly demonstrate the efficiency and accuracy of PaxNet\\xspace.\n\n\n\n\\bibliographystyle{ACM-Reference-Format}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":" \nThey're not alone in the house.\n\nWith a baby on the way and a brand new house, it seems Ginny and her husband, Sean, are on their way to a fresh start. But strange occurrences and financial strain seem determined to keep Ginny and Sean stuck in the past. Ginny begins to believe the house may be haunted...or that her husband might be trying to trick her into thinking so. As Ginny researches the house's former owner and the tragedy that happened there, it becomes clearer than ever that something is in the house with them. The question is, who...or what...is it?\nLittle Secrets\n\nMegan Hart\nDedication\n\nThis book is for the ventriloquist dummy sitting on my office shelf. Please don't come to life and kill me in my sleep.\nChapter One\n\nA fresh start.\n\nThat's what they needed. A new house, new jobs, new hobbies, new friends. Even her haircut was new, shorn to the shoulders instead of hanging halfway down her back. And her body was certainly getting newer all the time.\n\nGinny Bohn stepped out of her old car into her new driveway. She put her hand on the roof for a minute, the metal chilly under her fingertips, and looked at the house. Brick with black shutters. There'd been a garage at one time, according to the realtor who'd sold them the house, but a fire had destroyed it years ago, and the previous owner had never rebuilt. It had been replaced with a small fenced garden, bare now, but Ginny could easily imagine it filled with flowers\u2014assuming she became the sort of woman who took the time to plant seeds and make things grow.\n\n\"Honey, just got a call from the movers. They'll be here in about half an hour.\" Sean had pulled up next to her and was out of his car. He stretched, cracking his back and neck in the way that always made her cringe. He came around the front of his car to hold out his hand to her. \"Careful. The walk's a little wet. It could be slippery.\"\n\nIt was sweet, his hesitation, his desire to make her something fragile. It almost made Ginny want to grab his arm and have him guide her along the slightly buckled sidewalk as though she were somehow incapable of navigating it without his help. Almost.\n\nShe did take his hand, though, linking their fingers. They kissed matter-of-factly. That's how it had always been with them. Matter-of-fact, familiar kisses they shared almost every time they were close enough for their lips to meet. This time, she put her arms around his waist when he tried to get away and hooked her fingertips into his belt loops. Ginny held her husband closer, just for a moment, her head tipped back so he could kiss her more thoroughly, though Sean was too busy laughing to really do justice to the smooch.\n\n\"You want to give the neighbors a show, huh?\"\n\nGinny smiled, not giving a damn about the neighbors, just wanting to feel the pressure of his mouth on hers again. Taste him. Feel his breath on her face and maybe the beat of his heart against her. She stroked the fringes of hair that always fell in front of his ears, no matter how he had it cut. His bangs were longer now too, but she kept herself from ruffling them because she knew it irritated him when she messed with his hair. \"I love you.\"\n\n\"Love you too.\" His kiss lingered this time, not as long as she'd have liked, but with a little more heat. His hands fit against her hips. He nuzzled her nose with his, a gesture she'd told him a hundred times she hated, but that he never seemed to remember.\n\nFrom the backseat came a sudden, irritated yowl. Sean ducked to look through the back window. \"Noodles sounds pissed.\"\n\nGinny sighed. \"She doesn't like the carrier. Can you grab the litter pan and stuff? I'm going to put her in the upstairs bathroom until we can get everything settled.\"\n\nSean nodded and got them from the trunk as Ginny pulled out the carrier and looked at Noodles's furious, furry face. \"Shhh. You act like you're getting killed in there.\"\n\nNoodles, unimpressed with Ginny's scolding and Ginny in general, yowled again. Ginny balanced the carrier, which rocked in her arms as Noodles shifted, and followed Sean up the curving brick path to the cute-as-a-button front porch with its white-painted railing and double swing hung on chains from the roof. The front door was black, sober, the gold handle surprisingly ornate and old-fashioned, compared to the rest of the outside decor. It was the kind of handle that looked as though it used a fancy skeleton-type key, not the small and freshly cut generic silver sort they'd copied from the equally small but tarnished key the realtor had given them.\n\nA fresh key for their fresh start.\n\nIt slipped into the lock with a faint clinking sound, and Sean glanced at her over his shoulder before he twisted his wrist. The click was louder this time, but the door didn't open. Sean took the key out. Blew on it, which made Ginny laugh. Tried again. Waiting, the carrier too heavy, she set it down and held out her hand. Noodles yowled from inside, then went quiet.\n\n\"Want me to try?\"\n\nSean handed her the key, which slipped into the lock without effort. The handle creaked when she turned it, and then the door was opening with a squeal of hinges. Before she could step inside, Sean put out an arm to stop her.\n\nGinny had no time to protest or even prepare herself before Sean bent to sweep her into his arms, one beneath her legs and the other around her back. She let out a startled squeak and clung to him. He took two normal steps over the threshold before his foot hit a throw rug that had been placed a little too far away from the threshold. It started to skid out from under him.\n\nFor a sickening moment, Ginny was sure they were going to fall. Her body tensed, muscles going tight in anticipation of the pain. She bit her tongue and let out a yelp. Her hands slipped on Sean's back as he grappled and kept her from toppling onto the floor...just barely. He put her down, stepping on her foot as the pair of them stumbled forward another few steps, dancing to some routine neither had practiced.\n\nHeart pounding, breath catching sharp in the back of her throat, Ginny ended up straddling the errant throw rug, one of her hands on the wall and the other still clutching at the back of Sean's shirt. Her nails dug so deeply into the plaster they'd have broken if she still kept them long, but she'd given up her manicures months ago. It was the first time she'd looked at her hands with clipped nails and been grateful for it.\n\n\"You okay?\" Sean laughed nervously. His hand shifted on her waist, and he looked over his shoulder at the rug. \"We should throw that away. It could be dangerous.\"\n\nGinny took a slow, calming breath and rubbed the toe of her ballet flat along the polished wood floor. She found her balance, caught her breath. \"It's ugly anyway.\"\n\nIt was. Hideous, in fact. Fluffy yarn of alternating green and brown and orange stripes, with a contrasting flowery border. Nothing like she'd ever have chosen. Now that she looked around the hall, she saw a lot of things she didn't recognize, including an ugly telephone table. \"I thought Bonnie said she was going to have someone come and get rid of all this stuff, except what we specifically requested to be left in the contract.\"\n\nGeorge Miller, the former owner, had died\u2014in the hospital, thank God. Ginny didn't consider herself superstitious, but she wasn't sure she'd have been able to get past the idea of someone actually expiring in her house. They'd never met his son, Brendan, who'd inherited the house but whose lawyer had handled all the details. Brendan Miller had offered to let them take anything they wanted. Everything in the house, as a matter of fact. She'd asked only to keep the ugly but still serviceable barstools in the kitchen, a few of the lighting fixtures and the custom-made drapes in all the rooms, which were as hideous as the throw rug but better than the nothing she owned to replace them with.\n\n\"Yeah...she was supposed to.\"\n\nThe rug and the telephone table hadn't been on the list, Ginny knew that for sure. She tugged the small handle on a drawer meant to hold a pen and paper or a phone book and found instead a jumble of junk. Coins, paper clips, a bunch of old buttons, some tiny carved wooden figures. She lifted one, a thick wooden peg in a vaguely female shape without a face or limbs, the hash marks from the whittling knife still cutting deep into the wood. George Miller had been a carpenter. She tossed the peg back into the drawer. Cell phones and cordless handsets made tables like this irrelevant and unnecessary, though Ginny could vividly remember one similar to this that had been in her grandparents' front hallway. Different phones over the years, but that same table. With her finger, she pushed against the wood and the table rocked, one leg slightly shorter than the others.\n\n\"You okay?\"\n\nGinny looked up at Sean. Found a smile that quirked her lips without much true humor. \"Fine. Just a little shaky.\"\n\nMaybe it was from almost being squished within the first five minutes of entering her new home. More likely it was because she hadn't eaten in a few hours. Low blood sugar. She couldn't decide if she was hungry or on the verge of nausea.\n\n\"Do you feel sick? Should I get you something to eat? Some saltines? Shit. I don't think we have any.\"\n\nThere was that sweet concern again. In the first weeks of her pregnancy she'd spent more time hunched over the toilet than anywhere else, and Sean had been right there with her, holding her hair. Rubbing her back. Bringing her flat ginger ale and crackers and wet cloths to press along the back of her neck. He'd meant his concern as comfort, though she'd told him over and over that she hated any kind of attention on her when she was sick, and had suffered it only because it had taken too much energy to keep asking him to leave her alone.\n\n\"No.\" She shook her head. The dizziness wasn't fading, and the thick, throat-closing nausea was rising. \"I just need to sit down. Get a drink and something to eat. I'll be fine.\"\n\nHe hovered over her all the way down the hall and into the kitchen, where she let him ease her onto one of the barstools she had wanted to keep. She let him chafe her hands too. But beyond that, there wasn't much Sean could do to help her since the brand-new fridge they'd already had delivered was empty and so were the cupboards.\n\nSean opened one anyway and pulled out a vintage Looney Tunes glass adorned with Tweety Bird. \"Hey. Look at this? You want some ice water, honey? I think the fridge's hooked up.\"\n\nTheir old fridge had an ice maker but not one that dispensed through the door. The new one had both cubed and crushed ice, and also filtered water. They'd really splurged. Sean rinsed the glass at the sink, then pressed it against the dispenser. Nothing happened.\n\n\"Shit.\" He opened the fridge to reveal an empty interior, shadowed without the automatic light to illuminate it. \"Damn it.\"\n\nHe flicked the switch on the wall on and off a couple times, then looked at her. \"I thought you said you'd handled the electricity.\"\n\nIf her head hadn't been spinning, Ginny would've shaken it. As it was, all she could do was blink. \"I did. I called and told them we'd be moving in today, and that everything should be turned on.\"\n\nHe flicked the switch again, then once more. \"Well. They didn't. Are you sure you talked to the right person?\"\n\nShe could argue with him about it, or she could keep her lunch in her stomach. Ginny chose the latter. She breathed slowly in through her nose and out through her mouth, closing her eyes to push away the dizziness. She heard running water and looked up to see Sean at the sink.\n\nHe gave her a triumphant look, then offered the glass with a flourish. \"At least we have running water. Walla.\"\n\nThe term was actually voil\u00e0, but Ginny didn't point that out. She just took the glass of water and sipped it, grateful for the way the liquid slipped down the back of her throat, nice and cool. This house had a well and septic system, not like the city water and sewer at their old place, and the water actually had a flavor.\n\nSean rubbed her shoulder briefly. \"Be right back. I'll grab you a snack from the car.\"\n\n\"Oh! Bring Noodles too!\"\n\nShe was a lucky woman. Sean was a good husband. She breathed slowly, fighting another wave of sickness. She should get up, call the electric company and figure out what happened with the power. But for now, all Ginny could manage was to sit and sip the water her husband had poured for her.\n\nSean came in again with a handful of granola bars and the cat carrier. \"Where do you want her?\"\n\nShe took one of the snacks from him and tore it open, eyeing the carrier in which the cat was now growling. \"Oh Lord. She's going to pee on everything. Can you put her in the upstairs bathroom, with the litter box, and the door closed? And oh, put a sign on it so nobody opens it. I'll take care of getting her set up later.\"\n\nGinny heard the sound of a big vehicle in the drive. The movers must've arrived early. She'd have to get up in a minute to greet them herself, direct them where to put the boxes and furniture. \"I hear a truck, good. I was thinking we wouldn't be finished before it got dark, and I really want to at least get our bedroom set up before night.\"\n\n\"Yeah,\" he said. \"Especially since we won't have any lights.\"\n\nIt wasn't her fault, Ginny wanted to say. She'd spoken to the power company and made the arrangements for the electricity to be turned on today. Cable television, phone and Internet too. Trash next week. She had all the information in a folder still in the car, with the dates and times she'd called, the names of the customer service reps she'd spoken to. It was not her fault that someone, somewhere, had forgotten to do their job and flip a switch.\n\nShe said nothing, biting her tongue and hiding her grimace with another sip of water.\n\nSean started toward the stairs with the carrier. \"I'll get them started. You sit.\"\n\nShe did for another few minutes, not out of laziness or some sort of Lady of the Manor attitude, but simply because pushing herself off the barstool still required more of an effort than she felt capable of making. Her head still threatened to keep spinning if she didn't take some deep breaths, and she wasn't certain she wouldn't puke. And no wonder, she thought as she nodded and smiled at the first of the movers who brought in a box of dishes and settled it on the counter before heading back out for another load. She was exhausted, operating on little to no sleep over the past few days as she'd fretted over the final details of the sale that had seemed too good to be true. She'd stayed up late packing and making lists, paying bills, making sure everything in their new life was going as smoothly as possible. No sleep, not enough to eat... Her hands went protectively to her belly for a minute.\n\nThe doctor had assured her that a history of miscarriage didn't necessarily mean she couldn't carry to term. That everything about this pregnancy looked healthy and normal, no cause for alarm. Still, Ginny thought of the almost fall and the subsequent dizziness, and she forced herself to nibble away on the granola bar and sip water until it passed.\n\nBy five o'clock the truck was unloaded, the movers paid and tipped and sent on their way. The electricity had been restored, the culprit a blown circuit breaker, as it turned out, which Ginny discovered when she called the power company and was told that, indeed, their house was supposed to be fully supplied. Sean hadn't apologized for doubting her, but he had gone into the basement to fiddle with the fuse box, and the fact they had lights was enough to make it easy for Ginny to forget that she'd been angry. Noodles had been petted and loved up and generally soothed with a can of fancy cat food and fresh litter, though she was still confined to the bathroom until Ginny could get more of the house arranged and find a place to put the litter box.\n\nPizza was ordered and delivered. Ginny flipped open the cardboard box and breathed in the pepperoni-scented steam as Sean popped the top on a can of caffeine-free cola for her, a beer for him. She'd found a candelabra that had been a wedding gift, never used, along with a set of semi-melted red candles and a half-dozen votive holders. They were eating on paper plates since she hadn't yet unpacked the dishes, but Sean had discovered another couple of those Looney Tunes glasses in the cupboard.\n\n\"Cheers.\" Sean tipped his glass to hers. \"To our first, but not last, romantic dinner in our new house.\"\n\n\"Salut.\" Ginny sipped bubbly cola, relishing the sting of the carbonation in the back of her throat and the sweetly spreading glow of sugar. It would've been even better with caffeine, but Sean had read an article that said pregnant women shouldn't touch it.\n\nEven with the candles, it was far from the most elegant dinner they'd ever had. No flowers in crystal vases, no glittering silverware or gold-rimmed china. But he was right, it was romantic. Ginny downed a third slice of pizza without guilt over calories and listened to Sean wax philosophical on the benefits of hiring a landscaping service in the spring versus trying to get the yard in decent shape all by themselves. He was making plans for the future, she thought. And that was good.\n\nAll of this was good.\nChapter Two\n\nAll of this was bad.\n\nGinny woke, eyes wide, heart pounding, coughing at the sting of bile in her throat. She swallowed hard and pressed a fist between her breasts in a futile attempt to relieve some of the pressure as she sat up against the headboard. She'd gone to bed in a soft flannel granny gown, the house chilly enough to warrant the heavier pajamas since she'd been unable to find the boxes containing either the extra blanket or the flannel sheets. Slick sweat coated her now. She tossed back the comforter and let the cool air cover her instead.\n\nHeartburn. Nobody to blame but herself since she'd been the one to shove that third slice of pizza down her gullet. Ginny pressed her fist harder against her chest. Beside her, Sean slept the sleep of the guiltless, arms akimbo and one leg hooked outside the blankets. Despite the sweat still running rivers between her breasts and down her spine, Ginny was now cold. Her teeth chattered a few times as she forced herself to swallow, then again, hoping to at least get rid of some of the burning taste at the back of her throat.\n\nWhat had woken her? The heartburn, yes, but before that she'd been happily dreaming. Perhaps not so happily, considering the content of her dreams\u2014they'd been full of running and searching. Lots of losing, but not so much finding. Sometimes Ginny clung to dreams, but tonight she was more than happy to have been pulled from them.\n\nBut there. There it was again, the faint scritch-scratch of something in the ceiling. She strained, listening, but didn't hear it again. Shit. Mice. It wasn't entirely unexpected. They lived in an old house in an old neighborhood that backed up onto a farmer's field. There could easily be mice in the house.\n\nThis sounded much bigger than a mouse.\n\nThere were squirrels in the backyard, she'd seen them. They made a weird chittering sound she'd thought at first was birds. The scritch-scratch came again, softer and farther away.\n\nThe home inspector hadn't turned up any evidence of rodent infestation, but as she'd discovered when she tried to shower before bed, the guy had also completely somehow missed that the hot-water heater wasn't capable of providing enough water for a quick shower, much less a luxurious one. Forget about filling the old clawfoot tub in the master bathroom. He'd also passed the fuse box, which was obviously not operating up to standards since the power had gone out once more during the move. Ginny supposed she wouldn't be surprised if the entire house turned out to be overrun with Mickey and his buddies.\n\nThe noise didn't come again, but between it and the heartburn and the lingering dreams, Ginny was in no way going to return to slumberland. She stuck a tentative bare toe off the bed and felt for the edge of the vent in the floor, hoping for a tickle of hot air. Nothing. The house had been empty long enough that the heat had probably been set at a minimum, and because the temperature had been fine during the day, Ginny hadn't touched the thermostat before coming to bed. Sean probably hadn't either. It would've been fine if she were still snuggled up under the comforter, but not wide awake with the pizza sitting like a stone in her gut and reflux climbing her throat.\n\nGinny got out of the bed, which was placed much closer to the wall in this house. Their old bedroom had been bigger, laid out differently, though it lacked the attached master bath and dressing room that had sold her on this house. This room had more windows, one of them a cute dormer she intended to somehow transform into a reading nook or something equally as interesting, if only she could figure out how. Now her hand tapped along the wall and hit open space\u2014the dormer. She peeked into it but saw only darkness and felt an even cooler puff of air on her cheeks. Maybe the window at the end of the narrow space was open or needed some better insulation. She eased past the opening until her hand found the wall on the other side and oriented herself toward the door to the hall, blinking rapidly as though that would make her see better in the blackness.\n\nShe hadn't slept with a night-light since childhood, but wished for one now. All the small details of this house that she hadn't yet learned threatened to trip her worse than the slippery throw rug. She moved blindly, hands outstretched and feet shuffling along the wooden floor. The hardwood had seemed so appealing in the light of day, easy to keep clean and giving the house such a classic feeling. In the middle of the night, with cold bare feet, all she could think of was putting a rug big enough for the bedroom at the top of her mental \"gotta get\" list.\n\nShe bumped into a tall cardboard box, hitting with both her hand and foot at the same time. It moved when she hit it, lightweight. The wardrobe box, then, one of the few she'd managed to completely empty since the closet here was bigger than their old one and far better equipped with built-in shelving and rods. Sean let out a series of snuffling snorts and shifted, making the bed creak, but didn't wake. If she remembered correctly, the box was in front of the still-open closet door, which was between the bathroom door and the bedroom door, both of which should also be open.\n\nShe'd remembered wrong, obviously, because though she took a couple cautious steps, hands out, she found nothing but air. One more step and her fingertips grazed the wall, found the doorframe. She oriented herself again and discovered she'd also been wrong about the bedroom door being open, when she rammed into it with her face. She'd reached with both her hands, so the muffled thud of her nose hitting the wood wasn't loud, and she managed to bite back her cry of pain so neither noise woke Sean.\n\nWith her palms flat on the wood, Ginny pressed her forehead to the door, eyes watering. Cautiously, she felt her nose, but there was no blood and the pain faded rapidly. Her hand slid down the cool wood to the knob\u2014this one, like the one on the front door, was elaborately constructed of an ornate metal plate, a crystal knob and a real keyhole the realtor had promised was for show, since the door actually locked with a small button on the inside of the handle. Her fingers toyed with it as she twisted the knob, the hinges stuttering.\n\nSean muttered and snorted again, rolling around and probably tearing all the blankets up from the foot of the bed. Ginny froze, waiting to see if he'd woken. She really didn't want to wake him. Sean had to work in the morning, which felt like it could be a million or only a few hours in the future, she had no idea since she hadn't looked at the clock. And more than that, if she woke him, he'd want to make sure she was all right. He'd want her to stay in bed while he went downstairs and checked the thermostat, or brought her some water, or rubbed her feet, or rummaged around in the boxes in the bathroom to find her some antacid.\n\nThe last thing in the world Ginny wanted right now was to have Sean hovering over her, no matter how good his intentions.\n\nMore cool air washed over her as she eased herself into the hall. She took a second or two just to relish the fact they had an upstairs with a hallway\u2014their townhouse had been two-story but the second floor had been completely made up of their bedroom, a tiny guest room and shared bathroom. This house was so much bigger, so much a real house, not some rinky-dink starter home. Ginny took the time to savor this, until her teeth chattered harder and she had to wrap her arms around herself to keep warm.\n\nIt seemed they'd been looking at houses forever, unable to make the leap, until the housing market made it impossible not to go for it. They'd looked at this house four times before making their offer. The seller had accepted immediately without haggling, though they'd come in low the way the realtor had suggested. Four times Ginny had toured it, making sure she could imagine herself in just this spot, before she felt she could commit to it for what she knew didn't have to be the rest of her life, but surely felt like it.\n\nStanding outside her bedroom door now, Ginny could imagine the placement of every other door. Six of them\u2014four other bedrooms, empty because they didn't have enough furniture to fill them. A bathroom, inside which she could still hear Noodles's faint, annoyed meowing. A linen closet. In front of her, the railing surrounded the open space around the stairs, with the entrance an equidistant walk to the left or the right of her, depending on which way she felt like going. The movers hadn't closed the doors after putting the labeled boxes in them, and some light filtered in through the one directly to her left. That would be the nursery, and like the master, faced the street as well as the yard on the other side of the house. Blinking, her eyes adjusted to the dark. She could make out the faint shapes of open doors, a window at the other end of the hall.\n\nAnd something else.\n\nOn the far side of the hall, something in the shadows moved. She was sure of it. Something low to the ground, but not Noodles, because not only was the cat still locked up, this was bigger than a cat. Which meant it was bigger than a mouse. Bigger than a squirrel too. Way bigger. Oh God. What if it was a raccoon or something, come in from the attic? Didn't raccoons carry rabies? Ginny reached for the railing, smooth and cool beneath her fingers. The wood creaked as she squeezed. Whatever it was, whatever she'd seen, didn't move again.\n\nShe blinked hard, then again, eyes straining against the darkness. Something took shape in the place where she'd seen movement. A box, a pile of sheets and towels on top of it. She remembered it then. The movers had asked her if she wanted the box in one of the empty bedrooms, but she hadn't wanted to move it again. They'd left it by the linen closet, right there\u2014that's all it was. A stack of linens that probably hid her flannel sheets and the extra blanket she was missing.\n\nGinny had been holding her breath but let it out now on a long, low hiss that became a self-conscious laugh. Silly. Seeing things. New house, new life, all of it new, and of course she wasn't used to getting up in the night and seeing things move that should be still. And she was still cold, no heat sifting up from the registers.\n\nGinny went to the left, past the nursery and toward the linen closet, her hand on the railing to keep her from stumbling. Shuffling, one foot sliding in front of the other, bare toes on bare wood, without ever really lifting fully. She didn't want to bump into anything or step on something unexpected.\n\nJust before she got to the stairs, her sliding foot encountered what felt like a splinter. A faint, small sting had her pause and hop to rub it, her toes cold against equally chilled fingers. She couldn't find the splinter in her sole, and when she put her foot down gingerly the sting seemed to have gone.\n\nShe added a hall runner to the \"gotta get.\"\n\nThe thermostat was in the front hall, just at the bottom of the stairs. Unlike most of the house's fixtures, the thermostat was brand-new. Electronic. It had been replaced, along with the furnace, shortly before the previous owner passed away. It lit when she flipped up the cover to get at the buttons, and the green light that would be barely visible in the daytime was like a beacon to her night-sensitive eyes.\n\nGinny squinted against the sudden glow and leaned forward to push the button once, twice, again. The digital numbers changed from sixty-two\u2014way too cold, to sixty-five. That wasn't that much better. She pushed it a few more times, bringing the temperature to sixty-eight. Sean would grumble about the heating costs, but, dammit, she was cold. At least until they had some sort of idea how much this house would cost to heat she could claim ignorance in her desire for comfort.\n\nShe'd changed the temperature, but there was no rumble from the basement of the furnace kicking on, no welcome burst of warmth from the vents. To the left of the front door, she went through an archway and into what the realtor had called the \"front room.\" It connected to one of equal size through a set of pocket doors. Both rooms had fireplaces, as did the small parlor across the hall, though only the fireplace in the front room still worked. In the months leading up to this night, Ginny had imagined a hundred different ways she'd furnish this place, but for now the movers had simply put their old furniture and all the boxes marked \"Living Room\" into the front room.\n\nMost of the rooms had no overhead fixtures, just switches that operated certain outlets, and she hadn't yet figured out exactly where she wanted the lamps or which outlets worked from the switch. There was enough light from the front windows to direct her to the leather couch pushed up near the front window. She wove around the boxes and chairs to snag the afghan thrown over the back and wrap herself in it, then took a minute to peer through the front windows hung with lace curtains she'd never have chosen but had asked to keep anyway. Those curtains were hers now.\n\nThis place was hers.\n\nShe ran her fingers along the leather couch. The back of the overstuffed armchair. The cardboard boxes still crammed full and taped shut, guarding everything they owned. This house, this brand-new place, this fresh start...it began in chaos and confusion but would settle. It would. Ginny stood for a minute in the middle of the room, eyes closed, breathing in the smell of this new place. Her nose was cold. Her toes were cold. She hugged the afghan tighter around herself and waited to get tired. Waited to be warm.\n\nWell, the afghan was helping with the latter, but though every muscle ached, weighted with exhaustion, she was very far from sleepy. Mindful of her splinter-stuck foot, this time she didn't slide her feet while walking. So of course, two steps away from the couch, she stepped on something hard enough to make her gasp and hop with pain. The heavy thud of whatever it was moved along the floor as she kicked it, and though she tried to see what it was, shadows consumed it.\n\nDamn, it hurt, her sole bruised. Ginny added a pair of slippers to that ever-growing mental list\u2014clearly, she couldn't be shuffling around in her bare feet until everything had been unpacked and cleaned up. Instead of traversing the maze of boxes and possibly stubbing her toes or stepping on some other bit of moving detritus, Ginny decided to head for the kitchen via the hallway.\n\nFrom behind her, when she got to the doorway, came another sound of something clinking on the wood floor, so faint that though she turned her head toward it, she could easily convince herself that she'd imagined it. Or maybe that it had been her rings clicking on the wall as she gripped it to pause.\n\nShe listened.\n\nNothing. Only the sounds of an old house settling; sadly, still no huff and puff of hot air from the furnace vents. She felt a little warmer, though, under the blanket like a cape. Ginny's gran hadn't crocheted it. She wasn't the sort to spend her time in what she called \"handicrafts.\" But the blanket reminded Ginny of her. Gran had been a late-night wanderer, often with a blanket wrapped around her just like this. Checking the doors and windows to make sure they were locked. The stove to make certain all the burners were off. Gran had been one to bend over her sleeping children and grandchildren, holding her fingers under their nostrils to make sure they were still breathing. Ginny had woken many times in the night to see the shadowy figure looming over her, to hear the sharp pant of breath or Gran's grunt of approval when she figured out Ginny was still alive.\n\nGinny shivered at the memory, more from emotion than the chill. Gran's house had been similar to this one, full of nooks and crannies and delightful built-ins. A great place for kids to play hide and seek. A wonderful house for parties with neighbors, family and friends gathered around to eat, drink and be merry. Gran's Christmas Eve parties had been legendary, but only those closest to her knew how much effort and frustration she'd had in the planning and execution of even the simplest bits of the planning, how she'd worked herself into a frenzy with the preparations and how hard she'd crashed after. Those manic highs and lows had been the reason why Ginny's own mom had never thrown any parties, not even for birthdays. Maybe it was why her mom had never crept into her bedroom at night either, to make sure her children hadn't died in their sleep. Ginny couldn't say she minded that bit, but she had always loved helping Gran get ready for the Christmas Eve celebration.\n\nMaybe she'd have her own Christmas Eve party, Ginny thought as she moved through the dark hall past the small parlor. It didn't matter that she and Sean had always celebrated the holiday with no more fuss than a tabletop tree, a few gifts and dinner at the Chinese buffet\u2014unless they went to his mother's house. They'd get a real tree, floor to ceiling. Garland and tinsel and twinkling lights, with mistletoe in all the doorways. Gran's eggnog in the big crystal punch bowl she'd given Ginny when she moved into the nursing home and had distributed all of her \"treasures.\" It would be a lot of work, but Christmas was a couple months away and, besides, after that, the baby would be here. There'd be no time for parties after that, at least not for a while.\n\nAnd as for the other...would she be the sort of mother to creep around at night on cat-soft feet, always afraid? Or the other sort, so determined to let her kids go she didn't cling to them fiercely enough? Only time would tell.\n\nGinny made a pit stop in the powder room, her five-months-pregnant bladder reminding her that even without heartburn, sleeping through the night had become a thing of the past. She had to squint from the bright-white glare of the single, hideous fluorescent fixture. A new bathroom light went on the list. She peed forever, the jingle of urine splashing in the bowl sounding extra loud in the tiny room.\n\nFrom her spot in the light, the hall seemed extra dark, the shadows even thicker. There was an alcove beneath the rise of the stairs and in it, something moved. No flash of light to indicate eyes, no pale glimpse of a face, but nevertheless, the shape moved with a purpose. No random shift of light and dark, nor a mimic of her movements spilling into the hall and making the illusion of something else there.\n\nIn the dark, something moved.\n\nGinny froze, a crumpled wad of toilet paper in her fist, her bladder already loose and now looser with fear. She wouldn't have been able to stop the flow even if she wanted to, but the mortifying horror of being found dead on the toilet, with her faded and stretched-out panties around her ankles, forced a sudden startled laugh out of her. It also sounded extra loud. She cut off the noise by closing her mouth tight; in the next moment she finished peeing and that noise stopped too.\n\nShe waited, listening, trying to see what would loom up out of the shadows to get her. Of course, nothing did. She waited another thirty seconds before finishing on the toilet, pulling up her underpants and washing her hands while she stared hard into the mirror to make sure nothing came up behind her. And of course, nothing did.\n\nIn the doorway she put a hand out into the hall, searching for a light switch. None to the left. None to the right. There was one in the kitchen, she knew that. And one by the front door. But none here. She could stand here in the light forever, or she could venture out and face her fears.\n\nIt would be another box, she thought. A pile of something in the shape of a person, maybe a slowly toppling pile of curtains or tablecloths that had finally given in to gravity just when she happened to be looking at them. She stepped forward, into the hall, the alcove just a step or two in front of her and slightly to the left.\n\n\"It's just a jump to the left,\" she murmured and laughed again. \"And then a step to the right.\"\n\nShe didn't need a time warp. Two or three more steps in the direction would take her to the kitchen and the light switch, but for now she just blinked to let her eyes adjust again to the darkness. And then...she remembered. When they'd looked at this house, the alcove had been used to store the broom, a mop with its bucket and the vacuum cleaner. The basement door was there too.\n\nJust a leaning mop that had probably lost its precarious balance from her thundering weight as she walked. That was all. Nothing else.\n\nStill, that was twice in one night that Ginny'd let herself be scared into seeing things that weren't there. Understandable, she guessed. New house, still so unfamiliar. Everything in her life had been turned upside down. Nothing was the same. She let her hand rest on the slight bump beneath her nightgown. Nothing would ever be the same.\n\nBut this house was their new start, hers and Sean's. This was where they'd raise a family. God willing, they'd grow old together, still interested in sitting on the fabulous front porch in rocking chairs, holding hands. They'd make every clich\u00e9 and stereotype of a happy family come true, she thought with grim determination as she stared fiercely into the alcove, daring the shadows to shift in front of her again.\n\nOf course, nothing did.\n\nIn the kitchen, Ginny shuffled across the linoleum to open the fridge, squinting again as she pulled out a carton of orange juice they'd picked up from the convenience store. She really wanted a cup of hot tea but had no clue where either the kettle or the mugs were. Or the tea bags, for that matter. She poured a paper cup of juice and sipped at it as she looked around the humped shapes of stacked boxes on counters and the floor. They had put their old dining room table in here. In the townhouse, the table had seemed immense, but here the kitchen dwarfed it. She ran a finger along the clean modern edge, thinking maybe she'd hunt for something antique for the new dining room. Or at least something that looked antique, not another sleek but flimsy thing Sean would have to put together using one of those little wrenches that came in the box.\n\nSean had made some noise about turning at least one of the bedrooms into an office, but so far anything that might someday go into such a thing as a home office was still packed up in boxes that had been distributed to other rooms. However, her laptop was plugged in and charging on the table, she knew that for sure, because earlier she'd gone online to make sure the Internet was working and to check her email. She'd never had a desk. She'd always done all of her computer work at this very table in their minuscule dining room in the townhouse.\n\nInvestigating insurance fraud was never as exciting as most people would imagine, but there had been some occasional physical stress to it. Sometimes a stakeout, watching the man who'd claimed his back hurt too much to stack boxes do the tango with his wife in a backyard they hadn't secured from every line of sight. Or asking that seemingly innocent young woman to help her reach something on the top shelf of the grocery store, when the woman had claimed her injuries kept her from doing her job as a retail clerk. But most of Ginny's job was spent writing reports and making sure her facts were straight, that she had proof of the fraud that couldn't be disputed, that all her t's were crossed and i's dotted. She'd spent lots of late nights bathed in the glow of the laptop, a mug of hot tea and a plate of cheese and crackers next to her if she was being good, a piece of cake or some cookies or a bowl of ice cream if she wasn't.\n\nMost of those late nights, she wasn't being good. But that was all over now, and though she let her fingers drift across the laptop's polished-metal lid, detecting the smoother outline of the sticker she'd applied\u2014a zombie Snow White with her hands posed to hold the Apple logo\u2014Ginny didn't open the lid. She didn't check her email or surf the gossip sites or buy things she wanted but didn't need and couldn't really afford. She didn't log on to her Connex account, and she had no reason any longer to snoop around figuring out what other people tried to keep hidden. She was finished with that. This was their new start, and she wasn't going to ruin it by old habits.\n\nGinny climbed the stairs in the dark, one hand on the railing to ensure she didn't trip or fall and plummet to her sprawling, awkward death. She moved through the upstairs hall, also in the dark, found her bedroom door and crept into her bed, all without turning on the lights, all without anything jumping out at her. By the time she wiggled under the blankets and curled onto her side, the first edges of morning light were beginning to peek through the windows. When Sean's alarm went off, she sighed, dreaming, and snuggled back down beneath the comforter, finally able to sleep.\nChapter Three\n\nThe screams of children woke her. Ginny's eyes flew open and she clawed out at the air, her fingertips not close enough to skim the curtains but moving them a bit in the breeze her startled motion left behind. She gasped.\n\nKids playing outside, that was all. She could hear the shush-shush of feet rustling in leaves and the singsongy chant of some childhood rhyme she'd recognize when she woke up a little more, or it would drive her crazy until she could remember it. Like the night before, she was coated in sweat, her nightshirt sticking to her back. Her mouth tasted sour.\n\nShe didn't force herself to get up right away. She rolled onto her back for a minute or so, remembering with a bit of melancholy how once, not so long ago, she'd spent every night flat on her back with her hands crossed over her chest. Vampira, Sean sometimes called her, but that had been her favorite sleeping position since childhood. She missed it. Though her belly was barely bumped, her muscles and joints had already started going loose, and she needed a complicated arrangement of pillows and positions to keep her back from hurting. Her inevitable aches eased briefly when she switched positions, but in another few minutes they'd start up again and worse, if she didn't move. For now, she let herself relax into the mattress as she stared up at the ceiling.\n\nWatermarks, faint beneath a coat of paint, but still there. Some cracks in the plaster. This wasn't a new house, so she shouldn't be surprised, but after the townhouse's smooth, pristine and flawless ceiling, this view was far more interesting. Ginny thought again of her gran's house. For all her eccentricities, Gran had loved having her grandchildren come to stay, and Ginny and her female cousins had always bunked up in the \"rose room.\" Floral-patterned wallpaper and sheets, two double beds along one wall and a twin tucked under the eaves. It had been Ginny's aunt Patty's room growing up and still bore the marks inside the closet door where she'd used a pen to keep track of how tall she and her siblings had grown. The ceiling in the rose room had been plastered in swirls that made faces if you looked just right, and Ginny's cousin Dana had been the queen of telling stories about them.\n\nThis ceiling didn't have those swirls, but Ginny looked for faces anyway. One pattern of cracks and shadow made a man with a walrus mustache. Another a lady in a floppy hat. A smaller configuration looked more like letters, though she couldn't quite make out what word they spelled. Staring, she started to doze before another series of childish shrieks slammed her awake again.\n\n\"Brats,\" she muttered, without real anger, just before a patter of something like gravel pinged the window over the bed. Then she scowled and rolled herself upright to pull back the curtain and stare down into the front yard.\n\nA little boy and a little girl, both blond and in matching red coats, stared up at the window. When the curtain twitched, the boy screamed again, backing away so fast he tripped himself up and landed on his butt. The girl's mouth opened wide, her eyes wider, but she didn't move. Frozen by fear, maybe, Ginny thought as she shook a finger at them. The glass had cracked, cold air seeping in, but remembering the chill from the night before, she wasn't sure she could blame the kids for breaking it. It looked old, if it were possible to tell how old a window crack was, but the lines of it were dirty. Still, why the hell were they tossing pebbles at her window in the first place? She shook her finger again, shooing them, and the girl reached down to pull the boy by the hand. The pair of them ran off across the yard and disappeared around the right-hand corner, leaving behind an assortment of toys. A wagon, some sort of trike. A ball.\n\nGinny sighed. \"Great.\"\n\nShe didn't exactly hate kids. She liked her nieces and nephews well enough, and her cousins' kids. And she'd love her own child, of course, she thought as she cradled her belly for a moment. More than love. She'd cherish her own child.\n\nBut random kids? Random, ill-behaved, screaming, rock-throwing and trespassing kids? Not a fan. Besides, what were they doing in her yard anyway? The house they'd run toward was bigger than this one, with a matching yard that looked manicured and maintained. She thought she remembered a swing set in the back too.\n\nHer own childhood hadn't been so long ago that she couldn't remember how exciting it was to do the very things your parents didn't want you doing. The house next door might have a swing set and perfectly mown grass, but Ginny's backyard had a giant tree with a rope that probably had once been a tire swing hanging from it. It had a gentle slope in the back, just right for rolling down, and the creek that edged their property took a sharp bend away from that other house. Ginny's backyard had knee-high grass that would need to be mowed sooner rather than later and a big collection of leaves perfect for jumping in, if anyone bothered to rake them. The appeal of her backyard wasn't so strange after all. But they hadn't been in the back, they'd been in the front, and even so, she didn't want strange children carousing on her property, especially if they were prone to screaming and throwing stones at her house.\n\nIn the hall bath, she took some time to give Noodles more food and fresh water, though she didn't scoop the box. Sean had read an article about how cat poop could be toxic to fetuses, and while Ginny was more than happy to give up the chore to him, he had to be reminded to do it on a daily basis. It was already starting to stink, worse than usual since she was keeping the bathroom door closed.\n\nThe cat had calmed, at least a little, though she nipped at Ginny's fingertips when Ginny tried to scratch behind her ears. Not that Ginny blamed her, at least not too much. She'd be cranky too if she'd been locked up for two days. Then again, if Ginny peed all over everything, she wouldn't blame anyone who locked her in the bathroom.\n\n\"You stay,\" she told the cat who tried to sneak out after her. \"Just another day or so, boo-boo, so I can get everything set up for you. So you don't pee on stuff.\"\n\nThe cat gave her a bland look that somehow also managed to convey how supremely insulting she found the insinuation that she ever voided her bladder inappropriately. Then she shook herself so the bell on her collar jingled. She turned and blatantly gave Ginny her back.\n\n\"Well,\" Ginny said as she got up, joints creaking, \"don't look at me like that. You know you do.\"\n\nIn the kitchen, she found a surprise\u2014a counter and floor sticky with juice dried into a gluey puddle. Her jaw clenched, and the giveaway throb in her temple told her that her blood pressure had spiked.\n\nGinny took a few calming breaths that did nothing to calm her. Muttering a few choice curse words made her feel better as she stood with her hands on her hips, staring down at the spill any reasonable adult would've cleaned up...but she knew what Sean would say if she confronted him about it. He was busy. Late for work. Didn't mean to do it.\n\n\"Whatever,\" she said aloud.\n\nGetting onto her hands and knees to scrub the spill was far easier than getting up. In fact, for a few strained minutes Ginny was sure she wasn't going to get up at all\u2014she tried to stand too fast and the world began to spin. She had to crawl to the table and push herself up on one of the chairs, where she let her head drop onto her hands. She breathed slowly in, slowly out. The dizziness abated, but then her stomach lurched and she barely made it to the sink in time to heave out a mouthful of bitter bile. Ginny clung to the sink for a minute, with the water running to wash away the sick. When she felt a little better, she bent to take a long, slow drink, letting the water fill and overflow her mouth to rinse it before she swallowed.\n\nShe lifted her head at the sound of childish laughter, but could see nothing through the window over the sink. By the time she managed to get herself to the back door to look out into the yard, they were gone. Maybe they'd run around to the front of the house again to collect what they'd left behind. Maybe she should go out and give them a scare, be the mean old lady from next door who shouted \"get off my lawn!\" and shook a broom. The one who gave out pennies and apples for trick or treat, instead of candy.\n\nShe laughed aloud at that thought and settled for opening the back door to lean out a little farther. Unlike the front of the house with its gorgeous porch, the back had only a set of cracked and crumbling concrete stairs and a crooked metal railing. The flagstone patio had been long overgrown with weeds, the stones themselves sunken into the dirt and covered with moss, but Sean had already announced plans for a big deck that would take advantage of the sloping yard and view of the creek, what there was of it. She'd only looked at it the first time they saw the house, and the summer had been dry enough to make it rather unimpressive. Enough to dip your toes in, maybe, no more than that. All she'd cared about was how far the water was from the house, and if the basement had a sump pump\u2014which it did. Also, according to the realtor, that creek hadn't flooded in over twenty years, and only then after an unusual inland hurricane.\n\n\"You'll never need to worry about water from that creek,\" Bonnie had said. \"But you have the sump pump downstairs in the basement, just in case.\"\n\nGinny took one step onto the concrete stairs before deciding she wasn't about to go tumbling down into the yard just to chase after a few noisy kids. If it kept up, she'd have to go over and talk to their parents, and \"bitchy\" wouldn't be the best introduction to the neighborhood. She listened for the sound of rustling leaves and the screaming laughter, but all she caught was the scamper of a squirrel in the tree and the sound of a truck in the street out front.\n\nShe lifted her face to the fresh autumn air. Took a long, deep sniff. This was her favorite time of year. Sean always said he didn't like fall, when the days got shorter. He said there was never enough time to do everything you wanted to in a day when the winter came, but Ginny'd always liked the sorts of things you did in the dark anyway.\n\nHer cell phone rang from her pocket, the ringtone telling her without even looking that it was Sean. She debated for half a second about not answering\u2014the call reminded her of the juice spill, which annoyed her all over again. But if she didn't answer, he'd call back or send a text, and if she didn't answer those... Well, she didn't want to find out what her husband would do if he couldn't get in touch with her at once. If there was any worse introduction to the neighbors than bitching to them about their wayward spawn, a police car, fire truck and\/or ambulance screeching up to the front of the house would be it.\n\nShe went back into the kitchen as she thumbed the screen to answer the call. \"Hi.\"\n\n\"Hey, it's me. What're you up to?\"\n\n\"Nothing. What's up?\"\n\nShe glanced at the clock. Nearly noon, and she hadn't yet eaten. That was no good, even if she had woken just past eleven. Typically, the nausea had passed and she was ravenous. The problem was, did they have any food? Pizza, after the heartburn of the night before, was completely unappealing. She tried to remember the few groceries she'd brought along and drew a blank. Baby brain, her sister had called it. The inability of a pregnant woman to retain information of current importance.\n\n\"...so do you need me to get you anything?\" This was Sean, who'd been talking for a good few minutes while Ginny stared into the fridge and mentally checked off everything inside it as unappealing.\n\nHer stomach rumbled. She reached for a string cheese and tore the plastic. Took a bite. \"We're out of everything. As soon as I get something to eat and clean up the kitchen, I'll run to the store.\"\n\n\"Don't work too hard,\" Sean warned. \"You should take it easy.\"\n\nMaybe if he hadn't spilled the juice and left it for her to clean, she wouldn't have been so immediately irritated by his admonition, but as it was, Ginny had to count to five before saying, \"All this stuff won't unpack itself, Sean.\"\n\n\"But, honey, you know you need to be careful.\"\n\n\"I'm careful.\"\n\n\"When we talked about you not going back to work, I didn't mean you had to take on everything at home, that's all.\"\n\nShe heard the concern in his voice, knew he was being sincere and not trying to be overbearing or patronizing. Just like she knew that if she didn't organize the house, it would not only take him forever to even get started, but once he did, he'd put everything in all the wrong places.\n\nGinny tried to keep her voice light. \"What am I supposed to do, Sean? Sit around all day eating bonbons and watching the soaps? Do they even have soaps on anymore?\"\n\nSilence ticked between them. She'd probably hurt his feelings. She tried again. \"I have to do something, okay? I'll go slow; I won't lift anything heavy. But if nothing else, I need to be able to get the kitchen organized so I can go to the store and buy us some food. We can't survive on takeout.\"\n\nWell. They could. Their budget was another story. Even as she thought about the grocery store, she remembered the fast-food burger joint in the parking lot. She could just about murder a burger, fries and milkshake. The worst food always made her feel the best.\n\n\"You could get your studio set up,\" Sean said after another half a minute. \"That's not too strenuous.\"\n\nGinny laughed softly. \"Not physically, no. Mentally...\"\n\n\"You're a great painter.\"\n\n\"Sure. Of walls.\" She looked around the kitchen. \"Speaking of which...\"\n\n\"No,\" Sean said. \"No way. Don't even think about it.\"\n\n\"I could just stop at the paint store...look at colors.\"\n\n\"I told you we could go this weekend.\" Now he sounded as irritable as she felt, which in turn only made her all the more annoyed. \"I don't want you lifting heavy paint cans, and besides, Ginny, I know you. You'll go to pick out colors and come back with all the stuff, and I'll find you up on a ladder when I get home.\"\n\nShe couldn't really argue with him, since that was true. Still, it didn't sit well with her to be lectured that way, even if he had good intentions. \"You worry too much.\"\n\nMore silence. Now she'd hurt his feelings for sure. He always clammed up when he was upset, and she stifled a sigh. She settled for a halfhearted apology instead. \"I'm sorry.\"\n\n\"I worry. Of course I do. Am I not allowed?\"\n\nShe rubbed the space between her eyes with the tip of her finger. \"Of course you are. But\u2014\"\n\n\"I'll stop and pick up something for dinner on the way home,\" he told her. \"It's Friday. We can just relax tonight, and tomorrow we'll hit the grocery store, the mall, the paint store, whatever you want. Okay? I'll be there to do the heavy lifting, and you can pick out the colors. I can paint Sunday, before anything gets unpacked. Easier that way, we won't have to cover the furniture or anything.\"\n\nWhen he put it that way, made it sound so reasonable she couldn't argue, there was no way for her to reply with anything but a murmured \"fine.\"\n\nIt was far from fine, but as usual, Sean didn't appear to hear the frustration in her voice. \"Good. What do you want for dinner?\"\n\n\"If you're going to pick it up,\" Ginny said in a clipped tone, \"you should just decide.\"\n\n\"Okay, good idea. I'll surprise you. I'll be home a little later than usual, then. Are you good for now, though? Should I have my mom\u2014\"\n\n\"No. If I need anything, I'll tell my sister to bring it over.\" She couldn't tell him enough times that she didn't need his mother to help her do the things she could do on her own, or with the help of her own mother or siblings. Or friends. Sean's mother meant well, but she was an easily flustered, flighty and sort of useless kind of woman with whom Ginny had never really connected. It was going to be bad enough that she had to tippy-toe around the mess in her house while she waited for her big, strong man to help her with tasks she could so easily accomplish twice as fast and perfectly well on her own. She didn't need a hovering, cooing mother-in-law there to second-guess and wring her hands fretfully over every choice.\n\n\"I gotta go. Lunch meeting. Paul's picking up Chinese.\"\n\n\"Well,\" Ginny said drily, because how could she stay angry in the face of Sean's clear delight, \"I guess I know what you won't be bringing home for dinner then.\"\n\nHe laughed, and after a moment she did too. They exchanged I-love-yous, his coming first but hers at least sincere, and she disconnected. It wasn't so much that she wasn't angry anymore as it was that she was making a deliberate effort\u2014a choice, if you will\u2014to tuck that annoyance away and focus on the positives. She didn't have to cook dinner. That was a plus. Her husband loved her. Another plus. They'd just bought a new house, and she could be a lady of leisure in it. No more job. Didn't have to unpack. Bonus, bonus, bonus. Everything she listed eased her irritation into a smaller, tighter package, until she was able to get rid of it altogether.\n\nThat was until she wanted to drink some juice and it was all gone, and when her stomach growled because she still hadn't eaten, and when she wanted to at least have a peanut butter sandwich and there was no bread, only saltines, and she had to use a spoon because she couldn't find a knife to spread it with.\n\n\"Fuck this,\" Ginny said aloud. Then again, just because she could, as loud as she wanted, no neighbors with their ears pressed against the walls to hear her. Or at least she presumed so, unless she screamed at the top of her lungs or those kids were hanging around on the porch, nobody would hear her. \"Fuck this with something hard and sandpapery.\"\n\nIt felt good to have let off just that little bit of steam, but when she had to resort to eating her saltine crackers over the sink because the table was too crowded to set down a plate, Ginny knew there was no way she could wait until the weekend to at least get some part of the kitchen unpacked and organized. She finished her lunch and dusted her hands free of crumbs, then lifted the lid of her laptop to turn the music on since, of course, she hadn't yet found either her iPod or the speakers that went with it.\n\nShe'd learned to be fastidious about shutting down her browsers, cleaning her cache, signing out of her email and social media sites. Sean had his own computer, but that never stopped him from \"just hopping\" on to hers if it was more accessible or faster or just plain nicer than his older model desktop\u2014and the laptop almost always was all of those things. No matter how many times she'd tried to explain that the entirety of her job was contained on that laptop, that not only was all the information she gathered confidential\u2014legally\u2014but that when he went in and tried to fiddle around, closing tabs she'd left open on purpose or signing her out of whatever she'd been doing, he was potentially screwing her investigations. She'd thought about setting up a user account for him, except she knew he'd never use it because he only ever intended to do something \"real quick\" and would see no point. And now, she supposed, since she was no longer working, it wouldn't matter. He'd been on here this morning though; she could tell because though the laptop powered up to the login screen, the lid itself was slightly sticky. Like from juice. Frowning, she ran her fingers over the metal.\n\nSticky.\n\nNot like someone had spilled juice right on it. More like someone with juice-coated fingers had touched it. She tried the keys, but they all seemed fine. The screen too, which was a good thing, because if Sean had not only spilled juice and left it for her to clean up, but also had gotten it all over her computer, paying hell would be less expensive than totaling up Ginny's expense report.\n\nMoments ago she'd been on fire to get something done, but the inertia of pregnancy settled her more firmly into her chair as she opened iTunes and clicked Shuffle to play through her entire music library. She didn't have to check her email or anything else...but she was going to. Why not? Sean had said she was supposed to take it easy, and fooling around on the Internet, not even pretending it was for work, was as easy as it got.\n\nShe checked her messages, answered one from her mom, marked one from her brother as unread so she'd remember to answer it later, deleted a slew of fluffy kitten glitter angels and urban legend warning forwards from Sean's mom, who obviously never visited Snopes.com. Ginny logged in to Connex and skimmed the updates, wished an old college friend a happy birthday and thought about resisting the allure of starting up a word challenge game with some random strangers, but didn't. She loved and hated those games because despite her large and eclectic vocabulary, she sucked spectacularly at the sort of strategizing necessary to make the most of the double-word and letter tiles. This time, she started off with a seventy-six point word that made her whoop aloud with glee.\n\nWith half the day already gone, her stomach momentarily at least sated if not full and a house full of boxes silently cajoling her into opening them, Ginny moved to log out of everything but the music program. And then...the way it always happens, a song shuffled up. At the first note she froze, fingers on the trackpad twitching so the cursor flew around the screen.\n\nThis song.\n\nOh, this song.\n\nShe hadn't let herself listen to it in months, though there'd been a time when she'd played it over and over again on Repeat, barely a break in between. On her laptop, in the car, through her headphones as she exercised or shopped or sat on the beach and pretended to read. That song had been a punch in the gut every single time, and yet she'd done that to herself on purpose. She'd let it cut her open so she had to sew herself back up again, over and over and over, and gained some sort of sick satisfaction from it. Some measure of...relief, she guessed.\n\nSome closure.\n\nThe problem with doing something you don't want anyone to know about is that when it all goes wrong and you sit at your kitchen table and put your face in your hands and cry because you're hurting even though you have no right to ache...you can't talk to anyone about your pain.\n\nThis pain was hers to carry alone, and that was all right. She deserved it to be that way. Heavy, with nobody to help her carry it. Sharp enough to cut.\n\nShe ought to have deleted the song from her music library entirely instead of just taking it off her playlists, but the fact was that even now, months later, a year later, maybe a fucking lifetime later, this song would always have the ability to cut her open. She would always find a way to sew herself up; that was who she was. It was what she did. So now when it started and she sat at her old table in its new place and stared all around her at the unfamiliar patterns of light streaming through the windows, all Ginny could do was sit and listen.\n\nAnd when the song ended, though she knew she needed to get up and get to work, Ginny hit Replay.\nChapter Four\n\n\"It's cold in here.\" Sean set the paper bags of takeout food on the table. \"Has it been this cold all day?\"\n\nGinny didn't turn from the sink where she was washing her hands. She'd spent the rest of her day wiping out the cabinets and cleaning the counters and cupboards so that when Sean unpacked the dishes he could put them on clean shelves. She'd given the floor a good mopping too. For a house that hadn't been lived in for a long time, there'd been a lot of dust. Now she ran the water as hot as it would go, scrubbing under her nails to get them clean of grime.\n\n\"I don't know. I guess so.\" She shrugged, still not looking at him, still a little tender from her earlier self-indulgent, emotional binge. \"It wasn't that cold out today, was it? Turn up the thermostat.\"\n\nSean disappeared into the hall and came back a few minutes later. \"It was already set at seventy. That's warm enough. Doesn't feel like seventy in here.\"\n\nHer hands clean, she crossed the kitchen to greet him with a kiss. Pushed all the residual emo down, way down, made it invisible. \"Mmmm. Your nose is cold.\"\n\nHe laughed, but his brow remained creased. \"Aren't you cold?\"\n\n\"I was busy working today. Want a sweatshirt?\"\n\n\"Yeah,\" he said, finally focusing on her face. When she tried to pull away, he kept her close with his hands on her hips. \"Hey, you. I thought you were going to take it easy.\"\n\n\"Sean...\"\n\nHis smile tipped the corners of his lips but left his eyes uncrinkled. \"I just want you to be careful. That's all.\"\n\nGinny sighed but didn't try to pull away from him. \"I know. And I was. I took my time. I didn't overdo anything. And look how nice the kitchen looks, you didn't even notice.\"\n\nSean didn't look around. He kissed her instead. \"You could've let me help.\"\n\nEnough. It was too much. With a sigh, Ginny pushed out of his embrace. \"It was fine. Let's eat.\"\n\nHe didn't move from behind her as she went to the cupboard to pull out a couple of plates. \"I don't like it when you walk away from me like that.\"\n\n\"Yeah, well, I don't like it when you lecture me.\" She turned. \"I couldn't wait for you to get home to help me clean the kitchen, Sean. Okay? It had to be done, especially since there was juice all over the place. You think I should've just, what, walked over it all day? C'mon. That's crazy.\"\n\nHe tilted his head just a little. \"What happened with the juice?\"\n\n\"What do you mean, what happened with the juice?\" She put the plates on the table, then went to the drawer for forks and knives. \"You spilled it all over the floor.\"\n\n\"I didn't spill it.\"\n\nShe paused, silverware in hand. She'd expected excuses, not outright denial. \"Well, I sure didn't.\"\n\n\"I didn't even come in here this morning, I was running late. Had to stop at the Green Bean for coffee.\" He glanced at the fridge, then her. \"What juice?\"\n\n\"Certainly not the juice I bought at the grocery store today, since I was forbidden from going.\" He gave her a blank look. Ginny sighed. She put the forks and knives on the table next to the plates, then her hands on her hips. \"The orange juice we picked up from the convenience store. I woke up in the night and came downstairs and had some juice...\"\n\nGinny trailed off, trying to remember if she'd put the juice back in the fridge. Sean smiled just a little. She frowned. \"I didn't spill it. I'd remember that.\"\n\n\"Like you remembered leaving your purse on top of the car? Or when you put your phone in the fridge and the butter in your tote bag?\"\n\n\"Totally different,\" Ginny said crossly.\n\nSean said nothing, but this time his smile did crinkle the corners of his eyes in that way she loved so much. It made her want to trace his eyebrows, so finely arched she sometimes teased him that he had a secret waxing habit. It made her want to pull him close and open his mouth with hers and taste him, and if nothing else, she was grateful for that sudden, trip-trapping beat of her heart. There'd been a long, long period of time when she feared she'd forgotten what it was like to want him that way.\n\n\"Maybe I left the juice on the counter,\" she conceded. \"But I didn't spill it.\"\n\n\"It was probably Noodles.\"\n\nGinny frowned. \"She doesn't jump up on the counters.\"\n\n\"So you say,\" Sean told her. \"She just doesn't do it when we're around.\"\n\n\"She's locked in the bathroom,\" Ginny pointed out just as a familiar, plaintive cry meeped out from under the door to the pantry.\n\nSean went to the door and turned the knob. Noodles slipped out nonchalantly, wound around his legs a few times and then sat to lick her paw. Sean looked at Ginny, who frowned.\n\n\"I didn't let her out,\" Ginny said. \"And I sure didn't lock her in there.\"\n\n\"Maybe she snuck in there while you were cleaning, and you closed the door.\" He opened it wider and looked inside.\n\nThe narrow space, about four feet wide but easily eight feet deep, had floor-to-ceiling wire shelving Ginny hadn't even touched. She peered around his shoulder at the inside, but aside from a few dust balls matching the dirt caught in Noodles's whiskers, she saw nothing. At least the cat hadn't crapped or peed on the floor. Sean was talking, but Ginny ignored him to push into the small room. She looked up, down, side to side. At last, admitting defeat, she turned.\n\n\"...just tired,\" Sean was saying. \"You know she's an escape artist. Remember when she got behind the furnace and wouldn't come out?\"\n\nThis was true. Since kittenhood, Noodles had found her way in and out of places where she wasn't supposed to be. But this time, it didn't feel right. Ginny had shut the bathroom door firmly behind her this morning, and she hadn't gone into the pantry at all...had she? She looked again. It was dirty. She hadn't cleaned it. But it was possible the door had been open, even the smallest amount, enough for a nimble and mischievous cat to slip inside and hide before Ginny pulled it shut during her kitchen makeover.\n\n\"Sure,\" Ginny said and bent to lift Noodles. She stroked the soft fur as the cat butted her head against Ginny's chin. She smelled musty, and the white patches of her calico coat were smudged with grime.\n\nThe cat let out a warning growl and struggled to get down, and Ginny let her go. Noodles ran out of the kitchen through the arched doorway into the dining room. Ginny thought about chasing her, but sighed, figuring she'd just open up the hall bathroom door until she could get the litter box into the basement. And, of course, convince the cat to use it there.\n\n\"I'm going to set up the TV and stuff so we can watch something. I think we're behind on a few episodes of Runner.\" Sean pulled her close to kiss the top of her head, and before she could protest, followed the cat into the dining room.\n\nGinny looked around the mess in the kitchen\u2014dirty dishes, rice and beans left over from the Mexican takeout spilled on the table. So much for not having to take care of everything around the house, she thought and went to the doorway to peek through to the living room, where Sean had started pulling open one of the boxes and had his hands full of wires. So much for not unpacking anything until they painted the walls.\n\nSo much, she thought, for all of that.\nChapter Five\n\nGinny runs.\n\nOne foot in front of the other, never faltering, she runs and runs, dodging and weaving through a landscape that looks like something out of The Terminator. Scorched earth. Dark skies.\n\nHer heart pounds, her fists pump, but this feeling is not harsh or bad. It's glorious, working her body this way, like she never does in her real life. Ginny runs. She jumps. She stretches her arms; she grows wings.\n\nShe flies.\n\nAnd she falls.\n\n* * * * *\n\nGasping, Ginny woke again coated in sweat, again with her throat burning from reflux. Mexican food. She'd stayed up too late trying to make sure she didn't go to bed with her stomach too full, and she'd even managed a long, relaxing bath, which was a pleasant surprise considering the water heater's sporadic offerings. Yet still her body rebelled against the spicy food and here she was, sitting straight up in bed with her eyes wide and the dream of all her bones breaking still vivid enough to keep her from wanting to go back to sleep.\n\n\"You okay?\"\n\n\"Yeah. Go back to sleep.\"\n\n\"What time is it?\" Sean pressed the button on his clock radio and bathed the room in cold white light. \"Shit. Late.\"\n\nIt felt like she'd gone to bed hours ago, but it had only been about forty-five minutes since she'd slipped under the covers beside him. Ginny yawned with her hand over her mouth, hoping nothing else came up and out. The light from Sean's clock dimmed, then went out, but not before she caught a flash of something in the hall. Something bright, reflective. Something like eyes.\n\n\"Turn that light on,\" she whispered.\n\nSean had already fallen back to sleep, or mostly, and he let out a muffled \"hmmph?\"\n\nGinny rolled over to turn her bedside light on, which made Sean grunt and throw an arm over his face. \"I saw something in the hallway.\"\n\nSean sat up at once. \"What?\"\n\nHer pumped-up heartbeat wasn't helping the reflux. \"I saw something like...eyes.\"\n\n\"The cat.\"\n\n\"No,\" she said. \"Unless Noodles is now the size of a Great Dane and standing on her back legs on a box.\"\n\nSean got up and went to the half-open door and pulled it all the way to show the empty hallway. \"Nothing there. Probably something reflecting off of something. Or the window at the end of the hall. Or the night-light, maybe it was that.\"\n\nShe knew he was right, of course he was. The quick, bright flash might've even been her imagination or a quirk of the shadows. It could've been some light coming in through the window, splitting shadows she hadn't yet come to know.\n\nSean's warm hand on her back reminded her of how chilly she was, and Ginny turned off her light to snuggle back under the blankets. The reflux was fading with every swallow, just like the memories of her dream. She rolled to face her husband, who was sprawled on his back with one arm flung over his head. His slow, even breaths soothed her. She put a hand on his belly, first on top of the soft T-shirt, then slipped her fingers underneath to lay them on the warm skin beneath. Then, lower.\n\n\"Oh yeah?\" Sean sounded sleepy, but amused. Beneath her fingers, he responded.\n\n\"Yeah,\" Ginny said. \"Definitely.\"\n\nShe moved to pull him on top of her, but Sean resisted, rolling her on top of him instead. \"Don't want to squish...you.\"\n\nThe baby. He meant the baby. Ginny sighed, straddling him. It felt awkward now. Unwieldy, when she'd simply wanted him to move inside her, but now she had to do some sort of gymnastics routine to get things going. She kissed him when he pulled her down to his mouth, but all at once she couldn't stop thinking about the landscape from her dream and the inconsistent amount of hot water and the errands they needed to do the next day, which would come way too early the longer they were awake.\n\n\"No?\" Sean murmured into her ear, his breath warm. He pushed a hand between them, but the position wasn't quite right and the pressure against her was more pain than pleasure. \"I thought you wanted to.\"\n\nBut it was lost, and she didn't want to tell him so because she'd been the one to start this. It would be unfair to back out now. So she took him in her hand and shifted to slide him inside her, and she moved the way she knew he liked her to move.\n\nAnd when it was over, in the dark, she listened to the sound of his breathing slow and deepen beside her. She listened to Noodles's rhythmic purring. She listened to the soft scritch-scratch of something inside the walls of her house, and it was that sound that finally soothed her into sleep.\nChapter Six\n\nThe exterminator was much cuter than he had any right to be. Over six feet tall, dark hair, ice-blue eyes. Dimples in both cheeks when he grinned, which he did the second Ginny opened the door and wished she'd put on something nicer than a pair of yoga pants and one of Sean's old college sweatshirts.\n\n\"Mrs. Murphy?\"\n\nTechnically, she was not Mrs. Murphy since she'd kept her maiden name, but it was easier to nod than explain. She stepped aside to let him in. \"Yeah. Come on in. Thanks for getting here so fast.\"\n\n\"No problem. It's my job, right?\" He turned in a slow half circle, looking up at the ceiling before focusing on her. \"I'm Danny.\"\n\nHe held out a hand for her to shake, surprising her. Ginny's hand was engulfed inside his fingers. \"Ginny.\"\n\n\"You have mice?\" Danny asked as he let go. He set down the tool bag in his other hand and put both hands on his hips. He wore a dark-blue coverall with his name on a patch over his heart. Big black boots.\n\n\"I think so.\" Ginny tried not to ogle his ass when he turned again to look up, but didn't manage very well. It was a pretty stellar butt, hard to ignore even in the baggy coveralls. \"I've heard things in the walls and ceiling. We just moved in\u2014\"\n\n\"Yeah, I thought you must be new. I've been doing this neighborhood for a few years now, but this is the first time I've been called to this house.\"\n\nSomething in the way he said \"doing this neighborhood\" made her think of housewives clad in leopard-print robes, martini glasses in hand, standing on top of chairs and screaming while cartoon mice ran around them. \"We've been in for not quite a week.\"\n\n\"Guy who owned this place before never had a contract with us, not even just the standard maintenance. Your husband signed you up for that, so I can take care of that for you today too. If you want.\" Danny's grin made Ginny think he'd take care of lots of things she wanted.\n\nShe shoved that idea away as absurd. She was easily at least ten years older than him, face bare of makeup and inadequately showered. And pregnant. And married, she reminded herself. Still, looking wasn't touching, and she admired Danny's dimples again.\n\n\"Did he?\" she remembered to say. \"What does that mean?\"\n\n\"Means I'll come out every three months to treat your place for spiders, bees, wasps, check for termites. Whatever you need. And if you have any other problems, I'm your guy to take of that too. Like mice.\" Danny winked.\n\nGinny blinked.\n\nDanny pulled a flashlight from the tool bag and shone it along the crown molding. \"Termites usually aren't a problem in this neighborhood, but you get a lot of spiders and silverfish. Being so close to the creek, you can get millipedes and stuff in your basement too.\"\n\n\"But...you'll take care of all that. Right?\"\n\nDanny clicked off the flashlight and gave her another of those grins. Ginny wondered if he practiced them. \"Yep. All of it. If you want to show me into the basement first, I can take care of you from the ground up.\"\n\nOh, I bet you could, Ginny thought. Top to toe too. She bit the inside of her cheek to keep a straight face and pointed down the hall. \"Basement door's to the right there, in that alcove. I have a cat. You won't put anything down that's poisonous to animals, right?\"\n\nDanny shook his head. \"No, ma'am.\"\n\nJesus. Ma'am. It should've made her feel old and possibly respected, but instead made her flash on a vision of herself in librarian glasses and a pencil skirt, with her hair in a bun and a ruler in her hand.\n\nShe pointed the way toward the basement, but didn't follow him down. She was in the middle of not only a cutthroat game of online Scrabble, but also finishing up some final files she needed to send in to the company she no longer worked for. The word game was winning her attention, because while the files were important, they were also the last link she had to her job. As soon as she turned them in, she'd have no more reasons to think of herself as a working woman.\n\nShe got so engrossed in trying to figure out where to use her Q, U and Z tiles for the best results that she didn't hear Danny until he appeared in the kitchen suddenly enough to make her scream. \"You scared me,\" Ginny said unnecessarily, one hand on her heart. \"God.\"\n\n\"Sorry.\" Danny looked serious. \"I need to talk to you about your basement.\"\n\nThat sounded bad. Ginny hadn't actually been in the basement since they'd moved in. She remembered it as being unfinished and dry, the only thing she'd really cared about. Sean had talked about making it into a rumpus room, a place for a home theater. Sean talked about a lot of things.\n\n\"My husband has big plans for it,\" she said. \"Aside from that, what's the problem?\"\n\n\"It's your ductwork. Oh hey, puss.\" Danny crouched to offer a hand to Noodles, who sniffed it with disdain but let him pet her. He stroked the cat's fur, then looked up at Ginny. \"It's all over the place down there.\"\n\n\"Umm...?\" Ginny had no idea what ductwork was supposed to look like.\n\n\"Plenty of places for rodents and pests to hide. Basically, like a little superhighway for mice. But it's okay. I put some glue traps in there, and I'll add some bait traps in the attic. That's where you heard them, right? But you'll have to be sure your cat doesn't go up there.\"\n\nGinny looked at Noodles, busy licking her paw, and then at him. \"She doesn't go in the attic. Hell, right now she doesn't even go into the basement. The door's always closed.\"\n\nDanny frowned. \"You'll have to check the glue traps, which, honestly, I don't love. Bait traps are more effective, for sure. But you do risk the chance then that they might not go all the way outside to...\"\n\n\"Die?\"\n\nHe nodded. For an exterminator, Danny seemed awfully delicate.\n\nGinny sighed. \"So what, then? They'd get stuck inside the wall someplace and...rot?\"\n\nDanny nodded again. \"But, you know, mice. They're small. It would only stink for a little while. If you had a squirrel or something bigger, a raccoon, say...\"\n\n\"Jesus,\" Ginny muttered. \"Do you think we might?\"\n\n\"I won't know until I go up and check. This close to the creek, you might get rats.\" Danny made a pow-pow gesture with his fingers. \"But it's probably just mice. You heard noises in the walls?\"\n\n\"And ceiling. Yeah.\" For a moment she thought about mentioning the shape she thought she'd seen that first night here. The eyes. \"Rats...they can get pretty big?\"\n\nDanny smiled. \"Sure. Huge, some of them. When I was working in the city, I'm not even kidding you, I once saw a rat that was bigger than a Chihuahua.\"\n\n\"Jesus.\"\n\n\"Yeah.\" Danny nodded, eyes wide and serious for a moment before the grin was back. \"But around here? Nah. You'll get field mice and squirrels, sure. Sometimes a raccoon. Sometimes bats. But even if you do get rats, they're probably not that big.\"\n\nNot Great Dane size anyway. That was only in the movies. Ginny frowned. \"But you'll put out enough poison to take care of whatever it might be?\"\n\n\"Yep. If you'll show me the attic, I can get up there and take a look around. Do I need a ladder?\"\n\n\"No. There's a pulldown.\"\n\nUpstairs, she showed him the door in the ceiling and he pulled the cord, dropping the stairs. Ginny had left Sean to inspect the attic, but watching Danny's feet disappear into her ceiling, she decided to follow him. She stayed on the stairs, and stuck only her head through the hole. There was no floor up there, just a few boards placed across the beams, and the steeply pitched roof made it impossible for anyone but a child\u2014or a super tiny adult\u2014to stand upright except in the center.\n\n\"I thought it would be bigger,\" Ginny said. \"Like the size of the house.\"\n\n\"You have those gables and dormer things.\" Danny shone his flashlight into the corners, but didn't move more than a step or two from the hole in the floor. The beams creaked under his boots. \"Crawl spaces, right?\"\n\n\"Oh. Yeah. But still...\" Ginny gestured, though he wasn't looking, \"...I guess this isn't what I was expecting.\"\n\nHe grinned at her over his shoulder. \"Could be worse, right? You could've come up here and found something really creepy.\"\n\n\"Yeah, like Gary Busey.\"\n\nDanny obviously missed the reference to the 80s horror classic Hider in the House. Busey had played a crazy guy who built a secret room in Mimi Rogers's attic, then stalked her. Ginny had meant it as a joke, and if her brother, Billy, were here he'd have laughed. He'd watched that movie with her so many times they could both quote the dialogue back and forth. Ginny thought briefly of explaining, but couldn't muster the effort. Danny shone his light around the room, pausing to spotlight a round vent at one end. Then one on the other side, identical except for the mesh covering.\n\n\"There's part of your problem. That vent should have the same protection on it. That's how stuff can get in.\"\n\nGinny had a hard time believing mice would climb up the side of the house and in through a vent, but she supposed squirrels could. Or bats. She grimaced. A bat had flown into her college apartment once. All she could remember was some drunken frat boy her roommate was screwing going after it with a tennis racket.\n\n\"Can you fix that?\"\n\nDanny shook his head and pointed the light at the beams under his feet. \"Sorry. I can only spray for bugs and put down bait traps. But I'll do that up here. I see some droppings, not a huge amount, looks like mice. Kinda old, but, still, definitely some evidence that you have something going on. I'll put some glue traps in here too, along with the bait, okay?\"\n\n\"Sure. Yes. Perfect, thanks.\" Ginny backed down the rickety stairs carefully.\n\nDanny was there for about another hour, during which time Ginny managed a crushing win in her word game, but also completed all her final files and sent them off to where they needed to go. All done. All gone. She was officially unemployed for the first time since she was fifteen years old.\n\nWhen he came in one more time, Danny found her at the kitchen table, staring at her laptop's blank screen. A tiny bouncing icon alerted her to her sister Peg's instant message, but Ginny was ignoring it for now. She had her hands folded in front of her, not touching the keyboard or even the mouse she used because the laptop's small trackpad drove her crazy.\n\n\"Ma'am?\"\n\nGinny turned. \"Finished?\"\n\n\"I put the traps out. I found a few places out here your husband will want to check out too. Your house has a brick front, but the rest is aluminum siding. The places in the corners of the house where the siding comes together, those are open on the bottom. Sometimes they're capped, but the caps fall off. Rodents climb up in there, that's how they get in the walls. Just have him shove some steel wool up there. That should solve it.\"\n\nGinny pushed her chair back with a long screech on the linoleum. \"Thanks.\"\n\n\"You writing a book or something?\"\n\nTaken aback, she looked at the laptop. \"Huh? Oh. No. I'm not a writer.\"\n\n\"Oh. So you're the painter?\"\n\nThis took her even more aback, literally, as she stepped back. Her calves hit the chair. \"What?\"\n\n\"The easel and stuff, I saw it upstairs when I was putting some traps in your crawl space. You sure your cat won't get into them, right?\" Danny, bless his pretty face, looked worried. \"Even if she gets into the crawl space, she shouldn't be able to get into the bait, but...\"\n\n\"It'll be fine.\" Ginny wasn't sure how to feel about him not only helping himself to her crawl space, which, honestly, was what he was there to do, but also noticing her...stuff.\n\n\"My girlfriend took a couple art classes this summer. She's really into watercolors. What do you paint?\"\n\n\"Nothing,\" Ginny said too fast. \"Well. Nothing lately.\"\n\n\"Did you do the ones in the basement?\"\n\nThis stunned her so much she sat down. \"What?\"\n\n\"There were a bunch of canvases in the basement. Mostly flowers and stuff, big flowers.\" Danny demonstrated by holding his hands apart, shoulder width.\n\n\"Yes,\" Ginny said after a moment. \"Yes. Those are mine.\"\n\n\"They're pretty good. You ever think about showing them in a gallery or something?\"\n\n\"No. They were just for fun.\"\n\nDanny nodded like this made total sense, then paused to look at her more solemnly. \"So...it's not fun for you anymore, or...?\"\n\n\"I just haven't had time. That's all. I'll get back to it,\" Ginny told him, but from his look she thought he didn't believe her, any more than she did herself.\n\nAs soon as she closed the front door behind him, cutting off his cheerful reminder that he'd be back in three months to \"squirt for bugs,\" Ginny went to the basement door. Noodles was suddenly at her feet, winding around her ankles and meowing plaintively\u2014the cat had put on too much weight when they let her eat at her leisure, and she was reminding Ginny it was overdue time for her to be fed now. Ginny bent to pick her up and scratch her under the chin, then tossed her gently toward the kitchen and opened the basement door to squeeze through it. Noodles managed to run after her anyway, barely missing getting the tip of her tail caught. Bell collar jingling, she rocketed down the stairs in front of Ginny, who at least wanted to wait until she'd turned on the light.\n\nShe didn't have to go far to see the paintings. Someone had stacked them up against the wall closest to the foot of the basement stairs. Carelessly, without so much as a single sheet of bubble wrap or anything between them to protect the canvases from getting scratched. The movers must've done it, she thought, since she surely wouldn't have packed them that way. She hadn't packed them at all.\n\nFor a moment, Ginny saw herself crossing the short distance and flipping through those paintings, which represented not only hours of her time, but an entire rainbow of memories in every single shade of gray. She saw it so clearly\u2014how her hands would shift and move them, separating them into chronological order or by color scheme or theme. She closed her eyes and saw every single painting she'd ever done that had been worth saving.\n\nThe problem was, just like she hadn't packed them, Ginny hadn't saved them, either.\nChapter Seven\n\nDon't forget, I have class tonight.\n\nSean had sent the text that morning, but though once not so long ago Ginny'd carried her phone around with her like it was attached by an umbilical cord, she'd gotten out of the habit now that she wasn't working and also home full time, where she could be reached on the landline. She'd have remembered about his class when six o'clock arrived and he didn't come through the door, but it was good he'd reminded her. She could have a dinner of leftovers while she watched TV and tried to ignore the mess in the living room. At least the boxes made it easy for her to find a place to put her plate.\n\nPasta with roasted red peppers, hearts of palm and a sprinkling of olive oil and grated parmesan cheese. That was her dinner, along with a couple of crusty French rolls. Her craving for a nice glass of red wine was like a physical punch, but Ginny didn't even give the bottle a longing glance. Her previous doctor had told her a glass of wine a day was perfectly fine; this time around, she hadn't asked the ob-gyn's opinion. Better to just avoid it, the way she avoided caffeine and secondhand smoke, and baths that were too hot or showers that were too cold, artificial sweeteners and caustic chemicals. She cleaned with organic products or old home remedies like vinegar and baking soda. She looked both ways at least twice before crossing the street.\n\n\"Because I'm not taking any chances,\" Ginny told the cat, who couldn't be bothered to even glance her way. \"Nope. Not a single one.\"\n\nHer eyes had been bigger than her stomach, no small feat these days. With only half her dinner finished, Ginny took her plate to the kitchen and set it on the counter, intending to scrape it before putting it in the dishwasher, but discovering instead that once again, Sean had taken out the trash without replacing the garbage bag.\n\nA quick search of the cupboard beneath the sink turned up no box of trash bags, and she'd already learned from sad experience that this older-model dishwasher couldn't handle dishes that weren't already mostly clean. But Ginny, mindful that anger raised her blood pressure, didn't fume. Instead, she took her daily dose of vitamins, minerals, anti-nausea remedies and a potent cocktail of homeopathic tinctures that were supposed to guarantee her optimum health and that of her unborn child. Some went down easier than others, no doubt about that, but she forced herself to take each pill or liquid slowly.\n\nShe wasn't taking any chances.\n\nNoodles's bell collar jingled, and Ginny turned to greet that cat. \"What do you think, Noodles? Should I have that...chocolate cake... Noodles?\"\n\nGinny scanned the doorway to the hall, then the arched one leading to the dining room, expecting to see the cat sitting there, giving her the normal bored look. She'd heard the collar, the jingle of bells she suspected Noodles, should she ever find a voice, would disdain in favor of a collar with spikes. Yet...no cat.\n\n\"Noodles?\"\n\nSlowly, Ginny moved toward the hall, ears cocked for any sound of the bell. There it was, but far away and faint now. Upstairs. Ginny heard the sound of something rattling against the still-bare wooden floors up there, and the soft patter of paws. She climbed the stairs, one at a time, her hand there to pull her along like she was eighty years old and a hundred pounds overweight, which was sort of how she felt most of the time. At the top, she psh-pshed for the cat and listened again for the sound of the bell.\n\nNothing.\n\nShe called the cat's name again, but unlike a dog or even a cat with a pleasant personality, Noodles had never come when called. The sound of a can of food opening would bring her running as fast as her legs could carry her too-tubby body, but to the sound of her own name she was deliberately deaf.\n\nFrom the room Sean liked to call the office but Ginny thought of as the library came another faint jingle and the rattle of something on the wood floor. Ginny peeked in the doorway, but the cat was gone. The room was warm, though\u2014that was a bonus. Inside it on the floor, she found what the cat had been playing with. Bending to pick it up made her head spin a little, dammit, so she made sure to stand very, very carefully.\n\nIt was a small wooden figure. A lady. The paint had worn off, but her carved features were pretty. She wore forties fashion, an animal stole and peplum jacket. She was the size of Ginny's pinky and matched the ones she'd found that first day in the telephone table drawer. Ginny looked her over. The wood was warm in her palm.\n\nThis room had a fireplace that matched the one in the parlor below it, though like the one in the dining room, it had been blocked off, unusable. It also had a set of beautiful, floor-to-ceiling built-in bookcases that went from the fireplace to the opposite wall and around the corner. Gorgeous crown molding. On the other side of the fireplace was a dormer window like the one in her bedroom, but much larger. Inside the dormer area, one of those crawl-space doors.\n\nHer easel leaned against the wall in that dormer, surrounded by the boxes of her painting supplies. She hadn't put it there. Like the paintings in the basement, Ginny hadn't even packed any of this stuff. The last she'd seen it all was in their garage in the townhouse after not even looking at it for months before that.\n\nGinny rolled the figure in her palms, back and forth. Whoever had carved it was an artist, of sorts. How long had it taken him to create this tiny figure? To carve the details in the fox stole, the dress, the expression on the woman's face that, the longer Ginny looked at it, seemed to be a smirk? Had he loved this work, or had he spent the time on it because it was better than facing something else?\n\nGinny put the figure on one of the shelves. \"Noodles?\"\n\nStricken by the thought that the cat had somehow wormed her way into the crawl space and found the bait Ginny had promised exterminator Danny she'd keep her away from, Ginny moved toward the small door. It was a little ajar, cold air blowing in around the edges. Ginny swore she heard Noodles's bell jingling. With a low cry, she tugged the door open to find...\n\nNothing.\n\nWell, not totally nothing. She found some mouse turds and a few of those glue traps, along with a black bait box. Tattered pink insulation. Some weathered cardboard boxes she didn't recognize and refused to open. Oh, and a shit ton of frigid air wafting up from the open spaces under the eaves.\n\n\"Shit.\" If the cat got in there, she could easily get hurt. Ginny shoved the door closed extra tight and made a mental note to ask Sean about weather stripping around the door. Maybe even putting a lock on it to make sure it didn't blow open again.\n\nIn the kitchen, she found a smug-looking Noodles on top of the kitchen table and shooed her off, then thought better of it and picked up the protesting cat to give her a snuggle. Noodles might be a bit of a bitch, but they'd had her since she was a kitten.\n\n\"If something happened to you...\" Ginny kissed the cat's head, ignoring for a minute the way Noodles squirmed. At least until the cat made that low, warning growl that meant she was going to bite. Then Ginny put her down fast.\n\nAffection turned to annoyance quickly enough when she went to put her plate in the dishwasher, though. The cat had helped herself to all of Ginny's leftovers, even licking the plate clean in wide stripes. Great. Now she not only would pee on stuff they left lying around, she'd probably puke too. And in just the right spot for Ginny to find it with her bare feet in the middle of the night.\n\n\"Brat cat,\" Ginny said aloud, but Noodles had once more disappeared.\nChapter Eight\n\nGinny had fallen in love three and a half times in her life. Well, two half times, so maybe that counted as one whole time? So. Four times. She'd fallen in love four times, with five different men, one of whom she'd married.\n\nAs soon as she saw the crimson-upholstered fainting couch with carved wooden legs and accents of gold thread, she knew she had to have it. She'd never wanted anything so much at first glance, not ever.\n\nWell. Maybe once before, but that had been a man and not a piece of furniture.\n\nThis was the first time an inanimate object had moved her to such instant, almost-feral desire. She touched it with reverent fingers, testing the upholstery. It was old, not in the best shape. It didn't even look comfortable, really, unless maybe you were a Victorian lady used to corsets and sitting stiffly upright. It was definitely not the sort of couch you were supposed to loll upon.\n\n\"I want it,\" she said.\n\nSean turned from where he'd been looking at a display of old Looney Tunes glasses in a locked cabinet. \"Hey, look. Like the ones we found in the house. Jesus, they're like five bucks apiece. My mom had the whole set of these. I bet we could get them from her.\"\n\n\"Good luck with that.\" Sean's mom had lots of things tucked away in her cupboards, on shelves, stored in boxes. She wasn't apt to give anything up, though she was fond of making lists about who was going to get what when she inevitably passed away. Which, according to his mother, could be at any moment.\n\nIgnoring the sign that said Please Do NOT Sit, Ginny lowered herself onto the couch, testing the firmness. The legs didn't wobble. A puff of dust came out, tickling her nose. She looked at him. \"I want it.\"\n\nSean's mouth pursed. \"That? Why?\"\n\n\"It's perfect,\" Ginny said simply. There was no other answer. This couch was perfect, she wanted it. It didn't matter the cost or how they were supposed to get it home.\n\nSean scratched his head and cupped the back of his neck with a hand while he gave her a squint-eyed look. \"Where would you put it?\"\n\n\"The library.\" Already she could imagine just how she'd angle it in front of the bookshelves. The pendent lamp from the living room, an end table, a warm and cozy throw. She stroked a hand over the upholstery. \"It'll be my Christmas present.\"\n\nHer husband held out a hand to help her up. He pulled her into his arms, held her close and nuzzled her nose before kissing her lightly. \"It's old. Since when do you like old furniture?\"\n\n\"Newsflash,\" she said. \"We live in an old house now. The kind that antiques look good in. And it's good for the environment. You know. Reduce, reuse, recycle and all that.\"\n\nSean looked dubious. \"Okay. If you really want it.\"\n\n\"I do. I want it. It's so gorgeous, and it will be perfect in the library.\" She ran a hand over the fabric again.\n\n* * * * *\n\n\"I thought you were going to use it as a studio,\" he said in the car after they'd made arrangements for the couch to be delivered.\n\nGinny'd been staring out the window, thinking about telling him she wanted to stop someplace for an early dinner even though she'd just blown their budget, and something in his tone of voice kept her looking through the glass instead of at his face. \"I never said that.\"\n\n\"I thought, when we looked at the house, you said what great light that room had, how it would make a great place to paint.\"\n\n\"I don't remember saying that.\"\n\n\"I do,\" Sean said.\n\nGinny looked at him. \"It's not like I couldn't paint in a library, if I wanted to. What difference does it make what we call it? It's got all those bookshelves, it seems like a library. There's room for a desk in there for you, if you want to make it an office, which is what I thought you wanted it to be.\"\n\n\"No. The little room is going to be the office.\"\n\nShe distinctly remembered him calling the room with the bookshelves the office. She supposed it was too late to convince him to take her out for dinner too, since they were pulling up the street to the house. Sean parked, then peered upward through the windshield.\n\n\"Wish we had a garage.\" He pointed, frowning. \"Wonder why he never rebuilt it after the fire.\"\n\nGinny also looked through the glass toward the empty space where Sean's coveted garage had once stood. It was strange to see a house bare of a garage in a suburban neighborhood like this. Every other house on the block had one. Even their townhouse, tiny as it was, had had one. Here there was plenty of room in the driveway for at least four cars, but the trip from the car to the house wasn't even sheltered by a breezeway. It was on the ten-year plan, along with landscaping and finishing the basement.\n\n\"Who knows. Money, probably. Isn't everything always about money?\"\n\n\"Not everything,\" her husband said.\n\nGinny got out of the car. She went to the house and straight to the fridge to get a snack. She'd been looking forward to something ooey-gooey\u2014cheese sticks or chili fries, something like that. Instead, she had a choice of organic yogurt and granola or a handful of almonds. Totally not satisfactory. She found some ice cream in the freezer, contemplated a bowl, decided to eat it straight from the carton. Because she was only going to have a bite or two...right?\n\nSean must've had the same idea, because the carton was almost empty. Ginny frowned but scraped the cardboard sides with her spoon. So, she'd have to finish all of it; that was no big deal. She had the spoon in her mouth when Sean came into the kitchen behind her.\n\n\"What's for dinner?\"\n\nGinny, metal still tucked against her tongue, looked at him for a long half a minute before she slowly removed the spoon. \"I don't know. What is for dinner?\"\n\n\"I don't know.\" Sean opened the fridge. Looked inside. Looked at her. Closed the fridge. \"Are you going to make something or...?\"\n\n\"There's plenty of lunchmeat and deli rolls in there. Macaroni salad.\" Ginny scraped the last bit of ice cream and licked the spoon clean, then went to the trash to dump the carton. Through the back door, she caught a glimpse of two red coats. Blond hair. Those kids from next door were in her yard again.\n\n\"I thought you'd cook something.\"\n\n\"Did you?\" she said absently, trying to see exactly what the kids were doing.\n\nSean was quiet for another minute. \"No.\"\n\n\"Have a sandwich,\" Ginny told him and went out the back door.\n\n\"Where are you going?\"\n\n\"I'm going to chase those kids out of our yard,\" she told him and let the door close behind her.\n\nIn the side yard she found a soft-foam football on the grass and a battered wooden wagon half-hidden in the leaves. No kids, though. They must've ducked through the hedge. The yard itself was a mess even without the abandoned toys. Their yard had only the two big trees, but both had completely shed their leaves and none of them had blown away. The grass was also ankle high, brown and dead instead of lush and green, but overgrown just the same. Ginny scuffed through the piles made by the wind shoving the leaves against the hedge and the house, kicking her feet the way she had as a kid.\n\nThis was the side of the house with windows into the basement, and lined up along the window well, Ginny found a set of carved wooden figures. Like the lady in the fur stole, the figures wore clothes of past fashions, and their sizes ranged from the length of her pinky to match the rest of her fingers too. The weather had worn some of these harder than the ones she'd already found. How long had they been here? Ginny bent as best she could to peek into the window, but could see only dirty glass until she reached to rub a circle clean. Then, still nothing. She stood without moving the figures, wondering what sort of game the kids next door had been playing.\n\nIn the yard under the big tree, she tugged the rope that had once been a tire swing. It needed to come down. The tree itself needed a good pruning. And the leaves... Oh God. All the leaves.\n\nGinny looked toward the house. Sean had promised her he'd take care of the yard. Just like inside the house, he'd made her promise not to unpack the boxes, not to move the furniture, not to exert herself with all the tasks he was going to do. Yes, they'd had a lot of rain that made it hard to work outside. Yes, he worked long hours and had the responsibilities of school too. But he'd not only promised, he'd insisted that she promise him she wouldn't rake or mow or put stuff away. And now what? They were living out of boxes, never able to find anything when they needed it. The yard was a trash heap.\n\nShe would not be angry.\n\nShe would not lose her temper.\n\nShe would not say things that could not be taken back.\n\nInstead, she went to the shed and found a rake and the box of garbage bags she'd bought especially for the leaves. She took them into the yard and started raking. It would be dark soon, and her hands were cold, but so far Sean hadn't bothered to come out and see what she was doing, so she was going to keep doing it. It wasn't hard work, not even for a pregnant lady. Holding the bag open while she put the leaves inside and fought the frigid wind that had sprung up proved to be a lot harder. Frustrated, she looked toward the house, the windows in the parlor glowing faintly golden. Upstairs, the light in the master bedroom was on, and also the one in the hall, which touched the windows in the library\u2014it was a library, she thought fiercely.\n\nGinny's nose had begun to run with clear snot she didn't want to wipe with her bare hands. She dug in her pocket for a tissue and found only a few crumpled receipts. The dark had truly fallen by this time, and the temperature had dropped. She had no mittens or scarf, and the wind bit at her. This suddenly seemed a fool's task. She'd half filled one bag.\n\nLeaning on the rake, Ginny looked toward the house. Sean stood in the library, silhouetted in the window. How long had he been watching? Ginny waved, certain it was too dark now for him to see her, but after a moment he returned the gesture.\n\nScrew this, she thought. Tomorrow she'd remind him again about the yard and insist he let her help him, even if was just to hold the bag open for him. They'd halve the job and would be spending time together. Two birds, one stone, all that.\n\nShe did want to put the rest of this small pile into the bag, though, if only so the wind wouldn't scatter the leaves she'd already taken the time to gather. Ginny leaned the rake against a tree and bent to scoop the leaves, shoving them into the bag as fast as she could with numbed fingers. Her teeth chattered, more snot ran, and she was just about to give it up entirely when her fingers sank into something soft.\n\nThe leaves had been crunchy, rustling, brittle, but whatever this was clutched and clung to her fingers. Ginny yanked her hand back, shaking it, but it was coated in...something. She screamed and scraped her hand along the grass. When she turned, her shadow turned with her, no longer blocking the light. She didn't have a clear view of what she'd just grabbed, but it was clear enough.\n\nTemperatures were cold, but not yet freezing. The past few weeks had seen rain, a lot of it. The squirrel into which she'd just sunk her fingers was both mushy and saturated, its belly bloated. It stunk. Its eyes were gone, she could see that much, and its little mouth gaped with teeth that were too long. Ginny backpedaled with her hands out in front of her, her shriek locked in her throat.\n\nIn the kitchen, Sean sat at the table with a towering Dagwood sandwich in front of him. She pushed past him to the sink, where she ran the hot water aggressively and scrubbed at her hands. More soap. More scrubbing. Oh God, so gross. So gross.\n\n\"What the hell are you doing?\"\n\n\"I was bagging up leaves, and guess what I found. A dead squirrel. A rotten dead squirrel. And guess what I found it with, Sean? My hands!\" She shuddered, repulsed, but laughed too at the absurdity of it. \"It must've gotten into the bait and gone outside to die, just like the exterminator said.\"\n\nSean crossed to the sink to grab her hands. \"What? Jesus Christ, Ginny, what the hell were you thinking? Are you okay?\"\n\n\"I didn't puke, so yes, I guess so.\" She shuddered again, but the disgust was fading as the hot water and soap washed her clean. She tried to tug her hands from Sean's grip, but he held her tight.\n\n\"Get something,\" he said.\n\n\"Something like what?\"\n\n\"Hand sanitizer. Alcohol. Something! No, I'll get it.\" Incredibly, he grabbed a bottle of whiskey from the lower cabinet and spun the lid.\n\nGinny stared. \"What are you going to do with that? Light my hands on fire?\"\n\nSean looked at the bottle, sighed, put the cap back on. \"Maybe I should drink it.\"\n\n\"Sean...\" Ginny turned off the water and dried her hands on a towel. In the yard, knuckle deep in rotting squirrel, she'd screamed. Here in the bright kitchen with her hands clean, she didn't want him to keep fretting. \"It's not that big a deal. Really. It was gross and startling, but that's all.\"\n\n\"You shouldn't have been out there at all. Raking leaves? In the dark?\"\n\n\"I didn't start when it was dark,\" she pointed out. \"And the leaves needed raking. It's not a big deal.\"\n\nExcept that it sort of was.\n\n\"I thought I heard you go upstairs,\" Sean muttered. \"I didn't know you were still out there, I'd have come out and helped you.\"\n\nGinny pulled a bottle of hand lotion from the cupboard and rubbed it carefully into her skin. \"What do you mean, you didn't know I was out there? You saw me go out the back door.\"\n\n\"I thought maybe you came in the front.\"\n\n\"You thought I came in the front door,\" Ginny said. \"How on earth could you think I came in the front door? You saw me from the window upstairs.\"\n\n\"I didn't see you.\" Sean went back to the table and scraped his chair back to sit again in front of the sandwich.\n\nGinny's fingers curled so tight into her palm her fingernails dented the skin. Not quite painful\u2014not yet. But the potential was there. \"I saw you. In the window upstairs. I waved. You waved.\"\n\n\"I was in here making myself something to eat since you weren't cooking dinner.\"\n\nOh. No. He. Didn't.\n\n\"Yeah well, I was outside raking the leaves since you weren't doing that.\" She eyed the table. \"I notice you didn't make me a sandwich.\"\n\n\"How was I supposed to know you wanted one? You were upstairs. I figured you were pissed off at me, so you went upstairs to...you know. Be pissed.\"\n\nIncredulity kept her speechless for a minute as her mouth worked without finding words. And, ah yes, there was the bright sting of pain in her palms. Ginny focused, eased her grip. \"Sean. Seriously? You are kidding, right?\"\n\nHe looked at her, eyes narrowed. She couldn't tell if he was challenging her or genuinely oblivious. \"No.\"\n\n\"I saw you,\" Ginny said. \"Upstairs. In the window.\"\n\n\"I wasn't up there.\" He bit into the sandwich, spoke around a mouthful of sloppy lettuce and mayo. \"You want some of mine?\"\n\nShe didn't want any of his. She'd starve before she ate any piece of that dripping mess. Plunging her hands into a dead and bloated squirrel hadn't made her vomit, but that sandwich might.\n\nIn times past, she'd have raged at him until he turned his back and gave her the silent treatment, but this was supposed to be a fresh start. Right? All of this, everything new and fresh and different than it had been before. Every fucking piece of it.\n\nSean cut off a piece of the sandwich and held it aloft, ignoring the slide of tomato seeds and mayonnaise over his hand. \"Here. It's good. Eat some.\"\n\nGinny sat at the table across from him. \"Yes, okay,\" she said. \"Thanks.\"\nChapter Nine\n\nThis would be the baby's room.\n\nGinny had never wanted to know the gender of her child in advance. So few surprises in life were truly wonderful, and she'd always imagined the moment of birth to be the perfect time to discover if she was the mother of a daughter or a son. Because of her previous problems, this time around she'd been subjected to every possible test and what felt like an insane number of ultrasounds. You could get them in 3-D now, a scarily vivid image of your unborn child's face presented to you on a piece of photo paper or on disc to upload to the Internet and show off to all your friends. It was hard to make sure the techs didn't slip up and give away the baby's sex, even though most of them were genuinely eager to help keep the secret. Ginny thought Sean knew, though. She'd heard him murmuring to the tech once when she was getting dressed. Something about \"if you could make a guess.\"\n\nShe thought it was a boy.\n\nThat didn't mean she was decorating this room in any shade of blue, though. Both her sister Peg and sister-in-law, Jeannie, had done up their nurseries in pastels, with babycentric designs, and then complained when the kids got older and the rooms needed to be redecorated. Ginny had decided to go with a fun jungle theme, using vinyl stickers against brightly painted walls. The stickers could be pulled off later and replaced. The bedding could be changed as a child of either gender graduated from a crib to a bed, and the colors she'd chosen would work for even an older child. Two walls lime green, two a rich chocolate brown. This room also had a dormer. She envisioned benches with cushions, a cozy place to read and play. Maybe some curtains to make a cave or castle, depending on the kid's personality.\n\nFor now, it needed a thorough cleaning and all of the fixtures and trim taped off. Sean had forbidden her from doing any painting, but he hadn't been able to argue too strenuously against her taping things. At least so long as she promised not to get up on a ladder.\n\nThis pregnancy had been hard on her physically, but mentally Ginny had taught herself to feel better than she ever had. Less worried, for one thing. The more she'd learned about the possibilities of mental and physical defects, the less frightening having a child with special needs seemed.\n\nFor Sean, on the other hand, the more he knew, the more uneasy he became. The facts and statistics she devoured unsettled him. Sean didn't think he could balance a checkbook or clean a strange stain off a shirt or organize a surprise weekend away, and in the fourteen years they'd been married, Ginny'd never been able to convince him otherwise. There was no way she could convince him he could be a good father to a baby with problems. All she could do was not take chances. Reassure him. Let him fuss over her. Ginny had learned not to tell him what she discovered from her online research.\n\nShe'd learned not to tell him lots of things.\n\nWith her music playing, this time from a carefully chosen playlist that contained nothing to sneak-attack her emotions, Ginny gathered her bucket of supplies, all organic or nonchemical based because Sean had insisted he didn't want her breathing in toxic fumes. They'd had a cleaning service come through the house before they moved in, but she hadn't been particularly impressed with the job they'd done. It was bad enough that more of the former owner's belongings had been left behind than they'd wanted, but knowing the dust balls and finger smudges were someone else's grime...gross.\n\nSean's mom, Barb, had offered to come over and help, but that would've been more of a nightmare than trying to get water stains out of a tub without using bleach. Besides, Ginny had been looking forward to cleaning. Maybe it was all that time spent with Gran, who'd not only had a \"hired woman\" for most of her life but also spent more hours on her knees scrubbing the floors than she ever had in church.\n\nGinny's mom had been a terrible housekeeper, in direct response to her upbringing. Just like she'd never host a party, Christmas or otherwise, Ginny's mother also had never done more than the bare minimum when it came to keeping them from living in squalor. Beds unmade, dishes in the sink, dust everywhere\u2014that was Ginny's house growing up. It wasn't exactly like living in the pigsty Gran called it, but it wasn't quite as tidy as most of her friends' houses either, with moms who stayed home and presumably spent their days making use of old toothbrushes to keep their grout from going gray.\n\n\"I have better things to do with my time,\" Ginny's mom always said, and as a working woman herself, Ginny had often found that to be true. The kind of clean Gran had demanded took a lot of time and\/or money to maintain, and though Ginny always made sure to make the bed in the morning and put dirty dishes in the dishwasher, she also sometimes left her laundry in the basket for days on end before folding it and putting it away.\n\nNow, of course, she wasn't working and had nothing but time, and the combined clutter of a just-moved-into house with her being home all the time to actually see the mess...well, she understood now what her gran had meant when her fingers itched to \"get to fixing things.\"\n\nThis room wasn't in terrible shape. Mostly it bore the touches of time and disuse. Cobwebs in the corners of the ceiling\u2014she got those down easily enough with a broom. Something that looked like shavings in a couple of the corners, possibly from the insects Danny was supposed to have taken care of. She swept the room quickly, then drew a bucket of hot water from the bathroom and mopped the wood floor. They hadn't yet shopped for furniture, though she'd gone through a couple catalogs and picked out what she wanted. They'd need a carpet in here too, or maybe a couple fun throw rugs.\n\nShe looked around the room, envisioning it painted and furnished. There in the corner, a rocking chair. Her sister swore by a glider, but Ginny had seen a bent-cane rocker at one of the Amish markets a few summers ago. When she pictured herself with a baby in her arms, it was always in that chair. The crib could go here, she thought, away from the windows and the drafty cubbyhole door. The dresser, there. A bookcase in that corner, because her child would certainly be a reader. And in the closet...\n\n\"Ugh.\" She stopped, nose wrinkling when she opened up the double wooden doors. The dust in there was palpable, along with the acrid tang of mouse droppings. Old mouse shit was still shit, she thought with a grimace, and grabbed the broom.\n\nIn the closet, when it was clean, she planned to install a set of wire shelves and hangers to store toys and clothes. Like most of the others in the house, including hers, the one in this room was deep but narrow. Not really a walk-in. Almost like an afterthought. This one had nothing but a single bar running from front to back and a shallow wooden shelf above it. Lots of functional space made useless by that old-fashioned setup. She'd make it better.\n\nFor now, Ginny concentrated on sweeping out the dust bunnies and mouse poop and dead beetle shells. It was a good thing Barb wasn't here to help her out. Sean's mom would've screamed and tossed up her hands at even the hint of a little poop, no matter how ancient. Ginny laughed a little guiltily at the image. Barb meant well; it wasn't her fault Ginny found her mother-in-law as useless as the shelf in this closet.\n\nHer broom bumped that shelf as she worked, rattling the wood on the pegs holding it up. As it turned out, the shelf wasn't one long piece of wood, but three or four foot-long sections. She figured that out when she bumped it again and only part of it lifted, sending down a drift of more dust and pellets.\n\nGross.\n\nShe'd promised Sean she wouldn't exert herself or climb on ladders or lift heavy paint cans, but there was no way she could leave the filth. From downstairs, she brought the pantry step stool and set it up. That was only two steps; surely he wouldn't get on her case about that. Armed with a roll of paper towels and squirt bottle of vinegar and water, Ginny stepped onto the bottom step. Then the next. The ceilings in this house, including the closet, were a luxurious nine feet, but that meant that even with the stool, she wasn't quite tall enough to see all the way to the back of the shelf, or all the way to the end.\n\n\"A little at a time, baby, just a little at a time.\" Gran's words again, used for all manner of tasks that had seemed too daunting. One jar at a time while canning the endless supply of tomatoes from her garden, one pan of cookies at a time when she got to baking for the Christmas party, one stitch at a time when she was working on her embroidery. Ginny supposed the same held true for cleaning dirty closets. One wipe at a time.\n\nThe first paper towel she swiped across the top came away black with dirt and speckled with dead ants. Her swipe also brought along a couple ant traps. That explained the shavings in the corners. She dropped the paper towel. Spritzed more cleanser. Wiped again. This time was better.\n\nGinny used six paper towels to clean that section of the shelf, another four for the next. Then she had to get down and move the stool over a few inches. As she did, above her, the wood creaked. She straightened, looking, but saw nothing. Heard nothing. She looked around the closet and saw nothing there, either.\n\nGinny took a step back, looking upward again. Set into the ceiling was an attic access panel similar to the one in the hallway, though in here there was no room for a pull-down ladder. Ginny held her breath, listening, but the sound didn't repeat.\n\n\"Right,\" she said aloud, slapping her dirty hands against each other and wishing she'd thought to grab some rubber gloves. \"Come out, Gary Busey, wherever you are.\"\n\nThe joke didn't seem so funny when she herself had just moved into an old house with a sort of creepy crawl-space attic. And she was alone. And she'd heard a weird noise.\n\nGinny wasn't in the habit of talking to herself, at least not out loud, but more words slipped from her lips as though speaking them instead of just thinking them gave them the power to be true. \"It was nothing. Just like the nothing you saw in the hallway. Just like the nothing you saw in a window. Shadows, that's it. And old houses creaking. That's it, Virginia Eloise.\"\n\nThat's what her gran had called her, ever so formally. Nicknames, she said, were for people who didn't have beautiful names. She'd never referred to Ginny's mom by her name, always calling her honey or darling or sweetheart, affectionate terms that had curled Ginny's mom's lip because she said they weren't meant fondly but as a way of pointing out how unbeautiful her name was. Gertrude, named after Gran's mother-in-law. Ginny's mom went by Trudy and had been careful not to name her own daughters after any relatives.\n\nSaying it aloud that way settled something inside Ginny. Almost like Gran had said it herself, which was impossible, both because Gran wasn't here, and also because Gran had been incapable of saying much of anything that made sense for the past few years. She didn't talk much at all, in fact. She mostly sat and stared out the windows of the nursing home, even when Ginny came and brought her special treats that she wasn't supposed to have on her diet.\n\nFor the second time since moving into this house, emotion blindsided her. Ginny put a hand on her heart, mindless of the dirt on her white shirt, although it left a smudge she'd have to fuss with later and which would never quite come entirely clean. She closed her eyes, her other hand on the back of the stool for balance as she bent over with the force of her sorrow. Gran was old and had always been a little wacky, but to see her losing her mind...to watch her actually lose all sense of not just the people around her and where she was, but of herself...\n\nGinny'd been ignoring it for a long time. Her monthly visits were filled with forced cheer and lots of bright, one-sided conversation, and she pretended the headaches that always came home with her were from the stink of the cleaning chemicals in the home, not from the stress of smiling so hard her face hurt. She hadn't allowed herself to think fully on Gran's condition, or how it made her feel, and now it all swung up and hit her in the face like a stepped-on rake.\n\nBam. Slam. Right between the eyes, the pain sudden and fierce enough to make her gasp as she fought against tears and lost. Heat slipped down her cheeks and clogged her throat. She pressed her eyes closed tighter and shook her head, muttering aloud.\n\nNo, she wouldn't do this. She wouldn't let this happen again, this slip-slide into melancholy for which there was no cure. Hormones, circumstances, her own choices, the vagaries of fate\u2014whatever it was, she wasn't going to let herself give in to this burst of anxiety again. She just. Would. Not.\n\nBlinking rapidly, Ginny used a clean paper towel to scrub at her face. She drew in a few sharp breaths until the dust made her sneeze. She blew her nose, cleared her throat, got herself under control. She had a closet to clean and lots of other things this house needed. No time to break down over what she couldn't change.\n\nBolstered, determined, she got back up on the stool and gave the shelf a swift sweep with a handful of paper towels. Something skidded along the wood with her hand, but she was moving too fast, had been too hasty. More than dust and a few dead ants or some dried turds tipped over the edge of the shelf this time.\n\nThis time, a dead mouse hit her in the face.\n\nShe knew what it was at once, the desiccated corpse with its remnants of fur and dried worm of a tail. She was already crying out in revulsion, not fear, as it bounced off her mouth\u2014oh God, her mouth!\u2014and hit the floor without a noise. Ginny flailed and clutched first at the shelf for support, but the section she grabbed shifted and moved, tipping under the sudden weight of her fist. Something deeper toward the back moved and shifted too, but she lost sight of it as the section slid off its supports and cracked against the light fixture, first blotting the light into shadow and then distinguishing it entirely when the filaments in the bulb broke.\n\nShe fell.\n\nGinny braced herself for the pain of hitting the floor with her ass but managed to save herself, just a little, by stepping backwards and landing on one foot. Her ankle twisted, and she'd have gone all the way down except that the closet was too narrow to allow her the space. She hit the wall with her shoulder instead and left a small dent in the plaster.\n\nWith another shuddering cry of disgust, Ginny pushed backwards out of the closet. Panting, she spat the taste, real or imagined, of dead mouse and didn't dare lick her lips or scrub at her mouth with her bare hands. She got to her feet and went to the hall bathroom, where she washed her hands several times under the hottest water she could stand, then scrubbed her mouth.\n\nMouth dripping, face still twisted from the gross-out, Ginny caught sight of her reflection. Her throat worked\u2014she wasn't sure if she meant to cry again until a deep, low and grinding clutch of laughter pushed past her lips.\n\nOh God. Oh gross.\n\nNow she was even more happy Barb hadn't come over to help, because if she'd come across a dead mouse in any form, especially one that had touched her face, she'd have gone catatonic and had to be sedated. As it was, Ginny half thought she might puke, but a few sips of water settled her. So did some breathing.\n\nIt wasn't the mouse itself, since it was harmless and sad, a caricature of a rodent that had been squashed flat by some chasing tomcat's mallet. It must've died in the closet and dehydrated or mummified. No, it was the fact it had landed on her mouth, her lips... Ginny shuddered and washed her face again.\n\nOf all the gross things that had ever happened to her, including the dead squirrel, she thought this might be the worst. And as far as unexpected contact with deceased rodents went, Ginny'd had her lifetime allotment. Still, the trauma was fading by the time she finished in the bathroom and went back to the baby's room to gather up the poor thing and dispose of it along with all the dirty paper towels.\n\nThat task finished, it would've been easy enough for her to abandon the rest of the closet cleaning for another day. Her ankle hurt, though it didn't seem to be swelling, and her heart was still beating a little too fast. Her head felt a little spinny. But if she didn't finish now, she wasn't sure who would.\n\nPlus...something had moved when the shelf shifted. A box, she thought. Or a suitcase. Something solid, definitely not any kind of dead thing. At least she hoped not.\n\nThe light bulb in the fixture hadn't shattered, thank God, so she didn't have to clean up or explain broken glass. But she had no idea where to get another one in the mess of boxes downstairs. She'd have to take one from another fixture, though they too were burned out, she discovered when she pulled the chain in the other two bedroom closets, wondering what might be lurking on their shelves. Finally, she took the one from her bedroom closet, mentally adding light bulbs to the list that never seemed to get any shorter.\n\nFinally, light bulb replaced, stool settled firmly on the floor so it wouldn't tip, Ginny climbed up again to look at what was on the shelf. It was a suitcase, what her gran had called a \"train case.\" Her mom had used one as a makeup case when Ginny was small. Hers had a mirror inside and a removable shelf to separate the top from the bottom. It was blue and bore the initials of some dead aunt.\n\nThis case was of a similar size. Olive green, though the dust on it meant the color might indeed be brighter. Ginny pulled it gingerly toward her, careful not to tip this section of shelf in case it was as unsecured as its neighbor had been. The bulb she'd replaced was brighter than its predecessor, bright enough to chase away all the shadows even in the farthest depths of the closet. Even so, she was so focused on the case that she didn't notice the bones until she'd taken it by the handle and was half-turned to step down from the stool.\n\nTiny piles of bones, at least three, with some random bones scattered in between. Tiny skulls with long teeth. Beside the piles, tucked a little farther back on the shelf and on the opposite side of where she'd found the mouse, the light glinted off several plastic sandwich bags with misshapen forms inside them. Fur, bones, the spread of what must've once been the goo of blood and other fluids but which time had dried.\n\nThe smell, she thought, must've been atrocious.\n\nCarefully, she got down to set the case in the middle of the bedroom. Then, armed with the garbage bag and paper towels, the bottle of cleanser close by, she scraped the first pile of bones toward her. She had to stand on her tiptoes to get to the last set. Again, she wished for rubber gloves when her fingertips touched the plastic, and she half expected it to stick to the shelf, but the bags slid without resistance. Hamsters, she thought with a glance inside. Orange and white fur.\n\nShe put everything in the garbage bag and went back to the bathroom to scrub her hands. Then she took the garbage bag outside and stuffed it in the can. Back upstairs, she finished cleaning the shelf until the entire length of it, every section, gleamed with the cleanser and the closet smelled of nothing but vinegar.\nChapter Ten\n\nGinny told Sean about the case, though not about the bones or the dead mouse, or falling off the step stool. She did tell him about the light bulb, since he was sure to notice the one missing in their closet, though all she said was that it had burned out, not that it had broken. She tempered her disclosure over a full dinner of roast beef she'd done in the Crock-Pot with some onion-soup mix and a little red wine, baked potatoes, a nice salad decorated with dried cranberries and almonds and a sprinkling of bleu cheese. Plus, she waited until his mouth was full before she told him she'd found something while cleaning, so by the time he'd chewed and swallowed she could focus the conversation on the discovery and not her actions during it.\n\n\"It was there for a long time,\" she told him, picking at her own salad. It had seemed like a good idea to make it with all the extras, but she'd become so sensitive to smells and flavors that everything was jumbling together in a sensory overload. \"It's a girl's case, though. So I don't think it belonged to the owner or his son.\"\n\n\"How do you know it belonged to a girl?\" Sean speared another fork of meat. He sighed as he chewed, closing his eyes briefly in an almost-sexual expression of delight.\n\nIt amused her, that expression. She knew him so well, after all this time, it felt almost unfair to be so manipulative at keeping his attention directed on something else. But only almost.\n\n\"Because,\" she said with a point of her fork toward him, \"it just is. Boys don't use train cases. The kind with a liner and a mirror and stuff inside.\"\n\nHe drank slowly from his glass of wine, savoring it with another of those sighs. Despite an occasional craving, Ginny wasn't a big wine drinker, but he made it seem so delicious her mouth watered in envy. Of course that was her way, wanting what she couldn't have, even though she knew she wouldn't like it if she got it.\n\n\"They could,\" he said.\n\nShe laughed a little, though it faded quickly when she thought of the tiny skeletons, corpses that had been kept in baggies. That seemed more like a boy thing, if you were going to go by stereotypes. Puppy-dog tails and all that. \"I guess so. But I doubt it. It's a girl's case, I know it.\"\n\n\"What's in it?\"\n\n\"I don't know.\"\n\nSean paused with a crescent roll halfway to his mouth. Both brows lifted. \"Why not? You find this grand, secret treasure and you don't open it?\"\n\n\"I don't know; it didn't seem right.\" Ginny shrugged and reached for the bread basket. The crescent rolls, at least, seemed appealing, which made sense since of all the food she'd made, they were the only thing she hadn't made from scratch. Full of preservatives, she thought and buttered one anyway, before tucking it into her mouth. Just as she'd thought, delicious.\n\n\"After dinner.\"\n\n\"No.\" She shook her head. \"I told you, it doesn't seem right. It's not ours.\"\n\n\"Everything in this house is ours. Bought and paid for.\" Sean gestured with his fork, looking around the dining room, where Ginny had set up a card table with a fancy cloth and the good dishes, since she'd finally found them in an unmarked box.\n\nShe studied him around the wedding-gift candelabra. They'd never used it in their townhouse and here they were, using it again. \"Yeah, even the things we didn't want.\"\n\nHe laughed and drank more wine. He wiped his mouth with a cloth napkin that had also been a wedding gift. All these things they'd never really used before, yet seemed so right now. Wine made Sean horny, she remembered that now. She poured him another glass.\n\n\"C'mon. It's not that many. The telephone table, that throw rug. The case. That's not so much, when you think about how much stuff was in here when we looked at it, remember?\"\n\nShe did. The house had been fully furnished, top to bottom, though just the one old man lived in it. He'd had some nice pieces she sort of wished she'd thought to ask for, now that she knew how accommodating the son had been about unloading everything, but at the time her mind had still been awash with modern lines and small rooms. She hadn't been able to think ahead to what it would be like to actually live here and fill all the spaces that cried out for an ornate hand.\n\n\"Well. Who knows what else we might find?\" she pointed out, meaning that the cleaning service had done a woeful job.\n\nAs wasn't uncommon, her husband wasn't on the same track. He woo-wooed with his fingers and made the accompanying sound. \"Yessss, we might find something scarrrrry...like bonesssss...\"\n\n\"What?\" Startled, Ginny's fingers twitched and knocked her fork off the table. \"Why would you say that?\"\n\nSean gave her a curious glance. \"Like in that movie you and Billy are always quoting.\"\n\n\"There weren't any bones in that movie. He killed people and buried them in the backyard.\"\n\n\"But they found bones.\"\n\nGinny had to remind herself that Sean had never seen the movie; he knew of it only from what she and Billy said. Still, she hated the way he talked about it like he knew it and she didn't, like she was wrong and he was right. \"No. I don't think so.\" Dammit, now she couldn't really remember if there'd been bones in the movie. Maybe from the dog... Shit, she couldn't remember.\n\n\"Anyway,\" Sean said dismissively, \"you knew what I meant. So if we have Gary Sinise living in our attic...\"\n\n\"Jesus, Sean,\" she snapped, irrationally annoyed now. He'd listened to her and Billy talk about the movie for years. \"Gary Busey. It's Gary fucking Busey.\"\n\nSilence fell between them as she bit back her anger. He gave her that look she also hated. The one that said she was being an irrational bitch, but he forgave her. He always forgave her.\n\n\"Sorry.\" She sipped from her glass of lemonade, the taste sour enough to pucker her lips but still not as bad as the taste of her anger.\n\nAt first he said nothing, but then he shrugged. \"If you're not going to open it, what are you going to do with it?\"\n\n\"Throw it away.\"\n\nHe looked shocked. \"You can't just throw it away! What if there's something important in it?\"\n\n\"Like what? Money?\"\n\n\"There could be. That would be awesome.\" Sean grinned.\n\nGinny's mouth pursed, not quite an answering smile but easing toward one. \"A treasure map?\"\n\n\"Yeah!\"\n\n\"A winning lottery ticket, never cashed in. Keys to a safe deposit box in Switzerland? Oh, I know.\" She snapped her fingers. \"The Hope Diamond.\"\n\n\"That's in a museum somewhere, and, besides, it's supposed to be cursed.\"\n\n\"Okay, just some other big-ass diamond, then.\"\n\nHe laughed again. \"That would be good, huh? C'mon, babe, you can't just throw it away. At the very least, if you're not going to see what's inside, you should see if the son wants it.\"\n\nHe was right, she couldn't pretend otherwise.\n\nBut they were both wrong about what was inside the case. No money, no jewels, no map leading to a trove of buried treasure. There was a key, though, the tiny kind that was meant to fit into a diary. Ginny'd had one of those as a kid, the cover powder blue and fake leather. Her brother had broken the lock with one sharp tug, and she'd never written in it after that.\n\nThe diary itself was tucked beneath a sheaf of photos, most of them Polaroids yellowed with age. Some had scrawled descriptions in the white space along the bottom, but most were unidentified. There were a few more of those carved wooden figures, a whole set. There were also a number of childish drawings of typical things: a girl with a pony, a princess, a family, a tree, a rainbow. None of them were signed.\n\n\"Hey, look.\" Sean pulled the diary out and tugged the cover, but the lock held. \"Give me the key.\"\n\nGinny snatched it up before he could get it. \"No! You can't... Jesus, Sean. You can't read it. This belonged to someone.\"\n\nHis puppy eyes did nothing for her. \"I told you, it's ours now.\"\n\n\"You obviously never kept a diary.\" She scowled and curled her fingers tight around the key.\n\n\"No. Did you?\" He paused, gave her a head tilt, an up-and-down glance. \"Do you?\"\n\n\"I don't now.\"\n\nHe gave up on the diary and sifted through more of the stuff. \"Nothing in here looks very interesting.\"\n\n\"Don't sound so disappointed.\" The key warmed in her palm, and Ginny put it in her pocket, thinking she would have to remind herself to take it out or it would end up getting caught in the washing machine filter. \"You know if it really had been money, we'd have felt compelled to give it back to the rightful owner.\"\n\nSean reached to tug her onto his lap. He nuzzled her neck, and she let him. \"And I told you, babe, we are the rightful owners.\"\n\n\"Pffft.\" His hands on her hips felt good. So did the brush of his slightly scratchy beard on her throat. She arched into his touch.\n\n\"Look. If something happens with the roof or the heating system...\" his brows lifted when she pulled away to look at him, \"...yeah, I know, don't jinx it. But if something does happen, if that tree out there falls in the yard, we have to cut it up, right? That's our responsibility now.\"\n\n\"Well...\"\n\nHe settled his mouth back against her, teeth pressing gently as he spoke, \"So, I say, whatever we find is ours to keep. If we take the bad, we have to take the good too.\"\n\n\"Isn't it usually the other way around?\" She'd said it lightly, meaning nothing, but Sean was silent for a few seconds too long.\n\nThen he said, \"Yeah. Usually.\"\n\nHis hands drifted up her back. Pulled her a little closer. He kissed her throat and her slightly exposed collarbones. His breath was warm but made her shiver. When he shifted against her, the bulge of his erection pressing her hip, a flare of unexpectedly strong desire made her draw in a breath.\n\nGinny looked down at him, her hands caught in his hair as she tipped his face to hers. She kissed him long and slow, before pulling away a little. \"Come upstairs with me.\"\n\nHis gaze flickered, the corners of his mouth dipping low. Once more she was both a little sad and a little annoyed that after all this time and all these years, she knew him so well he hardly had to say a word, but he barely knew her at all, even when she was spelling it out.\n\n\"I will. Later. I have some homework I need to finish up.\"\n\nShe looked at the clock. \"Now?\"\n\n\"Yeah.\" He nuzzled against her throat again, but Ginny pulled away.\n\nShe got off his lap and started clearing the table. Sean got up and went into the living room. She watched him from the dining room as he rummaged in his bag. He pulled out the leather-bound notebook she'd bought for his birthday the year he started his coursework, and a handful of pens. She watched him tuck his phone into his pocket. When he straightened he threw her a smile she had to force herself to return.\n\n\"Hey, beautiful,\" her husband said.\n\nGinny cleared the table by herself again.\nChapter Eleven\n\nGinny didn't tell Sean about the bones in the closet, but that didn't mean she'd forgotten them. For the first week after her discovery, as she puttered around the house and barely managed to accomplish anything, since her husband still insisted she wait for him to \"help\" her, Ginny kept imagining opening another closet, a drawer, a cupboard, and finding another pile of dead-rodent remains. As she worked slowly, cleaning each cupboard with wet cloths and the baking soda and vinegar mixture that never seemed to clean as well as bleach, she never let her fingers trace the deep places, full of shadow, without thoroughly checking them first with a flashlight.\n\nShe hadn't forgotten the case or the diary within. Long ago, before she met Sean, Ginny'd had a college boyfriend. They'd met at a party in the first few months of their sophomore year, and it had been instant love. Well, lust. With the maturity of time behind her, she was willing to amend it to that. At any rate, she'd fallen hard and deep, her feelings for Joseph an abyss with walls so high she could barely see the sky. And with this love came the ever-present knowledge that she loved him more than he loved her. That she wanted him more than he wanted her. That somewhere, someday there would be something that would take him from her, whether it was a sports game, a drunken night out with his buddies...a girl.\n\nGinny'd become adept at sifting through his drawers without Joseph knowing. In the days before email and texting or even Connex, all she had was the shoebox full of love letters she found on his top closet shelf. The ones from Ginny, and the ones from someone else. Her discovery had led to an argument, the vehemence of which seemed to surprise him, and though it broke her heart to do it, Ginny was the one to end the relationship.\n\nNow, in the life she'd come to after leaving Joseph, Ginny didn't regret the breakup. She never wanted to feel that way about anyone, ever again, that lost and sinking feeling, that drowning. She never wanted love to be a prison. But some part of her did regret, for a very long time, the fact that she'd snooped.\n\nThat was probably what had led her to her job of investigating people who lied. Her work had confirmed how easily caught most people are, how little they suspect their secrets will ever see the light of day. It had taught her an unforgettable lesson about the importance of keeping secrets.\n\nGinny didn't know the owner of the suitcase or the diary or the pictures within it. There was no reason to think that reading the diary would affect her life in any way beyond satisfying her curiosity. Yet even though she'd put the case on one of the built-in bookcases in the dining room and passed it several times a day as she wove through the maze of boxes she wasn't allowed to lift, and though she stopped sometimes to look at it, to touch it with one fingertip, Ginny didn't open the case again.\n\nSean seemed to have forgotten it, at any rate. He got up in the morning and was off to work just as the light was hitting the sky. In their old place, their morning interaction had been limited to a mumbled \"morning\" as she stumbled past him on the way to the shower. Here, without a job to force her to wake up, she should've lounged in bed at least until the sun rose, but guilt forced her to head downstairs while her husband shaved and dressed. She made coffee. She made eggs. She made toast and bacon, even waffles, sometimes with a slightly curled slice of orange on the side he never ate and she rescued before the plate hit the trash, tucked away into a plastic storage container and saved in the fridge for the next morning. He never noticed if it started looking a little wilted. Frankly, she doubted Sean noticed much of anything that early in the morning, even with the mug of strong coffee she didn't drink herself yet had perfected the art of brewing.\n\nHe noticed her, though. His gaze followed her as she served him his food, sometimes at a place at the table, sometimes pressed into his hand along with a packed lunch as he rushed toward the door. He always took the time to kiss her in the morning, no matter how late he was running.\n\n\"How do you do it?\" Sean asked, his hands on her hips pulling her closer to press her belly between them.\n\n\"A little vanilla in the batter.\"\n\nHe shook his head. \"No. Not the waffles, well, I mean, not just the waffles. How do you know what I want to eat, or if I'll have time? You always have it ready just in time.\"\n\nGinny smiled and kissed him lightly, pushing up on her toes. The truth was, she had no idea what her husband wanted for breakfast, any more than she had a clue what she should make him for lunch. She guessed, that was all, and figured if it wasn't right he'd tell her. But then Sean had never been the one to press for decisions, happy to go along with whatever pushed him. If she made waffles, he'd eat waffles. If she made an egg sandwich, he'd eat that just as happily.\n\n\"I pay attention,\" she said.\n\n\"To what?\"\n\n\"To everything,\" Ginny said, and saw this wasn't an answer Sean totally understood. \"To what time you get out of bed, and how long you spend in the shower. I can tell if you're on time or not.\"\n\n\"I can barely tell if I'm on time.\" Sean kissed her again, slower this time, though it was one of the days when he'd lingered with his pillow through at least two snooze cycles. He pressed his forehead to hers, eyes closed for a second before he looked at her. \"You're amazing, you know that?\"\n\n\"Ah. But am I awesomazing? That's the question.\" She could taste the hint of coffee on his lips. The cream and sugar. Coffee had never appealed to her, ever, but now the flavor sent a tingle through her taste buds. Pregnancy made her crave strange things.\n\n\"You are awesomazing,\" he agreed. \"My awesomazing wife.\"\n\n\"You'd better go, or you'll be awesomazingly late.\"\n\nSean made a face, but backed away from her. \"Remember, I have class tonight.\"\n\nShe'd forgotten, but nodded as though she hadn't. \"Oh. Right.\"\n\n\"Need me to pick anything up on the way home?\"\n\n\"No. I need to get out today anyway. Do you want anything special from the grocery store?\"\n\nShe could see by the way his brows started to knit that he was going to protest, so she shut him up with another kiss. \"There'll be plenty of time for me to be stuck at home, Sean. I can still drive. I can still push a cart.\"\n\n\"Have someone load the groceries for you.\"\n\nShe sighed, but nodded again. \"Yes. Yes, I'll go to the fancy-pants grocery store that's twenty extra minutes away, so I can have a bag boy carry my shit for me. Oh, and spend twice as much money.\"\n\nHe laughed, but just a little. \"They'll all be fighting over who gets to carry your bags.\"\n\n\"Riiiiight. 'Cuz this is some hot piece of action going on here.\" She rolled her eyes and stepped back to show him the growing expanse of her abdomen.\n\nHer foot came down on something that would've dug hard into her sole had she not learned her lesson and started wearing hard-bottomed slippers. Instead, her foot slipped forward as her ungainly balance tipped her back, hands flailing. She'd have fallen if he hadn't grabbed her.\n\nHeart pounding, blinking fast as the slight red haze filtered around her vision, Ginny gripped Sean tight as he set her upright. Then the counter, thankfully, was solid under her touch. She shifted her feet cautiously, making sure the floor was solid too.\n\n\"Be careful,\" Sean said like a lecture. \"Christ, Ginny!\"\n\nAs if she'd done it on purpose. Frowning, she looked to see what she'd stepped on. Something metal gleamed. Small and round. She bent to get it before he could stop her, though regretted when she stood too fast and had to again grip the counter to keep herself from wobbling.\n\n\"What is it?\"\n\nShe opened her palm to show him. \"It's a button.\"\n\n\"Not one of mine.\"\n\nAs if she'd accused him of dropping it, she thought with a slight curl of her lip. \"Not unless you've started wearing things with flowers on them.\"\n\nOf course he hadn't. This was a girl's button. It looked vintage, tarnished metal with a raised flower design and a small hasp in the back where the thread attached it to the garment. Too small for a coat. A sweater maybe, or a blouse.\n\nShe closed her fingers around it. \"I'm fine, Sean. I wish you'd go. You're going to be late.\"\n\n\"You should stay home today, take it easy.\"\n\n\"All I've done since we moved in is take it easy,\" she reminded him with a gesture at the kitchen, still a mess with various boxes full of things he didn't want her to put away but hadn't done himself.\n\n\"I have to go.\"\n\n\"Have a good day,\" she called out after him from the doorway, with a wave he returned from the rolled-down car window.\n\nShe thought he was going to say he loved her, but instead he shouted, \"Buy organic!\"\n\nShe watched until the car pulled out of the drive and had disappeared down the street before she shut the front door on a swirl of cold wind. The hall had turned chilly, even through the bulky knit of the sweater she wore over her nightgown, but there'd be little sense in turning up the thermostat since she was going to leave for the store soon anyway.\n\nUpstairs, she stopped in the library and settled the button next to the tiny wooden lady in the fur stole on the otherwise empty bookshelf. In her bathroom, she ran the water hot until steam filled the room, thicker than she expected.\n\nNaked, she stood in front of the mirror and cupped her breasts. Ginny's sister Peg had taken after their mother and been blessed with what she called \"an abundancy of bosoms,\" but Ginny'd drawn the tiny-titty card. At least until now. She'd gone up two cup sizes. Her areolas, always a pale and almost-translucent pink, had darkened to a deep rose that still startled her when she saw it.\n\nHer belly too was a constant surprise. Rounded and firm, still hidable under baggy shirts but straining at the waistband of her \"fat\" jeans, crosshatched with the first faint silvering of stretch marks she rubbed faithfully with expensive cream but had no hope of avoiding entirely. Her body was changing, and there was nothing she could do about it. No way to stop it. And it would never be the same.\nChapter Twelve\n\n\"Ginny?\"\n\nGinny turned with an out-of-season cantaloupe in her hands. She'd been feeling for soft spots and debating if she should put the melon in her basket; she hated the smell of them and always had, but even more so now that she was pregnant. Sean liked them, though, and sometimes that seemed what marriage was all about. Buying things you didn't like for the sake of someone who did.\n\n\"Louisa. Hi.\" Ginny's smile felt a little forced, though it shouldn't. Louisa'd never been anything but kind. \"How are you?\"\n\n\"I'm grand, just grand, doll. But you...look at you.\" Louisa tilted her head to look her up and down. \"My goodness, I guess it's been a long time, hasn't it?\"\n\nGinny could count how long it had been since she'd seen Louisa, down to the day. She let the melon fall gently back amongst its brothers, and her hands went automatically to the mound of her stomach. \"Yeah. I guess it has.\"\n\n\"We've missed you. But I see you've been busy.\"\n\nGinny's fingers twitched on the front of her coat, thinking she must look gargantuan for Louisa to have noticed. Unless she hadn't, of course, and had meant nothing by her comment. Making small talk. Or she thought Ginny had just gained some weight, maybe, though it wasn't like she'd been skinny before.\n\n\"I'm pregnant.\"\n\nLouisa laughed. \"Yes, doll, I see that.\"\n\nGinny let out a slow breath that ended in a rueful chuckle as she shook her head. There was no reason for her to avoid Louisa's eyes, but she found herself cutting her gaze anyway. \"So. Yeah. I have been busy.\"\n\n\"You know, I still have a few of your pieces left from the show. I've been storing the ones that didn't sell, along with some others, but I'll be happy to get them back to you. They're lovely work. You could think about putting them in the next show, if you want.\" Louisa looked again at Ginny's belly, then her face. \"Though of course it might not be that easy for you, huh?\"\n\nGinny smiled faintly. \"No. It wouldn't be. You could just get rid of them, I guess.\"\n\n\"You don't want them?\" Louisa looked shocked, like Ginny'd talked about giving away a baby instead of a couple of self-indulgent oil paintings.\n\n\"Not really. I could always paint more. If I wanted to, I mean.\"\n\nLouisa chewed her bottom lip for a moment. \"Well...I hope you do. You made lovely pieces, Ginny. You had a real gift for quirky, fun things. And such great color choices.\"\n\n\"Yeah. Well...\" Ginny shrugged, trying to remember what it had been like back in those days when she'd stand in front of a blank canvas and imagine it into a picture, \"...it was only ever a hobby.\"\n\n\"We all need hobbies, and if yours is making something beautiful...\" Louisa shrugged carelessly and let her fingers flutter in the air. \"Nobody said you had to try to make a living at it\u2014\"\n\n\"Good thing, since I doubt I'd be able to.\"\n\n\"\u2014but you can make beauty, Ginny,\" Louisa continued. \"I hate to see you give it up. You should stop by the shop sometime, pick them up. Or I could bring them to your house. You're over by the high school, right? In that darling little townhouse community. It looks like a European village.\"\n\n\"We've moved.\" The words sounded too sharp. Biting. Ginny softened them with a smile. \"And sure, I'll stop by the shop sometime after we get settled in. Okay?\"\n\nShe had no intention of doing that. She didn't want the paintings Sean had stashed in the basement; she sure didn't want those other ones back. She didn't care about the pieces she'd spent so much time on. They'd only remind her of disappointment.\n\n\"That would be great. Come take a few classes again, if you want. On the house. Before your life changes forever,\" Louisa advised, with another knowing look and a grin that said she knew just how it would be. \"After that baby comes, you won't have much time for hobbies.\"\n\n\"No,\" Ginny told her. \"But I will have created something beautiful.\"\n\nLouisa looked faintly surprised before she nodded again, more solemnly this time. \"Yes. Amen to that. Well, listen, doll, you take care, okay? Stop by the shop. Like I said, we've missed you.\"\n\nGinny didn't doubt that was true. The people who hung around the Inkpot were mostly the sort to drink fancy coffee and wear sandals with socks and necklaces of beads they'd made themselves. Ren Faire types, Sean had called them. In to clean living and role-playing games, that sort, though of course there were a few like her, who were totally different. It didn't matter. The only rule in Louisa's classes was that every effort be praised. She encouraged the more talented members to help those less blessed, and even the people who couldn't learn to throw a pot that didn't collapse on the wheel, or sketch anything that looked like what it was supposed to be, discovered they were able to make something by the end of the four-week sessions. People came back to Louisa's classes for more than the instruction\u2014it was a social affair, with everyone often stopping by the coffee shop next door after class to spend another hour hanging out.\n\nGinny hadn't missed them at all.\n\nSuddenly, she felt worse about that than anything else. Impulsively, she reached for Louisa's hand and squeezed the cold fingers. \"I will. Okay? I will.\"\n\nLouisa squeezed back. \"I hope so, doll.\"\n\nAfter a few more minutes of chitchat, Louisa pushed off with her cart and Ginny checked her watch. She was in no rush. Sean would be late from class. But it would be getting dark soon, and she didn't want to carry groceries into the house in the dark.\n\n* * * * *\n\nBy the time she got home, the sun had dipped low enough that her driveway was cloaked in shadows when she pulled in. In the too-cold house she checked the thermostat, which was, as it had been every other time she checked, set to the right temperature. She bumped it up a few degrees anyway. Scarlett O'Hara refused to be hungry? Well, Ginny Bohn had a bug up her ass about being cold.\n\nShe hadn't bought much at the store, but, thinking of Sean's command she not carry her own bags, she'd asked the bagger to parcel out her items into all six of her reusable totes, when three would've sufficed. Unloading the car took twice as long as normal that way but left her only a little winded, so grudgingly she had to admit that maybe he'd been at least a little right.\n\nBy the time she got everything inside, the house had warmed up, but only a little, and only in weird places. The front hall was cold but the kitchen stifling, and so was the powder room. The living room had a few warm spots but the other rooms were still chilly\u2014probably because there were still boxes and junk heaped over the floor vents, she discovered. Instead of actually unpacking things, they'd been taking what they needed out of the boxes and shifting them around.\n\nThere was definitely something wrong with their \"almost brand-new\" heating system, and it was pissing her off. Not that her moods were the best anyway. Up and down, up and down. Seeing Louisa hadn't helped that much; now Ginny was thinking of things she'd rather have pushed aside.\n\nTea might help, both with the chill and the mood swings, but as she leaned over the sink to fill the kettle, Ginny caught a flash of motion out in the yard. Something white fluttered just below the edge of the window, or so she thought. With the light inside reflecting, all she could really see was her own face. It caught her attention for a second or two, her eyes wide and looking like dark hollows in her skull, her dark hair loosing from the pony tail and falling around her face. The pale slash of her lips.\n\nShe looked like a ghost.\n\nDisturbed, she turned off the faucet and set the kettle on the stove, but didn't turn on the burner. The high-pitched giggling screams of children wafted to her ears, along with the rattle and rustle of something alongside the house. She looked again out the window and saw nothing, but, dammit, those kids were messing around in the yard again, and it was getting dark. What in the hell were they doing anyway?\n\nShe went to the back door and strained to see them, but the only thing within eyesight was that wagon, overturned in the grass. When she opened the door, she meant only to ask them what the heck they thought they were doing\u2014in a nice way, though, not trying to be mean. However, at the first creak of the hinges, they took off in a rustle of leaves and a few dozen whispered giggles and shrieks.\n\n\"Hey...!\" She let her cry trail off, defeated.\n\nKids.\n\nShe thought of her nieces and nephews, those delightful little hellions who'd turned Peg's hair gray and were going to send Billy, according to him, to the poor house. She put her hands on her belly and rubbed in slow circles against the kick of tiny feet and punch of little fists. She let her fingers press a little harder against the lumps and bumps of her body. She imagined a little bum, the curve of a head or an elbow.\n\nWith a sigh, Ginny went out to confront them, but again found nothing but the remnants of childish games instead of the children themselves. The figures that had been lined up along the window well were still there, and she couldn't tell if they were in the same order or not. Something had been added, though. A plastic baggie of Goldfish crackers and an unopened juice box. Kids' snacks, but they didn't look like they'd been lost. They looked like they'd been set there deliberately. Frowning, Ginny took them in the house and tossed them in the trash, thinking again of her gran.\n\nGran had always been a woman of superstitions, even as at the same time she was utterly derisive of what she called \"flights of fancy.\" In her high phases Gran had left a bowl of milk out on the counter for the Brownies, who were something like leprechauns, from what Ginny remembered, and who'd make mischief in your house if you didn't feed them. Sometimes that bowl of milk went sour and chunky in the bowl, but woe to anyone who touched it. Other times it stayed empty on the counter, a reminder that Gran could be as stingy as she was generous. Ginny's mom had hated that tradition and the story that went along with it, but Ginny had a fondness for the memory.\n\nShe had no milk and wouldn't have left a bowl of it on the counter anyway, but something about the way the crackers and juice had been left reminded her of Gran's offering to the Brownies. It seemed like the sort of thing Ginny would like to pass down to her child, one of those weird family things that outsiders might not understand but that meant a lot to the people who did it.\n\nThe cat bumping around her ankles was a good reminder that whatever she left had to be more symbolic than nutritious, or at least not tempting to a cat on a diet, with a bad attitude. Ginny bent to scratch behind Noodles's ears, then lifted her. She wasn't any thinner. If anything, Noodles felt even heavier. Ginny rubbed the soft fur under the cat's chin. Usually this jingled the bell collar, but today her fingers found nothing but fur.\n\n\"Oh, you bad kitty, what did you do with it?\" Ginny sighed and set her down. \"Noodles, why are you so much trouble?\"\n\nThe cat meowed implacably, wound herself around Ginny's ankles some more, and when it became obvious that no food was forthcoming, sauntered away with her tail in the air. Ginny shook her head. In the townhouse, Noodles had always been around, but this house was so much bigger the cat was forever disappearing and showing up again unexpectedly, usually under someone's feet. The bell collar had kept her from getting stepped on more than a dozen times in the past few days alone. Without it, she was likely to get more than her feelings hurt.\n\nAdding a new collar to her ever-present mental shopping list, Ginny pulled a small crystal candy bowl from the cupboard. Also a wedding gift they'd never used, it of course had been one of the first things she pulled out from the packing boxes. Couldn't have been something practical, she thought, like her measuring cups and spoons. Nope, she had to find all the gaudy, expensive things they should've sold at a yard sale instead of bringing along. Well, it had a use now.\n\nShe filled it with peanuts, a snack Noodles would leave alone, for sure, and also wouldn't spoil. She added some chocolate-covered raisins too after a moment's thought, because although she was a sucker for most things chocolate, this particular treat was a present from Barb, probably purchased from a fundraising schoolkid. They had left a sour taste in Ginny's mouth, and she had no intentions of finishing them. As a symbolic offering to a mythical group of tiny, sometimes-vengeful creatures, it seemed perfect.\n\nUpstairs in the library, she looked at the boxes shoved against the wall and at her easel propped next to them. Her paints were in those boxes. Brushes, palette, some of the smaller canvases she'd bought months and months ago. Carefully wrapped and packaged solvents and brush cleaners. The paints would surely be dried up by now. The canvases stained, maybe. She remembered packing these boxes months ago, not for the move. Just to put them away. She'd emphatically labeled them not \"Studio\" but \"Art Supplies.\" She didn't need to open them to find out what was inside, and, really...she didn't want to.\n\nInstead, Ginny threw them all away.\nChapter Thirteen\n\nThe kettle screamed.\n\nGinny got up from the kitchen table to bring it back so she could pour her mug full of hot water. Peg had already helped herself to a mug of Sean's coffee and was setting out the platter of homemade cinnamon buns she'd brought. Ginny chose a tea bag from the box on the table and dunked it in the water, eyed the buns and ran a finger through the thick goo of icing on the bottom of the plate.\n\n\"God,\" she said as she sucked the sweetness, \"so good. Why didn't Gran ever teach me how to make these?\"\n\n\"I was the only lucky one, I guess.\" Peg rolled her eyes, but fondly. \"You know Dana asked her a hundred times for the recipe. Gran never let her have it.\"\n\n\"Has Dana asked you for it?\"\n\n\"Nope.\" Peg grinned. \"Should I give it to her?\"\n\n\"I guess if she asks you. I mean, there's no real reason to keep it to yourself, right? Unless you like being the only one who can make them.\" Ginny shrugged and stirred a spoonful of sugar into her tea.\n\n\"I don't. It's not that. It's just that Gran gave it to me, only me, and I feel like it would maybe be dishonoring her or something. I mean, maybe she had a reason for not giving it to Dana, right?\"\n\nGinny snorted softly and blew on her tea. \"Yeah, spite.\"\n\n\"Cold.\" Peg shook her head but laughed softly. \"Speaking of cold...why is it so freaking hot in here?\"\n\n\"I know. I know!\" Ginny tossed up her hands. \"The thermostat's set to seventy. I don't even want to imagine our next power bill.\"\n\n\"Seventy! Good heavens, Ginny. Why so hot?\"\n\n\"Because if I don't set it that high,\" Ginny said, \"the rest of the house is freezing.\"\n\n\"Have you checked the vents? Maybe something's blocking them.\"\n\nGinny gave her sister a look. \"Have you seen my house? Of course stuff is blocking them. I hope once I get everything put away...\"\n\nShe trailed off with a sigh. They'd been in the house for a month and it still looked like they'd just moved in yesterday.\n\n\"You know I told you I'd help you.\" Peg cut one of the buns in half, then again, to put just a quarter of it on her plate.\n\n\"I know. But you're busy; you have your own stuff to do.\" Ginny broke hers into smaller pieces but intended to eat them all. She watched her sister stir artificial sweetener in her coffee and add a splash of skim milk. \"Diet?\"\n\nPeg looked up, a little startled. \"I can't fit into my jeans. Unlike you, I'm not eating for two. It just looks like I've been.\"\n\nPeg's youngest, Luke, was eighteen. Her oldest, Jennifer, was twenty-eight. It had indeed been a long time since Peg was in Ginny's condition, but she was far from overweight. Ginny watched her sister pluck at the pieces of cinnamon bun without actually eating them. She studied the lines around Peg's eyes. The corners of her mouth. If she looked in a mirror, she'd see those same lines, just a little fainter but unmistakable. When had they started getting so old?\n\n\"What's going on?\"\n\nPeg sighed. \"I'm not supposed to say anything about it...\"\n\nGinny reached to put a hand on Peg's to keep her from further worrying the bun into scraps. \"About what? What's going on? Is Dale okay? The kids?\"\n\n\"He's fine. It's Billy.\"\n\n\"What's wrong with him?\" Ginny hadn't spoken to her brother in a few weeks, her move and his work keeping them from connecting. This wasn't unusual, since though she loved her brother, they didn't keep in touch as often as she did with Peg. \"Is he okay? Is he sick?\"\n\n\"No. It's Jeannie.\"\n\nGinny frowned. \"Spit it out, Peg, I'm not a mind reader.\"\n\n\"Jeannie's going to leave him.\"\n\nThis set Ginny back in her chair. She blinked rapidly, processing this. \"What? Why?\"\n\nPeg sighed and wiped her fingers with a paper napkin before cupping her mug. She lifted it to let the steam from the hot coffee bathe her face for a few seconds before answering, \"She's met someone else.\"\n\nGinny's stomach lurched into her throat. She took a bit of bun, too big, and it lodged in her throat so she had to gulp too-hot tea to swallow it. The tea burned her tongue.\n\n\"I asked how long it had been going on,\" Peg went on. \"I mean, my God, we just went to the beach with them this past summer, and things seemed fine. I mean, if anything, she was nicer than I'd ever seen her. And, you know, it's not that I don't like Jeannie,\" Peg added hastily, like she had to explain to Ginny, who knew very well the sort of woman her sister-in-law was.\n\nGinny stayed quiet for a moment. She and Sean hadn't gone to the beach this summer because all their vacation time and resources had been taken up with buying the house. She hadn't seen Jeannie since the spring, months and months ago. \"What does Mom say about it?\"\n\n\"Oh...I don't think she knows. Billy didn't even want me to tell you, because of...you know.\" Peg gestured vaguely at Ginny's belly, hidden by the table.\n\n\"Oh, for fuck's sake, I wish everyone would stop tiptoeing around me with stuff because I'm pregnant.\" Ginny tore another bite of cinnamon bun. This one went down much easier.\n\nPeg nibbled at a bite of her own, more daintily. \"Sorry. I think he was just being, you know. A good brother.\"\n\n\"Why'd he tell you, if not Mom? Or me?\"\n\n\"I...\" Peg looked caught, eyes shining, glittery.\n\n\"You what?\"\n\n\"I saw her. With him. The other guy.\"\n\nGinny paused with her mug halfway to her mouth, then set it down gently. \"Oh. God. Peg...you're the one who told him?\"\n\n\"Well, what would you have done?\" Peg frowned. \"I saw her out, bold as brass, with some other man. It was clear as anything what they were up to\u2014\"\n\n\"Why? Were they holding hands or making out or what? Were they dry humping in the middle of the grocery store?\" Ginny got up from the table on the unspoken pretense of washing her hands clean of sticky icing, but the real reason was so she could get her reaction under control. At the sink she turned on the hot water and ran it hard, letting it burn her hands while she drew in breath after shaky breath.\n\n\"They were at the movies together, in a matinee. Sitting close. I could tell, Ginny. I mean, anyone could've.\"\n\nGinny turned from the sink to look at her sister. Peg, the oldest, had always been a little mother to her siblings. It had annoyed the ever-loving shit out of her and Billy when they were younger, and sometimes still did, even though Ginny had come to appreciate her sister's concern, especially since their own mother so infrequently seemed to have any. Still...this...she shook her head slightly. She rubbed her burnt tongue on the roof of her mouth, the pain better than trying to find words. Safer.\n\n\"Did you talk to her first?\"\n\n\"No.\"\n\nGinny rubbed at her mouth with the back of one hand. \"You just went to Billy and told him.\"\n\n\"He's my brother!\"\n\nAnd Jeannie, as much as Peg might protest she liked their sister-in-law \"just fine,\" had always been a little standoffish. A little...not snotty, exactly, but she'd definitely never tried to ingratiate herself with Peg\u2014or Ginny, who'd decided long ago she didn't care. She didn't need Jeannie for a bestie, or even a sister. However, Peg had always felt a little snubbed.\n\n\"But it's not your business.\"\n\n\"How can you say that?\" Peg frowned again. \"What would you have done?\"\n\nGinny took a cloth from the sink and feigned an intense interest in wiping down the countertops, though she'd done a complete kitchen tidy in anticipation of Peg coming over.\n\n\"You're telling me you'd have just let her go on with it? Not told Billy at all? I don't believe you, Ginny.\"\n\n\"You don't know what was really going on, that's all I'm saying. And now, by telling him, Billy knows and Jeannie's leaving him, and maybe if you'd just left it alone...\" Ginny bit back her words. Her sore tongue ached as she rubbed it firmly against the back of her teeth. \"Forget it. How's Billy?\"\n\n\"How do you think he is? He's devastated.\"\n\n\"Yeah. I bet he is.\" Ginny sagged a little, then turned to say something placid, vapid, to turn her sister's attention away from an argument, but found she had nothing to say. Instead, she took her now-cool mug and popped it into the microwave and hit the button to warm it for a minute.\n\nThe overhead light flickered rapidly. The clocks on the microwave and stove beeped. The light went out. Came back on. Went out again while Peg looked up goggle-eyed and Ginny sighed a curse. At four thirty, there was just enough late afternoon sunshine to keep the kitchen from being totally dark, but in another ten minutes or so they'd be in complete shadow.\n\nPeg made a face. \"What's up with that?\"\n\n\"Circuit breaker.\" Ginny glanced at the coffeemaker, still on. \"There's something funky with the wiring in the kitchen. Or something. I guess it's the coffeemaker and the microwave together, I don't know. It was blown the day we moved in and has happened a bunch of times since then; we haven't been able to figure out exactly what trips it.\"\n\n\"You should have it looked at. And the furnace too.\"\n\n\"Yes, Mother.\" Ginny said it with a roll of her eyes, a familiar tease.\n\n\"Where's the fuse box? Are you going to go take care of it?\"\n\n\"Well. Sure. Of course. It's in the basement.\" Ginny paused. \"It will be dark down there.\"\n\n\"You have a flashlight?\"\n\n\"Oh sure. Somewhere.\" Ginny laughed and waved a hand toward the dining room, the living room beyond, overflowing with boxes. \"Probably ten of 'em.\"\n\n\"Candles?\"\n\n\"Sure, I have a candle.\" Ginny opened the drawer but found nothing, no candles, not even the mostly melted stubs of the ones she'd used in the candelabra. \"Here, I know. I have one of those lighter things. We'll use that.\" She found it in the drawer and went with her sister to the basement door.\n\nIt opened with a spine-tingling creak. Ginny paused and flicked the light switch on and off a few times, just in case. Of course, nothing. She held the lighter high, but the light didn't shine more than a few feet. The steps, she noticed, were dirty, the wood splintered on the sides. Rubber matting had been stapled to the center of each riser, presumably to keep people from slipping, and dust had collected in all the threads.\n\nPeg clutched at Ginny's shirt. \"You should let me go first.\"\n\n\"Because I'm heavy with child?\"\n\n\"Sarcasm suits you so nicely, sister dear. No. Because I'm older.\" But Peg didn't move, just peeked around Ginny's shoulder.\n\nEven as kids, Ginny had always been the one who had to go first in the haunted houses at Halloween time. Everyone had always strung behind her while she felt her way in the dark, hands out, ready to be the first in line for the monster leaping out. If she could do that, she should be able to go into her own basement where there was no masked creature waiting to grab her.\n\nProbably.\n\nShe had to let the lighter go out for a second to give her finger a rest from holding the trigger. Peg gasped when the alcove went dark. There was still light coming in from the hall behind her, but the stairs were pitch black.\n\n\"No windows in the basement?\"\n\n\"A couple,\" Ginny said. \"Not near the stairs, though.\"\n\nShe lit the lighter again. The shadows withdrew. She put her free hand on the railing, her foot on the top stair. Peg shuffled along behind her until Ginny sighed and stopped three stairs down.\n\n\"You're going to knock me down the stairs if you're not careful. Jesus, Peg. It's a basement. It's not quite five o'clock in the afternoon. Serial killers don't come out until at least five thirty.\"\n\n\"Funny.\" Peg backed up a step. \"It's still dark. Be careful! Do you know where the fuse box is?\"\n\nShe didn't, actually, since Sean had always been the one to go down and fix things. \"How hard can it be to find?\"\n\nThey found out in the next few minutes when they got to the bottom of the stairs and the lighter's glow didn't extend more than a foot or so beyond them. The paintings were gone, Ginny saw that at once. Sean had moved them. She waved the lighter around and caught sight of some of the empty boxes he'd brought down. Too few of them, of course. She had to look away, or be annoyed.\n\n\"It's got to be along a wall.\" Ginny moved toward the wall directly across from her.\n\nConcrete blocks, pale gray. One of the walls, the one on the side that had once been the garage, had been plastered over, probably after the fire. Otherwise, it was nothing special. The walls looked like any other basement she'd ever been in, long and bare and strung with cobwebs along the ceiling, which was open to the beams and hung with wires, tubes and ducts. She understood what Danny the exterminator had meant\u2014the ducts went everywhere.\n\nNo fuse box, though. Ginny, Peg shuffling behind, still clinging to her shirt, moved to the left. This wall, the one facing the backyard, also had no windows and no fuse box. The one to the left of that had two windows set deep into wells, through which only the faintest glimpse of failing daylight crept. One of them was the one where she'd found the figurines.\n\nBy this time they'd gone around to the back of the stairs, which had been walled off to make a sort of closet beneath them. Another short wall jutted out a few feet, not long enough to reach the far wall. Beyond this, at last, Ginny spotted the fuse box.\n\n\"Of course it's on the wall farthest away from where we started. I have to let the light go out again, hold on a second.\" Her finger had cramped from holding the trigger, and she let it go with a sigh of relief.\n\nPeg let out a squeak as the room went dark.\n\nGinny laughed and let out a low, groaning ghost noise.\n\n\"Stop it, Ginny.\"\n\n\"Where's my golden arrrrrrm?\" Ginny moaned, bringing back the old ghost story that always ended with screams. \"Where's my\u2014\"\n\n\"You're a jerk!\"\n\nStill laughing, Ginny squeezed the lighter's trigger and headed for the fuse box. Inside, a flash of orange showed her exactly which switch had been tripped. She clicked it over to the right position.\n\nThree things happened at the same time.\n\nOne, the furnace kicked on with a whoosh and a flash of blue light. Two, a hulking figure loomed up from around the wall at the bottom of the stairs. And three, Ginny dropped the lighter and plunged them into blackness.\nChapter Fourteen\n\nPeg shrieked; Ginny did too, more in response to her sister's scream than her own fear. In the dark, something shuffled in front of them. The shadows shifted. Ginny reacted instantly, hands fisting, then pistoning out as the dark shape got within reach.\n\nShe connected with something soft and felt the familiar skid of corduroy against her hand, but too late\u2014her other fist was striking like she thought she was some sort of Muhammad Ali. Only the sting like a bee part, though, no floating like a butterfly. She'd punched her husband in the face, and he went down to the concrete floor with a muffled shout.\n\nShe knew it was Sean because of the jacket and the sound of his muffled grunt, and because, who else would have come down the stairs to find her? With Peg still hollering and Sean letting out a few choice words of his own, Ginny tried to think about how far she was from the dangling light cord and if she dared risk trying to find it.\n\n\"Shut up, everyone!\" she shouted. \"Peg, it's Sean. Sean, honey, are you okay?\"\n\n\"You busted me in the eye!\"\n\n\"I'm sorry.\" She bit back a laugh, her hands still out, still blind, though now she could make out things a little better in the faint blue light from the furnace pilot light. \"Baby, I'm so sorry.\"\n\nSean got to his feet, more of a felt presence than seen. \"What happened to the light?\"\n\n\"I dropped it. Can you find the cord?\" Ginny didn't dare move.\n\nPeg had stopped screaming but her breath was shuddery and she was clutching Ginny's arm so hard she was going to leave a bruise. Peg was lucky Ginny hadn't punched her in the face with that roundhouse strike, and the thought of it made Ginny laugh all the harder. It was terrible, and her laughter definitely had a sobbing edge to it, but there it was.\n\n\"Hold on. Don't move,\" Sean said.\n\nIn the next minute, the light from the bare bulb blinded them. Ginny winced. Peg let out a mutter of gratitude. Sean put his hands on his hips and glared at them both.\n\n\"The fuck were you doing down here?\"\n\n\"The circuit breaker popped again,\" Ginny said.\n\n\"You should've waited for me to fix it.\"\n\n\"Uh...look, I really need to get going,\" Peg said.\n\nSilence. Ginny's sister hugged her and patted Sean's shoulder. She gave Ginny a sympathetic look but ducked away, up the stairs, leaving her alone. Ginny couldn't blame her, really. The temperature in the basement had dropped by about fifty degrees at Sean's statement, and it had nothing to do with the furnace.\n\nHe waited until the sound of Peg's footsteps upstairs went out the front door. \"You can't be coming down here in the dark, Ginny.\"\n\n\"What was I supposed to do, sit and wait for you to get home? I thought you had class tonight anyway.\"\n\n\"It was cancelled.\"\n\nShe made a face. \"And...you told me that? I was supposed to just somehow intuit it? I should've just waited upstairs in the dark and the cold for you to get home?\"\n\n\"You could've sent Peg down here\u2014\"\n\nGinny pushed past him, heading for the stairs. \"It's my house, not hers. I wasn't going to ask my sister to fix my circuit breaker. Besides, it's not like I'm stupid.\"\n\n\"I never said you were stupid,\" he told her in the kitchen when he caught up to her. \"Hey. Stop. Look at me.\"\n\nShe did, glaring.\n\n\"I was worried when I got home and you weren't here. That's all. I worry.\"\n\nShe softened a little toward him. Just a little. \"I was fine. If you hadn't come down just then, it would've been even better.\"\n\nHis smile quirked. \"I really scared the shit out of Peg, huh?\"\n\n\"No kidding. I'm surprised she didn't pee herself.\" Ginny smiled too.\n\n\"Why didn't you use a flashlight, at least?\"\n\n\"Why didn't you?\" she pointed out and waited for him to get it.\n\n\"I didn't have... Ah.\"\n\nGinny lifted a brow and made a show of peering around the arched doorway into the dining room. \"Yeah. We have one, huh? Right?\"\n\n\"I think so.\" Sean sighed. \"You want me to unpack those boxes.\"\n\n\"It would be nice. C'mon, we can do it together. You can lift all the heavy things. I'll tell you where they go.\" She smiled again, watching his face work as he tried to think of a way to get out of the task.\n\n\"Fine.\" He sighed again. \"Fine, fine, fine.\"\n\nBut though they unpacked four or five boxes and even managed to shift the furniture around at least sort of the way she wanted it to stay permanently, they found no flashlights. She did convince Sean he needed to run the vacuum cleaner, though, to get up all the bits and pieces of packing paper and dust that had been inevitably kicked up.\n\n\"It's way too strenuous,\" she told him with a very wicked grin as she plopped onto the couch with a book in her hand. \"You really should do it.\"\n\nGrumbling, Sean looked like he meant to argue, but wisely thought better of it. He plugged in the vacuum, a pricey model he'd given her as a gift one year for her birthday\u2014her birthday, for the love of all things holy! Which she'd never really let him live down.\n\nThe instant he toed the On switch, the lights went out.\n\n\"Shit,\" Sean said.\n\nIn the dark, Ginny laughed.\nChapter Fifteen\n\n\"Let me go first. You're gonna love it, I know it. Happy anniversary.\" Sean pressed a square package into Ginny's hands and stepped back with a grin. \"Go on. Open it.\"\n\nShe studied the wrapping carefully. She loved surprises. Loved getting presents. And though they'd both agreed that this year they wouldn't spend any money on each other for their anniversary since they'd just moved into the house, she'd been unable to help herself from buying him something she knew, absolutely knew he wanted. This package in her hands was just the right size for an iPad, which she'd been none too subtly hinting she wanted since her last birthday, when he'd presented her with a vacuum cleaner instead.\n\nGinny smoothed her fingers over the paper, enjoying the anticipation but feigning concern. \"I thought we said we weren't going to do anything for each other.\"\n\n\"I know, I know, but I saw this and knew you had to have it. Go on,\" Sean said. \"Open it up. Let's see.\"\n\nSo she did, sliding her fingers beneath the tape and pretending she meant to fold back the paper in one piece, the way she knew drove him crazy, because Sean liked to tear into gifts and leave the wrapping strewn all over in shreds. She laughed when he danced forward to take it from her, holding it back and away.\n\n\"Mine,\" she told him, and ripped the paper while holding her breath with excitement.\n\nShe had to stare at what was inside for a full minute before she remembered to let out the breath. Then another, her eyes not quite connecting with her brain. The paper slipped from her suddenly nerveless fingers and drifted to the floor where Sean crunched it under his feet as he shifted to lean over her shoulder.\n\n\"See? You plug it in and it charges up automatically. You can set it to act like a night-light or just when the power goes out, it will come on. And it's a two-pack, so we can put one down here and one upstairs.\"\n\nHe looked so proud, so gleeful all she could do was nod and smile like this two-pack of flashlights from the warehouse club was indeed the very best anniversary gift she'd ever received. \"Wow. Yeah. It's...wow.\"\n\nSean, clearly pleased with himself, took the package from her to open it, demonstrating how she could plug it into any outlet, how the light-sensitive function worked. How the flashlight itself fit neatly into the holder in any direction, not needing to be lined up any special way.\n\n\"It's great,\" Ginny said around her disappointment. She reminded herself they had said they weren't going to buy each other gifts. And that would've been better, she thought. To get nothing because they'd said they were doing nothing, than to get...\n\n\"I just want to make sure you're never left in the dark again,\" Sean said.\n\nOnly a selfish shrew could be angry or disappointed when he said something like that. And only a vindictive bitch could take pleasure in seeing his face when she took him down the hall and pulled the sheet off the present she'd had to be so careful to keep a surprise. Ginny guessed she must be both, because she was still a little bitter when she gave Sean his gift.\n\n\"What the...\" He stared at it. Then her. Jaw dropped. Then he grinned and went to his knees in front of it like a little kid. \"Holy shit. How did you do this?\"\n\n\"I had it delivered. Do you like it?\"\n\nHe put a hand on the 42-inch flat screen's box and looked down with a marveling shake of his head. \"Like it? I fucking love it.\"\n\nOf course she'd known he'd love it\u2014he'd talked with longing about replacing their ancient television for at least a few years. Sort of the way she'd been talking about wanting an iPad. But hearing the joy in his voice, Ginny could no longer be upset that she'd made such an effort when he hadn't made one of equal size...or expense, she admitted to herself, thinking of his reasons for choosing the flashlights. And when it came right down to it, money should be no measure of someone's affection.\n\n\"I'm glad,\" she said, and meant it. \"Let's get it set up, okay?\"\n\n\"Yeah, yeah.\" He looked up at her, eyes alight, then got to his feet to hug her hard. \"You're the best.\"\n\nShe was far from that, and nobody knew that better than she did. \"I'll go clean up in the kitchen. Then we can watch a movie or something?\"\n\n\"Yeah.\" Sean was already turning to the TV stand, unhooking all the cords and wires from the old set. \"That would be great.\"\n\nThe best. She was the best. Ginny cleaned off the table and loaded the dishwasher. She refilled the Brownies' bowl with wrapped mini candy bars this time, since Sean had seen fit to eat the peanuts and chocolate-covered raisins. She took the fine-linen cloth off the cheap card table and tossed it in the washer, and she set the candelabra back on the old dresser she used as a buffet. She put away the leftovers, making sure to package them in lunch-sized portions for Sean to take to work, because she was the best wife.\n\nThe best wife.\nChapter Sixteen\n\nThere's a lot of music at this party. Steady thumping. Lots of bass. Someone's strung up colored lights, orange and purple and green. Some are hung with pumpkins or ghosts. Some blink, though not in time with the music.\n\nGinny knows this place. This party. If she catches a glimpse of herself in a mirror, she'll see she's wearing a Little Red Riding Hood outfit. Blue-checkered dress, short, with a flouncy skirt and petticoat. White anklet socks. Red cape with a hood. She wore it because her roommate, Tina, convinced her that dressing slutty was the only way to attend a college Halloween party. Ginny'd planned to go dressed as a baseball player, in her brother's old uniform.\n\n\"Not cool,\" Tina says when Ginny tells her. \"Baseball players are only fuckable by gay dudes.\"\n\nGinny sees the logic in this, although the truth is she's not necessarily looking to get laid. She dated Roy, her last boyfriend, from her senior year in high school and all through her freshman year of college, but they broke up over the summer. She slept with another guy after that, just once or twice, to get over Roy, but the truth was that sex had always been something messy, kind of awkward, never the big deal it was all cracked up to be.\n\nStill, it's always better to be fuckable and have the choice of turning a guy down than be the girl at the party, standing alone by the punch bowl. Not that a party like this would have a punch bowl, necessarily, more like a keg, along with various bottles of cheap liquor that are mostly used for shots instead of fancy mixed cocktails, although there are a couple jugs of orange juice and cola on the makeshift bar.\n\nSo there she stands in her short, slutty dress and shoes that are pinching her toes. She has an empty basket over one arm, and it keeps getting in the way. Before the night's over she'll lose the basket and never see it again, which is terrible because she borrowed it from Tina, whose grandmother had bought it for her. Tina will always say she doesn't care, but some part of Ginny understands that Tina really does.\n\n\"Let me guess,\" says the frat boy. He's dressed like a baseball player, and now Ginny totally gets why Tina was right. Baseball player as a Halloween costume? Not fuckable. Or maybe it's the guy's beer belly or the rash of pimples across his forehead or the drunken, bleary way he's stroking his gaze all over her cleavage. \"Dorothy? There's no place home, right? If I get you wet, are you gonna melt?\"\n\n\"I'm Little Red Riding Hood, and that was the wicked witch, you asshole,\" this Ginny says, because this one is dreaming about that long-ago night when she wasn't quick enough with the comeback and had to fend off this loser's pawing grasp for another twenty minutes.\n\nThat's the beauty of dreams. If you know what you're doing, you can control them. Ginny's in charge of this dream now, so she can walk away from the horny baseball player and make her way right into the best part of things.\n\nShe sees him across the room, like the crowd parted just for her. Just so she can get to him without having to elbow her way past the plethora of tits and ass on display. Instead, she glides. Her shoes don't pinch because her feet don't touch the ground. She floats or maybe swims through the air, thick as water; she pushes it out of the way, scooping it with her hands.\n\nHe's the most beautiful man she's ever seen.\n\nHe wears a dark suit, a blue- and white-striped shirt, a red tie. He wears a clear-plastic raincoat, and his hair's slicked back. He carries an ax.\n\n\"Who's that guy?\" Tina says. \"Some huntsman.\"\n\n\"It's Patrick Bateman.\"\n\nPatrick Bateman, played by Christian Bale in the movie version that Ginny's seen, oh, about a dozen times. He's a serial killer, murderer of whores and vapid trust-fund bitches and dudes who get more expensive haircuts than his. He's dangerous.\n\nHe's perfect.\n\nHe's leaning against the wall with his ax at his side. As she gets closer, Ginny sees it's a real one, though with some sort of shiny metal tape over the edges, presumably to keep it from actually chopping anyone apart. He has no drink in his hand, and his fingers just barely tap his thigh in time to the music. He scans the room like he's looking for someone.\n\nThen, he finds her.\n\n\"Do you like Huey Lewis and the News?\" Patrick Bateman asks Ginny when she walks up to him in her suddenly ridiculous costume.\n\nThe real-world Ginny sputtered an answer to him, trying to be coy, trying to get him to laugh. And he had, the flash of interest in his eyes clear when she bent to set down her basket.\n\nBut this is a dream, this is different. Though in the real world his name had turned out to be Joseph and he was nowhere near as smooth...or rich...as the fictional Patrick Bateman, in the dream he stares at her assessingly before he gives her a faint, supercilious smile that widens into a broad, smarmy grin when she says, \"Their early work was a little too New Wave for my tastes, but when Sports came out in eighty-three...\"\n\nHe takes her, the way he did then, away from the party. They walk down dark alleys, past stumbling revelers. Drunken, groping couples. One or two sour-faced townies with the bad luck to be out and about in a college town on Halloween night. And then, back to his place. It's nicer in the dream than it had been in real life. Modern. Expensive. He feeds her frozen yogurt from the carton and when she moves to put the spoon on the table, he chastises her quickly.\n\nThe dream is jumbled now between real life and American Psycho and her own psyche. Patrick Bateman kisses her hard enough to bruise her mouth. She tastes blood. He strips her out of the costume and leaves it on the floor, then takes her up against the white wall in front of a mirror big enough to show the entire room. She can see over his shoulder, his firm ass pumping as he thrusts inside her, the skin of his back marked with swirling lines of black and green. A tattoo. Definitely a product of the dream.\n\nHe bites her shoulder.\n\nGinny stiffens with pleasure, her eyes closing as the room tips and spins. Climax bursts through her. Then again when he pulls away to show his mouth, teeth and lips dripping red with her blood and when he holds up a hand to show off the still-beating heart clutched in his fingers. Her heart.\n\n* * * * *\n\nGinny woke, panting, the aftermath of her orgasm still rippling through her. She lay on her side with her body pillow tucked between her legs, and though her mind still whirled from the strange, disturbing turn the dream had taken and from her climax, the only evidence of her reaction was the fast beat of her heart. She sipped in some air, slowly, letting her heartbeat slow.\n\nWow.\n\nYes, it was weird at the end. Definitely gross and a little scary. But also incredibly sexy. She hadn't been turned on like that since...well. A long time. She shifted under the blankets, suddenly more aware of Sean's light snores. Suddenly aware of the pressure in her bladder and how cold her nose was, how the faint ticking of the air from the vent that had soothed her to sleep was no longer there.\n\nGod dammit, the room was cold again. She'd had Sean set and reset the thermostat, even change the batteries. Now her face was like ice. Her toes too, way down at the bottom of the bed. Blinking, Ginny tried to see what time it was, but she'd turned the light completely off on her alarm clock. The clock had a design flaw that set the light from its brightest to dimmest levels, instead of the other way around, which meant hitting the button now would bathe the entire room in an eye-searing arc of blue-white light.\n\nWithout the time, all she could do was convince her body to ignore the insistent need to urinate, or risk getting out of bed to pee and discover that the alarm would be going off in about five minutes. Ginny shifted again, pulling her knees up as far as she could to tuck her feet together and try to warm them. This didn't help her bladder at all, and eventually with a small groan she flipped off the covers and eased herself out of bed.\n\nAt least in the bathroom the night-light had a clock on it. Even though she felt like she'd been sleeping for hours, it was still only just past midnight. Another six hours before Sean would get up and she'd be up too, making his breakfast and pretending she had important things to occupy her time. She peed forever. The bathroom floor was, if it was possible, colder than the bedroom. Her teeth chattered.\n\nShe heard the cries just as she pushed the handle to flush. They were lost immediately in the rush of water, which seemed to echo even more loudly in the dark. Ginny's head went up, eyes wide. She imagined herself as a gazelle, nostrils flaring at the scent of a cheetah in the grass. Ridiculous, and yet she strained to hear as the noise eased.\n\nNothing.\n\nBut she had heard it, she was sure. The faint but audible and unmistakable sound of sobs. Now all that reached her were Sean's snuffling snores and then the welcome rumble of the furnace kicking on...except that the air pushing up from the vent next to her wasn't hot. It was far from cold, the way air-conditioning would be, but it was barely tepid.\n\nShe was wide awake now. Even with the promise of a few more hours of sleep, even knowing she ought to relish this time before the baby came and interrupted nights would become her life for the next, oh, twenty years or something like that, there was no way she could get back to sleep. Only out in the hall did Ginny remember she hadn't put on her slippers, and winced in advance of whatever it was that she would eventually step on.\n\nGinny's mom had been a huge fan of warm milk to aid sleep, usually with a liberal dose of cocoa and vanilla sugar added to it. Of course the sugar and caffeine negated any benefit of the milk itself, but that homemade sleep remedy had always been a treat. Ginny made hers the way her mom had, stirring it slowly on the stove so the milk didn't burn or get a skin on the top. A perfect, creamy blend of sweetness and warmth. In her kitchen the light over the stove did little to chase away the dark, and she turned it off as soon as she'd finished making the cocoa.\n\nMaybe she ought to be scared in the dark, she thought with her hands wrapped around the mug as she let the heat from it bathe her face. It was still too hot to drink. She listened to the creaks and groans that were starting to become familiar. She listened for the faint sounds of sorrow, and convinced herself she'd imagined it. Maybe she should be afraid, but there was something comforting instead about standing with her back against the counter, sipping sweetness while the wind rustled the bushes outside.\n\nThe brush of soft fur on her ankles made her jump a little, scraping the legs of her chair on the floor. Then, the jingle of Noodles's collar. Sean must've found it, wherever the cat had lost it, and put it back on.\n\n\"Hey, puss. C'mere.\" Ginny reached to pet the cat but got only a waft of air as Noodles ducked out of reach. Her belly made bending under the table too awkward, and, besides, if she grabbed out in the dark, Noodles was just as likely to nip Ginny's fingers as she was to accept the caress. \"Fine. Be that way.\"\n\nShe took the mug upstairs with her, but instead of going into the master bedroom, she ducked into the library. Someday, after the garage and the landscaping and the dozens of other things they'd planned, she wanted to add bookshelves to the other two walls to match the built-ins already there. Get some comfy chairs with footstools that matched the Victorian sofa, and good lamps for reading. Maybe they could fix the fireplace in here. She imagined building a crackling fire, the smell of wood burning. Or even a gas insert, Ginny thought as she stood in the center of the room and made a slow circle. That would be cool too.\n\nOr hot, which would be even better, she thought with a shiver before realizing that this room, at least, was much warmer than the hall had been. Or her own room. She moved toward the window next to the fireplace and pressed her fingers to the glass. Definitely cold. But on the other side of the fireplace, the bookcase had warm air puffing out around the edges. Not just warm. Tropical. It felt so good she pressed her palm to the back of the bookcase and let the wood warm her.\n\nBut why was it so warm? That was the question. This room didn't face the street, but light from the streetlamps did cut across the side yard here, so with night-adjusted eyes she could make out the hardwood floor. It was also bare, with no rugs, and all the boxes had been stacked along the opposite wall. She found the floor vents easily enough, one on the left side of the fireplace beneath the window. And the other one...\n\n\"Here,\" she murmured and nudged the bookcase with her toe. The other one should be there, but they'd built the bookcase over top of it.\n\nShe set her mug on the fireplace mantle to explore. Yep, there was a notch there in the molding along the bottom shelf. When she put her hand there, the air was so hot it almost burned her fingers. She wanted to get down on her hands and knees and bask in it. She wanted to put down a beach towel and pretend she was in the hot summer sunshine, wearing a bikini and tanning oil, frying herself on some beach someplace, instead of getting up to pee at midnight and standing in the dark, freezing, with her belly big and round and full of child.\n\nInstead, she contented herself with sticking one foot against the notch and warming her toes until it felt like they might start to sizzle like bacon. Then the other. Back and forth she shifted with her hands on the bookshelf to help support her weight, and her eyelids grew heavier. Like she was slow dancing, she thought sleepily. Slow dancing with the bookcase.\n\nRidiculous...\n\nGinny snapped awake, trying to remember if she'd heard another set of those sobbing cries or if something else had woken her. It took her a good half a minute to realize it was just her body's way of protecting her from falling over, because she'd dozed on her feet, which were toasty warm at least. Stifling a yawn, she shuffled toward the doorway where her foot connected firmly with something furry and angry.\n\n\"Noodles,\" she scolded as the cat ran on silent feet down the hall, making shadows in front of the night-light. \"Don't you know better...\"\n\nGinny paused. She'd kicked Noodles because she was sitting, quiet. But when she ran, the telltale jingle of her collar had once again been silenced.\n\nHuh.\n\nBack in bed and coordinated around her multiple pillows, Ginny's lovely sleepiness had vanished. Every time she thought she might actually be able to fall back to sleep, Sean let out another grunt or a snuffle, or he shifted in the bed and pulled the covers off her and made her cold again. He'd always been a restless sleeper, but it seemed to be worse now. Or maybe she was just more sensitive to it. At any rate, at last Ginny had to resort to her old trick of counting backwards from a hundred in order to see if she could trick her brain into shutting down.\n\nShe got to eighteen before Sean coughed and her eyes flew open.\n\nStarting over, she got to thirty-seven before he let out a long, ripping fart that had her gritting her teeth.\n\nThis time, she got to fifty before he rolled onto his back and started snoring in earnest. Ginny sat up. She leaned over and poked him. Hard.\n\n\"Honey,\" she said, her tone making the endearment a lie. \"Roll over.\"\n\nHe did with another snort and snuffle, and she lay back and stared at the ceiling, even though she knew sleeping on her back was going to be impossible and she could only last a few minutes before she'd have to move. When she rolled onto her side to look out the windows, something like the first gleam of sunrise teased her.\n\nShe was still awake when the alarm went off.\nChapter Seventeen\n\nWith Sean's hands over her eyes, all Ginny could do was laugh as he guided her. \"What's the surprise?\"\n\n\"If I just told you, it wouldn't be much of a surprise, would it?\" His breath, warm against her cheek, made her want to nuzzle against him. \"Okay. Hold on a second. Don't look until I tell you. Okay?\"\n\n\"Okay, okay.\"\n\nShe heard shuffling. The scrape of wood against wood. She knew they were in the library because he'd brought her upstairs before making her close her eyes. But what on earth could he be about?\n\n\"Open them.\"\n\nShe did, expecting maybe a chair to match the sofa. Instead, Ginny faced her easel\u2014which had been set up in the dormer, with a fresh new canvas in the size she liked best. Beside it was the ugly telephone table. On it rested a new palette. Tubes of paint. A cup bristling with brand-new brushes.\n\n\"I know you threw all your old stuff away. I figured you needed some new things.\"\n\n\"Sean...\"\n\n\"Look, the light's great right there. I mean, I'm not an artist or anything, but the sun shines in that dormer almost all day long.\"\n\nShe knew it did. She'd noticed. Ginny moved forward to look at the setup, then at him. \"You didn't need to do this.\"\n\n\"Sure I did. Consider it a late anniversary present.\" Sean smiled. \"Since what I got you was lame, I know.\"\n\nShe didn't want him to feel like he had to make anything up to her. She especially hadn't wanted him to feel like he had to do it by buying her art supplies. Yet, faced with all of this new equipment, the pretty colors, the fresh and untainted brushes, Ginny's fingers did twitch. Just a little.\n\n\"I got you special paints and some nontoxic brush cleaner.\" He sounded so proud. \"Natural pigments and stuff. So...you don't have to worry about the chemicals.\"\n\nIt was too much. She should weep, but no tears sparked her eyes. Ginny stroked her fingertips over the canvas gently. Then she kissed him. \"Thank you.\"\n\n\"You like it?\"\n\n\"It's...so nice.\" The lie hardly tasted bitter at all. \"So unexpected.\"\n\n\"Now you can paint again,\" Sean said. \"I know you've been missing it. I can tell. You want to get started on something now?\"\n\n\"Later,\" Ginny said. \"I'll do it later.\"\nChapter Eighteen\n\n\"I know I have it. I know I have it in a box of things I packed specifically for the house.\" Ginny put her hands on her hips and looked around the smallest of the rooms on the second floor. No more than a closet really, it would be just big enough to hold a desk and a chair, maybe a set of shelves. Sean's office. If they ever moved everything out of it anyway.\n\nSean sighed and rubbed at his hair, that bad habit that always left him looking rumpled. She wanted to smooth it, to stroke down the sleek bits that always fell just in front of his ears and stroke it away from his forehead. Too many boxes blocked her.\n\n\"What kind of a box?\"\n\n\"A file box.\" She indicated the size with her hands, then pointed. \"Like one of those. Like any of those, but it's not one of those. I labeled all of them. It should be marked \"Linwood.\" Anything we got from Bonnie would be in there.\"\n\n\"Do we even know if we have anything about the furnace?\"\n\nShe shook her head and bit her lip. \"No. I don't know. Bonnie gave us that entire accordion file of stuff, warranties and receipts and all that. I'd imagine if there was something, it would be in there.\"\n\n\"Dammit, Ginny, the furnace guy needs to know this stuff.\"\n\nShe frowned at his tone. \"He can't figure out what's wrong on his own? I mean, isn't that supposed to be what he does? Figure out what's wrong?\"\n\n\"He says he needs a ductwork schematic so he can compare it to the rooms that aren't getting hot air. He thinks we might have a blockage somewhere.\" Sean cocked his head and leaned out of the doorway to listen. \"I think he's hollering for me, let me go check.\"\n\n\"What kind of blockage?\" she asked with a grimace, thinking of the sounds in the walls and imagining some sort of nest.\n\nBut Sean was already gone, leaving her among the mess of boxes he'd made when he started tearing things apart. If he'd just looked at the labels, she thought with a sigh, bending to put back a handful of manila envelopes filled with tax returns. But of course he hadn't. He'd expected her to magically hand him what he needed, and when she couldn't, he'd gone willy-nilly trying to find it. And couldn't be bothered to clean any of it up, either, she thought as she put the lid back on the file box.\n\nIt didn't solve the mystery of where the box had gone, though. She couldn't even remember seeing it, to be honest, but then why would she have looked for it? The real estate agent had presented them with an enormous, in Ginny's opinion, amount of trash related to the house. Ginny never looked at instruction manuals for any appliances she ever bought, so the chances of her ever reading through those for ones she'd inherited was equally as unlikely.\n\nThe repairman had been there for an hour already. Ginny tried not to think about the cost as she went downstairs to the kitchen to finish the brownies she'd been baking before Sean interrupted her for the wild-goose hunt for the schematics. The company charged by the hour, plus for parts, so even if he couldn't find and fix the problem, they were still going to be out some cash.\n\n\"Let me just pull it out of the air,\" she muttered, stirring the batter. Beating it, actually, though the recipe didn't call for such abuse. She slowed her motion, moving the thick, gooey liquid with the wooden spoon.\n\nThe kitchen, as usual, was blistering. She tasted sweat on her upper lip. She poured the batter into a baking pan and settled it in the oven, then set the timer. It took her only a few minutes to clean the mess she'd made, and she finished just as Sean and the repairman came up the basement stairs.\n\nGrunting.\n\nWhy were they grunting?\n\nGinny leaned out the kitchen doorway, astounded at the sight of her husband and some stranger wrestling with a table that looked too big to get around the sharp corner. In the way of men, they huffed and puffed commands at each other. \"Turn it...tilt it...tip it...yeah, that's it.\" It sort of sounded dirty, which might've made her giggle if she weren't so astonished.\n\n\"What the...what are you doing?\"\n\n\"Take it down the hall,\" Sean said to the repairman. With a grin over his shoulder at her, he said, \"I got you a dining room table.\"\n\n\"From the basement?\" Unable to make the angle in this direction, the men had taken the table down the hall and through the living room, but Ginny ducked into the dining room through the other doorway. \"Sean?\"\n\nThey settled the table in the middle of the room, under the stained-glass light fixture. Her husband looked at her proudly while the repairman-cum-furniture hauler dusted his hands on his coveralls. Ginny could only stare.\n\n\"That's a nice piece,\" the repairman said. \"Looks like cherry.\"\n\nSean, still grinning like he'd brought her a diamond ring wrapped in a rainbow shat from a unicorn, slapped a hand on it. \"Totally solid. Look at it. It was down in a corner, under a tarp.\"\n\n\"I remember seeing it when we looked at the house.\" It was not one of the pieces Ginny'd asked to keep. She bit her tongue in the presence of the repairman, who looked like he might be easily scandalized by a string of inventive invectives.\n\n\"Thanks for the help,\" Sean told him.\n\nThe other guy nodded. \"No problem. So, like I was saying, the rooms that are too hot are going to stay too hot so long as you've got your thermostat set so high. That's the furnace doing what it's meant to do.\"\n\nSean stepped aside to let the guy pass, heading for the front door, and Ginny peeked around the doorway to watch them. \"We set the thermostat so high to keep the other rooms, the cold rooms, at least bearable.\"\n\n\"I hear ya, fella. I hear ya. But these old houses, you know, sometimes the heat just goes right out the windows. Through chinks in the insulation, whatever. But I tuned it up and reset the system. You've got a good unit there.\" The repairman paused at the front door to look at Sean. \"It would've been better if I knew who installed it or had a better idea about where some of those ducts went, if they installed all new ducts or if they used the old ones, or maybe piggybacked...whatever. I mean, some of those ducts look like they're not even functioning to me, like maybe they were put in with the old unit or something. If I could tell if they put in new ducts with the new unit, that would help. But you should see an improvement anyway.\"\n\n\"Thanks.\" Sean shook the guy's hand and closed the door behind him, then turned to the thermostat and fiddled with it before looking down the hall at her. \"Hey. He says it should be fixed.\"\n\n\"I hope so.\" The smell of brownies had begun to permeate the air. She'd been careful not to lick the batter, no matter how tempting it had been, because of the raw eggs. Her stomach rumbled now.\n\n\"If it doesn't, we'll have him come back out,\" Sean said. \"Hey. Brownies?\"\n\n\"They'll be done in a few minutes.\"\n\n\"We can eat them on our new table.\"\n\nShe looked at the new-old dining room table, which was both exactly and nothing like what she'd wanted. \"It's filthy.\"\n\n\"You don't like it? I thought you'd love it. You wanted an antique table, you said.\"\n\nShe had mentioned it, yes. After they'd bought the fainting couch she'd looked at a few of the dining room sets in the same antique shop. They'd been sleek, Art Deco, with designs of inlaid wood and matching buffets that had beautiful and ornate drawer handles. Nothing like this square, sharp-edged, utilitarian wooden monstrosity. This table wasn't an antique, really. It was just...old. And well used, she thought, noticing the carved initials along one short side. CMM. Someone had been naughty.\n\n\"From the basement, Sean? Really?\"\n\nHe looked at it. \"You can polish it up, it'll be great. But if you don't like it\u2014\"\n\n\"No,\" she interrupted him. \"No. It's fine. We need a dining room table, and this way I can take my time looking for something. Someday.\"\n\nUntil then, they could use this table, even if it was ugly. Even if it wasn't what she wanted. She could make the best of things.\n\nHe settled his hands on her hips to pull her close for a kiss. \"Have I told you how awesome you are?\"\n\nHe had, many times, which only made her feel worse about hating the table he'd so obviously been happy to bring her. Ginny pushed onto her tiptoes, just a little, to kiss him back. \"Hmm. Because I ply you with sweets?\"\n\n\"You'll make me fat.\"\n\n\"Then I won't have to worry about any sexy, young chicks chasing after you,\" she teased, patting his flat, hard stomach. Sean never had to work out.\n\nHe looked at her seriously. \"You never have to worry about that, Ginny.\"\n\nShe'd meant it only as a joke, but his reply was so solemn it set her back. She cupped his face in her hands and kissed him again. \"I love you.\"\n\nHe nodded, eyes searching hers. \"You do?\"\n\n\"Of course I do,\" she told him, uneasy with the intensity of his expression. \"Of course.\"\n\n\"Good,\" Sean said. \"Let's eat brownies.\"\nChapter Nineteen\n\nThe smell. It was repulsive. Thick and cloying, the unmistakable stink of rot.\n\nGinny sniffed the air, then again. \"Ugh. God. Sean, you need to check the glue traps.\"\n\n\"I did, yesterday. Nothing.\" He rattled the paper instructions for the TV stand with a growl. \"Insert rod A into slot B. What the hell? There is no slot B.\"\n\nGinny sniffed again, walking slowly around the living room, which was still cluttered with a few boxes, even though it felt like they'd been unpacking forever. Sean had decided this was the weekend to put together the new television stand he'd insisted they needed for the new flat screen. He'd been cursing at it for the past hour and a half.\n\n\"Can't you smell that?\"\n\nHe sifted through a bag of small metal parts and plucked out a screw, then cursed some more when it didn't fit into the right hole. \"No. Smell what?\"\n\n\"I smell it.\" She sniffed again, nosing along the wall and over the vent. \"I can't believe you can't smell it. Something died in the walls, Sean. I'm sure of it.\"\n\n\"I thought you told me the exterminator said mice wouldn't smell that bad.\"\n\n\"I know what he said, but I'm telling you, I smell something disgusting. It's...\" She leaned over the vent blowing warm air that was nowhere near hot, and grimaced. She pulled her sweatshirt sleeve over her fingers and held it over her mouth and nose. \"It's stronger when the heat's on.\"\n\n\"Could be something in the ductwork.\" Sean shrugged, clearly unconcerned as he struggled with the TV stand's legs. \"God dammit. Why do they have to make these things so hard to build.\"\n\n\"We could've paid someone to put it together in the store,\" she reminded him, and wished she hadn't when she saw the set of his shoulders.\n\n\"They wanted to charge a hundred bucks for set-up and delivery.\"\n\n\"I know they did.\" But if they'd done that, they could now be watching a movie together or doing something else instead of this.\n\n\"I can do it anyway.\"\n\nShe sighed. \"I know you can.\"\n\nThe smell, thank God, had faded. Or she'd become immune to the stench. Either way, she could breathe with the filter of her sweatshirt. She watched him for a few more minutes, but knew better than to offer her help.\n\nThe next time the heat kicked on, though, the smell was back. She coughed from it, and Sean gave her a curious look. Ginny waved a hand in front of her face.\n\n\"You really don't smell that?\"\n\nSean stood and took a long, deep breath. \"Yeah. I smell something. It's faint, though.\"\n\n\"Please check those glue boards again. I'm sure something's dead on one!\"\n\nHe sighed. \"Sure, babe. Can I finish this first?\"\n\nHer look must've been answer enough, because Sean let out another sigh and hung his head. Without another word, he left the living room. She heard the slow tread of his feet on the stairs, in the hall, and finally into the nursery. She heard the creak of the cubbyhole door opening. More footsteps in the hall, then in their bedroom. She couldn't hear the cubbyhole door in their closet opening or closing. He came down a few minutes later with empty hands.\n\n\"I told you. Nothing. I mean, the guy said he didn't see any signs of anything, right?\"\n\n\"I still hear things,\" Ginny said stubbornly. \"In the walls. I told you.\"\n\nSean sighed and came closer, rubbing her upper arms to soothe her. \"I'm sure you do...\"\n\n\"I just heard it the other night,\" she pointed out. She did not add that she'd heard it while she was wakeful, unable to sleep, and he was snoring away.\n\nHe hugged her, stroking her hair. His shirt was damp. He smelled of sweat; she had to turn her head.\n\n\"All I can say is, I checked the traps. He said he'd be back to check the bait boxes. Right?\"\n\n\"Yes.\"\n\n\"So,\" Sean said, \"the next time he comes, ask him if he can smell it.\"\n\n\"Fine,\" she said, though it wasn't fine at all.\n\nHe worked in silence while she flipped through a couple of magazines. When he'd finished, he stood and waved at it. \"All done.\"\n\n\"Looks good.\"\n\nShit, now they'd been reduced to single syllables. Ginny sighed. \"You want some help hooking up the TV and stuff?\"\n\n\"No. I got it.\"\n\nShe went to the kitchen while he worked and made him an ice-cream sundae as a peace offering. She took it to him in the living room, then stood and shivered while he ate it. He offered her some, but she shook her head.\n\n\"I'm freezing.\" Ginny rubbed her arms and went to the vent in the floor, feeling a waft of lukewarm air. \"The kitchen's sweltering. I don't get it.\"\n\nSean sighed and handed her the empty ice-cream bowl. \"I'll call the repairman again tomorrow. Okay?\"\n\nGinny looked at the bowl, then at him. \"Yeah. That would be great.\"\n\nSometimes, he did get it. Sean got to his feet and hugged her, acting like he didn't notice that she'd turned her face when he tried to kiss her. \"Can't have my honey being cold, can I?\"\n\n\"It's just that it should work,\" Ginny said. \"It's supposed to be an almost-brand-new system, right? We just had the guy out here to check it out. It should just work.\"\n\n\"Lots of things should just work, but they don't.\" Sean looked at the TV stand.\n\nGinny knew that was certainly true. Marriage was one of them. Or maybe it was the other way around; marriage shouldn't work but did.\n\nShe looked toward the kitchen, then the bowl. \"You want anything else?\"\n\nSean, engrossed in his task, just grunted.\n\nGinny took the bowl into the kitchen and put it into the dishwasher. She stretched, slowly, droplets of sweat pearling on her forehead. The kitchen was still so stinking hot. The clock on the microwave blinked from their last power outage, and as she set it to the correct time, she noticed two things. The first, that it was getting late and she was getting tired. Second, Noodles had not yet been fed.\n\nThe reason she hadn't noticed was because the cat, who normally made her demands well known with a variety of vocal yowlings, had not seen fit to demand Ginny's services as head can opener. This was definitely not normal, but not entirely unheard of. Noodles could be cranky and sometimes suspicious, and Ginny was convinced the cat could also hold a grudge. If being shooed off Ginny's pillow this morning had sufficiently put her little pink nose out of joint, it was possible she was still hiding upstairs, even at the expense of her empty belly.\n\nOpening the can would bring her running, at least it usually did. Not this time. Ginny opened the can and scraped the gloopy, stinky contents into a bowl and set it on the special mat by the back door. No Noodles.\n\n\"Noodles! Ssss, ssss, sss!\"\n\nNo cat. In the sweltering kitchen, Ginny licked the sweat from her lip and fought off a wave of unease. She went through the dining room to the living room, where Sean still fought with the TV stand. She didn't bother asking him if he'd seen the cat. She went past him, into the front hall, up the stairs, into the bedroom. She got on her hands and knees and looked under the bed.\n\nNothing.\n\nGinny sat up, her breath coming a little too short in her lungs. She closed her eyes, trying to remember the last time she'd seen Noodles. This morning. The cat had been making herself at home, not just on Ginny's side of the bed, but on her pillow, and Ginny had clapped her hands and shouted to chase her away. She hadn't seen her since.\n\n\"Shit. Shit, shit...\" Ginny rubbed her face, got up and went downstairs.\n\n\"Sean. Have you seen Noodles?\"\n\nHe looked up at her, the building instructions crumpled in his fist. \"Huh? No.\"\n\nGinny sat slowly on the couch, which was still nowhere near where she wanted it to be. \"She's missing.\"\n\n\"What do you mean, missing?\" Sean wasn't paying attention to her, his focus still on the TV stand, which now at least looked as though it could hold the TV. \"I'm sure she's around. Did you call her?\"\n\n\"Yes. Of course. I opened her food, she didn't come running. I'm worried she got out of the house, maybe when I went out to get the mail.\"\n\n\"I'm sure she'll turn up.\" Sean turned back to fussing with cords and wires.\n\nThis answer didn't suit Ginny, who went through the house, calling the cat's name over and over with an increasing amount of desperation. Ginny looked in every closet, every crawl space, under every piece of furniture, behind every door. No Noodles. She opened the front door and called out into the night, thinking that if the cat had run out, she'd be more than eager to run back inside to her safe, warm house, but the only answer was a car passing by, splashing up a puddle from the day's earlier rain.\n\n\"She's gone,\" she told Sean in the living room, where he'd finally finished his project. \"I can't find her anywhere.\"\n\nHe looked up with a frown. \"She'll turn up. Even if she ran outside\u2014\"\n\n\"I called for her outside. She didn't come.\"\n\n\"Someone will find her.\" Sean looked at the TV, then at her. She could see the struggle on his face, the desire to finish getting his new toy set up and the knowledge he should somehow comfort her.\n\nAt least he thought he should. Ginny wasn't interested in being comforted. She backed up a step when it looked like her husband meant to come and hold her. She wanted to find her cat. Not be woo-wooed and petted.\n\n\"She'll come home,\" Sean said. \"If she's outside, someone will find her, or she'll come home.\"\n\nThe floor vibrated beneath Ginny's bare toes as the furnace kicked on. The curtains blew gently. Ginny took a breath, her hormone-enhanced sense of smell working hard. The faint scent of rot swirled around her, and she clapped a hand over her mouth and nose.\n\n\"What if she's not outside?\"\n\n\"Then she's fine,\" Sean soothed. He moved toward her but the cords tangled on his foot, and the television leashed him in place.\n\n\"No, no. I mean, what if she's in the house, but...trapped somewhere? Don't you smell that?\" Ginny cried, shuddering. \"Christ, Sean. Tell me you smell it.\"\n\nHe took a long, deep breath. \"I smell it. But it's hardly anything, honey. It's a mouse, like the guy said. Got stuck in the walls. It's not Noodles.\"\n\nSean dropped the cords. This time, Ginny didn't move away when he came to hold her. She pressed herself against him, her eyes closed, as he rubbed her back. Her belly made a bigger distance between them than she was used to, really noticeable for the first time.\n\n\"We'll find her. I promise.\"\n\nShe knew he couldn't promise anything of the sort, but she let him anyway.\nChapter Twenty\n\nGinny's gran might not always be completely locked in the present day, but she never looked anything less than her best. Compared to Ginny's mom, who wore a sweatshirt with a stain on it, her hair frowzy, her jeans out of style, Gran was the epitome of a classy lady. From her carefully coiffed perm to the shoes that matched her bag that matched her belt, Gran looked like she'd just stepped out of a salon run by fashion elves slaving to dress her. There was no such thing, of course. Ginny's mom had been the one to make sure Gran had everything she needed, and she wasn't afraid to make sure Ginny and everyone else knew it.\n\n\"Hours,\" she said. \"Hours it took us to get ready.\"\n\n\"Well, Gertrude, if you took some care with your own appearance, perhaps mine might not be such an offense,\" Gran said with a sniff and held her hand out for Sean to take as she hobbled through the front door.\n\nHe looked at it as though he wasn't sure if she meant for him to kiss it, and Ginny stifled a laugh. Gran had never approved of Sean, who'd shown up to his first family function wearing a black-leather jacket and a week's worth of beard scruff, riding a Harley-Davidson motorcycle. That he'd sold the bike and kept the jacket in the back of a closet now wouldn't make a difference. Gran could forget what she had for breakfast that morning, but she'd never forget that Ginny's \"boy\" was of the bad sort.\n\n\"Virginia. You're looking fat.\"\n\n\"Thanks, Gran.\" Ginny stepped aside so Sean and her mom could ease Gran into the living room and onto the couch, which stuck out awkwardly in the middle of the room because they'd stacked up all the still-unpacked boxes behind it.\n\n\"Mother, Ginny's not fat. She's pregnant,\" Ginny's mom said too loudly. \"Remember? We're here for Ginny's baby shower!\"\n\n\"And I'm old, not deaf,\" Gran retorted. She gave Sean an up-and-down look before turning back to Ginny. \"Virginia, have your boy bring me something to drink. Two fingers of Scotch, no ice.\"\n\n\"Mother,\" Ginny's mom began, but Ginny waved her quiet.\n\nSean backed up a step. \"I'll get it.\"\n\nHe already knew to make the drink only half the alcohol Gran requested, the other half water. She complained it was too strong the other way and wouldn't finish more than a few sips.\n\nGinny gestured for her mom to sit in the armchair while she took the other. They both looked at Gran, who stared back.\n\n\"Are you going to give me food, or what?\" Gran asked suddenly. \"It's mashed-potatoes-and-meat-loaf night at home. What are you going to feed me?\"\n\n\"There's plenty of food, Gran.\" Ginny smoothed her shirt over her belly, feeling self-conscious though she wasn't any fatter now than she'd been this morning when she got dressed.\n\nGran snorted as Sean brought the drink and pressed it into her hand. She stared up at him. \"You look much better clean-shaven.\"\n\nSean passed a hand over his cheeks and chin, then gave Ginny a brow-raised glance and a quick smile. \"Thanks.\"\n\n\"I liked his scruff,\" Ginny blurted. \"I liked it when he looked a little ragged.\"\n\nSean looked surprised. Then pleased. Gran, however, sniffed and daintily sipped from her glass before making a face and holding it out for him to take.\n\n\"This drink tastes like a ruffian made it.\"\n\n\"Can I be a ruffian if I'm clean-shaven?\" Sean said it with a straight face but a twinkle in his eyes.\n\nBefore Gran could reply, the doorbell rang and Sean went to answer it. Soon the living room overflowed with guests, and Ginny wished, hard, she'd been more demanding about unpacking the boxes Sean had insisted would be fine shoved up along the wall under the windows. She cringed as Billy's daughter Kristen, running after her brother, ran into a box that made a definite jingling sound when bumped.\n\nThat made her think of Noodles and the bell on her collar, and how the cat would've hated this crowd, and how they'd have had to lock her in a room to keep her from darting out the front door. How she'd probably have peed on something out of spite. Maybe, Ginny thought suddenly, with the door opening so often, Noodles would make her way back home, reappearing underfoot like the triumphant small queen the cat had always been.\n\nShe didn't, though, and playing hostess gave Ginny little time to think about it further. The present opening was an extravaganza of paper and ribbon, onesies and booties and diapers and plush toys. Such generosity moved Ginny to tears more than once, but it seemed quite acceptable for the expectant mother to cry when she opened a handmade and embroidered quilt it must've taken her mother more than a year to make. Ginny didn't point out what that meant, not when they were all there to celebrate. But she knew.\n\nIt was a big party, bigger than Ginny wanted, but her mom and Peg had insisted on inviting everyone who could possibly be invited, plus some people Ginny wasn't even sure she knew. Sean's mother hovered around making sure the food was all set up and drinks poured, which was nice but gave Ginny nothing to do but sit and hold court. Barb had also brought along all the paper products and baked a cake with plastic babies riding carrots on the top that looked like something out of a horror movie. Now she nursed a glass of white zinfandel and laughed too loudly at whatever Ginny's mom was saying to her.\n\nGinny had filled a plate with beef barbecue on a roll, macaroni salad, red beet eggs. The same food that had been served at every baby shower she'd ever been to. She'd been starving and ate too fast; now her stomach rumbled and ached, and she wondered if it would be bad form to ask everyone to leave so she could take a nap. She leaned in the parlor doorway, looking at the stacks of gifts she'd opened. Babies needed so many things.\n\n\"Look at all the loot.\" Sean appeared beside her.\n\n\"Yeah, I know.\" She leaned against him. \"Don't be a ruffian and run off with all of it.\"\n\nHe put an arm around her shoulders and kissed her temple. \"That's me, hooligan to the core. Maybe I should grow my beard again, you think?\"\n\n\"Beard. Ha. You never had a beard.\" She turned to face him. \"You only ever had that three days' worth of stubble. I could never figure out if you thought it made you look like a badass, or you were just too lazy to shave.\"\n\n\"Now you know, huh?\"\n\n\"Yeah. Too lazy,\" she teased, and kissed his mouth.\n\nHe tasted of Scotch and, more faintly, of cigarettes, but that had to be her imagination because Sean had quit smoking a few years ago. The first time she got pregnant. She kissed him again and slid her hands, flat, up the soft corduroy of his jacket to cup his shoulders.\n\n\"Were you...smoking?\"\n\nHe looked briefly guilty, then...not defiant. More like daring. \"I was out back with your brother, showing him where I wanted to put the shed.\"\n\nSurely the size or expense of a shed wasn't that important or stressful that a discussion necessitated cigarettes. \"Sean.\"\n\n\"Yeah. We were smoking. Drinking a little too. It's a party, baby.\" He nuzzled her, his mouth finding the sensitive spots on her neck and below her ear, making her shiver. \"Can't a guy get a little happy at his wife's baby shower?\"\n\nGinny had never been the one to tell Sean he had to quit smoking, he'd done that on his own. Of course she'd been glad he quit. Of course she had. But now, his kisses flavored with liquor and smoke, his hands sliding up her hips to cup her waist, Ginny had a flashback to the first night they'd fucked. You couldn't even have called it making love. No bed, no soft music or candles or slow, sweet undressing.\n\nThe first time she'd had him inside her, they'd been at a party thrown by someone she didn't know. A house she'd never been inside before. There'd been loud music, a table laden with food, the air thick with smoke and even the tang of marijuana. Sean had introduced her to all his friends as \"his girl,\" though they'd only been seeing each other for a week or so and Ginny wasn't even sure she meant to keep going out with him. His fingers had linked with hers, his other hand free to hold his cigarette, but he was so careful to never blow the smoke at her. Considerate that way, and also in how he made sure she always had a fresh drink, a plate full of goodies if that's what she wanted. He remembered that she liked pepperoni and not shrimp, that she preferred her drinks without ice. He'd paid attention to her back then.\n\nThe revelation had hit her, watching him laugh with a guy whose name she still didn't know. Sean's hand in hers, his focus on someone else, the sheen of colored lights from the Christmas tree making pretty patterns on his face and against the soft fringes of the hair he always, always, always wore in front of his ears. She'd tugged his hand and he'd turned to her at once, making her the most important person in the room to him.\n\nThey'd fucked in a tiny powder room, her ass slipping on cold porcelain, his jeans around his ankles. His hand over her mouth when she started to cry out. She could still taste his skin.\n\n\"Come with me,\" she whispered now and took his hand, but when she tugged him toward their powder room, Sean hung back.\n\n\"Ginny. What are you doing?\"\n\nNo, she wanted to say. Don't speak. Don't refuse me. Just come into the bathroom with me and shut the door, and put yourself inside me and put your hand over my mouth so nobody hears me cry out your name.\n\nShe laughed instead, shaky and self-conscious, her humor insincere. \"Nothing. Just playing.\"\n\n\"Later,\" he told her. He kissed her temple.\n\nBut there would be no later, because it wouldn't be the same. And that was the problem, wasn't it, Ginny thought as she watched her husband walk away from her. Things changed. Nothing stayed the same.\n\n\"Time for caaaake,\" Barb trilled, appearing in the arched doorway between the kitchen and dining room. \"Everyone, come and have some cake!\"\n\nGinny put on a smile. Sean's mom, for all her useless fluttering, did make a mean red velvet cake, with icing to die for, and Ginny fully intended to take advantage of her eating-for-two status. Despite the weird plastic babies\u2014riding carrots, of all things\u2014the cake was pretty, all white and red and presented on the special cake platter Barb used for every special occasion.\n\n\"You cut it,\" Barb said. \"I tried something new, a special recipe, because I know how much you like cherry pie, Ginny.\"\n\nThey had a cake knife, but of course it was still in a box somewhere, so Ginny took the cleaver Sean handed her. Someone made a joke about never getting between a pregnant woman and cake. Everyone laughed. Ginny pressed the blade slowly through the white icing, the dense red cake...and into something else.\n\nAs she lifted the first piece of cake, the insides oozed and dripped with sticky red goo. Thick clots of it clung to the cleaver. One plopped onto the table. Ginny recoiled.\n\nBlood.\n\nSo much blood.\n\n\"It's a cherrvelvet!\" Barb clapped her hands. \"A cherry pie inside a red velvet cake! I thought if it worked out, I'd make a cherrpumple for Thanksgiving. That's a cherry pie inside a pumpkin pie inside an apple pie.\"\n\nGinny swallowed against a thin sting of bile. \"It's...great.\"\n\n\"Looks like something that got squashed on the road,\" Gran said. \"And why do those cherubs have such gigantic private bits?\"\n\nGod bless Gran for distracting the crowd. Ginny handed the knife to Sean. She backed away from the oozing, dripping cake.\n\n\"We'll need some more plates,\" she said faintly. \"I'll get some.\"\n\nIn the still-unorganized pantry, Ginny let the spring-loaded door close behind her. The small room was blisteringly hot, and the stench that had so plagued the rest of the house earlier still lingered here. She put a hand over her mouth and nose and gripped the shelving with the other as she sagged.\n\n* * * * *\n\nBlood.\n\nThere's so much blood.\n\nShe goes to the toilet, her back aching, her belly cramping. Ginny knows what this means. She's always known. She could tell from the beginning. Something didn't feel right, it never had. She never told Sean. She didn't want him to worry. He's been so excited.\n\nWhen she goes to the toilet and puts her hand down between her legs, her fingers come away covered in blood. Dark, thick blood. Clots of it cling to her skin. Another rush of cramping squeezes her, and Ginny cries out. There's a woman in the next stall, so Ginny bites her lip against another cry.\n\nThe pain is worse than it was the other times. Like her insides are tearing. Shredding. Which...they are, she thinks as another wave of pain washes over her.\n\nShe is losing her baby.\n\nEverything inside her goes tight, tangled, twisted; her belly tenses. Hard like a rock. She's not even in maternity clothes yet, just wearing a size larger, and she lifts the hem of her T-shirt to press her hands against her bare skin. The blood smears on her pale skin. There is so much of it, it's everywhere.\n\nShe needs to find her phone. She needs to call for help. She needs to have this not be happening, but it is. Even as she fumbles for the phone and presses the emergency number, another series of contractions push through her. She can't speak through gritted teeth. She can only groan.\n\n\"Are you...okay?\" Someone outside the stall raps softly.\n\n\"No,\" Ginny manages to say. \"I'm losing my baby. Please call for help.\"\n\nThe sound of running feet. The slamming of the restroom door. Oh, how she wishes she'd stayed home today instead of going to the grocery store. Then she would be at home, in her own bathroom. But she's not, she's here, and the blood keeps coming.\n\nSomething soft and loose happens between her legs. Something tries to slip out of her, and Ginny holds it back with one hand while she leans to unlock the stall door with the other. There are paramedics there on the other side, one young man, one older woman.\n\n\"We're here, hon,\" says the woman and locks her gaze with Ginny's. \"We're here now. You can let go.\"\n\nAnd Ginny does.\n\nShe lets her daughter go, and there is more blood. Always more blood. And they take her away to the hospital, where Sean doesn't show up for hours and hours. When he does, his face is pale, his hair is mussed, he stinks of cigarettes and alcohol, maybe even the faintest hint of perfume. Ginny doesn't even care. She can't look at him when he's there at last, because he'd been so convinced it would all be okay, and she'd known it wouldn't, but she'd let him believe it.\n\n* * * * *\n\nHours have passed in her memory, but only minutes in the pantry. She needed to hurry before someone who meant well came in here after her and she had to blow off her tears as more sentimentality. Ginny scanned the shelves for paper plates and napkins and found a plastic bag she remembered buying at the store. The paper goods were inside.\n\nAs she moved to pull the bag from the shelf, her toe nudged against a bulk bag of rice she'd brought from the townhouse and dropped in the pantry without using once since they'd moved. The bag shifted, revealing the vent it had been covering, and fell on its side with the contents spilling. With a curse, Ginny bent to sweep up the grains with her fingers, too aware of the party noises from outside and expecting Sean's mom to poke her head inside at any minute.\n\nSome of the rice skittered across the floor and into the vent pulsing hot air. More spilled as she tried to lift the bag and close it. Ginny grabbed a handful, debating about just tossing it back in the bag and throwing the whole thing away\u2014the chances of them ever eating any of it seemed pretty slim at this point.\n\nSomething moved in her hand.\n\nStartled, Ginny looked down at her palm. Some of the white grains of rice were...moving. Wiggling. Too stunned to even drop it at first, the low, angry buzz of something else distracted her. As she watched, a fly forced its way out of the vent. Bobbing on the currents of hot air, it tumbled drunkenly toward her.\n\nGinny dropped the rice and maggots to swat at it, but the fly dive-bombed her. Disgusted, she backed up, still crouching. More flies came out of the vent, at first one by one, then in twos and threes. Twenty flies circled her. Then more.\n\nGinny scrambled backwards and hit the door, which opened inward and made it impossible for anyone to open, though by now she'd started screaming. Covering her face against the flies' assault, she tried to find the doorknob with her other hand, but her fingers skidded on the wood and missed the metal handle. She heard muffled shouts. The door bumped behind her, moving her toward the flies. She had to move forward into the thick of the buzzing swarm still pouring out of the vent so the door could open, but in her terror found it almost impossible to do it.\n\n\"Ginny!\" Sean hollered, pounding then shoving on the door hard enough to force her forward a few steps.\n\nThe flies swept past her and into the kitchen, where the much larger space dispersed them from a thick black cloud to a more widespread swarm. Party guests screamed and ducked, running. Someone ran into the table, shaking it hard enough to topple the cake stand onto its side, spattering red velvet cake and cherry pie all over. Billy, always a quick thinker, grabbed a swatter from the hook on the side of the cabinet, and started flailing.\n\n\"You okay?\" Sean looked into her eyes, holding her upright.\n\nGinny nodded, swallowing her disgust. \"Yes. Gross. What the hell?\"\n\nHe looked past her into the pantry. \"They must've been breeding in that bag of rice or something. You sure you're okay?\"\n\nGinny nodded again, straightening. \"Yes. Go help my brother. Get rid of them.\"\n\nGinny's mom opened the back door, letting in a swirl of icy wind, but allowing the men to shoo the flies toward it. Some fell dead under Billy's swatter and the rolled-up catalog Sean grabbed from the counter. Others flew off into the house, God only knows where, the thought of finding them later making her shudder.\n\nIn just a few minutes, everything had calmed down, except for Sean's mom, who sobbed over the broken cake like she'd given birth to it instead of her son. Peg and Dale made their goodbyes, while Billy wrangled his kids into the powder room to get them clean from eating the cake with their bare hands. Ginny's mom took Barb to the living room so she could get herself under control. Sean went to the alcove to find the mop and bucket, leaving Ginny and her gran standing in the middle of the now-empty, but at least flyless, kitchen.\n\nGran still clutched her glass of Scotch. Her lipstick had smudged, her carefully styled hair a little rumpled. She looked smaller than she ever had, her shoulders and back hunched. She lifted her glass in Ginny's direction, and Ginny waited for the scolding or the accusation that Ginny had a filthy house.\n\n\"That girl did it,\" Gran said.\n\n\"What girl? Kristen?\"\n\n\"Who's Kristen?\"\n\nGinny sighed. \"Billy's daughter, Gran. You know Kristen. She's ten? Blonde?\"\n\n\"Looks like her mother, oh that one.\" Gran nodded. \"Same sour face.\"\n\n\"Oh, Gran.\" Ginny bit the inside of her cheek to keep from laughing.\n\n\"Not that one. The other girl. The one with the dark hair. Like yours when you were small. I saw her upstairs.\" Gran sipped from her glass with a grimace and shuffled to the sink to pour away the liquid. \"Nobody knows how to make a decent drink anymore.\"\n\nAll of Billy's kids were as blond as Kristen, all taking after their mother, as Gran had pointed out. Peg's daughter Maria had dark hair, but she was away at college, not at the party. Ginny moved to take the empty glass before Gran could drop it.\n\n\"You saw a picture of Maria? Upstairs?\"\n\nGran looked contemptuous. \"Not a picture, Virginia. Listen to me. That girl. Upstairs.\"\n\nGran stabbed a gnarled finger, the nail painted bright red, at Ginny. \"That girl looked like a hobo. Hair a mess. Wearing rags. Shameful, really. A girl like that would bring filth with her.\"\n\n\"I don't understand.\" Unease dried Ginny's throat, so she filled Gran's glass with water from the tap and drank it, tasting a hint of Scotch. \"There was no girl, Gran.\"\n\nGran sighed. \"I saw her, just like I'm seeing you right now. You mark my words, Virginia. She's trouble.\"\n\n\"Who's trouble, Mother?\" Ginny's mom came into the kitchen and gave Ginny a sympathetic glance. \"I came to get Barb a cold compress.\"\n\n\"Oh. God.\" Ginny grimaced and moved aside so her mom could pull a clean dishcloth from the drawer. \"She's that bad, huh?\"\n\nGinny's mom lowered her voice. \"I swear she almost passed out.\"\n\nAgain, Ginny bit the inside of her cheek to keep from laughing as Sean reappeared with the cleaning supplies. He held them aloft triumphantly, then caught sight of her. His brows raised.\n\n\"What? They weren't where I thought they'd be.\" He looked from Ginny to Gran, to her mom. \"Where's my mom?\"\n\n\"She's calming down in the living room,\" Ginny's mom said.\n\nSean sighed, shoulders slumping, and set the mop and bucket down. \"Right. Okay. I'll be back to take care of this.\"\n\n\"You never mind,\" Gran said firmly. \"The day a man can clean a kitchen floor better than a woman can is the day we all get taken up to heaven on the back of a unicorn farting rainbows.\"\n\n\"Mom.\" Ginny's mom sighed and shook her head. \"For God's sake.\"\n\n\"No, Gran. You're not cleaning my floor. It's time for Mom to take you home anyway. You go.\" Ginny shooed her. \"I'll take care of this.\"\n\nAt first, Gran didn't move, but then she nodded and allowed Ginny's mom to shuffle her toward the front door. There she hung back to look askance into the living room and mutter something about \"ridiculous biddies,\" before Ginny's mom helped her into her coat and tied the scarf around her throat.\n\nIn the doorway, the cold air making Ginny shiver, Gran paused and wouldn't be moved along, even by her daughter's arm-tugging. \"You listen to me, Virginia. Get yourself a priest.\"\n\n\"Mom. What does Ginny want a priest for?\"\n\n\"That girl is trouble, Virginia. You get yourself a priest and get her out of your house.\"\n\nThen Ginny's mom was moving Gran off the porch and along the sidewalk toward the car, Sean's mom was up and in the kitchen, insisting on getting on her hands and knees to take care of the mess, and Sean was pouring himself a full glass of Scotch.\n\n\"What the hell was your grandmother talking about?\" he said in an aside as Ginny tried her best to just stay the hell out of Barb's way.\n\n\"I have no idea. Can you get rid of that rice?\"\n\n\"Sure.\" He drained the glass and set it in the sink. Then he took Ginny in his arms. \"Hey. You okay? You look pale.\"\n\n\"I feel a little woozy,\" Ginny told him. \"I'm going upstairs to lie down, okay?\"\n\n* * * * *\n\nThe nursery was still bare, though soon enough all the gifts they'd received today would be brought up to fill it. Ginny stood in the empty room, remembering how she'd imagined what it would look like when it was finished. When they had a baby cooing and crying in the crib Sean had not yet put together. When she would rock her child in the chair they'd not yet bought. She put her hands on her belly.\n\n\"A girl...with dark hair...like yours when you were small.\"\n\nGinny closed her eyes and whispered, \"Baby, are you here?\"\n\nBut when she opened them, the room was still as empty as it had always been.\nChapter Twenty-One\n\n\"Have you seen my mug?\"\n\nSean didn't even look up from his iPhone, where he was busy tapping away at some zombie game he'd become obsessed with. \"No.\"\n\nGinny looked again into the cupboard. She ran her fingers along the collection of mugs. None of them matched, which had never bothered her before but suddenly irritated her. Their plates matched. Their silverware matched. Their glasses even matched, a full set of tumblers, drinking glasses and wineglasses in a pattern she'd picked out for their wedding registry and sometimes regretted because it had been the most expensive one. They'd had to spend a fortune to finish the set after getting only a few pieces, and the cost to replace any that broke was ridiculous.\n\nGinny started pulling out the motley collection of freebies from banks and charities, lining them up on the counter until Sean finally bothered to look up and ask what she was doing. \"I'm looking for my mug. I told you. Have you seen it?\" A sudden uncharitable thought made her eyes narrow. \"Did you take it to work and leave it there?\"\n\n\"No.\"\n\n\"Think hard.\" She kept her tone as pleasant as she could, as nonconfrontational, but she couldn't keep it entirely sweet. \"Did you take my mug?\"\n\n\"I don't even know which mug you're talking about.\" Sean stood. \"I gotta run.\"\n\nAnd run he tried, without bothering to put his dish in the dishwasher. Or even the sink, which would still have been an affront, but would've at least been something of an effort. Ginny stared at the plate, the fork still soaking in the mess of fried eggs and last bit of jelly toast he hadn't eaten.\n\nShe'd be damned if she cleaned it up. She'd cooked him that breakfast when the very smell of frying eggs still made her want to heave. She'd even spread that toast with jelly, grape, which she also loathed, because he'd been running late in the shower and she didn't want him to have to rush. And now he not only got up without bothering to pretend he intended to clean up after himself, like any adult would, but to add another insult, he was ducking away from her inquiries about her mug.\n\n\"Hey!\" she cried, stopping him at the front door. The cold swirled in, but she didn't care just then. The house was going to be too damned cold anyway. A few minutes of wintery air pummeling her hardly mattered. \"My mug.\"\n\nSean sighed and turned. \"Which one?\"\n\n\"The one with the pink skull and crossbones on it. The one my sister bought for my birthday.\" She eyed him, still suspicious. \"You know, the tall, skinny one?\"\n\nShe saw a flicker of recognition in his eyes. It made her frown. She liked that mug because her sister had picked it out for her on a weekend trip away a few years ago. They'd gone to a bed-and-breakfast and done some outlet shopping, eaten in nice restaurants that weren't kid friendly. It had been the last time they'd done anything like that\u2014life had gotten in the way. Ginny had spent her next birthday in the hospital, miscarrying.\n\nShe liked it because it reminded her of good times. Sean liked it because the tapered bottom fit neatly into his cup holder. She'd bought him a travel mug, but he still took hers. He didn't see the problem, after all. There were plenty of mugs for her to use, and of course her insistence that he leave \"hers\" alone made her out as some unreasonable shrew.\n\n\"I didn't leave it at work.\"\n\n\"Did you use it?\" The accusation rang out, too loud, too harsh for this early in the morning and the enormity of the offense. Or lack of.\n\nHis gaze skittered from hers. \"I...if I did, I put it in the dishwasher. Look, I have to go. I'm going to be late.\"\n\n\"Fine. Go.\" She flapped a hand at him, already turning to swallow her anger, to shove it down deep so it couldn't come out in another outburst.\n\n\"Maybe you left it somewhere,\" he said from the doorway, but was gone before she could reply.\n\nLeft it somewhere?\n\nIt could've been a dig. At least, in the mood she was in, Ginny wanted to take it that way. It was true; she was more apt to be the one leaving her belongings strewn about. Her shoes, car keys, a sweater draped over the railing instead of hung in the closet. It was a flaw, she knew it, but because she knew how it irritated him to find her stuff all over the place, she'd been trying harder to make sure she was better about it.\n\nBecause she listened to him, she thought bitterly as she yanked open the dishwasher. The mug wasn't in there, and she wasn't surprised. She hadn't used the fucking thing. Yesterday she hadn't made tea because even the decaf seemed to be wreaking havoc with her sleep. She'd been up every night for the past three. Counting backwards from one hundred did nothing. Neither did lavender on her pillow, though it did give her varied vivid and intriguing dreams during the few hours she did manage to sleep.\n\nGinny closed the dishwasher and moved again to the cupboards to search them all in case someone had put it away in the wrong place. Nothing. Anger simmering, she drew in a slow breath and let it out, reminding herself to keep her blood pressure from rising. She could drink her herbal, decaf tea from a different mug. No big deal.\n\nExcept that it was, and she wanted to cry when she filled another mug with hot water and let the tea bag steep. Even as she swiped her tears with the back of her hand, Ginny knew she was being ridiculous. But that was the deal with pregnancy and lack of sleep, wasn't it? Emotions running high and close to the surface, ready to spill over.\n\nTonight if she didn't sleep she'd think about making an appointment with the doctor, she decided when she took her tea into the living room to look through the old issues of her magazine subscriptions that had finally caught up to the address change and arrived in bulk. She'd suffered this before. Not quite insomnia. She had no trouble getting to sleep when she went up to bed. Hell, there were some nights that if she hadn't had to wait for Sean to get home from class so they could have dinner and spend some time together, she'd have put her pj's on and hit the sheets by eight. No, her trouble wasn't falling asleep, but staying asleep, then getting back to sleep once she woke up. Every night between midnight and 2:00 a.m. She told herself her body was preparing for the baby. It didn't make the mornings come any later.\n\nIn the living room, the slick pile of magazines slipped from her fingers and the tea scalded her when she tried to keep from dropping them. She stubbed her toe on a box that had been nudged out of place, even though she'd specifically shoved it up against the wall, hard, last night. Ginny let out a muttered curse and set her mug down on the end table that should've been an inch or so to the left but instead had also been shifted so the mug toppled to the floor and soaked the magazines. It didn't break, at least there was that.\n\n\"God damn it,\" she said, then louder, \"son of a bitch.\"\n\nShe yanked a roll of paper towels from the cupboard and got on her hands and knees to blot up the mess. Her magazines were salvageable. She could make more tea. But, damn it, Ginny thought as she looked around the chaos of her living room, when the hell was Sean going to finish unpacking all these boxes the way he'd promised he would weeks ago? Months, now. It had been more like a couple of months.\n\n\"Screw this,\" Ginny muttered as she got to her feet. Her knees hurt, and so did her hand from the hot tea. For the first time in weeks, she'd been planning on just sitting with her feet up, the way her husband insisted, now that she had something to do while she sat, since everything else she might've occupied her time with was mostly still packed away in boxes. All she'd wanted was to read through the accumulated weeks of gossip from the celebrity magazines and maybe check out a few new recipes. Hell, learn a few things from Popular Science or the news magazines she'd ordered from her nephew's school fundraiser.\n\nBut nope. Instead, she looked around at the mess and could no longer ignore it. Couldn't avoid it. She was done waiting for him to \"get around to it.\"\n\nBy lunchtime, Ginny'd managed to unpack every box in the living room and move the ones that still needed to go upstairs into the hall. She'd been ruthless. If she took something out of the box, it either found a place in the living room or was designated for some other specific place in the house...or put into the trash. She'd hauled two full trash bags out to the curb and half filled another.\n\nShe even moved the furniture. Slowly, a little bit at a time, but she did it. It helped that they didn't have much. The furniture that had filled the living room in the townhouse left plenty of space when divided between the living room and dining room, and she wasn't quite sure that everything was placed exactly where she wanted it, but it would do. At least they could freaking use the room, she thought as she took a few minutes' breather by settling on the couch, her feet on the ottoman, the now-dry magazines on her lap and a fresh mug of tea on the end table.\n\nYes. This. She looked around the room with satisfaction, ignoring for the moment the boxes in the hall that would need to be carried upstairs and her rumbling stomach, which would be soothed only for so long by the liquid. For now, she was going to sit and enjoy the results of her hard work.\n\nShe woke up an hour later, her neck stiff, the magazine article only three-quarters read and the tea long cold. Blinking, wincing, Ginny stretched and rubbed her furry tongue on the roof of her mouth. She looked outside, where the skies had gone gray enough to make it seem later than it was.\n\nShe'd needed the rest, that was for sure, but it would've been nicer to take a nap in her bed. Or at the very least, lying down on the couch instead of sitting up. Now everything ached, joints popped, and she didn't feel very rested at all. She was hungry, though. Starving, in fact, which was a nice change from the intermittent nausea that had plagued her with enough frequency that even when she didn't feel sick to her stomach, she worried enough about feeling sick that she kept herself from eating too much.\n\nNow she felt like she could down an entire twelve-inch hoagie, a whole pizza, a couple of cheeseburgers with an order of fries and a thick, creamy milkshake. Chocolate, she thought as moved through her now completely uncluttered living and dining rooms toward the kitchen. No, mint chocolate chip. Yes. Maybe there was some ice cream in the freezer, and she'd treat herself to a scoop. Or two.\n\nShe hadn't felt this good, aside from the creaking joints, in ages. Even the nagging loss of Noodles wasn't weighing on her. She actually hummed under her breath as she pulled out the makings for a sandwich and lined it up on the kitchen table. Bread, turkey, roast beef, lettuce, pickles, mayo. She sliced some tomatoes and added them to the growing tower of lunchy goodness. No true Dagwood sandwich would be complete without some good spicy mustard\u2014her mouth watered at the thought\u2014and a few slices of Swiss cheese.\n\nExcept that when she looked for the mustard, the spot where it should be on the fridge door was ostentatiously empty.\n\nHuh.\n\nGinny looked again. Then at the other shelves. Then at last she found it, shoved way to the back behind the bottle of lemon juice and an expired carton of half-and-half she took out to toss in the trash. She pulled out the deli package of cheese too, frowning. Sean didn't use mustard. He liked mayo or, shudder, margarine on his sandwiches. She'd even known him to spread white bread with ketchup before adding bologna, a combination that had made her gorge rise even when she wasn't fighting the pregnancy nausea. He didn't use mustard, so she couldn't blame him for putting it away in the wrong place, because how hard was it, exactly, to put things back in the place where they'd been found. Right? Even Ginny, who admittedly sometimes left her shoes by the front door until she had more pairs there than in her bedroom closet, knew enough to replace the mustard in its slot on the door. Next to the ketchup and mayo and salad dressings, that's where the mustard went, and since it wasn't there, she had to assume she'd been the one who hadn't put it back.\n\nUneasily, thinking of her lost mug, Ginny opened the mustard jar and found it so empty she could barely scrape enough out of it to spread on her bread. This was annoying, but not tragic. What she found when she pulled the cheese out of the package, though, was enough to make her throw it down on the table with a low cry of outrage.\n\nIt was bitten.\n\nSomeone, and it could only have been Sean, because who else would've done it? Someone had taken a bite out of the entire block of sliced Swiss cheese and put it back in the package. The tooth marks were clear, rippled around the edge of one of the larger natural holes in the cheese. The entire package was ruined, and why? To what freaking purpose?\n\n\"He doesn't even like Swiss cheese!\" Ginny cried aloud.\n\nWhich probably explained why he'd taken only one bite, though it didn't come close to making sense of why he'd bitten it in the first place. Grumbling, Ginny threw the cheese and the half-and-half into the garbage, then finished her sandwich. Sans cheese it was still good, but her appetite had been cut in half. She finished only part of the sandwich and wrapped up the rest for later, tucking it back into the deeper realms of the fridge so Sean wouldn't accidentally take it for his lunch tomorrow.\n\n\"He'd take it and complain about how it had mustard on it,\" she groused to her sister when she called a few minutes later while Ginny was cleaning her mess. \"Gah. Peg, I'm so annoyed.\"\n\n\"I hear that.\" Peg's sigh filled up the phone. \"It's like a war zone in my house right now. Between indoor lacrosse, Dale's triathlon training and work, oh yeah. Work, 'cuz everything else we do is 'leisure.' We barely have time to breathe.\"\n\nGinny's hands drifted over the mound of her belly. \"It was different when I worked. I didn't notice the mess or care about it as much, I guess. Or maybe he was just more in to helping out, we both pitched in. But now that I'm home full time...\"\n\nPeg snorted a chuckle. \"Oh, just wait until the sprout arrives. You'll look back on this time as your glory days.\"\n\n\"You're not helping.\"\n\n\"Sorry.\" Peg sounded anything but contrite, but she did try again. \"I guess my best big-sister advice to you is to take it easy. Let Sean do the unpacking, like he wants to.\"\n\n\"Like he says he wants to. But then never does. I understand he's tired when he gets home from work, and school's been harder than he thought it would be. I get that too. But, Jesus Christ, I'm the one who has to maneuver around everything in boxes, try to find stuff...\" She trailed off.\n\n\"You need to take care of yourself. Should I have Dale talk to him?\"\n\n\"No,\" Ginny said, thinking of how little Sean would appreciate a lecture from his brother-in-law. \"You guys are busy enough. Anyway, I did it all down here, and he won't be home for a few more hours. I'll do some more and it won't even be an issue.\"\n\n\"It sounds like it is an issue, though. And, Gin, you need to be careful...\"\n\nIrritation flared. \"Sean says the same thing. You're supposed to be on my side.\"\n\nMore silence.\n\nGinny sighed. \"I had an appointment just last week. The doctor said everything was fine. I'm not on any restrictions. I'm not high risk. There's absolutely no indication of any problems. At. All. I'm not running a marathon or lifting barbells, for crying out loud, Peg. I'm just putting away books and knickknacks.\"\n\nShe didn't mention the huffing and puffing of pushing the couch and chairs into place, or the boxes of books that were meant to be taken upstairs. Her sister's pregnancies, all six of them, had gone off without a hitch. By the last one, she gave birth at home in her bed, with all the kids around her, cheering on the birth of their new baby brother, and it was over before the midwife even arrived.\n\nHer sister didn't have any real idea what it was like to know that the life inside her had died and was decaying, or what it was like to crouch in a public restroom, knowing there was nothing she could do to stop her body from rejecting it. Her sister understood grief and could probably understand the loss of a child more fully than Ginny could yet grasp, but she couldn't really understand what it was like to lose one in the womb.\n\nPeg also couldn't understand how important it was to Ginny that her loss did not define her. Not her life, and not this pregnancy. This child was still clinging to life inside her and had not yet shown any indication of giving up, and Ginny refused to surround herself and this baby with fear.\n\n\"I'm not overdoing it,\" she told her sister. \"Trust me.\"\n\n\"I just worry for you. That's all. I remember how devastated you were\u2014\"\n\n\"I'm fine.\" Ginny cut her sister off. \"Really. But I have to go. I've got to defrost some things for dinner and stuff.\"\n\n\"I guess you won't be serving any Swiss cheese,\" Peg said, and Ginny found some laughter at what just a short time ago had made her so angry.\n\n\"I should serve only Swiss cheese,\" she said. \"And mustard.\"\n\nThey said their goodbyes the way sisters do, without a lot of mush and gush, but a lot of love nevertheless.\n\nThen Ginny went back to the task she'd set herself.\nChapter Twenty-Two\n\nBy the time Sean got home that night, bringing flowers for no reason he'd admit to but which Ginny suspected had something to do with her missing mug, she'd carried all the books upstairs and put them away in the bookcases. She'd cleaned them all first, of course, and swept the floors. Dusted the mantel, that sort of thing. But what had transformed the room was the addition of her collection of hardcovers and paperbacks, many she'd owned since childhood, all arranged and displayed. Those books adorned the shelves like they'd been meant for them.\n\nShe took him by the hand after feeding him a slow-cooker beef bourguignon and homemade biscuits. When she led him upstairs, his confusion was obvious. And when she pushed him gently forward, inside, to show off her efforts, his face first fell. Then he scowled.\n\n\"What the hell is this? I thought maybe you were going to show me something you painted.\"\n\n\"It's my library,\" Ginny said calmly. \"I know you said you wanted it to be a studio, but I can paint in here just as well this way...\"\n\n\"That's not what I meant.\"\n\nShe crossed her arms, aware of how they rested now on the shelf of her belly in a way they hadn't even a few weeks ago. \"What did you mean, then?\"\n\nHe gestured. \"This. The books. Where are the boxes? How did you get them all up here?\"\n\n\"I carried them.\" The flavor of sarcasm was bittersweet. She didn't tell him that she'd done it a few at a time rather than lifting each heavy box. It had been easier to make many trips with lighter loads. It had taken her most of the rest of the day and left her sweaty, but satisfied.\n\n\"Christ, Ginny. I told you I'd do this.\"\n\nFrustration boiled out of her. About the books and the boxes, the unkept promises. The mug. The mustard. The cheese. And other things, months and years of things that had eaten away at her and been shoved down or pushed aside because it was always easier that way, because she owed him something greater than her anger.\n\n\"But you didn't, did you, Sean? You didn't do it! You promised and promised and promised, but every night you come home and you eat dinner, and then you disappear conveniently into the bathroom for your nightly dump while I take care of the cleanup, and then you have to read the mail, and watch some TV or do your homework, and by then it's time for bed, so you never get around to it.\"\n\n\"I'm tired when I get home! What do you think, I work all day and then can just come home and have all this energy left over to do your projects for you?\"\n\nShe seethed, her fists clenched. \"They're not my projects, Sean. They're part of living in this house, together, which I could easily do by myself, as you can see, because I did. And the only reason I didn't before was because you insisted that you'd do it. But you didn't. So I did. Why are we even fighting about this?\"\n\nBefore he could answer, she turned on her heel to leave the room, relentless in her desire to get away from him before she said something she regretted. She'd done it before, used her tongue to cut him, and he didn't forget. Sean might forgive, but he never, ever forgot.\n\nShe thought he wouldn't follow her. He didn't like confrontation, which was why they hardly ever fought, why whenever they did argue it was because of something she said, she did, her choice. She was the one who fought, never Sean.\n\nSo when he reached out to tap her shoulder, she whirled, startled. \"What?\"\n\n\"Don't walk away from me,\" he said.\n\nHer back stiffened. \"I'm tired, Sean. I want to take a hot shower and go to bed.\"\n\nFor another second or two, she thought he was really going to keep up with it, but he just shook his head.\n\nWhen she came out of the shower, though, he'd already turned down the sheets and sat on the edge of the bed, still fully dressed. He looked up at her when she came in, her skin still damp and flushed from the heat, her hair pulled on top of her head.\n\n\"I just don't want anything to happen to you, Ginny. That's all.\"\n\nShe sat beside him and thought about taking his hand, but the effort at that moment was too great. She couldn't tell what was raw between them, just that something was, and she didn't have the energy to deal with it.\n\n\"I can't sit around here doing nothing all day long.\"\n\n\"You could paint.\"\n\nShe sighed and rubbed at her eyes. \"It's not that easy, you know. Besides, painting was just a hobby.\"\n\nPainting had only ever been an excuse, something she'd taken up to fill in the long and lonely hours she'd wanted to spend with a husband consumed with work and school and things he wouldn't talk to her about. That she'd discovered she loved it was a bonus, something unexpected but lovely. But it hadn't started out that way. It hadn't been something she'd dreamed of doing as a kid. To see that she had some small talent for it had never pushed it beyond anything but a hobby. It had never quite become a passion.\n\n\"You could work on the baby's room.\"\n\nGinny said nothing.\n\n\"We could get a kitten,\" he said suddenly.\n\nGinny flinched. \"What? No? The last thing I want is another kitten, right before...I mean...you can't just replace Noodles! You can't just get another one because the one we had ran away.\"\n\nSilence, this time from him. Her words hung in the air between them, uncomfortable. Awkward. Bad memories threatened, of similar but more horrible conversations, and she pushed them away.\n\nSean looked at his hands, clasped lightly in his lap. \"You should take advantage of this time to rest and relax. Because you won't have this free time in a few months. If we're lucky.\"\n\nBile scratched at her throat and a burning pressed in her chest. Heartburn. Stress. The taste of long-simmering anger.\n\n\"If we're lucky,\" she repeated in a low voice. Then, louder, \"Lucky? Don't you think we'll be lucky? You think I'm going to lose this baby.\"\n\n\"Don't you think...\" he hushed himself, then turned to look at her, \"...don't you think I have a right to worry? Even the doctor said\u2014\"\n\n\"The doctor said there was no way to know if anything I did would've made a difference. She said that the body knows what's necessary, even if the mind and heart don't agree, do you remember that?\"\n\nHe huffed. \"Yes. She was a jerk.\"\n\n\"She was maybe a little brusque.\" Ginny had appreciated the obstetrician's assessment of her miscarriage, better than if she'd joined them in the hand wringing and breast beating. \"She also said that there's no reason to think this time is the same, or that we'll have any problems. We haven't, Sean. This baby is healthy. I'm healthy. I'm seven months along, and there are no signs of any problems like the last time.\"\n\nThere'd been genetic abnormalities. Ginny'd had every test possible this time through and been given the all clear for Down's, spina bifida, everything else. She and Sean had been poked and pinched and prodded, their DNA scanned, the probabilities of their conceiving a child with abnormalities factored, and the results had all come up the same, just like the doctor had said.\n\n\"Is that why you won't decorate the nursery, then?\" he challenged her suddenly. \"Because you're so convinced there's nothing wrong, that it will all be all right? Is that why you've left every single thing we got in the wrapping with the receipts attached...just in case?\"\n\nGinny got up, looked with longing at her pillow and the warm blankets, then at her husband. She lifted her chin. \"I'm going to read,\" she shot at him before he could say a word. \"With my feet up. I'm not tired now.\"\n\nWhich was a lie. She was exhausted. She was melting with it, the desire to sink into her bed and pull the blankets up, to lose herself in vivid dreams of old flames and movie actors. She wanted to sleep and end this fight, erase the knowledge that her husband did not believe in this child. That he didn't believe in her.\n\n\"Ginny. Wait.\"\n\nIn the doorway, Ginny paused. \"We have no reason to think there will be any problems, Sean.\"\n\n\"We have every reason,\" he said in a hard, low voice totally unlike his normal tone. \"Don't you get that? We have every reason.\"\n\nThe terrible thing was, no matter how hard she tried to convince herself otherwise, she knew he was right.\nChapter Twenty-Three\n\nShe was cold again.\n\nThe house had been warm when she went to sleep, and with her flannel pajamas, the pregnancy hormones and the knitted afghan, Ginny'd fallen asleep toasty warm. Now she woke, cold and disoriented. The blanket had bunched beneath her, pressing her flesh into an ache.\n\nShe was on the Victorian couch. She put a hand down to touch her book on the floor beside her. The lights were out, she realized as her fingertips brushed the smooth hardcover. She'd found an old favorite, Clive Barker's Imajica, in one of the boxes, and the dust cover had raised letters she could trace.\n\nThe lights were out.\n\nShe hadn't turned them out before falling asleep, she knew that much. She'd been reading by the light of the pendent lamp, which was too dim to make any dent in the shadows but cast a perfect pool of brightness for reading. She'd been on her side, the book propped on a pillow and the couch's firm, upholstered back providing a delightful pressure against her back. She remembered letting the book close, even recalled setting it gently on the floor. But she had not turned out the lights.\n\nSean must have come to check on her, seen her sleeping and turned off the lamp to leave her in peace. Ginny smiled at this thought of domestic kindness, until she remembered they'd been fighting. And at any rate, it wouldn't have been like him to leave her sleeping on the couch. He'd have insisted she come to bed to be more comfortable.\n\nBut if it wasn't Sean, who was it? The boogeyman, she thought with a small laugh. A cool gust of air swirled from beneath the couch and tickled her fingers, and suddenly the idea of the boogeyman didn't seem so laughable. She didn't quite snatch her hand up and out of the dark...but almost.\n\nGinny sat, pushing at the tangle of the afghan in frustration and not quite able to get herself free. She had no idea what time it was, but it felt like her normal midnight or 1:00 a.m. rising. Her bladder was telling her that anyway.\n\nSomething scratched and rustled inside the wall behind the bookshelves.\n\nGinny paused, thinking it was her own shuffling, but a second later the noise came again. The distinctive scritch-scratch of claws or nails against the inside of the drywall. She held her breath. Faintly, faintly, came the far-off jingle of a bell.\n\n\"Noodles?\" It came out as a whisper; she couldn't force her voice any louder than that. \"Noodles, sweetie...\"\n\nThrough the darkness came the sound of breathing, soft and fast and faint. Ginny froze, unable to move, incapable of closing her eyes though she was straining them so hard into the blackness that small, bright flowers had begun blooming around the edges of her vision. She'd put one foot on the floor a minute or so ago and now drew it up under the blanket she clutched to her neck. She had to pee so bad she could taste it, as her gran would've said.\n\nGinny swallowed, then again, her throat so dry she thought she might choke. What had shifted in the walls, what was now breathing so close to her? Or maybe not what. Maybe...who.\n\n\"A girl...with dark hair...like yours when you were small.\"\n\n\"Maeve,\" she whispered, because that was the name she'd wanted to give her daughter, the one she'd lost. \"Maeve, honey? Is it you?\"\n\nGinny tensed, waiting for something to touch her. Weeping, she reached into the darkness and found nothing but the night. She listened in the dark, but the bell didn't jingle again.\nChapter Twenty-Four\n\n\"A blank canvas is only intimidating if you don't let yourself imagine what you might paint on it.\"\n\nLouisa had been fond of saying that as she walked around the room, looking at what they were working on. Ginny knew she was trying to be helpful, but it was never that she'd found the blankness intimidating. If anything, she always had too many ideas about what to paint and never enough skill to re-create what was in her mind's eye. Not intimidating, frustrating.\n\nThe canvas in front of her was now frustratingly and intimidatingly blank. She'd dutifully kept it set it up on the easel near the window, where the light was best. She'd sketched out on a notepad the shapes of what she thought to paint, her pencil making a scritch-scratch noise that was uncomfortably like the noises she'd heard in the walls.\n\nShe looked around the room, waiting for inspiration to strike. And it did, but not to create anything. The fainting couch called to her. That, and a mug of tea and the rest of her book. That was a much better prospect.\n\nThere was nothing that said she had to paint. It should never have become something forced upon her, the way she felt forced now. But Ginny didn't feel like painting.\n\nThe truth was, Ginny was sure she'd never really feel like painting again.\n\nChildren's laughter caught her attention, and she looked out the window to search for them. As she watched, the boy pushed through the hedge and bent to pick up what looked like a badminton birdie. It must've flown over the bushes, though why those kids would be outside playing badminton at this time of year... Ginny shrugged. Not her kids, not her problem. Well, except for the fact they were in her yard again. The boy ran through the pile of leaves just down the hill, his sister following, and she lost sight of them.\n\nThe sky had gone gray again. Ginny shifted her weight from foot to foot and dabbed her paintbrush without enthusiasm into a blob of a color called Azure Sky, which was nothing like the color she could see through her window. Despite the chill in this room, it wasn't cold enough outside for snow, which meant the clouds covering the sun meant rain. Cold November rain, just like in that Guns N' Roses song. The faint rumble of thunder proved her right a minute or so later.\n\nAutumn thunder always felt strange, not like summer storms that sprang up after a hellish day and broke the heat. Storms in the fall were creepy, not sexy. The rain would be cold, not warm.\n\n* * * * *\n\nWarm rain. The sound of thunder. Ginny closed her eyes, remembering how the sky had opened, how everything had come down. Steady, unyielding rain, hard enough to sting her bare arms as she ducked between the tents, wishing she'd thought to bring a sweater or an extra outfit. She would be soaked by the end of the day, her carefully styled hair flattened, her makeup smeared, her pretty shoes bloated with wet.\n\nHer paintings hadn't looked the same in the glare of strung light bulbs instead of sunshine; nobody's had. They'd strung some on wires in the tent. Others leaned on unsteady easels, posts sinking into the rain-softened ground.\n\nThe crowds came, despite the rain. They splashed in the puddles and trekked through the mud. They carried shopping bags of crafts and paintings carefully wrapped in plastic to shield them from the ceaseless downpour. They waved giant tubs of French fries and apple dumplings and ignored the ice-cream truck and cotton-candy vendors\u2014cotton candy dissolves in the rain before it can be eaten. It leaves behind a sticky film as the only reminder of its sweetness. Ginny knew this because she'd watched a young mother pushing her toddler in a stroller, unaware of his tears as his treat washed away before he could get it to his little mouth.\n\nLots of people had come to the art show, and it had seemed to Ginny that at least most of them took a turn through the tent she shared with the rest of Louisa's students. Lots of strangers took the time to ask about her work, some even complimenting it. A couple even bought a piece. Lots of faces, lots of smiles, but no matter how often she looked up hopefully to the tent entrance, waiting, none of them had been his.\n\n* * * * *\n\nBlinking at the next rumble of thunder, farther away now, Ginny came back to herself. Her back ached, and so did her fingers from clutching at her paintbrush so hard. She'd been standing there so long the small dabs of paint she'd put on her palette had started to dry, a thin sheen of darker color covering the brightness beneath. She took a deep breath and put everything down.\n\nShe didn't want to paint today, or maybe never any other day, but she left her supplies where they were so that perhaps when she returned they might taunt her into feeling creative.\n\nGinny went downstairs to make some tea. As she filled the kettle at the sink, the first fat drops of rain splattered the glass. She leaned to crane her neck, trying to get a glimpse of the sky but couldn't quite make it.\n\nThe kids from next door ran across the yard, kicking at the leaves and shrieking with laughter. The wind picked up strands of the girl's blonde hair as she twirled and fell down into a pile, her arms and legs moving like she was making a snow angel. Her brother tossed the birdie at her, hitting her in the forehead. She sat up, indignant, and shouted something at him Ginny couldn't make out. Then she got up to chase him, and they ran out of sight again, down toward the creek.\n\nThe kitchen was broiling hot again, even as the library had been chilly. Her memories had left her flushed, but the temperature in here was unbearable, too hot for hot tea. She settled the kettle on the stove but didn't turn on the burner, and instead went to the fridge for a cold glass of orange juice.\n\nThe carton was empty, not even a swig left. Fuming, she tossed it in the trash and wrote \"orange juice\" on the list clinging beneath a magnet, advertising window treatments, on the fridge. She drank some water instead, then wet a paper towel and used it to dab at her face and the back of her neck, the swell of her cleavage.\n\nIt was too damned hot here. Too cold in other rooms. The damned furnace, Ginny thought, needed a kick. She didn't care what the guy said about it, there was something wrong.\n\nDetermined, she pulled the rechargeable flashlight from the wall dock. Sean would be happy to know she'd found a use for it, even if she didn't want to admit that it had been an eminently practical present.\n\nThe basement door creaked as she opened it. Behind it was the set of spindly, creaking and open wooden stairs with a splintery railing she gripped firmly. The floor below was concrete and unforgiving of a tumble. For a moment she considered taking off her slippers, but she wasn't wearing socks with them and the thought of going in bare feet into the filthy cellar was as unappealing as taking the time to go upstairs and put on socks. Instead, Ginny just made sure to settle each foot firmly on the stairs before she took another step.\n\nThe bulb swayed a little when she pulled the chain, and the pool of light at the bottom of the stairs shuddered. Just beyond it was another pull chain, though, and a few feet more, another. They were hung all over the basement, none connected by a light switch but all close enough that you never needed to pass through any darkness to get to the next...unless, as she discovered now, several of the bulbs had burned out.\n\nBut that's why she had the flashlight. Ginny clicked the button. The blue-white light flashed rapidly, like a strobe, before she clicked it again to put it on the regular setting. A steady beam of bright light shone ahead of her, looking almost solid with the dust she'd kicked up swirling in it.\n\nGinny held the flashlight in her palm and slowly waved the light from side to side. \"Whommmmm. Whommmm.\"\n\nLightsaber.\n\nThe effect was ruined when the light bent along any objects in the way, but still it was good for a giggle as she oriented herself. She swept the room with the flashlight. All the corners the light from the bulbs wouldn't reach, even if they were all lit.\n\nThe furnace was in the far corner, in a little jig-jog that upstairs was part of the dining room. Ginny flashed the light above her, into the ceiling joists. She remembered too late the sound of nails and claws, and screamed when the shivering shadows tossed a pair of bright eyes at her, a flash of teeth.\n\nSeconds later, of course, with her heart pounding and palms sweating, she had to laugh. It wasn't a raccoon\u2014or worse, a rat\u2014but a child's stuffed toy shoved into the space between the rafters and close to the silver ductwork.\n\nThat was creepy and gross, but not terrifying and not as weird as some people might think. She could remember as a kid hanging out in her grandma's basement with Peg and Billy and their cousins, trying to scare each other with stories or playing hide and seek. As a teenager, Ginny's uncle Rick had built a rec room of sorts in the basement, using spiffy 1970s paneling and cast-off furniture he and his friends had salvaged from the garbage. There'd been plenty of weird things tucked into the rafters of Gran's basement, including the gape-mouthed plastic face of a decapitated blow-up doll Peg and their older cousins had convinced her was a \"princess mask.\" Compared to that, a kid's teddy bear was hardly strange at all.\n\nTo her right was the concrete wall that had been repaired in the long-ago fire. It was just as dirty and hung with cobwebs as the other walls, but of slightly different brick. Sean had stacked their ski equipment along it but left bare the metal shelving unit she'd meant him to use. Ginny shook her head, biting her tongue and refusing to let herself get worked up over it.\n\nThe furnace kicked on with a rumble when she approached. That had to be a good sign, right? She didn't know much about furnaces, other than how to change the filter, but the repairman had mentioned that Reset button and whatever he'd done to it had worked, at least for a little while, so it seemed like it was worth a try again.\n\nOver in this corner, the bulb had not just blown, it was missing entirely. Ginny touched the chain and set the fixture swaying anyway as she passed, but she used the flashlight to look over the furnace. She regretted, now, playing the role of little wifey while Sean went with the repairman to check things out. She'd always made it a point in the past to be aware of everything, the basics of what she considered necessary adult skills. How to change a tire, balance a checkbook, change a fuse, mix a basic cocktail. In the days before she'd gone to a Mac, how to defrag a computer. Yet when the repairman was here, she'd hidden herself away in the kitchen like some sad parody of June Cleaver, complete with apron and bare feet.\n\nWhat the hell had happened to her?\n\nA sudden sob threatened to strangle her, but Ginny forced it back. No crying. Not here in the dim and dirty basement. Christ no. She wasn't going to lose it. She bit her tongue and rubbed the sore spot against her teeth until the urge to cry went away enough to ignore.\n\nThe Reset button. Where would it be? She shone the light over the entire furnace, but there was no helpful marking. The furnace itself still rumbled comfortingly but also deceptively.\n\n\"Let me heat your house,\" the furnace's rumble said. \"Or, you know, make you think I'm going to. But then you wake up freezing your tits off while chocolate melts in your cupboards.\"\n\nJust to the left of the furnace was a small window set high in the wall. It opened into a well framed with a half circle of metal and a patch of gravel at the bottom. She could see nothing through it, but some pale light filtered through, enough that when she went around the side of the furnace she could click off the flashlight.\n\nAnd there it was. At least, she assumed the switch on the side of the furnace, tucked between two sections and just above the place the filter nestled, was the right one. What else could it be?\n\nWhat's the worst that could happen, Ginny thought, and flicked it off.\n\nAt that moment, the light from the window cut out completely, leaving her in shadow. Ginny turned, the flashlight tumbling from her fingers as she spun. Framed in the window was a face, eyes wide and mouth yawing.\n\nIt screamed, shrill and high and piercing. The sound ripped at her eardrums and set the hair on the back of her neck on end. She dropped the flashlight, screaming herself, louder and more frantically than when the stuffed bear had startled her.\n\nThe face disappeared. More screams echoed. She heard the faint rustle of feet in the leaves.\n\nGinny collapsed against the furnace, no longer rumbling, and let herself dissolve into relieved laughter. She pressed a hand to her heart to slow the beating. The other went between her legs, praying she hadn't lost control of her bladder. She seemed safe enough there, though there'd been a moment when she was sure she was going to piss herself. At least she'd managed to avoid that.\n\nIt was one of the kids from next door, the face in the window. For whatever reason, they'd been peeking in her windows. Well, they'd had a scare, hadn't they? Maybe it would keep them away from her house, she thought, even as she laughed again at how they'd terrified her too.\n\nShe bent slowly, carefully, to find her flashlight, but it had rolled away somewhere. It was gone. She'd have to look for it later, in the brighter daylight or at least when she'd replaced all the burnt-out bulbs, but for right now her bladder was protesting the strain she'd put on it. She'd be lucky to make it upstairs.\n\nSomehow, with her knees knocked together, Ginny made it to the bathroom in time to avoid an accident. Washing her hands, she caught sight of her reflection and laughed again at the memory of the neighbor boy's terrified face. Of her own fear. She laughed, loud and long.\n\nAnd then she was weeping, both hands gripping the porcelain while she bent forward, helpless against the onslaught. Her shoulders heaved. Sobs racked her. She opened her mouth and almost expected to puke, that was how fierce the tears burned, but, instead, snot and saliva dripped into the sink. Fat, hot tears splashed. Her fingers curled and gripped, tight and tighter, because if she let go of the sink she was surely going to fall onto the floor.\n\nIt didn't pass with ease, this sudden burst of grief, this madness. It didn't fade or taper off into sniffles. It ended abruptly, like someone had slapped the hysteria out of her, and it didn't leave her feeling better, the way tears were supposed to. Everything about her face felt hot and swollen, and when she dared to face her reflection again, she looked how she felt.\n\nUgly.\n\nWith a determined shake of her head, Ginny gathered up the sorrow and the crazy, and she folded it like origami. She pushed it away, pushed it aside. Pushed it inside.\n\nDeep inside.\nChapter Twenty-Five\n\n\"What the hell were you doing in the basement anyway?\"\n\nAs predicted, Sean hadn't been amused by the story she spun as humorous so she could forget how it had ended. Ginny sighed and pushed her fork through the spaghetti noodles and sauce. It was a lackluster dinner, at best. Overcooked pasta and sauce from a jar, garlic bread she'd cobbled together from some leftover hamburger buns and garlic powder. She wasn't hungry anyway.\n\nShe looked at him across the dining room table. He seemed so far away, compared to their seats at the old table in the kitchen, but she figured if he was going to surprise her with this ugly table, she could insist they use it. \"I told you. I was looking at the furnace.\"\n\n\"If the furnace isn't working,\" Sean said, \"tell me about it, and I'll call the repair guy to come back. We paid him enough, he should make good.\"\n\nGinny frowned. \"I can call the repair guy. I'm not helpless.\"\n\nSean said nothing, just dug his fork into the pile of noodles on his plate and slurped them up. Sauce splattered. He washed down the mouthful with a swig from his glass, then forked a bite of salad. He chewed. Loudly.\n\nHe'd always eaten that way, openmouthed, slurping and smacking and crunching. It suddenly repulsed her. Stomach twisting, Ginny broke off a piece of bread and forced herself to nibble it. She hadn't eaten all day.\n\n\"And you should've told me yesterday.\" He pointed at her with his fork.\n\nHe had sauce around his mouth. Once upon a time, she'd have leaned to wipe it with the corner of her finger and tucked it in her own mouth. The thought of that, tasting something from his skin, suddenly repulsed her more. Ginny swallowed convulsively.\n\n\"You weren't home until late last night. I didn't think about it.\"\n\nHe'd come home after she was already in bed. He'd smelled again of cigarettes and liquor. He'd wanted to make love. They'd done it in the dark, without words, his hands roaming over the changing mountains and valleys of her body. It had been like making love to a stranger, which was why she'd been able to come and why she'd gone to sleep with silent tears soaking into her pillow after.\n\n\"Did you turn the furnace back on?\"\n\n\"I...\" Ginny hesitated, \"...I'm not sure. I don't know.\"\n\n\"No wonder it's so frigging cold in here.\" Sean frowned. \"I'll take care of it after dinner.\"\n\n\"Fine. Thank you.\" The words sounded stiff and ungrateful, but what did she have to be grateful for, exactly? Making her feel like an idiot? Or that he might actually complete this chore, instead of merely promising to do it?\n\n\"So,\" he said after a few minutes of nothing but the sound of silverware clinking on the plates, \"how was your day?\"\n\n\"Fine.\"\n\nHe leaned back in his chair and wiped his mouth with a paper napkin; bits of paper clung to the scruff of his goatee. He didn't notice. Ginny had to look away.\n\n\"Just fine?\"\n\n\"Just fine,\" she told him. \"How else would you expect it to be?\"\n\n\"Did you paint today?\"\n\nShe frowned. \"No.\"\n\n\"Why not?\"\n\nSaid that way, it could be a genuinely curious inquiry. Or it could be a dig. Either way, she had no answer for it, no good answer that wouldn't lead to more questions.\n\n\"I didn't feel like it.\"\n\nSean wiped at his mouth again, this time with his fingers. It was the gesture of a man trying to hold back words with his hand, but he didn't try hard enough. \"Why not?\"\n\n\"Christ, Sean. I don't know. I just didn't.\" Irritated, Ginny stood to take her plate into the kitchen.\n\nShe'd baked cookies earlier today when she was not painting. She'd dusted the living room and changed out the towels in the bathroom She'd run a couple loads of laundry. She'd read half a book. She'd napped. She'd watched a couple hours of television. She'd paid some bills. She'd done a lot of things that were not-painting, so why was she letting him make her feel like she'd squandered her day, when he was the one who kept telling her to relax?\n\n\"I just thought, you know. With all your time you'd be getting back into it by now.\"\n\nShe turned from the dishwasher. \"All my time?\"\n\n\"Yeah. All your time. I don't have all the time you have.\"\n\nBoth her brows went up. \"Everyone has the same amount of time, actually.\"\n\n\"Not actually.\" His half smile was supposed to charm her, but it didn't look genuine. \"Some of us die before the rest. So technically\u2014\"\n\n\"I was busy,\" she cut in. \"And I didn't want to paint. Okay? I just didn't feel like it. You act like it's some kind of crime.\"\n\n\"It's just that I wouldn't have bought all that stuff for you, if I'd known you weren't going to use it. I'd have spent the money on something else. In case you didn't notice,\" he added in a tone more sarcastic than she'd ever heard from him, \"we're not exactly floating in extra money.\"\n\n\"I never asked you to buy it!\"\n\nSean didn't say anything at first. They stared at each other across the kitchen until he dropped his gaze. Scuffed his toe on the floor she'd spent a good forty minutes sweeping today. Muttered something.\n\nHer senses of touch, smell, taste had heightened in this pregnancy, but her hearing apparently hadn't. \"What?\"\n\nHe looked up at her, somehow defiant. \"I said, 'you used to like it'. You used to love it. But when I try to help you with it, all of a sudden, you don't want to have anything to do with it.\"\n\nHer jaw dropped. She turned to the sink to wash her hands so he wouldn't see them shaking. \"Wow. What a selfish, self-absorbed thing to say.\"\n\n\"What's that supposed to mean?\"\n\nShe ran the water too hot, but refused to flinch as she rinsed away the soap. Her skin went red. She didn't turn to look at him. \"Why would you think my painting has any goddamned thing to do with you at all?\"\n\n\"That's the trouble,\" Sean said. \"It doesn't have any goddamned thing to do with me, and it never did.\"\n\nTime ticked past, one second at a time, the way it always did. An eternity passed in the span of one, two, three breaths. Ginny swallowed her words and refused to let his become bruises.\n\n\"I'm going to take these cookies over to the neighbors,\" she said finally when it became apparent that Sean was neither going to speak nor leave the kitchen until she did. \"It's about time we met them anyway, and I want to make sure the kids are okay.\"\n\n\"Tell them to stay the hell out of our yard.\"\n\nThe vehemence was so unlike him, it almost made her turn. Ginny kept herself still. She didn't want to look at him. Didn't want to see his face. She was afraid of what he'd see on hers.\n\n\"I'm going to take care of the furnace.\"\n\nShe waited for the sound of his footsteps moving away before she turned. Then she pressed her fists to her eyes until colored sparks danced. She drew in a few deep breaths. She didn't cry.\n\nGinny took the cookies out the back door, across the grass but not through the hole in the hedge. She walked all the way around to the sidewalk, then up the neighbors' driveway. She knocked on the front door, her knuckles stinging in the cold air.\n\nThe woman who opened the door looked surprised and a little suspicious, even when she saw the platter in Ginny's hand. She looked younger than Ginny by about five or six years, her hair in a messy topknot and faint lines of exhaustion around her eyes. She had a baby gnawing its fist on her hip, a spit rag tossed over her shoulder. \"Can I help you?\"\n\n\"Hi. I'm Ginny Bohn. From next door.\" Ginny twisted to point.\n\nThe woman looked at her belly, obviously just noticing the bump. \"Oh God. Yes, of course, come in, please! I'm Kendra. This is Carter.\"\n\nShe waved the baby's fist in Ginny's direction. \"Oh, cookies. Yum, thanks.\"\n\nGinny followed her into the house and down a hall to the kitchen. This house was newer than theirs, laid out more like a raised ranch than an expanded bungalow. It was bright and airy, decorated with furniture from IKEA and plenty of childish artwork. Also, toys all over the place she discovered when her toe nudged a couple of matchbox cars.\n\n\"Oh God. Sorry. Sorry,\" Kendra said as she settled Carter into a heavy plastic high chair and set a rubber hammer in front of him. \"No matter how many times I tell him... CARSON!\"\n\nShe turned to Ginny, who was still holding the cookies. \"Oh, let me take them. Can I get you something to drink? Coffee? Oh no. You're probably off that for now. I had to quit it when I was pregnant, even decaf gave me heartburn, I couldn't stand it. I could make some hot tea?\"\n\n\"That would be great, thanks.\" Ginny slid into a chair at the table across from the baby.\n\nPeg had been frazzled like this when her kids were smaller, and Ginny had no trouble imagining herself equally so. She put the cookies on the table and smiled at baby Carter, who gave her a solemn, somehow accusatory stare in return.\n\n\"You have three?\" Ginny asked.\n\nKendra put a kettle on the stove, then turned. \"Oh yeah. Three monsters. CARSOONNNNNN!\"\n\nThere came the pounding of feet on the stairs. A minute later, Carson skidded into the kitchen on sock feet. Red cheeks, tousled blond hair\u2014that was the face she'd seen in the window, all right. Close on his heels was his sister, hair in pigtails but otherwise a smaller version of her brother.\n\n\"You can't tell they're related at all,\" Ginny said lightly.\n\nKendra laughed. \"They take after their dad. I swear we had a third just so I'd get one that looked like he belonged to me.\"\n\nThe baby in the high chair didn't seem excited by this. His face scrunched up in silent despair. His mother sighed and rubbed the top of his head, waiting a full five seconds before the first wail hit. She looked at Ginny with a faint smile of apology.\n\n\"He's...cranky.\"\n\n\"He's a cutie. And these two,\" Ginny said, still keeping it light, well aware of how protective her sister could be about her children, \"I think I already know. Carter and...?\"\n\n\"Kelly,\" their mother offered with a curious look that became suspicious again, this time leveled at the children. \"What did you guys do?\"\n\nGinny shook her head. \"Nothing bad. I don't want you to think that. They've been playing in the yard, but\u2014\"\n\n\"Carson James Wood! Kelly Madison Wood! What did I tell you about that?\"\n\nIn his high-chair prison, Carter let out a longer, louder wail and pounded his fists until Kendra scooped him out with a sigh. Plunked on her hip, the baby stopped crying out loud, though the silent tears of outrage still coursed down his fat little cheeks. His mother kept her attention on the other two.\n\n\"But, Mama...\"\n\n\"Don't you 'but Mama' me. I told you both to stay in our yard... Oh God.\" She turned to Ginny. \"Were they down by the creek? They were down by the creek, weren't they?\"\n\n\"I don't...know,\" Ginny hedged, annoyed as she'd been by their repeated trespassing, but not willing to send the kids down the river, so to speak. Yet she also felt something like a mother's kinship with Kendra, like there was some sort of code she'd be breaking if she didn't take the mother's side. \"And really, it's okay. I know the leaves can be tempting. And honestly, if my husband actually put them in bags instead of just raking them into piles, they probably wouldn't even be tempted to come over and jump in them. Right, guys?\"\n\nKelly's eyes had gone wide, her lower lip atremble. She shook her head. \"We were looking for the little girl. We were leaving her snacks because she's\u2014\"\n\nCarson elbowed her. \"Shhh!\"\n\n\"Oh, for crying out loud.\" Kendra clapped a hand to her forehead. Behind her, the kettle started to whistle. \"Both of you. To your rooms. I don't want to hear it,\" she said before either of them could protest. \"Go. Now!\"\n\nShe turned off the burner with one hand, twisting her body expertly to keep the baby away from the heat. She took a couple of mugs from the cupboard and put them in front of Ginny, then a small box of loose tea bags. Finally, the kettle, which she also kept far away from the baby. She put the kettle on a trivet and took the seat across from Ginny.\n\n\"I'd pour,\" she said, \"but this guy here has grabby hands.\"\n\n\"I can get it,\" Ginny assured her. \"What would you like?\"\n\nKendra shook her head. \"Nothing for me, thanks.\"\n\nAn awkward silence fell as Ginny prepared the tea, but only for a couple seconds because Kendra snagged a cookie and bit into it. \"Oh wow. Fantastic! No, baby boy, not for you.\" She grinned at Ginny. \"He's nursing, doesn't even have a tooth yet, but he keeps trying to get at our food. Monkey see, monkey do. I guess that's how it is for the youngest.\"\n\n\"I guess so.\" Ginny dunked a tea bag and let it steep, then took a cookie. The dinner she hadn't eaten seemed delicious now.\n\n\"How many do you have? I mean, is this your first?\"\n\n\"Ah...\" Ginny paused, caught off guard though she knew she shouldn't be. It was a complicated question with a simple answer. \"Yes. Our first. But my sister has six, so I've got my auntie card.\"\n\n\"Not the same as having your own, let me tell you that.\" Kendra blew out a breath and looked tired again. \"When you can't give 'em back...\"\n\nAnother few seconds of strained silence. Ginny sipped her tea. Kendra bounced Carter on her knee.\n\n\"I am sorry about Kelly and Carson,\" she said finally. \"I've told them to stay out of your yard. I don't want them down by the creek. I know it's not deep or anything, but kids can drown in just a few inches of water, you know? And besides, they shouldn't be in your yard. I'm sorry.\"\n\n\"It's okay, really.\" Hearing Kendra apologize over and over made it harder for Ginny to be irritated by the kids coming through the hedge. \"Really, they mostly just jump in the leaves.\"\n\nShe thought about mentioning the stones they'd tossed at the house, the noise of their shrieking, the offerings they left in the window well, but in the end decided that would just get them into trouble and make her sound like exactly the sort of cranky neighbor she didn't want to become known as. Besides, she reasoned, someday not so far in the future she might have the kid who ran into the neighbors' yards and did naughty things.\n\n\"Still. I'm sorry. It's just that they're sort of obsessed with it.\"\n\nGinny's brow furrowed. \"The house?\"\n\n\"Yeah. Your house. My husband, Clark...\" Kendra sighed and shook her head. \"Men. They can be such idiots, right?\"\n\n\"Umm...sure. Right.\"\n\nKendra laughed a little. \"Yeah. Anyway, Clark was trying to get them into bed one night when I was out with some of my girlfriends for ladies' movie night... Oh hey, you should totally come! Stacy from down the street, that blue Colonial on the corner? They fixed up their basement pretty nice, big-screen TV and stuff, and her husband likes to have the guys over for poker and beer and sports or whatever, so she told him if he was going to do that, she was having chick-flick-and-wine nights, and it's been great. Maybe your husband would be interested?\"\n\n\"In...chick flicks and wine?\" It took her a second, but she laughed once she caught up to Kendra's train of thought. \"Oh. Sports and poker. He might be. I don't know. He's pretty busy.\"\n\n\"Oh. Travels a lot? I thought I saw that he was gone a lot. Not,\" Kendra said, \"that I'm a Sneaky Pete or anything, just that, you know, you guys being the new neighbors and all, in that house, we just all wondered what sort of people you were.\"\n\nThis pricked at Ginny's ears. \"That house? What do you mean?\"\n\n\"Oh...just that most of these places in the neighborhood have sold two, three, four times over in the past few years. I mean, we've been in here...five? Yeah, five years. And we're old-timers. But Mr. Miller from next door, he built that house before he got married. It was one of the first in the development. And he was the only owner. Until now, you guys. That's all I mean.\" Kendra looked vaguely guilty. \"I mean, it's not like someone died in the house or anything.\"\n\nSomething in the way Kendra said it sent up a flare. Ginny leaned forward. \"They told us he died in the hospital.\"\n\n\"Oh, he did, he did, totally,\" Kendra assured her. From upstairs came the sneaky, sly sliding of feet. \"You'd better be in your rooms!\"\n\nThe noises stopped. Carter giggled suddenly, as though pleased by his siblings' punishment. Kendra chucked him under the chin.\n\nGinny's head spun a little from trying to keep track of Kendra's conversation, but then she was used to Sean's mom, who was the queen of non sequiturs. \"So...your husband?\"\n\n\"Is an idiot? What?\" Kendra laughed again, looking confused.\n\n\"He was trying to get them to go to bed...?\"\n\n\"Oh, right, right. Yeah, so he was trying to get them into bed, and he told them that if they didn't go to sleep, he was going to tell the ghost next door about them, and she'd come over and get them.\" Kendra's mouth twisted, her smile rueful. \"Like I said, he's an idiot. Those kids didn't sleep for a week.\"\n\nNoises in the walls.\n\nCold spots.\n\n\"A girl...with dark hair...like yours when you were small.\"\n\n\"The...ghost?\"\n\nKendra laughed again. \"Oh, he just made that up. It's not true at all. I mean...\" she looked suddenly doubtful. \"It's just that Mr. Miller's daughter, Caroline. Well. You know about her, right?\"\n\n\"I don't.\" Ginny sipped more tea and took another cookie, suddenly ravenous.\n\n\"Oh. Well, she went missing when she was about fourteen or so. They never found her, so far as I know.\" Kendra's eyes gleamed for a moment. Her expression suggested she was both disturbed and solemn with the responsibility of being the one to share this story, but her gaze told a different, gleeful truth. \"Mr. Miller had an older son, Brendan. He'd be the one who sold you the house, right?\"\n\n\"Right. Yes.\"\n\n\"You know he built it himself? Old Mr. Miller?\"\n\n\"I didn't know that.\"\n\nKendra nodded. \"Yep. Custom-built. That's why it has so many nice features. I mean, all those built-ins, right?\"\n\nThe house did have nice features, but Ginny didn't care about them now. \"So...what happened?\"\n\n\"Oh right. Right. Well, old Mr. Miller's son and his wife, not the son's wife. Mr. Miller's wife. They moved out after about a year of not being able to find her. The story is that both of them refused to ever come back or set foot in the house again.\" Kendra lowered her voice, conspiring, though Ginny had no desire to share secrets. \"Mr. Miller lived there all alone for the past fifteen years. Nice guy. Always seemed super sad, though. And no wonder, right?\"\n\n\"No wonder,\" Ginny murmured. The tea had a flat taste she didn't like, but she sipped it anyway to give herself something to do. \"So...if they never found the girl...?\"\n\n\"I heard, just a rumor, you know, but I heard that someone found her on one of those Internet sites, living out in California or something. Runaway.\"\n\n\"So...she definitely didn't die.\"\n\nKendra looked guilty again. \"No. I mean, I don't think so. And surely not in the house. It was before we moved in, of course, but stories like that...\"\n\nGot exaggerated, Ginny thought. Inflamed. But that was exactly the reason why she should find it easy to believe the girl hadn't died in the house. \"Something like that would surely have gotten around.\"\n\n\"Well...yeah. I mean, like I said, Clark was being an idiot. He was just trying to scare the kids, which was stupid. I'm sorry they were in your yard, though. I'll have a talk with them.\"\n\n\"They were peeking in the basement windows. They scared me. I scared them, pretty bad, I think.\"\n\n\"Ohhhhh Gawd. Well, that explains why Carson came tearing in here the other day like wolves were chasing him. I'm sorry!\"\n\nGinny found a smile. \"Really, it's okay. I think they learned a lesson, didn't they?\"\n\n\"Still, I'm going to make them come down here and apologize.\" Before Ginny could stop her, Kendra had gone to the bottom of the stairs to shout up, \"Carson! Kelly! Get your butts down here, now!\"\n\nThe pounding of feet came again. The children looked caught and guilty, scuffing their feet on the linoleum. Kendra took them both and shoved them forward in front of Ginny. Carter grabbed a handful of Carson's hair and pulled hard enough to twist the older boy's head. There was a kerfuffle.\n\nGinny was already exhausted.\n\n\"You tell Mrs. Bohn you're sorry for going in her yard and for peeking in her windows.\"\n\n\"We just wanted to see the little girl,\" Kelly said through a lisp Ginny hadn't noticed before. \"She lives in the basement.\"\n\n\"Kelly, enough stories! Apologize right now.\"\n\n\"I'm sorry.\" Kelly's lower lip trembled.\n\n\"Me too,\" Carson added. \"We just wanted to see her, because sometimes she looks so sad.\"\n\n\"That's enough,\" Kendra snapped. \"Enough stories.\"\n\nGinny held up a hand. \"Wait a second. What do you mean, Carson?\"\n\n\"We always ask her to come out and play, but she\u2014\"\n\n\"Enough! Up to your rooms!\" Kendra yanked his sleeve to turn him toward the stairs, even as her son tried to finish his sentence. She cut him off with a swat to the rear, earning her a sullen glare, but both children obeyed. Kendra turned with a heavy sigh, Carter still bouncing on her hip. \"Honest to God, I'm going to kill my husband.\"\n\n\"It's fine. Really.\" Ginny stood. \"I have to get going anyway.\"\n\n\"I'll let you know about girls' night, okay? And I'll have Clark give your husband... What's his name?\"\n\n\"Sean.\"\n\nKendra nodded as though she'd already known. \"Right, right. I'll have Clark give Sean a call about boys' night. Okay? Sound good?\"\n\n\"Sounds...great,\" Ginny said, though her head was already awhirl from just forty minutes in this woman's presence. She couldn't imagine how it would be to spend any longer amount of time. She was nice and all, but...scattered.\n\nKendra led Ginny to the front door. \"Thanks for the cookies. I'll make sure to keep the kids out of the yard.\"\n\nGinny paused in the doorway to look over her shoulder. \"Oh, you know what? Don't worry about that. I like seeing them there. They can come over anytime they want.\"\n\n\"Are you sure?\" Kendra looked doubtful.\n\nGinny thought of Carson's earnest face, his sister's lispy lilt. Both of them had seemed pretty convinced there was a little girl in her basement. \"Yes. Absolutely.\"\nChapter Twenty-Six\n\nCMM\n\nGinny traced the letters on the diary. At least now she knew who the diary belonged to. It stayed locked, though. Caroline Miller might be dead, or she might be alive and living on a commune in California with a husband and a bunch of kids. But this was her private diary, and Ginny still didn't feel right breaking into it.\n\n\"You should just read it,\" Peg said. \"Maybe there are some clues!\"\n\n\"Like what kind of clues?\" Ginny put the diary back in the train case and closed the lid.\n\nPeg rolled her eyes. \"About what happened to her, duh.\"\n\n\"She obviously hid the diary away so nobody could find it because she didn't want anyone reading it, Peg. I feel like maybe I have to honor that.\"\n\n\"So her ghost doesn't get you?\" Peg lifted a brow and sliced into the chocolate cake Ginny had put in front of her. \"God, so good. When did you get in to baking?\"\n\nWhen I stopped being in to anything else, Ginny thought but didn't say. She pressed her fork into the crumbs on her plate. \"I told you there've been some strange things happening.\"\n\n\"You think it's a ghost, really? Because of what Gran said? C'mon, Ginny. That's just...stupid.\" Peg gave herself a quick sign of the cross.\n\n\"Gee, thanks,\" Ginny said dryly. \"Glad to know my older sister's on my side.\"\n\nPeg sighed and reached to pat Ginny's hand. \"Everything you told me about could be explained. Noises in the walls? The exterminator came, right? Cold spots? The furnace is fixed.\"\n\n\"Not really.\"\n\n\"So, it's still broken. Besides, it's more like your house is cold with hot spots, not cold spots, right?\" Peg pulled the front of her blouse away from her throat and fanned it back and forth. \"I mean, your kitchen's like a sauna.\"\n\n\"Yes. I know.\"\n\nShe hadn't told Peg about the other things. The mug that went missing. The figurines that turned up in random places, like the ones lined up in the window well. The distant sound of a jingle-bell collar from a cat that had been missing for weeks.\n\n\"Listen...\" Peg squeezed her hand again and waited until Ginny looked at her before continuing, \"...how are...things? With you and Sean.\"\n\nGinny gently extricated herself and got up from the table, ostensibly to carry her plate to the dishwasher, but really so she didn't have to look at her sister's face. \"Fine. They're fine.\"\n\n\"You know, when I was pregnant with Jennifer, I thought I might actually end up in prison for killing Dale. It was that bad.\"\n\nGinny turned. \"You never told me that.\"\n\n\"What could I tell you? That I was crazy? That I dreamed, actually dreamed...\" Peg paused, her voice cracking just a little. A glimmer of tears was swept away before they could fall. Peg cleared her throat. \"Jesus help me, Ginny. I dreamed about taking a knife to his throat and just slitting it open like he was a pig.\"\n\nStunned, Ginny leaned back against the counter, gripping it with her hands on either side. \"Jesus.\"\n\n\"I know, I know. It was terrible. I'd have these explicit, vivid dreams like I've never had before. And sometimes, in some of them...\"\n\nGinny watched her sister blush. Peg, an emergency room nurse, dealt with everything imaginable. Bad language, bodily fluids, objects inserted and stuck in orifices that were meant to be exit only. She had a bawdy sense of humor, so long as it wasn't sacrilegious. She'd been the one to tell Ginny about the intricacies of blow jobs and warn her about possibly crapping herself on the delivery table, so what could possibly be so bad, so strange, that it could fluster her?\n\n\"Sometimes, I'd dream that I'd...you know,\" Peg said in a low voice.\n\nGinny had no clue. \"You'd kill him?\"\n\nPeg shook her head, then nodded. Shook it again. \"Yes. I mean, no, yes, I dreamed that, but there were other dreams, not the same ones, usually. But sometimes. Yes, okay? I admit it. Once or twice, I dreamed that sort of dream, that violent dream, a big knife, lots of blood squirting. Once I woke up laughing.\"\n\nGinny thought of her own dreams, those slick, slippery, delicious visions of lust that had been visiting her almost nightly. \"Don't worry, I think that's normal. I mean, not the part about killing your husband, but if you want to analyze it, that big knife probably wasn't a knife. If you know what I mean.\"\n\n\"Sometimes,\" Peg said darkly, \"a cigar really is just a cigar, Ginny.\"\n\n\"Are you saying you really wanted to kill Dale? I don't believe it.\" Ginny pulled a glass from the cupboard and filled it with orange juice from the fridge.\n\nPeg waved away the glass Ginny offered. \"God help me, Ginny. I really did want to...well, maybe not kill him. But I sure did want him out of my life, away from me. I couldn't stand the sight of him. Everything about him drove me insane. The way he breathed at night, I'll tell you the truth, I had visions of putting a pillow over his face.\"\n\nGinny stifled another laugh, though it sounded slightly horrified. \"Oh, Peg.\"\n\nPeg shrugged. \"I had Jenny, and after that everything was fine. And it didn't happen any of the other times, just with that first. So I thought, you know, maybe...\"\n\n\"It's not my first time,\" Ginny said gently.\n\nPeg flushed again. \"I know. I'm sorry, that was stupid.\"\n\n\"No. It's okay.\" Ginny was tired of having to feel sorry for other people's grief surrounding her pain, but this was her sister, and she was trying to help. \"You want to know if I'm having crazy dreams too?\"\n\n\"Are you?\"\n\n\"I've been having my share of sexy ones,\" she admitted. \"Honestly, I don't mind. A nice wet dream once in a while? What's wrong with that?\"\n\n\"And the...other?\"\n\n\"No,\" Ginny said firmly. \"I don't dream of slicing my husband open with a knife, Peg. I think that was just your particular brand of crazy.\"\n\nPeg didn't seem offended. \"I just wanted you to know that if you were having some problems, I'd be willing to listen. Pregnancy can put a strain on any marriage, and so can a big move. And especially, hon, when you have the considerations you do\u2014\"\n\nGinny threw up a hand to stop her. \"Honestly, Peg, I'm completely burned out on this type of discussion. Yes, I had some problems before. But that was then, this is now. And I'm damned tired of everyone waiting with bated breath for me to lose this baby. That's it, that's all.\"\n\n\"I'm sorry,\" her sister said quietly. \"That's not what I'm doing. But can't you understand why Sean might be worried?\"\n\nGinny looked at her with narrowed eyes. \"What's he said?\"\n\n\"Nothing to me. He talked to Dale. Said he thought you weren't really facing things, that you wanted to forget what happened, and he was worried it was affecting how you were dealing with this pregnancy.\"\n\n\"I haven't forgotten!\" Ginny cried. \"As if I could ever forget!\"\n\n\"He says you won't decorate the nursery.\"\n\nGinny shook her head, refusing to get herself worked up over this. \"That's not forgetting. That's being practical. The baby will come home from the hospital and stay in our room for the first few months anyway. And we don't know if it's a boy or a girl, so why should we spend a lot of time and effort on something that might turn out to be...wrong? And babies have personalities,\" she continued. \"What if we go with a jungle theme and our daughter turns out to be a princess sort of girl? What if we have a boy who'd really like trains? Or, hell, the other way around, I don't care. I just want to focus on the baby.\"\n\nShe stared at Peg, who stared back.\n\n\"I just want to think about the baby,\" Ginny said quietly. \"Not the color of the fucking walls. Okay?\"\n\n\"Okay,\" came Peg's equally quiet answer. \"Okay.\"\nChapter Twenty-Seven\n\nGinny wears gartered stockings.\n\nHigh heels, black patent leather, red soles. Shoes she'd never buy in real life, but here they fit her perfectly. Here she walks like a queen in expensive stilettos, and she never thinks of falling.\n\nThe stockings swish, swish as she walks. The bare skin of her thighs rubs together. She wears silk panties. Red, she thinks. Yes, red. A matching bra of red and black, pushing her breasts forward, giving her cleavage that turns men's heads.\n\nOh, she wants to be that woman. The one who makes men stare. She wants to walk into a room and have them stop whatever they are doing. She wants their eyes to bulge out, their tongues to loll, to make them cartoon wolves with an AROOOOGA noise. She wants to be desired.\n\nThere's power in this. These clothes, this walk. The sway of hips, the jiggle of her breasts. She is all woman. She is not just desirable, she is desire.\n\nWhat will he think when he sees her? When her shirt unbuttons, one at a time, to reveal the slope and swell of her breasts? When she unzips her straight black skirt and lets it fall on the floor? When she steps out of her clothes and stands in front of him dressed like a wet dream, will he get hard for her right away, or will she have to kiss him first?\n\nOh, how she wants to kiss him.\n\nIt's all she can think about. That first kiss. How his mouth will taste, how his tongue will feel stroking hers. There have been nights when she can't stop from touching herself to the idea of that mouth, those lips and teeth on her flesh. He will kiss her mouth. He will kiss her face. He will kiss her jaw, her throat, the slope of her shoulder.\n\nHe will kiss her all over, every place she wants him to, and his hands will follow along the path his mouth makes. He will part her thighs and find her heat with fingers, tongue, cock, and she'll take him into all of her body's secret deep places.\n\nShe will take him all in.\n\n* * * * *\n\nGinny woke without opening her eyes, pleasure coursing through her and the shreds of the dream slipping away. She moaned before she could stop herself. When she opened her eyes, Sean was staring at her over the edge of his iPad.\n\n\"Nightmare?\"\n\nA nod made the lie easy. It would've been harder with words, but since she couldn't manage to find any, she settled for sitting up and scrubbing at her face.\n\n\"Must've been a doozy,\" Sean said. \"You were really wriggling around.\"\n\nEmbarrassed, Ginny let her fingers pressing her eyes keep her from meeting his gaze. \"I don't really remember it. What are you still doing up?\"\n\n\"Reading.\" He tipped the screen to show her, though she couldn't possibly read the text from this far away. \"You've only been asleep for about an hour.\"\n\n\"It felt like a lot longer.\"\n\nHe looked concerned and pulled her closer, though it wasn't comfortable for her to snuggle up to him the way he wanted her to. \"You want to talk about it?\"\n\n\"No.\"\n\n\"Was it about the baby?\"\n\n\"Sean, no.\" Ginny sighed and pushed away. \"I have to pee.\"\n\nShe padded to the bathroom, her habit, as always, to leave the light off and needing it even less with the light from the bedroom. She peed with her elbows resting on her knees, her face in her hands.\n\nShe remembered that lingerie. The bra and panties, expensive and sexy. An impulse purchase. They'd fit her perfectly. She'd tried them on about a dozen times but worn them only once. They were still in her drawer. She'd never wear them again, she knew that, but she hadn't been able to get rid of them, either.\n\nAt the sink she washed her hands and looked up without intending to see anything. Usually she averted her eyes anyway in case the irrepressible urge to shout out \"Bloody Mary\" three times overtook her. That old story had scared the bejesus out of her as a kid. Tonight, though, she was more awake than usual and there was more light. She saw her silhouette, the flash of her eyes and flip of her hair as she bent to the sink.\n\nAnd then, something else.\n\nA figure, a shadow, drifted past the bathroom door. Indistinct, but definitely there. Startled, Ginny whirled, ready to confront a sneaky Sean trying to scare her, but there was nothing there. She was frozen for a minute or so, waiting to see it again. She stared, hard, but saw nothing. Ginny cocked her head to listen for the sounds of anything from downstairs. Was that the creak of a footstep? The squeak of a cupboard. Ah. A late night snack. Satisfied she'd solved the mystery, Ginny headed back to bed.\n\nHe'd turned the light out before going down, at least he'd done that. Ginny crawled into bed with a sigh and settled into her pillows. The dream had faded but the memories lingered. Closing her eyes might bring them even more into focus, so she stared instead at the wall, the darkness...\n\nSomething touched her.\n\nNot something, someone. A hand stroked down her back, cupping her rear. Ginny shrieked and fought against the blankets and the touch, trapped by the tangled sheets at her feet and the pillows she'd propped herself up with.\n\n\"Hey, hey! Honey! Ginny, babe, it's me!\"\n\nSean snapped on the bedside lamp, but it took her a few seconds to stop fighting him. Ginny blinked rapidly. \"When did you get into bed?\"\n\n\"What do you mean, get into bed? I've been here the whole time. I got tired while you were in the bathroom. I figured I'd\u2014\"\n\n\"You turned off the light!\" she accused. \"I couldn't see that you were in here!\"\n\nSean frowned. \"Where else would I be?\"\n\n\"I saw you go into the hallway. I heard you go downstairs. I heard you in the kitchen,\" Ginny insisted when she saw the incredulity of his expression.\n\n\"Honey, you had a nightmare, maybe you just...\"\n\n\"No.\" She shook her head. \"No, I saw it, Sean. And more than that, I heard someone downstairs walking. I heard cupboards opening.\"\n\nSean reached behind the bed to pull out a golf club. He never played golf, but he'd kept that club behind the bed as long as she'd known him. \"Stay here.\"\n\nShe clutched his arm. \"No! What, are you crazy? Are you going to go downstairs? Sean, no. Call the police first.\"\n\nThe second she said it, she knew why he didn't want to do that. Then the police would come, and if there was nothing... Ginny let out a frustrated sigh. He wanted to reassure her, but he didn't believe her enough to call the cops.\n\nOn the other hand, he believed her enough to take the golf club.\n\n\"I'm going with you.\"\n\n\"No you're not. You stay here.\"\n\nThey could fight about it all night, or she could let him go. Ginny let him go, but as soon as he went down the stairs, she was up and out of bed, looking over the railing. Not that she could see anything, but she had her cell phone clutched in her hand, ready to dial 911.\n\n\"Sean!\"\n\nNo answer.\n\nOh God. It was like that story from when she was a kid, one from that same Bloody Mary book, about the headless roommate in the fur collar. She'd hide up here in the dark and reach out to feel him when he came back to bed, and instead of her husband, she'd find a decapitated corpse...\n\n\"SEAN!\"\n\nNo answer again. Her finger swiped her phone to unlock it, her finger hovering over the keypad. \"If you don't answer me, I'm calling the police!\"\n\nHe appeared so suddenly at the bottom of the stairs that she screamed. He waved the golf club at her. Ginny tried to remember to breathe.\n\n\"Nothing down here,\" he said.\n\n\"Are you sure?\"\n\n\"Babe, I'm positive. All the doors are locked and everything. There's nothing, I promise. It was just a bad dream.\"\n\nHe had one foot on the stairs, but she was already heading down. \"I want to see.\"\n\nSean sighed but stepped aside, humoring her. \"I'm telling you, honey, there's nothing.\"\n\nIn the kitchen, she looked around suspiciously, but he was right. Nothing. No sign of anything anyway. No door hanging open, no bloody splatters on the wall, no butcher knife missing from the block. Even the chairs weren't out of place.\n\n\"See?\" He came up behind her and kissed her hair. \"I told you.\"\n\nShe sagged against him, finally calming down. His hands traced circles on her belly. Warm, slow, soothing circles. Within a few seconds, she was giggling at the memory of how she'd jumped and screamed. Ridiculous.\n\n\"My hero,\" she murmured, turning in his arms to kiss his mouth.\n\n\"That's me.\" Sean's hands drifted lower to cup her butt again and pull her close. She could feel him through the thin material of his pajama bottoms. They rocked a little slowly together.\n\nThe body didn't separate the source of pleasure the way the mind did. When he kissed her neck, it felt good. When he rubbed against her, that felt good too. With the remnants of the dream lingering, Ginny was more than willing to see where this might go, but first she needed a drink.\n\n\"All that screaming,\" she explained.\n\n\"I'll be happy to make you scream some more,\" her husband offered with a naughty grin.\n\nShe was laughing too when she opened the cupboard to pull out a glass, but the laughter stopped like it had been slapped from her. So did every bit of desire. She turned to show him what she'd found.\n\nHer mug, the favorite black-and-pink one with the skulls.\n\n\"The least you could do is put it back where it belongs,\" Ginny said coldly.\n\nSean looked confused. \"I didn't put that there.\"\n\nShe gave him her back, too angry to even look at him. \"Someone did, and it wasn't me.\"\n\n\"I'm telling you, Ginny, it wasn't me.\"\n\n\"I didn't do it!\" she shouted and slammed the cupboard door hard enough to make the dishes inside rattle. She whirled on him. \"Why don't you just admit it, Sean? You took my mug to work and you forgot it, and then you brought it home and tried to sneak it back before I noticed. Well. I noticed. Why don't you just admit it?\"\n\n\"Because I didn't do it!\"\n\n\"So, who did?\"\n\nHe was silent for a while, staring. He shrugged and cut his gaze from hers, his expression sullen. Stubborn. Then he looked back at her almost boldly, like a challenge.\n\n\"Maybe you did and just forgot.\"\n\nGinny's lip curled. \"I didn't forget. How could I forget something like that?\"\n\nShe knew by his next silence he was thinking of all the times she had forgotten things, all her absentminded times. Ginny turned back to the sink so she didn't have to look at him. \"That's not fair, Sean.\"\n\nShe wanted to pull away, to storm from the room and ignore him, but his fingers insisted on soothing her. Working at the aches and knobs of pain and tension in her neck and back. He knew all the places to touch her. He always knew.\n\n\"All I'm saying is, you're distracted. You're pregnant. Pregnant women can be forgetful.\"\n\n\"Did you read that on the Internet?\" She sounded bitchy and knew it, but couldn't force herself to care.\n\n\"It's true, isn't it?\"\n\nWith a sigh, Ginny let herself relax against him. His lips pressed her temple and his hands slid down to rub her belly. She closed her eyes and put her hands over his, feeling the baby press upward against the touch. \"Yes. It could be.\"\n\nBut it wasn't. She hadn't seen that mug in weeks. The last time she remembered using it was the night she'd made tea and fallen asleep in the library. Surely if she'd used it since then and put it back in the wrong cupboard she'd remember, and it wasn't like she'd have done it on purpose to fuck with her own head.\n\n\"Things happen, that's all. Give yourself a break,\" Sean said.\n\nWhat he really meant was give him a break. Wasn't that what he always meant? And, because that's what she owed him, that's what Ginny did.\nChapter Twenty-Eight\n\nThey'd never had a lot of extra cash, even when she was working, but they'd always had enough to go on vacation, eat out a few times a week when neither of them felt like cooking, stop for an expensive coffee drink on the way to work instead of brewing it at home. Lots of little conveniences, that's what having enough money brought. And now...well, they weren't in debt, at least not beyond what Ginny considered \"normal\"\u2014mortgage, one car payment with the other vehicle paid off. She and Sean both tried hard to pay off their credit cards every month so they didn't carry a balance.\n\nThey weren't in debt, but they simply didn't have anything extra.\n\nNext month might be better, without the unexpected expense of the furnace repair and an electrician who'd looked at the fuse box but hadn't been able to figure out why they kept blowing fuses. She'd have to start holiday shopping in earnest, a task that always fell squarely on her for both sides of the family, and which she'd usually have finished by now. The move had taken up so much time over the summer she just hadn't had time. Or felt like it either, she thought as the baby moved inside her.\n\nShe tallied up the income and expenses, balancing the accounts. Then she logged into their credit union's online banking site and transferred some money from their savings, cringing at the remaining balance. It had been at least a week since she'd gone online, too busy with unpacking and everything to even think of it, but since dinner was already in the oven and Sean wasn't going to be home until after class, she had some time.\n\nShe looked up instructions on how to reupholster dining room chairs, then how to make slipcovers, then local places that would do it for her. The DIY sites made it seem deceptively easy, but she knew better. The businesses that specialized all charged too much for her budget, at least for now. She shifted on the seat, listening to it creak.\n\nShe flitted around the Internet, browsing some of the email shopping offers that had come in, watching a video her sister had sent a link to. Somehow, without thinking much of it, she ended up logging in to Connex. It had been weeks since she'd checked it, and she didn't even try to get caught up on the entire contents of her updates list. The site had changed a bit too, she noticed with a frown. She couldn't seem to find her inbox or how to upload a photo without stumbling around and hitting something by accident, and a bunch of status updates referenced opting out of new settings she didn't have the time or energy to care about.\n\nHer sister had uploaded a bunch of new pictures, though, and Ginny looked through them absently. Her fingertips found the edge of the table, tracing the letters over and over as she scrolled through her nieces and nephews mugging for the camera, modeling new outfits, whatever. One link led to another, to another and another, until she ended up on the page belonging to a friend of her niece Maria. The girl had posted a lot of personal stuff, much of it TMI. It was like a train wreck\u2014Ginny knew she shouldn't look, but did anyway.\n\nPeople used to put that stuff in journals. Now they put it on the Internet. She thought of the box upstairs in the library, the diary inside it. Her fingers traced the letters carved into the table, over and over.\n\nCMM\n\nCaroline. Her diary. Her name on this table. That girl had lived in this house, and disappeared from it too.\n\nGinny's fingers tap-tapped on her keyboard, running a Connex search for Brendan Miller. Of course, about two dozen names came up, but when she narrowed her search, she came up with just three names. She could tell nothing from the profile photos, since she had no idea what Brendan Miller looked like, but she knew he lived in Lancaster somewhere.\n\n\"There we go,\" she murmured to the screen. She brought up his profile and found an email address but no phone number.\n\nOne thing she'd learned in her business was that even people who should know better often didn't appropriately privacy-protect their online information. A quick further search discovered Miller's address, which was current and confirmed by the most recently updated white pages database. In under ten minutes, she had everything she needed to know to get in touch with him. It was one of the easiest searches she'd done in a long time, but now Ginny sat back in her seat and wondered what she would do with that information.\n\nHe had a right to know. He obviously didn't care about the furniture or other things in his father's house, but surely he'd have wanted something personal of his sister's. Even if he hadn't wanted anything to do with his dad when he was still alive, he at least deserved the chance to have what his sister had left behind.\n\nShe dialed the number as she went upstairs. A woman answered just as Ginny entered the library to look at Caroline's box.\n\n\"Hi, can I speak to Brendan Miller, please?\"\n\nThe beat of silence lasted way too long. \"Who's calling?\"\n\n\"This is Ginny Bohn, I\u2014\"\n\n\"Who?\"\n\nGinny paused. Brendan Miller hadn't come to the settlement. She had no idea if he was married, but this woman sounded like a suspicious wife. She tried again. \"My name is Ginny Bohn. I bought his father's house?\"\n\nAnother long pause. \"Yes? What about it?\"\n\n\"Is he there?\"\n\n\"No. He's not. Can I help you?\" The woman's clipped tone didn't sound the least bit helpful.\n\nGinny tried anyway. \"I've found something in the house.\"\n\n\"He doesn't want anything from there, he told me.\" Another pause, then a resentful sniff. \"I told him there were a lot of lovely things in that house that we could use, but he refused to have any of it. I said that even if he didn't want it, maybe the kids would, someday. I mean, it was family heirloom stuff.\"\n\n\"Oh, honestly...I don't know anything about that, really. There were only a few things we asked remain in the house anyway.\" She thought suddenly of the ugly table she didn't like, but the thought of offering it to this unhelpful and snide-sounding stranger was suddenly unpalatable. \"But I'm not talking about\u2014\"\n\n\"Whatever. It was his father's stuff. I guess if he didn't want it, who am I to say a word?\"\n\nGinny had been on this ride before, the up-and-down roller coaster of marital resentments. She understood how it felt to feel marginalized, she totally did. Yet nothing in this woman's attitude made Ginny sympathetic.\n\n\"It's not furniture,\" she said quickly before the woman could continue complaining. \"It's something personal.\"\n\nThe pause this time was longer. \"Like what?\"\n\n\"I'd really like to talk to him about it, if you don't mind.\"\n\nThat was the wrong thing to say. An audible, choking gasp poked Ginny's eardrum. The woman spat her words like bullets, \"I do mind, as a matter of fact. What did you say your name was again? What sort of personal business do you have with my husband?\"\n\n\"It's his sister's suitcase,\" Ginny said before the woman could go off on her some more. \"I thought he'd want it.\"\n\n\"My husband doesn't have a sister.\"\n\n\"No, well...um, so far as I know, she's...gone.\"\n\n\"Who did you say this was again?\"\n\nIrritated, Ginny sighed. \"Is there a better time I can reach him?\"\n\n\"No. Don't call here again. My husband didn't have a sister.\"\n\nWith that, Mrs. Brendan Miller hung up and left Ginny's jaw hanging open.\n\nWhat a bitch.\nChapter Twenty-Nine\n\nThe Ouija board had been spectacularly easy to get. Ginny simply ordered it online, and it arrived within a couple days. It was different than the one she remembered from slumber parties as a kid. This one was smaller, with glow-in-the-dark letters and planchette. Still, it had the same setup with YES, NO, GOODBYE and the alphabet curving across it.\n\nPeg, on the other hand, could not be convinced to use it. \"No. No way.\"\n\n\"Peggy, c'mon. We used to do it all the time as kids. Remember?\"\n\n\"I remember. But I'm not doing it now.\" Peg shook her head and then her finger. \"And you shouldn't, either. What are you thinking, bringing that into your house? Don't you know that you could attract...something?\"\n\n\"There's already something.\" Ginny put the box aside and set the board in the middle of the dining room table.\n\nPeg huffed. \"Oh, Ginny. Come on.\"\n\nGinny paused to look at her sister. \"I'm serious.\"\n\n\"You have mice or squirrels in your attic, that's all\u2014\"\n\n\"Mice and squirrels might get into the food in the pantry,\" Ginny said. \"But they wouldn't use a mug and then put it back in the cupboard.\"\n\n\"Well, neither would a ghost, for crying out loud.\"\n\nGinny raised a brow. \"How do you know?\"\n\n\"Ghosts are spiritual entities or whatever. They don't need to use a mug.\" Peg waved at the Ouija board. \"And this is just asking for trouble.\"\n\n\"You used to love this game!\"\n\n\"It's not a game,\" Peg said seriously. \"I mean it, Ginny. Father Simon spoke about it a few months ago, because of the kids getting in to stuff like that for Halloween. It's a bad tool. It invites bad things.\"\n\nGinny sighed and pulled the board toward her. She put her fingertips on the plastic planchette and moved it experimentally around the board. \"Father Simon is kind of hysterical.\"\n\nPeg made a snuffling noise of protest, but her expression said she knew Ginny was right. \"Don't be disrespectful.\"\n\n\"Didn't he also give a sermon about how couples should watch more television together because it prevented arguments?\"\n\nPeg's mouth worked on tamping back a smile. \"Yes. He did.\"\n\n\"And how's that working for you and Dale?\"\n\nPeg didn't answer for a moment, then said reluctantly, \"We fought over the remote.\"\n\nGinny slapped the table in triumph. The planchette bounced and slid, skewed toward the board's grinning sun. \"See?\"\n\nPeg shook her finger at Ginny again. \"He's the authority on spiritual matters, and if Father Simon says Ouija boards are dangerous and bad, I believe him. Besides, didn't you see The Exorcist?\"\n\n\"It's a movie, not real life.\"\n\n\"Paranormal Activity?\" Peg asked, as though that somehow was less fictional.\n\n\"Did they even use a Ouija in that?\" Ginny scoffed. \"Also, just a movie.\"\n\nPeg frowned, her expression shadowed. \"You don't remember, do you?\"\n\n\"What, the movie? Not much. I fell asleep, except for the last couple of minutes. Which were some of horror cinema's finest,\" Ginny admitted. \"But no, I don't remember the rest.\"\n\n\"Not the movie. The Ouija board. The one at Gran's house.\"\n\n\"I do remember it. That's why I got this one.\"\n\nPeg sighed. \"But you don't remember what happened with it. Obviously, you don't. Or else you wouldn't be sitting there with that thing in front of you.\"\n\nThis had the flavor of a story. Ginny perked, leaning closer to her sister. \"What happened?\"\n\n\"I don't really want to talk about it.\"\n\nGinny tossed a ball of the plastic wrapping that had been on the box at her. \"Bitch!\"\n\nPeg laughed, though it trailed into an uneasy sigh. \"It was some creepy, scary stuff, Gin. You were maybe...ten? Eleven?\"\n\nWhich would've made Peg seventeen. Ginny remembered being ten, vaguely. It was the year of the short haircut. Being mistaken for a boy was the most traumatic thing for Ginny that year, certainly no strange paranormal thing.\n\nUnless she'd just forgotten about it.\n\n\"We were at Gran's for Christmas. The big party, and we were staying over for the rest of the week, along with Roberta and Dana and Carla.\" Peg paused. \"Or maybe Dana wasn't there, I don't remember. No, I think she was on some youth group trip. Or had she already gone off to college? No, she had to be there, because\u2014\"\n\n\"Is it important?\"\n\nPeg shook her head, focusing. \"Oh no. Anyway, we were all staying over until New Year's.\"\n\n\"Gran had another party, didn't she?\" Now Ginny was remembering a little more. \"She always had the Christmas Eve party, but she had that New Year's thing one year.\"\n\n\"Just the one,\" Peg said ominously.\n\nGinny rolled her eyes. \"She had other New Year's Day things. I remember going there for pork and sauerkraut.\"\n\n\"Sure, just the family. But not a big party with neighbors and stuff. She always said it was because having two big parties so close together was too much work.\" Peg gave Ginny a significant look. \"But I think it was because of what happened.\"\n\n\"Jesus, Peggy. What happened?\"\n\nPeg frowned. \"Ginny, please.\"\n\n\"Sorry.\" Her apology was automatic but sincere; she didn't have to drink the Kool-Aid to understand that her sister liked the flavor.\n\n\"We were all playing with that thing.\" Peg gestured at the board. \"Roberta found it in one of the closets upstairs. It was probably Uncle Jimmy's.\"\n\n\"What, along with the collection of biker mags?\" Ginny snickered.\n\nPeg laughed too. \"Yeah. Anyway, she brought it downstairs after Gran went to bed. We were all supposed to be in bed too, but you know we never were.\"\n\nNighttime in Gran's house during Christmas vacation had created some of Ginny's favorite childhood memories. Sneaking cookies, playing endless games of Monopoly and Clue by the light of the tree. Everyone had new pajamas, the fabric still stiff. The older kids played their portable boom boxes, fighting over who got to pick the next cassette.\n\n\"So Roberta brings this thing down into the living room, and we set it up near the tree. By the train set. Dana and Carla went first. They held it on their knees, sitting crisscross applesauce.\"\n\nGinny laughed at the term. She hadn't heard that in years. Her cousins, fraternal twins, were usually the first to try anything new. \"You're taking a long-ass time to tell this story.\"\n\n\"Do you want to hear it or not?\" Ginny nodded, only slightly chastised, but Peg was determined to draw it out. \"I need a refill.\"\n\n\"Oh, Peg. C'mon!\" Ginny's protests were met with Peg's unwavering stare and her raised coffee mug. \"I'm the pregnant one.\"\n\n\"And you know I don't know how to use that fancy coffee thing you have.\"\n\nGinny narrowed her eyes, but got up. In the kitchen, she fussed with the coffee pods to brew her sister another cup. On a whim, she opened the cupboard to see if any other of her dishes had gone missing or been moved around, or if any other gifts had been left in their place. Today there was nothing.\n\nMaybe she really was just crazy.\n\nPeg came into the kitchen. \"You're a terrible hostess. I need something to eat.\"\n\nGinny removed Peg's full mug and replaced it with hers while she brewed some cocoa from another pod. Then she took both mugs and sat at the table while she watched her sister dig into the fridge. \"If I didn't know any better, I'd say you were trying to stall.\" Ginny gave Peg a significant look over the rim of her mug. \"So come on. Spill it. What, you think I'm going to get too scared or something?\"\n\nGinny had always been the kid who stayed up late reading scary books under the covers with a flashlight, who snuck downstairs to watch horror movies after her parents went to bed. Halloween had never seen her in a princess tiara or a cowgirl hat. Halloween was for zombies and witches and vampires. One year she was a werewolf, complete with ears and furry paws, a bloody and mangled red cape around her neck like a scarf. She'd won first prize in the parade. There wasn't much Peg could tell her about supernatural things that would scare her.\n\n\"You're the one who thinks she has a ghost,\" Peg pointed out. \"I don't want to freak you out.\"\n\n\"I've faced worse fears than ghosts.\" Ginny meant to say the words lightly, teasing, but they came out solemn and serious and heavy. She swallowed hot cocoa too fast, and it burned her tongue. She turned her face from her sister's.\n\nPeg reached for her hand anyway. \"Right. Sorry. I wasn't thinking.\"\n\nGinny shrugged again and blew on her cocoa, though now she no longer had the taste for it. \"So. This story?\"\n\nPeg grimaced. \"Fine. So we were all downstairs playing with the board, and you'd been upstairs sleeping or reading or something, I don't know. But you came down when we were doing it, and you wanted to play. So, you sat down across from Carla and put your fingers on that plastic thing\u2014\"\n\n\"The planchette.\" Fascinated by this story she couldn't remember, Ginny leaned forward.\n\n\"And it started to move.\"\n\nGinny waited, but Peg didn't say more. \"And?\"\n\n\"It started to fly all over the board, all wild. Carla accused you of moving it, but you said you weren't! And it went all over; it started spelling out words and stuff. We all got freaked out. You took your fingers off it, and it stopped. You put your hands on it, and it went again, crazy, until it flew right off the board.\"\n\n\"Wow.\" Ginny laughed. \"Sounds like I was pulling a fast one on you all.\"\n\nPeg was silent for a couple seconds. \"Maybe.\"\n\n\"Or what? Maybe it was Satan?\"\n\n\"You were a devil,\" Peg told her.\n\nGinny tossed a balled-up paper napkin at her. \"Hyuck, hyuck, hyuck. Very funny.\"\n\n\"You don't remember?\"\n\n\"Vaguely. But probably only because you just told me about it.\" Ginny turned her mug around and around in her hands, pressing her palms to the heated porcelain. \"I'm not sure what was so freaky about it.\"\n\nPeg hesitated, but pressed on. \"You said it wasn't you doing it, making it move. That it was the lady in the blue dress.\"\n\n\"Sounds terrifying.\"\n\n\"You insisted it was the lady in the blue dress, and she wanted to tell Carla something important, but Carla was so scared by that time that she wouldn't put her hands on the planchette. You were sitting there and it started moving again, we assumed you were pushing it of course. Trying to scare everyone. But it was going slower, more controlled, and we could read what it was saying. Over and over, 'Don't go to Boston'.\"\n\n\"Wow. Weird.\"\n\nPeg nodded and took a minute to sip from her coffee like it gave her strength. \"You know what happened, right?\"\n\n\"If I knew,\" Ginny pointed out, \"would I be asking you?\"\n\n\"Carla was supposed to go to Boston for a dance competition. Which you probably knew. I mean, everyone was talking about it. But she got so freaked out she didn't go.\"\n\n\"And what happened? Let me guess, the plane she was supposed to go on crashed or something.\" Despite herself, Ginny got a shiver.\n\nPeg shook her head. \"Nope. Nothing happened.\"\n\n\"Well...what on earth? Why would that mean anything then? I was probably just messing with her!\"\n\n\"She didn't go to Boston that weekend, and, instead, she went to Buffalo with her mom, shopping. They were in a car accident. Carla was driving.\"\n\nGinny wrinkled her nose. \"I remember that.\"\n\n\"The car was totaled but they were okay. But, one thing they never talked about, Carla only told me once and it was just after Peter and June got married, and they had that reception in that creepy inn, remember? And everyone was talking about how it was haunted, and what weird things happened, and Carla refused to stay there, she went and stayed at the Holiday Inn?\"\n\nThere were benefits to having a large family\u2014all the stories that started and ended with \"you remember?\" being one of them\u2014but not right now. Ginny waved an impatient hand. \"Yes, I remember.\"\n\n\"Carla told me that a steel rod from the truck that hit them went all the way through their windshield, all the way through the car. If her mom had been driving\u2014because her mom's a few inches taller, right?\u2014she'd have been decapitated. But Carla was driving because\u2014\"\n\n\"She didn't go to Boston.\"\n\n\"Yes!\" Peg crowed, triumphant, and slapped the table hard enough to make their mugs jump. \"Because you told her the lady in the blue dress said not to!\"\n\n\"Oh, Peg.\"\n\n\"The lady in the blue dress, by the way, is in a picture in Gran's house. It used to hang up in the dining room, but she took it down when she remodeled.\"\n\n\"So, I saw a picture of a lady in a blue dress and wanted to freak out Carla, and I made up a story that happened to have some weird repercussions.\" Ginny waved her hand again. \"You can't say it had anything to do with it. I mean, you can never know what might've happened. Only what did.\"\n\n\"I thought you of all people would be excited to hear that story.\"\n\nGinny laughed. \"But you're the one saying it's the devil's tool! Sounds to me like it saved Aunt Dina's life.\"\n\nPeg clearly hadn't thought about it that way, because she opened her mouth to say something, but stopped. Then started. Stopped again, while Ginny laughed. \"It's still not a toy,\" she said finally. \"And I think if you use it to contact whatever's in this house, you'll regret it.\"\n\n\"So you believe me, there's something here.\"\n\nPeg hesitated again. \"I don't know.\"\n\n\"Fine. Don't believe me. But if there's not a ghost or spirit or something in here...\" Ginny paused to draw a shallow breath that wanted very much to be a sob, \"...then it must be my husband trying to gaslight me. Which do you think I'd rather have it be?\"\n\nPeg didn't have an answer for that, at least not one that came out of her mouth. Ginny's sister's thoughts rolled over her face in a series of twitches and glances that required a lifetime of interpretation to understand. Fortunately, Ginny'd known her sister for her entire life. Peg didn't believe Sean was trying to drive Ginny crazy, but she didn't quite believe Ginny's stories, either.\n\nGinny frowned and got up to dump her cooling cocoa in the sink. \"I don't care if you believe me or not. I'm telling you, weird things are going on in this house.\"\n\n\"Maybe you should have Father Simon come over.\"\n\nGinny turned, brows raised. \"What? Why... Oh, Peg. Really? Look, I know you're trying to get me back into the fold and all that, but this is a really crappy way of doing it.\"\n\nPeg had the grace to look guilty, but then defiant. \"It wouldn't kill you to get a little religion.\"\n\n\"I don't go to church. Period. Why on earth would Father Simon come over to investigate my missing mug?\"\n\n\"He'd just come over to talk to you, that's all. If you and Sean are having issues...\"\n\n\"I'm not taking marital advice from a man who's not only not married, but will never be married and in fact has never, in all likelihood, ever even had sex,\" Ginny added. \"And besides that, it's not his business if Sean and I are having problems. I didn't say we were having problems!\"\n\nShe rinsed her mug and put it in the dishwasher, refusing to look at Peg. Her sister's mug scraped on the table. Then her chair skidded on the floor.\n\n\"It was just a suggestion,\" Peg said.\n\nGinny turned, not wanting to fight with her sister. \"I know.\"\n\n\"I'm not saying you're having problems. Just that it wouldn't be a surprise if you were. I mean, this is a stressful time. A move, new house, you're not working. The baby,\" Peg said quietly. \"I wouldn't be surprised at all if it was affecting your sleep or making you susceptible to strange ideas.\"\n\n\"Did Sean tell you that? That I haven't been sleeping?\"\n\nPeg looked caught. \"He said he was worried.\"\n\n\"I'm fine. And it's not my lack of sleep, I'm not imagining these things, and we aren't having any problems.\" Ginny scowled. \"I can't believe you're going to believe him over me anyway.\"\n\n\"Because you think he's gaslighting you? Oh, Ginny. Really? Sean?\"\n\nGinny stabbed a finger in the air. \"I don't want to believe it, no! But knowing he went and talked to you behind my back about how crazy I am only makes that seem more likely, doesn't it?\"\n\n\"I'd believe in a ghost before I'd believe your husband was trying to drive you crazy.\"\n\n\"Aha!\" Ginny cried triumphantly. \"Yes! You see what I'm talking about?\"\n\n\"I'd also believe you were crazy before I believed you had a ghost.\" Peg smirked and took her cup to the dishwasher. She studied her sister up close, seriously. \"Talk to Father Simon, Ginny. Do it as a favor to me.\"\n\n\"Do me a favor and stop trying to foist him on me.\"\n\nPeg sighed and closed the dishwasher. \"Fine, fine. Whatever. But don't ask me to do that devil board with you, and get rid of it. Don't have it in your house. Even if you think it can't do anything bad, it's a bad influence. And don't do it by yourself!\"\n\n\"Oh, like in Witchboard?\" Ginny hadn't thought about that, she'd been focused on the idea that two people were required to use it.\n\n\"I don't know what that is, but it sounds bad. Promise me you won't use it alone. That's how the demonic influences get inside you. Promise me!\"\n\nPeg seemed so serious that Ginny nodded. \"I promise. Of course. Fine. I'll toss it in the trash as soon as you leave. I promise, Peg.\"\n\nPeg eyed her suspiciously, but then nodded. \"Good. Call me later, okay?\"\n\n\"Fine. Yes.\" Ginny ushered her sister to the door, accepted her hugs and more advice, because that was what her sister did. But when the door closed, she didn't call her sister's priest to come over for dinner, and she didn't dump the Ouija board in the garbage either.\n\nIt wasn't the first time she'd broken a promise.\nChapter Thirty\n\nCandles were supposed to be romantic, Ginny thought. But when the power went out and your husband had misplaced the rechargeable flashlight and blamed you for it, candlelight was only annoying. In the kitchen she'd lit a series of tea lights in every single holder she could find. In the living room she'd lit a couple of jar candles, and in the dining room, collected in the center of the old, scarred table on a porcelain platter, she'd lit three huge pillar candles. The scents\u2014pine needles, vanilla, cinnamon and something called Misty Memories\u2014warred with one another. With her stomach too.\n\nThe light was pretty, golden and flickering over the old, polished wood. It hid all the flaws in the plaster, the dust in the grooves of the carved wood. It made shadows, though. Moving, deceitful shadows.\n\nGinny looked at the board in front of her. She should put it away, get in the car and go anywhere until the power came back on. But she didn't have the strength to face the Christmas mall crowds, frenzied by all the sales. She wasn't hungry and didn't want to sit by herself in a restaurant. And coffee shops had soured for her. Besides, Sean would be home in an hour or two and by then she'd be ready to eat. He could take her out to dinner if the electricity wasn't back on. Buy her a new rechargeable flashlight. Hell, a generator, if this was the sort of thing that was going to keep happening.\n\nFor now, she studied the board in front of her. Her fingers curled and pressed the cool plastic. She closed her eyes and drew a breath, anticipation tingling in her fingertips as she waited for the planchette to move.\n\nNothing.\n\nMaybe she had to say something. Some sort of greeting? \"Hello.\"\n\nStill nothing. She inched her fingers toward her and the planchette moved easily enough on the little felt pads at the ends of its legs. No friction or resistance. Ginny moved it around in a circle, then a figure eight, but she could tell it was her making it go. Not spiritual forces.\n\n\"Is there...anyone here?\"\n\nFor a second it seemed like the planchette twitched, but as she waited, breathless, nothing else happened. With a frown, Ginny sat back and stared at it. In the flickering light, the shadows beneath the planchette made it look like it was wiggling, just a little, but when she touched it, she could feel nothing.\n\nShe tried again, letting her fingertips rest so lightly on the plastic she was almost not touching it at all.\n\n\"Caroline? Caroline Miller. I'm talking to you.\" Ginny hadn't meant to let her voice drop so low and growly, but speaking at full volume seemed silly. She took a breath and held it for a second before letting it seep out through her nostrils. The smells of the melting wax made her want to sneeze.\n\nThe Ouija board did nothing. Frustrated, Ginny thumped her fist on the table. The planchette jumped and skewed a little, the pointer facing her.\n\n\"C'mon. I know there's someone here. I know you're in this house. Caroline,\" Ginny said. \"I have your suitcase. With your things in it. With your diary. I haven't read it yet, out of respect, but maybe...maybe you want me to read it? Why did you put those things up there in the closet? Why didn't you want anyone to find them?\"\n\nThat was a dumb question, she realized. Of course Caroline wouldn't want anyone to find her box of secrets. Ginny wouldn't have wanted anyone reading her diary when she was thirteen or fourteen either. She thought of her own little secrets. She wouldn't want anyone knowing of them now, though hers had been much more easily hidden and erased.\n\nBut not forgotten.\n\n\"Do you want me to read this journal?\" Ginny put her fingertips back on the planchette. \"Will it tell me what happened to you?\"\n\nThe candles flickered as though someone had blown a breath across the flames. Ginny froze, catching sight of motion from the corner of her eye. She turned, slowly, slowly and saw only shadows growing longer and shorter as the candle flames moved. Still, there was a presence here. She felt it, didn't she? The weight of someone's gaze. She strained to hear the sound of breathing, the creak of floorboards. The back of her neck prickled.\n\nShe whipped her head around, preparing to scream at the sight of some ghostly figure floating toward her with its mouth yawning wide to eat her up.\n\nNothing.\n\nWith a shuddery sigh, Ginny pushed the board away from her. Maybe Peg had been right. It was dangerous to do this alone, if only because of the things it made her imagine.\n\nShe was sliding the box back into its place on the shelf in the living room when the lights splashed across the front windows and tires crunched in the drive. She pulled the curtain enough to peek out, then met Sean at the door, opening it wide and stepping back out of the way so he could come in.\n\n\"Creepy.\" He unwrapped his scarf and hung it on the coatrack, then set his briefcase at the foot of the stairs where he'd forget it repeatedly until she bugged him about it.\n\n\"What's creepy?\" Ginny rubbed her arms against the chill breeze he'd brought in with him.\n\n\"The way you opened the door, like nobody was there.\" He leaned to kiss her, but absently. \"What's with the candles? Did I miss something? It's not our anniversary...\"\n\n\"The power's out.\"\n\nHe grinned at her, his teeth flashing white in the dim light. \"You're kidding, right? Here I thought you were planning some romantic dinner surprise. I got excited there for a minute, thinking about heart-shaped meat loaf.\"\n\n\"I couldn't cook anything without power.\" Guilt groped at her like a drunken prom date. It had been a long time since she'd surprised him with anything like that. The fact she hadn't even thought about it felt worse than the fact she hadn't done it.\n\nSean looked over his shoulder. \"How long's the power been out?\"\n\n\"About an hour.\"\n\n\"Well...\" he shrugged and gave her a confused look, \"...are you sure? I mean, did you check it?\"\n\n\"It's dark outside, Sean. I think I'd notice if the lights suddenly came back on.\"\n\n\"No, I mean...did you call the electric company or anything? Because everyone else on the street seems to have power.\"\n\nThis first stunned her, then annoyed her. She went automatically to the wall switch, already squinting against the flare of light she expected when she flipped it. But, like the Ouija board's reluctance to perform, the lights didn't respond either.\n\nShe couldn't keep the triumph from her voice. \"No. See? Out. Just like all those other times.\"\n\n\"Shit.\" Sean reached around her to flip the switch on and off several times. \"The whole rest of the street's lit up. Nobody else is dark. Must be another fuse.\"\n\n\"Of course it is.\"\n\n\"Still, why'd you sit here in the dark?\"\n\n\"Because the last time I went downstairs to look at the fuse box you told me I shouldn't. So I didn't,\" Ginny said carefully. So carefully.\n\nSean said nothing.\n\n\"I'm getting really tired of this, Sean.\"\n\nHe rubbed her arms gently. \"Yeah, yeah. I know.\"\n\n\"I mean, the power goes on and off if a squirrel sneezes in the yard. It's winter; it's dark a lot earlier and cold.\" She was getting ramped up and wanted to stop, but couldn't. \"It's ridiculous, Sean! We just bought this house; we paid more for the extra-detailed home inspection. Why does this keep happening?\"\n\nTo give her husband credit, he took her near hysteria in stride. He pulled her close and pressed his lips to her hair. With her cheek against the front of his jacket, the heat in her face faded. She fisted her hands in the material and closed her eyes, willing herself to remember this was not that big a deal.\n\n\"I'll get an electrician in here.\"\n\n\"We can't afford it,\" she muttered against him. \"Those guys charge an arm and a leg just to come through the front door, not to mention, God forbid, he finds something wrong.\"\n\nSean's back straightened. \"I wish you'd quit worrying so much about money.\"\n\n\"I wish we had more,\" Ginny said. \"Then I wouldn't worry.\"\n\nFor a moment, his hands tightened around her. Then he let her go, stepped back and headed for the kitchen. Ginny followed.\n\n\"I know you don't want me picking up a part-time job\u2014\"\n\nHe stopped in the kitchen so fast she almost ran into him. He turned, and even in the shaky, dim light from three-dozen tea lights, she could see his look of scorn. \"Ginny, get real. Get a part-time job? Like that?\" He gestured at her jutting belly.\n\n\"Pregnant women work all the time.\"\n\nHe put his hands on her upper arms, but unlike the soft caress of earlier, this time his fingers pinched a little too hard. \"I don't care.\"\n\n\"You're hurting me.\"\n\nHe softened his grip and rubbed to take away the sting. \"I'm sorry.\"\n\nShe knew he hadn't done it on purpose, but moved away from him anyway. \"I've always worked part time during the Christmas season. And it would help, Sean. A lot.\"\n\n\"I don't want anything to happen to you. That's all. Can't you understand that? Jesus,\" he added so suddenly, so fiercely, she jumped. \"I can't believe we even have to go over this again.\"\n\n\"I understand why you wouldn't want me to work doing what I was doing. I get that. But a few shifts here and there at the bookstore, or\u2014\"\n\n\"I'm going to check the fuse box.\"\n\n\"Fine.\" Ginny pushed past him and into the dining room, ostensibly to get her now-cold tea, but mostly to avoid snapping at him. She braced herself for the question about the flashlight, already biting her tongue, but all she heard was the thud of his shoes on the basement steps.\n\nA couple minutes after that, the lights came on.\n\nThe sudden brightness stung her eyes, the perfect excuse for tears. Ginny blinked them away, went to the kitchen to pour her tea in the sink. She wet a paper towel and dabbed her face with it too.\n\n\"It was a fuse again,\" Sean said from behind her.\n\nShe hadn't heard him come up, and startled, a hand on her heart. \"God. You scared me.\"\n\nHe had the flashlight in one hand, unlit now. \"Sorry.\"\n\n\"Where'd you get the flashlight?\"\n\nSean gave her a funny look as he ducked out of the doorway, presumably to hang it back on the nail and plug it in. \"What do you mean?\"\n\n\"It was gone.\" He had to remember the argument. Or maybe he didn't. Sean had a way of pushing things to the back of his mind that he didn't want to remember.\n\n\"I just figured you put it back.\"\n\n\"I didn't.\"\n\nSean sighed, looking weary. \"Ginny, I'm starving and exhausted. Can we just...not do this now?\"\n\n\"Do what?\" Tears threatened again. Knowing it was pregnancy hormones didn't help. She hated this up and down, this constant topsy-turvy. \"Talk? Discuss how this is the same flashlight that went missing and you accused me of taking it, or, what, I don't know, hiding it from you or something, but when it was my mug that was gone, you wouldn't even\u2014\"\n\n\"Enough!\" he cried too loud. \"Christ. Just...enough. Okay?\"\n\n\"Fine.\" She looked away from him and breathed in. Somehow, she found it within her to reach for his hands, to link their fingers. She looked at him, her husband, the man she'd bound herself to until death did them part. \"Want me to make you something to eat? I can heat up some pot roast. I know there's some in the freezer. Why don't you go take a hot shower and I'll make dinner. Then maybe we can watch a movie or something?\"\n\n\"I have some homework...\" He hesitated. She let go of his hands, but he grabbed hers back. \"No, pot roast sounds good. And a movie. That sounds great.\"\n\nHe went upstairs, and Ginny went into the kitchen to pull leftovers from the freezer and put them in the microwave. She hesitated before pushing the button, but this time the lights stayed on. No blown fuses.\n\nShe rinsed the empty Brownie bowl and held it to the light for a moment. She hadn't filled it for a few days because she'd been meaning to pick up some of the licorice treats Sean liked, but hadn't remembered the last time she was at the store. She did have some wrapped butterscotch candies\u2014Sean didn't like them, Ginny thought with a small smile, but maybe the Brownies did. They were in the pantry, and she steeled herself out of habit before she went in.\n\nGinny had avoided the pantry closet since the baby shower, unloading her weekly loads of groceries and cleaning supplies as fast as she could or making Sean do it. Every time she went inside it, she couldn't stop her skin from crawling with the memory of the flies swarming all over her. Worse now was the fact that the closet was so narrow\u2014with her increased bulk she could go in but could barely turn around enough to get out without backing out.\n\nIt was hot, as usual, though, thank God, the smell had gone away and there was no evidence at all of flies of any kind. Still, Ginny kept a foot propped against the door to keep it from swinging shut as she reached for the bag of candy on the shelf. It was right where she'd left it. But there was something else too.\n\nA red collar with a jingle bell.\n\nWith a cry, Ginny clutched it, mindless that the door shut behind her. She pressed it to her heart. \"Oh, Noodles.\"\n\nThe cat had gone, and unlike the folk song, it didn't seem like she was coming back. But there was this, at least. Ginny closed her eyes against tears for a moment. Then she opened them. She kissed the collar and rubbed the soft leather for a moment, making the bell jingle. Then she tucked it away inside an unused kitchen crock and tapped the lid gently.\n\nSean wouldn't understand. He might even think she put the collar there, which would raise uncomfortable questions about where the cat had gone. What had happened to her. What part Ginny had played in her disappearance.\n\n\"Thank you,\" Ginny said aloud. \"Thank you, Caroline. And...please...keep Noodles safe with you. Love her a lot...for me.\"\nChapter Thirty-One\n\n\"The house looks amazing. How did you do it?\" Kendra bounced Carter on her hip as the baby looked goggle-eyed at the lights strung along the molding.\n\n\"A little at a time. That's all.\" That was the truth too. A little bit every day for the past three weeks. A few strings of lights, a couple batches of cookies. Nothing too strenuous, nothing Sean could chastise her about.\n\nNothing, in fact, that he'd noticed. He'd come home from work and class every night, sometimes too late to even eat dinner. Sometimes he smelled of the bar. Sometimes he didn't. When he did get home early, he watched TV and went to bed. He hadn't so much as put a hand on her in weeks. Barely kissed her in the mornings. It was just like it had been...before, only this time she didn't have the Inkpot to distract her.\n\n\"Well, it looks great. Thanks so much for inviting us.\"\n\nGinny found a smile. \"Glad you could make it. Help yourself to food, drink, whatever.\" She looked at Carson and Kelly. \"I have games set up in the den, if you want to play them. There are some other kids in there.\"\n\nWatching them scamper off as Kendra took the baby toward the buffet table, Ginny scanned the room for sight of her husband. He'd gone into the carport to put some beer in the giant tub of ice. She needed him to get some more napkins down from the high pantry shelf.\n\n\"Talk about a white Christmas.\" This was Peg, a plate of food in her hands and a set of festive reindeer antlers on her head.\n\n\"Are you kidding? It's been snowing since the end of November. I'll be surprised if we ever have spring.\"\n\n\"Pessimist.\" Peg bit into a sugar cookie. \"Ooh. Gran's recipe?\"\n\n\"Yes. Of course.\"\n\nHer sister laughed. \"You've become ridiculously domestic.\"\n\n\"Wow, and you look ridiculously festive.\" Ginny looked at her own maroon, dropped-waist dress. It had a white-lace collar. She felt ridiculous. Like a pregnant toddler.\n\n\"You look gorgeous,\" Peg said. \"That dress is the perfect color for you.\"\n\nGinny rolled her eyes. \"Yeah. Thanks.\"\n\nPeg laughed. \"Oh stop. Great party, by the way. Who are all these people?\"\n\n\"Beats me. Word must've spread around the neighborhood.\" Ginny waved a hand toward the dining room, where she'd set the table with the vast display of food. \"Neighbors. Some people Sean works with. You guys. Friends. You know, the usual Christmas open house crowd.\"\n\n\"Who knew you were so popular?\"\n\nGinny laughed and shifted, wishing she'd gone with her sneakers instead of these too-tight black flats. Her dogs were barking. \"Not me.\"\n\nThe door opened, bringing in a swirl of frigid air and whirlwind of pine needles from the decaying wreath on the front door. More people. She knew them. Louisa from the Inkpot, along with Tiffany, Michele and Becky. Ginny would've stepped back in surprise, but the wall was at her back and she had no place to go.\n\n\"I'll let you go play hostess. I'm going to find Dale,\" Peg said and abandoned her.\n\nNot that Ginny should've felt abandoned. These people had been her friends once, or at the very least acquaintances. Louisa, in fact, had made quite the effort to stay in touch. Guilt stabbed her. She'd never returned Louisa's calls or messages, and though she'd promised her that day in the grocery store, Ginny had never gone back to the Inkpot.\n\n\"Ginny! Oh my God, you look so great!\" Tiffany had a loud voice and a bright smile, and she was a hugger. She came for Ginny with both arms open wide, engulfing her before patting and making baby goo-goo noises at her belly. \"Congratulations!\"\n\n\"Thanks. Umm...\"\n\n\"Your husband invited us,\" Tiffany said. \"Came by the Inkpot with a cute, little printed-out invitation and said to make sure that the whole gang came. All of us.\"\n\nOf course Sean hadn't mentioned it. He probably thought he was doing her a favor. Ginny smiled and nodded, accepted their hugs and congratulations, directed them toward the food and drink and where to hang their coats. She moved to close the door behind the last person, but someone pushed it open from the other side.\n\nAnd there he was.\n\n\"Hi, Ginny,\" Jason said. \"Merry Christmas.\"\n\nThe world whirled out from beneath her, but Ginny didn't fall, and she didn't spin. She would never do that again for him. Never.\n\n\"I came with Becky,\" he said quietly. Standing a little too close. Voice a little too low so she had to strain to hear him. His gaze held hers a little too long. \"I didn't mean to surprise you.\"\n\nGinny blinked rapidly. She took a breath. \"Oh, Jason. Yes you did.\"\n\nThen she turned on the flat heel of her pinching shoe and went upstairs, leaving him behind. It was quiet up there, the sounds of the party far enough away to make it clear to her she had responsibilities as a hostess to get down there and make sure everyone had enough to eat, enough to drink. Enough to make merry.\n\n\"Fa la la la fucking la,\" she whispered with the bedroom door shut tight behind her back. She put her hands on her belly as the baby inside squirmed and kicked. Her heart hurt, and it hurt to breathe, but, no, she was not going to cry. She was not going to lose her shit here and now. Not tonight, not because of him.\n\nHe'd come with Becky. Of course he did. Becky had always had her sights set on him. Becky, with her low-cut shirts and tits hanging out, her tiny, tight ass and skinny jeans tucked into knee-high boots, her blonde hair hanging down from underneath her trendy little caps.\n\nGinny's jealousy was huge and ugly and unrepentant. It came with sharp teeth and jagged claws and venom, and it tore her up from the inside out. She panted with the effort of keeping her tears locked up tight, but her mouth opened in a silent, yawning scream she stopped by biting the meaty part of her palm. She closed her eyes and breathed.\n\nShe breathed.\n\nIn the darkness by her closet, something moved.\n\nWith the lights off, the push of air as whatever it was moved gave it away, rather than anything she could actually see. The only light came from beneath the door, and it crept only an inch or so along the floor around her feet. Then, the flash of something.\n\nEyes.\n\nNow was not the time to fuck with her. Ginny stepped forward. \"Who's there? What are you doing in here?\"\n\nAnd then, more softly, \"Caroline?\"\n\nIt moved again, dark on dark. A hint of swirling hair, an outstretched hand. Ginny reached suddenly frigid fingers into the darkness. The room had gone so cold she was sure that in the light she'd have been able to see her breath.\n\n\"Ginny?\" The door bumped into her, closing at once. She stepped away as Sean pushed it again. Light spilled into the room, bright enough to make her squint. \"Are you\u2014 Hey. What's wrong?\"\n\n\"Nothing. It's too hot downstairs. I came up to change.\" She tore at the hideous dress, moving toward the open closet door. She turned on the light, looked up and down, but whatever it had been was gone. If anything, it was colder in here than in the bedroom, and in a moment she saw why.\n\nThe cubbyhole door was open. Not a lot. Just a crack, but it was enough to let the tendrils of icy outside air seep into the closet. Ginny closed it, hard. She traced the outline of the door with her fingertips and turned to look at her husband who'd come in behind her.\n\n\"I'll be down in a few minutes. I just want to put on something more comfortable.\"\n\n\"I thought maybe you weren't feeling good. Peg said you just disappeared.\"\n\nGinny straightened, looking him in the eye, searching his face, that lovely face, for any sign. Anything at all, any glimmer or hint that he'd known what he was doing when he invited her \"friends\" from the Inkpot.\n\nBut no matter how well she'd always been able to read him, Sean had never been transparent. For all the times she'd spent wishing he knew her, there'd been an equal number in which she'd been unable to understand him. Her husband stared back at her, his expression concerned but otherwise implacable.\n\n\"No. Just too hot. And this dress, I never should've put it on. It makes me look like Laura Ashley puked all over me.\" She swiped at the collar and then the casual French twist of her hair.\n\n\"You look beautiful.\"\n\n\"I look enormous and blotchy and disgusting!\" Ginny shouted.\n\nThe words rang in the closet, softened only a little bit by the hanging clothes. They echoed. They stung.\n\n\"You look,\" her husband told her, \"beautiful.\"\n\nThen he backed out of the closet and left her there.\n\nShe was going out of her mind. That was all. The simplest and easiest explanation\u2014she was crazy. There was nobody in the room; there had been nobody in the room. There was no ghost haunting this house.\n\nBut there was a ghost haunting her.\n\nWith shaking, swift fingers, Ginny used the mirror over her dresser to put on some lipstick and swipe some shadow on her eyelids. She dabbed perfume on her wrists and at her throat\u2014lilac and vanilla, a special-blended scent she ordered from an online parfumerie. She put on a pair of black-velvet pants, not maternity but of material stretchy enough they still fit. Then a dark-green sweater from the back of her closet, also not maternity but cut with a swing to the hem that meant she could leave it untucked to her thighs and have it look like it was meant to be worn that way. She twisted her hair up again, but left a few loose curls to frame her face.\n\nThen she went downstairs.\n\nThe party had grown while she was gone. More neighbors, her brother and Jeannie, hesitantly reconciled. A few more people from Sean's office. It was too much to hope that Jason had gone. Ginny saw him standing next to Becky in the far corner of the living room, next to the Christmas tree. Becky was laughing, tossing her hair. Jason was looking everywhere but at her, both his hands full of crystal punch glasses.\n\nGinny moved through the guests and played hostess, ignoring them both.\nChapter Thirty-Two\n\nSince finding the collar, Ginny no longer feared the pantry. She wasn't tall enough to reach the napkins Sean had not yet managed to get down for her, though she'd reminded him a couple times. People were pulling paper towels off the holder, not that it was a big deal, but she was going to run out of those soon too. Now Ginny stretched up onto her tiptoes, reaching, her fingertips skating along the plastic-wrapped package of napkins she'd purchased specifically for the Christmas party because they had snowmen on them. Festive napkins, perfect for a party, except by the time she managed to get them down the party would be over and she'd be stuck using holiday napkins until the Fourth of July.\n\nThe door opened, then shut.\n\n\"I can get that for you.\"\n\nShe knew that voice without turning, and there was scarcely enough room for her to whirl around. So she didn't. With her back to him, Ginny said, \"I can get it.\"\n\n\"You can't. Let me.\"\n\nShe could feel him against her back, a heat she'd imagined a hundred times. No, a thousand. He didn't touch her, but then he didn't have to, did he?\n\n\"I can get it,\" she repeated firmly, voice neutral, like she was speaking to a stranger.\n\nJason cleared his throat. Ginny's fingers curled in the wire shelving in front of her. She willed herself to stay still as he reached over her to grab the napkins down and hand them to her. When it was done, he didn't move away.\n\nHe said her name in a low voice, full of longing. More heat. Her cheeks flushed with it, but this was not the fire of lust kindling in her breastbone.\n\nIn all the times she'd imagined seeing him again, all the ways she'd thought about how this would go, never in one of them had she faced him with her belly pushed out in front of her as proof she was tied to another man. She'd never pictured him in her house. Her own fucking house.\n\nGinny turned, the expanse of her body pushing him back a step. \"What are you doing here, Jason? Really. What the hell are you doing?\"\n\n\"I wanted to see you.\"\n\n\"It's been almost a year.\"\n\nHe nodded, his eyes fixed firmly on her face like he was afraid of looking anywhere else. He was looking at her, but not really seeing her. \"I know. But I wanted to see you, and when Becky asked me...I just...\"\n\nOne of the things she'd always liked best about him was the way they could talk to each other. Whatever else had failed, it had never been their words. Watching him now, his mouth working with nothing coherent coming out, Ginny wanted nothing more than for him to shut up before he ruined every memory she had of every conversation they'd ever shared.\n\n\"You shouldn't be here, Jason.\"\n\n\"I know.\"\n\nThey stared in silence beneath the bright, bare bulb of the pantry. It was unforgiving, that light, shadowing his face and make the lines of it harder than she remembered. It highlighted the glint of silver in his hair, the lines at the corners of his eyes and mouth. He'd aged since she'd seen him last. Well. She supposed she'd changed more than he had, and in a far less flattering manner.\n\nIncredibly, he moved a step closer. Once, standing this close would've sent electricity arcing between them, a palpable spark. Now, there was nothing for her except the slow and dull throb of the grief she'd tried her very best to shed. The anger, rising.\n\nJason's voice rasped when he spoke, \"I'm sorry, Ginny.\"\n\nSorry he wanted to see her? Or for that other thing she'd worried over and picked apart until it had left her raw? Or for something else she'd never figure out, she thought, looking him over. It didn't matter. It was over. That cake was baked.\n\nThat door was closed.\n\nShe couldn't squeeze by him. \"Move.\"\n\nJason backed up, hands out but not touching her, to let her pass. He snagged her sleeve before she could totally get away. She stopped, but didn't turn. There wasn't enough room.\n\n\"I'm sorry, Ginny.\"\n\nShe couldn't keep her voice steady, though she tried. \"You should have come to Philadelphia.\"\n\n\"I wanted to.\"\n\n\"Not enough.\" She shrugged to show him how little she cared, and it must've worked because from behind her, Jason let out a small sigh.\n\n\"Well...you look great. And...I guess I'm glad to see that you decided to...you know. Stay with your husband.\"\n\nAt this she turned, her incredulity passing in a heartbeat.\n\n\"Jason,\" Ginny said with something close to pity. Something more like condescension. \"I never, ever intended to leave my husband.\"\n\nThat was it. All she had to say. He let her go, and she took the napkins in her trembling hands, into the dining room, where she opened the package and filled the basket she'd placed there for just that purpose. She focused, focused, eyes blinking to keep them dry, swallowing hard against the tightness in her throat that could've been tears or a scream, she couldn't tell the difference.\n\n\"Great party.\" Becky lifted her glass of wine. \"And you, wow, Ginny. You look so...happy.\"\n\nThere might've been smug satisfaction on the woman's face, or it might've been Ginny's imagination. Either way, she straightened with a smile. \"Thanks.\"\n\n\"I mean it. You're really glowing.\" Was that something wistful in Becky's tone?\n\n\"It's hot in here.\" Ginny made a show of fanning her face. \"But thanks.\"\n\nBehind her, Sean appeared with a platter of mini corn dogs from the oven. He put them on the table with a nod toward Becky and looked at Ginny. \"I was going to get the napkins for you. But I saw you already got them.\"\n\nBefore she could answer, Kelly and Carson pounded through the dining room with Ginny's nephew Luke fast on their heels, though at eighteen, Luke was old enough to know better. She reached as he thundered past, and brought him up short.\n\n\"No running, it's too crowded in here.\"\n\nLuke looked sheepish. \"Sorry, Aunt Ginny. Just playing with the little kids.\"\n\n\"Play something else,\" she suggested, but not unkindly. \"I don't need my whole house torn apart.\"\n\nBecky was gone by the time she finished scolding, but Sean was still there. He watched Luke head out after the kids, then looked at her with an expression so odd her heart thumped in response. But if there was something he wanted to say, he wasn't going to do it there.\n\nThe party went on. Food eaten, beverages drunk. Some people danced in the living room. Some kissed beneath the mistletoe. The front porch became the haven for the smokers, including Jason, whose profile Ginny could see through the kitchen window when she filled the kettle for tea. Sean was there too.\n\n\"Kelly? Carson?\" Kendra came into the kitchen without her little baggage on her hip. She wore her coat and a worried expression. \"Ginny, hi. Have you seen my kids?\"\n\n\"I saw them earlier, playing with my nephew.\" She turned from the sink and put the kettle on the burner but didn't turn it on. A look at the clock told her it was late, probably past the kids' bedtime. Close to her own, actually.\n\n\"I can't find them.\" Kendra gave her a weary smile. \"I think they're playing hide and seek.\"\n\n\"Oh.\" Ginny chewed the inside of her cheek. \"Umm...well. Let's find Luke and see if he knows where they are.\"\n\nLuke had abandoned the play of little kids in favor of facing off with his dad in Guitar Hero. The rapt audience of freshman girls from down the street made the reason for his decision obvious; nevertheless, a thin, helpless bite of annoyance stung her.\n\n\"I don't know where they are, sorry,\" Luke told her with an apologetic shrug. \"I haven't seen them in a couple hours.\"\n\nIt wasn't like Kendra was the queen of keeping an eye on her children, Ginny thought uncharitably at the sight of Kendra's panicked expression. Still, having once misplaced Luke in a department store when she was supposed to be taking him to pick out a birthday present for Peg, Ginny knew the rush of panic. She put a hand on Kendra's arm.\n\n\"We'll find them. They have to be in the house somewhere. Right?\"\n\nKendra nodded, looking doubtful. She pulled a cell phone out of her pocket and pressed a few numbers. \"Clark. Are the kids there with you? No, not Carter, I know he's there. No, well...they didn't go with you? No,\" she snapped. \"They're not missing. They're just...they're here somewhere. I just wanted to make sure they hadn't gone with you before I started hollering for them. Yes, we'll be home soon. Just put him in the crib, for God's sakes! Give him a bottle, whatever!\"\n\nShe disconnected and gave Ginny an embarrassed shrug. \"You'd think he'd never put the kid to bed before.\"\n\n\"We'll find them,\" Ginny repeated so she didn't have to comment on Kendra's marital discord. God knows, Ginny shouldn't judge. \"Let's look upstairs.\"\n\nSean had come back inside the house, his nose and cheeks flushed pink with cold. He smelled of smoke. Ginny pulled her husband aside to explain the situation. He didn't get it, she saw that clearly enough.\n\n\"I'm going to take Kendra upstairs and check around. Keep an eye out down here.\"\n\n\"Carson!\" Kendra cried, because at that moment her son ambled out of the dining room with a cookie in each hand and chocolate around his mouth. \"Where have you been? I've been looking for you everywhere! Where's your sister?\"\n\n\"Don't know.\" Carson lifted a cookie to his mouth but didn't bite it. He looked suddenly alarmingly green. \"Don't feel good.\"\n\nOh hell no, Ginny thought. \"Bathroom's down the hall.\"\n\nGod help Kendra if her son puked all over Ginny's brand-new area rug, she thought, but Kendra at least had this part of parenting down. She hustled Carson down the hall so fast he dropped both cookies before he could toss them. The bathroom door slammed behind them.\n\nGinny looked at Sean. \"I'll go look upstairs.\"\n\nHe lifted his eyes to the ceiling at the sound of pattering footsteps. \"Someone's up there.\"\n\nRunning. Back and forth along the hall and in the baby's room. Also the library. Ginny and Sean tracked the noises with their eyes for a minute before she gestured at the fallen cookies. \"Can you get that? I'll be back.\"\n\nHe looked first surprised, then annoyed, but said nothing as he bent to clean up the crumbled mess. Ginny left him to it, not interested in an argument, though his response had rubbed along her skin like a shark's scales being stroked the wrong direction. She climbed the stairs with one hand on the railing to help heave her bulk along, and by the time she reached the top she had to stop and catch her breath.\n\nShe flicked the switch on the wall, but the ceiling fixture stayed dark. Colored light from the outside Christmas bulbs came in through the window facing the street, but it was filtered even further by the stained glass. Some other light drifted into the hall from the bathroom night-light, but that was it.\n\nShadows moved, whispering and giggling. The patter of feet slapped the floor beyond the railing across from her and slipped around the corner into the nursery. The door shut with a click, cutting off the childish laughter.\n\n\"Kelly?\" Ginny put her left hand on the railing, but didn't move toward the nursery.\n\nShe could go to the left, past the bathroom and her bedroom, to get there, or she could go to the right and pass the library. The door was closed there too. If Kelly came out of the nursery, she could jog either left or right and avoid Ginny altogether, depending which way she ran. Ginny didn't feel like chasing her, especially not in the dark. As she hesitated, her decision was made for her.\n\nThe door to the library opened, and Kelly tumbled into the hallway on a couple stumbling steps before she stopped with her back to Ginny.\n\n\"Come out, come out, wherever you are!\"\n\nGinny put out a hand to snag the girl's dress. \"I'm right here.\"\n\nKelly whirled with a shriek that startled Ginny into echoing it. Her scream scared Kelly into another wail, and the girl ran back into the library. Ginny managed to gather her wits, though, and pursued her. The light in this room worked, though both of them put their hands up against the sudden brightness when she turned it on.\n\n\"Enough,\" Ginny snapped, her patience worn thin by the unpleasantly behaved neighbor children. \"Kelly, stop it.\"\n\nKelly's screams trailed into a whimper. Her terror faded visibly, though at the sight of Ginny's face, her eyes widened and she looked like she might start to sob instead of scream. \"You scareded me!\"\n\n\"You scared me too. You shouldn't be running around up here in the dark, especially alone.\"\n\n\"I'm not alone. I'm playing with the little girl.\"\n\n\"Which little girl? From the party? What did she look like?\"\n\nKelly looked evasive. \"Umm...she has dark hair.\"\n\n\"What's her name?\"\n\n\"I don't know her name, I think it's Carrie.\"\n\nGinny froze. Her gaze narrowed as her heart set up a quickened, nervous thumping in her wrists and the base of her throat. \"It's not nice to tell stories that aren't true, Kelly.\"\n\n\"I'm not telling stories.\" Kelly looked chastened, but unrepentant. She scuffed at the floor with one of her patent-leather shoes. \"Carson quit with us because he wanted some more cookies, but she kept playing.\"\n\n\"Kelly,\" Ginny said warningly. \"Don't make me tell your mom that you were lying to me.\"\n\n\"But I'm not! I was playing with the little girl, I was. You can ask her yourself!\"\n\nGinny looked around the room, half expecting to see a floating ghostly form. \"Where is she?\"\n\nKelly looked shifty-eyed again and shuffled her feet. \"She probably ran away.\"\n\n\"Where would she go?\"\n\n\"I don't know.\" Kelly shrugged.\n\n\"She was in here with you? Playing hide and seek?\"\n\nKelly nodded, looking a little relieved that Ginny's tone had gone a little softer. \"Yeah, she was hiding in the cubby, and I found her, but she ran through.\"\n\n\"In the...\" Stunned, Ginny looked toward the small door set into the wall. There was another identical door in the nursery, but she hadn't known the storage space connected the rooms.\n\nIt made sense, of course, but another surge of irritation rose. The one peek she'd taken had shown her a slanted, narrow space with nails sticking through the roof and a gap in the floorboards under the eaves, stuffed with insulation. Not the place to be running and playing.\n\n\"Where is she now, Kelly?\"\n\n\"I don't know, I told you, she ran through...\"\n\nThe girl kept talking for a minute or so, but Ginny was looking past her to the easel and paints she'd set up weeks ago and had been ignoring ever since. The painting on the easel had only been a few strokes, just a couple blobs of color and some light pencil sketching. She hadn't touched it since then, or even looked at it. But one thing Ginny knew for sure\u2014she most definitely had not been finger-painting.\n\nAnger, real anger now, instead of just annoyance, bubbled out of her. Stalking past the sniffling child, Ginny looked at the painting. Jaw set, she glared at Kelly.\n\n\"What did you do?\"\n\n\"I didn't...I mean, me and Carson...and the little girl...\"\n\n\"Why would you touch this? What made you think this would be okay, to go into someone else's house and touch their stuff?\" Ginny snapped a hand toward the mess on her canvas.\n\nSomeone had streaked it with color in broad, thick lines. There was a pattern to it, but no shape, no form. It wasn't a painting that meant to look like anything specific, not a dog or horse or person. Not even a rainbow. If anything, it resembled some of the most offbeat pieces of modern art she'd ever studied, and Ginny loathed modern art.\n\n\"Answer me!\" She took a step toward Kelly, who must've rightly assumed it was threatening though Ginny hadn't so much as raised a hand.\n\n\"I t-t-told you, it wasn't me or Carson. It was the little girl, she painted it!\"\n\n\"Let me see your hands.\"\n\nKelly tucked them at once behind her.\n\nGinny's mouth twisted, but she kept herself from shouting. She took another step toward Kelly. \"Kelly, I don't want to have to tell your mommy you've been naughty.\"\n\nReluctantly, Kelly held out a hand.\n\nGinny snagged her wrist to study the girl's palm for telltale signs of paint. That was how Kendra found them.\n\n\"Kelly? What's going on?\"\n\n\"Mommy!\" Kelly screamed and ran into her mother's arms, burying her face against Kendra's stomach. Her shoulders shook with sobs.\n\nKendra gave Ginny a narrow-eyed, frowning glare. Peg had once revealed to her younger sister that no matter how terrible her children had behaved, if another adult was the one disciplining them, something protective and feral reared its head. Ginny saw it in Kendra's gaze now, the sharp glance at Ginny's hands and then how she took up her daughter's to go over the wrist. Checking for bruises, maybe.\n\nGinny didn't care. She'd seen the evidence on Kendra's hands. She kept her voice sickly sweet, though. Concerned, not accusatory. She felt more than able and happily willing to take on Mama Bear, but the tiny, still-rational part of her held her tongue.\n\n\"She got into my paints. Be careful she doesn't get it on your pretty blouse.\"\n\nKendra yanked Kelly's grasping hands away from her at that news, peering harder at her daughter's palms. She sagged, then looked at Ginny, embarrassed. \"Oh God. Kelly. What on earth?\"\n\n\"You should take her home,\" Ginny said firmly enough that Kendra blinked. \"Now.\"\n\n\"Right. Yeah, I'm sorry. I'm really sorry, Ginny.\" Kendra paused. \"I cleaned your bathroom for you, though. Carson's okay now, but...yeah, I took care of it.\"\n\nAs though Ginny should be, what, grateful? She smiled thinly but kept her voice as sweet as sugar cookies. \"Thanks for coming to the party.\"\n\nKendra nudged Kelly forward; the girl went as reluctantly as she'd offered Ginny her hand. \"Say thank you to Mrs. Bohn.\"\n\nKelly hesitated.\n\nGinny smiled at her. She couldn't really get down on her knees to be eye to eye, but she bent forward so the girl had no choice but to stare her in the face.\n\n\"Merry Christmas, Kelly.\"\n\n\"Thankyouforhavingus.\" Kelly'd rushed the words and ducked back behind her mother, who gave Ginny another apologetic look.\n\n\"Merry Christmas. Thanks for the party, it was really lovely. Umm...yeah.\" Kendra fumbled when Ginny didn't return the smile. \"We'll just...okay. Good night.\"\n\nIn the doorway, Kelly looked over her shoulder at Ginny. \"Merry Christmas, Carrie!\"\n\nKendra shushed her, shoving her along, but Ginny turned to look behind her. The cubby door was cracked open, just a little, and she couldn't remember if it had been that way before or if that squeak of Kelly's patent-leather shoes had masked the creak of it opening.\n\nGinny stared at it for a long time as the sounds of merriment drifted up from downstairs. It wouldn't take much for her to cross the room and open that door. At the very least, she should shut it tightly to prevent drafts.\n\nShe opened the door the whole way, her fingers reaching for the pull chain on the light she remembered was inside. It, like the light in the hall, didn't come on. Ginny muttered a curse for ancient light bulbs. She looked along the low space, into the darkness, forcing herself not to be uneasy. There was nothing in there but dust and the nose-tickling smell of pink insulation, and maybe a mouse or two that had been smart enough to avoid the bait left by Danny, the exterminator.\n\nExcept there was something back there, in the deep and shifting shadows. Something small and crouched. And something on the floor too\u2014spots of green, the same color that had been on Kelly's palms and splashed across Ginny's canvas. She looked hard into the darkness and willed her eyes to find a face, hands, toes, but the figure remained solid and lifeless, without a glimmer of eyes or teeth. It remained a nothing.\n\nShe leaned in just a little to take another look at the floor, the small curved imprint of a bare foot. Unmistakable as anything else, just this one, the others more spattered and uneven. She was reaching to touch the paint and check to see if it was wet, when someone called her name. Startled, she jumped and smacked her head on the inside of the doorframe. With a muffled curse, she backed out of the cubby.\n\n\"Your friends are getting ready to leave. I thought you'd want to say goodbye.\" Sean leaned in the library doorway. \"What are you doing?\"\n\n\"Oh. That little heathen from next door was messing with my stuff. She was inside here, running from room to room.\"\n\n\"In the crawl space?\" Sean came up behind her to look over her shoulder. \"In the dark?\"\n\n\"Apparently.\"\n\nHe snorted soft laughter. \"Brave kid.\"\n\n\"Stupid kid. She could've fallen through, broken a leg. Or worse. Not to mention that she got into my paints and...\" Ginny broke off with a sigh and a shake of her head. \"Never mind. Just remind me, next year, when I say I want to have a party...that I don't.\"\n\nHis laugh sounded a little more like his usual, this time. \"Right. You say that now, but...\"\n\n\"Next year we'll have a little one of our own. I probably won't even have time to plan a party.\"\n\nHe gave her a curious look and put a hand on her belly, rubbing. A tiny foot, a little hand, pressed against his touch. He put his other hand next to it, his strong fingers curling against her flesh. \"You'll want to see everyone.\"\n\nGinny put her hands over his. \"Honestly? The only people I need to see will be the ones I already see all the time. I don't need to host a big shindig and invite all kinds of people over and feed them and stuff.\"\n\nSean said nothing for a moment or two. Then he leaned to kiss her. \"Your friends,\" he reminded. \"They said they were leaving. I thought you'd want to say goodbye.\"\n\nGinny knew then whom he meant, and why exactly he'd gone to such lengths to invite them. It was a relief, in a way, not to wonder anymore if he knew. She said nothing at first, letting the sounds from downstairs drift up to them...but not come between them.\n\nFinally, she pushed away just enough to look at him. \"They can find their own way out.\"\n\nAfter a moment he nodded. Then he hugged her again, his hands moving in small circles over her back as her belly made a bridge between them.\n\nIt wasn't until later in bed, when she'd woken once again in the night with wide eyes and the idea that someone had spoken her name, that Ginny recognized what had been so odd about the paint in the crawl space. It had looked like the print of a bare foot, a child's foot. But not Kelly's she remembered, because Kelly had been wearing patent-leather shoes.\nChapter Thirty-Three\n\nMore snow. Inches of it on top of the mountains they already had. Sean had made it to work, even though the radio stations were all warning motorists to stay off the road. His class would be cancelled, or so she thought, but he'd told her he had to make it in or forfeit some of the paid time off he wanted to take after the baby came. She had to weigh the anxiety of imagining him swerving into a ditch against knowing how much help it would be having him home, and in the end Ginny'd had no choice anyway. If her husband said he was going to work, he was going to work. It was stupid to make them all come in when they could just close the offices. It served no purpose to have people risk their lives to come in and pound away on keyboards, entering data.\n\n\"Not all of us have the benefit of working for ourselves,\" he'd reminded her this morning, early, when she'd rolled over in bed with a groan at the sound of his alarm.\n\nHe hadn't pointed out that Ginny wasn't working and hadn't been for months. She said nothing about it either, just got up and made him breakfast while he showered and the lights flickered, and she hoped the power wouldn't go out again.\n\nIt did, of course, blipping on and off a few times before finally cutting out altogether. Ginny sighed, frustrated and thinking of the three loads of laundry she had yet to do, the full dishwasher. Her rumbling stomach and the soup she'd intended to heat up for her lunch.\n\nHer cell phone rang, Peg on the other end inviting her over for lunch and to watch movies. Peg's house had power. Peg had a four-wheel-drive vehicle and wasn't afraid of driving in the snow, because, as Peg said, she'd survived teaching six teenagers how to drive.\n\n\"I'll come pick you up. We'll hang out. I'll bring you home when Sean gets home, or he can stop here on his way home from work. I'll make goulash. It'll be good.\" Peg paused. \"I never see you anymore.\"\n\nThat wasn't true\u2014Peg had stopped by on her way to what she called the \"fancy\" grocery store on Ginny's side of town just two weeks ago.\n\n\"You're just not used to having all your kids out of the house, that's all. Makes you feel like you're missing stuff you aren't.\"\n\n\"Not true,\" Peg said. \"What, I can't miss my baby sister? And besides, you know...I worry about you, a little bit. In that big house alone.\"\n\nGinny was silent for a second or two. \"You don't have to worry about me.\"\n\n\"But I do.\"\n\nAbove her, the lights flickered, but immediately dimmed and went dark. Ginny sighed. \"Yes. Please come get me. I can't stand it here without power; it's so gray today I can't even get any decent light to read.\"\n\n\"I'll be there in half an hour.\"\n\nIt took forty minutes, actually, the roads worse than Peg would admit at first. Her SUV handled everything okay, but it was eerie riding the snow-covered streets, the wind so bad the traffic lights swung. Not many other people were on the road.\n\n\"The smart ones,\" Ginny teased.\n\n\"Yeah, yeah. I can take you home, let you sit in the dark eating cold cereal.\"\n\nBut Ginny knew Peg wouldn't do anything like that, even if they'd been closer to Ginny's house and it would've made sense for Peg to turn around. As it was, by the time they got into Peg's driveway, the wind had kicked up further and the snow had become so thick it turned the afternoon to dusk. Peg pulled into the garage and shook her head.\n\n\"I hope we don't lose electricity.\"\n\nGinny made a face. \"You'd better not. The only reason I came with you was for the food and the warmth.\"\n\n\"And the entertainment,\" Peg pointed out. \"C'mon. Let's get inside.\"\n\nIt hadn't been as long Peg seemed to think, but it had been long enough that nostalgia crept over Ginny as she sat at her sister's kitchen table with a mug of Peg's homemade cocoa in front of her, along with a plate of shortbread. Instead of watching movies, they played cards and laughed about old stories. They filled themselves with cocoa and cookies, and Peg never did get around to making the goulash, which was okay since every time Ginny called Sean's number she got a message saying his number was unavailable.\n\nShe called her house four or five times as the afternoon wore on, becoming night, but each time the phone rang and rang without the answering machine picking up. That meant the power was off. The sixth time, she was able to leave a message for Sean, saying she was on her way home and would be there soon.\n\n\"The power's back on. Can you give me a ride home?\"\n\nPeg looked at the clock. \"You're not staying for dinner?\"\n\n\"Couldn't get ahold of Sean,\" Ginny said, already gathering her coat. Her back ached from sitting in the hard kitchen chair, but she wasn't going to tell her sister that. \"He should be almost home by now. At least the lights will be on for him.\"\n\n* * * * *\n\nAll the lights were on, blazing forth from every window, when Ginny got home. She waved as Peg backed out of the driveway. The snow had tapered off drastically an hour or so before stopping, so the roads were passable.\n\nHumming, pleased with the afternoon despite how it had begun, Ginny let herself in the front door. Sean, who seemed to come from nowhere, his face a strained mask and his hair askew, swooped down on her immediately.\n\n\"Where the hell were you?\"\n\nTaken aback, Ginny shrugged out of her coat and hung it on the coatrack by the front door before looking at him. \"I was with Peg. I left you a message.\"\n\n\"I didn't get any messages!\"\n\n\"I tried your phone,\" she said patiently, biting back her own instant anger that had been jerked upright by his tone. \"I kept getting an error message. But I left one on the answering machine.\"\n\nShe saw at once by his expression that he hadn't even checked, but before she could point that out or make an explanation, he'd launched into her again.\n\n\"God dammit, Ginny! You pissed and moaned about me going in to work today because you said the roads were too bad, then I come home to find you gone, every freaking light in the house on...your car still in the driveway. What the hell was I supposed to think? You couldn't leave me a goddamned note?\"\n\nOh, how tempted she was to snap back at him, to curl her lip. Instead, she pushed past him and down the hall to the kitchen, keeping her pace steady. Not running, not giving him that effort. In the kitchen, she looked at the answering machine, blinking merrily with one message. Hers, it had to be.\n\nShe turned as he came in after her. \"I did leave you a message. It's not my fault you didn't listen to it. And I told you, your cell phone\u2014\"\n\n\"There's nothing wrong with my fucking phone.\"\n\nGinny blinked, slowly. She took a step back from him. \"Are you calling me a liar?\"\n\nHis shoulders heaved and his fists clenched, lightly, but clenched just the same. He said nothing. Her gut twisted. Her heart became stone. She had a thousand words to say to him, perhaps many more than that, but she bit them all back, chewed and swallowed them. Ginny shoved them deep inside her to join everything else she'd pushed down.\n\nSean scowled. \"I came home and you were gone. All the lights were on, and you weren't here. No note, nothing.\"\n\nHer chin lifted, just a little. \"So...what? You thought I just ran off and left you? With all the lights on? Without my car?\"\n\n\"Someone could have come for you.\"\n\n\"Someone did come for me,\" she told him. \"My sister.\"\n\nSean shook his head, just barely.\n\nGinny sighed and shifted to ease the ache in her back. Her belly rippled with Braxton Hicks and the baby's kicking. She winced as she tucked her hands beneath it, wishing she could lift it in a sling and somehow relieve at least some of the weight.\n\nShe pointed at the blinking answering machine. She sounded weary, because that was how she felt. \"I left you a message. I'm sorry you were worried.\"\n\n\"I thought you'd gone off with him.\"\n\nShe'd had a hard-enough time catching her breath lately, but at this every molecule of oxygen left her. She sagged, fingers clutching at the countertop. Her tongue had gone thick, her lips numb. \"Oh, Sean. No.\"\n\n\"I thought, well, this is it. She's gone. She finally did it. But the least...\" his voice broke, and she saw with growing alarm that her stoic husband was close to tears, \"...the least she could fucking do is leave a note!\"\n\n\"I went to Peg's, that's it!\" Her own voice rose, razor-edged with hysteria. \"I didn't go off anywhere. I wouldn't. Sean, I\u2014\"\n\nHe turned from her when she tried to touch him, and Ginny let her hand fall to her side. She had no right to force him to let her soothe him. She had no right to anything, really, and she would take none.\n\n\"Just tell me something, because it's been driving me crazy thinking about it.\"\n\nShe swallowed hard, prepared to give him all the details, the few there were. \"What?\"\n\nSean scraped at his eyes with the fingertips of one hand, the other's fingers tucked into his belt loop on his hip like he was afraid what it would do if he didn't keep it tethered. \"Did you love him?\"\n\nIt wasn't the question she'd expected, but as soon as he asked, she knew she'd been foolish to think he'd ask anything else. \"No.\"\n\nHe looked at her, his expression horridly naked. \"Then why?\"\n\n\"Because you wouldn't touch me,\" she told him simply. She spread her fingers and gave a half shrug, her words the truth, with nothing to redeem them but that. \"You stopped hugging me, kissing me, touching me. You stopped all of it, Sean. It was like living with my brother...no, worse than that, because I've never doubted, even when he was being mean to me, that my brother loved me. And for a long time, with you, I wasn't sure.\"\n\n\"I never stopped loving you,\" he said hoarsely.\n\nShe hated the sound of tears in his voice. She hated that she'd done this, broken him somehow. Tearing off the scab was supposed to help an infected wound leak its poison, but this...she'd never wanted this.\n\n\"I wanted my husband there for me. I needed you. And you just...went away.\" She wanted to touch him, but mindful of his last reaction, kept her hands at her sides. \"I know you were grieving. But you wouldn't talk to me.\"\n\n\"I was. I couldn't bear it, how much you'd gone through. How I almost lost you too.\" He shook his head and began to pace. \"Seeing you in the hospital, knowing I'd done it to you...\"\n\n\"You didn't do anything to me.\" She wanted to empathize with him, or at the very least find an edge of sympathy for his agony, but all she heard was him burdening himself with what had been her pain. Making what happened to her, somehow...his, not even theirs. But just his.\n\nTypical, she thought. Turning the loss of their baby somehow, into his private torture. That he'd somehow been responsible, or that he could've done anything differently. It was typical of his need to fix things, regardless of what had broken them or how much worse his efforts at tinkering made them.\n\n\"I got pregnant. We both wanted kids. Something was wrong. We lost the baby.\" She took a deep breath, hating that she had to be the one, once again, to walk him through this. \"It happens to a lot of people. We're far from the only ones. And it was a blessing, I think, because we weren't prepared to have a child with special needs, or even to lose a child after it was born.\"\n\nHe shuddered, still pacing. Every so often his hand crept up to tug through his hair. \"I couldn't stand to think about it happening again.\"\n\n\"So...you just stopped...\" she shook her head, trying to piece this together, to make sense of it, \"...you stopped touching me.\"\n\n\"I had to.\"\n\n\"Despite what the nuns might've told you in the third grade, Sean, hugging doesn't lead to pregnancy.\"\n\nHe stopped his pacing, thank God. He looked at her. \"You just...you were at me. All the time.\"\n\nThis was not how she remembered it, not at all. Ginny remembered needing to hold him and be held. To grieve together. And later, when her body had healed enough for it, to make love to him again and find that closeness that had always been so much a part of their marriage.\n\n\"I was at you?\" She tried hard to think on it, to recall if she'd been some sort of vamping siren, skimping around in lingerie or accosting him in dark corners. \"You think wanting to have sex with my husband is...what...wrong? Weird? Abnormal?\"\n\n\"You were at me about having another baby.\"\n\nWith the counter behind her, Ginny couldn't move backwards. Her fingers slipped and gripped on the laminate. One nail bent and broke. Wincing, she held it in front of her to watch a small bead of blood form beneath the nail she pulled off and tossed into the sink drain. She sucked at the blood, and it was only half as bitter as all the things she wanted to say.\n\n\"And I couldn't do it,\" Sean said. \"I just...couldn't.\"\n\nAgain, Ginny racked her brain to think if she'd been the one pushing for another child. All she could think of was telling him, at the time, it would all be okay. They could try again. They could always try again. She remembered saying it from the hospital bed. She remembered saying it in the shower with the hot water pounding over her aching body as the blood leaked down her legs and into the drain. She'd said it to comfort him.\n\n\"I thought you wanted a baby,\" she told him and reeled with the idea that all this time she'd been trying for something he didn't even want.\n\n\"Not enough to risk losing you.\"\n\nShe'd been wrong to think Sean had never feared losing anyone, she saw that now. But, try as she might, she couldn't make herself sympathize with his choices.\n\nHe started pacing again.\n\nThis time, she reached out and grabbed his sleeve, hard enough to stop him. \"Stand still. Look at me.\"\n\nShe waited until he did, his gaze naked and horrid and hard to face, but she forced herself to do it. \"I am fine. This baby is fine. Nothing is going to happen to this baby, or to me. You're not going to lose me.\"\n\nHis sharp bark of laughter surprised her enough to let go of his sleeve. Her husband's mouth twisted\u2014not quite a smile, not quite a sneer. He closed his eyes for a long couple of seconds, and when he looked at her again, his gaze was flat and hard and assessing.\n\n\"I already did.\" He gestured at her belly. His shoulders sagged. He let himself sink into one of the kitchen chairs and buried his face in his hands, squeezing his head like a man in the grip of a migraine. \"Didn't I?\"\n\nShe sat across from him and took his hands, forcing him to let her hold them. She squeezed his fingers. He didn't look at her, but that made it easier and she was grateful for that.\n\n\"No. Never.\" There'd been times when Ginny was uncertain of what would happen and where her path would lead, but of this she had never been unsure. \"You did not lose me.\"\n\nHe looked up at her. \"I didn't want to watch you go through that again, and I was right, wasn't I?\"\n\nDisturbed, she squeezed his fingers again. \"No. You'll feel differently when the baby comes. You'll see, it will be all right, and we'll be okay\u2014\"\n\n\"I don't know if I can,\" he said quietly.\n\nHer heart started breaking. \"You don't know if you can what?\"\n\n\"Love it. Be a father to it. I figured I'd try, you know? I'd do my best. But I'm not sure I can.\"\n\n\"Oh, Sean. Of course you can. You'll love this baby. Lots of people are scared about becoming parents, but...you'll see. You'll be an amazing father.\"\n\n\"But I won't,\" he told her steadily, \"be the father.\"\n\n\"Of course you\u2014\" He wasn't talking about being afraid. Being incompetent. \"What? Oh\u2014 What the hell?\"\n\nShe snatched her hands from his. It would've been her turn to pace, if she'd been able to push herself away from the table, to get her bulk moving quickly enough. Instead, she sat, trapped by her inertia.\n\n\"I wondered if you were going to tell him,\" he continued conversationally, like they were discussing the day's mail. \"But he didn't seem to know. I saw him look at you at the Christmas party when they came in. He was surprised. Why didn't you tell him?\"\n\n\"Because it was none of his business,\" Ginny snapped. Her heart raced. Her stomach churned. She thought she might vomit or pass out; her hands trembled. \"Sean, you are the father of this baby.\"\n\nHe shook his head, pushing away from the table, and she grabbed for him, snagged his sleeve but couldn't keep him there. He jerked from her touch and stood so fast the chair tipped over behind him. She jumped at the crash and put her hands on her ears for a moment. She closed her eyes.\n\n\"You are the father of this baby,\" she told him, not wanting to see him not believing her. \"I swear to you.\"\n\n\"I can't be.\"\n\n\"You are!\" Ginny shouted, glaring at him.\n\nThey stared at each other. She shuddered. He backed up to the doorway as though he meant to flee, but stayed.\n\n\"I can't be,\" he said again. \"I had a vasectomy.\"\n\nGinny tried to draw a breath and could find nothing but dust in her throat and lungs. She choked on it, gasping. This was...this was...\n\nShe became aware of him shaking her. Her head lolled. Sean cradled her, slapping lightly at her cheeks until she shoved him away from her. She struggled to her feet, facing off with him.\n\n\"What in the actual fuck do you mean, you had a vasectomy?\"\n\nHe looked as she imagined she had when confronted with the acknowledgment of her infidelity. Guilty and somehow defiant too. The world tipped, but she refused to slide with it.\n\n\"I couldn't take the chance of getting you pregnant again. I couldn't.\" His tone had turned pleading, and he held out his hands to her.\n\nShe refused to take them. \"I can't believe you'd do something like that without even telling me!\"\n\n\"I knew you'd say no.\"\n\nShe spit to the side, her disgust so thick it was like some living thing she needed to eject. \"You should have told me. We should have talked about it. We could've talked about lots of things!\"\n\n\"I knew you wanted a baby\u2014\"\n\n\"No!\" The word slashed at her throat. \"You wanted to fix the situation, Sean, the way you always do. You wanted to fix it so you didn't have to worry about me getting pregnant, so you didn't have to worry about anything. It was all about you, like it always is!\"\n\n\"That's not true!\"\n\nShe shook her head and made a shoving gesture with her hands, though she was nowhere near him. \"When? When did you do it?\"\n\n\"The weekend you were gone to the art show.\"\n\n\"In Philly.\" She laughed dully. \"Oh Christ, Sean. Jesus Christ.\"\n\n\"I know you were with him. And it was almost eight months ago. You do the math.\"\n\n\"You don't know shit,\" she told him flatly, her rage undiminished, but finding herself incapable of raising her voice. \"And you do the math, you son of a bitch. I came home from Philly and we made love. Remember? After we went to that Italian place. You couldn't even wait until we got upstairs. We did it in the living room. Remember that?\"\n\nThe sex had been some of the best they'd ever had, and, now, even the memory of that pleasure was destined to become pain. Ginny pressed her hand to her breast, pushing on her heart, willing it not to burst. She swallowed around the lump in her throat.\n\nWithout looking at him, she shook her head. \"You know it takes a few weeks after a vasectomy to be completely sterile, right? They did tell you that, didn't they?\"\n\nHe didn't answer at first, and when she forced herself to look at him, she saw another combination of guilt and defiance.\n\n\"Did you even read the paperwork or pay any sort of attention, Sean? Did you? No,\" she said before he could, \"because you never do. You never pay any fucking attention!\"\n\n\"I went back a week later, like they said. I was all good.\"\n\n\"Yeah, well, we fucked before that, didn't we?\" She straightened. \"This is your baby, and you want to know how I'm so sure?\"\n\nHe said nothing.\n\n\"I didn't fuck Jason in Philadelphia, that's how I know. In fact,\" she added as though it were a casual admission, when it was anything but, \"I never slept with him. Not once. I never even kissed him, so how's that for being absolutely sure he's not the one who knocked me up?\"\n\nSean blinked. She held up a hand to stop him from speaking. His mouth opened, then closed.\n\n\"I did not have sex with him. Never. But I did have sex with you and, vasectomy or not, you are the one who fathered this child. So I would suggest you pull up your big-boy pants and start dealing with it. Because this is...this is how it is,\" she gasped, breaking down at last. \"This is how it fucking is!\"\n\nSomehow, he was holding her, rocking and stroking her back. She didn't want to be in his arms, but didn't have the energy to push him away. He comforted her, but it was his own comfort he sought, and Ginny was just too damned tired to stop him.\n\n\"But I thought...\" Sean murmured. \"I mean, I saw some of your emails...\"\n\nShe didn't have the strength to even confront him about his snooping; at any rate, she'd sent the emails and had been wrong to do it. She always knew she would own up to what she'd done, if she had to. But that was before she found out about Sean's betrayal, which was somehow bigger than what she'd done. Or maybe it wasn't, maybe it only felt that way to her and would be the opposite to him. It didn't matter.\n\n\"You didn't sleep with him,\" he said happily, as though that made everything okay. \"Good. Oh good.\"\n\nShe pushed away from him and went on clumsy feet to the sink to splash some cold water on her face and the back of her neck.\n\n\"I'm glad,\" Sean said from behind her. \"God, Ginny. You don't know how glad I am.\"\n\n\"Well,\" she said coldly without turning, \"don't be that fucking happy. I didn't sleep with him, but I wanted to. And I was going to. He didn't show. That's the only reason I didn't.\"\n\nThe lights went out again.\nChapter Thirty-Four\n\nInstant darkness fell over them.\n\nGinny heard the scrape of a chair on the linoleum and Sean's muttered curse. The table bumped, the glass salt and pepper shakers in the center of it rattling. Sean cursed again.\n\n\"Don't move,\" she said crossly. \"You're going to get hurt.\"\n\n\"Where's the flashlight?\"\n\n\"I don't know. It should have turned on when the power went out.\"\n\n\"Did you use it today?\"\n\nShe could sense where this argument was going and her answer came out clipped, \"No. I didn't have to. It was daylight when Peg came for me.\"\n\n\"Are you sure you didn't use it and then not put it back?\"\n\nThat he could accuse of her this, of all things, while they stood together in the dark, after having the worst fight of their marriage...Ginny lost it. She started to laugh, at first low and then louder. Her laughter cycled up and up, becoming a series of hiccuping, frantic guffaws that hurt her throat.\n\n\"Jesus, Ginny! Stop it!\" His hand passed by her close enough to brush her sleeve, but he missed her.\n\n\"I didn't use it!\" she shouted into the shadows. \"You're the one who used it last. Remember? You took it downstairs to check the fuse box the last time it blew. That's the last time I saw it. You left it down there, probably.\"\n\n\"Well shit.\" Sean sounded miserable.\n\nShe didn't want to take glee in it, but she did. If she'd been able to caper with it, she would've. She wasn't proud of that, but it was the truth. The full and awful truth. She was glad he'd been the one to lose the flashlight, that his accusation had bounced back and hit him in the face.\n\n\"You left it down there,\" she crowed. More laughter, this time cut off with her hand over her mouth because the sound of it disturbed her. It would've been better to vomit than keep laughing that way.\n\n\"I'd better go get it.\"\n\n\"How are you going to do that in the dark?\" she said derisively. Disgusted by this, by everything. And so suddenly tired all she wanted to do was lie down and close her eyes and maybe not wake up.\n\n\"Let me use your phone.\"\n\nThe baby moved inside her, a reminder of why Ginny couldn't give in to selfishness. She sighed and fished in her pocket for her phone. When she thumbed the screen, the dull blue gleam provided at least a little illumination. A thought snagged her; she pulled her hand back before he could take the phone.\n\n\"Where's yours?\" A beat of silence proved her right again. Ginny sighed, no longer gleeful. No longer glad to be proven right. \"You lost it?\"\n\n\"I dropped it,\" Sean said. \"I think I did something to the battery. It won't turn on.\"\n\n\"But there's nothing wrong with your goddamned phone,\" she quoted. \"Wow.\"\n\n\"Just give me yours. I'll go get the flashlight and we'll figure out what to do next.\" Sean took the phone from her limp fingers. \"You stay here. Don't move.\"\n\nGinny said nothing. She watched the blue light move out of the kitchen. She heard the basement door open, then the creak of Sean's feet on the stairs.\n\nShe sat in the darkness.\n\nShe sat for a long time.\n\nShe wasn't sure when the tears began, only that they started in silence. They burned in her eyes and slid down her cheeks. She tasted salt. She put her face in her hands and sobbed, shoulders heaving, body racking.\n\nIn the darkness, Ginny broke.\n\nAnd in the darkness, a small hand touched her.\n\nShe didn't startle from it, because somehow she'd been expecting this to happen. Small fingers curled over her shoulder, then slipped down her arm. A small hand held hers. The fingers were cold and slender, the nails ragged when they pressed lightly into Ginny's palm.\n\n\"Carrie?\" Ginny whispered, and got no answer but a squeeze.\n\nThen the hand withdrew as the sound of Sean's feet came up the stairs. The cellar door opened with a familiar creak, and in the next moment the white glare of the flashlight cut through the dark and pierced her eyes. Shadows danced behind her.\n\nSomething crashed.\n\n\"Ginny?\"\n\n\"I'm right here.\"\n\n\"I told you to stay put.\" Sean shone the light around the kitchen.\n\nGinny put up a hand to block the glare, but not before she saw a glass of water she'd left on the edge of the counter had fallen and smashed on the floor. She twisted in her chair, but whatever had knocked it off the counter had disappeared. She closed her fingers over the residual feeling of that cold touch.\n\n\"You could've cut yourself,\" Sean said.\n\n\"I didn't do that.\" Outside, the wind howled and a spatter of snow hit the windows over the sink.\n\nIt startled Sean, who crunched glass under his feet. He muttered another invective and swept the light over her. \"It just fell?\"\n\nGinny shrugged.\n\n\"Don't tell me it was a ghost.\"\n\nShe said nothing, told him nothing. His shoes crunched more glass and he swept the light around the room again. He grunted.\n\nThe lights came on as all the appliances beeped. Sean clicked off the flashlight. Ginny didn't move.\n\n\"It's broken,\" Sean said unnecessarily, and she didn't know if he meant the glass or everything else.\nChapter Thirty-Five\n\n\"Mr. Miller?\"\n\nThe man getting ready to cross the street paused, brow furrowed as he tightened his scarf around his neck. \"Yes?\"\n\nGinny had been good at her job, back when she did it. Good at getting people to talk to her without alarming them. Something in her face, she thought. Something pleasant and deceptively innocent. She wasn't investigating Brendan Miller for insurance fraud, so, really, he had nothing to fear from her, but he looked at her warily anyway.\n\n\"I'm Ginny Bohn.\"\n\nIt took him a second or two, but he figured it out. From his expression, Ginny thought he might bolt, and what would she do then? Waddle after him? The thought was laughable.\n\nHis gaze fell to the bump of her belly beneath her puffy coat. \"If there's a problem with the house, you need to talk to the realtor about it. We signed papers; you took it as is...\"\n\n\"It's not about the house. Well. It sort of is about the house.\" She stepped to the side, in front of him, when he moved to go around her. She'd stopped working because she was pregnant, but now realized something\u2014he might've shoved her aside if not for her belly. She held up the train case. \"I have this.\"\n\nMiller stopped trying to get around her. \"What is it?\"\n\n\"It belonged to your sister.\"\n\nThe cold wind had rubbed two pink spots in his cheeks. Now they grew brighter. His mouth worked for a few seconds before he actually spoke. \"Impossible.\"\n\n\"I found it in the house. I thought you might want it.\"\n\nHe fixed her with an angry, sullen glare. \"I don't want it. I don't want anything to do with anything from that house. Okay?\"\n\n\"Okay. Okay.\" She knew how to soothe, how to gentle. Like taming a skittish horse or a strange, tooth-baring dog. \"I'm sorry. I just thought you might like to read it.\"\n\n\"Read it?\"\n\n\"There's a diary,\" Ginny told him. \"At least, I think that's what it is. It has your sister's initials on it. Look, can I buy you a cup of coffee or something? Get in out of the cold?\"\n\nShe pointed down the street. \"My treat.\"\n\nFor a moment, she thought he'd say yes. She was so certain, in fact, that she'd already taken a few steps in that direction. But Miller shook his head and backed up.\n\n\"No. I don't think so.\"\n\nGinny paused. There was an art to working people, getting them to agree to things they didn't want to do, or to admit to what they'd prefer to keep a secret. \"I know what happened to your sister.\"\n\nIt was the wrong thing to say. His gaze flickered. His mouth thinned. \"Nobody knows what happened to my sister.\"\n\nThis time, he did push past her. Ginny managed to snag his sleeve, not hard enough to stop him but enough to give him pause. \"There's something in the house, Mr. Miller.\"\n\nHe'd just stepped off the curb, and stood between two parked cars. Everything about him vibrated the urgency to flee across the street. He looked at her over his shoulder.\n\n\"I told you. You bought it as is. If you have a problem, talk to your realtor.\" Then he stepped out into the street.\n\n\"I think it's your sister!\"\n\nThat stopped him. He turned. He wasn't much older than Ginny, though the lines on his face and gray in his hair made him seem so. He was good-looking, behind the scowl and the furrowed brow, but there was nothing welcoming about him. Not at all.\n\nHis gaze dropped to her belly. \"What is it that you want, exactly?\"\n\n\"I just want to know what happened. That's all.\"\n\nShe'd gotten to him. Everything about him sagged. He put a hand over his eyes, briefly, before looking down the street to the coffee shop.\n\n\"Fine. I'll give you half an hour, but that's it.\"\nChapter Thirty-Six\n\n\"She was always scribbling.\" Miller looked at the case Ginny had pushed across the table in front of him, but he didn't touch it. \"My dad called her Doodle. She'd draw for hours in these big notepads, that soft white paper...what was it called? Newsprint. She'd draw cartoons and landscapes and people, animals, just whatever. And when she got older, she'd write stories to go with them.\"\n\n\"Did she keep a diary?\"\n\nHe shrugged. \"I don't know. I didn't pay attention.\"\n\nSomething in this sentence seemed to break him again, because he put his elbow on the table and his face into the comfort of his hand. Ginny had learned the trick of seeing when someone was faking pain, but everything about this guy's anguish was real. Her fingers crept over the table toward him, but she stopped herself from touching his sleeve. He didn't seem the sort to welcome it.\n\n\"I didn't pay attention,\" he repeated. \"I didn't want to know. I didn't want to see.\"\n\n\"See what?\" Ginny asked gently.\n\n\"My mother didn't either, though that's not really an excuse.\" He shuddered. His fingers dug into his skin. His hand covered his eyes, so she couldn't see if he was crying, but his voice had gone rough and hoarse.\n\nGinny was silent, giving him time.\n\nWhen he finally looked at her, his eyes were rimmed red but dry. \"She was always his favorite. I was the son, right? He should've been taking me out to play ball in the backyard, toss around the pigskin. He should've been taking me fishing, camping, all that shit. But nope, Caroline was his favorite. They'd spend hours tucked up together in his big chair, reading books or looking at those stories she wrote. She could do anything and get away with it. Once she carved her initials in the dining room table, can you believe it? I thought my mom was going to murder her, but my dad just told her to leave it alone. Caroline was a daddy's girl, all the way. My mother called her 'the little wife.' Like she was joking, but I don't think she was.\"\n\nA worm of nausea twisted inside her. Ginny's fingernails scraped at the table as she withdrew her hand. She linked her fingers, squeezing them tightly together on the table in front of her.\n\n\"Yeah,\" Miller said, though Ginny hadn't spoken. \"You know what I'm talking about. It was cute when she was little, right? But not when she started to get boo...breasts. She shot up a few inches, started to get curves and wear makeup.\"\n\n\"How old was she?\"\n\n\"Thirteen. And, Christ...\" He pressed the heel of his hand to his forehead for a moment, then shook his head. \"I have two daughters. Seventeen and fourteen. I know what it's like, you know, when they start developing. It gets weird. They get weird.\"\n\nDistaste rippled across his face, and he gave her a look of such loathing that she'd have recoiled if it had been directed at her instead of his memories.\n\n\"You have to make some distance between you. You just have to. They're not your little girls anymore, you can't be having them sit on your lap and whisper in your ears. It's just...that's not what a dad does.\" His lips drew back to reveal perfectly straight but yellowed teeth. \"I have daughters. I know.\"\n\nGinny had a flash of a summer night years ago. She'd been what, twelve? Thirteen? She'd worn a thin, sleeveless nightgown with flowers on it, and it had shrunk in the wash. More likely, she'd grown. It hit her midthigh instead of reaching her knees, and the fabric had grown faded in the wash. It had been her favorite for a long time. She'd come out of the shower, her hair still wet, and gone to sit with her dad on the couch to catch a rerun of some comedy program. He'd been annoyed, told her to go change her clothes or stay in her room. She hadn't understood then and maybe not ever, until just now.\n\n\"What did your mother do?\"\n\nMiller grimaced and looked down at his hands, which had linked much the same as Ginny's. \"She had her favorite too, I guess. Even if by default.\"\n\n\"And then your sister disappeared.\"\n\n\"She never came home from school.\" He kept his voice low and his gaze on his hands. His fingers twisted tighter, knuckles going white with the pressure. \"She went in the morning and stayed all day. Her friend Laura was the last person to see her, and she was crossing the park, heading for home. She never made it.\"\n\n\"And they never found out what happened?\"\n\nAnother shrug. He looked out the window, toward the street. \"No. Someone reported they saw her getting into a white van, but, really, that's such a clich\u00e9; who knows if it's true. We had some other reports of her being spotted in different places across the country. California, Montana. None of them ever panned out. She was just...gone.\"\n\n\"What did your parents do?\"\n\nHe looked at her. His smile was terrible, thin-lipped and more like a snarl. \"They fell apart. They'd been on rocky ground for a while before that.\"\n\n\"Fighting a lot?\"\n\nHe shook his head. \"Oh no. They never fought, at least not where I heard them. No. They stopped talking to each other before Caroline went missing. After she was gone for a while and the news started to taper off and the investigation went cold, they stopped even seeing each other. It was like they were invisible to each other. They'd pass in the hall, not even look at each other.\"\n\n\"It must've been horrible for you. I'm sorry.\"\n\nHis laughter was worse than the terrible smile. \"It could've been worse. I could've been in the back of someone's van, right?\"\n\n\"How old were you?\"\n\n\"Sixteen.\" The cup in front of him had stopped steaming, and now he drank. \"We stayed there for another year after she went missing. We never talked about her. I mean, I tried, but I wasn't allowed to.\"\n\n\"They told you that?\"\n\n\"They didn't have to. I wasn't stupid, and I wasn't heartless. I could see it upset them. So it was sort of like...I'd never had a sister in the first place. They took down her pictures. They put away her things. I wanted to save some of them...I tried.\"\n\nHe cleared his throat once, then again. He looked at her, his gaze naked. \"She always wanted a cat, but my dad hated them. Said he was allergic, but he wasn't. He just didn't like them. It was maybe the one thing he didn't give her that she wanted, you know? But he let her have hamsters instead. A whole bunch of them, they just kept having babies. Well, after she was gone, nobody remembered to take care of the hamsters. They started eating each other.\"\n\nGinny swallowed, thinking of the plastic bags in the closet. The bones. The fluffs of fur. \"Oh.\"\n\n\"Yeah.\" Brendan Miller gave a short, sharp laugh. \"I didn't want them to take them away, though. I kept thinking, hey, maybe she'll come back. If she ran away, right? She'll come back for them. She loved those stupid hamsters. So I took them, and I hid them. But they died anyway.\"\n\nShe didn't have to ask what he'd done with them; she already knew. But she didn't know why. \"You hid them in the closet?\"\n\nHe nodded, looking guilty, then lifting his chin as though daring her to accuse him of something. What, she didn't know. \"Yeah. I was a kid. A dumb kid. I thought maybe, even though they were dead, she'd want them. So I put them away for her until she could come back.\"\n\n\"But she never came back.\"\n\n\"No. Then my mom left him, and I went with her.\"\n\nGinny's cup was long empty and her bladder had begun its protests. \"And you never went back? Or had anything to do with your dad ever again?\"\n\nMiller shook his head. \"No. And that bastard...he never...\"\n\nShe waited for him to compose himself, embarrassed not just by his story but by the somehow confessional nature of it. She shifted in her chair, needing the bathroom and not wanting to get up before he was finished. Miller swallowed hard and gave her another of those bright, hard stares.\n\n\"He never bothered trying to see me, I mean. We left, he never bothered or cared about us after that. He sent checks on time; that was all my mother cared about. And that was it.\" He drew in a shaky breath and let it out. \"Like I said. He had his favorite, and it wasn't me.\"\n\nBefore Ginny could say something, even to offer sympathy, Miller had pushed the train case back toward her.\n\n\"So I don't really care to read my sister's journal or whatever it is. Throw it in the trash. It's old news. She's gone, and she's never ever coming back.\"\n\n\"You're sure of that?\"\n\nHe shrugged. Then nodded. Stood. He tossed a couple dollar bills on the table, though she'd offered to pay for the coffee he'd barely touched. Miller stared down at her.\n\n\"I'm sure of it. All those stories about seeing her in California or wherever are bullshit, and you think so too, or else you wouldn't have come after me so hard. You think my sister's haunting your house, Mrs. Bohn? Well. You might be right. But, guess what, I don't care. And I don't want to know about it. Never contact me again, or I'll have no problem going to the authorities. And if you find anything else...\" he paused to give the box a sneer, \"...consider it yours. I don't want it back.\"\n\nShe didn't want to shout after him, but it took her two or three steps for every one of his just so she could catch up. Ignoring her suddenly violent urge to pee, Ginny caught him again at the curb. \"You think your father did something to her, don't you?\"\n\nMiller gave her the side eye. \"I don't think it. I know he did.\"\n\nThen he stepped off the curb and crossed the street, leaving her behind.\nChapter Thirty-Seven\n\nThis was it. She was going to open it and read it. Ginny plucked the key from beneath the stack of faded pictures, and fit it into the lock. She turned it. She opened Caroline Miller's diary and started to read.\n\nFour pages of random thoughts scrawled in an uneven hand. Nothing important, no revelations. Then...nothing. Blank pages.\n\n\"Wow. So that's kind of a letdown, huh?\" Sean leaned over her shoulder. \"Bummer.\"\n\nGinny had told him the entire story, come clean about everything. The things she'd seen and heard and felt and thought. The time for secrets had ended, for both of them. Sean didn't believe her about there being a ghost in the house, but he did believe that Ginny thought it was true. That was enough for her; it had to be.\n\nIn the past, Ginny hadn't asked him to quit smoking or sell his motorcycle. He'd decided those things on his own. She didn't ask him to stay, either. She hadn't asked Sean if he'd thought of leaving her, but she'd faced the idea of leaving him and hadn't done it; now she waited. Some things that were broken could be repaired, good as new. Others, even if they worked again, would always bear evidence of the damage that had been done. That was their marriage. Broken, repaired, working somehow. But not unscathed. Perhaps someday they'd both stop tiptoeing around each other, or maybe they'd always hold what each had done between them for the rest of their lives. The baby would make a difference. Or not.\n\n\"I love you\" was all she'd said, over and over, as Sean paced and ran his hands through his hair and when he'd asked her if it was his fault, any of this. \"I love you,\" she'd told him. \"That's what matters.\"\n\nShe wasn't sure if he believed in her love, any more than he believed in the ghost of Caroline Miller, but it seemed as though he believed Ginny believed in it. And that also had to be enough.\n\nNow, he squeezed her shoulders before withdrawing. \"I'm sorry. I know you hoped it would give you some clues. Or something.\"\n\n\"It's okay. I'm not sure I'd have learned anything I didn't already know. Or couldn't guess.\" Ginny closed the diary and put it back into the train case, along with the pictures and all the other bits and pieces of things that had once belonged to Caroline Miller. She looked at him. \"And maybe...maybe someone in a white van really did carry her off. Or maybe she's okay.\"\n\n\"We'll just never know,\" Sean said.\n\nAnd maybe it was better that way.\nChapter Thirty-Eight\n\nThe worst winter in twenty years was turning into the worst spring. While usually an early thaw would've been welcome, this year the massive amounts of snow that had accumulated were melting too fast for the earth to handle it. Water rushed day and night, down the street and into the drains, which overflowed and made lakes across the sidewalks and into the yards. Kids sloshed in it and ran paper boats down it.\n\nGinny looked for Pennywise, the clown.\n\nNot really, though she did avoid the drains when she went out to get the mail. Debris had clogged many of them, making the problem worse. So did the gray skies dumping more rain every day. She was sick of the sound it made on the roof. Sick of sloshing. Sick of everything being damp.\n\nShe was sick of waiting, waiting, for things to change.\n\nShe could see the star of her due date on the calendar now. The obstetrician told her she could go early. She could go late. This wasn't her first pregnancy, but it was her first to go to term and there was no predicting what her body was going to do. The baby would come when it was ready, the doctor told her. That was the way babies worked.\n\nFrom the nursery window, Ginny watched Kelly and Carson duck through the hedge and into her yard. The snow had gone, replaced by mud and the grass that had been overlong before winter came. They ran through it in their rain boots and slickers, slashing at the overgrowth with long sticks. Then back through the hedge, widening the hole for a moment before the brush closed behind them again.\n\nThe rocking chair Sean had struggled so hard to put together still squeaked, but rather than finding this annoying, Ginny took comfort in the consistent noise. She closed her eyes and rocked in the nursery she'd finally allowed herself to decorate, both hands on her big belly. She listened to the squeak and the creak of wood on wood.\n\nShe listened for the sound of footsteps.\n\nSomething like a sigh brushed past her on a swirl of cool air. They'd turned off the heat at the beginning of last week, and even in an unseasonably warm late February, it was not yet time for air-conditioning. She wanted to open the windows at least, but the rain prevented it. Instead, Sean had turned on the house fan.\n\nGinny rocked and breathed. Another whisper, another sigh, the soft pad of footsteps in the hallway. If she opened her eyes, she'd see nothing but shadows. Ghosts never showed themselves in the daytime. But she felt a presence, eyes watching. She felt a hopefulness. It was the only way to describe it.\n\nSo, she sang.\n\nHer gran had sung this lullaby to her when she was small. Like the best of Grimm's fairy tales, the song about a pair of children lost in the woods had delighted her childhood love of all things macabre, and it wasn't until Ginny got older that the horror of the lyrics had become clear. She sang it now anyway, to the baby in her womb and for those she'd lost.\n\nFor the one who seemed unable to leave.\n\nIn the yard, Kelly and Carson chased each other again. Their screams snapped open Ginny's eyes. She sighed herself, annoyed that they couldn't seem to stay in their own yard, that they had to be so loud. Her child would never, she thought mildly. Never, never.\n\nThey disappeared around the side of the house, heading back toward their yard again. Ginny's eyes drooped. She drifted. She'd been sleeping better at night, but now it seemed all she wanted to do was sleep. Her body's way of getting ready, she supposed.\n\nSleeping, she dreamed.\n\n* * * * *\n\nShe stood next to the crib Sean had so painstakingly put together. It had been dressed in pale-green-and-yellow linens decorated with jungle animals. She recognized it as the pattern someone had given her at her baby shower\u2014though, as was the way of dreams, she couldn't remember who'd given it. Cartoon lions and elephants and monkeys capered next to palm trees all over the sheets and bumpers. Over the crib hung a mobile, and it spun lightly and tinkled out a classical tune she couldn't recall the name of.\n\nThere was nothing scary in this dream, but a sense of unease twisted inside her, getting strong as she crossed the room to look into the empty crib. She touched the mobile with one finger to stop its turning, and the music ground to a sudden, unlovely halt. One, two, final notes plucked at the air and fell silent.\n\nGauzy curtains blew inward, reminding her again this wasn't real. She'd never have such long curtains. They were dangerous. Could tangle around a curious toddler's neck. And ugly, she thought. Pink and purple, sheer, tab tops.\n\nThese were not her curtains.\n\nAh, this, then, was not her baby's room. Ginny turned in half a circle, looking around. Again, in the way of dreams, she looked down at her flat belly and mourned the loss of it, even as she knew it had to mean she'd given birth. That meant there was a baby here, somewhere. Not in this room, which clearly belonged to an older child. A girl who favored pink and purple, a white-painted bed hung with netting Ginny would also never allow because it could strangle a child in the night.\n\nEverywhere she looked, she saw danger. A tangle of too many electric cords plugged into a socket. A bookcase laden with toys and books, unsecured to the wall, just begging to be toppled over onto a soft head. In the closet, a pull string, too long, and an empty socket, tempting a child's curious fingers to probe. Too many things that could cause harm. Her heart pounded. She clutched at the air, but touched nothing.\n\nSomewhere, far away, her baby cried for her. Ginny's nipples got tight and hard, burning. She covered them with her palms and felt the sharp points, but the pressure didn't relieve the tingling and hot fluid leaked through her shirt. An answering pull tugged in her womb, deep inside. Like menstrual cramps, but harder and more painful.\n\nShe turned around and around, but this room had no door. She was trapped here, and she ran her hands over the walls, now a garish pink. Like the inside of an organ. Like a wound. The steady thud-thud of a heartbeat throbbed in her ears as the floor beneath her shifted and pulsed.\n\nThe crying didn't stop. Desperate, Ginny pushed at the walls, feeling for the door or a window, and found nothing. She turned around and around. She punched at the walls and felt an answering pulse of pain deep inside her. She kicked, she punched, she swore and got nothing but more pain for her efforts.\n\nGinny quieted. Listening. The baby\u2014her baby, she had no doubts\u2014still wailed and wept. Ginny pressed her fingers to her forehead. This was a dream. She knew it, yet couldn't quite control it.\n\nShe was sitting in a rocking chair by the window but couldn't wake herself. She tried, with a leap and a howl, but the room stayed the same. She heard the creaking of the rocker on the wood and imagined her foot pressing. Releasing. Her breathing, soft and slow. But she could not wake herself.\n\nThe baby was still crying.\n\nThere was a door in this room, she remembered that. A small door, child-sized. It went from one room to the next. She didn't need to get out another way, if she could get through there...and suddenly it was there, that little door. That tunnel strewn with mouse shit and insulation, the hazard of nails poking through the roof, ready to score her scalp and make her bleed.\n\nThen she was in the library. She still heard the baby, though it was farther away now. Harder to hear.\n\nGinny's baby was crying, it was somewhere in the house, it was lost and she had to find it. Ginny had lost her baby. She'd lost her baby. She had lost her baby, and it was crying, it was screaming for her, it was crying and there was a pounding...a pounding...\n\nA pounding at the front door, and the bell rang. Steadily, insistently demanding she wake up, wake up, wake up.\n\n* * * * *\n\nGinny woke, startled, gasping, her hands pushing out instinctively to ward off the touch of whoever was shaking her awake. She was alone. She blinked and ran her tongue over her sleep-sticky teeth. She'd been asleep in the rocking chair, her head drooping, and now her neck hurt.\n\nShe rubbed it as she listened. No baby crying, but the doorbell rang again. Then a moment after it. Then there was a knocking.\n\nWith a sigh, she heaved herself up from the chair, ready to pound whoever was banging so fiercely. It had better be a special delivery package of something expensive, she thought as she lumbered down the stairs and shuffled to the front door. But it wasn't.\n\nIt was Kendra, hair wild, eyes wide. \"Ginny. I knew you were home, I saw your car.\"\n\n\"I was sleeping.\" It was a pointed response, but Kendra didn't get it.\n\n\"Are Kelly and Carson over here?\"\n\nGinny stifled a yawn with the back of her hand, noticing that Kendra looked soaked. Her jeans were black to the knees from wet, her shoes sloshing. She had no umbrella and her pale-pink T-shirt had gone semitransparent. She needed a sweatshirt.\n\n\"No.\"\n\n\"No?\" Kendra looked past her into the hall. \"Are you sure?\"\n\nAnnoyed, Ginny leaned against the doorway to relieve some of the pressure on her back. Her entire pelvis ached, along with her hips. Her joints were separating, according to the doctor. She was lucky she could stand upright at this point, that's how it felt.\n\n\"Yeah. I'm sure.\" Ginny paused. \"I saw them earlier, in the backyard.\"\n\nThis was also a pointed statement Kendra didn't seem to get. She shook her head, water splattering. \"I told them to stay in the yard. Not to come over here. But they just really like your yard, I guess...\"\n\n\"I guess they do, but they're not inside, I'm sure I'd have noticed that. Did you go around the back of the house and check? Maybe they're peeking in my windows again.\"\n\n\"They're not there.\" Kendra heaved a breath and burst into tears.\n\nGinny watched her impassively for a moment. Surely this was still a dream. Part of a nightmare that was worse than the one she'd been having. But it got worse, because Kendra let out a low, sobbing breath.\n\n\"OHMYGODTHECREEKTHEYWENTDOWNTOTHECREEK!\"\n\nThe words came out in a rush, one butted up against the other so that Ginny needed a half minute to parse out what the other woman had said\u2014but when she did, her resentment and snark fled as swiftly as a hare chased by nipping hounds. She reached for Kendra and found her wrists. Tugged them.\n\n\"Stop.\" She sounded calmer than she felt. \"Stop, we'll find them. We will go look, okay? I'm sure they're fine.\"\n\nKendra, panicked, shook her head so fast her hair stuck and clung to her cheeks. She moaned, a brittle, fragile noise that set the hair on the back of Ginny's neck standing upright. She bore down on Kendra's wrists, pinching the skin and maybe even crunching the bones. Kendra cried out, but looked at her.\n\n\"We will find them. Okay? Come on. Let's go look.\"\n\nKendra resisted. \"Noooooo...\"\n\nGinny wanted to slap her. \"If they're really down there, don't you think you'd better haul your ass and get to them?\"\n\n\"What if they are there?\" Kendra cried through chattering teeth. She jerked free of Ginny's grip and crossed her arms over her stomach, bending as though she might puke. \"What if they are there, and we find them?\"\n\nGinny stepped out of the way, no patience left for this woman whose terror she could empathize with, but whose uselessness she could not. \"We want to find them\u2014\"\n\nKendra shook her head again and looked at Ginny with wide but somehow sightless eyes. \"WHAT IF THEY'RE DEAD?\"\n\n\"They aren't dead,\" Ginny said solidly and looked past Kendra to the pouring rain. \"I'm sure they're wet and cold and scared of getting into trouble, but you won't ever know if you don't move your ass.\"\n\nKendra still didn't move, not until Ginny grabbed up her coat from the rack and pushed past her. She'd forgotten boots and had none anyway, her feet too swollen for anything but these slippers. Sodden grass squelched under her feet as she stepped off the porch and sidewalk, with Kendra finally following. They should've gone through the house and out the back, but too late now. Ginny hunched against the rain, which sent chill, dripping fingers trickling down the back of her neck.\n\nThe yard sloped gently but in her ungainly state she needed support and grabbed Kendra's arm to get it. Fortunately the woman didn't let her topple. She gripped Ginny by the arm, and then with a handful of her jacket in the back. She barely kept them both upright when Ginny slipped in the mud, though. The sudden shift in her weight sent pain rippling through her thighs and deep inside.\n\nGinny groaned. \"Oh God. Kendra, seriously, I'm like eleven months pregnant here; you're going to have to go down the bank yourself.\"\n\nThe creek had already risen up and over, flooding the lower yard. Ginny hadn't realized how close to the house it had crept, even in the few short hours since last she'd looked out the window. The grass floated like seaweed in the cold brown water. Rippling waves pushed some debris, newspapers or junk mail, it looked like, against Ginny's calf and she wrinkled her nose in disgust.\n\n\"I can't!\"\n\nGinny stood as straight as she could and backhanded Kendra across the face. The slap wasn't nearly as hard as she'd meant it to be, but it still cracked Kendra's cheek and left a red imprint.\n\nKendra gasped and put a hand to her face.\n\n\"Look at me!\" Ginny demanded. \"Should I be out here going after your kids? No! What the hell is wrong with you? Get your ass down there and look for them!\"\n\nKendra stepped back, and Ginny wavered in the squishy muck. More rippling water washed over her legs, splashing the hem of her nightgown. She looked to where the creek had once burbled and chuckled along over smoothly polished stones, as cheerful and dangerous as a fluffy kitten. It rushed and roared now.\n\nJust across it, in a pond like the one in which Ginny and Kendra stood, was a figure in a red hooded coat. It turned at the sound of Ginny's shouts, and Kendra let out a wail of relief.\n\n\"Kelly! Kelly! Where's Carson?\"\n\nHe was there too, a little farther on. Both children waved at their mother, who set off at a run through the overflowing creek toward them. She went down in a minute as her foot plunged into a hole or something. Kendra face-planted into the water and flailed.\n\nGinny wanted to laugh in that horrid way she always wanted to chuckle when someone fell, not because she really thought it was funny, but because there was no helping it. She cut it off into a strangled yelp and moved a step or two forward. She stopped herself from going farther.\n\nShe was pregnant, for God's sake, and already this was a bad idea. Kendra had fallen into the water, and Ginny didn't want to do the same. She'd never get up. She'd drown in her own backyard.\n\nBesides, Kendra had made it to her feet, soaked and muddy, but seemingly unscathed. Until she started to scream, that was. She screamed and flailed some more, backpedaling and falling again. She rolled, desperate to get up and out, while the kids stared at her with goggle eyes. Kelly got too close to the edge, where the water was rushing instead of just rippling, and her feet got taken out from under her.\n\nGinny watched in horror as Kelly went under, nothing but the red hood showing. In seconds she was downstream a few feet and sputtering up out of the water, but she couldn't get to her feet. Kendra screamed. Carson had put his hands over his face.\n\nGinny was moving without thinking, one hand on her belly and the other held out to balance herself like a tightrope walker with a pole. She slipped in the soggy grass but made it to water up to her knees before Kelly went under again. The girl had snagged on a tree branch that had fallen into the water, and it was the only thing keeping her from being pushed farther downstream. It was also pressing her under the water.\n\n\"OHMYGODMYBABY!\" Kendra shrieked.\n\nGinny was actually closer to the girl. Her slippers had come off. Her toes dug into the mud and grass, which gave her a better grip. Just a foot or so in front of her was where the creek normally ran, and she had no idea how deep it was. Just that it was fast. Beside her was a part of another branch that had fallen during one of the snowstorms, thick and heavy enough that she had to strain to lift it.\n\n\"Help me!\" she ordered Kendra.\n\nMiraculously, the other woman moved without fuss. She grabbed the end of the branch and lifted it as Ginny guided it toward Kelly. The girl grabbed at it. Missed. Grabbed again.\n\nThis time, she caught it. Her mother yanked it, moving Kelly just a few feet closer so she could grab the red coat.\n\nIt was over in minutes. Ginny found herself clutching a sobbing and shaking Kelly, both of them higher up on the lawn and away from the creek, while Kendra waded across using the branches and grabbed up Carson. By the time she got back to Ginny's yard, all of them had blue lips.\n\nBut they were alive.\n\n\"Oh my God, oh my God!\" Kendra clutched at both kids. \"What were you doing! What were you thinking?\"\n\n\"We...we...we wanted to see the...\" Carson could barely speak through his chattering teeth.\n\nWhatever it was he'd wanted to see, his mother wasn't interested. She shook him, then grabbed at his sister and started hauling them away from the water.\n\nGinny, exhausted and shivering, followed. Every part of her ached, and all she wanted was a hot shower, clean clothes and the comfort of her bed. Instead, she slipped on the sopping grass and went to her knees. Her fingers dug into the soft mud.\n\nAhead of her Kendra wasn't even looking back as she bustled her kids home. The three of them were sobbing and screaming. Ginny thought she should be offended her neighbor wasn't even bothering to look back and check on her, but that would mean she had to deal with Kendra and her two spawn. And didn't she understand, anyway, the force that moved a mother to forget everything else but her children?\n\nThe baby moved inside her, protesting the position and kicking out so hard the little feet stole Ginny's breath. She gasped, then coughed as she pushed herself along the muddy slope and struggled to get herself upright. Her fingers dug again into the soil and grass and slipped, scraping along a tree branch. Or something. Not a tree branch.\n\nGinny dug a little deeper and came up with something shorter than the length of her forearm. Knobbed on the end. It must once have been white, but time in the earth had turned it dark. She cradled it against her for a moment and looked behind her at the rushing water that had eroded so much of her yard. Then at the thing in her hand. She knew what had so captivated Kelly's and Carson's attentions, what they'd been looking at.\n\nIt was a bone.\nChapter Thirty-Nine\n\nShe'd showered and changed her clothes and tied her wet hair up on top of her head, but that only left the back of her neck exposed and vulnerable. Sean's fingers squeezed, squeezed her there. He meant the touch to be gentle and soothing, but every press against her sent a ripple of irritation down her spine, until finally she shrugged her shoulders to squirm away.\n\nThe officer who'd come to the house was young and fresh scrubbed. His uniform had been crisp and polished when he came to the door, but now shoes were spattered with mud and his trouser legs, sodden. He'd gone down to the creek, but in the dark and with the rain coming down more heavily than before, he'd come back in the house.\n\n\"How long will it take?\" Ginny asked. \"Until you know if it's human?\"\n\nThe officer looked uncomfortable. \"I'm not really sure. I've never had to deal with anything like this before. It could be a...week? Maybe?\"\n\n\"A week?\" Ginny sounded as outraged as she felt and turned away to keep him from seeing her face.\n\n\"Well...forgive me, but it's not like it's an emergency,\" the officer said quietly. \"We'll do the best we can and let you know, okay? In the meantime, if you find anything else, you let us know.\"\n\n\"We're not going to find anything else.\" Sean said this to soothe her, not the cop. He rubbed her shoulders again. \"Thanks for coming out, Officer.\"\n\nWhen he'd gone, and they were in bed, trying to sleep, Sean kissed her shoulder. \"It's going to be okay, Ginny. All of this will be okay. I promise you.\"\n\nShe knew about his promises, and wanted to believe him. Or at least that he meant it. But how could he promise her something like that?\n\n\"Maybe now you can just forget about all this stuff.\" Sean's voice rose beside her in the dark. He sounded sleepy.\n\nGinny was anything but tired. She breathed in the scent of lavender, faint now but still lovely. She needed to buy some more before the baby came. She had to do a lot of things before the baby came.\n\n\"It's over,\" Sean said. \"You have to know that, right?\"\n\nSean loved horror movies as much as she did, maybe more. He had to know that finding the source didn't make the ghosts go away. Not if you hadn't yet figured out what they wanted. Ginny breathed. She breathed.\n\nThe bed dipped as he rolled toward her. His hand rested on her hip, then slipped around to caress her belly. His heat covered her back as he nuzzled at her shoulder.\n\n\"You don't believe me,\" she said. \"Even after what we found.\"\n\n\"We don't know yet what you found.\"\n\n\"Bones, Sean.\"\n\n\"They could be anything. Animals. Someone's pet cemetery, probably.\"\n\nGinny shivered, though her flannel pj's, the blankets and her husband's warmth had made her anything but cold. \"They're not animal bones.\"\n\nSean sighed. \"Honey...even if they're not...\"\n\n\"What?\" She'd have turned to face him, but her body was too unwieldy. \"If they're not...what?\"\n\n\"Even if they're not animal bones, they're not hers.\"\n\n\"And that's supposed to reassure me?\" Ginny huffed and tossed his arm off her so she could heave herself upright. The floor felt very far away when she sought it with her toes. \"Jesus, Sean. If they're not Caroline's bones, whose are they? Do you think that makes me feel better?\"\n\nHe shifted to sit next to her, his legs long enough that his feet had no trouble finding the floor. He took her hand. \"I just think you've let yourself get too worked up about this Caroline Miller business. You said yourself there were reports of her being sighted in California and Idaho and a whole bunch of places.\"\n\n\"Her own brother thinks something happened to Caroline here, in this house. Something her dad did. Well, I think her dad killed her.\"\n\n\"And buried her down by the creek.\"\n\n\"Yes.\" Ginny shook her head. Her bladder panged. She might as well get up and go to the bathroom. She couldn't sleep anyway, and she'd be up in an hour, even if she peed now.\n\n\"That bone was small. It was really small.\"\n\nA Braxton Hicks contraction rippled across her belly, and she bit back a gasp as she put a hand on the bedpost to keep herself from doubling over. It wasn't painful, just surprising. Her fingers gripped the wood.\n\n\"It didn't belong to a fourteen-year-old girl. That's all I'm saying.\"\n\nGinny put her other hand on the lump of her belly. The baby was quiet, not moving. Did it dream, she wondered suddenly. Or did it share her dreams the way it shared what she ate?\n\n\"It was a baby's bones.\"\n\n\"We don't know that either.\"\n\nBut she knew it. She felt it. She rubbed her belly, but the baby didn't move. When was the last time she'd felt it move? She'd lost track. So close to the end now, it seemed the baby was always squirming, never still.\n\n\"I know it, Sean. I know it.\"\n\nShe didn't realize she was weeping until he pulled her close to stroke her hair and his T-shirt felt damp under her cheek. Ginny knuckled her eyes, pushing hard enough to make brightness bloom behind her eyelids. She drew one hitching breath after another but couldn't seem to get enough air. She rocked, and he rocked with her.\n\n\"It's a baby. I know it's a baby...\"\n\nHe hushed her, but she couldn't be soothed. Tiny bones in the closet, bigger bones in the yard. So much death in this house, and the baby hadn't moved, hadn't moved in hours.\n\nShe realized she'd said it aloud when he answered her.\n\n\"Does it hurt anywhere?\" Sean rubbed her back. \"Ginny. Are you...bleeding?\"\n\nShe shook her head, then wondered if he could see her in the dark or just feel the motion of her body. \"No. It's not that.\"\n\nShe'd spent the last few months telling him over and over that this was all going to be fine, but no matter how much she'd pummeled him, Sean had never quite believed it. Had she been wrong, and he right? Her hands moved over her belly again and again, seeking any motion beyond the tightening of her muscles.\n\n\"It doesn't hurt. They're not regular. It's Braxton Hicks, not regular contractions.\"\n\n\"It's too early for you to be in labor.\"\n\n\"I'm not in labor, Sean.\" Ginny swallowed snot and salt; the taste nauseated her.\n\n\"Should we call the clinic?\"\n\nAt two in the morning, calling the doctor for a little hysteria seemed unreasonable, unless you were a pregnant woman nearly due to give birth, one who'd found bones in her backyard.\n\n\"I'm just upset. Give it a few minutes.\"\n\nTogether, they waited in the dark, their hands linked, but the cramps and contractions eased.\n\n\"It's going to be okay, Ginny. I promise you,\" Sean said again as morning light eased through the windows.\n\nShe didn't believe him, but she believed he thought it was true, and this also had to be enough.\nChapter Forty\n\nGinny didn't want to look out into the backyard. Caution tape sectioned off the lower portion of the yard where the creek still ran deep and fast. The rain hadn't stopped, either. She didn't want to look out and see the churned mud and torn-up grass, or all the places the investigating team had dug. It was only a yard, the damage could be repaired.\n\nAnd they'd found nothing.\n\nThey were going to keep looking. Moving downstream, as well as starting farther upstream. There wouldn't be a single infant bone buried in her yard; that just didn't make sense. But it could've come from anywhere, not her yard, or its companions might've been swept away.\n\nIf only the rain would stop. Off and on for the past week and a half, sometimes a steady downpour and others a faint drizzle, the water refused to sink into the ground already oversaturated from the melting snow. Everything was wet. Nothing would dry.\n\nGinny didn't really want the tea she'd made\u2014even thinking about drinking it made her feel like she had to pee. The never-ending patter of rain on the roof didn't help. Her back hurt. Her feet hurt. All her joints ached and creaked. Sitting for too long made her rear end numb, but lounging on the couch left her too lethargic to finish all the tasks that had suddenly become so important. The books called it \"nesting,\" and Peg laughed when Ginny told her on the phone how she'd gotten the overwhelming urge to run every single washable item in the house through the laundry.\n\nGinny hadn't laughed. The need to make everything clean for the arrival of her baby was pathological and embarrassing, but she was also helpless against it. The problem had become lugging the baskets up and down the stairs, difficult enough earlier, before she'd become so huge and clumsy. Now impossible.\n\nShe comforted herself with folding and refolding the tiny shirts and pajamas. With stacking diapers and packages of wipes. She arranged and rearranged the plastic shelves full of baby items and organized the selection of stuffed toys on the wooden bookcase. She rocked in the chair, which, with its hard wooden back, should've been uncomfortable, but which offered her just the right amount of support.\n\nShe rocked a lot.\n\nShe dreamed.\n\nShe woke to darkness, sorry at once she'd allowed herself to nap in the chair instead of in bed or at least on the couch. Her head spun for a second or two when she stood, gripping the arm of the chair, and blinking to orient herself. The night-light in the hall reminded her where she was, but for a good few minutes she had to think about it.\n\nThe nursery. She was in the nursery. In her house. Sean was gone; he'd be home later, after work, after class. Later. Everything would be later.\n\nDiscomfort rippled across her lower belly. She stood for a moment, legs planted slightly more than shoulder-width apart. Something twinged inside her, and she winced.\n\nShe needed a drink. Her mouth had gone dry. Her stomach was a little upset, and the thought of a cold glass of ginger ale was all at once the most perfect thing she could imagine.\n\nShe shuffled toward the door out of habit and was glad for it when her foot kicked something small and metal. There was no way she could bend to pick it up, but she heard it skitter across the floor and land somewhere close to the soft throw rug. When she turned on the light switch near the door, the lights seemed to take forever to turn on. Those stupid fluorescent bulbs she hated. Too dim at first to see what she'd kicked, and she didn't want to wait until they got bright enough.\n\nIt was a button. Struggling, she managed finally to grab it and tucked it in her pocket. By the time she got downstairs, she had to pee. Again. The sound of rain had infiltrated everything, so that the splash of urine in the bowl was drowned out. She flushed, washed her hands, and in the kitchen poured herself a tall glass of ginger ale. The first sip went down easy, cool and sparkling. The second tasted bitter. She poured the rest down the sink.\n\nFrom the basement came the juddering sound of the sump pump going. The floor rumbled a little under her feet. Ginny put her hand on the counter for a moment, feeling off-balance. Her hair fell in her face, and she brushed it away.\n\nThe phone rang, and she thought about ignoring it, but answered anyway. \"Hello.\"\n\n\"Hey, sweetheart. It's me.\"\n\nThe sound of Sean's voice relieved her more than she knew she needed to be relieved. \"When are you coming home?\"\n\n\"Soon. The professor was late, and he's going over stuff that will be on the final. I was going to duck out early, but...\"\n\n\"No. You shouldn't miss it.\" This class was breaking him. He needed all the help he could get. Ginny took a deep breath, but her lungs barely filled. Too much baby, not enough room for anything else.\n\n\"What's going on there? You okay?\"\n\n\"It's still raining. And it's dark,\" she added.\n\nHe laughed. \"Turn the lights on?\"\n\n\"I don't know, I kind of like it in the dark. It's peaceful, with the sound of the rain.\"\n\n\"I thought you'd be sick of that by now. I am. How's the sump pump?\" Sean coughed lightly, then murmured something to someone on the other end of the line. \"Sorry, honey, I have to go. The break's over.\"\n\n\"Go, go.\" Ginny made a flapping gesture with her hand, though of course he couldn't see it. \"I'm fine. Everything's okay here. So long as the power stays on.\"\n\n\"Right.\" Sean paused. \"Shit. I should come home.\"\n\nShe laughed. \"No, no. I'm fine. Really. Be careful coming home, the rain's horrible.\"\n\n\"And you...the baby? You're still okay?\"\n\nHis question annoyed her less now that she was so close to the end, because he was asking something she could answer with full confidence and have him believe her.\n\n\"A few Braxton Hicks. That's all. I love you. Be careful,\" she added, trying to get him before he could disconnect, but he already had.\n\nThe floor vibrated again as the sump pump went off. Ginny peeked out the back door at the glitter of rain slanting down. The entire backyard had become a swamp. They wouldn't have to bother going to the beach this summer, they could simply pitch an umbrella off their back patio.\n\nHer stomach had settled a bit, which was good, though now the first tingles of heartburn were tickling the base of her throat. Ginny took a handful of saltines and a glass of milk with her upstairs to the library, where she picked up the book she'd been trying to finish for the past few weeks. There was nothing wrong with the book, it was all her. Easily distracted and having difficulty focusing. She thumbed the pages and settled her milk on the small wooden table she'd set up next to the chaise lounge. Then she tucked herself under the knitted throw and tried to concentrate.\n\nThe problem was the rain. Constant, ceaseless, drumming. She'd turned the heat up and now the room had become almost stifling. Ginny couldn't stand against the triple threat of heat, white noise and pregnancy. The book dropped from her fingers and she knew it as she let her head fall back against the chaise, but she was simply too drowsy to care.\n\nShe woke up when the electricity went out. The sudden black pounded her eyelids as thoroughly as if she'd been in the dark and the lights had snapped on. Ginny pushed herself upright, forgetting about the book, which crashed to the floor. She looked automatically toward the windows to see if theirs was the only house without power, but the outside was as dark as in. So, the whole neighborhood then.\n\nFortunately, she'd grown so used to moving around this house in the dark that it was no big deal for her to make her way downstairs, though she made sure to take her time and place each foot solidly and carefully on the step, to be sure she didn't tumble down. The rechargeable flashlight had turned on when the power cut out, so she had no trouble seeing her way to grab it from the dock.\n\nBeneath her feet, the floor did not vibrate. The sump pump wouldn't run without electricity, of course. Which meant water in the basement. Ginny moved to the back door again, but the light from inside reflected on the glass and she could see nothing but her own silhouette. She shielded the light and tried to see outside.\n\nThat couldn't be the creek, could it? Lapping at the edges of the flagstone patio? Ginny opened the back door, heedless of the rain, and held up her light.\n\n\"Oh shit.\" It was the creek, or maybe it was just that the saturated swamp of her backyard had joined with the still-rushing creek waters to make one vast pond that seemed to be creeping toward her house at an exponentially fast rate. \"Shit, shit.\"\n\nHer phone rang from her pocket. \"Sean? Where are you?\"\n\n\"I'm stuck. The roads are closed. The bridge is out.\" Sean sounded panicked. \"The state police are redirecting everyone; they said almost all the back roads are flooded out. I don't know how I'm going to get home. Are you okay?\"\n\nOther than the nausea now rumbling through her? \"Yes. I'm fine. But the power's out. And the water...Sean, the water's going to come into the basement.\"\n\n\"Don't you go down there!\" he warned. \"Not in the dark. It'll be fine.\"\n\nGinny moaned, imagining the water pouring in, rising. Ruining. \"But the water...\"\n\n\"It's too dangerous for you to go down there. Promise me you won't. In fact, you should go across to Kendra's house.\"\n\n\"They don't have power either,\" she told him. \"It's not just us. And frankly, I would rather drown.\"\n\nThis forced a laugh from both of them, and the small bit of humor made her feel better. Ginny looked into the backyard again, then went to the front of the house to look out the front door. In the driveway, the water was up to the middle of her car's tires. Fear, real fear, plucked at her.\n\n\"Everything's flooding here, Sean.\"\n\n\"You stay put. I'm going to ask the cop what to do. But you...don't you go out and try to drive in it.\" Sean sounded fierce, determined. Helpless. \"You stay there, Ginny. Get upstairs if you think it's going to come in the house.\"\n\n\"Oh, it won't, will it? Just the basement. It's not going to come in the house, Sean.\" But even as she said it, a vision of gray waves of water washing over her kitchen floor and down the hall filled her head.\n\nHer stomach muscles went tight. Hard. Ginny put a hand on the table, bending forward with her legs slightly spread. This felt different than the other times. This felt more...real.\n\n\"Ginny! Are you okay? What's going on?\" Sean had been talking to her, but she hadn't heard.\n\nGinny shook herself as the contraction passed. She blew out a slow breath. \"I'm fine. Just a contraction.\"\n\n\"Shit! What?\"\n\n\"Nothing to worry about,\" she soothed him, though in fact she was starting to wonder if there would be something to worry about very soon. She put a hand on her belly, low, then over her crotch. She ached down there. Inside, deep. \"Just get home when you can. Be careful\u2014\"\n\nBut again, she was cut off, this time not because her husband had a quick-draw disconnect thumb, but because the signal had been dropped. Ginny breathed in, breathed out, until the pain in her belly eased. She still ached inside, something like the worst sort of menstrual cramps. But when it passed she could stand upright and breathe easily again, at least as easily as was possible with her child doing the cha-cha against her ribs.\n\nIn the bathroom, though, the paper slid too easily against her flesh after she urinated. Something thick and wet spilled over the sides of the paper, and with a grimace Ginny maneuvered the light so she could see what it was. The doctor had warned her about the mucus plug, but truthfully Ginny hadn't quite understood what it was or what it would look like until she saw it there.\n\n\"Oh. Shit,\" she said softly and let the paper fall into the toilet. She flushed and struggled back into her panties. She washed her hands, forgetting that with the electricity off, the water pump wouldn't work until the tap sputtered and burped, going dry, and the toilet tank didn't refill.\n\nAnother contraction hit her at the base of the stairs. Still mild, she almost could've walked through it, but she didn't want to take a chance on the stairs. Ginny held the newel post until the pain passed and tried to remember how long it had been since the last one. She'd better start keeping track, no easy task with all the clocks stopped and no clue where her watch was.\n\nUpstairs. Sean had told her to go upstairs. She heaved herself up one at a time and went into the library, where she set the flashlight on the table. It wasn't bright enough to read by, but it cast enough light to show her the plate she'd left there earlier was empty. Ginny touched the crumbs with her fingertip and put them to her tongue. Salt. She wasn't hungry, which meant she'd probably eaten them. Right?\n\nBut she knew she hadn't.\n\nShe looked into the shadows, toward the small door of the crawl space. Of the closet. She'd have bent to look beneath the fainting couch if her body would bend that way, which it wouldn't. Instead, she sat, her knees slightly spread and her hands on top of them. There was nothing to do here but wait. She thumbed her phone and brought up Peg's number, but that call rang on and on without answer. Ginny typed in the net address for the local news channel, but the page stuck halfway and refused to load so that all she could read was \"Roads closed, police direct traffic to alternate\".\n\nShe waited for another contraction, but none came. She waited for the lights to come on, and that didn't happen either. In limbo, waiting, starting to get chilled as the house cooled, Ginny curled up under the blanket and let herself float in the darkness. Not quite dreaming, but not really awake.\n\n* * * * *\n\nIn front of her, a child.\n\nA girl. Long dark hair, bedraggled and tangled. Big dark eyes shadowed with darker circles in a pale, pale face. She wore a dirty, shapeless sack of a dress that had been cut from a flowered pillowcase and a tattered cardigan, the buttons of which were missing or hanging by threads. Scabs and dirt encrusted her legs and bare feet. She stared at Ginny as her mouth worked, but nothing came out.\n\nGinny blinked and sat, slowly. The flashlight had dimmed a little. She should have turned it off to conserve the charge. How long had she been sleeping?\n\n\"Come. Please.\" The girl held out a hand.\n\nThis was not the girl in the photos Ginny'd found, though there was something of her in the shape of her face. And this girl was too young. Ginny had never seen this girl, but she knew who it was, just the same. \"Caroline.\"\n\nThe girl said nothing.\n\n\"What do you want?\" It seemed the right question to ask a ghost who'd at last been brave enough to show her face. \"Are you here because I found...I found you?\"\n\nThe girl stepped forward. The floor creaked. Ginny frowned. Why would the floor creak beneath something that had no weight?\n\n\"Come. Please. Please?\" The girl held out her hand, the tiny fingers dirty, the nails ragged. She opened her palm, faceup. There was a button there, like the one Ginny had found earlier. It fell from her hand and hit the floor with a clink before rolling under the lounge.\n\nGinny was up and off the lounge, her arms pinwheeling, as fast as she could move. It wasn't fast at all. Her breath sharp in her lungs, she put the chaise between her and this wraith, this vision, this shade that was no ghost.\n\nOh God.\n\nIt wasn't a ghost.\n\n\"Who are you?\" Ginny's voice came out high, strained, not like her own voice at all.\n\n\"Water.\" The girl looked toward the bookcase, which made no sense. Then at Ginny. She pointed down. \"Water is down there.\"\n\n\"In...the basement?\"\n\nThe girl took another delicate, hesitant step toward Ginny. In the flashlight beam, her skin looked almost translucent, her dark eyes somehow filmed. Ginny was reminded of pictures she'd seen of cave fish and mushrooms, things grown in the dark.\n\n\"Down there. It's coming in. It's coming up!\" The girl shuddered and jerked. She turned and ran for the cubbyhole door while Ginny stared in stunned silence. \"Please come! Please come help!\"\n\n\"We want to see if the little girl can come and play. The little girl in the basement. I was playing with the little girl.\"\n\nGinny thought of what Kelly and Carson had said. She remembered what they'd called her. She spoke, her voice on the edge of a cough, but loud enough. \"Carrie.\"\n\nThe girl turned. Incredibly, she smiled. Her teeth were black and skewed, but the smile was sweet and shy. She held out her hand. \"Please come. Hurry. Come!\"\n\nYou think your father did something to her, don't you?\n\nI don't think it. I know he did.\n\nShe thought about things going missing and being returned. Of a waving silhouette in a window. Food set out in a bowl and being eaten.\n\nShe thought of the sound of footsteps in the night.\n\nShe thought about the place where the garage had once been, and the basement wall that looked different from all the others.\n\nAnd then she knew what had happened all those years ago.\nChapter Forty-One\n\nCarrie gestured frantically and ducked into the cubbyhole. Ginny bent to look inside. The rain sounded much louder here, and she shivered at the waft of damp, cold air.\n\n\"I can't fit in here, honey.\" The endearment slipped naturally from her lips. That was how you addressed a child. \"I'm too big. And I don't understand...\"\n\nCarrie disappeared. Only for a second, because after that her small head peeked out from a dark place in the corner. Ginny strained to look. It was an air duct.\n\nNot all of it made sense, but this did. Still, she shook her head and waved at the girl to come closer. Carrie's wide eyes blinked when Ginny shone the light too closely on her, and she put up a hand to block it.\n\n\"I can't fit in there, honey. Listen, you need to come out here, okay? I'll...I'll call someone. We'll go across the street, or next door.\" Ginny hesitated, thinking of the water rushing down the street, as swift and deep as the creek in the back. \"I will help you.\"\n\nCarrie let out a low, moaning cry of desperation. She wriggled from the air duct and dropped almost soundlessly to the dirty plywood. She moved toward Ginny, who backed up as instinctively as if a raccoon or a rat was coming at her, though the girl had shown no signs of antagonism.\n\n\"I'll take care of you,\" Ginny whispered.\n\nAnother contraction hit her. She groaned at the suddenness, one hand on her belly and the other grabbing at the easel. It tipped, spilling her unfinished canvas.\n\nCarrie cried out and covered her face. A second later she peeked through her fingers and moved closer. She put her small hands on Ginny's belly, and Ginny didn't have the concentration just then to recoil.\n\n\"Water, coming up. Down there.\" Carrie pointed again to the floor, to her bare feet, to something Ginny couldn't see. \"Mama says, come get help.\"\n\nGinny choked. Everything inside her seemed to be tensing, and the sharp ache inside her intensified. She counted the seconds\u2014one, two, three, four\u2014before everything eased. Her fingers had gouged the inside of her palms. She swallowed, hard.\n\n\"Your mama is down there?\"\n\nCarrie nodded and tugged at Ginny's sleeve. \"Please come. Mama says, the lady is nice. I tell her, the lady is nice. She says, you help us.\"\n\nCarrie looked pleading, then terrified. She shook so hard her teeth chattered and looked around the room, though there didn't seem to be anything to see. She looked back at Ginny.\n\n\"Mama says, he is gone!\"\n\n\"Who? Who's... Oh God, oh God,\" Ginny muttered. She managed to stand up straight. \"Your mama and you in the basement. And he...yes, honey. He's gone. And I'm going to help you. But I can't get into that air duct, I have to call someone. We'll call, okay?\"\n\nBut her cell phone still wouldn't pick up a signal, and the power was out, and though she knew there was a landline phone somewhere around, she didn't know where it was or even if it would work if she plugged it in.\n\n\"How high is the water, Carrie? How far up?\"\n\nCarrie hesitated, then ducked low to the ground and held her hand to just above Ginny's ankle. Then she put it higher. Higher. To Ginny's thigh.\n\nSurely it couldn't be that high, could it? If the basement was flooding that much, it would come into the first floor, wouldn't it? Ginny didn't know. All she could think of was a foundation where there'd once been a garage and the concrete wall in the basement below it.\n\n\"You come.\" Carrie tugged her sleeve, pulling Ginny toward the bookcase to the right of the fireplace.\n\nAll the pieces were falling into place. The figure she'd seen in the window upstairs\u2014not the one by the easel, but another window on the other side of the fireplace, one that she'd assumed was built for symmetry. The one behind the bookcase.\n\nCarrie reached along the molding and pressed something with her small fingers. The bookcase creaked. It shifted.\n\nIt moved.\n\nGinny cried out, not so much in surprise as a twisted sense of victory. She was not crazy. She was not crazy. And this house was not haunted, at least not by a ghost.\n\nCarrie grunted, shoving at the bookcase until Ginny hooked her fingers into the now-open slot and pulled. Even with the weight of the books, the case moved smoothly, on oiled tracks like a sliding glass door. It was an ingenious design, but the space behind it proved even more so.\n\nThere was a window here, identical to its sister on the other side of the fireplace, though this one was hung with a cloak of spiderwebs, the sill decorated with a multitude of dead flies. The narrow space between the outside wall and the back of the bookcases was just wide enough for Ginny to stand in, though even without the baby bulk, she'd have brushed both sides with her shoulders. Directly to the left of the opening was the brick side of the chimney, while about four feet to the right was an opening in the floor.\n\nJust a hole, no safety railing or anything to keep an unwary stumbler from falling into it. But then, Ginny thought, anyone entering this space would've had to know exactly where they were going. Carrie ducked around her and leaped toward the hole, too fast for Ginny to catch her.\n\nGinny, imagining the girl plummeting into an empty space, cried out.\n\nCarrie looked over her shoulder and stepped into the darkness. She didn't fall. She gestured for Ginny to come closer.\n\nIt was a metal spiral staircase, the first step a few inches below floor level. It circled into darkness, though when she hung the light into the opening, Ginny could see that the stairs continued without a break down to a small landing.\n\nShe'd never have thought she could fit into it, but she could. And did. There wasn't enough room for her to fall either forward or back, so even without a railing to grab she could put one foot at a time down on each step. She couldn't see Carrie below her, but she could hear her. When Ginny got to the landing, there wasn't much more room. She held the flashlight as high as she could to look around the narrow column of space.\n\nShe was behind the pantry that should've been an extra foot or two wider, but wasn't. \"Oh God. He did this. He built all of this...\"\n\nShe didn't have time to break down or dwell on the sickness that had led George Miller to build this house with a prison inside it, though that was what she was sure she'd find. Ginny gave herself a shake that set her head spinning. She braced for another contraction. Everything about her pelvis felt loose and wobbly, like her hips weren't hinged quite right. A contraction eased and passed, and she drew in a few shaky breaths.\n\nCarrie had already gone down the first few steps of the other spiral staircase, moving without fear or pause into the inky blackness that reached beyond the dimming glow from Ginny's flashlight. She spoke over her shoulder, \"Come.\"\n\nGinny followed, slowly and carefully. Her hand shook, which sent the light tipping back and forth. Shadows, light and dark. She eased down, step-by-step. The air smelled damp, thick with must that wanted to make her cough, if only she could draw a breath deep enough.\n\n\"Wait,\" Ginny whispered. Then louder, \"Carrie. Wait for me.\"\n\nThe narrow corridor bent at a right angle and ran along what must've been the back of the house. The exterior concrete wall was black with wet, and water was actually trickling in fast streams down from the ceiling. Water on the floor too, a couple inches that got deeper the farther she went, as though the passage was on a downward slope.\n\nGinny put one hand on the inside wall, also of concrete. George Miller had really done his work well. There was no sign from the basement that this section was two feet shorter than it should've been. She shuddered, stopping, her gorge rising. Her muscles tensed again as she leaned against the wall to let the contraction pass. Ginny swallowed bitter saliva, trying not to puke. Still, no pain. Just discomfort. Yet there was no denying it\u2014she was in labor.\n\n\"Carrie. Wait.\" The spasm passed, Ginny straightened and pulled out her phone again. Upstairs she'd fluctuated between one and two bars. Down here, with multiple layers of concrete and earth between her and the sky, there was nothing.\n\nThe water had risen another inch while she stopped. Carrie danced in it, impatient and frantic, while Ginny slogged toward her, grateful for her slippers. Even heavy and soaked, they were protection against whatever might be on the floor under the water.\n\nThe flashlight dimmed drastically before rallying and returning at half its former brightness.\n\nGinny shook it, knowing even as she did the effort was silly. It didn't have batteries the way old flashlights did. You couldn't rattle it into another burst of energy. Instead, the beam of light shook around and went even dimmer.\n\n\"Shit,\" she breathed.\n\nWhat the hell was she doing? Nine and a half months pregnant, no power, no phone, a faulty light, following some feral child into a basement that was flooding. She didn't even have a weapon. It was the classic stupid move from every horror movie, and suddenly she was choking with laughter. Bent with it, shoulders heaving as the flashlight swung dangerously close to the water and she fought the grip of another contraction.\n\nNo, no. Don't fight. Don't fight it.\n\nGinny tried to breathe through it the way the classes taught, but the laughter wouldn't let her. All of this, so ridiculous. So surreal. She'd have gone to her knees right there in the water if the passage weren't so narrow there wasn't room for her to fall.\n\nThe laughter and the contraction passed. Ginny wiped her face\u2014tears or sweat, she couldn't tell. Carrie had moved even farther down the passageway, around another corner. Ginny followed with the lamp, the light now the strength of a guttering candle, held high.\n\nThe corridor came to a dead end. Not a concrete wall here, but something shiny. Metal? Shit, the wall was metal, smudged with handprints and splashed with water where Carrie must have kicked. The girl turned as Ginny rounded the corner. She gestured.\n\n\"Come. Please.\" She tugged at a metal handle and the wall moved the way the bookcase had, sliding like a pocket door into a recess in the wall. Carrie put her back to the door and braced her feet against the wall on the other side. She grunted with the effort.\n\nGinny sloshed closer. The flashlight swung in her hand, back and forth. Dark and light. She could see nothing beyond the sliding metal door.\n\n\"Honey, I won't be able to get through there. There's not enough room.\"\n\nCarrie pushed harder, forcing the metal door farther into the recess. Her legs shook, and she bit on her lower lip. There was still no way Ginny could step over Carrie's legs and shove her bulk through the opening. Ginny put a hand on the door just above Carrie's head. She could feel it threatening to move the moment the little girl ceased her counterpressure.\n\nGinny wedged herself into the space as far as she could, her breasts and belly crowding against the girl. \"I got it. Go.\"\n\nCarrie moved at once, slipping from the space and letting the door move against Ginny's weight. Ginny had a vision of an elevator door slamming shut on an unwary passenger\u2014but somehow she guessed this door wouldn't spring open. It would cut your fucking fingers off instead.\n\nThe door sprang shut behind her the instant she left off the pressure, but she wasn't all the way through. It would've snapped her ankle had she not shoved the flashlight between the door and the wall just long enough to get her foot out. That was the end of the flashlight, which cracked with a snap and plunged them into perfect darkness.\n\nGinny closed her eyes and tried to catch her breath. Sounds became magnified without sight to counter them, she knew that. She heard the soft swish of Carrie's feet in the water and felt the push of it against her calves. Carrie's chilly fingers slipped into hers and squeezed. She probably could see much better than Ginny could, but even cats needed some amount of light in order to see in the dark. Carrie was still human.\n\nWasn't she?\n\nGinny grabbed her phone from her pocket and pressed the Home button to bring up the menu. Instant bright-white light. She slid her thumb along the screen to unlock it and tapped quickly, looking for a flashlight app. \"I can't believe I didn't think of this before.\"\n\nShe found it. More bright light, using her phone's LED flash set to a permanent glare. It lit the corridor, even narrower on this side of the door, and threw giant shadows on the walls. The ceiling was lower, hung with shiny ductwork and wires that cast weird patterns of shadow. Ginny ran her hand along the door. This side had no handle. Nothing but smooth, cold metal reflecting the light from her phone.\n\nThen, up near the top, higher than a child could reach, higher than Ginny herself could reach, she saw a small dark circle. She stretched to touch it, and her fingertip felt a rough edge before dipping just barely inside. A hole. Probably for some sort of key, which she did not have.\n\nShe was trapped down here.\n\nBut she would not be afraid. She would not let herself give in to terror. All the creepy, scary things that had happened since moving into this house had not sent her gibbering, and neither would this, especially since she knew the cause was nothing supernatural. She would not be afraid.\n\nShe was a liar.\n\nShe followed Carrie down the corridor and the ceiling got lower and lower. The light from her phone was eye-achingly bright and far-reaching, yet like the high beams of a car, limited in scope. They'd only gone about four feet when the corridor jogged again, this time to the right.\n\nThe smell hit her first, hard as a fist, thick like smoke. The musty, earthy smell of the water had been nose-tickling but normal. Natural. This stench, of unwashed bodies and rotten teeth, of food left to spoil...of human waste... Ginny retched, turning her head and certain she was going to vomit. She hadn't eaten anything in hours. Nothing came up but bile she spit into the water that was now up to her knees. She heaved again, then stood to shine the light.\n\nThe room was no larger than her bedroom and built on a cant that left her thinking of those haunted house rooms with the strobe lights, usually painted black and white, the kind with tilted floors to screw up your perspective. The ceiling was so low she had to bend her head, and the light showed her there were at least two slanted fun-house doorways. A small, domed refrigerator took up space in one corner, with what looked like a tiny two-burner stove next to it and a spindle-legged sink. Beside that, a child-sized wooden table perfect for tea parties in which every cup was poisoned. The furniture wasn't all she saw in this nightmare room.\n\nThere was something else too.\nChapter Forty-Two\n\nCarrie splashed through the water to reach a low metal bed frame and stained mattress shoved against the far wall. The water licked at the mattress's bottom edge. It would be sodden in minutes. There were four of them. Or five. Ginny could not yet look at whatever they were, those crouched and shivering things with pale faces and grasping, skeletal hands.\n\nGinny switched the phone from one hand to another. The light shifted. Shadows loomed and swooped from the shaking of her hands. She closed her eyes. Pain, this time fierce and deep and seemingly unrelenting until then...it was gone. She opened her eyes.\n\nWhatever was on the other side of the room had moved closer.\n\nGinny couldn't speak. Her breath came short and sharp as dizziness assailed her. She almost dropped the phone, but then clutched it so tight her fingers hurt.\n\nShe gave a breathless, gasping scream. They were in front of her, two of them, with Carrie hanging back with the other three still huddled on the bed. One stood, up and up, curving and hunched, its neck and head tilted at a strange angle to show her a smooth, pale face with dark eyes. It held out a hand to Carrie, who linked her fingers with it. They stood in front of Ginny, saying nothing. Just watching.\n\nAnd... Oh God. They were not monsters, they were not things made of shadow and fear. They were children. A boy of maybe thirteen, grown too tall for this cramped space. A girl a little older than Carrie but not much bigger, her dark hair as tangled and dirty but pulled back from her face with a length of frayed ribbon. And there too, thinner and without her collar, but purring as she was cuddled\u2014Noodles.\n\n\"Oh,\" Ginny said in a broken, hesitant whisper. \"Oh, you poor things.\"\n\nFrom beyond one of the dark doorways came a low, rattling hiss that turned the heads of all the children, those in front of her and the ones who'd hung back on the bed. Something slithered through the water, and Ginny saw a snake before she recognized it as something more sinister. A length of chain coiled and moved, disappearing into the water, now just past Ginny's knees.\n\n\"Mama, I brought the lady.\" Carrie turned with another shy smile, and the shadows shifted.\n\nThe figure in the doorway moved toward her, one shoulder higher than the other as the weight of the shackled wrist kept one hand closer to the floor. The shaved skull and hollow cheeks made it impossible to guess its gender, though the worn dress gave a hint. Ginny knew at once who she was.\n\n\"Caroline. Oh my God. Caroline Miller.\"\n\nCaroline gave a hoarse croak. She was weeping, Ginny saw, though it was too dark to see if there were any tears on her wasted cheeks. She drew Carrie to her and kissed her head. She looked at Ginny.\n\n\"Is he gone? He's really gone?\"\n\nGinny pushed closer, carefully, through the water, though the small room was mostly bare. \"He's gone,\" Ginny said, and let Caroline's withered hand take hers. \"I'm here.\"\n\nCaroline's body racked with silent sobs, but when she raised her face to Ginny's again, her eyes were dry. She licked her cracked lips. \"The water's coming in.\"\n\n\"Yes. I know.\" Ginny tried not to shudder. She failed.\n\nCaroline might look weak, but her grip was so tight it became painful. She leaned closer, her breath sour but her eyes bright. \"I sent Carrie for you. She said you were a nice lady, that you left things for her.\"\n\nGinny nodded, though that wasn't quite the truth. \"Yes. I didn't know... Oh my God. I'm so sorry. I didn't know.\"\n\nCaroline's bright gaze didn't dim as she shook her head. \"Nobody knew. It's all right. It's all right. You're here now. You can help us. You can help us?\"\n\n\"The door closed behind me.\" Ginny tried to breathe shallowly so she could avoid the stink. \"How do you open it?\"\n\nA rough and grating noise came from Caroline. Incredibly, a laugh. \"From this side? You don't.\"\n\nGinny watched as Caroline's children gathered silently around her, staring with wide eyes and open mouths. There were four children altogether. The oldest, a girl, hunched like her brother, put a hand on Caroline's shoulder and murmured something Ginny didn't understand.\n\nCaroline replied, her voice pitched low and mumbling, the words not only indistinct but not quite right. Strung together in a pattern Ginny couldn't recognize, like tuning in unexpectedly to a foreign radio station. Whatever Caroline said seemed to satisfy her daughter, because the girl nodded and stepped back, just outside the circle of light.\n\nGinny's fingers cramped, and she switched hands again. She looked at her phone's battery indicator. The water had risen impossibly higher, up to her thighs. \"My light won't last forever. We have to get you out of here. All of you. How do we open that door, Caroline?\"\n\n\"He had a key. Always a key. It was shaped like a...\" Caroline's mouth worked, her expression momentarily blank before her eyes focused again, \"...a T. It was shaped like a T and that's how he opened the door. There is no other door.\"\n\n\"Of course not.\" If they'd been able to get out through that door, wouldn't they have done it long ago? \"Carrie. She was small enough to fit through the ducts\u2014\"\n\nBefore Ginny could speak again, a swift and hot gush of fluid ran down her legs. She clutched her belly and held out the phone, blindly, hoping someone would take it from her before she dropped it. Someone did, and she put both her hands between her legs, terrified she'd feel a baby's head bulging against her giant cotton panties.\n\n\"The baby,\" she said, and a groan took the place of any words she might've said next.\n\n\"Your baby is coming,\" Caroline said flatly. \"Here and now? Don't worry. Don't worry.\"\n\nWorry was not the name for this vast and roaring terror, this overwhelming fury of anxiety and fear. Ginny gasped as the contraction peaked. She breathed with it, incapable of embarrassment though she'd bent over to put her hands on her knees and was panting like a dog on the street in August. Everything inside her pressed down, down. Only the rising water kept her from squatting right there, because she would not, could not push her baby out into that filthy wetness.\n\nShe became aware of Caroline on one side of her, the oldest daughter on the other. They walked Ginny in mincing crab steps, the largest she could take, through the doorway and into a smaller room the size of a closet. The light from her phone shone over their shoulders, illuminating a full-size bed and a dresser. As the light slanted crazily along the walls, Ginny saw they'd been hung with countless childish drawings like the one that had been left on her easel. The stick figures seemed to dance, and, nauseated, she closed her eyes while they took her to the bed.\n\nShe put both her hands on the damp mattress and bent forward. She breathed. And finally the contraction passed.\n\n\"My water broke. I'm having contractions,\" she said without opening her eyes. \"The baby is coming, yes. Here and now.\"\n\nTears came, shaking her. A firm hand squeezed her shoulder. She looked up to see Caroline.\n\n\"I will help you. Linna will help you. We know...we know about babies, lady.\"\n\n\"My name is Ginny.\"\n\n\"Ginny,\" Caroline said, and smiled. \"Ginny, we'll help you. It's going to be all right.\"\n\nNothing could be all right about this. Water up to their thighs, the stink overpowering everything else. Her baby born in this hovel? No. No. Ginny shook her head, but another contraction ripped its way through her, and there was nothing but pain to think about just then.\n\nIt passed.\n\nThere was something amazing about how much it hurt and how suddenly the pain ceased, and she'd have been able to think about it more if only the world would stop spinning. It was because she wasn't breathing, Ginny knew, and braced herself to take a long, stinking gulp of fetid air. It helped only a little.\n\n\"You need to get on the bed. Lie back. Lift your dress, and let us look.\" Caroline gestured. \"Deke. The light.\"\n\nGinny'd prepared herself mentally for knowing that in the hospital it was likely she'd have a whole slew of strangers staring at her vagina while she pushed this kid out. In no realm of her imagination had one of those staff been a teenage boy. There was no shielding of her body, though, no way even for her to protest, because she was too busy bearing down against the sudden pressure in her womb.\n\nHands helped her back onto the disgusting mattress, propped her with pillows. Hands lifted her nightgown and pulled her panties off. Ginny wept, not from pain or shame, but from knowing there was no way her baby could survive this filthy place.\n\n\"Mine did,\" Caroline told her, and Ginny realized she'd spoken aloud. \"The ones who got born alive, they all made it. Yours will too.\"\n\nGinny opened her eyes. \"My phone. I have to see if...there's a signal.\"\n\nDeke handed over her phone without argument. Ginny thumbed the screen, with no luck. She tried anyway, first with a call that wouldn't even send and then a text. After that, another. Each time, a small, angry red exclamation mark showed up next to her message, proving it didn't go through. Her fingers tightened on the phone with the next contraction, and Caroline gently pried it from her hand.\n\n\"Let Deke hold the light, Ginny.\"\n\nGinny panted and gasped, then gave in to a shriek. This embarrassed her more than anything else had, but Caroline patted her shoulder.\n\n\"It's okay to scream,\" she said. \"Nobody can hear you down here.\"\n\n\"You...all...can.\"\n\nCaroline gave another rusty laugh, as cutting as knuckles on a grater. \"Screams don't bother us.\"\n\nThe pain faded. Ginny drew a breath. She shifted on the bed, against the pillows, and tried to see the floor. \"The water?\"\n\n\"Still coming in.\" Caroline turned her head and coughed, then spit.\n\nThe chain on her wrist jangled. She didn't use that arm at all, Ginny saw. The fingers of that hand looked curled and useless. Caroline saw her looking and gave her a smile that showed straight, even teeth that could only be the product of expensive orthodontia.\n\nThis detail drove home the horror of this girl's...no, she was a woman now...this woman's life more than anything else could have. Ginny remembered that smile, flashing bright with metal. The curly permed hair, the fashionable clothes. Caroline had been smart and bright and beautiful once. All of that had been stolen from her, replaced with...this.\n\n\"When the pains get worse, one on top of each other, it will be almost time. You'll feel like you have to push the baby out. Like you have to use the toilet. You won't be able to help it. But until then, you should rest and try to get through each pain as it comes.\" Caroline said this calmly, as though it were the most normal thing in the world to be giving birth in a flooding basement, in the dark. \"Linna and I will be here to help you.\"\n\nGinny waited for the next pain, but her body seemed to have gone quiet. \"I can't feel it moving.\"\n\nCaroline said nothing at first. Then, softly, she put her good hand on Ginny's belly. \"It doesn't matter. Your body will push it out, no matter what.\"\n\nMore tears came, but Ginny forced them back. She clutched at Caroline's hand. \"I'm so sorry. So, so sorry I didn't figure this out sooner. I didn't put the pieces together, they were all right there. I should've known. I should've...\"\n\n\"How could you have known anything?\" Caroline looked surprised, then sad. \"My mother didn't know. My brother didn't know. Nobody ever did. But you're here now. And this will all be okay. But first, your baby\u2014babies don't wait for anything.\"\n\nMore contractions came and went, some longer than others. Ginny waited for them to form some sort of pattern, but they refused. Some were short, others long. She tried counting the minutes between them and lost track with their irregularity.\n\nGinny had no idea how much time had passed and asked to see her phone. Only half an hour since she'd come into this room. An hour, maybe a little longer since she'd last spoken to Sean. Not soon enough for him to worry. Not soon enough.\n\n\"Mama. Water,\" the girl a little bigger than Carrie said.\n\n\"Yes, Trixie. I see it.\" Caroline never lost her calm, flat tone. She sat on the bed next to Ginny and took her hand. \"It won't be long now.\"\n\n\"Until the baby comes?\"\n\n\"Or the water.\" She'd proven her ability to laugh, but there was nothing like humor in Caroline's voice now.\n\nGinny tried to think, to focus, to fixate on something her brain wanted to tell her was important. Something about...Carrie. Something about...the water.\n\nAnd then she couldn't think of anything but the agony. It tore at her. It consumed her. As though from far away, she heard Caroline murmuring to Linna, something about blankets. Something about a basin. But then the all-encompassing urge to bear down took over, and all Ginny could think about was pushing.\n\nHands moved her again, lifting her hips, in the almost nonexistent break between contractions, to slide something beneath her. Hands pressed on her knees, parting and pushing them back toward her hips. Hands cradled her feet, bare, the slippers lost.\n\nIt had never been like this. The pain had been a familiar echo, vaster but still not entirely foreign. But this grinding, desperate need to strain and push and expel...Ginny was helpless against it. She couldn't stop it. Her body did what it was meant to do. If she died, she thought, her body would continue to birth this child.\n\nShe felt the baby move down the birth canal, inch by agonizing inch. Too slow, and too fast at the same time. A bright, red-hot center of pain burned between her legs as she screamed. The sound spiraled up and up until her voice broke.\n\n\"The head. I see the head,\" Caroline told her. \"The baby's almost here.\"\n\nGinny looked down between her legs to see Caroline and Linna both poised there, hands ready. The light from her phone had no delicacy. It made everything harsh. Ginny saw the flash of the other children's faces, but Deke was solid and steady; he held the light focused firmly between Ginny's thighs.\n\nThe inexorable need to push cycled back, and Ginny rode it. Her fingers clutched the bare mattress, digging into the now-sodden material. She growled with her efforts, her teeth gritting so tightly she thought a tooth cracked.\n\nThe baby was born.\n\nFor a breathless, eternal moment, the emptiness inside confused her and Ginny sagged against the pillows. There was no cry, no wail. She struggled upright, desperate, and croaked out a wordless plea.\n\nCaroline bent over the child, then lifted it. The baby hung limply in her hands. Then it moved. Then it screamed.\n\nGinny had never been so happy to hear anything in her life.\nChapter Forty-Three\n\nGinny dozed with her baby tucked up close to her naked skin, both of them covered in a blanket and Noodles purring by her side. She was too exhausted to care how dirty it was. Caroline and Linna pressed on her belly, eliciting a fresh burst of pain and another hot slosh of something from inside her. They talked to each other in that mumbled, mangled language, ignoring her, while the other children were dispatched on errands Ginny didn't understand.\n\nShe didn't care.\n\nFor now, her son was safe. They were warm, though anything but dry. And Ginny was tired...so tired...\n\nShe didn't want to, but forced her eyes open. Her vagina burned and ached. Someone had pushed a folded bundle of material between her legs. The baby snuffled against her, then went quiet. Ginny was bleary, but awake. Her mouth tasted sour, dry. She'd have given almost anything for a drink of cold water.\n\nGinny smoothed her hand over her baby's soft head and marveled at the hair there. It was pale, not dark like hers. He took after Sean. Ginny wanted to know the color of his eyes, but didn't want to shine the bright light directly on him. \"You have electricity down here.\"\n\n\"Yes. Sometimes. But it was better to use a flashlight with batteries, when we could. It's so good to have light.\" Caroline's low voice drifted through the darkness. \"We have two lamps, but the outlets down here are bad. When we try to use the stove, sometimes it blows the bulbs.\"\n\nGinny thought of all the times the power had gone out. \"It blew our fuses too.\"\n\n\"He said he'd fix it. That and the heat, he got that directed in to us because it was so cold. But it never worked as well as he wanted. He said he could build things just fine, but he didn't understand electricity. We have a couple candles, but the matches got wet. They were your candles...\" Caroline paused and sounded almost shy. \"He stopped letting us have them, and no matches, either. Because we might start another fire. And that would've been very, very bad.\"\n\n\"Caroline, why did you wait so long before you sent Carrie to get me?\"\n\nThere was a long, long silence.\n\n\"We had to know,\" Caroline said quietly, \"if we could trust you.\"\n\nThe simple way she said it broke Ginny's heart. She found the other woman's hand, the one weighed down by the chain and shackle. \"You can trust me.\"\n\n\"He didn't know we were saving food. He didn't know we'd eat only half of what he brought us, eat the things that would go bad. I made us put the rest away, in case. He was getting old, you see. He said he'd be around forever, that no matter what happened he'd be there, but I knew better. So I made us put the food away. Batteries. The light bulbs, though he'd yell about how careless we were to break so many.\" Caroline coughed again. The noise was rattling and thick. She spit to the side, into the water. \"He always threatened not to bring more. To let us starve, or sit in the dark. But I always knew he wouldn't. He didn't want us to die, you know. He didn't want us dead.\"\n\nGinny shivered. \"But...I don't understand, Caroline. Why would you stay here? If Carrie could get out through the ducts to get food, to find me, why wouldn't you send her earlier? I would've helped you sooner.\"\n\nThe bed dipped as the children gathered around. Their mother looked around at them, then at Ginny. \"You're here now.\"\n\nA pulse of hot fluid leaked from between Ginny's legs. She grimaced, but there was nothing to do for it. \"My husband is on his way home, but the roads are closed. He might not get here for a while, but if he doesn't hear from me, he will worry. He'll send someone or do something. But, Caroline...we can't wait for that. You have to send Carrie out again.\"\n\nCaroline said nothing at first. She looked around at faces focused on hers. All of them were so quiet. They sat so still.\n\nShe looked at Ginny. \"Yes. Carrie should go out.\"\n\nCarrie let out a low wail and shook her head. She ran from the room. Caroline held up a hand when her sister moved to go after her. \"No, Trixie. I'll go after her.\"\n\nDragging her chain behind her, Caroline went into the next room, leaving the light behind. Ginny stared at the foot of the bed. Deke, the tall boy who'd held the phone while she gave birth. Linna, the oldest girl. Trixie, a little bigger than Carrie.\n\nThe circle of light stretched to the wall, and Ginny could clearly see a few of the drawings she'd noticed earlier. This one had six stick figures. One, the mother. Five children around her. She looked again at the group.\n\n\"One of you's missing.\"\n\n\"T-T-Tate.\" Trixie had a stutter, either nerves or a speech impediment. \"Huh-he got stuck. Huh-he was t-too big.\"\n\n\"He said bad things would happen if we tried to get away,\" Deke said flatly, the \"he\" in question clearly referring to George Miller, and not Deke's brother. \"Tate pushed him. He hit Tate's head, and Tate wasn't sure what happened. Then Tate said we need to try, Mama. We need to try. And he went\u2014\"\n\n\"Deke,\" Caroline said sharply, a dark silhouette in the doorway. \"That's enough.\"\n\nThe water, at least, seemed to have stopped rising. It sloshed against Caroline's thighs as she moved toward the bed. \"Carrie will go. But she's very afraid. She's not sure what to do.\"\n\n\"Do you have a pen? Paper? Something to write with?\"\n\n\"Everything's wet,\" Linna said. \"The crayons are broke. The paper was in the cabinet. It's ruined.\"\n\nTrixie let out a sob at that, and flung herself onto the bed while Linna rubbed her back.\n\nGinny was struck with inspiration. \"My phone. I'll type a note in it. She can take my phone and find someone. Go next door, give them the phone. They'll see it. If not, at the very least, they'll know it's my phone, or figure it out. And she can tell them, can't she? Where we are?\"\n\nCaroline hesitated. \"Carrie is...special.\"\n\n\"She's very special,\" Ginny said softly, seeing Carrie creeping up behind her mother. \"Very special and very brave. She'll know just what to do. Won't you, Carrie?\"\n\nCarrie stepped out from behind her mother, and into the feathery edges of the light circle. \"Scared.\"\n\n\"Bad things happen!\" Deke cried suddenly. \"Bad things happen when we try to go outside! Don't let her, Mama. Don't let her go!\"\n\n\"Deke, hush!\" Caroline shook him by the shoulders. \"You're going to scare your sisters. Stop. It will be fine. We have to do this. You don't understand, but we do.\"\n\nHorribly, Ginny thought she understood. Caroline remembered a life on the outside, but none of these children did. After living their entire lives down here, was it any wonder they might be afraid of what waited for them in the outside world? What horror stories had George Miller told them to keep them willingly imprisoned?\n\n\"Tate went out! Tate went out and he never came back! He got lost! Tate didn't come back!\"\n\nThe back of Caroline's hand cracked across Deke's face hard enough to send him splashing to his hands and knees in the water. He came up sputtering, backing away, across the room.\n\nGinny's stomach churned at the violence, and she cringed, covering her baby as best she could.\n\nNobody else seemed surprised at Caroline's actions. She spoke calmly, \"Deke. I don't want to make you go to the corner. Not now, like this. But if you can't get yourself under control, you will go to the corner even if you have to sit up to your ears in this water. Do you understand me?\"\n\nIn the white light, Deke looked extremely pale. Water sluiced over his face, mimicking tears. He nodded after a moment. \"Yes.\"\n\nCaroline sighed. \"Carrie, can you take this lady's phone up through the ducts. Out of the house? Can you go outside?\"\n\nCarrie shuddered visibly. She hadn't been splashed with water; her tears were real. She shook her head.\n\nAgain, Ginny was inspired. \"Carrie. You know the other children? The ones you were playing with here in the house? Kelly and Carson, the night of the party.\"\n\nCarrie looked fearful, then nodded.\n\n\"They live in the house next door to me. You can go to their house and see them. Their mother is very nice, she's a nice lady. Like me. You can trust her too.\"\n\nCaroline pushed the hair from Carrie's forehead and cupped her cheeks. The chain clanked. \"You have to be brave and strong, like Miss Ginny said. You have to do this for us, baby. It's...well. You just have to. Okay?\"\n\nCarrie nodded again. Ginny gestured for the phone. Carefully, so she didn't disturb her sleeping, still-unnamed baby, Ginny tapped a message into her phone. She saved it as a note, sent it as a text that failed, and added it to an email that also failed. It didn't matter, once Carrie got the phone into a place where there was a signal, all of the messages would be sent. She checked the time too.\n\nFive hours since the last time she'd spoken to Sean.\n\nIt seemed so impossible that all of this had happened in so little time, but she didn't need a pinch to prove she wasn't dreaming. She had the pain between her legs and the child snuffling lightly in her arms. She had the splash of water and five pale, wide-eyed faces staring at her.\n\n\"Do you have a plastic bag?\" Ginny asked. \"The kind for sandwiches. One that closes at the top. In case she drops it, so it doesn't get wet.\"\n\n\"No. We have a few plastic bags, but we used them to line the toilet.\" Caroline shook her head.\n\n\"Never mind. She's going up, right?\" Ginny took a deep breath, determined to be positive. \"Carrie. You hold it tight. Is there something else we can put it in for her? A purse or a bag we can tie to her?\"\n\n\"I have a stocking,\" Linna piped up shyly. \"It got a hole. We didn't mend it yet. She can use the match for it. It's long enough that we can tie the end to her shirt somehow.\"\n\n\"Yes. That will work.\" Ginny held up the phone. \"But once she takes it...we won't have any light.\"\n\nCaroline smiled. \"We've sat in the darkness before. We can do it again.\"\n\nGinny cringed at the idea of sitting here in the pitch black with four feet of water all around her, the bed like a boat in an unstable sea. But it was the only option. She needed to get her baby out of here.\n\nShe checked the message one last time, hoping something had managed to get through, but nothing had. She thumbed the screen and tilted it to show Carrie. \"See? Like this. You just push this button when you get there, you show it to them. Okay? That's all you do.\"\n\nThe phone looked twice as big in Carrie's tiny hand. Ginny lost her breath when it looked like the girl was going to drop it over the side of the bed, but she caught it. Carrie pushed the button and held up the phone for Ginny to see.\n\n\"Yes. Like that.\" Ginny took the phone, then the sock Linna handed her. \"Okay. Are we ready for the dark?\"\n\n\"Yes. We are,\" said Caroline.\n\nIt was instant and total and unyielding, that darkness. But it wasn't unfamiliar, and it was no longer terrifying. Carrie found her way into the ductwork. The rest of them huddled on the damp mattress, piled with equally damp blankets, trying to stay warm. The baby woke, wailing, and found Ginny's breast. The sharp pull of his lips and tongue on the sensitive nipple stung, and Ginny had no milk yet, but the baby sucked anyway and seemed content.\n\nGinny blinked, eyes straining, but the darkness here was total. It was something of a comfort, actually. She closed her eyes and let herself drift for a minute or so before forcing herself back to consciousness.\n\nCaroline's rusty laughter gritted out of the darkness. \"Ginny. I lied.\"\n\nGinny roused, trying to push through the wall of her weariness to understand. \"About what?\"\n\n\"Being ready for the darkness,\" Caroline said. \"Nobody is ever ready for it.\"\nChapter Forty-Four\n\n\"I was thirteen when I figured out I could get my daddy to give me anything I wanted.\" Caroline's words drifted out of the darkness.\n\nGinny had no idea how long they've been sitting in silence. Someone had moved onto the bed to cuddle next to her. She thought it must be Trixie by the size and the flow of hair. Trixie snored lightly.\n\n\"Caroline, you don't have to tell me this now.\"\n\n\"Now,\" Caroline said, \"might be the only time I ever tell it.\"\n\nThat would not be true; Ginny knew that. Caroline would have to tell her story a lot of times very soon. To the police, certainly. To her brother. To the media, if she wasn't careful, or if she wanted to earn something for her pain. Still, Ginny thought she knew what the other woman meant. What better time to tell a story like this, but in the dark?\n\n\"All my friends liked my dad the best because he was always around to do whatever we needed. He could always drive us to the mall, or he'd drop us off at the pool and pick us up. And he'd sometimes stop on the way home to treat us to ice cream. He didn't try to talk to us like the other dads did. He just listened. He let me pick the music. He was...cool. He was the cool dad.\n\n\"When I was thirteen, I wanted a bikini for the summer. My mother said no. She wanted me to wear a one-piece. But all my friends had them. She made me buy a babyish suit. I was the only one. The only one without a cool swimsuit. And my dad noticed.\"\n\nCaroline's voice was low and warm, like melting butter. Or maybe Ginny was melting, dissolving into the darkness and shadows. She put a hand between her legs and it came away wet, the smell of copper strong on her fingers. She was bleeding. A lot.\n\n\"He took me to the mall and gave me money to buy a new suit. He saw all my friends wearing them. He wanted me to have one. To be like them. They fought about it, really loud. Brendan hid in his room with his music playing loud, but I could hear them in their bedroom. He won the fight. He always won the fights. And he brought the suit to me; he tossed it down on the bed, and told me to try it on. To show him how it fit. So I did.\"\n\nPause. A breath. Silence, but for the sound of water, trickling.\n\n\"I figured out then that I could get whatever I wanted out of him. Out of boys, in general. With just a little show of T and A.\"\n\nThe phrase seemed oddly innocent, but fitting.\n\n\"Do you know,\" Caroline asked suddenly, \"if she knew there was something going on? My mother, I mean. She had to have known. Didn't she?\"\n\nGinny couldn't tell if Caroline was desperate for affirmation or denial. \"I don't know.\"\n\n\"She knew,\" Caroline whispered. Then, even softer, \"Do you think she misses me? Will she be happy to see me again?\"\n\n\"Oh, Caroline. I'm so sorry. But your mom passed away.\"\n\nMore silence. The baby made a meeping grunt, but settled. Ginny's thighs stuck together when she moved.\n\n\"So...I'm an orphan.\" Rusty, grating laughter became a sob. \"Just as well. Just as well.\"\n\nGinny had lost track of time long ago. She listened for the sound of shouts or voices, but heard nothing. \"These rooms...soundproofed?\"\n\n\"Yes. He built this house like this, you know. Back before he even met my mom. He built this house himself.\"\n\nGinny's lips pulled back from her teeth. \"Jesus Christ.\"\n\n\"He told me he was taking me to the beach for the weekend. As a special treat. We weren't supposed to let Mom know. He said nobody could know, or they'd want to come along. That's why he picked me up on the way home from school. I was walking. I was just scuffing my feet along in the leaves and thinking about going to the football game the next day. He pulled up in a van. I didn't know it, and I would never ever have gone inside.\" Caroline paused. Coughed. She coughed for a long time, and when she stopped, her voice had gone rough and thick with phlegm. \"But I saw it was my daddy, so I got in.\"\n\n\"Someone reported it. The van. They saw you get into it.\"\n\n\"But they didn't know it was him. Obviously. They thought a stranger took me.\"\n\n\"Yes,\" Ginny said.\n\nCaroline snorted. \"They were right.\"\n\nA sound like distant thunder rumbled, then stopped. Ginny felt the vibration in the bed, or maybe just thought she did. She strained, listening. They all did. But nothing else happened.\n\n\"When he showed me the bookcase, the hidden stairs, I thought it was amazing. So cool. And then when he showed me this little space, like a playroom, I thought it was so much fun. He told me this was our place, our special place, that nobody would ever know about it, and until he left me there and didn't let me out, I thought it was going to be great.\" Caroline sounded tired. \"Also, he lied. Because it wasn't just our special place. Not until later. Not until after.\"\n\n\"After what?\"\n\n\"After she died,\" Caroline said wearily, casually. \"I wasn't the first one.\"\n\n\"Oh. Oh God.\" There should've been more words than that, but Ginny had none.\n\n\"Her name was Terry. She was jealous of me, right from the start, because I had long hair and pretty teeth. She didn't have teeth. He'd pulled them so she couldn't bite.\"\n\n\"Please. You don't have to tell me this.\"\n\n\"I have to tell you this!\" Caroline hissed. \"I have been waiting for too long to tell someone this!\"\n\nGinny shut up after that. Caroline talked. Ginny listened.\n\n\"He never touched me, not until she died.\" That seemed to be an important fact. \"He promised he'd let me out. But I saw what he did to Terry. I didn't want to lose my teeth, or have him cut my hair. I thought someone would find me. I mean, I was in the house. In the goddamned house, right? How could they not know? How could they not hear me screaming? I did try to get away. So he put on the chain. And then...a baby.\"\n\nAt every pause, Ginny hoped Caroline wouldn't say anything else. She prayed for someone to find them. But other than another rumble that sounded like distant thunder, there was only blackness and Caroline's voice.\n\n\"He took it away from me. It was a boy. He said I wasn't old enough to take care of a baby. I was fifteen by then, or...I think I was. I lost track of time. I made marks, for a while, on the wall. But he saw them and erased them. So he took the baby. It was small anyway. I think it would've died. I think...it did die. Didn't it, Ginny? Did my baby die?\"\n\nGinny thought of the bones in her backyard. \"I think so, honey. Yes.\"\n\nCaroline gave a shuddering sigh. \"Then came Tate, and he let me keep him. Said I needed something to keep me occupied when he couldn't visit me. Tell me something. When did my mother die?\"\n\n\"I'm not sure, Caroline. But I know that she and your brother moved out of this house about a year after you went missing.\"\n\n\"Oh. Oh. Oh,\" Caroline said. \"Oh. She didn't stay? So she didn't know. Really? She didn't know I was here?\"\n\nMrs. Miller might have suspected something, but it couldn't have been this. \"I don't think so, honey. I'm sure she didn't know. I think she really thought you were...gone.\"\n\n\"She thought I was dead.\"\n\n\"Yes. I think so.\"\n\nCaroline gave a barking sob. \"Oh. Okay. That's good. That's good, you know? Because she just thought I was dead, she didn't leave me here on purpose.\"\n\n\"No. I don't think so.\" Ginny reached blindly to find Caroline.\n\nCaroline turned and pressed herself to Ginny's shoulder. Her tears were hot and wet on Ginny's neck; the baby gave a startled cry. Ginny put her hand between Caroline and the baby, but didn't push the other woman away. She cradled her as best she could.\n\n\"No, Caroline. She didn't leave you here on purpose. I'm sure of it.\"\n\nThey sat that way for another interminable amount of time. Ginny had never known kids to be so quiet for so long, but thought perhaps they'd all fallen asleep. Her muscles were stiff, and her back ached. Every movement sent another hot pulse between her legs, and her head spun.\n\n\"I'm bleeding,\" Ginny said. \"Too much, I think.\"\n\n\"I thought once the children came, he'd leave me alone. He promised, after each one, he would take us all upstairs. After the ones that didn't make it, he always said he would take me to the hospital. But he never did.\"\n\n\"How many times?\"\n\n\"I don't know,\" Caroline said, but Ginny knew that was a lie. No mother who lost a child could ever forget it, no matter how many times it happened.\n\n\"Caroline, I feel really bad. I feel really sick. I'm losing too much blood.\"\n\n\"Wait.\"\n\nThe bed moved as Caroline got up and returned a few minutes later. She took away the material bunched between Ginny's legs and replaced it with another. Ginny forced her mind from thinking of germs. Sepsis. Her breath shuddered. Oh God. Where was Sean? Where were the police?\n\n\"Shh. Shhh, listen.\" Ginny sounded drunk. She wished she were.\n\nDeke spoke up. \"It's Tate!\"\n\n\"Tate's gone,\" Linna snapped. \"Don't be a dumb-bum. Tate went up and he didn't come back. He's dead!\"\n\n\"Tate knew he was getting old. Tate said, what happened if he died? What would happen when he didn't bring the food? So Tate tried to get the key, the special key, but he pushed Tate down so hard it cracked his head open. And then he went crazy.\" Deke was sobbing, shouting as Caroline and Linna tried to shush him.\n\nHis shouts startled the baby awake. He began to wail. Ginny rocked and soothed, but nothing would calm the infant.\n\n\"Tate said the only way out was through the walls! And we didn't have food anymore, it was gone, and the heat was off, the 'lectric was off, it was cold! Tate went up,\" Deke said, more quietly, \"and he didn't come back.\"\n\n\"When did Tate go up?\" Ginny asked. \"Not so long ago. After we moved in, right? It was after that. George died in the hospital, and we bought the house four months after that. Your food ran out. Your power ran out. And Tate went up, into the duct, and didn't come back.\"\n\nThe furnace. The flickering lights. Oh God.\n\nThe flies.\n\nGinny heard herself muttering, though her words were a slurred jumble of sounds she couldn't distinguish, even to herself. She struggled to move, to get up, and felt herself falling into a different sort of darkness. She clutched her screaming baby. She would not drop the baby. She would not drop her son.\n\nThe sound of thunder wasn't distant this time.\n\n\"Ginny! Wake up! You have to get up! Now!\"\n\nGinny fought the waves of black and red, the swirling descent of gray. Someone shook her. Someone tried to take her baby from her arms, and she screamed, fighting.\n\nThe roar grew louder. So did the sound of water. Someone pulled her, got her standing, pushed her toward a spindle-legged chair shoved in front of the dresser. Which she could see.\n\nThere was light.\n\nFaint, from the room outside this one, but after so long in the dark, any speck of light was bright as a star. Ginny put one hand on top of the dresser, the other cradling her shrieking child. Screams were good, better than silence. Better than a baby who made no sound. Someone shoved her from behind, and her muscles protested as she tried to lift her leg and get on top. Something ripped inside her, something slid down her thighs, something thick and tearing her on the way out.\n\nClarity hit her along with the pain. The rumble and roar had been the basement wall collapsing. The water was rushing in, frothing and foaming and rising.\n\nThe ceiling in these rooms was not high.\n\nTrixie was already crouched on top of the dresser. There was no room for Ginny there. Instead, Ginny put her baby there, one hand on her son to keep him from toppling off. The naked infant squalled in fury.\n\nThe water rose.\n\nCaroline and Ginny clung to the dresser. On the opposite side of the room, Linna, clutching the cat, and Deke had scrambled on top of the table. The water had already reached their shins and was still climbing. Their backs pressed to the ceiling, necks angled.\n\nImpossibly, the water rose.\n\nIt was up to Ginny's chest, and filled with rubble. Something sharp hit her leg and sent her swirling into the muck, her head under the water with barely enough time to grab a breath. She came up choking, flailing with one hand, the other hooked into the dresser's metal handle.\n\n\"Asher!\" Until that moment, she hadn't known what she wanted to call her baby. But now she knew there was nothing else to call him.\n\nGinny got to her feet and put a hand on top of the dresser, found her son and held him in place. She didn't need a chair to get to the top of it now, because the water buoyed her. Trixie shuddered, and Ginny took her hand. She looked into her face, realized she could make out the sight of the girl's dark eyes, her open mouth.\n\nThe water rose and pushed her upward, to the low-hung ceiling. She clung to the dresser. She kicked to keep herself from being swept away. She focused on Trixie's face.\n\n\"I'm here,\" she cried. \"I'm here. I'm here.\"\n\nSomeone shouted her name. More light poured into the room, bright and glaring and focused in beams. Flashlights. Men in boots and yellow overcoats.\n\nAnd Sean.\n\nHe grabbed her, but she wouldn't let go of the dresser without Asher safely cradled against her. Firemen, the other men were firemen. They shouted, pointing, grabbing at Linna and Deke. Sean, hair plastered to his face, reached for Trixie and pulled her from the dresser, into the water that was now neck high. Ginny held her son over her head and moved, following him, the front of her nightgown gripped tight in Sean's hand.\n\nSomehow, he pulled them out, over the remnants of the collapsed basement wall and the dirt that filled the space. Down the corridor that rose at the slight incline so that by the time they reached the metal door, propped open with a large metal shaft, the water was only to their knees. Then they were through the door, the stairs in front of them and the firemen with the other children close behind them.\n\nThen they went up.\n\nAnd they didn't come back down.\nChapter Forty-Five\n\nGinny rocked.\n\nAsher, warm and clean, sucked sporadically at one breast, rosebud mouth going periodically lax. Ginny smoothed her hand over the baby's soft, downy hair. She sang him a lullaby, soothing her son to sleep. In the next room, her husband slept too.\n\nIn another house, another place, Caroline Miller and her children were also warm and fed and safe. But did they sleep? Did they dream? Ginny didn't know. The police and Social Services and Brendan Miller and lawyers and the media had all taken their pieces of Caroline and the children from the basement. Ginny had visited them in the hospital just once, right after they were found, and then only because she and Asher had been admitted as well.\n\nShe and Sean had managed, so far, to keep the reporters camped out in front of the house from bothering them too much; she supposed at some point they'd have to talk to them, or they would all give up and go away. They bothered Sean more than they did her, if only because Ginny clung to the idea that somehow sharing what happened might help Caroline and her family more than keeping all of it as yet one more secret.\n\nGinny was tired of secrets.\n\nIn her arms, Asher stirred and let out one small cry, then fell further into sleep. Ginny rocked, her eyes closed, smiling at the sound of Noodles's collar jingling as the cat padded into the library and jumped onto the Victorian couch where she'd taken to spending most of her time. She was still too thin, but she never got underfoot anymore, and she never ran into places she wasn't supposed to go.\n\nGinny rocked, dozing in the dark. There came the creep of small bare feet, the whisper of cold air swirling, the soft brush of fingertips against her own. Maybe those things would always be in this house. They'd never go away. But Ginny didn't open her eyes, because she knew it was all a dream. That was all, just a dream.\n\nShe wasn't haunted any longer.\nAbout the Author\n\nMegan Hart writes books. Some of them use a lot of bad words, but most of the other words are okay.\n\nShe can't live without music, the internet, or the ocean, though she and soda have achieved an amicable parting of ways. She can't stand the feeling of corduroy or velvet, and modern art leaves her cold. She writes a little bit of everything from horror to romance.\n\nFind her at: www.meganhart.com, www.twitter.com\/megan_hart, and www.facebook.com\/megan.hart!\nLive fast, die young, and leave a bloodthirsty corpse!\n\nGhost Heart\n\n\u00a9 2016 John Palisano\n\nLive fast, die young, and leave a bloodthirsty corpse. That's the saying of a new pack of fiendish predators infesting a New England town. They're infected with the Ghost Heart, a condition that causes them to become irresistible and invincible...as long as they drink the blood of the living. But these vampires don't live forever, and as the Ghost Heart claims them, their skin loses color and their hearts turn pale. When a young mechanic is seduced by the pack's muse, he finds falling in love will break more than his heart.\n\nEnjoy the following excerpt for Ghost Heart:\n\nWe headed outside, toward the alley behind the bar; we walked into the nice, cool drizzle. I loved the change in climate. The bar had been so hot and stuffy, so being outside was welcome. Mike hiked up the collar of his leather jacket and couldn't walk fast enough.\n\nOnce we made it around the corner I spotted the Whistleville River just beyond the bridge. I thought about how, a few miles up, on the east side of town, the Jeep in my shop had plunged over that same bridge. I imagined the bodies of the kids stuck there, tangled in the water grass, decomposing, and slowly turning into fish food. I shook off the idea\u2013\u2013just chalked it up to collateral damage from a nice Anchor Steam buzz.\n\nWe rounded the old brick building and made our way toward the parking lot in back. Rain filtered through the street lamps, falling in curtains, shifting and moving in the crosswinds. A new spotlight lit the rain from the side. Headlights. Tires screeched a few hundred feet in front of us.\n\nThere was a Jeep. Another damn Jeep, I thought. It raced toward us, stopping with a hard jerk. The rainfall increased.\n\nA window rolled down, but I couldn't see inside.\n\nA raspy, deep voice, said, \"Hey.\"\n\n\"What do you need, man?\" I asked. \"Come on.\"\n\n\"You know exactly what you did,\" he said. \"Disrespecting me and my girl.\"\n\n\"I don't know you, man,\" I said. \"I think you're mistaking me for someone else.\"\n\n\"So you're calling me a liar?\"\n\nI pulled my collar up. \"Look, friend,\" I said. \"I don't have time for this. I'm drunk. I want to go home.\"\n\n\"Oh, sure,\" he said. \"You just want to go home and forget all of this.\"\n\nHis door opened. It was the big, bald guy that had been next to Minarette. \"Too late,\" he said. \"The damage is already done. Talking about me.\" Two others got out of the Jeep. They looked like normal, clean-cut guys in their 20s, only there was something wrong with their expressions. I thought they had to have been high on something. Their eyes seemed so vacant.\n\n\"Like I just told you, man\u2013\u2013I'd have no reason to. I don't even know you.\"\n\nOne of the cronies said, \"Come on, Damian. Get him already.\" I had a name.\n\nI looked over to Mike, who was staring at the other two guys\u2014guys we didn't know\u2014who'd found their way over toward him.\n\nI got shoved. Damian. He was in my face. \"What're you going to say now?\"\n\nMy head got real dizzy. Too much booze. I wasn't in any kind of shape to fight.\n\nHe shoved me again. I tried to stay up, and did so barely.\n\nA holler, and Mike was against a wall.\n\n\"Leave him alone,\" I said. \"This isn't his fault.\"\n\n\"Well,\" Damian said, \"I don't care.\"\n\nHe pushed me, but I stepped back.\n\n\"This is bull,\" I said. \"And I sure as hell ain't gonna apologize for something I didn't do.\"\n\nHe swung for my face\u2013and just barely hit my shoulder.\n\nHe was way stronger than me. I was in trouble. My shoulder thudded. There'd be a nasty bruise. The bastard.\n\nI put up my fists while Damian hovered.\n\nMike was on the ground. One of the goons was on top of him. Looked like they were robbing him.\n\n\"Get off him,\" I said. \"You bullies. What the hell?\"\n\nHis fist, my face.\n\nEvery color exploded\u2013\u2013like I was in the middle of a mortar shell.\n\nI tried to put up my arms to block him, but I'd been knocked for a loop.\n\nAnd he hit me again.\n\nI felt incredible pain.\n\nI felt nothing.\n\nMy feet failed first\u2014my legs followed.\n\nDown for the count, the others cheered.\n\nEverything felt like it'd turned slow motion. Reaching behind, I sat myself down. The world spun. The hit was too much for me to brush off.\n\nDamian kicked me in the chest. My breath forced out, the hit knocked me to the pavement.\n\nI heard more cheering, and more laughing.\n\nMy breath wouldn't catch.\n\nDamian leaned over me. His pug-like face looked inches away, then, miles away.\n\nHe said something, but it didn't register. Everything sounded like we were inside a tunnel.\n\nMore kicks to my middle.\n\nMy body and mind were separate. There was nothing grounding me.\n\nI turned my head to check for my friend.\n\nThe two goons were centered on his head. They'd leaned down and it looked as though they were eating him. What the hell?\n\nThey were on his neck. They held him down with their hands. The veins on the backs of their hands were puffed out, like they'd done steroids.\n\nOne lifted his head. He looked so normal, other than those vacant eyes. The ghoul could've been anyone. Blood rimmed his lips and dripped from his chin: it was Mike's blood. Our eyes met for a moment and from between the ghoul's lips a worm-like thing slipped out. The tip was pointed and sharp. It twitched just a bit and I knew it was his tongue, although it was different than any other I'd ever seen or heard of.\n\nHe lowered his head again. When he connected, Mike jerked. It seemed more like a reflex than a reaction. Mike's eyes were shut. His skin had gone pale.\n\nThey were draining him of his blood.\n\nOne of their shirts had come undone in the front in the scuffle. His chest looked so white it was almost clear. I swear I could see his insides moving...could make out the faint movement of a beating, translucent heart. A stream of red entered the chamber, blossomed, and then colored the cradle of veins surrounding the organ. Blood. Mikey's blood\u2013drained from him and taken inside the ghost-like heart of the ghoul kneeling over him. How could the blood get from its tongue to its own heart so damn fast? I thought.\n\nI tried to yell and scream, but nothing came out. I was in shock. Nothing worked. I looked up to see where Damian was. He'd walked away from me and stood a few feet away from his creeps; he watched them work. Damian's skin was the same pasty white as his consigliore.\n\nI turned my head and every nerve inside me seemed to explode at once. Mike laid still a dozen feet from me. Rain ran from his forehead, down his cheek. Only it wasn't just rain, I noticed. A good stream of blood ran within the rain. There were unnatural gaps in his throat where they'd fed. Moon-shaped bruises marked his flesh.\n\nThose bastards. I'd get 'em. Somehow, some way, I knew I would. They stood over him, both of them wearing goatee-shaped smears of blood. Formerly empty eyes glistened. The blood\u2013\u2013Mike's blood\u2013\u2013had reinvigorated them.\n\nThe sons of bitches.\n\nDamian looked lit up from the inside. That sounds funny, because people don't glow. It's just that the rain seemed not to touch him. His skin didn't look well at all. Maybe it was because he was so pumped from the fight that he just looked that way. Who knows?\n\nHe'd won. They'd won in no time.\n\nI considered myself a big guy, but I'd been outgunned. I was shocked at how fast Damian had taken me down. It was inhuman.\n\nDrugs. Had to have been drugs. Maybe coke. Definitely some kind of upper. That'd explain it.\n\nThen? One final insult\u2013a swift kick to my balls.\n\nThe blinding pain knocked me down from space, and back inside my beaten husk.\n\nBile rose. I turned my head, spewing hot, half-digested beer everywhere.\n\n\"You should take a picture,\" one of the goons said.\n\n\"Don't need to,\" Damian said, \"I won't forget this.\"\n\nNeither would I.\n\nDamian laughed and looked down at me. \"You loser,\" he said. \"You got something smart to say now?\" He spat at me, turned, and walked away. I heard them get inside the Jeep, turn it on, and drive away.\n\nHe and his crew were gone, leaving Mike and me lying broken in the rain.\n\nOnce I could no longer hear their Jeep leave, I pulled myself up. Everything hurt and stung. My wet clothes clung to me. I crawled over toward Mike.\n\nNudged him.\n\nGot nothing.\n\nKept pushing, tapping, and calling his name.\n\nNo response.\n\nI used my thumb to try and open an eye.\n\nNo reaction.\n\nI checked for a pulse.\n\nFound one, but it was faint.\n\nThe skin of his wrist was cold.\n\nI tried CPR, but never really learned how, so it was for naught.\n\nReaching inside my coat pocket, I found my phone in pieces. One of Damian's kicks must've smashed it. Still, I tried to turn it on. Just in case I could make one more call. No dice.\n\nI screamed at Mike.\n\nA few folks had made their way over, probably after hearing my yelling. I hollered for them to call an ambulance, and the cops. They didn't have to, though. Blue and red lights arrived moments later, just as the rain let up. I got myself to a bench, and sat.\n\nWhen I finally got my head on straight, I watched as the EMTs put my friend on a stretcher. They rushed, going as fast as they could. Mike was in big trouble.\n\nSo was I. \neBooks are not transferable.\n\nThey cannot be sold, shared or given away as it is an infringement on the copyright of this work.\n\nThis book is a work of fiction. The names, characters, places, and incidents are products of the writer's imagination or have been used fictitiously and are not to be construed as real. Any resemblance to persons, living or dead, actual events, locale or organizations is entirely coincidental.\n\nSamhain Publishing, Ltd.\n\n11821 Mason Montgomery Road Suite 4B\n\nCincinnati OH 45249\n\nLittle Secrets\n\nCopyright \u00a9 2016 by Megan Hart\n\nISBN: 978-1-61923-087-3\n\nEdited by Don D'Auria\n\nCover by Kanaxa\n\nAll Rights Are Reserved. No part of this book may be used or reproduced in any manner whatsoever without written permission, except in the case of brief quotations embodied in critical articles and reviews.\n\nFirst Samhain Publishing, Ltd. electronic publication: February 2016\n\nwww.samhainpublishing.com\n\n","meta":{"redpajama_set_name":"RedPajamaBook"}}
+{"text":"\n\n\n\nProduced by Chuck Greif and the Online Distributed\nProofreading Team at http:\/\/www.pgdp.net (This file was\nproduced from images available at The Internet Archive)\n\n\n\n\n\n\n\n\n                     [Illustration: E. F. BENSON.]\n\n\n\n\n                               A REAPING\n\n                                  BY\n                             E. F. BENSON\n\n                       [Illustration: colophon]\n\n                        THOMAS NELSON AND SONS\n                       LONDON, EDINBURGH, DUBLIN\n                             AND NEW YORK\n\n\n\n\n                                  TO\n                          LADY EVELYN LISTER\n\n\n\n\nCONTENTS\n\n\nJUNE                       9\n\nJULY                      35\n\nAUGUST                    63\n\nSEPTEMBER                 87\n\nOCTOBER                  113\n\nNOVEMBER                 149\n\nDECEMBER                 181\n\nJANUARY                  215\n\nFEBRUARY                 247\n\nMARCH                    281\n\nAPRIL                    319\n\nMAY                      353\n\n\n\n\nA REAPING\n\n\n\n\nJUNE\n\n\nOf all subjects under or over the sun, there is none perhaps, even\nincluding bimetallism, or the lengthy description of golf-links which\none has never seen, so utterly below possible zones of interest as that\nof health. Health, of course, matters quite enormously to the\nindividual, but nobody with good health ever gives two thoughts (far\nless one word) to the subject. Nobody, in fact, begins to think about\nhealth until his own begins to be inferior. But, then, as if that was\nnot bad enough, he at once clubs and belabours his unhappy friends with\nits inferiority. It becomes to him the one affair of absorbing\nimportance. Emperors may be assassinated, Governments may crumble, it\nmay even be 92 degrees in the shade, but he recks nothing of those\ncolossal things. He ate strawberries yesterday, and has had a bilious\nheadache almost ever since. And the world ceases to revolve round the\nsun, and the moon is turned to blood, or ashes--I forget which.\n\nBut the real invalid, just like the man who enjoys real health, never\ntalks about such matters. It is only to the amateur in disease that they\nare of the smallest interest. The man who is well never thinks about his\nhealth, and certainly never mentions it; to the man who is really ill\nsome divine sense of irresponsibility is given. He brushes it aside,\njust as one brushes aside any innate inability; with common courage--how\nlavishly is beautiful gift given to whomever really needs it--he makes\nthe best of other things.\n\nThese poignant though obvious reflections are the outcome of what\noccurred this evening. I sat between two friends at dinner, both of them\npeople in whom one's heart rejoices. But one of them is obsessed just\nnow with this devil of health-seeking. The other has long ago given up\nthe notion of seeking for health at all, for it is not for her. She\nfaces incurable with gaiety. So I have to record two conversations, the\nworse first.\n\n'Oh, I always have ten minutes' deep-breathing every morning. It is the\nonly way I can get enough air. You have to lie on your back, you know,\nand stop one nostril with your finger, while you breathe in slowly\nthrough the other; and you should do it near an open window. There is no\nfear of catching cold, or if you do I can send you a wonderful\nprescription.... Then you breathe out through the other nostril. I wish\nyou would try it; it makes the whole difference. No, thanks, caviare is\npoison to me!'\n\n'Well, so is arsenic to me,' I said. 'But why say so?'\n\n(It did not sound quite so brusque as it looks when written down, and\nnative modesty prevents my explaining how abjectly patient I had been up\ntill then.)\n\nThen there came the reshifting of conversation, and we started again,\nwith change of partners.\n\n'I do hope you will come to see us again in August,' said the quiet,\npleasant voice. 'I shall go up to Scotland at the end of the month. Your\nbeloved river should be in order: there has been heaps of rain.'\n\nBut I could not help asking another question.\n\n'Ah, then they let you go there?' I said.\n\nShe laughed gently.\n\n'No, that is just what they don't do,' she said. 'But I am going. What\ndoes it matter if one hastens it by a few weeks? I am going to shorten\nit probably by a few weeks, but instead of having six tiresome months on\nboard a yacht, I am going to have rather fewer months among all the\nthings I love. Oh, Dick quite agrees with me. Do let's talk about\nsomething more interesting. Did you hear \"Tristan\" the other night? No?\nRichter conducted. He is such a splendid Isolde! There is no one to\napproach him!'\n\nThere, there was the glory of it! And how that little tiny joke about\nRichter touched the heart! Here on one side was a woman dying, and she\nknew it, but the wonder and the pleasure of the world was intensely\nhers. There, on the other, was the excellent Mrs. Armstrong. She could\nnot think about the opera or anything else except her absurd\ndeep-breathing and her ridiculous liver. Nobody else did; nobody cared.\nEven now I could hear her explaining to her left-hand neighbour that\nnext to deep-breathing, the really important thing is to drink a glass\nof water in the middle of the morning. Slowly, of course, in sips. And\nshe proceeded to describe what the water did. Well, I suppose I am\nold-fashioned, but I could no more think of discussing these intimate\nmatters at the dinner-table than I should think of performing my toilet\nthere. Besides--and this is perhaps the most unanswerable objection to\ndoing so--besides being slightly disgusting, it is so immensely dull!\n\nHowever, on the other side there was a topic as entrancing as the other\nwas tedious, and in two minutes my other neighbour and I were deep in\nthe fascinating inquiry as to how far a conductor--a supreme\nconductor--identified himself with the characters of the opera.\nCertainly the phrase 'Richter is such a splendid Isolde' was an\nalluring theme, and by degrees it spread round the corner of the table\n(we were sitting close to it), and was taken up opposite, when a member\nof the Purcell Society gave vent to the highly interesting observation\nthat the conductor had practically nothing to do with the singers, and\nwas no more than a sort of visible metronome put there for the guidance\nof the orchestra. It was impossible not to retort that the last\nperformance of the Purcell Society completely confirmed the truth of\nthat view of the conductor. Indeed, the chorus hardly thought of him\neven as a metronome. Or else, perhaps, they were deaf, which would\naccount for their sinking a tone and a half; in fact there were flowers\nof speech on the subject.\n\nBut how extraordinary a thing (taking the view, that is to say, that a\nconductor conceivably does more than beat time) is this transference of\nemotion, so that first of all Wagner, by means of merely black notes and\nwords on white paper, can inspire the conductor with that tragedy of\nlove which years ago, he wove out of the sunlight and lagoons of\nVenice; that, secondly, the conductor can enter into that mysterious and\nmystical union with his band and his singers, and reflect his own mood\non them so strongly that from throat or strings or wailing of flutes\nthey give us, who sit and listen, what the conductor bade them read into\nthe music, so that all, bassoons and double-bass, flutes and strings,\ntrumpets and oboes and horns, become the spiritual mirror of his\nemotion. By means of that little baton, by the beckoning of his fingers,\nhe pulls out from them the music which is in his own soul, makes it\ncommunicable to them. Indeed, we need not go to the Society for\nPsychical Research for experiments in thought-transference, for here is\nan instance of it (unless, indeed, we take the view of this member of\nthe Purcell Society) far more magical, far further uplifted out of the\nsphere of things which we think we can explain. For the mere degrees of\nloud or soft, mere alterations in _tempo_, are, of course, less than the\nABC of the conductor's office. His real work, the exercise of his real\npower, lies remote from, though doubtless connected with them. And of\nthat we can explain nothing whatever. He obsesses every member of his\norchestra so that by a motion of his hand he gets the same quality of\ntone from every member of it. For apart from the mere loudness and the\nmere time of any passage, there are probably an infinite number of ways\nof playing each note. Yet at his bidding every single member of the band\nplays it the same way. It is his thought they all make audible with a\nhundred instruments which have all one tone; else, how does that unity\nreach us sitting in our stalls?\n\nThat is the eternal mystery of music, which alone of the arts deals with\nits materials direct. It is not an imitation of sound, but sound itself,\nthe employment of the actual waves of air that are the whistle of the\nwind, and the crash of breakers, and the love-song of nightingales. All\nother branches of art deal only second-hand; they but give us an\nimitation of what they wish to represent. The pictorial artist can do no\nmore than lay a splash of pigment from a leaden tube on to his canvas\nwhen he wishes to speak to us of sunlight; he can only touch an eye with\na reflection in its corner to show grief, or take a little from the\nsize of the pupil to produce in us who look the feeling of terror that\ncontracts it. Similarly, too, the sculptor has to render the soft swell\nof a woman's bosom in marble, as if it was on marble a man would pillow\nhis head. It is all a translation, a rendering in another material, of\nthe image that fills us with love or pity, or the open-air intoxication\nof an April morning. But the musician works first-hand; the intangible\nwaves of air, not a representation of them, are his material. It is not\nwith a pigment of sound, so to speak, that the violins shiver, or the\ntrumpets tell us that the gods are entering Valhalla. Music deals with\nsound itself, with the whisper that went round the formless void when\nGod said, 'Let there be light,' with all that makes this delicate\norchestra of the world, no copy of it, no translation of it, but it\nitself.\n\nAnd for the time being, while the curtain is up, the control of these\nforces, their wail and their triumph, belongs to the conductor. He gives\nthem birth in the strings and the wind; he by the movement of a hand\nmakes them express all that sound expressed to the magician who first\nmapped them on his paper. Indeed, he does more; he interprets them\nthrough his own personality, giving them, as it were, an extra dip in\nthe bath of life, so that their colours are more brilliant, more vital\nof hue. Or is the member of the Purcell Society right, and is the man\nwho gives us this wonderful Isolde only a metronome?\n\nIt is often said that the deaf are far more lonely, far more remotely\nsundered from the world we know, than are the blind. It is impossible to\nimagine that this should not be so, for it is not only the sounds that\nwe know we hear, but the sounds of which for the most part we are\nunconscious, that form the link between us and external things. It\ncommonly happens, as in the dark, that we are cut off from all exercise\nof the eyes, and yet at such moments we have not been very conscious of\nloneliness. But it is rare that we are cut off from all sound, and the\nloneliness of that isolation is indescribable. It happened to me once in\nthe golden desert to the west of Luxor, above the limestone cliffs that\nrise from the valley where the Kings of Egypt lie entombed.\n\nI had sat down on the topmost bluff of these cliffs, having tethered my\ndonkey down below, for the way was too steep for him, and for several\nminutes observed my surroundings with extreme complacency. Below me lay\nthe grey limestone cliffs, but where I sat a wave of the desert had\nbroken, and the immediate foreground was golden sand. Farther away, in\nall hues of peacock green, lay the strip of cultivated land, and beyond,\nthe steel blue of the ancient and mysterious river. It was early yet in\nthe afternoon, and the sun still high, so that the whole land glittered\nin this glorious high festival of light and colour. And, looking at the\nimperishable monuments of that eternal civilization, it seemed that one\ncould not desire a more convincing example of the kindliness of the\ncircling seasons, of the beneficence that overlooked the world from\ngeneration to generation, so that man might well say that this\ntreasure-house of the earth was inexhaustible. No breeze of any sort was\nstirring, but the air, pure, hot, invigorating, was absolutely still.\nBut at that moment I suddenly felt as if something was dreadfully wrong,\nthough I did not at once guess what it was. Then came the thought, the\nidentification of what was wrong: it seemed as if the world was dead;\nthen came the reason for it: it was because there was no sound. For a\nmoment I listened in order to verify this--listened with poised breath\nand immovable limbs. Yes, I was right: there was no sound of anything at\nall; for once the ears were deprived of the delicate orchestra that goes\nup, a hymn of praise, day and night from the earth. It was like a\ndreadful nightmare.\n\nI first tried coughing, to see if that would be companionable, but that\ndid not do; I coughed, and then silence resumed its reign. I lit a\ncigarette. I moved, rustled, even got up and walked a little, kicking\nthe pebbles that lay about in the sand. But that was no use, and I\nperceived where the defect was. I knew I was alive, and could make\nsounds, but what I wanted was some evidence that something else was\nalive. But there was none.\n\nSomehow this fact was so disquieting that I sat down again to think\nabout it. In my reasonable mind I knew that absolutely everything was\nalive, only there was at this moment nothing to tell me so. Not a fly\nbuzzed over the hot sand, not a kite was to be seen wheeling slow as if\nin sleep, a black speck against the inviolable blue that stretched from\nhorizon to horizon. I was the only thing alive as far as I had evidence.\nOr supposing--the thought flashed suddenly across me--supposing I, too,\nwas dead? And what was this--this dome of air and the golden sand? Was\nit hell?\n\nI cannot describe the horror of this. Momentary as was the sensation, it\nwas of a quality, a depth of surcharged panic, which comes to us only in\nnightmares. I was alone, I was not within touch, in this utter\nstillness, of any other consciousness, and surely that must be hell, the\nouter darkness of absolute loneliness, which not even the glorious\ngolden orb swung centre-high in the blue could ever so faintly\npenetrate. Indeed, it and this iridescent panorama at my feet only added\nsome secret bitter irony to the outer darkness. All the light, the\ncolour, the heat, which one had so loved was there still, but life was\narrested, and there was nobody.\n\nThen quite suddenly and unexpectedly the farcical happened, for from\nsome hundred yards away down below the steep cliff up which I had\nclimbed came a long discordant bray from my donkey, who perhaps felt\nlonely, too. But I have never heard a sound which was to the spirit so\noverpoweringly sweet. I heard that, and gave a long breath, and shouted,\n'Thank you very much!' for the whole glory of the noon, which silence\nhad blackened, was instantly restored.\n\n       *       *       *       *       *\n\nOne of the interesting things to which I have alluded, in contrast with\nthe tedium of Mrs. Armstrong's health, was occurring to-day, for the\nthermometer had indeed been up in the nineties, a fact which fills all\nproper-minded people with pride. Our dear, stuffy old London had\nregistered 92 degrees in the shade at Messrs. Negretti and Zambra's that\nmorning, and I with my own eyes had seen it. It was impossible not to be\nproud, just as it is impossible not to be proud when one is in a train\nthat is going over seventy miles an hour, a thing that may be timed by\nthe small white quarter-mile posts that are so conveniently established\nby the side of the line. Once I went in a train that did a mile and a\nhalf in seventy-three seconds. I have not got over my elation yet. Or\nwhen an extraordinarily vivid flash of lightning occurs, with a\ncongested angry spasm of thunder coming simultaneously with it, are you\nnot sorry for the nerveless soul that does not thrill with personal\nelation at power made manifest? Or when Madame Melba sings the last long\nnote of the first act of 'La Boh\u00e8me'? Or when the organist in King's\nCollege Chapel pulls out the tubas, making the windows to rattle in\ntheir leaded panes by the concussion of the astonished air? Or when a\nperfectly enormous wave rides in from the Atlantic, and is transformed\nsuddenly from the illustrious blue giant into a myriad cascades of snowy\nwhite, as, jovially dealing itself its own death, as it were, it is\ndashed against the brown steadfast rock of the land? Or when Legs (I\nshall speak of him soon), as he did to-day, sliced his drive very badly\nat the fourth hole at Woking, and hit the front of the engine of an\nup-train with extraordinary violence, and thereupon collapsed on the tee\nin speechless laughter for the sheer joy of the gorgeously improbable\nfeat?\n\nFor all these things, so I take it, are evidence of the splendid energy\nof things in general in which we, each of us, have our share. So that\nwhen our train goes very fast, or when thunder cracks very loudly, or\nwhen blue waves are turned to smoke, though we are not actually\nresponsible in any way for these encouraging facts, which are dependent\non pressure in a boiler, electricity in the air, and a disturbance in\nmid-Atlantic, yet as by some wireless telegraphy, the energy of them is\ncaught in the receiver of ourselves, and we throb back to it, feeling\nthe pulse of life, which is exactly the same life in boiler and cloud\nand wave as that pulse in ourselves, which beats at the wrist. Life!\nLife! Life! All one--all absolutely one!\n\nAnd to-night, too, though not in any of these particular ways, how it\nthrobs and beats in this hot darkness of June! For a moment I wished I\nwas in the country, to feel the pulse of the woodland and the garden.\nFor the green things of the earth are awake all June; they never sleep\nday or night; they hold their breath sometimes in the hour before dawn,\nand they hang their heads sometimes beneath some scurry of summer rain;\nbut day and night their eyes shine; they are growing and living, and are\nalways awake till autumn comes, when they doze, and winter comes, when\nthey sleep sound, day and night alike, dreaming, perhaps, of the spring,\nwhen from deep sleep they will slowly awake again, aconites first, and\nsoon after daffodils, and then the buds of the hawthorn, little green\nsquibs of leaf....\n\nBut I had not gone a hundred yards from the doors within which I had\ndined, when the mysterious joy of London summer night smote these\nthoughts of the country into silence. The whole town was awake, theatres\nwere pouring out into the streets, and boarding the giants of the\nroadway, the snorting smelling motor-buses, their trotting brothers, and\nthe inferior cabs and hansoms, where one could be alone and not stop on\nthe way, but be taken decorously and dully to one's destination. There\nwas news, too, in the evening papers--a horrible murder, I think it\nwas, but the nature of the incident mattered very little. It was\nincident, anyhow; something had happened. And without wishing to know\nexactly what it was, I felt extraordinarily pleased that something had\nhappened.\n\nThe dip of Piccadilly between Devonshire House and Hyde Park was\ncomparatively empty, and a sudden shudder of the mind came across me. I\nhad been sitting next a dear friend, condemned to death. How _could_ I\nhave forgotten that, for forgotten it I had, in this riotous summer of\nLondon. Then I knew why I had forgotten it. It was because she had been\nso superior (an odious word, but there is no other) to it herself. That\ncourage, that passionate interest in the dear things of the world, her\ncontempt (for this time there is no need of another word) of death, had\nbeen infectious. To her it was a mere incident of life. 'Things in\ngeneral' were no less real and delightful to her because this incident\nwas coming close, than they were to me, who had not yet, as far as I\nknew, to look it in the face.\n\nYet, after all, to any of the others sitting at that table, death, so\nsmall an incident to her who had steadfastly regarded it, might in\nreality be closer than to her. And she exulted in the things of life\nstill: they had lost no interest for her.\n\nI stopped for a moment at the bottom of the hill, as one must when\nsomething quite new to oneself strikes one. That was the ideal she had\nshown. Fearless, undismayed, full of summer. 'And with God be the rest.'\n\n       *       *       *       *       *\n\nAt Hyde Park Corner a coffee-stall and an ice-cream stall jostled each\nother. Each had its following. But both at the moment seemed to me to be\nheretical, and instead I turned into the Park to walk as far as the\nAlexandra Gate, whence I had to get into Sloane Street.\n\nIt was like coming out of the roar of a tunnel into the day again, and\none's eyes (though conversely) had to get accustomed to the dark after\nthe glare and noise of the dear streets. A little wind whispered\noverhead in the planes; a little odour of moist earth came from Rotten\nRow. Quiet, solitary figures passed, or figures in pairs, closely\nlinked, but for the most part silent. On benches underneath the trees\nthere were pairs of figures. In Heaven's name why not? To flirt, to make\nlove, to look into eyes, is an applauded, and rightly applauded, pursuit\nin sequestered corners, under palms, beneath the eaves of the staircase,\nwith the band blaring from the ball-room just beyond. But it doesn't\nseem to strike the fastidious, who write letters to papers about the\n'state' of the parks, that it is just possible that there are other\npeople in the world who haven't got ball-rooms and palms, and marble\nstaircases. What are they to do, then? The answer of these\nletter-writers is deplorably futile, for they talk about indigent\nmarriages! As if you could stop the life of the world by pointing with\nimpious hands towards the Savings Bank! God laughs at it!\n\nBut the people who most call attention to the state of the park are\nthose who have sat in the back drawing-room with their 'gurls,' while\nmamma has been Grenadier at the door, and papa has put a handkerchief\nover his broad face, when he has finished his glass of port after lunch\n(after lunch!), and smokes his cigar in the dining room. It really is\nso. Young men and maidens may sit on a plush sofa in the dreadful back\ndrawing-room and behave as young men and maidens should (and if they\nshouldn't, they will); and why in the name of all that is decent should\nthey not sit on a bench in the Park and kiss each other? Yet the person\nwho objects to their doing so, and who writes to the papers in\nconsequence, is exactly the man who, in his semi-detached villa at some\nnameless suburb, draws his handkerchief over his face, and obscenely\nsnores, while Jack, a respectable bank-clerk, kisses Maria in the back\ndrawing-room. Good luck to them all, except to the horrible man who\nsnores and writes to the papers when he is awake! He would be better\nsnoring.\n\nThe moon had risen and rode high in a star-kirtled heaven, making a\ndiaper of light and shifting shadow below the shade of the many-elbowed\nplanes. Even now, close on midnight, it was extraordinarily hot, and for\na little the grass and the trees made me long again for the true\ncountry, where the green things on the earth are native, not, as here,\noutcasts in the desert island of the streets. Yet, when there is, as in\nLondon, so large a colony of castaways, extending, you will remember,\nright down from beyond the Serpentine Bridge to Westminster, so that,\nexcept for the crossing at Hyde Park, one may walk on grass for all\nthese solid miles, one hopes that the trees and flowers are tolerably\ncheerful, and do not sigh much for the wild places away from houses.\nNever was there a town so full of trees as this, for walk as you may in\nit, you will, I think, with three exceptions only, never find a street\nfrom some point in which you cannot see a tree to remind you of shade at\nnoontide and grassy hollows. But the names of those streets shall not\nhere be stated; they must, however, consider themselves warned.\n\nThen the streets again, crowded still with moving figures, each an\nentrancing enigma to any passenger whose soul is at all alert, and swift\nwith the passage of those glorious motor-buses, pounding and flashing\nalong on their riotous ways, the very incarnation to me of all that\n'town' means! I cannot imagine now what London was like without them. It\nmust have been but half alive, half itself. It is impossible to be\npatient with these curious folk who consider them nuisances, who say (as\nif anyone denied it) that they both smell and clatter. That is exactly\nwhy they are so typical of London; indeed, one is disposed to think that\nthey were not made with hands, but spontaneously generated out of the\nSpirit of the Town.\n\nAnd how delightful to observe their elephantine antics if the streets\nare slippery, when they behave exactly like a drunken man, with\nappearance still portentously solemn, as if he had heard grave news, but\nafflicted with strange indecision and uncertainty on questions of the\ndirection in which he intends to walk. I was on one the other day which\ndid the most entrancing things, and had it all to myself, as everybody\nelse got down, not seeming to see that if a motor-bus has been\n'overtaken' it is far safer to be on it than anywhere else in the\nstreet, just as a drunken man may lurch heavily with damage to others,\nbut never hurts himself. It was in Piccadilly, too, a beautiful theatre\nfor its man\u0153uvres. Trouble began as we descended the hill by the Green\nPark: it had _vin gai_, and was boisterously cheerful; but it was\nextraordinarily uncertain about direction, and slewed violently once or\ntwice, so that hansoms started away from our vicinity as rabbits scuttle\nfrom you in the brushwood. Then my bus suddenly pulled itself together\nand walked quite straight for a lamp-post by the kerb. It felt tired, I\nsuppose, and leaned wearily against it, snapping it neatly off with as\nlittle effort as it takes to pluck a daisy. Then it hooted, moved\ngravely on again, and, thinking it was a member of the Junior Athen\u00e6um,\nmade straight for the door. But it forgot to lift its feet up to get on\nto the pavement, and stumbled. Then it saw a sister-bus, backed away\nfrom the pavement, and tried to make friends. But the other simply cut\nit and passed by. So it gave a heavy sigh, and began to mount the hill\ntowards Devonshire House. But it had scarcely gone twenty yards when the\nbehaviour of its sister so smote upon its heart that it could not go on,\nand turned slowly round in the street to look back at that respectable\nbut uncharitable relation with pathetic and appealing eyes. It might\nhappen to anybody, it seemed to say, 'to take a drop too much, and you\nshouldn't judge too severely.'\n\nThis sense of being misunderstood gave it _vin triste_ of the most\npronounced kind. I have seldom seen so despondent a drunkard. It moaned\nand muttered to itself, and I longed to console it. But beneficent\nNature came to its aid: laid her cool hand upon its throbbing head, and\nit slept. I got gently off, feeling, as Mr. Rossetti, I think, says (if\nit was not he, it was somebody else), that I must step softly, for I was\ntreading on its dreams.\n\nAnd all this for a penny, which the conductor very obligingly refunded\nto me, as I had not been taken where I wanted to go!\n\n       *       *       *       *       *\n\nSloane Street, and soon my dear house, into which I was towed by my\nwatch-chain. For my latchkey was on the end of it, and, having opened\nthe door, I could not get the latchkey out, and had to step on tiptoe,\nfollowing the door as it opened. Wild music came from the upstairs, and,\nhaving disentangled my key, I ran up, to find Helen and Legs trying with\nsingular ill-success to play the overture to the 'Meistersingers,' from\na performance of which they had just returned. They took not the\nslightest notice of my entry.\n\n'No!' shouted Legs. 'One, two; wait for two! Oh, do get on! Yes, that's\nit. Sorry; I thought it was a sharp.'\n\nThey were nearing the end, and several loud and unsimultaneous thumps\ncame.\n\n'I've finished,' said Helen.\n\nLegs had one thump more.\n\n'So have I,' he said. 'Isn't it ripping?'\n\n\n\n\nJULY\n\n\nHelen has gone to church, after several scathing remarks about\nSabbath-breakers, by whom she means me, and probably also Legs, as I\nhear the piano being played indoors. As a matter of fact, I have not the\nslightest intention of breaking anything--though Legs seems to have\ndesigns on the strings--for even here under the trees on the lawn it is\nfar too hot to think of such a thing. Several slightly disappointed dogs\nrepose round me, who hoped that perhaps, as I was not going to church, I\nwas going for a walk. This afternoon, I am afraid, they will be\ndisappointed again, for I propose to go to afternoon service in the\ncathedral, and they will think I am going for a walk. But on Sunday dogs\nhave to pay for the commissions and omissions of the week.\n\nThe bells have stopped, so Helen will quite certainly be late, and the\nsilence of Sunday morning in the country grows a shade deeper. Fifi\njust now, with an air of grim determination, sat up to scratch herself;\nbut she could not be bothered, and sank down again in collapse on the\ngrass. Legs, too, has apparently found the heat too much even for him,\nand has stopped playing. And I abandoned myself to that luxury which can\nonly be really enjoyed on Sunday morning, when other people have gone to\nchurch (I wish to state again that I am going this afternoon), of\nthinking of all the things I ought to do, and not doing them. On Monday\nand Tuesday, and all through the week, in fact, you can indulge in that\nsame pursuit, but it lacks aroma: it is without bouquet. But give me a\nchair under a tree on Sunday morning, and let my wife call me names for\nsitting in it, and then let the church-bells stop. Fifi wants washing.\nLegs said so yesterday, and we meant to wash her this morning. I must\ncarefully avoid the subject if he comes out, since I don't intend to do\nso. Then I ought to write to the Secretary of State--having first\nascertained who he is--to remind him that Legs is going up for his\nForeign Office examination in November, and that his (the Secretary of\nState's) predecessor in the late Government promised him a nomination.\nHow tiresome these changes of Government are! One would have thought the\nConservatives might have held on till Legs' examination. Then I should\nnot (1) have to consult Whitaker to find out who the present Secretary\nof State is, and (2) write to him, and--probably--(3) find that either I\nhaven't got a Whitaker, or else that it is an old one. This will entail\nexpense as well.\n\nHow the silence grew! I could not even hear any bees buzz among the\nflower-beds, and wondered whether bees do no work on Sunday. There was\nnot a sound or murmur of them. Probably this is quite a new fact in\nnatural history, which has never struck anybody before. It would never\nhave struck me if I had gone to church. Then Fifi pricked one ear, sat\nup, and snapped at something. It was a winged thing, with a brown body,\nrather like a bee. How indescribably futile!\n\nThen there came a little puff of wind from the end of the garden, and\nnext moment the whole air was redolent with the scent of sweet-peas.\nSweet-peas! How strangely, vastly more intimate is the sense of smell\nthan any other! How at one whiff of odour the whole romance of life, its\nbeautiful joys and scarcely less beautiful sorrows, the dust and\nstruggle and the glory of it, rises up, clad not in the grey robes, or\nstanding in the dim light of the past, but living, moving,\nbreathing--part of the past, perhaps, but more truly part of the\npresent. Like a huge wave from the immortal sea of life, cool and green,\nand speaking of the eternal depths, yet exulting in sunshine and\nrainbow-hued in spray, all the memories entwined about this house held\nand enveloped me. Here lived once Dick and Margery, those perfect\nfriends; here, when they had passed to their triumphant peace, came she\nwhom, when I first saw her, I thought to be Margery. From this house\n(where still in memory of Margery we plant the long avenue of\nsweet-peas, because she loved them) two years ago we were married, and\nhere I sit now drowned in the beautiful past that is all so essential a\npart of this beautiful present.\n\nBut it would be as well, perhaps, if this book is to be in the slightest\ndegree intelligible (a thing which I maintain is a merit rather than a\ndefect), to put together a few simple facts concerning these last two\nyears.\n\nIt was two years ago last April that we were married, and took a small\nhouse in town, though we still spent a good deal of time down here with\nHelen's father. But before the year was out he died, leaving everything\nto Helen, who was his only child. So, as was natural, we continued to\nlive in the house which was so dear to both of us.\n\nLegs is my first cousin, and he has lived with us for a year past, for\nhe has neither father nor mother; and since he was cramming for his\nForeign Office work in town, it was far the best arrangement that he\nshould make his home with us. Legs is the only name he is ever known by,\nsince he is one of those people who are almost unknown by their real\nname (which in this case is Francis Horace Allenby), and are alluded to\nonly by some nickname which is far more suitable. If, for instance, I\nsaid to somebody who knew him quite well, 'Have you seen Francis\nlately?' I should probably be favoured with an inquiring stare, and\nthen, 'Oh, Legs you mean!' while to his million acquaintances (he has\nmore than anyone I ever knew) he is equally Legs Allenby. The name, I\nneed scarcely add, is a personal and descriptive nickname, for Legs\nchiefly consists of them. When he sits down, he would be guessed to be\nwell on the short side of middle height; when he stands up he is seen to\nbe well on the farther shore of it. He was Legs at school, and his\nfamily, very sensibly, and all his friends, saw how impossible it was to\ncall him Francis any more. For the rest, he is just over twenty,\nsandy-haired, freckle-faced, and green-eyed, with a front tooth broken\nacross, a fact that is continually in evidence, since he is nearly\nalways laughing. It would be sheer nonsense to call him good-looking,\nbut it would be as sheer to call him ugly, since, when you have got a\nface like Legs', either epithet has nothing to do with it. But I have\nnever seen any boy with nearly so attractive and charming a face, and\nLegs, whose nature is quite as nice as his face, and extremely like it,\nhas the most splendid time.\n\nAnd that, to finish these tedious explanations, is our household. There\nis no other inmate of it--no little one, you understand.\n\nLegs is an enthusiast--a fanatic on the subject of life. Everything,\nincluding even his foreign languages, which he has to cram himself with,\nis the subject of his admiration, and he discovers more secrets of life\nthan the rest of the world put together. At one time it is a chord which\nis meat and drink to him; at another the romances of Pierre Loti; or,\nagain, golf is the only thing worth living for, while occasionally some\ngirl, or, as often as not, a respectable elderly married woman, usurps\nhis heart. Last week he discovered that there were only two people in\ntown the least worth talking to, but yesterday, when I asked him who the\nsecond one was, having forgotten myself, I found that he had forgotten\ntoo, for if the 'Meistersinger' overture was not enough for anybody, he\nwas a person of no perception.\n\n'Why, it contains all there is,' he had said, when he finished it the\nother evening with Helen. 'It's all there, the whole caboodle.'\n\nBut this morning, from the silence indoors, I imagine he must have found\nanother caboodle--a book probably. Or equally possible, Legs has an\nattack of acute middle-age, which occasionally takes him like a bad\ncold in the head. Then he wonders whether anything is worth doing, and\nis sorry for Helen and me, because we are so frivolous. Six months ago,\nI remember, he had such an attack, induced by reading a book about three\nacres and a cow, which raised in him the sense of injustice that all of\nus three had so much more than that. During this period he took no sugar\nin his tea, refused wine, and began to write a book which was called\n'Tramps,' contrasting the horror of indigence with the even greater\nhorror of extravagance. It was really directed against Helen and me, for\nwe had lately bought a small, snuffling motor-car. These outbursts of\nSocialism are generally coincident with Atheism. But they do not last\nlong: Legs soon feels better again.\n\nI was right, it appeared, about the conjecture that he had found a book,\nbut I was wrong about the attack of middle-age. Legs jumped out of the\ndrawing-room window with wild excitement.\n\n'Oh, I say!' he cried, 'why did you never tell me? I thought Swinburne\nwas an awful rotter! But just listen.'\n\nAnd he read: 'When the hounds of spring are in winter's traces.'\n\n'Did you ever hear anything like it?' he said. '\"Blossom by blossom the\nspring begins!\" Why, it's magic! Oh, don't I know it! Do you remember--I\nsuppose you don't--when all the daffodils came out together last year?'\n\n'Oh, Legs, what an ass you are!' I said. 'Because you never noticed them\ntill I showed you them.'\n\n'No, I believe that's true. Oh, don't argue! Listen!'\n\nAnd he began all over again.\n\nThen he lay back on the grass with his hands underneath his head,\nlooking up unblinking into the face of the sun. That, by the way, is\nanother peculiarity of his: he looks straight at the sun at noonday, and\nis not dazzled. His eyes neither blink nor water. He can't understand\nwhy other people don't look at the sun.\n\nThen--if by any chance you care to understand this quiet, delightful\nlife we lead, it is necessary that you understand Legs--then his mood\nsuddenly changed.\n\n'Oh, I'm wrong about the daffodils,' he said; 'you showed me them. But\nthis chap _is_ a daffodil. I suppose he's quite old, too. I wonder how\nyou can get old, if you have ever felt like that. What a waste of time\nit is to do anything if you can feel. I hate this Foreign Office affair:\nwhy shouldn't I do nothing?'\n\n'Because you can't,' I remarked.\n\n'What do you mean?'\n\nI had not been to church, and so had heard no sermon. Therefore, I\npreached one on my own account.\n\n'You will know in about fifteen years,' I said. 'Anyhow, you will find\nthat, unless you are brainless and absurd, you must do something. You\nare quite wrong. It isn't nearly enough to feel. The moment you \"feel,\"\nyou want to create. You not only want, but you have to; you can't\npossibly help yourself. You have just read that heavenly poem. You now\nwant to write something like it. You hear what spring once said to a\npoet, and you want to put down what spring says to you!'\n\n'Oh, you're quite wrong,' said Legs. 'He has said what spring means.\nThat's the last word on the subject. But summer now: this, to-day----'\n\n'So you want to create,' said I.\n\nA glorious trait about Legs is that he never admits conviction. He only\nchanges the subject. Thus, if the subject is changed by him, his\ncontroversialist is satisfied.\n\n'I don't believe in the highest of the shortest suit if your partner\ndoubles,' he said. 'What are you to do if you have two spades and two\nclubs all contemptible?'\n\n'Lead the less contemptible.'\n\nLegs turned slowly over on his side, and lay with his face against the\nshort turf of the lawn. '\"Blossom by blossom,\"' he said, '\"the spring\nbegins.\" I wonder if he meant more than that! Did he mean to tell of the\ntime when one is young oneself, and it is all blossom? Lord, how\npriggish that sounds! But it is all blossom, except for this beastly\nGerman. I hate German! It sounds as if you were gargling. Damn! I have\nto go up by the early train to-morrow, too! And you and Helen will stop\nhere till after lunch. Grind, grind--oh, I lead the life of a dog! And\nthen, if I am very successful, I shall have the privilege of sitting on\na stool in a beastly building in Whitehall, and writing a _pr\u00e9cis_ from\nsome silly old man in Vienna or Madrid, about nothing at all. It isn't\nworth it!'\n\nLegs and I, it will be observed, deal largely in contradictions.\n\n'Yes, it is,' I said. 'Everything almost that one does is worth it. As\nlong as you are actively doing anything with all your heart, you can't\nbe wasting time, nor can there be anything better worth doing. It is\nonly when you say that a thing isn't worth doing that it becomes so.'\n\nLegs sat up again.\n\n'Oh, I want nine lives at least!' he said. 'Or why can't one buy some of\nthe time that hangs so heavy on other people's hands? I know a man who\nreads the _Times_ all through every morning, and the _Globe_ every\nevening. Yet, after all, I dare say it is quite as improving as sitting\nhere and talking rot as we are doing. I shall go and put in half an hour\nover that accursed Teutonic language before lunch.'\n\n       *       *       *       *       *\n\nLegs had, as it seemed to me, run over most of the topics of human\ninterest in the few minutes he had been out, and since I was still\nirrevocably determined neither to wash Fifi, nor to write to the\nSecretary of State, nor, indeed, to open the very large book on the\ncrisis in Russia, which I had brought out with me (to bring out a book\non Sunday morning and not to open it is strictly in accordance with the\nspirit of the thing), my mind went slowly browsing, like a meditative\ncow, over the dazzling display he had spread before me. And\ninstinctively and instantaneously I found myself envying him, though why\nI envied him I did not immediately know. But it was soon obvious; I\nenvied his power of making soul-stirring discoveries; his rapture over\nthat magical spring song of the man he had thought 'an awful rotter.' I\nenvied him his ignorance of the perfectly patent fact that it is only\nfools who can go on doing nothing, and of the fact that it is infinitely\nbetter to sit on a stool and do arithmetic for stockbrokers than to do\nnothing at all. But youth does not know that, and I think I envied him\nhis youth. Yet--so often does one contradict oneself--I knew very soon\nthat I did not envy him any of these things. After all, I still went on\nmaking soul-stirring discoveries, and propose to do so until the very\nend of my life, when I shall make the most soul-stirring discovery of\nall, which is death. And to envy the fact of his having just discovered\nthe magic of Swinburne's spring song would be exactly the same as\nenvying the appetite of somebody who has just come down to breakfast,\nwhen you are half-way through. Your eggs and bacon were delicious, but\nthe fact that you have eaten them makes it impossible to wish for them\nagain. And it should make you only delighted that other people keep\ncoming down to breakfast--till the end of your life they will do that,\nunless the world comes to an end first--and, thank God, they will find\neggs and bacon delicious too, hungry and fresh in the morning of their\nlives.\n\nI was becoming slightly too active in mind for the proper observance of\nSunday morning (given, of course, that you have chosen not to go to\nchurch), for the real attitude is a state of tranquil bemusedness, but\nit was too late to stop now.... What, in fact, did I want? Did I want\nto be twenty again, and go through the days and hours of those fifteen\nyears once more?\n\nYes, I did. If the world could be turned back for fifteen years, I would\ngladly take my place there, and go through it all, good and bad\ntogether, just as it has happened. I would encore this delightful song,\nin fact, and be content that it should be sung again--it, not another\nsong. Of course, if one could start again at the age of twenty--or ten,\nfor that matter--and live it over again with the knowledge,\ninfinitesimal as it is, that one has gained now, I imagine that the vast\nmajority of the world would put the hands of the clock back. On all\nthose thousands of occasions on which one has acted stupidly, unkindly,\nevilly, and has probably suffered for it without delay (for it is\nmercifully ordained that we have not long to wait before our punishment\nbegins, especially if we have been foolish), we should now do\ndifferently, remembering that it did not pay--to put things at their\nlowest--to be asses and knaves. Apart from that, we should have the same\nbeautiful, flawless days again, when, so I cannot but think, the\nbeneficent power has somehow come very close to us and our surroundings,\nand by its neighbourhood has given us a series, again and again\nrepeated, of hours in which we have been unable to imagine anything\nbetter than what we have got. We have wanted, with all the eager\nhappiness that wanting gives, and we have obtained; but before any\nleanness of the soul has entered we have wanted again. We have had\nhappiness, not content (since that implies the end of wanting) but\nhappiness, the content that dwells not in the present only, but looked\nforward. I have no idea whether, on the whole, I am happier than the\naverage of other people, since there is no thermometer yet invented that\ncan register that. But I do know that I would choose to go back and live\nit all over again, _as it has been_. With the little experience, the\nlittle knowledge that must inevitably come with years, whether one is\nstupid or not, I imagine that everybody would choose to go back, but I\nwish to state distinctly that I would go back without that. I suppose it\nwas that which made me just now feel I envied Legs. But I don't do that\nreally for this reason.\n\nSupposing that what I should choose (because I really should) were given\nme, what then? I should arrive again eventually in the mere measure of\nyears at the point where I am now, no different, no better, no worse. I\nshould like to go back, because it has been _such fun_. But there is\nbetter than that ahead: of that I am completely convinced. There are as\nmany (if not more, and I think there are more) entrancing discoveries\nfrom middle age as there have been from youth, and I am convinced again\nthat if one happens to live to be old there will be as many more.\n\nAfter all, to re-read life again would be like re-reading the first\nvolume of an absorbing book. One has revelled in the first volume, and\nnaturally wants to revel again. But what is going to happen? There is\nnothing that interests me so much as that. To-day, even in this quiet\ndomestic life of ours, there are a hundred threads leading out into\nunknown countries, all of which, if one lives, one will follow up. And\nall, big and tiny alike, are so stupendous. If, to take the forward\nview, I could see in a mirror now what and where all those people--few\nof them, no doubt, but friends--those who really matter, would be in a\nyear's time, how I should seize the magic reflector, and gaze into it!\nIncomparable as has been the romance of life up till now, it is known to\nme. But to peep into the second volume!\n\nThe sun, in the full blaze of which Legs had laid, peeped over the top\nof the elm in shade of which I had seated myself, and, not being\nLeggish, I shifted my chair again to consider this point.\n\nIt is a question of scale that is here concerned, though the scale seems\nto me to be an unreal one. If I happened to be the Emperor of All the\nRussias, and the magic mirror were given me, I should look eagerly out\nfor my own figure, and see if I still wore a crown. I should scrutinize\nthe faces of those around me, to see if war and the hell-hag of\nrevolution had been shrieking through my illimitable country. But my\ninterests are not soul-stirring to any but me, and anyhow not of\nEuropean importance. So I should look to see who sat on this lawn a\nyear hence; I should ask for a short survey of the Embassy at Paris, to\nsee if Legs was attached; I should visit a dozen houses or so. But if I\nwas allowed to put the clock back fifteen years, I should have to wait\nlonger for this.... So I must reconsider my choice, and I am afraid I\nmust reverse it. But it must be understood that I choose not to be\ntwenty again, merely because it will take longer to be forty and fifty.\nI want the second volume so much.\n\n'Or....' Here Helen's voice broke in. She had come back from church, and\nhad seated herself on the grass, and I believe that half of what\nappeared to be soliloquy was actually spoken to her. But she is\nwonderfully patient.\n\n'It is youth you want,' she said, 'and you have got it till you cease to\nwant it. It is only people who don't care about it that grow old. Or is\nthere more than that? Is it wanting to go on learning that keeps one\nyoung?'\n\nA dreadful misgiving came over me.\n\n'Am I dreaming?' I said. 'Or did you tell me the other day that I showed\nsigns of wishing to teach?'\n\nShe laughed.\n\n'No; it is quite true. But I will tell you when you cease to wish to\nlearn. I shall say it quite, quite clearly.'\n\nShe took off her hat, and speared it absently with a pin.\n\n'We had an awful sermon,' she said, 'all about the grim seriousness of\nlife, and the opportunities that will never come back. It does seem to\nme it is most absolute waste of time to give a thought to that. I shan't\ngo to church next Sunday. I don't feel fortified by thoughts like that.\nIt's much better for me to know that you would put the clock back, live\nit all over again. But about looking forward. Oh, Jack, I think I\nshouldn't look in the magic mirror if I had the chance. What if one saw\noneself all alone? One would live in dread afterwards.'\n\n'Or what if you saw a cradle in the room?' said I.\n\nShe looked up at me quickly, and then put out her hands for me to pull\nher up.\n\n'Perhaps I should look in the mirror,' she said.\n\nPoor Legs, as he had said, left by a very early train next morning, and\nHelen, moved by a sudden violent attack of vague duty, went with him.\nThe access was quite indeterminate. She thought merely that one ought to\nget back to town early on Monday, so as to have the whole day there\ninstead of splitting it up. Personally I followed neither her reasoning\nnor example, and intended to spend the day in dignified inaction in the\ncountry, and not split it up by going to town till after dinner. But to\nthe owner of a motor-car the train appears a degraded sort of business,\nand, greatly daring, I meant to start about nine in the evening, and be\nthe monarch of the road; for when there is no other traffic, any car\nbecomes a chariot of triumph. Helen, I may remark, loves our motor when\nshe does not want to go anywhere particular. When she does she takes the\ntrain. I think, in fact, that it was my proposal that we should drive up\ntogether after dinner that was the direct parent of her sense of duty.\n\n       *       *       *       *       *\n\nSo, when I came down at the not unreasonable hour of nine to breakfast,\nI found that I had the house to myself, and--I am not in the least\nashamed of the confession--found that the prospect of an absolutely\nsolitary day was quite to my mind. I do not believe myself to be\nunsociable or morose, but every now and then I confess that I like a day\nin which I see nobody. It is not that one is busy, and wants to get\nthrough one's work, for, on the contrary, when I have a great deal to\ndo, I hugely desire the presence and the conversation of friends in the\nintervals of 'doing.' But occasionally it is a very good thing to chew\nand ruminate, to be surrounded by the quiet green things of the earth,\nwhich give you all their best without waking the corresponding instinct\nto exchange ideas, to give something of yours to meet theirs. For\nintercourse with one's fellow-men, especially with one's friends, is\nlike some rapid interchange of presents. Everybody (everybody, at least,\nwho has the smallest sense of sociability) searches in his mind for any\nlittle thing that may be there, and gives it his friend, while the\nfriend, accepting it, gives something back. From all that--we cannot\ncall it an effort since it is so completely spontaneous on both\nsides--it is well to be free occasionally, to lie, so to speak, under\nthe pelting rain of life that is ever poured out from the voiceless,\neloquent, bright-eyed happiness of Nature, to make no plan, to\ncontemplate no contingency, to drop that sort of fencing rapier that we\nall wield when we are with our fellow-men, and lie like a log, with one\neye open it may be, and be rained upon by the things that live, and are\nclothed and nourished without toil or spinning.\n\nI am aware that the great Strenuists, from Mr. Roosevelt downwards,\nwould hold up their toil-hardened hands at this, exclaiming: 'You mean\nit is better now and then to be a cow than a Man?' Precisely so, but\ncows are not nearly as inactive as Man on these occasions ought to be.\nThey eat too long, and they switch their tails, and stamp their feet.\nBut the long, stupid, bovine gaze is moderately correct. At least, I\nhave never detected a shadow of intelligence in a cow's eye. If there is\nany, the man who occasionally becomes a cow must be careful to get rid\nof it. Nor must he be a cow too often: that is fatal. If he is a cow for\none day in every six weeks, I think he will find the proportion is\nabout right.\n\nSo all day, literally all day, I sat, or, when sitting became too\nfatiguing, lay on the lawn, and nothing happened that did not always\nhappen, but all was worth observing in a purely bovine manner, without\nintelligence. Little brown twigs occasionally fell from the elms, and\nonce or twice a withered yellow leaf came spinning on its own axis, as\nif it was the screw of some unseen steamer. A stag-beetle walked slowly\ndown from the wooden paling, and came some ten yards across the lawn. It\nstopped there about an hour, I should think, doing nothing whatever.\nThen it turned and went back on to the paling again. A robin took about\nthe same length of time to make up his mind that I was quite harmless,\nand eventually pecked at my bootlace, which was undone. It took him an\nenormous time to decide, with his head cocked sideways, whether it\ntasted nice or not, but eventually he settled it did not, for he did not\npeck it again. Then a jackdaw sat on one of the poles of the tennis-net,\nand said 'Jarck' seventeen times after I began to count. He began to\nsay it the eighteenth time, but stopped in the middle and ate an\nincautious earwig.\n\nThat was almost too exciting, and I transferred not my attention,\nbecause I had not got any, but my bovine gaze to the big flower-bed\nopposite. All summer was there, dim, hot, blossoming summer in full\nluxuriance of growth, so that scarcely a square inch of earth was\nvisible. I did not even name the dear familiar flowers that grew there.\nOne was a spire of blue, one was a cluster of orange; there was an\norchestra of red trumpets, a mist of starry grey, and bits of sky caught\nin a web of green. And from beyond (I could not help naming that) the\nodour of sweet-peas. I lay and soaked in it.\n\n       *       *       *       *       *\n\nTo use a simile, do you know those mysterious things which are to be\nfound on the chalk downs, called dew-ponds? Often, of course, they are\nfed with rain, but even when for months no rain has fallen, you will\nstill find them full. They just lie open to the sky and that is all. And\nthe mind, so it seems to me, is something like them. Often it is fed in\nthe obvious way, as the dew-pond with rain, by conscious thought, by\nactive intercourse with others. But sometimes it is not a bad thing for\nit to be like the dew-pond, just to lie open to the sky, and drink in\nthe eternal wine of Nature, which fills its pond again. All that is\nrequired of it is to do nothing whatever, not to think even, but just to\nbe there, to be in existence, to let go of everything. It really is\nworth the experiment, though it is not quite so easy as it sounds, for\nthoughts, ideas of some kind, keep leaking in. They must be firmly\nexcluded.\n\n       *       *       *       *       *\n\nThe snuffling motor rose like a hero to the occasion, and came round\nthrobbing with excitement. Something in the idea of this drive by night\nhad evidently taken its fancy, and it positively burned to exceed the\nlegal limit, a wish that I was only too glad to gratify. When we started\nthe crimson of the sunset was still aflame in the west, but gradually\nthe colour was withdrawn, as if some unseen hand was pulling out scarlet\nthreads that ran through some exquisite fabric of dainty embroidery,\nleaving there only the soft transparent ground of it. Then more\ngradually, so that the eye could not trace the appearance of each, but\nonly knew that the number was being multiplied, behind the dark velvet\nof the sky were lit the myriad suns that make a flame of space, and sing\nin their orbits. Colours faded and disappeared, and soon the world was\nturned to an etching of black and white. The roads were empty of\ntraffic, and though July was here, still from dark coppice and leafy\nscreen there sounded the one eternal song, the rapture of nightingales.\nOften it seemed to me as if we were standing still, while the world in\nits revolution span by us; there was but a space of lamp-lit road by\nwhich, shadow-like, dream-like, the trees and open spaces ran. For a\nlong piece together, as over the Hartford Bridge flats, nothing marked\nour passage except this whirling of the world. It seemed in the darkness\nthat time had ceased, and that from its own impetus this globe and the\nthousand globes above were circling still.\n\nThen in front there began to shine, like the reflected light of some\ncomet coming nearer, the huge glow-worm of London. For a while it\nrested, like some remote befogged star on the horizon; then its light\nbrightened, and its little crawling caterpillars, the trains and buses,\nbegan to creep by us, reaching out, as it were, to the end of the leaf,\nthe greenest and most succulent parts.\n\nThen, like the opening of a photographer's shutter, so swift it was, we\nwere in the traffic of the town again, and all was familiar, all was\nhome. The country was home too, and here was another. Which was the\ntruer sense? The sense that claimed the jackdaw on the tennis-net as a\nbrother, or the sense that rejoiced in this fierce-beating pulse of\nlife?\n\nPerhaps, since they are both true, there is no question of comparison.\n\n\n\n\nAUGUST\n\n\nSomething of the primeval savage blood still beats in us, we must\nsuppose, else why is it that we, effete inhabitants of London, who love\nthe closeness and proximity of our fellow-men so much, feel no less\nkeenly the rapture of being miles and miles away from railways and the\nfolk who travel on them? How quick, too, is the transition from one mood\nto another, so that while a week or two ago we rushed insanely, it may\nbe, but with extraordinary pleasure, from party to party, jabbering with\nchildlike delight to myriad acquaintances, face to face on a blocked\nstaircase, or in the drawing-room unwillingly silent while somebody\nsang, we now take the same childlike pleasure in long days of solitude.\nBut we may take our solitude in pairs, in company with a friend who for\nthe time being is no friend at all, but a bitter (and, it is to be\nhoped, disappointed) golfer, or we may lie out all day in the heather\nwith a silent stalker, or, as has been my fortunate lot for the last ten\ndays, may spend long hours, with a sandwich and a fishing-rod and a\ngillie, in angling over coffee- streams or windswept lochs.\n\nThe oldest inhabitants never remember anything like this summer, but\nthey are bad evidence, because their memories are probably very\ndefective owing to their age; but, what is more convincing, younger\npeople, whose memories are less impaired, never remember anything like\nit. So there has been little of the coffee- streams for me\npersonally, but, instead, long quiet days by this wonderful loch,\nsupposed to hold trout of fabulous dimensions, which, as far as I can\nmake out, nobody has ever caught, though every one agrees that they are\nthere. Then came a wonderful day, with more than trout-wonder in it.\n\nI came up here to this remote lodge alone, for the trio of us usually go\nour own ways in holiday time. Legs, in any case, had to go to Germany to\nlearn that classic and guttural tongue, and Helen and I always make\nvisiting arrangements independently of each other, unless we are both\nbidden to a house to which we both want to go. But it stands to reason,\nso it seemed to us, that husband and wife probably do not have the same\nfriends, and it is as absurd for her to stay at a house because the host\nis a great friend of mine as it is for me to stay at a house because the\nhostess is a great friend of hers. Coincidences sometimes happen, in\nwhich we both go together. Otherwise we make our own arrangements. I\ncannot bear some of her friends; she finds it almost impossible to\ntolerate some of mine. And with shouts of laughter we agree to differ.\nThen in September or October the trio will come together again, and will\nall talk at once, describing simultaneously, while nobody listens, our\ndelightful adventures.\n\nI started from the lodge that morning after an early breakfast, the\ngillie having already gone on with lunch, and what we hoped would be the\napparatus of death; for, the first time during this last week, it was a\nsoft and cloudy morning, with a warm wind from the south-west,\nsufficient even in this cup of the hills, where the lodge stands, to\nset the trees tossing their branches, and to strip the red ripe\nrowanberries from their stalks. Upon the unsheltered tops, then, where\nlay the dark- loch with its fabled inhabitants, there should be\nripple enough for fishing purposes. I walked unencumbered but for the\nfield-glasses I always carry; for nothing, during periods of waiting or\nin the half-hour that follows the sandwich, is so fascinating as to spy\nout the busy animal life on these empty moors, or find some three or\nfour miles away two or three little human specks moving very gently up\nthe hillside after the deer, or sitting there patiently till some\nuntoward affair, suspicious hinds, or a foul wind are lulled into\ninactivity.\n\nBut first I had a mile of pine-wood to climb, up steep, slippery,\nneedle-strewn paths, with bracken already yellowing on each side, making\na sea of russet and green, while from overhead, in the thick arching\nboughs, there came, as it were, the noise of an aerial sea, the hiss of\nripples on a sandy shore as the wind whistled through the stiff springy\nfoliage. Now and then a rabbit scuttled through the ferns, and once I\nsaw quite close at hand a roe-deer with flicking ears and startled eyes,\nthat, as it caught sight of me, gave me one shy look of the woodland,\nand then galloped off, cutting its way through the tall bracken. The\npath sometimes led by the side of the stream that came out of the loch\nto which I was bound, but the dryness of the summer had hushed its\nvoice, and it but trickled down the ways it was wont to prance along in\nspring. Here and there a tree of the tamer woodland, a beech, or\nstripling elm, grew among the primeval firs, but it looked as if it had\nwandered here by mistake, had strayed, a member of some later\ncivilization, into a settlement peopled by those of the older world.\n\nAnd as I walked something of the same feeling of strangeness, of having\ngone back to the earlier ages of the world, came over me also. Like the\nlost beech, there were none of my kind here, and I felt, though in an\nimmeasurably greater degree, what one feels when one stands in the\nvalley of the tombs of the Egyptian Kings. But all round me here were\nthings far more ancient than they. \u00c6ons before Pharaoh oppressed the\nchildren of promise there stood here on this hillside the ancestors in\ndirect line of this woodland. The knowledge of the dawn of the world,\nwhen it was still but a little time since God had bidden the green\nthings to live upon the earth, had been transmitted to these citizens of\nthe hillside, and to them time had been but a little thing, and a\nthousand ages were but as yesterday.\n\n       *       *       *       *       *\n\nAs I ascended farther and more remotely into the heart of the wood, a\nsort of eager tremor, a desire to see that which I knew was there, and\nwhich must be so overpowering in its immensity, began to grow on me.\nWild silent life bubbled and hummed round me; eyes watched me from\nbeneath the fern, and looked down on me from the over-arching fans of\nthe pines; ears were pricked at my footstep; strange wild smiles\nbroadened into a laugh at the intruder, at this child of immeasurably\nlater ages. Sometimes it seemed to me as if this ancient consciousness\nof the woods was scornful and contemptuous, so that I quickened my pace\nand longed to get out of this dark room; at other moments, and truer\nones, I knew better, knowing that I, too, was of it all, a\nmanifestation of life, a piece of the pine-woods and brother of the\nbracken.\n\nThere is no myth that grew so close to the heart of things as the story\nof Pan, for it implies the central fact of all, the one fact that is so\nindisputably true, that all the perverted ingenuity of man has been\nunable to split into various creeds about it. For Pan is All, and to see\nPan or to hear him playing on his pipes means to have the whole truth of\nthe world and the stars, and Him who, as if by a twisting thumb and\nfinger, set them endlessly spinning through infinite Space, suddenly\nmade manifest. Flesh and blood, as the saying is, could not stand that,\nand there must be a bursting of the mortal envelope. Yet that,\nindisputably also, is but the cracking of the chrysalis. How we shall\nstand, weak-eyed still and quivering, when transported from the dusk in\nwhich we have lived this little life, into the full radiance of the\neternal day! How shall our eyes gain strength and our wings expansion\nand completeness, when the sun of which we have seen but the reflection\nand image is revealed! That is to see Pan. It killed the mortal body of\nPsyche--the soul--when she saw him on the hill-top by the river, and\nheard the notes of his reed float down to her; but she and every soul\nwho has burst the flimsy barrier of death into life joins in his music,\nand every day makes it the more compelling. Drop by drop the ocean of\nlife, made up of the lives that have been, rises in the bowl in which\nGod dips His hands. He touches every drop.\n\n       *       *       *       *       *\n\nThe wood in front had grown thin, and I was nearly out on the open\nheather of the hills. Just here the path crossed the stream bed; a great\ngrey cliff of rock was above me, in which a pattern of lichens had found\ncrevices for their roots; the pine-trees waved solemnly overhead; the\nmiracle of running water, perhaps the greatest miracle of all, chuckled\nand eddied as it slid into the brown pool. And quite seriously I waited\nto see Pan. The ferns would be pushed aside, and the merry face would\nsmile at me (for Pan, though he kills you, is kind), and he would put\nhis pipes to his lips, and the world, as I had hitherto seen it, would\nswim away from me. And just before he puts his pipes to his mouth, I\nhope I shall say: 'Yes, begin; I am ready!' Or shall I stop my ears, and\nshut my eyes to him? I hope not. But the fern waved only, and the water\nran, and ... and I was going a-fishing.\n\nI suppose I had not gone more than a hundred yards after this pause when\nexecrable events occurred. It seemed as if some dreadful celestial\nhousemaid suddenly woke up, and went on with her work. She shut the\nwindow (that is to say, the wind dropped), and began to dust. She dusted\nall the clouds away, and in ten minutes there was not one left. From\nhorizon to horizon there was a sky positively Egyptian, and an\nabominable sun shone with hooligan ferocity. And I was going a-fishing!\nI said what I should not say with such extraordinary distinctness and\nemphasis that I rapidly took out my field-glass, and swept the\nuntenanted fields of heather to see that there was no one within a mile\nor two. But I expect the roe-deer heard.\n\n       *       *       *       *       *\n\nSandy was waiting for me at the near end of the loch, when I arrived\nthere a quarter of an hour afterwards. Scotchmen are never cynical, but\nI should otherwise have suspected him of cynicism when I saw that he had\nbeen at pains to set up my rod, and was soaking a length of gut. The\nbrilliance of the sun from the polished and untarnished field of water\nwas a thing to make the eyes dazzle. So I was cynical in turn, and, from\npure cynicism and nothing else at all, I put on (for the sake of the\ncurious) an astonishing fly, with a green body bound with silver, and a\nZulu. It was a shade too cynical to go out in the boat, for I think\nSandy would have seen through that, as it was impossible that any fish\nshould rise at anything in this state of affairs, and I fished from the\nshore. Fishing at all was an idiotic proceeding, and so the incredible\nhappened. I wish to call attention to the incredibility of it, since it\nhappens to be true.\n\n       *       *       *       *       *\n\nHere was I, then, on a still and windless morning, with a blazing sun\noverhead, and a looking-glass loch in which were supposed to be\nmonstrous fish, whose shyness apparently increased in ratio to their\nweight, for nobody had ever seen them before, but had only heard about\nthem second-hand, like ghost-stories. Half a dozen casts carried out a\nconvenient length of line, which fell, so it appeared to me, on the\nglassy surface of the water like the cane of an angry schoolmaster,\nresonant and cruel. Then at the end of the cane, where the Zulu was,\nthere came a boil just underneath the looking-glass; my rod bent, and\nthe reel screamed. For one moment I knew, so I thought (for the boil\ncame just as I was preparing to cast again), that I had hooked some\nstalwart weed, or perhaps a snag of tree-trunk. Then I knew I had hooked\na fish. He was clearly insane to have taken a fly at all, but what\nmattered was that he was a large lunatic. I thought I knew also that\nthis was but the first act of what would turn out to be a tragedy. But\nthe tragedy was not for me.\n\nAgain, for the sake of the curious, I will give his weight. He turned\nthe scale at five pounds some six hours later. So I imagine he was about\nfive and a half when he came out of the water with the Zulu in his\nmouth. He was mad; he turned a fierce Bedlamite eye on me.\n\nI dare say I am more impatient than the true fisherman, but when I have\ncast my fly upon the waters for three hours without a hint of a rise, I\nsit down, and do not feel it incumbent on me to rise again unless\nconditions change. So when, at about two o'clock, nothing further had\nbroken the surface of the loch except the cane of the schoolmaster, I\nfelt, after eating my sandwich, that I was not unlikely, without\nincurring the contempt of Sandy, to prolong the interval. I wanted also,\nafter my mis-tryst with Pan that morning, vaguely also, after that day\nof bovine observance of Nature which I had spent a week or two ago in\nthe garden at home, to 'sit up and take notice.' Instead of nirvanic\ncontemplation, I wanted to focus all that surrounded me, not to see a\nstag-beetle advance ten yards, and then go back to the place he advanced\nfrom, but to see the activity of it all, to be alert and to collect, not\nto be lazy and to soak.\n\nYes; it was a wonderful day. Almost immediately I spied two little\nhuman figures on the adjoining forest creeping, creeping up a steep\nbrae. A mile below I saw their ponies. They moved so slowly that it was\nonly possible to see they moved at all, because they passed out of the\nfield of my glass; the deer I could not find.\n\nThen, after watching them for ten minutes more, I saw they stopped.\nStealthy movements went on. Then came the sharp crack of a rifle, but\nbefore the report reached me they had both jumped up, and ran into a\nhollow of the hills, where I lost them. It was like being at sea, and\nhaving news twitched out from the receiver of a Marconi apparatus.\n\nBut hardly had that drama been played to its curtain when another\nstarted. The call of a startled grouse, 'Come back, come back, come\nback!' sounded close at hand, and it was followed by another and yet\nanother. Sandy had remained by the edge of the loch when I climbed this\nhillock for my lunch, and since then I had been very quiet, so I could\nnot imagine what had caused this commotion on the hill, as the stalkers\nwere not on this beat at all to-day. I could account, in fact, for the\nmovements of any human being that could have disturbed grouse for a mile\nor two. Then I looked up to the enormous sky, and saw.\n\nAbove me, but close, so that I could see the outspread feathers of the\nwing, was a golden eagle. As I watched I saw he was not vaguely\ncircling, looking out for prey, but employed in his stalk, even as on\nthe other side of the valley ten minutes ago I had watched another\nstalk. He was sweeping wide circles of the moor, and driving up towards\na gully of the hills behind the fowls of the mountain, flying in low and\never narrowing semicircles, so that it must seem to the terrified grouse\nand black game that huge-winged danger threatened from every quarter but\nthat. Yet still I could not guess what his plan was when he had driven\nthem there.\n\nAnd then I saw. Straight down from the grey crag of cliff that rose on\nthe west of this gully, into which he had driven the birds, there\ndropped his mate, savage and hungry, seeking her meat from God. Aha, you\ngrand Mistress Eagle; it is dinner-time!\n\nMerrily and well has the old cock-grouse lived in the heather, lying\nwarm in the sun, and filling himself with the good things of the\nmoorland, but to-day Pan sends him to your table, and in the swift\nhissing down-rush of your wings he hears his pipes. Pan will play them\nfor you, too, some day, and the grey film will cover over your fierce\nyellow eye that was wont undazzled to behold the sun in his strength,\nand the strong hooked beak which gasped for one breath more of the\naromatic moorland air will close, and be hungry no more, and the\ncrooked, horny talon will relax, and next year, maybe, I shall find\nwhitened bones on the hillside, and perhaps, crumpled up under them, a\nfeather, an eagle's feather. But I shall not be so foolish as to say I\nhave found you, for do I imagine that that is all there is of you, that\nyour life, your spirit, has been blown out like a candle? I know better\nthan that.\n\nFor, indeed, there is no other explanation possible of the incessant\nwar, the death, the murder, the butchery in which Nature's fair hands\nare steeped and stained, except by this one supposition that the spirit\nof bird and beast escapes at the moment of death from the splendid\nsunlit prison of this beautiful world, which has the bright-eyed hours\nfor its bars. Otherwise the world becomes a mere intolerable shambles,\nviler than Chicago. I at any rate cannot believe otherwise, but should\nany sceptical reader at this point ask me to sketch out for him the\nsubsequent movements of the wasp he has just squashed in the tongs, or\nthe trout I have just landed, I hasten to assure him that I have not the\nslightest idea about them. But that does not invalidate the explanation,\nnor in the least disturb my complete belief in it. I do not know what\nthe weather will be this day year. But I make no manner of doubt that\nthere will be weather of some kind. I only insist that he with his\ntongs, and I with my Zulu-fly, cannot destroy life. One cannot even\ndestroy matter; how much less, then, the lord and master of matter!\n\n       *       *       *       *       *\n\nI think I have never been in a house where absurd gaiety--the gaiety of\nfriends, of health, of outdoor spirits--was so rampant as here; and she\nwhose house it was, and who was leader of the ludicrous, was she, as you\nmay have guessed, who in June had asked me to come here for the last\ntime. That evening when I got home I found her sitting out in the garden\nenjoying the last half-hour of sunset, and she beckoned to me across the\nlawn.\n\n'It's true,' I said. 'I have caught the original trout. He had gone mad\nfrom old age and riotous living, and came to the fly when the sun was\nbrightest and the winds were dead.'\n\n'I wish you wouldn't use such beautiful language,' she said. 'How much\ndoes he weigh?'\n\n'About a ton. He has gone to be weighed now.'\n\n'And anything else?'\n\n'Not a fin. No more bites, as somebody said last night. I chattered with\nrage.'\n\n'You did; and what have you been thinking about?' she asked.\n\n'Pan chiefly. No, to be honest, I think I have thought about the fish\nmost. But Pan next!'\n\nShe turned rather slowly on her long wicker-couch, the tired aching\nbody for the moment usurping the use of her eyes.\n\n'Ah, don't let us talk,' I said; 'you are tired and suffering.'\n\nAt that she laughed.\n\n'All the more reason for thinking about something less inferior than\none's own health,' she said. 'What cowards we are nowadays! Why, our\nforebears in Elizabeth's time used to go smiling to the rack for the\nsake of some small difference of dogma, and we snivel when we have the\nopportunity of showing, by our contempt for pain, the truth of things\nthat matter much more. If bravery in the abstract and cheerfulness are\nnot worth being brave and cheerful for, I don't know what is. In any\ncase, what conclusion did you come to about Pan? Oddly enough, I have\nbeen thinking of him, too. Let's compare notes, and see if we mean the\nsame person.'\n\n       *       *       *       *       *\n\nI told her more or less what I have already written down on the subject,\nand at the end she nodded at me with the quick eager gesture that was so\ncharacteristic of her.\n\n'Hurrah!' she said. 'I have guessed the same. So perhaps our guesses are\nright. But I put it to myself rather more personally, and, though it\nsounds conceited, so much more vividly than you. That is only natural,\nyou know; Pan concerns me much more immediately than he concerns you, we\nhope. And another image of him suggested itself to me, which appeals to\nme more than your figure of the ferns being pushed aside, and the hand\nwith the pipes in it being raised to the smiling lips. Listen!'\n\n       *       *       *       *       *\n\nThe sun had dropped behind the big trees to the west of the lawn,\nleaving us in shadow, though it still shone on the hills to the east of\nthe house. But evening was coming without any chill or whisper of autumn\nin it, and in this northern latitude nights were short in August. It was\nas if she already saw dawn.\n\n'Jim and I and our children,' she said, 'and you and all my friends are\nshipwrecked, or so it would seem to anyone who did not understand, on a\nlittle rock surrounded by infinite sea. Every one alive in the world is\nthere, too, as a matter of fact, but our friends somehow are so big to\nus, and strangers and acquaintances so small in comparison, that all\nthat really is seen by us is our own immediate circle. Huge thumping\nseas surround our rock, and, for some occult reason, we all have to sit\nexactly where we are, while the waves rush up, and every moment sweep\nsomebody away. We can't move our places, and go higher up on the rock,\nand we have to sit and look at the big waves, we poor shipwrecked people\n(so a man who does not understand would say), and know that this wave or\nthe next will wash us off. That is the ignorant view of the situation,\nand the most pessimistic, so we will answer it at once.\n\n'Even if it was right, what then? Supposing we were shipwrecked, and all\nround us was the howling sea of death, would it not be much better,\nuntil the wave swept us off, to make the best of it, to talk, and laugh,\nand be pleasant with our friends, instead of looking with\nterror-stricken eyes at the hungry sea? How much nicer even for\nourselves to be amused and talk a little while, instead of being\nfrightened, and how much nicer for our friends when we are swept off, as\nwe all certainly shall be, to know that before we were swept off we\nwere moderately cheerful, and picked up bits of seaweed, and played with\nshells! I say nothing of the moral aspect of it all, because if you once\nbring that in there is no question any more about the matter, since in\none case we are brave, and in the other merely cowardly. But given that\nwe are shipwrecked, that the sea of hungry death surrounds us, and will\nsoon pick us off, how much better, on the lowest possible view of the\naffair, to play about, to be kind and gentle, even if to-morrow there\nwill be an end of us, utterly and for ever!\n\n'Yes, I am using beautiful language too. But I am talking of beautiful\nthings.\n\n'Well, that view is the silliest and most incomprehensible possible. How\ndid we get on this absurd rock, if only death surrounds us? Did we come\nfrom death into life? That is impossible, since scientifically you can't\nproduce life out of dead things. Or did some ship founder on the sea of\ndeath, and did we swim to shore, where we shall live until a wave sweeps\nus off again? That is possible; but, then, what was that ship on which\nwe once were passengers, that for a time anyhow, until it foundered, if\nit did founder, rode over these waves? That is a serious question, but\nthere is only one answer to it. The ship must have been life in some\nform. But the image does not seem convincing, does it?\n\n'What is left, then? Only this, that the sea which surrounds us on our\nlittle rock is not death at all, but life. Just as some day without\ndoubt a wave will sweep us off our rock again, so there is no doubt that\nonce a wave of that sea put us on the rock where you and I now are. If\nthere is a wreck at all, it is a land-wreck, a wreck that puts us on\nshore. From the great sea of life we have been washed up for a little\nmoment on to our little rock. Soon we shall be received back into life\nagain!\n\n'In the interval, though in a new sense we are wrecked, how interesting\nis our rock, and how full of dear people, and pink shells, and divine\nthings of the sea that life, not death, casts up round us, and nourishes\nby the spent water of its waves! How utterly idiotic it would be not to\ncollect them eagerly, these little bits, for when we go back into life\nwe shall see the forests from which they come, the sapphire caves in\nwhich they really dwell. A little bit of life, that grouse that the\neagles ate, was cast up close to you to-day. I shall particularly ask,\nwhen the wave takes me off again, where it came from. And I shall go and\nsee the place. And certainly I shall see Mistress Eagle come back.'\n\nCourage, huge, natural courage like this, absolutely unassumed,\nabsolutely instinctive, may have one of two effects on the beholder of\nit. It may make him weep for the admiration of it, or it may make him\nlaugh out of joyousness of heart for the same admiration. At least I\nlaughed.\n\n'Oh, be sure to show me the place when I come,' I said. 'I am certain\nthat Mistress Eagle will have a nice house.'\n\n'They all have,' she said. 'There are many mansions.'\n\nShe looked at me in silence a moment.\n\n'But I was not so certain of all these things when first I knew that I\nwas so soon to see them all,' she said. 'At first, though I was never\nexactly frightened, I was dazed and stunned. I saw nothing clearly. I\nmust use another image for that, and say that days passed as one sees\nthe landscape pass through a railway-carriage window which is blurred by\nrain. I could see nothing clearly; it was all dim and rain-streaked. But\nthen, without any conscious effort on my part, except perhaps a little\nexercise of patience, we passed--the train and I--out of the scud again,\nand soon the glass cleared, and I saw the green valleys and the sunny\nhillside just as they had always been.'\n\nAgain she paused.\n\n'I have not told you anything of importance yet,' she said; 'all I have\nsaid is really quite obvious. But this now----\n\n'You think of Pan as the smiling face that peeps from the fern, the\npresence that assures all suffering things that he is kind when he pipes\nto them, even though the sound means death. But surely that is no more\nthan a sort of pagan mythical aspect of him. I always think that he\nsuffers too, that every pain which he seems to inflict is only the\nreflection of the pain in his own universal heart, although he still\nsmiles. It is from the cross that He smiles at us all.'\n\n\n\n\nSEPTEMBER\n\n\nThe 'Season of mists and mellow fruitfulness' has indeed been a close\nbosom friend of the maturing sun, and for the last three days before\nLegs went back to his crammer in town, he, Helen, and I spent a\nprostrated existence. Heat that in July invigorates, is utterly\nintolerable if it occurs at the end of September, just as the crisp\nwinter day, which would be so welcome in January, descending to the\nearth as it usually does in June, produces merely amazed horror at the\nweather, and probably a cold. The superficial view that we suffer\nbecause we are improperly clad for these climatic surprises (a view that\nHelen put forward the other night) is beside the point. During these\ndays, if I was improperly clad, it was only because I has so little on.\nIn fact, only ten minutes before she had said as much.\n\nThe state of Legs' affections, I am bound to add, aggravated the\nsultriness of the weather, and made me feel exactly 350 (three hundred\nand fifty) years old. To take it at its best, he was embarked on a\nviolent flirtation with a dreadful girl; to take it at its worst, he was\nfalling in love with her. She is the daughter of a neighbouring minute\nsquire, who owns three turnip-fields, and calls it shooting. Legs shot\nover it the other day, and after walking over the whole estate twice,\ngot back to The Grange in time for lunch. This was before I returned\nfrom Scotland, or I should have tried to prevent it. Probably I should\nnot have succeeded.\n\nThe neighbouring squire's name is Ampthump. I know quite well that it is\nnot his fault, but that, wedded to what he is and a German wife, makes\nme unable to like him. His wife makes incredible quantities of jam,\nwhich, again, is an innocent pursuit; and Charlotte, the daughter, talks\nGerman to Legs, who I wish was more like Goethe. The whole family, in\nfact, as may have been already perceived, appear to me to be simply\nintolerable.\n\nThe attachment also has already led to equivocation on the part of\nLegs. He pretends that he talks to Charlotte because it is so good for\nhis German. He knows that it is not so, and I know it is not so, and I\nthink he knows that I know it is not so. But it really looks at the\nmoment that unless they marry each other there will be a broken or, at\nany rate, a cracked heart. I only hope it will not be Legs'. I don't\ncare the least what happens to Charlotte's heart. It may, however, be\nonly a flirtation, in which case there probably will not even be a\ncrack. Legs will wake up one morning, and after handling some precious\nwithered flowers will wonder what on earth they ever meant to him, and\nthrow them in the fire. Or Charlotte will do something equally\ndesperate. That is my hope; my fear is that they are falling in love\nwith each other.\n\nThis narrative, it should be understood, is the gist of what I have been\nsaying fragmentarily to Helen. She considers it a cynical view, which\nalarms me, since I hold the creed that all cynics are properly and\nirretrievably damned. To-night Legs went to bed early, with dishevelled\nhair, a wakeful eye, and a gale of sighs, and I came upstairs to talk\nto Helen about it all while she brushed her hair.\n\n'You are quite ridiculous about it,' she said. 'Because you happen not\nto like the Ampses (we have agreed on that abbreviation), you think that\nthey are unlovable. Legs has proved the contrary. Besides, what on earth\ndoes her name matter, if she is going to change it?'\n\nI groaned intentionally, and in a graveyard manner.\n\n'Do you mean that you think Legs is in love with her?' I asked.\n\n'Yes; at least, I hope so. He had a long talk with me to-day. He said he\nfelt it was time he settled down. What a darling! Just twenty! I wish I\nwas.'\n\nMost of this was irrelevant. I tried to pick out pieces that were not.\n\n'Of course, her name doesn't matter,' I said. 'Her name might be----\nWell, you can't do worse than Ampthump, and it does happen to be exactly\nthat. But her face is like a ham----'\n\n'That is superficial,' said Helen. 'Beside, it isn't. It's oval.'\n\n'So is a ham. And she's a prig. Ampthump! Good Lord!'\n\nI am afraid I shouted this, because she said: 'Hush! Legs will hear.'\n\n'Not he. Or if he does, he will think it is only the wind whispering the\nbeloved name.'\n\n'Yes, but you didn't whisper it. Oh, do take the brush. You made me send\nmy maid away, so you must do it yourself. I can't brush from here,\nbecause my arms are in front.'\n\nNow in my heart I pity everybody who has not seen Helen with her hair\ndown. All such folk, in all their millions, lead impoverished\nexistences. There is a wave in it that is like the big unbroken billows\nwhich succeed a storm, when the clouds have passed and the sun shines.\nIt is lit from within, even as they seem to be irradiated from the\ndepths. Those billows must go over a sandy foreshore, for they are\nyellow, and the sun--I know not how--must be foggy, for there is a\nlittle red light in them. And brushing, as I did now, I held my hand\nover them, and the hair rose to it with a tiny cracking sound. Her hair\ncame to my hand, lifted towards it that unminted gold that framed her\nface, and covered her ears. And for a little while it was no wonder that\nI forgot about Legs and his Charlotte.\n\n       *       *       *       *       *\n\nI suppose every one knows the sensation of being lost. You can be lost\nall by yourself, as I was once, as I have said, in the western desert of\nEgypt, on which occasion the bray of a donkey was to me the trumpet of\nthe Seraphin. That was a dreadful experience, since it implied being out\nof touch with life. But I should be glad to know if there is anything\nthe world holds which is more enraptured than the sense of being lost\nwith one other person, to feel the world swim away, and be dissolved, so\nthat you and the comrade you are with are quite alone. To feel that\nthere is no existence except the existence of her who is lost with\nyou.... It was Helen's hair.\n\n'That's the world's side; there's the wonder!' That lover understood.\nEveryone saw Helen's hair.\n\n    '\"But the best is when I glide from out them,\n      Cross a step or two of dubious moonshine,\n      Come out on the other side....\"'\n\nI never could quote correctly. The point is that the beloved has another\nface, the face she turns to her lover. No one else sees it; it is 'blind\nto Keats, him even.'\n\nA moment ago I thought that no one but me must see Helen's hair. Now let\nthem all see it, the waves of the sunlit sea, not breaking, unless the\nbreak be where I put my hand an inch above them.\n\n'Thanks, dear,' she said soon. 'You brush it much better than my maid.\nNow shall we talk for five minutes? Then I must go to bed.'\n\n       *       *       *       *       *\n\nI had hideous accumulations of various fag-ends of work to do, and at\nthe end of the five minutes, or it might be ten, I went downstairs\nagain, to begin at any rate this dreadful patchwork of odds and ends. It\nwas still, I was almost sorry to observe, only just eleven, and since I\nhad with both eyes open deliberately and firmly wasted all the hours of\nthe day, my uneasy Conscience told me that I had better, if it was to\nhave the ease it craved, not think of leaving my chair for a couple of\nhours at least. I argued this point with it, and lost some minutes, for\nI told it that it was extremely bad for me to work at night; that it\ntook more out of one than work in the day; that work done under these\ncircumstances was never good work; that doctors recommended one never to\nwork at night, but go peacefully to bed before the evening\nfever--whatever that might be--set in. Then there ensued a short\nspirited dialogue.\n\n'Most sensible,' said Conscience. 'Give me your word that you will get\nup at six to-morrow, then, and work for two hours before breakfast, and\nyou have my leave to go to bed now.'\n\n'But I shan't wake at six,' said I, 'and the servants have gone to bed.'\n\n'I will wake you,' said Conscience. (Conscience is quite capable of the\nodious feat.)\n\n'But I can't work before breakfast,' I said. 'It makes me feel'--I could\nnot think of the word for the moment--'oh yes, faint.'\n\n'Well, feel faint, then,' said Conscience.\n\n'But I would sooner not; it implies weakness of the heart.'\n\n'Not to do your work implies weakness of character.'\n\n'Shut up,' said I, 'and let me begin, then.'\n\nAnd I could swear that my Conscience gave a self-satisfied chuckle.\n\nFor an hour I waded wearily, knee-deep only, so to speak, in work, like\na man who wants to swim, but has to trudge out over level sands. Most\npeople, I fancy, even the laziest of us, like working, when we get up to\nour necks, or, better even, out of our depths, in it, but the wading is\nweary work. The worst of it was that the fact that I had to wade so far\nwas entirely my own fault, for the whole of the last week I had never\ntaken the trouble to finish up any one job, and now there waited for me\nseveral bills to pay, since a few mornings ago I had sat down to pay\nbills, and had paid them all except two or three; several letters to\nwrite, all of which had to begin either falsely (_i.e._, 'I have just\nfound your letter of the 17th) or apologetically (_i.e._ 'I haven't\nanswered your letter before because----'). Then there was a\nhalf-corrected proof of an unfinished article, badly written originally,\nand, what is more, written without conviction. It was on a subject that\ndid not particularly interest me, and I had only written it because the\nmisguided editor of a magazine had offered me \u00a325 for it, and I very\nmuch wished to buy a seal-top spoon which cost exactly that sum, and\nwhich I knew perfectly well I had no right to buy. So, saying to myself\nthat I would write this article (which I should not otherwise have\ndone), I had bought it, and here was the dismal price that I had to pay\nfor it--namely, that this wretched article was a piece of literary\ndishonesty. I had to fudge and vamp over it, trying to conceal the\nnakedness of the land by ornamental expressions. That was brought home\nto me now. It was all bad cheap stuff, and though most of us are\ncontinually turning out bad cheap stuff, not knowing it is bad and\ncheap, such manufactures become criminal when we do know it. As long as\nwork is honest from the workman's point of view, it is only his\nmisfortune when he does not know its valuelessness; but when he does\nknow its valuelessness, he sins by intention, and is a forger. I was\none, and by my forgery I had bought a seal-top that was not. I thought\nthat when I tacitly agreed to work for two hours to-night, my tiresome\nConscience would put its head under its wing, and leave me alone; but I\nfound now that it was broad awake again, and chirping like a canary.\n\n'What are you going to do?' it chirped. 'Are you going to send out a\nrotten forgery which everybody who knows anything will detect? or are\nyou going to tear it up, and be left with a purchase that you know you\ncan't really afford? Remember that you must get a new dining-room carpet\ntoo; you promised Helen you would. Chirp, chirp, chirp!'\n\nI am bound to say that this enraged me.\n\n'What's the use of making that row?' I said. 'It's you, Conscience, who\nhas to settle.'\n\n'I haven't the slightest idea,' said Conscience. 'It's your fault; you\nwouldn't listen to me when I told you that you had no right to accept\n\u00a325 for your dreadful article.'\n\n'You didn't say it so loud, then,' said I.\n\n'No, but you heard all right,' said Conscience.\n\n'I hardly heard,' said I. 'You spoke so indistinctly.'\n\n'Yes, but you did hear,' it chirped, with a sort of devilish\ncheerfulness. 'You knew quite well what I meant. Now you suffer for it.\nHurrah!'\n\nI wonder if I am cursed in this matter of Conscience beyond the majority\nof mankind. Often and often (I will swear to this in the House of Lords\nif necessary) my Conscience is hardly audible at all at the time when I\ndo anything which I ought not to do, or omit to do anything which I\nought. To continue the simile of the canary, which really fits the case,\nwhen the actual choice comes, it is as if the canary had a thick\ngreen-baize cover round its cage, and only hoarse and muffled notes\nreach me. Very often, indeed, I am sorry to say, I don't attend to them,\nor say it is only the cat, and in consequence do what I should not. Then\nthe moment it is done the baize cover is whisked off, and the infernal\nand cheerful chirping, or so it sounds, succeeds to the wrong choice or\nthe weak omission. And the burden of the chirping is always the same.\n\n'I told you so; I told you so. Now you are in a mess! What are you going\nto do now? Chirp, chirp, chirp!'\n\nAnd a hurricane of dry and deafening notes follows.\n\n       *       *       *       *       *\n\nI sat there with this column of stupid twaddle in my hands, and\nConscience watched me with its bright bird-like eye. Much as I like\nbirds, I hate their eyes, because they remind me of Conscience. They\nare beady and absolutely unsympathetic, frightfully quick to see, and\nwithout a particle of pity in them. Conscience never pities one at all;\nit is the foe that is of a man's household. It always gloats over one's\nmistakes, and things that are more than mistakes, and only says:\n\n'Here comes the master with the whip. A new lash, I see, this time. And\nwhat a thin shirt you have got on!'\n\nNor, when the whipping is over, does Conscience sympathize.\n\n'I told you so; I told you so,' it says. 'No, there is no soothing\nointment of any kind in the house. I ate it all up. Wasn't that a\nbeautiful new lash?'\n\n       *       *       *       *       *\n\nWell, I tore that dreadful nonsense up, and wrote another apologetic\nletter. I am getting quite good at them. But to-morrow--this is what\nmakes Conscience mad--I shall tell Helen about it. The telling is not\npleasant; it never is. But as soon as Helen knows, Conscience has simply\nto retire. It does not understand why it suddenly becomes so\nunimportant, and that gives it a fit of impotent rage. Nor do I quite\nunderstand, though I am nearer to the explanation than Conscience is.\nBut she understands. At least, I suppose so, or else she would not be\nable to put the green-baize cover on again.\n\nAnd then, what with apologetic letters, and the drawing of two or three\ncheques, and the stupid attempts, in this matter of the dishonest\narticle, to produce something out of nothing, by covering up the\nnothingness by more ornamental expressions, and the eventual destruction\nof it all, I found that the two hours were gone, and that I had kept my\npromise to the idiotic canary. It had ceased chirping from experience\nwhen I told it I was going to confess to Helen.\n\nThe night was intensely hot, and through the long open windows of the\nroom in which I had been working no breeze entered. Though September had\nbut a quarter more of its course to run, it was like some sultry July\nmidnight, portending storm, for when I went out to take the night-breath\nthe sky was thickly overcast, so that no direct ray either of moonlight\nor of starshine, came earthwards. The serrated outline of the elms at\nthe end of the lawn was scarce distinguishable against the scape of the\nclouds, and the low land of the water-meadows was blanketed in a mist\nthat was only just visible by its whiteness against the black blot of\nthe hills behind. Fifi, who had very sensibly decided to sleep on the\nveranda, did not stir when I came out, though I heard the instinctive\nthump of her short tail on the tiles, the natural politeness of the dear\ndog, though she really could not stand on ceremony with me to the length\nof getting up. So, maliciously, I am afraid, since I thought this\nslightly cavalier conduct, I said 'Puss,' though there was no Puss of\nany sort, as far as I was aware. But my malice was again thwarted, for\nFifi just tapped again with her tail, in courteous recognition of a\nstale old joke, just to show that she appreciated my intention, but she\nmade not the smallest further effort towards activity.\n\nSo she was half asleep, and all the world, this dear, blessed world,\nwhich is so full of merriness and simple, innocent pleasure, despite the\nfulminations of fashionable priests, was quite asleep, not stirring,\nscarcely breathing, just sleeping, sleeping. It was not yet the hour\nwhen, just before the hold of the night begins to tremble and be\nweakened in the sky, all living things wake for a moment--that\nmysterious moment, when sheep take a bite of grass and cows twitch their\ngrave ears, and horses stand up for a minute before they settle down to\nthe light morning sleep which dissolves with day, and when even indoors,\nif you sleep with a dog in your room, and happen yourself to be awake,\nyou will hear a stretching of limbs on your bed or on the carpet, and a\nlong sigh breathed into the blankets. Plants and flowers, so I truly\nbelieve, feel the same thing; and though there may be no wind\nperceptible to you if you are abroad, as sometimes I am, at that hour,\nyou will hear, just at the moment when cattle move and sheep take their\nbite of grass, a stir go through the trees, and a hushed whisper lisp in\nthe flower-beds. At that moment, too (you need not credit this, though\nit is absolutely true), though it has rained all night till then, and\nwill rain thereafter, steadily, soakingly till morning, the rain ceases,\nas suddenly as if a tap was turned off. Time and again I have tested\nthat.\n\nBut, as I have said, that mysterious moment was not due yet. It was\nstill two hours short of it, and everything was still asleep. Even in\nthe last minute or two Fifi had fallen fast asleep, too, after I had sat\ndown in a wicker chair on the veranda, for when I called her there was\nno tap of response. To-night, too, the sleep of the world seemed to me\n(feeling it as one does by that sixth sense, which still exists dormant\nin us, and is most awake at night) to be extraordinarily deep. It was\nthe sleep of a world that was very tired with this long hot summer.\nThere seemed no pulse stirring in it at all, as you may find it stir in\nthe light sleep in which Nature indulges in June, or still more in the\ndark, wet nights of spring, when the secret boiling up of life begins\nagain from hidden root to budding tendril, so that if you lay your ear\nto the trunk of a tree it seems that the effervescence of the young year\nis audible, and sings within it, even as the telegraph poles are\nresonant with the wind that hums in the wires. Nor could I hear, when I\nrose and walked across the lawn, even though the dew was heavy on the\ngrass, the hiss of startled worms, withdrawing from the approaching\nfootfall. Black, too, and lifeless, was the oblong of the house except\nwhere the lights burned in the room in which I had been trying to be\nhonest. The long herbaceous hedge was black, the lawn was black, Helen's\nwindows and Legs' were black.\n\nI went back to the seat I had just left, and lit a cigarette, meaning to\ngo upstairs to bed when I had smoked it. Fifi still lay motionless,\nthough generally any excursion into the garden at any time of day or\nnight sets her scampering. And then, quite suddenly, quite unexpectedly,\nfor nothing was further from my thoughts, I became aware that, though\nthe physical world was asleep, there was some enormous stir and activity\ngoing on in the occult world which surrounds and permeates us. Yet that\nis perhaps a wrong expression, for the same activity and stir always\ngoes on in that unsleeping realm; and I must express it more accurately\nby saying that the part of me which was able to perceive it was suddenly\nquickened. It is possible, of course, since I confess to being able to\ngo to sleep whenever I choose, and often without delay, when I do not,\nthat at that moment I fell asleep. But whether I fell asleep or not,\ndoes not make the slightest difference, for there was clearly some part\nof my brain awake, and it made my eyes think that they saw, and my ears\nthink that they heard, that which immediately followed.\n\nAs far as I am aware, in any case, I sat down again in a rather creaky\nbasket-chair and lit a cigarette. The match with which I lit it, I threw\non to the gravel path in front of me, and, since I required it no\nfurther, it proceeded to burn prosperously. By its light I could see\nFifi with her nose between her paws. I saw, also, that my shoe-lace was\nuntied.\n\nAnd then I heard my name called from the garden, in a voice that was\nperfectly familiar to me, though for the moment I could not say, so\nelusive is the ear, whose voice it was that called. It was not Helen's,\nit was not Legs', it was not ... and then I remembered whose voice it\nwas. It called me by name, once only, in the voice that had said, 'It is\nfrom the cross that He smiles at us all.'\n\nI do not think I was frightened, but simply for the purely personal\nreason, that to me there was nothing to be frightened at. The match\nstill burned on the gravel path, so short had been the measurement of\nthis in the world of time, and I could still see Fifi's nose buried\nbetween her paws. Then she raised it, looked out into the garden with\nterrified scrutinizing eyes, focussing them on something, invisible to\nme, and gave one long howl. But there was no moon. It was at something\nelse she howled.\n\nThen, I confess, as if some bomb had burst within me, terror flooded my\nwhole mind, submerging it, and I sprang up. Simultaneously I heard a\nsort of strangled scream from the room above, and the scurry of unshod\nfeet overhead. Next moment the sound of an opening door came to my ears,\nand a quick stumbling tread on the stairs. I ran indoors, and reached\nthe door leading from my room into the hall, just as the handle was\nseized and shaken by someone on the other side of it, and Legs burst\ninto the room, his hair all tumbled and erect, and his face wearing such\na mask of terror that for the moment I recognized him only because it\nmust be he.\n\n'Who is that in the garden?' he said. 'Someone in white, who looked up\nat my window? And Fifi howled at her.'\n\nThis would never do. Nerves, terror are the most infectious things in\nthe world, and unless I took steps, there would, I knew, be standing\nhere two babbling lunatics.\n\n'I was dozing in the veranda,' I said, 'and Fifi woke me by howling. She\nwoke you, too! Legs, don't be an ass! Pull yourself together. If there\nhad been anything, I should have seen it.'\n\nLegs was as white as a sheet. The whiteness somehow showed through his\nfreckled sun-tanned skin. He was swaying to and fro on his feet, as if\nhe would fall, and I put my arm around him, and deposited him in a\nchair. Then I poured out a wineglassful of neat whisky.\n\n'Don't speak another word till you have drunk that,' I said. 'Then I\nshall count ten slowly, and then you may speak.'\n\nFifi had followed me in, and sat close to the door whimpering. With my\nheart in my mouth and a perspiring forehead, I went across to the\nwindow as I counted, shut and locked it, and pulled down the blind.\n\n'Nine, ten,' I said.\n\nA little colour had begun to come back to Leg's face. He had drunk the\nwhisky, a beverage which he detested, like water, and the frozen fear of\nhis eyes was less biting. And then, as suddenly as it had come on, my\nterror left me. Whatever it was that I had heard, whatever it was that\nLegs had seen and Fifi perceived, there was nothing to terrify. Besides,\nwithin myself, now that the cowardly disorder of my nerves had passed, I\nbelieved I knew what it was that had made its presence so strangely\nperceived by us all. The mortal suffering of a dear friend was over.\nAlready I was ashamed of having told Legs that I had been asleep and had\nneither seen nor heard anything.\n\n'Legs, I lied just now,' I said. 'I heard my name called from the garden\nin Margaret's voice.'\n\n'You mean she is dead?' asked he gently. 'The last accounts had been\nbetter, I thought.'\n\n'I'm sure she is.'\n\nThen for a moment, like a sudden squall, the white terror passed over\nLegs' face again.\n\n'It was not her I saw,' he said hoarsely; 'it was Death. I thought she\nhad come for me. Fifi saw her too.'\n\nI sat down on the arm of his chair.\n\n'Yes, old boy,' I said, 'I think that you and Fifi both saw some\nmanifestation of what I heard. But there is nothing to be frightened at.\nBut how was it you were at your window? You had gone to bed hours ago.'\n\n'I know, but I couldn't sleep, so I got up and sat by the window.'\n\n       *       *       *       *       *\n\nWe sat there for some time after that, and by degrees Legs recovered\nfrom his collapse, and soon, instead of terror, mere sleepiness invaded\nhis face. Once or twice he stifled a yawn, and at length he got up.\n\n'I am dead sleepy,' he said. 'I think I shall go to bed.'\n\n'You are not frightened any longer, are you?' I asked.\n\nLegs looked at me out of drooping eyelids, and he seemed puzzled.\n\n'Frightened? What about?' he said. 'Good-night.'\n\n       *       *       *       *       *\n\nI was very late down next morning, and found that Helen and Legs had\nnearly finished breakfast. As I came in he jumped up.\n\n'Ah, here he is!' he cried. 'Now, did you sit up very late last night?'\n\nWhen he asked that I began to have some suspicion of what was coming\nnext.\n\n'Yes, very. Why?'\n\n'Well, were you talking to yourself? Helen and I both woke in the night,\nand heard talking in your room. I had had some dream that frightened me,\nand I nearly came downstairs for human companionship.'\n\n'Why didn't you?'\n\n'I was too sleepy. But--were you talking?'\n\n'No. You were dreaming. So was Helen. I may have groaned now and then\nover proofs, but not more than that.'\n\n       *       *       *       *       *\n\nLegs nodded at Helen.\n\n'I told you it was ghosts,' he said.\n\n'And you heard voices too?' I asked Helen.\n\n'Yes; at least, I thought so. But I was very sleepy. I thought also I\nheard Fifi howl.'\n\n       *       *       *       *       *\n\nSo, you see, there is no corroboration of my story, and if I dreamed it\nat all, or made it up, there is no one to whom I can appeal for\nconfirmation of its verity. But there is just this little bit of\nevidence--namely, that though Legs had finished breakfast, he went on\ndrinking cup after cup of tea. When Helen left us he explained this to\nme.\n\n'I woke with a mouth like a lime-kiln,' he said--'just as if I had been\ndrinking that dreadful whisky of yours. I drank most of my jug, too, and\nthey had to bring me more water to wash in.'\n\nWhat happened last night, then, had been wiped clean off Legs' brain\nagain. Whatever it was that he had seen, that which made him stumble\nwhite-faced downstairs, had gone. But an hour or two later, while we\nwere out playing croquet in the garden, some faint echo of it, I think,\ncrossed him again. A telegram was brought out for me, which contained\nwhat I knew it would contain, and I handed it to him when I had read\nit. Then we went quietly indoors.\n\nJust as we got into my room again, he said:\n\n'How odd that sensation is of feeling that something has happened\nbefore! When you handed me the telegram, I felt I knew what was in it.\nAnd during the last week she had been rather better, had she not?'\n\n\n\n\nOCTOBER\n\n\nThe business of the dining-room carpet (a case of conscience makes the\nwhole world kin, so I confidently return to this matter) was settled\nmore beautifully than I had thought possible. I told Helen all about it,\nand she said:\n\n'Thank goodness you tore the thing up! Dear, you _are_ such a silly ass!\nThere's nothing whatever more to be said. You are, aren't you?'\n\n'There's nothing more to be said, I believe you remarked.'\n\n'Well, you may just say \"Yes,\"' said she.\n\nSo I said 'Yes.' It was a variant of the woman's last word, spoken by a\nman instead.\n\n'There, now we'll go and quarrel about the rose-garden,' said she.\n\nWe went and quarrelled. She was flushed with triumph over making me say\n'Yes,' and in consequence I got my way about several disputed points,\nwhich to-day the darling thinks she chose herself.\n\nThe rose-garden is a design of unparalleled audacity, and when it grows\nup, it will be nothing short of stupendous. For between us Helen and I\nare territorial magnates, and beyond this house and garden, which are\nhers, I am owner of two fields, and limitless possibilities. I bought\nthem a year ago, in a sudden flush of extravagance, and for six months\nwe maintained there (at staggering loss) a poultry-yard in one corner\nand a cow over the rest. The original design, of course, was to make a\nsound investment in land, which, in addition to the fathomless pleasure\nof owning it, would keep us in butter, eggs, chickens to eat (not to\nmention, as I hasten to do, savouries of chicken liver on toast), and\npossibly beef. If one considers the question closely, it is difficult to\nsee how a cow can (1) give milk, and (2) give beef; but Helen, in\nvisionary enthusiasm, said we should have oxen as well, and why not pigs\nin the farther corner? I did not at once see why not, and I bought the\ntwo fields with the same unconcern as I should have bought a box of\nmatches, which yield so sure an enjoyment in the matter of lighting\ncigarettes.\n\nThen we both began to learn that, though we might be gardeners, we were\nnot farmers. The poultry-yard was (mistakenly, no doubt) erected at the\ncorner of the field nearest the house, and morning after morning we were\nawakened at dead and timeless hours. Helen said that when a hen made a\nlong clucking noise, it meant she had laid an egg, and that, till the\nthing became incredible, consoled me. For if she was right, it was clear\nthat hens laid invisible eggs, or that they were doing tiresome\nconjuring tricks, and that the long-drawn crow meant, 'I have laid an\negg, but see if you can find it. I am the mother of this disappearing\negg.' We usually were not able to do so, but sometimes an egg was found\nin a hedge, or in a ditch, which when found was totally uneatable,\nexcept by the Chinese. Personally, I believe that by some unhappy\nmischance we had bought celibate and barren poultry, whose customs drove\nus daily nearer Bedlam; in fact, it the pig that was our hellebore.\n\nThe pig was not a pig, but a sow. She went mad, too--or so I must\nbelieve--jumped the pigsty in the opposite corner, made a bee-line for\nthe poultry-yard, went through our beautiful wire-fencing as if it had\nbeen a paper hoop in a circus, and ate two hens. The cock beat a\nmasterly retreat, and was never heard of again. The other four hens\nfollowed him. And the sow, dripping with gore, lay down in the hen-house\nand slept. Almost before she woke, she was sold for a song.\n\nThen the cow came. I do not wish to libel her, but I think I may safely\nsay that she was milkless and excitable, and had a wild eye. She roamed\nover my fields (mine, I had bought them) as if they were her own. Had\nnot Legs been so agile and swift, she might have tossed him. As it was,\nshe ran into the brick wall at the lower end of the garden, and made her\nnose bleed. As far as I know, that was the only liquor that she parted\nwith. She was probably mad also, for she used to low in the middle of\nthe night, when all proper cows are fast asleep. Asleep or awake,\nhowever, now she makes her fantasias elsewhere. I almost hope she is\ndead, for it requires a larger optimism than I possess to believe that\nshe will ever become a proper cow, for she was more of a steed for\nMazeppa. Perhaps she was a horse after all, a horned horse. I wish we\nhad thought of that at the time. As it was, we sold her at outrageous\nloss, as a cow. And with her we parted with any idea of keeping farmyard\nanimals for purposes of gain. Perhaps we were not serious enough about\nit, and the animals saw that.\n\n       *       *       *       *       *\n\nThrough last spring and summer the fields rested after this invasion of\noutrageous animals, and about the middle of May it struck Helen and me\nsimultaneously that we were going to have a crop of hay. That was\ndelightful, and much less harassing than hens. Hay would not wake one at\ntimeless hours, nor would it go mad, and have to be sold at a quarter of\nthe price we gave for it, since we gave nothing for it at all. It was\nthe pound of tea thrown in with the fields we had bought, or the _Times_\nnewspaper thrown in with your subscription to that extraordinary\nlibrary.\n\nFrom this there was born the scheme of giving a haymaking party, to\nwhich we originally planned to ask everybody we knew, amended that to\nasking all the children we knew, and afterwards (this was Helen's\namendment) decided not to ask anybody at all, partly because children\nwere so serious, but chiefly because there might not be enough hay to go\nround. We neither of us knew how many square yards of hay it was\nreasonable to supply to each person, and it would be dreadful if there\nwas not enough. Either Helen or I, or both of us, would have to go\nwithout, and it was safer to give the haymaking party to each other. We\nwere in town all May, and the first half of June, but had left word with\nthe gardener to send us a postcard when the hay was ready. The weather\nthroughout these weeks was gloriously sunny, and in our mind's eye we\nsaw the crop growing taller and thicker with each blazing day.\n\nThen one evening came the memorable postcard:\n\n'A reddy.'\n\nWe flew to the 'A.' In the middle of the largest field was a small\nhaycock like a penwiper. One not quite so large and round at the top,\nmore like a pincushion, was visible in the next field.\n\n       *       *       *       *       *\n\nIt was clear after this that the Powers that Are willed that our fields\nshould not be used for utilitarian purposes. Hence the inception of the\nrose-garden.\n\nA brick wall (the one against which the insane cow had blooded her nose)\nbounded the garden. From there the ground declined steeply away into the\nmiddle of the larger field, which was cup-shaped, the ground rising on\nall sides of it. (It was at the centre of the cup, where the sugar is,\nthat the penwiper had been raked together.) To-day a flight of steps\nmade of broken paving-stones--an entrancing material--led down the side\nof the cup from the garden-gate, and up the opposite <DW72>. Standing\nwhere the sugar is, therefore, you saw on every side of you rising\nground, which had been terraced, and walks of broken paving-stone,\ncommunicating with the two staircases, lay concentrically round. And the\nHerculean labour which had already occupied us so many rapturous\nafternoons was to plant the whole cup with rose-trees, so that,\nstanding in the centre, there was nothing visible except sky and roses.\nThat was practically done; and to-day what occupied us was the\nconsideration of the level remainder of the field, of which there was\nsome half acre. It was rough, coarse grass, starred with dandelion,\nwhich gave the first hint. We wanted to get rid of the dandelion,\nand----\n\nAt last I got Helen to agree, and I mixed together in a wheelbarrow an\ninfinity of bulbs, and other delectable roots. There were big onion-like\ndaffodils, neat crocuses with an impatient little yellow horn sticking\nup, fritillary roots, bottle-shaped tulips, the corms of anemones, and\nthe orris of the iris. Then, trowel in hand, each with a bag of bulbs\ntaken haphazard out of the wheelbarrow and with a bag of sand to make a\ndelectable sprouting-place for the roots, we started. Every dandelion\nencountered was to be dug up with honesty and thoroughness, and where\nthe dandelion had been there was to be planted a bulb taken at random\nout of the bag. Helen said it would take ten years. Personally, when I\nlooked, I thought longer, but I did not say so, for I practice reticence\non discouraging occasions.\n\nI wonder how many people know the extraordinary delight of doing a thing\nfor oneself, starting from the beginning. I do not say that it gives me\nthe smallest pleasure to black my boots or brush my clothes, since\nsomebody has already made those boots and woven the cloth. But there is\nnothing more entrancing than to deal first-hand with Nature, to make\nholes in the earth, and put in them roots, the farthest back that we can\ngo with regard to vegetable life. Rightly or wrongly, it seems to me a\npleasure as clean and as elemental as the joy of creation itself.\nWhether we write a book, or paint a picture, or carve a statue, we,\nthough we do not really create, but only arrange what is in existence\nalready, are going back as far as we can, taking just the root-thoughts\nand translating them to song or shape. And though we do not really\ncreate at all, but only use and arrange, as I have said, the already\nexisting facts of the world, passing them, it may be, through the\ncrucible of the mind, we get quite as near to Nature, if not nearer,\nwhen we go a-bulb-planting. The bulbs are our thoughts, our pigments,\nwhat you will, and when in spring-time we shall see them making a meadow\nof Fra Angelico, it will be because we have actually planted these\nthings ourselves that the joy of creation will be ours. Not to do that\nwould be as if an artist laid no brush on the canvas himself, but merely\ndictated to a dependent where such a colour should be spread. But given\nthat he had a slave so intelligent and so obedient that he could follow\nto a hair's-breadth the directions given him, can you imagine the artist\nfeeling the possessive joy of creation in the result, even though it\nrealized the conception to the uttermost? Not I; nor, in the garden, do\nI care, like that, to see what others have done. It is not sufficient to\ndirect; one has to do it oneself.\n\nI love, too, and cannot conceive not loving, getting hot and dirty over\nthe wrestling with the clean, black earth. A great deal of nonsense is\ntalked about the dignity of labour, but it is chiefly talked by those\nwhose labour lies indoors, who, excellent craftsmen as they may be, go\nspudding about in the intangible realms of the mind. I doubt, indeed,\nwhether any market-gardener has ever spoken of the dignity of labour. We\nleave that to those who only know it by repute. But I long to put down\nthe manner of the transaction. I do not in the least think it dignified,\nbut it is such fun.\n\n       *       *       *       *       *\n\nThe green had mostly faded from the grass, leaving the meadow, as is\nalways the case in October, far more grey than green. Certain plants,\nhowever, were still of varnished brightness, and the dandelion leaf was\none. There was no need to pick and choose, and without moving a step, I\ndug the trowel down into the earth, loosened it all round the vegetable\nenemy, and lifted it. An ominous muffled snap came from inches down in\nthe earth, which I tried to pretend I had not heard. But one could not\ncheat the eye also. There, at the bottom of my excavation, was a milky\nroot, showing a danger-signal of white against the brown loam. I had to\ngo deeper yet: the whole of the tap-root must be exhumed. Another dig,\nanother snap, a raw-looking worm recoiled from the trowel, only just in\ntime, and eventually up came the remotest fibre. How good the earth\nsmelt! How reeking with the life of the world! Cold, clammy, rich earth,\never drawn upon by the needs of the Bank of Life, ever renewed by that\nwhich life paid back to it. A thousand years had gone to the formation\nof my trowelful, and a few inches below was the chalk, where a million\nlives a million years ago had spent themselves on the square inch of it.\nSlowly, by work of the myriad sea-beasts, this shoulder of chalk was\nheaved from the sea, the myriad lives became a myriad myriad, and here I\nhad the little lump of chalk borne up on the end of the trowel which\ntold of the labourers of the unnumbered years. Then, in a spoonful of\nsand, I put the sign, the evidence of another decade of millions on the\ntop of them, and stuck thereon an onion-like daffodil root that was born\nlast year. In a fortnight's time that child of to-day will have reached\ndownwards, feeling with delicate, pleased touch the sand of a thousand\nyears ago, will delve through the time of the pyramids of Egypt, will\ndraw moisture from the chalk that was old when our computation of time\nwas not yet born, and will blossom next April, feeding its sap on the\nprimeval years. And for what? To make Helen and Legs and me say, 'Oh,\nwhat a beautiful Horsfeldii!' Then we shall look at the fritillary that\nprospers a yard away.\n\nThe eternal romance of it all! To the right-minded there is nothing that\nis not a fairy-story. Like children, we crowd round the knees of the\nwonderful teller of it, and say, 'Is it true? Is it all true?' And He\ncan't tell lies. Sometimes, when we have a sort of moral toothache, we\nsit apart, and sniff. We say that scientifically we have proved there is\nno God. So said the fool in his heart. But nowadays the fools write it\ndown in their damned books, and correct the proofs of it, and choose the\nbindings of it, and read, with gusto, the thoughtful reviews of it. And,\nGod forgive them, they think they are very clever people, if I may be\nexcused for mentioning them at all.\n\nBut fairy-stories! How surprising and entrancing are even those which\npeople make up and put in books, while round us every day a fairy-story\nfar more wonderful is being told not only for us to read, but enacted\nfor us to see. It is only familiarity with it which robs us of the\nsense of its wonder, for imagine, if we could make ourselves ignorant\nagain of what happens to bulbs when we put them in the earth, how the\npossibilities of flying-machines would grow flat and stale before the\nopening of the daffodil. For a man's capacity for happiness is in great\nmeasure the same as his capacity for wonder and interest, and\nconsidering that there is absolutely nothing round us which does not\nteem with wonder if only we had the sense to see it, it argues very ill\nfor our----\n\nA wild shriek from the hillside opposite (distance forty yards)\ninterrupted me.\n\n'I didn't mean to,' cried Helen; 'but I cut a centipede in half. They\nare going in opposite directions.'\n\n'Dig another hole!' I shouted. 'Then go back when the halves have gone\naway. Yes, very distressing, but you can't avoid everything.'\n\n'Murderer!' said Helen.\n\nThis was feminine logic. I had not cut the centipede in half!\n\n       *       *       *       *       *\n\nIt was one of those golden October days of which we have now had some\nhalf-dozen. Every night there is a little frost, so that morning both\nlooks and smells exquisitely clean, and it is hardly possible to regret\nthe turn of the year; though dahlias are blackened, the trees blaze with\ncopper and gold, for in this week of windless days scarce a leaf has\nfallen, and the stems are as thick with foliage as they were in the\nsummer, and to my mind doubly beautiful. And this work of bulb-planting\nseems to bridge over the winter, for we are already at work on spring.\nBut in November, Helen and I mean to turn our faces townwards again, for\nit is possible there to be unaware of the transition to winter, which is\nso patently before one's eyes in the country, and which, with the best\nwill in the world, it is impossible not to find rather depressing. Some\npeople, I know, label the squalls of February March as execrable, and\nflee the country then. But we both love them. These are the last\ndespairing efforts of winter. His hand is already loosed from the earth;\nhe strikes wildly, knowing that there are but few blows left in him. But\nin the autumn he is gaining strength every day: it is life whose hold is\nbeing loosed. And that is not exhilarating to watch. True, it is only a\nmimic death-bed, but personally we don't want to sit by the bedside. In\nLondon there is no bedside. The shorter the day, the earlier the lamps\nare lit. Those avenues of shining eyes, which are not shocked whatever\nthey see.... And the fogs--the mysterious fogs! I suppose we are\nCockneys.\n\n       *       *       *       *       *\n\nHelen gave out first in the matter of bulbs, and came and sat by me.\n\n'How very dirty you are!' she said. 'And have you been planting bulbs\nwith your nose?'\n\n'Not at present. But it tickled, and so I rubbed it.'\n\n'Well, let's stop now. I want to go for a walk. My back aches with\nbending, and though I haven't got toothache, I feel as if I might have,\nand the kitchen-maid has given notice, and I don't think anybody loves\nme, and if Legs marries that awful girl, I will never speak to you\nagain. And they are coming to dinner to-night! I pray Heaven that Legs\nmay miss his train, and not get here till late.'\n\n'So do I. Yes; let's go for a real tramp on the downs. Hadn't I better\ngo and wash my face first?'\n\n'Oh no; what does it matter? But are you sure you don't want to go on\nbulbing?'\n\n'Quite sure. I think we won't go by the road, do you know. We can strike\nacross the meadows and up the beacon.'\n\nHelen gave a little purr--a querulous rumble of the throat.\n\n'I have the blues,' she said, with great distinctness. 'I was as happy\nas possible till ten minutes ago, and then they came on like--like a\nthunderstorm. Everything ached. I groaned aloud: my mind hurt me like\nlumbago. It hurts still. Oh, do rub something on it.'\n\nThat is one of the heavenly things about Helen. If she 'feels bad,' she\ncomes and tells me about it like a child. She scolds me for all sorts of\nthings of which I am perfectly innocent, because she knows I don't mind\none scrap (I love it, really, but I don't tell her that), and it makes\nher feel better. She scolded now, even when we had passed the\nwater-meadow and began a really steep ascent of the flanking hills.\n\n'I knew the kitchen-maid wouldn't stop,' she said, 'because those London\ngirls hate the country. So do I. And it was all your fault. You engaged\nher; I had nothing to do with it. And we never _had_ such a\nkitchen-maid. She cooks better than the cook, and does everybody else's\nwork as well. You might have known she wouldn't stand the country.'\n\n'Go on,' said I. 'My fault entirely. So is the toothache, isn't it?'\n\n'I haven't got one, but I might have. And that's your fault, too. I\nwanted to go to the dentist as I passed through London, and you\npersuaded me to come down here without stopping. It did ache just\nthen--it did.'\n\nThe hill got rather steeper.\n\n'Go on,' said I. 'How slowly you walk!'\n\n'Yes, but I have to do all the talking. You have no conversation. Oh\ndear, what a devil I am! Aren't I?'\n\n'Yes.'\n\n'There! I told you nobody loved me. Oh, look! we are going to have a\nreal red sunset. All the hills are getting molten, as if they were\nred-hot and glowing.'\n\nShe was feeling a little better--not much, but a little. We had come up\nthe two hundred feet of steep down-side as if we had been storming a\nbreach. To walk very fast up a hill makes all proper people feel better,\nunless they have heart disease, in which case they die, and so, we hope,\nfeel better also. But for those who have not heart disease, and want to\nfeel better, the prescription is confidently recommended.\n\n'And then that awful girl!' she went on. 'You insisted on being\nneighbourly, as you call it, with the Ampses, and this is the result.'\n\n'There has been none at present.'\n\n'No; but you tell me to ask the family to dinner on the very day that\nLegs comes down. Oh dear, what a heavenly evening! I should so enjoy it\nif everything wasn't wrong. Look at the sky! Fifty thousand little pink,\nfluffy angels floating about in it! Do you want to go right to the top\nof the hill?'\n\n'Yes, right to the top. Then I shall begin to answer you back.'\n\nHelen laughed.\n\n'Oh no, don't,' she said. 'It is no fun plaguing you if you dispute my\nfacts. So tell me quickly: isn't everything your fault, and not mine?\nPlease pull me, if you intend to go that pace.'\n\nSo I pulled her, she holding the end of my stick, and we arrived at the\nvery top of all. Sunset was below us, evening stars were above us, and\non the huge expanse of down there was no one else. It was the loneliness\nI love.\n\n'The devil has gone,' she said, after a while. 'You are rather nice to\nme. And I don't think I have toothache, and--well, you thought that\nCharlotte was a little Ampsy before I did. And even if nobody loves\nme--oh, how dirty your nose is!'\n\nThat was true, anyhow.\n\nAn extraordinary phenomenon in country towns is that, though nobody has\nanything to do, everyone feels extremely busy; whereas in town, though\nyou have got an enormous deal to do, you never feel busy at all, and\ncan, without fail, find time for anything else. I think there must be\nsome microbe which cannot live in London, but thrives elsewhere, which\nproduces the illusion of being rushed. Personally, I know it well: it is\nnot an old enemy of mine, nor is it an old friend, but it is a pleasant\nold humbug, which I am afraid I rather encourage. This evening, for\ninstance, when I went to my room after tea, I encouraged it, and argued\nthat one never had a moment to oneself. I had two hours in front of me\nnow, as a matter of fact, in which I should be undisturbed; but the Old\nHumbug said that it was all very well to think about the future. All he\nknew was that he--that is, I--had been rushed--yes, rushed--all day and\nall yesterday, and ever since we came down to this dear, sleepy old\ntown. To-day was Tuesday, and people were coming to dinner. We had gone\nout to lunch yesterday, and had dined out twice last week. Also, there\nwas the garden to attend to, and a little golf (almost every day, as a\nmatter of fact) was necessary for the health, and what with letters to\nwrite and cigarettes to smoke, and the Meistersinger overture to learn,\nin order to play it with Legs, I was a victim of this hurrying, bustling\nmode of life, which in a generation or so more would assuredly send\neverybody off their heads.\n\nI made myself quite comfortable in my chair, and proceeded to think\nabout it seriously, because I had two hours in front of me. It was all\nquite true (I was encouraging the Old Humbug, you will understand), and\nthe modern mode of life was insane. London, anyhow, was insane, and in a\nlittle while I should probably get to agree with the Old Humbug that I\nwas rushed and driven in the country also. But, to encourage credulity,\nI took London first. There one certainly was busy--all the hours, that\nis to say, of a day that began quite early and ended next morning were\nfull, and I reconstructed one such as I often spend, and hope to spend\nmany times more. I do not give it, because it seems to me the least\nedifying, and all stern moralists (the Old Humbug is an awfully stern\nmoralist) would--as, indeed, they have done--shake their heads over it,\nand say, 'To what purpose?' I will tell you that afterwards.\n\nI was called, let me say, at half-past seven, and after a few\nincredulous groans got up. I shaved, washed a little--not much, for\nreasons that will appear--drank some tea, and in a quarter of an hour\nwas wildly bicycling towards the Park. When things flourished very much,\nand money flowed, Helen and I rode champing steeds; but just now things\nwere what is called fluctuating, and I rode a bicycle, and she stayed\nin bed. An hour and a half of frantic pedalling on a hot June morning\nproduces excellent physical results; and at half-past nine I was in the\nswimming-bath at the Bath Club, where I became cool and clean. I changed\ninto another suit of flannels there, rode sedately home, and had\nbreakfast at precisely a quarter-past ten. By eleven I had eaten\nbreakfast, read the _Daily Mail_, and smoked a cigarette, and was about\nto spend a quiet, studious morning until half-past one (for we were\nlunching out at two), when Helen came in.\n\n'Do come to Lord's,' she said; 'it's Gentlemen and Players, and we can\nsit there till lunch. We can't go this afternoon, and you are playing\ngolf at Woking to-morrow.'\n\n'I can't. Not time.'\n\n'Oh, just this once.'\n\nJust this once, then, we went. It was too heavenly, and we were late for\nlunch.\n\nIt was one of the rather long lunches, and it was nearly four when we\nleft the house. Then, as we had neither of us seen the Sargents at the\nAcademy, we went there, since the afternoon was already gone, and got\nhome about six; and as we had been given a box at the opera for\n'Tristan,' which began at half-past seven, it was necessary to dine at\nhalf-past six--a terrible hour, but true. At the opera Legs picked Helen\nup to go to a ball, and I went home to answer my morning's post, which I\nhad not yet read.\n\nBut, it will be objected, Gentlemen and Players, and the one necessary\nvisit to the Academy, and 'Tristan' does not occur every day. Quite\ntrue; but something else always does, and the Old Humbug, who had got\nquite large and important during this short survey, said in those\ncanting tones which I knew so well: 'You are wasting your life over this\ninsensate rush and hurry. And you do no better down here. What have you\ndone to-day? Planted bulbs, and written two or three pages of your silly\nbook. What will you do to-morrow? You won't even write your silly book,\nbecause you are going to play golf with Legs in the morning, and you say\nyou can't work after lunch. And the days will make themselves into\nmonths, and the months into years' (here he dropped into poetry), 'and\nyou will ever be a name of scorn--at least, you would if ten minutes\nafter you were dead anybody remembered what your name was. But you will\nhave gone to your account.'\n\nWell, I join issue with the Old Humbug over this. For my part, I assert\nthat it was perfectly right for me to go to the Gentlemen and Players,\nand to the opera, and to plant bulbs, and to play golf with Legs\nto-morrow morning if fine. And as for his objection to what he calls\n'rush,' why, I fling it in his face, since I must rush. If I set apart a\ncertain time every day for private meditation, I should be simply bored.\nI should get--I suppose this must be the proposed practical effect of\nthe plan--no great and ennobling thoughts out of my solitary\nmeditations, and instead of feeling that I had spent the morning to some\nserious purpose, I should feel, and I think rightly, that I had merely\nwasted it. But if I have planted bulbs all morning, I haven't wasted it.\nI will assert that on the Day of Judgment; for I have been busy walking\nalong the path I feel sure I was meant to walk on. There are a thousand\nother paths all leading to the central and celestial light, and they\nare for other people to walk on. It would, of course, be a terrible\nwaste of time for one who by nature was a meditative recluse to go to\nthe match between the Gentlemen and Players, or for a deaf man to go to\n'Tristan,' or for a blind one to lie on his back and look at the\nfiltering sunlight between the leaves of beech-trees in June. But the\npoint for everybody is to get into touch with life as continually as he\ncan, and at as many points as he can. This is gospel. I would I had the\npalate of a wine-taster to get into touch with life there; the\nprehensile toe and sense of balance of a tight-rope walker to get into\ntouch there, the mathematical head of the astronomer to learn the orbit\nof a star that has never been seen, but only conjectured; or I wish very\nmuch indeed that I had the missionary spirit. Indeed, then I would go to\nthe nearest cannibal islands and (probably a good thing, too) be\ncheerfully devoured; or, again, if I had it in a lesser degree, I would\ngo and teach in the Sunday-school, and have a class for boys in the\nevening. I did try the Sunday-school when first I lived here, and for\nfour unhallowed Sundays I passed a feverish hour surrounded by mystified\ninfants and intolerable lithographs. You never saw such a failure as I\nwas: I dreaded those hours so much that I thought my reason would be\nunhinged. And the children used to regard me, I am sure, as they would\nhave regarded some queer, though harmless, creature of the menagerie. I\ncouldn't do that sort of thing.\n\n       *       *       *       *       *\n\nI neither made them happy nor could I teach them anything. That latter\nwas quite proved when, on the Sunday succeeding my fourth lesson, an\nArchdeacon came round and examined all the classes in turn. I think I\nshall never get over the nightmare horror of that scrutiny when he sat\nin my arm-chair at the desk, and I, the trembling instructor, stood by\nthe side while he asked my idiot flock who Adam and Eve and Cain and\nAbel ware, and other really elementary things. One child said that Eve\nwas God's wife, and I wished the earth might open and swallow me up.\nThen he came to the Catechism, and it really seemed as if nobody knew\nhis own name. And it was for that nightmare that I had spent four\nfeverish Sunday afternoons and a parody of days between, for every\nmoment Sunday was coming nearer.\n\nNo; I give more happiness to Legs by being soundly beaten by him at\ngolf, or by wasting (so says the Old Humbug) a morning in taking Helen\nup to Lord's to see the Gentlemen and Players. Also--I hasten to\nforestall criticism--I like it much better myself; and though you may,\nif you like, call it selfish, I hereby state that to like doing anything\nis a very good and Christian reason for doing it. Behold the gauntlet!\n\nFor we poor folk who really cannot teach in Sunday-schools, and are not\nemployed in making discoveries which will alleviate painful diseases,\nand do not serve our constituency or our King, and sneakingly throw\npamphlets about the Education Bill into the fire without reading them,\nbecause we know we don't actually care one pin what happens, and are in\nevery single respect quite unsatisfactory and useless and unornamental,\nhave yet, somehow (there can be no doubt of this), to add if we can to\nthe happiness, anyhow, of those dear folk among whom our lot has been so\ngraciously cast. We have no great gifts of any description; we are\nneither wise nor witty, and there has been only one talent given us,\nwhich is the power of enjoyment. Well, that is a very little one, you\nmay say, and a very selfish one to cultivate, but if we have nothing\nelse at all, had we not better try to make some use of that?\n\nFor the fact remains that it can be made some use of. Every one feels\nbetter for seeing one of these drones, who are neither soldiers, nor\nsailors, nor politicians, nor teachers, enjoy himself. Enjoyment in the\nair is like oxygen in the air: it quickens everybody, and in its way\nmakes them happier. The poor drones can neither teach nor fight, nor\nmake anybody good, but they can in their humdrum way make people a\nlittle more cheerful for a few minutes. For they have--this is what I\nmean by drones--a happy temperament, and as they are no good at all in\nany other direction, it is indeed time that they should be done to\ndeath by the workers of the hive if they do not exert themselves in the\nmere exercise of their temperament. And just as the drone of the hive\nlives immersed in the honey of his flowers and in the garnered stuff\nthat the workers have brought home, so the drone man must continue to\ntake active and continual pleasure in all the delightful things of this\nworld. He must pounce on enjoyment with eager zeal, and glut himself on\nit till he reels with the stupefaction of pleasure; he must keep himself\nkeen and alert for the smallest humorous or engrossing detail that is\nwithin his horizons: it is shameful if he does not go to bed every night\ntired with his own laughter and enjoyment. And woe to him if he invests\nhis pleasures with the serious garb of duty! The leader of the\ndelectable life who says that he plays golf because he finds exercise so\nimportant for his health, or who sits out all afternoon to watch other\npeople playing other games, and explains that his doctor (his doctor,\nforsooth!) tells him to have plenty of fresh air, or who drinks his\ndelicious wine and says that it is good for his digestion, is a mere\nscampish hypocrite. He plays games because they are such fun; he\nwatches other people play because it amuses him; he drinks wine because\nit tastes so nice.\n\nAnd he must never falter on his primrose path; the high gods have given\nhim but one little talent, and all that is asked of him is that he\nshould enjoy life enormously. He has got to do that, then. The soldier\nand sailor may not, perhaps, enjoy life, but they are useful in other\nways. The drone is only useful in this one. He must never remit his\nefforts, and must never want to; he must 'rush,' as the Old Humbug said,\nall the time, for if he ceases to rush he ceases to justify his\nexistence at all. And--a heavenly destiny, one, too, beyond all desert\nof his--he does, if he is at all a conscientious drone, make other\npeople a shade more cheerfully disposed than they would otherwise have\nbeen.\n\nThis breathless dissertation on drones requires at this point, as\nprinters say, 'paragraphing.' In other words, I began to talk about one\nthing, and without pause talked about another. It was really the fault\nof the Old Humbug, who said that I wasted those days in which I didn't\ndo something for somebody. I then justified my position on those days by\npleading the desire to be a drone--a life which, as I have sketched it\nout, seems to me to be wholly admirable. I wish to Heaven I could be in\nthe least like those adorable people. Misbegotten industry stands in my\nway, and a deep-rooted, but equally misbegotten, idea that if I am very\nindustrious I shall one day write a good story. Also, I have not the\ndrone temperament necessary for dronage. I am not, in fact, any longer\ndefending myself, but extolling other people.\n\n       *       *       *       *       *\n\nLoafers there are in plenty in this world, but personally I have no use\nfor them. They lead the same external lives as the lover of life leads,\nbut how different is the spirit that animates them! The loafer may have\nbeen side by side with the life-loving drone all day, at the same\nparties, at the same games, at the same music, but the one goes to all\nthese things in order to get through the hours without boredom, while\nthe other wishes the hours infinitely multiplied so that he might go to\nmore. The one sucks enjoyment of but a stupefied sort from them; the\nother catches the iridescent balls and bubbles of joy that are cast like\nsea-spray over the tides of time, only to throw out double of what he\nhas received. He is like some joyful juggler: a stream of objects pours\ninto the air from his flashing hands; he catches them and hurls them\ninto the air again, so that the eye cannot follow the procession of\nflying joys. And at the end, at the close of each day, he stands still\nfor a moment, his hands full of them, his memory stored with them, eager\nfor the next day.\n\nHow different is the loafer! Have you ever seen the chameleon feed on\nflies? It is just so that the loafer, who wants only to get through the\nhours, feeds on the simple, silly joys of life. In expression the\nchameleon is like a tired old gentleman with the face-ache, though the\nimpression of face-ache is chiefly produced by cheeks swollen in other\nways, for he rolls up his tongue in a ball in his mouth when he is going\nto feed. Then, with an expression of bored senility, he moves very\ncautiously to where a fly is sitting. When he is within range, he shoots\nout his tongue, and the fly sticks to the adhesive tip of it. There is a\nslight swallowing motion, and the chameleon again rolls about his greasy\neye, looking for the next victim. The loafer, in a metaphysical sense,\nhas got just such an adhesive tongue as the chameleon. He puts it out,\nand pleasures stick to it like postage stamps. Then he swallows them.\nObserve, too, when he has to make occupations for himself, how heavily,\nand stupidly he passes the hours! He will read the morning paper till\nmidday, then totter out into his garden, sadly remove one weed from the\npath, and totter back to the house to throw it in the fire. Then he will\nre-read a page of his paper, and write an unnecessary note with\nunnecessary care, probably wiping his pen afterwards. It will then be\nlunch-time. How different would the drone's morning have been! Even if\nhe had been compelled to spend it on the platform of Clapham Junction,\nhe would have constructed some 'dome in air' out of that depressing\nsuburb. The flashing trains would have allured him (especially the\nboat-trains), and his mind would have gone long journeys to the sunny\nSouth. He would have built romance round the signals, and found a\nfairy-tale in the advertisements.\n\n       *       *       *       *       *\n\nAnd what is the practical side of all this? for is it not temperament\nwhich makes the magic of these wonderful persons, and temperament is a\nthing which is supposed to be quite outside the power of its possessor\nto alter or amend?\n\nBroadly speaking, I suppose that is true, and we who do not possess the\nmagic would bungle terribly if we attempted to rival the flashing hands\nof the true conjurer. I do not suppose, at any rate, that it is worth\nwhile for the meditative recluse to spend his days and nights at festive\ngatherings, since he will never enjoy them himself, and, what is more\nimportant, he will, in his small way, eclipse the gaiety of those\nparties on which he sheds the gloom of his depressing countenance. Yet,\nsince I believe with my whole heart that joy and simple pleasure, so\nlong as they hurt nobody, are things wholly and entirely good, it\nbehoves every one to look sedulously in the garden of his mind to see\nwhether he cannot find there a few little seedlings of that species of\ntemperament which I have tried to indicate. His garden may be the most\nstrenuous and improving plot--a regular arboretum of high aspirations\nand earnest endeavours with the most beautiful gravel paths of cardinal\nvirtues leading by the thickets and shrubberies of spiritual strivings,\nbut, should he happen to find a few of these seedlings, and be able to\nraise them, they will not spoil the effect of the wholly admirable grove\nof moral purpose. To be quite candid, I think a little colour 'sets\noff,' as they say, the grandeur of high endeavour. It--well, it\nbrightens it up.\n\n\n\n\nNOVEMBER\n\n\n'I' remarked Helen, 'am the rose of Sharon, _and_ the lily of the\nvalley.'\n\nShe laid great stress on the 'and,' which gave a perfectly new\nsignificance to the verse.\n\n'The French for lily of the valley is _muguet_,' said Legs, with an\nintolerably superior air.\n\n'Oh, don't show off!' said she. 'The great thing in walking along a rail\nis to keep your balance.'\n\n'Through the looking-glass,' said I. 'Upon which the White Knight fell\nhead foremost into a hole----'\n\n'And kept on saying \"Plenty of practice,\"' said Legs.\n\n'It's easier if you wave your arms,' said she. 'Oh, there's a train\ncoming. Where's the gum?'\n\nLegs had the gum--a small penny bottle--and Helen hastily gummed a\npenny to the rail, and we all retired to the side of the line.\n\nIf you merely lay a penny on the rail, the chances are that the first\nwheel that goes over it causes it to jump, and it falls off, whereas if\nyou gum it----\n\nThere was a wild maniac shriek from the engine, suddenly dropping an\noctave as it passed us, and the huge train, towering high above us,\nthundered by with rattle of wheels and the throbbing oscillation of very\nhigh speed. A dozen bits of paper came trundling and dancing after it.\nThe rear of the van telescoped itself into a tiny square, and the signal\njust above us, which had been down to let the train pass, shot up a\nwarning, right-angled arm.\n\n'Oh, well over sixty,' said Legs, with deep appreciation, 'and there's\nthe penny sitting as tight as, as--I don't know. Lord, how hot!'\n\nThe penny had already been under half the wheels of four trains, and was\nso flattened that it was of knife-edge sharpness.\n\n'If you stropped it a little, you could shave with it,' said Helen.\n'What babies you are!'\n\nLegs was already busy on the up-line, arranging two pins crossways on\nthe gummed rail, so that they should be flattened and welded together,\nmaking an entrancing object closely resembling a pair of scissors.\n\n'The up-train will be through in five minutes,' he said. 'Chuck me the\npenny, Helen.'\n\nI had another object of interest--namely, a threepenny piece with a hole\nin it. I had tied a long string to this, the far end of which I held in\nmy hand. The reason for this was that the coin was beginning to crack,\nbut it would stand a wheel or two more, though it was already bigger\nthan a sixpence; after a wheel or two I could pull it away.\n\n'Gum!' said I.\n\nWe moved to the far side of the up-line, and waited. Soon from the\ntunnel a hundred yards below came a wreath of smoke, and the\nblack-fronted engine raced towards us. Everything went right on that\ndivine afternoon, and after four wheels had passed I jerked my\nthreepenny piece away. The scissors were adorable also, and it would be\nscarcely necessary to strop the penny. Of course, we made a _cache_ of\nthese objects, burying them in a small tin box with the addition of a\npiece of paper recording our names, weights, and ages. Legs also wrote a\nshort confession of how he murdered his two infant children, and hid the\nbodies in a bramble-bush ten paces from the _cache_. There was no such\nbramble-bush really, which would make it more puzzling to the inquiring\nmind. He represented himself as being perfectly impenitent, and ready to\ncommit similar crimes should opportunity occur. He signed it, Benjamin\nYates of 21A, Park Lane, W. Then we went home to tea.\n\n       *       *       *       *       *\n\nLegs had been in for his Foreign Office examination, and had come down\nto spend the next two days with us, before we all moved up to town;\nalso, to our deep-felt and secret joy, he had shown no desire to visit\nthe Ampses, or talk any more German with Charlotte. The process of\ndisillusionment began, I think, on the evening in October when he was\nhere last, and the Ampses dined with us; for I saw him overhear Mrs.\nAmps ask Helen who 'was the heir to all our beautiful property.' At that\nmoment I almost pitied Mrs. Amps. She had begun by making jam, but I\nfelt that she had gone on to cook Charlotte's goose. Legs, anyhow,\nstopped in the middle of a sentence, and took a couple of seconds to\nrecover himself. I am sure I don't wonder. You require to recuperate\nafter that sort of remark. I felt that I knew all about Mrs. Amps when\nshe asked that simple question. I felt as if I had known her parents and\ngrandparents, and could prophesy about her children and grandchildren;\nand Legs' eyes, which till that moment had been quite shut, began to\nopen, just to blink.\n\nNext day, however, he lunched with them. What happened I do not know,\nsince he has not told me; but he was rather silent in the evening, ate\nlittle, but drank four glasses of port after dinner. I think the\ninstinct of the drowning of care was there, and he was slightly cynical\nand Byronic afterwards. I love Legs.\n\nI hasten to add, lest I may appear unfeeling, that Charlotte has for the\nlast week or two been kind and encouraging to another young man, who is\nthe heir to far more beautiful property. I saw him at the golf-links\nyesterday in a bunker. He was arranging her hands so as to grasp her\nniblick properly. They seemed to want a great deal of arranging.\nBy-and-by they allowed my opponent and me to pass. Charlotte seemed not\nto recognize me, or else she was really so much employed in making her\nhat stay on that she did not see me. I mentioned, however, to Legs that\nI had seen her, but that she had not seen me. It seemed to interest him\nvery little.\n\nBut this morning, as Legs and I played golf over the grey back of the\nhuge down that rises from our happy valley, it seemed a sheer insanity\nthat we should all go up to London the next day, so blithe was the air,\nso invigorating to the whole sense. The short, springy turf seemed to\nput its own vitality into one's feet; they were shot forward\nautomatically without conscious effort. And--ah, the rapture!--(it\noccurred more frequently to Legs than to me) of seeing a clean white\nball scud for a hundred yards or so low over the ground, and then rise\nswallow-like against the ineffable blue! Golfers, I am told, reck\nnothing of their surroundings, provided only they drive far, approach\ndead, and hole their puts; and so I must conclude--indeed, I conclude\nit for other reasons--that I am no golfer. But I am an epicure in my\nsurroundings when I go a-golfing, and though the grey dunes and sandy\nhollows of the seaside course are most to my mind, I place very near\nthose perfect joys the hugeness of scale which you get only on the\nuplands. To-day no whiff of vapour flecked the whole field of the\nshining heavens, and the country, grey and green, with fire of autumn\nbeech-wood here and there, stretched map-like round us. But to the west\nthe view was even more stirring to that desire of the infinite which\nlies so close to the heart of man. There fold after fold of downs, the\nknitted muscles of the huge, kind earth lay in unending interlacement.\nAnd it was all empty. There were no trees, no lines of hedgerows to\nbreak the void, and lend a scale to the eye. From the immediate green\nforeground <DW72> after <DW72> melted into grey, and from grey to the blue\nof distance, which fused itself into the tender azure of the sky's\nhorizon, so that the line between earth and air indistinguishable. It\nwas as empty us the desert, yet one knew that from every inch of it a\nthousand lives rejoiced in the sun of November. Yes, even the knowledge\nthat there would be but few more of such hours before winter hurled its\narmoury of squalls on to the earth added, perhaps, to their joy. None\ncould have expected such a November as has been ours. We have snatched\nit from winter; it is our possession.\n\nYet the colour of the grass, no less than the underlying keenness of the\nair, savour of the sunless months. It is scarcely green; it has been\nbleached by the torrid months, and Nature is too wise to let it shine\nforth in a fresh coat of colour when so soon it will sleep, waiting for\nthe spring. High up in the liquid blue, too, of the sky there is the\nsparkle of frost, for all the warm strength of the sunlight. It is not\nsummer that floats above our heads, soon to descend earthwards, but the\nfrost and cold. Yet they bless the Lord also.\n\nBut though I feel all this, feel it in every bone and fibre of my body,\nI know that I feel it more when I am doing something else--as, for\ninstance, playing golf. I think it must be that one pleasure quickens\nothers. The fact of attempting to keep one's eye on the ball as one\nhits it makes the whole of one's perceptions more alert. If I was taking\na solitary walk here, with no occupation except that of walking, I know\nquite well that I should not be conscious of the same rapturous\nwell-being as I am now, when the object of my walk is to hit a small\npiece of indiarubber for three or four miles, hitting it, too, as seldom\nas possible. So it is not the mere hitting it that gives rapture, else\nthe rapture would be increased by the frequency of the operation. Oh, I\nhave been talking on the stroke! This will never do. But it was my own\nstroke, and Hampshire flew about in fids in consequence.\n\nThis was at the twelfth hole, and it made the match square. Legs, I need\nhardly remark, was playing a pitiful game for him. But on the\nmoment--this is one of the inexplicable things about those foolish\npeople who play games--- my whole mood changed. I cared no more at all\nfor the empty, glorious downs. I did not mind whether the grass was\nblue, or grey, or green, or magenta. I saw no more flaring beech-woods,\nno more mapped counties. There was one desire only in the entire\ncontents of my soul, and that was to beat Legs. I did not feel as if I\neven wanted anything so much as that, and if Mephistopheles had appeared\nat that moment to bargain for my salvation as the price of my victory, I\nshould have signed in my blood or any other blood that was handy. But\nMephistopheles was probably otherwise engaged. At any rate, after being\nstill all square at the seventeenth, I drove into a silly irrational\nbunker that ought never to have been there at the eighteenth. I took\nthree to get out. But we had a heavenly morning. If only ... well, well.\nAnd Legs told Helen that he only just won, because he was completely off\nhis game! The tongue is an unruly member. Mine is. Had I won, I should\nhave certainly told Helen that Legs played a magnificent game and I had\nonly just won. That sounds more generous than his remark, but if you\nthink it over, you will see that it comes to exactly the same thing.\n\nYes; it seems an insanity to leave the country just now, especially\nsince there is no earthly reason for our doing so. Divine things, it is\ntrue, are going on in town, for our matchless Isolde is conducting\nsymphony concerts, and a perfect constellation of evening stars are\nsinging together at the opera; but, after all, Legs and I play the\n'Meistersinger' overture arranged for four hands on the piano; while,\nfor the rich soup of Sloane Street mud and the vapour-ridden sky, we\nhave here the turf of the downland and the ineffable blue. In fact, I am\nsorry to go, but should be rather disappointed if I was told that I was\nnot going. Helen characterizes this state of mind as feeble, which it\nundoubtedly is, and says that she is perfectly willing either to go\nto-morrow or to stop on another week, if I will only make up my mind\nwhich I want to do. But there is the whole difficulty: I haven't the\nslightest idea which I want to do. You might as well say to a dog which\nis being called from opposite quarters by two beloved voices: 'Only make\nup your mind which of us you like best.' If it knew, the question would\nbe solved.\n\nWell, the question was solved by tossing up, and then, of course, doing\nthe opposite to what we had decided the arbitrament of the coin should\nindicate. If it was heads, we were to stop in the country; and since it\nwas heads, that helped us to decide that we would go to town. That, too,\nmay seem a feeble proceeding, but I do not think it really is. To do\nanything as irrational as tossing causes the mind to revolt from the\nabsurdity of abiding by the result. The consequence is that a weighty\nfactor for doing what the coin did not indicate is supplied; for you\nnever toss unless you are quite unable to decide.\n\nSo for the last afternoon the garden claimed me, for not only is the\ngarden the symbol and embodiment of the country, but to me it is a sort\nof diary almost, since the manual acts of planting and tending have got\nso interwoven with that which made one's mind busy while the hands were\nthus occupied that the sight of this plant or that, of a new trellis, or\nthe stacked sticks of the summer's sweet-peas, are, when one looks at\nthem as now, retrospectively, on the eve of departure, retranslated\nback, as are the records of a phonograph, into the memories that have\nbeen pricked and stamped into them. All I see--croquet hoops,\nflower-beds--without ceasing to be themselves, have all become a secret\ncipher. By some mysterious alchemy, something of oneself has passed into\nthem. Secret fibres of soul-stuff are woven into them. Through the touch\nof the hands that tended them, something from the being of that which\ndirected those hands has entered into their life, so that next year, it\nmay be, some regret belonging to an autumn day will flower in the\ndaffodils of our planting. Hope, I am sure, will flower too; and with\nhow vivid a wave of memory do I know what silent resolve went into the\ncutting back of that Gloire de Dijon! Thus, when in June its fragrance\nstreams in the air, one must trust that some fragrance not its own, but\nof a fruit-bearing effort, will be spread about the garden.\n\nThere, for instance, are the croquet-hoops still standing, though it\nmust be a month since we had played. A few withered leaves of the plane\nhave drifted against the wires, and the worms have been busy on the\nneglected lawn, that speaks only of November. But that corner hoop has a\nsignificance beyond paint and wire. It is the record of the telegram\nthat came out to me one morning in late September which I showed Legs.\nAfter that we abandoned the game, and went to the house. It may have\nmore for us yet, that corner hoop--more, I mean, than that memory of\nwhich I have spoken. Joy or sorrow may be so keen, so poignant on some\nday yet hid behind the veil of the future, when I shall be looking at\nit, that till the day of my death it will never again be seen by my\nmortal eye without rousing an immortal and imperishable memory. It is\nthus, in a manner antimaterialistic, so to speak, that men, material\nthings, are woven into the psychical web of life, so that, almost before\nthe eye has seen them, they have sent the message of their secret\nsignificance to the brain.\n\nEverywhere, wherever I look, the tangle of these subtle threads is\nspread, even as on summer dawns the myriad spinning of gossamer makes\nnetwork on the grass, so that each is crossed and intertwined with a\nwoof of others. There is the bank where I lay all one hot July day doing\nnothing, thinking nothing, just lapped in the tide of living things.\nThat has gone home. That bank and the hours I passed there are part of\nme now, even as I feel that I am part of it, and I have but to look at\nit now to bask again in the absorbing stupor of the midsummer. There, in\nits blades of grass and shadowed turf, is written my doing for the day.\nThe bank holds it in kind, safe keeping, so that when God inquires of me\nwhat I have done with that day that He gave me, the bank will be able to\nanswer for me. Nor does it tell my secret to the croquet-hoop that holds\nanother, nor to the clematis that on that day was a heaven full of\npurple stars spread over the trellis. There was nothing in all the\ntreasure of the summer so beautiful, so triumphant as that; but what to\nme now is the memory of the clematis? The memory of a friendship that is\nover. At least, I was looking at it when I know that somebody I had\nloved and trusted was neither trustworthy nor lovable. It was as if a\nfriend had pushed back the carpet from the boards of the room where he\nand I had so often been merry and intimate together, and showed me, with\na sort of secret hideous glee, that a sewer flowed beneath the floor.\nPoor clematis! it is sick at heart. Its thin, bare stalk shivers\nmournfully as this golden afternoon begins to turn a little grey with\nthe chill wind of evening.\n\nAh, if only he had said he was sorry! If only he had said that he knew\nit was wrong, but that the flesh was weak! If only he had even\ncontemplated the step, which to some extent undoes the wrong that has\nbeen done, I do not think the clematis would have shed a single one of\nits purple stars. All of us, saints or sinners, do dreadful things, the\nmemory of which is sufficient to make us long to sink into the earth for\nshame. But he only smiled behind his hand, and with whispered gusto told\nme about it, licking the chops of memory. It is _that_ which matters.\n\n       *       *       *       *       *\n\nThat corner of the garden had delayed me long, and it was already\ngetting dark when I had gathered up and fingered the gossamer threads\nthat lay so thickly down the border that led to the gate from which\ndescend the steps of the rose-garden. There were so many messages there.\nThe bare stalks of phloxes and campanulas, Oriental poppies and\nhollyhocks, Japanese anemone and iris--all had something to say. Some\nmemories were a little vague, faint, and dim even as the odour of the\nphloxes; some were tall and resplendent like the hollyhocks; some were\nvivid as the poppies. And then I went through the gate of the\nrose-garden and stood there. There was nothing there but the rose-trees;\nthere was no one there but Helen.\n\n       *       *       *       *       *\n\nSo the tale of the garden was told, and by the time it was finished dusk\nhad begun to deepen, and cheerful beckoning lights were gleaming from\nthe house. It was time to go in to take up, and with what love and\nalacrity, the pleasant hour of the present again; for it is not ever\ngood to linger too long over memories, or for however short a time to\nindulge in regrets, unless those regrets are to be built into the fabric\nof the present, making it stronger and more courageous. All other\nregrets, all other regarding of the past, which says, 'It is past; it is\nirretrievably done,' is enervating and poisonous, and but paralyzes our\nenergies. Indeed, it is better not to be sorry at all for the unwise,\nunkind, and mistaken things we have done if our sorrow tends to unfit\nus for doing better in the future.\n\nBut just as I crossed the lawn, going towards the house, another memory\nstarted up out of the dusk so clearly that I almost thought that I heard\nmy name called from the garden, and almost expected, when I got indoors,\nto hear again the sound of shuffling, unshod feet on the stairs. The\nmemory of that mysterious midnight hour, though I have not spoken of it\nagain, is seldom out of my thoughts. It does not sit, so to speak, in\nthe front row, but in the dimness that lies at the back of one's mind,\nout of which come those vague vapours which are, if they have body\nenough, eventually condensed into thought, just as out of thought is\ncoined speech and action. There in that dark kitchen of the mind I know\nthat the thought of that night has ever been simmering on the fire.\nSomething within me is not content with the fact that even at the moment\nthat the voice cried from the garden, at the moment when Legs saw the\nwhite face smiling at him, that dear soul passed to the other side.\nThere is more to come yet. Else--here is the vapour taking the shape of\nthought at last--else why did Legs, who scarcely knew her, receive that\nwarning? No echo of any memory of that night, strangely also, has ever\ncome back to him. He knows no more about it now than he did the next\nmorning, when he asked me if I had been sitting up late talking.\n\nI have told Helen all about it; I have told her too--for there is\nnothing so wild and fantastical that I would not tell it her--that there\nis some uneasy guest sitting at my hearth who stays in the shadow, so\nthat I cannot see his face. And she answered with a serenity that was\nalmost reassuring, saying that, if something more was coming, there was\nstill, whatever it was, nothing to fear; if otherwise, the uneasy guest\nwas moonshine of the imagination. That seems to cover the whole ground.\nBut the fact is that I am afraid of my fear--a thing for which it is\nidle to try to find excuses.\n\n       *       *       *       *       *\n\nWe are leaving quite early to-morrow morning, so, when I entered the\nhouse that evening after the tour of the garden, I had definitely\nfinished with the country for some weeks to come. So, too, had Helen\nand Legs, for tea had already gone into the drawing-room. And even as I\nlocked the garden-door behind me, I heard a sudden gust of wind come and\nshake the panes, as if this calm, golden day had been sent just for us,\nand that the moment we had finished with it the winds, overdue, but\nkindly waiting for us, began to drive their cloud-flocks out of the\nsouth-west. Nor was the coming of the rain long delayed. Even while we\nsat at tea, a sheet of it was flung with a sudden wild tattoo against\nthe panes, and there hissed on to the logs of the open hearth a few\nstray drops. Legs paused, with his mouth full of crumpet.\n\n'It makes me feel twice as comfortable as I was before,' he said. 'It\nmust be so beastly out of doors.'\n\nLegs had just uttered this thoroughly Lucretian sentiment, when--\n\nThe door opened, and Mr. Holmes was announced. I have refrained from\nmentioning Mr. Holmes before because I expected he would come in about\nnow, big with purpose. He is a kind little gentleman, about forty-five\nyears old, who lives with his sister, and does not do anything\nwhatever. He is generally known as the Bun-hander, because no tea-party\nhas ever been known to take place for miles round at which Mr. Holmes\nwas not handing refreshments to the ladies. That is his strength, his\nforte. His weakness is just as amiable--though, perhaps, hardly so\nuseful--for his weakness is Rank.\n\nHe constantly comes to see Helen--about once a fortnight, that is to say\n(for in the autumn he is very busy going to tea-parties)--for the\nreason, so Legs and I believe, that she is the daughter of the younger\nson of a peer. Helen will have none of this, and maintains that he comes\nto see her for Herself. Personally, I can behave beautifully when Mr.\nHolmes finds Helen and me alone, but I am rather nervous if Legs happens\nto be in the room, for he is quite unable to take his eyes off Mr.\nHolmes, but stares at him in a sort of stupor of wonderment. Once (that\nis a year ago now) he left the room very suddenly. Choking and muffled\nsounds were heard from the hall and the stamping of feet. Helen and I\ntalked very loud to overscore this, and I trust Mr. Holmes did not\nhear. But when Legs is there, I am afraid (it is a sort of nightmare)\nthat I shall be overtaken, too, with helpless giggling. If I begin,\nHelen will go off, and I can imagine no way of satisfactorily\nterminating the interview. Because if once I began laughing at Mr.\nHolmes, I do not see how I could ever stop. His appearance, his voice,\nhis conversation, are all quite inimitable.\n\nHe is small and inclined to stoutness, and has a fierce little\nmoustache, so much on end that it looks as if it had just seen a ghost.\nNot long ago he had no teeth to speak of; now they are as dazzling and\ncontinuous, as Mr. Wordsworth said, as the stars that shine. He has\nrather thin brown hair, which I will swear used to be streaked with\ngrey, but is so no longer, and he wears three rings with stones in them.\nOne is an emerald, so magnificent that it is almost impossible to\nbelieve in it. He is dressed in the very height and zenith of provincial\nfashion, and would no more be seen in shabby clothes than he would be\nseen without stays. Yes; I maintain it, and even Helen, who was a\nperfect St. Thomas about it for long, has admitted that occasional\ncreaks proceed from Mr. Holmes's person for which it is difficult to\noffer any other explanation. It was a creak, in fact, more than usually\nloud that made Legs leave the room on the occasion I have referred to.\nDown his trousers he has the most beautiful creases, and all his clothes\nnearly fit perfectly. He wears brown boots with cloth tops, above which\nwhen he sits down you can see socks with clocks on them stranglingly\nsuspended. In the winter he wears a hat with a furrow in it, and in the\nsummer a panama. He wears a knitted tie (just now it is rather the\nfashion here for young men to have ties knitted for them by their\nfriends), which Helen says is certainly machine-made, with a pin in it.\nHis shirt always has some stripe or colour in it, and his links are\ninvariably the same colour as the stripe. To-day the links were\nturquoise and the stripe light blue. And from top to toe it is all a\nlittle wrong, though since I do not know how clothes are made, I cannot\ntell you what is wrong. The effect, however, is that, though so\ncarefully arrayed, Mr. Holmes looks like a rather elderly shop-assistant\ngoing out on Sunday afternoon.\n\nMr. Holmes goes out much oftener than that, for he may be seen in the\nwindow of the club every morning from about half-past eleven till one. I\nhave often seen him sitting in the window there looking at illustrated\npapers, and smoking a cork-tipped cigarette, ladies' size. Then he goes\nhome to lunch, and after lunch either drives with his sister in a hired\nfly, or else, if it is very fine, goes round the ladies' golf-links,\nwhich are a good deal shorter than the men's. He has tea at the club and\nsits there till dinner. Then, after a blameless day, he goes home to\ndine and sleep. I suppose no one in the world has ever done less of any\ndescription.\n\nI have alluded to his weakness--rank; he has another, which is gossip.\nHe knows who was dining at the Ampses last Wednesday, and who lunched\nwith the Archdeacon on Sunday, and how the Bishop's wife is. It is he\nfor whom also the fashionable intelligence is written in the daily\npapers, and, though he never goes there, he knows who is in town, and\nwho lunched at Prince's last Sunday, or walked in the Park, and how the\nMarquis of God-knows-what is after his operation. (He always refers to\na Marquis _as_ a Marquis, to an Earl _as_ an Earl.) But, best of all,\nperhaps, he loves infinitesimal intrigue, especially if it concerns\nRank.\n\nAnd here my portentous secret must burst from me. For the fact is that\nfor the last three days the town has been convulsed, and I have been\nholding it all back, assuming an unnatural calm, so that it might all\ncome in a deluge. For three days ago a Duchess came here to open a\nwindow, or shut a door in the town-hall, which had been put up in memory\nof something, and was entertained to luncheon afterwards by the\nCorporation. And on this eye-opening occasion Helen was sent in before\nthe wife of the younger son of a Baronet. And in consequence the wife of\nthe younger son of the Baronet cut her afterwards, as with a knife; yet\nknife was no word for it: the averted eye was more like a scimitar.\nBefore the assembled company, when Helen went to shake hands with her\nafter lunch, she cut her, and she turned from her, revolving on her own\naxis like the eternal stars. Upon which, very properly, after two days'\nheated discussion, and a great demand for Debrett, public opinion sided\nwith the wife of the younger son of the Baronet, on the ground that\nHelen took her husband's rank, which in this case happened to be none at\nall. What made it worse was that the Duchess, who should have known\nbetter, being an old friend of Helen's, came to tea with her afterwards\nin a motor-car covered with coronets for all the world to see.\n\nYou may imagine that the fat was in the fire after that. Helen had no\nidea why the wife of the Baronet's younger son had cut her, and perhaps\nmight never have known had not Mr. Holmes dropped in only yesterday and\ntold her, adding that he was sure he could clear it up. I was not at\nhome when this interview took place, but when he entered the room this\nafternoon, after having called only yesterday, it was certain that he\nmust have come on this subject. He had a book in his pocket, which made\nan unusual bulge.\n\nLegs was steeped in wide-eyed contemplation as Mr. Holmes had his tea.\nFrom time to time I glanced at him, and saw that the corners of his\nmouth were faintly twitching. His eye travelled from Mr. Holmes's face\nto his jewelled hands; it lingered about his clothes, but came back,\nloverlike, to his face. In a few minutes we had learned about\neverybody--how the Lord-Lieutenant of the county had driven through in\nhis motor--not the Daimler, but a new Panhard--yesterday afternoon,\nstopping only at the fishmonger's, and taking the London road\nafterwards; how there had been a party at the barracks last night, at\nwhich there was music; but not very good music, Mr. Holmes was afraid;\nhow the Bishop had not influenza at all, but only a bad cold; how The\nPines had been taken by the Hon. Alice Accrington, who had a cork\nfoot--so sad. A rhinoceros had trodden on the original one.\n\nI had ceased to be able to look at Legs, but here I heard him give a\nlittle whimper, as a dog does when it wants a door to be opened for it.\nHelen all the time had been of impeccable behaviour. She had asked just\nthe right questions, and appeared so genuinely interested that I felt I\nhad never known before of what depth of hypocrisy she was capable. Then\nMr. Holmes's wealth of information began to grow thin, even as the\nstars burn thin at daybreak, and I knew that he was going to dawn, and\nthat the true reason for which he came was going to break forth. He put\ndown his cup on the tea-table, took a cigarette, and suddenly creaked.\n\nIf you can imagine a sneeze, a cough, a spit, the strangled wheeze\ncaused by a fish-bone in the throat, and the noise an empty siphon of\nsoda-water makes when you press the handle, all combined, you will\nfaintly grasp what Legs did. His effort to swallow the whole of this\nmixed convulsion was most praiseworthy, though I should think dangerous,\nand it came to my ears only as if someone had done it half a mile away.\nMr. Holmes, I am sure, heard nothing this time, and Legs left the room\nwith his handkerchief to his mouth in the manner of mourners in the\nsecond coach at a funeral. There was no sound outside, but soon after a\nmuffled tread overhead, where is his bedroom. Then for a moment I caught\nHelen's eye. She looked so inexpressibly grave that I nearly asked her\nwho was dead. Then dawn came. Mr. Holmes has a high cackling voice, and\nthe bulgy volume in his pocket was 'Whitaker's Almanack.'\n\n'I should have come before,' he said, 'but I wanted to come to you last,\nand really the afternoon has flown. About Tuesday now. Dear lady, you\nonly took your right place. There is no question about it. I have been\nto the Mayor, I have been to the Archdeacon. Look.'\n\nHe found a page in Whitaker, and gave Helen the volume. It was a table\nof precedence. I saw 'Eldest sons of younger sons of peers' underlined.\n\n'Look at the next column,' he said. 'The sister takes the rank of her\nhusband _or_ her elder brother. Now see where younger sons of Baronets\nand their wives come!'\n\nFar away below eldest sons of younger sons of peers, in an outer\ndarkness below even members of the fifth class of the Victorian Order, I\nsaw that obscure relationship. My emotions of various kinds almost\nsuffocated me. Helen was justified before all the world. It was _her_\nturn to cut the wife of the younger son of the Baronet if she chose.\n\nSo we talked very pleasantly for a quarter of an hour about the\nmovements of the aristocracy, and then Mr. Holmes 'rose to go.' His cab\nwas waiting, and I helped him on with a very magnificent fur coat in the\nhall, which in the somewhat indistinct light seemed to be made of the\npurest rabbit skin. In the dimness of the landing above I thought I\ncould see an obscure shadow leaning over the banisters which resembled\nLegs.\n\n'I hope, after this, your wife will take her proper place,' said Mr.\nHolmes. 'Of course, everyone knows the Duchess came here to tea.'\n\nHe lit a cigarette, and I heard the banister tremble slightly, as if\nfrom an infinitesimal earthquake.\n\n'It is so kind of you to have taken so much trouble,' said I firmly.\n\n'It was nothing. I am sure you need have no further anxiety.'\n\n       *       *       *       *       *\n\nI went back to the drawing-room. Helen's face was buried in a\nsofa-cushion, and Legs came downstairs in three jumps.\n\nSo we laughed till it was time to dress for dinner. Occasionally we\nseemed to be recovering, but then somebody said 'Creak,' or 'Baronet,'\nand a fresh relapse took place.\n\nI pity all poor souls who do not know Mr. Holmes. It is so sad for\nthem--sadder than the lady with the cork foot. Oh, think of it! This\ntriumphant vindication of Helen (which is all wrong, by the way) will\nlast him a long, long time. It has been a campaign, triumphantly\nconcluded, and I should not in the least wonder if he has half a bottle\nof champagne to-night. And after a time the excitement will die away,\nfading like a golden sunset, and he will settle down to his ordinary\nlife again, and read the paper in the morning, and go for a little drive\nin the afternoon, and have tea and toast at the club afterwards. And in\nthe spring the Panama hat will come out, and the rich fur coat be put\naway, and he will hand strawberries instead of buns, and iced coffee\ninstead of tea, and perhaps play a little croquet. But this week has\nbeen a great week for him--it really has. If you want to understand the\ngloriousness of Mr. Holmes, you must take my word for it that nothing so\nengrossing has happened to him for months.\n\n\n\n\nDECEMBER\n\n\nThis once-happy family has suddenly returned to the pit whence it was\ndigged, and it is impossible to imagine any more depressing spectacle\nthan we present. Dawn in faint flickers is beginning to shine on the\nwreck, and occasionally for a moment or two, though we may be\nover-sanguine, Helen and I can dimly imagine being happy again. Legs\ncannot do that yet; it is still midnight gloom with him.\n\nThe intelligent reader will scarcely need to be told that it is the\ninfluenza that has blackened the world like this. Helen began, and Legs\nand I followed within twenty-four hours. That, somehow, is a relief to\nher, since she feels she did not give it us. As if it mattered where it\ncame from! Besides, personally I would rather catch it from her than\nanyone else. Legs has had the worst visitation, because, after it was\nquite certain he had got it, he persisted in attending the last night of\nthe autumn opera season, did not enjoy it at all, of course, by reason\nof a splitting headache, and was really ill for a day or two. I was\ninfinitely wiser. As soon as the nymph touched me with her fairy hand I\nwent firmly to bed, turned my face to the wall like Hezekiah, and\nstopped there till the fever was over. After five days I tottered\ndownstairs to find an old, old woman sitting by the fire. It was Helen.\n\nI think that was the most dreadful day I ever remember. She told me\nagain and again how ill I looked until I was goaded into a sort of\ndepressed frenzy, and said I couldn't possibly look as ill as she. We\nboth had beef-tea in the middle of the morning, and to my horror, when\nit was brought, it was brought not by Raikes, my man who is as\nindispensable to this house as is the carburetter to a motor-car (for it\nwon't run without), but by an Awful Thing that I never saw before. In\nanswer to an inquiry, I was told that Raikes felt very ill, and had\nasked the Awful Thing to bring us our beef-tea instead of him. So I sent\nher back to Raikes with a thermometer that he was to be so good as to\nput under his tongue for one minute, and then return. It came back\nrecording 102 degrees. I gave the Awful Thing the thermometer to wash,\nand she instantly dropped it on the floor. It was, of course, broken\ninto twenty million fragments, but I remembered that, though I was a\nworm, I was a Christian worm, and said: 'Never mind. Please tell Raikes\nthat he is to go to bed instantly.' I then picked up the twenty million\nfragments, and cut myself severely. I said 'Damn!' quite softly.\n\nHelen winced, which was merely intended to annoy me, and it succeeded\nadmirably.\n\nSo there we sat exactly like that awful picture called 'Les Frileux,' in\nwhich an old man and an old woman sit apart under a leafless tree. The\nground is covered with the dead leaves. Soon they will die, too.\n\nIt is impossible to depict the dreariness of that morning. Outside a\nsort of jaundiced day showed the soupy mud that flooded Sloane Street,\nthrough which motor-buses, which once I thought so fine, splashed their\nway. A few sordid people under umbrellas bobbed by the windows, and as\nthe darkness increased a man with a long stick began to turn up the\nlamps. Then it instantly got rather lighter, and another man (not the\nsame one) with another long stick came and turned them down again. Upon\nwhich Egyptian darkness settled down over the town, and I must suppose\nthat the first man had caught the influenza, for he never turned them up\nany more.\n\nHelen was not reading; she was sitting by the fire looking mournfully at\nthe coals. This would not do at all, and in the intervals of a paroxysm\nof coughing I asked:\n\n'How is Legs this morning?'\n\n'Worse,' said Helen.\n\nI took up the _Daily Telegraph_, and read the list of the people who\nwere dead. I knew one of them slightly. Then my cut finger began to\nbleed again, which reminded me of the Awful Thing.\n\n'Servants are so ridiculous and tiresome,' I said. 'I should think your\nmaid might have found time to bring up our beef-tea, instead of that\ndreadful girl. I don't know where you get your servants from.'\n\n'Barton went to bed yesterday with influenza,' said Helen wheezily. 'She\nis very feverish--worse than Legs.'\n\nI can't say why, but this news made me feel rather better, so I lit a\ncigarette. It tasted exactly as if it had been made of the green weed\nwhich grows on stagnant horse-ponds. I felt much worse again at once,\nand was quite sure my temperature was going up. But I could not have the\nmournful satisfaction of knowing that this was true, because the\nthermometer was broken. And my finger continued to bleed. The blood was\nvery bright red--probably arterial. Yet, whatever was happening, it\nseemed impossible that things were as desperate as I thought them, and I\nmade the excellent determination to do something.\n\n'Will it disturb you if I play the piano?' I asked Helen.\n\n'Not the least.'\n\nI attempted to play the 'Etudes Symphoniques,' beginning with the last\nvariation, by reason of the sky-scraping spirits of it. I don't think I\nplayed any correct notes at all, and Helen (again to annoy me) made the\nnoise which tiresome people make to show that a wrong note gets on their\nhighly sensitive nerves. It consists of a whistling intake of the\nbreath. Though I had only played a dozen bars, the white notes in the\ntreble were spotted with blood, as if I was a Jew and the piano was the\nlintel of the door on Passover night. It was absurd to go on playing on\na blood-boltered piano, even if I could play the right notes, which I\ncould not. So again, with the laudable idea of doing something, I\nstaggered upstairs, brought down a moistened towel, and proceeded to\nclean the keys. I struck notes from time to time, and Helen kept on\nwincing.\n\n'Is that necessary?' she asked at length.\n\n'Yes, because I have bled over the piano. Besides, I'm cleaning it with\nthe soft pedal down.'\n\nThe door was flung open, and the Awful Thing appeared.\n\n'Dinner,' she said, and left the door open.\n\nWe went downstairs. 'Dinner' in Raikes' indisposition was huddled on to\nthe table. There were pieces of moist fish under one cover. There was a\nginger pudding under another. There were large potatoes under a third;\nand under the fourth a rich and red beef-steak. Then despair descended\non me.\n\n'Is the cook ill, too?' I asked of the Awful Thing.\n\n'Yes, sir.'\n\n'Who cooked this? Or, rather, didn't?'\n\n'Please, sir, I did.'\n\nThen quite suddenly, both for Helen and me, dawn began to break for a\nlittle. Here was three-quarters of the establishment incapacitated, and\nthe Awful Thing was calmly doing everybody's work as well as her own,\nwhich was that of a housemaid. Helen cheered up at once.\n\n'Please give me some fish,' she said to me. 'It looks quite excellent.'\n\nI helped her largely and sumptuously. We both understood each other at\nthis moment, and I put a thumping helping on to my own plate.\n\nHelen, greatly daring, took a greedy mouthful, and spoke to the Awful\nThing, who was beginning to beam largely on us.\n\n'Delicious,' she said to her. 'I had no idea you could cook so\nbeautifully. You needn't wait; we will ring. And you must have help in\nat once. Will you telephone to Mrs. Watkins' agency, asking for a--(she\npaused, and I know she was going to say 'cook')--a housemaid?'\n\nThe Awful Thing smiled from ear to ear, and a moment afterwards we heard\nthe insane ringing of the telephone.\n\n'Oh, I _couldn't_ send for a cook just this moment,' said Helen, when\nthe girl had left the room. 'She was bursting with pride at having\ncooked this. But if I eat it I shall be sick. What are we to do?'\n\nThe girl in her enthusiasm had built the fire three-quarters of the way\nup the chimney, though the day was muggy and warm beyond all telling.\nInto the heart of the blaze we stuffed large pieces of fish, which\nburned with a blue and oily flame.\n\n'Now ring,' said Helen.\n\nThe girl returned after a long pause.\n\n'Please 'm, Mrs. Watkins hasn't a housemaid to send, by reason of so\nmuch illness. But she can send a cook,' she said, and her face fell.\n\n'It's such a pity, when you can cook so well,' said Helen; 'but we must\nhave somebody. You can't do all the work.'\n\n'A char and I could manage, 'm,' she said, changing the plates with an\nawful clatter.\n\n'Oh, not with Mr. Legs ill,' said Helen. 'We shall have you knocked up\nnext, and where should we be then?'\n\nThe radiant smile returned to the girl's face.\n\n'Give me some steak, Jack,' said Helen, 'and a potato. How delicious it\nsmells!'\n\nThe Awful Thing again left the room, leaving, as it were, the fragrance\nof her smile behind her.\n\nWe made no attempt to eat any of the second course, but put two large\nslices of steak, two potatoes, and a big spoonful of perspiring\ncauliflower into the fire. Pieces of ginger-pudding followed it to the\nburning ghaut, and soon the door again opened, and coffee was brought\nin. This was an after-thought, I fancy, though ill-inspired and gritty.\nBut there was a coal-scuttle.\n\nI am afraid we both relapsed again after lunch, though for a time the\nshining example of the housemaid who had done the work of everybody else\ninspired us to attempt to play piquet, bezique, and the piano. But these\nwere all hopeless: it did not seem worth while dealing, and, in point of\nfact, the attempt at a duet came to a conclusion at the end of the first\npage, for Helen only groaned and said:\n\n'I can't turn over.'\n\nBut that, I am thankful to say, was our low-water mark.\n\n       *       *       *       *       *\n\nSunshine began to shine more strongly on the wreck when Legs, two days\nafterwards, came downstairs, with the cheering remark that he felt so\nill that he was sure he couldn't be as ill as he felt. Soon after he\nburst into hoarse laughter.\n\n'I shall cheer up when I have counted ten,' he remarked.\n\nWell, on the whole, when it was put simply and firmly like that, it\nseemed the best thing to do. Legs took change of the cheering process,\nand ordered a basin, soap, and three churchwarden pipes, and we blew\nsoap-bubbles, which, though it may not be in itself a work of high\nendeavour, had at least the result of making us do something, which is\nalways a good thing. So, when that was over, in order to contribute to\nthe wholesome atmosphere of employment, I brought in and read to him and\nHelen what I had written that morning, and had designed to appear in the\nbook you I are now reading. It was--I will not deceive you--a string (a\nlong one) of cheap and gloomy reflections on the mutability of life, the\nreality of suffering, and the certainty of death. I had taken some\ntrouble with it, but the most poignant and searing sentences made Legs\nsimply roll in his chair with laughter that was noiseless merely because\nhis throat was in such a state of relaxation that it could not make\nsounds. But with eyes streaming and in a strangled whisper he said:\n\n'Oh, do stop a moment till I don't hurt so much with laughing, and then\nread it again.'\n\nI looked at Helen. She had a handkerchief to her face, and her shoulders\nshook with incontrollable laughter.\n\n'It's much the funniest thing you ever wrote,' she said. 'Isn't it, now?\nBegin again at \"All the pain and sorrow with which we are\nsurrounded\"--oh, no, before that--something about \"It is when we are\nracked with suffering ourselves.\" Oh, Legs, isn't it heavenly?'\n\nLegs had recovered himself a little, but still drummed with his feet on\nthe carpet.\n\n'I never knew I could feel so much better so quickly,' he said. 'I felt\na mere worm when I proposed soap-bubbles. I want it all again from the\nbeginning, where what you thought was sunlight was barred with strange\nshadows. O Lor!'\n\nSo I gave them this intellectual--or should I say spiritual?--treat once\nmore, and then threw the manuscript into the fire, amid the shrill\nexpostulations of the others. Legs made heroic attempts to save it, but\nfruitlessly, or, indeed, I would print it here, as a warning to those\nwho do not feel very well to postpone their meditations upon life and\ndeath until they feel a little better. Also, I do not think that one's\nreflections on any subject are likely to be of much value unless they\nare founded on some sort of experience, and, to be quite honest, I had\nfounded my views that morning on the mutability of life and the anguish\nof the world on the depression which was the result of a feverish cold.\nThey were depressing enough, but I do not think that they were of\nsufficiently solid foundations. They proved, it is true, extraordinarily\ncheering to Helen and Legs, but one cannot be certain that the rest of\nthe world would be equally exhilarated. They might be taken seriously,\nthough Helen says I need not have been afraid of that.\n\nEvery man, even a pessimist, is supposed to have a perfect right to form\nhis own opinions, but if I had my way (there is not the least likelihood\nof it) I should establish a censorship of the press, which should be in\nthe hands of six young and cheerful optimists, who should decide whether\nsuch opinions were fit for publication. Quite rightly literature of an\nindecent nature, and work which may be supposed to have a tendency\nencouraging to criminals, is not allowed to be disseminated. I should\nput a similar prohibition on the dissemination of discouraging books,\nbooks which might be expected to suggest or foster the opinion that the\nworld is a poor sort of place, and that God isn't in His heaven at all.\nEven if this was proved to be true, I would count it criminal to attempt\nto convince anybody of it; it would be a murderous assault on the\nhappiness of private individuals. The law does not allow one to poison a\nman's bread with impunity, so how much more stringently should it forbid\nthe poisoning of the inward health of his soul! Nothing but harm ever\ncame from the dissemination of depressing truths, nothing but good from\nthe dissemination of innocent and joyful beliefs, even should it be\nproved that they had no foundation whatever. For if the world is a\ndreary and painful place, so much more need is there of courage and a\nhigh heart to render it the least tolerable, and if we are to be snuffed\nout like candles when we come to the end of our few and evil years, how\nmuch more is it the part of wisdom to snatch a little happiness out of\nthe circumambient annihilation!\n\nAnd to think that only this morning I had actually tried to commit this\ncrime, and was only saved from it by Legs' unutterable laughter. To be\ntruthful, I felt a little offended when he first began to laugh, and\ninwardly hoped that he would soon grow depressed and thoughtful as I\ncontinued to tell my rosary of discouraging things. But I need not have\nindulged that hope; it was forlorn from the beginning.\n\nInstead, it made both him and Helen feel much better. I am so content to\nleave it at that. I had hoped--I had, indeed--when I wrote those\ndepressing pages (which I wish to Heaven I had not burned) that possible\nreaders might see part of the serious side of things under the\ndiscouragement of my winged words. But now--two days later--I am far\nmore content that those two darlings should have laughed at what was\nwritten with such seriousness, than that all those into whose hands the\nprinted record of that manuscript might have fallen should have sighed\nonce over my jaundiced views about life and death, and sickness and\nmutability.\n\nOf course, death is an extremely solemn affair, but it seems to me\nnow--we are all recovering fast, and are drinking hypophosphates, and\nbeginning to be greedy again--that the solemnity of it ought to have\nbeen discounted long ago, if it is going to be solemn at all. Everyone,\nof course, is at liberty to take life solemnly from the time he begins\nto think at all. But whatever our attitude towards life is, the same\nought to be our attitude towards death, whether we believe that there is\na continuance of life afterwards, or whether we are so unfortunate as to\nbelieve that there is the quenched candle. For in the one case death is\nbut the opening of a door into a fuller light, a thing, it is true, that\nmay affect one for the moment, since from the weakness of the flesh we\ncling to what we know, while in the other death is just extinction, a\nconsummation which no pessimist should fear, since while he lived he\nhad held so poor an opinion of life. So whether we regard life as a\npleasant interlude in something else, or whether we regard death--a\nthing unthinkable to me--as the extinction of consciousness, I cannot\nbelieve that he is not a guest who is welcome when he comes. Personally\nI do not want him to come for a long time, since I am delighted with the\nworld, and it would be most annoying to die now when one is just\nrecovering from influenza, and hopes to go to the Richter concert\nto-morrow. But whatever one's belief about the future is, I cannot see\nthat there is an essential horror about death. I can conjure up horror\nof some kind about going to the dentist, about looking up trains in a\nBradshaw, since the print is so execrable and the connections so unruly,\nbut I go my journey, or I go to the dentist, and get to my destination,\nor am relieved of a troublesome tooth. Life does not seem to me the\nleast troublesome, it is true, but let us take it that by death I get to\nmy destination, or in any case get nearer it.\n\nBesides, how frightfully interesting!\n\nI did not die, but went to the Richter concert instead. Legs wished to\ngo, too, but that was clearly idiotic, and so Helen and I tossed up as\nto which of us should go, and which remain at home. I won, and went.\n\nThere was Isolde in his high chair. (Probably an intelligent critic will\nsay that Isolde was a woman, and I mean Tristan. But I don't.) He waved\na little wand, and the spirit of the Meistersingers filled the hall. It\nwas not, so it struck me, a remembrance only of their harmonious\njoviality, a mere picture of them; it was they who rollicked and made\nprocessions in the great thumping triads of their march. There they sat,\neach with his business, town clerk, and vintner burgomaster, and lawyer,\nand, best of all, the old tender-hearted shoemaker, on whose kindly face\nupturned to the sky one feather of the bird of love had fallen, though\nit had never come and nestled in his bosom. But it was not with\nbitterness that so great a loss had filled him; it had but refined him\nto a mellow kindliness that made all young things love him. There they\nall sat, so the band told me, over their songs and their sober\ncarousing, till the others went home, and Sachs was left alone with\nmusic yet unsung echoing in his kind old head, and throbbing in his\nyouthful heart. But he knew that such Divine melody was not to be\nrealized by him; some master of music had yet to come and put into notes\nand audible harmony that which existed but in the temple of his dreams,\nin the garden of things a man may conceive, but may not realize. Then\ncame there the gracious young knight, and Sachs heard that of which he\nhad dreamed, the song taught by the birds and the choirs of Nature to\nthe ardent heart of youth.\n\nThe triumph took wings and soared, lifting Sachs with it, him and his\nyearnings, and that fine old music, too, which was his. Inextricably\nmingled, they were knit one into each other, soaring into the sunrise.\n\n       *       *       *       *       *\n\nThereafter we were taken to the bleak mountain, where should gather the\nmaidens of storm, who did the will of Wotan. It was high and exposed\nabove the region of the trees, and shrill blew the winds over it, and\nthe heavens streamed above it. Fast and thick rode the army of menacing\nclouds, for the tempest in which the Valkyries rejoice, riding their\nuntamed steeds down the swift roadway of the winds, was broken out in\nmad fury. Yelling and screaming, it drove in mad circles of wrath round\nthe place where the nine maidens should foregather that evening, each\nwith the fruit of her day's quest slung across her saddle, each with a\nhero who should drink that night of the wine of the gods, which should\npour into his veins the fire of eternal life in place of the faint\nmortal blood that had beaten there before. Yet it was not love the\nmaidens sought. It was danger and death and heroic enterprise that bore\nthem so swiftly on their errands, and lit in them a fire brighter than\nlove has ever kindled. Their wine was the buffet of the tempest, their\nmeat the strong winds of God.\n\nThen there was heard, faint at first, the beating of the immortal hoofs\nin the rush of flying steeds; from east and west there shone out remote\nfires in the bedlam of the clouds, increasing, getting nearer and more\nblinding, till through the darkness of the tempest could be seen the\nfigures of the maidens gathering to their trysting-place, some at the\ngallop, some flying, and all drunk with adventure and swift deeds. Each\nthat day had prospered, each had a hero at her saddle, swooning now in\ndeath, but soon to be restored to the fuller life.\n\nSo gathered they, but as yet one was still missing--Br\u00fcnnhilde, the\nswiftest and best of them all, the dearest to the heart of Wotan, for,\nindeed, she was none other than his heart and his inviolable will. And\nwhile yet the others wondered at her tarrying, she came. But no hero had\nshe. She but led a woman into the midst of her sisters, for pity had\ntouched her fierce heart with so keen and intimate a pang that she had\ndisobeyed the behest of Wotan, and saved her of the race which he had\ndoomed to destruction.... The sorrow and the pain of the world had\nentered into her. Henceforth no more there would be for her the starry\nsplendour of Valhalla, throned on the thunder and rosy with the light of\neternal dawn. Soon for this her deed should another light shine on\ntower and palace wall--the light of the flames that consumed it.\n\n       *       *       *       *       *\n\nTempest, and love, and sorrow, and the doom of the immortal gods all\nmade audible in the eternal kingdom of the air! How is it that, when\nonce one has heard a miracle like this, one can ever so far forget it as\nto go back to the meanness of little miry ways? There are so many big\nthings in the world, and though one knows that, and has, according to\none's scale, seen and understood their size, yet we can still be so\ngross of perception that one can sit down, blear-eyed of vision, to\nwrite two-penny-halfpenny reflections about sorrow and mutability! (And\nbe rather pleased with them, too, until Legs and Helen laughed\nthemselves all out of shape.)\n\nHow large a place, too, in that which makes for size and the breeziness\nof living, does Art in some form or other occupy for most of us! Music\nand painting, literature and drama, are great doors flung wide to admit\none to the sunshine of God. Often, even to the spiritually-minded, the\navenues of prayer and directer communion seem somehow blocked; to\nothers, the majority, they are never wholly open. But to any who have an\nappreciation at all of what is beautiful, it must be a dark hour indeed\nwhen that approach is altogether shrouded and black, when neither\nAngelo, nor Velasquez, nor Shelley, nor Wagner, has a candle to give one\nto light the way. Millions of beautiful minds have their approach here.\nTo millions all idea of a personal God, to be approached directly, seems\ninconceivable, but it seems to me to be one of the perfectly certain\nthings in this very uncertain world that the passionate worship of\nbeauty, in whatever sort manifested, is no less a direct invocation than\nprayer and the bent knee. The study and the love for 'whatsoever things\nare lovely' is as royal a road, perhaps, as the other, for the passion\nfor what is beautiful is no less than the passion for the only\nBeautiful, and by such as feel that, all that is filthy is as unerringly\ncondemned as it is by those who call 'filthy' by another name--'sinful.'\nFor the perception of anything beautiful has to the perceiver a force of\npurging, while to the gross sense it is a sealed thing.\n\n    'O world as God has made it, all is beauty;\n     And knowing this is love, and love is duty,\n     What further can be sought for or declared?'\n\nAnd to that I say 'Amen.'\n\n       *       *       *       *       *\n\nThe 'kennel,' as that same magician of words said, is 'a-yelp' at this.\nArtists, of whatever sort, are supposed to be loose of life. Where that\nextraordinary delusion arose I have no idea, unless it had its origin in\nsome superficial observer of the manners and ways in the Latin quarter\nof Paris. That things not technically parochial may have occurred there,\nwho would deny? But for my part I think it just as un-Christian to nag,\nand to vex, and to be unkind as to be anything else under the sun. In\nfact, to put it broadly, I would as soon be a drunken and kind man as be\na sour and total abstainer. Sour and total abstainers will turn on me\ntheir eyes of smiling pity and horror, but perhaps it is only a matter\nof taste.\n\nBut to be 'nice' to people seems so immensely important. You may lecture\non the Lamentations of Jeremiah for hours together, with a battery of\nhistorical facts to help you, and yet do no particular good; but if you\nhelp a lame dog, canine or human, over a stile, you have been a far\nbetter Christian. I dare say that word offends some people, so I will\ncancel it, and say that you have been of far greater service in a world\nthat has fortuitously come into being, and will as fortuitously go out\nof being. Whatever may be the truth about things seen and unseen,\nhappiness is quite certainly better than misery, and laughter is better\nthan the most edifying tears.\n\nThe finger of the gloomy moralist is pointed at me. I knew it was going\nto be pointed--and in a sepulchral voice he says: 'What about death?'\n\nThe fact is that I don't know (nor does he), and it is not my affair.\nWhile I am alive I prefer to drink deep of the joy of life than to\nspeculate about what may come next. I can conjure up my death-bed as\noften as I choose, and make it a scene of moving pathos and dim vexed\ndoubts. There is nothing so easy. I can without the slightest effort\nadvance really profound problems as to 'what it all means,' since there\nis nothing so easy as asking unanswerable questions. What of the death\nof the wasp which I killed gleefully last August with a tennis-racquet?\nI haven't the slightest idea. All I know is that if next August another\nventures to buzz round my head when I am having tea on the lawn after a\nperspiring set, I shall, if possible, kill it again.\n\nIf only the gloomy moralist could give me a reasonable theory to show\nwhy I could not exterminate wasps, I would accept it. But he can't. He\nonly says it puzzles him. It puzzles me, too, but in the interval I kill\nthe wasp.\n\nThe fact is (degrading though it may sound) that I do not really believe\nthat we are any of us capable of understanding the mind of the Infinite\nGod. Philosophers try to explain little bits of it, and in their\nexplanation of the little bit of it bang their heads together like\nchildren playing hide-and-seek in the dark. _Hinc ill\u00e6 lacrim\u00e6_. The\npoor children have terrible headaches. I am extremely sorry, but it is,\nafter all, their fault. Instead of playing hide-and-seek in the dark,\nthey should go out and play in the light; then no heads would be hit\ntogether.\n\nIt is quite maddening to think of the energy expended over this\nhide-and-seek, when all the time the garden of the world's beauty is\nready waiting outside the door. If you have the instincts of a beast,\nperhaps it is better to grope in the dark; but if you have the rudiments\nof any other condition, go and play. All the beauty that the world holds\nis at your command. All that really matters in this world is to be\nenjoyed very cheaply. Most things worth reading can be bought for a\nshilling or two, and if that is not 'handy,' look at a tree instead, and\nabsorb the life that shines in each growing twig of it. Or if you are\nmusically minded, hear, as I have just heard, the glories of the maidens\nof the storm.\n\nOf course, no one thing is the least more wonderful than any other. All\nthat happens, if we look at it at all closely, is a marvellous\nconjuring-trick. Why don't ducks come out of hen's eggs? Is it not\nmarvellous that chickens invariably issue? If you go a step farther\nback, and learn something about the continuance of type, it becomes even\nmore wonderful. 'How' can be told us, but never 'why.' And so I am\nconfident in the unanswerableness of my riddle. Why do sounds like those\nof the violin and the brass in the 'Ride of the Valkyries' convey the\nessence of storm and tempest?\n\n       *       *       *       *       *\n\nAnother conjuring-trick of the most delightful kind occurred next\nmorning. At twelve o'clock last night the streets of London had, without\nasking (thereby reversing the sad tale of Oliver Twist), been given a\nsecond helping of brown porridge. It was ankle-deep on the roadway of\nSloane Street, thick brown porridge of mud; then during the night the\ntemperature went down, and it froze. The result is that for the copious\nsoup we are given a clean, dry roadway. There is no mud of any kind, not\neven frozen mud. The street is clear and dry, as if Oliver Twist had\nlicked it. But where has gone that two inches of obfusc lather? Has the\nwood-pavement drunk it in? Has it gone into the air? Has some celestial\nhousemaid, like the Awful Thing, been set to sweep the streets, even as\nshe has swept the sky, and given us the invigoration of frost in\nexchange for the wet blanket of chilly cloud? Coming back from Richter\nlast night, the streets were swimming; eight hours later (or it may be\nnine) one might walk barefoot across the road, or spread one's dinner\nthere, and get no taint. How it will be sparkling on the grasses and\nbrave evergreens at home, turned to diamond spray by the red sun of\nfrosty mornings!\n\n       *       *       *       *       *\n\n'O world as God has made it!'... How often involuntarily, as if coming\nfrom without, that line rings in my head! And how very little we, with\nall our jealousies, and depressions, and bickerings, and follies, are\nable to spoil or dim the beauty that is cast so broadly there. Puny as\nare our efforts for good, it really seems to me that our attempts at\nbeing evil are even more impotent and microscopic. We are often as\ntiresome and unpleasant as we know how to be, yet all the time we are\nswimming against that huge quiet tide of the beauty of the world as God\nmade it, the knowledge of which is love, and beyond which there is no\nfurther declaration possible. Sometimes, if we are very active indeed,\nand exert ourselves very much, we can stand still or even move a little\nway in opposition to the great tide, but soon our efforts must relax,\nand we are swept down again with the current that eternally flows from\nthe heart of the Infinite, and returns there again in those pulsations\nthat are the life and the light of the world.\n\nIt is impossible, indeed, unless we say that evil is the vital principle\nof the world, to think otherwise. War there is between the two huge\nforces, but it is just Satanism, and nothing else whatever, that makes\npeople say that the world is going from bad to worse. If you are so\nunfortunate as to be a Satanist, there is nothing more to be said, and I\nhope the devil will give you your due; but if otherwise, there can be no\nother conclusion than that good, all that is lovely and fine, is\nsteadily gaining ground. For it does not seem reasonable to suppose that\nGod contemplates some swift heady man\u0153uvre which shall suddenly take\nevil in the rear, and in a moment rout the antagonism. At any rate, as\nfar as we can possibly judge, it is by quiet processes that He deals\nwith the sum of the world, even as He deals with the units that make\nit. For just as nobody has any right to expect that the evil in his\nnature will be suddenly expunged, even though the moment should be one\nof blinding revelation, so we should acquiesce in the slow progress of\nthe sum-total. For there are only three possible alternatives--the first\n(namely, that the progress is from bad to worse), which is Satanism; the\nsecond, that there is now in the world (and will be) exactly the same\namount of evil and good as there has always been, in which case you are\nconfronted with the absurd proposition of two absolutely equal forces\nhaving made this scheme of things, which will war to all eternity; and\nthe third, that good is stronger than evil, and is quietly gaining\nground.\n\nThe objection to the first alternative is that it is Satanism--a very\nfatal objection. The objection to the second is that it is so\nstupendously dull. There cannot possibly be any point in anything if the\ntwo forces are equal. There can be no struggle in the mind as to whether\none ought or ought not to do certain things, if whatever you do or don't\ndoes not make any difference. There remains the third alternative. The\nobjection to that is ... well, I can't see there is any.\n\n       *       *       *       *       *\n\nHours ago this house has been asleep, the house in which I write on this\nearly morning of the New Year, the house which is home to me, even as my\nown is; for it is the house--you will have guessed--where lives she who\nis neither dearer nor less dear than Helen, and where we always spend\nthe week and a little more that begins before Christmas and finishes a\nlittle after the New Year has been swung from the voices of mellow\nbells. Before midnight we sat in the oak-panelled room and played the\nmost heavenly games, charades, and insane gymnastic exercises, and\ntable-turning, with terror when the dreadful table turned in a really\nunaccountable manner, all consecrated by love and laughter; and then,\nwhen the Old Year was to be numbered by minutes that the fingers could\nreckon, we drew nearer to the log fire and wished each other that which\nwe all wanted for each. Legs' triumphant entry into the Foreign Office\nwas no longer capable of a wish, since it was already accomplished, so\nhe was wished a wife; and--you will understand that we were all very\nintimate--my mother was wished freedom from all anxiety of whatever\nkind; and the old nurse of ninety years who had acted charades with us\nwith astonishing power was wished her century; and I was wished the\nholding of the frost, so that I might skate--they were flippant\nagain--and two cousins were respectively wished a microscope--one is of\ntender years--and a motor-car; and then, just as the clock jarred,\ntelling us there was but a minute more to the New Year, it was Helen's\nturn to be wished, and somebody said, 'Your heart's desire'; and she\nunderstood.\n\nImmediately afterwards the clock struck, and everybody kissed everybody\nelse, and said 'Happy New Year,' and no more. For you must not say\nanything more than that: you _must not_ even say 'Good-night,' else the\ncharm is broken. So in dead silence we lighted bedroom candles, for the\nritual was well known, and separated. And who knows but that all about\nthe house, as in the 'Midsummer Night's Dream,' the dances of the\nfairies circled up and down by the light of drowsy fires?\n\n\n\n\nJANUARY\n\n\nA hundred pounds have suddenly and unexpectedly appeared on the horizon.\nPeople who are very rich have not the slightest idea what that means to\nus. People who are very poor have not the slightest idea either, because\nthey would probably buy a public-house, or goodwill, or something of\nthat nature, and never have any fun out of it at all. But to people who\n'jog along' a hundred pounds is a treat which neither rich people nor\npoor can form any conception of. To those who just pay their way, as we\ndo, it means several weeks somewhere. The only question is 'Where?' At\nthis point in our argument it was impossible to proceed. Helen and I\nwere both being so unselfish that we couldn't go on. She said she longed\nto have two or three weeks in Switzerland; I said that what I really\nwanted was to go to the Riviera for a fortnight. Then, as always\nhappens, these subterfuges broke down, and we both confessed that we\nneither of us really wanted to go where we said we did. She wanted to go\nto Nice; I wanted to go to the high altitudes. So, with the\nunderstanding that we were to go where the coin said we should, and not\notherwise, we tossed up. It was high altitudes.\n\nHis country put in a claim for Legs at the Foreign Office,\nunfortunately, and he should not come with us; but we felt, when we\nobserved the urbanity of the French customs-house officials, who\nobligingly shut their eyes to the presence of large quantities of\ntobacco, and the politeness of the railway officials, that Legs had\nprobably made himself felt in our foreign relations already, and that he\nwas responsible for all this very civil behaviour. At B\u00e2le, however,\nwhere we had to change at the awful hour in the morning which is neither\nnight nor day, we found that Legs' diplomacy had not yet had time to\nmake itself felt, for we were subjected to a searching scrutiny. Luckily\nI had had experience of the manners and customs-house officials of B\u00e2le\nbefore, and had transferred my tobacco into my coat pockets, thus\nfrustrating the baffled Teuton. But I am afraid it gave certain secret\nglee to observe that my travelling companion of the night before--a\nstout white man, with a name on his labels so long that I could not read\nit, who had snored all the time--was caught, and his rich stores of\ncigarettes taken from him, to be sent, I suppose, to Berne, for the\ndelectation of the President of the Republic.\n\nSwitzerland is a land that always arouses curiosity as to how it came\nabout that a country in which the people are so small, so 'toy,' should\nin itself be on so gigantic and marvellous a scale. Is it that the\nliving among these stupendous surroundings has somehow dwarfed the\npeople, or has Nature, by one of her inimitable contrasts, made the\nhuman part of Switzerland so insignificant in order to set off the\nvastness of peak and snowfield? Certainly the glib commonplace that\nnational character is influenced and formed by national surroundings is\nhere gloriously contradicted, since, as far as I am aware, no Swiss has\never attained to eminence in anything. They are a little toy people,\nwho live in little toy towns, and make excellent chocolate, and run\ninnumerable hotels on the most economical principles. But even then they\ndo not (as one would expect) get very rich. They are never 'very'\nanything. 'But the chocolate is excellent,' said Helen to these\nspeculations.\n\nIt requires faith this morning to believe that in a few hours we shall\nbe crunching the dry, powdery snow beneath our feet, and before sunset\nbe skating or gliding down the white frozen road, with puffs of snow\ncoming from the bows of the toboggan, for here all down the shore of the\nLake of Thun the country is brown and grey, with scarce a streak of\nwhite to show that it is winter. Low overhead are fat masses of\ndirty-looking cloud, but between them (and this is the door where faith\nenters) are glimpses of the perfect azure which we expect up above. Now\nand then the sun strikes some distant hillside, or, like a flashlight,\nis turned on to the waters of the lake, making of them a sudden\naquamarine of luminous green. But the weather is undoubtedly mild; the\neaves of the wooden toy-stations drip with discouraging moisture, and\nInterlaken, when we reach it, wears a dreadful spring-like aspect, and\npeople are sitting out of doors at the caf\u00e9s, and appear to find it\nrelaxing.\n\nThen the first of these wonderful winter miracles happened. There was\nthe flat alluvial land at the end of the lake, across which ran the\nfussy little light railway which should take us above (so we hoped) the\nregion of cloudland. Grey and puddle-strewn was it, with here and there\na patch of dirty snow stained through with the earthy moistness beneath.\nA low-lying mist was spread over the nearer distance, which melted into\nthe thicker clouds of the sky itself. It was just such a view as you\nshall see anywhere in the English fen-land during February.\n\nWe were looking at this with, I am bound to say, a certain despondency.\nIt seemed almost certain that we should find dull weather (which means\nthaw) up above, when a sudden draught from some funnel of the hills came\ndown, making agitation and disturbance both among the low-lying mist and\nthe higher clouds. The former was vanquished first, and, torn to ribbons\nby the wind, and scorched up by a sudden divine gleam of sun that smote\ndownwards, disclosed in its vanishing the long, piney sides of an\nupward-leading gorge. The higher clouds, being thicker, took longer to\ndisperse, I suppose, for at its farther end the gorge was still full of\nscudding vapours. Then suddenly they cleared, and high, high above, a\nvignette of fairyland--the Jungfrau herself, queen of the snows--stood\nout in glacier, and snowfield, and peak, against a sky of incredible\nblue. There she stood in full blaze of sunshine, the silver-crystal\nmaiden, donned in blue, enough to open the eyes of the blind and make\nthe dumb mouth sing.\n\nThen afterwards, as the little Turkish bath of a train went heavenwards,\nhow magical and divine a change happened! Inside the steamy carriages,\nsmelling of railway-bags, and rugs, and forgotten sandwiches, it was not\npossible to see through the condensation on the window-panes, but the\nblood that trots through the body knew the change, and took a more\nstaccato note. Then--I suppose that travelling stupidity had seized us\nboth--it suddenly occurred to Helen that we might, without fear of\nprosecution, put the windows down, though by a printed notice of\nby-laws of the railway it was still defended that we should not agitate\nourselves out of it. Once a ticket-puncher, exactly like a figure out of\nNoah's ark, put them scowlingly up again; but with the boldness that\nthis whiff of mountain-air supplied, we again lowered them, after a\nfurther consultation of the by-laws.\n\nThe ineffable change had begun. Soon for the moistness of the lowland\nthere was exchanged a hint of frost--something that made outlines a\nlittle more determinate, a little crisper. Then, as we mounted higher,\nthere was further change. For dripping twigs of the trees there were\ntrees that showed a hard, white outline of frost; for the sullen muddy\nstream there was clearer water, that went on its way beneath half-formed\nlids of ice; and thinner and thinner above our heads grew the grey\nblanket of cloud.\n\nThen that, too, was folded away, and above is was the sun and the\nsparkling of the unending firmament. Below it had been like a London\nfog, when you cannot see the tops of the shrouded houses; now we saw the\nroofs of the world, the Queen Anne's mansion of Europe, all clean, all\nclear, just as they were when I saw this land three years ago. No tile\nhad slipped, no chimney-pot required repairs. The top of the world was\ngood. Oh, how good!\n\nThe clear dry air, the sunset lights on the peaks, the liquid twilight\n(keen as snuff to the nostril), from which the sun had gone! There was\nthe rose-tinted Wetterhorn, black Eiger, flaming finger of\nFinster-Aarhorn; or, on more human plane, the hiss of skates over the\nperfect ice, the passage of a toboggan, with a little Swiss girl holding\nin front of her a baby sister, and steering with her heels, and shrilly\nshouting '_Achtung!_' There was 'Madame' who keeps a restaurant (I do\nnot know her name), standing to see the train-passengers come in, and\nshaking hands, and saying, 'You shall have wings to-morrow, no legs'\n(alluding to an amiable altercation of three years ago, when I drew a\nkind but firm sort of line about eating chickens' legs for lunch on four\nconsecutive days); and there was the beerman, whose admirable beverage I\nalways drank at 11.30 a.m., being thirsty with skating; and there was a\nskater I knew, who attempted a rather swift back-bracket for the\nadmiration of the new arrivals by the train to see, and fell down in a\nparticularly complicated manner in the middle of it; and there was the\nbarrack of an hotel which always smells of roasting leather, because\npeople put their skates and boots on the hot-water pipes, and right\nabove it was the Mettelhorn; and to the left was the Lady Wetterhorn;\nand to the right the smooth, steely-looking toboggan-run down into the\nvalley. 'Oh, world----' I beg your pardon.\n\nI have omitted to mention the magic word on our luggage-labels,\n'Grindelwald.'\n\nThree years ago, I must tell you, among other foolish and futile deeds,\nI made a _cache_ underneath a particular tree on the path leading to the\nScheidegg, consisting, as far as I remember, of chocolate, coins, and\nmatches. These insignificant facts I published in another place, and\nsince then I have received every winter mysterious letters from\nGrindelwald, showing that other people are as absurd as myself. My\n_cache_, in fact, has been found (I gave directions which I hoped would\nbe sufficient), and it has been, so these letters tell me, enriched by\nother secret and beautiful things. There has been placed there, on\nseparate occasions, by separate passionate pilgrims, all manner of\nstore, and the very next morning, instead of going to skate, Helen and I\nskulked off with a toboggan to see what we should find. A poem on the\nWetterhorn, so I had been informed, was there, to form the nucleus of a\nlibrary; there were a tin of potted meat and some caramels for the\nlarder; and furniture had been added by a third person in the shape of a\nlead soldier and an ink-bottle; while the exchequer, I knew, also had\nbeen enriched by at least half a franc in nickel pieces. We had debated\nearnestly last night as to what to add to the establishment, if we found\nit, and eventually decided on a handkerchief, which is to be regarded by\npassionate pilgrims as a tablecloth, a reel of cotton, and a copy of\n'Shirley' in the sixpenny edition, to swell the library shelves. This\nlatter was in a small linen bag, to keep it from the wet.\n\nOf course, we did not expect to find all the objects that I had been\ninformed had been placed there from time to time, for the rule of the\n_cache_ is that you may use what you find there, provided only you\nreplace it with something else. The potted meat, for instance, one could\nnot expect to go undiscussed, and I cannot personally conceive leaving\ncaramels uneaten. But in place of those, if only passionate pilgrims had\nplayed the game, we should find other objects. Thus the _cache_ becomes\na sort of exchange and mart--a reciprocal table laid in the wilderness,\nwhere you take one dish and replace it with another.\n\nHow it all savours of romance to the childish mind! With agitated\nfingers you scoop away the earth and moss which form the entrance to the\n_cache_, under a pine tree on the empty, frozen hillside, and you know\nyou will find treasure of some kind, but what it is you cannot possibly\ntell. And inviolable secrecy must surround and embellish your man\u0153uvres;\nthe _cache_ should not be mentioned at all except discreetly to the\nelect, for it partakes of Freemasonry, the masons of which are those who\ndelight in idiotic proceedings. But just as three years ago I gave the\ninventory of the _cache_ as it was then, so in the minds of the idiotic\nthere may be felt some interest as to its inventory when the founder\nagain revisited it. _Caches_, of course, are socialistic in spirit, and\nanybody may appropriate whatever he chooses; but I should be glad if the\ncopy of 'Shirley' is left there. It is such a pleasant book to read\nafter lunch, if you are tobogganing alone. A book, at any rate, is\nrather a good thing to have in a _cache_, and the wishes of the founder\nwill be satisfied if another book is put there instead. But let us have\na book. I should prefer that it should not be the 'Encyclop\u00e6dia\nBritannica.'\n\nThe morning, I think, must have been ordered on purpose, for I can\nimagine nothing so exquisite being served up in the ordinary way, _\u00e0 la\ncarte_; such weather must have been specially chosen. Not a single\nripple of air stirred; an unflecked sky was overhead, and the sun, as we\nset off, just topped the hills to the south-east, and sat like a huge\ngolden bandbox on the rim of them. The frost had been severe in the\nnight, but in this windlessness and entire absence of moisture no\nfeeling of cold reached one. There was in the air a briskness of\nquality more than magical; it was as if made of ice and fire and wine,\nand in a sort of intoxication we slid down into the valley. Then,\ncrossing the stream, since there was water about, it suddenly seemed\ndesperately chill; but no sooner had we mounted a dozen yards of ascent\nagain than the same dry kindling of the blood reasserted itself.\nToboggans will not run of their own accord uphill, so I put ours under\nmy arm, and for a hundred yards we danced a _pas de quatre_ up the\ntrodden snow. We both sang all the time, different tunes, when suddenly\nwe saw a clergyman observing us from a few yards ahead. He had a wildish\nand severe eye, and we stopped. David before the Ark would have stopped\nif he had unexpectedly come on that man. He was sitting in the snow, and\nwore a black hat, black coat, and black trousers, but he had yellow\nboots. He kept his eye on us all the time that we were within sight, and\nseemed to have no other occupation. We neither of us dared to look round\ntill we had left him some way behind, neither did we dare to dance\nagain. Eventually I turned my head to look at him from behind a tree.\nHe was still sitting in the snow, not on a rug, you understand, nor on a\ntoboggan, nor on any of the things upon which you usually sit in the\nsnow. He was not breakfasting or lunching or looking at the view. He was\nsitting in the snow, and that was all. I have no explanation of any kind\nto offer about this unusual incident. Helen thinks he was mad. That very\nlikely is the case, but it is an interesting form of mania. Perhaps\nby-and-by we shall have an asylum for snow-sitters. Or is it a new kind\nof rest-cure?\n\nIt is astonishing how you can argue about things of which you know\nnothing. Indeed, I think that all proper arguments are based on\nignorance. If you know anything whatever on the subject of which you are\ntalking, you produce a fact of some kind, which knocks argument flat. It\nis only possible to reason rightly on those subjects concerning which no\nfact, except the phenomenon itself, is ascertainable. Had we asked the\nclergyman why he sat in the snow, he would probably have told us, and\nthe subject would have ceased to interest us conversationally. As it\nwas, we held heated debate upon him, just as if he was the Education\nBill, for a long time. But the unusualness of it merited attention and\nconjecture. And think how divine an opening for conversation at\ndinner-parties, if you know nothing of your neighbour, and have not\ncaught her name.\n\n'Did you ever see a clergyman sitting in the snow?'\n\nThat, in fact, was the outcome of our argument. No theory about him\nwould really hold water. He was probably a conversational gambit, which\nmight lead to much. For instance, in answer to your question, your\ninterlocutor might reply in five obvious ways:\n\n1. 'I once saw a clergyman, but he was not sitting in the snow.'\n\n2. 'I have seen snow, but I never saw a clergyman sitting in it.'\n\n3. 'I once saw a clergyman being snowballed.'\n\n4. 'Yes. What are your views about the best treatment for the insane?'\n\n5. 'Such strange things happen at Grindelwald. Did you know----'\n\nYes; he was probably a conversational opening made manifest to mortal\neyes. Anyhow, when we returned he was not sitting there. If he had been\nreal, he probably would have been--at least, if you once sit in the snow\nthere is no reason why you should ever get up. Obviously it is your\n_m\u00e9tier_.\n\nNow, everybody who lives in fogs and rainy places will fail to\nunderstand anything of these last deplorable pages. But if they go to\nthe thin clear air of Alps in winter, they will know that this sort of\nthing (given you have the luck to see a clergyman sitting in the snow)\nis invested with supreme importance. When the hot sun shines on ice, it\nproduces some kindly confusion of the brain; there is no longer any\npoint in trying to be clever or well-informed, or witty, or any of those\nthings that are supposed to convey distinction down below to their\nfortunate possessors: you go back to mere existence and joy of life. It\nis a trouble to be consecutive or conduct a reasonable argument;\ninstead, you open your mouth and say anything that happens to come out\nof it. Most frequently what issues is laughter, but apart from that, the\nonly conversation you can indulge in is preposterous and the only\nbehaviour possible is childish. That is why I love these roofs of the\nworld. The intoxication of interstellar space is in the air. Everything\nis so light--you, your body, your mind, your tongue, your aims and\nobjects. The only things that you take seriously are the things that do\nnot matter: the snow-sitter was one, the _cache_ was another. But as we\ngot nearer the _cache_, we became even more solemn than on the question\nof the snow-sitter. There was no telling what we should find there, even\nif we found the place at all. The tree might have been cut down since\nlast year; the whole _cache_ might have been rifled by some imperceptive\nhand. There was no end to the list of untoward circumstances that might\nhave despoiled us.\n\nAnd so we went through the wood: we came to the end of it, and there was\na tree--'of many one,' as Mr. Wordsworth prophetically remarked. On its\nroots were cut my humble initials: it was certainly The Tree.\n\n'Oh, quick, quick!' said Helen; 'let us know the worst!'\n\nThe root had arched a little since I saw it last. Moss and snow were\nplastered on it in a manner scarcely natural. I plucked the bandage\naway with hands that trembled. We found:\n\n1. A pencil.\n\n2. Something sticky, which I believe to have been the caramels.\n\n3. An empty potted-meat tin, with a wisp of paper inside it, on which\nwas written: 'I ate it. Quite excellent.'\n\n4. A candle-end.\n\n5. The famous poem on the Wetterhorn done up in canvas. (How laudable!)\n\n6. A Jock-Scot, salmon-trout size.\n\n7. A paper on which was written: 'What's the point?'\n\n8. A cigarette, very sloppy.\n\n9. A five-franc piece, wrapped up in paper, on which was written: 'I\ntook 4.50 away.'\n\n10. A little wooden pill-box containing a very small moonstone.\n\nI think we were very moderate in our exchanges, which is right, since\nyou must always leave the _cache_ richer for your presence, and we\nmerely took away the pencil and the poem on the Wetterhorn, leaving our\nhandkerchief, the reel of cotton, and the copy of 'Shirley.' Below the\nquestion 'What's the point?' we wrote, 'None, if you can't see it,' and\nadded, 'The founder and his wife visited the _cache_ on January 12,\n1907. They saw a clergyman sitting in the snow. Selah.'\n\nThen an awful thing happened. Even while these treasures were openly and\nsumptuously spread round us, down the path there came a merry Swiss\npeasant about a hundred years old. He looked at us and the treasures\nwith curiosity and contempt, and then burst into a perfect flood of\nspeech, of which neither of us understood one single word. When he\nstopped, I said politely, 'Ich weiss nicht,' just like Parsifal, and he\nbegan it, or something like it, all over again, with gesticulations\nadded, and in a rather louder tone, as if he was talking to a deaf man.\nUntil this torrent of gibberish was let loose on me, I had no idea how\nmuch there was in the world that I did not know; so with the desire to\nreduce _his_ opinion of himself also, I addressed him in English. I said\n'God save the King' right through, as much as I could remember of 'To be\nor not to be' from the play called 'Hamlet,' and had just begun on 'When\nthe hounds of spring are on winter's traces,' when he suddenly turned\npale, crossed himself (though it was a Protestant canton), and fairly\nfled down the path. I make no doubt that he thought he had met the\ndevil. Anyhow, he had met his match at unintelligible conversation.\n\nBut it was clearly no use running risks, for more of the merry Swiss\nmight come down the path, who, it was conceivable, might not be so much\nimpressed by unintelligible sounds, and we hurriedly reburied the\ntreasure, ate our lunch, and turned the bow of the toboggan homewards,\nsince we proposed to skate all afternoon. It was a year since I had been\nshod with steel. I burned for the frozen surface. But it was right to\nsee to the _cache_ first. There are some things you cannot wait for.\n\nWe spent three weeks in these divine futilities, if anything so utterly\nenjoyable can be considered futile. For my part, I do not believe it\ncan, since, as I have already said, to enjoy a thing very much,\nsupposing always that it does not injure anybody else, is a gilt-edged\ninvestment of your time; for enjoyment is not (as is falsely supposed)\nfinished with when the thing itself is done and over, for it is just\nthen that the high interest of it (though gilt-edged) begins to be paid.\nUntil one forgets about it (and by a merciful dispensation one remembers\nwhat is pleasant far longer and far more keenly than what is painful),\nsubsequent days and hours are all enriched, and therefore made more\nproductive, by these pleasurable memories. It is here, I think, that a\nwonderfully fresh and vivid student of the human mind--namely, R. L.\nStevenson--goes all wrong when he says that the past is all of one\ntexture. It seems to me--one is only responsible for one's own\nexperience--to be of two textures, one strong and the other weak; and\nthe strong one is the memory of things you have enjoyed, of happy days;\nthe other of times when, for some reason or other--pain, or anxiety, or\nfear--the lights have been low, and the sound of the grinding not low,\nbut loud. The human mind, in fact, is more retentive of its pleasures\nthan of its pain. In the moment of the happening either may seem the top\nnote of acuteness, but the echoes of the one indisputably live longer\nthan the echoes of the other; and though our consciousness, if you care\nto look at it that way, is largely a haunted house of the dead hours,\nyet happy ghosts are in preponderance, and seem solider than the shadows\nof its dark places; also (and this, I think, too, is indubitable) the\nanticipation of happiness is more acute than the anticipation of a\ncorresponding pain. In the future there are two textures also, as in the\npast.\n\nSince our return this contrast has been rather markedly brought before\nme. There are many things I much look forward to; at the same time,\nthere is something ahead which I am dreading. What it is I do not know.\nI think I should dread it less if I did. But it is, though quite\ncertain, quite vague. I connect it, however, with that evening in\nSeptember when I heard my name called, and when Legs saw something which\nhas since been expunged from his memory. And here is the contrast: the\nhappiness that lies stored for me in the hive of the future is more\npotent than the bitterness that is there. Both are coming--of that I am\nsure--and among the many very happy things which I know and expect, I\nfeel there is something I do not yet know which is happier than any. It\nis futile to guess at it. One might make a hundred guesses, and each\nwould seem feasible of accomplishment. But there, at the back of my\nmind, are these two transparencies, so to speak--one sunlit, the other\nstormy--and it is through them that the events of the day are seen by\nme. They colour--both of them--all I do; but the happy one is the\npredominant one. They do not neutralize each other; they are both there\nto their full. But I despair at giving coherently so elusive a picture\nas they make in my own mind. But, though elusive, it is intensely real,\nand for the first time I neither can, nor do I desire to, speak to Helen\nabout this thing which is so often in my mind. It is incommunicable.\n\nBut after these Swiss weeks there was not much time for me to think\nabout this, as it was imperatively demanded, by reasons over which I\nhave no control, that I should exercise my mind on the extremely\ndifficult art of the composition of English prose, which incidentally\nimplies doing two things at once; for not only have you to invent your\nlively and inspiring tale, but you have to tell it in a certain way. You\nmay choose at the beginning any way of the hundreds that there are of\ntelling it; but in the key in which it is originally pitched, in that\nkey it has to remain all the time. As a matter of fact, it probably does\nnot, and goes wandering about in other modes and scales; but every book\nought to be in the one key in which it opens, just as a picture ought to\nbe in one key. It is within the writer's liberties, of course, to write\nother books in other keys, and I think he is perfectly justified in\nlargely contradicting in one work what he has unhesitatingly affirmed in\nanother, but in each his point of view has to be consistent throughout.\n\nThe thing is not quite so easy as it sounds, and it is further\ncomplicated by a very real difficulty. Every story that is worth reading\nat all is bound to record change in the characters and general attitude\nof the people with whom it deals. The jaded author has to keep his eye\non each, and see that he behaves after some atrocious battering with\nwhich fate has visited him in a different manner than before this\nvisitation took place. If he is living in any sense of the word, the\nevent will have altered him. He will view things differently, and\ntherefore behave differently. Yet all the time he is the same\npersonality. It were better for him that he should be as adamant to the\nblows of circumstance than that the inner essence which is individuality\nshould be uncertainly rendered; and, like the dexterous Mr. Maskelyne\nwith his spinning-plates, the scribe has to keep his eye on all his\npuppets to see that none lapse into stagnation, and to poke them up with\nhis industrious pen.\n\nIt is here that the complicated question of consistency comes in which\njust now is worrying me to bewilderment. Dreadful and stinging events\nare happening to a most favourite puppet of mine. Providence is dealing\nwith her in a cruelly ironical manner, in a way that makes the poor\ndistracted lady take quite fresh views of a world she thought so warm\nand kindly. Yet it must be the same personality which has to be shown\nsitting behind these changed feelings and directing them all. That is\nthe consistency that has to be observed. Otherwise it ceases to be one\nstory, but becomes a series of really unconnected short stories, with\nthe technical absurdity that the heroine in each has the same name.\n\nYet there is this also: it takes all sorts to make a world (at least, a\nworld otherwise constructed would be an extremely dull one), but It, It\nitself, Life, lies somewhere in the middle of us all, and is the centre\nto which we approach. We, the all sorts which make the world, view it\nvery differently, though we are all looking at the same object. And here\na simile, a thing usually unconvincing, may assist. What if in the\ncentre there is something like a great diamond, blazing in the rays of\nthe sun? I, from the south, see soft blue lights in it; you, from the\nwest, see a great ruby ray coming out of the heart of it; another on the\nnorth says, 'This diamond is emerald green'; while from the east it\nseems of transcendent orange. So far, it is quite certain that we are\nall right, for the world, so to speak, refracts God, making Him\nmany-hued, even as white light is refracted by the triangle of a prism.\nAnd then let us suppose circumstances enter and shift me, who have been\non the south, where I saw blue, to the west, where I see red. The whole\ncolour of the world is changed to me, and yet there is no\ninconsistency. The same Ego honestly sees a changed colour. There would,\non the other hand, when my place was shifted by circumstance, be grave\ninconsistency if I continued to declare that I still saw blue. I do not.\nMy eyes tell me it is red. Just now my eyes told me it was blue. But _I_\nhave not changed, nor has the great diamond changed; it is merely that\nthe refracted light has taken another colour.\n\nIt is just that which one must perceive in the telling of a story. A\nperson who sees blue all his life probably sees nothing at all, nothing,\nanyhow, in the least worth recording. He is bound as the wheel of\ncircumstances goes round to see things in other lights. But that is not\ninconsistency; it is the truly consistent. Who wants, after all, for\never to draw the same conclusion from the same premises? Only fossils,\nand possibly molluscs.\n\nBut pity the sorrows of the story-teller! The _quality_ of the red has\nto be of the same quality as the blue. The same fire which strikes to\nthe south will indubitably strike to all other points of the compass,\nand when X is wheeled north, he will not see the same green as Y sees\nthere. He saw it through the alchemy of his own mind; it will be green,\nbut nobody else's green. Or if it is, he has no individuality to speak\nof. At least he belongs to a type that sees everything through the eyes\nof others. That is generally labelled conventional, and there seems no\nreason to change the name.\n\nHow I laboured during those last ten days of January, and how little\nresult there seems to be! Only--I console myself with this--the real\nlabour of writing does not chiefly consist in the effort of putting\nthings down, but in the moral effort of rejecting them. There is nothing\neasier than to fill pages and pages with improving reflections or\ninspiring events. But having done that, it is necessary to sound the\ntuning-fork and see if, as I said at first, the story is in tune, if the\nkey is kept. Usually it is not. On which the fire ought to make to\nitself a momentary beacon, or the waste-paper basket be replete. But the\npile of numbered pages should in any case be starving. That, as a matter\nof fact, is my sole argument that I have justified my existence during\nthese ten days. I have really worked a great deal, and the waste-paper\nbasket could say how generous has been its diet. I have really left out\na very great deal, and I hasten to forestall the critic who will say\nthat I should, in order to act up to this excellent standard, have left\nout the rest. I do not agree with him.\n\nThe key of which I have spoken has to be preserved, not only in matters\nof consistency in character-drawing, but in style as well. If you lead\noff with verbiage from the Orient, the East must continue, I submit, to\ndye your paragraphs till the last page is turned. Though you may have\nalso at your command pure wells of the most limpid simplicity, you will\nhave to reserve them for some other immortal work; they will not mix\nwith the incense and heady draughts from the East. Or should you fancy a\nmysterious Delphic mode of diction, Delphic you must be to the end.\nBut--as if all this was not so difficult, that, like Dr. Johnson, we\nalmost wish it was frankly impossible--interwoven in your Delphic or\nOriental narrative there must be a totally different woof--namely, the\nthread of the spoken word, the speeches that you put into the mouths of\nyour various characters. And the written word, be it remembered, is\nnever like the spoken word: the two vocabularies, to begin with, are\ntotally distinct, and though I would not go so far as to affirm that the\nspoken word ought to be ungrammatical, it should, if it is to recall\nhuman speech, be colloquial, conversational. In interchange of ideas by\nmeans of the mouth real people do not use fine language, especially when\ntheir emotions are strongly aroused. Then, instead of becoming\nhigh-flown and ornate in their speech, real people go to the opposite\nextreme, and instinctively use only the very simplest words. When this\nis stated, it seems natural enough, but you will find it very seldom\npractised. Novelists have a tendency to let their puppets employ\nmagnificent high-sounding words to express the intensity and splendour\nof great emotion; in fact, you may gauge the strength of their emotions,\nas a rule, by the sonorous quality of their adjectives. I believe the\nvery opposite to be the truth of the matter: people in the grip of\npassion do not use beautiful or highly- words; above all, they\ndo not, like Mr. Wegg, 'drop into poetry.' Yet nothing is commoner than\nto find prose degenerating into blank verse in the spoken records of\nemotional crises, as if blank verse was a sublime form of prose. Little\nNell is continually half-way between prose and poetry, so also is\nNicholas Nickleby when his indignation is roused. In fact, in some of\nhis scenes with Ralph they both forget themselves so much in their\npassion that torrents of decasyllabic lines flow from their lips. But,\non the other hand, the language of narrative should undoubtedly grow\nmore , more vivid in such descriptions as are the setting of\nsome very emotional scene. Yet it should not depart from its original\nkey.... Well, as Mr. Tulliver said, 'It's puzzling work talking.'\n\n       *       *       *       *       *\n\nBut though the days have been so full, I have seen everything,\neverything through the two transparencies that seem drawn between\nexternal happenings and me.\n\n\n\n\nFEBRUARY\n\n\nThe seasons, according to the literary and artistic view of things, have\nbeen rather out of joint this year. The autumn was not a time of mellow\nfruitfulness at all, because all the green things upon this earth had\nexhausted themselves in the long hot summer, and had no more spirit left\nto be fruitful with. Then January in England had been of the usual warm\nmugginess and mist which poets say are characteristic of autumn, but\nwhich in reality characterize winter. Indeed, I doubt if winter was ever\na time of hard frosts and sparkling snow, which is the artistic ideal,\nand I am disposed to believe that that version of it was really brought\nfrom Germany by the Prince Consort, and popularized by Charles Dickens.\nThen after the mists came the mellow fruitfulness, for I myself saw\nstrawberries in flower on February 2, and on February 9 Helen came in\nsaying she had found a real strawberry. That was strange enough, though\nperhaps the finding of an unreal strawberry would have been stranger\nstill, so I said, 'Where?' and she said, 'On the strawberry beds,\nsilly.'\n\nTherefore I started up, leaving a most important and epoch-making\nsentence unfinished (and I have never been able to remember what the end\nof it was going to be), because I wanted to see the strawberry, and\nwrite to the _Field_ about it. So she said, 'Are you going out already?'\nand I said, 'Yes, just to see the strawberry, and write to the _Field_,\nsaying I have.'\n\nThen she pointed to half-way down her person (since we are so abstemious\nof words that indicate the anatomy below the throat), and said:\n\n'Would X rays help?'\n\nBeing extremely clever that morning, of course I understood, and reviled\nher for eating an unnatural phenomenon. It was criminal; she might as\nwell have found the sea-serpent or the North Pole, and eaten it. But as\nusual she was artful, and led the conversation away to daffodils, which\nwere behaving in a manner nearly equal to that of the strawberry-plant.\nOne, indeed, was in bud (a thing incredible, but true), and I supposed\nshe had eaten that, too. That led us back to the strawberry again, which\nshe was not even sorry about, for she said it was far more interesting\nto be able to write to the _Field_ to say she had eaten a strawberry on\nFebruary 9 than that I should be able to say I had seen it. So I very\nkindly gave her my pen, and said:\n\n'Write quickly.'\n\nShe said:\n\n'Oh, but I am only a woman; I can't. They wouldn't put it in.'\n\n'I wish you hadn't put the strawberry in,' said I.\n\n'I think I shall wish that, too, before long,' said she.\n\nI only mention this in order to show the utter unreasonableness of my\nwife. If I want to write to the _Field_, and say there was a strawberry\nin my garden on February 9, she will allow me to say that though I did\nnot see it, she ate it. (She certainly would not have eaten it if I\n_had_ seen it.) But she will not write to say she ate it, like a true\nwoman. She says it does not matter, but added with a changed voice that\nshe was afraid it might. It did, for the fruitfulness of the season was\nnot so mellow as might have been wished.\n\n       *       *       *       *       *\n\nYes, once again spring has begun to stir in the fiery heart of the\nworld; once again the breath of Life blows the embers that seemed all\nwinter to be but grey and lifeless cinders, and from the centre the glow\nspreads, till that grey surface of ash is alive with flame again. And as\nthe flames shoot upwards they are like rockets, rising from over the\nwhole face of the world. At present they are but going upwards, those\nslender lines of flame, which are the sap that is rising through branch\nand leafless stem until it reaches the very ends of the twigs. Then\nthese rockets will burst in stars of leaf and opening flower, till the\nvast illumination is again complete. But in the warm soft February\nmorning, though I feel and know that this is so, I cannot help my\nthoughts going back to the other side of things. What of the\nillumination of last year? It is quenched dead, and even while the world\nis getting ready for the next one there still lie broadcast the ashes\nand fallen sticks of the last rocket-shower. However many more gladden\nthe world, even though to all infinity life was incessantly and\nbeautifully renewed, yet I cannot forgive the perishing of a single\nflower. I know well that the material is indestructible, that of life\nand the death of it is born fresh life, so that we are quite right to\nsay that life cannot be destroyed. But what of the individual rose, what\nof that one purple star of clematis that twinkled on the end of the stem\nI hold in my hand? Though it may be transformed, and will be\ntransformed, into a myriad other things, so that by its death it is\ntransfused into a hundred other flowers, and courses through the veins\nof life for ever, yet it, that individual object, will be seen no more.\nIts individuality is completely lost; it figures in new forms, not its\nown.\n\nIt is quite certain also that the same things happen to our bodies. The\ngrass grows thick on the graves of those we have loved, and the roots of\nthe roses penetrate deep. I saw once on the crumbling, sea-devoured East\nCoast of England the thing itself under my very eyes, which made it real\nto me in a way that nothing had ever done before. For a churchyard\nstood there on the very edge of the sandy cliff, and one night, with\nnoise of huge murmurous thunder, an acre of it slid down into the sea.\nNext morning I visited the place, and there, sticking out of the cliff,\nwere the bones of the dead that had been buried there. A ruin of roses\nthat had sprawled and trumpeted over the churchyard gate, which had been\nplucked in half by the fall, lay on the ground, and I wondered how the\ntrees had not slipped with the rest of the landslide, until I saw. Their\nroots had lain just where the fracture of the earth occurred, and in the\nexposed face of the new cliff I saw their anchorage. One was wrapped\nround a thigh-bone, another had made a network among ribs ... it was all\nhorrible and revolting. And that has happened to the million dead who\nhave lived and loved, whose limbs have been swift to move, who have\ndrawn rapturous, long breaths of this keen sea-scented air, whose eyes\nhave been bright and mouths eager when they met, lover and beloved. This\nis all--this ruin of red roses on the grass.\n\nThere is nothing in the world more certain than this, and one may as\nwell face it. Helen will die, and I shall die, and one of us will die\nfirst. And the other will sometimes see a grave with the grass green\nover it, and roses triumphant thereon. For we have settled most things\nat one time or another, she and I, and the manner of our funerals and\nwhat happens after has passed under discussion. We have decided\ndefinitely against cremation, because it seems such a waste of tissue,\nand we are both of us going to be properly buried, the one close to the\nother so that the same rose may bloom from us both. But she _will_ have\nroses and strawberries on her grave, so that the Sunday-school children\nmay pluck and eat them, while I, on the other hand, am going to be a\nspring-man, and have daffodils, for I feel no leaning, as I have said,\ntowards Sunday-schools. Here lies the difficulty: she wants a rich\nclayey soil for her roses and strawberries, and my daffodils will demand\nnot clay but sand. Also she is going to plant purple clematis by my\nhead, and clematis likes sand too. We have not yet perfectly decided\nwhere we are going to die, but it seems probable that the survivor will\nstay in the same place as the survived. But I want purple clematis,\nsince it was when I saw that that I knew somebody whom I had thought to\nbe a friend was false. Indeed, I have done all I could to forgive, but I\nthink a clematis that feeds on me may make it surer.\n\nOur funerals will shock the neighbourhood, I am afraid. I am going to\nhave the A flat Fugue and Prelude blared on the organ (it is time\nsomebody began to learn to play) at that distressing moment when my\ncoffin is wheeled out of the church, simply to show that I have enjoyed\nmyself enormously. Great Heaven! I should as soon think of having a dead\nmarch of whatever kind played over me as I should let them play the\nworks of Mr. Mendelssohn. I shall have had (whatever happens) an\nimmensely good time. It seems to me much fitter to return thanks for\nthat than to remind people that my poor body is dead, which they knew\nalready, or why did they come to my funeral service? As for requiems, I\nwill have none of them. Whatever happens, _I_, my body at least, cannot\npossibly lie quiet in my grave. The dear flowers planted there will see\nto that.\n\nOh, my God, my God, what unanswerable riddles you set us! Even this\nbody, and what happens to it, is so occupying a subject. I don't really\ncare what happens to mine: it may be set up in an anatomical museum if\nit will teach anybody anything; but Helen's.... Somehow, when I come out\nof the valley of the shadow, something of that must wait for her; or, if\nshe has gone through that passage first, I shall not know myself unless\nat the end of it, when the darkness lifts a little, I shall see grey\neyes looking at the procession of those passing over, and meeting mine,\nand saying somehow, 'I am here.' She must be there (is it not so?)\nwaiting on the eternal shore for me.\n\nThere she must be. I can't help what I believe; that is the one thing in\noneself which one can never change. And Dick will be there, and Margery\n... what a splendid day!\n\n       *       *       *       *       *\n\nThen the one horrible certainty descended on me again. In so few years\nwe shall all--our bodies, I mean, the appearance by which we recognize\neach other--not be our bodies at all, but part of the fibre of other\nliving things which are having their day, even as we have had ours. It\nis so now with Dick and Margery, so how shall I _know_ them? Are they to\nbe just voices in the air, presences that are felt? Is that all? Shall I\nnever see again that quiver on Margery's mouth, which means that a smile\nis ready to break from it? I don't want incorporeal presences. I want\nDick and his crooked nose, and Margery's smile....\n\nThen, on this warm February morning I must suppose that I went down into\nHell. Dead leaves and flowers, it was certain, were transformed into\nfresh living forms, the bones, too, and flesh of dead animals, and of\nmen and women, passed again into the great machine of life, and were\nserved up in new transformations, so that of the individual body nothing\nat all was left. That is bad enough; I shall never see Margery and Dick\nagain as I used to see them. Helen will pass, too, into other forms ...\nthat is bad enough. But this is infinitely worse. What of the individual\nsoul, the spirit that we love? Will that, too, as analogy grimly\ninsists, be put back again into the principle of eternal life from which\nit came, so that its identity, too, is lost, and lives but only as the\nautumn leaves of last year live in the verdure of the next spring? With\neverything else that happens; the bodies of those we love even, a cruel\nthing surely, but certainly true, are used up again to make fresh forms\nof life. Why should we suppose that God makes any exception in dealing\nwith the souls of men, the individuals? Every other form of life He uses\nand re-uses ... the world is but a lump of modelling clay, with which He\nbeguiles the leisure of eternity, making now one shape, then crushing it\nall up and making another.\n\nSo this is all that the promise of Eternal Life amounts to, that we\nshall pass back into the crucible, and issue forth again as bits of\nsomebody else! It seems to me a very mean affair; frankly, it seems a\nswindle. It is a poor trick to make us puny little creatures love one\nanother, and try to be kind, and console ourselves for the evil days and\nthe sorrows of the world with thoughts of the everlasting day that shall\ndawn for us all, if that everlasting day is nothing more than the day\nthat is here already; if the souls whom we have believed are at rest in\nsome ineffable peace and content, or, on the other hand, through further\nsuffering are getting nearer, ever nearer, to the perfection and flower\nof their being, have already passed into other forms of life, so that\nDante and Beatrice are themselves no longer (as we should call\n'themselves'), but have been infinitely divided into soldiers, sailors,\ntinkers, and tailors. In that sense they may be said to be alive still,\nbut it is a very paltry sense. _They_ (what we mistakenly call 'they')\nare as dead as if they had never been.\n\nIt is all very well to say that Dante is immortal by reason of his\ndeathless verse; that is all very well for us, but how is it for that\nfiery soul which is split up into a thousand other bodies? When he\nthought to open his eyes on the Mystical Rose as the dark waves of death\nslowly drew back from his emancipated spirit, it was all a dismal\nmistake. No Beatrice awaited him; she, too, is split into a million\nother forms of life. They were absorbed back into the central fire, and\na spark of Dante's soul went into this man, and another into that, so\nthat in this sense there is eternal life for him. But in no other; the\nDante which we mean was formed out of other lives, and into other lives\nhe went. The man is there no more, and there is no Beatrice. There will\nbe nothing of us either, unless you mean that at some future time I am\nalive because part of me has become perhaps a murderer, and another part\na politician, and another a housemaid, for all I know.\n\nThe February sun was warm; you might almost call it hot. A little wind\npregnant with spring moved through the bushes; the snowdrops, those pale\nheralds of the triumphant march of the new year, were thick in the grass\nwhere we had planted them, Helen and I, last autumn, so that they should\ngive us the earliest news of the returning tide of life. And to me this\nmorning they brought but bitter news, for they spoke not of the\nreturning of life, but of the thousand deaths which made them alive.\nThey pointed not forwards towards the glory of the many- summer,\nbut back to the innumerable decay of the autumn. And the quiet garden\nwhich I loved, the tiled mossy roof which I had called home, became the\nplace of death, even as last autumn death had called to me from it, and\nhad been seen by Legs, and had made the dog howl. Was it this that was\nhinted at by those dim forebodings which for months had never been\nabsent from me? Was the fear that crouched in the shadow ready to spring\ntaking form now? It seemed to me that the logic which had turned the\nworld to hell was irrefutable; I expected some shattering stroke that\nshould blot out sunshine and sensation from me for ever, proving that I\nand my logic were right. I had guessed the horrid secret of the world; I\nwas like a spy found with the plans of the enemy's fortress on me, and\nmust die, lest I should communicate them. I said that to myself; I said\n'Enemy's fortress,' meaning the world where I had loved and been loved.\n'Enemy,' mark you; I knew what I meant. The world was the enemy's\nfortress.\n\nAnd then, thank God--oh! thank God!--before that which was impending\nhappened, I said to myself that I was wrong. I did not at the moment see\nwhere I was wrong, but I knew that I must have made some gross and awful\nmistake. Things could not be as I had imagined them. And the moment I\nsaid that to myself the darkness lifted a little. It was all dark\nstill, but the quality of the darkness changed. And then, unbidden as a\ntune that suddenly rings in one's head, a few words made themselves\nrecollected. And they were, 'If I go down into hell, Thou art there\nalso.'\n\nAt that I caught a glimpse again of this dear garden and house, as I had\nseen and known them. I do not suppose that this blackness and loneliness\nof spirit which I have tried to indicate could have lasted more than a\nfew minutes, as measured in the world of time, but time has nothing to\ndo with the spirit. In a second, as computed by the unmeaning scale of\nhours and days, the soul may live a thousand lifetimes or die a thousand\ndeaths. Redemption may be wrought there in an infinitesimal fraction of\na moment, or in that same fraction a soul may damn itself. For it is not\nthe moment which is anything: it is the instantaneous choice which\ntherein sums up the infinite series of deeds which one has already done,\nand thoughts which one has harboured. And the message that leaps round\nthe world on electric wires is a sluggard to choice. My choice at this\nmoment was between the truth of what I had been elaborately thinking out\nand the truth of the words that rang in my head. There was reason on one\nside; there was just It on the other. And what was 'It'? Just that\nwhich, very faintly, but quite audibly, said that I had come near to\nblasphemy. There are many names for it: we all know its visitation,\nthough it is obscured sometimes because we encourage the Devil, who\ncomes to us all in many forms, and can take the most respectable\ndisguises, like those of intellect and mind. But perhaps the simplest\nname and the truest for It is the Grace of God.\n\nThen, in the same moment (I am lumbering in words, and trying to express\nwhat I know cannot be said), I saw that Helen was already half-way\nacross the grass, coming towards me. She held a telegraphic sheet in her\nhand, and there was in her face a gravity infinitely tender, and quite\nquiet, and quite normal. I had seen it there once before, when the news\ncame of her father's death, which was sudden.\n\n'Legs won't come down this afternoon,' she said gently. 'We have got to\ngo up to him.'\n\nAnd then she showed me the telegram.\n\n       *       *       *       *       *\n\nIt was not many hours before we knew all there was to be known. Legs had\nstarted to ride down from town, and turning into the King's Road from\nSloane Square his motor bicycle had skidded, and he had fallen under an\nomnibus. A wheel had passed over him.\n\nHe had a letter or two, which identified him, in his pockets, and he had\nbeen taken, since it was so near, back to the house in Sloane Street.\nWhen we got there he was still alive.\n\nHis room was at the back of the house, and we were allowed to go in at\nonce. He lay there, quite unconscious, and in no pain, for the only\nthing that could be done for him was to keep him like that. The\nbedclothes were not allowed to touch him, and a round wooden frame was\nunder them. There was no hope at all.\n\nHis bed ran out into the middle of the room, and Helen and I sat one on\neach side of it, while a little distance off was the doctor, who just\nwatched him. Sometimes he got up and looked at him, sometimes he softly\nleft the room, returning as quietly. And in those hours of waiting, for\na long time I was conscious of nothing except the trivial details of the\nroom itself. I suppose I had been there before--ah! yes, of course, I\nhad, when Legs had the influenza in the winter--but it was not familiar.\nYet it was just like what I should have expected Leg's room to be, and\nin a moment I found I knew it as well as I knew him. There was a pile of\nletters on the writing-table, a bag of golf-clubs in the corner, an\nenormous sponge on the washing-stand, and on the dressing-table a most\nelaborate shaving apparatus--a metal bowl, a little Etna for hot water,\na half-dozen razor blades in a neat case, with a sort of mowing-machine\nhandle. He had not packed them, since he was only going to be with us\nfor a couple of days, and he could never have used all those blades once\neach on that smooth chin....\n\nHe had been, as I remembered now, to a fancy-dress ball the night\nbefore, and his wardrobe, gaping open, showed the hose and ruffles of\nthe Elizabethan period, while hanging up by them was a small pointed\nbeard and a high head-top, with long and rather scanty brown hair. 'For\nthe point is,' Legs had said rather shrilly, 'everyone will say,\n\"Shakespeare, I presume?\" and I shall say, \"How dare you! I am Hall\nCaine!\" And if some people are a little cleverer and say, \"'The\nBondman,' I suppose?\" I shall say, \"You seem to have forgotten William\nShakespeare.\" Perhaps you don't think it funny. But then, you see, you\nare not going to the ball.'\n\n       *       *       *       *       *\n\nNo; we had not thought it very funny, and Legs had been rather ruffled.\nHe told us we had spoiled his pleasure, but if so, it must have very\nquickly become unspoiled again, for--it was only a week ago that he had\nconceived that idea--he spent a boisterously hilarious evening\nafterwards. But, how I wish we had not spoiled his pleasure even for\nthat moment! As if it mattered whether it was funny or not, so long as\nit amused him. Helen had said it was rather a cheap sort of joke.... And\njust then her eyes, too, saw the fancy dress hanging up in the wardrobe,\nand the moment afterwards she looked across to me. And then she left\nthe room for a little while. She, too, I am sure, had thought of that.\n\n       *       *       *       *       *\n\nI had a friend once who was killed in a railway accident. A year\nafterwards I was staying with his mother, and one evening, when we were\nalone, she began crying gently. 'Jim took his lunch with him to eat in\nthe train that day,' she said to me soon, and he had asked me to put him\nup an orange. But I forgot.'\n\nThat is the pathos of little things. Yes, you dear soul, weep a little\nover the forgotten orange, and let Helen weep a little because she said\nLeg's joke was cheap. And then let us think of the bigger things--the\nlove and the loving-kindness that have been ours, that bright, boyish\nspirit that made mirth in the home. Even now let us try to thank God for\nwhat has been. You know what Legs was to us--a sort of son, a sort of\nbrother.\n\nAll that afternoon we sat there, hearing London rumble distantly around\nus, and little stirrings and creakings came from different parts of the\nroom. Now the blind flapped, now a curtain sighed, or, as often happens\nin spring-time, a board of the flooring gave a little sharp rap, some\ninfinitesimal particle of sap still lingering in it, perhaps, and\nhearing the heralds of spring blowing their horns outside. Only from the\nbed there came no sound at all: he was still sunk deep in that sleep\nwhich the doctor hoped would join and be one with death. If he woke at\nall, there was a chance that he would suffer blinding, excruciating\npain. On the other hand, he might come to himself, just at the last\nmoment of all, when pain would be already passed.\n\nThe doctor was saying this in the hushed whisper with which we speak in\nthe chamber of death, though there may be no real reason why we should\nnot speak openly, when I heard a little stir from the bed, and, looking\nround, I saw that Leg's eyes were open, and that he was moving them this\nway and that, as if in search of something. Helen had seen, too, and\nnext moment she was by him. He recognized her, for there was welcome in\nhis eyes, and then, turning his head a little, he saw me. The doctor\nmeantime had moved to the head of the bed and looked at Leg's face very\nintently. Then he made a little sign to me that I should come up to the\nbed, and he himself went and stood by the window, looking out.\n\nAnd I understood.\n\nThen Legs spoke in his ordinary voice.\n\n'Wasn't it bad luck?' he said. My bicycle skidded, and the omnibus----\n\n'What is happening to me?' he asked quickly. 'Is it----'\n\nHelen laid her hand on his head.\n\n'Yes, my darling,' she said. 'But you are not afraid, are you?'\n\nFor a moment the pupils of his eyes contracted; then they grew quite\nnormal again.\n\n'No,' he said quickly. 'I've had an awfully good time. Oh, and it was a\ngreat success--Shakespeare, you know.'\n\nThen a shadow seemed to pass over his face and his eyelids fluttered.\n\n'Now? Is it coming now?' he said.\n\n'Yes, my darling,' said she again, and kissed him.\n\nLegs lay quite still for a moment with closed eyes. Then he quickly\nopened them again, and made as if he would raise his head.\n\n'Buck up, you two, won't you?' he said.\n\nFrom outside there came the dim roar of London, and little noises crept\nabout the room. But from the bed came no sound at all.\n\n       *       *       *       *       *\n\nTwo days afterwards we went down home again, arriving in the evening,\nand the body rested that night in his own room down here, to be taken\nnext day to the churchyard, which the sun blesses more than any other\nplace I have ever seen, and over which the grey Norman tower keeps\nwatch. His last charge to us had been to 'buck up,' and I do not know\nhow it was, but it seemed to us both as if he was still liking us to\n'buck up.' So, in so far as we found it possible, we did what Legs\nwished us to do.\n\nBut to-night he would have been here, making the third of a merry table,\nand when the servants had come in for the last time, bringing us coffee,\nit was not possible not to remember that, and Helen rose. And when she\nspoke, her voice trembled.\n\n'Is it very foolish of me?' she asked. 'And do you think Legs will mind?\nBut I feel as if I can't face to-morrow, unless I go and look at the\nplace where we shall put him. It is quite warm outside, Jack. Oh, let us\ngo out and look at it. It will seem more natural then. I think I shall\n\"buck up\" better if I see it first.'\n\nSo we went across the garden, and through the place of roses, and\nthrough the gate on the far side, and through the field which bounded\nthe churchyard. There was a great yellow moon just risen, and shadows\nwere sharp-cut, so that there was no doubt when we came to the place\nthat had been so newly dug. His uncle, Helen's father, lay there; the\ntwo graves were side by side.\n\nSo we sat there in silence for some time, very still, for a rat ran on\nto the mound of earth by the graveside, and sat there, smartening itself\nup, brushing its face and whiskers with nimble paws. The shadow of the\ntower swung just clear of the place, and sharp-cut in the light was that\noblong hole in the ground. There was nothing as yet to be said, for\nHelen was crying quietly to herself, and I could not stay those loving\ntears. Once she said to me: 'Oh, let us buck up!' But then she silently\nwept again.\n\nYou see, I know Helen. I knew that there was nothing of bitterness in\nher crying. Tears of that sort were not opposed to the bucking up. Legs\ndid not mean that he wanted us not to miss his dear companionship. He\nonly wanted us to stand up and be cheery, not be bitter or broken. But\nsince Helen felt she could face to-morrow better if she faced the scene\nof it, why, that was all right; it was bucking up.\n\nThen in a few little sentences we talked of the next day. There should\nbe the A flat Fugue--no funeral march--and we would have no funeral\nhymns, but just one Psalm, 'The Lord is my Shepherd,' and one hymn after\nall that had to be done was over; so then we would sing 'Adeste\nFideles,' Helen thought, for it is always Christmas since the first\nChristmas Day.\n\nHelen just moved as she sat there on the edge of his grave when we had\nsettled this as if to go home again, but----\n\nAnd then I told her all that I had thought three mornings ago--all the\ndoubts that merged into certainty, all the logical conclusions. Whether\nI then at that moment inclined more to the side of the Devil or of God\nI do not know, but in any case I told her all; and then she put her arms\nround me.\n\n'Yes, dear,' she said, 'but in hell He is there also. And we are all\nthere sometimes, and it is but the lowest step of the beautiful stair to\nheaven.'\n\nThe moon had swung behind the tower, and we sat in the darkness of its\nshadow.\n\n'It is all so simple,' she said. 'It all depends upon what you believe,\nnot what you think or what you reason about. Do you believe that we bury\nLegs to-morrow? Do you believe that he is dead, or that he has ceased to\nbe an individual? You may reason about it, and ask me, as you asked\nyourself, how you will recognize him if his body has become grass and\nflowers? I am quite content to say that I have no idea. You see, one\ndoesn't know all God's plans quite completely, and sometimes we are apt\nto think that if one doesn't know plans about a certain thing He hasn't\ngot one. We put our intelligence above His. That is a mistake.'\n\nAnd we sat in silence again; then Helen spoke asking me an extremely\nsimple question.\n\n'What does faith mean if you are right about it?' she said.\n\n'It means nothing. It is without meaning.'\n\n'And are you prepared to abide by that?'\n\nAgain there was silence. She sat a little apart from me, so that her\nquestions came from the darkness; they were put impersonally, so to\nspeak, not by Helen, but just by a voice.\n\n'Do you believe that Margery and Dick are nothing now except grass and\nflowers, and perhaps a little bit of the lives of other people? Do you\nreally believe it? And is Legs nothing now?'\n\nIt was quite still. We had come to a very sequestered corner of the\ngreat house of life to talk about these things. In front was the shadow\nof the grave, and over it now lay the shadow of the tower. Once from the\ngrave's side a few pebbles detached themselves and fell rattling to the\nbottom, and I had no answer to this. Three days ago I had asked myself\nthe same questions, and what I call my brain answered them; but now it\ngave no answer. Something, I suppose, had made it uncertain.\n\n'How can the wheel of an omnibus hurt Legs?' she asked. 'It can do no\nmore than hurt his body.'\n\nThen she came closer to me again.\n\n'And what does love mean?' she said.\n\n       *       *       *       *       *\n\nI think Legs must have enjoyed his funeral next day, because it was so\nextremely funny, and I think by this time that you know enough about him\nand Helen and me to allow us all to be amused at it. We had sent a note\nto our Vicar saying that we should like the A flat Prelude, and the\nPsalm, and the hymn which I have mentioned. He came in person, not to\nremonstrate, but to put on to us the correcter attitude. Death was a\nsolemn occasion. There was none so solemn, and the Hymns Ancient and\nModern provided some very suitable verses to be sung--'Now the\nlabourer's task is o'er,' for instance. (Legs a labourer, who was the\nmost gorgeous player at life that has ever been seen!) Besides, surely a\nChristmas hymn was out of place, when it would be Ash Wednesday in no\ntime. I said feebly that a Christmas hymn was surely always in place;\nbut dear Mr. Eversley looked pained, and Helen at once yielded. She was\nsure that the 'labourer's task' was most suitable.\n\nThen about the Psalm. There were two Psalms already provided for the\nBurial Service, and surely '\"The Lord is my Shepherd\" struck a different\nnote.' So said our Vicar. That was undeniable. And when should we sing\nthat Psalm? Then Helen was firm, and said that we thought we should go\nback into church at the end of the service, and--well, just sing it. It\nwas rather good to end with. But Mr. Eversley looked even more pained\nthan before. He had never heard of such a thing being done. That point\nwas left undecided for the moment, for there was clearly something even\nmore crucial to come.\n\nIt came.\n\nEver since the organist had heard of Legs' death he had been most\ndiligent at Chopin's Funeral March, of which he had of his own\ninitiative bought a copy in order to be able to perform it. The organist\nin question, who was also the schoolmaster, had had a sort of distant\nadoration for Legs ever since a year ago he had seen him drive a\ngolf-ball two hundred and sixty measured yards. Since then Legs had\nplayed with him once or twice, giving him enormous odds, and the distant\nadoration had ripened into a nearer one. 'He was such a pleasant young\ngentleman,' was the upshot of it. And the dear man had bought Chopin's\nFuneral March, since he wanted to play something 'more uncommon' than\nthe Dead March in 'Saul'!\n\nHere Helen and I were completely at one. There should be no A flat\nPreludes; it was to be Chopin's Funeral March.\n\nThere remained the question of the Twenty-third Psalm. Oh yes, it would\nstrike a different note, that was quite true; so there would be no going\nback into church, but we should have Chopin's Funeral March and 'Now the\nlabourer's task is o'er.'\n\nThe Vicar did not exactly beam when these things were settled, but he\nwas visibly relieved. He shook hands with us both, and said:\n\n'Terribly sudden, terribly sudden. At two precisely.'\n\n(Oh, Legs, how you would have enjoyed that! We did, too, for you told us\nto buck up. And it was so funny, after all we had planned!)\n\nThe Vicar's call had been made quite early, and it was scarcely twelve\nwhen he went away; but to us both it seemed as if Legs had been waiting\nsomewhere upstairs till he went in order to laugh over it with us. It\nwas as if he had been waiting on the landing, fresh from his bath, with\njust a dressing-gown on, so that he could not appear when other people\nwere there, but might come down barefooted when they had gone. He must\nhave been so amused at it. How he would skip into the drawing-room,\nafraid of prowling housemaids, to find us alone, and say, 'Sorry I\nhaven't got much on, but I had to come down after my bath.' Yes, after\nhis bath. It was so that it seemed to us. That wholesome spirit had been\nwashed, we thought, by what is called death. It was fresher, more\njubilant than ever. And on the Vicar's departure down he came to join us\nagain. I have no other words for it.\n\n       *       *       *       *       *\n\nThere was more to come, for hardly had the Vicar gone when it was\nannounced to us that Mr. Holmes had called, and might he see one of us\nfor a moment only. I felt that Legs was cornered now. He would have to\nstop here, hide behind the piano or something. I hoped he would behave\nhimself, and not make me laugh. So Mr. Holmes came in.\n\nI never saw anybody so wonderfully attired. He was all in black,\nincluding his gloves and his stick, and above his small neat buttoned\nboots when he sat down I saw a black sock. That may only have been\naccidental, but no accident would account for the fact that his cuffs\nhad a neat black border about half an inch wide. I wondered if he had\nblacked himself all over like the enthusiastic impersonator of Othello.\n\nHe had ventured to intrude on our grief, but only for a moment. Here\nHelen dropped her handkerchief, and they both bent down to pick it up\nand knocked their heads together, and I almost thought I heard a little\nstifled gasp from behind the piano. But Mr. Holmes had received no\nnotice of the funeral, which he had understood was to be to-day, and did\nnot know if we wished it to be quite private; if not, he would esteem it\na privilege to be allowed to pay his last respects. And here little Mr.\nHolmes gave a great gulp, and could not get on.\n\n'I did like him so much,' he said, after a moment. 'Two. Thank you, I\ncan let myself out!'\n\nAnd he walked away on tiptoe, as if it was most important not to make a\nnoise.\n\n       *       *       *       *       *\n\nIt was one of those sparkling February days, sunny and windless, and the\nair was full of the chirruping of birds. There was a moment's pause at\nthe gate of the churchyard, a moment's silence. Inside the church the\norgan ceased; then came great simple words:\n\n'I am the Resurrection and the Life.'\n\n\n\n\nMARCH\n\n\nHelen and I have a failing, though you may not have thought that such a\nthing was possible. It is a foolish weakness for old bits of rubbish. We\ncan neither of us without anguish and unutterable rendings bear to throw\nold and useless things away. The weakness has to be got over sometimes,\nbut we keep putting the work of destruction off, just as one puts off a\nvisit to the dentist, with the result that when it comes to pass we find\nthat it would have been far better to have done it long ago. However, if\nwe did not occasionally tear things up, and throw things away, the house\nwould become uninhabitable, so this morning we vowed to each other to\nspend the hours till lunch in the work of destruction. Our rubbish\ncollects chiefly in the room that is called mine, where she has a\nknee-hole table with nine drawers. She opened these one after the\nother. They were all full, and despair seized her.\n\n'I can't,' she said. 'Here are nine drawers all quite full of heart's\nblood. O Jack, look!'\n\nAnd she brought across to me a photograph I had taken of Legs jumping\nthe lawn-tennis net. He was sitting in the air apparently in an easy\nattitude. One knee seemed crossed over the other, and his mouth was wide\nopen.\n\n'It will be harder than ever this year,' she said, half to herself. 'And\nthere are nine drawers full!'\n\n'Circumscribe the drops of heart's blood as they come,' said I. 'Don't\nthink there are nine drawers full. Only keep thinking of the particular\nthing that has to be kept or thrown away.'\n\n'Oh, but it's only the fact that there are nine drawers full that makes\nit possible to throw anything away at all,' said she.\n\n'Hush, woman!' said I.\n\nPersonally, I am extremely methodical over the work of destruction. I\nclear a table and dump upon it a pile of heart's blood. This I sort\ninto three heaps, one of which is for destruction, one for preservation,\nand one for further consideration. I proceeded to do so now.\n\nThere were many pieces of string. Throughout the year I keep pieces of\nstring, because I know I shall use them. As a matter of fact, when I\nwant a piece of string I cut it off Helen's ball, and never use any of\nthe bits that I have saved, because I don't know where they are, and\nthey would prove to be the wrong length if I did. So on the day of\ndestruction I consign them to the dust-bin, and begin to collect again\nimmediately. Then there was a pill-box full of soft yellow powder, which\nLegs and I had collected from the little cedar-cones at some house where\nwe were staying in the autumn. That I put on to the heap of destruction,\nbut transferred it to the heap of consideration. Then there were a dozen\nlittle bits of verd-antique which I had picked up years ago on the beach\nat Capri, and which I had periodically tried to throw away. But I never\ncould manage it, and this morning, knowing it was useless to strive\nagainst the irresistible, I made no attempt whatever to steel myself to\ntheir destruction, but put them at once into the pile that was\npredestined unto life. There was a chunk of amber that I had picked up\nat Cromer, equally imperishable; yards and yards of indiarubber tape\nthat is the filling of a rubber-cored golf-ball; a small bottle with a\nglass stopper, clearly impossible to throw away, since it might come in\nuseful any day, and how foolish I should feel if this afternoon I wanted\na bottle with a glass stopper, and had to send into the town for one,\nwhereas, if I had been less iconoclastic, I might have airily produced\nthe exact thing needed out of the left-hand top drawer. Then came a\nlittle tin box full of pink powder, which I concluded was rouge. This\nwas puzzling.\n\n'When did I use rouge?' I asked Helen.\n\n'I don't know. Was it Legs', do you think, when he acted the Red Queen\nlast year?'\n\nNo, I couldn't throw that away. The Red Queen had been a piece of\ngenius. And next came the telegram from him to me saying that he had\npassed into the Foreign Office. Then there was a vile caricature of\nmyself at the top of my so-called swing at golf--quite unrecognizable,\nI assure you, but....\n\nThen came a mass of letters, receipted bills, and accounts rendered.\nAccounts rendered always fill me with suspicion, and I have to hunt\namong unpaid bills to find the items of the account rendered, as I feel\na moral certainty that this is an attempt to defraud me. But they are\ninvariably correct. But these and the receipted bills, which had to be\ndocketed and tied up together in a bundle, took time. Probably, however,\nI could tie them up with one of those many pieces of string which I had\nso diligently collected. By a rare and happy chance I found one that\nwould do exactly, and tied them up with a beautiful hard knot, and put\nthem on the predestination heap. A moment afterwards I found several\nmore to join the same packet, split my nail over trying to untie my\nbeautiful knot, and had to go upstairs for nail-scissors to cut it\nsmooth, and brought them down to cut the knot. No other piece of string\nin my collection would do, and so I cut a piece off Helen's ball, for\nshe had left the room for the moment.\n\nThen I came upon a large quantity of boxes of fusees, all partly empty.\nHow it happens is this: I go to play golf on a windy day, and, of\ncourse, have to buy at the club-house a box of fusees. These, on my\nreturn, or what remains of them, I methodically put in a drawer on\nreaching home. By an oversight I forget to take them out again when I\nplay next day, and so buy another box, which I similarly place in a\ndrawer. And if you play golf four or five times a week on these downs,\nwhere there is almost always a high wind, it follows that in the course\nof a year the amount of partly filled boxes of fusees which you collect\nabout you is nothing short of prodigious. I did not know how great a\nsupporter I was of home industries.\n\nMy methodical mind saw at once how these had to be treated. Of course,\nthrowing them all away was out of the question, and the right thing to\ndo was to produce out of every dozen of partly filled boxes some eight\nor nine completely full. This plan I began to put into practice at once.\n\nIt was necessary, of course, to find how many matches a full fusee-box\ncontained, but they are awkward to pack, and some seemed to hold ten and\nothers only seven; so when Helen came back, the table was covered, among\nother things, with fusees. So I waved my arms violently, and said: 'You\nshall not!' This was because the female nose, and the male nose if it is\nunaccustomed to tobacco-smoke, likes, positively likes, the smell of\nfusees; but to anyone who smokes tobacco the smell of them is, for some\nreason, perfectly nauseating, and that is why we only use them in the\nopen air.\n\nThen Helen's mean nature asserted itself. She said, 'Oh, I forgot you\ndon't like the smell,' and soon after (not at once, mark you) called my\nattention to some non-existent object of horticultural interest out of\nthe window. I turned, and in a moment she had lit a fusee, and\npositively inhaled the sickening perfume of it. I only wished she had\ninhaled it all.\n\nThe upshot was that we took a turn on the lawn, while the room with open\ndoor and windows recovered from its degrading odour.\n\n'How were you getting on?' she asked.\n\n'Not very well. I decided to destroy some string. I nearly destroyed a\npill-box with some cedar-flower dust in it. But I reserved that. At\nleast, I think I did.'\n\n'Why?'\n\n'Legs and I collected it, and I know Legs wouldn't have thrown it away,\nso I can't.'\n\nHelen was silent a moment; then,\n\n'Do you miss Legs very much?' she asked. 'His bodily presence, I mean,\nof course.'\n\n'Of course I do, just as you do. I miss him all the time. Oh, he is in\nthe room, and he laughs at us, or with us. I know that.'\n\n'Then what do you miss?' she asked.\n\n'The young body about the house.'\n\nThen Helen said: 'Oh, you darling!'\n\nThat sort of remark is always extremely pleasant, but I had no notion of\nher artfulness. I am glad to say that she has often said it before, so\nthat it was not particularly stupid of me not to guess that it meant\nanything especial. And with her artfulness she changed the subject to\nthat which I happened to be thinking about, thus making no transition.\n\n'I gave up,' she said. 'I found all my things were so connected with\nLegs that I couldn't destroy them. It is just what you said. We want to\nkeep the young thing in the house, since we are getting old--yes, it's\nno use saying \"Pouf!\"--and I can't destroy anything connected with him.\nSo shall we move our rubbish straight into Legs' room, and make a sort\nof young museum? Then, when we feel particularly middle-aged, we can go\nup there and sit among the young things. If we don't do that, we must\nclear out his room as well, and I can't see how we can. There are rough\ncopies of letters to that dreadful Charlotte; there is a letter in his\nhandwriting, there on his table, beginning----'\n\n'Beginning \"You're a damned fool!\"' said I, '\"but I don't intend to\nquarrel with you.\" Did you mean that one?'\n\n'Then you have been there, too?' she said.\n\n'Why, of course, every day. I go when you attend to household affairs\nafter breakfast; you go when you say you are going to bed. Didn't you\nknow?'\n\n'Certainly I did, but I thought you didn't know that I went there,' she\nsaid.\n\n'Ditto,' said I.\n\nThere was a huge rushing wind out of the south-west, and we stood a\nlittle while inhaling the boisterousness of it. All spring was in it,\nall the renewal of life.\n\n'How Legs is laughing at us!' she said.\n\n'I don't care. Let's have the museum of young things. Let's put there\nall the things we can't throw away. Oh, Helen, there are photographs,\ntoo! There is one of him in his last half at Eton.... There is one of\nyou and me when the Canadian canoe sank gently, and as we stood dripping\non the shore he photographed us. And I photographed him and you when you\nsaid you would skate a rocking-turn together, and fell down. Heart's\nblood, heart's blood! There ought to be a law which makes it a penal\noffence to keep photographs,'\n\nI suppose I had got excited, for Helen took my arm and said:\n\n'There, there!'\n\nBut even that did not do.\n\n'Oh, the pity of it,' I cried--'the pity of it! Why didn't he take a\ntrain to come down? Why didn't that omnibus pull up? He was ours, and\nhe would have married, and still been ours, and there would have been\nyoung things about the house again.'\n\nI suppose I had torn away from her, for now we were apart, facing each\nother, at the end of this; and she smiled so quietly, so serenely.\n\n'Do you think that I don't feel that, too?' she asked. 'Can't you see\nthat the wife who is mother of nothing must feel it more than the\nhusband who is father of nothing? Besides, you make your books--you are\nfather to them. What do I do? I order dinner.'\n\nAnd yet--it seems to me so strange now--I did not see. There was\nbitterness in her words, but all I thought was that there was no\nbitterness in her voice, or her face, or her smile. I did not quite\nunderstand that, I remember, but Helen has told me since that she did\nnot mean me to. She wanted--well, her plan evolves itself.\n\nAnd then she took my arm again.\n\n'It is nearly a month since dear Legs went away,' she said, 'since we\nhave actually heard and seen him. The last we heard was that he wanted\nus to buck up. Do you know, I think we have bucked up. But we have been\ndoing that singly; we have somehow lived rather apart, dear. Surely it\nis better to buck up together. I think the idea of a young museum is a\nvery good one. Let us put all the things we can't throw away into his\nroom. We have never used the room before, because Legs might always rush\ndown and want a bed; and so let us keep it like that. We might call it\nthe nursery.'\n\nAnd so the young museum was started. Helen had all manner of tender\ntrifles for it, all connected with Legs. She had all sorts of things I\nhad known nothing of: little baby garments, Legs' bottle, some baby\nsocks. Then there were child things as well: 'Alice in Wonderland,' the\ndepressing Swiss family called Robinson, a far better Robinson called\nCrusoe.\n\n       *       *       *       *       *\n\nAnd thus the nursery grew. 'Treasure Island' went there; a\nrocking-horse, which I remembered of old days, was brought down from an\nattic. Oh, how well, when I saw him again, I remembered him! He had a\ngreen base, nicely curved, on which he pranced to and fro, and my foot\nhad once been under it when he pranced, so that I lost a toenail, and\nwas rewarded with sixpence for stopping crying. He had a hollow\ninterior, the only communication with which were the holes of the\npommels, and on another dreadful day my sister had dropped a\nthree-penny-bit into one of them, with some idea of making a bank. A\nbank it was, but the capital was irrecoverable. The coin was still\nthere, for now I took up the whole horse with ease, that steed which had\nso often carried me, and heard a faint chink from his stomach. He had a\nwild eye, too, and flaming red nostrils, and the paint smelt just the\nsame as ever. And Helen produced a Noah's ark, in which the paint was of\nfamiliar odour, but different, and there was Ham without a stand, and\nMrs. Noah in a neat brown ulster, and Noah with a beard, and one good\nfoot, but the other was a pin. Elephants were there with pink trunks (I\nnever could understand why), and enormous ducks with pink bills (which\nnow threw a light on the colour of the elephants' trunks, since I\nsuppose that a brush full of pink was indiscriminately bestowed), and\nsmall spotted tigers, and nameless beasts which we called lynxes,\nchiefly because we did not know what they were, and did not know what\nlynxes were, so they were probably the ones. The ark itself had Gothic\nwindows, and a mean white bird, with a piece of asparagus in its mouth,\npainted on the roof, probably indicated the dove and the leaf.\n\nWe must have spent two days over the nursery, and during those days we\nconcentrated there all the young things of the house, and when it was\nfinished it was a motley room. There were photographs of Legs\neverywhere; all his papers were kept; everything that had any connection\nwith Legs and with youth was crammed into it. And when it was finished\nwe found that we sat there together, instead of paying secret visits to\nthe room, and we played at Noah's ark, sitting on the carpet, and played\nat soldiers, clearing a low table which had been Helen's nursery-table\n(for you cannot play soldiers on the floor, since they stagger on a\ncarpet), and peas from pea-shooters sent whole rows of Grenadiers down\nlike ninepins. But we could neither of us ride the rocking-horse, so\ninstead we tilted him backwards and forwards, and pretended he was\ncharging the foe.\n\nOf course, all reasonably-minded readers will say we were two absurd\npeople. We both of us disagree altogether. For you have to judge of any\nproceedings by its effects, and the effect in this case was that Legs'\ninjunction that we should 'buck up' became a habit. That inimitable\nyouth which Legs gave the home, he, his bodily presence, had gone. But\nsomehow the atmosphere was recaptured. We played at youth, at childhood,\ntill it became real again. For a household without youth in it is a dead\nhousehold; a puppy or a kitten may supply it, or an old man of eighty\nmay supply it. But youth of some kind must be part of one's environment.\nElse the world withers.\n\nAnother thing has happened to me personally. I have said that at the\nbeginning of the year I looked forward into the future through two\ntransparencies, one sunlit, the other dark. But now the dark one (I can\nexpress it in no other way) had been withdrawn. Dear Legs' death was not\nquite identical with it, for it was not withdrawn then. But during the\nmonth that followed it gradually melted away. I can trace just two\ncauses for it.\n\nThe first was this: In ineptitude of spirit I had reasoned to myself\nthat the death of the body logically implied the merging of the life\ninto the one central life. But after his death Legs became to my spirit\nmore individual than ever. And the second cause was this establishment\nof the nursery. Though youth might have passed for oneself, it still\nlived. One was wrong, too (at least I was), in thinking it had passed\nfrom oneself. Else how did I feel so singularly annoyed when Helen shot\ndown with a wet pea a whole regiment of my Life Guards? I was annoyed; I\nam still. It was a perfect fluke that the Colonel on horseback fell in\nsuch a way that he more than decimated his own regiment. And I am sure\nHelen shook the table, else why should the Brigadier-General, posted in\nthe extreme rear, have fallen off the table altogether? She won.\n\nMeantime in this first week of March the winds were roaring out of the\nsouth-west, and for a while, days together sometimes, squalls which the\nValkyrie maidens might have bridled to make steeds for their swift\ngoing came in unbroken procession from the Atlantic. Helen is a lover of\nthe sea, and these gales coming out of the waste of waters touch\nsomething within her as mysterious as the sixth sense of animals, who\nfeel and are excited by things that the five-sensed mortal is unaware\nof. To-day, however, was quiet and calm, and we stormed the steep ascent\nof the downs till we stood on the highest point of the Beacon, which\nlooks down on all other land towards the south-west, so that the river\nof wind that flows from the Atlantic comes here unbreathed and untamed\nby traverse of other country, and you get it fresh and salt as it was\nwhen it left the ocean.\n\nIn that interval of quiet weather there was nothing to be perceived by\nthe ordinary sense, but she sniffed the air like a filly at grass.\n\n'Wind is coming,' she said, 'the great wind from the sea. I don't care\nwhether your little barometer has gone up or not; what does it know of\nthe winds? We shall be at home before it comes, but I will tell you\nthen, as we sit close to the fire, what is happening in the big\nplaces.'\n\nShe was quite right; though the silly barometer had gone up, we were but\nhalf through dinner when the wind, which had been no more than a breeze\nall afternoon, struck the house as suddenly as a blow. The wood fire on\nthe hearth gave a little puff of smoke into the room, and then, thinking\nbetter, suddenly sparkled as if with frost, as the passage of the air\nabove the chimney drew it up. At that Helen's eyes were alight. She ate\nno more, but sat with her elbows on the table, while I, who have not the\nsixth sense, went gravely through mutton and anchovies on toast and an\norange. Then they brought in coffee, and she shook her head to that.\nMeantime that first warning of the wind had been justified; a Niagara of\nair poured over us, screaming and hooting, and making a mad orchestra of\nsound. At times it ceased altogether--the long pause of the\nconductor--and then, before one heard the wind at all, a tattoo of the\ndrums of rain, sounded on the window-pane. Then, heralded by those\ndrums, the whole mad orchestra burst into a great _tutti_ of screaming,\nhooting, sobbing. So much I could hear, but Helen was _of_ it somehow.\nSomething secret and sensitive within her vibrated to the uproar.\n\nI have seen her in the grip of the wind, as she expresses it, perhaps\nhalf a dozen times, and it always makes me vaguely uneasy. It is no less\nthan a possession, and yet I can think of no one whom I would have\nimagined less liable to such a thing. I can imagine her surrounded by\nthe terrors of fire or shipwreck, or any catastrophe that overthrows the\nreason, and makes men mere panic-stricken maniacs, keeping absolutely\ncalm, and infecting others by her self-possession. But now and then the\nwind takes possession of her, and she becomes like the Pythian\nprophetess.\n\n'Oh, to be alone with the sea and the gale to-night!' she said. 'Jack,\nwhat splendid things are happening in the great empty places of the\nworld! This has been brewing out on the Atlantic for a couple of days by\nnow, and there are thousands of miles of great white-headed waves rising\nand falling in the darkness, and calling to each other, and dancing\ntogether. Up above them, as in the gallery of the ball-room, is the\ngreat mad band of which we hear a little in our stuffy house, and it\nwill play to them all night and all to-morrow, and the waves will dance\nwithout ceasing, growing bigger as they dance, like some nightmare. Oh,\nyou can imagine nothing! But I see so clearly Mr. and Mrs. Wave and all\ntheir family dancing, dancing, all young, though white-headed, and\ngrowing bigger as they dance. They are cannibals, too, and a big wave\nwill eat up a little one, which makes it bigger yet. The wind loves to\nsee that. He gives a great blare of trumpets when he sees a cannibal\nwave. Oh, it must have happened this moment! That scream meant, \"Well\ndone, wave! That was a big one you swallowed!\"\n\n'Sometimes they see a ship coming along, and they love playing with\nships, because all proper ships like being out in the Atlantic\nball-room, and the waves crowd towards it, seeing which can lift it\nhighest. Whiz! Can't you hear the screw racing, as the wave that lifted\nthe stern runs away from under it? How the masts strike right and left\nacross a thousand stars, for the sky is quite clear! The winds have\nturned out the clouds as you turn out the chairs and tables from a room\nwhere you dance.'\n\nWe had gone up to Legs' room after dinner, and as she talked she went\nquickly from place to place, now pausing for a moment to look at a\nphotograph, now putting coal on the fire, or drawing aside the curtain\nto look into the night.\n\n'Oh, there is the eternal youth of the world,' she said--'the song of\nthe winds and the dance of the waves. I think all the souls of the\nlittle babies that are born come to land in the blowing from the sea. It\nis by that that vitality burns higher, and the fruitfulness of the world\nis renewed. Millions of blossoms of life are rushing over the land\nto-night, ready to drop into lonely homes----'\n\n'Ah, don't, don't,' I said. 'Helen, come and sit down and be quiet.'\n\nShe paused for a moment opposite me, looking at me with her wonderful\nshining eyes.\n\n'Not I, not I,' she said.\n\nShe still paused, still looking at me, still waiting for me to join her,\nas it were. And in that pause a sudden faint far-away light broke on\nme. She had said words which must have awoke in her, even as they awoke\nin me, the most keen and poignant sorrow that can touch those who love\neach other, and yet she was still smiling, and her eyes shone.\n\nI got up. Something of that huge joy that transfigured her was wrapping\nme round also. The thrill, the rapture in which she was enveloped, began\nto encompass us.\n\n'What do you mean?' I asked.\n\n'It is for you to tell me,' she said. 'It must be done that way.'\n\n'You said \"ready to drop into lonely homes,\"' I said.\n\n'\"So that they are filled with laughter,\"' said she.\n\nThen I knew.\n\n'It is here,' I said--'the nursery.'\n\nAnd at that the excitement, the exultation slowly passed from the face\nof my beloved, for there was no room there for more than motherhood.\nThough the wind still bugled and trumpeted outside, she heard it no\nmore; the wildness of the dancing waves, grey-headed, growing waves,\npassed by outside her.... The blossom ready to drop filled her heart\nwith the tenderness of the infinite deep love of the mother that shall\nbe.\n\nShe sat there on the floor at my feet, with her arms round my knees and\nher head pillowed there.\n\n'I have got to confess, too,' she said, 'though I am not ashamed of my\nconfession. But don't allow yourself to be hurt, Jack. Just hold on for\na minute without being hurt, and you will find that you are not. Now I\nshall hide my face, and speak to you like that. I have known it quite a\nlong time: before Legs died I knew it.'\n\nWell, I had to hold on for a minute or two, and not be hurt. If you\nthink it over, you, will agree it was rather a hard task that I had been\nset. On the other hand, about big things, about things that really\nmatter, you must take my word for it that Helen is never wrong. But I\nhad not been forbidden to ask a question.\n\n'Then why did you not tell me?' I said.\n\nHer head with the sunlit billows just stirred a moment, but she did not\nlook up, but spoke with a hidden face.\n\n'Because through all these weeks, my darling, you have been struggling\nagainst some bitterness of soul. You have made light of it to me, but I\nhad to be quite sure it had gone from you before I told you this. I know\nwhat it was--it was the doubts you talked about to me when we sat one\nnight at the edge of dear Legs' grave--when it was dug, but empty. And I\nhad to be quite sure it had all passed from you before I told you this.\nI have not been sure till now, and--and I wanted you so much to guess.\nYou nearly guessed, I felt, when we arranged this heavenly nursery.'\n\nThen again there was silence, and I think I never knew till then how\ndesperately difficult it is to be honest with oneself. It is so much\neasier to be honest with other people. At the first glance I told myself\nI had got over the bitterness and blindness of which she had spoken when\nwe talked together over Legs' grave, but gradually I became aware that I\nhad not. Somewhere deep down, so that while the days passed it concealed\nitself from me, that bitterness had still been there. In this book,\nwhich has tried to be honest, you will, I dare say, find no trace of it\nsince that night, but I had not probed deep enough. It had been there,\nand I think the days when we arranged the nursery finally expelled it.\nTo-night, at least, I believed it was gone, and since Helen believed so,\ntoo, perhaps we are right about it. She, the witch, the diviner, had\nknown me so much better than I had known myself all along.\n\nAll this took time, for the processes of honesty with me are slow. But\nthere is no difficulty about the matter, perhaps, if the head you love\nbest in all the world is pillowed on your knee. That is a stimulant, one\nmust imagine. So at last I said:\n\n'Yes, it's done.'\n\nShe came closer yet, and, like Mr. Holmes, we talked below our breath,\nin whispers, as if afraid of disturbing this great joy that had come\nfloating down on us, borne on the sea-spray, borne on the wind-tide,\nborne as you will, so that only it came here.\n\nThen, very soon after, she went to bed, and I was left sitting in the\nnursery, with its new significance. Yet it was not quite new. I had, as\nHelen said, 'half guessed before,' and I but wondered, now I knew, how\nmy imagination had halted half-way, and had not clearly seen the star on\nwhich Helen's eyes were fixed. Yet who would have known? She had been so\nfull of art in her wording; even that master-word she had used,\n'nursery,' seemed but to have slipped in, and I had thought she meant\nonly--as, indeed, she had said--that it was to be the room of young\nthings, where she should sit when the shadow of childlessness was chill,\nand with the aid of the memories of youth and play keep the mists of\nmiddle age from closing round us, and the frosts of old from settling\ntoo stiffly on the later years of our travel. The room was to be but a\npalliative or a tonic, as you will, a consolation for the things that\nwere not to be for us, and now it showed another face. It was not the\npast of which it spoke, but the future.\n\n       *       *       *       *       *\n\nI suppose I sat long over the embers of the fire, but these were hours\nthat had escaped from the hand of Time, and were not to be computed by\nhis scale. Sometimes I threw a log into the open hearth of the fireplace\n(ah, but that open hearth must be altered now; it would never do in the\nnursery), and sometimes I plied an industrious pair of bellows, but for\nthe most part I sat idle, looking into the fiery heart of the blaze; for\nthe news that Helen had made me guess was at first unrealizable. Though\nI knew it to be true, I had to absorb, digest it, since a great joy is\nas stunning a thing as the stroke of sorrow. And gradually, as gradually\nas the workings of the process of beauty, I began to feel, and not only\nto know, the name of the room where I sat. It was the nursery.\n\n       *       *       *       *       *\n\nBut Helen was wrong about one thing. She had said that the wind would\nplay to the dancing of the waves all night and all next day, but before\nI went to bed that wild orchestra of the storm had ceased. Its work was\ndone for us. It had blown the bud of the blossom of life into the house\nthat so longed for it.\n\n       *       *       *       *       *\n\nIt is strange how quickly the events of life become part of one. Next\nmorning I woke in full possession of the new knowledge. There was no\nquestion or uncertainty as to what that was which made a rapture of\nwaking. And with the same suddenness all real knowledge of what life had\nbeen before I knew this had passed from me. I could no longer in the\nleast realize what I had felt like before the moment came when Helen had\nmade me guess. Though that moment was so few hours away, yet I could no\nmore conceive existence without it than one can form any mental picture\nof what life would be without the gift of sight or hearing. It is not\nthat any huge event destroys all that went before it, but it so stains\nback through the turned pages of the past that they are all  and\nsuffused with it.\n\nHow the blackbirds and thrushes sang on that March morning! I had awoke\nbefore dawn to hear the early tuning-up going on in the bushes, and\nbefore long, since I was too happy to sleep, I got up, dressed quietly,\nand went out. The tuning-up was just over, and the birds were all busy\nwith breakfast, for you must know, as soon as they wake, they get in\nsinging-trim for the day before their bright-eyed quest, listening, with\nhead cocked as they scuttle over the lawn, for the sound of a worm\nmoving. They are so close to the ground themselves that they can\nlocalize this to within a fraction of an inch, and then in goes the\nspear-like beak, and the poor thing is dragged out of the soft,\ndew-drenched earth. They are not quite tidy eaters, these dear minstrels\nof the garden, for the point is to get your breakfast inside you beyond\nrecall, with the least possible delay. Swallow, gulp, swallow, and the\nthing is done. Then you give one long flute-like note of satisfaction,\nand listen again for the second course. But one cannot exactly say that\nthey have bad manners at table, for the extreme sensibleness of the plan\nexcludes all other considerations. Also, bad manners at table\nirresistibly suggest greediness, and no bird is ever greedy. They have\nexcellent appetites, and when they have had enough they stop eating, and\ninstantly begin to sing.\n\nIt was just at the end of birds' breakfast that I got out--that is to\nsay, it still wanted some minutes to sunrise. The lawn was all\ngossamer-webbed and shimmering with dew, as if some thin layer of\nmoonstone or transparent pearl had been veneered over emerald, and I\nfelt it almost a vandalism to walk over it, removing with my clumsy\nfeet whole patches of thin inimitable jewellery. The three-hour gale of\nthe night before had vanished to give place to a morning of halcyon\ncalm, and I augured one of those rare and exquisite days which March\nsometimes gives us--days of warm windlessness and the promise of spring.\nStraight in front of me rose the Beacon, still submerged in clear dark\nshadow, but high in the heavens above dawn had come, for it made a\ngolden fleece--one such as never Jason handled--of the little cirrhus\nclouds that the gale had forgotten to sweep away. Dawn would soon strike\nthe Beacon, too, but before that I hoped to stand on its top, and see\nthe huge embrace of day and night, the melting and absorption of\ndarkness into light. Even the river, with its waving water-weeds and\naqueous crystal, did not detain me, and I gave but ten minutes to the\nascent, for I wanted to welcome the dawn from a high place, to stand on\nthe roof of the hills to greet it.\n\nSlowly dawn descended from the sky, quivering and palpitating with\nlight. The great golden flood came nearer and nearer the earth, which as\nyet caught but the reflection from the radiant heavens. It hung a moment\nhovering, the bright-winged iridescent bird of dawn, just above my head,\nand then the sun leaped up, vaulting above the eastern hills. The level\nshafts of light swept across the land, a mantle of gold, while in the\nvalleys below the clear dusk still lay like tideless waters. But down\nthe hill-sides strode the day, throwing its bright arms about the night,\nenfolding and encompassing it in miraculous embrace, and I looked to\nwhere home was. Already the big elms in the garden were pillars of\nflame, then the roof burned, and suddenly the windows blazed\nsignal-like. Dawn had come.\n\nThat was not half the miracle. Light had awoke, the hills were gilded\nwith the sun, but at the touch of the gilding larks innumerable sprang\nfrom the warm tussocks of down-grass and aspired. A hundred singing\nspecks rose against the sky, each infinitesimal, so that they seemed but\nlike the little motes that swim across the eyeball, but these were\nliving things with open throat that hailed the sunrise. Perpendicularly\nthey rose, wings quivering, and throat a-tremble with song, till the\neye lost them against the dazzling azure of day, and only enraptured\nvoices from the air made the heavens musical, as if the morning stars\nsang together. Heaven made holiday. Its company of sweet singers and the\ngold of sunrise were one thing--the dawn.\n\nDear God, dear God, how I thank You for that indestructible minute! I\nknew now what the sunlit curtain that lay between the future and me was,\nand the very morning after I had known You let me see from this high\nplace the birth of day. In this physical world there was reproduced that\ngolden sunlit curtain. You made visible to me what my heart knew. And to\nme on the top of the Beacon the windows of my home flashed a beacon to\nme. And all was of Your making--the sun and the mounting skylarks, and\ndown below the trees of the garden, and the beaconing, flushing window\nof my beloved, and the fruit of the womb. When I come to die, I want to\nremember all that. Truth and Life were there, and the Way also. And what\nis the sum of those three things?\n\nYet was I content even then? Good heavens, no! There were many beautiful\nthings yet to be, and the glory of His gifts just lies in this--that\nthere is always something better to come. This great bran-pie of the\nearth never gives to our little groping hands its best present. There is\nalways something more. Your heart's desire is given you, but at the\nmoment of giving your heart is enlarged, and you ask for something\nbetter yet. And if you want it enough, you get it. The only difficulty\nis to want enough. For you are not given, so I take it, things that you\nhave not really desired. All sorts of bonuses come in, pleasant\nsurprises, but the solid dividend is for the man who wills. There are\nfluctuations, of course, but to look upwards, without doubt, is a\ngilt-edged affair. I correct that. The edge is gilt, and so is the rest\nof it, and the gilt is laid over gold.\n\n       *       *       *       *       *\n\nIt was thus that I looked from the top of the Beacon, with the mist of\nthe song of the invisible skylarks all round, and the blazing reflection\nof the windows of our room in the valley; and there among the skylarks\nit seemed that Legs joined me. It was of no use to deny he was there,\nsimply because it was silly to deny it. There is a French\nword--_revenant_--to express his presence, but even the solidity of that\nword failed to do justice. He had never gone away, and so he could never\nhave come back. He was with us all the time, and rejoiced in the\narrangement of the nursery, even as he had been so hopelessly amused at\nthe correctness of Mr. Holmes on the morning of his funeral.\n\nAnd at the moment of this I expected the 'open vision.' Life, and death,\nand birth, the three great facts, were so near realization. Again I\nexpected to see Pan peep over the brow of the Beacon, and to hear a\nflute-like song that was not of skylarks. I was ready--dear God, I was\nready.\n\nSo I thought for the moment, but before the next had beaten I knew I was\nnot. I wanted more--more of this divine world, more of what the next few\nmonths will bring. Should all be well when summer comes, I think I would\nchoose to die now. And the moment I thought that I knew its unreality. I\nwant to live through the beautiful years that will come. I want to have\na son at Eton or a daughter who turns the heads of eligible youths. I\nwant both, and more than both. Die! Who talked of that? I want to have a\nfull nursery. I want to see Helen old and grey-headed, with\ngrandchildren round her, and herself the youngest of them all. I want to\nlive through the whole of this beautiful life till old age; and though\nthat is called the winter of life, there is no need that it should be\nso. The last day of a man of eighty should be the most luxuriant of\nautumn, before the touch of winter has blackened the flowers; for it is\nonly the thought of death that makes us think of old age and winter\ntogether, and the thought that does that conceives falsely of death.\n\nSo, anyhow, it seemed to me on this midsummer morning of March. I knew\nthat all that was was kind. Pan smiled without cruelty, and if he smiled\nfrom the cross, it was from the throne of ineffable light that he smiled\nalso.\n\nOne by one the skylarks, sated with song, dropped down again to the\nsunlit down. Dawn had passed, and day had come, and--oh, bathos of\nbathos!--I was so hungry. If I had given but ten minutes to the ascent,\nI made but five of the reversed journey, and designed an early\nbreakfast to make existence possible till Helen came down; for it was\nyet not long after seven, and a Sahara of starvation lay between me and\nbacon. Yet, though I have said that this was bathos, I do not know that\nI really think so, since in this delightful muddle of life everything is\nso inextricably intertwined that bathos of some kind invariably is the\nsequel of all high adventure. The great scene is played, the sublime\nthing said, and then you have tea or take a ticket for somewhere. So I\nconfess only to literary bathos, and to disarm the critic I may state\nthat these quiet chronicles are not supposed to be literary at all, but\nmerely the plain account of quiet things as they happened.\n\nSo I lingered for a moment after the knee-shaking descent was over to\ntalk for a little, but not for long, with the river. There was a great\ntrout just below the bridge, and I am sure he knew it was still March,\nfor he wagged his impudent head at me, saying: 'I am perfectly safe. I\nshall eat steadily till April, and then observe your silly flies with a\ncontemptuous eye.' And though he was a three-pounder at least, I bore\nhim no grudge. I don't think I wanted to kill anything that morning.\n\nThen I crossed the further field, and came down into the rose-garden,\nstill meditating on the immediate assuagement of hunger. But then I saw\nwho stood there, and I meditated on this no more; for she was there.\n\n'I got up early,' she said, 'and found you had already gone. Oh,\ngood-morning! I forgot.'\n\n'I shall never forget the goodness of this morning,' said I.\n\nThen I saw that her eyes were brimming.\n\n'Ought I to have told you before?' she said. 'Forgive me if I ought.'\n\nIn that first hour of day we came closer to each other than ever before.\nMy beloved was mine, and the time of the singing-birds had come.\n\n\n\n\nAPRIL\n\n\nI must remind the indulgent reader, lest Helen and I should appear\ntediously opulent, that our Swiss trip in the winter was due to a\nwindfall of a hundred pounds--a thing which may conceivably happen to\nanybody, and in this instance happened to us. Consequently, the fact\nthat we went abroad again in April does not, if it is considered fairly,\nargue aggressive riches. In any case, refuse to stoop to degrading\njustifications. We did not go because it was good for our healths, which\nwere both excellent, nor because foreign travel improves and expands the\nmind. As a matter of fact, I do not believe it does, for the majority of\ntravellers are always comparing the foreign scenes they visit with spots\nin their native land, vastly to the advantage of the latter, and the\nfarther and more frequently they go, the more deep-rooted becomes their\ninsularity. We went merely because we enjoyed it, and had formed a\ncareful plan of retrenchment afterwards, being about to let the Sloane\nStreet house for the three summer months. That was rather a severe\ndecision to come to, since we both hate the idea of strangers using 'our\nthings' and sleeping in our beds; but by these means this expedition to\nGreece became possible, and when once it was possible it had already\nbecome necessary.\n\n       *       *       *       *       *\n\nSo here we sat this morning on the steps of the little temple of\nWingless Victory, wingless, as the old sunlit myth said, because, when\nthe nymph lighted on the sacred rock of the Acropolis, she stripped off\nher wings, which were henceforward useless to her, since she would abide\nhere for ever, just below the great house of defence that the Athenians\nhad raised to the Wisdom of God, Athene, who was born full-grown and in\npanoply of shield, and helmet, and spear, from the head of Zeus. Out of\nhis head she sprang in painless birth, with a cry that was heard by Echo\non Hymettus, and rang back in Echo's voice across the plain, the shout\nof the wisdom of God incarnate.\n\nAnd then Poseidon, the lord of the sea, who coveted these fair Attic\nplains, challenged Athene for the ownership thereof. Each must produce a\nsign of godhead, and the most excellent should win for its manifestor\nall the plain of Attica. There, high on the rock, where the great birth\nhad taken place, were the lists set, and with his trident Poseidon\nstruck the mountain-top, and from the dent there flowed a stream of the\nsalt sea, which was his kingdom; and then the grey-eyed goddess of\nwisdom laid aside her spear, and from the waving of her white hands\nthere sprang an olive-tree, the sign of peace and of plenty. So Poseidon\nwent down to his realm again, where no man may gather the harvest; for\nnone could question which was the more excellent sign.\n\nIt was after this, after the Athenians had raised the great house to the\nWisdom of God, that Wingless Victory came to abide here. It was not fit,\nfor all her greatness, to build her a house on the ground that had been\ngiven to Athene, so just outside the gates they made this platform of\nstone, and raised on it the shrine that looks towards Salamis.\n\nFables, so beautiful that they needed no further evidence of their\ntruth, sprang from ancient Greece, as flowers from a fruitful field.\nWhether they were true or not, whether that peerless woman's form that\nstands now in stone in the Louvre, alighting with rush of windy\ndraperies on the ship's prow, ever was seen here by mortal eye, or\nwhether the myth but grew from the brain of this wonderful people,\nmatters not at all. Beauty, according to their creed, was one with\ntruth, just as ugliness was falsehood. They denied ugliness: they would\nhave none of it, and it was from the practice of that conviction that\nthere rose the flawless city of art. Never, so we must believe, during\nthat wonderful century and a half, when from the ground, maybe, of the\nlifeless hieratic Egyptian art there shot up that transcendent flower of\nloveliness, of which even the fragments that remain to us now, battered\nand disfigured as they are, are in another zone of beauty compared to\nall that went before or has come afterwards, was anything ugly produced\nat all, except as deliberate caricature. It was no Renaissance--it was\nNaissance itself--the birth of the beautiful. On every side shot out\nthe rays of the miraculous many- star: from the marble of\nPentelicus flowed that torrent of statues which make all others look\ncoarse and unlovely, for the speed of the Greek eye was such that they\nsaw attitudes which pass before we of slower vision have perceived them.\nSometimes they saw things that were in themselves ungraceful, but how\nPheidias must have laughed with glee when, among the seventy horses of\nthe great procession on the frieze, he put in one that, cantering, stood\nupon one leg, while the other three were bunched underneath it. Taken by\nitself, it is a grotesque; taken with the others, it gives to the\njubilant procession of youths and horses the one perfect touch. More\nthan two thousand years ago a Greek saw that; two thousand years later\nwe with our focal planes in photography can say he was right.\n\nIn all arts the Greeks were right; they cut through the onyx of the\nsardonyx, leaving the lucent image in the sard; in the less eternal clay\nthey made the statuettes of Tanagra--those sketches of attitudes so\nnatural and momentary that, looking, we can scarcely believe that they\ndo not move: where a woman has already made up her mind to take a step\nforward, but has just not taken it; where she is in act of throwing the\nknuckle-bones, but has yet not thrown them; where a boy has determined\nto push back his chiton (for the day is hot), but has just not made the\nmovement. You cannot hope to understand the Greek genius, unless you\nrealize that our eyes are snails as compared with theirs. They saw with\nthe naked eye what our instantaneous photograph now tells us is the\ncase.\n\nAnd of their paintings! We have none left (and there's the pity of it)\nwhich even reflect the Greek master at his best. But corresponding to\nour English paintings on china, we have the Greek vases of the fourth\nand fifth centuries. They were made by journeymen in potters' shops, but\nthere is not one that lacks the supremacy of knowledge and observation.\nIt is as if a china-shop in the Seven Dials suddenly displayed in its\nwindow examples of the nude figure which showed a perfect knowledge not\nonly of anatomy, but of the romance of movement. The sculptors and\npainters of Greece saw perfectly. Even our academicians themselves\nappear to us to be not flawless. But in Greece we are not dealing with\nthese great lords of colour and drawing: we deal only, as far as drawing\ngoes, with little people in back streets. The noble church of St. Paul\nin the City of London, which so few people visit, was lately decorated.\nAt this moment I look on a sketch of a fragment of pottery.... It is by\none like whom there were thousands. It happens to be perfect in\ndraughtsmanship.\n\nTo think of one day in ancient Athens! In the morning I went up (I feel\nas if I must have done this) to see the new statue of Athene Promachos,\nwhich Pheidias had just finished. We knew little then about his work,\nexcept that he had been chosen to decorate the Parthenon, and those who\nhad seen his sketches for the frieze (which we can see now in the\nBritish Museum) said that they were 'not bad.' So after breakfast my\nfriend and I strolled towards the Acropolis, talking, as Athenians\ntalked, of 'some new thing'--in fact, we talked of several new things,\nand, being Athenians, we got quite hot about them, since we had (being\nAthenians) that keenness of soul that never says 'I don't care about\nthat,' or 'I take no interest in this.' Everything was intensely\ninteresting. It was a hot morning, and the plane-trees by the Ilyssus\nlooked attractive, and there was a company of people there whose talk\nmight be stimulating, but to-day we were too busy: we had to see the\nAthene Promachos, a bronze statue by Pheidias, forty feet high, and\nafter lunch (lunch was going to be rather grand, because a new play was\ncoming out, and Pericles was going to be there, and perhaps Aspasia) we\nwere going to \u00c6schylus's new tragedy, called the 'Agamemnon.' And my\nfriend, who was Alcibiades, was giving a supper-party in the evening.\nSocrates was coming, and a man who was really very pleasant, only he\nlistened and made notes, but seldom talked. His name was Plato.\n\nAlcibiades was rather profane sometimes, and spoke of the great gods as\nif he did not really believe in them. I, knowing him so well, knew that\nhe did, and that it was only his Puck-like spirit which made him in talk\nmake light of what he believed. All up the steps of the Propyl\u00e6a he was,\nthough amusing, rather profane, and then we came through the central\ngate, which was yet unfinished, and straight in front of us was the\nstatue. And some jest--I know not what--died on my friend's lips, and\nhis great grey eyes suddenly became dim with tears at the sight of\nbeauty, and his mouth quivered as he said:\n\n'Mighty Lady Athene, my goddess!'\n\nAnd with that he knelt down on the rock in front of where she stood, and\nprayed to the wisdom of God.\n\nHe refused to go to the grand lunch after this, and insisted on our\nremaining up here till it was time to get to the theatre, quoting\nsomething that Socrates had said about the cleansing power of beauty;\n'so we will not soil ourselves just yet,' quoth he, 'with the intrigues\nwe should hear about at lunch, but go straight from here to the\ntheatre.' So we bought from a peasant some cheese wrapped up in a\nvine-leaf, and a bottle of wine, and a loaf of bread and some grapes,\nand then went down the rock to the theatre. And still that divine vision\nhad possession of Alcibiades, for he paid no attention to the greeting\nof his friends, and bade them be silent. And soon the actors were come,\nand the watchman went up to the tower, and looked east, and saw the\nbeacons leap across the land, to show that the ten-year siege was over,\nand that Troy had fallen. Then slowly began to be unfolded the tale of\nthe stupendous tragedy. Home came Agamemnon, with his captive, the\nPrincess Cassandra, riding behind him in his chariot of triumph.\nClytemnestra, his wife, met him at the palace door, and with feigned\nobeisance and lying words of love welcomed him in, leaving Cassandra\noutside. Then there descended on the Princess the spirit of prophecy,\nand in wild words she shrieked out the doom that was coming. Quickly it\ncame: from within we heard the death-cry of the King, and the palace\ndoors swung open, and out came the Queen, fondling the axe with which\nshe had slain him.... The doom of the gods was accomplished.\n\nThen afterwards we went round to the green-room, and found \u00c6schylus\nthere, and Alcibiades, in his impulsive way--I tell him he has the\nfeelings of a woman--must kneel and kiss the hand that wrote this\nwonderful play. Socrates was there, too, putting absurd questions to\neverybody about the difference between the muse of tragedy and the muse\nof comedy; as if anybody cared, so long as \u00c6schylus wrote plays like\nthat! However, he got Plato to listen to him, and soon made him\ncontradict himself, which is what Socrates chiefly cares about. Pericles\ncame in, too, with Aspasia, to whom he kindly introduced me. Certainly\nshe is extraordinarily beautiful, and has great wit. But she called\nattention to her physical charms too much, which is silly, since they\nare quite capable of calling attention to themselves.\n\nAfterwards, since only Alcibiades and I had seen the wonderful statue,\nwe all strolled up to the Acropolis again to look at it and the sunset.\nSocrates came, too, and after we had examined and admired the bronze\ngoddess again, we went and sat on the steps of the temple of Athene. He\ntried his usual game of asking us questions till we contradicted\nourselves, but before long all of us refused to answer him any more,\nsaying that we were aware that we were totally ignorant of everything,\nand that there was no longer any need for him to prove it to us. And\nthen--exactly how it arose I don't know, but I think it was from the\nquestions and answers that had already passed--he began to weave us the\nmost wonderful fable, showing us how all that we thought beautiful here\non earth was but the reflection, the pale copy, of the beauty which was\neternal. Round the outer rim of the earth and the stars, he said, ran\nthe living stream of a great river, which, indeed, was heaven, and\neverything that we thought beautiful here had its archetype there, and\nall day and all night the gods drove round and round on this river of\nbeauty in their chariots. It was our business, then, here on earth, to\nlook for beauty everywhere, and never falter in the quest of it, for so\nwe prepared ourselves for the sight of that of which these things were\nbut the shadow, so that the greater would be the initiation which would\nbe ours after death. More especially we must seek for the beauty of\nspiritual things, which was the real beauty, and so order our bodies,\nour words, and actions, that they were all in tune with it, with the\nbeauty of prudence, and temperance, and kindness, and wisdom, for it\nwas of these that heaven itself and the living stream was composed, and\nthese shone from the eyes of the immortal gods.\n\n'So there is my prayer,' said he, rising and stretching out his hands to\nthe great statue, while we all rose with him. 'O Athene, give me inward\nbeauty of soul, and let the inward and the outward man be at one.'\n\nSo the sun set, but on the violet crown of Athens--the hills there,\nHymettus, Pentelicus, and Parnes--the light still lingered, and shone\nlike the river of beauty Socrates had told us about, till it faded also\nfrom the tops, and above the deep night was starry-kirtled.\n\n       *       *       *       *       *\n\nHelen is the most delightful person in the world to tell stories to.\nHowever lamely you tell them, she is absorbed in them, and never asks\nabout the weak points, as other children do. She might, for instance,\nhave asked if I was correct about my dates; did the 'Agamemnon' come out\nin the year that the 'Promachos' was made? Instead----\n\n'And who was I?' she asked. 'Don't tell me I was Aspasia, because I\ndon't like what you told me about her.'\n\n'No; you were not Aspasia,' I said rather hurriedly; 'and I rather think\nyou had had your turn in Greece at some other time. I didn't know you\nthen, except, perhaps, in the myths, for I am not sure that you were not\nElectra.'\n\n'Was she nice?' asked Helen.\n\n'She was very nice to Orestes.'\n\n'Oh, don't! Who was Orestes? What a nice name!'\n\n'You were his sister. That's all about mythology just now.'\n\n       *       *       *       *       *\n\nThe plain quivered under the sunlit haze of blue. To the south the dim\nsea was in tone like two skies poured together, and the isles of Greece\nfloated in it like swimmers asleep. Below, to the left, lay the theatre\nwhere I had seen the 'Agamemnon,' empty, but ready as if the play was\njust going to begin. Who knew what ghosts of those supreme actors were\nthere, what audience of the bright-eyed Greeks followed the drama? And\nabove us stood the presiding genius of Athens, the beautiful house\nbuilt for the virgin who sprang from the brain of God. A little more,\nand it would be her birthday again, and we should hear the sound of\nhorse-hoofs coming up the hill, and see the procession of the Athenian\nyouths, and the men with the bulls for sacrifice, and the wine-carriers,\nand the incense-bearer, and the priests of the great goddess. Another\ncompany would be there, too--the hierarchy of Olympus--come down on\nAthene's birthday to visit her in her beautiful home. With Zeus would be\nthe mother of the gods; and Aphrodite would be there, the spirit of love\nthat renews the earth; and Apollo, who makes it bright with sunshine;\nand Demeter, the mother of the cornfields; and Persephone, radiant, and\nreturned from the gate of death; and Hermes, the swift messenger whose\nfeet were winged; and Iris, who was rainbow, the sign of the beneficent\nseasons.\n\nAnd ... though we saw them not, there was not one missing. Love was\nhere, and below were the ripening cornfields, on which the sun shone;\nand beyond was the realm of Poseidon, and a squall of spring rain, that\npassed like a curtain in front of Hymettus, showed us Iris.\n\nThen it was time to go down townwards again, for the morning was passed;\nbut Helen paused at the doorway at the gate of the Acropolis, and looked\ntowards the temple.\n\n'Best of all, I like Socrates' prayer,' she said; 'and I must say it to\nmyself.'\n\n       *       *       *       *       *\n\nSpring had been rather late this year, and a week ago, when we drove out\nto the foot of Pentelicus, to have a country ramble, the rubbish of last\nyear's autumn was still in evidence. Then the spring began to stir, and\ntwo days ago, when we had gone out again, all the anemones except one\nkind were in full flower. They are heralds, those mauve and violet and\npink and white chalices of blossom, to tell us that the great procession\nof Primavera has begun. But last of all come the trumpeters, the scarlet\nanemones, and if the sun has been warm, and no north wind has delayed\nthe procession, they blow their blasts over the land just two days after\nthe heralds have appeared. So to-day after lunch we went out to hear\nthe trumpeters; to-morrow we shall see Primavera herself.\n\nSpring herself, the goddess Primavera, was very near to-day, for on\nthicket and brake and over the flank of the hill-side her trumpeters\nwere blowing their shrill blasts of scarlet. Two days before, the land\nwas sober-; now, wherever you looked, the wonderful anemone,\nlast to flower, stood high with full-blown petals. The movement and stir\nof the new life was hurrying to its climax. To-morrow, instead of the\nmyriad buds of the cistus and the pale stalks of orchid, the flowers\nwould be unfurled at the final touch of the spring, at the advent of the\ngoddess herself. To-day a myriad folded bells hung from the great bushes\nof southern heath, like stars still cloaked in mist; to-morrow, with one\nnight more of warm wind and a morning of sun, they would blaze and peal\ntogether; for it is thus in this wonderful Southern land that spring\ncomes: a few heralds go before, and then the army of trumpeters. After\nthis, She crosses the plain with the ardour of hot blood, so that all\nflowers blossom together, and every bud and beast goes suddenly\na-mating. Here there is none of our limitative February, our pinched\nhopes of March; all is quiet till the heralding of the anemones and the\ntrumpets of their scarlet brethren. Then, in full panoply of blossom,\nPrimavera and summer, too, are there together. For a week or two the\nland is aflame with flower, and then already the maturing of fruit-trees\nhas begun.\n\nNortherners though we are, both Helen and I claimed some strain of\nSouthern blood in the ecstasy of those days. That for which we wait and\nwatch for patient weeks in the shy approach of spring in England was\nhere done with a flame and a shout. There was no hesitancy or delay; no\nweak snowdrop said that winter was coming to an end weeks before spring\ncame, to die before the crocuses endorsed its message. Here all was\nasleep together till all woke together. Ten days ago there was no hint\nof spring save in the strong sunshine: the wilderness of winter still\nspread its icy hands. Then faster than the melting of the snow on the\ntop of Parnes came the heralds in the wilderness, and spring was there.\nIt was like the winter of Kundry's soul, to whom one morning Gurnemanz\nsaid: 'Auf! Der Winter floh, und Lenz ist da.' And on that day came\nParsifal and her redemption, and the ransomed of the Lord returned with\njoy and singing.\n\nI have no skill to tell of those days: for the past, all that I knew of\nthe history of this wonderful land, and the present, all that love\nmeant, and the future, the dear event that was coming closer, were so\ninextricably mingled that no coherence is possible. But if you love a\nplace, and are there with your beloved, and know that she will bear a\nchild to you before many weeks are over, you may make a paradise of\nClapham Junction, and find the joy of it a thing incommunicable. And how\nmuch more difficult a material is the magic of this land to work\nin--this little Attic plain, peopled with the ghosts of that wonderful\nage, which are not dead at all, but instinct with life to-day, at this\nmoment when spring has come, so forcibly that even the slow tortoises on\nthe side of Pentelicus hurried breathlessly about, with deep sighs (I\nassure you) till they found a congenial lady. Then they ran--positively\nran--round her in ever-narrowing circles, still sighing. There were\ngrasshoppers, too--green gentlemen and brown ladies. The brown ladies\ngenteelly ran away, but they never ran far. The great hawks sought each\nother in the sublime sky, and the young men and maidens of Athens as we\ndrove back were taking discreet walks together into the country. And\nfrom the Acropolis the maiden goddess, who is the Wisdom of God, looked\ndown, and was well pleased.\n\nFor, thank Heaven! the Wisdom of God in no prude. To all has it given a\nsoul, and to all souls is desire of some sort given--to one the\nperfection of form, to another the perfection of wit, to another the\nperfection of colour, to another the perfection of truth. For each there\nis a way; each has got to follow it; and for many there are various\nways, and these many must follow them all. If a thing is lovely and of\ngood report, we all have to hunt it home. It is no excuse to say you\nhave no time, for you have all the time there is. Search, search: there\nis the Way everywhere.\n\nIndeed, this is no mystical affair: it is the plainest sense. Whatever\nhappens, God is somehow revealed. But, being blind, we cannot always see\nthe revelation.\n\n       *       *       *       *       *\n\nTo-night, as Helen and I sit on deck of the steamer that takes us back\nagain to Marseilles, we wonder what gives Greece its inalienable magic.\nWe saw the fading of its shores in the dusk, and though the\nphosphorescence of the sea was a thing to marvel at, it was no longer\nthe phosphorescence of Greek waters. That little fig-leaf-fingered land\nhas sentiment somehow in its soil; it cannot fail to move anybody. Its\nhistory since the Great Age--it is no use to deny it--has been tawdry\nbeyond description. It yielded to the Romans, it scarcely resisted the\nAlbanians; and though some flickering spirit of its old grandeur flamed\nagain when its people rose against the Turkish rule in the early part of\nlast century, what are we to say of the spirit of the people when,\ntwelve years ago, they again fought their ancient and ancestral enemy?\nThe Turks strolled slowly southwards from the North of Thessaly, and\nonly the intervention of the Powers prevented Greece again becoming a\nTurkish province. The Hellenic battle-cry went shrilly up to Heaven, but\nthe Hellenic army trotted like a flock of sheep before the foe, until\nthe Powers said that the war must cease. Only the year before there had\nbeen revival of the Olympic games, and there had been a race from\nMarathon to Athens in memory of Pheidippides, who bore the news of that\nstupendous victory, and died as he reached Athens, saying, 'Greece has\nconquered the Persians.' A Greek won that peaceful race from Marathon;\nthe same Greek won the peaceful race home, and arrived back in Attica in\nthe very van and forefront of the retreating army. The 'host of hares'\nwas the Turkish name for the foes they never had occasion to meet, who\nstarted from their fortresses like hares from their forms, and galloped\nquietly away. Meantime the Greek fleet cruised in the Adriatic, and sank\na fishing-boat. When the war was over, they came home with the spoils\nof their victory--a hat, a fish, a net. Perhaps it is best to say that\nthere was no war at all: the Turkish armies made peaceful man\u0153uvres over\nThessaly, until they came to Volo. Then the Powers of Europe said: 'We\nthink your man\u0153uvres have extended far enough: kindly go home.'\n\n       *       *       *       *       *\n\nYet, somehow, the tragic futility of all this does not really touch\nGreece or the sentiment that the lovers of the lovely land feel for it.\nSupposing a Greek army, or a regiment of it, had met the Turk, and died\nin the cause of patriotism, that could not have added to the compelling\ncharm of Greece, and so the fact that none of these patriotic events\nhappened does not diminish it. In Greece, whatever may be done or left\nundone, you are in the country where once beauty shot up like the\naloe-flower, so that all else is inconsiderable beside that, since\nwhatever the world has achieved afterwards, whether in painting, or\nsculpture, or drama, or poetry, or in that eagerness of life which is\nthe true romance of existence, is measured, if only it be fine enough,\nby the standard set then. That is the haunting, imperishable charm of\nthis country, and, missing that, even the phosphorescence of waters by\nnight, divided by the swift keel of the lonely ship, was for a time a\nsoulless firework.\n\nThe magic of it--the magic of it!\n\nThereafter we staggered across the Adriatic, over the ridge and furrow\nof a grey and unquiet sea, till we found quiet below the heel of Italy.\nSoon to the south-west the horizon lay in skeins of smoke, and it was\nnot for hours afterward that the cone of Etna, uprearing itself, showed\nwhence the trouble came. Narrower grew the straits, till we passed out\nbeside Messina, and for the pillar of smoke which Etna had raised all\nday we sighted Stromboli, a pillar of fire by night. Next morning we\nwere in the narrows between Corsica and Sardinia, and saw the little\nvillages, tiny and toy-like, in the island whence sprang the brain that\nwas to light all Europe with the devouring flame of its burning. If the\ndead return, I think it is not in Elba of St. Helena, nor even in the\npomp of Paris, nor on the battle-field, that we must guess that Napoleon\nwanders. He sees the impotence of his destructive and untiring genius.\nThe lines of his new map of Europe have been gently defaced again by\ntime, and he sits quiet enough by the little house, where still the\ndescendants of his old nurse dwell, and sees the innocent campaigning of\nher grandchildren in their childish games. And when the time comes for\nunflinching justice to be done to that unflinching spirit, who spared\nnone, nor had pity so long as by any sacrifice the realization of his\nruthless imaginings came true, will not the spirit of his old nurse\nstand advocate, and remind Justice that, even in the midst of his\ngigantic schemes, he remembered her who had given him suck, and provided\nfor her maintenance? Somewhere in that iron soul was the soft touch of\nchildish days: he was kind who was so terrible, and that pen so unfacile\nand so bungling that he hated to write at all put a little paragraph of\nscarcely decipherable words to his will that showed (what would\notherwise have been incredible) how a certain gentleness of heart\nunderlay the iron.\n\nThough all these sights--the chimney of Etna, the furnace of Stromboli,\nthe island of Napoleon--were but milestones, passed before, to show us\nnow how far we were travelling from the magic land, yet each brought us\nnearer in time and space to the magic of home, and of the day, yet\nunnamed, which must already, like some peak of an unknown range, be\nbeginning to rear itself up in the foreground of the future.\n\nThen, as the magnet of Greece grew more remote, the magnet of home\ngained potentiality, until there was no question which was the stronger.\nWe had intended--that is to say, more than half intended--to stay a day\nor two in Paris; instead, we fled through Paris as if it had been a spot\nplague-ridden, meaning to pass the night in London. But even as we\nscurried from Gare de Lyon to Gare du Nord, so, too, we scurried from\nVictoria to Waterloo, with intention now fully declared to get down to\nthe dear home without pause. As far as I remember, we sustained life on\nthick brown tea and a Sahara of currant-cake; but at the end there was\nthe snorting motor waiting at the station, and a mile of sleeping\nstreets, cheered by the vision of Mr. Holmes going somewhere in a neat\nInverness cape and buttoned boots, a mile of spring-scented country\nroad, and then the little house, discreet behind its shrubbery, where\nwas the rose-garden, among other things, and among other things the\nnursery.\n\nThe night was very warm, and lit by the full moon of April, so, after we\nhad dined, and run like two children from room to room in the house,\nfirst to greet all the precious things of home, with Fifi, like an\nanimated corkscrew, performing prodigies of circular locomotion round\nus, we found that there was still a large part of home to greet, and so\nwent out into the garden, to see what April had brought forth there. No\nsudden riot or conflagration of leaf and flower, like that which we had\nseen blaze over the lower <DW72>s of Pentelicus, was there, but April day\nby day had done his gentle work, so that where we had left a bed still\nwinter-naked it was now mapped out into the claims of the plants.\nTo-morrow there would be disputes to be settled, for the day-lily had\npegged out more than her share, and between her and the iris a\ndelphinium would be crowded out of existence. But every plant--such is\nour rule--may claim all the ground it can get until the end of April;\nthen come round the judges of the court of appeal, and if any plant\ndistinctly says, 'I have not room to grow, because of these\nencroachers,' his appeal, if he promises at all well, is usually upheld,\nand the encroacher is shorn of his unreasonable encroachments. Even by\nthe moonlight it was quite certain that the court of appeal had a heavy\nday in front of it: there were lawsuits regarding land to settle, which\nwould require most careful adjustment, for the court hates depriving a\nrightful possessor of that which his vigor has appropriated. On the\nother hand, the slender aristocracy of the bed (for the aristocrat grows\nupwards rather than sideways) must not be elbowed out of existence. One\nplant only is allowed to do exactly what it pleases and when it\npleases--the <DW29>, which is 'for thoughts' that are always sweet, and\nso may roam unchecked and welcome, for who would set limits to the\nwanderings of so kindly and humble a soul? It but touches the ground,\ntoo (to be absolutely honest, I must confess that this has something to\ndo with the liberties we give it), as a moth still hovering and on the\nwing draws from the flower the sustenance it needs. It does not, so to\nspeak, sit down to make a square meal, or burrow with searching roots\ndeep into the earth, and drain it of all its treasure, but it is ever on\nthe move, like some bright-eyed beggar-girl, to whom none but the\nchurlish would grudge the wayside halfpenny. She will not linger and\nsettle and sponge on your bounty, but be off again elsewhere next\nmoment, just turning to you a smiling face, and whispering a murmured\nthanks in the bright language of flowers. So she is privileged to wander\neven in the sacred territory of the roses, where I hope she has already\nwandered wide. There, however, we did not penetrate to-night, for it and\nthe meadow we kept for the morrow. But on the top margin of the field\nagainst the sky I saw shapes that were unmistakable. To-morrow our\nhearts will go dancing with the daffodils.\n\nBut to-night we are content with the thoughts that the <DW29>s have\ngiven us, and can even forgive Milton for speaking of them as 'freaked\nwith jet.' Freaked with jet!--when Ophelia had said that they were 'for\nthoughts'! But, then, Milton speaks of the 'well-attired woodbine,'\nwhich is almost as bad. Imagine looking at <DW29>s, and finding it\nincumbent on one to say: 'I perceive they are freaked with jet'! But, as\none who had the highest appreciation of Milton remarked, to appreciate\nMilton is the reward of consummate scholarship, which was certainly a\nvery pleasant reflection for himself, and perhaps if I were a better\nscholar I should think with appreciation of the <DW29> 'freaked with\njet.' As it is, I merely conclude that Milton was flower-blind--a sad\naffliction.\n\n       *       *       *       *       *\n\nHelen is absolutely ultra-Japanese in her observance of the\nflower-festivals, of which she marks some dozen of red-letter days in\nthe year. They cannot, of course, be celebrated on any fixed day, since,\nowing to the vagaries of climate, there might not be a single lily to be\nseen, for instance, this year on the actual day which was Lily-day a\nyear ago. She waits instead, like the Japanese, until the particular\nflower is in the zenith of its blossoming, and then proclaims the\nfestival. Other flowers, naturally, sometimes are at their best on the\nred-letter day of another, but this, as she observes, is canonically\ncorrect, since St. Simon and St. Jude, and St. Philip and St. James, are\ncelebrated together. I was not, therefore, the least surprised next\nmorning, when, after a short excursion to the garden, she came in to\nbreakfast, saying:\n\n'It is Daffodil-day, and the day of its sisters of the spring.'\n\n'But we had the sisters of the spring in Greece,' said I.\n\n'Yes; that is the advantage of going to Greece: the Greek calendar is\ndifferent to ours. We had Easter Day before we started, and another\nEaster Day when we got there. Besides, it was Anemone-day, and the day\nof its sisters of the spring. The anemone's sisters were not the same\nas the daffodil's.'\n\nThis was convincing (even if I needed conviction, which I did not), and\nDaffodil-day it was.\n\nAfter the early heats of February the year had had a long set-back in\nMarch, and though April was nearly over, I doubt whether there had been\nany more gorgeous decoration in our absence than that which we found\nwaiting this morning in the church of the daffodils and its sisters of\nthe spring. It was not in vain that we had dug and delved last autumn\nwith such strenuous patience, for that half-acre of field beside the\nrose-garden was a thing to make the blind see. A rainbow of blossom lay\nover it all: the early tulips had opened their great chalices of gold\nand damask; the blue mist of forget-me-nots seemed as if a piece of the\nsky had fallen, and lay mutely under the trees; brown-speckled\nfritillaries crouched shyly in the grass, and their white-belled sister\nnestled beside them; narcissus was there, all yellow, and narcissus with\nthe eye of the pheasant; primroses still lingered, waiting for Helen's\nproclamation to take part in the festival; while some bluebells had\nhurried to be here in time; crocuses in the grass were like the dancing\nof the sun on green waters, or purple as the deep-sea caves; and\nanemones, greedy for more festivals, had hurried overland from Greece to\nbe here before us; and clumps of iris were like banners carried in\nprocession. These were the sisters of the spring. It was their day; but\nfirst it was Daffodil-day. Slender and single, tall and yellow, it was\nas if through the web of them, the golden net that they had laid over\nthe field, that you perceived their sisters. And the sun shone on them,\nand the great blue sky was over them, and the warm wind made them dance\ntogether.\n\nAfter a long time, Helen spoke.\n\n'Oh, oh!' she said.\n\nThat about expressed it.\n\n'My heart with pleasure fills,' she added.\n\n\n\n\nMAY\n\n\nIt always seems to me a matter for wonder why the astronomers, or Julius\nC\u00e6sar, or whoever it was who took the trouble to divide time up into\nmonths and years, should have made the day of the New Year come in the\nmiddle of winter. Probably it has got something to do with the solar\neclipse, or the lunar theory, or movements and motions quite\nunintelligible to the ordinary mind, which would easily have the point\nof beginning the New Year in spring--for instance, on May-day--when the\nseason is clearly suitable for beginning again. But to make a fresh\nstart by candlelight in a fog on the first of January implies a more\nvivid effort of the imagination and a sterner resolve of the spirit than\nmost of us able to manage. You might as well try to make up for misspent\nyears by selecting Blackfriars or Baker Street Station as a place to\nstart afresh in.\n\nPersonally, though I think the 1st of May would be a quite reasonable\noccasion on which to begin a New Year, I should prefer a rather later\ndate, when summer is more certain, and it was for this reason that when\nI formed this (I hope) harmless little project of putting down the quiet\nhappenings of a year of life, I began in June. Month by month I kept\nthis diary, and you will see when you come to the end of this month of\nMay that my plan was endorsed by what happened then, and that New Year\nmust, in the future, always begin for Helen and me on the first of June.\n\n       *       *       *       *       *\n\nEven with the early days of May summer descended on us, and Mr. Holmes's\nPanama hat and a neat new suit of yellowish flannel made their due\nappearance to confirm the fact. Soon, if this goes on, he will be\nhanding ices instead of buns at tea-parties, and I have often seen him\nlately on the ladies' links playing golf in his little buttoned boots.\nHe came to call yesterday, and told me of Charlotte's engagement, and\nannounced the fact that my Archdeacon (I call him mine because of what\nhappened at that dreadful Sunday-school) was giving a garden-party on\nthe 11th, and the wife of the younger son of our Baronet had not been\ninvited. The fact of the garden-party on the 11th was not new to us,\nbecause We Had Been Invited. Oh, revenge is sweet, and we gloated over\nthe discomfiture of the foe. Her mother had been a governess, too. That\nwas a new fact that Mr. Holmes had gathered in the last half-year--just\na governess, and not in a noble family even, but in the employment of a\nretired tradesman. That accounted for the fact that her daughter spoke\nFrench so well; no wonder, since the mother had to teach it. Her\nknowledge of that language, scraps of which she constantly introduced\ninto her conversation, had always puzzled Mr. Holmes; now he knew how it\nhad been acquired. Indeed, she had come rightly by it, poor thing! We\nnone of us grudged it her. And it was no wonder now to Mr. Holmes that\nshe looked so thin; probably she had never had enough to eat when she\nwas a child, and that indescribable air of commonness about her was\nperfectly accounted for. Indeed, Mr. Holmes became so sardonic that you\nwould have thought that his family was one (as I dare say it is)\ncompared to which the Plantagenets were parvenus; and Helen changed the\nsubject, which I thought was a pity, as I wanted to hear ever so much\nmore about the lady's obscure origin.\n\nWe chatted very pleasantly for a long time, and learned all that the\n_Morning Post_ had said in little paragraphs during the past week, and\nall that the Close and the County (I recommend that expression) and the\nMilitary were doing here. We were going to be very gay indeed; there was\nalready an absolute clash of entertainments during a week of cricket\nnext month, so that the Mayor was forced to give a luncheon-party one\nday instead of a mere tea, which he would probably not like at all,\nsince if ever there was a Mayor who collected candle-ends, this was the\none. Did I remember that which was called champagne at the famous lunch\nwhich has already been spoken of?\n\nIn fact, Mr. Holmes shook his head over the general trend of affairs,\nand spoke quite bitterly about the wave of Radicalism which was passing\nover the country. The County Club, so he said, which had always prided\nitself on being a little exclusive, was tainted with commonness now, and\nhad positively disgraced itself at the last election by letting in those\nthree new members. They were nobodies--local nobodies--one the son of a\ndoctor, another the father of a doctor; the third nobody at all.\nAnd--would I believe it?--there had been a veterinary surgeon up for\nelection as well. Luckily, the club had pulled itself together over him,\nand given him a smart shower of black-balls. No doubt the club was in\nwant of funds, but why, then, have built a new billiard-room? How much\nbetter to poke the butt-end of our cues into the chimney-piece, as we\nhad always done when playing from over the left-hand middle pocket, than\npurchase increased cue-room at the sacrifice of our standing as a County\nClub? If we did not draw the line somewhere, where were we to draw the\nline? That was unanswerable. We all said what is written, 'Tut!' and\nlooked very proud. Helen, I consider, looked prouder than Mr. Holmes,\nbut she disagrees with me, having seen her own face in the looking-glass\nover the mantelpiece. True, she had not the natural advantage that Mr.\nHolmes's aquiline nose conferred upon him, but the assumed curl of her\nlip was superb: she looked like a Duchess in her own right.\n\n       *       *       *       *       *\n\nHow slowly these beautiful days of May passed, for when one is very\nhappy and very expectant, time seems to stop. Exactly the opposite\nhappens when one is spending days that are full of pleasures, and living\nentirely in the moment, for then hours and days pass on unregarded, so\nthat it is Saturday again before you know the week has really begun. But\nhappiness--I but bungle with words over a thing that is obvious to\neverybody who knows the difference between happiness and pleasure--is a\nthing quite detached from the present moment, just as the sunlight which\nfloods these downs is not _of_ them. Happiness ever broods on the wing,\nand swings high above the things of the earth, like some poised eagle,\nor like the sun itself. It illuminates what it looks on, turning dew to\ndiamond, and striking sapphires into the heart of what has been a grey\nsea, but it is independent of material concerns; and were the world to\nbe withdrawn and extinguished, it would shine still. True, it shines on\nthe dewdrop and turns it into wondrous prismatic colours, and thus the\ncommon surface of life is always iridescent when we are happy. But\nhappiness--that golden, high-swung sun--does not, I think, particularly\nregard the jewels he makes out of common things: his own bright shining,\nperhaps, weaves a golden haze between him and what he shines upon.\n\n       *       *       *       *       *\n\nIt was somehow thus, I think, that things were with us during that first\nfortnight of May. Below the golden haze were these entrancing facts\nwhich I have just recorded about the Archdeacon's party, the frightful\ndisclosures concerning the mother of the wife of the younger son of the\nBaronet, and the growing plebeianism of the County Club; but neither\nHelen nor I could focus our attention on them; for though, as I have\nsaid, time went so slowly, yet there was not time enough to regard them:\nthey belonged to a different plane to that on which we were living. We\ncould penetrate down into it and giggle, but then our attention\nwandered, and before we knew it, we had swum up again like bubbles\nthrough water to the sunlit surface.\n\nThere took place, in fact, a revision in our list of joyful and dreadful\naffairs. No one could appreciate the humour of Mr. Holmes more than\nHelen did, but, as I have said, she could not attend to him now. Nor\ncould she attend to the perfectly hideous fact that the greater part of\nthe ceiling in the dining-room in Sloane Street had fallen, and that our\ntenants had (quite reasonably) demanded to be released from their\ntenancy, of which there was still six weeks to run, since the house was\nuninhabitable. Nor did I think she would have cared if the ceiling had\nsmothered them as they sat at dinner. And the dreadful earthquake in\nChina failed to move her, and so did the church crisis in France. But\nfor certain other things she cared more than ever, though you would\nhave said they were little enough. All the growth of the spring-time\nmade her eyes brighten and ever grow dim again, and she would dream over\nthe tiny buds of the rose-garden with smiles that were sped to her mouth\nfrom the inmost spring of happiness. She spread fat Heliogabalian feasts\nfor the birds, since they wanted nourishment now that they were so busy\nover their nests, and many dyspeptic bachelors and spinsters, I expect,\nreeled daily from their table laid on the lawn to sleep off the results\nof their excess. She loved the sun, too, more than she had ever loved\nit, and the shade also, and day and night, and all the firm, great\nforces of the world.\n\nNot less, too, did she love the little things of little rooms, and now\nwe never sat in the drawing-room, with its Reynolds' prints, but went\nalways to the nursery, with its rocking-horse and its Noah's ark, and\nits lead soldiers, and its play-table. But when there--when playing\nthese silly games of soldiers, which Helen had been wont to play as if\neternal salvation depended on the nice adjustment of a small tin\ncannon, which, when you pulled a string, shot a pea--she had a change of\nmood most disconcerting at first. Now and again she shot down my\nGeneralissimo, posted, as he should be, out of possibility of attack\nalmost, in the very rear of my army, by some inconceivable ricochet\nwhich would a few weeks ago have filled her mouth with laughter. But\nnow, when these unspeakable flukes occurred, and she upset the heaviest\nsoldiers in my brigade, instead of being delighted, she was sorry, and\napologized. To injury, which was bad enough, she added insult, which was\nworse, and said: 'I am afraid I must win now.'\n\n       *       *       *       *       *\n\nThere is another curious thing (Helen looks over my shoulder as I write,\nand agrees) that, though she still loves to play soldiers, she wants me\nto win. Consider it: whoever before wanted to play a game (and the more\nchildish the game, the less worth while you would have thought to play\nit), if he did not care about winning? Besides, it is so exceedingly\nunlike her--she is looking over my shoulder no more--not to play any\ngame as if life and death depended on it. But now she applauds my skill\nand my luck, and apologizes for her own.\n\n       *       *       *       *       *\n\nAnd then, when the game is over, and the Duke of Wellington on one side\nand Julius C\u00e6sar on the other lie dead, she still sits on the ground\nbeside the low play-table, and looks round the room with wandering,\nhappy eyes. There are the playthings I have told you of--the Noah's ark,\nthe rocking-horse, the great dolls'-house, the front of which, windows\nand door and all, is unfastened by a neat latch in the wall of the\nsecond story, and swings open altogether, so that you must be careful\nnot to unlatch it early in the morning or late at night, else you would\nsee all the ladies and gentlemen at their toilet in an embarrassing\nstate of undress. I found Helen the other morning playing at dolls all\nby herself. She had laid a banquet in the dining-room, and had arranged\nthe ladies and gentlemen on the stairs, so that one could see at once\nthat they were going down to dinner. From their attitudes, and a\ntendency to lean against each other or the wall, you might have thought\nthat they were trying to get upstairs after the banquet. But that, Helen\ntold me, was foolish, since their faces were all turned in the direction\nof downstairs. The answer was that they had indulged even more freely\nthan I had supposed, and were trying to get upstairs backwards.\n\nYes; we did all these extremely childish things, and so far from being\nashamed of them, I set them all down here for you to laugh at if you\nlike, or merely to be bored with. Things like these--playing at soldiers\nor at dolls--retained their interest, just as did the spirit of the\nblossoming summer, when Mr. Holmes's discoveries or the fall of the\nceiling in Sloane Street lacked the calibre to interest us. And, if you\ncome to think of it, though I thought an explanation would be difficult,\nnothing in the world could be more simple. Things about children, and\nbirth, and growth were clearly the only affairs that could concern us.\nOne morning, I remember, it was found that the foundations of the\ncathedral were in a dreadful state, and that it would probably fall\ndown. I told Helen this as she was engaged on preparing a Gargantuan\nbreakfast for the birds. She only said:\n\n'Oh, what a pity!'\n\nThat was all she cared for the historic Norman pile, with all kinds of\nKings and Queens buried inside it!\n\nThere is nothing more to be recorded of this month, since the only\nthings that seemed to us to have any real importance were just the\nchildishnesses of which I have already given you such amplitude of\nspecimens, until the morning of the last day of May.\n\n       *       *       *       *       *\n\nThe rule of the house was that there was no rule of any sort as regards\nbreakfast. Anybody who came into the dining-room at most hours of the\nmorning would find the breakfast perennials (bread, butter, sugar, milk,\nthe morning paper and marmalade) on the table, and would, on ringing a\nbell, be given the annuals--_i.e._, fresh tea and a hot dish. Similarly,\nanybody who did not come into the dining-room was supposed to be\nbreakfasting either elsewhere or not at all. So on this last morning of\nMay, on coming down, I rang the bell, and read the paper till bacon\ncame. An hour before I had just looked into Helen's room, and seen that\nshe was still asleep.\n\n       *       *       *       *       *\n\nThe bacon was rather long coming that morning--I try to reconstruct the\nday exactly as it happened--and I had already skimmed the news, and\nfound there was not any, and in default of it was reading a superb\naccount of the visit of a member of the Royal Family to Naples, who in\nthe afternoon had 'honoured' (so said the loyal press) the volcano of\nVesuvius with a visit. How gratifying for the immortal principle of\nfire! One hoped it would not become swollen in the head. This fortunate\nvolcano, whose cone had been blessed----\n\nAt the moment I heard a step outside. It was not from the kitchen: it\nwas coming from upstairs, and it came very quickly. Then,\ninstantaneously, terror seized me, for time and place were no longer now\nand here, but it was the evening when I heard my name called in the\ngarden, and thereafter heard Legs running downstairs. And quickly as the\nsteps came, they seemed to me to go on for ever; yet I had only just\ntime to get up, when there came a fumbling hand on the door, and Helen's\nmaid came in.\n\n'If you please, sir, would you send at once,' she began. 'The nurse----\n'\n\nThere were quicker ways than sending, and next minute I was flying up\nthe road on my bicycle. My mind, as I think must always happen with any\nmind in such moments, seemed curiously inactive, though somewhere there\nwas inside me a little bit of tissue, so to speak, that agonized, and\nhoped, and prayed. But for the most I only thought of one thing--that\nonce before I had gone on just the same errand, from this same house, up\nthe same road, to fetch the doctor for her, my dearest friend. O\nMargery! I go quickly to God and tell Him.... We want Him.\n\nAnd then the tissue that agonized and prayed sank out of sight again,\nand I was just speeding up the sunny, dusty road, on which, as I got\nnearer the town, the traffic became denser. Once a butcher's cart pulled\nsuddenly out into the middle of the road in front of me, and I thought\ncollision was inevitable, except that I knew that it was not possible\nthat I should be stopped when going on such an errand as this, and\nseveral times I passed people I knew, yet, though I knew them, their\nfaces were meaningless: they conveyed names, but nothing whatever more.\nAnd then--whether very soon or countless ages later, I had no idea--I\nwas at the doctor's door in the quiet, decorous street, which also was\nmeaningless--neither strange nor familiar, but purely without\nsignificance. Everything I saw was detached; nothing had any relation to\nlife, except just one thing: his dog-cart, which was at the door,\nconcerned me.\n\nHe had not yet started on his rounds, and it was not five minutes before\nhe was ready. He had only to pick up a little bag, into which he put a\ncase of some kind, and something bright, that I turned my eyes from, and\na bottle which he wrapped up--it seemed to me very neatly and\nslowly--which clinked against that which was already in the bag.\n\nThen he turned to me.\n\n'Now, if you take my advice,' he said, 'you won't come back with me, but\nwill go for a ride on this beautiful morning. You will not see your\nwife, and for the next hour or so it is not possible that I should have\nanything to tell you. We don't want you in the house: we don't want to\nbe bothered with you.'\n\nHe got briskly into his dog-cart, nodded to me over his shoulder, and,\ninstead of driving himself, gave his servant the reins. I know I shouted\nsomething after him, telling him, I think, to be careful, and so found\nmyself on the doorstep, looking at a bicycle which was leaning against\nthe pillar of the porch, and was evidently not mine. But, like the\ndog-cart, it was not meaningless, for it was Helen's, which I must have\nused by mistake. I must take it back; it was careless of me.\n\nThen his advice occurred to me, but it sounded ridiculous, as senseless\nas some nursery-rhyme. And at the thought there suddenly started in my\nhead the first two lines of 'Humpty-Dumpty.' I could not remember the\nlast two lines, but the first went round and round in my brain, keeping\ntime to my pedalling.\n\nSoon after I was home again, only a moment behind him, for he was just\ngetting out when I came to the gate, and I waited till he had gone in,\nso that he should not know I had failed to follow his advice--at least,\nI believe that was the reason, but I am not sure.\n\nI went round by the back way into the garden, and sat down in the\nveranda outside my own room, where Fifi was lying in the sun. But I had\nto coax her silently indoors, for I could not bear that she should lie\nthere, lest suddenly she should again look out into the garden, and howl\nat something she saw there. She would not come in at first, and once she\npricked her ears at something she saw outside, and I stopped mine, lest\nI should hear her howl. And all the time 'Humpty-Dumpty'--the first two\nlines of it--went on and on. It was so terribly lonely, too--just that\nsilly rhyme, and I all alone. If only Legs were here, or\nanybody--anybody. You see, this was not expected to-day, nor for weeks\nyet. My mother was coming to stay with us next week, until....\n\nThen I heard the muffled sound of steps in the room just above my\nhead--Helen's room--and at that for a little the babble and confusion of\nmy troubled brain cleared, and 'Humpty-Dumpty' ceased, and I was not\nafraid of Fifi howling, for there was no room for anything except the\nthought of Helen, who lay there, and of the life yet unborn. And I could\nnot help--I could not bear any of it for her. I could not even be with\nher: birth was as lonely as death.\n\nOutside the garden lay basking in the heat of the early summer, and\neverywhere the expansion of life, which had seemed to us so wonderful\nand glorious a thing through all these weeks of May, suddenly became\nsinister and menacing. What travail may not go to the opening of a\nsingle flower, or the maturing of its casket of seeds? It would all be\nof a piece with the cruelty and the anguish that runs through life like\na scarlet, bleeding thread, beginning, as now, even before birth, and\nnot even ending with death, since those who remain have the wound of\nthat yet to be healed. Right through life goes the scarlet thread,\nknotted on the farther side at each end, so that it shall not slip.\nAnd--'Humpty-Dumpty sat on a wall.' Ah, yes! I had it all now. 'The\nKing's horses' was what I could not remember. And at that the crowd of\ntrivialities again came between my mind and me.\n\nWe had set up the croquet-hoops again only last week, and had argued\nover the position of that particular corner one by which my ball had\nrested when last autumn a telegram had been brought me from the house.\nHelen had said it was square with the corresponding corner; I knew it\nwas not, and from here it was perfectly easy to see that she had been\nwrong. I hate an awry disposition of hoops. 'All the King's horses' ...\nthey really should bring these rhymes up to date; it ought to be\nmotor-cars instead of horses.\n\nThese things passed very slowly through my mind, for it acted as if it\nwas numbed and half-paralyzed, and the croquet-hoop occupied the\nforeground of it for a considerable time. I had let Fifi out again, and\nshe was racing about the lawn in the attempt to catch swallows, a feat\nof which she never realized the unreasonableness, and I had left the\ndoors into my room, both from the hall and from here outside, open. And\nthen, with the same rapidity as they had come, all these nonsense\nthings passed away again, for I heard steps on the stairs, and, going\nin, saw the doctor standing on the landing above, talking in low tones\nto the nurse. He saw me, made a little movement of his hand as if to\ndetain me, and when he had finished what he had to say to her, came\ndownstairs.\n\n'I will have a word with you,' he said gravely; and we went into my\nroom. I saw him looking at me rather curiously, and was wondering why,\nwhen he suddenly seemed to lean up against me. Then I perceived that it\nwas I who was swaying on my feet. He put me in a chair.\n\n'I suppose you have not had breakfast,' he said. 'You are to eat\nsomething immediately; I will ring the bell. And now listen. It is going\nto be difficult, and, I am afraid, dangerous, and it is better that you\nshould know it now.'\n\nAnd then the dear, kind man just laid his hand on my arm.\n\n'I'm awfully sorry,' he said; 'you can't think how I hate to tell you\nthis. I hope it will be all right; there is nothing yet that forbids me\nto hope that. Please God, we shall pull her through, but--well, well.'\n\nHe broke off as the door opened, and a servant came in.\n\n'Just bring a tray in here,' he said. 'Tea? Yes, tea, and an egg and a\ncouple of bits of toast. Thank you.'\n\n'Remember, I still hope it will be all right,' he said. 'And even\nif--well, you are both young still. Now I shall be back here in an hour\nat the outside.'\n\n'You are not going,' I said. 'You mustn't.'\n\n'Yes, yes. I know what you feel,' he said. 'But there is nothing for me\nto do here yet, and I have to make arrangements so that I can come back\nand remain here till all--is satisfactory.'\n\n'You don't stir from this house,' I said.\n\n'Do you think I should go if there was the slightest possibility of your\nwife needing me?' he said quietly.\n\n'No; I beg your pardon.'\n\n'That's all right. Now when your breakfast comes, eat it, and read a\nbook if you can, or go and garden. I am sure those roses of yours want\nlooking after, and I tell you it's a hard thing for a man in your\nposition, and a thing which we doctors respect, to go and occupy\nhimself. If you can't, you can't, but you might have a try.'\n\nThe servant brought in a tray before many minutes, and with it the\nmorning paper. When I had eaten, I took it up and looked at it. There\nwas no news, but the middle page contained an account of a visit to\nVesuvius by an English Prince. He 'honoured' the volcano with a visit.\nAnd then I knew that I had seen the paper before. But when? Years and\nyears ago, or this morning?\n\nWhat the doctor had said to me needed no time or thought for realizing\nit. I felt as if I had known it all along--known it all my life.\nBut--what happened next, if that all happened long ago? Was the room\noverhead the chamber of death or the chamber of birth? Next door to it\nwas the nursery, with its Noah's ark and its soldiers and its\nrocking-horse. Who going to ride on that? And the dolls'-house, with its\ntottering inhabitants--who next was to play with those, and open the\nwall? Oh, Helen, Helen, you and your child, will it be? Or will it be\nyou and I again, but after a long time, hoping once more? Or--dear God,\nno, not that!\n\n       *       *       *       *       *\n\nDaffodil-day, and its sisters of the spring! And Rose-day will come next\nmonth. Roses ... heaped for the beloved's bed. Dear God, not that: it\ndoes not mean that bed. Indeed--indeed it does not. You have so many\nsouls already in Your house of many mansions. Give us a few more years\ntogether, for they are so sweet, and a thousand years in Your sight are\nbut as yesterday. And we should so like a young thing, one of our own,\nin the house. But ... thank You very much for the years that have been\nso sweet. They have been--they have been. And, please don't let her\nsuffer or be frightened.\n\nThen I went across the lawn and into the rose-garden. Though we had been\nvery industrious there, I never saw yet the rose-tree on which there is\nnothing to be done, and for a little my hands made themselves busy. Then\nquite suddenly it all became impossible, and there was nothing in the\nworld except what the doctor had told me, and floating on the top of\nthat 'Humpty-Dumpty, Humpty-Dumpty.'\n\nSo it was within the hour that I got back again to the house, and the\ndoctor had not yet returned. I missed something familiar on the lawn,\nwithout at once knowing what it was, and then I saw that the birds'\nbreakfast was not there. That took me to the dining-room, where I found\nlunch was already laid, and with bread-crumb and little bits of cheese,\nand cold meat mixed, I made a plateful for them, though, as you know, it\nwas the last day of May, and I suppose it was but pauperism among the\nthrushes that I encouraged. But Helen all these days had done so. I knew\nshe would not like them to miss their provision.\n\nSoon after--so soon that the news of their belated meal had not yet\nbecome public among the birds--the doctor returned. I heard him go\nupstairs, and after that I crept into the hall, and sat down on the\nlowest step of the seventeen that led to the landing. Legs used to jump\ndown them in two bounds, taking eight steps first, and then nine, and\nget up (with a run) in three--two sixes and a five.... What am I\nmaundering about? And before very long I must have been sitting higher\nup the stairs, for I could see out of the window on the staircase. The\ndog-cart had drawn away from the door into the shade, and the groom had\ngot down, and was gently stroking the mare's nose. Then he laid his\nsmooth young cheek against it, and she stood quite still, liking it. I\nexpect he is kind to her.\n\nThe sun had swung round farther to the west, and it came in through the\nwindow. But now I was nearly at the top of the stairs; there were but\nthree above where I sat. The house was very still; below me on the\nground-floor there had been no step or sign of life, and there was\nnothing from behind the second door to the left just above me. Then came\nthe sharp tingle of an electric bell. There was only one room from which\nit could have come.\n\nI tapped very gently, though my heart beat so that I thought it must\nhave been a hammer-noise to those inside. The door opened a chink, and a\nlevel, quiet voice said: 'Some hot water, please--very hot.' Perhaps a\nminute afterwards I tapped again, and a hand took the can of hot water\nfrom me.\n\nI went back again, this time to the top step, and still waited. Since I\nhad done something, though it was but the handing of a can of hot water\ninto the room, that nightmare of incoherent thoughts began to clear more\ncompletely, and, like some remembered sunlight breaking clouds, and\nshining with the serene quietude of eventide, Helen--she herself, no\nintercepted vision, no vision even of remembrance only or\nanxiousness--shone out. Whatever happened, she was I, and I was she, and\nthe Will of God, whatever It might ordain for us, could not alter that.\nShe and I, I think, have never feared anything when we were together,\nand surely of all days that life or death could hold for us, we could\nnever be more together than to-day. So, surely, of all hours this is the\none when fear should be farthest from us, for never have we been\ntogether like this. Yet, O my God, my God, since Christ was born of a\nwoman, let Him go in there, the second door....\n\nAnd the next door, You know, is the nursery.... No, not the farther one,\nbut the one this side. Yes, yes, of course You know, but You might have\nforgotten. There's the Noah's ark there, and the dolls'-house, and the\nlead soldiers. We had hoped....\n\n       *       *       *       *       *\n\nRed light came in through the window on the stairs--light of sunset.\nOnce more the stinging sound of the electric bell came to me; once more\nI took up a can of hot water.\n\nThen it grew dark; in the hall below the lamp had been lit, and from the\nwindow, after the last red of sunset had faded, there came the distant\nshining of stars, endlessly remote. Then the door opened again, and the\nnurse came hurrying out, forgetting to close it. From within came the\ncry of a child.\n\n       *       *       *       *       *\n\n_June 1._--I overstep the bounds of the year, but you may like to know.\nQuite early this morning I was allowed to go in and look. They were\nsleeping, both of them--she and he.\n\nAfterwards I went into the nursery.\n\n\n                               THE END.\n\n                   *       *       *       *       *\n\n                           ESTABLISHED 1798\n\n                            [Illustration]\n\n                               T. NELSON\n                               AND SONS\n\n                             PRINTERS AND\n                              PUBLISHERS\n\n                   *       *       *       *       *\n\n                                  THE\n\n                            NELSON LIBRARY\n\n                           OF NOTABLE BOOKS.\n\n              _Uniform with this Volume and same Price._\n\n                           _ALREADY ISSUED._\n\n\nLIFE AT THE ZOO.            C. J. Cornish.\n\n     The Zoo is one of our great national playgrounds, and Mr. C. J.\n     Cornish, who had few rivals as a naturalist, provides in this\n     volume a most instructive and fascinating guide.\n\nTHE FOUR MEN.      Hilaire Belloc.\n\n     What \"The Path to Rome\" did for Central Europe Mr. Belloc's new\n     book does in equally pleasant fashion for the south country of\n     England.\n\nTHE CRUISE OF THE \"FALCON.\"       E. F. Knight.\n\n     Mr. E. F. Knight's tale of his cruise in distant South American\n     waters in a small yacht is one of the classics of sea adventure.\n\n\n\n\n\nEnd of the Project Gutenberg EBook of A Reaping, by E. F. Benson\n\n*** ","meta":{"redpajama_set_name":"RedPajamaBook"}}
+{"text":"\n\nHealth Communications, Inc.\n\nDeerfield Beach, Florida\n\nwww.hcibooks.com\nThis book contains general advice and is not intended to be, nor should it be, used as a substitute for specific medical advice from a physician.\n\nLibrary of Congress Cataloging-in-Publication Data\n\nJames, Genie.\n\nThe fountain of truth : outsmart hype, false hope, and heredity to recalibrate how you age \/ Genie James, MMSc, with C.W. Randolph Jr., MD.\n\npages cm\n\nISBN 978-0-7573-1715-6 (pbk.)\n\nISBN 0-7573-1715-4 (pbk.)\n\nISBN 978-0-7573-1716-3 (ePub)\n\nISBN 0-7573-1716-2 (ePub)\n\n1. Longevity\u2014Popular works. 2. Aging\u2014Prevention\u2014Popular works.\n\n3. Medical misconceptions\u2014Miscellanea. 4. Medicine, Popular\u2014Miscellanea.\n\nI. Randolph, C. W. II. Title.\n\nRA776.75.J35 2013\n\n613.2\u2014dc23\n\n2013004587\n\n\u00a92013 Genie James.\n\nAll rights reserved. Printed in the United States of America. No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without the written permission of the publisher.\n\nHCI, its logos, and its marks are trademarks of Health Communications, Inc.\n\nPublisher:\n\nHealth Communications, Inc.\n\n3201 S.W. 15th Street\n\nDeerfield Beach, FL 33442-8190\n\nCover images \u00a9Fotolia.com, 2013\n\nInterior design and formatting by   \nLawna Patterson Oldfield\n\nE-PUB formatted by Dawn Von Strolley Grove\nBecause I want them to know now what\n\nI wish I had known then,\n\nthis one's for the girls, especially:\n\nShelley\n\nSarah\n\nDanika\n\nLulu\n\nAshley\n\nRilyn\n\nMeghan Grace\n\nMadelyn Hope\n\nBella\n\nOlivia\n\nJulie\n\nTovah\n\nand\n\nGirls Inc. of Jacksonville, FL, Nashville, TN and Northwest Oregon\nContents\n\nAcknowledgments\n\nPrologue\n\nIntroduction\n\n[PART ONE: Your Age   \nof Opportunity](PART_ONE.html#PART-ONE--Your-Age--of-Opportunity)\n\nCHAPTER ONE: How Old Are Your Ovaries?\n\nCHAPTER TWO: Step 1: Get Rid of Belly Fat for Good\n\nCHAPTER THREE: Step 2: Move and Groove\n\nCHAPTER FOUR: Step 3: Detox Your Home\n\n[CHAPTER FIVE: Step 4: Find a Doctor   \nWho Can Help](Chapter_5.html#CHAPTER-5--Step-4---Find-a-Doctor-Who--Can-Help)\n\n[PART TWO: More Natural Tips   \nto Dial Back Your Age](PART_TWO.html#PART-TWO--More-Natural-Tips-to-Dial-Back-Your-Age)\n\nCHAPTER SIX: Four More Belly-Blasting Secrets\n\nCHAPTER SEVEN: Super Supplements\n\n[CHAPTER EIGHT: What's Good for Your Waist   \nIs Great for Your Face](Chapter_8.html#CHAPTER-8--What-s-Good-for--Your-Waist-Is-Great--for-Your-Face)\n\nCHAPTER NINE: Pump Up Your Pelvic Power\n\n[PART THREE: A Fresh Look   \nat Anti-aging Truths   \nAs Old As Dirt](PART_THREE.html#PART-THREE--A-Fresh-Look-at-Anti-aging--Truths-As-Old-As-Dirt)\n\nCHAPTER TEN: Faith\u2014A Proven Path to Wellness\n\nCHAPTER ELEVEN: Girlfriends Are Next to God\n\nCHAPTER TWELVE: Lighten Up and Play More\n\nCHAPTER THIRTEEN: Play More Sports\n\nCHAPTER FOURTEEN: Partner with Care\n\nCHAPTER FIFTEEN: Matter, No Matter Your Age\n\nEpilogue\n\nAppendix A: Favorite Belly Flat Recipes\n\nBreakfast\n\nMain Dishes: Perfect for Lunch or Dinner\n\nSalads and Side Dishes\n\nAppendix B: Resources\n\nReferences\nAcknowledgments\n\nWith Gratitude\n\nMy mother, Betty Frank Sandusky Williams, always believed\u2014and told me emphatically\u2014that I could be and do anything I wanted. I wanted to please her, to make her proud, but for decades I waffled through more ambitions than boyfriends, never quite finding myself or purpose until my mid-thirties. Today at fifty-three, I have found my passion: writing for and speaking to women about health and aging. I am also increasingly jazzed about running a healthcare company while championing the growth of women-owned businesses. My life, my work and this book are embellished with my mother's fingerprints. I like to believe all are legacies she would relish.\n\nWriting this book has been an especial privilege. Conversations with hundreds of women created the framework for a fresh dialogue on aging. From that platform, many busy physicians and medical researchers took time to painstakingly walk me through how emerging optimal aging science can translate into day-to-day lifestyle choices. Foremost among these game-changing healers is my husband and business partner C.W. (Randy) Randolph, Jr., MD. Heart and hats off to you, babe!\n\nYet this book would still be only wishful thinking had it not been for the belief and tenacity of Jill Marsal, founding partner of the Marsal Lyon Literary Agency. Then, Health Communications, Inc. (HCI)\u2014our publisher for From Hormone Hell to Hormone Well and From Belly Fat to Belly Flat\u2014stepped up to move my idea from concept to print. Much gratitude goes to Peter Vegso, president of HCI, for believing this book could make a difference in women's lives. Deep felt thanks also to Allison Janse, gifted editor and wise counsel, for guiding me away from the banal towards new information women need to hear now.\n\nKudos too to Kim Weiss, HCI's director of Communications, and Nannette Noffsinger, our media and public relations consultant. For close to a decade, Nannette has proven to be a rare blend of media coach, cheerleader and friend . . . one I would have thought out-of-my-league as she produced NBC's Today Show for over ten years and continues to work extensively around the world for NBC News. Thank heavens she is so down-to-earth lovely and, also, believes in this message.\n\nFinally, my deepest gratitude goes to our medical professionals and staff of both Dr. Randolph's Ageless and Wellness Medical Center and our Natural Medicine Pharmacy. Each day your skills and commitment translate theory and academic medical science into a life-changing healing ministry for the patients we serve. You all are incredible.\nPrologue\n\nAt the supposed midpoint of my life, two events forced me to internalize a duo of heart-nuking truths. The first: Laura Randolph's death. The second: my first gray pubic hair. Both snuck up on me. Both sucked.\n\nI should have seen Laura's death coming. We were uncommon best friends, she being my husband's first wife. Nevertheless, from the very first encounter, I was enchanted by her bubbly laugh, off-cuff Nora Ephron wit, and wicked cool style. Laura struggled with a rare hybrid of ovarian cancer off and on for twenty-five years, yet her seemingly boundless tenacity buoyed hopefulness, even optimism in those who loved her. Those last months I refused to see my friend leaving us millimeter by millimeter.\n\nThe pubic hair, the gray pubic hair, is shallow by comparison. But that's me\u2014both deep and shallow. I sat on the toilet completely flummoxed. I rarely wear glasses when going to the bathroom. When I do, I seldom scrutinize down there.\n\nHow long has it been there? Do I pluck? Has Randy seen it?\n\nThere was no way around these two magnitudinous facts. I was aging; someday I would die.\n\nAt some juncture in time, all of us are brought to our knees by the undeniable truths that we are no longer young and we will someday die. However, my personal come-to-Jesus moment was ironic: as chief executive officer of Dr. Randolph's Ageless and Wellness Medical Center, I had skin in the game of our culture's American-Idolish search for the latest and greatest fountain of youth. Actually, I had more than financial security and a career path invested: Dr. Randolph is my husband, Randy.\n\nRandy's Ageless and Wellness Medical Center is internationally recognized as a center of clinical excellence in anti-aging medicine. Our medical professionals treat more than ten thousand patients each year. As CEO, information on the latest and greatest anti-aging procedures and products swamped my desk; I sorted it into one of three buckets:\n\n  * Mind-bending research written by highly credentialed physician pioneers typically housed in respected academic medical centers;\n  * Questionable studies written by medical researchers with financial ties to the fountain-of-youth potions they plug;\n  * Borderline-trashy marketing hype.\n\nA fact: In this millennium, medical miracles have the potential to reduce our risk of chronic disease while positively impacting our long-term health, sexuality, and longevity. These radical advances are retooling the healthcare industry. What this means to you, me, our children, and generations to come is that we can increasingly get a heads-up before trouble hits.\n\nDiagnostic tests can tell us whether or not we have heart disease, breast cancer, or Alzheimer's genes. New scanning technology displays if our brains are snappy-fit or in danger of turning to mush. Genetic blueprints sharply reveal how youthful debauchery or years of poor lifestyle choices can result in long-term damage to DNA, setting us up to age more rapidly.\n\nMore news (if you, like me, have a vain streak): The fields of cosmetic and dermatologic medicine have shifted into spheres of supreme artistry. Unlike our mothers and grandmothers, if you or I decide to have \"some work\" done, we are unlikely to come out from under the knife (or the end of a needle) looking like tightly pulled Saran Wrap or a squishy-cheeked Persian cat. Better yet, we have multiple options for a face-lift on a budget.\n\nStill, this exploding anti-aging industry isn't always something to Hula-Hoop about. According to Global Industry Analysts, a 70- to 80-million-member-strong consumer base, \"seeking to keep the dreaded signs of aging at bay,\" will push the US market for anti-aging products from about $80 billion in 2012 to more than $114 billion by 2015. Sadly, many of the glitzy antiaging products we shell out big bucks for are just plain junk; worse, some can be harmful.\n\nWhen marketing gets ahead of medical science, bad things can happen. Hopes are preyed upon and wallets drained. Some supposed \"fountains of youth\" have been associated with serious side effects. At greatest risk for disappointment and despair: women like you and me, devastated by cottage-cheese thighs, a lengthening neck waddle, and an increasingly dry vagina\u2014or a gray pubic hair.\n\nWhen marketing gets ahead of medical science,   \nbad things can happen.\n\nThere I was, overseeing a medical center called \"Ageless and Wellness\" while silently grappling with grief, my own fears about aging, and a growing sense of purposelessness. Then one day as I sat flipping through the latest oversized magazine crossing my desk, something shifted. The magazine's header read \"The Future of Beauty and Anti-aging.\" Four gorgeous twenty-something females (two in bikini tops) hovered on its cover. I cringed. Photoshopped beauty shouldn't be our standard for aging. That kind of beauty doesn't even look real. And, in fact, it isn't real.\n\nPhotoshopped beauty shouldn't be   \nour standard for aging.\n\nI am fifty-three years old. Sure, I miss my cellulite-free rump, smoking energy, and pie-eyed ambitions, but I don't want to go back in time. Like any woman living more than two decades, I have suitcases full of sadnesses, wish-I-hads, and would-never-agains. I've lost loved ones, chose career over childbearing, and made the mistake of mixing men I love with my money. Still, each time something has gone wrong or fallen apart, I've had to learn how to make different, better choices. The decades I have lived are the reason I have increasingly grown more alive, more real.\n\nI looked again at the magazine cover and thought, What exactly does anti-aging mean anyway? If you aren't aging, aren't you . . . well . . . dead? I wondered, What if every woman could know right now what only the experience of the next two to four decades could teach? What if it were possible to glean and transfer that wisdom? What if I could facilitate a forum where women in their thirties and forties got a one-up on positive aging and living? What if I could enable women fifty, sixty, seventy, and beyond to internalize \"now or never\" and act accordingly? What if I could find myself again in the process?\n\nI made a decision to write a book that cut through the clutter and confusion surrounding this anti-aging movement. I committed to entwine new, mind-popping medical science with stories of women who are exquisitely traversing the decades. I imagined a future where this book would ignite a movement, shifting the paradigm of what it means to cultivate a more beautiful interior with age. What it means to grow more real.\n\nBut what is the truth about aging? How can we cut through the hype, false hope, and even the baggage of our heredity to discern a future that women can desire and even be excited about? I wrote this book to find out. By interviewing dozens of pioneering medical researchers, I discovered:\n\n  * Which medical science advances have the potential to reverse or retard aging at a cellular level;\n  * The clinically proven \"superconditions\"\u2014for example, lifestyle choices\u2014that most impact health, well-being, and longevity;\n  * How much of this emerging medical science on aging is readily accessible and actionable for the everyday woman on a budget.\n\nI also personally interviewed hundreds of women ranging from age nineteen to ninety-three. I asked them all:\n\n  * What do you fear most about aging?\n  * What are your regrets?\n  * What excites you now?\n  * What are you most looking forward to?\n  * What is the most counterintuitive advice on aging you have to offer?\n\nIn just a few months I had a database of fabulous, make-it-happen women: women thumbing their noses at the notion that we must grow increasingly fat, dull, dried-up, and disconnected as we age; women in their twenties and thirties caught up in the debate of whether women really can have it all, all at once, or in stages; and older women considering elastic-waist jeans, plucking chin hairs, and suffering heartbreaking losses yet determinedly becoming more vital, vibrant, and relevant every decade.\n\nIntersecting science with real-life stories spawned something special: a fresh approach to recalibrating how we age boomeranging many preconceived notions. I became increasingly excited as I researched and wrote. Then something unforeseen occurred\u2014a collision of knee-buckling disappointments and sadness caused me to lose my sense of purpose, question the faith that has always been my compass, and avoid my face in the mirror. It took me a while to find my way back. I didn't do it alone. I needed the help of new wisdom and fresh tools. I share this with you, too\u2014my story\u2014from my heart to yours.\n\nTell me a fact and I'll learn.\n\nTell me a truth and I'll believe.\n\nBut tell me a story and it will live\n\nin my heart forever.\n\n\u2014Native American proverb\nIntroduction\n\nI'm imagining you picked up this book for one of three reasons:\n\n  1. You don't feel as vibrant as you once did. When you look in the mirror, you are not thrilled with the face and body staring back at you. The adage \"You are as young as you feel\" makes you nauseous.\n  2. You've had a wake-up call. Either you or a woman you love has had a brush with something feared\u2014cancer, heart disease, multiple sclerosis, Alzheimer's\u2014and you are newly motivated to do everything possible to protect your health and to live longer.\n  3. You are at that tipping point, probably somewhere in your thirties, when things you once took for granted in your twenties don't come easy, if they come at all: quickly losing three pounds after a weekend binge of margaritas, chips, and salsa; having enough unquestioned energy to cheerfully work, go for a run, hang out with friends, and\/or cook dinner and take care of preschoolers; wanting and having sex all the time to preferring an extra hour of sleep over frisky bedroom moves.\n\nYou wonder, Is there something, anything, I can do to slow down or turn back my aging?\n\nIs it possible my getting older could somehow equate to my becoming better, becoming someone more versus less as the years go by?\n\nMy answer is a resounding \"Yes!\" We can't stop aging, and we shouldn't want to. But the idea that we can age differently, with greater self-reliance and ever-swelling joy, is not the wine talking\u2014it's a medical fact.\n\nPhysically, biochemically, mentally, and sexually, women peak in our twenties, plateau in our thirties, and begin a sharp descent in our forties. According to aging expert Mark Houston, MD, between forty and fifty is the decade of vulnerability. During these years, women age the equivalent of 18.6 years, while men age 15.2. From fifty on, we can make choices that positively impact our health and how we age; however, the older we are, the harder and longer it takes to slow down the clock or undo damage from decades before.\n\nFrom conversations with thousands of women, I have grouped the female life stages like so:\n\nStage 1: The Age of Arrogance\n\nYou are between twenty and thirty and you are in good shape, have high energy, think like a lightning rod, and have a sizzling sex drive. You're not particularly motivated to shake up lifestyle choices that work for you now, and you feel sorry for older women with droopy boobies, big bottoms, and lackluster\n\nlives. You are certain your future will never morph into such a clich\u00e9.\n\nStage 2: The Age of Reality\n\nYou are between thirty and forty. You have gained five to fifteen pounds, feel a bit sluggish, and have sex once or twice a week but fake it more than you want to admit. You wonder if you need a prescription for an antidepressant. You have started reading health magazines and following Dr. Oz on Twitter. You call your mom for advice.\n\nStage 3: The Age of Vulnerability\n\nYou are between forty and fifty. Everything seems to be spinning out of control: your periods, weight, hot flashes, workload, family responsibilities. You doubt yourself a little to a lot, especially when scrutinizing your reflection in the mirror. You go to doctors for help and can recite your mom's health history as if it were your own.\n\nStage 4: The Age of Now or Never\n\nYou are over fifty. Sure, you would like to be thinner and look younger, but something more profound has your attention: mortality, yours and that of those you love. A corner of your mind recognizes that now is your make-it-or-break-it time. If you don't thrust forward immediately to reprogram lifestyle choices, your health and dreams may fall away for good. You regard your mom's foibles with awakened empathy.\n\nWhatever age you might be, sit up and take notice. Right now is your \"Age of Opportunity,\" your crucial moment. While results may come easier and quicker if you are in your thirties, it is never too late to reap the benefits. My recommended approach integrates Randy's expertise in the emerging science of age-management medicine with enviable women's perspectives on \"fountains of truths\" as old as dirt.\n\nPart 1 describes why lifestyle medicine, the science surrounding the cellular-level impact of everyday choices on health and hormone balance, must become your personal manifesto. The biggest variable accelerating your aging: estrogen-related belly fat. Randy and I offer hope and solutions for naturally rebalancing unhealthy hormone levels, which will allow you to lose unwanted pounds and inches once and for all.\n\nPart 2 offers more natural tips to burn off your fat and turn back your age. Super news: What is good for your waist is also great for your face. Even better, follow my guidance and your sexual nature will transform you into a cocreative powerhouse.\n\nPart 3 draws from conversations with hundreds of women, sharing six proven pointers for filling up and spilling over with the best your life has to offer.\nPART ONE:\n\n[Your Age   \nof Opportunity](Content.html#PART-ONE--Your-Age--of-Opportunity)\n\nWomen today have the opportunity to age very differently than our mothers and grandmothers. We are no longer doomed to become increasingly fatter, sexless, and senile as hormone levels diminish and we move through and beyond menopause. Emerging science, called \"lifestyle medicine,\" proves beyond a shadow of a doubt that daily habits influence not only how fast and well we age but whether or not our genes express health or disease, thinness or fatness, contentment or depression. More exciting, breakthrough medical research surrounding the still-controversial topic of hormone replacement is so revolutionary you probably have not heard of it before\u2014and quite possibly your busy primary care physician hasn't either.\n\nWith awareness comes opportunity for new action. This means you have to take the lead. If you want the next twenty, thirty, or forty years of life to be a hell of a lot better than your first twenty, thirty, or forty years of life, you must assume personal responsibility for understanding your options and making informed decisions. Begin right now to make choices that will recalibrate your chronological age and override any hardwired, negative genetic coding. Again, I am here to help.\n\nFor the last five years I have worked with Randy to refine a holistic approach to turning back your inner clock at a cellular level. We begin with a methodology for determining the age of your ovaries, or how much older you are without them, then provide a four-step plan to recover years while also decelerating ongoing aging. Our four steps are:\n\n  1. Our Belly Flat Diet\u2014including foods and supplements\u2014nutritionally engineered to eliminate age- and hormone-related belly fat and keep it off for good;\n  2. A fun exercise strategy to decrease the overabundance of estrogen and to counter fat-packing stress hormones;\n  3. A strategy for eliminating toxins in your home that dangerously mimic estrogen;\n  4. And, when lifestyle choices alone are not enough, a natural, medically supervised approach to restoring needed hormone levels eroded by age and stress.\n\nIf you stick to this plan indefinitely, you will feel and look better as the years roll by. It will improve your health, flatten your tummy, give your face a more youthful glow, and increase your zest for what's yet to come. Is there any reason you should not get started today?\nCHAPTER 1:\n\n[How Old Are Your Ovaries (or How   \nMuch Older Are You Without Them)?](Content.html#CHAPTER-ONE--How-Old-Are-Your-Ovaries-)\n\nWhen the second edition our book, From Hormone Hell to Hormone Well, was awarded the 2010 Bronze National Consumer Health Information Award, I was instantly a speaker in demand. At one private girls' high school alumnae luncheon, the audience nodded, laughed, passed tissues, took notes, and gave me a standing ovation. I was preparing to head to the airport when Hannah, dean of students, asked if I had a few minutes. The door to her office barely shut before she broke down.\n\n\"I'm only forty-two but I feel\u2014and know I look\u2014much older. I've been dean for seven years, every year setting an enrollment record. Early on I secured the opportunity for our school to participate in a national study examining the benefits and pitfalls of learning in a single-gender environment. Released in 2009, the results put this school on the map.\n\n\"The study showed girls graduating from single-gender schools expressed more confidence in their math, computer, writing, and communication skills than girls attending independent coeducational high schools. Overnight we shifted from being a well-respected local girls' school to one of the most sought-after educational destinations for young women across the United States.\n\n\"Now should be the high point of my career. Instead, I feel stale, and my performance is lethargic. At home, I'm short-tempered and unenthused. My mother was a gifted pediatrician, but once she turned forty, she got fat and depressed, eventually losing all steam about her life's purpose. Is that my future?\"\n\nMany women share this woman's fears. Your mother's experience does not have to dictate yours. Hormones are no slam-dunk anecdote for what you are going through, but based on medical science and clinical evidence, a combination of lifestyle choices and natural hormone replacement will likely be a critical variable in turning you around.\n\nHow Our Ovaries Age\n\nChronological age is not necessarily the age of our ovaries. This is an important differentiation because ovaries are the chief production plant for our sex hormones, that is, estrogen, testosterone, and progesterone. The \"older\" our ovaries are, the fewer sex hormones they can produce and the more rapidly we age.\n\nClinical studies evidence how hormones are chemical messengers needed for life. At different ages and life cycles, hormones foster puberty, contribute to fertility, and support reproductive health. While less well understood within the traditional medical community, breakthrough clinical research out of such respected medical schools as Harvard and Emory University prove that optimum hormone balance should be considered a critical variable in a lifetime strategy for breast, heart, bone, and brain health.\n\nInsufficient hormone levels do more than trigger the onset of menopause. When hormone levels begin to wane and sputter as early as a woman's thirties, the negative impact is physical, mental, and emotional. Ovaries age more rapidly, and therein begin to slow down hormone production earlier, if you had:\n\n  * Your first menstrual cycle before age twelve,\n  * Your first child in your late teens or twenties,\n  * Multiple children,\n  * Long-term exposure to xenohormones, for example, synthetic, biochemically foreign hormones commonly found in our environment.\n\nAging accelerates even more when your aging body's lagging hormone levels shift you into natural menopause, or if you experience the shock of sudden menopause after a partial or complete hysterectomy.\n\nHow Many Damn \"Pauses\" Are There?\n\nThirty-six-year-old Lily sat down in Randy's office. \"I feel as if I am having an out-of-body experience. I snap at the kids, don't give a crap about work, and daydream about dark chocolate bars with sea salt rather than when my husband and I can have an evening alone. My oldest sister suggested I get my hormones checked, but she is fifty-two and through menopause. My periods are still regular, but a girlfriend said I might be premenopausal or perimenopausal. Just how many damn 'pauses' are there?\"\n\nAfter scrutinizing Lily's lab results, Randy revealed that her hormone profile and symptoms validated that she had moved from her \"reproductive years\" to a cycle medically termed \"premenopause.\" Many women and their doctors misdiagnose this life cycle because periods remain regular. In five to ten years, a shift in Lily's hormone level profile, along with irregular menstrual cycles, will shift her from \"premenopause\" into \"perimenopause.\"\n\nWhether you are a regularly menstruating, slightly pudgier, more irritable, low-libido-suffering premenopausal gal; a hot flashing, night sweating, irregular bleeding perimenopausal lady; an increasingly depressed menopausal woman experiencing loss of bone density, painful vaginal dryness, and increased risk of heart disease; or a postmenopausal matriarch doing crossword puzzles to ward off dementia, it is essential you understand how cellular-level shifts in hormone balance dictate your health and well-being.\n\nAs women, we physically, biochemically, mentally, and sexually peak in our twenties, plateau in our thirties, and begin a sharp descent in our forties. Recall, between age forty and fifty is \"the decade of vulnerability.\" Age fifty and beyond, the harder we have to work to reverse negative trends already in motion. Now let's examine the role our hormones\u2014or lack of them\u2014play.\n\nIn the Age of Arrogance, our twenties, hormone production is typically jamming, meaning levels of all sex hormones\u2014estrogen, progesterone, and testosterone\u2014are optimum. You start the negative aging trend when, as you move into your thirties, your ovaries are unable to produce as many needed hormones as they once could.\n\nAge of Reality thirty-something women are particularly at risk for a medical condition termed \"estrogen dominance.\" Because progesterone levels are the first to slow down, declining 120 times more rapidly than estrogen levels, the ovaries are producing way more estrogen than can be balanced by diminished progesterone levels. Common symptoms of estrogen dominance in the thirty-something woman include worsened premenstrual syndrome (PMS); weight gain around the belly, butt, hips, and thighs; decreased energy, loss of sex drive, and onset of headaches or migraines around the period. Clinical studies show lack of progesterone also negatively impacts breast, heart, bone, and brain health.\n\nPremenopausal women in their thirties and early forties having regular periods are most likely to be misdiagnosed. Rather than recognize the symptoms, test hormone levels, and treat hormone imbalances accordingly, many medical professionals miss the boat. Instead they Band-Aid the underlying issue with a prescription for an antidepressant or diet pill. In the United States, ob-gyns\u2014not mental health professionals\u2014are the number-one prescriber of antidepressants.\n\nDuring the Age of Vulnerability, the forties, hormones act in a stop-go, up-down hormone popcorn popper. Women in their forties having irregular periods are perimenopausal, literally meaning \"around menopause.\" During these years, progesterone levels decline even more significantly while estrogen levels become sporadic, sometimes high peaks, other times low troughs. Without stable estrogen, menstruation doesn't occur regularly and vaginal tissue can atrophy (weaken), making lubrication difficult and intercourse painful.\n\nTestosterone levels also begin to decline in our \"decade of vulnerability.\" When testosterone production sputters, usually mid- to late forties, women suffer loss of energy, decreased sex drive, loss of muscle tone, and less experience of sexual pleasure.\n\nFifty-one, the official launch of our \"Now or Never\" years, is the average age women in the United States enter menopause. Clinically, menopause is defined as not having had a period for a year or more. Our aging ovaries may still sporadically sputter, but levels of all three sex hormones\u2014estrogen, progesterone, and testosterone\u2014are in the toilet. We do not look or feel as good as we once did. Skin becomes dry and wrinkly; vaginas do not lubricate easily. Worse, health risks increase. Ongoing\/untreated hormone-level decline is clinically linked with an increased risk of breast and endometrial cancers, heart attack, stroke, osteoporosis, and Alzheimer's disease.\n\nDo nothing to boost lagging hormone levels and you stack the cards for a fatter, more disease-prone, emotionally labile, increasingly sexless, and potentially senile future.\n\nLike Mother, Like Daughter   \n. . . or Not?\n\nLike Hannah, I spent years worrying my menopause would mimic my mother's. Mother was the baby in a family of five children. Her daddy, Franklin Kingsley Sandusky, ran mills, first in Alabama and later Florida, that manufactured the tops of wooden barrels. At some point there was an accident. The fingers of his right hand were cut off on the assembly line. He was soon out of work. Granddaddy Sandusky then took up farming outside a little hiccup town called Bonifay, Florida. There was not much money. As a teenager, my mother was a hot one to handle, smoking cigarettes at school and sneaking out of her bedroom window at night.\n\nMother wanted to go to college but they couldn't afford it. She got a scholarship to a secretarial training program and lined up a waitress job to cover expenses. Sadly, Granddaddy Sandusky wouldn't allow it. I'm told Grandmother Sandusky didn't have much say. At nineteen, Betty Frank married her high-school sweetheart, June Terry.\n\nAt twenty, Betty Frank Terry gave birth to a six-pound gumdrop named Sheila. At twenty-four, she learned June had testicular cancer. At twenty-five she was a widow . . . with a five-year-old and a $110-per-month life insurance check.\n\nNow what? Not much money, no education or skills, living in a town without a stoplight. What were her options?\n\nYou guessed right. At twenty-seven, my mother wed my father, Angus Douglas Williams, Jr., becoming Betty Frank Williams. Four years later, I was born.\n\nFor the next ten years, we were a happy family. We moved to Panama City, Florida, where Daddy worked as an Allstate Insurance salesman. I was severely asthmatic in my early childhood, and my pediatrician advised Mother that I shouldn't play any running games or sports. I was so overprotected, I had few playmates but entertained myself with a menagerie of pets\u2014including a chicken named Bambam and a skunk named Violet\u2014in our backyard.\n\nSometime in her early forties, something inside Mother soured. She began complaining of bruising headaches and had frequent, extreme mood swings and violent outbursts. After some of the worst, she seemed dazed and perplexed by the havoc in her wake. I've since learned, sadly from personal experience, that something of an amnesic fugue can occur with severe migraines.\n\nTo the outside world, we continued to be that \"happy family\" involved in the community and attending church. Yet from about ten years old until I escaped to college at seventeen, my life was a terror-breathing house of mirrors. Some days Mother was my most loving advocate, instructor of beauty, and strategic life coach. Too often, however, she was an irrational, raging shrew . . . a woman of whom I was afraid.\n\nIn Mother's early fifties, she calmed down and came back to herself, as if an exorcism had occurred. At first shyly, then diligently, she made peace with the women she loved most: Sheila, me, and her sister. There were emotional scars, but over time and with love we rediscovered faith in family and the magic of memory-making. Mother and I even went on a three-week girl-trip cruise to Greece. Then Mother, at only sixty-four, had a sudden heart attack and died.\n\nToday I fully believe my mother was literally a victim, a marionette, of her hormones\u2014or lack thereof. It is also clinically probable that an underlying hormone imbalance contributed to her heart disease. All evidence points to her suffering an extreme hormone disorder at a cellular level that jettisoned with the onset of perimenopause. The fact that her extreme behavior abated after menopause supports this premise.\n\nI am determined that my life will be calmer and longer and have a happier ending. Mother's story is my motivation to understand all I can about how hormone levels shift with age and what can be done to turn that around. The next chapters share with you what I know now.\nCHAPTER 2:\n\n[Step 1: Get Rid of   \nBelly Fat for Good](Content.html#CHAPTER-TWO--Step-1--Get-Rid-of-Belly-Fat-for-Good)\n\nYour belly fat is aging you right now. It is also increasing your risk of developing a chronic disease. If you are overweight, advice to \"eat less and exercise more\" probably makes you want to either cry or spit. Sadly, if you are older than thirty-five, you are likely in this crowd. Medical statistics show that women will gain an average of one to two pounds every year from the age of thirty-five to fifty-five.\n\nConsider both Wendy's and Almeda's frustration: Forty-three-year-old Wendy sobbed in the shower. Tonight she and Saj were attending an Athena Awards ceremony where Saj's mother Almeda was an honoree, but despite starving herself for weeks, Wendy still couldn't zip up a single pre-baby outfit. If she wore a maternity dress, people were bound to ask, \"When are you due?\" She would feel the fool telling them, \"My baby's two years old.\"\n\nWomen will gain an average of one to two pounds every year from the age of thirty-five to fifty-five.\n\nA few miles away, sixty-one-year-old Almeda stepped into her evening-out uniform, a plus-size 18 Eileen Fisher black pantsuit. She looked in the mirror, sighed, and reached for a brightly colored shawl. Maybe this shawl will hide my rolls of back fat. A few more years of unstoppable weight gain and I'll be buying clothes from a custom tent store.\n\nWendy and Almeda are not alone. A study funded by the Centers for Disease Control (CDC) and released May 2012 indicates that more than 30 percent of American adults are now overweight and 34 percent are obese. By 2030, a projected 42 percent of American adults will be obese and 11 percent will be severely obese.\n\nWhat is the link between getting older and fatter? More critically, is there anything to be done to reverse this trend? Our medically proven premise is this: Those extra pounds creeping up and cementing around your middle have little to do with your inability to diet properly, limit carbs, walk an extra mile, or do more crunches; they have everything to do with an age-related shift in hormone production. And, yes, this book can help you turn that around.\n\nHormone Imbalance   \nSabotages Your Waistline\n\nOdds are you were most fit and svelte in your twenties. Your \"reproductive years\" is the season of life when your hormones are in optimum balance, biologically stimulating you to procreate and conceive. The mix of essential hormones includes the sex hormones (estrogen, progesterone, and testosterone), as well as FSH (follicle-stimulating hormone), LH (luteinizing hormone), and GnRH (gonadotropin-releasing hormone). These interact as if in an intricate symphony, each playing a unique yet interdependent function within your body and brain.\n\nWithin the ovaries are follicles that store eggs, or ova. In her early twenties, a woman has approximately four hundred thousand follicles. By the mid-thirties, the number of follicles has fallen from four hundred thousand follicles to approximately twenty-five thousand. Each follicle releases another chemical messenger, Inhibin B, whose function is to literally \"inhibit\" the ovaries' production of estrogen. The fewer follicles, the less Inhibin B in your system, the more estrogen produced by the ovaries, sometimes up to 30 percent more than what was produced by the ovaries in the reproductive years. Simultaneously, progesterone production begins to decline at a rate 120 times faster than estrogen levels. As described in Chapter 1, this imbalance between rising estrogen and declining progesterone levels is clinically termed \"estrogen dominance.\"\n\nEstrogen dominance fosters weight gain around the belly, butt, hips, and thighs, and then\u2014double whammy\u2014body fat produces even more estrogen.\n\nThis cycle continues throughout the thirties, forties, fifties, and beyond. Even during and after menopause, estrogen dominance is still a concern.\n\nMany women make the mistake of thinking that if they are no longer having regular menstrual cycles, they no longer have to worry about sabotaging hormone levels or the lack thereof. This is wrong. It is a misconception to believe that once you stop having periods, your ovaries turn off like a light switch.\n\nAlthough menopause indicates a drastic shift in your body's hormonal equilibrium, it does not mean that your sex hormones have completely evaporated from your body. The ovaries of a menopausal woman are still quite active, producing 40 to 60 percent of the estrogen produced in younger years. Progesterone and testosterone production, however, will continue to decline.\n\nIf you are a woman who has entered an abrupt, artificial menopause as a result of a complete or partial hysterectomy, you can still be estrogen dominant. Even though you no longer have ovaries, your body fat is still producing estrogen.\n\nAs your body's progesterone production decreases with age and estrogen becomes dominant, your body releases insulin more rapidly and more often. Insulin is the hormone within your body primarily responsible for metabolism and fat storage. Most of the actions of insulin are directed at metabolism (control) of carbohydrates (sugars and starches), lipids (fats), and proteins. When fluctuating hormones unnaturally stimulate insulin release, you get hungry faster and often crave sugar. In fact, insulin-triggered food cravings can sometimes be uncontrollable, causing you to ingest more calories and pack on even more pounds.\n\nEstrogen Dominance   \nTies Up Testosterone\n\nAge-related hormonal discordance gets worse for your waistline. Estrogen levels that are too high stimulate an increase of sex hormone binding globulin (SHBG) two to three times the normal levels. SHBG then circulates through your bloodstream, tying up free testosterone like Velcro, therein negatively impacting healthy weight and body mass composition. Sexual desire and experience of pleasure also decrease. In your late forties and early fifties, the ovaries further decline in testosterone production, predisposing you to an even bigger, bulging middle.\n\nEstrogen Dominance   \nDisrupts Thyroid Function\n\nThe thyroid gland is best known for its metabolic function, which affects weight. Statistics show that one in eight women between the ages of thirty-five and sixty-five, and one in five women older than sixty-five, have some form of thyroid disease. Hyperthyroidism results from the body producing too much thyroid hormone, but far more common is hypothyroidism, the result of not making enough thyroid hormone. Of similar impact at a cellular level, estrogen dominance renders the thyroid dysfunctional, causing your body's metabolism to slow down. The resulting condition is called relative hypothyroidism.\n\nUndetected and untreated, thyroid disorders can not only cause weight gain, they can lead to heart disease, osteoporosis, and other serious health problems. About 26 percent of women in their mid- to   \nlate forties are diagnosed with some type of thyroid condition. Younger women are also at risk. Ten percent of women will have thyroid dysfunction following pregnancy. New mothers often ignore their symptoms, attributing them to postpartum depression\u2014just as middle-age women attribute their symptoms to menopause. Consider thirty-two-year-old Cassie and fifty-one-year-old Ruth: six months after Olivia was born, Cassie continued to feel low, not like herself. Looking in the mirror made it worse. Her once-shapely form remained in a pudgy caterpillar shape. She cried continuously but snapped Duane's head off when he asked how he could help. Not sure what to do, Duane called his mother to come.\n\nInstead of helping Cassie with the baby, however, Ruth ran between unfinished projects and the bathroom, where she constantly filled diaper pails with blood-soaked maxipads. She was so irritable that Cassie kept Olivia in a portable crib in her bedroom and only came out to fight with Ruth over the thermostat.\n\nDuane confided to his best friend, \"The baby is great, but when it comes to Mom and Cassie, I don't know whether to call a doctor, personal trainer, or a shrink.\"\n\nBottom line: age-related hormone imbalance, particularly estrogen dominance, is fueling the elastic jeans industry. Five pounds can quickly morph to twenty, thirty, or more. If this is happening to you, you've got bigger issues than elastic versus zippers. Your health is in jeopardy.\n\nEstrogen Dominance Is a Health Risk\n\nThe trouble with estrogen-related belly fat is that it not only makes you unhappy with how you look, it makes you feel lousy, puts your health at risk, and accelerates your aging. Estrogen-related belly fat is not limited to the extra layer of padding located just below the skin (subcutaneous fat). It also includes visceral fat, which lies deep inside your abdomen, surrounding your internal organs. Although subcutaneous fat poses cosmetic concerns, visceral fat is associated with far more dangerous health consequences. This is because an excessive amount of visceral fat produces hormones and other substances that can raise blood pressure, negatively alter good and bad cholesterol levels, and impair the body's ability to use insulin (insulin resistance). All of this can increase the risk of serious health problems, including:\n\n  * Breast cancer\n  * Cardiovascular disease\n  * Stroke\n  * Type 2 diabetes\n  * Colorectal cancer\n\nRecent research also shows that women in their forties who have belly fat are more likely to get Alzheimer's and other forms of dementia in their seventies. Other studies link belly fat to an increased risk of premature death\u2014regardless of overall weight.\n\nSpeaking of Aging Too Fast . . .   \nEstrogen Dominance and   \nPrecocious Puberty\n\nAbout 1 in 5,000 children experience early puberty. Studies suggest that, on average, kids are starting puberty earlier than they once did. Could the rise in obesity be playing a role? Many experts think so, at least when it comes to girls.\n\n\"I think it's quite clear that some of the early puberty we're seeing is related to obesity,\" says Paul Kaplowitz, MD, PhD, chief of the division of endocrinology at Children's National Medical Center in Washington, DC. \"It's not the whole story, but it's a factor.\"\n\nThe excess fatty tissue produces higher estrogen levels, which leads to greater insulin resistance, causes girls to have more body fat, and then produces more estrogen. Just as with older women, the estrogen-belly-fat-estrogen cycle feeds on itself, speeding up aging for preadolescent girls.\n\nBelly Flat Diet Eliminates   \nEstrogen Dominance\n\nOver the last decade, I have worked with Randy to hone in on the link between an underlying condition of estrogen dominance and age-related belly fat to offer women a solution. Our Belly Flat Plan begins with our Belly Flat Diet. Our clinically vetted, nutritional approach is biochemically engineered to eliminate the fat-generating extra estrogen in your system.\n\nOur Belly Flat Diet is a unique version of the Mediterranean way of eating, with extra emphasis on foods proven to decrease your extra estrogen load. The traditional Mediterranean diet emphasizes healthy fats from nuts, fish, and vegetable oils; fruits, veggies, whole grains, beans, nuts, legumes, olive oil, and flavorful herbs and spices; eating fish and seafood at least a couple of times a week; enjoying poultry, eggs, cheese, and yogurt in moderation; and saving sweets and red meat for special occasions. Like Randy, most doctors agree it is fine to top off with a splash wine in moderation (if you want).\n\nWe recommend adding to the Mediterranean diet specific \"belly-blaster\" foods proven to melt away unwanted pounds and inches. Follow our diet and you can lose those extra pounds around your middle once and for all. If you are in your thirties and just beginning to worry about a muffin top or love handles, our Belly Flat Diet can put your fears to rest for life.\n\nYour smaller waist will be one very noticeable benefit. Another less obvious but very real bonus of losing those pounds once and forever will be decreasing your risk of the previously named chronic diseases and very likely extending your life span.\n\nDaily Belly Blaster Foods\n\nCruciferous Vegetables and Indole-3-Carbinol (I3C)\n\nEating a variety of cruciferous vegetables is key to losing and keeping off belly fat. How estrogen is metabolized in the body is determined by an individual's biochemical makeup, with some people producing more 2-hydroxy derivatives (the \"good\" estrogens) and others producing more 4- and 16-hydroxyesterone (the \"bad\" estrogens). Consuming large amounts of cruciferous vegetables, such as broccoli, asparagus, cauliflower, spinach, Brussels sprouts, celery, beet root, kale, cabbage, parsley root, radish, turnip, and collard and mustard greens, has been shown to improve the production of \"good\" estrogen and foster an optimum 2\/16 estrogen ratio.\n\nBiochemically, here's what happens: cruciferous vegetables contain a phytonutrient called indole-3-carbinol (I3C). I3C has been shown to act as a catalyst to pull estrone down a benign pathway to 2-hydroxy estrone, thus decreasing levels of the carcinogenic 4- and 16-alpha-hydroxyestrone. To put it simply, this means that cruciferous vegetables can help decrease the body's load of unhealthy estrogens and reduce an overall unhealthy condition of estrogen dominance.\n\nCitrus Fruits: D-Limonene\n\nD-limonene, found in the oils of citrus fruits, has been shown to promote detoxification of estrogen. Common citrus fruits include oranges, grapefruit, tangerines, lemons, limes, and tangelos. Research also found that, when administered an extract of D-limonene, both male and female lab mice evidenced lower body weight.\n\nInsoluble Fiber\n\nThere are two types of fiber: soluble and insoluble. Soluble fiber dissolves in water and is degraded by bacteria in your colon. It forms a gel in your intestines, which regulates the flow of waste material through your digestive tract. This type of fiber is found in oatmeal, oat bran, dried peas, beans, lentils, apples, pears, strawberries, and blueberries. Soluble fiber is good for you, but no matter how much of it you eat, it won't influence your hormonal equilibrium.\n\nInsoluble fiber, on the other hand, can directly impact your hormone balance by helping decrease estrogen overload. Insoluble fiber binds to extra estrogen in the digestive tract. This extra estrogen is later eliminated in the body through the feces. According to the Harvard School of Public Health, sources of insoluble fiber include whole grains, whole-wheat breads, barley, couscous, brown rice, whole-grain breakfast cereals, wheat bran, seeds, carrots, cucumbers, zucchini, celery, and tomatoes.\n\nLignans\n\nGround or milled flaxseed, sesame seeds, and flaxseed oil are part of a food group called lignans. The friendly bacteria in our intestines convert plant lignans into the \"human\" lignans, primarily enterolactone, that have a weak estrogenlike activity. When there are low estrogen levels in the body, these weak lignan \"estrogens\" make up some of the insufficiency. When the body is estrogen dominant, however, the lignan \"estrogens\" bind to the human body's estrogen receptors, thereby reducing human estrogen activity at a cellular level.\n\nOily Fish\n\nFish oil, rich in omega-3 fatty acids, assists with the production of many hormones. Oily fish, such as salmon, sardines, and herring, all have high levels of omega-3 fatty acids, which have been shown to have a weak estrogenic effect. Cellular binding of omega-3 fatty acids \"tricks\" the body into reducing an underlying condition of estrogen dominance.\n\nThe role of omega-3 fatty acids is of particular note in the production of testosterone. According to a 2011 study published in the American Journal of Clinical Nutrition, eating a diet rich in oily fish can improve body-fat ratio by promoting muscle protein synthesis.\n\nYou will find tasty recipes in Appendix A. Follow our Belly Flat Diet Plan for just four weeks for a tighter tummy, rebalanced body, and recharged life.\n\nBLAST AWAY YOUR BELLY FAT: Foods that Reduce Your Extra Estrogen Load\n\nCRUCIFEROUS VEGETABLES: Eat 2 to 3 servings a day (broccoli, asparagus, cauliflower, spinach, Brussels sprouts, celery, alfalfa, beet root, kale, cabbage, parsley root, radish, turnip, collard and mustard greens)\n\nCITRUS FRUITS: Eat 1 serving a day (oranges, grapefruit, tangerines, lemons, limes, and tangelos)\n\nINSOLUBLE FIBER: Eat 2 servings a day (whole grains, whole wheat breads, barley, couscous, brown rice, whole-grain breakfast cereals, wheat bran, seeds, carrots, cucumbers, zucchini, celery, and tomatoes)\n\nLIGNANS: Eat 2 to 3 tablespoons a day (ground or milled flaxseed, sesame seeds, or flaxseed oil)\n\nOILY FISH: Eat 2 to 3 servings per week (salmon, herring, mackerel, and sardines)\nCHAPTER 3:\n\n[Step 2:   \nMove and Groove](Content.html#CHAPTER-THREE--Step-2--Move-and-Groove)\n\nStep 2 of the Belly Flat Plan requires you to get up and get moving. While you are likely well aware of how exercise burns calories, you may be startled to review the science proving how regular, strenuous exercise helps reduce estrogen dominance, decrease health-risky stress hormones, and decelerate aging at a DNA level.\n\nExercise Decreases   \nEstrogen Dominance\n\nIn 2007, the Penn Ovarian Aging Study showed that the more women exercised, the healthier their estrogen-progesterone ratio. Studies in such respected medical publications as the Journal of the National Cancer Institute and Cancer Epidemiology Biomarkers Preview have shown that after twelve months of regular exercise (thirty minutes\/five days per week), unhealthy estrogen levels drop an average of 13 percent.\n\nIn another 2010 study, postmenopausal women who regularly did intense exercise for a year had lower levels of estrogen compared to women who didn't exercise. Before joining the study, most of the 320 postmenopausal women ages fifty to seventy-four were overweight and did very little or no exercise. The women were randomly split into two groups: half the women kept up their usual level of activity during the one-year study; half the women did intense aerobic exercise for about 225 minutes (4 hours and 45 minutes) each week during the study.\n\nThe women in the exercise group usually worked out for about forty-five minutes three to five days per week. At least three sessions each week were done with trainers at a fitness center and the rest of the exercise was done at home. The exercise was intense and raised the women's heart rates to a target level above resting heart rate.\n\nThe researchers measured the women's levels of several hormones and a related protein at the beginning of the study, in the middle of the study (six months), and at the end of the study (one year). At the end of one year, the levels of estrogen were 7 to 9 percent lower in the women in the exercise program compared to the women who kept up their usual slack activity level.\n\nExercise Counters   \nFat-Packing Stress Hormones\n\nStress can make you tense and sick. It can also make you fat. Exercise counters all three.\n\nYour adrenal glands produce three stress hormones: adrenaline, cortisol, and DHEA. Short-term, urgent stress\u2014such as seeing your five-year-old reach for a hot skillet or having your husband ask you to watch him skydive\u2014triggers a rush of adrenaline. Long-term, chronic stress has a different impact at a cellular level.\n\nChronic stress is defined as a circumstance that exists for three months or more. Some more common chronic stressors for women include ongoing financial pressures, single motherhood, caring for an ill and aging parent, attempting to juggle a heavy workload and home life, or attempting to discipline an irascible teenager. Chronic stress causes the adrenal glands to first produce an overabundance of cortisol, then, once this supply is exhausted, cortisol levels plummet. Too high or too low cortisol levels pack even more pounds around your waist.\n\nNumerous studies in the last several decades have confirmed that regular physical activity relieves feelings of stress and, at a cellular level, regulates cortisol production. Being active on a regular basis helps eliminate the threat that surging cortisol levels will contribute to an ever-expanding waistline.\n\nExercise Releases   \nFat-Burning Hormone\n\nA hormone released by the muscles during exercise transforms white cells into brown cells, according to research findings published in the June 2012 Harvard Health Letter. The hormone, known as irisin, also appears to overcome insulin resistance, a condition leading to type 2 diabetes. White cells simply store fat. Brown cells, in contrast, actually burn fat. And brown cells keep burning fat even after you have stopped exercising.\n\n\"Irisin travels throughout the body in the blood and alters fat cells,\" explains Dr. Anthony Komaroff, editor of the Harvard Health Letter. \"If your goal is to lose weight and keep it off, you want to exercise with the objective to increase the number of brown fat-burning cells and decrease the number of white fat-storing cells.\"\n\nExercise Turns Back   \nYour Cellular Clock\n\nIn heartening research published in 2011 in the Proceedings of the National Academy of Sciences, exercise reduced or eliminated almost every detrimental effect of aging in mice. The study followed a group of mice from birth. By the time they reached eight months, or their early sixties in human terms, the animals were extremely frail and decrepit, with spindly muscles, shrunken brains, enlarged hearts, shriveled gonads, and patchy, graying fur. Listless, they barely moved around their cages. All were dead before reaching a year of age\u2014except the mice that exercised.\n\nHalf of the mice were allowed to run on a wheel for forty-five minutes three times a week, beginning at three months. These rodent runners were required to maintain a fairly brisk pace, the equivalent of a person running a fifty- or fifty-five minute 10K (a 10K race is 6.2 miles). The mice continued this regimen for five months.\n\nAt eight months, when their sedentary lab mates were bald, frail, and dying, the running rats remained youthful. They had full pelts of dark fur, no salt-and-pepper shadings. They also had maintained almost all of their muscle mass and brain volume. Their genitalia had not shriveled, and their hearts were as good as new.\n\nEnough about rodents . . . what about women? What does new science say about exercise and aging? A study published in the January 2008 issue of Archives of Internal Medicine confirmed the beneficial impact of exercise at the cellular level. The London-based study was founded on the observation that telomeres (regions of repetitive DNA at the end of a chromosome) in white blood cells erode and shorten during the aging process. Thus, their length and quality are biological indicators of human aging, sort of an internal lifeline. Researchers compared the length and quality of the telomeres in 1,200 sets of twins; within each set, one twin exercised regularly and the other was sedentary.\n\nResearchers found that the longer, healthier telomeres of the active twin indicated a younger biological age\u2014sometimes by as much as nine years\u2014when compared to the biological age indicated by the shorter, degraded telomeres of the sedentary twin.\n\nSix Tips to Move More\n\nDo a Rooski\n\nElaine's morning routine of ten minutes of yoga followed by a brisk four-mile walk had kept her trim for twenty-plus years, but between forty-seven and fifty-four, her waist ballooned from size 8 to size 12. She tried leaving jam off her toast, forgoing afternoon cheese and cracker snacks, and walking an extra mile every day\u2014all to no avail. Her dejection was palpable.\n\n\"Please tell me there is something else I can do. I can't stand the idea of being stuck with this swollen body for the rest of my life.\"\n\nMetabolism slows down by 5 percent each decade. Women also lose 6 to 8 percent of muscle mass per decade. This is bad news because muscle burns the most calories. The final kicker: If, like Elaine, you always have the same workout routine, your muscles will adapt. This means that over time your favorite exercise routine will burn fewer and fewer calories.\n\nMy approach to outsmarting my muscles is rooski play. \"Rooski\" is a football term meaning \"trick\" play. By constantly switching up exercise routines with a variety of new muscle-building, strength-training, and heart-pumping activities, you can keep your muscles constantly guessing\u2014and those calories consistently churning and burning. Sure, you can continue to walk and run, but with options like water aerobics, Zumba dancing, hot yoga, paddleboarding, and rollerblading, there is even more fun to be had.\n\nBuddy Up\n\nLet's face it. Exercising can be boring. Add a friend, spouse, partner, or lover to the mix and suddenly being on the treadmill becomes a lot more interesting. Working out together is not only a great way to catch up, but it makes your calorie-burning time go by a lot faster. Whether you're trying out a kickboxing class, running on the track, or logging time on the elliptical trainer, doing it next to a friend means you have someone to talk and laugh with while you sweat.\n\nTake Up a New Sport\n\nIn grammar school, no one ever picked me for their kickball team, and my college tennis instructor told me I would be better off playing board games. At fifty-three, could there still be hope that I might find a sport that motivated me more toward fitness? Consider these role models: Academy Award\u2013winning actress Geena Davis has stated that she was not an athlete growing up and that her introduction to archery was in 1997. Two years later, Davis was one of three hundred women who vied for a semifinals berth in the US Olympic archery team to participate in the Sydney 2000 Summer Olympics. Don't think of archery as stationary. While it may not be an aerobic sport, it is a great workout for shoulders, back, and abdominal core muscles.\n\nKathy Martin, a busy working mother who had never competed at a track meet, started running in her late forties. In September 2011, she turned sixty. In the year that followed, Kathy ran in thirteen highly competitive races, including the Chicago Marathon and a cross-country championship in Seattle, and she set nine American and two world records.\n\nIn 2012, ninety-three-year-old Tao Porchon-Lynch was awarded the Guinness Book of World Records title of world's oldest yoga instructor. Tao's been teaching yoga for sixty-one years, and it has kept her in amazing shape. She loves doing it, saying, \"I am going to teach yoga until I can't breathe anymore. Then it's going to carry me to the next planet. I love yoga. It brightens my day, and it makes everybody smile.\" Even after having a hip replacement, Tao's still more nimble than many of her students. Oh, if you think yoga is not a \"team sport,\" Tao has more to offer. She's also a competitive dancer in tango competitions and partners with a man who is a whopping sixty-nine years younger.\n\nPost Goals for Friends and Followers\n\nTelling your mirror about your exercise plan may not keep you as focused as posting the proclamation to all 248 of your Facebook friends or tweeting it to dozens of followers. For New Jersey resident Colleen Lange, forty-one, what started out as a little weight-loss contest between a few close friends on Facebook grew into a group of more than twenty participants from four states\u2014some of whom she's never met. \"We post daily\u2014sharing recipes, supporting one another, even talking a little smack for extra motivation,\" she says.\n\n\"Connecting with people who share the same goals is like having your very own cheering section,\" says Rachel Meltzer Warren, RD, a New York\u2013based nutritionist. \"There's always someone there to celebrate when you drop those first ten pounds or help you get back on track if you regain three.\" Personal stories show it works, and the scientific community is paying attention: government-funded studies are under way looking at how technologies like social networking can help young adults achieve healthy weights.\n\nStay in Your PJs\n\nWhen I used time as an excuse, my cardiologist, Pam Rama, MD, said, \"Work out in your pajamas. Forgo the time and trip to the gym by plugging in a DVD or checking out the On-Demand channel on your television to access a variety of fun, fit exercises. I do it all the time.\"\n\nWhile you are in your pjs, consider . . .\n\nSexercise\n\nSex burns calories and can fuel weight loss. The average lovemaking session burns between 50 and 100 calories. Having sex three times a week burns 7,500 calories per year. That's the equivalent of jogging seventy-five miles. The more intense the sex, the more calories burned: up to 15,000 calories annually, which equals more than four pounds of fat burned away. In addition, certain sexual positions will strengthen your core, increase flexibility, and tone tummy, tush, and thighs. For details on a few frisky bedroom moves, check out Chapter 9.\nCHAPTER 4:\n\n[Step 3:   \nDetox Your Home](Content.html#CHAPTER-FOUR--Step-3--Detox-Your-Home)\n\nDisturbing studies show that everyday toxins in your home can disrupt hormone balance, accelerate your aging, and make you and your family sick. Exposure to environmental toxins, called xenohormones, is also emerging as a possible contributor to our fat epidemic.\n\nHow Toxins Make You Fat\n\nThe body accumulates weight-inducing toxins from air and water pollution, toxic foods, medications, and many other elements with which we come into contact. When toxins accumulate in the body, the liver and other organ systems try to filter them out of the body as rapidly as they come in. When the liver is overloaded, the body resorts to using fat to insulate the toxins from damaging the body's tissues.\n\nWhen a person tries to lose weight, it becomes difficult to remove fat that the body is using to protect itself. If a person does find a way to lose the weight, they often will see it come back more quickly, as the body will immediately try to conserve more fat where it was lost. Those who consume the most toxic foods, take medications, and exercise the least will gain weight the fastest.\n\nToxins Are Stealthy Dangers\n\nDanika, Randy's daughter and mother of our granddaughter Lulu, called, sounding worried: \"I thought breast-feeding Lulu was the best way to get our baby off to a healthy start, but I finished reading this book today. Now I'm scared.\"\n\nThe book, Breasts: A Natural and Unnatural History, by Florence Williams, tells the author's story of discovering her own breast milk was loaded with toxins. She says many toxins, including the flame retardants found in her breast milk, leach in from ordinary household items like couches and electronics, which often contain flame retardants. Animal studies have shown certain types of flame retardants interact as xenohormones.\n\nToxins in Danika's breast milk?\n\nThe idea floored me. Nevertheless, I suspected it could be true. While the possibility greatly concerned me, Randy and I encouraged Danika to examine this new information with pragmatism. The short- and long-term health benefits of breast-feeding have long been clinically established. The disturbing reality that xenohormones abound in our everyday environment, potentially accumulating in our bodies and particularly our breasts, should be a call to action. Rather than stop breast-feeding and forgo the proven benefits thereof, a better option could be to determine how women like Danika (and you and me) can decrease exposure to these toxic chemicals on a daily basis.\n\nHidden Estrogen Mimickers\n\nPersons living in the United States and Western Europe have been found to have much higher estrogen levels than persons living in underdeveloped countries. Environmental estrogen-like hormones in the foods we eat and the chemicals we use are often hidden causative agents of estrogen dominance. Estrogen mimickers in the form of chemicals (xenoestrogens), and foods and plants (phytoestrogens), mimic the action of estrogen produced in cells.\n\nAccording to Virginia Hopkins, coauthor of Dr. John Lee's What Your Doctor May Not Tell You About Menopause, \"Most of our exposure to xenohormones comes in very small amounts and any one tiny dose won't have a significant effect. The problem is that most of us are exposed to many tiny doses every day, all day and that has a cumulative effect.\" The additive effect of years of chronic exposure to environmental estrogens can contribute to a condition of estrogen dominance.\n\n\"Xeno\" literally means \"foreign,\" therefore xenoestrogens equal foreign estrogens. Xenoestrogens can be found in certain pesticides, herbicides, plastics, fuels, car exhaust, and drugs. Over time, these substances can increase the estrogen load in the body. Xenoestrogens can also be found in many meats and dairy products in the form of chemicals and growth hormones given to the animals.\n\nSome Beauty and Cleaning Products May Harm\n\nMany general hygiene consumer products\u2014such as creams, lotions, soaps, shampoos, perfume, hair spray, and room deodorizers\u2014contain petrochemicals. These compounds often have chemical structures similar to estrogen and can act like estrogen when introduced into the body. Some shampoos targeted at the African American community even advertise their estrogen content. A 2007 cosmetic industry's chemical safety assessments revealed that 57 percent of baby soaps, 34 percent of body lotions, and 22 percent of all personal care products contained petrochemicals. Need I say more?\n\nSimilarly, industrial solvents are another source of xenoestrogens. Industrial solvents are commonly found in cosmetics, fingernail polish, fingernail polish remover, glues, paint, varnishes, cleaning products, carpet, fiberboard, and other processed woods.\n\nDon't Drink the Water\n\nRandy and I reported in From Hormone Hell to Hormone Well how just drinking water from a faucet can raise your body's level of xenohormones. According to Marcelle Pick, ob-gyn NP, \"Researchers worldwide have observed that fish in our lakes and rivers are actually switching gender due to the high levels of effluent estrogens. Even though mainstream media has only begun to recognize this as 'news,' experts have been discussing the problem of pharmaceutical pollution for more than twenty-five years and have known about 'gender-bent' fish for more than ten years now! Some surmise these changes to be caused in part by excessive levels of steroids\u2014largely excreted by humans using birth control pills and synthetic hormonal replacement therapy (HRT). Our water treatment facilities are not designed to remove hormonal pollutants.\"\n\nPlastic Disrupts Hormone Balance\n\nBisphenol A (BPA) is another insidious danger. BPA is a weak synthetic estrogen found in many rigid plastic products, food and formula can linings, dental sealants, and on the shiny side of paper cashier receipts (to stabilize the ink). Its estrogen-like activity makes it a hormone disruptor, like many other chemicals in plastics. Hormone disruptors can affect how estrogen and other hormones act in the body by blocking them or mimicking them, which throws off the body's hormonal balance.\n\nBPA also seems to affect brain development in the womb. In 2011, a study found that pregnant women with high levels of BPA in their urine were more likely to have daughters who showed signs of hyperactivity, anxiety, and depression. The symptoms were seen in girls as young as three.\n\nLimit Consumption of Genetically Modified Organism (GMO) Foods\n\nMost corn and soybean crops grown in the United States are genetically modified, meaning their DNA has been altered to make them more resistant to viruses, bacteria, or insects. GMOs allow farmers to produce larger, healthier crops and make them available at more affordable prices. The uptick in crop production is needed to support rising food demand for a burgeoning populace. The downside is that several studies suggest that GMO foods can have a long-term toxic health impact.\n\nA December 2012 Kaiser Permanente newsletter spoke out against GMOs. The article cautioned that consumers should limit exposure to them by avoiding processed food and, when possible, by choosing organic or \"non-GMO\" labeled produce.\n\nWhat Now?\n\nA few years ago Randy and I   \ntrademarked Protect the Girls\n\nwith the goal of setting forth straightforward steps every woman could do to support her optimum hormone balance, healthy weight, and breast health. Our excellent news is that these same lifestyle actions also support healthy weight and reproductive development for little \"girls\" in our lives.\n\n  * Stick with our Belly Flat Diet for life.\n  * Exercise more with your daughter(s) and all the little girls you love.\n  * Buy organic and non-GMO foods when at all possible. Avoid hormone-treated meats and \u00adpoultry at all costs.\n  * Drink filtered water. Researchers worldwide have observed fish switching gender due to the high level of effluent estrogen in the water. The problem is attributed to high levels of synthetic hormones excreted by women using synthetic HRT or birth control pills. Our water treatment facilities are not designed to remove hormonal pollutants.\n  * Drink water from BPA-free containers.\n  * Never microwave in plastic containers. Avoid using plastic wrap to cover food for \u00admicrowaving.\n  * Detox your diet, home, and beauty regimen. \"Natural\" alternatives can sometimes be pricey. You'll find several terrific resources for money-saving tips for natural and organic living in Appendix B.\n\nCHAPTER 5:\n\n[Step 4:   \nFind a Doctor Who   \nCan Help](Content.html#CHAPTER-FIVE--Step-4--Find-a-Doctor--Who-Can-Help)\n\nForty-nine-year-old Esmeralda was confused. Over the last ten years her once-fit runner's physique had become mushy around the middle. Revving up for an extra mile and chomping on more celery sticks had no sustainable impact on her midriff flab. Now, constantly miserable with hot flashes and night sweats, she was beginning \"The Change.\" She knew she didn't like how she looked or felt. What she didn't know was what to do about it.\n\nDr. Ellison, her family's primary care physician for more than two decades, encouraged her to sweat it out but said, \"If you must have something, I can write a prescription for a synthetic conjugated estrogen with medroxyprogesterone acetate, but I would want you take it the shortest amount of time possible because of potential side effects.\"\n\nLizbeth, her forty-two-year-old sister, thought she had a better answer: \"Stay away from those drugs. I just read an article in a magazine while getting my nails done. It said take a supplement called Mexican Wild Yam.\"\n\n\"All bunk,\" declared best friend Phoebe. \"You need to see a doctor who understands and prescribes bioidentical hormones. I've been on them for seven years and have never felt better. But don't take my word for it. Go get your hormone levels tested.\"\n\nWhat Our Mothers Couldn't Tell Us and Many Doctors Still Don't Know\n\nNo wonder Esmeralda's frustrated. At some point in her life, many women will experience a reality where no matter how committed they are to our Belly Flat Diet, how much they exercise, or how conscientiously they detox their home, unwanted body fat glams back onto the belly and backside, and other uncomfortable symptoms emerge. When this occurs, it means the balance of sex, thyroid, and adrenal hormones has become so off-kilter that lifestyle alone is no longer enough to keep your hormones in balance. It's time for professional medical intervention. If this is your current situation, you must carefully vet the physician or medical professional you choose to help.\n\nFor the most part, mainstream medicine continues to view imbalances of life-sustaining hormones as \"normal.\" Most doctors rarely test for hormone levels and blindly reject the idea that restoring hormone profiles to youthful ranges is preventive health and a sound wellness strategy. Years before irregular bleeding, hot flashes, and\/or night sweats, symptoms of estrogen dominance can include anxiety, depression, fatigue, headaches (including migraines), worsened premenstrual syndrome (PMS), fuzzy thinking and\/or memory loss, breast tenderness, and low libido. As previously described, the consequence of this uninformed viewpoint is that many women are misdiagnosed and prescribed diet pills, diuretics, and\/or antidepressants. Others suffer discomforts and diseases largely correctable and preventable.\n\nThe reason for the confusion is this: Physicians are not adequately educated in medical school about hormone issues associated with the aging ovary. They have been trained to prescribe synthetic hormone replacement therapy drugs (HRT) that multiple studies in the last ten years have been shown to increase a woman's risk of breast cancer, heart attack, stroke, and dementia. Many physicians are unaware that they have other options to prescribe for their patients.\n\nIn the last several years, hormone replacement therapy (HRT) has frequently been in the news, sometimes as the grim reaper of death, other times as the panacea for ageless youth and sexuality. Magazine articles, television shows, and celebrities bandy about terms like \"natural,\" \"bioidentical,\" \"synthetic,\" or \"pharmaceutical\" as if they were interchangeable. They are not.\n\n\"Natural\" hormones are those hormones produced within the body by the ovaries, the testes, the adrenal glands, and the hypothalamus. These hormones travel through the bloodstream to fit into specific hormone-receptor sites located throughout the body and brain. Each hormone receptor site will recognize the specific molecular structure of a single type of hormone. This means that a receptor site for progesterone will not recognize estrogen or testosterone; it will only recognize the molecular structure of progesterone.\n\nHormones produced within your body attach to their receptor sites like keys fitting into locks. The chemical term for this key-and-lock phenomenon is called relative binding affinity (RBA). The hormones your ovaries make have a 100 percent RBA for their respective receptor sites.\n\nWhy Bioidentical Hormones   \nAre Safe and Synthetic   \nHormones Are Dangerous\n\nBioidentical hormones are derived from plants (soybeans or wild yams) using biochemistry processes. The biochemical process assures that the molecular structure of bioidentical hormones is identical to that of the natural human hormones once produced by your body. Just like the hormones your ovaries once made, bioidentical hormones have a 100 percent RBA for those hormone-receptor sites throughout your body and brain. When they plug in perfectly, your body and brain once again receive essential chemical messages that only hormones can deliver. Because our bodies recognize, accept, and respond to bioidentical hormones just as they would to hormones from our ovaries, bioidentical hormone replacement therapy (BHRT) is not only safe, it is extremely effective.\n\nIt is important to realize that the molecular structure of natural human hormones cannot be patented. Consequently, neither can the identical molecular structure of bioidentical hormones be patented. Without a patent, how could a pharmaceutical company protect its formulation and, most important (to the company), its profits? The answer: They can't. Consequently, for almost three quarters of a century, pharmaceutical companies have been developing, patenting, and marketing hormones that have a slightly different molecular structure from natural human hormones and bioidentical hormones. The pharmaceutically produced and patented hormones should be referred to as synthetic hormones. The list of synthetic hormones on the market today includes such brand names as Premarin, Prempro, Menest, Orthoest, Activella, and Femhrt, among many others.\n\nSynthetic hormones have shapes not seen in nature. Synthetic hormones' poor fit with the body's hormone receptors produces unnatural chemical reactions and striking alterations in biologic activity. As a result, their RBA is less than 100 percent, resulting in side effects and health risks. Premarin, for instance, is metabolized horse estrogen with a low affinity for binding with any human hormone receptor. Premarin is also composed of 49.3 percent of the cancer-promoting estrogen estrone (E1). This is almost ten times the ratio that occurs naturally within the body.\n\nIn July 2002, the National Institutes of Health (NIH) halted a large, in-progress study examining the effects of the widely used synthetic hormone replacement therapy medication Prempro, which combines the altered molecular structures for both estrogen and progesterone. (Note: synthetic progesterone is referred to as \"progestin.\") The study, one of five major studies that made up the large clinical trial called the Women's Health Initiative (WHI), was discontinued because the synthetic hormones were found to increase a woman's risk of breast cancer, as well as heart disease, blood clots, and stroke. Later findings linked synthetic hormone replacement to an increased risk for Alzheimer's disease. Findings were published in the Journal of the American Medical Association (JAMA).\n\nOver the last decade, irrefutable data substantiating the dangers of synthetic hormone replacement has continued to mushroom. The following is excerpted from Prempro's product label warning:\n\n\"Using [synthetic, conjugated from horse urine] estrogens with [synthetic] progestins may increase your chance of getting heart attack, stroke, breast cancer or blood clots.\n\nUsing [synthetic, conjugated from horse urine] estrogens with [synthetic] progestins may increase your chance of dementia. Using [synthetic, conjugated from horse urine] estrogen alone, e.g., Premarin, may increase your chance of getting cancer of the uterus. What is the most important information I should know about PREMPRO and PREMPHASE (combinations of estrogens and a progestin)?\n\nDo not use estrogens with progestins to prevent heart disease, heart attacks, strokes, or dementia (decline of brain function)\n\nUsing estrogens with progestins may increase your chances of getting heart attacks, strokes, breast cancer, or blood clots\n\nUsing estrogens with progestins may increase your chance of getting dementia, based on a study of women age 65 years or older\n\nDo not use estrogen alone to prevent heart disease, heart attacks, or dementia\n\nUsing estrogen alone may increase your chance of getting cancer of the uterus (womb)\n\nUsing estrogen alone may increase your chances of getting strokes or blood clots\n\nUsing estrogen alone may increase your chance of getting dementia, based on a study of women age 65 years or older\n\nYou and your healthcare provider should talk regularly about whether you still need treatment with PREMPRO or PREMPHASE.\n\nIf bioidentical hormone replacement is safe and has been proven to have multiple, positive, long-term health benefits, you may wonder why more physicians continue to recommend and prescribe synthetic hormones versus BHRT. The answer is a mix of ignorance, confusion, and marketing hype.\n\nRandy's Story\n\nLike every other physician of his generation, Randy was trained in medical school to prescribe synthetic hormones, the then-popular Premarin and Prempro, and he did for his first few years after he opened his medical practice. Early on, however, he became greatly concerned by the side effects he observed in his patients, for example, the weight gain, fibrocystic breasts, and increased mood swings. Convinced these could be early road signs for later health disasters, he began to research a safer alternative.\n\nPrior to attending medical school, Randy was a compounding pharmacist specializing in pharmacognosy, that is, natural, plant-based medicines. He drew on this background and also reached out to physician researchers, including Dr. John Lee (author of What Your Doctor May Not Tell You About Menopause and What Your Doctor May Not Tell You About Breast Cancer) and Dr. Joel Hargrove, then chief of the Department of Reproductive Medicine at Vanderbilt University.\n\n\"After reading Dr. Lee's books and reviewing Dr. Hargrove's research, I was fully convinced BHRT was the safest and most efficacious therapeutic option for my patients suffering symptoms of hormone level decline. This was 1996. I was certain that in a year or two, BHRT would be as common a protocol among my physician peers as performing routine pap smears or washing hands before and after a patient's physical exam.\"\n\nUnfortunately, the traditional medical community was slow to wake up. For more than a decade, Randy's peers ostracized him for prescribing \"that bioidentical snake oil.\" Their unfounded derision cost him more than his pride. In 2003, Randy called me one night from his truck. He was crying. \"They said if I don't stop prescribing BHRT, I'm out.\"\n\nRandy had just left the board meeting of North Florida OB\/GYN, a physicians' group-practice-without-walls of which he was a partner. These types of organized physician groups benefit members by centralizing business functions (such as billing, human resources, and payroll), using their combined purchasing strength to negotiate lower costs of medical supplies and negotiating higher reimbursement, for example, payment from insurance companies.\n\nIn the 1980s I served as executive vice president for PrincipalCare, a company that bought and sold ob-gyn practices, so I was readily familiar with how these organizations work. I understood immediately that if Randy exited this group and went solo, the financial impact would be humongous.\n\n\"Randy, this is a big deal. Maybe you should reconsider. If you have to take on responsibilities for all those business functions, I can guarantee your overhead costs are going to increase by a minimum of 30 percent, and how much insurance companies will pay you will decrease by 40. That's a 70 percent negative financial hit.\"\n\nHe did not hesitate.\n\n\"My patients matter more than money. Synthetic hormones harm women, put their lives at risk. BHRT is literally life-giving. I don't have a choice. I'll have to figure the business stuff out on my own.\"\n\nBy choosing to continue to prescribe BHRT for his patients despite looming detrimental business and financial implications, Randy stayed true to the spirit of the Hippocratic Oath by not harming his patients.\n\nIn 2008\u20139 two unlikely champions shifted the science of BHRT into the spotlight. Soon, many of those same physicians who had shunned Randy were calling for advice.\n\nTwo Unlikely Bedfellows\n\nThe about-face in how the traditional medical community diagnoses and treats hormone imbalances has been galvanized by two unlikely bedfellows: respected Republican senator Charles Grassley and revered Democrat Oprah Winfrey.\n\nIn December 2008, Senator Grassley wrote pharmaceutical giant Wyeth about the company's practice of medical ghostwriting. The senator then initiated an investigation and obtained documents from recent lawsuits involving Wyeth's synthetic hormone therapy products, for example, the Premarin and Prempro family of drugs, showing how Wyeth hired a medical communications and education company, DesignWrite, to draft review articles regarding the breast cancer risk of hormone therapy. In other words, Wyeth paid DesignWrite to produce legitimate-appearing medical articles saying only those things the company wanted doctors to believe.\n\nDid you just gasp? If you didn't, consider this: because doctors across the United States were fed inaccurate information about the safety of synthetic hormone drugs, millions of women's lives were put at risk\u2014very possibly your own or that of a woman you love.\n\nSenator Grassley had the gonads to take on Big Pharma. He and the US Senate Special Committee on Aging chairman and Wisconsin Democrat Herb Kohl drafted the Physician Payments Sunshine Act, signed into law in 2011. This law now requires public disclosure of all financial relationships between physicians and the pharmaceutical, medical device, and biologics industries, a big step toward pharmaceutical company transparency and truth in advertising.\n\nIn 2009, Oprah brought the conversation to a crescendo. \"The veil has been lifted,\" she declared on air, describing her personal experience with BHRT. \"If you are planning on living past thirty-five, this is something you need to know.\"\n\nOprah interviewed medical expert Christiane Northrup, MD, board-certified ob-gyn and internationally recognized author of multiple trailblazing books, including Women's Bodies, Women's Wisdom and The Wisdom of Menopause, and Robin McGraw, author of What's Age Got to Do with It? Women across America took note and called their doctors. Those doctors pooh-poohing BHRT soon discovered once-loyal patients hanging up and dialing a doctor who would listen.\n\nHow to Avoid a   \nThelma and Louise\u2013style Plunge\n\nWe have now firmly established that with age, hormone levels shift and decline. This does not happen in one fell swoop. It takes years and sometimes decades. Without help, you set yourself up to be a living metaphor of Whack a Mole. If you do not frequent a Chuck E. Cheese pizza parlor, you may not know this game. Here's how it's played: A child holds a mallet over a tabletop with multiple holes the size of a large coffee can. When the motorized mole pops up through a hole, the child squeals, attempting to slam its head. If the mallet hits the mole, the child scores a point. The metaphor: Age is the mallet. Without the right mix of the right kinds of hormones, you are the mole. BHRT can be your escape from this fiendish cycle.\n\nBHRT is a personalized medical approach to restoring optimum hormone balance. Specimens of saliva, blood, or urine can be analyzed to determine exactly which hormones you are deficient in. Then, based on your personal hormone level profile, a prescription can be individualized to give you back exactly what you are missing. These one-size-does-NOT-fit-all prescriptions are compounded onsite in a special compounding pharmacy.\n\nEven with BHRT, what works for you today will likely not work for you sometime in the future. Why? Because with age, levels will continue to shift. The following is a synopsis of my personal experience.\n\nI was forty-three when Randy and I met. I knew nothing about BHRT and was skeptical. Since turning forty, I had gained two to three pounds per year, making me ten pounds over my preferred weight. I was also having difficulty sleeping.\n\nRandy tested and analyzed my hormone levels, quickly diagnosing me as \"estrogen dominant.\" He recommended his over-the-counter bioidentical progesterone cream. Within a month, I dropped the extra pounds and was sleeping through the night. I must admit, I was surprised.\n\nAt forty-five, my periods had become irregular, my energy low, and I had once again moved my skinny jeans to the back of my closet. More advanced blood analysis determined progesterone levels had dipped even lower. I now needed prescription-strength bioidentical progesterone. I had now also become deficient in estrogen and testosterone, so Randy also prescribed a bioidentical prescription for each. With my new triumvirate of hormones, I thought I would be \"fixed\" for good.\n\nWrong.\n\nThree months following my forty-eighth birthday, I morphed up two pant sizes without any change in eating or exercising habits. New blood work analysis diagnosed a kerflooey thyroid system. A prescription for natural thyroid was added to my regimen.\n\nToday, at fifty-three, I no longer have periods, meaning I am officially postmenopausal. Over the last year, forgetfulness has nagged me, and, as my daddy would say, \"My get-up-and-go got up and went.\" Recent hormone level testing identified a pregnenolone deficiency.\n\n\"What the heck is pregnenolone?\" I asked Randy.\n\n\"Pregnenolone is an essential hormone for women of all ages. The average young adult produces about 14 mg per day. As with other hormones, however, pregnenolone production declines with age. At age seventy-five, the body produces about 60 percent less pregnenolone than it did at age thirty-five. This has led scientists to consider pregnenolone supplementation as a way to turn back the clock on aging and counter the consequences of this dramatic drop in hormone levels.\"\n\nI now have a bottle of pregnenolone next to my toothpaste and swallow a capsule every morning.\n\nPassionate About Progesterone\n\nCan you identify one hormone consistently deficient in all women over thirty-five? The answer is progesterone. Like a middle child, this second hormone in your sex hormone trilogy is often overlooked and its importance underestimated. I am passionate about helping women understand the critical importance of bioidentical progesterone supplementation. Here's why:\n\nProgesterone is often called the \"feel good\" hormone because of its calming, positive effect on moods. It also has a relaxation response that helps you sleep better. Wondering about your always-pooching tummy? Too much estrogen causes abdominal bloating while progesterone is a natural diuretic.\n\nAt a cellular level, progesterone does much more for our overall health and well-being. Evocative clinical studies by Drs. Kenna Stephenson and Helene Leonetti give evidence of progesterone's benefit to both heart and bone health. Researchers at Emory University have found a link between progesterone levels and long-term, optimum cognitive functioning.\n\nMost electrifying are the scientific studies investigating progesterone's role in cancer protection. Progesterone balances, or neutralizes, estrogen's propensity to promote cell growth. Unchecked cell growth is a precursor to cancer. Many studies validate how restoring optimum progesterone levels to eliminate an underlying cellular condition of estrogen dominance can have a cancer-protective effect.\n\nIf you want to support your body's optimum hormone balance for life, I recommend you consult a medical professional trained in the intricate endocrinology of hormone balance, preferably one board-certified by the American Academy of Antiaging Medicine (A4M).\n\nHowever, if you do nothing else, review all the clinical evidence underscoring the long-term health benefits of progesterone and consider using an over-the-counter bioidentical progesterone cream. Never, ever, ever take any form of estrogen\u2014even bioidentical\u2014without pairing it with bioidentical progesterone.\n\nTrust Your Instincts\n\nWhen you need a doctor or medical professional skilled in restoring hormone balance, do your due diligence regarding their training and board certifications in age-management medicine, as well as their clinical reputation for patient satisfaction. Ultimately, trust your gut: Is this someone who will listen to you? Is this someone you want on your team?\n\nMore advice on finding a medical professional trained and certified in hormone-level analysis, a compounding pharmacy, and a listing of vetted over-the-counter bioidentical progesterone creams are also included in Appendix B.\nPART TWO:\n\n[More Natural Tips to   \nDial Back Your Age](Content.html#PART-TWO--More-Natural-Tips--to-Dial-Back-Your-Age)\n\nOur Belly Flat Plan described in Part 1 is a great start to looking and feeling younger. Now I want to help you see and feel more positive impact more quickly. Chapter 6 describes four more belly-blasting secrets (I am betting three out of four of these you are really going to like). Chapter 7 provides a list of super supplements to accelerate weight loss at a cellular level and fill in any remaining nutritional gaps. Chapter 8 explains why what is good for your waist is also great for your face. Chapter 9 reveals why sexually vital women have decisively better quality of life and then gives options for rejuvenating aging private parts, restoring sexual vitality, and tapping into your innate feminine creative energy.\nCHAPTER 6:\n\nFour More Belly-Blasting Secrets\n\nYou love spending time with your girlfriends, but did you realize the power you have over each other when it comes to your weight? Your chances of being overweight or obese increase half a percent with every friend in your network who is obese, finds a November 2010 study from Harvard. That more than adds up: Your chances of obesity double for every four obese friends you have, say researchers. Even if that friend lives thousands of miles away, your chances of gaining weight still go up, according to a 2007 New England Journal of Medicine study. That may be because your perception of being overweight changes\u2014living larger seems acceptable since the heavy person is a friend.\n\nHang Out with Fit Girlfriends\n\nExperts believe your fit girlfriend's lifestyle and behaviors subconsciously rub off on you, but you don't have to ditch overweight friends to lose weight. In fact, research from Oxford finds making a pact to get fit and trim together is a bonus. Once a friend starts to lose weight, you have a greater chance of losing your unwanted pounds as well. According to Holli Thompson, women's health author and founder of NutritionalStyle.com, \"Some interesting studies have been released lately about how integrative those gal-pal relationships are and how they affect our size and ultimately our health.\"\n\nStudies out of Massachusetts Institute of Technology (MIT) confirm that common lifestyle, body weight, and body mass index are consistent in groups of girlfriends. Healthy friends tend to stick together and support one another's healthy lifestyle choices through exercise, better food choices, and even their optimistic outlook. Makes sense, doesn't it?\n\nHarvard University published studies in 2007 confirming that obesity spreads among friends, and that your chances of becoming obese or gaining weight is in large part dependent upon who you're hanging out with. Arizona State University recently went further with Harvard's study and published more findings, investigating three different pathways toward behavior that might attempt to explain this phenomenon. They found the \"monkey see, monkey do\" behavior to be the strongest; people don't necessarily think about body weight when they make decisions based on what their group of friends is doing. It seems to be very much a pack mentality when it comes to food, which is why you often see people of the same size together.\n\nYou've seen that in action. Your friend says, \"Forget the diet; I'm diving in for the ribs and chocolate cake.\" You're pulled. She's doing it\u2014why can't you? You want to have fun too. You find yourself eating more and more, \"having fun,\" and twenty pounds later, wondering what happened.\n\nThe MIT study shows that the opposite can be true. Your girlfriends can support you and lift you up to better choices. Use the power of the \"monkey see\" mentality to make a date with your girlfriends\u2014go for a power walk, check out the local vegetarian restaurant, meet at Barre class, and try not to make it all about the food. You need those girlfriends, and they need you. Realize the importance of their lifestyles and work toward healthier foods and activities for yourself and all your BFFs.\n\nPour a Small Glass of Wine\n\nSadie adjourned her last meeting as Temple board chair by inviting the entire board to lunch. Seated outside at the Sunset Grille, the women kvetched about the latest market tumble, gubernatorial fund-raising, and the pluses and minuses of long-wearing nail polish. Overhearing Sadie request a wine-by-the-glass list, conversation screeched to a halt.\n\n\"Celebrating?\" Kendra queried, disapproval in her eyebrows.\n\n\"Just sticking to my diet,\" Sadie smiled.\n\n\"Your diet?\" Neta squealed, tugging at her too-tight knit top.\n\n\"Yes, my diet. A few months ago our daughter Amy attended a medical conference in Boston on obesity and health risks. A paper was presented showing that women who were light to moderate drinkers were less likely to gain weight over time. Amy knows how frustrated I've become watching my weight creep up so she e-mailed the link to the study and urged me to give it a try. For two months now I've been having a little bit of wine every day. Without doing anything else different, I'm down eight pounds.\"\n\nStaying well hydrated catalyzes weight loss and staves off poor brain function, lethargy, and low moods. Water, herbal teas, broth-based soups, low-fat milk, and fresh produce all boost hydration.\n\nWine, though not hydrating, provides antioxidants and has been clinically correlated with less weight gain for women as they age. As stated earlier, Randy and most doctors agree that, a \"splash of wine\" is a healthful component of our Belly Flat Diet. Studies back them up: In 2010 researchers at Brigham and Women's Hospital in Boston published the first long-term study on women's drinking habits and weight gain. The study involved 19,220 women over the age of thirty-eight who were of normal weight. After thirteen years, women who consistently consumed a moderate amount of alcohol per day were 30 percent less likely to be overweight and nearly 70 percent less likely to be obese than nondrinkers.\n\nOnce again, there's a hormone connection. The relationship between alcohol consumption and insulin resistance evidences a U-shaped curve: insulin resistance was shown to be minimal in women with regular mild to moderate (3 oz.) alcohol consumption and higher in both heavy drinkers and women who never had a drink.\n\nFinally, for ladies loving the vine, a weight loss solution to toast!\n\nEat Some Chocolate\n\nA 2012 study found that people who frequently ate chocolate had a lower body mass index (BMI) than people who didn't. Is it time to ditch fat-free for fudge?\n\nFor the study, published in the March 26 issue of Archives of Internal Medicine, researchers examined more than one thousand healthy men and women who were free of heart disease, diabetes, and cholesterol problems. They were all enrolled in another study that measured the effects of cholesterol-lowering statin drugs, but for this study researchers assigned them questionnaires that gauged how often participants chowed down on chocolate.\n\nThe researchers found that the participants\u2014who were an average age of fifty-seven\u2014ate chocolate an average of twice a week and exercised roughly 3.5 times per week. But the more frequent chocolate-eaters had smaller BMIs, a ratio of height and weight that's used to measure obesity.\n\nWhat explains the effect? Even though chocolate can be loaded with calories, it's full of antioxidants and other ingredients that may promote weight loss. Research also shows that a little bit of chocolate can be good for your health. Compared with people who rarely ate chocolate (about one bar per month), the people who ate the most chocolate (slightly more than one bar per week) had a 27 percent and 48 percent reduced risk of heart attack and stroke, respectively.\n\nPlease don't read this section and believe I am giving you carte blanche to eat a chocolate bar a day. As with the recommendation on wine, the key is moderation, but researchers' definition of \"moderation\" can vary. After reviewing the scientific literature and clinical studies, Randy recommends an average amount of 6.7 grams of chocolate per day, corresponding to a small square of chocolate twice or three times a week.\n\nChew\n\nDr. Pam Rama is a superb cardiologist who has risen to the zenith of her career as the medical director of Baptist Health System's HeartWise program while also raising four children. I don't think I could tie her shoelaces. Add that to the fact that she is drop-dead gorgeous, and it would be easy to be intimidated, except she is down-to-earth and fun.\n\nWhen we scheduled dinner to talk about this book, I didn't realize until the menu was in my hands that I would be intimidated about what to order when eating with my cardiologist. I chose a green salad, grilled salmon, and steamed broccoli.\n\nActually, I thought, that is exactly what I would have ordered anyway.\n\nOur food arrived. I took a couple of bites, still chatting away. Pam stopped me.\n\n\"Chew.\"\n\n\"Huh?\"\n\n\"Chew more. Chew each bite at least thirty times, forty is better.\"\n\n\"Really?\" I responded, imagining with this new directive we could be at the restaurant until dawn. \"Why?\"\n\n\"New research shows chewing to have many health benefits, including weight control. Check it out.\"\n\nResearchers from Iowa State University (ISU) presented findings at April 2012's Experimental Biology Conference demonstrating that Dr. Rama's tip of taking your time by chewing longer and slower during a meal really can help you eat less. This weight loss tip is based on the fact that the longer you chew, the more the hunger hormone, called ghrelin, is suppressed, while another hormone, called leptin, whose role is to stimulate feelings of fullness, increases.\n\nIn the study, twenty ISU students were given a metronome and told to chew every time it ticked. The students were divided into two groups where the metronome ticked for fifteen times with one group and forty times with the other group before the two groups were allowed to swallow their chewed food. During the study, plasma glucose and hormone levels were measured, as well as the students' sense of appetite. What the researchers discovered was that the students who chewed more actually ate less than the students who chewed less.\n\n\"When people chewed the pizza forty times before swallowing, there was a reduction in hunger, preoccupation with food, and a desire to eat,\" said James Hollis, an Iowa State assistant professor of food science and human nutrition who coauthored the study.\n\nThe reason why chewing longer is healthier is this: The longer you chew your food, the more the food will be exposed to saliva, and as a result more nutrients will be absorbed. This is especially true for nuts and seeds, as well as fruits and vegetables, as they contain hard cellulose fibers that cannot be broken down anywhere but in the mouth. This is why when you eat nuts, corn, or other vegetables, they just seem to pass through your system if they are not properly chewed.\n\nWhen your body is able to absorb all the nutrients from the foods that you eat, you will have much higher energy levels. On top of that, no energy will be wasted on eliminating foods that the body cannot digest or break down.\nCHAPTER 7:\n\nSuper Supplements\n\nIn cages at McMaster University in Hamilton, Ontario, sits a group of mice who don't act their age. Unbelievably, they don't seem to be aging at all. For several years, the Canadian mice have been drinking a cocktail of thirty dietary supplements and vitamins. The concoction seems to be keeping the rodents young.\n\nScientists say the mice that were given the anti-aging cocktail had an unusual spring in their step. They had no loss of physical activity compared to a group of mice who didn't imbibe. In fact, those mice showed a 50 percent decrease in physical activity. Astonishingly, the combination of vitamins and supplements lengthened the life span of the rodents that took it by 11 percent. Did the Canadian scientists discover the legendary Fountain of Youth? If you are a mouse, they did.\n\nForget the mice, what about us girls? Can certain supplements keep us healthier? Can swallowing some pill truly help balance our hormones? Is there solid medical science to validate how nutritional supplements slow down or reverse our aging, or could our health and wallets be preyed upon by some vitamin and supplement manufacturer promising to turn back our inner clock?\n\n\"Yes\" is the correct response to all the questions above. Succinctly, yes, there is solid medical science to support the benefits of nutritional supplementation\u2014and yes, you could easily spend way too much money on anti-aging supplements that provide no true health benefits. This chapter provides a framework to help you sort through the hype and clutter to help you make your best informed choice about whether or not supplements are right for you.\n\nWho Needs Supplements?\n\n\"In an ideal world, no one would need dietary supplements,\" says Randy. \"Our Belly Flat Diet would provide all the vitamins, minerals, and nutrients your body needs no matter your age. Unfortunately, even women who do their best to 'eat right' will likely need some nutritional supplementation, because with age our bodies lose the ability to metabolize and store needed nutrients.\"\n\nStatistics indicate that dietary supplement use is widespread among US adults age twenty and older. The percentage of the US population who use at least one dietary supplement increased from 42 percent in 1988\u201394 to 53 percent in 2003\u20136 (Figure 1). Women were found to be more likely to use one or more dietary supplements than men. More recent industry survey results, many of which point to usage in 2011, settle around the 70 percent mark. Vitamins and dietary supplements (VDS) enjoy broad acceptance in the United States, where nearly half of all users reported using more than one supplement daily, and nearly 10 percent took five or more supplements a day.\n\nOne might argue that the growing trend of supplement use is a result of a baby-boomer consumer-driven frenzy to ward off the Grim Reaper, but guess what? Odds are, like Randy, your doctor and nurse are not betting on the food on the table to provide all the nutrients he or she needs.\n\nAn online survey administered in October 2007 to 900 physicians and 277 nurses by Ipsos Public Affairs for the Council for Responsible Nutrition (CRN), a trade association representing the dietary supplement industry, found that 72 percent of physicians and 89 percent of nurses used dietary supplements regularly, occasionally, or seasonally. Regular use of dietary supplements was reported by 51 percent of physicians and 59 percent of nurses.\n\nThe most common reason given for using dietary supplements was for overall health and wellness (40 percent of physicians and 48 percent of nurses), but more than two-thirds cited more than one reason for using the products. When asked whether they \"ever recommend dietary supplements\" to their patients, 79 percent of physicians and 82 percent of nurses said they did. Also, almost all said they recommend supplements to their patients, whether or not they took them themselves.\n\nMelt Belly Fat More Quickly\n\nDrawing on his original training as a compounding pharmacist specializing in pharmacognosy (plant-based medicine), Randy continuously researches and tests hordes of nutritional supplements to ascertain their true mechanism for influencing estrogen metabolism. In concert with our Belly Flat Diet, he recommends a select group of supplements that he has found to expedite the elimination of too much circulating estrogen, thereby enhancing healthy hormone balance and promoting weight loss and weight management more quickly. They are:\n\nCalcium D-Glucarate\n\nCalcium D-glucarate is a natural substance that promotes the body's detoxification process and supports hormonal balance. Calcium D-glucarate facilitates the detoxification process by inhibiting the reabsorption of estrogen-like toxins into the bloodstream, allowing them to leave the body and be excreted in the feces. In animals, calcium D-glucarate has been found to lower unhealthy estrogen levels and thereby inhibit the development or progression of cancer.\n\nDiindolymethane (DIM)\n\nDIM is a phytonutrient akin to the indole-3-carbinol (I3C) found in cruciferous vegetables. DIM has unique hormonal benefits. It supports the activity of enzymes that improve estrogen metabolism. DIM helps support PMS symptoms, fat loss, and healthy estrogen metabolism.\n\nThe B Vitamins\n\nThe B vitamins, such as B1, B2, B3, B5, B6, B12, and folate, do a lot within your body to support estrogen detoxification. Conversely, if your body is deficient in B vitamins, you will have higher levels of circulating estrogens. By now, you definitely know that increased estrogen levels lead to estrogen dominance, and estrogen dominance will most certainly lead to weight gain and the inability to get that weight off.\n\nB vitamins also impact estrogen activity for the hormone receptors at the cellular level. Clinical studies have shown that high levels of intracellular (e.g., within the cell) B6 can decrease the binding response at the estrogen hormone receptor site. What happens at the cellular level is sort of like an internal game of musical chairs: if the music stops and B6 sits down in the \"estrogen chair,\" then the estrogen molecule is out of the game.\n\nVitamin B complex supplements also may help with a variety of health conditions, including anxiety, depression, fatigue, heart disease, premenstrual syndrome, and skin problems. In addition, many people take a vitamin B complex to increase energy, enhance mood, improve memory, and stimulate the immune system.\n\nBecause the B vitamins work together to perform such vital tasks at the cellular level, Randy recommends you take the entire B complex, not just one or two of the B vitamins.\n\nVitamin E\n\nVitamin E has been shown to reduce PMS-related breast tenderness, nervousness, depression, headache, fatigue, and insomnia. Low vitamin E levels were linked to estrogen dominance. Furthermore, vitamin E deficiency has been found to inhibit estrogen detoxification.\n\nCalcium-Magnesium Combo\n\nMagnesium is another supplement that helps the body eliminate excess estrogen. For women, magnesium levels tend to fall at certain times during the menstrual cycle. These shifts in magnesium levels can upset an optimum calcium-magnesium ratio. In proper balance, the body better absorbs and assimilates the calcium it needs and allows calcium to migrate out of tissue and organs where it doesn't belong.\n\nWithout magnesium, calcium may not be fully utilized. Underabsorption of calcium can lead to menstrual cramps. Similar to a vitamin E deficiency, when the body does not have enough magnesium to support calcium absorption, many women report PMS symptoms, such as mood swings, fatigue, headaches, and sleeplessness.\n\nPremenstrual chocolate craving is a phenomenon that has puzzled a great many physicians. They have been unable to explain why some women have this overwhelming urge to eat lots and lots of chocolate right before their periods, yet at other times of the month, their chocolate cravings remain under control. Women find their PMS-related chocolate cravings abate when taking a calcium-magnesium combo.\n\n7- Keto Dehydroepiandrosterone (DHEA)\n\nDehydroepiandrosterone (DHEA) is one of the hormones produced by the adrenal glands. After being secreted, it circulates in the bloodstream as DHEA-sulfate (DHEAS) and is converted as needed into other hormones. Since it is a precursor to testosterone, DHEA may help build muscle. It is very unusual for anyone under the age of thirty-five or forty to have low DHEA levels. As we age, however, the body's production of DHEA declines, so people older than forty can most definitely become DHEA deficient.\n\nWhile many anti-aging enthusiasts are familiar with DHEA, far fewer are likely to be as aware of its metabolite, 7-Keto DHEA, which functions within the body to safely boost immune function and help reduce body fat. The term 7-Keto DHEA is, in fact, a brand name for the chemical compound 3-acetyl-7-dehydroepiandrosterone. Human blood levels of both 7-Keto DHEA and DHEA tend to rise and fall in a similar pattern with age: increasing until the twenties, beginning to decline in the thirties, and continuing to decline until the levels are reduced by about 50 percent by age fifty. Clinical studies have shown that as 7-Keto DHEA levels go down in middle age, body weight tends to go up.\n\n7-Keto DHEA stimulates weight loss through a process called thermogenesis. This term refers to the creation of heat at a cellular level. The more thermogenesis, the higher the metabolic rate and the more fat that is literally burned up as energy. Studies have also demonstrated that 7-Keto does not accumulate in the body over time and is free of unhealthy side effects.\n\nBecause 7-Keto DHEA is a natural hormone metabolite, it benefits the body in two ways: (1) it helps restore hormone balance while, at the same time, it (2) works internally to melt away those unwanted pounds.\n\nChitosan\n\nChitosan is processed from the shells of crustaceans such as shrimp, lobster, and crabs. Basically stated, chitosan acts as a superfiber. The swelling action of chitosan creates a sensation of feeling full, thereby serving to suppress the appetite. In addition, the superfiber characteristics of chitosan foster a natural cleansing process that is extremely vital to weight loss.\n\nChitosan is also able to absorb between six and ten times its weight in fat and oils. It then converts the fat molecules into a form that the human body does not absorb. When chitosan causes less fat to enter the body, the body has to turn to previously stored body fat to burn for energy. The net result: weight loss.\n\nSix Vitamins and Supplements to   \nSupport Hormone Balance\n\nCalcium D-glucarate: Take 1,000 mg twice per day\n\nDiindolymethajne (DIM): Take 200 mg per day\n\nB-Complex: Take 1 capsule per day\n\nVitamin E: Take 400 IU per day\n\nCalcium-magnesium combo: Take a ratio of two parts calcium (1,400 mg) to one part magnesium (750 mg)\n\n7-Keto DHEA: Take 100 mg per day\n\nChitosan: Take 750 mg three times a day\n\nEvery Woman Also Needs . . .\n\nI take more than a dozen supplements a day. That sounds like a lot to swallow, but my approach to nutritional supplementation is not hit or miss. I rely on my team of physician experts\u2014Randy and Drs. Lori Leaseburge and Pam Rama for guidance. Their shared medical opinion is that every woman older than thirty-five should also take the following supplements:\n\nMultivitamin\n\nThe Harvard School of Public Health suggests a daily multivitamin, calling it a \"great nutrition insurance policy.\" But what about the Wall Street Journal article \"Study Finds Multivitamins Don't Cut Risk of Heart Attack\" (November 5, 2012)?\n\n\"No single pill should be intended to replace diet and exercise,\" says Dr. Leaseburge. \"There is no magic bullet, but a good multivitamin will help even out the ups and downs of women's sometime irregular diet and lifestyle habits. And, with regard to that article and the studies cited within, I would question the quality of the multivitamin used in the study.\"\n\nNot all vitamins are manufactured equally (see more on how to choose quality supplements in the pages that follow).\n\nVitamin D\n\nAccording to Randy, \"Most people\u2014and even their physicians\u2014are unaware of the important role that vitamin D plays in retarding aging, promoting energy and health weight maintenance, boosting the immune system, and reducing your risk of cancer. And, unfortunately, most people are unaware that their body is deficient in needed vitamin D levels. Though vitamin D is naturally produced by the human body when exposed to sunlight, most American adults spend most of their time indoors, and when they are outside, they usually wear sunscreen. Even though my medical practice is in the sunny state of Florida, I routinely test my patients' vitamin D levels.\"\n\nRandy is not alone in his medical opinion. Dr. Oz states that \"it is commonly believed that over 50 percent of Americans are vitamin D deficient.\" When unrecognized and untreated, vitamin D deficiency can lead to more serious conditions, including osteoporosis, high blood pressure, heart and inflammatory problems, multiple sclerosis, and chronic pain, among others.\n\nCoQ10\n\nOur bodies produce CoQ10, a substance that's necessary for cells to function. It helps produce an important molecule known as adenosine triphosphate, or ATP. ATP is the fuel that helps power the energy-producing center of the cell known as mitochondria. As we get older, our body produces less CoQ10, and as a result, our cells don't function as they should. According to the Mayo Clinic, those who suffer from chronic diseases such as Parkinson's, cancer, diabetes, and cardiovascular disease have lower CoQ10 levels than healthy people.\n\nResearchers also believe that taking CoQ10 as a supplement can help prevent heart disease. It helps prevent blood from clotting and may act as an antioxidant, which protects our cells against the effects of free radicals that can damage cells and cause heart disease. Moreover, researchers suspect CoQ10 supplements may improve the health of diabetics by managing blood sugar, cholesterol, and blood pressure.\n\nOmega-3 Fish Oil\n\nFish oil helps reduce internal inflammation in the body. According to the American Heart Association, omega-3 fatty acids decrease cardiac arrhythmias, reduce risk of sudden heart attack, and lower blood pressure, triglycerides, and cholesterol levels. The essential omega-3 fatty acids help to alleviate pain, depression, and uterine cramps and pain at the onset of menstruation. One of the main causes of PMS appears to be excess production of pro-inflammatory eicosanoids. Fish oil contains high quantities of eicosapentaenoic acid (EPA), which helps the body produce anti-inflammatory eicosanoids, providing relief.\n\nOmega-3 fish oil helps to energize and hydrate cells of the body, adding shine, gloss, and health to skin and hair. It also protects the skin against the harmful ultraviolet rays of the sun. Consumption of omega-3 fish oil has also been found to act as a stimulant for hair growth.\n\nWhen patients complain of burping up a fishy taste when taking fish oil supplements, Dr. Rama says, \"Store them in your freezer. The temperature won't change your ability to swallow or digest them, but the fishy taste goes away.\"\n\nProbiotics\n\nProbiotic products contain bacteria and\/or yeasts that assist in restoring the balance in our gut. Probiotics are believed to protect us in two ways. The first is the role that they play in our digestive tract. We know that our digestive tract needs a healthy balance between the good and bad bacteria, so what gets in the way of this? It looks like our lifestyle is both the problem and the solution. Poor food choices, emotional stress, lack of sleep, antibiotic overuse, other drugs, and environmental influences can all shift the balance in favor of the bad bacteria.\n\nWhen the digestive tract is healthy, it filters out and eliminates things that can damage it, such as harmful bacteria, toxins, chemicals, and other waste products. On the flip side, it takes in the things that our body needs (nutrients from food and water) and absorbs and helps deliver them to the cells where they are needed.\n\nThe idea is not to kill off all of the bad bacteria. Our body needs both bad and good bacteria. The problem occurs when the balance is shifted and we have more bad than good. An imbalance has been associated with diarrhea, urinary tract infections, muscle pain, and fatigue.\n\nThe other way that probiotics help is the impact that they have on our immune system. Some physicians believe that this role is the most important. Our immune system is our protection against germs. When it doesn't function properly, we can suffer from allergic reactions, autoimmune disorders (for example, ulcerative colitis, Crohn's disease, and rheumatoid arthritis), and infections (for example, infectious diarrhea, Helicobacter pylori, skin infections, and vaginal infections).\n\nCaution: Be Certain of Quality\n\nDietary supplements are regulated by the FDA, but not as drugs or as foods. The rules for dietary supplements are as follows: Manufacturers don't have to seek FDA approval before putting dietary supplements on the market. In addition, companies can claim that products address a nutrient deficiency, support health, or are linked to body functions\u2014if they have supporting research and they include a disclaimer that the FDA hasn't evaluated the claim.\n\nManufacturers must follow good manufacturing practices to ensure that supplements are processed consistently and meet quality standards. These regulations are intended to keep the wrong ingredients and contaminants, such as pesticides and lead, out of supplements, as well as make sure that the right ingredients are included in appropriate amounts.\n\nOnce a dietary supplement is on the market, the FDA is responsible for monitoring its safety. If the FDA finds a product to be unsafe, it can take action against the manufacturer or distributor or both, and it may issue a warning or require that the product be removed from the market.\n\nThese regulations provide assurance that dietary supplements meet certain quality standards and that the FDA can intervene to remove dangerous products from the market. The rules do not, however, guarantee that supplements are safe for everyone. Because many supplements contain active ingredients that have strong effects in the body, these products can pose unexpected risks. For example, taking a combination of herbal supplements or using supplements together with prescribed medications could lead to harmful, even life-threatening results. For this reason, it's important to talk with your doctor about the supplements you intend to take.\n\nAnd not all supplements are of equal quality. Manufacturers of supplements are responsible for ensuring that the claims they make about their products aren't false or misleading and that they're backed up by adequate evidence. However, they aren't required to submit this evidence to the FDA.\n\nAccording to Randy, \"Supplement manufacturers often add in a variety of fillers to their vitamin and mineral supplements for numerous reasons: easier and faster production, colorants to make products more eye-appealing, coatings to make pills and capsules easier to swallow. The reasons for using fillers aside, the real problem lies in how these fillers impact your body and health. Simply put, it's not good. Just as processed foods are loaded with additives and fillers, the same goes for vitamins. Any supplement that has additives and fillers may be harmful to your health in the same way those processed foods are. Some unhealthy fillers include hydrogenated oils, artificial colors, and magnesium stearate, which is used as a lubricant so that the vitamins don't stick to one another or the equipment being used. The controversy surrounds a few studies on magnesium stearate. One study links this compound to creating a suppressed immune system. Other studies show that this 'chalk' creates a biofilm in the body. This biofilm blocks the body from absorbing any of the needed nutrients. Titanium oxide is another filler widely used as a pigment in vitamins. However, the research is now showing that exposure to this metal (along with other metals) can lead to problems with the immune function in the body.\"\n\nHow Do You Choose?\n\nWhen it comes to choosing which and what brand of supplement, I want you to be a smart consumer and do your homework. Don't just rely on a product's marketing. Look for objective, research-based information to evaluate any health or medical claims. To get reliable information about a particular supplement, I recommend you:\n\n  * Ask your doctor or pharmacist. Even if they don't know about a specific supplement, they may be able to point you to the latest medical guidance about its uses and risks. Today there are proven higher-quality supplement companies whose integrity is enhanced because they choose to only be sold through medical professional offices. We offer Randy's signature brand in our National Medicine Pharmacy and on our website, www.agelessandwellness.com. Metagenics, Xymogen, and OrthoMolecular brands are three others in this league.\n  * Look for scientific research findings. Two good sources include the National Center for Complementary and Alternative Medicine (NCCAM) and the Office of Dietary Supplements. Both have websites that provide information to help consumers make informed choices about dietary supplements.\n  * Contact the manufacturer. If you have questions about a specific product, call the manufacturer or distributor. Ask to talk with someone who can answer questions, such as what data the company has to substantiate its products' claims.\n\nCHAPTER 8:\n\nWhat's Good for   \nYour Waist Is Great   \nfor Your Face\n\nMy grandmother used to say that a smiling, kind, and well-mannered woman will dazzle no matter the age or shape of her features. Good news: That truism did not go out with hats and gloves. When I consider all the women interviewed for this book, the one that stands out to me as hands-down most beautiful is Susan Eister Estes of Richmond, Virginia. A glorious blond with an enigmatic smile and ready laugh, Susan will likely continue turning heads no matter her age. What makes Susan's beauty especially astonishing, however, is that she maintains such a countenance while devotedly caring for a husband with Alzheimer's disease in their home.\n\n\"I try to never feel sorry for myself or slump around,\" says Susan. \"I get up, get dressed, get busy and always put on lipstick.\"\n\nBut, as we age, we may need more than a great smile and bright tube of lipstick to look our prettiest. Good news: our Belly Flat Plan will not only help you lose those pounds and have more energy, it can help turn back the clock on your face! Here's how:\n\nBelly Flat Diet   \nRejuvenates Your Face\n\nA study published in March 2010 in the Journal of the American Academy of Dermatology supports the idea that certain vitamins help to protect our skin. Better yet, many of the same foods that can boost your defenses against skin cancer (the most common type of cancer) will also help keep your skin looking younger and smoother and ward off wrinkles. Our bonus: they are all belly-blaster foods! Here's how they promote youthful beauty from the inside out:\n\nVitamin C\u2013Rich Foods\n\nIn 2007, research published in the American Journal of Clinical Nutrition showed that eating more vitamin C\u2013rich foods, such as citrus fruits, strawberries, broccoli, and red peppers, may help ward off wrinkles and age-related dryness. Vitamin C's skin-smoothing effects may be due to its ability to mop up free radicals produced from ultraviolet rays and also its role in collagen synthesis. Collagen is fibrous protein that keeps skin firm, and vitamin C is essential for collagen production.\n\nPink grapefruit is a vitamin C\u2013rich, super skin-beautifying food. Pink grapefruit gets its pink-red hue from lycopene, a carotenoid that may help to keep your skin smooth. In a study published in 2008 in the European Journal of Pharmaceutics and Biopharmaceutics, researchers found that of the twenty individuals studied, those who had higher skin concentrations of lycopene had smoother skin.\n\nOmega-3-Rich Fatty Fish\n\nOmega-3-rich fish may help keep your skin looking youthful and prevent skin cancer. The omega-3 fatty acids DHA and EPA (docosahexaenoic and eicosapentaenoic acids, respectively) found in salmon, tuna, sardines, and mackerel may shield cell walls from free-radical damage caused by UV rays, according to a 2009 study in the American Journal of Clinical Nutrition. Researchers followed the eating habits of more than 1,100 Australian adults for approximately five years and found that for those who ate a little more than 5 ounces of omega-3-rich fish (such as salmon) each week, the development of precancerous skin lesions decreased by almost 30 percent.\n\nEPA has been shown to preserve collagen, a fibrous protein that keeps skin firm. EPA in combination with DHA helps to prevent skin cancer by reducing inflammatory compounds that can promote tumor growth, according to Homer S. Black, PhD, professor emeritus in the Department of Dermatology at Baylor College of Medicine in Houston.\n\nWhole Grains\n\nWhole-grain bread, pasta, and cereal are surprisingly high in antioxidants (whole-wheat cereal, for example, contains a comparable amount to most fruits and vegetables), and eating additional antioxidants is key to youth-proofing your complexion, says Lisa Drayer, RD. \"The levels of the body's natural antioxidants decrease with age, so adding them to your diet becomes even more important.\"\n\nAlmonds\n\nEating a handful of almonds (a one-ounce serving is about 20 to 24 whole nuts) every day boosts levels of vitamin E, one of the most important antioxidants for skin health.\n\nRed Wine and Dark Chocolate\n\nThe magic effect red wine and dark chocolate can have on our skin comes from antioxidants. The dark skin and seeds of the grapes in red wine are rich in polyphenols, a type of antioxidant that includes resveratrol. The antioxidants in red wine soak up damaging free radicals that play a role in aging. Dark chocolate also has a high concentration of polyphenols and provides the same benefits of protecting cell membranes with anti-inflammatory properties.\n\nConsider Farm to Face\n\nWe've talked about why you should choose organic foods when possible and, also, the importance of detoxing your home from chemicals that can silently sabotage your health while accelerating your aging and putting children at risk for precocious puberty. Now consider the potential benefits of 100 percent natural skin care.\n\nWhile the previously quoted statistics linking petrochemicals in common skin care and beauty products are disturbing, I must admit I have personally struggled for years to find an organic skin-care line that not only feels and smells good but works to decrease lines and signs of aging. My good news is that today there are a growing number of excellent organic lines on the market with dynamic women leading this skin-care revolution. Several excellent companies are listed in Appendix B, but here I want to give a shout-out to two women who started their companies motivated by personal experience.\n\nKelly Teegarden created her organic line after a long struggle to regain her health after a near-terminal bout of thyroid cancer. Kelly says, \"I am convinced that Kelly Teegarden Organics (KTO) is my 'why' for getting cancer. I thank God I am still here to help as many people as I can on their crusade to health.\" KTO products are available online and at Whole Foods.\n\nSimilarly, while pregnant and helping care for her stepfather with cancer, Tata Harper determined to do everything she could to keep toxic petrochemicals out of her beauty regimen. \"Beauty was and is a priority for me, but health is an even bigger one. Nevertheless, I did not think women like me should have to sacrifice one for the other.\"\n\nTata drew from her training as an industrial engineer to work with a team of chemists, biologists, botanists, agriculturists, and aromatherapists from around the globe to create her proprietary 100 percent natural products.\n\nTata says, \"Beauty is a way of living, not a product. What a woman puts on her face and the skin of her body should match who she is on the inside, as well as how she chooses to live. If our eyes are the mirrors of our souls, then our faces and skin are a reflection of our choices. Why wouldn't every woman choose natural, healthy skin-care products that can help increase our visible beauty versus those that can make us sick? Why wouldn't we make natural beauty an inside-out nonnegotiable priority?\"\n\nExercise Helps You Look Younger\n\n\"Anything that promotes healthy circulation also helps keep your skin healthy and vibrant,\" says dermatologist Ellen Marmur, MD, author of Simple Skin Beauty: Every Woman's Guide to a Lifetime of Healthy, Gorgeous Skin and associate professor of dermatology at Mount Sinai School of Medicine. By increasing blood flow, exercise helps nourish skin cells and keep them vital.\n\n\"Blood carries oxygen and nutrients to working cells throughout the body, including the skin,\" says Marmur. \"In addition to providing oxygen, blood flow also helps carry away waste products, including free radicals, from working cells. Contrary to some claims, exercise doesn't detoxify the skin. The job of neutralizing toxins belongs mostly to the liver. But by increasing blood flow, a bout of exercise helps flush cellular debris out of the system. You can think of it as cleansing your skin from the inside.\"\n\nThe boost in blood flow and oxygen to the skin cells also carries nutrients that improve skin health. Also, when you exercise, your skin begins to produce more of its natural oils, which helps skin look supple and healthy.\n\nGet More Beauty Rest\n\nMost people have experienced sallow skin and puffy eyes after a few nights of missed sleep. But it turns out that chronic sleep loss can lead to lackluster skin, fine lines, and dark circles under the eyes. When you don't get enough sleep, your body releases more of the stress hormone cortisol. In excess amounts, cortisol can break down skin collagen, the protein that keeps skin smooth and elastic.\n\nA person may look a decade older in response to stress-induced changes in facial tissues that often accompany insomnia. Few people are aware, however, that chronic insomnia inflicts significant damage to skin tissues, from premature aging to disorders like eczema, psoriasis, and atopic dermatitis.\n\nLack of sleep also upsets the balance of two more hormones: ghrelin and leptin. When you sleep less than eight hours, ghrelin levels go up and leptin levels go down. Ghrelin is the hormone that makes you feel hungry while leptin signals the brain that you are full. Sleep-deprived women frequently say, \"I am hungry all the time,\" but their drive to eat has nothing to do with a big appetite or weak willpower and everything to do with their hormones.\nCHAPTER 9:\n\n[Pump Up   \nYour Pelvic Power](Content.html#CHAPTER-NINE--Pump-Up-Your-Pelvic-Power)\n\nAging research indicates sexually vital women have decisively better quality of life indicators and self-report a more positive aging experience. Many women make the mistake of equating sexual vitality to sexual activity and carnal mirth, particularly intercourse. Don't! No matter your age, sexual vitality is our inner, God-given energizer for joy and creative juiciness. If you have fun in the sack, enjoy the bonus while burning the calories.\n\nMultiple factors dim sexual vitality and dampen creative juiciness, including aging private parts. The good news: we can fix those.\n\nWhen There's a Drought Down There\n\nMary Nell parked her car across the street and looked in the rearview mirror, making certain her hat, scarf, and sunglasses obscured most of her face, then took a deep breath to wind up her courage. A few years back, she would never have imagined stepping into a shop that sold adult movies and sex toys, but she was desperate. Intercourse had become painful, often with skin tearing and bleeding. She had embarrassingly bought a lubricant from the drugstore, but it itched and burned, and the day after using it, she woke up with a nasty urinary tract infection. That night Nigel began sleeping in the guest room down the hall.\n\nIronically, the next week Pastor Duncan asked her and Nigel to host a small group of five couples in their home every Wednesday. \"I can think of no better twosome to lead this six-session video-based study on how intimacy in marriage is part of God's plan. Nigel and Mary Nell, will you do it?\"\n\nMary Nell wasn't sure if God looked less favorably on hypocrites or loose women, but crossing the street and walking into a store with a purple neon penis in its window, she felt like both.\n\nLack of vaginal moisture and lubrication does not mean you are entering the life stage of the dried-up crone, nor does it correlate to lack of love or lust. If your vagina is so dry that intercourse is uncomfortable or painful, no lingerie, sexy video, hours of foreplay, or amount of prayer or meditation will get your juices flowing.\n\nVaginal dryness is more common for perimenopausal and menopausal women but is often a surprising and depressing issue for younger, regularly menstruating women. Studies cited by the National Institutes of Health also show that, because estrogen levels fall after childbirth and remain suppressed while breastfeeding, vaginal dryness can be a problem for new moms. Also, women with premature ovarian failure, polycystic ovarian syndrome (PCOS), vulvar dystrophy (a condition where the outer part of the vagina becomes dry and the skin thickens) and vulvodinyia (a condition where there is pain around the opening of the vagina where there is no identifiable cause) are at risk.\n\nAnd if you think vaginal dryness is an issue for only the sexually active, think again. Women not having intercourse or engaging in manual or oral stimulation of the genital area often complain of vaginal dryness. Some say the soreness, burning, and itching makes them uncomfortable sitting, standing, exercising, or urinating.\n\nIf your vagina is as dry as an emery board, stop feeling damaged or just plain old. Evaluate recommended lifestyle changes and potential treatment options, and work with a knowledgeable medical professional as needed. Tips for improving vaginal lubrication include:\n\nReplace what's missing: Synergistic bioidentical estrogen, progesterone, and\/or testosterone replacement can reinvigorate and nourish fragile vaginal tissue.\n\nClean up carefully: Certain personal hygiene products, such as soaps, bubble baths, deodorants, and even douching with water, can contribute to vaginal dryness.\n\nWake up your clitoris: Arousal stimulates lubrication; however, a woman's clitoris (her primary genital organ of arousal) can become less sensitive with age, hormonal changes, or the aftereffects of a hysterectomy. Wake up your clitoris by embracing the miracle of masturbation. If you enjoy some variety, a vibrator can also be very handy. Good news for the accessory-loving diva: vibrators come in a variety of styles and colors. (Check out the Deluxe Rabbit Pearl Vibrator in pink.)\n\nPlay with toys: The vaginal wall can atrophy (weaken), causing foreplay and intercourse to become uncomfortable to painful. Try a sex toy such as a vaginal dilator or dildo to increase the diameter of your vagina.\n\nAdd foreign moisture wisely: Over-the-counter lubricants may be helpful, but read the labels before you buy. For instance, there is a difference between a lubricant to increase vulvo-vaginal comfort and a vaginal moisturizer hydrating the mucous membrane lining of the vaginal canal. Also, some ingredients in certain over-the-counter products are absolute no-no's because they create a medium for bacteria transfer and growth, thereby increasing risk of urinary tract infection. Recommended resources for over-the-counter lubricants are included in Appendix B.\n\nUrine Is Never an   \nAttractive Accessory\n\nFour blocks into her walk, Terri stopped, cagily adjusted her underwear under dark, baggy sweat pants, then sighed and turned home. Despite an absorbent liner, her panties were dangerously damp. Continuing around the golf course would, at best, mean uncomfortable chafing and, at worst, a full-fledged accident before she reached the clubhouse just six blocks down on her carefully mapped-out route of accessible restrooms. Tears seeped under her sunglasses.\n\nWalking four miles with my neighborhood gang used to be my favorite part of the day, she cried inside. Now even walking alone risks humiliation. This bathroom business is ruining my life. I'm afraid to go to church or a basketball game with Thomas. What if I leak and people around me smell pee pee? And when it comes to sex, Thomas has been patient, but how long can that last?\n\nUrine is unattractive, and damp panties are uncomfortable. Still, one in six women experience overactive bladder (OAB). While more common in older adults, OAB can strike women of all ages. For women like Terri, OAB puts the brakes on a great deal of joy and quality of life. But it doesn't have to be this way.\n\nDid you know that there is more than one kind of leaky bladder? \"Stress incontinence\" is when small amounts of urine escape when you sneeze, pee, laugh, or exert pressure on the bladder by lifting or exercising. \"Urge incontinence\" is when you have to go now and urine escapes before you reach the toilet.\n\nWhy urine leakage happens during sex is easy enough to understand. Sexual activity can place extra pressure on the abdomen, causing urine to leak. This causes many women to avoid sex, as it makes them feel unclean or unattractive. Women suffering from stress incontinence usually can tell when during intercourse they are most likely to leak. But urge incontinence occurs unpredictably, making it difficult to feel sensual and\/or fully enjoy sex. The chance of embarrassment is also greater\u2014much more urine leaks during an episode of urge incontinence compared with stress incontinence, and women with urge incontinence often leak during orgasm.\n\nWhichever type of incontinence you might suffer, an integrated approach can help get you out of the bathroom sans the absorbent pads in your panties. New surgical techniques (bladder slings, bladder neck suspension, the injection of bulking agents such as collagen, or sacral nerve stimulation via a small, pacemakerlike device surgically placed under the skin) may very well give you back your life. Before you make that decision to go under anesthesia, talk with your doctor about nonsurgical approaches, including physical therapy and pelvic-floor yoga exercises.\n\nYou and your doctor have most likely heard how Kegel exercises help strengthen the pelvic floor. I would wager that neither you nor your doctor knows that the National Institutes of Health recognizes an ancient and sacred yoga practice called \"Mula Bandha\" as a confirmed method to treat and even prevent stress incontinence, or that spring-loaded sex toys have been clinically proven to strengthen the interior pubococcygeal (PC) muscle surrounding the urethra and control the flow of urine.\n\nHave It Your Way\n\nYouthful romps in the hay are typified by easy erections, voluminous lubrication, surefire penetrative sex, and explosive orgasms. As the decades roll by, this sequence frequently loses its sizzle. Reasons may vary. Perhaps you just aren't as \"into it\" as you used to be. Maybe his erections aren't as quick or hard. Perhaps your lovemaking routine has gone stale. Never you mind. It's high time you move past male-oriented, wham-bam-thank-you-ma'am sexual encounters anyway.\n\nWhile women have been conditioned to regard lack of desire or any degree of sexual dysfunction as not-to-be-discussed disappointments, I encourage you to throw back the covers and consider this: The time it takes you to get in the mood or for his equipment to get into gear can be all yours. Most women need about twenty minutes of thinking about sex before their bodies are primed for full response. Now your precoitus warm-up can benefit you both. Ask for and receive all the snuggling, caressing, and oral sex you have craved for years and he'll have plenty of time to gladly oblige.\n\nConsider the following tips to spark a fresh attitude toward friskiness:\n\nRead a book or watch a video on tantric sex. The goal of tantric sex is not the big \"O\"; instead, the techniques are intended to prolong the sexual experience, increase sexual energy, and heighten the experience of intimacy with your partner. Some women report that by channeling the sexual energy that would normally leave their bodies during orgasm, they achieve a state akin to temporary enlightenment. According to Christiane Northrup, MD, author of Women's Bodies, Women's Wisdom, \"Through the prolonged, intimate connection with another, women's nasty stress hormone levels lower while health- and life-enhancing oxytocin and serotonin levels shoot through the roof.\"\n\nTurn it off with a flare. Look him in the eye, kiss him fully on the lips (tongue preferred), then pull back and put your finger on the off button of your cell phone. Next, walk your phone into another room and leave it. Return and reach for his. Repeat. Then undress as you walk back. Odds are he won't worry about the messages he might be missing.\n\nExplore new regions. The base of your spine is erotica waiting, so turn over and let him touch and taste. Also ask him to gently stroke and kiss your belly just above the pubic hairline.\n\nGo to new heights. Hike to the top of a fire tower, book a room on the twenty-fifth floor with a floor-to-ceiling window, or make out in a hot-air balloon. The point is, when you have little between the two of you but sky, your sense of limitation falls away. And that breathless feeling of being almost off the edge will heighten the urgency and excitement of every touch.\n\nHave a mint. Place a jigger of peppermint schnapps or cr\u00e8me de menthe on your bedside table. Put your finger in the liquor, then in his mouth. Have him do the same with you. Then experiment with edible finger painting. The cool mint sensation plus breath and tongue on breasts and genitals guarantees a new kind of buzz.\n\nTeach him the joyful game of start and stop. This is a tried-and-true method of helping men last longer in bed. Pay attention to when he's approaching his point of no return, then use body language (or ask) to change positions. He'll calm down a bit and most likely oblige you with a few more minutes of oral or manual stimulation until he is fully aroused again.\n\nTrade report cards. So what if you can't get pregnant? You are never too old to contract sexually transmitted diseases (STDs). Recent statistics show a growing rate of STDs in the over-fifty crowd, even among the nursing home population. So, anytime you are sexually active with a new partner, don't be coy. Buy the condoms until he provides current lab work validating negative results for all STDs. And be fair\u2014give him yours.\n\nNo Sex, No Worries:   \nYou Are a Cocreative Powerhouse\n\nI hate bugs, particularly roaches and swarming things that bite. Consequently, a mid-August trip to a home on the banks of North Florida's Ichetucknee River was a guaranteed descent into personal purgatory.\n\n\"Tell me again why you're going to see some old lady in the woods?\" Randy asked.\n\n\"Kathleen told me this woman Dailey is an unbelievable artist who makes exquisite masks and figures out of dead animal bones. I'm interviewing her for my chapter on cocreation. You know, about how when you partner with the Divine, you can create new life even from things broken, decaying, or thought dead.\"\n\nRandy rolled his eyes. \"Sounds like modern-age voodoo, but whatever. If you're going to spend the night, you should take my pistol. What do you really know about this woman? Who knows what kind of weirdo she might be?\"\n\nForgoing the pistol, I loaded up on insect repellant, pen, paper, and a tape recorder. After two and a half hours on isolated roads, I stopped to pee behind a bush. Didn't occur to me to fear snakes, and, for the moment, bugs were on hiatus, so I walked through dense trees down to the riverbank. A white-tailed doe raised her head to stare at me before returning to drink from crystalline water. Unnerved, I hurried back to my ordinary car, turned on National Public Radio, and determined to find Dailey's house before sunset.\n\nSoon after, I pulled into a dirt driveway next to a petite red cabin nestled under oaks and pines. A note on the door read, \"Ran out for wine. Be back in a jiff. Come on in.\"\n\nI cracked the door and stood transfixed.\n\nWith light golden walls, a dusk blue ceiling, and sheer peach silk curtains, the last sunbeams through the windows transformed Dailey's studio into a veritable sunset. On the walls hung intricately jeweled, hand-painted animal skulls, as large as bison and as small as a cardinal. They, too, sent prisms of color across walls and floor. The fattest cat I had ever seen raised its head from a crimson pillow, blinked, then lowered back to sleep.\n\nA rich chuckle broke my reverie. \"I'm so glad you got here in time. Was afraid you would miss my nightly rainbow.\"\n\nI turned to face a wiry woman with short gray hair and dancing eyes\u2014one of the most enigmatic, sensual women I have ever encountered.\n\nAs with Dailey, art can be sensual outlet and expression for many women. Others express their innate sexuality through dance and movement. In fact, clinical data validates that women often go beyond intercourse to enliven their sexuality in a number of creative modalities.\n\nA survey of more than eight hundred women between the ages of forty to one hundred found that almost half of the women surveyed had not had sex in a month, yet they declared themselves \"sexually satisfied.\" On the surface, these findings in the January 2012 edition of the American Journal of Medicine appear contradictory, a conundrum. Then the fine print comes into focus.\n\nThe study's authors suggest that, for older women, the sexual experience moves far beyond literal intercourse. They suggest the experience of nurturing, affirming, and sustaining relationships become female-enlivened sexual activities. This new definition of sexuality embraces our ageless and ever-present creative power; in other words, our divinely derived, cocreative power.\n\nWomen are natural-born incubators and creators. In our reproductive years, our innate sexual vitality, plus copulating with a male, plus fertilization of the ovum (egg) can lead to children. This sequence, termed \"procreation,\" works properly for a limited number of decades. Also, a man\u2014or at least his sperm\u2014is requisite.\n\nOn the other hand, sexual vitality, plus an openness to partner with your Higher Power, combined with inspired action unifies Spirit and matter. This \"cocreative\" phenomenon can occur at any age but, typically, becomes richer and deeper with age.\n\nSometimes a woman's cocreative energy becomes tangible via art, music, writing, gardening, leadership development, new business ventures, sports enthusiasm, or philanthropic passion. Other times, a more subtle, interior birthing occurs, one evidenced by deeper joy and greater serenity regarding life's inevitable beauty and brutality.\n\nI was in my early thirties when my professional functioning in a male-dominated healthcare business environment had become flat, stale, and lackluster. A friend suggested I try something new to recharge my creative batteries: a drumming circle. The idea was way out of my comfort zone, but I decided to give it a roll. Intent on always being fashion-appropriate, I donned calf-high boots (in case of snakes or varmints), bought a flowered peasant dress from a vintage store, and put my hair in braids.\n\nI arrived at a farmhouse on the outskirts of Nashville looking like a past-my-prime, wannabe flower child. Thankfully, the other women in pressed jeans and button-down shirts didn't comment. They simply opened their circle a little wider. Then a beautiful redhead named Suchi Waters Benjamin (who now lives in Maui and is the founder of the Center for Co-Creative Living) handed me a small drum and began to sing. My master's level training in neural processing, for example, using right-brain rhythms to reprogram linear left-brain thinking, immediately (and for the very first time) shifted from academic to personal.\n\nWriting is something I never imagined doing or had any formal training in. Still, the sequence between inspiration and transcription merges my inborn feminine creativity with my more masculine inclination to get things done. Like a mother holding a newborn, I behold each word, page, and book a miracle. I fully believe Fountain of Truth was incepted two decades ago in my drumming circle on that Tennessee farm.\nPART THREE:\n\n[A Fresh Look at Anti-aging   \nTruths As Old As Dirt](Content.html#CHAPTER-THREE--Step-2--Move-and-Groove)\n\nForty-something Hollis Wilder appears to effortlessly bridge superbeautiful, supersavvy and supermogul. A wife, mother of two, successful businesswoman, and owner of cupcake stores Sweet! by Holly, Hollis is best known as two-time winner of Cupcake Wars and star of the Cooking Channel. She has also written a sure-to-be-bestseller cookbook, Savory Bites: Meals You Can Make in Your Cupcake Pan. Talk about a paradigm shifter: cupcake pans as our template for healthy portion control!\n\n\"Getting older is not easy but it's worth it. I like myself more every decade, and I'm told I look better and better with time,\" Hollis said firmly. \"I attribute my positive aging to three things. The first is that I am clear on the fact that I am a product, a consequence, of my choices. You know, the whole 'reap what you sow' philosophy. I clearly own the truth that who I am, who I become, is up to my attitude, my preparation, and my effort. My life continues to improve because I choose to be tenaciously positive regardless of setbacks or sad things that happen, and I work hard at the other stuff too . . . health and lifestyle choices upping my ante of being sharp, svelte, and super-fun at fifty, sixty, seventy, and beyond. The second is I have a tight group of girlfriends. They know me. They both have my back and call me on my crap. My girlfriends are my go-to lifeline. The third is I am honest about the squirrelly stuff.\"\n\n\"What does that mean?\" I put my pen down and leaned in.\n\n\"Let's first clarify what it means to get better, not older. Enviable women have lives that, on most days, overflow with meaning, love, joy, and good health. I say 'most days' because we all have bad days, live on a flawed planet, and know sad things happen to good people. For the most part, however, women who feel and look great no matter their age experience life on the upswing. My experience is that women aging poorly have individual versions of sad, squirrelly little secrets. They constantly blame their less-than-optimum personal circumstance on others or bad luck. They play life out as victims. They whine. Women aging poorly might say they want a fit body and good health, but their actions tell another story. Maybe it's drinking too much or food bingeing. Maybe it's choosing mindless television over an hour of exercise. Maybe it's staying in a dysfunctional marriage. Maybe it's overspending. It doesn't matter. I am convinced that the lies we tell ourselves and act out every day accelerate our decay.\"\n\nHollis mirrored what I heard from hundreds of women aging in enviable fashion: We are the consequence of our individual choices and actions. This mind-set is psychologically termed a \"strong internal locus of control.\" Conversely, people with an \"external locus of control\" believe the quality of their life depends on luck and other people. They are the whiners and victims.\n\nStuff Happens\n\nLive long enough and stuff happens. First, and sometimes second and third, loves fizzle. Career aspirations go flat, or there is downsizing at work. People we thought we could count on disappoint. Children get in trouble, break our hearts. Sometimes retirement funds blow away like a dandelion in the wind. You, or someone you love, gets really sick. Then there is death, as immutable as gravity but oh-so-much-more heartsickening when the loss is someone you love.\n\nWomen with a strong internal locus of control refuse to be buckled by stuff, including inevitable mistakes, disappointments, heartaches, and the physical marks of years passing by. They learn\u2014and intrinsically grow more desirable, beautiful, and resilient\u2014as a result. Women lacking a strong internal locus of control tend to squander precious time and energy grieving both what happened to them and what never did. Women lacking a strong internal locus of control also tend to use food as an emotional buffer.\n\nFood for Thought\n\nClare Lavendar, PhD, holistic nutritionist and dear friend, recently introduced me to the concept of primary foods versus secondary foods. The philosophy is the foods you eat are secondary to all the other things that feed you\u2014your relationships, career, spirituality, and exercise routine. Those things are regarded as primary foods. Secondary foods are what you put into your mouth and swallow. The premise is, if your primary foods are in balance and satisfying, you won't crave foods that sabotage your waistline, your health, and how quickly you age.\n\nI reached out to Nan Allison, MS, RD, LDN, coauthor with Carol Beck of Full and Fulfilled: The Science of Eating to Your Soul's Satisfaction. Nan's ongoing clinical work and research examines why we choose to eat the way we do. She explained: \"Most people, and particularly women, use food to celebrate, comfort, nurture, distract, and numb out. Anxiety, loneliness, boredom, and anger all send triggers to our brain that associate with eating. Most women then use food like a drug in an effort to quell their core feelings. Some women do the reverse. They don't eat, literally attempting to starve their feelings away.\"\n\nAccording to Evelyn Tribole, MS, RD, and Elyse Resch, MS, RD, FADA, authors of Intuitive Eating, \"Food won't solve the problem. If anything, eating for an emotional hunger will only make you feel worse in the long run. You'll ultimately have to deal with the source of the emotion, as well as the discomfort of overeating.\"\n\nWhat You Need Now\n\nA strong internal locus of control is not for purchase, and no one can give it to you. It is a deeply ingrained thinking pattern that typically starts early in life and then develops over a period of years, but it's never too late to learn how to think differently. No matter how old or young you are, you can start right now to build your internal locus of control muscles. The following will help:\n\n  * Faith in a Higher Power\n  * A good group of girlfriends\n  * A commitment to laugh often and play more\n  * Taking up a sport\n  * A good marriage\n  * A purpose to give back now and leave a legacy once gone.\n\nCHAPTER 10:\n\nFaith\u2014A Proven Path to Wellness\n\nIn my twenties and thirties I was an overachieving, stressed-to-the-max, whirling-dervish workaholic. My astonishing evolution into a more calm, joyful, and purposeful person began accidentally. On a lark with a girlfriend, I found myself naked in a hot tub overlooking Big Sur, conversing with an elder Benedictine monk (Brother David Steindl-Rast) about the meaning of my life.\n\nDon't spritz your knickers thinking I am going to tell you that to decelerate your aging, you too have to go cross-country and get into a hot tub with a monk. As we get older, there is less need for extreme drama; moreover, active faith does not require a kickoff event. Millions of women of faith confirm that an ingrained and intentional belief in a Higher Power works best when quietly radiating through each moment of every day.\n\nLike the mystical Sufis, I personally believe there are as many paths to God as there are breaths of children. Accordingly, when I speak of a Higher Power, I am not endorsing a specific religiosity. Whatever name you call your God or Higher Power, I am rallying you to press into your faith.\n\nI have grown very intentional about nurturing my faith. I start every morning with a \"coffee with God\" hour. I pray, journal, and read my devotionals. I also surround myself with Spirit-enlivened women.\n\nPhyllis Tousey is right down the road, reading her Bible as the sun rises over the Atlantic Ocean. I can close my eyes and feel the sparks from Debbie Austin's prayers igniting her ministry. Rhonda Marko tells me she prays aloud in her car. With the advent of Bluetooth technology, people pulling alongside think she is talking on the phone. Stacey Graham shares an inspirational quote on her voice-mail message. Passing time during chemo treatments, Linda Cunningham texts me daily meditation messages. Marjean Coddon nudges me to bridge from my Christian heritage to also embrace the wisdom of the Kabbalah. Sue Fort White and I both read Daily Word and God Calling. I know if I call her at any time to ask advice, we'll begin our conversation from the same spiritual context.\n\nConsider the following: spiritually active women are healthier, have longer life expectancies, and experience enhanced quality of living.\n\nThe Science of Spirituality\n\nAscending the steps of the red-brick, white-columned house of eighty-five-year-old Velma Morris, I cursed the airline that lost my bag. Mrs. Morris was the South's first recognized female religious writer. It had taken me more than three years to finagle an interview. Now, sans black suit with pearls, I would be on her doorstep wearing none-too-clean stretch jeans, an old sweatshirt, and a thirty-year-old pair of clogs (yes, clogs; they slide off easy in the airport security line and keep my feet warm on the plane). So much for first impressions.\n\nI knocked, expecting a maid to open the door, take one look, and shoo me off the premises. Instead Mrs. Morris greeted me in black sweats and a white apron, looking like a plump, blue-haired penguin. A stereo blaring Frankie Valli's \"Big Girls Don't Cry\" drowned out her greeting.\n\nOver deviled eggs, celery stuffed with homemade pimento cheese, and ham sandwiches on oven-warm bread, we got down to business. Our conversation would leave an inestimable mark on me.\n\n\"Please don't call me a 'religious writer.' That label makes me feel like I am selling God as if he were a good used car. I simply write about everyday people asking the same questions humans have asked since the beginning of time: Why do men fight wars? Why do children get cancer and other terminal illnesses? Why are some people blessed with prosperity so stingy? Why haven't you caused the penis to drop off every man who participates in or condones female genital mutilation? What's really going to happen to me when I die?\n\n\"Honey, I'm a Christian and Reverend Morris was an ordained Presbyterian minister, but if he were alive he'd tell you the same thing I'm about to. The common denominator in the sacred texts of all faiths is this: Choosing to be at peace regardless of what's going on in your life or our world is feet-hit-the-street faith in action. In the daily battle between good and evil, sickness and health, it's our ultimate trump card.\"\n\nChoosing to be at Peace regardless of what's going on   \nin your life or our world is feet-hit-the-street-faith in action.   \nIn the daily battle between good and evil, sickness and health,   \nit's our ultimate trump card.\n\nMayo Clinic backs up Mrs. Morris's premise. \"Nurture your spirit no matter what you call your source of inspiration,\" a Mayo Clinic health letter advises, then cites research for credibility. Duke University studied four thousand people for four years and found that those who attended church weekly had a 28 percent lower mortality rate overall when compared to those who didn't belong to a church community. The researchers also considered income, education, chronic diseases, other illnesses, health habits, exercise, smoking, drinking, body fat, social participation, and psychological status. None of these factors explained the results.\n\nOther research shows people who are regularly involved in religious and spiritual activities statistically live longer than those who are not. Church attendance was the strongest predictor of longevity. Various theories have been put forth to explain this spiritual dimension of longevity. Physical explanations include the fact that people who are involved in religious groups benefit from the social networks they form. If they get sick, others look out for them. Religious beliefs may also lead to less risky behavior.\n\nIn addition, a well-developed sense of spirituality may help people better cope with life's tough psychological demands. In a later study done by Duke University of 1,700 older Americans, researchers at Duke University Medical Center found that those who attend religious services had stronger immune responses. About 60 percent of the men and women surveyed attended religious services at least once a week. Blood tests showed that regular attendees were less likely to have a high level of an immune system protein involved in age-related diseases. This study suggests a direct positive effect.\n\nPrayer, Meditation, and the Relaxation Response\n\nSurveys indicate that nearly 90 percent of patients with serious illness will engage in prayer for the alleviation of their suffering or disease. Among all forms of complementary medicine, prayer is the single-most widely practiced healing modality. Prayer is the second-most common method of pain management (after oral pain medication) and the most common nondrug method of pain management.\n\nA well-noted study by Dr. Herbert Benson, a cardiovascular medicine specialist at Harvard Medical School, documented the potential healing benefits of spiritual practices, such as prayer and meditation (as well as hypnosis and other relaxation techniques). Benson demonstrated that the body responds to these practices with what he calls the relaxation response, which consists of \"a lowering of the heart rate, blood pressure, and breathing rate; a reduced need for oxygen; less carbon dioxide production.\" In effect, the relaxation response is the opposite of the stress response and can be consciously used to modulate the impact of stress.\n\nSome of the effects of prayer may be due to the larger context within which prayer occurs, which is usually one of religious commitment and social support.\n\nMedical researchers believe prayer improves health in the following ways:\n\nThe relaxation response: prayer elicits the relaxation response, which lowers blood pressure and other factors heightened by stress.\n\nSecondary control: prayer releases control to something greater than oneself, which can reduce the stress of needing to be in charge.\n\nThe placebo response: prayer can enhance a person's hopes and expectations, and that in turn can positively impact health.\n\nHealing presence: prayer can bring a sense of a spiritual or loving presence and alignment with God or an immersion into a universal unconsciousness.\n\nPositive feelings: prayer can elicit feelings of gratitude, compassion, forgiveness, and hope, all of which are associated with healing and wellness.\n\nMind-body-spirit connection: when prayer uplifts or calms, it inhibits the release of cortisol and other hormones, thus reducing the negative impact of stress on the immune system and promoting healing.\n\nFaith in Action\n\nDr. Pam Chally, dean of Brooks College of Health at the University of North Florida, and a beautiful blond with enviable skin and kickstart brain, shares: \"My core belief is we are all here to serve. I grew up on a farm. My high school graduating class had only forty-six people in it. You might have thought my youth would give me a small view of the world. But, you see, faith was the fabric of both my family and my community. And faith has no bounds. Even as a little girl, I believed with God all things are possible. I continue to live that out today.\"\n\nGod is doing a lot with Dean Chally. She was awarded HealthSource magazine's 2012 Celebration of Nurses award and has received the EVE award for Education, UNF Distinguished Professor, Desmond Tutu Peace and Reconciliation Award, the Transformational Leadership and Collaborative Engagement Award, and was honored as a Woman of Influence by the Jacksonville Business Journal. Despite the hoopla, she contends, \"My greatest accomplishment will always be to stay present, not get preoccupied, so I can treat everyone\u2014whether the janitor or the president\u2014with a high level of respect, love, and honor.\"\n\nCarla Harris is a managing director in the Institutional Advisory Group at Morgan Stanley Investment Management. She is also the author of Expect to Win and a gospel singer who sells out Carnegie Hall. Carla has been on the following lists: Fortune magazine's \"The Most Powerful Black Executives in Corporate America\" and \"The Most Influential List\" (2005), Black Enterprise magazine's \"Top 50 African Americans on Wall Street,\" Essence magazine's \"The 50 Women Who Are Shaping the World,\" Ebony magazine's \"15 Corporate Women at the Top,\" and the Network Journal's 2005 list of \"25 Most Outstanding Women in Business.\" She was also named \"Woman of the Year 2004\" by the Harvard University Black Men's Forum. When it comes to how faith has shaped her aging, Carla says, \"My faith is the center of my life; it is part of everything I do, including my business. My faith is what keeps me positive. I could say about aging, 'Oh, boy, I'm aging and things don't work the way they used to,' or I can say, 'One of the great benefits of aging is the cultivation of a discerning spirit. I get quiet for a few minutes, check in with prayer, and quickly know if I should be doing X versus Y. Now, in my fifties, I have a way of discerning things that was not possible in my twenties or thirties.\"\n\n\"Note: I believe there is a world of difference between wisdom and discernment. Wisdom evolves from my learning from the experiences God puts in my way. Discernment takes wisdom a step farther down a spiritual path. When I really don't know, don't have a clue what the right answer is, I can tune in and let God tell me what is right. You see, I would never have had the spiritual muscles for that much trust in my youth.\"\n\nListening to God saved Laura Bergman's (Laura B.) life. Laura B. shared how, in 2006, she resigned from the board of a large national philanthropy company, knowing that her best work was done there: \"I wasn't sure what was next, but I felt a pregnant sense of purpose. I prayed and prayed and would get guidance like 'clean out your closets,' 'straighten out yours and Tom's personal accounting record,' and 'fix up your home a bit.' It was tempting to be frustrated because I had this sense that my purpose was bigger, but I disciplined myself to obey. Better to follow God's nudges with a cheerful heart. This went on for a year.\"\n\n\"Then, in April 2007, after a routine mammogram, I was told there was a suspicious area in my breast that could 'possibly one day, maybe, turn into breast cancer.' I prayed for guidance and God told me to go for a second opinion. I did immediately and soon learned I definitely had breast cancer not in one, but both, breasts. If I hadn't prayed, listened, and acted on Divine Guidance, who knows if I would be alive today.\"\n\nTalking About Trusting Your Instincts\n\nLaura B. shared how, immediately upon her diagnosis, she asked a question. The answer to her question would lay out God's purpose for the next several years of her life: \"I said 'I would like to speak to a woman who has the exact kind of breast cancer I have so she can tell me from her own experience what I might expect.' The doctors at Mayo told me they didn't have any kind of patient-to-patient sharing system. I knew right then and there the mission God had been preparing me for: to start a breast cancer advocacy program. I couldn't have done it, or done it with peace of mind, if I hadn't spent the year before getting my house in order.\"\n\nLaura B. is founder of Pink Sisters and Friends advocacy group at Mayo Clinic in Florida. The program serves hundreds of women each year, matching the diagnosis and protocol of each new breast cancer patient with a Pink Sister advocate with the same. The two women stay together, in constant contact, throughout treatment. Together they tackle the gritty stuff: doctor and hospital appointments, dealing with surgeries, chemotherapy, radiation and its effects, and home care after surgery, as well as psychological, familial, financial, and sexual challenges. Laura B. has been nominated for the L'Or\u00e9al Paris 2012 Women of Worth Award, which recognizes and honors women making an exceptional difference in their communities.\n\nUnbelief on the Uptick\n\nAccording to a 2012 report from the Pew Center, the number of people in the United States who say they believe in \"nothing in particular\" is on the rise. Barry Kosmin, coauthor of three American Religious Identification Surveys, theorizes why \"None\" has become the \"default category.\" He says, \"Young people are resistant to the authority of institutional religion, older people are turned off by the politicization of religion, and people are simply less into theology than ever before.\"\n\nKosmin's surveys were the first to brand the \"Nones\" in 1990 when they were 6 percent of US adults. By the 2008 survey, \"Nones\" were up to 15 percent. By 2010, another survey, the biannual General Social Survey, bumped the number to 18 percent.\n\nLadies, I am not here to evangelize, but these statistics raise grave concerns. If faith in a Higher Power is the single most defining variable of our aging and quality of life, then even a slight trend in growing disbelief is an insidious risk factor. It raises the bar for all women of faith: Are you\u2014am I\u2014living a life that other women and girls want to emulate? Do you and I look, act, and smile like we have something they should want?\nCHAPTER 11:\n\nGirlfriends Are   \nNext to God\n\nAt forty-two, I met an older woman who became my friend and, through her love and counsel, filled in many of the blank spaces my mother left behind. She would also open the door to new directions for my life\n\nIt was 2002 and I was worn out from a corporate turnaround, so I decided to take a sabbatical. I leased my house in Nashville, packed up Bodhisattva (Bodhi, my shih tzu), Namaste (Nami, my marmalade-colored kitty), several pairs of cutoffs, and my computer to head to Apalachicola, a tiny town in Florida's panhandle. My girlfriends thought I was crazy.\n\n\"It doesn't have a traffic light. You're kidding, right?\"\n\n\"Does FedEx even deliver there?\"\n\n\"The grocery store is called the what? The Piggly Wiggly?\"\n\n\"I've been there. It's pretty, but you know there are no single men, at least none who still have their own teeth.\"\n\nBut I was undeterred, fixated. Today I look back and wonder why. I can only conclude it was a God thing.\n\nMy move wasn't seamless. I fractured my foot and the moving truck went missing for three days. Geoff, one of my new neighbors, knocked on the door of the house I had rented to warn, \"You know, gators are right out there in the bay at the end of your yard. They're likely to think that puppy and kitty of yours are fancy hors d'oeuvres.\"\n\nYikes! For the first time I wondered if I might have made a huge mistake.\n\nMy third day I woke at dawn and watched the sunrise over the bay. I remember feeling exquisitely peaceful, then jumping up, panicking. Nami wasn't on my bed. Or in the house. I threw on a pair of cutoffs, put on my protective boot, pulled my hair into a ponytail, and hobbled out in search.\n\nI called and called. No Nami. I looked all over my yard, even tiptoeing down to the edge of the bay. No Nami. I could hear Bodhi frantically barking in the house. I started crying. Then I saw her, all five pounds of yellow fur, scampering into a hedge in the yard next door. I hadn't yet met this neighbor, but I had been told she was an older woman whose husband was in the nursing home\u2014and also that she had a very sophisticated security system. Seeing a light in the window, I decided best to knock on the door and announce myself before foraging for Nami in her bushes.\n\nI knocked. The door opened.\n\n\"Hi, I'm Genie James and I just moved next door. I am sorry to bother you so early, but my cat is hiding in your hedge. I need to catch her and take her home, but I didn't want you to be scared if you looked out your window and saw a strange woman in a boot going through your bushes.\"\n\nTwinkling eyes looked me up and down. The woman pursed her lips for a second then seemed to relax. She smiled.\n\n\"Do you play bridge, honey? Our fourth died last week.\"\n\n\"No, ma'am. I am sorry, but I don't.\"\n\n\"Oh well. Do you drink wine?\"\n\n\"Yes, ma'am, I do.\"\n\nShe brightened. \"Now that's nice. I'll see you over here this afternoon at five.\"\n\nWith that, Smiles Randolph, my new best friend, Randy's mother, and my future mother-in-law, closed her door.\n\nThe Hormone Connection\n\nClinical studies show girlfriends give more than psychological and emotional benefit. A cohort of supportive women can have a positive impact on the immune system, cardiac and breast health, pain response, and even reduce the chances of catching a cold. According to UCLA researchers Drs. Laura Klein and Shelley Taylor, bonding with women stimulates the release of a hormone called \"oxytocin,\" which has a calming, peaceful effect. The positive effects of oxytocin are enhanced by the presence of estrogen, which circulates in the system of even non-menstruating women. This calming response doesn't happen in men because testosterone, the male dominant hormone, blocks oxytocin's positive neurological stimulation.\n\nGirlfriends Shore Up   \nInner Locus of Control\n\nI am certain that if it were not for my girlfriends and my sister Sheila, I would be either loony or dead. They help me through life's rough patches, celebrate my successes, and nudge (or pull) me into a better future. Most of all, my girlfriends hold me accountable for my choices.\n\nStacey was with me the last time I saw Mother alive and the first time I dared tell a large audience stories from my heart. Pat has helped me negotiate multiple life contracts, from business to personal, ensuring I hang on to what's my due and come out ahead. Suebee was the first to nudge me down an authentic spiritual path and continues to counsel my sometimes knee-jerk personality to \"pray, listen, wait, then act.\"\n\nShirley bailed me out of a jam, then helped me get a job. Aleene proves that those you love in junior high can sometimes grow up and old with you. She and Fran are strong threads weaving back into my memory what was good from our childhood. Kay shows me by example how our God is a God of second and third new beginnings.\n\nVicki never judges when I land in deep doo-doo. She cheerfully shows up with a shovel and helps me get a plan. Marjean always listens, then creates a safe place for me to be scared before I can be strong. By juggling sickness, a business, and single motherhood, Linda teaches me how courage, tenacity, and tenderness never go out of style.\n\nGirlfriends Can Be a Lifeline\n\nI saw Linda yesterday. Only a week and she looked at least ten pounds thinner, weight she didn't have to lose. And she was befuddled. I've seen it before, too many times: chemo brain. I feel angry: Why her? Helpless: What can I do? My homemade four-cheese macaroni casserole won't help.\n\nLinda has chutzpah, tenacity, and a tight girlfriend group who have known her and loved her actively for decades. I'm new to this sisterhood. Though they might be wary of a stranger in their midst, these women understand Linda is going to need all the help she can get. So they include me. We'll show up, do what\u2014no all\u2014we can. Thankfully, Linda's long-term prognosis is excellent. Still, Smiles and Laura taught me the hardest life lesson: No matter how much we pray, love and give of ourselves we can't cancel out sickness. Or cheat death when it's time. But girlfriends can sometimes tip the balance back in our favor. Consider Maggie's story:\n\nIn November of 1998 I was diagnosed with late-stage ovarian cancer\u2014stage 3-C\u2014which is one step away from stage 4 (e.g., worst-case scenario). The cancer hadn't yet spread to my lungs, but it had spread to other parts of my body. I was forty-five, recently widowed (my husband had been killed the year before in a car accident), and the mother of seven-year-old twin boys.\n\nMy gynecologist told me there were factors in my favor: my relatively young age and the fact that I was fit and had never smoked. Still, the first oncologist said I had only a 20 to 30 percent chance of being alive in five years. For seconds I sat stunned; then I stood up.\n\n\"Your statistics will not define my future,\" I said, walking out his door.\n\nI got into my parked car and cried, then started calling and calling. My girlfriends. There are twelve of us who have loved each other since junior high. All were at my home the following evening, three flying cross-country to show up.\n\nWe made a pact. I would choose to live. And if\/when I wanted to give up, my girlfriends would take turns standing by my side and whispering in my ear until I was strong enough to once again choose life.\n\nThat first year was the hardest. For starters I underwent something called \"debulking\" surgery. The surgeon was able to get only 75 percent of the cancer out. Then I suffered through six months of aggressive chemotherapy. Never a day went by that I didn't hear from all eleven girlfriends. One or two were always with me, holding my head over the toilet or doing the day-to-day chores of mothering my boys.\n\nWhen I missed midget football games, one girlfriend videotaped them. Another sewed a flannel nightgown in school colors, making me a legitimate bed-bound cheerleader. Delicious breakfasts, lunches, and dinners appeared like room service. Weekly movie night became a new tradition. Requisite: only movies with very happy endings.\n\nWhen I lost my hair, eyebrows, and lashes, we took over the wig salon, poured champagne, and determined I shun sandy brown locks for flamboyant, curly tresses. Orange, I was told, radiates warmth and energy while stimulating appetite, activity, and socialization. My demure, no-frills self somersaulted. \"Orange curly hair? Why not?\"\n\nMy new oncologist enrolled me in a study following fifty women with advanced ovarian cancer. The study was designed to evaluate correlation between social support\u2014or lack thereof\u2014and levels of an inflammatory protein called interleukin-6, or IL-6. High levels of IL-6 are linked to a poorer prognosis for women with ovarian cancer.\n\nEvery three months for the first year and every six months for the past eleven, I have sat down with a psychologist. Absent a husband or lover, I am cynically probed as to how \"only girlfriends\" could adequately meet innate needs for closeness, intimacy, and nurturance. Despite multiple interviewers' skepticism, I unfailingly score highest in social support. Other numbers illuminate my scores' significance: that first year my IL-6 scores were consistently 70 percent lower than my peers. Remember: the lower, the better. Even better, my lower than average IL-6 scores have remained stable for more than a decade. The most bittersweet number: of the original fifty women in the study, I am the only one remaining.\n\nLocus of Control Makes a Friend,   \nFinds Hope, and a Future\n\n\"It is because of Kristi that I am here, well and working. Kristi also helped my whole family have a better life,\" said Habiba.\n\nHabiba, a Bosnian immigrant, first came to the United States after suffering heinous atrocities during her country's civil war. Through near-starvation, untreated illness, untended pregnancy, barbaric living conditions, not knowing if her young husband was dead or alive, and being separated from her infant daughter, Habiba says she never considered giving up.\n\n\"It is hard to describe the kind of fear that lived in my heart, but I never forgot that I had to do everything I could to stay strong, to be positive. I had to look forward.\"\n\nObviously, this woman has an internal locus of control to envy, but what exactly was it that Habiba looked forward to?\n\nIn 1999, reunited with her husband and daughter, Habiba came to the United States. Though educated, she spoke little English, so she got the only job she could, working for an office building cleaning service in Jacksonville, Florida. One night while emptying a trash can, she met Kristi.\n\n\"When I met Kristi I was already very sick,\" Habiba shared, \"but I tried not to focus on that. Instead, I would come into this important woman's office in the early evening and try to be quiet so as to not disturb her work. But she talked to me. She asked me questions about my life and my family. She genuinely seemed to care.\"\n\nHabiba, only in her late twenties, was experiencing painful joint symptoms. She was soon diagnosed with rheumatoid arthritis and told by her doctor that, unless she began aggressive therapy immediately, she would be in a wheelchair before age forty. But the treatment was expensive and she had no insurance.\n\n\"I could not allow myself to focus on that bad news,\" Habiba told me, \"so I would spend hours before work prying my fingers open and massaging my hands and feet. Every movement became painful, even driving. Still, I believed God would not have helped me survive the war unless I was needed to live to be a strong and good mother for my beautiful daughter. My choice was to give up, stay home and die, or keep showing up. I chose to show up. One thing that helped was that almost every night I worked, I would see Kristi.\"\n\n\"Kristi\" is a petite blond daughter of a Baptist minister. She is also president of a Jacksonville Citibank site and a state officer for Florida, responsible for community and government relations, site work environment, and the overall culture of more than 4,800 employees. I asked about her hand in Habiba's life.\n\n\"I watched Habiba clean my office with more than meticulous dedication; she exuded something akin to reverence. I was struck by the way this warmly friendly yet surprisingly dignified young woman would quietly pick up and dust every one of my family pictures, particularly those of my son. I noticed she would look at those closely. I would learn my son is the same age as her daughter.\"\n\nDespite language and barriers of class and station, Kristi and Habiba became friends. Their bond bore fruit. In 2002 Habiba became a Citibank employee. Today she works in the Quality Department for Transaction Services, where she was awarded a 2012 Quality Excellence Award because, during her entire tenure, she has made zero errors processing credit cards and serving customers.\n\nHabiba's resilient internal locus of control is awe-inspiring. Nevertheless, it is Kristi's story I find most life-impacting. I ask myself, Would I even have noticed Habiba emptying my trash can? Would I have bridged our differences to begin and sustain a conversation, much less foster a friendship? Would I have taken the risk of championing to my superiors the hiring of a cleaning woman with broken English and health issues?\n\nThe old me would have said, \"Probably not.\" Because of Habiba and Kristi's story, however, I would like to believe my answer today might very possibly be \"Yes.\"\nCHAPTER 12:\n\nLighten Up and Play More\n\nPicture two tables of women eating lunch. At the first table sit two friends smiling, laughing, and sharing an appetizer. At the second table sit two scowling women robotically inhaling their salads while checking text messages. At which table are the women radiating health and well-being? Now, be honest. At which table would you likely be sitting?\n\nUnplug and Chill Out\n\nForty-two year-old Velma knows she shouldn't text while driving, so she compromises, putting her thumbs to work only at stop lights or during traffic jams. Today a blaring horn and middle-finger-out-the-window tells her she missed her light. Guiltily, Velma pockets the phone in her purse.\n\nShe comes home, slips off her pantsuit, puts on sweats, and looks at the elliptical in the den off their bedroom. She has about an hour before Connor shows up with the kids. They had agreed to take turns carpooling to soccer practice to trade off an hour for working out every other night. Velma looks at the elliptical again and considers her increasingly smushy thighs. Then she plops on the bed, pulls her phone out of her purse, and begins checking e-mails. She's just logged onto Facebook when she hears Connor's car. Quickly, she stashes the phone, throws water on her face, and grabs a towel\u2014just as she would have had she been working up a sweat.\n\n\"Web-based technology is no longer a convenience; it's my cocaine. My urge to log on is more irresistible than taking care of myself, spending time with my children, or having sex with Connor. Worse, I feel constantly anxious when unplugged.\"\n\nUnfortunately, our multitasking, information-overloaded, instant-results-driven \"Information Generation\" is increasingly at risk of being chronically harried and exhausted as stalking to-do lists evoke gnawing guilt and growing feelings of inferiority. New research indicates heavy Internet use leads to something termed \"popcorn brain,\" a condition where your brain becomes so accustomed to the constant stimulation of electronic multitasking that you become uncomfortable and unfit for life offline. Worse, other studies show insistent information fatigue can have harmful cognitive impact, leading women to use poorer judgment and make life-impacting choices they quickly regret.\n\nDr. Nora Volkow, director of the National Institute on Drug Abuse, explains that constant stimulation can activate dopamine cells in the nucleus accumbens, a main pleasure center of the brain. This simulates the addictive pleasure-seeking compulsion to never unplug. The looming doom is that over time, and with enough Internet usage, the structure of our brains can actually physically change.\n\nResearchers in China did MRIs on the brains of eighteen college students who spent about ten hours a day online. Compared with a control group who spent less than two hours a day online, these students had less gray matter, the thinking part of the brain. A 2012 Newsweek review of findings from more than a dozen other countries showed similar trends.\n\nWhile it may seem counterintuitive, one of the best health and beauty tips for any age is to chill out more. At a minimum, learn how to be \"unavailable\" for at least ten minutes three times a day. Get out of your house or office, or go to a room where you won't be disturbed. Leave your phone; at the very least, turn it off completely. Get practical to become unavailable.\n\nI advise overbusy, high-demand women to do their best to chill out more. If nothing else, spend more time in the bathroom and take more showers, places you are less likely to be interrupted. Block out all noise and distractions, center yourself, discern between what you have the power to impact and what you do not, and cultivate calm.\n\nLaughter Is Good Medicine\n\nLaughing is a neurotransmitter aphrodisiac. It also has an ecstatic ripple effect. According to my friend and \"joyologist\" Kathleen Halperin, \"We've long known that the ability to laugh is helpful to those coping with major illness and the stress of life's problems. But researchers are now saying laughter can do a lot more\u2014it can basically bring balance to all the components of the immune system, which helps us fight off diseases.\n\n\"Laughter provides a safety valve that shuts off the flow of stress hormones and the fight-or-flight compounds that swing into action in our bodies when we experience stress, anger, or hostility. These stress hormones suppress the immune system, increase the number of blood platelets (which can cause obstructions in arteries), and raise blood pressure.\n\n\"When we're laughing, natural killer cells that destroy tumors and viruses increase, as do gamma interferon (a disease-fighting protein), T cells (which are a major part of the immune response), and B cells (which make disease-destroying antibodies).\n\n\"Laughter may lead to hiccupping and coughing, which clears the respiratory tract by dislodging mucous plugs. Laughter also increases the concentration of salivary immunoglobulin A, which defends against infectious organisms entering through the respiratory tract.\n\n\"What may surprise you even more is the fact that researchers estimate that laughing one hundred times is equal to ten minutes on the rowing machine or fifteen minutes on an exercise bike. Laughing can be a total body workout! Blood pressure is lowered, and there is an increase in vascular blood flow and in oxygenation of the blood, which further assists healing. Laughter also gives your diaphragm and abdominal, respiratory, facial, and leg and back muscles a workout. That's why you often feel exhausted after a long bout of laughter\u2014you've just had an aerobic workout!\"\n\nThe psychological benefits of humor are quite amazing, according to doctors and nurses who are members of the American Association for Therapeutic Humor. People often store negative emotions, such as anger, sadness, and fear, rather than expressing them. Laughter provides a way for these emotions to be harmlessly released.\n\nLaughter can also be cathartic. That's why some people who are upset or stressed out go to a funny movie or a comedy club, so they can laugh the negative emotions away (these negative emotions, when held inside, can cause biochemical changes that can affect our bodies). Increasingly, mental health professionals are suggesting \"laughter therapy,\" which teaches people how to laugh\u2014openly\u2014at things that aren't usually funny and to cope in difficult situations by using humor.\n\nLaugh at Yourself\n\nThe ability to laugh at ourselves allows us the opportunity to embrace our flaws and promotes self-acceptance. It does not include harmful put-downs, ridicule, or negative sarcasm. Nor are we advertising that we are defective; rather, we are demonstrating that we are human.\n\nHumor is a positive coping mechanism that not only improves our mood, it builds our self-esteem. Unfortunately, we often resort to all kinds of unhealthy coping mechanisms like drinking, smoking, eating, overworking, and so on to make ourselves feel good. While these habits offer temporary boosts, they further undermine our self-esteem. Choose to laugh instead!\n\nTips for Laughing More Every Day\n\nWatch a funny TV show every day (old syndicated sitcoms are a great source of reliable laughs).\n\nRead the daily comics online or in the newspaper.\n\nLearn an age-appropriate joke and tell it to a youngster. Making children laugh is one of the best ways to bring more joy and humor into your life.\n\nSubscribe to a joke, quip, or quote of the day and have the laughs delivered right to your in-box.\n\nShare funny articles with friends and family members. Sometimes the thought of making someone else laugh can give us the same sense of well-being we get when we laugh at a joke. Reaching out to others with humor is a wonderful way to strengthen connections and emotional intimacy.\n\nThe Cheeky Science of Play\n\nIn March of 2012, my friend Pat Shea and I journeyed to Sante Fe to attend a workshop on transformational speaking led by Gail Larsen. The stated objective of the workshop was to help attendees discover their \"original medicine,\" for example, our singular gifts and talents that define our unique, individual roles in influencing change. I was skeptical that I had an original medicine, believing my best talent was regurgitating helpful information and other people's good ideas. Regurgitate is synonymous with vomit. \"Vomiter of Other People's Stuff\" didn't strike me as an original medicine name people would gravitate toward.\n\nGail paired me with twenty-something Alexandra Franzen, who sported several mesmerizing tattoos and surprisingly red hair. Alexandra introduced herself: \"I am half Jewish, half Swedish, a former lesbian, a recovering socialist, and a onetime helicopter pilot. I've moved through my strumpet years and my cubicle season to most recently establish myself as a quite successful entrepreneur.\"\n\nReally? Partnering with this nascent, maverick beauty was supposed to help me become a better speaker and more intentional agent of change?\n\nI looked at Gail. Raised my eyebrows. She smiled and nodded, pleased how well the two of us were bonding.\n\n\"Do you prefer Alex to Alexandra?\"\n\n\"Only if you think you don't have time to say my whole name.\"\n\nOh boy. Fumbling for common ground, I asked \"Alexandra\" about her entrepreneurial business.\n\n\"Mostly, I play with words. In fact, play is my work.\"\n\nWhoosh went my comfort zone. For the next five days Alexandra turned my thinking inside-out, explaining why playing every day is as critical to health, productivity, and creativity as sleep or food. Alexandra would later put her thinking to paper in her adorable book 50 Ways to Say You're Awesome published by Sourcebooks, spring 2013.\n\nNeurological science backs up Alexandra's premise. Our brains are made up of billions of neurons with connections between them. These neurons are bundled into groups called neural pathways. We have neural pathways for memory, attachment, emotions, language, motor control, each of our senses, and many more. When children and adults play, many parts of the brain are activated at the same time.\n\nThe brain \"lights up\" during a child's dramatic play: fine motor, gross motor, language, emotion, and memory; abstract concepts like \"good guys\" and \"bad guys\"; scientific concepts like cause and effect; social concepts like \"taking care of baby\"; and big themes like aggression, affection, loyalty, and power\u2014each with its own neural pathway\u2014are all seen during children's play.\n\nGary Landreth, professor at the University of North Texas, states that children act out their life experiences and feelings in a natural, dynamic, and self-healing manner. Play is the symbolic language children use for self-expression. When adults play, we begin to feel more positive because of a release of endorphins, and our stress levels go down. It calms and relaxes us while stimulating our brains and bodies. It helps us connect with others and increases our energy level and excitement. It is the antidote for loneliness, isolation, anxiety, and depression. It helps us feel happy, creative, and resilient.\n\nPlay is also the single best way for adults to simultaneously activate multiple neural pathways. If you want to know more, there is an Institute of Play and multiple books on the subject. My new fave is Stuart Brown's Play: How It Shapes the Brain, Opens the Imagination, and Invigorates the Soul. If, like me, you are hardwired to be a workaholic on overdrive, you also need to know:\n\nPlay has been scientifically proven to be good for the brain. All animals play. Grizzly bears that play the most survive the longest. Rats that socialize develop bigger, more complex brains. Play stimulates nerve growth in the portions of the brain that process emotions and executive function.\n\nPlay teaches us to use our imaginations. Imagination is perhaps the most powerful human ability, letting us create simulated realities we can explore without abandoning the real world. As kids grow older, the line between pretend and real becomes more solid, but imaginative play continues to nourish the spirit.\n\nRough-and-tumble play teaches us how to cooperate and play fair. Research in humans and animals has shown that roughhousing is necessary for the development of social awareness, cooperation, fairness, and altruism. If young rats are denied rough-and-tumble play, they develop serious social problems in adulthood and aren't able to mate.\n\nPlay helps us learn to be friends. When children are four to six years old, they start \"mutual play,\" listening to other kids' points of view and incorporating them into imaginative games. This mutual play is the basic state of friendship that sustains us throughout our lives.\n\nSometimes the best way to learn a complicated subject is to play with it. That's why kids often learn computer systems faster than adults; they aren't afraid to just try stuff out and see what works. Kids don't fear doing something wrong. If they do, they learn from it and do it differently the next time.\n\nPhysical play delays mental decline in old age. Research on this is still in the early stages, but older people who get regular exercise are less likely to suffer cognitive decline. Doing crossword puzzles, brainteasers, and other thinking games seems to help, too.\n\nA little play can help solve big problems. Play is nature's great tool for creating new neural networks and for reconciling cognitive difficulties. When we play, dilemmas and challenges naturally filter through the unconscious mind and work themselves out. Even a few hours spent doing something you love can make you new again.\n\nWhen we get play right, all areas of our lives go better. One of the hardest things to teach kids is how to make it past difficulty or boredom to find the fun. Making all of life an act of play occurs when we recognize and accept that there may be some discomfort in play, and that every experience has both pleasure and pain.\n\nMy friend Alexandra has a black belt in play. Her work is her play and her play is her work, consequently, she has a hard time telling the difference between the two. I am still learning, but I do know this: Play means more than unplugging. It means freeing your mind to have fun. If you have forgotten how to play, borrow a three-year-old. Get on the floor or go outside. Let them reteach you. Even better, consider taking up a sport.\nCHAPTER 13:\n\nPlay More Sports\n\nElizabeth Nye, director of Girls, Inc., in northwest Oregon, and I were on the phone. She said, \"My life, my aging experience, is nothing like my grandmother's or my mother's. My grandmother grew up on a farm and never finished high school. In her adult years, she got up at four in the morning and walked to the nursing home where she worked as a kitchen aide. My mother got her high school diploma, married, worked as a secretary, and smoked her entire life. Relentless economic insecurity wore them down. Both were determined to create a different reality for me.\"\n\nBoy, did they.\n\nA graduate of Georgetown University's School of Foreign Service, Elizabeth started her career at Population Services International (PSI) in Washington, DC. While there, she worked on a variety of social marketing projects for malaria prevention, clean water, and HIV\/AIDS prevention in the developing world. She then devoted the next five years to the pharmaceutical industry at Bristol-Myers Squibb. As product manager in the company's Virology Division, she marketed HIV\/AIDS products in the United States. She was also responsible for leading the Virology Division's Women's Initiative and African American Initiative to better understand and address the impact of HIV\/AIDS on these populations. In 2006, Elizabeth relocated to her hometown of Portland, Oregon, with her husband and two daughters.\n\n\"So, when you think of the circumstances and perspectives associated with how you grew up compared to those surrounding your daughters today, what do you believe will be the most significant game changer?\" I asked Elizabeth.\n\nI assumed one of two responses to my question: education or more financial security. She surprised me by saying, \"Sports. I think the fact that both my daughters are actively involved in sports teams will make the biggest difference.\"\n\nThough I fully understood that playing a sport is a calorie-burning exercise with great social dynamics, it took me a while to buy into the power of sports as a vehicle for positive aging. Reflect back that I was the puny, asthmatic child kept indoors. When I did go to physical education, no one ever chose me to be on their kickball team. Not only was I sickly, I was\u2014and continue to be\u2014comically uncoordinated. I rang Donna Orender.\n\nDonna, who lives near me in Jacksonville Beach, Florida, is a game changer of national scope. From 2003 to 2010 she served as president of the Women's National Basketball Association (WNBA). In 2005, Donna was named one of the ten most powerful women in sports by FOXSports.com.\n\n\"The data is undeniable. According to a survey of executive women, 81 percent played sports growing up and 69 percent said sports helped them to develop leadership skills contributing to their professional success.\" Many other women echoed Elizabeth's and Donna's thinking: \"I believe my basketball experience can be directly linked to my business experience and success,\" said Mary Claire Bonner, who retired in 2009 from Aetna. Before retiring, the New York resident was senior vice president of local and regional business for Aetna, working directly for the company's president and \"running a large business that concentrated on small and midsize employer health benefit needs in more than thirty states.\" Reflecting on the position, Bonner discovered that her favorite part about leading others was creating a strong team. The skills to be good at this, Bonner noted, started in fifth grade when she joined the basketball team.\n\n\"Playing basketball changed me as a person,\" said Bonner, who played from grade school through her junior year at one of Penn State's Commonwealth campuses.\n\nAccording to the Hall of Fame Network magazine, the Women's Sports Foundation (started by female tennis champion Billie Jean King) became established because, \"In an economic environment where the quality of our life is dependent on two-income families, our daughters cannot be less prepared for the highly competitive workplace than our sons.\"\n\nDigging into the data, I was astounded by the correlation between sports and success among women, especially high-profile leaders. The key contributor was Title IX, a landmark legislation that celebrated its fortieth anniversary this year. A 2010 study by Betsey Stevenson, chief economist of the US Department of Labor, found that Title IX accounted for a 20 percent increase in women's education and for about a 40 percent rise in employment for women ages twenty-five to thirty-four.\n\nIn a 2012 column for CNN Money, Sanyin Siang, executive director of the Coach K Leadership and Ethics Center at Duke University, wrote, \"Well documented are IMF managing director Christine Lagarde competing in synchronized swimming; PepsiCo CEO Indra Nooyi playing cricket; Kraft Foods CEO Irene Rosenfeld engaging in four varsity sports in high school and basketball in college; and HP CEO Meg Whitman playing lacrosse and squash.\n\n\"This is not surprising considering the mental discipline and social skills honed through playing sports. At the Coach K Center, we hear often from leaders such as Fidelity President Kathy Murphy, Walmart EVP of People Division Susan Chambers, and Morgan Stanley Managing Director Carla Harris about how sports had fostered the resilience, agility, and team-orientation that advanced their leadership. Also, engaging in and appreciating sports gives women an understanding of business parlance that is often steeped in sports analogies and provides another avenue for social inclusion in the workplace.\"\n\nA Platform for Social Change\n\nIn the 2012 Summer Olympics, for the very first time, women athletes from Saudi Arabia were represented, and the US Olympic team had more women delegates than men. Women also made up 40 percent of the total number of athletes at the 2012 games.\n\nIn June 2012, Hillary Clinton announced the Global Sports Mentoring Program, saying, \"We want to find ways to get more women and girls on the field, the court, the track, in the pool, the mat, wherever their interests and talents take them so they can discover their strengths, develop their skills, experience that special satisfaction that sports can bring, win or lose. And we believe in the positive effects that can flow out of that experience with girls and women across their lifetimes and, by extension, for their families and communities.\"\n\nThe 1984 Olympic gold medalist Nawal El Moutawakel says, \"Sport is one of the best tools for social change because it is a large part of cultures around the world and reaches into every socioeconomic class of society. Sport helps develop self-esteem and confidence, improve physical and mental well-being, serve as a medium of communication, and empower women to improve themselves and their communities.\"\n\nI called my niece Shelley and asked her to chime in. Shelley is director of development for Explore Austin, a six-year committed mentoring program for underserved girls and boys. Think leadership development meets Outward Bound\u2013type vision quests, including strenuous hiking and rock climbing to get a sense of Explore Austin's model: \"What we have seen with our youth in these programs is that moving through the realms of physical exertion, collaboration, and teamwork creates deep bonds of trust difficult to ignite through only conversation. Taking young women out of their comfort range to stretch them physically, mentally, and emotionally is a dynamic engine for personal growth, particularly when they experience how we, their adult mentors, are right there in it with them.\n\n\"What has been just as exciting as watching our girls mature as individuals and leaders is the response of the women mentors. They tell us that, while they originally believed committing to mentor these young girls for six years would be something of a chore, a long-term community obligation, they find they grow as much if not more than the girls they are mentoring. They say that what becomes most transformative is the opportunity to not only lead, but participate. As the years go by, girls who were 'students' evolve into the life-teachers.\"\n\nDon't Wait Until You Retire\n\nForbes columnist and retirement activist Robert Laura warns: \"Too often people think there is a magical world they will enter once they officially retire, which will help them do things they didn't do before they retired . . . walking every day, eating healthy, writing memoirs that turn into New York Times bestsellers, and so on. But there is nothing magical about retirement. In fact, it's my belief that retirement will only magnify what you already are. If you frequent the couch, prefer fatty foods, or indulge in an evening cocktail before bed, retirement will only provide more time for you to reinforce your existing habits.\"\nCHAPTER 14:\n\nPartner with Care\n\nMy personal opinion is that girlfriends and pets are nonnegotiable companions, but most women also long for a Prince Charming with whom to float into forever. Is marriage worth the effort?\n\nWhen It Works\n\nPhyllis Tousey was explaining to me why she is convinced women should regard \"breakfast and sleep as a holy disciplines,\" when her phone buzzed.\n\n\"Oh my goodness,\" she girly-gushed, \"it's Chip.\"\n\nChip is Phyllis's husband of forty-something years. He is also a top-notch estate lawyer. I should know. He is ours.\n\nNothing about Phyllis makes you think \"girly-gush.\" As a nurse epidemiologist, her research in how nutrition and lifestyle choices impact the immune system is rubber-hits-the-road stuff. As a woman, her faith frames her as a no-nonsense, rock-solid pillar of her church community. So perhaps you can imagine my surprise when I watched her scramble to answer her phone, then blush as she and Chip set the time and place for their dinner date. She hung up, looked at me, and said, \"He is such a blessing.\"\n\nDr. Cathy Christie shared, \"I am married to the love of my life. When we are separate during our days, I can't wait until the time we are together again.\"\n\nDonna Orender took the theme of happily ever after to a whole new level: \"MG is my beloved.\"\n\nThese women, all role models for an aging experience any woman would aspire to, include a happy marriage as part of their health and beauty regimen. Medical science says they are on the right track.\n\nHealth Benefits of Marriage\n\nResearch suggests that married people enjoy significantly better health than the unmarried. In other words, marriage is good for your health and your heart, in more ways than one. In fact, one sociologist suggested the health benefits of marriage are as significant as the benefit from giving up smoking. Here are top findings:\n\nMarried adults have a greater likelihood of living longer than their unmarried counterparts. Married people live longer because they are more likely to enjoy better physical health. This is partly due to the fact they're more likely to recognize symptoms, seek medical treatment, avoid risky behavior, recover quicker, and eat a healthier diet. Spouses are intimately aware of and impacted by their spouse's choices. In a sense, couples have a significant vested interest in watching out for one another and encouraging healthy choices and behavior. Researchers found emotional support from a spouse can help people recover from both minor and major illnesses and even help cope with chronic diseases. Some studies even suggest that marital relationships actually boost the immune system, making sickness less likely in the first place.\n\nMarried women are less likely to suffer from any form of mental illness. Married people have significantly lower rates of severe depression and at least half the likelihood of developing any psychiatric disorder than never-married, cohabiting, and divorced people. According to Tara Parker-Pope, the Well columnist for the New York Times and author of For Better: The Science of a Good Marriage, some of today's most interesting research on the relationship between marriage and health is being led by a pair of researchers at Ohio State University College of Medicine. The duo, Ronald Glaser and Jan Kiecolt-Glaser, are also, fittingly, married to each other. Glaser and Kiecolt-Glaser's scholarly collaboration has its roots in a chance encounter during a faculty picnic in October 1978 on the Ohio State campus. Glaser and Kiecolt eventually met for lunch at the university's hospital cafeteria. They married a year later, in January 1980.\n\nRecruiting seventy-six women\u2014half of whom were married and the other half separated or divorced\u2014the Glasers used marital-quality scales, types of questionnaires that ask couples to indicate agreement or disagreement with statements like \"If I had to do it over again, I would marry the same person\" or \"We often do things together.\"\n\nNext, using blood tests, the Glasers measured the women's immune-system responses, tracking their levels of antibody production and other indicators of immunity strength. The results showed that the women in unhappy relationships and the women who remained emotionally hung up on their ex-husbands had decidedly weaker immune responses than the women who were in happier relationships (or were happily out of them).\n\nAn Unhappy Marriage   \nCan Make You Sick\n\nResearch shows that women in an unhappy marriage have a weaker immune system than women who are happily married or divorced. Marriage stress and heart problems have a strong connection with women. Women who face the highest level of stress in marriage increase the possibility of a heart attack and bypass.\n\nRelationship stress can also affect the body's ability to heal. In the research, wounds took a day longer to heal with partners who quarreled compared to those who didn't. With couples who showed a higher tension and a high level of hostile behavior, the wounds took two days longer to heal.\n\nHappily Ever After Hits a Snag   \nfor the Over-Fifty Crowd\n\nMarried women should face the fact that odds are that they will someday be alone. Across the industrialized world, women still live five to ten years longer than men. Among people older than 100 years old, 85 percent are women. And while widowhood might loom, new research from the National Center for Family and Marriage Research at Bowling Green University in Ohio shows that one in four people getting divorced is older than fifty. In 1990, the statistic was less than one in ten.\n\nAccording to the Modeling Income in the Near Term (MINT) data system, divorced and widowed baby boomer women are one of the fastest growing economically vulnerable population segments. Women who choose to think their husbands, or any man, will take care of them financially for life are engaged in highly risky thinking.\n\nThe scenarios for risk and na\u00efvet\u00e9 vary. There are women who continue to rely on a spouse as their sole financial safety net, other women who consider how best to merge investment portfolios should they marry again, and a burgeoning subset of breadwinning women who need to protect personal assets in the event of divorce. Yes, breadwinning women.\n\nA 2012 survey of 1,410 American women and 604 men\u2014between the ages of twenty-five and sixty-eight\u2014found that 53 percent of women make more money than their male counterparts, with an increasing number of women assuming this role as a result of partners losing jobs during the financial crisis, divorce, and marrying later in life.\n\nKeep Dollars in Your Own Pocket\n\nIn 2006, Kristen Walter was living her dream life. Her college-sweetheart-turned-husband Scott adored her. Their two children, Nick and Lexi, were healthy, good kids. Scott's executive level job allowed a nice house in friendly community with country club amenities. Kristen loved being the mom who kept their home sparkling with good energy and home-cooked meals while also serving as room mom, softball coach, and Bible study leader.\n\nThen disaster struck. Scott, only thirty-seven at the time, woke up having a massive heart attack. Blessfully, Scott recovered but a few months later he rocked Kristen's world again saying:\n\n\"Honey, we need to think about you doing something.\"\n\nKristen says, \"I remember that moment vividly. I was standing at our kitchen sink washing dishes. My gut hurt. The idea of working outside our home was as foreign to me as standing on a street corner hustling. But, though I wanted to ignore him, I knew in my heart Scott was right. I prayed and asked for guidance. Because family was and is my priority, I asked God to help me plan my life around my work, not my work around my life.\"\n\nSoon after, Kristen met Molly Geil, executive national vice president for Arbonne International, in the grocery line. One friendly conversation spawned more. Soon Kristen was in the business of marketing beauty, health, and wellness products to women. In eight months, she had earned an Arbonne Mercedes-Benz. Amazingly, her team was able to do this in Nevada, which at the time was the hardest-hit state in the recession.\n\nAvoid the Gilded Cage\n\nFifteen minutes late, Lauren swept into the Palm wearing a vintage cream-colored Chanel confetti skirt suit and gold Fendi kitten heels. Heads turned. Zigzagging through white tablecloths, handing out business cards right and left, Lauren was her own best advertisement.\n\nSpotting Sandy, Lauren shimmied into a corner booth. \"How are you, sweetie?\" she asked, leaning in for a kiss while surreptitiously scrutinizing her oldest friend.\n\n\"Fine, good,\" Sandy answered, then nervously looked at her watch.\n\nAntennae up, Lauren schmoozed on. \"And how's Dan? Any news from his last interview?\"\n\n\"Dan? Well . . . Dan's fine. Just a little depressed. But that's normal, you know. He's been chief operating officer of a multimillion dollar company for years. How do you stop that one day and start interviewing the next?\"\n\nTheir waiter interrupted. Lauren ordered a bottle of Groth Chardonnay. Sandy demurred at first but, when the wine was poured, gulped almost the entire glass in one swallow.\n\n\"Honey,\" Lauren cooed, \"you look a little thin. Eat some bread.\"\n\n\"Oh, I've been running more. With Dan in the house all the time, he keeps the television blaring and it makes me nervous. So I go running two to three times a day now. Dan doesn't mind. He likes me thinner.\"\n\nDespite layers of silver and diamond bangles, Lauren noticed a purplish smudge around Sandy's right wrist. \"Is Dan drinking more?\" she probed gently.\n\nSandy's nose flared like a spooked thoroughbred. \"Not really. I mean, he never has a scotch before five. An occasional Bloody Mary in the morning and maybe some wine with lunch, but that doesn't count when you are essentially on eternal vacation. I mean, look at us.\" She tipped her wineglass in Lauren's direction.\n\nLauren had gleaned enough. Seven years earlier, she had walked out with her three teenage girls, cash in wallet, and as much clothes and jewelry as could be stuffed into the back of her BMW sedan. Desperate for funds, she staged a couture consignment sale at a friend's house four states away. To her amazement, she scored four thousand dollars, which she immediately reinvested shopping at estate sales and, later, rifling through European flea markets.\n\nToday she owns Couture Closets in five cities, a wonderful riverfront home in town, and a dream-retreat cabin outside of Asheville. Best of all, all three daughters were in college on her dime, not a penny required from their scumbag father.\n\n\"Sandy,\" she said now, \"I could truly use your help. Please become one of my buyers. You would be great. We would have such fun traveling to estate sales all over the world. It would be like old times when we stewed for Delta. Best of all, think how great it would be making your own money finding fabulous clothes!\"\n\nSometimes Prince Charming Farts\n\n\"Genie, are ya'll hiring at the medical practice? I have a friend who really needs a job.\"\n\nUghhh. I hate being buttonholed by friends with unskilled children, relatives, or chums needing work. Nevertheless, every so often a pearl lands in my lap.\n\n\"We are recruiting for a position in our front office, Sue. Tell me about your friend.\"\n\n\"She is absolutely amazing. She has chaired four events for the school where my Brian is and each one blew the roof off fund-raising. I don't know how she does it with three children in three different schools, but she is equally active with all of them and also last year chaired the board of Girls, Inc. I know from experience that she is fanatically organized, always working off lists and spreadsheets. And she is an excellent writer. You wouldn't believe the quarterly fund-raising updates she sends out mixing facts and figures with true-to-life stories. But she's no nerd. She has a super laugh and a really sweet way of encouraging others to do their best. When Marilynne chairs an event, the other moms can't move fast enough to sign up for her committees.\"\n\nHmmm. Goal-oriented lists and spreadsheets, chairing a non-profit board, strong writing skills, proven motivational talent, a good sense of humor and\u2014evidently\u2014superior time management abilities . . . I was intrigued.\n\n\"What's her name again?\"\n\n\"Marilynne Ford. Her husband was Chris Ford.\"\n\n\"Chris Ford the Ponte Vedra stockbroker who supposedly put on an Armani suit, drank a bottle of Dom Perignon, and drove his Porsche off the Dames Point Bridge?\"\n\n\"Uh-huh.\"\n\nThree days later I interviewed Marilynne. She was candid that, while she had not worked for pay in close to twenty years, her family's recent tragedy meant she now had to. She was poised and articulate. Most impressive was her clear determination to provide her children with an alternate example of how to cope when times are tough.\n\nI didn't hire her. Instead, I passed her r\u00e9sum\u00e9 onto the chair of a search committee. Within six weeks, Marilynne was named development coordinator for Experience Works, a nonprofit organization committed to helping older adults re-enter the workforce.\n\nWhen the Going Gets Tough,   \nRemember Why You Fell in Love\n\nWhen Smiles first introduced me to Randy on Christmas Eve 2002, I hardly noticed him. He was nice enough, and I loved his two-year-old lab, Rigger, but I was relishing my time alone and not interested in men or dating.\n\nAfter that, Randy came back and forth from Jacksonville to Apalachicola almost every weekend. I would see him when the three of us\u2014Smiles, Randy, and I\u2014would have dinner. I slowly became intrigued by this gentle, gifted healer who paradoxically was also a good old boy who loved to hunt and fish.\n\nOn May 7, 2003, we went to lunch. Alone. As Smiles used to say, \"Randy came home from that lunch thirty-six hours later.\"\n\nNot long after, I moved to Jacksonville. Three years later, we married.\n\nWhat's funny about Randy and me is that our paths have always been very close. I grew up in Panama City, Florida, only an hour and a half away from Apalachicola. We shopped at the same store for our Easter outfits, our parents celebrated special occasions at the same small Italian restaurant, and we believe we vacationed on St. George Island only blocks away from each other at least three times.\n\nWhat's real about the two of us is that we share a deep faith in God. We also both understand the importance of family, blood and chosen. We are quirky, jointly spoiling our blended pet family\u2014Rigger, Bodhi, and Nami\u2014as if they are baby angels. And, not to be understated, we have both been healthcare mavericks, ahead of our time in our individual and shared commitment to a more integrated, holistic model of care. But these last several years we have hit some rough waters.\n\n\"Rough as in the size of a puddle?\" my friend Deirdre asked.\n\n\"More like the size of the Atlantic Ocean. But we hung in there.\"\n\n\"What helped you weather the storm?\"\n\n\"After another same-theme fight, I was majorly upset, stressed, and sleepless. About three in the morning I got up and started working on my computer. Work is my go-to drug to dull anything I don't want to feel or can't figure out. Randy came downstairs, and instead of ridiculing me for being an over-the-top workaholic or starting up our fight again, he popped a bottle of champagne, poured me a glass, turned on that Cajun redneck reality show Duck Dynasty, and suggested I curl up with him to watch. I was so surprised I did.\n\n\"Something happened that night. As we laughed, sipped, and cuddled, I remembered why I fell in love with this man\u2014not as a doctor or my business partner but as a guy. Just a goofy, sometimes selfish, yet pretty sexy guy who happens to really love me.\n\n\"The next day I determined to shift my attention away from what Randy wasn't doing for me and focus more on what I needed to do for myself. I decided to practice what I was preaching in my writing. I got back into the routine of working out regularly and scheduling time with my girlfriends. I also decided to stop bitching at Randy and hire a handyman and an executive assistant.\"\nEpilogue\n\n\"Ineed matching bras and panties. Really. Stop rolling your eyes and send someone. Come on. Matching bras and panties are every woman's secret empowerment.\"\n\nPause.\n\n\"Don't tell me yours don't match now?\"\n\nTrite as it sounds, I didn't know whether to laugh or cry. It was January 5, 2012. Laura had been admitted to the ICU the night before. We\u2014Randy and I\u2014rushed over as soon as we heard. She was supposedly \"better,\" now in a step-down unit.\n\nAfter weeks of dinosaur-heavy business stress compounded by a raw, draining, and, ultimately, pointless fight with Randy the night before, I was exhausted. Laura's hospital bed beckoned like a lemon-cr\u00e8me-stuffed, raspberry-iced cupcake. My foggy brain registered that coveting someone's hospital bed\u2014particularly someone you adored\u2014was perverse and had probably sent some weird reverberations out into the Universe.\n\nI asked the nurse for directions to the nearest awful coffee machine and rallied, sending someone out for underwear. Excuse me, lingerie: pastel, ultrasoft, and, of course, matching bras and panties.\n\nI thought back to a night years earlier when Laura was staying with us so I could take her back and forth to Mayo for treatment. After a day of chemo-induced, dignity-stealing nausea and poopy problems, I popped a chick flick into the DVD, hoping to do something, anything, to make her smile. The movie was Kissing Jessica Stein. Randy was in our living room interviewing a new doctor.\n\nIf you haven't seen it, this movie is about a straight Jewish girl who becomes smitten with an elfin free spirit, who happens to also be a girl. The unlikely protagonist navigates the horror-heartbreak-acceptance of her family, as well as an unexpected rekindled attraction to her college boyfriend. It is funny. Laura and I laughed so hard Randy poked his head in.\n\n\"Can you two please be a little less loud? I am in a very serious meeting.\"\n\nWe put our hands over our mouths. Laura's giggles spilled like jelly beans out of an open bag. I chortled so hard wine spewed up and out my nose. Peeking around Randy, I saw Rigger, our yellow lab, licking the doctor's left ear while Nami attacked his shoelaces as if they were squirming earthworms. I barely made it to the bathroom.\n\nIt's a family trait, not incontinence . . . Sheila and I both tinkle in our panties when we belly laugh.\n\nRandy banged the door.\n\nLater, part bewildered, part petulant, he said, \"Sometimes you act like you love Laura more than me.\"\n\nNow if any weird thought just flipped across your smutty little mind, squash it.\n\nTruth is, I did love Laura powerfully, in deep ways I cannot explain. I suppose I should have been jealous because of her history with Randy, and also because she was a showstopper: a mix of Julianne Moore and Diane Lane good looks with a petite, sculpted figure Barbie would have envied. But Laura was so much more.\n\nLaura was a spiritual rule-keeper, an extravagant mother, and the kind of optimist who believed Lassie would always come home before the credits rolled. To me, she was always abundantly kind, unswervingly encouraging. Laura told me I was smart, I was loved, I was on the right path. On good days, I believed her.\n\nWhen I first moved in with Randy, it was Laura who counseled me on shifting from singledom to couplehood with a sense of humor in tow. She taught me mnemonics to keep straight the names of everyone in his large Catholic family. She also grooved on my work, taking notes when I described my latest team-building initiative, clipping articles on customer service and proffering stringent feedback when she thought I was too soft\u2014or too hard\u2014a leader.\n\nAlways feminine and gleefully accessorized, Laura teased me to be more girly. She told me to wear a lace camisole under my pinstripe suit and try honey-warm highlights around my face. She would dare me to skinny-dip and surprise Randy when he came out to walk our dogs.\n\nOur last afternoon together, Laura urged me to learn to speak Italian, try dark chocolate with caramel and sea salt, and\u2014of all things\u2014make sure our medical practice did not give up converting from paper-based medical charts to electronic health records (EHR).\n\n\"I know it's expensive, Genie,\" she said, \"and surely it's going to be a hassle, but look . . .\" Laura opened a four-inch, three-ring binder showing me pages and pages of crisp, texted health records, most highlighted, some with her lusciously penciled notes and intricate drawings in the margin.\n\nNew government regulations were requiring that all hospitals and doctors' offices shift to EHR or be subject to financial penalties. But the transition was a huge investment and, understandably, Randy preferred his tried-and-true pen and paper chart to a computer keyboard and screen. I sighed.\n\n\"And this new book you are writing on aging, how's it going?\"\n\n\"I am thinking of punting the whole project. There are so many books about aging already out there. Health Communications Inc. has agreed to publish, but I'll be funding a lot of the marketing and media out of pocket.\n\n\"I sit at the computer and stare at the screen for hours. I'm beginning to think the whole idea is just another ego trip, my wanting to pretend I have answers I don't. I mean, what fresh thing do I have to say about aging? It's not like I'm the poster woman for getting it right.\"\n\nLaura's prednisone-puffy face sharpened. Her near-skeletal little hands reached out and grabbed mine.\n\n\"You must write this book. You have to. For Danika. For my granddaughters yet to be born.\"\n\nTears rolled down my face and off my chin.\n\n\"Promise me,\" she demanded.\n[Appendix A:   \nFavorite Belly Flat Recipes](Content.html#Appendix-A--Favorite-Belly-Flat-Recipes)\n\nBreakfast\n\nBroiled Grapefruit Serves 2\n\n1 grapefruit\n\n1 teaspoon ground ginger\n\n1 tablespoon honey\n\nPreheat the broiler.\n\nCut the grapefruit in half and use a small serrated knife to cut out the sections. Spoon the sections and juice into a bowl. Scrape out all of the remaining thick skin and pulp and discard. Spoon the sections from the bowl back into the halves. This is best done one half at a time.\n\nSprinkle the ginger and drizzle the honey over the top of each grapefruit half. Place the halves on a baking sheet.\n\nBroil for 3 to 5 minutes, until the honey begins to bubble.\n\nBanana in Bark Serves 1\n\n1 tablespoon ground almonds\n\n1 tablespoon ground or milled flaxseed\n\n1 whole banana\n\nMix almonds and flaxseeds on a large plate.\n\nPeel banana and roll in mixture until fully covered.\n\nEither eat immediately or place on wax paper and put in refrigerator to chill before eating.\n\nAsparagus Omelet Serves 2\n\n4 egg whites\n\n\u00bc teaspoon salt\n\n4 egg yolks\n\nDash of pepper\n\n1 cup 1% milk-fat cottage cheese\n\n1-1\/2 tablespoons extra-virgin olive oil\n\n6 asparagus spears, trimmed and slightly steamed\n\nPreheat the oven to 350\u00b0 F.\n\nBeat egg whites until frothy. Add salt and beat until stiff.\n\nBeat yolks until thick and lemon colored; add pepper and cottage cheese and beat until well blended. Fold egg whites into yolks.\n\nHeat olive oil in a 10-inch iron skillet; pour in omelet and cook approximately 3 minutes, until the bottom is lightly browned.\n\nTop with asparagus spears and finish cooking in the oven for 15 minutes or until a knife inserted in the center comes out clean.\n\nCitrus Ambrosia Serves 2 to 3\n\n1 11-ounce can mandarin oranges, drained\n\n1 pink grapefruit (or grapefruit sections in a jar, packed in juice,   \nnot syrup), cut into \u00bd-inch pieces\n\n2 tangerines, cut into \u00bd-inch pieces\n\n1 cup fresh cherries, pitted and halved\n\n1\/8 cup silvered almonds\n\nToss all the ingredients together and chill for 1 hour.\n\nEgg in a Nest Serves 2\n\n2 slices whole-grain bread\n\nExtra-virgin olive oil\n\n2 eggs\n\n2 tablespoons grated Parmesan cheese\n\nRed pepper flakes to taste\n\nTurn a small glass upside down to cut a circle out of each bread slice.\n\nLightly coat a skillet with olive oil, place the bread slices in the skillet, and break an egg into each hole.\n\nCook on medium heat, turning once until the eggs are the desired firmness.\n\nRemove from the skillet and garnish with the Parmesan cheese and red pepper flakes.\n\nFlaxseed and Fruit Smoothie Serves 1\n\n1 cup skim milk\n\n1 cup frozen berries\n\n1 banana\n\n1 cup plain low-fat yogurt\n\n1 tablespoon flaxseed oil\n\n1 tablespoon ground flaxseed\n\nCombine all ingredients except flaxseed in blender. Blend well.\n\nPour into large glass and stir in flaxseed.\n\nOrange Smoothie with Flaxseeds Serves 4\n\n1 banana\n\n1 6-ounce can frozen orange juice concentrate\n\n2 cups vanilla soy milk\n\n1 teaspoon ground ginger\n\n2 tablespoons milled flaxseeds\n\nPlace the banana, orange juice concentrate, soy milk, and ginger in a blender. Process until the ingredients are blended and smooth.\n\nStir in the flaxseeds and serve.\n\nSMILEY TOAST Serves 2\n\n1 tablespoon sesame seed butter   \n(may substitute almond or peanut butter)\n\n2 slices whole-grain or whole-wheat bread, toasted\n\n\u00bd banana, sliced\n\n2 dozen raisins\n\nSpread the sesame seed butter on the warm toast.\n\nUse the banana slices to make eyes.\n\nArrange the raisins in a semicircle like a smile.\n\nYogurt and Fruit Parfait Serves 1\n\n1\/3 cup ground almonds\n\n2 tablespoons ground or milled flaxseed\n\n1 cup plain low-fat yogurt\n\n3\/4 cup berries\n\nRaw honey (optional)\n\nIn small bowl, mix almonds with flaxseed.\n\nIn parfait glass, alternately layer yogurt, berries, and almond-flaxseed mixture.\n\nIf desired, drizzle a small amount of honey on top.\n\nCrustless Cruciferous Quiche Serves 6\n\n1 cup skim or fat-free milk\n\n6 eggs\n\n1 cup frozen chopped spinach, thawed, drained and squeezed dry\n\n1 cup frozen chopped broccoli   \n(thawed, well drained)\n\n1 finely chopped small white onion\n\n1\/2 cup grated mozzarella cheese\n\nSalt and pepper to taste\n\nPreheat oven to 425 degrees. Beat eggs and milk together until frothy. Pour into well-oiled, medium-size iron skillet. Add in veggies and onion and top with cheese. Bake for 30 minutes until top is crusty. Add salt and pepper if needed.\nMain Dishes:\n\n[Perfect for  \nLunch or Dinner](Content.html#Main-Dishes--Perfect-for-Lunch-or-Dinner)\n\nTurkey and Asparagus Wraps Serves 4\n\n16 asparagus spears\n\n1\/3 cup plain low-fat yogurt\n\n1 teaspoon lemon juice\n\n1 teaspoon curry powder\n\nSalt and pepper to taste\n\n1 pound deli-style turkey breast, sliced\n\nWash, trim, and steam asparagus spears until tender. Set aside to cool.\n\nMix yogurt, lemon juice, curry powder, salt, and pepper in a small bowl.\n\nPlace 2 or 3 slices of turkey on a plate. Spread with the yogurt mixture. Put asparagus spears on one end and roll up. Secure with toothpick if necessary.\n\nServe immediately or refrigerate.\n\nVegetable Barley Soup Serves 8\n\n2 quarts vegetable broth\n\n1 cup uncooked barley\n\n2 large carrots, chopped\n\n2 stalks celery, chopped\n\n1-14.5-ounce can diced tomatoes with juice\n\n1 cup cauliflower, chopped\n\n1 cup broccoli, chopped\n\n1 onion, chopped\n\n3 bay leaves\n\n1 teaspoon garlic powder\n\n1 teaspoon salt\n\n1\/2 teaspoon ground black pepper\n\n1 teaspoon dried parsley\n\n1 teaspoon curry powder\n\n1 teaspoon paprika\n\n1 teaspoon Tabasco (optional)\n\nPour the vegetable broth into a large pot. Add the barley, carrots, celery, tomatoes, cauliflower, broccoli, beans, onions, and bay leaves.\n\nSeason with the garlic powder, salt, pepper, parsley, curry powder, paprika, and Tabasco.\n\nBring to a boil, then cover and simmer over medium-low heat, stirring occasionally, for at least 1 hour.\n\nSavory Spinach and   \nSalmon Salad Serves 4\n\n3 tablespoons orange juice\n\n2 tablespoons low-sodium soy sauce\n\n2 teaspoons minced fresh ginger\n\n1 teaspoon raw honey\n\n1-1\/2 teaspoons extra-virgin olive oil\n\n4 (6-ounce) salmon fillets\n\n2 cups fresh spinach, washed\n\n1\/2 cup mandarin orange slices\n\n1\/3 cup chopped green onions (scallions)\n\n2 hard-boiled eggs, sliced\n\n1\/4 cup ground almonds mixed with 1 tablespoon ground or milled flaxseed\n\nIn a small bowl, combine the orange juice, soy sauce, ginger, and honey. Whisk, gradually adding the olive oil, until well blended.\n\nGrill, bake, or poach the salmon fillets.\n\nMake a bed of spinach on each of four plates. Place salmon in the center of each one.\n\nGarnish each with mandarin orange slices, scallions, and eggs. Drizzle the orange-soy sauce dressing over the entire plate and sprinkle with almond-flaxseed mixture.\n\nGrilled Fish   \nwith Citrus Marinade Serves 4\n\n1\/4 cup extra-virgin olive oil\n\n2 tablespoons grapefruit juice\n\n3 tablespoons lime juice\n\nSalt and pepper to taste\n\n4 (6-ounce) fish fillets\n\nCombine olive oil, grapefruit juice, lime juice, salt, and pepper. Pour over fish and marinate in refrigerator for 2 or more hours.\n\nGrill and serve.\n\nNot Your Ordinary   \nTuna Salad Serves 4\n\n1 large (12-ounce) can white albacore tuna in water\n\n1 red pepper, chopped fine\n\n1 yellow pepper, chopped fine\n\n1\/4 cup grated green apple\n\n1\/3 cup shredded fresh cabbage\n\n2 tablespoons plain low-fat yogurt\n\n1\/2 teaspoon lemon juice\n\n1\/2 teaspoon rice wine vinegar\n\nSalt and pepper to taste\n\nMix all ingredients together in a large bowl.\n\nChill for at least 1 hour before serving.\n\nCauliflower Crab Cakes Serves 6\n\n2 cups crabmeat\n\n2 cups cooked and mashed cauliflower\n\n1\/3 cup minced celery\n\n1\/3 cup minced onion\n\n1 tablespoon parsley\n\n2 eggs, beaten\n\nExtra-virgin olive oil or grapeseed oil (for saut\u00e9ing)\n\nCombine all ingredients in a large bowl, except the olive oil.\n\nForm into 6 patties and chill in refrigerator for at least 1 hour.\n\nBrown in skillet lightly coated with olive oil.\n\nChicken and Cold Rice Salad Serves 10\n\n2 cups cooked brown rice (you may use whole-grain minute brown rice)\n\n2-1\/2 cups cooked chicken, cubed\n\n1 cup broccoli florets, steamed al dente\n\n\u00bd cup shredded almonds\n\n2 tablespoons minced parsley\n\n1 cup plain low-fat yogurt\n\nSpinach\n\n4 tangerines\n\nToss everything together in a large bowl except the spinach and tangerines. Chill for at least 2 hours.\n\nServe on a bed of spinach and surround with tangerine slices.\n\nGrilled Salmon   \nwith Dill and Lemon Serves 4\n\n2 tablespoons dill\n\n1 teaspoon garlic powder\n\n4 tablespoons extra-virgin olive oil\n\n2 tablespoons white wine vinegar with tarragon\n\nJuice of 2 lemons\n\n4 (1-inch thick) salmon steaks, 4 ounces each\n\nMix dill, garlic powder, olive oil, vinegar, and lemon juice. Pour mixture over salmon steaks and marinate in the refrigerator for at least 2 hours, turning at least once.\n\nPlace salmon on hot grill and cook 3\u20135 minutes on each side.\n\nTurkey, Broccoli,   \nand Cashew Roll-Ups Serves 2\n\n1\/2 pound thinly sliced roast turkey breast\n\n1 tablespoon honey\n\n1 teaspoon lemon juice\n\n1\/2 teaspoon grated fresh ginger\n\n1\/2 cup finely chopped steamed broccoli\n\n1\/4 cup crushed cashews\n\nOn a cutting board or on wax paper, separate the turkey slices into two piles.\n\nIn a small bowl, whisk together the honey, lemon juice, and ginger, then add the broccoli and cashews.\n\nSpread the mixture on the turkey slices, roll up, and secure with toothpicks.\n\nQuick Turkey-Stuffed   \nCabbage Serves 8\n\n1 pound ground turkey\n\n1 large onion, chopped\n\n1 cup brown rice, uncooked\n\n1\/4 cup fresh mint, chopped, or 1 tablespoon dried mint\n\n1 egg, beaten\n\n2 (14-ounce) cans chicken broth\n\nSalt and pepper to taste\n\n2 tablespoons extra-virgin olive oil\n\n1 medium cabbage\n\n2 tablespoons flaxseed\n\nPreheat oven to 350\u00b0 F.\n\nCombine the turkey, onion, rice, mint, egg, broth, salt, and pepper and saut\u00e9 in olive oil until browned.\n\nIn another pot, steam the cabbage until it is slightly soft.\n\nAfter the cabbage cools, cut off the bottom, peel off the leaves, and place them in a baking dish one by one.\n\nSpoon a helping of turkey mixture onto each leaf and then roll it up.\n\nSprinkle with flaxseed and bake for 15\u201320 minutes.\n\nVegetable-Stuffed   \nFlank Steak Serves 6\n\n2 pounds flank steak\n\n1\/2 cup red wine vinegar\n\n3 cloves garlic, finely chopped\n\n2 teaspoons dried thyme\n\n1 cup fresh spinach\n\n4 large carrots, boiled and cut in 1\/2 inch pieces\n\n1 cup sun-dried tomatoes soaked in red wine\n\n1 small onion, thinly sliced\n\n3 teaspoons red pepper flakes\n\n2 tablespoons extra-virgin olive oil\n\n1-1\/2 cups beef broth\n\nArrange the steak on a cutting board so that the long side is parallel to you. Using a long knife, butterfly the steak to within \u00bd inch of the far edge so that it opens like a book.\n\nPound the steak with a mallet until about \u00bc inch thick. Transfer to a baking dish, poke holes in the meat, and sprinkle with the red wine vinegar, garlic, and thyme. Cover and marinate overnight.\n\nPreheat the oven to 375\u00b0 F. Put the meat on the cutting board and top with a layer of the spinach, carrots, sun-dried tomatoes, and onions. Sprinkle evenly with the red pepper flakes.\n\nStarting with the edge closest to you, roll the meat forward to form a tight cylinder. Using kitchen twine, tie the meat at 1-inch intervals.\n\nHeat the olive oil in a large, deep skillet or Dutch oven and sear the meat on all sides over high heat until brown.\n\nPour in the beef broth, then add enough water to reach about one-third of the way up the sides of the meat.\n\nCover, transfer to the oven, and bake until tender, approximately 2 hours.\n\nRemove the meat and place on a clean cutting board. Let stand 10 minutes before slicing. Spoon a tablespoon of the beef broth mixture over the meat before serving.\n\nVeggie Burgers Serves 4\n\n1 medium onion, chopped very fine\n\n1 medium bell pepper, chopped very fine\n\nExtra-virgin olive oil\n\n2 cups cooked fresh spinach, drained and chopped fine\n\n1 cup mashed cooked cauliflower\n\n1 can black beans, drained\n\n1 tablespoon garlic powder\n\n1 egg, beaten well\n\n2 pieces whole-grain toast, crumbled fine\n\nSalt and pepper to taste\n\nSaut\u00e9 onion and bell pepper in olive oil until onion is translucent.\n\nWhen mixture has cooled, place it in a large bowl and add all other ingredients, mixing well.\n\nForm patties and place on cookie sheet. Chill for at least 2 hours before browning in olive oil in a skillet on top of the stove, or bake at 400\u00b0F for 30 minutes.\nCHAPTER 15:\n\nMatter, No Matter Your Age\n\nThere is a special place in hell for women who do not help other women.\n\n\u2014Madeleine K. Albright\n\nEighty-one-year-old Carol Baras lives in San Diego. Over a forty-year career, Carol and her husband Bill started and ran multiple companies; some expanded into global markets. She also founded a nonprofit telephone hotline for troubled youth for which she was featured in People magazine and on Good Morning America. Over a chocolate martini, Carol told me she has \"so much more left to give\" and is \"looking forward to what the next ten years might surprise me with.\"\n\nLike Carol, you and I can choose to believe that we can make a difference. Choosing to matter means we stay needed, interesting, and relevant. No matter our age, life can matter more.\n\nVolunteer\n\nIn December 2011, I sat down with Crissy Haslam, the First Lady of Tennessee. I went to Emory University with Crissy and Bill (Governor Haslam); in fact, Crissy was a Kappa Alpha Theta sorority sister, and I was also a little sister in Bill's Sigma Chi fraternity, but it had been decades since we connected. Prettier than Sissy Spacek, with girl-next-door charm belying an astute, nimble brain, Crissy was born to be a First Lady. I asked my old friend how she hoped to use their conjoined political platform to make a difference for women and girls in Tennessee.\n\nCrissy talked about Bill's linked commitment to improving education and adding jobs across the state. She described her own passion for early literacy improvement, the importance of coaching parents as to the exponential benefits of reading with their children, and how she hoped to help catalyze more parental involvement in schools. Crissy was articulate and eloquent. Then I asked, \"How do you make it real? Your philanthropic interests, I mean?\"\n\nCrissy's eyes lit up. She leaned forward, becoming immediately animated, talking with her hands and punctuating certain sentences with a tap of her foot. She told me about delivering meals for the Love Kitchen, a nonprofit in Knoxville, Tennessee, that provides meals, clothing, and emergency food packages to the homebound, homeless, and unemployed. She beamed, explaining how the Love Kitchen was founded by two retired grandmothers and sisters, Helen Ashe and Ellen Turner, who felt God placed in their hearts a mission to serve via food for the hungry.\n\n\"For decades I served on multiple nonprofit boards, and, of course, Bill and I regularly tithed to church and many charitable organizations,\" Crissy shared, \"but it wasn't until I personally picked up those meals, drove, delivered them, and looked into the eyes of those I was serving that I truly felt it\u2014you know, the reality of what my time commitment on boards really meant, and also the pay-it-forward value of the checks we write. My experience made me humble. It made my privilege to give intensely, intimately real.\"\n\nGive Wisely\n\nWomen have for centuries been the fuel of our nation's volunteer efforts; now we are also the rising force in philanthropy. According to Betsy Brill, founder of Strategic Philanthropy Ltd., and a regular contributor to Forbes.com, \"It's not just who gives that is changing\u2014there is, after all, a rich history of high-profile women contributing generously to significant causes\u2014it's how women and to whom women are giving that is redefining contemporary philanthropy.\n\n\"To a great degree, the charitable giving by women, directly or through women's funds, focuses on improving the quality of life and opportunity for girls and women. The exponential growth of women's funds suggests an increasing acceptance of the idea that philanthropic investments in women and girls will increasingly fuel positive change in communities. It also suggests a growing interest in philanthropic models that allow donors to leverage and pool their charitable dollars to achieve maximum impact.\"\n\nIn compassionate hands, tools of influence and power can be life-giving and world-changing. Since our grandmothers marched as suffragettes and our mothers dipped their toes in the women's movement by choosing to work and demanding separate checking accounts, women across the globe have emerged as an economic force to be reckoned with. In 2009, women controlled an estimated 27 percent, about $20 trillion, of the world's wealth. Women now own more than 17 million US businesses and control $4.3 trillion of the $5.9 trillion in US consumer spending. Projections are that women-controlled wealth in the United States will grow at an average of 8 percent through 2014\u2014making US women the largest single economic force in the world.\n\nAbsurdly, women living in the United States today are poorer than men in all racial and ethnic groups. In 2008, more than half the 37 million Americans living in poverty were women. More recent surveys tell us that 38 percent of women thirty to fifty-five years old are worried they will live at or near poverty because they cannot adequately save for retirement. Thirteen percent of women older than seventy-five are poor, compared to 6 percent of men. More bad news: poverty rates for males and females living in the United States are the same throughout childhood but increase for women during their childbearing years and again in old age.\n\nFrankly, these facts about intractable female poverty across the globe should startle those of us who are more fortunate to action. We will be\u2014we must be\u2014the force narrowing this wealth-to-poverty gap.\n\nIn the United States, key findings of a 2009 report released by the Foundation Center and the Women's Funding Network included:\n\n  * Private and community foundations increased their giving for activities targeting women and girls from an estimated $412.1 million in 1990 to nearly $2.1 billion in 2006.\n  * Women's funds are guided by the principle that women catalyze and lead the way to change in neighborhoods and communities; 98 percent of the women's funds surveyed indicated that achieving social change was a high priority for their fund.\n  * Excitedly, women-led philanthropy is seeding sustainable change for women and girls, not only in this nation but across the globe. At the September 2011 Asia-Pacific Economic Cooperation summit, Secretary of State Hillary Clinton declared a tipping point for women. She said, \"When we liberate the economic potential of women, we elevate the economic performance of communities, nations, and the world. There is a stimulative ripple effect that kicks in when women have greater access to jobs and the economic lives of our countries: Greater political stability. Fewer military conflicts. More food. More educational opportunity for children. By harnessing the economic potential of all women, we boost opportunity for all people.\"\n\nOrganizations such as the International Network of Women's Funds, the Global Fund for Women, and Women for Women International are driving a more comprehensive approach to social change by focusing philanthropic giving on initiatives impacting human rights, health, and economic empowerment\u2014one girl or woman at a time.\n\nChoosing where and to whom to give your hard-earned (or inherited) money, as well as where you choose to invest your precious time and individual talents, is a really big deal. I choose based on the following four criteria:\n\n  1. A cause for women, girls, or animals about which I am personally passionate,\n  2. Confidence\/trust in the administration and leadership of that cause,\n  3. The pressing needs in my community that I can somehow personally impact\/influence and, most important,\n  4. What need touches my heart as exceptionally \"real.\"\n\nI believe women's fingerprints on the changes we impact must be sealed with a commitment far beyond charity. Our giving is more than an academic, do-good, slam-dunk exercise. Our philanthropy must be a conscious, intellectually vetted, yet heart-driven investment.\n\nI urge you to find a cause you are passionate about, then touch and feel that need. Like the Velveteen Rabbit, then and only then will your giving become real. Then and only then will we, as women, become the turbochargers for the change our world needs now.\n\nLog On with Purpose\n\nNo matter her age, a woman wearing jeans, sitting on her patio, and typing on her iPad has the potential to have as much\u2014if not more\u2014impact as a herd of Capitol Hill lobbyists or a billion-dollar multimedia advertising campaign. Let men play video games. Women will tune in, log on, and make a difference. Internet and social media platforms put exponential power at our fingertips. Applications such as Facebook, Twitter, Pinterest, and blogs increasingly catapult everyday people into dynamo activists. The Dragonfly Effect, by Jennifer Aaker and Andy Smith, is a stimulating and inspirational resource for any woman wanting to log on, join the conversation, and have impact locally, nationally and\/or globally.\n\nOne of my favorite examples of one person tapping the power of the Internet to circumvent the status quo is Chase Adams, cofounder of Watsi (watsi.org). Chase was a Peace Corps volunteer in Costa Rica when a woman on a bus approached him and asked for money to help fund her son's needed medical treatment. Chase told me he \"had an epiphany\" and soon returned to the States with the mission of developing a crowd-funding community to connect people like you and me with patients in serious need of medical care and treatments.\n\nIn summer 2012, only two years after Chase's original epiphany, his vision went live, relying fully on volunteer talent. Neera was one of the first patients funded on Watsi. She is a twenty-four-year-old Nepali woman who had suffered from seizures since she was a child. Her mother died when she was young and her father remarried, but Neera was never taken to seek treatment for her condition. Neera was later married and then widowed, a position that carries major social stigma in Nepal. One day, while Neera was cooking (a task that Nepali women are expected to perform regardless of ability) she had a seizure and suffered severe burns on her face and body.\n\nWhen Neera's profile was posted on Watsi, she needed skin grafting and reconstructive surgery to repair severe damage caused by burns. Neera comes from a district in Nepal where the average per capita income is less than a dollar a day. She had no financial support from her family and could not afford the $975 medical treatment she needed. She was literally out of options.\n\nThen, something amazing happened. Twenty-two Watsi donors stepped in and fully funded Neera's treatment. Now, for the first time in four years, Neera has regained the use of her hand. Her burns have been treated and her skin is healing. She is on track to live a normal, happy, healthy life.\n\nCecelia is a Watsi donor. When asked, \"Why do you support Watsi and why did you share it on Facebook?\" here's what she said: \"I'm a two-time breast cancer survivor. I have the wonderful First World benefit of insurance, a good job led by supportive people, a husband who makes enough to give me the room to take care of myself . . . in short, I'm rich beyond measure compared to almost everyone else in the world. I have so much to be grateful for.\"\n\n\"Every day I get piles and piles and piles of mail from nonprofits looking for money. I don't blame them\u2014I know times are tough. I've worked for nonprofits before and I know how things work. But it hasn't stopped my becoming inured to the pleas. It's like walking down a street with hundreds of hands in my way, and I've started to feel resentful. Because of my experience, I know that the mailers I'm getting cost money, and that some of these organizations are only passing a small percentage of the donations they receive to the actual people they're purporting to help.\"\n\n\"Along comes Watsi, allowing me to donate directly to someone in need. And there she is\u2014a sixty-year-old woman facing cancer and the debilitating treatments on her own, without all my resources. And all my money goes to her treatment. Alleluia! Thank you for allowing me to share.\"\n\nUnfortunately, the misconception persists that technology-based communication is primarily a playground for the young-in-years. This is dead-wrong thinking. A growing composite of data from such respected sources as the American Association of Retired Persons (AARP) and the Pew Research Foundation indicates that the over-fifty crowd is one of the fastest growing segments using mobile technology. And, according to Facebook, since 2008 the number of women over fifty-five on the site has tripled and makes up the social network's fastest-growing age-group.\n\nSlow to enter the world of social media, I attended the 2012 BlogHer conference in New York. I hated to miss the wave of our future but, like many other women, my concern was time. With everything else on my to-do list, when was I supposed to tweet or whatever? Deanna Zandt, author of Share This! shot back: \"Social networking is much more than a way to find your high school friend or learn what your favorite celebrity had for breakfast. It's how women can connect, share our stories, and build empathetic relationships that change our world. But the world needs many different kinds of voices to make these conversations work. We need you!\"\n\nSoon after, while speaking to a group of seventy-something women, I asked them how many had an opinion regarding pro-choice movements.\n\n\"Do you mean how many of us are for it?\" one woman asked crisply.\n\n\"No, how many of you have an opinion one way or another?\"\n\nAll forty-two hands shot up.\n\n\"You may have read recently that thousands of Turkish women rallied in Istanbul amid growing fears that the Turkish Islamic government might ban abortion. Whether you are pro-choice or pro-life, you can have a voice in this controversy. All you have to do is log on.\"\n\nThree days later, seventy-nine-year-old Anne sent me this e-mail: \"After hearing you speak, I logged on and joined in several blog conversations. I also asked my thirty-four-year-old granddaughter to help me set up a Twitter account and teach me how to tweet. She got my thirteen-year-old-great-granddaughter involved. For the first time in decades, they looked at me as if I was interesting, as if I had something to say. Now my fifty-six-year-old daughter is envious of their attention and praise. She asked me if I would show her how to log on and join in too. Thank you for reminding me that my voice matters, that someone out there still cares to hear what I have to say.\"\n\nVote and\/or Run\n\nTwenty-three-year-old Lacey walked from the den where her father Lester was pumping his fists in the air watching Fox News talking heads on his wide-screen TV into the sunroom where her mother, Ronnie, sat highlighting \"The National Women's Political Caucus Guide for How to Run Low-Budget Campaigns.\" Exasperated, she interrupted.\n\n\"Doesn't it seem like both of you are wasting tons of energy? I mean, both of you are passionate about opposing political agendas. Why get so worked up? At the end of the day, you will just cancel out each other's vote. Why even bother?\"\n\nRonnie faced her daughter square on.\n\n\"Why vote?\" She spoke quietly but with fire, \"Because my opinion matters to the future of the United States of America as well as our country's global agenda. So does your dad's. So does yours. A person's vote is not right or wrong. It is simply their voice, their personal power of truth to share and, in the process of voting, conglomerate with all the other voices in our nation. A vote is how each of us can exercise our freedom and also fulfill our responsibilities as a citizen. One person's voice can and never will cancel out another's.\"\n\n\"Lacey, honey, political agendas and their associated candidates will come and go. Sometimes we elect clowns, charlatans, and suited-up bags of gas, but in this nation and in our world, new voices are emerging\u2014voices of activism, entrepreneurship, and compassionate capitalism. Voices with the power to change our world. Increasingly, these are the voices of women coming from the grass roots.\"\n\n\"Your dad's and my disparate political views have strengthened, rather than frayed, our mutual esteem of each other's intelligence and experience. Again, if you think our individual votes cancel each other out, you are dead wrong. Standing firm in our own truths while respectfully agreeing to disagree is the stuff of democracy. I suggest you reframe your experience of our family political landscape and realize that the essence of world peace is modeled in your very own home!\"\n\nYes, your vote matters, so never, ever fail to fill out that ballot. But some of us may choose to do more. The number of women stepping up and running for office is increasing. Wonderful news: more and more of us are winning. In 2011, US women held 90, or 16.8 percent, of the 535 seats in the 112th US Congress; 17, or 17.0 percent, of the 100 seats in the Senate; and 73, or 16.8 percent, of the 435 seats in the House of Representatives. In addition, three women served as delegates to the House from Guam, the Virgin Islands, and Washington, DC. Around the world, 19.2 percent of all national parliamentary positions (including seats in the upper house or Senate, where applicable) are held by women. Of the 181 countries for which data was provided, the United States ranked almost exactly in the middle: 92nd, just above Turkmenistan.\n\nWorried that there are not enough hours in the day for you to take care of your own health, attend to your family, grow your business or expand your career, manage your financial well-being, and also run for office? Listen to someone who is doing it to tell you that, if this is the role the Universe is calling you to fill in our world today, you can:\n\nLast year, I gave birth to my son Joaqu\u00edn\u2014who is now nine months old. It put me in an exclusive group; I became only the eighth member of Congress to give birth while serving in office. The first woman to give birth while in office was Yvonne Braithwaite Burke in 1973. From 1973 to 1996, there were no women in Congress who gave birth! It's obvious that having a child while being responsible to thousands of constituents is a daunting task. But it's not impossible.\n\nNot only was I able to continue serving the interests of my constituents during my pregnancy, I even made it down to the House floor\u2014maternity gear and all\u2014to cast votes the night before I gave birth. I'm not going to tell you it was easy. I have a great staff and a wonderful family who made it easier. If anything, this opportunity showed me how much women can accomplish.\n\nLast fall, Stanford and the University of Chicago conducted a study comparing female lawmakers to their male counterparts. The study focused on three specific measurements of the effectiveness of each member of Congress: (1) the number of pieces of legislation each member introduced, (2) the number of members of Congress who cosponsored each piece of legislation, and (3) the amount of discretionary spending each member was able to direct to his or her district.\n\nThe conclusion of the study was clear: women are more effective legislators than men. Quantitatively, women introduce more legislation and procure more resources for their districts. Qualitatively, the legislation women introduce receives greater support from their colleagues.\n\nThe results reaffirm a truth I've seen again and again in my life and my time in Congress: women get things done and don't take no for an answer. That's not to diminish the impact of my male colleagues, but there are times when women show greater fortitude and a stronger commitment than our male counterparts. We've had to work harder to get where we are, which means we keep digging and trying new methods until we get the results we want.\n\nWomen often view issues with a 360-degrees lens\u2014by examining parts of the debate the way it relates to the whole. Moms, aunts, sisters, or mothers in Congress are good for our government and good for our nation. It leads to better policies, better laws, and better governance. It's why our nation's Capitol now has more bathrooms for women and a breast-feeding room.\n\nIn your own day-to-day lives, you will bring out your passion for people and the world around you. Don't just sit on the sidelines. Don't just take the world as you find it. Use your voice to stand up to injustice. Empower the powerless.\n\nYou can write the laws rather than react to them. You can ensure the next generation of women legislators have an easier road and more role models than your generation does\u2014and than my generation had.\n\nI look forward to hearing more voices of women in the debates that will help shape the world in which my son grows up.\n\nExcerpted from Congresswoman Linda T. S\u00e1nchez's, February 25, 2010, blog. Congresswoman S\u00e1nchez represents the 39th Congressional District of California.\n\nThis One's for the Girls\n\nI don't have children of my own. I would have particularly loved to have had a daughter but not especially one with any of the men I hung out with during my procreating years. While this fact saddens me it also deepens my commitment to positively contribute to girls' lives, those belonging to women in my circle of friends, my community, and our country. How else will I live on? What better legacy could I leave?\n\nWhenever I have a \"George Bailey moment\" (the character in the movie It's a Wonderful Life who doesn't think his life matters and considers suicide only to be interrupted by a guardian angel), I stop and consider:\n\nI have a few fingerprints on Sarah White's life, and she many on mine. At eighteen, Sarah is already proving herself to be a committed engine for equality and social justice.\n\nMy niece Shelley recently sent me a card inscribed with the following Marianne Williamson quote:\n\nOur deepest fear is not that we are inadequate. Our deepest fear is that we are powerful beyond measure. We ask ourselves, who am I to be brilliant, gorgeous, talented, fabulous? Actually, who are you not to be? We were born to make manifest the glory of God that is within us. And as we let our own light shine, we give other people permission to do the same.\n\n\"Genie,\" Shelley wrote inside, \"you teach me this. I would not be who I am or be excited about the woman I will continue to grow into without you in my life.\" Hardly a better legacy than that, except possibly . . .\n\nI adore Randy's daughter Danika and am goo-goo for our granddaughter Lulu. I am sure I will spoil her, but more than that, I want to do everything I can to help her grow into a woman who is smart, strong, kind, and bold.\n\nBesides being a super aspiration for Lulu, \"Smart, strong, and bold\" is the motto for Girls Inc. This nonprofit organization works in communities across the United States to inspire girls to be strong, smart, and bold through life-changing programs and experiences that help girls navigate gender, economic, and social barriers. I am on the board of the Girls Inc. chapter in Jacksonville, Florida. It, and the Girls Inc. chapters in Nashville, Tennessee, and Northwest Oregon are three of my personal choice philanthropies.\n[Salads   \nand   \nSide Dishes](Content.html#Salads-and-Side-Dishes)\n\nCruciferous Couscous Serves 8\n\n1 cup minced broccoli\n\n1 cup fresh spinach\n\n1 cup minced cauliflower\n\n2 tomatoes, cubed\n\n2 tablespoons minced garlic\n\n2 tablespoons extra-virgin olive oil\n\n3 cups water\n\n2 cups quick-cook couscous\n\n2 tablespoons balsamic vinegar\n\nSalt and pepper to taste\n\nPreheat oven to 350\u00b0 F.\n\nMix broccoli, spinach, cauliflower, and tomatoes with garlic and 1 tablespoon of olive oil and place on a baking sheet. Bake for 15\u201320 minutes, until tender.\n\nIn a pot, bring water and remaining tablespoon of olive oil to a boil, add couscous and return to a boil, then remove from heat and let stand for 5 minutes. Fluff with a fork and let cool.\n\nPlace couscous on a plate and top with roasted vegetables.\n\nDrizzle with balsamic vinegar before serving.\n\nBeets and Brussels Sprouts Serves 6\n\n4 medium beets\n\n10\u201312 Brussels sprouts\n\nExtra-virgin olive oil\n\n1 small white onion, peeled and sliced thin\n\n3 tablespoons frozen orange juice concentrate\n\n2 teaspoons grated ginger\n\nSalt and pepper to taste\n\nBoil beets for 45 minutes or until tender by touch with a fork; drain when done.\n\nRemove and discard the outer leaves of the Brussels sprouts and boil them in a separate pan for 5 minutes. Drain and cut in half.\n\nPut olive oil in skillet, saut\u00e9 onion, and add beets and Brussels sprouts. Cook until warm, about 2\u20133 minutes, then stir in orange juice concentrate and ginger. Salt and pepper to taste.\n\nCreamy Coleslaw Serves 10\n\n4 cups shredded cabbage\n\n1 cup grated carrots\n\n1-1\/2 cups plain low-fat yogurt\n\n1-1\/2 tablespoons finely chopped celery\n\n1 teaspoon grated onion\n\n3 tablespoons white vinegar\n\n1 tablespoon raw honey\n\n3\/4 teaspoon salt\n\nDash of pepper\n\nMix all ingredients in a large bowl and toss well.\n\nRefrigerate for at least 2 hours before serving.\n\nBroccoli and Cauliflower   \nin Lime Dressing Serves 4\n\n1 tablespoon low-sodium soy sauce\n\n2 teaspoons raw honey\n\n3 tablespoons fresh lime juice\n\n1 cup broccoli florets\n\n1 cup cauliflower florets\n\nSalt and pepper to taste\n\nRed pepper flakes (optional)\n\nMix soy sauce, honey, and lime juice. Set aside.\n\nBoil broccoli and cauliflower for about 5 minutes, till they are tender yet firm. Drain and toss immediately with soysauce, lime mixture. Season with salt and pepper to taste.\n\nSprinkle with red pepper flakes if desired.\n\nSpicy Kale and Beans Serves 8\n\n2 cups dried black-eyed peas\n\n1 bunch kale (about 2 pounds)\n\n1 large onion, diced\n\n1 tablespoon extra-virgin olive oil\n\n2 tablespoons white vinegar\n\n1\/4 teaspoon crushed red pepper (optional)\n\n2 hard-boiled eggs, chopped\n\nSoak black-eyed peas overnight.\n\nThe next day place the drained peas in a large saucepan, cover with water, and bring to a boil over high heat. Boil for 3 minutes.\n\nRemove the pan from the heat, cover tightly, and let stand for 1 hour.\n\nWash kale, remove large stem ends, and coarsely chop the leaves.\n\nSaut\u00e9 the onion in a large skillet. Add the kale and cook for about 5 minutes, until the leaves are wilted but still bright green. Stir in black-eyed peas, vinegar, and crushed red pepper until the entire mixture bubbles with heat.\n\nTop with the eggs before serving.\n\nSpinach and Feta Brown Rice Serves 6\n\n1 cup brown rice\n\n1 cup fresh spinach leaves\n\n2 teaspoons minced garlic\n\n4 tablespoons low-fat feta cheese\n\nSalt and pepper to taste\n\nCook the brown rice according to directions on the package, or bring to a boil 2 parts water to 1 part rice in a medium-size pot. Cook on low for 30\u201345 minutes until the rice is soft.\n\nBefore serving, add the spinach and garlic, stirring until the spinach wilts.\n\nPut on a plate and garnish with feta cheese.\n\nBeet and Orange Salad Serves 4\n\n1 jar pickled beets\n\n2 cans mandarin oranges in water, or 2 tangerines, peeled, sectioned, and seeded\n\n1\/2 cup orange juice\n\n1 cup fresh spinach\n\n1\/4 cup slivered almonds\n\nToss beets and mandarin orange or tangerine sections in orange juice. Chill for at least 1 hour.\n\nServe on bed of spinach.\n\nGarnish with slivered almonds.\n\nMashed Cauliflower   \nwith Turkey Bacon Serves 4\n\n1 medium head cauliflower\n\nSalt and pepper to taste\n\n1\/4 cup skim milk\n\n1\/4 cup instant low-carb mashed potatoes\n\n1 teaspoon garlic paste or garlic powder\n\n2 slices cooked turkey bacon, crumbled\n\nCut cauliflower into florets and steam with salt and pepper until very tender.\n\nPlace in blender and add all ingredients except turkey bacon. Blend until smooth.\n\nReturn to pot or stove to reheat if necessary.\n\nGarnish with crumbled turkey bacon.\n\nCabbage-Apple Salad Serves 6\n\n1 cup shredded cabbage\n\n1 cup diced apple\n\n1 cup chopped celery\n\n1\/4 cup plain low-fat yogurt\n\n2 tablespoons flaxseed\n\nMix all ingredients in a large bowl. Toss, chill, and serve.\n\nSkillet Broccoli or   \nAsparagus with Sesame Seeds Serves 6\n\n2 large stalks fresh broccoli or 12\u201314 asparagus spears\n\n2 tablespoons extra-virgin olive oil\n\n2 tablespoons sesame seeds\n\n2 or 3 cloves garlic, peeled and sliced\n\nSalt and pepper to taste\n\nWash broccoli and cut the florets into medium to large pieces. Peel the tough outer layer from the stems and slice the inner tender, juicy portion in half.\n\nOr, if using asparagus, wash the spears and trim off the ends.\n\nIn a large skillet with a lid, heat the oil over high heat. Add sesame seeds and saut\u00e9, stirring until lightly toasted; be careful not to over-cook, because they burn quickly. Also, keep a lid handy because the sesame seeds might start to pop.\n\nAdd the broccoli (or asparagus) and garlic, and stir for a few seconds. Add salt and pepper to taste, and stir.\n\nCover the skillet and remove from heat; let sit about 15 minutes. The broccoli (or asparagus) will retain its color and be tender and crisp.\nAppendix B: Resources\n\nBest Resource: Yourself\n\nYou want to make smart, belly-flat-friendly, hormone-healthy choices but may not be sure where to start. While this section provides a listing of suggested resources, I want to take a minute to remind each reader that you are the one in charge of determining what is right for you, your body, and ultimately, your health.\n\nI strongly encourage you to do your research, ask informed questions, reach out to other women for input and vet health and medical professionals to discern their credentials, training, experience, and reputation. Ultimately assemble a health coaching and care team you trust, but remember: Even with your team in place, you are the common variable in every health and wellness conversation you will have. Therein, you must always take the lead.\n\nChoosing Organic\n\nFoods\n\nMany people initially complain that buying organic meat and produce is too expensive and that sorting through food product labels to determine GMO versus non-GMO foods can be confusing. The good news is that, because of growing consumer demand, almost every grocery store in the nation now stocks organic foods and also has a health food aisle. Some progressive grocery store chains in larger metropolitan markets even have a nutritionist on the floor available to help you sort through questions, labels, and foods to match special dietary needs.\n\nI applaud the grocery store industry for increasingly responding to shoppers wanting more natural options. My hat is off to two companies who continue to raise the bar:\n\nWhole Foods Markets: Founded in 1980 as one small store in Austin, Texas, Whole Foods Markets is now the world's leading retailer of natural and organic foods, with more than 340 stores in North America and the United Kingdom. This is a mission-driven company whose popularity and growth deserves celebration. To find a Whole Foods Market near you, go to www.wholefoodsmarket.com.\n\nTrader Joe's: I particularly like the clear labeling of all Trader Joe's private labeled products clearly promising NO artificial flavors, colors or preservatives, NO monosodium glucarate (MSG), NO genetically modified (GMO) ingredients, NO added trans fats. There are currently 375 Trader Joe's grocery stores in the United States. To find one near you, go to www.traderjoes.com\/stores.\n\nTo learn more about the \"why\" of organic food choices, or to locate a source for organic foods near you, go to: www.organicfoods.com, www.organicconsumers.org, www.eatwell.org, and www.NutritionStyle.com. Local farmer's markets can also be a great source for organic foods, often at lower prices than you will pay in the supermarket or health food store.\n\nHousecleaning Products\n\nRemember it is important to decrease your exposure to environmental estrogens, or xenoestrogens. I recommend using organic, eco-friendly housecleaning products when at all possible. One excellent resource is Billee Sharp's book Lemons and Lavender. Also check out www.greenworkscleaners.com and www.eartheasy.com.\n\nSafe Cosmetics\n\nI urge you to make sure your cosmetics are safe. Many are not. An excellent website for detailed information on beauty products is www.safecosmetics.org. The Campaign for Safe Cosmetics is a coalition of public health, educational, religious, labor, women's environmental and consumer groups. The coalition's goal is to require the health and beauty industry to protect consumer health by phasing out all chemical ingredients linked to cancer, birth defects, and other health problems, and replace them with safer alternatives.\n\nIf you want to learn more and then take informed action, Ellen Murmur's book Simple Skin Beauty: Every Woman's Guide to a Lifetime of Healthy, Gorgeous Skin is a wonderful resource. Whole Foods Premium Body Care line of products are among the most exceptional and affordable personal care products available. They meet the strictest standards for quality sourcing, environmental impact, results and safety. Whole Foods Premium Body Care is an excellent choice for being good to your whole body.\n\nIn addition, many websites inform and offer naturally terrific products. Some favorites include:\n\nwww.kellyteegardenorganics.com\n\nwww.tataharperskincare.com\n\nwww.aubrey-organics.com\n\nwww.burtsbees.com\n\nwww.dermae.com\n\nwww.pangeaorganics.com\n\nwww.weleda.com\n\nNote: Some of these products are more wallet-friendly than others. Do your homework, check out reviews, ask for samples and, ultimately, choose the ones that work best for your body, face and budget.\n\nVaginal Lubricants\n\nOver-the-counter vaginal lubricants can be a good choice to self-treat a condition of vaginal dryness, but read the label before you buy. Some ingredients in certain products are absolute no-no's because they create a medium for bacteria and growth, thereby increasing risk of urinary tract infection. Others contain potentially harmful petrochemicals. Safe sources for vaginal lubricants include:\n\nwww.bewellstaywell.com\/Sylk-Natural\n\nwww.goodcleanlove.com\n\nwww.yesyesyes.org\n\nIf a condition of vaginal dryness and\/or painful intercourse persists, consult your doctor.\n\nVitamins and Supplements\n\nAccording to a recent survey of nearly 1,000 supplements conducted by ConsumerLab.com, a product-certification company, one out of four supplements has quality problems, such as contamination or a failure to include an ingredient listed on the label.\n\nI own the Natural Medicine Pharmacy that adjoins Randy's Ageless and Wellness Medical Center. Drawing on Randy's expertise as a physician specializing in age-management medicine, as well as his training as a compounding pharmacist specializing in pharmacognosy (plant-based medicine), I have established one criterion for the vitamins and supplements we carry that is simple yet non-negotiable: Research-driven dietary supplements with a proven track record. This means digging deep to evaluate quality, truth in labeling, and company adherence to stringent manufacturing guidelines. These guidelines include:\n\n  * Raw materials testing\n  * Potency testing\n  * Product traceability\n  * Purity testing\n  * Product freshness\n  * Microbiology testing\n\nThe bottom line is truth in packaging. People have the right to know what they are putting in their body and they deserve to get the amount of active ingredient they are paying for. Randy has developed a signature-formulation, private-label line of vitamins and supplements available online at www.hormonewellstore.com.\n\nRandy has also vetted and recommends the following brands. The following three are available only through a physician's office. For more information on their products or to find a doctor in your area who carries them, go to:\n\n  * Metagenics, www.metagenics.com\n  * Orthomolecular, www.orthomolecularproducts.com\n  * Xymogen, www.xymogen.com\n\nLife Extension is a direct-to-consumer source for vitamins and supplements that Randy and I have high confidence in. In addition to its natural product offerings, Life Extension publishes a fantastic scientifically rooted monthly magazine. For more information on Life Extension, go to www.lef.org.\n\nHormone Health\n\nOver-the-Counter Progesterone Creams\n\nMany women find that using an over-the-counter (OTC) bioidentical progesterone cream is an excellent first step toward eliminating estrogen dominance and restoring hormone balance. You can purchase bioidentical progesterone creams in most health foods stores. The good news is that it is available. The bad news is that some products are better than others. The reason for this discrepancy: There is no regulatory body that oversees the production or standardization of product manufacturing for so-called \"natural\" products. What this means to the average consumer is that there is great variation among the many OTC progesterone creams on the market today.\n\nWhile I cannot reveal Randy's exact formula for his Natural Balance Cream, I can share here several of the critical variables that define its integrity of ingredients and proven efficacy:\n\n  * Dr. Randolph's Natural Balance Cream is a bioidentical formulation of progesterone. This means that the molecules of progesterone suspended in the cream have exactly the same molecular structure as those produced by the human body. The body recognizes, receives, and utilizes these molecules. The parent molecule for progesterone comes from a substance known as diosgenin, which is found in soy or Mexican wild yam. Many products on the market today containing soy or Mexican wild yam claim to be a \"natural progesterone\"; however, until the diosgenin is converted from its original molecular structure, the body will not recognize it. Consequently, soy or Mexican wild yam it the raw state will not generate the same clinical response.\n  * Dr. Randolph's Natural Balance Cream contains the maximum concentration of bioidentical progesterone that can be mixed in an OTC product. Some creams have a \"little\" progesterone in their mix but not enough to generate a consistent and positive user response.\n  * The progesterone in Dr. Randolph's Natural Balance Cream meets the United States Pharmacopoeia gold standards for quality and purity. In addition, the laboratory used to produce Dr. Randolph's Natural Balance Cream compounds this product under strict guidelines approved by the National Association of Compounding Pharmacists. This is not required by law so not all product manufacturers go to the trouble or expense.\n  * The progesterone molecule in Dr. Randolph's Natural Balance Cream is encased within a liposomal delivery system. This is critical because, as the many layers of the oily globule of liposome melt away like a snowball, the hormones are dispersed continuously through the skin for up to twelve hours. This means that the underlying issue of hormonal imbalance is eliminated, hormonal balance is restored, and my patients have continuous relief of their symptoms throughout the day. In contrast, other over-the-counter progesterone creams are not formulated for sustained release. When the progesterone is immediately absorbed through the skin, a quick spike in progesterone levels occurs causing a temporary vs. constant relief of symptoms. Therein a formulation that is time-released is much more clinically effective.\n\nDr. Randolph's Natural Balance Cream is available at www.hormonewellstore.com. Randy has also reviewed the following OTC progesterone creams and verified their quality and truth in advertising:\n\nArbonne International\n\nPhytoProlief and Prolief Natural Balancing Creams\n\nwww.arbonne.com\n\nEmerita\n\nPro-Gest progesterone cream\n\nwww.progest.com\n\nSeeking Medical Professional Help\n\nIf you have committed to the lifestyle changes outlined in this book and tried an OTC progesterone cream for more than one month with no relief of symptoms, this is a signal that your body needs more. It is time for you to seek professional medical help to diagnose and treat a more advanced or more multifactorial underlying condition of hormone imbalance.\n\nCaution: treating hormone imbalances is serious medicine. Do your homework before choosing a medical professional. Find out:\n\n  * Board certification(s)\n  * Training in hormone health, what organization, how long. Excellent organizations offering continuing medical education (CME) programs on natural hormone health are the American Academy of Anti-Aging Medicine (A4M), www.a4m.com and the Institute of Functional Medicine, www.functionalmedicine.org\n  * Protocol for routinely analyzing hormone levels via blood work or saliva testing.\n  * Years of experience\n  * Reputation among patient community, then\n  * Trust your instincts. Is this someone you can partner with and trust?\n\nWhen looking for a physician or medical professional in your area, an excellent resource can be your local compounding pharmacist. If you need help finding a compounding pharmacy in your area, contact one of the two organizations listed here.\n\nThe International Academy of Compounding Pharmacists (IACP)\n\nP.O. Box 1365\n\nSugar Land, TX 77487\n\nPhone: 281-933-8400\n\nFax: 281-495-0602\n\nWebsite: http:\/\/www.iacprx.org\n\nProfessional Compounding Centers of America (PCCA)\n\n9901 South Wilcrest Drive\n\nHouston, TX 77099\n\nPhone: 877-798-3224\n\nFax: 877-765-1422\n\nWebsite: http:\/\/www.pccarx.com\n\nFinally, patients come to Dr. Randolph's Ageless and Wellness Medical Center from across the nation and around the globe. If you are interested in scheduling an appointment with Randy or one of the medical professionals trained by him practicing within our clinic, call (904) 249-3743 or reach us through our website, www.agelessandwellness.com.\n\nTesting Hormone Levels\n\nMeasuring hormone levels is essential for the proper diagnoses of perimenopause, menopause, andropause, or other disease states such as hypothyroidism and adrenal exhaustion (chronic fatigue syndrome). Hormone level testing also enables your physician to closely monitor hormone levels during treatment to ensure they all remain adequately balanced and within the optimal physiological range. Hormone levels can be analyzed through blood, saliva, or urine.\n\nPhysicians on the cutting edge of optimal aging medicine are increasingly adding specialized lab services into the clinics. At Dr. Randolph's Ageless and Wellness Medical Center we have recently partnered with Atherotech Diagnostic Lab, www.atherotech.com, and Genova Diagnostics, www.gdx.net. An emerging, and still controversial, trend is lab companies offering tests directly to the consumer. Two very credible companies offering this service are:\n\n  * MyMedLab, www.mymedlab.com\n  * ZRT, www.zrtlab.com\n\nRecommended Reading\n\nHormone Health\n\nIf you have not already done so, I strongly recommend you read the first two books Randy and I co-authored: From Hormone Hell to Hormone Well and From Belly Fat to Belly Flat. The first will provide you a more comprehensive treatise of the history and science behind bioidentical hormone replacement. The second expounds more fully on the links between estrogen dominance, belly fat, and what you can do about it. Other excellent hormone health resource books include:\n\nLee, John R., MD, with Virginia Hopkins. What Your Doctor May Not Tell You About Menopause. New York: Warner Books, 1996.\n\nLee, John R., MD, with Jesse Hanley, MD, and Virginia Hopkins. What Your Doctor May Not Tell You About Perimenopause. New York: Warner Books, 1999.\n\nLee, John R., MD, with David Zava, PhD, and Virginia Hopkins. What Your Doctor May Not Tell You About Breast Cancer. New York: Warner Books, 2002.\n\nMorgentaler, Abraham, MD, Testosterone for Life: Recharge Your Vitality, Sex Drive and Overall Health. New York: McGraw Hill, 2009.\n\nNorthrup, Christiane, MD, Women's Bodies, Women's Wisdom: Creating Physical and Emotional Health and Healing. New York: Bantam Books, 1994.\n\nNorthrup, Christiane, MD, The Wisdom of Menopause: Creating Physical and Emotional Health and Healing During the Change. New York: Bantam Books, 2001.\n\nSchwartz, Erika, MD, The Hormone Solution. New York: Warner Books, 2002.\n\nSeaman, Barbara. The Greatest Experiment Ever Performed on Women: Exploding the Estrogen Myth. New York: Hyperion Books, 2003.\n\nSomers, Suzanne, Ageless: The Naked Truth About Bioidentical Hormones. New York: Crown Publishers, 2006.\n\nSomers, Suzanne, The Sexy Years, Discover the Hormone Connection: The Secret to Fabulous Sex, Great Health, and Vitality for Women and Men. New York: Crown Publishers, 2004.\n\nTaylor, Eldred, MD, and Bell-Taylor, Ava, MD, Are Your Hormones Making You Sick? Physicians Natural Medicine, Inc., 2000.\n\nWilson, James L., ND, DC, PhD, Adrenal Fatigue. Petaluma, CA: Smart Publications, 2003.\nReferences\n\nPrologue\n\nCrary, D. (2011, August 11). Boomers will be spending billions to counter aging. Boston. Retrieved from http:\/\/articles.boston.com\/2011-08-20\/lifestyle\/29909735_1_anti-aging-retirement-age-dietary-supplements.\n\nIntroduction\n\nRandolph, C. W., & James, G. (2009). Chapter 1: Decades of Desperate Women Dangerously Duped. In From Hormone Hell to Hormone Well. (pp. 5\u201322). Deerfield Beach, Florida: Health Communications, Inc.\n\nChapter 1\n\nRandolph, C. W. (2009). From Hormone Hell to Hormone Well. (2nd ed.). (pp. 1\u2013240). Deerfield Beach, Florida: Health Communications, Inc.\n\nBaron, P. (2005, June). \"Hormone Testing for Optimal Health.\" Life Extension Magazine.\n\nHormone treatment options (2012). Women in Balance. Retrieved July 23, 2012, from http:\/\/womeninbalance.org\/choices-in-therapy\/hormone-treatment-options\/.\n\nRandolph, C. W. (2011). Women's Hormone Health. Dr. Randolph's Natural Hormone Institute. Retrieved July 23, 2012, from http:\/\/www.hormonewell.com\/manage_moods.html.\n\nMayeaux, E. J. (2005, August 30). The menopausal patient and hormone replacement therapy. LSU. Retrieved August 25, 2012, from http:\/\/www.sh.lsuhsc.edu\/fammed\/outpatientmanual\/menopause-hrt.htm.\n\n\"Hormone Balance.\" (2012, March 29). Life Extension.\n\nChapter 2\n\nWeight loss in middle age. (2012). Livestrong. Retrieved August 3, 2012, from http:\/\/www.livestrong.com\/weight-loss-in-middle-age\/.\n\nGann, C. (2012, May 7). Fat Forecast: 42% of Americans Obese by 2030. ABC News.\n\nBelly fat in women: Taking\u2014and keeping\u2014it off, http:\/\/www.mayoclinic.com\/health\/belly-fat\/WO00128.\n\nWhitmer RA, Gustafson DR, et al. \"Central obesity and increased risk of dementia more than three decades later.\" Neurology. 2008 Sep 30;71(14):1057\u201364. doi: 10.1212\/01.wnl.0000306313.89165.ef. Epub 2008 Mar 26.\n\nR. Morgan Griffin, Obesity and Early Puberty: What's the Risk?, Retrieved from http:\/\/children.webmd.com\/features\/obesity.\n\nSmith GI, Atherton P, Reeds DN, et al. \"Dietary omega-3 fatty acid supplementation increases the rate of muscle protein synthesis in older adults: a randomized controlled trial.\" Am J Clin Nutr February 2011 vol. 93 no. 2 402\u2013412.\n\nObesity and Early Puberty: What's the Risk? http:\/\/children.webmd.com\/features\/obesity (June, 2010).\n\nChapter 3\n\nSchmitz KH, Lin H, Sammel MD, Gracia CR, Nelson DB, Kapoor S, DeBlasis TL, Freeman EW. \"Association of physical activity with reproductive hormones: the Penn Ovarian Aging Study\". Cancer Epidemiol Biomarkers Prev, 2007 Oct; 16(10):2042\u20137. Epub 2007 Sep 28.\n\nBreast Health. (2011). Natural Hormone Institute. Retrieved August 26, 2012, from http:\/\/www.hormonewell.com\/breastHealth_diet.html.\n\nExercise Lowers Estrogen Levels in Older Women, Retrieved from http:\/\/www.breastcancer.org\/research-news\/20100216.\n\nSafdar A, Bourgeois JM, Ogborn DI, et al. \"Endurance exercise rescues progeroid aging and induces systemic mitochondrial rejuvenation in mtDNA mutator mice\". PNAS, (2011): 201019581, http:\/\/www.pnas.org\/content\/early\/2011\/02\/18\/1019581108.full.pdf+html.\n\nCherkas LF, Hunkin JL, Kato BS, et al. \"The Association Between Physical Activity in Leisure Time and Leukocyte Telomere Length\". Arch Intern Med, 2008; 168(2): 154\u2013158. Doi:10.1001\/archinternmed2007.39.\n\nPadycula, J. (2012, April 18). Buddy Up for Workout. She Knows.\n\nActress goes archer: Geena Davis in Olympic archery semifinals, http:\/\/sports  \nillustrated.cnn.com\/olympics\/news\/1999\/08\/05\/davis_archery_ap\/ (August 1999).\n\nAfter Late Start, Runner Is Speeding Through Records, http:\/\/www.nytimes.com\/2012\/04\/02\/sports\/runner-kathy-martin-60-is-speeding-through-records.html?pagewanted=all (April 2012).\n\n7 Ways Technology Helps You Lose Weight: The New Digital Diet, http:\/\/www.  \nprevention.com\/weight-loss\/weight-loss-tips\/7-ways-technology-helps-you-lose-weight.\n\nChapter 4\n\nEWG Research Shows 22 Percent of All Cosmetics May Be Contaminated With Cancer-Causing Impurity, http:\/\/www.ewg.org\/release\/ewg-research-shows-22-percent-all-cosmetics-may-be-contaminated-cancer-causing-impurity (Feb 2007).\n\nExposure to Chemicals in Plastic, http:\/\/www.breastcancer.org\/risk\/factors\/plastic (September 2012).\n\nBraun JM, Kalkbrenner AE, et al. Impact of Early-Life Bisphenol A Exposure on Behavior and Executive Function in Children. Pediatrics. 2011 Nov;128(5):873\u201382. doi: 10.1542\/peds.2011\u20131335. Epub 2011 Oct 24.\n\nKaiser Permanente Says GMO Controversy Misleading, Retrieved from http:\/\/news.health.com\/2012\/12\/03\/kaiser-permanent-says-gmo-controversy-misleading\/ (December 2012).\n\nChapter 5\n\nRisks and Benefits of Estrogen Plus Progestin in Healthy Postmenopausal Women: Principal Results From the Women's Health Initiative Randomized Controlled Trial. (2002). Journal of the American Medical Association, 288(3), 321\u2013333.\n\nWyeth Pharmaceuticals, Inc: Prempro. (2009, May). US Food and Drug Administration. Retrieved July 23, 2012.\n\nWinfrey, O. (2009, February). Is Hormone Replacement Therapy Right for You?. The Oprah Show. Retrieved July 23, 2012, from http:\/\/www.oprah.com\/health\/Is-Hormone-Replacement-Therapy-Right-for-You\/9.\n\nNorthrup, C. (2009, January 29). Hormone Replacement Therapy Q&A Webcast. The Oprah Winfrey Show. Webcast retrieved from http:\/\/www.oprah.com\/health\/The-Hormone-Replacement-Webcast-with-Dr-Christiane-Northrup.\n\nMcGraw, R. (2009). Retrieved August 25, 2012, from http:\/\/www.robinmcgraw.com\/books.html.\n\nL. Leaseburge, Personal communications. 2011.\n\nRenee, J. (2011, March 28). Information on the Pregnenolone Supplement. Livestrong. Retrieved July 23, 2012, from http:\/\/www.livestrong.com\/article\/408509-information-on-the-pregnenolone-supplement\/.\n\nLeonetti, H. B. (2005). Transdermal Progesterone Cream as an Alternative Progestin in Hormone Therapy. Alternative Therapies in Health & Medicine, 11(6), 36\u201338.\n\nStein, D. G., Sayeed, I., & Atif, F. (2012). Progesterone Prodrugs and Analogs as Neuroprotective Agents. Emory Institute for Drug Development. Retrieved July 23, 2012, from http:\/\/eidd.emory.edu\/progesterone-prodrugs-and-analogs.\n\nSkelly, L., & Korschun, H. (2011, July 13). Progesterone Inhibits Growth of Neuroblastoma Cancer Cells. Emory Woodruff Health Sciences Center. Retrieved July 23, 2012, from http:\/\/shared.web.emory.edu\/whsc\/news\/releases\/2011\/07\/progesterone-inhibits-growth-of-neuroblastoma-cancer-cells-.html.\n\nChapter 6\n\nThompson, Holli, Your Girlfriends and Your Weight, Retrieved from http:\/\/askmissa.com\/2011\/12\/16\/your-girlfriends-and-your-weight\/.\n\nCohen, Emma, Ejsmond-Frey, Robin, Knight, Nicola and Dunbar, R.I.M. \"Rowers high: behavioural synchrony is correlated with elevated pain thresholds.\" Biol. Lett. Rsbl20090670. Published online September 15, 2009 doi: 10.1098\/rsbl.2009.0670,http:\/\/rsbl.royalsocietypublishing.org\/content\/early\/2009\/09\/14\/rsbl.2009.0670.full.pdf+html.\n\nJaslow, R. (2012, March 27). Eating lots of chocolate helps people stay thin, study finds. CBS News.\n\nBrian Buijsse, Cornelia Weikert, Dagmar Drogan, Manuela Bergmann, and Heiner Boeing. Chocolate consumption in relation to blood pressure and risk of cardiovascular disease in German adults. European Heart Journal, DOI: 10.1093\/eurheartj\/ehq068.\n\nHollis, J. (2012, April 16). Chew on this: study finds additional chewing reduces food intake in young adults. In Iowa State University. Retrieved from http:\/\/archive.news.iastate.edu\/news\/2012\/apr\/chewing.\n\n5 Benefits of Properly Chewing Food. (2008). 3 Fat Chicks. Retrieved August 26, 2012, from http:\/\/www.3fatchicks.com\/5-benefits-of-properly-chewing-food\/.\n\nChapter 7\n\nDiscovery Health, 5 Anti-Aging Supplements That Really Work, http:\/\/health.howstuffworks.com\/wellness\/aging\/anti-aging-tips\/5-anti-aging-supplements.htm (Nov 2012).\n\nVadim Aksenov, Jiangang Long, Sonali Lokuge, Jane A Foster, Jiankang Liu and C David Rollo, \"Dietary amelioration of locomotor, neurotransmitter and mitochondrial aging\" Experimental Biology and Medicine (2010, 235): 66\u201376, http:\/\/ebm.rsmjournals.com\/content\/235\/1\/66.full.pdf+html.\n\nGahche J, Bailey R, Burt V, et al. Dietary Supplement Use Among U.S. Adults Has Increased Since NHANES III (1988\u20131994). NCHS data brief, no 61. Hyattsville, MD: National Center for Health Statistics. 2011.\n\nBailey RL, Gahche JJ, Lentino CV, et al. Dietary supplement use in the United States, 2003\u20132006. J Nutr. 2011; 141(2):261\u2013266.\n\nDickinson A, Boyon N, and Shao A. Physicians and nurses use and recommend dietary supplement: report of a survey. Nutrition Journal. 2009; 8:29.\n\n\"11 Proven Benefits of Omega-3 Fish Oil for Women,\" Retrieved November 27, 2012 from http:\/\/www.ehow.com\/about_5445018_proven-omega-fish-oil-women.html.\n\nBetty Kovacs, MS, RD, \"Probiotics,\" Retrieved November 17, 2012 from http:\/\/www.medicinenet.com\/probiotics\/article.htm.\n\n\"Herbal Supplements: What to Know Before You Buy,\" Retrieved November 17, 2012 from http:\/\/www.mayoclinic.com\/health\/herbal-supplements\/SA00044.\n\n\"4 Dangerous (and Common) Vitamin Fillers You Must Avoid,\" Retrieved November 17, 2012 from http:\/\/www.draxe.com\/4-dangerous-and-common-vitamin-fillers-you-must-avoid\/.\n\nChapter 8\n\n\"Vitamin D Levels and oral supplementation in patients with skin cancer\", Journal of American Academy of Dermatology Vol 62, Issue 3, Supplement 1 (March 2010): Page AB66.\n\nBrierley Wright, \"Anti-aging foods for your skin,\" http:\/\/www.eatingwell.com\/blogs\/health_blog\/anti_aging_foods_for_your_skin (May 2012).\n\nMaria Celia B Hughes, Gail M Williams, Anny Fourtanier, and Adele C Green, \"Food intake, dietary patterns, and actinic keratoses of the skin: a longitudinal study\", The American Journal of Clinical Nutrition Vol 89 no. 4, (April 2009): 1246\u20131255.\n\nPeter Jaret, \"Coping With Acne: Your Care Plan,\" Retrieved from http:\/\/www.webmd.com\/skin-problems-and-treatments\/acne\/acne-care-11\/exercise.\n\nRobert Haas, MS, \"How Chronic Insomnia Destroys Skin Health,\" Retrieved October 30, 2012 from http:\/\/www.lef.org.\n\nChapter 9\n\nVaginal Dryness, Retrieved January 13, 2013 from http:\/\/www.nlm.nih.gov\/medlineplus\/ency\/article\/000892.htm.\n\nChapter 10\n\nKoenig, Harold G, Hays, Judith, et al, Does Religious Attendance Prolong Survival? J Gerontol A Biol Sci Med Sci (1999) 54 (7): M370-M376. doi: 10.1093\/gerona\/54.7.M370\n\nUniversity of Colorado at Boulder (1999, May 17). Research Shows Religion Plays a Major Role in Health, Longevity. ScienceDaily, Retrieved January 23, 2013, from http:\/\/www.sciencedaily.com\/releases\/1999\/05\/990517064323.htm\n\nKoenig, H.G., Cohen H.J., George L.K., Hays J.C., Larson D.B., Blazer D.G. (1997) \"Attendance at religious services, interleukin-6 and other biological indicators of immune function in older adults.\" International Journal of Psychiatry in Medicine 27 233\u2013250\n\nJonas, W.B., Crawford, C.C. (2003) Healing Intention and Energy Medicine. New York: Churchill Livingstone.\n\nGlazer, S. (January 14, 2005). \"Can Spirituality Influence Health?\" CQ Researcher. Vol. 15, no. 2: 1\u201335.\n\nPuchalski, C., MD, (2004). Spirituality in health: the role of spirituality in critical care. Critical Care Clinics. Vol. 20: 487\u2013504.\n\nDossey, L. MD (1993). Healing Words. San Francisco: Harper Collins Publishers.\n\nGeneral Social Surveys, 1972\u20132010. Conducted by the National Opinion Research Center at the University of Chicago.\n\nChapter 11\n\nLee, H. (2009, June). Oxytocin: The great facilitator of life. Progress in Neurobiology. 8(22), 127\u2013151.\n\nChapter 12\n\nCohen, E. (2011, June 23). Does life online give you \"popcorn brain?\" CNN.\n\nBe Well: Laugh. (2012). UCF Wellness and Health Promotion Services. Retrieved August 26, 2012, from http:\/\/bewellucf.com\/2011\/11\/17\/be-well-laugh\/\n\n\"Is Laughter Part of Your Healthy Lifestyle?\" (2012). Healthy Living.\n\nGail Larsen. (2012). \"Real Speaking.\" Retrieved August 5, 2012, from http:\/\/www.realspeaking.com\/.\n\nVessels, J. (2011, March 1). \"Play: It's Just Good for You!\" Health & Wellness.\n\nChapter 13\n\nMassMutual Financial Group and Oppenheimer Funds, \"From the Locker Room to the Boardroom: A Survey on Sports in the Lives of Women Business Executives,\" 2002.\n\nStephanie Wilcox, How Do Team Sports Help Develop Girls Into Future Leaders?, http:\/\/www.theglasshammer.com\/news\/2010\/12\/08\/how-do-team-sports-help-develop-girls-into-future-leaders\/ (December 2010).\n\nStevenson, Betsey. Beyond the Classroom: Using Title IX to Measure the Return to High School Sports. (Cambridge, MA: NBER Working Paper Series, National Bureau of Economic Research, 2010) , http:\/\/www.nber.org\/papers\/w15728.\n\nErnst & Young, Women Make All the Difference in the World, http:\/\/www.ey.com\/GL\/en\/Issues\/Driving-growth\/Growing-Beyond\u2014-High-Achievers\u2014-Women-make-all-the-difference-in-the-world (2012).\n\nHillary Rodham Clinton, U.S. Secretary of State, announcing the Global Sports Mentoring Program, 21 June 2012.\n\nChapter 14\n\nParker-Pope, T. (2010, April 14). Is Marriage Good for Your Health? The New York Times.\n\nPerls, Tom. New England Centenarian Study, Boston University School of Medicine, http:\/\/www.bumc.bu.edu\/centenarian\/overview\/.\n\nResearch, Statistics, and Policy Analysis. (2012, August 2). \"Social Security.\" Retrieved August 5, 2012, from http:\/\/www.ssa.gov\/policy\/docs\/workingpapers\/index.html.\n\nStudy: Majority of Women Becoming Primary Breadwinners. (2012, July 16). CBS News. Retrieved August 27, 2012, from http:\/\/washington.cbslocal.com\/  \n2012\/07\/16\/study-majority-of-women-becoming-primary-breadwinners\/.\n\nChapter 15\n\nExcerpted from Congresswomen Linda T. S\u00e1nchez, February 25, 2010 blog. Congresswomen S\u00e1nchez represents the 39th Congressional District of California.\n","meta":{"redpajama_set_name":"RedPajamaBook"}}
+{"text":" \nThe Great Tapestry of Scotland\n\n**Panel 1 stitched by:**\n\n_KA Two_\n\nLinda McClarkin\n\nCarol Whiteford\n\n**Stitched in:**\n\nBeith, Kilwinning\n\n_For all those who have worked on The Great Tapestry of Scotland_\nThis eBook edition published in 2013 by  \nBirlinn Limited  \nWest Newington House  \n10 Newington Road  \nEdinburgh  \nEH9 1QS  \n_www.birlinn.co.uk_\n\nCompleted panels from The Great Tapestry of Scotland  \ncopyright \u00a9 Great Scottish Tapestry Charitable Trust\n\nPhotographs copyright \u00a9 Alex Hewitt  \nPanel designs copyright \u00a9 Andrew Crummy\n\nForeword \u00a9 Alexander McCall Smith  \nIntroduction and other text \u00a9 Alistair Moffat\n\nThe moral right of Alistair Moffat to be identified as the author of this work have been asserted by him in accordance with the Copyright, Designs and Patents Act 1988.\n\nAll rights reserved. No part of this publication may be reproduced, stored or transmitted in any form without the express written permission of the publisher.\n\nISBN: 978-1-78027-160-6  \neBook ISBN: 978-0-85790-656-4\n\nBritish Library Cataloguing-in-Publication Data  \nA catalogue record for this book is available from the British Library\n\n# Contents\n\nForeword\n\nIntroduction\n\nThe Tapestries\n\nPANEL 2 The Ceaseless Sea\n\nPANEL 3a The Formation of Scotland\n\nPANEL 3b The Collision\n\nPANEL 4 Scotland Emerges from the Ice\n\nPANEL 5 The Wildwood\n\nPANEL 6 Tents and Tipis\n\nPANEL 7 The First Farmers\n\nPANEL 8 Brochs, Crannogs and Cairns\n\nPANEL 9 Pytheas the Greek\n\nPANEL 10 The Coming of the Legions\n\nPANEL 11 Ninian at Whithorn\n\nPANEL 12 Dalriada\n\nPANEL 13 Cuthbert and the Gospels\n\nPANEL 14 The Crosses and the Angles\n\nPANEL 15 Dunnichen\n\nPANEL 16 The Vikings\n\nPANEL 17 Dumbarton Rock\n\nPANEL 18 Constantine Climbs the Hill of Faith\n\nPANEL 19 Carham\n\nPANEL 20 King Macbeth\n\nPANEL 21 St Margaret of Scotland\n\nPANEL 22 The Flowers of the Borders\n\nPANEL 23 David I and the Wool Trade\n\nPANEL 24 St Andrews Cathedral\n\nPANEL 25 Duns Scotus and the Schoolmen\n\nPANEL 26 Somerled, Lord of the Isles\n\nPANEL 27 Haakon at Kyleakin\n\nPANEL 28 The Death of Alexander III\n\nPANEL 29 William Wallace and Andrew Moray\n\nPANEL 30 Bannockburn\n\nPANEL 31 The Rain at Carlisle\n\nPANEL 32 The Black Death\n\nPANEL 33 The University of St Andrews\n\nPANEL 34 The Ancient Universities\n\nPANEL 35 Orkney, Shetland and Scotland\n\nPANEL 36 Rosslyn Chapel\n\nPANEL 37 Chepman and Myllar\n\nPANEL 38 Blind Harry\n\nPANEL 39 Waulking\n\nPANEL 40 Flodden\n\nPANEL 41 The Thrie Estaitis\n\nPANEL 42 The Court of Session\n\nPANEL 43 The Scottish Reformation\n\nPANEL 44 Mary, Queen of Scots\n\nPANEL 45 The Reivers and the Rescue of Kinmont Willie\n\nPANEL 46 Robert Carey's Great Ride\n\nPANEL 47 The Making of the King James Bible\n\nPANEL 48 The Dawn of the Ulster Scots\n\nPANEL 49 Witches\n\nPANEL 50 The National Covenant\n\nPANEL 51 Droving\n\nPANEL 52 Philiphaugh, 1645\n\nPANEL 53 The Killing Times\n\nPANEL 54 The Massacre at Glencoe, 1692\n\nPANEL 55 The Bank of Scotland Founded\n\nPANEL 56 The Darien Scheme\n\nPANEL 57 The Act of Union, 1707\n\nPANEL 58 The Jacobite Rising of 1715\n\nPANEL 59 The Kilt\n\nPANEL 60 The Jacobite Rising of 1745\n\nPANEL 61 The Ordnance Survey\n\nPANEL 62 English Advances, Gaelic Retreats\n\nPANEL 63 The Royal and Ancient Golf Club\n\nPANEL 64 The First School for Deaf and Dumb Children\n\nPANEL 65 James Small and the Swing Plough, 1770\n\nPANEL 66 Enlightenment Edinburgh\n\nPANEL 67 Edinburgh's New Town\n\nPANEL 68 James Watt and the Steam Engine\n\nPANEL 69 The Tobacco Lords\n\nPANEL 70 Adam Smith\n\nPANEL 71 David Hume and Jean-Jacques Rousseau\n\nPANEL 72 The Highland and Lowland Clearances Gather Pace\n\nPANEL 73 Weaving and Spinning\n\nPANEL 74 James Hutton's Theory of the Earth\n\nPANEL 75 James Boswell and Smoked Fish\n\nPANEL 76 The Forth and Clyde Canal, Burke and Hare\n\nPANEL 77 Scotland and the Drive for Empire\n\nPANEL 78 Robert Owen and New Lanark\n\nPANEL 79 Robert Burns and 'Tam o' Shanter'\n\nPANEL 80 The False Alarm\n\nPANEL 81 Henry Raeburn\n\nPANEL 82 Walter Scott\n\nPANEL 83 Fingal's Cave\n\nPANEL 84 The Scotsman, Founded 1817\n\nPANEL 85 George Smith and the Glenlivet\n\nPANEL 86 Borders Tweed\n\nPANEL 87 The Growth of Glasgow\n\nPANEL 88 Sheep Shearing\n\nPANEL 89 The First Reform Act\n\nPANEL 90 Kirkpatrick Macmillan\n\nPANEL 91 Queen Victoria at Balmoral\n\nPANEL 92 The Scots in India\n\nPANEL 93 The Disruption\n\nPANEL 94 David Octavius Hill and Robert Adamson\n\nPANEL 95 The Railway Boom\n\nPANEL 96 The Caithness School, 1851\n\nPANEL 97 Fitba\n\nPANEL 98 Irish Immigration after the Famine\n\nPANEL 99 James Clerk Maxwell\n\nPANEL 100 Scots in Africa\n\nPANEL 101 Highland Games\n\nPANEL 102 Scottish Rugby\n\nPANEL 103 Shinty and Curling\n\nPANEL 104 Scots in North America\n\nPANEL 105 The Paisley Pattern\n\nPANEL 106 The Battle of the Braes\n\nPANEL 107 Mill Working\n\nPANEL 108 Robert Louis Stevenson\n\nPANEL 109 Workshop of the Empire\n\nPANEL 110 The Scottish Trades Union Congress Forms\n\nPANEL 111 Keir Hardie\n\nPANEL 112 The Herring Girls\n\nPANEL 113 The Discovery Sails from Dundee\n\nPANEL 114 Dundee, Jute, Jam and Journalism\n\nPANEL 115 Shetland, the Isbister Sisters\n\nPANEL 116 Charles Rennie Mackintosh\n\nPANEL 117 The Munros\n\nPANEL 118 The 1914\u20131918 War\n\nPANEL 119 The Building of HMS Hood, the Battle of Ypres 1917\n\nPANEL 120 Elsie Inglis\n\nPANEL 121 The Sinking of HMY Iolaire off Stornoway, 1919\n\nPANEL 122 Eric Liddell\n\nPANEL 123 Women Get the Vote\n\nPANEL 124 Whaling\n\nPANEL 125 The General Strike, 1926\n\nPANEL 126 Fair Isle\n\nPANEL 127 Hugh MacDiarmid\n\nPANEL 128 Ramsay MacDonald and the Rise of the Labour Party\n\nPANEL 129 The Great Depression\n\nPANEL 130 Tenement Life\n\nPANEL 131 The Second World War\n\nPANEL 132 The Clydebank Blitz\n\nPANEL 133 War Defences\n\nPANEL 134 D-Day, 1944\n\nPANEL 135 The First Edinburgh Festival\n\nPANEL 136 East Kilbride and the New Towns\n\nPANEL 137 The National Health Service\n\nPANEL 138 Television Arrives\n\nPANEL 139 The Washer Women\n\nPANEL 140 Cumbernauld New Town\n\nPANEL 141 North Sea Oil\n\nPANEL 142 Aberdeen\n\nPANEL 143 Linwood and the Hillman Imp\n\nPANEL 144 Pop Music Booms\n\nPANEL 145 Glenrothes\n\nPANEL 146 The Upper Clyde Shipbuilders\n\nPANEL 147 Stop Yer Ticklin', Jock!\n\nPANEL 148 The Rise of the Scottish National Party\n\nPANEL 149 Scotland at the Movies\n\nPANEL 150 Scotland's World Cup in Argentina\n\nPANEL 151 The Miners' Strike\n\nPANEL 152 Gaelic Resurgent\n\nPANEL 153 Glasgow \u2013 European City of Culture\n\nPANEL 154 Dolly the Sheep\n\nPANEL 155 The Scottish Parliament Reconvenes, 1999\n\nPANEL 156 AND 157 Parliament of the Ancestors, Parliament for the Future\n\n# Foreword\n\nThe tapestry illustrated in this book tells the story of Scotland. As one might imagine, to illustrate the history of a country, even with some 165 panels at one's disposal, is no easy task. Nor is it easy to tell it in a way that brings centuries of a nation's existence to life in an entertaining and vivid way. The Great Tapestry of Scotland does all this, and it does it in a way that was instantly recognised and appreciated by the public when it first went on show in the Scottish Parliament in the late summer of 2013. People came in their thousands \u2013 so many, in fact, that at times the building was thought to have reached capacity and faced the hitherto unthought-of possibility of being closed for a few hours.\n\nI was part of the group of people behind the tapestry project, and I went to the opening with my heart in my mouth. What would the public judgement be on a project that had such large ambitions? Would people find fault with our selection of subjects? Would the artist's style fail to resonate with the people of Scotland? I need not have worried for a moment on any of these scores. People stood before the tapestry with wonderment and delight on their faces. Some cried with emotion \u2013 the greatest tribute, I think, that any work of art can be given. I saw young children gazing at the panels with rapt attention. I saw elderly people recognise images and references and share them with each other with all the joy that goes with discovering something one has long known about but perhaps forgotten or not thought about for a long time. Everybody standing before the tapestry seemed transported by the experience.\n\nThe tapestry is the work of more than a thousand people from all over Scotland and beyond. The central artistic vision, though, is that of three people: Alistair Moffat, who selected the subject matter and wrote the text; Andrew Crummy, who drew the designs; and Dorie Wilkie, who supervised and inspired the stitching. All three made this object, but Andrew is the one who must be particularly celebrated as the artist. A man of great modesty, he never seeks praise, but it must be said here that he has, quite simply, wrought a masterpiece.\n\nThen there are the many stitchers who have themselves put their own artistic stamp on the tapestry. Although Andrew designed the main images, he generously left a great deal of scope for individual stitchers to make their own contribution. And they have done so magnificently, adding numerous touches throughout, giving yet more life and colour to this lovely object.\n\nWhen we started this project, which began with a telephone call I made to Andrew Crummy proposing that we do it, I had no idea that the result would be so lovely and so affecting. I had no idea, too, that in bringing the tapestry to completion so many people would be brought together in friendship. But that has all happened, and now the Scottish nation has something that it can treasure for many years \u2013 centuries, we hope \u2013 to remind us all of who we are and all the love, suffering, hope, disappointment, and triumph that makes up the life of a people. I hope, too, that we shall be able to share that with others from other nations, who will see the universal human story written on these panels of linen and perhaps reflect on the ideals of brother- and sisterhood that have always been so important in Scotland and remain so today.\n\nAlexander McCall Smith\n\n# Introduction\n\nOn a summer afternoon in the 1890s, Donald MacIver drove his pony and trap up the track on the west side of Loch Roag on the Atlantic shore of the Isle of Lewis. A teacher at the school at Breascleit, he found himself completing the last stage of a long journey back into the past. Beside Donald sat an old man. His uncle, Domnhall Ban Crosd, had sailed across the ocean from Canada so that, before he died, he could at last return home. As Donald flicked the reins and the pony trotted up the rise at Miabhig, the vast panorama of the mighty Atlantic opened before them. Below spread the pale yellow sands of Uig at low tide and around the bay lay a scatter of townships.\n\nHis by-names hint at Domnhall Ban's demeanour. Common enough when used to distinguish the owner of a frequently found name, _ban_ means 'fair headed' but _crosd_ is unusual \u2013 probably deriving from an English adjective, it is somewhere between 'obstinate' and 'ill natured'. Perhaps the old man set a characteristic stone-face when he gazed on the heartbreakingly beautiful bay at Uig, the water glinting in the summer sun, a sparkle, a sight he had not beheld for fifty years, not since the white-sailed ships slipped over the horizon bound for a new life in Canada.\n\nAs Donald guided his pony gently over the rutted tracks, grass growing green on their crests, they came at last to journey's end, the place Domnhall Ban Crosd had seen only in his dreams. It was the township of Carnais, his birthplace, where crofting families had been cleared off the land by the agents of an absent and uncaring aristocracy in 1851. Like many, Domnhall Ban and his people had found good and even prosperous lives in Canada but, in their hearts, there was _ionndrainn_. It means 'something that is missing', 'an emptiness', and, before he grew too old for a long sea voyage, the child of Carnais wanted to see his home place once more. But, when Donald pulled on the reins and braked the trap, there was nothing. Nothing at all to see. The croft houses had been tumbled down, the fences and fields opened to sheep pasture. What had been a busy, living landscape, home to the chatter of children and the day-in, day-out labour of farming families, had simply been obliterated. As he looked around at the desolation, the old man's face at last crumpled and he wept, tears falling for all that experience in one place, all lost and gone, memories that would die with him. ' _Chaneil nith an seo mar a bha e, ach an ataireachd na mara_ ,' he said to his nephew \u2013 'There is nothing here now as it was, except for the surge of the sea.'\n\nMuch moved by his uncle's sadness, Donald MacIver wrote his great lyric, ' _An Ataireachd Ard_ '. In memory of loss and change, of the wash of history over Scotland, it begins:\n\n_An ataireachd bhuan_\n\n_Cluinn fuaim na h'ataireachd ard_\n\n_That torunn a'chuain_\n\n_Mar chualas leams' e 'nam phaisd_\n\n_Gun mhuthadh, gun truas_\n\n_A' sluaisreadh gainneimh na tragh'd_\n\n_An ataireachd bhuan_\n\n_Cluinn fuaim na h'ataireachd ard._\n\nThe ceaseless surge\n\nListen to the surge of the sea\n\nThe thunder of the ocean\n\nAs I heard it when I was a child\n\nWithout change, without pity\n\nBreaking on the sands of the beach\n\nThe ceaseless surge\n\nListen to the surge of the sea.\n\nThe Great Tapestry of Scotland begins and ends with images of the ceaseless surge of the sea. The thunder of the ocean, its belly-hollowing elemental beauty is where our story of Scotland begins and it may well be where it ends, aeons into the future. But what made the rhythm and swirl of Donald MacIver's words resonate was his uncle's grief at the unflinching passage of time and the destruction of the way of life of ordinary people. And despite his tears, the old man's instinctive understanding that it could not be other. Lying on the edge of beyond, the land of Scotland, the hard, ancient rocks of Lewisian gneiss, the old red sandstone and the shales and coal of the Midland Valley, endure but the story of the people who lived on its straths, in its river valleys and glens changes constantly and sometimes brutally.\n\nNo mere backdrop, Scotland is utterly singular. Most who see a photograph or a sequence on TV of a landscape without road signs or a familiar landmark can recognise it instantly and intuitively as a place in Scotland. Not sure exactly where but it's definitely Scotland. There is a mixture of atmosphere, look and something very simple. The colours of Scotland are like no other and, above all, the Great Tapestry and its thousand makers have perfectly captured the blues, greens, reds, greys and browns of our distinctive geography.\n\nThe sorrow of Domnhall Ban Crosd is also characteristic. The old man did not weep for the passing of empires, the boasts of Ozymandias, he did not stand in the magnificent ruins of greatness, he wept for the scatter of once-snug cottage walls, for the passing of a way of life, for the body-warmth of a dying culture, the humanity of shared ills and privations, the clear and unambiguous sense of a community, the ghosts of people who would help each other when the waves crashed on Mangersta Head and storms blew in off the ocean. On that summer afternoon, Domnhall Ban knew that night had fallen on Carnais.\n\nIf those who look at its panels also listen closely to the Great Tapestry, they will hear something of what it sounds like, not only like the thunder of the ocean but also the distant march of armies, the jingle of harness, the roar of cannon and the echo of oratory. But much louder and more insistent will be the tread of ordinary people, the _craic_ by the fireside, the whispers of hunters in the wildwood, the chatter of plough teams, the whistle of a shepherd, the clatter and rattle of mills, foundries and shipyards. And the singing, the waulking songs, the pawky melodies of Harry Lauder, the psalms, the keening and the grief. The sounds of a community, what Domnhall Ban knew at Carnais. But, despite the efforts of the powerful, the disengaged and outsiders, Scotland is still, just, a community, and a series of connected communities. Scots still believe in a collective sense of ourselves and that society should take responsibility for the old, the weak and the poor. Lest this sound like a manifesto, it should be pointed out that it is not a proposal for the future but an observation of what happened in the past \u2013 and in the present. The Great Tapestry is, after all, the creation of a community \u2013 the thousand stitchers who came together in groups to make it.\n\nAnd, lest our sense of ourselves becomes self-congratulatory, the ceaseless surge of the sea and the ungraspable millions of years it took to form the geology of our nation remind us that the history of Scotland is but a speck bobbing on the ocean of time. Eleven thousand years may seem like an epic sweep but it is as nothing when the night sky is clear and the light of dead stars shines down on our tiny planet. And, being Scots, we should be further chastened by the knowledge that, if we have somehow managed to retain a worthwhile, workable sense of community, there were plenty of times when we behaved extremely badly and continue to.\n\nThe Great Tapestry of Scotland is, then, a people's history of a people, made by a thousand of those people. That says much about its content, its cultural bent, but why exactly is it so beautiful, so evocative and how did it come into being?\n\nAlexander McCall Smith is the simple answer to the last question. He made it happen. Having seen the remarkable Prestonpans Tapestry, a beautiful embroidered record of the battle fought near the town in 1745, Sandy was impressed not only with the panels and their design but also with the impact they had on those who looked at them. They were mesmerised. An idea leapt into Sandy's head. Why not make a tapestry that tells all of Scotland's story and do it for 2014, the Year of Homecoming, the Commonwealth Games in Glasgow, the Ryder Cup at Gleneagles, the 700th anniversary of Bannockburn? A tapestry of Scotland for Scotland's year. The artistic inspiration behind the Prestonpans project was Andrew Crummy and, once Sergeant McCall Smith had recruited him, he called me. What should be in it? How should the narrative run? Would I write it? Within ten seconds, I was enlisted \u2013 Corporal Moffat, sir. Our platoon was completed when Dorie Wilkie, the wonderful Head Stitcher on the Prestonpans Tapestry, agreed to oversee the making of what we decided to call The Great Tapestry of Scotland. Gillian Harte became our excellent administrator and, finally and vitally, Jan Rutherford and Anna Renz agreed to do the publicity and raise the funds needed. They did a superb job.\n\nNow we needed an army, a thousand volunteer stitchers to transform my narrative and Andrew Crummy's superb drawings into the tapestry. And, once the call went out, they enlisted within months. Groups of stitchers sat down all over Scotland, from Shetland to Galloway, and they began the hard, detailed, fiddly, inspiring work of making all come alive.\n\nPrincipal amongst the pleasures of working on this unique project has been my collaboration with Andrew Crummy. In the best Scottish tradition, I kent his mother \u2013 the remarkable Helen Crummy who founded the Craigmillar Festival Society \u2013 and, with his passionate inherited interest in community arts, Andrew is certainly his mother's son. From the outset, we agreed that if the tapestry was to speak clearly of Scotland, then it had to tell a story of our people, of all who lived here for all those who live here now.\n\nWhat made collaboration a joy was Andrew's abundant talent. He knows how to make bold statements, his drawing is absolutely assured and it proceeds from a clear grasp of the essential narrative. No messing about, arrow-straight to the point. The lyrical opening panel shows Andrew's talent perfectly as sheep, fish and birds dance in the hair of the woman who looks up at the hand with the needle while, around her, the arms of others enfold scenes and objects of Scotland \u2013 standing stones, tenements, shipyards, books, musical instruments and a football. And, moving briskly through the picture plane, trains puff across viaducts, ships sail, fully rigged, to unknown destinations while an oil drilling platform is planted in the sea. It is nothing less than the singular, all-encompassing, wholly original work of a great artist.\n\nCrucially, Andrew also understood what stitchers liked to stitch and his drawings deliberately involved them directly in the creative process, inviting them to add images, even alter the original. We both talked of Renaissance altarpieces and how painters appended predella panels below the Virgin and Child, the Crucifixion or the Adoration of the Magi or whatever the main image was. In these, painters sometimes added street scenes, landscapes or a portrait of the donor and, by leaving spaces in the drawings, Andrew and I hoped that stitchers would take the initiative. And they did! We also hoped that each would attach an impresa and perhaps initials. And they did. They made it their own.\n\nWhat further encouraged additional bits of history in each panel was geography. Not only did the tapestry have to tell a story of the Scots, it had to include as much of the land of Scotland as was feasible. Panels were set in the Northern Isles, the Highlands, the cities, the Hebrides, the Midland Valley, Galloway and the Borders. And then, very importantly, those linked to a particular area were usually sent to groups of stitchers who lived in the same area. And that process of matching was wonderfully fruitful.\n\nThe needs of a wide geographical spread did not inhibit my major task \u2013 what to include and what to exclude. Over the eleven millennia of our story, Scotland has changed enormously. Cities did not always exist and places now sparsely populated were once important. Before 3,000 BC, very few hunter-gatherers lived in the wildwood and it seems that the islands of the Atlantic coast supported a much larger proportion of the population than they do now. Before 1700, the vast majority of Scots lived in the countryside while after 1900 most had moved to the Lowland cities. Without bending the narrative out of shape, it was possible to locate Scotland's story right across the nation.\n\nDrama is naturally eye-catching and it would have been easy to arrange our past around a series of battles, wars, martyrdoms and coronations and the other familiar set pieces. And some did insist on inclusion. But, generally, it seemed better to talk of how change affected the many rather than leaders or elites of various sorts. For those who worked the land for generations, invasion and political change often meant the replacement of one set of masters with another. The great problem is silence. Written records usually noted the doings of the mighty, the notorious and the saintly \u2013 and rarely have anything to say about ordinary people. Andrew and I therefore decided to show something of the lives of the many with generic panels, images of people working, walking their lives under Scotland's huge skies. These are simple and eloquent, needing little explanation.\n\nEarly Scotland is also \u2013 conventionally \u2013 a story of men. My own work in DNA studies and elsewhere has led me to believe that, in the millennia before the last two centuries, the status of women was little better than that of informal slavery. And so Andrew and I took every opportunity to include female figures, wherever figures were needed, both in the specific and generic panels. _Clann-Nighean an Sgadain_ , the Hebridean Herring Girls who followed the fishing fleets in the 19th and early 20th centuries to gut and barrel the herring catch as it came ashore around Scotland's ports, is a good example. We did not distort our national story to include women where they were not actors in events, but we recognised that always they were there, giving life to the nation, and we tried never to forget that.\n\nNot that we would have been allowed to. Almost all of the stitchers are women and they would not have let us do anything less. That fact remains an absolutely determinant influence.\n\nThe completion of this epic project is also the end of an equally epic, long, emotional and unexpected journey. And the tapestry itself is unexpected. Two years ago, I had no idea that it would look like it does. The Great Tapestry of Scotland sparkles, glows and surprises. It is not only the work of Andrew Crummy, Dorie Wilkie and myself, inspired by Sandy McCall Smith, it is a unique expression \u2013 a history of a nation written and made by a thousand people.\n\n**Panel 158 stitched by:**\n\n_The Red Lichties_\n\nEvelyn Chaplain\n\nBob Chaplain\n\nRena Freeburn\n\nJanette Nairn\n\nEileen Shepherd\n\nAlice Sim\n\nJessy Smart\n\nMary Stephen\n\nLinda Walker\n\nMargaret Wynne\n\n**Stitched in:**\n\nInverkeilor, Arbroath\n\n# The Tapestries\n\n_Almost all of the images and the panels are easy to understand and some have text to help identify who is who and what is what. They also run in a broadly chronological order but what follows adds a context, the briefest summary of the story of Scotland, background music for this stunning procession of the past._\n\n# PANEL 2  The Ceaseless Sea\n\nIn the beginning was the sea. Hundreds of millions of years ago, when the Earth was very young and vast primordial landmasses rose and were submerged, what would become Scotland was unrecognisable. Different parts of our familiar and beloved geography lay far distant from each other, some were attached to huge continents, some were splintered fragments, still others lay submerged on the bed of an ancient ocean. Lying between three palaeo-continents, Laurentia, Baltica and Avalonia, the great expanse of the Iapetus Ocean was beginning to shrink as tectonic movement shaped and reshaped the crust of the Earth. An unimaginably long time ago, about 410 million years, much of what became Scotland was waiting to rise out of the prehistoric seas and be welded together into one of the most geologically distinct places on our planet. Scotland was to be the deposit of a series of ancient collisions. And, throughout our prehistory and in more modern times, these collisions would remain central to an understanding of our nation and its people. Our history is written in our rocks just as surely as it is in monastic chronicles, census returns or the stones and bones of archaeological digs.\n\n**Panels stitched by:**\n\nHelen Nairn\n\nFrances MacLean\n\nDebbie Muir\n\nNino Stewart\n\nMarjorie Watters\n\n**Stitched in:**\n\nKinlochmoidart\n\n# PANEL 3a The Formation of Scotland\n\nAs powerful tectonic forces bulldozed the great continents across the face of the Earth, splinters were sometimes sheared off. Known as terranes, four very different rock formations were ground together to make the map of Scotland. To the north-west, Lewisian gneiss made the Western Isles, Coll, Tiree, Iona, the western peninsula of Islay known as the Rhinns and parts of the Atlantic coastline. Tremendously hard, the gneiss is igneous, forged deep in the molten crust of the Earth and forced upwards as the palaeo-continents moved, buckling and corrugating the strata. To the east and south of the Lewisian formations, thick layers of Old Red Sandstone predominate and their most spectacular monuments are the Torridon Mountains and the singular peak of Suilven. Below the Great Glen lie Dalriadan rocks, mostly strata of sandstone, shale and limestone. The mountains and glens of such as the Cairngorms were moulded and changed as volcanoes roared and ice ages froze the land. Below the abrupt frontier of the Highland Boundary Fault, the geology of the Midland Valley, better known as the Central Belt, was originally formed in tropical latitudes. Many millions of years ago, vast and dense forests grew where Glasgow, Stirling, Perth and Edinburgh now stand. It was swampland and, as the trees died and fell, they made layers of carbon. These strata would much later become Scotland's coalfields. The Southern Upland Fault was formed by the fourth terrane. Following a line from Ballantrae in Ayrshire to Dunbar on the Firth of Forth, its hills were once the bed of a primordial ocean.\n\n# PANEL 3b The Collision\n\nSome way to the south of the Southern Upland Fault lies a fascinating geological relic. On the west coast of the Isle of Man, near the hamlet of Niarbyl, the cliffs of a small cove have running diagonally across them a thin, greyish-white seam of rock. It is visible for only a hundred metres or so before it disappears into the waters of the Irish Sea but it is a memorial to the making of Scotland. Known as the Iapetus Suture, it marks the precise place where the vast continents of Laurentia and Avalonia collided, having welded the four terranes together. And as an entertaining footnote, the harder rocks of the Southern Upland Fault caused the strata of the softer rocks of the North of England to buckle and push the coal and iron ore seams nearer to the surface. In this way, Galloway and the Borders made West Cumbria and Tyneside a gift of their traditional industries. The angle of the ancient collision is remembered in the north-east to south-west slant of Scotland's geography, the Great Glen, the Highland Boundary Fault, the Midland Valley and the Southern Upland Fault, to say nothing of the compass direction of the modern border between England and Scotland. The grain of the land has been immensely influential on our history. Upland and lowland follows it, lochs and rivers run that way, tracks and roads are forced to take its direction. Geology formed Scotland and the land and the sea formed the character of the people.\n\n**Panel stitched by:**\n\n_North Berwick Creative Embroidery Group_\n\nCarole Bailey\n\nAngeniet Black\n\nAudrey Brown\n\nTertia Crawford\n\nMargaret Dinning\n\nMargaret Holm\n\nVi Jones\n\nPat Lucas\n\nBunty McInroy\n\nIsobel Russell\n\nMargaret Struth\n\nLyn Vaughan\n\nSandra Walker\n\n**Stitched in:**\n\nDunbar, Gullane, East Linton, North Berwick\n\n# PANEL 4  Scotland Emerges from the Ice\n\nThe last ice age ended very much more recently. Some time around 9,000 BC, glaciers began to groan, splinter and grind over the landscape of Scotland like prehistoric sandpaper. Bulldozing boulders, gravel and other deposits, they shaped familiar landmarks like Stirling and Edinburgh's castle rocks, they scarted out glens and sea lochs, rolling hills and mountains. And, when the ice at last drew back, people came \u2013 the Pioneers who first settled Scotland. They came from the south \u2013 some from the ice-age refuges, the famous painted caves on either side of the Pyrenees, others walked to Scotland from the east. Until c. 6,000 BC, the southern basin of the North Sea was dry land. What first drew the Pioneers, our earliest ancestors, northwards was probably the migration of the animals they depended on. Many were cold adapted, like reindeer, and, as the ice melted very quickly in a process known as climate flickering, the animals could not evolve fast enough and so they were forced to chase the cold. And our ancestors followed them north. Although they looked like us, these hunter-gatherers will have seemed young to a modern eye. Few lived beyond the age of thirty, women perhaps dying young because of complications in childbirth. Since the bone needle had been invented, the Pioneers were able to make warm and close-fitting garments from skins, their look probably most resembling that of the Forest Indians of the eastern United States. When they came to Scotland, they may have come to a land of plenty, a place where roots, fruits, nuts and berries grew and fish and prey animals thrived. And their arrival reminds us of a basic historical truth \u2013 every Scot is an immigrant. The only question is when our ancestors arrived.\n\n**Panel stitched by:**\n\n_Pioneer Panelbeaters_\n\nEileen Henderson\n\nLaura Henderson\n\nAnnette Hunter\n\nKirsten Hunter\n\n**Stitched in:**\n\nLinlithgow, Edinburgh\n\n# PANEL 5  The Wildwood\n\nAfter the ice, willow scrub was the first to colonise, growing in sheltered, damp places, and then came aspen, birch, hazel, elm, oak and finally lime. As they grew and the leaves of a thousand autumns fell and enriched the soil, Scotland was carpeted by a vast wildwood \u2013 a green and temperate jungle that stretched away on every horizon. By the end of the eighth millennium BC, deciduous trees could thrive and survive at 2,500 feet above sea level and only in the windblown Highland ranges would the wildwood fail to grow. With the trees and the dense canopy came animals which nested, burrowed and browsed in the shade at ground level, living off a plenitude of insects, seeds, leaves, grasses, roots and shoots. But there was also drama. The aurochs, giant wild cattle with seven-foot horn-spreads, thrashed through the undergrowth and wild boar, especially sows with young, could be a fearsome sight on the charge. Red and roe deer and elk also flitted amongst the shadows of the trees. Streams, rivers and lochs were home to fish, waterfowl, otters and beavers while all were closely followed by their predators \u2013 wolves, lynxes and bears. And, principal amongst them, the first Pioneers to arrive after the ice. It may well be that the animals of the wildwood saw people so rarely that they were more curious than afraid. They soon learned to be.\n\n**Panel stitched by:**\n\n_The Edinburgh Tapestry Tenners_\n\nJo Avery\n\nJoy Dunn\n\nJoan Houston\n\nJoan Leslie\n\nMorag Macleod\n\nElizabeth McCall Smith\n\nElizabeth McGuigan\n\nGill Salvesen\n\nKatherine Shaw\n\nLiz Sutherland\n\nMarjory Watson\n\nSusan Wexler\n\nNikhat Yusaf\n\n**Stitched in:**\n\nLinlithgow, Edinburgh\n\n# PANEL 6  Tents and Tipis\n\nThe Pioneers at first left only gossamer traces. At Cramond, near Edinburgh, the postholes left by the whippy green rods needed to make a bender tent encampment have been found and dated to the ninth millennium BC. And more spectacular, larger postholes were excavated at Barns Ness in East Lothian and interpreted as the earliest substantial house yet found in Scotland. This remarkable find closely resembled a similar structure unearthed at Howick, in Northumberland, and, in 2013, another was found at Echline during the construction of the Queensferry Crossing, on the south side of the Forth. Each had been made from whole tree trunks rammed into postholes and canted inwards to form a tipi shape. This frame was then covered by turf or perhaps thatch made with bracken and ferns. The living space was an oval shape and could accommodate a family of seven or eight. But the importance of these ancient houses was that they signified ownership of the land around them or at least the exercise of customary rights. Why expend all that energy if others could fish and hunt nearby? The most important resource was firewood for heat, cooking and light and many bands of hunter-gatherers will have ranged wide to gather it and possibly been forced to move when wood became scarce. Very few people lived in Scotland at the time the houses at Barns Ness and Echline were built \u2013 perhaps no more than a thousand or so. If these houses were occupied at the same time, the family bands who lived in them may have known each other.\n\n**Panel stitched by:**\n\n_Halflinbarns Schoolhouse Weaving Group_\n\nVi Bathgate\n\nFrances Fergusson\n\nFrances Gardiner\n\nBeth Orr\n\nIrene Rendells\n\nDiane Rendells\n\nMuriel Stuart\n\n**Stitched in:**\n\nNorth Berwick, East Linton\n\n# PANEL 7  The First Farmers\n\nIn the years around c. 3,000 BC, the greatest revolution in Scotland's history took place. Farmers crossed the North Sea from Europe and they brought new techniques of cultivating crops and domesticating animals. Life changed utterly and amongst many spectacular monuments to the productivity of the farmers were the great stone circles of Orkney and the large timber halls built on the banks of rivers in eastern Scotland at places such as Balbridie, Claish and Kelso. The population grew very quickly particularly as a consequence of growing cereal crops. It turns out that the invention of porridge changed the world. In hunter-gatherer societies infants with soft baby teeth found the wild harvest of roots, fruits, nuts and berries difficult and consequently they were breastfed for much longer. While nursing, women generally cannot conceive and so the birth interval was long. When cereal cultivation began, the dried or charred grains could be mashed into a protein-rich porridge and fed to infants. This allowed mothers to stop breast-feeding earlier and it greatly reduced the birth interval so the first farmers began to have much larger families. And the production of food surpluses in good years allowed these growing communities the time to do work not associated with agriculture \u2013 the building of the great religious monuments of prehistoric Scotland.\n\n**Panel stitched by:**\n\nJo Constantine\n\nFrances Gardiner\n\nRosalind Neville-Smith\n\n**Stitched in:**\n\nOrkney, North Berwick\n\n# PANEL 8  Brochs, Crannogs and Cairns\n\nWith no windows, only one door and a double skin of thick drystane walls, the remarkable structures known as brochs may seem like a response to Scotland's weather. Most were built towards the end of the first millennium BC in the Northern Isles and around the Atlantic coasts although isolated examples were found near Galashiels and in Berwickshire. Brochs were almost certainly status symbols probably built by specialist teams of masons for important individuals. They stand as what archaeologists call 'statements in the landscape'. No doubt warm and relatively spacious with two or three floors, they were not easily defended. Crannogs were. Raised on wooden piles driven into the beds of lochs, just offshore, these houses were surrounded by water and accessible only by a single wooden causeway. Loch Tay had several along its banks. Crannogs were a watery variant on the roundhouse, a design that evolved in later prehistory in Scotland. Conical in shape, the roof beams usually rested on a low circular stone wall and the house was made weather tight by turf or bracken thatch. Some were large with diameters of 33 to 36 feet and had two storeys. Cairns were more enigmatic. Often found on hilltops, they may have been memorials or places where the worship of sky gods took place. Such beliefs may have been more common amongst the Carnonacae of Wester Ross and the coastal glens down to Skye. Their presence was noted on a map made in the second century AD and their name means the People of the Cairns.\n\n**Panel stitched by:**\n\nDianne Laing\n\nFrances Gardiner\n\n**Stitched in:**\n\nNorth Berwick\n\n# PANEL 9  Pytheas the Greek\n\nThe first traveller to leave any sort of a record of his journey to Scotland was Pytheas, a Greek traveller and explorer from the mercantile colony of Massalia, modern Marseilles. Some time around 320 BC, he came north to visit Britain, possibly because the tin miners of Cornwall exported the metal to the Mediterranean for the production of the alloy known as bronze. Or perhaps he was just curious. In any case, Pytheas gave our country its name. He called it Pretannike, the land of the Pretannikoi, the People of the Tattoos. Probably conferred by the kindreds who lived on the southern coasts of the Channel, it referred to the habit of body decoration, something that had probably died out on the Continent and made the peoples of Britain different. Almost certainly transported in curraghs, sea-going hide boats sailed by experienced local merchants and mariners, Pytheas circumnavigated Britain. Using a measuring stick called a gnomon, he took a reading of latitude on the Isle of Lewis, probably at the ancient stone circle at Calanais. Intrepid and inquisitive, the Greek explorer travelled further north to Orkney and Shetland and perhaps even beyond. The record of his journey, _On the Ocean_ , has been lost but it was so widely referred to \u2013 and mocked \u2013 by other writers in antiquity that large parts of it can be reconstructed. When the Romans came to conquer, almost four centuries later, they altered Pretannike to Britannia. If they had not, we might all be called the Prits rather than the Brits.\n\n**Panel stitched by:**\n\nMargaret Macleod\n\nMary Macleod\n\n**Stitched in:**\n\nIsle of Lewis\n\n# PANEL 10  The Coming of the Legions\n\nIn AD 43, the Emperor Claudius entered Colchester in triumph at the head of his victorious legions, their eagle standards glinting, hob-nailed boots tramping. To impress the defeated British kindreds, they had even brought war elephants to plod past the crowds. In the following forty years, the Romans pushed the edges of their Empire farther and farther northwards until the governor of the province of Britannia, Julius Agricola, led an invasion force into Scotland. Having brigaded together three legions and regiments of auxiliaries, he marched his men up through the Tayside and Angus glens to a place the Romans called the Graupian Mountain. Probably at the foot of Bennachie in Aberdeenshire, a pivotal battle was fought. A confederacy of Caledonian kindreds commanded by Calgacus, the first Scot to be named in the historical record, was defeated by the discipline, determination and organisation of the legions and auxiliaries. But the victory was not consolidated, Scotland was soon abandoned and Agricola recalled to Rome. Some time around 122, the Emperor Hadrian ordained that a wall should be built. It stretched from near the mouth of the Tyne, through the Hexham Gap to the Solway coast beyond Carlisle. It was as much a clear limit to conquest as a means of controlling the frontier between the province of Britannia and the barbarians of the north, an early definition of Scotland. The south of Scotland was briefly part of the Empire in the middle of the second century when Hadrian's successor, Antoninus Pius, commanded a turf wall to be raised between the Forth and Clyde. But it was a short-lived occupation. Several Roman commentators grumbled that Scotland was simply not worth the bother.\n\n**Panel stitched by:**\n\nCaroline M Buchanan\n\nAnne Hamill\n\nSusan Lindsay\n\n**Stitched in:**\n\nLarbert, Falkirk\n\n# PANEL 11  Ninian at Whithorn\n\nCarlisle appears to have survived as a functioning Roman town for at least two centuries after the fall of the Western Roman Empire. When St Cuthbert visited in 685, he was shown a working fountain, something that implied a working water supply. Early Christianity in Britain flourished principally amongst town dwellers and there seems to have been a well-established church in the town served by ordained priests. Carlisle may have been the place where Ninian himself was instructed and ordained. His dates are a matter for constant revision but he probably preached some time in the fifth century. At the ancient church of Whithorn in Galloway, what is known as the Latinus Stone was found. It seems not to have been a tombstone but rather the commemoration of the foundation of a shrine or a monastic refuge of some kind. Perhaps it remembers the mission of Ninian who, according to a slightly confused account from Bede of Jarrow written in the early eighth century, had been sent to convert 'the Southern Picts'. This last is probably a vague reference, from a monk who never left the twin monasteries of Monkwearmouth and Jarrow, to the native kindreds of the Novantae and the Selgovae, the early peoples of Galloway and Dumfries. Ninian caused a church to be built at Whithorn and it was notable because it was made in stone 'which was unusual amongst the British' sniffed the sainted Bede. It was known as Candida Casa, the White House, and the aura of sanctity around Ninian's foundation has endured for more than 1,500 years.\n\n**Panel stitched by:**\n\nShirley McKeand\n\n**Stitched in:**\n\nDumfries\n\n# PANEL 12  Dalriada\n\nIn the fifth century AD and perhaps for some time before, Irish war bands had been crossing the North Channel to invade and to settle. DNA evidence supports the notion of colonisation by Northern Irish kindreds, in particular those known as the Ui Neill. They seized the Atlantic coastlines of Argyll (the name means the Coast of the Gael), several southern Hebridean islands and the shores of Lorne and further north. In their mouths, these warriors and settlers brought the Gaelic language to replace the Pictish dialects of those they subdued. For many centuries afterwards, Gaelic was described as Irish or Erse. Dalriada became the collective name, according to Bede, for the territory of the incomers and eventually its kings became powerful right across Scotland. Recent research has discovered a frontier between the British kingdom of Strathclyde and the Dalriadan Gaels. It is marked by the Clach nam Breatainn, the Stone of the Britons, at the head of Loch Lomond in the north and in the south in the Firth of Clyde by the sentinel Cumbraes, literally the islands of the British. One of the earliest Scottish historical documents to survive is the _Senchus Fer n-Alban_ , the _History of the Men of Alba_ , now the Gaelic name for all Scotland but then a reference only to Dalriada. It is a naval muster roll and it describes a seaborne warrior culture, something that would endure for a millennium or more. More peacefully, the Irish invaders eventually brought Columba to Scotland and to Iona, surely the spiritual heart of the West.\n\n**Panel stitched by:**\n\nFelicity Blackburn\n\nCatherine Borthwick\n\nHelen Brodie\n\nMorag Keenan\n\nEileen MacPhie\n\nLinda McLean\n\nHeather Simpson\n\n**Stitched in:**\n\nArisaig, Malliag, Edinburgh\n\n# PANEL 13  Cuthbert and the Gospels\n\nSome time in the late seventh century, a young man rode from the hills above Melrose to the gate of the old Celtic monastery that lay in a loop of the River Tweed. Cuthbert wished to become a monk and lead an exemplary life of prayer, contemplation and preaching. He became Bishop of Lindisfarne and, shortly after his death on the tidal island, he was canonised. And immediately claimed by the English. The reality is more complicated. Cuthbert is an Anglian name but, in the seventh century, the Borders and the Lothians formed part of the glittering kingdom of Northumbria and would remain so for more than three centuries. A northern dialect of early English would develop into Scots and it was introduced by the Angles of the Tweed Basin and the Lothians. Surely the most beautiful and dramatic in Britain, Durham Cathedral was raised on Cuthbert's bones and his saintly cult brought pilgrims, gifts and money to his tomb and those who tended it. His name supplied an early definition of Englishness when the people who lived to the south of the Tweed began to call themselves the Haliwerfolc, the People of the Holy Man. But, in truth, these claims and counterclaims pale into insignificance beside the glorious achievements of the church in the north between the seventh and ninth centuries. The Book of Kells was essentially a production of the Columban monastery at Iona and the Lindisfarne Gospels were made by the island monks in their windblown refuge off the Northumberland coast. Perhaps one day these stunning artistic and spiritual achievements will come back home where they belong.\n\n**Panel stitched by:**\n\n_Stitchers o' Stow_\n\nAnna Houston\n\nHelen Houston\n\nLorna Lyons\n\nPatricia McMahon\n\nDiana Muir\n\nKaren Nelson\n\nSerpil Renton\n\nAmanda Runciman\n\nKathleen Runciman\n\nLibby Runciman\n\nElizabeth Simm\n\nDorothy Small\n\nDeborah Wood\n\nwith stitches from the children of Stow Primary School\n\n**Stitched in:**\n\nStow, Galashiels, Lauder\n\n# PANEL 14  The Crosses and the Angles\n\nWhen the ambitious and ruthless kings of Northumbria pushed west with their war bands in the late seventh and eighth centuries, they obliterated the ancient British kingdom of Rheged. Hinged on Carlisle, it stretched as far west as Dunragit, or the Fort of Rheged, near Stranraer, and perhaps as far south as Lancashire. In the wake of the warriors came holy men and artists. Two of the most beautiful and intricate examples of early Christian sculpture in Britain stand at Bewcastle, on the moors east of Carlisle, and at Ruthwell, in Dumfriesshire. Scenes from scripture were carved on the cross shafts and arms and brightly painted. The Ruthwell Cross also has a place in the story of British literature. Carved in Northumbrian runes cut on the lower, narrow sides of the shaft are two extracts from one of the earliest surviving poems composed in English, _The Dream of the Rood_. It is part of a very sophisticated scheme of words and images for reading the cross and its central conceit is that the rood or cross upon which Christ was crucified had a personality. This is the haunting second stanza:\n\nI [lifted up] a powerful king \u2013\n\nThe Lord of Heaven I dared not tilt.\n\nMen insulted both of us together;\n\nI was drenched with blood poured from the man's side.\n\n**Panel stitched by:**\n\n_The Angle Stitchers_\n\nHelen Allan\n\nMargaret Boe\n\nGeorgina Chapman\n\nBarbara Downie\n\nAmelia Little\n\nYvonne Tweedie\n\nFrea Webster\n\n**Stitched in:**\n\nEdinburgh, Haddington\n\n# PANEL 15  Dunnichen\n\nFew battles genuinely alter the course of history and even fewer in the early history of Scotland were commemorated by an artist. In the kirkyard at Aberlemno in Angus, there stands a remarkable Pictish stone. Carved on one side are four scenes from the Battle of Dunnichen, fought in 685 only six or so miles from Aberlemno. The sculptor shows a genuine understanding of warfare. There is a shield wall of infantry with the man in the front rank waiting for the shock of a cavalry charge and he holds up a shield with a prominent boss at an angle that might meet a downward cut from a cavalry warrior. Behind him is a soldier holding a long spear that projects well beyond his comrade and behind him stands a third rank ready to support and plug gaps in the line. Warriors on horseback may represent the two great kings whose armies clashed at Dunnichen. The Picts were led by Bridei and the Angles by Ecgfrith. Having conquered the Tweed Valley, the Lothians and probably Fife, the Anglian juggernaut had rumbled northwards to be met by Bridei and his war bands at Dunnichen, near Forfar, where it was brought to a juddering halt. Ecgfrith was killed and the Angles retreated south of the Forth. Bridei's achievement was not only a decisive victory; it made Scotland possible. If the Northumbrians had won and gained control of fertile Pictland, consolidating as they did in the south, England and the English language might have stretched all the way to Shetland.\n\n**Panel stitched by:**\n\n_Angus Embroidery and Textile Artists_\n\nPat Beaton\n\nNorman Beaton\n\nDot Chalmers\n\nLinda Clark\n\nBetty Fotheringham\n\nElspeth Foxworthy\n\nHelen Fulford\n\nElizabeth Hill\n\nIsobel Hyslop\n\nDeborah Kenward\n\nJanette Nairn\n\nEna Norrie\n\nPatricia Rae\n\nJoan Robb\n\nIolanta Robertson\n\nMary Stephen\n\nSandra Taylor\n\nLinda Walker\n\nMargaret Wynne\n\n**Stitched in:**\n\nCarnoustie, Forfar, Arbroath\n\n# PANEL 16  The Vikings\n\nIn 793, fell portents were seen. Thunder and lightning rent the air and dragons flew. On the shingle beach of the holy island of Lindisfarne, fearsome warriors rasped their ships up above the high-tide line and raced for the doors of the monastery before the terrified monks could bolt them. The Vikings had sailed into history. Even across thirteen centuries the shock of the first attacks is still palpable. The outrage of the church was expressed in an early description: the Vikings were known as the Sons of Death. Their _dreki_ or dragon ships were sleek, fast and very versatile, able to deliver warriors to targets that lay inland but up navigable rivers, their shallow draught taking the keels clear of reefs of sand or rocks and their double-ended hulls allowing them to reverse in narrow channels. After the first raids, Vikings began to colonise parts of Scotland. Orkney and Shetland retain the clearest Scandinavian cultural and genetic legacies while almost all the place names on the Isle of Lewis are Norse in origin. Some names recall how the Viking sea lords saw Scotland as a landmass articulated by a series of sea roads. The place name Tarbert \u2013 there is a Tarbert in Kintyre and another between Loch Lomond and Loch Long (and the Firth of Clyde) \u2013 means 'an overbringing', a place where crews trundled their longships overland, probably by pulling them on log rollers.\n\n**Panel stitched by:**\n\n_Gaels Stitchers_\n\nSheila Forrest\n\nHeather Forrest\n\nCatherine Harrison\n\nMadalene Lee\n\nAnn MacGilp\n\nMaureen McKellar\n\nMaureen Robinson\n\nMargaret Smith\n\nMary Swift\n\n**Stitched in:**\n\nDunoon\n\n# PANEL 17 Dumbarton Rock\n\nIn the summer of 870, lookouts on the ancient fortress of Dumbarton Rock saw a bone-chilling sight. Rounding the headland and gliding into the upper Clyde were the sails of more than two hundred Viking dragon ships. Standing in the prows of the most splendid were sea lords and the Norse kings of York and Dublin. They had come to lay siege to the Rock of the Clyde, Alcluith, the seat of the Old-Welsh-speaking kings of Strathclyde, Ystrad Clud. After four months of bitter attrition, the well on the Rock ran dry and the garrison was forced to surrender. Instead of slaughter, the Strathclyde aristocracy faced slavery. The Vikings carried on a widespread and lucrative trade and elite captives will have fetched high prices at the slave market in Dublin. King Artgal appears to have evaded capture but he had the greater misfortune to fall into the hands of King Constantine of Alba or Scotland, the son of Kenneth MacAlpin. The last independent king of Strathclyde was executed. The siege and the failure of its dynasty appeared to spell the end of the ancient kingdom as a separate polity but its name and crown lingered in the records for more than a century after Artgal's death. At the Battle of Carham in 1018, there was a final flourish. King Owain of Strathclyde rode south with his war band to join the forces of Malcolm II of Scotland in an attack on the Angles of Northumbria. But, by the middle of the 11th century, no more is heard of the rulers who held court on the spectacular Rock of the Clyde.\n\n**Panel stitched by:**\n\n_Artgal's Rocks_\n\nLouise Foster\n\nAshley Holdsworth\n\nPatricia Livingston\n\nSarah Muir\n\nJulie Robertson\n\n**Stitched in:**\n\nGlasgow, Edinburgh\n\n# PANEL 18  Constantine Climbs the Hill of Faith\n\nThe unrelenting pressure exerted by Viking sea lords in the Hebrides and along the Atlantic shore helped push the focus of an emerging Scottish kingship eastwards. Scone became important and, when Constantine succeeded to the throne, he and his Bishop of St Andrews, Cellach, made an important public declaration:\n\n[They] pledged themselves upon the Hill of Faith near the royal city of Scone, that the laws and disciplines of the Faith, and the rights in churches and gospels, should be kept in conformity with [the customs of] the Scots.\n\nConstantine may have been the first to call himself King of Alba, still the Gaelic name for Scotland. This choice and the clear will to mould a national church after a Scottish\/Gaelic model can be seen as an explicit rejection of Pictishness \u2013 Alba not Pictavia \u2013 but it can also be interpreted as a declaration of unity, a conscious and public attempt to weld together disparate parts into a new kingdom. In the 10th century, Scotland spoke the dialects of at least six languages \u2013 Norse, Gaelic, Pictish, Old Welsh, Scots and, for the cultured, Latin. The name of Scotia for the lands north of the Forth was current in the 11th century but it was not until the late Middle Ages that Scotland began to be widely used for the whole nation. But it may be said that the idea of Scotland was much in Constantine's mind as he and Cellach climbed the Hill of Faith at Scone.\n\n**Panel stitched by:**\n\n_Constant Stitchers_\n\nKatie Antonio\n\nHelen Huxley\n\nDorothy Maingot\n\n**Stitched in:**\n\nPerth, Glencarse\n\n# PANEL 19  Carham\n\nIn the summer of 1018, on the banks of the River Tweed, Scottish axemen swung their weapons to brutal effect as they cut a Northumbrian war band to pieces. Led by Malcolm II and King Owain of Strathclyde, the Scots broke the ranks of the Northumbrian spearmen of Uhtred, Earl of Northumbria. Carham is now a tiny, sleepy hamlet on the English bank of the river but, to the emerging Scottish kings, it was a place of great significance. The Tweed Valley had been part of the kingdom and earldom for three centuries, it spoke English and shared a common culture, from matters of religion to agriculture. But Malcolm's axemen chopped through these ancient ties to make an emphatic statement \u2013 that the Tweed Valley was now part of the kingdom of Scotland. English chroniclers attached much less importance to the battle. Some indeed believed that this fertile swathe of territory had been granted to Kenneth II of Scotland by the English King Edgar in 981. But Malcolm II may not only have been fighting a force mustered from the lands south of the Tweed, he may also have been asserting himself over people to the north who still saw themselves as Anglian. It took more than a century before the Tweed Basin thought of itself as not English and it may be that Carham was the bloody beginning of a transition.\n\n**Panel stitched by:**\n\nJane Cole\n\nVal Fairbairn\n\nMig Moore\n\nCarolyn Scott\n\nIsa Scott\n\nMargaret Waller\n\nMoira Wilson\n\n**Stitched in:**\n\nTweedbank, Galashiels\n\n# PANEL 20  King Macbeth\n\nThe famous play by William Shakespeare may be seen as five acts of unrelenting defamation. Guilty of none of the foul crimes enacted on stages all over the world, Macbeth was, in reality, a competent monarch whose kingdom was sufficiently peaceful for him to go on pilgrimage to Rome. There, he was said to have scattered money like seed and no doubt sought papal forgiveness for only the most routine of sins. Almost never given his royal rank, Macbeth was a king twice over. From the early 1030s, he was King of Moray, an ancient polity around the shores of the Moray Firth, and, when King Duncan of Alba invaded in 1040, he was killed \u2013 in battle not in bed. Aside from resistance from Crinan (lay abbot of Dunkeld), the Abbot of Iona and young (not old) Duncan's father, King Macbeth of Alba ruled unchallenged and legitimately for 14 years. In 1054, Siward, Earl of Northumbria, raided deep into Scotland and began a process of destabilisation which ended in King Macbeth's death in 1057. He was succeeded by his stepson, Lulach. Almost erased from the regnal lists, King Lulach reigned for only a year before being deposed by Malcolm III. Far from being the ruthless, wicked, witch-obsessed tyrant of the Scottish play, King Macbeth was lauded as 'the renowned' after his death and, in _The Prophecy of Berchan_ , the author sang of 'the generous king' and 'the red, tall, golden-haired one, he will be pleasant to me among them; Scotland will be brimful west and east during the reign of the furious red one'. King Macbeth was famous as _Ri Deircc_ , 'the Red King'. Perhaps his red hair was traduced into red blood. Whaur's yer Willie Shakespeare noo?\n\n**Panel stitched by:**\n\nSandra Leith\n\nGlennie Leith\n\nIngrid McGown\n\nPaddy McGruer\n\nRhea Scott\n\n**Stitched in:**\n\nMoray, Portsoy\n\n# PANEL 21  St Margaret of Scotland\n\nBorn in Hungary, a princess of the English royal house of Wessex, Margaret fled to Scotland in 1066 after the Norman Conquest. Four year later, she married Malcolm III Canmore (not because of his head, big or otherwise; the nickname was, in fact, a title for, in Gaelic, _Ceann Mhor_ means something like 'Great Leader\/Ruler'). She was a paragon of piety and fertility. Canonised in 1250, she deserved more immediate recognition for mothering no fewer than five sons and two daughters, all of whom lived to adulthood. This ensured that the MacMalcolm dynasty would not lack heirs. Three of Margaret's sons reigned as kings of Scotland \u2013 Edgar, Alexander and David \u2013 but it is perhaps for her faith that St Margaret will be remembered. Pilgrimage was becoming increasingly popular in 11th-century Scotland and the queen laid plans to make it easier to visit the great shrine of St Andrew. A ferry for pilgrims ran from South Queensferry to North Queensferry, where the road and rail bridges now stand, and, at North Berwick, a longer crossing could be made to Earlsferry in the East Neuk. In addition, Queen Margaret also worked to bring the Scottish church closer to the papacy in Rome but also did not ignore the sacred sites of the Celtic church when she ensured that Iona's monastery was restored. Many churches (and an excellent university in Edinburgh) are dedicated to her, the oldest being St Margaret's Chapel in Edinburgh Castle, the most venerable building in the city. It was endowed and founded by her equally pious son, David I. Margaret's other signal achievement was to have been one of the few women to take a leading role in our early history.\n\n**Panel stitched by:**\n\n_St Margaret's Stitchers_\n\nBrenda Borrows\n\nRebecca Brown\n\nVonny Burke\n\nBeryl Butcher\n\nSally Clark\n\nAgnes Mabon\n\nKate McDonald\n\n**Stitched in:**\n\nDalgety Bay, Aberdour, Dunfermline, Cairneyhill\n\n# PANEL 22  The Flowers of the Borders\n\nThe immense power of religious belief is sometimes underestimated in modern, secular Scotland. The landscape of the 12th-century Border country was punctuated by its products. Eight hundred years ago, it would have looked entirely natural to our eyes. Almost all buildings were made from wood and roofed with bracken or turf. Only the smoke of cooking fires would have drawn our attention. The tallest landmarks on the horizon were trees, roads were no more than narrow tracks and the only substantial man-made structures were the stockaded motte and bailey fortresses being thrown up by local lords \u2013 and they were few. But, in four places in the Borders, within a radius of little more than ten miles, piety raised up four mighty churches. Awe-inspiring and vast, the abbeys at Kelso, Jedburgh, Melrose and Dryburgh were a stunning architectural response to a profound depth of belief. Inspired by David I and all founded in the first half of the 12th century, communities of reformed monks came to the south of Scotland to build churches and monasteries. The first and perhaps the most significant were the Tironensians from Picardy in France. After a false start near Selkirk, they moved to a bend in the River Tweed at Kelso. There the construction of a vast cathedral-scale church began. With a double crossing and a collection of conventual buildings around it, Kelso Abbey was a wonder of the age. Now, only part of the west end still stands but it is a truly monumental fragment. At Jedburgh, most of the nave survives and Melrose has a roof as it carried on being used as a parish church. But perhaps the prettiest and most atmospheric of all is Dryburgh, the last resting place of Sir Walter Scott and Earl Haig. The abbeys were lavishly patronised and sometimes wealthy and anxious noblemen who knew they were dying gave gifts that involved them becoming novice monks, thus qualifying for burial inside the sacred precincts. This taking of holy orders was _ad succurrundum_ , done in a hurry. They believed that the soil of the precinct would cleanse their bodies of sin.\n\n**Panel stitched by:**\n\n_Scottish Borders Embroiderers' Guild_\n\nAlison Delaney\n\nCatherine Edmundson\n\nBarbara Farquhar\n\nSusan Gray\n\nAli Halley\n\nCathy McCulloch\n\nJenni Young\n\n**Stitched in:**\n\nGalashiels, Hawick, St Boswells, Jedburgh, Melrose\n\n# PANEL 23  David I and the Wool Trade\n\nProsperity as well as piety raised the Border abbeys, St Andrews and Glasgow Cathedrals and the other great churches of early medieval Scotland. Much wealth was accumulated by the rapid development of the wool trade. As well-organised and semi-industrial textile production intensified in Flanders and Italy, the demand for raw wool grew. And, in the relatively benign climate of the 12th century, the monasteries in the Borders ran huge sheep ranches in the Cheviots (and there were others in the likes of the Lammermuirs \u2013 the Lambs' Moors) and a busy inland market was established at the royal burgh of Roxburgh. This enormously important medieval town has been entirely effaced; not one stone has been left standing upon another, a casualty of centuries of border warfare with England. But it saw the beginning of urban life in Scotland. Once the spring clip had taken place and the fleeces had been baled into woolpacks, sold at Roxburgh and either carted or possibly floated on rafts down the Tweed to Berwick, they were loaded at the quays on to ships bound for the Low Countries and further afield. Berwick became a very wealthy town, the location of groups of foreign merchants and the largest contributor of customs revenues to the crown. David I and his successors created royal burghs to stimulate trade and bring Scotland into the mainstream of the European economy. Until the Wars of Independence destroyed Roxburgh and ultimately severed the arterial link with Berwick, Scotland prospered on the backs of its sheep.\n\n**Panel stitched by:**\n\n_Broomlands\/Roxburghshire WRI_\n\nKathleen Binnie\n\nElizabeth Bruce\n\nVal Horsburgh\n\nCathy Simpson\n\nJoan Turnbull\n\nVal Van Der Reijden\n\nHazel Woodsell\n\n**Stitched in:**\n\nKelso\n\n# PANEL 24  St Andrews Cathedral\n\nOnly two shrines in Western Europe housed the relics of the Apostles, men who had known Jesus. One was St James at Santiago de Compostela in Northern Spain and the other was St Andrew at St Andrews. According to legend, St Regulus, a Greek monk from Patras, brought three finger bones of the saint's right hand, the upper bone of an arm, a kneecap and a tooth. By 1070, these had been housed in St Regulus' Church, now in the precincts of St Andrews Cathedral, its 33-metre-high tower a famous landmark. By the 13th century, and probably earlier, St Andrew was seen as the patron saint of Scotland and the possession of his relics was a powerful buttress for the emergence of a national church. He had been crucified on the crux decussata, the diagonals of the St Andrews Cross, the saltire flag of Scotland. In the middle of the prosperous 12th century, the construction of a grand cathedral to hold the relics was begun and it continued for more than a century. It was finally dedicated in 1318 in the presence of King Robert the Bruce. The town plan of St Andrews reflected the cathedral's central importance with North and South Streets laid out as processional ways leading to and from the West Door. By the time of the Scottish Reformation in the middle of the 16th century, the great church was stripped of its altars and images and the relics of St Andrew lost. Stone robbers removed much of the fabric and now the east gable stands almost alone as the most impressive reminder of the cathedral's medieval grandeur.\n\n**Panel stitched by:**\n\n_Hill of Tarvit Textile Conservation Volunteers NTS_\n\nChristeen Anderson\n\nSoan Cairns\n\nAlison Docherty\n\nUrsula Doherty\n\nAnne Halford McLeod\n\nMargaret May\n\nAnn Miller\n\n**Stitched in:**\n\nCupar, Auchtermuchty, Dairsie, Upper Largo, Newport on Tay, Ceres\n\n# PANEL 25  Duns Scotus and the Schoolmen\n\nThe local and correct pronunciation of the Berwickshire town of Duns is 'dunce' and the name may not be mocked for it arose in its other usage from the philosophy of one of the greatest intellectuals of the Middle Ages. Probably born in or near the town c. 1266, John Duns (or Dunce) Scotus was ordained in the priesthood in the Order of the Friars Minor in Northampton. They were popularly known as the Franciscans. Towards the end of 1302, John began teaching at the University of Paris. In his copious and hugely influential writings, he produced elegant arguments for the existence of God, the reality of the Immaculate Conception and much else. Scotus's work was widely admired and his followers were sometimes known as Schoolmen or Scholastics. His work remains important and, in the 20th century, there was a significant resurgence of interest and admiration. Philosophers such as Peter King, Gyula Klima, Paul Vincent Spade and others acknowledged his immense contribution to western thought. But, in the 16th century, John Duns Scotus's reputation was much dented. When his supporters opposed the work of Renaissance humanists and their fascination with classical \u2013 and pagan \u2013 philosophers such as Aristotle and Plato, and in particular the creation of the King James Bible, they were derided as Dunsmen or Dunces \u2013 people incapable of good scholarship. But modern dunces should be much consoled by their close association with one of Scotland's greatest minds and one of our most beautiful counties.\n\n**Panel stitched by:**\n\n_Embroiderers' Guild Kelso Branch_\n\nAnn Bacon\n\nMary Bonsor\n\nElizabeth Cuthbert\n\nStephanie Dempsey\n\nKay Gardiner\n\nColina Harris-Burland\n\nTricia Marshall\n\nAlison Minter\n\nMargaret Mitchell\n\nPat Nicol\n\nTricia Reynolds\n\nAnita Rhind\n\nRineke Sangster\n\nAnne White\n\nHelen Williams\n\n**Stitched in:**\n\nColdstream, Tweedmouth, Hawick, Ednam, Gordon, Kelso, Duns\n\n# PANEL 26  Somerled, Lord of the Isles\n\nTechnological change often drives politics in unexpected directions. The Hebridean warship known as the birlinn developed from the Viking _dreki_ , the 'dragon ships'. _Naibhig_ in Gaelic, meaning 'the little ships', they were smaller than the longships and much more manoeuvrable as a result not only of scale but a simple innovation. The pilots of the _dreki_ used a steer board (the derivation of starboard because the steer board was usually on the right-hand side) attached to the side of the ship but the birlinns were fitted with a hinged rudder fixed to the keel and therefore in the centre. Over the shallow and rocky coastal waters of the Hebrides, these fast and nimble little ships could go places larger boats with a deeper draught dared not. The birlinns were central to the power of Somerled. Sometimes known as Somerled the Viking, his name is from Sumar-lidi or Summer Raider. Rising to prominence through a mixture of conquest and judicious marriage, he established himself as Lord of the Isles, ruling over an Atlantic principality that included many of the islands of the Southern Hebrides and Argyll. In 1156, Somerled and his captains won a great naval battle against Godred Olafsson and by 1158 he had seized Godred's kingdom of Man and the Northern Hebrides. But Somerled's ambitions were not satisfied. In 1164, he sailed a huge army up the Clyde to attack Malcolm IV's kingdom of Scotland. The expedition ended in disaster, Somerled died, the Islesmen retreated and his vast territories broke into smaller lordships. He is seen as the progenitor of Clan Donald and his descendants succeeded to the lordship until the late Middle Ages.\n\n**Panel stitched by:**\n\nKatherine MacLean\n\n**Stitched in:**\n\nLochaber\n\n# PANEL 27  Haakon at Kyleakin\n\nIn the summer of 1263, King Haakon of Norway mustered a huge fleet. Alexander II and his son, Alexander III, were actively attempting to incorporate the Atlantic coastlands and islands into their kingdom of Scotland. The gathering of the vast Norwegian fleet after it had crossed the North Sea must have been a memorable sight for it is remembered in a well-known place name. Kyleakin on the Isle of Skye is from the Kyles or Narrows of Haakon, the place where the king's fleet dropped anchor, gathered intelligence and made plans. By October, the fleet had moved south to occupy the Firth of Clyde, using the shelter of the Cumbraes as an anchorage and a source of water and supplies. Negotiations opened but soon failed and, on a stormy night, several Norwegian ships were driven onshore at Largs. A Scottish force arrived and fierce fighting followed on the beach. It was inconclusive but nevertheless the season and the weather were against Haakon, October being late in the year for a large fleet to be at sea, and they set sail for Norway. Three years later, the Treaty of Perth was agreed and the ancient dispute over the sovereignty of the Hebrides, the Atlantic shore and the Isle of Man was settled in Scotland's favour. But Orkney and Shetland were to remain outside the realm of the MacMalcolm kings and their heirs for another two centuries.\n\n**Panel stitched by:**\n\n_South Skye Stitchers_\n\nMargaret Scott\n\nMargaret Govier\n\nLaila Hall\n\nAnn Hickey\n\nJulie Mace\n\nEmma Morrison\n\nFlora Struthers\n\n**Stitched in:**\n\nIsle of Skye\n\n# PANEL 28  The Death of Alexander III\n\nThe night of 19 March 1286 was stormy. Strong winds whipped up the spindrift off the Firth of Forth, rain spattered the royal feasting hall at Edinburgh Castle. But King Alexander III of Scotland was unabashed \u2013 he was celebrating his marriage to the beautiful Yolande de Dreux and, perhaps flushed with wine, he was determined to share her bed that very night. The problem was that his new bride was at the royal manor at Kinghorn, across the storm-tossed Forth. Despite entreaties from his anxious courtiers, Alexander called for his horse to be saddled, rode to South Queensferry and crossed safely. Perhaps all would be well. But, somewhere along the cliff path to Kinghorn, the king became detached from his retinue and his horse probably lost its footing in the wind and the dark. The following morning search parties found Alexander III's body on the beach, at the foot of a steep and rocky slope. His neck was broken. Scotland was immediately plunged into a dynastic crisis. All three of the king's children had died and the heir presumptive was Margaret, the Maid of Norway, his granddaughter. She died before she reached Scotland. Alexander's rashness was to cost his realm dear. After the wind and rain of 19 March 1286, much greater storms burst over Scotland as the kingdom without a king began the nightmare of the Wars of Independence.\n\n**Panel stitched by:**\n\n_Fab Four Fifers on the Forth_\n\nDorothy Balfour\n\nJean Boath\n\nDilys Campbell\n\nChris Fair\n\n**Stitched in:**\n\nCairneyhill, Dalgety Bay, Aberdour\n\n# PANEL 29  William Wallace and Andrew Moray\n\nAfter the death of Alexander III, what became known as 'The Great Cause' developed momentum as several noble families claimed and contended for the vacant throne. In 1293, Edward I of England judged that John Baliol had the strongest claim but he immediately began to manipulate the new king. When Baliol renounced his homage to Edward, the English immediately invaded and forced the King of Scots to abdicate in a humiliating ceremony where the royal arms were ripped off his surcoat. Edward is said to have commented, 'A man does good business when he rids himself of a turd.' In 1296, Andrew Moray led a rebellion in the north and he was soon joined by a force commanded by William Wallace, a minor nobleman from Ayrshire. His name is written as Le Waleis and may have denoted a family who spoke Old Welsh, the ancient tongue of Strathclyde. At Stirling Bridge, the allies scored a stunning victory in 1297. By allowing only part of a much larger English army to cross the narrow bridge before attacking them, they caused chaos. Retreating English soldiers were driven into the ranks of those behind, the bridge broke down and many drowned. After the battle, Wallace and Moray were proclaimed Guardians of Scotland but it seems that Moray died of his wounds soon afterwards. A year later, fortunes reversed as the English triumphed at Falkirk and Wallace spent seven years evading capture. But, in 1305, he was betrayed and taken to London to suffer the appalling agonies of a traitor's death. Having been dragged naked through the streets, he was hanged and, while still alive, emasculated and eviscerated before being beheaded. Knowing what his fate would be, Wallace asserted at his trial that, since he had never been the subject of Edward I, he could be no traitor. It is a stirring story of defiance to the last.\n\n**Panel stitched by:**\n\n_Perth Embroiderers' Guild_\n\nLorna Morrison\n\nGladys Anderson\n\nDorothy Lewin\n\nHeather Moir\n\nAnthea Pawley\n\nMichelle Peet\n\nJanice Reid\n\nWendy Rosier\n\nMary Ross\n\nPat Scales\n\nLoretta Whitcomb\n\nJeanette Yates\n\nMargaret Young\n\n**Stitched in:**\n\nPerth, Blairgowrie, Methven, Scone\n\n# PANEL 30  Bannockburn\n\nIn the Middle Ages and long before, Stirling was the pivot of Scotland. To the west of the great castle rock, treacherous Flanders Moss stretched and it forced those armies that wished to march north to cross the River Forth below the castle walls. In 1314, Stirling was held by an English garrison besieged by the Scots. Edward II mustered a huge army of more than 2,000 armoured knights, the medieval equivalent of modern tanks, and 16,000 infantry and archers. The English king was looking to confront the man who had had himself crowned King of Scots, Robert the Bruce. His army was probably less than half the size of Edward's but he had the immense advantage of choosing the battleground. He would move his army around it brilliantly. And he forced the squadrons of English heavy cavalry to ride over boggy and tussocky ground, probably at the Carse of Balquiderock, about a mile and a half east of the traditional site. Over two days, a remarkable conflict took place. It began with a famous single combat when Sir Henry de Bohun charged with his lance levelled at King Robert \u2013 who flicked the reins of his pony to turn him aside at the last moment and with a backhanded cut, felled the English knight in an instant. A roar went up from the Scottish ranks. It was an omen. Formed up in flexible, mobile squares bristling with spears, the Scots attacked the English heavy cavalry as it struggled on the soft terrain. The squadrons broke and the first day ended with momentum swinging to Bruce. Disarray and disagreement in the English ranks allowed the Scottish squares, or schiltrons, to drive deep into the mass of the English army. Archers could not fire for fear of hitting their own men. Buckling under intense pressure, they retreated and were then routed. Edward fled and the slaughter began. It was an unlikely, telling and famous victory.\n\n**Panel stitched by:**\n\n_Two Toxophilists_\n\nCaroline M Buchanan\n\nMargaret Martin\n\n**Stitched in:**\n\nFalkirk, Stirling\n\n# PANEL 31  The Rain at Carlisle\n\nFlushed with victory at Bannockburn, King Robert led an invasion of England and, in the spring of 1315, laid siege to the old Roman city of Carlisle and its dour castle. Heroic resistance was directed by Sir Andrew de Harcla, a local lord who was created Earl of Carlisle for his efforts. But the more compelling reason for the city's survival was not dogged defence \u2013 it was rain. That spring, unusually heavy and persistent rain fell all over Europe and, around Carlisle's stout walls, Bruce's siege engines became bogged down and were easily toppled in the soaking ground, his army encampment was washed out and little food could be commandeered locally. It was the beginning of the Little Ice Age, a series of periods of bad and cold weather that was to grip Europe for more than five hundred years. The rain not only drove Bruce from Carlisle in 1315, it continued into 1316 and there was widespread famine across Europe. Crops failed, bread was hugely expensive, no winter forage could be cut and herds and flocks had to be slaughtered. By the summer of 1317, the rain had relented but hunger had depleted and weakened the population. In Scotland, between 10 and 25 per cent of the population had died and food production did not return to normal until the mid 1320s. Medieval governments simply could not deal with the crisis caused by climate change and famine and criminal activity became common as people were forced to steal to live. But it was religious faith that may have been rocked most severely. Prayer and intercession had no effect and the 14th century saw the rise of new beliefs across Europe \u2013 what were immediately cast as heretical sects.\n\n**Panel stitched by:**\n\n_Galloway Broderers_\n\nAnne Ackerley\n\nLorraine Challis\n\nHelen Keating\n\nJane McCandlish\n\nLinda Murtough\n\nMarion Owen\n\nSusie Seed\n\nMargaret Surplice\n\nRuth Williams\n\nwith stitches by Children of Kirkbean Primary School\n\n**Stitched in:**\n\nGatehouse-of-Fleet, Kirkbean, Castle Douglas, Melrose, Carsphair\n\n# PANEL 32  The Black Death\n\nAt Caddonfoot near Galashiels, a Scottish army gathered for a raid into England. Their captains saw what they called 'the foul death of the English' \u2013 the devastating arrival of the Black Death in the summer of 1348 \u2013 as an opportunity. Believing that this English disease would not affect them, the army attacked Durham and brought the contagion back north. By 1350, it was raging through towns, villages and farm places. Originating in China, the Black Death swept west to Europe and spread quickly to England where it killed between 30 and 50 per cent of the inhabitants \u2013 between 3 and 5 million people. Those figures may have been a little less disastrous in Scotland but not by much. The plague was carried by fleas and infected human beings through contact with rats. Also known as bubonic plague, it attacked the lymph glands to cause swelling and then made its deadly way into the bloodstream. A variant was pneumonic plague and it was transmitted through the air rather than by touch. Death was rapid \u2013 only three or four days after the symptoms appeared. The consequences of such a pandemic were dramatic and long lasting. A shortage of farm labour drove up wages but caused output to decline, while the death of many priests through contact with their parishioners contributed to a slackening in the certainties of belief. But, in one way, the Black Death did benefit Scotland. In 1346, two years before its arrival in England, Edward III's army had defeated the Scots at the Battle of Neville's Cross, near Durham and the young king seemed set to realise his grandfather's dream of subjugating Scotland. Disease diverted him and the danger of English domination passed.\n\n**Panel stitched by:**\n\nMichael Blacklock\n\nCatherine Hamilton\n\nLiz Sanderson\n\n**Stitched in:**\n\nMelrose, St Boswells\n\n# PANEL 33  The University of St Andrews\n\nThe oldest in Scotland and the third oldest in the English-speaking world, the university at St Andrews was formally founded by papal bull in 1413. Bishop Henry Wardlaw, commemorated in an imposing new statue near St Mary's College, had licensed a small group of teachers and students two years previously but the blessing of the papacy allowed degrees to be awarded. By the middle of the 16th century, the town, castle and university became embroiled in the Scottish Reformation and Patrick Hamilton and George Wishart were burned at the stake. By the 18th century, the university had declined and, when Samuel Johnson visited in 1773, there were only a hundred or so students. A century later, St Andrews was much strengthened by the incorporation of University College Dundee with its bias towards science and, in 1894, Agnes Blackadder graduated, becoming the first woman in Scotland to gain a similar degree to those awarded to men. Dundee became a separate university in 1967 and St Andrews was so anxious to increase its numbers in 1968 that the entrance qualifications were so low as to admit a number of unsuitable undergraduates. In 1413, Lawrence of Lindores was made rector of the university, the first in an unbroken line of distinguished rectors.\n\n**Panel stitched by:**\n\nElizabeth Bracher\n\nThelma Grieg\n\nKate Scorgie\n\n**Stitched in:**\n\nKirkcaldy, Burntisland, Kilconquhar\n\n# PANEL 34  The Ancient Universities\n\nWhile England languished with only two universities, at Oxford and Cambridge, until the 18th century, Scotland had four \u2013 at St Andrews, Glasgow, Aberdeen and Edinburgh. Indeed, for a time Aberdeen had two university colleges. The historical effects of this imbalance have long been obvious. But the impulse behind these foundations is less clear. One historian has described the creation of the ancient Scottish universities as nothing less than a national programme for higher learning. But the output of graduates was small. St Andrews and Glasgow, founded in 1451, were producing perhaps only 30 graduates a year. The need to train clerics for the Reformation church certainly accelerated growth in the second half of the 16th century but what lay behind these remarkable foundations may have been something like a sense of the value of education \u2013 something that is still discernible, just, in modern Scotland. The pity is that it is now both difficult and very expensive for young people from modest backgrounds to gain entry to university and to taste something of one of Scotland's greatest cultural and historical glories.\n\n**Panel stitched by:**\n\nWendy Ewart\n\nLindsey Fraser\n\nVivian French\n\nLesley Kerr\n\nElizabeth Laird\n\nDavid McDowall\n\nVikki Reilly\n\nAnna Renz\n\nKathryn Ross\n\nJan Rutherford\n\nLiz Short\n\nGill Small\n\nEleanor Updale\n\nLesley Winton\n\nAnna Winton\n\n**Stitched in:**\n\nWest Linton, Aberdour, Edinburgh\n\n# PANEL 35  Orkney, Shetland and Scotland\n\nSince the Vikings began to sail 'westoversea' from Scandinavia in the eighth century, Orkney and Shetland had been under their control. Powerful Earls of Orkney such as the wonderfully named Sigurd the Stout sometimes posed a threat to the mainland. Kenneth II could do nothing to prevent Sigurd annexing Caithness and extending his reach as far south as the Moray Firth by the end of the 10th century. It was not until the 15th century that what became known as 'the matter of Norway' was resolved. In 1468, a Scottish embassy was despatched to negotiate the marriage of Margaret, the daughter of King Christian I of Denmark and Norway, to James III of Scotland. Terms were agreed and they brought to an end a long-running dispute over the payment of an annual tribute to the Scottish crown for the Western and Northern Isles. Until the full amount of Princess Margaret's dowry was paid, all of the historic rights and lands of the kings of Norway and Denmark in Orkney and Shetland were to be ceded to the Scottish crown. The dowry was never paid and, in 1472, the Northern Isles formally became part of the kingdom of Scotland. At last, the nation's frontiers reached their modern extent. And the Scandinavian character of the archipelagos began to change. The language of Norn had died out by the end of the 19th century and recent DNA studies have shown the extent of a Scottish takeover since the late 15th century. Amongst men with old Orkney surnames, such as Flett, Foubister and Linklater, Viking DNA markers are found in 35 per cent while, in the general population, it declines to only 20 per cent.\n\n**Panel stitched by:**\n\n_Shetland Needleworkers_\n\nSheila Peterson\n\nPatricia Brown\n\nHelen Burgess\n\nRita Fraser\n\nMaureen Harkness\n\nMali Hewamanage\n\nSylvia Jamieson\n\nJuliet Nicolson\n\n**Stitched in:**\n\nShetland\n\n# PANEL 36  Rosslyn Chapel\n\nThis gorgeously decorated and much visited chapel was founded by William Sinclair, the Earl of Orkney and Caithness, in the middle of the 15th century. Only the choir of what was intended as a family church was completed. It stands on a small hill near the dramatically sited Sinclair Castle at Roslin. The sculptural decoration of the interior is dazzling and no expense appears to have been spared in making one of the most beautiful churches in Scotland. But, after the Reformation, the chapel was closed and only reopened in 1861 under the aegis of the Scottish Episcopal Church. Many writers have attempted to make links between Rosslyn Chapel and freemasonry and the Knights Templar but none of these interpretations have so far appeared convincing. Famously, the novelist Dan Brown brought his protagonists in his _Da Vinci Code_ to a denouement at Rosslyn and part of the subsequent film, starring Tom Hanks, was shot at the chapel \u2013 all of which has helped bring many thousands of visitors to an otherwise quiet corner of Midlothian.\n\n**Panel stitched by:**\n\n_The Apprentice Stitchers_\n\nFiona McIntosh\n\nAnne Beedie\n\nMargaret Humphries\n\nJean Lindsay\n\nJinty Murray\n\nPhilippa Peat\n\nBarbara Stokes\n\n**Stitched in:**\n\nRoslin\n\n# PANEL 37  Chepman and Myllar\n\nIn 1507, a royal licence was granted to Walter Chepman and Andrew Myllar to print books in Scotland. For what was essentially a monopoly, the licence specified that these should be 'bukis of our lawis, actis of parliament, croniclis, mess bukis, and portuus efter the use of our Realme, with addiciouns and legendis of Scottis sanctis' and the King would decide what reasonable prices for these books should be. This seems to have been a conscious attempt at modernisation and even nation building. Sadly, the press only lasted two or three years but it did mark the beginning of a long, honourable and occasionally profitable tradition of printing and publishing in Edinburgh. Set up in the Cowgate, Chepman and Myllar did have some success and it may be that their edition of Blind Harry's _The Wallace_ was a best-seller. With the _Aberdeen Breviary_ , a compilation of the rituals and ceremonies of the Scottish Church intended to show how different they were from the English, the output of the new press is tinged with propaganda. But, by 1600, others had set up in business and their catalogues brimmed with popular works in Scots as well as grammars and Latin texts.\n\n**Panel stitched by:**\n\n_The Bobbin Stitchers_\n\nAnne Mackinnon\n\nMeriel Tilling\n\n**Stitched in:**\n\nLoanhead, Dalkeith\n\n# PANEL 38  Blind Harry\n\nThe full title of the epic poem composed by the minstrel known as Blind Harry is _The Actes and Deidis of the Illustre and Vallyeant Campioun Schir William Wallace_. It tells the tale of the great hero and royal accounts record that Harry sang 'a ballad' for King James IV on 2 January 1492 accompanied by two Gaelic clarsach players. More of a historical novel rather than a history, _The Wallace_ played powerfully to patriotic sentiment and, at twelve volumes, it could not fail to make an impression. Robert Burns admired Blind Harry but others, such as the historian John Major, attacked him for inventing battles and fictional episodes in William Wallace's life. Perhaps the greatest debt is owed by Hollywood and Mel Gibson's memorable film _Braveheart_. Also ahistorical and inventive, it nevertheless excited great patriotic sentiment. Gibson also knew, as no doubt did Blind Harry, that it was principally entertainment and, when the film premiered at the Macrobert Arts Centre in the University of Stirling in 1995, the bulb in the projector blew. The star of the film leapt up onstage in his kilt and suggested he tap-danced while repairs were made. But _Braveheart_ and Blind Harry's great poem both succeeded in showing just that \u2013 Wallace's extraordinary bravery.\n\n**Panel stitched by:**\n\n_Sangstream Stitchers_\n\nAnne Rowe\n\nJane Angel\n\nSheila Capewell\n\nNancy Davis\n\nShauna Dickson\n\nKate Frame\n\nMaureen Morris\n\nMarion Mullins\n\nKatharine Proudfoot\n\nGill Simpson\n\nDonna Watt\n\nHelen Wyllie\n\n**Stitched in:**\n\nEdinburgh\n\n# PANEL 39  Waulking\n\nFor many centuries, urine was not wasted in the West Highlands or the Hebrides. It was needed for the process known as waulking cloth. Two processes were involved. The first involved scouring or removing oils and impurities from the cloth and that was where urine came in. The ammonia it contains helped in cleaning cloth. The second stage was the thickening of what had been cleaned and in Gaelic culture this gave rise to a particular music. The groups of women who sat around a table pounding and moving a bolt of cloth around would sing a very rhythmic song so that everyone worked at the same pace and in unison. Individual verses are usually sung by one person and then all join in the chorus. This last is often a string of nonsense words, the Gaelic equivalent of tra-la-la, but they can be very beguiling, almost hypnotic. ' _Coisich a Ruin_ ', 'Come on, my love', is one of the most famous and it played to a much wider audience through the crystal voice of Karen Matheson when the band Capercaillie released a version that climbed into the UK music charts. Waulking songs are now only performed for the sake of the music since waulking itself began to be done mechanically in the 1950s.\n\n**Panel stitched by:**\n\n_Wester Ross Waulkers_\n\nLiza Adam\n\nMelinda Christmas\n\nLennie Cole\n\nIsobyl la Croix\n\nFiona Macintyre\n\n**Stitched in:**\n\nGairloch\n\n# PANEL 40  Flodden\n\nThe reign of James IV, often seen as Scotland's first renaissance monarch, is also remembered for military disaster. In 1513, the king was killed on Flodden Field as he led the downhill charge of a large Scottish army against an English force under the command of the Earl of Surrey. It was an impulsive and foolhardy action since it removed any ability to command. Pinned in the ruck of hand-to-hand combat, James could see nothing of the course of the battle and could not affect matters once the armies had joined. By contrast, the old warrior, Surrey, stayed at the rear and was mounted so that he had some sense of where he needed to reinforce, retreat or push forward. When dawn broke on the morning after the battle, the landscape of hell was revealed. On the gently undulating northern ridge of Branxton Hill, more than 10,000 men lay dead or dying. Through a long dark night the battlefield had not been a silent graveyard. Trapped under lifeless comrades, crippled, hamstrung or horribly mutilated, fatally wounded men still breathed. Bladed weapons rarely kill outright and they were often used to bludgeon men to their knees or into unconsciousness. Some will have been put out of their misery by parties of English soldiers scouring the field by torchlight for plunder, others will have bled to death, maimed, lacerated by vicious cuts, screaming, fainting and screaming once more in their death agonies. The slaughter was unprecedented, especially thinning the ranks of the Scottish nobility, but losses were also great on the English side and no invasion of Scotland followed immediately. But Flodden ushered in a century of instability in the south \u2013 the age of the Border Reivers.\n\n**Panel stitched by:**\n\n_Flodden Embroiderers_\n\nHelen De Le Mar\n\nCaroline Proctor\n\nDiane Skene\n\nBelinda Trustram Eve\n\n**Stitched in:**\n\nTongue\n\n# PANEL 41  _The_ _Thrie_ _Estaitis_\n\nWritten by Sir David Lyndsay, this satire on the three estates of Scotland \u2013 the clergy, the nobility and the burgesses \u2013 was first performed in the open air outside Cupar in 1552. In essence an attack on these interest groups, the play was written against a background of rising tension in Scotland as the Reformation developed momentum. The Scots language used by Lyndsay is often ripe and the text contains what may be the first appearance of the word 'fuck'. It is a robust piece in another sense because its form still entertains \u2013 a production of _The Thrie Estaitis_ formed the dramatic centrepiece of the 1948 Edinburgh Festival and it was adapted in 1996 by John McGrath who cast Sylvester McCoy in a major role.\n\n**Panel stitched by:**\n\n_The Schroders_\n\nLynne Schroder\n\nJim Schroder\n\n**Stitched in:**\n\nCallander\n\n# PANEL 42  The Court of Session\n\nIn 1532, the Scottish Parliament passed an act to establish the College of Justice and it may be seen as the founding moment of the institutions of Scots law \u2013 the point at which it was made formally clear that it was different from English law. This was not the culmination of a process. Scots and English law during the Middle Ages had been largely similar. It was the outcome of a political deal. After the disaster at Flodden, James V was broke and he appealed to Pope Paul III for a portion of the Scottish church revenue that went to Rome. A sizable cut \u2013 about 10,000 gold ducats \u2013 was agreed but on one condition \u2013 Scotland had to set up a College of Justice based on the principles of canon law, complete with senators and other Roman paraphernalia. As the Reformation began to take significant hold in northern Europe, Paul III wanted to create as many institutional links as possible with Scotland. Surprisingly, these newfangled innovations survived the Scottish Reformation and, from that point on, they began to grow away from the principles of English common law. And so, for advocates, procurators fiscal and all of the other lucrative differences, we have a cash-strapped king and an astute pope to thank. When people remark that Scots law is based on Roman law, as though it adds some spurious dignity and antiquity, they should remember that it was in effect canon law. And the cost? A snip at 10,000 ducats.\n\n**Panel stitched by:**\n\nMhairi MacDonald-Greig\n\n**Stitched in:**\n\nEdinburgh\n\n# PANEL 43  The Scottish Reformation\n\nCompared with the tremendous upheavals in Europe and the blood that was spilled in France and Germany in particular, the Reformation in Scotland was relatively peaceful and quick. While there was conflict and martyrdom on both sides, a formal break with the papacy and the Catholic Church was achieved quickly. After the death of Mary of Guise, the Catholic mother of Mary, Queen of Scots, the so-called Reformation Parliament passed acts in 1560 abolishing the old faith. _The First Book of Discipline_ , partly written by John Knox, set education as a priority. So that the mass of people could read the Bible, the Word of God, for themselves, there would be a school in every parish. This took much sacrifice and many years but it was eventually achieved. This set literacy in Scotland at uniquely high levels and it would form the basis of many achievements to come. But the new kirk was impoverished as secular lords and the Crown grabbed as much of the property of the old church, of the monasteries and convents as it could. The patrimony of the Dukes of Roxburgh, for example, reflects much of the estate of the Abbey of Kelso. The reformed church received only a sixth of the income of the old and even by 1562 there were only 257 ministers for 1,067 kirks. The balance was made up by more than 600 readers. John Knox is often seen as the embodiment of the Scottish Reformation as well as a by-word for dour joylessness. In reality, he was a brilliant man and a great preacher possessed of immense courage. He survived two years as a galley slave, chained to a rowing bench, to come back to Scotland to lead the forces for change.\n\n**Panel stitched by:**\n\n_St. Thomas's_\n\nGladys Bennett\n\nAudrey Axon\n\nAlison Bogie\n\nSue Brown\n\nIrene Brydon\n\nFiona Campbell\n\nWilma Finlayson\n\nMark Gilmour\n\nKay Hush\n\nVerity Macfarlane\n\nMargaret Mitchell\n\nAgnes Murray\n\nBelinda Petherick-Kerr\n\nMhairi Taylor\n\nChris Young\n\n**Stitched in:**\n\nEdinburgh\n\n# PANEL 44  Mary, Queen of Scots\n\nFeckless, impulsive, romantic, beautiful, deceitful \u2013 adjectives that could all be applied to Mary, Queen of Scots. One of half a dozen instantly recognisable figures from Scottish history, she lived, reigned and was executed at a time of tremendous change \u2013 and virtually every move she made was the wrong move. The only legitimate child of James V, she succeeded to the throne at the tender age of six days. Having spent most of her childhood in France, she married the Dauphin, Francis, and, when he briefly became king, she was queen consort. On his death in 1560, Mary made plans to return to her realm of Scotland and, in 1565, she married her cousin, Henry Stuart, Lord Darnley. Two years later, he was murdered and his house at Kirk o' Field in Edinburgh blown up \u2013 probably by James Hepburn, Earl of Bothwell with whom Mary fell in love and married. After an uprising against her, Mary was forced to abdicate in favour of her son, James VI. At which point in this soap opera, she fled south to seek the protection of her cousin, Elizabeth I of England. Even though Mary had claimed her throne and was a Catholic. She was placed under stately home arrest and after eighteen and a half years in captivity, Mary was beheaded for plotting against Elizabeth. Feckless, impulsive, romantic, beautiful deceitful \u2013 and probably very unlucky, her 45 years were certainly eventful but there is a sense that the history of Scotland happened without or despite her.\n\n**Panel stitched by:**\n\n_Hopetoun Tapestry Conservation Volunteers_\n\nMorag Austin\n\nMargot Baird\n\nJane Beth Brown\n\nSallie Bryson\n\nMartha Creasey\n\nAlison Docherty\n\nCharlotte Docherty\n\nRos Duffy\n\nGillian Gyte\n\nMaureen Johnson\n\nBeryl Johnston\n\nHelen Kelly\n\nIsobel Potts\n\nRita Poulter\n\n**Stitched in:**\n\nHopetoun, Edinburgh, Lancashire\n\n# PANEL 45  The Reivers and the Rescue of Kinmont Willie\n\nFor almost a century after Flodden, royal authority was either weak or remote on either side of the border and a thoroughly criminalised society came into being \u2013 what is known as the age of the Border Reivers. What mattered was not the rule of law but family loyalty and, as armies crossed and recrossed the Tweed Valley or took the western route to Dumfriesshire, theft became an easier way of life than farming. Kinmont Willie Armstrong was typical. A ruthless, violent thief, he changed sides, did deals and survived. But, at a truce day in 1596, he was captured and imprisoned in Carlisle Castle. This act was in contravention of the rules of truce where those who attended cross-border hearings on complaints were immune from arrest. But the English warden, Lord Scrope, ignored this. Walter Scott of Buccleuch, on whose land Armstrong had been captured, organised a rescue. It was to be the last great raid and it was brilliantly successful \u2013 a superb example of the reivers' skills. It was the last raid because the world was about to change. Queen Elizabeth of England was old and without an heir. James of Scotland had been assured that he would succeed and no longer would thieves be able to hop over the border and play off one jurisdiction against another. The days of the Border Reivers were coming to a welcome close.\n\n**Panel stitched by:**\n\n_Smailholm Stitchers_\n\nIsabel Atkinson\n\nAvril Blown\n\nFiona Brown\n\nDenise Hunter\n\nDerrick Jowett\n\nRobyn Kinsman Blake\n\nSusan Mason\n\nVeronica Ross\n\nSally Scott Aiton\n\nMargaret Shaw\n\nMargaret Skea\n\nCatherine Tees\n\n**Stitched in:**\n\nSmailholm\n\n# PANEL 46  Robert Carey's Great Ride\n\nOn 24 March 1603, Sir Robert Carey began a journey he hoped would bring him favour and gain. He attended the dying Queen Elizabeth in her last days and when the old queen expired, his sister, Philadelphia, pulled a ring from her finger. It had been a gift from James VI of Scotland. She gave it to her brother who rode like the wind to Edinburgh to give James the news he had been waiting for for almost 20 years. Carey had organised a relay of post horses and he reached Berwick after only 48 hours. But somewhere north of the town he fell off his horse and, while he lay on the ground, it kicked him. Spattered with blood and mud, he clattered into the inner courtyard of Holyrood-house and demanded that King James be woken. When Carey gave him the old queen's ring, he knew that his life and the history of his nation had turned in a new direction. The Union of the Crowns would surely lead to ever-closer ties. But James's new courtiers in London were not impressed with Carey. His conduct was condemned as 'contrary to all decency, good manners and respect' and he was dismissed from the post James had given him as a Gentleman of the Bedchamber. But he persisted and was eventually made Earl of Monmouth by Charles I in 1626. He wrote a memoir which became an excellent source for the story of the Border Reivers and its second edition was annotated by Sir Walter Scott.\n\n**Panel stitched by:**\n\n_Ageing Well Edinburgh_\n\nRobina Brown\n\nCaroline M Buchanan\n\nLinda Garcia\n\nLaura Kempton Smith\n\nKath Laing\n\nSheila McFarland\n\nSheila Miller\n\nMary Stewart\n\nRuth Watson\n\n**Stitched in:**\n\nEdinburgh\n\n# PANEL 47  The Making of the King James Bible\n\nBegun in 1604 and completed seven years later, what is known as the King James Bible was one of the greatest achievements of his reign. Completed by 47 scholars, it set new standards of accuracy and is often hailed as the most influential piece of literature in history. Also known as the Authorised Version, it was intended to reflect the structure and particular beliefs of the Church of England. But, in truth, the brilliance of the translation from Greek for the New Testament and from Hebrew for the Old Testament has transcended that original purpose. Spelling has been modernised but the glory of the language is unaltered. Here are the first three verses from 1 Corinthians 13:\n\nThough I speak with the tongues of men and of angels, and have not charity, I am become as a sounding brass, or a tinkling cymbal.\n\nAnd though I have the gift of prophecy and understand all mysteries, and all knowledge, and though I have all faith, so that I could move mountains, and have not charity, I am nothing.\n\nAnd though I bestow all my goods to feed the poor, and though I give my body to be burned, and have not charity, it profits me nothing.\n\nVast numbers of phrases in common use come from the King James Bible as well as the text of the Lord's Prayer and the glorious translation that is Genesis 1 to 11.\n\n**Panel stitched by:**\n\n_Heirs of 1843_\n\nFiona Anderson\n\nMarie Austin\n\nWinifred Cumming\n\nLilias Finlay\n\nMary Godden\n\nNan Laird\n\nDorothy MacKenzie\n\nJean Mackinlay\n\nChristine MacPhail\n\nDeborah Miller\n\nElizabeth Mitchell\n\nMaggie Morley\n\nJean Morrison\n\nJudith Pickles\n\nMaggie Romanis\n\nwith stitches from 30 members of St Andrew's and St George's West Church, Edinburgh\n\n**Stitched in:**\n\nEdinburgh\n\n# PANEL 48  The Dawn of the Ulster Scots\n\nIn 1606 two ambitious Ayrshiremen launched a venture that would change the course of Irish and British history. Hugh Montgomery of Braidstane, and his near-neighbour James Hamilton from Dunlop, set up Scotland's first 'colony', across the North Channel in Ulster.\n\nMontgomery and Hamilton both had connections at the court of King James VI\/I in London. In 1605, through opportunism and influence, they managed to dispossess the Irish chieftain Con O'Neill of two-thirds of his huge landholdings in Down and Antrim. O'Neill was facing execution for alleged rebellion against the crown, and in exchange for a pardon agreed that Montgomery and Hamilton would each take one-third of his land \u2013 to be settled by Scots loyal to King James.\n\nThe following year thousands of Scots, mainly Protestant farming families from the southwest, began moving to eastern Ulster to inhabit and cultivate the lands given up by O'Neill. They and their descendants became the Ulster Scots. The success of the Hamilton\/Montgomery Settlement directly inspired two further ventures blessed by King James: the Jamestown colony founded in Virginia in 1607, and the Protestant Plantation of the rest of Ulster beginning in 1610. Ulster-Scots later contributed mightily as migrants to many other countries; for example, seventeen Presidents of the United States have had Ulster Scots, or 'Scotch-Irish', ancestry.\n\n**Panel stitched by:**\n\n_Crewel Chicks 'n' Dave_\n\nKate Edmunds\n\nShona McManus\n\nElizabeth Raymond\n\nMary Richardson\n\nDave Richardson\n\n**Stitched in:**\n\nTranent, Cockenzie, Dunbar\n\n# PANEL 49  Witches\n\nOne of the greatest disfigurements in Scotland's history, one which banishes objectivity, was the pursuit and appalling torture and murder of women (and some men) as witches. In 1563 the interest in witchcraft stirred when the Reformation Parliament passed an act outlawing it on pain of death. By the 1590s, witch trials had begun and Agnes Sampson of North Berwick was dragged before James VI and some of his nobles at Holyrood to be interrogated. She denied all charges and was sent back to prison 'there to receive such torture as has been lately provided for witches'. Agnes was hanged by the neck, choking and vomiting for an hour before finally confessing. Her crimes were trivial and probably linked to the native tradition of folk healing and natural remedies. But the enthusiasm for such dreadful treatment was connected to the notion that Scotland was a godly commonwealth and that witches polluted it. Burning was the most common form of execution because it was seen as the most effective method of extirpating witches and witchcraft. Many women and men were strangled before fires were lit around them but others were burned alive. In Edinburgh in 1608, a group of perhaps eight women were tied to stakes to be burned alive. The fires burned through the bonds of three who ran out of the inferno with horrific burns. And the crowd threw them back in.\n\n**Panel stitched by:**\n\n_The Coven_\n\nJill Brennan\n\nVal Cowan\n\nJoan Doig\n\nEmma Mackenzie\n\nElizabeth Smith\n\n**Stitched in:**\n\nNorth Berwick, Athelstaneford, East Saltoun\n\n# PANEL 50  The National Covenant\n\nSigned in Greyfriars Kirkyard in 1638, this remarkable document was, in essence, a covenant between God and Scotland or Christ's Kingdom of Scotland. Many noblemen and thousands of relatively humble people attached their names, pledging themselves to defend their rights to a national church. It was a response to the efforts of Charles I to bring the church in Scotland into conformity with the Church of England. Written by Archibald Johnston and Alexander Henderson, it demanded a Scottish Parliament and a General Assembly free from royal interference and the abolition of bishops in the kirk. The Covenanters raised an army and, in what are known as the Bishops' Wars, they defeated Royalist forces. These conflicts sparked the War of the Three Kingdoms, what is usually miscalled the English Civil War. Until 1650 and the arrival of Oliver Cromwell's New Model Army, the Covenanters effectively ruled Scotland. When Charles II was restored, the Killing Times began. Covenanters were hunted down and persecuted. Unable to congregate in towns, they held field conventicles \u2013 open-air services \u2013 and sometimes thousands gathered. Covenanters were powerful in Galloway but gradually, with some lenience on offer from James II, these fiercely held beliefs died away and by the end of the 17th century only tiny sects were active.\n\n**Panel stitched by:**\n\n_3 in EH3_\n\nMhairi MacDonald-Greig\n\nElizabeth Mason\n\nMargo Mason\n\nBlanka Peters\n\n**Stitched in:**\n\nEdinburgh\n\n# PANEL 51  Droving\n\nBefore the transport revolutions of the 19th and 20th centuries, animals walked to market. Sheep, even geese (who were made to walk through tar and then sand so that their feet could cope with a journey) and other animals were driven to be sold but by far the greatest traffic was in cattle. The Highland economy in particular bred a tough, small breed that was in great demand to supply salt beef to the British army as the drive for empire got under way in the 18th century. Notably, drovers from Skye swam their cattle across the straits from Kylerhea to Glenelg, tying them nose to tail, before herding them through the mountain passes and down to the market known as the Falkirk Trysts. In Highland glens, particularly brilliant patches of green grass show where the stances were, the overnight stops of the herds moving through the mountains. Each summer, beasts would stop, graze and muck the ground of the stances before moving on. Rob Roy MacGregor was perhaps the most famous, or notorious, drover and it was the cattle business which caused him to become an outlaw. When the railways penetrated Scotland and the North in the later 19th century, droving came to an end. But the roads where the beasts walked can still be seen.\n\n**Panel stitched by:**\n\n_The Highland Stitchers_\n\nBarbara Campbell\n\nAgnes Greig\n\nPenny Stevenson\n\nPat Thornton\n\n**Stitched in:**\n\nMuir of Ord, Culloden Moor, Beauly\n\n# PANEL 52  Philiphaugh, 1645\n\nHaving fought a brilliant campaign in the Highlands, James Graham, the Marquis of Montrose, the Royalist commander in Scotland, found himself at Philiphaugh, near Selkirk. He was pursued by a Covenanter army led by a brilliant soldier, General David Leslie, and, on 13 September 1645, plans were laid. It was misty and Montrose's scouts failed to report the advance of the enemy. The first he knew of an attack was the sound of gunfire and he arrived to find his army in some confusion. But the Royalist musketeers managed to beat off the initial assault. What Montrose did not realise was that Leslie had divided his forces and sent his cavalry along the Hartwoodburn and, screened by Howden Hill, they arrived at the battlefield and attacked the rear of the Royalist army. Brave and even reckless, Montrose charged 2,000 Covenanter dragoons with only 100 cavalry. His captains urged him to flee, telling their general that the king's cause in Scotland would be lost without him. And so, with 30 men, he cut his way out. But the cause was lost anyway and Montrose would suffer an ignominious death in Edinburgh on 21 May 1650, hanged on the Burgh Muir with a favourable biography round his neck. It was a sad end for a brilliant military tactician, a daring commander, one of several to serve the undeserving Stewart dynasty.\n\n**Panel stitched by:**\n\n_Ettrick & Yarrow Stitchers_\n\nAlison Blackadder\n\nAnn Hardie\n\nSandra Howat\n\nMargaret Robinson\n\n**Stitched in:**\n\nSelkirk\n\n# PANEL 53  The Killing Times\n\nIn the 1680s, the conflict between Covenanters and the government forces of Charles II and James II and VII reached its bloody crescendo. Driven out of towns and villages by powerful disapproval and the attentions of patrolling dragoons, Covenanting congregations took to the hills so that they could worship in peace. These services were known as field conventicles and sometimes thousands attended. At Irongray, near Dumfries, a huge conventicle was protected from the dragoons by a company of armed sympathisers. The Covenanting resistance to the restored Stewart monarchy's efforts to establish the Church of Scotland they wanted was at its stiffest in the south-west. At Wigtown a remarkable martyrdom took place. Passionate Covenanters, two young girls, Margaret Wilson and her sister, took to the hills to hear services and avoid persecution. But Margaret was arrested, imprisoned and, with an older woman, condemned to die a cruel death. In Wigtown Bay, they were tied to stakes at low tide and left to drown as the tide came in. As efforts to extirpate their interpretation of the Protestant faith intensified in the 1680s, Covenanters were killed or executed. Preachers were forced to go about the countryside in disguise and one of the more unlikely attempts at anonymity was the mask allegedly worn by Alexander Peden. More likely to draw attention than deflect it, the mask appears nevertheless to have been effective. In at least two places in the south of Scotland, there are rocks known as Peden's Pulpit. By the end of the 17th century, Covenanting zeal was waning \u2013 but, in contemporary Scotland, some of that thrawn spirit mercifully lives on.\n\n**Panel stitched by:**\n\n_EH12 group_\n\nAlison Bruce, Lady Marnoch\n\nVivienne Cameron\n\nJennifer Harding-Edgar\n\nSue Kerridge\n\nSteve Shillito\n\nLeona Thomas\n\nChris Young\n\nwith stitches from some members of the EAL and other ASL services at East Neighbourhood Centre, Edinburgh\n\n**Stitched in:**\n\nEdinburgh\n\n# PANEL 54  The Massacre at Glencoe, 1692\n\nWith the departure of the Stewarts in 1688, William of Orange and his advisors kept a weather eye on the Highlands, already simmering as an area of continued support for what would be called Jacobitism. Rebellion and a victory at Killiecrankie (where the Jacobite general, John Graham, 'Bonnie Dundee', was killed) had not been followed through and resistance finally fizzled out in 1690. These events had persuaded the London government to demand oaths of allegiance in return for a pardon for any part Highland chiefs may have taken in the uprising. Many sought permission from the exiled James VII and II and, typically, he dithered. Messages of agreement were finally sent to the Highlands in the middle of December 1691. Winter snows held up Alastair MacIain, the 12th chief of the MacDonalds of Glencoe and he was late in swearing allegiance. A plot began to form. The sworn enemy of the MacDonalds, John Campbell, Earl of Breadalbane, was in London and he persuaded the Lord Advocate, John Dalrymple, the Master of Stair, that MacIain's oath was both late and not properly sworn. Stair then convinced King William to sign an order for the execution of a den of thieves in Glencoe. The massacre now had the royal seal of approval and orders were despatched north. In late January 1692, Robert Campbell of Glenlyon led 120 men into the glen where they were billeted with families. Two weeks later, probably with the arrival of fresh orders carried by a Captain Drummond, the killing suddenly began. MacIain was stabbed to death as he attempted to get out of bed and, in all, 38 men died as blades grew bloody and muskets fired. About 40 women and children subsequently perished of exposure in the snows after their houses were burned. It was a disgraceful episode but, 50 years later, much worse was to follow in the Highlands.\n\n**Panel stitched by:**\n\n_Ben Nevis_\n\nNorma Callison\n\nSandra Casey\n\nKaren Lees\n\nDaphne MacLean\n\nJohan Morton\n\nGillian Oram\n\n**Stitched in:**\n\nFort William, Roy Bridge\n\n# PANEL 55  The Bank of Scotland Founded\n\nIn 1695, a year after the Bank of England, the Bank of Scotland was established by an act of the Scottish Parliament. It was the first in Europe to issue its own banknotes and it continues to do so. Unlike its counterpart in London, the Bank of Scotland was founded to support and finance business enterprise. Its first Chief Accountant was George Watson and a leading director was John Holland, an Englishman. Suspected of sympathy for the exiled Stewart kings, the Bank found itself in competition with an upstart rival, the Royal Bank of Scotland. It was set up by royal charter in 1727. There followed 25 years of intense competition \u2013 what were known as the Bank Wars, an enmity that has never entirely faded. Into the 20th century, mergers and acquisitions took place. The Royal Bank took over the National Commercial Bank of Scotland in 1969 and the British Linen Bank became part of the Bank of Scotland in 1971. Both had issued their own banknotes and, sadly, these fell out of circulation. In 1999, rivalry ignited once more when the Bank of Scotland attempted to take over the much larger National Westminster Bank and this prompted a bid from the Royal Bank which was ultimately successful. Throughout three centuries of business, the major Scottish banks acquired a hard-won reputation for caution and probity, for steadiness and solidity. All of which exploded in 2008 when both institutions began to fall apart in a tangle of incompetence and mismanagement. Catastrophe and immediate closure were only just averted by the prompt action of the government in guaranteeing both banks' debts. But the reputational damage will take generations to repair.\n\n**Panel stitched by:**\n\n_Blacket Stitchers_\n\nAlison Cunningham\n\nAileen Gardiner\n\nJen McDowell\n\nRoz Preston\n\nSue Ross Stewart\n\nRie Stevenson\n\n**Stitched in:**\n\nEdinburgh\n\n# PANEL 56  The Darien Scheme\n\nAs Scotland saw its southern neighbour lay down the foundations of a great empire, the Company of Scotland was formed with the aim of fostering Scottish imperial projects. It raised a staggering sum, \u00a3400,000, about a fifth of the wealth of Scotland, in order to colonise part of the Isthmus of Panama, a place known as Darien. An original aim was to dig a canal to link the Atlantic and the Pacific. The first expedition of five ships set out from Leith in July 1698 with 1,200 people on board. Having dropped anchor in the Bay of Darien on 2 November, the settlers christened their new home Caledonia. But they were entirely unprepared for the stifling heat of the following summer, their supplies of food rotted and people began to die. The mortality rate reached ten a day. After almost a year, only 300 settlers had survived the tropical, insect-infested conditions and one ship limped back to Scotland with the terrible news. But it did not reach port in time to prevent a second expedition setting sail with a thousand hopefuls on board. Disease struck once more and the Spanish attacked the colonists' Fort St Andrew. The Darien Scheme had turned into a disaster. Of the 2,200 settlers who sailed from Scotland, only a few hundred survived and the nation lost a fortune. The episode laid a ground of failure and near-bankruptcy that pushed powerful individuals and interests closer to the idea of political \u2013 and economic \u2013 union with England.\n\n**Panel stitched by:**\n\n_Lasy Daisy_\n\nLiz Cameron\n\nLesley Evans\n\nMargaret Lyon\n\nSheena MacDonald\n\nIsobel Potts\n\nRita Poulter\n\nJean Thomson\n\n**Stitched in:**\n\nEdinburgh, Ceres\n\n# PANEL 57  The Act of Union, 1707\n\nThe Darien Disaster led indirectly to the Union of the Scottish and English Parliaments. It was claimed that union would help Scotland recover. A sum of cash, known as the 'Equivalent', was paid by the English Exchequer. The amount was \u00a3398,085, almost exactly equivalent to the \u00a3400,000 lost at Darien. It was viewed as compensation by many who had invested in the scheme. From an English point of view, the deal was straightforward \u2013 they wanted to ensure that a different monarch could not reign in Scotland. By contrast, there was tremendous opposition in Scotland, especially in Edinburgh where the novelist Daniel Defoe was operating as an English spy. 'For every Scot in favour, there is 99 against!' he reported to London. But despite its unpopularity, the terms of the Act of Union were agreed in 1706 and enacted the following year. The independence of the Church of Scotland (and with it, matters of education) and the legal system were guaranteed but, in most other senses, the union was comprehensive. In addition to a unified legislature in London, there was both a customs and a monetary union. It was a hostile merger but, by the middle of the 18th century, Scotland's economy was beginning to thrive and many Scots were able to make a career in the sprawling British Empire. After the death of Queen Anne in 1714, the last of the Stewarts had gone \u2013 but their ambitions were undimmed.\n\n**Panel stitched by:**\n\n_The Craiglockhart Crew_\n\nMargaret Beck\n\nMoira Davidson\n\nMary Gillespie\n\nMarysia Holmes\n\nMaureen Johnson\n\nMargot McDowall\n\nDorothy Morrison\n\nDeborah Pearce\n\nMargo Taylor\n\nAlison Wardlaw\n\nAnn Williams\n\n**Stitched in:**\n\nEdinburgh\n\n# PANEL 58  The Jacobite Rising of 1715\n\nWhen the Act of Union of 1707 applied the Act of Settlement and assured the succession of the Protestant House of Hanover, the exiled Stewarts were compelled to act. Queen Anne, the last representative of the dynasty, had died in 1714 and George I acceded. When James VII and II wrote to the Duke of Berwick in August 1715, he knew that history was waiting: 'I think it is now more than ever Now or Never.' But chronic indecision and poor communications made success ever more unlikely. Despite receiving no commission from the exiled king, John Erskine, the Earl of Mar, raised the standard of rebellion at Braemar on 27 August. Within weeks, his 12,000-strong army had control of the north and had captured Perth. Meanwhile, John Campbell, Duke of Argyll, mustered a much smaller force of 4,000 in support of George I. At Sheriffmuir, near Dunblane, the armies met and, with three times more soldiers, it looked a certain victory for Mar. But the battle was a botched, confused affair. Argyll's left wing was far shorter than the Jacobite right wing opposite them and should have been quickly outflanked, rolled up and defeated but orders were unclear. Argyll seized the initiative and, attacking the Highlanders' left wing, drove them back before turning to help his outnumbered left wing. Mar refused to commit his entire army in an all-out attack and Argyll claimed victory. A traditional song captured the confusion:\n\nThere's some say that we wan and some say that they wan\n\nAnd some say that nane wan at a', man,\n\nBut one thing is sure that at Sheriff Muir\n\nA battle was fought on that day, man,\n\nAnd we ran and they ran and they ran and we ran,\n\nAnd we ran and they ran awa' man.\n\n**Panel stitched by:**\n\n_Dunblane Group_\n\nJudith Abbott\n\nHeather Bovill\n\nCaroline B Buchanan\n\nMaud Crawford\n\nJenny Haldane\n\nMavis Oldham\n\nMary Storrar\n\nLysbeth Wilson\n\n**Stitched in:**\n\nDunblane, Stirling, Auchterarder\n\n# PANEL 59  The Kilt\n\nScottish weddings and other formal occasions now resemble a Highland Games. Within a generation, kilts \u2013 and, indeed, full tartan fig \u2013 have become de rigeur for grooms, best men and most guests. Fifty years ago Lowlanders would not have been seen dead in a kilt. The origins of all of this Highland splendour are, of course, disputed. The big kilt or plaid \u2013 a term used, confusingly, by Americans for tartan \u2013 was worn as a kilt in the modern sense but was also big enough to act as a cloak, hood or even a blanket. All was kept decent by a belt and the word kilt appears to derive from tucking in or kilting up a plaid round the waist. Many hypotheses have been advanced for the shrinking of the big kilt to the modern, small kilt. But the most likely, certainly the most entertaining, was that it was invented by an Englishman. Thomas Rawlinson was a Quaker from Lancashire who ran a charcoal burning and foundry business in Lochaber in the 1720s. Noticing that the big kilt encumbered his foresters as they felled trees and his foundry men as they smelted iron, he promoted the wearing of the small kilt. It was probably a natural development encouraged by the Englishman because illustrations of men wearing a version have been found to predate the 1720s. What was worn under the kilt remains a secret but occasionally, when Highlanders fought in hot weather, they cast off their plaids and charged in their shirts. All was on resplendent show at _Blar na Leinne_ , the Field of the Shirts, a clan battle between Frasers and McDonalds near Fort William in the summer of 1544. It is not known how long the shirts were.\n\n**Panel stitched by:**\n\n_Smailholm Group_\n\nIsabel Atkinson\n\nAvril Blown\n\nFiona Brown\n\nDenise Hunter\n\nDerrick Jowett\n\nRobyn Kinsman Blake\n\nSusan Mason\n\nVeronica Ross\n\nSally Scott Aiton\n\nMargaret Shaw\n\nMargaret Skea\n\nCatherine Tees\n\n**Stitched in:**\n\nSmailholm, Gordon\n\n# PANEL 60  The Jacobite Rising of 1745\n\nIn the summer of 1745, Bonnie Prince Charlie was rowed up Loch Shiel to join the muster of his Highland army. But only a handful of men greeted him. As hopes faded, the sound of the pipes of Clan Cameron was heard. About a thousand clansmen were led to Glenfinnan by Locheil and the rising had begun. Whatever faults he had \u2013 and they appear to have been many \u2013 Charles Edward Stuart had charisma and the surprise is not that he failed but that he very nearly succeeded. Unlike the Earl of Mar and the Old Pretender, the Prince and his generals moved forward with real intent. As clansmen flocked to the standard, they moved south quickly, reaching Edinburgh in September and taking the city without loss. At Prestonpans, the Highland charge tore into the terrified ranks of a government army commanded by Sir John Cope. The battle was a rout, lasting no more than 15 minutes and Prince Charles's victory sent shock waves throughout Britain. On 8 November, the Jacobites invaded England and took Carlisle. But, by the time they reached Derby, the three government armies in England were closing in. Even if the Highlanders won again, they would lose between 1,000 and 1,500 casualties and that would make a fighting retreat back to Scotland impossible. Meanwhile London was in uproar, the stock market was falling and nervous preparations were made. The argument for retreat prevailed and it led all the way to Culloden and defeat in April 1746. The Stewart cause was at last lost. It was the beginning of a long end for the clans and an appallingly punitive campaign of killings and rapes followed the departure of the Prince. He died drunk, dissolute and friendless in Italy.\n\n**Panel stitched by:**\n\n_EH41_\n\nKathleen Bain\n\nCandy Richardson\n\nAvis Moore\n\nMarjory Smith\n\nCindy Sykes\n\nwith stitches by 32 delegates to the Episcopalian Conference 'Ta See Oursels'\n\n**Stitched in:**\n\nHaddington\n\n# PANEL 61  The Ordnance Survey\n\nOne of the reasons the Jacobite army penetrated as far south as Derby in 1745 was the lack of good mapping and decent roads. The far superior government forces were unable to react, to move quickly enough to position themselves and their artillery in the path of a fast-moving Highland army. That is why the wonderfully detailed and accurate maps of Britain are called the Ordnance Survey. In 1747, Lt Col. David Watson proposed that reliable maps of the Scottish Highlands be made to ensure that rebellious clansmen could not simply disappear into an uncharted wilderness or use local geographical knowledge to tactical advantage. The resulting maps were scaled on one inch to a thousand yards. War, or the threat of war, stimulated the mapping of the South of England when William Roy, one of Watson's assistants in Scotland, began to work on Kent and Sussex. Fear of a French invasion was the prompt. By the 1820s, the first complete surveys of England and Wales were produced. The Ordnance Survey is an adornment to life in Britain, despite the recent abandonment of the green Pathfinder series, a set of superbly detailed maps easily handled out of doors. By comparison, maps of European countries are poor but, during the First World War, the Ordnance Survey began work on maps of Belgium, France and Italy. Would that they were more easily available.\n\n**Panel stitched by:**\n\n_Inverness & Area_\n\nJudie Holliday\n\nAnne Omand\n\n**Stitched in:**\n\nFortrose, Dochgarroch\n\n# PANEL 62  English Advances, Gaelic Retreats\n\nThe defeat at Culloden in 1746 was the beginning of a long end for the Highland clans, their culture and their language. Gaelic had crossed the North Channel from Ireland with the establishment of the kingdoms of Dalriada and, at its zenith, the language was spoken all over the Highlands and the Western Isles but, as the brutal aftermath of the Jacobite Risings merged into the Clearances, people left the Highlands with Gaelic, a language that they rarely passed on to their children. Now it is spoken by less than 1 per cent of all Scots. The original native tongue of Scotland was what might be best described as Old Welsh or Cumbric. Now entirely effaced, its only traces are to be found in the landscape, in natural features and place names. Peebles is from _pebyll_ , the Old Welsh word for tents and it probably meant a shieling. Altcluit meant the Rock of the Clyde, now Dumbarton Rock, and Penicuik is from _Pen y Cog_ , Cuckoo Hill. Early versions of English invaded with the Angles as they overran the Tweed Basin and the Lothians in the seventh century and, in the medieval period, it spread with the establishment of towns and trade. But, until the 19th century, Gaelic reached down the Perthshire, Angus and Aberdeenshire glens and it is likely that those who lived on either side of the linguistic frontier understood each other well enough. Now the fate of Gaelic is very perilous and, if it is lost, it will be all Scotland's loss.\n\n**Panel stitched by:**\n\n_In Stitches_\n\nJean Gowans\n\nTina Hammond\n\nCarolyn Irvine of Drum\n\nHelen Jackson\n\nDiana Munro\n\nMairi Skinner\n\n**Stitched in:**\n\nBanchory, Letham, Potarch\n\n# PANEL 63  The Royal and Ancient Golf Club\n\nGolf is a Scottish invention. All other claims are entirely insubstantial and at St Andrews and around Edinburgh the modern game began to take shape. The Royal and Ancient Golf Club of St Andrews is probably not the oldest in the world. That honour could probably be claimed by the Royal Burgess Golfing Society of Edinburgh which was founded almost 20 years before, in 1735. But the R&A is the most powerful because it makes the rules. But it does not own a golf course. The Old Course and the others around it are the property of the citizens of the town of St Andrews. Regarded as the home of golf, the links (i.e. seaside) courses are also closest to the game's origins. The strip of land behind the dunes of the West Sands, what is now the world-renowned Old Course and its sister courses, was too sandy for cultivation and was used for grazing. In the bitter winter winds, sheep scraped out bields or shelters and these sandy indentations became known as bunkers. The fairways of the Old Course were not laid out or created but are the well-mown folds of naturally undulating land. And the rough is very rough. On a calm summer's day, the Old Course seems to present few difficulties to the professionals who contend for the Open Championship. But, when the wind blows, it becomes evil, unfair, impossible \u2013 a conspiracy of nature and the R&A. In reality the Old is the greatest golf course in the world. Bar none.\n\n**Panel stitched by:**\n\n_Tee'd Off_\n\nPat Freeth\n\nJennifer Link\n\nEddie Link\n\nShona Morrison\n\nLis Smith\n\nSheila Tunstall-James\n\nMairi Wheeler\n\n**Stitched in:**\n\nKinross, Glenlomond, Auchtermuchty, Forgandenny, St Monans\n\n# PANEL 64  The First School for Deaf and Dumb Children\n\nThomas Braidwood founded the first school in Britain designed to teach deaf and dumb children. It was set up in a house in the Canongate in Edinburgh in 1780. Developing out of what he called the 'combined system', Braidwood codified the earliest version of British Sign Language. It was based on the use of the hands, the shapes they made, how they were turned, were placed and moved. C is made by curving the thumb and index finger into a semicircle and D by adding the straight index finger of the other hand to the semicircle. Deaf and dumb people talk in a mesmerising flurry of hand movements and British Sign Language has developed regional dialects. Some Scottish signs are not understood in the south of England. Braidwood's first pupil was Charles Sheriff, the son of Alexander Sheriff, a wealthy wine merchant working in Leith. The school tended to cater for those with the means to pay its fees but Joseph Watson, a relative of Braidwood, founded the first public school for the deaf and dumb. Based in Bermondsey in 1792, the London Asylum for the Deaf and Dumb was very successful. By 1783, Braidwood himself had moved to London to found a new school and his grandson set up a school in Virginia in the USA. Scots pioneered this field of education and Thomas Braidwood would have smiled to see British Sign Language recognised alongside English, Welsh, Gaelic and others as one of Britain's official languages and also to see that his original school in the Canongate is remembered in the nearby district of Edinburgh, the Dumbiedykes.\n\n**Panel stitched by:**\n\n_Cammo Quilters_\n\nKatherine Forsyth\n\nRosemary Gordon-Harvey\n\nAvril Green\n\nElizabeth Reekie\n\nGillian Swanson\n\nNorma Watkins\n\nCaroline Watson\n\n**Stitched in:**\n\nEdinburgh\n\n# PANEL 65  James Small and the Swing Plough, 1770\n\nScottish agriculture had long been hampered by poor technology. The Auld Scots Ploo was built mostly of wood tipped by iron and it took a team of four powerful oxen to pull it through the ground. It often broke down when it hit big stones or roots. And because it did not turn the sod completely, the auld ploo needed an army of plough followers to break up big clods and pull out weeds. In the 1770s, a Berwickshire blacksmith, James Small, perfected the swing plough. Cast all in iron at the Carron Ironworks, it had a screwed shape that turned the sod over completely, could be pulled by one strong horse and guided by one skilled man. Because it ploughed a deeper furrow, the swing plough improved drainage and brought more land into cultivation. Small was a perfectionist who worked on his prototypes endlessly and even spent time in prison as a debtor. He did not patent his design and consequently it was imitated very widely. This in turn accelerated the speed of change on the land and its adoption on the prairies of the USA and Canada made these regions into the breadbaskets of the world. Even now James Small is little known and his huge contribution to the modern world badly understood. He died in 1793 of overwork and in great poverty. The swing plough changed cultivation radically and, by doing that, it changed the world.\n\n**Panel stitched by:**\n\nChristine Covell\n\nSandra Douglas\n\nLinda Jobson\n\n**Stitched in:**\n\nPeebles, Eskbank\n\n# PANEL 66  Enlightenment Edinburgh\n\nPerhaps the most glittering period in Scotland's history was the second half of the 18th century \u2013 what is known as the Enlightenment. Between 1768 and 1771, Encyclopaedia Britannica was established in Edinburgh by William Smellie, a printer, editor and antiquary. At Anchor Close, just off the High Street, work began on setting down all human knowledge \u2013 it was an age when such a feat was thought possible, the organisation of all there is to know. The Encyclopaedia appeared in 100 weekly instalments and could be pithy. The entry for Woman was four words long \u2013 'the female of man'. But it proved popular and a second edition was soon put in train. It was published at a time of great intellectual ferment in Scotland and in Edinburgh in particular. An English visitor, a chemist called Amyat, left a famous observation, 'Here I stand at what is called the Cross of Edinburgh [the Mercat Cross in the High Street] and can, in a few minutes, take 50 men of genius and learning by the hand.' It was an intense environment. The medieval tenements of the Old Town piled people on top of each other and intellectual clubs met to discuss and dispute in the taverns off the High Street and the Canongate. The Select Society was founded by the painter Allan Ramsay, and the philosophers David Hume and Adam Smith. The Poker Club was intended to 'poke things up a bit'. Publishing was busy but this remarkable period of intellectual achievement was underpinned and understood by a society where, at 75 per cent, mass literacy was the highest in the world.\n\n**Panel stitched by:**\n\n_Shepherd House Group_\n\nJean Cameron\n\nAnn Fraser\n\nSarah Hynd\n\nMarianne More Gordon\n\nFrances Stevens\n\n**Stitched in:**\n\nInveresk, Musselburgh, North Berwick\n\n# PANEL 67  Edinburgh's New Town\n\nOvercrowding and building collapse forced the city council to extend Edinburgh's boundaries to include the farmland beyond the Nor Loch, an unpleasant body of water that lay at the northern foot of the Castle Rock. It is now Princes Street Gardens. A design competition for what would be known as the New Town was won by 26-year-old James Craig and his simple layout of a grid of three main streets linking two squares and crossed by five north-to-south thoroughfares is essentially what survives today. The street and square names reflected contemporary politics. George Square and St Andrew Square were to represent the union of the two nations. But someone remembered that there was already a George Square on the south side of the city and so Queen Charlotte was commemorated in Charlotte Square as well as Queen Street. And the royal princes were immortalised in Princes Street, although few can remember their names. Intended as a residential suburb, the New Town has a very pleasing symmetry. There were problems at each end in the south which were never quite resolved. Craig intended George Street to have a church in the squares at each end to finish the vistas but Sir Lawrence Dundas decided to build his house in St Andrew on the site intended for the church and St Andrew and St George's is in George Street. The connection with the Old Town down the Mound was not properly planned, resulting in the chicane into Hanover Street but perhaps the most glaring misfit is Queensferry Street and the way it is awkwardly attached to the south side of Charlotte Square and the west end of Princes Street. But the whole scheme is a wonderfully harmonious and concrete testament to the Edinburgh Enlightenment and a time when the city council could get things right.\n\n**Panel stitched by:**\n\nCatherine Harlick\n\n**Stitched in:**\n\nRhiconich\n\n# PANEL 68  James Watt and the Steam Engine\n\nBorn in Greenock in 1736, James Watt became a distinguished engineer and inventor \u2013 a Scot whose fame remains worldwide. But he did not invent the steam engine \u2013 rather, he made it work much more efficiently. While employed at Glasgow University as an instrument maker, Watt saw how wasteful of energy designs for steam engines were. They repeatedly cooled and reheated their cylinders. Instead, Watt invented the separate condenser and this hugely improved the power and efficiency of pumps in particular. And it made them cheaper to manufacture. When Watt conceived a means of producing rotary motion, the use of the steam engine broadened well beyond pumping. The brilliance of what he achieved could be seen on old steam trains where the huge iron wheels were very obviously driven by rotary power. In partnership with Matthew Boulton, Watt became a wealthy man and, throughout his life, he continued to work on inventions. Not only did he come up with the concept of horsepower as a means of measurement, the unit of electricity known as a watt is named after him.\n\n**Panel stitched by:**\n\n_Campsie Stitchers_\n\nNancy Bailey\n\nIsobel Shaw\n\nJacky Young\n\n**Stitched in:**\n\nGlasgow\n\n# PANEL 69  The Tobacco Lords\n\nThe Act of Union may have been deeply unpopular in Edinburgh but it converted Glasgow into a very busy port. Its position on the Clyde was enormously important in the age of sailing ships. The trade winds first hit Europe in the west of Scotland and this gave ships sailing out of Glasgow for the American colonies a two-to-three-week advantage. From 1710 until c. 1760, the city boomed because of the trade in tobacco. The wealthiest and most enterprising merchant was John Glassford and he ran a fleet of 25 merchant ships. Known as the Tobacco Lords or sometimes the Virginia Dons, Glasgow merchants built huge mansions that gave their names to the city's streets \u2013 Thomas Buchanan, Archibald Ingram and Glassford amongst them. The American Revolution of 1776 caused great difficulties. The Glasgow merchants had lent vast sums to the planters of Virginia and Maryland and, after the break with Britain, few of these debts were ever repaid. But, ever versatile, the Tobacco Lords switched their trade to cotton in the British West Indies. Their most lasting legacy is in Glasgow's Georgian architecture.\n\n**Panel stitched by:**\n\n_The Last Gasp Group_\n\nMarie Abbott\n\nJenny Barnett\n\nSheila Bruges\n\nElma Muir\n\nSally Nairn\n\nAnne North\n\nPamella Roberts\n\nHilda Stewart\n\n**Stitched in:**\n\nBlairgowrie, Perth, Dunkeld\n\n# PANEL 70  Adam Smith\n\nPerhaps the most influential thinker of the Scottish Enlightenment, Adam Smith was born in Kirkcaldy in 1723. The old harbour fascinated him as he watched ships load and unload at the quaysides. Having studied at Glasgow University and Oxford, he lectured at Edinburgh before being appointed Professor of Moral Philosophy at Glasgow. He is rightly renowned for two enormously influential pieces of scholarship. _The Theory of Moral Sentiments_ was published in 1759 but the work he is most famous for is _An Inquiry into the Nature and Causes of the Wealth of Nations_. It is considered the first modern treatment of economics and it has been enormously influential since its publication in 1776. Many scholars see Smith's dictum of the play of self-interest in economic life as being summarised in this famous sentence:\n\nIt is not from the benevolence of the butcher, the brewer or the baker, that we expect our dinner, but from their regard to their own interest. We address ourselves, not to their humanity but to their self-love, and never talk to them of our own necessities but of their advantages.\n\nAs with many great thinkers, Smith has been interpreted by politicians of many different persuasions but his clear-eyed thinking makes his work highly accessible almost 250 years after _The Wealth of Nations_ was first published.\n\n**Panel stitched by:**\n\nLouise Foster\n\nMeg Murray\n\nChristine Simm\n\nJean Taylor\n\nFiona Wemyss\n\nFiona Wilkie\n\n**Stitched in:**\n\nForgandenny, Edinburgh\n\n# PANEL 71  David Hume and Jean-Jacques Rousseau\n\nFamously decent and mild-tempered, the great philosopher of the Edinburgh Enlightenment David Hume was known as Le Bon David. This title was conferred on him in Paris where he acted as the undersecretary to the British ambassador, Lord Hertford, between 1763 and 1765. He was lionized for his work, writing to a fellow historian that 'I can only say that I eat nothing but ambrosia, drink nothing but nectar, breathe nothing but incense and tread on nothing but flowers'. This idyll was disturbed by the case of the philosopher, Jean-Jacques Rousseau. After the publication of his radical tract, _The Social Contract_ , with its famous first line, 'Man is born free and everywhere he is in chains', he was persecuted. Hume offered to take Rousseau to safety in England despite warnings that the refugee was not a man to be trusted. And they soon quarrelled. Rousseau suspected a plot against him and began to abuse him and draw friends and influential people into the row. In less than a year, their relationship had descended into mutual loathing. Such was Hume's fury that his reputation for moderation and goodness was severely endangered when he wrote of Rousseau as 'surely the blackest and most atrocious villain, beyond comparison, that now exists in the world'. Oh dear.\n\n**Panel stitched by:**\n\n_Coastal Stitchers_\n\nSheila Baird\n\nJean Dawson\n\nChristina Dougan\n\nAlison Sanchez-Ruiz\n\nHilary Williams\n\n**Stitched in:**\n\nAberlady, Ballencrieff, Gullane\n\n# PANEL 72  The Highland and Lowland Clearances Gather Pace\n\nIn the late 18th and throughout the 19th century, both the Highlands and Lowlands of Scotland slowly emptied of people. Landlords wanted to make profits from sheep or sporting estates and, in the Lowlands, smallholders were cleared as farms grew larger and more cost-effective. The growing cities of the Central Belt offered work for many but significant numbers reached the quaysides of Glasgow, Greenock and elsewhere and kept going, seeking new lives in North America, Australia, New Zealand and other developing countries. It may be that the Highland Clearances are better understood because of the actions of what seemed like a brutal aristocracy. There was resistance and, at the Battle of the Braes in Skye in 1882, crofters fought with the police. But the Napier Commission met and a reform of the law allowed security of tenure and limited the ability of landlords to evict families.\n\n**Panel stitched by:**\n\n_Links Needlework_\n\nSusie Alexander\n\nBetty Beazley\n\nIsabel Bryce\n\nMargaret Cormack\n\nHelen Deasy\n\nSheena Esson\n\nLiz Farmer\n\nEdwina Fraser\n\nLynn Fraser\n\nSheena Fraser\n\nCath Fraser\n\nMargaret Gowans\n\nAda Grant\n\nKathleen Grant\n\nMarion Hailstone\n\nSheila Hamilton\n\nCath Hay\n\nStephanie Hoyle\n\nCathy MacGillivray\n\nJanet Mackenzie\n\nAnne Marie Mackenzie\n\nMargaret MacLennan\n\nEvelyn Main\n\nMary McBean\n\nMairi Neilan\n\nEvelyn Reid\n\nAnn Sutherland\n\nDorothy Sutherland\n\nIngrid Wallace\n\nMary Williams\n\n**Stitched in:**\n\nNairn\n\n# PANEL 73  \nWeaving and Spinning\n\nFor countless generations, ordinary people had made their own yarn by spinning raw wool. But, by the Middle Ages and probably before, weaving had become a specialist craft. Looms were operated by men because of the strength needed in the process, while women generally made yarn on a spinning wheel. Individuals created their own designs but the undyed wool limited what those might be. The characteristic tartans with all their differences are really an invention of the 19th century. Most cloth was 'hodden grey' woven from home-reared wool and entirely utilitarian.\n\n**Panel stitched by:**\n\nJacqueline Walters\n\n**Stitched in:**\n\nThurso\n\n# PANEL 74  James Hutton's Theory of the Earth\n\nOne of the great figures of the Enlightenment, James Hutton, was born in Edinburgh in 1726. He was a farmer in Berwickshire and experimented with the new practices of the Agrarian Revolution. But, like many great scientists, Hutton was a superb observer. Looking at rock formations, and especially those at Siccar Point on the Berwickshire coastline, he became convinced that the Earth was an organism \u2013 that it was being perpetually formed by natural processes. In 1785, he coined the word superorganism for the Earth. These processes that ensured constant change included sedimentation, erosion and volcanic activity. Hutton was the originator of geology as a scientific discipline, cutting thought about the making of the Earth loose from religious dicta and tradition. But, like many men of the Enlightenment, he was a polymath. Interested in the project to build a Forth and Clyde canal, Hutton was a shareholder and site inspector as well as a campaigner for the relief of excise duty on the carrying of coal \u2013 the sort of traffic the canal might carry. He was also a partner in a chemical company that made ammonia from domestic soot and he dabbled in property development in Edinburgh. His biography describes the life of a busy intellect with an insatiable curiosity about the world.\n\n**Panel stitched by:**\n\n_Crewel Chicks 'n' Dave_\n\nKate Edmunds\n\nShona McManus\n\nElizabeth Raymond\n\nMary Richardson\n\nDave Richardson\n\n**Stitched in:**\n\nTranent, Cockenzie, Dunbar\n\n# PANEL 75  James Boswell and Smoked Fish\n\nIn the late 18th century, James Boswell gained a literary reputation as the biographer and companion of Dr Samuel Johnson but he also had something to say about Finnan haddies. Being very partial to the lightly smoked haddock from the village of Findon, near Aberdeen, Boswell was pleased to see them available in London but less happy when he tasted them. So that they made the sea journey unspoiled, the Finnan haddies had been much more heavily smoked and were, frankly, a bit tough. Arbroath smokies are the other famous product of the east coast. Also haddock, they are first salted overnight and then tied in pairs with hemp twine. Left to dry, they are hung over a triangular piece of wood and set over a fire of hardwood in a barrel. This is then sealed with a wet jute sack (so that it won't catch fire) and within only an hour, the haddocks are cooked. A beautiful rich brown, Arbroath smokies are a genuine delicacy \u2013 although not delicate but strong-tasting. Boswell's views are not recorded \u2013 nor do they matter a jot.\n\n**Panel stitched by:**\n\n_The Coburg Ladies_\n\nCatherine Aitken\n\nSarah McCabe\n\nMargaret McCabe\n\nAnnie Wright\n\n**Stitched in:**\n\nPrestonpans\n\n# PANEL 76  The Forth and Clyde Canal, Burke and Hare\n\nJames Hutton's investment in a canal linking the east and west of Scotland was worthwhile and, in the early 19th century, work began on digging a canal between Fountainbridge in Edinburgh and Port Dundas in Glasgow. The men doing the labouring work were known as navigators or navvies and many of these were Irish. The most notorious were William Burke and William Hare who committed a series of unusual murders in 1828, the details of which were very memorable. They sold the corpses of their 16 victims to Dr Robert Knox, an anatomist at Edinburgh University who needed fresh bodies for dissection. He charged fees for his lectures and, although he was never arrested or brought to trial, the Edinburgh mob and the press believed that Knox was complicit in the murders. When Burke was convicted and sentenced to death (Hare turned King's evidence and was granted immunity from prosecution), the judge specified that his body be publicly dissected. This famous case did not spark any anti-Irish rioting but it did spawn many films, TV programmes, plays, novels and songs.\n\n**Panel stitched by:**\n\n_Whippity Stouries_\n\nChristine Simm\n\nJean Taylor\n\nFiona Wilkie\n\n**Stitched in:**\n\nBo'ness\n\n# PANEL 77  Scotland and the Drive for Empire\n\nAs the British Empire expanded in the second half of the 18th century, Scots were at the forefront. Out of all proportion to the relative populations, there were many Scots serving in the British Army, especially Highlanders. Sorely in need of troops to prosecute the Seven Years War against the French, the Prime Minister, William Pitt, encouraged the recruitment of two regiments of clansmen. Justifying putting arms into the hands of men who had rebelled against the crown only ten years before, Pitt wrote to the king, saying:\n\nI sought for merit wherever it was to be found. It is my boast that I was the first minister who looked for it and found it in the mountains of the north. I called it forth and drew into your service a hardy and intrepid race of men . . . [T]hey served with fidelity as they fought with valour and conquered for you in every part of the world.\n\nThe last Highland charge was probably made at the battle for control of Canada on the Heights of Abraham. Much of the British Empire was bought by the blood of the clans. More peacefully, Scots became involved centrally in the East India Company, especially after it came under the control of Henry Dundas. Many Orcadians in particular worked in Canada for the Hudson's Bay Company as it opened up the vast interior of the north. And what cannot be ignored is the involvement of many Scots in the infamous triangular trade. Merchant ships took manufactured goods to West Africa where they were traded for slaves. They were then shipped in 'the middle passage' to the Caribbean and to the USA where so-called cash crops such as sugar, rum, molasses, tobacco, hemp and cotton were loaded for the return voyage to Britain.\n\n**Panel stitched by:**\n\n_The Meikle Family_\n\nGraham Meikle\n\nFiona Meikle\n\nAnnabel Meikle\n\n**Stitched in:**\n\nPathhead, Edinburgh\n\n# PANEL 78  Robert Owen and New Lanark\n\nA child of the Enlightenment, Robert Owen was a mill manager in Manchester who became interested in the welfare of his workers. During a visit to Glasgow, he fell in love with Caroline Dale, the daughter of David Dale, the owner of the mills at New Lanark. The mills there derived their power from the Falls of the Clyde. Owen persuaded his partners to buy New Lanark and he proceeded to make a series of reforms. First he opened a mill shop for his workers that sold quality goods at little more than wholesale prices and, when bulk buying created profits, these were passed on to customers. These principles became the basis for the Co-op. Owen also founded infant childcare at New Lanark and promoted the education of the children of his workers. After a time, Owen's mind turned to more political and philosophical activities. He began to lobby for changes in legislation but was unhappy with the Factory Act of 1819. Owen wrote _The Revolution in the Mind and Practice of the Human Race_ in which he argued that character is formed by nature and individual circumstances. By 1828, after some disagreements, Owen resigned all connection with New Lanark and, based in London, he formed the Association of all Classes of all Nations and it was he who coined the term 'socialism'.\n\n**Panel stitched by:**\n\n_New Lanark_\n\nLea Barrie\n\nJanice Glover\n\nNancy Howat\n\nTamara Jones\n\nJulia McMurray\n\nLesli Paterson\n\nMarjorie Romer\n\nPat Strong\n\nLiz Young\n\nwith stitches by members of the New Lanark Trust staff\n\n**Stitched in:**\n\nLanark, Glasgow\n\n# PANEL 79  Robert Burns and 'Tam o' Shanter'\n\nThe genius of Robert Burns lay in his ability to create poetry that is complex, tremendously beautiful and accessible to all. At Burns Suppers the world over, the recital, usually from memory, of 'Tam o' Shanter' is a highlight. Pure entertainment, it is probably one of the most rollicking, energetic poems ever composed. Written in 1790, the poem first appeared a year later in the _Edinburgh Magazine_ before being published in Francis Grose's _Antiquities of Scotland_. Burns asked Grose to include a drawing of Alloway Kirk and Grose agreed so long as the poet wrote something to go with it. The glorious result was 'Tam o' Shanter'. The opening is unmatched:\n\nWhen chapman billies leave the street,\n\nAnd drouthy neibors, neibors meet;\n\nAs market days are wearing late,\n\nAnd folk begin to tak the gate,\n\nWhile we sit bousing at the nappy,\n\nAn' getting fou and unco happy,\n\nWe think na on the lang Scots miles,\n\nThe mosses, waters, slaps and stiles,\n\nThat lie between us and our hame,\n\nWhere sits our sulky, sullen dame,\n\nGathering her brows like gathering storm,\n\nNursing her wrath to keep it warm.\n\n**Panel stitched by:**\n\n_Blister Sisters_\n\nMary Doherty\n\nMaureen Finlay\n\nMargaret Mitchell\n\nMargaret Potter\n\n**Stitched in:**\n\nEast Calder\n\n# PANEL 80  The False Alarm\n\nAs the French invasion force assembled at Boulogne in 1803\u201304, Napoleon itched to add Britain to his conquests but the weather was against him. Nevertheless, the threat was taken very seriously and nowhere more so than in the Scottish Borders. The old early warning system used in the days of English invasion was revived as balefires were piled up on hills and high points. If the French landed on the North Sea coast, a relay of beacons would be lit and the forces of volunteers recruited in the countryside would muster. Walter Scott, attached to the Midlothian Yeomanry Cavalry, was amongst these. On 31 January 1804, the system went spectacularly into action. A volunteer sergeant new to the Borders peered into the darkness from the balefire at Hume Castle and, yes, he was certain he could make out the yellow flames of a fire lit on Dowlaw in north Northumberland. Having lit the Hume balefire, the alarm crackled rapidly through the Borders as beacons were lit on Peniel Heugh, the Dunion and Crumhaugh Hill at Hawick. These triggered others to be lit in Teviotdale, Ettrick and Yarrow. When volunteers saw the fires, adrenalin must have pumped as they pulled on their boots and buckled on their sabres. Walter Scott was in Cumberland when the Hume bale was fired and he rode 100 miles to join his regiment at Dalkeith. The Borders was in uproar and rumours ricocheted. Regiments of Napoleon's cuirassiers were expected to come clattering along the Berwick road at any moment. But they did not. No one did. What the unfortunate sergeant at Hume had seen was not a balefire but the everyday work of some Northumbrian charcoal burners. Phew.\n\n**Panel stitched by:**\n\n_Lady Stitchers_\n\nSonya Anderson\n\nPhyllis Hogg\n\nBarbara Plevin\n\n**Stitched in:**\n\nBroxburn, Port Seton\n\n# PANEL 81  Henry Raeburn\n\nThe Rev. Robert Walker's day out skating on Duddingston Loch was immortalised by Henry Raeburn in the 1790s. It was only one stunning portrait from many painted by, arguably, Scotland's first great master. In a busy life, he produced hundreds of superb works, amongst them beautifully observed portraits of Walter Scott and Alastair Ranaldson Macdonell of Glengarry. These two illustrate Raeburn's great subtlety. A handsome man and one of the most famous people in Europe during his lifetime, Scott liked to portray himself as a straightforward sort of man. But, with a wonderful handling of light and the ability to capture expression, Raeburn shows an altogether more complex man, with sadness as well as creativity seeming to emanate from his gaze into the middle distance. Macdonell of Glengarry was a well-known clan chief and he was the model for Scott's creation, Fergus Mac-Ivor in his first novel, _Waverley_. Dressed in full Highland fig and clutching a musket, he looks the very epitome of the clansman whose genealogy and history reach back into the mists of time. But his features and expression are telling. He was known as a haughty and ruthless man. Far from being the protector of his people, he cleared Glengarry of crofters, felled timber for sale and exploited his patrimony for his own profit. Such is the quality and variety of Raeburn's portraits between the 1790s and 1820s that they can be read as a commentary on the age.\n\n**Panel stitched by:**\n\nJennifer Myles\n\n**Stitched in:**\n\nSouth Queensferry\n\n# PANEL 82  Walter Scott\n\nScott was a phenomenon \u2013 a figure who helped form many perceptions of modern Scotland. Born in Edinburgh in 1771, he was educated to follow his father into the law but Scott's lameness, perhaps caused by a bout of polio, sent him to recuperate in the Borders with relatives. These visits influenced the 'wee sick laddie' very much. At his grandparents' farm near Kelso, he listened to his aunt recite the Border ballads and they fired his imagination. As a young man, he began to make written copies of the ballads, what had been mainly an oral tradition. In 1802 and 1803, Scott had published _The Minstrelsy of the Scottish Border_ and the three volumes were very successful. But, in 1805, this collection was spectacularly overshadowed by Scott's own composition, _The Lay of the Last Minstrel_. This long narrative poem, set in the Borders, became a huge best-seller and it established the author as a figure of real renown. He brought out two more long poems, _Marmion_ and _The Lady of the Lake_. With his novel _Waverley_ and a string of subsequent successes, Scott became a worldwide name, the first author to be feted during his own lifetime. His novels also turned Scotland into a destination for tourists, especially when the railways came after the 1860s. And his fame had all sorts of other effects. When Scott organised the state visit of George IV to Scotland, he wrapped it in tartan, making the whole thing a Highland affair. This, in turn, caused the Borders textile mills to clack and rattle as demand for tartan boomed. Tragically, a series of unhappy business deals impoverished Scott and he killed himself with overwork to pay back his debts.\n\n**Panel stitched by:**\n\nCatherine Guiat\n\nEileen Henderson\n\nAnnette Hunter\n\n**Stitched in:**\n\nEdinburgh\n\n# PANEL 83  Fingal's Cave\n\nJames Macpherson had style \u2013 or, at least, nerve. In 1760, he was induced to have published _Fragments of Ancient Poetry Collected in the Highlands of Scotland_ , a set of translations he claimed to have made from Gaelic. A year later, he announced the discovery of something astonishing, something to rival the classical epics of Homer and Virgil. Composed by Ossian, at some time in the distant and mist-strewn past, this was _Fingal, an Ancient Epic Poem in Six Books_. Wildly popular and deeply suspect at the same time, the work of Ossian\/Macpherson helped create a vision of a romantic Highlands where a noble past had created tales to rival anything in Latin and Greek. Goethe adored the poems and Napoleon reputedly kept a copy by him on campaign. The real Gaelic name for the remarkable sea cave on the uninhabited Hebridean island of Staffa is _An Uamh Bhin_ , 'the Singing Cave', presumably after the echoic sound made by the tides. But the Ossian poems persuaded Sir Joseph Banks to rename it Fingal's Cave. The romantic composer, Felix Mendelssohn, visited this haunting place, with its perpendicular basalt columns, and wrote an overture, _The Hebrides_ , based on the echoes.\n\n**Panel stitched by:**\n\n_Scoraig Stitchers_\n\nAlison Barr\n\nJill Beavitt\n\nAggie Brudenell\n\nKath Bush\n\nCatherine Dagg\n\nAnthea Douglas\n\nNick Lancaster\n\nJoany McGuire\n\nSusan McSweeney\n\nJessie Millard\n\nHugh Piggott\n\nDjinni Van Slyke\n\n**Stitched in:**\n\nDundonnell\n\n# PANEL 84  _The Scotsman_ , Founded 1817\n\n_The Scotsman_ newspaper, founded as a weekly by a lawyer, William Ritchie, and a customs official, Charles Maclaren, was based in Edinburgh. Apparently a reaction to 'the unblushing subservience' of other papers to the Edinburgh establishment, its motto was 'impartiality, firmness and independence'. It quickly grew popular in the east of Scotland as the newspaper that would become _The Herald_ , founded in 1783, dominated in Glasgow and the west. When newspaper stamp tax was abolished in 1850, _The Scotsman_ moved to a daily circulation. It has a distinguished history under several notable editors such as Alastair Dunnett, Eric MacKay and Magnus Linklater. _The Scotsman_ 's coverage of the arts and the Edinburgh Festival and Fringe in particular has been crucial in the development of Scotland's culture. When Allen Wright, the arts editor, instituted the Fringe Firsts Awards for original drama on the Fringe in 1973, it single-handedly stimulated new work in Britain. More recently, the newspaper has come under pressure as, in line with the sector, its circulation has fallen and valued staff have been laid off. Nevertheless, _The Scotsman_ was named Newspaper of the Year in Scotland for 2012.\n\n**Panel stitched by:**\n\n_The Penicuik Team_\n\nJoan Cape\n\nMary Darling\n\nDeborah Hall\n\nFiona Hutcheson\n\nIsobel Ritchie\n\nJan Young\n\n**Stitched in:**\n\nWest Linton, Penicuik, Roslin\n\n# PANEL 85  George Smith and the Glenlivet\n\nWhisky production in Scotland was mostly illicit until the early 19th century, made by those anxious to avoid paying excise duty. When George Smith founded the Glenlivet Distillery in 1824, it was unusual for being entirely legal. This made Smith unpopular and he was in the habit of never travelling without a pair of pistols in his belt. He had also to protect his brand from imitators. As it does now, Glenlivet had an excellent reputation. Malt whisky may be said to be one of the few Scottish products that combines the best of Highland and Lowland. Near the distillery at Ballindalloch, Josie's Well supplies pure water from the foothills of the Grampian massif while the barley malt comes from Portgordon in Banffshire. When George Smith died in 1871, his famous brand was being appropriated by others who attached Glenlivet to their own names. Legal action was not entirely successful. Others were permitted to hyphenate Glenlivet but only the product distilled by the Smiths could be called _The_ Glenlivet. It has phenomenal sales, being the most popular in the USA and the second best-selling worldwide. Global sales currently run at 6 million bottles. It is to be hoped that Josie's Well never runs dry.\n\n**Panel stitched by:**\n\n_Granite Quoins_\n\nZuzana Banicova\n\nPamela Brice\n\nPamela Cook\n\nBruce Duncan\n\nJosephine Duncan\n\nDilly Emslie\n\nCoral Goldfarb\n\nHeather Hutton\n\nEwan Jeffrey\n\nOma Kapilla\n\nRudra Kapilla\n\nCarole Keepax\n\nMaria Mirick\n\nMichelle Morgan\n\nTracy Nelson\n\nSarah Richardson\n\nCatherine Stollery\n\n**Stitched in:**\n\nAberdeen, Edinburgh, Monymusk, Laurencekirk\n\n# PANEL 86  Borders Tweed\n\nWalter Scott and his friend, James Hogg, both had a writer's eye for telling detail. Both men and their characters sometimes wore a cloth called the Shepherd's Check or the Shepherd's Plaid. Border weavers had worked the undyed whites, blacks and browns of yarn into the warp and weft to create a rectilinear pattern. Portraits of Scott and Hogg exist of them draped with the check or plaid and it became fashionable. After the visit of George IV in 1822, the manufacture of tartan saw the mills in the Borders become very busy but what really promoted the manufacture of tweed was the wearing of trousers. Until the early 19th century, most men rode horses and wore breeches. But, in 1829, a diarist, Archibald Craig, recorded men wearing trousers made from Shepherd's check with a black coat. Colours were introduced and, in 1830, a London tailor, an expatriate Scot called James Locke, visited Galashiels to encourage the mill owners to create pattern books of tweed and tartan so that he could show it to customers. Reluctantly, they agreed. At the same time, suits were becoming fashionable \u2013 jackets and trousers made from the same cloth and pattern. Locke coined a new name for the output of the Border mills when he sold it as tweed and tweed suits became the acme of good taste. The rest is sartorial history.\n\n**Panel stitched by:**\n\n_Melrose Group_\n\nElizabeth Chalmers\n\nRuth Dall\n\nMary Wilson\n\n**Stitched in:**\n\nMelrose, St Boswells\n\n# PANEL 87  The Growth of Glasgow\n\nBy the 1820s, Glasgow's population had outgrown Edinburgh's. The Tobacco Lords had had a deep water port dredged as far upriver as Port Glasgow but, into the 19th century, the volume of trade persuaded investment in allowing merchant ships to dock in the heart of the city. In 1795, the Forth-Clyde Canal opened with its terminus at Port Dundas and this brought the raw materials of Lanarkshire into the city. Glasgow was growing very fast and became one of the earliest European cities to reach a population of one million. By the end of the 19th century, it was known as 'The Second City of the Empire' and the city's motto, 'Let Glasgow Flourish', was never more apposite. Shipbuilding and heavy engineering clustered close to the river but, across the city, textile and carpet manufacture, cigarette making, printing and publishing and all manner of industry were providing employment and making fortunes. The 19th century saw the building of the Kelvingrove Art Gallery, the Mitchell Library and the City Chambers. There was a palpable sense of civic pride and one of its exemplars was William Burrell. A ruthless ship owner who had the nerve to buy at the bottom of the market and sell at the top, he was also a near-obsessive art collector. After his death in 1958, he bequeathed his fabulous collection to Glasgow and the building that houses it in Pollok Park is a shared delight.\n\n**Panel stitched by:**\n\n_Glasgow Society of Women Artists_\n\nHelen Abdy\n\nSusan Black\n\nLyn Dunachie\n\nAdrianne Foulds\n\nNetta Hunter\n\nMargaret Murphy\n\nIngrid Parker\n\nAnn Rennie\n\nAnn Wilson\n\n**Stitched in:**\n\nGlasgow\n\n# PANEL 88  Sheep Shearing\n\nWool was pulled or combed out of fleeces before sheep were sheared and as the animals shed winter wool, this could be relatively easy. But, as wool production intensified in the Middle Ages, fleeces had to be sheared in the spring clip, often before lambing if the weather was not too cold. Washing the wool was a significant part of the process and there is a Scots song that remembers oily or tarry _oo_ , the word for wool:\n\nTarry Oo, Tarry Oo,\n\nTarry Oo is ill tae spin,\n\nCaird it weel, caird it weel,\n\nCaird it weel ere ye begin.\n\nWhen 'tis cairded, row'd and spun,\n\nThen the wark is halflins done,\n\nBut when woven, drest and clean,\n\nIt may be cleading for a queen.\n\n**Panel stitched by:**\n\n_Pentland Stitches_\n\nAli Cameron\n\nSara-Jayne Donaldson\n\nHilda Ibrahim\n\nAngela E Lewis\n\nMeg Macleod\n\nAnn Mair\n\nCarmel Ross\n\n**Stitched in:**\n\nThurso, Dunnet, Caithness\n\n# PANEL 89  The First Reform Act\n\nIn the brief reign of William IV, this long overdue piece of legislation redistributed parliamentary seats to reflect the great changes wrought by the Industrial Revolution. Separate acts were passed for Scotland, Ireland, and England and Wales but the effects were similar. Rotten boroughs such as Grampound in Cornwall or Dunwich in Suffolk elected two MPs even though only a handful of men had the vote in each village. Steered through parliament by Earl Grey (also famous for tea), the acts ensured new constituencies were created for burgeoning cities such as Manchester and Glasgow. Counties could elect two members, known as Knights of the Shires, and universities retained representation in the House of Commons. Until 1918, Edinburgh and St Andrews, in a unique cooperation, elected a joint MP, as did Glasgow and Aberdeen. In 1832, the electorate was all male and consisted only of a tiny proportion of the population at around 500,000. Voters were property owners whose land or houses was reckoned to be worth a certain sum and those who owned two or more houses in different constituencies were allowed to vote more than once. The Reform Act increased the franchise to 813,000. In Scotland, the rise was much more dramatic. Before the Act, only 4,239 could vote and this increased to more than 65,000. Aberdeen, Dundee and Perth were awarded MPs for the first time.\n\n**Panel stitched by:**\n\n_Pentland Stitchers_\n\nTish Alderson\n\nAnne Chater\n\nLois-May Donaldson\n\nAileen Gardiner\n\nDiana Gordon Smith\n\nViv Henderson\n\nAnthea Johnston\n\nJeannie Laidlaw\n\nCaroline Pearson\n\nMary Warrack\n\n**Stitched in:**\n\nEdinburgh, Milnathort, Eskbank, Dalkeith\n\n# PANEL 90  Kirkpatrick Macmillan\n\nA blacksmith born in 1812 at Keir in Dumfries and Galloway, Kirkpatrick Macmillan invented the machine that his countryman, Sir Chris Hoy, cycled to Olympic glory. A plaque on the smiddy wall at Courthill records that '[h]e builded better than he knew'. In 1839, Macmillan made a wooden prototype with iron-rimmed wheels, something a blacksmith will have been skilled at, and a steerable front wheel. But crucially the back wheel could be powered by pedals connected to it by rigid rods, following the principles laid down by James Watt. Macmillan's relative, James Johnston, set out in the 1890s 'to prove that to my native country of Dumfries belongs the honour of being the birthplace of the invention of the bicycle'. Such an upfront statement invites scepticism but this should all be ignored. Like James Small, Kirkpartick Macmillan was clearly a talented inventor, one of a long line of Scots who added to the enjoyment and convenience of all. And how else can Sir Chris Hoy's phenomenal, natural, intuitive talent be explained? Bicycles are part of Scotland's heritage, then and now.\n\n**Panel stitched by:**\n\n_Pans People_\n\nSusan Findlay\n\nFrances Glynn\n\nAvril Harris\n\n**Stitched in:**\n\nPrestonpans, Longniddry\n\n# PANEL 91  Queen Victoria at Balmoral\n\nQueen Victoria and Prince Albert fell in love with Scotland. After their first visit in 1842, they came back often, particularly to the Highlands. Preferring the slightly less rainy climate of Deeside, Albert eventually bought the Balmoral Estate and spent much of his summers shooting deer and other game. Where royalty led, the gentry followed and many imitations of Balmoral were built \u2013 so-called shooting lodges sprang up on many lochsides, some of them vast piles that lay empty for most of the year. Highland balls were a royal favourite and, once again, the court set the pace for the British aristocracy as they reeled, strathspeyed and sweated around dance floors in kilts, plaids and thick woollen hose. Tartan and all things Highland became fashionable. Balmorality was born and, with the annual holiday of the royal family, it continues. Bafflingly Queen Victoria announced that, at heart, she was a Jacobite. Her diary extracts from 1848 to 1861, _Leaves from the Journal of Our Life in the Highlands_ , was a best-seller and, in the railway age, it brought many tourists north. Just as they were arriving, many Highlanders were leaving, some of them cleared off their land by force.\n\n**Panel stitched by:**\n\n_Glenisla_\n\nHeather Berger\n\nClaire Broadhurst\n\nSheila Bruce\n\nCatriona Campbell\n\nPippa Clegg\n\nOlive Duncan\n\nJeanette McGill\n\nMary Ogilvie\n\nChristine Palmer\n\nKirsty Palmer\n\n**Stitched in:**\n\nBlairgowrie, Abernyte, Edinburgh, Kirriemuir\n\n# PANEL 92  The Scots in India\n\nAfter the victories of the Seven Years War and the prowess of Robert Clive and the armies of the East India Company, India became a vast source of raw materials for British industry as well as a market for manufactured goods. Until the middle of the 19th century, the subcontinent was controlled by this private company. It could conclude treaties, fight wars and defy governments. Henry Dundas was appointed President of the Board of Control and he oversaw the recruitment of many Scots in India. By 1792, one in nine working for the Company was Scots and a third of all the officers in its armies. Many fortunes were made. Scots invested so heavily in the India tea trade that production outstripped that of China. Dundee became a centre for the processing of jute. A vegetable fibre, it was used for making sacking, carpet and linoleum backing and many other purposes. So many Scots were resident in Calcutta, now Kolkata, that a regiment known as the Calcutta Scots was raised for the Indian army.\n\n**Panel stitched by:**\n\n_Wardie Church Stitchers_\n\nAnn Bell\n\nKaren Bowman\n\nSusan Dyer\n\nRhona Else\n\nSusan Fraser\n\nJean Jenkins\n\nFrances Mackinnon\n\nFiona Mauritzen\n\nJane Prowse\n\nBarbara Purdie\n\nJanet Rust\n\nSusie Standley\n\nJean Temple\n\n**Stitched in:**\n\nEdinburgh\n\n# PANEL 93  The Disruption\n\nBy 1843, tensions within the Church of Scotland had reached breaking point. The central issue concerned who had the right to appoint a minister to a parish \u2013 the laird or the kirk session. It struck chords with a bloody past, when battles had been fought over the related issue of independence for the kirk. Led by Thomas Chalmers, the Evangelical Party opposed the right (enshrined in an act of 1712) of lairds to 'prefer' their candidates while the Moderates supported it \u2013 because it was the law. At the General Assembly of the Church of Scotland, held in St Andrew and St George's in Edinburgh, a staged walkout took place. The retiring Moderator, Dr David Welsh, led 121 ministers and 73 elders out of the established church and down to Tanfield Hall in Canonmills. There they constituted themselves as The Free Church of Scotland. Eventually 474 ministers out of about 1,200 came out. It was an act of real bravery. Most lost their manses and kirks immediately since they were the property of the Church of Scotland and a frantic building programme began that almost doubled the number of churches, church halls and manses in Scotland. By a quirk of history, the Disruption inadvertently created the circumstances in Edinburgh where the Festival Fringe could find performance spaces and flourish. The Disruption was also the first great public event to be recorded by photography. After parliament rescinded the act of 1712 in 1929, the kirk was reunited, almost.\n\n**Panel stitched by:**\n\n_Heirs of 1843_\n\nFiona Anderson\n\nMarie Austin\n\nWinifred Cumming\n\nLilias Finlay\n\nMary Godden\n\nNan Laird\n\nDorothy MacKenzie\n\nJean Mackinlay\n\nChristine MacPhail\n\nDeborah Miller\n\nElizabeth Mitchell\n\nMaggie Morley\n\nJean Morrison\n\nJudith Pickles\n\nMaggie Romanis\n\n**Stitched in:**\n\nEdinburgh\n\n# PANEL 94  David Octavius Hill and Robert Adamson\n\nA painter, Hill, joined Adamson, an engineer, to create the first photographic studio in Scotland. It was based at Rock House at the western end of Calton Hill in Edinburgh. Most of the early photographs, known as calotypes, were portraits, many of them shot out of doors. Not only did they provide a record of the Disruption, the two also began to photograph outdoor scenes, landscapes and working people. They particularly enjoyed shooting the work of the fishermen of Newhaven and the fishwives who walked up to Edinburgh to sell the fresh catch round the doors. Even though the studio and the partnership only lasted five years, the output was prodigious, the earliest photographic record of life in Scotland.\n\n**Panel stitched by:**\n\n_Floral Ring_\n\nAnnalise McBride\n\nMargot Miller\n\nHazel Stewart\n\nHeather Young\n\n**Stitched in:**\n\nEdinburgh\n\n# PANEL 95  The Railway Boom\n\nRailways were built in Scotland from the 1830s onwards but, for some time, there was no network. The earliest were industrial, transporting coal and other raw materials between Glasgow, Dundee and Edinburgh. But, by 1842, investment was pouring in as the Edinburgh to Glasgow line opened and the Caledonian Company linked with the English rail network at Carlisle in 1848. Its great rival was the North British Railway and, in order to challenge and dominate the lines in the east of Scotland, it had to build bridges. The first spanned the Firth of Tay and opened in 1878 only to collapse a year later in a storm with the loss of 75 lives. The Forth Bridge was deliberately designed to look much stronger and it is the first structure in the world to be built of steel. Opened in 1890, its elegant cantilever construction has become an icon and, in any competition to decide the most famous manmade structure in Scotland, the Forth Bridge would surely win. Its completion opened up the north of Scotland to rapid rail travel. Contrary to popular belief, the bridge does not need constant repainting. A contract was completed in 2011 that should ensure that no repainting is needed until 2046 at the earliest.\n\n**Panel stitched by:**\n\n_Fifan Leddies_\n\nMargaret Caldwell\n\nJennifer di Folco\n\nMargaret Ewan\n\n**Stitched in:**\n\nBurntisland, Anstruther\n\n# PANEL 96  The Caithness School, 1851\n\nBy the 19th century, the dream of John Knox and the reformers had been realised. There was a school in almost every parish in Scotland and one of the consequences was a high level of mass literacy. At between 70 and 77 per cent, it was highest in Britain in the counties of Caithness and Berwickshire. But the picture was patchy. In Lanarkshire, attendance at school was below 50 per cent. But, when school boards came into existence in 1872 and they took over much of education from the church, standards rose. Scotland's universities, especially Glasgow, pioneered the needs of students drawn from an urban and middle-class background rather than simply providing an education for the gentry. It offered degrees in law, medicine, engineering, science, and divinity. Entrance qualifications were standardised and women were admitted in the 1890s, with St Andrews taking the lead. Education continues to be set at a premium in Scotland \u2013 at least in theory. But the brief window that allowed children to progress from school to university with fees paid by the local authority and a maintenance grant from the Scottish Education Department is now closed. This obviously restricts the number of talented young people who can benefit from Scotland's great educational tradition.\n\n**Panel stitched by:**\n\n_Caithness Textile Artists_\n\nValerie Barker\n\nJoan Dancer\n\nLouise Hunt\n\nDorothy Johnston\n\nElla Lawrence\n\nShirley MacLeod\n\nCelia More\n\nCatherine Swanson\n\n**Stitched in:**\n\nLybster, Wick, Thurso, Halkirk\n\n# PANEL 97  Fitba\n\nFuteball is recorded as early as 1424, when it was outlawed. But what might seem like football now was played all over Scotland. Ancient versions, like the Jethart Hand Ba' game, do not involve much kicking \u2013 of the ball, at least \u2013 but other variants do. It was not until the 1860s and 1870s that football's laws were codified and the first association of clubs was in England. Queens Park in Glasgow was founded in 1867 and it played against English opposition, reaching the FA Cup final twice. The first international match between Scotland and England took place in 1872. It was a 0\u20130 draw but the first time the Scottish football team played in navy blue jerseys with a thistle embroidered on the breast was two years later. They borrowed the design from the national rugby team. Scottish clubs began to form, mainly in the west at first, and the second oldest cup competition in the world began in 1873 with the Scottish Cup. Vale of Leven and Queens Park dominated the early years. Players were paid almost from the outset and Scotland's largest and wealthiest clubs have traditionally been Rangers and Celtic (although Rangers have recently fallen on hard times) and the first Old Firm match was played on Glasgow Green. After the Second World War, Scottish football enjoyed some glorious episodes. In 1967, Celtic became the first British side to win the European Cup and, in 1972, Rangers won the Cup Winners' Cup. But, since the mid 1970s, there has been a general decline \u2013 apart from Alex Ferguson's achievements at Aberdeen in the early 1980s \u2013 and the national side has fallen far down the world rankings. But hope springs eternal.\n\n**Panel stitched by:**\n\n_The Glasgow Banner Group_\n\nMarilyn Caddell\n\nClare Hunter\n\nMary McCarron\n\nGrace Pratt\n\nNorma Ventisei\n\nAgnes Wylie\n\n**Stitched in:**\n\nBishopbriggs, Uddington, Balquhidder, Lanark\n\n# PANEL 98  Irish Immigration after the Famine\n\nDuring the devastating potato blight in Ireland and the famine that followed between 1845 to 1851, emigration to Scotland accelerated rapidly. At its peak in 1848, the average number of immigrants disembarking weekly at the quays in Glasgow was estimated at more than a thousand and between January and April of that year, 42,860 came. Even in a city expanding as fast as Victorian Glasgow, this influx was dramatic \u2013 probably the most intense episode of immigration into Scotland for a thousand years. Many quickly found jobs in the heavy industries clustering around the coalfields of North Lanarkshire. It was hard, menial and frustrating work. Many Irish Catholics found it difficult to rise through the ranks of skilled workers but, after the horrors of the Great Famine, these jobs put bread on the table and roofs over the heads of families. Founded in 1887 by the Irish Marist Brother Walfrid, Celtic Football Club had a social as well as a sporting purpose and the club soon began to prosper. In 1888, the first Old Firm match with Rangers took place. Celtic won 5\u20132 but many of their players had been signed from Hibernian Football Club in Edinburgh. The two clubs became emblematic of the tensions between Catholics and Protestants in Scotland but mercifully most of the violence is now usually confined to the football pitch.\n\n**Panel stitched by:**\n\n_Trinity Stitchers_\n\nJoyce Ager\n\nMuriel Cleland\n\nDoreen Guy\n\n**Stitched in:**\n\nEdinburgh, North Berwick\n\n# PANEL 99  James Clerk Maxwell\n\nBorn in Edinburgh in 1831, this brilliant but sometimes underestimated scientist brought a series of equations, experiments and observations about electricity, optics and magnetism together into a consistent theory. He argued convincingly that all three phenomena are manifestations of the electromagnetic field. In 1865, Maxwell showed that electrical and magnetic fields move through space as waves that travel at the speed of light. At the age of only 25, he was appointed Professor of Natural Philosophy at Aberdeen University. The Adams Prize, given by St John's College, Cambridge, chose as its topic the apparent stability of the rings around Saturn. Maxwell postulated that they were composed of small particles, each of which independently orbited the planet. Not only did he win the prize but, in 1980, the Voyager space exploration programme proved him right. Interested in optics and the study of colour vision, Maxwell's research made colour photography possible. Despite his pioneering work across many fields of science, James Clerk Maxwell seems not to have taken himself too seriously. Based on the song 'Comin' Through the Rye' by Robert Burns, he composed his own version to reflect his interest in physics:\n\nGin a body meet a body\n\nFlyin' through the air.\n\nGin a body hit a body,\n\nWill it fly? And where?\n\nMaxwell sadly died young of cancer at the age of 48 but his contribution to modern scientific thought was immense.\n\n**Panel stitched by:**\n\n_The Gladsmuir Group_\n\nPatricia Coupe\n\nPru Irvine\n\nSusanne Lowe\n\nCelia Williams\n\n**Stitched in:**\n\nGladsmuir, West Garleton, Pencaitland, Pathhead\n\n# PANEL 100  Scots in Africa\n\nDespite their involvement with the infamous slave trade, Scots made a positive contribution to the story of Africa in the 19th and early 20th centuries. Perhaps the most famous was David Livingstone, a medical missionary and explorer who sought to discover the sources of the River Nile. His meeting on 10 November 1871, with H. M. Stanley is immortalised in the quote, 'Dr Livingstone, I presume?' Since the explorer's own account does not mention this amusing piece of irony and Stanley later tore out the relevant pages of his journal, the phrase may be a fabrication. But it is a good line anyway. Mary Slessor was a missionary who penetrated deep into the Calabar region of West Africa. It was difficult for a woman but, in some ways, her gender made her less threatening. She preached against witchcraft, human sacrifice and the abandonment of twins. Slessor died of malaria in 1915. Alexander Gordon Laing was the first European to reach the fabled city of Timbuktu. On an expedition across the Sahara Desert, from north to south, in 1826 and despite being wounded in Tuareg attacks, he entered the city. But the intrepid Laing was murdered soon afterwards and his valuable papers lost. Tragedy was mixed with Scottish dourness. Mungo Park discovered the middle reaches of the great River Niger. He was born on the farm of Foulshiels, near Selkirk, in 1771 and, when he came back home in 1793 after many years abroad, his mother and brothers heard a knock at the door. 'Aye,' said one of his brothers, 'that'll be Mungo.' 'How do you know?' asked his mother. 'I saw him get off the coach in Selkirk.' was the reply. No matter how renowned, no one gets above themselves in Scotland.\n\n**Panel stitched by:**\n\n_St Blanes Group_\n\nJudith Abbott\n\nCaroline B. Buchanan\n\nMaud Crawford\n\nSarah Gammell\n\nFiona Gibson\n\nJenny Haldane\n\nDorothy Morton\n\nLibby Taylor\n\nAnne Thomson\n\n**Stitched in:**\n\nMurthly, Callander, Port of Menteith, Duns\n\n# PANEL 101  Highland Games\n\nThese tartan-clad summer celebrations of caber tossing, Highland dancing, bagpiping, heavy and light events and general conviviality are mostly a Victorian invention \u2013 a cheery facet of Balmorality. But some of the events or individual sporting contests are clearly much older. Perhaps the most idiosyncratic is _Maide Leisg_ , Gaelic for 'Lazy Stick'. It involves two men sitting on the ground with the soles of their feet pressed against each other. Between them, they hold a stick in their hands and attempt to pull each other up until one is raised off the ground. It is generally believed to be the oldest event at the Carloway Highland Games on the Isle of Lewis. There are 14 major Highland Games held each year in Scotland, most of them north of the Highland Line. In the USA there are 147 and one in Switzerland. The greatest athlete ever to compete in the heavy events was probably Bill Anderson. He won 16 Scottish Championship titles from 1959 to 1974 and was American champion from 1976 to 1980. The Scots Hammer, wooden shafted and thrown from a standing position, was Anderson's most successful event and his world record of 123 feet, 5 inches still stands unbeaten.\n\n**Panel stitched by:**\n\n_Strathpeffer Craft & Craic Group_\n\nCarole Bancroft\n\nMary Bethune\n\nJanet Bowen\n\nKit Bowen\n\nPatricia Haigh\n\nClara Hickey\n\nMorag Hickey\n\nPat Justad\n\nLaura Lee McWhinney\n\nAlison Munro-White\n\nAudrey North\n\nDenise Page\n\nGrace Ritchie\n\n**Stitched in:**\n\nStrathpeffer, Garve\n\n# PANEL 102  Scottish Rugby\n\nThe Scottish Rugby Union was at first a Glasgow and Edinburgh affair with all the founding clubs except one coming from the cities. And, even though the game's heartland was in the Borders, it continued to be run by clubs of former pupils of Edinburgh and Glasgow schools for many years. Selection for international matches could be idiosyncratic and, until the era of league rugby in the early 1980s, Borders players had to have irresistible talent to be selected. The first match between England and Scotland was played in 1871 at Raeburn Place in Edinburgh and the Scots triumphed by a try to nil. It has been downhill ever since. Nevertheless, the amateur era was attractive, even romantic. Players could be heroes at Murrayfield on an international Saturday but they returned to their work on Monday as though very little had happened. One gnarled Borders prop forward scored a try against France in Paris but, as he cycled to his work as a railwayman on the Monday morning, he was accosted in the street by two old ladies who demanded to know why his wits had deserted him \u2013 why had he not run behind the French posts to make the conversion easier? In the days of the amateur game, Scotland could occasionally field a team to beat the best (except New Zealand), with victories over England being especially sweet. However, now that players are paid, Scotland rarely punches above its demographic weight. But, as with football, there is hope \u2013 especially at the beginning of each new season. Maybe this time . . .\n\n**Panel stitched by:**\n\n_The Mascots_\n\nMargaret Ferguson Burns\n\nMaeve Greer\n\nKathleen Grigor\n\nMargaret Hill\n\nJanet Speirs\n\n**Stitched in:**\n\nEdinburgh\n\n# PANEL 103  Shinty and Curling\n\nIn addition to golf and, arguably, rugby, Scotland has given two other distinctive games to the world. Shinty is a robust form of hockey, with its most recognisable descendant probably being ice hockey. Unlike the genteel English version, shinty involves shoulder-charge tackling, tackling with the stick or caman, playing the ball in the air (the source of a few head injuries) and other moves only occasionally recognised in the rule book. Now it is played mostly in the Highlands where famous teams like Kingussie, Newtonmore and Fort William have dominated. In the 19th and early 20th centuries, the game was popular in England, especially in Lancashire and Nottinghamshire. Some modern players have suggested that, for her Harry Potter books, J. K. Rowling was inspired by shinty to invent quidditch. Curling is more international and the game is firmly established in Canada, the USA, Switzerland, Sweden and elsewhere. It is also an Olympic sport. With immense precision, curling stones are slid down a long rectangular rink of ice to settle in a circular target zone or to knock a rival's stone out of contention. Made from granite, much of it quarried from Ailsa Craig in the Firth of Clyde, the stones make a noise when sliding down the rink. This has given rise to the phrase 'the roaring game'. In a hard Scottish winter, when lochs and rivers freeze, many very occasional curlers dust off ancient stones and take to the ice and they often roar.\n\n**Panel stitched by:**\n\nSusie Finlayson\n\nLinda Jobson\n\n**Stitched in:**\n\nEskbank, Dalkeith\n\n# PANEL 104  Scots in North America\n\nIn the decades immediately following the American Revolutionary War of 1775\u20131783, Scots, seen as loyalists, were unpopular. But into the 19th century, as emigration accelerated, several expatriate Scotsmen played leading roles in shaping the USA. Born in Dunfermline in 1835, the son of a weaver, Andrew Carnegie emigrated with his father and family in 1848. Fanatically hard-working and self-improving (through the use of libraries), the young Carnegie rose quickly through the ranks of the Pennsylvania Railroad Company. He began to borrow money to invest in railway-related companies and, after the American Civil War (1861\u201365), made huge profits. In 1901, he formed US Steel and it was the first company to be worth more than a billion dollars. Carnegie's philanthropy was also immense and perhaps his greatest legacy is the network of libraries he funded in Scotland. John Muir was very different. A naturalist and an advocate for the active preservation of the natural world, he helped set up the Yosemite and Sequoia national parks. The Sierra Club, which he founded, is one of the most important conservation organisations in the USA. According to his biographer, Donald Worster, Muir's mission was 'saving the American soul from total surrender to materialism'. Known as _Aglooka_ , 'Long Strider', by the Inuit, John Rae explored Northern Canada in the mid 19th century. Originally from Orkney, he understood that the best way to survive Arctic conditions was to imitate the indigenous peoples in their clothing and way of life. Canada's northern regions were mapped by Rae and his work was consolidated by Sandford Fleming, a Scottish-born surveyor and map maker. He also designed the first Canadian postage stamp and proposed a system of worldwide standard time. After missing a train in Ireland, he produced a plan for a series of time zones based on the Greenwich meridian and, by 1919, it had been adopted. Scots pioneered much in and beyond North America.\n\n**Panel stitched by:**\n\nSally Wild\n\nFrances Fettes\n\n**Stitched in:**\n\nEskbank\n\n# PANEL 105  The Paisley Pattern\n\nIn the late 18th century, soldiers returning from the wars in India brought back silk and woollen Kashmir shawls decorated with the _boteh_ , a droplet-shaped vegetable motif of Persian or Indian origin. From c. 1800 onwards, weavers in Paisley began to produce shawls with this design. Their looms were sufficiently sophisticated to allow them to weave with five colours of yarn. The _boteh_ quickly became known as the Paisley pattern and, in the 19th century, it was printed on cotton and woollen shawls and headsquares. Printed Paisley pattern was much cheaper than woven and almost as fashionable.\n\n**Panel stitched by:**\n\n_C plus 3_\n\nLorna Chapman\n\nCatherine Hughes\n\nJulie MacNaughton\n\nHazel Pert\n\n**Stitched in:**\n\nGlasgow\n\n# PANEL 106  The Battle of the Braes\n\nIn 1882, after more than a century of oppression following the defeat at Culloden, crofting communities on Skye refused to accept it anymore and violence flared. When tenants were faced with eviction from farmland in the Braes district south of Portree, they withheld payment of rent and also herded their sheep to graze on the slopes of Ben Lee, an area expressly forbidden to them. It had formerly been common land but it was seized by Lord Macdonald's men after the Clearances. When a Sheriff Officer arrived from Portree to serve the eviction notices, an angry mob burned them in front of him. Women were especially active and vocal. Fifty policemen were summoned from Glasgow to enforce Lord Macdonald's will but, after they arrested the men responsible for burning the notices of eviction, they were attacked by a large mob. Three hundred men and women fought in the Battle of the Braes and seven women were badly injured by truncheon blows. Prisoners were tried and convicted without a jury. Newspapers expressed outrage and MPs called for an inquiry. Unrest spread to Glendale where more violence erupted. Astonishingly, a negotiator was sent to Skye, on board a navy gunboat, and it was agreed that, if the Glendale men stood trial, a Royal commission would be set up. Known as the Napier Commission, it proposed mild reforms but, after four crofter MPs were elected in 1885, the Crofters' Act was passed. Much more radical, it gave every crofter security of tenure. Immune from eviction, they were now able to pass on their crofts to their heirs. It was a remarkable and long overdue victory.\n\n**Panel stitched by:**\n\n_Skye Quilt Studio_\n\nSue Cooper\n\nIrene Curren\n\nLin Leighton\n\nJennifer Lewis\n\nLiz Macleod\n\nAnn Nicolson\n\nIrene Owen\n\nBarbara Rutterford\n\nAnne Trimmer\n\nShirley Urquhart\n\n**Stitched in:**\n\nSkye\n\n# PANEL 107  Mill Working\n\nAs Paisley Pattern and other distinctive designs grew more popular, spinning and weaving were increasingly industrialised in Scotland. The outworkers who drove the looms and knitting frames at home, often with their wives and family carding and spinning yarn, were usually men. But as demand grew and entrepreneurs invested in scale, weavers and spinners moved into mills. At first water-driven and then steam-powered, with the coming of the railways, the mills also took on more and more women. The larger mechanical looms often needed delicate fingers to tease out problems and the muscle-power that drove the foot pedals of the domestic frames and looms became less important. Many of the beautiful shawls and fabric woven in Paisley and the west of Scotland were exported through Glasgow.\n\n**Panel stitched by:**\n\nFrances Gardiner\n\nGillian Hart\n\nYvonne Murphy\n\nJeannie Roberts\n\n**Stitched in:**\n\nPort Seton, North Berwick\n\n# PANEL 108  Robert Louis Stevenson\n\nFew writers have written masterpiece after masterpiece but _Treasure Island_ , _Kidnapped_ , _The Strange Case of Dr Jekyll and Mr Hyde_ and _The Master of Ballantrae_ have left an indelible mark. And, for TV watchers and filmgoers of a certain age, _The Black Arrow_ , _The Wrong Box_ and _St Ives_ were an early and thrilling introduction to the work of Robert Louis Stevenson. Unfussy in his language, brilliant in plot structure and characterisation, he was a genius, an original, a truly great writer. And yet the _Oxford Anthology of English Literature_ managed no mention of him or his work in more than 2,000 pages and, for decades of the 20th century, so-called critics consigned Stevenson to the ghetto of children's literature. Born in Edinburgh, he began writing when very young but, by the age of 23, his health began to become a problem. Stevenson spent much of his short life looking for a climate in which he might thrive. Between 1880 and 1887, his most famous books were published. By 1888, the famous author was in the Pacific and, in 1890, he bought land in Samoa. There he took the name of _Tusitala_ , 'the Teller of Tales'. In 1894, Stevenson collapsed and died, only 44 years old. Here is the well-known and much misquoted epitaph he wrote for himself:\n\nUnder the wide and starry sky\n\nDig the grave and let me lie.\n\nGlad did I live and gladly die,\n\nAnd I laid me down with a will.\n\nThis be the verse you grave for me:\n\nHere he lies where he longed to be;\n\nHome is the sailor, home from sea,\n\nAnd the hunter home from the hill.\n\n**Panel stitched by:**\n\n_Stitchers wi' Smeddum_\n\nAnn Dickson\n\nLynn Fraser\n\nJanet Macaulay\n\nLiz Neilson\n\n**Stitched in:**\n\nTranent, Pencaitland\n\n# PANEL 109  Workshop of the Empire\n\nThe Lanarkshire coalfield and the deposits of iron ore that lay nearby combined with Glasgow's status as a busy Atlantic port to spark the city's great industrial development. Once it became possible to transport these raw materials in bulk by canal and then by rail, shipbuilding, railway locomotive and carriage building and other heavy engineering flourished. And what these industries produced could be readily exported through the Glasgow quays. Parkhead Forge may now be a shopping centre but in the 19th and early 20th centuries it was the largest steelworks in Scotland, employing more than 20,000 men at its height. Four railway manufacturers had plants in the Springburn District. St Rollox grew to be the largest and it is still operational as a rolling stock repair and maintenance facility. Heavy industry may have predominated in the 19th century in the west of Scotland where its output found ready markets in the developing infrastructure of the colonies of the Empire but other sorts of manufacturing were also important. Textiles, especially the spinning and weaving of cotton, flourished until the American Civil War cut off supplies in the 1860s. Coal mining powered much of Scotland's industry until the second half of the 20th century but by that time heavy engineering was declining. And the gradual disappearance of the vast market that was the British Empire was a fundamental reason for the shrinkage of Scottish manufacturing, one of its workshops.\n\n**Panel stitched by:**\n\n_Linlithgow Stitchers_\n\nChristine Anderson\n\nHazel Briton\n\nGloria Fleming\n\nAileen Rasberry\n\nValerie Spence\n\nPatricia Swan\n\n**Stitched in:**\n\nLinlithgow, Falkirk\n\n# PANEL 110  The Scottish Trades Union Congress Forms\n\nRepresenting 630,000 members of 39 affiliated trades unions, the STUC is separate from the TUC, the British Trades Union Congress. In the voting for the first Parliamentary Committee 1897 (later the General Council), Miss M. H. Irwin gained the most support but, on the grounds that such a thing was politically premature, she declined to be chairman. Margaret Irwin was raised in St Andrews and took an early form of degree from the university. Involved with the Women's Suffrage movement, she was secretary of the Scottish Council for Women's Trades and central to the setting up of the STUC. The first woman to occupy the role of chairman was Miss Bell Jobson in 1937. The STUC was seen as more radical than the English TUC with many of its leaders being members of the Independent Labour Party. Its goal was 'to secure the collective ownership of the means of production, distribution and exchange' \u2013 a dictum only recently removed from the philosophy of the modern Labour Party.\n\n**Panel stitched by:**\n\n_Clydeside 5_\n\nRosalind Jarvis\n\nJane Logan\n\nJacky Mackenzie\n\nJim Mulrine\n\nCarol Woodward\n\n**Stitched in:**\n\nGlasgow\n\n# PANEL 111  Keir Hardie\n\nBorn in 1856 in a one-room cottage near Holytown in Lanarkshire, Keir Hardie had a meteoric rise. From being a miner, he became a union organiser and was then elected as MP for West Ham South in London in 1892. A year later, he and others formed the Independent Labour Party. After making a speech in parliament attacking the monarchy, Hardie lost his London seat but was later elected MP for Merthyr Tydfil and Aberdare, a constituency he served for the rest of his life. In the 1906 Liberal landslide, Labour won 29 seats and began to grow into a powerful political force. Two years later, Hardie resigned as Labour leader and became closely involved with campaigning for votes for women. Highly principled, he was a pacifist and was appalled by the slaughter of the First World War. He contacted socialist colleagues in other countries in order to organise an international general strike to bring the fighting to an end. Deeply unpopular, Hardie nevertheless persisted and made anti-war speeches wherever and whenever he could. But, a year after the outbreak of hostilities, he suffered a series of strokes and died in hospital in Glasgow on 26 September 1915.\n\n**Panel stitched by:**\n\n_Smith Stitchers_\n\nRita Smith\n\nShirley Smith\n\nAudrey Smith\n\n**Stitched in:**\n\nFalkirk, Glasgow\n\n# PANEL 112  The Herring Girls\n\nKnown as _Clann-Nighean an Sgadain_ , many of the Herring Girls came from the Hebrides and their seasonal work followed the herring catch around the coasts of Scotland. Particularly from the 1840s onwards, they moved from the ports at Lerwick on Shetland down to Wick, Fraserburgh, Peterhead, all the way down to Eyemouth and then further south to the quaysides of Yarmouth and Lowestoft. Their job was to gut the herring catch and pack it into barrels where the fish was preserved in brine. Working at tremendous speed, the _Clann-Nighean_ worked in crews of three with two gutting and a third, usually the tallest, packing because she could reach the bottom of the barrel more easily. The guts were also saved and used as a noxious fertiliser by farmers. The women worked incredibly quickly and the average number of fish they gutted in a minute was 40 but it could be much higher. Of course, they often sang as they worked.\n\n**Panel stitched by:**\n\n_Herring Gulls_\n\nAlison Dickson\n\nJanet Raeburn\n\n**Stitched in:**\n\nNorth Berwick\n\n# PANEL 113  The _Discovery_ Sails from Dundee\n\nLaunched in 1901, the _Discovery_ was the last traditional wooden, three-masted ship to be built in Britain. Designated RRS, Royal Research Ship, she was designed for the British National Antarctic Expedition led by Ernest Shackleton and Robert Falcon Scott. The hull was built of wood so that it would not be frozen into the ice and the bow was designed in such a way that it would rise up over the pack ice and break it by its deadweight. The expedition sighted the Antarctic coastline on 8 January 1902 and dropped anchor in McMurdo Sound. For the next two years, locked into the ice, the ship would remain there as the crew surveyed and mapped the coastline. The _Discovery_ expedition was reckoned a success since its crew were able to show that Antarctica was a continent, they relocated magnetic south and made excellent maps and charts. Using controlled explosives, the ship was freed from the ice in February 1904 and it docked at Spithead seven months later. Despite its success, the British Antarctic Expedition found itself in financial difficulties and the ship was sold to the Hudson's Bay Company. After a long career in various roles, the RRS _Discovery_ returned to the city that made her and, on 3 April 1986, she docked in Dundee to a very warm welcome. Since then she has become the focus of a superb museum at Discovery Point.\n\n**Panel stitched by:**\n\nBarbara Bell\n\nUrsula Doherty\n\nTessa Durham\n\nCatherine Jones\n\nDawn White\n\n**Stitched in:**\n\nNewport on Tay\n\n# PANEL 114  Dundee, Jute, Jam and Journalism\n\nDundee is very old and its name probably derives from _Dun-Deagh_ , 'Fire Hill'. This may have been a reference to the early habit of fires being lit on hills on the old Celtic quarter days. It is also appropriate since it was the fire of the industrial revolution that made Dundee grow into the fourth largest city in Scotland. Jute and whales were the stimulus. A vegetable fibre imported from India, jute was tough but hard to spin. Whale oil, a by-product of the city's other industry, was found to lubricate this process very well and the manufacture of many jute-based products such as canvas, sacking and twine began. In 1797, Janet Keiller invented marmalade. Without doubt. Other unlikely sources of early recipes for this excellent product have been put forward but none convince. But marmalade and jam, made from the fruits of the nearby berry fields, made Dundee famous \u2013 as did Oor Wullie and Desperate Dan. Generations were raised on the characters of _The Beano_ , _The Dandy_ and _The Sunday Post_. All were owned by DC Thomson, an enormously prolific and influential publishing company based in the city. One of the best but least remembered creations of the artists and writers at DC Thomson was Black Bob. This strip about a Border collie sheepdog and shepherd Andrew Glen, his alleged owner, ran in _The Weekly News_ until 1967. Both are still missed.\n\n**Panel stitched by:**\n\n_3J_\n\nEleanor Arthur\n\nMargaret Purvis\n\nMarilyn Rattray\n\nAlister Rutherford\n\nJessie Sword\n\n**Stitched in:**\n\nBroughty Ferry, Dundee\n\n# PANEL 115  Shetland, the Isbister Sisters\n\nIn rural communities all over Scotland women worked hard, doing menial, muscular tasks that would perhaps surprise modern sensibilities. On farms men did everything associated with horses \u2013 ploughing, carting, harvesting \u2013 while women did the back-breaking work of weeding, shawing, milking and bringing fuel for the fire. This everyday portrait of two sisters shows them with peat creels on their backs, chatting as they walk, knitting without a glance at what their practised fingers are doing. The exodus from the land of the last century has rendered much of what used to be thought of women's work redundant as well as changing its traditional meaning. Washing the dishes does not compare to weeding a field of turnips in the rain.\n\n**Panel stitched by:**\n\n_Pentland Stitches_\n\nAli Cameron\n\nSara-Jayne Donaldson\n\nAngela E. Lewis\n\nMeg Macleod\n\nAnn Mair\n\nCarmel Ross\n\n**Stitched in:**\n\nThurso, Dunnet, Caithness\n\n# PANEL 116  Charles Rennie Mackintosh\n\nBuilt between 1897 and 1908, the Glasgow School of Art made the international reputation of Charles Rennie Mackintosh, Scotland's greatest artist. Trained as an architect, he, nevertheless, designed virtually every element of the houses and institutions he had built, from the cutlery to the door furniture and the fabrics used for curtains. Later in life, he also revealed himself as a superb watercolourist of landscapes and flowers. Mackintosh worked closely with his wife, Margaret Macdonald, and she occasionally created gesso panels for his domestic commissions. The Glasgow School of Art is his greatest achievement. Built on an awkward sloping site and difficult to see from any distance, it is a triumph of invention and appropriateness. With the support of the school's headmaster, Francis Henry 'Fra' Newbery, Mackintosh was given great freedom and he expressed himself with brilliance. The library is a masterpiece. But, aside from two major domestic commissions, a school and a church, Mackintosh's prodigious talents were not used to the full partly because the First World War halted most building projects and partly for personal reasons. He left Glasgow to live first in London (and occasionally in Suffolk) and then in south-west France. Recalling his remarkable talent, the daughter of Fra Newbery, Mary Newbery Sturrock said, in the 1980s, that she wept at the waste of such an artist. 'I'd like to have seen his fiftieth house \u2013 it wouldn't have been a bit like the first.'\n\n**Panel stitched by:**\n\n_The Dotty Stitchers_\n\nIrene Mitchell\n\nLindsay Morrison\n\nMarion Nimmo\n\nChristine Rettig\n\nKate Ross\n\nDorothy Stalker\n\nSue Whitaker\n\nSusan Wylie\n\n**Stitched in:**\n\nErskine, Glasgow, Clydebank, Renfrewshire\n\n# PANEL 117  The Munros\n\nNon-Gaelic speakers may have difficulty in pronouncing the names of Scotland's mountains, but thanks to Sir Hugh Munro, we know how many of them rise to over 3,000 feet \u2013 282. But there is also a classification known as a Munro Top, a summit that is not seen as a separate mountain, perhaps part of a ridge. There are 227 of them. Munro bagging has become a popular pastime. More than 4,000 have climbed them all but the number who have scaled more than a few must be much higher. Ben Nevis is probably the most famous Munro, being the highest mountain in Britain, but, with the constant traffic up and down, is it being worn down?\n\n**Panel stitched by:**\n\nJoan Kerr\n\n**Stitched in:**\n\nFort William\n\n# PANEL 118  The 1914\u20131918 War\n\nThe slaughter of the First World War is commemorated on memorials all over Scotland. Impossibly long lists of the names of young men who died between 1914 and 1918 remember an unparalleled death toll. Few understood why Britain was at war with Germany, Austria and Turkey but any dissent was buried by patriotic zeal. Young men not in uniform, for whatever reason, risked having a white feather pinned to them by women who called them cowards. Men who had been wounded were given special badges to avoid such humiliation. As the war stagnated in the mud of Flanders and the death toll spiralled, Scotland suffered disproportionately with perhaps as many as a quarter of all Scots who fought being killed. The commander of the British Expeditionary Force from 1915 to 1918 was Field Marshal Douglas Haig, the 1st Earl Haig of Bemersyde. Born in Edinburgh in 1861, he became a regular soldier who rose through the ranks to great eminence. But, since the 1960s, his reputation has been attacked, with some historians recalling the label 'Butcher Haig' and criticising him for the needless slaughter of soldiers. Others have defended Haig, arguing that his strategy was ultimately successful. Certainly the experience of the trenches changed politicians' view of what a conscript army would and would not do, especially in the Second World War. There would be no repeat of the casualty lists of 1914\u20131918 where a staggering 4 per cent of British men were killed.\n\n**Panel stitched by:**\n\n_EH6 Group_\n\nSorrell Bentinck\n\nAlison Black\n\nAndrea Bloomfield\n\nLucinda Byatt\n\nCherry Campbell\n\nFiona Campbell Byatt\n\nSarah Conlon\n\nKaren Howlett\n\nNaomi Jennings\n\nYoshiko Nakano\n\nAlison Roarty\n\n**Stitched in:**\n\nEdinburgh\n\n# PANEL 119  The Building of HMS _Hood_ , the Battle of Ypres 1917\n\nPasschendaele has become a byword for the miseries of fighting in the trenches of Flanders. Also known as the Third Battle of Ypres, it was an offensive launched by the Allies to gain control of the ridges to the south and east of the Belgian city of Ypres. Lloyd George, the prime minister, was against the operation as was General Foch, the French commander. They wanted to wait for American reinforcement. But Field Marshal Haig persisted with the strategy of attacking in the west and approval was granted in July 1917. The human cost was vast and the gains negligible. Wet weather bogged down the advance, the German Fourth Army fought back strongly and ultimately British and French troops had to be deployed to Italy to shore up that front after an Austrian victory at Caporetto. If the land war was a stalemate, Britain hoped to win the war at sea and, on 1 September 1916, the keel of HMS _Hood_ was laid down on Clydeside. Designated as a battlecruiser, _Hood_ was huge and very heavily armed, outgunning the _Mackensen_ -class German equivalents. She carried eight 42-calibre guns that could launch a 1,920-pound shell over a range of 17 miles at maximum elevation. Known as the _Mighty Hood_ , she was not commissioned until 1920. More than 20 years later, this huge battlecruiser fought in one of the most famous naval engagements of the Second World War, the Battle of the Denmark Strait. The _Hood_ was sunk by the _Prinz Eugen_ and the _Bismarck_ with an explosion in her magazines causing enormous loss of life. Out of a crew of 1,148, only three survived.\n\n**Panel stitched by:**\n\n_The Appin Stitchers_\n\nJoy Blakeney\n\nJean Breckenridge\n\nHelen Currie\n\nDoreen Evans\n\nJanet Fairbairn\n\nMidge Gourlay\n\nFiona Hunter\n\nMargaret Rayworth\n\nMorag White\n\nAnne White\n\nPat Wyeth Webb\n\n**Stitched in:**\n\nAppin, Port Appin, Benderloch, Barcaldine\n\n# PANEL 120  Elsie Inglis\n\nOne of the first female doctors to qualify, Elsie Inglis set up in medical practice in 1894 and opened a maternity hospital in Edinburgh. She was very unhappy with the standard of medical care for women and that propelled her into the women's suffrage movement. Elsie Inglis did pioneering work during the First World War but died in 1917 on her return to Britain. The Elsie Inglis Memorial Maternity Hospital was founded in Edinburgh and it continued to innovate in midwifery. It seemed that the consideration, warmth and humanity of its inspiration lasted for generations. The closure of the Elsie Inglis in 1988 may have made sense to some but its passing was much mourned by many.\n\n**Panel stitched by:**\n\nFiona Kirton\n\nJo Macrae\n\nDeborah Ramage\n\n**Stitched in:**\n\nEdinburgh\n\n# PANEL 121  The Sinking of HMY _Iolaire_ off Stornoway, 1919\n\nOn the evening of 31 December 1918, the Admiralty yacht, HMY _Iolaire_ (the name means 'Eagle' in Gaelic) sailed out of Kyle of Lochalsh bound for Stornoway on the Isle of Lewis. On board were 280 sailors returning home, having survived the horrors of the First World War. Within sight of the twinkling lights of the harbour, the yacht struck the infamous Beasts of Holm, a reef only twenty yards from the shore. In heaving seas, the yacht began to break up and 205 men were drowned, 181 of them Lewismen. In a small island community, the impact of this tragedy was huge. Only 75 survived and many of them because of the bravery and strength of John F. MacLeod of Ness. He swam ashore with a line, wrapped it around himself and about 40 men pulled themselves to safety along it. No satisfactory explanation for the disaster was ever produced and the legacy of the _Iolaire_ is still felt. In 1958, a memorial was erected at Holm and a stone pillar built on the Beasts.\n\n**Panel stitched by:**\n\n_Sea-Mistresses_\n\nTracey MacLeod\n\nMoira MacPherson\n\nGillian Scott-Forest\n\n**Stitched in:**\n\nHarris, South Uist\n\n# PANEL 122  Eric Liddell\n\nThe romantic days of amateur sport have no better exemplar than Eric Liddell. The Flying Scotsman won a gold medal for the 400 metres at the Paris Olympic Games of 1924, having refused, on religious grounds, to run on a Sunday in the heats for the 100 metres, his better event. His idiosyncratic running style, with his head flung back, his mouth wide open and his arms flailing, was remembered by all who saw him but it was his blistering speed that dazzled. His rival, Harold Abrahams, defended Liddell, saying, 'People may shout their heads off about his appalling style. Well, let them. He gets there.' The year before the Olympic Games, he had played rugby for Scotland. And, at the age of 23, he retired from sport so that he could become a missionary in China. Interned by the Japanese, Liddell died in 1945, probably of a brain tumour and malnourishment. But his achievements were never forgotten and they were immortalised in the Oscar-winning film, _Chariots of Fire_. When the great Scottish sprinter, Allan Wells, won Olympic gold in Moscow in 1980, he dedicated his race to the memory of Eric Liddell.\n\n**Panel stitched by:**\n\n_The Liberton Connection_\n\nRosemary Leask\n\nSheila MacIsaac\n\n**Stitched in:**\n\nEdinburgh\n\n# PANEL 123  Women Get the Vote\n\nElsie Inglis became a founder member of the Scottish Women's Suffragette Federation in 1906 and worked hard to see the franchise extended. Others were even more militant and Flora Drummond, a postmistress, went to prison nine times as a leader of the Women's Social and Political Union. Some poured acid into pillar boxes, slashed portraits of the king and set fire to buildings such as Ayr Racecourse, Leuchars Railway Station and the Whitekirk in East Lothian. In prison, many went on hunger strikes. The First World War supplied unexpected support as suffragettes suspended campaigning. While men fought and died on the Western Front in their millions, many women worked in the munitions industries, joined the Women's Land Army or drove buses and ambulances. All were seen in a different light as a result. New legislation was enacted in 1918 to give all women over the age of 30 the right to vote in general elections and to stand as MPs. But pressure for equality did not abate and, in 1928, women 21 and over were allowed to vote.\n\n**Panel stitched by:**\n\n_Soroptimist International of Edinburgh_\n\nEdith Elliot\n\nJane Green\n\nClephane Hume\n\nWinifred Keeves\n\nIsabel Smith\n\n**Stitched in:**\n\nDalkeith, Edinburgh\n\n# PANEL 124  Whaling\n\nDemand for whale oil for use in lighting drove a major expansion in whaling. Out of ports such as Dundee and Peterhead, whaling ships sailed the Arctic but overfishing seriously depleted stocks. In the early 20th century, Christian Salvesen established a whaling station in Shetland as well as one in the Antarctic on the island of South Georgia. Salvesen became the largest whaling company in the world in part because of their pioneering use of the entire carcass of the whale. By the end of the 20th century, stocks had dwindled so alarmingly that a complete moratorium had to be imposed to allow species to recover.\n\n**Panel stitched by:**\n\nRebecca Fish\n\nAlasdair Fish\n\nRosalind Neville-Smith\n\nJoyce Peace\n\nHazel Shearer\n\nMolly Shearer\n\nLeah Shearer\n\n**Stitched in:**\n\nOrkney\n\n# PANEL 125  The General Strike, 1926\n\nLasting only nine days, the strike was called by the Trades Union Congress in an attempt to force the government to halt a downward spiral of wages and conditions for 800,000 miners. More than 1.7 million workers downed tools. In the face of well-laid preparations and the support of the middle classes, students and others in maintaining services, the General Strike failed. Instead of building a land fit for heroes after the First World War, there was unemployment and deprivation, a broken promise to those who survived the trenches or lost relatives. In Glasgow, huge crowds gathered and soldiers were mobilised. The Riot Act was read from Glasgow City Chambers and the police charged demonstrators. But the left-wing MPs, Mannie Shinwell and Willie Gallacher, had no coherent political plan apart from protest and nothing of any real consequence was achieved. The strike was called off without any guarantees, terms or even a written statement. Such abject surrender was greeted with a mixture of fury and disgust in Glasgow.\n\n**Panel stitched by:**\n\n_Big County Gals_\n\nJune McEwan\n\nKaren Phillpot\n\nGill Tulloch\n\n**Stitched in:**\n\nPitlochry\n\n# PANEL 126  Fair Isle\n\nMidway between Shetland and Orkney, Fair Isle is small, sparsely populated and famous. In 1900, the population was 400 but now it numbers only 70 and most live in the crofts on the southern part of the island. It is famous in the shipping forecast and for a traditional style of knitting. The latter became popular when the Prince of Wales, later to reign briefly as Edward VIII, took to wearing Fair Isle knitted sleeveless jumpers to play golf in the early 1920s. Very colourful and with finely worked horizontal geometric patterns, these designs consequently became the acme of fashion. And they are still are.\n\n**Panel stitched by:**\n\nMarietta di Ciacca\n\nEdna Elliott-McColl\n\nSusan Finlayson\n\n**Stitched in:**\n\nPort Seton\n\n# PANEL 127  Hugh MacDiarmid\n\nHugh MacDiarmid's masterpiece, _A Drunk Man Looks at the Thistle_ incorporates a passage on the General Strike \u2013 a pessimistic response to its failure. The great lyric is a montage of invective and humour and a collection of themes dealt with in what seem like distinct poems. But it has an authentic voice in Scots \u2013 something MacDiarmid imbibed in his youth in the Borders town of Langholm. It begins:\n\nI amna fou sae muckle as tired \u2013 deid dune.\n\nIt's gey and hard wark coupin' gless for gless\n\nWi' Cruivie and Gilsanquhar and the like,\n\nAnd I'm no' juist as bauld as aince I wes.\n\nChristened Christopher Murray Grieve, MacDiarmid was born in Langholm in 1892 and he led a deeply individual life. During the 1930s, he was expelled from the Communist Party for being a Scottish Nationalist and then expelled from the Scottish National Party for being a communist. And, in 1956, when Soviet tanks invaded Hungary, he rejoined the Communist Party. Living in abject poverty, scraping a living with journalism, MacDiarmid worked tirelessly as a poet and writer and happily admitted that his work was of variable quality. But he was undoubtedly a Scottish literary genius, enormously influential, thrawn and indefatigable. When MacDiarmid died in 1978, his fellow poet, Norman MacCaig, suggested that each year the great man's passing be celebrated by two minutes' pandemonium.\n\n**Panel stitched by:**\n\n_The Albyn Stitchers_\n\nBarbara Gregor\n\nLinda Herd\n\nDiana Herriot\n\nSamantha Townsend\n\n**Stitched in:**\n\nLivingston\n\n# PANEL 128  Ramsay MacDonald and the Rise of the Labour Party\n\nThe first Labour Prime Minister was a Scot, Ramsay MacDonald. The illegitimate son of a farm labourer, John MacDonald and a housemaid, Anne Ramsay, he was raised and educated in Lossiemouth. In 1906, he was elected to the House of Commons as a Labour MP for Leicester and, in 1911, he became party leader. Implacably opposed to the First World War, he lost his seat in 1918 but returned to parliament in 1922 in the general election that saw Labour replace the Liberals as the second largest party. And, into the 1920s, MacDonald's pacifism and his oratorical skills earned him the respect of the electorate. By the early 1920s, Labour had become the main opposition party to the Conservatives and, in 1924, King George V called on him to form a minority government with the support of the Liberal Party. It only lasted nine months but showed that Labour had the ability to govern credibly. In 1929, MacDonald and Labour returned to government as the largest party but were overwhelmed by the onset of the Great Depression. When he formed a National Government, MacDonald was expelled from the Labour Party but carried on as prime minister. In 1931, he won a huge majority and Labour's Westminster representation was reduced to a rump of only 50 seats. Prime minister on three separate occasions, he stood down in 1935, losing his seat in the election (but becoming MP for the Combined Scottish Universities in a by-election a year later) and two years later MacDonald resigned as his physical and mental health collapsed. From a single-parent family living in a cottage in Lossiemouth to 10 Downing Street \u2013 it was an astonishingly meteoric life course for the time.\n\n**Panel stitched by:**\n\nJenny Allan\n\nPolly Atkinson\n\nJoan Bell\n\nAriane Burgess\n\nAmanda Cooper\n\nCorinne Davies\n\nRian Davies\n\nIsabel di Sotto\n\nJenny Doig\n\nDaphne Francis\n\nAnne Gavin\n\nMary Haslam\n\nJacqui Hassan\n\nKatyi Hassan\n\nLinda Jones\n\nWilma MacBain\n\nFiona MacDonald\n\nMarjorie Macleod\n\nJoyce MacNaught\n\nShiela McCourt\n\nDiane McGregor\n\nAnne Milligan\n\nKatherine Murray\n\nJudi O'May\n\nFrances Powell\n\nKathleen Purmal\n\nLinda Robertson\n\nKathleen Ross\n\nAnne Skene\n\nWendy Springett\n\nYvonne Stuart\n\nDavina Thomas\n\nJune Watson\n\nJames Watson\n\n**Stitched in:**\n\nBurghead, Forres, Findhorn\n\n# PANEL 129  The Great Depression\n\nWhen Wall Street crashed and confidence evaporated in the American economy in late 1929, the effects ricocheted around the world. And, unlike in 2008, there was no international consensus on how to effect recovery. Britain's world trade fell by half, unemployment soared to 3.5 million and the output of heavy industry plunged by a third. Scotland was particularly affected by the latter. But, because there had been no post-war boom to speak of in the 1920s, the slump was less extreme than it was in North America. However, the Scottish coalfields saw many men laid off as demand dried up and shipbuilding was badly hit on the Clyde. In 1931, unemployment benefit was paid according to need rather than contributions made but it was contingent on a means test and this was the cause of bitter resentment. A deep north\u2013south split became apparent in Britain. A recovery began slowly in the south where house-building offered employment but the heavy industries and mining in Scotland, on Tyneside and in Yorkshire and Lancashire and Wales remained depressed. And the widespread resentment at Conservative economic policies began to store up a wish for change that erupted in the surprising general election result of 1945. But what began to send men and women back to work in the heavy industries of the north was the government decision to rearm in the face of Nazi aggression.\n\n**Panel stitched by:**\n\n_Kelly's Cats_\n\nAudrey Anderson\n\nPat Balfour\n\nMary Bibby\n\nAlison Black\n\nHeather Bramwell\n\nNan Chalmers\n\nHeather King\n\nMargaret Moir\n\nGail Neiman\n\nKay Paul\n\nOlive Pauline\n\nAlison Purvis\n\nMargaret Ruddiman\n\nMaureen Stuchbury\n\nJune Willox\n\nVerna Wilson\n\n**Stitched in:**\n\nAberdeen\n\n# PANEL 130  Tenement Life\n\nAs a solution to the density of urban living, tenements are ancient. In Rome and the great cities of the ancient world, multi-occupancy in buildings of several storeys was common. In industrialising England, the term soon acquired a pejorative gloss since it meant overcrowding and slum conditions, but in Scotland tenement life could be very attractive to working people. Even though shared toilets on the stair-heid were the norm, and the drying green and entrance were communal, the neighbourliness and mutual support, especially in hard times, usually more than compensated. Densely packed communities produced their own entertainment and social and sporting clubs thrived. As did political discourse. One of the greatest products of the Paisley tenements was Willie Gallacher, a founder of the Communist Party of Great Britain who was MP for West Fife between 1935 and 1950. In the slum clearances of the 1960s and 70s tenements were demolished and replaced with tower blocks. Which in turn were demolished and replaced with \u2013 tenements. This time with their own toilets.\n\n**Panel stitched by:**\n\n_Paisley Patter_\n\nMarie Connelly\n\nGladys Connolly\n\nCarla Corneli\n\nNatalie Elliott\n\nLiz Gardiner\n\nChristine Gilmour\n\nIrene Harvey\n\nLesley King\n\nAga Kulet\n\nCatherine Lappin\n\nMorven McAlister\n\nMargaret McBride\n\nPaula McKeown\n\nMargaret Muir\n\nAnne Ross\n\nGrant Scott\n\nJan Walker\n\nRita Winters\n\nMichaela Wright\n\n**Stitched in:**\n\nPaisley, Elderslie, Greenock, Glasgow\n\n# PANEL 131  The Second World War\n\nThe first action of the Second World War was fought in the skies above the Firth of Forth as German aircraft attacked the naval base at Rosyth. Several were shot down. Scapa Flow in Orkney was also strategically vital for the North Atlantic and Shetland's proximity to Norway gave rise to the Shetland bus, a series of voyages by fishing boats to help Norwegians flee the Nazi invasion and aid their resistance efforts. Scots were notable in operations with Air Chief Marshal Hugh Dowding from Moffat leading Fighter Command and Robert Watson Watt's invention of radar giving defenders the edge in the Battle of Britain in 1940. Col. David Stirling founded the SAS, the Special Air Service. He was eventually captured and sent to Colditz Castle. But perhaps the most significant incarceration was the surrender and capture of the 51st Highland Division at St Valery. After the defeat of the British Expeditionary Force and the evacuation from Dunkirk, bad weather forced General Fortune to capitulate. For most of the duration of the war, they were held at Stalag XX-A in Poland and, in 1945, forced to march 450 miles west to Luneberg Heath, north of Hanover. On the home front, Prof. John Raeburn from Aberdeen was an agricultural economist who organised the 'Dig For Victory' campaign that fed Britain between 1939 and 1945. He encouraged people to convert lawns and flowerbeds into allotments and to keep chickens and pigs in back gardens. More than 900 'pig clubs' sprang up and, with rationing, it is said that Britain has never had as healthy a diet as it did during the Second World War.\n\n**Panel stitched by:**\n\n_JHEMMS_\n\nMarion Harkins\n\nJoan Marsh\n\nSusan Matthew\n\nHeather Neal\n\nEileen Rennie\n\nMary Woodward\n\n**Stitched in:**\n\nEdinburgh, Musselburgh\n\n# PANEL 132  The Clydebank Blitz\n\nIn March 1941, the Luftwaffe mounted two devastating raids on Clydebank and its shipbuilding yards. The town was destroyed and 528 people lost their lives, a further 617 were seriously injured and many were hurt by blast debris. Out of approximately 12,000 houses, 4,000 were completely destroyed and 4,500 severely damaged, making 35,000 people homeless. But the major targets of the John Brown shipyard and Beardmore's Diesel Works were not badly hit and were able to continue to function. The German raid was vast, with 439 bombers dropping more than 1,000 bombs. Only two were shot down by the RAF but the guns of the Polish destroyer, _Piorun_ , helped defend the town from the docks as it fired at the aircraft. There is a war memorial in Clydebank dedicated to the bravery of the Polish sailors. The attack took place over two nights and the damage to housing in the first raid was collateral since most of the workers lived close to the yards and the plants. But the following night seems to have been a terror raid aimed at civilians in an effort to damage not only their houses but also their morale. It failed.\n\n**Panel stitched by:**\n\n_Nervous Needles_\n\nShona Glenn\n\nClaire McDonald\n\nCarole Ross\n\nMarjorie Sinclair\n\n**Stitched in:**\n\nEdinburgh\n\n# PANEL 133  War Defences\n\nSeveral generations after the German surrender, the marks of war can still be found in Scotland. When an invasion fleet sailed up the Oslo Fjord in 1940, one of its strategic aims was to open up a thousand-mile-long start line for an invasion of Britain. That, in turn, exposed the east coast of Scotland to a direct threat of invasion. Concrete obstacles to inhibit tank movement were hurriedly built on beaches and pillboxes set up at strategic vantage points. But, while the nearest and clearest threat was to the south-east of England, Scotland had few soldiers to defend its shores. In many places, a German invasion force could have landed unchallenged. The Moray Firth coastlands were thought to be particularly vulnerable and so the Cowie Stop Line was constructed. Near Stonehaven and following the line of the Cowie Water, a series of earthworks and concrete obstacles was built between the North Sea coastline and the foothills of the Grampians. This narrow neck of land was the gateway to southern Scotland and, while invaders might have been able to land in Moray or Banff, they would have had to round that corner of Scotland's geography. In truth, the Cowie Stop Line would have slowed down an advance rather than halt it. In the 1950s and 1960s, Scotland's concrete defences against the Nazis supplied schoolchildren with a setting for war games with toy guns that had more than a shiver of authenticity. They were built for a real war and might really have been attacked.\n\n**Panel stitched by:**\n\nMay Bowie\n\nPatsy Brown\n\nFrances Fettes\n\n**Stitched in:**\n\nNorth Berwick, Eskbank\n\n# PANEL 134  D-Day, 1944\n\nOn 6 June 1944, the largest amphibious invasion ever mounted landed on the beaches of Normandy to begin the reconquest of Nazi Europe. More than 73,000 American soldiers, 61,715 British and 21,400 Canadians either splashed ashore under heavy fire or were parachuted in the night before. Known as Operation Overlord, the landings saw some of the fiercest fighting of the Second World War. Scotland had not only been involved in the preparations and training for the landings, many Scottish soldiers attempted to fight their way up the beaches. Simon Fraser, Lord Lovat, commanded 1 Special Service Brigade and he instructed his personal piper, Bill Millin, to play as the troops landed. Under heavy German fire, he played 'The Road to the Isles' and 'Hielan' Laddie'. The piper was the only soldier in the Normandy Landings to wear a kilt \u2013 a Cameron tartan worn by his father in the trenches in the First World War. Apart from his sgian-dubh sheathed in his socks, Millan carried no weapons. Later, he talked to captured German snipers who said that they did not shoot at him because they thought he was crazy. Lovat's brigade advanced to Pegasus Bridge, bravely defended by the Ox and Bucks Light Infantry, with Millin leading and playing the pipes. Twelve men wearing berets were shot through the head and commandos then raced across the bridge with helmets on. In 2009, Piper Millin was decorated by the French government, receiving the Croix d'Honneur for gallantry.\n\n**Panel stitched by:**\n\nMargaret Burgess\n\nOlive McCrone\n\nAnne Ratigan\n\nCaroline Scott\n\nNicki Slater\n\nAlison Wood\n\n**Stitched in:**\n\nEdinburgh, Winchburgh\n\n# PANEL 135  The First Edinburgh Festival\n\nGrim austerity followed the end of the Second World War but one sparkling innovation lit the gloom. In 1947, the first Edinburgh Festival took place. The brainchild of Rudolf Bing, the director of Glyndebourne Opera and Lord Provost Sir John Falconer, it brought orchestras, ballet, theatre and exhibitions to the city each August. The official festival programme was immediately joined by the Festival Fringe \u2013 eight groups who wanted to add their largely amateur performances. Apparently, 1947 saw a warm and sunny summer, something which added greatly to the atmosphere of optimism. The Fringe has come to dominate the Edinburgh Festival and it is now the largest and best arts festival in the world. It operates like no other. Performers may join the Festival Fringe Society, hire a venue and put on a show. There is no central artistic control and this openness has been the stimulus for the successful beginning of many brilliant careers, from the playwright Tom Stoppard, through the Cambridge Footlights of 1981, with Stephen Fry, Hugh Laurie, Emma Thompson and Tony Slattery, to the brilliant impressionist Rory Bremner. Cultural entrepreneurs have begun to exercise a degree of selection and few have had the impact of Bill Burdett-Coutts over the last 30 years with Assembly Productions. Collectively, as Beethoven is played alongside stand-up comedy and the Scotsman Fringe First Awards for new drama are up for grabs, Edinburgh is simply the best place to be in August \u2013 in the world. No question.\n\n**Panel stitched by:**\n\n_Festival Group_\n\nJo Allen\n\nAnn Campbell\n\nMairi Campbell\n\nHeather Davidson\n\nAlice Henderson\n\nEleanor Horton\n\nJackie Kemp\n\nJan Kerr\n\nLe-Anne Koh\n\nSue Lougheed\n\nDelia Marriott\n\nJanis McGravie\n\nMoira Nelson\n\nKirstin Norrie\n\nSigridur Oladottir\n\nFfion Reville\n\nFiona Roche\n\nPatsy Seddon\n\nHeather Swinson\n\nLinda Swinson\n\nMargareta Thomson\n\nCatriona White\n\nLois Yelland\n\n**Stitched in:**\n\nDalkeith, Ontario, Port Seton, Edinburgh, Gorebridge, Eddleston\n\n# PANEL 136  East Kilbride and the New Towns\n\nOn 6 May 1947 East Kilbride was designated as Scotland's first new town. Overcrowded housing in Glasgow was a stimulus for the Clyde Valley Regional Plan and it envisaged satellite new towns and peripheral housing estates to relieve pressure. Planners created neighbourhoods in East Kilbride, each with local shops, services and primary schools, and in the centre was a series of linked malls. When the first residents came, it must have seemed like the New Jerusalem. The old, decrepit tenements of Glasgow, with shared toilets, vermin and a cheek-by-jowl existence were replaced with houses with proper bathrooms and heating. East Kilbride has grown to a population of almost 74,000. In 1948, Glenrothes was founded around a mine that never fully functioned, but that source of employment was replaced by the introduction of major electronics companies. To the east of Glasgow, Cumbernauld was founded in 1956 and, near Edinburgh, Livingston in 1962. Irvine completed Scotland's five new towns in 1964.\n\n**Panel stitched by:**\n\nSandy Andrew\n\nAnn Arnot\n\nSandra Douglas\n\nJoanna Young\n\n**Stitched in:**\n\nEskbank, Edinburgh, West Linton\n\n# PANEL 137  The National Health Service\n\nBritain's greatest post-war achievement was the foundation of the National Health Service. This key element of the incoming Labour government's manifesto was difficult to deliver. Many doctors had been used to charging for their services and were unwilling to cooperate but Aneurin Bevan, the Minister of Health, later said he had 'stuffed their mouths with gold'. In his book, _In Place of Fear_ , Bevan wrote that 'no society can legitimately call itself civilised if a sick person is denied medical aid because of a lack of means'. That remains the guiding principle of the NHS. The National Health Service (Scotland) Act brought the new service into being with the intention of care being free at the point of need. Some prescription charges were brought in and have continued in England but, in Scotland where healthcare is devolved, there are no such charges. With an ageing population living longer, the cost of the NHS in Scotland is rising and stood at \u00a311.35 billion for 2010\u20132011. But the service is also Scotland's largest employer. In 2007, the Scottish government announced its opposition to partnerships between the NHS and the private sector and, a year later, it abolished the much-resented car parking charges at all hospitals except those funded by a private finance initiative scheme in Dundee, Edinburgh and Glasgow. The history of the NHS in Scotland is filled with remarkable statistics. In the year it was founded, around 500,000 Scots who needed them were prescribed spectacles and, partly due to consistent campaigning, the number of smokers in Scotland has declined from 80 per cent in 1954 to 23 per cent in 2011.\n\n**Panel stitched by:**\n\n_Friends in Fine Embroidery_\n\nLydia Lawson\n\nJune McAleece\n\nIrene Wood\n\n**Stitched in:**\n\nDunfermline, Dalgety Bay, Kirkcaldy\n\n# PANEL 138  Television Arrives\n\nIn the 1950s, the BBC's monopoly was broken with the introduction of ITV in Scotland. STV was first, launching in 1957, and its programmes were what one regulator called 'distressingly popular'. The _One O'Clock Gang_ developed a devoted following and the programmes STV took from the other ITV companies soon out-rated the BBC's. Based in Glasgow, STV served the Central Belt and most of the population. Grampian Television set up in Aberdeen in 1961 and, to serve southern Scotland, Border Television began broadcasting from Carlisle in the same year. Diversity was the keynote as three separate daily news services were added to the BBC's national bulletins. And, as brands, STV, Grampian and Border were well recognised and well loved. They also supplied a local training ground for many behind and in front of the cameras. Many famous faces first appeared on _Scotland Today_ , _North Tonight_ or _Lookaround_. These three stations supplied a huge volume of local features, news and drama for about 40 years. But, when Prime Minister Margaret Thatcher decided that the ITV companies should make cash bids to renew their franchises and takeovers would be allowed, the old system began to disintegrate. Most of the invaluable regional coverage supplied by STV, Grampian and Border was swept away in a series of mergers and regulatory changes after 2008.\n\n**Panel stitched by:**\n\n_Musselburgh Stitchers_\n\nGaynor Allen\n\nSusan Finlay\n\nSue Henderson\n\nRosemary Taylor\n\n**Stitched in:**\n\nMusselburgh\n\n# PANEL 139  The Washer Women\n\nAs Scotland became more affluent and the cost of white goods in particular came within the reach of ordinary people, society shifted a little. Washing machines installed and operated at home effected one of the most striking changes. The 'steamies' or the communal washhouses began to close down. Before the coming of cheaper appliances, most women had taken their weekly wash to the steamie and these buildings, still recognisable in many towns and cities, were social hubs, part of the fabric of a neighbourhood. Also, professional washerwomen became redundant. And clothes probably became a little less well cleaned and pressed.\n\n**Panel stitched by:**\n\nGail Hughes\n\nKate MacKenzie\n\n**Stitched in:**\n\nMontrose\n\n# PANEL 140  Cumbernauld New Town\n\nThe new town was originally designed without any pedestrian road crossings and was widely believed to be dominated by the needs of drivers rather then walkers. In 2002, Cumbernauld was voted the worst town in Scotland and, in 2005, its Town Centre won an unwanted public nomination for demolition in a Channel Four TV series. But action was taken. Parts of the Town Centre were indeed demolished and the housing stock's high quality was emphasised. And far from being a concrete jungle, Cumbernauld has space and greenery and the Campsie Fells can be seen from many places. In 2010, the new town was voted the most improved town in Scotland.\n\n**Panel stitched by:**\n\n_Boulton_ \\+ _Conley_\n\nElizabeth Boulton\n\nHelen Conley\n\n**Stitched in:**\n\nCumbernauld\n\n# PANEL 141  North Sea Oil\n\nIn September 1965, a British Petroleum drilling rig found gas under the bed of the North Sea and, four years later, oil was also discovered. As the huge Brent and Forties fields came on stream, Britain began to enjoy an unlikely oil boom. Aberdeen became a city of business and high property prices while construction further north at Nigg Bay and terminals at Sullom Voe on Shetland have supplied employment. Linked by a 230km-long pipeline to the Piper and Occidental fields, Flotta on Orkney is also a major focus. Grangemouth is the centre of the petrochemical industry. Scotland became and remains the largest oil producer in the European Union and it is estimated that the industry employs more than 100,000, about 6 per cent of the total workforce. Production has begun to decline but vast reserves, perhaps 20 billion barrels, still remain under the chill waters of the North Sea. Most oil fields will remain in production until 2020 and, as oil prices rise, companies are very motivated to explore and exploit further. Teams have already been exploring more of the north-east Atlantic Basin around the Hebrides and to the west of Shetland \u2013 areas that were previously thought to be uneconomic.\n\n**Panel stitched by:**\n\n_Lundie_\n\nCatharina Dessain\n\nJudy Drysdale\n\nKatherine Ellvers\n\nSusan Houstoun\n\nSylvia Learoyd\n\nSusan Macgregor\n\nFiona Macphie\n\nGeorgie Middleton\n\nGeorgie Sampson\n\nCaroline Southesk\n\nMargaret Taylor\n\nGriselda Thornton-Kemsley\n\nPatsy Walker\n\n**Stitched in:**\n\nForfar, Angus, Brechin, Laurencekirk\n\n# PANEL 142  Aberdeen\n\nFor many centuries, Aberdeen has been a vibrant urban centre. Made a royal burgh by King David I between 1124 and 1153, it quickly became busy and, for the times, populous. Its situation at the mouth of the River Dee, with a harbour, good sea and land communications and facing a North Sea trading area all helped to make the city the third largest in Scotland by the 17th century. Uniquely, Aberdeen had two universities. King's College was founded in 1495 by Bishop William Elphinstone and, in 1593, Marischal College was endowed by George Keith, the Earl Marischal of Scotland. The two universities only united in 1860 and they are the fifth oldest in the English-speaking world. In the first half of the 19th century, Aberdeen redeveloped its harbour as the fishing and shipbuilding industries became busier. In the later 20th century, the oil industry once again filled the quaysides with ships that serviced the rigs and most oil companies run offices in the city. The impact on house prices has been marked. Perhaps one of the most glorious chapters in Aberdeen's history has been the phenomenal success of its football team under the management of Alex Ferguson. During Ferguson's reign, they won the Scottish League Premier Division three times, the Scottish Cup four times, The Scottish League Cup once and both the European Cup Winners' Cup and the European Super Cup in 1983.\n\n**Panel stitched by:**\n\n_Inverdon Stitchers_\n\nNeil McMillan\n\nMary Middleton\n\nDiane Stanley\n\nUrsula Thompson\n\n**Stitched in:**\n\nBridge of Don\n\n# PANEL 143  Linwood and the Hillman Imp\n\nAmongst the wreckage of Britain's once-vibrant car industry, the story of the Hillman Imp stands out. To compete with the Mini and find a niche as a second car for more affluent families, the Rootes Group took a huge gamble. Without much experience of building smaller cars, they built a huge new, computerised plant at Linwood, west of Glasgow, which opened on 2 May 1963. It lay close to a rail depot and its output could be transported quickly to showrooms all over Britain. But an early problem was that the railway was also needed to bring parts north from the Rootes plant at Ryton, near Coventry. Affordable, good-looking and innovative, the Imp initially sold well but it proved to be unreliable and sales dropped off. Half of the whole output of Imps were sold in the first three years of production. But it was seen as a Scottish car and continued to sell well north of the border. Rootes were bought by the American car maker, Chrysler, in 1967 but, despite price discounting, sales of the Imp declined. Production finally ceased in 1976 and, five years later, the Linwood plant closed. But Imps live on and models are lovingly cared for, many of them polished in Scottish garages.\n\n**Panel stitched by:**\n\n_Tillicoultry 'Needles & Gins'_\n\nMargaret Callander\n\nMorag Clark\n\nShirley Galletly\n\nJoan Gibson\n\nMyra Legge\n\nGill Pritchard\n\nLesley Thornton\n\n**Stitched in:**\n\nDevonside, Alva, Tillicoultry, Menstrie, Cambusbarron\n\n# PANEL 144  Pop Music Booms\n\nWhen the Beatles, the Rolling Stones and many other groups broke on to the popular music scene in the early 1960s, their success had at least two effects. First, it quickly became clear that young people were a separate market for much more than music. Fashion changed and very large numbers of teenagers and men and women in their twenties suddenly appeared to have spending power \u2013 enough money to create many new markets. The second effect was the worldwide success of British pop. It inspired a generation of Scottish musicians and singers. In contrast to the more traditional appeal of Kenneth McKellar and Moira Anderson, the likes of Donovan and Jack Bruce were genuine innovators. Born Donovan Leitch in Maryhill in Glasgow, his 'Catch the Wind' reached number 4 in the UK charts in 1965 and was followed by a string of hits on both sides of the Atlantic. Cream was the name of the band that included Jack Bruce, Eric Clapton and Ginger Baker. Their third album, _Wheels of Fire_ , was the world's first platinum-selling double album. Others added to the variety of pop music coming out of Scotland, ranging from Lulu and the Bay City Rollers to the Incredible String Band.\n\n**Panel stitched by:**\n\n_Jacquie & The Juniors_\n\nImogen Allen\n\nJacquie McNally\n\nCharlie McNally\n\n**Stitched in:**\n\nMusselburgh\n\n# PANEL 145  Glenrothes\n\nHaving been established in central Fife to house miners who were to work at the new Rothes Colliery, this new town grew out of failure, prospered and became the administrative capital of the region. Opened by the Queen and designed as the first super-pit with state-of-the-art technology, the new colliery was supposed to produce 5,000 tons of coal a week but, beset by geological problems and flooding, it lasted only four years and closed in 1963. But Silicon Glen came to the rescue. A Scottish variant of Silicon Valley south of San Francisco, this was a collective name for the establishment of hi-tech industry in Scotland. In 1960, Hughes Aircraft opened its first factory outside of the USA to make semiconductors. It was followed by Elliott Automation in 1965 and several others. Rodime of Glenrothes pioneered the 3.5-inch hard disk drive in 1983 and it collected royalties from many manufacturers. The town was well laid out with open spaces and good access from the A92 dual carriageway. It had merged well into the countryside and the other villages and small towns around it.\n\n**Panel stitched by:**\n\n_The Coo's Tail_\n\nRuby Henderson\n\nJan MacArthur\n\nPatricia Macindoe\n\nMairi Stewart\n\n**Stitched in:**\n\nOld Kilpatrick, Helensburgh, Glasgow\n\n# PANEL 146  The Upper Clyde Shipbuilders\n\nUCS was formed in 1968 from a group of shipbuilders on the Clyde to become more competitive and achieve economies of scale. But very quickly, in 1971, the company went into receivership after the Conservative government of Edward Heath refused a loan of \u00a36 million. The consortium had a full order book and forecast profits for 1972. Instead of going on strike, the workforce voted to continue working and fulfil the orders. The UCS work-in was led by Jimmy Reid, Jimmy Airlie, Sammy Gilmore and Sammy Barr. All were members of the Communist Party and Reid was a gifted orator. In a speech to the workforce, he emphasised that their demeanour and image were vital. The world was watching and there would be 'no hooliganism . . . no vandalism . . . [and] no bevvying'. The work-in successfully stirred public sympathy, cash was raised and, when John Lennon sent a cheque for \u00a35,000, some wag feigned amazement, saying, 'But Lenin's deid.' In 1972, the government agreed to restructure the yards around two new companies \u2013 Govan Shipbuilders and Scotstoun Marine Ltd \u2013 and to inject \u00a334 million. It was a victory and shipbuilding continued on the Clyde.\n\n**Panel stitched by:**\n\n_West of Scotland Guild of Weavers, Spinners and Dyers_\n\nSusan Black\n\nMargaret Cameron\n\nAlison Christie\n\nLyn Dunachie\n\nBron Ellis\n\nMaryel FitzRandolph\n\nJean Mabon\n\nLiz MacKinlay\n\nMargaret McBlane\n\nJoan McDowall\n\nChristina McLachlan\n\nMarlen McMaster\n\nKatie Shirley\n\nFlora Smith\n\nCaroline Thomson\n\nMary Wilkinson\n\n**Stitched in:**\n\nKilmacolm, Glasgow, Paisley, Fintry, Johnstone\n\n# PANEL 147  Stop Yer Ticklin', Jock!\n\nPrincipally known as a singer and songwriter, Harry Lauder was also very funny. He was the first British artiste to sell a million records and his songs, such as 'Keep Right on to the End of the Road', 'Roamin' in the Gloamin'' and 'A Wee Deoch-an-Doris' are still popular. Lauder came out of the music hall traditions and music has long been associated with Scottish comedy. Perhaps the greatest of Scottish comedians, Billy Connolly, began his performing life with the Humblebums and played the banjo. But there have been other strands. Chic Murray had a surreal brand of humour and Scottish sketch comedy with the likes of Rikki Fulton has been sublime. The Rev. I. M. Jolly is an immortal character. There appears to be such a phenomenon as Scottish humour and what is cheering is that, through the success of Connolly and others, it is understood worldwide.\n\n**Panel stitched by:**\n\n_Liz & Marilyn_\n\nBarbara Bell\n\nUrsula Doherty\n\nElizabeth Duke\n\nTessa Durham\n\nCatherine Jones\n\nMarilyn Nicholson\n\nDawn White\n\n**Stitched in:**\n\nDalkeith, Dunbar\n\n# PANEL 148  The Rise of the Scottish National Party\n\nArguably the most dynamic force for change in post-war Scottish politics has been the Scottish National Party. It has grown from little more than an irrelevant fringe to become the majority party of government in the new Scottish Parliament. Its rise began with Winnie Ewing's stunning victory at the Hamilton by-election of 1967 and continued with the victories of 11 MPs in the 1974 general election. Ewing has been a central figure in the success of the SNP ever since, winning election after election. Her party forced serious consideration and eventual implementation of Scottish devolution and the resulting Scottish Parliament is now dominated by the SNP. The first referendum on the issue was held in 1979 but hi-jacked by a London Labour MP, George Cunningham. He successfully argued that, if majority of Scots voted yes, then it would have to reach 40 per cent of the electorate. In the event, there was a small majority but it fell short of that percentage. When Labour came to power in 1997, a second referendum was held and the result was an overwhelming majority, 74%, in favour of a Scottish Parliament.\n\n**Panel stitched by:**\n\n_Stitchers Ecosse_\n\nFrances Cohen\n\nRuth Currie\n\nFrances Gardiner\n\nRhona MacKenzie\n\nLinda Watson\n\n**Stitched in:**\n\nLongniddry\n\n# PANEL 149  Scotland at the Movies\n\n'The name's Bond, James Bond.' Immortal words, uttered by Sean Connery, they established him as one of the most famous film actors of all time. Other Scots, such as Ewan McGregor, Brian Cox, Kelly Macdonald, Tom Conti and Tilda Swinton, have followed. Behind the camera, Alexander MacKendrick, Bill Forsyth and Lynne Ramsay have directed memorable work. And one of the most commercially successful films of all time, _Braveheart_ , told the story of a Scottish hero \u2013 William Wallace. Definitions can be elusive. Many films have been made in Scotland but can Alfred Hitchcock's wonderful _The 39 Steps_ be called a Scottish film? It was adapted from a novel by John Buchan, a Border Scot but directed by an Englishman and had English actors in lead roles. Even Peggy Ashcroft played a crofter's wife. Perhaps one of the very best was _Local Hero_ , written and directed by Bill Forsyth in 1983. It starred Americans, Peter Riegert and Burt Lancaster, but also Denis Lawson and Fulton Mackay. But what it portrayed was very Scottish \u2013 a Lowlander's use of imagined Highland attitudes to create a modern fairy tale. And, with its superb dialogue and acting and a haunting theme by Mark Knopfler, it was very successful.\n\n**Panel stitched by:**\n\n_Boatie Blest Stitchers_\n\nJackie Berg\n\nJohn Berg\n\nCarmel Daly\n\nJon Gerard\n\nBernie Goslin\n\nAgnes Greig\n\nMarion Harkin\n\nLucy Hyde\n\nShelly Jones\n\nGareth Jones\n\nJennifer Nesbit\n\nBill Peach\n\nMartine Robertson\n\nJoyce Souness\n\n**Stitched in:**\n\nPort Seton\n\n# PANEL 150  Scotland's World Cup in Argentina\n\nDashed hopes and overblown optimism characterise one of the most calamitous episodes in Scotland's sporting history. Throughout the 1960s and 1970s, Scotland's football team had occasionally been inspired and, when they left for the 1978 World Cup in Argentina, crowds filled Hampden Park and lined the route to Prestwick Airport. They believed they could win the trophy. An opening defeat by Peru made that seem unlikely and a dismal draw against Iran meant that they had to defeat a brilliant Dutch team by three clear goals to avoid going out of the competition. Finally, playing the football of which they were capable, they beat Holland 3\u20132, with Archie Gemmill scoring one of the greatest goals of all time. From the ridiculous to the sublime, as so often with Scottish sport. At the time of writing, national football languishes in mediocrity, with Scotland ranked 50th in the world, one place below the Cape Verde Islands but above Panama. Fans must sometimes pine for the crazy, heady days of 1978 and marching with Ally's Army.\n\n**Panel stitched by:**\n\n_Allanwater Stitchers_\n\nAnn Gambles\n\nLibby Hughes\n\nFrances Rankin\n\nCatriona Whitton\n\n**Stitched in:**\n\nDunblane\n\n# PANEL 151  The Miners' Strike\n\nThe Miners' Strike of 1984\u201385 seemed like the end of an era. In the 1970s, the National Union of Mineworkers had humbled a Conservative government but Margaret Thatcher was determined that history would not repeat itself. After stockpiling fuel and deploying police forces all over the country, she succeeded in forcing the miners back to work and pushing through a programme of closure. Secret papers recently released show that she was willing to involve the army in an industrial dispute. Thatcher was fortunate to face Arthur Scargill, a miners' leader who polarised opinion but whose predictions of the demise of deep mining turned out to be accurate. In Scotland, many famous collieries, such as Monktonhall, Bilston Glen, Polmaise, Seafield and Longannet, shut down in the year following the unsuccessful strike. There are no working deep mines left in Scotland and heavy industry has also been drastically reduced. The strike not only changed the face of Scottish industry, it also altered the course of Scottish politics and some would argue not for the better.\n\n**Panel stitched by:**\n\nAgnes Greig\n\nPauline O'Brien\n\nLibby O'Brien\n\n**Stitched in:**\n\nTranent, Prestonpans\n\n# PANEL 152  Gaelic Resurgent\n\nIn a very surprising announcement, the Conservative Secretary of State for Scotland, Malcolm Rifkind, announced the creation of a fund to pay for Gaelic television programmes. On 18 December 1989, he outlined a plan to provide \u00a38 million for programmes to be broadcast by the Scottish ITV companies, STV and Grampian. Two hundred hours of new programming was to be commissioned by the Gaelic Television Committee. It proved a massive stimulus and, with the success of the likes of _Machair_ , a Gaelic-language soap opera, it put Gaelic back at the centre of Scottish life. At the same time, Gaelic rock music was becoming more and more popular through the phenomenon of Runrig. Formed in Skye, this remarkable band developed a hybrid Gaelic\u2013English form of rock music which often used the narratives of Highland history as its subject matter. The songs were written by the brothers Rory and Calum Macdonald, and the lead singer, Donnie Munro, possessed a sublime voice. Others, such as Capercaillie, were also popular bands. But the twin impact of Gaelic television and Runrig prevented Gaelic from fading from the national agenda. The number of speakers is tiny but there appears to exist a political will to ensure survival of some sort.\n\n**Panel stitched by:**\n\nChristine Haynes\n\nPauline Elwell\n\n**Stitched in:**\n\nMorar\n\n# PANEL 153  Glasgow \u2013 European City of Culture\n\nReinvention lay at the heart of Glasgow's successful bid to become European City of Culture in 1990. An idea formed by the actress Melina Mercouri, the Greek Minister of Culture, and her French counterpart Jack Lang, the intention was to highlight both the diversity and commonality of European culture. Athens, Florence, Amsterdam, Berlin and Paris all preceded Glasgow and all were perhaps more obvious choices. But the old industrial heart of Scotland rose brilliantly to the challenge. All of the city's galleries and theatres took part and some were specially created. At a huge Harland & Wolff engine shed, a play called _The Ship_ recalled great industrial traditions. Written by Bill Bryden, it was enormously popular. The celebrations ran for a year and Glasgow capitalised on the exposure to promote itself as a tourist destination.\n\n**Panel stitched by:**\n\nFiona Hamilton\n\n**Stitched in:**\n\nEast Renfrewshire\n\n# PANEL 154  Dolly the Sheep\n\nBorn in July 1996, Dolly, the first mammal to be cloned, became the world's most famous sheep. At the Roslin Institute near Edinburgh, Ian Wilmut, Keith Campbell and their colleagues succeeded in creating Dolly from an adult somatic cell using the process of nuclear transfer. The donor cell was taken from a mammary gland and the birth of the sheep proved that a cell taken from a specific part of the body could be used to create a complete individual. When Ian Wilmut was asked where the name of Dolly came from her replied, 'Dolly is derived from a mammary gland cell and we couldn't think of a more impressive pair of glands than Dolly Parton's.' Having lived her whole life at the Roslin Institute and given birth to six lambs, Dolly died in 2003. Since she was born, many other large mammals, such as bulls and horses, have been successfully cloned.\n\n**Panel stitched by:**\n\nYvonne Beale\n\n**Stitched in:**\n\nOrkney\n\n# PANEL 155  The Scottish Parliament Reconvenes, 1999\n\nWhen Sheena Wellington sang Robert Burn's great lyric, 'A Man's a Man for A' That' at the opening of the Scottish Parliament in 1999, her beautiful, crystal voice caught the mood of the nation perfectly. All rejoiced on 1 July, Scotland's day. And setting party allegiances aside, all agreed that Winnie Ewing MSP should preside over the opening session. Her memorable words were:\n\nThe Scottish Parliament, adjourned on the 25th day of March in the year 1707, is hereby reconvened.\n\nAnd despite initial scepticism that Scotland was the most over-governed country in the world and the appalling cost overruns on the construction of the parliament building, the new institution has been a success. It is now difficult to imagine Scotland without MSPs and the debates at Holyrood. No doubt the Scottish Parliament will change, as all vibrant political institutions do, but it is unlikely to disappear as it did in 1707.\n\n**Panel stitched by:**\n\n_Scissor Sisters_\n\nLinda Jobson\n\nIsobel Reilly\n\nDorie Wilkie\n\n**Stitched in:**\n\nEskbank\n\n# PANEL 156 AND 157  Parliament of the Ancestors, Parliament for the Future\n\nHistory is about change and change always involves loss. But moments such as the opening of the Scottish Parliament may be seen as a pause and the freeze-frame of a tapestry panel is an attempt to catch that moment and hold it still. The Great Tapestry of Scotland may never end, may be added to over the coming times, but this version pauses with a Parliament of the Ancestors, the men and women who helped make Scotland, and a Parliament for the Future. Two panels show leaders \u2013 the four Presiding Officers and the four First Ministers who have held office since 1999. Other living Scots are there and our ancestors below and beside them. They are all flanked by the stitchers \u2013 the women who not only made the tapestry but also made a version of Scotland when they first picked up their needles and thread.\n\n**Panel 156 stitched by:**\n\n_Cupar Needles_\n\nIshbel Duncan\n\nLisbeth Kervell\n\nJoyce MacRae\n\n**Stitched in:**\n\nCupar, Ladybank, St Andrews\n\n_Liberton Ladies_\n\nShirley Dawson\n\nSheila Farquhar\n\nDorothy Morrison\n\nSylvia Robertson\n\nAnn Weir\n\nMargaret Wilson\n\n**Stitched in:**\n\nEdinburgh\n\n_Firth of Forth Stitchers_\n\nSheila Chambers\n\nCelia Mainland\n\nJenny Mayor\n\nIsabel Weaver\n\n**Stitched in:**\n\nPrestonpans, Port Seton\n\nSylvia Robertson\n\n**Stitched in:**\n\nPitlochry\n\n**Panel 157 stitched by:**\n\n_Hands Sewlo_\n\nSheila Chambers\n\nSheena Dolan\n\nCelia Mainland\n\nJenny Mayor\n\nIsabel Weaver\n\n**Stitched in:**\n\nPrestonpans, Port Seton, Edinburgh\n\n_Strathmore Stitchers_\n\nDoris Black\n\nNancy Craig\n\nMary Daun\n\nValona Gouck\n\nSusan Greaves\n\nSheila Hawick\n\nNanette Henderson\n\nVal McDonald\n\nTessa Mendez\n\nIsobel Ovens\n\nAlison Robertson\n\nMargaret Stanford\n\nIssy Valentine\n\nIrene Ward\n\n**Stitched in:**\n\nAlyth, Forfar, Letham, Guthrie, Broughty Ferry\n\nDoreen Scotland\n\nNan Duffy\n\nDilly Emslie\n\nMaggie Sturrock\n\nTeresaWallace\n\n**Stitched in:**\n\nEdinburgh\n\n_Sewster Sisters_\n\nAnnabelle Broadhurst\n\nAlexa Dewar\n\nCatherine Gerrard\n\nWillie Grieve\n\nFanny Grieve\n\nJane Jowitt\n\nIrene Martin\n\nNorma McCaskill\n\nKaren Skilling\n\nElspeth Turner\n\n**Stitched in:**\n\nEdinburgh, Lasswade\n\n**Panel stitched by:**\n\n_Strathendrick Stitchers_\n\nMargaret Burgess\n\nLyn Dunachie\n\nMargaret Gibb\n\nMargaret Harrison\n\nChristina McLachlan\n\nMoira Murray\n\nCarol Omand\n\nMorag Proven\n\nSusan Rhind\n\nBette Scott\n\nMarion Tyson\n\nPam Waller\n\nHeather Wright\n\n**Stitched in:**\n\nGlasgow, Mingavie, Fintry, Bearsden, Drymen, Blanefield, Killearn\n\n**Panel stitched by:**\n\nMeg Porteous\n\n**Stitched in:**\n\nEdinburgh\n\n# Endnotes\n\n. Panel to be completed after publication.\n\n. Panel to be completed after publication.\n","meta":{"redpajama_set_name":"RedPajamaBook"}}
+{"text":"\n# **Making European Breads**\n\n**Glenn Andrews**\n\n## **CONTENTS**\n\n**Introduction**\n\n**Basic Breadmaking Techniques**\n\n**Using the Best Ingredients**\n\n**Recipes for European Breads**\n\nGrambrot \u2014 Hearty German Wheat Bread\n\nSicilian Semolina Bread\n\nPumpernickel\n\nKatie's Bulgur Bread\n\nLimpa \u2014 Swedish Rye Bread\n\nBasque Sheepherders' Bread\n\nFrench Bread\n\nCiabatta\n\nFocaccia\n\nAlsatian Mixed-Grain Bread\n\nPortuguese Sweet Bread\n\nIrish Soda Bread\n\nChristiania Kringle\n\nBrioche Bread\n\nGerman Farmhouse Graham Bread\n\n**Using European Breads **\n\nBruschetta\n\nHerbed Bread\n\nGarlic Bread\n\nAmerican Fondue\n\nThe Alpine Sandwich\n\nCroute au Fromage\n\nPanzanella \u2014 Italian Bread Salad\n\nOyster Loaf \u2014 A Pacifier\n\nNew Orleans Bread Pudding\n\nProvence Spread\n\n## **Introduction**\n\nThe breads of Europe are wonderful! They're honest, healthful, and greatly loved around the world. These days you can find a variety of European breads not only at bakeries but also at your local grocery store.\n\nYou'll find two important chapters immediately following this introduction. The first gives detailed instructions for basic bread-making techniques, and the second describes the ingredients used. Please read these sections carefully before you begin, and refer back to them as you bake (unless you're already a semipro at bread making) \u2014 the recipe instructions will assume that you know the information contained in the techniques section (below).\n\nAfter the actual bread recipes, you'll find some great ideas for using what you've made \u2014 Bruschetta, Garlic Gorgonzola Bread, Alpine Sandwiches, American Fondue, Italian Bread Salad, and many others.\n\nHave fun! \n\n## **Basic Breadmaking Techniques**\n\nMany of the same techniques are used for making different kinds of yeast breads. The most basic steps include proofing, combining the ingredients, kneading, rising, shaping, baking, and cooling.\n\n**Proofing.** Proofing has two functions: It activates the yeast, and it's a way to be sure that the yeast is still good (it may have expired while on the store or kitchen shelf). To proof your yeast, stir it into warm (95\u2013110\u00b0F) water, usually along with a little sugar (the exact amount will be specified in each recipe). Allow the mixture to sit for five minutes, or until it is foamy. If the mixture does not foam after five minutes, then you know that your yeast is no longer active \u2014 throw it out and start again with a fresh supply.\n\n**Combining the ingredients.** Mixing is done differently for different breads, so refer to the individual recipes for complete instructions. However, if you are using a food processor to combine the ingredients and knead the dough, put all of the dry ingredients into the bowl first and mix them well before adding and processing the other ingredients.\n\n**Kneading.** Here's where you finish up the combining process and give your bread texture. Many bread bakers do this in a food processor, pulsing off and on until a nice dough is formed, then giving the dough a final turn or two by hand.\n\nTraditional kneading is done by hand on a floured board. Keeping your hands well-floured, form the dough into a pancake shape. Use the heels of your hands to push the dough away from you. Then fold the dough over and rotate it a quarter turn. Keep doing this, adding more flour to the board and your hands as necessary, until the dough is elastic and has lost its stickiness. A proper dough has been accurately described as feeling like a baby's bottom! (Doughs containing rye flours, however, will always seem a little sticky.)\n\n**Rising.** After the dough has been kneaded, place it in a greased bowl and rotate it until the dough is greased on all sides. Cover the bowl with a damp towel and let it sit in a warm place (between 80\u00b0 and 85\u00b0F, preferably) until the dough has roughly doubled in size (usually about an hour). A gas oven with a pilot light is often the perfect temperature for rising. If you have an electric oven, maintaining the proper temperature may be difficult. The best way to achieve the proper temperature for rising is to keep the heat in the oven off, boil a pan of water and place it on the lower rack of the oven, and put the covered bowl on the higher rack.\n\nThe dough of yeast breads will often need to rise more than once. See the recipe instructions for the appropriate number of rising times.\n\n**Punching down.** Punching down is the more emphatic term for deflating risen dough. Just bang your fist into the center of the risen dough \u2014 this is fun! A thorough punching down also involves turning the dough out of the bowl onto a floured board and kneading it just a few turns.\n\n**Shaping.** Before it can be baked, the dough needs to be shaped into a loaf form. For an elongated loaf, flatten the dough into a rectangle; the shorter end represents the length of the bread you want or the pan you're using. Roll up the dough tightly from this short end, then pinch the ends and tuck them under.\n\n**Baking.** Breads can be baked on baking sheets or, if they're in pans, right on the oven's racks. But, especially for French bread, focaccia, or any other free-form bread, it really is nice to use a baking stone, which approximates an authentic baker's brick oven. These are rather expensive, though. You might want to try an easy substitute: Purchase some unglazed terra-cotta or quarry tiles from a store that sells ceramic tile. They're usually inexpensive, and come in 6-inch squares \u2014 if you buy nine of them, you'll have an area 18 inches by 18 inches, about the size of a baking stone, and you'll have spent about one-sixth as much money. Use these tiles to line a regular oven rack and bake your bread on top of them.\n\n**Testing for doneness.** The classic way to find out if a loaf has baked long enough is to remove it from the pan and rap it with your knuckles. If the bread sounds hollow, it's done.\n\n**Finishing the bread.** Sometimes it's nice to rub the top of a warm baked loaf with a little soft butter or milk. With or without that final touch, put the finished loaf on a rack until it's completely cool. If the bread was baked in a bread pan, let it cool in the pan for about 15 minutes before removing the loaf and setting it on the rack.\n\n**Keeping bread on hand.** When the bread is thoroughly cool, put it into a plastic bag. Keep it at room temperature for immediate use or freeze it for later use. Bread can be frozen and thawed any number of times \u2014 but if you want the bread sliced, it's best to do so before you freeze it.\n\n## **Using the Best Ingredients**\n\nIt's truly amazing how such a limited number of ingredients can turn out so many different \u2014 and wonderful \u2014 breads. The quality of ingredients you use can make the difference between spectacular breads and those that are just ho-hum. So go for the best!\n\n**Flour.** Ideally, you should try to find whole-grain flours (including cornmeal) that are freshly milled. If you're lucky, you'll have a local food co-op or health-food store that grinds different types of flour in-store and keeps them refrigerated. Otherwise, buy the best brands you can find. White flour (also called all-purpose) can be unpredictable. If possible, use flour that is labeled bread flour, which will be especially rich in gluten, an important ingredient for high-rising breads. Other than bread flour, by far the best flour to use is unbleached white, but measuring the proper amount to use is sometimes difficult. Start by using a little less than the recipe calls for, then keep adding more, spoonful by spoonful, until you get a proper dough (see page 3 for a description of a proper dough).\n\n**Yeast.** Long ago, yeast was only available in cake form. Today, you'll find cakes of yeast occasionally, but usually what you'll see is called active, or dry, yeast. You can buy it in packets from your local grocery store or in bulk at health-food stores and co-ops. There's also \"Rapid Rise\" yeast, which seems very vigorous. (Ignore the package instructions and use it just as you would any other dry yeast.) One package of dry yeast contains about 2\u00bc teaspoons.\n\n**Liquid.** Use the liquid called for in the recipe instructions \u2014 but if you're feeling inventive, you can experiment with other liquids such as beer or orange juice. Water is the most common liquid used in European breads, but be sure it's good clean water. If you have to buy bottled springwater to ensure purity, it's worth the cost to avoid any \"off\" tastes that impure water may impart to your finished bread.\n\nIt's also important that the liquid used in the proofing process (see page 2) be at the proper temperature. If it's too cool, the bread making will be slowed down. If it's too hot, the yeast will be killed and your bread won't rise properly. The temperature you want the liquid to be is between 95\u00b0 and 110\u00b0F, just a comfortable lukewarm.\n\n**Sweetener**. Yeast thrives on sugar, so most breads contain at least a little bit of it, usually added when the yeast is proofing. Honey can be substituted for small amounts \u2014 up to a tablespoon \u2014 of cane sugar, but on no account should you use artificial sweeteners!\n\n**Fat.** If butter is called for, use just that \u2014 butter. Margarine is not suitable for bread making (though I have heard that it's widely used in Denmark in their pastries). If it's olive oil that's called for, use the extra-virgin kind, which is the purest and has the best taste.\n\n**Salt.** Any salt will do, but sea salt is especially lovely. Don't skimp here \u2014 undersalted bread is terrible, and the amount of salt that actually ends up in each slice of bread is quite small.\n\n## **Converting Recipe Measurements to Metric**\n\nUse the following formulas for converting U.S. measurements to metric. Since the conversions are not exact, it's important to convert the measurements for all of the ingredients to maintain the same proportions as the original recipe.\n\nWhile standard metric measurements for dry ingredients are given as units of mass, U.S. measurements are given as units of volume. Therefore, the conversions listed above for dry ingredients are given in the metric equivalent of volume.\n\n## **Recipes for European Breads**\n\nThere are so many recipes for European breads that it would be impossible to list them all in one place. Here is a sampling of some of my favorite varieties \u2014 experiment at will with the flavoring ingredients to create new and wonderful variations of some time-honored recipes.\n\n**GRAMBROT**\n\n**_Hearty German Wheat Bread_**\n\n_This is an old German recipe that suits today's tastes \u2014 there's no fat and no sugar!_\n\n**1 package dry yeast**\n\n**1\u00bd cups warm water**\n\n**1 teaspoon salt**\n\n**3 cups whole-wheat flour**\n\n**1 cup white flour**\n\n1. Proof the yeast in the warm water. Use oil or nonfat margarine spread to grease the inside of a large bowl.\n\n2. In another large bowl, combine the salt, whole-wheat flour, and white flour. Add the proofed yeast mixture and mix well until you've achieved a proper dough.\n\n3. Knead the dough for 10 minutes on a floured surface, then place it in the greased bowl, rotating the dough until it is greased all over. Cover the bowl with a moist cloth, set it in a warm place, and allow it to rise until roughly doubled.\n\n4. Punch down the dough, then shape it into a loaf and place it in a roughly 8 x 4-inch bread pan. Set in a warm place to rise. Preheat the oven to 375\u00b0F.\n\n5. When the loaf has roughly doubled in size, place it in the oven. Bake for 25 to 30 minutes. Remove from the oven and set on a rack to cool.\n\n**SICILIAN SEMOLINA BREAD**\n\n_The flour you want here is the very fine, high-gluten semolina used for making pasta. It's as soft to the touch as bath powder, and makes a yellowish, dense (but light), quite engaging bread._\n\n**1 package dry yeast**\n\n**1 cup warm water**\n\n**1 teaspoon honey**\n\n**1\u00bd cups semolina flour**\n\n**2 teaspoons salt**\n\n**1\u20132 cups white flour**\n\n**1 tablespoon olive oil**\n\n**Egg wash (1 egg beaten with 1 tablespoon water)**\n\n**\u00bc cup sesame seeds (optional)**\n\n1. In a large bowl, proof the yeast in the warm water with the honey. Grease the surface of a large bowl with butter or oil.\n\n2. Stir in first the semolina flour and the salt, then I cup of the white flour.\n\n3. Place \u00be cup of the white flour on a bread board. Place the dough onto the board and knead the flour into it, adding more as necessary. Then put the dough in the greased bowl, cover, and let it rise until it has roughly doubled in size.\n\n4. Punch down the dough and allow it to rise again. Shape into one large round loaf or two oval ones. Use the olive oil to grease a baking sheet. Place your bread on this. Cover loosely and set it in a warm place to rise. Preheat the oven to 375\u00b0F.\n\n5. Let the dough rise only until it's almost doubled (about half an hour). Slash diagonally, two or three times per loaf, across the top(s). Then brush with the egg wash and sprinkle on the sesame seeds, pressing them in lightly. Bake for 35 to 40 minutes, then remove and cool on a rack.\n\n**PUMPERNICKEL**\n\n_Slices of pumpernickel spread with sweet butter were my favorite snack when I was young. Later I learned that most pumpernickel contains cocoa \u2014 now I realize why I love it so much._\n\n**1 package dry yeast**\n\n**3\/4 cup warm water**\n\n**\u00bc cup molasses**\n\n**1\u00bd teaspoons salt**\n\n**1 tablespoon dry cocoa**\n\n**2 teaspoons caraway seeds (optional)**\n\n**1 cup rye flour**\n\n**1 cup white flour**\n\n**1 cup whole-wheat flour**\n\n**2 tablespoons cornmeal**\n\n**Egg wash (1 egg beaten with 2 tablespoons flour)**\n\n1. Proof the yeast in the warm water with the molasses. Grease the surface of a large bowl with butter or oil.\n\n2. In a large bowl, combine the salt, cocoa, caraway seeds (if you're using them), and rye flour. Add the yeast mixture and mix well.\n\n3. In another bowl, combine the white flour and the whole-wheat flour. Add about 1\u00bd cups of this mixture to the rye-flour mixture and mix well until you've achieved a proper dough.\n\n4. Turn out the dough onto a floured board and knead in the rest of the white and wheat flour mixture. Place in the greased bowl and allow the dough to rise, covered, until it has roughly doubled in size \u2014 this may take an hour or even two.\n\n5. Punch down the dough and let it rise again. Then shape the dough into a round loaf. Sprinkle the cornmeal on a baking sheet, place the loaf on it, cover, and set it in a warm place to rise. Preheat the oven to 375\u00b0F.\n\n6. When the loaf has roughly doubled in size, cut across its top in two or three places with a single-edged razor-sharp knife. Brush with the egg wash. Bake for 40 to 45 minutes.\n\n**KATIE'S BULGUR BREAD**\n\n_Here's a robust and tasty middle-European type of bread, with a few additions divined by a clever cook._\n\n**1 package dry yeast**\n\n**I cup warm water**\n\n**1 cup bulgur wheat**\n\n**3 tablespoons butter**\n\n**1\u00bd cups hot broth (chicken, beef, or vegetable)**\n\n**1\u00bc cups cold water**\n\n**2 teaspoons salt**\n\n**\u00bc cup honey**\n\n**\u00bc cup molasses**\n\n**1 teaspoon caraway seeds**\n\n**3 cups whole-wheat flour**\n\n**3 cups white flour**\n\n1. Proof the yeast in the warm water. Grease the surface of a large bowl with butter or oil, and grease two 8 x 4-inch bread pans.\n\n2. In a large bowl, combine the bulgur wheat and butter. Bring the broth to a boil, stir it in, and let the mixture sit for half an hour. Add the cold water, salt, honey, molasses, caraway seeds, and proofed yeast. Mix well.\n\n3. Now stir in the whole-wheat and white flours. Mix until you've achieved a proper dough, then remove from the bowl and knead well.\n\n4. Place the dough in the greased bowl, cover, and let it rise. Then remove the dough from the bowl, punch it down, and shape it into two large loaves. Put the loaves into the greased bread pans and set them in a warm place to rise. Preheat the oven to 375\u00b0F.\n\n5. When the loaves have risen so that the dough reaches the tops of the pans, place them in the oven. Bake for 40 to 45 minutes, remove from the oven, and allow them to cool on a rack.\n\n**LIMPA**\n\n**_Swedish Rye Bread_**\n\n_Limpa, a rye bread from Sweden, is extraordinarily good. The aromatic touch of orange peel and spice makes it a bread you won't soon forget._\n\n**2 packages dry yeast**\n\n**\u00bc cup warm water**\n\n**\u2153 cup brown sugar**\n\n**\u00bc cup molasses**\n\n**4 tablespoons butter, cut in 4 pieces**\n\n**1 tablespoon salt**\n\n**Grated zest of 1 large orange**\n\n**2 teaspoons anise seeds**\n\n**1 teaspoon fennel seeds (optional)**\n\n**2 teaspoons caraway seeds (optional)**\n\n**13\/4 cups boiling water**\n\n**3 cups rye flour**\n\n**3 cups white flour (or a little less)**\n\n**3 tablespoons milk**\n\n1. Proof the yeast in the warm water along with a pinch of the brown sugar you'll be using later. Grease the surface of a large bowl with butter or oil, and lightly grease a large baking sheet.\n\n2. In another large bowl, combine the molasses, the rest of the brown sugar, butter, salt, orange zest, and seeds. Add the boiling water and stir. When this mixture cools to lukewarm (around 100\u00b0F), add the proofed yeast mixture and mix well. Then stir in the flour, beginning with all of the rye flour, then adding the white until you have achieved a proper dough.\n\n3. Knead the dough for about 10 minutes, adding the rest of the white flour as needed. Put the dough into the buttered or oiled bowl, turning it so that all sides will be greased. Cover the bowl with a damp cloth and set it in a warm place.\n\n4. When the dough has doubled in size, punch it down, cover, and let it rise again.\n\n5. When the dough has doubled in size again, remove it from the bowl. Shape it into two oval loaves, each about 12 to 14 inches long. Place the loaves on the lightly greased baking sheet. Cover and place in a warm spot to rise. (Don't place the loaves too close together, as they will spread.) Preheat the oven to 375\u00b0F.\n\n6. When the loaves have fully risen, slash their tops diagonally in several places, French-Bread style. Bake for 25 to 30 minutes. Remove from the oven, brush the tops with the milk, and allow them to cool on a rack.\n\n**BASQUE SHEEPHERDERS' BREAD**\n\n_The Basque country, high in the Pyrenees between Spain and France, is the ancestral home of this monumental loaf, but it's also made by the Basque sheep-herders of the American West. One loaf will feed \u2014 and make happy \u2014 a good-size crowd!_\n\n**3 cups water**\n\n**\u00bd cup (1 stick) butter**\n\n**\u2154 cup sugar**\n\n**1 tablespoon salt**\n\n**2 packages dry yeast**\n\n**8\u00bd cups white flour**\n\n1. In a large saucepan, combine the water, butter, sugar, and salt. Heat gently just until the butter melts. Cool to lukewarm, then add the yeast and let it proof for 10 minutes.\n\n2. Mix in 5 cups of the flour, then turn out the dough onto a floured board. Knead in the remaining flour. Form the dough into a ball, place it in the greased bowl, and let it rise, covered, until doubled.\n\n3. Punch down the dough and form it into a ball.\n\n4. Let the dough rest while you prepare the pan. Use a 10-inch Dutch oven. Grease it well, including its cover. Put a circle of foil in the bottom of the pan and grease that, too. Then put in your ball of dough. Put on the lid and set in a warm place to rise. Preheat the oven to 375\u00b0F.\n\n5. Let the dough rise until it pushes the lid up about \u00bd inch. Then bake, with the lid still on, for 12 or 13 minutes. Remove the lid and bake for another 30 to 35 minutes. Turn out onto a rack, remove the foil, and let it cool.\n\n**FRENCH BREAD**\n\n_This is as close as you can come to the perfect bread turned out by French bakeries. It's the large, fat loaf used most often for Garlic Bread (see page 25). I make this in a food processor, putting the flour and salt into the bowl, then adding the yeast mixture followed by the combination of the rest of the ingredients._\n\n_Many recipes for French Bread contain no fat, but the loaves made that way turn stale in about half a day._\n\n**1\u00bd packages (or a scant 3\u00bd teaspoons) dry yeast**\n\n**\u00bc cup warm water**\n\n**\u00bc teaspoon sugar**\n\n**1 cup cold water**\n\n**2 tablespoons butter**\n\n**3\u00bd cups white flour**\n\n**1 teaspoon salt**\n\n**2 tablespoons cornmeal**\n\n1. Proof the yeast in the warm water along with the sugar. Grease the surface of a large bowl with butter or oil.\n\n2. In a small saucepan, mix the cold water and the butter and slowly heat until the butter has melted.\n\n3. In a large bowl, combine the flour, salt, proofed yeast, and melted butter and water. Mix until you have achieved a proper dough. Then knead well, form the dough into a ball, and allow it to rise, covered, in the greased bowl.\n\n4. Punch down the dough, knead it briefly, and let it rise again in the greased bowl.\n\n5. Punch down the dough again and form it into a loaf about 13 inches long. Sprinkle the cornmeal on a baking sheet. Place the loaf on the baking sheet and set it in a warm place to rise. Preheat the oven to 450\u00b0F.\n\n6. When the loaf has roughly doubled in size, use a very sharp knife or a single-edged razor blade to cut three long diagonal slashes in its top. Slide the loaf onto tiles, if you're using them, or put the baking sheet in the preheated oven. Immediately throw 4 ice cubes onto the oven floor. (This will create the steam that would be present in a French baker's oven.)\n\n7. After 5 minutes, throw in 4 more ice cubes. Ten minutes after that, turn the oven down to 400\u00b0F and bake for about 20 minutes more. Then remove the loaf from the oven and allow to cool on a rack.\n\n**Boules:** A boule is simply a round loaf of French bread. Make French Bread dough as directed, letting it rise twice. Then divide the dough in two. One at a time, form the halves into round balls. You want to create a tight skin on each ball, so stretch it tight in all directions, tucking it under as you go. Slash an X in the top of each loaf and bake as directed.\n\n**CIABATTA**\n\n_Most of the Italian bread you see in this country is pretty indistinguishable from French Bread, but not this one! Olive oil makes the difference and gives it a subtle tang. Note also that there's only one rising._\n\n**1 package dry yeast**\n\n**1 teaspoon sugar**\n\n**I cup warm water**\n\n**3\u00bd cups white flour**\n\n**\u00bd teaspoon salt**\n\n**\u2154 cup olive oil (plus a bit more for the top of the loaf)**\n\n1. Proof the yeast with the sugar in the warm water. Grease a baking sheet.\n\n2. In a large bowl, combine the flour and salt. Add the olive oil and the yeast mixture and mix well until you achieve a proper dough.\n\n3. Knead, but not as thoroughly as with other breads \u2014 stop when the dough is still a little rough. Form it into an oval loaf and place it on the greased baking sheet. Coat the top lightly with more olive oil. Cover and set it in a warm place to rise. Preheat the oven to 400\u00b0F.\n\n4. Allow the dough to rise for about an hour. Then sprinkle the top of the loaf with flour. Bake for 25 to 30 minutes, remove from the oven, and allow it to cool on a rack.\n\n**FOCACCIA**\n\n_Focaccia, from Italy, is an example of the exceptional European breads that we have recently discovered in this country. It's a flat bread often served these days in upscale restaurants. Here's the basic recipe, followed by a few variations. The dough also makes a perfect pizza crust._\n\n_This amount of dough makes two Focaccias. You can, if you wish, freeze half of it, before shaping, for future use._\n\n_I once ran out of regular unbleached white flour when making this, and substituted self-rising flour for_ H _cup of it. The result was exceptional!_\n\n**1 package active dry yeast**\n\n**1 \u2154 cups warm water**\n\n**2 teaspoons sugar**\n\n**4 cups unbleached white flour (or 3\u00bd cups unbleached white flour and \u00bd cup self-rising flour)**\n\n**1 teaspoon salt**\n\n**8 tablespoons olive oil**\n\n**Kosher or coarse sea salt**\n\n1. Proof the yeast in the water along with the sugar.\n\n2. In a large bowl, combine the flour and salt. Slowly stir in the proofed yeast and 2 tablespoons of the olive oil. Knead the dough on a floured board for just a few minutes until it's satiny, then cover it, place it in a warm spot, and let it rise for 40 minutes.\n\n3. Punch down the dough, then divide it into two parts. Form each into a flat disk, then stretch each one with your hands into a 12-inch circle. Put onto one or two baking sheets. Cover and set them in a warm place to rise. Preheat the oven to 425\u00b0F.\n\n4. Let the dough rise for 30 to 45 minutes. Then \"dimple\" the dough by pressing gently all over it with your fingertips. Drizzle on the remaining olive oil, then sprinkle lightly with the Kosher salt. Bake for about 20 minutes, or until light brown. Serve warm, cut into wedges. Some people also brush \u00bd cup of melted butter over the Focaccias the moment they come out of the oven. This will disappear into the bread after just a few minutes.\n\n**Rosemary Focaccia:** Before baking the Focaccia, combine \u00bd cup olive oil, 2 tablespoons minced fresh rosemary, and 3 teaspoons minced garlic, mix well and drizzle on top of the dough. Don't try this with dry rosemary \u2014 there just isn't enough moisture in the top-ping or enough time to reconstitute it.\n\n**Sage Focaccia:** Add 2 tablespoons minced fresh sage to the dough as you knead it.\n\n**ALSATIAN MIXED-GRAIN BREAD**\n\n_As the Alsatians have done for many years, here you can add varying combinations of ingredients. Make a three-grain bread, a five-grain bread, or whatever suits your taste._\n\n**1 package dry yeast**\n\n**2 cups water (or 1 cup water and 1 cup milk)**\n\n**2 tablespoons honey**\n\n**1\u00bd cups (total) of any combination of: cornmeal, barley flour (or pearl barley, soaked or parboiled), raw oats, millet, triticale, quinoa, rice flour, or soaked or sprouted wheat berries \u2014 whatever you want! You can also try using an all-natural mixed-grain cereal from a health food store or co-op.**\n\n**1 tablespoon salt**\n\n**2 cups whole-wheat flour**\n\n**3 cups white flour**\n\n1. In a large bowl, proof the yeast in the warm water with the honey. Grease the surface of a large bowl with butter or oil, and grease two 8\u00bd x 4H-inch bread pans.\n\n2. Stir the combination of grains and the salt into the proofed yeast. Then add the whole-wheat flour and 2 cups of the white flour. Turn out the dough onto a floured board and knead in the rest of the white flour.\n\n3. Let the dough rise in the greased bowl, covered, until doubled. Punch down. Shape it into two loaves, place in the greased bread pans, and set them in a warm place to rise. Preheat the oven to 350\u00b0F.\n\n4. When the loaves have roughly doubled again, place them in the oven and bake for about 1 hour. When done, place them on a rack to cool.\n\n**PORTUGUESE SWEET BREAD**\n\n_If you live in New England, you can probably buy Portuguese sweet bread in a supermarket, since so many Portuguese sailors and their families ended up living there. (Your own will be better, though.) It's not the sort of sweet bread you might expect \u2014 just a light and delicious egg bread._\n\n**1 package dry yeast**\n\n**\u00bc cup lukewarm water**\n\n**\u00bd cup plus 1 teaspoon sugar**\n\n**4 tablespoons soft butter**\n\n**\u00bc cup warm milk**\n\n**2 eggs, lightly beaten**\n\n**1\u00bd teaspoons salt**\n\n**2\u00bc cups (or more) white flour**\n\n1. In a large bowl, proof yeast in the warm water with 1 teaspoon of the sugar. Grease the surface of a large bowl with butter.\n\n2. In a small bowl, combine the butter with the warm milk. Then add the rest of the sugar and mix well. Add this to the yeast mixture and stir well. Reserve a small portion (about one-quarter of the total amount) of the beaten eggs and add the rest, along with the salt, to the mixture. Mix well.\n\n3. Then add 2 cups of the flour, 1 cup at a time, kneading it in with your hands, right in the bowl. Move the dough to a floured board and keep on kneading, adding more flour as needed, until the dough is smooth and elastic. Place the dough in the buttered bowl, cover, and allow it to rise until doubled.\n\n4. Punch down the dough and shape into a ball. Put it into a 9-inch buttered ovenproof skillet or a large bread pan. Cover (but not tightly) and set it in a warm place to rise. Preheat the oven to 350\u00b0F.\n\n5. When the dough has risen, brush the top of the loaf with the remaining beaten egg. Bake for about half an hour, remove from the oven, and allow it to cool on a rack.\n\n**IRISH SODA BREAD**\n\n_This is probably the easiest bread you could ever make. Luckily, it's also one of the best, with a smooth texture and great taste._\n\n**4 cups white flour**\n\n**1 tablespoon salt**\n\n**I teaspoon baking powder**\n\n**1 teaspoon baking soda**\n\n**\u00bd cup raisins or dried currants (optional)**\n\n**1\u00bd\u20132 cups buttermilk (or use regular milk combined with 2 teaspoons white or cider vinegar)**\n\n1. Preheat the oven to 375\u00b0F. Butter a baking sheet or cake pan.\n\n2. Mix together the dry ingredients, then stir in the raisins or currants (if you're using them). Add enough buttermilk to make a soft, kneadable dough.\n\n3. Turn out the dough onto a floured board and knead it briefly until you have a proper dough. Shape it into a round loaf and place it on the baking sheet or cake pan. Cut a cross in the top of the loaf with a single-edged razor blade or very sharp, floured knife.\n\n4. Bake for 35 to 40 minutes. Remove from the oven and allow it to cool on a rack.\n\n**Irish Whole-Wheat Soda Bread:** Follow the same recipe as for Irish Soda Bread, above, but use 3 cups whole-wheat flour and 1 cup white flour instead of the 4 cups white flour, and change the amount of baking soda to I teaspoon.\n\n**CHRISTIANIA KRINGLE**\n\n_A friend with a Danish background gave me the recipe for this good-anytime-of-day treat. It's a piecrust sort of dough, topped with a cream puff pastry and lightly glazed and flavored with almond. You could serve it for dessert, but it's at its shining best for breakfast or with morning coffee or afternoon tea._\n\n**_For the pastry base:_**\n\n**\u00bd cup ( 1 stick) butter**\n\n**1 cup white flour**\n\n**1 tablespoon cold water**\n\n**_For the cream puff topping:_**\n\n**\u00bd cup ( 1 stick) butter**\n\n**1 cup water**\n\n**1 cup white flour**\n\n**3 eggs**\n\n**\u00bd teaspoon almond extract**\n\n**_For the almond glaze:_**\n\n**1 cup confectioner's sugar**\n\n**1 tablespoon soft butter**\n\n**\u00bd teaspoon almond extract**\n\n**Milk (enough to make the glaze spreadable)**\n\n1. **The pastry base.** Cut the butter into the white flour, add the cold water, and mix well. Pat the pastry dough out onto a baking sheet, making two long strips, each 3 inches wide and roughly 16 inches long.\n\n2. **The cream puff topping.** Preheat the oven to 400\u00b0F. In a medium saucepan, combine the butter and the water and heat until boiling. Remove from the heat and add the white flour; stir until smooth. Add the eggs 1 at a time, beating after each addition, then stir in the almond extract. Spread the topping over the pastry strips. Bake for 45 to 50 minutes.\n\n3. **The almond glaze.** Combine the confectioner's sugar, soft butter, almond extract, and enough milk to make the mixture spreadable. Frost the strips with the glaze while they're still warm. Serve at room temperature, cut into crosswise slices about 1 inch wide.\n\n**BRIOCHE BREAD**\n\n_The brioche dough of France can be made into a marvelous loaf bread. A food processor makes this remarkably simple._\n\n**1 package dry yeast**\n\n**\u00bc cup warm milk**\n\n**1 tablespoon sugar**\n\n**2\u00bd cups white flour**\n\n**1\u00bd teaspoons salt**\n\n**\u00bc pound cold butter, cut into 8 pieces**\n\n**2 eggs**\n\n1. Proof the yeast in the warm milk with the sugar. Grease the surface of a large bowl with butter or oil, and grease a 5-cup loaf pan (a Pyrex bread pan is this size).\n\n2. Combine the flour and salt in a food processor bowl. Buzz for a few seconds. Add the butter and process until the mixture looks like coarse cornmeal. Pour in the proofed yeast and process very briefly. Then add the eggs and process until the dough forms a ball.\n\n3. Remove the dough from the food processor bowl and place it in the greased bowl. Let it rise, covered, until it's doubled in size (about 1\u00bd hours).\n\n4. Punch down the dough, knead it a time or two, and shape it into a loaf. Put into the greased bread pan, cover, and set it in a warm place to rise. Preheat the oven to 350\u00b0F.\n\n5. Allow the loaf to rise until it has roughly doubled in size (about 1 hour). Then place it in the oven and bake for about 40 minutes. Remove from the oven and allow it to cool on a rack.\n\n**GERMAN FARMHOUSE GRAHAM BREAD**\n\n_There's something about this good bread that seems to give you instant strength and energy \u2014 especially if you make it with freshly ground whole-wheat or graham flour._\n\n**1 cup whole-wheat or graham flour**\n\n**1 cup white flour**\n\n**1 teaspoon baking soda**\n\n**1 tablespoon sugar**\n\n**\u00bd teaspoon salt**\n\n**1 cup milk mixed with 1 teaspoon vinegar**\n\n**1 egg, lightly beaten**\n\n**\u00bc cup molasses**\n\n1. Preheat the oven to 325\u00b0F. Butter a 5-cup loaf pan.\n\n2. In a large bowl, combine the wheat or graham flour, white flour, baking soda, sugar, and salt. Stir in the milk, egg, and molasses. Mix well and put it into the loaf pan.\n\n3. Bake for 1 hour, then remove from oven. Allow it to cool on a rack.\n\n## **Using European Breads**\n\nEuropean breads are the foundation of many a wonderful dish. Try any of the following recipes using your homemade European bread for mouthwatering treats.\n\n**BRUSCHETTA**\n\n_In Italy, this is the only garlic bread served. These little toasts are just right as an appetizer or to serve with soup or salad. You can make them with slices of large French bread, but they're at their best when made from baguettes, sliced on the diagonal._\n\n**Bread**\n\n**Garlic cloves, sliced in half**\n\n**Extra-virgin olive oil**\n\n**Salt and freshly ground black pepper**\n\n1. Slice your bread however you prefer \u2014 from \u00bd to 1\u00bd inches thick. Toast (preferably over charcoal, if you can).\n\n2. While the toast is still warm, rub each slice with the cut face of the garlic cloves. Drizzle on quite a bit of extra-virgin olive oil, and top with salt and pepper to taste.\n\n**Variation 1:** The Italians often cut a tomato in half and rub the cut side into the toasts, along with the garlic. The garlic and tomato disappear into the bread.\n\n**Variation 2:** You can also mince the garlic and finely dice the tomatoes and just spread them on the toasts. A little chopped fresh basil is a fine addition (and traditional) here, too.\n\n**American Bruschetta:** In typical fashion, Americans have fun with their bruschetta. We put anything and everything on it \u2014 cheese, capers, sun-dried tomatoes, and more.\n\n**HERBED BREAD**\n\n_Herbed Bread is a good place to start your exploration of the wonderful world of long-loaved bread. Not only is it good on its own, but it can serve as a base for many other variations._\n\n**1 tablespoon butter (or more)**\n\n**1 tablespoon olive oil (or more)**\n\n**Fresh or dried herbs (any combination)**\n\n**1 loaf French Bread (see page 14)**\n\n1. In a small saucepan, melt the butter. You may need to use more butter and oil for larger loaves of bread.\n\n2. In a small bowl, mix together the melted butter and olive oil. Then add herbs to the mix. Which ones? How much? It doesn't matter! You can use any herbs you want (fresh or dried), and in any amounts. Some days I'm in the mood for just tarragon; other times it's a mixture of tarragon and rosemary; often it's basil or oregano with a liberal sprinkling of fresh Italian parsley chopped finely.\n\n3. Cover a baking sheet with foil. Slice a loaf of French Bread lengthwise, as though you were making a giant sub, and lay it flat on the foil.\n\n4. Spread the herbed mixture onto the open face of the bread. Just before you're ready to eat, put the baking sheet about 3 inches beneath the broiler unit in your stove. Broil until the edges just start to brown. Now close the bread back up so it looks like a loaf again and slice through it at 1- to 1H-inch intervals. Or, if you'd rather, you can slice without closing up the loaf. It's up to you.\n\n**Cheese Bread:** You can make a fantastic Cheese Bread by top-ping an Herbed Bread creation with your choice of cheese. Try using grated or shredded Parmesan, cheddar, Swiss, or whatever comes to mind.\n\n**GARLIC BREAD**\n\n_Some meals \u2014 and some people \u2014 scream for garlic bread. It couldn't be simpler to make. (For a true Italian garlic bread, see Bruschetta on page 23.) For larger loaves, increase the amounts of the ingredients listed below._\n\n**1 tablespoon butter**\n\n**1 tablespoon olive oil**\n\n**1 teaspoon minced garlic**\n\n**Fresh or dried herbs (optional)**\n\n**1 loaf French Bread (see page 14)**\n\nProceed as directed in the instructions for Herbed Bread (page 24), but add the minced garlic and herbs to taste to the olive oil and butter mixture before spreading it on the bread. Depending on the garlic tolerance of the people who'll be eating, you may want to increase the amount of garlic proportionally. You can also substitute garlic powder or garlic salt, but fresh is best.\n\n**Garlic Gorgonzola Bread:** Before broiling the garlic bread, top it with some Gorgonzola cheese, sliced or crumbled. Keep an eye on it as it broils to make sure the cheese doesn't burn, and don't put the loaf back together after broiling \u2014 it's meant to be served open faced.\n\n**AMERICAN FONDUE**\n\n_American Fondue contains the cheese and bread used for a traditional Swiss fondue, but is much simpler to make and is served from a baking dish rather than around a fondue pot. It's the perfect dish for many occasions, and can be made to seem totally informal, rather hearty, or quite elegant, depending on what bread, mustard, and liquid you use._\n\n**_Homestyle Fondue_**\n\n_This is what I always made when my husband, trying to spare me work, said, \"Let's just have a cheese sandwich.\"_\n\n**4 tablespoons soft butter**\n\n**1 tablespoon prepared mustard (your favorite)**\n\n**6 slices Ciabatta (see page 15) or other firm white bread**\n\n**3 eggs**\n\n**2\u00bc cups milk**\n\n**3\/4 pound cheddar cheese, shredded or in small cubes**\n\n1. Preheat the oven to 350\u00b0F. Thoroughly butter a large baking pan.\n\n2. Combine the butter and mustard and spread the mixture on the sliced bread. Then cut the slices into roughly 1H-inch squares and put them into the buttered baking pan.\n\n3. Run the eggs, milk, and cheese together in a blender or food processor until well blended. Pour over the bread squares.\n\n4. Bake for 35 minutes, or until golden brown.\n\nMAKES 4\u20136 SERVINGS\n\n**_Hearty Fondue_**\n\n_This version is perfect for a cold, blustery day._\n\nFollow the recipe for Homestyle Fondue, but with these changes:\n\nUse a hearty bread such as pumpernickel instead of white, and spread it with a mixture of butter and German mustard. Use beer (light or dark) in place of the milk.\n\n**_Elegant Fondue_**\n\n_This one comes closest in flavor to the Swiss original. To make it even more so, you can add a tablespoon or two of Kirsch to the liquid mixture._\n\nFollow the recipe for Homestyle Fondue, but with these changes:\n\nUse Gruy\u00e9re or Emmentaler cheese instead of the cheddar; Dijon mustard; and a mixture, in place of the milk, of white wine and milk or cream. Add a dash or two of nutmeg on top.\n\n**THE ALPINE SANDWICH**\n\n_For me, this is a calming, soothing combination, reserved for moments of stress and chaos. It came into my life as a restaurant sandwich served alongside a tarragon-laced salad, and is an example of the fact that you can add absolutely anything to bread and come up with a winner._\n\n**1 loaf French Bread (see page 14)**\n\n**Wine vinegar**\n\n**Extra-virgin olive oil**\n\n**Tarragon and other herbs of choice**\n\n**Swiss cheese (or another variety)**\n\n**1 small onion, thinly sliced**\n\n1. Preheat the oven to 400\u00b0F.\n\n2. For each sandwich, cut and split a 4- or 5-inch piece of French Bread. Sprinkle a little wine vinegar on the opened-up rolls, followed by a little olive oil. Top with tarragon and\/or other herbs to taste.\n\n3. Slice the Swiss cheese, or any other cheese you want (I usually use cheddar), thinly, and layer over the herbs on the bread. Finally, top the sandwich with slivers of the onion.\n\n4. Because you don't want to burn the onion, this open-faced sandwich is best baked rather than broiled. Bake for 10 minutes, or until the cheese has melted.\n\n**CROUTE AU FROMAGE**\n\n_I first had this luscious dish in a Swiss restaurant. My attempts to obtain the recipe got me nowhere. (All they would tell me was that the bread was marinated. Marinated? Yes, with wine.) So I figured it out myself, with a few embellishments. If you're ever looking for a Sunday-night supper or festive brunch dish, this one's perfect._\n\n_To serve three, cut the recipe in half and divide everything between three ovenproof plates, then broil._\n\n**2 medium onions, chopped**\n\n**4 tablespoons butter, divided**\n\n**18 slices French Bread (see page 14)**\n\n**\u00bd cup white wine**\n\n**2 medium tomatoes, chopped**\n\n**6 slices bacon, diced and cooked (optional)**\n\n**1 pound Gruy\u00e9re or Swiss cheese, grated**\n\n**8 ounces cream cheese or Neufch\u00e2tel**\n\n**Freshly ground black pepper**\n\n1. In a small saucepan, brown the onion lightly in 2 tablespoons of the butter.\n\n2. Place the bread in a 9 x 13-inch baking pan. Drizzle on the white wine, and strew the onion, tomatoes, and bacon (if you're using it) over the bread.\n\n3. In a small saucepan, melt the rest of the butter, add both cheeses, and stir until melted. Add pepper to taste and spoon the mixture over the bread slices, being careful to leave some edges of the bread exposed.\n\n4. Broil briefly until the cheese is bubbly and the exposed edges of the bread are brown.\n\nMAKES 6 SERVINGS\n\n**PANZANELLA**\n\n**_Italian Bread Salad_**\n\n_To be authentic, you should use an Italian bread to make this unusual Florentine salad, but any of the yeast breads in this bulletin will do very nicely. This is a somewhat untraditional version \u2014 the more common method in Italy is to soak stale bread in water, then squeeze it hard._\n\n**6 slices Ciabatta (see page 15) or other good bread, in 3\/4-inch cubes**\n\n**6 tablespoons extra-virgin olive oil, divided**\n\n**1 large ripe tomato, peeled and cut into 1\/2-inch cubes**\n\n**1 small sweet pepper (any color), in 1\/2-inch cubes**\n\n**1 small red onion, quartered and sliced thin**\n\n**1 tablespoon chopped fresh basil leaves**\n\n**1 tablespoon red wine vinegar**\n\n**Salt and freshly ground black pepper**\n\n1. Brown the bread cubes in 4 tablespoons of the olive oil, stirring constantly. Drain on paper towels.\n\n2. Combine the tomato, sweet pepper, onion, and basil leaves in a salad bowl. Top with the bread cubes.\n\n3. Make a simple dressing from the remaining 2 tablespoons of olive oil, the vinegar, and salt and pepper to taste. Add to the salad bowl and toss. Chill for half an hour before serving\n\nMAKES 4\u20136 SERVINGS\n\n**OYSTER LOAF**\n\n**_A Pacifier_**\n\n_New Orleans and San Francisco share an old custom of using the Oyster Loaf as a pacifier. When a husband had been out late, he brought his wife one of these as a peacemaker._\n\n**1 loaf French Bread (see page 14) or Ciabatta bread (see page 15)**\n\n**2 eggs**\n\n**20 frying-size oysters (save their liquor)**\n\n**1 cup cracker crumbs**\n\n**6 tablespoons butter, divided**\n\n**Salt and freshly ground black pepper**\n\n**3 tablespoons cream**\n\n1. Preheat the oven to 375\u00b0F.\n\n2. Cut the bread in two lengthwise, making the cut near the top of the loaf. Hollow out the insides by removing most of the soft bread and setting it aside.\n\n3. Beat the eggs with 2 tablespoons of the oyster liquor. Dip the oysters in this and then in the cracker crumbs. Saut\u00e9 them in 3 tablespoons of the butter for about 3 minutes, or until their edges curl. Place them in the hollowed bread shell. Now fill in any large gaps between the oysters with some of the bread you removed from the shell.\n\n4. Melt the remaining 3 tablespoons of butter and mix it with 3 more tablespoons of the oyster liquor. Sprinkle about three-quarters of this over the oysters and bread. Put the top of the bread back on and brush it with the rest of the butter-and-oyster liquor mixture.\n\n5. Bake for 10 minutes, or until hot.\n\nMAKES 2\u20134 SERVINGS\n\n**NEW ORLEANS BREAD PUDDING**\n\n_I've seen a recipe that instructed me to make bread pudding from the worst bread I could find. I tried that once. No! Now I know that the better the bread, the better the dessert._\n\n**1 large loaf slightly stale bread**\n\n**3 eggs, lightly beaten**\n\n**1\u00bd cups white sugar**\n\n**\u00bc cup brown sugar**\n\n**1 small can evaporated milk**\n\n**1 tablespoon pure vanilla**\n\n**\u00bd teaspoon nutmeg**\n\n**1 cup raisins**\n\n1. Preheat the oven to 350\u00b0F. Butter a 9 x 9-inch baking pan.\n\n2. Tear the bread into small pieces. Put the pieces in a bowl, cover with cold water, and let sit for a minute or two. Squeeze out the water and add the eggs, white sugar, brown sugar, evaporated milk, vanilla, nutmeg, and raisins. Mix well.\n\n3. Put the mixture in the baking pan, and bake for 50 minutes. Serve warm, with hard sauce.\n\nMAKES 6\u20138 SERVINGS\n\n**PROVENCE SPREAD**\n\n_This recipe was excerpted from_ Herbal Breads, _by Ruth Bass (Storey Publishing, 1996). Top quality olive oil is the ticket here. Don't look at the unit price, just keep tasting until you find a fine olive oil that you like. Some chefs swear by Spanish olive oil, others only use Italian. Saying they are all alike is like saying all potatoes are the same._\n\n**\u00bd cup pitted black olives**\n\n**\u00bd cup pitted green olives**\n\n**I cup extra virgin olive oil**\n\n**6 garlic cloves**\n\n**2 tablespoons capers**\n\n**Juice of \u00bd small lemon (2 teaspoons)**\n\n**Salt and freshly ground pepper**\n\n**Crusty Italian or French bread**\n\n1. Finely chop enough black olives to make 1 tablespoon. Do the same with the green, mix the two, and set aside.\n\n2. Pour the oil mixture into a blender and add the garlic, capers, and lemon juice. Blend until almost smooth.\n\n3. Put the oil into a small bowl and stir in the chopped olives. Add salt and pepper to taste.\n\n4. Spread on thin slices of toast made with crusty bread.\n\n1 CUP\n_The mission of Storey Publishing is to serve our customers by publishing practical information that encourages personal independence in harmony with the environment._\n\nEdited by Nancy Ringer  \nIllustrations by Mary Rich  \nCover design by Carol J. Jessop (Black Trout Design)  \nText production by Nancy Lamb  \nProduction assistance by Sarah Crone\n\n\u00a9 1997 by Storey Publishing, LLC.\n\nAll rights reserved. No part of this bulletin may be reproduced without written permission from the publisher, except by a reviewer who may quote brief passages or reproduce illustrations in a review with appropriate credits; nor may any part of this bulletin be reproduced, stored in a retrieval system, or transmitted in any form or by any means electronic, mechanical, photocopying, recording, or other without written permission from the publisher.\n\nThe information in this bulletin is true and complete to the best of our knowledge. All recommendations are made without guarantee on the part of the author or Storey Publishing. The author and publisher disclaim any liability in connection with the use of this information. For additional information please contact Storey Publishing, 210 MASS MoCA Way, North Adams, MA 01247.\n\nStorey books and bulletins are available for special premium and promotional uses and for customized editions. For further information, please call 1-800-793-9396.\n\nPrinted in the United States by Walsworth Publishing Company\n\n**Library of Congress Cataloging-in-Publication Data**\n\nAndrews, Glenn  \nMaking European breads \/ Glenn Andrews  \np. cm  \nISBN 978-0-88266-998-4 (alk. paper)  \n1. Bread. 2. Cookery (Bread). I. Title.  \nTX769.A67 1997  \n641.8'15\u2014dc21\n\n97-13175  \nCIP\n\n##  **M ORE COUNTRY WISDOM BULLETINS YOU WILL ENJOY!**\n\nThese and other Storey Country Wisdom Bulletins are available for $3.95 at your local bookstores, garden centers, farm stores, and gift shops. Use the order numbers listed under each bulletin to make your requests. You can also order directly from Storey Publishing by writing to us at 210 MASS MoCA Way, North Adams, MA 01247 or by calling 1-800-441-5700. For more information about our books and bulletins, visit our Web site at _www.storey.com._\n","meta":{"redpajama_set_name":"RedPajamaBook"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzsvrq b/data_all_eng_slimpj/shuffled/split2/finalzzsvrq
new file mode 100644
index 0000000000000000000000000000000000000000..af881f3e017872b39985d3885e22edf4c3bf40e1
--- /dev/null
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@@ -0,0 +1,5 @@
+{"text":"In today's brave new world, data and information of all sorts are rapidly proliferating. Many companies who have used traditional databases such as the Oracle database or Microsoft SQL Server for their data requirements since the 1990s have now started to explore new and different databases \u2013 primarily because their database license costs have been spiraling over the years, with more database servers and more CPU core licenses being required.\nOne strong database contender to replace the traditional large corporate database is PostgreSQL (also simply called Postgres, as a nickname of sorts). It has a double advantage in that it is provided as an open source community edition, and it is compatible with the Oracle database. This is quite attractive to companies who have been using Oracle databases, since some of their applications may install a lot of database-level code in the form of PL\/SQL procedures, functions, and packages, and if the database-level code can be moved to an Oracle-compatible database, well and good. Of course, there are some applications created by Oracle such as Oracle E-Business Suite (EBS) that cannot be moved to other databases like PostgreSQL, due to their deep integration with the Oracle database.\nIn this article series, we will explore why and how enterprises are starting to use PostgreSQL. In particular, we will look at enterprise-powered versions of the PostgreSQL database, such as EnterpriseDB Postgres Advanced Server (EDBPAS), that are even more closely compatible with Oracle and make it easier to start using this low-cost database, either on premises or to the cloud. In the coming parts of this series, we will look more closely at the features of EDBPAS and explore installation and other features.\nFirst, let us recap a few basic facts about PostgreSQL. Essentially, it is an open-source object-relational database that is more than 20 years old, and was inspired by the white paper \"A Relational Model of Data for Large Shared Data Banks\" by Edgar Frank Codd; the same paper that also influenced Oracle to create its initial relational database management system (RDBMS). PostgreSQL is owned by the Postgres foundation, and has an open source license. Because the database code is open source it can be forked and modified. It runs on the major operating systems \u2013 Linux, UNIX, and Windows. UNIX variants that can be used are IBM AIX, HP-UX, Solaris, Tru64, BSD, SGI IRIX, and macOS. The database includes most ANSI-SQL:2008 data types, and supports many of the features of the SQL:2011 standard.\nPostgreSQL is supported by a large international community, mainly in postgresql.org and other countries in the world. The PostgreSQL Global Development Group regularly releases updates. On 21st September 2017, PostgreSQL 10 RC 1 was released, which is the first release candidate of the new version 10 of PostgreSQL. The database is ACID (Atomicity, Consistency, Isolation, Durability) compliant, object-oriented, and with geospatial features. There are some production PostgreSQL databases worldwide that have more than 4TB of data.\nPostgreSQL, even in its open source version, has sophisticated capabilities suitable for large enterprises. One example is Multi-Version Concurrency Control (MVCC). This feature enables concurrent access to the database and avoids table locks when data is being updated by database writer sessions, with database reader sessions being able to read the previous version of the data at the same time. MVCC thus allows the database to scale to a large number of concurrent sessions without succumbing to contention between readers and writers.\nAs another example of its enterprise-level features, the PostgreSQL database performs Write Ahead Logging (WAL) to achieve fault tolerance and write efficiency. Due to MVCC and WAL, the PostgreSQL database can also enable online (hot) backups. The WAL feature also makes Point in Time Recovery (PITR) possible from a database backup along with the WAL log, and also enables streaming replication (asynchronous by default) between the primary and standby databases.\nNested transactions can be performed via database savepoints, and the database also boasts a powerful internal query planner (optimizer). PostgreSQL is a database that is considered to be \"compatible\" with the Oracle database, one reason being its own PL\/SQL-compatible language. Stored procedures can be created in 12 languages, including Java, Perl, Python, Ruby, Tcl, C\/C++, and PL\/pgSQL, which is like Oracle's PL\/SQL. Tablespaces can be created, and the database allows partitioning to manage large tables effectively and with better performance, with a maximum table size of 32TB.\nThe database allows datatypes such as varchar, integer, numeric, XML, and JSON\/B. The database can also enforce referential integrity with foreign keys. There are out-of-the-box functions such as to_char, to_date, and constraints such as check, not null, primary. Foreign Data Wrappers (FDW) can be used to connect to external data sources, similar to the database links available in Oracle, except that PostgreSQL can connect to different databases such as MongoDB.\nUser Defined Functions (UDF) can be created in open source PostgreSQL, but an important consideration is that there are no packages (stored collections of procedures and functions) like those in the Oracle Database, and, as for reports, this has stopped a number of Oracle Database users from moving to the open source version of PostgreSQL. However, the EnterpriseDB Postgres Advanced Server distribution does allow the creation of database packages.\nNow, let's consider a large cloud company and its database offerings. Many of us are aware of Amazon AWS and the Relational Database Service (Amazon RDS) that it offers to customers around the world. Amazon allows the setup of multiple third-party databases in RDS (such as MySQL, Oracle, SQL Server, PostgreSQL, and MariaDB). Amazon's own Aurora database was released initially with MySQL compatibility, followed by a PostgreSQL-compatible version in late 2016.\nAs per Amazon, this was due to customer demand and interest in PostgreSQL along with AWS optimizations for the cloud. The idea was to allow this database as an option for new greenfield deployments, or even migration from other existing databases in the company such as Oracle or SQL Server. Amazon now offers Aurora as PostgreSQL 9.6 on the Amazon Aurora cloud-optimized storage.\nBut what are the benefits of EnterpriseDB Postgres Advanced Server (EDBPAS) as compared to the Aurora version? We continue in the next part of this article series, when we will look more closely at the features of EDBPAS, and then explore the installation and other aspects of this distribution, which is fast becoming popular around the world.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Has a lot of nice features, including long filenames, minimize to system tray and presets similar to those on a car stereo system. Lets you organize your MIDI files either by directory or by creating a playlist. Displays MIDI song lyrics, including soft karaoke files. Lets you enter \"notes\" about each of your MIDI files.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"A former Federal Commissioner for Works and Housing, Lateef Femi Okunnu, says it is \"rubbish\" for members of the Ohanaeze Ndigbo to ask for slots in the cabinet of the Lagos State government.\nA former Federal Commissioner for Works and Housing, Lateef Femi Okunnu, says it is \"rubbish\" for members of the Ohanaeze Ndigbo to ask for slots in the cabinet of the Lagos State government.\n2019 Elections:We will obey: Nigeria Army responds to Buhari's statement.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Soft and high quality fur material and lustrous satin lining on both sides.\nSoft, furry and so very snuggly! This plush piece is truly remarkable with soft and high quality fur material and lustrous satin lining on both sides. Your sweetheart will simply adore its softness and relish the oodles of cuteness it exudes. This shoulder wrap will not only add a touch of elegance and classiness to your girl's special occasion dresses, but will also keep her warm and cozy on chillier evening parties. Gorgeous flower at the top closure features amazing details of hand crafted petals and multi toned shades with charming satin ribbons dangling enticingly, enhancing the charismatic appeal of this swish piece! Pair it with matching flower girl dresses, holiday dresses and hair accessories to complete the diva look of your dainty damsel!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Russell Moore, president of the Southern Baptist Convention's Ethics and Liberty Commission (ERLC), recently warned couples not to consider themselves equal partners.\n\"Often, I think, the gospel is obscured because God has designed a picture of the gospel in the one-flesh union of husband and wife, when that is broken down, you have a false gospel that is being preached,\" stated Moore in an interview with Mark Dever of 9Marks Ministries, reports ABPNews.com.\nWhen it came to his own marriage, Moore said that he spends most of his time at ERLC's Washington D.C. office, while his wife and five children live in Nashville, Tenn.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzswki b/data_all_eng_slimpj/shuffled/split2/finalzzswki
new file mode 100644
index 0000000000000000000000000000000000000000..a0cc3c4f14f792bedbf7940b989cc1444d6ca183
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzswki
@@ -0,0 +1,5 @@
+{"text":"OLC provides CEUs for CLEAR\/ROAYL meetings.\n* FREE for 1 employee of Supporting Member Institutions with $400,000 or below in total operating revenue. Member schools\/school districts receive one free CLEAR or ROYAL membership.\n** $50 per year for 1 employee in our Ohio service area with $400,000 or below in total operating revenue. Non-member schools\/school districts in our Ohio service area receive one $50 CLEAR or ROYAL membership.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The serial supports serial communication between the BBC micro:bit and another computer. Basically, this allows you to send data from the micro:bit to your own computer. This is very useful for debugging purposes: you can add write line statements in your code and see them display on your computer as the program executes.\nThe code below shows a simple script that sends a line when the BBC micro:bit starts and another line each time the button A is pressed.\nData is also automatically streamed to serial by the bar graph block and picked up by the editor. This data can be streamed to the cloud as well.\nUnfortunately, using the serial library requires quite a bit of a setup.\nIf you are running a Windows version earlier than 10, you must install a device driver (for the computer to recognize the serial interface of the micro:bit).\nAlso, if you don't see the serial port as one of your computer's devices, you might need to update the firmware on the micro:bit. Find the device name for the attached serial port in the following instructions for your operating system.\nInstall the terminal emulator Tera Term. At the time of this writing, the latest version is 4.88 and can be downloaded from here. Follow the instructions from the installer.\nOnce both the driver and the terminal emulator are installed, plug in the micro:bit and wait until the device is fully setup. Then, open TeraTerm.\nCheck \"Serial\"; in the dropdown menu, pick the COM port that says \"mbed Serial Port\". Hit Ok.\nIn the menus, hit Setup > Serial Port and set the baud rate to 115200.\nYou should be good. Feel free to hit Setup > Save Setup in the menus to erase the default configuration file with a new one so that you don't have to type in the settings again.\nPlease note that Windows will assign you a different COM port if you plug in another micro:bit. If you're juggling between micro:bits, you'll have to change the COM port every time.\nIf you prefer another terminal emulator (such as PuTTY), here are some instructions.\nOpen PuTTY; on the main screen, use the following settings: Serial \/ COM14 \/ 115200. Replace COM14 with the COM port number you wrote down previously. Feel free to type in a name and hit \"Save\" to remember this configuration.\nInstall the program screen if it is not already installed.\nFind which device node the micro:bit was assigned to with the command ls \/dev\/ttyACM*.\nIf it was \/dev\/ttyACM0, type the command screen \/dev\/ttyACM0 115200. If it was some other device node, use that one in the command instead. Note: You may need root access to run screen successfully. You can probably use the command sudo like this: sudo screen \/dev\/ttyACM0 115200.\nTo exit screen, type Ctrl-A Ctrl-D.\nAlternative programs include minicom and so on.\nscreen \/dev\/cu.usbmodem1422 115200 will open up the micro:bit's serial output. To exit, hit Ctrl-A Ctrl-D.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"One of the best acne cleansers you can make at home to save money on acne cleansers\u2026.\nPurify oily, acne-prone skin with this gentle, astringent tea tree oil facial wash recipe. This aromatherapy recipe is one of the best acne cleansers I've found \u2013 powerful, but gentle enough for daily use.\nThis aromatherapy recipe for acne includes lavender, lemongrass and petitgrain to reduce oil production, plus aloe-vera and vegetable glycerin to repair skin and soften blemishes.\nTea tree essential oil kills breakout-causing bacteria and chamomile reduces redness and soothes skin irritation.\nPour everything into a 200ml (or larger) dark glass or PET plastic bottle. Shake well to combine. Wait 24 hours before using to let the mixture \"cure\".\nAn aromatic bath two to three times per week is especially helpful for depression. Sometimes the aromatic baths are effective enough on their own as a remedy for mild bouts of depression or feeling blue.\nMix the essential oils in the honey. Fill the bath tub with warm water and then add the aromatic honey mixture. Stir well using your hands. Soak for 20 to 30 minutes.\nthen pour into the bottle.\nVariation: Instead of using olive oil, you can use lighter oils, such as coconut, vitamin E, apricot or jojoba.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The Travel Colours City Guides are for design-loving travellers who like to explore the trendiest places in each city.\nBerlin. While Berlin has always attracted visitors for its nightlife and modern art scene, now its the booming food scene.With countless number of good bars, restaurants and cafes to choose from, this Berlin Guide will show you a curated selection of our favorite places to \"sleep, eat, drink, shop and explore\".\nFrom cozy caf\u00e9s and coffee roasters to restaurants with menus boasting products from local artisans and growers to Berlin's most prestigious fashion shops hidden in a backyard, this guide will lead you away from the tourist hubs to some of Berlin's hidden local gems.\nShipping rates: $9 EU; $9 Worldwide.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Join Crash Bandicoot as he, his friends, and enemies, set off on a racing frenzy across different environments.\nAfter the disappointment of not being able to save any of my progress on Crash Bandicoot: Nitro Kat 3D, I thought I would try my luck with Nitro Kart 2.\nAlso priced at \u00a32.99 from the iTunes Store, I was surprised to find that although it is a predecessor to Nitro Kart 3D, Nitro Kart 2 actually has far more features.\nThe racing game has several modes of play, which include: Cup, Time Attack, Skill, Multiplayer, and Mission.\nCup mode is the main aspect of the game, in which you race across three tracks in succession to earn points and try to be the overall trophy winner. Each trophy you win, unlocks another set of courses. Some of the track environments will be very familiar to Crash Bandicoot fan, such as N. Sanity Beach (rechristened Bandicoot Beach), but others, such as the Haunted Mansion are new. Personally, I don't think the Haunted Mansion environment fits Crash Bandicoot and is more at home in a Mario video game.\nIn Time attack you race alone along a track and try to get the best time. There are no pre-set time targets, so you are only competing against your previous score.\nMultiplayer mode is self-explanatory, but I found that you could wait in the antechamber for a long time, waiting for other players to come along, but no one ever does. It appears that not enough people play - or play multi-mode to be able to race against someone else, which is a shame.\nSkill and Mission modes are similar. In the former you get points for making 'wild jumps', while in the latter you score points for using drag around corners.\nThe graphics look great, and the music is fun (a remix of original Crash Bandicoot music), and the tracks have depth (there are different routs you can take, which makes it more interesting).\nUnfortunately I fell into the same problem with Nitro Kart 3D. Since the developers have not updated the game in a few years, it is not completely compatible with the latest generation of iPod Touch. I was therefore unable to save any progress. Other Apple mobile devices should be fine, but if not, you can easily get a refund.\nDo you prefer monster or mythology stories in The X-Files?","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzwhpx b/data_all_eng_slimpj/shuffled/split2/finalzzwhpx
new file mode 100644
index 0000000000000000000000000000000000000000..e70dc8f420d9ba379b92e900ce5e35ac6bd7ac12
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzwhpx
@@ -0,0 +1,5 @@
+{"text":"When you do something for 10 years, you're bound to get noticed. Especially when that something is axe throwing. Naturally, BATL has been featured in countless articles and news segments over the years. Global News, Vice Magazine, The Guardian \u2013 you name it, we've shown them how to throw an axe.\nBeyond that we've also had many notable people come to our venues, for a variety of reasons. 3 time NBA Champion Rick Fox was in recently to film a segment leading up to the NBA all-star game in Toronto. Mike Cammalleri of the New Jersey Devils came for a charity event we had. And actor Gerard Butler just came to have a private party for his birthday.\nThat said, you don't need to be famous to enjoy the simple pleasure of throwing an axe. Book your private party today!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"A heating engineer in Devon by training, he admits to knowing absolutely zilch about the Sport of Kings. He does not begin each morning with a detailed assessment of the runners and riders in Sporting Life. Indeed he denies ever having seen a copy of the horse racing person's Bible. He goes to the races maybe once or twice a year. That is all. Last month he went to Devon and Exeter races at Haldon, outside Exeter. He did have a look at the card for the day. He made two selections for each of the eight races, intending to place \u00a32 to win on each of his choices.\nThen, like all good engineers, being a whizz on maths he calculated two times \u00a32 per race was four. Do the same for eight races, and that ends up as an outlay of \u00a332. So instead he plumped for just a single \u00a32 rollover stake, and just one horse in each race. He then watched in disbelief as in turn each of his horses romped home. And his initial \u00a32 stake rolled over, to accumulate some 700,000 times. After the victory of his final winner, a 12-1 outsider that had never won a race before, friend Steve was moved to admit: \" I still know nothing about horses. I am just a heating engineer. At least, I was a heating engineer. Up until now.\" Some guys have all the luck.\nWhen did I really know the nuclear accident at Fukushima had changed the world? When international stars appearing with the world's finest opera company, the Metropolitan of New York, refused to fulfil contractual obligations in Japan.\nThe exquisite Ukranian soprano Anna Netrebko was due to perform in Nagoya and Tokyo. She is best known in the UK for her stunning performance at the Albert Hall proms - she raised the roof at the last night of the 2007 proms. Known as La Bellisima, she explained her walk out with reference to family members touched by the 1986 nuclear accident at Chernobyl, in her native country . Whilst she herself would have been still a child then, the echoes from Fukushima were simply too much for her.\nConversely the magnificent Maltese tenor Joseph Calleja had no such childhood memories. Malta is one of the majority of EU states which have eschewed any nuclear construction. Calleja, a Verdi and Puccini specialist who has frequently wowed Covent Garden audiences, simply refused to travel to Japan whilst the radiation leaks continued.\nPrima donnas throwing wobblies? Perhaps. After all, the shows did go on, albeit with understudies, and with the permanent staff at the Met reminded firmly they had contractual obligations to be there, regardless of any perceived dangers. But in my view, such violent reactions from two of the world's greatest opera stars made it plain. Attitudes to nuclear electricity generation have changed after Fukushima. Not just in Germany, or in Switzerland. But all around the civilised world.\nGermany's largest power company, E.On, has announced with great fanfare it will sue the German government for maintaining a nuclear tax on spent fuel rods in place, despite ordering it to close all its nuclear power plants. But will it?\nAfter all, chancellor Angela Merkel is merely doing what the previous Social Democratic government had intended to do. Which is to close all 17 nuclear plants some time between 2020 and 2025.\nOf course E.On, and the other three German nuclear generators, would like to sweat their assets for as long as possible. But, in practice, the last of them could never have staggered on much into the 2030s. And certainly all are set to be generating power long after their original official design life, just as is happening with the equivalent aged stations in Britain.\nI suspect, in the end, and for all its' bluster, E.On has absolutely no intention of ending up challenging such a popular government decision in the courts. Especially if it continue to link the closure decision to the spent fuel rod tax. When the tax was introduced last year, the four generators quietly acquiesced, knowing they had been onto a good tax-free thing for rather longer than they might have anticipated. Probably a period of discrete silence from E.On might serve everybody's interests well. Including its own.\nWhen the troubled Sellafield mixed oxide (Mox) plant was built in the 1990s, at a cost of \u00a31.34 bn, it had to wait several years before it was given an operating licence. The principal justification for awarding that licence in 2001 was the confident belief it would supply hundreds of tonnes of Mox fuel to Japanese reactors. And so make oodles of money for the UK.\nIn the end, only one Japanese company - Chubu Electric - signed a definite contract. But it operates the Hamoaka plant. And this sits on two major geological faults. In the post-Fukushima antipathy in Japan to any nuclear risks, it is likely to be forced to close. So bang goes the sole source of funds.\nIt will cost British taxpayers a further \u00a3800m to run the Mox plant for the rest of the decade. And then a further \u00a3150m to decommission this expensive heap of radioactive metal. Even in nuclear terms, compared with uranium ,Mox fuel is astronomically expensive.\nThank God Sellafield is way up in remote Cumbria. If such a complete white elephant had been built anywhere vaguely near where the TV companies and the chattering classes are based, it would be notorious as one of the worst wastes of public money on record. Geographical obscurity should not obscure the truth.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Here you have been trapped in very dangerous tribal hut nothing is going well here. So go ahead play this escape game. You can find useful stuffs and interesting puzzles clues symbols to solve and finally escape from the tribal hut. Good luck and have fun!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"\"We are all thieves when it comes to fine food.\"\nSo says the chef at Melbourne's newest addition to the cafe scene.\nQuaintly positioned in the remains of an old chocolate factory, the Breakfast Thieves cafe offers brunch-hunters a welcome resite from the constant bustle of Smith Street. Since its opening earlier this April, mouthwatering scents of modern Australian cuisine with a tantalizing Asian edge have wafted out onto the sidewalk and infiltrated the neighbourhood.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Exclusive: Trainer Richie Woodhall Says Rio Ferdinand Is A Natural Talent - \"He's Picked Up A Fair Bit From Watching Boxing\"\nRichie Woodhall has given himself 12 weeks to turn ex-football star Rio Ferdinand into a prizefighter.\nFerdinand this week announced his intention to turn pro \u2013 and he will be trained by Richie Woodhall.\nWoodhall won bronze in 1988 Seoul Olympics and the WBC super-middleweight title in his own ring career and he now helps coach the Great Britain amateur squad at the English Institute of Sport in Sheffield and works as a TV pundit.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzwzox b/data_all_eng_slimpj/shuffled/split2/finalzzwzox
new file mode 100644
index 0000000000000000000000000000000000000000..5caab099a5c43ae5c9e7d11992c32a512c237a0b
--- /dev/null
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@@ -0,0 +1,5 @@
+{"text":"Does the PSN Store offer more quality than Xbox 360 Marketplace?\nWell it looks like the PS3 might have snuck past the 360 without anyone noticing, while its online functionality is still in its infancy when comparing the upcoming downloadable games the PS3 easily has the more exciting titles.\nWarhawk alone is a title that is impossible for Xbox to compete against in its marketplace due to the Core having no hard drive and the premium only having 30Gb.\nProbably the best Arcade title is Geometry wars which is sad now because that was released a long time ago and nothing else of any innovation has appeared since then.\nThe PS3 now has Warhawk (which was so popular it virtually crashed the servers on release) and if we look into the future we see great games like WipEout HD, SOCOM, PAIN and GT5 to name a few.\nThanks ocelot for the tip.\nPrevious article TimeShift commercial looking for attention?\nNext article Can Ace Combat 6 Save Xbox 360 in Japan?","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The quickest way to determine what build or version of the Apache webserver you are running on your Ubuntu Server is simply to call up a terminal and run the -v switch on the apache2 command.\nThis will output both the version as well as a date of when this server version was first built.\nPleasingly enough, this will also work if you want to find out what version of PHP you are running on your Ubuntu Server, as well as what version of MySQL (both client and server) you are running.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Liposuction, also known as lipoplasty and suction lipectomy, is a surgical procedure designed to improve the body shape by removing unwanted fat deposits from specific areas of the body. Liposuction is commonly used on the abdomen, buttocks, back, chin, hips, knees, neck, thighs, and upper arms.\nThe procedure is often performed along with other procedures, such as abdominoplasty (tummy tuck) or facelift surgery. There are several fairly new techniques in liposuction which offer many benefits over the traditional techniques. Ultrasound-assisted liposuction and tumescent liposuction are two of these techniques and they typically involve quicker recovery times and more precise results.\nThe best candidate for liposuction is a person who is physically healthy and of normal weight but with areas of excess fat that are resistant to regular diet and exercise. Patients should also be psychologically stable and have realistic expectations of what the procedure can offer. Patients with firm, elastic skin tend to achieve the best results, while older patients with diminished skin elasticity may not achieve the same results as a younger patient.\nLiposuction can successfully improve the shape of the body and, like many cosmetic procedures; it may improve one's self confidence as well. Dr. Hardaway has indicated that after surgery, you're likely to discover you're able to wear a wider variety of clothing than before. However, liposuction is not a substitute for regular exercise and a healthy diet. By following a healthy diet and exercise regime, you can help to maintain your liposuction results.\nThe amount of time needed for liposuction will vary due to several factors, including the amount of fat being removed, the technique used, and the size of the area being treated. In general, however, liposuction typically takes between one and two hours to complete. The procedure is commonly performed on an outpatient basis and typically involves general anesthesia, or IV sedation.\nDr. Hardaway usually begins the procedure with small incisions made on the areas of the body being treated. A narrow, blunt-tipped tube called a cannula is then inserted through these incisions. She then manipulates the cannula disrupting the fat cells before they are suctioned out. The incisions are then stitched closed.\nDr. Hardaway determines the technique she will use based on her assessment of the patient. For ultrasound-assisted liposuction, or Vaser Liposuction, an ultrasound probe is inserted beneath the skin to liquefy the fat before it is suctioned out. For tumescent liposuction, a fluid containing saline and lidocaine are injected into the fatty tissues, making them swollen and firm, before they are suctioned out. Both of these techniques offer several benefits over traditional liposuction, including quicker recovery times and more precise results, though the surgical times can be significantly longer.\nAfter surgery, you may experience some discomfort, stiffness, burning or temporary numbness. Fluid drainage from the incisions, swelling and bleeding is rare.\nDr. Hardaway will prescribe pain medication to help with any discomfort that you feel, as well as antibiotics to prevent infection. Patients should start walking around as soon as possible, or as recommended, to help prevent blood clots in the legs.\nA compression garment similar to a girdle is typically worn for several weeks to support the skin and control swelling. Stitches are typically removed within seven to ten days. Most of the swelling and bruising subsides within three weeks, though some slight swelling can persist for more than six months. You will notice an immediate change in the shape of your body after surgery, though the final results will be apparent when all of the swelling has completely faded. Many of the newer techniques, such as Vaser liposuction, typically involve a decrease in post-operative bruising and swelling.\nThe amount of time it takes to recover after liposuction will vary for each patient. Generally, most patients are able to return to work within a few days after surgery, though more strenuous activity will need to be postponed for two to four weeks. Dr. Hardaway will discuss all factors of the surgery and recovery with you during you consultation.\nDr. Hardaway was interviewed on Channel 2's News.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"copper ore crusher in zambia and copper mining ... Crushers business in zambia, stone and rock crushing ... Zambia Copper Processing Plant Manufacturerstone crusher .\nrock quarry crusher in zambia. rock quarry crusher in ... For Sale In Zambia) manufacturer, Professional Stone ... and cement business in zambia ore crusher.\nStone Crusher in Zambia,Copper Ore Crusher in Zambia,Rocks Crusher Manufacturer Copper Crushing Plant in Zambia Project in Zambia is typica.\ncrushers of stone in south africa\u2013 Rock Crusher Mill. ... Zambia Copper Ore Crusher manufacturer in Shanghai, China. Zambia Copper Ore Crusher is manufactured from .\nZambia copper ore crusher manufacturer in Shanghai, ... Used Fixed Stone Crusher Portable Crusher. ... Portable rock crusher is a new kind of crushing machine.\n... work Solution for ore miningmini rock ... At Zambia. copper stone crushing plant in zambia ... can crushing many crusher manufacturer in india,zambia .","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"This entry was posted on Sunday, September 1st, 2013 at 8:17 pm\tand is filed under Opinion. You can follow any responses to this entry through the RSS 2.0 feed. You can leave a response, or trackback from your own site.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzxzke b/data_all_eng_slimpj/shuffled/split2/finalzzxzke
new file mode 100644
index 0000000000000000000000000000000000000000..eb9e2dceda7ca9bdf62878683579707c77e0dfa8
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@@ -0,0 +1,5 @@
+{"text":"Paper mache is a basic craft technique learned at a young age. Just because young artists learn to work with paper mache doesn't mean it's just for kids, though. On the contrary, paper mache can be as big, involved and detailed as you would like. While it will take more time and material, you can make life-sized human sculptures with paper mache by using some of the basic techniques you learned as a kid.\nDraw your idea for a sculpture on paper. This drawing will serve as a reference throughout your project.\nBuild your sculpture base. You can construct a base for a life-sized human by making the body from cardboard and balloons that have been taped or tied together. For a sturdier sculpture, construct a skeleton from one-inch by two-inch lumber that has been screwed together or use PVC pipe that has been glued. Fill out the base with additional materials, such as empty soda bottles, shipping tubes and even aluminum foil. Work slowly and get your base as close to life-like as possible.\nWrap your sculpture base in lightweight chicken wire. The chicken wire should be flexible enough to squeeze and form around the limbs and other places where the body will need definition. Use wire cutters to cut various points of the chicken wire to help form the sculpture base.\nStand up your base to test whether it can stand on its own. If the structure can stand before paper mache has been added, it will also likely stand afterward.\nPrepare your work area by laying down a large sheet of plastic or drop cloth to protect the floor and other close objects.\nMake the paper mache paste by combining three cups of water with 2\u00bd cups of flour. Mix it until the flour has been completely incorporated. Paper mache paste should be the consistency of pancake batter.\nTear strips of newspaper. Strips should be approximately two inches wide, though lengths can vary.\nDip strips of newspaper into the paper mache. Use your fingers like a squeegee to slide down the strip and remove excess mixture. Lay the strip across your sculpture. Smooth the strip to make it as flat as possible.\nOverlap your strips and continue to paper mache until the sculpture is completely covered. Four coats of paper mache are recommended. Make sure the final strips are smooth.\nAllow the sculpture to dry thoroughly. This can take up to 13 hours.\nFix any spots with which you are not completely happy. Allow the sculpture to dry again.\nPaint your finished sculpture with house paint, spray paint or craft paint.\nIf you are having difficulty getting strips to stick to the chicken wire, you can press one end through the holes in the wire.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Background: The national coverage of exclusive breastmilk did not reach its target in 2014. One reason behind this was due to the swelling of the breast to mastitis condition. Hence the baby was not getting enough breastmilk. World Health Organization WHO estimates the mastitis case between 2.6 and 33 %, with global prevalence around 10% of all breastfeeding women. In 2014, Panembahan Senopati Bantul Yogyakarta Hospital documented 5.3% women with mastitis and 4.2% in 2015.\nObjective: This research aims to evaluate the correlation of ages, education, employment status, parity, and mastitis history with mastitis case in Panembahan Senopati Bantul Yogyakarta Hospital.\nMethods: This research was an observational analytic with a case-control design. Data analytic was performed using the Chi-Square test.\nResults: The results of bivariate analysis showed that age (OR = 0.673; 95% CI; 0.345-1.314), education (OR = 1.288; 95% CI; 0.678-2.44), employment status (OR = 1.368; 95% CI; 0.749 -2.497), and parity (OR = 0.829; CI 95%; 0.455-1.511) did not have a significant correlation with the incidence of mastitis, but the history of mastitis showed a significant association with the incidence of mastitis (OR = 2.280; 95% CI; 1.112-1.674 ).\nConclusion: There was a correlation between the history of mastitis with the recurrence of mastitis in Panembahan Senopati Bantul Yogyakarta Hospital.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"If you are considering an addition to your home, a canopy over your deck or a patio offers a long-term solution to needed space at a lower cost. Your deck\/patio will be shaded from the hot sun rays as well as provide outdoor protection during a rainy day for your entire family.\nYou could even add drop down curtains\/roll up curtains to provide protection from the cold on top of still being able to enjoy the extra room and the beauty of the outdoors. Call 904-388-8770 to start customizing your canopy!\nBoree Canvas Unlimited can custom build a carport or a boathouse to shield your vehicles and boats from the damaging effects of the sun and rain. This will preserve your investment and save you money\u2026.guaranteed!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"At MBSL, we have been providing outsourced finance and accounting solutions to our clients for over twenty years.\nWith a wide range of experience of many sectors, we have demonstrated the ability to deliver a tailored solution to each client we work with. We do not impose a standard solution on any client. We listen ! We deliver what our clients want and work with them to achieve their business goals.\nOur ethos sets us apart from other accountancy firms with our dedicated approach to both technology and security. Always forward thinking, we have operated in a paperless environment for over twelve years and we are one of a select number of accountancy firms globally to attain ISO 27001 Certification for information security management, ensuring our business systems and controls meet best practice standards.\nBe a forward thinking business and talk to us to see how we can reduce costs, increase quality of information and help you achieving your strategic aims.\nThe focus of MBSL is to not simply serve clients but to build an exceptional relationship by working in partnership with them. We empower business owners to accomplish their goals, both business and personal.\nTogether with clients we develop a set of mutually agreed objectives. These objectives form the basis of a contract with our client. This process allows us to individually tailor our services to meet their outcomes.\nAt regular review meetings, we ensure that our client's expectations are being met and that our services to them are meaningful and achievable. This gives us a more informed understanding of the actual processes involved in the production of our client's products and\/or services.\nIn an age when data breaches are becoming more and more of a threat, forward thinking companies are taking the necessary steps to protect their clients' information. MBSL recognised this need and in February 2010, achieved ISO 27001:2005 Certification. In February 2015, MBSL maintained this sector leading approach when it was awarded ISO27001:2013 Certification.\nWith this certification, we can provide something that no other accountancy firm in Ireland can offer - independent certification that its business and systems controls are such that they meet international best practice standards.\nInformation Technology is a critical part of MBSL's service offering. We pride ourselves on capitalising on the latest technology for the most efficient delivery of services. With cloud-based accounting systems we tailor the right solution to the right client, to ensure that each client gets the solution that best suits them. This may see the introduction of new software or MBSL taking responsibility for the maintenance of accounting records in the clients existing software.\nAlways forward thinking, MBSL has operated in a paperless environment for over 11 years. This was a significant step to take for an accountancy firm, however it was also key to the philosophy of the organisation with regard to sustainable business practices. This move has been embraced by staff and clients alike and more importantly, MBSL has been able to bring clients along on this journey.\nYes we are accountants, but our focus is supporting your business.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Web traffic is good, but leads are better.\nIf there is a steady stream of traffic to your website, but only a few people are filling out conversion forms you are wasting your marketing, letting precious leads slip through your fingers.\nLanding pages are the first critical step in a process designed to shorten your sales cycle. Every element on the page must work together to drive conversions.\nSimplify the layout so there is only one purpose on the page.\nTell visitors they have arrived at the right place.\nMake the case for why visitors need what you are offering.\nLearn how to create buttons that DEMAND attention.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzabypb b/data_all_eng_slimpj/shuffled/split2/finalzzzabypb
new file mode 100644
index 0000000000000000000000000000000000000000..8925f7d87a973f1b2ff7cb646f85957be0b08405
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzabypb
@@ -0,0 +1,5 @@
+{"text":"The skilled personal injury lawyers and accident attorneys at Schwartz Zweben LLP have successfully represented many victims of dog bite injury. They understand the complexity of investigating, litigating, negotiating, and resolving dog bite injury claims and are fully knowledgeable of the liability issues, medical ramifications, and legal options involved.\nJames Vaught Law Office, PLLC is available to handle many types of personal injury cases and lawsuits. Common types of personal injury cases include: Wrongful Death, Serious Injuries, Medical Malpractice, Defective Pharmaceutical, Dog Bites, Tractor Trailer Crashes, and Car Wrecks.\nThe Lexington Kentucky Personal Injury Lawyers of Rosenbaum & Rosenbaum, P.S.C. have represented the injured throughout Kentucky since 1945. These injury Attorneys are devoted to providing aggressive representation to those who have been seriously injured as a result of someone else's negligence regardless of whether it was caused by an individual, big corporation or insurance company.\nAt our Lexington personal injury office, our attorneys handle personal injury claims involving the following: Medical malpractice, Car accidents, Slip and falls, Truck accidents, Motorcycle accidents, Dog bites, Premises liability, and Defective products.\nAt the Becker Law Office we take your injury personally. For over 25 years we\ufffdve been dedicated to working for injured people who need help. We handle many types of cases, ranging from accidents to dog bites, and we always vow to fight for you.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Located on Interstate 5, Kelso is at the confluence of the Cowlitz, Coweeman and Columbia Rivers. Kelso is 48 miles north of Portland, Oregon and 125 miles south of Seattle. Kelso is also only 80 miles east of the Pacific Ocean. Kelso is a city in southwest Washington State and is the county seat of Cowlitz County. At the 2010 Census, the propulation was 11,925. Kelso shares its long western border with Longview, Washington.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Limit stress: strive for 20 minutes of daily prayer, meditation, tai chi, yoga, guided imagery, focused breathing, etc. Click anywhere on this summary to read the article.\nTaiChiUSA is holding two healing open houses this weekend so you can get started on those New Year's Resolutions.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"LEADTOOLS provides developers with the LEAD Resize DirectShow Filter and Media Foundation Transform for use in their .NET (C# & VB) and C\/C++ applications. This filter is used to resize videos with various subsampling algorithms, such as Bicubic, Bresenham, and Resample, with the purpose of improving playback quality on different screens or to reduce bandwidth requirements by streaming smaller video sizes. Video Resize can also be permanently applied during live capture or post-processing and is a key step in saving disk space.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Paulo was very nice and informatve.His knowledge about Portugal was very impressive. He made our tour very enjoyable. We had a wonderful day!\nTOUR GUIDE RESPONSE: Thank you for your comment and for joining this tour. I am glad you enjoyed the tour and wish you a nice stay in Portugal.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzacodi b/data_all_eng_slimpj/shuffled/split2/finalzzzacodi
new file mode 100644
index 0000000000000000000000000000000000000000..144461f12509c3542b057e92c8407631bb5fa638
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzacodi
@@ -0,0 +1,5 @@
+{"text":"While developers continue to build out the infrastructure of the crypto economy, many others are turning the bear market into ground zero in preparations for the next bull run. In their Q3 report, for instance, Grayscale investments described this inflow of institutional money pouring in and making positions in the market. Another smaller demographic, made up by firms like Wilshire Phoenix, are also making similar bets and shifting much of their focus toward the crypto verse.\nThe most significant recent news to come from the small investment group was their October 31, 2018, announcement to partner with Coinbase Custody. The two will reportedly work towards a \"mid-December\" deadline before launching a statutory trust for institutional clients. CEO William Herrmann told BTCManager that everything is on track and only a \"meteoric\" change of events could disrupt the timeline.\nA November 8, 2018 press release also outlined the hire of Alexander Chang and William Cai who both bring nearly 20 years of experience working at J.P. Morgan. In sum, Herrmann assembled a team made up of high-caliber financial experts and actionable crypto experience, along with partnering with one of the biggest names in the space in two months.\nCome December; it's likely that many in the industry will be paying very close attention to Wilshire Phoenix. Their success could spell the onboarding of anxious institutional capital and the ability for old heads to take bitcoin seriously.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"This 2019 UK Full Sovereign and 2019 UK Britannia 1oz Gold Coin Collection consists of the following coins: 1 x 2019 UK Full Gold Sovereign and 1 x 2019 UK Britannia 1oz Gold Coin.\nThe Sovereign coin weighs 7.98g and is struck in 22ct gold, equalling 7.31g of Fine Gold Content.\nThe Britannia coin weighs 1 Troy Ounce and is struck in 999.9 Fine Gold.\nThese coins are minted at The Royal Mint in Llantrisant, South Wales.\nThese coins are VAT free and Capital Gains Tax exempt.\nOne of each coin supplied.\nThe 2019 UK Full Sovereign Gold Coin features Pistrucci's 'St. George and the Dragon' design on the reverse. The 2019 UK Britannia 1oz Gold Coin features the iconic image of Britannia. Details of the coin's weight, fineness and year-date also feature.\nThe obverse design of each coin depicts the fifth portrait of Queen Elizabeth II.\nThis collection offers you a cheaper way to buy these 2 coins. If you are looking to buy greater quantities you may find the Mix & Match section offers you the best price for these coins.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The Police are on the heels of the hoodlums who pulled the triggers first from the residence of the National Democratic Congress (NDC) candidate, forcing the police to react during the by-election in Ayawaso West Wuogon Constituency last Thursday.\nThe confrontation resulted in injuries and a restricted security breach.\nDAILY GUIDE can confirm that but for the ill-fated firing of gunshots, which originated from the residence of the NDC candidate, Delali Kwesi Brempong, the isolated security breach whose outcome was largely exaggerated particularly on social media would not have occurred.\nAs many as six deaths were reported by some mainstream media outlets in what was regarded by some observers as part of a grand scheme to give the polls a bloody coloration.\nHowever, details from the law enforcement agency hinged upon hospital records negated the death reports.\nA 1st February 2019 press release from the Ghana Police Service's Public Affairs Directorate pointed at the hunt for the shooters which led to the sustaining of injuries by a little over a dozen persons; some of them receiving treatment at the 37 Military Hospital.\nThe injured persons were part of party activists who had been deployed to the candidate's residence as they awaited further directives.\nThe unexpected development scuttled the intended operations for which they were assembled, according to sources in the security agencies.\nAn earlier report by DAILY GUIDE presaged a nasty occurrence during the polls which was intended to rob the by-election of its shine and possibly disrupt it.\nThe Police release, signed by Sup\/Ms Shiella Kessie Abayie-Buckman, appealed to anybody with information on the incident to call 0244623866, 0242814863.\nOne of the vehicles which conveyed the NDC activists for the Thursday assignment bore the registration number GT 9829-11.\nThe plan was a complex operation which was to end in a scuttled by-election and give the managers of the polls a bad name, thereby inciting the people of Ghana against them.\nThe resounding success of the polls even after the failed withdrawal by NDC when the end of voting was barely three hours away did not inure to the interest of the planners of the complex operation which never was.\nThe display of an AK 47 during an NDC press conference by the grouping's Chairman as being a firearm seized from an NPP activist during the confusion at the residence of the MP has provoked varied questions.\nMany wondered why the weapon was not handed over to the Police whose forensic experts can trace its ownership using its serial numbers.\nAK 47s are part of firearms which are restricted to the security agencies and would not be licensed to private persons.\nAn NDC top official has also boasted that their party has weapons; assertion which triggers questions about its authenticity and whether the law enforcement agency would be interested in finding out more from this source if it was not mere political talk or hot air.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"We do not need another social network with 1 billion users. Part of the problem is having so many users and so much power concentrated in one place. And setting out to achieve 1 billion users means it's an ad-based platform that will inherently revisit many existing problems.\nAgreed. And yet I am still bothered by the barriers of entry to alternatives, including Micro.blog. Apart from the usually-discussed technical and know-how barriers to entry (which honestly I find dubious on their own), what of social barriers? I see a lot of what faces whenever I check in on Micro.blog. How do we avoid revisiting this existing problem?","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"UKSA We created the identity and guidelines for this charity, an independent maritime education facility based in Cowes on the Isle of Wight. The charity focuses on youth development and professional training. It's fresh, breezy appearance encapsulates a soaring metaphor in a simple, instantly recognisable and infinitely adaptable identity.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzacsch b/data_all_eng_slimpj/shuffled/split2/finalzzzacsch
new file mode 100644
index 0000000000000000000000000000000000000000..6ae0cb4493470fe210f9d4ff3bea9aa3d4f82e7b
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzacsch
@@ -0,0 +1,5 @@
+{"text":"AHS DECA team members were just awarded and are now headed to State!\nCongratulations to Alcoa High School DECA members as 19 of 23 students will be advancing on to the State Competition in Chattanooga, TN. A special shout out to Valerie Fryer who took home 1st place, Iman Coffin 2nd place and Andrea Velasquez 3rd place.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Dont forget to visit the site homepage after reading this FIAT 501 spares page.\nSpares for 501s - find the bits you need here.\nThis page has been added to enable anyone to advertise for 501 spares they want, or list parts they no longer need, online and for FREE!\nRemember to Bookmark this page, and call back again to view the latest 501 adverts in this, the oldclassiccar online swapmeet.\nSimply go to the relevant page for your FIAT - see the menu on the Classics Free Ads homepage if this isn't it - and post your ad for free.\nDon't miss: A live snapshot of 501 eBay items available to bid on today!\nTo place your free ad, please complete the form further down the page - No restored FIATs for sale, only complete examples if they're scrap and being sold for parts only.\nI recommend never, ever, sending payment for any 501 car part without seeing it first, and verifying the identity of the seller, and the FIAT item, in person. Sadly, as in all walks of life, not everyone online is as honest as we'd like (Caveat Emptor).\n501 parts and memorabilia being sold in the UK and Internationally right now on eBay. To see complete cars, please scroll down.\nSpares Wtd Ad. (Malta) (16\/11).\nSaloon Spares Wtd Ad. (Australia) (7\/6).\nParts F\/S Ad. (LINCOLNSHIRE) (6\/1).\nParts F\/S Ad. (Netherlands) (12\/10).\nFiat 501 parts, ask me what You looking for.\nSpares Wtd Ad. (Philadelphia PA USA) (11\/11).\nSpares Wtd Ad. (England) (10\/11).\nPlace your free FIAT 501 advert.\nFREE FIAT up for grabs!\nYou can buy general titles, and search for model specific 501 titles in their online catalogue. If you want to find FIAT-related gift ideas, say for Christmas or a birthday, its well worth a look.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"When it comes to NYC entertainment, WestHouse Hotel is nestled among the epicenter of the arts and culture of New York City. With many of the city's finest entertainment establishments only a stone's throw away, guests have unprecedented access to the best things to do in NYC.\nIndulge in the celebrated The New York Theater District, Radio City Music Hall, New York City Ballet, New York Philharmonic, and The Metropolitan Opera during your stay in NYC. For a close to the evening, enjoy socializing and stunning views of the city on The Terrace.\nOur in-house team of expert lifestyle curators will personalize your experience in New York City to meet your every desire.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Happy Saturday y'all! I have a question for you all today. Have you ever been excited for a book and then start reading only to be massively disappointed? Is it just me?! Today I'm going to share with you one of those books. I don't typically review books I don't finish. For that matter, I rarely stop reading a book, but sometimes the book just isn't a good fit and it's better for me if I just set it down. Project Pandora by Aden Polydoros was that kind of book for me.\nI really, really, REALLY wanted to like Project Pandora by Aden Polydoros. First off, did you see that cover?! It's drool worthy! Plus assassins, government experiments, the whole cloak and dagger feel of the blurb\u2026.yeah I was super excited to read it. However, it just didn't live up to my expectations, and after reading to just pass the halfway mark I set it down, unfinished \ud83d\ude41 Where do I even start?\nPlease note that since I technically didn't finish the book, this review is my opinion and reasons why I didn't finish, not a reflection of the author's writing.\nThe format for Project Pandora was really interesting. In fact it was the book's saving grace for me. As the reader, you are dropped into the middle of the story with not much background on the characters, what they are doing or even how they are doing what they are doing. It helps to build the suspense. Between the chapters though are case notes about the characters which give insight into what they've been trained to do. While I didn't finished the book, I did skim through to read all the case notes. I found them more interesting than the actual story.\nHades \ud83d\ude00 This kid is supposed to be the dark horse of the book, the creep. However I fell in love with him. He's a very dark character, twisted, and the only one who seems to remember and be immune to the directives.\nThe head hopping!!!! Oh my goodness, y'all I had no clue whose POV I was in half the time I was reading. The characters blurred together as their voices (aside from Hades's) just weren't distinct enough for me. I lost count of the times I had to go back to the beginning of the chapter to figure out who was narrating at the time. For me this was a big problem. There were too many characters to try to keep them straight, not to mention how each character had two separate lives (the normal teen and the assassin).\nThe whole two people in one body thing was weird for me. Technically they were the same person, but whenever a specific sentence was said that would trigger them to become an assassin and know things\/people they didn't know before. It was all very strange and not well explained\u2026.granted I didn't finish the story. Maybe it's explained later on?\nThe plot was nonexistent. At least I never picked up on it. AND I was pass the midway point of the book! I should have at least known what the assassins were doing at this point, right?!\nUnderage drinking and drugs. Look I get it, you think I'm a prude and you're right. I know 16 and 17 year olds drink and some get their hands on marijuana, but I don't like to see it in books geared towards the young adult crowds, ESPECIALLY when peer pressure is involved. Big turn off for me. I nearly set the book down at that point.\nOverall, Project Pandora isn't a bad read. The author's writing is decent and the characters were intriguing, even if I never really picked up on the plot. However, it just wasn't my kind of book. I enjoy science fiction, but I lean more towards robots, aliens, and dystopia \ud83d\ude09 If you enjoy thrillers and suspense with a dash of sci-fi, I recommend you give it try.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"AveSeena - Honeyactive Beauty Mask | Natural Me Beauty\u2122 \u2013 Natural Me Beauty\u2122 \u00a9 2019. | All Rights Reserved.\nUniquely thick, decadently rich beauty treatment like no other, designed to offer new levels of natural nourishment leaving the skin looking illuminated and full of vitality. Delicious pure honey-based mask enriched with sublime botanicals smells sumptuous upon application, while gently releases the face from impurities with a sense of creamy smoothness and silky comfort.\nIn just 10 minutes, rich and delicious Honeyactive Beauty Mask delivers a bouquet of pure honey, botanical oils, delicate floral extracts and ultra fine white kaolin, while it liberates from impurities with its very gentle exfoliation feature. The face, neck and d\u00e9collet\u00e9 look visibly smooth, while natural radiance and clarity are restored. Ultra nourishing texture leaves the face with a sense of silky soft delight radiating with natural purity.\nThe thick waterless texture is where the secret is hidden, which absorbs all impurities in itself, while honey and rest of the botanicals provide their reviving energies. This mask is suggested to anyone who is looking for all-in-one non-drying cleansing, soothing, softening beauty mask.\nDeep Detoxing Facial-at-Home: Take a dollop of the mask and warm between your fingers. After warming, apply a generous layer on your dry face and let it sit for 10 minutes. Wash off with lukewarm water with gentle movements. It works as a face mask and exfoliating detoxer, leaving skin cleansed, nourished and glowing. We recommend this application for especially rough feeling days.\nLight Refreshment: Mix a dollop of the mask in your hand with 1-2 drops of water before application and then apply to dry face. Remove the mask with warm water after 10 minutes. This application is ideal for an instant glow and a quick pick me up.\nPowerhouse Facial: Add a drop of (a pump full of) Essential Radiance Serum to the mask and mix it well before application. Apply to dry face and let it do its magic for 10-15 minutes. Wash with lukewarm water. Add Essential Radiance Serum \u2013 beauty equivalent of a shot of espresso,- into your mask turn it into a high-performance facial to deliver radiance and smoothness gently to reveal a fresher, more vibrant skin \u2013 instantly.\nThe mask does not dry\/harden. Avoid the immediate eye area and lips. Mix with facial oil for personalized application. Follow with serum, eye product, and moisturizer. Use it 1-3 times a week.\nNote: pH is between 5.5-6.5. Vegetarian. Contains natural and\/or organic ingredients, please store at room temperature and dry place.\nSensible Cautions: Avoid direct eye contact. Store away from heat and light. Some people may have allergic or other reactions to some ingredients. Please read the labels and ingredient lists carefully and follow appropriate directions for use. Once opened, best used within 12 months.\n*Please note that your mask is made with pure, real Honey. Sometimes Honey may crystallize, yet it is still nutritious as ever! In fact, the crystallization (granules) proves that the Honey in your Mask is of highest quality, not filtered out of nutritious peptides, nectar and enzymes. Temperatures lower than 50 degrees Fahrenheit (12 degrees Celsius) can lead to crystallization of Honey. During cold weather, the Mask during shipping or just sitting in your cold cabinet might lead to the beginning of crystallization because of the lower temperatures. Please do not refrigerate your Mask. If you experience crystallization and want your Mask slippery smooth, please fill a bowl with hot water and let your jar rest until the crystals dissolve. Please do not microwave your Mask because the heat will destroy many of your nutrients, precious nutrients, enzymes and vitamins in the Honey and botanicals.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzaczyv b/data_all_eng_slimpj/shuffled/split2/finalzzzaczyv
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+{"text":"Let's do a simple test. Take a piece of A4 size paper and draw a vertical line in the middle of the page. On the left hand column, list down your assets, and take note of your emotion as you write them. On the right hand column, list down your debts (i.e. liabilities), and take note of your emotion as you write them.\nDo you feel good as you write your debts? Especially if your debts list is longer than your assets? It is therefore human nature to avoid talking about debts because of the negative emotions attached to it. As a result, most people underestimate the level of their debt when they die. It is not uncommon to see an estate greatly diluted by outstanding estate debts.\nLike it or not, the debts need to be settled. Even death does not wipe out debts. Settlement of debts has first priority over estate distribution. It is only when all estate debts have been settled, or in rare occasion been forgiven, that the residual estate can be distributed amongst beneficiaries.\nThe creditors will usually lodge a caveat on the estate, until they are satisfied that there are sufficient funds in the estate and an undertaking from the estate legal representative to settle the debt before distribution is present. Before the caveat is removed, the letter of administration or grant of probate will not be issued. This will delay the estate settlement process.\nThe hospital from which the deceased was receiving medical treatment before will require an undertaking by the family members to settle the last medical expenses. Arrangements can be made with the hospital to use the deceased person's CPF Medisave account to settle this. If the amount remains unsettled, the hospital will recover the debt from the guarantor, who usually is one of the deceased person's family members. If this fails, the hospital will file a caveat on the deceased estate, or sue the estate (and the guarantor) to recover the outstanding debt.\nThe bank will view the death of a debtor as default of loan. When a bank is informed that a borrower has died, it will write to the surviving borrower(s) to seek the death certificate, letter of administration (or grant of probate), and the latest income documents from the surviving borrower(s). This is to assess the credit-worthiness of the surviving borrower(s). If the income of the borrower is not enough to support the loan, additional borrowers need to be brought in. Otherwise, the bank will do a mortgagee sale on the property. If the mortgagee sale is not enough to cover the outstanding debt, the bank will recover the remaining debt from the estate. The bank has the right to net off the outstanding debt from the deceased person's bank saving accounts.\nIf the deceased is the principal card member, the bank will cancel the principal and the supplementary card, and recover the outstanding debt from the estate (the bank will normally file a caveat on the estate). The surviving supplementary card member will be liable for his\/her own debts incurred in the supplementary card. The bank has the right to net off the outstanding debt from the deceased bank saving accounts.\nIf the deceased is the supplementary card member, the bank will cancel the supplementary card, and the principal card holder will be liable for all the transactions incurred by the supplementary card member.\nSome credit cards are tied to an overdraft facility. The outstanding amount will also be recovered.\nThe bank will close the car loan account and recover the car. The car will be sold in open market. If the amount is insufficient to cover the outstanding loan, the bank will recover the outstanding amount from the estate. The bank has the right to net off the outstanding debt from the deceased person's bank savings accounts.\nThe bank will close the loan account and recover from the estate by filing a caveat. If there is more than one personal guarantor on the account, the bank will recover the loan from the rest of the surviving guarantors.\nIt is not uncommon for loans to be incurred privately. For example, a sole proprietor would borrow money from a relative or friend to keep a business going. The issue is that if there is no formal IOU, it is very difficult for the creditor to recover from the estate. The estate legal representative is under no legal obligation to recognise such loans, let alone settle it. However, if there is admissible evidence, like a bounced cheque issued by the debtor shortly before he dies, the creditor can file a caveat onto the estate.\nIf the estate has insufficient cash to settle the debts, the legal representative will have to forcedly sell the estate assets to raise the funds. If the forced sale amount is still not sufficient, the estate will be insolvent, and there will be NOTHING left to distribute to the beneficiaries, however beautifully the will was written.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Will was featured in Men's Fitness magazine as one of \"Hollywood's Top Trainers\". He has earned over 10 certifications Including; USA Boxing Certified Coach, USA Track & Field Certified Coach, Medical Exercise Specialist and Master of Fitness Sciences. And has been awarded commendations from the City of Los Angeles for his \"diligent efforts in the field of Fitness & Health\". Will's dedication to fitness is reflected in his professional accolades, as well as in his personal life.\nWill, was additionally one of the Fitness Experts for Men's Fitness magazine's \"Ask Our Experts\" column where he helped to answer the personal fitness and training questions of their over one million readers.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"racking as well as mounting brackets.\nAs a distributor for innovative global leaders, DPA Solar gives solar installers & electricians access to the latest technology by manufacturers who have stood the test of time.\nDPA Solar gives you access to the most up-to-date technologies backed by manufacturers who have stood the test of time, guaranteed to be around as long as the systems you install.\nWhether you specialise in stand alone power or hybrid\/grid tie, if it's solar we're the team for you. Talk to us about SimpliPhi Lithium Batteries and how they can be teamed with the Inverter\/Charger of your choice.\nChoosing the right products from the start saves you time and money. We are a dedicated team of engineers & solar experts ready to give you the best product advice and support all the way.\nWhen trouble arises, you need to know your supplier is at the end of the phone. We're here to assist in whatever way we can. As well as access to our engineers Monday to Friday, you can find technical information & tools online 24\/7.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"You are currently browsing the daily archive for January 11, 2017.\nThe seven weeks after the press conference at which Maithripala Sirisena announced his candidature were hectic and tense. During the conference itself, I had a telephone call to say that the Presidential Secretariat had called to demand that the vehicle I was using be returned. This struck me as petty, and foolish given that Chandrika Kumaratunga had just announced that those of us who had come out in favour of the common candidate would be persecuted.\nI am aware that Mahinda Rajapaksa felt he had been betrayed by Maithripala Sirisena since, even when they had had dinner together the night before, the latter had given no hint that he was going to contest. But the manner in which I was deprived of my vehicle, even while I was still technically Adviser to the President on Reconciliation, indicated the manner in which anyone who was open in their actions would be treated.\nIn my case the President had no reason at all to feel betrayed, since I had written to him clearly in October to say we could not support him if he did not proceed with some of the reforms he had pledged earlier. And over the last few months I had made clear the need for reform, both Vasantha and I even proposing Private Members Bills with regard to burning issues such as reducing the size of the Cabinet. Interestingly enough, Vasantha told me that the President had called him and said that he was being unduly influenced by me, but he did not bother to speak to me himself. It was only just before the common candidate declared himself that one of his confidantes, Sarath Wijesinghe, called me and said that he assumed I would support the President. But even Sarath had no answer when I mentioned what worried me, such as the appalling treatment of Chris Nonis.\nI have no hard feelings though about Mahinda Rajapaksa, because I believe he was grossly misled by a small coterie around him who cared neither for him nor for the country. What was surprising was that a man of such capacity, and sensitivity to the needs of the country, should have allowed himself to be dominated by a bunch of callous rascals. I should note that, though I have never had any high regard for Basil Rajapaksa, I do not include him in the category of those with undue influence, since he was undoubtedly a man of ability. And he achieved much in terms of development, even though he was not capable of twinning this with human development, which was essential if the fruits of development were to be equitably distributed. And of course he was largely responsible for alienating the President from the senior members of his party, since the impression they had, indicated to me vividly by one of the most decent members of the Cabinet, John Seneviratne, was that he was usurping the powers of all other ministries.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"According to Christoph Bartneck's 2009 scientometric analysis of CHI conference proceedings , Asian researchers' activity at the CHI conference was very small at that point; between 1982 and 2008, only 3.61 percent of papers in the main conference proceedings were from Asian countries, and Japan was the only country listed in the top 10 most active countries. Six years have passed since Bartneck's report. During that time, 2,214 papers have been published in the CHI proceedings. This is larger than the total number of papers that Bartneck analyzed (only 1,945 papers were published in CHI proceedings from 1982 to 2008, according to the ACM Digital Library).\nThis surprisingly rapid growth of the CHI community has had a significant effect on the proceedings' statistics over the past six years. Here, we aim to share insights from an analysis of the CHI proceedings from 2009 to 2014. Bartneck's analysis was more broadly focused on both quantity and quality, including numbers of citations and best paper awards. Due to limited space and resources, we focus only on the quantitative analysis, including geography, organization, and author statistics.\nGenerally, our analysis follows Bartneck's method. We employ the idea of credit for the analysis, in which one paper equals one credit. Each author receives an equal share of the credit. For example, for a paper written by four authors, each coauthor receives 0.25 credits . This method is also effective at determining how much a country or an organization contributed to a paper. We analyzed 2,214 papers, so the total amount of credits was 2,214 for each category. We compiled author statistics by using another method; all authors of a paper receive one credit because the analysis is aimed to determine how many papers were authored or coauthored by each author.\nOur analysis focused on analyzing main conference proceedings\u2014not extended abstracts. Main conference proceedings consist of papers that have been submitted to the \"Papers and Notes\" category, which has the most competitive reviewing process and represents the latest important findings in HCI. All data was taken from the ACM Digital Library (ACM DL). For this analysis, we wrote a PHP data-scraping script to retrieve data for the analysis of table-of-contents data in the ACM DL. Data for the analysis included the paper title, author name and affiliation, and the number of coauthors (country name is included in the author affiliation). Data was saved as an XML file. When we retrieved the data, 334 author affiliations were missing, so we manually fixed them. Likewise, if the organization name in the author affiliation was inconsistent (e.g., many authors used double affiliations, making the organization names inconsistent), we also manually fixed these entries. All data processing was done with PHP scripts.\nThere is limited space here to present the results of our analysis. Details, including the complete list of countries, organizations, and authors, can be found at: http:\/\/daisukesakamoto.jp\/articles\/Asian-Researchers-at-the-CHI-Conference\/.\nGeography. In general, we determined that 44 countries from all over the world contributed to the CHI conference from 2009 to 2014. The overview distribution is shown in Figure 1. When Bartneck analyzed the CHI proceedings from 1982 to 2008, the U.S. received 67.34 percent of all credits in the main conference proceedings, and the top four countries (the U.S., the U.K., Canada, and Japan) accumulated 88.16 percent of all credits. In our analysis, the U.S. received 49.22 percent of all credits, and the top countries were reshuffled. The top five countries were the U.S., the U.K., Canada, Germany, and France. These five countries received 79.3 percent of all credits. Among the Asian countries, Japan was the highest ranking at sixth (Table 1). Looking at the other Asian countries, South Korea was in eighth place, China was in 12th place, and Singapore was in 13th place (Figure 2). Finally, Asian countries received 183.93 (8.35 percent) of all 2,214 credits, which is an almost 5 percent increase from the previous analysis. This suggests that Asian countries are becoming more active in the CHI community.\nOrganizations. Next, we analyzed author affiliations. This analysis covered only Asian countries. Our results show that 106 organizations contributed to the CHI conference. The Korean Advanced Institute of Science and Technology (KAIST) in South Korea was the most productive organization in Asia (Table 2). Even though the other South Korean organizations were not in the top 10 list, the 25.43 credits from South Korean organizations indicate that South Korea is a rapidly growing country in the field of HCI research.\nThe National University of Singapore was in second place, and Nanyang Technological University was in seventh. These two universities received 89.68 percent of Singapore's credits, contributing highly to Singapore's move up in the geographical analysis list (Table 1). Likewise, National Taiwan University received 75.5 percent of Taiwan's credits. Just five Taiwanese organizations contributed to CHI, but this time Taiwan's rank rose from 34 to 17. Interestingly, four Japanese organizations were in the top 10 list, and 10 organizations were in the top 20 list, but its final rank in the geographical analysis dropped from four to six. Actually, Japan is the highest-ranking Asian country. The largest number of organizations came from Japan (34 organizations; Table 3), but the small contributions from each one might have affected the geographical result.\nOne thing we need to be concerned with is that many authors have double affiliations. For example, in the case of the \"JST ERATO Igarashi Design Interface Project\" (ranked 10 in Table 2), usually the affiliation would include the author's university (e.g., the University of Tokyo) and this organization. In this case, the same amount of credit goes to both organizations so that the total amount of credits is larger than in the geographical analysis. This is a worldwide trend, not a characteristic of Asian countries specifically. In summary, organizations in Asia received 194.1 credits in total, and the top 10 organizations received 109.64 credits (56.48 percent). The average credit was 1.83 (SD = 3.54).\nAuthors. A large number of authors from Asia contributed to the CHI conference. In fact, 518 unique authors whose affiliation contained an Asian country's name were found in the proceedings. The average number of publications per author was 1.50, and the median was 1; 388 authors (74.9 percent) published just one paper in six years. Table 4 shows the list of authors who published at least six papers in the six-year period. Further, we determined that 50 authors (9.65 percent) published at least three papers. These results indicate that the active organizations employ productive researchers. However, by considering this data along with the organization results, it shows there is much room for growth. Almost half the credits go to organizations that are ranked below 11. This indicates that there are organizations with the potential to grow in the future. Our analysis targeted a narrow range of years, and we are in the process of rapid growth, so some authors are not included in this analysis. An appropriate method for author analysis might be necessary to understand the stats more deeply.\nLimitations. These days, it is common for researchers to leave their home country and work in other countries and on other continents. One researcher may get a tenured position in another country, while another researcher has an internship in a foreign country but will return to his or her home country when finished. The situation of each individual researcher is different, and it is impossible to know all of them from the data we had from the proceedings. A major limitation of this analysis is that it was done with a limited data source. We aimed to understand the research activity of Asian researchers at the CHI conference; however, we can say for sure that this is not a comprehensive analysis.\nAnother concern is our lack of confidence in the handling of double affiliations. The problem is that the credit should go to the affiliation where the research had primarily been conducted. However, our limited resource for analysis does not allow us to do this. It may be possible to identify where the research was done by carefully reading the papers, but this is unrealistic. We need more collaborators or better methods to conduct this kind of meta research.\nFinally, we used a quick-and-dirty method for the analysis. Some data was fixed manually, and some part of the data processing was done manually (copying data to a spreadsheet, etc.), so this analysis may contain errors. Let us know if you find any problems. We will revise the appendix Web page according to your error report.\nIn this article, we presented a brief analysis of Asian researchers at the CHI conference. Asia is a rapidly growing region in HCI research, and the results indicate that there is room for more growth. We hope many Asian researchers contribute to the CHI 2015 conference. Likewise, we expect that even more Asian researchers will actively participate the next time the CHI conference is held in Asia.\nAs noted, we used a quick-and-dirty method for the analysis. Please be advised this analysis may contain errors, so we caution use of these results outside this discussion.\nThanks to Christoph Bartneck for his help.\nDaisuke Sakamoto is a project lecturer at the University of Tokyo. His research interests include human-computer interaction, human-robot interaction, user experience, and interaction design.\nFigure 1. Overview distribution of credit for all countries.\nFigure 2. Credit distribution among Asian countries.\nTable 2. Top 10 organizations in Asia.\nTable 3. Number of organizations represented from each country.\nTable 4. Authors who published at least six papers in six years.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzaedyz b/data_all_eng_slimpj/shuffled/split2/finalzzzaedyz
new file mode 100644
index 0000000000000000000000000000000000000000..5f5df13eab70c4920a37f098960e54a979dd714d
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@@ -0,0 +1,5 @@
+{"text":"Department of English has been introduced at Benazir Bhutto Shaheed University, Lyari, Karachi on 1st March, 2012.The department has been rendering invaluable services in achieving the academic goals of the deprived and less educated community, rural in general and urban in particular, in order to uplift the noble cause of education.\nDepartment of English has been introduced at Benazir Bhutto Shaheed University, Lyari, Karachi on 1st March, 2012.The department has been rendering invaluable services in achieving the academic goals of the deprived and less educated community, rural in general and urban in particular, in order to uplift the noble cause of education. Being aware of the present needs of the learners and their real challenges of the future, the department has designed skill oriented and learner based four years BS (HONS) Program in English Linguistics and Literature. Linguistics focuses on various aspects of language such as phonetics, morphology, generative grammar and syntax. Another specially designed area for raising the language competence of the learner is through practice of authentic and advanced textual use of language in context. Literature focuses on unfolding the hidden capabilities of a learner through literary criticism, fictions, dramas, literary essays and critical based learning.\nThe department not only maintains regularity in academics but offers the students, in every semester, a variety of exciting co-curricular and cultural activities.\nTo generate excellent undergraduate and graduates in a learner centered environment. To create scholars who will endeavor for further knowledge and awareness, to endorse peace, freedom, democracy and acceptance in our students through development of critical thinking.\nOur intention is to provide quality education through latest teaching strategies in order to make students understand the essence of education and develop their critical thinking. Moreover; the aim to mould the students into better human beings, so that they become capable of contributing effectively and efficiently towards national development.\nTo develop clarity, precision and maturity in spoken and written English. To focus attention on the importance of cultural and social values in understanding the forces that have shaped civilization that are shaping the contemporary world.\nTo prepare students to investigate language related problem and issues in the everyday world. This can best be achieved by providing students with a broad background of knowledge about the nature of language and language use as situated in social discursive and interactional social context.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"January 31, 2014 \u2022 Ectromagnetic theory says there should be a bit of matter that is unipolar: neither north or south. So far no one has found a magnetic monopole, but physicists at Amherst college have come close.\nJanuary 21, 2014 \u2022 Computer scientist Peter Stone has taken his passion for soccer into the lab. He's developing robots that can play soccer. The work requires expertise in computer vision, robotics and understanding about how autonomous agents work together.\nJanuary 21, 2014 \u2022 The Rosetta spacecraft has awakened. It was put in hibernation for 31 months while its orbit took it nearly half a billion miles from the sun, too far for its solar arrays to keep the spacecraft operational. But now it's close enough, and European Space Agency mission managers will start preparing for Rosetta's rendezvous with a comet later this year.\nJanuary 3, 2014 \u2022 Students at Rice University in Houston are finding low-cost solutions to big global health problems. One particularly successful device that helps infants breathe has already been tested in Malawi.\nJanuary 1, 2014 \u2022 Some metal alloys will \"remember\" a shape when you heat them to the same temperature they were originally shaped at. So a straight wire made from one of these \"shape memory alloys\" might change back into a spring when heated, or vice versa. But the alloys that exist today change shape at low temperatures. Materials scientists at Sandia National Laboratory have developed new alloys that don't change shape until they reach hundreds of degrees, opening the door to thousands of new applications.\nDecember 27, 2013 \u2022 Anyone who has waited endlessly at a red light at an empty four-way intersection knows traffic signals are not the most efficient way to keep traffic flowing. Computer scientists have figured out a way to utilize the space in intersections far more efficiently. The problem is the solution will have to wait for a day when cars can drive themselves.\nDecember 24, 2013 \u2022 Astronauts went on a space walk to fix a broken cooling system aboard the International Space Station on Tuesday. A pump that sends coolant through the system broke earlier this month. It's a critical element and it had to be repaired for the space station to resume normal operations.\nNovember 6, 2013 \u2022 A research chemist applied his analytical smarts to his son's eye cancer. By analyzing family photos starting with some taken just a few days after birth, the dad found that signs of retinoblastoma, a rare eye cancer, could be detected quite early.\nSeptember 12, 2013 \u2022 Cut a tumor from a child's brain and you may save a life. But surgery can hurt the child if healthy brain cells are removed. A Seattle doctor is working on a substance that might help. It binds tightly to cancer cells and makes them glow, so they're easier to distinguish from healthy tissue.\nSeptember 10, 2013 \u2022 NASA's Lunar Atmosphere and Dust Environment Explorer is on its way to the Moon. It lifted off on time Friday night from NASA's Wallops Flight Facility in Virginia. Some early software problems have been successfully resolved.\nSeptember 6, 2013 \u2022 A satellite is scheduled to take off for the Moon Friday \u2014 carrying an instrument that could represent the future of deep space communication. Instead of sending data back to earth using radio waves, the Lunar Laser Communications Demonstration will use pulsed light waves.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Policy issue: Is the condition regarding eutrophication of Europe's seas improving?\nNo general trend is observed in summer surface chlorophyll-a concentrations either in the Baltic Sea area, Greater North Sea or coastal waters of Greece. The majority of stations show no trend. A few stations in Danish estuaries and Danish North Sea show a decreasing trend, whilst a few stations in Belgian, Finnish, Lithuanian and Swedish coastal waters show an increasing trend. No trend is observed in the Dutch coastal waters which receive high nutrient inputs from the Rhine river basin, nor in Norwegian and Greek coastal waters. Finland reports that the increasing trend of chlorophyll-a in the eastern Gulf of Finland and in the Archipelago Sea is due to increase in phosphorus released from hypoxic\/anoxic sediments (Kauppila & B\u00e4ck, 2001; H\u00e4nninen et al. 2000). Due to the information available for this indicator the efforts in the reduction of nutrient loads didn't face success yet.\nFor references, please go to https:\/\/www.eea.europa.eu\/data-and-maps\/indicators\/chlorophyll-a-concentrations-in-transitional or scan the QR code.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Composite & Steel Frame Kit For Building Lightweight Pistols Ready-to-build, high-tech, double stack frame can be fashioned into your customer's dream, competition or self-defense pistol. The 2011's modular grip frame is made from fiber-filled, nylon composite that's married to a CNC machined, 4140 steel upper, for the very best combination of durability, precision fit and lighter weight. Features: 20 lpi checkering on the front strap and triggerguard; heavy-duty dust cover for scope mounting; pre-contoured to accept high-ride beavertails and aftermarket grip safeties with .250\u2033 radius; integral magazine well, oversized rails. Ready for gunsmith fitting. Fra Mfg: Sti SPECS: Grip Frame \u2013 Fiber-filled polymer composite, black. Upper Frame \u2013 4140 steel, in-the-white, matte finish. Kit includes checkered, nylon mainspring housing and pin; titanium\/composite trigger (long) and blued mag catch (complete). Frames accept most 1911 Auto slide\/barrel assemblies. FFL re.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"In his lifelong process of research and experimentation some fundamental opportunities became manifest for his own personal development in the social realm. Through a long process of practicing the \"social questioning\", with a wish to support his clients in a meaningful way, he found a path of initiation in the social realm, which enables him to develop an authentic personality. It is his intention in this book to outline this path of initiation for everyone who has the desire to go out and deal with the social question in today's world. This path has the opportunity to develop an organ of observation for social reality as well as the ability to handle the social reality one is confronted with, both in the organisation one works in and in one's personal life. It also develops an organ of observation for the forces that work between us, in which we can grow as a conscious spiritual person. This is important for our own spiritual being as well as for the whole of humanity, living in the Divine creation given to us. The studying of this book can be the introduction to the path of initiation described in it. Still more important is to practise it in one's personal practical life, for which various tips are given. What he describe here is to be seen as a lens through which to observe one's social reality. On the basis of these conscious observations and their connection to one's own personality, one can then come to a conscious, meaningful deed.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzaelbu b/data_all_eng_slimpj/shuffled/split2/finalzzzaelbu
new file mode 100644
index 0000000000000000000000000000000000000000..0fbe3c390a337024da6b9d4f849dd5e4cd271c8c
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzaelbu
@@ -0,0 +1,5 @@
+{"text":"The main part of the PR formula is your vote. For m-dionis to get your vote, click on the arrow near the number.\nYou can note that you are a fan by clicking on a star \u2014 in this case m-dionis receive a b\u00f3igger PR.\nAdditionally, you can also add them to friends and observe in the section Wazzup every action of m-dionis.\n\u0411\u043b\u0430\u0433\u043e\u0434\u0430\u0440\u044e \u0437\u0430 \u0432\u0438\u0437\u0438\u0442, \u043d\u0430\u0434\u0435\u044e\u0441\u044c \u0431\u044b\u043b\u043e \u0438\u043d\u0442\u0435\u0440\u0435\u0441\u043d\u043e!\n\u041f\u0440\u0438\u0432\u0435\u0442! \u041e\u0431\u044f\u0437\u0430\u0442\u0435\u043b\u044c\u043d\u043e \u0431\u0443\u0434\u0443 \u043a \u0442\u0435\u0431\u0435 \u0437\u0430\u0445\u043e\u0434\u0438\u0442\u044c. \u0417\u0430 \u043f\u0440\u0435\u0434\u043b\u043e\u0436\u0435\u043d\u0438\u0435 \u0434\u0440\u0443\u0436\u0431\u044b \u0441\u043f\u0430\u0441\u0438\u0431\u043e!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"With Kajabi, users may create digital products such as membership sites, online courses and training portal sites efficiently and effectively. In order to design engaging and aesthetic product content, users can create customizable themes, stream videos, add file downloads and upload pictures and audio. Users may also have all of their products together onto one site using one login, a personalized URL and an integrated blog to help get subscribers.\nFurther to digital content production, Kajabi simplifies sales, marketing and customer engagement. Since Kajabi integrates with Stripe, users may collect straightforward electronic payments. To simplify marketing strategies, Kajabi allows users to build custom forms and landing pages with not trouble. Users may also send email broadcasts and then track their performance with Kajabi's email analytics. Kajabi embraces a lot of built-in integrations with applications like Google Analytics, MailChimp and Infusionsoft allowing users to handle their online products and share their digital content effortlessly.\nKajabi has great tools for creating sales funnels and landing pages, and converting website visitors into clients. You may also handle all of your e-mail marketing within Kajabi \u2013 such as automated follow-up sequences \u2013 that means, there's no need for a third party platform (although it does integrate quite easily with Aweber, Mailchimp, and Active Campaign). In general, Kajabi is the best most other online course software in the native advertising and selling capabilities it offers.\nKajabi gives several variations on a core site theme that looks modern and professional and additionally, it provides very appealing, functional templates for creating your landing pages and sales pipeline pages. There is also a market for custom, premium themes if you do not desire to go with what Kajabi provides.\nWhen it is about actually creating your website, Kajabi has pre-designed themes which you can install with a single click. The themes dictate the overall appearance and style of your site and help you quickly and easily create a nice website.\nThey have advanced the Premier Theme into a super flexible framework and they use it to create all the new website themes. These themes are now called Layout Presets and now there are 6 unique Presets accessible in their library, all of which are built on the\"Premier Framework\".\nAll the Layout Presets have a minimal and contemporary design and they do look great. As soon as you install a Preset, you can further modify it using the webpage builder to give your site the look and feel you desire.\nYou may still use any of the 10 themes that have now been relocated to the Legacy Themes archive if you want. Although, as Kajabi won't be working to refine them any longer, I will advise that you use just the Premier Theme for your site.\nLastly, Kajabi also extends you the option to import and use custom themes. So, if you are a power user, you may hire a programmer to build a completely custom theme for your site.\nKajabi utilizes Wistia \u2013 among the all-around top video hosting programs \u2013 fully integrated for uploading and streaming your videos. Because of this you can trust that your videos will deliver as great as possible for whatever screen or bandwidth any individual customer is using \u2014 and that they will be as secure as possible.\nKajabi has been around for quite some time. Because of this, there are a number of businesses out there that have designed custom themes for it (as mentioned previously) and that provide a range of services to support it. Additionally, there are plenty of articles and blog posts out there addressing just about anything you need to do with Kajabi. If you can't find what you would like through the business' own training and support \u2013 which are quite good \u2013 you will most likely be able to find it out on the net.\nGenerally speaking, Kajabi will provide you a lot more power than most course platforms over developing a website where you are able to offer your classes but also manage the whole Web presence for your business. In other words, you don't need to set up a separate site on WordPress or another platform \u2013 a Kajabi site can become your home base.\nWhen you look at Kajabi, it is certainly going to look like the pricing is a little off the charts. The cheapest of these plans is $103 a month, and that's when you pay yearly. It is not exactly what online course startups want to hear, but it is obvious who Kajabi caters to.\nAn Enterprise plan is also sold, but you have to contact the Kajabi company in order to receive more info on that. I'd assume Kajabi is glad to support any enterprise size, so you can customize your plan.\nThe live chat and email support are provided 24\/7. So, when you stumble upon a situation with your course website, you can contact one of the live chat representatives, no matter what time it is. I asked a question and they got back to me in 20 minutes. Needless to say, it doesn't always happen so quickly and they don't have a guarantee to answer you within 20 minutes. But for me, this reveals a good support.\nAnd you know what, Kajabi does quite a great job at the bulk of those functions, and they're constantly developing and adding more features.\nBut what almost always astonishes me is the live customer care. They offer 24\/7 live chat, and the longest I have ever had to wait (and I've chatted at all times of night and day!) Is three or four minutes, once. Every other occasion (and there have been many!) I'm almost instantly connected to someone. And these people are awesome. They will get you the answer, assist you troubleshoot your issue, and also tinker around with a few codings or options on the page to make it work if you are trying something a little more custom and can not quite get it to work right. Seriously. The support team is top-notch!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"In my opinion, since the nature of the assistance is openly acknowledged, public assistance in health care is superior to the public option.\nThe goal of the current health care reform is intellectually dishonest. The current political effort is about finding ways to shove those with known expensive medical conditions into pools to cover their costs. To make up for the deficit, politicians wish to force others to buy insurance that is multiples of their expected expenses.\nA far better approach is to create a structure where individuals have a mechanism to properly account for the care they receive, and to handle the exceptions with openly acknowledged assistance.\nThe intellectual dishonesty of pooled insurance diminishes our health care system. It leads to corruption and erodes the foundations of our health care.\nThe problem is that we are trying to design our entire health care system around exceptions rather than designing the system around the known and real needs of all people. All people have expected health expenses of several hundred thousand dollars in a full life time. Our system should be designed to help people save these resources.\nAs the distribution of income follows the distribution of wealth, if we had a system where people owned their health care resources, we would immediately eliminate the growing disparity in income caused by the foolish system of employer based health care.\nA system based on intellectual honesty will always lead to better results than one based on intellectual dishonest.\nThe Medical Savings and Loan is an intellectually honest alternative to pooled insurance. The system creates a savings and lending account for all policy holders. People place money in their savings in times of health, and withdraw funds in times of need. When need exceeds savings, people can take out loans. If they need more money than they are capable of repaying, the system has them turn to public assistance.\nPeople are actually wonderful creatures who take great pains to help each other. With the medical savings and loan helping identify those with extraordinary needs, people will be happy to find to find ways to extend such assistance.\nThe problem with an intellectually dishonest system (like the bills before congress) is that hard working people become victims of those gaming the system ... by following the inherently dishonest route, people will degenerate into petty jealousies, into conspiracies, and conclude that to survive they must become manipulators themselves.\nPlease contact your congressman and tell them that this path of dishonesty will lead us to ruin. The honest path would lead to a bright future.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"In the latest issue, Diana is back home after besting Ares, Steve Trevor is recovering and Aphrodite, the goddess of love, is a houseguest. The mystical creatures on her side are also without a home and are refugees in our country, nodding to the fiercely debated immigration debate ripped from media headlines.\nThe foil this arc is Greg Rucka's foe Veronica Cale, who is holding captive Nemesis\u2014the goddess of grudges\u2014to go after Wonder Woman in revenge for the loss of her daughter.\nIn Virginia, pegasus Cadmus returns to warn Diana that supernatural games are afoot, as a giant burning sinkhole appears in Washington DC. There, Diana faces Nemesis in giant bug form for three pages of non-stop action.\nNow I must take some time to praise the skill of Jesus Marino on art, his storytelling and acting only rivaled by his own draftsmanship. A full page pinup of Wonder Woman riding the soaring Cadmus is knockout gorgeous. The fight scenes are epic and his work is complimented by Andy Owen's inks and Romulo Fajardo's colors.\nBack to the action, Cale is not only a powerful business woman, but one who can sway the press to her influence. She also seeks to have superheroes register for fighting the good fight much like in Marvel's Civil War. Wonder Woman is forced to relent in front of the cameras, at least for now.\nCale also reveals that a Cosmic paradox has disappeared Themyscira and Olympus, and like her griffin and Pegasus friends is now an exile here on Earth.\nHere's to hoping Diana can find out what happened to her home-world and Olympus as she soars off into the sky pondering her fate.\nI really do hope Wilson has the answers and they will be revealed in the coming issues, a Wonder Woman without her mystical home would be a very lonely existence indeed.\nBack to The Trinity, perhaps Batman and Superman will be brought in to help solve the mystery. I would love to hear Wilson write dialogue for a Bruce, Clark and Diana scene. She has a great line in this book about how Steve is self-conscious about his manhood and abilities as a lover under the watchful eye of the couch surfing Aphrodite.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Description Physical therapy is the art and science of rehabilitation, restoring independence and a maximal level function to individuals with physical an psychological disorders. Physical therapists use exercise, heat, water, cold, ice, sound and electricity as well as assistive devices and artificial limbs to help individuals regain maximal functional potential. The physical therapist is a health professional who contributes to the multidisciplinary team through patient evaluation, treatment planning and delivery, education, research and consultation in clinics, industry and the community.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzaewnb b/data_all_eng_slimpj/shuffled/split2/finalzzzaewnb
new file mode 100644
index 0000000000000000000000000000000000000000..4257b93571f499e4861ec91a1e0ea0efcef0aee1
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzaewnb
@@ -0,0 +1,5 @@
+{"text":"RG Herald: Ramirez Elem., 1st Place at UIL Meet.\nRamirez Elem., 1st Place at UIL Meet.\nDr. Mario E. Ramirez Elementary competed in their first UIL Meet on Saturday, December 11, 2010 at La Union Elementary. Dr. Mario E. Ramirez Elementary earned a sweapstakes for 2nd, 4th and 5th Grade and were the first place Champions for the meet. Congratulations go out to all the students, parents, coaches and administrators for their excellent performance. GO DIAMONDBACKS!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Seattle Police Department reports pertaining to compliance with Ordinance #108333.\nAudits and reports from the Seattle Police Department's Police Intelligence Auditor and Chief of Police showing compliance with the Seattle Police Intelligence Ordinance (Ordinance 108333), which limited the collection of political and religious information except in very limited circumstances where written authorization had been granted on a case-by-case basis. These audits and reports reference the types of authorizations granted, as well as the number granted in that year. Additional documentation in these folders includes copies of the constitution and bylaws for both the Law Enforcement Intelligence Unit (LEIU) and Western States Information Network (WSIN).\n[Item and date], Police Intelligence Audits, Record Series 1802-N2. Box [number], Folder [number]. Seattle Municipal Archives.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"It's Official! Prince Harry and Meghan Markle are Engaged to Be Married!\nGet ready for a royal wedding everyone. Prince Harry and American actress Meghan Markle are getting married!\nPrince Charles made the official announcement on behalf of his youngest son and the bride-to-be this evening. And Meghan has also revealed how Harry went down on one knee to pop the question.\nHis Royal Highness The Prince of Wales is delighted to announce the engagement of Prince Harry to Ms. Meghan Markle,\" Prince Charles said, in an official statement from Clarence House.\nHarry, 33, proposed to Meghan, 36, in London earlier this month while the couple were cooking a roast chicken dinner at his Kensington Palace apartment. \"It was just an amazing surprise. It was so sweet, and natural and very romantic. He got down on one knee,\" Meghan reveals.\nThe spectacular engagement ring includes two diamonds from Harry's mother's collection and another diamond from Botswana, where the couple enjoyed their first holiday together. Harry says Princess Diana would have adored his bride-to-be.\n\"I think she would be over the moon, jumping up and down, so excited for me,\" Harry says.\nThe couple shared the happy news with the Queen and other family members soon after over afternoon tea. The engagement comes 18 months after they met on a blind date in Toronto, where Meghan was acting in the television show, Suits.\nPrince Harry has also sought and received the blessing of Ms Markle's parents,\" Prince Charles said.\nMeghan's parents are Doria Ragland, a 61-year-old yoga instructor and Thomas Markle, an Emmy award-winning lighting director who lives in Mexico. The couple are divorced and Meghan, who recently moved to London with her two dogs, Bogart and Guy, has also been married before.\nHer parents wished the couple \"a lifetime of happiness\" in an official statement, released via the palace.\nKensington Palace followed the announcements with confirmation of its own, sharing the news on Instagram, with a casual snapshot of the Prince and his soon-to-be royal bride.\nNo firm details yet on the wedding, except that it is likely after the arrival of older brother Prince William's third child with wife Kate Middleton, due in April. We do expect that Prince George and Princess Charlotte will star in the bridal party as the world's cutest royal page boy and flower girl.\nKensington Palace tweeted that Prince William and Kate are \"very excited\" at the engagement.\nDon't miss any royal wedding news, make sure to follow Mum Central on Facebook.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The second trailer to Murakami Takashi's debut live-action film with GAGA Corporation, \"Mememe no Kurage\"\u300c\u3081\u3081\u3081\u306e\u304f\u3089\u3052\u300d(Jellyfish Eyes), has been released on YouTube. Along with the new trailer, the theme song of the renowned Japanese artist's first movie is also played in the trailer: livetune's \"Last Night, Good Night (Re:Dialed)\", the remake of the popular hit song featuring Hatsune Miku which is also one of the tracks in livetune's compilation album, \"Re:Dial\".\nAs if the announcement of Re:Dial was not enough for kz to start the year, another news arrived to Twitter\u2014announcing that he also took part in the second season \"Ore no Imouto ga Konna ni Kawaii Wake ga Nai\" as the producer of ClariS' new single, \"reunion\", which will be used as the show's opening theme song.\nAnother great upcoming work has been announced to be released on 2013, which is kz(livetune)'s latest album. This album will feature lots of popular tracks from his five-year long career as a Vocaloid producer, and even a couple of new tracks to boot.\nOne of the most amusing and intriguing parts of the growing phenomenon of Vocaloid culture originating from Nico Nico Douga is the fact that the songs are all composed by the users of the software, regarded as \"VocaP\" (Vocaloid Producer). There are so many great, talented VocaP's, each of them having their own way of using the famous voice synthesizer to produce masterpieces that captivated millions of people in Japan, and even around the world. In this post, I'm going to talk about one of my top four favorite Voca-Ps\u2014besides supercell, DECO*27, and 40mP\u2014all of which have influenced my music preference and my passion to be a musician myself until now: kz, or as most of us refer more often as \"band\" or \"group\", livetune.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Sphero Support and Knowledge Base - Does SPRK+ work with other Sphero apps?\nAll Categories \u200b>\u200b Sphero Products \u200b>\u200b Sphero SPRK+ \u200b>\u200b Does SPRK+ work with other Sphero apps?\nDoes Spider-Man work with other Sphero apps?","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzafqgw b/data_all_eng_slimpj/shuffled/split2/finalzzzafqgw
new file mode 100644
index 0000000000000000000000000000000000000000..f51bf5339cf23cbb498830c45948bb0c2d59b467
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzafqgw
@@ -0,0 +1,5 @@
+{"text":"The Cooperative Sheet Music program enables participating school districts to share the cost of purchasing choral, band, and orchestral music. Schools will be able to share a collection that contains over 3,600 titles with selections targeted to students from kindergarten through Grade 12.\nThe Cooperative Sheet Music program is an optional RWADA based service under the MultiMedia Service. Districts that are not members of MultiMedia Service can participate in the Cooperative Sheet Music program at the non-member rate.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Here we will try to provide you with all information needed to setup a SiteFusion server and connect to it using the SiteFusion client.\n$ php -m | grep -e \"json\\|sockets\\|posix\\|pcntl\"\nIt should show all four module names, if anything is missing it's not installed.\nInstall SiteFusion by downloading a release package or by using Git. Where you see 5.x.x below, replace for the version of your choice.\nIn my case, the user is called 'dev' and I extracted the package to \/home\/dev\/sitefusion, although a more suitable location for production environments might be \/usr\/local\/sitefusion or something like that, depending on conventions on your operating system or distribution.\nThe file daemon.conf holds the configuration for the SiteFusion daemon.\nStart as root (default and recommended): Leave the runtimeDirectory and logfile locations to the default system locations, set changeIdentity to TRUE and fill in a safe user (systemUser) and group (systemGroup) for the daemon to change to (like a separate user owning the sitefusion installation directory, or like the nobody or apache\/www-data users who should have at least reading permissions in the sitefusion installation directory). Then start as root. The daemon will then change to the safe user and automatically make sure it can use the log and runtime locations as that user.\nStart as unpriviledged user: Create new runtime and log directories in the sitefusion installation directory (mkdir run && mkdir log) and change the runtimeDirectory and logfile configuration options to have them point to these locations. Set the changeIdentity option to FALSE. Then start as the unpriviledged user.\nNote that in neither scenario the daemon will be running as root. In the first it will only start as root and then change to the safe user as soon as runtime and log locations are setup correctly.\nMake sure the logfile and runtimeDirectory configuration settings contain a path to a logfile (doesn't have to exist yet) and a path to a directory respectively. When you're done editing save and close the file. The second file you have to edit is webfrontend.conf. This one concerns the communication between the webserver and the daemon.\nNote: From SiteFusion Server 6.0 onward, the default setup is to use sqlite3 instead of mysql for the database used by the daemon to store process and service information. Use the field databaseDSN to set the database protocol, host and credentials instead of using the deprecated fields databaseHost, databaseUsername and databasePassword. The default databaseDSN will use the pre-packaged sqlite3 database in the 'db' directory.\nThe rest of the installation directory needs to have the same permissions if you plan to run the daemon as root and have it change to a safe user. At least it needs to be readable by the user running the daemon.\nThe last thing you have to to is add an alias or vhost to the webserver configuration and point it to the SiteFusion www directory. In the case of apache, either add the following configuration to an existing vhost or create a new one.\nNotice that you have to point to the www directory in the location you extracted the SiteFusion files to earlier.\nAfter this, reload or restart the webserver to activate your new alias. You can test if things are working by opening a browser and navigating to http:\/\/yourip\/sitefusion\/login.php. If you don't get a 404 or 403 error, you will be fine. Depending on your browser you should see a piece of JSON text indicating an input error, or a download dialog for login.php resulting in a textfile with the same contents.\nIf no errors occur, you can shut down the daemon by pressing Ctrl+C and restart without the -d and -f options for production environments. If you are planning to debug instead of running production applications, you can choose to run with the option -d. Debug information is then made available through the log, and the debugger in SiteFusion Admin will be enabled to connect to running processes.\nThe server setup is now completed. Assuming you've set up the client (if not, read here), it will prompt you for a number of options. Server URL is the URL of the server we just set up. So http:\/\/yourip\/sitefusion. The example application is the administration application so in the application field fill in 'sitefusion\/admin'. You can leave arguments empty. Finally, the default username is 'admin' and the password is 'secret'. This password can be changed by editing the file sfadminPasswd.php.inc in the app\/sitefusion\/ directory.\nMost error messages will be reasonably clear as to what's causing the problem. If you get errors concerning undefined functions, you're probably missing one of the required PHP modules as listed in the Requirements section above.\nIf the error refers to the logfile path or the runtime directory path, make sure that these paths are correct and writable to the user running the daemon.\nupdate 32-9-09: I've added runtimeDirectory as a necessary change since it needs to be changed according to the distribution your server is running.\nCurrently the SiteFusion daemon doesn't work on non-POSIX systems, however you could simply run a virtual machine with Linux to use SiteFusion.\nwhat are you recommended linux dist. ? Debian, Ubuntu, etc. ?\nAnd ... has vmware appliance for sitefusion download link.\nAnd ... If you not recommend linux dist. I'm TURNKEY linux to counsel.\nWe recommend both Debian and Ubuntu, but any distribution should be fine. We have no ready-made vmware appliance but following this tutorial you should be able to create one without much trouble.\nThe 'sfdatabase'-script requires the readline-extension. If you don't have it, the readline() manual page has a substitute function, place it at the top of the script.\nCan't find daemon\/start.php. Please check your 'sitefusionPath' configuration option in conf\/daemon.conf, the start.php file it's on sitefusion\/daemon\/ but i don't know why don't work!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Our sweet boy started in the 2 year old class at school this year and he is LOVING it!\nHe was nervous about going to school and being in the \"big kid class\", but he smiled big and was so brave!\nBreakfast of champions before school!\nWe took a trip to Austin for a few days and had fun at the Children's Museum one afternoon. He loved the train table!\nThe water room was a huge hit!\nThis kid became a sno cone pro!\nSweet boys eating snacks together!\nSilly friends with their ice cream cones!\nA lovely duet played by 2 adorable kiddos!\nDays at the pool followed by Torchy's Tacos? Yes, please!\nAedan & Parker after playing & swimming together one morning.\nWhat a great summer we had!\nEarly summer playdate with friends! We love summer time when we get to see our friends more often.\nRoad trip lunch \u2013 headed to Abilene!\nTrampoline fun with Logan in Abilene!\nIce cream cone in the car? Why not?\nOne of my favs from our beach photo shoot!\nEating dinner at Steve's Landing on one of our last nights at the beach.\nMore pics from our summer to come!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Padded crescent bar stool black atlantic shopping, bar stool: new 617 round padded bar stool cushions. Poundex set of 2 metal frame bar height bar stools with. Unique shape stainless steel bar stool with simple.\nFurniture chrome metal counter stool using round white, round padded bar stools black rocket events. White swivel bar stool with cross back and round padded. Furniture chrome metal counter stool using round white.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"It is the ideal complex to enjoy unforgettable holidays in the cosmopolitan zone of COSTA ADEJE and#8211; TENERIFE, located 700 meters to the beaches Torviscas and Fa\u00f1abe and near the shopping centres. There are many leisure and sport possibilities around, such as golf course, Aqua Park and skypark. The complex consists of studios and apartments, renovated in 2004. Distance to the international airport is 18 km.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzafspk b/data_all_eng_slimpj/shuffled/split2/finalzzzafspk
new file mode 100644
index 0000000000000000000000000000000000000000..ed12598407aea76f3051a850e1b2e203e77f1f68
--- /dev/null
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@@ -0,0 +1,5 @@
+{"text":"You can expect a lot from the 2019 Subaru Legacy. This 4 door, 5 passenger sedan leads among competitors in its segment. It features a continuously variable transmission, all-wheel drive, and a 2.5 liter 4 cylinder engine. A wealth of standard features means that you no longer have to sacrifice. Such as cruise control, power front seats, an automatic dimming rear-view mirror, a blind spot monitoring system, turn signal indicator mirrors, lane departure warning, remote keyless entry, and a split folding rear seat. Features such as automatic climate control and leather upholstery prove that economical transportation does not need to be sparsely equipped. Rear passengers enjoy the seat heating functionality, keeping them warm during the winter months. Audio features include a CD player with MP3 capability, steering wheel mounted audio controls, and 12 speakers, providing excellent sound throughout the cabin. Passengers are protected by various safety and security features, including: head curtain airbags, front side impact airbags, traction control, brake assist, a panic alarm, an emergency communication system, and 4 wheel disc brakes with ABS. This car was designed with safety in mind, allowing you to drive with even greater assurance. We pride ourselves in the quality that we offer on all of our vehicles. Stop by our dealership or give us a call for more information.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Filing for bankruptcy is never an easy process and having a myriad of questions about the ordeal is more than normal. It is one of the more stressful experiences that you can go through, especially when you have to decide between Chapter 7 and Chapter 13 bankruptcy. These are some hard questions that you need to ask and doing so without the proper council could lead to some more headaches. There, are, however, a few things that you should know about Chapter 13 bankruptcy and common questions that we will get into here with the help from the best Chapter 13 Bankruptcy Lawyer Los Angeles.\nIf you are in a position to file for Chapter 13 bankruptcy, you should do your best to investigate the option. This filing method allows you to keep your assets while you discharge your unsecured debt.\nIn short, this means that you can keep your belongings while paying off your debt. These agreements usually stretch between a three or five-year payment plan. In the case where you are able to pay off your debt on time, the rest of your debt can be removed.\nAt this stage, it is also worth mentioning that student loans, alimony, and child support do not count as dischargeable debts.\nChapter 13 bankruptcy is probably your best choice if you have a regular income. In these cases, you will be able to avoid foreclosure, make up payments and avoid accruing interest.\nAlthough there are a number of benefits to filing for Chapter 13 bankruptcy. There are also a couple of disadvantages. Chief among these being that you have a three to five-year payment plan.\nAt the time, it might seem bad, but you also won't be able to hold any credit cards, although it can also be a blessing in disguise. The other problem is that you will have great difficulty obtaining a mortgage, seeing that the banks see you as a high-risk candidate. This is mainly due to the fact that your credit score will be next to nothing.\nWhen you file for bankruptcy, it is your last resort. This means that you have too much debt and you are sinking. However, to file for Chapter 13, you may have no more than $394 725 worth of unsecured debt which includes personal loans and credit card debt.\nFurthermore, you may not have more than $1 184 200 worth of secured debt like a home loan. These figures can fluctuate according to the consumer price index.\nYou don't have to go through the process alone, you can get professional help from us here at Karine Karadjian. We have the best Chapter 13 Bankruptcy Lawyer Los Angeles that will guide you every step of the way. All you need to do is to give us a call on any of these numbers (323) 426 8300\/(949)565-4707 or (818)574-1202 for a free consultation and we will help you make the difficult decisions.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"At Rempel Insurance Brokers, our policy is to serve you. Your insurance broker is as important to you or your business as a competent accountant or lawyer. Proper insurance coverage in Manitoba is critical. We believe that an Independent Insurance Broker has the knowledge and experience to guarantee you the best coverage, service and rates available in Manitoba.\nOur reputation and relationship that we have developed with insurance companies, gives you the assurance that you receive the best coverage at the best rate possible. Your insurance will help get you back to where you were before a claim.\nRempel Insurance Brokers provides you with professional service representing most major insurance companies and we will consider the potential hazards to make recommendations on how to improve your rates and coverages.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Learn more about Travel Accessory Bundle.\nThis is a graphical representation to help you design your new purse. Due to the handmade nature of this product, actual fabric scale, placement and pattern direction may vary.\nPremier fabrics are superior texture or Designer brands.\nRSD supports the Feisty Fighters with a $10 donation for all Woven Hope purse designs.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"When we first hit the road, the very idea of staying overnight in a Walmart free camping spot made me cringe. Eleven years later, we found ourselves spending the night at the Whitehorse Walmart camping spot in the middle of town. With a lovely view of the wetlands outside our door, the scenery wasn't nearly as horrendous as some Wallyworlds we have chosen.\nEvery RVer who travels the Alaska Highway will go through Whitehorse at some point. As the capitol of the Yukon, it's the last place to stock up for at least 500 miles in every direction. The store itself really stinks compared to most Walmarts, but up here you get what you get.\nAs a bonus, there is a free dump station, fuel and propane right next door.\nThe Whitehorse Walmart had everything RVers need.\nBut now, Whitehorse Walmart free camping is coming to an end. Last week the Canadian Broadcasting Corp announced that Walmart is banning overnight RV parking in Whitehorse. The store has caved in to pressure from locals, who say that RVers are trashing the parking lot, creating parking mayhem and forcing campgrounds to close.\nOur Fountain of Youth friends who happened to be there at the same time we were, Jim and Patti Ballou, are quoted in this Yukon News article about the Whitehorse Walmart free camping ban.\nMost of what the media reports is true: RVers pack into that Walmart. And I'll admit, it was a bit of a shock to see RVers pulling out barbecues and lawn chairs in the parking lot. It often had the festive feel of a tailgate party. And some rigs looked like they had been there far too long. We spotted this notice on a few rigs.\nBut overall we saw much less trash and sketchy characters there than any other Walmart that allows RV camping.\nWalmart is blaming the camping ban on RVers. But the real truth is probably that local parks have put pressure on the store. Old news stories say park owners are pissed because travelers are passing up their establishments in favor of this place.\nWhitehorse Walmart RV parking is now over.\nHonestly after seeing what's available in Whitehorse, I can't blame RVers for choosing an asphalt parking lot over the so-called \"parks\" in town. We paid for one night at a park and it was one of the most crowded places we've stayed outside of Las Vegas.\nIf local Whitehorse RV parks would do some upgrades so that guests weren't crowded in like sardines, and have more full hookup campsites available, more of us would be willing to spend the $35 \u2013 $45 per night at their shoddy establishments. I know I would.\nRVers shop at this Walmart, even though the selections stink.\nIn the Pioneer RV Park quoted in the article, the sites are so close together, that our stairs almost hit the neighbor's slide out. Poor Wyatt bonked his head on their fender when he stepped outside. Our friends Jim and Patti tried to get into this particular park, and couldn't get in. That's the only reason they stayed at Wallyworld that night.\nWhere will all the RVers go?\nWe're sad to hear that another Walmart RV camping spot has bitten the dust. We wonder how many more will follow?\nThe restaurants benefited from our free Walmart parking. In the end we contriibuted generously to the Whitehorse economy.\nTotally agree Patti. We ate out three times during our visits to Whitehorse to and from Alaska, and shopped way more than we would have if we had paid the going rate on rent for our stay.\nWe passed though in late June this year while RV'ing to Alaska. We would never bookdock there. Very dirty and a lot of people just living there in the parking lot. Did not feel safe at all and we were hit up for money 3 times walking to and from the store. Maybe Walmart could do more to keep people limited to staying just 1 night and not all summer.\nI have to completely agree! If the RV parks would keep their rates reasonable and their sites spacious, then there wouldn't be a need for RV'ers to stay in Walmarts lot. But as a full-time RVer as well, I can see the value of just a quick overnight when you are just passing through.\nI think the blame is on both sides here; 1) You have RV'ers not adhering to Walmart etiquette but pulling out BBQs and slides & 2) Like you mentioned, the campgrounds stink. And Walmart isn't going to become a campground or campground management so their only option is to enforce a no overnight policy.\nSad to see another one go.\nHey Ryan, we are totally with you on that. Sounds like you've traveled up here before so you know what we mean about the state of the campgrounds in the Yukon. Not worth the money. Thanks for reading.\nHi, I've seen some pretty inappropriate behavior in Walmart parking lots. People forget they are not camping; they are parking. I always go in and ask permission and buy something to pay for the privilege to park. It's a shame that a few ruin it for the majority.\nWhen I traveled I only used Walmart as a last resort. I remember one Friday evening panicking because I was so tired and every place I saw was full and I couldn't see well at night to drive. There are so many more RVs on the road than there were years ago. I was never so glad to find any place in my life. Walmart saved me.\nI think you're right about nearby parks and the pressure. I think that's what happened in Durango, CO., too. I have also stayed in some pretty \"questionable\" \"RV parks\".\nHi Nancy! I do agree, Walmarts and retail lots are not campgrounds. The situation at Whitehorse was probably the worst example we'd seen in our travels.\nThanks for reading. I'm enjoying poking around on your blog, you've got a lot of adventures under your belt! I would love to interview you some time for an article.\nHow nice of you to write. I'll be poking around your blog while you are poking around mine! LOL Things have changed a lot for me, but finding your blog is like finding a surprise.\nThat is a sweet thing to say Nancy, likewise!","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"The Madera County School Boards Association (MCSBA) task force developed a criteria for board members to be eligible for the scholarship program. An individual member of the MCSBA could submit a request for the purpose of participating in board member development programs such as offered through CSBA, SSDA, or other academy type programs. The purpose of the request would be to provide an opportunity to enhance the skills of the school board member in performance of his or her elected duties.\nBelow is a link to the scholarship form.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"My dear sister and her endowed with the most fantastic sense of humor husband are coming down from Bangalore to be with the kids for 3 days. This while Ravi and I take 6 hours off each consecutive day to read\/talk\/discuss\/delve into\/play with\/explore UNSCHOOLING in all its glory \ud83d\ude42 We plan to read Sandra, Ren, Pam, Kelly, and so many others who dish it out well cooked, seasoned perfectly.\nWe will give the newly married couple the evening off each day. And the morning\/afternoon will be ours\u2026 about 9-3. Living this life involves a tremendous effort in deschooling ourselves. In focusing on healing ourselves of past issues. Our inner voices shape our lives and if i am today able to smile at chaos (i used to be a stickler for list\/order\/plans etc) it is because i could let go of internal issues, could heal myself and allow chaos \ud83d\ude42 So while i may still want to make lists and have a plan in place.,.. its okay if the kids make changes, cancel things or if the lists change. I\"m able to go with the flow mostly. Kids are already perfect in a way\u2026 they are happy, learning, connected deeply with their inner voices\u2026 i just need to get out the way. So this conference is for me and Ravi to look within and learn to get out the way\u2026 and allow ourselves freedom to just \"be\". We need to baby ourselves. We need to listen to our needs\u2026 if i want to explore eating lots of snickers bars\u2026 then so be it. LOA is also helping me here. I no longer associate food with guilt, boredom, gain in weight etc.\nRavi often has issues with Raghu's hair falling over his eyes. Raghu has longish shoulder length hair. A while ago Ravi and Raghu played with styling hair\u2026 and Raghu found that he could use oil and water and get his hair to stay up. I coudl see that it was Ravi's issue. Raghu was fine the way he was. But at some point Raghu foudn his hair bugged him when he was trying to concentrate on something. And that day he decided to use gel, oil, water to style his hair off his eyes. So while it was nice to give Raghu a way to help himself\u2026. Ravi can now see that if its his own issue\u2026 he needs to be able to talk without coercing. And if Raghu feels the same\u2026 as he indeed did the day his hair bothered him\u2026.. then he will ask\/use our help. Ravi is happier when Raghu's hair is off his forehead\u2026. but he is only now beginning to see the implications of coercion and how it affects the relationship. And at the same time he sees the beauty in simple non-coercive conversation, offering solutions\/help and backing off.\nMeanwhile Raghu had a wish for home-made pizzas with the works\u2026 fresh mushroom, olives, mozzarella, home-made sauce, etc\u2026 for his little party of 7 kids! Zoya, my 3 nephews and nieces down for a longish stay during their school hols, Aarohi and her brother Rushabh and Devesh. Wow it was quite something today. For about 2 hours the house was in chaos\/laughter\/some tears\/jumping on the bed\/crazy conversations about babies coming out of stomachs.. correction from Raghu\u2026 uterus etc\u2026 It was a blast. but at the end of 2 hours i requested the 3 building guests to make their way home\u2026 moms were ready to pick them up. The remaining 3 cousins and my 2 are now chilling out with individual games\/interests.\nRaghu in the middle of his pizza looks up and says,\"Amma\u2026 3 glasses of iced tea with one ice cube each please\". and a smile that made me feel so happy to be able to comply and give him such pleasure. He loves sharing his home, food and loves planing elaborate meals.\nIt has been quite an experience for Raghu to have his cousins here. He seems to find it easier to switch to Hindi from time to time so his cousins understand him. They do speak English but are more fluent in Hindi and probably think in Hindi and Marathi. They live in Indore, way north of here.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The hallmark of leadership is to be remembered for your accomplishments during and after your work is done.\nBut it seems no matter how wealthy or well achieved some leaders are, they're unable to get as much recognition as they deserve. And even when they do, they're often quickly forgotten as soon as they hand over the baton of leadership.\nHowever, as a leader, you know you deserve better.\nPresence is inevitable for leaders who want to be recognized and unforgettable. Steve Jobs had it, was recognized and now he is simply unforgettable. For Steve, it wasn't just about being a top leader; it was about having a compelling presence. This meant that he had to be available to share his convictions and values to impact more people.\nThis is why with presence you can leverage your influence to make yourself heard, understood, followed and unforgettable.\nHere's how to build a strong presence and become an unforgettable leader.\nIntegrate your character into your brand.\nLeaders who went on to be unforgettable achieved that feat by not only leading their brands but by letting their thoughts, character, values and vision shine through.\nThis gave them a unique presence.\nFrom Coca-Cola's Asa Griggs Candler to Apple's Steve Jobs, it is their character integrated into their brand that has made them and their brands unforgettable.\nThey never held back from sharing their thoughts and visions so much so that they're remembered not only for what they did but for what they said in such a way that they still wield a compelling presence.\nHaving social presence gives you a reach bigger than your original sphere of influence. It goes beyond your influence in the office or city and puts you on a global map. But it is not enough to be on social media, you also want to be personable.\nToday's brands are becoming increasingly personable on social media so should their leaders too.\nPeople are looking to be influenced and inspired on social media and only top leaders can have as much impact. And this often requires leaders to be personable by sharing a real vision so people can follow what they do and love them for it.\nIt is also about leaders being firmly rooted in the now to change the future. That is leaders are able to connect with what's happening in the present, feel it and provide insight on how to move people forward from there.\nOne sure way to stay present online and offline is by syncing your personality offline with your online presence seamlessly through personal branding.\nPersonal branding involves having a well-tailored social media presence to share your convictions and values unhindered.\nPersonal branding requires you to show up at high profile events as a guest or a speaker as often as possible to share ideas and network with other leaders. Where there's non that matches your profile, you can also create an event of your own.\nYou can stay present through personal branding PR campaigns across different media with pointers to how people can follow your work.\nYou could also write a book to stay present and unforgettable by sharing your knowledge, insights and experiences.\nYour leadership achievements may be remarkable but there's more to becoming an unforgettable leader.\nIt all comes down to your presence.\nWhether it is reflecting your character through your brand, being personable on social media or staying present online and offline, consider making presence a lifestyle.\nGet started with presence here.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Founded over fifty years ago, Barton Music is an independent music publisher with a world class catalog containing \"timeless standards written by the likes of Sammy Cahn, James Van Heusen, Jule Styne, Ken Lane, Irving Taylor, Robert Wells and many more legendary writers.\n'Because Editions' was founded in Paris 2006 and it quickly became a distinctive platform where artists have the freedom to exchange, cross over and express themselves. 'Because' has welcomed a variety of artists, writers and styles from Justice, Drum 'N Bass, Ragga from La Phaze, and The Plasticines to the catalogue.\nIndependent music publishing company Boss Level is home to the likes of Paces, Jonathan Boulet and Chela.\nBoosey & Hawkes are one of the largest, specialist classical music publishing companies in the world. Their unrivalled catalogue includes works from some of the 20th centuries most distinguished composers, as well as many of the leading international classical composers of today.\nOver the last 60 years, Budde has become one of the most successful independent publishers in Europe with offices in London, Berlin and Paris. Distinguished by brilliant administration, superb licensing and combined with a young and active A&R team, Budde focuses on new talented writers and producers, as well as on new interesting labels that are looking for a serious and experienced partner for their publishing.\nAn NYC-based independent music publishing agency. Founded in 2012, Big House has become known for its \"Brill Building\" approach to writer and artist development. Services include transparent administration, crafted collaboration, cross-brand sync, and for creating bridges to overall international market exploitation and artist-side deals. With an indie pop focus and edgy sync-friendly catalogue, Big House also prides itself on unparalleled client-side services.\nIn the past 3 years Cloud 9 has been very successful with already over 100 single hits of which Cloud 9 is the publisher, varying from dance to rock and from kids to hip hop as well as being active as music supervisor for several successfully released movies in Holland.\nCRC is a new generation music company fundamentally built around artist management culture, invested in long-term artist career development. The team have been responsible and integral in the development and success of huge numbers of international artists careers, with No. 1 singles, No. 1 albums, Brit Awards, Mercury nominations, international festival headliners and Radio 1 A list records to name but a few.\nOne of the biggest independent publishers in Scandinavia. Cosmos Music (Scandinavian Songs) represent several Swedish writer\/producers, administers many Swedish catalogues and is the proud sub-publisher of around 50 foreign catalogues.\nDrive Music Publishing is an Los Angeles based independent company, founded by Ana Ruiz and Joe Rausch. Working with legacy catalogs to new songwriters, DRV takes a dynamic approach across all genres of music to provide the best opportunities worldwide.\nDomino Publishing Company is the music publishing arm of Domino Recording Company, the UK based independent which for over 20 years has been one of the must influential labels in contemporary music.\nElevate Music is Ireland's most progressive and creative publishing company with a growing catalogue that reflects the modern face of Irish songwriting ranging from Electronica to Rock to World.\nFounded in 1965 (at the instigation of Benjamin Britten) as sister company to the distinguished book publisher Faber & Faber, Faber Music is now one of the leading independent British publishers of contemporary classical and non-classical music. The company enjoys a reputation as one of the most influential publishers of serious contemporary classical music in the world. It also represents some of the UK's most successful film and TV composers as well as a small, elite roster of alternative artists and songwriters.\nWith more than 50 years of publishing experience, Fermata do Brasil specialise in renowned Brazilian writers, as well as a wide range of international copyrights.\nAround 15,000 music rights are at the centre of Filmkunst-Musikverlag's licensing and copyright administration business based in Munich. The FKM catalogue includes timeless film classics (\"People On Sunday\") as well as world famous TV series, such as \"Heidi\", \"Scarlett\", \"Captain Future\", \"Pippi Longstocking\", \"Vickie the Viking\" and \"Maya the Bee\".\nThe Imagem Music banner incorporates heritage catalogues such as Cahn, and CP masters and, proudly represents over 100,000 outstanding copyrights from some of the biggest names in pop and rock, works by classical masters of the 20th and 21st centuries, and timeless favourites of musical theatre.\nImperial is a New Zealand publisher signing masters and publishing on heritage NZ acts.\nLonely Planet TV's flagship series, Six Degrees, screens in over 100 countries. Playful, sassy, streetsmart and unexpected, Lonely Planet Six Degrees connects its viewers to what makes each city special \u2013 the people that live in it.\nLucky Number is the London-based music publishing arm of Lucky Number Records. Lucky Number currently represents FTSE, EyeDress, Jake Bullit and Oscar.\nManners McDade Music Publishing Limited is a UK music publisher established in 2006. Their catalogue includes the work of top film and TV composers as well as a select roster of European composers, many of whom come from a classical background.\nMedian M\u00fczik Edisyon is an Istanbul based music publishing company who administer publishing rights of Turkish, Kurdish, Arabic writers worldwide. Median has about 25,000 local titles and are adding new songs with new signing & creative projects like co-writing, scorings, adaptations, and sample marketing. The roster consists of back catalogs since 60s, covering pop, rock, folk and hip-hop songs, and huge TV series & film scorings.\nMe Gusta Music is a Nashville-based full service copyright administration company. Established in 2010, it provides publishing administration, record licensing and synch clearance to clients who span a variety of genres \u2013 from country and Americana to rap and soul.\nMeisel Music's extensive experience and successful national & international relationships with authors, producers and sub publishers help them to connect composers and songwriters. Meisel have had the honour to work together with artists like Modern Talking, David Bowie, Peter Maffey, U2 and many more.\nNext Decade is more than just a music publisher. With a hands-on and service oriented approach, the company share years of experience and knowledge about music and licensing. Founded by Stu Cantor in 1989, the roster of songwriters and catalogues is growing constantly.\nNatural History NZ is one of the largest factual program producers globally and has built a substantial catalogue of musical scores in a publishing joint venture with Native Tongue. These works are featured in programs screened on National Geographic, Discovery, Animal Planet and numerous other networks around the world.\nA London-based independent music publishing, production and artist development company, founded in 2006. Phrased Differently has managed over 1,500 song placements. These include over 60 No. 1's, 250 Top 10 records, 50 Platinum Records and 60 Gold records around the world.\nNative Tongue are proud to work with the Orbison Publishing companies. From developing new songwriters and artists to working classic catalogues of timeless songwriters and performers. Including contemporary songwriters like Liz Rose who co-wrote 'You Belong With Me' with Taylor Swift and Tommy Lee James who has written for artists as diverse as Delta Goodrem, Reba McEntire and the Pussycat Dolls to name but a few.\nNative Tongue and Outpost Publishing team together to help develop and work Bertie Blackman's catalogue.\nPulse is home to more than 70 of today's top songwriters, producers and artists, including hitmakers Bonnie McKee (Katy Perry, Britney Spears, Cher, Ke$ha), Christian \"Bloodshy\" Karlsson (Madonna, Britney Spears, Galantis, Miike Snow), Tim Pagnotta (Neon Trees, Walk The Moon), Oliver Goldstein (Jennifer Lopez, Kelly Clarkson, Christina Aguilera), Tony Maserati (Beyonce, P!nk, Lady Gaga, Robin Thicke), and Anne Preven (Madonna, Miley Cyrus).\nRaleigh Music Group provides outstanding creative and administrative service to a select number of music publishing clients including the catalogs of Elvis Presley and George Gershwin. The company, founded in 2015 by music industry veterans Peter Raleigh and Steve Storch, utilizes a high touch approach in order to maximize royalty revenue and protect the long term value and ownership rights of the songs they represent.\nReformation administers the songs of Gary Kemp, the writer behind the hits of Spandau Ballet. From the late 70s Spandau Ballet notched up a run of 20 hit singles and spent more than 500 weeks in the UK charts selling in excess of 20 million albums along the way. Few bands are truly timeless, but the continued popularity of their music is also in part a testament to the legacy of Gary Kemp's lyrics and melodies.\nCanadian based catalogue primarily containing production music and TV Cues. Highly concentrated on both film and children's TV music content.\nCasablanca Media Publishing\/ Red Brick Songs is a leading Canadian independent publisher since 2001 and is home to a growing roster of songwriters including The Rural Alberta Advantage, The Fast Romantics, Said The Whale, The Dirty Nil, Dan Davidson, Zanski, Folly & The Hunter, Jeremy Fisher, Aidan Knight, Young Rival, Cuff The Duke, Jeen O'Brien, Nygel Asselin, Swim Good Now, Teen Daze, Psychic Twin, Field Mouse, Sleep Millennium and many more.\nRound Hill Music is an independent, high-end music publisher with a mission to provide exceptional music. As a boutique size publisher they oversee a small catalogue but including many quality song writers and songs. Including Andreas Carlsson, Charlie Midnight & Randy Goodrum.\nSchedler Music is one of the most successful publishers in the German-speaking area. Schedler Music represents many successful international publishing companies and popular catalogues in GSA and Italy.\nSouthern Crossroads Music is an Americana, Alt. Country, Folk \/ Accoustic Blues and Country focused music company that owns and controls both master rights and publishing rights. That roster currently comprises producers, artists and songwriters based in the UK and the US. It has presence and\/or joint-ventures in London, Nashville, Los Angeles and New York and is represented in Canada, Australia, Germany, France, Scandinavia, Italy and Spain.\nSince the beginning of its publishing history in far 1962, Targa-Musik stands for world famous German pop and folk hits, such as those by Elke Sommer. With the creation of its two film music editions \"Musical Magic\" and \"Roxy-Sound\", today Targa also publishes cutting edge film scores from film productions made in Germany.\nCloud 9 Music has started the new publishing company 'TousEnsemble Publishing' (TEP) together with CNR Records Belgium and Regi Penxten (aka Regi). The company will represent amongst others Regi's copyrights worldwide. Regi is known as a producer, writer and one of Belgium's hottest exports under past aliases Sylver and Milk Inc. Wim Coryn (managing director TEP), formerly working for a.o. BMG Publishing, will be working alongside Regi.\nSpawning from a number of ventures in the Licensing world & with offices in both the US & Uk. Think Sync Music Publishing is a wide-ranging catalogue consisting of various contemporary genres, geared towards getting placements in TV, Film & Gaming.\nTrafalgar13 Music House is a progressive studio specialised in original music production and sound design. Since 2004 they have composed and produced innovative music for feature films, TV productions with an individual, artistic and personal language of its own.\nTuneCore is an online music distribution service enabling musicians and other rights-holders the opportunity to sell their music while also giving writers the chance to have their publishing interest represented worldwide.\nT.Y Songs (aka Thomas & Young) are a British songwriting production team based between Los Angeles, CA and London, UK. The creative duo, consisting of Owen Thomas and Lily-May Young, write and record for both recording artists and TV & Film on both sides of the Atlantic.\nUnited Future Creators Inc. is a Japanese music publisher founded in June 2012. Producing and creating music mostly for J-POP and K-POP artists and also doing management and rights business for songwriters, composers and DJs. We are releasing numerous hit songs from our worldwide music creators team \"Future Unison\".\nUltra International Music Publishing (UIMP) is the music publishing arm of Ultra Music that represents artists, songwriters, producers and other publishing rights holders from around the world. While Ultra Music is key player in Electronic Dance Music, the publishing company expands genres and has continued success in the Pop, Urban, Dance and Reggae arena. Consistently, Ultra has been placed among the Top 10 Quarterly Airplay chart, released by Billboard each quarter. Even surpassing publishing companies affiliated with major record labels, Ultra's presence in the market is prevalent.\nFounded in 1984, Westbury Music Ltd is an independent publisher of contemporary British music. Currently looking after some 70,000 works spread over several hundred catalogues, it is now one of the most respected independent publishing companies in the world.\nFounded in 1945 by the inimitable song-writing duo Rodgers & Hammerstein, the Williamson catalogue includes such famous and well-loved works as 'The Sound of Music', 'Oklahoma' & 'The King & I'. In addition to representing their classic copyrights, nurtures new talent with their current roster including writers John Bucchino, Joe DiPietro & Ann Hampton Callaway.\nWixen Music Publishing, Inc. was formed in 1978 as an alternative channel for publishing administration, copyright management, and royalty compliance review. Founder Randall Wixen has a degree in economics from UCLA and has taken several professional-level courses on law and business aspects of music publishing.\nBased in Nashville, Words & Music have been providing a superior catalogue focused on active Country & Western works for several years. With a catalogue boasting many prominent writers within the genre such as Dierks Bentley, Brett Beavers & Waylon Jennings, Words & Music have become one of the leading catalogues in this field.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The Women's Boston Flight Jacket is a fresh take on the classic bomber jacket. This wow-worthy wearable is wind and water-resistant, made of soft River Tec\u2122 nylon taffeta and fully lined with lightweight nylon. It is available in several colors and sizes, and its features include a modern silhouette, a full-zip styling with metal zipper, snap-closure pockets and rib-knit detailing on the hem, collar and cuffs. Customize with an imprint of your brand logo to maximize corporate exposure on a fashion-forward product.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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@@ -0,0 +1,5 @@
+{"text":"You can reduce water runoff from your yard by planting a rain garden. A rain garden is a small depression planted to flowers and ornamental grasses. It is designed to temporarily hold and soak in rain water from a roof, driveway or open area. A rain garden is not a pond or wetland. It is dry most of the time and holds water after a rain. Water collected in the rain garden slowly soaks into the soil and disappears in less than 48 hours.\nA rain garden is a beautiful and functional addition to your yard. Beautiful because of the plants you grow, functional because it helps control stormwater runoff from your property. Watch our rain garden animation by clicking the picture below and see how a rain garden works, both above and below ground. You can make it rain and watch the water flow. After you have viewed the animation, please come back and take a short (3 question) survey to give us your feedback.\nPractical Rain Garden video from Backyard Farmer on Extension TV.\nNebraska Bioretention and Rain Garden Plants Guide, Extension Circular EC1261. Information on a broad range of plants suited to bioretention and rain gardens in Nebraska and the Central Plains region. 64 pages. Cost: $15.00.\nSee the Publications page for more rain garden and stormwater resources.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Big Oak Flat, Ca \u2014 Earlier today, the Tuolumne Narcotics Team made its first marijuana eradication of the year as 12,800 plants were discovered off Marshes Flat Rd. in the South County.\nLt. Dan Bressler says the plants were located in three separate grows, and were spotted by authorities during an aerial over flight.\nNo suspects were located at the gardens during the time of the eradication, but several items were seized, including a 22 caliber rifle.\nDue to the evidence on scene, the Sheriff\u00b4s Office says it was determined to be the work of a Mexican Drug Trafficking Organization.\nThe marijuana eradication season for the Tuolumne Narcotics Team begins July 1st and continues through early Fall.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Drawstring hood, branded side inserts, long sleeves, relaxed fit.\nComposition and Care: Fabric 1: 80% cotton, 20% polyamide. Fabric 2: 100% cotton. Fabric 3: 85% viscose, 15% polyester. Dry clean.\nCelebrated for intricate cuts and a sporty aesthetic, Stella McCartney takes on cosy casualwear with the Logo Hoodie. A departure from the designer's polished vision, it offers an ultra-relaxed fit for a laid-back vibe, heightened by branded inserts that drape to the sides for added movement. Providing comfort in a cotton blend, inject a refined element by adding trousers and heeled sandals.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"First American company to deliver full portfolio powered by virtualized software and IP to the 5G edge architecture delivering on LTE and 5G for both NSA and SA networks.\nJMA Wireless, a global leader of wireless communications solutions, will be exhibiting at MWC Barcelona where they will showcase their complete ultra-wideband LTE and 5G solutions architecture built around XRAN, TEKO, and PHAZR technologies. JMA Wireless has the first complete architecture that spans the entire spectrum, providing a smooth transformation to 5G services. Its 5G technologies address the full range of spectrum, from 600 MHz up to 39 GHz, and is the only company that has delivered RAN as 100 percent virtualized software technology. The result is the only fully integrated architecture that addresses both non-standalone and standalone 5G systems for in-building, large venues, and outdoor 5G and LTE, while addressing the emerging demand to move RAN assets toward software virtualization and continuing to drive intelligence to the edge of the network.\nThe JMA Wireless PHAZR mmWave small cells are ideal for environments such as offices, airports, stadium suites, and other areas that can take advantage of 5G spectrum performance. Combining PHAZR small cells with JMA's XRAN provides a complete LTE\/5G NSA in-building\/venue solution. XRAN is a 100 percent virtualized LTE RAN that utilizes JMA's TEKO DAS or Cell Hub for RF distribution. The combination of XRAN, PHAZR and TEKO create the broadest spectrum support with the highest level of virtualized software technology resulting in a solution that delivers the lowest overall cost of ownership in the industry.\nJMA Wireless is exhibiting at MWC Barcelona in Hall 5, Stand 5D41 this week. Registered attendees can request to experience live demonstrations of JMA Wireless solutions.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Fixed mv error message when copy\/extracting from tar-compressed archives.\nFixed extraction of lzma\/bzip2 compressed files.\nFixed crashes with cut\/copy\/paste operations.\nFixed truncated name when opening bzip2 compressed files.\nRestored recurse option with zip archives.\nHave the progress bar when using the switches from cmd-line to move nicely.\nFixed crash when pressing the stop button after using the Multi-Extract dialog.\nFixed crash when opening arj archives with hardlinks.\nFixed missed error handling when using the switches from cmd-line.\nGUI when an entry is protected.\ndeleting\/adding\/renaming an entry in the archive.\nFixed missed archive reload when using cut popupmenu option.\nOverride DEL key when renaming an entry to avoid delete dialog to appear.\nFixed crash when renaming entries in the archive.\nMade the sidepane to select the corresponding row when using the arrow buttons.\nFixed sensitivenes of menu and buttons when operating on the archive.\nRemoved switch code for opening or reloading the archive.\nFixed issues when using \"Open with\" with only one file selected.\nFixed unneeded call to xa_watch_child() when using the cmd-line switches.\nRemoved the message dialog when converting rar archives to sfx executables.\nMade the \"Open With\" dialog more user-friendly.\nApplied patch from Enrico Troger for storing a 64 bit pointer into a string.\nFixed filename with dot in the main window when opening lzma compressed files.\nFixed missed extraction of lzma compressed files.\nFixed extraction of only first file with tar when extracting without the path.\nMade the stop button to stop extraction from Multi-Extract Dialog too.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzaiehs b/data_all_eng_slimpj/shuffled/split2/finalzzzaiehs
new file mode 100644
index 0000000000000000000000000000000000000000..dbd1f72e96aedd4b87ebb5cf89839e2b2c8957da
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzaiehs
@@ -0,0 +1,5 @@
+{"text":"One of the major sections of the photographic archive, this collection of pictures of road cars of normal construction, photographed on the special occasions of their respective periods (for example, the official introduction of models at the leading motor shows) or in photographic coverage especially commissioned by the publisher over the years.\nThe acquisition of \"Style Auto\", published from the start of the Sixties until the end of the Seventies, brought unification that was also relative to the photographic archive, made up of thousands of pictures that document the golden era of Italian design by stylists including Pininfarina, Bertone, Zagato and Giugiaro. This collection also includes numerous original sketches and designs.\nThe Novafoto-Sorlini collection, acquired from the publisher in 1989, is worth a chapter on its own. It is a really important photographic source (about 8,000 pictures) taken by Alberto Sorlini, the official photographer of the Mille Miglia from 1947 to 1957. In that decade, Sorlini built up a patrimony of pictures that show almost all the drivers, crews and cars that competed in the celebrated Brescian marathon. A collection of great historic-cultural importance that documents the most extraordinary road race of all time.\nThe most recent acquisition by Giorgio Nada Editore is a series of photographic archives devoted specifically to the world of motor racing, with particular reference to the period between the Sixties and today: this is a collection containing thousands of photographs of Formula 1, the world sports car and prototype championships, touring car races, Formula Junior and, in some cases, rallying.\nIn 2016, the publisher's photographic archives were further enhanced by yet another acquisition: over 300,000 photographs of the Franco Villani collection. For many years, the noted Bologna photographer worked for the major motor sport magazines \u2013 top of the list AutoSprint - from the second half of the '50s until the early 2000s. He took pictures at all the Formula 1 and sports car racing events, as he did world championship motorcycling, now MotoGP. Whether at the wheel of an F1 single seater or on a motorbike, all the top motor sport champions were photographed by him during the second half of the 20th century.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"To enquire about booking Alana please complete the basic form below and we will contact you to discuss your requirements.\nI really enjoy being in front of the camera for still modelling and video work. During the process I feel excited, confident and in the moment, and no matter what it always improves my mood. I volunteered and helped organise a catwalk training session coming up with themes and matching models by height and the more experienced with the less experienced, to support and challenge them. I enjoy new challenges and I'm open to difficult shoots to get the right shot, such as previously shooting in cold water.\nIf you are interested in booking Alana for a production please click the make enquiry button and complete the simple form. We will then contact you to discuss your project and it's requirements.\nAlana is a registered young performer on Spotlight. Click the Spotlight button to view Alana's profile page.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Don't miss out on your chance to save 25% off brands including Converse, Dr. Martens, TOMS and more! Simply enter code TREAT at the basket to redeem.\nAct fast, this offer ends at midnight tonight!\nPrevious: IKEA: Special Offers for IKEA FAMILY Members with Up to $30 OFF!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"SIMply Prairie is a two-week interdisciplinary project for grades 5-8 and includes science, language arts, math and social studies. This project is best conducted in the fall when the prairie plants are more easily identified. It might also be conducted in May. Access to a prairie habitat and Internet access are required.\nA description of how the authors taught the project.\nA tutorial for teachers new to problem-based learning.\nAuthors: Bill Fraccaro, Johnson School, Wheaton, IL; Larry Cwik and Pat Franzen, Madison Junior High School, Naperville, IL; and Bernie Jokiel, Gary D. Jewel Middle School, North Aurora, IL.\nCreated for the NTEP II Fermilab LInC program sponsored by Fermi National Accelerator Laboratory Education Office and Friends of Fermilab, and funded by United States Department of Energy, Illinois State Board of Education, North Central Regional Technology in Education Consortium which is operated by North Central Regional Educational Laboratory (NCREL), and the National Science Foundation.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Does plan A make sense?\nBud Dietrich, AIA 2. Establish a budget. While a budget should be in any goal statement, it's such an important piece that it's included here as a separate task. When making your budget, of course you'll begin with what you can afford, and how the cost of your house fits in with your overall plans for the future. When you're ready to get down to details, include everything that will go into the project: the cost of the land, local fees and taxes, design and engineering fees, construction of not just the home but the landscape, plus furniture and decorating.\nWhere do you want to be? How do you want to live? What are you looking for? Maybe you want that house in the mountains or with the ocean view, but it's not in the cards right now, for economic or other reasons. No matter; you'll likely be able to reinvent yourself later. For now, it's the burbs with the good schools or some other place. The point is, find a spot on the globe that you can claim as your own and build what will be a home. And maybe that land isn't a few acres that's never been trampled on. Maybe it's an existing house that's just old and tired and has suffered some neglect. The house whispers to you that it really does want to shed those avocado-colored appliances, that shag carpeting and those single-pane windows, and you know you're the person to do that. So take heart if you decide to transform that sow's ear into a silk purse. You'll be amazed at the transformation that can take place.\nWhile you might think you can go it alone, assembling a team of tried and true professionals is the better approach. After all, you wouldn't represent yourself in court. So why wouldn't you entrust your single largest investment to an experienced team that won't be learning on your dime? An architect and a builder (if not one and the same) are going to be your most important team members. These people will act as guide, therapist, advocate and counselor throughout the journey that creating your home is. And, as with all good professionals, the right guide can ensure that the journey is all the more enjoyable.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzajduc b/data_all_eng_slimpj/shuffled/split2/finalzzzajduc
new file mode 100644
index 0000000000000000000000000000000000000000..97bdbbe7cb213a0a4723a555912b33d2f5c24b0b
--- /dev/null
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+{"text":"There isn't any doubt in most individuals's thoughts that technology has, general, increased our quality of life. Founded in 1846, the company has turn into one of the mainstays of American information reporting, and it has been the recipient of many awards all through the years. Whereas apps like Fb Paper will stream both private-social and common news content, most of the different apps like Pulse, AP Cell, Digg and Simply, will syndicate basic purpose news feeds. If exhausting core information is for you, then AP from Associated Press is the app of alternative. Digital expertise does have many benefits and has our trendy society constructed around it, however we can not ignore the unfavourable sides. Improved expertise and new skills would possibly encourage the papers to strive AR again.\nThere aren't any a lot agricultural challenges within the United States of America because of science and know-how. Expertise is the appliance of scientific information for practical functions (Oxford Advanced Learners Dictionary). Associated Press (AP) is a not-for-revenue news agency which is owned by a cooperative of media outlets. As the title counsel, Merely app is easy and a fast to make use of app that aggregates all news you'd in all probability ever want. This generation, greater than every other, has been utterly enveloped in technology.\nKids immediately are prepared to use the unimaginable power of technology as it's and how it will be. They aren't constrained by limitations that have been extreme but which have now been largely eliminated. The Tokyo Shimbun is a Japanese newspaper that has used AR to vary articles geared toward adults into simply understandable articles for kids. Although advantageous, the advancement of digital expertise also has many drawbacks, amongst which is the lack of safety. The Millennium Era can't be understood with out additionally understanding of the know-how of the 21st century. In truth, the Millennium Era cannot start to know a time when technology was not a vital part of almost every thing that goes on in the world.\nOver-reliance upon technology might be seen in virtually every aspect in life, and may eventually leads to dependancy. Their knowledge of pc know-how might be like that of automotive racers, fighter jet pilots or astronauts. The importance of science and technology when critically and analytically defined can fill more than two thousand pages of a textbook. The significance of self-employment is just not detachable from the good expertise has done to man.\nThe most important pitfall of digital technology is the constant exposure of youths in the direction of unhealthy and worldly values. Therefore, this piece covers the significance of science and know-how on information, transportation, banking, agriculture, education, security, marketing and on employment generation. It gives simple and quick access to the newest news stories, sports activities scores, and climate. The consolation introduced by digital know-how has additionally developed a robust sense of dependency.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The VLA9 project idea was formed from careful analysis of the data in the ISG report on Randomised Badger Culling Trials (RBCT). The name of the VLA9 comes from the spoliotype mapping carried out on the type of TB present in cattle populations across the UK. An area of North, North West Devon and North Cornwall was shown to have only a high level of Spoligotype VLA9 8555321.\nTo prove the level of support for a targeted badger de-population exercise to reduce levels of TB in both cattle and badgers over a sustained period (five years) or until vaccines are made available.\nTo prove to Defra and Government that farmers or their appointed agents would provide a unique local answer to a problem that has been so far difficult to fund and execute.\nTo learn the lessons from the ISG work - ensuring that pertubation is mitigated by using hard and soft boundaries (coast, rivers and main roads), managing the edge effect by effective process, managing from the 'outside in' on a concerted and consistent basis. Most importantly that any cull will have to be proactive as a reactive cull has been proven to be ineffective from the trials.\nTo provide a solution that is deliverable, cost effective, safe, and humane and delivers disease reduction in badgers and cattle.\nTo empower farmers under strict guidelines and supervision to be able to tackle the disease that threatens their businesses and livelihoods in the absence, currently, of an effective eradication policy for boine T.B.\nTo allow badger culling licences to be granted to control a notifiable disease that all present methods have failed to deliver (lifting the moratorium that continues to be currently in place for the purposes of the duration of the Krebs trials - which have not concluded).\nOver 1,200 completed and signed licence application forms are held by the VLA9 project team which represents 70% of the farmers in the area and 80% of the land mass discounting Forestry Commission and National Trust land.\nParishes with the higher levels of TB infection in cattle showed the highest level of supportfor the project.\nMore farmers would join the project when issues of confidentiallity, security and method are resolved.\nVLA9 project demonstrates that farmers would contribut fully to a TB eradication plan and take ownership of the disease annually and could provide an effective method of controlling TB as recommended by Sir David King.\nThe VLA9 project gained universal approval from veterinary surgeons practicing in the area.\nThe project could be rolled out on a wider scale to all affected TB areas.\nMethod and security are critical to success; any population control of wildlife must take out entire social groups to avoid perturbation. Farmers or their appointed agents must be sure their confidentiallity is preserved.\nThe project provides a logical structure, local knowledge and a committed partnership with Defra to give a long term solution to the spread of TB.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Board Exams for Class 11 in Tamil Nadu \u2013 A Good Idea?\nThe very premise that students will study and teachers will teach only if there is a board exam is RIDICULOUS.\nForget the question of whether we need the newly introduced XIth Std board exams inTN. Do we actually need such teachers and students?\nHere are some good points for and against this decision.\nIf you ask me, we don't even need an education system which drives students towards mugging up\/memorizing the course content, vomit it verbatim, and clear their exam papers in \"flying colours\".\nUntil we work on an understanding-based\/practical-oriented curriculum right from Class I and stop the scramble to obtain \"lucrative\" engineering\/medical college seats that promise riches irrespective of the students' aptitudes and interests, it won't matter even if there are board exams for all twelve years starting from Class I.\nUntil then, we will keep producing useless engineers who don't have a clue about engineering (me included) and doctors whose only focus will be to maximize their Return on Investment.\nWe reap what we sow. Karma.\nCongratulations on achieving a significant milestone. A thousand posts is no mean job and that too on varied subjects retaining the interest of readers always sustained.\nBoard or no board, examinations or no examinations, what is essential is an understanding of the subject and the concepts. This is lackingand needs to be tightened up.\nCongratulations on completing 1000 blog posts. Looking forward to reading the next 1000 posts. I have no idea about the subject. So I will learn something new from your post and comments.\n1000th post wooooow unbelievable achievement according to me. Congratulations.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Guest blog by Sarah Hollister, RN, PHN, IBCLC: As a nurse and an International Board Certified Lactation Consultant (IBCLC), I have the opportunity to work with nearly every pregnant woman and new mom and baby at a group of four primary care health centers in Northern California. I would like to share my experience, concerns and request for collaboration to closely examine the new practice of placenta encapsulation, as it has grown to become a component of the postpartum experience for the new moms who I work with and throughout the United States. I have encountered assumptions that placenta consumption increases milk production, is a prevention for postpartum depression, and has existed in history as an ancient human practice. I will provide a summary here of the work I do and what I have found with my clients involving this practice.\nDisclaimer: The following blog is merely a collection of notes and not a serious scientific research paper. There is obviously a pressing need for more research. My intention with this blog post is not to make any conclusive statements about the practice of placenta encapsulation or placentophagy, which I am not qualified to do anyway, but merely to offer the classical Chinese perspective as an urgently-needed correction to some misinformation promoted in popular and Chinese medicine circles.\nPerhaps because I am teaching a gynecology class right now while dealing with the very final last-minute revisions and the release drama of my new book, the \"Divine Farmer's Classic of Materia Medica,\" it has struck me lately how similar the production of a book is to the conception, pregnancy, labor, and birth of a real child, and then the postpartum recovery.\nWhile I was really hoping to announce the release of my forthcoming (and over-due) translation of the Divine Farmer's Classic of Materia Medica, we are still fine-tuning it. It is a precious text that needs to be treated with reverence and presented correctly. So please be patient and accept my apologies for the delay. Any day now though....literally!\nIn the meantime, I have a sweet little story to share with you: A friend recommended that I check out the work of Jane English, translator, publisher, shaman, nature photographer, and apparently just somebody whose work I would really relate to.\nIn conclusion, the simple answer to the seemingly simple question on how to map modern gestational age to the ancient Chinese way of counting pregnancy in ten months is therefore anything but simple and frustratingly vague: While modern biomedicine begins counting the gestational age of the embryo or fetus from the ONSET of the last menstrual period, classical Chinese texts begin counting roughly a week later, ideally one or two days after the CONCLUSION of the last menstrual period. In other words, the first month of pregnancy in traditional China would start from what we count as the second week of gestational age, or a week before biomedicine would consider ovulation to have taken place. How exactly you translate this information into your clinical practice shall be left to your discretion.\nA treatment for women suffering from vacuity cold in the Sea of Blood, which is exploited externally by wind and cold. These struggle and bind [with the blood] and fail to disperse, instead collecting and gathering to form lumps, or possibly to form hardenings and conglomerations. [It is also indicated] for blood and q\u00ec pouring out aggressively, with soreness and pain in the abdomen and rib-sides, tightness and distention in the lesser abdomen, possibly intermittent vacuity rumbling, retching and vomiting of phlegm and foam, spinning head and flowery vision, heaviness and pain in the thighs and knees, a withered yellow facial complexion, swollen and heavy limbs, when the menstrual period is about to start flowing a feeling as if she were critically ill, possibly excessive, possibly scant [menstrual flow], red or white vaginal discharge, unstoppable [vaginal] flooding and leaking, panic palpitations with forgetfulness, frequent urination, possibly descent of white fluid, intermittent eruption of vacuity heat, night sweats, and marked emaciation.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"I'm a feature writer for an award-winning agency called Talk to the Press and I have been researcing TMAU.\nI would love to get in contact with a female sufferer to talk about their experiences.\nDo you have any case studies or know of anyone who would be willing to talk to me?\nIt would just be informal at first and off-the-record, but if they decided they were happy to go ahead with a story I would send it out to the national newspapers and women's magazines.\nThey would receive a fee for their time and it could be a great way to raise awareness in the national media.\nWould be interested in helping to raise awareness if admin could forward me a telephone number or she could pm me.\nI'm sure she would sen her phone number for a chat.\nI responded to this also - although I've been interviewed before I thought I'd offer in case no one was comfortable to come forward, as I think it's important our story gets out there.\nSo it's on p.27 of the Mirror today. On balance the article is ok - to some degree it's what you would expect, but it's fairly balanced. However, and I made this very clear - I do not, like many of us, smell of fish. And it's in the headline even though I said no. However, that's a compromise I suppose I'm ok with - it's out there now. And perhaps the people in my local newsagents will be a little more polite to me when I buy the paper. And perhaps not!\nHope you are all well - I've not posted for ages - work is so busy - hope you are all keeping your spirits up - don't let this define who you are. You are all amazing people. Fight for your happiness because you deserve it. We all do.\nEllie has not only done the above article for The Mirror but she is also on BBC Radio 5 live tomorrow (Monday 7th April) at 8.40. Tune in if you can!\nGreat job, Ellie. Brilliant work on our behalf!!\nwell done and massive thanks Ellie....it was also in the daily mail paper today, cant find link online at moment..I also have more media interest in the back of this (ann diamond radio show?) - will keep you all updated.\nWell done Ellie, you are an inspiration! Good to see you looking so healthy and well in the Mail pic too.\nCongratulations, Ellie, for a job well done. We owe you much for your hard work and courage.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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@@ -0,0 +1,5 @@
+{"text":"Who Will Come Out On Top Of VH1's 100 Greatest Artists Of All Time?\nStarting Monday, September 6 with back-to-back episodes at 10\/9c and 11\/10c, VH1 will count down its 100 Greatest Artists of All Time (the five-part countdown will continue every day at 10\/9c through Thursday, Sept. 10).\nVH1 has revealed the Top 5 (although their specific ranking will remain a secret until the final show airs).\nHow can Guns N' Roses be higher than Fleetwood Mac! This list is so wrong! Who's with me?\nBob Dylan has the worst voice in recorded human history.\nFM is long gone in the minds of most, a relic of the late 70s and the 80s, I think that is why the low ranking. They basically died after Tango in 87 and only resurfaced briefly with The Dance in 97. If they'd been more productive over the decades, then maybe it would be different. Ask any younger person these days about FM and they won't know who it is!\nAerosmith and Tina Turner and Arethra Franklin are ahead of Fleetwood Mac. Really? I guess Fleetwood Mac needs new songwriters since that is how the other 3 have survived. Ridiculous.\nI think you guys are taking this all a little too serious :) It's all subject to opinion at the time the poll was taken... It's all for fun.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Coating Tech Slot Dies has an immediate opening for a Machinist-CNC grinder operator. This position's primary function is operating CNC surface grinder(s) with attention to detail and precise flatness and finish tolerances. In addition to operating surface grinders, the position responsibilities include polishing, deburring, assembly and general shop cleaning and maintenance. Experience in operating manual and\/or CNC mills and lathes is beneficial. To be successful in this position the team member must have the ability to work independently as well as in team-based environments to meet deadlines within all specifications.\nCoating Tech Slot Dies is a privately-owned precision die manufacturing company located in Eau Claire, WI. Our products serve customers throughout the world in the adhesives, aerospace, construction, energy, filtration, graphics, medical and optical film markets. As a leader in slot die technology we value innovation, ethics and relationships.\nDue to our continued growth, we are seeking skilled team members who value working in a team-based environment and have a passion for obtaining excellence in serving our customers, process, precision manufacturing, and product performance.\nWe offer a highly competitive wage and benefit package. As a small, privately owned company, we understand that our employees are our most valuable resource and we reflect that in our 'employee first' philosophy.\nRequired to stand a majority of the time, ability to lift a minimum of 50 pounds is required.\nPlease visit www.slotdies.com\/about\/careers for the complete position description.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"wedding band canada. But for men who might consider themselves experts at buying jewellery, and know exactly what they have to like , nor like buying a music band can nevertheless be tricky. Wedding ceremony ring is definitely an item they're going to wear almost every day. Because of this their ring is required to be practical and functional additionally it needs to fit the man's individual style, and be something which he's happy and comfortable to consider throughout the day, everyday.\nBasically, you'll find definite positives and negatives to each metal, it's due to exactly what lifestyle the Groom leads, and his individual preferences. No person metal is ideal for everyone. Once you have decided which metal you need, you possibly can filtering the search results by material, and rapidly eliminate all rings that aren't constructed from that metal.\nWhen you have a basic idea of what kind of ring you'd like, then it's time for that enjoyable aspect can be trying them out for. Allow her at the least two to three months prior to the wedding date to be able to engagement ring shop. You will require now to be able to surf, research prices and also take another look at rings which catch your current eye. When you've got your heart collection over a custom band, you'll likely want all the more time. Though you shouldn't be afraid of becoming trendy, ensure that the type you ultimately choose is something you'll want to use pertaining to, say, another 40 years.\nwedding band canada. Selecting the most wedding band canada are a wide and important decision for each and every couple, but it is often a particularly daunting job for men. For many men, their wedding band might be the sole item of jewellery they've ever really bought for their own end, as well as it sometimes the sole part of jewellery that they will ever use.\nIf selecting a Wedding Band invest in a style you truly really like. Don't choose a style determined by trends. This is certainly an item of jewelry you may wear for years to come, so it's always best to look at the long term. Remember that having a matching wedding ring with your spouse seriously isn't necessary. Your partner may prefer an alternative shade metal or even a contrasting style. There's no formal guideline book in picking wedding band canada.\nRituals are different, wedding vows are often special, nevertheless the traditions in the wedding band has made it through the years and is a normal symbol of long lasting love in several nationalities and belief systems. At matvuk.com we've wedding rings in various designs, sizes, weight loads, shapes and sizes, each of them spectacular plenty of that you should adore for an existence time.\nSearching out the ring that matches doing this criteria is often tricky, so we've made a guide to finding the ideal wedding band canada, with many clues, ideas and inspiration from our personal beautiful selection of wedding bands. Hopefully, this will help make the job a little bit easier.\nChoose a style that can your everyday activities in thought. A wonderful ring is significant, but so is actually a durable ring that is worn daily. With many metals available, it's recommended that you know which metals won't only suit your tastes, but additionally your lifestyle. Precious metals have 100's of years of tradition behind them, and may be resized, but modern metals like titanium and tungsten are hardier in relation to everyday life. No person metal is preferable to all the others, as every one has strengths and weaknesses. Solve these questions pick which metal is ideal for you.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Vr wedding organizer added 21 new photos \u2014 with Nuri Nur Hasanah.\nSo who are VR Wedding Design?\nYou mean the World to Me - For more, please visit www.thethemewedding.\nVR Wedding Planner is pioneer in Celebrity Management. We offer celebrity Co-ordination and Management Services in India and Overseas.\nSan Diego All American Red Wine and Blue edding in Temecula-5-4.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Two scholarships are available for postgraduate research in Scots Private Law at the University of Edinburgh for the 2019-2020 academic year.\nTuition Fees at the UK\/EU rate and a small maintenance award.\nOne award is available for the LLM by Research in Law and one award is available for the PhD in Law.\nApplicants must have applied for admission for the LLM by Research in Law or PhD in Law.\nThe Edinburgh Legal Education Trust Scholarship application can be found on the School of Law website.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzajvrv b/data_all_eng_slimpj/shuffled/split2/finalzzzajvrv
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+{"text":"Alaska Premier Gutters provides a wide variety of gutter repair services and gutter installation, including every component of your system.\nAfter reviewing your site we will come out and professionally install your gutters to your complete satisfaction.\nAlaska Premier Gutters will come out to your home on a one-time basis or, why not let us manage your gutter system?\nDo I Need Gutters on the Upper Story?\nAlthough there is no patent answer, there are signs to look for that might suggest guttering your upper roof.\nIs there splash-back on your siding? Splash-back happens when rain water draining off the upper roof hits the roof or ground below and water splashes back onto your house. The immediate effect is that you will get dirt stains on your siding from the ground or the roof below. The long-term effect is that your paint will start to deteriorate in these areas and then your siding will start to deteriorate. Many times upper story windows will be near the lower roof and they will get hit with the splash back. These windows will definitely get dirty (and they aren't usually easy to get to and clean). Also, more importantly, if you have older wood-framed windows the water can and will ruin your window frame. This is a very expensive problem as you are usually looking at replacing some siding as well as the window. Roofs with steeper slopes tend to get less splash-back than lower sloped roofs. So, if you do not see dirt stains on the bottom of your siding and the bottom of your siding is not wet after a rain, then you can wait and see. Upper sections can always be done later if the evidence starts to appear.\nAlaska Premier Gutters will be happy to come out and assess your needs-just give us a call.\nHow Many Downspouts Do I Need?\nDo I Need Drip Edge?\nDo I Need to Replace My Gutters?","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Three and more decades ago, when the Libertarian Alliance kept \"The Alternative Bookshop\" in Covent Garden, we used to print badges that said useful things to people: rather as if we were Marxists-Turned-Upside-Down \u2013 in the words of one of my very perceptive and incisive University chums.\nI call this badge to mind [I have kept in the Main Lower Library's Archive Of Objects an example of all the best ones we made] in view of the events of Thursday et-seq. If the Liberal Capitalist West was properly at \"Defence-Stations\" \u2013 and it is not \u2013 then it is quite inconceivable that the Russian dictatorship-Junta would even dare to contemplate thinking even privately of destabilizing Ukraine to chew off bits of it \u2013 let alone (worse) inveigling traitorous Marxist-sympathisers within Ukraine to do so as its catspaws.\nIncidentally we also wouldn't have more than a light regional but nugatory difficulty with \"Islam\": which it is believed is a sort of mysogynistic pre-capitalist desert-survival-guide, but which most of its tacit adherents resignedly accept the Fatwa that it is a \"Religion\". For the individual human costs of trying to \"leave\" it, as prescribed in its Book, don't bear thinking about.\nA major and exact historical parallel, in the same continent, is in front of our noses. In 1938 as you all know, this is when the Third Reich privately egged on the Sudeten-\"Germans\" under the fascistleftoid Nazi Conrad Henlein, in their efforts to dismember Czechosolvakia. In that instance a major reason was the intended confiscation of Europe's third largest military organisation, plus the hijacking of Czech and Slovak heavy industry like the Skoda armaments-complex. The Czechoslovak Army alone fielded 43 divisions in that year, not counting its armour-capability.\nEastern Ukraine, as you all know, contains the major part of that country's industrial and coal mining areas.\nI leave you all to draw your own conclusions.\nIn the meantime, as War Secretary, I'll ensure that all Anglosphere Nations that wish to \"travel with us on the Rad Map For Peace\" \u2013 the proper one, not the US Democratic Party one \u2013 at the very least, are armed to the teeth, without any sort of restriction.\nI'll also be tearing up the Ottawa Treaty and denouncing it on behalf of the UK, for which I will have defence responsibility. It will be my decision, taken in the UK's best interests. It's no other nation's damn business whether we choose to deploy \"airfield denial weapons\" or not, for example.\nThanks for this David. I had been quite offended by Sean's weasel words on the subject.\nChris: Sean and I as officers on the LA's Bridge, disagree sometimes \u2013 even about strategic libertarian policy, if indeed there is such a thing \u2013 but we are always agreed that it shall be \"never in front of the Sailors\". Then we allow each other to say what we say, provided that it doesn't get the LA or us into trouble with the Gestapo.\nFor example, I don't entirely share Sean's insistence that Tony Blair be strung up from a lamp-post after a War Crimes Trial, although I would go for the trial which would be plenty bad enough for is foul wife and spoilt children as it would stand.\nWhat weasel words? I always make myself clear.\nI don't think Sean actually uttered in his post on the subject. any \"weasel words\", whatever those might be. Apologies, I missed the context of that when I was replying as the phone rang and it was the beighbours about something. I was indicating that while I might not sometimes agree totally with what he says \u2013 he can say what he likes. It's our bolg after all!\nEr, well, Chris, I've just read Sean's piece again, and I can't seem to find where he was being mealy-mouthed about anything, if that's what you meant.\nWell Mr Blair and Dr Gabb do not always disagree \u2013 for example they both utterly misunderstand what sort of person Mr Putin is, and think he can be worked with (he can not \u2013 and not because he is a Marxist, which he is not).\nIt is a good post David \u2013 it reminds us all that the M16 has usually been on the right side in conflicts (as has the FN FAL) and the AK47 has usually been on the wrong side.\nA rather basic truth that this site \u2013 with its visits by article writers of the down-with-big-business down-with-the-rich school of thought tends to forget.\nOn the specific point about the Ukraine\u2026..\nThe pro E.U. social democrat types in Kiev are hardly my cup of tea (to put the matter mildly \u2013 very mildly), but they are better than the mixture of vicious drunken criminals and deluded Marxists that Mr Putin has been arming (and sending in fighters to back) in eastern Ukraine.\nAs for the Americans arming the Kiev government \u2013 apart from the practical problems (American weapons need training to use \u2013 for example giving a AK47 to a random thug will work, give him an M16 and he will jam it) there is also the Obama problem.\nWith hide sight too little work was done on the financial corruption of Comrade Barack and Comrade Michelle in the Chicago days.\nSome dissenting Marxists (the Marxist movement in Chicago is split into factions) insisted that both Comrade Barack and Comrade Michelle were more interested in corruptly looting money for themselves than in \"The Cause\" \u2013 however most (although not all) people looking into the Obamas from the other side (including me) were obsessed with their political background and looked at their potential criminality as an after thought (it at all).\nFor this neglect we are guilty.\nA lot of hard evidence of criminality against Comrade Barack and Comrade Michelle (the sort of stuff that could send them to prison) would be useful.\nNot just in relation to the Ukraine \u2013 but in all sorts of matters, foreign and domestic.\n\"But that is blackmail\" \u2013 blackmail is a traditional tactic in such matters. Even morally it can be the lesser-evil.\nAlthough I am still wary of getting involved in the Ukraine \u2013 not just the Crimea (which is lost \u2013 although no Ukrainian politician is going to admit it) but in the Ukraine in general.\nIt does seem to be a place where the other side has the advantage.\nUnless the people that Mr Putin has employed drink too much \u2013 odd as he is a sober man (although a vicious criminal) himself.\nThe unstable coalition of various social democrat type parties has fallen again in the Ukraine.\nPeople with long (ish) memories will remember that this has happened before.\nAt least the puppets of Moscow are united \u2013 the alternative is (and always has been) like a \"herd\" of cats (try herding cats).","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Euro Motoren was the first full line automotive core supplier in Europe in the eighties and nineties.\nFounded in 1979 and under the ownership of founder Wil Linders since.\nOperating from two warehouses, totaling 5,000 sq. mtr.\nCustomer base included Valeo, GKN,GM-Vauxhall, Renault, TRW, Delco and many more reputable companies.\nMember of APRA since 1989 and Board Member from 1990 to 2005.\nIn 2000 Euro Motoren became European Master Distributor for Sunair AC products, USA.\nMember of the MACPartners Europe since 2005 and Board Member from 2005 to 2012.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Aim for at least 5 or more cups of fruits & veggies\/day: Fruits and vegetables are the number one source of anti-inflammatory properties, vitamins, minerals, nutrients, antioxidants and fiber, which has shown in research to be helpful in the prevention of certain disease states.\nA colorful diet ensures that you are getting a balance and variety of vitamins and minerals that your body needs.\nRed Fruits and Vegetables (such as beets, radishes, raspberries, tomatoes ) contain nutrients such as lycopene, quercetin, ellagic acid, which are nutrients that have been shown to reduce tumor growth, the risk of prostate cancer, LDL or the \"bad\" cholesterol levels, and lower blood pressure.\nOrange and Yellow Fruits and Vegetables (such as apricots, carrots, sweet potatoes) contain nutrients such as beta-carotene, flavonoids, zeaxanthin, vitamin C, and potassium, which may help to reduce the risk of prostate cancer, age-related macular degeneration, lower blood pressure and LDL cholesterol, promote healthy joints and work with calcium and magnesium to build healthy bones.\nGreen Fruits and Vegetables: (such as asparagus, spinach, green grapes) contain nutrients such as chlorophyll, calcium, folate, fiber, lutein, vitamin C, zeaxanthin, and beta-carotene, which may help to lower blood pressure and LDL cholesterol levels, reduce the risk for certain types of cancer, boost the immune system, support vision health and digestion.\nBlue and Purple Fruits and Vegetables: (such as blueberries, prunes, eggplant) contain nutrients such as vitamin C, lutein, quercetin, zeaxanthin, fiber, resveratrol, flavonoids, and ellagic acid, which may help to lower LDL cholesterol levels, boost immune function, support healthy digestion and vision health, reduce tumor growth, fight inflammation, improve calcium and other mineral absorption, and limit the activity of cancer cells.\nWhite Fruits and Vegetables: (such as garlic, onions, brown pears, white nectarines) contain nutrients such EGCG, beta-glucans, lignans, and SDG, which may help to enhance immune functioning, reduce the risk of breast, prostate and colon cancer, hormone-related cancers and balance hormone levels.\nFresh: all fresh produce is a great choice, however when selecting produce make sure you are choosing fruits and vegetables that are in season, ripe, and the best in flavor.\nFrozen: requires very little preparation, make sure to choose fruits and vegetables in their own \"natural\" state (meaning no added butter or sauces or sugars).\nCanned: canned fruits and vegetables are a great thing to have on hand for a quick meal or snack and for meal preparation. Choose low-sodium canned vegetables and canned fruit that are packed in water or 100% juice.\nDried: dried fruit is great to keep on hand for adding to cereals, salads, trail mix, yogurt, etc. Keep in mind that \u00bd cup is considered a serving size and watch for added sugars like honey, sugar, corn syrup. Only the fruit itself should be on the ingredients list.\n100% Fruit or Vegetable Juice: Stick to an 8oz serving size or less of 100% fruit juice. While it is better to eat whole fruits and vegetables because of the fiber, 100% fruit and or vegetable juice can also be a healthy option. Just remember to limit yourself to an 8oz portion.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"MY COMMENTS: I enjoyed the book very much, and it was great fun to see some of our favorite fairy tale characters come to life.\nAnd if you discover that you love the Storybook Valley books, be sure to join Stacy's street team, the Storybook Valley Sweethearts on Facebook for book launch activities and exclusive sneak peeks. Members will get to hear the latest Storybook Valley news before anyone else, and even read excerpts of works-in-progress and give input on cover design.\nThanks so much for sharing Sleeping Beauty on your blog, Maryann!\nYou're welcome. I am still working on finishing the book \u2013 too much business interfering with reading time \u2013 but it is good so far. And I do love the idea of visiting these childhood fairy-tale friends in a new light. Quite fun.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"Digital IC Design Engineer - APN Software Services Inc.\nStrong digital design basics, RTL design, exposure to the designs like ADC\/DAC.\nGood understanding of Synthesis, DFT and Timing.\nResponsible for the entire digital design flow of complex integrated mixed signal ICs.\nAble to go from spec to GDSII\/LAFF handoff for import into Analog top level.\nDefine architecture and execute design of critical digital system functions such as supply sequencing, peripheral control, serial interface, local memory and external uP interface\/control.\nDesign of complex digital state machines including analog\/digital domain synchronization.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Disability insurance offers your employees income replacement (typically 60% of salary) in the event they are unable to work due to illness or injury. Disability Insurance plans can include bonuses.\nBoth Short-Term and Long-Term disability payment options are available. Short Term Disability generally has a shorter waiting period and, depending on the policy, is payable for up to 2 years (often less.) Long-Term Disability has a longer waiting period and is payable for longer periods of time \u2013 several years, or even a lifetime \u2013 again, depending on the policy.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"What part of your portfolio is in stocks or funds of foreign companies?\nRe: Foreign investing - how much?\nAre you asking out of curiousity, or to gain some insight? What are you *really* asking?\nYou might be better off asking... what %age truly *national*, multinational, and *international* - where national and international would be the smaller cap companies that only operate in their local markets.\nThe majority of my investments are in mutual funds in Vanguard. So what I'm really asking is, what percentage of that should be in foreign investments?\nThink simple. You all are quite knowledgeable but my query is rather basic.\nI can tell you what mine is, but I don't claim it's what yours should be: About 25% of my \"account\" investments (that is, excluding rental property, for me) are in ex-US equity index funds and a couple of percent in ex-US corporate bonds that happen to be picked up in one of my bond index funds (vs. about 55% in US equity index funds).\nMy stocks are held such that 70% are in Vanguard's total stock (U.S.) market index (VTSAX) and the remaining 30% are in Vanguard's total international stock market index (VTIAX).\nI think Jack Bogle has suggested as little as 20% international is fine, while market cap for international would mean a 45:55 ratio in favor of international. Most large companies - foreign and domestic - do business in both the United States and outside. The above distinction is really just where they're based.\nAnd a reminder: if you treat all your retirement assets as one large portfolio, put the international stocks in taxable so they're eligible for the foreign tax credit.\nand noticed that a Canadian recommendation for a lazy portfolio holds a ratio of 33%Canadian stocks to 66% US and international.\nYup. I invest 100% in NZ real estate. You are all foreign.\nI should really start buying equities or indexes or any of that stuff. I just know absolutely nothing about it, having spent the last 5 years hustling to get good at property. I only like paying people for results... are their any funds who take nothing if they don't make money but a gradually higher percentage of profit if they do well?\nWhat do you mean, 'if they make money'? The markets don't always go up, and if you're not overpaying for your mutual fund then the fund is making the bare minimum to meet its costs whether the fund goes up or down. Are you proposing that the expense ratio should be higher in years that the stock market is good and lower in the others?\nYou know this is quite dangerous? Having all your eggs in one basket , especially an over valued one like Australasian real estate, and a soft currency like nz$.\nBy investing in a low cost index you \" buy the market\" . It will go up and down. Generally up.\nYou need to do some figuring on your own on this. Vanguard suggests anywhere from 20% up to the market weighting of 55%. I've personally opted for a 50\/50 US\/International split. I like the currency diversification and I don't have a US-centric worldview like many people do. It seems to be working fine for me so far.\nYou know this is quite dangerous? Having all your eggs in one basket, especially an over valued one like Australasian real estate, and a soft currency like nz$.\nI know it's quite dangerous. I buy\/renovate\/sell property heavily and from that and a couple of other business ventures I have a fair bit of free cash now.\nThe thing is, property is the devil I know and through it I'm financially free. More than that I've worked in finance\/banking and don't trust the people shilling investments in the global market past the end of my arm.\nI've started to talk to a contact about buying into the US real estate market and with the high NZ dollar I'm thinking about parking money in USD.\n50% of my equities are investing internationally.\nThat's a nice idea, but it just ain't possible in publicly-traded equities. So kick back, pop open a frosty beverage, and buy the market.\nOut of my stocks, I aim for 25% foreign, 75% US. It's pretty simple and easy to stick to.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"I'm happy to speak on the phone at any time between 9am and 9pm, 7 days a week. I can also organise meetings by Skype.\nIf a problem arises at anytime, you can email me and I will get back to you by email or by phone ASAP.\nI'm a strong and robust barrister, and I will work very hard to get the best result for you.\nI'm a tough negotiator, and I will fight your corner at all times.\nI'm just a phone call or email away, and I'm happy discuss your case whenever is best for you.\nI'm very clear and straightforward in my advice, and I'll explain everything until you understand it properly.\nYou can trust me to work hard for you.\nI've been practising as a barrister since 2006. Before that, I worked in a solicitors' firm.\nI have a strong reputation in family law cases, and I always put my clients first.\nI've worked alongside many law firms throughout the UK over the last 10 years.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Author Melissa WilcockPosted on January 23, 2016 January 23, 2016 Categories Art, Blogging, family, Kids, Light Painting, Melissa, Missy Norman Photography, Photograph, photography, Technology, UncategorizedTags art, Child, edit, lightroom, Melissa Norman, Missy Norman, Missy Norman Photography, MissyNormanPhotography, Photo, photo-shoot, Photographer, PhotographyLeave a comment on Godzilla Light Painting.\nIt has been a while since I have done any light painting. I truly miss it. With this light painting, I did Halloween props that I got on sale. I also used a texture with the background. I think that the image came out beautifully.\nAuthor Melissa WilcockPosted on January 16, 2016 January 23, 2016 Categories Art, Books, Editing, Halloween, Light Painting, Missy Norman Photography, Photograph, photography, Props, UncategorizedTags art, creative, lightroom, Melissa Norman, Missy Norman, Missy Norman Photography, MissyNormanPhotography, Photo, photo-shoot, Photographer, PhotographyLeave a comment on Light Painting Time Again!\nSelfie of the New Year.\nThis year is starting off with a bang. Once again I find myself sitting here and writing 2015 on every single thing. Christmas clean up has happened. A birthday has happened. (Not mine) School started back from Christmas break. Personally break was all too short. I absolutely love having my kiddo home. A new beginning in my chapter began on Christmas. I am learning how to sew. Actually I am not bad at it, I have no idea how I went my existence without a sewing machine. A new year marks new hair, I updated my hair color and I totally love it. I moved my studio, it is smaller, but that is okay. The things we do for our kids. I am looking forward to an amazing new year. I am excited to see what is in store.\nAuthor Melissa WilcockPosted on January 9, 2016 January 23, 2016 Categories Blogging, Me, Melissa, Missy Norman Photography, More, photography, UncategorizedTags Melissa Norman, Missy Norman, Missy Norman Photography, MissyNormanPhotography, Photo, Photographer, Photography, SelfLeave a comment on Selfie of the New Year.\nA New Shoot for ME.\nOver Christmas I was doing a dear friends photography. It was of her two beautiful little girls. After the shoot I decided to do some maternity photos for my friend. She was due anytime. The shoot happened early, due to me rescheduling the shoot. Thankfully I did move the shoot up. The next day she went into labor. I would have not been able to do the girls Christmas shoot or maternity photos. I love her belly, it is such a perfect baby belly. I was honored to have shot these images for her and her last pregnancy.\nIs she not totally beautiful!?!?!?!\nAuthor Melissa WilcockPosted on January 2, 2016 January 23, 2016 Categories Handmade, Maternity, Melissa, Missy Norman Photography, More, Photograph, photography, Skin, UncategorizedTags Maternity, Missy Norman, Missy Norman Photography, MissyNormanPhotography, Photo, Photographer, PhotographyLeave a comment on A New Shoot for ME.\nMissy Norman Photography Create a free website or blog at WordPress.com.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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index 0000000000000000000000000000000000000000..f3bfde71f65ca438c9d1fff420e4e45a0bd3f646
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+{"text":"It lists the hardware platforms supported by FreeBSD as well as the various types of hardware devices ( storage controllers, network interfaces, so on) along with known working instances of these devices. Canon LBP 6000 driver is available on this web page without cost. Canon LBP 6000 Free Download Driver. Canon driver printer lbp 6000.\nCashback Epson: Acquista una delle stampanti Epson selezionate per l' iniziativa singolarmente o insieme a una confezione multipack di cartucce Epson dal 15\/ 12\/ al 28\/ 02\/. Jadi kedua tombol tersebut posisinya Anda tekan.\nHome \u00bb Canon LBP6000\/ LBP6018 Use the links on this page to download the latest version of Canon LBP6000\/ LBP6018 drivers. High- performance Controller: Canon' s high- performance controllers optimize the processing time on the PC printers that ensure a higher level of compression faster data transfer. The new canon series can be purchase on online store or you can buy this LBP6000\/ LBP6018 at the computer & printers shop at.\nCanon driver printer lbp 6000. ; Page 3 Manuals for the Printer The manuals for this printer are arranged as follows.\nT\u00e9l\u00e9charger driver \/ pilote imprimante HP Laserjet p1102 gratuit pour windows 7, windows 8 win 8. A seguire l' elenco completo comprensivo di Software & Driver compatibili con la Stampante Multifunzione Epson XP- 247 Expression Home. Toner Saving Mode - When the imageCLASS LBP6000 is switched into the toner saving mode, it allows for up to a 20% more economical toner usage.\nVi\u1ec7c c\u00e0i \u0111\u1eb7t \u0111\u00fang driver Canon 2900 tr\u00ean m\u00e1y t\u00ednh s\u1ebd gi\u00fap cho vi\u1ec7c k\u1ebft n\u1ed1i gi\u1eefa m\u00e1y t\u00ednh v\u00e0 m\u00e1y in \u0111\u01b0\u1ee3c th\u00f4ng su\u1ed1t, gi\u00fap cho. 50( 32bit & 64bit) \u30bd\u30d5\u30c8\u6982\u8981: \u3053\u306e\u30c9\u30e9\u30a4\u30d0\u30fc\u306f\u3001 \u30ad\u30e4\u30ce\u30f3\u30c7\u30d0\u30a4\u30b9\u306b\u5bfe\u5fdc\u3057\u305f\u3001 Linux OS \u7528\u306e\u30d7\u30ea\u30f3\u30bf\u30fc\u30c9\u30e9\u30a4\u30d0\u30fc\u3067\u3059\u3002. Service manual download Link for some printers like Epson Printers Brother Printers, Canon Printers, Lexmark Printers , Hewlett Packard > Printers Xerox.\nCanon driver printer lbp 6000. T\u00e9l\u00e9charger driver \/ pilote Epson Stylus SX235w gratuit. 3) \u30bd\u30d5\u30c8\u6982\u8981: \u672c\u88fd\u54c1\u306f\u3001 Apple\u793e\u306eMac OS \u74b0\u5883\u304b\u3089\u5370\u5237\u3059\u308b\u305f\u3081\u306e\u30bd\u30d5\u30c8\u30a6\u30a8\u30a2\u3067\u3059\u3002. \u540d\u79f0: LIPS4 Printer Driver for Linux Ver.\nTIP OF THE DAY Know this: aiming to optimally configure the system drivers through the Windows device manager could quite possibly slow the system device . Bad drivers have a tendency to affect many layers on your personal machine, as well as the application they' re exclusively pertaining to. A large number of challenges may bring about expired Syn- Tech USB Board- Prokee ( along with other driver) the effected computer is in risk of becoming slower perhaps. Toner Cartridge Recycling Program - Canon was the first in the printing industry to create a toner cartridge recycling program.\nDriver Canon LBP 2900\/ 2900B gi\u00fap m\u00e1y t\u00ednh k\u1ebft n\u1ed1i v\u1edbi m\u00e1y in Driver Canon LBP 2900\/ 2900B c\u0169ng gi\u00fap kh\u1eafc ph\u1ee5c c\u00e1c l\u1ed7i li\u00ean quan \u0111\u1ebfn m\u00e1y in Canon LBP 2900\/ 2900B, \u0111\u1ec3 th\u1ef1c hi\u1ec7n nh\u1eefng thao t\u00e1c in mong mu\u1ed1n h\u1ed7 tr\u1ee3 in h\u1ea7u h\u1ebft c\u00e1c t\u1eadp tin \u0111a ph\u01b0\u01a1ng ti\u1ec7n tr\u00ean \u1ed5 c\u1ee9ng m\u00e1y t\u00ednh c\u1ee7a b\u1ea1n. - Silahkan printer Anda harus dalam keadaan mati - Selanjutnya tekan dan tahan tombol stop atau reset yang diikuti dengan menekan tombol power.\nVi\u1ec7c c\u00e0i \u0111\u1eb7t \u0111\u00fang driver Canon 2900 tr\u00ean m\u00e1y t\u00ednh s\u1ebd gi\u00fap cho vi\u1ec7c k\u1ebft n\u1ed1i gi\u1eefa m\u00e1y t\u00ednh v\u00e0 m\u00e1y in \u0111\u01b0\u1ee3c th\u00f4ng su\u1ed1t, gi\u00fap cho b\u1ea1n c\u00f3 th\u1ec3 in c\u1ea5n c\u00e1c t\u00e0i li\u1ec7u t\u1eeb m\u00e1y t\u00ednh ra m\u00e1y in. TIP OF THE DAY Know this: aiming to optimally configure the system drivers through the Windows device manager could quite possibly slow the system device alternatively cause your computer to fail.\nIf an internal link led you here, you may wish to change the link to point directly to the intended 1547 Canon Printer PDF manuals. Good news from the owner thus those who look for the download link for all in one printer driver , full software of Canon i- SENSYS LBP 6000, the download link is accessible no worry for broken link as any link for any platform works properly.\nSo whether you are looking for a Stylus cartridge color Epson printer cartridge, any black is your best online resource. Page 2 LBP3460 Laser Printer User' s Guide. This document contains the hardware compatibility notes for FreeBSD 11. Canon i- Sensys LBP 6000 Laser Printer.\nThe limited warranty set forth below is given by Canon U. Canon LBP 6000 Driver download for Windows 7 Windows 10 8. L' iniziativa \u00e8 soggetta a Termini e Condizioni. This disambiguation page lists articles associated with the title Canon. User manuals Canon Printer Operating guides Service manuals. Phi\u00ean b\u1ea3n 64bit c\u1ee7a ph\u1ea7n m\u1ec1m Driver Canon LBP 2900\/ 2900B \u0111\u01b0\u1ee3c ch\u00fang t\u00f4i c\u1eadp nh\u1eadt \u0111\u1ec3 cho b\u1ea1n d\u1ec5 d\u00e0ng download, vi\u1ec7c download v\u00e0 c\u00e0i \u0111\u1eb7t l\u00e0 quy\u1ebft \u0111\u1ecbnh c\u1ee7a b\u1ea1n. Generally for Postscript printers, you will not need a driver as all applications produce PostScript.\nCanon LBP 6000 driver is available on this site at no charge. Canon toner cartridges are 100% recovered with zero landfill waste. Change Bit switch 1 bit 0 ( might display as the first one on the left using number keys) status 0 to 1).\nDriver che troviamo suddivisi per sistemi operativi Windows e Mac, con funzionalit\u00e0 complete oppure di base. \u0627\u06cc\u0646 \u067e\u0631\u06cc\u0646\u062a\u0631 \u06cc\u06a9\u06cc \u0627\u0632 \u067e\u0631\u06a9\u0627\u0631\u0628\u0631\u062f\u062a\u0631\u06cc\u0646 \u067e\u0631\u06cc\u0646\u062a.\nFor the printing system getting access to the printer- specific features the manufacturer supplies a PPD file for every PostScript printer. Logiciel imprimante pour windows 7, windows 8 win 8.\nPage 1 Laser Beam Printer User' s Guide User' s Guide IMPORTANT: Read this manual carefully before using your printer. Canon LBP 6000 driver is a software pertaining to Canon LBP 6000 printer device to connect with a laptop computer. Canon Europe digital SLR cameras, inkjet printers & professional printers for business , leading provider of digital cameras home users. \u540d\u79f0: LIPSLX Printer Driver & Utilities for Mac Version 10.\nCanon LBP 6000 driver is a computer software in support of Canon LBP 6000 printer device to connect with a computer. Canon driver printer lbp 6000.\nThis utility was recognized by many users all over the world as a modern convenient alternative to manual updating of the drivers also received a high rating from known computer publications. Canon LBP6000\/ LBP6018 Driver Download \u2013 The Canon LBP6000\/ LBP6018 Driver Canon LBP6000\/ LBP6018 Driver windows 7 64bit, Canon LBP6000\/ LBP6018 Driver Mac, Canon LBP6000\/ LBP6018 Driver series OS X & linux 32 bit & 64 bit.\nSave this manual for future reference. All drivers available for download have been scanned by antivirus program.\nA large number of challenges may bring about expired Syn- Tech USB Board- Prokee ( along with other driver) the effected computer is in risk of becoming slower perhaps rendered ineffective. Canon i- SENSYS LBP 6000 Driver & Software Download For Windows Mac Os Linux. Driver Printer Canon MX497 menjadi sesuatu hal yang sangat penting sekali dimana ketika anda akan memakai printer tersebut dalam rangka untuk mencetak dokumen dan foto atau mau copy dan scan, maka sistem operasi terlebih dahulu akan mendeteksi Printer anda. Canon LBP 6000 driver is also an efficient laser quality printer that can deliver 19 webpages in 1 minute.\nCANON imageCLASS LBP6000 LIMITED WARRANTY. 0( Mac OS : 10. Canon driver printer lbp 6000. Please refer to them for detailed information.\n( \" Canon USA\" ) with respect to your Canon imageCLASS LBP6000 packaged with this limited warranty ( the \" Product\" ). 1 Windows 10 Vista ( 32bit. \u0633\u0644\u0627\u0645 \u0627\u06af\u0631 \u0634\u0645\u0627 \u0647\u0645 \u0645\u0627\u0646\u0646\u062f \u0628\u0633\u06cc\u0627\u0631\u06cc \u0627\u0632 \u0627\u0641\u0631\u0627\u062f \u067e\u0631\u06cc\u0646\u062a\u0631 Canon LBP6000 \u0628\u0631\u0627\u06cc \u0686\u0627\u067e \u06a9\u0631\u062f\u0646 \u0627\u0633\u062a\u0641\u0627\u062f\u0647 \u0645\u06cc \u06a9\u0646\u06cc\u062f \u062d\u062a\u0645\u0627 \u0634\u0645\u0627 \u0647\u0645 \u0628\u0631\u0627\u06cc \u062f\u0631\u06cc\u0627\u0641\u062a \u062f\u0631\u0627\u06cc\u0648\u0631 \u0627\u06cc\u0646 \u067e\u0631\u06cc\u0646\u062a\u0631 \u0628\u0627 \u0645\u0634\u06a9\u0644 \u0631\u0648\u0628\u0631\u0648 \u0634\u062f\u0647 \u0627\u06cc\u062f.\n1 Windows 10 Vista ( 32bit & 64- bit. Offers a huge selection of Epson ink and toner cartridges at steep discounts!\n1 Vista, XP 32 & 64 bits Mac. Jika telah diinstal Driver maka bisa dipakai apabila belum maka sebaliknya, dimana OS Windows tidak mau menerima hal ini. The Driver Update Tool \u2013 is a utility that contains more than 27 million official drivers for all hardware, including canon lbp 6000 driver. Use less memory than conventional printer files provided PC , prepared latest drivers , notebook, softwares for your printers other gadgets Free printer software.\nCanon LBP 6000 driver is also an efficient laser quality printer which could deliver 19 pages in 1 minute.\nAug 07, \u00b7 Canon LBP 6000 download driver for windows, mac os and Linux software installer. First Canon quick print unique technology that saves time, effort and money by applying technology to the printer:. Canon LBP6000\/ LBP6018 Driver Download Software IJ Printer Driver Start Canon Scan Utility for windows 10 \/ 8. 1 \/ 8 \/ 7 \/ XP 32 bit \/ 64 bit, Mac OS X.\nIn computers, a printer driver or a print processor is a piece of software on a computer that converts the data to be printed to a format that a printer can understand. The purpose of printer drivers is to allow applications to do printing without being aware of the technical details of each printer model.\nPrinter drivers should not be confused with print spoolers, that queue print jobs and.\nIncludes links to useful lect a Product Series to begin and then choose your Product Model from the listings below. \u0414\u0440\u0430\u0439\u0432\u0435\u0440\u0430 \u0434\u043b\u044f \u043f\u0440\u0438\u043d\u0442\u0435\u0440\u043e\u0432 Canon - \u0441\u043f\u0438\u0441\u043e\u043a \u0432\u0441\u0435\u0445 \u043c\u043e\u0434\u0435\u043b\u0435\u0439 \u0434\u043b\u044f \u0434\u0430\u043d\u043d\u043e\u0433\u043e \u043f\u0440\u043e\u0438\u0437\u0432\u043e\u0434\u0438\u0442\u0435\u043b\u044f, \u043f\u0440\u0435\u0434\u0441\u0442\u0430\u0432\u043b\u0435\u043d\u043d\u044b\u0445 \u043d\u0430 \u0441\u0430\u0439\u0442\u0435 ic Postscript \" driver\".","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Well this week was a little hard because we couldn't find very many people, but it was a really cool to see people at church yesterday. We are not sure if that family is going to get married this week or the next, things are weird here in Colombia, you can't just get married, you have to plan a cita (appointment) way in advance so we are looking to see if we can have someone do us a favor or we'll have to wait a week longer. I hope not because this week is combos (transfers) again. Who knows what will happen this time! Only another 2 1\/2 transfers before I will be home. It's kind of weird and scary to think about all at the same time. Well I gotta go. I love you all!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Premier Pavers Dropfaces give a stylish, modern edge to your steps and stair treads, as opposed to the more classical look of the bullnose finish. The modern looking L-shape of our dropfaces will give your paved staircase a clean, crisp, sharp finish that will beautifully compliment your pavers.\nIf you're looking for something that spans the gap between modern design and a timeless classical finish, why not take a look at our rounded edge dropfaces. With their rounded edge, these dropfaces combine the softer look of bullnose pavers with the clean, modern lines of an L-shaped dropface.\nAdding a dropface to your paved staircase not only adds a stylistic element, it also makes your staircase much more comfortable to traverse. We've all gone through the awful pain of catching a toe on the sharp edge of a paver when rushing down the stairs to get to work on time, or chasing the kids around the backyard. Dropfaces look sharp, but an incredible amount of effort has been placed into crafting the dropfaces so they are completely smooth whilst still retaining those beautiful, sharp lines.\nLandscaping: Creating a beautiful and interesting garden is all about building an exciting visual spectacle. Constructing gardens with varying levels is a popular practice, which is often achieved by using paved retaining walls to hold garden beds. Using dropfaces on the uppermost pavers gives garden walls a smooth, yet perfectly finished look and feel, and hides any imperfections on exposed edges.\nPool areas: our paver make perfect drop face pool copings for pool edges, creating a beautiful neat look and ensuring no one gets a cut foot when they jump in.\nHousing entrances: Stepped entries to houses look amazing; nothing creates a sense of grandeur like a beautifully crafted staircase leading up to the front door. If you want to sharpen the look and make it a little more modern, go for an L-shaped dropface and create uniformity throughout your staircase. Use Bluestone pavers and Bluestone dropfaces for a quintessentially Melbourne look.\nIndoor staircases: Nothing says sophistication and an eye for detail like dropfaces on indoor, stone staircases. Visitors will be drawn to your staircase and be amazed by the combination of natural materials and modern lines.\nOur L-shaped or rounded dropfaces give you the ability to top off your pavers with either a sleek, modern edge or a more classical, rounded edge. Modern aesthetics place a high standard on uniformity and our dropfaces ensure that your pavers look exactly the same from every vantage point. You'll have a smooth finish that won't cause any injuries to either you, or your visitors. So, if you want to give your pavers an edge, Premier Pavers has the answer.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Rio Rancho Mayor Hull in the House!\nMayor Hull Updates us on the \"State of the City\"\nWe were fortunate to have Rio Rancho Mayor Greg Hull speak to us during our December 13th meeting. He shared a video with us and updated us on the many projects going on in Rio Rancho. The city of vision is growing in the right direction!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Anti-slip profiles - excellent slip resistance and comfort.\nLarge Holes for drainage - good drainage for spills and swarf's.\nRobust construction, made from an extremely strong natural rubber compound.\nProvides good level of comfort in addition to slip resistance.\nHeavy weight - non shifting.\nConnectors join mats end to end or for Octomats sideways also.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzakcat b/data_all_eng_slimpj/shuffled/split2/finalzzzakcat
new file mode 100644
index 0000000000000000000000000000000000000000..0786ca0f70d63e5c05dfdc8c609a4651ff785a3c
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzakcat
@@ -0,0 +1,5 @@
+{"text":"Metallic is a trend we definitely didn't mind seeing on the Fall 2012 runways. At Ralph Lauren, golds appeared on knits, piping on blazers, and brocade embroideries, while Midas touched upon the model's toe caps and brocade patterned suits at Marni. We also spotted the trend at Prabal Gurung, Elie Saab, Tracy Reese and Barbara Bui, to name just a few. Our favourite way to incorporate this trend into your fall wardrobe is with a metallic pattern, but we also think gold or silver accessories can add a luxe touch perfect for the office or after five.\n1. Preen Eva metallic cotton-blend and lace skirt, $1,030, net-a-porter.com.\nThis pencil skirt combines two major trends for fall \u2014 metallic and lace. Chic and feminine, it's a perfect option for the office paired with a simple white button up or a brocade patterned top.\n2. Zara jacquard loom trousers, $80, zara.com.\nIf you're going to pick up any style of trousers this fall, make it ankle cropped ones. Seen all over the Fall 2012 runways, we love that this pair incorporates the metallic and print trend so beautifully.\n3. H&M silver loafers, $60, hm.com.\nA dressy loafer? We think so! While we usually see silver shoes in the sandal department, we're really loving them in a loafer style. Try pairing them with a black leather skirt, lace straight leg pants or even a cute printed dress.\n4. French Connection Galaxy crepe pipe detail top, $118, canada.frenchconnection.com.\nGo for instant sparkle with a simple metallic top that's easy to wear. We suggest wearing with anything from a pair of black skinny pants to a dark wash jean.\n5. Clare Vivier foldover clutch, $185, shopbop.com.\nAdd a burst of glamour to your going-out look with a metallic clutch. The great thing about a folder over clutch like this one is it's spacious, meaning you can fit your evening essentials in it.\n6. Mango glitter leather gloves, $50, shop.mango.com.\nWe know it seems early to start talking gloves, but it is mid-August and before you know you're going to be digging through your closet for a pair. Trade up your basic black this season and try a metallic pair for a touch of luxe.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"An average driving game at best when originally released, Roadsters hasn't aged well and is unlikely to win any new fans, even with the fresh lick of paint the emulators can give.\nDemul 0.5.8.2 (DirectX 10) \u2013 Menu items are illegible if the \"Disable auto sort\" option is turned on. Otherwise the game runs well with no noticeable glitches.\nThe game also released on the Nintendo 64 and Game Boy Colour consoles, as well as on the Sony PlayStation console.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"What's a better way to announce a run of tour dates together than to release an EP of tunes those two artists collaborated on? Up that ante a bit and these two dance music heavy hitters have done that plus some as they've dedicated the entire project to one of their pets. You read that right. Premiering here on DAD today, Lilou's Theme is a three track beast mode of a release from Sinden and Astronomar dedicated to Sinden's French bulldog Lilou. While the tunes themselves don't have much to do with Lilou on the surface, the sense of humor with the project is absolutely nailed as well as the music itself. Both producers have a fairly prolific history of making house oriented music that completely fucks with your head while moving your feet and this release definitely maintains the tradition. The title track is a chaotic jackin tune in the spirit of Green Velvet's classic, \"Percolator\". \"Bondeye\" is a complete mindfuck that's as twisted as it is banging. The closer \"Roarshack\" is the most warehouse techno flavored of the three and definitely would fit right at home in a peak time set from Boys Noize. All three tunes are out now for your listening pleasure and make sure to peep these immensely talented dudes if they are hitting your city in the coming days.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Want to learn even more about Arduino ?\nArduino is a prototype platform (open-source) based on an easy-to-use hardware and software. It consists of a circuit board, which can be programed (referred to as a microcontroller) and a ready-made software called Arduino IDE (Integrated Developmen t Environment), which is used to write and upload the computer code to the physical board.\nBefore you start proceeding with this tutorial, we assume that you are already familiar with the basics of C and C++ . If you are not well aware of these concepts, then we will suggest you go through our short tutorials on C and C++. A basic understanding of microcontrollers and electronics is also expected.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Feel free to call us right now at 417-437-9345. That is the phone number to the fitness center Topeka where we will answer any questions that you may have for us. We are the best gym that you have ever seen or will ever find. If you're looking for a gym, then you can look no further. We are the best because there are so many reasons why we are the best. Our members love us and we want you to see why. That is why you need to come to her gym today and see everything we have to offer for yourself. It will be the best decision that you have ever made because we are so great. It is such a great offer of that did not mean to pass up on anymore.. Waiting in come in today to get started on your journey with us.\nWanted to get in shape recently then We are the best place for you. We have everything you could ever ask for to get into the best shape of your life. Are free trainers withhold you with any questions that you may have about this. They will help you get into the shape of your life. On that you have to do is follow the programs that we have designed for you. Everything we have to offer is free of charge and included in your membership. This is why the Fitness Center Topeka is the best Jim of all time. We give you so much for free because we want you to get the best value for your money. That is why we only charge you $5 a month four 4 access to our entire gym.\nThis is the best deal that you will ever see, because there is no one else doing this. It is hard to believe that art memberships really are five dollars a month, but They really are. Our are so low because we want you to be able to afford going to the gym. If you come to the Fitness Center Topeka, you will change your life forever. We want you to change your life forever for just a few dollars because you are worth it.\nEveryone that comes to your gym loves how cheap it is because they get to save money and spend it elsewhere. If you comes the gym 12 days a month, you will pay the low price of $5. This is the cheapest dealership that you will ever find anything the Entire life. You will love coming to the gym because you will pay less the more that you come.\nOnce you come to her gym and see for yourself everything that we have to offer you are going to want to sign up. The fitness center Topeka is beautiful, everyone that comes to her gym absolutely loves us. ur equipment is state-of-the-art, and it is the best gym equipment that you will ever find. You will love our free training schedule because we help you with everything that you need to know to get into shape. Our staff was very friendly, and they love fitness because that is their passion. Everyone that comes to our free training sessions leaves much smarter and more knowledgeable about fitness. We want you to be one of these people, so that you can help others. Using our equipment is not hard at all, and you will look how easy it really is. This is because our equipment is very user-friendly.\nThere are so many reasons why we stand out at the fitness center Topeka because we work really hard to be the best. Therefore, we are voted the best of Topeka. If you come and see for yourself what makes it so great, you were going to want to be a part of what we have cough. One thing that are members really love about the Fitness Center Topeka is our free trainer and nutrition instruction.\nWe have done all of the hard work for you, so all that you have to do is follow our programs. We guarantee that if you follow our programs you are going to see the results that you want to see you. So come on and saying get started with our free trainer. You are going to love how easy we make it for you, because to us you are worth it. We want nothing more than for you to succeed at your goals.\nIt doesn't matter what your goals are, because we want to help you achieve matter. Even if you do not have any goals, we want to help you set goals for yourself in the future. We have helped so many people with their goals in the past, we want to help you next. The Fitness Center Topeka is the absolute best place to come to get better. There is nowhere like us, and that is why you need to come to her gym and sign up today. You will go to love everything we have to offer because we are the best. So coming today and sign up with us get started And the rest of your life.\nIf you still have questions for us, and how much should cost at our gym, you can call us at anytime. Our phone number is 417 437-9345.We are open and staffed 24 seven. We are always there for you because you matter to us and other gyms are not there for you at all times. We want you to be able to understand everything that we have to offer clearly. That is why we have made everything so simple at our gym. Once you come in today and see for yourself everything that we have to offer you will see that we have the best option for you. You were going to love everything that we have in our gym, and you will want to be apart of it. Stop waiting in come in today and get started with us. We want to help you change your life forever.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzaklde b/data_all_eng_slimpj/shuffled/split2/finalzzzaklde
new file mode 100644
index 0000000000000000000000000000000000000000..9d9468a3c3cd294d917b4b96bd62c7a5df1a5041
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzaklde
@@ -0,0 +1,5 @@
+{"text":"It lowers the level of bad cholesterol in you body. Boosts body metabolism. Helps regulating blood pressure. 7 'Healthy' Items You Need To Throw Out Right Now Helps to lose weight.\nEmily Rowlandson 14 May 2018 marie claire 565 Views \u00b7 View Upvoters Green tea has a substance called Tannin. The stronger the green tea, then higher the content of tannins in it. This substance can increase the acidity in the stomach, causing stomach ache, constipation, and nausea. So, it is best NOT to consume it first thing in the morning on an empty stomach\u2026 And don't consume it with food either because it green tea mixed with food can block the body's absorption of iron potentially causing an iron deficiency!\n168 Views Boosts up the immunity. A number of studies have found that green tea can combat some of the biggest risk factors for cardiovascular disease \u2013 reducing levels of cholesterol, LDL cholesterol and triglycerides. A study that followed more than 40,000 Japanese people over an 11 year period showed that participants who drank at least five cups of green tea per day had a significantly reduced risk of dying \u2013 especially from cardiovascular disease \u2013 compared to people who didn't drink it.\n1k Views \u00b7 View Upvoters \u00b7 Answer requested by Augustine Skhosana Helps in boosting the metabolism rate of the body.\nAsk New QuestionSign In Though caffeine contents in green tea is not as much as you would find in coffee, and since water contents do hydrate the body; it is considered healthier than drinking coffee when considered in drinking on empty stomach.\nAnswered Dec 9 2017 Have you ever googled yourself? Do a \"deep search\" instead. * @license Licensed under MIT license Contact Us Though caffeine contents in green tea is not as much as you would find in coffee, and since water contents do hydrate the body; it is considered healthier than drinking coffee when considered in drinking on empty stomach.\nPromotes healthier bones. What are the benefits of drinking green tea over black tea? Supports healthier way to lose weight.\nWeight Loss What are the benefits of green tea?\nEating Out Increases Levels of Phthalates In The Body, Study Says Sales For Coconut Oil Have Apparently Plummeted Green tea is one of the hardest working beverages in the biz with a huge range of impressive, research-backed health benefits. We take a deep dive into the science behind the brew to find out exactly what wellness perks it offers.\nWeight Loss * https:\/\/github.com\/csnover\/TraceKit Now, to be fair to green tea and its amazing benefits, which outway its minor side effects for healthy individuals, I have also put together a post called 17 Health Benefits Of Green Tea (And The Studies Behind Them). It will show how amazing green tea is for all sorts of conditions including asthma, weight loss, blood pressure, diabetes and many others.\nAsk New Question Lower high blood pressure. Helps treating ear infection.\nWhat are the benefits of drinking green tea in the early morning with an empty stomach? New Idea Jamey Gaylord Green tea is alkaline; so for people with weak stomach, it is suggested to eat something before drinking green tea on empty stomach; however, for people with healthy stomach who wishes to increase metabolism to lose weight it would be good to drink first thing in the morning.\n* @namespace TraceKit Subscribe Green tea is well known for providing one with maximum benefits for the overall health of the body. It works well for providing one with the best benefits for the overall health of the body. Thus, here are some of the benefits that one may experience from drinking green tea.\nPrivacy Statement Beauty Crew Answered Dec 6 2016 Helps treating ear infection.\nLauren Williamson Answered Dec 6 2016 Is green tea a fat burner? Does it help with weight loss? How much should I drink? What is the most effective time of day to drink it? Helps lower the level of bad cholesterol from the body.\nWhat are health benefits of drinking green tea in the morning? \u00a9 2018 Pacific Magazines Green tea has a substance called Tannin. The stronger the green tea, then higher the content of tannins in it. This substance can increase the acidity in the stomach, causing stomach ache, constipation, and nausea. So, it is best NOT to consume it first thing in the morning on an empty stomach\u2026 And don't consume it with food either because it green tea mixed with food can block the body's absorption of iron potentially causing an iron deficiency!\nBeauty Crew Subscribe Today Subscribe Jamey Gaylord Prevents from food poisoning.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Chrome Marks \"Not Secure\" to HTTP sites from July 2018 \u2013 Why?\nYES! You read it right. From July starting, Google Chrome will start marking as \"NOT SECURE\" to HTTP sites, according to a google chrome marks blog post released by Chrome's security product manager Emily Schechter. Chrome currently displays a neutral information icon, but starting with version 68, the browser will warm users with an additional notification in the address bar. Chrome currently marks HTTPS-encrypted sites with a green lock icon and \"Secure\" sign.\nGoogle has being poking users away from unencrypted sites for years, but this one is the most forceful poke yet. Google search began down-ranking unencrypted sites in 2015, and the following year, the chrome team instituted a similar warning for unencrypted passwords fields.\nHow will this affect your browsing experience?\nOur straight answer would be; it won't. You will still be able to browse non-HTTPS websites as previously you used to do. The idea behind this step is to let users know that their personal and private information is at a higher risk of getting access to unauthorized people while browsing non-HTTPS sites.\nWill this affect websites that do not adopt HTTPS?\nYES, if websites does not use HTTPS might witness a fall in traffic, as a yellow triangle with \"NOT SECURE\" plastered next to it has the potential to scare away some of the more categorized users. However, it is a good thing for the long run, as it will incentivize more developers to use HTTPS on their websites.\nSeriously, there is no reason developers shouldn't adopt it. It is cheaper and simpler than ever before. Let us Encrypt and Google's SSL for Google App Engine makes it a lot easier for HTTPS to be implemented. There are also tools such as lighthouse, which identifies websites resources that are ready to be upgraded to HTTPS.\nWhat Should I Be Doing to Secure My Pages?\nIf you believe as we do that HTTPS is going to be the new standard, then I can bet you're thinking about what you actually need to do to ensure your corporate web pages and landing pages are secured. The three steps given below are meant to get you started on the right direction.\nWhile the title of this step sounds like you need to have mad technical skills- you really don't. The thing you need to do is to type your corporate website URL into any browser. For us, it's opening up chrome and typing, www.subsign.co. You are looking to see if the address bar says HTTP: or HTTPS: and\/or if using chrome, you are searching for green lock and the word \"Secure\" when you load. If you see HTTPS, your corporate site is secure. It is that easy.\nIf your business site is not secured, you may want to do a quick audit of your pages. You want to note the address of any page that has a form or collects information. With the recent update to chrome, it's those HTTP page that collect data that will display the \"not secure\" warning in the address bar. This list will also give you a starting point for the discussion you will need to have with your IT and web teams in the next step.\nIt is now time to enlist your experts. Make a schedule and set a time to meet the person who is responsible for your corporate website or others you trust and have a proven record of accomplishment. You will want to be sure they know of the changes to chrome browser. & you want to know of any plans that may already be in place to secure your corporate pages.\nGoogle has shared web traffic stats.\n81 of the top 100 sites on web use HTTPS by default.\nIt is also being said, even HTTPS isn't 100% waterproof, however, it is a step in the right direction to ensure a more secure browsing experience for everyone.\nAre You Aware of Affiliate Marketing?","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Backup all your important things. External drives are not expensive and many have easy to use backup functions. I have heard of many people losing their photos, videos, school papers, etc. when customer support or the local shop reloads Windows.\nGood passwords on all accounts and deactivating accounts not used.\nConvince users to use accounts without admin access for day to day use.\nYears and years ago in the dark ages, after the advent of the Apple ][ but before the coming of the IBM PC, I had a summer job doing data entry in the evenings. There was a room with 4 terminals connected to a small computer of some sort, the size of a large white-box PC, sitting in the corner. It was used for collecting the entered data and transmitting it via modem to the corporate mainframe in another state.\nThe side cover was kept off the computer because I guess it had cooling issues. Fortunate that it was, because one evening I noticed some flashes of light out of the corner of my eye. I investigated and found that a large capacitor in the power supply had caught fire! The p\/s was of discreet components not encased in a closed box like today's PCs, so I was able to blow out the fire after unplugging the computer.\nI've often wondered what the chances of something like that happening to the PCs we now have all over the house which are left on 24x7. Is there a small rate of defect in power supplies et al. that poses such a danger? Have houses burned down from PCs spontaneously combusting?\nRegarding the UPS, while your laptop may have it's own battery, if it is plugged in, make sure it's on a good surge protector.\nBackups are so overlooked they should be first on the protections list. They won't keep the bad stuff out, but when you have to reimage due to bad stuff, you want those precious documents saved somewhere. \"How do I get all my stuff back?\" is probably the most common family-tech support call... There are a few more 'purpose built' external drives out there lately with integrated software for backup, but you still have to get Mom to use them.\nFor a multi-PC family, this gets trickier. Who put which document on what computer makes recovery (and day-to-day homework tracking!) tricky. A NAS is probably beyond most home setups, but a centralized backup to an external drive on one machine probably isn't.\nAt the end of the day, backup has to be INVISIBLE, something that just happens when the computer is on the home network. Apple's Time Machine is a great example of this.\nAs people rely on digital storage for keepsakes (photos, movies, etc), the Realities of Backups need to be thought out, more than ever.\nSetting up a reciprocation with a neighbor, aka store your backups in their house and vice-versa, can work well if you have such a candidate. The caveat appears if an event of larger scope takes place, for example a wildfire or tornado, etc.\nOnline backups are great. However, as today's headlines in NY show, anything on the internet is subject to civil discovery no matter how \"private\" you think it is. In truth, you have zero control over the data with any online type of backup, and vendor claims of \"privacy\" are 100% feel-good. Probably not an issue for most, though, unless you get divorced or your teenager gets implicated in something (civil or criminal).\nStoring backups in your car is not so great. If your car gets robbed, they're gone. And if your house catches fire, as a firefighter I can tell you, it's common that the cars in the driveway go with it. Or with flooding, etc.\nTradeoffs abound, so make sure everyone is on the same page with them.\n@Ken: I've actually had power supply failures like that in modern PCs. There was smoke but the physical damage was limited to components inside the power supply casing. The ability to contain this kind of failure without setting fire to the house is necessary to achieve a UL listing.\nand OpenDNS configured appropriately too.\nBetter yet - Ubuntu. Honestly:- try to help them switch, it's is likely to be a excellent investment in your time.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"You can find our selection of Masonic Necklaces here. All our Masonic Necklaces are of high quality and double checked before shipping to our customers. We have a FREE international shipping and 45 days money back guarantee for all Masonic Necklaces.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"ROAD BIKES ARE BORN FOR COMPETITIONS AND ARE VERY APPRECIATED BY THE RUNNERS OR PASSIONATE AMATEURS FOR THEIR LIGHTNESS, SLIDING AND SPEED.\nRoad\/Gravel bike. The 250 G Yamaha Giant SyncDrive Sport 80 Nm built-in motor offers outstanding power support that, combined with the quality of the Aluxx SL aluminum frame, allows easy and fast pedaling. The 2 \u00d7 10 gearbox and disc brakes ensure the driver an easy driving and a soft braking.\nThe carbon frame with PRO technology makes this bike very light and fast, so much that it has won the first place in the 100th Giro d'Italia. The Shimano 2 \u00d7 11 52-36 11\/28 gearbox ensures comfort and efficiency during driving.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"Dr. Oz advises Garcinia cambogia extract quite highly, and it's easy to view why. Garcinia Cambogia has been getting a great deal of interest from the media and from blog writers recently in Beqaa Lebanon.\nGarcinia cambogia extract is being proclaimed as one of the most highly effective supplement offered for aiding folks in Beqaa Lebanon to burn fat properly and completely. It assures to do all this with totally organic, completely safe components, and without causing the individual to experience any sort of side effects.\nSo are all of these things true? Considered that Dr. Oz suggests Garcinia Cambogia, it would certainly stand to factor that individuals in Beqaa Lebanon could rely on Garcinia Cambogia to help you realize your weight loss dreams.\nA terrific numerous products and wellness suggestions are featured on Dr. Oz's tremendously well-liked television program. Some of them acquire his suggestion, and several of them do not.\nOffered his level of know-how and interest for helping people in Beqaa Lebanon lead healthier, healthier lives, it stands to reason that his suggestions should bring a bunch of importance. If you're looking for a significant way to burn fat, and to have that weight stay off also, the truth that Dr. Oz recommends Garcinia cambogia extract ought to be exceptionally encouraging to individuals in Beqaa Lebanon.\nGarcinia cambogia extract is basic, yet exceptionally effective. Taken from the skins of the Southeast\/South Asian fruit of the exact same name, this extract has actually been revealed through a massive array of researches to produce extremely attractive results most of the time.\nMaking use of HCA as its key element, Garcinia cambogia extract is both a fat deposits burner and an appetite suppressant. Individuals in Beqaa Lebanon are going to burn fatty tissue, constructing leaner muscle in the deal, yet you're visiting also do a great deal more.\nGarcinia cambogia extract works to make individuals in Beqaa Lebanon really feel fuller with smaller dishes, while all at once increasing things like your serotonin degrees. Without side effects to stress over (merely see to it to avoid Garcinia Cambogia items which contain points like caffeine), you're visiting feel much better in general. You'll have much more electricity, and you're not going to be drawn to give in to points like stress-eating.\nIt's again very easy to view why Garcinia cambogia extract in Beqaa Lebanon would get such a beautiful review from someone like Dr. Oz. It does specifically just what it assures it will do, and it does this with no risks to your personal health. Exactly what can be much better than that?\nHow Did Garcinia cambogia extract Obtain So Popular in Beqaa Lebanon?\n2 years ago this fruit was a mostly unidentified fruit utilized by the locals of South East Asia. After some medical research studies and a scientific test, plus a prominent medical professional's recommendation, the road to popularity began. However I'm getting ahead of myself.\nRegarding a year ago Dr. Oz offered Garcinia Cambogia extract to the world. All of us know Dr. Oz\u2013 he's the TV medical professional whose objective is to help individuals live a healthier life. No surprise Garcinia made its debut on this program.\nWhat I have actually picked up from this program is as adheres to: there was a medical test conducted just before the launch and it passed with flying recommendations. All the participants managed to slim down without any side effects, and they didn't also have to make changes to their diet or workout routine.\nThe main guest on the program was Dr. Chen, that detailed exactly what Garcinia cambogia extract is and just how it works. Besides that, 2 arbitrary people from the audience volunteered to attempt the extract. The outcomes were incredible to say the least.\nBoth females that checked it handled to lose between 5-7 pounds in merely one week. A lot better yet, they didn't do a severe eleventh hour diet regimen or pump iron at the fitness center. They lived their regular lives and took care of to shed a lot of weight\u2013 securely and normally.\nWith Garcinia cambogia extract individuals in Beqaa Lebanon could lose weight without any hard exercises or insane weight loss. Everyones in Beqaa Lebanon have to do is take it regularly and delight in the results.\nThere have actually been a lots of favorable assessments on Garcinia cambogia extract Extract. From health specialists to blog writers, everybody appears to be claiming the very same aspect of Garcinia cambogia extract. If you're serious about slimming down, and you're tired of using items that just do not work, this could be the solution you've wanted all along.\nDr. Oz is one positive reviewer from several others, yet when you once more consider his performance history, it's a referral that carries a massive level of reliability. Instead of purchasing around for items with unverified credibilities, learn for yourself just what Garcinia Cambogia can do for individuals in Beqaa Lebanon.\nIt seems as though the followers in Beqaa Lebanon in this extract are raising every day. Offered just how rapidly you're visiting outcomes, there's a likelihood that you're going to become a significant believer in your very own right.\nThinking about just how economical this product is, there actually is no reason not to try it. Trying to find the best nutritional supplement can be tough. It doesn't take long for a search to develop into an incredibly expensive, continuously annoying undertaking.\nQuit wasting your time on possibilities that may end up taking your cash, and probably even compromising your health somehow. Dr. Oz waits every product or suggestion he advises. If that's not nearly enough for you, then nothing is.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Funnel chart is released! Use it to compress more information into one chart and understand conversions in a more intuitive way.\nUpdated Facebook Pages data source to latest Facebook API version.\nIssue with table column width adjustment.\nIssue with Y-axis scale display for line\/column charts.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Dragon Concept Art by Jaemin Kim.\nWatch little video demonstration on how to use the rig in the above video.\nDragon_lighting: this is where you animate the dragon, the lighting was setup already (mental ray). Use Render_cam for the best results.\nTexture folder: all the texture files (color, normal, displacement, specular).\n* This rig is for non-commercial use only. Animation practice, personal projects, 11s club contests and demo reel are totally fine.\nPs: If you like the rig, please share him to your friends.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Penicillin (PCN or pen) is a group of antibiotics which include penicillin G (intravenous use), penicillin V (oral use), procaine penicillin, and benzathine penicillin (intramuscular use). Penicillin antibiotics were among the first medications to be effective against many bacterial infections caused by staphylococci and streptococci. Penicillins are still widely used today, though many types of bacteria have developed resistance following extensive use. About 10% of people report that they are allergic to penicillin; however, up to 90% of this group may not actually be allergic. Serious allergies only occur in about 0.03%. All penicillins are \u03b2-lactam antibiotics. Penicillin was discovered in 1928 by Scottish scientist Alexander Fleming. People began using it to treat infections in 1942. There are several enhanced penicillin families which are effective against additional bacteria; these include the antistaphylococcal penicillins, aminopenicillins and the antipseudomonal penicillins. They are derived from Penicillium fungi.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Best fruit trees for Tucson. Learn what fruit trees grow and thrive in our climate. Which varieties work here, how do you plant and take care of them? Come and learn how you can harvest peaches, figs, pomegranates, and much more right from your own yard.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"Today: we leave church in silence; except for those attending the PCC meeting.\nPCC: meets after Mass to receive and approve the 2017 accounts.\nSt Alban's Singers: many thanks to Charles, John, Nadia and Gwen for their musical and culinary efforts to make last Saturday's concert such a successful and enjoyable occasion.\nToday: we welcome Fr Paul Benfield SSC as our Preacher. Fr Paul is the Registrar of the Diocese of Blackburn and the Chairman of the Catholic Group in General Synod.\nIlford Hospital Chapel: Fr Stuart is celebrating the 12.35pm Eucharist at the Chapel this Thursday 22nd March. Please join us if you are able.\nPCC: meets next on 25th March after Mass to receive and approve the 2017 accounts.\nPCC: the PCC meets briefly after Mass today to arrange dates for the APCM and for receiving and approving the 2017 accounts.\nLent Appeal: this Lent we are raising money for the Nando's appeal for the relief of malaria.\nCoach Trip to Rochester Cathedral: we plan a visit to Rochester Cathedral for Mass and a tour on Saturday 24th March. The cost is \u00a330, which includes admission, a guided tour and the return coach journey. We will leave Ilford at 9.15am and depart from Rochester at 4.00pm. Places still available. Please see Gwen Dodkins.\nSt Alban's Singers: the next concert will be on Saturday 17th March 2018. Admission will be \u00a35 on the door.\nThis week: no midweek masses this week as Fr Stuart is on leave.\nPCC: today's PCC meeting has been postponed until Sunday 11th March. We will then meet briefly to arrange dates for the APCM and for receiving and approving the annual accounts.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Now that we have gotten most of the NAMM 2011 coverage out of our system, we go back to doing what we do best... DJ Gear Reviews! As promised, KRK has sent us a pair of their Best Buy Exclusive KRK VXT4-W (white) professional studio monitors so we could take them in the lab for a Full Written Review. Our very own D-Macli put these speakers through the paces and was very impressed with the overall sound quality and durability of these powered monitors. Check inside for the review and product specs!\nThe VXT4 is the most compact speaker in KRK's VXT series of studio monitors. Whereas its big brothers - the VXT6 and the VXT8 - are better suited for studio use, the VXT4's small stature and reasonable price tag allows audiophiles to bring the pristine sound quality that KRK is known for to their desktop or home entertainment system.\nKRK provided us with a pair of VXT4w monitors which are special white editions that are meant to match a Mac or iPod. They are being sold exclusively at Best Buy for $249.99 each \u2013 so a pair would run you just under $500. The yellow woofer provides a nice contrast to the white speaker cabinet and gives the VXTs a modern, yet professional look.\nSetup was easy \u2013 simply plug the unit in and connect an audio source. The special Neutrik Combo Connector gives you the ability to connect XLR, 1.\/4\" TRS or 1\/4\" phone plugs into one connection point. This is a unique feature but also limiting because you can have only one source connected at a time. Also since the VXT is targeted towards home entertainment\/computer use I would have liked to see the ability to connect an RCA or 1\/8 inch connection. However KRK does include a cable specific to typical computer setups: A stereo mini (1\/8 inch) plug split out to two \u00bc\" plugs.\nAlthough the VXT4w is is targeted towards home use, some features seem targeted towards a sophisticated studio or DJ setup. For example, the output volume for each speaker is found on the back of the unit and controlled by a very, very small knob. Personally for home use I prefer to have the volume in the front for easy access, but this is a small gripe. The VXT4 also lacks a headphone connection, which is an option I have found to be a useful when using my Alesis M1 Active 520s.\nWhat the VXT4 lacks in convenience, it more than makes up for it in sound quality. With a Kevlar woofer, silk tweeter, and specially designed cabinet, the KRK is built to provide pristine, undistorted sound at high volumes. I compared the sound of the VXT4s to my Alesis 520s and could instantly tell that the VXT4s provide crisper highs and deeper lows. I was amazed by how much full-sounding base could come out of a 4\" woofer. Even more impressive was the performance of the tweeter which seemed to breathe new life into songs by allowing me to hear every single element of the music clearly regardless of how high I set the volume.\nHere at DJBooth.net we were amazed at how the VXT4 can pump out clean sound at high volumes. The KRK VXT4 Monitors deliver incredible performance in a small package and are perfect monitors for perfectionist DJs and Producers looking for a combination of portability and crisp clear sound. However, people looking for the flexibility to have multiple inputs hooked up at once or a headphone output may be slightly disappointed. Even so, in the end even those users may be glad to sacrifice some convenience in exchange for incredible sound as long as they are not frightened by the high price tag.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"So, we all know the Xbox One app on Windows 10, and how it lets you stream games from your console to your PC. It's great, even if it plays like balls over wifi. We also know the Xbox Play Anywhere, that lets you kind of swap between console and PC. We even know all about Backwards Compatibility that lets you play 360 games on your Xbox One. This December, a new app is coming that promises to change the way you game \u2013 a literal game changer.\nThat app is the Xbox One Streaming to Oculus Rift app that's being released on December 12. You can get the full official pitch on what that'll mean for you over on Xbox Wire, but let's be honest. This is weird.\nYou're in a geometric base looking out into space, playing a game on a big screen, that's inside your little Oculus screens, via Windows 10, connecting to your Xbox One.\nOr you're inside a bare, creepy building \u2013 again, looking at a screen, and presumably hoping that there's nothing beside you. Yes, that's not an environment I want to hang out in the ambience of \u2013 everything about games has steered me toward the belief that something bad is going to happen in a place like that.\nThe angle is that using your Oculus Rift, you'll be able to stream these games from the Xbox One to your Rift, doing so in a virtual environment. Think of it as a special 3D theatre that gives you a separate screen to look at. If you like screens, good for you \u2013 you'll get a screen inside your virtual Oculus-provided environment for you to look at.\nAs environments go, they look interesting- kind of like a strange, lonely take on the old Playstation Home environments we saw back on the PS3.\nThe lead environment looks good as well, but seems to be a place I'd rather explore than chill in while playing a different game altogether.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Irish: reduced Anglicized form of Gaelic \u00d3 Rodaigh 'descendant of Rodach', a personal name probably derived from rod 'spirited', 'furious'.\nWhere is the Roddy family from?\nYou can see how Roddy families moved over time by selecting different census years. The Roddy family name was found in the USA, the UK, Canada, and Scotland between 1840 and 1920. The most Roddy families were found in the USA in 1880. In 1891 there were 89 Roddy families living in Lancashire. This was about 28% of all the recorded Roddy's in the UK. Lancashire had the highest population of Roddy families in 1891.\nUse census records and voter lists to see where families with the Roddy surname lived. Within census records, you can often find information like name of household members, ages, birthplaces, residences, and occupations.\nWhat did your Roddy ancestors do for a living?\nIn 1881, the most common Roddy occupation in the UK was Coal Miner. 8% of Roddy's were Coal Miners. A less common occupation for the Roddy family was Shoemaker. Coal Miner, Bricklayers Labourer and Labourer were the top 3 reported jobs worked by Roddy.\nCensus records can tell you a lot of little known facts about your Roddy ancestors, such as occupation. Occupation can tell you about your ancestor's social and economic status.\nWhat Roddy family records will you find?\nThere are 3,000 census records available for the last name Roddy. Like a window into their day-to-day life, Roddy census records can tell you where and how your ancestors worked, their level of education, veteran status, and more.\nThere are 642 immigration records available for the last name Roddy. Passenger lists are your ticket to knowing when your ancestors arrived in the UK, and how they made the journey - from the ship name to ports of arrival and departure.\nThere are 1,000 military records available for the last name Roddy. For the veterans among your Roddy ancestors, military collections provide insights into where and when they served, and even physical descriptions.\nYou've only scratched the surface of Roddy family history.\nWhat is the average Roddy lifespan?\nBetween 1943 and 2004, in the United States, Roddy life expectancy was at its lowest point in 1943, and highest in 2003. The average life expectancy for Roddy in 1943 was 43, and 74 in 2004.\nAn unusually short lifespan might indicate that your Roddy ancestors lived in harsh conditions. A short lifespan might also indicate health problems that were once prevalent in your family. The SSDI is a searchable database of more than 70 million names. You can find birthdates, death dates, addresses and more.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Finance MMC Funzela Ngobeni tabled his R59 billion budget on Thursday.\nJOHANNESBURG - The City of Johannesburg says it's disappointed that Treasury cut its Urban Settlements Development Grant but is still allocating a whopping R1 billion to housing for the 2018\/2019 financial year.\nHe says the city faces a massive housing challenge and he's disappointed that this grant was cut by R336 million.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"Relax on beautiful Lake Martin in this 4 BR\/2Full\/2Half Bath lake cottage with fantastic waterfront views and spacious covered wraparound porch. This is a New Listing that is newly furnished, with additional outdoor furniture and indoor decor coming soon. It has been updated with fresh paint throughout, keyless entry and all safety detectors. Outdoor areas are a dream. Wraparound porch has sectionals, rockers and dining tables with seating for 16. Lower porches, landings and decks include benches, swing and hammock. There is a large covered boat house, dock and floating dock with a boat ramp next to the driveway to launch your own boat. Vacant lot across street for additional parking.\nEnjoy the porches in the mornings with your cup of coffee or a good book, water activities during the day, hammock and swing for afternoon nap or enjoying the beautiful sunset and the gas grill for dinner.\nFirst floor includes family room with plenty of seating on sofa and 4 large swivel chairs, TV, fully equipped kitchen, laundry room with washer\/dryer, master BR with King bed, private bath with garden tub, rainfall shower and walk in closet. There is also a half bath on main floor. Seating for 8 at the dining table and 3 additional bar seats.\nUpstairs has 2 BR with 1 King and 1 Queen bed and another full bath with shower\/tub combo. Both bedrooms have lake view.\nBoth King rooms in home have room for a blow up mattress or crib if you need.\nDownstairs has a large bonus BR with 2 single beds and 1 bunk. Also includes another half bath.\nThis cottage is centrally located to Birmingham, Auburn, Montgomery or Atlanta and is perfect for a large family, 2 families, ladies getaway, men's golf weekend or Auburn football in the fall. Family friendly neighbors on either side.\nThis is truly a Southern Charm!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Beautiful 2nd Floor Corner Unit at Serene Kaibo Kai in Rum Point.\nEnjoy the privacy of the 2nd floor corner unit, while the large windows give a picturesque view of the beautiful sunset and a tranquility of Kaibo Beach and North Sound.\nCorner Kai has been recently furnished with with no expense sparred. Perfect for families with children as well as couples.\nThe large Master Suite has a king sized bed, with en-suite bathroom, walk-in closet, and direct access to the large lanai. It is perfect for the couple that wants to get away from it all.\nEach bedroom and the living area have 42\" flat panel TVs with cable.\nKorner Kai is located in beautiful Rum Point and is in the Kaibo Yacht Club condo Phase II complex. There is a beautiful walking beach on premise, with large fresh water pool and heated jacuzzi.\nYou also have your own private BBQ grill at the unit.\nOur condo was clean, comfortable and well maintained. Location is good, - mostly quiet, safe and management was friendly and responsive. It met and exceeded our expectations.\n\u2605 \u2605 \u2605 \u2605 \u2605 The condo worked out very well for our family of five. The owner has clearly put a lot of thought into stocking the unit with anything you might need from cookware to beach chairs, cooler, etc. The views from the deck and master bedroom are fantastic especially at sunset. The yacht club, which is set in the middle of the buildings serves good food and drink but is a bit expensive. The sand and swimming right off the condo aren\\'t like seven mile beach but will work for a quick dip or kayaking, etc. Rum Point Beach is a one minute drive and was absolutely fantastic. It has areas that are shaded from sun by palm trees, free beach chairs, several bars, good food, volleyball and other water activities if you want to pay. The beaches are mostly sand with some vegetation at the bottom.The only negative to this unit may be location depending on your circumstances. Nothing on the island compares to the water and beaches of Seven Mile Beach. Plan on a good 45 minutes to an hour drive or a fairly pricey water shuttle from the yacht club, which will put you on a schedule. We drove to seven mile five days and weren\\'t bothered by it. Popping in a couple stops along the way made it even easier. At Smith\\'s Barcadere we snorkled with thousands of fish only twenty feet off shore. At Spott\\'s Beach we swam with a huge sea turtle in the wild. I could go on and on about our experiences here but don\\'t miss those two. Thanks to the owner for meeting our expectations and I\\'m sure we\\'ll be back to the island.\nGreat Stay at Korner Kai!\nGreat condo with beautiful location! All the amenities needed, including linens, towels, coffee pot, & blender. They even had great extras like life jackets, beach toys & a kayak!\nOur family of four loved this condo complex. The drive in from the airport to this peaceful beautiful district was a happy surprise. We loved our choice of staying in this area with a rental car. We explored all of the island and had such fun discovering stops like the blow holes the lighthouse 7 mile of course and Georgetown and up to hell lol. If we had stayed on 7 mile we probably wouldn\\'t have seen near as much. Loved the botanical gardens and st Pedro\\'s castle. This is the most low key relaxed condo complex and had everything we needed and more. No stress! Easy dinners right out the door on the beach at the yacht club Perfect vacation. Only regret was the beautiful full moon didn\\'t let us explore the bio bay. Maybe next time.\nWe chose this condo as we wanted to getaway for a quiet relaxing retreat from a stressful work life. This place ticked off all the boxes. It\\'s everything we wanted - quiet, beautiful views, serene beach to stroll, fantastic pool and jacuzzi. Plus we really enjoyed going over to the Kaibo for lunch, drinks and dinner. We would definitely recommend this for someone wanting to get away from it all.\nLoved this \\\"corner kai\\\" unit as it\\'s obviously on the corner of the complex - quiet and great views of the north sound, beach, pool; and short walk over to the Kaibo. Loved going around the corner to Rum Point beach and hanging out at the Wreck Bar for lunch, snoozing in the hammocks and having fun snorkeling in gin clear shallow waters. What\\'s not to love? We\\'ll be back!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"But that's not even the half of it. Added to everything else is the fact that Moore County is halfway between the beautiful Blue Ridge Mountains to the west, and the pristine shorelines of North Carolina's Outer Banks. Hiking, mountain biking and snowboarding are just as easy as surfing, sailing, or deep-sea fishing. But don't think for a second that outdoor enthusiast have to travel to enjoy the wonders of nature. That's not even close to true. Residents enjoy strolling through Weymouth Woods, boating and wakeboarding on any number of local lakes, hunting, fishing, and let's not forget the golf! With the Pinehurst Resort and dozens of other world-class courses in close reach, Moore County is truly the home of American golf.\nIncludes Cities\/Towns: Aberdeen, Pinehurst, Southern Pines.\nMoore County's sandy soil has long been a powerful draw for horse enthusiasts. From steeplechases to dressage, there are all kinds of events to keep spectators busy. And for those not content to let others have all the fun, there are plenty of places to get out and gallop. A few of our larger attractions include the McLendon Hills Equestrian Center, and the 4,000-acre Walthour-Moss Foundation land preserve.\nAs the Home of American golf, you should expect Moore County to offer access to some incredible courses\u2014and that's definitely the case. Having hosted several US Open Golf Championships, most anyone with a set of clubs knows we're home to the world-famous Pinehurst Resort with eight outstanding options for all skill levels. But in addition, Moore County has 30 other incredible courses that keep residents guessing and visitors coming back.\nWith a name like the Sandhills, you might not expect to find many lakes, maybe not even any lakes. But as with many things, Moore County is a bit unexpected. For water lovers like fisherman, kayakers, wake boarders and sunbathers, there are dozens of lakes with an incredible range of shapes and sizes. Not all allow powerboats, but that's actually a really good thing!\nFrom visual and performing arts to theater and film, our area boasts a thriving arts community. The Arts Council of Moore County organizes events ranging from art galleries to musicals to jazz concerts by world-renowned musicians. Film connoisseurs frequent the Sunrise Theater, which showcases independent films. And for the musically inclined, check out the The Carolina Philharmonic, a local symphonic orchestra with a wide stylistic range that includes favorites like Bach and Beethoven. The Carolina Philharmonic is also a Carnegie Hall Link Up partner in bringing music education to 3-5th grade students. Multiple venues for live music provide evenings for music lovers of all types - bluegrass, jazz, folk, you name it from listening rooms to large auditoriums.\nAmtrak station located in Southern Pines.\nThe healthcare industry in Moore County is not only big \u2014 it's the area's largest employer\u2014 it also rates incredibly well for quality of care. A primary reason for such a dynamic healthcare infrastructure is the region's affluent retiree population. But retirees alone aren't enough to draw top-notch physicians to a community, and have them stay. Quality of life and access to first-class educational opportunities play a major role in the recruitment and retention of healthcare professionals.\nFirstHealth is the Carolinas regional system serving 15 counties in North and South Carolina that includes four hospitals with a total of 661 beds. With the system, Moore Regional Hospital is the flagship institution. Located in Pinehurst, Moore Regional is home to 395 beds and offers access to a full range of advanced healthcare services.\nJan 30.9 F\u00b0 40.3 F\u00b0 49.7 F\u00b0 4.6 \"\nFeb 33.1 F\u00b0 43.6 F\u00b0 54.1 F\u00b0 3.6 \"\nMar 40.3 F\u00b0 51.5 F\u00b0 62.7 F\u00b0 4.6 \"\nApr 48.3 F\u00b0 60.2 F\u00b0 72.1 F\u00b0 3.2 \"\nMay 57.0 F\u00b0 68.0 F\u00b0 79.0 F\u00b0 3.5 \"\nJun 65.1 F\u00b0 75.5 F\u00b0 85.9 F\u00b0 4.2 \"\nJul 69.3 F\u00b0 79.3 F\u00b0 89.2 F\u00b0 5.1 \"\nAug 67.9 F\u00b0 77.7 F\u00b0 87.4 F\u00b0 4.4 \"\nSep 62.0 F\u00b0 72.0 F\u00b0 82.1 F\u00b0 4.3 \"\nOct 49.7 F\u00b0 60.9 F\u00b0 72.0 F\u00b0 3.8 \"\nNov 41.8 F\u00b0 52.1 F\u00b0 62.4 F\u00b0 3.3 \"\nDec 33.9 F\u00b0 43.4 F\u00b0 52.9 F\u00b0 3.3 \"\nSo the question becomes; what means so much to you that you're willing to give freely of your time, energy and expertise?","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"All samba schools that will perform on the Sambadrome in Carnival 2018 have already chosen their hymns and released the official recordings. That means Carnival season has already arrived. Another hint is the reopening of the Terreir\u00e3o do Samba project, a venue that will be really important for those who want a Carnival warm-up since the samba schools won't be rehearsing at the Sambadrome for the next parades.\nTerreir\u00e3o is located at a historic site. The renovations that created the grand Presidente Vargas Avenue destroyed Pra\u00e7a Onze, a place with symbolic importance for Rio's carnival. It was at Pra\u00e7a Onze in 1932 where the very first samba school contest were organized by a newspaper. That was the place were poor people used to spend Carnival while the wealthier population used to go to events at the fancier Avenida Rio Branco, built with a B\u00e9lle \u00c9poque inspiration.\nAfter Pra\u00e7a Onze was torn down in the beginning of the 1940's samba schools events were taken to the new avenue. But the site never was lost in memory, being constantly remembered in many themes as the birthplace of Samba Schools.\nTerreir\u00e3o project was conceived in 1991. The grand opening was on February 7th of that year. The objective was to preserve the memory of old composers, give space to the new ones and also serve as an alternative for those who couldn't afford tickets to watch the samba schools main show at the Sambadrome.\nThis year's season already received famous samba singers like Jorge Arag\u00e3o, Dudu Nobre and Mart'Nalia. The event list is long including the Friday happy hour.\nOn December 2nd, Samba National Day, the activity will be intense with Lecy Brand\u00e3o, many other samba bands and also the old composers of Salgueiro samba school, one of the most traditional in Rio. During the month the happy hour session will receive other samba stars.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Easy Medical Practice Funding\/Financing - Get Your Loan Today!\nHiring more staff to meet increased demand.\nPurchasing more equipment to help get more service.\nHelping your cash flow while waiting through repayment cycles.\nExpanding or remodeling your practice.\nAdvertisement to help increase your customer base.\nWhen you're looking to take on business capital, banks may be the first thing that comes to mind. This isn't a surprise, as many businesses do just that, but bank loans aren't an option for everyone. Banks are becoming wary of supporting small businesses. Many now require collateral, strong credit histories, as well as detailed business plans. If you're just starting out or have prior issues that have hurt your credit, getting funding will become a lot more difficult. Things aren't perfect if you do qualify, either. Banks take quite a bit of time before initial approval and release of funds. For many reasons, you may need funds quickly. Perhaps you have an influx of patients and need more staff on hand to accommodate them. Maybe an important piece of equipment has broken and you need to repair it quickly. Sometimes, reasons may be a bit more universal. For example, the business can be slow during a portion of the year, and you need extra cash flow to stay afloat. In all of these situations, waiting for a bank loan may take too long.\nHowever, there are alternative funding methods out there that can provide easy and quick medical practice funding for your business. At eBusiness Funding, we use merchant cash advances to help businesses that need operating capital but may have issues getting funding from banks. How do they work? First and foremost, the money you get is not based on your credit history, but the strength of your business currently. Read more about how we determine the amount of medical practice funding you can get.\nFill out our simple contact form if you're interested in learning more or getting started. Our approval rate is 95% for small businesses that meet our minimum standards. What are these standards? As long as your medical practice has operated for at least six months with over $10,000 in monthly revenue, your business is a strong candidate for our alternative business funding solutions.Don't take any more time while your business needs outside capital. Choose eBusiness Funding and apply for your merchant cash advance today!","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"CTFishTalk.com: Delicious Crappie - Connecticut Fishing Recipes ( CT Fishing Recipes ) - A Community Built for Connecticut Fisherman.\nA few months ago I made myself 4 crappie fillets and loved it. All fillets came from 2 slab 13\" crappie weighing over a pound. Here's how it went.\nThe crappie seemed to have a bit of a smokey flavor to them. This will have you keeping bigger crappie to bake and throwing smaller crappie that you'd fry back in the water.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"As a self-managed NDIS participant you probably value choice, flexibility and control when it comes to managing your NDIS plan.\nIf you are looking for a platform that offers choice, flexibility and control when it comes to choosing and managing a support worker or a team of support workers, look no further than Careseekers.\nCareseekers makes it easy to find, manage and pay support workers online and has been custom built for NDIS self managed participants. Follow these 5 steps to create a team of support workers ready to go!\nChoose your own team of support workers through Careseekers.\nA \"great support worker\" means something different to everyone. It may be someone who\u2026.\nCareseekers is a diverse support worker community, operating across Australia. You can search for workers; post your job and invite potential support workers to apply for your job or message them directly.\nOnce you have found a support worker\/s to engage you can fill in the online service agreement and award them your job. The service agreement allows you to state important information about the role including the rates, types of support, emergency contacts. You can also download the service agreement as needed. The service agreement on Careseekers is based on the NDIS service agreement template.\nWhen a support worker submits their invoice, you will be sent an email notification letting you know that you have an invoice to approve and pay (using your NDIS funds). Invoices remain on your dashboard giving you a record of all the support worker payments you have made.\nCreate a payment schedule that suits you!\nYou have the control to make the payments when it suits you and your support workers, you can request that workers submit the invoices daily, weekly or fortnightly.\nCareseekers is more than just a platform. Our customer support team is here to support participants create a team of support workers or can help in the case of urgent or unusual requests. Please get in touch with our team and you will be amazed by our ability to help you find the perfect support worker match \u2013 no matter what the requirement.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Royal Festival Hall is located in Central London and the address is Southbank Centre, Belvedere Road, SE1 8XX.\nFind full information on travel by public transport and parking.\nWhen graduates and guests book their tickets they are asked for access information. Please provide this during the booking process when asked. We can make appropriate arrangements for seating during the ceremony. If you would like to discuss access needs for any of your party please email us as we are on hand to make your day as enjoyable and as smooth as possible.\nFacilities at Royal Festival Hall are accessible to all and full information can be found on their website.\nRefreshments are available to purchase at numerous outlets throughout Royal Festival Hall.\nComplimentary drinks will be provided at the reception after the ceremonies.\nThere is a cloakroom situated on level 1 of Royal Festival Hall for graduates and guests to store their belongings in. There is a charge of \u00a31 per item.\nGraduates will not be permitted to bring any bags or coats into the auditorium with them. Please make sure you collect your belongings on the same day they are stored. Items left in the cloakroom are at the owner's risk and Royal Festival Hall cannot accept any responsibility for loss or damage.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Personal info and history about the Ibiza villa rental with breathtaking sea views.\nThen \"Wu Wei\" is the choice!\nWe are the proud owners of a fabulous Spanish villa \"Wu Wei\", situated at the wonderful island of Ibiza. For the name of the villa we got our inspiration in 'Taoism', it means 'action without action'. Whilst staying in our fantastic villa, you can just enjoy and relax, but you do not have to do anything for it.\nThe villa is an oasis of peace where you can escape the stress of daily life, to chillout with at the background an unbelievable land- and seascape with enchanting Ibiza sunsets and sunrises.\nThe Spanish villa is situated on a hill with extraordinary views to the Northwest of the Mediterrenean Sea, called the San Vicente bay. Cala San Vicente is in the Northeast of the island, at approx 25 min driving from Ibiza town and about 15 min to the nearest Ibiza golf course. The villa itself, is surrounded by a natural garden of more than 5,000m2 (approx 1.5 acres) and to the left and right there is a wildlife zone of 40,000m2 (10 acres). With only higher and lower on the mountain some neighbors.\nIt is just an oasis of peace and quiet.\nWe offer this Ibiza rental villa for many years and have received many positive reactions that you can read on Zoover.\nThe smiling faces on our page Happy People is more than what words can describe, a stay in our Spanish villa is just fun!\nDo you want to view before renting?\nIf you are in Spain or even closer on the island of Ibiza and you would like to see what we offer, you can always contact us for an appointment, but I hope you will understand that we have to respect the privacy of our clients (if the villa is occupied).\nThe address of our villa is not shown on our website for these particular reasons. We believe you also, would appreciate privacy and peace, when you would be renting the villa. Our clients always comment that the images on the website have given them a clear understanding of the different rooms and outdoor space, and feel very familiar with the surroundings when they arrive.\nPlease click on the images on the side of the 'window frame' to see the different parts of the villa.\nWe are on Facebook and have a profile 'Cala San Vicente' , you are welcome to join this fast growing group. There you can view more photographs, video's and popular hotspots of the Cala san Vicente area.\nThe Spanish authorities really do their uppermost to fight fraud, in the world of Tourism, and so they should. Since a few years it is illegal in the Balearic Islands to offer property for rent for a short period (1 to 3 weeks) WITHOUT the official rental licence.\nWe have a licence number VT-0272-E, issued by the Tourism Association of Ibiza. This number is registered with \"Tourism-Ibiza and Formentera\". And like in any hotel, there will be a Complaints Form of the Consulate for Tourism to hand, but we can guarantee that none of our clients have ever needed to use this Form.\nGuarantee that your payments are secure.\nThis Spanish villa had a full check-over and is adapted to the required standards by the Tourist Inspector and has obtained an inspection booklet for yearly check-ups. This villa is equipped to guarantee the safety for the tenants, especially children.\nEach room or open space has its own fire alarm which is, as the security system, connected to the central computer of the security company and the local police. We, as owners, will also be notified immediately by automatic telephonic message. The highly user-friendly security alarm keeps unwelcome intruders at bay. This way we try to guarantee you a problem and stress-free holiday.\nAvailable periods can be checked on Ebay.\nAny other questions will be answered with pleasure, by contacting us via the contact form on this site.\nWe try to accommodate you as much as possible, in your search to that special holiday on Ibiza.\nThis official complain document you will receive together with the keys of the Villa.\nWe certainly do not wish for you to experience holiday problems.\nbut you must have heard of very unpleasant situations like double bookings or villas\/houses which are so badly damaged that they are just not suitable anymore to rent out. This could be the case if there is a bush fire, like in Beniras in 2010, where several acres (with houses) were burned down.\nPhysical or material damage due to a villa\/house in bad state of repair, could also happen.\nWithout insurance your holiday could be ruined.\nAs official license holders, we are obliged to have an insurance for all eventualities.\nBecause pictures say more than a 1000 words ...Click on the pictures to take a look inside the villa.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Marking its 21st birthday, World Egg Day celebrates the health benefits of the humble egg\u2014which offers the highest quality of protein available. An important dietary component for fulfilling the nutritional requirements of people living in both developed and developing communities, the egg is a versatile food, able to be eaten on its own or as a necessary ingredient in many recipes. Eggs are essential for fetal development, healthy brain development, maintaining concentration, aiding the immune system, and more. Today, enjoy eggs your favorite way!\nWhen Sylvia goes out to the barn to collect the chicken's eggs, something's up with Trudy. She squawks and screeches when Sylvia tries to reach under her, but Sylvia takes it in stride and with humor: \"'Don't get so egg-cited!'\" she says \"'I'll get your egg tomorrow.'\" But the next day Sylvia is met with the same reaction. Trudy isn't acting like the other chickens in Grandma's backyard; if fact, she isn't even acting like Trudy! \"'What's up with this chicken?'\" Sylvia wonders.\nSylvia decides Trudy must be hungry and tries to lure her off her nest by offering chicken feed, but while all the other hens \"wolfed it down like chocolate,\" Trudy remains firmly on her roost. Sylvia tries everything she can think of to move Trudy, but nothing works. That night she and Grandma consult The Big Book About Chickens. Here they discover that \"'Trudy is broody!'\" Grandma reads on: \"'Broody hens stay on their eggs so they will hatch into chicks.'\" But Sylvia and Grandma know that Trudy's eggs are not the kind that hatch.\nTo view a gallery of artwork plus more books for readers of all ages by Peter J. Welling, visit his website!\nToday, I'm pleased to talk to Jane Sutton about her books, her journey as a writer, her family, and the joys and inspiration of being a new grandmother!\nAs someone who loves humor and was voted class comedienne in high school, what were some of the books you most enjoyed as a child and young adult?\nTwo of my childhood favorites were Dr. Seuss's Horton Hatches an Egg and Robert Louis Stevenson's A Child's Garden of Verses. As a young adult, I was drawn to Virginia Woolf novels (I know\u2013not exactly humorous, but most comedians are prone to depression).\nYou write both picture books and books for older children. What inspires or influences your stories?\nMy childhood memories have been the basis of many of my books\u2014experiences and feelings. What's Up with This Chicken? was inspired by a true story my friend-since-we-were-11 Fay told me about one of her backyard hens who refused to get off her eggs. I said, \"This has to be a children's book!\" So I invented characters, had the child protagonist solve the problem, and snuck in a subtle message about the importance of empathy.\nAs someone who always wanted to write and who achieved the goal of becoming a published author, can you briefly describe your journey?\nAt a young age, I was encouraged by my teachers. They would be impressed by something I wrote and send me to show it to another teacher, which really pumped me up. I was an editor of my high school newspaper and after graduating from college had a job writing for a newspaper, wrote ads and press releases, and sold some stories for reading comprehension tests. My first book, What Should a Hippo Wear? was published when I was 29. I've had periods where everything I wrote was selling, and periods when nothing I wrote was selling. It's a tough market!\nWhat's the best part about writing books for kids?\nWell, I never wanted to grow up, and when I realized it was happening whether I liked it or not, I vowed to always remember what it felt like to be a child. Writing for kids helps me do that.\nYou conduct school presentations and workshops for kids from kindergarten age through grade 5, can you describe a funny or poignant anecdote from one of your events?\nOne school had a wonderful program that paired parents and their children as writing partners. The culminating event was a presentation by me about how to make writing come alive. Then the parent-children pairs displayed their books, and it was so lovely to see how much the experience meant to the adults and the children. They also loved showing me\u2014the big famous author\u2014what they'd written and I could sincerely point out parts of their writing that were especially effective. The whole thing made me ferklempt.\nI read that you love elephants and collect elephant-inspired items. Can you tell me about one of your favorites?\nOne of my favorite elephants is gray and plastic (about the size of a toaster) and lives outside, in view of our kitchen window. My mother, who died in 2004, gave him to me and named him \"Sabu.\" He has survived about 18 New England winters so far, sometimes getting totally buried by snow and then poking out the tip of his trunk as the snow starts to melt. My mom asked me once if the elephant made me think of her, and yes, he surely does.\nTalking about your mom makes me think of the strong relationship between Sylvia and her grandmother in What's Up with This Chicken. Can you tell me about your own family?\nMy husband, Alan, taught a variety of grade levels spanning grades 1-6. He served as a science coordinator, curriculum developer, and teacher mentor. He's written and co-authored six books for educators, four focused on science instruction and two about systems thinking. Currently, he coordinates the systems thinking program at a grades 5-12 public school and also presents workshops at meetings and conferences. We met in college and have been married for 41 years!\nMy son, Charlie, works for a coalition made up of organizations pushing for a better transportation system in Massachusetts. He's worked in Massachusetts public policy since graduating from college in 2007. He married the wonderful Amberly, a nurse, in 2014, and they recently had a baby!\nMy daughter Becky is the director of an SAT tutoring program. Her company tries to make SAT tutoring as fun and effective as possible, so they try to match the kids with tutors who have the right personality for the student's learning style. The SAT has completely changed in the last year, so they have had to retrain all of their tutors and rewrite all their curricula. For any tutoring program, building students' confidence is key. So much of standardized testing is psychological.\nI understand Becky also writes a blog for all of us grammarians who like a laugh once in awhile called Apostrophe Catastrophes: The Worlds' Worst. Punctuation. Can you tell me a little about how she got started?\nYou're a new grandmother! Can you tell me a little about your grandson?\nThat's a dangerous question! How much time do you have? Caleb is an adorable, cuddly little person. And he's now a month old!\nAs a new mother, I was very anxious and questioned all my decisions. But as a grandma, there's none of that anxiety. Being with my grandson is pure joy. And seeing how loving, confident, and tender my son and daughter-in-law are with the baby fills me even more love. And seeing him in my husband's arms as he gazes down at him\u2026I'm getting ferklempt again!\nHave you thought about how being a grandmother might influence or inspire your future work?\nMy next book, a Passover-themed picture book, is scheduled for publication by Kar-Ben in the spring of 2018.\nI love Mother's Day and Father's Day because they are occasions for my husband and me to get together with our children. And there's good eating involved.\nThe first time I hosted Thanksgiving, rather than attending one at my parents' house, my mother gave me very, very specific instructions about what to buy. For example, there had to be 2 Butterball frozen turkeys, both between 11 and 13 pounds. The reason for 2 is so that each of the 4 grandchildren could have a drumstick. My daughter and I were rummaging through the supermarket frozen case trying to find the exact acceptable weight for exacting Grandma. Our fingers were half frozen, and I admit that I kind of dropped a turkey on my daughter's finger. The turkeys, by the way, were quite delicious.\nHow has a holiday influenced your work?\nThe festive, joyous celebration of Chanukah shows up in my 2 Chanukah picture books: Esther's Hanukkah Disaster (Kar-Ben) and Aiden's Magical Hanukkah (Hallmark).\nThanks so much for chatting, Jane! It's been wonderful getting to know you. I wish you all the best with all of your books, and am looking forward to seeing your next book!\nConnect with Jane Sutton on her website and catch up with her events and other fun activities on her blog!","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"Larceny, stealing and theft all refer to the act of taking something from someone without their consent and with the intention to permanently deprive them of that something.\nThe crime is not made out until the offender has taken away the item from the owner.\nStealing from a ship in distress or wreck.\nAn offence of Larceny is what is known as a Table 1 offence under the relevant legislation, which means it is to be dealt with in the Local Court unless an election is made for trial on indictment by the Department of Public Prosecutions (DPP).\nThe nature and elements of the offence of Larceny were considered in Ilich v R (1986) 162 CLR 110 wherein it was held a person who is lawfully in possession of something cannot be guilty of larceny of it.\nwhosoever commits larceny \u2026 shall, except in the cases hereinafter otherwise provided for, be liable to imprisonment for five years.\nSince Larceny is a criminal offence, the burden of proof lies on the Prosecution.\nIf you are charged with the offence of Larceny what are your options?\nNational Criminal Lawyers have been successful in defending a number of Larceny charges where the prosecution could not establish each of the elements of Larceny. We have also achieved a number of non-convictions for Larceny charges.\nLarceny can be punished with a prison sentence of up to 2 years (if heard in the Local Court) or up to 5 years (If heard in the District Court).\nCriminal Lawyers have defended thousands of people charged with Larceny and are experts at these hearings.\nFor Larceny (over $5,000 and up to $15,000) 35% of offenders received full time imprisonment, whereas 31% received a good behaviour bond and 9% were fined.\nIn NSW, a court can impose any of the following penalties for a Larceny charge.\nThe alleged victim had her mobile phone stolen at a shopping mall food court. Our client was identified from the CCTV footage. The court found that the prosecution had not made out their case and the larceny charge was dismissed.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Wheen using this website, some information may be collected automatically using 'cookies'. These are small text files that facilitate the processing of your data and enable us to analyse how the website is being used. Cookies can be temporary or permanent.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"While we could recognize just how important our health and wellness is and we could understand just how difficult it could be to control it with all the many some other worries about our heads, it becomes safer to say in which health monitoring does not work properly. After almost all, we're just planning to become sick even as age, so you'll find nothing we are capable of doing about that, right? Though this process does demand less work folks as an individual, it also can change the results of our health and wellness, causing people illnesses becoming a sort regarding self-fulfilling prophesy. You think you don't need to monitor your quality of life, so you never, and you obtain sick.\nYour well being is something changes on a regular basis, even in case you are not alert to it. Every time you do one thing to or along with your body, you might be changing just how it works, both today and in the foreseeable future. As you be sure decisions in your lifetime, you can impact your quality of life, increasing or perhaps decreasing just how your physique feels and also behaves. In order to make sure that you are certainly not doing problems for your physique, you should monitor your quality of life on an even more regular schedule. Whether you determine to simply tune in to your physique or to employ a health overseeing system, you must make sure you understand your physique now in order to see any time changes take place.\nEven one of the most terrible diseases began small. If someone could actually spot these kinds of problems just before they started to be larger, chances are usually good they might have had the oppertunity to do something about them. When you've got an proven health overseeing system, you'll be able to pay attention to your physique now so that you can compare that to upcoming days. Once you see in which things are usually 'off, ' it is possible to head with a doctor or it is possible to change yourself to see when you can turn your system around. Most of the time, even an awful disease may be halted inside its tracks if it is found eventually.\nThere are a variety of well being monitoring systems designed for everyday visitors to use just like Health Picture, Stress Keep an eye on, Biological Age group Test\u2026 Some applications are an easy task to purchase also to use because the day movements forward. Because of this, you can register with your quality of life every evening, or once or twice a evening, to notice where your quality of life levels are usually. In in this way, you will make changes immediately to modify your well being monitoring final results.\nNo matter everything you decide, your quality of life is continually changing. It's your responsibility to do something about it once you notice the particular changes, so it could help to attempt to notice these. With Well being Snapshot, you can start to change the method that you look your health. As opposed to simply like a person which waits to have sick, you usually takes charge today preventing illness to start with by basically monitoring your quality of life.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"No matter what part of Canada you're in, there are a plethora of amazing natural, scenic areas to explore. Hiking and camping areas are abundant here, as a true Canadian appreciates nature and all it has to offer the mind, body, and soul. Whether you're from Canada or just Canadian at heart, check out some of the most beautiful areas to hike and explore nature. Here are some of our favorite picks from all around the country.\nBanff National Park is Canada's oldest national Park and the second oldest in all of North America. Located in the Rocky Mountains, Banff is a winter wonderland when it comes to beautiful glaciers and snow-capped mountain views. This park is a great place for viewing wildlife, such as bighorn sheep and elk. Even in summer, the temperature at Banff can drop fairly low, so pack your warmed gear for hiking in subarctic and arctic climates. After your hike or before you begin your day, relax at the famous hot springs flowing out of Sulphur Mountain, open year-round.\nThe North Shore Mountains in Vancouver have a great selection of hiking trails, such as the popular Dog Mountain trail. Hiking at Dog Mountain, which will give you one of the most beautiful views of the Vancouver area. Make your trip more memorable and visit Mount Seymour, where Dog Mountain begins, and turn your hiking journey into an overnight adventure by getting an RV rental in Vancouver and camping out for the night.\nThe Bruce Peninsula is one of the most popular destinations in Ontario for hiking, camping, and fishing. The Bruce Peninsula itself offers Fathom Five National Marine Park (a marine conservation area) as well as Bruce Peninsula National Park. The latter is not only a phenomenal hiking destination, but a great place to birdwatch and observe other wildlife, such as red foxes and snowshoe hares. The highlight of this destination is the grotto, a scenic cave along the shoreline filled with a pool of the bluest water you'll see in all of Ontario. It's a must-see on every visit to the peninsula.\nCape Breton offers options for hikers who are beginners as well as expert hikers. The Skyline trail is one of the most popular and is accessible for hikers of all abilities. It boasts a breathtaking view of the coast, so stay for a while and enjoy the sights. You may even see a whale in the Gulf of St. Lawrence! This may not be the best hike for children, as the cliffs are steep and could be potentially dangerous if not careful.\nAlbion Falls, located in the city of Hamilton, offers gorgeous self-guided trails with a stunning view of their waterfalls. This is a great park for novice hikers, as it is not incredibly difficult. A new law in effect forbids you from hiking down to the falls for safety purposes, but this doesn't mean you can't enjoy the sight from afar.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"After a whitening appointment, dentists can provide patients with a maintenance program that effectively removes staining daily. The goal is to help the patient stay continuously within the realm of a bright, natural smile. Many \"natural\" maintenance options circulating on the Internet may catch a patient's eye. Dentists should understand the science behind these products and objectively identify which products work or do not work. This article reviews various whitening products, such as activated charcoal and blue covarine, with its optic blue illusion instant whitening effect, and examines how to strengthen teeth and maintain healthy smiles through solutions such as nano-hydroxyapatite and fluoride.\nEvaluate the science behind natural teeth whitening products and staining mechanisms.\nAssess whitening and strengthening options, such as blue covarine and nano-hydroxyapatite.\nDescribe best practices for limiting abrasiveness and enamel damage from tooth-brushing products.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"Q: How do you pronounce the name of the Foundation?\nQ: What are the basic requirements to be eligible for a grant?\nThe Foundation makes grants only to organizations that are tax-exempt under Section 501(c)(3) of the Internal Revenue Code. The Foundation does not make grants to individuals. The Foundation's primary geographic focus in grantmaking is the Chicago, Minneapolis, and Washington, DC. With the exception of archaeological-related projects, we do not provide funding outside these geographic areas.\nQ: When are decisions made?\nThe Board of Directors meets twice a year to make funding decisions, once in the autumn and once in the spring. The Board occasionally will defer a final decision on a request to a subsequent Board meeting even if an application has passed initial review; this delay is intended to ensure a thoughtful, informed decision.\nQ: Why can't I access the online application system to submit a new funding inquiry?\nThe online application system is accessible for limited periods of time. Deadlines will be stated in a text box on the Welcome Page.\nQ: What type of support does the Foundation provide?\nThe Foundation supports direct service efforts, as well as advocacy at the local, state, and \/or federal levels. The Foundation provides grants for (1) general operating support, (2) new projects, and (3) some capital campaigns (limited).\nQ: Why can't I find guidelines that fit my area of work?\nWe believe that the Foundation can be most effective by concentrating its resources in a few programmatic and\/or geographic areas. Inevitably this means we cannot support many worthy projects outside our program interests.\nQ: Can I apply for a job, internship, or volunteer work at the Luther I. Replogle Foundation?\nThe Foundation operates with a staff of one. We appreciate the interest, but the Foundation does not accept resumes; does not offer volunteer or internship opportunities; and does not maintain employment or internship information about our grantees or other organizations.\nQ: Is the Luther I. Replogle Foundation operated by the Replogle Globe Company?\nNo. Although Mr. Replogle was the founder of the Replogle Globe Company, the Foundation is independent. The Replogle Globe Company is currently owned and operated by the Herff Jones Company.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Shalu shukla is a passionate dancer and keep on enhancing his skills. Here he has prepared a dance cover of famous song KAAM 25 by DIVINE. In DREAMIL we are helping hidden talents to show their art to the world.\nPlease comment and tell us about the artist and where to improve. It will help Shalu Shukla to go to above level.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Topic: Join us today with our favorite bespeckled guest, Phil \"Just like it Sounds\" Gerbyshak \u2013 and the first return guest to our show. In this episode, we will discuss the importance of effective communication and we will discuss the impact of Social Media in all its forms on ourselves and our society; the good, the bad, the ugly \u2013 what works, what doesn't work, and why. You will find no \"fake news\" in this episode.\nPhil's first visit can be found here.\nAbout Phil: I am a former VP of information technology at a full service financial services company, where my team and I were responsible for supporting over 500 applications used by advisers, investment bankers, research analysts and support staff. I also worked as a director of social strategy for a compliance software vendor focused on the financial services vertical, where I coached, consulted with and trained customers and prospects how to integrate social media and social selling into their daily sales practices. Currently I do programs, training, and coach people how to build their business by maximizing their use of technology and social media. I have written 5 books, more than 2,500 articles, and been quoted in the Wall Street Journal, Daily Globe and Mail, Forbes, Fortune, Financial Times, Investor's Business Daily, Inc. and many other publications. I live in Tampa, Florida where I spend my free time reading business books, listening to podcasts, and playing pinball. I have been on the cover of and writer for Speaker Magazine for the NSA \u2013 not the ones that listen, the ones who talk \u2013 a lot! I currently I have a network of approximately 3 million on social media.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Merlot Verdelho Residences offer a luxurious, indulgent and romantic accommodation experience in the historic town of Penola South Australia \u2013 gateway to the Coonawarra wine region.\nMerlot Verdelho Residences are ideal for couples seeking a private, peaceful and romantic getaway. And with the added benefit of a discrete interconnecting passageway, both Residences can be opened up for group bookings of up to twelve guests.\nMerlot Verdelho Residences are ideally located in Penola town centre. Walk to restaurants, shops and caf\u00e9s. Coonawarra is renowned as one of one of Australia's finest wine regions and is particularly known for producing world class red wines.\nWhen you book directly with us, your reservation provides us with more of the resources necessary to deliver you the perfect getaway. This is why we guarantee you will always find the best and fairest prices on our website, or by calling\/emailing us directly.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Colombia Farc Peace Agreement | UU.\nLuis Moreno Ocampo, the Argentine former chief of the International Criminal Court, says the accord signed on Wednesday by Juan Manuel Santos, Colombia's president, is \"like a work by Van Gogh that only gained public recognition after he died\".\nOn September 26 2016, Uncommon Union accompanied Luis Moreno Ocampo to Colombia to witness the signing of the peace agreement between the FARC guerrillas and the government. This historic event followed our previous work on March 2016 where Uncommon Union travelled to Cuba with Ocampo to help document and manage communications during his negotiations between the FARC and the Colombian government. Building on this work, we continued to document the historic proceedings, manage communications, and conduct outreach for Mr. Ocampo who has spent over a decade advising leadership on both sides of the conflict.\nLeading up to and during the signing, Uncommon Union facilitated press relations with global and regional outlets. This was an exceptional and unique opportunity for Mr. Ocampo to explain his years of work, address the Colombian people directly, and express his support for the agreement; which was voted on in a public referendum only 5 days later.\nNoticias: Luis Moreno Ocampo celebra que la Uni\u00f3n Europea suspenda los efectos de estar en la lista de organizaciones a la guerrilla de las Farc por los pr\u00f3ximos seis meses.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzawvoi b/data_all_eng_slimpj/shuffled/split2/finalzzzawvoi
new file mode 100644
index 0000000000000000000000000000000000000000..58f2d91cf79141f42c13d0cc930e69053f5b1450
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+{"text":"I've been sitting here for about an hour, trying to figure out what to write. I've started this post four or five times now, and each time, I've deleted the entire draft. I finally figured out why I was having so much trouble.\nI began drafting another post while in the waiting room at my OB's office a few days ago. I'm hoping to share that post later this week, so I guess I didn't save anything of substance for this post.\nRather than worrying about it, why don't we get right to this week's menu plan?\nAs you can see, even this week's menu plan is light on content. I'll be celebrating my birthday on Thursday, so Caroline and I are [likely] going to treat ourselves to a special dinner out, just the two of us. Then we'll be spending time with family, so no cooking on Friday or Saturday night either.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"In 2012, the first issue of a new international quarterly ECONTECHMOD \u2013 an international quarterly journal on economics of technology and modelling processes was issued jointly by the Polish Academy of Sciences Branch in Lublin, University of Engineering and Economics in Rzesz\u00f3w and the Lviv Polytechnic National University. The journal, edited in English, is devoted to the economics of technology and process modeling.\nprof. Juri Bobalo \u2013 rector of the Lviv Polytechnic National University.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The global animal healthcare market is observing significant growth due to increasing adoption of pet animals and growing global consumption of meat and milk. The increasing number of health concern of animals has led to an increase in global healthcare expenses in farms. The animal farms are increasing at a rapid pace in global scenario due to increasing demand for protein-rich foods such as eggs, meat and milk. To meet the ever-increasing demand of these products it leads to problematic farming practices for instance, pushing animals into small spaces which enable the contagious diseases to spread rapidly. The farmers are increasingly adopting feed additives and administering vaccines to get rid of such harmful diseases.\nAdditionally, searching for innovative solutions for animals' health have become an important component for getting rid with the disadvantage of modern industrial farming and growing diseases among animals. Thus, in turn, increase the growth of the animal healthcare market.\nThe growth of generic and OTC drugs for animals are the key trends observed in the growth of global animal healthcare market. The declining research and development productivity is leading to the growth of animal pharmaceutical product portfolio. The low productivity of research and development increases competition among the key manufacturers.\nThe pets carry various bacteria, viruses, parasites and fungi that cause infections once transmitted in the human body. It has been suggested by many veterans that the people need to get the pets with desired vaccinations in order to prevent them and their families from any animal-borne diseases. According to a recent survey, it was observed that the global animal healthcare market is much greater than the expenditure incurred on human medicines. The companies are increasingly developing and improvising vaccines, parasiticides, anti-infective, drugs for pain, cancers and sedation for pet and farm animals.\nBased on the product, the animal healthcare market is categorized into feed additives, vaccines, parasiticides, anti-infectives, anti-inflammatories, others. Feed additives is further segmented into antibiotics, vitamins, amino acids, enzymes, antioxidants, prebiotics & probiotics, minerals. Parasiticides is further segmented into ectoparasiticides, endoparasiticides, endectocides. Of all the product categories, feed additives held the largest share in the animal healthcare market owing to the increasing demand of animal-derived products and increasing animal feed production, worldwide.\nOn the basis of animal type, the animal healthcare market is categorized into farm animals, companion animals. Of all animal type categories, farm animal held the larger share in the global animal healthcare market owing to the increase in the production of the farm animals to meet the demand of animal-based products.\nFarm animals is further segmented into fish, swine, poultry, ruminants. Companion animals is further segmented into canine, feline, equine. Among all farm animals, ruminants held the largest revenue share in the market due to the ever-growing demand of the meat and dairy products worldwide.\nCompanion animals is further segmented into canine, feline, equine. Among all companion animals, the canine category held the largest revenue share in the market owing to the increasing expenditure on animal health and wellness.\nBase on the distribution channel, the animal healthcare market is categorized into veterinary clinics, veterinary hospitals, pharmacies and drug stores, online and retail stores. Among all, veterinary hospitals held the largest share in the global animal healthcare market owing to the growing preference of pet owners for veterinary hospitals as these hospitals offered complete health check-up at one place and also saves the time of pet owners, which is also one of the major factors for opting these hospitals for their pets. This, in turn, increases the market growth of this segment in comparison to all other facilities.\nThe company is increasingly investing in the animal health industry as the treatments can be used for both the commercial and companion animal markets and is driving the growth of global animal healthcare market. The development of animal health vaccines requires less investment as compared to developing the drugs and therapeutics for humans. The animal health products do not involve spending large amount of investments as on human trials and involve greater longevity and less competition. The development of new animal diseases in a frequent manner is creating opportunities for the animal healthcare market to grow globally.\nThe adoption of western and changing lifestyles in the developing countries is also driving the growth of the animal healthcare market. The countries in the Asia-Pacific region such as China and India have increased the adoption of pet animals and this supports the growth of the global animal healthcare market. The ownership of pets is considered to be a status symbol in these developing countries and pets such as cats and dogs are adopted by the people for therapeutic and psychological benefits in the country.\nThe increasing awareness among the people for using vaccination for pets is creating opportunities for the global animal healthcare market.\nThe huge investments incurred for the manufacturing of animal healthcare drugs are the major restraint observed in the growth of global animal healthcare market. Animal healthcare are used for treating and preventing infectious diseases. The vaccines and pharmaceuticals are used for keeping animals healthy. The vaccinations that are available are required for treating diseases and infections that can be harmful for the humans.\nGlobally, North America was the largest animal healthcare market as there is a large number of established players in the market that focus on promoting investment opportunities in the animal healthcare industry and these pharmaceutical companies are based out in U.S. The increasing trend amongst the consumers for pet is driving the demand for North American animal healthcare market.\nAsia-Pacific is expected to witness the fastest growth in the global animal healthcare market. Australia is one of the countries in this region which has the highest rate of pets. The increasing awareness related to livestock-related diseases is the key reason due to which the demand for animal healthcare is growing in this region.\nSome of the major players operating in the global animal healthcare industry are Zoetis Inc., Merck & Co. Inc., Bayer AG, Virbac Group, Ceva Sant\u00e9 Animale, C.H. Boehringer Sohn AG & Co. KG, Elanco Animal Health, Phirbo Animal Health, Kyoritsu Seiyaku, Dechra Pharmaceuticals, China Animal Husbandry Industry Company, Nippon Zenyaku Kogyo, Hipra, Jinyu Group, Ouro Fino Saude Animal, and Vetoquinol S.A.\nZoetis Inc., Merck & Co. Inc.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"To claim \"MOXSH OVERSEAS EDUCON PRIVATE LIMITED\", please review and complete the checklist below.\nYour Email DOES NOT match \"Company contact Email\" <moksh.germany@gmail.com> as in MCA records.\nYou CANNOT claim \"MOXSH OVERSEAS EDUCON PRIVATE LIMITED\" yet. Please review failed steps in the above checklist, rectify them by following the instructions provided. Refresh this page after rectifying all the failed steps to claim this Company.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Here is an excellent opportunity for a turnkey business. This 41+\/- acre property includes a massive sand pit bearing A3 sand, the type in demand for current road improvements on the nearby Starke\/Bladwin bypass as well as for use in other building projects and septic systems. Plus, a 2004 Hitachi EX 200, an LC 5 Trackhoe and an EZ Screen 1500 XL for screening sand are included with the land. Road agreements and a current customer list are available for review by prospective buyers. You could also consider rezoning this property from its current Timber status, giving you a massive tract of land to develop, or build your own private estate paradise.\nListing courtesy of Re\/Max Connects.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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@@ -0,0 +1,5 @@
+{"text":"During these joyful holidays we write to you, our Christian neighbors all over the world, in order to thank you truly for the beautiful and gracious responses that we Muslims have been receiving from the very first day we issued our invitation to come together to 'A Common Word' based on 'Love of God and love of neighbor' (see http:\/\/www.acommonword.com for the document and the responses).\nWe thank you and wish you all a joyous and peaceful Christmas Holiday Season commemorating the birth of Jesus Christ \uf075.\nWe pray, during these blessed days, which happen to coincide with the Muslim feast of the Hajj or Pilgrimage, commemorating the faith of the Prophet Abraham \uf075, that the New Year may bring healing and peace to our suffering world. God's refusal to let Abraham \uf075 sacrifice his son\u2014granting him instead a ram\u2014is to this day a Divine warrant and a most powerful social lesson for all the followers of the Abrahamic faiths, to ever do their utmost to save, uphold and treasure every single human life and especially the lives of every single child. Indeed, it is worthy of note that this year Muslim scholars issued a historical declaration affirming the sanctity of human life\u2014of every human life\u2014as an essential and foundational teaching in Islam upon which all Muslim scholars are in unanimous agreement (see details at: http:\/\/www.duaatalislam.com ).\nMay the coming year be one in which the sanctity and dignity of human life is upheld by all. May it be a year of humble repentance before God, and mutual forgiveness within and between communities. Praise be to God, the Lord of the worlds..","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"With almost 20 years experience in aesthetic and cosmetic skin rejuvenation, and daily \"on the ground\" experience with our team at SkinAssist Clinic in Denver we've seen a lot come and go. Here's what we're seeing for 2018 as we relax and take a breath during the summer, and look forward to a busy fall season.\nMicroneedling appears to have now become the most popular service in non-surgical skin rejuvenation ever. We get more calls for microneedling these days than for any other service. And with recent FDA clearance of a microneedling device imported from overseas, the field of competitors is set to explode as numerous players file applications for their own FDA clearances. We use The Made in America Procell Therapies Microchanneling and Microdermabrasion Multi-Tool, and have found nothing better when it comes to outcomes, comfort, safety, downtime, pain and side effects, (lots of the former and none of the latter)! And from what we see on a daily basis, there has never been a more effective tool for anti-aging than Procell's microchannel delivery of cytokine based topicals. And with the stimulative effect of microchanneling and collagen induction we see real corrective results as well!\nLED Light Therapy is powering the next generation of regenreative skincare products and immuno-stimulative treatments. With the advent of cytokine based topicals that stimulate biologic response, and all the other things we do to use the inflammatory response to rejuvenate, (meaning microderm, chemical peels, microneedling, RF, LASER & IPL), LED has become an essential modality to drive the healing response required to see the best results. We've seen recovery time go from hours and days, to minutes and hours since we added POLY Red Light Therapy for hair regrowth and skin rejuvenation, Blue Light for Acne and Infrared for pain. And the new hand held POLY Go LED pods are a wonderful retail product that helps our clients take the healing home!\nMicrodermabrasion continues to delight clients with an effective and stimulating mechanical exfoliation. Impersonators come and go, but our DiamondTome has produced more happy clients over the past decade than any other device we have in the office. And ours has returned more than 30 Times our initial investment in that time! And with microderm continuing to appear in the top 5 most requested services (according to the annual ASAPS survey) our DiamondTome has become one of our most reliable sources of joy and revenue. It's both a great way to introduce clients to aesthetic skin rejuvenation, and it keeps them coming back with the satisfaction and stimulation they feel with a monthly or quarterly treatment for years or even decades!\nMicroblading is fast becoming the industry standard for enhancing eyebrows. We recently started offering it at SkinAssist, and the ability to produce a natural textured look, combined with versatile shaping of the brow mean more satisfaction for the client, and more fun for the practitioner! We're seeing all ages for uneven eyebrows in terms of shape and thickness. And we're able to help clients who have experienced a loss of brow due to simple pre-mature aging, injury, medical treatment, and even chemotherapy.\nRF Microneedling continues to develop an impressive body of work for our clientele. We use the INTRAcel device to power TOTAL FACIAL RF, which works in conjunction with POLY LED and cytokine based topicals to produce faster results in our older clients. The ability to do what used to require a LASER and a week or two of downtime with just one day of mild redness and swelling has truly revolutionized what we can do for deep wrinkles and laxity.\nPDO Threads have come into their own as we have discovered just where and when they work best. Forget the \"used car lot\" promise of a non-surgical face lift, but keep in mind the ability to achieve significant facial volumization across broad areas for far less money than traditional fillers, reduce sagging and crepiness in the neck, and redefine sagging jowels. Our NovaThreads practice continues to grow. Our clients continue to see beautiful results, and we continue to help other practitioners learn the tricks and pearls that make PDO Thread practice successful.\nDermaplaning has become our \"go to\" service for that same day\/next day need to look fabulous. Removal of peach fuzz and a gentle exfoliation are combined at our shop with LED Red Light Therapy and a cytokine\/hyaluronic acid based topical moisturizer to give clients the immediate glow they want for that big event!\nFeel free to comment below and let us know what your seeing trend in 2018! Contact Us or call (800) 551-5582 to learn more about the services we offer at SkinAssist, or the services and technologies for which we offer devices, technologies and the most comprehensive training and certification.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Paragon Software Group has released Microsoft NTFS for Mac 17 by Paragon Software (beta). With the new 17th edition, you can back up the entire macOS system to an external drive formatted under Windows. You can write, edit, copy, move and delete files on Microsoft NTFS volumes directly from your Mac. Users can mount, unmount, verify, format or set any of your Microsoft NTFS volumes as a startup drive. The software is targeted towards Mac home users, forensic experts, IT managers and original equipment manufacturers.\nMicrosoft NTFS for Mac 17 by Paragon Software is part of File System Link (FSL) technology that solves incompatibility issues between file systems and operating systems. Paragon's FSL technology provides rapid, transparent, full read and write access to non-native file systems such as exFAT, NTFS, FAT32, HFS+, APFS, ReFS under Windows, Mac, Linux, iOS, Android, QNX, and other platforms. Paragon File System Link provides a valuable enhancement for OEMs, benefiting external storage support end users. For more information on Paragon's File System Link offerings, visit https:\/\/www.paragon-software.com\/technologies\/file-system-link-business\/.\nThis entry was posted on April 4, 2019 at 7:41 am and is filed under Commentary with tags Paragon.\tYou can follow any responses to this entry through the RSS 2.0 feed. You can leave a response, or trackback from your own site.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"1. Access and use of this site is provided by the Derby RSPCA. References in these Terms to \"we\" or \"us\" or \"our\" refer to the Derby RSPCA.\n2. The Derby RSPCA reserves the right to change these Terms at any time. Please check these Terms on a regular basis for changes. Your continued use of the site after changes are posted will mean you accept the Terms as modified by the posted changes.\n5. The material on this site is provided for general information only and is not intended as specific advice. If you have any concerns about your pet (or an animal in your charge)'s health you should always consult your local vet. Whilst we try to ensure that the information on the site is accurate, we can accept no responsibility for the accuracy of the site content. Users who rely on the information in the site do so at their own risk. This site is made available for public viewing solely on the basis that the Derby RSPCA excludes to the extent lawfully permitted all liability whatsoever for any loss or damage arising out of use of this site, or reliance upon its contents.\nPlease note that we do not control external websites. We are not responsible for the contents of external websites or the availability of linked pages. Inclusion of a link to an external website does not imply any endorsement by or affiliation with the RSPCA of that site, its content, any product or service its operators may provide.\n11. These Terms and use of this site shall be governed by English law. All claims or disputes arising out of or in connection with this site shall be subject to the exclusive jurisdiction of the courts of England. However, the Derby RSPCA shall have the right to commence and prosecute any legal or equitable action or proceeding before any court of competent jurisdiction to obtain injunctive and other relief in respect of the infringement of intellectual property rights.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Around the Spiral with Edward Lifegem: Wizard101: Nightmare Before Christmas Tree Winners!\nWizard101: Nightmare Before Christmas Tree Winners!\nAll combinations of Wizard101 (a treant like creature) Christmas (Christmas lights) and Halloween (a spooky theme and if you look closely the tree actually looks alive). I love this image and I'm glad you submitted your wonderful graphic to me. I hope you have a great Halloween (and Christmas when it arrives)! :) I'll send your 5,000 Crowns & W101 Vampire and Mummy Outfit codes to your email soon.\nAnother Halloween Christmas Tree by Charles. This wonderful piece of artwork combines Wizard101 (The Awesome Wizard hat that thre tree is wearing), Halloween (Do I see candy corn? :O) and Christmas (the candycane wand) together. Thank you for submitting your art for my contest. I hope you have a nice and safe Halloween next week! :) I'll send your 5,000 Crowns & W101 Vampire and Mummy Outfit codes to your email soon!\nGabriel did an amazing job including everything that I wanted in this contest. Wizard101 ( I love that the tree topper is a W101 logo) Halloween (Look at all of the Halloween ornaments, amazing!) and Christmas (the presents under the tree and the bow on the side of the tree). You did an amazing job, thank you for submitting your wonderful artwork. :) I'll send your 5,000 Crowns to your email soon!\nHere's another Halloween Christmas tree from Tabitha that also combines Wizard101 (our Wizard101 teachers as ornaments, that's actually a pretty cool idea- KI please sell them), Halloween (I love that you have ghost and pumpkin ornaments and the pumpkins in the front) and Christmas (Christmas lights and Christmas presents plus Felix Navidad). Thank you for your amazing tree, you'll recieve your 5,000 crowns through email soon!","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"Captain Marvel Wallpaper Tumblr, Download this Images for free in high resolution. Captain Marvel Wallpaper Tumblr was posted in March 3, 2019 at 12:03 pm and This Captain Marvel Wallpaper Tumblr Images has viewed by 17 users. If you wanna have it as yours, please click full size and you will go to page download in full size, so you just choose the size above the images that you want in \"Download\", Click it and download the Captain Marvel Wallpaper Tumblr Images.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Rhino specializes in short and medium run digital color printing with outstanding quality.\nWire binding, coil binding, perfect binding, scoring, folding, die cutting, round corners and tabs, or ask us about your specialty project!\n8-Color up to 41\u2033 wide by any length. Perfect for trade show presentations, posters, and signs.\nWholesale greeting cards, so you can create your own Etsy or Online Greeting Card Shop!\nNeed an Estimate? We are happy to send you a quote for all your printing needs!\nReceived your Estimate? Ready to Order? Easily upload and\/or send us your files.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Walt disney Coloring Pages - Abu. Walt disney Coloring Page of Abu from \"Aladdin\" (1992). HD Wallpaper and background images in the personajes de walt disney club tagged: walt disney characters walt disney coloring pages aladdin abu pruefever.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Distinguish upper motor neuron and lower motor neuron pathology on physical exam.\nClassify neuropathies as mono vs polyneuropathy, radiculopathy, motor vs sensory, or mononeuropathy multiplex.\nHave a differential and conduct a diagnostic workup for each neuropathy syndrome.\nKnow how to workup carpal tunnel syndrome, and recognize indications for surgery.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"NEW MILFORD, N.J. -- The parents of a New Milford man killed in last month's lower Manhattan terror attack filed a notice of intent Tuesday to sue the City of New York, among others, for \"creating and maintaining a hazardous condition\" on the pedestrian and bike path along the West Side Highway.\nDarren Drake, 32, was among those mowed down on the path by a rented truck driven by as Sayfullo Saipov of Paterson on Oct. 31 because the city and its related agencies failed to take the necessary steps to protect them, Barbara and James Drake contend in their notice of a wrongful-death claim.\n\"The manner in which the claim arose was that [the] vehicle was permitted, allowed and able to enter the bike path,\" says the notice, filed with the city Comptroller's Office and the Hudson River Park Trust, both in lower Manhattan, by attorney Rosemarie Arnold of Fort Lee.\n\"The public reasonably believed\u2026and relied on the fact that [the path] would be free from vehicular traffic,\" the notice says.\nAuthorities should have taken into account \"the recent stream of terrorists using vehicles as weapons,\" the \"reasonably foreseeable probability of people using cars as weapons,\" and prepared for \"foreseeable\" terrorism in downtown Manhattan, the notice filed by Drake's parents says.\nThey and their son's estate are seeking an unspecified amount of damages that include, but aren't limited to, the \"extensive trauma and significant disfigurement\" he sustained, the funeral, burial and other associated expenses and the \"loss of [his] services and society.\"","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"On October 2, 2018 Reputation Institute kicked off its annual Global Reputation Summit in Amsterdam. This member-exclusive event was made up of the Reputation Leaders Network \u2014 a knowledge-sharing, peer-to-peer network for senior executives working to advance the practice of Reputation Intelligence.\nIn addition to workshops, client and expert presentations on topics like CEO Activism, Engaging Stakeholder Dialogues, Integrating Reputation as a KPI, RI Founder Dr. Charles J. Fombrun was awarded a Lifetime Achievement in Reputation.\nCharles accepted this award in front of colleagues, peers, and clients at an event marking tremendous growth at Reputation Institute at a time when the global economy is increasingly reputation-focused.\nThis thoughtful speech was written and presented by Charles' colleague Dr. Cees van Riel. It is a testament to the historical impact of corporate reputation and its growing importance for the future of business.\nIt was exactly 22 years ago today that we met at Stern Business School in New York. At the time, I had read a number of your articles, particularly your AMJ piece entitled \"What is in a Name.\" In that article you had combined reputation data with media and financial data. Your fact-based conclusion was that a strong reputation is evoked by two key points: money spent on corporate messaging and the amount and nature of media coverage about the organization in question.\nSerendipity brought us together, or I should say more modestly, enabled me to get to know you at an important crossroads in our careers. You were on the brink of finishing the last details of your Harvard Business Book on Reputation and I had just published Principles of Corporate Communication with Prentice Hall. The aim of my visit was to persuade you to create a new academic journal together with me. The CORPORATE REPUTATION REVIEW became a reality already one year later. This was launched at the same time as the annual Reputation Institute Conference that we organized between 1997 and 2014.\nEverybody knows you from the research you have done on reputation. However, as of the early eighties your research was focused around three interrelated topics: human resources, consultancy and, of course, reputation.\nAfter your bachelor study in physics, Wharton professor Noel Trichy inspired you to understand more about the political structure of organizations. This resulted in various academic articles and your first book called Strategic Human Resource Management, published in 1984. In that year you moved from Wharton to Stern Business School where you remained until 2004. Many renowned professors at US business schools were fortunate to have you as their PhD coach. To mention a few: Violina Rindova, Naomi Gardberg, and Michael Barnett. At least five articles \u2014 co-authored with them \u2014 and all focusing on reputation, were the result of this investment.\nReputation became the key topic of interest in your research throughout the nineties. However, you did more. One of the most famous courses at Stern during your tenure was an elective course in how to build and manage a consultancy firm. This resulted in another book which you published in 2003 called The Advice Business.\nAfter you had left Stern, you took a sabbatical and bought an apartment in Paris. Our cooperation continued and resulted in the Fame & Fortune book that provided proof points about the financial impact of having a sound reputation \u2013 something you already had referred to in the AMJ article in 1990. The Fame & Fortune book became a best-seller that could be found at airport bookstalls. We traveled around the world to evangelize the reputation management religion. You actively promoted the idea at an early stage that fame is rooted in what you say. And that what you say has to be 100% embedded in what you do. That basic notion still is the core reason why companies embrace the necessity to safeguard their reputation. Happily enough, you convinced managers and many others that reputation management is not only a defensive strategy aimed at preventing a crisis. On the contrary, a positive reputation brings prosperity not only to an individual organization but also to society at large. The more highly reputed companies we have in society, the better it is for employees, investors, regulators and NGOs \u2014 and for society as a whole.\nIn stimulating this core idea, you built the Reputation Institute. The Reputation Institute did not start with the aim of becoming a successful business with activities in 40 countries around the globe. The objective to start with this platform was much more modest and can be summarized as coming from two driving forces. First, the intention to build a new discipline with the help of both top academics in Business Schools and top practitioners. Second, the desire to have reputation data on a global scale that is academically valid and practically useful, inspiring academic research and helping practitioners to increase the performance of their firms.\n\"Great things are never done by one person. They are done by a group of people\" is the opening quote in the introduction of the latest version of your reputation book. That is no doubt the case. However, it is clear to me \u2014 and to all the people with whom you have collaborated over the past decades \u2014 that you are the key pioneer in developing initiatives that resulted in a new measurement instrument applied on a global scale, resulting in a database that consists of millions and millions of data revealing an important mechanism in the public discourse around companies. It was also you who started as early as the late nineties with an initiative to add media data to the perception data we were already gathering with RepTrak. It must be great for you to see that as of today the Reputation Institute is now mining both perception and media data to provide a more profound understanding about why companies have a high or a low reputation.\nCharles, this is just a short overview of all the contributions you have given to us in the academic world, in the business world, and of course at the Reputation Institute in creating a world vision around reputation management. Today, we would like to say publicly, \"THANK YOU, CHARLES.\" Thank you for your innovative ideas, perseverance, and occasional stubbornness. But above all for the world-class THOUGHT LEADERSHIP that you have shown around your favorite topic, REPUTATION.\nIt is an honor and a privilege to present you this sculpture today as a symbol of recognition and appreciation for everything you have done in the last three decades through your thought leadership in reputation management. Congratulations, and thank you for everything.\nIf you'd like to learn more about Dr. Charles J. Fombrun and his research, we invite you to take a look at the recently republished 20th-Anniversary edition of his book Reputation: Realizing Value from the Corporate Image.\nConnect with us anytime to learn more about the Reputation Leader's Network or how your company scores on reputation.\nWhat's on the Minds of Corporate Communications Execs?\nDiscover what CCOs around the world care most about \u2014 their challenges and opportunities and how corporate reputation impacts their roles and the companies they represent.\nReputation must be a KPI.\nUse industry to your advantage.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Hopefully here are the answers to some of the questions you may have. If you have any other questions please don't hesitate to contact a member of our team. We would be happy to help.\nHow old can you be to join?\nYou can join in our martial arts classes from the age of 6. There is no limit to how old you can be. Our system is designed for everyone.\nThere is no joining fee for either our MMA fitness, or our martial arts classes, although if you wish to do the martial arts there is a yearly membership and insurance cost. This is \u00a320.00 for U16's and \u00a325.00 for adults.You do not need to pay this right away and have 3 lessons to decide if it's for you.\nWhat do I need to train in?\nFor both classes you can just wear shorts\/ tracksuit, which ever you feel more comfortable in and a t-shirt. For the martial arts you can purchase a training suit (a Gi) from us but this is not essential right away.\nWill I get hurt or injured?\nAs with any physical activity there is always a chance to get hurt but safety is our number one priority. Since being founded in 2011 we have had no major injuries. Students are put through warm up exercises to prevent minor injuries such as pulled muscles. Any light sparring takes place with gloves and full head protection.\nAbove you can find our Blue coat venue. Check the map for location and directions.\nAbove you can find our Walton venue. Check the map for location and directions.\nWe aim to respond to you as quick as possible, please contact us using one of the methods below.\nStop by at one of our classes and talk to one of our qualified instructors. We want to see you soon!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Painkiller addiction: Is it really an epidemic?\nOxyContin Addiction: Sales Of Painkillers Soar; How Many Overdose A Year?\nBy IBTimes Staff Reporter: Subscribe to IBTimes's RSS feed Sales of popular prescription painkillers have increased dramatically nationwide, leading experts to believe a new addiction epidemic is taking over the United States.\nBy Noelle Hunter - Staff Writer The Morehead News SelfRefind, currently located at 224 E. Main Street, offers medication-assisted treatment (MAT) that helps addicts transition from addiction to recovery, according to the company's website.\nALCOHOLICS Anonymous has been slammed for its decision to serve booze at its national conference gala dinner in Melbourne this weekend. Victorian members have spoken out in disgust that alcohol will be for sale tomorrow night.\nPainkiller Addiction Rapidly Spreads Across the U.S.\nSales of popular prescription painkillers in the US have shown dramatic rises in new parts of the country, the Associated Press reports, and experts believe the push to relieve patients' suffering is producing an addiction epidemic.\nBrandi Tomko has been arrested for impersonating a veterinarian in order to fuel her heroineaddiction. A Summit County Ohio woman has been arrested and is facing a 33-count criminal indictment for pretending to be a veterinarian.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"This latte was not a planned recipe; I just threw the ingredients together and then realized that the early morning sun was too perfect to be true - and too perfect not to be used for a photo.\nSo I went out and took a photo of my beautiful mug - a present from my even more beautiful friend Mel from Virginia - without even knowing if the liquid in it would be worth the shot.\nIt was. I'm not going to say that this is the best latte that ever saw the morning dew, but I definitely liked it and I'm definitely sure that it's worth to share.\nSince I ran into a box of coconut sugar about two weeks ago, I have started to experiment with it. Not because I'm totally convinced that it really is better than any other sugar which is 80% sucrose, but because I really like the light caramel flavour of it.\nIf you want to read my musings about coconut sugar, you can do so on my German blog (I think Google might translate it).\nHeat milk with cinnamon, clove and mace or nutmeg.\nStir coconut sugar into coffee.\nRemove cinnamon stick and clove from milk and either froth or steam milk, if it's a kind of milk that can be frothed or steamed, or just combine coffee and hot milk.\nHave you tried coconut sugar yet?","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Riven: The Sequel to Myst has just popped up on the iOS App Store. Now, I don't want to blow anyone's mind or anything, but there's one very important detail the game's name fails to make clear: this is the sequel to Myst. Pretty crazy, right?\nExcellent news, adventure fans! An iPad-optimized version of Riven: The Sequel to Myst has just popped up on the iOS App Store. Now, I don't want to blow anyone's mind or anything, but there's one very important detail the game's name fails to make clear: this is the sequel to Myst. Pretty crazy, right?\nMyst probably holds a special place in a lot of your hearts. Or maybe you just remember it as a stupid-hard adventure game about an island covered with obtuse puzzles. Neither would surprise me, really, and I fall somewhere in the middle. I first played the game on the Sega Saturn, and while I enjoyed my time with it, I never managed to actually finish it.\nRiven begins directly where Myst ends, and it invites players to solve a series of puzzles across five different islands. Think you have what it takes to handle that level of islandry? Is \"islandry\" even a word? I can answer the second question (no, it's not), but only you can answer the first.\nAnd while we're on the topic, I have to confess that I've never actually had the chance to play Riven. I have every intention of fixing that now, though. What about everyone else?","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"Nothing changed but the fence.\nIt's clear to see that this homeowner had set out to create a beautiful outdoor living space for themselves. New patio, steps, landscaping, beautiful pool, the works! There was just one thing left to add. The fence!\nThe \"Before\" photo here was sent to us to show the difference a Grand Illusions Vinyl WoodBond install can make in a yard.\nWe were stunned to see the huge difference a fence can make when it's properly chosen to match the surrounding decor. The warmth of the wood tone was the perfect compliment to the earth-tone of the patio and the cool blue of the pool. The color and realistic wood grain of the fence drew all the pieces of the puzzle together to create the outdoor masterpiece in the \"After\" photo.\nHow did you hear about your Grand Illusions product?\nThe contracter had it displayed in their office.\nHow did you choose your particular color\/woodgrain for your property? Were you matching any home accents?\nIt was pretty consistent with the furniture in our home and kitchen. The uniqueness of the product is what really sold us. It's just something you don't see every day. We also love the color for our own personal tastes. This may sound funny, but I'm actually working with an architect to start updating the front of our house. We're going to base the look of it on the fence.\nWe're incredibly satisfied. I have strangers and neighbors walking up all the time. I've had people walk up and knock on it. People stopping to say how beautiful it is. Before we did all of this yard work, we didn't have anything in the back yard. Just grass and an old wood deck with a stockade fence. Now with that fence, we feel like we're not in New York. We feel like we went away some where on vacation. It provides such beauty and privacy.\nWe absolutely love it. I could literally sit back here and look at my fence all day. It really came together very nicely. We put a ton of time and effort into the whole backyard design and it was beautiful. But until the fence went up, the yard just wasn't complete. The fence completed the whole look we were looking for.\nI certainly would and I do. I recommend it to everybody.\nPeople stopping their car and touching my fence. It's the funniest thing. They'll just stop dead in the street and get out of their cars to see if it's wood or not. It says a lot about the product.\nWhite fences are very sterile to me. The Grand Illusions Mahogany has a natural look and feel. You're getting the best of a man made structure that will stand the test of time and a visually natural looking product. I always wanted the look of a wood fence, just without the maintenance. I now have a wood fence that will stand the test of time.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Our Father who art in Heaven is connected to the Third Eye and the Pituitary . This energy center is in our forehead and head and is a healing center. It is the site of the golden bowl what can be tipped filling us full with the rising of our attention to the highest in us.\nHoly is Thy Name is in the center in the brain and is connected with the Pineal Gland and the highest chakra.The name of God is seated deep in the center of the brain, a pearl of great price. It is always there, never is lost. Name of God given, imprinted in everyone. We are all the same through this Holy Name. Faithful and True part of us.\nThy Kingdom Come, Thy Will be done on earth as it is in Heaven \u2013 Throat Chakra, 5th also. Bend the head to bend the proud and stiff necked in me to let a Kingdom come of God. There are those, under the Altar, who say \"How long oh Lord, holy and true must we remain under the Altar.\" To hold back the bent and stiff necked and let out those clothed in white raiment.\nGive us this day, our daily bread, the chakra of that built into the floor of the pelvic basin, connected to the sacred bones of the Sacrum. We have been given white light here. How will I color it? Is it for me or for others? The earth of us. A place to grow a Tree of Life, Fruits twelve, one for every month, for healing what ever comes up, Leaves for the healing of the Nations in us. So what are we thinking and what are we growing in the sacred place.\nForgive us our debts, as we forgive our debtors. First Chakra. The debt we owe to God for ignoring the path of God for us, in us. So asking for that forgiveness as well as forgiving others. In the Solar Plexus, allowing that infinity to embrace both these forgivenesses.\nLead me not into temptation, the seat of the emotions in us. Let me not be tempted to keep the energy of the Sacrum, that white light, for selfishness. Less and Less of me and more and More of God. Give it back up to the highest, watered for growth, for filling the overflowing golden bowl, coming down again to grow that Tree of Life. Choice of moving from this to that.\nBut Deliver us from the evil in our own Hearts. The Heart is the place of Christ, Love and compassion and warmth. Holding that basket. Which gives us the morning star, which shines out of the growing Tree of Life. That bursts in our chest like a great heart shaped balloon when we work it for Love. Or the choice for evil, what is wicked, envy and jealousy and all the intensity of storm and twist. It is all ours, for self versus for others. Who will we let lean on us as we are on the journey, who will we love and help.\nFor Thine is the Kingdom, the Power and the Glory.The three higher pull the four lower together, for the Holy Spirit, for the Water of Life, For the Tree of Life to be in us.\nSaw in the night that this whole question of light is important.We live by the light of the sun. In New Grange, there is that place where the light shines into a chamber. There used to be a golden bowl in there and when the light hit it, it looked like golden light flowed out of the bowl. Light travels. So too can the light given to us in the first and lowest chakra. If we arrange ourselves correctly, or if we are arranged as in New Grange, the light can travel from the lowest to the highest in us and back down again. Waiting for the light to blaze up.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"my dog asleep can do this advert but this video is up side down.\nCadbury Dairy Milk, Chocolatey, Alien, Martian, Chocolate, Space ship, Dancing, Party, Interstellar, Interstellar Party.\nAt Coldwell Banker, our calling is to help people find a place to call home, and if love is what makes a house a home, this campaign highlights how even the .\nPETA's dog milk advert has been making its rounds on social media, and it's pretty clear why. Are you disgusted by it? Why not cows milk too?\nWelcome to Cadbury's official YouTube channel. Packed with cinematic treats and snackable video content, it's the perfect place to free the joy!\nTitle: Cadbury Dairy Milk Medley HP: New Cadbury Dairy Milk Medley is a deeply luxurious and indulgent experience that tastes like how this blissful dog feels\".","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Enzo Crotti announced on the web that is now available the new blog on the subject of classical guitar, with articles, videos, tab, lessons and music for classical guitar.\nSeptember 2014 \u2013 The Italian guitarist Enzo Crotti has a blog for fans of classical guitar. On this blog you'll find new and classic pieces for classical guitar, video, lessons, ensemble music, and other useful and pleasant news about music for classical guitar. There will also be free music and mp3 of classical guitar. A perfect place for those who love music for classical guitar.\nSure, you'll see that on this site you will find many interesting songs to listen to and to play!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"If you are in the market to buy or sell a home, you have come to the right spot. Whatever your real estate needs, I can help you reach your goals with confidence. While visiting my website, you can access the most current real estate data and search for homes throughout the Western U.S. Then, when the time is right, feel free to contact me so that I can put the power of Windermere to work for you.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzazxrl b/data_all_eng_slimpj/shuffled/split2/finalzzzazxrl
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@@ -0,0 +1,5 @@
+{"text":"Explore access to water with this simple game.\n1) Have a discussion with your young people about the importance of water in their everyday lives and ask them to imagine what it would be like not to have access to clean, running water. Talk about World Water Day, an international day of awareness that encourages people to learn about how the lack of water impacts people's lives, and how to reduce water use and wastage.\n2) Divide the young people into pairs and give each pair a piece of paper, some tape, two straws and one cup with a little water in it. Alternatively, if making your own wax paper, distribute normal paper and crayons instead, asking the young people to completely colour one side of the paper using the crayons, to make it waterproof.\n3) Within their pairs, ask each young person to take turns using their straw to put a droplet of water at one end of a piece of wax paper that is taped to a flat surface, like a table or the floor.\n4) They should then take turns using their straws to try and blow the droplet from one end of the paper to the other, without it breaking up into smaller droplets. Discuss with the young people that this symbolises the long journeys some people have to make carrying water to their homes and families, without spilling it on the way.\n5) The bigger the droplet, the more difficult it is to keep intact. Ask the young people to experiment using different strengths of breath and holding the straws at different angles to find the most efficient way of transporting the water from one end to the other.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Found it interesting that I saw how many members there were in dip in my alliance by a %.\nThis gives you a good idea on what is happening.\nHave I mooned you today?\nThere is a couple new add-ons to the war declaration system. One of them is that there is now a pulldown box where you can set (or choose not to) a percentage of members in diplo mode that equals an automatic loss of the war. (Choices are 25, 50, 75 and disabled). You can also choose to have inactives count, or not.\nThat's pretty cool then. I don't declare war anyways, so I was not aware of that feature.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Feasting at the Palace is an experience worthy of crossing the seas for, with sumptuous cuisine served in elegant environments fit for royalty. Explore our curated array of epicurean restaurants, where flavours from European banquets mingle with ethereal Eastern flair, to create delectable dishes served with a luxurious touch that is unique to the Palace. Discover some of the best restaurants in Dubai here at Emerald Palace Kempinski.\nEnjoy sophistication like never before at Emerald Palace Kempinski Dubai's bars and lounges. Offering some of the best bars in Dubai, sip on a refreshing beverage courtesy of our resident mixologists as you catch up with loved ones, celebrate a special occasion or simply soak up the sun as you relax by the pool.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Federal Board of Intermediate and Secondary Education Islamabad has the mandate to hold the Inter class annual exams in the attached colleges located in Islamabad city and in all cantonment areas of the country. FBISE Islamabad announces the HSSC annual exams result 2018 during the month of July or August. Federal Board starts checking of the papers of FA\/FSc-I after finishing of the papers. FBISE Islamabad assigns the task of checking of the papers to the senior teachers of the Board and prepares the Intermediate class result 2018 within three months.\nIslamabad Board manages the HSSC-I annual exams for the students, who want to appear in the 11th class annual exams and pass their papers while FBISE Islamabad prepares the 12th class result 2018 at the same time but the names of the position holder students are announced one day before the announcement of the FBISE Islamabad Inter-II result 2018. After the announcement of the Islamabad Board FA\/FSc class result 2018, the result of Intermediate class in uploaded on the site with complete information for the facility of the students.\nYou can check BISE Islamabad Board Inter result 2018 online at www.result.pk here online.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"8 drawer dresser white ikea bedroom furniture dressers stores in dubai mall.\nwhite desk for teenage girl teen girls desktop room furniture dubai karama des.\nlarge size of teen lounge chairs home remodel inspiration and designing fancy in amazing small house furniture uae abu dhabi pin by on things room chair.\nsectional sofa bed to buy black furniture with ottoman sleeper ikea used abu dhabi so.\nboys room decor baby furniture sets and boy ideas image desk for online philippines.\nteen sleeper sofa chair couch bed futon dorm lounge home ideas easy furniture dubai cheap small in quartz blue homemade fat.\ninterior design idea essentials for kids homework stations share the space desk furniture stores in dubai.\nkids classroom desk desks for rooms furniture online chennai.\ncool lounge chairs bedroom magnificent brilliant modern chair seating for teen furniture abu dhabi dubizzle.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbacry b/data_all_eng_slimpj/shuffled/split2/finalzzzbacry
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index 0000000000000000000000000000000000000000..55c18a2cc1252bc9ccbd6ce125fa34b1277a9b14
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@@ -0,0 +1,5 @@
+{"text":"Fire up your engines and compete in new Bikini Bottom challenges with SpongeBob's pet, Gary. This is no snail's pace race! Build spelling skills on 4 wacky new tracks! You must have the SpongeBob SquarePants Clam Prix game to play with this Expansion Pack.\nImportant: This game expansion pack requires the SpongeBob SquarePants Clam Prix game.\n\u00a9 2011 Viacom International Inc. All rights reserved. Created by Stephen Hillenburg.\nFire up your engines and compete in new Bikini Bottom challenges with SpongeBob's loyal pet, Gary.\nBuild spelling skills on four wacky new tracks . No time for snail's pace\u2014sound out letters and get in the race!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"At Cajigas & Co. we advise and guide on the implementation and development of your daily basic business operations in Panama.\nWe can accompany you from the beginning of your business by forming a company for your personal security and commercial facility, obtaining an operation notice to start local activities, guiding you with all interchanges the business may confront with providers or local authorities.\nLikewise, we provide first or second legal opinions with respect to contracts, transactions, investments or general inquiries that commonly occur in a wide range of commercial activities. We have advised people in a diverse range of national and international companies focused on issues such as real estate, agriculture, technology, commerce and banking.\nThe due legal attention will guarantee the trust and continuity of your business.\nPanama has an economy based on services,a solid and competitive financial center and the second largest free zone in the world, in addition more than 6% of world trade is done through the Panama Canal.\nIt is the center of operations for Latin America and the Caribbean of many of the major companies of the S & P 500, because it has proven to be a global logistics hub with an enviable political and social stability for a country in the region.\nPanamanian legislation seeks to attract and promote investments, create jobs and transfer knowledge and technology, with with the purpose of being a competitive country taking full advantage of its geographical position, physical infrastructure and international services.\nOur team is formed by lawyers who are experts in international legal issues and commercial law who advise a diverse group of foreign and local clients, as well as companies of different categories that have or plan to have a representation in Panama.\nBy knowing your needs we are sure that in Cajigas & Co we can be your legal support for your business operations in Panama.\nDo you need a expert on Business Operations in Panama?","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Rating: 5 4 3 2 1 4.4\/5 from 9366 votes.\nC 24 Ne Service Manuals can be very useful guide, and c 24 ne service manuals play an important role in your products. The problem is that once you have gotten your nifty new product, the c 24 ne service manuals gets a brief glance, maybe a once over, but it often tends to get discarded or lost with the original packaging.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"This kitchen remodeling, while modern in its planning, incorporates traditional English detailing such as the mantel-style range recess. A display shelf over the cooking area houses treasured serving pieces, and an appliance garage near the sink conceals the electric teapot for daily afternoon tea. The adjacent family room includes large windows to provide sweeping views of the home's generous grounds. This space also provides access to outdoor dining and recreation spaces and offers a comfortable place to relax. The project also includes a spacious and functional mudroom area for the family's four boys and a light-filled home office. Hidden in the center of the layout are a small half-bath and a pantry. The remodeling bridges the style of the new with the elegance and dignity of the original 1950's brick home. The remodel consists of 1,290 sq ft.\nWe love the window bench in the family room, where lots of guests are able to sit during gatherings.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"To her intimate friends she was known as 'Pearl', to her enthralled public she was Janis Joplin, the greatest female singer in the history of rock and roll.\nCatch Pearl \u2013 the Janis Joplin Story at The Basement, The Arts Centre Gold Coast on June 27, 2015.\nPearl - The Janis Joplin Story, delivers in words and music her tumultuous outlook on life, love and being the first lady of rock who got a little dirty!\nFeaturing, Liza Ohlback one of Australia's premier Jazz and Blues singers. Including hits such as 'Cry Cry Baby','Move Over', 'Piece of My Heart', 'Bobby McGee' and more.\nAlso featuring the influences of greats such as Bessie Smith, Billie Holiday, Bob Dylan & Leonard Cohen. 'Pearl' transports the audience into a smokey, jazz blues club feel\u2026 mixed with a good dose of pure acid rock and bohemian vintage freedom!\nIf you're planning to watch the show, you'll be glad to know that our Self Contained Apartments Gold Coast are just a few minutes away from The Arts Centre Gold Coast.\nWe offer a selection of studio apartments, one bedroom and two bedroom apartments. Our studio apartments can accommodate a maximum of three guests. Perfect for couples and small families, the stylish and cosy studio apartments feature basic kitchen facilities, full laundry facilities, one bathroom, two single beds, cable television, a private balcony and more. We also have one bedroom apartments and two bedroom apartments. The one bedroom apartments can accommodate three guests as well while the two bedroom apartments are suitable for five guests. Whether you are going on a family holiday, a business trip, or a holiday with your mates, we have apartments to satisfy your needs.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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@@ -0,0 +1,5 @@
+{"text":"What kind of support will I get from Camber Outdoors to help my company act on the CEO Outdoor Equity Pledge?\nCamber Outdoors has convened an online, invite-only forum to help member companies activate on their commitment to equitable leadership and inclusion within the CEO Outdoor Equity Pledge. This online forum is a space for one delegate or representative from each member company that has signed the CEO Outdoor Equity Pledge to access curated resources and guidance, ask questions, and participate in dialogue and discussion surrounding best practices, challenges, and successes.\nAre there resources available to help guide us in activation of the CEO Outdoor Equity Pledge?\nYes, Camber Outdoors has adapted guidance on best and promising practices for fostering greater diversity and inclusion within companies situated in the active-outdoor industries into a Handbook. The first module of this Handbook is focused on Executive Commitment and Accountability and curates the evidence behind these promising practices, and guidance for action steps your company can take today, this week, and this month.\nWhat happens if the leadership at my company changes? Should our new Executive\/CEO sign onto the CEO Outdoor Equity Pledge?\nYes. One of the keys to ensuring that commitments to diversity and inclusion within your company continue to progress, is executive commitment. As such, it's important for your company's new Executive\/CEO to establish their visible commitment by re-signing the CEO Outdoor Equity Pledge on behalf of your company.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Design, commercialization and promotion of tourist products.\nMethods for the analysis of tourism and the influence of new technologies on consumption habits.\nPlanning of tourist destinations and sustainable development.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"I graduated from South Gate High School 3 months after my 17th birthday in 1970. I met my future husband that summer, who was also in my graduating class. I had purchased my class ring and had my name engraved, but he never got one. He wanted to wear mine after we married in 1971 and I let him. We went to the beach one day, he went in the water with it and came out without it. I've always wondered if someone finally found it buried under the sand. I hope so!\nHere are two pictures of me with my then-future husband, Eddie. If you look closely, my forever-lost 1970 class ring is on his baby finger on his right hand.\nWhat a sweet & sad memory. I'm so glad you at least have these lovely photos of your lost ring & of Eddie & you--I love how candid the photos are & you are such a beauty! I lost my black specs in the ocean in Delaware a few years back--I'm hoping it's good luck to lose something to the sea.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"What Do You Think YOUR Beloved is Googling?\nI just read in The Week magazine that about a thousand times a month, the question \"How do I hide a body\" is Googled; About 1,900 times a month people ask on Google, \"How do I get away with murder?\" and a whopping 40, 500 times a month, someone is Googling, \"Why did I get married?\"\nWhat do you think YOUR loved one might be Googling?!?","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"A quick view of the most influential metrics in Pearland.\nPearland is a city in the U.S. state of Texas, within the Houston\u2013The Woodlands\u2013Sugar Land metropolitan area. Most of the city is in Brazoria County, with portions extending into Fort Bend and Harris counties. As of the 2010 census, the city's population was 91,252, up from a population of 37,640 at the 2000 census. Pearland's population growth rate from 2000 to 2010 was 142 percent, which ranked Pearland as the 15th-fastest-growing city in the U.S. during that time period, compared to other cities with a population of 10,000 or greater in 2000. Pearland is the third-largest city in the Houston MSA, and from 2000 to 2010, ranked as the fastest-growing city in the Houston MSA and the second-fastest-growing city in Texas. As of 2015 the population had risen to an estimated 108,821.\nView our selection of available properties in the Pearland area.\nCheck out the latest demographic information available for Pearland.\nGet to know Pearland better with our featured images and videos.\nLearn more about schools near Pearland complete with ratings and contact information.\nBrowse through the top rated businesses in the most popular categories Pearland has to offer.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbdixq b/data_all_eng_slimpj/shuffled/split2/finalzzzbdixq
new file mode 100644
index 0000000000000000000000000000000000000000..31fab0651a946bc9d45ff6152ece48acf54d8723
--- /dev/null
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@@ -0,0 +1,5 @@
+{"text":"Whether you're trying to blackout an entertainment room from the sunshine outside or just provide a little privacy for a bathroom window, custom fit window treatments are a must. Window Imagery specialise in custom made, from roman-shades to wooden blinds and roller-shades to mini-blinds and EVERYTHING in between, we have them all. We even carry specialized wooden shutters and decorative hardware so that your windows have a truly unique look and feel.\nWe work close with our suppliers and in shop workroom to give you exactly what you need. Our team of professional designers and installation crews will custom measure and install your items for you. No job to big or to small, we've got the tools and knowledge. Meet with one of our designers and pick from a wide variety of custom-made shades or blinds today and we'll have them ordered and installed in no time!\nWe work with popular and trusted brands such as; Charles Parsons, Wilson Fabrics, Maurice Kain, Slender Morris, Ken Bimbler, Warwick, Zepel Fabrics, Gummerson Fabrics \u2013 just to name a few. Take the hassle out of the job and let us do all the leg work for you.\nThank you so much for the fantastic curtains and blinds fitted. They are incredible and surpassed all our expectation. Thoroughly recommend the service and quality of the curtains.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"So we've finally come to the end of one of the most painful animes I've watched, and I sadly don't mean that in a good way. If you haven't guessed already I'll be talking about the anime, where and how things went wrong as well as a few other thoughts I've had.\nSo to be blunt Devil Survivor 2: The Animation if bad. It's the worst kind of bad too. The kind where it wasn't just mess it was the fact that there was so much wasted potential. It was directed by the same person who did the Persona 4: The Animation, so you would think that it was in good hands, but in every way that Persona 4 translated well into an anime, Devil Survivor 2 fails miserably.\nNow one of the main issues, and I'll say this outright, is that this series simply didn't have enough time. It had 13 episodes, each about 24 minutes long, you can do the math and that sure doesn't add up to a whole lot of time to tell such an expansive story. And Devil Survivor 2 is expansive, being about 30 hours in length. So right out of the gate The Animation falls right at the start with two broken kneecaps, but it's all about how they tell the story that matters right? Well they managed to make an absolute mess of that too.\nAnother problem is that The Animation can't seem to decide on a tone. It seems to be going for incredibly serious most of the time, but it randomly decides to pull a complete 180 at the worst possible times to inject comic relief, which is also one of the worst continuous I've seen in a while. Daichi being weak and having weak demons is pulled out a minimum of 3 times, each time fitting in as well as a square peg into a circular hole. This wouldn't be such an issue if The Animation didn't try to pull off such a bleak and hopeless atmosphere, which it does anyway, regardless of it fits or not.\nDevil Survivor 2 is extremely character driven. While there certainly is a a plot outside of that, the real story is told through the struggles of a wide variety of people trying to survive in some rather extreme situations. The Animation successfully captures essentially none of that. Characters are introduced, then either discarded for entire episodes or simply killed off. Any characterization that does occur is almost always comically late and lacking in any detail that makes the characters more relatable. Devil Survivor 2 has some of the best character arcs I've seen in a game. You have characters like Joe, a carefree slacker who's easy lifestyle eventually costs him his girlfriends life. Entire arcs like this are converted to brief 5 second cameos that haven't a hope at managing to capture what makes the characters interesting.\nPerhaps the thing about the characters is the personality prescribed to Hibiki, the main character. In the game you simply made his choices and choose what he would say, but that obviously wouldn't work in an anime so of course they would have to define his character. It's a shame that they pretty much choose the most boring, clich\u00e9d and overused personality the could think of. There is absolutely nothing interesting about his character, which is made all the more irritating with how it goes about showcasing it.\nThen there's his relationship with the main antagonist, Yamato. He worked well in the game because he wasn't all powerful, while he did have his own goes, in the end he was just trying to survive like everyone else. That's all sadly stripped away and he is instead made into some sort of omnipotent demi-god that only Hibiki can oppose. Yamato's relationship with the Anguished One is also a time consuming waste, as it tries to make us feel pathos for Yamato, while in the next scenes he's practically killing kittens for how much we feel sorry for him.\nDemon battles are also both weirdly underused, and shockingly unpolished. Person 4: The Animation's battles may not have blown anyone away, but at the very least they functioned, which sadly can't be said about Devil Survivor 2: The Animation's. The battles either drag on for far too long while doing far too little, or they're cut into boring segments that do a horrible job at conveying what's going on. I'll use the Mizar fight as an example. In the game it's probably one of the coolest battles, you fight a giant regenerating blob on a dragon which is entwined around two high rise buildings. In the show the entire battle is literally wrapped up in less then a minute with no one so much as summoning a demon to fight it.\nThe game also features a great soundtrack, which is almost entirely left out of The Animation. Instead we got some rather bland orchestral pieces, while songs that would have fit it perfectly, like Break Out, get left by the wayside.\nSo in closing Devil Survivor 2: The Animation is a disappointing mess that I can't recommend to anyone. So here's my personal plea; play the game. I don't think I can articulate how much better it is, and I'm sure I sound like a broken record, because I've written about the game before, but it really is an amazing experience.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Cheltenham Minster with St Matthew's is a church community at the heart of Cheltenham. Together we worship and explore what it means to follow Jesus Christ and share his hope. We have two church buildings within close proximity which gives us two venues of different styles in which to pursue our vision. We are a church family of all ages, backgrounds and situations in life. What unites us is our hope through Jesus Christ, a commitment to the Bible as God's Word, and a desire to share this with others. So whoever you are, we'd like to think you can find a home amongst us.\nCome and see us one Sunday, it would be great to meet you.\nDuring Lent we are listening to the 'seven words' that Jesus spoke as he was being crucified. These are some of the most significant and moving words that Jesus spoke. They will touch our hearts, help us to understand the meaning of his death for us and move us to love him more. Come and join us on our journey towards Easter.\nPromised from centuries past. Planned before the beginning of time. Enacted in human history. Demanding a response. All focused on Jesus the suffering servant. Sweet delight for those who receive. Events with a nasty twist for those who do not. Join us at 6.30 as we journey the road from Easter promise to fulfilment, and beyond.\nMidweek Prayer 12:30pm Mon-Fri, at Cheltenham Minster with a half hour midweek communion on Wednesday.\nThroughout the week Small groups, including Bible study, support, prayer and music meet at various times during the week. Please contact the office for more information.\nThe Messenger is a publication brought to you by the Communications Team at the Diocese of Gloucester.\nClick the button below to visit the Gloucester diocese parish resources page. There you will find the most recent edition of The Messenger.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Back to the blog and ready for Kickstarter!\nComments Off on Back to the blog and ready for Kickstarter!\nI had no idea this project would take the form it did. My life\u2013and my family's life\u2013changed a lot as week after week of late nights and all-weekend benders hunched over the computer became a sort of normal. The final result? Roughly 225 pages and 10 chapters later, Love Under Repair: How to Save Your Marriage and Survive Couples Therapy is born!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Turn your average haircut into something spectacular with this tool. It's simple to use. Don't get left behind in a style that is crushing the internet.\nEverywhere you look you see naturally twisted and textured cuts. Even star rappers and athletes are rocking this new style. It just looks good. If you're in a rush and just want to check out our top pick on Amazon\u2026click here!\nIf you want to bring out the best in your amazing hair, then the right tools are incredibly important to have around. Hair sponges in particular can take your hair to the next level with twisted and naturally textured curls, giving you confidence knowing your look is on point. When shopping around you'll find that there are many hair sponges to choose from, which can make it hard to determine which you should invest in.\nHow do you know which hair sponge is going to make your hair look better than ever?\nWhile you could try out each and every option, that could get expensive (and result in styling disasters). To make things easier, we've reviewed the top 10 sponges right here so you can find for yourself which is worth investing in!\nIf you're looking for smaller 7mm holes for perfect smaller curls (like for an afro), then this is a great sponge to consider. Not only does it come with a 1 year replacement warranty, but it's also made with high quality materials that may last longer than inferior quality materials. It works best on damp hair like most other sponges, and is designed to be used in small circular motions with a light hand. The design of this is black with blue writing on the front, and it has a large surface area to make it easy to grab.\nClick here to check it out on Amazon right away!\nAlthough this may look like other hair sponges you see, it's important to realize it's different because of the materials it is made with. If you want something that's going to last longer than most others, than this is one that's worth thinking about. You will pay more for this, but it will be worth it.\nThis sleek looking sponge is made with small 7mm holes that are strategically placed to ensure perfect spacing between each of the twists of your hair. Made with ethylene vinyl acetate, it's strong and easy to maneuver around your head no matter what style of hair you're working with. What's even better is that this material lasts much longer than others, so although it's more expensive, you'll be enjoying it a lot longer than you would other sponges.\nTo use it, all you have to do is lightly brush it along damp hair in a circular motion; it's that easy! You can even use it with styling gel if you prefer to have certain product in your hair while you use it. This can take hours out of styling time, which is why so many people use it on a daily basis to refine their style. The price point on this is higher, but it's definitely worth it if you're looking to invest in a tool rather than just buy one to replace every few weeks.\nThis is one of the more affordable brushes on the market, but that doesn't mean the quality isn't there. In fact, it has really great reviews from those who own it, and is definitely worth checking out if you want to save some money. While it may not last as long as others, it does have a chance at being around for a while if you're good about taking care of it. Quick cleaning and gentle use can take this from a tool you have a few days to one that you have for many months.\nWhen using this brush you'll want to make sure your hair is slightly damp, just like with most other sponge curl products. You can achieve that dampness with water or gel, whichever you prefer. Once it's ready, brush the sponge in a circular motion around your head, making sure to only use a light touch. Most reviewers say this works particularly well for dreads, as the holes are quite deep and can hold a lot of hair in them.\nThe name of this alone will catch your eye if the bright purple color doesn't! This sponge is made for natural hair, including afros, so it's perfect if you have hair that isn't exactly easy to manage. It can even be used on men, women and children, making it a tool that everyone in the household can use. What's unique about this brush is that it has a really thick top layer to help ensure a solid grip, which means less slipping around in your hands. It also has 9mm holes uniformly placed throughout the brush for defined coils, curls and twists.\nClick here to check it out on Amazon now!\nBy simply brushing this in a circular motion across your head, you'll be on your way to defined texture in no time. You can also use it with regular styling gel or slightly damp hair, whichever you prefer. This is one of the top options that is made to last, but keep in mind the materials should be cared for gently. This means only lightly brushing it along your head so the holes don't stretch out.\nThis is another great brush to consider if you're looking to add texture to an afro or need to start your dreads more easily. However, it also helps create beautifully defined curls in just minutes, making it an easy choice for anyone who just doesn't have a lot of time to style their locks.\nMuch like other brushes, you have to use this one in a circular motion on damp hair, but it's unique in the fact that it works well on fine hair in particular. If you use a light touch, then this sponge is likely to last you a while, although it will probably need to be replaced after about 8 uses. It can help you create those hot defined curls that everyone has in just minutes and the best part is that they will stay that way for a long time!\nThe reversible design on this brush makes it very unique and incredibly useful. In fact, because it has two different designs in one brush, you'll save money from having to buy two different brushes. On one side you will have 7mm holes for curls or twists that you'd like to put in your short hair. Just like with other brushes, you'd lightly brush this on your damp hair to make curls in minutes.\nOn the other side of the brush is a wavy sponge that's perfect for use with afros and dreads. Another unique feature of this product is that it's made from recyclable materials, so it's an eco-friendly option if you're looking to \"go green\". It also has a money back guarantee for a full refund, which is great in the world of hair products!\nThe small size of this makes it the perfect choice for traveling, and it won't take up much space in a bag or even counter. It might be more difficult to grip than other options, but is nice for kids who are doing their own hair.\nAnother 2 in 1 option for anyone looking for a hair tool that they can easily use for short hair just as well as longer hair. This is a bit different than other 2 in 1 hair sponges because it isn't perfectly round. Instead it's elongated to make it easier to hold onto as you brush. It's also a bit larger than traditional sponges, so it's perfect for anyone with larger hands that need a solid grip. This can be used for many different styles of hair, from dreads to afros, making it versatile for the entire family.\nOne side of this sponge features small holes that will grab onto the hair to make textured and defined curls incredibly quickly. The other side has wavy foam spikes that can help perfect all types of hair styles in seconds of it being used.\nIt can be difficult to continue styling your hair if it grows quickly or you constantly change your mind about how you want it to look. This is why a 3-pack of different curl sponges is so worth having around. While it might not be as compact as the brushes that come 2-in-1, it is worth that extra space. One brush has 8mm holes for tighter curls, another brush has 10mm holes for a little bit looser curls and the last brush has foam waves for defined and textured dreads or other types of curls. If you let the sponges do the work for you (meaning you use a light touch and use it is directed), they should last you for a long time.\nHave trouble keeping hold? These have easy to hold designs that are great for any size hands, from teens to adults. You can also use these as your hair grows from very short to long, making each of them items that you'll use on a regular basis.\nThis sponge isn't as popular as other options, but it does work well if you want to style your hair in just a few minutes. It's one of the best choices to have for traveling, as it's affordable and can be tossed out after just a few uses. Just like other hair sponges, you will have to use this on slightly damp hair in a circular motion.\nJust keep in mind that because this is a cheaper option, it does need to be handled with care. This is why it's a popular choice for traveling or times when you just need something quick to take with you. The holes on this brush will help add texture to your hair and can be used on dreads, an afro and many other hair styles.\nYou can use this sponge for dreads, locs, twists, an afro or many other hair styles. What's unique about it is the thickness of the actual sponge, which helps make it easier to grip and easier to use on any length of hair. If you've had trouble keeping hold of these in the past, then this is something that will help you save a lot of time as well as frustration.\nThe material will last a long time if you take good care of it and always use it with light to medium pressure rather than firm pressure.The holes on the bottom of the sponge will grip onto your hair to create texture and defined curls in just a few minutes. If you've never used one of these before and aren't sure about whether or not you'll like it, this is a great one to start with. Not only is it affordable, it's also easy to travel with and keep packed away.\nHair sponges are incredibly useful, easy to find and can save you a lot of time when you're doing your hair. And because they can be used on so many different styles, it's great for your entire family or anytime you want to change up your look. If you haven't used one of these before, then try one of those listed above right now. You'll likely find that it's something you never want to be without again because your hair looks so hot while using it.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"German Packaging Award 2015. Ready, steady, go!\nthe German Packaging Award is the top European award for the best solutions within the packaging industry. It puts your services in the spotlight and also provides the perfect showcase for innovative companies.\nA nomination for \u2013 or achieving \u2013 the German Packaging Award has a far-reaching and sustained effect, not only on customers and competitors, employees and professionals, but also the general public.\nDesigned as a performance showcase, the German Packaging Award has been generating a fair and exciting competition of ideas and solutions since 1963. Over the years, the competition has developed into the most prestigious European award in the packaging arena.\nThis is of course due to the quality of the packaging solutions entered, while the excellent work of our judges is also a very important factor. In 2015, the jury is once again made up of experts from trade associations, leading research institutes and universities as well as members from the industry, trade and professional press.\nIn addition, as the host of the competition the German Packaging Institute is also working constantly on improvements. We have for instance improved the evaluation process in 2015 by introducing even more precise target criteria, so as to better reflect each entry in each of the six categories.\nYou now have time until the end of June to submit your products, ideas and solutions for the German Packaging Award. Take advantage of this opportunity and secure your place at the top!\nTo get everything you need in just a few clicks, take a look here.\nI'm looking forward to your entries and would be happy to welcome you at the festive ceremony at the end of September in Nuremberg.\nGet ready for this prestigious European competition all about packaging!\nUnder the auspices of the Federal Minister for Economic Affairs and Energy we are looking for the best and most innovative packaging solutions by 30th June, 2015.\nThe German Packaging Award is international, spans the entire industry and range of material and is awarded in five categories. Judging is carried out by an independent panel of experts according to criteria that will do your innovation justice.\nReal PR and recognition for your company, your brand, your products and services. Win a trophy, certificate and seal to recognise your innovation and best practice solutions.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Kartagener Syndrome is a topic covered in the 5-Minute Clinical Consult.\nKartagener syndrome (KS) is a rare disorder characterized by defects in the ultrastructure of cilia impairing their motility. It is also referred to as primary ciliary dyskinesia (PCD).\nSome geographically isolated communities, such as the Amish, may have a higher incidence due to consanguinity.\nIncidence of KS is ~1\/15,000 to 30,000 live births.\nKS stems from a disturbance or loss of ultrastructural components of cilia, more specifically dynein arms that help motor cilia.\nIt causes cilia and flagella to lose their normal and fast synchronous beat and waveform patterns, which affect ciliary-dependent areas in the body, such as respiratory epithelia, brain ependyma, embryonic node, oviduct, and sperm.\nChronic upper and lower respiratory tract infections result from lack of mucociliary clearance of mucus and bacteria leading to chronic sinusitis, bronchitis, and permanent lung damage.\nImpairment of ependymal ciliary function within the brain may lead to decreased cerebral spinal flow and thus result in hydrocephalus that may be seen in these patients.\n40\u201350% of patients with PCD present with lateralization defects, such as situs inversus totalis, affecting left\u2013right organogenesis due to absent or affected motility within embryonic nodal cilia, which results in random orientation of thoracoabdominal contents that is not genetically predetermined (1).\nReduced motility of flagellated sperm as well as cilia within oviducts of the fallopian tubes in females may lead to decreased fertility and possible sterility in both genders.\nStephens, Mark B., et al., editors. \"Kartagener Syndrome.\" 5-Minute Clinical Consult, 27th ed., Wolters Kluwer, 2019. Medicine Central, im.unboundmedicine.com\/medicine\/view\/5-Minute-Clinical-Consult\/816266\/all\/Kartagener_Syndrome.\nKartagener Syndrome. In: Stephens MB, Golding J, Baldor RA, et al, eds. 5-Minute Clinical Consult. 27th ed. Wolters Kluwer; 2019. https:\/\/im.unboundmedicine.com\/medicine\/view\/5-Minute-Clinical-Consult\/816266\/all\/Kartagener_Syndrome. Accessed April 25, 2019.\nKartagener Syndrome [Internet]. In: Stephens MB, Golding J, Baldor RA, Domino FJ, editors. 5-Minute Clinical Consult. Wolters Kluwer; 2019. [cited 2019 April 25]. Available from: https:\/\/im.unboundmedicine.com\/medicine\/view\/5-Minute-Clinical-Consult\/816266\/all\/Kartagener_Syndrome.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Hire 3 Blacque Chix for a Corporate Event or Performance Booking.\nAs actors, writers, producers and women, 3 Blacque Chixs mission is to broaden awareness of the sexuality of and increase positive images in the media for the aging female population, especially (but not exclusive to) women of color, who are the most under-represented. HEROTIQUE-AAHH...\" (90-minute theater production with music)-- A \"sexistential comedy\" celebrates the sexual vibrancy and viability of women over 40; and all those planning on LIVING to be, and the men (and sometimes the women) who love them. 3 Blacque Chix Productions was formed in Los Angeles, CA in November 2004 by Mariann Aalda, Iona Morris, and Lola Love. \"HEROTIQUE-AAHH...,\" their first production, has been critically and commercially successful in LA. .\nLearn how to hire 3 Blacque Chix or research availability and pricing for booking 3 Blacque Chix to perform at a corporate event, fundraiser, private party, social media campaign, fair or festival, tradeshow or conference, endorsement project, or hire 3 Blacque Chix as a guest speaker, please submit the form to our agency.\nFrequently Asked Questions about booking 3 Blacque Chix.\nIs 3 Blacque Chix available to work my event?\n3 Blacque Chix's current work schedule and location in relation to your event's type, date, budget, and locality will play a primary factor in the acceptance of booking 3 Blacque Chix for your party. As long as your funding is commensurate to the price range to book 3 Blacque Chix, CTI will propose your event to the official agency representing 3 Blacque Chix. After we consult with them, we can circle back with you to provide a personal and accurate quote that includes: final costs, availability, and actual interest in being hired for your specific event or party.\nHow much is 3 Blacque Chix charging for events?\nCTI can give you the 3 Blacque Chix booking price once we submit your event details to their camp. \"How much does it cost to book 3 Blacque Chix for an event?\" is a frequent question we get, but the final 3 Blacque Chix booking price is contingent on many variables and the booking fee we show is based on a range derived from our past experience with what will 3 Blacque Chix charge for an event. Also, thier speaking fee might be different than the fee shown for the cost to perform or to just appear. Popularity, career stage, along with current demand will cause fluctuations in their speaking price too. The final booking price may be more than what is shown, but you should at least have a budget within that range for them to consider an appearance, performance or speaking engagement.\nHow do I contact the 3 Blacque Chix agent or manager?\nCTI negotiates on behalf of professional event planners and talent buyers for paid appearances! We cannot give out direct contact information of the 3 Blacque Chix agent or help distribute fan mail, personal messages, gifts, requests for free appearances, or donations of time. The 3 Blacque Chix manager and agent will only work with us if we present them with funded offers for their client to work your event.\nCan I book 3 Blacque Chix for an event at a price less than shown?\nThere is low probability to book 3 Blacque Chix for a party with anything substantially less than the published booking cost price range. We will negotiate with the 3 Blacque Chix agent on your behalf, but if your funding is not in place then it will more difficult to book 3 Blacque Chix, because they get many offers to work events that are at or above asking price. It is also important to consider the travel costs when researching the 3 Blacque Chix booking fee, as additional expenses like this, as well as an entourage of staff and equipment, are not necessarily built-in to the booking fee range shown. If your budget does not factor that in, then there is no benefit for them to want to be hired. We do not personally speak for 3 Blacque Chix, but in our experience in booking talent for over 25 years, this is generally the outcome of inadequate funding.\nCALL US TODAY (760) 729-2000 and speak with a CTI 3 Blacque Chix booking agent to get direct availability and pricing for having this talent to work your event.\nWith almost 30 years of industry experience working with professional event planners and talent buyers like you, we can successfully work to hire 3 Blacque Chix for a corporate event, personal appearance, corporate entertainment, speaking engagement, endorsement, private party, or wedding. For complete information on booking 3 Blacque Chix for projects like branding and promotional marketing, product or service launches, print advertising campaigns, media events, fundraisers, social marketing campaigns, and a guest appearance on TV\/Movie specials including, documentaries, infomercials or voice over in video games, please call us at (760) 729-2000.\nCTI will contact the 3 Blacque Chix agent, manager or representative on your behalf to inquire about the current 3 Blacque Chix booking fee and availability, while negotiating the lowest cost on your behalf. Choose CTI as your booking agency for 3 Blacque Chix to hire at corporate events, conventions, trade shows, business retreats or for television and radio commercials, voice overs, and charity events. Be advised that the 3 Blacque Chix booking price may drastically change based where spokesperson campaigns, speeches, fairs and festivals; and even a shoutout, birthday party, or private concert is located when factoring domestic or international travel. CTI agency will help book 3 Blacque Chix if the buyer has the budget and if they are interested in the event. CTI can also assist on the production requirements for 3 Blacque Chix, when needed.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The number of wearable payment technology devices worldwide will increase by 38% by the end of 2016, a report from the International Data Corporation states. The Worldwide Quarterly Wearable Device Tracker report found that the demand for wearable payment technology is rising, with over 30 million extra units estimated to be ordered this year.\nWearable payment technology uses the established contactless payment system to allow consumers to make payments in shops without using their credit or debit card or mobile phone.\nThe majority of the 110 million unit estimate consists of smartwatches and wristbands, with only 9.8 million wearable payment technology units expected to be sales in clothes, eyewear and 'hearable' devices.\nThe popular Apple Watch is expected to have the greatest market share amongst the smartwatch category, totalling almost 50% in 2016. A smartwatch developed by Android will have the second largest total of devices sold in 2016, predicted to be 21.4% of units worldwide.\nIt was also announced that the number of wearable payment technology devices will total 200 million units worldwide in just three years, and in 2020 the figure is expected to exceed 237 million devices.\nThe report notes the demand for 'smarter watches', traditional mechanical style watches with some technology capability, such as sleep or fitness trackers. The IDC believes that simpler technologies built into well-known branded watches will have a strong uptake by consumers who don't require the full app suite encased on the current smartwatches.\nResearchers from the IDC believes that smartwatches are still in their early stages of capability, and expect to see increased functionality over the next few years.\nIs your company equipped to accept payments from wearable payment technology devices? All of Handepay's card terminals have contactless capabilities including Apple Pay as standard, allowing customers to pay using a wide range of devices. Get a quote for one of our card terminals, and make sure your business can take payments from the latest technology.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"SPORTS FLOORING, INC. is a factory direct distributor of Everlast Sports Surfacing with products manufactured by Ecore International.\nWith over 30 years of combined experience in the sports flooring industry our staff has the knowledge and experience to help you make the right decisions regarding flooring for your facility.\nWe are a customer service driven company believing that the sale is not complete until the customer is satisfied.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbdnho b/data_all_eng_slimpj/shuffled/split2/finalzzzbdnho
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+{"text":"This questioning happens almost every time I meet someone new in a professional capacity. I'd like to solve the mystery for you today of exactly what the heck 'sales enablement' is!\nMarketing (after careful planning with the technical and c-level teams) drives the branding and awareness of our messaging.\nThen the sales team, with the support of sales enablement, begins outreach in support of that marketing message. Of course, the sales team needs to have a deep understanding of why marketing created their content in the way that they did. That communication is part of sales enablement.\nAs certain leads evaluate our message and offerings through discussion with the sales team, sales enablement needs to support that evaluation with content and collateral so that the sales team can guide the lead into purchasing.\nContracts are then transitioned from sales into project management prior to work kicking off.\nI think of enablement in two ways. First is enabling the sales staff to execute their outreach in the most efficient manner and provide them with the support they need to gain meetings. Things like outreach and follow-up schedules, or a dedicated informational sheet for putting together messaging. Second is providing the sales staff with buyer enablement tools. Things such as article links, content one-pagers, infographics or educational research to share with prospects. Zirous believes that enabling our buyers to make an informed decision puts us in a competitive position and allows us to add value to our prospects even before a contract is signed. This belief is rooted in the Challenger Sale approach.\nestablishes that you understand what the customer is going through.\nprompts sales reps to reframe their problem and propose another way to solve it than they originally thought of.\ntells you to almost overwhelm them with the problems they will face if they ignore this problem \u2013 this is called rational drowning.\nfinally, it prompts you to show them the path to success using your solution.\nDone correctly, this leads to a prospect who appreciates the value and new perspective you've brought to their pain point, setting the stage for a faster route to the purchasing decision.\nThe reality is, all of us are wearing sales hats at one point or another in our job roles. Whether sales is your primary duty, or you're looking out for a potential project at the next networking event you attend, utilizing this challenger selling model could push your prospect, customer, or networking colleague into purchasing from your company.\nWhat sales enablement processes does your company have in place? If you're a sales professional and want to know more about the challenger selling approach, I recommend reading the Challenger Sale by Matthew Dixon and Brent Adamson to get you set on the right path.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"As long as you are conscious of sound effects, basic grammar, whole words instead of just kanji, and the role of loan words and colloquialisms used by even young children, you're good to go. From this point onward, it's all about practice, but notably, the right kind of practice.\nRead something you enjoy. You're never going to be committed to finding unfamiliar kanji and looking up unfamiliar terms, and putting them together in real, coherent sentences, if you are not emotionally committed to what you are reading.\nUnderstand the genre. If you're reading a sappy romance, expect to encounter a lot of subtle terms about love, emotions, and general feelings. Treat \"love\" as a core concept, not just a character, and it'll make things easier.\nRe-read as you make more progress. Some of the greatest joy I have ever obtained comes from reading something a couple of years later and understanding far more of it. This is so that you can feel your progress in your own heart.\nConnect what you learn to your own thoughts. You won't retain nearly as much if you aren't thinking actively. If you remember a particular phrase because of a particular manga situation, all the better. You have likely formed a long-term memory.\nIn addition to all of this, Japanese is truly reinforced by using it!! I cannot stress this enough. This is something I have learned the hard way. The best way to turn what you learn from reading manga into active, living Japanese is to find a friendly environment to use it, to receive positive, constructive criticism and reinforcement, and to get a greater feel for the language through using it.\nOf course, this is something I emphasize in my own Japanese tutoring at Learn Out Live. I seek to extend to others the knowledge I have gained from hard personal experience and a great deal of experimentation. I did not have enough help when I was learning. I hope I have done my small part to help you.\nArticle first published as How To Read Manga Better, Part 5 on Technorati.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"drywall screw for metal studs dry wall screw can you use drywall screws in metal studs self drilling drywall screws for metal studs.\n5300 kit b kit 2 x registered 3m 5300 series motor termination kit d5300 kit 18 140vr.\nketer stronghold resin storage shed 8 x storage shed keter stronghold 10 ft x 8 ft resin storage shed reviews keter stronghold resin storage shed accessories.\nilluminated light switch led lighted light switch led toggle dimmer switch led toggle dimmer switch light almond led compatible toggle.\nhumidifier for big room best humidifier for large room best large room humidifier best humidifier big room best humidifier for large room room humidifier big w humidifier big room.\nstick and peel tile ash peel and stick tile adhesive lowes peel stick tile lowes.\ntool chest set metal tool box set hand tools home tool set matco tool box setup tool box setups on utes.\nwood burning stove thermostat wood stove with blower wood burning stoves freestanding stoves the home depot 1 wood wood stove wiring thermostat to wood burning stove wood burning stove temperature con.\nwood grain tile bathroom inspirations wood grain tile with wood grain tile wood grain and on wood grain tile floor in bathroom wood grain tile bathroom ideas.\nblack floating wall shelf floating black shelves floating wall shelf black white black floating shelf for sky box black floating corner wall shelf black gloss floating wall shelves.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The IP address 123.22.241.153 is provided by VNPT Corp, it's belong to the CDIR (Classless Inter-Domain Routing) 123.22.192.0\/18 (range 123.22.192.0 to 123.22.255.255). The autonomous system number (ASN) is 45899 and the numerical IP for 123.22.241.153 is 2065101209. You can ping 123.22.241.153 or do a traceroute by clicking on the button.\nWe use several IP database to locate 123.22.241.153. You can find the differents ip locations our Google map, coordinates 105.98 - 9.6033.\nSPAM database lookup for adresse IP 123.22.241.153. Check if a website or an IP is blacklisted on major databases.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"This is a slip latex appliance. Apply with Pros-Aide, Pro Adhesive or Beta Bond. Blend in edges with creamed Pros-Aide and\/or stipple latex over it. Use a Rubber Mask Grease paint, such as Graftobian you'll find on our site, and add blood. Remove with Super Solv, or Bond Off!","meta":{"redpajama_set_name":"RedPajamaC4"}}
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@@ -0,0 +1,5 @@
+{"text":"If a week is a long time in politics, eight years is an epoch in the modern papacy.\nThe shock resignation - the first in more than 600 years - of the 85-year-old Pope Benedict XVI comes at the end of an eight-year rule of the Holy See that has seen the foundations of the Catholic Church shaken as never before.\nA man most defined by his strict adherence to the oldest and most challenged portions of Church doctrine was always going to struggle to unite a global faith that was already reeling from what His Holiness himself had called the \"cloud of filth\" of a decades-long sexual abuse scandal that rose to the very feet of the Vatican's cosseted leadership.\nIn fact, the allegations involved the man himself. As Joseph Ratzinger, he served as the Archbishop of Munich and Freising in the late 70s and early 80s and was said to have personally approved the transfer of a priest accused of molestation to his diocese in order to receive treatment and therapy.\nBut at a time when faith and its influence of the lives on non-believers was being questioned amid clashes between Muslims and Christians in Asia, the Middle East, Africa and elsewhere - not to mention the existential nature of its threat following the attacks on the World Trade Centre in 2001 - Pope Benedict chose to define his papacy with the challenge of attacking what he called \"aggressive\" forms of secularism around the world.\nYet despite his reputation - largely earned during the many years he worked closely with his predecessor, Pope John Paul II, as head of the Congregation for the Doctrine of the Faith - Pope Benedict showed a surprising flexibility when it came to dealing with the secular world.\nCynics may argue he had little choice, but it has to be conceded that his efforts to bind the Vatican bank to European Union rules on money laundering after a series of financial scandals at the Holy See's bank were historic departures from previous Vatican \"Droit de Seigneur\".\nEqually, his views on some forms of contraception showed signs of secular influence when he told the German journalist Peter Seewald that condoms, if used to prevent the spread of Aids, may be considered \"a first step in a movement toward a different way, a more human way of living sexuality.\"\nAs mild as it may seem in print, the mere reference to a change in Church doctrine with respect to contraception is perhaps one of the most significant developments in papal thinking for many decades. Not to mention a seemingly powerful contraction to his earlier views on the \"dictatorship of relativism\".\nHe was also unafraid to criticise his own faith - and by extension challenge the notion of papal infallibility - when we castigated officials for \"not being vigilant enough\" during the Church's response to allegations of clerical sexual abuse.\nIt was an extraordinary move, particularly given the pontiff's earlier letter, De delictis gravioribus, written when he was Cardinal Bishop of Velletri-Segni, clarified the Church's view that internal investigations remain confidential.\n\"In the name of tolerance, tolerance is being abolished,\" His Holiness told Seewald. \"This is a real threat we face.\"\nThe reluctant pontiff faced that secular challenge in a remarkably liberal, if agonisingly slow, way. And for a man who had a geninue hatred of moral relatavism, there's more than a touch of irony in the fact that the sucess of his legacy can only be measured in terms relative to those of his more conservative predecessors.\nWhat remains to be seen is whether his slow steps towards a flexible approach will be taken up by his successors.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Here are highlights from the second half action in Old Bridge's win over Sayreville on day one of the 2016 East Brunswick Team Camp played over at Hammarskjold Middle School in East Brunswick.\nIn the fourth game of the evening at the 2016 East Brunswick Team Camp at Hammarskjold Middle School, Sayreville took on Old Bridge in another edition of the Battle of Route 9. The Knights, led the the likes of Brandon Millings, Rich Calandrino, and Kyle Paris stormed out to a 37-16 lead several minutes into the second half, and then put together an 14-2 spurt for a commanding 51-18 lead, and then coasted to a 61-41 victory.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Nano4Imaging has different options for business-to-business collaboration. These are closely related to our core products. For more information please send a mail at info@nano4imaging.com to get our teaser for collaboration opportunities.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"It Happened One Night: Screwball Comedy?\nFrank Capra's It Happened One Night would seem to have all the earmarks of the great Hollywood genre the screwball comedy, but the jury has long been out on whether it can be classified that way, as it was released in 1934, a couple of years before the concept was really established. For our new release of the film, we turned to the quick-witted duo of critics Molly Haskell and Phillip Lopate for a snappy \"she said, he said\" discussion of the matter. Here's an excerpt from that highly entertaining conversation.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The group says it was documenting cruel conditions at the farm.\nPETALUMA - Authorities have arrested 40 animal rights activists who entered a Northern California organic egg farm to document alleged cruel conditions and rescue chickens.\nThe Sonoma County Sheriff's Office says the trespassing arrests were made Tuesday in Petaluma after several hundred people gathered at Sunrise Farms. Those detained were to be cited and released.\nAuthorities say they answered a report that activists had broken doors to some coops and removed birds. Protesters deny any property was damaged.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbgmgc b/data_all_eng_slimpj/shuffled/split2/finalzzzbgmgc
new file mode 100644
index 0000000000000000000000000000000000000000..43f0de3e0babb2b28c6e30fcb77e5d6cc8eef653
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzbgmgc
@@ -0,0 +1,5 @@
+{"text":"HMCS Terra Nova (DDE 259) was a Restigouche-class destroyer that served in the Royal Canadian Navy and later the Canadian Forces from 1959-1997.\nShe was the sixth ship of her class and the first Canadian war ship to bear the name HMCS Terra Nova. The ship honours the Terra Nova River in Newfoundland as well as an earlier civilian ship the Terra Nova, which gained fame during a scientific exploration voyage to Antarctica. Both the river and the Antarctic (symbolized by a penguin) are featured on the ship's badge.\nTerra Nova was laid down on 11 June 1953 at Victoria Machinery Depot Ltd., Victoria and launched on 21 June 1955. She was commissioned into the RCN on 6 June 1959 with the pennant number 259.\nTerra Nova was selected by the Canadian Forces for the Improved Restigouche (IRE) project and completed this refit in 1968. She was also selected as one of ten destroyers in the Destroyer Life Extension (DELEX) project and completed this refit on 9 November 1984.\nTerra Nova was decommissioned on 11 July 1997.\nAfter being paid off Terra Nova appeared as an American destroyer in the movie \"K-19: The Widowmaker\".\nOn 20 November 2009, along with her sister ship Gatineau, Terra Nova departed Halifax Harbour by tow and was taken to Pictou, NS to be dismantled for salvage.\nModified as Improved Restigouche Class 1967-1968. DELEX refit in 1985.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Thailand might be considered a cheap travel destination but it has some of the best hotel\/resorts in the world which not only provide luxurious stays but also have some of the most wonderful experiences that you could have while staying at a resort or hotel. The amazing variety of landscapes and a very nice tropical setting makes Thailand an irresistible place and there are these certain resorts which have certainly made good use of the natural backgrounds and came up with some really nice places for the travellers with ample fun and complete comfort.\nTo come up with only a few of all the many options that are available is not easy and it is really harsh to say that others are any lesser in comparison to the following hotels. But ultimately, it is about individual tastes and preferences that a person has and when you have to make choices, this helps in determining the one which makes a better choice for an individual. You can always book your hotels online before you visit Thailand as it is really congested travel destinations with lots and lots of travellers coming and leaving each day and this might land you into some sort of problem in getting the desired option. Online booking is also a great option as you can use Klook voucher codes to get your hotels booked at lowest prices by availing the exclusive discounts.\nThis hotel is the best place for those who want a peaceful and tranquil location to stay. This is located at Koh Kood island which is the least populated but 4th largest island in terms of area. This hotel can be reached through a 90-minute flight from the resort's own private plane available at Bangkok. This resort offers a very great experience as it is exclusively unrivalled in terms of the locations and isolation without compromising on luxury and comfort. This hotel is designed beautifully with the help of eco-friendly methods and offers the true Thai experience in terms of its interiors. There are 36 resorts villas out of which 11 are completely private residences and these villas are undoubtedly the largest in the world. The beachside view and the sunny side view of the rainforest make it an exotic location. There are private butlers provided to every visitor. The hotel offers a wide range of exclusive facilities with its huge and stunning swimming pools, the mangrove eatery, the treetop dining tables, the open air cinema and what not?\nThe location of this resort at the Pharang Peninsular makes it a delightful stay for the visitors as they can stroll around to witness the extraordinary scenic beauty of the surrounding Krabi Marine National Park. This 5-star hotel offers one of the best cuisines in Thailand along with its equally popular relaxation spa with incredible services. The breathtaking visual that covers the landscapes all around the place along with the lush greenery and clear sea view makes it an extraordinary place altogether. This hotel is considered to be one of the best places for wedding celebrations and honeymoon destination and could be easily found on most travel blogs amongst the best-rated resorts ever.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The tool window is available only when the Dart plugin is installed and enabled. The plugin is not bundled with PhpStorm, but it can be installed from the JetBrains plugin repository as described in Installing, Updating and Uninstalling Repository Plugins and Enabling and Disabling Plugins.\nPhpStorm integrates with the Dart Analyzer tool that performs static analysis of your Dart source code. All inconsistencies and potential problems are reported in the Dart Analysis tool window with the possibility to navigate to the fragment of the source code where the problem was detected.\nReanalyze Dart Sources Click this button to run the analysis of the Dart source code of the project without stopping the Dart Analysis server.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Fresh Energy's science policy director, J. Drake Hamilton, was recently recognized as one of the Top 10 Climate Leaders of 2015-2016 by the Climate Reality Project.\nJ. Drake Hamilton's actions on climate leadership led her to receiving the #6 award out of nearly 11,000 global leaders. At Fresh Energy, J. continues to inspire and encourage diverse audiences to support climate and energy policies that combat the threat of global warming and create a clean energy future. If you would like J. to speak at your event, please contact her at 651-726-7562 or Hamilton@fresh-energy.org.\nJ. Drake Hamilton runs the Global Warming Solutions program at Minnesota-based Fresh Energy, an independent nonprofit that supports climate solutions that maximize economic development in the Midwest. J. estimates that she has given presentations to over 70,000 people, including to audiences at the US Environmental Protection Agency, Target Corporation, and 3M. She represented Fresh Energy at the international climate negotiations in Paris last year, and was invited to meet with President Obama at the White House for the roll out of the EPA's Clean Power Plan. She was recognized by Minneapolis-St. Paul magazine as one of 100 Minnesotans \"who make things happen.\" J. has completed 84 acts of leadership since the beginning of 2015.\nSince its establishment in 2011, the Climate Reality Project has trained over 11,000 individuals representing 126 countries on education initiatives related to climate change as well as continuing to develop a grassroots network to address climate change.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Chevrolet\u00ae trucks have been a staple of the American dream for 100 years. Generations of hard-working Americans have relied on their trusty Chevy\u00ae to haul everything under the sun to enrich their way of life. It could be countless bushels of this year's sweet corn harvest, or their son or daughter's furniture as they move into a college dorm, or even a sweet 454 crate motor going into the next award- winning custom hot rod. Chevy\u00ae trucks have been delivering greatness since 1918.\nTo commemorate the anniversary of Chevy\u00ae trucks, Auto World is honored to replicate this 1957 Chevy\u00ae Cameo\u2122 3124 pickup in factory-matched Jet Black paint. The interior is properly detailed with matching black seats with white inserts. This truck is a fine addition to the American Muscle line of 1:18 die-cast replicas.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbhcsa b/data_all_eng_slimpj/shuffled/split2/finalzzzbhcsa
new file mode 100644
index 0000000000000000000000000000000000000000..a8ca1bc7dc668a8d0be0ca25c95cf631e05dd970
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzbhcsa
@@ -0,0 +1,5 @@
+{"text":"FL 2014-2015 COST COMPARISONS FOR CAPITAL COMMON.Grand Total per 100 sf $2097.86 Square Foot Cost $20.98 (Quantities and costs are presented in thousands except the Square Foot Cost) Notes on interior finishes: 1. ..\nCourtyard Aluminum Fence - MMC Fencing & Railing.DSI Columns & Aluminum Posts Polyrail Vinyl Railing Continuous Handrails Courtyard Aluminum Fence Lighting Specials & Clearance Get A Quote Blog FAQs Contact Us Courtyard Aluminum Fence CourtYard Aluminum Fence* represents elegance and ..\nShop Fence Panels at Lowes - Lowe's Home Improvement Hardware Store.Shop fence panels in the fencing & gates section of Lowes . Find quality fence panels online or in store. .. Ironcraft (Actual: 4-ft x 6.025-ft) Ironcraft Black Powder-Coated Aluminum Decorative Fence Panel Enter your location OK Compare Multiple Sizes (4) ..\nMinnesota Sustainable Housing Initiative - University of Minnesota.siding, fiber-cement, aluminum, vinyl siding, hardboard, and wood-resin\/plastic composite .. (not considering maintenance costs). Fiber-cement board's cost per year of expected life is significantly less than other options. Life Cycle Analysis Analysis ..\nPanel Siding costs Forum Archinect.(not the fancy aluminum proprietary hanging system which is about 34-36 per sf.) Feb 6, 08 2:39 pm jorgeouvina History \u00b7 Contact There is another panel manufacturer called royal that is the same as Trespa. Trespa has taken the market and therefore since has ..\nSmooth Aluminum Concrete Forms for Concrete Construction by Precise Forms, Inc.Ledger System Pour Ready Process Smooth Aluminum Concrete Forms Precise Forms provides a complete line of standard and decorative aluminum forms as well as supporting accessories and wall ties. Custom concrete aluminum forms are available for ..\nSee Metal Fa\u00e7ade Panels - Flexibility and Tremendous Design Potential; Citadel Architectural Products.Metal Fa\u00e7ade Panels Offer Flexibility and Tremendous Design Potential Metal fa\u00e7ade panels are available in a variety of durable materials: composite aluminum, stainless steel, Rheinzink, copper, and many others. These high-quality, exterior wall system metal ..\nAverage Wall Cost Per Square Foot - - Ambrico Thin Brick EZ-WALL Thin Brick System Brick Veneer.Average Wall Cost Per Square Foot In Selected Cities (amounts in dollars) Wall Description (click wall type to view detail) Birmingham AL Jacksonville FL Atlanta GA ..\n4x8 sheets vinyl panels - Outside Wpc Deck,Wpc Products,Outdoor Wpc Fence & Wall Panel-Seven Trust.Luan Panels covered with decorative vinyl. Exact colors may vary from those .. NOTE 2: Pricing is PER sheet. Up to 10 .. Wall Covering - Rigid Sheet & Rub Rail - Wallguard ..\nShop Fence Panels at Lowes - Lowe's Home Improvement Hardware Store.Freedom (Actual: 2.83-ft x 7.63-ft) Pre-Assembled Newport White Vinyl Panel Enter your location for pricing and availability. OK ZIP Code Compare Compare ..\nsquare foot price composite wall panel system.Where to select wall panels with attractive and reasonable price .. having a modern wall panel system .. What is the cost per square foot to build a composite ..","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Added by sos4nt (Stefan Sch\u00fc\u00dfler) over 2 years ago. Updated almost 2 years ago.\nKernel provides the global conversion functions Integer(), Float() and Rational(). Each one convert its argument to the respective type.\nParsing errors are signaled by raising a TypeError or an ArgumentError.\nI often use these methods to parse numeric string values, but I find the need for a rescue ArgumentError block quite annoying an distracting.\nI therefore request a new feature: Integer(), Float() and Rational() should take an optional block that serves as a callback when parsing failed.\nThis resembles the way Hash#fetch or Hash#fetch_values work when a block is given.\nAbsolutely, I've mentioned that in the last sentence.\n+1 I like this idea. It must also be backwards compatible.\nOnly \"invalid value for Integer()\" from a string?\nInteger can raise exceptions other than ArgumentError against a string, for example.\nI don't have a strong opinion on this.\nWe looked at this issue at yesterday's developer meeting.\nPeople there argued that explicit keyword argument is much easier to read than passing a block. They say it is not obvious what the block is going to do. So if the OP is OK we would like to to merge this issue with #12732. How do you feel, Stefan?\nwe would like to to merge this issue with #12732. How do you feel, Stefan?\nI'm fine with with that.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"This 9.5\" x 13\" original antique map was recovered from Volume Ten of \"The World's Progress, with Illustrative Texts From Masterpieces of Egyptian, Hebrew, Greek, Latin, Modern European and American Literature.\" The book was published in 1915 by the Delphian Society and edited by J. K. Brennan. The Delphian Society was a national organization that promoted the education of women in the United States. This organization was founded around 1910 in Chicago.\nThis map illustrates the territorial growth of the United States. The paper shows a bit of browning and soil, but is generally clean and crisp. Center crease. Title page from the original book shown for informational purposes and is not included with the sale.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"BERKELEY, Calif. - California won three out of seven events and leads the field with 364.5 points after the first day of the Pacific Tiger Invitational on Friday afternoon at the Chris Kjeldsen Pool Complex on the campus of the University of the Pacific. Cal junior Jonas Tilly won the 50-yard freestyle in a time of 20.76, junior teammate Graham Lentz won the 100-yard breaststroke in a time of 57.61 and the Cal A squad won the 200-yard medley relay in a time of 1:35.14.\nThe Golden Bears are followed by second-place USC with 317.5 points, Denver is third with 174 and the host Tigers are fourth with 121. Stanford and UC Davis did not swim today but will compete tomorrow at the Pacific Tiger Invitational, which also has a women's competition.\nCal took second in three events: the 200-yard freestyle (sophomore Dominik Meichtry, 1:38.94), the 400 individual medley (sophomore Louis Vayo, 4:03.12) and the 50 free (sophomore Joe Whittington, 20.95). Tilly also took third in the 200 free (1:41.94), sophomore teammate Justin Pollard was third in the 400 IM (4:04.14) and the Cal A squad was third in the 400 free relay (3:13.57).\nNo Bears swam in the night's final event, the 1650 freestyle. Also, there is no diving competition at this meet.\nCal returns to action at the Pacific Tiger Invitational at noon on Saturday.\n1. Cal A 1:35.14 (40 points).\n2. Dominik Meichtry 1:38.94 (17). 3. Jonas Tilly 1:41.94 (16). 5. Garrett Wagner 1:42.09 (14). 6. Daniel Lysaught 1:42.16 (13). 21. Spencer Rodman 1:46.88. 24. Ryan Lean 1:47.03. 26. William Copeland 1:47.16. 27. Michael Jafari 1:47.21. 41. Nate Rogers 1:51.05. 42. Dash Rothberg 1:51.80. 44. Joe Whittington 1:52.12. 45. Evan Lane 1:52.25. 49. Eric Dunipace 1:54.86. 50. Matt Lyon 1:55.71.\n2. Louis Vayo 4:03.12 (17). 3. Justin Pollard 4:04.14 (16). 4. Miguel Molina 4:06.90 (15). 5. Patrick O'Neil 4:08.17 (14). 6. Ian Lentz 4:08.81 (13). 8. Graham Lentz 4:12.22 (11). 9. Mark Eckert 4:13.41 (9). 10. Jernej Godec 4:13.69 (7). 11. Chris Gibson 4:13.74 (6). 14. Dominic Cathey 4:14.51 (3). 16. Henrique Barbosa 4:14.83 (1). 17. David Russell 4:15.65. 18. Milorad Cavic 4:15.98. 22. Paul Hernandez 4:19.76. 27. Richard Hunter 4:24.62. 28. Ryan Emanuel 4:25.68. 30. Graeme Baldwin 4:29.79.\n1. Jonas Tilly 20.76 (20). 2. Joe Whittington 20.95 (17). 4. William Copeland 21.24 (15). 6. Matt Lyon 21.61 (12.5). 9. Evan Lane 21.75 (9). 11. Dash Rothberg 21.79 (7). 14. Eric Dunipace 22.14 (3). 25. Daniel Lysaught 22.87. 28. Spencer Rodman 22.98. 34. Ryan Lean 23.64. 35. Michael Jafari 23.66.\n1. Graham Lentz 57.61 (20). 5. Henrique Barbosa 58.44 (14). 13. Richard Hunter 1:01.09 (4). 17. Graeme Baldwin 1:02.48. 18. Paul Hernandez 1:03.39.\n3. Cal A 3:13.57 (32).","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"I am a computer scientist, mathematician, and developer who is fascinated about the interplay of complex mathematical theory and practical applications. My research falls under the scope of Optimization, Computer Vision, Computational Geometry, and Robotics. Problems within these fields typically comprise a geometrical component that makes them highly visual and allows to derive and to understand conceptually demanding algorithms.\nIn my undergraduate and graduate studies in Mathematics with minors in Computer Science and Engineering Science at the University of Bayreuth, Germany, I completed my diploma and PhD thesis at the chair for Numerical Mathematics, Optimization, and System and Control Theory and Robotics and Embedded Systems. I developed a technique for the optimal placement of multiple cameras in three-dimensional space in the context of safe human-robot cooperation in my doctoral work. I achieved a doctoral and post-doctoral scholarship. The first allowed a five months research stay at the University of Cambridge, UK, where I developed an optimization solver at the Cambridge Image Analysis group. In my post-doctoral work, I collaborate with the Algorithms and Data Structures group on the analysis of vision functions.\nIn my spare time, I am a passionate amateur photographer, I like travels and sport activities, such as soccer, snowboarding, volleyball, and dancing. Additionally, I have gathered some practical experience in graphics, layout, C++, databases, CAD, and web development.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbikqs b/data_all_eng_slimpj/shuffled/split2/finalzzzbikqs
new file mode 100644
index 0000000000000000000000000000000000000000..7668dc4ed1a6518068f127e4abe554c3172e88c3
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzbikqs
@@ -0,0 +1,5 @@
+{"text":"Harry Kewell \/\u02c8kju\u02d0.\u0259l\/ (born 22 September 1978) is an Australian football coach and former player who is currently coaching Watford's Under 21s. Kewell played for Leeds United, Liverpool, Galatasaray, Melbourne Victory, Al-Gharafa and Melbourne Heart. Internationally he has received 58 caps, and scored 17 goals, while playing for the Australian national team. A left winger also capable of playing as an attacking midfielder or second striker, he is often regarded within the media as \"Australia's finest football export\", despite his career being blighted with injury.\nKewell scored a goal against Croatia which took Australia through to the knockout stages of the 2006 World Cup, the Australian national team's second World Cup. He is a member of the Executive Committee of the Australian Professional Footballers' Association. Kewell also has a British passport through his father's heritage. Former Middlesbrough midfielder-turned pundit Robbie Mustoe named Kewell as one of the greatest players he had played against but questioned his consistency and attitude after his initial injuries. Former German international Michael Ballack has also highlighted Kewell's ability and inconsistency.\n\"You can't explain what the feeling is like when you're out there. We've made a big step forward now, and now we've just got to look forward to the World Cup.\"\n\"I will also try my best to break it and hope to produce enough good football to keep them voting for me.\"\n\"Yes, the referee made a decision which no one can believe. He has gifted them the game as it turned out, but you have got to get on with it when something like that happens.\"\n\"We are playing well and creating chances but can't get that second goal. But sometimes one is enough.\"\n\"Just one of those days.\"\n\"Teams are coming here and working hard. We're creating chances but we can't get the second goal - but we're doing enough.\"\n\"To win was a massive achievement. We had to dig in. We knew it would be tough, even with 11 men, but we pulled through.\"\n\"It was crucial for us to score the first but we stung them with the quick second, although they came back well.\"\n\"We knew we were up against it from the first minute, even with 11 men.\"\n\"The first goal was crucial for us because it is always difficult to score against ten men.\"","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"You can put most sorts of debris into a dumpster rental in Highland Mills. There are, however, some exceptions. For instance, you cannot put substances into a dumpster. That includes motor oil, paints, solvents, automotive fluids, pesticides, and cleaning agents. Electronics and batteries are also prohibited. If something presents an environmental risk, you likely cannot place it in a dumpster. Get in touch with your rental company if you're uncertain.\nSpecific kinds of okay debris, however, may require additional fees. In case you plan to throw away used tires, mattresses, or appliances, you should ask the rental company whether you need to pay another fee. Adding these to your dumpster may cost anywhere from $25 to $100, determined by the thing.\nPlanning ahead for your dumpster rental in Highland Mills will make your project easier and safer to end on time. When renting a dumpster in Highland Mills, follow these tips to help you plan ahead.\n2. Prepare the items you wish to remove. In the event you are cleaning out a waterlogged basement, for example, try to remove as much of the damaged stuff before the dumpster arrives.\nPaying a flat rate normally means that you simply get to prevent any hidden fees and charges. Businesses which usually do not charge flat fees may contain additional charges for basic services like dropping off the dumpster. Keep in your mind, nevertheless, which you still might need to pay extra if you need to include specific items and materials in the dumpster.\nHaving a flat rate also means you can prepare your payment in advance. This is especially useful when you need someone else to make the payment for you. When you understand the exact quantity, you can compose a check in advance. Afterward, anyone can pay the rental company when its driver drops off the dumpster.\nDetermining where to place your dumpster can have a big effect on how fast you finish endeavors. The most efficient option would be to pick a location that's near the worksite. It's important, nevertheless, to consider whether this location is a safe alternative. Make sure the area is free of obstructions that could trip individuals while they carry heavy debris.\nMany individuals choose to place dumpsters in their driveways. This really is a convenient alternative as it usually means you can avoid asking the city for a license or permit. If you have to place the dumpster on the road, then you definitely need to contact your local government to inquire whether you are required to get a license. Although many municipalities will let people keep dumpsters on the road for short levels of time, others will ask you to complete some paperwork. Following these rules will help you stay away from fines that will make your job more costly.\nDumpster rental firms commonly have set costs, but the fine art of negotiation is consistently in play. Should you'd like to dicker on the cost, call around to other companies first so you understand what the going rental rate is. When you are talking to the business you would like to utilize, merely mention other costs which are better than theirs. Some companies will not just fit that lower cost, they'll really defeat it to satisfy their customers.\nYou can also ask about discounts when you are renting a dumpster in Highland Mills. Some companies will supply special discounts to active members of the military, veterans, seniors, members of the neighborhood Chamber of Commerce and more. It never hurts to ask; you can occasionally save up to 10 percent or more with this approach.\nYou may also save on the cost by asking about the best approach to prevent extra expenses (such as tipping fees or overage charges) and making sure you order the right size container; you do not want to pay for more container than you really desire.\nThere are lots of things to consider when picking a local or national dumpster rental firm in Highland Mills. Follow these guidelines to assist you determine which choice is better for you.\nA local dumpster rental firm in Highland Mills may offer better customer services that help you finish your project while keeping prices low. A lot of them, nevertheless, have a limited number of dumpsters to rent. Should you not schedule an appointment in advance, you may not have the choice you need.\nA national dumpster rental firm in Highland Mills will usually have more sizes and designs to satisfy the exceptional needs of your endeavor. National firms are likewise a good choice for building crews that work in a number of cities. Some people, however, complain that national firms aren't as adaptable as locally owned firms.\nIf you rent a 30 yard dumpster, you will be getting a container that can hold 30 cubic yards of waste or debris. Your 30 yard container will measure about 22 feet long by 8 feet wide by 6 feet high. These numbers could vary somewhat determined by the dumpster rental firm in Highland Mills you pick. A 30 yard dumpster will hold between 9 and 15 pickup truck loads of waste, therefore it is an excellent choice for whole-house residential cleanouts as well as commercial clean-up projects.\nDo I require a permit to rent a dumpster in Highland Mills?\nIf this is your very first time renting a dumpster in Highland Mills, you may not know what's legally permissible in regards to the placement of the dumpster. If you plan to put the dumpster totally on your own property, you are not normally required to get a permit.\nIf, however, your job requires you to set the dumpster on a public road or roadway, this will under normal conditions mean you have to try to qualify for a permit. It's always advisable to consult your local city or county offices (perhaps the parking enforcement department) if you own a question regarding the demand for a permit on a road.\nShould you don't get a permit and find out later that you were required to have one, you'll likely face a fine from your local authorities. In most dumpster rental in Highland Mills cases, though, you should be just fine without a permit as long as you keep the dumpster on your property.\nMost residential areas allow roll-off dumpsters. When you own a drive, then you could usually park the dumpster there so you do not annoy your neighbors or cause traffic issues by putting it on the street.\nSome jobs, though, will require putting the dumpster on the road. If this applies to you, then you definitely should speak to your city to find out whether you are required to get any permits before renting the dumpster. Generally, cities will allow you to keep a dumpster on a residential street for a brief period of time. If you think you will need to be sure that it stays on the street for several weeks or months, though, you might need to get a license.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The NJIT Business Club is a professional development club that aims to encourage business majors, minors and any students with an interest in business to become more familiar with the business culture. Professionalism will not only help them in their future careers but also during their current on-going studies. We strive to highlight the importance of networking and aim to establish connections between members and future potential employers. The Business Club also works to provide students with the business tools and skills necessary to succeed in the \"real world\".\nIf you are interested in joining the Business Club, contact the Business Club at businessclubnjit@gmail.com for more information.\nStudent-led and run user group at NJIT's Martin Tuchman School of Management. Our mission is to support students as they learn the skills to get hired into the Salesforce ecosystem. Join our group to ask questions, build your network, and learn about upcoming meetings.\nThe NJIT Investment Club is a student-run association that is committed to educating its members about investing in the stock market and investing for the long term. By managing a real stock portfolio, students will be able to develop critical skills that will prepare them for the real world. Any student is welcome to join the club regardless of their abilities. For those members that are not familiar with investing, every semester the club teaches Investing 101, which is an introduction to the markets and principles of value investing.\nThe IAC brings students together who want to focus on Innovation and the Commercialization of Technology. Each year the club has a focus project and in past years we have all learned how to do app development, conduct lean startup practices, and enhance commercialization of military technology in partnership with Lakehurst Naval Air Station. The club attracts students from throughout NJIT to create business-like teams with technical and business students working together. Both undergraduate and graduate students are welcome to join.\nView the IAC website, or contact Dr. Michael Ehrlich, a club advisor, for more information.\nAlpha Kappa Psi was founded on the principles of educating its members and the public to appreciate and demand higher ideals in business and to further the individual welfare of members during college and beyond. It is a unique, prestigious association of students, professors, graduates and professionals with common interests and goals. Alpha Kappa Psi provides valuable educational, friendship and networking opportunities.\nBeta Gamma Sigma is the international Honor Society for AACSB accredited schools. For membership requirements and meeting information, contact Michael T Sweeney at michael.t.sweeney@njit.edu, the MTSM Undergraduate Advisor. You can also visit the Beta Gamma Sigma website to learn more about the benefits of membership.\nOur next induction ceremony will be held during the spring semester of 2017. Last year, NJIT inducted over 50 new members into their chapter of Beta Gamma Sigma at the Martin Tuchman School of Management Awards Ceremony.\nWant to be a Financial Guru? You should join the NYSSA, NJIT Martin Tuchman School of Management Chapter. NYSSA is the New York Society of Security Analysts and our chapter aims to promote networking, outreach, and programming among students of finance and finance-related disciplines. You will be able to enjoy access to a unique and powerful networking platform, free events, value investing archives, and much more.\nThe CFA (Chartered Financial Analyst) Institute sponsors a competition each year where students from around the world are invited to present a professional-quality equity research report to a team of CFA judges. The winners from our region (NYSSA) around NYC go on to compete in the national and international levels. The NJIT team is nominated by the finance and accounting faculty and has has been a local finalist (one of the final four teams) several times. Past team members have gone on to Wall Street careers.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"This is a set of (4) NEW AVID 06 in a 17x8 and 17x9 combination. It is required to lightly roll your rear fenders if your car is lowered. Recommended tire size is 215\/40\/17 for the front wheel and 245\/40\/17 for the rear wheels.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Chelsea won the 2016\/17 Premier League title this month.\nPremier League champion Chelsea cancelled its victory parade in London on Sunday following the Manchester terror attack.\n\"Everyone associated with Chelsea Football Club offers our heartfelt condolences to those affected by Monday's terror attack in Manchester,\" an official club statement read.\n\"Our thoughts go out to all the victims, and their families and friends.\"\nSalman Abedi, the 22-year-old suicide bomber, killed 22 and injured at least 64 when he detonated an improvised bomb in the foyer of Manchester Arena in central Manchester on Monday.\n\"In light of these tragic events, we feel it is inappropriate to go ahead with the victory parade in London on Sunday given the heightened security threat announced by the government, and recognising that this is a developing situation,\" Chelsea added.\n\"Having consulted with the Metropolitan Police, Hammersmith & Fulham Council, and other authorities, we know the emergency services would have been as professional as ever, but we would not want in any way to divert important resources by holding an additional, non-ticketed event on the streets of London.\"\nAs a mark of respect, Chelsea will wear black armbands during its FA Cup final match against Arsenal at Wembley Stadium on Saturday. Chelsea will also \"make a donation to a fund supporting the victims of this horrific attack.\"\nThere will be increased security at Wembley at the weekend, the Football Association confirmed. \"Fan safety is of paramount importance and we have robust security measures in place at Wembley Stadium,\" a spokesman said on Tuesday.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbjped b/data_all_eng_slimpj/shuffled/split2/finalzzzbjped
new file mode 100644
index 0000000000000000000000000000000000000000..adeecc35a32c070f11ef9b91a83e8c92eaafe1a7
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzbjped
@@ -0,0 +1,5 @@
+{"text":"Do you ever want to see style that's different from the regular trend-nazi's of NY?. If so, Craig Altamira is the go-see man who has an incredible eye for fashion. He knows style when he sees it, I'm always amazed by the people I see on his blog, so I'm excited that I got the chance to meet him and have my picture taken, one day on the lovely vintage streets of Soho. He is not only an amazing photographer but an awesome well spoken person. It's always good to bump into him, especially on my lunch break;).\nI saw the pics on there!! You look gorgeous of course.\nI adore this coat. Style Maven you are.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"It is almost a year since the New Turkish Commercial Code (\"TCC\") came into force on 1st July, 2012 providing fundamental changes and solutions to problems that caused difficulties in the past.\nOne of the issues that divided the legal community under the previous code related to the jurisdiction of the Turkish Courts over foreign flag vessels using the Bosphorus and the Dardanelles Straits (the \"Turkish Straits\") in transit without calling at a Turkish port. Prior to the TCC the Turkish Courts frequently held themselves out to have jurisdiction over foreign flag vessels in transit and issued warrants of arrest regardless of the clear provisions of the Montreux Convention Regarding the Regime of the Straits dated 20\/07\/1936 (\"Montreux Convention\") giving foreign flag vessels a right of innocent and uninterrupted passage.\nThe TCC purported to overcome that problem by limiting the jurisdiction of the Turkish Courts to order arrest of foreign flag vessels in Turkish waters; in effect prohibiting arrest of such vessels while in transit in the Turkish Straits. It is obvious the legislator wanted to underline that the Montreux Convention should be respected.\nThis did not end the debate since some argued, that Article 1355 of the TCC which vests jurisdiction in the Turkish Courts to issue arrest warrants against foreign flag vessels that drop anchor in its area of jurisdiction could still be relied upon to continue ordering arrest of such vessels regardless that they have no intention of calling at a Turkish port and are in effect, in transit. The question that arose is, will the transit of a foreign flag vessel be broken bringing it within the jurisdiction of the Turkish courts and will it lose the protection of the Montreux Convention if it drops anchor in a Turkish anchorage area for the sole purpose of replenishing bunkers and\/or stores.\nThe background to this debate is that there is currently a tunnel construction under the Bosphorus that will connect Asian and European Istanbul by rail. While the Bosphorus has always been open to two-way traffic, so that northbound and southbound vessels could use the waterway at the same time, the construction works necessitated a limitation to the use of the Strait. For several years now and until the project is concluded the waterway is open to one way traffic northbound or southbound alternating twice a day and passage is in convoys. This inevitably results in build up of shipping traffic on both ends of the Bosphorus and the anchorage areas in the Black Sea and Marmara Sea entrances are continuously crowded with vessels waiting to join the next convoy. Some of these vessels intending to use the waterway in transit and having no intention to call at a Turkish port for cargo operations or repairs, take the opportunity to replenish bunkers and\/or stores while in the anchorage area. Are these vessels exposed to warrants of arrest issued by the Turkish courts merely because they dropped anchor and replenished bunkers and\/or stores?\nThe Turkish Court of Appeal addressed that issue in a recent decision in effect holding that (i) the foreign flag vessel in question was enjoying the right of innocent passage under the Montreux Convention; (ii) the Turkish courts do not have jurisdiction over a foreign flag vessel using the Turkish Straits in transit and (iii) the fact that the vessel dropped anchor at one of the anchorage areas merely for the purpose of replenishing bunkers did not mean that her transit was broken vesting jurisdiction in the Turkish Courts to issue a warrant for her arrest.\nThis is a welcome ruling and brings clarity to the subject of the jurisdiction of the Turkish courts over foreign flag vessels intending to use the Turkish Straits in transit alone but which are forced to anchor due to congestion and to await their turn to use the waterways.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Your due date is June 28, 2019.\nCongratulations! You are 31 weeks pregnant.\nYour baby is the size of a coconut.\nAt 31 weeks pregnant, your baby is the size of a coconut and weighing in at 3-plus pounds. Your baby is sleeping more, making faces, hiccupping, swallowing, breathing and pedaling with little hands and feet.\nThis creamy-white opalescent gem symbolizes purity and innocence.\nHenry VIII of England was a monarch of Tudor dynasty known for separating the Church of England from the Roman Catholic Church and marrying six different women in an attempt to have a son as heir to the throne.\nWhat are y'all drinking besides water???\nAnyone Announcing on Thanksgiving? Ideas?!\nThere are 31,236 active discussions happening now with other June 2019 parents-to-be.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Angelman syndrome is a neurogenetic disorder that is characterized by severe intellectual disability, developmental delay, exuberant behavior with a happy demeanor, absent speech, epilepsy, and motor impairment.\nIt is caused by a deficient UBE3A gene expression which may be due to various abnormalities of chromosome 15.\n#3 In the United States, it affects about 1 in 15,000 people. In Sweden, there is a prevalence of about 1\/12,000 in children ages 6 to 13 years.\n#4 The most common genetic defect leading to AS is a maternal deletion of DNA from chromosome 15. This leads to an absence of expression of the UBE3A gene in the paternally imprinted brain regions.\nChromosomes, which are present in the nucleus of human cells, transport the genetic information for each individual.\n#5 These genetic changes occur as random events in early embryonic development or during the formation of eggs and sperm (reproductive cells).\n#6 In a small percentage of cases, AS occurs when an individual inherits 2 copies of chromosome 15 from his or her father instead of one copy from each parent.\n#8 Clinical features of the AS do not become manifest until after the first year of life.\ndelayed or slow growth in head circumference, typically resulting in relative or absolute microcephaly (a birth defect where a baby's head is smaller than expected) by age 2.\nnormal hematologic, metabolic, and chemical laboratory profiles.\nfeeding difficulties, like \u2013 problems swallowing and sucking, may occur during the first year of life.\nchromosome analysis \u2013 to see if any parts of the chromosomes are missing (deletions).\n#15 Most children with the symptoms will have seizures and will need antiepileptic therapy.\n#16 More importantly, they have a higher risk of early death due to aspiration pneumonia (occurs when you inhale stomach acid, food, or saliva into the lungs), seizures, and chronic lung diseases.\nDave Henderson \u2013 he was an American professional baseball player who spent 6 seasons in Oakland, featuring a World Series Championship and 3 consecutive American League Championships. Dave died in December 2015 of a heart attack possibly linked with a recent kidney transplant. He had two sons and was married twice.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"In the ancient Chinese practice of feng shui, one area of your home is considered the energetic center of money or wealth. This is a good area for conducting business or managing household finances. But where this area is located depends on which bagua form, or school, of feng shui you practice. Classical, or traditional, feng shui defines the money area differently than the Western, or BTB, feng shui school.\nThe bagua is essentially an energy map used by feng shui practitioners to help define areas of the home that have specific meaning and usefulness. In the classical school of feng shui, the bagua map is defined according to the compass direction of the home's front door, while in the BTB (Western) feng shui school, the front door's compass direction is not relevant. Which school you follow is up to you, although many people find the principles of the Western school easier to follow.\nIn the classical feng shui school, the wealth or money area is considered to be the southeast corner of your home or office space. In the Western or BTB feng shui school, the money area is the upper left area of your space as viewed on a floor plan.\nWhenever possible, the best place to locate a home office or conduct your financial management tasks is in the overall money center of your home. But if this is not possible, remember that each individual room has an area that is best suited for wealth energy. If an entire room in the appropriate location is not available, then the southeast corner of any room (for the classical school) or the upper left of your floor plan (for the BTB school) will be most suitable for a wealth area.\nWhile defining the money area of your home, or the money area in an individual room, is easy enough, simply defining the space will do nothing to enhance prosperity and wealth. You must also take steps to manage the energy flow by making intelligent decisions regarding decor and by avoiding the most common mistakes that are made in money areas.\nWood is the feng shui element of the money or wealth area, and the color blue and images of water will nourish wood. A small water feature, such as a fountain, can greatly enhance a money center.\nProperly placed mirrors can multiply the prosperity-power of a money area, especially if the mirrors reflect natural scenes or blue colors.\nThe right type of growing plant is conducive to a wealth area. The common jade plant is one good choice.\nSmall statures or icons depicting symbols of wealth can enhance prosperity.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbktvh b/data_all_eng_slimpj/shuffled/split2/finalzzzbktvh
new file mode 100644
index 0000000000000000000000000000000000000000..a53138277f65a0f2c2de7c1f30a8f2db7b5af2fa
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzbktvh
@@ -0,0 +1,5 @@
+{"text":"The Mega Moolah Slot Jackpot is at a record breaking high of well over $9 million dollars. This progressive slot jackpot is ready to burst, and it could pay out at any time. The last Mega Moolah jackpot was won back in April 2015 by Mark A. from Canada. He walked away with $7,562,019 CAD when he won the progressive Mega Moolah Slot jackpot. This lucky player had been a player for just 7 days at Royal Vegas Casino when he hit the big one.\nTo anyone who thinks you need to bet big to walk away with a jackpot this size, you are wrong. Actually, the most common triggering bet that has won the Mega Moolah progressive jackpot has been just $0.50. Can you image walking away with well over $9,000,000 from just a 50 cent bet? It has been done, and can be done again, so be sure to try your luck now before someone else walks away a multi-millionaire.\nPlay the Mega Moolah Progressive Jackpot Slot now!\nThis entry was posted in Articles, Gambling, Microgaming, News, Online Casinos by bestfreecasinochips. Bookmark the permalink.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Shannon Babcock Real Estate -Century 21 Creekside Realty (Proudly Serving Buyers and Sellers in Chilliwack, Agassiz, Harrison Hot Springs and Hope BC) : **OPEN HOUSE THIS SUNDAY NOON- 2** at Lake Country Estates in #HopeBC This Brand New Home is moments #KawkawaLake LOTS STARTING from 105k!!\n**OPEN HOUSE THIS SUNDAY NOON- 2** at Lake Country Estates in #HopeBC This Brand New Home is moments #KawkawaLake LOTS STARTING from 105k!!\nKawkawa Lake, Hope - Come View this Wonderful Home and Awesome Neighbourhood. Step out of the Ordinary into this 'Brand New Cut Above' Spacious Split level home with full walk-out basement absolutely brimming with exquisite detailing.\nSituated in desirable Lake Country Estates just \u00bd block to beach and boat launch. Corner lot with A+ curb appeal, commanding mountain views from awesome cedar deck, 45 ft long double driveway & side parking for large RV.\nFrom the Soaring covered entry, custom quality finishings and superior craftsmanship to the abundance of windows and 9' ceilings on both floors, hardwood, extensive tile, stone & woodwork this one of kind home is DRESSED TO IMPRESS! Airy flexible layout offers Chef's Delight Gourmet Kitchen,3 bedrooms + den on main with flex room down. (Framed in for rec room, 2 bedrooms and plumbed for 3rd bathroom. Picturesque setting, with endless year round recreation. Call Today!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Economical and easy to install covers for IBC's.\nIdeally used in conjunction with IBC spill sumps. Perfect for keeping rainwater out of spill sump.\nTarp designed to fit most IBC and standard Containment Type Products. This tarp helps prevent contamination of drums and sump area, permit easy visual inspection, and keeps rainwater and debris out when stored outdoors. Elasticized bottom with adjustable straps. Attached document pouch for holding chemical information, MSDS sheets and other documents.\nDimensions: 56.5\u2033L x 56.5\u2033W x 44\u2033 H.\nOften referred to as dust covers, these Lighter Weight IBC Pullover Covers serve to protect IBC's from industrial debris, dust, occasional splashes and other contamination.\nMultiple options, plus custom are available to meet most requirements. Be sure to check if cover has UV Inhibitors for outdoor use.\nNote: When shown in details, UV Inhibitors are often rated by percentage to provide a good idea of how long a cover could last outdoors. IF UVI not shown, outdoor usage is not recommended.\nOn custom covers with UV Inhibitors there are often minimum purchase quantities for non-stock types.\nBest practices are recommended to ensure covers are compatible with chemicals in the plant.\nDo you need a light weight clear full size cover, an IBC heater or blanket and any number of other options for covering your IBC.\nCall or email us with your specific requirement. We most likely have what you are looking for.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"I have just recently sold this listing at 702 - 528 Beatty Street, Vancouver.\nI have just recently sold this listing at # 403 5211 GRIMMER ST, Burnaby.\nI have just recently sold this listing at # 116 7388 MACPHERSON AV, Burnaby.\nI have just recently sold this listing at 8075 NELSON AV, Burnaby.\nI have just recently sold this listing at 2627 W 34TH AV, Vancouver.\nI have just recently sold this listing at 3435 W 15TH AV, Vancouver.\nI have just recently sold this listing at # TH2 5782 BERTON AV, Vancouver.\nI have just recently sold this listing at # 1505 989 NELSON ST, Vancouver.\nI have just recently sold this listing at # 208 3580 W 41ST AV, Vancouver.\nI have just recently sold this listing at # 408 969 RICHARDS ST, Vancouver.\nI have just recently sold this listing at 6045 TYNE ST, Vancouver.\nI have just recently sold this listing at 4480 COLLINGWOOD ST, Vancouver.\nI have just recently sold this listing at # 250 2565 W BROADWAY BB, Vancouver.\nI have just recently sold this listing at # 605 1408 STRATHMORE MEWS BB, Vancouver.\nI have just recently sold this listing at 2304 VINE ST, Vancouver.\nI have just recently sold this listing at 2151 MAPLE ST, Vancouver.\nI have just recently sold this listing at # 315 55 E CORDOVA ST, Vancouver.\nI have just recently sold this listing at 2713 PRINCE EDWARD ST, Vancouver.\nI have just recently sold this listing at # PH1 141 WATER ST, Vancouver.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"styles and influences from punk to jazz, progressive to sadcore, and modern rock to thrash into meter-bending, riff-driven heaviness fueled by abrasively honest lyrics.\nFollowing the release of their 2015 debut on Belgium's Mausoleum Records to critical worldwide response, the band set to work on their follow-up album, now set for release in early 2019 on new label Ripple Music.\nLook for A Feast for the Sixth Sense, the new album from The Ghost Next Door, coming in early 2019 on deluxe vinyl, CD and digital from Ripple Music!","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzblibx b/data_all_eng_slimpj/shuffled/split2/finalzzzblibx
new file mode 100644
index 0000000000000000000000000000000000000000..e28dfee341cd78bfb1eaf46a60a2137c6f0ff75f
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzblibx
@@ -0,0 +1,5 @@
+{"text":"A real number object interpreted in computable mathematics (in realizability) yields the computable real numbers (often just called computable numbers). This is a core object in computable analysis\/exact analysis.\nTypically a computable real number is an algorithm that successively produces a sequence of rational numbers that approximate a given real number ever more closely.\nLast revised on March 3, 2014 at 23:58:28. See the history of this page for a list of all contributions to it.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Solomon Baba Akumba, a level 300 student of the Kwame Nkrumah University of Science and Technology (KNUST) has emerged the winner of the 2018\/19 Malta Guinness Maltavator Challenge, Season 2 held on Saturday at the Lizzy Sports Complex in Accra.\nAkumba braved sweltering temperatures and a field of 40 athletes to complete three rounds of a challenging obstacle course en route to claiming a GH\u20b55000 cash prize and the opportunity to represent Ghana at the Pan-African Maltavator Championship slated for February 24 in Johannesburg, South Africa.\nA breathless Akumba in an interview after he was presented with a dummy cheque said he was delighted with his win and looking forward to conquering Africa.\nAccording to him, he was not going to get complacent in his training as he readies for the contest in South Africa.\nHe will be joined by five others to compete in Johannesburg including Samuel Osei Mensah and Emmanuel Gyeta who completed the course in second and third place respectively. The pair received GH\u20b53000 and GH\u20b52000 for their performances.\nThe course included obstacles such as weave run through cones, high-knee runs through tyres, scrambler or low crawls, hurdles, human maze, tire flips, balance run on a log, archery, monkey bars and box jumps.\nThe Brands Manager of Malta Guinness, Mr Roland Ofori said he was hopeful that the winner of the Pan-African Maltavator Challenge will emerge from the six finalists.\nHe also commended all the participants especially those who participated during the regional qualifiers in Accra, Takoradi, Kumasi and Tamale.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The pinnacle of German road bike tyre technology from Schwalbe. Faster and lighter than the best clincher tyres.\nYou can have Schwalbe Pro One Tubeless tyres fitted, with sealant, to your HUNT wheels prior to dispatch. Normal suggested retail price (SRP) for these tyres is \u00a3134 per pair.\nSchwalbe have launched the new 70g lighter Pro One Tubeless. Now only 230g for a 23mm and lower rolling resistance than the clincher version!\nThe Schwalbe tubeless road tyres consistently win plaudits from journalists and customers alike. We have been testing them for a long time and have been very impressed. The One Pro's are Schwalbe's top of the range tyres and feature their triple compound tread so you get extremely low rolling-resistance combined with excellent grip and durability. In fact the Tubeless version of this tyre only has 17w rolling resistance against the clincher version which has 19w.This all comes with the added benefits of tubeless including great puncture protection. The tubeless beads on Schwalbe tyres have a flat edging and the new Mirco-Skin carbon tubeless layer makes them incredibly good at sealing with the rim bed and sidewall and very easy to fit.\nThe highly respected German Magazine TOUR said \"Unbeatable with punctures... \u2013 the \"self-repair\" with sealant means that no tube can compare ... The Schwalbe PRO ONE (preformed ideally and can be installed without tools and is immediately sealed) weighs approx. 290 grams including 40 millilitres of sealant, while the Conti folding tyre with tubes weighs 320 grams.\"\nBicyclerollingresistance.com | \"The new Pro One is the lowest rolling resistance tire I have tested to date.\"\nBikeradar.com | \"Light weight, suppleness and ease of installation makes for the best road tubeless we have tested\"\nThe new TUBELESS tyre is already being used by professionals. Martin Elmiger, Swiss champion of WorldTour Team IAM Cycling, and Sylvain Chavanel, have already put the tubeless tyre to the test in extreme conditions. There is no tougher challenge for either cyclists or tyres than the cobblestones of Paris-Roubaix. Their initial scepticism gave way to huge enthusiasm for the tubeless tyres. Both cyclists were amazed at the speed and comfort with which the tubeless tyres guided them over the hellish cobblestones. They didn't experience a single snake-bite, by far the most frequent cause of blow-outs on the Paris-Roubaix route. Both cyclists completed the race without a single puncture. Martin Elmiger came in at fifth place.\nRoger Kluge, also from the IAM Team, has already tested the new tyres Down Under: \"This tubeless tyre has rolling properties like a track tyre!\" From a track specialist like Roger Kluge, this is a substantial compliment.\nWhich size to run 23, 25, or 28mm?\nAny of these three tyre sizes will work excellently with our wheels and provide great performance but advantages can be gained by selecting the tyre width to match your intended routes and types of riding.. While 23mm wide tyres are a great choice for many riders and have become the status quo over the past decade the consensus is shifting towards 25mm and in some cases even 28mm (even for pro-riders) as having lower rolling resistance, puncture protection and of course comfort\/grip.\nWeight is obviously a factor and if you can save some on the tyre that will make a bit of difference to climbing, so our view is if we're going to be doing rides\/races with lots of climbing, where the descending speed\/grip is not so important, then we would consider the 23mm option. We would also use 23mm tyres with our Race Season wheels for any short aero focused events such at time-trials and sprint\/Olympic distance triathlons on smooth roads, as there is reasonable consensus that a narrower tyre than rim improves aerodynamics. For pretty much any normal events\/riding we would go for 25mm to improve rolling resistance\/speed, grip and comfort. We would use 28mm tyres with a lower pressure (around 85-95psi) for training, rough roads, cobbles, or super long events where reducing fatigue is essential.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"One of Mali's top jihadist leaders has been killed in a raid by French forces, the Malian army says.\nIt says Amadou Koufa died in Friday's operation in the central Mopti region.\nFrance earlier suggested that Koufa might be among about 30 Islamists \"put out of action\" in the raid.\nThe radical preacher is described as a senior member of the Jamaat Nusrat al-Islam wal-Muslimin (JNIM) group that has carried out frequent attacks in Mali and neighbouring Burkina Faso.\n\"I confirm that Amadou Koufa was killed during the operation,\" Malian army spokesman Col Diarran Kone told Reuters news agency on Saturday. He declined to provide any further details.\nThe French army had earlier said that its operation targeted a base controlled by Koufa.\nJNIM has so far made no public comments on the latest reports.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"What do you blog about when there's nothing to blog about?!\nI ended up back in NM recently due to an illness in the family, and just haven't felt much like doing anything creative. Also, work has been quite busy (same for the Roommate); I had 2 all-day conferences last week, one of which I presented at, and that stuff takes more time than you think it will. We've also had freezing cold weather, and I do a lot of walking outside. Sometimes I get home & really can't bear to do much other than sit on the couch & get warm! I'm not complaining...I love winter & the chance to wear my turtlenecks & woolen suits & eat hearty soups, but I do believe the cold takes it out of you. The snow has been fun--haven't seen any in a long time; it was the first time in our house!\nRight now I'm fixing again a too-strong magnet snap on the front of a lovely green leather purse. It tore through the lining about the 4th day I used it. I repaired it once, but sort of over-repaired it: in trying to weaken the magnet a bit, I over-weakened it & it wouldn't stay shut at all. Now I think I've got the magnet just the right amount of weakened, and am hand-sewing it in via an interfacing-lined contrasting patch. I hope it works because I really like the purse...lots of pockets & shiny green leather. Yum!\nWhile sewing, I usually watch stuff on hulu as we don't have a TV. Right now I'm thrilled that Project Runway 11 is on, and especially excited because there's a Native American woman (from Taos Pueblo) on this year & she's good. Patricia Michaels, has great turquoise jewelry, too, which I envy. I don't really want to be a designer, I really just wish I could zoom back to Abu Dhabi & get Mohammed to copy some stuff for me!\nSome exciting news: My friend Tilly has just released her first pattern. I know she's been dreaming about this for a while, so it's exciting to see a friend's dream come to fruition. She runs most days, sews a lot, has a full-time job & travels for work, so I don't see how she does it. Check out the new pattern, the Mathilde blouse! Not only do you get a cool pattern, you get all sorts of helpful info, such as using contrasting fabric, how to deal with digital patterns & what the heck to do if your measurements don't fit into one size or another. Tilly is an excellent teacher, so check out the blouse & her blog to get learning.\nI also just discovered, via a tweet from DIYCouture, The People's Print. I love the idea of designing my own fabric (Spoonflower is another place, too). This is The People's Print's manifesto: The People's Print pioneers new concepts for textile design combining traditional and digital technologies, open source, participatory and co-design, aiming to create a community of like minded people with a passion for designing their own textiles.\nWork continues anon on Simplicity 3994, from 1941, when Bella isn't assisting. But it's slow (I had to seriously re-design the princess seams & even add a dart; all thanks to Allison @ BitsofThread.com). That type of fix\/re-fix\/fix again really takes it out of me & I end up getting quite sick of the project & want to never look at it again. So 3994 & I have been avoiding each other the last few weeks.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzblynt b/data_all_eng_slimpj/shuffled/split2/finalzzzblynt
new file mode 100644
index 0000000000000000000000000000000000000000..0545657c18818f0fcbc51cc43fd0693db5276d43
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@@ -0,0 +1,5 @@
+{"text":"Beautifully engraved certificate from the Texaco Oil Company issued from 1974 - 1978. This historic document was printed by the American Banknote Company and has an ornate border around it with a vignette of oil derricks, factories, and a seaport with a ship next to the dock. This item has the printed signatures of the Company\ufffds President and Secretary.\n1901 \ufffd Joseph S. \"Buckskin Joe\" Cullinan and Arnold Schlaet found The Texas Fuel Company.\n1902 \ufffd The Texas Fuel Company opens its first office in Beaumont, Texas.\n1902 \ufffd As a result of The Texas Fuel Company's need for greater capital to increase operations, Cullinan and Schlaet found The Texas Company, officially located in Corsicana, Texas. A month later, The Texas Company absorbs The Texas Fuel Company and inherits the latter's office in Beaumont, Texas.\n1903 \ufffd The Texas Company's third oil well drilled at Sour Lake, Texas, comes in as a gusher, saving the company from bankruptcy.\n1911 \ufffd The Texas Company opens its first filling station in Brooklyn, New York, and introduces Texaco Auto Gasoline.\n1912 \ufffd The Texas Company introduces Texaco Motor Oil.\n1931 \ufffd Through the acquisition of Indian Refining Company, The Texas Company gains the rights to the Havoline brand name.\n1932 \ufffd The Texas Company replaces Texaco New and Better Gasoline with Texaco Fire Chief Gasoline, which has an octane rating higher than government standards for fire engines.\n1936 \ufffd The Texas Company and Standard Oil Company of California (later Chevron) form the California Texas Oil Company, Limited (later Caltex) to unite The Texas Company's extensive eastern hemisphere marketing network with Standard's production operations in the Middle East.\n1959 \ufffd The Texas Company officially changes its corporate name to Texaco Inc.\n1984 \ufffd Texaco purchases Getty Oil Company, acquiring 1.9 billion barrels of proven crude oil reserves and natural gas liquids.\n2000 \ufffd Chevron Corporation and Texaco Inc. announce a merger that will create ChevronTexaco Corporation.\n2001 \ufffd Chevron Corp. and Texaco Inc. complete their merger following stockholder approvals of both companies. ChevronTexaco enters the global marketplace as the second-largest U.S.-based energy company and the fifth largest in the world.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"\"Though we are approaching it, we haven't quite reached the end of the road for The Children Of The Night world tour. We know many of you have been waiting for it, so we are now very happy to announce that we'll make one final tour through North America, this time as a headliner! It's going to be a pleasure to be able to give you a full set in our very own way. See you this fall!\"\nFollowing the completion of their worldwide touring behind The Children Of The Night, TRIBULATION is expected to begin work on the follow up to the widely praised release. Please see below for the full list of dates on the upcoming tour in North America, as well as summer festivals in Europe and a string of Australian dates.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The new KBOX Systems Management and Deployment Appliance now works with current and forthcoming versions of Windows and Mac OS, including Windows 7 and Snow Leopard. A network administrator would use this product to get the network in compliance and to install patches and new software at the network level, rather than on each individual computer.\nPros: KBOX allows complete control of a network in one box. This is less expensive than software solutions, and the Web browser interface look and feel is familiar to the end user.\nCons: Like all of the products in this category, this networking appliance is expensive and takes time to learn (though KBOX appliances cost between $10,000 and $50,000 compared to competing products priced between $150,000 and $250,000). Kace recommends eight hours of training, which is less than most but does take time.\nOverall evaluation: With the complexities facing school networks, a product like KBOX can be very helpful for IT staff to manage the network. This makes it a good value, despite the higher price tag.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Mark Perez - It's the classic board game miniature turned into a life-size spectacle !  \"A visually stunning work of science inspired art and performance!\" Check out the Mousetrap on KRON 4! press Here!\n25 ton...Rube Goldberg machine !\nAt the Henry Ford museum!\nPeople who make the Mousetrap go!\nturned into a life-size spectacle !\nCheck out the Mousetrap on KRON 4! press Here!\nThe Mousetrap has combined Newtonian physics and large-scale art with performance to create a unique brand of engineering fun packaged into a traveling road show!\naudiences of all ages via science centers, museums, and festivals across the United States!\nWe are excited to announce the release of Esmerelda Strange's full length album!\nHow to Defy Gravity with Six Simple Machines!\nCopyright 2013 Mark Perez. All rights reserved.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Indie documentary \"Life, Animated\" tells the story of Owen Suskind, a young man with autism who grew up understanding emotions and relationships through the lens of Disney's animation catalogue.\nOur design for the film's titles and interstitials are inspired by home movies and memories of Owen's childhood. In a nod to Owen's extensive collection of sketchbooks, we gave cel animation a paper treatment and sketched over home video stills to create pencil portraits of Owen and his family. Paper textures were shot with a slider to create backplates for the animated line work.\n\"Life, Animated\" premiered at the Sundance Film Festival in 2016, director Roger Ross Williams won the award for Best Director - U.S. documentary. The film has since been bought by The Orchard for wider theatrical release later this year.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbmfjs b/data_all_eng_slimpj/shuffled/split2/finalzzzbmfjs
new file mode 100644
index 0000000000000000000000000000000000000000..dd97c2b4ddb1f83e63bf5fb04cde3636b58ec161
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzbmfjs
@@ -0,0 +1,5 @@
+{"text":"Spring Capital Group (\"SCG\") has EXTENSIVE EXPERIENCE in Real Estate, Construction, Project Development, Venture Capital, Technology, Energy Services, Manufacturing, Hospitality, Timber, Agriculture and Gaming.\nSCG seeks high value added investment potentials in growth industries and real estate, and selective debt opportunities in asset backed circumstances. SCG makes investments directly as a Principal Operator and in Joint Venture with Operating Partners.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The Suunto Ambit3 Peak Black Watch with Heart Rate Monitor is the ultimate GPS watch for training, sports, and exploring. The Ambit3 Peak features route navigation, track back, weather information, and a compass to keep you safe every step of the way during your adventure. When training, you can record speed, distance, pace and use the heart rate monitor to review your body's expenditure. The Ambit3 watch can be connected to your phone, which can be used as a second watch display. From your watch see calls, messages and push notifications. The Suunto Ambit3 Peak isn't just a watch, it's a lifestyle decision.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Today, Lenka \u010eul\u00edkov\u00e1, a Czech, has written to me and asked, \"I'd like to know how to say I have no money.\" Well, Lenka, perfect timing. And you've come to an expert in this because English teachers really have no money. We're very poor.\nAnd there are many ways in English to say that you're poor.\nYou can say you're broke.\nYou can say you're skint.\nDown to my last dime.\nI'm bust. I'm as poor as a church mouse.\nI haven't got a bean.\nI'm boracic. That's Cockney rhyming slang. Boracic lint, skint, skint means poor. Get it? So, I'm skint. I haven't got two pennies to rub together.\nAnd that's about the lot for English teachers. We don't earn a lot of money. And the fact, Lenka, that you're a Czech reminded me that. If we receive a cheque, especially our pay-cheque, then we're no longer quite as poor as we were.\nAnyway, I hope you've enjoyed this Daily Dose of English and I'll see you again tomorrow. Goodbye for now.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The Jefferson County Board of Election Commissioners will hold two meetings, according to a news release.\nA special meeting will be held at 6 p.m. Friday, May 25, at the election center, 123 Main St. The purpose is to address a recount request of the county judge's race from Dutch King. Three absentee ballots will also be adjudicated.\nA special meeting will be held at 6 p.m. Wednesday, May 30, at the election center to announce results of a recount of the county judge's race.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"There is no railway station in Coalville. The nearest station is in Loughborough, around 8 miles away, with regular bus services providing connections into Coalville town centre.\nThere are plenty of buses running into the centre of Coalville from the surrounding towns and villages. The number 11 connects to Greenhill and Agar Nook, while the 120 runs to Markfield and Leicester. If you're travelling into Coalville by bus, you should get off on London Road, which turns into High Street in the town centre. From here, turn onto Whitwick Street, follow this road for a few moments and you will see the job centre office on your left.\nThe job centre in Coalville is just moments from the main A511, which links the town with Ashby-de-la-Zouch, Burton-on-Trent and the M1 motorway. So the best way to get to the job centre if you're travelling by car is to follow the A511 into the town then, at the main roundabout, turn onto Thornborough Road. From here, turn right onto High Street and then left onto Whitwick Road.\nShort stay parking (up to three hours) is available in the three car parks located in Coalville town centre.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbpayq b/data_all_eng_slimpj/shuffled/split2/finalzzzbpayq
new file mode 100644
index 0000000000000000000000000000000000000000..522fa4e022f5ebd5cfcae37c75039472d4b8c6d6
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzbpayq
@@ -0,0 +1,5 @@
+{"text":"The first in a thrilling new mystery series set around the sacred historical sites of the British Isles Recently retired teacher Hilary decides that the best way to stop herself worrying about her husband, who's away doing voluntary work in war-torn Gaza, is to distract herself with a holiday.\nShe invites her good friend Veronica, a recent widow, to accompany her on a trip to Glastonbury, to see the ancient sights.\nBut even they are unprepared for the shocking discovery Hilary makes at the Chalice Well: an abandoned bag, containing a bomb ...Who is to blame?\nA foreign tourist? An eccentric pagan author? Or an angry local who resents that a Christian place has been 'overun' by other beliefs?\nHilary and Veronica just want to be on holiday, but they can't help but be sucked in to the investigation.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Wheelers Luxury Gifts values your custom and prides itself on offering a prompt and professional service to our clients.\nAll sizes and details specified per product are approximate and correct at the time of including on the site. There may be slight colour variations between the image displayed and the actual item. We update our product listing on a regular basis to reflect changes in our product lines. Please check product details to ensure you are choosing the item you require. All items are subject to availability and promotional offers are available whilst stocks last.\nWe cannot confirm the price of a Product until your order has been accepted in accordance with our Order acceptance policy. Order acceptance and the completion of the contract between you and us will take place when we despatch Products to you that you have ordered, unless we have notified you that we do not accept your order. Please only order from us if you agree with our terms and conditions, thank you.\nIf you have any questions or need further details on a product, please feel free to contact us. Our staff team are trained to be familiar with our product lines and will be able to offer the assistance you require.\nIn the unlikely event that you receive an item broken or faulty, we will replace it Free of Charge. We may require the item to be photographed and emailed or returned to us and we will refund you the cost incurred. Items broken on arrival must be notified within 3 Days of receipt.\nIf you decide that the Goods are not appropriate you may return them at your cost for a refund on the Goods. The Goods must be returned in perfect resaleable condition, including the original Presentation\/Gift Boxes where applicable. We reserve the right not to refund or offer a partial refund on returned Goods that do not comply with the above. We normally allow you up to 28 days within which to return items. We are unable to accept returns on Jewellery, Perfumes and Cosmetics, Drinks and Food Gifts.\nWe will not offer an exchange or refund on items that have been damaged or scratched if you have not notified us within three days of receipt, or that have been used, or that are not in the original packaging. In this situation, the item(s) will be returned to you at your cost.\nDue to hygiene reasons, we, unfortunately, cannot accept returns, refunds or exchanges on earrings and clothing items including swim shorts, boxer shorts, pyjamas, or socks, unless faulty.\nIn your original package will be an invoice\/receipt. Please return a copy of this with your item together with a note stating the reason for the return. We advise all customers to return any item via a traceable method, as we accept no liability for items that may become lost during transit. Customers are expected to meet the cost of returning items, except in the case of faulty or damaged goods. Alternatively, you may bring the item into our store, where a trained advisor will be on hand to assist you.\nIf you have used a promotional code to receive a free promotional item and have decided to return the items that enabled you to qualify for the free promotional item, you must return all of the items purchased on the order which included the promotional item and the free promotional item in an unused and sealed condition to receive a full refund. If you have opened or used the promotional item then the retail price of the promotional item will be deducted from the money originally spent before a refund is issued.\nThe FREE Laurent-Perrier Champagne Bottle promotion can only be redeemed against full priced products, not applicable to products already on promotion or offer. If you return your order, you will be required to return your free bottle of champagne or have the retail cost of the champagne deducted from your refund. Postage costs will also be the customer's responsibility. Please note that we can't ship alcohol outside of the E.U.\nIf you have not received your order after 14 days of placing your order you must inform us of non-delivery of goods. If after 21 days we have not been informed of the non-delivery of goods we cannot be held responsible.\nIf there is a packing discrepancy with your delivery we must be informed in writing within 3 days of you receiving your order. If your order shows any breakages you must inform us in writing within 3 days of your receiving your order. Otherwise, we cannot be held responsible for any packing discrepancies with your order.\nOur delivery services detailed on our delivery page are listed to try to ensure you receive your delivery at the allocated time selected. We will despatch on time, but then we are in the hands of our couriers and therefore cannot be held responsible for delays beyond our control which may result in your parcel not being delivered at a specified time. If the delivery time selected at the point of order has not been met by our third party couriers, then all postage costs will be refunded.\nIf you are exchanging goods for an item that is of a lower value than that originally purchased, we will refund the difference via your original payment method.\nIf you are exchanging goods for an item that is of greater value than that originally purchased, we will require payment for the difference, prior to despatch.\nThe majority of our items come with a manufacturers guarantee. This varies depending upon the manufacturer (Please see your item's guarantee certificate where relevant). Please telephone us in the first instance so that we can advise you further. Most of our suppliers also offer a repairs service if your item is outside of its guarantee. Please give us a call, and a trained advisor will be happy to obtain a quote for you. We will need the item back to get a quote for your repair. You will need to pay the postage costs in getting the item to us. The manufacturer will need the item to assess the repair costs and will advise us of the costs. We in turn will advise you before sanctioning any work. Obviously, if the item is within the manufacturers guarantee terms then there will be no charge for the repair. If however, they consider the damage to be fair wear and tear then full costs may be recovered.\nAny refunds will be given via your original payment method. Refunds will be issued within 5-7 working days of processing and should show up on your account within 10-14 days, although some card providers can take a little longer.\nChanges and Cancellations - Under the United Kingdom Distance Selling Regulations you are entitled to cancel your contract between the time of placing your order up until the item is delivered. Anything cancelled after delivery will be treated as a return.\nPersonalised items cannot be refunded or exchanged. If you have had a gift that has been personalised to your specifications we cannot offer a refund or exchange on these items. However, we will always endeavour to help you the customer and if the engraving has been done on an exchangeable part, we will try to obtain a replacement part for you. There would be a charge for the replacement part.\nThe 24-hour engraving service does not include Sundays or Bank Holidays.\nWe cannot ship alcohol to destinations outside of Europe.\nCustomers are responsible for paying any import tax and duties that are requested from their Customs Office when deliveries are shipped abroad.\nWe do not store personal and\/or financial information, nor do we share this information with 3rd parties.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"A powerful dark presence of an intense nature, connected with the local nature and with its spirit, ascending and descending, shaping through the clouds of the present. You can sometimes feel it is there, but you cannot grab it. Only your clean mind and soul can came close of its presence. From the very beginning we shaped ourselves into relating with it, so that is now part of us. That's how we understand Negur\u0103 Bunget.\nNegur\u0103 Bunget is a black fog coming from a deep dark dense forest. The name tries to picture somehow the kind of atmosphere, both musical and spiritual we'd want to create through our music. It has also a symbolical nature, standing for the inexpressible parts of our ideology. The two words are also from the Tracic substrate of the Romanian language (the oldest one, containing about 90 words) as the interest for our local history and spirituality is something of crucial importance and meanings for us as a band.\nImmortally rising from the spirituality of immortal lands, Negur\u0103 Bunget began their journey through the universe of Black Metal in early 1995, with Hupogrammos (guitars\/vocals\/keyboards) and Negru (drums). The duo's first demo, From Transilvanian Forests, was recorded at Magic Sound Studio in Bucharest, November 1995, and released at the end of the same year, under the band's original name, Wiccan Rede. The release of this demo was preceded by the band's first live performance, in Buzau (June 1995) and was then followed by concerts all around Romania.\nOnce their conceptual perspective shifted towards the exploration of the universe and spirituality of their ancestors, the band's name changed to Negur\u0103 Bunget. The new approach became clearly evident on the debut album, Z\u00eernindu-s\u0103, released by the newly founded Romanian label Bestial Records (late 1996).\nDecember 1998 brought Negur\u0103 Bunget's second release, Sala Molksa, a mini-album also released by Bestial Records. Sala molksa introduced the band's third official member, Spurcatu, on guitars\/bass. The 4 tracks unveil a much more straight-forward sound, complex, aggressive, atmospheric, fast \u2013 The Music of Transilvanian Spirituality. The band played in July 1999 at the Open Hell Festival in Volyne (Czech Republic).\nM\u0103iastru sfetnic is the third opus of Negur\u0103 Bunget. The album was released by Bestial Records in 1999 in a special digipak format (designed and handmade by Negur\u0103 Bunget). More than 56 minutes, yet only 6 epic tracks, united in haunts, evilness, descent, atmospheres, speed and aggression\u2026 displaying the band's vision on how a Black Metal-based world should have looked like.\nM\u0103iastru sfetnic also brought the band a 3-album contract with Italian label code666 Records. A new track was recorded specially for code666's compilation Better Undead Than Alive, and the band's first video was shot for the same track (V\u0103zduh) in the mountains of Transilvania.\n'N crugu bradului album, the first one to be released by code666 Records, can be seen as a spiritual endeavor though the essences of the Transilvanian traditional spirituality. Just 4 tracks, symbolizing the evolution of a year's 4 seasons, yet 54 minutes of conceptual Black Metal. The album was released in February 2003, and got positive reviews all over the media, making the band better known on the international scene.\nIn April\/March 2004 Negur\u0103 Bunget embarked on its first European tour, with gigs in Portugal, Germany, Belgium, Slovakia and Hungary. In the following summer the band also played at Under The Black Sun Festival in Germany and on Wanted Festival in Hungary. Finally, few gigs in Romania ended the year.\n2005 saw the release of the limited edition Inarborat Kosmos MCD. With lyrics in English, the 4-track EP offers an apparently traditional Black Metal, but turned complex by intricate and slightly chaotic guitar sounds, which have by now become a landmark of the band's constant quest for the proper rendition of Transilvanian Spirituality. The CD also contains a multimedia section with 2 live recordings. A second European Tour followed in October-November 2005, in anticipation of the following album, OM. 23 gigs all over Europe showed the band live potential is no less than its studio one.\nOM is finally released by code666 Records in November 2006 in a special double digipak edition featuring a CD and a full DVD. With OM Negur\u0103 Bunget proposes once again a new complete opus, focused around the symbolism and signification of one unitary concept. Romanian language offers a vast and complex signification: OM means MAN (the human being), but OM is also the main contemplative state for elevation in traditional esotericism.\nA new extended European Tour took place in April\/May 2007, with the band performing more than 40 concerts throughout the Europe. Another tour was done at the end of 2007, focused more on Netherlands and Germany. In October 2007 Negur\u0103 Bunget signed a contract with German label Lupus Lounge \/ Prophecy Production followed by the release in 2008 of the early Negur\u0103 Bunget albums on a special digipak edition. A live DVD was also recorded at the beginning of 2008 and is planned to be released by Prophecy Productions in early 2011.\nThe begging of 2008 marked yet another European tour for Negur\u0103 Bunget, this time along The Vision Bleak and Farsot. This was followed by a Romanian Tour in April. In the summer of 2008 Negur\u0103 Bunget played some of the most important Metal Festivals: Wacken Open Air and Summerbreeze, and also Artmania Festival, Kaltenbach Open Air, Rockmania Festival, Paalisare Rock Fest. Another extended European Tour was completed in October\/November 2008.\nThe work on M\u0103iestrit, a re-interpretation of the original Negur\u0103 Bunget album M\u0103iastru sfetnicwas finalized before the end of 2009, and the album was released in March 2010, by Prophecy Productions.\n2010 was one the most intense year in the history of Negur\u0103 Bunget. Two albums released, followed by a long European tour as headliners (Spirit of the Land Tour), another tour in autumn (Black Trolls Over Europe \u2013 10 dates), some festivals in between (including Hellfest) and a lot of work, all make this year special and unforgettable. Figures for 2010: 72 gigs played, 2 European Tours as headliners, more than 50.000 km of driving by van, 66 answered interviews, more than 10.000 CD's sold.\n2011 marked 15 years of Negur\u0103 Bunget, so the band activities celebrated this anniversary. April brought the release of the Live DVD \u2013 Focul Viuand of the MCD \u2013 Poart\u0103 de dincolosymbolically marking the end of the 10 years cooperation with Code666 Records. April also meant a tour along Norwegian legends Enslaved. Some summer festivals were played \u2013 Exitand Party San. The second part of 2011 saw the band on the Transilvanian Legacy Tour, traveling through Europe and Romania, playing more than 35 dates. In 2011 Negur\u0103 Bungetstarted developing its most ambitious project, The Transilvanian Trilogy, a complex musical and visual trilogy.\n2012 was another busy year for the band. March brought Negur\u0103 Bungeton Paganfest Europein an amazing line-up along Eluveitie, Primordialand Solstafir. April\/May was time for the very first US and Canadian tour \u2013 almost 40 dates were set for this amazing endeavor \/ Transilvanian Legacy. Some summer festivals followed. The first part of the visual \/ musical trilogy was developed and a complex production and recording process started.\nSome media recognition \u2013 the band was voted Best Romanian Metal band in 2012 by www.metalhead.ro, Romanian leading metal Portal and Metal Artist of 2012 by www.czb.ro.\nThe 2013 started again intensely, with a complete reset of the band studio and live concept, meant to take the Negur\u0103 Bunget on some brand new paths. A small tour in spring, some festivals in the summer, all prepared for an intense end of the year, with the band performing over 70 gigs in 3 months, throughout 3 different tours (including an extensive one with Greek legends Rotting Christ).\nT\u0102U, the first part of the Transilvanian Trilogy is finally seeing the light of day in 2014. Negur\u0103 Bunget prepared an extensive promotion for it, including an European spring, followed by one in US and Canada in May and June.\nWith more than 500 live appearances up to date, Negur\u0103 Bungetis eager to unleash the power of their Black Metal, promoting its musical and spiritual vision and, as always\u2026 The Transilvanian Spirituality.\nIn 2002 Noctem records its first demo \"Unholy blood\" which was commercialized for 2 years.\nAfter several line-up changes the band continues its way performing in national shows and composing new tracks, totally immersed in the seek of a more personal music style.\nThe band commercializes their live mcd, entitled \"Live 2004\" (self- produced and recorded at their natal city) for two years.\nIn 2007 it's time for the release of the last demo of the band, entitled \"God Among Slaves\". This opus received a good success from the online specialized press and metalheads.\nThe band, composed by its new members, fulfills different tours through their country. In July 2008 the band signs with Noisehead Records (Austria) and carries out the \"Noctem European Tour 08\" where the band performs through different countries: Spain, Portugal, France, Andorra, UK, Belgium and Italy.\nIn January of 2009 Noctem enters the studio to record their debut LP \"Divinity\" at Vienna (At) which counts with the collaborations of Christos Antoniou (Septic flesh and Chaos star) and Leal (ex-Forever Slave).\nNoctem performs in different presentations of the album \"Divinity\" being included in festivals and several tours, like the ones at Spain or Portugal in 2009, sharing stage with headliners such as Malevolent Creation (USA),Vomitory (Sweden) and Napalm Death.\nAfter a few shows at Spain, and the recording process of the third and most ambitious video extracted from \"Divinity\", in May 2010 the band is included in the \"Collectors of the King Tour \", supporting Ragnarok(Norway) through Europe: Austria, Germany, Poland, Belgium and The Netherlands.\nIn September 2010, the third video extractes from \"Divinity\", entitled with the same name, was worldwide released; Including the collaboration of Christos Antoniou (Septic Flesh) in the orchestral arrangements and a team formed by more than 80 members.\nIn march of 2011 the band announces their signing with the record label Rising Records for the European release of their new album: \"Oblivion\".\nThe album was released in Spain in April 2011 and in Europe in June 2011 and licensed in USA through Metal Blade Records in October 2011. \"Oblivion\" received an outstanding feedback in Europe and North America.\nThe \"Oblivion Spanish Tour 2011\" started in the end of April and Noctem performed on the national stages during May, June, August and the end of September.\nIn summer 2011 and after receiving a great response from the international specialized metal press and fans, Noctem announced their appearance in the \"Lux Mundi Europe Tour\u2032\u2032, where the band shared stage with international well known acts as Samael, Melechesh and Keep of Kalessin.\nExilium was recorded in the summer of 2013 in theAGR Studios (Valencia) produced and mastered by Daniel Cardoso (Anathema). The album is be released by Art Gates Records in Europe in April 7th.\nIn Summer 2014 Noctem inks deal with NorthAmerican label Prosthetic Records to release \"Exilium\" September 16th in Australia and The Americas!!!\nAntagonist Zero was formed 2010 in Porvoo, Finland.\nThe band describes their music as Catatonic Metal, which is a mix of different types of metal genres, namely doom, death\/black and post metal.\nAntagonist Zero consists of seasoned musicians from such local bands as Adastra, Cavus, Coprolith and Lost in Tears.\nThey released their debut Nighttime Harmony March 2012.\nThis melodic, atmospheric and slow-burning heavy three song EP has been embraced by listeners around the world and has received excellent reviews. This was only a taste of the band's material, of which there was more than enough for a whole album.\nThe debut full-lenght album \"Doomed\" was released 28.9.2012 through Violent Journey Records.\nThe promoting of the album included live shows, festivals and tours across Europe, Baltics, Russia and China.\nNow the band is in studio recording their second full-lenght album which is set to be released early 2015.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"I've been Fortched! Thank you so much, brother. These tunes are excellent, and what a voice! I don't think I ever knew what you sounded like. I like what you're singing about, too...things we all deal with being human. I can't wait to listen again. \"Harmony\" is particularly fun to play along with on the mandolin. More to report later. You rock!\nThank you so much, brother. These tunes are excellent, and what a voice! I don't think I ever knew what you sounded like. I like what you're singing about, too...things we all deal with being human. I can't wait to listen again. \"Harmony\" is particularly fun to play along with on the mandolin. More to report later. You rock!\nHey Fortch! Luv the new site! Cool pictures!","meta":{"redpajama_set_name":"RedPajamaC4"}}
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